international journal of analysis and applications issn 2291-8639 volume 1, number 2 (2013), 123-127 http://www.etamaths.com fixed point theorem on multi-valued mappings j.maria joseph1,∗, e. ramganesh2 abstract. in this paper, we prove a common fixed point theorem for two multivalued self-mappings in complete metric spaces. 1. introduction and preliminaries: the study of fixed points for set valued contractions and nonexpansive maps using the hausdorff metric was initiated by markin. later, an interesting and rich fixed point theory for such maps has been developed. the theory of set valued maps has applications in control theory, convex optimization, differential inclusions and economics. following the banach contraction principle nadler introduced the concept of set valued contractions and established that a set valued contraction possesses a fixed point in a complete metric space. subsequently many authors generalized nadlers fixed point theorem in different ways[[1],[2]]. definition 1.1. let x and y be nonempty sets. t is said to be a multi-valued mapping from x to y if t is a function from x to the power set of y . we denote a multi-valued map by t : x → 2y . definition 1.2. a point x0 ∈ x is said to be a fixed point of the multi-valued mapping t if x0 ∈ tx0. example 1.3. every single valued mapping can be viewed as a multi-valued mapping. let f : x → y be a single valued mapping. define t : x → 2y by tx = {f(x)}. note that t is multi-valued mapping iff for each x ∈ x,tx ⊆ y. unless otherwise stated we always assume tx is non-empty for each x ∈ x. definition 1.4. let (x,d) be a metric space. a map t : x → x is called contraction if there exists 0 ≤ λ < 1 such that d(tx,ty) ≤ λd(x,y), for all x,y ∈ x. definition 1.5. let (x,d) be a metric space. we define the hausdorff metric on cb(x) induced by d. that is h(a,b) = max { sup x∈a d(x,b), sup y∈b d(y,a) } for all a,b ∈ cb(x), where cb(x) denotes the family of all nonempty closed and bounded subsets of x and d(x,b) = inf{d(x,b) : b ∈ b}, for all x ∈ x. 2010 mathematics subject classification. 54h25,47h10. key words and phrases. fixed point, multivalued map, hausdorff metric. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 123 124 joseph and ramganesh definition 1.6. let (x,d) be a metric space. a map t : x → cb(x) is said to be multi valued contraction if there exists 0 ≤ λ < 1 such that h(tx,ty) ≤ λd(x,y), for all x,y ∈ x. lemma 1.7. [3] if a,b ∈ cb(x) and a ∈ a, then for each ε > 0, there exists b ∈ b such that d(a,b) ≤ h(a,b) + ε. 2. main results theorem 2.1. let(x,d) be complete metric space and let s,t : x → cb(x) be multivalued maps satisfying h(tx,sy) ≤ ad(x,ty) + b(d(x,sy) + d(ty,tx)), where 0 < a + 2b < 1, a,b ≥ 0, for all x,y ∈ x. then f(t) = f(s) 6= ∅ and tx = sx = f(t), for all x ∈ f(t). proof. fix any x ∈ x. define x0 = x and let x1 ∈ tx0. by lemma(1.7), we may choose x2 ∈ sx0 such that d(x1,x2) ≤ h(tx0,sx0) + (a + b). now d(x1,x2) ≤ h(tx0,sx0) + (a + b) ≤ ad(x0,x1) + b(d(x0,x1) + d(x1,x2)) + (a + b) ≤ a + b 1 − b d(x0,x1) + a + b 1 − b . by lemma(1.7), there exists x3 ∈ tx2 such that d(x3,x2) ≤ h(tx2,sx0) + (a + b)2 1 − b . now d(x3,x2) ≤ h(tx2,sx0) + (a + b)2 1 − b ≤ ad(x2,x1) + b(d(x2,x2) + d(x1,x3)) + (a + b)2 1 − b ≤ a + b 1 − b d(x2,x1) + ( a + b 1 − b )2 ≤ ( a + b 1 − b )2d(x0,x1) + 2( a + b 1 − b )2 continuing this process, we obtain by induction a sequence {xn} such that x2n ∈ sx2n−2, x2n+1 ∈ tx2n, such that d(x2n+1,x2n+2) ≤ h(tx2n,sx2n) + (a + b)2n+1 (1 − b)2n , d(x2n,x2n+1) ≤ h(sx2n−2,tx2n) + (a + b)2n (1 − b)2n−1 , fixed point theorem on multi-valued mappings 125 now, d(x2n,x2n+1) ≤ h(sx2n−2,tx2n) + (a + b)2n (1 − b)2n−1 ≤ ad(x2n,tx2n−2) + b(d(x2n,sx2n−2) + d(tx2n−2,tx2n) + (a + b)2n (1 − b)2n−1 ≤ ad(x2n,x2n−1) + b(d(x2n−1,x2n) + d(x2n,x2n+1) + (a + b)2n (1 − b)2n−1 ≤ (a + b) (1 − b) d(x2n−1,x2n) + (a + b)2n (1 − b)2n also, d(x2n+1,x2n+2) ≤ h(tx2n,sx2n) + (a + b)2n+1 (1 − b)2n ≤ ad(x2n,tx2n) + b(d(x2n,sx2n) + d(tx2n,tx2n) + (a + b)2n+1 (1 − b)2n ≤ ad(x2n,x2n+1) + b(d(x2n,x2n+1) + d(x2n+1,x2n+2) + (a + b)2n+1 (1 − b)2n ≤ (a + b) (1 − b) d(x2n,x2n+1) + (a + b)2n+1 (1 − b)2n+1 therefore, d(xn,xn+1) ≤ (a + b) (1 − b) d(xn−1,xn) + (a + b)n (1 − b)n for all n ∈ n and let k = (a+b) (1−b) d(xn,xn+1) ≤ kd(xn−1,xn) + kn ≤ k(kd(xn−2,xn−1) + kn−1) + kn = k2(d(xn−2,xn−1)) + kk n−1 + kn ≤ ... ≤ knd(x0,x1) + nkn. since k < 1 , ∑ kn and ∑ nkn have same radius of convergence, {xn} is a cauchy sequence. since (x,d) is complete, there exists z ∈ x such that xn → z. d(tz,z) ≤ d(z,x2n+2) + d(x2n+2,tz) ≤ d(z,x2n+2) + h(tz,sx2n) ≤ d(z,x2n+2) + [ad(z,tx2n) + b(d(z,sx2n) + d(tx2n,tz))] ≤ d(z,x2n+2) + [ad(z,x2n+1) + b(d(z,x2n+2) + d(x2n+1,tz))] → ad(z,z) + b[d(z,z) + d(z,tz)] as n →∞. therefore d(tz,z)(1 − b) ≤ 0. hence d(tz,z) = 0 126 joseph and ramganesh h(tz,sz) ≤ ad(z,tz) + b(d(z,sz) + d(tz,tz))] = ad(z,tz) + bd(z,sz) ≤ ad(z,tz) + bd(z,tz) + bd(tz,sz) ≤ (a + b)d(z,tz) + bh(tz,sz) h(tz,sz) ≤ ( a + b 1 − b )d(z,tz) hence, h(tz,sz) = 0, z ∈ tz = sz and therefore z ∈ f(t) 6= ∅, z ∈ f(s) 6= ∅, to complete the proof, it is enough to show following four cases: (i) f(t) ⊆ tz and sx = tx for all x ∈ f(t). (ii) tz ⊆ f(t) (iii) tx = tz for all x ∈ f(t) (iv) f(s) ⊆ tz for any x ∈ f(t), d(x,tz) ≤ h(tx,sz) ≤ ( a + b 1 − b )d(x,tz) this shows that d(x,tz) = 0 and x ∈ tz.further h(sx,tx) ≤ ( a + b 1 − b )d(x,tx) = 0 and x ∈ sx = tx.for any x ∈ tz d(x,tx) ≤ h(sz,tx) ≤ ( a + b 1 − b )d(tz,x) = 0 this shows that x ∈ tx. now, we see thattz = f(t) ⊆ f(s)and sx = tx for all x ∈ f(t). for any x ∈ f(t), h(tx,sz) ≤ ( a + b 1 − b )d(x,tz) = ( a + b 1 − b )d(x,f(t)) = 0 hence, tx = sz = tz. it remains to show that f(s) ⊆ tz = f(t). for any x ∈ f(s), d(x,tz) ≤ h(tx,sz) ≤ ( a + b 1 − b )d(tx,z) ≤ ( a + b 1 − b )h(tx,sz) ≤ ( a + b 1 − b )2d(x,tz) hence, d(x,tz) = 0. then x ∈ tz and f(s) ⊆ tz. � in what follows, let ( denote multimap. corollary 2.1. let t : x ( x be a multivalued map with nonempty compact values and r ∈ [0, 1) such that h(tx,t2y) ≤ rd(x,ty), fixed point theorem on multi-valued mappings 127 for all x,y ∈ x. then, f(t) 6= ∅ and tx = f(t) for all x ∈ f(t). remark 2.2. let s be a self mapping (multi valued or single valued) defined on x, we denote f(s) the collection of all fixed points of s. if one of s and t in theorem 2.1 is single valued, then the set f(t) = f(s) is singleton and the maps s and t have a unique common fixed point in x. 3. acknowledgments the authors would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper. references [1] lai-jiu lin and sung-yu wang,common fixed point theorems for a finite family of discontinuous and noncommutative maps,fixed point theory and applications,vol.2011,article id 847170,19 pages. [2] j. maria joseph, m.marudai, common fixed point theorem for set-valued maps and a stationary point theorem, int. journal of math. analysis, 6, (2012), no. 33, 1615 1621. [3] s.b. nadler, multi-valued contraction mappings, pacific journal of mathematics, 30 (1969), 475–488. 1department of mathematics, st.joseph’s college,tiruchirappalli,tamil nadu,india 2department of educational technology, bharathidasan university,tiruchirappalli,tamil nadu,india ∗corresponding author local influence analysis of non-parametric regression model with random right censorship i. ii. y ( , , ) t t x z t ( ) t t y x β z α t ε   1 ( , , ) t n x x x 1n  1 ( , , ) t p β β β 1 ( ) ( ( ), , ( )) t q α t α t α t t ε ( | , , ) 0 , t t e ε x z t  2 ( | , , ) t t v a r ε x z t ζ y c y y c ( , , ) t t x z t m i n ( , )y c  ( )δ i y c  ( )i  { ( , , , , ) } t t k k k k k x z t δ ( , , , , ) t t x z t δ 1 ( , , ) t t k k k p x x x k y ( , ) k k δ m i n ( , ) k k k y c  ( ) k k k δ i y c  k  1 k δ  k  0 k δ  i f i y g i c in f { : ( ) 1} i f η t f t  ii ff  1 gg  1 1 1 1 ( ) ( ) p q i i i k i k i k i k k e δ g β x z α t        ， ni ,,2,1  1 0 ( ) ( ) ( ) 1 ( ) f gi η η i i i i y i y e δ g d g t d f y g y e y         1 1 ( ) p q i k i k i k i k k e y β x z α t      1 1 1 ( ) ( ) p q i i i k i k i k i k k e δ g β x z α t        ， ni ,,2,1  1 { ( ) , 1 } i i i δ g i n     1 1 ( ) ( ) p q i i k ik ik i i k ki δ β x z α t ε g          ni ,,2,1  i ε  0 i e ε   ， 2 ( ) i v a r ε ζ    g g ˆ g g ˆ [ 0 , ] ( ) 1 ( ) 1 ( ) ,ˆ ( ) 2 ( ) 0 , j j i δ t n j n j j n n i f t g t n i f t                         ( ) 1 2 m a x { , , , } , n n     1 ( ) [ ], 1 , 2 , , n j i ji n i j n        ( ) i i i i δ l g    ( ) , 1 , , t t i i i i i l x β z α t ε i n      iii. β 1 1 1 1 0 0ˆ( ) s u p , ( ) , , n n n i i i i i i i i l n p p p p               β η β  ˆ ˆ ˆ( ) ( ) ( ) t t i i i i i i x t l x z t       η β μ β α 1 ˆ ˆˆ ( ) ( ) ( )t s t g t      α 1 ˆ ( ) ( ) n t i i i i s t w t x x    1 ˆ ( ) ( ) n t i i i i g t w t x y     1 ˆ ( ) ( ) n i i i t w t x   μ 1 ( ) ( ) ( ) n i h i h i i w t k t t k t t     ( ) ( )hk k h   ( )k  ，h ( )ω [ , ]a b ，0 1a b   ， 1 1 1 , ˆ ( ) i t i p n  λ η β ， 1 1 1 ( ) . ˆ ( ) n t i i l    β λ η β β 1 1 ˆ( ) l o g ( ( ) ) , n t e i i l    β λ η β rλ 1 0 1 ˆ ( ) ˆ ( ) n i t i i    η β λ η β λ β 1 1 1 n ˆq ( , ) l o g ( ( ) ) . n t i i n     λ β λ η β ， β̂ λ̂ ：     1 1 1 1 1 1 1 0 1 0 n 1 , n n 2 , n q ( , ) ˆ ˆq ( , ) ( ) ( ) ˆ ( )q ( , ) ˆq ( , ) ( ) . n t i i i n ti i i n n                           λ β λ β η β λ η β λ η βλ β λ β λ λ η β β β iv. j ( , ) t j j x l ( ) t t i i i i i l x β z α t ε     ， .i j ( ) ˆ j β β j ( , , , ) t t j j j j x z t l β̂ ( ) ˆ j β ， ， β 1 1 1 ( ) 2 2 .1 2 1 1 1 ˆ ˆ ˆ ( ){1 (1 )} , j i p β β n s s s η β o       1 1 1 2 2 1 2 2 ˆ ( ) ˆ ˆ( ( ) ( ) ) ˆ ( ) 0 t t i f i i f i f η β e η β η β e s s β s s s η β e β                              1 2 2 . 1 2 1 1 1 1 2 s s s s    ( ) ( ) ˆ ˆ ˆ ˆ( ) ( ) ( ) , t j j j e c d m β β m β β   2 2 ˆ ( ) e β β l β m β     j ， ˆ( ) j v a r y  j ， 2ζ̂  2ˆ ( )ζ j   .( ) ˆ ˆ( ) 2 ( ) ( ) j e e j e l d m l β l β  v. n ω r ( | ) e l β ω , 1 ( | ) ( ) n e i e i i l β ω ω l β    0 ω ω 1n  0 ( | ) ( ) e e l β ω l β ˆ ˆ( ) 2[ ( ) { ( )} ] d e e e l ω l β l β ω  ˆ ( )β ω β ( | ) e l β ω 0 ( )ω a ω a h  0 ( 0 )ω ω 0 ( ) / | a d ω a d a h   h n r t t , ( ) e ω l d ω（ ） c 0 0 ( ) ( ) e t h l d ω ω h h h 0 0 0 2 2 1 ˆ( ) , ˆ{ ( ) } 2 2 { ( ) } e te β etl d ω ω β ω l d β ω h l β ω ω            2 ( , ) βω e l d β ω   p n ( , )k i , ( ) k β e i l β c 0 ( ) h ω 1 1 0 p p n λ λ λ λ        0 ( ) e l d ω h  1( , ) : 1 , t m m m n v v v m n  0 ( ) e m m ml d ω h v λ v 0 ( ) e l d ω h 0 ( ) 1 . e n t m m ml d ω m h λ v v    1 v 1 λ 2 1 j p e m m j m c λ v    je 1n  j 1 v ( , , ) t t j j j x z t j e c       1 1 2 2 .1 2 1 (1 ) 2 1 (1 ) 2 1 (1 ) , j e j p j p t j j p c e l d o e c d o n s o          1 2 1 1 1 , ˆ ˆ ( ) ( , ) (1 ) , ˆ ˆ1 ( ) i j β e j j ptβ β i s s η β l x β o λ η β         1 1 1 , 2 1 ˆ ˆ, ˆ ˆ( ) ( )1 , ˆ(1 ( ) ) tn i i λ n t i i β β λ λ η β η β s q n λ η β            1 2 1 , 2 1 ˆ ˆ, ˆ ˆ ˆ ˆ( ) ( ) ( ) ( 1 ( ) )1 , ˆ( 1 ( ) ) t t n i β i β i i β n t i i β β λ λ η β λ η β η β λ η β s q n λ η β                2 1 2 , 2 1 ˆ ˆ, ˆ ˆ ˆ ˆ( ) ( ) ( ) ( 1 ( ) )1 , ˆ( 1 ( ) ) t t n i β i β i i λ n t i i β β λ λ η β λ η β η β λ η β s q n λ η β                2 2 2 , 2 1 ˆ ˆ, ˆ ˆ( ) ( )1 , ˆ( 1 ( ) ) t t n β i β i β n t i i β β λ λ η β λλ η β s q n λ η β          1 2 2 . 1 2 1 1 1 1 2 s s s s    vi. 2p  1 2 ( ) 2 s i n 3 α t π t        ，   2 2 1 ( ) 1 .5 2 α t t  ， ~ ( 0 , 1 )t u 2q  [1 . 0 , 1 . 5 ] t β  ~ ( 0 , )x n  ~ ( 7 , )z n  5 5 5 5           ~ ( 0 , 1 )ε n ( ) t t y x β z α t ε   c i i i c y λ  ， 1 , 2 , , ,i n   1 ~ ( 0 , ) n i i λ u c  1 0 0 0n  5c  2 0 % 1 3 5 ( l o g ) n h n n   2 2 1 5 ( ) (1 ) ( 1 ) 1 6 k t t t   、 、 、 、 i ， ( ) i η β β λ ˆ ˆ6 . 6 1 7 1 0 . 0 0 7 2β λ、   ， 1 1 1 2 2 1 2 2 , , ,s s s s i e c i e c i e c i e c 、 、 、 、 ， ： ： international journal of analysis and applications volume 17, number 2 (2019), 226-233 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-226 a note on generalized indexed norlund summability factor of an infinite series b. p. padhy∗, p. tripathy and b. b. mishra department of mathematics, school of applied sciences, kiit, deemed to be university, bhubaneswar-24, odisha, india ∗corresponding author: birupakhya.padhyfma@kiit.ac.in abstract. in the present article, we have established a result on generalized indexed absolute norlund summability factor by generalizing results of mishra and srivastava on indexed absolute cesaro summabilty factors and padhy et.al. on the absolute indexed norlund summability. 1. introduction in 1930, j.m.whittaker [18] was the 1st to establish a result on the absolute summability of fourier series and in 1932, m. fekete [6] established a result on generalized indexed summability. later on the researchers like daniel [4] in 1964, das [5] in 1966, siya ram [15] in 1969, mazhar [11] in 1971, mishra and srivastava [13] in 1984, sulaiman [16] in 2011 etc. have established results on indexed summability factors of an infinite series. let ∑ an be a given infinite series with sequence of partial sums {sn} . let tnα be the nth (c,α) mean (with order α > −1) of the sequence {sn} and is given by tαn = 1 aαn n∑ k=0 aα−1n−ksk, n ∈ n,where a α n = γ(n + α + 1) γ(α + 1)γ(n + 1) , received 2018-11-21; accepted 2018-12-18; published 2019-03-01. 2010 mathematics subject classification. 40d15, 40f05, 40g99. key words and phrases. absolute summability; summability factors; infinte series. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 226 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-226 int. j. anal. appl. 17 (2) (2019) 227 then the series ∑ anis said to be summable |c,α|k,k ≥ 1, [7] if ∞∑ n=1 (n)k−1|tαn − t α n−1| k < ∞. let tn be the nth (c, 1)mean of the sequence {sn} and is given by tn = 1 n + 1 n∑ k=0 sk, then the series ∑ anis said to be summable |c, 1|k,k ≥ 1, [3] if ∞∑ n=1 (n)k−1|tn − tn−1|k < ∞. (1.1) suppose {qn} be a sequence of real numbers with qn > 0, such that qn = n∑ ν=0 qν →∞, as n →∞(q−i = q−i = 0, i ≥ 1) (1.2) the sequence to sequence transformation tn = 1 qn n∑ ν=0 qn−νsν (1.3) defines the sequence {tn} of the (n,qn)means of the sequence {sn} generated by the sequence of coefficients {qn} . the series ∑ an is said to be summable |n,qn| if the sequence {tn} is of bonded variation i.e; ∑ |tn −tn−1| is convergent. the series ∑ an is said to be summable |n,qn|k,k ≥ 1 ,if (see [8]) ∞∑ n=1 ( qn qn )k−1 |tn −tn−1| k < ∞ (1.4) clearly,|n,qn|k-summabiity is same as |c, 1|-summabiity,when qn = 1, for all values of n. further any sequence {αn} of positive numbers the series ∑ an is said to be summable |n,qn,αn|k,k ≥ 1 if ∞∑ n=1 (αn) k−1|tn −tn−1| k < ∞ (1.5) and is said to be summable |n,qn,αn; δ|k,k ≥ 1,δ ≥ 0 if ∞∑ n=1 (αn) δk+k−1|tn −tn−1| k < ∞ (1.6) for any sequence {µn}, ∑∞ n=1 anµn is an infinite series. we define ∆µn = µn −µn−1, |∆µn| = |µn −µn−1| also, for any sequence {µn}, by µn = o(n), we mean that the sequence {µnn } is bounded. int. j. anal. appl. 17 (2) (2019) 228 2. known theorems concerning with |c, 1| and |n,qn| summability kishore [10] has proved the following theorem: theorem 2.1. let q0 > 0,qn ≥ 0 and (qn) be a non-decreasing sequence. if ∑ an is summable |c, 1| then the series ∑ anqn(n + 1) −1 is summable |n,qn|. later on ram [15] has proved the following theorem related to absolute norlund factors of infinite series. theorem 2.2. let (qn) be a non-increasing sequence with q0 > 0,qn ≥ 0. if n∑ k=1 1 k |sk| = o(yn) as n →∞; where (yn) is a positive non-decreasing sequence and (µn) is a sequence such that ∞∑ n=1 n|∆2µn|yn < ∞; |µn|yn = o(1) as n →∞, then the series ∑ anqn(n + 1) −1 is summable |n,qn|. also verma [17] has proved the following summability factor theorem: theorem 2.3. let (qn) be a non-increasing sequence with q0 > 0,qn ≥ 0. if ∑ an is summable |c, 1|k then the series ∑ anqn(n + 1) −1 is summable |n,qn|k,k ≥ 1. in 1984, mishra and srivatava [13] proved the following theorem for |c, 1|k summability. theorem 2.4. let (yn) be a positive non-decreasing sequence and let there be sequnces {βn} and {µn} such that |∆µn| ≤ βn; (2.1) βn → 0 as n →∞; (2.2) |µn|yn = o(1) as n →∞; (2.3) ∞∑ n=1 n|∆βn|yn < ∞; (2.4) ∞∑ n=1 1 n |sn| k = o(ym) as m →∞, (2.5) then the series ∑∞ n=1 anµn is summable |c, 1|k,k ≥ 1. int. j. anal. appl. 17 (2) (2019) 229 very recently, padhy et al. [14] have proved a theorem on |n,qn|k-summability by extending theorem 2.4, in the following form: theorem 2.5. let for a positive non-decreasing sequence (yn),there be sequences {βn} and {µn} satisfying the conditions 2.1 to 2.5 and {qn} be a sequence with {qn}∈ r+ such that qn = o(nqn); (2.6) ∞∑ n=1 qn qn |sn| k = o(ym) as m →∞; (2.7) qn−r−1 qn = o ( qn−r−1 qn qr qr ) ; (2.8) m+1∑ n=r+1 ( qn qn )k−1 qn−r qn = o ( qr qr ) , (2.9) then the series ∑∞ n=1 anµn is summable |n,qn|k,k ≥ 1,where 0 ≤ r ≤ n. it should be noted that if we take qn = 1∀n then condition 2.7 will be reduced to 2.5. in what follows, we have generalized known theorems 2.4 and 2.5 to |n,qn,αn; δ|k summability in the form of the following theorem after studying [1] and [2] : 3. main theorem theorem 3.1. let (yn) be a positive non-decreasing sequence and there be sequences {βn} and {µn} such that the conditions 2.1 to 2.5 are satisfied.further let {qn} be a sequence of real numbers with qn > 0, such that qn = o(nqn); (3.1) ∞∑ n=1 qn qn |sn| k = o(ym) as m →∞; (3.2) qn−r−1 qn = o ( qn−r−1 qn qr qr ) ; (3.3) m+1∑ n=r+1 (αn) δk+k−1 qn−r qn = o ( qr qr ) , (3.4) then the series ∑∞ n=1 anµn is summable |n,qn,αn; δ|k,k ≥ 1,δ ≥ 0. we require the below mentioned lemma to prove our main theorem: int. j. anal. appl. 17 (2) (2019) 230 4. lemma [5] let (yn) be a positive non decreasing sequence and there be sequences {βn} and {µn} such that the conditions 2.1 to 2.5 are satisfied.then βnyn = o(1) as n →∞, (4.1) ∞∑ n=1 βnyn < ∞. (4.2) 5. proof of the main theorem suppose (τn) refers to the (n,qn)mean of the series ∑∞ n=1 anµn. then by definition, we have τn = 1 qn n∑ r=0 qn−r r∑ s=0 asµs = 1 qn n∑ s=0 asµs n∑ r=s qn−r = 1 qn n∑ s=0 asµsqn−s = 1 qn n∑ r=0 arµrqn−r thus τn − τn−1 = 1 qn n∑ r=1 qn−rarµr − 1 qn−1 n−1∑ r=1 qn−r−1arµr = n∑ r=1 ( qn−r qn − qn−r−1 qn−1 ) arµr = 1 qnqn−1 n∑ r=1 (qn−rqn−1 −qn−r−1qn)arµr = 1 qnqn−1 [ n−1∑ r=1 ∆{(qn−rqn−1 −qn−r−1qn) µr} ] n∑ ν=1 aν, with p0 = 0 = 1 qnqn−1 [ n−1∑ r=1 (qn−rqn−1 −qn−r−1qn) µrsr + n−1∑ r=1 (qn−r−1qn−1 −qn−r−2qn) ∆µrsr ] (by abel’s transformation) = tn,1 + tn,2 + tn,3 + tn,4 (say) now, to show ∑∞ n=1 anµn is summable |n,qn,αn; δ|k,k ≥ 1,δ ≥ 0, by 1.6,we need to show that ∞∑ n=1 (αn) δk+k−1|τn − τn−1| k < ∞. int. j. anal. appl. 17 (2) (2019) 231 i.e; to show that ∞∑ n=1 (αn) δk+k−1|tn,1 + tn,2 + tn,3 + tn,4| k < ∞. it will be enough to show that ∞∑ n=1 (αn) δk+k−1|tn,j| k < ∞ for j = 1, 2, 3, 4. to establish the main theorem by using the inequality given by minkowski. now we have m+1∑ n=2 (αn) δk+k−1|tn,1| k m+1∑ n=2 (αn) δk+k−1 | 1 qnqn−1 n−1∑ r=1 qn−rqn−1µrsr| k ≤ m+1∑ n=2 (αn) δk+k−1 1 qn ( n−1∑ r=1 qn−r|µr| k|sr| k )( 1 qn n−1∑ r=1 qn−r )k−1 (using holder′s inequality) = o(1) m∑ r=1 |µr| k|sr| k m+1∑ n=r+1 (αn) δk+k−1 ( qn−r qn ) = o(1) m∑ r=1 |µr| k|sr| k qr qr , by 3.4 = o(1) m∑ r=1 qr qr |sr| k|µr||µr| k−1 = o(1) m−1∑ r=1 ∆|µr| r∑ w=1 qw qw |sw| k + o(1)|µm| m∑ r=1 qr qr |sr| k = o(1) m−1∑ r=1 |∆µr|yr + o(1)|µm|ym , by 3.2 = o(1), as m →∞ (by the lemma and 2.3) next, m+1∑ n=2 (αn) δk+k−1 |tn,2| k = m+1∑ n=1 (αn) δk+k−1 | 1 qnqn−1 n−1∑ r=1 qn−r−1qnµrsr| k ≤ m+1∑ n=2 (αn) δk+k−1 1 qn−1 ( n−1∑ r=1 qn−r−1|µr| k|sr| k )( 1 qn−1 n−1∑ r=1 qn−r−1 )k−1 (5.1) int. j. anal. appl. 17 (2) (2019) 232 = o(1) m∑ r=1 |µr| k|sr| k m+1∑ n=r+1 (αn) δk+k−1 ( qn−r−1 qn−1 ) = o(1) m∑ r=1 |µr| k|sr| k qr qr = o(1), as m →∞, as in proof of the 1st part. further, m+1∑ n=2 (αn) δk+k−1 |tn,3| k = m+1∑ n=1 (αn) δk+k−1 | 1 qnqn−1 n−1∑ r=1 qn−r−1qn−1∆µrsr| k ≤ m+1∑ n=2 (αn) δk+k−1 1 qn ( n−1∑ r=1 qn−r−1|∆µr||sr| k )( 1 qn n−1∑ r=1 qn−r−1|∆µr| )k−1 since, ( 1 qn n−1∑ r=1 qn−r−1|∆µr| ) ≤ n−1∑ r=1 |∆µn| ≤ n|∆µr| ≤ nβn therefore, m+1∑ n=2 (αn) δk+k−1 |tn,3| k ≤ o(1) m∑ r=1 (rβr) k−1|∆µr||sr| k m+1∑ n=r+1 (αn) δk+k−1 qn−r−1 qn = o(1) m∑ r=1 |∆µr||sr| k qr qr ≤ o(1) m∑ r=1 βr|sr| k qr qr = o(1) m−1∑ r=1 ∆ (βr) r∑ w=1 qw qw |sw| k + o(1)(βm) m∑ r=1 qr qr |sr| k = o(1) m−1∑ r=1 |∆βr|yr + o(1)(βm)ym = o(1) as m →∞ now, m+1∑ n=2 (αn) δk+k−1 |tn,4| k = m+1∑ n=2 (αn) δk+k−1| 1 qnqn−1 n−1∑ r=1 qn−r−2qn∆µrsr| k (5.2) int. j. anal. appl. 17 (2) (2019) 233 ≤ m+1∑ n=2 (αn) δk+k−1 1 qn−1 ( n−1∑ r=1 qn−r−2|∆µr||sr| k ) 1 qn−1 n−1∑ r=1 qn−r−2|∆µr| k−1 = o(1) m∑ r=1 (rβr) k−1|∆µr||sr| k m+1∑ n=r+1 (αn) δk+k−1 ( qn−r−1 qn ) , (as above) = o(1) m∑ r=1 |∆µr||sr| k qr qr = o(1) as m →∞. (as above) this completes the proof of the theorem. 6. conclusion if (yn) is a positive non-decreasing sequence and there be sequences {βn} and {µn} such that the conditions 2.1 to 2.5 along with the conditions 4.1 and 4.2 are satisfied then the series ∑∞ n=1 anµn is summable |n,qn,αn; δ|k,k ≥ 1,δ ≥ 0, under the conditions 3.1 to 3.4.thus, our result generalizes the result of mishra and srivastava [13] and padhy et. al [14]. references [1] bor, h., a note on two summability methods, proc. amer. math. soc., 98(1986), 81-84. [2] bor, h., a note on absolute summability factors, int. j. math. and math. sci., 17(3), (1994), 479-482. [3] bor.h, on absolute summability factors of infinite series, rocky mt. j. math., 23(4)(1993), 1221-1230. [4] daniel, e.c., on absolute summability factors of infinite series, proc. japan acad., 40(2)(1964), 65-69.. [5] das, g, on absolute norlund summability factors of infinite series, j. lond. math. soc., 41(1966), 685-692. [6] f. fekete, m., on absolute summability of infinite series, proc. edinburgh math. soc., 3(2)(1932), 132-134. [7] flett, t.m., on an extension of absolute summability and some theorems of little wood and palay, proc. lond. math. soc., 7(1957), 113-141. [8] hardy, g.h., divergent series, oxford university press, (1949). [9] hsiang,,f.c., on absolute norlund summability of a fourier series, j. austral. math. soc., 7(1967), 252-256. [10] kishore, n., on the absolute norlund summability factors, riv. mat. univ. parma, 6(1965), 129-134. [11] mazhar,s.m., on absolute summability factors of infinite series, tohoku math. j., 23(3)(2008), 433-451. [12] misra, u.k., misra, m. and padhy, b.p., on the local property of indexed norlund summability of a factored fourier series, int. j. res. rev. app. sci., 5(1)(2010), 52-58. [13] mishra, k.n and srivastava, r.s.l., on absolute cesaro summability factors of infinite series, portugaliae math., 42(1)(1983-84), 53-61. [14] padhy, b.p., majhi, b., samanta, p., misra, m. and misra, u.k., a note on the absolute indexed norlund summability, new trends math. sci., 6(4), (2018), 54-59. [15] ram, s., on the absolute norlund summability factors of infinite series, indian j. pure appl. math., 2(1971), 275-282. [16] sulaiman, w.t., on some absolute summability factors of infinite series, gen. math. notes, 2(2)(2011), 7-13. [17] verma, r.s., on the absolute norlund summability factors, riv. math. univ. parma, 3(1977), 27-33. [18] whittaker, j.m., the absolute summability of fourier series, proc. edinburgh math. soc., 2(2)(1930), 1-5. 1. introduction 2. known theorems 3. main theorem 4. lemma cite5 5. proof of the main theorem 6. conclusion references int. j. anal. appl. (2022), 20:55 bipolar fuzzy magnified translations in groups u. venkata kalyani1,∗, t. eswarlal1, k. v. narasimha rao2, a. iampan3 1department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, vaddeswaram, ap, india 2department of mechanical engineering, koneru lakshmaiah education foundation, vaddeswaram, guntur, ap, india 3department of mathematics, school of science, university of phayao, maeka, mueang, phayao 56000, thailand ∗corresponding author: u.v.kalyani@gmail.com abstract. in this paper, we define a bipolar fuzzy magnified translation (bfmt) of a bipolar fuzzy subgroup (bfsg) of a group. based on this concept we have also developed some important results and theorems on bipolar fuzzy groups. 1. introduction the notion of a fuzzy set (fs) was introduced in 1965 by zadeh [16]. the fs theory has various expansions, such as intuitionistic fuzzy sets (ifs), interval-valued fuzzy sets (ivfs), vague sets (vs), and so on. with, the traditional fs representation it is not easy to explicitly express the difference of the irrelevant elements from the contrary elements. based on these observations, in 2000, lee [1] introduced a bipolar valued fuzzy set (bvfs), an extension of fss whose range of the membership degree (msd) is enlarged from [0,1] to [−1,1]. kalyani and eswarlal [6–10] have introduced and studied the bipolar vague cosets, normal groups, bipolar fuzzy sublattices,ideals and also gave the application of topsis and electre1 method on bipolar vague sets. the idea of fuzzy magnified translation (fmt) has been coined by majumder and sardar [15] in 2008. jun [14] explored the bipolar fuzzy translations (bft) in bck/bci-algebras in 2009. kumar [13] popularized bipolar valued fuzzy translations (bvft) in semigroups in 2012. the notion of an intuitionistic fuzzy magnified received: aug. 28, 2022. 2010 mathematics subject classification. 06d50, 06d72, 03e72. key words and phrases. bipolar fuzzy set; bipolar fuzzy sublattice; bipolar fuzzy ideal; homomorphism. https://doi.org/10.28924/2291-8639-20-2022-55 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-55 2 int. j. anal. appl. (2022), 20:55 translation (imft) in groups has been discussed by sarma [12] in 2012. the notion of translation (t), multiplication (m), and extension (e) applied in distinct aspects on different structures in algebra. in 2019, iampan [4] studied translation (t) and density (d) of a bipolar valued fuzzy set (bfs) in up-algebras. anggraenil [5] has given bft(bipolar fuzzy translation), bfe(bipolar fuzzy extension), and bfm(bipolar fuzzy multiplication) on bipolar anti-fuzzy ideals of k-algebras in 2019. in 2021, alshehri [2] studied fuzzy translation (ft) and multiplication (fm) in brk algebras, khamrot [3] studied on a right weakly regular semigroup (rwrsg) of generalized a bipolar fuzzy subsemigroup (bfssg). here in this paper, we introduce a bipolar fuzzy magnified translation in groups and obtained some interesting results. throughout the paper fs stands for a fuzzy set, fg stands for a fuzzy subgroup, bfs stands for a bipolar fuzzy subset, bfsg stands for a bipolar fuzzy subgroup, bfnsg stands for a bipolar fuzzy normal subgroup, bfmt stands for a bipolar fuzzy magnified translation, g is always a group and d is universe of discourse. 2. preliminaries here, we will review a few standard definitions that are relevant to this work. definition 2.1. [16] a mapping δ :z→ [0,1] is represented to as a fuzzy subset (fs) of a nonempty set z. definition 2.2. [1] a bfs b in d is an object having the form b= {< t ,bn(t ),bp(t ) >: t ∈d}, where bp : d → [0,1] and bn : d → [−1,0]. the positive membership degree (+ve msd) bp(t ) denotes the satisfaction degree of an element t to the property corresponding to b and the negative membership degree (-ve msd) bn(t ) denotes the satisfaction degree of t to some implicit counter property of b. for the sake of simplicity, we shall use the symbol b =< bn,bp > for the bfs b= {< t ,bn(t ),bp(t ) >: t ∈d}. definition 2.3. [11] a bfs b of g is said to be a bfsg of g if the following conditions are satisfied: (i) bp(ξη)≥min{bp(ξ),bp(η)}, (ii) bp(ξ−1)≥bp(ξ), (iii) bn(ξη)≤max{bn(ξ),bn(η)}, and (iv) bn(ξ−1)≤bn(ξ) for all ξ,η in g. definition 2.4. [11] a bfsg b of g is said to be a bfnsg of g if the following conditions are satisfied: (i) bp(ξη)=bp(ηξ) and (ii) bn(ξη)=bn(ηξ) for all ξ,η in g. definition 2.5. [11] let η be a mapping from a group k to a group k′ and let a=(k;an,ap) bfs in k and b=(k′;bn,bp) bfs in η(g)=k′ defined by bp(h)= sup{ap(g)} and bn(h)= inf{an(g)}, int. j. anal. appl. (2022), 20:55 3 where g ∈ η−1(h) for all g ∈k and h ∈k′. a is called the preimage of b under η and is denoted by η−1(b) and is defined by for g ∈k, (η−1(bp(g)))=bp(η(g)) and (η−1(bn(g)))=bn(η(g)). remark 2.1. [4] for any bfs b=< bn,bp > in d, we denote 5=−1− inf{bn(t ) : t ∈d} and 4 = 1− sup{bp(t ) : t ∈ d}. let b =< bn,bp > be a bfs in d and (θ,ϑ) ∈ [5,0]× [0,4]. by a bipolar fuzzy (θ,ϑ)-translation of b=< bn,bp >, we mean a bfs bt (θ,ϑ) =< bn (θ,t) ,bp (ϑ,t) >, where bn (θ,t) : d → [−1,0] defined by bn (θ,t) (t ) = bn(t )+ θ and bp (ϑ,t) : d → [0,1] defined by bp (ϑ,t) (t )=bp(t )+ϑ for all t ∈d. 3. bipolar fuzzy magnified translations in groups definition 3.1. let b =< bn,bp > be a bfs in d and (α,β) ∈ [0,1], (θ,ϑ) ∈ [5,0]× [0,4]. by a bfmt of b =< bn,bp >, we mean a bfs m = {< r,bn (α,θ) (r),bp (β,ϑ) (r) >: r ∈ d} or simply as m = {< r,bnm(r),b p m(r) >: r ∈ d}, where b n m = b n (α,θ) : d → [−1,0] and bpm(r) = b p (β,ϑ) : d → [0,1] defined by bnm(r)=b n (α,θ) (r)= αbn(r)+θ and bpm(r)=b p (β,ϑ) (r)= βbp(r)+ϑ for all r ∈d. example 3.1. let d= {1,ω,ω2} and let b= {< 1,−0.2,0.3 >,< ω,−0.3,0.4 >,< ω2,−0.1,0.5 > }. then θ ∈ [−0.9,0] and ϑ ∈ [0,0.5]. let α = 0.1,β = 0.2,θ = −0.8,ϑ = 0.2. hence, a bfmt m = {< 1,−0.8,0.26 >, < ω,−0.83,0.36 >,< ω2,−0.81,0.3 >}. theorem 3.1. let m be a bfmt of a bfsg b of g. then (i) bpt(r −1)=bpt(r) and b n t(r −1)=bnt(r), (ii) bpt(r)≤b p t(e) and b n t(r)≥b n t(e) for all r,e ∈ g. proof. (i) bpt(r −1) = bp (α,θ) (r−1) = αbp(r−1)+ θ = αbp(r)+ θ = bp (α,θ) (r). similarly, bnt(r −1) = bnt(r). (ii) bpt(e)=b p (α,θ) (e)= αbp(e)+θ ≥ αbp(r)+θ =bp (α,θ) (r). similarly, bnt(r)≥b n t(e). � theorem 3.2. let m be a bfmt of a bfsg b of g. then (i)bpt(ry −1)=bpt(e)⇒b p t(r)=b p t(y), (ii) bnt(ry −1)=bnt(e)⇒b n t(r)=b n t(y) for all r,y,e in g. proof. (i) bpt(r)=b p (α,θ)(r) = αbp(r)+θ = αbp(ry−1y)+θ ≥ αbp(r)+θ ≥ α{min{bp(ry−1),bp(y)}}+θ ≥min{α(bp(ry−1)+θ),α(bp(y)+θ)} 4 int. j. anal. appl. (2022), 20:55 =min{bp(α,θ)(ry −1),bp(α,θ)(y)} =min{bp(α,θ)(e),b p (α,θ)(y)} =bp(α,θ)(y). similarly, we can prove (ii). � theorem 3.3. let m be a bfmt of a bfsg b of g. then m is a bfsg of g. proof. let x,y ∈ g.then we have bpt(ry −1)=bp(β,ϑ)(ry −1) = βbp(ry−1)+ϑ ≥ βmin{bp(r),bp(y−1}+ϑ = βmin{bp(r),bp(y)}+ϑ =min{(β(bp(r)+ϑ)),(β(bp(y)+ϑ))} =min{bp(β,ϑ)(r),b p (β,ϑ)(y)} =min{bpt(r),b p t(y)}. theref ore,bpt(ry −1)≥min{bpt(r),b p t(y)}. similarly, bnt(ry −1)≤max{bpt(r), b p t(y)}. hence, m is a bfsg of g. � theorem 3.4. let m be a bfmt of a bfsg b of g. then h = {r ∈ g : bpt(r) = b p t(e) and bnt(r)=b n t(e)} is a subgroup of g. proof. let r,y ∈ h. by theorem 3.1, we have bpt(r −1)=bpt(r)=b p t(e). similarly, bnt(r −1)=bnt(r)=b n t(e). thus bpt(r −1)=bpt(e) and b n t(r −1)=bnt(e). so r −1 ∈ h. now, bpt(ry)=b p (β,ϑ)(ry) ≥min{bp(β,ϑ)(r),b p (β,ϑ)(y)} =min{bp(β,ϑ)(e),b p (β,ϑ)(e)} int. j. anal. appl. (2022), 20:55 5 =bp(β,ϑ)(e)=b p t(e). likewise bnt(ry)≤b n t(e). now, bpt(e)=b p t((ry)(ry) −1) geqmin{bpt(ry),b p t(ry)} =bpt(ry). similarly, bpt(e)≤b p t(ry). so ry ∈ h. hence, h is a subgroup of g. � the proof of the following two theorems is similar to the proof of theorem 3.4. theorem 3.5. let m be a bfmt of a bfsg b of g. then h = {< r,bpt(r) >:b p t(r)=b p t(e)} is a fuzzy subgroup (fsg) of g. theorem 3.6. let m be a bfmt of a bfsg b of g. then h = {< r,bnt(r) >:b n t(r)=b n t(e)} is an anti-fuzzy subgroup (afsg) of g. theorem 3.7. let g and g1 be any two groups. then the homomorphic image of a bfmt m of a bfsg b of g is a bfsg of g1. proof. let κ : g → g1 be a homomorphism. let v=κ(m),where m is a bfmt of a bfsg b of g. we shall show that v is a bfsg of g1. now, for κ(r) and κ(y) in g1, we have vp(κ(r)κ(y)−1)=vp(κ(r)κ(y−1)) =vp(κ(ry−1) ≥bpt(ry −1) = βbp(ry−1)+α ≥ βmin{bp(r),bp(y−1}+α = βmin{bp(r),bp(y}+α =min{(βbp(r)+α),(βbp(y)+α} ≥min{vp(κ(r)),vp(κ(y))}. 6 int. j. anal. appl. (2022), 20:55 thus vp(κ(r)κ(y)−1)≥min{vp(κ(r)),vp(κ(y))}. now, vn(κ(r)κ(y)−1)=vn(κ(r)κ(y−1)) =vn(κ(ry−1) ≥bnt(ry −1) = βbn(ry−1)+α ≤ βmax{bn(r),bn(y−1}+α = βmax{bn(r),bn(y}+α =max{(βbn(r)+α),(βbn(y)+α} ≤min{vn(κ(r)),vn(κ(y))}. thus vn(κ(r) κ(y)−1)≤max{vn(κ(r)),vn(κ(y))}. hence, v is a bfsg of g1. � theorem 3.8. let g and g1 be any two groups. then the homomorphic pre-image of a bfmt of a bfsg b of g1 is a bfsg of g. proof. let g and g1 be any two groups. let m =κ(b),where m is a bfmt of a bfsg b of g1. we shall show that b is a bfsg of g. now, for κ(r) and κ(y) in g, we have bp(ry−1) = bpt(κ(ry −1)) = bpt(κ(r)κ(y −1))) = bpt(κ(r)(κ(y)) −1) = βbp(κ(r)(κ(y))−1)+α ≥ βmin{bp(κ(r)),bp(κ(y))}+α = min{βbp(κ(r))+α,βbp(κ(y))+α} = min{bpt(κ(r)),b p t(κ(y))} = min{bp(κ(r)),bp(κ(y))} thus bp(ry−1)≥ min{bp(κ(r)),bp(κ(y))}. int. j. anal. appl. (2022), 20:55 7 now, bn(ry−1) = bnt (κ(ry −1)) = bnt (κ(r)κ(y −1))) = bnt (κ(r)(κ(y)) −1) = βbn(κ(r)(κ(y))−1)+α ≤ βmax{bn(κ(r)),bn(κ(y))}+α = max{βbn(κ(r))+α,βbn(κ(y))+α} = max{bnt (κ(r)),b n t (κ(y))} = max{bn(κ(r)),bn(κ(y))} thus bn(ry−1)≤ max{bn(κ(r)),bn(κ(y))}. thus b is a bfsg of g. thus the homomorphic pre-image of a bfmt of a bfsg b of g1 is a bfsg of g. � the proof of the following two theorems is similar to the proof of theorem 3.7, 3.8. theorem 3.9. let g and g1 be any two groups. then the homomorphic image of a bfmt m of a bfnsg b of g is a bfsng of g1. theorem 3.10. let g and g1 be any two groups. then the homomorphic pre-image of a bfmt of a bfnsg b of g1 is a bfnsg of g. acknowledgment: the authors wish to thank prof. k.l.n. swamy for his valuable suggestions. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] k.m. lee, bipolar-valued fuzzy sets and their operations, in: proc. int. conf. on intelligent technologies, bangkok, thailand, (2000), 307–312. https://cir.nii.ac.jp/crid/1573105975701596416. [2] h.a. alshehri, fuzzy translation and fuzzy multiplication in brk-algebras, eur. j. pure appl. math. 14 (2021), 737–745. https://doi.org/10.29020/nybg.ejpam.v14i3.3971. [3] p. khamrot, n. deetae, on right weakly regular semigroups of generalized bipolar fuzzy subsemigroups, missouri j. math. sci. 33 (2021), 195-205. https://doi.org/10.35834/2021/3302195. [4] n. udten, n. songseang, a. iampan, translation and density of a bipolar-valued fuzzy set in up-algebras, ital. j. pure appl. math. 41 (2019), 469-496. [5] r. anggraenil, bipolar fuzzy translation, extention, and multiplication on bipolar anti fuzzy ideals of k-algebras, amer. j. eng. res. 8 (2019), 69-78. [6] u.v. kalyani, t. eswarlal, homomorphism on bipolar vague normal groups, adv. math., sci. j. 9 (2020), 3315–3324. https://doi.org/10.37418/amsj.9.6.11. https://cir.nii.ac.jp/crid/1573105975701596416 https://doi.org/10.29020/nybg.ejpam.v14i3.3971 https://doi.org/10.35834/2021/3302195 https://doi.org/10.37418/amsj.9.6.11 8 int. j. anal. appl. (2022), 20:55 [7] u.v. kalyani, t. eswarlal, bipolar vague cosets, adv. math., sci. j. 9 (2020), 6777–6787. https://doi.org/10. 37418/amsj.9.9.36. [8] u.v. kalyani, t. eswarlal, j. kavikumar, a. iampan, bipolar fuzzy sublattices and ideals, int. j. anal. appl. 20 (2022), 45. https://doi.org/10.28924/2291-8639-20-2022-45. [9] u.v. kalyani, t. eswarlal, y. bhargavi, application of bipolar vague sets on mcdm problems, mater. today: proc. (2020). https://doi.org/10.1016/j.matpr.2020.10.088. [10] u.v. kalyani, t. eswarlal, bipolar vague electre 1 method for mcdm problems, aip conf. proc. 2375 (2021), 020024. https://doi.org/10.1063/5.0066365. [11] m.s. anitha, k.l. muruganantha prasad, k. arjunan, notes on bipolar-valued fuzzy subgroups of a group, bull. soc. math. services standards. 7 (2013), 40–45. https://doi.org/10.18052/www.scipress.com/bsmass.7.40. [12] p.k. sarma, on intuitionistic fuzzy magnified translation in groups, int. j. math. sci. appl. 2 (2012), 139-146. [13] s.k. sardar, s.k. majumder, p. pal, bipolar valued fuzzy translationin semigroups, math. aeterna, 2 (2012), 597-607. [14] y.b. jun, h.s. kim, k.j. lee, bipolar fuzzy translations in bck/bci-algebras, j. chungcheong math. soc. 22 (2009), 399-408. [15] s.k. majumder, s.k. sardar, fuzzy magnified translation on groups, j. math., north bengal univ. 1 (2008), 117-124. [16] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.37418/amsj.9.9.36 https://doi.org/10.37418/amsj.9.9.36 https://doi.org/10.28924/2291-8639-20-2022-45 https://doi.org/10.1016/j.matpr.2020.10.088 https://doi.org/10.1063/5.0066365 https://doi.org/10.18052/www.scipress.com/bsmass.7.40 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. bipolar fuzzy magnified translations in groups references international journal of analysis and applications volume 17, number 3 (2019), 388-395 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-388 some properties of geodesic strongly e-b-vex functions wedad saleh∗ department of mathematics, taibah university, almedina, saudi arabia ∗corresponding author: wed 10 777@hotmail.com abstract. geodesic e-b-vex sets and geodesic e-b-vex functions on a riemannian manifold are extended to the so called geodesic strongly e-b-vex sets and geodesic strongly e-b-vex functions. some basic properties of geodesic strongly e-b-vex sets are also studied. 1. introduction convexity and its generalizations play an important role in optimization theory, convex anlysis and minkowski space [3, 4, 6, 9, 10]. youness [17] defined e-convex sets and e-convex functions by relaxing the definitions of convex sets and convex functions, which have some important applications in various branches of mathematical sciences [1, 12, 13]. also, youness [18] extended the definitions of e-convex sets and e-convex functions to strongly e-convex sets and strongly e-convex functions. the b-vex functions which shares many properties with convex functions was introduced by bector and singh [2]. some reserchers studied some new generalizations of convex functions by relaxing definitions of e-convex functions and b-vex functions such as e-b-vex functions [15] and strongly e-b-vex functions [19]. also, generalization of convexity on riemannian manifolds were presented in ( [5], [8] , [14], [16]). received 2019-03-08; accepted 2019-04-05; published 2019-05-01. 2010 mathematics subject classification. 52a20, 52a41, 53c20, 53c22. key words and phrases. geodesic e-convex sets; geodesic e-convex functions; riemannian manifolds. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 388 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-388 int. j. anal. appl. 17 (3) (2019) 389 in this paper, a new class of sets on riemannian manifolds, called geodesic strongly e-b-vex sets, and a new class of functions defined on them, called geodesic strongly e-convex functions, have been proposed. also, some of their properties have been discussed. this paper divides into three sections. in section 2, some of definities and properties which will be used throughout this work are presented that can be found in many books on differential geometry such as [16] . in section 3, a geodedic strongly e-b-vex set and geodesic strongly e-b-vex function are studied with some of their properties. 2. preliminaries now, let ℵ is a c∞ n-dimensional riemannian manifold, also µ1,µ2 ∈ℵ and δ : [0, 1] −→ℵ be a geodesic joining the points µ1 and µ2 , which means that δµ1,µ2 (0) = µ2 and δµ1,µ2 (1) = µ1. strongly e-convex sets (sec) and strongly e-convex (sec) functions were introduced in [18] such as: definition 2.1. (1) a subset ω ⊆ rn is strongly e-convex (sec) set if there is a map ε: rn −→ rn such that δ(αµ1 + ε(µ1)) + (1 − δ)(αµ2 + ε(µ2)) ∈ b for each µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1]. (2) a function g : ω ⊆ rn −→ r is strongly e-convex (sec) function on ω if there is a map ε: rn −→ rn such that ω is a sec set and g(δ(αµ1 + ε(µ1)) + (1 −δ)(αµ2 + ε(µ2))) ≤ δg(ε(µ1)) + (1 − δ)g(ε(µ2)), ∀µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1]. definition 2.2. [5] (1) considering ε: ℵ −→ ℵ is a map. a subset ω ⊂ ℵ is geodesic e-convex iff there exists a unique geodesic ηε(µ1),ε(µ2)(δ) of length d(µ1,µ2), which belongs to ω, ∀µ1,µ2 ∈ ω and δ ∈ [0, 1]. (2) a functin g : ω ⊆ℵ−→ r where ω is a gec set in ℵ is geodesic e-convex if g(ηε(µ1),ε(µ2)(δ)) ≤ δg(ε(µ1)) + (1 − δ)g(ε(µ2)), ∀µ1,µ2 ∈ ω and δ ∈ [0, 1]. definition 2.3. [7] (1) considering ε: ℵ −→ ℵ is a map. a subset ω ⊂ ℵ is geodesic strongly e-convex(gsec) iff there exists a unique geodesic ηαµ1+ε(µ1),αµ2+ε(µ2)(δ) of length d(µ1,µ2), which belongs to ω, ∀µ1,µ2 ∈ ω, α ∈ [0, 1] and δ ∈ [0, 1] and . int. j. anal. appl. 17 (3) (2019) 390 (2) a functin g : ω ⊆ ℵ −→ r, where ω is a gsec set in ℵ, is geodesic strongly e-convex (gsec) funtion if g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δ)) ≤ δg(ε(µ1)) + (1 −δ)g(ε(µ2)), ∀µ1,µ2 ∈ ω and δ ∈ [0, 1]. 3. geodesic strongly e-b-vex sets and geodesic strongly e-b-vex functions in this part of work, a geodesic strongly e-b-vex (gse-b-vex) set and a geodesic strongly e-b-convex (gse-b-vex) function in a riemannian manifold ℵ are given and some of their properties are discussed. definition 3.1. a subset ω of ℵ is called a geodesic strongly e-b-vex (gse-b-vex) iff there exists a unique geodesic ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) of length d(µ1,µ2), which belongs to ω, ∀µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1]. remark 3.1. (1) every gse-b-vex set is a gsec set when b(µ1,µ2,δ) = 1. (2) every gse-b-vex set is a ge-b-vex set when α = 0. (3) when ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) = δb (αµ1 + ε(µ1)) + (1 −δb) (αµ2 + ε(µ2)) , then we have strongly e-b-vex set. now, some propertie of gse-b-vex sets are propoed. proposition 3.1. every convex set ω ⊂ℵ is a gse-b-vex set. the proof of the above proposition is direct that by taking ε: ℵ−→ℵ as the identity map,b(µ1,µ2,δ) = 1 and α = 0. proposition 3.2. let ω ⊂ℵ be a gse-b-vex set, then ε(ω) ⊆ ω. proof. since ω is a gse-b-vex set, then ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈ ω, µ1,µ2 ∈ ω, α ∈ [0, 1] and δ ∈ [0, 1]. let δb = 1 and α = 0, then ηε(µ1),ε(µ2) = ε(µ2) ∈ ω, then ε(ω) ⊆ ω. � theorem 3.1. suppose that a set {ωj}j=1,2,··· ,n is an arbitrary collection of gse-v-vex subsets of ℵ, then ∩j=1,2,··· ,nωi is a gse-b-vex set. proof. considering {ωj}j=1,2,··· ,n is a collection of gse-b-vex subsets of ω. if ∩j=1,2,··· ,nωj is an empty set, then the result is obvious. assume that µ1,µ2 ∈∩j=1,2,··· ,nωj, then µ1,µ2 ∈ ωj. hence, ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈ ωj,∀α ∈ [0, 1] and δ ∈ [0, 1]. hence, ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈∩j=1,2,··· ,nωj,∀α ∈ [0, 1] and δ ∈ [0, 1]. � int. j. anal. appl. 17 (3) (2019) 391 remark 3.2. however, the above theorem is not true for the union of gse-b-vex sets. now, we introduce the definition of a geodesic e-b-vex (gse-b-vex) function on ℵ. definition 3.2. assume that ω ⊂ℵ is a gse-b-vex set . a function g : ω −→ r is called a geodesic strongly e-b-vex (gse-b-vex) if g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ γg(ε(µ1)) + (1 −γ)g(ε(µ2)), (3.1) ∀µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1]. if the inequality (3.1) is strict, then g is called a strictly gse-b-vex function. example 3.1. assume that g : r −→ r such that g(µ) = −|µ|. aslo, assume that ε: r −→ r is defined as ε(µ) = αµ where 0 < α ≤ 1,∀µ ∈ r and the geodesic η is given as ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) =   1 2α [αµ2 + ε(µ2) + δb(αµ1 + ε(µ1) −αµ2 −ε(µ2))] ; µ1µ2 ≥ 0, 1 2α [αµ2 + ε(µ2) + δb(αµ2 + ε(µ2) −αµ1 −ε(µ1))] ; µ1µ2 < 0 =   µ2 + δb(µ1 −µ2) ; µ1µ2 ≥ 0, µ2 + δb(µ2 −µ1) ; µ1µ2 < 0, then g is gse-b-vex function. proposition 3.3. let g : ω −→ r be a gse-b-vex function on a gse-b-vex set ω ×ℵ, then g(αµ + ε(µ)) ≤ g(ε(µ)), µ ∈ ω and α ∈ [0, 1]. proof. since g is gse-b-vex function on gse-b-vex set ω, then g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg(ε(µ1)) + (1 −δb)g(ε(µ2)), then for δb = 1, we have g(αµ1 + ε(µ1)) ≤ g(ε(µ1)). � theorem 3.2. if g1 : ω −→ r is a gse-b-vex function on a gse-b-vex set ω ⊂ ℵ and g2 : u −→ r is a non-decreasing convex function such that rang(g1) ⊂ u, then the composite function g2og1 is gse-b-vex function on ω. int. j. anal. appl. 17 (3) (2019) 392 proof. by using the hypothesis, we can write all x1,x2 ∈ b,α ∈ [0, 1] and γ ∈ [0, 1], g1(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg1(ε(µ1)) + (1 −δb)g1(ε(µ2)), ∀µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1] and since g2 is a non-decreasing convex function, then we get g2og1(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) = g2 ( g2(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ) ≤ g2 (δbg1(ε(µ1)) + (1 −δb)g1(ε(µ2))) ≤ δbg2 (g1(ε(µ1))) + (1 − δb)g2 (g1(ε(µ2))) = δb(g2og1)(ε(µ1)) + (1 −δb)(g2og1)(ε(µ2)) hence, g2og1 is gse-b-vex on ω. moreover, g2og1 is a strictly gse-b-vex function if g2 is a strictly nondecreasing convex function. � theorem 3.3. considering gi : ω −→ r, i = 1, 2, ...,n are gse-b-vex functions. then, the function g = n∑ i=1 ξigi is also gse-b-vex geodesic on ω, ∀ξi ∈ r,ξi ≥ 0. proof. since gi, i = 1, 2, ...,n are gse-b-vex functions, then gi(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbgi(ε(µ1)) + (1 − δb)gi(ε(µ2)), ∀µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1], hence, ξigi(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbξigi(ε(µ1)) + (1 − δb)ξigi(ε(µ2)). this implies to, g(ηαµ1+ε(x1),αµ2+ε(µ2)(δb)) = n∑ i=1 ξigi(ηαµ1+ε(x1),αµ2+ε(µ2)(δb)) ≤ δb n∑ i=1 ξigi(ε(µ1)) + (1 − δb) n∑ i=1 ξigi(ε(µ2)) = δbg(ε(µ1)) + (1 − δb)g(ε(µ2)). then g is gse-b-vex function. � next, we show that a funcion is gse-b-vex iff its epigraph is a gse-b-vex set. definition 3.3. assume that ω ⊂ ℵ× r,e : ℵ −→ ℵ, b : ω × ω × [0, 1] −→ r+ and f : r −→ r. a set ω is called a geodesic strongly e ×f -convex ( gse ×f -b-vex ) if ( ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbf(ξ1) + (1 −δb)f(ξ2) ) ∈ ω, ∀(µ1,ξ1), (µ2,ξ2) ∈ ω, α ∈ [0, 1] and γ ∈ [0, 1]. int. j. anal. appl. 17 (3) (2019) 393 remark 3.3. from definition 3.3, we have found ω ⊆ℵ is a gse-b-vex set iff ω×r is a gse×f -b-vex set. now , the epigraph of a function g : ω ⊂ℵ−→ r is given as e(g) = {(µ,a) : µ ∈ ω,a ∈ r,g(µ) ≤ a} . (3.2) theorem 3.4. suppose that ω ⊆ℵ is a gse-b-vex set, g : ω −→ r is a function and f : r −→ r is a map such that f(g(µ)+a) = g(ε(µ))+a, ∀a ∈ r, a > 0. then, g is a gse-b-vex on ω iff e(g) is a gse×f -b-vex on ω ×r. proof. let (µ1,a1), (µ2,a2) ∈ e(g). since ω is gse-b-vex, then ηαµ1+ε(µ1),αµ2+ε(µ2)(δb) ∈ ω, ∀α ∈ [0, 1] and δ ∈ [0, 1]. when α = 0 and δb = 1, we have ε(µ1) ∈ ω also, when α = 0 and δb = 0 we get ε(µ2) ∈ ω. assume that f(a1) and f(a2) where g(ε(µ1)) ≤ f(a1) and g(ε(µ2)) ≤ f(a2). then (ε(µ1),f(a1)), (ε(µ2),f(a2)) ∈ e(g). considering g is a gse-b-vex on ω, then g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbg(ε(µ1)) + (1 −δb)g(ε(µ2)) ≤ δbf(a1) + (1 − δb)f(a2). this is leading to, ( ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbf(a1) + (1 − δb)f(a2) ) ∈ e(g), which means that e(g) is gse × é-b-vex on ω ×r. conversely, let us take e(g) is gse × é-b-vex on ω × r. assume that µ1,µ2 ∈ ω,α ∈ [0, 1] and δ ∈ [0, 1], then (µ1,g(µ1)) ∈ e(g) and (µ2,g(µ2)) ∈ e(g). in addition, ( ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbf(g(µ1)) + (1 − δb)f(g(µ2)) ) ∈ e(g) =⇒ g(ηαµ1+ε(µ1),αµ2+ε(µ2)(δb)) ≤ δbf(g(µ1)) + (1 −δb)f(g(µ2)) = δbg(ε(µ1)) + (1 − δb)g(ε(µ2)). hence, the result. � theorem 3.5. let {ωj}j=1,··· ,n be a family of gse×f -b-vex sets. then ∩j=1,··· ,nωj is also gse×f -b-vex set. proof. let (µ1,a1), (µ2,a2) ∈∩j=1,··· ,nωj, then (µ1,a1), (µ2,a2) ∈ ωj, ∀j. =⇒ ( ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbf(a1) + (1 − δb)f(a2) ) ∈ ωj, int. j. anal. appl. 17 (3) (2019) 394 ∀α ∈ [0, 1] and δ ∈ [0, 1] . hence, ( ηαµ1+ε(µ1),αµ2+ε(µ2)(δb),δbf(a1) + (1 − δb)f(a2) ) ∈∩j=1,··· ,nωj, ∀α ∈ [0, 1] and δ ∈ [0, 1]. this shows that , ∩j=1,··· ,nωj is gse ×f-b-vex set. � theorem 3.6. suppose that g : r −→ r such that g(g(x) + µ) = g(ε(x)) + µ,∀µ ∈ r,µ > 0. let {gi}i∈i be a family of real valued functions that is defined on a gse-b-vex set ω and bounded from above. then, g(x) = supi∈igi(x),x ∈ ω is gse-b-vex on ω. proof. let gi, i ∈ i be a gse-b-vex function on ω, then e(gi) = {(x,µ) : x ∈ ω,µ ∈ r,gi(x) ≤ µ} are gse ×f-b-vex on ω ×r. hence, ∩i∈ie(gi) = {(x,µ) : x ∈ ω,µ ∈ r,gi(x) ≤ µ,i ∈ i} = {(x,µ) : x ∈ ω,µ ∈ r,g(x) ≤ µ} is gse ×f-b-vex set. then, by theorem 3.4 g is a gse-b-vex function. � references [1] i. a. abou-tair and w. t. sulaiman, inequalities via convex functions, int. j.math. sci. 22(1999), 543-546. [2] c. r. bector and c. singh, b-vex functions, j. optim. theory appl. 71(2) (1991), 237-253. [3] v. boltyanski, h. martini and p.s. soltan, excursions into combinatorial geometry, springer, berlin, 1997. [4] l. danzer, b. grünbaum and v. klee, helly’s theorem and its relatives. in: v. klee (ed.) convexity. proc. sympos. pure math., vol.7, pp.101-180. amer. math. soc., providence, 1963. [5] a. iqbal, s. ali and i. ahmad, on geodesic e-convex sets, geodesic e-convex functions and e-epigraphs, j. optim. theory appl. 55(1)(2012), 239-251. [6] m. a. jiménez, g. r. garzón and a. r. lizana, optimality conditions in vector optimization. bentham science publishers, 2010. [7] a. kiliçman and w. saleh, on geodesic strongly e-convex sets and geodesic strongly e-convex functions, j. inequal. appl. 2015 (2015), 297. [8] a. kiliçman and w. saleh, on properties of geodesic semilocal e-preinvex functions, j. inequal. appl. 2018 (2018), 353. [9] h.martini and k.j. swanepoel, generalized convexity notions and combinatorial geometry, gongr. numer. 164 (2003), 65-93. [10] h. martini and k.j. swanepoel, the geometry of minkowski spacesa survey, part ii. expo. math. 22 (2004), 14-93. [11] f. mirzapour, a. mirzapour and m. meghdadi, generalization of some important theorems to e-midconvex functions, appl. math. lett. 24(8) (2011),1384-1388. [12] m. a. noor, fuzzy preinvex functions, fuzzy sets syst. 64(1994), 95-104. [13] m. a. noor, k. i. noor and m. u. awan, generalized convexity and integral inequalities, appl. math. inf. sci. 9(1)(2015), 233-243. int. j. anal. appl. 17 (3) (2019) 395 [14] t. rapcsak, smooth nonlinear optimizatio in rn, kluwer academic, 1997. [15] y. r. syau, lixing jia, and e. stanley lee, generalizations of e-convex and b-vex functions, comput. math. appl. 58(4) (2009), 711-716. [16] c. udrist, convex funcions and optimization methods on riemannian manifolds, kluwer academic, 1994. [17] e. a. youness, on e-convex sets, e-convex functions and e-convex programming, j. optim.theory appl. 102 (1999), 439-450. [18] e. a. youness and tarek emam, strongly e-convex sets and strongly e-convex functions, j. interdisciplinary math. 8(1)(2005), 107-117. [19] g.y. wang, some properties of strongly e-b-vex functions, sustainable development-special track within scet 2012(2012), 247. 1. introduction 2. preliminaries 3. geodesic strongly e-b-vex sets and geodesic strongly e-b-vex functions references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 34-41 http://www.etamaths.com geometric characterizations of the differential shift plus alexander integral operator rabha w. ibrahim∗ abstract. in this effort, we deal with a new integral operator in the open unit disk. this operator is formulated by the complex alexander operator and its derivative. furthermore, we introduce a new subspace of the hardy space containing the normalized analytic functions. we shall prove that the new integral operator is closed in the subspace of normalized functions. geometric characterizations are established in the sequel based on the maximality of jack lemma. 1. introduction the study of operators concerns with the many intersecting classes of functions on function spaces, imposed by functional operators (integral and differential). these operators can be formulated by the kinds of the operators directly or the usage of some process. the importance in this direction is to study the boundedness and the compactness of these operators. these formulations brought two sets of operators: linear operators and nonlinear operators. the information itself delivers the topological or geometrical characterizations of the spaces of functions. the data for this class of operators shows a dynamic role in mathematics, computer science and physics. to establish an operator utilizing the functional theory, and then study its characterizations, is one of the important goals of present studies in the geometric function theory and its connected areas. the aim of the current work is to impose a new operator in the open unit disk based on the complex alexander operator. in [1] (for recent work [2]), alexander introduced a first order integral operator a[f](z) = ∫ z 0 f(ξ) ξ dξ, f ∈ a(u), where a(u) is the set of all normalized analytic functions in the open unit disk u := {z ∈ c : |z| < 1.} note that the alexander integral operator is the inverse of the alexander differential operator given by the formula da[f](z) = zf′(z), z ∈ u. based on this operator, we shall propose a modified integral operator in the open unit disk. this modification leads to define some new classes of analytic functions, specialized by the normalized class of analytic functions in the open unit disk. our study is realized by the geometric characterizations and boundedness of the new operator. 2. processing let u := {z : |z| < 1} be the open unit disk of the complex plane and h(u) be the space of holomorphic functions on the open unit disk. a holomorphic function f(z) = ∞∑ n=0 anz n, z ∈ u received 8th december, 2016; accepted 6th march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 30c45. key words and phrases. univalent function; unit disk; analytic function; subordination and superordination. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 34 geometric characterizations 35 on the open unit disk belongs to the hardy space h2(u), if its sequence of power series coefficients is square-summable: h2(u) = {f ∈ h(u) : ∞∑ n=0 |an|2 < ∞}. consequently, it can be defined a norm on h2(u) as follows [3] ‖f‖2h2(u) = ∞∑ n=0 |an|2. since l2(u) is banach space, then h2(u) is also a banach space on u. in the sequel, we consider a subset of analytic function, which are normalized as follows: f(0) = 0 and f′(0) = 1. thus, f is defined as follows: f(z) = z + ∞∑ n=2 anz n, z ∈ u. we denote this class by a(u). it is clear that a(u) ⊂ h(u) satisfying the above norm. the space h∞ is known as the vector space of bounded holomorphic functions on u, satisfying the norm ‖f‖h∞ = sup |z|<1 |f(z)|, f ∈ h(u). it is clear that h2(u) ⊂ h∞, f ∈ h(u). we proceed to introduce a new operator. define the following operator da : a(u) → a(u) as follows: da[f](z) := zf′(z), z ∈ u, f ∈ a(u). this operator is called the differentiation shift operator. now for f ∈ a(u), we define the differentiation shift plus complex alexander operator as follows: (a) [f](z) : = 1 2 ( zf′(z) + ∫ z 0 f(ξ) ξ dξ ) , z,ξ ∈ u = z + ∞∑ n=2 αnz n, f ∈ a(u). (2.1) obviously, (a) [f] ∈ a(u). moreover, since da[f] is a linear isometry operator and the complex alexander operator is contraction, then the operator (2.1) is bounded in the hardy space h2(u). let s2(u) be the space defined by s2(u) := { f ∈ h(u) : f′ ∈ h2(u) } end with the norm ‖f‖2s2(u) = ‖f‖ 2 h2(u) + ‖f ′‖2h2(u). this space is subspace from h∞, banach algebra, and every polynomial is dense in it ( see [3], proposition 1). a direct application, we have the following proposition proposition 2.1. let f ∈ s2(u), then (a) : s2(u) → s2(u). moreover, let s20 (u) := { f ∈ s2(u) : f(0) = 0 } , then s20 (u) ⊂ s 2(u). 36 ibrahim it has been shown that the range of the alexander operator is equal to s20 (u). in this effort, we define a subspace s21 (u) as follows: s21 (u) := { f ∈ s2(u) : f(0) = 0, f′(0) = 1 } . then we have the following relation: s21 (u) ⊂ s 2 0 (u) ⊂ s 2(u). proposition 2.2. let (a) ∈ h2(u). then rang(a) ⊂ s21 (u). proof. let g(z) ∈ rang(a), then there exists a normalized function f(z) ∈ h2(u) such that g(z) = (a) [f](z) = 1 2 ( zf′(z) + ∫ z 0 f(ξ) ξ dξ ) , z,ξ ∈ u. then we obtain g′ ∈ h2(u) with the properties g(0) = 0, g′(0) = 1. hence g ∈ s21 (u). proposition 2.3. let f,g ∈ a(u). then ‖f ∗g‖2s21(u) ≤‖f‖ 2 s21(u) ‖g‖2s21(u), z ∈ u, where ∗ is represented the convolution product (f ∗g)(z) = (z + ∞∑ n=2 anz n) ∗ (z + ∞∑ n=2 bnz n) = z + ∞∑ n=2 an bnz n. proof. it is clear that (f ∗g)(0) = 0 and (f ∗g)′(0) = 1; thus (f ∗g) ∈ s21 (u). moreover, in view of the young’s inequality for convolutions, we have ‖f ∗g‖2s21(u) = ‖f ∗g‖ 2 h2(u) + ‖(f ∗g) ′‖2h2(u) ≤ 2 ∞∑ n=0 |anbn|2 ≤ 2 ∞∑ n=0 |an|2|bn|2 ≤ ∞∑ n=0 |an|2 + ∞∑ n=0 |bn|2 = ‖f‖2s21(u)‖g‖ 2 s21(u) . proposition 2.4. let (a) [f] ∈ h2(u), f ∈ a(u). then rangl(f)(z) := rang ( 2 (a) [f]) ∗ f(z) 2 ) ⊂ s21 (u). proof. let l(f)(z) := (2 (a) [f]) ∗ ( f(z) 2 ) then there exists a normalized function f(z) ∈ h2(u) such that l(f)(z) = z + ∞∑ n=2 `nz n ∈ a(u), f ∈ a(u) since f ∈ h2(u) then l(f)′ ∈ h2(u) with the properties l(f)(0) = 0, l(f)′(0) = 1. geometric characterizations 37 the function f ∈ a(u) is called starlike of order α ∈ [0, 1) if and only if < (zf′(z) f(z) ) > α, z ∈ u; this class is denoted by s∗(α). and f ∈ a(u) is called convex of order α ∈ [0, 1) if and only if < ( 1 + zf′′(z) f′(z) ) > α, z ∈ u; this class is denoted by k(α). finally, the function f ∈ a(u) is called bounded turning of order α if and only if < ( f′(z) ) > α; this class is symbolled by b(α). note that f ∈ s∗ ⇔ a[f] ∈ k. we need the following result in the sequel (see [4]) lemma 2.1. let h(z) be analytic in u with h(0) = 0. then if |h(z)| approaches its maximality at a point z0 ∈ u when |z| = r, then z0h′(z0) = �h(z0), where � ≥ 1 is a real number. in addition, we request the subordination idea, which is formulated as follows: suppose that f(ζ) and g(ζ) are analytic in the open unit disk u. then f(ζ) is called subordinate to g(ζ) if for analytic function φ(ζ) in u achieving φ(0) = 0, |φ(ζ)| < 1, (ζ ∈ u) and f(ζ) = g(φ(ζ)). this subordination is symbolled by f(ζ) ≺ g(ζ), ζ ∈ u. 3. findings in this section, we introduce sufficient conditions to study the geometric properties of the operator (2.1). theorem 3.1. let f ∈ h2(u). then (a) [f] is bounded on s21 (u). proof. consider the integral operator (1) as follows: (a) [f] = 1 2 ( zf′(z) + ∫ z 0 f(ξ) ξ dξ ) , z,ξ ∈ u, we have ‖(a) [f]‖s21(u) =‖(a) [f]‖h2(u) + ‖(a) [f] ′‖h2(u) ≤‖f‖h2(u) + ‖f‖h2(u) + 2 max |z|<1 |f(z)| ≤ 4‖f‖h2(u). thus the operator (1) acts from h2(u) onto s21 (u); which is bounded. remark 3.1 in 1960, biernacki showed that f ∈ s ⇒ a[f] ∈ s, but this brings out to be wrong (see [5], theorem 8.11). this leads that the alexander integral operator a[f] does not cover the class s. theorem 3.2. consider the operator (2.1). if < (z(a) [f]′′(z) (a) [f]′(z) ) < 0, z ∈ u, f ∈ a(u), then (a) [f] ∈ s∗. 38 ibrahim proof. let µ be a real positive constant satisfying z(a) [f]′(z) (a) [f](z) = 1 + µω(z) 1 −µω(z) , ω(z) 6= 1 µ , µ > 0, where ω(z), z ∈ u is a function in the open unit disk. obviously ω(z) is analytic in u such that ω(0) = 0. we aim to show that |ω(z)| < 1 in u. differentiating both side logarithmically, we obtain 1 + z(a) [f]′′(z) (a) [f]′(z) = 2µzω′(z) 1 −µ2ω2(z) + 1 + µω(z) 1 −µω(z) . thus, by the assumption we have < ( 1 + z(a) [f]′′(z) (a) [f]′(z) ) = < ( 2µzω′(z) 1 −µ2ω2(z) + 1 + µω(z) 1 −µω(z) ) < 1, z ∈ u, f ∈ a(u). if there exists a point z0 ∈ u such that max |z|≤|z0| |ω(z)| = |ω(z0)| = 1, then lemma 2.1 implies that ω(z0) = e iθ and z0ω ′(z0) = �ω(z0), � ≥ 1. thus, we obtain 1 + z0(a) [f]′′(z0) (a) [f]′(z0) = 2µz0ω ′(z0) 1 −µ2ω2(z0) + 1 + µω(z0) 1 −µω(z0) = 2µ�eiθ 1 −µ2e2iθ + 1 + µeiθ 1 −µeiθ since < ( 1 1 −µeiθ ) = 1 1 + µ , therefore, we conclude that < ( 1 + z0(a) [f]′′(z0) (a) [f]′(z0) ) = < ( 2µz0ω′(z0) 1 −µ2ω2(z0) + 1 + µω(z0) 1 −µω(z0) ) = < ( 2µ�eiθ 1 −µ2e2iθ + 1 + µeiθ 1 −µeiθ ) = 2µ� 1 + µ2 + 1 ≥ (1 + µ)2 1 + µ2 > 1. hence, < (z0(a) [f]′′(z0) (a) [f]′(z0) ) > 0, which contradicts the assumption of the theorem. this leads that there is no z0 ∈ u such that |ω(z0)| = 1 for all z ∈ u i.e z(a) [f]′(z) (a) [f](z) ≺ 1 + µz 1 −µz , z ∈ u, f ∈ a(u). this completes the proof. theorem 3.3. consider the integral operator (2.1). if for 1 < ℘ < 2, such that <{ z(a) [f]′′(z) (a) [f]′(z) } > ℘ 2 , , z ∈ u, f ∈ a(u), then (a) [f](z) ∈ b. geometric characterizations 39 proof. define a function ψ(z), z ∈ u as follows: (a) [f]′(z) = (1 −ψ(z))℘, z ∈ u, where, ψ(z) is analytic with ψ(0) = 0. we need only to show that |ψ(z)| < 1. from the definition of ψ, we have z(a) [f]′′(z) (a) [f]′(z) = ℘ −zψ′(z) 1 −ψ(z) . hence, we obtain <{ z(a) [f]′′(z) (a) [f]′(z) } = ℘<{ −zψ′(z) 1 −ψ(z) } > ℘ 2 , ℘ ∈ (1, 2). in view of lemma 2.1, there exists a complex number z0 ∈ u such that ψ(z0) = eiθ and z0ψ ′(z0) = �ψ(z0) = �e iθ, � ≥ 1. since < ( 1 1 −ψ(z0) ) = < ( 1 1 −eiθ ) = 1 2 then, we attain <{ z(a) [f]′′(z0) (a) [f]′(z0) } = ℘<{ −�ψ(z0) 1 −ψ(z0) } = ℘<{ −�eiθ 1 −eiθ } ≤ ℘ 2 , � = 1, which contradicts the assumption of the theorem. hence, there is no z0 ∈ u with |ψ(z0)| = 1, which yields that |ψ(z)| < 1. moreover, we have (a) [f]′(z) ≺ (1 −z)℘, which means that <[(a) [f]′(z)] > 0, equivalently, (a) [f]′(z) ∈ b. this completes the proof. theorem 3.4. consider the integral operator (2.1). if for ℘ > 1/2, such that <{ z(a) [f]′(z) (a) [f](z) } > 2℘− 1 2℘ , then (a) [f](z) z ≺ (1 −z)1/℘. proof. define a function w(z), z ∈ u as follows: (a) [f](z) z = (1 −w(z))1/℘, z ∈ u, where, w(z) is analytic with w(0) = 0. we need only to show that |w(z)| < 1. from the definition of w, we have z(a) [f]′(z) (a) [f](z) = 1 − zw′(z) ℘(1 −w(z)) . hence, we obtain < {z(a) [f]′(z) (a) [f](z) } = < { 1 − zw′(z) ℘(1 −w(z)) } > 2℘− 1 2℘ , ℘ > 1/2. 40 ibrahim in view of lemma 2.1, there exists a complex number z0 ∈ u such that w(z0) = eiθ and z0w ′(z0) = �w(z0) = �e iθ, � ≥ 1. therefore, we arrive at < {z0(a) [f]′(z0) (a) [f](z) } = < { 1 − z0w ′(z0) ℘(1 −w(z0)) } = < { 1 − �w(z0) ℘(1 −w(z0)) } = 1 −< { �eiθ ℘(1 + eiθ) } = 2℘− 1 2℘ , and this is a contradiction with the assumption of the theorem. hence, there is no z0 ∈ u with |w(z0)| = 1, which yields that |w(z)| < 1. this completes the proof. theorem 3.5. let f ∈ a(u) satisfied ∣∣∣zf′′(z) f′(z) − zf′(z) f(z) ∣∣∣ < 1. then (a) [f] ∈ b. proof. let f ∈ a(u). dividing (2.1) by f(z), z ∈ u \{0} and differentiating logarithmic, we have z(a) [f]′(z) (a) [f](z) = 1 + zf′′(z) f′(z) − zf′(z) f(z) + zda [f](z) a [f](z) . now let ϑ(z) = a [f](z) ⇒ ϑ′(z) = da [f](z). thus, we obtain zda [f](z) a [f](z) = zϑ′(z) ϑ(z) . in view of lemma 2.1, there exists a complex number z0 ∈ u such that ϑ(z0) = eiθ and z0ϑ ′(z0) = �ϑ(z0) = �e iθ, � ≥ 1. therefore, we conclude that∣∣∣z(a) [f]′(z) (a) [f](z) − 1 ∣∣∣ = ∣∣∣zf′′(z) f′(z) − zf′(z) f(z) + zda [f](z) a [f](z) ∣∣∣ ≤ ∣∣∣zf′′(z) f′(z) − zf′(z) f(z) ∣∣∣ + ∣∣∣zϑ′(z) ϑ(z) ∣∣∣ ≤ ∣∣∣zf′′(z) f′(z) − zf′(z) f(z) ∣∣∣ + ∣∣∣�eiθ eiθ ∣∣∣ = ∣∣∣zf′′(z) f′(z) − zf′(z) f(z) ∣∣∣ + � < 1 + � =: ρ. then in virtue of theorem 5.5g p299 in [2], we obtain∣∣∣(a) [f]′(z) − 1∣∣∣ < 1 ⇒ (a) [f] ∈ b. geometric characterizations 41 4. conclusion and discussion here, we provided a complex integral in the open unit disk based on the alexander operator ((a) [f]). the new operator is achieved the differential, shift plus operator (da[f](z) = zf′(z), f ∈ a(u). boundedness of the new integral operator is suggested in new extended space (s21 (u)). in addition, some geometric characterizations; such as univalent, starlike and bounded turning are studied. our main tool is based on the jack lemma. it has proven that if (a) : a(u) → a(u). for future work, one can use the new operator to define new classes of analytic functions. furthermore, for further investigations, one can study the subordination and superordination idea by employing the above integral. additionally, it can be studied the connection between closed ideals of a banach algebra together with closed invariant subspaces of the operator da[f]. references [1] j.w. alexsander, functions which map the interior of the unit circle upon simple regions, ann. math., 17(1915), 12–22. [2] s.s. miller and p.t. mocanu, differential subordinations: theory and applications, mrcel dekker inc., new york, 2000. [3] c. zeljko and b. paudyal, invariant subspaces of the shift plus complex volterra operator, j. math. anal. appl. 426(2)(2015), 1174–1181. [4] i.s. jack, functions starlike and convex of order α, j.london math. soc. 3(1971), 469–474. [5] p.l. duren., univalent functions, springer-verlag, 1983. faculty of computer science and information technology, university malaya, 50603, malaysia ∗corresponding author: rabhaibrahim@yahoo.com 1. introduction 2. processing 3. findings 4. conclusion and discussion references international journal of analysis and applications volume 18, number 1 (2020), 85-98 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-85 the optimal homotopy asymptotic method with application to second kind of nonlinear volterra integral equations h. ullah1,∗, s. mukhtar2, m. nawaz1, m. adnan3 1department of mathematics, abdul wali khan university mardan kpk, pakistan 2 department of basic sciences, deanship of preparatory year, king faisal university, hofuf 31982, al ahsa, saudi arabia 3 department of mathematics, islamia college (chartered university) peshawar kpk, pakistan ∗corresponding author: hakeemullah1@gmail.com abstract. in this paper, we solved some problems of nonlinear second kind of volterra integral equations by optimal homotopy asymptotic method(oham). we compared the results obtained by oham with the exact solutions of the problems. we find that the results obtained by oham are effective, simple and explicit from others analytical methods. we also showed the fast convergence of oham and list some examples to show the effectiveness of this method. in graphical analysis, we can see the exactness, accuracy and convergence of the method.the oham has mechanized steps that can be easily achieved with the help of mathematica. all computational work and graphs are obtained by mathematica 9. 1. introduction most of the problems are nonlinear in nature, especially in engineering and applied sciences. there are many applications of volterra integral equations (vie‘s) in applied field including bio-mechanics, fluid mechanics, demography and the study of viscoelastic materials. an italian mathematician and physicist vito volterra invented these equations in his mathematical physics research in 1908 [1]. there are several analytical and numerical methods, such as finite difference method, finite element method, perturbation received 2019-06-17; accepted 2019-07-22; published 2020-01-02. 2010 mathematics subject classification. 45d05. key words and phrases. nonlinear; volterra integral equation; oham; explicit; convergence; mathematica. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 85 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-85 int. j. anal. appl. 18 (1) (2020) 86 method, etc which can be used to obtain an approximate solutions of the nonlinear problems. however, there are several complications such as in grid modification, selection of stability conditions and selection of small and large parameters etc. in order to avoid these complications, decomposition method [2] was introduced, which is an exceptionally effective and powerful method for solving linear and nonlinear problems in various fields. the researchers introduced some others methods to deals such type of problems with easy way and less efforts. there are some analytical methods for solving such type of problems as we have; homotopy perturbation method (hpm) [3], group analysis method (gam) [4], differential transform method (dtm) [5], variational iterative method (vim) [6] and adomian decomposition method (adm) [7]. here we discussed some nonlinear volterra integral equations of the second kind. the general form of the nonlinear volterra integral equation is; ψ(x) = f(x) + λ ∫ x 0 k(x,t)g(ψ(t))dt (1.1) the function g(ψ(x)) is nonlinear in ψ(x) such as ψ2(x), ψ3(x), eψ(x), sinψ(x) and many others. in eq. (1.1), λ is a parameter and k(x,t) is the kernel of integral equation [8].the integration limit for volterra integral equations are function of ‘x‘ and not a constant value like in fredholm integral equations.the kernel k(x,t) in eq.(1.1) will be assuming a separable kernel. 2. optimal homotopy asymptotic method recently, engineers and scientists known the applications of oham in linear and nonlinear problems [9] and [10], because this method continuously deforms complex problems into simple problems which can be solved very easily. this method gives a quick way to the convergence of approximate series and keep more proficiency and high potentiality in science and engineering for solving nonlinear problems. several researchers have broadly studied different mathematical methods for integral equations such as [12] and [13]. here, we discuss oham which is proposed by marinca and herianu [11]. consider a general nonlinear problem [14]. τ{α(x)} + f(x) + ℵ{a(x)} = 0 (2.1) where τ is known as function which is called linear operator, f(x) is a given function, ℵ is a nonlinear operator and α(x) is unknown function. according to oham [12], we construct a homotopy: ω × [0, 1] −→ < for (2.1) which satisfy (1 −ρ)[τ{α(x,ρ)} + f(x)] = h(ρ)[τ{α(x,ρ)} + f(x) + ℵ{α(x,ρ)}] (2.2) where h(ρ) represents a nonzero auxiliary function for ρ 6= 0 and h(0) = 0. obviously, when, ρ = 0 then it holds that α(x, 0) = α0(x) (2.3) int. j. anal. appl. 18 (1) (2020) 87 and when, ρ = 1 then it holds that α(x, 1) = α1(x) (2.4) suppose that the auxiliary function h(ρ) can be expressed as; h(ρ) = σmj=1cjρ j (2.5) where cj, j = 1, 2, 3, ... are constant. putting ρ = 0 in eq.(2.2), it holds that τ{α0(x)} + f(x) = 0 (2.6) by taylor‘s series, the oham solution can be calculated as; α(x,ρ,cj) = α0(x) + σ m k=1αk(x,cj)ρ m (2.7) where j = 1, 2, 3, ... when ρ = 1 , then eq. (2.7) becomes α(x,ρ,cj) = α0(x) + σ m k=1αk(x,cj) (2.8) substituting eq. (2.8) into eq. (2.2) and equating the coefficient of the same power of ρ, we get; τ{α1(x)} = c1ℵ{α0(x)} (2.9) τ{αm(x)−αm−1(x)} = cmℵ{α0(x)}+ σ m−1 j=1 cj[τ{αm−j(x)}+ℵm−j{α0(x) + α1(x) + ... + αm−1(x)}] (2.10) where m = 2, 3, ... and ℵm{α0(x) + α1(x) + ... + αm−1(x)} are the coefficient of ρm in the expansion of n{a(x,ρ)} about ρ. ℵ{α(x,ρ,cj)} = ℵ0{α0(x)} + σ∞m=1ℵm{α0(x),α1(x), ... + αm(x)}ρ m (2.11) the result of mth order approximation are follow; αm(x,ci,j) = α0(x) + σ m k=1αk(x,cj),j = 1, 2, ...,m (2.12) substituting eq. (2.12) into (2.1), we get residual equation. <(x,cj) = τ{αm(x,cj)} + f(x) + ℵ{αm(x,cj)} (2.13) int. j. anal. appl. 18 (1) (2020) 88 if <(x,cj) = 0 then αm(x,cj) will be the exact solution. for finding the constants cj ,j = 1, 2, 3, ... using least square method, at first consider. =(cj) = ∫ b a <2(x,cj)dx (2.14) then the constants cj ,j = 1, 2, 3, ... can be identified as follow. ∂= ∂c1 = ∂= ∂c2 = ∂= ∂c3 (2.15) replacing the values of cj,j = 1, 2, 3, ... in eq. (2.13), we get the approximate solution. 3. some numerical examples of nonlinear volterra integral equations. in this section we used oham to solve some nonlinear volterra integral equations while the exact solution is also given. example 1. consider a nonlinear second kind of volterra integral equation with the exact solution ψ(x) = x2 [15] ψ(x) = x2 + x5 10 − 1 2 ∫ x 0 ψ2(t)dt. (3.1) we start from zero order solution and proceed similarly step by step. ψ0(x) = x 2 + x5 10 (3.2) which is the solution. ψ0(x) = 1 10 (10x2 + x5) (3.3) ψ1(x) = −x2 − x5 10 −x2c1 − x5c1 10 + ψ0 + c1ψ0 + 1 2 xc1ψ 2 0 (3.4) ψ1(x) = 1 200 x5(10 + x3)2c1 (3.5) ψ2(x) = −x2c2 − x5c2 10 + c2ψ0 + 1 2 xc2ψ 2 0 + ψ1 + c1ψ1 + xc1ψ0ψ1 (3.6) ψ2(x) = x5(10 + x3)2(10c1 + (10 + 10x 3 + x6)c21 + 10c2) 2000 (3.7) ψ3(x) = −x2c3 − x5c3 10 + c3ψ0 + 1 2 xc3ψ 2 0 + c2ψ1 + xc2ψ0ψ1 + 1 2 xc1ψ 2 1 + ψ2 + c1ψ2 + xc1ψ0ψ1 (3.8) int. j. anal. appl. 18 (1) (2020) 89 ψ3(x) = 1 16000 x5(10 + x3)2(16(10 + 10x3 + x6)c21 +(80 + 160x3 + 116x6 + 20x9 + x12)c31+ 16c1(5 + (10 + 10x 3 + x6)c2) + 80(c2 + c3) (3.9) the series solution is given as; ψ(x) = ψ0(x) + ψ1(x) + ψ2(x) + ψ3(x) (3.10) that is, ψ(x) = 1 16000 x2(10 + x3)(24x3(100 + 110x3 + 20x6 + x9)c21 + x 3 (800 + 1680x3 + 1320x6 + 316x9 + 30x12 + x15)c31 + 16x 3(10 + x3)c1(15 + (10 + 10x 3 + x6)c2)+ 80(20 + 2x3(10 + x3)c2 + x 3(10 + x3)c3) (3.11) for finding the values of ci, we use the least square method. c1 = −0.1677940548,c2 = 0.1114129522,c3 = 0.0386473756. by putting the constant values of ci in eq.(3.11), we get. ψ(x) = −2.95263×10−7x2(10 + x3)(−338681 + 33793x3 −2992.5x6 −274.362x9 + 236.282x12 + 30x15 + x18) (3.12) int. j. anal. appl. 18 (1) (2020) 90 table 1. in this table, we compared oham solution and exact solution of eq. (3.1), where λ represents the absolute error of oham. x oham solution exact solution λ 0.0 0.0 0.0 0.0 0.1 0.01 0.01 2.20146 × 10−9 0.2 0.0400001 0.04 6.79223 × 10−8 0.3 0.0900005 0.09 4.65761 × 10−7 0.4 0.160002 0.16 1.58682 × 10−6 0.5 0.250003 0.25 3.24191 × 10−6 0.6 0.360004 0.36 3.65805 × 10−6 0.7 0.49 0.49 2.79695 × 10−7 0.8 0.639996 0.64 4.44442 × 10−6 0.9 0.810002 0.81 1.57289 × 10−6 1.0 0.999986 1.0 0.0000142671 figure 1. shows the comparison of oham and exact solution of the eq.(3.1) int. j. anal. appl. 18 (1) (2020) 91 figure 2. shows the residual solution of the problem. figure 3. shows the comparison of zero order, first order, second order and third order of oham solution and exact solution of eq.(3.1) example 2. consider a nonlinear second kind of vie with exact solution ψ(x) = x [15] ψ(x) = x− x4 4 + ∫ x 0 tψ2(t)dt (3.13) we used oham to find analytical solution. ψ0(x) = x− x4 4 (3.14) ψ0(x) = 1 4 (4x−x4) (3.15) ψ1(x) = −x + x4 4 −xc1 + x4c1 4 + ψ0 + c1ψ0 − 1 2 x2c1ψ 2 0 (3.16) int. j. anal. appl. 18 (1) (2020) 92 ψ1(x) = − 1 32 x4(−4 + x3)2c1 (3.17) ψ2(x) = −xc2 + x4c2 4 + c2ψ0 − 1 2 x2c2ψ 2 0 + ψ1 + c1ψ1 −x 2c1ψ0ψ1 (3.18) ψ2(x) = − 1 128 x4(−4 + x3)2(4c1 + (−2 + x3)2c21 + 4c2) (3.19) ψ3(x) = −xc3 + x4c3 4 + c3ψ0 − 1 2 x2c3ψ 2 0 + c2ψ1− x2c2ψ0ψ1 − 1 2 x2c1ψ 2 1 + ψ2 + c1ψ2 −x 2c1ψ0ψ2 (3.20) ψ3(x) = − 1 2048 x4(−4 + x3)2(32(−2 + x3)2c21 + (64 − 128x 3 + 112x6 − 40x9 + 5x12)c31 +32c1(2 + (−2 + x3)2c2) + 64(c2 + c3) (3.21) the series solution is; ψ(x) = ψ0(x) + ψ1(x) + ψ2(x) + ψ3(x) (3.22) that is, ψ(x) = x− x4 4 − 1 32 x4(−4 + x3)2c1 − 1 128 x4(−4 + x3)2(4c1 + (−2 + x3)2c21 + 4c2)− 1 2048 x4(−4 + x3)232(−2 + x3)2c21 + (64 − 128x 3 + 112x6 − 40x9 + 5x12)c31+ 32c1(2 + (−2 + x3)2c2) + 64(c2 + c3) (3.23) for finding values of ci , using least square method. c1 = −0.8102578861,c2 = 0.5900091712,c3 = 0.2244268082. by putting these values in eq.(3.23), we get ψ(x) = x(1+0.0224215x3−0.161442x6+0.368418x9−0.337199x12+0.12507x15−0.0207792x18+0.0012987x21) (3.24) int. j. anal. appl. 18 (1) (2020) 93 table 2. in this table, we compared oham solution and exact solution of eq. (3.13), where λ represents the absolute error of oham. x oham solution exact solution λ 0.0 0.0 0.0 0.0 0.1 0.100002 0.1 2.22604 × 10−6 0.2 0.200034 0.2 0.0000338453 0.3 0.300148 0.3 0.000148429 0.4 0.400346 0.4 0.000345904 0.5 0.500461 0.5 0.000460563 0.6 0.600208 0.6 0.000207781 0.7 0.69962 0.7 0.000379802 0.8 0.799578 0.8 0.00042168 0.9 0.900738 0.9 0.000737938 1.0 0.997787 1.0 0.00221256 figure 4. shows the comparison of oham and exact solution of the eq.(3.13) int. j. anal. appl. 18 (1) (2020) 94 figure 5. shows the residual solution of the problem. figure 6. shows the comparison of zero order, first order, second order and third order of oham solution and exact solution of eq.(3.13) example 3. consider a nonlinear vie with exact solution ψ(x) = x2. [15] ψ(x) = x2 + x6 12 − 1 2 ∫ x 0 tψ2(t)dt (3.25) oham solution: ψ0(x) = x 2 + x6 12 (3.26) ψ0(x) = 1 12 (12x2 + x6) (3.27) ψ1(x) = −x2 − x6 12 −x2c1 − x6c1 12 + ψ0 + c1ψ0 + 1 4 x2c1ψ 2 0 (3.28) ψ1(x) = 1 576 x6(12 + x4)2c1 (3.29) int. j. anal. appl. 18 (1) (2020) 95 ψ2(x) = −x2c2 − x6c2 12 + c2ψ0 + 1 4 x2c2ψ 2 0 + ψ1 + c1ψ1 + 1 2 x2c1ψ0ψ1 (3.30) ψ2(x) = x6(12 + x4)2(24c1 + (24 + 12x 4 + x8)c21 + 24c2) 13824 (3.31) ψ3(x) = −x2c3 − x6c3 12 + c3ψ0 + 1 4 x2c2ψ 2 0 + ψ1 + c1ψ1 + 1 2 x2c1ψ0ψ1 (3.32) ψ3(x) = 1 1327104 x6(12 + x4)(192(24 + 12x4 + x8)c21 + (2304 + 2304x 4 + 912x8 + 120x12 + 5x16)c31 +192c1(12 + (24 + 12x 4 + x8)c2) + 2304(c2 + c3) (3.33) the series solution is given below; ψ0(x) = ψ0(x) + ψ1(x) + ψ2(x) + ψ3(x) (3.34) that is, ψ(x) = 1 1327104 x2(12 + x4)(288x4(12 + x4)(24 + 12x4 + x8)c21 + x 4(12 + x4)(2304 + 2304x4+ 912x8 + 120x12 + 5x16)c31 + 192x 4(12 + x4)c1(36 + (24 − 12x4 + x8)c2)+ 2304(48 + 2x4(12 + x4)c2 + x 4(12 + x4)c3)) (3.35) to find the values of ci, where ci = 1, 2, 3, ..., we use least square method. c1 = −0.2887286851,c2 = 0.1652477727,c3 = 0.0721620291. put the values of ci in eq.(3.35), we get. ψ(x) = −9.06849 × 10−8x2(12 + x4)× (−918933 + 76484.9x4 − 5863x4 − 5863x8 − 311.456x12 + 347.023x16 + 36x20 + x24) (3.36) int. j. anal. appl. 18 (1) (2020) 96 table 3. in this table, we compared oham solution and exact solution of eq. (3.25), where λ represents the absolute error of oham. x oham solution exact solution λ 0.0 0.0 0.0 0.0 0.1 0.01 0.1 1.01049 × 10−10 0.2 0.04 0.2 6.41391 × 10−9 0.3 0.0900001 0.3 7.04647 × 10−8 0.4 0.16 0.4 3.58157 × 10−7 0.5 0.250001 0.5 1.08877 × 10−6 0.6 0.360002 0.6 2.00225 × 10−6 0.7 0.490002 0.7 1.50262 × 10−6 0.8 0.639998 0.8 1.71305 × 10−6 0.9 0.809999 0.9 5.7885 × 10−7 1.0 0.999991 1.0 8.55913 × 10−6 figure 7. shows the comparison of oham and exact solution of the eq.(3.25) int. j. anal. appl. 18 (1) (2020) 97 figure 8. shows the residual solution of the problem. figure 9. shows the comparison of zero order, first order, second order and third order of oham solution and exact solution of eq.(3.25) 4. conclusion in this research article, we presented the application of (oham) by solving some examples of nonlinear volterra integral equations of the second kind. this technique is verified on three different problems.the technique showed to be an accurate and well-organized method for finding approximate solutions for the nonlinear volterra integral equations of the second kind. the (oham) is relatively simple to apply. it is shown that, with few terms, the method is capable of giving sufficient accuracy.this method can be a promising tool for solving strongly nonlinear problems. the convergence of (oham) to exact solution is very excellent and quick. int. j. anal. appl. 18 (1) (2020) 98 5. acknowledgment the authors would like to thanks the reviewers for the valuable comments and suggestion which help in the improvement of this paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] kumar, s., singh, omp. dixit, s., generalized abel inversion using homotopy perturbation method, appl. math. 2 (2011), 254-257. [2] jafari, h., gejji, v.d., revised adomian decomposition method for solving a system of nonlinear equations, appl. math. comput. 175 (2006), 1-7. [3] ghoreishi, m., ismail, a.m. and alomari, a.k., comparison between homotopy analysis method and optimal homotopy asymptotic method for nthorder integrodifferential equation. math. methods appl. sci. 34 (15) (2011), 1833-1842. [4] ayub m, haq s, siddiqui a.m, hayat t., group analysis of two dimensional flow of a non-newtonian fluid, islamabad j. sci. 13 (1) (2003), 47-59. [5] ganji d.d, afrouzi r. a., the application of differential transformation method to nonlinear equation arising in heat transfer, int. commun. heat mass transfer, 38 (6), 815-820 (2011). [6] avaji, m., hafshejani, j.s., dehcheshmeh, s.s. and ghahfarokhi, d.f., solution of delay volterra integral equations using the variational iteration method. j. appl. sci. 12 (2) (2012), 196-200. [7] feng, j.q. and sun, s., numerical solution of volterra integral equation by adomian decomposition method. asian res. j. math. 4 (1) (2017), article no. arjom.33105. [8] wazwaz, a. m. linear and nonlinear integral equations (vol. 639). heidelberg: springer. (2011). [9] yang, l. h., li, h. y., and wang, j. r. solving a system of linear volterra integral equations using the modified reproducing kernel method. abstr. appl. anal. 2013 (2013), art. id 196308. [10] wazwaz, a. m. linear and nonlinear integral equations (vol. 639). heidelberg: springer. (2011). [11] marinca, v. and herianu, n., application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. international commun. heat mass transfer, 35 (6) (2008), 710-715. [12] yang, l. h., li, h. y., and wang, j. r., solving a system of linear volterra integral equations using the modified reproducing kernel method. abstr. appl. anal. 2013 (2013), art. id 196308. [13] maleknejad, k., mahmoudi, y., taylor polynomial solution of high-order nonlinear volterra fredholm integro-diffrential equations, appl. math. comput. 145 (2-3) (2003), 641-653. [14] almousa, m.s.t., approximate analytical methods for solving fredholm integral equations, doctoral dissertation, university sains malaysia, 2015. [15] wazwaz, a.m., a first course in integral equations. world scientific, singapore, 2015. 1. introduction 2. optimal homotopy asymptotic method 3. some numerical examples of nonlinear volterra integral equations. 4. conclusion 5. acknowledgment references international journal of analysis and applications volume 17, number 3 (2019), 440-447 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-440 on the behaviors of rough fractional type sublinear operators on vanishing generalized weighted morrey spaces feri̇t gürbüz∗ hakkari university, faculty of education, department of mathematics education, hakkari 30000, turkey ∗corresponding author: feritgurbuz@hakkari.edu.tr abstract. the aim of this paper is to get the boundedness of rough sublinear operators generated by fractional integral operators on vanishing generalized weighted morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. also, rough fractional integral operator and a related rough fractional maximal operator which satisfy the conditions of our main result can be considered as some examples. 1. introduction and useful informations 1.1. background. the classical fractional integral (the classical fractional integral operator is also known as riesz potential.) was introduced by riesz in 1949 [6], defined by iαf(x) = (−∆) −α 2 f(x) 0 < α < n, = 1 γ (α) ∫ rn f(y) |x−y|n−α dy with γ (α) = π n 2 2αγ ( α 2 ) γ ( n 2 − α 2 ) , where γ (·) is the standard gamma function and iα plays an important role in partial diferential equation as the inverse of power of laplace operator. especially, its most significant feature is that iα maps lp(rn) received 2018-12-19; accepted 2019-01-07; published 2019-05-01. 2010 mathematics subject classification. 42b20, 42b25, 42b35. key words and phrases. fractional type sublinear operator; rough kernel; vanishing generalized weighted morrey space; a ( p s′ , q s′ ) weight. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 440 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-440 int. j. anal. appl. 17 (3) (2019) 441 continuously into lq(rn), with 1q = 1 p −α n and 1 < p < n α , through the well known hardy-littlewood-sobolev imbedding theorem (see pp. 119-121, theorem 1 and its proof in [7]) for iα. let ω ∈ ls(sn−1), 1 < s ≤ ∞, ω(µx) = ω(x) for any µ > 0, x ∈ rn \{0} and satisfy the cancellation condition ∫ sn−1 ω(x′)dσ(x′) = 0, where x′ = x|x| for any x 6= 0. we first recall the definitions of rough fractional integral operator tω,α and a related rough fractional maximal operator mω,α as follows: definition 1.1. define iω,αf(x) = ∫ rn ω(x−y) |x−y|n−α f(y)dy 0 < α < n, mω,αf (x) = sup r>0 1 rn−α ∫ |x−y|<r |ω(x−y)| |f(y)|dy 0 < α < n. next, we give the definition of weighted lebesgue spaces as follows: definition 1.2. (weighted lebesgue space) let 1 ≤ p ≤∞ and given a weight w (x) ∈ ap (rn), we shall define weighted lebesgue spaces as lp(w) ≡ lp(rn,w) =  f : ‖f‖lp,w =  ∫ rn |f(x)|pw(x)dx   1 p < ∞   , 1 ≤ p < ∞. l∞,w ≡ l∞(rn,w) = { f : ‖f‖l∞,w = esssup x∈rn |f(x)|w(x) < ∞ } . here and later, ap denotes the muckenhoupt classes (see [2]). now, let us consider the muckenhoupt-wheeden class a (p,q) in [5]. one says that w (x) ∈ a (p,q) for 1 < p < q < ∞ if and only if [w]a(p,q) := sup b  |b|−1∫ b w(x)qdx   1 q  |b|−1∫ b w(x)−p ′ dx   1 p′ < ∞, (1.1) where the supremum is taken over all the balls b. note that, by hölder’s inequality, for all balls b we have [w]a(p,q) ≥ [w]a(p,q)(b) = |b| 1 p −1 q −1‖w‖lq(b)‖w −1‖lp′(b) ≥ 1. (1.2) by (1.1), we have  ∫ b w(x)qdx   1 q  ∫ b w(x)−p ′ dx   1 p′ . |b| 1 q + 1 p′ . (1.3) int. j. anal. appl. 17 (3) (2019) 442 on the other hand, let µ (x) = w (x) s′ , p̃ = p s′ and q̃ = q s′ . if w (x) s′ ∈ a ( p s′ , q s′ ) , then we get µ (x) ∈ a (p̃, q̃). by (1.2) and (1.3), ‖µ‖lq̃(b)‖µ −1‖lp̃′(b) ≈ |b| 1+ 1 q̃ −1 p̃ (1.4) is valid. now, we introduce some spaces which play important roles in pde. except the weighted lebesgue space lp(w), the weighted morrey space lp,κ(w), which is a natural generalization of lp(w) is another important function space. then, the definition of generalized weighted morrey spaces mp,ϕ (w) which could be viewed as extension of lp,κ(w) has been given as follows: for 1 ≤ p < ∞, positive measurable function ϕ(x,r) on rn×(0,∞) and nonnegative measurable function w on rn, f ∈ mp,ϕ(w) ≡ mp,ϕ(rn,w) if f ∈ llocp,w(rn) and ‖f‖mp,ϕ(w) = sup x∈rn,r>0 1 ϕ(x,r) ‖f‖lp(b(x,r),w) < ∞. is finite. note that for ϕ(x,r) ≡ w(b(x,r)) κ p , 0 < κ < 1 and ϕ(x,r) ≡ 1, we have mp,ϕ(w) = lp,κ(w) and mp,ϕ(w) = lp(w), respectively. extending the definition of vanishing generalized morrey spaces in [3] to the case of generalized weighted morrey spaces defined above, we introduce the following definition. definition 1.3. (vanishing generalized weighted morrey spaces) for 1 ≤ p < ∞, ϕ(x,r) is a positive measurable function on rn × (0,∞) and nonnegative measurable function w on rn, f ∈ v mp,ϕ (w) ≡ v mp,ϕ(rn,w) if f ∈ llocp,w(rn) and lim r→0 sup x∈rn 1 ϕ(x,r) ‖f‖lp(b(x,r),w) = 0. (1.5) inherently, it is appropriate to impose on ϕ(x,t) with the following circumstances: lim t→0 sup x∈rn (w(b(x,t))) 1 p ϕ(x,t) = 0, (1.6) and inf t>1 sup x∈rn (w(b(x,t))) 1 p ϕ(x,t) > 0. (1.7) from (1.6) and (1.7), we easily know that the bounded functions with compact support belong to v mp,ϕ (w). on the other hand, the space v mp,ϕ(w) is banach space with respect to the following finite quasi-norm ‖f‖v mp,ϕ(w) = sup x∈rn,r>0 1 ϕ(x,r) ‖f‖lp(b(x,r),w), such that lim r→0 sup x∈rn 1 ϕ(x,r) ‖f‖lp(b(x,r),w) = 0, int. j. anal. appl. 17 (3) (2019) 443 we omit the details. moreover, we have the following embeddings: v mp,ϕ (w) ⊂ mp,ϕ (w) , ‖f‖mp,ϕ(w) ≤‖f‖v mp,ϕ(w). henceforth, we denote by ϕ ∈b (w) if ϕ(x,r) is a positive measurable function on rn × (0,∞) and positive for all (x,r) ∈ rn × (0,∞) and satisfies (1.6) and (1.7). the purpose of this paper is to consider the mapping properties for the rough fractional type sublinear operators tω,α satisfying the following condition |tω,αf(x)|. ∫ rn |ω(x−y)| |x−y|n−α |f(y)|dy, x /∈ supp f (1.8) on vanishing generalized weighted morrey spaces. similar results still hold for the operators iω,α and mω,α, respectively. on the other hand, these operators have not also been studied so far on vanishing generalized weighted morrey spaces and this paper seems to be the first in this direction. at last, here and henceforth, f ≈ g means f & g & f ; while f & g means f ≥ cg for a constant c > 0; and p′ and s′ always denote the conjugate index of any p > 1 and s > 1, that is, 1 p′ := 1 − 1 p and 1 s′ := 1 − 1 s and also c stands for a positive constant that can change its value in each statement without explicit mention. throughout the paper we assume that x ∈ rn and r > 0 and also let b(x,r) denotes x-centred euclidean ball with radius r, bc(x,r) denotes its complement. for any set e, χ e denotes its characteristic function, if e is also measurable and w is a weight, w(e) := ∫ e w(x)dx. 2. main results our result can be stated as follows. theorem 2.1. suppose that 0 < α < n, 1 ≤ s′ < p < n α , 1 q = 1 p − α n , 1 < q < ∞, ω ∈ ls(sn−1), 1 < s ≤∞, ω(µx) = ω(x) for any µ > 0, x ∈ rn\{0} such that tω,α is rough fractional type sublinear operator satisfying (1.8). for p > 1, w (x) s′ ∈ a ( p s′ , q s′ ) and s′ < p, the following pointwise estimate ‖tω,αf‖lq(b(x0,r),wq) . (w q (b (x0,r))) 1 q ∞∫ 2r ‖f‖lp(b(x0,t),wp) (w q (b (x0, t))) −1 q dt t (2.1) holds for any ball b (x0,r) and for all f ∈ llocp,w (rn). if ϕ1 ∈ b (wp), ϕ2 ∈ b (wq) and the pair (ϕ1,ϕ2) satisfies the following conditions cδ := ∞∫ δ sup x∈rn ϕ1 (x,t) (wq (b (x,t))) 1 q 1 t dt < ∞ (2.2) for every δ > 0, and ∞∫ r ϕ1 (x,t) (wq (b (x,t))) 1 q 1 t dt . ϕ2(x,r) (wq (b (x,t))) 1 q , (2.3) int. j. anal. appl. 17 (3) (2019) 444 then for p > 1, w (x) s′ ∈ a ( p s′ , q s′ ) and s′ < p, the operator tω,α is bounded from v mp,ϕ1 (w p) to v mq,ϕ2 (w q). moreover, ‖tω,αf‖v mq,ϕ2 (wq) . ‖f‖v mp,ϕ1 (wp) · (2.4) proof. since inequality (2.1) is the heart of the proof of (2.4), we first prove (2.1). for any x0 ∈ rn, we write as f = f1 + f2, where f1 (y) = f (y) χb(x0,2r) (y), f2 (y) = f (y) χ(b(x0,2r))c (y), r > 0 and χb(x0,2r) denotes the characteristic function of b (x0, 2r). then ‖tω,αf‖lq(wq,b(x0,r)) ≤‖tω,αf1‖lq(wq,b(x0,r)) + ‖tω,αf2‖lq(wq,b(x0,r)) . let us estimate ‖tω,αf1‖lq(wq,b(x0,r)) and ‖tω,αf2‖lq(wq,b(x0,r)), respectively. since f1 ∈ lp (wp,rn), by the boundedness of tω,α from lp (wp,rn) to lq (wq,rn) (see theorem 3.4.2 in [4]), (1.4) and since 1 ≤ s′ < p < q we get ‖tω,αf1‖lq(wq,b(x0,r)) ≤‖tω,αf1‖lq(wq,rn) . ‖f1‖lp(wp,rn) = ‖f‖lp(wp,b(x0,2r)) . rn−αs ′ ‖f‖lp(wp,b(x0,2r)) ∞∫ 2r dt tn−αs ′+1 ≈‖ws ′ ‖l q s′ (b(x0,r))‖w −s′‖l ( ps′ ) ′(b(x0,r)) ∞∫ 2r ‖f‖lp(wp,b(x0,t)) dt tn−αs ′+1 . (wq (b(x0,r))) 1 q ∞∫ 2r ‖f‖lp(wp,b(x0,t)) ‖w −s′‖l ( ps′ ) ′(b(x0,t)) dt tn−αs ′+1 . (wq (b(x0,r))) 1 q ∞∫ 2r ‖f‖lp(wp,b(x0,t)) [ ‖ws ′ ‖l ( qs′ ) (b(x0,t)) ]−1 1 t dt . (wq (b(x0,r))) 1 q × ∞∫ 2r ‖f‖lp(wp,b(x0,t)) (w q (b(x0, t))) −1 q 1 t dt. now, let’s estimate the second part (= ‖tω,αf2‖lq(wq,b(x0,r))). for the estimate used in ‖tω,αf2‖lq(wq,b(x0,r)), we first have to prove the below inequality: |tω,αf2 (x)|. ∞∫ 2r ‖f‖lp(wp,b(x0,t)) (w q (b(x0, t))) −1 q 1 t dt. (2.5) by [1] (see pp. 7 in the proof of lemma2:), we get |tω,αf2 (x)|. ∞∫ 2r ‖ω (x−·)‖ls(b(x0,t)) ‖f‖ls′(b(x0,t)) dt tn+1−α . (2.6) int. j. anal. appl. 17 (3) (2019) 445 on the other hand, by hölder’s inequality we have ‖f‖ls′(b(x0,t)) =   ∫ b(x0,t) |f (y)|s ′ dy   1 s′ ≤   ∫ b(x0,t) |f (y)|p |µ (y)|p̃ dy   1 p   ∫ b(x0,t) |µ (y)|−p̃ ′ dy   1 p̃′s′ ≤   ∫ b(x0,t) |f (y)|p |µ (y)|p̃ dy   1 p (wq (b(x0, t))) −1 q |b(x0, t)| 1 s′ + 1 q −1 p = ‖f‖lp(wp,b(x0,t)) (w q (b(x0, t))) −1 q |b(x0, t)| 1 s′ + 1 q −1 p , (2.7) where in the second inequality we have used the following fact: by (1.4), we get the following:   ∫ b(x0,t) |µ (y)|−p̃ ′ dy   1 p̃′s′ ≈ [ ‖µ‖lq̃(b(x0,t)) ]− 1 s′ [ |b (x0, t) |1+ 1 q̃ −1 p̃ ] 1 s′ = [( ‖ws ′ ‖lq̃(b(x0,t)) )−1 |b (x0, t) |1+ 1 q̃ −1 p̃ ] 1 s′ =     ∫ b(x0,t) |w (y)|q dy   −s ′ q |b (x0, t) |1+ s′ q −s ′ p   1 s′ = (wq (b(x0, t))) −1 q |b(x0, t)| 1 s′ + 1 q −1 p . (2.8) at last, substituting (3.10) in [1] and (2.7) into (2.6), the proof of (2.5) is completed. thus, by (2.5) we get ‖tω,αf2‖lq(wq,b(x0,r)) . (w q (b(x0,r))) 1 q × ∞∫ 2r ‖f‖lp(wp,b(x0,t)) (w q (b(x0, t))) −1 q 1 t dt. combining all the estimates for ‖tω,αf1‖lq(wq,b(x0,r)) and ‖tω,αf2‖lq(wq,b(x0,r)), we get (2.1). int. j. anal. appl. 17 (3) (2019) 446 now, let’s estimate the second part (2.4) of theorem 2.1. indeed, by the definition of vanishing generalized weighted morrey spaces, (2.1) and (2.3), we have ‖tω,αf‖v mq,ϕ2 (wq) = sup x∈rn,r>0 1 ϕ2(x,r) ‖tω,αf‖lq(wq,b(x0,r)) . sup x∈rn,r>0 1 ϕ2(x,r) (wq (b (x0,r))) 1 q × ∞∫ r ‖f‖lp(b(x0,t),wp) (w q (b (x0, t))) −1 q dt t . sup x∈rn,r>0 1 ϕ2(x,r) (wq (b (x0,r))) 1 q × ∞∫ r (wq (b (x0, t))) −1 q ϕ1 (x,t) [ ϕ1 (x,t) −1 ‖f‖lp(b(x0,t),wp) ] dt t . ‖f‖v mp,ϕ1 (wp) supx∈rn,r>0 1 ϕ2(x,r) (wq (b (x0,r))) 1 q × ∞∫ r (wq (b (x0, t))) −1 q ϕ1 (x,t) dt t . ‖f‖v mp,ϕ1 (wp) . at last, we need to prove that lim r→0 sup x∈rn 1 ϕ2(x,r) ‖tω,αf‖lq(wq,b(x0,r)) . lim r→0 sup x∈rn 1 ϕ1(x,r) ‖f‖lp(wp,b(x0,r)) = 0. but, because the proof of above inequality is similar to theorem 2 in [3], we omit the details, which completes the proof. � corollary 2.1. under the conditions of theorem 2.1, the operators mω,α and iω,α are bounded from v mp,ϕ1 (w p) to v mq,ϕ2 (w q). corollary 2.2. for w ≡ 1, under the conditions of theorem 2.1, we get the theorem 2 in [3]. references [1] a. s. balakishiyev, e. a. gadjieva, f. gürbüz and a. serbetci, boundedness of some sublinear operators and their commutators on generalized local morrey spaces, complex var. elliptic equ. 63(11) (2018), 1620-1641. [2] f. gürbüz, on the behaviors of sublinear operators with rough kernel generated by calderón-zygmund operators both on weighted morrey and generalized weighted morrey spaces, int. j. appl. math. stat. 57(2) (2018), 33-42. [3] f. gürbüz, a class of sublinear operators and their commutators by with rough kernels on vanishing generalized morrey spaces, j. sci. eng. res. 5(5) (2018), 86-101. [4] s. z. lu, y. ding and d. yan, singular integrals and related topics, world scientific publishing, singapore, 2006. [5] b. muckenhoupt and r.l. wheeden, weighted norm inequalities for singular and fractional integrals, trans. amer. math. soc. 161 (1971), 249-258. int. j. anal. appl. 17 (3) (2019) 447 [6] m. riesz, l´intégrale de riemann-liouville et le problème de cauchy, acta math. 81 (1949), 1-222. [7] e. m. stein, singular integrals and differentiability properties of functions, princeton n j, princeton univ press, 1970. 1. introduction and useful informations 1.1. background 2. main results references int. j. anal. appl. (2022), 20:56 on complete, horizontal and vertical lifts from a manifold with fλ (6,4) structure to its cotangent bundle manisha m. kankarej1,∗, jai pratap singh2 1department of mathematics and statistics, zayed university, dubai, uae 2b.s.n.v.p.g. college, lucknow university, lucknow, india ∗corresponding author: manisha.kankarej@gmail.com abstract. manifolds with fλ(6,4) structure was defined and studied in the past. later the geometry of tangent and cotangent bundles in a differentiable manifold with fλ(6,4) structure was studied. the aim of the present paper is to study complete, horizontal and vertical lifts from a manifold with fλ (6,4)structure to its cotangent bundle. 1. introduction the research on the properties of tensorial structure on manifolds and its extension to tangent and cotangent bundles is always gaining attraction from the researchers. yano [12], [13], [14] introduced the idea of horizontal and vertical lifts on the tangent bundles. kim [6] studied properties of f manifold. dube [5], upadhyay and gupta [11] studied integrability conditions of f 2v+4 + f 2 = 0; f 6 = 0 and of type (1; 1) and f(k;−(k −2)) structure satisfying fk − fk−2 = 0; (f 6= 0; i). srivastava [9], [10] studied complete lifts of (1,1) tensor field f satisfying structure fν+1 − λ2fν−1 = 0 and extended in mn to cotangent bundle. nivas and saxena [8] studied horizontal and complete lifts from a manifold with fλ(7,−1) structure to its cotangent bundles. li and krupka [7] discussed the properties of tangent bundles. cayir [1], [2] and [3] studied lifts of fν+1,λ2fν−1 structure. received: aug. 11, 2022. 2010 mathematics subject classification. 15a72, 47b47, 53a45, 53c15. key words and phrases. horizontal lift; vertical lift; complete lift; cotangent bundle; nijenhuis tensor. https://doi.org/10.28924/2291-8639-20-2022-56 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-56 2 int. j. anal. appl. (2022), 20:56 let m be a differentiable manifold of class c∞ and let ctm denote the cotangent bundle of m. then ctm is also a differentiable manifold of class c∞ and dimension 2n. throughout this paper we shall use the following notations and conventions: (i) the map n : ctm → m denotes the projection map of ctm onto m. (ii) suffixes a, b, c. . . .h, i, j. . . ..take value 1 to n and i = i +n. suffixes a, b, c, . . . ., take the value 1 to 2n. (iii) jrb (m) denote the set of tensor fields of class c ∞ and type (r,s) on m. similarly jrb ( ctm) denotes the set of such tensor fields in ctm. (iv) vector fields in m are denoted by x, y, z. . . .and the lie-derivative by lx. (v) the lie product of x, y is denoted by [x, y]. if a is a point in m and n−1 (a) is a fibre over a. any point p ∈ n−1 (a) is the ordered pair (a, pa), where p is 1-form in m and ‘pa’ is the value of p at a. let u be a coordinate neighborhood in m such that a ∈ u. then u induces a coordinate neighbourhood n−1 (u) in ctm and p ∈ n−1 (u) by [4]. 2. complete lift of fλ(6,4) structure let m be an n – dimensional differentiable manifold of class c∞. suppose there exists on m, a tensor field f (6=0) of type (1,1) by [6] and [10] we have f 6 − λ2f 4 =0 (2.1) where λ is a complex number not equal to zero. in such a manifold m, let us put l = f 4 λ2 , m = i − f 4 λ2 (2.2) where i denote the unit tensor field. then it can be easily shown that l2 = l, m2 = m, l +m = i and l ∗m = m∗ l =0 (2.3) thus, the operators ‘l’ and ‘m’ when applied to the tangent space m at a point are complementary projection operators. hence there exist complementary distributions l∗ and m∗ corresponding to the projection operators ‘l’ and ‘m’ respectively. if the rank of f is constant everywhere and equal to r, the dimension of l∗ and m∗ are r and (n-r) respectively. let us call such a structure on m as fλ (6,4) structure of rank r. let f hi be component of f at a in the coordinate neighbourhood u of m. then the complete lift f c of f is also a tensor field of type (1,1) in ctm, where components f a b in π −1 (u) are given by [4] int. j. anal. appl. (2022), 20:56 3 f h i = f h i ; f h i =0 ; f h i = pa ( ∂f ah ∂x i − ∂f ai ∂xh ) ; f h i = f i h (2.4) where (x1, x2,. . . ., xn ) are coordinates of a in u and pa has components ( p1, p2,. . . .,pn). thus we can, write f c = ( f a b ) = [ f hi 0 pa ( ∂if ah − ∂hf a i ) f ih ] (2.5) where ∂i = ∂/∂x i. if we put ∂if ah − ∂h f a i =2∂ [i f a h ] , then we can write f a b as f c = ( f a b ) = [ f hi 0 2pa∂ [ if ah ] f ih ] (2.6) thus, we have (f c ) 2 = [ f hi 0 2pa∂ [ if ah ] f ih ] [ f ij 0 2pt∂ [ jf ti ] f j i ] or (f c ) 2 = [ f hi f i j 0 2paf i j ∂ [ if ah ] + 2ptf i h∂ [ jf ti ] f j i f i h ] (2.7) if we put 2paf i j ∂ [if a h ]+ 2ptf i h∂ [ jf ti ] = lhj (2.8) (f c ) 2 = [ f hi f i j 0 lhj f j i f i h ] (2.9) squaring again from [4] we get (f c ) 4 = [ f hi f i j 0 lhj f j i f i h ][ f j k f kl 0 ljl f l kf k j ] or (f c ) 4 = [ f hi f i j f j k f kl 0 f j k f kl lhj + f j i f i h ljl f l kf k j f j i f i h ] (2.10) thus (f c ) 6 = [ f hi f i j f j k f kl 0 f j k f kl lhj + f j i f i h ljl f l kf k j f j i f i h ] [ f lmf m n 0 lln f n mf m l ] (2.11) 4 int. j. anal. appl. (2022), 20:56 or (f c ) 6 =   f hi f ij f jkf kl f lmf mn 0 f j k f kl f l mf m n lhj + f j i f i h f lmf m n ljl + f l kf k j f j i f i h lln f n mf m l f l k f k j f j i f i h   putting again f j k f kl f l mf m n lhj + f j i f i h f lmf m n ljl + f l kf k j f j i f i h lln (2.12) = λ2{f pq f q n lhp + f p r f r h lpn} thus, in view of the equations (2.12) and also (2.1), the above equation (2.11) takes the form (f c ) 6 =   λ2 f hpf pq f qr f rn 0 λ2 { f p q f q n lhp + f p r f r h lpn } λ2f n r f r qf q p f p h   = λ2   f hp f pq f qr f rn 0 f p q f q n lhp + f p r f r h lpn f n r f r qf q p f p h   or (f c) 6 − λ2 (f c)4 =0 hence the complete lift f c of f also has fλ (6,4) structure in the cotangent bundle ctm. thus, we have theorem 2.1. in order that the complete lift f c of a (1,1) tensor field f admitting fλ (6,4)-structure in m may have the similar structure in the cotangent bundle ctm, it is necessary and sufficient that f j k f kl f l mf m n lhj + f j i f i h f lmf m n ljl + f l kf k j f j i f i h lln = λ 2 { f pq f q n lhp + f p r f r h lpn } 3. nijenhuis tensor of complete lift of f 6 the nijenhuis tensor of (1,1) tensor field f on m is given by nf ,f (x,y )= [f x, f y ]− f [f x, y ]− f [x, f y ]+ f 2[x,y ] (3.1) also, for the complete lift of f 6, the nijenhuis tensor is given by n (f 6 ) c ,(f 6 ) c (xc,y c)= [(f 6 ) c xc , (f 6) c y c ] − (f 6) c [(f 6) c xc , y c] − (f 6) c [xc , (f 6) c y c ] +(f 6 ) c (f 6) c [xc , y c] (3.2) in the view of the equation (2.1), the above equation takes the form n (f 6 ) c ,(f 6 ) c (xc,y c)= [(λ 2 f 4) c xc , (λ 2 f 4) c y c ] −(λ2f 4) c [ (λ 2 f 4) c xc, y c ] −(λ2f 4) c [ xc, (λ 2 f 4) c y c ] +(λ 2 f 4) c (λ 2 f 4) c [xc , y c] (3.3) int. j. anal. appl. (2022), 20:56 5 = λ4{[(f 4)c xc , (f 4)cy c] − (f 4)c [ (f 4) c xc, y c ] −(f 4)c [ xc, (f 4) c y c ] +(f 4) c (f 4) c [xc , y c]} also, (f 4) c xc = (f 4x) c +ν(lxf 4) (3.4) where (νf ) has components (νf )= [ p0a f a i ] (3.5) in view of the equation (3.4), the equation (3.3) takes the form of a horizontal lift of fλ(6,4) structure. n( (f 4 ) c ,(f 4 ) c ) (xc, y c) = λ4 {[(f 4x)c, (f 4y )c] +[ν (lxf 4) , (f 4y )c] +[(f 4x)c,ν (ly f 4)] + [ ν ( lxf 4 ) , ν ( ly f 4 )] − (f 4) c [ (f 4 x) c , y c ] − (f 4) c [ ν ( lxf 4 )c , y c ] − (f 4) c [ xc,(f 4 y ) c ] − (f 4) c [ xc,ν ( ly f 4 )] + (f 4 ) c (f 4 ) c [xc, y c]} (3.6) let us now suppose that lxf 4 − ly f 4 =0 (3.7) the equation (3.6) takes the form n( (f 4 ) c ) ,(f 4 ) c ( xc, y c ) = λ4 { [(f 4 x) c , (f 4 y ) c ] }− (f 4) c [ (f 4 ) c , y c ] (3.8) − (f 4) c [ xc,(f 4 y ) c ] + (f 4 ) c (f 4 ) c [xc, y c] suppose further that the (1,1) tensor field f satisfies f 4 = λ2i (3.9) then in the view of the equation (3.8), the equation (3.7) takes the form of n( (f 4 ) c ) ,(f 4 ) c ( xc, y c ) = λ8 {[ xc, y c ] − [ xc, y c ] − [ xc, y c ] + [ xc, y c ]} =0. hence, we have theorem 3.1. the nijenhuis tensor of the complete lift of f 6 vanishes if the lie – derivatives of the tensor field f 4 with respect to x and y are both zero and the tensor field f 2 acts as gfstructure operator on m. 6 int. j. anal. appl. (2022), 20:56 4. horizontal lift of fλ (6,4)structure let f, g be the tensor fields of type (1, 1) of manifold m. if f h be the horizontal lift of f, we have by [4] and [14] f hgh + ghf h = (f g + gf ) h (4.1) equating f and g, we get (f h) 2 = (f 2) h (4.2) squaring equation (4.2) on both sides we get, (f h) 4 = (f 4) h (4.3) taking cube of (4.2) and using (4.2) itself and (4.3) we get, (f h) 6 = (f 6) h (4.4) since f gives fλ(6,4) – structure on m, so f 6 − λ2f 4 = 0 taking horizontal lift in the above equation we get, (f 6) h − λ2(f 4) h = 0 (4.5) in view of the equation (4.3) and (4.4), the above equation (4.5) takes the form (f 6) h − λ2(f h) 4 = 0 thus, we have the following theorem: theorem 4.1. let f be the tensor field of type (1, 1) admitting fλ(6,4) structure in m. then the horizontal lift f h of f also admits the similar structure in the cotangent bundle ctm. 5. vertical lift of fλ (6,4)structure let f, g be the tensor fields of type (1, 1) of manifold m. if f v be the vertical lift of f, we have f v gv + gv f v = (f g + gf ) v (5.1) equating f and g, we get (f v ) 2 = (f 2) v (5.2) squaring equation (5.2) on both sides we get, (f v ) 4 = (f 4) v (5.3) taking cube of (5.2) and using (5.2) itself and (5.3) we get, (f v ) 6 = (f 6) v (5.4) int. j. anal. appl. (2022), 20:56 7 since f gives fλ(6,4) – structure on m, so f 6 − λ2f 4 = 0 taking vertical lift in the above equation we get, (f 6) v − λ2(f 4) v = 0 (5.5) in view of the equation (5.3) and (5.4), the above equation (5.5) takes the form (f 6) v − λ2(f v ) 4 = 0 thus, we have the following theorem: theorem 5.1. let f be the tensor field of type (1, 1) admitting fλ(6,4) structure in m. then the vertical lift f v of f also admits the similar structure in the cotangent bundle ctm. 6. conclusion in this research, fλ(6,4) structure has been defined on an n-dimensional differentiable manifold of class c∞. further properties of complete, horizontal and vertical lifts of fλ(6,4) structure are defined on its cotangent bundle. the necessary and sufficient conditions for cotangent bundles to have the properties of m in complete, horizontal and vertical lifts are also discussed. properties of nijenhuis tensor of complete lift of f 6 is also a part of this paper. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] h. cayir, tachibana and vishnevskii operators applied to xv and xh in almost paracontact structure on tangent bundle t(m), new trends math. sci. 4 (2016), 105-115. https://doi.org/10.20852/ntmsci.2016318821. [2] h. cayir, lie derivatives of almost contact structure and almost paracontact structure with respect to xv and xh on tangent bundle t(m), proc. inst. math. mech. 42 (2016), 38-49. [3] h. cayir, some notes on lifts of the ((ν+1),λ2(ν−1)) structure on cotangent and tangent bundle, commun. fac. sci. univ. ank. ser. a1 math. stat. 70 (2021), 241–264. https://doi.org/10.31801/cfsuasmas.712861. [4] l.s. das, r. nivas, v.n. pathak, on horizontal and complete lifts from a manifold with fλ(7;1)-structure to its cotangent bundle, int. j. math. math. sci. 8 (2005), 1291-1297. https://doi.org/10.1155/ijmms.2005.1291. [5] k.k. dube, on a differentiable structure satisfying f 2v+4 + f 2 =0; f 6 =0 of type (1; 1), nepali math. sci. rep. 17 (1998), 99-102. [6] j.b. kim, notes on f-manifold, tensor n-s, 29 (1975), 299-302. [7] t. li, d. krupka, the geometry of tangent bundles: canonical vector fields, geometry. 2013 (2013), 364301. https://doi.org/10.1155/2013/364301. [8] r. nivas, m. saxena, on complete and horizontal lifts from a manifold with hsu-(4; 2) structure to its cotangent bundle, nepali math. sci. rep. 23 (2004), 35-41. [9] s.k. srivastava, r. nivas, on horizontal & complete lifts from a manifold with fλ(7,−1) structure to its cotangent bundle, j. tensor soc. india, 14 (1996), 42-48. https://doi.org/10.20852/ntmsci.2016318821 https://doi.org/10.31801/cfsuasmas.712861 https://doi.org/10.1155/ijmms.2005.1291 https://doi.org/10.1155/2013/364301 8 int. j. anal. appl. (2022), 20:56 [10] s.k. srivastava, on the complete lifts of (1, 1) tensor field f satisfying structure fν+1 −λ2fν−1 = 0, nepali math. sci. rep. 21 (2003), 89-99. [11] m.d. upadhyay, v.c. gupta, integrability conditions of a f(k;−(k −2)) structure satisfying fk −fk−2 =0; (f 6=0; i), rev. univ. nac. tucuman, 20 (1976), 31-44. [12] k. yano, on a structure defined by a tensor field f of type (1,1) satisfying f 3+ f =0, tensor n. s. 14 (1963), 99-109. [13] k. yano, c.s. houh, b.y. chen, structures defined by a tensor field φ of type (1,1) satisfying φ4 ± φ2 = 0, tensor n. s. 23 (1972), 81-87. [14] k. yano, s. ishihara, tangent and cotangent bundles, marcel dekker inc., new york, (1973). 1. introduction 2. complete lift of f(6,4) structure 3. nijenhuis tensor of complete lift of f6 4. horizontal lift of f(6,4)structure 5. vertical lift of f(6,4)structure 6. conclusion references international journal of analysis and applications volume 16, number 1 (2018), 117-124 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-117 permanently weak amenability of rees semigroup algebras hasan hosseinzadeh1 and ali jabbari2 1department of mathematics, ardabil branch, islamic azad university, ardabil, iran 2department of mathematics, payame noor university, tehran, iran ∗corresponding author: jabbari al@yahoo.com abstract. in this paper, we consider n-weak amenability of full matrix algebras and we prove that the rees semigroup algebra is permanently weakly amenable. 1. introduction let a be a banach algebra, and let x be a banach a-bimodule. then a linear map d : a −→ x is a derivation if d(ab) = a ·d(b) + d(a) · b for every a,b ∈ a. let x ∈ x, and set δx(a) = a · x − x · a for every a ∈ a. then δx is a derivation; these derivations are inner derivations. the space of continuous derivations from a into x is denoted by z1(a,x), and the subspace consisting of the inner derivations is n1(a,x); the first cohomology group of a with coefficients in x is h1(a,x) = z1(a,x)/n1(a,x). a banach algebra a is weakly amenable if h1(a,a∗) = {0}. for example, the group algebra l1(g) is weak amenable for each locally compact group g [7]. let k ∈ n; a banach algebra a is called k-weakly amenable if h1(a,a(k)) = {0}. dales, ghahramani and grønbæk brought the concept of k-weak amenability of banach algebras [5]. a banach algebra a is called received 23rd september, 2017; accepted 7th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. primary 43a07, secondary 46h25. key words and phrases. amenability; inverse semigroup; rees semigroup; weak amenability. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 117 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-117 int. j. anal. appl. 16 (1) (2018) 118 permanently weakly amenable if h1(a,a(k)) = {0}, for each k ∈ n. in [5], authors showed that for a locally compact group g, l1(g) is n-weakly amenable for all odd numbers n, but for even case this was open. this open problem solved in [4] and a new prove introduced by zhang [9]. the above mentioned problem open for semigroups and semigroup algebras. for rees semi group algebras, mewomo [8], proved that these algebras are (2k+1)-weakly amenable, in this paper, we investigate permanent weak amenability of n×n matrix banach algebras. finally, we prove that the rees semigroup algebras are permanently weak amenable. 2. characterization of derivations consider the algebra mn of n×n matrices. let a be a banach algebra. the banach algebra mn(a) is the collection of n×n matrices with components in a. we identify the dual of mn(a) with mn(a∗) and we have (a · λ)ij = n∑ s=1 ajs ·λis, (λ ·a)ij = n∑ s=1 λsj ·asi, (2.1) for each a = (aij) ∈ mn(a) and λ = (λij) ∈ mn(a∗). derivations from mn(a) into mn(a ∗) is studied in [1]. set eij which it is a n×n matrix, such that whose (i,j)th entry is 1 and other entries are 0. for each a ∈ a, the matrix a⊗eij is a matrix that whose (i,j)th entry is a and others entries are 0. lemma 2.1. let a be a banach algebra and let d : a −→ a∗ be a continuous derivation, then d induces a continuous derivation d : mn(a) −→ mn(a∗). moreover, if d is an inner derivation, then d is inner derivation. proof. define d : mn(a) −→ mn(a∗) by d((a)ij) = (d(aij)) or d((a)ij) = (d(aji)). clearly, continuity of d implies continuity of d. similar to argumentation in [6, pp. 17], we have d(ab) = a · d(b) + d(a) · b for every a,b ∈ mn(a). thus, d is a module derivation. as well as, if d is inner, by a similar method in proof of theorem 2.7 of [6], d is inner. � by (2.1), 〈λ⊗ekl, (λij) · (aij)〉 = 〈(aij) · (λ⊗ekl), (λij)〉 = 〈 n∑ s=1 (asl ·λ⊗ekl), (λij)〉 = n∑ s=1 〈asl ·λ, λks〉 = 〈λ, n∑ s=1 λks ·asl〉, (2.2) for each λ ∈ a∗, (λij) ∈ mn(a∗∗), (aij) ∈ mn(a) and 0 ≤ k,l ≤ n. hence, (2.2) implies that ((λij) · (aij))kl = n∑ s=1 λks ·asl, (2.3) int. j. anal. appl. 16 (1) (2018) 119 for each (λij) ∈ mn(a∗∗), (aij) ∈ mn(a) and 0 ≤ k,l ≤ n. similarly ((aij) · (λij))kl = n∑ s=1 aks · λsl, (2.4) for each (λij) ∈ mn(a∗∗), (aij) ∈ mn(a) and 0 ≤ k,l ≤ n. by induction on m, for each (aij) ∈ mn(a) and (λij) ∈ mn(a(m)) we have ((λij) · (aij))kl = n∑ s=1 λsl ·ask, ((aij) · (λij))kl = n∑ s=1 als ·λks, (2.5) when m is odd and in the case where m is even, we have the following actions: ((λij) · (aij))kl = n∑ s=1 λks ·asl, ((aij) · (λij))kl = n∑ s=1 aks ·λsl. (2.6) now; we are ready to prove the following lemma that plays an important role in our main results. lemma 2.2. let a be a unital banach algebra. then every derivation from mn(a) into mn(a (m)) (a(m) is the m-th dual of a) is the sum of an inner derivation and a derivation induced by a derivation from a into a(m). proof. let ea be the identity element of a. suppose that d : mn(a) −→ mn(a(m)) is a continuous derivation. for each i,j and k,l, define dklij : a −→ a (m) by dklij (a) := (d(a ⊗ eij))kl, for each a ∈ a. clearly, dklij is linear. we prove this lemma in two cases. case 1. let m be an odd positive number. for every a,b ∈ a and each 1 ≤ t ≤ n, we have ( [ d(a⊗eit) ] · (b⊗etj))kl = n∑ s=1 (d(a⊗eit))sl · (b⊗etj)sk = n∑ s=1 dslit (a) · bδtsδjk = d tl it(a) · bδjk, and ((a⊗eit) · [ d(b⊗etj) ] )kl = n∑ s=1 (a⊗eit)ls · (d(b⊗etj))ks = n∑ s=1 aδilδts ·dkstj (b) = aδil ·d kt tj (b), where δ is the kronecker’s delta. then dklij (ab) = aδil ·d kt tj (b) + d tl it(a) · bδjk. (2.7) thus, diiii is a derivation from a into a (m). from (2.5) and (2.7), the following statements hold d jl ij(a) = d il ii(ea) ·a (i 6= l), d ki ij (a) = a ·d kj jj (ea) (j 6= k), (2.8) int. j. anal. appl. 16 (1) (2018) 120 and again by (2.7) and for 1 ≤ i,j, l ≤ n, we have d jj jj (a) = d ij ji(ea) ·a + d ji ij (a) = d ij ji(ea) ·a + d li il(ea) ·a + d jl lj (a) = d ij ji(ea) ·a + d li il(ea) ·a + d ll ll(a) + a ·d jl lj (ea), (2.9) and d ij ji(a) = a ·d ij ji(ea) + d jj jj (a). (2.10) hence d ij ji(ea) = −d ji ij (ea) for every 1 ≤ i,j ≤ n, and consequently by (2.9), the following relation holds d ji ij (a) = d li il(ea) ·a−a ·d lj jl(ea) + d ll ll(a). (2.11) together with (2.9) and (2.10) we have d ji ij (a) = d ij ji(a) −d ij ji(ea) ·a−a ·d ij ji(ea), (2.12) for every a ∈ a. by (2.7) and (2.10) the following equality holds d ij kl(a) = d ij ki(ea) ·a + d ii il (a) = d ij ki(ea) ·a + d ji ij (ea) ·a + d ij jl (a) = d ij ki(ea) ·a + d ji ij (ea) ·a + a ·d ij jl (ea) + d jj jj (a) = d ij ki(ea) ·a + a ·d ij jl (ea) −d ij ji(ea) ·a−a ·d ij ji(ea) + d ij ji(a), (2.13) for every a ∈ a. then by (2.8), (2.12) and (2.13), we have (d(ars))ij = n∑ k,l=1 d ij kl(akl) = n∑ k=1 d ij ki(ea) ·aki + n∑ l=1 diiil (ail) = n∑ k=1 d ij ki(ea) ·aki + n∑ l=1 ajl ·d ij jl (ea) −dijji(ea) ·aji −aji ·d ij ji(ea) + d ij ji(aji) = n∑ k=1 d kj kk(ea) ·aki + n∑ k=1 ajk ·dikkk(ea) + d ji ij (aji), (2.14) for every (ars) ∈ mn(a). as well as, (d(ekkeii))ik = n∑ k=1 dskkk(ea)δsi + n∑ k=1 δksd is ii (ea) = d ik kk(ea) + d ik ii (ea) = 0. this shows that dikkk(ea) = −d ik ii (ea). now; for every 1 ≤ k,j ≤ n define dkj = d kj kk. by the above obtained results we have (d(ars))ij = n∑ k=1 dkj(ea) ·aki − n∑ k=1 ajk ·dik(ea) + d ji ij (aji) = ((drs(ea)) · (ars) − (ars) · (drs(ea)))ij + d ji ij (aji). (2.15) int. j. anal. appl. 16 (1) (2018) 121 set d(ea) =   dl11l(ea) . . . 0 ... dl22l(ea) ... 0 . . . dlnnl(ea)   n×n . then by (2.11) and (2.15) we have d((ars)) = ( dij(ea) + d(ea) ) · (aij) − (aij) · ( dij(ea) + d(ea) ) +(dllll(aij)), where (dllll(aij)) is a diagonal matrix. case 2. now; let m be an even positive number. then by (2.6) we have ( [ d(a⊗eit) ] · (b⊗etj))kl = n∑ s=1 (d(a⊗eit))ks · (b⊗etj)sl = n∑ s=1 dksit (a) · bδtsδjl = d kt it (a) · bδjl, and ((a⊗eit) · [ d(b⊗etj) ] )kl = n∑ s=1 (a⊗eit)ks · (d(b⊗etj))sl = n∑ s=1 aδikδts ·dsltj(b) = aδik ·d tl tj(b), for every a,b ∈ a. then dklij (ab) = aδik ·d tl tj(b) + d kt it (a) · bδjl. (2.16) thus, diiii is a derivation from a into a (m). by (2.6) and (2.16), the following equalities hold d kj ij (a) = d ki ii (ea) ·a (k 6= i), d il ij(a) = a ·d jl jj(ea) (j 6= l), (2.17) and for 1 ≤ i,j, l ≤ n, (2.16) follows diiii(a) = d ji ji(a) + d ij ij (ea) ·a = d ij ij (ea) ·a + d jl jl(ea) ·a + d li li(a) = d ij ij (ea) ·a + d jl jl(ea) ·a + d ll ll(a) + a ·d li li(ea), (2.18) and d ji ji(a) = d ii ii(a) + d ji ji(ea) ·a, (2.19) for every a ∈ a. therefore dijij (ea) = −d ji ji(ea), for every 1 ≤ i,j ≤ n. then (2.18) implies that d ji ji(a) = d jl jl(ea) ·a−a ·d il il(ea) + d ll ll(a). (2.20) as well as, d ij kl(a) = d ij kj(ea) ·a + d jj il (a) = d ij kj(ea) ·a + a ·d ij il (ea) + d ji ji(a), (2.21) int. j. anal. appl. 16 (1) (2018) 122 for every a ∈ a. by using the relations (2.17) and (2.21), for every (ars) ∈ mn(a), we have (d(ars))ij = n∑ k,l=1 d ij kl(akl) = n∑ l=1 d ij kj(ea) ·akj + n∑ k=1 d jj il (ail) = n∑ k=1 d ij kj(ea) ·akj + n∑ k=1 ail ·d ij il (ea) + d ji ji(aji) = n∑ k=1 dikkk(ea) ·akj + n∑ k=1 aik ·d kj kk(ea) + d ji ji(aji). (2.22) since (d(ekkeii))ik = n∑ k=1 dkskk(ea)δis + n∑ k=1 δisd sk ii (ea) = d ki kk(ea) + d ik ii (ea) = 0, (2.23) dkikk(ea) = −d ik ii (ea). now; define dkj = d kj kk for every 1 ≤ j,k ≤ n. then by the above obtained results we have (d(ars))ij = n∑ k=1 dik(ea) ·akj − n∑ k=1 aik ·djk(ea) + d ji ji(aji) = ((drs(ea)) · (ars) − (ars) · (drs(ea)))ij + d ji ji(aji). (2.24) similar to case 1, set d(ea) =   d1l1l(ea) . . . 0 ... d2l2l(ea) ... 0 . . . dnlnl(ea)   n×n . now; by applying (2.20) and (2.24) the following holds d((ars)) = ( dij(ea) + d(ea) ) · (aij) − (aij) · ( dij(ea) + d(ea) ) +(dllll(aij)). hence proof is complete. � weak amenability and (2k + 1)-weak amenability of mn(a) considered in [3, 8]. now; by above lemma we have the following result: theorem 2.1. let a be a unital banach algebra. then a is permanently weakly amenable if and only if mn(a) is permanently weakly amenable. proof. let mn(a) be permanently weakly amenable and let d : a −→ a(k) be a continuous derivation, k ∈ n. then by lemma 2.1, d induces a continuous derivation d : mn(a) −→ mn(a(k)). hence, by our assumption d is inner and lemma 2.1, implies that d is inner. conversely, suppose that a is permanently weakly amenable. let d : mn(a) −→ mn(a(k)) be a continuous module derivation, k ∈ n. then by lemma 2.2, it is equal to the sum of an inner derivation and a int. j. anal. appl. 16 (1) (2018) 123 derivation induced by a derivation from a into a(k). since a is permanently weakly module, d is equal to sum of two inner derivations. thereby, mn(a) is permanently weakly module amenable. � example 2.1. let g be a discrete group. then by [4], [5] and theorem 2.1, mn(` 1(g)) is permanently weakly amenable. example 2.2. let a be a unital c∗-algebra. then mn(a) is permanently weakly amenable. 3. rees semigroup algebras let g be a group, and m,n ∈ n; the zero adjoined to g is o. a rees semigroup has the form s = m(g,p,m,n); here p = (aij) ∈ mn,m(g) is the collection of n × m matrices with components in g. for x ∈ g, 1 ≤ i ≤ m and 1 ≤ j ≤ n, let (x)ij be the element of mm,n(go) with x in the (i,j)-th place and o elsewhere. as a set, s consists of the collection of all these matrices (x)ij. multiplication in s is given by the formula (x)ij(y)kl = (xajky)il (x,y ∈ g, 1 ≤ i,k ≤ m, 1 ≤ j, l ≤ n). it is known that s is a semigroup. now; consider the semigroup mo(g,p,m,n), where the elements of this semigroup are those of m(g,p,m,n), together with the element o, identified with the matrix that has o in each place (so that o is the zero of mo(g,p,m,n)), and the components of p are belong to go. the matrix p is called the sandwich matrix in each case. the semigroup mo(g,p,m,n) is a rees matrix semigroup with a zero over g. we write mo(g,p,n) for mo(g,p,n,n) in the case where m = n. as well as, p is called regular if every row and column contains at least one entry in g. the semigroup mo(g,p,m,n) is regular as a semigroup if and only if the sandwich matrix p is regular. according to [6] we have the following equalities as banach spaces `1(s) = mo(`1(g),p,m,n) = m(`1(g),p,m,n) ⊕cδ0. bowling and duncan proved that for any rees semigroup s, `1(s) is weakly amenable [3, theorem 2.5] and after them mewomo in [8], proved that `1(s) is (2k + 1)-weakly amenable where s = mo(g,p,n), for k,n ∈ n. now; we are completing them works as follows: theorem 3.1. let s = mo(g,p,n), n ∈ n. then `1(s) is permanently weakly amenable. proof. it is sufficient we show that `1(s) is (2k)-weakly amenable, for k ∈ n. for any locally compact group g, `1(g) is permanently weakly amenable ( [4, pp. 3179] and [5, theorem 4.1]). theorem 2.1 implies that mn(` 1(g)) is (2k)-weakly amenable. since mn(` 1(g)) = `1(s), `1(s) is (2k)-weakly amenable. � let s be a semigroup. the weak amenability of `1(s) is considered by blackmore in [2]. he proved that `1(s) to be weakly amenable whenever s is completely regular, in the sense that, for each s ∈ s, there exists int. j. anal. appl. 16 (1) (2018) 124 t ∈ s with sts = s and st = ts. suppose that s has a zero o. then s is o-simple if s[2] 6= {o} and the only ideals in s are {o} and s. the semigroup s is called completely o-simple if it is o-simple and contains a primitive idempotent. corollary 3.1. let s be an infinite, completely o-simple semigroup with finitely many idempotents. then `1(s) is permanently weakly amenable. proof. by corollary 4.2 of [8], it suffices to show that `1(s) is (2k)-weakly amenable, for k ∈ n. the semigroup s is isomorphic as a semigroup to a regular rees matrix semigroup with a zero mo(g,p,n), n ∈ n [6, theorem 3.13]. now; apply theorem 3.1. � acknowledgment. the authors wish to thank professor yong zhang for pointing out the reference [9] in response to an earlier version of this article. this work is supported by a grant from the ardabil branch, islamic azad university. thus; the first author would like to express his deep gratitude to the ardabil branch, islamic azad university, for financial supports. references [1] r. alizadeh and g. esslamzadeh, the structure of derivations from a full matrix algebra into its dual, iranian j. sci. tec, trans. a, 32(2008), 61–64. [2] t. d. blackmore, weak amenability of discrete semigroup algebras, semigroup forum, 55(1997), 196–205. [3] s. bowling and j. duncan, first order cohomology of banach semigroup algebras, semigroup forum, 56(1998), 130–145. [4] y. choi, f. ghahramani, y. zhang, approximate and pseudo-amenability of various classes of banach algebras, j. funct. anal., 256(2009), 3158-3191. [5] h. g. dales, f. ghahramani and n. grønbæk, derivatios into iterated duals of banach algebras, studia math., 128(1)(1998), 19–54. [6] h. g. dales, a. t-m. lau and d. strauss, banach algebras on semigroups and their compactifications, memoirs amer. math. soc., 205, american mathematical society, providence, 2010. [7] b. e. johnson, weak amenability of group algebras, bull. london math. soc., 23(1991), 281-284. [8] o. t. mewomo, on n-weak amenability of rees semigroup algebras, proc. indian acad. sci., 118(4)(2008), 547–555. [9] y. zhang, 2m-weak amenability of group algebras, j. math. anal. appl., 396(2012), 412-416. 1. introduction 2. characterization of derivations 3. rees semigroup algebras acknowledgment references international journal of analysis and applications volume 17, number 4 (2019), 586-595 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-586 common fixed point theorem for ćirić type quasi-contractions in rectangular b-metric spaces shu-fang li, fei he∗ and ning lu school of mathematical sciences, inner mongolia university, hohhot 010021, china ∗corresponding author: email address: hefei@imu.edu.cn abstract. the purpose of this paper is to give positive answers to questions concerning ćirić type quasicontractions in rectangular b-metric spaces proposed in george et al. (j. nonlinear sci. appl. 8 (2015), 1005-1013). 1. introduction and preliminaries in [1], george et al. introduced the concept of rectangular b-metric spaces as a generalization of metric space, rectangular metric space and b-metric space (see also [2, 3]). since then many fixed point theorems for various contractions were established in rectangular b-metric spaces (see [4–12]). definition 1.1. ( [1]) let x be a nonempty set and the mapping d : x ×x → [0,∞) satisfies: (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x) for all x,y ∈ x; (3) there exists a real number s ≥ 1 such that d(x,y) ≤ s[d(x,u) + d(u,v) + d(v,y)] for all x,y ∈ x and all distinct points u,v ∈ x\{x,y}. then d is called a rectangular b-metric on x and (x,d) is called a rectangular b-metric space (in short rbms) with coefficient s. received 2019-03-28; accepted 2019-04-30; published 2019-07-01. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. common fixed point theorem; ćirić type quasi-contractions; rectangular b-metric space. the research was supported by the national natural science foundation of china (11561049, 11471236). c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 586 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-586 int. j. anal. appl. 17 (4) (2019) 587 definition 1.2. ( [1]) let (x,d) be a rbms, {xn} be a sequence in x and x ∈ x. then (1) the sequence {xn} is said to be convergent in (x,d) and converges to x, if for every ε > 0 there exists n0 ∈ n+ such that d(xn,x) < ε for all n > n0 and this fact is represented by limn→∞xn = x or xn → x as n →∞. (2) the sequence {xn} is said to be cauchy sequence in (x,d) if for every ε > 0 there exists n0 ∈ n+ such that d(xn,xn+p) < ε for all n > n0 and p > 0. (3) (x,d) is said to be a complete rbms if every cauchy sequence in x converges to some x ∈ x. in the setting of rbms, limit of a convergent sequence is not necessarily unique and also every convergent sequence is not necessarily a cauchy sequence. for details, we can see [1]. however, we have that the following result. lemma 1.1. ( [3]) let (x,d) be a rbms with s ≥ 1, and let {xn} be a cauchy sequence in x such that xn 6= xm whenever n 6= m. then {xn} can converge to at most one point. george et al. [1] raised the following problems. problem 1.1. ( [1]) in [1, theorem 2.1], can we extent the range of λ to the case 1 s < λ < 1? problem 1.2. ( [1]) prove analogue of chatterjea contraction, reich contraction, ćirić contraction and hardy-rogers contraction in rbms. in [6], mitrović has given a positive answer to problem 1.1. in [7], mitrović et al. obtained an analogue of reich’s contraction principle in rbms and thus give a partial solution to problem 1.2. for further results, the reader can refer to [13, 14]. in this paper, we proved a common fixed point theorem for ćirić type quasi-contractions in rbms. it is well known that ćirić contraction is more general than other contractions in problem 1.2. thus, we give a complete solution to the above problem 1.2. 2. main results the following lemma is crucial in this paper. lemma 2.1. let (x,d) be a rbms with coefficient s ≥ 1 and f,g : x → x be two self maps such that f(x) ⊆ g(x). assume that there exists λ ∈ [0, 1 s ) such that d(fx,fy) ≤ λ max{d(gx,gy),d(gx,fx),d(gy,fy),d(gy,fx),d(gx,fy)}. (2.1) taking x0 ∈ x, we construct a sequence {yn} by yn = fxn = gxn+1. if yn 6= yn+1 for all n ∈ n+, then int. j. anal. appl. 17 (4) (2019) 588 (1) for m ∈ 0 ∪n+ and p ∈ n+, there exists 1 ≤ k(p) ≤ p such that δ(o(ym,m + p)) = d(ym,ym+k(p)), where o(ym,m + p) = {ym,ym+1, · · · ,ym+p},δ(a) = supx,y∈a d(x,y). (2) yn 6= ym whenever n 6= m. (3) δ(o(y0,n)) ≤ s1−sλ[d(y0,y1) + d(y1,y2)]. (4) δ(o(y0,∞)) ≤ s1−sλ[d(y0,y1) + d(y1,y2)], where o(y0,∞) = {y0,y1, · · · ,yn, · · ·}. (5) {yn} is a cauchy sequence. proof. (1) let m ∈ {0, 1, 2, · · · ,} and p ∈ n+. using (2.1), for any i,j ∈ n+ with m < i < j ≤ m + p, we have that d(yi,yj) = d(fxi,fxj) ≤ λ max{d(gxi,gxj),d(gxi,fxi),d(gxj,fxj),d(gxi,fxj),d(gxj,fxi)} = λ max{d(yi−1,yj−1),d(yi−1,yi),d(yj−1,yj),d(yi−1,yj),d(yj−1,yi)} ≤ λδ(o(ym,m + p)) < δ(o(ym,m + p)). this implies that max{d(yi,yj) : i,j ∈ n+ and m < i < j ≤ m + p} < δ(o(ym,m + p)). since δ(o(ym,m + p)) = max{d(yi,yj) : i,j ∈ n+ and m ≤ i < j ≤ m + p}, there exists k(p) with 1 ≤ k(p) ≤ p such that δ(o(ym,m + p)) = d(ym,ym+k(p)). (2.2) (2) suppose that yn = yn+p for some n,p ∈ n+. then, by (2.1) we obtain that δ(o(yn,n + p)) = d(yn,yn+k(p)) = d(yn+p,yn+k(p)) = d(fxn+p,fxn+k(p)) ≤ λ max{d(gxn+p,gxn+k(p)),d(gxn+p,fxn+p),d(gxn+k(p),fxn+k(p)), d(gxn+k(p),fxn+p),d(gxn+p,fxn+k(p))} int. j. anal. appl. 17 (4) (2019) 589 = λ max{d(yn+p−1,yn+k(p)−1),d(yn+p−1,yn+p),d(yn+k(p)−1,yn+k(p)), d(yn+k(p)−1,yn+p),d(yn+p−1,yn+k(p))} ≤ λδ(o(yn,n + p)), which implies δ(o(yn,n + p)) = 0. however, this is impossible because δ(o(yn,n + p)) ≥ d(yn,yn+1) > 0. therefore, yn 6= ym whenever n 6= m. (3) let n ∈ n+. then, using (2.1) and (2.2), we get that δ(o(y0,n)) = d(y0,yk(n)) ≤ s[d(y0,y1) + d(y1,y2) + d(y2,yk(n))] = s[d(y0,y1) + d(y1,y2)] + sd(fx2,fxk(n)) ≤ s[d(y0,y1) + d(y1,y2)] + sλ max{d(gx2,gxk(n)),d(gx2,fx2),d(gxk(n),fxk(n)), d(gx2,fxk(n)),d(gxk(n),fx2)} = s[d(y0,y1) + d(y1,y2)] + sλ max{d(y1,yk(n)−1),d(y1,y2),d(yk(n)−1,yk(n)), d(y1,yk(n)),d(yk(n)−1,y2))} ≤ s[d(y0,y1) + d(y1,y2)] + sλδ(o(y0,n)). this implies that δ(o(y0,n)) ≤ s 1 −sλ [d(y0,y1) + d(y1,y2)]. (2.3) (4) note that limn→∞δ(o(y0,n)) = δ(o(y0,∞)). thus, from (2.3) we see that δ(o(y0,∞)) ≤ s 1 −sλ [d(y0,y1) + d(y1,y2)]. (5) for any n,p ∈ n+, d(yn,yn+p) ≤ λδ(o(yn−1,n + p)) ≤ λ2δ(o(yn−2,n + p)) ≤ ··· ≤ λnδ(o(y0,n + p)) ≤ λnδ(o(y0,∞)) ≤ λn · s 1 −sλ [d(y0,y1) + d(y1,y2] → 0(n →∞). therefore, {yn} is a cauchy sequence in x. � int. j. anal. appl. 17 (4) (2019) 590 theorem 2.1. let (x,d) be a rbms s ≥ 1 and f,g : x → x be two self maps such that f(x) ⊆ g(x), one of these two subsets of x being complete. if there exists λ ∈ [0, 1 s ) such that d(fx,fy) ≤ λ max{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx)}, (2.4) for all x,y ∈ x, then f and g have a point of coincidence in x. moreover, if f and g are weakly compatible (i.e., they commute at their coincidence points), then they have a unique common fixed point. proof. let x0 be an arbitrary point of x. choose x1 ∈ x such that fx0 = gx1. now, we can construct a sequence {yn} defined by yn = fxn = gxn+1, for n = 0, 1, 2, . . . (2.5) if yk = yk+1 for some k ∈ n+, then fxk+1 = yk+1 = yk = gxk+1 and f and g have a point of coincidence. suppose, further, that yn 6= yn+1 for all n ∈ n+. by lemma 2.1, we can obtain {yn} is a cauchy sequence in x. suppose, e.g., that the subspace g(x) is complete (the proof when f(x) is complete is similar). then {yn} tends to some ω ∈ g(x), where ω = gu for some u ∈ x. suppose that fu 6= gu. then d(fu,yn) = d(fu,fxn) ≤ λ max{d(gu,gxn),d(gu,fu),d(gxn,fxn),d(gu,fxn),d(gxn,fu)} = λ max{d(gu,yn−1),d(gu,fu),d(yn−1,yn),d(gu,yn),d(yn−1,fu)}. note that d(gu,yn−1) → 0, d(yn−1,yn) → 0 and d(gu,yn) → 0 as n → ∞. then, for sufficiently large n ∈ n+, max{d(gu,yn−1),d(gu,fu),d(yn−1,yn),d(gu,yn),d(yn−1,fu)} = max{d(gu,fu),d(yn−1,fu)} and d(fu,yn) ≤ λ max{d(gu,fu),d(yn−1,fu)}. (2.6) denote m(xn,u) = max{d(gu,fu),d(yn−1,fu)} for n ∈ n+. then we can consider the following cases. case 1. if there exists a subsequence {m(xnk,u)} of {m(xn,u)} such that m(xnk,u) = d(gu,fu), then d(fu,ynk ) ≤ λd(gu,fu). note that d(yn,yn−1) → 0, d(yn,gu) → 0 and 1 s d(fu,gu) ≤ d(fu,ynk ) + d(ynk,ynk−1) + d(ynk−1,gu). (2.7) thus, taking upper limit as k →∞ in (2.7), we obtain that 1 s d(fu,gu) ≤ lim sup k→∞ d(fu,ynk ) ≤ λd(gu,fu). this implies that d(gu,fu) ≤ sλd(fu,gu), which is a contradiction with sλ < 1 and fu 6= gu. int. j. anal. appl. 17 (4) (2019) 591 case 2. if there exists n ∈ n+ such that m(xn,u) = d(yn−1,fu) for all n > n, then (2.6) implies that d(fu,yn) ≤ λd(yn−1,fu) ≤ λ2d(yn−2,fu) ≤ ···≤ λn−nd(yn,fu) = λn( 1 λn d(yn,fu)) → 0(n →∞), that is d(fu,yn) → 0 as n →∞. since d(gu,yn) → 0 as n →∞, by lemma 1.1 we have that fu = gu. this is a contradiction. thus, we prove that fu = gu = ω, that is u is a point of coincidence of f and g. if f,g are weakly compatible, then, by fu = gu = ω, we obtain that fω = fgu = gfu = gω, and hence that ω is a point of coincidence of f and g. let us prove that ω = fω = gω. using (2.1), we get that d(ω,fω) = d(fu,fω) ≤ λ max{d(gu,gω),d(gu,fu),d(gω,fω),d(gu,fω),d(gω,fu)} = λ max{d(ω,fω), 0, 0,d(ω,fω),d(fω,ω)} = λd(ω,fω). since λ < 1, we have that d(ω,fω) = 0, which implies that ω = fω = gω. therefore, ω is a common fixed point of f and g. let us prove that the common fixed point of f and g is unique. suppose that ω1 and ω2 are two common points of f and g, that is ω1 = fω1 = gω1 and ω2 = fω2 = gω2. using (2.1), we get that d(ω1,ω2) = d(fω1,fω2) ≤ λ max{d(gω1,gω2),d(gω1,fω1),d(gω2,fω2),d(gω1,fω2),d(gω2,fω1)} = λd(ω1,ω2). since λ < 1, we have that d(ω1,ω2) = 0, that is ω1 = ω2. thus, the common fixed point of f and g is unique. � taking g = ix (identity mapping of x) in theorem 2.1 we obtain the following. corollary 2.1. (ćirić type contraction) let (x,d) be a rbms with coefficient s ≥ 1 and f : x → x be a mapping. assume that there exists λ ∈ [0, 1 s ) d(fx,fy) ≤ λ max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)} for all x,y ∈ x. then f has a unique fixed point. from corollary 2.1, the following corollaries immediately follow. int. j. anal. appl. 17 (4) (2019) 592 corollary 2.2. (chatterjea type contraction) let (x,d) be a rbms with coefficient s ≥ 1 and f : x → x be a mapping. assume that there exists k ∈ [0, 1 s ) such that d(fx,fy) ≤ k 2 (d(x,fy) + d(y,fx)), for all x,y ∈ x. then f has a unique fixed point. corollary 2.3. (reich type contraction) let (x,d) be a rbms with coefficient s ≥ 1 and f : x → x be a mapping. assume that there exist λ,µ,δ ∈ [0, 1) with λ + µ + δ < 1 s such that d(fx,fy) ≤ λd(x,y) + µd(x,fx) + δd(y,fy), for all x,y ∈ x. then f has a unique fixed point. corollary 2.4. (hardy-rogers type contraction) let (x,d) be a rbms with coefficient s ≥ 1 and f : x → x be a mapping. assume that there exist αi ∈ [0, 1)(i = 1, 2, 3, 4, 5) with α1 + α2 + α3 + α4 + α5 < 1s such that d(fx,fy) ≤ α1d(x,y) + α2d(x,fx) + α3d(y,fy) + α4d(x,fy) + α5d(y,fx), for all x,y ∈ x. then f has a unique fixed point. remark 2.1. from corollary 2.1-corollary 2.4, we see that problem 1.2 has been fully answered. finally, we give an example to illustrate our main result. example 2.1. let x = a ⋃ b, where a = { 1, 1 2 , 1 4 , 1 8 } and b = {0, 2}. define d: x ×x → [0, +∞) such that d(x,y) = d(y,x) for all x,y ∈ x and d(x,y) =   0, x = y; |x−y|, x,y ∈ a; 13 6 , x,y ∈ b; 3 4 , x ∈ a\{1}, y ∈ b; 2, x = 1, y ∈ b. let f: x → x be a map defined by f(x) =   1, x ∈ b; x 2 , x ∈ a\{1 8 }; 1 8 , x = 1 8 . and g be an identity mapping on x. then the following hold: (a) (x,d) is a complete rectangular b-metric space with coefficient s = 4 3 ; int. j. anal. appl. 17 (4) (2019) 593 (b) (x,d) is neither a metric space nor a rectangular metric space; (c) all conditions in theorem 2.1 are satisfied with λ = 1 2 ; (d) f and g have a unique common fixed point x = 1 8 . proof. first, let us prove (a). clearly, conditions (1) and (2) of definition 1.1 hold. to see (3), for all x,y ∈ x and all distinct points u,v ∈ x \{x,y}, we consider the following three cases. case 1. if x,y ∈ a or x,y ∈ b, we only need to consider the case of x,y ∈ b with u,v ∈ a\{1}. in this case, d(u,v) ≥ d( 1 4 , 1 8 ) = 1 8 . so we have d(x,y) = 13 6 = 4 3 ( 3 4 + 1 8 + 3 4 ) ≤ 4 3 [d(x,u) + d(u,v) + d(v,y)]. case 2. if x ∈ a\{1} and y ∈ b, then d(x,y) = 3 4 . let us consider the following three cases. • if v ∈ b ⋃ {1}, then d(x,y) = 3 4 < d(v,y) ≤ d(x,u) + d(u,v) + d(v,y). • if u ∈ b, then d(x,y) = 3 4 = d(x,u) ≤ d(x,u) + d(u,v) + d(v,y). • if u,v ∈ a and v 6= 1, then d(x,y) = 3 4 = d(v,y) ≤ d(x,u) + d(u,v) + d(v,y). case 3. if x = 1 and y ∈ b, then we consider the following two cases. • if u ∈ b or v ∈ b, then d(x,u) = 2 or d(v,y) = 13 6 . so we have d(x,y) = 2 ≤ d(x,u) + d(v,y) ≤ d(x,u) + d(u,v) + d(v,y). • if u,v ∈ a, then v 6= 1. it follows that d(x,u) + d(u,v) ≥ d(1, 1 2 ) + d( 1 2 , 1 4 ) = 3 4 . so we have d(x,y) = 2 = 4 3 ( 3 4 + 3 4 ) ≤ 4 3 [d(x,u) + d(u,v) + d(v,y)]. additionally, in this case, we can also check that (b) holds. hence, from the above three cases, we prove that (x,d) is a rectangular b-metric space with coefficient s = 4 3 . since x is a finite set, we know that (g(x),d) = (x,d) is complete. now we prove (c). it is sufficient to prove that (2.4) holds with λ = 1 2 . since d(x,y) = d(y,x), we consider the following three cases. case 1. if x,y ∈ b. in this case, d(fx,fy) = 0. so (2.4) holds. int. j. anal. appl. 17 (4) (2019) 594 case 2. if x ∈ b and y ∈ a, then fx = 1, d(gx,fx) = 2 and fy ∈ a. in this case, we have d(fx,fy) ≤d(1, 1 8 ) = 7 8 < 1 2 d(gx,fx) ≤ 1 2 max{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(fx,gy)}. case 3. if x,y ∈ a, it is clear that d(fx,fy) = 1 2 d(gx,gy) for all x,y ∈ a\{1 8 }, which follows that (2.4) holds. so we assume that x = 1 8 . in this case, we have d(fx,fy) = 1 2 y − 1 8 < 1 2 ( y − 1 8 ) ≤ 1 2 max{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(fx,gy)}. from the above three cases, we show that (c) holds. obviously, f and g have a unique common fixed point fx = gx = x = 1 8 . � references [1] r. george, s. radenović, k. p. reshma and s. shukla, rectangular b-metric space and contraction principles, j. nonlinear sci. appl. 8 (2015), 1005-1013. [2] h. s. ding, v. ozturk and s. radenović, on some new fixed point results in b-rectangular metric spaces, j. nonlinear sci. appl. 8 (2015), 378-386. [3] h. s. ding, m. imdad, s. radenović and j. vujaković, on some fixed point results in b-metric, rectangular and b-rectangular metric spaces , arab. j. math. sci. 22 (2016), 151-164. [4] j. r. roshan, v. parvaneh, z. kadelburg and n. hussain, new fixed point results in b-rectangular metric spaces, nonlinear anal., model. control. 5 (2016), 614-634. [5] d. w. zheng, p. wang, n. citakovic, meir-keeler theorem in b-rectangular metric spaces, j. nonlinear sci. appl. 10 (2017), 1786-1790. [6] z. d. mitrović, on an open problem in rectangular b-metric space, j. anal. 25 (1) (2017), 135-137. [7] z. d. mitrović, r. george and n. hussain, some remarks on contraction mappings in rectangular b-metric spaces, bol. soc. paran. mat., in press. [8] z. d. mitrović and s. radenović, a common fixed point theorem of jungck in rectangular b-metric spaces, acta math. hungar. 153(2) (2017), 401-407. [9] p. sookprasert, p. kumam, d. thongtha and w. sintunavarat, extension of almost generalized weakly contractive mappings in rectangular b-metric spaces and fixed point results, afr. mat. 28 (2017), 271-278 [10] n. hussaina, v. parvanehb, badria a. s. alamria, z. kadelburg, f-hr-type contractions on (α,η)-complete rectangular b-metric spaces, j. nonlinear sci. appl. 10 (2017), 1030-1043. [11] f. gu, on some common coupled fixed point results in rectangular b-metric spaces, j. nonlinear sci. appl. 10 (2017), 4085-4098. [12] o. ege, complex valued rectangular b-metric spaces and an application to linear equations, j. nonlinear sci. appl. 9 (2015), 1014-1021. [13] z. d.mitrović, s. radenović, the banach and reich contractions in bv(s)-metric spaces, j. fixed point theory appl. 19(4) (2017), 3087-3095. int. j. anal. appl. 17 (4) (2019) 595 [14] s. aleksić, z. d.mitrović, s. radenović, a fixed point theorem of jungck in bv(s)-metric spaces, period. math. hung., 77 (2) (2018), 224–231. 1. introduction and preliminaries 2. main results references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 222-228 doi: 10.28924/2291-8639-15-2017-222 functional sequential and trigonometric summability of real and complex functions m.h. hooshmand∗ abstract. limit summability of functions was introduced as a new approach to extensions of the summation of real and complex functions, and also evaluating antidifferences. also, limit summand functions generalize the (logarithm of) gamma-type functions satisfying the functional equation f (x + 1) = f(x)f (x). recently, another approach to the topic entitled analytic summability of functions, has been introduced and studied by the author. since some functions are neither limit nor analytic summable, several types of summabilities are needed for improving the problem. here, i introduce and study functional sequential summability of real and complex functions for obtaining multiple approaches to them. we not only show that the analytic summability is a type of functional sequential summability but also obtain trigonometric summability (for functions with a fourier series) as another its type. hence, we arrive at a class of real and complex function spaces with various properties. thereafter, we prove several properties of functional sequential, and also many criteria for trigonometric summability. finally, we state many problems and future directions for the researches. 1. introduction in the theory of indefinite sum, antidifference and finite calculus, obtaining some special solutions of the difference functional equation ∇f(x) := f(x) −f(x− 1) = f(x) ; x ∈ e, (1.1) is very important, where e is the domain of a real or complex function f or a subset of c (analogously for 4f(x) := f(x + 1) −f(x) = f(x), where 4 is the forward difference operator, e.g., see [1]). but the author discovered an approach to the solution of the equation (on 2001) when he tried to generalize the bohr-mollerup theorem and gamma-type functions (see [2,3,5]). the following are its summary. let f be a real or complex function with domain df ⊇ n∗ = {1, 2, 3, · · ·}. put σf = {x|x + n∗ ⊆ df}, and then for any x ∈ σf and n ∈ n∗ set rn(f,x) = rn(x) := f(n) −f(x + n), fσn(x) = fσ`,n(x) := xf(n) + n∑ k=1 rk(x). the function f is called limit summable at x0 ∈ σf if the functional sequence {fσn(x)} is convergent at x = x0. the function f is called limit summable on the set s ⊆ σf if it is limit summable at all the points of s. now, put fσ(x) = fσ`(x) = lim n→∞ fσn(x) , r(x) = r(f,x) = lim n→∞ rn(f,x). therefore dfσ = {x ∈ σf|f is limit summable at x}, and fσ` = fσ is the same limit function of fσn with domain dfσ . the function f is called limit summable if it is summable on σf , r(1) = 0 and df ⊆ df − 1. in this received 17th september, 2017; accepted 21st october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 40a30; 39a10. key words and phrases. limit summability; analytic summability; antidifferences. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 222 functional sequential and trigonometric summability of functions 223 case the function fσ is referred to as the limit summand function of f. if f is limit summable, then dfσ = df − 1 and fσ(x) = f(x) + fσ(x− 1) ; ∀x ∈ df. therefore, if f is limit summable then its limit summand function fσ satisfies (1.1). the paper [2] states a framework for studying the limit summable functions including basic conditions, many criteria for limit summability, uniqueness conditions for its summand function and so on. example 1.1. if |a| < 1, then the complex function f(z) = az is absolutely limit summable and fσ(z) = a a−1 (a z − 1). also, if 0 < b 6= 1 and 0 < a < 1, then the real function g(x) = cax + logb x is limit summable and gσ(x) = ca a− 1 (ax − 1) + logb γ(x + 1). one can see the topic of limit summability in [2,3]. since some of the important special functions are not limit summable, recently, another type of summability entitled ”analytic summability” has been introduced in [4]. we obtain it again in this paper (as a type of functional sequential summability). 2. functional sequential summability let δ = {δn(x)}∞n=0 be a linearly independent functional sequence of complex (or real) functions on e ⊆ c. put se(δ) := span(δ) = span(δ0(x),δ1(x),δ2(x), · · ·). similarly, if δ = {δn(x)}∞n=0 and β = {βn(x)}∞n=0 are two functional sequences on e such that δ∪β is linearly independent, then we may set se(δ,β) := span(δ ∪β), (se(δ1, · · · ,δk) can be defined analogously). now, let δ = {δn(x)}∞n=0 be as mentioned and ∆ = {∆n(x)}∞n=0 a functional sequence satisfying ∆n(x) − ∆n(x− 1) = δn(x) ; x ∈ e , n ∈ n0 = {0, 1, 2, · · ·}. (2.1) since there are infinitely many such functional sequence ∆ (because (1.1) has the general solution f(x) = fp(x) + ϕ(x) where fp is an its solution and ϕ any 1-periodic function), we fix one of them. the equation implies each ∆n(x) is defined on e ∪ (e − 1). also, since ∆ is linearly independent (see proof of theorem 2.2), we may put sσe(δ; ∆) = s σ e(δ) := span(∆) = span(∆0(x), ∆1(x), ∆2(x), · · ·), and so sσe(δ) = se(∆). note that we use the notation s σ e(δ) when ∆ is well-known in the topic and there is no any risk of confusion. example 2.1. if δn(z) = z n, then δn(z) and ∆n(z) = bn+1(z + 1) − bn+1 n + 1 satisfies the conditions, where bn(z) is the bernoulli polynomial and bn = bn(1) (see [4]). here sσe(δ) is equal to the space of all polynomials with zero constant. note. in continuation, we consider δ, e, ∆, se(δ) and sσe(δ) with the mentioned conditions. lemma 2.1. we have f ∈se(δ) if and only if there exists a unique f ∈sσe(δ) satisfying the difference functional equation (1.1). (notation. hence we shall denote f by fσδ , or simply fσ when there is no any risk of confusion.) proof. if f ∈ se(δ) then there exist δn1, · · · ,δnr ∈ δ and coefficients cn1, · · · ,cnr such that f =∑r 1 cnkδnk. putting f = ∑r 1 cnk∆nk we have f ∈ s σ e(δ) and f satisfies the equation. also, if g ∈sσe(δ) satisfies (2.1) then g = ∑t 1 dnj ∆nj , for some coefficients dm1, · · · ,dmt, and r∑ k=1 cnkδnk = f(x) −f(x− 1) = f(x) = g(x) −g(x− 1) = t∑ j=1 dmjδnj (x) ; x ∈ e. 224 m.h. hooshmand hence, independence of δ implies t = r and {cn1, · · · ,cnr} = {dm1, · · · ,dmr}. therefore, g = r∑ 1 cnk∆nk = f. the converse is obvious (by (2.1)). � definition 2.1. let f(x) = ∑∞ 0 cnδn(x) be convergent on e. we say the function f is (δ, ∆)summable (or simply δ-summable) if the series ∑∞ 0 cn∆n(x) is convergent on e (the terms uniformly and absolutely δ-summable are defined analogously). moreover, if it is the case, then we may put fσδ (x) = fσ(x) := ∞∑ 0 cn∆n(x), and call it δ-summand function of f on e (this symbol agrees with the above notation , i.e., if f ∈se(δ) then the function fσ that is gotten from lemma 2.2 and this definition are the same). now, we set se(δ) := {f : e → c|f is δ-summable} s σ e(δ) := {fσδ|f ∈se(δ)} it is obvious that se(δ) ⊆se(δ) ⊆ the space of all functions with a fourier series on e, and sσe(δ) ⊆ s σ e(δ). theorem 2.1. the sets se(δ) and s σ e(δ) are functions spaces and the map σ = σδ : se(δ) →s σ e(δ), defined by σ(f) := fσ, is a surjective linear map with sσe(δ) ⊆ σ(se(δ)). also, we have (a) fσ satisfies the difference functional equation (1.1). (b) the surjective linear map σ is bijective (and so se(δ) ∼= s σ e(δ)) if and only if δ = {δn(x)}∞n=0 is infinitely linearly independent (i.e., ∑∞ 0 cnδn(x) = 0 on e implies cn = 0, for all n ∈ n0, e.g., δn(x) = x n or δn(x) = e inx). therefore, if this is the case then sσe(δ) = σ(se(δ)). proof. it is obvious that se(δ) and s σ e(δ) are vector spaces and σ is a surjective linear map from se(δ) to s σ e(δ). then, lemma 2.2 implies sσe(δ) ⊆ σ(se(δ)). now, note that linearly independence (resp. infinitely linearly independence) of δ on e implies linearly independence (resp. infinitely linearly independence) of ∆ on e ∪ (e − 1). because if ∑ cnj ∆nj (t) = 0 for all t ∈ e ∪ (e − 1) then∑ cnj ∆nj (x) = 0 , ∑ cnj ∆nj (x− 1) = 0 ; ∀x ∈ e. therefore ∑ cnj (∆nj (x) − ∆nj (x− 1)) = ∑ cnjδnj (x) = 0 ; ∀x ∈ e, and so all cnj are zero. now, if f,g ∈ se(δ) then f(x) = ∑∞ 0 cnδn(x), g(x) = ∑∞ 0 dnδn(x), for all x ∈ e, and fσ(x) =∑∞ 0 cn∆n(x), gσ(x) = ∑∞ 0 dn∆n(x), for all x ∈ e ∪ (e − 1). so fσ(x) −fσ(x− 1) = ∞∑ 0 cn(∆n(x) − ∆n(x− 1)) = ∞∑ 0 cnδn(x) = f(x) ; ∀x ∈ e. also, if σ(f) = σ(g), then ∑∞ 0 (cn −dn)∆n(x) = 0 on e ∪ (e − 1) and so cn −dn = 0 (because δ is infinitely linearly independent), which means σ is injective. hence the proof is complete. � functional sequential and trigonometric summability of functions 225 2.1. analytic summability. let δn(z) = z n (z ∈ c) and consider all terms of example 2.1 on an open domain d = e ⊆ c. then sd(δ) is a subspace of all analytic functions on d. also, f(z) = ∑∞ 0 cnz n ∈sd(δ) is δ-summable if and only if the series ∞∑ 0 cn bn+1(z + 1) − bn+1 n + 1 is convergent on d, and this is the same ”analytic summability of f” from [4]. therefore (according to [4]) we denote it by fσa instead of fσδ and call this f ”analytic summable”, and also denote the space by sd(a). theorem 2.4 implies σa is a surjective linear map from sd(a) to s σ d(a), and so sd(a) ∼= s σ d(a) that is a new result about analytic summability. one can see many criteria, results and open problems about this type of functional sequential summability in it. example 2.2. the exp function is analytic summable in c and expσ(z) = ∞∑ n=0 1 n! σ(zn) = lim n→∞ n∑ n=0 n+1∑ k=1 1 n! n! k!(n + 1 −k)! bn+1−kz k = lim n→∞ n+1∑ n=1 n∑ k=n−1 1 n! bk+1−n (k + 1 −n)! zn = ∞∑ n=1 1 n! ( ∞∑ j=0 bj j! )zn = e e− 1 ∞∑ n=1 zn n! = e e− 1 (ez − 1). so expσa (z) = e e−1 (e z − 1) (details can bee seen in [4]). 3. trigonometric summability now, we can introduce another important type of functional sequential summability in two ways: (a) consider γ = {einx}∞n=−∞ (x ∈ r, n ∈ z). here, the indices set n0 is replaced by z. therefore, in the definition of functional sequential summability, f has the fourier series form f(x) = ∑∞ −∞ cne inx. putting γn(x) = ein ein−1 (e inx − 1) for n 6= 0 and γ0(x) = c0x , one see that γn satisfies the conditions (for all real numbers x). hence, f is γ-summable on e if and only if the following series is convergent ∞∑ −∞ cnγn(x) ; x ∈ e. (3.1) (b) put δ = {cos(nx)}∞n=0 and β = {sin(nx)}∞n=0 (x ∈ r). the sequence δ ∪β is infinitely linearly independent. for every positive integer n, put ∆n(x) = cos(nx) + cos(n) − cos(nx + n) − 1 2(1 − cos(n)) , bn(x) = sin(nx) + sin(n) − sin(nx + n) 2(1 − cos(n)) , and also ∆0(x) = 1, b0(x) = 0. it is easy to see that they satisfy the conditions. it is obvious that f is δ,β-summable on e (i.e., f ∈se(δ,β)) if and only if it has the fourier series of the form f(x) = a0 2 + ∑∞ n=1 an cos(nx) + bn sin(nx) and the following series is convergent ∞∑ n=0 an∆n(x) + bnbn(x). (3.2) now, let f(x) = ∞∑ −∞ cne inx = a0 2 + ∞∑ n=1 an cos(nx) + bn sin(nx). it is easy to see that the series (3.1) is convergent if and only if (3.2) is convergent, and so we arrive at the following definition. 226 m.h. hooshmand definition 3.1. let e, f, γ and δ,β be as the above. we call f trigonometric summable (on e) if the series (3.2) (or equivalently (3.1)) is convergent. also, we put fσtrg (x) = fσ(x) := a0 2 x + ∞∑ n=1 an∆n(x) + bnbn(x) = c0x + ∞∑ −∞ cnγn(x) ; x ∈ e, and call it trigonometric summand of f (on e). now, we prove many equivalent conditions for trigonometric summability and also several criteria for trigonometric summability of functions with a fourier series. theorem 3.1. (some criteria for trigonometric summability of real and complex functions) let f(x) = a0 2 + ∞∑ n=1 an cos(nx) + bn sin(nx) = ∞∑ −∞ cne inx is defined on e ⊆ r. for n 6= 0 put an = an − cot( n 2 )bn , bn = bn + cot( n 2 )an , cn = cn − i cot( n 2 )cn. also, set a0 = a0, b0 = c0 = 0. then (a) the following statements are equivalent: f is trigonometric summable on e; the series ∑∞ n=1 an cos(nx) + bn sin(nx) −an is convergent (to 2fσ(x) −a0x) on e ; the series ∑∞ −∞cne inx −cn is convergent (to fσ(x) − c0x) on e; the series ∑∞ n=1 csc( n 2 ) sin(nx 2 )(an cos(n x+1 2 ) + bn sin(n x+1 2 )) is convergent (to fσ(x) − 12a0x). (b) if the fourier series ∑∞ n=1 bn cos(n x 2 ) − an sin(nx2 ) (i.e., the series is gotten from the fourier series of f, replacing an, bn and x by bn, an and −x2 respectively) is absolutely convergent, then f is absolutely trigonometric summable and |fσ(x) − 1 2 a0x| ≤ ∞∑ n=1 |bn cos(n x 2 ) −an sin(n x 2 )|. (c) if the fourier series ∑∞ n=1 an sin(n 2 ) cos(nx+1 2 ) + bn sin(n 2 ) sin(nx+1 2 ) (i.e., the series is gotten from the fourier series of f, replacing an, bn and x by an sin(n 2 ) , bn sin(n 2 ) and x+1 2 respectively) is absolutely convergent, then f is absolutely trigonometric summable and |fσ(x) − 1 2 a0x| ≤ ∞∑ n=1 | an sin(n 2 ) cos(n x + 1 2 ) + bn sin(n 2 ) sin(n x + 1 2 )|. proof. by using the identity sin(n) 1−cos(n) = cot( n 2 ), we obtain (for all n ≥ 1) 2∆n(x) = cos(nx) − 1 + cot( n 2 ) sin(nx) = 2 sin( nx 2 ){cot( n 2 ) cos( nx 2 ) − sin( nx 2 )} = csc( n 2 ) sin(nx + n 2 ) − 1 = 2 csc( n 2 ) sin( nx 2 ) cos( nx + n 2 ), 2bn(x) = sin(nx) + cot( n 2 )(1 − cos(nx)) = 2 sin( nx 2 ){cos( nx 2 ) + cot( n 2 ) sin( nx 2 )} = cot( n 2 ) − csc( n 2 ) cos(nx + n 2 ) = 2 csc( n 2 ) sin( nx 2 ) sin( nx + n 2 ), 2γn(x) = (1 − i cot( n 2 ))(einx − 1). therefore, ∞∑ n=1 an∆n(x) + bnbn(x) = ∞∑ n=1 an cos(nx) + bn sin(nx) −an = ∞∑ −∞ cn(e inx − 1) = ∞∑ −∞,n6=0 cnγn(x), functional sequential and trigonometric summability of functions 227 and ∞∑ n=1 an(cos(nx) − 1) + bn sin(nx) = 2 ∞∑ n=1 sin(n x 2 ){bn cos(n x 2 ) −an sin(n x 2 )} = ∞∑ n=1 sin( nx 2 )( an sin(n 2 ) cos(n x + 1 2 ) + bn sin(n 2 ) sin(n x + 1 2 )). now one can obtain the results, directly. � the above theorem together with the identity a2n + b 2 n = (a 2 n + b 2 n) csc 2(n 2 ) imply the following corollary. corollary 3.1. let an and bn are two sequences of real or complex numbers. (a) if one of the series ∑∞ n=1 |an|+|bn| |sin(n 2 )| , ∑∞ n=1 √ a2n+b 2 n |sin(n 2 )| is convergent, then the function f(x) := a0 2 +∑∞ n=1 an cos(nx) + bn sin(nx) (is defined in r) is absolutely and uniformly trigonometric summable on whole r and fσ(x) = 1 2 a0x + 1 2 ∑∞ n=1 an cos(nx) + bn sin(nx) −an = 1 2 a0x + ∑∞ n=1 csc( n 2 ) sin(nx 2 )(an cos(n x+1 2 ) + bn sin(n x+1 2 )) = 1 2 a0x + ∑∞ n=1 sin(n x 2 )(bn cos(n x 2 ) −an sin(nx2 )) = c0x + ∑∞ −∞cne inx −cn and f′σ(x) = ∞∑ n=1 nbn cos(nx) −nan sin(nx). hence, f′σ(x) has a fourier series in r. note. if ∑∞ n=1 an − cot( n 2 )bn (= ∑∞ n=1 an) is convergent then trigonometric summability of the function f(x) = a0 2 + ∑∞ n=1 an cos(nx) + bn sin(nx) is equivalent to convergence of the fourier series∑∞ n=1 an cos(nx) + bn sin(nx) , and so fσ(x) − 1 2 a0x (if exists) has a fourier series. example 3.1. consider the real function f(x) := 1 + ∞∑ n=1 sin(n 2 ) n2 cos(nx) + sin(n 2 ) n2 sin(nx) it is defined on whole r and by using the above results, f is absolutely and uniformly trigonometric summable and fσ(x) = x− π2 12 + 1 2 ∞∑ n=1 ( csc(n 2 ) − cos(n 2 ) n2 ) cos(nx) + ( cos(n 2 ) + csc(n 2 ) n2 ) sin(nx) = x + ∞∑ n=1 csc2(n 2 ) n2 sin( nx 2 )(cos(n x + 1 2 ) + sin(n x + 1 2 )) = x + ∞∑ −∞ cne inx −cn. also f′σ(x) = 1 + ∞∑ n=1 ( cos(n 2 ) + csc(n 2 ) n ) cos(nx) + ( cos(n 2 ) − csc(n 2 ) n ) sin(nx) ; x ∈ r, and |fσ(x) −x| ≤ ∞∑ n=1 2 n2 = π2 3 ⇒|fσ(x)| ≤ π2 3 + |x| ; x ∈ r. 228 m.h. hooshmand at last, there are some important questions for future studies of trigonometric, analytic and limit summability of functions (similar to the open problems of [4]). open problem 1. let f be an analytic function defined on open domain d = e . if f is both analytic and trigonometric summable, then is it true that fσtrg = fσa on d? open problem 2. let f be a function with a fourier series on e with the property n∗ ⊆ e ⊆ σf . if f is both limit and trigonometric summable, then is it true that fσl = fσtrg on e? open problem 3. if f is trigonometric summable on e = df , then under what conditions is it a unique solution of the equation (1.1) with the initial property fσ(0) = 0 ? (compare to the uniqueness theorem 3.1, corollary 3.4 of [2] and theorem a, corollary 3.4 of [3]). finally, as another future direction for the researches, one may study and discover other functional sequential summability (by taking some appropriate functional sequences δ = {δn(x)}∞n=0 ), and also intersection of the spaces of limit, trigonometric and analytic summable functions. note that every constant function f(x) = c lies in the intersection and σ`(c) = σa(c) = σrg(c) = cx. references [1] p.m. batchelder, introduction to linear difference equations, dover publications inc., 1968. [2] m.h. hooshmand, limit summability of real functions, real anal. exch. 27(2001), 463-472. [3] m.h. hooshmand, another look at the limit summability of real functions, j. math. ext. 4(2009), 73-89. [4] m.h. hooshmand, analytic summability of real and complex functions, j. contemp. math. anal., 5(2016), 262-268. [5] r. j. webster, log-convex solutions to the functional equation f(x + 1) = g(x)f(x): γ -type functions, j. math. anal. appl., 209 (1997), 605-623. young researchers and elite club, shiraz branch, islamic azad university, shiraz, iran ∗corresponding author: hadi.hooshmand@gmail.com 1. introduction 2. functional sequential summability 2.1. analytic summability 3. trigonometric summability references international journal of analysis and applications volume 16, number 2 (2018), 254-263 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-254 curvature dependent energy of surface curves in minkowski space talat korpinar1, ridvan cem demirkol1,∗ and vedat asil2 1mus alparslan university, department of mathematics, 49250, mus, turkey 2firat university, department of mathematics, 23100, elazig, turkey ∗corresponding author: rcdemirkol@gmail.com abstract. in this paper, we firstly introduce kinematics properties of the moving particle lying on a surface s. we assume that the particle corresponds to a different type of surface curves such that they are characterized by using the darboux vector field w in minkowski spacetime. based on this result, we present geometrical understanding of the energy of the particle in each darboux vector fields whether they lie on a spacelike surface or a timelike surface. then, we also determine the bending elastic energy functional for the same particle on a surface s by assuming the particle has a bending feature of elastica. finally, we prove that bending energy formula can be represented by the energy of the particle in each darboux vector field w. 1. introduction the study of computing an energy for a given vector field depending on the structure of the geometrical spaces has earned such attention in the last couple years. it has been shown that tis type of computations has numerous applications in various fields. thus, multidisciplinary subjects have been evolved. for instance, wood [1] studied energy on the unit vector field firstly. gil-medrano [2] worked on a relation between energy and volume of vector fields. chacon et al. in [3], [4] investigated the energy distribution and corrected energy received 2017-10-17; accepted 2017-12-16; published 2018-03-07. 2010 mathematics subject classification. 53c41, 53a10. key words and phrases. energy; minkowski space; darboux vector field; surface curve. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 254 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-254 int. j. anal. appl. 16 (2) (2018) 255 of distributions on riemannian manifolds. altin [5] computed the energy of frenet vector fields for given non-lightlike curves. the corresponding theory for the functional of curvature-based energy is considered to be at its early stages of evolution. some prolific fields and pioneering studies for this theory can be found in mathematical physics, membrane chemistry, computer-aided geometric design and geometric modelling, shell engineering, biology and thin plate ( [6], [7], [8], [9], [10]). darboux frame is considered as a natural moving frame. it is implemented in surface geometry to characterize features of the curve lying on the surface. it is also used to understand characterization of functionals, which play a significant role to construct surface curves. some of the well-known functionals and related works can be given as follows. bending energy functional has appeared firstly bernoulli-euler elastica formulation for energy [11]. mvs energy functional is used for aesthetic surface design [12]. mvs cross energy functional computes the deviation of the surface from a cylinder or a perfect sphere [13]. the energy of infinitesimally small surface area can be defined by average energy of surface curves which pass through that surface. thus, we work on curves lying on appropriate surfaces to compute these functionals. moreover, total surface energy can be obtained by integrating the energy of the small area element over the entire surface [14]. in this study, we compute energy on the moving particle lying on the surface which is defined in minkowski spacetime. minkowski spacetime has a close connection between mass-energy and motion-energy concept, which are topics of special relativity ( [15], [16], [17], [18]). furthermore, we aim to present usefulness of geometrical perspective on the computation of the energy by calculating curvature-based energy for surface curves. moreover, we introduce the relation between energy on surface curves in each darboux vector field w and curvature-based bending energy functional. the method we use for computing the energy of darboux vector fields is that we consider a vector field as a map from manifold m to the riemannian manifold (tm, ρs), where tm is tangent bundle of a riemannian manifold and ρs is a sasaki metric induced from tm naturally. the designation of the paper is as follows. we firstly present fundamental definitions of darboux frame equations for different types of surface curves in minkowski space. then we give a geometrical interpretation of the energy for unit vector fields. guided by this fundamental information, we compute energy of the moving particle corresponding to a surface curve in minkowski space. 2. kinematics of the particle on a surface let γ be a particle moving on a surface s such that the precise location of the particle is specified by γ = γ (t) , where t is a time parameter. changing the time parameter describes the motion and trajectory. thus, the trajectory corresponds to a curve ζ in the surface for a moving particle. it is convenient to remind the arc-length parameter s, which is used to compute the distance traveled by a particle along its trajectory. int. j. anal. appl. 16 (2) (2018) 256 it is defined by ds dt = ‖v‖ , where v = v (t) =dζ dt is the velocity vector and dζ dt 6= 0. in particle dynamics, the arc-length parameter s is considered as a function of t. thanks to the arc-length, it is also determined serret-frenet frame, which allows us determining the characterization of the intrinsic geometrical features of the regular curve. this coordinate system is constructed by three orthonormal vectors e(α) and the curve ζ itself, assuming the curve is sufficiently smooth at each point. the index within the parenthesis is the tetrad index that describes a particular member of the tetrad. in particular, e(0) is the unit tangent vector, e(1), e(2) is the unit normal and binormal vector of the curve ζ, respectively. orthonormality conditions are summarized by e(α)e(β) = ηαβ, where ηαβ is a euclidean metric such that: diag(1, 1, 1) . thus, we have the following formulas for the frenet frame equations. de(0) ds = κe(1), de(1) ds = −κe(0) + τe(2), de(2) ds = −τe(1), where κ and τ are curvature and torsion of the curve, respectively. in addition to frenet frame, it can be defined a new frame called as darboux frame on the oriented surface s. for the trajectory of the moving particle, which corresponds to a curve ζ on the surface, the darboux vectors e(0), n, p = e(0) × n, are defined. they are the unit tangent of the curve, unit normal of the surface, and normal of the tangent, respectively. they satisfy following equations and properties. case 1. let s be an oriented spacelike surface and moving particle γ lying on s has a unit spacelike tangent vector e(0), then we have de(0) ds = κgp + κnn, dp ds = −κge(0) + τgn, (2.1) dn ds = κne(0) + τgp, where κg, κn, τg are geodesic curvature, normal curvature and geodesic torsion of the curve, [19]. case 2. let s be an oriented timelike surface and moving particle lying on s has a unit spacelike tangent vector e(0), then we have int. j. anal. appl. 16 (2) (2018) 257 de(0) ds = κgp −κnn, dp ds = κge(0) + τgn, (2.2) dn ds = κne(0) + τgp, where κg, κn, τg are geodesic curvature, normal curvature and geodesic torsion of the curve, [19]. case 3. let s be an oriented timelike surface and moving particle lying on s has a unit timelike tangent vector e(0), then we have de(0) ds = κgp + κnn, dp ds = κge(0) − τgn, (2.3) dn ds = κne(0) + τgp, where κg, κn, τg are geodesic curvature, normal curvature and geodesic torsion of the curve, [19]. since we identify e(0) as a unit vector as a tangent to the curve at each point on the curve, we have e(0) = dγ u/ds, where γu is the point on the trajectory of curve ζ. thus e(0), p and n generate the darboux frame w and equation 2.1, 2.2, and 2.3 are known as darboux equations for each case. 3. energy on the unit vector fields in space we first give the fundamental definitions and propositions which are used to compute the energy of the unit vector field. definition 3.1. for two riemannian manifolds (m,ρ) and (n,h) the energy of a differentiable map f : (m,ρ) → (n,h) can be defined as εnergy (f) = 1 2 ∫ m n∑ a=1 h (df (ea) ,df (ea)) v, (3.1) where {ea} is a local basis of the tangent space and v is the canonical volume form in m [1]. proposition 3.1. let q : t ( t1m ) → t1m be the connection map. the following two conditions hold: i) ω ◦q = ω ◦dω and ω ◦q = ω ◦ ω̃, where ω̃ : t ( t1m ) → t1m is the tangent bundle projection; ii) for % ∈ txm and a section ξ : m → t1m; we have q (dξ (%)) = d%ξ, (3.2) where d is the levi-civita covariant derivative [1]. int. j. anal. appl. 16 (2) (2018) 258 definition 3.2. for ς1, ς2 ∈ tξ ( t1m ) , we define ρs (ς1, ς2) = ρ (dω (ς1) ,dω (ς2)) + ρ (q (ς1) ,q (ς2)) . (3.3) this yields a riemannian metric on tm. as known ρs is called the sasaki metric that also makes the projection ω : t1m → m a riemannian submersion. 4. bending energy functional by darboux vector fields in the theory of relativity, all the energy moving through an object contributes to the total mass of the body that measures how much it can resist to acceleration. each kinetic and potential energy makes a highly proportional contribution to the mass [20], [21], [22], [23] . in this study not only we compute the energy on surface curves but we also investigate its close correlation with bending energy of elastica which is a variational problem proposed firstly by daniel bernoulli to leonard euler in 1744. euler bending elastic energy formula for a space curve in the 3-dimensional frenet curvature along the curve is known as hb = 1 2 ∫ ∥∥de(0) e(0)∥∥2 ds, (4.1) where s is an arclength, [10]. furthermore, we know that geodesic curvature and normal curvature are not independent and sum of their squares on a minimal surface gives kg = − ( κ2n + τ 2 g ) (4.2) where kg is gaussian curvature [10]. case 1. let s be an oriented spacelike surface and moving particle lying on s has a unit spacelike tangent vector e(0) theorem 4.1. let γ be a moving particle on surface s such that it corresponds to a curve ζ. then, energy on the particle in tangent vector field by using sasaki metric is stated by εnergye(0) = 1 2 (s + ∫ s 0 ( κ2g −κ 2 n ) ds). proof. from equation 3.1 and 3.2 we know εnergye(0) = 1 2 ∫ s 0 ρs ( de(0)(e(0)),de(0)(e(0)) ) ds. using also equation 3.3 we have ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ(dω(e(0)(e(0))),dω(e(0)(e(0)))) +ρ(q(e(0)(e(0))),q(e(0)(e(0)))). int. j. anal. appl. 16 (2) (2018) 259 since e(0) is a section, we get d(ω) ◦d(e(0)) = d(ω ◦ e(0)) =d(idc) = idtc. we also know q(e(0)(e(0))) = de(0) e(0)=κgp + κnn thus, we find from the equation 2.1 ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ ( e(0), e(0) ) + ρ ( de(0) e(0),de(0) e(0) ) = 1 + κ2g −κ 2 n so we can easily obtain εnergye(0) = 1 2 (s + ∫ s 0 ( κ2g −κ 2 n ) ds). this completes the proof. � conclusion 4.1. let ζ be a spacelike curve lying on spacelike surface s. then we have ∥∥de(0) e(0)∥∥2 = κ2g−κ2n. thus, we obtain following relation for the bending energy of elastica: hb = εnergye(0) − 1 2 s. proof. it is obvious from the equation 4.1 and theorem 4.1. � theorem 4.2. let γ be a moving particle on surface s such that it corresponds to a curve ζ. then, energy on the particle in vector field of the normal of the surface by using sasaki metric is stated by εnergy (n) = 1 2 (s + ∫ s 0 ( κ2n + τ 2 g ) ds). proof. if we follow the similar steps as in the theorem 4.1, the proof is obvious. � conclusion 4.2. let ζ be a spacelike curve lying on spacelike surface s.then we have for a gaussian curvature kg εnergy (n) = 1 2 (s− ∫ s 0 kgds). proof. it is obvious from equation 4.1 and theorem 4.2. int. j. anal. appl. 16 (2) (2018) 260 theorem 4.3. let γ be a moving particle on surface s such that it corresponds to a curve ζ. then, energy on the particle in tangent’ s normal vector field by using sasaki metric is stated by εnergy (p) = 1 2 (s + ∫ s 0 ( κ2g − τ 2 g ) ds). proof. if we follow the similar steps as in the theorem 4.1, the proof is obvious. case 2. let s be an oriented timelike surface and moving particle lying on s has a unit spacelike tangent vector e(0). theorem 4.4. let γ be a moving particle on surface s such that it corresponds to a curve ζ. then, energy on the particle in tangent vector field by using sasaki metric is stated by εnergye(0) = 1 2 (s + ∫ s 0 ( −κ2g + κ 2 n ) ds). proof. from equation 3.1 and 3.2 we know εnergye(0) = 1 2 ∫ s 0 ρs ( de(0)(e(0)),de(0)(e(0)) ) ds. using the equation 3.3 and knowing e(0) is a section we obtain that ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ(dω(e(0)(e(0))),dω(e(0)(e(0)))) +ρ(q(e(0)(e(0))),q(e(0)(e(0)))), and d(ω) ◦d(e(0)) = d(ω ◦ e(0)) =d(idc) = idtc. it is also true that q(e(0)(e(0))) = de(0) e(0)=κgp −κnn moreover we find from the equation 2.2 ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ ( e(0), e(0) ) + ρ ( de(0) e(0),de(0) e(0) ) = 1 −κ2g + κ 2 n. thus we can easily obtain εnergye(0) = 1 2 (s + ∫ s 0 ( −κ2g + κ 2 n ) ds). conclusion 4.3. let ζ be a spacelike curve lying on timelike surface s. then we have ∥∥de(0) e(0)∥∥2 = −κ2g + κ2n. thus, we obtain following relation for the bending energy of elastica: hb = εnergye(0) − 1 2 s. int. j. anal. appl. 16 (2) (2018) 261 proof. it is obvious from the equation 4.1 and theorem 4.4. � theorem 4.5. energy on the moving particle in surface’s normal and tangent’s normal vector field by using sasaki metric is stated by εnergy (n) = 1 2 (s + ∫ s 0 ( κ2n − τ 2 g ) ds), εnergy (p) = 1 2 (s + ∫ s 0 ( κ2g + τ 2 g ) ds). proof. if we follow the similar steps as in the theorem 4.4, the proof is obvious. case 3. let s be an oriented timelike surface and moving particle lying on s has a unit timelike tangent vector e(0). theorem 4.6. let γ be a moving particle on surface s such that it corresponds to a curve ζ. then, energy on the particle in tangent vector field by using sasaki metric is stated by εnergye(0) = 1 2 (−s + ∫ s 0 ( κ2g + κ 2 n ) ds). proof. from equation 3.1 and 3.2 we know εnergye(0) = 1 2 ∫ s 0 ρs ( de(0)(e(0)),de(0)(e(0)) ) ds. by using the equation 3.3 we have ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ(dω(e(0)(e(0))),dω(e(0)(e(0)))) +ρ(q(e(0)(e(0))),q(e(0)(e(0)))). since e(0) is a section, we also get d(ω) ◦d(e(0)) = d(ω ◦ e(0)) =d(idc) = idtc. moreover, it is clear that q(e(0)(e(0))) = de(0) e(0)=κgp + κnn. thus, we find from equation 2.3 ρs ( de(0)(e(0)),de(0)(e(0)) ) = ρ ( e(0), e(0) ) + ρ ( de(0) e(0),de(0) e(0) ) = −1 + κ2g + κ 2 n int. j. anal. appl. 16 (2) (2018) 262 and finally εnergye(0) = 1 2 (−s + ∫ s 0 ( κ2g + κ 2 n ) ds). conclusion 4.4. let ζ be a timelike curve lying on timelike surface s. then we have ∥∥de(0) e(0)∥∥2 = κ2g +κ2n. thus, we obtain following relation for the bending energy of elastica: hb = εnergye(0) + 1 2 s. proof. it is obvious from the equation 4.1 and theorem 4.6. � theorem 4.7. energy on the moving particle in surface’s normal and tangent’s normal vector field by using sasaki metric is stated by εnergy (n) = 1 2 (−s + ∫ s 0 ( −κ2n + τ 2 g ) ds), εnergy (p) = 1 2 (−s + ∫ s 0 ( −κ2g + τ 2 g ) ds). proof. if we follow the similar steps as in the theorem 4.6, the proof is obvious. � 5. conclusion in this study, we studied energy on the particle in the darboux vector fields in minkowski spacetime considering kinematics of the particle. furthermore, we set a connection between energy on the particle in these vector fields and elastica of bending functional. this is important for our future work since a simple characterization on the energy of a vector field can be described as it is up to constants, in other words, it is square l2 norm of the vector field’s covariant derivative. thanks to this definition, not only we will correlate the concept of the energy with a volume for the moving particle in these vector fields in space. as is known, elastic energy may occur by applying different forces besides bending such as twisting and stretching. in our next studies, we also determine the correlation between energy on the particle in each darboux vector field and stretching and twisting energy functional. computing the energy on the moving particle has a wide range of application in the theoretical and applied physics. therefore, it will also be investigated the energy on the moving particle in different force fields thanks to classical mechanics by obtaining dynamics of the particle in space including work done and force acting on the particle besides the energy. we believe that this study also will lead up to further research on the relativistic dynamics of the particle in different spacetimes in terms of computing the energy on a particle in different force fields. int. j. anal. appl. 16 (2) (2018) 263 references [1] c.m. wood, on the energy of a unit vector field, geom. dedicata. 64 (1997), 319-330. [2] o. gil medrano, relationship between volume and energy of vector fields, differ. geom. appl. 15 (2001) 137-152. [3] p.m. chacon, a.m. naveira and j.m. weston, on the energy of distributions, with application to the quaternionic hopf fibrations, monatsh. math. 133 (2001) 281-294. [4] p.m. chacon and a.m. naveira, corrected energy of distribution on riemannian manifolds, osaka j. math. 41 (2004) 97-105. [5] a. altin, on the energy and pseduoangle of frenet vector fields in rv?, ukr. math j. 63 (2011) 969-975. [6] g. kirchhoff, ber das gleichgewicht und die bewegung einer elastichen scheibe, crelles j. 40 (1850) 51-88. [7] e. catmull and j. clark, recursively generated b-spline surfaces on arbitrary topological surfaces, comput.-aided des. 10 (1978), 350-355. [8] t. lopez-leon, v. koning, k.b.s. devaiah, v. vitelli and a.a. fernandez-nieves, frustrated nematic order in spherical geometries, nature phys. 7 (2011) 391-394. [9] t. lopez-leon, a.a. fernandez-nieves, m. nobili and c. blanc, nematic-smectic transition in spherical shells, phys. rev. lett. 106 (2011) 247802. [10] j. guven j, d.m. valencia and j. vazquez-montejo, environmental bias and elastic curves on surfaces, phys. a: math. theory. 47 (2014) article id 355201. [11] l. euler, additamentum ‘de curvis elasticis’, in methodus inveniendi lineas curvas maximi minimive probprietate gaudentes, lausanne, 1744. [12] c.h. sequin, cad tools for aesthetic engineering, comput.-aided des. appl. 1 (2004) 301-309. [13] d. zorin, curvature-based energy for simulation and variational modelling, proceedings of the international conference on shape modelling and applications. smi’05 (2005) 196-204. [14] p. joshi and c. sequin, energy minimizer for curvature-based surface functional, cad conference, waikiki, hawaii. (2007) 607-617. [15] a. einstein, zur elektrodynamik bewegter k?rper, annalen der physik. 17 (1905), 891-921. [16] a. einstein, relativity:the special and general theory, henry holt, new york, 1920. [17] t. roberts, s. schleif and j.m. dlugosz, what is the experimental basis of special relativity? usenet physics faq, 2007. [18] a. einstein, does the inertia of a body depend on its energy content?, annalen der physik, 18 (1905) 639-641. [19] m.k. saad, h.s. abdel-aziz, g. weiss and m.a. soliman, relation among darboux frames of null bertrand curves in pseudo-euclidean space, 1st int. wlgk11, 2011. [20] r. capovilla, c. chryssomalakos and j. guven, hamiltonians for curves, j. phys. a. 35 (2002) 6571-6587. [21] m. carmeli, motion of a charge in a gravitational field, phys. rev. b. 138 (1965) 1003-1007. [22] j. weber, relativity and gravitation, interscience, new york, 1961. [23] g. napoli, l. vergori, extrinsic curvature effects on nematic shells, phys. rev. lett. 108 (2012), article id 207803. 1. introduction 2. kinematics of the particle on a surface 3. energy on the unit vector fields in space 4. bending energy functional by darboux vector fields 5. conclusion references international journal of analysis and applications volume 17, number 4 (2019), 659-673 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-659 slip impact on diffusion of a solute in creeping sinusoidal motion of a newtonian fluid with wall features mallinath dhange∗ and gurunath sankad b.l.d.e.a’s v.p. dr. p. g. halakatti college of engineering and technology, vijayapur, india ∗corresponding author: math.mallinath@bldeacet.ac.in abstract. this article is concerned with the effect of slip boundary condition on two-dimensional creeping movement of a newtonian fluid in the existence of pervious medium with wall features and heterogeneoushomogeneous chemical responses. the objective of this paper is to measure the performance of slip and wall feature constraints through graphs. it is observed that diffusion ascends with an increase in slip and wall constraints. the effective diffusion coefficient has been computed through long wavelength supposition and taylor’s condition for chemical responses. 1. introduction mathematical modeling is the illustration of a system using scientific observation and linguistic. it is applied to study the complications in medical science. bio-fluid dynamics is the branch of biomechanics which deals with the kinematics and dynamics of the fluids present in human beings, animals and plants. it spans from cells to organs, covering diverse aspects of functionality of systemic physiology, including cardiovascular, lymphatic, neurological, respiratory, reproductive, auditory and urinary systems. the biological systems are very complex and have defied all attempts at satisfactory mathematical solutions. these complicated systems are studied theoretically by means of approximated models whose simplified nature becomes amenable to mathematical analysis and give meaningful mathematical solutions. hence the mathematical analysis and understanding of bio-fluid dynamics seem to be extremely important and useful for diagnosis and clinical received 2019-01-28; accepted 2019-03-14; published 2019-07-01. 2010 mathematics subject classification. 76v05, 76z05, 76s05, 92c10. key words and phrases. chemical reactions; creeping flow; diffusion; newtonian fluid; wall features. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 659 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-659 int. j. anal. appl. 17 (4) (2019) 660 purposes. the vital mechanism for fluid transportation in bio-fluid dynamics is peristalsis. peristalsis is a coordinated response wherein a wave of construction preceded by a wave of relaxation passes down a hollow viscus. from the perspective of fluid dynamics, peristalsis is typified by the dynamic interface of the fluid flow and movement of the flexible boundaries of the conduit. the study of the mechanism of peristaltic motion was first experimentally examined by latham [1]. after this study, several experts have explored the creeping sinusoidal transportation of dissimilar liquids under various circumstances ( [2] [4]). mittra and prasad [5] analyzed the movement of newtonian liquid under peristalsis to know the effects of the viscoelastic behavior of walls. further, a few investigators have studied the wall sound effects on different fluids with peristalsis ( [6] [7]). the dispersion is the process by which material is transported from one portion of a system to another as a result of random molecular motion. dispersion of soluble matter in laminar flow has biological applications such as drug and nutrients distribution in the body. through dispersion, metabolites are swapped between a cell and its environment or among the tissues and bloodstream. due to its importance, many investigators explored the dispersion of a solute in newtonian and non-newtonian liquids under different limitations following taylors approach( [8] [12]). creeping stream with a pervious intermediate has attained significance in the current decade because of its practical applications chiefly in biomechanics and geophysical fluid dynamics. even in some pathological situation like: transportation of liquid in kidneys, in lungs, gallbladder with stones, small blood vessels and tissues and bones and allocation of fatty cholesterol can be well thought-out as a pervious medium. the proper functioning of these depends on the stream of blood, nutrients, etc., through them. hence, several authors studied influence of porosity in dissimilar liquids ( [13] [19]). in many situations like physiological and engineering, the fluid slips at the walls of the channel. in slip conditions the boundary and the fluid moves with different velocity. it is important in describing the macroscopic effect of certain molecular phenomenon where interaction between fluid and solid occurs. the slip boundary condition was initially proposed by beaver and joseph [20]. saffman [21] modified the periphery condition of beaver and joseph. the presence of slip phenomenon at the boundaries and interfaces has been observed in physiological streams, flows through pipes in which chemical responses take place at the walls. to the best of our knowledge, no attempt has yet been reported to discuss the impact of wall features and heterogeneous-homogeneous chemical responses on creeping flow of a newtonian fluid in a pervious medium with a slip condition through taylors approach. 2. mathematical formulation and methodology consider the creeping flow of a newtonian liquid through a pervious medium in the 2dimensional conduit and assumed that conduit is packed with pervious material. figure 1 displays the migrant waves. int. j. anal. appl. 17 (4) (2019) 661 figure 1. physical model of the issu. the migrant sinusoidal wave is given by the subsequent equation: ± [ a sin 2π λ (x− ct) + d ] = ±h = y, (2.1) the relating flow conditions of the present issue are: 0 = ∂v ∂y + ∂u ∂x , (2.2) − ∂p ∂x + µ∇2u− µ k̄ u = ρ [ ∂ ∂t + v ∂ ∂y + u ∂ ∂x ] u, (2.3) − ∂p ∂y + µ∇2v− µ k̄ v = ρ [ ∂ ∂t + v ∂ ∂y + u ∂ ∂x ] v, (2.4) referring [7], the condition of the springy wall movement is specified as: p−p0 = l(h), (2.5) where, −t ∂2 ∂x2 + m ∂2 ∂t2 + c ∂ ∂t = l. (2.6) here, c the coefficient of sticky damping force, m the mass per/area and t the tension in the membrane. applying long wavelength hypothesis, conditions (2.2) to (2.4) yield as: 0 = ∂v ∂y + ∂u ∂x , (2.7) 0 = − ∂p ∂x + µ ∂2u ∂y2 − µ k̄ u, (2.8) int. j. anal. appl. 17 (4) (2019) 662 0 = − ∂p ∂y . (2.9) following bhatt and sacheti [22], the allied border conditions are u = −d √ da γ ∂u ∂y , at y = ±h. (2.10) where, γslip constraint, dapervious constraint. it is presumed that there is no horizontal displacement of the wall, p0 = 0 and the channel walls are inextensible, µ ∂2u ∂y2 − µ k̄ u = ∂ ∂x l(h) at y = ±h, (2.11) where −t ∂3h ∂x3 + m ∂3h ∂x∂t2 + c ∂2h ∂x∂t = ∂ ∂x l(h) = ∂p ∂x . (2.12) solving eqns. (2.8), 2.9) with (2.10) and (2.11), we attain u(y) = 1 µm21 ( ∂p ∂x ) [a′1cosh(m1y) + a ′ 2cosh(m1y) − 1] , (2.13) the mean speed is specified and obtained as: ū(y) = 1 2h ∫ h −h u(y)dy = 1 µm21 ( ∂p ∂x )[ a′1 m1h sinh(m1h) − 1 ] . (2.14) referring [13], the relative liquid speed is given as: ux = u− ū = 1 µm21 ( ∂p ∂x ) [a′1 cosh(m1y) + a ′ 2 sinh(m1y) −a ′ 3] . (2.15) where a ′ 1 = 1( 1 − da γ2 d2m21 ) cosh(m1h) , a ′ 2 = − √ da γ dm1( 1 − da γ2 d2m21 ) cosh(m1h) , a ′ 3 = ( sinh(m1h)( 1 − da γ2 d2m21 ) m1h cosh(m1h) , p ′ = −t ∂3h ∂x3 + m ∂3h ∂x∂t2 + c ∂2h ∂x∂t = ∂p ∂x , m1 = √ 1 k . utilizing taylor [8], the scattering equation for the concentration c of the material in isothermal circumstances is d ∂2c ∂y2 −k1c = u ∂c ∂x + ∂c ∂t . (2.16) here, k1 the rate constant of first order chemical response , d the molecular diffusion coefficient , and c liquid concentration. int. j. anal. appl. 17 (4) (2019) 663 it is expected that ū≈c ( [13]), utilizing ū≈c , and consequent non-dimensional quantities, η = y d , θ = t t̄ , t̄ = λ ū , h = h d , p = d2 µcλ p ′, ξ = (x− ūt) λ ,da = k̄ d2 . (2.17) equations (2.15), (2.16) and (2.12) reduces to d2 µm2 ∂p ∂x [a1 cosh(mη) + a2 cosh(mη) −a3] = ux, (2.18) ∂2c ∂η2 − k1d 2 d c = − d2 λd ux ∂c ∂ξ , (2.19) − � [ −e3(2π)2 sin(2πξ) + (e1 + e2)(2π)3 cos(2πξ) ] = p, (2.20) where, e1−the rigidity e2−the stiffness, e3−the viscous damping force in the wall and �−the amplitude ratio. the dispersion with 1storder irreversible chemical response occur in the mass of the liquid and at the channel walls. referring [11], the wall conditions are specified as: 0 = fc + ∂c ∂y at y = [a sin 2π λ (x− ūt) + d] = h, (2.21) 0 = −fc + ∂c ∂y at y = −[a sin 2π λ (x− ūt) + d] = −h. (2.22) after non-dimensionalisation, the eqns. (2.21) and (2.22) yield as: 0 = βc + ∂c ∂η at η = [� sin(2πξ) + 1] = h, (2.23) 0 = −βc + ∂c ∂η at η = −[� sin(2πξ) + 1] = −h, (2.24) where the heterogeneous response rate constraint is β = fd, relating to catalytic response at the dividers. alluding eqns. (2.23) and (2.24), the primitive of (2.19) is attained as: c(η) = − d4 λµdm2 ∂c ∂ξ p [ a4 cosh(mη) + a5 sinh(mη) + a6 cosh(αη) + a7 sinh(αη) + a8 ] . (2.25) here, α = √ k1 d d, m = m1d = √ 1 da . the volumetric flow rate q is specified and attained as: q = ∫ h −h cuxdη = −2 d6 λµ2d ∂c ∂ξ g(α,β,�,e1,e2,e3,da,γ,ξ), (2.26) int. j. anal. appl. 17 (4) (2019) 664 where, g(α,β,�,e1,e2,e3,da,γ,ξ) = { p2 m4 [ a1a4 2 b1 + a1a5 2 b2 + (a1a8 −a3a4)b3 −a3a6b4 −b5 + a1a6b6 + a2a7b7 ]} , (2.27) a1 = 1( 1 − da γ2 m2 ) cosh(mh) , a2 = − √ da γ m( 1 − da γ2 m2 ) cosh(mh) , a3 = sinh(mh) mh ( 1 − da γ2 m2 ) cosh(mh) ,a4 = 1 (m2 −α2) ( 1 − da γ2 m2 ) cosh(mh) , a5 = − √ da γ m (m2 −α2) ( 1 − da γ2 m2 ) cosh(mh) , a6 = 1( 1 − da γ2 m2 ) cosh(mh)(α sinh(αh) + β cosh(αh)) {−(m sinh(mh) + β cosh(mh)) (m2 −α2) + β sinh(mh) mhα2 } , a7 = √ da γ m( 1 − da γ2 m2 ) cosh(mh) (m cosh(mh) + β sinh(mh)) (α cosh(αh) + β sinh(αh)) , a8 = sinh(mh) mhα2 ( 1 − da γ2 m2 ) cosh(mh) , b1 = sinh(2mh) + 2mh 2m , b2 = sinh(2mh) − 2mh 2m , b3 = sinh(mh) m , b4 = sinh(αh) α , b5 = a3a8h, b6 = m sinh(mh) cosh(αh) −α cosh(mh) sinh(αh) m2 −α2 , b7 = m cosh(mh) sinh(αh) −α sinh(mh) cosh(αh) m2 −α2 . looking at condition (2.27) with fick’s law of scattering, the dispersing coefficient d∗ was computed to such an extent that the solute disperses near to the plane moving with the typical speed of the flow and is specified as 2 d6 µ2d g(α,β,�,e1,e2,e3,da,γ,ξ) = d ∗. (2.28) let ḡ be the average of g, and is obtained by the following equation: ∫ 1 0 g(α,β,�,e1,e2,e3,da,γ,ξ)dξ = ḡ. (2.29) int. j. anal. appl. 17 (4) (2019) 665 3. discussion of outcomes the impact of various constraints on the mean effective scattering coefficient can be observed through the expression ḡ (α,β,�,e1,e2,e3,da,γ,ξ). using mathematica software the graphs are plotted for eqn. (2.29). figure 2. illustration of γ for ḡ at e1 = 0.1, e2 = 0.0, e3 = 0.06, � = 0.2, α = 1.0, da = 0.9 figure 3. illustration of γ for ḡ at e1 = 0.1, e2 = 4.0, e3 = 0.06, � = 0.2, β = 5.0, da = 0.9 the effects of slip constraint on ḡ have been illustrated through the figures 2 4. it is noticed that, scattering rises as slip constraint (γ) augments. as we already known that, heterogeneous response takes place more at the boundary of the conduit, due to slip boundary condition less heterogeneous response takes place at the boundary. hence the dispersion is more in the case of no-slip boundary as compared to slip int. j. anal. appl. 17 (4) (2019) 666 figure 4. illustration of γ for ḡ at e1 = 0.1, e2 = 4.0, e3 = 0.00, α = 1.0, β = 5.0, da = 0.9 figure 5. illustration of da for ḡ at e1 = 0.1, e2 = 0.0, e3 = 0.06, � = 0.2, α = 1.0, γ = 0.08 boundary. figures 5 7 indicates that ḡ enhances with a growth in the darcy number (da). this result concurs with the outcome of [6], [13], and [14]. consider the figures 8 16 for the effect of the rigidity (e1), stiffness (e2) and viscous damping force (e3) of the wall on the dispersal coefficient (ḡ). it is observed that ḡ ascends monotonically with an increase in e1,e2 and e3. int. j. anal. appl. 17 (4) (2019) 667 figure 6. illustration of da for ḡ at e1 = 0.1, e2 = 4.0, e3 = 0.06, � = 0.2, β = 5.0, γ = 0.08 figure 7. illustration of da for ḡ at e1 = 0.1, e2 = 4.0, e3 = 0.00, α = 1.0, β = 5.0, γ = 0.08 wall feature affects the increase in velocity of the liquid in conduit which reasons to enhance the scattering. these outcomes agree with the results of [7]. figures 2, 5, 8, 11, 14 display that scattering lessens with heterogeneous substance response rate β and also figures 3, 6, 9, 12, and 15 clear that scattering descends as homogeneous compound response rate α falls down. it is observed that the solution expression for ḡ be in agreement with sobh [15] when there is no wall features. further, it is noticed that ḡ be in agreement with that [13] and [17], if there is absence of slip boundary condition ( i.e. with no-slip boundary condition). int. j. anal. appl. 17 (4) (2019) 668 figure 8. illustration of e1 for ḡ at da = 0.9, e2 = 0.0, e3 = 0.00, � = 0.2, α = 1.0, γ = 0.08 figure 9. illustration of e1 for ḡ at da = 0.9, e2 = 4.0, e3 = 0.06, � = 0.2, β = 5.0, γ = 0.08 4. conclusions in this article, we inspected that the slip and wall effects on two-dimensional creeping flow of a newtonian fluid through a pervious medium in the existence of chemicals responses. identical behavior is noticed for wall features, slip constant γ and darcy number da on dispersion coefficient (ḡ) and it is also witnessed the opposite behavior of homogeneous response rate α and heterogeneous response rate β are observed on concentration profile. lastly, it concludes that wall feature constants, slip constraint, and darcy constant favor diffusion. this model may help in better understanding of the transport phenomena occurring in the intestine leading to absorption of nutrients and drugs. int. j. anal. appl. 17 (4) (2019) 669 figure 10. illustration of e1 for ḡ at da = 0.9, e2 = 0.0, e3 = 0.00, α = 1.0, β = 5.0, γ = 0.08 figure 11. illustration of e2 for ḡ at da = 0.9, e1 = 0.1, e3 = 0.06, � = 0.2, α = 1.0, γ = 0.08 int. j. anal. appl. 17 (4) (2019) 670 figure 12. illustration of e2 for ḡ at da = 0.9, e1 = 0.1, e3 = 0.06, � = 0.2, β = 5.0, γ = 0.08 figure 13. illustration of e2 for ḡ at da = 0.9, e1 = 0.1, e3 = 0.00, α = 1.0, β = 5.0, γ = 0.08 int. j. anal. appl. 17 (4) (2019) 671 figure 14. illustration of e3 for ḡ at da = 0.9, e1 = 0.1, e2 = 4.0, � = 0.2, α = 1.0, γ = 0.08 figure 15. illustration of e3 for ḡ at da = 0.9, e1 = 0.1, e2 = 4.0, � = 0.2, β = 5.0, γ = 0.08 int. j. anal. appl. 17 (4) (2019) 672 figure 16. illustration of e3 for ḡ at da = 0.9, e1 = 0.1, e2 = 4.0, α = 1.0, β = 5.0, γ = 0.08 int. j. anal. appl. 17 (4) (2019) 673 references [1] t. w. latham, fluid motions in a peristaltic pump, masters thesis, massachusetts institute of technology, cambrige, 1966. [2] y. c. fung and c. s. yih, peristaltic transport, asme transactions: j. appl. mech. 35(4)(1968), 669-675. [3] m. y. jaffrin, a. h. shapiro and s. l. weinberg, peristaltic pumping with long wavelengths at low reynolds number, j. fluid mech. 37(1969), 799-825. [4] d. takagi and n. j. balmforth, peristaltic pumping of viscous fluid in an elastic tube, j. fluid mech. 672(2011), 196-218. [5] t. k. mittra and n. s. prasad, on the influence of wall properties and poiseuille flow in peristalsis, j. biomechan. 6(1973), 681-693. [6] a. v. ramana kumari and g. radhakrishnamacharya, effect of slip and magnetic field on peristaltic flow in an inclined channel with wall effects, int. j. biomath. 5(6)(2012), 1250015. [7] g. c. sankad and g. radhakrishnamacharya, effect of magnetic field on peristaltic motion of micropolar fluid with wall effects, j. appl. math. fluid mech. 1(2009), 37-50. [8] g. i. taylor, dispersion of soluble matter in solvent flowing slowly through a tube, proc. royal soc. lond. 219(a)(1953), 186-203. [9] d. padma and v. v. ramanarao, effect of homogeneous and heterogeneous reactions on the dispersion of a solute in laminar flow between two parallel porous plates, indian j. technol. 14(1976), 410-412. [10] p. s. gupta and a. s. gupta, effect of homogeneous and heterogeneous reactions on the dispersion of a solute in the laminar flow between two plates, proc. royal soc. lond. 330(a)(1972), 59-63. [11] p. chandra and r. p. agarwal, dispersion in simple microwfluid flows, int. j. eng. sci. 21(1983), 431-442. [12] d. philip and p. chandra, effect of heterogeneous and homogeneous reactions on the dispersion of a solute in simple microwfluid, indian j. pure appl. math. 24(1993), 551-561. [13] h. alemayehu and g. radhakrishnamacharya, dispersion of solute in peristaltic motion of a couple stress fluid through a porous medium, tamkang j. math. 43(4)(2012), 541-555. [14] g. ravikiran and g. radhakrishnamacharya, effect of homogeneous and heterogeneous chemical reactions on peristaltic transport of a jeffrey fluid through a porous medium with slip condition, j. appl. fluid mech. 8(3)(2015), 521-528. [15] a. m. sobh, effect of homogeneous and heterogeneous reactions on the dispersion of a solute in mhd newtonian fluid in an asymmetric channel with peristalsis, br. j. math. computer sci. 3(4)(2013), 664-679. [16] k. n. mehta and m. c. tiwari, dispersion in presence of slip and chemical reactions in porous wall tube flow, defence sci. j. 38(1988), 1-11. [17] g. sankad and m. dhange, peristaltic pumping of an incompressible viscous fluid in a porous medium with wall effects and chemical reactions, alexandria eng. j. 55(2016), 2015-2021. [18] d. pal, effect of chemical reaction on the dispersion of a solute in a porous medium, appl. math. model. 23(7)(1999), 557-566. [19] j. c. misra and s. k. ghosh, a mathematical model for the study of blood flow through a channel with permeable walls, acta mech. 122(1997), 137-153. [20] g. s. beaver and d. d. joseph, boundary conditions at a naturally permeable wall, j. fluid mech. 30(1967), 197-207. [21] p. g. saffman, on the boundary conditions at the surface of a porous medium, stud. appl. math. 1(1971), 93-101. [22] b. s. bhatt and n. c. sacheti, on the analogy in slip flows, indian j. pure appl. math. 10(1979), 303-306. 1. introduction 2. mathematical formulation and methodology 3. discussion of outcomes 4. conclusions references international journal of analysis and applications volume 16, number 1 (2018), 25-37 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-25 equivalence of sturm-liouville problem with finitely many δ-interactions and matrix eigenvalue problems abdullah kablan∗, mehmet aki̇f çeti̇n faculty of arts and sciences, department of mathematics, gaziantep university, gaziantep, 27310, turkey ∗corresponding author: kablan@gantep.edu.tr abstract. the purpuse of this article is to show the matrix representations of sturm-liouville operators with finitely many δ-interactions. we show that a sturm-liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former sturm-liouville operator. moreover an example is also presented. 1. introduction acording to classical spectral theory, a sturm–liouville problem (slp) consisting of the equation −(py′)′ + qy = λwy, on j = (a,b) and boundary conditions has infinite spectrum under some assumptions. atkinson in his book [1] suggested that if the coefficients of slp satisfy some conditions, the problem may have finite eigenvalues. then in [2], kong, wu and zettl obtained the following result: for every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint slp with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals. recently, there has been much attention paid to the slps with finite spectrum. for a comprehensive treatment of the subject we refer the reader to the book by zettl [3], and the papers by kong, wu and zettl [2], ao, sun, and zhang [4], [5] and ao, bo and sun [6], [7]. in 2009, the equivalence of slp with received 7th september, 2017; accepted 20th october, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 34b24, 47a10, 15a18. key words and phrases. one-dimensional schrödinger operator; finite spectrum; point interactions. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 25 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-25 int. j. anal. appl. 16 (1) (2018) 26 a matrix eigenvalue problem was first constructed by volkmer and zettl in [8]. by equivalance of matrix eigenvalue problems for the slps with finite spectrum we mean to construct a matrix eigenvalue problem with exactly the same eigenvalues as the corresponding slp. then, the matrix representations of slps with finite spectrum are extended to various problems. for the slps see [8][11] and for fourth order boundary value problems see [12][16]. the goal of this paper is to find the matrix representation of the following sturm-liouville problem with finitely many δ-interactions: − (py′)′ + ∞∑ n=1 αnδ(x−xn)y + qy = λwy, on j = (a,b), (1.1) where j = (a,x1) ∪ (x1,x2) ∪ ...∪ (xn,b), x1, ...,xn ∈ (a,b) with −∞ < a < b < ∞, αj’s are real numbers, δ(x) is the dirac delta function and λ ∈ c is a spectral parameter. sturm-liouville equations with dirac delta function potentials often appear in quantum mechanics. for example, such an equations had been used for modelling of atomic and molecular systems including atomic lattices, quantum heterostructures, semiconductors, organic fluorescent materials, solar cells etc. (see [17], [18], [19] and citations of them). recently, we generalize the finite spectrum result to the problem (1.1) in [20]. the equation (1.1) is equivalent to the many-point boundary value problem, (see [19]). so we can understand problem (1.1) as studying the equation − (py′)′ + qy = λwy, on j, (1.2) and n transmission conditions cjy (xj−) = y (xj+), y =   y py′   , j = 1, 2, ...,n (1.3) where xj’s are inner discontinuity points and cj =   1 0 αj 1   . additionally, let us consider the boundary conditions of the form ay (a) + by (b) = 0, a,b ∈ m2(c) (1.4) where a = (aij)2×2, b = (bij)2×2 are complex valued 2 × 2 matrices and m2(c) denotes the set of square matrices of order 2 over c. here, the coefficients fulfill the following minimal conditions: r = 1 p , q, w ∈ l(j, c), (1.5) where l(j, c) denotes the complex valued functions which are lebesgue integrable on j. int. j. anal. appl. 16 (1) (2018) 27 the bc (1.3) is said to be self-adjoint if the following two conditions are satisfied: rank(a,b) = 2, aea∗ = beb∗ with e =  0 −1 1 0   . (1.6) it is well known that under the condition (1.5), the bcs (1.3) fall into two disjoint classes: seperated and coupled. the seperated boundary conditions have the canonical representation: cos αy(a) − sin α(py′)(a) = 0, 0 ≤ α < π (1.7) cos βy(b) − sin β(py′)(b) = 0, 0 < β ≤ π. the real coupled boundary conditions have the canonical representation: y (b) = ky (a) with k = (ks,t)2×2, ks,t ∈ r, det(k) = 1. (1.8) let u = y and v = (py′). then we have the system representation of equation (1.2) u′ = rv, v′ = (q −λw)u, on j. (1.9) 2. matrix representations of slps with finitely many δ-interactions definition 2.1. a sturm-liouville equation with finitely many δ-interactions (1.1) or equivalently the equation (1.2) with transmission condition (1.3) is said to be of atkinson type if, for some integers mj ≥ 1, j = 0, 1, ...,n, there exists a partition of the interval j a = x00 < x01 < x02 < ... < x0,2m0+1 = x1, (2.1) x1 = x10 < x11 < x12 < ... < x1,2m1+1 = x2, ... xn−1 = xn−1,0 < xn−1,1 < xn−1,2 < ... < xn−1,2mn−1+1 = xn, xn = xn0 < xn1 < xn2 < ... < xn,2mn+1 = b such that for each j ∈{0, 1, ...,n} r = 1 p = 0 on (xj,2k; xj,2k+1] , k = 0, 1, ...,mj − 1 and [ xj,2mj ; xj,2mj+1 ) , xj,2k+1∫ xj,2k w 6= 0, xj,2k+1∫ xj,2k q 6= 0, k = 0, 1, ...,mj, (2.2) and q = w = 0 on [xj,2k+1; xj,2k+2] , xj,2k+2∫ xj,2k+1 r 6= 0, k = 0, 1, ...,mj − 1. (2.3) int. j. anal. appl. 16 (1) (2018) 28 our main aim in this section is to constract matrix eigenvalue problems in such a way that its eigenvalues are exactly the same as those of the corresponding slps with finitely many δ-interactions of atkinson type. definition 2.2. a slp with finitely many δ-interactions of atkinson type is said to be equivalent to a matrix eigenvalue problem if the former has exactly the same eigenvalues as the latter. we begin by stating some additional notation. for each j ∈{0, 1, ...,n} given (2.1)-(2.3), let pjk =   xj,2k∫ xj,2k−1 r   −1 , k = 1, 2, ...,mj; qjk = xj,2k+1∫ xj,2k q, wjk = xj,2k+1∫ xj,2k w, k = 0, 1, ...,mj. (2.4) and let introduce the notation m = n∑ j=0 mj. (2.5) we note from (2.2) and (2.3) that pjk, wjk ∈ r�{0} , and no sign restrictions are imposed on them. from (2.2) and (2.3) we can make the following observation: for any solution u, v of (1.9), u is constant on the intervals where r is identically zero and v is constant on the intervals where both q and w are both identically zero. let u0k = u(x), x ∈ [x0,2k; x0,2k+1] , k = 0, 1, ...,m0 − 1, (2.6) u0m0 = u(x), x ∈ [x0,2m0 ; x0,2m0+1), uj0 = u(x), x ∈ (xj0; xj1] , j = 1, 2, ...,n ujk = u(x), x ∈ [xj,2k; xj,2k+1] , k = 1, 2, ...,mj − 1, j = 1, 2, ...,n− 1 ujmj = u(x), x ∈ [xj,2mj ; xj,2mj+1), j = 0, 1, ...,n− 1 unk = u(x), x ∈ [xn,2k; xn,2k+1], k = 1, 2, ...,mn vjk = v(x), x ∈ [xj,2k−1; xj,2k), k = 1, 2, ...,mj, j = 0, 1, ...,n and set vj0 = v(xj0+), vj,mj+1 = v(xj,2mj+1−), j = 0, 1, ...,n. (2.7) lemma 2.1. assume eq. (1.2) is of atkinson type. then for each j = 0, 1, ...,n and for any solution u, v of eq. (1.9), we have pjk(ujk −uj,k−1) = vjk, k = 1, 2, ...,mj, (2.8) vj,k+1 −vjk = ujk(qjk −λwjk), k = 0, 1, ...,mj. (2.9) int. j. anal. appl. 16 (1) (2018) 29 conversely, for any solution ujk, k = 0, 1, ...,mj and vjk, k = 0, 1, ...,mj + 1 of system (2.8), (2.9), there is a unique solution u(x) and v(x) of eq. (1.9) satisfying (2.6) and (2.7). proof. relying on the first equation of (1.9), for k = 1, 2, ...,mj, we have ujk −uj,k−1 = u(xj,2k) −u(xj,2k−2) = xj,2k∫ xj,2k−2 u′ = xj,2k∫ xj,2k−2 rv = xj,2k∫ xj,2k−1 rv = vjk xj,2k∫ xj,2k−1 r = vjk�pjk. this establishes (2.8). similarly, from second equation of (1.9), for k = 0, 1, ...,mj, we have vj,k+1 −vjk = v(xj,2k+1) −v(xj,2k−1) = xj,2k+1∫ xj,2k−1 v′ = xj,2k+1∫ xj,2k−1 (q −λw)u = xj,2k+1∫ xj,2k (q −λw)u = ujk xj,2k+1∫ xj,2k (q −λw) = ujk(qjk −λwjk), which gives (2.9). on the other hand, if ujk, vjk satisfy (2.8) and (2.9), then we define u(x) and v(x) according to (2.6) and (2.7), and then extend them continuously to the whole interval j as a solution of (1.9) by integrating the equations in (1.9) over subintervals. � first, we consider slp with transmission condition(1.2)-(1.4) with seperated bc (1.7). theorem 2.1. assume α ∈ [0,π) , β ∈ (0,π]. define an (m + 1) × (m + 1) tridiagonal block matrix pαβ =   m0 n1 m1 n2 m2 . . . . . . nn mn nn+1 mn+1   and diagonal matrices qαβ = diag (q00 sin α, q01, ...,q0,m0−1, q0m0 + q10, q11, ...,qn,mn−1, qnmn sin β) , wαβ = diag (w00 sin α, w01, ...,w0,m0−1, w0m0 + w10, w11, ...,wn,mn−1, wnmn sin β) . then slp with transmission conditions (1.2), (1.3), (1.7) is equivalent to matrix eigenvalue problem (pαβ + qαβ) u = λwαβu, (2.10) int. j. anal. appl. 16 (1) (2018) 30 where m is as defined in (2.5), u = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] t and the matrices mj’s and nj’s are defined as follows: m0 = [ p01 sin α + cos α, −p01 sin α ] , (2.11) for each j = 0, 1, ...,n the mj × 2 matrices nj+1 =   −pj1 pj1 + pj2 0 −pj2 0 0 ... ... 0 0   , (2.12) for each j = 0, 1, ...,n− 1 the mj ×mj matrices mj+1 =   −pj2 pj2 + pj3 −pj3 −pj3 pj3 + pj4 −pj4 . . . . . . . . . −pj,mj−1 pj,mj−1 + pj,mj −pj,mj −pj,mj pj,mj + pj+1,1 + αj+1 −pj+1,1   , (2.13) and the mn × (mn − 1) matrix mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pn,mn −pn,mn −pn,mn sin β pn,mn sin β − cos β   . (2.14) proof. for each j = 0, 1, ...,n and k = 1, 2, ...,mj − 1, there is one-to-one correspondence between the solutions of system (2.8), (2.9) and the solutions of the following system: pj1(uj1 −uj0) −vj0 = uj0(qj0 −λwj0), (2.15) pj,k+1(uj,k+1 −ujk) −pjk(ujk −uj,k−1) = ujk(qjk −λwjk), (2.16) vj,mj+1 −pjmj (ujmj −ujmj−1 ) = ujmj (qjmj −λwjmj ) (2.17) int. j. anal. appl. 16 (1) (2018) 31 therefore, by lemma 2.1, any solution of equation (1.9), and hence of (1.2), is uniquely determined by a solution of system (2.15)-(2.17). note that from boundary condition (1.7), we have u00 cos α = v00 sin α (2.18) unmn cos β = vn,mn+1 sin β and for each j = 0, 1, ...,n− 1 from the transmission condition (1.3), we have uj+1,0 = ujmj (2.19) vj+1,0 = αj+1ujmj + vj,mj+1. additionally, for each j = 0, 1, ...,n− 1 from the equations (2.15)-(2.19), we have pj+1,1 ( uj+1,1 −ujmj ) −pjmj ( ujmj −uj,mj−1 ) −ujmj ( qjmj −λwjmj ) (2.20) = αj+1ujmj + ujmj (qj+1,0 −λwj+1,0) pj+1,2 (uj+1,2 −uj+1,1) −pj+1,1 ( uj+1,1 −ujmj ) = uj+1,1 (qj+1,1 −λwj+1,1) . (2.21) then the equivalence follows from (2.15)-(2.18) and (2.20), (2.21). � corollary 2.1. assume α, β ∈ (0,π) . define the (m + 1) × (m + 1) tridiagonal block matrix pαβ =   m0 n1 m1 n2 m2 . . . . . . nn mn nn+1 mn+1   and diagonal matrices qαβ = diag (q00,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qn,mn−1,qnmn ) wαβ = diag (w00,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wn,mn−1,wnmn ) . then slp with transmission conditions (1.2), (1.3), (1.7) is equivalent to matrix eigenvalue problem (pαβ + qαβ) u = λwαβu (2.22) where u = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] t , int. j. anal. appl. 16 (1) (2018) 32 and the matrices mj’s and nj’s are defined as in theorem 2.1 except m0 and mn+1. in this case, m0 is 1 × 2 matrix m0 = [ p01 + cot α −p01 ] , and mn+1 is mn × (mn − 1) matrix mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pn,mn −pn,mn −pn,mn pn,mn − cot β   . proof. if we divide the first and the last rows of system (2.10) by sin α and sin β respectively, then we obtain (2.22). � theorem 2.1 and its corollary show that the problem (1.2)-(1.4), (1.7) of atkinson type have representations by tridiagonal matrix eigenvalue problems. now, we will show that the problem (1.2)-(1.4), (1.8) of atkinson type also have representations. theorem 2.2. consider the boundary condition (1.8) with k12 = 0. define the m × m matrix which is tridiagonal except for the (1,m) and (m, 1) entries p1 =   m0 −k11pnmn n1 m1 n2 m2 . . . . . . nn mn −k11pnmn nn+1 mn+1   and diagonal matrices q1 = diag ( q00 + k 2 11qnmn,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qnmn ) , w1 = diag ( w00 + k 2 11wnmn,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wnmn ) . then slp with transmission conditions (1.2), (1.3), (1.8) is equivalent to matrix eigenvalue problem (p1 + q1) u = λw1u (2.23) where u = [ u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,un,mn−1 ]t , int. j. anal. appl. 16 (1) (2018) 33 and the elements of the matrix p1 are defined as follows: the 1 × 2 matrix m0 = [ −k11k21 + p01 + k211pnmn −p01 ] , for each j = 0, 1, ...,n− 1 the mj × 2 and for j = n the (mn − 1) × 2 matrices nj+1 =   −pj1 pj1 + pj2 0 −pj2 0 0 ... ... 0 0   , for each j = 0, 1, ...,n− 1 the mj ×mj matrices mj+1 =   −pj2 pj2 + pj3 −pj3 −pj3 pj3 + pj4 −pj4 . . . . . . . . . −pj,mj−1 pj,mj−1 + pj,mj −pj,mj −pj,mj pj,mj + pj+1,1 + αj+1 −pj+1,1   , and the (mn − 1) × (mn − 2) matrix mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−2 pn,mn−2 + pn,mn−1 −pn,mn−1 −pn,mn−1 pn,mn−1 + pnmn   . proof. as mentioned before, the transmission condition (1.3) is the same as (2.19). on the other hand, since k12 = 0, the boundary condition (1.8) is represented as follows: unmn = k11u00 (2.24) un,mn+1 = k21u00 + k22v00 where k11k22 = 1. we find out that for each j = 0, 1, ...,n − 1 and k = 0, 1, ...,mj − 1 there is one-to-one correspondence between the solutions consisting of system (2.8), (2.9), (2.19), (2.24) and the solutions of the int. j. anal. appl. 16 (1) (2018) 34 following system: [ −k11k21 + k211 ( pj+1,mj+1 + qj+1,mj+1 −λwj+1,mj+1 )] uj0 (2.25) = (λwj0 −pj1 −qj0)uj0 + pj1uj1 + k11pj+1,mj+1uj+1,mj+1−1 pj,k+1 (uj,k+1 −ujk) −pjk (ujk −uj,k−1) = ujk (qjk −λwjk) (2.26) pj+1,1 (uj+1,1 −uj+1,0) −vj+1,0 = uj+1,0(qj+1,0 −λwj+1,0) (2.27) pj+1,mj+1 ( k11uj0 −uj+1,mj+1−1 ) −pj+1,mj+1−1uj+1,mj+1−1 (2.28) = pj+1,mj+1−1uj+1,mj+1−2 + uj+1,mj+1−1(qj+1,mj+1−1 −λwj+1,mj+1−1) then, by lemma 2.1, any solution of system (1.9), hence of (1.2), is uniquely determined by a solution of system (2.25)-(2.28). � theorem 2.3. consider the boundary condition (1.8) with k12 6= 0. define the (m + 1) × (m + 1) matrix which is tridiagonal except for the (1,m + 1) and (m + 1, 1) entries p2 =   m0 1 k12 n1 m1 n2 m2 . . . . . . nn mn 1 k12 nn+1 mn+1   and diagonal matrices q2 = diag (q00,q01, ...,q0,m0−1,q0m0 + q10,q11, ...,qn,mn−1,qnmn ) w2 = diag (w00,w01, ...,w0,m0−1,w0m0 + w10,w11, ...,wn,mn−1,wnmn ) then slp with transmission conditions (1.2), (1.3), (1.8) is equivalent to matrix eigenvalue problem (p2 + q2) u = λw2u (2.29) where u = [u00,u01, ...,u0m0,u11, ...,u1m1, ...,un1, ...,unmn ] t , and the elements of the matrix p3 are defined as follows: for each j = 0, 1, ...,n the matrices nj+1 ’s and for each j = 0, 1, ...,n−1 the matrices mj+1 ’s are defined as in theorem 2.2. on the other hand, the 1×2 matrix m0 = [ p01 − k11k12 −p01 ] , int. j. anal. appl. 16 (1) (2018) 35 and the mn × (mn − 1) matrix mn+1 =   −pn2 pn2 + pn3 −pn3 −pn3 pn3 + pn4 −pn4 . . . . . . . . . −pn,mn−1 pn,mn−1 + pnmn −pnmn −pnmn pnmn − k22 k12   . proof. the boundary condition (1.8) can be represented as follows: unmn = k11u00 + k12v00, vn,mn+1 = k21u00 + k22v00. since k11k22 −k12k21 = 1, we have from the this condition that v00 = − k11 k12 u00 + 1 k12 unmn, vn,mn+1 = − 1 k12 u00 + k22 k12 unmn. on the other hand, if we consider the transmission condition (2.19), the proof is similar with theorem 2.2. � 3. example in this section, we give an example to illustrate that a slp with finitely many δ-interactions and it’s equivalent matrix eigenvalue problem, we will construct it, have same eigenvalues. consider the slp with δ-interactions on j = (−3, 0) ∪ (0, 6), − (py′)′ + δ(x− 0)y + qy = λwy. (3.1) this equation is equivalent to the following slp − (py′)′ + qy = λwy (3.2) with transmission condition   y(0−) −y(0+) = 0y(0−) + py′(0−) −py′(0+) = 0. (3.3) by choosing α = 0 and β = π, we consider the following boundary conditions  y(−3) = 0y(6) = 0. (3.4) int. j. anal. appl. 16 (1) (2018) 36 in this case, the matrices in (1.3) and (1.4) become c1 =   1 0 1 1   , a =   1 0 0 0   , b =   0 0 1 0   respectively. now, let’s take a partition of the interval j as follows: a = −3 < −2 < −1 < 0 = x1 (3.5) x1 = 0 < 2 < 3 < 4 < 5 < 6 = x2 = b. this yields that m0 = 1, m1 = 2 and define the piecewise constant functions p, q, w are as follows: p(x) =   ∞, (−3,−2) 1, (−2,−1) ∞, (−1, 0) ∞, (0, 2) 1 2 , (2, 3) ∞, (3, 4) 1 4 , (4, 5) ∞, (5, 6) q(x) =   0, (−3,−2) 0, (−2,−1) 1, (−1, 0) 2, (0, 2) 0, (2, 3) 3, (3, 4) 0, (4, 5) 4, (5, 6) w(x) =   1, (−3,−2) 0, (−2,−1) 3, (−1, 0) 4, (0, 2) 0, (2, 3) 1, (3, 4) 0, (4, 5) 2, (5, 6) (3.6) by using the similar method as given in [4], [5] or [20] we have the following two eigenvalues λ1 = 0.67442, λ2 = 3.75739. (3.7) on the other hand, if we find the values pjk, qjk, wjk from (2.4), and use theorem 2.1 we get the matrices p0π =   1 0 0 0 −1 5 2 −1 2 0 0 −1 2 3 4 −1 4 0 0 0 −1   , q0π =   0 0 0 0 0 5 0 0 0 0 3 0 0 0 0 0   , w0π =   0 0 0 0 0 11 0 0 0 0 1 0 0 0 0 0   , (3.8) and so the matrix eigenvalue problem (p0π + q0π) u = λw0πu, (3.9) which is equivalance of slp with finitely many δ-interactions in (3.1). indeed, if we find the eigenvalues of the matrix eigenvalue problem (3.9) we obtain the eigenvalues in (3.7). int. j. anal. appl. 16 (1) (2018) 37 references [1] f.v. atkinson, discrete and continuous boundary problems, academic press-new york, london, 1964. [2] q. kong, h. wu and a. zettl, sturm–liouville problems with finite spectrum, j. math. anal. appl. 263 (2001), 748–762. [3] a. zettl, sturm-liouville theory, amer. math. soc., math. surveys monographs, no. 121, 2005. [4] j.j. ao, j. sun and m.z. zhang, the finite spectrum of sturm-liouville problems with transmission conditions, appl. math. comput. 218 (2011), 1166–1173. [5] j.j. ao, j. sun and m.z. zhang, the finite spectrum of sturm-liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, result. math. 63(3-4) (2013), 1057-1070. [6] j.j. ao, f.z. bo and j. sun, fourth-order boundary value problems with finite spectrum, appl. math. comput. 244 (2014), 952-958. [7] f.z. bo and j.j. ao, the finite spectrum of fourth-order boundary value problems with transmission conditions, abstr. appl. anal. 2014 (2014), art. id 175489. [8] q. kong, h. volkmer and a. zettl, matrix representations of sturm-liouville problems with finite spectrum, result. math. 54 (2009), 103–116. [9] j.j. ao, j. sun and m.z. zhang, matrix representations of sturm–liouville problems with transmission conditions, comput. math. appl. 63 (2012), 1335–1348. [10] j.j. ao. and j. sun, matrix representations of sturm–liouville problems with eigenparameter-dependent boundary conditions, linear algebra appl. 438 (2013), 2359–2365. [11] j.j. ao. and j. sun, matrix representations of sturm–liouville problems with coupled eigenparameter-dependent boundary conditions, appl. math. comput. 244 (2014), 142-148. [12] j.j. ao, j. sun and a. zettl, matrix representations of fourth-order boundary value problems with finite spectrum, linear algebra appl. 436 (2012), 2359-2365. [13] j.j. ao, j. sun and a. zettl, equivalance of fourth-order boundary value problems and matrix eigenvalue problems, result. math. 63 (2013), 581–595. [14] s. ge, w. wang and j.j. ao, matrix representations of fourth-order boundary value problems with periodic boundary conditions, appl. math. comput. 227 (2014), 601–609. [15] a. kablan and m. d. manafov, matrix representations of fourth-order boundary value problems with transmission conditions, mediterranean j. math. 13(1) (2016), 205-215. [16] j.j. ao. and j. sun, matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions, linear multilinear algebra 63(8) (2015), 1590-1598. [17] l.n. pandey and t.f. george, intersubband transitions in quantum well heterostructures with delta-doped barriers, appl. phys. lett. 61 (2016), 1081. [18] r.k. willardson and a.c. beer, semiconductors and semimetals, academic press, london, 1984. [19] s. albeverio, f. gesztesy, r. høegh-krohn and h. holden, solvable models in quantum mechanics, springer-verlag, berlin/new york, 1988. [20] a. kablan, m.a. çetin and m.d. manafov, the finite spectrum of sturm-liouville operator with finitely many δinteractions, pre-print. 1. introduction 2. matrix representations of slps with finitely many 0=x"010e-interactions 3. example references international journal of analysis and applications volume 17, number 5 (2019), 838-849 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-838 anemia modelling using the multiple regression analysis murat sari∗, and arshed a. ahmad department of mathematics, yildiz technical university, istanbul 34220, turkey ∗corresponding author: sarim@yildiz.edu.tr abstract. the aim of this article is to forecast anemia from a population through biomedical variables of individuals using the multiple linear regression model. the study is conducted in terms of dataset consisting of 539 subjects provided from blood laboratories. a multiple linear regression model is produced through biomedical information. to achieve this, a mathematical method based on multiple regression analysis has been applied in this research for a reliable model that investigate if there exists a relation between the anemia and the biomedical variables and to provide the more realistic one. for comparison purposes, the linear deep learning methods have also been considered and the current results are seen to be slightly better. the model based on the variables and outcomes is expected to serve as a good indicator of disease diagnosis for health providers and planning treatment schedules for their patients, especially predict of the type of anemia. 1. introduction a mathematical model is an essential tool for analyzing pathological characteristics and it can be used for various reasons as in the literature [1-10]. to assess situations seen in hospitals, any disease condition has several effects for a single disease. so, most outcomes in real life problems are affected by multiple input variables. as signified in the literature [11-14], the anemia of chronic inflammation and it was initially thought to be associated primarily with the infectious, inflammatory diseases. anemia is a lower blood hemoglobin level below normal limits determined by the world health organization (who) [11]. this received 2019-05-15; accepted 2019-06-19; published 2019-09-02. 2010 mathematics subject classification. 93a30. key words and phrases. anemia; medical modelling; mathematical modelling; regression model. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 838 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-838 int. j. anal. appl. 17 (5) (2019) 839 decrease in the level of hemoglobin leads to the lack of access to the organs of the body enough amount of oxygen and therefore appear in the symptoms of a headache, fatigue, and inability to focus and attention. as pointed out by hbert et al [12], anemia is one of the most common cases among blood diseases worldwide. this article aims at predicting pathological subjects from a population through physical biomedical variables (eight blood variables, sex, and age) and output (anemia types). it is important to predict the type of anemia because there has been an increase in the incidence of anemia among different segments of society. to make the best biomedical decisions, medical predictions play a very important role in the process of diagnosis and planning treatment for health providers. so, our goal is to develop a new mathematical model to study the effect of the blood variables, sex, and age on the types of anemia. our model, different from the mathematical models given in the literature [15-19], has also been successfully used in the prediction of several types of anemia through a large group of blood variables, sex, and age. in the literature, many studies were carried out [16,18,21-23] by using relatively less number of input variables to predict the type of anemia. the methods used in the corresponding studies produced relatively less accurate results. for the blood variables, hemoglobin (hb), red blood cell (rbc), mean corpuscular volume (mcv), mean corpuscular hemoglobin (mch), and red cell distribution width (rcdw) were analyzed by sirachainan et al [16]. they used the mcv and mch [18]. jimnez [21] used the rbc, hb, and hematocrit (hct). another researcher [22] considered the mcv, mch, hct, and hb. piplani et al [23] used the hb, rbc, mcv, and mch. despite all those pioneering advances in these fields, the corresponding studies used a limited number of blood variables. to the best knowledge of the authors, more general models representing the behaviour closer to nature have been produced for the first time. the more number of input variables makes the derived model more realistic in the biomedicine. thus, for such a realistic model, for such a large number of input variables a study has been accomplished here. therefore, this study is believed to be an important contribution to predict the types of anemia. despite very effective, striking and frontier studies in the literature, researchers have used models with limited number of variables. therefore, the present study focuses on the determination of the type of anemia through a very large number of the observational variables, more realistic one. since many researchers have commonly considered the multiple regression analysis among the modelling techniques to deal with various problems including anemia [19,23-32], the multiple analysis is taken into account in modelling the current biomedical problem. the remainder of the paper is organized as follows: section 2 highlight the study samples, explain linear regression analysis procedure and test the model. building the linear model of data by the regression analysis has been given in section 3. regression model has been analyzed and discussed in section 4. finally, conclusions and future research directions have been detailed. int. j. anal. appl. 17 (5) (2019) 840 2. materials and methods 2.1. study samples. the data were collected from observations of blood variables in order to identify a healthy or infected person and involved 539 subjects provided from blood laboratories in iraq. individuals between 6-56 years old have been taken into consideration and included 248 males, 291 females. subjects are consisting of 211 healthy ones and of 328 anemic ones to build the model. the number of variables studied and selected for building the model is eleven, the independent variables identified are ten and a dependent variable. the dependent variable consists of six different outputs are healthy (0) and five blood diseases are iron deficiency anemia (1), deficiency vitamin b12 (2), thalassemia (3), sickle cell (4) and spherocytosis (5). here the samples for people and for each subject readings of blood variables are [11,12] hemoglobin (hb), red blood cells (rbc), mean corpuscular hemoglobin (mch), white blood cell (wbc), hematocrit (hct), mean corpuscular hemoglobin concentration (mchc), platelets (plt), mean corpuscular volume (mcv) and sex and age. the corresponding blood variables can be briefly introduced as follows. the hb is a portable protein inside the rbc and contains iron atoms, and that carries oxygen from the lungs to the body’s tissues and returns carbon dioxide from the tissues back to the lungs. the rbcs are concave cells are useless nucleus contains the hb. the mch is the calculated value derived from the hb measurement and a number of red cells. the wbcs are the cells of the immune system that are involved in protecting the body against infectious disease. the hct is percentage of the rbcs volume of total blood volume. the mchc is the calculated concentration of hb in a specific volume of rbc. the plt is an irregular, disc-shaped element in the blood that assists in blood clotting. the plts are usually classed as blood cells as well. average size of the red cells in a sample is measured by the mcv. the other biophysical variables, sex and age, are considered. because natural hb in the body varies from male to female, and thus male: 1, female: 2. yet, natural hb in the body varies according to age. the anemia types and blood variables for our data are displayed in table 1. int. j. anal. appl. 17 (5) (2019) 841 table 1. some samples from the data hb rbc mch wbc mcv hct mchc plt sex age anemia type 17.5 5.55 31.6 14.1 92 50.9 34.5 318 2 23 0 16.3 6.07 26.9 8.16 80.9 49.1 33.2 349 1 23 0 11.1 4.38 25.3 5.8 81 35.6 31.1 227 1 11 1 11.1 4.85 22.8 10 81 39.4 28.1 274 2 16 1 9 3.47 25.8 2.3 88 30.4 29.5 148 1 11 2 1.46 4.4 30.4 59.8 108 15.8 28.2 330 2 29 2 8.1 3.6 22.4 12 78 28.1 28.7 472 1 15 3 3.92 6.6 16.8 8.3 60 23.7 27.9 443 2 17 3 8.3 2.58 31.9 12.4 103 26.7 30.9 458 1 11 4 7.9 2.88 27.4 17.55 83 23.9 33.1 703 1 16 4 6.8 5.77 11.7 11.9 49 28.4 23.8 573 2 11 5 2.2. multiple linear regression model. consider a multiple linear regression (mlr) model with k predictor variables, independent observations y = b0 + b1x1 + b2x2 + ... + bkxk + � = b0 + k∑ i=1 bixi + �. (2.1) the observations recorded for each of these n levels can be expressed in the following way y1 = b0 + b1x11+b2x12 + ... + bkx1k + �1 y2 = b0 + b1x21+b2x22 + ... + bkx2k + �2 ... yi = b0 + b1xi1+b2xi2 + ... + bkxik + �i ... yn = b0 + b1xn1+b2xn2 + ... + bkxnk + �n (2.2) the dependent observations y1,y2, ...,yn, and the independent observations x1,x2 , ...,xk, have n levels. then xij represents the ith level of the jth predictor variable, xj. system (2) can be represented as follows: y = bx + �, (2.3) int. j. anal. appl. 17 (5) (2019) 842 y =   y1 y2 ... yn   ,x =   1 x11 x12 . . . x1k 1 x21 x22 . . . x2k ... ... ... ... 1 xn1 xn2 . . . xnk   ,b =   b0 b1 ... bk   ,� =   �1 �2 ... �n   (2.4) where y,x,b and � stand for the observations, the regression coefficients and an unobserved random variable that adds noise to the linear relationship between the dependent variable and regressors, respectively. to obtain the regression model, b should be known. therefore, b is estimated by using the least square estimates as follows b̂ = (xt x)−1xt y, (2.5) where xt represents the transpose of the matrix x while (xt x)−1 represents inverse of the matrix (xt x). knowing the estimate b̂, the mlr model can now be expressed as [33,34] ŷ = b̂x, (2.6) where ŷ is the estimated value for y from the regression. 2.3. test for the model. the linear regression model estimation is selected and the sum of square tests. the computation formula can be given as follows: sst = n∑ j=1 (yj − ȳ)2,ssr = n∑ j=1 (ŷj − ȳ)2,sse = n∑ j=1 (yj − ŷj)2 = n∑ j=1 e2j. (2.7) the coefficient of determination is a measure showing the rate of the contribution of the independent variables in the interpretation of the change in the dependent variable as known from the literature [35,36]. it is given as follow: r2 = ssr sst = 1 − sse sst . (2.8) a terminological difference arises in the expression mean squared error (mse). the mse of a regression is a measure of the average of the sum of squared error and how the concentration of data around the regression model. the smaller the mse, whenever the results are more accurate [35,36]. then it is given by mse = 1 n n∑ j=1 e2j. (2.9) 3. building linear regression analysis model the currently produced mlr model is a linear equation determined as previously mentioned in section 2.2. the obtained model is as follows: y = b0 + b1hb + b2rbc + b3mch + b4wbc + b5mcv +b6hct + b7mchc + b8plt + b9sex + b10age + � (3.1) int. j. anal. appl. 17 (5) (2019) 843 where y is type of the anemia and bi, 0 ≤ i ≤ 10, are the parameters to be determined. the linear regression model, as explained in section 2.2, is estimated as ŷ = 6.377 − 0.224hb − 0.224rbc − 0.029mch + 0.001wbc +0.0005mcv − 0.016hct + 0.007mchc + 0.001plt −0.311sex − 0.009age. (3.2) here the coefficient values of the linear model have been obtained through the multiple regression approach, to find the model that is more realistic (see table 4). as previously mentioned, the model can be represented in matrix form as follows: ŷ = b̂x (3.3) where ŷ =   y1 y2 ... y539   ,x =   1 hb11 rbc12 . . . age110 1 hb21 rbc22 . . . age210 ... ... ... ... 1 hb539,1 rbc539,2 . . . age539,10   ,b =   6.377 −0.224 −0.224 −0.029 0.001 0.0005 −0.016 0.007 0.001 −0.311 −0.009   (3.4) here ŷ and x represent the estimates for output (types of the anemia) and the independent observations matrix, respectively. 4. results and discussion different strategies of mathematical methods are implemented to analyze blood variables, as in the literature [16,18,22,37]. the multiple regression analysis have been taken into account by many researchers [19,23-32] while dealing with various anemia problems at different levels. however, they used a limited number of blood variables and they did not study a relationship for the prediction of the types of anemia. therefore, the current study concentrates on the investigation of the relationship between a very large number of blood variables and the types of anemia. various versions of models, based on the variables, are derived (see table 2). int. j. anal. appl. 17 (5) (2019) 844 table 2. various forms of the multiple linear models: blood variables, sex, and age models r2 mse model 1 for (hb, sex and age) 0.568 0.935 model 2 for (rbc, sex and age) 0.174 1.787 model 3 for (mch, sex and age) 0.255 1.611 model 4 for (wbc, sex and age) 0.229 1.667 model 5 for (mcv, sex and age) 0.190 1.752 model 6 for (hct, sex and age) 0.649 0.759 model 7 for (mchc, sex and age) 0.243 1.637 model 8 for (plt, sex and age) 0.271 1.577 model 9 for (hb, rbc, sex and age) 0.686 0.680 model 10 for (mch, wbc, sex and age) 0.304 1.509 model 11 for (mcv, hct, sex and age) 0.649 0.760 model 12 for (mchc, plt, sex and age) 0.314 1.486 model 13 for (wbc, mcv, hct, mchc, sex and age) 0.668 0.723 model 14 for (hb, rbc, mch, plt, sex and age) 0.698 0.656 the models produced in terms of larger number of blood variables show better correlation than the models produced in terms of less number of blood variables for predicting the types of anemia in equation (3.2). however, naturally some of the variables are of more effect than others. after the essential requirements have been verified for the multivariate analysis in equation (3.2), the variables have been included for the mlr analysis. those variables consist of regression coefficients b, the blood variables (hb, rbc, mch, wbc, mcv, hct, mchc, plt), sex, and age. therefore, the mlr shows the synergistic effect of predicting the types of anemia better than the ones used fewer blood variables. the enter method of the mlr has been used in the current analysis. all the variables were introduced into the regression model as selected by the enter method of the mlr. in the outcome of the current analysis, it has been found that there is a more significant relation (r2=0.699) of the mlr model. it means that 69.90% of the change in the relationship between all blood variables, sex, and age for the types of anemia is explained. thus, it is concluded that the regression model with the blood variables, sex, and age are seen to be significant (p < 0.000). that means simultaneous consideration of the blood variables, sex, and age has a significant effect on the relationship on the determination of the types of anemia (see table 3). int. j. anal. appl. 17 (5) (2019) 845 table 3. analysis of variance for the correlation in equation (3.2) sum of squares degrees of freedom mean square f-stat p-value regression 809.354 10 80.935 122.838 0.000 residual 347.889 528 0.659 total 1157.243 538 the standardized coefficient (beta) compares the effect force of each individual blood variables, sex, and age to the types of anemia. it is thus given by standardizedbetaj = bj ∗ sd(xj)/sd(y ) . the hb absolute value of the beta coefficient is (−0.663) has the strongest relationship with the types of the disease comparison to the other variables rbc (−0.345), sex (−0.106), hct (−0.100), mch (−0.090), plt (0.080), age (−0.065), wbc (0.016), mchc (0.016) and mcv (−0.001). the interpretation of the beta value for the hb signifies that for every change in the hb, the dependent variable will be changed by the beta coefficient value (see table 4). the t-test was used to measure the partial effect of the variables hb, rbc, mch, wbc, mcv, hct, mchc, plt, sex, and age on the types of anemia. notice that these variables have been seen to affect the types of anemia but in varying rates (see table 4). the histogram of the residuals which confirm that the data are distributed according to a normal distribution with a mean of zero and a standard deviation of 0.991 (see figure 1). to find out the extent of spread the random error around the linear regression model, the mlr use the mean square residuals, mse=0.659 (see table 3). small values of the mse indicate the concentration of data around the linear regression model (see figure 2). in this study, comparing criteria are constructed on the principle of whether the technique provides a suitable prediction or not. this task is achieved by comparing with the deep learning method (lstm). the results demonstrate that the linear regression has the best fit to the initial dataset comparing to the deep learning method (lstm) (see table 5). therefore, the present study provides an accurate model for prediction of the types of anemia. int. j. anal. appl. 17 (5) (2019) 846 table 4. analysis of the multiple regression coefficients given in equation (3.2) unstandardized coefficients standardized coefficients b std. error beta t-stat p-value (const.) 6.377 0.552 11.563 0.000 hb -0.224 0.062 -0.663 -3.581 0.000 rbc -0.224 0.066 -0.345 -3.392 0.001 mch -0.029 0.015 -0.090 -1.931 0.054 wbc 0.001 0.003 0.016 0.549 0.583 mcv 0.0005 0.008 -0.001 -0.015 0.988 hct -0.016 0.028 -0.100 -0.575 0.565 mchc 0.007 0.016 0.016 0.464 0.643 plt 0.001 0.000 0.080 2.637 0.009 sex -0.311 0.074 -0.106 -4.191 0.000 age -0.009 0.004 -0.065 -2.303 0.022 table 5. comparison of the mlr results with the results of the linear deep learning method methods sse mse r2 linear regression analysis 347.889 0.659 0.699 linear deep learning methods (lstm) 349.869 0.665 0.695 lstm: long short term memory 5. conclusions and recommendation this study has forecasted the types of anemia through biomedical information under the consideration of eight different blood variables, sex, and age of individuals. multiple linear regression model, for the first time, have been derived in forecasting the types of anemia. the results revealed that the regression model is very promising and is capable of making the prediction. in the analysis of the current anemia problem, the multiple regression method has been found to be more accurate than linear deep learning methods. it has been concluded that the model is expected to be helpful for diagnosis of the types of anemia to health providers and designing an appropriate treatment programs for their patients. for future research, these mathematical model may be improved under the consideration of various computational methods. int. j. anal. appl. 17 (5) (2019) 847 figure 1. histogram plot of the residuals figure 2. normal pp plot of regression standardized residual references [1] m. farman, z. iqbal, a. ahmad, a. raza and eu. haq, numerical solution and analysis for acute and chronic hepatitis b, int. j. anal. appl. 16 (2018), 842-55. [2] e. gulbandilar, a. cimbiz, m. sari and h. ozden, relationship between skin resistance level and static balance in type ii diabetic subjects, diabetes res. clin. pract. 82 (2008), 335-339. [3] m. sari, e. gulbandilar and a. cimbiz, prediction of low back pain with two expert systems, j. med. syst. 36 (2012), 1523-1527. [4] d. okuonghae, a mathematical model of tuberculosis transmission with heterogeneity in disease susceptibility and progression under a treatment regime for infectious cases, appl. math. model. 37 (2013), 6786-808. int. j. anal. appl. 17 (5) (2019) 848 [5] m. sari, c. tuna and s. akogul, prediction of tibial rotation pathologies using particle swarm optimization and k-means algorithms, j. clin. med. 7 (2018), 65. [6] s. conoci, f. rundo, s. petralta and s. battiato, in advanced skin lesion discrimination pipeline for early melanoma cancer diagnosis towards poc devices, 2017 european conference on circuit theory and design (ecctd), 2017, ieee, 1-4. [7] m. sari, relationship between physical factors and tibial motion in healthy subjects: 2d and 3d analyses, adv. ther. 24 (2007), 772-783. [8] m. sari and bg. cetiner, predicting effect of physical factors on tibial motion using artificial neural networks, expert syst. appl. 36 (2009), 9743-9746. [9] c. liddell, n. owusu-brackett and d. wallace, a mathematical model of sickle cell genome frequency in response to selective pressure from malaria, bull. math. biol. 76 (2014), 2292-2305. [10] m. sari and c. tuna, prediction of pathological subjects using genetic algorithms, comput. math. methods med. 2018 (2018). [11] world health organization. worldwide prevalence of anaemia 1993-2005: who global database on anaemia, 2008. [12] pc. hbert, g. wells, ma. blajchman, j. marshall, c. martin, g. pagliarello, m. tweeddale, i. schweitzer and e. yetisir, a multicenter, randomized, controlled clinical trial of transfusion requirements in critical care, n. engl. j. med. 340 (1999),409-417. [13] a. kim, s. rivera, d. shprung, d. limbrick, v. gabayan, e. nemeth and t. ganz, mouse models of anemia of cancer, plos one 9 (2014), e93283. [14] li xuejin, m. dao, g. lykotrafiti and ge. karniadakis, biomechanics and biorheology of red blood cells in sickle cell anemia, j. biomech. 50 (2017), 34-41. [15] j.c. mcallister, modeling and control of hemoglobin for anemia management in chronic kidney disease, doctoral dissertation, university of alberta, 2017 (2017). [16] n. sirachainan, p. iamsirirak, p. charoenkwan, p. kadegasem, p. wongwerawattanakoon, w. sasanakul, n. chansatitporn and a. chuansumrit, new mathematical formula for differentiating thalassemia trait and iron deficiency anemia in thalassemia prevalent area: a study in healthy school-age children, southeast asian j. trop. med. public. health. 45 (2014), 174-182. [17] a. ngwira and l. n. kazembe, analysis of severity of childhood anemia in malawi: a bayesian ordered categories model, open access med. stat. 6 (2016), 9-20. [18] il. roth, b. lachover, g. koren, c. levin, l. zalman and a. koren, detection of β-thalassemia carriers by red cell parameters obtained from automatic counters using mathematical formulas, mediterr. j. hematol. infect. dis. 10 (2018), 1-10. [19] f. habyarimana, t. zewotir and s. ramroop, structured additive quantile regression for assessing the determinants of childhood anemia in rwanda, int. j. environ. res. public health. 14 (2017), 652. [20] s. piplani, m. madaan, r. mannan, m. manjari, t. singh and m. lalit, evaluation of various discrimination indices in differentiating iron deficiency anemia and beta thalassemia trait: a practical low cost solution, ann. pathol. labor. med. 3 (2016), a551-559. [21] cv. jimnez, iron-deficiency anemia and thalassemia trait differentiated by simple hematological tests and serum iron concentrations, clin. chem. 39 (1993), 2271-2275. [22] n. soleimani, relationship between anaemia, caused from the iron deficiency, and academic achievement among third grade high school female students, procedia-soc. behav. sci. 29 (2011), 1877-1884. int. j. anal. appl. 17 (5) (2019) 849 [23] v. sharma and r. kumar, dating of ballpoint pen writing inks via spectroscopic and multiple linear regression analysis: a novel approach, microchem. j. 134 (2017), 104-113. [24] x.z. huang, y.c. yang, y. chen, c.c. wu, r.f. lin, z.n. wang and x. zhang, preoperative anemia or low hemoglobin predicts poor prognosis in gastric cancer patients: a meta-analysis, dis. markers 2019 (2019), article id 7606128. [25] f. al-hadeethi and m. al-safadi, using the multiple regression analysis with respect to anova and 3d mapping to model the actual performance of pem (proton exchange membrane) fuel cell at various operating conditions, energy, 90 (2015), 475-482. [26] am. saviano and fr. loureno, measurement uncertainty estimation based on multiple regression analysis (mra) and monte carlo (mc) simulationsapplication to agar diffusion method, measurement, 115 (2018), 269-278. [27] sm. paras, a simple weather forecasting model using mathematical regression, indian res. j. ext. educ. 12 (2016), 161-168. [28] j. daru, j. zamora, b.m. fernndez-flix, j. vogel, o.t. oladapo, n. morisaki, . tunalp, m.r. torloni, s. mittal and k. jayaratne, risk of maternal mortality in women with severe anaemia during pregnancy and post partum: a multilevel analysis, lancet glob. health, 6 (2018), e548-e554. [29] ho. kok hoe, p. joshua and o. kok seng, a multiple regression analysis approach for mathematical model development in dynamic manufacturing system: a case study, j. sci. res. develop. 2 (2015), 81-87. [30] p.h. nguyen, s. scott, r. avula, l.m. tran and p. menon, trends and drivers of change in the prevalence of anaemia among 1 million women and children in india, 2006 to 2016, bmj glob. health, 3 (2018), e001010. [31] aj. prieto, a. silva, j. de brito, jm. macas-bernal and fj. alejandre, multiple linear regression and fuzzy logic models applied to the functional service life prediction of cultural heritage, j. cult. herit. 27 (2017), 20-35. [32] m. little, c. zivot, s. humphries, w. dodd, k. patel and c. dewey, burden and determinants of anemia in a rural population in south india: a cross-sectional study, anemia 2018 (2018), article id 7123976. [33] jo. rawlings, sg. pantula and da. dickey, applied regression analysis: a research tool, second edition, springer science & business media, 2001. [34] m. srikanta and dg. akhil, chapter 4 regression modeling and analysis, applied statistical modeling and data analytics, a practical guide for the petroleum geosciences, 1st edition, elsevier, (2018), 69-96. [35] j.f. rudolf, j.w. william and s. ping, regression analysis: statistical modeling of a response variable, second edition, elsevier, 2006. [36] ac. cameron and pk. trivedi, regression analysis of count data, cambridge university press, 2013. [37] n. bessonov, a. sequeira, s. simakov, y. vassilevskii and v. volpert, methods of blood flow modelling, math. model. nat. phenom. 11 (2016), 1-25. 1. introduction 2. materials and methods 2.1. study samples 2.2. multiple linear regression model 2.3. test for the model 3. building linear regression analysis model 4. results and discussion 5. conclusions and recommendation references international journal of analysis and applications volume 19, number 2 (2021), 264-279 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-264 smoothing approximations for least squares minimization with l1-norm regularization functional henrietta nkansah∗, francis benyah, henry amankwah department of mathematics, university of cape coast, cape coast, ghana ∗corresponding author: hnkansah@ucc.edu.gh abstract. the paper considers the problem of least squares minimization with l1-norm regularization functional. it investigates various smoothing approximations for the l1-norm functional. it considers quadratic, sigmoid and cubic hermite functionals. a tikhonov regularization is then applied to each of the resulting smooth least squares minimization problem. results of numerical simulations for each smoothing approximation are presented. the results indicate that our regularization method is as good as any other non-smoothing method used in developed solvers. 1. introduction we consider the problem (1.1) min α g(α) = f(α) + λj(α) where f(α) is smooth, j(α) is non-smooth and λ > 0 is the regularization parameter. in particular, we examine f(α) = ‖xα− y‖22 and j(α) = ‖α‖1 . therefore, the problem becomes (1.2) min α g(α) = ‖xα− y‖22 + λ‖lα‖1 , and l is the p×n discrete approximation of the (n−p)-th derivative operator. received october 8th, 2019; accepted november 4th, 2019; published march 5th, 2021. 2010 mathematics subject classification. 68w25. key words and phrases. least squares minimization; regularization; smoothing approximations; over-determined systems. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 264 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-264 int. j. anal. appl. 19 (2) (2021) 265 in this paper, we focus on over-determined linear model of the form xα = y, where α ∈<p is the vector of unknowns, y ∈<m is the vector of observations and x ∈<m×p is the decision matrix. when m ≥ p and the columns of x are linearly independent, we can determine α by solving the least squares problem of minimizing the quadratic loss ‖xα− y‖22 , where ‖v‖2 = ( ∑ i v 2 i ) 1 2 denotes the l2-norm of v and ‖α‖1 = ∑ |αi| . when m is not large enough compared to p, a simple least-squares problem leads to over-regularization. 1.1. background. the l1-norm minimization has attracted attention in data fitting as an effective technique for solving over-determined systems of linear equations and has also been proposed for regularization ([7], [10], [11]). the l1norm is a matrix norm that penalizes the sum of maximum absolute values of each row. this regularizer encourages row sparsity, that is, it encourages the entire rows of the matrix to have zero elements. in essence, this type of regularization aims at extending the l1 framework for learning sparse models to a setting where the goal is to learn a set of sparse models. learning algorithms based on l1 regularized loss functions have had a relatively long history in machine learning, covering a wide range of applications such as sparse sensing ([5], [6]), l1-logistic regression [8] and structure learning of markov networks [9]. a well known property of l1 regularized models is their ability to recover sparse solutions. because of this, they are suitable for applications where discovering significant features is of value and where computing features is expensive. in addition, it has been shown that in some cases, l1 regularization can lead to sample complexity bounds that are logarithmic in the number of input dimensions, making it suitable for learning in high dimensional spaces [9]. [11] developed an interior point algorithm for optimizing a twice differentiable objective regularized problem with an l1norm. one of the limitations of this approach is that it requires the exact computation of the hessian of the objective function. this might be computationally expensive for some applications both in terms of memory and time. an alternative approach was proposed by [10], who combined a gradient-descent method with independent l∞ projections. for the special case of a linear objective function, the regularization problem can be expressed as a linear programme [12]. while this is feasible for small problems, it does not scale to problems with large number of variables. ([13], [16]) also proposed an l1 projection algorithm which is a special case of the algorithm where m = 1. the derivation of the general case for l1 regularization is significantly more involving, as it requires reducing a set of l∞ regularization problems tied together through a common l1 -norm to a problem that can be solved efficiently. similar to the l1norm, the l2 -norm has also been proposed for sparse approximation. this norm penalizes the sum of the l2norms of each row ([14], [15],[17]). int. j. anal. appl. 19 (2) (2021) 266 a standard technique to prevent over-regularization is tikhonov regularization or the l2-norm [3] given by (1.3) min α g(α) = ‖xα− y‖22 + λ‖α‖ 2 2 . the l2regularized least squares problem (lsp) has an analytic solution of αl2 = (x t x + λi)−1xt y. solutions to the lsp can either be computed by direct methods or by nondirect method, that is, applying iterative methods to the linear system of equations (xt x + λi)α = xt y. iterative methods are efficient especially when there are fast algorithms for the matrix-vector multiplications with the decision matrix x and its transpose xt . singular value decomposition of x in component form is (1.4) αreg = p∑ i=1 fi uti y σi vi where fi are the filter factors and is given by fi = σ2i λ + σ2i . if λ << σ2i , fi ≈ 1, which indicates that the filter factors has no effect on the solution. thus, equation (1.4) is without regularization. however, if λ >> σ2i , fi ≈ σ2i λ , and σ2i λ → 0. this indicates that the filter factors has effect on the solution. in this case, it reduces the effect of the smaller singular values. thus, equation (1.4) gives the solution with regularization. another technique is the l1-regularized least squares in which we substitute a sum of absolute values for the sum of squares used in the l2-norm regularization, to obtain equation (1.2). this problem in equation (1.2) always has a solution but it needs not be unique. the first part of g(α) is smooth but the second part is non-smooth. in this paper, we explore three smoothing approximations that can be used to replace the l1-norm regularized term thereby enabling us to apply the tikhonov regularization method. these approximations are the quadratic approximation of a function [2], sigmoid function approximation [1] and the cubic hermite approximation. these three approximations are used to obtain a regularized solution to the least-squares problem in the case where l = ip, which is the tikhonov regularization of order zero. in each case, we will compare the solution from our regularization method with that of the modified newton’s method, which is mostly used in the int. j. anal. appl. 19 (2) (2021) 267 literature. we begin by implementing the modified newton’s method used by lee et al.(2006) for solving an unconstrained optimization problem in order to ascertain the challenges associated with the method. 2. smoothing approximations 2.1. quadratic approximation. lee et al. (2006), proposed a method for transforming the non-differentiable l1-norm function into a differentiable function by replacing it with a differentiable approximation. for a one dimensional case, the approximation to the absolute value function is given by |x| ≈ √ x2 + �. to determine the best approximate solution, we first examine the nature of the plot for various values of �. approximation of the absolute value function for different values of �, is given in figure 1. figure 1. quadratic approximation of |x|� for various values of the approximation parameter, �. figure 1 indicates that lim �→0 |x|� = |x| . thus, we choose � = 0.0001 for the subsequent implementation. the gradient, ∇(|x|�), and the hessian, ∇2(|x|�), of the smoothing approximation of the absolute value function given in single variable form are derived as follows: ∇(|x|�) = x √ x2 + � and ∇2(|x|�) = �(√ x2 + � )3 . for x ∈<p, ‖x‖1 = p∑ i=1 |xi| ≈ p∑ i=1 |xi|� . int. j. anal. appl. 19 (2) (2021) 268 the loss function given in equation (1.2) therefore becomes (2.1) g(α) ≈‖xα− y‖22 + µ p∑ i √ α2i + �. the regularized solution of (2.1)is given as αµ = (x t x + µi)−1xt y, where µ = 1 2 λ�− 1 2 . the regularized solution αµ written in terms of singular value decomposition (svd) is given in component form as αµ = p∑ i=1 σi σ2i + µ (uti y)vi, where σi are the singular values of the matrix x. 2.1.1. derivation of analytic solution using lee-quadratic approximation. let k(x) = p∑ i √ x2i + � = ‖x‖� ≈‖x‖1 . since k(x) is differentiable at x = 0, a taylor’s expansion about x = 0 is given as k(x) ≈ k(x0) + ∇k(x0)t (x − x0) + 1 2 (x − x0)t∇2k(x0)(x − x0) + · · · ≈ k(0) + ∇k(0)t x + 1 2 xt∇2k(0)x + · · · , where k(0) = p� 1 2 , ∇k(0) = ∇k(x) = xi√ x2i + � ∣∣∣∣∣ x=0 = 0, and ∇2k(0) = ∇2k(x) = �(√ x2i + � )3 ∣∣∣∣∣ x=0 =   �− 1 2 0 · · · 0 0 �− 1 2 · · · 0 ... · · · . . . · · · 0 0 · · · �− 1 2   = �− 1 2 ip. therefore, k(x) = p� 1 2 + 1 2 xt�− 1 2 xip. thus, g(α) in (2.1) becomes (2.2) g(α) = ‖xα− y‖22 + λ ( p� 1 2 + 1 2 αt�− 1 2 α ) int. j. anal. appl. 19 (2) (2021) 269 which is now differentiable. finding the gradient of g(α) and equating to zero gives 2xt xα− 2xt y + λα�− 1 2 = 0 xt xα + 1 2 λα�− 1 2 = xt y( xt x + 1 2 λ�− 1 2 i ) α = xt y (2.3) αµ = ( xt x + µi )−1 xt y, where µ = 1 2 λ�− 1 2 . the method usually considered in the literature after obtaining a smoothing approximation to replace the l1-norm functional is an unconstrained optimization method known as the modified newton’s method. to compare our results using regularization with the modified newton’s method, we now implement this method. the algorithm based on the implementation of modified newton’s method is formulated as xk+1 = xk −βkh(xk)−1∇g(xk), where xk+1 is the next iterate, xk is the current iterate, ∇g(xk) is the gradient at the current iterate xk, βk > 0 is the step size and h(xk) is the hessian at the current iterate. from (2.1), the gradient of g(α) is given as ∇g(α) = 2xt (xα− y) + λg(α), where g(α) = [ α1(α 2 1 + �) −1 2 , α2(α 2 2 + �) −1 2 , · · · , αp(α2p + �) −1 2 ]t , and the hessian is also given as h(α) = 2xt x + �λh(α), where h(α) = diag [ (α21 + �) −3 2 , (α22 + �) −3 2 , · · · , (α2p + �) −3 2 ] . we now consider the implementation of the algorithm. int. j. anal. appl. 19 (2) (2021) 270 2.1.2. numerical experiment. to illustrate our results, we make use of the 12 × 7 hilbert submatrix of the 12 × 12 hilbert matrix which constitute an overdetermined system. hilbert matrices are known to be very ill-conditioned because the coefficient matrix xt x is almost near zero. y is chosen such that the true solution is α = [1, 1, 1, 1, 1, 1, 1]t . we want to find α ∈<p such that xα = y. numerical simulations are performed to obtain an optimal regularization parameter µ which will hopefully give a solution close to the true solution. we define µ = 10−16 for the modified newton’s method and µ = 10−30 for the regularization method after several iterations. script-files are created in octave 4.0 to compute the solutions at various iterations. in the implementations, we define � = 0.0001, βk = β = 2 and set initial guess x0 = 0.25 ∗ ones(7, 1). we also initialize x := ones(7, 1). numerical simulations are performed with the best approximate solution occurring at the 81st iteration. the result of the implementation of the algorithm based on lee’s approximation is given in table 1 along with its regularized solution. table 1 shows the solutions corresponding to the optimal regularization parameter of the two methods. table 1: modified newton’s method(mnm) vrs regularization method(rm) with quadratic approximation method µ α̂ ‖αexact − α̂‖ mnm 10−16 1.000000680934881 0.999975996291894 1.000210550708668 0.999239773222684 1.001312540621827 0.998920937855809 1.000339707272387 1.31254062182729e− 003 rm 10−30 0.999999999998873 1.000000000038512 0.999999999666076 1.000000001202600 0.999999997920514 1.000000001715320 0.999999999457774 2.07948647190648e− 009 from table 1, by increasing the value of µ from 10−16, the solution seems to deteriorate. the iterations show that as we move away from the 81st iterate, there is not much difference between the solutions from the 82nd to the 100th iteration. the accuracy in the computed solution of mnm corresponding to µ = 10−16 is just about 3 digits. the loss in the accuracy of the solution is due to the fact that the coefficient matrix xt x of the normal equations is ill-conditioned, with a condition number κ ≈ 2.31648078701200e + 015. int. j. anal. appl. 19 (2) (2021) 271 these results verify the findings of lee et al. (2006) that gives a slow convergence which is due to the ill-conditioning of xt x. in order to overcome the undesirable effects of ill-conditioning, we make use of regularization method. it is found that the best approximate solution for the mnm occurred at µ = 10−16, with the step size β = 2, and at the 81st iteration. for the rm, the best approximate solution occurred at µ = 10−30 with nine digit accuracy. 2.2. sigmoid function approximation. in this section, we consider the sigmoid function approximation to the l1-norm functional. it takes advantage of the non-negative projection operators (x)+ = max(x, 0) and (−x)+ = max(−x, 0). this projection function can be smoothly approximated by the integral of a sigmoid function [1] given as (x)+ ≈ p(x,κ) = x + 1 κ log(1 + e−κx) and (−x)+ ≈ p(−x,κ) = −x + 1 κ log(1 + eκx). the functions p(x,κ) and p(−x,κ) are members of a class of smoothing functions presented in [1]. these smoothing approximations of the projections have been used to transform the standard l1-norm formulation into an efficiently solved unconstrained problem [2]. by combining p(x,κ) and p(−x,κ), we obtain the identity |x| = (x)+ + (−x)+ where (x)+ + (−x)+ are the left and right parts of abs(x) = |x| . we arrive at a smoothing approximation for the absolute value function that consists of the sum of the integral of two sigmoid functions given by |x| ≈ (x)+ + (−x)+ = p(x,κ) + p(−x,κ) = 1 κ [log(1 + e−κx) + log(1 + eκx)] def = |x|κ a graph of different values of the parameter κ in approximating the absolute value function is given in figure 2. int. j. anal. appl. 19 (2) (2021) 272 figure 2. sigmoid approximations of |x|κ for various values of κ. figure 2 indicates that |x|κ →|x| as κ →∞. that is, lim κ→∞ |x|κ = |x| . thus, it would be suitable to choose κ = 1000 for the subsequent implementation. given the smoothing approximation, the gradient ∇(|x|κ) and the hessian ∇ 2(|x|κ) in single variable form is derived as follows: |x|κ = 1 κ [ log(1 + e−κx) + log(1 + eκx) ] ∇(|x|κ) = 1 κ [ −κe−κx 1 + e−κx + κeκx 1 + eκx ] = (1 + eκx) − (1 + e−κx) (1 + e−κx)(1 + eκx) = (1 + eκx) (1 + e−κx)(1 + eκx) − (1 + e−κx) (1 + e−κx)(1 + eκx) = 1 1 + e−κx − 1 1 + eκx . therefore, ∇(|x|κ) = (1 + e −κx)−1 − (1 + eκx)−1, and ∇2(|x|κ) = (1 + e −κx)−2κe−κx + (1 + eκx)−2κeκx = κeκx (1 + eκx)2 [ (1 + eκ)2 (1 + e−κx)2 (e−κx)2 + 1 ] = κeκx (1 + eκx)2 [(e−κx(1 + eκx)2) (1 + e−κx)2 + 1 ] int. j. anal. appl. 19 (2) (2021) 273 = κeκx (1 + eκx)2 [(e−κx + 1)2 (1 + e−κx)2 + 1 ] = 2κeκx (1 + eκx)2 . therefore, ∇2(|x|κ) = 2κeκx (1 + eκx)2 . for x ∈<p, ‖x‖1 = p∑ i=1 |xi| ≈ p∑ i=1 |xi|κ . the loss function in (1.2) therefore becomes (2.4) g(α) = ‖xα− y‖22 + λ p∑ i 1 κ [ log(1 + e−καi ) + log(1 + eκαi ) ] . which is now differentiable. the gradient of g(α) is given as ∇g(α,κ) = 2xt (xα− y) + λ p∑ i=1 (1 + e−καi )−1 − (1 + eκαi )−1. a linear approximation to ∇(|α|κ) = (1 + e −κα)−1 − (1 + eκα)−1 is obtained as k(α) = 1 2 ( κα + (κα)3 3! + · · · ) after some expansion and simplification. by ignoring terms of higher order, we obtain the linear approximation k(α) = 1 2 κα. using this linear approximation and equating ∇g(α) to zero, we solve the equation to obtain xt xα + 1 2 λk(α) = xt y, which gives αµ = (x t x + µi)−1xt y where µ = 1 4 λκ. using the linear approximation for k(α), we obtain the minimization of g(α) as xt xα + 1 4 λκα = xt y. thus, (2.5) αµ = ( xt x + µi )−1 xt y, int. j. anal. appl. 19 (2) (2021) 274 where µ = 1 4 λκ. therefore, the singular value decomposition solution in component form is given as αµ = p∑ i=1 σi σ2i + µ (uti y)vi. to implement the modified newton’s method for sigmoid approximation, we want to find α ∈ <p such that xα = y. from (2.4), the gradient of g(α) is given by ∇g(α) = 2xt (xα− y) + λg(α) where g(α) = [ (1 + e−κα1 )−1 − (1 + eκα1 )−1, · · · , (1 + e−καp )−1 − (1 + eκαp )−1 ]t and hessian ∇2(g(α)) = 2xt x + 2κλ h(α), where h(α) = diag [ eκα1 (1 + eκα1 )2 , eκα2 (1 + eκα2 )2 , · · · , eκαp (1 + eκαp )2 ] . in the implementation, we define κ = 300, βk = β = 3 and set initial guess α0 = 0.25∗ones(7, 1). we also initialize α := ones(7, 1). numerical simulations are performed with the best approximate solution occurring at the 84th iterate. the results of the implementation based on sigmoid approximation is given in table 2 for the modified newton’s method and the regularization method. table 2: modified newton’s method(mnm) vrs regularization method(rm) with sigmoid approximation method µ α̂ ‖αexact − α̂‖ mnm 10−16 1.000005092022318 0.999820688361910 1.001571753611182 0.994327784550849 1.009789408764376 0.991954381867874 1.002532280742229 9.78940876437595e− 003 rm 10−30 0.999999999998873 1.000000000038512 0.999999999666076 1.000000001202600 0.999999997920514 1.000000001715320 0.999999999457774 2.07948647190648e− 009 from table 2, the accuracy in the computed solution of mnm corresponding to µ = 10−16 is just about 3 digits. the accuracy in that of the rm is up to about 9 digits. int. j. anal. appl. 19 (2) (2021) 275 2.3. cubic hermite approximation. the cubic hermite approximation is a spline where each piece is a third-degree polynomial specified in hermite form: that is, by its values and first derivatives at the end points of the corresponding domain interval. the hermite form of a cubic polynomial defines the polynomial p(x) by specifying two distinct points [−γ, γ], and providing values for the following four equations   0 1 2γ 3γ2 0 1 −2γ 3γ2 1 γ γ2 γ3 1 −γ γ2 −γ3     a0 a1 a2 a3   =   1 −1 γ γ   .(2.6) solving for the unknown parameters in equation (2.6), gives a0 = γ 2 , a1 = 0, a2 = 1 2γ , a3 = 0. therefore, p(x) = γ 2 + 1 2γ x2. to determine the best approximate solution, we first examine the nature of the plot of the absolute value function for various values of γ. the graph of the abs(x) is given in figure 3 for various values of the parameter γ. figure 3. cubic hermite approximation of |x|γ for various values of approximating parameter, γ. figure 3 indicates that lim γ→0 |x|γ = |x| . int. j. anal. appl. 19 (2) (2021) 276 the implementation shows that γ = 0.05 is a more suitable choice. thus, the scalar cubic hermite approximation to the absolute value function is given as |x|γ ≈ γ 2 + 1 2γ x2. the gradient ∇(|x|γ) and the hessian ∇ 2(|x|γ) are derived as follows: ∇(|x|γ) = x γ and ∇2(|x|γ) = 1 γ . for x ∈<p, ‖x‖1 = p∑ i=1 |xi| ≈ p∑ i=1 |xi|γ . the loss function in equation (1.2) therefore becomes, (2.7) g(α) = ‖xα− y‖22 + λ p∑ i (γ 2 + 1 2γ α2i ) , which is now differentiable. the gradient of g(α) is given as ∇g(α) = 2xt (xα− y) + λg(α), where g(α) = (α1 γ , α2 γ , · · · , αp γ )t . equating ∇g(α) to zero and solving gives αµ = (x t x + µi)−1xt y, where µ = 1 2γ λ. therefore, the singular value decomposition solution in component form is given as αµ = p∑ i=1 σi σ2i + µ (uti y)vi. to implement the modified newton’s method for cubic hermite approximation, we want to find α ∈<p such that xα = y. now, the hessian of g(α) is ∇2(g(α)) = 2xt x + 1 γ λip. in the implementation, we define γ = 0.05, βk = β = 3 and set initial guess α0 = 0.25∗ones(7, 1). we also initialize α := ones(7, 1). numerical simulations are performed with the best approximate solution occurring at the 87th iterate. the result of the implementation based on cubic hermite approximation is given in table 3 for the modified newton’s method and the regularization method. int. j. anal. appl. 19 (2) (2021) 277 table 3: modified newton’s method(mnm) vrs regularization method(rm) method µ α̂ ‖αexact − α̂‖ mnm 10−16 0.999983772069521 1.000576215743305 0.994919717190969 1.018414062557087 0.968111442843396 1.026280481466755 0.991709642462265 3.18885571566037e− 002 rm 10−30 0.999999999998873 1.000000000038512 0.999999999666076 1.000000001202600 0.999999997920514 1.000000001715320 0.999999999457774 2.07948647190648e− 009 we note from tables 1, 2 and 3 that the mnm solutions vary for the three smoothing approximations considered. however, the regularized solutions are the same for each of the three smoothing approximations at µ = 10−30. we now compare the three smoothing approximations using regularization with a non-smooth method which makes use of truncated newton interior-point method described in [4]. in that paper, they developed a matlab solver for large-scale l1-regularized least squares problems called l1 ls. using our value of the parameter µ = 10−30 in the l1 ls, we display in table 4 the result of all three regularization methods (rm) and that of the l1 ls. table 4 : summary of methods and their solutions at µ = 10−30 method solution corresponding to µ = 10−30 ‖α− α̂‖ rm 0.999999999998873 1.000000000038512 0.999999999666076 1.000000001202600 0.999999997920514 1.000000001715320 0.999999999457774 2.07948647190648e− 009 l1 ls 1.000000000000292 0.999999999990242 1.000000000081881 0.999999999715937 1.000000000472646 0.999999999624568 1.000000000114463 4.72645700355656e− 010 int. j. anal. appl. 19 (2) (2021) 278 from table 4, it is seen that the solutions from the three smoothing approximations by regularization method is as good as the non-smooth method, which is accurate to about 10 digits. 3. conclusion we have considered three different methods for approximating the absolute value function, which is nondifferentiable and used in the l1-norm minimization problem. under each of these three methods namely, the quadratic, sigmoid and cubic hermite approximations, we have obtained optimal approximate solutions by means of the newton’s method and by regularization. it is observed that the results of the newton’s method under all three methods show visible differences and produce solutions that are accurate only to at most 3 digits. however, the regularized solution using smoothing approximations produce almost the same results that are accurate to 9 digits at the same parameter value of µ = 10−30. this value of µ is obviously much smaller than the usual machine representation of 10−16 for 0. the results obtained are as good as the non-smooth methods used in developed solvers. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] c. chen, o.l. mangasarian, a class of smoothing functions for nonlinear and mixed complementarity problems, comput. optim. appl. 5 (1996), 97–138. [2] s.i. lee, h. lee, p. abbeel, a.y. ng, efficient l1 regularized logistic regression. https://www.aaai.org/papers/aaai/ 2006/aaai06-064.pdf (2006). [3] a. neumaier, solving ill-conditioned and singular linear systems: a tutorial on regularization, siam rev. 40 (1998), 636–666. [4] k. koh, s.-j. kim, s. boyd, an interior-point method for l1-regularized logistic regression, j. mach. learn. res. 8 (2007), 1519-1555. [5] w.j. fu, penalized regressions: the bridge versus the lasso, j. comput. graph. stat. 7 (1998), 397–416. [6] d.l. donoho, denoising via soft-thresholding, ieee trans. info. theory, 41 (1995), 613–627. [7] d.l. donoho, i.m. johnstone, ideal spatial adaptation by wavelet shrinkage, biometrika. 81 (1994), 425–455. [8] b. efron, the estimation of prediction error: covariance penalties and cross-validation, j. amer. stat. assoc. 99 (2004), 619–632. [9] b. efron, t. hastie, i. johnstone, r. tibshirani, least angle regression, ann. stat. 32 (2004), 407–499. [10] m. schmidt, g. fung, r. rosales, fast optimization methods for l1 regularization: a comparative study and two new approaches, in: j.n. kok, j. koronacki, r.l. de mantaras, s. matwin, d. mladenič, a. skowron (eds.), machine learning: ecml 2007, springer berlin heidelberg, berlin, heidelberg, 2007: pp. 286–297. [11] j.a. tropp. just relax: convex programming methods for subset selection and sparse approximation. ieee trans. inform. theory, 51 (3) (2006), 1030–1051. [12] b.a. turlach, shape constrained smoothing using smoothing splines, comput. stat. 20 (2005), 81–104. [13] b.t. polyak, introduction to optimization. optimization software inc. publication division, new york, 1987. https://www.aaai.org/papers/aaai/2006/aaai06-064.pdf https://www.aaai.org/papers/aaai/2006/aaai06-064.pdf int. j. anal. appl. 19 (2) (2021) 279 [14] y. nesterov, a method of solving a convex programming problem with convergence rate o(1/k2), sov. math. dokl. 27 (2) (1983), 372-376. [15] s.-j. kim, k. koh, m. lustig, s. boyd, d. gorinevsky, an interior-point method for large-scale l1-regularized least squares, ieee j. sel. top. signal process. 1 (2007), 606–617. [16] p. zhao, b. yu, on model selection consistency of lasso. j. mach. learn. res. 7 (2006), 2541-2563. 1. introduction 1.1. background 2. smoothing approximations 2.1. quadratic approximation 2.2. sigmoid function approximation 2.3. cubic hermite approximation 3. conclusion references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 69-76 http://www.etamaths.com sufficient conditions for the oscillation of odd-order neutral dynamic equations başak karpuz∗ abstract. in this paper, we study oscillation and asymptotic behaviour of odd-order delay dynamic equations. we first state an oscillation test for odd-order nonneutral equations, then by comparison we provide sufficient conditions for all solutions of neutral equations to be oscillatory or tend to zero depending on two main ranges of the neutral coefficient. 1. introduction in this paper, we will study the oscillation of solutions to the higher-order delay dynamic equations of the form [ x(t) + a(t)x ( α(t) )]∆n + b(t)x ( β(t) ) = 0 for t ∈ [t0,∞)t, (1.1) where n ∈ n, t is a time scale unbounded above, t0 ∈ t, a ∈ crd([t0,∞)t,r), b ∈ crd([t0,∞)t, [0,∞)r), and α,β ∈ crd([t0,∞)t,t) are unbounded nondecreasing functions such that α(t),β(t) ≤ t for all t ∈ [t0,∞)t. we also assume when necessary that α ∈ c([t0,∞)t,t) has an inverse α−1 ∈ c([α(t0),∞)t, [t0,∞)t). we will confine our attention to the following ranges of the coefficient a. (r1) a ∈ crd([t0,∞)t, [0, 1]r) with lim supt→∞a(t) < 1. (r2) a ∈ crd([t0,∞)t, [−1, 0]r) with lim inft→∞a(t) > −1. the oscillation problem for dynamic equations on time scales has attracted a lot of attention immediately after the discovery of time scale calculus. although there are several such works in the literature, the majority is focuses on second-order equations. an important reason for this is due to lack of technical inequalities which combines higher-order derivatives with lower-order ones. in this paper, we will generalize and improve the technique by das [7] employed for differential equations. for some works on higher-order dynamic equations we refer the readers to [8–14, 17, 18, 20, 22, 23]. we will be giving comparison tests for the oscillation of higher-order delay dynamic equations with first-order delay dynamic equations. as we will be making comparison with first-order delay dynamic equations, we find useful to cite the following references [2, 3, 6, 16, 19, 24], where the authors study oscillation of all solutions of first-order delay dynamic equations. to give an exact definition of a solution of delay dynamic equation (1.1), we need to define t−1 := min{α(t0),β(t0)}. definition 1.1 (solution). a function x : [t−1,∞)t → r, which is rd-continuous on [t−1, t0]t and x+a·x◦α is n-times rd-continuously ∆-differentiable on [t0,∞), is called a solution of (1.1) provided that it satisfies the functional delay equation (1.1) identically on [t0,∞). it can be shown as in [15] that (1.1) admits a unique solution, which exists on the entire interval [t−1,∞)t, when an initial function ϕ : [t−1, t0]t → r, which is n-times rd-continuously ∆-differentiable, is provided. more precisely, we mean in the equation that x∆ j (t) = ϕ∆ j (t) for t ∈ [t−1, t0]t and j = 0, 1, · · · ,n. definition 1.2 (oscillation). a solution x of (1.1) is called nonoscillatory if there exists s ∈ [t0,∞)t such that x is either positive or negative on [s,∞). otherwise, the solution is said to oscillate (or is called oscillatory). received 12th january, 2017; accepted 22nd march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. primary: 34n05; secondary: 34k11, 39a12, 39a21. key words and phrases. oscillation; odd-order; delay dynamic equations. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 69 70 karpuz this paper is constructed in the following setting. § 2 includes some fundamental results on time scale polynomials and nonoscillatory functions (§ 2.1). in this section (§ 2.2), we also quote some results from the recent paper [17]. in § 3.1, we provide some comparison theorems on the qualitative behaviour of higher-order delay dynamic equations when the neutral term is absent (i.e., a(t) ≡ 0 for t ∈ [t0,∞)t), and in § 3.2, we extend these results to higher-order neutral delay dynamic equations together with some illustrative general examples. 2. auxiliary lemmas in this section, we will form the background for the proof of our main result. 2.1. time scales. this section is dedicated to some general results on time scales. in the sequel, we introduce the definition of the generalized polynomials on time scales (see [1, lemma 5] and/or [4, § 1.6]) hk ∈ c(t×t,r) as follows: hk(t,s) :=   1, k = 0∫ t s hk−1(η,s)∆η, k ∈ n for s,t ∈ t. (2.1) note that, for all s,t ∈ t and all k ∈ n0, the function hk satisfies h∆1k (t,s) = { 0, k = 0 hk−1(t,s), k ∈ n for s,t ∈ t. (2.2) in particular, for t = z, we have hk(t,s) = (t− s)(k)/k! for all s,t ∈ z and k ∈ n0, where (·) is the usual factorial function, and for t = r, we have hk(t,s) = (t−s)k/k! for all s,t ∈ r and k ∈ n0. property 2.1 ([14, property 1]). by using induction and (2.1), it is easy to see for all k ∈ n0 that hk(·,s) ≥ 0 on [s,∞)t and (−1)khk(·,s) ≥ 0 on (−∞,s]t. in view of (2.2), for all k ∈ n, hk(·,s) is increasing on [s,∞)t, and (−1)khk(·,s) is decreasing on (−∞,s]t. below, we give two lemmas related to the generalized polynomials on time scales. lemma 2.1 (taylor’s formula [4, theorem 1.113]). if n ∈ n, s ∈ t and f ∈ cnrd(t,r), then f(t) = n−1∑ k=0 hk(t,s)f ∆k (s) + ∫ t s hn−1 ( t,σ(η) ) f∆ n (η)∆η for t ∈ t. lemma 2.2 ([17, lemma 2]). if k ∈ n, ` ∈ n0 and s ∈ t, then hk+`(t,s) = ∫ t s hk−1 ( t,σ(η) ) h`(η,s)∆η for t ∈ t. as an immediate consequence of lemma 2.2, we can give the following alternative definition of the generalized polynomials: hk(t,s) :=   1, k = 0∫ t s hk−1 ( t,σ(η) ) ∆η, k ∈ n for s,t ∈ t. (2.3) the following is the main tool for studying qualitative properties of higher-order dynamic equations. lemma 2.3 (kiguradze’s lemma [1, theorem 5]). let sup t = ∞, n ∈ n and f ∈ cnrd([t0,∞)t,r + 0 ). suppose that either f∆ n ≥ 0(6≡ 0) or f∆ n ≤ 0( 6≡ 0) on [t0,∞)t. then, there exist s ∈ [t0,∞)t and m ∈ [0,n)z such that (−1)n−mf∆ n (t) ≥ 0 for all t ∈ [s,∞)t. moreover, the following assertions hold. (i) f∆ k (t) > 0 holds for all t ∈ [s,∞)t and all k ∈ [0,m)z. (ii) (−1)m+kf∆ k (t) > 0 holds for all t ∈ [s,∞)t and all k ∈ [m,n)z. a basic result on higher-order derivatives of a function is quoted below. lemma 2.4 ( [1, lemma 7]). if sup t = ∞, n ∈ n and f ∈ cnrd([t0,∞)t,r), then the following conditions are true. (i) lim inft→∞f ∆n (t) > 0 implies limt→∞f ∆k (t) = ∞ for all k ∈ [0,n)z. oscillation of odd-order neutral dynamic equations 71 (ii) lim supt→∞f ∆n (t) < 0 implies limt→∞f ∆k (t) = −∞ for all k ∈ [0,n)z. the following can be easily obtained from the previous result. corollary 2.1 ([10, corollary 2.10]). if sup t = ∞, n ∈ n and f ∈ cnrd([t0,∞)t,r + 0 ), then lim t→∞ f∆ k (t) = 0, ∀k ∈ (m,n)z, where m ∈ [0,n)z is the key number in kiguradze’s lemma. the following result is the key tool of this paper. lemma 2.5 (cf. [7, lemma 1]). assume that sup t = ∞, n ∈ n be odd and f ∈ cnrd([t0,∞)t,r + 0 ) with f∆ n ≤ 0( 6≡ 0) on [t0,∞)t. if kiguradze’s lemma holds with m ∈ [0,n)z, then f(r) ≥ hm(r,s)hn−m−1(r,t)f∆ n−1 (t) for all t ∈ [r,∞)t, (2.4) where r ∈ [s,∞)t. proof. by lemma 2.3 (i), r ∈ [s,∞)t implies f∆ j (r) > 0 for all j ∈ [0,m)z and f∆ m+1 (r) ≤ 0, where m ∈ [0,n)z is even. using property 2.1, lemma 2.2 and taylor’s formula, we get f(r) = m−1∑ k=0 hk(r,s)f ∆k (s) + ∫ r s hm−1 ( r,σ(η) ) f∆ m (η)∆η ≥ (∫ r s hm−1 ( r,σ(η) ) ∆η ) f∆ m (t) = hm(r,s)f ∆m (r) (2.5) for all r ∈ [s,∞)t. by lemma 2.3 (i), t ∈ [r,∞)t implies (−1)m+kf∆ k (t) > 0 for all k ∈ [m,n)z and f∆ n (t) ≤ 0. using property 2.1, (2.3) and taylor’s formula, we get f∆ m (r) = n−m−1∑ k=0 hk(r,t)f ∆m+k (t) + ∫ r t hn−m−2 ( r,σ(η) ) f∆ n−1 (η)∆η = n−m−1∑ k=0 (−1)khk(r,t)(−1)m+kf∆ m+k (t) + ∫ t r (−1)n−m−2hn−m−2 ( r,σ(η) ) f∆ n−1 (η)∆η ≥ (∫ t r (−1)n−m−2hn−m−2 ( r,σ(η) ) ∆η ) f∆ n−1 (t) = (∫ r t hn−m−2 ( r,σ(η) ) ∆η ) f∆ n−1 (t) = hn−m−1(r,t)f ∆n−1 (t) (2.6) for all t ∈ [r,∞)t. using (2.5) and (2.6), we arrive at (2.4), which completes the proof. � 2.2. recent results. below, we quote two fundamental results from [17], which will be required in the sequel. theorem 2.2 ( [17, theorem 2 (ii)]). assume that (r1) holds and n ∈ n is odd. if (1.1) has a nonoscillatory solution x with lim supt→∞ |x(t)| > 0, then so does x∆ n (t) + [ 1 −a ( α(t) )] b(t)x ( β(t) ) = 0 for t ∈ [t0,∞)t. theorem 2.3 ( [17, theorem 3]). assume that (r2) holds and n ∈ n. if (1.1) has a nonoscillatory solution x with lim supt→∞ |x(t)| > 0, then so does x∆ n (t) + b(t)x ( β(t) ) = 0 for t ∈ [t0,∞)t. 3. main results now, we can study oscillation and asymptotic behaviour of higher-order delay dynamic equations. 72 karpuz 3.1. nonneutral equations. in this section, we will focus on the higher-order delay dynamic equation x∆ n (t) + b(t)x ( β(t) ) = 0 for t ∈ [t0,∞)t. (3.1) our first comparison result is the following, which presents results on oscillation. theorem 3.1. assume that n ∈ n is odd with n ≥ 3, and there exist two functions δ,γ ∈ crd([t0,∞)t,t) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that x∆(t) + b(t) min k∈[0,n)z k=even { hk ( β(t),δ(t) ) hn−k−1 ( β(t),γ(t) )} x ( γ(t) ) = 0 for t ∈ [t0,∞)t (3.2) is oscillatory. then, every solution of (3.1) is oscillatory. proof. assume the contrary that x is an eventually positive solution of (3.1). then, there exists t1 ∈ [t0,∞)t such that x(t), x ( β(t) ) > 0 for all t ∈ [t1,∞)t. by kiguradze’s lemma, there exist t2 ∈ [t1,∞)t and an even integer m ∈ [0,n)z such that for all t ∈ [t2,∞)t, we have x∆ k (t) > 0 for all k ∈ [0,m)z and (−1)m+kx∆ k (t) > 0 for all k ∈ [m,n)z. in particular, we have x∆ n−1 > 0 on [t2,∞)t. it follows from lemma 2.5 that x ( β(t) ) ≥ hm ( β(t),δ(t) ) hn−m−1 ( β(t),γ(t) ) x∆ n−1( γ(t) ) for all t ∈ [t3,∞)t, (3.3) where t3 ∈ [t2,∞)t satisfies δ(t) ≥ t2 for all t ∈ [t3,∞)t. substituting (3.3) into (3.2), we see that x∆ n−1 is an eventually positive solution of y∆(t) + b(t)hm ( β(t),δ(t) ) hn−m−1 ( β(t),γ(t) )} y ( γ(t) ) ≤ 0 for all t ∈ [t3,∞)t or y∆(t) + b(t) min k∈[0,n)z k=even { hk ( β(t),δ(t) ) hn−k−1 ( β(t),γ(t) )} y ( γ(t) ) ≤ 0 for all t ∈ [t3,∞)t. (3.4) by [5, theorem 3.1 and corollary 4.2], (3.2) admits an eventually positive solution too. this is a contradiction and the proof is complete. � depending on the oscillation test to be applied for the first-order delay equation, the functions δ and γ can be chosen appropriately as it is illustrated by an example below. before we proceed, let us recall a well-known result due to myškis [21], which ensures that the first-order delay differential equation x′(t) + b(t)x ( β(t) ) = 0 for t ∈ [t0,∞)r is oscillatory if lim sup t→∞ ( t−β(t) ) < ∞ and lim inf t→∞ ( t−β(t) ) lim inf t→∞ b(t) > 1 e . now, we can give a simple example to explain how to choose appropriately the functions δ and γ. example 3.1. let t = r, and consider the following odd-order delay differential equation x(n)(t) + b(t)x(t−β0) = 0 for t ∈ [t0,∞)r, (3.5) where n ≥ 3 is an odd integer, β0 ∈ r+ and b ∈ c([t0,∞)r,r+0 ). choosing δ(t) = t − δ0 and γ(t) = t−γ0, where δ0 ≥ β0 ≥ γ0, we get min k∈[0,n)z k=even {( (t−β0) − (t−δ0) )k k! ( (t−β0) − (t−γ0) )n−k−1 (n−k − 1)! } = min k∈[0,n)z k=even { (δ0 −β0)k(β0 −γ0)n−k−1 k!(n−k − 1)! } = ( min{δ0 −β0,β0 −γ0} )n−1 (n− 1)! . the last expression is maximized if δ0 − β0 = β0 − γ0. so, by letting δ0 = β0 + α and γ0 = β0 − α, where β0 ≥ α ≥ 0, we obtain the dynamic equation x′(t) + b(t)αn−1 (n− 1)! x ( t− (β0 −α) ) = 0 for t ∈ [t0,∞)t. (3.6) oscillation of odd-order neutral dynamic equations 73 due to the result by myškis (quoted above), every solution of ( (3.6) and hence) (3.5) is oscillatory if lim inf t→∞ b(t)αn−1(β0 −α) (n− 1)! > 1 e . (3.7) now, define the function f(λ) = λn−1(β0 −λ) for β0 ≥ λ ≥ 0. we compute that f′(λ) = λn−2 ( β0(n− 1) −nλ ) . then, f′ (n− 1 n β0 ) = 0 and f′′ (n− 1 n β0 ) = −n ( n− 1 n β0 )n−2 < 0, which yields max 0≤λ≤β0 { f(λ) } = f (n− 1 n β0 ) = (n− 1)n−1 nn βn0 . therefore, the oscillation condition (3.7) is optimized as lim inf t→∞ b(t) > n! eβn0 ( n n− 1 )n−1 . 3.2. neutral equations. in the previous section, we have stated some oscillation conditions for nonneutral equations, however, for neutral equations this phrase replaces with the so-called “almost oscillation”, i.e., every solution oscillates or tends to zero at infinity. our main tools in this section will be the comparison tests theorem 2.2 and theorem 2.3, and the oscillation test theorem 3.1. our first result for neutral equations investigates (1.1) when the neutral coefficient is in the range (r1). theorem 3.2. assume that n ∈ n is odd with n ≥ 3 and (r1) holds. further, assume that there exist two functions δ,γ ∈ crd([t0,∞)t,t) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that x∆(t) + [1 −a ( β(t) ) ]b(t) min k∈[0,n)z k=even { hk ( β(t),δ(t) ) hn−k−1 ( β(t),γ(t) )} x ( γ(t) ) = 0 for [t0,∞)t (3.8) is oscillatory. then every solution of (1.1) is oscillatory or tends to zero asymptotically. proof. the proof follows from theorem 2.2 and theorem 3.1. � as an immediate consequence of the theorem above, we can give the following corollary by combining theorem 3.2 by [2, theorem 1] and [19, theorem 2]. corollary 3.1. assume that n ∈ n is odd with n ≥ 3 and (r1) holds. if there exist two functions δ,γ ∈ crd([t0,∞)t,t) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that lim inf t→∞ inf −λϕ∈r+([γ(t),t)t) λ>0 { 1 λe−λϕ ( t,γ(t) )} > 1 or lim inf t→∞ ∫ t γ(t) ϕ(η)∆η > m and lim sup t→∞ ∫ σ(t) γ(t) ϕ(η)∆η > 1 − ( 1 − √ 1 −m )2 , where ϕ(t) := [1 −a ( β(t) ) ]b(t) min k∈[0,n)z k=even { hk ( β(t),δ(t) ) hn−k−1 ( β(t),γ(t) )} for t ∈ [t0,∞)t, then every solution of (1.1) oscillates or tends to zero asymptotically. our next result treats (1.1) when the neutral coefficient is in the range (r2). theorem 3.3. assume that n ∈ n is odd with n ≥ 3 and (r2) holds. further, assume that there exist two functions δ,γ ∈ crd([t0,∞)t,t) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that (3.2) is oscillatory. then, every solution of (1.1) is oscillatory or tends to zero asymptotically. proof. the proof follows from theorem 2.3 and theorem 3.1. � 74 karpuz corollary 3.2. assume that n ∈ n is odd with n ≥ 3 and (r2) holds. if there exist two functions δ,γ ∈ crd([t0,∞)t,t) satisfying δ(t) ≤ β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that lim inf t→∞ inf −λψ∈r+([γ(t),t)t) λ>0 { 1 λe−λψ ( t,γ(t) )} > 1 or lim inf t→∞ ∫ t γ(t) ψ(η)∆η > m and lim sup t→∞ ∫ σ(t) γ(t) ψ(η)∆η > 1 − ( 1 − √ 1 −m )2 , where ψ(t) := b(t) min k∈[0,n)z k=even { hk ( β(t),δ(t) ) hn−k−1 ( β(t),γ(t) )} for t ∈ [t0,∞)t, then every solution of (1.1) oscillates or tends to zero asymptotically. consider now the following additional assumptions∫ ∞ t0 b(η)∆η = ∞ (3.9) and lim sup t→∞ a(t) < 0 (3.10) under which we will state an oscillation test for (1.1). theorem 3.4. assume that n ∈ n is odd with n ≥ 3, (3.9) and (r2) hold with (3.10). further, assume that there exist k0 ∈ n and a function γ ∈ crd([t0,∞)t,t) satisfying β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that x∆(t) + k0−1∑ `=0 [ `−1∏ ν=0 a ( αν(t) )] b(t)hn−1 ( β(t),γ(t) ) x ( γ(t) ) = 0 for t ∈ [t0,∞)t (3.11) is oscillatory. then, every solution of (1.1) is oscillatory. proof. assume the contrary that x is an eventually positive solution of (1.1). then, there exists t1 ∈ [t0,∞)t such that x(t), x ( α(t) ) , x ( β(t) ) > 0 for all t ∈ [t1,∞)t. set y(t) := x(t) + a(t)x ( α(t) ) for t ∈ [t1,∞)t. (3.12) therefore, we have y∆ n (t) + b(t)x ( β(t) ) = 0 for t ∈ [t1,∞)t (3.13) showing that for j ∈ [0,n)z the functions y∆ j are monotonic on [t2,∞)t for some t2 ∈ [t1,∞)t. (c1) let y > 0 on [t2,∞)t. by kiguradze’s lemma, we learn that there exist t2 ∈ [t1,∞)t and an even integer m ∈ [0,n)z such that t ∈ [t2,∞)t implies y∆ k (t) > 0 for all k ∈ [0,m)z and (−1)m+ky∆ k (t) > 0 for all k ∈ [m,n)z. (a) let m ∈ [2,n)z, i.e., y∆ > 0 on [t2,∞)t. therefore, limt→∞y(t) > 0. we also have x ≥ y on [t2,∞)t, i.e., lim inft→∞x(t) > 0. this implies by (3.9) that limt→∞y∆ n−1 (t) = −∞, which yields limt→∞y(t) = −∞. (b) let m = 0, i.e., y∆ < 0 on [t2,∞)t. then, by recursively substituting x into (3.12), we get for k ∈ n and all t ∈ [t3,∞)t that x(t) = k−1∑ `=0 [ `−1∏ ν=0 a ( αν(t) )] y ( α`(t) ) + [ k−1∏ ν=0 a ( αν(t) )] x ( αk(t) ) ≥ k−1∑ `=0 [ `−1∏ ν=0 a ( αν(t) )] y(t), where αk(t) ≥ t2 for all t ∈ [t3,∞)t. the rest of the proof follows from the proofs of [17, theorem 3] and theorem 3.1. oscillation of odd-order neutral dynamic equations 75 (c2) let y < 0 on [t2,∞)t. in this case, we see that x(t) ≤ a(t)x ( α(t) ) for all t ∈ [t2,∞)t, which implies boundedness of the function x. therefore, y is bounded too. applying kiguradze’s lemma for the function (−y), we learn that there exist t2 ∈ [t1,∞)t and an odd integer m ∈ [0,n)z such that t ∈ [t2,∞)t implies y∆ k (t) < 0 for all k ∈ [0,m)z and (−1)m+ky∆ k (t) < 0 for all k ∈ [m,n)z. in particular, we have y∆ < 0 on [t2,∞)t, i.e., limt→∞y(t) < 0. then, we obtain y(t) ≥−a(t)x ( α(t) ) for all t ∈ [t2,∞)t, which yields x(t) ≥− y ( α−1(t) ) a ( α−1(t) ) for all t ∈ [t2,∞)t. thus, we have lim inft→∞x(t) > 0 by (r2). proceeding as in the case (c1a), we obtain limt→∞y(t) = −∞, which contradicts the boundedness of y. thus, the proof is complete. � corollary 3.3. assume that n ∈ n is odd with n ≥ 3, (3.9) and (r2) hold. if there exist k0 ∈ n and a function γ ∈ crd([t0,∞)t,t) satisfying β(t) ≤ γ(t) ≤ t for all t ∈ [t0,∞)t such that lim inf t→∞ inf −λψk0∈r +([γ(t),t)t) λ>0 { 1 λe−λψk0 ( t,γ(t) )} > 1 or lim inf t→∞ ∫ t γ(t) ψk0 (η)∆η > m and lim sup t→∞ ∫ σ(t) γ(t) ψk0 (η)∆η > 1 − ( 1 − √ 1 −m )2 , where ψk(t) := k−1∑ `=0 [ `−1∏ ν=0 a ( αν(t) )] b(t)hn−1 ( β(t),γ(t) ) for t ∈ [t0,∞)t and k ∈ n, then every solution of (1.1) oscillates. remark 3.1. if a is a constant function, i.e., a(t) ≡ −a0, where a0 ∈ (0, 1)r, then ψk0 in corollary 3.3 can be replaced by b(t) 1 −a0 hn−1 ( β(t),γ(t) ) for t ∈ [t0,∞)t. references [1] r. p. agarwal and m. bohner, basic calculus on time scales and some of its applications, results math. 35 (1999), no. 1-2, 3–22. [2] m. bohner, some oscillation criteria for first order delay dynamic equations, far east j. appl. math. 18 (2005), no. 3, 289–304. [3] m. bohner, b. karpuz and ö. öcalan, iterated oscillation criteria for delay dynamic equations of first order, adv. difference equ. 2008 (2008), art. id. 458687. [4] m. bohner and a. c. peterson, dynamic equations on time scales an introduction with applications, birkhäuser boston, inc., boston, ma, 2001. [5] e. braverman and b. karpuz, nonoscillation of first-order dynamic equations with several delays, adv. difference equ. 2010 (2010), art. id. 873459. [6] y. şahiner and i. p. stavroulakis, oscillations of first order delay dynamic equations, dynam. systems appl. 15 (2006), no. 3-4, 645–655. [7] p. das, oscillation criteria for odd order neutral equations, j. math. anal. appl.,188 (1994), no. 1, 245–257. [8] l. erbe, r. mert, a. c. peterson, and a. zafer, oscillation of even order nonlinear delay dynamic equations on time scales, czechoslovak math. j. 63 (2013), no. 138(1), 265–279. [9] l. h. erbe, g. hovhannisyan, and a. c. peterson, asymptotic behavior of n-th order dynamic equations, nonlinear dyn. syst. theory 12 (2012), no. 1, 63–80. [10] l. h. erbe, b. karpuz, and a. c. peterson, kamenev-type oscillation criteria for higher-order neutral delay dynamic equations, int. j. difference equ. 6 (2011), no. 1, 1–16. [11] s. r. grace, r. p. agarwal, and a. zafer, oscillation of higher order nonlinear dynamic equations on time scales, adv. difference equ. 2012 (2012), art. id. 67. [12] s. r. grace, r. mert, and a. zafer, oscillatory behavior of higher-order neutral type dynamic equations, electron. j. qual. theory differ. equ. 2013 (2013), art. id. 29. [13] b. karpuz, asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations, appl. math. comput. 215 (2009), no. 6, 2174–2183. 76 karpuz [14] b. karpuz, unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, electron. j. qual. theory differ. equ. 2009 (2009), art. id. 34. [15] b. karpuz, existence and uniqueness of solutions to systems of delay dynamic equations on time scales, int. j. math. comput. 10 (2011), no. m11, 48–58. [16] b. karpuz, li type oscillation theorem for delay dynamic equations, math. methods appl. sci. 36 (2013), no. 9, 993–1002. [17] b. karpuz, sufficient conditions for the oscillation and asymptotic behaviour of higher-order dynamic equations of neutral type, appl. math. comput. 221 (2013), 453–462. [18] b. karpuz, comparison tests for the asymptotic behaviour of higher-order dynamic equations of neutral type, forum math. 27(2015), no. 5, 2759–2773. [19] b. karpuz and ö. öcalan, new oscillation tests and some refinements for first-order delay dynamic equations, turkish j. math. 40 (2016), no. 4, 850–863. [20] r. mert, oscillation of higher-order neutral dynamic equations on time scales, adv. difference equ. 2012 (2012), art. id. 68. [21] a. d. myškis, linear homogeneous differential equations of the first order with retarded argument, uspehi matem. nauk (n.s.) 5 (1950), no. 2(36), 160–162. [22] x. wu, t. x. sun, h. j. xi, and c. h. chen, kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales, adv. difference equ. 2013 (2013), art. id. 248. [23] j. yang, s. liu, and x. k. hou, oscillation and existence of nonoscillatory solutions of forced higher-order neutral dynamic systems on time scales, pure appl. math. (xi’an) 25 (2009), no. 4, 665–670. [24] b. g. zhang and x. h. deng, oscillation of delay differential equations on time scales, math. comput. modelling 36 (2002), no. 11-13, 1307–1318. dokuz eylül university, tınaztepe campus, faculty of science, department of mathematics, buca, 35160 i̇zmir, turkey. ∗corresponding author: bkarpuz@gmail.com 1. introduction 2. auxiliary lemmas 2.1. time scales 2.2. recent results 3. main results 3.1. nonneutral equations 3.2. neutral equations references international journal of analysis and applications volume 16, number 6 (2018), 868-881 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-868 integral inequalities via generalized geometrically r-convex functions muhammad aslam noor1,∗, khalida inayat noor1 and farhat safdar2 1department of mathematics, comsats university islamabad, islamabad, pakistan 2sardar bahadur khan women’s university, quetta, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we introduce and investigate a new class of generalized convex functions, called generalized geometrically r-convex functions. some new hermite-hadamard integral inequalities via generalized geometrically r-convex functions have been established. results proved in this paper can be viewed as new significant contributions in this area of research. 1. introduction several branches of mathematical and engineering sciences has been developed by using the crucial and significant concepts of convex analysis and hence it becomes one of the most interesting and useful concept of mathematics for last few decades. there are mainly two aspects of the convex functions which have played very important and crucial part in the developments of various branches of pure and applied sciences. first aspect is concerned with differentiable convex functions. it is known that if a function f is differentiable on the convex set k, then the f is a convex function, if and only if, it satisfies the inequality 〈f′(u),v −u〉≥ 0, ∀v ∈ k, (1.1) received 2018-06-10; accepted 2018-08-13; published 2018-11-02. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. generalized convex functions; generalized geometrically r-convex functions; hermite-hadamard type inequalities. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 868 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-868 int. j. anal. appl. 16 (6) (2018) 869 where f′(.) is the frechet differential of f, which is called the variational inequality. variational inequalities were introduced and studied by stamapcchia [35] in potential theory. variational inequalities can be viewed as natural generalization of the variational principles. it is remarkable that a wide class of unrelated problems, which arises in every branch of pure and applied sciences can be studied in the unified and general framework of variational inequalities and their variant forms. for the applications, formulation, dynamical systems, sensitivity analysis, numerical methods, error bounds and other aspects of the variational inequalities and optimization see [3, 4, 8, 16–21, 35]. hermite [13] and hadamard [12] proved that a function f is convex function on the interval [a,b], if and only if, f satisfies the inequality a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 , ∀a,b ∈ [a,b], (1.2) which is known as hermite-hadamard inequality and is one of the most important inequality. in recent years, much attention has been given to derive the hermite-hadamard type inequalities for various types of convex functions, see [6, 11, 14, 15, 22–33]. the concept of convexity has been extended and generalized in several directions using new and innovative techniques, see [1, 2, 3, 5, 7, 11, 12, 13]and the references therein. pearce et. al [11] introduced the class of r-convex functions. several authors have derived hermite-hadamard type inequalities for various classes of r-convex functions, see [22, 28, 31–33]. gordji et al. [9] introduced an important class of convex functions involving the bifunction, which is called generalized(ϕ-convex) convex function. these generalized convex functions are nonconvex functions. for recent developments, see [5, 6, 7, 14, 15, 16, 17, 18, 20, 21, 22] and the references therein. inspired and motivated by the ongoing research in this field, we introduce a new class of generalized convex function, which is known as generalized geometrically r-convex function. we derive some new hermitehadamard integral inequalities for these nonconvex functions. some special cases are discussed, which can be obtained from our new results. using the technique and ideas of this paper, one may obtain hermitehadamard type integral inequalities for other classes of convex functions and their variant forms. 2. preliminaries let i = [a,b] be an interval in real line r. let f : i → r be a continuous function and η(·, ·) : r×r → r be a continuous bifunction. first of all, we have the following well known and new concepts. int. j. anal. appl. 16 (6) (2018) 870 definition 2.1. [9]. a function f : i = [a,b] → r is said to be generalized convex function with respect to a bifunction η(·, ·) : r×r → r, if f((1 − t)a + tb) ≤ (1 − t)f(a) + t[f(a) + η(f(b),f(a))],∀a,b ∈ i,t ∈ [0, 1]. definition 2.2. [32]. a function f : i = [a,b] → r is said to be generalized r-convex function with respect to a bifunction η(·, ·) : r×r → r, if ∀a,b ∈ i, t ∈ [0, 1] f((1 − t)a + tb) ≤   { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r , r 6= 0, [f(a)]1−t[f(a) + η(f(b),f(a))]t, r = 0. if η(f(b) −f(a)) = f(b) −f(a), then, the definition 2.2 reduces to definition 2.3. [11]. a function f : i = [a,b] → r is said to be r-convex function, if f((1 − t)a + tb) ≤   { (1 − t)[f(a)]r + t[f(b)]r }1 r , r 6= 0, [f(a)]1−t[f(b)]t, r = 0. note that for r = 1, we have classical convex functions and for r = 0, we have log-convex functions. definition 2.4. [33]. the set i ⊂ r+ is said to be geometrically convex set, if a1−tbt ∈ i, ∀a,b ∈ i,t ∈ [0, 1]. definition 2.5. [33]. a function f : i ⊂ r+ =(0,∞)→ r is said to be geometrically convex, if f(a1−tbt) ≤ (1 − t)f(a) + t(f(b)), ∀a,b ∈ i,t ∈ [0, 1], we now define a new concept of generalized geometrically r-convex functions. definition 2.6. . a function f : i = [a,b] → r is said to be generalized geometrically r-convex function with respect to a bifunction η(·, ·) : r×r → r, if ∀a,b ∈ i, t ∈ [0, 1] f(a1−tbt) ≤   { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r , r 6= 0, [f(a)]1−t[f(a) + η(f(b),f(a))]t, r = 0. if t = 1 2 , then definition 2.6 reduces to f( √ ab) ≤   { [f(a)]r+[f(a)+η(f(b),f(a))]r 2 }1 r , r 6= 0, √ [f(a)][f(a) + η(f(b),f(a))], r = 0. int. j. anal. appl. 16 (6) (2018) 871 the function f is called generalized geometrically jensen r-convex function. if η(f(b),f(a)) = f(b) −f(a), then definition 2.6 reduces to a new concept. definition 2.7. . a function f : i = [a,b] → r is said to be generalized geometrically r-convex function with respect to a bifunction η(·, ·) : r×r → r, if ∀a,b ∈ i, t ∈ [0, 1] f(a1−tbt) ≤   { (1 − t)[f(a)]r + t[f(b)]r }1 r , r 6= 0, [f(a)]1−t[f(b)]t, r = 0. if t = 1 2 , then f( √ ab) ≤   { [f(a)]r+[f(b)]r 2 }1 r , r 6= 0, √ [f(a)f(b)], r = 0. the function f is called generalized geometrically jensen r-convex function. the generalized logarithmic means of order r of positive numbers a,b is defined by lr(a,b) =   r r+1 ar+1−br+1 ar−br , r 6= 0,−1,a 6= b, a−b log a−log b, r = 0,a 6= b, ab log a−log b a−b , r = −1,a 6= b, a, a = b. definition 2.8. the beta function, also called the euler integral of the first kind, is defined as β(x,y) = ∫ 1 0 tx−1(1 − t)y−1dt = γ(x)γ(y) γ(x + y) , x,y > 0 where γ(.) is a gamma function. 3. main results in this section, we establish several new integral inequalities of hermite-hadamard type for generalized geometrically r-convex functions. theorem 3.1. let f : i → r be generalized geometrically r-convex function on i. then for 0 < r ≤ 1, we have 1 log b− log a ∫ b a 1 x f(x)dx ≤ ( r r + 1 ){( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r . int. j. anal. appl. 16 (6) (2018) 872 proof. let f be generalized geometrically r-convex function on i. then, ∀a,b ∈ i,t ∈ [0, 1], we have f(a1−tbt) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r . using minkowski’s inequality and the fact that f is generalized geometrically r-convex function, we have 1 log b− log a ∫ b a 1 x f(x)dx = ∫ 1 0 f(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r dt ≤ {(∫ 1 0 (1 − t) 1 r [f(a)]dt )r + (∫ 1 0 t 1 r [f(a) + η(f(b),f(a))]dt )r}1 r = {( r r + 1 )r( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r = ( r r + 1 ){( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r , which is the required result. � corollary 3.1. if η(f(b),f(a)) = f(b)−f(a), then, under the assumptions of theorem 3.1, we have a new result. 1 log b− log a ∫ b a 1 x f(x)dx ≤ ( r r + 1 ){( [f(a)]r + [f(b)]r )}1r . theorem 3.2. let f : i → r be generalized geometrically r-convex function on i. then for 0 < r ≤ 1, we have 2[f( √ ab)]r − 1 (log b− log a) ∫ b a [ 1 x f(x) + η(f( ab x ), 1 x f(x))]rdx ≤ 1 log b− log a ∫ b a [f(x)]rdx ≤ { [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r + [f(b) + η(f(a),f(b))]r ) , proof. let f be generalized geometrically r-convex function on i. then, ∀a,b ∈ i,t ∈ [0, 1], we have f(a1−tbt) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r . using (2.1) and substituting x = a1−tbt and y = atb1−t, we have [f( √ ab)]r ≤ { [f(a1−tbt)]r + [f(a1−tbt) + η(f(atb1−t),f(a1−tbt))]r 2 } . int. j. anal. appl. 16 (6) (2018) 873 integrating the above inequality with respect to t on [0,1], we have [f( √ ab)]r ≤ 1 2 ∫ 1 0 { [f(a1−tbt)]r + [f(a1−tbt) +η(f(atb1−t),f(a1−tbt))]r } dt = 1 2(log b− log a) ∫ b a 1 x { [f(x)]r + [f(x) + η(f( ab x ),f(x))]r } dx. this implies 2[f( √ ab)]r − 1 (log b− log a) ∫ b a 1 x [f(x) + η(f( ab x ),f(x))]rdx ≤ 1 (log b− log a) ∫ b a 1 x [f(x)]rdx. (3.1) consider [f(a1−tbt)]r ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r } . (3.2) [f(atb1−t)]r ≤ { (1 − t)[f(b)]r + t[f(b) + η(f(a),f(b))]r } . (3.3) adding (3.2) and (3.3), we have [f(a1−tbt)r + [f(atb1−t)]r ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r } + { (1 − t)[f(b)]r + t[f(b) + η(f(a),f(b))]r } . integrating the above inequality with respect to t on [0,1], we have 2 log b− log a ∫ b a fr(x)dx ≤ ∫ 1 0 { (1 − t) ( [f(a)]r + [f(b)]r ) + t ( [f(a) + η(f(b),f(a))]r +[f(b) + η(f(a),f(b))]r )} dt, which implies that 1 log b− log a ∫ b a [f(x)]rdx ≤ {{ [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r +[f(b) + η(f(a),f(b))]r )} . (3.4) combining (3.1) and (3.4), we have 2[f( √ ab)]r − 1 (log b− log a) ∫ b a [ 1 x f(x) + η(f( ab x ), 1 x f(x))]rdx ≤ 1 log b− log a ∫ b a [f(x)]rdx ≤ { [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r + [f(b) + η(f(a),f(b))]r ) , int. j. anal. appl. 16 (6) (2018) 874 which is the required result. � corollary 3.2. if η(f(b),f(a)) = f(b)−f(a), then, under the assumptions of theorem 3.2, we have a new result. [f( √ ab)]r ≤ 1 log b− log a ∫ b a [f(x)]rdx ≤ { [f(a)]r + [f(b)]r 2 } . theorem 3.3. let f : i → r be generalized geometrically r-convex function on i and r ≥ 0, then 1 log b− log a ∫ b a 1 x f(x)dx ≤   ( r r+1 ){( [f(a)+η(f(b),f(a))]r+1−[f(a)]r+1 ) [f(a)+η(f(b),f(a))]r−[f(a)]r } , r 6= 0 [f(a)+η(f(b),f(a))]−[f(a)] log[f(a)+η(f(b),f(a))]−log[f(a)], r = 0. proof. first, let r > 0 and f be generalized geometrically r-convex function on i. then ∀a,b ∈ i,t ∈ [0, 1], we have 1 log b− log a ∫ b a 1 x f(x)dx = ∫ 1 0 f(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r dt. (3.5) substituting u = [(1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r] in (3.5), we have 1 log b− log a ∫ b a 1 x f(x)dx ≤ 1 [f(a) + η(f(b),f(a))]r − [f(a)]r ∫ [f(a)+η(f(b),f(a))]r [f(a)]r u 1 r du = ( r r + 1 ){( [f(a) + η(f(b),f(a))]r+1 − [f(a)]r+1 ) [f(a) + η(f(b),f(a))]r − [f(a)]r } for r = 0, we have f(a1−tbt) ≤ { [f(a)]1−t[f(a)η(f(b),f(a))]t } . hence we have, 1 log b− log a ∫ b a 1 x f(x)dx = ∫ 1 0 f(a1−tbt)dt ≤ ∫ 1 0 { [f(a)]1−t[f(a) + η(f(b),f(a))]t } dt = [f(a)] ∫ 1 0 { [f(a) + η(f(b),f(a))] [f(a)] }t dt = [f(a) + η(f(b),f(a))] − [f(a)] log[f(a) + η(f(b),f(a))] − log[f(a)] , which is the required result. � int. j. anal. appl. 16 (6) (2018) 875 corollary 3.3. if η(f(b),f(a)) = f(b)−f(a), then, under the assumptions of theorem 3.3, we have a new result. 1 log b− log a ∫ b a 1 x f(x)dx ≤   ( r r+1 ){ [f(b)]r+1−[f(a)]r+1 [f(b)]r−[f(a)]r } = lr ( f(a),f(b) ) , r 6= 0{ [f(b)−f(a)] log[f(b)]−log[f(a)] } = l ( f(a),f(b) ) , r = 0. theorem 3.4. let f : i → r be generalized geometrically r-convex function on i and r ≥ 0, then 1 log b− log a ∫ b a 1 x f(x)dx ≤   f(a),r 6= 0,f(a) = f(b),[ log[f(a)+η(f(b),f(a))]−1−log[f(a)]−1 ] [f(a)+η(f(b),f(a))]−1−[f(a)]−1 ,r = −1. proof. first, let r > 0, f be generalized geometrically r-convex function on i and f(a) = f(b). then ∀a,b ∈ i,t ∈ [0, 1], we have 1 log b− log a ∫ b a 1 x f(x)dx = ∫ 1 0 f(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(a),f(a))]r }1 r dt. = f(a) for r = −1 and f(a) 6= f(b), we have 1 log b− log a ∫ b a 1 x f(x)dx = ∫ 1 0 f(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]−1 + t[f(a) + η(f(b),f(a))]−1 }−1 dt = 1 [f(a) + η(f(b),f(a))]−1 − [f(a)]−1 ∫ [f(a)+η(f(b),f(a))]−1 [f(a)]−1 1 u du = [ log[f(a) + η(f(b),f(a))]−1 − log[f(a)]−1 ] [f(a) + η(f(b),f(a))]−1 − [f(a)]−1 , which is the required result. � corollary 3.4. if η(f(b),f(a)) = f(b)−f(a), then, under the assumptions of theorem 3.4, we have a new result. 1 log b− log a ∫ b a 1 x f(x)dx ≤   f(a),r 6= 0,f(a) = f(b), f(a)f(b) log[f(b)]−log[f(a)] f(b)−f(a) = l−1(f(a),f(b), r = −1. int. j. anal. appl. 16 (6) (2018) 876 theorem 3.5. let f,g : i → r be generalized r1-convex function and generalized r2-convex functions respectively on i. then for r1 > 0,r2 > 0 >, we have 1 log b− log a ∫ b a 1 x f(x)g(x)dx ≤ ( r r + 1 ){( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + ( r r + 1 ){( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 . proof. let f,g : i → r be generalized r1-convex function and generalized r2-convex functions respectively on i with (r1 > 0,r2 > 0). then ∀a,b ∈ i,t ∈ [0, 1], we have f(a1−tbt) ≤ { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 , g(a1−tbt) ≤ { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 . using cauchy’s and minkowski’s inequalities and the fact that f and g are generalized r1 and r2-convex functions, we have 1 log b− log a ∫ b a 1 x f(x)g(x)dx = ∫ 1 0 f(a1−tbt)g(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 dt ≤ 1 2 ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 2 r1 dt + 1 2 ∫ 1 0 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 2 r2 dt ≤ 1 2 {(∫ 1 0 (1 − t) 2 r1 [f(a)]2dt )r1 2 + (∫ 1 0 t 2 r1 [f(a) + η(f(b),f(a))]2dt )r1 2 } 2 r1 + 1 2 {(∫ 1 0 (1 − t) 2 r2 [g(a)]2dt )r2 2 + (∫ 1 0 t 2 r2 [g(a) + η(g(b),g(a))]2dt )r2 2 } 2 r2 int. j. anal. appl. 16 (6) (2018) 877 = {( r r + 1 )r1 2 ( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + {( r r + 1 )r2 2 ( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 = ( r r + 1 ){( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + ( r r + 1 ){( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 , which is the required result. � corollary 3.5. if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.5, we have 1 log b− log a ∫ b a 1 x f(x)g(x)dx ≤ ( r r + 1 ){( [f(a)]r1 + [f(b)]r1 )} 2r1 + ( r r + 1 ){( [g(a)]r2 + [g(b)]r2 )} 2r2 . theorem 3.6. let f,g : i → r be generalized r1-convex function and generalized r2-convex functions respectively on i. then for r1 > 0,r2 > 0 and 1 r1 + 1 r2 = 1, we have 1 log b− log a ∫ b a 1 x f(x)g(x)dx ≤ 1 2 {( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 ) 1 r1 ( [g(a)]r2 + [g(a) + η(g(b),g(a))]r2 ) 1 r2 } . proof. let f,g : i → r be generalized r1-convex function and generalized r2-convex functions respectively on i with (r1 > 0,r2 > 0). then ∀a,b ∈ i,t ∈ [0, 1], we have f(a1−tbt) ≤ { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 g(a1−tbt) ≤ { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 . int. j. anal. appl. 16 (6) (2018) 878 using holder’s inequality and the fact that f and g are generalized r1 and r2-convex functions, we have 1 log b− log a ∫ b a 1 x f(x)g(x)dx = ∫ 1 0 f(a1−tbt)g(a1−tbt)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 dt ≤ {∫ 1 0 (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 dt } 1 r1 {∫ 1 0 (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 dt } 1 r2 = {( [f(a)]r1 ∫ 1 0 (1 − t)dt + [f(a) + η(f(b),f(a))]r1 ∫ 1 0 tdt )} 1 r1 {( [g(a)]r2 ∫ 1 0 (1 − t)dt + [g(a) + η(g(b),g(a))]r2 ∫ 1 0 tdt )} 1 r2 = 1 2 {( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 ) 1 r1 ( [g(a)]r2 + [gr2 (a) + η(g(b),g(a))]r2 ) 1 r2 } , which is the required result. � corollary 3.6. if η(f(b),f(a)) = f(b)−f(a), then, under the assumptions of theorem 3.6, we have a new result. 1 log b− log a ∫ b a 1 x f(x)g(x)dx ≤ {( [f(a)]r1 + [f(b)]r1 ) 1 r1 ( [g(a)]r2 + [g(b)]r2 ) 1 r1 } 2 . theorem 3.7. let f,g : i → r be generalized geometrically r-convex function on i. then for r > 0, we have ( 1 log b− log a ∫ b a 1 x f(x)g(x)dx )r ≤ { m(a,b) ( r r + 2 )r + n(a,b) ( β( 1 r + 1, 1 r + 1) )r} . where m(a,b) = ( [f(a)]r[g(a)]r + [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r ) n(a,b) = ( [f(a)]r[g(a) + η(g(b),g(a))]r + [g(a)]r[f(a) + η(f(b),f(a))]r ) , int. j. anal. appl. 16 (6) (2018) 879 and β(·, ·) is the beta function. proof. let f,g be two generalized geometrically r-convex functions on i. then ∀a,b ∈ i,t ∈ [0, 1], we have f(a1−tbt) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r g(a1−tbt) ≤ { (1 − t)[g(a)]r + t[g(a) + η(g(b),g(a))]r }1 r . using minkowski’s inequality and the fact that f and g are generalized geometrically r-convex functions, we have ( 1 log b− log a ∫ b a 1 x f(x)g(x)dx )r = (∫ 1 0 f(a1−tbt)g(a1−tbt)dt )r ≤ {∫ 1 0 ( (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r (1 − t)[g(a)]r + t[g(a) + η(g(b),g(a))]r )1 r dt }r = {∫ 1 0 ( (1 − t)2[f(a)]r[g(a)]r +t(1 − t) ( [f(a)]r[g(a) + η(g(b),g(a))]r + [g(a)]r[f(a) + η(f(b),f(a))]r ) +t2 ( [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r )}r ≤ { [f(a)]r[g(a)]r (∫ 1 0 (1 − t) 2 r dt )r + ( [f(a)]r[g(a) + η(g(b),g(a))]r +[g(a)]r[f(a) + η(f(b),f(a))]r )(∫ 1 0 [t(1 − t)] 1 r dt )r + ( [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r )(∫ 1 0 t 2 r dt )r = ( [f(a)]r[g(a)]r + [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r )(∫ 1 0 t 2 r dt )r + ( [f(a)]r[g(a) + η(g(b),g(a))]r +[g(a)]r[f(a) + η(f(b),f(a))]r )(∫ 1 0 [t(1 − t)] 1 r dt )r = { m(a,b) ( r r + 2 )r + n(a,b) ( β( 1 r + 1, 1 r + 1) )r} , int. j. anal. appl. 16 (6) (2018) 880 which is the required result. � acknowledgements the authors would like to thank the rector, comsats university islamabad, pakistan, for providing excellent research and academic environments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl, 335(2007), 1294-1308. [2] m. alomari, m. darus and s. s. dragomir, new inequalities of simpson’s type for s-convex functions with applications, rgmia res. rep. coll, 12 (4)(2009). [3] c. baiochi and a. capelo, variational and quas-variational inequalities, wiley, new york, (1984). [4] j. crank, free and moving boundary problems, clarendon press, osford, uk, (1984). [5] g. cristescu, l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrechet, holland,(2002). [6] m. r. delavar and s. s. dragomir, on η-convexity, math. inequal. appl, 20(1)(2017), 203-216. [7] s. s. dragomir and c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university, australia, (2000). [8] r. glowinski, j. l. lions and r. tremolieres, numerical analysis of variational inequalities, north-holland, amsterdam, (1981). [9] m. e. gordji, m. r. delavar and m. d. sen, on ϕ convex functions, j. math. inequal, 10(1)(2016), 173-183. [10] m. e. gordji, m. r. delavar and s. s. dragomir, an inequality related to η-convex functions (ii), int. j. nonlinear anal. appl, 6(2)(2015), 27-33. [11] p. m. gill, c. e. m. pearce , j. pecaric, hadamards inequality for r-convex functions, j. math. anal. appl, 215(1997), 461470. [12] j. hadamard, etude sur les proprietes des fonctions entieres et en particulier dune fonction consideree par riemann, j. math. pure. appl, 58(1893), 171-215. [13] c. hermite, sur deux limites d’une integrale definie, mathesis, 3(1983), 82. [14] d. h. hyers and s. m. ulam, approximately convex functions, proc. amer. math. soc, 3(1952), 821-828. [15] c. p. niculescu and l. e. persson, convex functions and their applications. springer verlag, new york, (2006). [16] m. a. noor, on variational inequalities, phd thesis, brunel university, london, uk, (1975). [17] m. a. noor, general variational inequalities, appl. math. letters, 1(1988), 119-121. [18] m. a. noor, new approximation schemes for general variational inequalities, j. math. anal. appl, 251(2000), 217-230. [19] m. a. noor, some developments in general variational inequalites, appl. math. comput. 152(2004), 199-277. [20] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inform. sci. 10(5)(2016), 1811-1814. [21] m. a. noor, k. i. noor and th. m. rassias, some aspects of variational inequalities, j. comput. appl. math. 47(1993), 285-312. [22] m. a. noor, k. i. noor and m. u. awan, some new estimates of hermite-hadamard inequalities via harmonically convex functions, le mathematiche, lxxi(ii)(2016), 117-127. int. j. anal. appl. 16 (6) (2018) 881 [23] m. a. noor, k. i. noor, m. u. awan and f. safdar, on strongly generalized convex functions, filomat, 31(18)(2017), 5783-5790. [24] m. a. noor, k. i. noor and f. safdar, generalized geometrically convex functions and inequalities, j. inequal. appl, 2017(2017):22. [25] m. a. noor, k. i. noor and f. safdar, integral inequaities via generalized convex functions, j. math. computer, sci, 17(4)(2017), 465-476. [26] m. a. noor, k. i. noor, s. iftikhar, f. safdar, integral inequaities for relative harmonic (s,η)-convex functions, appl. math. comp. sci, 1(1)(2015), 27-34. [27] m. a. noor, k. i. noor, s. iftikhar and s. safdar, generalized (h,r)-harmonic convex functionsand inequalities, inter. j. math. anal. 16(4)(2018),542-555. [28] m. a. noor, k. i. noor and f. safdar, integral inequaities via generalized (α,m)-convex functions, j. nonlinear. func. anal, 2017, (2017), article id: 32. [29] m. a. noor, k. i. noor, s. iftikhar, inequaities via (p,r)-convex functions, rad, (2018). [30] m. a. noor, k. i. noor and f. safdar, new inequalities for generalized log h-convex function, j. appl. math. inform, 36(3-4)(2018), 245-256. [31] m. a. noor, k. i. noor, f. safdar, m. u. awan and s. ullah, inequaities via generalized log m-convex functions, j. nonlinear. sci. appl, 10(2017), 5789-5802. [32] m. a. noor, k. i. noor and f. safdar, generalized r-convex functions and integral inequalities. int. j. anal. appl, 16(2018). [33] c. p. niculescu, convexity according to the geometric mean, math. inequal. appl, 3(2)(2000), 155-167. [34] n. p. n. ngoc, n.v. vinh, p. t. t. hien, integral inequalities of hadamard type for r-convex functions, int. math. forum, 4 (35)(2009), 1723-1728. [35] g. stampacchia, formes bilineaires coercivities sur les ensembles convexes, c. r. acad. sci. paris, 258(1964), 4413-4416. . 1. introduction 2. preliminaries 3. main results acknowledgements references international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 113-122 http://www.etamaths.com fixed point and tripled fixed point theorems under pata-type conditions in ordered metric spaces zoran kadelburg1,∗ and stojan radenović2 abstract. in this paper, we first prove a version of the fixed point theorem obtained in [v. pata, a fixed point theorem in metric spaces, j. fixed point theory appl. 10 (2011) 299–305], adjusted for monotone mappings in ordered metric spaces, as well as some generalizations. then we apply them to obtain results of this type for tripled fixed points in two cases—for monotone and mixed-monotone mappings with three variables. an example is given to show the difference between some of these results. 1. introduction a very interesting extension of the banach contraction principle was recently obtained by v. pata in [1]. some researchers followed this approach and already several other fixed point results in the spirit of pata have appeared, see, e.g., [3, 4, 2, 5]. on the other hand, fixed points of monotone mappings in ordered metric spaces have been a matter of investigation ever since the first results given by ran and reurings in [6]. this includes so-called coupled and tripled fixed points. generally speaking, fixed point results in ordered spaces use weaker contractive conditions (restricted to comparable pairs of points), but at the expense of an additional assumption that the given mapping is monotone. some coupled fixed point results with pata-type conditions have been recently obtained in [3, 4]. in this paper, we first prove “ordered versions” of the basic pata’s result, as well as some generalizations. then we apply them to obtain results of this type for tripled fixed points in two cases—for monotone and mixed-monotone mappings with three variables. an example is given to show the difference between some of these results. 2. preliminaries we begin with some notation and preliminaries. throughout the paper, (x ,d,�) always denotes a partially ordered metric space, i.e., a triple where (x ,�) is a partially ordered set and (x ,d) is a metric space. for x,y ∈ x , x � y will denote that x and y are comparable, i.e., either x � y or y � x holds. 2010 mathematics subject classification. primary 47h10; secondary 47h09. key words and phrases. ordered metric space; pata-type contraction; tripled fixed point. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 113 114 z. kadelburg and s. radenović recall that the space (x ,d,�) is said to be regular if it has the following properties: (i) if for a non-decreasing sequence {xn}, xn → x as n → ∞, then xn � x for all n; (ii) if for a non-increasing sequence {xn}, xn → x as n → ∞, then xn � x for all n. throughout the paper, ψ : [0, 1] → [0,∞) will be a fixed increasing function, continuous at zero, satisfying ψ(0) = 0. 3. pata-type fixed point results in ordered metric spaces in this section, x0 will be an arbitrary stable point in the given ordered metric space (x ,d,�), and ‖x‖ will be defined by ‖x‖ = d(x,x0). it will be clear that the results do not depend on the particular choice of point x0. theorem 3.1. let the space (x ,d,�) be complete and let λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. let f : x → x be a non-decreasing map such that there exists x0 satisfying x0 � fx0 and suppose that the inequality (3.1) d(fx,fy) ≤ (1 −ε)d(x,y) + λεαψ(ε)[1 + ‖x‖ + ‖y‖]β is satisfied for every ε ∈ [0, 1] and all x,y ∈ x with x � y. if f is continuous or (x ,d,�) is regular, then f has a fixed point z ∈x. moreover, (i) the set of fixed points of f is a singleton if and only if it is totally ordered; (ii) the set of fixed points of f is a singleton if for every two points u,v ∈ x there exists w ∈x, comparable with u, v and fw. proof. as remarked before the formulation of theorem, we can take that the point for which x0 � fx0 is the same as the one for which ‖x‖ is defined to be equal to d(x,x0). also, without loss of generality, we can assume that x0 � fx0. then the sequence {xn} defined by xn+1 = fxn, n = 0, 1, . . . , is non-decreasing. suppose further that xn 6= fxn for each n (otherwise there is nothing to prove). since any two terms of {xn} are comparable, the inequality (3.1) can be used in the same way as in the proof of [1, theorem 1] to obtain that: (1) the sequence {d(xn,xn+1)} is strictly decreasing and tends to some d∗ ≥ 0 as n →∞; (2) the sequence {cn} is bounded, where cn = ‖xn‖; (3) d∗ = 0. (4) {xn} is a cauchy sequence, thus converging to some z ∈ x. if the mapping f is continuous, then fxn → fz = z since fxn = xn+1. if the space (x,d,�) is regular, then for the non-decreasing sequence {xn} we have that xn � z for each n. now, for ε = 0, we get from (3.1) that d(fxn,fz) ≤ d(xn,z) → 0 wherefrom fxn → fz, i.e., fz = z. uniqueness of the fixed point. (i) if the set of fixed points fix(f) is a singleton, then it is totally ordered. conversely, assume that fix(f) is totally ordered and that u,v are two (comparable) fixed points of f. applying (3.1), we get d(u,v) = d(fu,fv) ≤ (1 −ε)d(u,v) + kεαψ(ε), fixed point theorems under pata-type conditions 115 where [1 + ‖u‖ + ‖v‖] β = k > 0, i.e., εd(u,v) ≤ kεαψ(ε), for each ε ∈ [0, 1], and it follows that u = v. (ii) suppose now that for every two points u,v ∈x there exists w ∈x , comparable with u, v and fw. assume that u and v are distinct fixed points of f. if they are comparable, we get a contradiction as in (i). if not, choose w as stated. then, u = fnu � fnw and v = fnv � fnw; we will prove that d(u,fnw) ↓ u∗ = 0 and d(v,fnw) ↓ v∗ = 0. indeed, for ε = 0, we get from (3.1) that d(u,fnw) = d(ffn−1u,ffn−1w) ≤ d(fn−1u,fn−1w) = d(u,fn−1w) and, similarly, d(v,fnw) = d(ffn−1v,ffn−1w) ≤ d(fn−1v,fn−1w) = d(v,fn−1w), i.e., d(u,fnw) ↓ u∗ and d(v,fnw) ↓ v∗. it remains to prove that u∗ = v∗ = 0. we will prove first that the sequence cn = ‖fnw‖ is bounded. we have (3.2) cn = d(f nw,x0) ≤ d(fnw,fn+1w) + d(fn+1w,fw) + d(fw,x0). since, by assumption, w � fw, we have that the sequence d(fnw,fn+1w) decreases. indeed, taking again ε = 0 in (3.1), we get that d(fnw,fn+1w) = d(ffn−1w,ffnw) ≤ d(fn−1w,fnw) ≤ ···≤ d(w,fw). then it follows from (3.2) that cn ≤ d(w,fw) + d(ffnw,fw) + d(fw,w) + d(w,x0) = 2d(w,fw) + d(w,x0) + (1 −ε)d(w,fnw) + λεαψ(ε)[1 + ‖fnw‖ + ‖w‖] β ≤ 2d(w,fw) + d(w,x0) + (1 −ε)d(w,x0) + (1 −ε)d(x0,fnw) + λεαψ(ε)[1 + cn + d(w,x0)] β. hence, we get that εcn ≤ aεαψ(ε)cαn + b, for some constants a,b > 0. now, in the same way as in [1, lemma 2.1], it follows that {cn} is a bounded sequence. we are now able to prove that, e.g., u∗ = 0. indeed, d(u,fnw) = d(ffn−1u,ffn−1w) ≤ (1 −ε)d(fn−1u,fn−1w) + λεαψ(ε)[1 + ‖u‖ + ‖fn−1w‖] β, i.e., d(u,fnw) ≤ (1 −ε)d(u,fn−1w) + kεαψ(ε), for some k > 0. passing to the limit as n →∞, we get that u∗ ≤ (1 −ε)u∗ + kεαψ(ε), i.e., u∗ = 0. in the same way, v∗ = 0 is proved. it follows that d(u,v) ≤ d(u,fnw) + d(fnw,v) → 0 + 0 = 0, i.e., u = v. � 116 z. kadelburg and s. radenović remark 3.1. theorem 3.1 is strictly stronger than [6, theorem 2.1]. on the one side, the hypotheses of [6, theorem 2.1] imply those of theorem 3.1, which follows in the same way as it was proved in [1, §3] that the classical banach’s contractive condition implies pata’s condition (3.1). on the other side, the example of function f : [1, +∞) → [1, +∞), f(x) = −2 + x− 2 √ x + 4 4 √ x (see [1, example, p. 303]) shows that condition (3.1) can be satisfied when banach’s condition is not. it is also an example of the situation when condition (ii) for the uniqueness of fixed point (in the previous theorem) is fulfilled (since the given space is totally ordered). it is well known that there are a lot of generalizations of banach contraction principle, obtained by modifying the basic contractive conditions (see, e.g., [7]). some of them already have their pata-type versions (see [4, 2]). we shall present here a result of pata-type for so-called generalized contractions, in the “ordered” version. theorem 3.2. let the space (x ,d,�) be complete and let λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. let f : x → x be a non-decreasing map such that there exists x0 satisfying x0 � fx0 and suppose that the inequality d(fx,fy) ≤ (1 −ε) max { d(x,y),d(x,fx),d(y,fy), d(x,fy) + d(y,fx) 2 } (3.3) + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]β is satisfied for every ε ∈ [0, 1] and all x,y ∈ x with x � y. if f is continuous or (x ,d,�) is regular, then f has a fixed point z ∈x. moreover, (i) the set of fixed points of f is a singleton if and only if it is totally ordered; (ii) the set of fixed points of f is a singleton if for every two points u,v ∈ x there exists w ∈x, comparable with u, v and fw. proof. 1. as usual, starting with the given point x0, construct the sequence {xn} by xn+1 = fxn, n = 0, 1, . . . similarly as in the proof of theorem 3.1, this sequence is monotone, hence condition (3.3) can be used for its elements. suppose that xn 6= xn+1 for each n. in order to prove that the sequence {d(xn,xn+1)} is decreasing, suppose, to the contrary, that d(xk,xk+1) = max{d(xk−1,xk),d(xk,xk+1)} for some k ∈ n. then, applying (3.3) with x = xk−1, y = xk, we get that d(xk,xk+1) = d(fxk−1,fxk) ≤ (1 −ε) max { d(xk−1,xk),d(xk,xk+1), 1 2 d(xk−1,xk+1) } + λεαψ(ε)[1 + ‖xk−1‖ + 2‖xk‖ + ‖xk+1‖]β = (1 −ε)d(xk,xk+1) + kεαψ(ε), for some k > 0. it follows that d(xk,xk+1) = 0, a contradiction! hence, {d(xn,xn+1)} is a (strictly) decreasing sequence, thus tending to some d∗ ≥ 0. 2. denote cn = ‖xn‖. we will prove that the sequence {cn} is bounded. fixed point theorems under pata-type conditions 117 we have that cn = d(xn,x0) ≤ d(xn,xn+1) + d(fxn,fx0) + c1 ≤ 2c1 + (1 −ε) max { d(xn,x0),d(xn,xn+1),d(x0,x1), 1 2 (d(xn,x1) + d(xn+1,x0)) } + λεαψ(ε)[1 + ‖xn‖ + ‖xn+1‖ + ‖x1‖] β ≤ 2c1 + (1 −ε) max{cn,c1,cn + c1} + λεαψ(ε)[1 + cn + c1 + cn + c1] β ≤ 2c1 + (1 −ε)(cn + c1) + λεαψ(ε)[1 + 2c1 + 2cn] α (it was used that d(xn,xn+1) ≤ c1, d(xn,x1) +d(xn+1,x0) ≤ d(xn,x0) +d(x1,x0) + d(xn+1,xn)+d(xn,x0) ≤ 2(cn+c1) and ‖xn+1‖≤ d(xn+1,xn)+d(xn,x0) ≤ c1 +cn). finally, we get that εcn ≤ aεαψ(ε)cαn + b for some constants a,b > 0. in the same way as in the proof of [1, lemma 3], it follows that the sequence {cn} is bounded. 3. now we use the boundedness of {cn} to prove that d∗ = 0. indeed, we have that d(xn+1,xn) = d(fxn,fxn−1) ≤ (1 −ε)d(xn,xn−1) + λεαψ(ε)[1 + ‖xn‖ + 2‖xn−1‖ + ‖xn+1‖] β ≤ (1 −ε)d(xn,xn−1) + kεαψ(ε), for some k > 0. passing to the limit as n →∞, it follows that d∗ = 0. 4. in order to prove that {xn} is a cauchy sequence, suppose the contrary. then, using the standard procedure (see, e.g., [8, lemma 2.1]), we get that there exist δ > 0 and two increasing sequences of integers {m(k)} and {n(k)}, such that nk > mk > k and the sequences d(xn(k)+1,xm(k)) and d(xn(k),xm(k)−1) tend to δ as n →∞. putting x = xn(k), y = xm(k)−1 in (3.3), and using the boundedness of {cn}, we get that d(xn(k)+1,xm(k)) ≤ (1 −ε)d(xn(k),xm(k)−1) + kεαψ(ε). passing to the limit as k →∞, we get that δ = 0, a contradiction! hence, {xn} is a cauchy sequence, and it converges to some z ∈ x. 5. the proof that fz = z in either of the given cases is the same as for theorem 3.1. 6. the uniqueness of the the fixed point under one of the assumptions (i) or (ii) can be proved similarly as in theorem 3.1. � remark 3.2. similarly as in the classical situation, treated in [7], it can be proved that theorem 3.2 contains as special cases several other pata-type results in their order versions. in particular, this includes kannan, chatterjea, reich, zamfirescu and hardy-rogers results. since the exact formulations and proofs are obvious, we omit the details. 4. tripled fixed point results for monotone and mixed-monotone mappings we will use the following terminology. definition 4.1. let f : x3 →x be a mapping. 118 z. kadelburg and s. radenović (1) f is called non-decreasing if it is non-decreasing in all three variables. (2) f is called mixed-monotone if it is non-decreasing in the first and third variables, and non-increasing in the second variable. (3) a point y = (x,y,z) ∈x3 is called a tripled fixed point of the first kind (or borcut kind [9]) if (4.1) f(x,y,z) = z, f(y,x,z) = y, f(z,y,x) = z. (4) a point y = (x,y,z) ∈x3 is called a tripled fixed point of the second kind (or berinde-borcut kind [10]) if (4.2) f(x,y,z) = z, f(y,x,y) = y, f(z,y,x) = z. remark 4.1. in what follows, tripled fixed point results of the first kind will be proved for monotone mappings, while those of the second type will be connected with mixed-monotone mappings. it will be clear in the sequel that part (3) of the previous definition can be modified in several ways. in fact, any three combinations of elements x,y,z can be taken instead of (x,y,z), (y,x,z) and (z,y,x) in (4.1), with the only condition that the first entry of each triple matches the right-hand side. in particular, the “cyclic” case, i.e., the condition f(x,y,z) = x, f(y,z,x) = and f(z,x,y) = z can be considered. it will also be clear which modifications should be made to the results that follows, so we will not state them explicitly. moreover, the same treatment can be applied in the case of arbitrary number of variables. it is important to notice that this considerably differs from the case of “mixedmonotone situation”. namely, as was shown in [11], in this case only some particular combinations are possible (in particular, the cyclic case cannot be treated in this way). the following lemma is easy to prove. lemma 4.1. (i) if relations v1 and v2 are defined on x3 by y v1 v ⇔ x � u∧y � v ∧z � w, y = (x,y,z), v = (u,v,w) ∈x3 and y v2 v ⇔ x � u∧y � v ∧z � w, y = (x,y,z), v = (u,v,w) ∈x3, and d : x3 ×x3 → r+ is given by d(y,v ) = d(x,u) + d(y,v) + d(z,w), y = (x,y,z), v = (u,v,w) ∈x3, then (x3,d,vi), i = 1, 2 are ordered metric space. the space (x3,d) is complete if and only if (x ,d) is complete. moreover, the spaces (x3,d,vi) are regular if and only if (x ,d,�) is such. (ii) if f : x3 →x is non-decreasing (w.r.t. �), then the mapping t 1f : x 3 →x3 given by t 1fy = (f(x,y,z),f(y,x,z),f(z,y,x)), y = (x,y,z) ∈x 3 is non-decreasing w.r.t. v1. (iii) if f : x3 → x is mixed-monotone, then the mapping t 2f : x 3 → x3 given by t 2fy = (f(x,y,z),f(y,x,y),f(z,y,x)), y = (x,y,z) ∈x 3 is non-decreasing w.r.t. v2. fixed point theorems under pata-type conditions 119 (iv) the mappings tif , i = 1, 2 are continuous if and only if f is continuous. (v) the mapping f has a tripled fixed point of the first (resp. of the second) kind if and only if the mapping t 1f (resp. t 2 f ) has a fixed point in x 3. if what follows, y0 = (x0,y0,z0) will be a fixed element in x3 and for y = (x,y,z) ∈x3, we will denote ‖y‖ = ‖x,y,z‖ = d(y,y0) = d(x,x0) + d(y,y0) + d(z,z0). it will be clear that the obtained results do not depend on the particular choice of the point y0. we will prove first some results for monotone mappings and tripled fixed points of the first (borcut) kind. theorem 4.1. let f : x3 → x be a non-decreasing mapping, and suppose that there exist x0,y0,z0 ∈ x such that x0 � f(x0,y0,z0), y0 � f(y0,x0,z0), z0 � f(z0,y0,x0). let, for some fixed constants λ ≥ 0, α ≥ 1 and β ∈ [0,α], the inequality d(f(x,y,z),f(u,v,w)) + d(f(y,x,z),f(v,u,w)) + d(f(z,y,x),f(w,v,u))(4.3) ≤ (1 −ε)(d(x,u) + d(y,v) + d(z,w)) + λεαψ(ε)[1 + ‖x,y,z‖ + ‖u,v,w‖] β holds for all ε ∈ [0, 1] and all x,y,z,u,v,w ∈x with (x � u, y � v and z � w) or (x � u, y � v and z � w). finally, suppose that f is continuous or that the space is regular. then f has a tripled fixed point y ∗ = (x∗,y∗,z∗) ∈x3 of the first kind. proof. consider the space (x3,d,v1) and the mapping t 1f : x 3 →x3, as defined in lemma 4.1.(i) and (ii). the mapping t 1f is non-decreasing w.r.t. v1. let y = (x,y,z) and v = (u,v,w) be comparable w.r.t. v1, i.e., let (x � u, y � v and z � w) or (x � u, y � v and z � w) hold. then, the condition (4.3) holds, which can be written as d(t 1fy,t 1 fv ) ≤ (1 −ε)d(y,v ) + λε αψ(ε)[1 + d(y,y0) + d(v,y0)] β. in other words, t 1f satisfies condition of the type (3.1) in the space (x 3,d,v1). applying theorem 3.1, we obtain that t 1f has a fixed point y ∗ = (x∗,y∗,z∗) ∈x3, which is, by lemma 4.1.(v), a tripled fixed point of the first kind of mapping f. � corollary 4.1. let f : x3 → x be a non-decreasing mapping, and suppose that there exist x0,y0,z0 ∈ x such that x0 � f(x0,y0,z0), y0 � f(y0,x0,z0), z0 � f(z0,y0,x0). let, for some fixed constants λ ≥ 0, α ≥ 1 and β ∈ [0,α], the inequality d(f(x,y,z),f(u,v,w))(4.4) ≤ 1 −ε 3 (d(x,u) + d(y,v) + d(z,w)) + λεαψ(ε)[1 + ‖x,y,z‖ + ‖u,v,w‖] β holds for all ε ∈ [0, 1] and all x,y,z,u,v,w ∈x with (x � u, y � v and z � w) or (x � u, y � v and z � w). finally, suppose that f is continuous or that the space is regular. then f has a tripled fixed point y ∗ = (x∗,y∗,z∗) ∈x3 of the first kind. proof. suppose that y = (x,y,z),v = (u,v,w) ∈ x3 are comparable w.r.t. v1. applying (4.4) to the triples (x,y,z) and (u,v,w), we get that d(f(x,y,z),f(u,v,w))(4.5) ≤ 1 −ε 3 d(y,v ) + λεαψ(ε)[1 + d(y,y0) + d(v,y0)] β. 120 z. kadelburg and s. radenović applying the same inequality to the triples (y,x,z) and (v,u,w), we obtain d(f(y,x,z),f(v,u,w))(4.6) ≤ 1 −ε 3 d(y,v ) + λεαψ(ε)[1 + d(y,x0) + d(x,y0) + d(z,z0) + d(v,x0) + d(u,y0) + d(w,z0)] β ≤ 1 −ε 3 d(y,v ) + λεαψ(ε)[1 + d(y,y0) + d(v,y0) + 4d(x0,y0)] β. finally, applying (4.4) to the triples (z,y,x) and (w,v,u), we get d(f(z,y,x),f(w,v,u))(4.7) ≤ 1 −ε 3 d(y,v ) + λεαψ(ε)[1 + d(z,x0) + d(y,y0) + d(x,z0) + d(w,x0) + d(v,y0) + d(u,z0)] β ≤ 1 −ε 3 d(y,v ) + λεαψ(ε)[1 + d(y,y0) + d(v,y0) + 4d(x0,z0)] β. adding up the inequalities (4.5), (4.6) and (4.7), and writing temporarily a = d(y,y0) + d(v,y0), we get the following estimate: d(t 1fy,t 1 fv ) ≤ (1 −ε)d(y,v ) (4.8) + λεαψ(ε){[1 + a]β + [1 + a + 4d(x0,y0)]β + [1 + a + 4d(x0,z0)]β} now, [1 + a]β + [1 + a + 4d(x0,y0)] β + [1 + a + 4d(x0,z0)] β = [1 + a]β[1 + (1 + 4d(x0,y0) 1 + a )β + (1 + 4d(x0,z0) 1 + a )β] ≤ [1 + a]β[1 + (1 + 4d(x0,y0))β + (1 + 4d(x0,z0))β] = c[1 + a]β, where c is a constant (not depending on y , v and ε). hence, putting λ1 = λc, (4.8) can be written as d(t 1fy,t 1 fv ) ≤ (1 −ε)d(y,v ) + λ1ε αψ(ε)[1 + d(y,y0) + d(v,y0)] β, which means that all the conditions of theorem 4.1 are fulfilled. � the following example shows that theorem 4.1 is strictly stronger than corollary 4.1. fixed point theorems under pata-type conditions 121 example 4.1. let x = r be equipped with the usual metric and order. the mapping f : x3 → x defined by f(x,y,z) = 1 8 (5x + y + z) is obviously nondecreasing. it is easy to obtain that d(f(x,y,z),f(u,v,w)) + d(f(y,x,z),f(v,u,w)) + d(f(z,x,y),f(v,u,w)) = | 5x + y + z 8 − 5u + v + w 8 | + | 5y + x + z 8 − 5v + u + w 8 | + | 5z + y + x 8 − 5w + v + u 8 | ≤ 5 8 |x−u| + 1 8 |y −v| + 1 8 |z −w| + 5 8 |y −v| + 1 8 |x−u| + 1 8 |z −w| + 5 8 |z −w| + 1 8 |y −v| + 1 8 |x−u| = 7 8 [d(x,u) + d(y,v) + d(z,w)], i.e., d(t 1fy,t 1 fv ) ≤ λd(y,v ), where λ = 7 8 . similarly as in [1, §3], it can be proved that also (4.3) holds for appropriate λ, α and β, all ε ∈ [0, 1] and all comparable y,v ∈x3. on the other hand, suppose that the condition (4.4) of corollary 4.1 holds, i.e., | 5x + y + z 8 − 5u + v + w 8 | ≤ 1 −ε 2 [|x−u| + |y −v| + |z −w|] + λεαψ(ε)[1 + ‖x,y,z‖ + ‖u,v,w‖]β is satisfied for each ε ∈ [0, 1] and all comparable y,v ∈ x3. taking ε = 0, y = v and z = w, we obtain that 5 8 |x−u| ≤ 1 2 |x−u| which obviously cannot hold (except when x = u). consider now mixed-monotone mappings and tripled fixed points of the second (berinde-borcut)) kind. theorem 4.2. let f : x3 → x be a mixed-monotone mapping, and suppose that there exist x0,y0,z0 ∈ x such that x0 � f(x0,y0,z0), y0 � f(y0,x0,z0), z0 � f(z0,y0,x0). let, for some fixed constants λ ≥ 0, α ≥ 1 and β ∈ [0,α], the inequality d(f(x,y,z),f(u,v,w)) + d(f(y,x,y),f(v,u,v)) + d(f(z,y,x),f(w,v,u)) ≤ (1 −ε)(d(x,u) + d(y,v) + d(z,w)) + λεαψ(ε)[1 + ‖x,y,z‖ + ‖u,v,w‖] β holds for all ε ∈ [0, 1] and all x,y,z,u,v,w ∈x with (x � u, y � v and z � w) or (x � u, y � v and z � w). finally, suppose that f is continuous or that the space is regular. then f has a tripled fixed point y ∗ = (x∗,y∗,z∗) ∈ x3 of the second kind. proof. the proof is similar to the proof of theorem 4.1, using the mapping t 2f of lemma 4.1.(iii) in the space (x3,d,v2). � corollary 4.2. let f : x3 → x be a mixed-monotone mapping, and suppose that there exist x0,y0,z0 ∈ x such that x0 � f(x0,y0,z0), y0 � f(y0,x0,z0), 122 z. kadelburg and s. radenović z0 � f(z0,y0,x0). let, for some fixed constants λ ≥ 0, α ≥ 1 and β ∈ [0,α], the inequality d(f(x,y,z),f(u,v,w)) ≤ 1 −ε 3 (d(x,u) + d(y,v) + d(z,w)) + λεαψ(ε)[1 + ‖x,y,z‖ + ‖u,v,w‖] β holds for all ε ∈ [0, 1] and all x,y,z,u,v,w ∈x with (x � u, y � v and z � w) or (x � u, y � v and z � w). finally, suppose that f is continuous or that the space is regular. then f has a tripled fixed point y ∗ = (x∗,y∗,z∗) ∈x3 of the first kind. a similar example as example 4.1 can be constructed to show that theorem 4.2 is strictly stronger than corollary 4.2. acknowledgement the authors are thankful to the ministry of education, science and technological development of serbia. conflict of interests the authors declares that there is no conflict of interests regarding the publication of this article. references [1] v. pata, a fixed point theorem in metric spaces. j. fixed point theory appl. 10 (2011) 299–305. [2] m. chakraborty, s. k. samanta, a fixed point theorem for kannan-type maps in metric spaces, arxiv:1211, 7331v2 [math. gn] 16 dec 2012. [3] m. eshaghi, s. mohseni, m. r. delavar, m. de la sen, g. h. kim, a. arian, pata contractions and coupled type fixed points. fixed point theory appl. 2014 (2014) article id 130. [4] z. kadelburg, s. radenović, fixed point theorems for pata-type maps in metric spaces, submitted. [5] m. paknazar, m. eshaghi, y. j. cho, s. m. vaezpour, a pata-type fixed point theorem in modular spaces with application, fixed point theory appl. 2013(2013), article id 239. [6] a. c. m. ran, m. c. b. reurings, a fixed point theorem in partially ordered sets and some application to matrix equations, proc. amer. math. soc. 132 (2004), 1435–1443. [7] b. e. rhoades, a comparison of various definitions of contractive mappings, trans. amer. math. soc. 226 (1977) 257–290. [8] s. radenović, z. kadelburg, d. jandrlić, a. jandrlić, some results on weak contraction maps, bull. iranian math. soc. 38(3) (2012), 625–645. [9] m. borcut, tripled fixed point theorems for monotone mappings in partially ordered metric spaces, carpathian j. math. 28, 2 (2012), 207–214. [10] v. berinde, m. borcut, tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, nonlinear anal. tma 74 (2011), 4889–4897. [11] a. roldán, j. mart́ınez-moreno, c. roldán, multidimensional fixed point theorems in partially ordered complete metric spaces, j. math. anal. appl. 396 (2012), 536–545. 1university of belgrade, faculty of mathematics, studentski trg 16, 11000 beograd, serbia 2university of belgrade, faculty of mechanical engineering, kraljice marije 16, 11120 beograd, serbia ∗corresponding author international journal of analysis and applications volume 18, number 1 (2020), 50-62 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-50 some inequalities for n-time differentiable mappings using a multi-step kernel with applications sofian obeidat∗ department of basic sciences, deanship of preparatory year, university of hail, hail 2440, saudi arabia ∗corresponding author: obeidatsofian@gmail.com abstract. in this paper, we develop a new multi-step kernel and use it to establish new ostrowski’s type inequalities for n-time differentiable mappings, whose n-th derivatives satisfy convexity and quasi-convexity conditions. applications of our findings to random variables and approximation of integrals are given. 1. introduction the classical ostrowski inequality (1938, see [8]) is given as follows: if x ∈ [x1, x2] then∣∣∣∣∣∣g(x) − 1x2 −x1 x2∫ x1 g(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + (x− x1+x2 2 )2 (x2 −x1)2 ] (x2 −x1)‖g′‖∞ , (1.1) where g is a differentiable function defined on a finite interval [x1, x2], whose derivative is integrable and bounded over [x1, x2]. the constant 1/4 is the best possible. when x = x1+x2 2 , inequality 1.1 reduces to the midpoint version ∣∣∣∣∣∣g( x1 + x22 ) − 1x2 −x1 x2∫ x1 g(t)dt ∣∣∣∣∣∣ ≤ (x2 −x1)4 ‖g′‖∞ . the importance of ostrowski’s type inequalities is due to their applications in different aspects. for generalizations and variants of ostrowski’s type inequalities, we refer the reader to [2] and [4]. received 2019-10-26; accepted 2019-11-22; published 2020-01-02. 2010 mathematics subject classification. primary 26d15, 26d20; secondary 26d99. key words and phrases. convex functions; hölder inequality; quasi-convex function; ostrowski inequality. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 50 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-50 int. j. anal. appl. 18 (1) (2020) 51 recently, several authors have derived ostrowski’s type inequalities for n-time differentiable mappings whose n-th derivatives satisfy different types of convexity conditions, see for example [?], [3], [5], [7], [10] and [12]. in particular, ozdemir and yildiz, in [9], obtained the following three theorems for n-time differentiable mappings . theorem 1.1. [9] suppose that n is a positive integer, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣ is convex on [x1, x2], then∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0 ( 1 + (−1)k ) (x2 −x1) k+1 2k+1 (k + 1)! g (k) ( x1 + x2 2 )∣∣∣∣∣∣ (1.2) ≤ (x2 −x1) n+1 2n (n + 1)! [∣∣g(n) (x1)∣∣ + ∣∣g(n) (x2)∣∣ 2 ] . theorem 1.2. [9] suppose that n is a positive integer, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is convex on [x1, x2] , where q > 1, then∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0 ( 1 + (−1)k ) (x2 −x1) k+1 2k+1 (k + 1)! g (k) ( x1 + x2 2 )∣∣∣∣∣∣ (1.3) ≤ (x2 −x1) n+1 2n+1n! ( 1 1 + pn )1 p  (∣∣g(n) (x1)∣∣q + 3 ∣∣g(n) (x2)∣∣q 4 )1 q + 3 ∣∣g(n) (x1)∣∣q + ∣∣g(n) (x2)∣∣q 4 ] , where 1 p + 1 q = 1. theorem 1.3. [9] suppose that n is a positive integer, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is convex on [x1, x2] , where q ≥ 1, then∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0 ( 1 + (−1)k ) (x2 −x1) k+1 2k+1 (k + 1)! g (k) ( x1 + x2 2 )∣∣∣∣∣∣ (1.4) ≤ (x2 −x1) n+1 2n+1(n + 1)! [( n + 1 2n + 4 ∣∣∣g(n) (x1)∣∣∣q + n + 3 2n + 4 ∣∣∣g(n) (x2)∣∣∣q)1q + ( n + 3 2n + 4 ∣∣∣g(n) (x1)∣∣∣q + n + 1 2n + 4 ∣∣∣g(n) (x2)∣∣∣q)1q ] . in this paper, we generalize the inequalities obtained by ozdemir and yildiz in [9] for convex and quasi convex functions using a multi-step kernel. then we introduce some applications of our findings to random variables and approximation of integrals. throught this paper, r denotes the set of all real numbers and j ⊂ r denotes an interval. the concepts of convex and quasi convex functions, which are well known in the literature, are given as in the following two definitions. int. j. anal. appl. 18 (1) (2020) 52 definition 1.1. [6] a function g : j ⊂ r → r is said to be convex if the inequality g (tx1 + (1 − t) x2) ≤ tg (x1) + (1 − t) g (x2) , holds for all x1, x2 ∈ j and t ∈ [0, 1]. definition 1.2. [11] a function g : j ⊂ r → r is said to be quasi convex if the inequality g (tx1 + (1 − t) x2) ≤ max{g (x1) , g (x2)} holds for all x1, x2 ∈ j and t ∈ [0, 1]. 2. main results we start this section with the following two lemmas. lemma 2.1. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. suppose that g (n) ∈ l1 ([x1, x2]). then for every s1, s2 ∈ [0, 1] the identity ∫ s2 s1 g (x2 + t (x1 −x2)) dt (2.1) = n−1∑ k=0 [ (s2 −r) k+1 g(k) (x2 + s2 (x1 −x2)) − (s1 −r) k+1 g(k) (x2 + s1 (x1 −x2)) ] (x2 −x1) −k (k + 1)! + (x2 −x1) n n! ∫ s2 s1 (t−r)n g(n) (x2 + t (x1 −x2)) dt, holds for each r ∈ r. proof. using integration by parts repeatedly, we get that 1 n! ∫ s2 s1 (t−r)n g(n) (x2 + t (x1 −x2)) dt (2.2) = − n−1∑ k=0 (t−r)k+1 g(k) (x2 + t (x1 −x2)) (x2 −x1) n−k (k + 1)! ∣∣∣∣∣ s2 s1 + 1 (x2 −x1) n ∫ s2 s1 g (x2 + t (x1 −x2)) dt. multiplying (2.2) by (x2 −x1) n , and substituting the upper and lower integral bounds, the result follows. � int. j. anal. appl. 18 (1) (2020) 53 lemma 2.2. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g (n) ∈ l1 ([x1, x2]) then∫ x2 x1 g (x) dx (2.3) = 2m−1∑ l=1 n−1∑ k=0 [ 1 + (−1)k ] 2−m(k+1) (x2 −x1) −k−1 (k + 1)! g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m ) + (x2 −x1) n+1 ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt, where pn,m (t) = 1 n!   tn, if t ∈ [ 0, 1 2m ] ( t− 21−ml )n , if t ∈ ( 2l−1 2m , 2l+1 2m ] , l = 1, 2, · · ·, 2m−1 − 1 (t− 1)n , if t ∈ ( 1 − 1 2m , 1 ] . proof. using lemma 2.1, we have ∫ 1 2m 0 g (x2 + t (x1 −x2)) dt (2.4) = n−1∑ k=0 [( 1 2m )k+1 g(k) ( x1+(2 m−1)x2 2m )] (x2 −x1) −k (k + 1)! + (x2 −x1) n n! ∫ 1 2m 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt, ∫ 1 (1− 12m ) g (x2 + t (x1 −x2)) dt (2.5) = n−1∑ k=0 − (−1 2m )k+1 g(k) ( (2m−1)x1+x2 2m ) (x2 −x1) −k (k + 1)! + (x2 −x1) n n! ∫ 1 (1− 12m ) pn,m (t) g (n) (x2 + t (x1 −x2)) dt, and ∫ 2l+1 2m 2l−1 2m g (x2 + t (x1 −x2)) dt (2.6) = n−1∑ k=0 [ g(k) ( (2l+1)x1+(2 m−2l−1)x2 2m ) + (−1)k g(k) ( (2l−1)x1+(2m−2l+1)x2 2m )] 2m(k+1) (x2 −x1) −k (k + 1)! + (x2 −x1) n n! ∫ 2l+1 2m 2l−1 2m pn,m (t) g (n) (x2 + t (x1 −x2)) dt, for l = 1, 2, · · ·, 2m−1 − 1. combining (2.4), (2.5) and (2.6), the result follows. � int. j. anal. appl. 18 (1) (2020) 54 theorem 2.1. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣ is convex on [x1, x2], then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.7) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2mn (n + 1)! [∣∣g(n) (x1)∣∣ + ∣∣g(n) (x2)∣∣ 2 ] . proof. using convexity of ∣∣g(n)∣∣ on [x1, x2], we get that ∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.8) ≤ 1 n! ∫ 1 2m 0 tn [ (1 − t) ∣∣∣g(n) (x2)∣∣∣ + t ∣∣∣g(n) (x1)∣∣∣]dt + 1 n! 2m−1−1∑ l=1 ∫ 2l+1 2m 2l−1 2m ∣∣∣∣t− l2m−1 ∣∣∣∣n [(1 − t) ∣∣∣g(n) (x2)∣∣∣ + t ∣∣∣g(n) (x1)∣∣∣]dt + 1 n! ∫ 1 1− 1 2m (1 − t)n [ (1 − t) ∣∣∣g(n) (x2)∣∣∣ + t ∣∣∣g(n) (x1)∣∣∣]dt = 1 2mn (n + 1)! [∣∣g(n) (x1)∣∣ + ∣∣g(n) (x2)∣∣ 2 ] . using identity 2.3 and inequality 2.8, the result follows. � remark 2.1. in theorem 2.1, (1) if m = 1, inequality 2.7 reduces to inequality 1.2. (2) if m = 2, inequality 2.7 reduces to ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0 ( 1 + (−1)k ) 2−2(k+1) (x2 −x1) −k−1 (k + 1)! [ g (k) ( 3x1 + x2 4 ) (2.9) +g (k) ( x1 + 3x2 4 )]∣∣∣∣ ≤ (x2 −x1) n+1 22n (n + 1)! [∣∣g(n) (x1)∣∣ + ∣∣g(n) (x2)∣∣ 2 ] . int. j. anal. appl. 18 (1) (2020) 55 (3) if m = 3, inequality 2.7 reduces to ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0 ( 1 + (−1)k ) 2−3(k+1) (x2 −x1) −k−1 (k + 1)! [ g (k) ( 7x1 + x2 8 ) (2.10) +g (k) ( 5x1 + 3x2 8 ) + g (k) ( 3x1 + 5x2 8 ) + g (k) ( x1 + 7x2 8 )]∣∣∣∣ ≤ (x2 −x1) n+1 23n (n + 1)! [∣∣g(n) (x1)∣∣ + ∣∣g(n) (x2)∣∣ 2 ] . theorem 2.2. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is convex on [x1, x2] , where q > 1, then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.11) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2m(n+1)n! ( 1 1 + pn )1 p × [( 1 2m+1 ∣∣∣g(n) (x1)∣∣∣q + (1 − 1 2m+1 )∣∣∣g(n) (x2)∣∣∣q)1q + (( 1 − 1 2m+1 )∣∣∣g(n) (x1)∣∣∣q + 1 2m+1 ∣∣∣g(n) (x2)∣∣∣q)1q +2 2m−1−1∑ l=1 ( l 2m−1 ∣∣∣g(n) (x1)∣∣∣q + (1 − l 2m−1 )∣∣∣g(n) (x2)∣∣∣q)1q   , where 1 p + 1 q = 1. proof. using holder’s inequality, we get that ∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.12) ≤ 1 n! (∫ 1 2m 0 tpndt )1 p (∫ 1 2m 0 ∣∣∣g(n) (x2 + t (x1 −x2))∣∣∣q dt )1 q + 1 n! 2m−1−1∑ l=1 (∫ 2l+1 2m 2l−1 2m ∣∣∣∣t− l2m−1 ∣∣∣∣pn dt )1 p (∫ 2l+1 2m 2l−1 2m ∣∣∣g(n) (x2 + t (x1 −x2))∣∣∣q dt )1 q + 1 n! (∫ 1 1− 1 2m (1 − t)pn dt )1 p (∫ 1 1− 1 2m ∣∣∣g(n) (x2 + t (x1 −x2))∣∣∣q dt )1 q . int. j. anal. appl. 18 (1) (2020) 56 using convexity of ∣∣g(n)∣∣q on [x1, x2] , we find that∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.13) ≤ 1 2m(n+1)n! ( 1 1 + pn )1 p × [( 1 2m+1 ∣∣∣g(n) (x1)∣∣∣q + (1 − 1 2m+1 )∣∣∣g(n) (x2)∣∣∣q)1q + (( 1 − 1 2m+1 )∣∣∣g(n) (x1)∣∣∣q + 1 2m+1 ∣∣∣g(n) (x1)∣∣∣q)1q +2 2m−1−1∑ l=1 ( l 2m−1 ∣∣∣g(n) (x1)∣∣∣q + (1 − l 2m−1 )∣∣∣g(n) (x2)∣∣∣q)1q   . using identity 2.3 and inequality 2.13, the result follows. � remark 2.2. in theorem 2.2, (1) if m = 1, inequality 2.11 reduces to inequality 1.3. (2) if m = 2, inequality 2.11 reduces to∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0   ( 1 + (−1)k ) 2−2(k+1) (x2 −x1) −k−1 (k + 1)! ( g (k) ( 3x1 + x2 4 ) (2.14) +g (k) ( x1 + 3x2 4 ))]∣∣∣∣ ≤ (x2 −x1) n+1 22(n+1)n! ( 1 1 + pn )1 p  2 (∣∣g(n) (x1)∣∣q + ∣∣g(n) (x2)∣∣q 2 )1 q + ( 7 ∣∣g(n) (x1)∣∣q + ∣∣g(n) (x2)∣∣q 8 )1 q + (∣∣g(n) (x1)∣∣q + 7 ∣∣g(n) (x2)∣∣q 8 )1 q   . theorem 2.3. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is convex on [x1, x2] , where q ≥ 1, then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.15) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2m(n+1) (n + 1)! × [( 2−m (n + 1) (n + 2) ∣∣∣g(n) (x1)∣∣∣q + (1 − 2−m (n + 1) (n + 2) )∣∣∣g(n) (x2)∣∣∣q)1q int. j. anal. appl. 18 (1) (2020) 57 +2 2m−1−1∑ l=1 ( 21−m ∣∣∣g(n) (x1)∣∣∣q + (1 − 21−ml)∣∣∣g(n) (x2)∣∣∣q)1q + (( 1 − 2−m (n + 1) (n + 2) )∣∣∣g(n) (x1)∣∣∣q + 2−m (n + 1) (n + 2) ∣∣∣g(n) (x2)∣∣∣q)1q ] . proof. using convexity of ∣∣g(n)∣∣q on [x1, x2], we get that∣∣∣∣∫ 1 0 pn,m (t) g (n) (x2 + t (x1 − x2)) dt ∣∣∣∣ (2.16) ≤ 1 n! (∫ 1 2m 0 t n dt )1 p (∫ 1 2m 0 t n ∣∣∣g(n) (x2 + t (x1 − x2))∣∣∣q dt )1 q + 1 n! (∫ 1 1− 1 2m (1 − t)n dt )1 p (∫ 1 1− 1 2m (1 − t)n ∣∣∣g(n) (x2 + t (x1 − x2))∣∣∣q dt )1 q + 1 n! 2m−1−1∑ l=1  (∫ 2l+12m 2l−1 2m ∣∣∣∣t − l2m−1 ∣∣∣∣n dt )1 p × (∫ 2l+1 2m 2l−1 2m ∣∣∣∣t − l2m−1 ∣∣∣∣n ∣∣∣g(n) (x2 + t (x1 − x2))∣∣∣q dt )1 q   ≤ 2−m(n+1) (n + 1)! ( 2−m (n + 1) (n + 2) ∣∣∣g(n) (x1)∣∣∣q + (1 − 2−m (n + 1) (n + 2) )∣∣∣g(n) (x2)∣∣∣q)1q + 2−m(n+1) (n + 1)! (( 1 − 2−m (n + 1) (n + 2) )∣∣∣g(n) (x1)∣∣∣q + 2−m (n + 1) (n + 2) ∣∣∣g(n) (x2)∣∣∣q)1q + 2−m(n+1)+1 (n + 1)! 2m−1−1∑ l=1 ( 2 1−m ∣∣∣g(n) (x1)∣∣∣q + (1 − 21−ml)∣∣∣g(n) (x2)∣∣∣q)1q . using (2.3) and (2.16), the result follows. � remark 2.3. in theorem 2.3, (1) if m = 1, inequality 2.15 reduces to inequality 1.4. (2) if m = 2, inequality 2.15 reduces to∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− n−1∑ k=0   ( 1 + (−1)k ) 2−2(k+1) (x2 −x1) −k−1 (k + 1)! ( g (k) ( 3x1 + x2 4 ) (2.17) +g (k) ( x1 + 3x2 4 ))]∣∣∣∣ ≤ (x2 −x1) n+1 22(n+1) (n + 1)! (( n + 1 4n + 8 )∣∣∣g(n) (x1)∣∣∣q + (3n + 7 4n + 8 )∣∣∣g(n) (x2)∣∣∣q)1q + 2 (x2 −x1) n+1 22(n+1) (n + 1)! (∣∣g(n) (x1)∣∣q + ∣∣g(n) (x2)∣∣q 2 )1 q + (x2 −x1) n+1 22(n+1) (n + 1)! (( 3n + 7 4n + 8 )∣∣∣g(n) (x1)∣∣∣q + ( n + 1 4n + 8 )∣∣∣g(n) (x2)∣∣∣q)1q . int. j. anal. appl. 18 (1) (2020) 58 theorem 2.4. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣ is quasi-convex on [x1, x2], then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.18) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2mn (n + 1)! max {∣∣∣g(n) (x1)∣∣∣ ,∣∣∣g(n) (x2)∣∣∣} . proof. using quasi-convexity of ∣∣g(n)∣∣ on [x1, x2], we get that ∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.19) ≤ max {∣∣g(n) (x1)∣∣ , ∣∣g(n) (x2)∣∣} n! [∫ 1 2m 0 tndt + 2m−1−1∑ l=1 ∫ 2l+1 2m 2l−1 2m ∣∣∣∣t− l2m−1 ∣∣∣∣n dt + ∫ 1 1− 1 2m (1 − t)n dt   = 1 2mn (n + 1)! max {∣∣∣g(n) (x1)∣∣∣ , ∣∣∣g(n) (x2)∣∣∣} . using identity 2.3 and inequality 2.19, the result follows. � theorem 2.5. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is quasi-convex on [x1, x2], where q > 1, then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.20) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2mnn! ( 1 1 + pn )1 p ( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q , where 1 p + 1 q = 1. int. j. anal. appl. 18 (1) (2020) 59 proof. using holder’s inequality and quasi-convexity of ∣∣g(n)∣∣q on [x1, x2], we get that∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.21) ≤ 2−m(n+1)+1 n! ( 1 1 + pn )1 p [( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q + ( 2m−1 − 1 )( max {∣∣∣g(n) (x1)∣∣∣q ,∣∣∣g(n) (x2)∣∣∣q})1q ] = 1 2mnn! ( 1 1 + pn )1 p ( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q using (2.3) and (2.21), the result follows. � theorem 2.6. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and x1, x2 ∈ j◦ with x1 < x2. if g(n) ∈ l1 ([x1, x2]) and ∣∣g(n)∣∣q is quasi-convex on [x1, x2] , where q ≥ 1, then ∣∣∣∣∣∣ ∫ x2 x1 g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (x2 −x1) −k−1 (k + 1)! (2.22) ×g (k) ( (2l − 1) x1 + (2m − 2l + 1) x2 2m )]∣∣∣∣ ≤ (x2 −x1) n+1 2mn (n + 1)! ( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q proof. using quasi-convexity of ∣∣g(n)∣∣q on [x1, x2], we get that∣∣∣∣ ∫ 1 0 pn,m (t) g (n) (x2 + t (x1 −x2)) dt ∣∣∣∣ (2.23) ≤ 2 2mn+m (n + 1)! ( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q + 2 2mn+m (n + 1)! ( 2m−1 − 1 )( max {∣∣∣g(n) (x1)∣∣∣q , ∣∣∣g(n) (x2)∣∣∣q})1q = 1 2mn (n + 1)! ( max {∣∣∣g(n) (x1)∣∣∣q ,∣∣∣g(n) (x2)∣∣∣q})1q . using (2.3) and (2.23), the result follows. � 3. some applications we start this section with an application to approximation of an integral. recall that a partition d of a finite interval [c, d] , c < d, is a finite sequence of numbers c = c0 < c1 < · · · < cn = d. proposition 3.1. suppose that n and m are positive integers, g : j ⊆ r → r is n-time differentiable mapping on j◦ and c, d ∈ j◦ with c < d. let d : c = c0 < c1 < · · · < cn = d be a partition of [c, d]. if int. j. anal. appl. 18 (1) (2020) 60 g(n) ∈ l1 ([c, d]) and ∣∣g(n)∣∣ is convex on [c, d], then d∫ c g (x) dx = a (g, d) + e (g, d) , (3.1) where a (g, d) (3.2) = n−1∑ j=0 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (cj+1 − cj) −k−1 (k + 1)! ×g (k) ( (2l − 1) cj + (2m − 2l + 1) cj+1 2m )] , and |e (g, d)| (3.3) ≤ 1 2mn+1 (n + 1)! n−1∑ j=0 (cj+1 − cj) n+1 (∣∣∣g(n) (cj)∣∣∣ + ∣∣∣g(n) (cj+1)∣∣∣) . proof. for each j = 0, 1, · · ·, n− 1, applying theorem 2.1 over the interval [cj, cj+1] ,we have∣∣∣∣∣∣∣ cj+1∫ cj g (x) dx− 2m−1∑ l=1 n−1∑ k=0   ( 1 + (−1)k ) 2−m(k+1) (cj+1 − cj) −k−1 (k + 1)! (3.4) ×g (k) ( (2l − 1) cj + (2m − 2l + 1) cj+1 2m )]∣∣∣∣ ≤ (cj+1 − cj) n+1 2mn (n + 1)! [∣∣g(n) (cj)∣∣ + ∣∣g(n) (cj+1)∣∣ 2 ] . note that e (g, d) = d∫ c g (x) dx−a (g, d) = n−1∑ j=0     cj+1∫ cj g (x) dx− 2m−1∑ l=1 n−1∑ k=0 ( 1 + (−1)k ) 2−m(k+1) (cj+1 − cj) −k−1 (k + 1)! ×g (k) ( (2l − 1) cj + (2m − 2l + 1) cj+1 2m )]) . using the triangle inequality and inequality 3.4, we get that |e (g, d)| ≤ 1 2mn+1 (n + 1)! n−1∑ j=0 (cj+1 − cj) n+1 (∣∣∣g(n) (cj)∣∣∣ + ∣∣∣g(n) (cj+1)∣∣∣) . � the second application will be devoted to random variables. int. j. anal. appl. 18 (1) (2020) 61 proposition 3.2. let x be a random variable taking its values in the finite interval [c, d] , where 0 < c < d, with a probability density function g : [c, d] → [0, 1]. if g is n-time differentiable, g(n) ∈ l1 ([c, d]) and∣∣g(n)∣∣ is convex on [c, d] , then for any positive integer m,∣∣∣∣∣∣e (x) −  d− 2m−1∑ l=1 n∑ k=1   ( 1 + (−1)k ) 2−m(k+1) (d− c)−k−1 (k + 1)! (3.5) ×g (k−1) ( (2l − 1) c + (2m − 2l + 1) d 2m )) − 2−m+1 (d− c)−1 2m−1∑ l=1 pr ( x ≤ (2l − 1) c + (2m − 2l + 1) d 2m ) ∣∣∣∣∣∣ ≤ (d− c)n+2 2m(n+1) (n + 2)! [∣∣g(n) (c)∣∣ + ∣∣g(n) (d)∣∣ 2 ] , where e (x) is the expectation of x. proof. let g (x) = ∫ x c g (t) dt for x ∈ [c, d]. using integration by parts and the facts that g′ = g and g (d) = 1, we get that e (x) = d− ∫ d c g (t) dt. since g(k+1) = g(k), 0 ≤ k ≤ n, applying theorem 2.1 on g, we have∣∣∣∣∣∣ ∫ d c g (t) dt− 2m−1∑ l=1 n∑ k=1   ( 1 + (−1)k ) 2−m(k+1) (d− c)−k−1 (k + 1)! (3.6) ×g (k−1) ( (2l − 1) c + (2m − 2l + 1) d 2m )] − 2−m+1 (d− c)−1 2m−1∑ l=1 pr ( x ≤ (2l − 1) c + (2m − 2l + 1) d 2m )∣∣∣∣∣∣ ≤ (d− c)n+2 2m(n+1) (n + 2)! [∣∣g(n) (c)∣∣ + ∣∣g(n) (d)∣∣ 2 ] . thus, ∣∣∣∣∣∣e (x) −  d− 2m−1∑ l=1 n∑ k=1   ( 1 + (−1)k ) 2−m(k+1) (d− c)−k−1 (k + 1)! ×g (k−1) ( (2l − 1) c + (2m − 2l + 1) d 2m )) − 2−m+1 (d− c)−1 2m−1∑ l=1 pr ( x ≤ (2l − 1) c + (2m − 2l + 1) d 2m ) ∣∣∣∣∣∣ ≤ (d− c)n+2 2m(n+1) (n + 2)! [∣∣g(n) (c)∣∣ + ∣∣g(n) (d)∣∣ 2 ] . � int. j. anal. appl. 18 (1) (2020) 62 remark 3.1. in proposition 3.2, if m = 1 then identity 3.5 reduces to∣∣∣∣e (x) −d + (d− c) pr ( x ≤ c + d 2 ) (3.7) + n∑ k=1 ( 1 + (−1)k ) 2−(k+1) (d− c)−k−1 (k + 1)! g (k−1) ( c + d 2 )∣∣∣∣∣∣ ≤ (d− c)n+2 2(n+1) (n + 2)! [∣∣g(n) (c)∣∣ + ∣∣g(n) (d)∣∣ 2 ] . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. cerone, s.s. dragomir, j. roumeliotis and j. sunde, a new generalization of the trapezoid formula for n-time differentiable mappings and applications, demonstr. math. 33 (4) (2000), 719–736. [2] x. l. cheng, improvement of some ostrowski-grüss type inequalities, computers math. appl. 42 (2001), 109-114. [3] l. chun and f. qi, integral inequalities of hermite-hadamard type for functions whose third derivatives are convex, j. inequal. appl. 2013 (2013), art. id 451. [4] s. s. dragomir and s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and for home numerical quadrature rules, computers math. appl. 33 (11) (1997), 15-20. [5] d. y. hwang, some inequalities for n-time differentiable mappings and applications, kyung. math. j. 43 (2003), 335-343. [6] j. l. w. v. jensen, on konvexe funktioner og uligheder mel lem middlvaerdier, nyt. tidsskr. math. b. 16 (1905), 49-69. [7] a. i. kechriniotis and y. a. theodorou, new integral inequalities for n-time differentiable functions with applications for pdfs, appl. math. sci. 2 (8) (2008), 353-362. [8] a. ostrowski, uber die absolutabweichung einer differentierbaren funktion von ihren integralmittelwert, comment. math. helv., 10 (1938), 226-227. [9] m. e. ozdemir and c. yıldız, a new generalization of the midpoint formula for n-time differentiable mappings which are convex, arxiv:1404.5128v1, 2014. [10] b.g. pachpatte, new inequalities of ostrowski and trapezoid type for n-time differentiable functions, bull. korean math. soc. 41 (4) (2004), 633-639. [11] j. e. pečarič, f. proschan, y.l. tong, convex function, partial ordering and statistical applications, academic press, new york, 1991. [12] s. h. wang and f. qi, inequalities of hermite-hadamard type for convex functions which are n-times differentiable, math. inequal. appl. 16 (4) (2013), 1269–1278. 1. introduction 2. main results 3. some applications references international journal of analysis and applications volume 18, number 2 (2020), 194-211 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-194 nonlinear (m,p)-isometric and (2,p)-concave mappings on complex normed spaces el moctar ould beiba1, aydah mohammed ayed al-ahmadi2,∗ 1department of mathematics and computer sciences, faculty of sciences and techniques, university of nouakchott al aasriya, p.o. box 5026, nouakchott, mauritania 2mathematics departement, college of sciences and arts, in al-qurayyat, jouf university, saudi arabia ∗corresponding author: aydahahmadi2011@gmail.com abstract. let s be a self mapping on a complex normed space x. in this paper, we study the class of mappings satisfying the following condition ∑ 0≤k≤m (−1)m−k (m k )∥∥skx − sky∥∥p = 0, for all x, y ∈ x, where m is a positive integer. we prove some of the properties of these classes of mappings. 1. introduction let h be a complex hilbert space and let b(h) be the algebra of all bounded linear operators on h. in the 1990s, agler and stankus [1] studied the following operator. for an operator t ∈ b(h) and a positive integer m, define bm(t) := m∑ k=0 (−1)m−k ( m k ) t∗ktk. (1.1) an operator t ∈ b(h) is said to be m-contractive ( respectively, m-expansive and m-isometric ) if bm(t) ≤ 0 ( respectively, bm(t) ≤ 0 and bm(t) = 0 ) for some positive integer m. clearly, t is an received 2019-12-12; accepted 2020-01-13; published 2020-03-02. 2010 mathematics subject classification. 54e40, 62h86. key words and phrases. (m, p)-isometries; m-isometric operators; expansive and contractive operator. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 194 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-194 int. j. anal. appl. 18 (2) (2020) 195 m-contractive if and only if m∑ k=0 (−1)m−k ( m k )∥∥tkh‖2 ≥ 0, ∀ h ∈h, t is an m-expansive if and only if m∑ k=0 (−1)m−k ( m k )∥∥tkh‖2 ≤ 0, ∀ h ∈h, and t is an m-isometry if and only if m∑ k=0 (−1)m−k ( m k )∥∥tkh‖2 = 0, ∀ h ∈h. agler and stankus [1–3] developed a theory for m-isometric operators with rich connections to toeplitz operators, classical function theory and nonstationary stochastic processes. the topics related to m-isometries are currently being studied intensively (see e.g., [4, 17, 22]). in [5, 11, 14, 20] certain types of operators (composition, multiplication, shift) were considered and some conditions under which these operators are m-isometries were given; let l2(t) be the set of square integrable measurable functions on of the unit circle t = ∂d and h2(t) be the corresponding hardy space. let l∞(t) be the set of bounded measurable functions on t and let h∞(t) := l∞∩h2(h). for φ in l∞(t) of the unit circle t = ∂d, the toeplitz operator tφ with symbol φ on the hardy space h2(t) is given by tφf := p(φf) , (f ∈ h2(t)), where p denotes the orthogonal projection of l2(t) onto h2(t). for φ ∈ l∞(t), a toeplitz operator tφ is m-expansive if and only if m∑ j=0 (−1)m−j ( m j )∥∥tjφk∥∥2 ≤ 0, ∀ k ∈ h2(t), is m-contractive if and only if m∑ j=0 (−1)m−j ( m j )∥∥tjφk∥∥2 ≥ 0, ∀ k ∈ h2(t), and m-isometric if and only if m∑ j=0 (−1)m−j ( m j )∥∥tjφk∥∥2 = 0, ∀ k ∈ h2(t). a generalization of m-isometries to operators on general banach spaces has been presented by several authors in the last years. f. bayart introduces in [7] the notion of (m,p)-isometries on general (real or complex) banach spaces. an operator t on a banach space x is called an (m,p)-isometry if there exists an integer m ≥ 1 and a p ∈ [1,∞), with m∑ k=0 (−1)m−k ( m k ) ‖tkx‖p = 0 (x ∈h). (1.2) int. j. anal. appl. 18 (2) (2020) 196 bayart showed that all basic properties of m-isometries on hilbert spaces (which we should now refer to as (m, 2)-isometries) carry over to (m,p)-isometries on banach spaces and, further, that (m,p)-isometries are never n-supercyclic if x is of infinite dimension and complex. in [18] the authors took of the restriction p ≥ 1. they considered the equation (1.2) for p > 0 real and studied the role of the second parameter p and also discussed the case p = ∞. most results in the literature remain valid, with their existing proofs, for p in this extended range. let x and y be metric spaces. a mapping s : x −→ y is called an isometry if it satisfies dy (sx,sy) = dx(x,y), for all x,y ∈ x, where dx(., .) and dy (., .) denote the metrics in the spaces x and y , respectively. in the paper [9], the authors introduced the concept of (m,q)-isometry for maps on a metric space (x,dx) as : a mapping s : x → x is called an (m,q)-isometry for (m ≥ 1, integer, q > 0 real) if it satisfies m∑ k=0 (−1)k ( m k ) dx ( sm−kx,sm−ky )q = 0, ∀ x,y ∈ x. in [12] the author consider a(m,p)-isometries, where, for an operator a ∈ b(x), t ∈ b(x) (the algebra of bounded linear operators) is a(m,p)-isometric if β(p)m (t,a,x) := m∑ k=0 (−1)m−k ( m k ) ‖atkx‖p = 0 (x ∈x). (1.3) evidently, an i(m,p)-isometry is an (m,p)-isometry; if x = h is a hilbert space, then m∑ k=0 (−1)m−k ( m k ) ‖atkx‖p = 0 ⇐⇒ m∑ k=0 (−1)m−k ( m k ) ‖|a|tkx‖p = 0 (x ∈x). if β (p) m (t,a,x) ≤ 0 ( resp. β (p) m (t,a,x) ≥ 0 ) ,∀ x ∈ x , t is said to be (a,m,p)-expansive ( resp. (a,m,p)contractive ) . we refer the interested reader to [13, 19] for complete details. a mapping s (not necessarily linear) on a normed space x ( [16]) is an (m,p)-isometry ( m ≥ 1 integer and p > 0 real) if, for all x,y ∈x , ∆pm(s; x,y) := m∑ k=0 (−1)m−k ( m k )∥∥skx−sky∥∥p = 0. (1.4) when m = 1, (1.4) is equivalent to ‖sx−sy‖ = ‖x−y‖, ∀ x,y ∈x , and when m = 2, (1.4) is equivalent to ‖s2x−s2y‖p − 2‖sx−sy‖p + ‖x−y‖p = 0, ∀ x,y ∈x . in [15] it was observed that a sequence ( an ) n∈n in r is concave if an+2 − 2an+1 + an ≤ 0, ∀n ∈ n (1.5) int. j. anal. appl. 18 (2) (2020) 197 or equivalently if an ≥ an−1 + an+1 2 , ∀n ∈ n. (1.6) a sequence ( an ) n∈n in r is log concave if a2n ≥ an−1an+1, ∀n ∈ n (1.7) remark 1.1. (1) from ( 1.6 ) , we get that any concave sequence of non negative numbers is log concave. (2) if (an)n∈n is a log concave sequence of non negative numbers then an+1 an ≤ an an−1 , ∀n ∈ n. (1.8) therefore the sequence an+1 an is decreasing sequence of non negative numbers. after a short introduction and some connections with known results in this context, the main results of the paper are presented as follows. in section 2, we will introduce and study some properties of (2,p)concave mappings. the main results of this section are proposition 2.1, proposition 2.3, proposition 2.4 and corollary 2.2. in section,3, a parallel study of the classes of nonlinear (m,p)-isometric mappings are presented. exactly we will give conditions under which:a self mapping s is (m,p)-isometry it becomes (k,p))-isometry for 1 ≤ k ≤ m − 1 (proposition 3.1, proposition 3.2 and corollary 3.1. the product of a (m,p)-isometry and a (k,p)-isometry is a (m + k − 1,p)-isometry for k = 1, 2, 3. a power of (2,p)-isometry is again an (2,p)-isometry.. 2. (2,p)-concave mappings in this section, let (x ,‖.‖) be a complex normad space, s : x −→x is a map. definition 2.1. let s be a self map (not necessary linear) on complex normed space §. s is said to be a (2,p)-concave if s satisfy ∥∥s2x−s2y‖p − 2∥∥sx−sy∥∥p + ∥∥x−y∥∥p ≤ 0, for all x,y ∈x . remark 2.1. it is easy to check that a self map s on complex normed space x is (2,p)-concave if and only if the sequence an = ∥∥snx−sny∥∥p is concave. note the following proposition, which lists some general properties of (2,p)-concave mappings. int. j. anal. appl. 18 (2) (2020) 198 proposition 2.1. let s be a self map on complex normed space x. if s is an (2,p)-concave. then the following statements hold: (1) ∥∥snx−sny∥∥p + (n− 1)∥∥x−y∥∥p ≤ n.∥∥sx−sy∥∥p, x,y ∈x , n = 0, 1, 2, ... (2) ∥∥sx−sy∥∥p ≥ n− 1 n ∥∥x−y∥∥p, n ≥ 1, x,y ∈x . (3) ∥∥sx−sy∥∥p ≥ ∥∥x−y∥∥p for all x,y ∈x . (4) ∥∥sx−sy∥∥ ≤ 2 1p ∥∥x−y∥∥ ∀ x,y ∈ s(x) (the range of s). (5) s is injective. proof. (1) from the assumption that s is a (2,p)-concave, we get ∥∥s2x−s2y∥∥p −∥∥sx−sy∥∥p ≤ ∥∥sx−sy∥∥p −∥∥x−y∥∥p. replacing x by skx and y by sky leads to ∥∥sk+2x−sk+2y∣∣p −∥∥sk+1x−sk+1y∥∥p ≤ ∥∥sk+1x−sk+1y∥∥p −∥∥skx−sky∥∥p, for k ≥ 0. thus means that ∥∥snx−sny∥∥p = ∑ 1≤k≤n ∥∥skx−sky∥∥p −(‖sk−1x−sk−1y∥∥p) + ∥∥x−y∥∥p ≤ n (∥∥sx−sy∥∥p −∥∥x−y∥∥p) + ∥∥x−y∥∥p ≤ n ∥∥sx−sy∥∥p + (1 −n)∥∥x−y∥∥p. hence, ∥∥snx−sny∥∥p + (n− 1))∥∥x−y∥∥p ≤ ∥∥sx−sy∥∥p. (2) form the inequality in (1), it follows ∥∥sx−sy∥∥p ≥ n− 1 n ∥∥x−y∥∥p. (3) by taking n →∞ we get ∥∥sx−sy∥∥p ≥ ∥∥x−y∥∥p. (4) from the fact that s is a (2,p)-concave we get ∥∥s2x−s2y∥∥p ≤ 2∥∥sx−sy∥∥p −∥∥x−y∥∥p ≤ 2∥∥sx−sy∥∥p, ∀ x,y ∈x . int. j. anal. appl. 18 (2) (2020) 199 this means that ∥∥s2x−s2y∥∥p ≤ 2 1p ∥∥sx−sy∥∥p, or equivalently ∥∥sx−sy∥∥p ≤ 2 1p ∥∥x−y∥∥p, ∀ x,y ∈ s(x). (5) the injectivity of s follows immediately from the statement (3). � remark 2.2. from the injectivity of a (2,p)-concave map s, we get, by the assertion (2) of the remark 1.1, that for two elements of x x 6= y the sequence (∥∥sn+1x−sn+1y∥∥p∥∥snx−sny∥∥p ) n is decreasing. corollary 2.1. let s be (2,p)-concave on the normed space x and let s−1 be the inverse map of s as a bijection from x on s(x) (the range of s). then, both s and s−1 are continuous on s(x). proof. the continuity of s and of s−1 on s(x) follow, respectively, from the assertions (4) and (3) in proposition 2.1. � proposition 2.2. let s be (2, p)-concave on the normed space x and r be an isometric mapping of x such that s and r commute. then, rs is a (2, p)-concave mapping on x. proof. the statement follow immediately from the fact ∥∥(rs)2x− (rs)2y∥∥ = ∥∥s2x−s2y∥∥ and ∥∥(rs)x− (rs)y∥∥ = ∥∥sx−sy∥∥, ∀ x,y ∈x . � the proof of the following proposition is taken from the statements (1) and (2) of remark 1.1. for the reader ’s convenience. here we give a proof. proposition 2.3. let s be a self map on complex normed space x such is a (2,p)-concave..then the following assertions holds: (1) ∥∥sx−sy∥∥2p ≥ ∥∥x−y∥∥p∥∥s2x−s2y∥∥p for all x,y ∈x . (2) for each n and x,y ∈ x such that x 6= y,the sequence (∥∥sn+1x−sn+1y∥∥p∥∥tnx−tny∥∥p ) n≥0 is monotonically decreasing to 1. int. j. anal. appl. 18 (2) (2020) 200 proof. (1) since s is a (2,p)-concave map, it follows d ∥∥sx−sy∥∥2p ≥ (∥∥x−y∥∥p + ∥∥s2x−s2y∥∥p 2 )2 ≥ (∥∥x−y∥∥p2 ∥∥s2x−s2y∥∥p2 )2 ≥ ∥∥x−y∥∥p∥∥s2x−s2y∥∥p, ∀ x,y ∈x . (2) by observing that the (2,p)-concavity of s implies that ∥∥sn+1x−sn+1y∥∥p − 2∥∥snx−sny∥∥p + ∥∥sn−1x−sn−1y∥∥p ≤ 0. (2.1) on the other hand, since (∥∥sn−1x−sn−1y∥∥p2 −∥∥sn+1x−sn+1y∥∥p2 )2 ≥ 0, we obtain ∥∥sn−1x−sn−1y∥∥p2 ∥∥sn+1x−sn+1y∥∥p2 ≤ ∥∥sn+1x−sn+1y∥∥p + d∥∥sn−1x−sn−1y∥∥p 2 ≤ ∥∥snx−sny∥∥p (by (2.1)). consequently, ∥∥sn−1x−sn−1y∥∥p∥∥sn+1x−sn+1y∥∥p ≤ ∥∥snx−sny∥∥2p. hence, ∥∥sn+1x−sn+1y∥∥p∥∥snx−sny∥∥p ≤ ∥∥snx−sny∥∥p∥∥sn−1x−sn−1y∥∥p , so the sequence is monotonically decreasing. to calculate its limit in view of the statement (3) of proposition 2.1, divide (2.1) by ∥∥sn−1x−sn−1y∥∥p to get 1 − 2 ∥∥snx−sny)p∥∥sn−1x−sn−1y∥∥p + ∥∥sn+1x−sn+1y∥∥p∥∥snx−tny∥∥p ∥∥snx−sny∥∥p∥∥sn−1x−sn−1y∥∥p ≤ 0. by tanking n tend to infinity we obtain that ∥∥snx−sny∥∥p∥∥sn−1x−sn−1y∥∥p −→ 1 as n →∞. � proposition 2.4. let s be a self map on a normed space x . if s is an bijective (2,p)-concave, then , s−1 is an (2,p)-concave. int. j. anal. appl. 18 (2) (2020) 201 proof. since s is an (2,p)-concave, we have ∥∥s2x−s2y∥∥p − 2∥∥sx−sy∥∥p + ∥∥x−y∥∥p ≤ 0, ∀ x,y ∈x . under the assumption that s is bijective, it follows by replacing x by s−2x and y by s−2y that ∥∥x−y∥∥p − 2∥∥s−1x−s−1y∥∥p + ∥∥s−2x−s−2y∥∥p ≤ 0, ∀ x,y ∈x . therefore s−1 is a (2,p)-concave. � corollary 2.2. let s be a self map on a normed space x . if s is an bijective (2,p)-concave, then , s is isometric mapping. proof. since s is an (2,p)-concave , we have in view of the statement (2) of proposition 2.1 that ∥∥sx−sy∥∥p ≥ ∥∥x−y∥∥p. moreover, since s is a bijective (2,p)-concave, we have s−1 is (2,p)-concave, hence ∥∥s−1u−s−1w∥∥p ≥ ∥∥u−w∥∥p ∀ u,w ∈x . letting u = sx and w = sy, this implies ∥∥sx−sy∥∥p = ∥∥x−y∥∥p, for all x,y ∈x . this means that s is isometric. � 3. some properties of nonlinear (m, p)-isometric mapping in this section, let (x ,‖.‖) be a complex normad space, s : x −→x is a map, m ∈ n and p ∈ (0,∞) is a real number. we define the quantity ∆pm(s; x,y) := ∑ 0≤k≤m (−1)m−k ( m k )∥∥skx−sky∥∥p, (3.1) for all x,y ∈x . definition 3.1. ( [16]) let s : x −→x be a map (not necessary linear). s is said to be an (m,p)-isometric mapping for some positive integer m and p ∈ (0,∞) if s satisfying ∑ 0≤k≤m (−1)m−k ( m k )∥∥skx−sky∥∥p = 0, for all x,y ∈x. int. j. anal. appl. 18 (2) (2020) 202 remark 3.1. (1) a self mapping s on x is an (1,p)-isometry if ‖x−y‖ = ‖sx−sy‖ ∀ x,y ∈x . (2) a self mapping s on x is an (2,p)-isometry if ‖s2x−s2y‖p − 2‖sx−sy‖p + ‖x−y‖p = 0, ∀ x,y ∈x . (3) a self mapping s on x is an (3,p)-isometry if for all x,y ∈x , ‖s3x−s3y‖p − 3‖s2x−s2y‖p + 3‖sx−sy‖p −‖x−y‖p = 0, ∀ x,y ∈x . remark 3.2. the following remarks are obvious consequence of definition 2.1. (1) a (1,p)-isometry is an isometry and vice versa. (2) every isometric mapping is (m,p)-isometric mapping for all m ≥ 1 and p ∈ (0,∞).. theorem 3.1. let s be a self map on complex normed space x. then following statements hold: (1) ∆pm(s; x,y) = ∆ p m−1(s; x,y) − ∆ p m−1(s; sx,sy), (3.2) for all x,y ∈x and m ∈ n. (2) if s is an (m,p)-isometry, then s is an (k,p)-isometry for all integer k with k ≥ m. proof. (1) in view of the standard formula ( m k ) = ( m−1 k ) + ( m−1 k−1 ) for binomial coefficients, it follows that ∆pm(s; x,y) = ∑ 0≤k≤m (−1)k ( m k )∥∥skx−sky‖p = ∥∥x−y∥∥p+ ∑ 1≤k≤m−1 (−1)k ( m k )∥∥skx−sk∥∥p+(−1)m∥∥smx−smy∥∥p = ∥∥x−y∥∥p+ ∑ 1≤k≤m−1 (−1)k ((m− 1 k ) + ( m− 1 k − 1 )∥∥(skx−sky∥∥p + +(−1)m ∥∥smx−smy∥∥p = ∆ p m−1(s; x,y) − ∆ p m−1(s; sx,sy). (2) the statement (2) follows from the statement (1). � int. j. anal. appl. 18 (2) (2020) 203 lemma 3.1. let s be a self map on a complex normed space x, then s is a (2,p)-isometry if and only if ∥∥skx−sky∥∥p −k∥∥sx−sy‖p + (k − 1)∥∥x−y∥∥p = 0, ∀ k = 0, 1, · · · ,x,y ∈x . proof. the proof follows by using a mathematical induction, so we omit it. � following [10], we say that an m-isometry s is strict if m = 1, or m ≥ 2 and s is not an (m− 1)-isometry. examples of strict m-isometries for any m ≥ 2 are provided in [6, proposition 8] (see also [23, example 2.3]). proposition 3.1. let s be a self map on a complex normed x that is a (m,p)-isometric and a (2,p)-concave mapping. then s is a (2,p)-isometric mapping. proof. since s is an (m,p)-isometry, ∆pm(s; x,y) = 0 for all x,y ∈x . by (3.2), we have ∆ p m−1(s; x,y) = ∆ p) m−1(s; sx,sy). therefore ∆ p m−1(s; x,y) = ∆ p) m−1(s; sx,sy) = ∆ p m−1(s; s nx,sny); for n = 1, 2, .... from the assumption that s is a (2,p)-concave it follows in view of the statement (2) of proposition 2.1 that   ∥∥sx−sy∥∥p −∥∥x−y∥∥p ≥ 0 ∥∥x−y∥∥p − 2∥∥sx−sy)p + ∥∥s2x−t2y∥∥p ≤ 0 this means that the sequence (∥∥sn+1x − sn+1y∥∥p −∥∥snx − sny∥∥p) n≥0 is monotonically non-increasing and therefore bounded so, is converges. hence there exists a constant β such that ∥∥sn+1x−sn+1y∥∥p −∥∥snx−sny∥∥p −→ β as n −→∞. note that ∆ p m−1(s; s nx,sny) = ∑ 0≤k≤m−−1 (−1)k ( m− 2 k )[∥∥sn+kx−sn+k∥∥p−∥∥sn+1+kx−sn+1+ky∥∥p]. letting n →∞ in the preceding equality leads to ∆ p m−1(s; s nx,sny) → ∑ 0≤k≤m−2 (−1)k ( m− 2 k ) β = 0. we obtain that ∆ p m−1(s; x,y) = 0. applying the corresponding results of the (m− 1,p)-isometry and (2.p)concave mapping, we obtain that ∆ p m−2(s; x,y) = 0. continue these processes we get ∆ p 2(s; x,y) = 0. consequently, s is a (2,p)-isometric mapping. � int. j. anal. appl. 18 (2) (2020) 204 proposition 3.2. let s be a self mapping on a complex normed space x that is a contractive (∥∥sx−sy∥∥ ≤∥∥x−y∥∥, ∀ x,y ∈x). in addition if s is an (m,p)-isometry then s is an (m− 1,p)-isometry for m ≥ 2. proof. under the assumption that s is a contractive mapping, it follows that, ∥∥sn+1x−sn+1y∥∥p ≤ ∥∥snx−sny∥∥p, ∀ x,y ∈x and n ∈ n. this means that (∥∥snx−sny∥∥p) n∈n is deceasing sequence, so convergent. from the fact that s is an (m,p)-isometry and together (??), we get ∆ p m−1(s; x,y) = ∆ p m−1(s; sx,sy) = ... = ∆ p m−1(s; s nx,sny). on the other hand, we have ∆ p m−1(s; s nx,sny) = ∆pm−2(s; s nx,sny) − ∆pm−2(s; s n+1x,sn+1y), so that ∆ p m−1(s; s nx,sny) = m−2∑ j=0 (−1)j ( m− 2 j )[∥∥sn+jx−sn+jy∥∥p −∥∥sn+1+jx−sn+1+jy∥∥p]. by taking the limit n →∞ in the preceding equality leads to ∆ p m−1(s; s nx,sny)to0. consequently,, ∆ p m−1(s; x,y) = 0 and hence, t is an (m− 1,p)-isometry. � as a consequence of this proposition, we have the following corollary. corollary 3.1. let s be a self mapping on a complex normed space x which is a contractive. then s is an (m,p)-isometry if and only if s is an isometry. lemma 3.2. let s be a self mapping on a complex normed space x . then s is an (2, p)-isometry if and only if ∥∥snx−sny∥∥p −∥∥sn−1x−sn−1y∥∥p = ∥∥sx−sy∥∥p −∥∥x−y∥∥p, for all integer n ≥ 2 and x,y ∈x . proof. the proof is obvious by mathematical induction. � the author in [21] show that if s is a 2-isometric operator on a hilbert space, then sn is a 2-isometric operator. int. j. anal. appl. 18 (2) (2020) 205 theorem 3.2. let s be a self map on complex normed space x . if s is an (2,p)-isometry, then so is tn for all n ∈ n. proof. we will induct on n, the result obviously holds for n = 1. suppose then the assertion holds for n ≥ 2, i.e ∥∥s2nx−s2ny∥∥p − 2∥∥snx−sny∥∥p + ∥∥x−y∥∥p = 0, ∀x,y ∈x . then ∥∥s2n+2x−s2n+2y∥∥p − 2∥∥sn+1x−sn+1y∣∣p + ∥∥x−y∥∥p = ∥∥s2s2nx−s2s2ny∥∥p − 2∥∥sn+1x−sn+1y∥∥p + ∥∥x−y∥∥p = 2 ∥∥s2n+1x−s2n+1y∥∥p −∥∥s2nx−s2ny∥∥p −2 ∥∥sn+1x−sn+1y∥∥p + ∥∥x−y∥∥p = 2 ( 2 ∥∥sn+1x−sn+1y∥∥p−∥∥sx−sy∥∥p)−∥∥s2nx−s2ny∥∥p −2 ( ‖sn+1x−sn+1y ∥∥p+∥∥x−y∥∥p = 2 ∥∥sn+1x−sn+1y∥∥p −∥∥s2nx−s2ny∥∥p −2 ∥∥sx−sy∥∥p + ∥∥x−y∥∥p = 2 ∥∥sn+1x−sn+1y∥∥p−(2∥∥snx−sny∥∥p−∥∥x−y∥∥p) −2 ∥∥sx−syw�p+∥∥x−y∥∥p ≤ 2 ∥∥sn+1x−sn+1y∥∥2 − 2∥∥snx−sny∥∥p − 2∥∥sx−sy∥∥p + 2∥∥‖x−y∥∥p = 2 (∥∥sx−sy∥∥p−∥∥x−y∥∥p)−2∥∥sx−sy∥∥p + 2∥∥x−y∥∥p (by lemma3.2) = 0. thus means that sn is (2,p)-isometric mapping. � the following theorem gives a characterization of (3,p)-isometric mappings. theorem 3.3. let s be a self mapping for a normed space x. then s is an (3,p)-isometric mapping if and only if s satisfies ∥∥snx−sny∥∥p = ∥∥x−y∥∥p + nψ1(s,x,y) + n2ψ2(s,x,y) (3.3) where ψ2(s,x,y) = 1 2 (∥∥s2x−s2y∥∥p − 2∥∥sx−sy‖p + ∥∥x−y‖p) (3.4) and ψ1(s,x,y) = 1 2 ( − ∥∥s2x−s2y∥∥p + 4∥∥sx−sy∥∥p − 3∥∥x−y‖p) (3.5) int. j. anal. appl. 18 (2) (2020) 206 proof. we prove the if part of the theorem. assume that s satisfies (3.3). for n = 3 we obtain ∥∥s3x−s3y∥∥p = ∥∥x−y∥∥p + 3ψ1(s,x,y) + 9ψ2(s,x,y) = ∥∥x−y∥∥p + 3 2 ( − ∥∥s2x−s2y∥∥p + 4∥∥sx−sy∥∥p − 3∥∥x−y∥∥p) + 9 2 (∥∥s2x−s2y‖p − 2∥∥sx−sy∥∥p + ∥∥x−y∥∥p) = ∥∥x−y‖p − 3∥∥s2x−s2y∥∥p − 3∥∥sx−sy∥∥p. it follows that ∥∥s3x−s3y‖p − 3∥∥s2x−s2y∥∥p + 3∥∥sx−sy∥∥p −∥∥x−y∥∥p = 0, so that, s is an (3,p)–isometry. we prove the only if part. assume that s is an (3,p)-isometry. we prove (3.3) by mathematical induction. for n = 1 it is true. assume that (3.3) is true for n and prove it for n + 1. indeed, for all x,y ∈x we have ∥∥sn+1x−sn+1y∥∥p = ∥∥snsx−snsy∥∥p = ∥∥sx−sy‖p + nψ1(s,sx,sy) + n2ψ2(s,sx,sy) = ∥∥sx−sy∥∥p + n 2 ( − ∥∥s3x−s3y∥∥p + 4∥∥s2x−s2y∥∥p − 3∥∥sx−sy∥∥p) + n2 2 (∥∥s3x−s3y∥∥p − 2∥∥s2x−s2y∥∥p + ∥∥sx−sy∥∥p) = ( n2 −n 2 )∥∥s3x−s3y∥∥p −(n2 − 2n)∥∥s2 −s2y∥∥p + ( n2 − 3n + 2 2 )∥∥sx−sy∥∥p. now, using the fact that t is an (3,p)-isometry we can obtained ∥∥sn+1x−sn+1y∥∥p = ( n2 −n 2 )(∥∥x−y∥∥p + 3∥∥s2x−s2y∥∥p − 3∥∥sx−sy∥∥p) + − ( n2 − 2n )∥∥s2x−s2y∥∥p + ( n2 − 3n + 2 2 )∥∥sx−sy∥∥p = ( n2 + n 2 )∥∥s2x−s2y∥∥p + (−2n2 + 2 2 )∥∥sx−sy∥∥p + ( n2 −n 2 )∥∥x−y∥∥p int. j. anal. appl. 18 (2) (2020) 207 = ( n2 + n 2 )(∥∥x−y∥∥p + 2ψ1(s,x,y) + 4ψ2(s,x,y)) + ( −2n2 + 2 2 )(∥∥x−y∥∥p + ψ1(s,x,y) + ψ2(s,x,y)) + (n2 −n 2 )∥∥x−y∥∥p = ∥∥x−y∥∥p + (n + 1)ψ1(s,x,y) + (n + 1)2ψ2(s,x,y). � theorem 3.4. let t,s be a self mappings on a complex normed space x such that ts = st . the following properties hold for all x,y ∈x. (1) if s is an (1,p)-isometry, then ∆pm(ts; x,y) = ∆ p m(t ; x,y). (3.6) (2) if s is a (2.p)-isometry, then ∆ p m+1(ts; x,y) = (m + 1)∆ p m(t ; tsx,tsy) − (m + 1)∆ p m(t, .tx,ty) +∆ p m+1(t ; x,y). (3.7) (3) if s is an ((3,p)-isometry, then ∆m+2(ts; x,y) = ∆m+2(t ; x,y) + (m + 2)(m + 1) 2 [ ∆pm(t; t 2s2x,t2s2y) +∆pm(t ; t 2sx,t2sy) + ∆pm(t ; t 2x,t2y) ] + 5(m + 2) 2 ∆m+1(t ; tsx,tsy) − 1 2 ∆m(t ; tx,ty). (3.8) proof. (1) from the fact that s is an (1,p)-isometry,i.e.; ∥∥skx−sky∥∥p = ∥∥x−y∥∥p ∀ x,u ∈x , and the fact ts = st it follows by elementary calculation that ∆pm(ts; x,y) = ∑ 0≤k≤m (−1)m−k ( m k )∥∥(ts)kx− (ts)ky∥∥p = ∑ 0≤k≤m (−1)m−k ( m k )∥∥sktk −sktky∥∥p = ∑ 0≤k≤m (−1)m−k ( m k )∥∥tkx−tky∥∥p = ∆pm(t; x,y). (2) assume that s is an (2,p)-isometry. then we have by using [19, lemma3.4] that ∥∥skx−sky∥∥p = k∥∥sx−sy∥∥p + (1 −k)∥∥x−y∥∥p, ∀ x,y ∈x , k = 0, 1, · · · . int. j. anal. appl. 18 (2) (2020) 208 a simple calculation shows that βam+1(ts; x,y) = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )∥∥(ts)kx− (ts)ky∥∥p = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )∥∥sktk −sktky∥∥p = ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k )[ k ∥∥stkx−stky∥∥p + (1 −k)∥∥tkx−tky∥∥p] = ∑ 1≤k≤m+1 (−1)m+1−kk ( m + 1 k )∥∥tksx−tksy∥∥p + ∑ 0≤k≤m+1 (−1)m+1−k ( m + 1 k ) (−k + 1) ∥∥tkx−tky∥∥p = (m + 1) ( ∑ 0≤k≤m (−1)m−k ( m k )∥∥tk+1sx−tk+1sy∥∥p)− (m + 1)∆pm(t ; tx,ty) +∆ p m+1(t ; x,y) = (m + 1)∆am(t ; tsx,tsy) − (m + 1)∆ p m(t ; tx,ty) + ∆ p m+1(t ; x,y). (3) assume that s is an (a, 3)-isometry and ts = st. in view of theorem 3.3 we have that βam+2(ts) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥(st)kx− (st)ky∥∥p = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥sktkx−sktky∥∥p = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )[∥∥tkx−tky∥∥p + kψ1(s,tkx,ty) + k2ψ2(s,tkx,ty)] = { ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k )∥∥tkx−tky∥∥p ︸ ︷︷ ︸ i + ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) kψ1(s,t kx,tky) ︸ ︷︷ ︸ j + ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2ψ2(s,t kx,tky) ︸ ︷︷ ︸ k } . int. j. anal. appl. 18 (2) (2020) 209 clearly i = ∆ p m+2(t ; x,y). j = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) kψ1(s,t kx,tky) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k 1 2 ( − ∥∥s2tkx−s2tky∥∥p + 4∥∥stkx−stky∥∥p − 3∥∥tkx−tky‖p) = − 1 2 (m + 2)∆m+1(t ; ts 2x,ts2y) + 2(m + 2)∆m+1(t; tsx,tsy) − 3 2 ∆m+1(t; tx,ty). k = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2ψ2(s,t kx,tky) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥s2tkx−s2tky∥∥p − 2∥∥stkx−stky‖p + ∥∥tkx−tky‖p) = ∑ 0≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥tks2x−tks2y∥∥p − 2∥∥tksx−tksy‖p + ∥∥tkx−tky‖p) = (−1)m+1(m + 2) (∥∥tks2x−tks2y∥∥p − 2∥∥tksx−tksy‖p + ∥∥tkx−tky‖p) + ∑ 2≤k≤m+2 (−1)m+2−k ( m + 2 k ) k2 1 2 (∥∥tks2x−tks2y∥∥p − 2∥∥tksx−tksy‖p + ∥∥tkx−tky‖p) by observing that k2 = k(k − 1) + k and k2 ( m + 2 k ) = (m + 2)(m + 1) ( m k − 2 ) + (m + 2) ( m + 1 k − 1 ) , ; k ≥ 2. k = (m + 2)(m + 1) 2 [ ∆pm(t; t 2s2x,t2s2y) + ∆pm(t ; t 2sx,t2sy) + ∆pm(t; t 2x,t2y) ] + (m + 2) 2 [ ∆ p m+1(t; ts 2x,ts2y) + ∆ p m+1(t ; tsx,tsy) + ∆ p m+1(t; tx,ty) ] . by combining i,j and k we obtain ∆m+2(ts; x,y) = ∆m+2(t ; x,y) + (m + 2)(m + 1) 2 [ ∆pm(t; t 2s2x,t2s2y) + ∆pm(t; t 2sx,t2sy) +∆pm(t; t 2x,t2y) ] + 5(m + 2) 2 ∆m+1(t ; tsx,tsy) − 1 2 ∆m(t; tx,ty). this completes the proof of the theorem. � the proof of the following corollary follows by combing theorem 3.4 and theorem 3.1. corollary 3.2. let t,s be a self mappings on a complex normed space x such that ts = st . if t is an (m,p)-isometry and s is an (n,p)-isometry for n ∈ {1, 2, 3}, then ts is an (m + n− 1,p)-isometry for n ∈{1, 2, 3}. int. j. anal. appl. 18 (2) (2020) 210 theorem 3.5. let s be a self map on a complex normed space x. if s is bijective (m,p)-isometry, then s−1 is an (m,p)-isometry. proof. since s is an (m,p)-isometry it follows that ∆pm(s; x,y) = 0; ∀ x,y ∈ x . taking into account the fact that s is a bijective map we get by direct calculation 0 = ∆pm(s; s −mx,s−my) = ∑ 0≤k≤m (−1)m−k ( m k )∥∥sk−mx−sk−my∥∥p = (−1)m∆pm(s −1; x,y). therefore s−1 is an (m,p)-isometry. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. agler and m. stankus, m-isometric transformations of hilbert space i, integral equations oper. theory, 21 (4) (1995), 383–429. [2] j. agler, m. stankus, m-isometric transformations of hilbert space ii, integral equations oper. theory, 23 (1) (1995), 1–48. [3] j. agler, m. stankus, m-isometric transformations of hilbert space iii, integral equations oper. theory, 24 (4) (1996), 379–421 [4] j. agler, a disconjugacy theorem for toeplitz operators, amer. j. math. 112 (1990), 1-14. [5] d. alpay, h. t. kaptanoglu, toeplitz operators on arveson and dirichlet spaces, integral equations oper. theory, 58 (2007), 1-33. [6] a. athavale, some operator theoretic calculus for positive definite kernels, proc. amer. math. soc. 112 (1991), 701-708. [7] f. bayart, m-isometries on banach spaces, math. nachr. 284 (2011), 2141–2147. [8] t. bermúdez, a. martinón , j. a. noda, products of m-isometries , linear algebra appl. 438 (1) (2013), 80–86. [9] t. bermúdez, a. martinón and v. müller, (m, q)-isometries on metric spaces, j. oper. theory, 72 (2) (2014), 313-329. [10] f. botelho, j. jamison, isometric properties of elementary operators, linear algebra appl. 432 (2010), 357-365. [11] m. cho, s. ota, k. tanahash, invertible weighted shift operators which are m-isometrie; proc. amer. math. soc. 141 (2013), 4241-4247. [12] b. p. duggal, tensor product of n-isometries iii, funct. anal. approx. comput. 4 (2012), 61-67. [13] c. gu, on (m, p)-expansive and (m, p)-contractive operators on hilbert and banach spaces. j. math. anal. appl. 426 (2015), 893–916. [14] k. hedayatian, a class of four-isometries on function spaces, italian j. pure appl. math. 16 (2004), 193-200. [15] f. qi and b. n. guo, monotonicity of sequences involvung convex function and sequence, math. inqual. appl. 9 (2) (2006), 247–254. [16] h. khodaei and a. mohammadi, generalizations of alesandarov problem and mazur-ulam theorem for two-isometries and two-expansive mappings, commun. korean math. soc. 34 (3) (2019), 771-782. [17] e. ko, j. lee, on m-isometric toeplitz operators, bull. korean math. soc. 55 (2018), 367-378. int. j. anal. appl. 18 (2) (2020) 211 [18] p. hoffman, m. mackey and m. ó searcóid, on the second parameter of an (m, p)-isometry, integral equations oper. theory, 71 (2011), 389-405. [19] s. a. mahmoud, on a(m, p)-expansive and a(m, p)-hyperexpansive operators on banach spaces-ii. j. math. comput. sci. 5 (2) (2015), 123–148. [20] s. panayappan, s. k. latha, it some isometric composition operators, int. j. contemp. math. sci. 5 (2010), 615-621. [21] s.m. patel, 2-isometric operators, glasnik mat. 37 (2002), 143–147. [22] p., l., robins, m. composition operators that are m-isometries, houston j. math. 31 (1) (2005), 255-266. [23] v. m. sholapurkar, a. athavale, completely and alternatingly hyperexpansive operators, j. oper. theory, 43 (2000), 43-68. 1. introduction 2. (2,p)-concave mappings 3. some properties of nonlinear (m,p)-isometric mapping references int. j. anal. appl. (2022), 20:11 nonconvex vector optimization and optimality conditions for proper efficiency e. kiyani1,2,∗, s. m. vaezpour1, j. tavakoli2 1department of mathematics and computer science, amirkabir university of technology, 424, hafez avenue, 15914 tehran, iran 2school of arts and sciences, university of british columbia, okanagan 333 university way, kelowna, b.c., canada ∗corresponding author: kiyani-e@ut.ac.ir abstract. in this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. we further investigate q-proper efficiency in a real vector space, where q is some nonempty (not necessarily convex) set. the necessary and sufficient conditions for q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. the scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the q-proper efficiency in vector optimization with and without constraints. 1. introduction many works have been done with vector optimization problems under real linear spaces without any particular topology [1,2,4–14]. however, only a few authors focus on nonconvex vector optimization problems [5,7,13–15]. inspired by this fact, the main purpose of this paper is to study some optimality conditions on q-proper efficiency of general nonconvex optimization problems in a real linear vector space without topology, by using nonlinear scalarization functions. received: aug. 7, 2019. 2010 mathematics subject classification. 90c26, 90c29, 90c30. key words and phrases. vector optimization; algebraic interior; vector closure; gerstewitz’s function. https://doi.org/10.28924/2291-8639-20-2022-11 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-11 2 int. j. anal. appl. (2022), 20:11 efficiency is one of the most important concepts in vector optimization. this concept has been studied in many papers [1,2,4,5,8]. kuhn and tucker and geoffrion [16,17], introduced the concept of proper efficiency. since then, different definitions of proper efficient points have been introduced by the other authors. wang and li [18, 19] studied the benson and borwein proper efficiency in finite-dimensional euclidean spaces. borwein [20] has proposed a definition for extending geoffrion’s concept of proper efficiency to the vector maximization problem in which the domination cone could be any nontrivial, closed convex cone. adán and novo [1,2,8] used the vector closure to define the concept of benson proper efficiency of vector optimization problems and they proved scalarization theorems. also, they investigated weak and proper efficiency of vector optimization problems with generalized convex set-valued maps involving relative algebraic interior and vector closure of ordering cone in linear spaces. ha [21] presented the notion of q-minimal solution of vector optimization problems via topological concepts, where q is some nonempty open (not necessarily convex) cone. q-minimal points were characterized by the hiriart-urruty function. scalarization techniques play a vital role in sketching the numerical algorithms and duality results [5,7,8,10,12,22,23]. during the last three decades, many authors have been interested in extending scalarization approaches in vector optimization. nonlinear scalarization approaches have been widely used as efficient methods to study several optimization problems in recent years. in vector optimization, two types of nonlinear scalarization functions are most widely used, the gerstewitz function [24] and the oriented distance function [21]. the gerstewitz function is a nonlinear scalarization function most commonly used in optimization problems with vector-valued or set-valued maps [5, 13, 15, 22, 25]. this function was introduced by different names such as the gerstewitz function, nonlinear scalarization function, shortage function, and smallest strictly monotonic function, [25–28]. the properties of the gerstewitz function in a topological vector space with a closed convex (solid) cone, have been studied in [26–29]. hernández and rodríguez-marín [30] presented an extension of the gerstewitz function and characterized some topological properties to obtain a nonconvex scalarization and optimality conditions for set-valued optimization problems. the nonconvex separation functional in real linear spaces without considering a topology has been presented by la torre, popovici, and rocca [31,32] . they showed that weakly cone-convex vector-valued functions can be characterized in terms of weakly convexity and weakly quasiconvexity of the gerstewitz scalarization functions. the authors in [13,15] extend the gerstewitz function from the topological spaces to real linear via algebraic concepts. beside the gerstewitz function, the oriented distance function is a common scalarization function in vector spaces introduced by hiriart–urruty [33]. the oriented distance function has been used to study well-posedness and stability for vector optimization problems in [35–38]. the generalized version int. j. anal. appl. (2022), 20:11 3 of the oriented distance function introduced by crespi et al. [35] can be used to characterize optimality conditions of set-valued vector optimization. in this paper, we propose a new definition of distance function by using vector closure. for this a new definition of the boundary set is presented which can be used to define the new form of the oriented distance function. the aim of this work is to provide a necessary and sufficient conditions of q-global borwein vectorial proper efficient (q-gbov) solutions in a real vector space. we use algebraic concepts such as algebraic interior and vectorial closure to define and characterize q-gbov. the necessary and sufficient conditions for q-proper efficient solutions of nonconvex optimization problems are obtained via scalarization by oriented distance function and nonlinear scalarization function in a vector space. as the reader sees, some arguments developed for global borwein’s proper efficiency are still valid for q-gbov. some results in this paper, are the generalization of several results given in [1,2,5,8,13,39]. the remainder of the paper is organized as follows: in section 2, we introduce an algebraic boundary set in a vector space and study its properties. we discuss the notion of q-proper efficiency where q is not necessarily a convex set in section 3. section 4 is devoted to the scalarization functions; we describe a new nonlinear scalarization functions and explain how to use these functions to obtain optimality conditions. finally, constrained problems in real vector spaces have been discussed in section 5. we use the results of previous sections to obtain optimality conditions for the constrained problems without convexity assumption. the results of this paper can be also used to develop a vector optimization on vector spaces, which can be applied to any numerical and theoretical scalar optimization. 2. preliminaries throughout the paper, x and y are real spaces and a is a subset of x. furthermore, we consider k ⊆ y be a pointed convex proper cone which introduces a partial order on y by the equivalence relation y1 ≤ y2 ⇔ y2 − y1 ∈ k. k is called pointed if k ∩ (−k) = {0}. the cone generated by a is denoted by cone(a). moreover, a nonempty set f ⊂ y is said to be free disposal with respect to a convex cone k ⊂ y if f + k = f. the algebraic interior of a and the vectorial closure of a are denoted by cor(a) and vcl(a), respectively and these are defined as follows [1] cor(a) = {x ∈ a : ∀x ′ ∈ x , ∃λ ′ > 0; ∀λ ∈ [0,λ ′ ], x + λx ′ ∈ a}, vcl(a) = {b ∈ x : ∃x ∈ x ; ∀λ ′ > 0, ∃λ ∈ [0,λ ′ ]; b + λx ∈ a}. when cor(a) 6= ∅ we say that a is solid, a is algebraically open if cor(a) = a and a is called vectorially closed if a = vcl(a). it is known that, if cor(k) 6= ∅, then cor(k)∪{0} is a convex cone, in addition cor(k) + k = cor(k) and cor(cor(k)) = cor(k) for solid nontrivial convex cone k [8]. for each q ∈ y , q-vector closure of a in the direction q is denoted by vclq(a) and define as follows: vclq(a) = {x ∈ x : ∀λ ′ > 0, ∃λ ∈ [0,λ ′ ] ; x + λq ∈ a}. 4 int. j. anal. appl. (2022), 20:11 in fact it can be shown that vclq(a) = {x ∈ x : ∃λn ≥ 0, λn → 0 ; x + λnq ∈ a, ∀n ∈ n}. obviously, a ⊂ vclq(a) ⊂∪q∈y vclq(a) = vcl(a). now, for an arbitrary functional h : y →<∪{±∞}, define s(h,r,r) := {y ∈ y : h(y) rr}, ∀r ∈ re, ∀r ∈{≤,<,>,≥}, s(h,r, =) := {y ∈ y : h(y) = r}, ∀r ∈ re ∪{±∞}, for a free disposal q. the following proposition shows that there is e ∈ cork such that vcle(q) + (0, +∞)e = cor(q). proposition 2.1. [15] suppose that q is free disposal with respect to an algebraic solid convex cone k. then vcle(q) + cor(k) = vcle(q) + (0, +∞)e = cor(q), where e ∈ cor(k). the boundary of a subset a of a topological space x is the set of points which can be approached both from a and from the outside of a. more precisely, it is the set of points in the closure of a not belonging to the interior of a. the algebraic boundary of a set a in a vector space can be defined by using algebraic type of interior and closure. definition 2.1. the algebraic boundary of a set a denoted by bd(a) and is defind as follows bd(a) = vcl(a)\cor(a). which is the set of points in the vectorial closure of a not belonging to the algebraic interior of a. it is clear that bd(ta) = tbd(a) for t > 0 because vcl(ta) = tvcl(a) and cor(ta) = tcor(a), cor(ta) = {x ∈ ta : ∀x ′ ∈ x , ∃λ ′ > 0; ∀λ ∈ [0,λ ′ ], x + λx ′ ∈ ta}, cor(ta) = {1 t x ∈ a : ∀x ′ ∈ x , ∃λ ′ > 0; ∀λ ∈ [0,λ ′ ], 1 t x + λ t x ′ ∈ a} which lead to 1 t x ∈ cor(q) for a free disposal q. also, we have vcl(ta) = {b ∈ x : ∃x ∈ x ; ∀λ ′ > 0, ∃λ ∈ [0,λ ′ ]; b + λx ∈ ta}. vcl(ta) = {1 t b ∈ x : ∃x t ∈ x ; ∀λ ′ > 0, ∃λ ∈ [0,λ ′ ]; 1 t b + λx t ∈ a}. proposition 2.2. for q ⊂ y , we have vcl(y\q) = y\cor(q). proof. assume by contradiction that x ∈ vcl(y\q) and x ∈ cor(q). x ∈ vcl(y\q) implies that for all λ ′ > 0 there exist x ′ ∈ x and λ ∈ [0,λ ′ ] such that x + λx ′ ∈ y\q. (2.1) int. j. anal. appl. (2022), 20:11 5 on the other hand, for x ∈ cor(q) we have ∀x ′′ ∈ x, ∃λ ′′ > 0, ∀λ ∈ [0,λ ′′ ], x + λx ′′ ∈ q. (2.2) if we consider x ′′ = x ′ and λ ′ = λ ′′ then we can write ∃λ ′ = λ ′′ > 0, ∀λ ∈ [0,λ ′′ ], x + λx ′′ = x + λx ′ ∈ q, (2.3) which contradicts 2.1. therefore x /∈ cor(q). � from proposition 2.2, it is now clear that bd(y\q) = bd(q). 3. proper efficiency definition of global borwein vectorial proper efficient solutions in vector spaces, for the first time, introduced in [8] where the concept of vector closure has been used. the notion of q-minimal points which q is some nonempty open (not necessarily convex) cone was presented by ha [21]. necessary and sufficient conditions for these q-minimal points were characterized by the hiriart-urruty function. now, we are in the position to introduce a general concept of weak proper efficiency, proper efficiency, and global borwein proper efficiency via algebraic concepts. let y and z be two real spaces that are partially ordered by nontrivial ordering convex cones k and m, respectively. let f : x → y and g : x → z be two maps on x. consider the following unconstrained and constrained problems: (up ) min{f (x) : x ∈ x}, (cp ) min{f (x) : x ∈ x, g(x) ∈ (−m)}, and the following vector optimization problem: (p ) min{f (x) : x ∈ s}, where the feasible set s can be either s = x or s = {x ∈ x; g(x) ∈ (−m)}. definition 3.1. a point x0 ∈ s is called a q-proper efficient solution (q-ef) of (p) if (f (s) − f (x0)) ∩ (−q\{0}) = ∅. (3.1) if q is a solid set and 0 /∈ cor(q), then x0 is called q-weak proper eficient solution (q-wef) of (p) when (f (s) − f (x0)) ∩ (−cor(q)) = ∅. (3.2) 6 int. j. anal. appl. (2022), 20:11 definition 3.2. a point x0 is a q-gbov for (p) with respect to q if vcl(cone(f (s) − f (x0))) ∩ (−q\{0}) = ∅. (3.3) it is easy to see that if x0 is a q-gbov for (p), then x0 is also a (q-ef) and a (q-wef) for (p). 4. scalarization in this section, we will present the necessary and sufficient optimality conditions for q-global borwein vectorial proper efficient solutions of vector optimization problems. a useful approach for solving a vector problem is to reduce it to a scalar problem. in general, scalarization means the replacement of a vector optimization problem by a suitable scalar problem which tends to be an optimization problem with a real valued objective function. the main idea of this section obtained from [13, 15]. in [13] the gerstewitz function is generated by a general convex cone in a real space and the authors investigated some properties of this function such as sub-additive and positively homogeneous. however, similar to [15], in this section we consider the nonconvex separation function which is an extension of the gerstewitz function and generated by a subset of a linear space instead of a convex cone. the main properties of the nonconvex separation functional were extended from the topological framework to the linear setting via suitable algebraic counterparts [15]. now, let e ∈ cor(k). the gerstewitz function heq(y) : y −→r is defined by heq(y) := inf{t ∈r : y ∈ te −q}, (4.1) where q ⊂ y . it has been proved that heq is finite [31, remark 2.3], whenever q is a vectorially closed and algebraic solid proper convex cone, and in this case, one has heq(y) = sup{h(y) : h ∈ q +,h(e) = 1}, ∀y ∈ y , where q+ denotes the positive polar cone of q [5]. in the following, some properties of the gerstewitz function are addressed. one can find [15, theorem 4.1, theorem 4.2]. specifically, these theorems are the generalization of [13, lemma 2.8, lemma 2.9]. theorem 4.1. [15] consider e ∈ y \{0} and ∅ 6= q ⊂ y . we have the following properties of heq i) s(heq, 0,≤) = (−∞, 0]e −vcleq, ii) s(heq, 0,<) = (−∞, 0)e −vcleq, iii) s(heq, 0, =) = (−vcleq)\((−∞, 0)e −vcleq), iv) s(heq, 0,≥) = y\((−∞, 0)e −vcleq). theorem 4.2. [15] consider e ∈ y \{0} and ∅ 6= q ⊂ y . if vcle(q) is a cone, then heq is positively homogeneous. it means that heq(αy) ≤ αh e q(y), where y ∈ y and α > 0. int. j. anal. appl. (2022), 20:11 7 in theorem 4.3, we prove heq is sub-additive whenever q is closed under addition. this theorem will be used in the sequel. theorem 4.3. consider e ∈ y \ {0} and let ∅ 6= q ⊂ y be closed under addition and heq is finite. then heq(y1 + y2) ≤ h e q(y1) + h e q(y2), for all y1,y2 ∈ y , except for these make it indeterminate form ∞−∞. proof. from definition of heq given in ( 4.1) and lemma 3 in [15], we have yi ∈ heq(yi )e −vcle(q), i = 1, 2. we can use the fact that he vcle(q) = heq to obtain yi ∈ hevcle(q)(yi )e −vcle(q), i = 1, 2. then obviously, y1 + y2 ∈ (hevcle(q)(y1) + h e vcle(q) (y2))e −vcle(q), which implies he vcle(q) (y1 + y2) ≤ hevcle(q)(y1) + h e vcle(q) (y2), and this yields heq(y1 + y2) ≤ h e q(y1) + h e q(y2). � definition 4.1. let q ⊂ y . a distance function is defined by d(y,q) = inf{λ ∈ r≥0 : y ∈ vcl(λq)}, and d(y,∅) = +∞. for a set q ⊂ y let the oriented distance function 4q : y → r∪{±∞} be defined as 4q(y) = d(y,q) −d(y,y \q). one can find the main properties of the oriented distance function in topological spaces in [33, 40]. however, here we recall them for conveniences. if y ∈ cor(q), then there exists a sequence λn → 0 such that y ∈ vcl(λnq), thus we get d(y,q) = 0 and then 4q(y) < 0. also, if d(y,q) = 0, then y ∈ cor(q). therefore, we can write y ∈ cor(q) if and only if 4q(y) < 0. moreover, 4q(y) > 0 if and only if y ∈ vcl(q). since bd(y\q) = bd(q) and bd(q) = vcl(q)\cor(q), we have 4q(y) = 0 if and only if y ∈ bd(q). furthermore, it is obvious that 4y\q = −4q . theorem 4.4. for t ∈ (0, +∞), we have 8 int. j. anal. appl. (2022), 20:11 ∆q(ty) = t∆q(y). proof. consider y ∈ y . thus d(ty,q) = inf{λ ≥ 0; ty ∈ vcl(λq)}, from definition of vcl(q), for all µ ′ > 0, there exist x ∈ y such that d(ty,q) = inf{λ ≥ 0; ∃µ ∈ [0,µ ′ ], ty + µx ∈ λq} = inf{λ ≥ 0; ∃µ ∈ [0,µ ′ ], y + µ t x ∈ λ t q} = tinf{ λ t ≥ 0; ∃µ ∈ [0,µ ′ ], y + µ t x ∈ λ t q}. (4.2) therefore, ∆q(ty) = t∆q(y). � in the following theorem, we use the gerstewitz function to obtain the sufficient condition for qgbov. theorem 4.5. let e ∈ cor(k), ∅ 6= q ⊂ y be an algebraically open set which is closed under addition, and let vcle(q) be a cone. if x0 satisfies the following condition heq(f (x) − f (x0)) ≥ 0 ∀x ∈ s, then x0 is a q-gbov for (p). proof. for x ∈ s we have heq(f (x) − f (x0)) ≥ 0. thus, by theorem 4.1, one has f (x) − f (x0) ∈ y\((−∞, 0)e −vcle(q)) ∀x ∈ s. now, let 0 6= y ∈ vcl(cone(f (s) − f (x0))). by definition of vectorial closure, there exist x ∈ y and a sequence of positive real numbers λn such that λn → 0 and y + λnx ∈ cone(f (s) − f (x0)). (4.3) therefore, there are sequences αn ≥ 0 and yn ∈ f (s) such that y + λnx = αn(yn − f (x0)). it is obvious that there exist an n ∈ n such that αn > 0. since heq(yn − f (x0)) ≥ 0, then it implies that heq(y + λnx) ≥ 0. int. j. anal. appl. (2022), 20:11 9 and 0 ≤ heq(y + λnx) ≤ h e q(y) + λnh e q(x). by taking limit n →∞, we have heq(y) ≥ 0. now, by applying theorem 4.1, we conclude that y ∈ y\((−∞, 0)e −vcle(q)). on the other hand, for q ∈ cor(q) there exists λ > 0 such that q − [0,λ]e ⊆ q. it means that q ∈ λe + q ⊆ (0,∞)e + vcle(q). therefore, cor(q) ⊆ (0,∞)e + vcle(q). since q is an algebraically close set, we have q ⊆ (0,∞)e + vcle(q), and −q ⊆ (−∞, 0)e −vcle(q), which implies that y ∈ y\(−q). hence, vcl(cone(f (x) − f (x0))) ∩ (−q\{0}) = ∅ ∀x ∈ s. thus, x0 is a q-gbov for (p). � the following theorems state the necessary condition for a point to become a q-gbov for problem (p). in theorem 4.6, we use the nonconvex separation function to obtain the necessary condition while in theorem 4.7, the oriented distance function has been used. we would like to point out that the oriented distance function is a simple tool to work, thus optimality conditions can be obtained with simple calculations by using the properties of the oriented distance function without any condition on the set q. theorem 4.6. let ∅ 6= q ⊂ y is free disposal with respect to an algebraic solid convex cone k. suppose that there exists e ∈ cor(k) such that x0 is a q-gbov for problem (p), then x0 satisfies the following condition 10 int. j. anal. appl. (2022), 20:11 heq(f (x) − f (x0)) ≥ 0 ∀x ∈ s. proof. let us suppose x0 does not satisfy the condition and we have heq(f (x) − f (x0)) < 0, for some x ∈ s. hence, from theorem 4.1 we get f (x) − f (x0) ∈ (−∞, 0)e −vcle(q). on the other hand, since q is free disposal with respect to k, then by proposition 2.1, one has f (x) − f (x0) ∈ (−∞, 0)e −vcle(q) = −cor(q) ⊂−q\{0}. therefore, f (x) − f (x0) ∈ (f (s) − f (x0)) ∩ (−q\{0}). but as one can see, this contradicts the definition of q-gbov. � theorem 4.7. let e ∈ cork and x0 ∈ s such that x0 is a q−gbov for (p), then 4−q(f (x) − f (x0)) ≥ 0, x ∈ s. (4.4) proof. by contrary suppose that for e ∈ cork there exists x ∈ s such that 4−q(f (x) − f (x0)) < 0. thus, we can write f (x) − f (x0) ∈−cor(q) ⊂−q\{0}, which means f (x) − f (x0) ∈ (f (s) − f (x0)) ∩ (−q\{0}). furthermore, we have f (x) − f (x0) ∈ vcl(cone(f (s) − f (x0))) ∩ (−q\{0}) = ∅, ∀x ∈ s, which contradicts the assumption that x0 is q−gbov. therefore, 4−q(f (x) − f (x0)) ≥ 0. � from theorems 4.5 and 4.6 we conclude the following corollary. corollary 4.1. let e ∈ cor(k), ∅ 6= q ⊂ y is free disposal with respect to an algebraic solid convex cone k and be closed under addition such that cor(q) = q, and let vcle(q) be a cone. then x0 is a q-gbov for problem (p) if and only if x0 satisfies the following condition heq(f (x) − f (x0)) ≥ 0 ∀x ∈ s. int. j. anal. appl. (2022), 20:11 11 in the following theorem, we present necessary and sufficient conditions of q-wef for the problem (p). theorem 4.8. let us assume x0 ∈ s. the point x0 is a q-wef if and only if 4−q(f (x) − f (x0)) > 0, ∀x ∈ s. (4.5) proof. assume that x0 is not q-wef. then one has (f (s) − f (x0)) ∩ (−cor(q)) 6= ∅. hence, there exists f (x) ∈ f (s) such that (f (x) − f (x0)) ∈ (−cor(q)), which implies 4−q(f (x) − f (x0)) < 0. � then this proofs the necessary condition. the sufficient condition follows easily. 5. scalarization and constrained problems constrained problems in real vector spaces were originally studied in [5,8]. in [8, theorem 4.3], the authors showed a relation between hurwicz victorial proper efficient solutions and benson vectorial proper efficient solutions in unconstrained and constrained problems. moreover, the relation between ε-benson vectorial proper efficient solutions in unconstrained and constrained problems studied in [5, theorem 4.12]. here, we use the results of previous theorems to obtain optimality conditions for the constrained problems without convexity assumption. in addition, the relation between solutions of constrained and unconstrained problems will be discussed. definition 5.1. we say that the slater constraint qualification for constraint problems (cp) holds if there exists x ∈ s such that g(x) ∈ (−cor(m)). hereafter, the set of all linear operators from z to y is denoted by o(z,y ), and γ is denoted by γ = {t ∈ o(z,y ) : t (m) ⊆ cor(k)}, where m and k are as above. it is important to know lagrangian mapping l : x × γ −→ y , corresponding to the constrained vector optimization problem is defined by l(x,t ) = f (x) + t (g(x)), where f and g are defined in section 3 and t ∈ γ . by the map l, one can convert (cp) to an unconstrained vector optimization problem min{f (x) + (t ◦g)(x) : x ∈ x}. (5.1) in the following theorem, we discuss q-gbov in corresponding problems. 12 int. j. anal. appl. (2022), 20:11 theorem 5.1. in a constrained vector optimization problem, assume that e ∈ cor(k), ∅ 6= q ⊂ y be an algebraically open set which is free disposal with respect to the algebraic solid convex cone k, and closed under the addition that 0 ∈ vcl(q). let the convex cones k and m are pointed and the slater constraint qualification holds, assume that t (g(x0)) = 0 for x0 ∈ s and t ∈ γ. if x0 is a q-gbov for problem given in 5.1, then x0 is a q-gbov for (cp). proof. as discussed in theorem 4.6, since x0 is a (q-gbov) for problem 5.1, one has heq(f (x) + (t ◦g)(x) − (f (x0) + (t ◦g)(x0))) ≥ 0. from theorem 4.3, we have heq(f (x) − f (x0)) + h e q((t ◦g)(x) − (t ◦g)(x0))) ≥ heq(f (x) + t ◦g)(x) − (f (x0) + (t ◦g)(x0))) ≥ 0. since t (g(x0)) = 0, this implies heq(f (x) − f (x0)) + h e q(t ◦g)(x))) ≥ 0. (5.2) on the other hand, 0 ∈ vcl(q), then by proposition 2.1, we have (t ◦g)(x) ⊆−cor(k) −vcl(q) = (−∞, 0)e −vcle(q) = −cor(q). from theorem 4.1, we deduce that heq((t ◦g)(x)) ≤ 0. (5.3) therefore 5.2 and 5.3 yield heq(f (x) − f (x0)) ≥ 0. thus, by theorem 4.5 it follows that x0 is a (q-gbov) for (cp). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. adán, v. novo, efficient and weak efficient points in vector optimization with generalized cone convexity, appl. math. lett. 16 (2003), 221–225. https://doi.org/10.1016/s0893-9659(03)80035-6. [2] m. adán, v. novo, weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness, eur. j. oper. res. 149 (2003), 641–653. https://doi.org/10.1016/s0377-2217(02)00444-7. [3] x.-h. gong, optimality conditions for henig and globally proper efficient solutions with ordering cone has empty interior, j. math. anal. appl. 307 (2005), 12–31. https://doi.org/10.1016/j.jmaa.2004.10.001. [4] e. hernández, b. jiménez, v. novo, benson proper efficiency in set-valued optimization on real linear spaces, lecture notes in economics and mathematical systems 563, springer, berlin, (2006) 45-59. https://doi.org/10.1016/s0893-9659(03)80035-6 https://doi.org/10.1016/s0377-2217(02)00444-7 https://doi.org/10.1016/j.jmaa.2004.10.001 int. j. anal. appl. (2022), 20:11 13 [5] e. kiyani, m. soleimani-damaneh, approximate proper efficiency on real linear vector spaces, pac. j. optim. 10 (2013), 715-734. [6] e. kiyani, m. soleimani-damaneh, algebraic interior and separation on linear vector spaces: some comments, j. optim. theory appl. 161 (2014), 994–998. https://doi.org/10.1007/s10957-013-0416-3. [7] e. kiyani, s.m. vaezpour, j. tavakoli, optimality conditions for approximate solution of set-valued optimization problems in real linear spaces, twms j. appl. eng. math. 11 (2021), 395-407. [8] m. adán, v. novo, proper efficiency in vector optimization on real linear spaces, j. optim. theory appl. 121 (2004), 515–540. https://doi.org/10.1023/b:jota.0000037602.13941.ed. [9] z.-a. zhou, x.-m. yang, j.-w. peng, �-optimality conditions of vector optimization problems with set-valued maps based on the algebraic interior in real linear spaces, optim. lett. 8 (2014), 1047–1061. https://doi.org/ 10.1007/s11590-013-0620-y. [10] z.-a. zhou, x.-m. yang, scalarization of �-super efficient solutions of set-valued optimization problems in real ordered linear spaces, j. optim. theory appl. 162 (2014), 680–693. https://doi.org/10.1007/ s10957-014-0565-z. [11] e. hernández, b. jiménez, v. novo, weak and proper efficiency in set-valued optimization on real linear spaces, j. convex anal. 14 (2007), 275–296. [12] j. jahn, vector optimization, theory, applications, and extensions, springer, berlin, 2011. [13] j.h. qiu, f. he, a general vectorial ekeland’s variational principle with a p-distance, acta. math. sin.-english ser. 29 (2013), 1655–1678. https://doi.org/10.1007/s10114-013-2284-z. [14] j.-h. qiu, a pre-order principle and set-valued ekeland variational principle, j. math. anal. appl. 419 (2014), 904–937. https://doi.org/10.1016/j.jmaa.2014.05.027. [15] c. gutiérrez, v. novo, j.l. ródenas-pedregosa, t. tanaka, nonconvex separation functional in linear spaces with applications to vector equilibria, siam j. optim. 26 (2016), 2677–2695. https://doi.org/10.1137/16m1063575. [16] a.m. geoffrion, proper efficiency and the theory of vector maximization, j. math. anal. appl. 22 (1968), 618–630. https://doi.org/10.1016/0022-247x(68)90201-1. [17] h.w. kuhn, a.w. tucker, nonlinear programming, in proceedings of the second berkeley symposium on mathematical statistics and probability, university of california press. berkeley. ca. (1951) 481-492. [18] s. wang, z. li, scalarization and lagrange duality in multiobjective optimization, optimization. 26 (1992) 315–324. https://doi.org/10.1080/02331939208843860. [19] s.j. li, y.d. xu, s.k. zhu, nonlinear separation approach to constrained extremum problems, j. optim. theory appl. 154 (2012), 842–856. https://doi.org/10.1007/s10957-012-0027-4. [20] j. borwein, proper efficient points for maximizations with respect to cones, siam j. control optim. 15 (1977), 57–63. https://doi.org/10.1137/0315004. [21] t.x.d. ha, optimality conditions for several types of efficient solutions of set-valued optimization problems, in: p.m. pardalos, t.m. rassias, a.a. khan (eds.), nonlinear analysis and variational problems, springer new york, new york, ny, 2010: pp. 305–324. https://doi.org/10.1007/978-1-4419-0158-3_21. [22] s. khoshkhabar-amiranloo, m. soleimani-damaneh, scalarization of set-valued optimization problems and variational inequalities in topological vector spaces, nonlinear analysis: theory meth. appl. 75 (2012), 1429–1440. https: //doi.org/10.1016/j.na.2011.05.083. [23] z.a. zhou, j.w. peng, scalarization of set-valued optimization problems with generalized cone subconvexlikeness in real ordered linear spaces, j. optim. theory appl. 154 (2012), 830–841. https://doi.org/10.1007/ s10957-012-0045-2. [24] c. gerth, p. weidner, nonconvex separation theorems and some applications in vector optimization, j. optim. theory appl. 67 (1990), 297–320. https://doi.org/10.1007/bf00940478. https://doi.org/10.1007/s10957-013-0416-3 https://doi.org/10.1023/b:jota.0000037602.13941.ed https://doi.org/10.1007/s11590-013-0620-y https://doi.org/10.1007/s11590-013-0620-y https://doi.org/10.1007/s10957-014-0565-z https://doi.org/10.1007/s10957-014-0565-z https://doi.org/10.1007/s10114-013-2284-z https://doi.org/10.1016/j.jmaa.2014.05.027 https://doi.org/10.1137/16m1063575 https://doi.org/10.1016/0022-247x(68)90201-1 https://doi.org/10.1080/02331939208843860 https://doi.org/10.1007/s10957-012-0027-4 https://doi.org/10.1137/0315004 https://doi.org/10.1007/978-1-4419-0158-3_21 https://doi.org/10.1016/j.na.2011.05.083 https://doi.org/10.1016/j.na.2011.05.083 https://doi.org/10.1007/s10957-012-0045-2 https://doi.org/10.1007/s10957-012-0045-2 https://doi.org/10.1007/bf00940478 14 int. j. anal. appl. (2022), 20:11 [25] d.t. luc, theory of vector optimization. lecture notes in economics and mathematical systems 319, springer, berlin, (1989). [26] g.y. chen, x.x. huang, x.g. yang, vector optimization: set-valued and variational analysis, springer-verlag, berlin, (2005). [27] a. göpfert, chr. tammer, c. zălinescu, on the vectorial ekeland’s variational principle and minimal points in product spaces, nonlinear anal.: theory meth. appl. 39 (2000), 909–922. https://doi.org/10.1016/s0362-546x(98) 00255-7. [28] s.j. li, x.q. yang, g.y. chen, nonconvex vector optimization of set-valued mappings, j. math. anal. appl. 283 (2003), 337–350. https://doi.org/10.1016/s0022-247x(02)00410-9. [29] a.göpfert, h. riahi, c. tammer, et al. variational methods in partially ordered spaces, springer-verlag, new york, (2003). [30] e. hernández, l. rodríguez-marín, nonconvex scalarization in set optimization with set-valued maps, j. math. anal. appl. 325 (2007), 1–18. https://doi.org/10.1016/j.jmaa.2006.01.033. [31] d. la torre, n. popovici, m. rocca, scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions, nonlinear anal.: theory meth. appl. 72 (2010), 1909–1915. https://doi.org/10.1016/j.na.2009. 09.031. [32] d. la torre, n. popovici, m. rocca, a note on explicitly quasiconvex set-valued maps, j. nonlinear convex anal. 12 (2011), 113-118. [33] j. b. hiriart-urruty, new concepts in nondifferentiable programming, bull. soc. math. france, 60 (1979) 57-85. [34] j. jahn, scalarization in multi objective optimization, in: p. serafini (ed.), mathematics of multi objective optimization, springer vienna, vienna, 1985: pp. 45–88. https://doi.org/10.1007/978-3-7091-2822-0_3. [35] g.p. crespi, i. ginchev, m. rocca, first-order optimality conditions in set-valued optimization, math. meth. oper. res. 63 (2006), 87–106. https://doi.org/10.1007/s00186-005-0023-7. [36] g.p. crespi, a. guerraggio, m. rocca, well posedness in vector optimization problems and vector variational inequalities, j. optim. theory appl. 132 (2007), 213–226. https://doi.org/10.1007/s10957-006-9144-2. [37] g.p. crespi, m. papalia, m. rocca, extended well-posedness of quasiconvex vector optimization problems, j. optim. theory appl. 141 (2009), 285–297. https://doi.org/10.1007/s10957-008-9494-z. [38] g.p. crespi, m. papalia, m. rocca, extended well-posedness of vector optimization problems: the convex case, taiwan. j. math. 15 (2011), 1545-1559. https://doi.org/10.11650/twjm/1500406363. [39] q. qiu, x. yang, some properties of approximate solutions for vector optimization problem with set-valued functions, j glob optim. 47 (2010) 1–12. https://doi.org/10.1007/s10898-009-9452-9. [40] j.b. hiriart-urruty, tangent cones, generalized gradients and mathematical programming in banach spaces, math. oper. res. 4 (1979), 79–97. https://doi.org/10.1287/moor.4.1.79. https://doi.org/10.1016/s0362-546x(98)00255-7 https://doi.org/10.1016/s0362-546x(98)00255-7 https://doi.org/10.1016/s0022-247x(02)00410-9 https://doi.org/10.1016/j.jmaa.2006.01.033 https://doi.org/10.1016/j.na.2009.09.031 https://doi.org/10.1016/j.na.2009.09.031 https://doi.org/10.1007/978-3-7091-2822-0_3 https://doi.org/10.1007/s00186-005-0023-7 https://doi.org/10.1007/s10957-006-9144-2 https://doi.org/10.1007/s10957-008-9494-z https://doi.org/10.11650/twjm/1500406363 https://doi.org/10.1007/s10898-009-9452-9 https://doi.org/10.1287/moor.4.1.79 1. introduction 2. preliminaries 3. proper efficiency 4. scalarization 5. scalarization and constrained problems references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 129-144 http://www.etamaths.com analysis of discrete mittag leffler functions n.shobanadevi1,∗ and j.jagan mohan2,∗ abstract. discrete mittag leffler functions play a major role in the development of the theory of discrete fractional calculus. in the present article, we analyze qualitative properties of discrete mittag leffler functions and establish sufficient conditions for convergence, oscillation and summability of the infinite series associated with discrete mittag leffler functions. 1. introduction & preliminaries fractional calculus is a mathematical branch investigating the properties of derivatives and integrals of fractional orders. many scientists have paid lot of attention due to its interesting applications in various fields of science and engineering, such as viscoelasticity, diffusion, neurology, control theory and statistics [24]. like the exponential function in the theory of differential equations, mittag leffler function plays an important role in the theory of fractional differential equations. the definition for one parameter mittag leffler function was given by gösta mittag leffler [22]. later, agarwal [1] defined the two parameter mittag leffler function. definition 1. let t ∈ r and α,β ∈ r+. the one and two parameter mittag leffler functions are defined by eα(t) = ∞∑ k=0 tk γ(αk + 1) ,(1.1) eα,β(t) = ∞∑ k=0 tk γ(αk + β) .(1.2) the analogous theory for nabla discrete fractional calculus was initiated by miller & ross [21], gray & zhang [10] and atici & eloe [6], where basic approaches, definitions, and properties of fractional sums and differences were discussed. a series of papers continuing this research has appeared recently [6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 23, 25, 26]. throughout this article, we shall use the following notations, definitions and known results of nabla discrete fractional calculus [6, 25]. for any a, b ∈ r, na = {a,a + 1,a + 2, ...........}, na,b = {a,a + 1,a + 2, ...........,b} where a < b. 2010 mathematics subject classification. 39a10, 39a99. key words and phrases. fractional order; nabla difference; convergence; oscillation; summability; periodicity. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 129 130 shobanadevi and mohan definition 2. for any α, t ∈ r, the α rising function is defined by tα = γ(t + α) γ(t) , t ∈ r\{......,−2,−1, 0}, 0α = 0. we observe the following properties of rising factorial function. lemma 1. assume the following factorial functions are well defined. (1) tα(t + α)β = tα+β. (2) if t ≤ r then tα ≤ rα. (3) if α < t ≤ r then r−α ≤ t−α. definition 3. let u : na → r, α ∈ r+ and choose n ∈ n1 such that n −1 < α < n. (1) (nabla difference) the first order backward difference or nabla difference of u is defined by ∇u(t) = u(t) −u(t− 1), t ∈ na+1, and the nth order nabla difference of u is defined recursively by ∇nu(t) = ∇(∇n−1u(t)), t ∈ na+n. in addition, we take ∇0 as the identity operator. (2) (fractional nabla sum) the αth order fractional nabla sum of u is given by (1.3) ∇−αa u(t) = 1 γ(α) t∑ s=a+1 (t−ρ(s))α−1u(s), t ∈ na where ρ(s) = s− 1. also, we define the trivial sum by ∇−0a u(t) = u(t) for t ∈ na. (3) (r l fractional nabla difference) the αth order riemann liouville type fractional nabla difference of u is given by (1.4) ∇αau(t) = ∇ n [ ∇−(n−α)a u(t) ] , t ∈ na+n. for α = 0, we set ∇0au(t) = u(t), t ∈ na. (4) (caputo fractional nabla difference) the αth order caputo type fractional nabla difference of u is given by (1.5) ∇αa∗u(t) = ∇ −(n−α) a [ ∇nu(t) ] , t ∈ na+n. for α = 0, we set ∇0a∗u(t) = u(t), t ∈ na. the unified definition for fractional sums and differences is as follows. definition 4. let u : na → r, α ∈ r+ and choose n ∈ n1 such that n −1 < α < n. then (1) the αth order nabla fractional sum of u is given by ∇−αa u(t) = 1 γ(α) t∑ s=a+1 (t−ρ(s))α−1u(s), t ∈ na. analysis of discrete mittag leffler functions 131 (2) the αth order r l fractional difference of u is given by (1.6) ∇αau(t) = { 1 γ(−α) ∑t s=a+1(t−ρ(s)) −α−1u(s), α /∈ n1, ∇nu(t), α = n ∈ n1, for t ∈ na+n. theorem 2. (power rule) let α > 0 and µ > −1. then, (1) ∇−αa (t−a)µ = γ(µ+1) γ(µ+α+1) (t−a)µ+α, t ∈ na. (2) ∇αa (t−a)µ = γ(µ+1) γ(µ−α+1) (t−a) µ−α, t ∈ na+n . definition 5. a function u is said to be slowly oscillating if u(t) − u(s) → 0 for any t,s ∈ na, whenever s →∞, t > s, ts → 1. definition 6. a function u is said to be t periodic if u(t + t) = u(t) for all t ∈ na. the positive integer t is called the period of the function u. further t is said to be the basic period if there does not exist a smaller period t1 ∈ z+ such that t1 < t. definition 7. a continuous and bounded function u is said to be s asymptotically periodic if there exists t > 0 such that u(t + t)−u(t) → 0 as t →∞. in this case, we say that t is an asymptotic period of u and that u is s asymptotically t periodic. 2. qualitative properties of discrete mittag leffler functions the definitions for one and two parameter discrete mittag leffler functions are given by atsushi nagai [2] and atici & eloe [8] respectively. definition 8. let t ∈ n0, λ ∈ (−1, 1) and α,β ∈ r+. the one and two parameter discrete mittag leffler functions are defined by fα(λ,t) = ∞∑ k=0 λk tαk γ(αk + 1) ,(2.1) fα,β(λ,t α) = ∞∑ k=0 λk tαk γ(αk + β) .(2.2) lemma 3. we observe the following properties of (2.1) and (2.2). (1) fα,1(λ,t α) = fα(λ,t). (2) f1,1(λ,t 1) = f1(λ,t) = (1 −λ)−t. (3) fα,β(λ,t α) ≥ 0. (4) fα,β(λ,t α) ∼ eα,β(λtα) (t →∞). (5) fα(λ,t) ≥ [1+(α−1)λ (1−λ)2γ(t), t ∈ n2. (6) fα,β(λ,t α) ≤ (1 −λ)−1, 2 ≤ t ≤ β. (7) fα,β(λ,t α) ≥ 1 (1−λ)γ(t), 2 ≤ β ≤ t. proof. the proofs of (1), (2), (3) and (4) follow from (2.1) and (2.2). to prove (5), we consider (2.1). clearly, γ(t + αk) γ(αk + 2) ≥ 1, t ∈ n2. 132 shobanadevi and mohan hence fα(λ,t) = ∞∑ k=0 λk tαk γ(αk + 1) ≥ 1 γ(t) ∞∑ k=0 λk(αk + 1) = [1 + (α− 1)λ (1 −λ)2γ(t) . now we consider (2.2) to prove (6) and (7). clearly, 1 γ(t) ≤ 1 and γ(t + αk) γ(αk + β) ≤ 1, 2 ≤ t ≤ β and γ(t + αk) γ(αk + β) ≥ 1, 2 ≤ β ≤ t. hence fα,β(λ,t α) = ∞∑ k=0 λk tαk γ(αk + β) ≤ ∞∑ k=0 λk = 1 1 −λ and fα,β(λ,t α) = ∞∑ k=0 λk tαk γ(αk + β) ≥ 1 γ(t) ∞∑ k=0 λk = 1 (1 −λ)γ(t) . � theorem 4. the two parameter discrete mittag leffler function has the following properties. (1) fα,β(λ,t α) is monotonically increasing on n0. (2) fα,β(λ,t α) is slowly oscillating on n0. (3) fα,β(λ,t α) is not a periodic function. (4) fα,β(λ,t α) is s asymptotically periodic function on n0,b. proof. let t,s ∈ n0 such that t > s. then t−s = t ∈ z+. consider fα,β(λ,t α) −fα,β(λ,sα) = ∞∑ k=0 λk γ(αk + β) [ tαk −sαk ] = ∞∑ k=0 λk γ(αk + β) [γ(t + αk) γ(t) − γ(s + αk) γ(s) ] = ∞∑ k=0 λk γ(αk + β) [γ(s + t + αk) γ(s + t) − γ(s + αk) γ(s) ] = ∞∑ k=0 λk γ(αk + β) γ(s + αk) γ(s) [(s + t − 1 + kα s + t − 1 )(s + t − 2 + kα s + t − 2 ) ... (s + kα s ) − 1 ] = ∞∑ k=0 λk γ(αk + β) γ(s + αk) γ(s) [( 1 + kα s + t − 1 )( 1 + kα s + t − 2 ) ... ( 1 + kα s ) − 1 ](2.3) > 0. thus, we have s < t ⇒ fα,β(λ,sα) < fα,β(λ,tα). analysis of discrete mittag leffler functions 133 further, letting s →∞ in (2.3), we get fα,β(λ,t α) −fα,β(λ,sα) → 0. hence we have (1) and (2). let t be any positive integer and consider fα,β(λ, (t + t) α) = ∞∑ k=0 λk (t + t)αk γ(αk + β) = ∞∑ k=0 λk γ(t + t + αk) γ(t + t)γ(αk + β) 6= ∞∑ k=0 λk γ(t + αk) γ(t)γ(αk + β) = fα,β(λ,t α). thus, fα,β(λ,t α) is not a t periodic function. letting s →∞ in (2.3), we get (2.4) fα,β(λ, (s + t) α) −fα,β(λ,sα) → 0. since fα,β(λ,t α) is continuous and bounded on n0,b, the proof of (4) is complete. � lemma 5. let α, β and γ ∈ r+. the following are valid. (1) ∇fα(λ,t) = λtα−1fα,α(λ, (t + α− 1)α), t ∈ n0. (2) ∇ [ tβfα,β+1(λ, (t + β) α) ] = tβ−1fα,β(λ, (t + β − 1)α), t ∈ n0. (3) ∇−β0 fα(λ,t) = t βfα,β+1(λ, (t + β) α), t ∈ n0. (4) ∇−γ0 [ tβfα,β+1(λ, (t + β) α) ] = tβ+γfα,β+γ+1(λ, (t + β + γ) α), t ∈ n0. (5) ∇−γ−1 [ (t + 1)β−1fα,β(λ, (t + β) α) ] = (t + 1)β+γ−1fα,β+γ(λ, (t + β + γ) α), t ∈ n0. (6) ∇β0∗fα(λ,t) = λt α−βfα,1(λ, (t + α−β)α), 0 < β < 1, t ∈ n1. (7) ∇γ0 [ tβfα,β+1(λ, (t + β) α) ] = tβ−γfα,β−γ+1(λ, (t + β −γ)α), t ∈ nn . (8) ∇γ−1 [ (t + 1)β−1fα,β(λ, (t + β) α) ] = (t + 1)β−γ−1fα,β−γ(λ, (t + β − γ)α), β 6= γ, t ∈ nn−1. (9) ∇β−1 [ (t+1)β−1fα,β(λ, (t+β) α) ] = λ(t+1)α−1fα,α(λ, (t+α) α), t ∈ nn−1. 134 shobanadevi and mohan proof. consider (1). ∇fα(λ,t) = ∞∑ k=0 λk γ(αk + 1) ∇tαk = ∞∑ k=0 λk γ(αk + 1) [γ(t + αk) γ(t) − γ(t− 1 + αk) γ(t− 1) ] = ∞∑ k=0 λk γ(αk + 1) γ(t− 1 + αk) γ(t− 1) [t− 1 + αk t− 1 − 1 ] = ∞∑ k=1 λk γ(αk) γ(t− 1 + αk) γ(t) = λ ∞∑ k=0 λk γ(αk + α) γ(t− 1 + αk + α) γ(t) = λ ∞∑ k=0 λk γ(αk + α) tαk+α−1 = λtα−1 ∞∑ k=0 λk γ(αk + α) (t + α− 1)αk = λtα−1fα,α(λ, (t + α− 1)α). consider (2). ∇ [ tβfα,β+1(λ, (t + β) α) ] = ∞∑ k=0 λk γ(αk + β + 1) ∇[tβ(t + β)αk] = ∞∑ k=0 λk γ(αk + β + 1) ∇tαk+β = ∞∑ k=0 λk γ(αk + β + 1) [γ(t + αk + β) γ(t) − γ(t− 1 + αk + β) γ(t− 1) ] = ∞∑ k=0 λk γ(αk + β + 1) γ(t− 1 + αk + β) γ(t− 1) [t− 1 + αk + β t− 1 − 1 ] = ∞∑ k=0 λk γ(αk + β) γ(t− 1 + αk + β) γ(t) = ∞∑ k=0 λk γ(αk + β) tαk+β−1 = tβ−1 ∞∑ k=0 λk γ(αk + β) (t + β − 1)αk = tβ−1fα,β(λ, (t + β − 1)α). analysis of discrete mittag leffler functions 135 consider (3). ∇−β0 fα(λ,t) = ∞∑ k=0 λk γ(αk + 1) ∇−β0 t αk = ∞∑ k=0 λk γ(αk + 1) γ(αk + 1) γ(αk + β + 1) tβ+αk (using power rule) = tβ ∞∑ k=0 λk (t + β)αk γ(αk + β + 1) (using lemma 1(1)) = tβfα,β+1(λ, (t + β) α). consider (4). ∇−γ0 [ tβfα,β+1(λ, (t + β) α) ] = ∞∑ k=0 λk γ(αk + β + 1) ∇−γ0 [t β(t + β)αk] = ∞∑ k=0 λk γ(αk + β + 1) ∇−γ0 t αk+β = ∞∑ k=0 λk γ(αk + β + 1) γ(αk + β + 1) γ(αk + β + γ + 1) tαk+β+γ = tβ+γ ∞∑ k=0 λk (t + β + γ)αk γ(αk + β + γ + 1) (using lemma 1(1)) = tβ+γfα,β+γ+1(λ, (t + β + γ) α). consider (5). ∇−γ−1 [ (t + 1)β−1fα,β(λ, (t + β) α) ] = ∞∑ k=0 λk γ(αk + β) ∇−γ−1 [(t + 1) β−1(t + β)αk] = ∞∑ k=0 λk γ(αk + β) ∇−γ−1 (t + 1) αk+β−1 = ∞∑ k=0 λk γ(αk + β) γ(αk + β) γ(αk + β + γ) (t + 1)αk+β+γ−1 = (t + 1)β+γ−1 ∞∑ k=0 λk (t + β + γ)αk γ(αk + β + γ) = (t + 1)β+γ−1fα,β+γ(λ, (t + β + γ) α). consider (6). ∇β0∗fα(λ,t) = ∇ −(1−β) 0 [ ∇fα(λ,t) ] (using definition 3(4)) = ∇−(1−β)0 [ λtα−1fα,α(λ, (t + α− 1)α) ] (using (1)) = λtα+1−β−1fα,β+1−β(λ, (t + α + 1 −β − 1)α) (using (4)) = λtα−βfα,1(λ, (t + α−β)α). 136 shobanadevi and mohan (7) and (8) are obtained by replacing γ by −γ in (4) and (5) respectively. consider (9). ∇β−1 [ (t + 1)β−1fα,β(λ, (t + β) α) ] = ∞∑ k=0 λk γ(αk + β) ∇β−1[(t + 1) β−1(t + β)αk] = ∞∑ k=0 λk γ(αk + β) ∇β−1(t + 1) αk+β−1 = ∞∑ k=1 λk γ(αk + β) γ(αk + β) γ(αk) (t + 1)αk−1 = λ ∞∑ k=0 λk γ(αk + α) (t + 1)αk+α−1 = λ(t + 1)α−1 ∞∑ k=0 λk (t + α)αk γ(αk + α) (using lemma 1(1)) = λ(t + 1)α−1fα,α(λ, (t + α) α). � remark 1. from lemma 5(6), we have (2.5) ∇α0∗fα(λ,t) = λfα(λ,t), 0 < α < 1, t ∈ n1, implies fα(λ,t) is an eigenfunction of the operator ∇α0∗. in other words, fα(λ,t) is a nontrivial solution of the fractional nabla difference equation ∇α0∗u(t) = λu(t), t ∈ n1. remark 2. from lemma 5(9), we have ∇α−1 [ (t + 1)α−1fα,α(λ, (t + α) α) ] = λ(t + 1)α−1fα,α(λ, (t + α) α), t ∈ n1, implies (t+1)α−1fα,α(λ, (t+α) α) is an eigenfunction of the operator ∇α−1. that is, (t+ 1)α−1fα,α(λ, (t+α) α) is the solution of the riemann liouville type fractional nabla difference equation ∇α−1f(t) = λf(t), t ∈ n1. now, we prove that (t + 1)α−1fα,α(λ, (t + α) α) is also slowly oscillating on n0 and s asymptotically periodic on n0,b. for this purpose, let t,s ∈ n0 such that t > s. then t−s = t ∈ z+. now consider analysis of discrete mittag leffler functions 137 u(t) −u(s) = (t + 1)α−1fα,α(λ, (t + α)α) − (s + 1)α−1fα,α(λ, (t + α)α) = (t + 1)α−1 ∞∑ k=0 λk (t + α)αk γ(αk + α) − (s + 1)α−1 ∞∑ k=0 λk (s + α)αk γ(αk + α) = ∞∑ k=0 λk (t + 1)αk+α−1 γ(αk + α) − ∞∑ k=0 λk (s + 1)αk+α−1 γ(αk + α) = ∞∑ k=0 λk γ(αk + α) [ (t + 1)αk+α−1 − (s + 1)αk+α−1 ] = ∞∑ k=0 λk γ(αk + α) [γ(t + αk + α) γ(t + 1) − γ(s + αk + α) γ(s + 1) ] = ∞∑ k=0 λk γ(αk + α) [γ(s + t + αk + α) γ(s + t + 1) − γ(s + αk + α) γ(s + 1) ] = ∞∑ k=0 λk γ(αk + α) γ(s + αk + α) γ(s + 1) [(s + t − 1 + αk + α s + t ) ... (s + αk + α s + 1 ) − 1 ] = ∞∑ k=0 λk γ(αk + α) γ(s + αk + α) γ(s + 1) [( 1 + αk + α− 1 s + t ) ... ( 1 + αk + α− 1 s + 1 ) − 1 ] . (2.6) letting s → ∞ in (2.6), we get u(t) − u(s) → 0, i.e., u(s + t) − u(s) → 0. further, (t + 1)α−1fα,α(λ, (t + α) α) is continuous and bounded on n0,b. hence the proof. finally, we show that (t + 1)α−1fα,α(λ, (t + α) α) is also not a t periodic function. let t be any positive integer and consider u(t + t) = (t + t + 1)α−1fα,α(λ, (t + t + α) α) = (t + t + 1)α−1 ∞∑ k=0 λk (t + t + α)αk γ(αk + α) = ∞∑ k=0 λk (t + t + 1)αk+α−1 γ(αk + α) 138 shobanadevi and mohan = ∞∑ k=0 λk γ(t + t + αk + α) γ(t + t + 1)γ(αk + α) 6= ∞∑ k=0 λk γ(t + αk + α) γ(t + 1)γ(αk + α) = ∞∑ k=0 λk (t + 1)αk+α−1 γ(αk + α) = (t + 1)α−1 ∞∑ k=0 λk (t + α)αk γ(αk + α) = (t + 1)α−1fα,α(λ, (t + α) α) = u(t). 3. convergence & oscillation in the present section we establish sufficient conditions on convergence and divergence of the infinite series (3.1) ∞∑ k=0 λk tαk γ(αk + β) associated with discrete mittag leffler function. the following theorem discusses the convergence of (3.1) using d’alembert’s ratio test. theorem 6. the infinite series (3.1) converges absolutely for each t ∈ na, λ ∈ r and α,β ∈ r+ such that |λ| < 1. proof. consider (3.1). here ak = λ k t αk γ(αk + β) = λk γ(t + αk) γ(t)γ(αk + β) = λk 1 γ(t) γ(αk + t) γ(αk + β) . then ak+1 ak = λ γ(αk + t + α) γ(αk + t) γ(αk + β) γ(αk + β + α) . as k →∞, ∣∣∣ak+1 ak ∣∣∣ = |λ|(αk)(t+α−t)(αk)(β−(β+α)) = |λ|. using d’alembert’s ratio test, the proof is complete. � remark 3. since absolute convergence implies convergence, the infinite series (3.1) converges for |λ| < 1 and diverges for |λ| ≥ 1. divergent series are often classified further into properly divergent, oscillate finitely and oscillate infinitely series [18]. in the following theorems we discuss these properties for (3.1). theorem 7. the infinite series (3.1) diverges properly for each t ∈ na, λ ∈ r and α,β ∈ r+ such that t ≥ β ≥ 1 and λ ≥ 1. proof. consider the infinite series (3.1) with λ ≥ 1. here ak = λ k t αk γ(αk + β) = λk γ(t + αk) γ(t)γ(αk + β) . analysis of discrete mittag leffler functions 139 let sn be the n-th partial sum of the series (3.1). then sn = n∑ k=0 ak = n∑ k=0 λk γ(t + αk) γ(t)γ(αk + β) . since λ ≥ 1 and γ(t+αk) γ(αk+β) ≥ 1 for t ≥ β ≥ 1, we have sn ≥ n∑ k=0 1 γ(t) = (n + 1) γ(t) and hence lim n→∞ sn = +∞. thus, the infinite series (3.1) diverges properly for t ≥ β ≥ 1 and λ ≥ 1. � let λ ∈ r+. replacing λ by −λ in (3.1), we get an alternating series of the form (3.2) ∞∑ k=1 (−1)k−1λk−1 tαk−α γ(αk −α + β) . a general criterion for infinite oscillation and a different version of leibnitz test for alternating series are given in theorems 8 and 9 respectively. theorem 8. [4] if a = ∑∞ k=1(−1) k−1ak, where ak > 0 and ak+1 ak → λ > 1 as k →∞, then a oscillates infinitely. theorem 9. [3] given an alternating series a = ∑∞ k=1(−1) k−1ak, ak > 0, if ak ak+1 can be expressed in the form (3.3) ak ak+1 = 1 + µ k + o ( 1 kp ) , p > 1 then a is convergent if µ > 0, oscillatory µ ≤ 0. using theorems 8 and 9, we now discuss the oscillatory behaviour of (3.2). theorem 10. the alternating series (3.2) oscillates infinitely for each t ∈ na, λ ∈ r and α,β ∈ r+ such that λ > 1. proof. consider (3.2). here ak = λ k−1 t αk−α γ(αk −α + β) = λk−1 γ(t + αk −α) γ(t)γ(αk −α + β) = λk−1 1 γ(t) γ(αk + t−α) γ(αk + β −α) > 0. then ak+1 ak = λ γ(αk + t) γ(αk + t−α) γ(αk + β −α) γ(αk + β) . as k →∞, ak+1 ak = λ(αk)(t−(t−α))(αk)((β−α)−β) = λ. using theorem 8 the proof is complete. � theorem 11. the alternating series (3.2) oscillates finitely for each t ∈ na and α,β ∈ r+. proof. consider (3.2) with λ = 1. here ak = tαk−α γ(αk −α + β) = γ(t + αk −α) γ(t)γ(αk −α + β) = 1 γ(t) γ(αk + t−α) γ(αk + β −α) > 0. 140 shobanadevi and mohan then ak ak+1 = γ(αk + t−α) γ(αk + t) γ(αk + β) γ(αk + β −α) . for large k, ak ak+1 = (αk)((t−α)−t) [ 1 + o ( 1 αk )] (αk)(β−(β−α)) [ 1 + o ( 1 αk )] = 1 + o ( 1 k2 ) . using theorem 9 the proof is complete. � combining all these results, we have corollary 1. let t ∈ na, λ ∈ r and α,β ∈ r+. the infinite series (3.1)  converge, for −1 < λ < 1; diverge properly, for λ ≥ 1 and t ≥ β ≥ 1; oscillate finitely, for λ = −1; oscillate infinitely, for λ < −1. since the radius of convergence of (3.1) is 1, we have the following result on the uniform convergence of (3.1). corollary 2. let t ∈ na, λ ∈ r and α,β ∈ r+. for any 0 < r < 1, the infinite series (3.1) converges uniformly for each λ ∈ [−r,r]. 4. preliminaries on summability the present section contains some basic definitions and results concerning summability theory [11, 20, 3] which will be useful in section 4. the method of convergent series is simply a particular method of associating a definite number called sum denoted by s with the series and using this number in place of the convergent series in calculations. but for the divergent series, this sum does not exist. the problem of divergent series is to associate a number with such a series called sum denoted by s so that it can be used in place of the divergent series in calculations. any definite method by which we can associate a sum with a given divergent series is called the method of summation. the methods of summation are designed primarily for the oscillating series. we now discuss three important summability methods given by abel, borel and le roy [3, 11, 20]. let k ∈ n0 and ak,λ ∈ c. consider a divergent series (4.1) ∞∑ k=0 ak. abel’s method: [3, 11, 20] the series (4.1) is said to be abel summable (a summable), if the series (4.2) ∞∑ k=0 akλ k converges in the disk d = {λ : |λ| < 1} and (4.3) lim λ→1− ∞∑ k=0 akλ k = s. analysis of discrete mittag leffler functions 141 then s is called a-sum of the series (4.1) and is denoted by (4.4) ∞∑ k=0 ak = s (a). borel’s method: [3, 11, 20] the series (4.1) is said to be borel summable (b summable), if the series (4.5) e−λ ∑ ak λk k! converges to s and (4.6) ∫ ∞ 0 e−λ ∞∑ k=0 ak λk k! dλ = lim λ→∞ ∫ λ 0 e−λ ∞∑ k=0 ak λk k! dλ = s. the complex number s is called b-sum of the series (4.1) and is denoted by (4.7) ∞∑ k=0 ak = s (b). le roy’s method: [3, 11, 20] the series (4.1) is said to be le roy summable (l summable), if (4.8) lim α→1 lim n→∞ n∑ k=0 ak γ(αk + 1) γ(k + 1) = s, 0 < α < 1. here s is called l-sum of the series (4.1) and is denoted by (4.9) ∞∑ k=0 ak = s (l). finally we conclude this section with two important theorems on a and l summabilities. theorem 12. [3, 11, 20] every convergent series is summable (a, l) with sum equal to sum i.e. abel and le roy methods are regular. theorem 13. [3, 11, 20] a properly divergent series is not summable (a, l) with finite sum. 5. summability of the infinite series (3.1) in this section we discuss summability of (3.1) using the results obtained in section 3 and preliminaries described in section 4. the following corollary is a consequence of theorems 12, 13 and corollary 1. corollary 3. for each t ∈ na and α,β ∈ r+, the infinite series (3.1) is (1) (a, l)summable for −1 < λ < 1. (2) not (a, l) summable with finite sum for λ ≥ 1. (3) uniformly (a, l) summable for each λ ∈ [−r,r] such that 0 < r < 1. theorem 14. for each t ∈ na and α,β ∈ r+, the infinite series (3.1) is a summable for λ = −1. 142 shobanadevi and mohan proof. from corollary 1 we know that (5.1) ∞∑ k=0 (−1)k tαk γ(αk + β) oscillates finitely. also, (5.2) ∞∑ k=0 (−1)k tαk γ(αk + β) λk converges absolutely in the disk d = {λ : |λ| < 1} and (5.3) lim λ→1− ∞∑ k=0 (−1)k tαk γ(αk + β) λk exists and is finite. so, the infinite series (3.1) is a summable with finite sum for λ = −1. � theorem 15. for each t ∈ na, 0 < α < 1 and β = 1, the infinite series (3.1) is l summable for λ < 1. proof. consider the alternating series (5.4) ∞∑ k=0 (−1)k(−λ)k tαk γ(αk + β) with λ ≥ 1. clearly it oscillates finitely for λ = 1 and oscillate infinitely for λ > 1. here ak = (−λ)k tαk γ(αk + 1) . from (4.8), we have s = lim α→1 lim n→∞ n∑ k=0 ak γ(αk + 1) γ(k + 1) = lim α→1 lim n→∞ n∑ k=0 ( t + αk − 1 k ) (−λ)k = ( 1 1 + λ )t exists for each λ > −1. thus (3.1) is l summable for λ < 1. hence the proof. � 6. conclusion taking α = β = 1 in (3.1), we get (6.1) ∞∑ k=0 λk γ(t + k) γ(t)γ(αk + 1) = ∞∑ k=0 ( t + k − 1 k ) λk = ∞∑ k=0 ( −t k ) λk. we know that, theorem 16. for each t ∈ na, the infinite series (6.1) (1)   converges to (1 −λ)−t, for −1 < λ < 1; diverges properly, for λ ≥ 1; oscillates finitely, for λ = −1; oscillates infinitely, for λ < −1. (2) is a summable for −1 ≤ λ < 1 and b summable for λ < 1 (3) is not (a, b) summable with finite sum for λ ≥ 1 (4) is uniformly (a, b) summable for each λ ∈ [−r,r] such that 0 < r < 1. analysis of discrete mittag leffler functions 143 theorem 16 gives a proper justification to corollaries 1, 2, 3, theorems 14 and 15 for α = β = 1. here we note that le roy’s definition coincides with borel’s, whenever the later is convergent [3, 11, 20]. so one can replace b summability by l summability in (2), (3) and (4) of theorem 16. references [1] agarwal, r.p., a propos d’une note de m.pierre humbert, c. r. acad. sci., 236 (1953), 2031 2032. [2] atsushi nagai, discrete mittag leffler function and its applications, publ. res. inst. math. sci., kyoto univ., 1302 (2003), 1 20. [3] bromwich, t.j., an introduction to the theory of infinite series, macmillan, london, 1908. [4] butterworth, i.b., infinite oscillation of alternating series, the mathematical gazette, 34 (1950), number 310, 298 300. [5] elaydi, s., an introduction to difference equations, undergraduate texts in mathematics, 3rd edition, springer, new york, 2005. [6] ferhan m.atici and paul w.eloe, discrete fractional calculus with the nabla operator, electron. j. qual. theory differ. equat., special edition i (2009), number 13, 12 pages (electronic). [7] ferhan m.atici and paul w.eloe, gronwall’s inequality on discrete fractional calculus, computers and mathematics with applications, 64 (2012), 3193 3200. [8] ferhan m.atici and paul w.eloe, linear systems of nabla fractional difference equations, rocky mountain journal of mathematics, 41 (2011), number 2, 353 370. [9] george a.anastassiou, nabla discrete fractional calculus and nabla inequalities, mathematical and computer modelling, 51 (2010), 562 571. [10] gray, h.l. and zhang, n.f., on a new definition of the fractional difference, mathematics of computaion, 50 (1988), number 182, 513 529. [11] hardy, g.h., divergent series, oxford press, london, 1949. [12] hein, j., mc carthy, s., gaswick, n., mc kain, b. and spear, k., laplace transforms for the nabla difference operator, pan american mathematical journal, 21 (2011), number 3, 79 96. [13] jagan mohan, j. and deekshitulu, g.v.s.r., solutions of nabla fractional difference equations using n transforms, commun. math. stat., 2 (2014), 1 16. [14] jagan mohan, j., deekshitulu, g.v.s.r. and shobanadevi, n., stability of nonlinear nabla fractional difference equations using fixed point theorems, italian journal of pure and applied mathematics, 32 (2014), 165 184. [15] jagan mohan, j. and shobanadevi, n., stability of linear nabla fractional difference equations, proceedings of the jangjeon mathematical society, 17 (2014), number 4, 651 657. [16] jagan mohan, j., variation of parameters for nabla fractional difference equations, novi sad j. math., 44 (2014), number 2, 149 159. [17] jagan mohan jonnalagadda, solutions of perturbed linear nabla fractional difference equations, differential equations and dynamical systems, 22 (2014), number 3, 281 292. [18] james m.hylop, infinite series, interscience publishers, london, 1959. [19] jan cermak, tomas kisela and ludek nechvatal, stability and asymptotic properties of a linear fractional difference equation, advances in difference equations 2012 (2012), article id 122. [20] lloyd leroy smail, some generalizations in the theory of summable divergent series, dissertation, columbia university, 1913. [21] miller, k.s. and ross, b., fractional difference calculus, proceedings of the international symposium on univalent functions, fractional calculus and their applications, 139 152, nihon university, koriyama, japan, 1989. [22] mittag leffler, g.m., sur la nouvelle fonction eα(x), c. r. acad. sci. paris, 137 (1903), 554 558. [23] nihan acar, ferhan m.atici, exponential functions of discrete fractional calculus, applicable analysis and discrete mathematics, 7 (2013), 343 353. [24] podlubny, i., fractional differential equations, academic press, san diego, 1999. [25] thabet abdeljawad, fahd jarad and dumitru baleanu, a semigroup like property for discrete mittag leffler functions, advances in difference equations, 2012 (2012), article id 72. 144 shobanadevi and mohan [26] thabet abdeljawad and ferhan m.atici, on the definitions of nabla fractional operators, abstract and applied analysis, 2012 (2012), article id 406757. 1fluid dynamics division, school of advanced sciences, vit university, vellore 632014, tamil nadu, india 2department of mathematics, birla institute of technology and science pilani, hyderabad campus, hyderabad 500078, telangana, india ∗corresponding author international journal of analysis and applications volume 16, number 6 (2018), 921-948 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-921 received 2018-07-29; accepted 2018-09-09; published 2018-11-02. 2010 mathematics subject classification. 91b26. key words and phrases. strategic alliance; grey forecasting model (gm); data envelopment analyses (dea); vietnam fertilizer industry. ©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 921 an application of grey system theory and dea in strategic alliance in vietnamese agricultural industry thanh-tuyen tran* scientific research office, lac hong university, no. 10 huynh van nghe, bien hoa city, dong nai province, vietnam *corresponding author: copcoi2@gmail.com abstract. collaboration is at the heart of every business success [1]. indeed, every aspect of a business is dependent on a partnership one way or another. however, successful partnerships require a lot of factors and efforts from both sides in order to assure the necessary cooperation needed to harness the respective potency of each partner ([2]; [3]; [4]). therefore, this study aims to develop tools which are grey theory and dea models generate the effectiveness of enterprises in vietnamese agricultural industry then offer an effective way to figure out the most suitable strategic partners. the most influenced enterprises are selected to collect realistic data from financial reports of vietnam issued stock market in four consecutive financial years. the targeted decision making unit (dmu) has some potential partner for collaboration in the future, but they are also advised to stay away with some dmus, which may make them even weaker after doing alliance. although this research is specifically applied to the fertilizer industry, the proposed method could also be applied to other manufacturing industries. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-921 int. j. anal. appl. 16 (6) (2018) 922 i. introduction the fertilizer industry development relies on low labor costs, efficiency, large system of foreign exchange, an easy import and export procedures for exporters and the open policies for foreign investors ([5]; [6]). currently, the fertilizer industry is facing more challenges such as how to maintain their competitiveness in today’s fierce market, to diversify products, and divert from processing into other forms which can bring more advantages for the industry ([7]; [8]). in specific, there are three major problems: equipment and modern technology selection, maintaining a stable and capable workforce and floating capital. the problems cannot be overcome when firms are doing individually [9]. we would recommend finding the alliance partners for companies to solve those existing problems by combining data envelopment analysis (dea) and grey theory. since errors in information are unavoidable, consequently, grey theory and dea model are hired to forecast the business in the future and productively evaluatethe performance in firm’s efficiency ranking [10]. the purpose of this research is to provide an assessment model based on grey theory gm (1, 1) and data envelopment analysis (dea) and suggest an appropriated establishment of partnership after many thoughtful considerations. ii. reseach methodology 2.1 grey forecasting model and data envelopment analysis in grey system theory, gm (n, m) denotes a grey model, where n is the order of the difference equation and m is the number of variables ([11]; [12]). although various existing types of grey models can be applied for forecasting, most of researchers, lecturers have paid focused on gm (1, 1) models in their prediction method due to its computational efficiency ([13]; [14]). it should be noted that in real time applications, with the complex data sets, the reduction in the computing time is even more important than the rest of parameters ([15]; [16]; [17]; [18]). int. j. anal. appl. 16 (6) (2018) 923 gm (1, 1) is applied with the purpose of a forecasting for a series of time. and it can only been applied in non-negative data sequences, in this analysis, future values of the original data points can be predicted by grey model because they are positive. during recent years, some models have been presented to solve negative data in dea models. however, they do not discriminate between efficient dmus and only evaluate them as being efficient. in this part, we propose a model by which we discriminate between such dmus it is “slacks – based measure of efficiency” (smb) introduced by tone [19]. then, we extend the “slack – based measure of supper – efficiency” (super – sbm) for dea model with positive and negative inputs and outputs. in this model with n dmus with the input and output matrices ( ) m nijx x r  =  and ( ) s nijy y r  =  , respectively.  is a non-negative vector in nr . the vectors m s r −  and ss r+  indicate the input excess and output shortfall respectively. sbm model in fractional form is as follows [19]: 01 01 1 1/ mmin = 1 1 / m i ii s i ii s x s y s  − = − = +   s.t 0 ,x x s − = + 0 ,y y s + = − +0, s 0, s 0.    let an optimal solution for sbm be * * * *( , , , )p s s − + . a dmu 0 0 ( , )x y is sbm-efficient, if * 1p = . this condition is equivalent to * 0s − = and 0s + = , no input excesses and no output shortfalls in any optimal solution. sbm is non-radial and deals with input/output slacks directly. the sbm returns and efficiency measure between 0 and 1. the top onehave the full effective status indicated by unity. according to super-sbm model by tone [20], assuming that the dmu 0 0 ( , )x y is sbm-efficient, 1p  = , super-sbm model is as follows: int. j. anal. appl. 16 (6) (2018) 924 01 01 1 / mmin = 1 / m i ii s r rr x x y y s  = =   s.t 1, 0 , n j j j x x =    1, 0 , n j j j y x =    0 0 and y ,y x y  0 y , 0.y y   comparable to other dea models, determine how to deal with negative outputs in model efficiency evaluation is fairly important [21]. but the properly role of negative data is effectiveness measurement, therefore dea-solver pro 4.1 manuel had new change as below let us suppose 0. ro y  it is defined and rry y + + − by   1,..., max 0 , rj rjr j n y y y + = =    1,..., min 0 . rj rjr j n y y y + = =  (1) 1 rr y y + + − = = , the term is replaced by ( ) 0 / r rr r rr y y y s y y + + + − − + + − − ( ) ( ) 2 0 / r r rr y s b y y + −+ + − , where b is a large positive number, (in dea-solver b=100). 2.2 development of research in this study, grey theory and dea model are combined in a group of methodical evaluation models. the development of research in this paper is implemented by the data information of vietnamese fertilizer industry and also selected all related documentations as references. then after subject confirming and proceeding industrial analysis, the development of this study is presented in figure 1 as below: int. j. anal. appl. 16 (6) (2018) 925 figure 1: study development int. j. anal. appl. 16 (6) (2018) 926 iii. applicable case result and analysis 3.1 data collection to apply the research on grey forecasting model and dea literature review, three main participations are selected as fixed assets, cost of goods sold, operating costs which are essential to the sources of fertilizer industry. and we select the net sales, operating profit, net profits as our output factors owing to the essential index to analyze the company’s financial effectiveness. we show the realistic data of 2016 which are gained from the financial statement that they are selected at vietnam issued stock market website with the vietnam currency unit. the companies are listed in table 1. table 1: companies list number order code companies 1 a petrovietnam fertilizer and chemicals corporation 2 b petrovietnam ca mau fertilizer jsc 3 c binhdien fertilizer jsc 4 d lam thao fertilizers and chemicals jsc 5 e the southern fertilizers jsc 6 f quang binh import and export jsc 7 g ninhbinh phosphate fertilizer jsc 8 h central petrovietnam fertilizer and chemicals jsc 9 i south-east petrovietnam fertilizer & chemicals jsc 10 k van dien fused magnesium phosphate fertilizer jsc 11 n south-west petrovietnam fertilizer and chemicals jsc to apply the research on grey forecasting model and dea literature review, three main participations are selected as fixed assets, cost of goods sold, operating costs which are essential to the sources of fertilizer industry. and we select the net sales, operating profit, net profits as our output factors owing to the essential index to analyze the company’s financial effectiveness. we show the realistic data of 2016 which are gained from the financial statement that they are selected at vietnam issued stock market website with the vietnam currency unit. int. j. anal. appl. 16 (6) (2018) 927 table 2: input and output factors of companies in fertilizer industry in 2016 company input (units: volume million, $thousand) input (units: volume million, $thousand) fix assets cost of goods sold operating cost net sales net profits operating profit a 1,910,477 5,528,946 1,248,517 7,924,787 1,164,775 1,385,216 b 8,754,407 3,595,508 963,306 4,910,171 624,340 632,709 c 742,125 5,038,820 489,927 5,942,917 350,100 421,064 d 193,750 3,233,437 562,608 3,964,661 138,150 171,686 e 150,386 2,105,100 149,510 2,338,362 90,589 102,510 f 272,675 4,300,199 224,435 4,495,270 13,561 16,690 g 9,559 447,691 75,801 546,139 19,334 23,145 h 45,939 1,910,249 60,932 1,997,252 25,168 31,289 i 35,167 2,071,763 69,801 2,165,958 23,353 26,457 k 16,853 689,058 176,225 907,609 44,432 54,398 n 31,797 2,153,810 56,339 2,237,995 28,117 35,149 s o u r c e s : f i n a n c i a l s t a t e m e n t s o f c o m p a n i e s the grey model (1, 1) is utilized to predict the input and output factors values for each decision making unit in 2016 and 2017. in the table 2, we take the total deposits of dmu1 as an example to explain how to calculation. other variables are calculated in the same way. in this research, we use 5 periods of data (2012-2016) to forecast the input and output variables value in 2017 and 2018. here, we select the fixed assets of company a as example to calculate in detail the procedure as following (table 3 and table 4). int. j. anal. appl. 16 (6) (2018) 928 table 3: inputs and outputs data of all dmus in 2017 company fixed assets cost of goods sold operating costs net sales net profits operating profit a 1,685,963.9 0 5,458,073.0 4 1,328,062.5 6 7,787,165.4 2 898,571.13 1,119,514.3 5 b 8,276,456.7 4 3,173,383.2 0 1,125,406.6 9 4,671,239.0 4 711,194.13 736,011.19 c 895,353.32 4,770,331.4 5 529,771.77 5,687,218.0 4 366,951.03 437,892.40 d 202,773.07 3,262,600.7 2 592,196.17 3,969,420.5 3 156,268.90 194,096.28 e 107,026.31 1,991,256.2 2 129,892.69 2,187,819.0 1 77,099.39 84,327.45 f 343,824.05 5,616,309.6 7 318,580.84 5,883,841.4 6 42,496.76 52,313.94 g 7,996.72 381,027.75 65,632.55 461,389.60 12,934.22 16,429.11 h 41,678.48 1,902,502.9 5 64,176.59 1,983,763.9 2 22,869.33 27,362.72 i 38,872.59 1,858,105.7 1 70,642.69 1,948,927.1 1 21,974.00 24,062.58 k 3,458.66 684,922.27 188,073.55 906,731.96 43,991.94 50,259.14 n 37,166.07 2,043,969.4 9 58,719.15 2,127,340.1 8 29,031.92 33,935.78 source: calculating by author int. j. anal. appl. 16 (6) (2018) 929 table 4: inputs and outputs data of all dmu s in 2018 company fixed assets cost of goods sold operating costs net sales net profits operating profit a 1,545,418.0 9 5,075,411.34 1,352,662.8 3 7,232,792.69 740,254.75 926,069.79 b 7,625,272.6 4 2,821,149.52 1,208,493.7 2 4,320,868.44 727,436.45 769,271.25 c 1,083,730.4 0 4,517,465.95 590,576.54 5,483,607.79 401,330.76 475,474.11 d 207,478.12 3,135,209.31 602,198.50 3,747,812.28 117,641.82 144,764.92 e 70,160.26 1,937,956.48 118,505.13 2,113,179.50 70,160.84 74,911.50 f 430,583.44 7,382,882.89 536,768.63 7,716,047.30 41,852.48 51,236.70 g 6,545.77 343,641.55 56,991.30 410,420.12 9,649.57 12,546.64 h 37,782.09 1,782,638.32 69,989.40 1,860,623.01 19,980.84 23,313.15 i 36,493.75 1,622,388.64 69,521.73 1,706,539.05 20,017.98 20,720.02 k 1,636.32 673,631.40 202,180.74 894,452.32 37,039.20 41,643.87 n 39,446.23 1,892,235.76 63,977.69 1,975,691.83 28,427.04 32,268.06 source: calculating by author 3.2 evaluating process table 5 indicated that the forecasting value of dmus are good because most of mape of dmu less than 10% and the mape average of all thirty commercial banks is 10.48% (less than 20%) which confirm gm (1, 1) model suitable in this case study. therefore, this means the results in table 5 have a good reliability. int. j. anal. appl. 16 (6) (2018) 930 table 5: average mape error of dmus company fixed assets cost of goods sold operating costs net sales net profits operating profit average mape of dmus a 4.36 4.87 3.71 4.02 17.88 12.17 7.84 c 2.48 1.05 8.41 2.32 14.30 15.14 7.28 d 8.88 0.70 6.63 0.66 4.61 4.90 4.40 e 4.73 4.45 3.75 5.11 21.41 21.47 10.15 f 24.93 4.73 5.39 4.17 4.66 5.59 7.28 g 53.28 1.26 14.29 1.80 101.01 104.12 45.96 h 4.10 4.59 3.34 4.69 14.97 14.69 7.73 i 0.15 4.58 4.70 4.29 8.43 8.49 4.40 k 12.75 2.79 4.48 2.80 2.38 4.06 4.88 n 37.71 0.98 2.35 1.44 11.15 5.94 9.93 source: calculating by author 3.3 alliance setting-up stages dea expects that the input and output factors must be metis tonicity ([22]; [23]). prior to the procedure of dea analysis, we have to ensure the connection between input and output factors and tonicity ([24]; [25]; [26]; [27]; [28]). therefore, in this paper, we employ pearson correlation analysis to see if our data fits the assumption of dea. correlation coefficient between input and output variables are high than 0.6, which exhibits a highly positive correlation and well complies with the prerequisite condition of the dea model. here, we run the software of super-sbm-i-v by choosing the realistic data of 2016 to rank the companies’ effectiveness before alliances. the empirical results are obtained in the below table. int. j. anal. appl. 16 (6) (2018) 931 table 6: efficiency, ranking before strategic alliances rank dmu score 1 g 1.875656 2 k 1.703822 3 f 1.377278 4 n 1.321511 5 d 1.268671 6 c 1.213142 7 a 1 8 i 0.94212 9 e 0.937486 10 h 0.86823 11 b 0.612298 source: calculating by author here, company eis chosen as target company for alliance considering to the outcome of data ranking of 2016 before strategic alliance by reason of couple of reasons. firstly, company e acquired the point less than 1 all of the period from 2012 2016, implying that they did not have good business performance. subsequently, they should boldly develop their effectiveness by alliance model. secondly, company e is in major position in the fertilizer industry. to implement our empirical research, we combine e with the rest of dmus to reach 21 virtual alliances. finally, we use the software of dea-solver for calculation of super-sbm-i-v model for 21 dmus. table 7 shows the score and ranking results of virtual alliance in 2018. int. j. anal. appl. 16 (6) (2018) 932 table 7: performance ranking of virtual alliance rank dmu score group 1 k 4.44656 2 g 1.887691 3 e + f 1.675027 1 4 n 1.189458 5 e + d 1.178774 1 6 b 1.127153 7 e + k 1.11175 2 8 a 1.098635 9 e + n 1.090146 2 10 e 1.076125 11 e + c 1.053355 2 12 c 1.012596 13 d 1.006937 14 e + a 1 2 15 e + g 0.991406 2 16 e + i 0.976105 3 17 e + h 0.967737 3 18 i 0.932377 19 h 0.918227 20 f 0.902863 21 e + b 0.681111 3 source: calculating by author in this examination, enterprise e is established as the objective enterprise which was positioned as the ten in comparison to the other 11 dmus in 2016.the southern fertilizer jsc (sfg) takes a hand in manufacturing, sale of fertilizer and other chemical products. the company’s main products include nitrogen-phosphorous-potassium (npk) fertilizer, organic int. j. anal. appl. 16 (6) (2018) 933 npk fertilizer, solid and liquid yogen fertilizer, phosphorous fertilizer, sulfuric acid, and agricultural organic minerals among others.the southern fertilizer jsc looks for strategic alliances. as indicated by the positioning of virtual cooperation, the examinations of observational outcomes split into three gatherings and translate as underneath: first, the companies, which acquires brighter outcome after strategic alliance and also put their partnership more effectively, are the first prioritized candidate. both corporationf and d helped the e to develop the result into a higher level after strategic alliance, which can be observed in table 8. table 8: the first priority in alliance strategy rank dmu score group 3 e+f 1.675027 1 5 e+d 1.178774 1 source: calculating by author second, the dmu which increases performance after strategic alliance while other dmu gets worst is the second priority. total five companies in this group are shown in table 9. table 9: the second priority in alliance strategy rank dmu score group 7 e + k 1.11175 2 9 e + n 1.090146 2 11 e + c 1.053355 2 14 e + a 1 2 15 e + g 0.991406 2 source: calculating by author thirdly, the dmus which become worse and worse after strategic alliances are not suggested in this study. it is unnecessary to put in any effort for partnership because no advantages between both candidates and target candidates. table 10 presented 3 companies in the group as below int. j. anal. appl. 16 (6) (2018) 934 table 10: the third priority in alliance strategy rank dmu score group 16 e + i 0.976105 3 17 e + h 0.967737 3 21 e + b 0.681111 3 source: calculating by author the importance of strategic alliance has been consistently emphasized as the key factors of business survival in the era of globalization. it helps companies to reduce risk and easily penetrate into the market. however, it is a big challenge to have a successful strategic alliance. application of a strategic alliance can give rise to less than competitiveness or cause large enterprises to become even larger and small enterprises even smaller. iv. recommendations and conclusions at this moment, more and more competition dramatically arises in fertilizer industry. according to the viet nam fertilizer association, the domestic fertilizer industry has experienced a growth in output, but lacking of competitive ability. the industry still continues to widely apply the usage of old-fashioned production technology while the world’s fertilizer industry uses many modern technologies to reduce production costs. in long term, local fertilizer factories will lose their market shares or even have to dissolve if they do not embrace new creation advancement in technology.although the industry counts around600 companiesbut most of them are small-medium sized.products made in vietnam are low-tomedium quality.supplementary to this, like any existing market, one of the essential challenges is operating the management of the supply chain, in-depth understanding the import requirements and ensuring that the product can be delivered to the customer and/or consumer.input/ output factors fluctuate in different periods, which makes “business future” in uncertain success. therefore, in this research, we propose a new methodology which combines the gm (1, 1) model and dea model to find the right alliance partners for target company under several inputs and outputs. many related subjects of strategic alliance have been already done research by many scholars and experts. however, this study provides firms with a method tolimit the possibilities of risks, int. j. anal. appl. 16 (6) (2018) 935 creates the mode of penetration. but how strategic alliance opens up for firms to be roaring successful is the enormous challenge. this research concentrates on the connection between key collusion and firms' execution of vietnamese fertilizer by using gm (1, 1) model and dea model. this study reaches some conclusions through a series of literature reviews and empirical results. 1. the gm(1, 1) model helps the enterprises to predict what will happen in the future regarding particular elements: fixed assets, cost of goods sold, operating costs, net profits, operating profit ,which are important to the firm’s efficiency in doing business based on the realistic data and information in the past time. however, there are alwaysexistent errors in predicting processes, thus the mape is utilized to ensure whatever collection of inputs or outputs is almost precise or not. in this examination, the range of mape values from 2% to 20%, whichguarantee that gm (1, 1) delivers high accurateness. 2. this study shows that the dea model is based on the resource-based theory. the supersbm model was used to assess the11 firms separately and calculate the operational performance of 21simulated decision making units for strategic alliances. thanks to this methodology, we can simply divide 11 candidates into three groups. in this study, company e, among famous fertilizer companies in vietnam, is an objective company for strategic alliance with the others 10 firms. we observe the two companies which are the best candidates because profits are generated for both sides: target company e and 2 candidate companies due to the effective alliance. this factled to the outstanding efforts from both: collaborative innovation agreement and renewal products. the second priority is a group of companies with five companies and target company should carefully consider when implementing alliance because they can get the risk after strategic alliances. the third group includes companies: e, h & b, which are unnecessarily to be cared because there is no advantage for two alliances. int. j. anal. appl. 16 (6) (2018) 936 references [1] z. s. hassanzadeh, s. r. hosseini and f. honarbakhsh, study of the educational factors contributing to realization of the objectives of entrepreneurial university. int. j. adv. appl. sci., 2(10) (2015), 1-12. [2] j. c. anderson and j. a. narus, a model of distributor firm and manufacturer firm working partnerships. j. market., 54(1990), 42-58. [3] d. t. wilson, an integrated model of buyer-seller relationships. j. acad. market. sci., 23(4) (1995), 335-345. [4] a. walter, t. a. müller, g. helfert and t. ritter, functions of industrial supplier relationships and their impact on relationship quality. industr. market. manag., 32(2) (2003), 159-169. [5] r. pomfret, growth and transition: why has china's performance been so different?. j. comparative econ., 25(3) (1997), 422-440. [6] f. yuruk and p. erdogmus, finding an optimum location for biogas plant: a case study for duzce, turkey. neural comput. appl., 29(1) (2018), 157-165. [7] t. t. tran, forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci., 4(9) (2017), 86-95. [8] t. t. tran, a strategic alliance study by performance evaluation and forecasting techniques: a case in the petroleum industry. int. j. adv. appl. sci., 5(2) (2018), 136-147. [9] m. mendola, agricultural technology adoption and poverty reduction: a propensity-score matching analysis for rural bangladesh. food policy, 32(3) (2007), 372-393. [10] n. t. nguyen and t. t. tran, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci., 5(9) (2018), 73-81. [11] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci., 3(12) (2016), 5-20. [12] n. t. nguyen and t. t. tran, optimizing mathematical parameters of grey system theory: an empirical forecasting case of vietnamese tourism. neural comput. appl., (2017), https://doi.org/10.1007/s00521-0173058-9. [13] s. liu, j. forrest and y. yang, a brief introduction to grey systems theory. grey syst., theory appl., 2(2) (2012), 89-104. [14] y. y. tai, j. y. lin, m. s. chen, and m. c. lin, a grey decision and prediction model for investment in the core competitiveness of product development. technol. forecast. social change, 78(7) (2011), 1254-1267. [15] s. chtourou, m. chtourou and o. hammami, a hybrid approach for training recurrent neural networks: application to multi-step-ahead prediction of noisy and large data sets. neural comput. appl., 17(3)(2008), 245254. [16] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci., 3(12) (2016), 5-20. [17] n. t. nguyen and t. t. tran, mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dynamics in nature and society, 2015. int. j. anal. appl. 16 (6) (2018) 937 [18] n. t. nguyen and t. t. tran, raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl., (2018). https://doi.org/10.1007/s00521018-3639-2. [19] k. a. tone, slacks-based measure of efficiency in data envelopment analysis, eur. j. oper. res., 130(2001), 498-509. [20] k. a. tone, a slacks-based measure of super-efficiency in data envelopment analysis, eur. j. oper. res., 143(2002), 32-41. [21] j. tongzon, efficiency measurement of selected australian and other international ports using data envelopment analysis. transport. res. part a: policy practice, 35(2)(2001), 107-122. [22] w. fan, j. xu, y. wu, w. yu, j. jiang, z. zheng and c. tian, parallelizing sequential graph computations. in proceedings of the 2017 acm international conference on management of data, 2017, (pp. 495-510). acm. [23] m. sozio and a. gionis, the community-search problem and how to plan a successful cocktail party. in proceedings of the 16th acm sigkdd international conference on knowledge discovery and data mining (2010), (pp. 939-948). acm. [24] a. hailu and t. s. veeman, non-parametric productivity analysis with undesirable outputs: an application to the canadian pulp and paper industry. amer. j. agric. econ., 83(3) (2001), 605-616. [25] g. j. morton, d. e. cummings, d. g. baskin, g. s. barsh and m. w. schwartz, central nervous system control of food intake and body weight. nature, 443 (2006), 289–295. [26] j. liu, q. xu, h. chen, l. zhou, j. zhu and z. tao, group decision making with interval fuzzy preference relations based on dea and stochastic simulation. neural comput. appl., (2017), https://doi.org/10.1007/s00521-017-3254-7. [27] a. al-refaie, c. w. wu and m. sawalheh, dea window analysis for assessing efficiency of blistering process in a pharmaceutical industry. neural comput. appl., (2018), https://doi.org/10.1007/s00521-017-3303-2. [28] r. mahmoudi, a. emrouznejad amd m. rasti-barzoki, a bargaining game model for performance assessment in network dea considering sub-networks: a real case study in banking. neural comput. appl., (2018), https://doi.org/10.1007/s00521-018-3428-y. int. j. anal. appl. 16 (6) (2018) 938 apendix input and output factors of target dmus in 2012 companies fix assets cost of goods sold operating cost net sales net profits operating profit a 2,371,392 8,997,366 1,318,093 13,321,852 3,067,647 3,574,740 b 12,436,315 2,967,940 410,051 4,076,182 736,671 730,296 c 425,142 6,869,767 366,813 7,422,968 158,867 197,199 d 219,612 3,495,007 502,080 4,494,851 394,091 509,684 e 563,219 2,544,853 202,360 2,840,282 98,79 119,108 f 31,870 2,302,833 92,291 2,391,848 2,038 2,382 g 19,403 551,271 142,841 770,310 63,697 79,316 h 65,240 2,347,980 43,250 2,440,980 43,649 53,177 i 30,373 3,546,253 64,083 3,649,449 40,376 50,015 k 42,682 659,152 153,577 875,652 68,800 90,547 n 26,853 3,087,222 43,145 3,178,573 53,324 66,068 int. j. anal. appl. 16 (6) (2018) 939 input and output factors of target dmus in 2013 companies fix assets cost of goods sold operating cost net sales net profits operating profit a 2,368,444 7,011,191 1,194,639 10,363,418 2,179,191 2,586,225 b 11,209,745 5,065,121 830,907 6,263,118 531,710 495,135 c 466,150 5,895,935 379,110 6,585,110 261,684 318,832 d 173,294 3,668,449 531,103 4,768,477 446,820 580,370 e 537,410 2,343,321 185,888 2,638,857 115,398 140,514 f 57,542 1,861,569 57,992 1,939,946 21,348 25,760 g 17,316 579,585 112,844 735,370 40,451 46,074 h 61,656 2,447,841 48,183 2,542,168 36,380 48,519 i 44,993 3,218,254 76,630 3,336,440 31,409 42,486 k 76,255 731,509 145,336 959,652 80,543 105,472 n 23,564 2,811,818 44,765 2,890,025 30,394 40,606 int. j. anal. appl. 16 (6) (2018) 940 input and output factors of target dmus in 2014 companies fix assets cost of goods sold operating cost net sales net profits operating profit a 2,295,454 7,121,096 1,276,866 9,548,850 1,134,458 1,557,395 b 11,004,157 4,586,281 840,164 6,044,143 820,887 798,534 c 426,608 5,696,732 336,303 6,377,225 288,549 356,146 d 207,529 3,856,523 587,436 4,985,068 438,723 553,003 e 519,572 1,962,180 180,825 2,237,982 100,898 115,315 f 299,256 2,503,864 72,899 2,655,043 64,419 84,372 g 15,787 533,179 106,828 682,933 36,468 44,322 h 56,177 2,252,616 46,832 2,348,012 40,198 51,855 i 52,297 2,712,487 69,519 2,821,395 29,570 39,741 k 19,131 713,894 144,563 929,122 85,211 87,618 n 38,205 2,470,498 42,643 2,548,198 31,887 40,914 int. j. anal. appl. 16 (6) (2018) 941 input and output factors of target dmus in 2015 companies fix assets cost of goods sold operating cost net sales net profits operating profit a 1,853,676 6,612,424 1,355,133 9,764,947 1,522,461 1,855,678 b 9,848,606 3,950,628 1,145,494 5,582,239 712,460 712,527 c 652,335 5,278,378 425,014 6,037,884 280,234 337,002 d 191,584 3,673,450 590,926 4,651,235 306,285 391,334 e 159,206 2,118,099 141,198 2,337,950 86,046 99,837 f 171,237 3,319,407 113,278 3,516,965 77,278 93,609 g 11,508 427,693 83,221 532,533 17,638 21,653 h 50,728 2,369,227 51,944 2,452,136 27,958 34,392 i 50,050 2,562,297 78,202 2,673,131 27,286 33,264 k 17,974 722,029 164,166 956,801 65,183 81,636 n 34,790 2,519,510 46,905 2,600,069 32,042 37,566 int. j. anal. appl. 16 (6) (2018) 942 input and output factors of target dmus in 2016 companies fix assets cost of goods sold operating cost net sales net profits operating profit a 1,910,477 5,528,946 1,248,517 7,924,787 1,164,775 1,385,216 b 8,754,407 3,595,508 963,306 4,910,171 624,340 632,709 c 742,125 5,038,820 489,927 5,942,917 350,100 421,064 d 193,750 3,233,437 562,608 3,964,661 138,150 171,686 e 150,386 2,105,100 149,510 2,338,362 90,589 102,510 f 272,675 4,300,199 224,435 4,495,270 13,561 16,690 g 9,559 447,691 75,801 546,139 19,334 23,145 h 45,939 1,910,249 60,932 1,997,252 25,168 31,289 i 35,167 2,071,763 69,801 2,165,958 23,353 26,457 k 16,853 689,058 176,225 907,609 44,432 54,398 n 31,797 2,153,810 56,339 2,237,995 28,117 35,149 int. j. anal. appl. 16 (6) (2018) 943 forecasting results for dmus from 2013 to 2016 dmu s fixed assets cost of goods sold operating costs net sales net profits operating profit year s a 2,388,126.86 7,299,812.9 9 1,234,054.8 1 10,463,402.5 1 1,950,911.0 8 2,390,966.2 6 2013 2,189,047.14 6,788,028.1 8 1,256,913.7 3 9,718,506.95 1,607,186.3 0 1,977,823.3 6 2014 2,006,563.15 6,312,124.2 5 1,280,196.0 8 9,026,640.92 1,324,021.2 8 1,636,068.7 7 2015 1,839,291.45 5,869,585.6 1 1,303,909.6 9 8,384,029.22 1,090,746.2 1 1,353,367.0 8 2016 b 11,486,844.5 3 5,080,520.1 3 846,388.72 6,380,816.98 649,771.36 616,742.81 2013 10,583,070.0 2 4,516,601.3 7 908,876.29 5,902,217.90 664,610.90 644,613.17 2014 9,750,403.67 4,015,275.4 9 975,977.22 5,459,516.59 679,789.36 673,742.99 2015 8,983,250.76 3,569,594.9 1 1,048,032.1 0 5,050,020.50 695,314.46 704,189.16 2016 c 417,145.17 5,931,482.1 3 343,035.68 6,580,120.63 256,466.57 315,013.78 2013 504,910.07 5,617,066.4 1 382,407.74 6,344,543.24 280,494.99 342,049.55 2014 611,140.18 5,319,317.2 3 426,298.75 6,117,399.84 306,774.63 371,405.63 2015 739,720.48 5,037,351.1 2 475,227.36 5,898,388.48 335,516.41 403,281.17 2016 d 184,995.92 3,826,074.9 9 553,820.83 4,994,870.56 486,537.02 627,238.47 2013 int. j. anal. appl. 16 (6) (2018) 944 189,288.48 3,676,682.1 8 563,175.00 4,716,012.59 366,273.13 467,820.04 2014 193,680.64 3,533,122.5 6 572,687.15 4,452,722.96 275,736.48 348,919.26 2015 198,174.71 3,395,168.3 6 582,359.97 4,204,132.48 207,578.99 260,238.22 2016 e 579,545.95 2,219,522.9 9 187,488.58 2,513,688.16 112,428.28 135,410.05 2013 379,916.84 2,160,113.2 6 171,051.66 2,427,931.32 102,310.31 120,290.25 2014 249,051.53 2,102,293.7 4 156,055.74 2,345,100.16 93,102.91 106,858.71 2015 163,263.79 2,046,021.8 7 142,374.50 2,265,094.85 84,724.13 94,926.93 2016 f 139,782.22 1,880,837.6 4 39,532.09 1,989,414.54 45,174.60 56,854.20 2013 175,054.39 2,472,442.7 4 66,606.59 2,608,910.65 44,489.72 55,683.47 2014 219,227.03 3,250,133.3 3 112,223.73 3,421,315.50 43,815.23 54,536.84 2015 274,546.03 4,272,441.3 6 189,082.87 4,486,700.12 43,150.96 53,413.83 2016 g 17,812.17 575,917.57 115,441.56 736,926.97 41,751.20 48,301.58 2013 14,580.27 519,408.91 100,242.40 655,519.01 31,148.48 36,887.10 2014 11,934.78 468,444.85 87,044.38 583,104.14 23,238.31 28,170.06 2015 9,769.29 422,481.34 75,584.03 518,688.91 17,336.94 21,513.00 2016 h 61,718.53 2,468,162.4 5 45,368.49 2,563,400.06 39,247.53 51,926.86 2013 55,948.66 2,312,659.2 49,477.75 2,404,278.59 34,290.40 44,241.91 2014 int. j. anal. appl. 16 (6) (2018) 945 1 50,718.19 2,166,953.2 4 53,959.20 2,255,034.49 29,959.38 37,694.30 2015 45,976.70 2,030,427.2 8 58,846.56 2,115,054.62 26,175.39 32,115.70 2016 i 50,042.98 3,196,925.4 3 75,310.21 3,315,225.38 31,905.42 43,767.33 2013 46,980.56 2,791,367.2 9 74,115.19 2,902,910.81 29,065.35 37,687.57 2014 44,105.56 2,437,257.7 6 72,939.13 2,541,875.81 26,478.10 32,452.35 2015 41,406.49 2,128,070.1 5 71,781.73 2,225,742.73 24,121.15 27,944.36 2016 k 69,034.53 732,010.20 140,824.91 957,559.67 87,542.56 106,627.95 2013 32,660.79 719,943.09 151,388.03 944,591.68 73,706.82 88,350.10 2014 15,452.08 708,074.91 162,743.48 931,799.31 62,057.77 73,205.39 2015 7,310.51 696,402.38 174,950.69 919,180.19 52,249.80 60,656.74 2016 n 29,289.43 2,782,729.9 6 41,666.27 2,859,617.14 31,582.90 41,514.28 2013 31,086.36 2,576,154.4 7 45,397.65 2,655,768.12 30,924.88 39,474.13 2014 32,993.53 2,384,914.0 7 49,463.20 2,466,450.57 30,280.56 37,534.24 2015 35,017.71 2,207,870.3 8 53,892.83 2,290,628.60 29,649.67 35,689.69 2016 int. j. anal. appl. 16 (6) (2018) 946 mape calculating for dmus from 2013 to 2016 dmus fixed assets cost of goods sold operating costs net sales net profits operating profit year a 0.831046126 4.116590049 3.299390862 0.9647832 10.475444 7.549951686 2013 4.635547643 4.677198847 1.562597015 1.7767265 41.669969 26.99561505 2014 8.247781831 4.541447342 5.529857586 7.5607792 13.034141 11.83444692 2015 3.726061535 6.161022481 4.436678796 5.7950103 6.3556301 2.29920229 2016 b 2.471952086 0.304022899 1.863231306 1.8792393 22.204088 24.56053657 2013 3.82661733 1.519305739 8.178437474 2.3481426 19.037468 19.27542536 2014 0.997119035 1.636385084 14.7985746 2.1984442 4.5856106 5.443163906 2015 2.61404069 0.720707464 8.795346636 2.8481595 11.367918 11.29747875 2016 c 10.51267363 0.60290911 9.515527255 0.0757674 1.9937899 1.197563902 2013 18.35457038 1.398443681 13.70928726 0.5124762 2.7912122 3.95805446 2014 6.314979658 0.775602403 0.302283726 1.3169487 9.4708804 10.20873198 2015 0.324004609 0.029151344 3.000373581 0.7492704 4.1655485 4.223307378 2016 d 6.75264103 4.296802082 4.277481516 4.747712 8.8888184 8.075619689 2013 8.789383771 4.663289108 4.129982405 5.3972264 16.513808 15.40370719 2014 1.094371267 3.82004489 3.086485535 4.2679427 9.9738885 10.83849993 2015 2.283723542 5.001840549 3.51078815 6.0401502 50.256233 51.57800809 2016 e 7.840559703 5.283015274 0.861046281 4.7432974 2.5734587 3.632341309 2013 26.87888539 10.08741602 5.404861424 8.4875268 1.3997435 4.314484942 2014 56.43350477 0.746200265 10.52263082 0.3058301 8.201324 7.033176797 2015 8.56315681 2.806428744 4.772587155 3.1332679 6.4741546 7.397391298 2016 f 142.9220754 1.035075386 31.83182459 2.5499955 111.61045 120.707288 2013 41.50346437 1.25491082 8.631675732 1.7375368 30.936956 34.00243411 2014 28.02549984 2.086929224 0.93069136 2.7196603 43.301808 41.73974439 2015 0.686175888 0.645496541 15.75161309 0.1906421 218.19896 220.0349318 2016 g 2.865376084 0.632767883 2.30190748 0.2117258 3.2142614 4.834787255 2013 7.643815581 2.58263864 6.164671664 4.0141558 14.586828 16.7747376 2014 int. j. anal. appl. 16 (6) (2018) 947 3.708543236 9.52829413 4.594251804 9.4963393 31.751415 30.09770429 2015 2.199958954 5.631039699 0.286241911 5.0262094 10.329263 7.051216505 2016 h 0.101422786 0.830178572 5.841293482 0.8351949 7.8821559 7.023767879 2013 0.40646951 2.665488071 5.64944064 2.3963502 14.696244 14.6814982 2014 0.019338597 8.537542344 3.87956054 8.0379518 7.1585402 9.601931657 2015 0.08207085 6.291236393 3.422569495 5.8982354 4.0026662 2.64214953 2016 i 11.22391796 0.662737159 1.72228776 0.6358459 1.5804908 3.015896799 2013 10.16585478 2.908042972 6.611411841 2.8892025 1.7066201 5.167039374 2014 11.87701298 4.879966566 6.729845073 4.9101668 2.9608697 2.440034955 2015 17.7424437 2.717837439 2.837677776 2.7601978 3.2892842 5.621788834 2016 k 9.468852659 0.068515274 3.103902509 0.2180303 8.6904612 1.095976026 2013 70.72182241 0.847337559 4.721148677 1.6649781 13.500811 0.835561257 2014 14.03090946 1.932621349 0.866510581 2.6130502 4.7945484 10.32707351 2015 56.62193163 1.065858078 0.723112981 1.2749086 17.594975 11.50545964 2016 n 24.29736247 1.034492368 6.922215208 1.0521661 3.9116337 2.236813993 2013 18.63274851 4.27672745 6.459803005 4.2214194 3.0172939 3.519253898 2014 5.163756206 5.342147014 5.453997959 5.1390339 5.4972881 0.084534393 2015 10.12896285 2.50998816 4.341881054 2.3518195 5.451032 1.538273265 2016 int. j. anal. appl. 16 (6) (2018) 948 mape results for dmus from 2013 to 2016 dmus fixed assets cost of goods sold operating costs net sales net profits operating profit average mape of dmus a 4.36010928 4.87406468 3.70713106 4.02432479 17.88379591 12.16980399 7.83653829 b 2.47743229 1.04510530 8.40889750 2.31849642 14.29877108 15.14415115 7.28214229 c 8.87655707 0.70152663 6.63186796 0.66361569 4.60535774 4.89691443 4.39597325 d 4.73002990 4.44549416 3.75118440 5.11325783 21.40818698 21.47395873 10.15368533 e 24.92902667 4.73076508 5.39028142 4.16748058 4.66217019 5.59434859 7.28214229 g 4.10442346 4.59368509 3.33676821 4.68710757 14.97044169 14.68961141 7.73033957 h 0.15232544 4.58111134 4.69821604 4.29193307 8.43490155 8.48733682 4.39597325 i 12.75230736 2.79214603 4.47530561 2.79885326 2.38431622 4.06118999 4.87735308 k 37.71087904 0.97858307 2.35366869 1.44274180 11.14519889 5.94101761 9.92868152 n 14.55570751 3.29083875 5.79447431 3.19110971 4.46931193 1.84471889 5.52436018 international journal of analysis and applications volume 17, number 3 (2019), 448-463 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-448 fixed point results for φ− (γ,η,n,m)−contractions with applications to nonlinear integral equations hasanen a. hammad1 and manuel de la sen2,∗ 1department of mathematics, faculty of science, sohag university, sohag 82524, egypt https://orcid.org/0000-0001-8724-9367 2institute of research and development of processes university of the basque country 48940leioa (bizkaia), spain ∗corresponding author: manuel.delasen@ehu.eus abstract. the aim of this paper is to introduce a new class of pair of contraction mappings, called φ − (γ,η,n,m)-contraction pairs, and obtain common fixed point theorems for a pair of mappings in this class, satisfying a weakly compatible condition. as an application, we use mappings of this class to find the existence of solutions for nonlinear integral equations on the space of continuous functions and in some of its subspaces. moreover, some examples are given here to illustrate the applicability of these results. 1. introduction and preliminaries the banach contraction mapping plays an important role in solving nonlinear problems. then a lot of publications are devoted to the study and solutions of many practical and theoretical problems by using this condition ( [1][7]). continuity in this line, we establish some common fixed point results for a class of contraction mappings wherein contractive inequality is controlled by a positive function satisfying a stability condition at 0. after that, we use the class of mappings to establish the existence and uniqueness results for solutions of nonlinear received 2019-01-02; accepted 2019-02-12; published 2019-05-01. 2010 mathematics subject classification. 47h10, 47h05, 47h04. key words and phrases. φ − (γ,η,n,m)-contraction pairs; weakly compatible mappings; nonlinear integral equations. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 448 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-448 https://orcid.org/0000-0001-8724-9367 int. j. anal. appl. 17 (3) (2019) 449 integral equations. finally, we present some concentrated examples to illustrate the usability of the obtained results. definition 1.1. let (x,d) be a metric space. a pair of self-mappings (s,t) is said to be: icompatible [8] iff limn→∞d(stxn,tsxn) = 0, whenever {xn}⊂ x is such that lim n→∞ sxn = lim n→∞ txn = r for some r ∈ x. iinon-compatible [9] if there exists at least one sequence {xn}⊂ x such that limn→∞sxn = limn→∞txn = r for some r ∈ x such that limn→∞d(stxn,tsxn) is either non-existent or nonzero. iiisatisfies the property (e.a) [10] if there exists a sequence {xn}⊂ x such that lim n→∞ sxn = lim n→∞ txn = r for some r ∈ x. iv-satisfies the common limit in the range of t property (clrt ) [11] if there exists a sequence {xn} ⊂ x such that lim n→∞ sxn = lim n→∞ txn = tr for some r ∈ x. vsatisfies non-trivially weakly compatible (wc) condition [12] if they commute at their coincidence points, whenever the set of coincidences is nonempty. definition 1.2. a point x ∈ x is called: 1a coincidence point (cp) of s and t if sx = tx, the set of coincidence points of s and t will be denoted by c(s,t) and if x ∈ c(s,t), then w = sx = tx is called a point of coincidence (poc) of s and t . 2common fixed point of s and t if sx = tx = x. remark 1.1. it may be noted that: 1non-trivial weak compatibility is a necessary, hence minimal condition for the existence of common fixed points of contractive type mapping pairs. 2commutativity at coincidence points of s and t is a coincidence point, whenever x is a coincidence point of s and t . 3non-trivially weakly compatible mappings may equivalently be called as coincidence preserving mappings. 4compatible mappings are necessarily coincidence preserving since compatible mappings commute at each coincidence points. however, the converse need not be true. the following lemma will be used to prove our results: lemma 1.1. [13] let (x,d) be a metric space and {xn} be a sequence in x such that lim n→∞ d(xn,xn+1) = 0. int. j. anal. appl. 17 (3) (2019) 450 if {xn} is not a cauchy sequence in x, then there exist ε > 0 and sequences of integers positive (m(k)) and (n(k)) with m(k) > n(k) > k such that, d(xm(k),xn(k)) ≥ ε, d(xm(k)−1,xn(k)) < ε and (1) limk→∞d(xm(k)−1,xn(k)+1) = ε, (2) limk→∞d(xm(k),xn(k)) = ε, (3) limk→∞d(xm(k)−1,xn(k)) = ε. 2. φ− (γ,η,n,m)-contraction pairs and their coincidence point as in [3], we will use functions γ,η : r+ → [0, 1) satisfying that γ(r) + η(r) < 1 for r ∈ r+ and  lim sup s→0+ γ(s) < 1, lim sup s→r+ η(s) 1−γ(s) < 1, ∀r > 0. (2.1) now, we present our contraction mappings. definition 2.1. let (x,d) be a metric space and let s,t : x → x be mappings. the pair (s,t) is called a φ− (γ,η,n,m)-contraction pair if for all x,y ∈ x φ(d(sx,sy)) ≤ γ(d(tx,ty))φ(n(x,y)) + η(d(tx,ty))φ(m(x,y)), where φ : r+ → r+ is a continuous function satisfying that φ(rn) → 0 ⇐⇒ rn → 0, (2.2) and n(x,y) = max { d(tx,ty), d(sx,tx).d(sy,ty) 1 + d(tx,ty) } , m(x,y) = min{d(sx,tx),d(sy,ty),d(sy,tx),d(sx,ty)} . (2.3) proposition 2.1. let s,t : x → x be mappings on a metric space x with s(x) ⊂ t(x). if the pair (s,t) is a φ− (γ,η,n,m)-contraction pair, then for any x◦ ∈ x, a sequence {yn} defined by yn = sxn = txn+1 n = 0, 1, .. satisfies: (i) limn→∞d(yn,yn+1) = 0, (ii) {yn}⊂ x is a cauchy sequence in t(x). int. j. anal. appl. 17 (3) (2019) 451 proof. let x◦ ∈ x be an arbitrary point. since s(x) ⊂ t(x), there exists x1 ∈ x such that sx◦ = tx1. by continuing this process inductively we get a sequence {xn} in x such that yn = sxn = txn+1. now, φ(d(txn+1,txn+2)) = φ(d(sxn,sxn+1)) ≤ γ(d(txn,txn+1))φ(n(xn,xn+1)) (2.4) +η(d(txn,txn+1)))φ(m(xn,xn+1)), where n(xn,xn+1) = max { d(txn,txn+1), d(sxn,txn).d(sxn+1,txn+1) 1 + d(txn,txn+1) } = max { d(txn,txn+1), d(txn+1,txn).d(txn+2,txn+1) 1 + d(txn,txn+1) } ≤ max{d(txn,txn+1),d(txn+2,txn+1)} and m(xn,xn+1) = min{d(sxn,txn),d(sxn+1,txn+1),d(sxn+1,txn),d(sxn,txn+1)} = min{d(txn+1,txn),d(txn+2,txn+1),d(txn+2,txn),d(txn+1,txn+1)} = 0. if n(xn,xn+1) = d(txn+2,txn+1), then from (2.4) we get φ(d(txn+1,txn+2)) ≤ γ(d(txn,txn+1))φ(d(txn+2,txn+1)) < φ(d(txn+2,txn+1)), which is a contradiction. so n(xn,xn+1) = d(txn,txn+1), then from (2.4) and using the properties of the function γ, we have φ(d(txn+1,txn+2)) ≤ γ(d(txn,txn+1))φ(d(txn,txn+1)) < φ(d(txn,txn+1)). thus {vn} = {φ(d(txn,txn+1))} is a decreasing sequence of positive numbers bounded below by zero, and so converges to b ≥ 0. now if b > 0, then by taking lim sup on both sides of the above inequality we have a contradiction. thus, lim n→∞ vn = lim n→∞ φ(d(txn,txn+1)) = 0. consequently, from the stability condition at zero (2.2) we conclude that lim n→∞ d(yn,yn+1) = lim n→∞ d(sxn,sxn+1) = lim n→∞ d(txn+1,txn+2) = 0. (2.5) this proves (i). int. j. anal. appl. 17 (3) (2019) 452 to prove (ii) we are going to suppose that {yn} ⊂ t(x) is not a cauchy sequence. then there exist an ε > 0 and sequences of integers positive (m(k)) and (n(k)) with m(k) > n(k) > k such that, d(xm(k),xn(k)) ≥ ε, d(xm(k)−1,xn(k)) < ε. from lemma 1.1 and the continuity of φ we have φ(ε) = lim sup k→∞ φ(d(txm(k)+1,txn(k)+1)) = lim sup k→∞ φ(d(sxm(k),sxn(k))) ≤ lim sup k→∞ γ(d(txm(k),txn(k)))φ(n(xm(k),xn(k))) + lim sup k→∞ η(d(txm(k),txn(k)))φ(m(xm(k),xn(k))), (2.6) where n(xm(k),xn(k)) = max   d(txm(k),txn(k)),d(sxm(k),txm(k)).d(sxn(k),txn(k)) 1+d(txm(k),txn(k))   = max   d(txm(k),txn(k)),d(txm(k)+1,txm(k)).d(txn(k)+1,txn(k)) 1+d(txm(k),txn(k))   (2.7) and m(xm(k),xn(k)) = min   d(sxm(k),txm(k)),d(sxn(k),txn(k)),d(sxn(k),txm(k)),d(sxm(k),txn(k))   = min   d(txm(k)+1,txm(k)),d(txn(k)+1,txn(k)),d(txn(k)+1,txm(k)),d(txm(k)+1,txn(k))   (2.8) letting k →∞ in (2.7), (2.8) and by (2.5), we can write lim k→∞ n(xm(k),xn(k)) = max{ε, 0} = ε, lim k→∞ m(xm(k),xn(k)) = min{0, 0,ε,ε} = 0. therefore, (2.6) is now φ(ε) ≤ lim sup k→∞ γ(d(txm(k),txn(k)))φ(ε) < φ(ε). which is a contradiction, hence {yn}⊂ x is a cauchy sequence. � lemma 2.1. let s and t be self-mappings on a metric space (x,d) and the pair (s,t) is a φ−(γ,η,n,m)contraction pair. if s and t have a poc in x then it is unique. int. j. anal. appl. 17 (3) (2019) 453 proof. let z ∈ x be a poc of the pair (s,t). then there exits x ∈ x such that sx = tx = z. suppose that y ∈ x, sy = ty = u with u 6= z . then φ(d(z,u)) = φ(d(sx,sy)) ≤ γ(d(tx,ty))φ(n(x,y)) + η(d(tx,ty))φ(m(x,y)) ≤ γ(d(z,u))φ(n(x,y)) + η(d(z,u))φ(m(x,y)). (2.9) using (2.3) we have n(x,y) = max { d(tx,ty), d(sx,tx).d(sy,ty) 1 + d(tx,ty) } = max { d(z,u), d(z,z).d(u,u) 1 + d(z,u) } = d(z,u), and m(x,y) = min{d(sx,tx),d(sy,ty),d(sy,tx),d(sx,ty)} = min{d(z,z),d(u,u),d(u,z),d(z,u)} = 0. substituting it into (2.9) we get φ(d(z,u)) ≤ γ(d(z,u))φ(d(z,u)) < φ(d(z,u)), which is a contradiction, therefore z = u. � theorem 2.1. let s and t be self-mappings on a metric space (x,d) such that (i) s(x) ⊂ t(x), (ii) t(x) ⊂ x is a complete subspace of x, (iii) the pair (s,t) is a φ− (γ,η,n,m)-contraction pair. then, the pair (s,t) has a unique poc. proof. let yn = sxn = txn+1, n = 0, 1, .. be a cauchy sequence defined by proposition 2.1 satisfies {yn} = {txn+1}⊂ t(x). since t(x) ⊂ x is a complete subspace of x, then there exists v ∈ t(x) such that lim n→∞ yn = lim n→∞ sxn = lim n→∞ txn+1 = v, thus we can find u ∈ x such that tu = v. now, we shall prove that tu = su, then φ(d(v,su)) = φ(d(sxn+1,su)) ≤ γ(d(txn+1,tu))φ(n(xn+1,u)) + η(d(txn+1,tu))φ(m(xn+1,u)), (2.10) int. j. anal. appl. 17 (3) (2019) 454 where n(xn+1,u) = max { d(txn+1,tu), d(sxn+1,txn+1).d(su,tu) 1 + d(txn+1,tu) } = max { d(v,tu), d(v,v).d(su,tu) 1 + d(v,tu) } = 0 (2.11) and m(xn+1,u) = min{d(sxn+1,txn+1),d(su,tu),d(su,txn+1),d(sxn+1,tu)} = min{d(v,v),d(su,tu),d(su,v),d(v,tu)} = 0. (2.12) applying (2.11) and (2.12) in (2.10), we have φ(d(v,su)) ≤ 0, this holds only if d(v,su) = 0. so tu = su = v, therefore v is a poc of s and t. from lemma 2.1 we conclude that v is a unique poc. � 3. common fixed point for φ− (γ,η,n,m)-contraction pairs in this section general common fixed point results for a pair of mappings belonging to the φ−(γ,η,n,m)contraction class, under a minimal commutativity condition are given. theorem 3.1. let s and t be self-mappings on a metric space x satisfying the conditions of theorem 2.1, if the pair (s,t) is non-trivially weakly compatible pair, then there are a unique common fixed point of s and t . proof. since the pair (s,t) is non-trivially weakly compatible, then they commute at their unique coincidence point. hence, ssu = stu = tsu = ttu , using uniqueness of the poc, we obtain that v = su is a common fixed point of (s,t). uniqueness of the common fixed point can be proved using the same reasoning as above. � now, we omit the condition s(x) ⊂ t(x) from the above theorem and obtain the following result: theorem 3.2. let s,t : x → x be mappings on a metric space (x,d) satisfying the property (e.a). consider the pair (s,t) is non-trivially weakly compatible φ− (γ,η,n,m)-contraction pair. if t(x) ⊂ x is closed, then s and t have a unique common fixed point. proof. since the pair (s,t) satisfies the property (e.a), there exists a sequence {xn}⊂ x such that lim n→∞ sxn = lim n→∞ txn = v, for some v ∈ x. since t(x) is closed, so v ∈ t(x) and v = tu for some u ∈ x. as in the proof of the theorem 2.1, we can prove that v = tu = su and that v is a unique poc of s and t . the existence of the unique common fixed point follows as in the proof of theorem 3.1. � int. j. anal. appl. 17 (3) (2019) 455 remark 3.1. since noncompatible mappings on a metric space (x,d) satisfy the property (e. a). therefore, conclusion of theorem 3.2 still valid if we consider s and t noncompatible mappings. we can replace conditions (i) and (ii) of theorem 2.1 by a single condition and obtain the following result. here s(x) denotes the closure of the range of the mapping s. theorem 3.3. let s and t be self-mappings on a metric space (x,d) such that (i) s(x) ⊂ x is a complete subspace of x, (ii) the pair (s,t) is a φ− (γ,η,n,m)-contraction pair. then the pair (s,t) has a unique poc. furthermore, if the pair (s,t) is nontrivially weakly compatible, then s and t have a unique common fixed point. in the next result, we drop the closeness of the range of mapping and replace the property (e. a) by clrt property. theorem 3.4. let (x,d) be a metric space and s,t : x → x satisfying the clrt property. let us suppose that the pair (s,t) is φ− (γ,η,n,m)-contraction pair. if the pair (s,t) is non-trivially weakly compatible, then s and t have a unique common fixed point. proof. since the pair (s,t) satisfies the clrt property, there exists a sequence {xn}⊂ x such that lim n→∞ sxn = lim n→∞ txn = tv, for some v ∈ x. the rest of the proof runs with similarities to the proof of the previous results. � remark 3.2. notice that by considering particular functions, as constants, for the functions γ, η as well as by considering φ = id (the identity mapping), or by choosing a particular form for m(x,y) and n(x,y) in the class of φ− (γ,η,n,m)-contraction pair, we can obtain known several subclasses of mappings. 4. application to a class of nonlinear integral equations in this section, we will study the existence of solutions for a class of nonlinear integral equations by using the existence of coincidence and common fixed points for mappings belonging to the φ− (γ,η,n,m)contraction class. let x = c([0,t],r) denote the space of all continuous functions on [0,t], it is a complete metric space equipped with the uniform metric d d(u,v) = sup t∈[0,t] {|u(t) −v(t)|}, u,v ∈ x. (4.1) int. j. anal. appl. 17 (3) (2019) 456 now, following the idea in [4], we discuss an application of fixed point techniques to obtain the solution of the nonlinear integral equation: x(t) = f1(t) −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,x(s))ds + λ ∫ t 0 h2(t,s)k2(t,s,x(s))ds, (4.2) where t ∈ [0,t], δ,λ real numbers, f1,f2 ∈ c([0,t],r) are known, f1(t) ≥ f2(t) and k1,k2,h1,h2 are continuous real-valued functions in [0,t] ×r. to attain our aim, we will use some functional associated with h-concave and quasilinear functions [14]. let c be a convex cone in the linear space x over r and let l 6= 0 be a real number. a functional φ : c → r is called l-superadditive on c if φ(x + y) ≥ l(φ(x) + φ(y)), for any x,y ∈ c. let g be a real non-negative function and φ a functional satisfying φ(tx) ≤ g(t)φ(x), for any t ≥ 0 and x ∈ c, is called g-positive homogeneous. notice that necessarily g(1) = 1. the following lemma is very important in the sequel: lemma 4.1. [14] let u,v ∈ c and φ : c → r be a non-negative, l-superadditive and g-positive homogeneous functional on c. if m ≥ m > 0 are such that u−mv and mv −u ∈ c, then lg(m)φ(v) ≤ φ(u) ≤ 1 l g(m)φ(v). now, our theorem concerned with the existence solution of system (4.2) become affordable. theorem 4.1. suppose that the following conditions are satisfied: (i) t∫ 0 supt∈[0,t] |hi(t,s)| ≤ 1, i = {1, 2}, (ii) for each s ∈ [0,t] and for all x,y ∈ x, there is mi ≥ 0 such that |ki(t,s,x(s)) −ki(t,s,y(s))| ≤ mi |x(s) −y(s)| ≤ mi‖x−y‖ , i = {1, 2}, (iii) lemma 4.1 holds for φ being a non-negative, l-superadditive with l > 0 and g-positive homogeneous functional. then the integral equation (4.2) has at least one solution in x, provided that |δ|m1 + |λ|m2 = 1. (4.3) proof. we define the following operators, for each x ∈ x sx(t) = −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,x(s))ds (4.4) int. j. anal. appl. 17 (3) (2019) 457 and tx(t) = x(t) −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,x(s))ds. (4.5) clearly, s and t are self operators on x. now, for all x,y ∈ x by using (i)-(ii), we have |sx(t) −sy(t)| ≤ |δ| ∫ t 0 |h1(t,s)| |k1(t,s,x(s)) −k1(t,s,y(s))|ds ≤ |δ| ∫ t 0 sup t∈[0,t] |h1(t,s)| |k1(t,s,x(s)) −k1(t,s,y(s))|ds ≤ |δ| ∫ t 0 sup t∈[0,t] |h1(t,s)|m1 |x(s) −y(s)|ds ≤ |δ|m1 ‖x−y‖ ∫ t 0 sup t∈[0,t] |h1(t,s)| ≤ |δ|m1 ‖x−y‖ . this implies that ‖sx−sy‖ = |sx(t) −sy(t)| ≤ |δ|m1 ‖x−y‖ . (4.6) by a similar way we get ∣∣∣∣∣λ ∫ t 0 h2(t,s)k2(t,s,x(s))ds−λ ∫ t 0 h2(t,s)k2(t,s,y(s))ds ∣∣∣∣∣ ≤ |λ| ∫ t 0 |h2(t,s)| |k2(t,s,x(s)) −k2(t,s,y(s))|ds ≤ |λ| ∫ t 0 sup t∈[0,t] |h2(t,s)|m2 |x(s) −y(s)|ds ≤ |λ|m2 ‖x−y‖ . consequently, it follows that ‖tx−ty‖ ≥ ‖x−y‖− ∣∣∣∣∣λ ∫ t 0 k2(t,s,x(s))ds−λ ∫ t 0 k2(t,s,y(s))ds ∣∣∣∣∣ ≥ (1 −|λ|m2)‖x−y‖ , (4.7) since condition (4.3) implies that |λ|m2 < 1, so (4.7) yields ‖x−y‖≤ 1 (1 −|λ|m2) ‖tx−ty‖ . (4.8) using (4.6), (4.8) and condition (4.3), we obtain that ‖sx−sy‖≤ |δ|m1 (1 −|λ|m2) ‖tx−ty‖ = ‖tx−ty‖ . moreover, there exists 0 ≤ m < 1 depending of x and y such that m(x,y)‖tx−ty‖≤‖sx−sy‖≤‖tx−ty‖ . (4.9) int. j. anal. appl. 17 (3) (2019) 458 now, by (iii), φ is a non-negative, continuous, 2-superadditive and g-positive homogeneous functional on the cone r+ satisfying (2.2). for u = ‖sx−sy‖, v = ‖tx−ty‖ and the inequality (4.9), the lemma 4.1 allows us to conclude that, φ(‖sx−sy‖) ≤ 1 2 φ(‖tx−ty‖). let γ,η : r+ → [0, 1) satisfying (2.1) with γ(t) ≥ 1 2 for any t ∈ r+. hence we obtain φ(‖sx−sy‖) ≤ 1 2 φ(‖tx−ty‖) ≤ γ(‖tx−ty‖)φ(n(x,y)) + η(‖tx−ty‖)φ(m(x,y)). therefore, (s,t) is a φ− (γ,η,n,m)−contraction pair. since s is a continuous mapping and x is a complete, s(x) is a complete subspace of x, therefore from theorem 3.3, the pair (s,t) has a unique poc (say p◦); i.e., p◦ = sx ∗(t) = tx∗(t). thus, −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,x ∗(s))ds = x∗(t) −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,x ∗(s))ds, or equivalently, x∗(t) = f1(t) −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,x ∗(s))ds + λ ∫ t 0 h2(t,s)k2(t,s,x ∗(s))ds. therefore, x∗ ∈ x is a solution of the nonlinear integral equation (4.2). � under the notion of non-trivial weak compatibility of the pair (s,t) given in (4.4) and (4.5), the next result shows that there exists a (unique) solution of the equation (4.2) satisfying a certain integral equation. proposition 4.1. under the hypotheses of theorem 4.1, if the pair of mappings (s,t) defined in (4.4)(4.5) is non-trivially weakly compatible, then there is a unique solution ζ of the equation (4.2) satisfying the integral equation f1(t) = −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds. proof. since the pair (s,t) is non-trivially weakly compatible, from theorem 3.1 there is a unique solution ζ satisfying that sζ(t) = tζ(t) = ζ(t), moreover stζ(t) = tsζ(t), where stζ(t) = −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,ζ(s))ds, tsζ(t) = −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,ζ(s))ds−f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds. from this, we obtain −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,ζ(s))ds = −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s,ζ(s))ds −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds. this implies that f1(t) = −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds. int. j. anal. appl. 17 (3) (2019) 459 this completes the proof. � remark 4.1. in view of the proof of proposition 4.1, one can observe that the only solution which satisfies the equation f1(t) = −λ ∫t 0 h2(t,s)k2(t,s,ζ(s))ds, is a unique common fixed point of the pair (s,t) defined in (4.4) and (4.5). 5. the equation (4.2) on a compact subspace of (x,d) in that follows by (µ,d) we denote a compact subspace of x endowed with the induced uniform metric d defined in (4.1). to establish the existence result in this case, we will use the operator s given in (4.4) and the next auxiliary mapping: hx(t) = κx(t) −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds, 0 ≤ κ < 1. (5.1) theorem 5.1. under conditions (i)-(iii) of theorem 4.1, if s,h defined in (4.4) and (5.1) are non-trivially weakly compatible self-mappings of (µ,d), then for all x ∈ µ the equation (4.2) has a unique solution ζ ∈ µ satisfying f1(t) = −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds, provided that |δ|m1 + |λ|m2 = κ. holds. proof. we claim that (s,h) has the property (e. a) if it is non-trivially weakly compatible. in fact, let ζn → ζ a sequence of functions on µ converging to ζ, where the function ζ is a unique point of coincidence of the weakly compatible pair (s,h). from the continuity of the function ki(t,s) we have lim n→∞ sζn(t) = −f2(t) + δ ∫ t 0 h1(t,s)k1(t,s, lim n→∞ ζn(s))ds = sζ(t), lim n→∞ hζn(t) = qζn(t) −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s, lim n→∞ ζn(s))ds = hζ(t). then, we conclude that (s,h) has the property (e. a). on the other hand, it is easy to check that the operator h is continuous on (µ,d). since µ is a compact and hausdorff space, the closed map lemma implies that h(µ) is a closed. thus, from theorem 3.2, h and s have a unique common fixed point ζ ∈ µ. the existence of a unique solution satisfying the above relation is obtained from the proof of theorem 4.1, replacing the mapping t by h. the representation for the solution follows from the proof of proposition 4.1, upon replacing t by h. � int. j. anal. appl. 17 (3) (2019) 460 6. the equation (4.2) on non-complete metric space the existence theorem 4.1 was proved by applying theorem 3.3, since s(x) is a complete subspace. however, if equation (4.2) is posed in a non-complete metric subspace (χ,d) of (x,d), we are not able to apply such theorem. by imposing an extra condition we obtain the following existence result for this case. theorem 6.1. suppose the following conditions are satisfied: (i) t∫ 0 supt∈[0,t] |hi(t,s)| ≤ 1, i = {1, 2}, (ii) for each s ∈ [0,t] and for all x,y ∈ χ, there is mi ≥ 0 such that |ki(t,s,x(s)) −ki(t,s,y(s))| ≤ mi |x(s) −y(s)| , i = {1, 2}, (iii) λ ∫t 0 h2(t,s)k2 ( t,s,δ ∫ s 0 h1(s,κ)k1(κ,s,x(κ))dκ + f1(s) −f2(s) ) ds = 0. then the integral equation (4.2) has a unique solution ζ ∈ χ, satisfying f1(t) = −λ ∫ t 0 h2(t,s)k2(t,s,ζ(s))ds, provided that |δ|m1 + |λ|m2 = 1. proof. from the proof of theorem 4.1, it is sufficient to show that the pair (s,t) defined in (4.4)-(4.5) has a poc in χ. to obtain this, we will apply theorem 2.1, thus we prove that s(χ) ⊆ t(χ). in fact, adopting the same reasoning as in [4], by assumption (iii), for x(t) ∈ χ we have t(sx(t) + f1(t)) = sx(t) + f1(t) −f1(t) −λ ∫ t 0 h2(t,s)k2(t,s,sx(s) + f1(s))ds = sx(t) −λ ∫ t 0 h2(t,s)k2 ( t,s,δ ∫ s 0 h1(s,κ)k1(κ,s,x(κ))dκ + f1(s) −f2(s) ) ds = sx(t). thus, from theorem 2.1 s and t have a unique poc, so all coincidence point related with the poc is a solution of the integral equation (4.2) in χ. as the proof of proposition 4.1, the formula for the solution is a consequence of the non-trivially weakly compatibility and the existence of a unique common fixed point. � 7. illustrative examples in this section we are going to consider some nonlinear integral equations on c([0, 1],r) defined in (4.2). the existence of solutions will be established as an application of the previous results. example 7.1. let us consider the following nonlinear integral equation: x(t) = f1(t) −f2(t) + 5 4 ∫ t 0 ( 4s 5 )( 5 4 tx(s))ds + 5 4 ∫ 1 0 ( 5ts 6 )( −6t2 5 x(s))ds = f1(t) −f2(t) + 5t 4 ∫ t 0 sx(s)ds− 5t3 4 ∫ 1 0 sx(s)ds, t ∈ [0, 1]. (7.1) int. j. anal. appl. 17 (3) (2019) 461 taking δ = λ = 5 4 , the kernels functions ki(t,s,x(s)) and hi(t,s), i ∈{1, 2} given by k1(t,s,x(s)) = 5 4 tx(s), k2(t,s,x(s)) = −6t2 5 x(s), h1(t,s) = 4s 5 , h2(t,s) = 5ts 6 . notice that the functions ki(t,s,x(s)), i ∈{1, 2} satisfy |ki(t,s,x(s)) −ki(t,s,y(s))| ≤ 5 4 |x(s) −y(s)| , for all x,y ∈ c([0, 1],r), and the functions hi(t,s) satisfy 1∫ 0 sup t∈[0,1] |hi(t,s)| ≤ 1, i = {1, 2}. thus, theorem 4.1 guarantees that the equation (7.1) has at least one solution, hence, the solution is the cp of the mappings s and t, which defined as follows: sx(t) = −f2(t) + 5 4 ∫ t 0 tsx(s)ds, tx(t) = x(t) −f1(t) + 5 4 ∫ 1 0 t3sx(s)ds. now, let κ be a coincidence point of (s,t), and we assume that the following system is satisfied  κ(t) = f1(t) −f2(t),5 4 ∫ 1 0 t3sκ(s)ds = 5 4 ∫ t 0 tsκ(s)ds, for all t ∈ [0, 1]. (7.2) since t = 0 obviously holds, we assume t 6= 0. notice that the second equality of the system is equivalent to t2 ∫ 1 0 sκ(s)ds = ∫ t 0 sκ(s)ds. differentiating with respect to t, equality above is equivalent to 2 ∫ 1 0 sκ(s)ds = κ(t). that means, the constant functions are the only coincidence point of (t,s) satisfying (7.2), provided f1(t)− f2(t) is also constant. let κ(t) = ζ ∈ r, we obtain sζ = −f2(t) + 5 4 ∫ t 0 tsζds = −f2(t) + 5 8 t3ζ, tζ = ζ −f1(t) + 5 4 ∫ 1 0 t3sζds = ζ −f1(t) + 5 8 t3ζ. therefore, the system (7.1) has a solution at the constant function κ(t) = ζ. on the other hand, notice that the pair (s,t) is not weakly compatible. in fact, stζ = ssζ = −f2(t) + 5 4 ∫ t 0 ts ( −f2(s) + 5 8 s3ζ ) ds = −f2(t) + 5ζ 32 t6 − 5t 4 ∫ t 0 sf2(s)ds and tsζ = −f2(t) −f1(t) + 45ζ 32 t3 − 4t3 5 ∫ 1 0 sf2(s)ds = ζ + 45ζ 32 t3 − 4t3 5 ∫ 1 0 sf2(s)ds. int. j. anal. appl. 17 (3) (2019) 462 therefore, the solution κ(t) = ζ does not satisfy the integral equation given in proposition 4.1. example 7.2. we will consider the following nonlinear integral equation:  x(t) = (sin(e2)−sin(e))e−t e+1 − e 2t+2−et+2 e2+e + et+1 + 2 (e2+e) ∫ t 0 e2t−2s+1 x(s) 2 ds + 1 e+1 ∫ 1 0 es−t+1 cos(x(s))ds, t ∈ [0, 1]. (7.3) equation (7.3) is of the form (4.2), for k1(t,s,x(s)) = e t+1 x(s) 2 , k2(t,s,x(s)) = e cos(x(s)), h1(t,s) = e t−2s, h2(t,s) = e s−t, f1(t) = (sin(e2) − sin(e))e−t e + 1 , f2(t) = e2t+2 −et+2 e2 + e −et+1, and δ = 2 e2−e, λ = − 1 e+1 . notice that m1 = e 2 , m2 = e and |δ|m1 + |λ|m2 = 1. let the mappings (s,t) given in this case by sx(t) = et+1 − e2t+2 −et+2 e2 + e + 2 (e2 + e) ∫ t 0 e2t−2s+1 x(s) 2 ds, tx(t) = x(t) − (sin(e2) − sin(e))e−t e + 1 + 1 e + 1 ∫ 1 0 es−t+1 cos(x(s))ds. we are going to find the coincidence points of (s,t). a point κ(t) ∈ x is a cp of (s,t) if e2t [ e1−t − e2 −e2−t (e2 + e) + 2e (e2 + e) ∫ t 0 e−2s κ(s) 2 ds ] = e−t [ κ(t)et − sin(e2) − sin(e) e + 1 + e e + 1 ∫ 1 0 es cos(κ(s))ds ] , equivalently, e3t [ e1−t − e2 −e2−t (e2 + e) + 2e (e2 + e) ∫ t 0 e−2s κ(s) 2 ds ] = κ(t)et − sin(e2) − sin(e) e + 1 + e e + 1 ∫ 1 0 es cos(κ(s))ds. (7.4) since the term − sin(e2) − sin(e) e + 1 + e e + 1 ∫ 1 0 es cos(κ(s))ds is constant and the left side of equality (7.4) depends of t, necessarily we have that sin(e2) − sin(e) e + 1 = e e + 1 ∫ 1 0 es cos(κ(s))ds, or equivalently, sin(e2) − sin(e) = ∫ 1 0 es+1 cos(κ(s))ds, whose solution is κ(t) = et+1. notice that equality (7.4) is satisfied for this function and set+1 = tet+1 = et+1, tset+1 = stet+1 = et+1. int. j. anal. appl. 17 (3) (2019) 463 therefore, κ(t) = et+1 is a unique coincidence and fixed point of (t,s). also the pair (t,s) is non-trivially weakly compatible, so from the proposition 4.1, equation (7.3) has a solution satisfying the integral equation f1(t) = −λ ∫ 1 0 h2(t,s)k2(t,s,ζ(s))ds (sin(e2) − sin(e))e−t e + 1 = 1 e + 1 ∫ 1 0 es−t+1 cos(x(s))ds, whose solution is x(t) = et+1. question. in theorem 4.1, we consider φ is 2-superadditive i.e., l = 2 > 0. are the results still true if we take l < 0? references [1] m. berzig, s. chandok and m. s. khan, generalized krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem, appl. math. comput., 248 (2014), 323–327. [2] d. gopal, m. abbas and c. vetro, some new fixed point theorems in menger pm-spaces with application to volterra type integral equation, appl. math. comput., 232 (2014), 955–967. [3] z. liu, x. li, s. minkan and s.y. cho, fixed point theorems for mappings satisfying contractive condition of integral type and applications, fixed point theory and appl., 2011 (2011), art. id 64. [4] h. k. pathak, m.s. khan and r. tiwari, a common fixed point theorem and its application to nonlinear integral equations, comput. math. appl., 53 (2007), 961–971. [5] s. radenović, t. došenović, t. a. lampert and z. golubov́ıć, a note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations, appl. math. comput., 273 (2016), 155–164. [6] n. shahzad, o. valero and m.a. alghamdi, a fixed point theorem in partial quasi-metric spaces and an application to software engineering, appl. math. and comput., 268 (2015), 1292–1301. [7] n. hussain and m. a. taoudi, krasnoselskii-type fixed point theorems with applications to volterra integral equations, fixed point theory appl., 2013 (2013), art. id 196. [8] g. jungck, compatible mappings and common fixed points, int. j. math. math. sci., 9(4) (1986), 771–779. [9] r. p. pant, discontinuity and fixed points, j. math. anal. appl., 240 (1999), 284–289. [10] m. aamri and d. el moutawakil, some new common fixed point theorems under strictly contractive conditions, j. math. anal. appl., 270 (2002), 181–188. [11] w. sintunavarat and p. kumam, common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, j. appl. math., 2011 (2011), art. id 637958. [12] g. jungck, common fixed point for noncontinuous nonself maps on nonmetric spaces, far east j. math. sci., 4(2) (1986), 199–215. [13] g. u. r. babu and p. d. sailaja, a fixed point theorem of generalized weakly contractive maps in orbitally complete metric spaces, thai j. math., 9(1) (2011), 1–10. [14] l. nikolova and s. varošanec, properties of some functionals associated with h-concave and quasilinear functions with applications to inequalities, j. inequal. appl. 2014 (2014), art. id 30. 1. introduction and preliminaries 2. -(,,n,m)-contraction pairs and their coincidence point 3. common fixed point for -(,,n,m)-contraction pairs 4. application to a class of nonlinear integral equations 5. the equation (4.2) on a compact subspace of (x,d) 6. the equation (4.2) on non-complete metric space 7. illustrative examples question. references int. j. anal. appl. (2023), 21:53 doubt m-polar fuzzy sets based on bck-algebras bayan albishry, sarah o. alshehri∗, nasr zeyada depatment of mathematics, faculty of science, university of jeddah, jeddah, saudi arabia ∗corresponding author: soalshehri@uj.edu.sa abstract. doubt m-polar subalgebras (ideals) were introduced and some properties were investigated. also, doubt m-polar positive implicative (commutative) ideals were defined and related results were proved. 1. introduction the main problem in fuzzy mathematics is how to carry out the ordinary concepts to the fuzzy case. the difficulty lies in how to pick out the rational generalization from the large number of available approaches. it is worth noting that fuzzy ideals are different from ordinary ideals in the sense that one cannot say which bck-algebra element belongs to the fuzzy ideal under consideration and which one does not. the concept of fuzzy sets was introduced by zadeh [1]. since then these ideas have been applied to other algebraic structures such as semigroups, groups, rings, modules, vector spaces and topologies. in 1991, xi [5] applied the concept of fuzzy sets to bck-algebras which are introduced by imai and k.iseki [2]. in [10], a.al-masarwah and a. ghafur ahmad introduced the concept of doubt bipolar fuzzy subalgebra and ideals in bck/bci algebra. in this paper we introduced the notion of doubt m-polar fuzzy subalgebras and ideals of bck -algebras.moreover, we define the notion of doubt m-polar fuzzy positive implicative (commutative) ideal of bck -algebras, and investigate some related properties. we show that in a positive implicative(commutative) bck-algebra, a fuzzy subset is a doubt m-polar fuzzy ideal if and only if it is a doubt m-polar fuzzy positive implicative ideal. we show that m-polar fuzzy subset of a bck-algebra is a doubt m-polar fuzzy positive implicative received: mar. 27, 2022. 2010 mathematics subject classification. 08a72. key words and phrases. bck-algebras; fuzzy subalgebras; m-polar fuzzy subalgebras; m-polar fuzzy ideals; doubt m-polar fuzzy subalgebras; doubt m-polar fuzzy ideals. https://doi.org/10.28924/2291-8639-21-2023-53 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-53 2 int. j. anal. appl. (2023), 21:53 (commutative) ideal if and only if the doubt σ-level cut set of this m-polar fuzzy subset is an doubt m-polar fuzzy positive implicative (commutative) ideal. 2. preliminaries first, we recall some elementary aspects which are used to present the paper. throughout this paper, x always denotes a bckalgebra without any specifications, and for details about the theory of these algebras we may refer to [2–4,7]. an algebra (x;∗,0) of type (2,0) is called a bck-algebra if it satisfies the following axioms for all x,y,z ∈ x : i: ((x ∗y)∗ (x ∗z))∗ (z ∗y)=0, ii: (x ∗ (x ∗y))∗y =0, iii: x ∗x =0, iv: x ∗y =0 and y ∗x =0 imply x = y. v: 0∗x =0. a partial ordering ≤ on a bck-algebra x can be defined by x ≤ y if and only if x ∗y =0. any bck-algebra x satisfies the following axioms for all x,y,z ∈ x: (i1) x ∗0= x, (i2) (x ∗y)∗z =(x ∗z)∗y, (i3) x ∗y ≤ x, (i4) (x ∗y)∗z ≤ (x ∗z)∗ (y ∗z), (i5) x ≤ y ⇒ x ∗z ≤ y ∗z,z ∗y ≤ z ∗x. a bck-algebra x is said to be positive implicative if it satisfies the following equality: (∀x,y,z ∈ x)((x ∗z)∗ (y ∗z)= (x ∗y)∗z). a bck-algebra x is said to be commutative if it satisfies the following equality: (∀x,y ∈ x)(x ∧y = y ∧x), where x ∧y = y ∧x. definition 2.1 [5]a non-empty subset i of a bck-algebra x is called a subalgebra of x if x ∗y ∈ i for any x,y ∈ i. definition 2.2 [5]a non-empty subset s of a bck-algebra x is called an ideal of x if (s1) 0∈ s, (s2) x ∗y ∈ s and y ∈ s,then x ∈ s for all x,y ∈ x. int. j. anal. appl. (2023), 21:53 3 definition 2.3 [7] a non-empty subset s of a bck-algebra x is called a positive implicative ideal of x if if it satisfies (s1) and (s3) (x ∗y)∗z ∈ s and y ∗z ∈ s,then x ∗z ∈ s for all x,y ∈ x. definition 2.4 [6] a non-empty subset s of a bck-algebra x is called a commutative ideal of x if if it satisfies (s1) and (x ∗y)∗z ∈ s and z ∈ s,then x ∗ (y ∧x)∈ s for all x,y ∈ x. lemma 2.5 [7]an ideal s of a bck-algebra x is commutative if and only if the following assertion is valid (∀x,y ∈ x)(x ∗y ∈ s ⇒ x ∗ (y ∧x)∈ s). definition 2.6 [1]a fuzzy set in a bck-algebra x is a function µ : x → [0,1]. definition 2.7 [5]a fuzzy set µ in a bck-algebra x is called a fuzzy subalgebra of x if µ(x ∗y)≥ min{µ(x),µ(y)} for all x,y ∈ x. definition 2.8 [5]a fuzzy set µ in a bck-algebra x is called a fuzzy ideal of x if µ(0)≥ µ(x) and µ(x)≥ min{µ(x ∗y),µ(y)} for all x,y ∈ x. introduced the definition of a doubt fuzzy subalgebra and a doubt fuzzy ideal in bck-algebras, which are as follows: definition 2.9 [8]a fuzzy set a = {(x,µa(x))|x ∈ x} in x is called a doubt fuzzy subalgebra of x if µa(x ∗y)≤ max{µa(x),µa(y)} for all x,y ∈ x. definition 2.10 [8]a fuzzy set a = {(x,µ(x))|x ∈ x} in x is called a doubt fuzzy ideal of x if µa(0)≤ µa(x) and µa(x)≤ max{µa(x ∗y),µa(y)} for all x,y ∈ x. the proposed work is done on m−polar fuzzy sets.the formal definition of an m-polar fuzzy set is given below: definition 2.11 [9]an m-polar fuzzy set q on a nonempty set x is a mapping q : x → [0,1]m. the membership value of every element x ∈ x is denoted by q(x)= (p1 ◦q(x),p2 ◦q(x), ...,pm ◦q(x)). where pi ◦q : [0,1]m → [0,1] is defined the i-th projection mapping. note that [0,1]m(m-th−power of [0,1]) is considered as a poset with the point wise order ≤, where m is an arbitrary ordinal number (we make an appointment that m = {n | n < m} when m > 0), ≤ is defined by x ≤ y ⇔ pi(x) ≤ pi(y) for each i ∈ m (x,y ∈ [0,1]m), and pi : [0,1]m → [0,1] is the i-th projection mapping (i ∈ m). it is easy to see that 0= (0,0, ...,0)is the smallest value in [0,1]m and 1= (1,1, ...,1) is the largest value in [0,1]m. 4 int. j. anal. appl. (2023), 21:53 definition 2.12 [11]an m-polar fuzzy set q in x is called an m-polar fuzzy subalgebra of x if it satisfies the following conditions for all x,y ∈ x : q(x ∗y)≥ inf{q(x),q(y)}. thats is (∀ x,y ∈ x) (pi ◦q(x ∗y)≥ inf{pi ◦q(x),pi ◦q(y)}) for each i =1,2, ...,m. definition 2.13 [11]an m-polar fuzzy set q is called an m-polar fuzzy ideal of x if it satisfies the following conditions for all x,y ∈ x : (f1) q(0)≥ q(x), (f2) q(x)≥ inf{q(x ∗y),q(y)}. thats is (∀ x,y ∈ x) (pi ◦q(x)≥ inf{pi ◦q(x ∗y),pi ◦q(y)}) for each i =1,2, ...,m. definition 2.14 [12] an m-polar fuzzy set q in x is called a m-polar fuzzy positive implicative ideal if it satisfies (f1) and (f3) q(x ∗z)≥ inf{q((x ∗y)∗z),q(y ∗z)} for all x,y,z ∈ x. thats is (∀ x,y ∈ x) (pi ◦q(x ∗y)≥ inf{pi ◦q(x),pi ◦q(y)}) for each i =1,2, ...,m. definition 2.15 [11]an m-polar fuzzy set q in x is called a m-polar fuzzy commutative ideal if it satisfies (f1) and (f4) q(x ∗ (y ∧x))≥ inf{q((x ∗y)∗z),q(z)} for all x,y,z ∈ x. thats is (∀ x,y ∈ x) (pi ◦q(x ∗ (y ∧x))≥ inf{pi ◦q((x ∗y)∗z),pi ◦q(z)}) for each i = 1,2, ...,m. 3. doubt m-polar fuzzy subalgebras in this section, we introduce doubt m-polar fuzzy subalgebras in bck-algebras and investigate some of their properties. definition 3.1 let q be an m-polar fuzzy subset of x, then q is called a doubt m-polar fuzzy subalgebra of x if it satisfies the following conditions: (for all x,y ∈ x)(q(x ∗y)≤ sup{q(x),q(y)}). thats is (∀ x,y ∈ x) (pi ◦q(x ∗y)≤ sup{pi ◦q(x),pi ◦q(y)}) for each i =1,2, ...,m. example 3.2 consider a bck-algebra x = {0,a,b,c} with the following cayley table: ∗ 0 a b c 0 0 0 0 0 a a 0 0 a b b a 0 b c c c c 0 int. j. anal. appl. (2023), 21:53 5 defined a 4-polar fuzzy set q : x → [0,1]4 by: q(x)=   (0.1,0.3,0.4,0.5), if x =0, (0.2,0.4,0.6,0.7), if x = a, (0.3,0.5,0.7,0.8), if x = b, (0.4,0.6,0.8,0.9), if x = c.   by routine calculation, we know that q is a doubt m-polar fuzzy subalgebra of x. for any m-polar fuzzy set q on x and σ =(σ1,σ2, ...,σm)∈ [0,1]m, the set q[σ] = {x ∈ x : q(x)≤ σ}, is called the doubt σ -level cut set of q and the set qs[σ] = {x ∈ x : q(x) < σ}, is called the doubt strong σ -level cut set of q. theorem 3.3 let q be an m-polar fuzzy set over x and σ = (σ1,σ2, ...,σm) ∈ [0,1]m. if q is a doubt m-polar fuzzy subalgebra of x, then the nonempty doubt σ -level cut set of q is a subalgebra of x. proof. suppose that q is doubt m-polar fuzzy subalgebra of x and q[σ] 6= φ. for any x,y ∈ q[σ] we have q(x)≤ σ and q(y)≤ σ. it follows from definition (3.1) that q(x ∗y)≤ sup{q(x),q(y)}≤ σ, therefore, x ∗y ∈ q[σ]. hence, q[σ]is a subalgebra of x. � theorem 3.4 let q be an m-polar fuzzy set over x and let σ = (σ1,σ2, ...,σm) ∈ [0,1]m. if q is a doubt m-polar fuzzy subalgebra of x, then the nonempty doubt strong σ -level cut set of q is a subalgebra of x. proof. suppose that q is doubt m-polar fuzzy subalgebra of x and qs [σ] 6= φ. for any x,y ∈ qs [σ] we have q(x) < σand q(y) < σ. it follows from definition (3.1) that q(x ∗y)≤ sup{q(x),q(y)} < σ, therefore, x ∗y ∈ qs [σ] . hence, qs [σ] is a subalgebra of x. � theorem 3.5 let q be an m-polar fuzzy set over x and q[σ] 6= φ is subalgebra of x for all σ =(σ1,σ2, ...,σm)∈ [0,1]m. then q is a doubt m-polar fuzzy subalgebra of x. 6 int. j. anal. appl. (2023), 21:53 proof. assume the contrary, that there exist a,b ∈ x such that q(a∗b) > sup{q(a),q(b)} thus, there is σ =(σ1,σ2, ...,σm)∈ [0,1]m such that q(a∗b) > σ ≥ sup{q(a),q(b)}. so, one can conclude that a,b ∈ q[σ] andt a∗b /∈ q[σ]. but this contradicts that q[σ] is subalgebra of x. therefore, q(x ∗ y) ≤ sup{q(x),q(y)} for all x,y ∈ x. hence, q is a doubt m-polar fuzzy subalgebra of x. � theorem 3.6 let q be an m-polar fuzzy set over x and qs [σ] 6= φ be subalgebra of x for all σ =(σ1,σ2, ...,σm)∈ [0,1]m. then q is a doubt m-polar fuzzy subalgebra of x. proof. suppose that there exist a,b ∈ x such that q(a∗b) > sup{q(a),q(b)}. so, there exists σ =(σ1,σ2, ...,σm)∈ [0,1]m such that q(a∗b) > σ > sup{q(a),q(b)}. consequently, a,b ∈ qs [σ] and a∗b /∈ qs [σ] . but this contradics that qs [σ] is subalgebra of x. hence, q is a doubt m-polar fuzzy subalgebra of x. � proposition 3.7 if q is a doubt m-polar fuzzy subalgebra of x, then q(0)≤ q(x)for all x ∈ x. proof. for any x ∈ x, we have q(0) = q(x ∗ x) ≤ sup{q(x),q(x)} = q(x) for all x ∈ x. this completes the proof. � proposition 3.8 if every doubt m-polar fuzzy subalgebra q of x satisfies q(x ∗y)≤ q((y) for all x,y ∈ x, then q is constant. proof. note that in a bck-algebra x, x ∗ 0 = x for all x ∈ x, since q(x ∗ y) ≤ q((y), we haveq(x) = q(x ∗0) ≤ q(0), it follows from proposition (3.7) that q(x) = q(0) for all x,y ∈ x. therefore, q is constant. � int. j. anal. appl. (2023), 21:53 7 for elements x and y of a bck-algebra x, let us write x ∗yn for (...((x ∗y)∗y)∗ ...)∗y and xn ∗y for x ∗ (...∗ ((x ∗ (x ∗y))...) where y and x occur n times respectively. proposition 3.9 let q be a doubt m-polar fuzzy subalgebra of x and n ∈ n. then for any x ∈ x, we have (1) q(xn ∗x)≤ q(x) , if n is odd. (2) q(xn ∗x)= q(x) , if n is even. proof. 1. if n is odd, then n = 2k − 1 for some positive integer k. let x ∈ x, then q(x ∗ x) = q(0)≤ q(x). now assume that q(x2k−1 ∗x)≤ q(x) for some positive integer k. then, q(x2(k+1)−1 ∗x) = q(x2k+1 ∗x) = q(x2k−1 ∗ (x ∗ (x ∗x))) = q(x2k−1 ∗ (x ∗0)) = q(x2k−1 ∗x) ≤ q(x) this proves (1). similarly, we can prove (2). � 4. doubt m-polar fuzzy ideals. in this section, we introduce the notions of doubt m-polar fuzzy ideals in bck-algebras. several fundamental properties and theorems related to these concepts are also studied and investigated. definition 4.1 an m-polar fuzzy set q in x is called a doubt m-polar fuzzy ideal if it satisfies the following conditions for all x,y ∈ x: (1) q(0)≤ q(x), (2) q(x)≤ sup{q(x ∗y),q(y)}. thats is (∀ x,y ∈ x) (pi ◦q(x)≤ sup{pi ◦q(x ∗y),pi ◦q(y)}) for each i =1,2, ...,m. example 4.2 consider a bck-algebra x = {0,a,b,c} which is given in example (3.2) defined a 4-polar fuzzy set q : x → [0,1]4 by: q(x)=   (0.1,0.2,0.3,0.4), if x =0 (0.3,0.5,0.6,0.8), if x = a,b (0.5,0.6,0.7,0.8), if x = c   . by routine calculation, we know that q is a doubt m-polar fuzzy ideal of x. proposition 4.3 let q be a doubt m-polar fuzzy ideal of x. if ≤ is a partial ordering on x, then q(x)≤ q(y)for all x,y ∈ xsuch that x ≤ y. 8 int. j. anal. appl. (2023), 21:53 proof. assume that ≤ is a partial ordering on x defined by x ≤ y if and only if x ∗ y = 0 for all x,y ∈ x. then q(x) ≤ sup{q(x ∗y),q(y)} = sup{q(0),q(y) = q(y). this completes the proof. � proposition 4.4 let q be an m-polar fuzzy ideal of x. if xsatisfies the following assertion: (∀x,y,z ∈ x)(x ∗y ≤ z), then q(x)≤ sup{q(y),q(z)} for all x,y,z ∈ x. proof. assume that x ∗y ≤ z that valid in x. then q(x ∗y)≤ sup{q((x ∗y)∗z),q(z}=sup{q(0),q(z)}= q(z), for all x,y,z ∈ x. it follows that q(x)≤ sup{q(x ∗y),q(y)≤ sup{q(y),q(z)}, for all x,y,z ∈ x. this completes the proof. � proposition 4.5 let q be a doubt m-polar fuzzy ideal of x. then q(x ∗y)≤ q((x ∗y)∗y)⇔ q((x ∗z)∗ (y ∗z))≤ q((x ∗y)∗z), for all x,y,z ∈ x. proof. note that ((x ∗ (y ∗z))∗z)∗z = ((x ∗z)∗ (y ∗z))∗z ≤ (x ∗y)∗z for all x,y,z ∈ x. assume that q(x ∗y)≤ q((x ∗y)∗y) for all x,y,z ∈ x. it follows from (i2) and proposition (4.3) that q((x ∗z)∗ (y ∗z)) = q((x ∗ (y ∗z))∗z) ≤ q(((x ∗ (y ∗z))∗z)∗z) ≤ q((x ∗y)∗z), for all x,y,z ∈ x. conversely, suppose that q((x ∗z)∗ (y ∗z))≤ q((x ∗y)∗z), (4.1) int. j. anal. appl. (2023), 21:53 9 for all x,y,z ∈ x. if we substitute z for y in equations (4.1). then q(x ∗z) = q((x ∗z)∗0) = q((x ∗z)∗ (z ∗z)) ≤ q((x ∗z)∗z) for all x,z ∈ x by using (iii) and (i1) � proposition 4.6 let q be a doubt m-polar fuzzy ideal of x. then q(x ∗y)≤ sup{q(x ∗z),q(z ∗y)}, for all x,y,z ∈ x. proof. note that ((x ∗ y)∗ (x ∗ z)) ≤ (z ∗ y) for all x,y,x ∈ x. it follows from proposition (4.3), that q((x ∗y)∗ (x ∗z))≤ q(z ∗y). now, by definition (4.1), we have q(x ∗y) ≤ sup{q((x ∗y)∗ (x ∗z)),q(x ∗z)} ≤ sup{q(x ∗z),q(z ∗y)}, for all x,y,z ∈ x. this completes the proof. � proposition 4.7 let q be a doubt m-polar fuzzy ideal of x. then q(x ∗ (x ∗y))≤ q(y), for all x,y ∈ x. proof. let q be a doubt m-polar fuzzy ideal of x. then for all x,y ∈ x, we have q(x ∗ (x ∗y)) ≤ sup{q((x ∗ (x ∗y))∗y),q(y)} = sup{q((x ∗y)∗ (x ∗y)),q(y)} = sup{q(0),q(y)} = q(y). this completes the proof. � theorem 4.8 let q be a m-polar fuzzy set over x and let σ ∈ [0,1]m. if q is a doubt m-polar fuzzy ideal of x, then the nonempty doubt σ -level cut set of q is an ideal of x. 10 int. j. anal. appl. (2023), 21:53 proof. assume that q[σ] 6= φ for σ ∈ [0,1]m. clearly, 0∈ q[σ]. let x ∗y ∈ q[σ] and y ∈ q[σ]. then q(x ∗y)≤ σ and q(y)≤ σ. it follows from definition (4.1) that q(x)≤ sup{q(x ∗y),q(y)}≤ σ. so, x ∈ q[σ]. therefore q[σ] is an ideal of x. � theorem 4.9 let q be an m-polar fuzzy set over x and let σ ∈ [0,1]m. if q is a doubt m-polar fuzzy ideal of x, then the nonempty doubt strong σ -level cut set of q is an ideal of x. proof. assume that qs [σ] 6= φ for σ ∈ [0,1]m. clearly, 0∈ qs [σ] . let x ∗y ∈ qs [σ] and y ∈ qs [σ] . then q(x ∗y) < σ and q(y) < σ. it follows from definition (4.1) that q(x)≤ sup{q(x ∗y),q(y)} < σ. so, x ∈ qs [σ] . therefore qs [σ] is an ideal of x. � theorem 4.10 let q be a m-polar fuzzy set over x and assume that q[σ] 6= φ is an ideal of x for all σ ∈ [0,1]m. then q is a doubt m-polar fuzzy ideal of x. proof. assume that q[σ] 6= φ is an ideal of x for all σ ∈ [0,1]m. if there exist h ∈ x such that q(0) > q(h) then q(0) > σh ≥ q(h), for some σh ∈ [0,1]m. then 0 /∈ q[σh].which is contradiction. hence q(0) ≤ q(h), for all x ∈ x. now, assume that there exist h,q ∈ x such that q(h) > sup{q(h∗q),q(q)}. then there exist β ∈ [0,1]m such that q(h) > β ≥ sup{q(h∗q),q(q)}. it follow that h∗q ∈ q[β] and q ∈ q[β], but h /∈ q[β]. this is impossible, and so q(x)≤ sup{q(x ∗y),q(y)}, for all x,y ∈ x. therefore, q is a doubt m-polar fuzzy ideal of x. � theorem 4.11 let q be an m-polar fuzzy set over x and assume that qs [σ] 6= φ is an ideal of x for all σ ∈ [0,1]m. then q is a doubt m-polar fuzzy ideal of x. proof. assume that qs [σ] 6= φ is an ideal of x for all σ ∈ [0,1]m. if there exist h ∈ x such that q(0) > q(h) then q(0) > σh > q(h) for some σh ∈ [0,1]m. then 0 /∈ qs[σh], which is a contradiction. hence q(0) ≤ q(h), for all x ∈ x. now, assume that there exist h,q ∈ x such that q(h) > sup{q(h∗q),q(q)}. then there exist β ∈ [0,1]m such that q(h) > β > sup{q(h∗q),q(q)}. it follow that h∗q ∈ qβ and q ∈ qs[β] , but h /∈ q s [β] . this is impossible, and so q(x)≤ sup{q(x ∗y),q(y)}, for all x,y ∈ x. therefore, q is a doubt m-polar fuzzy ideal of x. � proposition 4.12 let q be a doubt m-polar fuzzy ideal of x. if the inequality x ∗y ≤ z holds in x, then q(x)≤ sup{q(y),q(z)}, for all x,y,z ∈ x. int. j. anal. appl. (2023), 21:53 11 proof. let q be a doubt m-polar fuzzy ideal of x and let x,y,z ∈ x be such that x ∗ y ≤ z. then (x ∗y)∗z =0, and so q(x) ≤ sup{q(x ∗y),q(y)} ≤ sup{sup{q((x ∗y)∗z),q(z)},q(y)} = sup{sup{q(0),q(z)},q(y)} = sup{q(y),q(z)}. this completes the proof. � theorem 4.13 in a bck-algebra x, every doubt m-polar fuzzy ideal of x is a doubt m-polar fuzzy subalgebra of x. proof. let q be a doubt m-polar fuzzy ideal of a bckalgebra x. for any x,y ∈ x, we have q(x ∗y) ≤ sup{q((x ∗y)∗x),q(x)} = sup{q((x ∗x)∗y),q(x)} = sup{q(0∗y),q(x)} = sup{q(0),q(x)} ≤ sup{q(x),q(y)}. hence, q is a doubt m-polar fuzzy subalgebra of a bck-algebra x. � example 4.14 in example (4.2), q is a doubt m-polar fuzzy ideal of x, so that q is a doubt m-polar fuzzy subalgebra of x. the converse of theorem (4.13) is not true in general as seen in the following example. example 4.15 the doubt m-polar fuzzy subalgebra q in example (3.2) is not a doubt m-polar fuzzy ideal of x, since q(b)= (0.3,0.5,0.7,0.8)� (0.2,0.4,0.6,0.7)= sup{q(b∗a),q(a)}. we give a condition for a doubt m-polar fuzzy subalgebra to be a doubt m-polar fuzzy ideal in a bck-algebra. theorem 4.16 let q be a doubt m-polar fuzzy subalgebra of x. if the inequality x ∗ y ≤ z holds in x, then q is a doubt m-polar fuzzy ideal of x. 12 int. j. anal. appl. (2023), 21:53 proof. let q be a doubt m-polar fuzzy subalgebra of x. then from proposition (4.5), q(0)≤ q(x) for all x ∈ x. as x∗y ≤ z holds in x, then from proposition (4.12), we get q(x)≤ sup{q(y),q(z)} for all x,y,z ∈ x. since x ∗ (x ∗ y) ≤ y for all x,y ∈ x, then q(x) ≤ sup{q(x ∗ y),q(y)}. hence, q is a doubt m-polar fuzzy ideal of x. � definition 4.17 let (x,∗,0)and (x ′ ,∗ ′ ,0 ′ ) be two bck-algebras, a homomorphism is a map f : x → x ′ satisfying f (x ∗y)= f (x)∗ f (y) for all x,y ∈ x. definition 4.18 let f : x → x ′ be a homomorphism of bck-algebras and let q be a m-polar fuzzy set in x ′ , then the m-polar fuzzy set qf in x define by qf = q◦ f (i.e.,qf (x) = q(f (x) for all x ∈ x) is called the preimage of q under f . theorem 4.19 an onto homomorphic preimage of a doubt m-polar fuzzy ideal is a doubt m-polar fuzzy ideal. proof. let f : x → x ′ be an onto homomorphism of bck-algebras, q be a doubt m-polar fuzzy ideal in x ′ , and qf be preimage of q under f . for any x ′ ∈ x ′ there exist x ∈ x such that f (x)= x ′ . we have qf (0)= q(f (0))= q(0 ′ )≤ q(x ′ )= q(f (x))= qf (x). let x ∈ x and y ′ ∈ x ′ , then there exist y ∈ x such that f (y)= y ′ . we have qf (x) = q(f (x)) ≤ sup{q(f (x)∗y ′ ),q(y ′ )} = sup{q(f (x)∗ f (y)),q(f (y))} = sup{q(f (x ∗y)),q(f (y))} = sup{qf ((x ∗y)),qf ((y))}. hence, qf is a doubt m-polar fuzzy ideal of x. � proposition 4.20 let q be a doubt m-polar fuzzy ideal of x. then the sets j = {x ∈ x : q(x)= q(0)} is an ideal of x. proof. obviously, 0 ∈ j. hence, j 6= φ. now, let x,y ∈ j such that x ∗ y,y ∈ j. then q(x ∗ y) = q(0) = q(y).now, q(x) ≤ sup{q(x ∗ y),q(y)} = q(0) since q be a doubt m-polar fuzzy ideal of x,q(0) ≤ q(x). therefore, q(0) = q(x). it follows that x ∈ j, for all x,y ∈ x. therefore, j is an ideal of x. � int. j. anal. appl. (2023), 21:53 13 for any elements ω ∈ x, we consider the sets: xω = {x ∈ x | q(x)≤ q(ω)}. clearly, ω ∈ xω. so that xω is a nonempty set of x. theorem 4.21 let ω be any element of x. if q is a doubt m-polar fuzzy ideal of x, then xω is an ideal of x. proof. clearly, 0 ∈ xω. let x,y ∈ x be such that x ∗ y ∈ xω and y ∈ xω. then q(x ∗ y) ≤ q(ω),q(y)≤ q(ω) it follows that from definition (4.1), that q(x)≤ sup{q(x ∗y),q(y)}≤ q(ω) hence,x ∈ xω. therefore xω is an ideal of x. � theorem 4.22 let ω ∈ x and let q be an m-polar fuzzy set over x. then if xω is an ideal of x, then the following assertion is valid for all x,y,z ∈ x (a1) q(x)≥ sup{q(y ∗z),q(z)}⇒ q(x)≥ q(y), if q satisfies (a1) and (a2) q(0)≤ q(x) for all x ∈ x. then xω is ideal for all ω ∈ im(q). proof. (1) assume that xω is ideal of x for ω ∈ x. let x,y,z ∈ x be such that q(x)≥ sup{q(y ∗ z),q(z)}. then y ∗ z ∈ xω and z ∈ xω where ω = x. since xω is an ideal of x, it follows that y ∈ xω for ω = x.hence, q(y) ≤ q(ω) = q(x). (2) let ω ∈ im(q) and suppose that q satisfies (a1) and (a2).clearly, 0 ∈ xωby (a2). let x,y ∈ x be such that x ∗ y ∈ xω and y ∈ xω. then q(x ∗y)≤ q(ω) and q(y)≤ q(ω), which implies that sup{q(x ∗y),q(y)}≤ q(ω). it follows from (a1) that q(ω)≥ q(x). thus, x ∈ xω, and therefore xω is ideal of x. � 5. doubt m-polar fuzzy positive implicative ideals. in this section, we introduce doubt m-polar fuzzy positive implicative ideals in bckalgebra, several fundamental properties and theorems related to this concept are and theorems related to this concept are also studied and investigated. definition 5.1 an m-polar fuzzy set q in x is called a doubt m-polar fuzzy positive implicative ideal if it satisfies the following conditions for all x,y,z ∈ x: (1) q(0)≤ q(x). (2) q(x ∗z)≤ sup{q((x ∗y)∗z),q(y ∗z)}. 14 int. j. anal. appl. (2023), 21:53 thats is (∀ x,y ∈ x) (pi ◦q(x ∗z)≤ sup{pi ◦q((x ∗y)∗z),pi ◦q(y ∗z)}) for each i =1,2, ...,m. example 5.2 consider a bckalgebra x = {0,1,2,3,4} with the following cayley table: ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 0 0 2 2 2 0 0 2 3 3 3 3 0 3 4 4 4 4 4 0 defined a 4-polar fuzzy set q : x → [0,1]4by: q(x)=   (0.3,0.2,0.1,0.3), if x =0, (0.6,0.3,0.8,0.6), if x =1, (0.5,0.2,0.6,0.5), if x =2, (0.4,0.2,0.3,0.4), if x =3, (0.7,0.4,0.9,0.7), if x =4.   by routine calculation, we know that q is a doubt m-polar fuzzy positive implicative ideal of x. theorem 5.3 any doubt m-polar fuzzy positive implicative ideal of x is a doubt mpolar fuzzy ideal of x. proof. let q be a doubt mpolar fuzzy positive implicative ideal of x. then q(0)≤ q(x). by taking z =0 in definition(5.1) we have , q(x)≤ sup{q(x ∗y),q(y)}. hence, q is a doubt m-polar fuzzy ideal of x. � the converse of theorem (5.3) is not true in general as seen in the following example. example 5.4 consider a bckalgebra x = {0, f , j, l} with the following cayley table: ∗ 0 f j 1 0 0 0 0 0 f f 0 0 f j j f 0 j 1 1 1 1 0 defined a 4-polar fuzzy set q : x → [0,1]3 by: q(x)=   (0.3,0.3,0.3), if x =0, (0.5,0.5,0.8), if x = f , j, (0.3,0.3,0.3), if x = l.   int. j. anal. appl. (2023), 21:53 15 by routine calculation, we know that q is a doubt m-polar fuzzy ideal of x. but it is not a doubt m-polar fuzzy positive implicative ideal of x. since q(j ∗ f ) = q(f ) = (0.5,0.5,0.8) � sup(q((j ∗ f )∗ f ),q(f ∗ f )} = sup{q(0),q(0)} = q(0) = (0.3,0.3,0.3). we now give the condition for a doubt m-polar fuzzy ideal to be a doubt m-polar fuzzy positive implicative ideal of x. theorem 5.5 an m-polar fuzzy set of x is a doubt m-polar fuzzy positive implicative ideal of x if and only if it is a doubt mpolar fuzzy ideal of x and the following condition is valid for all x,y ∈ x. q(x ∗y)≤ q((x ∗y)∗y). (5.1) proof. suppose q is a doubt m-polar fuzzy positive implicative ideal of x. by theorem (5.3), q is a doubt mpolar fuzzy ideal of x. if z is replaced by y in definition (5.1) , then q(x ∗y) ≤ sup{q(x ∗y)∗y),q(y ∗y)} = sup{q((x ∗y)∗y),q(0)} = q((x ∗y)∗y). for all x,y ∈ x. conversely, let q be a doubt m-polar fuzzy ideal of x. then, q(0) ≤ q(x) for all x ∈ x. also, since ((x ∗z)∗z)∗ (y ∗z)≤ (x ∗z)∗y =(x ∗y)∗z. for all x,y ∈ x, it follow by proposition (5.3) that q(((x ∗z)∗z)∗ (y ∗z))≤ q((x ∗y)∗z). now, by (5.1) q(x ∗z) ≤ q((x ∗z)∗z) ≤ sup{q(((x ∗z)∗z)∗ (y ∗z)),q((y ∗z)} ≤ sup{q((x ∗y)∗z),q(y ∗z)}. 16 int. j. anal. appl. (2023), 21:53 hence, q is a doubt m-polar fuzzy positive implicative ideal of x. � theorem 5.6 in positive implicative bck-algebra x, every doubt m-polar fuzzy ideal is a doubt mpolar fuzzy positive implicative ideal. proof. let q be a doubt m-polar fuzzy ideal of a positive implicative bck-algebra x, we have ((x ∗z)∗ ((x ∗y)∗z))∗ (y ∗z) = ((x ∗z∗)∗ (y ∗z))∗ ((x ∗y)∗z) = ((x ∗y)∗z)∗ ((x ∗y)∗z) = 0. and so, ((x ∗z)∗ ((x ∗y)∗z))∗ (y ∗z)=0, i.e.,((x ∗z)∗ ((x ∗y)∗z))≤ (y ∗z), for all x,y,z ∈ x. since q is a doubt m-polar fuzzy ideal, it follow from proposition (4.4) that q(x ∗z)≤ sup{q((x ∗y)∗z),q(y ∗z)}. hence, q is a doubt m-polar fuzzy positive implicative ideal of x. � theorem 5.7 let q be an m-polar fuzzy set of a bckalgebra x. then q is a doubt m-polar fuzzy positive implicative ideal of x if and only if it satisfies (∀σ ∈ [0,1]m)(q[σ] 6= φ ⇒ q[σ]is a positive implicative ideal of x for all σ ∈ [0,1] m). proof. since q is a doubt m-polar fuzzy positive implicative ideal of x, then q is a doubt mpolar fuzzy ideal of x and so every σ-level cut set q[σ] of q is an ideal of x. let x,y,z ∈ x be such that (x ∗y)∗z ∈ q[σ] and (y ∗z)∈ q[σ]. then q((x ∗y)∗z)≤ σ and q(y ∗z)≤ σ. it follow that : q(x ∗z)≤ sup{q((x ∗y)∗z),q(y ∗z)}≤ σ. so that (x ∗z)∈ q[σ]. hence q[σ] is a positive implicative ideal of x. conversely, assume that q[σ] 6= φ is a positive implicative ideal of x for all σ ∈ [0,1]m. if there exist h ∈ x such that q(0) > q(h), then q(0) > σh ≥ q(h) for some σh ∈ [0,1]m. then 0 /∈ q[σh], which is contradiction. hence q(0)≤ q(h), for all x ∈ x. now, assume that there exist h,k,q ∈ x such that q(h∗q) > sup{q((h∗k)∗q),q(k ∗q)}. then there exists β ∈ [0,1]m such that q(h∗q) > β ≥ sup{q((h∗k)∗q),q(k ∗q)}. it follow that (h ∗ k) ∗ q ∈ q[β] and k ∗ q ∈ q[β], but h ∗ q /∈ q[β]. this is impossible, and so q(x ∗ z) ≤ sup{q((x ∗ y)∗ z),q(y ∗ z)} for all x,y,z ∈ x. therefore, q is a doubt m-polar fuzzy positive implicative ideal of x. � int. j. anal. appl. (2023), 21:53 17 corollary 5.8 if q is a doubt m-polar fuzzy positive implicative ideal of a bck-algebra x, then qs [σ] 6= φ is a positive implicative ideal of x for all σ ∈ [0,1]m. proof. straightforward. � theorem 5.9 let ω be an element of a bck-algebra x. if q is a doubt m-polar fuzzy positive implicative ideal of x, then xω is a positive implicative ideal of x. proof. let q is a doubt m-polar fuzzy positive implicative ideal of bck-algebra x, then it is a doubt mpolar fuzzy ideal of x and so xω is an ideal. thus 0 ∈ xω. now, assume that (x ∗ y)∗ z ∈ xω and y ∗z ∈ xω for any x,y,z ∈ x. then q(((x ∗y)∗z)≤ q(ω) and q(y ∗z)≤ q(ω). it follow from definition (5.1) that q(x ∗z)≤ sup{q((x ∗y)∗z,q(y ∗z)}≤ q(ω). thus q(x∗z)≤ q(ω). hence , x∗z ∈ xω and therefore xω and therefore xω is a positive implicative ideal of x. � 6. doubt m-polar fuzzy commutative ideals. in this section, we introduce doubt m-polar fuzzy commutative ideals in bckalgebra, several fundamental properties and theorems related to this concept are and theorems related to this concept are also studied and investigated. definition 6.1 an m-polar fuzzy set q in x is called a doubt m-polar fuzzy commutative ideal if it satisfies the following conditions for all x,y,z ∈ x: (1) q(0)≤ q(x). (2)q(x ∗ (y ∧x))≤ sup{q((x ∗y)∗z),q(z)}. that is, (∀ x,y ∈ x) (pi ◦q(x ∗ (y ∧x))≤ sup{pi ◦q((x ∗y)∗z),pi ◦q(z)}) for each i = 1,2, ...,m. example 6.2 consider a bckalgebra x = {0,a,b,c} which is given in example (3.2) define an m-polar fuzzy set q : x → [0,1]m by: q(x)=   α =(α1,α2, . . . . . . ,αm), if x =0, β =(β1,β2, . . . . . . ,βm), if x = a, γ =(γ1,γ2, . . . . . . ,γm), if x = b,c.   where α,β,γ ∈ [0,1]m and α < β < γ. by routine calculation, we know that q is a doubt m-polar fuzzy commutative ideal of x. 18 int. j. anal. appl. (2023), 21:53 theorem 6.3 every doubt m-polar fuzzy commutative ideal of bck-algebra x is a doubt mpolar fuzzy ideal of x. proof. let q be a doubt mpolar fuzzy commutative ideal of bck-algebra x. then q(0)≤ q(x). now, for any x,y,z ∈ x, we have q(x) = q(x ∗ (0∧x)) ≤ sup{q(x ∗0)∗z),q(z)} = sup{q(x ∗z),q(z)}. hence, q is a doubt m-polar fuzzy commutative ideal of x. � the converse of theorem (6.3) is not true in general as seen in the following example. example 6.4 consider a bckalgebra x = {0,1,2,3,4} with the following cayley table: ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 0 0 3 3 3 3 0 0 4 4 4 4 3 0 defined an m-polar fuzzy set q : x → [0,1]m by: q(x)=   α =(α1,α2, . . . . . . ,αm), if x =0, β =(β1,β2, . . . . . . ,βm), if x =1, γ =(γ1,γ2, . . . . . . ,γm), if x =2,3,4.   where α,β,γ ∈ [0,1]m and α < β < γ. by routine calculation, we know that q is a doubt m-polar fuzzy ideal of x.but it is not a doubt m-polar fuzzy commutative ideal of x. since: q(2∗ (3∧2))� sup{q(2∗3)∗0),q(0)} . now we give the condition for a doubt m-polar fuzzy ideal to be a doubt m-polar fuzzy commutative ideal of x. theorem 6.5 let q be a doubt m-polar fuzzy ideal of a bck-algebra x. then q is a doubt m-polar fuzzy commutative ideal of x if and only if the following condition is valid for all x,y ∈ x q(x ∗ (y ∧x))≤ q(x ∗y). (6.1) int. j. anal. appl. (2023), 21:53 19 proof. assume that q is a doubt m-polar fuzzy commutative ideal of a bck-algebra x. we have: q(x ∗ (y ∧x))≤ sup{q(x ∗y)∗z),q(z)}. by taking z =0; then we get q (x ∗ (y ∧x))≤ q(x ∗y). conversely, suppose that a doubt m-polar fuzzy ideal q of a bckalgebra x satisfies the condition (6.1). then q(x ∗y)≤ sup{q((x ∗y)∗z) ,q(z)} . (6.2) for all x,y ∈ x. using (6.1) and (6.2), we have q(x ∗ (y ∧x))≤ sup{q(x ∗y)∗z),q(z)}. therefore, q is a doubt m-polar fuzzy commutative ideal of x. � theorem 6.6 in commutative bck-algebra x, every doubt m-polar fuzzy ideal is a doubt mpolar fuzzy commutative ideal. proof. let q be a doubt m-polar fuzzy ideal of a commutative bck-algebra x, we have ((x ∗ (y ∧x))∗ ((x ∗y)∗z))∗z = ((x ∗ (y ∧x))∗z)∗ ((x ∗y)∗z) ≤ (x ∗ (y ∧x))∗ (x ∗y) = (x ∧y)∗ (y ∧x) = 0. and so ((x ∗(y ∧x))∗((x ∗y)∗z))∗z =0, i.e.,((x ∗(y ∧x))∗((x ∗y)∗z))≤ z for all x,y,z ∈ x. since q is a doubt m-polar fuzzy ideal, it follow from proposition (4.4) q(x ∗ (y ∧x))≤ sup{q((x ∗y)∗z),q(z)}. hence, q is a doubt m-polar fuzzy commutative ideal of x. � theorem 6.7 let q be an m-polar fuzzy set of a bckalgebra x. then q is a doubt m-polar fuzzy commutative ideal of x if and only if it satisfies (∀σ ∈ [0,1]m)(q[σ] 6= φ ⇒ q[σ] is a commutative ideal of x for all σ ∈ [0,1] m). (6.3) proof. let q is a doubt m-polar fuzzy commutative ideal of x. then q is a doubt m-polar fuzzy ideal of x and so every σ-level cut set q[σ] of q is an ideal of x. let x,y,z ∈ x be such that (x ∗y)∗z ∈ q[σ] and z ∈ q[σ]. then q((x ∗y)∗z)≤ σ and q(z)≤ σ. it follow that : q(x ∗ (y ∧x))≤ sup{q((x ∗y)∗z),q(z)}≤ σ. so that x ∗ (y ∧x)∈ q[σ]. hence q[σ] is a commutative ideal of x. 20 int. j. anal. appl. (2023), 21:53 conversely, suppose that (6.3) is valid, q(0) ≤ q(h) for all x ∈ x. let q((x ∗ y) ∗ z) = α = (α1,α2, ...,αm) and q(z) = β = (β1,β2, ...,βm) for all x,y,z ∈ x. then (x ∗ y) ∗ z ∈ q[β] and z ∈ q[β]. without loss of generality, we may assume that β ≤ α. then q[α] ⊆ q[σ], and so z ∈ q[σ]. since q[σ] is a commutative ideal of by hypothesis, we obtain that (x ∗ (y ∧ x)) ∈ q[α], and so q(x ∗(y ∧x))≤ α =sup{q((x ∗y)∗z),q(z)}. therefore, q is a doubt m-polar fuzzy commutative ideal of x. � corollary 6.8 if q is a doubt m-polar fuzzy commutative ideal of a bck-algebra x, then qs [σ] 6= φ is a commutative ideal of x for all σ ∈ [0,1]m. proof. straightforward. � theorem 6.9 let ω be an element of a bck-algebra x. if q is a doubt m-polar fuzzy commutative ideal of x, then xω is a commutative ideal of x. proof. if q is a doubt m-polar fuzzy commutative ideal of bck-algebra x, then it is a doubt mpolar fuzzy ideal of x and so xω is an ideal. thus 0 ∈ xω.now let x ∗ y ∈ xω for any x,y ∈ x. then q(x ∗y)≤ q(ω). it follows from theorem (6.5) that q(x ∗ (y ∧x))≤ q(x ∗y)≤ q(ω). thus, x ∗ (y ∧ x) ∈ xω and therefore x ω. hence, xω is a commutative ideal of x by lemma (2.5). � 7. conclusions we discussed the notion of doubt m-polar fuzzy subalgebras and ideals of bck-algebras.also, we introduced the notion of doubt m-polar fuzzy positive implicative (commutative)ideal of bck -algebras. our defintions probably can be applied in other kinds of doubt m-polar fuzzy ideals of bck-algebras. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [2] y. imai, k. iséki, on axiom systems of propositional calculi, i, proc. japan acad. ser. a. math. sci. 41 (1965), 436-439. https://doi.org/10.3792/pja/1195522378. [3] k. iséki, an algebra related with a propositional calculus, proc. japan acad. ser. a. math. sci. 42 (1966), 26-29. https://doi.org/10.3792/pja/1195522171. https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.3792/pja/1195522378 https://doi.org/10.3792/pja/1195522171 int. j. anal. appl. (2023), 21:53 21 [4] k. iseki, s. tanaka, an introduction to the theory of bck-algebras, math. japon. 23 (1978), 1-26. [5] o.g. xi, fuzzy bck-algebras, math. japon. 36 (1991), 935–942. [6] j. meng, commutative ideals in bck-algebras, pure appl. math. 9 (1991), 49-53. (in chinese). [7] j. meng, y. b. jun, bck-algebras, kyung moon sa co. seoul, korea 1994. [8] y.b.jun, doubt fuzzy bck/bci-algebras, soochow j. math. 20 (1994), 351–358. [9] j. chen, s. li, s. ma, x. wang,m-polar fuzzy sets: an extension of bipolar fuzzy sets, sci. world j. 2014 (2014), 416530. https://doi.org/10.1155/2014/416530. [10] a. al-masarwah, a.g. ahmad, doubt bipolar fuzzy subalgebra and ideals in bck/bci-algebras, j. math. anal.9 (2018), 9-27. [11] a. al-masarwah, a.g. ahmad, m-polar fuzzy ideals of bck/bci-algebras, j. king saud univ. sci. 31 (2019), 1220–1226. https://doi.org/10.1016/j.jksus.2018.10.002. [12] a. al-masarwah, a.g. ahmad, g. muhiuddin, d. al-kadi, generalized m-polar fuzzy positive implicative ideals of bck-algebras, j. math. 2021 (2021), 6610009. https://doi.org/10.1155/2021/6610009. https://doi.org/10.1155/2014/416530 https://doi.org/10.1016/j.jksus.2018.10.002 https://doi.org/10.1155/2021/6610009 1. introduction 2. preliminaries 3. doubt m-polar fuzzy subalgebras 4. doubt m-polar fuzzy ideals. 5. doubt m-polar fuzzy positive implicative ideals. 6. doubt m-polar fuzzy commutative ideals. 7. conclusions references international journal of analysis and applications volume 17, number 2 (2019), 208-225 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-208 fixed points for triangular α−admissible geraghty contraction type mappings in partial b-metric spaces haitham qawaqneh1,∗, mohd salmi noorani1, wasfi shatanawi2,3 and habes alsamir1 1school of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, 43600 ukm, selangor darul ehsan, malaysia 2department of mathematics and general courses, prince sultan university, riyadh 11586, saudi arabia 3department of medical research, china medical university hospital, china medical university, taichung 40402, taiwan ∗corresponding author: haitham.math77@gmail.com abstract. in this paper, we introduce the notion of generalized c−class functions for geraghty contraction type mappings on a set x. we utilize our new notion to prove fixed point results in the setting of triangular weak α−admissible mappings with respect to η in partial b-metric spaces. our results modify and improve many exciting results in the literature. also, we introduce an example and an application to show the validity of our main result. received 2018-09-26; accepted 2018-11-20; published 2019-03-01. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. c-class functions; α−admissible mapping; fixed point; b−metric spaces; partial metric spaces; partial b−metric spaces. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 208 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-208 int. j. anal. appl. 17 (2) (2019) 209 1. introduction and preliminaries one of the most important tools in fixed point theory is banach contraction principle. a lot of authors have extended or generalized this contraction and proved the existence of fixed and common fixed point theorems (for example see [19][28]). in this sequel, bakhtin [7] and czerwik [10] introduced b-metric spaces as a generalization of metric spaces. they proved the contraction mapping principle in b−metric spaces that generalized the famous banach contraction principle in such spaces. after that, several papers have dealt with fixed point theory for single-valued and multi-valued operators in b-metric spaces (for example see [11], [27], [29], [32]). on the other hand, matthews [21] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the contraction mapping principle [8] can be generalized to the partial metric context for applications in program verifications. b−metric spaces [7] and partial metric spaces [21] are two well known generalizations of usual metric spaces. also, the banach contraction principle is a fundamental result in the fixed point theory, which has been used and extended in many different directions. recently, shukla [35] introduced a generalization and unification of partial metric and b-metric spaces as the concept of partial b-metric space. in this section, we recall some useful definitions and auxiliary results that will be needed in the sequel. throughout this paper, n and r denote the set of natural numbers and the set of real numbers, respectively. definition 1.1. ( [7], [10]) let x is a nonempty set and let s ≥ 1 be a given real number. a function d : x × x → [0,∞) is said to be a b−metric space on x if and only if for all x,y,z ∈ x, the following conditions hold: (1) d(x,y) = 0 if and only if x = y, (2) d(x,y) = d(y,x), (3) d(x,z) ≤ s[d(x,y) + d(y,z)]. the triplet (x,d,s) is called a b−metric space. it is well known that the class of b-metric spaces is larger than the class of metric spaces when s = 1, the concept of b-metric space coincides with the concept of metric space. example 1.1. consider the set x = [0, 1] endowed with the function d : x × x → [0,∞) defined by d(x,y) = |x−y|2 for all x,y ∈ x. clearly, (x,d, 3) is a b−metric space but it is not a metric space. int. j. anal. appl. 17 (2) (2019) 210 example 1.2. let x = r and let the mapping d : x ×x → [0,∞) be defined by d(x,y) =| x−y |2 for all x,y ∈ x. then (x,d) is a b-metric space with coefficient s = 2. definition 1.2. [21] let x be a nonempty set. a function p : x ×x → [0,∞) is called a partial metric space if for all x,y,z ∈ x, the following conditions are satisfied: (p1) x = y ⇔ p(x,x) = p(x,y) = p(y,y), (p2) p(x,x) ≤ p(x,y), (p3) p(x,y) = p(y,x), (p4) p(x,y) ≤ p(x,z) + p(z,y) −p(z,z). the pair (x,p) is called a partial metric space(pms). the sequence {xn} in x converges to a point x ∈ x if limn→∞p(xn,x) = p(x,x). also the sequence {xn} is called p−cauchy if the limn,m→∞p(xn,ym) exists. the partial metric space (x,p) is called complete if for every p-cauchy sequence {xn}n∞, there is some x ∈ x such that p(x,x) = lim n→∞ p(xn,x) = lim n,m→∞ p(xn,xm). a basic example of a partial metric space is the pair (r+,p), where p(x,y) = max{x,y} for all x,y ∈ r+. definition 1.3. [35] let x be a nonempty set. a function b : x ×x → [0,∞) is called a b−partial metric space if for all x,y,z ∈ x, the following conditions are satisfied: (pb1) x = y if and only if b(x,x) = b(x,y) = b(y,y), (pb2) b(x,x) ≤ b(x,y), (pb3) b(x,y) = b(y,x), (pb4) there exists a real number s ≥ 1 such that b(x,y) ≤ s[b(x,z) + b(z,y)] − b(z,z). remark 1.1. [35] in a partial b−metric space (x,b) if x,y ∈ x and b(x,y) = 0, then x = y, but the converse may not be true. remark 1.2. [35] it is clear that every partial metric space is a partial b−metric space with coefficient s = 1 and every b−metric space is a partial b−metric space with the same coefficient and zero self-distance. however, the converse of this fact need not hold. example 1.3. [35] let x = r+, p > 1 is a constant and b : x ×x → r+ be defined by b(x,y) = [max{x,y}]p −|x−y|p int. j. anal. appl. 17 (2) (2019) 211 for all x,y ∈ x. then, (x,b) is a partial b−metric space with coefficient s = 2p > 1, but it is neither a b-metric nor a partial metric space. proposition 1.1. [35] let x be a nonempty set such that p is a partial and d is a b−metric with coefficient s > 1 on x. then the function b : x ×x → r+ defined by b(x,y) = p(x,y) + d(x,y) for all x,y ∈ x is a partial b-metric on x, that is, (x,b) is a partial b−metric space. definition 1.4. [35] let (x,b) be be a partial b−metric space with coefficient s. let {xn} be any sequence in x and x ∈ x. then: (i) a sequence {xn}⊆ x converges to a point x ∈ x if limn→∞ b(xn,x) = b(x,x), (ii) a sequence {xn}⊆ x is said to be a cauchy sequence in (x,b) if, for every given � > 0, there exists n(�) ∈ n such that limn,m→∞ b(xn,xm) exists and is finite for all m,n ≥ n(�), (iii) (x,b) is said to be complete partial b−metric space if cauchy sequence {xn}⊆ x there exists x ∈ x such that lim n,m→∞ b(xn,xm) = lim n→∞ b(xn,x) = b(x,x). note that in a partial b−metric space the limit of convergent sequence may not be unique. samet el al. [31] introduced the notion of α−admossible mapping and studied many fixed point theorems. after that several authors used the notion of α-admissible to prove and construct many fixed and common fixed point theorems (see [14][1]). samet et al. [31] presented the notion of α-admissible mapping as follows: definition 1.5. [31] let f : x → x and α : x ×x → [0,∞). then f is called α-admissible if ∀x,y ∈ x with α(x,y) ≥ 1 implies α(fx,fy) ≥ 1. definition 1.6. [17] let t : x → x and α : x ×x → [0,∞). then t is called a triangular α-admissible mapping if (1) t is α-admissible; (2) α(x,z) ≥ 1 and α(z,y) ≥ 1 imply α(x,y) ≥ 1. sintunavarat [32] presented the notion of weak α-admissible mappings as follows: definition 1.7. [32] let x be a nonempty set and let α : x×x → [0,∞) be a given mapping. a mapping f : x → x is said to be a weak α-admissible mappings if the following condition holds: x ∈ x with α(x,fx) ≥ 1 ⇒ α(fx,f2x) ≥ 1. int. j. anal. appl. 17 (2) (2019) 212 remark 1.3. [32] it is customary to write a(x,α) and wa(x,α) to denote the collection of all αadmissible mappings on x and the collection of all weak α-admissible mappings on x. one can verify that a(x,α) ⊆wa(x,α). qawaqneh et al. [23] presented the notion of α-admissible with respect to another function η for the pair of self-mappings s and t on a set x as follows: definition 1.8 ( [23]). let s,t : x → x be two mappings and α : x ×x → [0, +∞) be a function such that the following conditions hold: (1) if α(x,y) ≥ 1, then α(sx,ty) ≥ 1 and α(tsx,sty) ≥ 1; (2) if α(x,z) ≥ 1 and α(z,y) ≥ 1, then α(x,y) ≥ 1. then we say that the pair (s,t) is triangular α-admissible. definition 1.9 ( [23]). let s,t : x → x be two mappings and α,η : x ×x → [0, +∞) be two functions such that the following conditions hold: (1) if α(x,y) ≥ η(x,y), then α(sx,ty) ≥ η(sx,ty) and α(tsx,sty) ≥ η(tsx,sty); (2) if α(x,z) ≥ η(x,z) and α(z,y) ≥ η(z,y), then α(x,y) ≥ η(x,y). then we say that the pair (s,t) is triangular α-admissible with respect to η. lemma 1.1 ( [23]). let s,t : x → x be two mappings and α,η : x × x → [0, +∞) be two functions such that the pair (s,t) is triangular α-admissible with respect to η. assume that there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). define a sequence {xn} in x by sx2n = x2n+1 and tx2n+1 = x2n+2. then α(xn,xm) ≥ η(xn,sxm) for all m,n ∈ n with n < m. in 2014, ansari [4] defined the concept of c-class function as the following: definition 1.10. [4] a mapping f : r+ ×r+ → r is called a c-class function if it is continuous and for s,t ∈ [0,∞), f satisfies the following two conditions: (1) f(s,t) ≤ s; and (2) f(s,t) = s implies that either s = 0 or t = 0. the family of all c−class functions is denoted by c. example 1.4. [4] the following functions f : r+ ×r+ → r are elements in c. (1) f(s,t) = s− t for all s,t ∈ [0,∞). (2) f(s,t) = ks for all s,t ∈ [0,∞), where 0 < k < 1. (3) f(s,t) = s (1+t)r for all s,t ∈ [0,∞), where r ∈ [0,∞). (4) f(s,t) = (s + l)(1/(1+t) r) − l for all s,t ∈ [0,∞), where r ∈ (0,∞), l > 1. int. j. anal. appl. 17 (2) (2019) 213 (5) f(s,t) = s logt+a a for all s,t ∈ [0,∞), where a > 1. (6) f(s,t) = s− ( 1+s 2+s )( t 1+t ) for all s,t ∈ [0,∞). (7) f(s,t) = sβ(s) for all s,t ∈ [0,∞), where β : [0,∞) → [0, 1) is continuous. (8) f(s,t) = s−ϕ(s) for all s,t ∈ [0,∞), where ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 if and only if t = 0. (9) f(s,t) = sh(s,t) for all s,t ∈ [0,∞), where h : [0,∞) → [0,∞) is a continuous function such that h(s,t) < 1 for all s,t ∈ [0,∞). (10) f(s,t) = s− ( 2+t 1+t )t for all s,t ∈ [0,∞). (11) f(s,t) = n √ ln(1 + sn) for all s,t ∈ [0,∞). in 2016, ansari and kaewcharoen [6] gave the definition of a generalized α−η −ψ −ϕ−f-contraction type mapping and proved same fixed point theorems for such mappings in complete metric spaces. definition 1.11 ( [6]). let (x,d) be a metric space and α,η : x×x → [0,∞) be two functions. a mapping t : x → x is said to be a generalized α−η−ψ−ϕ−f -contraction type mapping if α(x,y) ≥ η(x,y) implies ψ(d(tx,ty)) ≤ f(ψ(m(x,y)),ϕ(m(x,y))), where m(x,y) = max{d(x,y),d(x,tx),d(y,ty)}. hussain et al. [15] introduced the concepts of α−η-complete metric spaces and α−η-continuous functions. definition 1.12 ( [15]). let (x,d) be a metric space and α,η : x×x → [0,∞) be two functions. then x is said to be an α,η-complete metric space if every cauchy sequence {xn} in x with α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n converges in x. definition 1.13 ( [15]). let (x,d) be a metric space and α,η : x × x → [0,∞) be two functions. a mapping t : x → x is said to be an α,η-continuous mapping if each sequence {xn} in x with xn → x as n →∞ and α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n implies txn → tx as n →∞. theorem 1.1 ( [6]). let (x,d) be a metric space. assume that α,η : x × x → [0,∞) are two functions and t : x → x is a mapping. suppose that the following conditions are satisfied: (1) (x,d) is an α,η-complete metric space; (2) t is generalized α−η −ψ −ϕ−f -contraction type mapping; (3) t is triangular α-orbital admissible mapping with respect to η; (4) there exists x1 ∈ x such that α(x1,tx1) ≥ η(x1,tx1); (5) t is an α,η-continuous mapping. int. j. anal. appl. 17 (2) (2019) 214 then t has a fixed point x∗ ∈ x. khan et al. [20] introduced the notion of an altering distance function as follows: definition 1.14. [20] a mapping ψ : r+ → r+ is called an altering distance function if the following properties are satisfied: (1) ψ is monotone and nondecreasing; (2) ψ(t) = 0 if an only if t = 0. the set of all altering distance functions is denoted by ψ. in the rest of this paper, we let φ be the set of all functions ϕ : r+ → r+ such that (1) ϕ is continuous. (2) ϕ(t) = 0 if and only if t = 0. 2. main result in this section, we introduce the concept of generalized c−class functions for geraghty contraction type mappings on a set x and we prove fixed point results for self mappings on α,η− partial b−metric space. now, we present the notion of triangular weak α-admissible with respect to another function η for the self-mapping s on a set x. definition 2.1. let s : x → x be a mapping and α,η : x ×x → [0, +∞) be two functions such that the following conditions hold: (1) if α(x,snx) ≥ η(x,snx), then α(snx,sn+1x) ≥ η(snx,sn+1x), (2) if α(x,z) ≥ η(x,z) and α(z,y) ≥ η(z,y), then α(x,y) ≥ η(x,y), for all n ∈ n. then we say that s is triangular weak α-admissible with respect to η. now, we introduce the following example to illustrate our new definition. example 2.1. let x = [0, +∞). define s : x → x by sx = x2. also, define the functions α,η : x×x → [0, +∞) by α(x,y) = ex+y and η(x,y) = ey−x. then s is triangular weak α-admissible with respect to η. proof. if α(x,sx) ≥ η(x,sx), then ex+x 2 ≥ ex 2−x. so x + x2 ≥ x2 − x. so 2x ≥ 0. hence x ≥ 0. since x ≥ −x, then x + x4 ≥ x4 − x. so ex+ 4 ≥ e 4−x. hence α(x,4 ) ≥ η(x,4 ). so α(sx,ty) ≥ η(sx,ty). also, since x2 ≥ −x2, then x2 + y2 ≥ y2 − x2. so ex 2+y2 ≥ ey 2−x2 . hence α(x2,y2) ≥ η(x2,y2). so α(sx,s2x) ≥ η(sx,s2x). also, if α(x,z) ≥ η(x,z), and α(z,y) ≥ η(z,y), then x+z ≥ z−x and z+y ≥ y−z. so x ≥−x and hence x + x2 ≥ x2 −x. therefore ex+y ≥ ey−x. therefore α(x,sx) ≥ η(x,sx). � int. j. anal. appl. 17 (2) (2019) 215 by taking a special case of lemma 1.1and generalize with is triangular weak α−admissible with respect to η, we present a lemma that will be helpful for us to achieve our main result. lemma 2.1. let s : x → x be a mappings and α,η : x×x → r are a functions such that s is triangular weak α−admissible with respect to η. assume that there exist x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). define a sequence {xn} in x by sxn = xn+1. then α(xn,xm) ≥ η(xn,xm) for all m,n ∈ n with n < m. proof. since α(x0,sx0) ≥ η(x0,sx0) and s is weak α−admissible, we get  α(x0,x1) = α(x0,sx0) ≥ η(x0,x1), then α(sx0,sx1) = α(sx0,s 2x0) = α(x1,x2) ≥ η(x1,x2). by triangular α−admissibility, we get  α(sx0,sx1) = α(x1,x2) ≥ η(x1,x2), then α(s2x0,s 2x1) = α(x2,x3) ≥ η(x2,x3) and α(s2x1,s 2x2) = α(x3,x4) ≥ η(x3,x4). again, since α(x3,x4) ≥ η(x3,x4), then α(s2x3,s 2x4) = α(x4,x5) ≥ η(x4,x5) and α(s2x4,s 2x5) = α(x5,x6) ≥ η(x5,x6). by continuing the above process, we conclude that α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n∪{0}. now, we prove that α(xn,xm) ≥ 1, ∀m,n ∈ n with n < m. given m,n ∈ n with n < m. since  α(xn,xn+1) ≥ η(xn,xn+1), α(sxn,s 2xn) = α(xn+1,xn+2) ≥ η(xn+1,xn+2), then, we have α(xn,xn+2) ≥ η(xn,xn+2). again, since   α(xn,xn+2) ≥ η(xn,xn+2) α(sxn+1,s 2xn+1) = α(xn+2,xn+3) ≥ η(xn+2,xn+3), we deduce that α(xn,xn+3) ≥ η(xn,xn+3). int. j. anal. appl. 17 (2) (2019) 216 by continuing this process, we have α(xn,xm) ≥ η(xn,xm) for all n ∈ n with m > n. � in order to facilitate our subsequent arguments, we introduce the notion of generalized c−class functions for self mappings on a set x. definition 2.2. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1, s : x → x be a geraghty contraction type mapping and α,η : x × x → r be a function. let f ∈ c, ψ ∈ ψ and ϕ ∈ φ. then s is called generalized c−class function with α(x,y) ≥ η(x,y), then ψ(b(sx,sy)) ≤ λf(ψ(m(x,y)),ϕ(m(x,y))), (2.1) where m(x,y) = max{b(x,y),b(x,sx),b(y,sy), b(x,sy) + b(y,sx) 2 } (2.2) and λ ∈ [0, 1 s ). theorem 2.1. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be geraghty contraction type mapping on x. assume that α,η : x × x → [0, +∞) are a functions. suppose that the following conditions hold: (1) s is generalized c−class function. (2) s is a triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. (4) s is α,η−continuous mappings. then s has a unique fixed point. proof. we divide the proof to three steps: step 1. let x0 ∈ x be such that α(x0,sx0) ≥ η(x0,sx0). define a sequence {xn} in x such that xn+1 = sxn for all n ∈ n. if xn0 = xn0+1 for some n0 ∈ n, then it is very easy to show that s has a fixed point. now, since the pair s is α−admissible, then α(x1,x2) = α(sx0,s 2x0) ≥ η(x1,x2) and α(x2,x3) = α(sx1,s 2x1) ≥ η(x2,x3). int. j. anal. appl. 17 (2) (2019) 217 again, by using the property of weak α−admissible and repeating the above process for n-times, we have α(xn,xn+1) ≥ η(xn,xn+1) and α(xn+1,xn) ≥ η(xn+1,xn). using the property of triangular weak α−admissible, we can deduce that for any n,m ∈ n with m > n, we have α(xn,xm) ≥ η(xn,xm) and α(xm,xn) ≥ η(xm,xn). suppose xn 6= xn+1 for all n ∈ n, by lemma 2.1, we have α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n. since s is a generalized c−class function, we have ψ(b(xn+1,xn)) = ψ(b(sxn,sxn−1)) ≤ λf(ψ(m(xn,xn−1)),ϕ(m(xn,xn−1))) ≤ λψ(m(xn,xn−1)), (2.3) for all n ∈ n, where m(xn,xn−1) = max{b(xn,xn−1),b(xn,sxn),b(xn−1,sxn−1), b(xn,sxn−1) + b(xn−1,sxn) 2 } = max{b(xn,xn−1),b(xn,xn+1),b(xn−1,xn), b(xn,xn) + b(xn−1,xn+1) 2 } = max{b(xn,xn−1),b(xn,xn+1)}. (2.4) if m(xn,xn−1) = b(xn,xn+1), then ψ(b(xn+1,xn)) ≤ λf(ψ(m(xn,xn−1),ϕ(m(xn,xn−1))) ≤ λψ(m(xn,xn−1)) = λψ(b(xn+1,xn)), < ψ(b(xn+1,xn)). (2.5) which is contraction. thus we conclude that m(xn,xn−1) = b(xn,xn−1). by (2.2), we get that ψ(b(xn+1,xn)) ≤ λψ(b(xn,xn−1)) for all n ∈ n. on repeating this process, we obtain ψ(b(xn+1,xn)) ≤ λnψ(b(x1,x0)) (2.6) for all n > 0. since ψ is nondecreasing, we have b(xn+1,xn+2) ≤ b(xn,xn+1) for all n ∈ n. similarly, we can show that b(xn,xn+1) ≤ b(xn−1,xn). for all n ∈ n∪{0}. it follow that the sequence {b(xn,xn+1)} is nonincreasing for all n ∈ n. therefore there exists r ≥ 0 such int. j. anal. appl. 17 (2) (2019) 218 that limn→∞ b(xn,xn+1) = r. we claim that r = 0. now, we have ψ(b(xn+1,xn+2)) ≤ λf(ψ(b(xn,xn+1)),ϕ(b(xn,xn+1))) < f(ψ(b(xn,xn+1)),ϕ(b(xn,xn+1))). taking n →∞ , we obtain that ψ(r) ≤ λf(ψ(r),ϕ(r)) < f(ψ(r),ϕ(r)). this implies that ψ(r) = 0 or ϕ(r) = 0 which yields lim n→∞ b(xn,xn+1) = 0. (2.7) step 2. to prove that {xn} is a cauchy sequence, there exist � > 0 and two subsequences {xm(k)} and {xn(k)} of {xn} with mk > nk > k such that: d(xn(k),xm(k)) ≥ �,d(xn(k),xm(k)−1) < �. then, using the triangular inequality we get b(xn,xm(k)) ≤ s[b(xn(k),xn(k)+1) + b(xn(k)+1,xm(k))] − b(xn(k)+1,xn(k)+1) ≤ sb(xn(k),xn(k)+1) + s2[b(xn(k)+1,xn(k)+2) + b(xn(k)+2,xm(k)) −sb(xn(k)+2,xn(k)+2) ≤ sb(xn(k),xn(k)+1) + s2b(xn(k)+1,xn(k)+2) + s3b(xn(k)+2,xn(k)+2) + ... + sm−nb(xm(k)−1,xm(k)). using (2.6) in the above inequality b(xn,xm(k)) ≤ sλnb(x1,x0) + s2λn+1b(x1,x0) + s3λn+3b(x1,x0) + ... + sm−nλm−1b(x1,x0) ≤ sλn[1 + sλ + (sλ)2 + ...]b(x1,x0) = sλn 1 −sλ b(x1,x0). as λ ∈ [0, 1 s ) and s > 1, it follows from the above inequality that lim n,m→∞ b(xn,xm) = 0. therefore, {xn} is a cauchy sequence in the complete b−partial metric space x step3. we now prove that s has a fixed point. since {xn} is a cauchy sequence in the complete b−partial metric space x and by completeness of x, then there exists x∗ ∈ x such that lim n,m→∞ b(xn,x ∗) = lim n,m→∞ b(xn,xm) = b(x ∗,x∗). (2.8) int. j. anal. appl. 17 (2) (2019) 219 we will show that x∗ is a fixed point of s. for any n ∈ n, we have b(x∗,sx∗) ≤ s[b(x∗,xn+1) + b(xn+1,sx∗)] − b(xn+1,xn+1)] ≤ s[b(x∗,xn+1) + b(sxn,sx∗)] ≤ sb(x∗,xn+1) + sλb(xn,x∗). using (2.8) in the above inequality, we obtain b(x∗,sx∗) = 0, that is, sx∗ = x∗. thus, x∗ is a fixed point of s. step4. let us show that the fixed point of s is unique. let u,v ∈ x be two distinct fixed points of s, that is, su = u and sv = v. it follows from (2.2) that ψ(b(u,v)) = ψ(b(su,sv)) ≤ λf(ψ(max{b(u,v),b(u,su),b(v,sv), b(u,sv) + b(v,su) 2 }),ϕ(max{b(u,v),b(u,su),b(v,sv), b(u,sv) + b(v,su) 2 })) ≤ λψ(max{b(u,v),b(u,su),b(v,sv), b(u,sv) + b(v,su) 2 }) = λψ(max{b(u,v),b(u,u),b(v,v), b(u,v) + b(v,u) 2 }) = λψ(b(u,v)), < ψ(b(u,v)). which is contraction. therefore, we must have b(u,v) = 0, that is, u = v. thus, the fixed point of s is unique. � the continuity of s in theorem 2.1 can be dropped. theorem 2.2. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be geraghty contraction type mapping on x. assume that α,η : x × x → [0, +∞) are a functions. suppose that the following conditions hold: (1) s is c−class function. (2) s is triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). (4) if {xn} is a sequence in x such that α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n and xn → x∗ ∈ x as n → ∞, then there exist a subsequence {xn(k)} of {xn} such that α(xn(k),x∗) ≥ η(xn(k),x∗) for all k ∈ n. then s has a unique fixed point. int. j. anal. appl. 17 (2) (2019) 220 proof. following the same proof as in theorem 2.1, we construct the sequence {xn} be defining xn+1 = sxn for all n ∈ n converging to x∗ ∈ x such that α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n. by condition (5), there exist a subsequence {xn(k)} of {xn} such that α(xn(k),x∗) ≥ η(xn(k),x∗) for all k ∈ n. therefore, ψ(b(xn(k)+1,tx ∗)) = ψ(d(sxn(k),tx ∗)), ≤ λf(ψ(m(xn(k),x∗),ϕ(m(xn(k),x∗))), ≤ f(ψ(m(xn(k),x∗))), (2.9) for all n ∈ n. now, m(xn(k),x ∗) = max{b(xn,x∗),b(xn(k),sxn(k)),b(x∗,sx∗), (2.10) b(xn(k),sx ∗) + b(x∗,sxn(k)) 2 }, = max{b(xn(k),x∗),b(xn(k),xn(k)+1),b(x∗,x∗), (2.11) b(xn(k),x ∗) + b(x∗,sxn(k)) 2 }, = max{d(xn(k),x∗),d(xn(k),xn(k)+1))}. (2.12) by taking n →∞ in (2.9) and using (2.7), we obtain ψ(b(x∗,sx∗)) ≤ λf(ψ(b(x∗,sx∗)),φ(b(x∗,sx∗))), which implies that b(x∗,sx∗) = 0, that is, sx∗ = x∗. � now, we use theorem 2.1 and theorem 2.2 to present many fixed point results: corollary 2.1. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be mapping on x. assume that α : x ×x → [0, +∞) is a function. also, suppose that the following conditions hold: (1) for all x,y ∈ x with α(x,y) ≥ 1), we have ψ(b(sx,sy)) ≤ λf(ψ(b(x,y)),ϕ(b(x,y)). (2) s is generalized c−class function. (3) s is a triangular weak α-admissible. (4) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. (5) s is α,η−continuous mappings. then s has a unique fixed point. proof. follows the same proof of the theorem 2.1 by defining η : x ×x → r via η(x,y) = 1. � corollary 2.2. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be mapping on x. assume that α : x ×x → [0, +∞) is a function. also, suppose that the following conditions hold: int. j. anal. appl. 17 (2) (2019) 221 (1) for all x,y ∈ x with α(x,y) ≥ 1, we have ψ(b(sx,sy)) ≤ λf(ψ(b(x,y)),ϕ(b(x,y)). (2) s is generalized c−class function. (3) s is a triangular α-admissible. (4) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. (5) if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n ∈ n and xn → x∗ ∈ x as n → ∞, then there exist a subsequence {xn(k)} of {xn} such that α(xn(k),x∗) ≥ 1 for all k ∈ n. then s has a unique fixed point. proof. follows the same proof of the theorem 2.2 by defining η : x ×x → r via η(x,y) = 1. � let β : [0, +∞) → [0, 1) be a continuous function. define s : [0,∞)× [0,∞) → [0,∞) via f(s,t) = sβ(t). then f ∈c. by theorem 2.1 and theorem 2.2, we have the following results: corollary 2.3. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be mapping on x. assume that α,η : x ×x → [0, +∞) are a functions. suppose there exist ψ ∈ ψ and a continuous function β : [0, +∞) → [0, 1) such that for all x,y ∈ x with α(x,y) ≥ η(x,y), we have ψ(b(sx,sy)) ≤ λf(β(ψ(b(x,y))),ϕ(b(x,y)). (2.13) also, suppose that the following conditions hold: (1) s is generalized c−class function. (2) s is a triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. (4) s is α,η−continuous mappings. then s has a unique fixed point. corollary 2.4. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be mapping on x. assume that α,η : x ×x → [0, +∞) are a functions. suppose there exist ψ ∈ ψ and a continuous function β : [0, +∞) → [0, 1) such that for all x,y ∈ x with α(x,y) ≥ η(x,y), we have ψ(b(sx,sy)) ≤ λf(β(ψ(b(x,y))),ϕ(b(x,y)). (2.14) also, suppose that the following conditions hold: (1) s is generalized c−class function. (2) s is a triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ η(x0,sx0). (4) if {xn} is a sequence in x such that α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n and xn → x∗ ∈ x as n → ∞, then there exist a subsequence {xn(k)} of {xn} such that α(xn(k),x∗) ≥ η(xn(k),x∗) for all k ∈ n. int. j. anal. appl. 17 (2) (2019) 222 then s has a unique fixed point. example 2.2. let x = [0, 1] and b : x × x → r define by b(x,y) = |x−y|2 for all x,y ∈ x. define ψ,φ : [0,∞) → [0,∞) by ψ(t) = t and φ(t) = 4 25 t. define the mapping s : r → r by sx = ln x 5 . also, we define the functionsα,η : x ×x → [0,∞) by α(x,y) =   ex+y if x,y ∈ [0, 1], 0 if otherwise, η(x,y) =   1 if x,y ∈ [0, 1], 0 if otherwise. and f(r,t) = r − t for all r,t,x,y ∈ x. firstly, it is easy to see that (x,b) is a complete partial b−metric space with s = 3. then s is a triangular weak α-admissible with respect to η. indeed, if α(x,sx) ≥ η(x,sx), then α(sx,s2x) ≥ η(sx,s2x), so α(x, ln x + 1) = ex+ln x > 1 = η(x, ln x),then α(ln x, ln (ln x)) = eln x+ln (ln x) ≥ e = η(ln x, ln (ln x)).so x ≥ 0 and hence sx ≤ 0. therefore, α(x,sx) ≥ η(x,sx). we will prove that s is a generalized c−class function. since α(x,sx) ≥ η(x,sx). then we have x,y ∈ [0, 1] and then ψ(d(sx,sy)) = ∣∣∣∣ln x5 − ln y5 ∣∣∣∣2 = 1 25 |ln x− ln y|2 = 1 25 b(x,y) ≤ 1 25 m(x,y) = m(x,y) − 24 25 m(x,y) = ψ(m(x,y)) −φ(m(x,y)) = f(ψ(m(x,y)),φ(m(x,y))). then s is a generalized c−class function and all assumptions of corollary 2.1 are satisfied. hence s has a unique fixed point. 3. applications in this section, we apply our results to construct an application on lebesgue-integrable. denote by γ the set of all functions γ : r+ → r+ satisfying the following conditions: int. j. anal. appl. 17 (2) (2019) 223 (1) γ is lebesgue-integrable on each compact of r+; (2) for each � > 0, we have ∫ � 0 γ(z)dz > 0 . theorem 3.1. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be geraghty contraction type mappings on x. also, let f ∈c and γ1,γ2 ∈ γ. assume that α,η : x ×x → [0,∞) be two functions such for all x,y ∈ x with α(x,y) ≥ η(x,y), we have ∫ d(sx,ty) 0 γ1(z))dz ≤ f (∫ max{d(x,y),d(x,sx),d(tx,ty), b(x,sy)+b(y,sx) 2 } 0 γ1(z)dz, ∫ max{d(x,y),d(x,sx),d(tx,ty), b(x,sy)+b(y,sx) 2 } 0 γ2(z)dz ) . also, suppose the following hypotheses: (1) s is generalized c−class function. (2) s is a triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. (4) s is α,η−continuous mappings. then s has a unique fixed point. proof. define the functions ψ,ϕ : r+ → r+ via ψ(t) = ∫ t 0 γ1(z))dz and ϕ(t) = ∫ t 0 γ2(z))dz. noting that ψ is an altering distance function and ϕ ∈ φ. so s is triangular weak α−admissible with respect to η. so s satisfies all the hypotheses of theorem 2.1. therefore s has a fixed point. � theorem 3.2. let (x,b) be a complete b−partial metric space with coefficient s ≥ 1 and s be geraghty contraction type mappings on x. also, let f ∈c and γ1,γ2 ∈ γ. assume that α,η : x ×x → [0,∞) be two functions such for all x,y ∈ x with α(x,y) ≥ η(x,y), we have ∫ d(sx,ty) 0 γ1(z))dz ≤ f (∫ max{d(x,y),d(x,sx),d(tx,ty), b(x,sy)+b(y,sx) 2 } 0 γ1(z)dz, ∫ max{d(x,y),d(x,sx),d(tx,ty), b(x,sy)+b(y,sx) 2 } 0 γ2(z)dz ) . also, suppose the following hypotheses: (1) s is generalized c−class function. (2) s is a triangular weak α-admissible. (3) there exists x0 ∈ x such that α(x0,sx0) ≥ 1. int. j. anal. appl. 17 (2) (2019) 224 (4) if {xn} is a sequence in x such that α(xn,xn+1) ≥ η(xn,xn+1) for all n ∈ n and xn → x∗ ∈ x as n → ∞, then there exist a subsequence {xn(k)} of {xn} such that α(xn(k),x∗) ≥ η(xn(k),x∗) for all k ∈ n. then s has a unique fixed point. proof. follow from theorem 2.2 by defining ψ(t) = ∫ t o γ1(z))dz and ϕ(t) = ∫ t o γ2(z))dz. noting that the mapping s satisfies all the hypotheses of theorem 2.2. � 4. acknowledgement the authors would like to acknowledge the grant: ukm grant dip-2014-034 and ministry of education, malaysia grant frgs/1/2014/st06/ukm/01/1 for financial support. references [1] t. abdeljawad, meir-keeler α-contractive fixed and common fixed point theorems, fixed point theory appl., 2013 (2013), art. id 19. [2] a. aghajani, m. abbas, j.r. roshan, common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, mathematica slovaka, (2013). [3] s. alizadeh, f. moradlou, p. salimi, some fixed point results for (α,β) − (ψ,φ)−contractive mappings, filomat, 28 (2014), 635-647. [4] a. h. ansari, note on ϕ − ψ− contractive type mappings and related fixed point, the 2nd regional conference on mathematics and applications, (2014), 377-380. [5] a.h ansari, w. shatanawi, a. kurdi, g. maniu, best promixity points in complete metric spaces with (p)−property via c-class fuctions, j. math. anal., 7 (2016), 54-67. [6] a. h. ansari, j. kaewcharoen, c-class functions and fixed point theorems for generalized α-η-ψ-ϕ-f-contraction type mappings in α-η-complete metric spaces, j. nonlinear sci. appl., 9 (2016), 4177-4190. [7] i.a. bakhtin, the contraction mapping principle in almost metric spaces, funct. anal., 30 (1989), 26-37. [8] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrals, fund. math, 3 (1922), 133-181. [9] a. branciari, a fixed point theorem of banach-caccioppoli type on a class of generalized metric spaces, publ. math. debrecen. 57 (2000), 31-37. [10] s. czerwik, contraction mappings in b-metric spaces, acta math. inform. univ. ostrav., 1 (1993), 5-11. [11] s. czerwik, nonlinear set-valued contraction mappings in b-metric spaces, atti sem. mat. univ. modena, 46 (1998), 263-276. [12] p. das, a fixed point theorem in a generalized metric space, soochow j. math., 33 (2007) 33-39. [13] h. isik, a. h. ansari,s. chandok, common fixed points for (ψ,f,α,β)−weakly contractive mappings in generalized matric space via new functions, gazi univ. j. sci., 4 (2015),703-708. [14] n. hussain, p. salimi and a. latif, fixed point results for single and set-valued a α−eta−ψ−contractive mappings, fixed point theory appl., 2013 (2013), art. id 212. int. j. anal. appl. 17 (2) (2019) 225 [15] n. hussain, m. a. kutbi, p. salimi, fixed point theory in α−complete metric spaces with applications, abstr. appl. anal., 2014 (2014), article id 280817. [16] e. karapinar, b. samet, generalized α−ψ−contractive type mappings and related fixed point theorems with applications, abstr. appl. anal., (2012), article id 793486. [17] e. karapinar, p. kumam and p. salimi, on α − ψ−meir-keeler contractive mappings, fixed point theory appl., 2013 (2013), art. id 94. [18] e. karapinar, α−ψ−geraghty contraction type mappings and some relatead fixed point results, filomat, 28 (2014), 37-48. [19] m. s. khan, a fixed point theorem for metric spaces, rend. inst. math. univ. trieste., 8 (1976), 69-72. [20] m. s. khan, m. swaleh, and s. sessa, fixed point theorems by altering distances between the points, bull. aust. math. soc., 30 (1984), 1-9. [21] s.g matthews, partial metric topology, proc. 8th summer conference on general topology and application. ann. new york acad. sci., 728 (1994), 183-197. [22] d. k. patel, th. abdeljawad, d. gopal, common fixed points of generalized meir-keeler α-contractions, fixed point theory appl., 2013 (2013), art. id 260. [23] h. qawaqneh, m. s. m. noorani, w. shatanawi, h. alsamir, common fixed points for pairs of triangular (α)−admissible mappings, j. nonlinear sci. appl., 10 (2017), 6192-6204. [24] h. qawaqneh, m. s. m. noorani, w. shatanawi, k. abodayeh, h. alsamir, fixed point for mappings under contractive condition based on simulation functions and cyclic (α,β)−admissibility, j. math. anal., 9 (2018), 38-51. [25] h. qawaqneh, m. s. m. noorani, w. shatanawi, fixed point results for geraghty type generalized f−contraction for weak alpha-admissible mapping in metric-like spaces, eur. j. pure appl. math., 11 (2018), 702-716. [26] h. qawaqneh, m.s.m. noorani, w. shatanawi, common fixed point theorems for generalized geraghty (α,ψ,φ)-quasi contraction type mapping in partially ordered metric-like spaces, axioms, 7 (2018), art. id 74. [27] h. qawaqneh, m.s.m. noorani, w. shatanawi, fixed point theorems for (α,k,θ)−contractive multi-valued mapping in b−metric space and applications, int. j. math. comput. sci., 14 (2018), 263-283. [28] h. qawaqneh, m.s.m. noorani, w. shatanawi, h. alsamir, some fixed point results for the cyclic (α,β) − (k,θ)−multivalued mappings in metric space, international conference on fundamental and applied sciences (icfas2018), 2018. [29] j. r. roshan, v. parvaneh, s. sedghi, n. shobkolaei, w. shatanawi, common fixed points of almost generalized (ψ,ϕ)scontractive mappings in ordered b-metric spaces, fixed point theory appl., 2013 (2013), art. id 159. [30] p. salimi, a. latif and n. hussain, modified a α − ψ−contractive mappings with applications, fixed point theory appl. 2013 (2013), art. id 151. [31] b. samet, c. vetro, p. vetro, fixed point theorems for a α − ψ−contractive type mappings, nonlinear anal., theory methods appl., 75 (2012), 2154-2165. [32] w. sintunavarat, nonlinear integral equations with new admissibility types in b−metric spaces, j. fixed point theory appl., 18 (2016), 397-416. [33] w. shatanawi and m. postolache, common fixed point results for mappings under nonlinear contraction of cyclic form in ordered metric spaces, fixed point theory appl., 2013 (2013), art. id 60. [34] w. shatanawi, m. noorani, h. alsamir and a. bataihah, fixed and common fixed point theorems in partially ordered quasi-metric spaces, j. math. computer sci., 16 (2016), 516-528. [35] s. shukla, partial b-metric spaces and fixed point theorems, mediterr. j. math., 11(2014), 703-711. 1. introduction and preliminaries 2. main result 3. applications 4. acknowledgement references international journal of analysis and applications issn 2291-8639 volume 1, number 1 (2013), 79-99 http://www.etamaths.com convergence theorems of an implicit iterates with errors for non-lipschitzian asymptotically quasi-nonexpansive type mappings g. s. saluja abstract. the aim of this paper is to study an implicit iterative process with errors for two finite families of non-lipschitzian asymptotically quasinonexpansive type mappings in the framework of real banach spaces. in this paper, we have obtained a necessary and sufficient condition to converge to common fixed points for proposed scheme and mappings and also obtained strong convergence theorems by using semi-compactness and condition (b′). 1. introduction let e be a real banach space and ue = {x ∈ e : ‖x‖ = 1}. e is said to be uniformly convex if for any ε ∈ (0, 2] there exists δ > 0 such that for any x,y ∈ue, ‖x−y‖≥ ε implies ∥∥∥∥x−y2 ∥∥∥∥ ≤ 1 − δ. in 1973, petryshyn and williamson [13] established a necessary and sufficient condition for a mann [12] iterative sequence to converge strongly to a fixed point of quasi-nonexpansive mapping. subsequently, ghosh and debnath [5] extended the results of [13] and obtained some necessary and sufficient conditions for an ishikawa-type iterative sequence to converge to a fixed point of quasi-nonexpansive mapping. in 2001, liu in [10, 11] extended the results of ghosh and debnath [5] to a more general asymptotically quasi-nonexpansive mappings. in 2003, sahu and jung [15] studied ishikawa and mann iteration process in banach spaces and they proved some weak and strong convergence theorems for asymptotically quasinonexpansive type mapping. in 2006, shahzad and udomene [17] gave the necessary and sufficient condition for convergence of common fixed point of two-step modified ishikawa iterative sequence for two asymptotically quasi-nonexpansive mappings in real banach space. recently, qin et al. [14] studied a general implicit iterative process for a finite family of generalized asymptotically quasi-nonexpansive mappings and established 2010 mathematics subject classification. 47h09, 47h10, 47j25. key words and phrases. asymptotically quasi-nonexpansive type mapping, general implicit iterative process with errors, common fixed point, strong convergence, uniformly convex banach space. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 79 80 saluja strong convergence theorem of the proposed iterative process in the framework of real banach space. the main goal of this paper is to establish the strong convergence of general implicit iterative process studied by qin et al. [14] which includes schu’s explicit iterative processes and sun’s implicit iterative processes as special cases for two finite families of non-lipschitzian asymptotically quasi-nonexpansive type mappings on a closed convex unbounded set in a real uniformly convex banach spaces. our results unify, improve and generalize many known results given in the existing current literature. 2. preliminaries and lemmas let c be a nonempty subset of a normed space e and t : c → c be a given mapping. the set of fixed points of t is denoted by f(t), that is, f(t) = {x ∈ c : t(x) = x}. the mapping t is said to be (1) nonexpansive if ‖tx−ty‖ ≤ ‖x−y‖(2.1) for all x,y ∈ c. (2) quasi-nonexpansive [2] if ‖tx−p‖ ≤ ‖x−p‖(2.2) for all x ∈ c, p ∈ f(t). (3) asymptotically nonexpansive [6] if there exists a sequence {un} in [0,∞) with limn→∞un = 0 such that ‖tnx−tny‖ ≤ (1 + un)‖x−y‖(2.3) for all x,y ∈ c and n ≥ 1. (4) asymptotically quasi-nonexpansive if f(t) 6= ∅ and there exists a sequence {un} in [0,∞) with limn→∞un = 0 such that ‖tnx−p‖ ≤ (1 + un)‖x−p‖(2.4) for all x ∈ c, p ∈ f(t) and n ≥ 1. (5) uniformly l-lipschitzian if there exists a positive constant l such that ‖tnx−tny‖ ≤ l‖x−y‖(2.5) for all x,y ∈ c and n ≥ 1. (6) asymptotically nonexpansive type [8], if lim sup n→∞ { sup x,y∈c ( ‖tnx−tny‖−‖x−y‖ )} ≤ 0.(2.6) convergence theorems of an implicit iterates with errors 81 (7) asymptotically quasi-nonexpansive type [15], if f(t) 6= ∅ and lim sup n→∞ { sup x∈c, p∈f (t ) ( ‖tnx−p‖−‖x−p‖ )} ≤ 0.(2.7) remark 2.1. it is easy to see that if f(t) is nonempty, then asymptotically nonexpansive mapping, asymptotically quasi-nonexpansive mapping and asymptotically nonexpansive type mapping are the special cases of asymptotically quasinonexpansive type mappings. the mann and ishikawa iteration processes have been used by a number of authors to approximate the fixed points of nonexpansive, asymptotically nonexpansive mappings, and quasi-nonexpansive mappings on banach spaces (see, e.g., [5, 7, 9, 10, 11, 13, 18, 23]). recall that the modified mann iteration which was introduced by schu [16] generates a sequence {xn} in the following manner: x1 ∈ c, xn+1 = (1 −αn)xn + αntnxn, n ≥ 1,(2.8) where {αn} is a real sequence in the interval (0, 1) and t : c → c is an asymptotically nonexpansive mapping. in 2001, xu and ori [24] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a hilbert space h. let c be a nonempty subset of h. let t1,t2, . . . ,tn be self-mappings of c and suppose that f = ∩ni=1f(ti) 6= ∅, the set of common fixed points of ti, i = 1, 2, . . . ,n. an implicit iteration process for a finite family of nonexpansive mappings is defined as follows, with {tn} a real sequence in (0, 1), x0 ∈ c: x1 = t1x0 + (1 − t1)t1x1, x2 = t2x1 + (1 − t2)t2x2, ... xn = tnxn−1 + (1 − tn )tnxn, xn+1 = tn+1xn + (1 − tn+1)t1xn+1, ... which can be written in the following compact form: xn = tnxn−1 + (1 − tn)tnxn, n ≥ 1(2.9) where tk = tk mod n . (here the mod n function takes values in n). and they proved the weak convergence of the process (2.4). in 2003, sun [19] extend the process (2.9) to a process for a finite family of asymptotically quasi-nonexpansive mappings, with {αn} a real sequence in (0, 1) 82 saluja and an initial point x0 ∈ c, which is defined as follows: x1 = α1x0 + (1 −α1)t1x1, ... xn = αnxn−1 + (1 −αn )tnxn, xn+1 = αn+1xn + (1 −αn+1)t21 xn+1, ... x2n = α2nx2n−1 + (1 −α2n )t2nx2n, x2n+1 = α2n+1x2n + (1 −α2n+1)t31 x2n+1, ... which can be written in the following compact form: xn = αnxn−1 + (1 −αn)tki xn, n ≥ 1(2.10) where n = (k − 1)n + i, i ∈n . sun [19] proved the strong convergence of the process (2.10) to a common fixed point, requiring only one member t in the family {ti : i ∈n} to be semi-compact. the result of sun [19] generalized and extended the corresponding main results of wittmann [21] and xu and ori [24]. in 2010, qin et al. [14] studied the following general implicit iteration process for two finite families of generalized asymptotically quasi-nonexpansive mappings {s1,s2, . . . ,sn} and {t1,t2, . . . ,tn}: x1 = α1x0 + β1s1x0 + γ1t1x1 + δ1u1, x2 = α2x1 + β2s2x1 + γ2t2x2 + δ2u2, ... xn = αnxn−1 + βnsnxn−1 + γntnxn + δnun, xn+1 = αn+1xn + βn+1s 2 1xn + γn+1t 2 1 xn+1 + δn+1un+1,(2.11) ... x2n = α2nx2n−1 + β2ns 2 nx2n−1 + γ2nt 2 nx2n + δ2nu2n, x2n+1 = α2n+1x2n + β2n+1s 3 1x2n + γ2n+1t 3 1 x2n+1 + δ2n+1u2n+1, ... which can be written in the following compact form: xn = αnxn−1 + βns h(n) i(n) xn−1 + γnt h(n) i(n) xn + δnun, n ≥ 1(2.12) where x0 is the initial value, {un} is a bounded sequence in c, and {αn}, {βn}, {γn} and {δn} are sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. since for each n ≥ 1, it can be written as n = (h − 1)n + i, where i = i(n) ∈ {1, 2, . . . ,n} = n , h = h(n) ≥ 1 is a positive integer and h(n) →∞ as n →∞. convergence theorems of an implicit iterates with errors 83 in this paper, motivated by [14], we study general implicit iterative process (2.12) for two finite families of non-lipschitzian asymptotically quasi-nonexpansive type mappings {s1,s2, . . . ,sn} and {t1,t2, . . . ,tn} in banach spaces. also we establish some strong convergence theorems for said scheme and mappings. we remark that implicit iterative process (2.12) which includes the explicit iterative process (2.8) and the implicit iterative process (2.10) as a special case in general. if si = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then the implicit iterative process (2.12) is reduced to the following implicit iterative process: xn = (αn + βn)xn−1 + γnt h(n) i(n) xn + δnun, n ≥ 1.(2.13) if ti = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n} = n , then the implicit iterative process (2.12) is reduced to the following explicit iterative process: xn = αn 1 −γn xn−1 + βn 1 −γn s h(n) i(n) xn + δn 1 −γn un, n ≥ 1.(2.14) denote the indexing set {1, 2, . . . ,n} by n . let {ti : i ∈ n} be n uniformly lt,i-lipschitzian asymptotically quasi-nonexpansive type self-mappings of c and {si : i ∈ n} be n uniformly ls,i-lipschitzian asymptotically quasi-nonexpansive type self-mappings of c. we show that (2.12) exists. let x0 ∈ c and x1 = α1x0 + β1s1x0+γ1t1x1+δ1u1. define w : c → c by: wx = α1x0+β1s1x0+γ1t1x1+δ1u1 for all x ∈ c. the existence of x1 is guaranteed if w has a fixed point. for any x,y ∈ c, we have ‖wx−wy‖ ≤ γ1 ‖t1x−t1y‖≤ γ1lt,1 ‖x−y‖ ≤ γ1lt ‖x−y‖ ,(2.15) where lt = max{lt,i : 1 ≤ i ≤ n}. now, w is a contraction if γ1lt < 1 or lt < 1/γ1. as γ1 ∈ (0, 1), therefore w is a contraction even if lt > 1. by the banach contraction principle, w has a unique fixed point. thus, the existence of x1 is established. similarly, we can establish the existence of x2,x3,x4, . . . . thus, the implicit algorithm (2.12) is well defined. the distance between a point x and a set c and closed ball with center zero and radius r in e are, respectively, defined by d(x,c) = inf y∈c ‖x−y‖ , br(0) = {x ∈ e : ‖x‖≤ r}.(2.16) in order to prove our main results, we need the following definition and lemmas. definition 2.1.(see [19]) let c be a closed subset of a normed space e and let t : c → c be a mapping. then t is said to be semi-compact if for any bounded sequence {xn} in c with ‖xn −txn‖→ 0 as n →∞, there is a subsequence {xnk} of {xn} such that xnk → x ∗ ∈ c as nk →∞. 84 saluja lemma 2.1.(see [20]) let {an} and {bn} be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ an + bn, n ≥ 1.(2.17) if ∑∞ n=1 bn < ∞, then limn→∞an exists. in particular, if {an} has a subsequence converging to zero, then limn→∞an = 0. lemma 2.2.(see [14]) let e be a uniformly convex banach space, s > 0 a positive number, and bs(0) a closed ball of e. then there exists a continuous, strictly increasing and convex function g : [0,∞) → [0,∞) with g(0) = 0 such that ‖ax + by + cz + dw‖2 ≤ a‖x‖2 + b‖y‖2 + c‖z‖2 + d‖w‖2 −abg(‖x−y‖)(2.18) for all x,y,z,w ∈ bs(0) = {x ∈ e : ‖x‖ ≤ s} and a,b,c,d ∈ [0, 1] such that a + b + c + d = 1. 3. main results we begin with a necessary and sufficient condition for convergence of {xn} generated by the general implicit iterative process (2.12) to prove the following result. theorem 3.1. let c be a nonempty closed convex subset of a banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasinonexpansive type mapping and let si : c → c be a uniformly ls,i-lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) ⋂ ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn +βn +γn +δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.12). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n}(3.1) and bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n} (3.2) such that ∑∞ n=1 an < ∞ and ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. then {xn} converges strongly to some point in f if and only if lim inf n→∞ d(xn,f) = 0. convergence theorems of an implicit iterates with errors 85 proof. the necessity is obvious and so it is omitted. now, we prove the sufficiency. for any p ∈ f, from (2.12), (3.1) and (3.2), we have ‖xn −p‖ = ∥∥∥αnxn−1 + βnsh(n)i(n) xn−1 + γnth(n)i(n) xn + δnun −p∥∥∥ = ∥∥∥αn(xn−1 −p) + βn(sh(n)i(n) xn−1 −p) + γn(th(n)i(n) xn −p) + δn(un −p)∥∥∥ ≤ αn ‖xn−1 −p‖ + βn ∥∥∥sh(n)i(n) xn−1 −p∥∥∥ + γn ∥∥∥th(n)i(n) xn −p∥∥∥ +δn ‖un −p‖ ≤ αn ‖xn−1 −p‖ + βn(‖xn−1 −p‖ + bn) + γn(‖xn −p‖ + an) +δn ‖un −p‖ = (αn + βn)‖xn−1 −p‖ + γn ‖xn −p‖ + βnbn + γnan +δn ‖un −p‖ = (1 −γn −δn)‖xn−1 −p‖ + γn ‖xn −p‖ + βnbn + γnan +δn ‖un −p‖ ≤ (1 −γn)‖xn−1 −p‖ + γn ‖xn −p‖ + an + bn +δn ‖un −p‖ .(3.3) since from restriction (a) γn ≤ d it follows from (3.3) that ‖xn −p‖ ≤ ‖xn−1 −p‖ + 1 1 −γn (an + bn) + δn 1 −γn ‖un −p‖ ≤ ‖xn−1 −p‖ + 1 1 −d (an + bn) + δn 1 −d ‖un −p‖ ≤ ‖xn−1 −p‖ + 1 1 −d (an + bn) + m 1 −d δn,(3.4) where m = supn≥1{‖un −p‖}, since {un} is a bounded sequence in c. this implies that d(xn,f) ≤ d(xn−1,f) + mn,(3.5) where mn = 1 1−d (an + bn) + m 1−dδn. since by assumptions of the theorem,∑∞ n=1 an < ∞, ∑∞ n=1 bn < ∞ and ∑∞ n=1 δn < ∞, it follows that ∑∞ n=1 mn < ∞. therefore, from lemma 2.1, we know that limn→∞d(xn,f) exists. since by hypothesis lim infn→∞d(xn,f) = 0, so by lemma 2.1, we have lim n→∞ d(xn,f) = 0.(3.6) 86 saluja next we prove that {xn} is a cauchy sequence in c. it follows from (3.4) that for any m ≥ 1, for all n ≥ n0 and for any p ∈ f, we have ‖xn+m −p‖ ≤ ‖xn+m−1 −p‖ + 1 1 −d (an+m + bn+m) + m 1 −d δn+m ≤ ‖xn+m−2 −p‖ + 1 1 −d (an+m−1 + bn+m−1) + m 1 −d δn+m−1 + 1 1 −d (an+m + bn+m) + m 1 −d δn+m ≤ ‖xn+m−2 −p‖ + 1 1 −d [(an+m + an+m−1) + (bn+m + bn+m−1)] + m 1 −d [δn+m + δn+m−1] ≤ . . . ≤ . . . ≤ ‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (ak + bk) + m 1 −d n+m∑ k=n+1 δk.(3.7) so, we have ‖xn+m −xn‖ ≤ ‖xn+m −p‖ + ‖xn −p‖ ≤ ‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (ak + bk) + m 1 −d n+m∑ k=n+1 δk +‖xn −p‖ = 2‖xn −p‖ + 1 1 −d n+m∑ k=n+1 (ak + bk) + m 1 −d n+m∑ k=n+1 δk. (3.8) then, we have ‖xn+m −xn‖ ≤ 2d(xn,f) + 1 1 −d n+m∑ k=n+1 (ak + bk) + m 1 −d n+m∑ k=n+1 δk, ∀n ≥ n0.(3.9) for any given ε > 0, there exists a positive integer n1 ≥ n0 such that for any n ≥ n1, d(xn,f) < ε 6 , n+m∑ k=n+1 (ak + bk) < (1 −d)ε 3 ,(3.10) and n+m∑ k=n+1 δk < (1 −d)ε 3m .(3.11) convergence theorems of an implicit iterates with errors 87 thus, from (3.9)-(3.11) and n ≥ n1, we have ‖xn+m −xn‖ < 2. ε 6 + 1 1 −d . (1 −d)ε 3 + m (1 −d) . (1 −d)ε 3m = ε.(3.12) this implies that {xn} is a cauchy sequence in c. thus, the completeness of e implies that {xn} must be convergent. assume that limn→∞xn = p. now, we have to show that {xn} converges to some common fixed point in f . indeed, we know that the set f = ∩ni=1f(ti) ⋂ ∩ni=1f(si) is closed. from the continuity of d(x,f) = 0 with limn→∞d(xn,f) = 0 and limn→∞xn = p, we get d(p,f) = 0,(3.13) and so p ∈ f, that is, {xn} converges to some common fixed point in f. this completes the proof. if si = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 3.1 is reduced to the following result: corollary 3.1. let c be a nonempty closed convex subset of a banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.13). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. then {xn} converges strongly to some point in f if and only if lim inf n→∞ d(xn,f) = 0. if ti = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 3.1 is reduced to the following result: corollary 3.2. let c be a nonempty closed convex subset of a banach space e. let si : c → c be a uniformly ls,i-lipschitz and asymptotically quasinonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} 88 saluja be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.14). put bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, such that ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. then {xn} converges strongly to some point in f if and only if lim inf n→∞ d(xn,f) = 0. we prove a lemma which plays an important role in establishing strong convergence of the general implicit iterative process (2.12) in a uniformly convex banach space. lemma 3.1. let c be a nonempty closed convex subset of a real uniformly convex banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping and let si : c → c be a uniformly ls,ilipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) ⋂ ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.12). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n} and bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞ and ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. then lim n→∞ ‖xn −trxn‖ = lim n→∞ ‖xn −srxn‖ = 0, ∀ r ∈{1, 2, . . . ,n}. convergence theorems of an implicit iterates with errors 89 proof. as in the proof of theorem 3.1, limn→∞‖xn −q‖ exists for all q ∈ f . it follows that the sequence {xn} is bounded. in view of lemma 2.2, we see that ‖xn −q‖ 2 ≤ αn ‖xn−1 −q‖ 2 + βn ∥∥∥sh(n)i(n) xn−1 −q∥∥∥2 + γn ∥∥∥th(n)i(n) xn −q∥∥∥2 +δn ‖un −q‖ 2 −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ + bn]2 + γn[‖xn −q‖ + an]2 +δn ‖un −q‖ 2 −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ 2 + k′n] + γn[‖xn −q‖ 2 + k′′n] +m1δn −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) = (αn + βn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnk ′ n + γnk ′′ n +m1δn −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) = (1 −γn − δn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnk ′ n + γnk ′′ n +m1δn −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ (1 −γn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + (k′n + k ′′ n) +m1δn −αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥),(3.14) where m1 is a appropriate constant such that m1 = supn≥1{‖un −q‖ 2} and k′n = b2n + 2‖xn −q‖bn and k′′n = a2n + 2‖xn −q‖an, since ∑∞ n=1 an < ∞ and∑∞ n=1 bn < ∞, it follows that ∑∞ n=1 k ′ n < ∞ and ∑∞ n=1 k ′′ n < ∞. this implies that αnβng (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) ≤ (1 −γn)[‖xn−1 −q‖2 −‖xn −q‖2 ] +(k′n + k ′′ n) + m1δn.(3.15) in view of restrictions (a), (b), ∑∞ n=1 k ′ n < ∞ and ∑∞ n=1 k ′′ n < ∞, we obtain that lim n→∞ g (∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥) = 0.(3.16) since g : [0,∞) → [0,∞) is a continuous, strictly increasing, and convex function with g(0) = 0, we obtain that lim n→∞ ∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥ = 0.(3.17) next, we show that lim n→∞ ∥∥∥th(n)i(n) xn −xn−1∥∥∥ = 0.(3.18) 90 saluja from lemma 2.2, we also see that ‖xn −q‖ 2 ≤ αn ‖xn−1 −q‖ 2 + βn ∥∥∥sh(n)i(n) xn−1 −q∥∥∥2 + γn ∥∥∥th(n)i(n) xn −q∥∥∥2 +δn ‖un −q‖ 2 −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ + bn]2 + γn[‖xn −q‖ + an]2 +δn ‖un −q‖ 2 −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) ≤ αn ‖xn−1 −q‖ 2 + βn[‖xn−1 −q‖ 2 + k′n] + γn[‖xn −q‖ 2 + k′′n] +m1δn −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) = (αn + βn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnk ′ n + γnk ′′ n +m1δn −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) = (1 −γn − δn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + βnk ′ n + γnk ′′ n +m1δn −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) ≤ (1 −γn)‖xn−1 −q‖ 2 + γn ‖xn −q‖ 2 + (k′n + k ′′ n) +m1δn −αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥),(3.19) this implies that αnγng (∥∥∥th(n)i(n) xn −xn−1∥∥∥) ≤ (1 −γn)[‖xn−1 −q‖2 −‖xn −q‖2 ] +(k′n + k ′′ n) + m1δn.(3.20) in view of restrictions (a), (b), ∑∞ n=1 k ′ n < ∞ and ∑∞ n=1 k ′′ n < ∞, we obtain that lim n→∞ g (∥∥∥th(n)i(n) xn −xn−1∥∥∥) = 0.(3.21) since g : [0,∞) → [0,∞) is a continuous, strictly increasing, and convex function with g(0) = 0, we obtain that (3.18) holds. notice that ‖xn −xn−1‖ ≤ βn ∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥ + γn ∥∥∥th(n)i(n) xn −xn−1∥∥∥ +δn ‖un −xn−1‖ .(3.22) in view of (3.17) and (3.18), we see from the restriction (b) that lim n→∞ ‖xn −xn−1‖ = 0,(3.23) which implies that lim n→∞ ‖xn −xn+j‖ = 0, ∀ j ∈{1, 2, . . . ,n}.(3.24) convergence theorems of an implicit iterates with errors 91 since for any positive integer n > n, it can be written as n = (h(n) − 1)n + i(n), where i(n) ∈{1, 2, . . . ,n} = i, observe that ‖xn−1 −tnxn‖ ≤ ∥∥∥xn−1 −th(n)i(n) xn∥∥∥ + ∥∥∥th(n)i(n) xn −tnxn∥∥∥ ≤ ∥∥∥xn−1 −th(n)i(n) xn∥∥∥ + lt ∥∥∥th(n)−1i(n) xn −xn∥∥∥ ≤ ∥∥∥xn−1 −th(n)i(n) xn∥∥∥ +lt (∥∥∥th(n)−1i(n) xn −th(n)−1i(n−n)xn−n∥∥∥ + ∥∥∥th(n)−1i(n−n)xn−n −x(n−n)−1∥∥∥ + ∥∥x(n−n)−1 −xn∥∥).(3.25) since for each n > n, n = (n−n)(mod n), on the other hand, we obtain from n = (h(n)−1)n +i(n) that n−n = ((h(n)−1)−1)n +i(n) = (h(n−n)−1)n +i(n−n). that is, h(n−n) = h(n) − 1, i(n−n) = i(n).(3.26) notice that∥∥∥th(n)−1i(n) xn −th(n)−1i(n−n)xn−n∥∥∥ = ∥∥∥th(n)−1i(n) xn −th(n)−1i(n) xn−n∥∥∥ ≤ lt ‖xn −xn−n‖ ,(3.27) and ∥∥∥th(n)−1i(n−n)xn−n −x(n−n)−1∥∥∥ = ∥∥∥th(n−n)i(n−n) xn−n −x(n−n)−1∥∥∥ .(3.28) substituting (3.27) and (3.28) into (3.25), we obtain that ‖xn−1 −tnxn‖ ≤ ∥∥∥xn−1 −th(n)i(n) xn∥∥∥ + lt(lt ‖xn −xn−n‖ + ∥∥∥th(n−n)i(n−n) xn−n −x(n−n)−1∥∥∥ + ∥∥x(n−n)−1 −xn∥∥). (3.29) in view of (3.18) and (3.24), we obtain that lim n→∞ ‖xn−1 −tnxn‖ = 0.(3.30) notice that ‖xn −tnxn‖ ≤ ‖xn −xn−1‖ + ‖xn−1 −tnxn‖ .(3.31) it follows from (3.23) and (3.30) that lim n→∞ ‖xn −tnxn‖ = 0.(3.32) notice that ‖xn −tn+jxn‖ ≤ ‖xn −xn+j‖ + ‖xn+j −tn+jxn+j‖ +‖tn+jxn+j −tn+jxn‖ ≤ (1 + lt)‖xn −xn+j‖ + ‖xn+j −tn+jxn+j‖(3.33) for all j ∈{1, 2, . . . ,n}. 92 saluja from (3.24) and (3.32), we obtain that lim n→∞ ‖xn −tn+jxn‖ = 0, ∀ j ∈{1, 2, . . . ,n}.(3.34) note that any subsequence of a convergent sequence converges to the same limit, it follows that lim n→∞ ‖xn −trxn‖ = 0, ∀ r ∈{1, 2, . . . ,n}.(3.35) letting ls = max{ls,i : 1 ≤ i ≤ n}, we have∥∥∥sh(n)i(n) xn −xn−1∥∥∥ ≤ ∥∥∥sh(n)i(n) xn −sh(n)i(n) xn−1∥∥∥ + ∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥ ≤ ls ‖xn −xn−1‖ + ∥∥∥sh(n)i(n) xn−1 −xn−1∥∥∥ .(3.36) in view of (3.17) and (3.23), we see that∥∥∥sh(n)i(n) xn −xn−1∥∥∥ = 0.(3.37) observe that ‖xn−1 −snxn−1‖ ≤ ∥∥∥xn−1 −sh(n)i(n) xn−1∥∥∥ + ∥∥∥sh(n)i(n) xn−1 −snxn−1∥∥∥ ≤ ∥∥∥xn−1 −sh(n)i(n) xn−1∥∥∥ + ls ∥∥∥sh(n)−1i(n) xn−1 −xn−1∥∥∥ ≤ ∥∥∥xn−1 −sh(n)i(n) xn−1∥∥∥ + ls(∥∥∥sh(n)−1i(n) xn−1 −sh(n)−1i(n−n)xn−n∥∥∥ + ∥∥∥sh(n)−1i(n−n)xn−n −x(n−n)−1∥∥∥ + ∥∥x(n−n)−1 −xn−1∥∥). (3.38) in view of∥∥∥sh(n)−1i(n) xn−1 −sh(n)−1i(n−n)xn−n∥∥∥ = ∥∥∥sh(n)−1i(n) xn−1 −sh(n)−1i(n) xn−n∥∥∥ ≤ ls ‖xn−1 −xn−n‖ ,(3.39) and ∥∥∥sh(n)−1i(n−n)xn−n −x(n−n)−1∥∥∥ = ∥∥∥sh(n−n)i(n−n) xn−n −x(n−n)−1∥∥∥ ,(3.40) we obtain that ‖xn−1 −snxn−1‖ ≤ ∥∥∥xn−1 −sh(n)i(n) xn−1∥∥∥ + ls(ls ‖xn−1 −xn−n‖ + ∥∥∥sh(n)−1i(n−n)xn−n −x(n−n)−1∥∥∥ + ∥∥x(n−n)−1 −xn−1∥∥). (3.41) in view of (3.17), (3.24) and (3.37), we obtain that lim n→∞ ‖xn−1 −snxn−1‖ = 0.(3.42) notice that ‖xn −snxn‖ ≤ ‖xn −xn−1‖ + ‖xn−1 −snxn−1‖ + ‖snxn−1 −snxn‖ ≤ (1 + ls)‖xn −xn−1‖ + ‖xn−1 −snxn−1‖ .(3.43) from (3.23) and (3.42), we see that lim n→∞ ‖xn −snxn‖ = 0.(3.44) convergence theorems of an implicit iterates with errors 93 on the other hand, we have ‖xn −sn+jxn‖ ≤ ‖xn −xn+j‖ + ‖xn+j −sn+jxn+j‖ + ‖sn+jxn+j −sn+jxn‖ ≤ (1 + ls)‖xn −xn+j‖ + ‖xn+j −sn+jxn+j‖ , ∀ j ∈{1, 2, . . . ,n}. (3.45) it follows from (3.24) and (3.45) that lim n→∞ ‖xn −sn+jxn‖ = 0, ∀ j ∈{1, 2, . . . ,n}.(3.46) note that any subsequence of a convergent sequence converges to the same limit, it follows that lim n→∞ ‖xn −srxn‖ = 0, ∀ r ∈{1, 2, . . . ,n}.(3.47) this completes the proof. now, we are in a position to prove our strong convergence theorems. theorem 3.2. let c be a nonempty closed convex subset of a real uniformly convex banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping and let si : c → c be a uniformly ls,ilipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) ⋂ ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.12). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n} and bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞ and ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if one of {s1,s2, . . . ,sn} or one of {t1,t2, . . . ,tn} is semicompact, then the sequence {xn} converges strongly to some point in f . proof. by lemma 3.1, it follows that lim n→∞ ‖xn −trxn‖ = lim n→∞ ‖xn −srxn‖ = 0, ∀ r ∈{1, 2, . . . ,n}.(3.48) without any loss of generality, we may assume that s1 is semi-compact. therefore, by (3.48), it follows that limn→∞‖xn −s1xn‖ = 0. since s1 is semi-compact, 94 saluja therefore there exists a subsequence {xnj} of {xn} such that xnj → x∗ ∈ c. for each r ∈{1, 2, . . . ,n}, we get that ‖x∗ −srx∗‖ ≤ ∥∥x∗ −xnj∥∥ + ∥∥xnj −srxnj∥∥ + ∥∥srxnj −srx∗∥∥ . (3.49) since sr is lipschitz continuous, we obtain from (3.48) that x ∗ ∈ ∩nr=1f(sr). notice that ‖x∗ −trx∗‖ ≤ ∥∥x∗ −xnj∥∥ + ∥∥xnj −trxnj∥∥ + ∥∥trxnj −trx∗∥∥ . (3.50) since tr is lipschitz continuous, we obtain from (3.48) that x ∗ ∈∩nr=1f(tr). this means that x∗ ∈ f . in view of theorem 3.1, we obtain that limn→∞ ‖xn −q‖ exists for all q ∈ f, therefore {xn} converges to x∗ ∈ f, and hence the result. this completes the proof. if si = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 3.2 is reduced to the following result: corollary 3.3. let c be a nonempty closed convex subset of a banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.13). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if one of {t1,t2, . . . ,tn} is semicompact, then the sequence {xn} converges strongly to some point in f . if ti = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 3.2 is reduced to the following result: corollary 3.4. let c be a nonempty closed convex subset of a banach space e. let si : c → c be a uniformly ls,i-lipschitz and asymptotically quasinonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.14). put bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, convergence theorems of an implicit iterates with errors 95 such that ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if one of {s1,s2, . . . ,sn} is semicompact, then the sequence {xn} converges strongly to some point in f. remark 3.1. theorem 3.2 extends and improves theorem 3.3 due to sun [19] to the case of more general class of asymptotically quasi-nonexpansive mapping and general implicit iterative process and without the boundedness of c which in turn generalizes theorem 2 by wittmann [21] from hilbert spaces to uniformly convex banach spaces. in 2005, chidume and shahzad [1]) introduced the following conception. recall that a family {ti}ni=1 : c → c with f = ∩ n i=1f(ti) 6= ∅ is said to satisfy condition (b) on c if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that for all x ∈ c max 1≤i≤n {‖x−tix‖}≥ f(d(x,f)).(3.51) based on condition (b), qin et al. [14] introduced the following conception for two finite families of mappings. recall that two families {si}ni=1 : c → c and {ti}ni=1 : c → c with f = ∩ n i=1f(si) ⋂ ∩ni=1f(ti) 6= ∅ are said to satisfy condition (b′) if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that for all x ∈ c max 1≤i≤n {‖x−six‖ + ‖x−tix‖}≥ f(d(x,f)).(3.52) note that condition (b′) defined above reduces to the condition (b) [1] if we choose si = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}. finally, an application of the convergence criteria established in theorem 3.1 is given below to obtain yet another strong convergence result in our setting. 4. application theorem 4.1. let c be a nonempty closed convex subset of a real uniformly convex banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping and let si : c → c be a uniformly ls,ilipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) ⋂ ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.12). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n} 96 saluja and bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞ and ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn, and c ≤ γn ≤ d < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if {s1,s2, . . . ,sn} and {t1,t2, . . . ,tn} satisfy condition (b′), then the iterative sequence {xn} converges strongly to some point in f. proof. as in the proof of theorem 3.2, (3.48) holds. taking lim inf on both sides of condition (b′) and using (3.48), we have that lim infn→∞ f(d(xn,f)) = 0. since f is a nondecreasing function with f(0) = 0 and f(r) > 0 for all r ∈ (0.∞), it follows that lim infn→∞d(xn,f) = 0. now by theorem 3.1, xn → p ∈ f, that is, {xn} converges strongly to a point in f. this completes the proof. if si = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 4.1 is reduced to the following result: corollary 4.1. let c be a nonempty closed convex subset of a banach space e. let ti : c → c be a uniformly lt,i-lipschitz and asymptotically quasi-nonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(ti) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.13). put an = max { 0, sup p∈f, n≥1 (∥∥∥th(n)i(n) xn −p∥∥∥−‖xn −p‖) : i ∈n}, such that ∑∞ n=1 an < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c ∈ (0, 1) such that a ≤ αn + βn and b ≤ γn ≤ c < 1/lt, where lt = max{lt,i : 1 ≤ i ≤ n}, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if {t1,t2, . . . ,tn} satisfies condition (b), then the iterative sequence {xn} converges strongly to some point in f . if ti = i, where i denotes the identity mapping, for each i ∈{1, 2, . . . ,n}, then theorem 4.1 is reduced to the following result: convergence theorems of an implicit iterates with errors 97 corollary 4.2. let c be a nonempty closed convex subset of a banach space e. let si : c → c be a uniformly ls,i-lipschitz and asymptotically quasinonexpansive type mapping for each 1 ≤ i ≤ n. assume that f = ∩ni=1f(si) is nonempty. let {un} be a bounded sequence in c. let {αn}, {βn}, {γn} and {δn} be sequences in (0, 1) such that αn + βn + γn + δn = 1 for each n ≥ 1. let {xn} be a iterative sequence generated in (2.14). put bn = max { 0, sup p∈f, n≥1 (∥∥∥sh(n)i(n) xn−1 −p∥∥∥−‖xn−1 −p‖) : i ∈n}, such that ∑∞ n=1 bn < ∞. assume that the following restrictions are satisfied: (a) there exist constants a,b,c,d ∈ (0, 1) such that a ≤ αn, b ≤ βn and c ≤ γn, for all n ≥ 1; (b) ∑∞ n=1 δn < ∞. if {s1,s2, . . . ,sn} satisfies condition (b), then the iterative sequence {xn} converges strongly to some point in f. remark 4.1. (i) our results extend the corresponding results of ud-din and khan [4] to the case of more general class of asymptotically quasi-nonexpansive mappings considered in this paper. (ii) our results also generalize and improve the corresponding results of sun [19], wittmann [21] and xu and ori [24] to the case of more general class of nonexpansive, asymptotically quasi-nonexpansive mappings and general implicit iterative process for two finite families of mappings considered in this paper. (iii) our results also extend the corresponding results of [1, 3, 7, 15] and many others. example 4.1. let e be the real line with the usual norm |.| and k = [0, 1]. define t : k → k by t(x) = sinx, x ∈ [0, 1], for x ∈ k. obviously t(0) = 0, that is, 0 is a fixed point of t, that is, f(t) = {0}. now we check that t asymptotically quasi-nonexpansive type mapping. in fact, if x ∈ [0, 1] and p = 0 ∈ [0, 1], then |t(x) −p| = |t(x) − 0| = |sinx− 0| = |sinx| ≤ |x| = |x− 0| = |x−p|, that is, |t(x) −p| ≤ |x−p|. thus, t is quasi-nonexpansive. it follows that t is asymptotically quasi-nonexpansive with the constant sequence {kn} = {1} for each n ≥ 1 and hence it is asymptotically quasi-nonexpansive type mapping (by remark 2.1). but the converse does not hold in general. 98 saluja 5. conclusion the class of asymptotically quasi-nonexpansive type mapping is more general than the class of nonexpansive, quasi-nonexpansive, asymptotically nonexpansive and asymptotically quasi-nonexpansive mappings. therefore the results presented in this paper are improvement and generalization of several known results in the existing literature. references [1] c.e. chidume, n. shahzad, strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, nonlinear anal., tma, 62 (2005), no. 6, 1149-1156. [2] j.b. diaz, f.t. metcalf, on the structure of the set of subsequential limit points of successive approximations, bull. amer. math. soc. 73 (1967), 516-519. [3] h. fukhar-ud-din, s.h. khan, convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings, int. j. math. math. sci. (2004), no. 37-40, 1965-1971. [4] h. fukhar-ud-din, a.r. khan, convergence of implicit iterates with errors for mappings with unbounded domain in banach spaces, int. j. math. math. sci. 10 (2005), 1643-1653. [5] m.k. ghosh, l. debnath, convergence of ishikawa iterates of quasinonexpansive mappings, j. math. anal. appl. 207 (1997), 96-103. [6] k. goebel, w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35 (1972), 171-174. [7] s.h. khan, w. takahashi, approximating common fixed points of two asymptotically nonexpansive mappings, sci. math. jpn. 53 (2001), no. 1, 143-148. [8] w.a. kirk, fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, israel j. math. 17 (1974), 339-346. [9] l.s. liu, ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces, j. math. anal. appl. 194 (1995), no. 1, 114-125. [10] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 259 (2001), 1-7. [11] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings with error member, j. math. anal. appl. 259 (2001), 18-24. [12] w.r. mann, mean value methods in iteration, proc. amer. math. soc. 4 (1953), 506-510. [13] w.v. petryshyn, t.e. williamson, strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, j. math. anal. appl. 43 (1973), 459-497. [14] x. qin, s.m. kang, r.p. agarwal, on the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings, fixed point theory appl. (2010), article id 714860, 19pp. (doi:10.1155/2010/74860) [15] d.r. sahu, j.s. jung, fixed point iteration processes for non-lipschitzian mappings of asymptotically quasi-nonexpansive type, int. j. math. math. sci. 33 (2003), 2075-2081. [16] j. schu, weak and strong convergence to fixed points of asymptotically nonexpansive mappings, bull. astra. math. soc. 43 (1991), no. 1, 153-159. convergence theorems of an implicit iterates with errors 99 [17] n. shahzad, a. udomene, approximating common fixed points of two asymptotically quasi-nonexpansive mappings in banach spaces, fixed point theory appl. (2006), article id 18909, 10pp. [18] t. shimizu, w. takahashi, strong convergence theorem for asymptotically nonexpansive mappings, nonlinear anal. 26 (1996), no. 2, 265-272. [19] z. sun, strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 286 (2003), no. 1, 351-358. [20] k.k. tan, h.k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178 (1993), 301-308. [21] r. wittmann, approximation of fixed points of nonexpansive mappings, arch. math. 58 (1992), 486-491. [22] h.k. xu, inequalities in banach spaces with applications, nonlinear anal. 16 (1991), no. 12, 1127-1138. [23] y. xu, ishikawa and mann iterative process with errors for nonlinear strongly accretive operator equations, j. math. anal. appl. 224 (1998), no. 1, 91-101. [24] h.k. xu, r.g. ori, an implicit iteration process for nonexpansive mappings, numer. funct. anal. optim. 22 (2001), no. 5-6, 767-773. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications issn 2291-8639 volume 1, number 2 (2013), 106-112 http://www.etamaths.com a note on: multi-step approximation schemes for the fixed points of finite family of asymptotically pseudocontractive mappings mogbademu, adesanmi alao abstract. in this paper, using an analytical technique we obtain a strong convergence for a modified threestep iterative scheme due to suantai [6] for asymptotically pseudocontractive mappings in real banach spaces. our result is an improvement and a correction of rafiq’s [4] results. 1. introduction let e be an arbitrary real banach space and let j : e → 2e ∗ be the normalized duality mapping defined by j(x) = {f ∈ e∗ :< x,f >= ‖x‖2 = ‖f‖2},∀x ∈ e where e∗ denotes the dual space of e and < .,. > denotes the generalized duality pairing between e and e∗. the single-valued normalized duality mapping is denoted by j. let k be a nonempty closed convex subset of e and t : k → k be a map. the mapping t is said to be uniformly llipschitzian if there exists a constant l > 0 such that ‖tnx−tny‖≤ l‖x−y‖ for any x,y ∈ k and ∀n ≥ 1. the mapping t is said to be asymptotically pseudocontractive if there exists a sequence (kn) ⊂ [1,∞) with limn→∞kn = 1 and for any x,y ∈ k there exists j(x−y) ∈ j(x−y) such that < tnx−tny,j(x−y) >≤ kn‖x−y‖2,∀n ≥ 1. the concept of asymptotically pseudocontractive mappings was introduced by schu [5]. ofoedu [3] used the modified mann iteration process introduced by schu [5] , xn+1 = (1 −αn)xn + αntnxn n ≥ 0, to obtain a strong convergence theorem for uniformly lipschitzian asymptotically pseudo-contractive mapping in real banach space setting. this result itself is a generalization of many of the previous results (see [3] and the references therein). 2010 mathematics subject classification. 47h10, 46a03. key words and phrases. noor iteration; uniformly lipschitzian; asymptotically pseudocontractive ; three-step iterative scheme ; banach spaces. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 106 a note 107 recently, rafiq [4] employed the iterative scheme introduced by suantai [6] to establish a strong convergence result for a modified three-step iterative scheme when dealing with asymptotically pseudocontractive mappings. in fact, he proved the following theorem : theorem 1.1 ([4]). let k be a nonempty closed convex subset of e, t : k → k be the asymptotically pseudocontractive mapping with t(k) bounded and the sequence kn ⊂ [1,∞), limn→∞kn = 1 such that f(t) = {x ∈ k : tx = x} 6= φ. further let t be uniformly continuous and {an}n≥,{bn}n≥0,{cn}n≥0,{a′n}n≥0, {b′n}n≥0,{c′n}n≥0,{a′′n}n≥0 be real sequences in [0, 1]; an +bn +cn = 1 = a′n +b′n +c′n satisfying the following conditions: (i) limn→∞(bn + c ′ n) = 0 = limn→∞ b ′ n = limn→∞ c ′ n = limn→∞a ′′ n (ii) ∑ n≥0(bn + an) = ∞ . for arbitrary x0 ∈ k, let {xn}∞n=1 be the iterative sequence defined by xn+1 = anxn + bnt nyn + cnt nzn (1.1) yn = a ′ nxn + b ′ nt nzn + c ′ nt nxn. zn = (1 −a′′n)xn + a ′′tnxn suppose for any ρ ∈ f(t) there exists a strictly increasing function ψ : [0,∞) → [0,∞) with ψ(0) = 0 such that < tnx−ρ,j(x−ρ) >≤ kn‖x−ρ‖2 −ψ(‖x−ρ‖) for all x ∈ k. then {xn}n≥0 converges strongly to a fixed point of t . we observed some mistakes in the proof of the theorem above. for instance, in equation (10) of rafiq [4] the author set; dn = ‖tnyn −tnxn+1‖, en = ‖tnzn − tnxn+1‖. he further obtained from equations (12) and (14) that limn→∞‖yn − xn+1‖ = 0, limn→∞‖zn − xn+1‖ = 0. and, using the uniform continuity of t , he concluded that dn = limn→∞‖tnyn − tnxn+1‖ = 0, en = limn→∞‖tnzn − tnxn+1‖ = 0. this conclusion is, however not correct. for example, let tx = 4x ∀x ∈ r and suppose xn+1 = (1 − 1n) ,yn = zn = (1 + 1 n ) for all n ≥ 1, obviously dn = limn→∞‖tnyn − tnxn+1‖ 6= 0, en = limn→∞‖tnzn −tnxn+1‖ 6= 0. thus, the result of rafiq [4] needs to be improve. in this paper, an improvement and a correction to the main result of rafiq [4] is presented. the following lemmas are needed. lemma 1.1 [3, 4] . let e be real banach space and j : e → 2e ∗ be the normalized duality mapping. then, for any x,y ∈ e ‖x + y‖2 ≤‖x‖2 + 2 < y,j(x + y) >,∀j(x + y) ∈ j(x + y). lemma 1.2 [7]. let {αn} be a nonnegative sequence which satisfies the following inequality αn+1 ≤ (1 −λn)αn + σn, where λn ∈ (0, 1),∀n ∈ n, ∑∞ n=1 λn = ∞ and σn = o(λn). then limn→∞αn = 0. 2. main results theorem 2.1. let k be a nonempty closed convex subset of e, t : k → k be asymptotically pseudocontractive and uniformly lipschitzian map with lipschitzian constant l > 0 and the sequence kn ⊂ [1,∞), limn→∞kn = 1 such that f(t) = 108 mogbademu {x ∈ k : tx = x} 6= φ. let {an}n≥0,{bn}n≥0,{cn}n≥0,{a′n}n≥0, {b′n}n≥0,{c′n}n≥0,{a′′n}n≥0 be real sequences in [0, 1] satisfying : (i) an + bn + cn = 1 = a ′ n + b ′ n + c ′ n (ii) limn→∞ bn = limn→∞ b ′ n = limn→∞ cn = 0 = limn→∞ c ′ n (iii) ∑ n≥0(bn + cn) = ∞. for arbitrary x0 ∈ k, let {xn} ∞ n=1 be the iterative sequence defined by (1.1). suppose for any ρ ∈ f(t) there exists a strictly increasing function ψ : [0,∞) → [0,∞) with ψ(0) = 0 such that < tnxn −ρ,j(xn −ρ) >≤ kn‖xn −ρ‖2 −ψ(‖xn −ρ‖) for all x ∈ k. then {xn}n≥0 converges strongly to a fixed point of t . proof: from lemma 1.2, the equation (1.1) and the definition of the asymptotically pseudocontractive and uniformly lipschitzian map, we have ‖xn+1 −ρ‖2 = ‖(1 − bn − cn)(xn −ρ) + bn(tnyn −ρ) + cn(tnzn −ρ)‖2 ≤ (1 − (bn + cn))2‖xn −ρ‖2 +2〈bn(tnyn −ρ) + cn(tnzn −ρ),j(xn+1 −ρ)〉 ≤ (1 − (bn + cn))2‖xn −ρ‖2 +2bn〈tnyn −tnxn+1,j(xn+1 −ρ)〉 +2bn〈tnxn+1 −ρ,j(xn+1 −ρ)〉 +2cn〈tnxn+1 −ρ,j(xn+1 −ρ)〉 +2cn〈tnzn −tnxn+1,j(xn+1 −ρ)〉 ≤ (1 − (bn + cn))2‖xn −ρ‖2 +2bn(kn‖xn+1 −ρ‖2 −ψ(‖xn −ρ‖)) +2bn‖tnyn −tnxn+1‖‖xn+1 −ρ‖ +2cn‖tnzn −tnxn+1‖‖xn+1 −ρ‖ +2cn(kn‖xn+1 −ρ‖2 −ψ(‖xn −ρ‖)) ≤ (1 − (bn + cn))2‖xn −ρ‖2 + 2kn(bn + cn)‖xn+1 −ρ‖2 −2(bn + cn)ψ(‖xn+1 −ρ‖) +2bn‖tnyn −tnxn+1‖‖xn+1 −ρ‖ +2cn‖tnzn −tnxn+1‖‖xn+1 −ρ‖ (2.1) ≤ (1 − (bn + cn))2‖xn −ρ‖2 + 2kn(bn + cn)‖xn+1 −ρ‖2 −2(bn + cn)ψ(‖xn+1 −ρ‖) +2bn‖tnyn −tnxn+1‖‖xn+1 −ρ‖ +2cn‖tnzn −tnxn+1‖‖xn+1 −ρ‖ ≤ (1 − δn)2‖xn −ρ‖2 + 2knδn‖xn+1 −ρ‖2 −2δnψ(‖xn+1 −ρ‖) + 2bnl‖yn −xn+1‖‖xn+1 −ρ‖ +2cnl‖zn −xn+1‖‖xn+1 −ρ‖, where 0 ≤ δn = bn + cn < 1. we note that a note 109 ‖yn −xn+1‖ = ‖xn+1 −yn‖ = ‖(1 − bn − cn)(xn −yn) + bn(tnyn −yn) +cn(t nzn −yn)‖ ≤ (1 − bn − cn)‖xn −yn‖ + bn‖tnyn −ρ + ρ−yn‖ +cn‖tnzn −ρ + ρ−yn‖ ≤ (1 − bn − cn)‖xn −yn‖ + bn(1 + l)‖yn −ρ‖ +cn(l‖zn −ρ‖ + ‖yn −ρ‖) = (1 − bn − cn)‖xn −yn‖ + bn(1 + l)(‖yn −xn + xn −ρ‖) +cnl(‖zn −xn + xn −ρ‖) + cn(‖yn −xn + xn −ρ‖) ≤ (1 − bn − cn)‖xn −yn‖ +bn(1 + l)(‖yn −xn‖ + ‖xn −ρ‖) +cnl(‖zn −xn‖ + ‖xn −ρ‖) +cn(‖yn −xn‖ + ‖xn −ρ‖) ≤ (1 + bnl)‖xn −yn‖ + δn(1 + l)‖xn −ρ‖ +cnl‖zn −xn‖ (2.2) = (1 + bnl)‖xn −yn‖ + δn(1 + l)‖xn −ρ‖ +cnl(‖zn −ρ + ρ−xn‖) ≤ (1 + bnl)‖xn −yn‖ + δn(1 + l)‖xn −ρ‖ +cnl(‖zn −ρ‖ + ‖ρ−xn‖) = (1 + bnl)‖xn −yn‖ + [δn(1 + l) + cnl]‖xn −ρ‖ +cnl‖zn −ρ‖ ≤ (1 + bnl)‖xn −yn‖ + [δn(1 + l) + cnl]‖xn −ρ‖ +cnl(1 + a ′′ nl)‖xn −ρ‖ = (1 + bnl)‖xn −yn‖ + [δn(1 + l) +cnl(2 + a ′′ nl)]‖xn −ρ‖. observe that (2.3) ‖xn −yn‖ = ‖yn −xn‖ = ‖(1 − b′n − c′n)xn + b′ntnzn + c′ntnxn −xn‖ = ‖b′n(tnzn −xn) + c′n(tnxn −xn)‖ ≤ b′nl‖zn −ρ‖ + (b′n + (1 + l)c′n)‖xn −ρ‖ ≤ b′nl(1 + a′′nl)‖xn −ρ‖ + (b′n + (1 + l)c′n)‖xn −ρ‖ = [b′nl(1 + a ′′ nl) + (b ′ n + (1 + l)c ′ n)]‖xn −ρ‖. substituting (2.3) into (2.2) then, (2.4) ‖xn+1 −yn‖ ≤ d1n‖xn −ρ‖ 110 mogbademu where d1n = (1 + bnl)b ′ nl(1 + a ′′ nl) + b ′ n + (1 + l)(c ′ n + δn) + cnl(2 + a ′′ nl). in a similar way (2.5) ‖zn −xn+1‖ = ‖xn+1 −zn‖ = ‖(1 − bn − cn)(xn −zn) + bn(tnyn −zn) +cn(t nzn −zn)‖ ≤ (1 − bn − cn)‖xn −zn‖ + bnl‖yn −ρ‖ + bn‖zn −ρ‖ +cn(1 + l)‖zn −ρ‖ = (1 −δn)‖xn −zn‖ + bnl‖yn −xn + xn −ρ‖ +bn‖zn −ρ‖ + cn(1 + l)‖zn −ρ‖ ≤ (1 − δn)(‖xn −ρ‖ + ‖zn −ρ‖) + bnl‖yn −xn‖ +bnl‖xn −ρ‖ + bn‖zn −ρ‖ + cn(1 + l)‖zn −ρ‖ ≤ (1 − δn)(‖xn −ρ‖ + (1 + a′′nl)‖xn −ρ‖) +bnl[b ′ nl(1 + a ′′ nl) + (b ′ n + (1 + l)c ′ n)]‖xn −ρ‖ +bnl‖xn −ρ‖ + bn(1 + a′′nl)‖xn −ρ‖ +cn(1 + l)(1 + a ′′ nl)‖xn −ρ‖ ≤ cn(‖xn −ρ‖ + (1 + a′′nl)‖xn −ρ‖) +bnl[b ′ nl(1 + a ′′ nl) + (b ′ n + (1 + l)c ′ n)]‖xn −ρ‖ +bnl‖xn −ρ‖ + bn(1 + a′′nl)‖xn −ρ‖ +cn(1 + l)(1 + a ′′ nl)‖xn −ρ‖ = d2n‖xn −ρ‖, where (2.6) d2n = cn(2 + a ′′ nl) + bn(1 + l[b ′ nl(1 + a ′′ nl) + (b ′ n + (1 + l)c ′ n)]) +(1 + a′′nl)(bn + cnl(1 + l)). substituting (2.3) and (2.5) into (2.1) we have the equation that follows (2.7) ‖xn+1 −ρ‖2 = (1 − δn)2‖xn −ρ‖2 + 2knδn‖xn+1 −ρ‖2 −2δnψ(‖xn+1 −ρ‖) + 2bnld1n‖xn −ρ‖‖xn+1 −ρ‖ +2cnld 2 n‖xn −ρ‖‖xn+1 −ρ‖ ≤ (1 − δn)2‖xn −ρ‖2 + 2knδn‖xn+1 −ρ‖2 −2δnψ(‖xn+1 −ρ‖) + bnld1n(‖xn −ρ‖2 + ‖xn+1 −ρ‖2) +cnld 2 n(‖xn −ρ‖2 + ‖xn+1 −ρ‖2) ≤ (1 − δn)2‖xn −ρ‖2 + 2knδn‖xn+1 −ρ‖2 −2δnψ(‖xn+1 −ρ‖) + δnld1n(‖xn −ρ‖2 + ‖xn+1 −ρ‖2) +δnld 2 n(‖xn −ρ‖2 + ‖xn+1 −ρ‖2) + 2δnld2n. setting, (2.8) an = (1 − (2knδn + δnl(d1n + d2n))) bn = ((1 −δn)2 + δnl(d1n + d2n)) (2.9) cn = (2(1 −kn) −δn − 2l(d1n + d2n)) dn = 2ld 2 n. suppose we set infn≥n ψ(‖xn+1−ρ‖) 1+‖xn+1−ρ‖2 = r. then r = 0. if it is not the case, we assume that r > 0. let 0 < r < min{1,r}, then ψ(‖xn+1−ρ‖) 1+‖xn+1−ρ‖2 ≥ r, i.e., (2.10) ψ(‖xn+1 −ρ‖) ≥ r + r‖xn+1 −ρ‖2 ≥ r‖xn+1 −ρ‖2. a note 111 since limn→∞knδn = 0, there exists a natural number n0 such that 1 2 < an < 1, for all n > n0. thus equation (2.7) becomes, (2.11) ‖xn+1 −p‖2 ≤ bnan‖xn −p‖ 2 − 2δn ψ(‖xn+1−ρ‖) an + δndn an ≤ (1 − δncn)‖xn −p‖2 − 2δnψ(‖xn+1 −ρ‖) + 2δndn. substituting (2.10) into (2.11), we have (2.12) ‖xn+1 −p‖2 ≤ 1−δncn1+2δnr ‖xn −p‖ 2 + 2δndn 1+2δnr ≤ (1 − δn (cn+2r) 1+2δnr )‖xn −p‖2 + 2δndn1+2δnr since limn→∞δn = limn→∞cn = 0, we choose n1 > n0 such that (cn+2r) 1+2δnr > r, for all n > n1. it follows from (2.12) that (2.13) ‖xn+1 −p‖2 ≤ (1 −δnr)‖xn −p‖2 + 2δndn1+2δnr for all n > n1. if we set bn = ‖xn−ρ‖, it follows from lemma 1.2 that, limn→∞ bn = 0, which is a contradiction. thus, there exists an infinite subsequence such that limn→∞ bnj0+1 = 0. next, we prove that limn→∞ bnj0+m = 0 by induction. let ∀ � ∈ (0, 1), choose nj0 > n such that bnj0+1 < �, cnj0+1 > ψ(�) 4 , dnj0+1 < ψ(�) 2 . first, we want to prove bnj0+2 < �. suppose it is not the case. then bnj0+2 ≥ �, this implies ψ(bnj0+2) ≥ ψ(�). using (2.11) we now obtain the following (2.14) b2nj0+2 ≤ b 2 nj0+1 − δn ψ(�) 4 �2 − 2δnψ(�) + 2δn ψ(�) 2 ≤ b2nj0+1 − δnψ(�) < �2, which is a contracdiction. hence bnj0+2 < � holds and inductively we can show that bnj0+i < �, ∀i ≥ 1 holds. this implies that limn→∞ bn = 0, i.e., limn→∞‖xn−ρ‖ = 0. references [1] s. s. chang, some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, proc. amer. math. soc., 129(2000), 845-853. [2] s. s. chang, y. j. cho, j. k. kim, some results for uniformly l-lipschitzian mappings in banach spaces, applied mathematics letters, 22(2009), 121-125. [3] e.u. ofoedu, strong convergence theorem for uniformly l-lipschitzian asymptotically pseudocontractive mapping in real banach space, j. math. anal. appl., 321(2006), 722-728. [4] a. rafiq, multi-step approximation schemes for the fixed points of finite family of asymptotically pseudocontractive mappings, general mathematics, vol.19 nos. 4(2011), 41c49. [5] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, j. math. anal. appl., 158(1999), 407-413. [6] s. suantai, weak and strong convergence criteria of noor iterations for asymptotically nonexpansive mappings, j. math. anal. appl., 311(2005), 506-517. [7] x. weng, fixed point iteration for local strictly pseudocontractive mappings, proc. amer. math. soc.113(1991), 727-731. 112 mogbademu [8] b. xu, m. a. noor, fixed point iterations for asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl., 267, 2002, 444-453. department of mathematics, university of lagos, lagos nigeria international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 231-247 http://www.etamaths.com fractional differential equations and inclusions with nonlocal generalized riemann-liouville integral boundary conditions bashir ahmad1,∗, sotiris k. ntouyas1,2 and ahmed alsaedi1 abstract. in this paper, we study a new kind of nonlocal boundary value problems of nonlinear fractional differential equations and inclusions supplemented with nonlocal and generalized riemannliouville fractional integral boundary conditions. in case of single valued maps (equations), we make use of contraction mapping principle, fixed point theorem due to sadovski, krasnoselskii-schaefer fixed point theorem due to burton and kirk, and fixed point theorem due to o’regan to obtain the desired existence results. on the other hand, the existence results for inclusion case are based on krasnoselskii’s fixed point theorem for multivalued maps and nonlinear alternative for contractive maps. examples illustrating the main results are also constructed. 1. introduction fractional order differential and integral operators play an important role in the mathematical modeling of several real world problems. it has been mainly due to the fact that such operators can describe the memory and hereditary properties of various materials and processes involved in the problem at hand. examples include physics, chemical technology, population dynamics, biotechnology, and economics [1–3]. in recent years, the study of initial and boundary value problems of fractional differential equations involving a variety of conditions have been investigated by several researchers, and the literature on the topic is now much enriched. for examples and details, see [4][18] and the references cited therein. nonlocal conditions are found to be more plausible than the standard initial conditions for the formulation of some physical phenomena in certain problems of thermodynamics, elasticity and wave propagation. as a matter of fact, such conditions become inevitable in a situation when the controllers at the boundary positions dissipate or add energy according to censors located at intermediate positions. further details can be found in the work by byszewski [19, 20]. integral boundary conditions also find decent applications in blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. in particular, the assumption of ‘circular cross-section’ throughout the vessels in the study of fluid flow problems is not always justifiable. in this situation, integral boundary conditions provide a more realistic (practical) approach. also, integral boundary conditions are found to be useful in regularizing ill-posed parabolic backward problems in time partial differential equations, see for example, mathematical models for bacterial self-regularization [21]. integral boundary conditions involve classical, riemann-liouville or hadamard or erdélyi-kober type integral operators. in [22], it has been discussed that riemannliouville and hadamard fractional integrals can jointly be represented by a single integral, which is called generalized riemann-liouville fractional integral. in this paper, we introduce a new class of boundary value problems of fractional differential equations and inclusions supplemented with nonlocal and generalized riemann-liouville fractional integral received 18th october, 2016; accepted 4th january, 2017; published 1st march, 2017. 2010 mathematics subject classification. 34a08, 34b10, 34a60. key words and phrases. caputo fractional derivative; generalized riemann-liouville fractional integral; nonlocal conditions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 231 232 ahmad, ntouyas and alsaedi boundary conditions. in precise terms, we consider the following nonlocal problems:  dαx(t) = f(t,x(t)), t ∈ [0,t], x(0) = g(x), x(t) = β ρ1−q γ(q) ∫ ξ 0 sρ−1 (ξρ −sρ)1−q x(s)ds := β ρiqx(ξ), 0 < ξ < t, (1.1) and { dαx(t) ∈ f(t,x(t)), t ∈ [0,t], x(0) = g(x), x(t) = β ρiqx(ξ), 0 < ξ < t, (1.2) where dα is the caputo fractional derivative of order 1 < α ≤ 2, f : [0,t] × r → r is a continuous function, g : c([0,t],r) → r is a continuous function, ρiq is the generalized riemann-liouville fractional integral of order q > 0, ρ > 0 (see definition 2.4) and f : [0,t] × r → p(r) is a multi valued function, (p(r) is the family of all nonempty subjects of r). we remark that g(x) in (1.1) and (1.2) may be represented as g(x) = ∑p j=1 αjx(tj), where αj,j = 1, . . . ,p, are given constants and 0 < t1 < ... < tp ≤ 1. the rest of the paper is organized as follows: in section 2 we present some useful preliminaries and lemmas. section 3 deals with the existence and uniqueness results for problem (1.1) which are established via contraction mapping principle and fixed point theorems due to sadovski, a krasnoselskiischaefer fixed point theorem due to burton and kirk, and a fixed point theorem due o’regan. in section 4, we discuss the existence of solutions for problem (1.2) by means of krasnoselskii fixed point theorem for multivalued maps and nonlinear alternative for contractive maps. examples illustrating the main work are also presented. 2. preliminaries in this section, we recall some basic concepts of fractional calculus [1, 2] and present known results needed in our forthcoming analysis. definition 2.1. the riemann-liouville fractional integral of order q > 0 of a continuous function f : (0,∞) → r is defined by jqf(t) = 1 γ(q) ∫ t 0 (t−s)q−1f(s)ds, provided the right-hand side is point-wise defined on (0,∞). definition 2.2. the riemann-liouville fractional derivative of order q > 0, n− 1 < q < n, n ∈ n, is defined as d q 0+f(t) = 1 γ(n−q) ( d dt )n ∫ t 0 (t−s)n−q−1f(s)ds, where the function f(t) has absolutely continuous derivative up to order (n− 1). definition 2.3. the caputo derivative of order q for a function f : [0,∞) → r can be written as cdqf(t) = d q 0+ ( f(t) − n−1∑ k=0 tk k! f(k)(0) ) , t > 0, n− 1 < q < n. remark 2.1. if f(t) ∈ cn[0,∞), then cdqf(t) = 1 γ(n−q) ∫ t 0 f(n)(s) (t−s)q+1−n ds = in−qf(n)(t), t > 0, n− 1 < q < n. definition 2.4. [22] the generalized riemann-liouville fractional integral of order q > 0 and ρ > 0, of a function f(t), for all 0 < t < ∞, is defined as ρiqf(t) = ρ1−q γ(q) ∫ t 0 sρ−1f(s) (tρ −sρ)1−q ds, provided the right-hand side is point-wise defined on (0,∞). fractional differential equations and inclusions 233 remark 2.2. we notice that the above definition corresponds to the one for riemann-liouville fractional integral of order q > 0 when ρ = 1, while the famous hadamard fractional integral follows for ρ → 0, that is, lim ρ→0 ρiqf(t) = 1 γ(q) ∫ t 0 ( log t s )q−1 f(s) s ds. the following lemma is obvious via definition 2.4. lemma 2.1. let q > 0 and p > 0 be the given constants. then ρiqtp = γ ( p+ρ ρ ) γ ( p+ρq+ρ ρ ) tp+ρq ρq . (2.1) lemma 2.2. [2] for q > 0, the general solution of the fractional differential equation cdqx(t) = 0 is given by x(t) = c0 + c1t + . . . + cn−1t n−1, where ci ∈ r, i = 1, 2, . . . ,n− 1 (n = [q] + 1). in view of lemma 2.2, it follows that iq cdqx(t) = x(t) + c0 + c1t + . . . + cn−1t n−1, (2.2) for some ci ∈ r, i = 1, 2, . . . ,n− 1 (n = [q] + 1). lemma 2.3. for any y ∈ c([0,t],r), the following linear fractional boundary value problem{ cdαx(t) = y(t), 1 < α ≤ 2, x(0) = g(x), x(t) = β ρiqx(ξ), 0 < ξ < t, (2.3) is equivalent to fractional integral equation: x(t) = jαy(t) + t λ { β ρiqjαy(ξ) −jαy(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), (2.4) where λ = t −β ξρq+1 ρq γ( 1+ρ ρ ) γ( 1+ρq+ρ ρ ) 6= 0. (2.5) proof. it is well known that the general solution of the fractional differential equation in (2.3) can be written as x(t) = c0 + c1t + j αy(t), (2.6) where c0,c1 ∈ r are arbitrary constants. using the first condition (x(0) = g(x)) given by (2.3) in (2.6), we get c0 = g(x). applying the generalized riemann-liouville fractional integral operator on (2.6) and using lemma 2.1, we obtain ρiqx(t) = ρiqjαy(t) + c0 tρq ρq 1 γ(q + 1) + c1 tρq+1 ρq γ( 1+ρ ρ ) γ( 1+ρq+ρ ρ ) , (2.7) which together with the second condition of (2.3) yields jαy(t) + c1t + c0 = β ρiqjαy(ξ) + βc0 ξρq ρq 1 γ(q + 1) + βc1 ξρq+1 ρq γ( 1+ρ ρ ) γ( 1+ρq+ρ ρ ) . (2.8) using c0 = g(x) in (2.8), we find that c1 = 1 λ { β ρiqjαy(ξ) −jαy(t) + βg(x) ξρq ρq 1 γ(q + 1) } . substituting the values of c0,c1 in (2.6), we get (2.4). conversely, it follows by direct computation that the integral equation (2.4) satisfies the problem (2.3). this completes the proof. � 234 ahmad, ntouyas and alsaedi for computational convenience, we introduce the notations: jzf(s,x(s))(y) = 1 γ(z) ∫ y 0 (y −s)z−1f(s,x(s))ds, ρizf(s,x(s))(y) = ρ1−z γ(z) ∫ y 0 sρ−1f(s,x(s)) (yρ −sρ)1−z ds, where z > 0 and y ∈ [0,t]. 3. existence and uniqueness results for problem (1.1) we denote by c = c([0,t],r) the banach space of all continuous functions from [0,t] → r endowed with a topology of uniform convergence with the norm defined by ‖x‖ = sup{|x(t)| : t ∈ [0,t]}. also by l1([0,t],r) we denote the banach space of measurable functions x : [0,t] → r which are lebesgue integrable and normed by ‖x‖l1 = ∫t 0 |x(t)|dt. in view of lemma 2.3, we define an operator p : c →c associated with problem (1.1) as (px)(t) = jαf(s,x(s))(t) + t λ { β ρiqjαf(s,x(s))(ξ) −jαf(s,x(s))(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), t ∈ [0,t]. (3.1) let us define p1,2 : c →c by (p1x)(t) = jαf(s,x(s))(t) + t λ { β ρiqjαf(s,x(s))(ξ) −jαf(s,x(s))(t) } , (3.2) and (p2x)(t) = [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x). (3.3) clearly (px)(t) = (p1x)(t) + (p2x)(t), t ∈ [0,t]. (3.4) in the sequel, we use the notations: p0 := tα γ(α + 1) ( 1 + t |λ| ) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) , (3.5) and k0 := 1 + t |λ| |β| ξρq ρq 1 γ(q + 1) . (3.6) theorem 3.1. let f : [0,t] ×r → r be a continuous function. assume that (a1) |f(t,x) −f(t,y)| ≤ l‖x−y‖,∀t ∈ [0,t], l > 0, x,y ∈ r; (a2) g : c([0, 1],r) → r is a continuous function satisfying the condition: |g(u) −g(v)| ≤ `‖u−v‖, ` < k−10 , ∀ u,v ∈ c([0, 1],r), ` > 0; (a3) γ := lp0 + k0` < 1. then the boundary value problem (1.1) has a unique solution on [0,t]. fractional differential equations and inclusions 235 proof. for x,y ∈ c and for each t ∈ [0,t], from the definition of p and assumptions (a1) and (a2), we obtain |(px)(t) − (py)(t)| ≤ sup t∈[0,t] { jα|f(s,x(s)) −f(s,y(s))|(t) + t |λ| jα|f(s,x(s)) −f(s,y(s))|(t) + |β|t |λ| ρiqjα|f(s,x(s)) −f(s,y(s))|(ξ) + ∣∣∣∣∣1 + β tλ ξ ρq ρq 1 γ(q + 1) ∣∣∣∣∣|g(x) −g(y)| } ≤ l‖x−y‖jα(1)(t) + l‖x−y‖ t |λ| jα(1)(t) + l‖x−y‖ |β|t |λ| ρiqjα(1)(ξ) + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) |g(x) −g(y)| ≤ l { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } ‖x−y‖ + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖x−y‖ = (lp0 + k0`)‖x−y‖. hence ‖(px) − (py)‖≤ γ‖x−y‖. as γ < 1 by (a3), the operator p is a contraction map from the banach space c into itself. hence the conclusion of the theorem follows by the contraction mapping principle (banach fixed point theorem). � example 3.1. consider the following fractional boundary value problem  cd3/2x(t) = sin2(πt) 2(et + 9) ( |x(t)| |x(t)| + 1 + 1 ) |x(t)| + √ 3 4 , t ∈ [0, 1], x(0) = 1 2 + 1 12 tan−1(x(1/8)), x(1) = 1 2 2/3i3/2x(3/4). (3.7) here, α = 3/2, t = 1, β = 1/2, ξ = 3/4, ρ = 2/3, q = 3/2, f(t,x) = sin2(πt) 2(et + 9) ( |x| |x| + 1 + 1 ) |x| + √ 3 4 , g(x) = (1/12) tan−1(x(1/8)). since |f(t,x)−f(t,y)| ≤ 1 10 ‖x−y‖, |g(x)−g(y)| ≤ 1 12 ‖x−y‖, therefore, (a1) and (a2) are respectively satisfied with l = 1/10 and ` = 1/12. using the given values, it is found that λ ≈ 0.8851733, p0 ≈ 1.6599468, k0 ≈ 1.563258. clearly γ = lp0 + k0` ≈ 0.2962661 < 1. thus, the conclusion of theorem 3.1 applies and the boundary value problem (3.7) has a solution on [0, 1]. our second existence result is based on sadovskii’s fixed point theorem. before proceeding further, let us recall some auxiliary material. definition 3.1. let m be a bounded set in metric space (x,d), then kuratowskii measure of noncompactness, α(m) is defined as inf{� : m covered by a finitely many sets such that the diameter of each set ≤ �}. definition 3.2. [23] let φ : d(φ) ⊆ x → x be a bounded and continuous operator on banach space x. then φ is called a condensing map if α(φ(b)) < α(b) for all bounded sets b ⊂ d(φ), where α denotes the kuratowski measure of noncompactness. lemma 3.1. [24, example 11.7] the map k + c is a k-set contraction with 0 ≤ k < 1, and thus also condensing, if 236 ahmad, ntouyas and alsaedi (i) k,c : d ⊆ x → x are operators on the banach space x; (ii) k is k-contractive, i.e., ‖kx−ky‖≤ k‖x−y‖ for all x,y ∈d and fixed k ∈ [0, 1); (iii) c is compact. lemma 3.2. [25] let b be a convex, bounded and closed subset of a banach space x and φ : b → b be a condensing map. then φ has a fixed point. theorem 3.2. let f : [0,t]×r → r be a continuous function and condition (a2) holds. in addition we assume that: (a4) g(0) = 0; (a5) there exists a nonnegative function p ∈ c([0,t],r) and a nondecreasing function ψ : [0,∞) → (0,∞) such that |f(t,u)| ≤ p(t)ψ(‖u‖) for any (t,u) ∈ [0,t] ×r; then the problem (1.2) has at least one solution on [0,t]. proof. let br = {x ∈c : ‖x‖≤ r} be a closed bounded and convex subset of c, where r will be fixed later. we define a map p : br →c as (px)(t) = (p1x)(t) + (p2x)(t), t ∈ [0,t], where p1 and p2 are defined by (3.2) and (3.3) respectively. notice that the problem (1.2) is equivalent to a fixed point problem p(x) = x. step 1. (px)br ⊂ br. let us select r ≥ ψ(r)‖p‖p0 1 − `k0 where p0 and k0 are defined by (3.5) and (3.6). for any x ∈ br, we have ‖px‖ ≤ jα|f(s,x(s))|(t) + t |λ| { |β| ρiqjα|f(s,x(s))|(ξ) + jα|f(s,x(s))|(t) } + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) |g(x) −g(y)| ≤ ψ(‖x‖)jαp(s)(t) + ψ(‖x‖) |β|t |λ| ρiqjαp(s)(ξ) + ψ(‖x‖) t |λ| jαp(s)(t) + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖x‖ ≤ ‖p‖ψ(r) { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `r = ψ(r)‖p‖p0 + k0`r < r, which implies that (px)br ⊂ br, step 2. p1 is compact. fractional differential equations and inclusions 237 observe that the operator p1 is uniformly bounded in view of step 1. let τ1,τ2 ∈ [0,t] with τ1 < τ2 and x ∈ br. then we obtain |(p1x)(τ2) − (p1x)(τ1)| ≤ |jαf(s,x(s))(τ2) −jαf(s,x(s))(τ1)| + |τ2 − τ1| |λ| jα|f(s,x(s))|(t) + |β||τ2 − τ1| |λ| ρiqjα|f(s,x(s))|(ξ) ≤ ψ(r) γ(α) ∣∣∣∣ ∫ τ1 0 [(τ2 −s)α−1 − (τ1 −s)α−1]p(s)ds + ∫ τ2 τ1 (τ2 −s)α−1p(s)ds ∣∣∣∣ + ψ(r)|τ2 − τ1| |λ| ( jαp(s)(t) + |β| ρiqjαp(s)(ξ) ) , which is independent of x and tends to zero as τ2 −τ1 → 0. thus, p1 is equicontinuous. hence, by the arzelá-ascoli theorem, p1(br) is a relatively compact set. step 3. p2 is continuous and γ-contractive. to show the continuity of p2 for t ∈ [0,t], let us consider a sequence xn converging to x. then, we have ‖p2xn −p2x‖ ≤ ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) |g(xn) −g(x)| ≤ ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖xn −x‖, which, in view of (a2), implies that p2 is continuous. also p2 is γ-contractive, since γ = ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) ` = k0` < 1. step 4. p is condensing. since p2 is continuous, γ-contraction and p1 is compact, therefore, by lemma 3.1, p : br → br with p = p1 + p2 is a condensing map on br. from the above four steps, we conclude by lemma 3.2 that the map p has a fixed point which, in turn, implies that the problem (1.2) has a solution. � example 3.2. consider the following boundary value problem  cd3/2x(t) = e−2t π √ 9 + t2 ( x tan−1 x + π/2 ) , 0 < t < 1, x(0) = 1 4 (1 − cos x) , x(1) = 1 2 2/3i3/2x(3/4). (3.8) observe that |f(t,x)| ≤ p(t)ψ(|x|) with p(t) = e−2t 2 √ 9 + t2 , ψ(|x|) = 1 + |x|, and g(0) = 0, ` = 1/4 as |g(u) −g(v)| ≤ (1/4)|u−v|. thus, all the conditions of theorem 3.2 are satisfied and hence by its conclusion, the problem (3.8) has at least one solution on [0, 1]. our next result relies on the following fixed point theorem due to burton and kirk [26]. theorem 3.3. let x be a banach space, and a,b : x → x be two operators such that a is a contraction and b is completely continuous. then either (a) the operator equation y = a(y) + b(y) has a solution, or (b) the set e = { u ∈ x : λa ( u λ ) + λb(u) = u } is unbounded for λ ∈ (0, 1). theorem 3.4. assume that f,g : [0,t] × r → r are continuous functions and conditions (a2) and (a4) hold. in addition we suppose that: 238 ahmad, ntouyas and alsaedi (a6) there exists a function p ∈ l1(j,r+) such that |f(t,u)| ≤ p(t), for a.e. t ∈ j, and each u ∈ r. then the boundary value problem (1.1) has at least one solution on [0,t]. proof. to transform the problem (1.1) into a fixed point problem, we consider the map p : c → c given by (px)(t) = (p1x)(t) + (p2x)(t), t ∈ [0,t], where p1 and p2 are defined by (3.2) and (3.3) respectively. we shall show that the operators p1 and p2 satisfy all the conditions of theorem 3.3. step 1. the operator p1 defined by (3.2) is continuous. let xn ⊂ br = {x ∈c : ‖x‖≤ r} with ‖xn−x‖→ 0. then the limit ‖xn(t)−x(t)‖→ 0 is uniformly valid on [0,t]. from the uniform continuity of f(t,x) on the compact set [0,t]×[−r,r], it follows that ‖f(t,xn(t)) −f(t,x(t))‖ → 0 uniformly on [0,t]. hence ‖p1xn −p1x‖ → 0 as n → ∞ which implies that the operator p1 is continuous. step 2. the operator p1 maps bounded sets into bounded sets in c. it is indeed enough to show that for any r > 0 there exists a positive constant l such that for each x ∈ br = {x ∈c : ‖x‖≤ r}, we have ‖p1x‖≤ l. let x ∈ br. then ‖p1x‖ ≤ jα|f(s,x(s))|(t) + t |λ| { |β| ρiqjα|f(s,x(s))|(ξ) + jα|f(s,x(s))|(t) } ≤ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t) := l. step 3. the operator p1 maps bounded sets into equicontinuous sets in c. let τ1,τ2 ∈ [0,t] with τ1 < τ2 and x ∈ br. then, for each x ∈ br, we obtain |(p1x)(τ2) − (p1x)(τ1)| ≤ |jαf(s,x(s))(τ2) −jαf(s,x(s))(τ1)| + |τ2 − τ1| |λ| jα|f(s,x(s))|(t) + |β||τ2 − τ1| |λ| |ρiqjα|f(s,x(s))|(ξ) ≤ 1 γ(α) ∣∣∣∣ ∫ τ1 0 [(τ2 −s)α−1 − (τ1 −s)α−1]p(s)ds + ∫ τ2 τ1 (τ2 −s)α−1p(s)ds ∣∣∣∣ + |τ2 − τ1| |λ| ( jαp(s)(t) + |β| ρiqjαp(s)(ξ) ) , which is independent of x and tends to zero as τ2 − τ1 → 0. thus, p1 is equicontinuous. step 4. the operator p2 defined by (3.3) is a contraction. this was established in step 3 of theorem 3.2. step 5. a priori bounds on solutions. now it remains to show that the set e = { u ∈c : λp2 ( u λ ) + λp1(u) = u } is unbounded for some λ ∈ (0, 1). let λ ∈ (0, 1) and x ∈e be a solution of the integral equation x(t) = λjαf(s,x(s))(t) + λ t λ { β ρiqjαf(s,x(s))(ξ) −jαf(s,x(s))(t) } +λ [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), t ∈ [0,t]. fractional differential equations and inclusions 239 then, for each t ∈ [0,t], we have |x(t)| ≤ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t) +λ [ 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ][∣∣∣g(x(s) λ ) −g(0) ∣∣∣ + |g(0)| ] ≤ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t) + [ 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ] `‖x‖, or (1 −k0`)‖x‖≤ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t). consequently we have ‖x‖≤ m := 1 (1 −k0`) [ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t) ] , which shows that the set e is bounded, since k0` < 1. hence, p has a fixed point in [0,t] by theorem 3.3, and consequently the problem (1.1) has a solution. this completes the proof. � finally, we show the existence of solutions for the boundary value problem (1.1) by applying a fixed point theorem due to o’regan in [27]. lemma 3.3. denote by u an open set in a closed, convex set c of a banach space e. assume 0 ∈ u. also assume that f(ū) is bounded and that f : ū → c is given by f = f1 + f2, in which f1 : ū → e is continuous and completely continuous and f2 : ū → e is a nonlinear contraction (i.e., there exists a nonnegative nondecreasing function φ : [0,∞) → [0,∞) satisfying φ(z) < z for z > 0, such that ‖f2(x) −f2(y)‖≤ φ(‖x−y‖) for all x,y ∈ ū). then, either (c1) f has a fixed point u ∈ ū; or (c2) there exist a point u ∈ ∂u and λ ∈ (0, 1) with u = λf(u), where ū and ∂u, respectively, represent the closure and boundary of u. in the next result, we use the terminology: ωr = {x ∈c : ‖x‖ < r}, mr = max{|f(t,x)| : (t,x) ∈ [0,t] × [−r,r]}. theorem 3.5. let f : [0,t] × r → r be a continuous function and conditions (a1), (a2), (a4) and (a5) hold. in addition we assume that: (a7) sup r∈(0,∞) r p0ψ(r)‖p‖ > 1 1 −k0` , where p0 and k0 are defined in (3.5) and (3.6) respectively. then the boundary value problem (1.1) has at least one solution on [0,t]. proof. by the assumption (a7), there exists a number r0 > 0 such that r0 p0ψ(r0)‖p‖ > 1 1 −k0` . (3.9) we shall show that the operators p1 and p2 defined by (3.2) and (3.3) respectively, satisfy all the conditions of lemma 3.3. step 1. the operator p1 is continuous and completely continuous. we first show that p1(ω̄r0 ) is bounded. for any x ∈ ω̄r0, we have ‖p1x‖ ≤ jα|f(s,x(s))|(t) + t |λ| { |β| ρiqjα|f(s,x(s))|(ξ) + jα|f(s,x(s))|(t) } ≤ mrjαp(s)(t) + mr |β|t |λ| ρiqjαp(s)(ξ) + mr t |λ| jαp(s)(t) ≤ ‖p‖mr { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } = mr‖p‖p0. 240 ahmad, ntouyas and alsaedi thus the operator p1(ω̄r0 ) is uniformly bounded. let τ1,τ2 ∈ [0,t] with τ1 < τ2 and x ∈ br. then |(p1x)(τ2) − (p1x)(τ1)| ≤ |jαf(s,x(s))(τ2) −jαf(s,x(s))(τ1)| + |τ2 − τ1| |λ| jα|f(s,x(s))|(t) + |β||τ2 − τ1| |λ| |ρiqjα|f(s,x(s))|(ξ) ≤ mr γ(α) ∣∣∣∣ ∫ τ1 0 [(τ2 −s)α−1 − (τ1 −s)α−1]p(s)ds + ∫ τ2 τ1 (τ2 −s)α−1p(s)ds ∣∣∣∣ + mr|τ2 − τ1| |λ| ( tα γ(α + 1) + |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) ) , which is independent of x and tends to zero as τ2 −τ1 → 0. thus, p1 is equicontinuous. hence, by the arzelá-ascoli theorem, p1(ω̄r0 ) is a relatively compact set. now, let xn ⊂ ω̄r0 with ‖xn − x‖ → 0. then the limit ‖xn(t) −x(t)‖→ 0 is uniformly valid on [0,t]. from the uniform continuity of f(t,x) on the compact set [0,t] × [−r0,r0], it follows that ‖f(t,xn(t)) −f(t,x(t))‖→ 0 uniformly on [0,t]. hence ‖p1xn −p1x‖ → 0 as n → ∞ which proves the continuity of p1. this completes the proof of step 1. step 2. the operator p2 : ω̄r0 → c([0,t],r) is contractive. this is a consequence of (a2). step 3. the set p(ω̄r0 ) is bounded. the assumptions (a2) and (a4) imply that ‖p2(x)‖≤ k0`r0, for any x ∈ ω̄r0. this, with the boundedness of the set p1(ω̄r0 ) implies that the set p(ω̄r0 ) is bounded. step 4. finally, it will be shown that the case (c2) in lemma 3.3 does not hold. on the contrary, we suppose that (c2) holds. then, we have that there exist λ ∈ (0, 1) and x ∈ ∂ωr0 such that x = λpx. so, we have ‖x‖ = r0 and x(t) = λjαf(s,x(s))(t) + λ t λ { β ρiqjαf(s,x(s))(ξ) −jαf(s,x(s))(t) } +λ [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), t ∈ [0,t]. using the assumptions (a4) − (a6), we get r0 ≤ ‖p‖ψ(r0) { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `r0, which yields r0 ≤ p0ψ(r0)‖p‖ + k0`r0. thus, we get a contradiction: r0 p0ψ(r0)‖p‖ ≤ 1 1 −k0` . thus the operators p1 and p2 satisfy all the conditions of lemma 3.3. hence, the operator p has at least one fixed point x ∈ ω̄r0, which is a solution of the problem (1.2). this completes the proof. � example 3.3. consider the following fractional order boundary value problem  cd3/2x(t) = e−2t 2π √ 9 + t2 ( x tan−1 x + π/2 ) , 0 < t < 1, x(0) = 1 4 (1 − cos x) , x(1) = 1 2 2/3i3/2x(3/4). (3.10) observe that |f(t,x)| ≤ p(t)ψ(|x|) with p(t) = e−2t 4 √ 9 + t2 , ψ(|x|) = 1 + |x|, and g(0) = 0, ` = 1/4 as |g(u) − g(v)| ≤ (1/4)|u − v|. with ψ(r) = 1 + r, ‖p‖ = 1/12, λ ≈ 0.8851733, p0 ≈ 1.6599468, k0 ≈ fractional differential equations and inclusions 241 1.563258 (as found in example 3.1), we have that (a7) holds, since sup r∈(0,∞) r p0ψ(r)‖p‖ ≈ 7.2291473 > 1.6415361 ≈ 1 1 −k0` . thus, all the conditions of theorem 3.5 are satisfied and hence by its conclusion, the problem (3.10) has at least one solution on [0, 1]. 4. existence results for problem (1.2) first of all, we introduce notions and recall some basic material on multivalued maps related to our work [28–30]. for a normed space (x,‖ · ‖), let pcl(x) = {y ∈ p(x) : y is closed}, pb(x) = {y ∈ p(x) : y is bounded}, pcp(x) = {y ∈ p(x) : y is compact} and pcp,c(x) = {y ∈ p(x) : y is compact and convex}. a multivalued map g : x →p(x) : (i) is convex (closed) valued if g(x) is convex (closed) for all x ∈ x; (ii) is bounded on bounded sets if g(b) = ∪x∈bg(x) is bounded in x for all b ∈ pb(x) (i.e. supx∈b{sup{|y| : y ∈ g(x)}} < ∞); (iii) is called upper semi-continuous (u.s.c.) on x if for each x0 ∈ x, the set g(x0) is a nonempty closed subset of x, and if for each open set n of x containing g(x0), there exists an open neighborhood n0 of x0 such that g(n0) ⊆ n; (iv) g is lower semi-continuous (l.s.c.) if the set {y ∈ x : g(y) ∩b 6= ∅} is open for any open set b in e; (v) is said to be completely continuous if g(b) is relatively compact for every b ∈pb(x); (vi) is said to be measurable if for every y ∈ r, the function t 7−→ d(y,g(t)) = inf{|y −z| : z ∈ g(t)} is measurable; (vii) has a fixed point if there is x ∈ x such that x ∈ g(x). the fixed point set of the multivalued operator g will be denoted by fixg. definition 4.1. a multivalued map f : [0,t] ×r →p(r) is said to be carathéodory if (i) t 7−→ f(t,x) is measurable for each x ∈ r; (ii) x 7−→ f(t,x) is upper semicontinuous for almost all t ∈ [0,t]; further a carathéodory function f is called l1−carathéodory if (iii) for each α > 0, there exists ϕα ∈ l1([0,t],r+) such that ‖f(t,x)‖ = sup{|v| : v ∈ f(t,x)}≤ ϕα(t) for all ‖x‖≤ α and for a. e. t ∈ [0,t]. for each x ∈ c([0,t],r), define the set of selections of f by sf,x := {v ∈ l1([0,t],r) : v(t) ∈ f(t,x(t)) for a.e. t ∈ [0,t]}. we define the graph of g to be the set gr (g) = {(x,y) ∈ x ×y,y ∈ g(x)} and recall two useful results on closed graphs and upper-semicontinuity. lemma 4.1. ( [28, proposition 1.2]) if g : x → pcl(y ) is u.s.c., then gr (g) is a closed subset of x ×y ; i.e., for every sequence {xn}n∈n ⊂ x and {yn}n∈n ⊂ y , if when n → ∞, xn → x∗, yn → y∗ and yn ∈ g(xn), then y∗ ∈ g(x∗). conversely, if g is completely continuous and has a closed graph, then it is upper semi-continuous. lemma 4.2. ( [31]) let x be a banach space. let f : [0,t]×r →pcp,c(x) be an l1− carathéodory multivalued map and let θ be a linear continuous mapping from l1([0,t],x) to c([0,t],x). then the operator θ ◦sf : c([0,t],x) →pcp,c(c([0,t],x)), x 7→ (θ ◦sf )(x) = θ(sf,x,y) is a closed graph operator in c([0,t],x) ×c([0,t],x). 242 ahmad, ntouyas and alsaedi lemma 4.3. ( [32], krasnoselskii’s fixed point theorem). let x be a banach space, y ∈ pb,cl,c(x) and a,b : y →pcp,c(x) two multivalued operators. if the following conditions are satisfied (i) ay + by ⊂ y for all y ∈ y ; (ii) a is contraction; (iii) b is u.s.c and compact, then, there exists y ∈ y such that y ∈ ay + by. definition 4.2. a function x ∈ c2([0,t],r) is a solution of the problem (1.2) if x(0) = g(x), x(t) = β ρiqx(ξ), and there exists a function f ∈ l1([0,t],r) such that f(t) ∈ f(t,x(t)) a.e. on [0,t] and x(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x). theorem 4.1. assume that (a2) holds. in addition we suppose that: (h1) f : [0,t] ×r →pcp,c(r) is l1−carathéodory multivalued map; (h2) there exists a function p ∈ c([0,t],r+) such that ‖f(t,x)‖p := sup{|y| : y ∈ f(t,x)}≤ p(t), for each (t,x) ∈ [0,t] ×r. then the boundary value problem (1.2) has at least one solution on [0,t]. proof. to transform the problem (1.2) to a fixed point problem, we define an operator n : c −→p(c) by n(x) =   h ∈c : h(t) =   jαf(s)(t) + t λ { αρiqjαf(s)(ξ) −jαf(s)(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x),   for f ∈ sf,x. next we introduce operators a : c −→c and b : c −→p(c) by ax(t) = [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), (4.1) b(x) = { h ∈c : h(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } . } (4.2) observe that n = a + b. we shall show that the operators a and b satisfy all the conditions of lemma 4.3 on [0,t]. first, we show that the operators a and b define the multivalued operators a,b : br → pcp,c(c) where br = {x ∈ c : ‖x‖ ≤ r} is a bounded set in c. first we prove that b is compact-valued on br. note that the operator b is equivalent to the composition l◦sf , where l is the continuous linear operator on l1([0,t],r) into c, defined by l(v)(t) = jαv(s)(t) + t λ { β ρiqjαv(s)(ξ) −jαv(s)(t) } . suppose that x ∈ br is arbitrary and let {vn} be a sequence in sf,x. then, by definition of sf,x, we have vn(t) ∈ f(t,x(t)) for almost all t ∈ [0,t]. since f(t,x(t)) is compact for all t ∈ j, there is a convergent subsequence of {vn(t)} (we denote it by {vn(t)} again) that converges in measure to some v(t) ∈ sf,x for almost all t ∈ j. on the other hand, l is continuous, so l(vn)(t) →l(v)(t) pointwise on [0,t]. fractional differential equations and inclusions 243 in order to show that the convergence is uniform, we have to show that {l(vn)} is an equicontinuous sequence. let t1, t2 ∈ [0,t] with t1 < t2. then, we have |l(vn)(t2) −l(vn)(t1)| ≤ |jαvn(s)(t2) −jαvn(s)(t1)| + |t2 − t1| |λ| jα|vn(s)|(t) + |β||t2 − t1| |λ| |ρiqjα|vn(s)|(ξ) ≤ 1 γ(α) ∣∣∣∣ ∫ t1 0 [(t2 −s)α−1 − (t1 −s)α−1]p(s)ds + ∫ t2 t1 (t2 −s)α−1p(s)ds ∣∣∣∣ + |t2 − t1| |λ| ( jqp(s)(t) + |β| ρiqjqp(s)(ξ) ) . we see that the right hand of the above inequality tends to zero as t2 → t1. thus, the sequence {l(vn)} is equicontinuous and hence, by the arzelá-ascoli theorem, we get that there is a uniformly convergent subsequence. so, there is a subsequence of {vn} (we denote it again by {vn}) such that l(vn) →l(v). note that l(v) ∈l(sf,x). hence, b(x) = l(sf,x) is compact for all x ∈ br. so b(x) is compact. now, we show that b(x) is convex for all x ∈c. let z1,z2 ∈b(x). we select f1,f2 ∈ sf,x such that zi(t) = j αfi(s)(t) + t λ { β ρiqjαfi(s)(ξ) −jαfi(s)(t) } , i = 1, 2, for almost all t ∈ [0,t]. let 0 ≤ λ ≤ 1. then, we have [λz1 + (1 −λ)z2](t) = jα[λf1(s) + (1 −λ)f2(s)](t) + t λ { β ρiqjα[λf1(s) + (1 −λ)f2(s)](ξ) −jα[λf1(s) + (1 −λ)f2(s)](s)(t) } . since f has convex values, so sf,u is convex and λf1(s) + (1 −λ)f2(s) ∈ sf,x. thus λz1 + (1 −λ)z2 ∈b(x). consequently, b is convex-valued. obviously, a is compact and convex-valued. the rest of the proof consists of several steps and claims. step 1: we show that a is a contraction on c. for x,y ∈c, we have |ax(t) −ay(t)| = ∣∣∣∣∣1 + β tλ ξ ρq ρq 1 γ(q + 1) ∣∣∣∣∣|g(x) −g(y)| ≤ ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) |g(x) −g(y)|, ≤ k0`‖x−y‖, which, on taking supremum over t ∈ [0,t], yields ‖ax−ay‖≤ l0‖x−y‖, l0 = k0`. this shows that a is a contraction as l0 < 1. step 2: b is compact and upper semicontinuous. this will be established in several claims. claim i: b maps bounded sets into bounded sets in c. let br = {x ∈ c : ‖x‖ ≤ r} be a bounded set in c. then, for each h ∈ b(x),x ∈ br, there exists f ∈ sf,x such that h(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } . 244 ahmad, ntouyas and alsaedi then, for t ∈ [0,t], we have |h(t)| ≤ jα|f(s)|(t) + t |λ| {|β|ρiqjα|f(s)|(ξ) −jα|f(s)|(t) } ≤ jαp(s)(t) + |β|t |λ| ρiqjαp(s)(ξ) + t |λ| jαp(s)(t) ≤ ‖p‖ { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } . thus, ‖h‖≤‖p‖ { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } . claim ii: b maps bounded sets into equicontinuous sets. let τ1,τ2 ∈ [0,t] with τ1 < τ2 and x ∈ br. then, for each h ∈b(x), we obtain |h(τ2) −h(τ1)| ≤ |jαf(s)(τ2) −jαf(s)(τ1)| + |τ2 − τ1| |λ| jα|f(s)|(t) + |β||τ2 − τ1| |λ| ρiqjα|f(s)|(ξ) ≤ 1 γ(α) ∣∣∣∣ ∫ τ1 0 [(τ2 −s)α−1 − (τ1 −s)α−1]p(s)ds + ∫ τ2 τ1 (τ2 −s)α−1p(s)ds ∣∣∣∣ +‖p‖ |τ2 − τ1| |λ| ( tα γ(α + 1) + |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) ) ≤ ‖p‖ γ(α + 1) [τα2 − τ α 1 + 2(τ2 − τ1) α] + ‖p‖ |τ2 − τ1| |λ| ( tα γ(α + 1) + |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) ) . obviously the right hand side of the above inequality tends to zero independently of x ∈ br as τ2 − τ1 → 0. therefore it follows by the ascoli-arzelá theorem that b : c → p(c) is completely continuous. by claims i and ii, b is completely continuous. by lemma 4.1, b will be upper semicontinuous (since it is completely continuous) if we prove that it has a closed graph. claim iii: b has a closed graph. let xn → x∗,hn ∈ b(xn) and hn → h∗. then we need to show that h∗ ∈ b(x∗). associated with hn ∈b(xn), there exists fn ∈ sf,xn such that for each t ∈ [0,t], h(t) = jαfn(s)(t) + t λ { β ρiqjαfn(s)(ξ) −jαfn(s)(t) } . thus it suffices to show that there exists f∗ ∈ sf,x∗ such that for each t ∈ [0,t], h∗(t) = j αf∗(s)(t) + t λ { β ρiqjαf∗(s)(ξ) −jαf∗(s)(t) } . let us consider the linear operator θ : l1([0,t],r) →c given by f 7→ θ(f)(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } . observe that ‖hn(t) −h∗(t)‖ = ∥∥∥jα(fn(s) −f∗(s))(t) + t λ { β ρiqjα(fn(s) −f∗(s))(ξ) −jα(fn(s) −f∗(s))(t) }∥∥∥ → 0, as n →∞. fractional differential equations and inclusions 245 thus, it follows by lemma 4.2 that θ ◦ sf is a closed graph operator. further, we have hn(t) ∈ θ(sf,xn ). since xn → x∗, we have that h∗(t) = j αf∗(s)(t) + t λ { β ρiqjαf∗(s)(ξ) −jαf∗(s)(t) } , for some f∗ ∈ sf,x∗ . hence b has a closed graph (and therefore has closed values). in consequence, the operator b is compact and upper semicontinuous. step 3: here, we show that a(x) + b(x) ⊂ br for all x ∈ br. suppose x ∈ br, with r > p0‖p‖ 1 −k0` and h ∈b are arbitrary elements. choose f ∈ sf,x such that h(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), t ∈ [0,t]. following the method for proof for claim i, we can obtain |h(t)| ≤ ‖p‖ { tα γ(α + 1) + t |λ| tα γ(α + 1) + t |λ| |β| γ(α + 1) ξα+ρq ρq γ(α+ρ ρ ) γ(α+ρq+ρ ρ ) } + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖x‖. thus ‖h‖≤ p0‖p‖ + k0`r < r. (4.3) hence ‖h‖≤ r, which means that a(x) + b(x) ⊂ br for all x ∈ br. thus, the operators a and b satisfy all the conditions of lemma 4.3 and hence its conclusion implies that x ∈a(x) +b(x) in br. therefore the boundary value problem (1.2) has a solution in br and the proof is completed. � to prove our next result, we make use of the following form of the nonlinear alternative for contractive maps [33, corollary 3.8]. theorem 4.2. let x be a banach space, and d a bounded neighborhood of 0 ∈ x. let z1 : x → pcp,c(x) and z2 : d̄ →pcp,c(x) two multivalued operators satisfying (a) z1 is contraction, and (b) z2 is u.s.c and compact. then, if g = z1 + z2, either (i) g has a fixed point in d̄ or (ii) there is a point u ∈ ∂d and λ ∈ (0, 1) with u ∈ λg(u). theorem 4.3. assume that (a2)and (h1) are satisfied. in addition we suppose that: (h3) there exists a continuous nondecreasing function ψ : [0,∞) → (0,∞) and a function p ∈ l1([0,t],r+) such that ‖f(t,x)‖p := sup{|y| : y ∈ f(t,x)}≤ p(t)ψ(‖x‖) for each (t,x) ∈ [0,t] ×r; (h4) there exists a number m > 0 such that m ψ(m) > 1 (1 −k0`) [ jαp(s)(t) + t |λ| { |β| ρiqjαp(s)(ξ) + jαp(s)(t) }] , (4.4) where k0 is defined in (3.6). then the boundary value problem (1.2) has at least one solution on [0,t]. proof. to transform the problem (1.2) to a fixed point, we define an operator n : c([0,t],r) −→p(c) and consider the operators a and b defined in the beginning of the proof of theorem 4.1. as in theorem 4.1, one can show that the operators a and b are indeed the multivalued operators a,b : br →pcp,c(c) where br = {x ∈ c : ‖x‖ ≤ r} is a bounded set in c, a is a contraction on c and b is u.s.c. and compact. 246 ahmad, ntouyas and alsaedi thus the operators a and b satisfy all the conditions of theorem 4.2 and hence its conclusion implies either condition (i) or condition (ii) holds. we show that the conclusion (ii) is not possible. if x ∈ λa(x) + λb(x) for λ ∈ (0, 1), then there exists f ∈ sf,x such that x(t) = jαf(s)(t) + t λ { β ρiqjαf(s)(ξ) −jαf(s)(t) } + [ 1 + β t λ ξρq ρq 1 γ(q + 1) ] g(x), t ∈ [0,t], and |x(t)| ≤ jαp(s)|ψ(‖x‖)(t) + t |λ| { |β| ρiqjαp(s)ψ(‖x‖)(ξ) + jαp(s)ψ(‖x‖)(t) } + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖x‖ ≤ ψ(‖x‖) [ jαp(s)(t) + t |λ| { |β| ρiqjαp(s)(ξ) + jαp(s)(t) }] + ( 1 + |β| t |λ| ξρq ρq 1 γ(q + 1) ) `‖x‖. thus (1 −k0`)‖x‖≤ ψ(‖x‖) [ jαp(s)(t) + t |λ| { |β| ρiqjαp(s)(ξ) + jαp(s)(t) }] . (4.5) if condition (ii) of theorem 4.2 holds, then there exists λ ∈ (0, 1) and x ∈ ∂bm with x = λn(x). then, x is a solution of (1.2) with ‖x‖ = m. now, by the inequality (4.5), we get m ψ(m) ≤ 1 (1 −k0`) [ jαp(s)(t) + t |λ| { |β| ρiqjαp(s)(ξ) + jαp(s)(t) }] , which contradicts (4.4). hence, n has a fixed point in [0,t] by theorem 4.2, and consequently the problem (1.2) has a solution. this completes the proof. � example 4.1. consider the following boundary value problem of fractional differential inclusions  d3/2x(t) ∈ f(t,x(t)), t ∈ [0, 1], 0 < t < 1, x(0) = 1 8 x(1/4), x(1) = 1 2 2/3i3/2x(3/4), (4.6) where f(t,x(t)) = [ 2 √ t2 + 64 ( |x(t)| 2 ( |x(t)| |x(t)| + 1 + 1 ) + 1 5 ) , e−t (10 + t) ( sin x(t) + 1 15 )] . clearly |f(t,x)| ≤ p(t)ψ(|x|), where p(t) = 2/ √ t2 + 64, ψ1(|x|) = |x| + 1/5 and ` = 1/8. using the values: λ ≈ 0.8851733, p0 ≈ 1.6599468, k0 ≈ 1.563258 (see example 3.1) and the condition (h4), we find that m > m1 ' 0.2130287. since the hypotheses of theorem 4.3 are satisfied, the problem (4.6) has a solution on [0, 1]. references [1] i. podlubny, fractional differential equations, academic press, san diego, 1999. [2] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [3] d. baleanu, k. diethelm, e. scalas, j.j. trujillo, fractional calculus models and numerical methods, series on complexity, nonlinearity and chaos. world scientific, boston, 2012. [4] r.p. agarwal, y. zhou, y. he, existence of fractional neutral functional differential equations, comput. math. appl. 59 (2010), 1095-1100. [5] d. baleanu, o.g. mustafa, r.p. agarwal, on lp-solutions for a class of sequential fractional differential equations, appl. math. comput. 218 (2011), 2074-2081. [6] b. ahmad, j.j. nieto, riemann-liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, bound. value probl. 2011 (2011), art. id 36. fractional differential equations and inclusions 247 [7] b. ahmad, s.k. ntouyas, a. alsaedi, new existence results for nonlinear fractional differential equations with three-point integral boundary conditions, adv. differ. equ. 2011 (2011), art. id 107384. [8] d. o’regan, s. stanek, fractional boundary value problems with singularities in space variables, nonlinear dynam. 71 (2013), 641-652. [9] b. ahmad, s.k. ntouyas, a. alsaedi, a study of nonlinear fractional differential equations of arbitrary order with riemann-liouville type multistrip boundary conditions, math. probl. eng. 2013 (2013) art. id 320415. [10] b. ahmad, j.j. nieto, boundary value problems for a class of sequential integrodifferential equations of fractional order, j. funct. spaces appl. 2013 (2013), art. id 149659. [11] l. zhang, b. ahmad, g. wang, r.p. agarwal, nonlinear fractional integro-differential equations on unbounded domains in a banach space, j. comput. appl. math. 249 (2013), 51-56. [12] x. liu, m. jia, w. ge, multiple solutions of a p-laplacian model involving a fractional derivative, adv. differ. equ. 2013 (2013), art. id 126. [13] s.k. ntouyas, s. etemad, on the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, appl. math. comp. 266 (2015), 235–243. [14] s.k. ntouyas, s. etemad, j. tariboon, existence results for multi-term fractional differential inclusions, adv. differ. equ. 2015 (2015), art. id 140. [15] r.p. agarwal, s.k. ntouyas, b. ahmad, a.k. alzahrani, hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments, adv. difference equ. 2016 (2016), art. id 92. [16] b. ahmad, r.p. agarwal, a. alsaedi, fractional differential equations and inclusions with semiperiodic and threepoint boundary conditions, bound. value probl. 2016 (2016), art. id 28. [17] b. ahmad, s.k. ntouyas, some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions, rev. r. acad. cienc. exactas fis. nat. ser. a math. racsam 110 (2016), 159-172. [18] h. dong, b. guo, b. yin, generalized fractional supertrace identity for hamiltonian structure of nls-mkdv hierarchy with self-consistent sources, anal. math. phys. 6 (2) (2016), 199-209. [19] l. byszewski, v. lakshmikantham, theorem about the existence and uniqueness of a solution of a nonlocal abstract cauchy problem in a banach space, appl. anal. 40 (1991), 11-19. [20] l. byszewski, theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, j. math. anal. appl. 162 (1991) 494-505. [21] r. c̆iegis, a. bugajev, numerical approximation of one model of the bacterial self-organization, nonlinear anal. model. control. 17 (2012), 253-270. [22] u.n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218 (2015), 860–865. [23] a. granas, j. dugundji, fixed point theory, springer-verlag, new york, 2005. [24] e. zeidler, nonlinear functional analysis and its application: fixed point-theorems, springer-verlag, new york, vol. 1 1986. [25] b.n. sadovskii, on a fixed point principle, funct. anal. appl. 1 (1967), 74-76. [26] t.a. burton, c. kirk, a fixed point theorem of krasnoselskii-schaefer type, math. nachr. 189 (1998), 23-31. [27] d. o’regan, fixed-point theory for the sum of two operators, appl. math. lett. 9 (1996), 1-8. [28] k. deimling, multivalued differential equations, walter de gruyter, berlin-new york, 1992. [29] sh. hu, n. papageorgiou, handbook of multivalued analysis, volume i: theory, kluwer, dordrecht, 1997. [30] g.v. smirnov, introduction to the theory of differential inclusions, american mathematical society, providence, ri, 2002. [31] a. lasota, z. opial, an application of the kakutani-ky fan theorem in the theory of ordinary differential equations, bull. acad. polon. sci. ser.sci. math. astronom. phys. 13 (1965), 781–786. [32] a. petrusel, fixed points and selections for multivalued operators, seminar on fixed point theory cluj-napoca 2 (2001), 3-22. [33] w.v. petryshyn, p. m. fitzpatric, a degree theory, fixed point theorems, and mapping theorems for multivalued noncompact maps, trans. amer. math. soc., 194 (1974), 1-25. 1nonlinear analysis and applied mathematics (naam)-research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia 2department of mathematics, university of ioannina, 451 10 ioannina, greece ∗corresponding author: bashirahmad−qau@yahoo.com 1. introduction 2. preliminaries 3. existence and uniqueness results for problem (1.1) 4. existence results for problem (1.2) references international journal of analysis and applications volume 17, number 5 (2019), 771-792 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-771 some fixed point theorems in menger probabilistic partial metric spaces with application to volterra type integral equation amir ghanenia, mahnaz khanehgir, reza allahyari∗ and mohammad mehrabinezhad department of mathematics, mashhad branch, islamic azad university, mashhad, iran ∗corresponding author: rezaallahyari@mshdiau.ac.ir abstract. in this paper, we introduce the notion of menger probabilistic partial metric space and prove some fixed point theorems in the framework of such spaces. some examples and an application to volterra type integral equation are given to support the obtained results. finally, we apply successive approximations method to find a solution for a volterra type integral equation with high accuracy. 1. introduction the concept of a menger probabilistic metric space (briefly, menger pm-space) was initiated by menger [15]. the idea of menger was to use a distribution function instead of a nonnegative number for the value of a metric. the study of this space was expanded rapidly with the pioneering works of schweizer and sklar [20], stevens [25] and some of their coworkers. in 1972, sehgal and bharucha-reid [23] obtained a generalization of the banach contraction principle on a complete menger space. since then, a number of mathematicians have made a substantial contribution to the theoretical development of menger pm-spaces (see [4,6–10,12,17,26]). received 2019-04-19; accepted 2019-08-14; published 2019-09-02. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. fixed point; menger probabilistic metric space; partial metric space; volterra type integral equation; successive approximations method. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 771 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-771 int. j. anal. appl. 17 (5) (2019) 772 on the other hand, matthews [14] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks. in recent years, some scholars have investigated the topological properties of partial metric spaces and have established some fixed point results in these spaces (see [1, 3, 5, 16, 19, 22]). in this work, using the concepts of menger probabilistic metric space and partial metric space, we establish a new concept of menger probabilistic partial metric space. we present some fixed point theorems in these spaces. some examples and an application to volterra type integral equation are given to illustrate the usability of our results. finally, we apply successive approximations method to find an approximate solution for a volterra type integral equation with high accuracy. first, we recall some basic definitions and facts which will be used further on. we denote by r the set of real numbers and r+ the set of nonnegative real numbers. definition 1.1. [14] a partial metric on a nonempty set x is a mapping p : x ×x → r+ such that for all x,y,z ∈ x: (p1) x = y if and only if p(x,x) = p(x,y) = p(y,y), (p2) p(x,x) ≤ p(x,y), (p3) p(x,y) = p(y,x), (p4) p(x,y) ≤ p(x,z) + p(z,y) −p(z,z). a pair (x,p) is called a partial metric space, if x is a nonempty set and p is a partial metric on x. it is clear that if p(x,y) = 0, then from (p1) and (p2), x = y. but if x = y, p(x,y) may not be 0. a basic example of a partial metric space is the pair (r+,p), where p(x,y) = max{x,y} for all x,y ∈ r+ (see [16]). each partial metric p on a nonempty set x generates a t0 topology τp on x whose base is the family of open p-balls {bp(x,ε) : x ∈ x,ε > 0}, where bp(x,ε) = {y ∈ x : p(x,y) < ε + p(x,x)} for all x ∈ x and ε > 0. definition 1.2. [14] let (x,p) be a partial metric space, {xn} be any sequence in x and x ∈ x. then: (i) the sequence {xn} is said to be convergent to x with respect to τp if lim n→∞ p(xn,x) = p(x,x). (ii) the sequence {xn} is said to be a cauchy sequence in (x,p), if lim n,m→∞ p(xn,xm) exists and is finite. (iii) (x,p) is said to be a complete partial metric space if for every cauchy sequence {xn} in x, x ∈ x exists such that lim n,m→∞ p(xn,xm) = lim n→∞ p(xn,x) = p(x,x). notice that in a partial metric space the limit of a convergent sequence may not be unique. int. j. anal. appl. 17 (5) (2019) 773 in the sequel, we recall the definition of n-th order t-norm. shi et al. [24] gave the following definition of n-th order t-norm. definition 1.3. a mapping t : πni=1[0, 1] → [0, 1] is called an n-th order t-norm if the following conditions are satisfied: (i) t(0, 0, . . . , 0) = 0, t(a, 1, 1, . . . , 1) = a for all a ∈ [0, 1], (ii) t(a1,a2,a3, . . . ,an) = t(a2,a1,a3, . . . ,an) = t(a2,a3,a1, . . . ,an) = . . . = t(a2,a3,a4, . . . ,an,a1), (iii) ai ≥ bi, i = 1, 2, 3, . . . ,n imply t(a1,a2,a3, . . . ,an) ≥ t(b1,b2,b3, . . . ,bn), (iv) t(t(a1,a2,a3, . . . ,an),b2,b3, . . . ,bn) = t(a1,t(a2,a3, . . . ,an,b2),b3, . . . ,bn) = t(a1,a2,t(a3,a4, . . . ,an,b2,b3),b4, . . . ,bn) = ... = t(a1,a2,a3, . . . ,an−1,t(an,b2,b3, . . . ,bn)). when n = 2, we have a binary t-norm, which is commonly known as t-norm. the following are three basic continuous 3-th order t-norms: (1) the minimum 3-th order t-norm, say tm , defined by tm (a,b,c) = min{a,b,c}. (2) the product 3-th order t-norm, say tp , defined by tp (a,b,c) = abc. (3) the lukasiewicz 3-th order t-norm, say tl, defined by tl(a,b,c) = max{a + b + c− 2, 0}. these t-norms are related in the following way: tl ≤ tp ≤ tm . definition 1.4. [21] a function f : r → r+ is called a distribution if it is increasing left-continuous with inf t∈r f(t) = 0 and sup t∈r f(t) = 1. the set of all distribution functions is denoted by d+. a special distribution function is given by h(t) =   0, t ≤ 0, 1, t > 0. definition 1.5. [21] a menger probabilistic metric space (briefly, menger pm-space) is a triple (x,f,t) where x is a nonempty set, t is a continuous t-norm, and f is a mapping from x ×x into d+ such that, if fx,y denotes the value of f at the pair (x,y), the following conditions hold: (pm1) fx,y(t) = h(t) if and only if x = y, (pm2) fx,y(t) = fy,x(t), (pm3) fx,y(t + s) ≥ t(fx,z(t),fz,y(s)) for all x,y,z ∈ x and s,t ≥ 0. int. j. anal. appl. 17 (5) (2019) 774 according to [21], the (ε,λ)-topology in a menger pm-space (x,f,t) is a family of neighborhoods nx of a point x ∈ x given by nx = {nx(ε,λ) : ε > 0,λ ∈ (0, 1)}, where nx(ε,λ) = {y ∈ x : fx,y(ε) > 1 − λ}. the (ε,λ)-topology is a hausdorff topology. definition 1.6. [21] let (x,f,t) be a menger pm-space. then: (i) a sequence {xn} in x is said to be convergent to x in x if, for every ε > 0 and λ > 0, a positive integer n exists such that for each n ≥ n, fxn,x(ε) > 1 −λ. (ii) a sequence {xn} in x is called a cauchy sequence if, for every ε > 0 and λ > 0, a positive integer n exists such that for each n,m ≥ n, fxn,xm (ε) > 1 −λ. (iii) a menger pm-space is said to be complete if every cauchy sequence in x is convergent to a point in x. example 1.1. let x = r+, t(a,b) = ab a+b−ab if a,b ∈ (0, 1] and t(a,b) = 0, if a = 0 or b = 0. define f : x ×x →d+ by fx,y(t) =   t t+|x−y|, if t > 0, 0, if t ≤ 0 for all x,y ∈ x. then (x,f,t) is a complete menger pm-space. definition 1.7. [10] a function φ : r+ → r+ is said to be a φ-function if it satisfies the following conditions: (i) φ(t) = 0 if and only if t = 0, (ii) φ(t) is strictly monotone increasing and φ(t) →∞ as t →∞, (iii) φ is left-continuous in (0,∞), (iv) φ is continuous at 0. from now on, we mean by φ the class of all φ-functions and by ψ the class of continuous functions ψ : r+ → r+ such that ψ(0) = 0 and ψn(an) → 0, whenever an → 0 as n →∞. 2. main result in this section, first we describe the new concept of menger probabilistic partial metric space. then we improve some fixed point results of gopal et al. [10], in the setup of menger probabilistic partial metric spaces. definition 2.1. a probabilistic partial metric space is an ordered pair (x,f) where x is a nonempty set, f : x×x →d+ is given by (x,y) 7→ fx,y, such that the following conditions are satisfied for all x,y,z ∈ x and t ∈ r+: int. j. anal. appl. 17 (5) (2019) 775 (ppm1) fx,y(t) = fx,x(t) = fy,y(t) if and only if x = y, (ppm2) fx,y(t) = fy,x(t), (ppm3) fx,x(t) ≥ fx,y(t), (ppm4) if fx,z(t1) = 1, fz,y(t2) = 1 and fz,z(t3) = 1 for t1, t2, t3 ∈ r+, then fx,y(t1 + t2 + t3) = 1. it is clear that every probabilistic metric space is a probabilistic partial metric space. however, the converse of this fact needs not hold. for example, x = y does not imply fx,y(t) = h(t). see the following example. example 2.1. let (x,p) be a partial metric space. if f : x ×x →d+ is a mapping defined by fx,y(t) = h(t−p(x,y)), ∀x,y ∈ x,t ∈ r, then (x,f) is a probabilistic partial metric space. obviously, x = y does not imply fx,y(t) = h(t). definition 2.2. a menger probabilistic partial metric space is a triple (x,f,t), where (x,f) is a probabilistic partial metric space, t is a continuous 3-th order t-norm and the following inequality holds: fx,y(t1 + t2 + t3) ≥ t(fx,z(t1),fz,y(t2),fz,z(t3)) (2.1) for all x,y,z ∈ x and all t1, t2, t3 ∈ r+. remark 2.1. let (x,f) be as example 2.1. then (x,f,tm ) is a menger probabilistic partial metric space induced by (x,p). example 2.2. let (x,p) be a partial metric space. define a mapping f : x ×x →d+ by fx,y(t) =   t t+p(x,y) , if t > 0, 0, if t ≤ 0 for all x,y ∈ x. then (x,f,tm ) is a menger probabilistic partial metric space. definition 2.3. let (x,f,t) be a menger probabilistic partial metric space. then: (i) a sequence {xn} in x is said to be convergent to x in x if, for each t > 0, lim n→∞ fxn,x(t) = fx,x(t). (ii) a sequence {xn} in x is called a cauchy sequence if, for each t > 0, lim m,n→∞ fxm,xn (t) exists. (iii) a menger probabilistic partial metric space is said to be complete if for every cauchy sequence {xn} in x, a point x ∈ x exists such that lim m,n→∞ fxm,xn (t) = lim n→∞ fxn,x(t) = fx,x(t). (iv) a sequence {xn} is called g-cauchy if for each p ∈ n and t > 0, lim n→∞ fxn,xn+p (t) exists. int. j. anal. appl. 17 (5) (2019) 776 (v) the space (x,f,t) is called g-complete if for every g-cauchy sequence {xn} in x, a point x ∈ x exists such that lim n→∞ fxn,xn+p (t) = lim n→∞ fxn,x(t) = fx,x(t). definition 2.4. (see also [10]) let x be a nonempty set, f : x → x be a mapping and β,γ : x × x × (0,∞) → (0,∞) be two functions. then f is said to be (β,γ)-admissible if for all x,y ∈ x and all t > 0 we have β(x,y,t) ≤ 1 implies β(fx,fy,t) ≤ 1 and γ(x,y,t) ≥ 1 implies γ(fx,fy,t) ≥ 1. definition 2.5. let (x,f,t) be a menger probabilistic partial metric space, f : x → x be a given mapping and β,γ : x×x×(0,∞) → (0,∞) be two functions. we say that f is a (β,γ)-admissible ψ-type contractive mapping if f is a (β,γ)-admissible mapping, satisfying in the following inequality γ(fx,fy,t) ( 1 ffx,fy(φ(ct)) − 1 ) ≤ β(x,y,t)ψ ( 1 fx,y(φ(t)) − 1 ) (2.2) for all x,y ∈ x and all t > 0 such that fx,y(φ(t)) > 0, where c ∈ (0, 1), φ ∈ φ and ψ ∈ ψ. according to gopal et al. [10, theorem 2.1], we present a new fixed point theorem in the menger probabilistic partial metric spaces. theorem 2.1. let (x,f,t) be a g-complete menger probabilistic partial metric space and f : x → x be a (β,γ)-admissible ψ-type contractive mapping satisfying the following conditions: (i) x0 ∈ x exists such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0, (ii) if {xn} is a sequence in x such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ n, and for all t > 0 and xn → x as n →∞, then β(xn,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ n and for all t > 0. then f has a fixed point. proof. let x0 ∈ x be such that condition (i) holds. we define inductively the sequence {xn} in x by xn+1 = fxn, for n = 0, 1, 2, . . . . we may suppose that xn+1 6= xn for each n, otherwise f has obviously a fixed point. we conclude from (β,γ)-admissibility of the mapping f, the condition (i), and by induction that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ n and all t > 0. from properties of the function φ, it is possible to find some t > 0 such that fx0,x1 (φ(t)) > 0. it implies that fx0,x1 (φ( t c )) > 0, too. from (2.2), we have that 1 fx1,x2 (φ(ct)) − 1 ≤ γ(fx0,fx1, t) ( 1 ffx0,fx1 (φ(ct)) − 1 ) ≤ β(x0,x1, t)ψ ( 1 fx0,x1 (φ(t)) − 1 ) ≤ ψ ( 1 fx0,x1 (φ(t)) − 1 ) . int. j. anal. appl. 17 (5) (2019) 777 repeating the above procedure successively r times (r < n), we obtain 1 fxn,xn+1 (φ(c rt)) − 1 ≤ ψn−r ( 1 fxr,xr+1 (φ( crt cn−r )) − 1 ) . (2.3) since ψn(an) → 0, whenever an → 0, then from (2.3), for any positive real number r we have fxn,xn+1 (φ(c rt)) → 1, as n →∞. (2.4) now, let ε > 0 be given and choose r > 0 so that φ(crt) < ε. regarding (2.4) we deduce fxn,xn+1 (ε) → 1, as n →∞, for every ε > 0. (2.5) on the other hand, we can write fxn,xn+p (ε) ≥ t ( fxn,xn+1 ( ε 3 ),fxn+1,xn+1 ( ε 3 ),fxn+1,xn+p ( ε 3 ) ) ≥ . . . ≥ t ( fxn,xn+1 ( ε 3 ),fxn,xn+1 ( ε 3 ),t(fxn+1,xn+2 ( ε 9 ),fxn+1,xn+2 ( ε 9 ), t(. . . ,t(fxn+p−2,xn+p−1 ( ε 3p−1 ),fxn+p−2,xn+p−1 ( ε 3p−1 ),fxn+p−1,xn+p ( ε 3p−1 ))) ) . on making n →∞ and in view of (2.5), for any positive integer p, we have fxn,xn+p (ε) → 1, as n →∞, for every ε > 0. it follows that {xn} is a g-cauchy sequence. since (x,f,t) is g-complete, {xn} is convergent and lim n→∞ fxn,xn+p (t) = lim n→∞ fxn,u(t) = fu,u(t) (2.6) for some u ∈ x. furthermore, we get ffu,u(ε) ≥ t ( ffu,xn+1 ( ε 3 ),fu,xn+1 ( ε 3 ),fxn+1,xn+1 ( ε 3 ) ) . (2.7) taking into account the continuity of φ at zero, t1 > 0 exists such that φ(t1) < ε 3 . from (2.6), n0 ∈ n exists such that fxn,u(φ(t1)) > 0 for all n ≥ n0. hence, applying (ii) we derive that 1 fxn+1,fu( ε 3 ) − 1 ≤ γ(fxn,fu,t1) ( 1 ffxn,fu(φ(t1)) − 1 ) ≤ β(xn,u,t1)ψ ( 1 fxn,u(φ( t1 c )) − 1 ) for all n ≥ n0. now, letting n →∞, since ψ(0) = 0 and by the continuity of function ψ, we obtain fxn+1,fu( ε 3 ) → 1, as n →∞. from (2.1) and (2.6) fxn+1,xn+1 ( ε 3 ) → 1 as n →∞, too. passing n to infinity in the relation (2.7) it follows that ffu,u(ε) = 1, for each ε > 0. thus fu = u. this completes the proof. � int. j. anal. appl. 17 (5) (2019) 778 example 2.3. let x = r+. define f : x ×x →d+ by fx,y(t) = t t + max{x,y} for all x,y ∈ x and for all t > 0. define the mapping f : x → x by fx =   x 2 , if x ∈ [0, 1), 2x, otherwise and the functions β and γ from x ×x × (0,∞) into (0,∞) by β(x,y,t) = t t + |x−y| , γ(x,y,t) =   t t+2(x+y) , if x,y ∈ [0, 1), 1 4 ( t t+x+y ), otherwise for all t > 0. now, consider self-mappings φ and ψ on r+ defined by φ(t) = ψ(t) = t and let c = 1 2 . obviously, f is (β,γ)-admissible. to show that f is a (β,γ)-admissible ψ-type contractive mapping, we have to check the condition (2.2). to do this, we distinguish three cases: case i: if 0 ≤ x ≤ y < 1, then γ(fx,fy,t) ( 1 ffx,fy(φ(ct)) − 1 ) = y t + |x + y| ≤ β(x,y,t)ψ ( 1 fx,y(φ(t)) − 1 ) = y t + |x−y| . case ii: if x ∈ [0, 1) and y /∈ [0, 1], then γ(fx,fy,t) ( 1 ffx,fy(φ(ct)) − 1 ) = 2y 2t + |x + 4y| ≤ β(x,y,t)ψ ( 1 fx,y(φ(t)) − 1 ) = y t + |x−y| . case iii: if x,y /∈ [0, 1), then γ(fx,fy,t) ( 1 ffx,fy(φ(ct)) − 1 ) = y t + |2x + 2y| ≤ β(x,y,t)ψ ( 1 fx,y(φ(t)) − 1 ) = y t + |x−y| . it can be easily verified that all conditions of theorem 2.1 hold, and therefore f has a fixed point. we denote by fix(f) the set of fixed points of f. in what follows, we give a sufficient condition for the uniqueness of the fixed point in theorem 2.1. (h): for all u,v ∈ fix(f) and for all t > 0 there exists z ∈ x such that β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1, int. j. anal. appl. 17 (5) (2019) 779 and γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1. similar to this condition was already considered in the paper [10]. theorem 2.2. adding condition (h) to the hypotheses of theorem 2.1, we obtain that f has a unique fixed point. proof. let u,v ∈ fix(f). from condition (h), z ∈ x exists such that β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1, and γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1. taking into account (β,γ)-admissibility of f, we obtain β(fz,f2z,t) ≤ 1, β(u,fz,t) ≤ 1, β(v,fz,t) ≤ 1 and γ(fz,f2z,t) ≥ 1, γ(u,fz,t) ≥ 1, γ(v,fz,t) ≥ 1. consequently, by induction, we get β(zn,zn+1, t) ≤ 1, β(u,zn, t) ≤ 1, β(v,zn, t) ≤ 1 and γ(zn+1,zn+2, t) ≥ 1, γ(u,zn+1, t) ≥ 1, γ(v,zn+1, t) ≥ 1 for all t > 0, where zn = f nz. then, using (2.2) we derive 1 fu,fnz(φ(ct)) − 1 ≤ γ(u,zn, t) ( 1 ffu,f(fn−1z)(φ(ct)) − 1 ) ≤ β(u,zn−1, t)ψ ( 1 fu,fn−1z(φ(t)) − 1 ) ≤ ψ ( 1 fu,fn−1z(φ(t)) − 1 ) . it follows that 1 fu,fnz(φ(ct)) − 1 ≤ ψn ( 1 fu,z(φ( t cn−1 )) − 1 ) for all n ∈ n. letting n → ∞, we obtain lim n→∞ fu,zn+1 = 1. a similar argument shows that lim n→∞ fv,zn+1 = 1. using these facts, it is easily can be shown that fu,v(t) = 1 for any t > 0. it implies that u = v, and the proof is complete. � int. j. anal. appl. 17 (5) (2019) 780 example 2.4. let x = r+ and fx,y(t) = h(t − max{x,y}) for all x,y ∈ x and for all t > 0. clearly, (x,f,tp) is a g-complete menger probabilistic partial metric space. define the mapping f : x → x by f(x) = x 2 and the functions γ and β from x ×x × (0,∞) into (0,∞) by β(x,y,t) =   2, if x,y ∈ (2,∞), t t+|x−y|, otherwise γ(x,y,t) =   1 3 , if x,y ∈ (5,∞), 5, otherwise for all t > 0. also suppose that φ,ψ : r+ → r+ defined by φ(t) = ψ(t) = t and let c = 1 2 . now, it can be easily shown that all the hypotheses of theorem 2.2 are satisfied and so f has a unique fixed point. in the sequel, we first introduce the concept of (β,γ)-contractive mapping of type (i) and then we describe a fixed point theorem concerned with these kinds of contractions in the framework of menger probabilistic partial metric spaces. definition 2.6. let (x,f,t) be a menger probabilistic partial metric space and f : x → x be a given mapping. we say that f is a (β,γ)-contractive mapping of type (i), if functions β : x×x×(0,∞) → (0,∞) and γ : x ×x × (0,∞) → (0,∞) exist such that β(x,y,t)ffx,fy(φ(t)) ≥ γ(fx,fy, t c ) min { fx,y(φ( t c )),fy,fy(φ( t c )),fy,fx(φ( t c )) } (2.8) for all x,y ∈ x and all t > 0, where φ ∈ φ and c ∈ (0, 1). example 2.5. let x = {0, 2, 4} and f be as in example 2.3. clearly, (x,f,tm ) is a complete menger probabilistic partial metric space. define the mapping f : x → x by f(0) = 0, f(2) = 4 and f(4) = 2. also, define two functions β and γ from x ×x × (0,∞) into (0,∞) by β(x,y,t) =   2, if x = y, 1 2 , otherwise γ(x,y,t) =   1, if x = y = 2, or x = y = 4, 2, if x = y = 0, 1 x+y+1 , otherwise. now, consider φ : r+ → r+ defined by φ(t) = t and let c = 1 2 . then f is a (β,γ)-contractive mapping of type (i). int. j. anal. appl. 17 (5) (2019) 781 two useful following lemmas help us to prove theorem 2.3. lemma 2.1. let (x,f,t) be a menger probabilistic partial metric space and φ be a φ-function. then the following statement holds. if for x,y ∈ x and c ∈ (0, 1) we have fx,y(φ(t)) ≥ fx,y(φ( tc )) for all t > 0, then x = y. proof. the proof is similar to [4, lemma 2.9]. � lemma 2.2. let (x,f,t) be a menger probabilistic partial metric space. then the function f is a lower semi-continuous function of points, i.e., for every fixed t > 0 and every two sequences {xn}, {yn} in x such that lim n→∞ fxn,x(t) = fx,x(t) = 1 and lim n→∞ fyn,y(t) = fy,y(t) = 1 it follows that lim inf n→∞ fxn,yn (t) = fx,y(t). proof. let t > 0 and ε > 0 be given. since fx,y is left-continuous at t, so h exists such that 0 < 2h < t and fx,y(t) − fx,y(t − 2h) < ε. put fx,y(t − 2h) = a. taking into account continuity of t and t(a, 1, 1) = a, there is a real number l in (0, 1), fulfills t(a,l, l) > a− ε 3 and t(a− ε 3 , l, l) > a− 2ε 3 . on the other hand, by our assumptions, an integer mh,l exists such that fxn,x( h 2 ) > l and fyn,y( h 2 ) > l, whenever n > mh,l. now, by (2.1) and (ppm3) fxn,yn (t) ≥ t(fxn,y(t−h),fy,yn ( h 2 ),fy,yn ( h 2 )) (2.9) and fxn,y(t−h) ≥ t(fxn,x( h 2 ),fx,y(t− 2h),fx,x( h 2 )) > t(a,l, l) > a− ε 3 . (2.10) thus, on combining (2.9) and (2.10), we have fxn,yn (t) ≥ t(a− ε 3 , l, l) > a− 2ε 3 > fx,y(t) −ε. this completes the proof. � now, we present a new version of [10, theorem 3.2] due to gopal et al. in the menger probabilistic partial metric spaces. theorem 2.3. let (x,f,t) be a complete menger probabilistic partial metric space, which satisfies t(a,a,a) ≥ a with a ∈ [0, 1]. let f : x → x be a (β,γ)-contractive mapping of type (i) satisfying the following conditions: (i) f is (β,γ)-admissible, (ii) there exists x0 ∈ x such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0, int. j. anal. appl. 17 (5) (2019) 782 (iii) if {xn} is a sequence in x such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ n, and all t > 0 and xn → x as n →∞, then β(xn−1,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ n and all t > 0. then f has a fixed point. proof. since t(a,a,a) ≥ a for all a ∈ [0, 1], then t ≥ tm . let x0 ∈ x be such that (ii) holds and define a sequence {xn} in x so that xn+1 = fxn, for n = 0, 1, . . . . we suppose xn+1 6= xn for all n = 0, 1, . . ., otherwise f has trivially a fixed point. from (i), (ii) and by induction, we get β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ n and all t > 0. by continuity of φ at zero, for each t > 0 we can find r > 0 such that t > φ(r) and therefore using (2.8) and (ppm3) we have fxn,xn+1 (t) ≥ β(xn−1,xn,r)ffxn−1,fxn (φ(r)) ≥ γ(xn,xn+1, r c ) min { fxn−1,xn (φ( r c )),fxn,fxn (φ( r c )),fxn,fxn−1 (φ( r c )) } ≥ min { fxn−1,xn (φ( r c )),fxn,xn+1 (φ( r c )) } . we will show that fxn,xn+1 (φ(r)) ≥ fxn−1,xn (φ( r c )). (2.11) if we assume that fxn,xn+1 (φ( r c )) is the minimum, then from lemma 2.1, we deduce that xn = xn+1. this is in contradiction with the assumption xn 6= xn+1 and so fxn−1,xn (φ( r c )) is the minimum i.e., inequality (2.11) holds. now, from (2.11), it follows that fxn,xn+1 (t) ≥ fxn,xn+1 (φ(r)) ≥ fxn−1,xn (φ( r c )) ≥ . . . ≥ fx0,x1 (φ( r cn )), that is, fxn,xn+1 (t) ≥ fx0,x1 (φ( r cn )), (2.12) for arbitrary n ∈ n. next, let m,n ∈ n with m > n, then by (2.1) and (ppm3) we get fxn,xm ((2(m−n) − 1)t) ≥ t(fxn,xn+1 (t),fxn,xn+1 (t),fxn+1,xm ((2(m−n) − 3)t)). by strictly increasing of φ and also making use of (2.12) we have fxn,xm ((2(m−n) − 1)t) ≥ min { fxn,xn+1 (t), . . . ,fxm−1,xm−2 (t),fxm−1,xm (t) } ≥ min { fx0,x1 (φ( r cn )), . . . ,fx0,x1 (φ( r cm−2 )),fx0,x1 (φ( r cm−1 )) } = fx0,x1 (φ( r cn )). since φ( r cn ) →∞ as n →∞, for fixed ε ∈ (0, 1), n0 ∈ n exists such that for each n ≥ n0, fx0,x1 (φ( r cn )) > 1 −ε. this implies that, for every m > n ≥ n0, fxn,xm ((2(m−n) − 1)t) > 1 −ε. int. j. anal. appl. 17 (5) (2019) 783 by the arbitrariness of t > 0 and ε ∈ (0, 1), we deduce that {xn} is a cauchy sequence in the complete menger probabilistic partial metric space (x,f,t). thus, u ∈ x exists such that lim m,n→∞ fxm,xn (t) = lim n→∞ fxn,u(t) = fu,u(t). (2.13) we are going to show that u ∈ fix(f). indeed, we have ffu,u(2t) ≥ t ( ffu,xn (φ(r)),fxn,u(2t− 2φ(r)),fxn,xn (φ(r)) ) ≥ min { ffu,xn (φ(r)),fxn,u(2t− 2φ(r)) } . we may assume that xn 6= fu for all n ∈ n, since otherwise if xn = fu for infinitely many values of n, then u = fu and hence the proof is finished. now, from (2.13), for any arbitrary ε ∈ (0, 1) and n large enough, we get fxn,u(2t− 2φ(r)) > 1 − ε. hence, ffu,u(2t) ≥ min{ffu,xn (φ(r)), 1 − ε}. since ε > 0 is arbitrary, it yields that ffu,u(2t) ≥ ffu,xn (φ(r)). next, using (2.8) we deduce fu,fu(2t) ≥ fxn,fu(φ(r)) ≥ β(xn−1,u,r)ffxn−1,fu(φ(r)) ≥ γ(fxn−1,fu, r c ) min { fxn−1,u(φ( r c )),fu,fu(φ( r c )),fu,fxn−1 (φ( r c )) } ≥ min { fxn−1,u(φ( r c )),fu,fu(φ( r c )),fu,xn (φ( r c )) } . by taking the limit infimum on both sides of the above inequality and applying lemma 2.2, we have fu,fu(2t) ≥ lim n→∞ inf fxn,fu(φ(r)) ≥ lim n→∞ inf min { fxn−1,u(φ( r c )),fu,fu(φ( r c )),fu,xn (φ( r c )) } ≥ min { 1 −ε,fu,fu(φ( r c )), 1 −ε } . finally, since ε ∈ (0, 1) is arbitrary, then ffu,u(φ(r)) ≥ fu,fu(φ( rc )). from lemma 2.1, we conclude that u = fu and so we achieve our desired goal. � example 2.6. let x = [0, 1], p(x,y) = max{x,y} and (x,f,t) be as in example 2.1. then (x,f,t) is a complete menger probabilistic partial metric space. define the mapping f : x → x by fx =   x 4 , if x ∈ [0, 1 2 ) ∪ ( 1 2 , 1], 0, if x = 1 2 , and the functions β and γ from x ×x × (0,∞) into (0,∞) by β(x,y,t) = t + max{x,y} t + min{x,y} , int. j. anal. appl. 17 (5) (2019) 784 γ(x,y,t) = t x + y + t . we consider φ : r+ → r+ defined by φ(t) = t and let c = 1 2 . it is routine to see that all the hypotheses of theorem 2.3 are satisfied, and therefore f has a fixed point. theorem 2.4. adding condition (h) to the hypotheses of theorem 2.3, we obtain that f has a unique fixed point. proof. let u,v ∈ x be such that u = fu and v = fv. from condition (h), z ∈ x exists such that β(z,fz,t) ≤ 1 with β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1, and γ(z,fz,t) ≥ 1 with γ(u,z,t) ≥ 1 and γ(v,z,t) ≥ 1. by virtue of the fact that f is (β,γ)-admissible and using induction, we derive β(zn,zn+1, t) ≤ 1, β(u,zn, t) ≤ 1, β(v,zn, t) ≤ 1, and γ(zn+1,zn+2, t) ≥ 1, γ(u,zn+1, t) ≥ 1, γ(v,zn+1, t) ≥ 1 for all t > 0, where zn = f nz (n ∈ n). by continuity of φ, r > 0 exists such that t > φ(r) and therefore we have fu,zn+1 (t) ≥ fu,zn+1 (φ(r)) ≥ β(u,zn,r)ffu,fzn (φ(r)) ≥ γ(fu,fzn, r c ) min { fu,zn (φ( r c )),fzn,zn+1 (φ( r c )),fzn,fu(φ( r c )) } ≥ min { fu,zn (φ( r c )),fzn,zn+1 (φ( r c )) } . now, we consider following cases: case i. if fzn,zn+1 (φ( r c )) is the minimum, then by (2.8) and (ppm3), it follows that fu,zn+1 (φ(r)) ≥ fzn,zn+1 (φ( r c )) ≥ β(zn−1,zn, r c )ffzn−1,fzn (φ( r c )) ≥ γ(zn,zn+1, r c2 ) min { fzn−1,zn (φ( r c2 )),fzn,fzn (φ( r c2 )),fzn,zn (φ( r c2 )) } ≥ min { fzn−1,zn (φ( r c2 )),fzn,zn+1 (φ( r c2 )) } . now, if fzn,zn+1 (φ( r c2 )) is the minimum for some n ∈ n, then by lemma 2.1, we deduce that zn = zn+1. applying (ppm3), we get fu,zn+1 (φ(r)) ≥ fzn,zn+1 (φ( r c2 )) ≥ fu,zn+1 (φ( r c2 )), then u = zn+1. consequently int. j. anal. appl. 17 (5) (2019) 785 β(v,u,t) ≤ 1 and γ(fv,fu,t) ≥ 1 for all t > 0 and thus we have fv,u(φ(t)) ≥ β(v,u,t)ffv,fu(φ(t)) ≥ γ(fv,fu, t c ) min { fv,u(φ( t c )),fu,fu(φ( t c )),fu,fv(φ( t c )) } ≥ fv,u(φ( t c )). again, by lemma 2.1, we have u = v. on the other hand, if fzn−1,zn (φ( r c2 )) is the minimum, then fzn,zn+1 (φ( r c )) ≥ fzn−1,zn (φ( r c2 )) ≥ . . . ≥ fz0,z1 (φ( r cn+1 )), and, letting n → ∞, we get fzn,zn+1 (φ( r c )) → 1. therefore lim n→∞ fu,zn+1 (t) = 1. a similar method shows that lim n→∞ fv,zn+1 (t) = 1. by virtue of these facts, we get fu,v(t) = 1 for each t > 0. hence, u = v. case ii. suppose that fu,zn (φ( r c )) is the minimum, then we get fu,zn+1 (φ(r)) ≥ fu,zn (φ( r c )) ≥ fu,zn−1 (φ( r c2 )) ≥ . . . ≥ fu,z0 (φ( r cn+1 )). a similar argument as above shows that u = v, and the proof is complete. � in the sequel, we first introduce the concept of (β,γ)-contractive mapping of type (ii) and then we describe a fixed point theorem concerned with these kinds of contractions in the setup menger probabilistic partial metric spaces. definition 2.7. let (x,f,t) be a menger probabilistic partial metric space and f : x → x be a given mapping. we say that f is a (β,γ)-contractive mapping of type (ii), if functions β : x×x×(0,∞) → (0,∞) and γ : x ×x × (0,∞) → (0,∞) exist such that β(x,y,t)ffx,fy(φ(t)) ≥ γ(fx,fy, t c ) min { fx,fx(φ( t c )),fy,fy(φ( t c )) } (2.14) for all x,y ∈ x, for all t > 0, where c ∈ (0, 1) and φ ∈ φ. now, we present a new version of [10, theorem 3.4] due to gopal et al. in the menger probabilistic partial metric spaces. theorem 2.5. let (x,f,t) be a complete menger probabilistic partial metric space and f : x → x be a (β,γ)-contractive mapping of type (ii). suppose that the following conditions hold: (i) f is (β,γ)-admissible, (ii) x0 ∈ x exists such that β(x0,fx0, t) ≤ 1 and γ(x0,fx0, t) ≥ 1 for all t > 0, (iii) for each sequence {xn} in x such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1, for all n ∈ n and all t > 0, k0 ∈ n exists such that β(xm−1,xn−1, t) ≤ 1 and γ(xm,xn, t) ≥ 1, for all m,n ∈ n with m > n ≥ k0 and for all t > 0, int. j. anal. appl. 17 (5) (2019) 786 (iv) if {xn} is a sequence in x such that β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n ∈ n and all t > 0 and xn → x as n →∞, then β(xn−1,x,t) ≤ 1 and γ(xn,fx,t) ≥ 1 for all n ∈ n and all t > 0. then f has a fixed point. in addition, if condition (h) holds, then f has a unique fixed point. proof. let x0 ∈ x be such that (ii) holds. define a sequence {xn} in x such that xn+1 = fxn for all n = 0, 1, . . . . we may suppose that xn+1 6= xn for all n = 0, 1, . . ., otherwise f has trivially a fixed point. by (i) and (ii), and applying induction, we get β(xn−1,xn, t) ≤ 1 and γ(xn,xn+1, t) ≥ 1 for all n and all t > 0. by continuity of φ at zero, for each t > 0, r > 0 exists such that t > φ(r), thus β(xn−1,xn,r) ≤ 1 and γ(xn,xn+1, r c ) ≥ 1. it follows from condition (2.14) and (ppm1) that fxn,xn+1 (t)) ≥ ffxn−1,fxn (φ(r)) ≥ β(xn−1,xn,r)ffxn−1,fxn (φ(r)) ≥ γ(xn,xn+1, r c ) min { fxn−1,xn (φ( r c )),fxn,xn+1 (φ( r c )), } ≥ min { fxn−1,xn (φ( r c )),fxn,xn+1 (( r c )) } . now, if fxn,xn+1 (φ( r c )) is the minimum, then fxn,xn+1 (φ(r)) ≥ fxn,xn+1 (φ( r c )) and so by lemma 2.1, xn = xn+1, which contradicts the assumption xn 6= xn+1. thus fxn−1,xn (φ( r c )) is the minimum, and so fxn,xn+1 (t) ≥ fxn−1,xn (φ( r c )) ≥ . . . ≥ fx0,x1 (φ( r cn )). letting n →∞, then fxn,xn+1 (t) → 1. (2.15) we claim that {xn} is a cauchy sequence. suppose the contrary. then there exist ε > 0, λ ∈ (0, 1) for which we can find subsequences {xm(s)} and {xn(s)} of {xn} such that n(s) is the smallest index for which s < m(s) < n(s), fxm(s),xn(s) (ε) ≤ 1 −λ, fxm(s),xn(s)−1 (ε) > 1 −λ. by the properties of φ, ε1 > 0 exists such that φ(ε1) < ε. we deduce that fxm(s),xn(s) (φ(ε1)) ≤ 1 − λ, so {xn} is not cauchy sequence with respect to φ(ε1) and λ. thus, increasing sequences of integers m(s) and n(s) exist such that n(s) is the smallest index for which s < m(s) < n(s), fxm(s),xn(s) (φ(ε1)) ≤ 1 −λ, fxm(s),xn(s)−1 (φ(ε1)) > 1 −λ. (2.16) take a real number η such that 0 < η < φ( ε1 c ) −φ(ε1). from (2.16) it follows that fxm(s),xn(s)−1 (φ( ε1 c ) −η) > 1 −λ. int. j. anal. appl. 17 (5) (2019) 787 then, for any 0 < λ1 < λ < 1, by (2.15) it is possible to find a positive integer n1 such that for all s > n1, we have fxm(s)−1,xm(s) (η) > 1 −λ1, fxn(s)−1,xn(s) (η) > 1 −λ1. (2.17) from (2.17) it follows that fxm(s)−1,xm(s) (φ( ε1 c )) ≥ fxm(s)−1,xm(s) (η) > 1 −λ1 > 1 −λ. (2.18) a similar relation holds when one substitutes xm(s)−1 and xm(s) with xn(s)−1 and xn(s), respectively. on the other hand, from (2.18) we observe that 1 −λ ≥ fxm(s),xn(s) (φ(ε1)) = ffxm(s)−1,fxn(s)−1 (φ(ε1)) ≥ β(xm(s)−1,xn(s)−1,ε1)ffxm(s)−1,fxn(s)−1 (φ(ε1)) ≥ γ(fxm(s)−1,fxn(s)−1, ε1 c ) min { fxm(s)−1,xm(s) (φ( ε1 c ))fxn(s)−1,xn(s) (φ( ε1 c )) } > γ(fxm(s)−1,fxn(s)−1, ε1 c ) min{1 −λ, 1 −λ} ≥ 1 −λ. this is a contradiction, therefore {xn} is a cauchy sequence in the complete menger probabilistic partial metric space. hence, u ∈ x exists such that lim m,n→∞ fxm,xn (t) = lim n→∞ fxn,u(t) = fu,u(t). now, we show that u is a fixed point of f. we have ffu,u(2t) ≥ t(ffu,xn (φ(r)),fxn,u(2t− 2φ(r)),fxn,xn (φ(r)). (2.19) by continuity of φ, r > 0 exists such that t > φ(r). furthermore, for arbitrary δ ∈ (0, 1), n0 ∈ n exists such that for all n ≥ n0, we get fxn,u(2t− 2φ(r)) > 1 − δ. (2.20) hence, from (2.15), (2.19) and (2.20), we obtain that ffu,u(2t) ≥ t(ffu,xn (φ(r)), 1 − δ, 1 − δ). since δ > 0 is arbitrary and t is continuous, we can write ffu,u(2t) ≥ ffu,xn (φ(r)). without loss of generality we may assume that xn 6= fu for all n ∈ n, otherwise if for infinitely many values of n, xn = fu, then u = fu, and hence the proof is finished. applying (2.14) and (iv), we derive fu,fu(2t) ≥ fxn,fu(φ(r)) ≥ β(xn−1,u,r)ffxn−1,fu(φ(r)) ≥ γ(fxn−1,fu, r c ) min { fxn−1,fxn−1 (φ( r c )),fu,fu(φ( r c )) } . int. j. anal. appl. 17 (5) (2019) 788 letting n → ∞ in the above inequality, we get ffu,u(φ(r)) ≥ fu,fu(φ( rc )). thus u = fu by lemma 2.1. hence f has a fixed point. moreover, if (h) holds, then by using a similar technique as in the proof of theorem 2.4 one can see that u is a unique fixed point of f. � 3. application to integral equation here, in this section, we wish to study the existence of a solution to a volterra type integral equation, as an application of our results. let k > 0 be an arbitrary fixed number. consider the following volterra type integral equation: x(t) = g(x(t)) + ∫ t 0 ω(t,s,x(s))ds, t ∈ [0,k] (3.1) where g ∈ c(r) and ω ∈ c([0,k] × [0,k] ×r,r). equip c([0,k],r) with the partial metric p(x,y) = max t∈[0,k] {|x(t)|, |y(t)|}, x,y ∈ c([0,k],r). now, we define the mapping f : c([0,k],r) ×c([0,k],r) →d+ by fx,y(t) = h(t−p(x,y)), t > 0, x,y ∈ c([0,k],r). (3.2) we know that (c([0,k],r),f,tm ) is a complete menger probabilistic partial metric space. theorem 3.1. let k > 0, g ∈ c(r), and ω ∈ c([0,k] × [0,k] ×r,r), satisfying the following conditions: (i) ‖ω‖∞ = sup t,s∈[0,k],x∈c([0,k],r) | ω(t,s,x(s)) |< ∞, (ii) c1,c2 ∈ (0, 1) exist such that for all x ∈ c([0,k],r), all r ∈ r, and all t,s ∈ [0,k] we have |g(r)| ≤ c1|r|, |ω(t,s,x(s))| ≤ c2 k |x(s)|, and c = c1 + c2 < 1. then, the volterra type integral equation (3.1) has a solution x ∗ ∈ c([0,k],r). proof. consider the complete menger probabilistic partial metric space (c([0,k],r),f,tm ) defined as (3.2). now, we define the mapping f : c([0,k],r) → c([0,k],r) by f(x)(t) = g(x(t)) + ∫ t 0 ω(t,s,x(s))ds. for each x,y ∈ c([0,k],r) we have p(f(x),f(y)) = max t∈[0,k] {|f(x)(t), |f(y)(t)|} ≤ max t∈[0,k] {|g(x(t))| + ∫ t 0 |ω(t,s,x(s))|ds, |g(y(t))| + ∫ t 0 |ω(t,s,y(s))|ds} ≤ c( max t∈[0,k] {|x(t)|, |y(t)|}) = cp(x,y) ≤ c max{p(x,y),p(y,f(y)),p(y,f(x))}. int. j. anal. appl. 17 (5) (2019) 789 applying (3.2), for any r > 0, we derive ff(x),f(y)(r) = h ( r −p(f(x),f(y)) ) ≥ h (r c − max{p(x,y),p(y,f(y)),p(y,f(x))} ) = min { fx,y( r c ),fy,f(y)( r c ),fy,f(x)( r c ) } for all x,y ∈ c([0,k],r). thus all conditions of theorem 2.3 are satisfied, when φ(r) = r for all r > 0 and β(x,y,t) = γ(x,y,t) = 1 for all x,y ∈ c([0,k],r) and t > 0. therefore, f has a fixed point x∗ ∈ c([0,k],r), which is the solution of the integral equation (3.1). � example 3.1. consider the following volterra type integral equation x(t) = 1 3 cos(x(t)) + ∫ t 0 ln(t + s + 1 2 ) sin(e−3tx(s)) 3 cosh 5 √ 2s + 1 ds. (3.3) observe that eq. (3.3) is a special case of the eq. (3.1) when g(r) = 1 3 cos(r) (r ∈ r), and ω(t,s,x(s)) = ln(t+s+ 1 2 ) sin(e−3tx(s)) 3 cosh 5 √ 2s+1 (t,s ∈ [0, 1]). obviously, all the conditions of theorem 3.1 are satisfied. then the volterra type integral equation (3.3) has at least one solution which belongs to the space c([0, 1],r). 4. numerical results numerical methods can help us to investigate the solutions of differential and integral equations (see for instance [2, 11, 18]). in this section, we use a numerical method to find an approximate solution for a volterra type integral equation. for this purpose, we use successive approximations method (sam) [2] to find a solution for the example 3.1. the sam, also called the picard iteration method provides a scheme that can be used for solving initial value problems or integral equations. this method solves any problem by finding successive approximations to the solution by starting with an initial guess as x0(t), called the zeroth approximation. as will be seen, the zeroth approximation is any selective real-valued function that will be used in a recurrence relation to determine the other approximations. the most commonly used values for the zeroth approximations are 0, 1, or t. of course, other real values can be selected as well. given the nonlinear volterra type integral equation x(t) = f(x(t)) + λ ∫ t a k(t,s,x(s))ds, where x is the unknown function to be determined, k(t,s,x(s)) is the kernel, and λ is a parameter. the successive approximation method introduces the recurrence relation x̃0(t) = any selective real valued function, (4.1) x̃n+1(t) = f(x̃n(t)) + λ ∫ t a k(t,s, x̃n(s))ds, n ≥ 0. int. j. anal. appl. 17 (5) (2019) 790 the solution is determined by using the limit x̃(t) = lim n→∞ x̃n+1(t). now, we discuss on the solutions of example 3.1. to this end, consider the following nonlinear equation x(t) = f(x(t)) + g(x(t)), (4.2) where f(x(t)) = 1 3 cos(x(t)), (4.3) and g(x(t)) = ∫ t 0 ln(t + s + 1 2 ) sin(e−3tx(s)) 3 cosh 5 √ 2s + 1 ds. (4.4) applying sam, we solve the nonlinear volterra type integral equation (3.3). to this aim, put t = 0 in equation (3.3). we get x(0) = 1 3 cos(x(0)) or equivalently α = 1 3 cos(α). it gives us α = 0.3176508287. now, we choose x̃0(t) = 0.3176508287 in (4.1), by doing two steps of successive approximations method we find x̃2(t) and consider it as an approximation for x(t) (see fig. 1). since equation (3.3) is nonlinear, it is difficult to proceed this method further. in order to see how good is this approximation, we put x̃2(t) in the left and right hand sides of equation (3.3) instead of x(t), and consider the difference of these values as error. we define errsam (t) := |x̃2(t) − (f(x̃2(t)) + g(x̃2(t)))|. in table (1), we have calculated x̃2(t) and errsam (t) at different values of t. the error graph of errsam (t) is also plotted in figure (2) in the interval [0, 1]. table 1. the values of x̃2(t) and error related to sam for different values of t t ũ2(t) errsam (t) 0 3.17e-1 e-10 0.1 3.14e-1 2.93e-5 0.2 3.15e-1 2.86e-5 0.3 3.16e-1 1.23e-5 0.4 3.17e-1 1.1e-5 0.5 3.19e-1 3.66e-5 0.6 3.21e-1 6.22e-5 0.7 3.23e-1 8.66e-5 0.8 3.24e-1 1.09e-4 0.9 3.25e-1 1.31e-4 1 3.27e-1 1.51e-4 int. j. anal. appl. 17 (5) (2019) 791 figure 1. graph of approximate solution x̃ in [0,1] figure 2. absolute error graph of approximate solution x̃ in [0,1] references [1] t. abdeljawada, e. karapinar and k. taş, existence and uniqueness of a common fixed point on partial metric spaces, appl. math. lett. (2011), 1–5. [2] w. abdul-majid, linear and nonlinear integral equations: methods and applications, springer, 2011. [3] h. aydi, e. karapinar and s. rezapour, a generalized meir-keeler contraction on partial metric spaces, abstr. appl. anal. 2012 (2012), article id 287127. [4] n.a. babaĉev, nonlinear generalized contraction on menger pm-spaces, appl. anal. discrete math. 6 (2012), 257–264. [5] m. bukatin, r. kopperman, s. matthews and h. pajoohesh, partial metric spaces, am. math. montly, 116 (2009), 708–718. [6] s. chauhan, s. bhatnagar and s. radenović, common fixed point theorems for weakly compatible mappings in fuzzy metric spaces, le mathematiche, lxviii (2013)-fasc. i, 87–98. [7] s. chauhan, m. imdad, c. vetro and w. sintunavarat, hybrid coincidence and common fixed point theorems in menger probabilistic metric spaces under a strict contractive condition with an application, appl. math. comput. 239 (2014), 422–433. [8] t. došenović, d. rakić, b. carić and s. radenović, multivalued generalizations of fixed point results in fuzzy metric spaces, nonlinear anal., model. control, 21(2) (2016), 211–222. [9] p.n. dutta, b.s. choudhury and k.p. das, some fixed point results in menger spaces using a control function, surv. math. appl. 4 (2009), 41–52. int. j. anal. appl. 17 (5) (2019) 792 [10] d. gopal, m. abbas and c. vetro, some new fixed point theorems in menger pm-spaces with application to volterra type integral equation, appl. math. comput. 232 (2014), 955–967. [11] l. grammont, nonlinear integral equations of the second kind: a new version of nyström method. numer. funct. anal. optim. 34(5) (2013), 496-515. [12] o. hadžić and e. pap, fixed point theory in probabilistic metric spaces, kluwer academic publishers, 2001. [13] f. hasanvand and m. khanehgir, some fixed point theorems in menger pbm-spaces with an application, fixed point theory appl. 2015 (2015), 81. [14] s.g. matthews, partial metric topology, proc. 8th summer conference on general topology and applications. ann. n.y. acad. sci. 728 (1994), 183–197. [15] k. menger, statistical metrics, proc. nat. acad. sci. usa, 28 (1942), 535–537. [16] z. mustafa, j. rezaei roshan, v. parvaneh and z. kadelburg, some common fixed point results in orderd partial b-metric spaces, j. inequal. appl. 2013 (2013), 562. [17] p. patle, d. patel, h, aydi and s. radenović, on h+type multivalued contraction and its applications in symmetric and probabilistic spaces, mathematics, 7 (2019), 144. [18] m. rabbani and r. arab, extension of some theorems to find solution of nonlinear integral equation and homotopy perturbation method to solve it, math. sci. 11(2) (2017), 87–94. [19] s. romaguera, a kirk type characterization of completeness for partial metric spaces, fixed point theory appl. 2010 (2010), article id 493298, 6 pages. [20] b. schweizer and s. sklar, statistical metric spaces, pacific j. math. 10 (1960), 313–334. [21] b. schweizer and a. sklar, probabilistic metric spaces, elsevier, north-holland, new york, 1983. [22] s. sedghi, n. shobkolaei, t. došenović and s. radenović, suzuki-type of common fixed point theorems in fuzzy metric spaces, math. slovaca 68(2) (2018), 451–462. [23] v.m. sehgal and a.t. bharucha-reid, fixed point of contraction mappings in pm-spaces, math. syst. theory, 6 (1972), 97–102. [24] y. shi, l. ren and x. wang, the extension of fixed point theorems for set valued mapping, j. appl. math. comput. 13 (2003), 277–286. [25] stevens, metrically generated pm-spaces, fund. math. (1968), 259–269. [26] y. su and j. zhang, fixed point and best proximity point theorems for contractions in new class of probabilistic metric spaces, fixed point theory appl. 2014 (2014), 170. 1. introduction 2. main result 3. application to integral equation 4. numerical results references international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 89-96 http://www.etamaths.com convergence to common fixed point for nearly asymptotically nonexpansive mappings in banach spaces g. s. saluja abstract. the purpose of this paper is to study modified s-iteration process to converge to common fixed point for two nearly asymptotically nonexpansive mappings in the framework of banach spaces. also we establish some strong convergence theorems and a weak convergence theorem for said mappings and iteration scheme under appropriate conditions. 1. introduction let c be a nonempty subset of a banach space e and t : c → c a nonlinear mapping. we denote the set of all fixed points of t by f(t). the set of common fixed points of two mappings s and t will be denoted by f = f(s) ∩f(t). the mapping t is said to be lipschitzian [1, 16] if for each n ∈ n, there exists a constant kn > 0 such that ‖tnx−tny‖ ≤ kn ‖x−y‖ for all x, y ∈ c. a lipschitzian mapping t is said to be uniformly k-lipschitzian if kn = k for all n ∈ n and asymptotically nonexpansive [4] if kn ≥ 1 for all n ∈ n with limn→∞kn = 1. it is easy to observe that every nonexpansive mapping t (i.e., ‖tx − ty‖ ≤ ‖x − y‖ for all x, y ∈ c) is asymptotically nonexpansive with constant sequence {1} and every asymptotically nonexpansive mapping is uniformly k-lipschitzian with k = supn∈n kn. the asymptotic fixed point theory has a fundamental role in nonlinear functional analysis (see, [2]). the theory has been studied by many authors (see, e.g., [6], [7], [10], [12], [21]) for various classes of nonlinear mappings (e.g., lipschitzian, uniformly k-lipschitzian and non-lipschitzian mappings). a branch of this theory related to asymptotically nonexpansive mappings has been developed by many authors (see, e.g., [4], [5], [9], [11], [12], [14], [15], [17]-[19]) in banach spaces with 2010 mathematics subject classification. 47h09, 47h10, 47j25. key words and phrases. nearly asymptotically nonexpansive mapping, modified s-iteration process, common fixed point, strong convergence, weak convergence, banach space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 89 90 g. s. saluja suitable geometrical structure. fix a sequence {an} ⊂ [0,∞) with limn→∞an = 0, then according to agarwal et al. [1], t is said to be nearly lipschitzian with respect to {an} if for each n ∈ n, there exist constants kn ≥ 0 such that ‖tnx − tny‖ ≤ kn(‖x − y‖ + an) for all x, y ∈ c. the infimum of constants kn for which the above inequality holds is denoted by η(tn) and is called nearly lipschitz constant. a nearly lipschitzian mapping t with sequence {an,η(tn)} is said to be nearly asymptotically nonexpansive if η(tn) ≥ 1 for all n ∈ n and limn→∞η(tn) = 1 and nearly uniformly k-lipschitzian if η(tn) ≤ k for all n ∈ n. in 2007, agarwal et al. [1] introduced the following iteration process: x1 = x ∈ c, xn+1 = (1 −αn)tnxn + αntnyn, yn = (1 −βn)xn + βntnxn, n ≥ 1(1.1) where {αn} and {βn} are sequences in (0, 1). they showed that this process converge at a rate same as that of picard iteration and faster than mann for contractions and also they established some weak convergence theorems using suitable conditions in the framework of uniformly convex banach space. we modify iteration scheme (1.1) for two nonlinear mappings. let c be a nonempty subset of a banach space e and s, t : c → c be two nearly asymptotically nonexpansive mappings. for given x1 = x ∈ c, the iterative sequence {xn} defined as follows: x1 = x ∈ c, xn+1 = (1 −αn)tnxn + αnsnyn, yn = (1 −βn)xn + βntnxn, n ≥ 1(1.2) where {αn} and {βn} are sequences in (0, 1). the iteration scheme (1.2) is called modified s-iteration scheme for two nonlinear mappings. if we put s = t, then iteration scheme (1.2) reduces to s-iteration scheme (1.1). the aim of this paper is to establish some strong convergence theorems and a weak convergence theorem of modified s-iteration scheme (1.2) for two nearly asymptotically nonexpansive mappings in the framework of banach spaces. 2. preliminaries for the sake of convenience, we restate the following concepts. a mapping t : c → c is said to be demiclosed at zero, if for any sequence {xn} in c, the condition xn converges weakly to x ∈ c and txn converges strongly to 0 imply tx = 0. nearly asymptotically nonexpansive mappings 91 a mapping t : c → c is said to be semi-compact [3] if for any bounded sequence {xn} in c such that ‖xn −txn‖ → 0 as n → ∞, then there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ c strongly. we say that a banach space e satisfies the opial’s condition [13] if for each sequence {xn} in e weakly convergent to a point x and for all y 6= x lim inf n→∞ ‖xn −x‖ < lim inf n→∞ ‖xn −y‖. the examples of banach spaces which satisfy the opial’s condition are hilbert spaces and all lp[0, 2π] with 1 < p 6= 2 fail to satisfy opial’s condition [13]. now, we state the following useful lemma to prove our main results. lemma 2.1. (see [20]) let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of nonnegative numbers satisfying the inequality αn+1 ≤ (1 + βn)αn + rn, ∀n ≥ 1. if ∑∞ n=1 βn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞αn exists. 3. main results in this section, we prove some strong convergence theorems and a weak convergence theorem for two nearly asymptotically nonexpansive mappings in the framework of banach spaces. theorem 3.1. let e be a banach space and c be a nonempty closed convex subset of e. let s, t : c → c be two nearly asymptotically nonexpansive mappings with sequences {a′n,η(sn)}, {a′′n,η(tn)} and f = f(s) ∩f(t) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)η(tn)−1 ) < ∞. let {xn} be the modified s-iteration scheme defined by (1.2). then {xn} converges to a common fixed point of the mappings s and t if and only if lim infn→∞d(xn,f) = 0. proof. the necessity is obvious. thus we only prove the sufficiency. let q ∈ f. for the sake of convenience, set anx = (1 −βn)x + βntnx and gnx = (1 −αn)tnx + αnsnanx. then yn = anxn and xn+1 = gnxn. moreover, it is clear that q is a fixed point of gn for all n. let η = supn∈n η(s n) ∨ supn∈n η(tn) and an = max{a′n,a′′n} for all n. 92 g. s. saluja consider ‖anx−any‖ = ‖((1 −βn)x + βntnx) − ((1 −βn)y + βntny)‖ = ‖(1 −βn)(x−y) + βn(tnx−tny)‖ ≤ (1 −βn)‖x−y‖ + βnη(tn)(‖x−y‖ + a′′n) ≤ (1 −βn)‖x−y‖ + βnη(tn)‖x−y‖ + βnanη(tn) ≤ (1 −βn)η(tn)‖x−y‖ + βnη(tn)‖x−y‖ +βnanη(t n) ≤ η(tn)‖x−y‖ + anη(tn).(3.1) choosing x = xn and y = q, we get ‖yn −q‖ ≤ η(tn)‖xn −q‖ + anη(tn).(3.2) now, consider ‖gnx−gny‖ = ‖((1 −αn)tnx + αnsnanx) − ((1 −αn)tny + αnsnany)‖ = ‖(1 −αn)(tnx−tny) + αn(snanx−snany)‖ ≤ (1 −αn)η(tn)(‖x−y‖ + a′′n) + αnη(s n)(‖anx−any‖ + a′n) ≤ (1 −αn)η(tn)(‖x−y‖ + an) + αnη(sn)(‖anx−any‖ + an) ≤ (1 −αn)η(tn)‖x−y‖ + αnη(sn)‖anx−any‖ +(1 −αn)anη(tn) + αnanη(sn).(3.3) now using (3.1) in (3.3), we get ‖gnx−gny‖ ≤ (1 −αn)η(tn)‖x−y‖ + αnη(sn)[η(tn)‖x−y‖ +anη(t n)] + (1 −αn)anη(tn) + αnanη(sn) ≤ (1 −αn)η(tn)η(sn)‖x−y‖ + αnη(tn)η(sn)‖x−y‖ +(1 −αn + 2αn)anη(tn)η(sn) ≤ η(tn)η(sn)‖x−y‖ + 2anη(tn)η(sn) ≤ [ 1 + ( η(tn)η(sn) − 1 )] ‖x−y‖ + 2anη2 = (1 + pn)‖x−y‖ + qn,(3.4) where pn = ( η(tn)η(sn)−1 ) and qn = 2anη 2. since by hypothesis ∑∞ n=1 ( η(sn)η(tn)− 1 ) < ∞ and ∑∞ n=1 an < ∞. it follows that ∑∞ n=1 pn < ∞ and ∑∞ n=1 qn < ∞. choosing x = xn and y = q in (3.4), we get ‖xn+1 −q‖ = ‖gnxn −q‖≤ (1 + pn)‖xn −q‖ + qn.(3.5) applying lemma 2.1 in (3.5), we have limn→∞‖xn −q‖ exists. nearly asymptotically nonexpansive mappings 93 next, we shall prove that {xn} is a cauchy sequence. since 1 +x ≤ ex for x ≥ 0, therefore, for any m,n ≥ 1 and for given q ∈ f, from (3.5), we have ‖xn+m −q‖ ≤ (1 + pn+m−1)‖xn+m−1 −q‖ + qn+m−1 ≤ epn+m−1‖xn+m−1 −q‖ + qn+m−1 ≤ epn+m−1 [epn+m−2‖xn+m−2 −q‖ + qn+m−2] + qn+m−1 ≤ e(pn+m−1+pn+m−2)‖xn+m−2 −q‖ +e(pn+m−1+pn+m−2)[qn+m−2 + qn+m−1] ≤ . . . ≤ e (∑n+m−1 k=n pk ) ‖xn −q‖ + e (∑n+m−1 k=n pk ) n+m−1∑ k=n qk ≤ e (∑∞ n=1 pn ) ‖xn −q‖ + e (∑∞ n=1 pn ) n+m−1∑ k=n qk = k‖xn −q‖ + k n+m−1∑ k=n qk(3.6) where k = e (∑∞ n=1 pn ) < ∞. since lim n→∞ d(xn,f) = 0, ∞∑ n=1 qn < ∞(3.7) for any given ε > 0, there exists a positive integer n1 such that d(xn,f) < ε 4(k + 1) , n+m−1∑ k=n qk < ε 2k ∀n ≥ n1.(3.8) hence, there exists q1 ∈ f such that ‖xn −q1‖ < ε 2(k + 1) ∀n ≥ n1.(3.9) consequently, for any n ≥ n1 and m ≥ 1, from (3.6), we have ‖xn+m −xn‖ ≤ ‖xn+m −q1‖ + ‖xn −q1‖ ≤ k‖xn −q1‖ + k n+m−1∑ k=n qk + ‖xn −q1‖ ≤ (k + 1)‖xn −q1‖ + k n+m−1∑ k=n qk < (k + 1) ε 2(k + 1) + k ε 2k = ε.(3.10) this implies that {xn} is a cauchy sequence in e and so is convergent since e is complete. assume that limn→∞xn = q ∗. since c is closed, therefore q∗ ∈ c. next, we show that q∗ ∈ f . now limn→∞d(xn,f) = 0 gives that d(q∗,f) = 0. since f is closed, q∗ ∈ f. thus {xn} converges strongly to a common fixed point of the mappings s and t. this completes the proof. 94 g. s. saluja theorem 3.2. let e be a banach space and c be a nonempty closed convex subset of e. let s, t : c → c be two nearly asymptotically nonexpansive mappings with sequences {a′n,η(sn)}, {a′′n,η(tn)} and f = f(s) ∩f(t) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)η(tn) − 1 ) < ∞. let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). let {xn} be the modified s-iteration scheme defined by (1.2). if either s is semi-compact and limn→∞‖xn−sxn‖ = 0 or t is semi-compact and limn→∞‖xn −txn‖ = 0, then the sequence {xn} converge strongly to a point of f. proof. suppose that t is semi-compact and limn→∞‖xn − txn‖ = 0. then there exists a subsequence {xnj} of {xn} such that xnj → q ∈ c. also, we have limj→∞‖xnj −txnj‖ = 0 and we make use of the fact that every nearly asymptotically nonexpansive mapping is nearly k-lipschitzian. hence, we have ‖q −tq‖ ≤ ‖q −xnj‖ + ‖xnj −txnj‖ + ‖txnj −tq‖ ≤ (1 + k)‖q −xnj‖ + ‖xnj −txnj‖→ 0. thus q ∈ f . by (3.5), ‖xn+1 −q‖≤ (1 + pn)‖xn −q‖ + qn. since by hypothesis ∑∞ n=1 pn < ∞ and ∑∞ n=1 qn < ∞, by lemma 2.2, limn→∞‖xn− q‖ exists and xnj → q ∈ f gives that xn → q ∈ f . this shows that {xn} converges strongly to a point of f. this completes the proof. as an application of theorem 3.1, we establish another strong convergence result as follows. theorem 3.3. let e be a banach space and c be a nonempty closed convex subset of e. let s, t : c → c be two nearly asymptotically nonexpansive mappings with sequences {a′n,η(sn)}, {a′′n,η(tn)} and f = f(s) ∩f(t) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)η(tn) − 1 ) < ∞. let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). let {xn} be the modified s-iteration scheme defined by (1.2). if s and t satisfy the following conditions: (i) limn→∞‖xn −sxn‖ = 0 and limn→∞‖xn −txn‖ = 0. (ii) there exists a constant a > 0 such that[ a1‖xn −sxn‖ + a2‖xn −txn‖ ] ≥ ad(xn,f) where a1 and a2 are two non-negative real numbers such that a1 + a2 = 1. then the sequence {xn} converge strongly to a point of f. proof. from conditions (i) and (ii), we have limn→∞d(xn,f) = 0, it follows as in the proof of theorem 3.1, that {xn} must converge strongly to a point of f. this completes the proof. nearly asymptotically nonexpansive mappings 95 theorem 3.4. let e be a banach space satisfying opial’s condition and c be a nonempty closed convex subset of e. let s, t : c → c be two nearly asymptotically nonexpansive mappings with sequences {a′n,η(sn)}, {a′′n,η(tn)} and f = f(s)∩ f(t) 6= ∅ is closed such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)η(tn)−1 ) < ∞. let {αn} and {βn} be sequences in [δ, 1 −δ] for some δ ∈ (0, 1). let {xn} be the modified s-iteration scheme defined by (1.2). suppose that s and t have a common fixed point, i −s and i −t are demiclosed at zero and {xn} is an approximating common fixed point sequence for s and t , that is, limn→∞‖xn − sxn‖ = 0 and limn→∞‖xn −txn‖ = 0. then {xn} converges weakly to a common fixed point of s and t. proof: let q be a common fixed point of s and t . then limn→∞‖xn−q‖ exists as proved in theorem 3.1. we prove that {xn} has a unique weak subsequential limit in f = f(s) ∩ f(t). for, let u and v be weak limits of the subsequences {xni} and {xnj} of {xn}, respectively. by hypothesis of the theorem, we know that limn→∞‖xn −sxn‖ = 0 and i −s is demiclosed at zero, therefore we obtain su = u. similarly, tu = u. thus u ∈ f = f(s)∩f(t). again in the same fashion, we can prove that v ∈ f = f(s) ∩f(t). next, we prove the uniqueness. to this end, if u and v are distinct then by opial’s condition, lim n→∞ ‖xn −u‖ = lim ni→∞ ‖xni −u‖ < lim ni→∞ ‖xni −v‖ = lim n→∞ ‖xn −v‖ = lim nj→∞ ‖xnj −v‖ < lim nj→∞ ‖xni −u‖ = lim n→∞ ‖xn −u‖. this is a contradiction. hence u = v ∈ f. thus {xn} converges weakly to a common fixed point of the mappings s and t. this completes the proof. remark 3.1. our results extend and generalize the corresponding results of [8], [14], [15], [17], [18], [20] and many others from the existing literature to the case of modified s-iteration scheme and more general class of nonexpansive and asymptotically nonexpansive mappings considered in this paper. references [1] r.p. agarwal, donal o’regan and d.r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, nonlinear convex anal. 8(1)(2007), 61-79. [2] f.e. browder, nonlinear operators and nonlinear equations of evolution, proc. amer. math. symp. pure math. xvii, amer. math. soc., providence, 1976. [3] c.e. chidume and b. ali, weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 330(2007), 377-387. 96 g. s. saluja [4] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1)(1972), 171-174. [5] j. gornicki, weak convergence theorems for asymptotically mappings in uniformly convex banach spaces, comment. math. univ. carolinae 30(1989), 249-252. [6] j.s. jung, d.r. sahu and b.s. thakur, strong convergence theorems for asymptotically nonexpansive mappings in banach spaces, comm. appl. nonlinear anal. 5(1998), 53-69. [7] j.s. jung and d.r. sahu, fixed point theorem for non-lipschitzian semigroups without convexity, indian j. math. 40(2)(1998), 169-176. [8] s.h. khan and w. takahashi, approximating common fixed points of two asymptotically nonexpansive mappings, sci. math. jpn. 53(1)(2001), 143-148. [9] w.a. kirk, a fixed point theorem for mappings which do not increase distances, amer. math. monthly 72(1965), 1004-1006. [10] w.a. kirk, fixed point theorem for non-lipschitzian mappings of asymptotically nonexpansive type, israel j. math. 17(1974), 339-346. [11] w.a. kirk, c. martinez yanez and s.s. kim, asymptotically nonexpansive mappings, nonlinear anal. 33(1998), 1345-1365. [12] t.c. lim and h. xu, fixed point theorems for mappings of asymptotically nonexpansive mappings, nonlinear anal. 22(1994), 1345-1355. [13] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc. 73(1967), 591-597. [14] m.o. osilike and s.c. aniagbosor, weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, math. and computer modelling 32(2000), 1181-1191. [15] b.e. rhoades, fixed point iteration for certain nonlinear mappings, j. math. anal. appl. 183(1994), 118-120. [16] d.r. sahu, fixed points of demicontinuous nearly lipschitzian mappings in banach spaces, comment. math. univ. carolinae 46(4)(2005), 653-666. [17] j. schu, weak and strong convergence to fixed points of asymptotically nonexpansive mappings, bull. austral. math. soc. 43(1)(1991), 153-159. [18] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, j. math. anal. appl. 158(1991), 407-413. [19] k.k. tan and h.k. xu, the nonlinear ergodic theorem for asymptotically nonexpansive mappings in banach space, proc. amer. math. soc. 114(1992), 399-404. [20] k.k. tan and h.k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178(1993), 301-308. [21] h.k. xu, existence and convergence for fixed points for mappings of asymptotically nonexpansive type, nonlinear anal. 16(1991), 1139-1146. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications issn 2291-8639 volume 2, number 2 (2013), 173-177 http://www.etamaths.com fixed point of order 2 on g-metric space animesh gupta abstract. in this article we introduce a new concept of fixed point that is fixed point of order 2 on g-metric space and some results are achieved. 1. introduction and preliminaries in 2003, mustafa and sims [4] introduced a more appropriate and robust notion of a generalized metric space as follows. definition 1.1. [4] let x be a nonempty set, and let g : x ×x ×x → [0,∞) be a function satisfying the following axioms: (1) g(x,y,z) = 0 if and only if x = y = z; (2) g(x,x,y) > 0, for all x 6= y; (3) g(x,y,z) ≥ g(x,x,y), for all x,y,z ∈ x; (4) g(x,y,z) = g(x,z,y) = g(z,y,x) = · · · (symmetric in all three variables); (5) g(x,y,z) ≤ g(x,w,w) + g(w,y,z), for all x,y,z,w ∈ x. then the function g is called a generalized metric, or, more specifically a g-metric on x, and the pair (x,g) is called a g-metric space. definition 1.2. suppose that (x,g) is a g-metric space, t : x → x is a function and x0 ∈ x is fixed point of t. we call x0 is a fixed pointof order 2 if it is not alone point and the following satisfies: lim x→x0 g(tx,tx,x0) g(x,x,x0) = 1(1.1) we remember the following definitions. we will show that for the case (a) there is not fixed point of order 2 but in two other cases there is fixed point of order 2. definition 1.3. suppose that (x,g) is a g-metric space, t : x → x is a function. (a) t is a contraction, if there exist k ∈ [0, 1) such that g(tx,ty,tz) ≤ kg(x,y,z) for all x,y,z ∈ x. (b) t is a contractive mapping, if g(tx,ty,tz) < g(x,y,z) for all x,y,z ∈ x which x 6= y 6= z. (c) t is non-expansive mapping, if g(tx,ty,tz) ≤ g(x,y,z) for all x,y,z ∈ x. in the following we consider first some properties for fixed point of order 2. 2010 mathematics subject classification. primary 47h10; secondary 54h25, 55m20. key words and phrases. fixed point, fixed point of order 2, contraction mapping, non expansive mapping. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 173 174 gupta 2. main results proposition 2.1. if x0 ∈ x is a fixed point of order 2 for t on x. then t is continuous at x0. proof. limn→∞g(tx,tx,x0) = limx→x0 g(t x,t x,x0) g(x,x,x0) g(x,x,x0) limx→x0 g(t x,t x,x0) g(x,x,x0) limx→x0 g(x,x,x0) = 0. � proposition 2.2. let (x,g) be a metric space and t : x → x be a function such that x0 ∈ x is a fixed point for t , not alone point for x and alone point for t(x). then x0 is not fixed point of order 2 for t . proof. according to assumption x0 is alone point for t(x). there is a neighborhood of x0, like n(x0) such that n(x0) ∩t(x) and each x ∈ n(x0) implies that g(tx,tx,x0) = 0. therefore, limx→x0 g(t x,t x,x0) g(x,x,x0) = 0, i.e; x0 is not a fixed point of order 2 for t . � proposition 2.3. suppose that x0 ∈ x be a fixed point for ti : x → x which i = 1, 2, ...,n where (n ∈ n) and also limx→x0 g(tix,tix,x0) g(x,x,x0) = λi. then x0 is a fixed point of order 2 for t1t2...tn if and only if λ1λ2...λn = 1. proof. ti is continuous at x0 for all i = 1, 2, ...,n by a simple change of variable that lim x→x0 g(tk(tk+1...tnx),tk(tk+1...tnx),x0) g(tk+1...tnx,tk+1...tnx,x0) = lim t→x0 g(tkt,tkt,x0) t,t,x0 and the last limit is equal with λk for k = 1, 2, ...,n. hence, lim x→x0 g(t1t2...tnx,t1t2...tnx,x0) g(x,x,x0) = lim x→x0 g(t1(t2...tn)x,t1(t2...tn)x,x0) g(t2...tn,t2...tn,x0) g(t2(t3...tn)x,t2(t3...tn)x,x0) g(t3...tn,t3...tn,x0) ... g(tnx,tnx,x0) g(x,x,x0) λ1λ2...λn � proposition 2.4. letx0 ∈ x be a fixed point for ti : x → x for i = 1, 2, ...,n and n ∈ n. (a) if x0 is fixed point of order 2 for all ti, then x0 is fixed point for t1t2...tn. (b) if x0 is fixed point order 2 for t1t2 and t2 , then x0 is fixed point of order 2 for t1. proof. (a) by proposition 2.1. (b) x0 is fixed point of order 2 for t1t2 and t2. thus, limx→x0 g(t1t2x,t1t2x,x0) g(x,x,x0) = 1, limx→x0 g(t2x,t2x,x0) g(x,x,x0) = 1. since t is continuous at x0 for t = t2x. 1 = lim x→x0 g(t1t2x,t1t2x,x0) g(x,x,x0) limx→x0 g(t2x,t2x,x0) g(x,x,x0) = lim x→x0 g(t1t2x,t1t2x,x0) g(t2x,t2x,x0) = lim t→x0 g(t1t,t1t,x0) g(t,t,x0) � fixed point of order 2 on g-metric space 175 proposition 2.5. suppose that x0 is not alone point and is a fixed point for ti : x → x for i = 1, 2, ...,n and n ∈ n. (a) if ti be a contractive mapping or non expansive mapping for i = 1, 2, ...,n and n ∈ n and limx→x0 g(tix,tix,x0) g(x,x,x0) = λi. then x0 ∈ x is a fixed point of order 2 for t1t2...tn if and only if x0 is a fixed point of order 2 for all ti. (b) if limx→x0 g(t1x,t1x,x0) g(x,x,x0) = λ then x0 is a fixed point of order 2 for t1 if and only if x0 be a fixed point of order 2 for t n 1 where n is arbitrary positive integer. (c) if t1 be a contractive mapping or non-expansive mapping, then x0 is a fixed point of order 2 for t1 if and only if there exist n ∈ n such that x0 be a fixed point of order 2 for tn1 . proof. (a) let ti be a contractive mapping for all i = 1, 2, ...,n. if x0 is a fixed point of order 2 for all ti then by proposition 2.3, x0 is a fixed point of order 2 for t1t2...tn. now assume that x0 is a fixed point of order 2 for t1t2...tn then by proposition 2.2, 1 = limx→x0 g(t1t2...tnx,t1t2...tnx,x0) g(x,x,x0) = λ1λ2...λn. but all ti are contractive mappings so g(t1x,t1x,x0) g(x,x,x0) < 1 which implies that λi ≤ 1 for all i = 1, 2, ...n. hence, λ1 = λ2 = ... = λn = 1. proof for non expansive is similar. (b) by proposition 2.2, limx→x0 g(t n1 x,,x0) g(x,x,x0) = λn. then λn = 1 if and only if λ = 1 because λ ≥ 0. (c) let t1 be a contractive mapping and there exists n ∈ n such that x0 is a fixed point of order 2 for tn1 .t1 is a contractive mapping. so g(tn1 x,t n 1 x,x0) < ... < g(t1x,t1x,x0) < g(x,x,x0) 1 = lim x→x0 g(tn1 x,t n 1 x,x0) g(x,x,x0 ≤ g(t1x,t1x,x0) g(x,x,x0 ≤ 1. therefore, limx→x0 g(t1x,t1x,x0) g(x,x,x0 = 1. � proposition 2.6. suppose that (x,g) is a metric space, t : x → x is a function and x0 is a fixed point of t . if t is contraction then x0 is not a fixed point of order 2 for t . proof. since t is a contractive mapping so there exists α ∈ [0, 1) such that g(tx,ty,tz) ≤ αg(x,y,z) for all x,y,z ∈ x. therefore g(t x,t x,x0) g(x,x,x0) ≤ α < 1 and x0 can not be a fixed point of order 2 for t. � proposition 2.7. suppose that x0 ∈ x be a fixed point of order 2 for t : x → x where t is one to one and g is left inverse of t . then x0 is also a fixed point of order 2 for g. proof. it is clear that x0 is a fixed point for g. on the other hand, since t is continuous at x0 for t = tx so 176 gupta 1 = lim x→x0 g(tx,tx,x0) g(x,x,x0) = lim x→x0 g(g(t(tx)),g(t(tx)),x0) g(gtx,gtx,x0) = lim t→x0 g(g(tt),g(tt),x0) g(gt,gt,x0) = lim t→x0 g(t,t,x0) g(gt,gt,x0) = lim t→x0 1 g(gt,gt,x0) g(t,t,x0) therefore, limt→x0 g(gt,gt,x0) g(t,t,x0) = 1. � in the following we give another condition for the fixed point of order 2. proposition 2.8. suppose that x0 is not alone point and is a fixed point for t : x → x. (a) if limx→x0 g(t x,t x,x) g(x,x,x0) = 0 then x0 is a fixed point of order 2 for t . (b) if limx→x0 g(t x,t x,x) g(t x,t x,x0) = 0 then x0 is a fixed point of order 2 for t . proof. (a) from the definition of g-metric space we have | g(x,x,x0) −g(tx,tx,x0) | ≤ g(tx,tx,x) 1 − g(tx,tx,x0) g(x,x,x0) ≤ g(tx,tx,x) g(x,x,x0) ≤ 1 + g(tx,tx,x0) g(x,x,x0) limx→x0 g(t x,t x,x0) g(x,x,x0) = 1. (b) prove of this part is similarly as prove of (a). � proposition 2.9. suppose that x0 is a fixed point for t : x → x and ψ : x → r+ is a real valued function. (a) if x0 be a fixed point of order 2 for t then limx→x0 g(t x,t x,x) g(x,x,x0) ≤ 2. (b) if g(tx,tx,x) ≤ 2ψ(x) −ψ(tx) ≤ g(x,x,x0) for all x ∈ x then x0 is a fixed point of order 2 for t if and only if limx→x0 g(t x,t x,x) g(x,x,x0) = 0. proof. (a) from the inequality g(tx,tx,x) ≤ g(tx,x0,x0) + g(x0,tx,x) ≤ g(tx,tx,x0) + g(x,x,x0) g(tx,tx,x) g(x,x,x0) ≤ g(tx,tx,x0) g(x,x,x0) + 1. therefore, limx→x0 g(t x,t x,x) g(x,x,x0) ≤ 2. (b) from inequality g(tx,tx,x) ≤ 2ψ(x) −ψ(tx) ≤ g(x,x,x0), g(x,x,tx) + g(tx,tx,t2x) + ... + g(tn−1x,tn−1x,tnx) ≤ σni=12ψ(t i−1x) −ψ(tix) = 2ψ(x) −ψ(tnx) fixed point of order 2 on g-metric space 177 and g(tn−1x,tn−1x,tnx g(x,x,x0) = g(tn−1x,tn−1x,tnx) g(tn−1x,tn−1x,tn−2x) g(tn−1x,tn−1x,tn−2x) g(tn−2x,tn−2x,tn−3x) ... = ... g(t2x,t2x,x0) g(tx,tx,x0) g(tx,tx,x0) g(x,x,x0 , since limx→x0 g(t n−1x,t n−1x,t nx) g(x,x,x0) = limx→x0 g(t x,t x,x) g(x,x,x0) and limx→x0 g(t n−kx,t n−kx,t nx) g(x,x,x0) = 1 which k = 1, 2, ...n − 1, so limx→x0 g(t n−1x,t n−1x,t nx) g(x,x,x0) = limx→x0 g(t x,t x,x) g(x,x,x0) . from inequality g(tx,tx,x) ≤ 2ψ(x) − ψ(tx) ≤ g(x,x,x0). it is clear that ψ(tnx) is strict decreasing. n g(tx,tx,x) g(x,x,x0) ≤ lim x→x0 2ψ(x) −ψ(tnx) g(x,x,x0) ≤ lim x→x0 2ψ(x) −ψ(tnx) 2ψ(x) −ψ(tx) ≤ lim x→x0 2ψ(x) −ψ(tnx) 2ψ(x) −ψ(tnx) = 1. hence, limx→x0 g(t x,t x,x) g(x,x,x0) = 1 n . since n is arbitrary positive integer, limx→x0 g(t x,t x,x) g(x,x,x0) = 0. � references 1. m. edelstein, an extension of banach;s contraction principle, proc. amer. math. soc. 12(1961), 7-10. 2. t. h. kim, e. s. kim and s. s. shin, minimization thorems relating to fixed point theorems on complete metric spaces, math. japon. 45(1997), no. 1, 97-102. 3. z. liu, l. wang, sh. kang, y. s. kim, on nonunique fixed point theorems, applied mathematics e-notes, 8(2008), 231-237. 4. z. mustafa and b. sims, a new approach to generalized metric spaces, j. nonlinear and convex anal. 7 (2006), no. 2, 289–297. 5. c. k. zhong, on ekeland’s variational principle and a minimax theorem, j. math. anal. appl. 205(1997), no. 1, 239-250. department of applied mathematics, sagar institute of science, technology & research, ratibad, bhopal 462043 int. j. anal. appl. (2022), 20:53 mathematical model of hepatitis b virus with effect of vaccination and treatments saif h. elkhadir1,∗, ali e. m. saeed2, abdelfatah abasher3 1mathematics department, omdurman islamic university, sudan 2mathematics department, alzaiem alazhari university, sudan 3mathematics department, jazan university, saudi arabia ∗corresponding author: safemath@gmail.com abstract. in this paper, a mathematical model of hepatitis b virus with vaccination and treatments is studied, stability analysis discussed and the disease-free equilibrium and endemic equilibrium points obtained, the basic reproductive number r0 determined and became the threshold for equilibrium points stability. the study showed when r0 < 1 the disease-free equilibrium point was stable, whereas r0 > 1 the virus is endemic and the endemic equilibrium point is stable. the sensitivity analysis for the parameters that could reduce the spread of hepatitis b virus is studied. finally the numerical simulation are established by using sagemath software package to show the effect of vaccination and treatments. we found that vaccination and also treatments give an effect on value of r0. increasing the value of the vaccine in the immunized compartment or in the suspected compartment may decrease the value of r0 which mean reduce the spread of the disease. 1. introduction infectious diseases are disorders caused by small organisms such as bacteria, viruses, fungi or parasites. some infectious diseases transmitted from infected person to another by direct physical contact, airborne droplets, water or food, disease vectors, or mother to newborn. hepatitis b is an infectious disease of the liver caused by the hepatitis b virus (hbv). in some people hepatitis b can become chronic leading to liver failure, liver cancer or cirrhosis. most healthy adults who are newly infected will recover without any problems. but babies and young children may not be able to successfully get rid of the virus. 90% of healthy adults will get rid of the virus and recover without any problems; 10% will develop chronic hepatitis b. young children – up to 50% of young children between 1 and 5 years received: aug. 31, 2022. 2010 mathematics subject classification. 92c60. key words and phrases. epidemic disease; hbv model; hepatitis b virus. https://doi.org/10.28924/2291-8639-20-2022-53 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-53 2 int. j. anal. appl. (2022), 20:53 who are infected will develop a chronic hepatitis b infection. infants – 90% will become chronically infected; only 10% will be able to get rid of the virus [1]. mathematical model of hepatitis b virus based on mseir model, the population divided into five compartments immunized(m), susceptible (s), exposed (e), infected (i) and recovered (r). there are many previous studies on the spread of hepatitis b, zhang (2015) study the application and optimal control for an hbv model with vaccination and treatments [2], aniji (2019) developed the model by adding the carrier compartment [3], beay (2017) developed the model by adding migration compartments [4], wiraningsih (2015) discussed the model with vaccination effect [5], cao (2015) studied the global stability of an epidemic model with carrier state in heterogeneous network [6]. in this paper we use seir model and apply it to hepatitis b virus. we extend the model by introducing the immunized class. the paper outline as follows, in section 2, we present the description and assumption of the model, in section 3 we discus the stability analysis for disease free equilibrium and endemic equilibrium, in section 4 we show the relation between the basic reproductive number and the model parameters, in section 5 the explore the numerical simulation to show the dynamical behaviour of our results, we finished the paper with a conclusion in section 6. 2. description of the model the population of our model is divided into five compartments: immunized (m), susceptible (s), exposed (e), infected (i) and recovered (r). the interaction between the five compartments is shown in the following diagram: figure 1. the transmission of hepatitis b virus model the basic assumptions of the model: • birth rate equal to the death rate. • the population size is constant n. • the rate of transmission from exposed to infective is η. • the rate of transmission from infected to recovered compartment is ψ. • the rate of transmission from susceptible to infected class is ν. • the individual who take vaccine transfer from susceptible class to recovered class with rate δ. int. j. anal. appl. (2022), 20:53 3 • the individual who lost immune transfer from recovered class to susceptible class with rate γ the differential equations that govern the model are dm dt = αb −φm −µm ds dt =(1−α)b +φm +γr− (νi +µ+δ)s de dt = νsi −ηe −µe (1) di dt = ηe −ψi −βi −µi dr dt = δs +ψi −γr−µr where, m +s +e + i +r = n (total population). the description of the parameters are shown in the following table parameter description α immunized newborns β hb induced mortality γ rate of transmission from r to s µ rate of natural mortality ν rate of transmission from s to e η rate of transmission from e to i δ rate of transmission from s to r ψ rate of transmission from i to r φ rate expiration of vaccine efficacy by assumption that the total population is constant, (the birth rate = the death rate) then the model can be expressed in a simpler term in the form dm dt = αµ−φm −µm ds dt =(1−α)µ+φm +γr− (νi +µ+δ)s de dt = νsi −ηe −µe (2) di dt = ηe −ψi −βi −µi dr dt = δs +ψi −γr−µr 4 int. j. anal. appl. (2022), 20:53 3. stability analysis 3.1. equilibrium solutions: let e(m,s,e,i,r) be the equilibrium point of the system (2), at the equilibrium state, we have dm dt = ds dt = de dt = di dt = dr dt =0 that gives, αµ−φm −µm =0 (1−α)µ+φm +γr− (νi +µ+δ)s =0 νsi −ηe −µe =0 (3) ηe −ψi −βi −µi =0 δs +ψi −γr−µr =0 3.2. disease-free equilibrium dfe:. let e0(m0,s0,e0, i0,r0) be a solution of the system (3), suppose both i and e must be zero, (i0 = 0,e0 = 0). by substitute (i = 0,e = 0) into the system (3) we get αµ− (φ+µ)m =0 (1−α)µ+φm +γr− (ν +δ)s =0 (4) δs +−(γ +µ)r =0 by solving these equations we get m0 = αµ φ+µ , r0 = δ(φ+(1−α)µ) φγ +δφ+µφ+µγ +µδ +µ2 , and s0 = φγ +µγ +αµγ +φµ+µ2 −αµ2 φγ +δφ+µφ+µγ +µδ +µ2 therefore the disease free equilibrium of the model is e0 ( αµ φ+µ , φγ +µγ +αµγ +φµ+µ2 −αµ2 φγ +δφ+µφ+µγ +µδ +µ2 ,0,0, δ(φ+(1−α)µ) φγ +δφ+µφ+µγ +µδ +µ2 ) int. j. anal. appl. (2022), 20:53 5 3.3. basic reproductive number (r0) for dfe:. according to calculation method of r0 [12], let x = [e,i]t and set f = ( νsi 0 ) , v = ( (η +µ)e −ηe +(ψ +µ+β)i ) then we can write dx dt =f −v, compute f and v where, f =   ∂f1 ∂e ∂f1 ∂i ∂f2 ∂e ∂f2 ∂i   e0 = ( 0 νs0 0 0 ) , v =   ∂v1 ∂e ∂v1 ∂i ∂v2 ∂e ∂v2 ∂i   e0 = ( η +µ 0 −η ψ +µ+β ) , the next generation matrix k = fv −1 k = 1 (η +µ)(ψ +µ+β) ( 0 νs0 0 0 )( ψ +µ+β 0 η η +µ ) , k = ( νηs0 (η+µ)(ψ+µ+β) νs0 (ψ+µ+β) 0 0 ) the eigen values of k are λ1 =0 and λ2 = νηs0 (η +µ)(ψ +µ+β) = νη(φγ +µγ +αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +µ+β)(φγ +δφ+µφ+µγ +µδ +µ2) , hence, r0 = ρ(k)= νη(φγ +µγ +αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +µ+β)(φγ +δφ+µφ+µγ +µδ +µ2) , (5) theorem 3.1. the disease free-equilibrium point e0 of the system (2) is asymptotically stable if r0 < 1 and unstable if r0 > 1. the value of r0 measures whether the disease will spread and become endemic or will disappear from the population. when r0 < 1, the disease will disappear and the exposed and infected compartments tend to zero as time goes on. when r0 > 1, the disease will spread and become endemic. this means that each compartments will be positive valued for a long time. in another words, in the case of r0 > 1 the endemic equilibrium point e∗ exists and stable. 6 int. j. anal. appl. (2022), 20:53 3.4. stability analysis of dfe:. we examine the behavior of the model near e0, linearize the system near e0, suppose the left hand side of system (2) be f1, . . . ,fn respectively, then the jacobin matrix is given as j ( f1,f2,f3,f4,f5 m,s,e,i,r ) =  −(φ+µ) 0 0 0 0 φ −(νi +µ+δ) 0 −νs γ 0 νi −(η +µ) νs 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   the jacobian matrix of the disease-free equilibrium is given by: j(e0)=   −(φ+µ) 0 0 0 0 φ −(µ+δ) 0 −νs0 γ 0 νi −(η +µ) νs0 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   now let j(e0)= [ j0 j2 j1 j3 ] , where j0 = [ −(φ+µ) 0 φ −(µ+δ) ] ,j2 = [ 0 0 0 0 −νs0 γ ] j1 =   o 0 0 0 0 δ   and j3 =   −(η +µ) νs0 0 η −(ψ +β +µ) 0 0 ψ −(γ +µ)   the eigen values given by: |j(e0)−λ|=0 implies that, ∣∣∣∣∣ −(φ+µ)−λ 0φ −(µ+δ)−λ ∣∣∣∣∣ =0 (6) and ∣∣∣∣∣∣∣∣ −(η +µ)−λ νs0 0 η −(ψ +β +µ)−λ 0 0 ψ −(γ +µ)−λ ∣∣∣∣∣∣∣∣ =0 (7) from (6) we obtain λ1 =−φ−µ, int. j. anal. appl. (2022), 20:53 7 λ2 =−µ−δ, and from (7), λ3 =−γ −µ, (η +µ+λ)(ψ +β +µ+λ)−νηs0 =0, that yields λ2 +(η +ψ +β +2µ)λ+(η +µ)(ψ +β +µ)−νηs0 =0, or, λ2 +ω1λ+ω2 =0 (8) where, ω1 = η +ψ +β +2µ, and ω2 =(η +µ)(ψ +β +µ)−νηs0 from routh-hurwitz criterion [13], equation (8) have negative real part if and only if: ω1 > 0 and ω2 > 0, since all the parameters η,ψ,β,µ are positive then ω1 > 0, and the condition (ω2 > 0) yields (η +µ)(ψ +β +µ) > νηs0 it can be seen that all the eigenvalues have negative real parts and therefore the disease-free equilibrium is locally asymptotically stable. 3.5. stability analysis of endemic equilibrium: denote the endemic equilibrium of the system (2) is e∗ =(m∗,s∗,e∗, i∗,r∗), which can obtain by putting dm dt = ds dt = de dt = di dt = dr dt =0 or, αµ−φm −µm =0 (1−α)µ+φm +γr− (νi +µ+δ)s =0 νsi −ηe −µe =0 (9) ηe −ψi −βi −µi =0 δs +ψi −γr−µr =0 by solving these equations we obtain, m∗ = αµ (φ+µ) = σ0, e∗ = (ψ +µ+β) η i = σ1i, s∗ = (η +µ)σ1 ν = σ2, 8 int. j. anal. appl. (2022), 20:53 r∗ = ψ (γ +µ) i∗ + δσ2 (γ +µ) = i∗σ3 +σ4, and i∗ = (µ+δ)σ2 −γσ4 − (1−α)µ−φσ0 (σ3γ −νσ2) where, σ0 = αµ (φ+µ) ,σ1 = (ψ +µ+β) η ,σ2 = (η +µ)σ1 ν ,σ3 = ψ (γ +µ) and σ4 = δσ2 (γ +µ) the jacobian matrix of the system (4.2) at e∗ j(e∗)=   −(φ+µ) 0 0 0 0 φ −(νi∗ +µ+δ) 0 −νs∗ γ 0 νi∗ −(η +µ) νs∗ 0 0 0 η −(ψ +β +µ) 0 0 δ 0 ψ −(γ +µ)   now let j(e0)= [ j0 j2 j1 j3 ] , where j0 = [ −(φ+µ) 0 φ −(νi∗ +µ+δ) ] ,j2 = [ 0 0 0 0 −νs∗ γ ] j1 =   0 νi∗ 0 0 0 δ   and j3 =   −(η +µ) νs∗ 0 η −(ψ +β +µ) 0 0 ψ −(γ +µ)   the eigen values given by: |j(e∗)−λ|=0 that implies that ∣∣∣∣∣ −(φ+µ)−λ 0φ −(νi∗ +µ+δ)−λ ∣∣∣∣∣ =0 (10) and ∣∣∣∣∣∣∣∣ −(η +µ)−λ νs∗ 0 η −(ψ +β +µ)−λ 0 0 ψ −(γ +µ)−λ ∣∣∣∣∣∣∣∣ =0 (11) from (10) we obtain λ1 =−φ−µ, and λ2 =−(νi∗ +µ+δ), since i∗ is positive implies that λ1,λ2 are negative. and from (11), −(η +µ+λ)[(ψ +β +µ+λ)(γ +µ+λ)]−νs∗(γ +µ+λ)η =0 int. j. anal. appl. (2022), 20:53 9 or, (γ +µ+λ)[(η +µ+λ)(ψ +β +µ+λ)+νηs∗] = 0, that yields , λ3 =−γ −µ, and λ4,5 are given from λ2 +ω1λ+ω2 =0, where ω1 = η +2µ+ψ +β ω2 =(η +µ)(ψ +β +µ)−νηs∗ =0 thus, λ2 +ω1λ =0, λ4 =0, λ5 =−ω1 it can be seen that all the eigenvalues have negative real parts and therefore the endemic equilibrium is locally asymptotically stable. 4. sensitivity analysis of r0 in this section, we show the relation between the basic reproductive number r0 and the parameters that can reduce the spread of the disease α,δ and ψ. the value of the parameters are shown in the table 1 parameter description value references α immunized newborns 0.05-1 [10] β hbv induced mortality 0.015 [10] γ rate of transmission from r to s 0.06 [10] µ rate of natural mortality 0.0121 [10] ν rate of transmission from s to e 0.08 assume η rate of transmission from e to i 0.75 assume δ rate of transmission from s to r 0.1-1 [10] ψ rate of transmission from i to r 0.1-1 [10] φ rate expiration of vaccine efficacy 0.05 [10] table 1. parameters values of the model 10 int. j. anal. appl. (2022), 20:53 4.1. the relation between r0 and α: we can write the basic reproductive number as r0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = k1 −αk2 where, k1 = νη(φγ+µγ+φµ+µ2) (η+µ)(ψ+β+µ)(φγ+δφ+µφ+µγ+µδ+µ2) and k2 = (µγ+µ2)νη (η+µ)(ψ+β+µ)(φγ+δφ+µφ+µγ+µδ+µ2) suppose the values of the parameters are given as β = 0.015,γ = 0.07,µ = 0.0125,ν = 0.08,η = 0.75,δ = 0.02,ψ =0.03 and φ =0.05. table 2 and figure 2 show the relation between α and r0 for six values of α namely α = 0.15,α = 0.25,α =0.35,α =0.45,α =0.55, and α =0.65. parameter α r0 free/endemic 0.15 1.068428 endemic 0.25 1.046399 endemic 0.35 1.024369 endemic 0.45 1.00234 endemic 0.55 0.98031 free 0.65 0.958281 free table 2. the relation between α and r0 figure 2. the relation between α and r0 4.2. the relation between r0 and δ: we can write the basic reproductive number as r0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = c1 c2 +δc3 where, c1 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2 c2 =(η +µ)(ψ +β +µ)(φγ +µφ+µγ +µ 2) c3 =(η +µ)(ψ +β +µ)(φ+µ). suppose that the values of parameter are given as α = 0.05,β = 0.015,γ = 0.07,µ = 0.0125,ν = 0.08,η = 0.75,ψ = 0.03 and φ = 0.05. table 3 and figure 3 show the effect of immunization for adults to the value of r0. for six values of δ, namely δ = 0.01,δ = 0.05,δ = 0.1,δ = 0.15,δ = 0.2, and δ =0.25. int. j. anal. appl. (2022), 20:53 11 parameter δ r0 free/endemic 0.01 2.265647 endemic 0.05 1.581678 endemic 0.1 1.148342 endemic 0.15 0.901386 free 0.2 0.741849 free 0.25 0.630293 free table 3. the relation between δ and r0 figure 3. the relation between δ and r0 4.3. the relation between r0 and ψ: we can write the basic reproductive number as r0 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2) (η +µ)(ψ +β +µ)(φγ +δφ+µφ+µγ +µδ +µ2) = a1 (a2 +a3ψ) where, a1 = νη(φγ +µγ −αµγ +φµ+µ2 −αµ2 a2 =(η +µ)(φγ +δφ+µφ+µγ +µδ +µ 2)(β +µ) a3 =(η +µ)(φγ +δφ+µφ+µγ +µδ +µ 2) we suppose that the values of parameter are given as α =0.05,β =0.015,γ =0.07,µ =0.0125,ν = 0.08,η =0.75,δ =0.02 and φ =0.05. table 4 and figure 4 show the effect of treatments on infected individuals(ψ) to the value of r0. for six values of ψ, namely ψ =0.01,ψ =0.05,ψ =0.1,ψ =0.15,ψ =0.2, and ψ =0.25. parameter ψ r0 free/endemic 0.01 2.42524 endemic 0.05 1.173503 endemic 0.1 0.713306 free 0.15 0.512375 free 0.2 0.399765 free 0.25 0.327735 free table 4. the relation between ψ and r0 figure 4. the relation between ψ and r0 12 int. j. anal. appl. (2022), 20:53 5. results and discussion 5.1. effect of immunization of newborns (α). we suppose that the values of parameter of the model are given as β=0.015, δ=0.05, γ=0.09, µ=0.0125 φ=0.05, ν=0.08 ,η=0.55 ,ψ=0.3. there are five values of α, 0.05, 0.25, 0.45, 0.65, 0.85 and 1. the figures 5 show the effect of increasing value of α gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) the population compartments α = 0.05 (b) the population compartments α = 0.25 (c) the population compartments α = 0.45 (d) the population compartments α = 0.65 (e) the population compartments α = 0.85 (f) the population compartments α = 1 figure 5. the variation of the parameter α int. j. anal. appl. (2022), 20:53 13 5.2. effect of immunization of adults (δ). we suppose that the values of the parameters are given as α=0.1,β=0.015, γ=0.09, µ=0.0125, φ=0.07, ν=0.08 ,η=0.5 and ψ=0.3, there are five values of δ are 0.1, 0.2, 0.3, 0.4, 0.5 and 0.6. the figures 6 show the effect of increasing value of δ gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) the population compartments δ =0.1 (b) the population compartments δ =0.2 (c) the population compartments δ =0.3 (d) the population compartments δ =0.4 (e) the population compartments δ =0.5 (f) the population compartments δ =0.6 figure 6. the variation of the parameter δ 14 int. j. anal. appl. (2022), 20:53 5.3. effect of treatment on infected to the spread of hepatitis b virus (ψ). we suppose that the values of parameter of the model are given as α = 0.1,β = 0.015, ,γ = 0.09,µ = 0.0125,φ = 0.07,ν =0.08,η =0.65 and δ =0.05, there are five values of ψ, are ψ =0.1,ψ =0.2,ψ =0.3,ψ = 0.4,ψ = 0.5 and ψ = 0.6. the figure 7 show the effect of increasing value of ψ gives an increasing in immunized and recovered curves and a decreasing in infectious curve. (a) the population compartments ψ =0.1 (b) the population compartments ψ =0.2 (c) the population compartments ψ =0.3 (d) the population compartments ψ =0.4 (e) the population compartments ψ =0.5 (f) the population compartments ψ =0.6 figure 7. the variation of the parameter ψ int. j. anal. appl. (2022), 20:53 15 6. conclusion in the model of hepatitis b virus using the standard seir model has been developed by considering vaccination and treatments in compartments. this strategy aims to reduce the spread of hepatitis b virus. the model was divided into five compartments which include immunized, exposed, infected, and recovered compartments. the model has two cases free equilibrium point and endemic equilibrium point. the existence of endemic equilibrium point depends on the value of reproductive number r0. from the table 2 and table 3 we notice that increasing the value of α and δ leads to decreasing in the value of r0, which its becomes from value greater than one to value less than one, that means giving more vaccines for newborns and adults change the disease from endemic to free condition. from table 4 we notice that increasing in the value of ψ leads to decreasing in r0, which its becomes from value greater than one to value less than one, which means the treatments on infected has a big effect on changing from endemic case to free case. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] hepatitis b foundation, https://www.hepb.org, accessed 20 august 2022. [2] j. zhang, s. zhang, application and optimal control for an hbv model with vaccination and treatment, discrete dyn. nat. soc. 2018 (2018), 2076983. https://doi.org/10.1155/2018/2076983. [3] m. aniji, n. kavitha, s. balamuralitharan, analytical solution of seicr model for hepatitis b virus using hpm, aip conf. proc. 2112 (2019), 020024. https://doi.org/10.1063/1.5112209. [4] l.k. beay, kasbawati, s. toaha, effects of human and mosquito migrations on the dynamical behavior of the spread of malaria, aip conf. proc. 1825 (2017), 020006. https://doi.org/10.1063/1.4978975. [5] e.d. wiraningsih, f. agusto, l. aryati et al. stability analysis of rabies model with vaccination effect and culling in dogs, appl. math. sci. 9 (2015), 3805-3817. [6] j. cao, y. wang, a. alofi, et al. global stability of an epidemic model with carrier state in heterogeneous networks, ima j. appl. math. 80 (2014), 1025–1048. https://doi.org/10.1093/imamat/hxu040. [7] n. scott, m. hellard, e.s. mcbryde, modeling hepatitis c virus transmission among people who inject drugs: assumptions, limitations and future challenges, virulence. 7 (2015), 201–208. https://doi.org/10.1080/ 21505594.2015.1085151. [8] j.m. haussig, s. nielsen, m. gassowski, et al. a large proportion of people who inject drugs are susceptible to hepatitis b: results from a bio-behavioural study in eight german cities, int. j. infect. dis. 66 (2018), 5–13. https://doi.org/10.1016/j.ijid.2017.10.008. [9] j. wong, m. payne, s. hollenberg, a double-dose hepatitis b vaccination schedule in travelers presenting for late consultation, j. travel med. 21 (2014), 260–265. https://doi.org/10.1111/jtm.12123. [10] a.v. kamyad, r. akbari, a.a. heydari, et al. mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis b virus, comput. math. methods med. 2014 (2014), 475451. https://doi.org/10.1155/2014/475451. [11] m.a.e. osman, i.k. adu, simple mathematical model for malaria transmission, j. adv. math. comput. sci. 25 (2017), jamcs.37843. https://www.hepb.org https://doi.org/10.1155/2018/2076983 https://doi.org/10.1063/1.5112209 https://doi.org/10.1063/1.4978975 https://doi.org/10.1093/imamat/hxu040 https://doi.org/10.1080/21505594.2015.1085151 https://doi.org/10.1080/21505594.2015.1085151 https://doi.org/10.1016/j.ijid.2017.10.008 https://doi.org/10.1111/jtm.12123 16 int. j. anal. appl. (2022), 20:53 [12] p. van den driessche, j. watmough, reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, math. biosci. 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02) 00108-6. [13] d.r. merkin, introduction to the theory of stability, springer new york, 1997. https://doi.org/10.1007/ 978-1-4612-4046-4. https://doi.org/10.1016/s0025-5564(02)00108-6 https://doi.org/10.1016/s0025-5564(02)00108-6 https://doi.org/10.1007/978-1-4612-4046-4 https://doi.org/10.1007/978-1-4612-4046-4 1. introduction 2. description of the model 3. stability analysis 3.1. equilibrium solutions: 3.2. disease-free equilibrium dfe: 3.3. basic reproductive number (r0) for dfe: 3.4. stability analysis of dfe: 3.5. stability analysis of endemic equilibrium: 4. sensitivity analysis of r0 4.1. the relation between r0 and : 4.2. the relation between r0 and : 4.3. the relation between r0 and : 5. results and discussion 5.1. effect of immunization of newborns () 5.2. effect of immunization of adults () 5.3. effect of treatment on infected to the spread of hepatitis b virus () 6. conclusion references international journal of analysis and applications volume 17, number 5 (2019), 722-733 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-722 characterizations of classes of harmonic convex functions and applications imran abbas baloch1,2,∗, manuel de la sen3 and i̇mdat i̇şcan4 1abdus salam school of mathematical sciences, gc university, lahore, pakistan govt. college for boys gulberg, higher education department, punjab, pakistan 3institute of research and development of processors, university of the basque, country campus of leioa (bizkaia), 48940 leioa, spain 4department of mathematics, faculty of arts and sciences, giresun university, 28100, giresun, turkey ∗corresponding author: iabbasbaloch@gmail.com, iabbasbaloch@sms.edu.pk abstract. in this paper, we consider classes of harmonic convex functions and give their special characterizations. furthermore, we consider hermite hadamard type inequalities related to these classes to give some non-numeric estimates of well-known definite integrals. 1. introduction the classical or the usual convexity is defined as follows: a function f : ∅ 6= i ⊆ r −→ r, is said to be convex on i if inequality f (tx + (1 − t) y) ≤ tf(x) + (1 − t) f(y) holds for all x,y ∈ i and t ∈ [0, 1]. received 2019-06-23; accepted 2019-08-01; published 2019-09-02. 2010 mathematics subject classification. 26d15, 26a51, 26d10, 26a15. key words and phrases. harmonic convex functions; harmonic (p,(s,m))-convex functions; hermite hadamard inequalities; well-known definite integrals. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 722 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-722 int. j. anal. appl. 17 (5) (2019) 723 a number of papers have been written on inequalities using the classical convexity and one of the most fascinating inequalities in mathematical analysis is stated as follows f ( a + b 2 ) ≤ 1 b−a b∫ a f(x)dx ≤ f(a) + f(b) 2 , (1.1) where f : i ⊆ r −→ r be a convex mapping and a,b ∈ i with a ≤ b . both the inequalities hold in reversed direction if f is concave. the inequalities stated in (1.1) are known as hermite-hadamard inequalities. for more results on (1.1) which provide new proof, significantly extensions, generalizations, refinements, counterparts, new hermite-hadamard-type inequalities and numerous applications, we refer the interested reader [1–4, 14] and the references therein. the usual notion of convex function have been generalized in diverse manners. some of them are the so called harmonically convex function, harmonically (α,m)-convex function and p-convex function, which are stated in the definitions below. definition 1.1. [9] a function f : i ⊆ r\{0}→ r is said to be harmonically-convex function on i if f ( xy tx + (1 − t)y ) ≤ tf (y) + (1 − t) f (x) holds for all x,y ∈ i and t ∈ [0, 1]. if the inequality is reversed, then f is said to be harmonically concave. definition 1.2. [10] a function f : i ⊆ r\{0} → r is said to be harmonically (α,m)-convex function on i if f ( mxy mty + (1 − t)x ) ≤ tαf (x) + m (1 − tα) f (y) holds for all x,y ∈ i and t ∈ [0, 1], where α ∈ [0, 1] and m ∈ (0, 1]. if the inequality is reversed, then f is said to be harmonically (α,m)-concave. definition 1.3. [11, 13] let i ⊆ (0,∞) be a real interval. a function f : i → r is p-convex, where p ∈ r/{0} if f ( [txp + (1 − t)yp] 1 p ) ≤ tf (y) + (1 − t) f (x) holds for all x,y ∈ i and t ∈ [0, 1]. if the inequality is reversed, then f is said to be p-concave. 2. main results the convexity of functions and their generalized forms, play an important role in many fields such as economic science, biology, optimization. in recent years, several extensions, refinements, and generalizations have been considered for classical convexity. in [7], i. a. baloch and i̇. i̇şcan defined a new class of functions which is defined as follow: int. j. anal. appl. 17 (5) (2019) 724 definition 2.1. [6, 7] a function f : i ⊆ r\{0}→ r is said to be harmonically (s,m)-convex function on i if f ( mxy mty + (1 − t)x ) ≤ tsf (x) + m (1 − t)s f (y) holds for all x,y ∈ i and t ∈ [0, 1], where s ∈ (0, 1] and m ∈ (0, 1]. if the inequality is reversed, then f is said to be harmonically (s,m)-concave. definition 2.2. [5] a function f : i ⊆ r\{0} → r is said to be harmonically (p, (s,m))-convex function, where p ∈ r/{0}, s,m ∈ (0, 1], if f ( mxy [t(my)p + (1 − t)xp] 1 p ) ≤ tsf(x) + m(1 − t)sf(y), for all x,y ∈ i with my ∈ i and t ∈ [0, 1]. if the inequality is reversed, then f is said to be harmonically (p, (s,m))-concave. now, first of all, we give an example of harmonic convex function which is not a convex function. x y f(x) = ln(x) here, some more examples of harmonic convex functions. x y f(x) = √ x int. j. anal. appl. 17 (5) (2019) 725 x y f(x) = 1 x2 x y f(x) =   1−x x if 0 < x ≤ 1 0 if 1 < x ≤ 2 x−2 x if x > 2 lemma 2.1. let i ⊆ r/{0} be a real interval. define i−1 = {y ∈ r,y = 1 x ,x ∈ i}. a function f : i → r is harmonically convex if and only if g : i−1 → r is convex, where g is defined as g(y) = f( 1 y ). proof. let f be harmonically convex function on i and consider a function g : i−1 → r defined as g(y) = f( 1 y ). now, ∀ y1,y2 ∈ i−1, ∃ x1,x2 ∈ i such that y1 = 1x1 and y2 = 1 x2 g(ty1 + (1 − t)y2) = f ( 1 ty1 + (1 − t)y2 ) = f ( 1 t 1 x1 + (1 − t) 1 x2 ) = f ( x1x2 tx2 + (1 − t)x1 ) ≤ tf(x1) + (1 − t)f(x2) = tf ( 1 y1 ) + (1 − t)f ( 1 y1 ) + tg(y1) + (1 − t)g(y2). int. j. anal. appl. 17 (5) (2019) 726 from above, we conclude the proof. � lemma 2.2. [12] let f be a real function defined on an interval i ⊆ r. then, f is convex if and only if ∣∣∣∣∣∣∣∣∣ 1 x f(x) 1 y f(y) 1 z f(z) ∣∣∣∣∣∣∣∣∣ /∣∣∣∣∣∣∣∣∣ 1 x x2 1 y y2 1 z z2 ∣∣∣∣∣∣∣∣∣ ≥ 0 for all three distinct points x,y,z of i; equivalently, if and only if ∣∣∣∣∣∣∣∣∣ 1 x f(x) 1 y f(y) 1 z f(z) ∣∣∣∣∣∣∣∣∣ ≥ 0 (2.1) for all x < y < z in i. lemma 2.3. let i ⊆ (0,∞) and i−1 has a similar definition as given in lemma 2.1. a function f : i → r is harmonically convex if and only if h : i → r is convex, where h is defined as h(z) = zf(z) . proof. from the lemma 2.1, a function f is harmonically convex function on i if and only if g(y) = f( 1 y ) is convex on i−1 and by lemma 2.2, a function g(y) is convex if and only if ∣∣∣∣∣∣∣∣∣ 1 y1 g(y1) 1 y2 g(y2) 1 y3 g(y3) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all y1,y2,y3 ∈ i−1 such that y1 < y2 < y3. equivalently∣∣∣∣∣∣∣∣∣ 1 y1 f( 1 y1 ) 1 y2 f( 1 y2 ) 1 y3 f( 1 y3 ) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all y1,y2,y3 ∈ i−1 such that y1 < y2 < y3. equivalently ∣∣∣∣∣∣∣∣∣ x1 1 x1f(x1) x2 1 x2f(x2) x3 1 x3f(x3) ∣∣∣∣∣∣∣∣∣ ≥ 0 int. j. anal. appl. 17 (5) (2019) 727 for all x1,x2,x3 ∈ i such that x1 > x2 > x3. now, by interchanging the first and third row, first and second column, and relabeling x3 = z1, x2 = z2 and x1 = z3, we arrive at∣∣∣∣∣∣∣∣∣ 1 z1 z1f(z1) 1 z2 z2f(z2) 1 z3 z3f(z3) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all z1,z2,z3 ∈ i such that z1 < z2 < z3, which is equivalent to the convexity of the function h(z) = zf(z) on i. � proposition 2.1. [9] let i ⊆ r/{0} be a real interval and f : i → r is a function, then ; 1) if i ⊂ (0,∞) and f is convex and nondecreasing function, then f is a harmonically convex function. 2) if i ⊂ (0,∞) and f is a harmonically convex and nonincreasing function, then f is convex function. 3) if i ⊂ (−∞, 0) and f is a harmonically convex and nondecreasing function, then f is convex function. 4) if i ⊂ (−∞, 0) and f is convex and nonincreasing function, then f is a harmonically convex function. proof. the proof is immediate by following inequality 0 ≤ t(1 − t)(x−y)2 for all x,y ∈ i. equivalently, xy tx + (1 − t)y ≤ ty + (1 − t)x for all x,y ∈ (0,∞), and xy tx + (1 − t)y ≥ ty + (1 − t)x for all x,y ∈ (−∞, 0). � examples: • let f : (0,∞) → r, f(x) = x and g : (−∞, 0) → r, g(x) = x, then f is a harmonically convex function and g is a harmonically concave function. • let f : (0,∞) → r, f(x) =   x, 0 < x < 24 − 4 x , x ≥ 2 is harmonically convex function on (0,∞), since xf(x) is convex on (0,∞). • let f : (0,∞) → r, f(x) = (x− 1)2 + 1 x is a harmonically convex function on (0,∞), since xf(x) is convex on (0,∞). int. j. anal. appl. 17 (5) (2019) 728 • by the proposition 2.1, increasing convex function is harmonically convex. this means that ex is harmonically convex. the following result of the hermite-hadamard type holds for class of harmonically convex functions. theorem 2.1. [9] let f : i ⊆ r/{0} → r be harmonically convex and a,b ∈ i with a < b. if f ∈ l[a,b], then following inequalities hold f ( 2ab a + b ) ≤ (ab) b−a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 . (2.2) the above inequalities are sharp. remark 2.1. since, f(x) = ln x is harmonic convex on (0,∞), therefore by inequality (2.2) we have 2ab a + b ≤ e [ ab ba ] 1 b−a ≤ √ ab now,using lemma 2.3, we get following interesting result. theorem 2.2. let i ⊆ (0,∞) be a real interval. if function f : i → r is harmonically convex and a,b ∈ i with a < b such that f ∈ l[a,b], then the following inequalities hold ( a + b 2 ) f ( a + b 2 ) ≤ 1 b−a ∫ b a xf(x)dx ≤ af(a) + bf(b) 2 . (2.3) proof. since, f(x) is harmonically convex on i, therefore by lemma 2.3 xf(x) is convex on i. hence, by the use of inequality (1.1), we get required result. � remark 2.2. since, f(x) = ln x is harmonic convex on (0,∞), therefore by inequality (2.3) we have ( a + b 2 )a+b 2 ≤ e− a+b 2 . [ bb 2 aa 2 ] 1 b−a ≤ √ aabb theorem 2.3. [8] let i ⊆ r be a real interval. if function f : i → r is p-convex, where p ∈ r/{0} and a,b ∈ i with a < b such that f ∈ l[a,b], then following inequalities hold f ([ ap + bp 2 ]1 p ) ≤ p bp −ap ∫ b a f(x) x1−p dx ≤ f(a) + f(b) 2 . (2.4) lemma 2.4. let i ⊆ r/{0} be a real interval and i−1 has similar definition as given in lemma 2.1. a function f : i → r is harmonically p-convex if and only if g(y) = f( 1 y ) is p-convex. int. j. anal. appl. 17 (5) (2019) 729 proof. let f be a function on i and consider a function g : i−1 → r defined as g(y) = f( 1 y ), is p-convex function. now, ∀ y1,y2 ∈ i−1, ∃ x1,x2 ∈ i such that y1 = 1x1 and y2 = 1 x2 f ( x1x2 [tx p 2 + (1 − t)x p 1] 1 p ) = f ( 1 [ty p 1 + (1 − t)y p 2 ] 1 p ) = g([ty p 1 + (1 − t)y p 2 ] 1 p ) ≤ tg(y1) + (1 − t)g(y2) = tf ( 1 y1 ) + (1 − t)f ( 1 y2 ) = tf(x1) + (1 − t)f(x2) from above, we conclude the proof. � lemma 2.5. let i ⊆ r be a real interval. a function f : i → r is a p-convex function if and only if∣∣∣∣∣∣∣∣∣ 1 xp f(x) 1 yp f(y) 1 zp f(z) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all x,y,z ∈ i such that x < y < z. lemma 2.6. let i ⊆ (0,∞) be a real interval and i−1 has similar definition as given in lemma 2.1. a function f : i → r is harmonically p-convex if and only if h : i → r defined by h(z) = zpf(z) is convex. proof. from the lemma 2.4, a function f is a harmonically p-convex function on i if and only if g(y) = f( 1 y ) is p-convex and by lemma 2.5, a function g(x) is p-convex if and only if∣∣∣∣∣∣∣∣∣ 1 y p 1 g(y1) 1 y p 2 g(y2) 1 y p 3 g(y3) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all y1,y2,y3 ∈ i such that y1 < y2 < y3. equivalently ∣∣∣∣∣∣∣∣∣ 1 1 x p 1 f(x1) 1 1 x p 2 f(x2) 1 1 x p 3 f(x3) ∣∣∣∣∣∣∣∣∣ ≥ 0 for allx1,x1,x1 ∈ i such that x1 > x2 > x3. now, by the multiplying the ith row of the determinant by x p i , interchanging the first and third row, first and second column, and relabeling x3 = z1, x2 = z2 and x1 = z3, we arrive at int. j. anal. appl. 17 (5) (2019) 730 ∣∣∣∣∣∣∣∣∣ 1 z p 1 z p 1f(z1) 1 z p 2 z p 2f(z2) 1 z p 3 z p 3f(z3) ∣∣∣∣∣∣∣∣∣ ≥ 0 for all z1,z2,z3 ∈ i such that z1 < z2 < z3, which is equivalent to the convexity of the function h(z) = zpf(z). � proposition 2.2. let i ⊆ r/{0} be a real interval, p ∈ r/{0} and f : i → r is a function, then; • if p ≥ 1 and harmonically convex and nondecreasing function, then f is harmonically p-convex function. • if p ≥ 1 and harmonically p-convex and nonincreasing function, then f is harmonically convex function. • if p ≥ 1 and harmonically p-concave and nondecreasing function, then f is harmonically concave function. • if p ≥ 1 and harmonically concave and nonincreasing function, then f is harmonically p-concave function. • if p ≤ 1 and harmonically p-convex and nondecreasing function, then f is harmonically convex function. • if p ≤ 1 and harmonically convex and nonincreasing function, then f is harmonically p-convex function. • if p ≤ 1 and harmonically concave and nondecreasing function, then f is harmonically p-concave function. • if p ≤ 1 and harmonically p-concave and nonincreasing function, then f is harmonically concave function. proof. since f(x) = xp, p ∈ (−∞, 0) ∪ [1,∞), is convex function on (0,∞) and f(x) = xp, p ∈ (0, 1], is a concave function on (0,∞), therefore the proof is obvious from the following power mean inequalities [txp + (1 − t)yp] 1 p ≥ tx + (1 − t)y, p ≥ 1 , which is equivalent to xy [txp + (1 − t)yp] 1 p ≤ xy tx + (1 − t)y , p ≥ 1 , int. j. anal. appl. 17 (5) (2019) 731 and [txp + (1 − t)yp] 1 p ≤ tx + (1 − t)y, p ≤ 1 , which is equivalent to xy tx + (1 − t)y ≤ xy [txp + (1 − t)yp] 1 p , p ≥ 1, for all x,y ∈ (0,∞) and t ∈ [0, 1]. � similar to inequality given in theorem 2.1, we proved the result for harmonically p-convex functions as given below theorem 2.4. [5] let i ⊆ r/{0} be a real interval and f : i → r be harmonically p-convex, a,b ∈ i with a < b. if f ∈ l[a,b], then following inequalities hold f ( 2 1 p ab [ap + bp] 1 p ) ≤ p(ab)p bp −ap ∫ b a f(x) xp+1 dx ≤ f(a) + f(b) 2 . (2.5) also, by using lemma 2.6 and theorem 3, we have another interesting result as theorem 2.5. let i ⊆ (0,∞) be a real interval and i−1 has similar definition as given in lemma 2.1. if function f : i → r is harmonically p-convex, where p ∈ r/{0} and a,b ∈ i with a < b such that f ∈ l[a,b], then following inequalities hold( ap + bp 2 ) f ([ ap + bp 2 ]1 p ) ≤ p bp −ap ∫ b a f(x) x1−2p dx ≤ apf(a) + bpf(b) 2 . (2.6) 3. applications • we have relation between harmonic, logarithmic and arithmetic means as follow 2ab a + b ≤ ab b−a (ln b− ln a) ≤ a + b 2 (hla inequality) whose proof is a direct consequence of the inequality of theorem 2.1 for f(x) = x,∀x ∈ (0,∞). furthermore, we have following consequence of the inequality of theorem 2 for f(x) = x,∀x ∈ (0,∞).( a + b 2 )2 ≤ 1 3 (a2 + ab + b2) ≤ a2 + b2 2 . even in more general setting for p ∈ r/{0, 1}, we have 2 1 p ab [ap + bp] 1 p ≤ p(ab)p bp −ap ( b1−p −a1−p 1 −p ) ≤ a + b 2 int. j. anal. appl. 17 (5) (2019) 732 which is a direct consequence of inequality of theorem 2.4 and we have ( ap + bp 2 )p+1 p ≤ p bp −ap . b2p+1 −a2p−1 2p + 1 ≤ ap+1 + bp+1 2 . which is a direct consequence of inequality of theorem 5 for f(x) = x , ∀x ∈ (0,∞). • for a,b ∈ (0,∞) and a 6= b, we have e 2ab a+b ≤ ab b−a ∫ b a ex x2 dx ≤ ea + eb 2 , whose proof is a direct consequence of inequality of theorem 2.1 and we have ( a + b 2 ).e a+b 2 ≤ (1 −a)ea + (b− 1)eb b−a ≤ aea + beb 2 . which is a direct consequence of inequality of theorem 2 for f(x) = ex which is not only a harmonic convex function, even it is a harmonic p-convex function too. therefore, for p ≥ 1, we have e 2 1 p ab [ap+bp] 1 p ≤ p(ab)p bp −ap ∫ b a ex xp+1 dx ≤ ea + eb 2 . from above discussion, it is easy to conclude that we have a good estimate of ∫ b a ex xn dx ∀n ∈ n • since f(x) = x2ex 2 is non-decreasing convex function on (0, 1), so it is harmonic convex function. therefore, by using inequality of theorem 2.1 for a,b ∈ (0,∞) and a 6= b, we have ( 2ab a + b )2 e( 2ab a+b )2 ≤ ab b−a ∫ b a ex 2 dx ≤ a2ea 2 + b2eb 2 2 for all a,b ∈ (0,∞). • since, f(x) = sin(−x) is convex and non-decreasing function in (0, π 2 ), therefore it is harmonic convex ∀x ∈ (0, π 2 ). therefore, by using inequality of theorem 2.1 for a,b ∈ (0,∞), we have sin a + sin b 2 ≤ ab b−a ∫ b a sin x x2 dx ≤ sin( 2ab a + b ) similarly, we can estimate ∫ b a sin x xn dx and ∫ b a cos x xn dx for all n ∈ n such that a,b ∈ (0,∞). 4. conclusion the harmonic convexity of a function is the basis for many inequalities in mathematics as you may see in this research paper. furthermore, harmonic convexity provide an analytic tool to estimate several known definite integral like ∫ b a ex xn dx, ∫ b a ex 2 dx, ∫ b a sin x xn dx and ∫ b a cos x xn dx ∀n ∈ n, where a,b ∈ (0,∞). we have discussed several important aspect of harmonic p-convex functions and encourage the interested researcher to explore more interesting results for this class of functions. int. j. anal. appl. 17 (5) (2019) 733 5. funding the authors are grateful to the basque government by its support through grant it1207/19. this research article is partially supported by higher education commission of pakistan too. 6. acknowledgement the authors wish to express their heartfelt thanks to the referees for their constructive comments and helpful suggestions to improve the final version of this paper. references [1] d-y. hwang, some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables, appl. math. comput. 217 (2011), 9598-9605. [2] d-y. hwang, some inequalities for differentiable convex mapping with application to weighted midpoint formula and higher moments of random variables, appl. math. comput. 232 (2014), 68-75. [3] s.s. dragomir, hermite-hadamard’s type inequalities for convex funtions of selfadjoint operators in hilbert spaces, linear algebra appl. 436 (2012), no. 5, 1503-1515. [4] s.s. dragomir and c.e.m. pearce, selected topics on hermite-hadamard type inequalities and applications, rgmia monographs, 2000. available online at https://rgmia.org/monographs/hermite hadamard.html. [5] i. a. baloch and i. iscan, some hermite-hadamard type integral inequalities for harmonically (p,(s,m))-convex functions, j. inequal. spec. funct. 8 (2017), 65-84. [6] i. a. baloch, i. iscan and s. s. dragomir, fejer’s type inequalities for harmonically (s,m)-convex functions, int. j. anal. appl. 12 (2016), 188-197. [7] i. a. baloch, i.işcan, some ostrowski type inequalities for harmonically (s,m)-convex functoins in second sense, int. j. anal. 2015 (2015), article id 672675, 9 pages. [8] z. b. fang and r. shi, on the(p,h)-convex function and some integral inequalities, j. inequal. appl., 2014 (2014), art. id 45. [9] i̇. i̇şcan, hermite-hadamard type inequaities for harmonically functions, hacettepe j. math. stat. 43 (6) (2014), 935-942. [10] i̇.i̇şcan, hermite-hamard type inequalities for harmonically (α,m)-convex functions. arxiv:1307.5402 [math.ca], 2015. [11] i̇. i̇şcan, hermite-hadamard type inequaities for p-convex functions, int. j. anal. appl. 11 (2016), 137-145. [12] c. p. niculescu and l. e. persson (2006). convex functions and their applications, springer-verlag, new york. [13] k.s. zhang and j.p. wan, p-convex functions and their properties, pure appl. math. 23(1) (2007), 130-133. [14] m. e. gordji, m. r. delavar and m. de la sen, on φ-convex functions, j. math. inequal. 10 (2016), 173-183. 1. introduction 2. main results 3. applications 4. conclusion 5. funding 6. acknowledgement references int. j. anal. appl. (2023), 21:3 received: nov. 21, 2022. 2020 mathematics subject classification. 62l10, 62e10. key words and phrases. control charts; tbe; exponential distribution; weighted exponential distribution; adjustment bias; arl. https://doi.org/10.28924/2291-8639-21-2023-3 © 2023 the author(s) issn: 2291-8639 1 identifying process deterioration in weighted exponentially distributed time between events taswar iqbal*, muhammad zafar iqbal, muhammad kashif, ghulam farid department of mathematics & statistics, university of agriculture faisalabad, faisalabad, pakistan *corresponding author: taswar.iqbal@njc.edu.pk abstract. in observational studies, the probability of selection of sampling units is not always equal. the recorded observations are biased in this scenario. the unweighted distributions in such situations are not useful until the inclusion probability of each item is same. the theory of weighted distributions offers a unifying approach for these types of conditions because it considers the adjustment bias. failure to comply with such adjustment may lead to inappropriate results. in this article, an efficient mentoring scheme (weighted-tbe chart) for time between events (tbe) using weighted exponential distribution has been proposed based on weighted variance (wv) method. a comparison has been established between cc based on weighted and unweighted probability distributions. the performance measure arl has been calculated using monte carlo simulations. the weighted-tbe chart has provided least values of arl in the presence of unwanted process variations and proved to be more effective than the existing scheme. further the proposed control chart has been applied to time between failures data to show its practical applicability. 1. introduction in any industrial sectors, the deviation of a characteristic of interest from its target has a key importance for its acceptability in the market. this variation can be categorized into natural and assignable variations. natural variations, often called common cause variations, are an unavoidable https://doi.org/10.28924/2291-8639-21-2023-3 mailto:taswar.iqbal@njc.edu.pk 2 int. j. anal. appl. (2023), 21:3 feature of every process that cannot be avoided. on the other hand, assignable variations are due to external sources like negligence of operators, failure of components of system and power breakdown etc. such variations may cause the working process to be shifted from its target [1]. in the manufacturing sector, cc are extensively used as an assessment tool that serve a crucial role in detecting unwanted variations. the main purpose of control charting techniques is to assist quality engineers in tracking unexpected changes in any manufacturing process. tbe cc are broadly used to supervise the occurrence of defects or non-conforming articles in an industrial process. apart from manufacturing processes, any process with inter-arrival time or tbe random variable can be monitored by the tbe cc. for example, time between monitoring of regular maintained system [2], consecutive radiation pulses [3], time between medical errors [4] and time between a non-parametric cusum scheme [5]. the concept of time between events control chart was first time presented by [6] and [7]. the tbe cc that is based on the inter-arrival of nonconforming articles follow the exponential random variables. due to this, the tbe cc are commonly known as exponential control charts [8]. the exponential distribution has been considered the most suitable model to access the lifespan of an element or a product when the failure of an element is possible at any time regardless the age of an item. in designing of any control chart for detecting the shift in a process, quality control researchers may be deceived by the assumption of normality. because the data gathered in the subgroups may be extremely skewed standard cc do not yield useful findings, hence alternative approaches should be used [9]. this issue is also discussed by [10] and [11]. to model the tbe, the exponential distribution is generally used and assumed to be the better model for skewed data sets. it is right to say that scientists are often unable to pick sampling units with the same probability in observational studies. for these issues, a unifying answer can be found in the theory of weighted distributions. adjustment bias is considered in weighted distributions. if researchers don't make this adjustment, the results are misleading. the idea of weighted distribution was first time presented by [12]. rao, on the other hand, explored weighted distributions in a unified manner. it was indicated by [13], we cannot argue that the recorded numbers are a random sample from the true distribution because of a variety of factors. for example, when certain occurrences are unobservable, the original distribution is damaged or 3 int. j. anal. appl. (2023), 21:3 uneven probability sampling is used. during the past few decades, different tbe cc using exponential distribution have been proposed. the theme of this research is to utilize weighted probability distribution of quality characteristics instead of using usual probability distribution that remained still unaddressed. if the quality characteristic of any product follows an asymmetric distribution, shewhart type cc might be misleading. even so, if we use shewhart cc the chances of type-i error increases due to increase in skewness because of the variability in the population. the most common ways to dealing with such skewed scenarios are heuristic cc, transformation and increasing the sample size. three methods are used to create heuristic cc. [14] proposed the semi variance approximation. [15] formed the weighted variance methodology based on semi variance approximation. this technique depends on standard deviation of sample ranges and means, they derived the skewed control limits for such distributions. a simple and useful method of constructing cc by using weighted variance method was established by [16]. skewness leads to biased arl, especially for shewhart control chart [17] and [18]. to overcome the problem, weighted variance method has been utilized to develop the control limits for weighted-tbe chart for weighted exponential distribution. 2. material and methods the design structure of the weighted-tbe chart is presented in this section. the weighted variance method has been adopted to address the skewness of weighted exponential distribution 2.1 control charts based on unweighted distribution assume that the variable of interest such as the tbe designated by x, follows an exponential distribution having scale parameter 𝜃. then the pdf of the distribution is by. 𝑓(𝑥) = 1 𝜃 𝑒 − 𝑥 𝜃, 𝑥 > 0, where 𝜃 represents the mean and standard deviation. according to [19] the transformed random variable 𝑋∗ = 𝑋 1 𝛽 follows the weibull distribution with two parameters, shape (𝛽) and scale (𝜃 1 𝛽) respectively. , the weibull distribution follows the approximately normal distribution when we have 𝛽 = 3.6 [20]. the cc for the exponential distribution 4 int. j. anal. appl. (2023), 21:3 were suggested by [9] and [21] by applying the same transformation. santiago and smith [9], suggested the following control limits of shewhart-type control chart. 𝐿𝐶𝐿 = 𝜃 1 3.6 [γ (1 + 1 3.6 )] − 𝑘√γ (1 + 2 3.6 ) − γ2 (1 + 1 3.6 ), 2.1 𝑈𝐶𝐿 = 𝜃 1 3.6 [γ (1 + 1 3.6 )] + 𝑘√γ (1 + 2 3.6 ) − γ2 (1 + 1 3.6 ) . 2.2 2.2 proposed weighted-tbe chart based on wv method. the weighted exponential distribution is a generalized form of exponential distribution. the probability density function of weighted exponential distribution with parameter 𝜃 > 0 is 𝑓(𝑥) = 1 𝜃2 𝑥𝑒 − 𝑥 𝜃 , 𝑡 ≥ 0. 2.3 the corresponding cdf is 𝐹(𝑋) = 1 − 𝑒 − 𝑥 𝜃[𝑥 + 𝜃] 𝜃 . 2.4 the mean and the variance of weighted exponential distribution are 𝐸(𝑋) = 2𝜃, 𝑉𝑎𝑟(𝑋) = 2𝜃2. the weighted variance method adjusts the control limits for skewed distribution according to the skewness of the underlying population with no assumptions. the concept behind the weighted variance method is to split the skewed distribution into two segments at its average, then use each segment to generate a new symmetric distribution. these symmetric distributions are used to set up the control chart limits in wv method. if we know the parameters of the process, then the control limits of �̅� control chart are given by 𝑈𝐶𝐿 = 𝜇 + 3 𝜎 √𝑛 √2𝑃𝑥 , 𝐿𝐶𝐿 = 𝜇 − 3 𝜎 √𝑛 √2(1 − 𝑃𝑥 )), where “𝑃𝑥" represents the probability of a random variable x will be less than or equal to its mean. it is important to note that the weighted variance �̅� chart reduces to the standard �̅� chart when 𝑃𝑥 = 0.5. here, n is the sample size while 𝜎 represents the standard deviation of x. 5 int. j. anal. appl. (2023), 21:3 the theoretical control limits of weighted exponential distribution by using its mean and standard deviation are given as 𝑈𝐶𝐿 = 2𝜃0 (1 + 𝑘 √2 × √2𝑃𝑥 ), 2.5 𝐶𝐿 = 2𝜃0, 2.6 𝐿𝐶𝐿 = 2𝜃0 (1 − 𝑘 √2 × √2(1 − 𝑃𝑥 )), 2.7 where 𝑘 specifies the width of the control limits, 𝜃0is the in-control value of scale parameter 𝜃. for out of control (ooc) process, the parameter shifts to another value 𝜃1 = 𝛿𝜃0. the control limits lcl, ucl and cl are the parameters of the proposed weighted-tbe chart. the charting statistic from the weighted exponential distribution will be plotted against the control limits given eq. (7-9). in case of natural variations if the sample points of charting statistic falling inside the ucl and lcl with random behavior, that the process is said to be a “stable process” otherwise it is known as “shifted process”. the arl and δarl performance measures used in proposed study. 3. results and discussion 3.1 simulation study based on r language 4.1.0 [22], the simulation study has been performed and necessary measures have been calculated given in tables (1-4). the in-control average run length (arl0) has been fixed at 200,300 and 370. the corresponding probabilities of first oocpoint are 0.005,0.0033 and 0.0027, respectively. the step-by-step algorithm to perform the simulations of proposed weighted-tbe chart is given by: 3.2 algorithm the following steps are used to carry out the simulation study. step 1: control limits of weighted-tbe chart are derived by utilizing known parameter of weighted exponential distribution. step 2: 30000 samples of size n = 1 is selected from weighted exponential distribution with pdf given in equation 3.1. step 3: each sample is tested against pre-determined control limits until it exceeds them. 6 int. j. anal. appl. (2023), 21:3 step 4: if a sample falls outside the control limits, the sample number is recorded as the value of random variable rl and loop goes to step 2 again. step 5: the above procedure is repeated 50000 times and the random variable rl contains 50000 values. step 6: on the last step mean, median, standard deviation, minimum and maximum of rl are computed. the percentiles (25th, 75th and 99th) of rl and percent decrease in arl are computed as well. 3.3 performance assessment of proposed weighted-tbe chart to determine the behavior of arl for the proposed chart in more detail, it is suggested to use some other beneficial measures of rl like standard deviation of run length (sdrl) median of run length (mrl), minimum run length (minrl), maximum run length (maxrl) and some percentiles of rl along with rl curves. all measures for recommended chart have been evaluated by using monte carlo simulations. the resulting measures for some selecting cases are presented in table-1. further from tables 2–4, the following trend has been observed in 𝐴𝑅𝐿𝑠 of the proposed weighted-tbe chart: 1. a decreasing trend has been observed in ooc value of average run length (arl1) as the 𝛿 increases. this indicates that as the change in mean is large the shifted process can be identified more quickly. 2. a faster decline in trend has been noted in arl1 as arl0 is set at higher value. 3. the difference between the decreasing trend in 𝐴𝑅𝐿1 is large when 𝛿 is set as a lower value. for example, arl1 decreases to 48.07% when arl0 = 200 and 𝛿 = 1.1, on the same value of shift parameter the arl1 decreases to 53.83% when we set arl0 = 370, but as the value of shift parameter increases the decreasing trend get smaller. for example, when the value of shift parameter 𝛿 = 2.50 arl1 decreases to 97.71% when 𝐴𝑅𝐿𝑜 = 200, on the same value of shift parameter the arl1 decreases to 98.26% and 98.51% when we set arl0 = 300 and 370 respectively. 7 int. j. anal. appl. (2023), 21:3 table 1 rl profile of weighted-tbe chart for shifted process (𝐴𝑅𝐿0=370) k=3.742 theta=1 arl0=370 shift arl mrl sdrl min rl max rl p25 p75 p99 1 370.56 254 371.16 1 2994 104 517 1677 1.1 170.84 120 168.12 1 1354 51 235.25 777.01 1.2 95 66 95.75 1 1062 29 130 448.01 1.3 60.23 41 59.86 1 592 18 83 270.01 1.4 40.77 28 40.68 1 381 12 55 191 1.5 29.75 21 29.14 1 366 9 41 133 1.6 22.55 16 22.35 1 221 7 31 102 1.7 17.93 12 17.35 1 165 6 25 80.01 1.8 14.47 10 13.79 1 134 5 20 64 1.9 11.81 8 11.31 1 119 4 16 53 2 10.1 7 9.54 1 83 3 14 45 2.5 5.53 4 5.04 1 51 2 7 24 3 3.7 3 3.17 1 27 1 5 15 the efficiency of any control chart is associated with its detection ability for the shifted process by holding in-control arl at the fixed point. lower values of 𝐴𝑅𝐿1 correspond to faster indication about the shifted process. so, any control chart with the smaller values of arl1 is considered superior in the detection of shift in the process. 3.3 comparative study for a fixed value of arl0, it is desired that the arl1 values for a control chart are as small as possible, so that when the process deviates from the targeted value, the suggested control chart spot that as quickly as possible. a comprehensive comparison of the proposed charting method and existing chart with the help of their arl values is shown in this section. for the stated value of 𝐴𝑅𝐿𝑜 the width of the control limit (𝑘) for the weighted-tbe chart will be computed, and the arl1 associated with various process shift (𝛿) values will be retrieved. table 2 given below represents different values of arl when the width of control limit k= 3.391 and arl0 = 200 ( risk of type − i error 0.005) for various values of the deviated process. table-3.5 and table-3.6 represents the situation of arl0 = 300 and 370 respectively. in tables 24, we have included the 𝐴𝑅𝐿𝑠 for the comparative t-chart. 8 int. j. anal. appl. (2023), 21:3 3.3.1 advantages of the weighted-tbe chart over the t-chart the advantages of the weighted-tbe chart over the t-chart [9] have been discussed in this section. in the existing chart, the power transformation 𝑋1/4 has been used and the control limits have been established. the results of simulation study of proposed weighted-tbe chart and the existing tchart have been compared at arl0 = 200,300 and 370 on the various values of shift parameter 𝛿. the arl1 values of t-chart proposed by [9] are reported in the tables 2–4 for various values of arl0. the tables indicate that the weighted-tbe chart provides smaller values of arl1for different shift sizes. it has been indicated that weighted-tbe chart is more effective than t-chart to identify the scale shift. it can be observed from table-2 that on the average t-chart noticed δ = 1.1 at 155th sample, whereas the proposed chart spotted it at 102nd sample, which shows about 53 samples fast detection by weighted-tbe chart. as long as δarl is concerned, at δ = 1.1, arl has decreased by 22.48% and 48.70% for t-chart and weighted-tbe chart respectively. similarly, it can be seen from table-4 that on average δ = 1.1 has been detected by existing t-chart on the 278th sample, whereas it has been detected by the weighted-tbe chart at 171st sample, which indicates about 93 samples earlier detection by weighted-tbe chart. at δ = 1.1, arl has decreased by 24.99% and 53.83% for t-chart and weighted-tbe chart, respectively. table 2 the arl when arl0 = 200 (k = 3.391) shift (δ) t-chart by santiago and smith proposed 1.00 200.00 200.00 1.10 155.05 (22.48) 102.61(48.70) 1.20 119.00 (40.50) 61.98 (69.01) 1.30 91.88 (54.06) 41.66 (79.17) 1.40 71.95 (64.03) 28.87 (85.57) 1.50 57.33 (71.34) 21.66 (89.17) 1.60 46.52 (76.74) 16.87 (91.57) 1.70 38.42 (80.79) 13.55 (93.23) 1.80 32.24 (83.88) 11.28 (94.36) 1.90 27.45 (86.28) 9.46 (95.27) 2.00 23.68 (88.16) 7.95 (96.03) 2.50 13.22 (93.39) 4.58 (97.71) 3.00 8.80 (95.6) 3.3 (98.35) 9 int. j. anal. appl. (2023), 21:3 table 3 the arl when arl0 = 300 (k = 3.629) shift (δ) t-chart by santiago and smith proposed 1.00 300.00 298.92 1.10 227.57 (24.14) 143.48 (52.17) 1.20 170.82 (43.06) 81.47 (72.84) 1.30 129.11 (56.96) 52.17 (82.61) 1.40 99.14 (66.95) 36.64 (87.79) 1.50 77.62 (74.13) 26.41 (91.20) 1.60 61.99 (79.34) 20.1 (93.30) 1.70 50.45 (83.18) 15.92 (94.69) 1.80 41.79 (86.07) 12.93 (95.69) 1.90 35.17 (88.28) 10.96 (96.35) 2.00 30.02 (89.99) 9.4 (96.87) 2.50 16.07 (94.64) 5.23 (98.26) 3.00 10.39 (96.54) 3.58 (98.81) table 4 the arl when arl0 = 370 (k = 3.742) shift (δ) t-chart by santiago and smith proposed 1.00 370.01 370.56 1.10 277.52 (24.99) 170.84 (53.83) 1.20 205.93 (44.34) 95.00 (74.32) 1.30 153.94 (58.39) 60.23 (83.72) 1.40 117.02 (68.37) 40.77 (88.98) 1.50 90.78 (75.46) 29.75 (91.96) 1.60 71.91 (80.56) 22.55 (93.91) 1.70 58.10 (84.30) 17.93 (95.15) 1.80 47.80 (87.08) 14.47 (96.09) 1.90 39.99 (89.19) 11.81 (96.81) 2.00 33.95 (90.82) 10.10 (97.27) 2.50 17.78 (95.19) 5.53 (98.51) 3.00 11.33 (96.94) 3.700 (99.00) 10 int. j. anal. appl. (2023), 21:3 3.4 performance comparison using arl graphs. the performance of the weighted-tbe chart is shown in figure 1-3 for arl0 = 200, arl0=300 and arl0=370. from figure 1-3, it has been observed that each arl curve of weighted-tbe chart is lower than the existing control chart. that leads to the conclusion that the suggested weightedtbe chart identifies the scale shift more rapidly than the existing technique. figure 1: arl curves of proposed and santiago and smith (2013) charts at 𝐴𝑅𝐿0 = 200 figure 2: arl curves of proposed and santiago and smith (2013) charts at 𝐴𝑅𝐿0 = 300 figure 3: arl curves of proposed and santiago and smith (2013) charts at 𝐴𝑅𝐿0 = 370 it can be comprehended from figure 1-3 and table 1-3 that the weighted-tbe chart outperforms the chart proposed by santiago and smith at all shifts levels. 11 int. j. anal. appl. (2023), 21:3 3.5 applications in this section, the practical application of weighted-tbe chart has been presented to the given data sets for the purpose of comparison with the existing control chart to identify process shift. in section 3.5.1, existing and weighted-tbe chart has been applied to the data set representing the time between failures for 30 repairable items for showing the performance of the weighted-tbe chart over the control chart based on unweighted distribution. in section 3.5.2, example of simulated data set has been presented to evaluate the performance of the proposed control chart. 3.5.1 data application in this section, we intend to provide an application example for illustration purposes. weighted exponential distribution has wide range of applications in the field of reliability engineering. the proposed weighted-tbe chart has been applied in this section to a real life data set showing the time between failures for 30 repairable items taken from [23]. the estimation of parameter has been performed in r 4.1.0 software [22]. for stable process, the sample size 𝑛 = 30 has generated the estimated value of scale parameter is 𝜃𝑜 = 0.771. to evaluate the detection ability of proposed and existing charts, the shift of size 𝛿 = 1.5 has been introduced in the stable value of scale parameter 𝜃𝑜 = 0.771 using 𝜃1 = 𝛿𝜃𝑜. the values of charting statistic have been plotted on control limits and the comparison has been made between proposed and existing charts. the process shift has not been noticed at any sample point by the existing control chart based on unweighted exponential distribution. whereas, the proposed weighted-tbe chart has detected the same shift at first sample after its occurrence. this reveals that using the weighted distribution instead of unweighted distribution may enhance the detection probability of control charts when tbe data is required to monitor. moreover, the production lot may be saved from producing the faulty items caused by unwanted variations. figure 4: weighted-tbe chart for time between failures of repairable items 12 int. j. anal. appl. (2023), 21:3 figure 5: santiago and smith control chart for time between failures of repairable items 3.5.2 simulated data example in this section simulated data have been used to show the practical application of weighted-tbe chart and the chart based on usual unweighted exponential distribution with scale parameter 𝜃 = 1. in this simulated data set, 20 values have been taken from both the weighted and unweighted exponential distribution with in-control process. twenty (20) more observations taken with shifted parameter using shift size 𝛿 = 2. in figures (6-7), blue vertical line divides the shifted and stable processes and the comparison between the proposed and existing charts has been established. the calculated values of the parameters are as follows: 𝐶𝐿 = 2, 𝐿𝐶𝐿 = −2.76867, 𝑈𝐶𝐿 = 7.76802 we have considered the existing control chart of santiago & smith (2013) the resulting values of parameters are ucl = 1.84332, lcl = -0.04111, and cl = 0.90111 shown in figure 7. it is noticeable that from figures 6-7 the unweighted-based chart did not detect the shift at any point from 20th to 40th sample. the weighted-tbe chart outperform the shift overall on 7 out of shifted 20 sample points at 24th, 25th, 27th, 28th, 30th, 35th and 38th samples. from these figures, it can be noted that the existing chart declared the process as in control. so, the chart did not detect the shift in the process. the weighted-tbe chart can be used to identify the unwanted variations. it can be helpful in preventing the lot from more defective items and to keep the process in control by taking essential measures. the above example clearly showed that weighted-tbe chart is more effective in identifying unwanted variations. it is effective in the process of tbe such as field of food industry, reliability engineering, medicine, and others. the utilization of weighted variance method to proposed chart 13 int. j. anal. appl. (2023), 21:3 leads to minimize the effect of skewness on arl. therefore, shewhart type structure of proposed control chart is workable in monitoring industrial processes related to tbe. figure 6: proposed weighted-tbe chart applied to simulated data figure 7: santiago and smith control chart applied to simulated data. 4. summary and future works in the field of quality control, one of the most challenging tasks is to identify the unnatural variations occurred in manufacturing processes. from the last many years number of studies have been conducted on monitoring tbe. many researchers have used different skewed distributions like exponential [24] gamma [25] weibull [26] to model the tbe data. traditional and existing studies used the unweighted distributions without considering the fact that weather or not the sampling units of production process have equal probability of selection. often the quality engineers are not able to choose sampling units with the same probability in observational studies. using unweighted distributions instead of weighted distributions may destroy the originality of the data and lead the process engineers in trouble to identifying process shifts due to misrepresentative results. in such conditions, the observations reported by the process are biased. 14 int. j. anal. appl. (2023), 21:3 such observations can only be modeled by the suitable choice of weighted distributions where adjustment bias is considered. in this article, a new weighted-tbe chart has been presented. extensive simulations study has been conducted on the different values of control chart parameters. the proposed control charting technique has been proved to be superior to the existing control chart as it provides the smaller values of arl. the proposed control charting scheme is recommended to extend to other probability distributions, such as weighted-weibull, weighted-gamma and weighted-erlang distribution etc. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.c. montgomery, introduction to statistical quality control, john wiley & sons, 2020. [2] m.b.c. khoo, m. xie, a study of time-between-events control chart for the monitoring of regularly maintained systems, qual. reliab. eng. int. 25 (2009), 805–819. https://doi.org/10.1002/qre.977. [3] p. luo, t.a. devol, j.l. sharp, cusum analyses of time-interval data for online radiation monitoring, health phys. 102 (2012), 637–645. https://doi.org/10.1097/hp.0b013e3182430106. [4] e. doğu, monitoring time between medical errors to improve health-care quality, int. j. qual. res. 6 (2012), 151-157. [5] z. he, y. gao, l. qu, z. wang, a nonparametric cusum scheme for monitoring multivariate timebetween-events-and-amplitude data with application to automobile painting, int. j. product. res. 60 (2021), 5432–5449. https://doi.org/10.1080/00207543.2021.1959664. [6] s. vardeman, d. ray, average run lengths for cusum schemes when observations are exponentially distributed, technometrics, 27 (1985), 145-150. [7] j.m. lucas, counted data cusum's, technometrics, 27 (1985), 129-144. [8] f.f. gan, designs of oneand two-sided exponential ewma charts, j. quality technol. 30 (1998), 55–69. https://doi.org/10.1080/00224065.1998.11979819. [9] e. santiago, j. smith, control charts based on the exponential distribution: adapting runs rules for the t chart, qual. eng. 25 (2013), 85–96. https://doi.org/10.1080/08982112.2012.740646. [10] e.g. schilling, p.r. nelson, the effect of non-normality on the control limits of x̄ charts, j. qual. technol. 8 (1976), 183–188. https://doi.org/10.1080/00224065.1976.11980743. https://doi.org/10.1002/qre.977 https://doi.org/10.1097/hp.0b013e3182430106 https://doi.org/10.1080/00207543.2021.1959664 https://doi.org/10.1080/00224065.1998.11979819 https://doi.org/10.1080/08982112.2012.740646 https://doi.org/10.1080/00224065.1976.11980743 15 int. j. anal. appl. (2023), 21:3 [11] z.g.b. stoumbos, m.r. reynolds jr., robustness to non-normality and autocorrelation of individuals control charts, j. stat. comput. simul. 66 (2000), 145–187. https://doi.org/10.1080/00949650008812019. [12] r.a. fisher, the effect of methods of ascertainment upon the estimation of frequencies, ann. eugenics. 6 (1934), 13–25. https://doi.org/10.1111/j.1469-1809.1934.tb02105.x. [13] c.r. rao, on discrete distributions arising out of methods of ascertainment, sankhyā: indian j. stat. ser. a, 27 (1965), 311-324. [14] f. choobineh, d. branting, a simple approximation for semivariance, eur. j. oper. res. 27 (1986), 364–370. https://doi.org/10.1016/0377-2217(86)90332-2. [15] f. choobineh, j.l. ballard, control-limits of qc charts for skewed distributions using weightedvariance, ieee trans. rel. r-36 (1987), 473–477. https://doi.org/10.1109/tr.1987.5222442. [16] d.s. bai, i.s. choi, x̄ and r control charts for skewed populations, j. qual. technol. 27 (1995), 120– 131. https://doi.org/10.1080/00224065.1995.11979575. [17] d. karagöz, c. hamurkaroğlu, control charts for skewed distributions, adv. meth. stat. 9 (2012), 95106. https://doi.org/10.51936/ghaa8860. [18] m. riaz, q.u.a. khaliq, s. gul, mixed tukey ewma-cusum control chart and its applications, qual. technol. quant. manage. 14 (2017), 378–411. https://doi.org/10.1080/16843703.2017.1304034. [19] n.l. johnson, continuous univariate distributions, houghton mifflin, 1970. [20] l.s. nelson, a control chart for parts-per-million nonconforming items, j. qual. technol. 26 (1994), 239–240. https://doi.org/10.1080/00224065.1994.11979529. [21] m. aslam, m. azam, c. h. jun, a new control chart for exponential distributed life using ewma, trans. inst. measure. control. 37 (2014), 205–210. https://doi.org/10.1177/0142331214537293. [22] r core team, r: a language and environment for statistical computing, version 4.0.3. r foundation for statistical computing, vienna, austria. 2020. url: https://www.r-project.org. [23] d.p. murthy, m. xie, r. jiang, weibull models, vol. 505, john wiley & sons, 2004. [24] m. aslam, m. azam, n. khan, c.h. jun, a control chart for an exponential distribution using multiple dependent state sampling, qual. quant. 49 (2014), 455–462. https://doi.org/10.1007/s11135-0140002-2. [25] v. alevizakos, c. koukouvinos, monitoring reliability for a gamma distribution with a double progressive mean control chart, qual. reliab. eng. int. 37 (2020), 199–218. https://doi.org/10.1002/qre.2730. [26] n. khan, m. aslam, s.m.m. raza, c.-h. jun, a new variable control chart under failure-censored reliability tests for weibull distribution, qual reliab engng int. 35 (2018), 572–581. https://doi.org/10.1002/qre.2422. https://doi.org/10.1080/00949650008812019 https://doi.org/10.1111/j.1469-1809.1934.tb02105.x https://doi.org/10.1016/0377-2217(86)90332-2 https://doi.org/10.1109/tr.1987.5222442 https://doi.org/10.1080/00224065.1995.11979575 https://doi.org/10.51936/ghaa8860 https://doi.org/10.1080/16843703.2017.1304034 https://doi.org/10.1080/00224065.1994.11979529 https://doi.org/10.1177/0142331214537293 https://www.r-project.org/ https://doi.org/10.1007/s11135-014-0002-2 https://doi.org/10.1007/s11135-014-0002-2 https://doi.org/10.1002/qre.2730 https://doi.org/10.1002/qre.2422 international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 178-184 http://www.etamaths.com some generalized notions of amenability modulo an ideal hosein esmaili and hamidreza rahimi∗ abstract. in this paper some generalized notions of amenability modulo an ideal of banach algebras such as uniformly (boundedly) approximately amenable (contractible) modulo an ideal of banach algebras are investigated. using the obtained results, uniformly (boundedly) approximately amenability (contractibility) modulo an ideal of weighted semigroup algebras are characterized. 1. introduction let a be a banach algebra and x be a banach a-bimodule, by a derivation d we mean a bounded linear map d : a → x such that d(ab) = a.d(b) + d(a).a, (a,b ∈ a). an inner derivation is a derivation d which there exists x ∈ x such that d(a) = adx(a) = a · x − x · a, (a ∈ a). a derivation d : a → x is called approximately inner if there exists a net (ξα) in x such that d(a) = lim α adξα(a) (a ∈ a) where the limit is taken in norm of x. if the above limit exists in the w∗topology (say, x is a dual module) then d is called w∗-approximately inner. a banach algebra a is called boundedly approximately amenable (contractible) if, for each banach a-bimodule x and each continuous derivation d : a → x∗ (d : a → x) there exist k > 0 and a net (ξα) in x∗ (in x) such that for each a ∈ a and α, ‖a.ξα − ξα.a‖ ≤ m.‖a‖ and d(a) = limα adξα(a), a is called uniformly approximately amenable (contractible) if for each banach a-bimodule x, each continuous derivation d from a to x∗ ( to x) is the limit of a sequence of inner derivations in the norm topology of the set of all bounded operators from a into x∗, i.e. b(a,x∗) (into x, i.e. b(a,x)). some characterizations of these concepts of amenability are investigated in [5–7]. the concept of amenability modulo an ideal for a class of banach algebras which could be considered as a generalization of amenability of banach algebra was introduced by the first author and amini in 2014 [1]. using this idea, it is shown that a semigroup s is amenable if and only if the semigroup algebra l1(s) is amenable modulo an ideal induced by appropriate congruence σ on s, for a large class of semigroups. in further researches, it was shown that amenability modulo an ideal can be characterized by the existence of virtual diagonal modulo an ideal and approximate diagonal modulo an ideal. to see the details of these results and more on this topic, we refer to [1, 10, 11]. in this paper we shall continue the investigation of amenability modulo an ideal, in particular that of boundedly approximate amenability modulo an ideal and uniformly approximate amenability modulo an ideal of banach algebras. afterward, for a large class of semigroups, we introduce some characterization of amenability modulo an ideal of weighted semigroup algebras. this paper is organized as follow; in section two, we give some basic notions of generalized amenability and amenability modulo an ideal of banach algebras and we show that the concepts approximately contractible modulo an ideal, approximately amenable modulo an ideal and w∗−approximately amenable modulo an ideal of banach algebras are equivalent. in section three, we investigate to the generalized notions of amenability modulo an ideal of banach algebras such as, uniformly approximately amenable (contractible) modulo an ideal and boundedly approximately amenable (contractible) modulo an ideal of banach algebras. in section four, we consider the generalized notions of amenability received 19th november, 2016; accepted 16th january, 2017; published 1st march, 2017. 2010 mathematics subject classification. 43a07, 46h25. key words and phrases. uniformly approximately amenable modulo an ideal; boundedly approximately amenable modulo an ideal; weight semigroup algebra. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 178 some generalized notions of amenability modulo an ideal 179 modulo an ideal for the weighted semigroup algebra l1(s) and we finish this section with give some examples. 2. preliminaries in this section we recall some basic notions which we need in this paper. to see more details, reader can refer to [1, 10–12]. definition 2.1. let i be a closed ideal of a. a banach algebra a is amenable (contractible) modulo i if for every banach a-bimodule x such that i ·x = x ·i = 0, and every derivation d from a into x∗ (into x) there is φ ∈ x∗ such that d = adφ on the set theoretic difference a\i := {a ∈ a : a /∈ i}. all over this paper we fix a and i as above, unless they are otherwise specified. theorem 2.1. ( [1, theorem 1]) the following assertions hold. i) if a/i is amenable and i2 = i then a is amenable modulo i. ii) if a is amenable modulo i then a/i is amenable. iii) if a is amenable modulo i and i is amenable, then a is amenable. let a be a banach algebra and i be a closed ideal of a. with the module actions a.b̄ := ab and b̄.a := ba, a i is a banach a-bimodule where ā is the image of a in a i . also a i ⊗̂a can be consider as a banach a−bimodule where the module actions are the linear extension of a.(b̄ ⊗ c) := ab⊗c and (b̄⊗c).a := (b̄⊗ca), (a,b,c ∈ a). by the diagonal operator we mean the bounded linear operator defined by the linear extension of π : (a i ⊗̂a) → a i by π(b̄⊗c) = bc. clearly, π is a a−bimodule homomorphism. definition 2.2. (i) by a virtual diagonal modulo i, we mean an element m ∈ (a i ⊗̂a)∗∗ such that; a ·π∗∗(m) − ā = 0 (a ∈ a) and a ·m −m ·a = 0 (a ∈ a\ i), (ii) an approximate diagonal modulo i, we mean a bounded net (mα)α ⊆ (ai ⊗̂a) such that; a.π(mα) − ā → 0 (a ∈ a) and a.mα −mα.a → 0 (a ∈ a\ i). (iii) a diagonal modulo i, we mean an element m ∈ (a i ⊗̂a) such that; a.π(m) − ā = 0 (a ∈ a), and a.m−m.a = 0, (a ∈ a\ i). we recall that a bounded net (uα)α ⊆ a is called approximate identity modulo i if lim α uα · a = lim α a ·uα = a (a ∈ a\i). if a is amenable modulo i then a has an approximate identity modulo i. it is shown that a banach algebra a is amenable modulo i if and only if a has an approximate diagonal modulo i,if and only if a has a virtual diagonal modulo i [10]. by appropriate modifications, the following theorem may be proved in much the same way as [4, theorem 1.9.21]. theorem 2.2. a is contractible modulo i if and only if a has a diagonal modulo i. definition 2.3. a banach algebra a is called approximately amenable (contractible) modulo i if for every banach a-bimodule x such that i · x = x · i = 0, every bounded derivation d : a → x∗ (d : a → x) is approximately inner on the set theoretical difference a\i := {a ∈ a : a /∈ i}. theorem 2.3. the following statements are equivalent; a) a is approximately contractible modulo i; b) a is approximately amenable modulo i; c) a is w∗-approximately amenable modulo i. proof. it is easily seen that (a → b) and (b → c), so we only need to show that (c → a). since a is w∗-approximately amenable modulo i, a] is w∗-approximately amenable modulo i ( by [11, theorem 3.2]). now [11, theorem 3.3], provide us to consider a net (mi) ⊆ (a ] i ⊗̂a])∗∗ such that a · mi − mi · a → 0 (∀a ∈ a]\i) and π∗∗(mi) → ē in the w∗-topology of (a ] i ⊗̂a])∗∗ and a∗∗, respectively. let � > 0 and consider finite sets f ⊆ a]\i, φ ⊆ (a]\i)∗ and n ⊆ (a ] i ⊗̂a])∗, so there exists j such that (a ∈f,φ ∈ φ,f ∈n), |〈a.f −f.a,mj〉| = |〈f,a.mj −mj.a〉| < � and | 〈φ,π∗∗(mj) − ē〉 |< � 180 esmaili and rahimi using the weak∗-continuity of π∗∗ and goldstine’s theorem, we can choose m ∈ (a ] i ⊗̂a]) such that | 〈f,a.m−m.a〉 |=| 〈a.f −f.a,m〉 |< �, and | 〈φ,π(m) − ē〉 |< �, for each a ∈f,φ ∈ φ and f ∈n . hence there exists (mi) ⊆ (a ] i ⊗̂a]) such that a.mi−mi.a → 0 (a ∈ a\i) and π(mi) → ē in the w-topology of (a ] i ⊗̂a]) and a], respectively. now for every finite set f = {a1,a2, , ...,an}⊆ a]\i, (a1.mi −mi.a1, ...,an.mi −mi.an,π(mi)) → (0, ..., 0, ē) weakly in (a ] i ⊗̂a]) ⊕ (a]\i). therefore (0, ..., 0, ē) ∈ c̄ow{(a1.mi −mi.a1, ...,an.mi −mi.an,π(mi))}. set p = {(a1.mi −mi.a1, ...,an.mi −mi.an,π(mi))}, so co(p) = {(a1.m −m.a1, ...,an.m −m.an,π(m)) ∈ co{mi}}. we have (0, ..., 0, ē) ∈ c̄ow(p) = c̄o‖‖(p). the hahn-banach theorem implies that for each � > 0 there exists u�,f ∈ co{mi} such that ‖a.u�,f −u�,f .a‖ < � and ‖π(u�,f ) − ē‖ < �, (a ∈ f). now by [11, theorem 3.8] proof is complete. � 3. uniformly and boundedly approximate amenability (contractibility) modulo an ideal of banach algebras definition 3.1. a banach algebra a is uniformly approximately amenable (contractible) modulo i if for every banach a-bimodule x such that i · x = x · i = 0 and every continuous derivation d : a → x∗ (d : a → x) there is a net (xα) ⊆ x∗ ((xα) ⊆ x) such that d(a) = lim α adxα(a) where the convergence is uniform for each a ∈ a\i such that ‖a‖≤ 1, lemma 3.1. a banach algebra a is uniformly approximately contractible modulo i if and only if a] is uniformly approximately contractible modulo i. proof. let a be uniformly approximately contractible modulo i, x be a banach a]−bimodule and d : a] → x be a bounded derivation. then there are ξ ∈ exe and d1 : a] → exe such that d = d1 +adξ. we have d1(e) = 0 and d1|a ∈z1(a,exe). since a is uniformly approximately contractible modulo i, there exits (ζn) ∈ exe such that d1(a,α) = lim n adζn(a), (a ∈ a\i,α ∈ c,‖a‖ + |α| ≤ 1). now if (a,α) ∈ (a\i) ⊕c = (a\i)] such that ‖a‖ + |α| ≤ 1, then d1(a,α) = d1(a, 0) + αd1(e) = d1(a) = adζa(a). hence d(a) = d1(a) + adξ(a) = lim n adζn(a) + adξ(a) = lim n adζn+ξ(a). conversely, let x be a banach a−bimodule and d : a → x∗ be a bounded derivation. defining (a,α).x = a.x + α.x and x.(a,α) = x.a + α.x (x ∈ x, (a,α) ∈ a]) makes a] into an a]−bimodule. define d̃ : a] → x by d̃(a,α) = d(a) ((a,α) ∈ a]). clearly d̃ is a bounded derivation. supposing a] is uniformly approximately contractible modulo i, there is (ξn) ⊆ x such that d̃ = lim n adξn on the unit ball of (a\i)]. now d̃|a, as required. � lemma 3.2. let x be an a-module and (en) ⊆ x be a sequence such that for each a ∈ a\i with ‖a‖ ≤ 1, a = lim n a.en. then a has a right identity modulo i, i.e. there exists u ∈ a such that a.u = a (a ∈ a\i). proof. let rf denote the right multiplication by f ∈ x. then there is (en) ⊆ x with ‖rf − id‖≤ 1, so rf is invertible. this implies that there is a g ∈ b(x,a) such that rf ◦ g = id. set u = g(f), so u.f = rf ◦ g(f) = f. then auf = af and for each a ∈ a\i, (au−a).f = 0. this means that u is a right identity modulo i. � lemma 3.3. suppose that a is uniformly approximately contractible modulo i. then a has an identity e on a\i, i.e. e.a = a.e = a (a ∈ a\i). some generalized notions of amenability modulo an ideal 181 proof. consider a as a a−bimodule where the module actions are defined by a.x = ax and x.a = 0 (a ∈ a,x ∈ x). let d : a → a∗∗ defined by d(a) = â be the canonical embedding. it is clear that d is a bounded derivation. since a is uniformly approximately contractible modulo i, there is (eα) ⊆ a∗∗ such that d(a) = lim aden(a) (a ∈ a\i,‖a‖ ≤ 1), so a = lim n a.en. using lemma 3.2, a has a right identity modulo i. the same argument is true for aop, and hence a has an identity e on a\i. � theorem 3.1. let a be uniformly approximately contractible modulo i. then a is contractible modulo i. proof. by lemma 3.3, we may suppose that a has an identity ”e” on a\i. define d : a → kerπ ⊆ (a i ⊗̂a) by d(a) = ā⊗e− ē⊗a. then d is a bounded derivation and ‖d‖≤ 2. since a is uniformly approximately contractible modulo i, there is (tn) ∈ kerπ such that adtn → d uniformly for a ∈ a\i, with ‖a‖ ≤ 1. suppose that tn = ∑ i x̄ni ⊗ y n i and s = ∑ i āj ⊗ bj ∈ kerπ. since π(s) = π(tn) = 0,∑ i āibi = ∑ i aibi = 0 and ∑ i x̄nj y n j = ∑ i xnj y n j = 0. hence, ‖stn −s‖ = ‖ ∑ i,j ājx̄ n i ⊗y n i bj − ∑ j āj ⊗ bj‖ = ‖ ∑ i,j ājx̄ n i ⊗y n i bj − ∑ i,j ājbjx̄ n i ⊗y n i − ∑ j āj ⊗ bj + ∑ j ājbj ⊗e‖ = ‖ ∑ j āj( ∑ i x̄ni ⊗y n i bj − ∑ i bjx̄ n i ⊗y n i − ē⊗ bj + bj ⊗e)‖ ≤ ∑ j ‖ ∑ i x̄ni ⊗y n i bj − ∑ i bjx̄ n i ⊗y n i −e⊗ bj + bj ⊗e‖‖āj‖ = ∑ j ‖ ∑ i x̄ni ⊗y n i bj ‖bj‖ − ∑ i bj ‖bj‖ x̄ni ⊗y n i −e⊗ bj‖bj‖ + bj ‖bj‖ ⊗e‖‖āj‖‖bj‖ ≤ ∑ j sup ‖c‖≤1 ‖tn.c− c.tn −e⊗ c + c⊗e‖‖āj‖‖bj‖. it implies that ‖stn − s‖ ≤ sup ‖c‖≤1 ‖adtn(c) −d(c)‖ on the unit ball of kerπ, hence stn → s uniformly on the unit ball of kerπ and by lemma 3.2, kerπ has a right identity modulo i, u. set v = ē⊗e−u, then π(v) = ē−π(u) and for each a ∈ a\i, a.v−v.a = 0. thus v is a diagonal modulo i and hence a is contractible modulo i( by theorem 2.2). � definition 3.2. a banach algebra a is boundedly approximate amenable (contractible) modulo i if for each banach a-bimodule x with x · i = i · x = 0 and each continuous derivation d : a → x∗ (d : a → x) there exist k > 0 and a net (ξα) in x∗ (x) such that for each a ∈ a\i and α, ‖a.ξα − ξα.a‖≤ m.‖a‖, and d(a) = limα adξα(a). theorem 3.2. then the following assertions hold; (i) if a is boundedly approximate amenable modulo i, then a i is boundedly approximate amenable. (ii) if a i is boundedly approximate amenable and i2 = i then a is boundedly approximate amenable modulo i analogous assertions satisfy for uniformly approximately amenable modulo an ideal banach algebras. proof. (i) suppose that x is a banach a i -bimodule and d : a i → x∗ is a bounded derivation. now x is a clearly banach amodule with the module actions defined by a.x = π(a).x , x.a = x.π(a), (a ∈ a,x ∈ x) where π : a → a i is the canonical quotient map. since i ·x = x · i = 0 and d ◦π : a → x∗ is a bounded derivation, there is a (ξα) ⊂ x∗ such that ‖a.ξα − ξα.a‖ ≤ m.‖a‖ (for some m > 0) and d◦π = lim α adξα on a\i. we have ‖π(a).ξα−ξα.π(a)‖ = ‖a.ξα−ξα.a‖≤ m.‖a‖ and d(π(a)) = d ◦π(a) = lim α adξα(a), (π(a) ∈ a i ). hence a i is boundedly approximate amenable modulo i. 182 esmaili and rahimi (ii) suppose that x is a banach a-bimodule such that x · i = i · x = 0 and d : a → x∗ is a bounded derivation. we can consider x as an a i -bimodule with the module actions a.x = π(a).x , x.a = x.π(a), (a ∈ a,x ∈ x). the equality i2 = i provide us to define the well-defined bounded derivation d : a i → x∗ by d(π(a)) = d(a) (a ∈ a). since a i is boundedly approximate amenable modulo i, there is a (ξα) ⊂ x∗ such that ‖π(a).ξα−ξα.π(a)‖≤ m.‖a‖ (for some m > 0) and d = limα adξα. it is not far to see that the net (adξα) is norm bounded in b(a,x∗) and d(a) = d(π(a)) = lim α adξα(a). � the proof of the following result is the same way as theorem 3.2. corollary 3.1. the following conditions are hold; (i) if a is boundedly approximate contractible modulo i, then a i is boundedly approximate contractible. (ii) if a i is boundedly approximate contractible and i2 = i then a is boundedly approximate contractible modulo i analogous assertions satisfy for uniformly approximately contractible modulo an ideal. for a banach algebra a, it is shown that a is uniformly approximately amenable if and only if it is amenable [6, theorem 3.1]. using theorem 3.2, we have the following result. corollary 3.2. suppose a is a banach algebra and i is a closed ideal of a such that i2 = i. then a is uniformly approximate amenable modulo i if and only if it is amenable modulo i. theorem 3.3. a banach algebra a is boundedly approximate amenable modulo i if and only if there exists a constant m > 0 such that for any banach a-bimodule x with x · i = i · x = 0 and any continuous derivation d : a → x∗ there is a net (ηi) ⊆ x∗ such that a) sup i ‖adηi‖≤ m‖d‖, b) d(a) = lim i adηi(a), (∀a ∈ a\i). proof. let assumptions (a) and (b) hold, then ‖adηi‖ ≤ m‖d‖ = m‖d‖ ‖a‖ (a ∈ a/i). therefore a is boundedly approximately amenable modulo i. conversely, let a be a boundedly approximately amenable modulo i. consider there is no such m. suppose that for every integer n ∈ n, mn is banach module such that mn · i = i · mn = 0 and dn : a → m∗n is a derivation with ‖dn‖ > n. now x = l1(mn) is a banach a−module with dual l∞(m∗n). put d = (dn), d : a → l∞(m∗n) is a continuous derivation and d(a) = (dn(a)) = lim i (adηn i (a)). since ‖dn‖ > n, ‖d‖ → ∞ which is contradiction. � the same argument of [12, theorem 3.2 and 3.3] and minor changes, we have the following theorems; theorem 3.4. a banach algebra a is boundedly approximately amenable modulo i if and only if a# is boundedly approximately amenable modulo i. theorem 3.5. let a be a banach algebra and i be a closed ideal of a. if a is boundedly approximately amenable modulo i then; (a) there is a net (mi)i ⊆ (a ] i ⊗̂a#)∗∗ and a constant l > 0 such that ā.mi−mi. ā → 0, π∗∗(mi) → ē, and ‖ā.mi −mi. ā‖≤ l‖ā‖ , for each ā ∈ (a # i ). conversely, if (a) holds and the net (π∗∗(mi)) is bounded then a is boundedly approximately amenable modulo i. 4. algebras related to discrete semigroups we generally follow [3,9] for definitions and basic concepts of semigroups. for a semigroup s, the set (possibly empty) of idempotents of s is denoted by e = e(s). a semigroup s is called an e-semigroup if e(s) is a sub-semigroup of s, e-inversive if for each x ∈ s, there exists y ∈ s such that xy ∈ e(s), regular if the set of inverses of a ∈ s, v (a) = {x ∈ s : a = axa,x = xax} 6= φ, inverse semigroup if moreover, the inverse of each element is unique, e-unitary if for each x ∈ s and e ∈ e(s), ex ∈ e(s) implies x ∈ e(s), semilattice if s is a commutative and idempotent semigroup and finally s is called eventually inverse if every element of s has some power that is regular and e(s) is a semilattice. some generalized notions of amenability modulo an ideal 183 by a group congruence ρ on semigroup s we mean a congruence ρ such that s/ρ is a group. the kernel of a congruence ρ on a semigroup s ”kerρ” is the set {a ∈ s : aρ ∈ e(s/ρ)} = {a ∈ s : (a,a2) ∈ ρ}. we denote the least group congruence on s (if exist) by σ. the least group congruence on semigroups have also been considered by various authors [8, 13]. it is shown that if s is an einversive e-semigroup such that e(s) is commutative (s is an eventually semigroup) then the relation σ = {(a,b) ∈ s × s |ea = fb for some e,f ∈ es} (σ′ = {(s,t) : es = et, for some e ∈ e(s)}) is the least group congruence on s [8, 13]. we recall that a function ω : g → (0,∞) such that ω(g1g2) ≤ ω(g1)ω(g2) (g1,g2 ∈ g) is called a weight on group g. the weight ω on group g is called symmetric if ω(g) = ω(g−1)(g ∈ g) and for any weight ω, by symmetrization of ω, we mean the weight defined by ωω(g) = ω(g)ω(g −1). the weighted semigroup algebra (or beurling algebra on semigroup s) l1(s,ω) = {f | f : s → c, ∑ s∈s |f(s)|ω(s) < ∞} with ‖f‖1,ω = ∑ s∈s |f(s)|ω(s) and convolution product is a banach algebra. in the case ω = 1, the weighted semigroup algebra l1(s,ω) is called semigroup algebra and is denoted by l1(s). we recall the following lemma, which is detailed in [12]. lemma 4.1. the following statements hold: (i) if s is a semigroup, ρ is a congruence on s and ω is a weight on s, then l1(s,ω) iρ ' l1(s/ρ,ωρ) where ωρ([s]ρ) = inf{ω(s) : s ∈ [s]ρ} is the induced weight on s/ρ and iρ is an ideal in l1(s,ω) generated by the set {δs − δt : s,t ∈ s with (s,t) ∈ ρ}; (ii) if s is an e-inversive semigroup with commuting idempotents or s is an eventually inverse semigroup, σ is the least group congruence on s and ω is a weight on s, then l1(s/σ,ωσ)) ' l1(s,ω) iσ where iσ is a closed ideal of l 1(s,ω) and i2σ = iσ. it is shown that for a locally compact group g and a weight ω on g, the beurling algebra l1(g,ω) is boundedly approximately contractible if and only if the beurling algebra l1(g,ω) is amenable, if and only if g is amenable and ω is bounded on g [7, corollary 2.2]. the same conclusion can be drawn for beurling algebra of a weighted semigroup as follow; theorem 4.1. suppose that ω is a weight on semigroup s. if s is an e-inversive semigroup with commuting idempotents or s is an eventually inverse semigroup, then the followings assertions are equivalent. (i) the semigroup s is amenable and ωωσ is bounded where ωσ is the induced weight on s/σ. (ii) the weighted semigroup algebra l1(s,ω) is boundedly approximately contractible modulo iσ. proof. the semigroup s is amenable if and only if s/σ is amenable [1, theorem 2], if and only if l1(s/σ,ωσ) is amenable (because s/σ is a group), if and only if l 1(s/σ,ωσ) is boundedly approximately contractible (because ωωσ is bounded on s/σ and by [7, corollary 2.2]), if and only if l 1(s,ω) is boundedly approximately contractible modulo iσ (by corollary 3.1). � for a loccaly compact group g and a symmetric weight on ω on g, if lim x→∞ ω(x) = ∞, then l1(g,ω) is not boundedly approximately amenable [7, corollary 2.8]. thus we have the following corollary for the weighted semigroup algebras; corollary 4.1. if s is a semigroup, ρ is a group congruence on s with kerρ is central and ω is a weight on semigroup s such that lim x→∞ ω(x) = ∞(x ∈ s/ρ). then l1(s,ω) is not boundedly approximately amenable modulo iρ. proof. since kerρ is central, the semigroup s is amenable if and only if s/ρ is amenable. on the other hand, s/ρ is a group and lim x→∞ ω(x) = ∞(x ∈ s/ρ), so l1(s/ρ,ωρ) is not boundedly approximately amenable and consequensly l1(s,ω) is not boundedly approximately amenable modulo iρ. � we end this paper to give some illustrative examples. example 4.1. (i) let s = {pmqn : m,n ≥ 0} be the bicyclic semigroup generated by p,q, then s/σ ' z where σ = {(s,t) ∈ s×s : se = te, for some e ∈ e(s)} is the least group congruence on s [1]. using theorem 4.1, amenability of s implies that l1(s) is boundedly approximately amenable modulo 184 esmaili and rahimi iσ. we note that l 1(s) is not boundedly approximately amenable because l1(s) is a not approximate amenable. (ii) let s = (n,∨) be the commutative semigroup of positive integers with maximum operation, then e(s) = s. set mσn if and only if km = kn, for some k ∈ e(s) (n,m ∈ n). obviously σ is the least group congruence on s and s/σ ' gs is the maximum group image of s. since gs is finite, l1(s/σ) is contractible and consequently l1(s/σ) is boundedly approximately contractible and boundedly approximately amenable [2, 6]. thus l1(s) is boundedly approximately contractible modulo iσ and boundedly approximately amenable modulo iσ. we note that l 1(s) is not contractible because l1(n) has not diagonal. (iii) let g = f2 be a free group with two generators a,b, t = (n0, +)×(n,max), where n0 = n∪{0} and s = g×t . then e(s) = {(1g,e) : e ∈ e(t)} is infinite. under the homomorphism φ : (g,t) 7→ g, g is the maximum group homomorphism image of s. suppose that s/σ ' g where σ is a group congruence on s. then l1(s) is not boundedly approximately amenable (contractible) modulo iσ, since otherwise l1(s) iσ ' l1(g) should be boundedly approximately amenable (contractible) which is contradiction. acknowledgement. the authors sincerely thank the referee(s) for their valuable comments and suggestions, which were very useful to improve the paper significantly. references [1] m. amini and h. rahimi, amenability of semigroups and their algebras modulo a group congruence, acta math. hung., 144 (2) (2014), 407-415. [2] y. choi and f. ghahramani, approximate amenability of schatten classes, lipschitz algebras and second duals of fourier algebras, quart. j. math. 62 (2011), 39-58. [3] a. h. clifford and j. b. preston, the algebraic theory of semigroups i, american mathematical society, surveys 7, american mathematical society, providence (1961). [4] h.g. dales. banach algebras and automatic continuity, clarendon press, oxford,(2000). [5] f. ghahramani, r. j. loy generalized notions of amenability, j. funct. anal, 208, (2004), 229-260. [6] f. ghahramani, r. j. loy, and y. zhang, generalized notions of amenability, ii, j. functional analysis 254 (2008), 1776-1810. [7] f. ghahramani, e. samei and y. zhang, generalized amenability properties of the beurling algebras, j. aust. math. soc. 89 (2010), 359-376. [8] r. s. gigon, congruences and group congruences on a semigroup, semigroup forum, 86 (2013), 431-450. [9] j. m. howie, fundamentals of semigroup theory, clarendon press, oxford (1995). [10] h. rahimi and kh. nabizadeh, amenability modulo an ideal of second duals of semigroup algebras, mathematics, 4 (3) (2016), art. id 55. [11] h. rahimi and e. tahmasebi, hereditary properties of amenability modulo an ideal of banach algebras, j. linear topol. algebra, 3 (2) (2014), 107114. [12] h. rahimi and a. soltani, approximate amenability modulo an ideal of banach algebras, u.p.b. sci. bull., series a, 78 (3) (2016), 233-240. [13] m. siripitukdet and s. sattayaporn, the least group congruence on e-inversive semigroups and e-inversive esemigroups, thai journal of mathematics, 3 (2005), 163-169. department of mathematics, faculty of science, central tehran branch, islamic azad university, p. o. box 13185/768, tehran, iran ∗corresponding author: rahimi@iauctb.ac.ir 1. introduction 2. preliminaries 3. uniformly and boundedly approximate amenability (contractibility) modulo an ideal of banach algebras 4. algebras related to discrete semigroups references international journal of analysis and applications volume 17, number 5 (2019), 686-710 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-686 estimation of different entropies via taylor one point and taylor two points interpolations using jensen type functionals tasadduq niaz1,2,∗, khuram ali khan1, d̄ilda pečarić3, josip pečarić4 1department of mathematics, university of sargodha, sargodha 40100, pakistan 2department of mathematics, the university of lahore, sargodha-campus, sargodha 40100, pakistan 3catholic university of croatia, ilica 242, zagreb, croatia 4rudn university, miklukho-maklaya str. 6, 117198 moscow, russia ∗corresponding author: tasadduq khan@yahoo.com abstract. in this work, we estimated the different entropies like shannon entropy, rényi divergences, csiszar divergence by using the jensen’s type functionals. the zipf’s mandelbrot law and hybrid zipf’s mandelbrot law are used to estimate the shannon entropy. further the taylor one point and taylor two points interpolations are used to generalize the new inequalities for m-convex function. 1. introduction and preliminary results in numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points for example in the situation when one obtained the number of data after experiment which actually represent the value of function for a limited number of value of the independent variable. it is usually require to interpolate which means that it has to be estimated the value of the function for an intermediate value of independent variable. there are many interpolating polynomial can be found in literature for example taylor polynomial, lidstone polynomial etc. received 2019-05-27; accepted 2019-07-16; published 2019-09-02. 2010 mathematics subject classification. 26d07, 94a17. key words and phrases. m-convex function; jensen’s inequality; shannon entropy; fand rényi divergence; taylor interpolation; entropy. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 686 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-686 int. j. anal. appl. 17 (5) (2019) 687 the most commonly used words, the largest cities of countries income of billionare can be described in term of zipf’s law. the f-divergence which means that distance between two probability distribution by making an average value, which is weighted by a specified function. as f-divergence, there are other probabilities distributions like csiszar f-divergence [15, 16], some special case of which are kullback-leiblerdivergence use to find the appropriate distance between the probability distribution (see [19,20]). the notion of distance is stronger than divergence because it give the properties of symmetry and triangle inequalities. probability theory has application in many fields and the divergence between probability distribution have many application in these fields. many natural phenomena’s like distribution of wealth and income in a society, distribution of face book likes, distribution of football goals follows power law distribution (zipf’s law). like above phenomena’s, distribution of city sizes also follow power law distribution. auerbach [2] first time gave the idea that the distribution of city size can be well approximated with the help of pareto distribution (power law distribution). this idea was well refined by many researchers but zipf [28] worked significantly in this field. the distribution of city sizes is investigated by many scholars of the urban economics, like rosen and resnick [26] , black and henderson [3], ioannides and overman [14], soo [27], anderson and ge [1] and bosker et al. [4]. zipf’s law states that: “the rank of cities with a certain number of inhabitants varies proportional to the city sizes with some negative exponent, say that is close to unit”. in other words, zipf’s law states that the product of city sizes and their ranks appear roughly constant. this indicates that the population of the second largest city is one half of the population of the largest city and the third largest city equal to the one third of the population of the largest city and the population of n-th city is 1 n of the largest city population. this rule is called rank, size rule and also named as zipf’s law. hence zip’s law not only shows that the city size distribution follows the pareto distribution, but also show that the estimated value of the shape parameter is equal to unity. in [17] l. horváth et al. introduced some new functionals based on the f-divergence functionals, and obtained some estimates for the new functionals. they obtained f-divergence and rényi divergence by applying a cyclic refinement of jensen’s inequality. they also construct some new inequalities for rényi and shannon entropies and used zipf-madelbrot law to illustrate the results. the inequalities involving higher order convexity are used by many physicists in higher dimension problems since the founding of higher order convexity by t. popoviciu (see [24, p. 15]). it is quite interesting fact that there are some results that are true for convex functions but when we discuss them in higher order convexity they do not remaind valid. in [24, p. 16], the following criteria is given to check the m-convexity of the function. if f(m) exists, then f is m-convex if and only if f(m) ≥ 0. in recent years many researchers have generalized the inequalities for m-convex functions; like s. i. butt et int. j. anal. appl. 17 (5) (2019) 688 al. generalized the popoviciu inequality for m-convex function using taylor’s formula, lidstone polynomial, montgomery identity, fink’s identity, abel-gonstcharoff interpolation and hermite interpolating polynomial (see [5–9]). in [23] t. niaz et al generalized the refinement of jensen’s inequality for m-convex function using abelgontscharoff green function and fink’s identity. in [18] k. a. khan et al used refinement of jensen inequality and introduced new functional based on an f-divergence functional, and estimate some bounds for the new functionals, the f-divergence and rényi divergence. they also constructed some new inequalities for réneyi and shannon estimates. they also generalized the new inequality for m-convex function using montgomery identity. further the used hybrid zipf mandelbrot law to estimate the shannon entropy. since many years jensen’s inequality has of great interest. the researchers have given the refinement of jensen’s inequality by defining some new functions (see [12, 13] ). like many researchers l. horváth and j. pečarić in ( [10, 13], see also [11, p. 26]), gave a refinement of jensen’s inequality for convex function. they defined some essential notions to prove the refinement given as follows: let x be a set, and: p(x) := power set of x, |x|:= number of elements of x, n:= set of natural numbers with 0. consider q ≥ 1 and r ≥ 2 be fixed integers. define the functions fr,s : {1, . . . ,q}r →{1, . . . ,q}r−1 1 ≤ s ≤ r, fr : {1, . . . ,q}r → p ( {1, . . . ,q}r−1 ) , and tr : p ({1, . . . ,q}r) → p ( {1, . . . ,q}r−1 ) , by fr,s(i1, . . . , ir) := (i1, i2, . . . , is−1, is+1, . . . , ir) 1 ≤ s ≤ r, fr(i1, . . . , ir) := r⋃ s=1 {fr,s(i1, . . . , ir)}, and tr(i) =   φ, i = φ;⋃ (i1,...,ir)∈i fr(i1, . . . , ir), i 6= φ. next let the function αr,i : {1, . . . ,q}r → n 1 ≤ i ≤ q int. j. anal. appl. 17 (5) (2019) 689 defined by αr,i(i1, . . . , ir) is the number of occurences of i in the sequence (i1, . . . , ir). for each i ∈ p({1, . . . ,q}r) let αi,i := ∑ (i1,...,ir)∈i αr,i(i1, . . . , ir) 1 ≤ i ≤ q. (h1) let n,m be fixed positive integers such that n ≥ 1, m ≥ 2 and let im be a subset of {1, . . . ,n}m such that αim,i ≥ 1 1 ≤ i ≤ n. introduce the sets il ⊂{1, . . . ,n}l(m− 1 ≥ l ≥ 1) inductively by il−1 := tl(il) m ≥ l ≥ 2. obviously the sets i1 = {1, . . . ,n}, by (h1) and this insures that αi1,i = 1(1 ≤ i ≤ n). from (h1) we have αil,i ≥ 1(m− 1 ≥ l ≥ 1, 1 ≤ i ≤ n). for m ≥ l ≥ 2, and for any (j1, . . . ,jl−1) ∈ il−1, let hil(j1, . . . ,jl−1) := {((i1, . . . , il),k) ×{1, . . . , l}|fl,k(i1, . . . , il) = (j1, . . . ,jl−1)}. with the help of these sets they define the functions ηim,l : il → n(m ≥ l ≥ 1) inductively by ηim,m(i1, . . . , im) := 1 (i1, . . . , im) ∈ im; ηim,l−1(j1, . . . ,jl−1) := ∑ ((i1,...,il),k)∈hil(j1,...,jl−1) ηim,l(i1, . . . , il). they define some special expressions for 1 ≤ l ≤ m, as follows am,l = am,l(im,x1, . . . ,xn,p1, . . . ,pn; f) := (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 pij αim,ij  f   l∑ j=1 pij αim,ij xij l∑ j=1 pij αim,ij   and prove the following theorem. theorem 1.1. assume (h1), and let f : i → r be a convex function where i ⊂ r is an interval. if x1, . . . ,xn ∈ i and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1, then f ( n∑ s=1 psxs ) ≤am,m ≤am,m−1 ≤ . . . ≤am,2 ≤am,1 = n∑ s=1 psf (xs) . (1.1) int. j. anal. appl. 17 (5) (2019) 690 we define the following functionals by taking the differences of refinement of jensen’s inequality given in (1.1). θ1(f) = am,r −f ( n∑ s=1 psxs ) , r = 1, . . . ,m, (1.2) θ2(f) = am,r −am,k, 1 ≤ r < k ≤ m. (1.3) under the assumptions of theorem 1.1, we have θi(f) ≥ 0, i = 1, 2. (1.4) inequalities (1.4) are reversed if f is concave on i. 2. inequalities for csiszár divergence in [15, 16] csiszár introduced the following notion. definition 2.1. let f : r+ → r+ be a convex function, let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) be positive probability distributions. then f-divergence functional is defined by if (r, q) := n∑ i=1 qif ( ri qi ) . (2.1) and he stated that by defining f(0) := lim x→0+ f(x); 0f ( 0 0 ) := 0; 0f (a 0 ) := lim x→0+ xf (a x ) , a > 0, (2.2) we can also use the nonnegative probability distributions as well. in [17], l. horv́ath, et al. gave the following functional on the based of previous definition. definition 2.2. let i ⊂ r be an interval and let f : i → r be a function, let r = (r1, . . . ,rn) ∈ rn and q = (q1, . . . ,qn) ∈ (0,∞)n such that rs qs ∈ i, s = 1, . . . ,n. then they define the sum as îf (r, q) as îf (r, q) := n∑ s=1 qsf ( rs qs ) . (2.3) we apply theorem 1.1 to îf (r, q) theorem 2.1. assume (h1), let i ⊂ r be an interval and let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) are in (0,∞)n such that rs qs ∈ i, s = 1, . . . ,n. int. j. anal. appl. 17 (5) (2019) 691 (i) if f : i → r is convex function, then îf (r, q) = n∑ s=1 qsf ( rs qs ) = a [1] m,1 ≥ a [1] m,2 ≥ . . . ≥ a [1] m,m−1 ≥ a [1] m,m ≥ f (∑n s=1 rs∑n s=1 qs ) n∑ s=1 qs. (2.4) where a [1] m,l = (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij  f   ∑l j=1 rij αim,ij l∑ j=1 qij αim,ij   (2.5) if f is concave function, then inequality signs in (2.4) are reversed. (ii) if f : i → r is a function such that x → xf(x)(x ∈ i) is convex, then( n∑ s=1 rs ) f ( n∑ s=1 rs∑n s=1 qs ) ≤ a[2]m,m ≤ a [2] m,m−1 ≤ . . . ≤ a [2] m,2 ≤ a [2] m,1 = n∑ s=1 rsf ( rs qs ) = îidf (r, q) (2.6) where a [2] m,l = (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij    ∑lj=1 rijαim,ij∑l j=1 qij αim,ij  f  ∑lj=1 rijαim,ij∑l j=1 qij αim,ij   . proof. (i) consider ps = qs∑ n s=1 qs and xs = rs qs in theorem 1.1, we have f ( n∑ s=1 qs∑n s=1 qs rs qs ) ≤ . . . ≤ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij∑ n s=1 qs αim,ij  f   l∑ j=1 qij∑n i=1 qi αim,ij rij qij l∑ j=1 qij∑n i=1 qi αim,ij   ≤ . . . ≤ n∑ s=1 qs∑n i=1 qs f ( rs qs ) . (2.7) on multiplying ∑n s=1 qs, we have (2.4). (ii) using f := idf (where “id” is the identity function) in theorem 1.1, we have n∑ s=1 psxsf ( n∑ s=1 psxs ) ≤ . . . ≤ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 pij αim,ij     l∑ j=1 pij αim,ij xij l∑ j=1 pij αim,ij  f   l∑ j=1 pij αim,ij xij l∑ j=1 pij αim,ij   ≤ . . . ≤ n∑ s=1 psxsf(xs). (2.8) now on using ps = qs∑ n s=1 qs and xs = rs qs , s = 1, . . . ,n, we get n∑ s=1 qs∑n s=1 qs rs qs f ( n∑ s=1 qs∑n s=1 qs rs qs ) ≤ . . . ≤ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij∑ n s=1 qs αim,ij     ∑l j=1 qij∑n s=1 qs αim,ij rij qij∑l j=1 qij∑n s=1 qs αim,ij  f   ∑l j=1 qij∑n s=1 qs αim,ij rij qij∑l j=1 qij∑n s=1 qs αim,ij   ≤ . . . ≤ n∑ s=1 qs∑n s=1 qs rs qs f ( rs qs ) . (2.9) int. j. anal. appl. 17 (5) (2019) 692 on multiplying ∑n s=1 qs, we get (2.6). � 3. inequalities for shannon entropy definition 3.1 (see [17]). the shannon entropy of positive probability distribution r = (r1, . . . ,rn) is defined by s := − n∑ s=1 rs log(rs). (3.1) corollary 3.1. assume (h1). (i) if q = (q1, . . . ,qn) ∈ (0,∞)n, and the base of log is greater than 1, then s ≤ a[3]m,m ≤ a [3] m,m−1 ≤ . . . ≤ a [3] m,2 ≤ a [3] m,1 = log ( n∑n s=1 qs ) n∑ s=1 qs, (3.2) where a [3] m,l = − (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij   log   l∑ j=1 qij αim,ij   . (3.3) if the base of log is between 0 and 1, then inequality signs in (3.2) are reversed. (ii) if q = (q1, . . . ,qn) is a positive probability distribution and the base of log is greater than 1, then we have the estimates for the shannon entropy of q s ≤ a[4]m,m ≤ a [4] m,m−1 ≤ . . . ≤ a [4] m,2 ≤ a [4] m,1 = log(n), (3.4) where a [4] m,l = − (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij   log   l∑ j=1 qij αim,ij   . proof. (i) using f := log and r = (1, . . . , 1) in theorem 2.1 (i), we get (3.2). (ii) it is the special case of (i). � definition 3.2 (see [17]). the kullback-leibler divergence between the positive probability distribution r = (r1, . . . ,rn) and q = (q1, . . . ,qn) is defined by d(r, q) := n∑ s=1 ri log ( ri qi ) . (3.5) corollary 3.2. assume (h1). (i) let r = (r1, . . . ,rn) ∈ (0,∞)n and q := (q1, . . . ,qn) ∈ (0,∞)n. if the base of log is greater than 1, then n∑ s=1 rs log ( n∑ s=1 rs∑n s=1 qs ) ≤ a[5]m,m ≤ a [5] m,m−1 ≤ . . . ≤ a [5] m,2 ≤ a [5] m,1 = n∑ s=1 rs log ( rs qs ) = d(r, q), (3.6) int. j. anal. appl. 17 (5) (2019) 693 where a [5] m,l = (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij    ∑lj=1 rijαim,ij∑l j=1 qij αim,ij   log  ∑lj=1 rijαim,ij∑l j=1 qij αim,ij   . if the base of log is between 0 and 1, then inequality in (3.6) is reversed. (ii) if r and q are positive probability distributions, and the base of log is greater than 1, then we have d(r,q) = a [6] m,1 ≥ a [6] m,2 ≥ . . . ≥ a [6] m,m−1 ≥ a [6] m,m ≥ 0, (3.7) where a [6] m,l = (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 qij αim,ij    ∑lj=1 rijαim,ij∑l j=1 qij αim,ij   log  ∑lj=1 rijαim,ij∑l j=1 qij αim,ij   . if the base of log is between 0 and 1, then inequality signs in (3.7) are reversed. proof. (i) on taking f := log in theorem 2.1 (ii), we get (3.6). (ii) since r and q are positive probability distributions therefore ∑n s=1 rs = ∑n s qs = 1, so the smallest term in (3.6) is given as n∑ s=1 rs log ( n∑ s=1 rs∑n s=1 qs ) = 0. (3.8) hence for positive probability distribution r and q the (3.6) will become (3.7). � 4. inequalities for rényi divergence and entropy the rényi divergence and entropy come from [25]. definition 4.1. let r := (r1, . . . ,rn) and q := (q1, . . . ,qn) be positive probability distributions, and let λ ≥ 0, λ 6= 1. (a) the rényi divergence of order λ is defined by dλ(r,q) := 1 λ− 1 log ( n∑ i=1 qi ( ri qi )λ) . (4.1) (b) the rényi entropy of order λ of r is defined by hλ(r) := 1 1 −λ log ( n∑ i=1 rλi ) . (4.2) the rényi divergence and the rényi entropy can also be extended to non-negative probability distributions. if λ → 1 in (4.1), we have the kullback-leibler divergence, and if λ → 1 in (4.2), then we have the shannon entropy. in the next two results, inequalities can be found for the rényi divergence. int. j. anal. appl. 17 (5) (2019) 694 theorem 4.1. assume (h1), let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) are probability distributions. (i) if 0 ≤ λ ≤ µ such that λ,µ 6= 1, and the base of log is greater than 1, then dλ(r,q) ≤ a[7]m,m ≤ a [7] m,m−1 ≤ . . . ≤ a [7] m,2 ≤ a [7] m,1 = dµ(r,q), (4.3) where a [7] m,l = 1 µ− 1 log  (m− 1)!(l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij     l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   µ−1 λ−1   the reverse inequalities hold in (4.3) if the base of log is between 0 and 1. (ii) if 1 < µ and the base of log is greater than 1, then d1(r,q) = d(r,q) = n∑ s=1 rs log ( rs qs ) ≤ a[8]m,m ≤ a [8] m,m−1 ≤ . . . ≤ a [8] m,2 ≤ a [8] m,1 = dµ(r,q), (4.4) where a [8] m,l =≤ 1 µ− 1 log  (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   exp   (µ− 1) l∑ j=1 rij αim,ij log ( rij qij ) l∑ j=1 rij αim,ij     here the base of exp is same as the base of log, and the reverse inequalities hold if the base of log is between 0 and 1. (iii) if 0 ≤ λ < 1, and the base of log is greater than 1, then dλ(r,q) ≤ a[9]m,m ≤ a [9] m,m−1 ≤ . . . ≤ a [9] m,2 ≤ a [9] m,1 = d1(r,q), (4.5) where a [9] m,l = 1 λ− 1 (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   . (4.6) proof. by applying theorem 1.1 with i = (0,∞), f : (0,∞) → r, f(t) := t µ−1 λ−1 ps := rs, xs := ( rs qs )λ−1 , s = 1, . . . ,n, int. j. anal. appl. 17 (5) (2019) 695 we have ( n∑ s=1 qs ( rs qs )λ)µ−1λ−1 = ( n∑ s=1 rs ( rs qs )λ−1)µ−1λ−1 ≤ . . . ≤ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij     l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   µ−1 λ−1 ≤ . . . ≤ n∑ s=1 rs (( rs qs )λ−1)µ−1λ−1 , (4.7) if either 0 ≤ λ < 1 < β or 1 < λ ≤ µ, and the reverse inequality in (4.7) holds if 0 ≤ λ ≤ β < 1. by raising to power 1 µ−1 , we have from all ( n∑ s=1 qs ( rs qs )λ) 1λ−1 ≤ . . . ≤  (m− 1)!(l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij     l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   µ−1 λ−1   1 µ−1 ≤ . . . ≤   n∑ s=1 rs (( rs qs )λ−1)µ−1λ−1  1 µ−1 = ( n∑ s=1 qs ( rs qs )µ) 1µ−1 . (4.8) since log is increasing if the base of log is greater than 1, it now follows (4.3). if the base of log is between 0 and 1, then log is decreasing and therefore inequality in (4.3) are reversed. if λ = 1 and β = 1, we have (ii) and (iii) respectively by taking limit, when λ goes to 1. � theorem 4.2. assume (h1), let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) are probability distributions. if either 0 ≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base of log is between 0 and 1, then 1∑n s=1 qs ( rs qs )λ n∑ s=1 qs ( rs qs )λ log ( rs qs ) = a [10] m,1 ≤ a [10] m,2 ≤ . . . ≤ a [10] m,m−1 ≤ a [10] m,m ≤ dλ(r,q) ≤ a [11] m,m ≤ a[11]m,m ≤ . . . ≤ a [11] m,2 ≤ a [11] m,1 = d1(r,q) (4.9) int. j. anal. appl. 17 (5) (2019) 696 where a[10]m,m = 1 (λ− 1) ∑n s=1 qs ( rs qs )λ (m− 1)!(l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij ( rij qij )λ−1 log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   and a[11]m,m = 1 λ− 1 (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   . the inequalities in (4.9) are reversed if either 0 ≤ λ < 1 and the base of log is between 0 and 1, or 1 < λ and the base of log is greater than 1. proof. we prove only the case when 0 ≤ λ < 1 and the base of log is greater than 1 and the other cases can be proved similarly. since 1 λ−1 < 0 and the function log is concave then choose i = (0,∞), f := log, ps = rs, xs := ( rs qs )λ−1 in theorem 1.1, we have dλ(r, q) = 1 λ− 1 log ( n∑ s=1 qs ( rs qs )λ) = 1 λ− 1 log ( n∑ s=1 rs ( rs qs )λ−1) ≤ . . . ≤ 1 λ− 1 (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   ≤ . . . ≤ 1 λ− 1 n∑ s=1 rs log (( rs qs )λ−1) = n∑ s=1 rs log ( rs qs ) = d1(r, q) (4.10) and this give the upper bound for dλ(r, q). since the base of log is greater than 1, the function x 7→ xf(x) (x > 0) is convex therefore 1 1−λ < 0 and int. j. anal. appl. 17 (5) (2019) 697 theorem 1.1 gives dλ(r, q) = 1 λ− 1 log ( n∑ s=1 qs ( rs qs )λ) = 1 λ− 1 (∑n s=1 qs ( rs qs )λ) ( n∑ s=1 qs ( rs qs )λ) log ( n∑ s=1 qs ( rs qs )λ) ≥ . . . ≥ 1 λ− 1 (∑n s=1 qs ( rs qs )λ) (m− 1)!(l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij     l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   = 1 λ− 1 (∑n s=1 qs ( rs qs )λ) (m− 1)!(l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij ( rij qij )λ−1 log   l∑ j=1 rij αim,ij ( rij qij )λ−1 l∑ j=1 rij αim,ij   ≥ . . . ≥ 1 λ− 1 n∑ s=1 rs ( rs qs )λ−1 log ( rs qs )λ−1 1∑n s=1 rs ( rs qs )λ−1 = 1∑n s=1 qs ( rs qs )λ n∑ s=1 qs ( rs qs )λ log ( rs qs ) (4.11) which give the lower bound of dλ(r, q). � by using the theorem 4.1, theorem 4.2 and definition 4.1, some inequalities of rényi entropy are obtained. let 1 n = ( 1 n , . . . , 1 n ) be a discrete probability distribution. corollary 4.3. assume (h1), let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) are positive probability distributions. (i) if 0 ≤ λ ≤ µ, λ,µ 6= 1, and the base of log is greater than 1, then hλ(r) = log(n) −dλ ( r, 1 n ) ≥ a[12]m,m ≥ a [12] m,m ≥ . . .a [12] m,2 ≥ a [12] m,1 = hµ(r), (4.12) where a [12] m,l = 1 1 −µ log  (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il) ×   l∑ j=1 rij αim,ij     l∑ j=1 rλij αim,ij l∑ j=1 rij αim,ij   µ−1 λ−1   . int. j. anal. appl. 17 (5) (2019) 698 the reverse inequalities holds in (4.12) if the base of log is between 0 and 1. (ii) if 1 < µ and base of log is greater than 1, then s = − n∑ s=1 pi log(pi) ≥ a[13]m,m ≥ a [13] m,m−1 ≥ . . . ≥ a [13] m,2 ≥ a [13] m,1 = hµ(r) (4.13) where a [13] m,l = log(n) + 1 1 −µ log  (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   exp   (µ− 1) l∑ j=1 rij αim,ij log ( nrij ) l∑ j=1 rij αim,ij     , the base of exp is same as the base of log. the inequalities in (4.13) are reversed if the base of log is between 0 and 1. (iii) if 0 ≤ λ < 1, and the base of log is greater than 1, then hλ(r) ≥ a[14]m,m ≥ a [14] m,m−1 ≥ . . . ≥ a [14] m,2 ≤ a [14] m,1 = s, (4.14) where a[14]m,m = 1 1 −λ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   log   l∑ j=1 rλij αim,ij l∑ j=1 rij αim,ij   . (4.15) the inequalities in (4.14) are reversed if the base of log is between 0 and 1. proof. (i) suppose q = 1 n then from (4.1), we have dλ(r, q) = 1 λ− 1 log ( n∑ s=1 nλ−1rλs ) = log(n) + 1 λ− 1 log ( n∑ s=1 rλs ) , (4.16) therefore we have hλ(r) = log(n) −dλ(r, 1 n ). (4.17) now using theorem 4.1 (i) and (4.17), we get hλ(r) = log(n) −dλ ( r, 1 n ) ≥ . . . ≥ log(n) − 1 µ− 1 log  nµ−1 (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il) ×   l∑ j=1 rij αim,ij     l∑ j=1 rλij αim,ij l∑ j=1 rij αim,ij   µ−1 λ−1   ≥ . . . ≥ log(n) −dµ(r, q) = hµ(r), (4.18) (ii) and (iii) can be proved similarly. � int. j. anal. appl. 17 (5) (2019) 699 corollary 4.4. assume (h1) and let r = (r1, . . . ,rn) and q = (q1, . . . ,qn) are positive probability distributions. if either 0 ≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base of log is between 0 and 1, then − 1∑n s=1 r λ s n∑ s=1 rλs log(rs) = a [15] m,1 ≥ a [15] m,2 ≥ . . . ≥ a [15] m,m−1 ≥ a [15] m,m ≥ hλ(r) ≥ a [16] m,m ≥ a[16]m,m−1 ≥ . . .a [16] m,2 ≥ a [16] m,1 = h (r) , (4.19) where a [15] m,l = 1 (λ− 1) ∑n s=1 r λ s (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rλij αim,ij   log  nλ−1 l∑ j=1 rλij αim,ij l∑ j=1 rij αim,ij   and a [16] m,1 = 1 1 −λ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 rij αim,ij   log   l∑ j=1 rλij αim,ij l∑ j=1 rij αim,ij   . the inequalities in (4.19) are reversed if either 0 ≤ λ < 1 and the base of log is between 0 and 1, or 1 < λ and the base of log is greater than 1. proof. the proof is similar to the corollary 4.3 by using theorem 4.2. � 5. inequalities by using zipf-mandelbrot law in probability theory and statistics, the zipf-mandelbrot law is a distribution. it is a power law distribution on ranked data, named after the linguist g. k. zipf who suggest a simpler distribution called zipf’s law. the zipf’s law is defined as follow (see [28]). definition 5.1. let n be a number of elements, s be their rank and t be the value of exponent characterizing the distribution. zipf ’s law then predicts that out of a population of n elements, the normalized frequency of element of rank s, f(s,n,t) is f(s,n,t) = 1 st∑n j=1 1 jt . (5.1) the zipf-mandelbrot law is defined as follows (see [21]). definition 5.2. zipf-mandelbrot law is a discrete probability distribution depending on three parameters n ∈{1, 2, . . . ,},q ∈ [0,∞) and t > 0, and is defined by f(s; n,q,t) := 1 (s + q)thn,q,t , s = 1, . . . ,n, (5.2) int. j. anal. appl. 17 (5) (2019) 700 where hn,q,t = n∑ j=1 1 (j + q)t . (5.3) if the total mass of the law is taken over all n, then for q ≥ 0, t > 1, s ∈ n, density function of zipfmandelbrot law becomes f(s; q,t) = 1 (s + q)thq,t , (5.4) where hq,t = ∞∑ j=1 1 (j + q)t . (5.5) for q = 0, the zipf-mandelbrot law (5.2) becomes zipf ’s law (5.1). conclusion 5.1. assume (h1), let r be a zipf-mandelbrot law, by corollary 4.3 (iii), we get. if 0 ≤ λ < 1, and the base of log is greater than 1, then hλ(r) = 1 1 −λ log ( 1 hλn,q,t n∑ s=1 1 (s + q)λs ) ≥ . . . ≥ 1 1 −λ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 1 αim,ij (ij + q)hn.q,t   log   1hλ−1n,q,t l∑ j=1 1 αim,ij (ij−q) λs l∑ j=1 1 αim,ij (ij−q) s   ≥ . . . ≥ t hn,q,t n∑ s=1 log(s + q) (s + q)t + log(hn,q,t) = s. (5.6) the inequalities in (5.6) are reversed if the base of log is between 0 and 1. conclusion 5.2. assume (h1), let r1 and r2 be the zipf-mandelbort law with parameters n ∈ {1, 2, . . .}, q1,q2 ∈ [0,∞) and s1,s2 > 0, respectively, then from corollary 3.2 (ii), we have if the base of log is greater than 1, then d̄(r1,r2) = n∑ s=1 1 (s + q1)t1hn,q1,t1 log ( (s + q2) t2hn,q2,t2 (s + q1)t1hn,q2,t1 ) ≥ . . . ≥ (m− 1)! (l− 1)! ∑ (i1,...,il)∈il ηim,l(i1, . . . , il)   l∑ j=1 1 (ij+q2) t2hn,q2,t2 αim,ij     ∑l j=1 1 (ij+q1) t1hn,q1,t1 αim,ij∑l j=1 1 (ij+q2) t2hn,q2,t2 αim,ij   log   ∑l j=1 1 (ij+q1) t1hn,q1,t1 αim,ij∑l j=1 1 (ij+q2) t2hn,q2,t2 αim,ij   ≥ . . . ≥ 0. (5.7) the inequalities in (5.7) are reversed if base of log is between 0 and 1. int. j. anal. appl. 17 (5) (2019) 701 6. shannon entropy, zipf-mandelbrot law and hybrid zipf-mandelbrot law here we maximize the shannon entropy using method of lagrange multiplier under some equations constraints and get the zipf-mandelbrot law. theorem 6.1. if j = {1, 2, . . . ,n}, for a given q ≥ 0 a probability distribution that maximize the shannon entropy under the constraints ∑ s∈j rs = 1, ∑ s∈j rs (ln(s + q)) := ψ, is zipf-madelbrot law. proof. if j = {1, 2, . . . ,n}. we set the lagrange multipliers λ and t and consider the expression s̃ = − n∑ s=1 rs ln rs −λ ( n∑ s=1 rs − 1 ) − t ( n∑ s=1 rs ln(s + q) − ψ ) just for the sake of convenience, replace λ by ln λ− 1, thus the last expression gives s̃ = − n∑ s=1 rs ln rs − (ln λ− 1) ( n∑ s=1 rs − 1 ) − t ( n∑ s=1 rs ln(s + q) − ψ ) from s̃rs = 0, for s = 1, 2, . . . ,n, we get rs = 1 λ (s + q) t , and on using the constraint ∑n s=1 rs = 1, we have λ = n∑ s=1 ( 1 (s + 1)t ) where t > 0, concluding that rs = 1 (s + q)thn,q,t , s = 1, 2, . . . ,n. � remark 6.2. observe that the zipf-mandelbrot law and shannon entroy can be bounded from above (see [22]). s = − n∑ s=1 f (s,n,q,t) ln f(s,n,q,t) ≤− n∑ s=1 f(s,n,q,t) ln qs where (q1, . . . ,qn ) is a positive n-tuple such that ∑n s=1 qs = 1. int. j. anal. appl. 17 (5) (2019) 702 theorem 6.3. if j = {1, . . . ,n}, then probability distribution that maximize shannon entropy under constraints ∑ s∈j rs = 1, ∑ s∈j rs ln(s + q) := ψ, ∑ s∈j srs := η is hybrid zipf-mandelbrot law given as rs = ws (s + q) k φ∗(k,q,w) , s ∈ j, where φj(k,q,w) = ∑ s∈j ws (s + q)k . proof. first consider j = {1, . . . ,n}, we set the lagrange multiplier and consider the expression s̃ = − n∑ s=1 rs ln rs + ln w ( n∑ s=1 srs −η ) − (ln λ− 1) ( n∑ s=1 rs − 1 ) −k ( n∑ s=1 rs ln(s + q) − ψ ) . on setting s̃rs = 0, for s = 1, . . . ,n, we get − ln rs + s ln w − ln λ−k ln(s + q) = 0, after solving for rs, we get λ = n∑ s=1 ws (s + q) k , and we recognize this as the partial sum of lerch’s transcendent that we will denote with φ∗n (k,q,w) = n∑ s=1 ws (s + q)k with w ≥ 0,k > 0. � remark 6.4. observe that for zipf-mandelbrot law, shannon entropy can be bounded from above (see [22]). s = − n∑ s=1 fh (s,n,q,k) ln fh (s,n,q,k) ≤− n∑ s=1 fh (s,n,q,k) ln qs where (q1, . . . ,qn ) is any positive n-tuple such that ∑n s=1 qs = 1 under the assumption of theorem 2.1 (i), define the non-negative functionals as follows. θ3(f) = a[1]m,r −f (∑n s=1 rs∑n s=1 qs ) n∑ s=1 qs, r = 1, . . . ,m, (6.1) θ4(f) = a[1]m,r −a [1] m,k, 1 ≤ r < k ≤ m. (6.2) int. j. anal. appl. 17 (5) (2019) 703 under the assumption of theorem 2.1 (ii), define the non-negative functionals as follows. θ5(f) = a[2]m,r − ( n∑ s=1 rs ) f (∑n s=1 rs∑n s=1 qs ) , r = 1, . . . ,m, (6.3) θ6(f) = a[2]m,r −a [2] m,k, 1 ≤ r < k ≤ m. (6.4) under the assumption of corollary 3.1 (i), define the following non-negative functionals. θ7(f) = a [3] m,r + n∑ i=1 qi log(qi), r = 1, . . . ,n (6.5) θ8(f) = a [3] m,r −a [3] m,k, 1 ≤ r < k ≤ m. (6.6) under the assumption of corollary 3.1 (ii), define the following non-negative functionals give as. θ9(f) = a [4] m,r −s, r = 1, . . . ,m (6.7) θ10(f) = a [4] m,r −a [4] m,k, 1 ≤ r < k ≤ m. (6.8) under the assumption of corollary 3.2 (i), let us define the non-negative functionals as follows. θ11(f) = a [5] m,r − n∑ s=1 rs log ( n∑ s=1 log rn∑n s=1 qs ) , r = 1, . . . ,m (6.9) θ12(f) = a [5] m,r −a [5] m,k, 1 ≤ r < k ≤ m. (6.10) under the assumption of corollary 3.2 (ii), define the non-negative functionals as follows. θ13(f) = a [6] m,r −a [6] m,k, 1 ≤ r < k ≤ m. (6.11) under the assumption of theorem 4.1 (i), consider the following functionals. θ14(f) = a [7] m,r −dλ(r, q), r = 1, . . . ,m (6.12) θ15(f) = a [7] m,r −a [7] m,k, 1 ≤ r < k ≤ m. (6.13) under the assumption of theorem 4.1 (ii), consider the following functionals. θ16(f) = a [8] m,r −d1(r, q), r = 1, . . . ,m (6.14) θ17(f) = a [8] m,r −a [8] m,k, 1 ≤ r < k ≤ m. (6.15) under the assumption of theorem 4.1 (iii), consider the following functionals. θ18(f) = a [9] m,r −dλ(r, q), r = 1, . . . ,m (6.16) θ19(f) = a [9] m,r −a [9] m,k, 1 ≤ r < k ≤ m. (6.17) int. j. anal. appl. 17 (5) (2019) 704 under the assumption of theorem 4.2 consider the following non-negative functionals. θ20(f) = dλ(r, q) −a[10]m,r, r = 1, . . . ,m (6.18) θ21(f) = a [10] m,k −a [10] m,r, 1 ≤ r < k ≤ m. (6.19) θ22(f) = a [11] m,r −dλ(r, q), r = 1, . . . ,m (6.20) θ23(f) = a [11] m,r −a [11] m,r, 1 ≤ r < k ≤ m. (6.21) θ24(f) = a [11] m,r −a [10] m,k, r = 1, . . . ,m, k = 1, . . . ,m (6.22) under the assumption of corollary 4.3 (i), consider the following non-negative functionals. θ25(f) = hλ(r) −a[12]m,r, r = 1, . . . ,m (6.23) θ26(f) = a [12] m,k −a [12] m,r, 1 ≤ r < k ≤ m. (6.24) under the assumption of corollary 4.3 (ii), consider the following functionals θ27(f) = s −a[13]m,r, r = 1, . . . ,m (6.25) θ28(f) = a [13] m,k −a [13] m,r, 1 ≤ r < k ≤ m. (6.26) under the assumption of corollary 4.3 (iii), consider the following functionals. θ29(f) = hλ(r) −a[14]m,r, r = 1, . . . ,m (6.27) θ30(f) = a [14] m,k −a [14] m,r, 1 ≤ r < k ≤ m. (6.28) under the assumption of corollary 4.4, defined the following functionals. θ31 = a [15] m,r −hλ(r), r = 1, . . . ,m (6.29) θ32 = a [15] m,r −a [15] m,k, 1 ≤ r < k ≤ m. (6.30) θ33 = hλ(r) −a[16]m,r, r = 1, . . . ,m (6.31) θ34 = a [16] m,k −a [16] m,r, 1 ≤ r < k ≤ m. (6.32) θ35 = a [15] m,r −a [16] m,k, r = 1, . . . ,m, k = 1, . . . ,m. (6.33) 7. generalization of refinement of jensen’s, rényi and shannon type inequalities via taylor one point and taylor two points interpolations in [5], the following functions are consider to generalized the popoviciu’s inequality, defined as (u−v)+ =   (u−v), v ≤ u;0, v > u, int. j. anal. appl. 17 (5) (2019) 705 and the well known taylor formula is as follows. let m be a positive integer and f : [α1,α2] → r be such that f(m−1) is absolutely continuous, then for all u ∈ [α1,α2] the taylor’s formula at point c ∈ [α1,α2] is f(u) = tm−1(f; c; u) + rm−1(f; c; u), (7.1) where tm−1(f; c; u) = m−1∑ l=0 f(l)(c) l! (u− c)l, and the remainder is given by rm−1(f; c; u) = 1 (m− 1)! ∫ u c f(m)(t)(u− t)m−1dt. the taylor’s formula at point α1 and α2 is given by: f(u) = m−1∑ l=0 f(l)(α1) l! (u−α1)l + 1 (m− 1)! ∫ α2 α1 f(m)(t) ( (u− t)m−1+ ) dt. (7.2) f(u) = m−1∑ l=0 (−1)lf(l)(α2) l! (α2 −u)l + (−1)m−1 (m− 1)! ∫ α2 α1 f(m)(t) ( (t−u)m−1+ ) dt. (7.3) we construct some new identities with the help of taylor polynomial (7.1). theorem 7.1. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. then we have the following identities: (i) θi(f) = m−1∑ l=2 f(l)(α1) l! θi ( (u−α1)l ) + 1 (m− 1)! ∫ α2 α1 f(m)(t)θi ( (u− t)m−1+ ) dt, i = 1, 2, . . . , 35. (7.4) (ii) θi(f) = m−1∑ l=2 (−1)lf(l)(α2) l! θi ( (α2 −u)l ) + (−1)m−1 (m− 1)! ∫ α2 α1 f(m)(t)θi ( (t−u)m−1+ ) dt, i = 1, 2, . . . , 35. (7.5) proof. using (7.2) and (7.3) in (1.3), we get the required result. � theorem 7.2. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. let f is m-convex function such that f(m−1) is absolutely continuous. then we have the following results: (i) if θi ( (u− t)m−1+ ) ≥ 0 t ∈ [α1,α2], i = 1, 2, . . . , 35, int. j. anal. appl. 17 (5) (2019) 706 then θi(f(u)) ≥ m−1∑ l=2 f(l)(α1) l! θi ( (u−α1)l ) , i = 1, 2, . . . , 35. (7.6) (ii) if (−1)m−1θi ( (t−u)m−1+ ) ≤ 0 t ∈ [α1,α2], i = 1, 2, . . . , 35, then θi(f(u)) ≥ m−1∑ l=2 (−1)lf(l)(α2) l! θi ( (α2 −u)l ) , i = 1, 2, . . . , 35. (7.7) proof. since f(m−1) is absolutely continuous on [α1,α2], f (m) exists almost everywhere. as f is m-convex therefore f(m)(u) ≥ 0 for all u ∈ [α1,α2]. hence using theorem 7.1 we obtain (7.6) and (7.7). � theorem 7.3. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. then the following results are valid. (i) if f is m-convex, then (7.6) holds. also if f(l)(α1) ≥ 0 for l = 2, . . . ,m− 1, then the right hand side of (7.6) will be non-negative. (ii) if m is even and f is m-convex, then (7.7) holds. also if f(l)(α1) ≤ 0 for l = 2, . . . ,m− 1 and f(l) ≥ 0 for l = 3, . . . ,m− 1, then right hand side of (7.7) will be non-negative. (iii)if m is odd and f is m-convex function then (7.7) is valid. also if f(l)(α2) ≥ 0 for l = 2, . . . ,m−1 and f(l)(α2) ≤ 0 for l = 2, . . . ,m− 2, then right hand side of (7.7) will be non positive. in [7, p.20] the green function g : [α1,α2] × [α1,α2] → r is defined as g(u,v) =   (u−α2)(v−α1) α2−α1 , α1 ≤ v ≤ u; (v−α2)(u−α1) α2−α1 , u ≤ v ≤ α2. (7.8) the function g is convex and continuous with respect to v, since g is symmetric therefore it is also convex and continuous with respect to variable u. let ψ ∈ c2 ([α1,α2]), then ψ (t) = α2 − t α2 −α1 ψ(α1) + t−α1 α2 −α1 ψ(α2) + α2∫ α1 g (t,v) ψ′′(v)dv. (7.9) theorem 7.4. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. then we have the int. j. anal. appl. 17 (5) (2019) 707 following results: (i) for i = 1, 2, . . . , 35, θi(f) = α2∫ α1 θi(g(t,v)) ( n−1∑ l=1 f(l)(α1)(v − α1)l−2 (l − 2)! ) dv + 1 (n − 3)! α2∫ α1 f (m) (s)   α2∫ α1 θi(g(t,v))(v − s)n−3dv  ds. (7.10) (ii) for i = 1, 2, . . . , 35, θi(f) = α2∫ α1 θi(g(t,v)) ( n−1∑ l=1 fl(α2)(v − α2)l−2 (l − 2)! ) dv − 1 (n − 3)! α2∫ α1 f (m) (s)   α2∫ α1 θi(g(t,v))(v − s)n−3dv  ds (7.11) proof. using (7.9) in θi, i = 1, 2, . . . , 35, we get θi(f) = α2∫ α1 θi (g(t,v)) f ′′(v)dv. (7.12) differentiate (7.2) twice, we get f′′(v) = n−1∑ l=2 f(l)(α1) (l− 2)! (v −α1)l−2 + 1 (m− 3)! α2∫ α1 f(m)(v −u)m−3du. (7.13) using (7.13) in (7.12) and using fubini’s theorem, we get (7.10). similarly use second derivative of (7.3) in (7.12) and apply fubini’s theorem, we get (7.11). � now we obtain generalization of refinement of jensen’s inequality for n-convex function. theorem 7.5. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. let f is m-convex function such that f(m−1) is absolutely continuous. then we have the following results: (i) if α2∫ u θi (g(t,v)) (v −u)n−3dv ≥ 0 u ∈ [α1,α2], i = 1, 2, . . . , 35, (7.14) then θi(f) ≥ α2∫ α1 θi (g(t,v)) ( n−2∑ l=2 f(l)(α1)(v −α1)l−2 (l− 2)! ) dv, i = 1, 2, . . . , 35, (7.15) and if u∫ α1 θi (g(t,v)) (v −u)n−3dv ≤ 0 u ∈ [α1,α2], i = 1, 2, . . . , 35, (7.16) then θi(f) ≥ α2∫ α1 θi (g(t,v)) ( n−2∑ l=2 f(l)(α2)(v −α2)l−2 (l− 2)! ) dv i = 1, 2, . . . , 35. (7.17) int. j. anal. appl. 17 (5) (2019) 708 proof. similar to the proof of theorem 7.2. � corollary 7.6. assume (h1), let f : [α1,α2] → r be a function where [α1,α2] ⊂ r be an interval. also let x1, . . . ,xn ∈ [α1,α2] and p1, . . . ,pn are positive real numbers such that n∑ i=1 pi = 1. then the following results are valid. (i) if f is m-convex, then (7.15) holds. also if n−1∑ l=2 f(l)(α1)(v −α1)l−2 (l− 2)! ≥ 0, (7.18) then θi (f) ≥ 0, i = 1, 2, . . . , 35. (7.19) (ii) if m is even and f is m-convex, then (7.17) holds. also if n−1∑ l=2 f(l)(α2)(v −α2)l−2 (l− 2)! ≥ 0, (7.20) then (7.19) holds. remark 7.7. we can investigate the bounds for the identities related to the generalization of refinement of jensen inequality using inequalities for the c̆ebys̆ev functional and some results relating to the gr̈uss and ostrowski type inequalities can be constructed as given in section 3 of [5]. also we can construct the nonnegative functionals from inequalities (7.6), (7.7), (7.15) and (7.17) and give related mean value theorems and we can construct the new families of m-exponentially convex functions and cauchy means related to these functionals as given in section 4 of [5]. funding the research of 4th author was supported by the ministry of education and science of the russian federation (the agreement number no. 02.a03.21.0008). competing interests the authors declares that there is no conflict of interests regarding the publication of this paper. authors contribution all authors jointly worked on the results and they read and approved the final manuscript. acknowledgements the authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. int. j. anal. appl. 17 (5) (2019) 709 references [1] anderson, g., & ge, y. the size distribution of chinese cities. reg. sci. urban econ., 35 (2005), 756-776. [2] auerbach, f. (1913). das gesetz der bevlkerungskonzentration. petermanns geographische mitteilungen, 59 (2005), 74-76. [3] black, d., & henderson, v. urban evolution in the usa. j. econ. geogr., 3 (2003), 343-372. [4] bosker, m., brakman, s., garretsen, h., & schramm, m. a century of shocks: the evolution of the german city size distribution 19251999. reg. sci. urban econ., 38 (2008), 330-347. [5] butt, s. i., khan, k. a., & pečarić, j. generaliztion of popoviciu inequality for higher order convex function via tayor’s polynomial, acta univ. apulensis math. inform., 42 (2015), 181-200. [6] butt, s. i., mehmood, n., & pečarić, j. new generalizations of popoviciu type inequalities via new green functions and fink’s identity. trans. a. razmadze math. inst., 171 (2017), 293-303. [7] butt, s. i., & pečarić, j. popoviciu’s inequality for n-convex functions. lap lambert academic publishing, (2016). [8] butt, s. i., & pečarić, j. weighted popoviciu type inequalities via generalized montgomery identities. rad hazu. mat. znan., 19 (2015), 69-89. [9] butt, s. i., khan, k. a., & pečarić, j. popoviciu type inequalities via hermite’s polynomial. math. inequal. appl., 19 (2016), 1309-1318. [10] horváth, l. a method to refine the discrete jensen’s inequality for convex and mid-convex functions. math. computer model., 54 (2011), 2451-2459. [11] horváth, l., khan, k. a., & pečarić, j. combinatorial improvements of jensens inequality / classical and new refinements of jensens inequality with applications, monographs in inequalities 8, element, zagreb. (2014). [12] horváth, l., khan, k. a., & pečarić, j. refinement of jensen’s inequality for operator convex functions. adv. inequal. appl., 2014 (2014), art. id 26. [13] horváth, l., pečarić, j. a refinement of discrete jensen’s inequality, math. inequal. appl. 14 (2011), 777-791. [14] ioannides, y. m., & overman, h. g. zipf’s law for cities: an empirical examination. reg. sci. urban econ., 33 (2003), 127-137. [15] csiszár, i. information measures: a critical survey. in: tans. 7th prague conf. on info. th., statist. decis. funct., rand. proc. 8th eur. meeting stat., vol. b (1978), 73-86. [16] csiszár. i. . information-type measures of difference of probability distributions and indirect observations. stud. sci. math. hungar. 2 (1967), 299-318. [17] horváth, l., pecaric, d. & pečarić, j. estimations of f-and rényi divergences by using a cyclic refinement of the jensen’s inequality. bull. malaysian math. sci. soc., 42 (2019). 933-946 . [18] khan, k. a., niaz, t., pečarić, d̄., pečarić, j. refinement of jensen’s inequality and estimation of fand renyi divergence via montgomery identity. j. inequal. appl., 2018 (2018), art. id 318. [19] kullback, s. information theory and statistics. courier corporation. [20] kullback, s., & leibler, r. a. (1951). on information and sufficiency. anna. math. stat., 22 (1997), 79-86. math. dokl. 4(1963), 121-124. [21] lovricevic, n., pecaric, d. & pecaric, j. zipfmandelbrot law, f-divergences and the jensen-type interpolating inequalities. j. inequal. appl., 2018 (2018), art. id 36. [22] matic, m., pearce, c. e., & pečarić, j. shannon’s and related inequalities in information theory. in survey on classical inequalities (pp. 127-164). springer, dordrecht. (2000). int. j. anal. appl. 17 (5) (2019) 710 [23] niaz, t., khan, k. a., & pečarić, j. on generalization of refinement of jensen’s inequality using fink’s identity and abel-gontscharoff green function. j. inequal. appl., 2017 (2017), art. id 254. [24] pečarić, j., proschan, f., & tong, y. l. convex functions, partial orderings and statistical applications, academic press, new york. (1992). [25] rényi, a. on measure of information and entropy. in: proceeding of the fourth berkely symposium on mathematics, statistics and probability, pp. 547-561. (1960). [26] rosen, k. t., & resnick, m. the size distribution of cities: an examination of the pareto law and primacy. j. urban econ., 8 (1980), 165-186. [27] soo, k. t. zipf’s law for cities: a cross-country investigation. reg. sci. urban econ., 35 (2005), 239-263. [28] zipf, g. k. human behaviour and the principle of least-effort. cambridge ma edn. reading: addison-wesley. (1949). 1. introduction and preliminary results 2. inequalities for csiszár divergence 3. inequalities for shannon entropy 4. inequalities for rényi divergence and entropy 5. inequalities by using zipf-mandelbrot law 6. shannon entropy, zipf-mandelbrot law and hybrid zipf-mandelbrot law 7. generalization of refinement of jensen's, rényi and shannon type inequalities via taylor one point and taylor two points interpolations references international journal of analysis and applications volume 17, number 3 (2019), 406-419 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-406 homotopy perturbation method combined with zz transform to solve some nonlinear fractional differential equations lakhdar riabi∗, kacem belghaba, mountassir hamdi cherif, djelloul ziane laboratory of mathematics and its applications (lamap), université oran1, oran 31000, algeria ∗corresponding author: mountassir27@yahoo.fr abstract. the idea proposed in this work is to extend the zz transform method to resolve the nonlinear fractional partial differential equations by combining them with the so-called homotopy perturbation method (hpm). we apply this technique to solve some nonlinear fractional equations as: nonlinear time-fractional fokker-planck equation, the cubic nonlinear time-fractional schrödinger equation and the nonlinear timefractional kdv equation. the fractional derivative is described in the caputo sense. the results show that this is the appropriate method to solve somme models of nonlinear partial differential equations with time-fractional derivative. 1. introduction fractional calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders. during the last decade, fractional calculus has found applications in numerous seemingly diverse fields of science and engineering. fractional differential equations are increasingly used to model problems in fluid mechanics, acoustics, biology, electromagnetism, diffusion, signal processing, and many other physical processes [14]. since the physical side is often associated with fractional differential equations as explained in the previous paragraph, many researchers have used existing methods to solve this type of equations, others are trying received 2019-01-30; accepted 2019-03-18; published 2019-05-01. 2010 mathematics subject classification. 26a33, 44a05, 34k37, 35f61. key words and phrases. caputo fractional derivative; homotopy perturbation method; zz transform; fokker-plank equation; schrödinger equation; kdv equation. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 406 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-406 int. j. anal. appl. 17 (3) (2019) 407 to discover new methods faster than existing methods, and all this to facilitate the resolution of this type of equations, especially non-linear ones. these efforts have strengthened this area of research through many methods, among them we find homotopy perturbation method (hpm). this method was established in 1998 by j.h. he ( [10], [11], [12]) and applied to various linear and nonlinear differential equations, its application was also extended to linear and nonlinear fractional differential equations. then, a new option emerged recently, includes the composition of laplace transform, sumudu transform, natural transform or elzaki transform with this method for solving linear and nonlinear differential equations. among which are the homotopy perturbation transform method [13], homotopy perturbation sumudu transform method [17], natural homotopy perturbation method [1], homotopy perturbation elzaki transform method ( [3], [8]). the objective of the present study is to combine two powerful methods, homotopy perturbation method and zz transform method to get a better method to solve nonlinear fractional partial differential equations. the modified method is called fractional homotopy perturbation zz transform method (fhpzztm) and we will apply them to solve the following nonlinear time-fractional fokker-planck equation, the cubic nonlinear time-fractional schrödinger equation and the time-fractional kdv equation. the present paper has been organized as follows: in section 2, some basic notions about fractional calculus and basic definitions and properties of the zz transform method. in section 3, we give an analysis of the proposed method. in section 4, we present solutions to proposed equations explaining how to apply the proposed method. finally, the conclusion follows. 2. basic definitions in this section, we give some basic notions about fractional calculus, zz transform and zz transform of fractional derivatives which are used further in this paper. 2.1. fractional calculus. we give some basic definitions and properties of the fractional calculus theory as the riemann–liouville fractional integrals and caputo fractional derivative (see [7], [15]). definition 2.1. let ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis r. the riemann– liouville fractional integral iα0+f of order α ∈ r (α > 0) is defined by (iα0+f)(t) = 1 γ (α) ∫ t 0 f(τ)dτ (t− τ)1−α , t > 0, α > 0 (i00+f)(t) = f(t). here γ(·) is the gamma function. int. j. anal. appl. 17 (3) (2019) 408 theorem 2.1. let α > 0 and let n = [α] + 1. if f(t) ∈ acn [a,b] , then the caputo fractional derivative (cdα0+f)(t) exist almost everywhere on [a,b] . if α /∈ n, (cdα0+f)(t) is represented by (cdα0+f)(t) = 1 γ (n−α) ∫ t 0 f(n)(τ)dτ (t− τ)α−n+1 , (2.1) where d = d dx and n = [α] + 1. remark 2.1. in this paper, we consider the time-fractional derivative in the caputo’s sense. when α ∈ r+, the time-fractional derivative is defined as (cdαt u)(x,t) = ∂αu(x,t) ∂tα =   1 γ(m−α) ∫ t 0 (t− τ)m−α−1 ∂ mu(x,τ) ∂τm , m− 1 < α < m, ∂mu(x,t) ∂tm , α = m, where m ∈ n∗. 2.2. definitions and properties of the zz transform. the zz transform was defined by zain ul abadin zafar [20] in 2016. in this section, we give some basic definitions and properties of this transform (see [20]). definition 2.2. let u(t) be a function defined for all t ≥ 0. the zz transform of u(t) is the function t(v,s) defined by z [u(t)] = t(v,s) = s ∫ ∞ 0 u(vt)e−stdt. (2.2) theorem 2.2. if u(t) is piecewise continuous in every finite interval 0 ≤ t ≤ k and of exponential order γ for t > k, then its zz transform t(v,s) exists for all s > γ, v > γ. proof. (see [20]) � 2.2.1. some properties of the zz transform . 1. the zz transform of the nth derivative of u(t) is given by z[u(n)(t)] = sn vn z[u(t)] − n−1∑ k=0 sn−k vn−k u(k)(0). (2.3) 2. some elementary functions and their transformations int. j. anal. appl. 17 (3) (2019) 409 u(t) z [u(t)] 1 1 t v s tn n!v n sn , n = 0, 1, 2, . . . tα γ(α + 1)v α sα , α > 0. theorem 2.3. the zz transform of the time-fractional derivative in the caputo’s sense is defined as z [ (cdα0+u)(t); (v,s) ] = sα vα z[u(t)] − n−1∑ k=0 sα−k vα−k u(k)(0) , n− 1 < α ≤ n, n = 1, 2, . . . (2.4) proof. (see [19]) � 3. fractional homotopy perturbation zz transform method to illustrate the basic idea of this method, we consider a general nonlinear fractional partial differential non homogeneous equation with initial conditions of the form cdαt u(x,t) + ru(x,t) + nu(x,t) = g(x,t), (3.1) where t > 0,x ∈ r, 0 < α ≤ 1 and the initial conditions u(x, 0) = h(x). (3.2) cdαt u(x,t), is the caputo fractional derivative of the function u(x,t), r is the linear differential operator, n represent the general nonlinear differential operator, and h(x,t) is the source terms. applying the zz transform on both sides of (3.1) and using the differentiation property of this transform (2.4), we obtain z[u(x,t)] = h(x) + vα sα z[g(x,t)] − vα sα z[ru(x,t) + nu(x,t)]. (3.3) taking the inverse zz transform on both sides of equations in system (3.3) and then by using initial conditions (3.2), we have u(x,t) = g(x,t) −z−1( vα sα z[ru(x,t) + nu(x,t)]), (3.4) where g(x,t) represents the terms arising from the nonhomogeneous terms and the prescribed initial conditions. now, we applying the classical perturbation technique, we can assume that the solution can be expressed as a power series in p , as given below int. j. anal. appl. 17 (3) (2019) 410 u(x,t) = ∞∑ n=0 pnun(x,t), (3.5) where the homotopy parameter p , is considered as a small parameter ( p ∈ [0, 1]). the nonlinear terms can be decomposed as nu(x,t) = ∞∑ n=0 hn(u), (3.6) where hn are he’s polynomials [5] of u0,u1,u2, ...,un, and they can be calculated by the formulas given below hn(u0, ...,un) = 1 n ∂n ∂pn [n( ∞∑ i=0 piui)]p=0,n = 0, 1, 2... (3.7) using (3.5) and (3.6), we can rewrite (3.4) as ∞∑ n=0 pnun = g(x,t) −p(z−1[ vα sα z[r ∞∑ n=0 pnun + ∞∑ n=0 pnhn(u)]]). (3.8) this is a coupling of the zz transform and homotopy perturbation methods (zzhpm) using he’s polynomials. now, equating the coefficient of corresponding power of p on both sides of (3.8), we get u0(x,t) = g(x,t), ... un(x,t) = −z−1(v α sα z[run−1(x,t) + hn−1(u)]), (3.9) where n ∈ n∗. we continue in this manner to obtain the general recursive relations. finally, the approximate solution is calculated by u(x,t) = lim n→∞ n∑ n=0 un(x,t). (3.10) the convergence of the series (3.10) has been proved in ( [4], [5]). 4. applications of fractional homotopy perturbation zz transform method in this section, we apply fractional homotopy perturbation zz transform method with the caputo fractional derivative to solve nonlinear time-fractional fokker-planck equation, nonlinear time-fractional schrödinger equation and nonlinear time-fractional kdv equation. example 4.1. we consider the following nonlinear time-fractional fokker-planck equation cdαt u = ( x 3 u)x − ( 4 x u2)x + (u 2)xx, 0 < α ≤ 1, (4.1) int. j. anal. appl. 17 (3) (2019) 411 with the initial condition u(x, 0) = x2. (4.2) by applying zz transform of (4.1), we obtain z[dαt u] = z[( x 3 u)x] −z[( 4 x u2)x] + z[(u 2)xx]. (4.3) using the differentiation property of the zz transform in (4.1), we get z[u(x,t)] = x2 + vα sα ( z[( x 3 u)x] −z[( 4 x u2)x] + z[(u 2)xx] ) . (4.4) taking the inverse zz transform on both sides of (4.4), we obtain u(x,t) = x2 + z−1[ vα sα ( z[( x 3 u)x] −z[( 4 x u2)x] + z[(u 2)xx] ) ]. (4.5) by applying the aforesaid homotopy perturbation method, we have ∞∑ n=0 pnun = x 2 + p(z−1[ vα sα ( z[( x 3 ∞∑ n=0 pnun)x] −z[ p∑ n=0 hn(u)] ) ]). (4.6) equating the coefficient of the like power of p on both sides in (4.6), we get p0 : u0(x,t) = x 2, ... pn : un(x,t) = z −1[v α sα ( z[(x 3 ∑∞ n=0 p nun)x] −z[ ∑p n=0 hn(u)] ) ], (4.7) where n ∈ n∗. the first few components of he’s polynomials, are given by h0 = −[ 4xu 2 0]x + u 2 0xx, h1 = −[ 4x2u0u1]x + 2(u0u1)xx, h2 = −[ 4x(2u0u2 + u 2 1)]x + 2(2u0u2 + u 2 1)xx, h3 = −[ 4x(2u0u3 + 2u1u2)]x + 2(2u0u3 + 2u1u2)xx, ... (4.8) using he’s polynomials (4.8) and the iteration formulas (4.7) we obtain int. j. anal. appl. 17 (3) (2019) 412 u0(x,t) = x 2 u1(x,t) = x 2 t α γ(α+1) , u2(x,t) = x 2 t 2α γ(2α+1) u3(x,t) = x 2 t 3α γ(3α+1) u4(x,t) = x 2 t 4α γ(4α+1) , ... (4.9) the first four terms of the decomposition series solution for (4.1) are given by u(x,t) = x2 + x2 tα γ(α + 1) + x2 t2α γ(2α + 1) + x2 t3α γ(3α + 1) + x2 t4α γ(4α + 1) + ... hence u(x,t) = x2 +∞∑ k=0 tkα γ(kα + 1) = x2eα(t α), where eα(t α) is a mittag-leffler function defined as: eα(t α) = ∑+∞ k=0 tkα γ(kα+1) . at special case, when α = 1 we obtain u(x,t) = x2et, (4.10) which is the exact solution to the fokker-planck equation of (4.1)-(4.2) as presented in [18] by adomian decomposition method (adm), in [16] by varational iteration method (vim) and in [6] by homotopy perturbation method (hpm). (a) (b) figure 1. (a) the exact solution, (b) the approximate solution when α = 1 of (4.1)-(4.2) int. j. anal. appl. 17 (3) (2019) 413 (c) (d) (e) figure 2. (c) the approximate solution when α = 0, 7, (d) the approximate solution when α = 0, 9 of, (e) plots of u(x,t) versus at x = 1 for different values for α = 1,α = 0, 9,α = 0, 7 and α = 0, 5 of (4.1)-(4.2) example 4.2. we consider the following cubic nonlinear time-fractional schrödinger equation icdαt u + uxx − 2|u| 2u = 0, 0 < α ≤ 1, (4.11) with the initial condition u(x, 0) = eix. (4.12) by applying zz transform of (4.11), we obtain iz[dαt u] + z[uxx] − 2z[|u| 2u] = 0. (4.13) using the differentiation property of the zz transform and initial conditions in (4.13), we get z[u(x,t)] = eit + i vα sα ( z[uxx] − 2iz[|u|2u] ) . (4.14) int. j. anal. appl. 17 (3) (2019) 414 taking the inverse zz transform on both sides of (4.14), we obtain u(x,t) = z−1[ vα sα ( iz[uxx] − 2iz[|u|2u] ) ]. (4.15) by applying the homotopy perturbation method, we have ∞∑ n=0 pnun = e ix + p(z−1[ vα sα ( iz[( ∞∑ n=0 pnun)xx] − 2iz[ p∑ n=0 hn(u)] ) ]). (4.16) equating the coefficient of the like power of p on both sides in (4.16), we get p0 : u0(x,t) = e ix, ... pn : un(x,t) = z −1[v α sα (iz[( ∑∞ n=0 p nun)xx] − 2iz[ ∑p n=0 hn(u)])], (4.17) where n ∈ n∗. the nonlinear term is given by n(u) = |u|2u, |u|2 = uu, n(u) = u2u. (4.18) the first few components of he’s polynomials, are given by h0 = u0u0, h1 = 2u0u0u1 + u 2 0u1, h2 = 2u0u0u2 + u 2 1u0 + 2u0u1u1 + u 2 0u2 ... (4.19) using he’s polynomials (4.19) and the iteration formulas (4.17) we obtain u0(x,t) = e ix, u1(x,t) = −3ieix t α γ(α+1) , u2(x,t) = e ix (3it α)2 γ(2α+1) u3(x,t) = −eix (3itα)3 γ(3α+1) u4(x,t) = e ix (3it α)4 γ(4α+1) , ... (4.20) then the approximate solution in a series form is given by u(x,t) = eix −eix (3itα) γ(α + 1) + eix (3itα)2 γ(2α + 1) −eix (3itα)3 γ(3α + 1) + u4(x,t) = e ix (3it α)4 γ(4α + 1) + ... hence int. j. anal. appl. 17 (3) (2019) 415 u(x,t) = eix +∞∑ k=0 (−1)k (3itα)k γ(kα + 1) = eixeα(−3itα). at special case, when α = 1 we obtain u(x,t) = ei(x−3t), (4.21) which is the exact solution to the cubic nonlinear schrödinger equation of (4.11) as presented in [2]. example 4.3. we consider the time-fractional kdv equation cdαt u− 3(u 2)x + uxxx = 0, 0 < α ≤ 1, (4.22) with the initial condition u(x, 0) = 6x. (4.23) z[dαt u] − 3z[(u 2)x] + z[uxxx] = 0. (4.24) using the differentiation property of the zz transform in (4.24), we have z[u(x,t)] = 6x + vα sα z[3(u2)x −uxxx]. (4.25) taking the inverse zz transform on both sides of (4.25), we obtain u(x,t) = 6x + z−1[[ vα sα z[3(u2)x −uxxx]]. (4.26) by applying the hpm method, we have ∞∑ n=0 pnun = 6x + p(z −1( vα sα z[3 ∞∑ n=0 pn(hn(u))x − ∞∑ n=0 pnunxxx])). (4.27) equating the coefficient of the like power of p on both sides in (4.27), we get p0 : u0(x,t) = 6x, ... pn : un(x,t) = z −1(v α sα z[3(hn−1(u))x − (un−1)xxx]), (4.28) where n ∈ n∗. the first few components of he’s polynomials are given by int. j. anal. appl. 17 (3) (2019) 416 h0 = u 2 0, h1 = 2u0u1, h2 = 2u0u2 + u 2 1, h3 = 2u0u3 + 2u1u2, ... (4.29) using he’s polynomials (4.29) and the iteration formulas (4.28) we obtain u0(x,t) = 6x u1(x,t) = 6x(36) 1 γ(α+1) tα, u2(x,t) = 6x(36) 2 2 γ(2α+1) t2α u3(x,t) = 6x(36) 3 4γ 2(α+1)+γ(2α+1) γ2(α+1)γ(3α+1) t3α u4(x,t) = 6x(36) 4 ( 8γ2(α+1)+2γ(2α+1) γ2(α+1)γ(3α+1) + 4 γ(α+1)γ(2α+1) ) γ(3α+1) γ(4α+1) t4α, ... (4.30) the first four terms of the decomposition series solution for (4.22) is given by u(x,t) = 6x + 6x(36) 1 γ(α + 1) tα + 6x(36)2 2 γ(2α + 1) t2α + 6x(36)3 4γ2(α + 1) + γ(2α + 1) γ2(α + 1)γ(3α + 1) t3α + 6x(36)4 ( 8γ2(α + 1) + 2γ(2α + 1) γ2(α + 1)γ(3α + 1) + 4 γ(α + 1)γ(2α + 1) ) γ(3α + 1) γ(4α + 1) t4α + ... the first four terms of the decomposition series solution, for the special case α = 1, is given by u(x,t) = 6x[1 + (36)t + (36)2t2 + (36)3t3 + (36)4t4 + ...] (4.31) that gives u(x,t) = 6x 1 − 36t , |36t| < 1, (4.32) which is an exact solution to the kdv equation (4.22)-(4.23) as presented in [21]. int. j. anal. appl. 17 (3) (2019) 417 (a) (b) figure 3. (a) the exact solution, (b) the approximate solution when α = 1 of (4.22)-(4.23) (c) (d) figure 4. (c) and (d) the approximate solutions of (4.22)-(4.23) when α = 0.5 and α = 0.9 respectively. int. j. anal. appl. 17 (3) (2019) 418 5. conclusion the coupling of homotopy perturbation method (hpm) and the zz transform method, proved very effective to solve nonlinear fractional partial differential equations. the proposed algorithm provides the solution in a series form that converges rapidly to the exact solution if it exists. from the obtained results, it is clear that the hpzztm yields very accurate solutions using only a few iterates. the results show that the homotopy perturbation zz transform method (hpzztm) can be further implemented to solve other physical models of nonlinear partial differential equations with time-fractional derivative. references [1] adio, a. k., a reliable technique for solving gas dynamic equation using natural homotopy perturbation method, glob. j. sci. front. res. (f), 17 (2017), 48-56. [2] ayati, z., biazar, j., and ebrahimi, s., a new homotopy perturbation method for solving linear and nonlinear schrödinger equations, j. interpolation approx. sci. comput. 2014 (2014), art. id jiasc-00062. [3] bhadane, p. k. g., and pradhan, v. h., elzaki transform homotopy perturbation method for solving porous medium equation, int. j. res. eng. technol. 2 (2013), 116-119. [4] biazar, j., and aminikhah, h., study of convergence of homotopy perturbation method for systems of partial differential equations, comput. math. appl. 58 (2009), 2221-2230. [5] biazar, j., and ghazvini, h., convergence of the homotopy perturbation method for partial differential equations, nonlinear anal., real world appl. 10 (2009), 2633-2640. [6] biazar, j., hosseini, k., and gholamin, p., homotopy perturbation method fokker-planck equation. int. math. forum. 19 (2008), 945-954. [7] diethelm, k., the analysis fractional differential equations, springer-verlag berlin heidelberg, 2010. [8] elzaki, t. m., and hilal, e. m. a., homotopy perturbation and elzaki transform for solving nonlinear partial differential equations, math. theory model. 2 (2012), 33-42. [9] ghorbani, a., beyond adomian polynomials: he polynomials, chaos solitons fractals, 39 (2009), 1486-1492. [10] he, j. h., a coupling method of homotopy technique and perturbation technique for nonlinear problems, int. j. non-linear mech. 35 (2000), 37-43. [11] he, j. h., application of homotopy perturbation method to nonlinear wave equations, chaos solitons fractals. 26 (2005), 695-700. [12] he, j. h., homotopy perturbation technique, comput. meth. appl. mech. eng. 178 (1999), 257-262. [13] kumara, s., yildirim, a., khan, y., and weid, l., a fractional model of the diffusion equation and its analytical solution using laplace transform, scientia iranica b. 19 (2012), 1117-1123. [14] mohamed, m. s., al-malki, f., and al-humyani, m., homotopy analysis transform method for time-space fractional gas dynamics equation, gen. math. notes. 24 (2014), 1-16. [15] podlubny, i., fractional differential equations, academic press, san diego, ca, 1999. [16] sadhigi, a., ganji, d., and sabzehmeidavi, y., a study on fokker-planck equation by variational iteration method. int. j. nonlinear. sci. 4 (2007), 92-102. [17] singh, j., kumar, d., and sushila.,homotopy perturbation sumudu transform method for nonlinear equations, adv. theor. appl. mech. 4 (2011), 165-175. int. j. anal. appl. 17 (3) (2019) 419 [18] tatari, m., dehghan, m., and razzaghi, m., application of adomain decomposition method for the fokker-planck equation. math. comp. model. 45 (2007), 639-650. [19] zafar, z. u. a., application of zz transform method on some fractional differential equations, int. j. adv. eng. global technol. 4 (2016), 1355-1363. [20] zafar, z. u. a., zz transform method, int. j. adv. eng. glob. technol. 4 (2016), 1605-1611. [21] ziane, d., belghaba, k., and hamdi cherif, m., fractional homotopy perturbation transform method for solving the time-fractional kdv ,k(2,2) and burgers equations, int. j. open probl. comput. math. 8 (2015), 63-75. 1. introduction 2. basic definitions 2.1. fractional calculus 2.2. definitions and properties of the zz transform 3. fractional homotopy perturbation zz transform method 4. applications of fractional homotopy perturbation zz transform method 5. conclusion references international journal of analysis and applications volume 16, number 1 (2018), 125-136 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-125 maximize minimum utility function of fractional cloud computing system based on search algorithm utilizing the mittag-leffler sum rabha w. ibrahim department of mathematics and computer science, modern college of business & science, 133, muscat, oman corresponding author: rabhaibrahim@yahoo.com abstract. the maximum min utility function (mmuf) problem is an important representative of a large class of cloud computing systems (ccs). having numerous applications in practice, especially in economy and industry. this paper introduces an effective solution-based search (sbs) algorithm for solving the problem mmuf. first, we suggest a new formula of the utility function in term of the capacity of the cloud. we formulate the capacity in ccs, by using a fractional diffeo-integral equation. this equation usually describes the flow of ccs. the new formula of the utility function is modified recent active utility functions. the suggested technique first creates a high-quality initial solution by eliminating the less promising components, and then develops the quality of the achieved solution by the summation search solution (sss). this method is considered by the mittag-leffler sum as hash functions to determine the position of the agent. experimental results commonly utilized in the literature demonstrate that the proposed algorithm competes approvingly with the state-of-the-art algorithms both in terms of solution quality and computational efficiency. 1. introduction the utility is a quantity of favorites over some set of properties; it signifies fulfillment skilled by the agent of a property. the notion is a significant under heading of the optimization theory in the economy and received 24th september, 2017; accepted 6th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 44a45, 34a08, 26a33. key words and phrases. fractional calculus; fractional differential equation; fractional differential operator; utility function; cloud computing system . c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 125 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-125 int. j. anal. appl. 16 (1) (2018) 126 the game theory: since the agent cannot straight measure benefit, fulfillment or gladness from the property or service, the economists have developed methods of signifying and measuring the utility in terms of the measurable economic selections. one of the challenges in ccs is how to dynamically distribute information (data) to the agents such that qualities of service constraints are fulfilled and operating costs are minimized. the adjustment between these two contradictory aims can be expressed by a utility function. the general formula of the linear utility function in ccs is as follows (see [1]) ui = n∑ j=1 ωjρij, where ρij is the rate of outcomes and ωj is the weight achieving ∑ j ωj = 1. the non-linear utility function is defined by the formula ( see [2]) ui = n∏ j=1 ρ ωj ij , 0 < ωj < 1. the author introduced a modification to the non-linear utility function by employing the tsalis entropy [3], wiener processing [4] and by using the solution of fractional differential equation [5]. in our discussion, we impose a new formula of the non-linear utility function involving fractional calculus. the new formula is based on the classic definition of the riemann-liouville operators, where the integral operator is given by (see [6][9]) iα(ν)(t) = 1 γ(α) ∫ t 0 (t−s)α−1ν(s) ds and the fractional differential operator (the classic fractional calculus) is formulated by 0d α t ν(t) = d dt i1−αν(t), α ∈ (0, 1). this formula is modified recent active utility functions. the principle of the definition is based on the diffeointegral diffusion fractional equation. experimental results commonly utilized in the literature demonstrate that the proposed algorithm competes approvingly with the state-of-the-art algorithms both in terms of solution quality and computational efficiency. 1.1. aims. • purpose: the paper schemes self-possessed of multiple networking agents within a location. it proposes a mathematical modeling of intelligence ccs management systems. the study aims to maximize minimum utility function of the system in term of the capacity of the cloud. the study of this problem is very limited and is not suggested widely in the ccs. • methodology: the paper suggested a multi-agent system based on the fractional calculus of a hybrid diffeo-integral system. the continuous dynamics are established by employing the fractional differential equation and the fractal is considered for a time, while the space is illustrated in ordinary cases. the technique is based on the mittag-leffler sum (diffusion processes). this process is the int. j. anal. appl. 16 (1) (2018) 127 net movement of information or data from a state of low observation in a state of high observation. this property is a basic realization in the multi-agent computing systems. moreover, this process yields a convergence to the equilibrium point (a point of maximal utility). also, can be suggested as a summation search solution (sss). • outcomess: the paper provides a large number of solutions to maximize the minimum utility function in ccs. the method is a proficient technique to solve various types of fractional differential systems. • research modulation/implications: researchers are encouraged to test and modify the proposed method, using any mathematical inclusion containing the capacity of the cloud. one can add a controller to the system, with less constrains. • practical implications: the paper involves inspirations for the expansion of a powerful for carrying and integrating the cloud system stability and movement. • originality/value: this paper satisfies an acknowledged need to study how to the utility function of cloud computing system can be maximized under the same capacity of the system. here, we do not formulate the dynamic system of the cloud as can be recognized in the formerly works. 2. algorithm our algorithm is divided into two parts. the first part deals with derivative the utility function from the fractional diffeo-integral equation. the second part contains the numerical computation, based on the summation search solution (sss). we introduce an effective sss algorithm for solving the problem mmuf. 2.1. the diffeo-integral equation. in this study, we consider the transport of information or data, in ccs in one-dimensional with the capabilities of real-time observing of the saturated line average. we derive a mathematical model based on data balance that incorporates dispersion and diffusion. let xi, i = 1, ...,n be the need (what the agent needs from ccs to receive information) of the user (agent) i in the sector λi from ccs, x = (x1, ...,xn) be the set of demand (acts or positions) for all users; the total cross-sectional area normally available for flow is ωλ = ∑n i=1 ωλi. now let c(x,t) be the capacity of ccs during time t with its flux q(x,t). we additionally assume that the sensor is produced or shattered with rate f(x,t). for example, f could be a reaction term or source term. finally, we denote by v the darcy velocity and υ = v /ω, ω ∈ (0, 1) the average velocity. all the above functions are in c[rn × j,r] (the space of real valued functions). to introduce the fractional diffeo-integral equation that imposes the utility function, we let xi ∈ [ai,bi] and ω := ∏n i=1[ai,bi] then we have the following equation: 0d α t (∫ ω c(x,t)ωλdx ) = q(x,t)λ −q(x̄, t)λ + ∫ ω f(x,t)λdx, (2.1) where x is the minimum value of x ∈ ω and x̄ is the maximum value of x ∈ ω. int. j. anal. appl. 16 (1) (2018) 128 the term on the left-side is the rate of change of the total amount of sensor in the sector, and the first two terms on the right compute the rate that the sensor flows into the sector at x and the rate that it flows out at x̄; the last term is the rate that the sensor is established in the sector. by the continuity of the functions, we have ∫ ω ( c (α) t ω + qx −f ) dx = 0, where c (α) t (x,t) := 0d α t c(x,t). consequently, we obtain c (α) t ω + qx = f. there are three generally accepted methods to determine the function q(x,t). it can be calculated by advection (the transfer of the flow of a fluid, especially horizontally in the atmosphere, the sea or the cloud), which incomes that users are simply approved by the ccs; this brings us to describe qa in the following formal: qa(x,t) = v c(x,t). another technique of utilization is by diffusion. this is the distribution initiated by the random signal and crashes by the users themselves. in this case the function qd is represented by qd(x,t) = −ωδcx(x,t), where δ is the diffusion coefficient. there is a third type of the function q, which is called dispersion. this case the flux will consider to be hybrid qh, taking the formula qh(x,t) = −ωϕcx(x,t), where ϕ is the dispersion coefficient. therefore, we have the general form of the flux as follows: q(x,t) = qa(x,t) + qd(x,t) + qh(x,t). this implies the fractional reaction-advection-dispersion equation ωc (α) t (x,t) = ( ω∆cx ) x (x,t) −v cx(x,t) + f(x,t), ∆ = ϕ + δ, which can be written as a fractional differential equation in the form: c (α) t (x,t) = ∆cxx(x,t) −υcx(x,t) + 1 ω f(x,t), ω ∈ (0, 1). hence, the general solution of the above equation is formulated by c(x,t) = u(x,�tα), α ∈ (0, 1], (2.2) where u is represented in the fractional utility function. given a set n = {1, 2, ...,n} of n− agents and the utility of ccs is denoted by uij ≥ 0 (i,j). a dispersion problem consists in choosing a set m of agents from int. j. anal. appl. 16 (1) (2018) 129 n, such that some objective function defined on the selected agents of m is maximized or minimized the utility of ccs. formally, the max-minsum utility function (mmuf) can be stated as follows: maximize minu(x,�tα) = ∑ j∈m uij subject to, θ := {m ⊂ n, |m| = m}. (2.3) ( α ∈ (0, 1], � ∈ (0, 1), t ∈ j = [0,t],t < ∞ ) . our aim is to find a suitable solution in terms x and tα. in other words, we look for formal power series. since our approach depends on the fractional calculus, therefore, we suggest applying the mittag-leffler sum. a system to formalize the notion of formal preferences is by transmission to each state x a real number u(s) that is called the utility of state x for that particular agent. officially, for two states x and x′ holds u(x) > u(x′) if and only if the agent selects state x to state x′, and u(x) = u(x′) if and only if the agent is unmoved between x and x′. next subsection describes the numerical approach of it. 2.2. the mittag-leffler sum. the mittag-leffler function occurs obviously in the solution of fractional integral equations and mainly in the investigation of the fractional generalization of the kinetic equation, random walks, lvy flights, and so-called super-diffusive transport. the ordinary and generalized mittag-leffler functions include between a purely exponential law and power-like performance of phenomena administered by ordinary kinetic equations and their fractional complements [10]. let u(x,t) = ∞∑ n=0 un (t αx)n be a formal power series in t and x. define the transform bαu of u by bαu(x,t) ≡ ∞∑ n=0 (tαx)n γ(1 + αn) . (2.4) then the mittag-leffler sum of u is given by lim α→0 bαu(x) if each sum converges and the limit exists. then the mittag-leffler sum (mls) of u is given by esαu(x) = ∫ ∞ 0 e−tbαu(tαx) dt. in other words, the utility function is maximized by using the transform bα. since the utility function is formalized by the capacity of the ccs, then the capacity of the cloud will not reduce. int. j. anal. appl. 16 (1) (2018) 130 2.3. the objective function. the basic support learning model contains the following facts: a set includes environmental data and agents’ positions s; a set contains the movements m; strategies of transitioning from the set of agents’ positions to the set of movements; procedures that control the scalar immediate objective (sio) of a move (shift); and procedures that define what the agent supports. the procedures are often stochastic. the observation naturally includes the sio related to the last change. in this work, the agent is also expected to detect the present environmental formal, in which situation we talk about full observability, whereas in the contradictory situation, we focus about partial observability (see fig.1). occasionally, the set of movements (m) existing to the agent is controlled. a support learning agent relates to its environment in discrete time steps. at each time t, the agent collects an observation ot, which naturally includes the sio (xt). it then selects a movement mt from the set of movements (m), which is consequently sent to the environment. the environment transfers to a new formal xt+1 and the object xt+1 related with the transition (xt,mt,xt+1) is calculated. the aim of a supportive learning agent is to gather as much reward as potential. the agent can select any movement as a function of the past and it can even randomize its achievement selection. when the agent’s act is associated with an agent which acts optimally for the beginning, the variation in act implies rise to the notion of regret. annotation is that in order to act near optimally, the agent must aim about the long period significance of its actions (i.e., maximize the forthcoming utility). in this position, we can define the formal of mmuf by the criteria u∗(x,t) = max t∈j bα u(x,t), (2.5) where bα is formulated in (2.4). 3. simulation jafari et al [11] and ashouraie and navimipour [12] utilized two types of algorithm to determine the utility function in ccs. they applied the genetic algorithms and heterogeneous resources, respectively. salih et al., [1, 2] introduced a method of employing the user preferences. fractional utility functions, for the first time are suggested by the authors in [3][5]. we deal with computational agents; communication involves several stages of abstraction. on the lower, ‘network’ stage, one would make sure that the messages that are communicated among the agents arrive safely and timely at their destination. a formal technique to designate communicate is by giving each communication as an action that updates the information of an agent about aspects of the state; like those described in fig.1. the communication primitives that are switched among agents are normally raised to as communicative acts or speech acts. each agent i ∈ m ⊂ n = {1, ,n} calculates its best-response function ui depending on the assumption that all local utility functions uij are int. j. anal. appl. 16 (1) (2018) 131 fig.1 an agent precedes actions (information) in ccs which is deduced into a payment and a demonstration of the state which is fed back into the agent. ccs action reward to agent observing int. j. anal. appl. 16 (1) (2018) 132 common information among agents. similarly, in the backward pass, where an agent notifies the other agents of its action choice, we have supposed that the actions of the agents are well-arranged and these orderings are common information. the result of these two common information assumptions is that each agent can track the procedure in parallel, selecting an arbitrary elimination order from the set m ⊂ n. we need the following description in the sequel: : a joint action x∗ = (x∗i ) is an equilibrium in dominant actions for agent i ∈ m if we have ui(xi, t) ≤ ui(x∗i , t) := u ∗ i (3.1) : for all joint actions (x−i,xi). proposition 3.1 for agent i satisfies (3.1) then u∗i is unique. proof. suppose there are two utility functions u∗i and u ∗∗ i for the same action x ∗ i such that ui(xi, t) ≤ ui(x∗i , t) = u ∗ i and ui(xi, t) ≤ ui(x∗i , t) = u ∗∗ i , which is a contradiction. � the complete algorithm is presented as follows: initialize n,α, (x,t) select m ⊂ n (agents with the minimum utility) with |m| = m for each agent i ∈ m in parallel if i 6= 1 wait until agent i− 1 sends ok. end let uij be all local utility (initial and communicated) that include agent i. compute bαu(x,t) ≡ ∞∑ n=0 (tαx)n γ(1 + αn) . compute u∗(x,t) = maxt∈j bα u(x,t), send u∗(x,t) to agent j ∈ m if i 6= m send ok to next agent i + 1. assign role j to agent i∗, (the agent who has maximum uij ) int. j. anal. appl. 16 (1) (2018) 133 wait until all x∗−i (the action of agent) are received. note that −i means all agents except the agent i. end the ok signal is required for synchronization, ensuring an execution order of the algorithm allowing to the rejection order of the agents. moreover, the initial distribution of utility functions of the agents in the selected set m ⊂ n can be done as follows: agent 1 in the removal order receipts all utility functions that include this agent, agent 2 revenues all functions that contain this agent and are not distributed to agent 1, and so on, until no more utility functions are left. the capacity of the ccs does not change, because the utility function is defined in terms of it. also, we have considered the actions xi, i = 1, ...,m of the agents i ∈ m are ordered and these orderings are common information. this application is preserved simultaneously by consuming the training red hat of hybrid it environments. this system provides robust devices for cloud organization with private or hybrid cloud organization capabilities, active visibility technologies and advanced virtualization controlling panels. proposition 3.2 if the agents run the same position xi, and it is common data in the recent state what actions the agents are considering, then the agents must be having the same utility. proof. the point that in the present state the action xi of an agent i is shared data among all agents indicates that there must be a set es that is common dart among all agents, and in all positions of which agent i would recive action xi, for all i ∈ e. since e is common data among the agents, it must be self-evident to all agents, therefore, we can write e = m for all i. this implies that all agents in m have take the same capacity from the ccs in the same time. but the utility function is in term of the capacity, therefore all ui must be equal. � 4. discussion applying a utility function would be informal if we had full observability of the state x. then we could just utilize x in (2.5) and calculate the desired optimal outcome (supposing of course that we have a controllable algorithm for achieving this). though, as we know that , xi is exposed only to agent i. one option would be to request each agent to tell us his position xi, but there is no promise that an agent will report his true status. recall that each agent i forms his own favorites over outcomes, specified by his utility function ui(t,xi) with xi his true status. if by reporting a false type x̃i an agent i assumes to get higher payoff than by reporting xi, then this agent may surely consider lying. consider the m contains m = 5 agents requesting one information (one item) in the same or different time t ∈ j = [0,t]. we assume a direct demonstration technique in which the agents are requested to report int. j. anal. appl. 16 (1) (2018) 134 their positions (xi) during a period of time ti ∈ j, and based on their reports, the ccs calculates an optimal outcomeu∗i that solves (2.5). the outcome can be formulated by the following equation: u(x,t) = max t∈j 5∑ i=1 ui(xi, ti). (4.1) by using the suggested transform bαu(x,t), α ∈ (0, 1]; each agent does not depend on the report of the other agents. a simple and capable technique for computing optimal utilities in ccs, when the transit method is available is value iteration. we initialize a random utility value u∗(x,t) for each status and then iteratively apply (2.5). we repeat until convergence, which is computed in a relative increase in u between two successive update steps. value iteration converges to the optimal u∗(x,t) for each state. the maximum utility appears, when the fractional power α takes its maximal value in (0, 1]. table 1 shows the experimental outcomes for m = 5 agents and different values of the fractional order α. the utility of the ccs is estimated throughout different time t ∈ [1, 5]. obviously, the maximum utility increases, whenever increasing α → 1. we compere our new algorithm with the previous works. let the capacity of the system is equal to 1; hence the maximum value of the utility function is also equal to 1. consider the set of the outcomes for 5-agent system xi = {0.1, 0.15, 0.2, 0.25, 0.3}. consequently, we obtain ui(x,t α) = {0.1 ∗ tα, 0.15 ∗ tα, 0.2 ∗ tα, 0.25 ∗ tα, 0.3 ∗ tα},( t = 1, 2, ..., 5, α ∈ (0, 1], xi ∈ (0, 1) ) . in [1], the authors suggested the values of ωj = 0.2. in our method theses values can be determined as follows: ωj = 1 γ(1 + jα) , j = 1, ..., 5. it is clear that the mmu is converging faster than the utility functions in [1] and [4] (see fig. 2). int. j. anal. appl. 16 (1) (2018) 135 table 1. fractional multi-agent system (α) time [1]α=1 [4] mmu 0.95 1 0.199 0.6111 0.25 2 0.39 0.6128 0.501 3 0.599 0.6137 0.751 4 0.789 0.629 1 5 0.989 0.638 0.75 1 0.656 0.338 2 0.657 0.676 3 0.6585 1 4 0.6743 5 0.680 0.5 1 0.6814 0.615 2 0.6827 1 3 0.6839 4 0.7001 5 0.7113 0 1 2 3 4 5 0 1 time t ∈{1, ..., 5} t h e u ti li ty o f c c s a t α = 0 .9 5 mmu [1] [4] 0 1 2 3 4 5 0 1 time t ∈{1, ..., 5} t h e u ti li ty o f c c s a t α = 0 .7 5 & 0 .5 mmu ( α = 0.75) [4] ( α = 0.75) mmu ( α = 0.5) [4] ( α = 0.5) fig.2: the utilty function for various values of α ∈ (0, 1] 5. conclusion we investigated, the study of what method to maximize min utility function, a process initiated by a fractional diffeo-integral system. the method was supplementary capable and exceeds those of mathematical int. j. anal. appl. 16 (1) (2018) 136 software recommendation and recreating the stable profit of exchangeable with the overall cost as well as task distribution. otherwise, the objective function is involved in the concept of the fractional function called the mittag-leffler sum. by utilizing this function, we familiar an adaptation to the objective function. the technique acknowledged two rewards: transforms the problem of constrained optimization into unconstrained and with an appropriate selection of the fractional order, the method is a good approximation. moreover, one may suggest multi-connection, by applying the above method. finally, the method can be stretched to higher-dimension, when the number of agents in the multi-agent system becomes large. competing interests : the author has no competing interests in the manuscript. references [1] y. k. salih et al., a user-centric game selection model based on user preferences for the selection of the best heterogeneous wireless network, ann. tlcommun. 70(5-6) (2015), 1–10. [2] y. k. salih et al., an intelligent selection method based on game theory in heterogeneous wireless networks, trans. emerg. telecommun. technol. 27 (12) (2016), 1641–1652. [3] r. w. ibrahim et al., perturbation of fractional multi-agent systems in cloud entropy computing, entropy 18 (1) (2016), 31. [4] r. w. ibrahim, a. gani, hybrid cloud entropy systems based on wiener process, kybernetes 45 (7) (2016), 1072–1083. [5] r. w. ibrahim, y. k. salih, on a fractional multi-agent cloud computing system based on the criteria of the existence of fractional differential equation, math. sci. (2017), 1–7. [6] i. podlubny, fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. vol. 198. academic press 1999. [7] a. a. kilbas, h. m. srivastava and j.j. trujillo, theory and applications of fractional differential equations. north-holland, mathematics studies, elsevier 2006. [8] d. baleanu, j. machado, and a. luo, fractional dynamics and control. springer science & business media, 2011. [9] r. w. ibrahim, fractional calculus of multi-objective functions & multi-agent systems. lambert academic publishing, saarbrucken, germany 2017. [10] r. hilfer, applications of fractional calculus in physics. singapore: world scientific, 2000. [11] n. jafari, et al., job scheduling in the expert cloud based on genetic algorithms, kybernetes, 43( 8) (2014),1262-1275. [12] m. ashouraie, n. navimipour, priority-based task scheduling on heterogeneous resources in the expert cloud, kybernetes: 44 (10) ( 2015), 1455-1471. 1. introduction 1.1. aims 2. algorithm 2.1. the diffeo-integral equation 2.2. the mittag-leffler sum 2.3. the objective function 3. simulation 4. discussion 5. conclusion references int. j. anal. appl. (2022), 20:52 on the frictional contact problem of p(x)-kirchhoff type eugenio cabanillas lapa∗, willy barahona martinez instituto de investigación, facultad de ciencias matemáticas-unmsm, lima, perú ∗corresponding author: cleugenio@yahoo.com abstract. in this article we consider a class of frictional contact problem of p(x)-kirchhoff type, on a bounded domain ω ⊆ r2 . using an abstract lagrange multiplier technique and the schauder’s fixed point theorem we establish the existence of weak solutions. furthermore, we also obtain the uniqueness of the solution assuming that the datum f1 satisfies a suitable monotonicity condition. 1. introduction the purpose of this work is to investigate the existence of weak solutions for the boundary value problem −m (∫ ω 1 p(x) |∇u|p(x)dx ) div ( |∇u|p(x)−2∇u ) = f1(x,u) in ω u = 0 on γ1 m (∫ ω 1 p(x) |∇u|p(x)dx ) |∇u|p(x)−2 ∂u ∂ν = f2(x) on γ2∣∣∣m(∫ ω 1 p(x) |∇u|p(x)dx ) |∇u|p(x)−2 ∂u ∂ν ∣∣∣ ≤ g(x), m (∫ ω 1 p(x) |∇u|p(x)dx ) |∇u|p(x)−2 ∂u ∂ν = −g u |u| , if u 6= 0 on γ3 (1.1) where ω ⊆ r2 is a bounded domain with smooth enough boundary γ, partitioned in three parts γ1, γ2, γ3 such that meas (γi ) > 0, (i = 1, 2, 3); f1 : ω ×r → r, f2 : γ2 → r, g : γ3 → r and m : [0, +∞[→ [m0, +∞[ are given functions, p ∈ c(ω). the study of the p(x)kirchhoff type equations with nonlinear boundary conditions of different class received: aug. 16, 2022. 2010 mathematics subject classification. 35j25, 46e35, 74g25. key words and phrases. p(x)-kirchhoff type equation; frictional contact condition; schauder fixed point theorem; uniqueness of solutions. https://doi.org/10.28924/2291-8639-20-2022-52 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-52 2 int. j. anal. appl. (2022), 20:52 have attracted expensive attention in recent years, we refer to some interesting works [1,6,13,16] and references therein. one reason of such interest is that various real fields require pde problems with variable exponent, for example, electrorheological fluids and image restoration. the other reason is that the nonlocal problems with variable exponent, in addition to their contributions to the modeling of many physical and biological phenomena, raise greater mathematical difficulties due to their nonlinearities; see for example [2,15,19]. cojocaru-matei [5] studied the unique solvability of problem (1.1) in the case m(s) = 1, f1(x,u) ≡ f1(x),p = constant ≥ 2, which models the antiplane shear deformation of a nonlinearly elastic cylindrical body in frictional contact on γ3 with a rigid foundation; see, e.g. [18]. they used a technique involving dual lagrange multipliers, this allow to write efficient algorithms to approximate the weak solutions; see [14]. for our situation, the behavior of the material is described by the hencky-type constitutive law: σ(x) = ktrε(u(x))i3 + µ(x)‖εd(u(x))‖ p(x)−2 2 εd(u(x)) where σ is the cauchy stress tensor, tr is the trace of a cartesian tensor of second order,σ(x) ε is the infinitesimal strain tensor, u is the displacement vector,i3 is the identity tensor, k,µ are material parameters, p is a given function;εd is the desviator of the tensor ε defined by εd = ε − 1 3 (trε)i3 where trε = 3∑ i=1 εii; see for instance [12]. if, the lamé coefficient is given by µ = m (∫ ω 1 p(x) |∇u|p(x)dx ) we obtain our mechanical problem (1.1). thanks to the above mentioned research articles, we consider the problem (1.1), under appropriate assumptions on m and f1, and establish the existence of a unique weak solution of this problem via lagrange multipliers and the schauder fixed point theorem. in this sense, we generalize the main result in [5]. also, we state a simple uniqueness result under suitable monotonicity condition on f1. the paper is designed as follows. in section 2, we introduce the mathematical preliminaries and give several important properties of p(x)-kirchhoff-laplace operator. we deliver a weak variational formulation with lagrange multipliers in a dual space. section 3, is devoted to the proofs of main results. 2. preliminaries for the reader’s convenience, we point out some basic results on the theory of lebesgue-sobolev spaces with variable exponent. in this context we refer the reader to [8,17] for details. firstly we state some basic properties of spaces w 1,p(x)(ω) which will be used later. denote by s(ω) the set of all measurable real functions defined on ω. two functions in s(ω) are considered as the same element int. j. anal. appl. (2022), 20:52 3 of s(ω) when they are equal almost everywhere. write c+(ω) = {h : h ∈ c(ω),h(x) > 1 for any x ∈ ω}, h− := min ω h(x), h+ := max ω h(x) for every h ∈ c+(ω). define lp(x)(ω) = {u ∈s(ω) : ∫ ω |u(x)|p(x) dx < +∞ for p ∈ c+(ω)} with the norm |u|lp(x)(ω) = |u|p(x) = inf{λ > 0 : ∫ ω | u(x) λ |p(x) dx ≤ 1}, and w 1,p(x)(ω) = {u ∈ lp(x)(ω) : |∇u| ∈ lp(x)(ω)} with the norm ‖u‖1,p(x) = |u|lp(x)(ω) + |∇u|lp(x)(ω). proposition 2.1 ( [11]). the spaces lp(x)(ω) and w 1,p(x)(ω) are separable reflexive banach spaces. proposition 2.2 ( [11]). set ρ(u) = ∫ ω |u(x)|p(x) dx. for any u ∈ lp(x)(ω), then (1) for u 6= 0, |u|p(x) = λ if and only if ρ( uλ) = 1; (2) |u|p(x) < 1 (= 1; > 1) if and only if ρ(u) < 1 (= 1; > 1); (3) if |u|p(x) > 1, then |u| p− p(x) ≤ ρ(u) ≤ |u|p + p(x) ; (4) if |u|p(x) < 1, then |u| p+ p(x) ≤ ρ(u) ≤ |u|p − p(x) ; (5) limk→+∞ |uk|p(x) = 0 if and only if limk→+∞ρ(uk) = 0; (6) limk→+∞ |uk|p(x) = +∞ if and only if limk→+∞ρ(uk) = +∞. proposition 2.3 ( [9,11]). if q ∈ c+(ω) and q(x) ≤ p∗(x) (q(x) < p∗(x)) for x ∈ ω, then there is a continuous (compact) embedding w 1,p(x)(ω) ↪→ lq(x)(ω), where p∗(x) =   np(x) n−p(x) if p(x) < n, +∞ if p(x) ≥ n. proposition 2.4 ( [11]). the conjugate space of lp(x)(ω) is lq(x)(ω), where 1 q(x) + 1 p(x) = 1 holds a.e. in ω. for any u ∈ lp(x)(ω) and v ∈ lq(x)(ω), we have the following hölder-type inequality∣∣∫ ω uv dx ∣∣ ≤ ( 1 p− + 1 q− )|u|p(x)|v|q(x). we introduce the following closed space of w 1,p(x)(ω) x = {v ∈ w 1,p(x)(ω) : γu = 0 a. e. on γ1} (2.1) where γ denotes the sobolev trace operator and γ1 ⊆ γ, meas (γ1) > 0, therefore x is a separable reflexive banach space. now, we denote ‖u‖x = |∇u|p(x), u ∈ x. 4 int. j. anal. appl. (2022), 20:52 this functional represents a norm on x. proposition 2.5 ( [3]). there exists c > 0 such that ‖u‖1,p(x) ≤ c‖u‖x for all u ∈ x. then, the norms ‖.‖x and ‖.‖1,p(x) are equivalent on x. we write l(u) = ∫ ω 1 p(x) |∇u|p(x) dx. proposition 2.6. the functional l : x → r is convex. the mapping l′ : x → x′ is a strictly monotone, bounded homeomorphism, and is of (s+) type, namely un ⇀ u and lim sup n→+∞ l′(un)(un −u) ≤ 0 implies un → u, where x′ is the dual space of x. proof. this result is obtained in a similar manner as the one given in [10], thus we omit the details. � now, we define the spaces s = { u ∈ w 1 p′(x) ,p(x)(γ) : ∃v ∈ x such that u = γv a.e on γ } (2.2) which is a real reflexive banach space, 1 p(x) + 1 p′(x) = 1 for all x ∈ ω, and y = s′, the dual of the space s. (2.3) let us introduce a bilinear form b : x ×y −→r : b(v,µ) = 〈 µ,γv 〉y×s , (2.4) a lagrange multiplier λ ∈ y , 〈 λ,z 〉 = − ∫ γ3 m (∫ ω 1 p(x) |∇u|p(x)dx ) |∇u|p(x)−2 ∂u ∂ν zdγ , ∀z ∈ s and the set of lagrange multipliers λ = { u ∈ y : 〈 µ,z 〉6 ∫ γ3 g(x)|z(x)| , ∀z ∈ s } . (2.5) from (1.1)4 we deduce that λ ∈ λ. let u be a regular enough function satisfying problem (1.1). after some computations we get (by using density results) m (∫ ω 1 p(x) |∇u|p(x)dx )∫ ω |∇u|p(x)−2∇u.∇vdx = ∫ ω f1(x,u)vdx + ∫ γ2 f2(x)γvdγ + m (∫ ω 1 p(x) |∇u|p(x)dx )∫ γ3 |∇u|p(x)−2 ∂u ∂ν γvdγ (2.6) int. j. anal. appl. (2022), 20:52 5 for allv ∈ x , where u satisfies (1.1)5 on γ3 now, we write problem (2.6) as an abstract mixed variational problem (by means a lagrange multipliers technique) we define the following operators: i) a : x → x′,given by 〈 au,v 〉 = m (∫ ω 1 p(x) |∇u|p(x)dx )∫ ω |∇u|p(x)−2∇u.∇vdx, u,v ∈ x. ii) f : x → x′,given by 〈 f (u),v 〉 = ∫ ω f1(x,u)vdx + ∫ γ2 f2(x)γvdx , u,v ∈ x. (2.7) so, we are led to the following variational formulation of problem (1.1) problem 1. find u ∈ x and λ ∈ λ such that 〈 au,v 〉 + b(v,λ) = 〈 f (u),v 〉 , ∀v ∈ x (2.8) b(u,µ−λ) ≤ 0 ∀µ ∈ λ ⊆ y to solve this problem, we will apply the schauder fixed point theorem. firstly, we "freeze" the state variable u on the function f, that is we fix w ∈ x such that f = f (w) ∈ x′. so, we are led to the following abstract mixed variational problem. problem 2. given f ∈ x′ find u ∈ x and λ ∈ λ such that 〈 au,v 〉 + b(v,λ) = 〈 f ,v 〉 , ∀v ∈ x b(u,µ−λ) ≤ 0 ∀µ ∈ λ ⊆ y. (2.9) the unique solvability of problem 2 is given under the following generalized assumptions. let (x,‖‖x) and (y,‖‖y ) be two real reflexive banach space. (b1): a : x → x′ is hemicontinuous; (b2): ∃h : x →r such that (a) h(tw) = tγh(w) with γ > 1 , ∀t > 0,w ∈ x; (b) 〈 au −av,u −v 〉x×x ≥ h(v −u), ∀u,v ∈ x; (c) ∀(xν) ⊆ x : xν ⇀ x inx =⇒ h(x) ≤ lim ν→∞ sup h(xν) (b3): a is coercive. (b4): the form b : x ×y es bilinear, and (i) ∀(uν) ⊆ x : uν ⇀ u in x =⇒ b(uν,λν) → b(u,λ) (ii) ∀(λν) ⊆ y : λν ⇀ y in y =⇒ b(vν,λν) → b(v,λ) (iii) ∃α̂ > 0 : inf µ∈i u 6=0 sup v∈x v 6=0 b(v,µ) |v|x|µ|y ≥ α̂ 6 int. j. anal. appl. (2022), 20:52 (b5): λ is a bounded closed convex subset of y such that 0y ∈ λ. (b6): ∃c1 > 0,q > 0 : h(v) ≥ c1‖v‖ q x , ∀v ∈ x. theorem 2.1. assume (b1) (b6). then there exists a unique solution (u,λ) ∈ x × λ of problem 2. proof. see [5]. to solve problem 1, we start by stating the following assumptions on m , f1 , f2 and g (a1) m : [0, +∞[→ [m0, +∞[ is a locally lipschitz-continuous and nondecreasing function; m0 > 0. (a2) f1 : ω ×r→r is a caratheodory function satisfying |f1(x,t)| ≤ c1 + c2|t|α(x)−1 , ∀(x,t) ∈ ω ×r, α ∈ c+(ω)with α(x) < p∗(x), α+ < p−. (a3) f2 ∈ lp ′(x)(γ2), g ∈ lp ′(x)(γ3), g(x) ≥ 0 a.e on γ3. we have the following properties about the operator a. proposition 2.7. if (a1) holds, then (i) a is locally lipschitz continuous. (ii) a is bounded, strictly monotone. furthermore 〈au −av,u −v〉≥ kp‖u −v‖ p̂ x where p̂ =  p − if ‖u −v‖x > 1, p+ if ‖u −v‖x ≤ 1. so, we can take h(v) = kp‖v‖ p̂ x . (iii) 〈au,u〉‖u‖x → +∞ as ‖u‖x → +∞. proof. (i) assume that m is lipschitz in [0,r1] with lipschitz constant lm, r1 > 0. we have, for u,v,w ∈ b(0,r1) 〈au −av,w〉 = [m(l(u)) −m(l(v))] ∫ ω |∇u|p(x)−2∇u.∇v dx + m(l(v)) ∫ ω ( |∇u|p(x)−2∇u −|∇v|p(x)−2∇v ) .∇w dx. using the lipschitz continuity of m, the holder inequality and the inequality 〈||x|α−2x−|y|α−2y,x− y〉| ≤ c|x −y|(|x| + |y|)α−2 , ∀x,y ∈rn, 2 ≤ α < +∞, we get |〈au −av,w〉| ≤ c‖u −v‖x‖w‖x, which implies ‖au −av‖x′ ≤ c‖u −v‖x. int. j. anal. appl. (2022), 20:52 7 ii)the functional s : x → x′ defined by 〈su,v〉 = ∫ ω |∇u|p(x)−2∇u.∇v dx, ∀u,v ∈ x, (2.10) is bounded (see [10]). then 〈su,v〉 = m(l(u))〈su,v〉 ∀u,v ∈ x. (2.11) hence, since m is continuous and l is bounded (see proposition 2.6), a is bounded. to obtain that a is strictly monotone we develop the same arguments to those in [7], we omit it. to establish the inequality in ii), we apply lemma 3 in [4] to obtain 〈au −av,u −v〉≥ ∫ ω ( m(l(u))|∇u|p(x)−2∇u −m(l(v))|∇v|p(x)−2∇v ) .(∇v −∇u) dx ≥m0 ∫ ω 1 p(x) (|∇u −∇u|p(x)) dx ≥ m0 p+ ∫ ω |∇u −∇u|p(x) dx ≥ m0 p+ ‖u −v‖p̂ x . iii)for u ∈ x with ‖u‖x > 1 we have 〈au,u〉 ‖u‖x = m (∫ ω 1 p(x) |∇u|p(x) dx )∫ ω |∇u|p(x) dx ‖u‖ ≥m0‖u‖ p−−1 x → +∞ as ‖u‖x → +∞. � proposition 2.8. the form b : x ×y →r defined in (2.4) is bilinear and, it verifies i), ii) and iii) in assumption (b4). moreover b(u,µ) ≤ ∫ γ3 g(x)|u(x)|dγ for all µ ∈ λ; (2.12) b(u,λ) = ∫ γ3 g(x)|u(x)|dγ (2.13) b(u,µ−λ) ≤0 for all µ ∈ λ. (2.14) moreover, λ is a bounded closed convex subset of y such that 0y ∈ λ. proof. the assertions i), ii), iii) and λ bounded are word for word as [5], theorem 3, pags 138-139. it is obvious to check (2.12). to justify (2.13), we have to show that, a.e. x ∈ ω −m (∫ ω 1 p(x) |∇u|p(x)dx ) |∇u(x)|p(x)−2 ∂u(x) ∂ν u(x) = g(x)|u(x)| in fact, let x ∈ ω . if |u(x)| = 0, then −m (∫ ω 1 p(x) |∇u|p(x) dx ) |∇u(x)|p(x)−2 ∂u(x) ∂ν u(x) = 0 = g(x)|u(x)| on γ3. 8 int. j. anal. appl. (2022), 20:52 otherwise, if |u(x)| 6= 0,then −m( ∫ ω 1 p(x) |∇u|p(x) dx)|∇u(x)|p(x)−2 ∂u(x) ∂ν u(x) =g(x) (u(x))2 |u(x)| =g(x)|u(x)| on γ3 furthermore, for all µ ∈ λ : b(u,µ−λ) = b(u,µ) −b(u,λ) = 〈 µ,γu 〉y×s −〈 λ,γu 〉y×s . (2.15) hence, thanks to (2.12), (2.13) and (2.15), we obtain (2.14). � 3. existence and uniqueness of solutions we are ready to solve problem 1. for this, we consider the banach spaces x and y given in (2.1) and (2.3) respectively, the form b : x ×y →r defined in (2.4) and the set λ in (2.5). theorem 3.1. suppose (a1) − (a3) hold. then problem 1 admits a solution (u,λ) ∈ x × λ. proof. we apply the schauder fixed point theorem. as has been said before, we "freeze" the state variable u on the function f , that is, we fix w ∈ x and consider the problem: find u ∈ x and λ ∈ λ such that 〈 au,v 〉 + b(v,λ) = 〈 f ,v 〉 , ∀v ∈ x (3.1) b(u,µ−λ) ≤ 0 ∀µ ∈ λ ⊆ y. (3.2) with f = f (w) ∈ x′. note that by the hypotheses on α and f1, given in (a2), we have f1(w) ∈ lα ′(x)(ω) ↪→ x′. by theorem (2.1), problem (3.1)-(3.2) has a unique solution (uw,λw ) ∈ x × λ. here we drop the subscript w for simplicity. setting v = u in (3.1) and µ = 0y in (3.2), using proposition 2.7 ii), we get kp‖u‖ p̂ x ≤ (2c1cα‖w‖σx + 2c2cα|ω| + cp|f2|p′(x),γ2 )‖u‖x (3.3) where σ =  α − if ‖w‖x > 1, α+ if ‖w‖x ≤ 1, and cχ is the embedding constant of x ↪→ lχ(x)(ω). then ‖u‖x ≤ [c(1 + ‖w‖x)] 1 p̂−1 . therefore, either ‖u‖x ≤ 1 or ‖u‖x ≤ [c(1 + ‖w‖x)] 1 p−−1 . (3.4) int. j. anal. appl. (2022), 20:52 9 since p− > α+ + 1, we have tp −−1 −ctσ −c → +∞ as t → +∞ hence, there is some r̄1 > 0 such that r̄1 p−−1 −cr̄1 σ −c ≥ 0 (3.5) from (3.4) and (3.5) we infer that if ‖w‖x ≤ r̄1 then ‖u‖x ≤ r̄1. thus there exists r1 = min{1, r̄1} such that ‖u‖x ≤ r1 for all u ∈ x. (3.6) for this constant, define k as k = {v : v ∈ lα(x)(ω),‖v‖x ≤ r1} which is a nonempty, closed, convex subset of lα(x)(ω). we can define the operator t : k → lα(x)(ω), tw = uw where uw is the first component of the unique pair solution of the problem (3.1)-(3.2), (uw,λw ) ∈ x × λ from (3.6) ‖tw‖x ≤ r1, for every w ∈ k, so that t (k) ⊆ k. moreover, if (uν)ν≥1 (uwν ≡ uν) is a bounded sequence in k, then from (3.6) is also bounded in x. consequently, from the compact embedding x ↪→ lα(x)(ω), (twν)ν≥1 is relatively compact in lα(x)(ω) and hence, in k. to prove the continuity of t , let (wν)ν≥1 be a sequence in k such that wν → w strongly inlα(x)(ω) (3.7) and suppose uν = twν. the sequence {(uν,λν)}ν≥1 satisfies 〈 auν,v 〉 + b(v,λν) = 〈 f (wν),v 〉 , ∀v ∈ x b(uν,µ−λν) ≤ 0 ∀µ ∈ λ. using (3.6)-(3.7) we can extract a subsequence (uνk ) of (uν) and a subsequence (wνk ) of (wν) such that uνk → u ∗weakly inx, uνk → u ∗ strongly in lα(x)(ω) and a.e. in ω, wνk → w a.e. in ω, l(uνk ) → t0, for some t0 ≥ 0, (3.8) and in view of continuity of m m(l(uνk )) → m(t0). (3.9) 10 int. j. anal. appl. (2022), 20:52 we shall show that u∗ = tw. to this end, by choosing uνk −u ∗ as a test function, we have〈 auνk,uνk −u ∗ 〉 + b(uνk −u∗,λν) = 〈 f (wνk ),uνk −u∗ 〉〈 au∗,uνk −u ∗ 〉 + b(uνk −u∗,λ∗) = 〈 f (w),uνk −u∗ 〉 . (3.10) then [m(l(u∗) −m(l(uνk )] ∫ ω |∇u∗|p(x)−2∇u∗.(∇uνk −∇u ∗) dx+ m(l(uνk )) ∫ ω (|∇u∗|p(x)−2∇u∗ −|∇uνk | p(x)−2∇uνk ).(∇uνk −∇u ∗) dx+ b(uνk −u ∗,λ∗ −λνk ) = 〈 f (w) −f (wνk ),uνk −u ∗ 〉 . (3.11) since b(uνk −u ∗,λ∗ −λνk ) ≥ 0, by the inequality |x| p−2x −|y|p−2y ≥ c|x −y|p, p ≥ 2, we obtain m0cp ∫ ω |∇uνk −∇u ∗|p(x) dx + [m(l(u∗) −m(l(uνk )] ∫ ω |∇u∗|p(x)−2∇u∗.(∇uνk −∇u ∗) dx ≤ | 〈 f (wνk ) −f (w),uνk −u ∗ 〉 | (3.12) but, using (3.8) we get |[m(l(u∗) −m(l(uνk )] ∫ ω |∇u∗|p(x)−2∇u∗.(∇uνk −∇u ∗) dx| ≤ ϑνk p− | ∫ ω |∇u∗|p(x)−2∇u∗.(∇uνk −∇u ∗) dx|→ 0 as k →∞, (3.13) where ϑνk = max{‖uνk‖ p− x ,‖uνk‖ p+ x } + max{‖u∗‖p − x ,‖u∗‖p + x } is bounded. also, by (a2), (3.8) and the compact embedding of x ↪→ lα(x)(ω) we deduce, thanks to the krasnoselki theorem, the continuity of the nemytskii operator nf1 : l α(x)(ω) → lα ′(x)(ω) w 7−→ nf1 (w), (3.14) given by (nf1 (w))(x) = f1(x,w(x)), x ∈ ω. hence ‖f1(wνk ) − f1(w)‖α′(x) → 0 it follows from the definition of f and the above convergence that | 〈 f (wνk ) −f (w),uνk −u ∗ 〉 |→ 0 (3.15) thus, from (3.12)-(3.15) we conclude that uνk → u ∗ in x since the possible limit of the sequence (uν)ν≥1 is uniquely determined, the whole sequence converges toward u∗ ∈ x therefore, from (3.7) and the continuous embedding x ↪→ lα(x)(ω), we get u∗ = tw ≡ uw. int. j. anal. appl. (2022), 20:52 11 on the other hand b(v,λ) ‖v‖x = 〈f (w),v〉−〈au,v〉 ‖v‖x ≤ 〈f (w),v〉 ‖v‖x + ‖au‖x′ ≤ 1 ‖v‖x [∫ ω f1(x,w)v dx + ∫ γ2 f2(x)γv dγ ] + la‖u‖x + ‖a0‖x′ ≤ c(‖f1(w)‖α′(x) + ‖f2‖p′(x),γ2 + ‖a0‖x′ + 1) (3.16) next, using the boundedness of the operator nf1 and the sequence (uν)ν≥1, and the inf-sup property of the form b, we get ‖λν‖y ≤ c. it follows that up to a subsequence λν → λ0 weakly in y for some λ0 ∈ y . so (u∗,λ∗) and (u∗,λ0) are solutions of problem (3.1)-(3.2).then, by the uniqueness λ0 = λ∗ ≡ λw. this shows the continuity of t. to prove that t is compact, let (wν)ν≥1 ⊆ k be bounded in lα(x)(ω) and uν = t (wν). since (wν)ν≥1 ⊆ k, ‖wν‖x ≤ c and then, up to a subsequence again denoted by (wν)ν≥1 we have wν → w weakly in x by the compact embedding xinto lα(x)(ω), it follows that wν → w strongly in lα(x)(ω). now, following the same arguments as in the proof of the continuity of t we obtain uν = t (wν) → t (w) = u strongly in x thus t (wν) → t (w) strongly in lα(x)(ω). hence, we can apply the schauder fixed point theorem to obtain that t possesses a fixed point. this gives us a solution of (u,λ0) ∈ x × λ of problem 1, which concludes the proof. � next, we consider the uniqueness of solutions of (2.8). to this end, we also need the following hypothesis on the nonlinear term f1. (a4) there exists b0 ≥ 0 such that (f1(x,t) − f1(x,s))(t − s) ≤ b0|t − s|p(x) a.e. x ∈ ω,∀t,s ∈r. our uniqueness result reads as follows. theorem 3.2. assume that (a1) − (a4) hold. if, in addition 2 ≤ p for all x ∈ ω̄, then (2.8) has a unique weak solution provided that kp b0λ −1 ∗ < 1, 12 int. j. anal. appl. (2022), 20:52 where λ∗ = inf u∈x\{0} ∫ ω |∇u|p(x) dx∫ ω |u|p(x) dx > 0. proof. theorem 3.1 gives a weak solution (u,λ) ∈ x × λ. let (u1,λ1), (u2,λ2) be two solutions of (2.8). considering the weak formulation of u1 and u2 we have 〈 aui,v 〉 + b(v,λi ) = 〈 f (ui ),v 〉 , ∀v ∈ x (3.17) b(ui,µ−λi ) ≤ 0 ∀µ ∈ λ ⊆ y i = 1, 2. by choosing v = u1 −u2, µ = λ2 if i = 1 and µ = λ1 if i = 2, we have 〈au1 −au2,u1 −u2 〉 + b(u1 −u2,λ1 −λ2) = 〈f (u1) −f (u2),u1 −u2 〉 ,∀v ∈ x b(u1 −u2,λ2 −λ1) ≤ 0 ∀µ ∈ λ ⊆ y. (3.18) it gives 〈au1 −au2,u1 −u2 〉 = 〈f (u1) −f (u2),u1 −u2 〉 + b(u1 −u2,λ2 −λ1) thus, using (3.18) and repeating the argument used in the proof of proposition 2.7, ii) we get kp ∫ ω |∇u1 −∇u2|p(x) dx ≤ |〈 f1(u1) − f1(u2),u1 −u2 〉 | ≤ | ∫ ω (f1(x,u1) − f1(x,u2))(u1 −u2) dx| ≤ | ∫ ω |u1 −u2|p(x) dx ≤ b0λ−1∗ ∫ ω |∇u1 −∇u2|p(x) dx consequently when kp b0λ −1 ∗ < 1, it follows that u1 = u2. this completes the proof. � acknowledgements: this research was partially supported by the vicerectorado de investigaciónunmsm-perú and is part of the doctoral thesis of the second author. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] g.a. afrouzi, m. mirzapour, eigenvalue problems for p(x)-kirchhoff type equations, electron. j. differ. equ. 2013 (2013), 253. https://www.emis.de/journals/ejde/2013/253/abstr.html. [2] g.a. afrouzi, n.t. chung, z. naghizadeh, multiple solutions for p(x)-kirchhoff type problems with robin boundary conditions, electron. j. differ. equ. 2022 (2022), 24. https://ejde.math.txstate.edu/volumes/2022/24/ abstr.html. [3] m.m. boureanu, a. matei, m. sofonea, nonlinear problems with p(·)-growth conditions and applications to antiplane contact models, adv. nonlinear stud. 14 (2014), 295–313. https://doi.org/10.1515/ans-2014-0203. [4] l.e. cabanillas, l. huaringa, a nonlocal p(x)&q(x) elliptic transmission problem with dependence on the gradient, int. j. appl. math, 34 (2021), 93-108. https://doi.org/10.12732/ijam.v34i1.4. https://www.emis.de/journals/ejde/2013/253/abstr.html https://ejde.math.txstate.edu/volumes/2022/24/abstr.html https://ejde.math.txstate.edu/volumes/2022/24/abstr.html https://doi.org/10.1515/ans-2014-0203 https://doi.org/10.12732/ijam.v34i1.4 int. j. anal. appl. (2022), 20:52 13 [5] m. chivu cojocaru, a. matei, well-posedness for a class of frictional contact models via mixed variational formulations, nonlinear anal.: real world appl. 47 (2019), 127–141. https://doi.org/10.1016/j.nonrwa. 2018.10.009. [6] n.t. chung, multiple solutions for a class of p(x)-kirchhoff type problems with neumann boundary conditions, adv. pure appl. math. 4 (2013), 165-177. https://doi.org/10.1515/apam-2012-0034. [7] g. dai, r. ma, solutions for a p(x)-kirchhoff type equation with neumann boundary data, nonlinear anal.: real world appl. 12 (2011), 2666–2680. https://doi.org/10.1016/j.nonrwa.2011.03.013. [8] l. diening, p. harjulehto, p. hästö, et al. lebesgue and sobolev spaces with variable exponents, springer berlin heidelberg, berlin, heidelberg, 2011. https://doi.org/10.1007/978-3-642-18363-8. [9] x.l. fan, j.s. shen, d. zhao, sobolev embedding theorems for spaces wk,p(x)(ω), j. math. anal. appl. 262 (2001), 749–760. https://doi.org/10.1006/jmaa.2001.7618. [10] x.l. fan, q.h. zhang, existence of solutions for p(x)-laplacian dirichlet problem, nonlinear anal.: theory methods appl. 52 (2003), 1843–1852. https://doi.org/10.1016/s0362-546x(02)00150-5. [11] x. fan, d. zhao, on the spaces lp(x) and wm,p(x), j. math. anal. appl. 263 (2001), 424–446. https://doi.org/ 10.1006/jmaa.2000.7617. [12] w. han, m. sofonea, quasistatic contact problems in viscoelasticity and viscoplasticity, american mathematical society, providence, r.i, 2002. [13] m.k. hamdani, n.t. chung, d.d. repovš, new class of sixth-order nonhomogeneous p(x)-kirchhoff problems with sign-changing weight functions, adv. nonlinear anal. 10 (2021), 1117–1131. https://doi.org/10.1515/ anona-2020-0172. [14] s. hüeber, a. matei, b.i. wohlmuth, efficient algorithms for problems with friction, siam j. sci. comput. 29 (2007), 70–92. https://doi.org/10.1137/050634141. [15] e.j. hurtado, o.h. miyagaki, r.s. rodrigues, multiplicity of solutions to class of nonlocal elliptic problems with critical exponents, math. methods appl. sci. 45 (2021), 3949–3973. https://doi.org/10.1002/mma.8025. [16] c. vetro, variable exponent p(x)-kirchhoff type problem with convection, j. math. anal. appl. 506 (2022), 125721. https://doi.org/10.1016/j.jmaa.2021.125721. [17] m. ruzicka, electrorheological fluids: modeling and mathematical theory, springer-verlag, berlin, 2002. [18] a. matei, m. sofonea, variational inequalities with applications: a study of antiplane frictional contact problems, springer, new york, 2009. [19] n. tsouli, m. haddaoui, e.m. hssini, multiple solutions for a critical p(x)-kirchhoff type equations, bol. soc. paran. mat. 38 (2019), 197–211. https://doi.org/10.5269/bspm.v38i4.37697. https://doi.org/10.1016/j.nonrwa.2018.10.009 https://doi.org/10.1016/j.nonrwa.2018.10.009 https://doi.org/10.1515/apam-2012-0034 https://doi.org/10.1016/j.nonrwa.2011.03.013 https://doi.org/10.1007/978-3-642-18363-8 https://doi.org/10.1006/jmaa.2001.7618 https://doi.org/10.1016/s0362-546x(02)00150-5 https://doi.org/10.1006/jmaa.2000.7617 https://doi.org/10.1006/jmaa.2000.7617 https://doi.org/10.1515/anona-2020-0172 https://doi.org/10.1515/anona-2020-0172 https://doi.org/10.1137/050634141 https://doi.org/10.1002/mma.8025 https://doi.org/10.1016/j.jmaa.2021.125721 https://doi.org/10.5269/bspm.v38i4.37697 1. introduction 2. preliminaries 3. existence and uniqueness of solutions acknowledgements: references paper title (use style: paper title) international journal of analysis and applications volume 18, number 1 (2020), 117-128 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-117 received 2019-05-01; accepted 2019-11-18; published 2020-01-02. 2010 mathematics subject classification. 82b31. key words and phrases. solar energy; solar cells; sps; sun; wireless power transmission. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 117 some aspects of future energy generation in using of solar power satellites muncho j. mbunwe1, udochukwu b. akuru1, hilary u. ezea2, ogbonnaya i. okoro3, m. ayaz ahmad4,* 1department of electrical engineering, university of nigeria, nsukka, nsukka 410001, nigeria 2department of electrical and electronics engineering, federal university, oye-ekiti, ekiti , nigeria 3department of electrical and electronic engineering, michael okpara university of agriculture, umudike, umuahia 440001, nigeria 4physics department, faculty of science, p.o. box 741, university of tabuk, 71491, saudi arabia *corresponding author: mayaz.alig@gmail.com abstract. energy, which is required to run space satellites, is as old as space technology itself. the location of these satellites has made it more applicable for unconventional means of energy generation to run them. the sun being the universal and greatest source of all forms of energy, as well as the closest energy source to space satellites, has been so appropriately trapped using solar cells. solar power satellites are designed to capture solar energy and transmit that energy to receiving stations using wireless power transmission mechanism. this paper is written to rehearse the immediate and associated usefulness of solar energy trapped from space as an alternative for electricity generation for the future; it is also an attempt to appraise the prospects of this scientific conviviality. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-117 int. j. anal. appl. 18 (1) (2020) 118 i. introduction space satellites are generally regarded as spacecraft that receive signals and send them back to earth. however, they are extremely complex and expensive because they have to work and survive in space for specified time periods, usually up to 15 years. to make this possible, a satellite has to produce its own power by generating electricity from sunlight falling on photovoltaic cells or solar panels. batteries are used to store the energy, so that the satellite can continue to work when the sun is eclipsed or far away. because space is not a friendly environment, space satellites have to survive temperature variations of more than 200°c in a situation described similarly to someone standing in front of a fireplace with a blazing fire while an air conditioner pumps freezing air onto his back. outside the protection of the earth’s atmosphere, the level of radiation (uv, x-rays, gamma-rays and all sorts of energetic particles) is much higher and more destructive than on the ground. before they can even begin to operate in space, satellites have to survive the bone-shaking launch. then the solar panels have to be opened and antennas, which are often stowed to take less space in the launcher, deploys before the satellite enters its operational orbit. this is responsible for the high amount of time and the great number of professionals needed in designing, building and checking for the correct functioning of a satellite. electrical power delivery to earth using space power satellites (sps) has a lot of similarities with space satellite technology. peter glaser, an american aerospace engineer, first proposed sps in the late 60s [1], [2]. during the energy crises in the 70s, the us government took a hard look at them. the studies generated by this inquiry essentially reported that sps were technologically possible but their cost and launch requirements were not. fifteen years later nasa conducted a study to determine if anything had changed [3]. the new study concluded that costs were still high but they were not as high as originally predicted and that there were no technological showstoppers. at present, utility companies and governments are taking a closer look at sps [4], [5]. not only are they looking at them as sources of power on earth but also, as sources of power for satellites to reduce their size and launch costs [6]. sps systems collect solar energy in space and transmit it via a microwave energy beam to an earth-based rectenna which converts the beam into electricity for use on earth, essentially defining the three stages of its operation (fig. 1). sps is thought to have several advantages over other forms of alternative energy, particularly over terrestrial implementation of solar power. int. j. anal. appl. 18 (1) (2020) 119 for instance, power supply system based almost entirely on renewable energy source will have to deal with intrinsic variability of the resources [7]. nevertheless, sps is not affected by weather, seasons, or by the earth’s atmosphere, which can otherwise act as a filter. consequently, there is bound to be more intense exposure to sunlight not hindered by weather; hence no need for expensive storage which would have been of use during periods of low supply. besides, it is harmless to the environment, and with less political interference, since it is environmental friendly, as there is no co2 problem associated with it. although, there have been contrary views that it is too early to implement this technology because of its huge investment requirement, safety concerns and an infantile technology, among others [8], [9]. as a result, this paper is another attempt at highlighting the feasibility and inevitability of the sps technology for future electricity supply. this paper has been organized as follows. section 2 is used to highlight the need for future energy sources. section 3 is used to briefly describe the sps technology as well as their challenges. section 4 is used to provide the state-of-the-art technology, while section 5 is used to give a conclusion. further, energy and power as used in this paper mean the same thing. similarly, solar power satellite (sps), space-based solar power (sbsp) or space solar power (ssp) as used correspondingly within this paper is without prejudice. fig. 1. operational diagram of an sps system int. j. anal. appl. 18 (1) (2020) 120 fig. 2. proposed change in supplies for fossil fuels ii. the need for future energy sources currently, the world relies on fossil fuels to generate electrical power. the reality of a projected but certain fall of the world’s oil reserve is confirmed in studies, e.g., [10]. coal, a major source of energy supply is faced with a similar threat. fig. 2 is meant to show the rate at which these energy sources are expected to decline in the coming years [11]. it is understandable that as the world population increases, the natural resources to produce energy decreases, and alternative methods to produce sustainable, environmentally friendly, and cost effective energy are required (table 1). meantime, fossil fuels are not only depleting, but altering local environment and the world climate [12]. in the same vein, all three factors – petroleum peaking and declining, growing concerns of fuel-derived ghgs, and growing global demand for energy – were supported as the driving force for recent advancements in sps technology [13]. unlike nuclear energy which is equally clean and abundant but very risky, there is no doubt that renewable energy sources give a huge prospect for future energy sources because of its large reservoir and lowering cost of energy, even as it remains the world’s fastest-growing source of electric power in the midst of ever increasing demand by end users [14]. in an era when new energy options are urgently needed, space solar power is an inexhaustible solution – int. j. anal. appl. 18 (1) (2020) 121 sps scores an edge over the ground-based (pv) solar power in three notable areas: increased availability, easy accessibility, higher efficiency [15] and the technologies to make it happen is now at an advanced stage [13]. in a paper, china academy of space technology (cast) grouped into three the major advantages of sps to china’s energy development as sustainable development, disaster prevention and mitigation, and retaining and cultivating talent [16]. stating the several advantages to sps, it was reported in ref. [17] that solar radiation can be more efficiently collected in space, where it is roughly three times stronger than on the surface of the earth and it can be collected 24 hours per day (since there are no clouds or night in high earth orbit). thus, ensuring sps does not use up valuable surface area on the earth by beaming to areas with the highest demand at any particular time. at the same time, current space missions are narrowly constrained by a lack of energy for launch and use in space. missions that are more ambitious will never be realized without new, reliable, and less-expensive sources of energy [18]. other benefits such as clean and safe energy source, limitless reserves and employment/investment opportunities make sps the future energy system to beat. however, like every new technology, the next section is used to highlight some major concerns being raised against space-based solar power technology. iii. technology development and challenges of sps space-based solar power is a system for the collection of solar power in space for use on earth. sps differs from the usual method of solar energy collection in that the solar panels used to collect the energy would reside on a satellite in orbit, rather than on earth's surface. in space, collection of the sun's energy is unaffected by the day/night cycle, weather, seasons, or the filtering effect of earth's atmospheric gases. the average solar energy per unit area in space, i.e., vicinity of the earth is 1366 w/m2 as compared to less than an average of 250 w/m2 obtainable on the surface of the earth. this drastic reduction in energy being as a result of atmospheric absorption and scattering, seasonal variations and weather [19]. int. j. anal. appl. 18 (1) (2020) 122 table 1. expected growth rate of supply and demand of energy sources [20] average projected energy demand growth absolute change (as percent of current supply) by 2025 necessary growth in renewables + nuclear (percent of current supply) by 2025 necessary annual growth in renewables + nuclear 1% 15% 22% 9.3% 2% 32% 39% 12% 3% 51% 58% 15.6% 4% 73% 80% 18.5% 5% 98% 105% 20% a. overview of sps technology space based solar power is comprised of two major technologies which have been studied in some form since the 1980s [21], these are: architecture of the satellite and receiver module; and, the means to beam energy back to earth. both possess challenges, with that of transmission technology being direr. to this, wireless power transmission has been identified as key to the development of sps [22]. these two main technologies are further processed as: 1) means of collecting solar power in space: solar power collection in space can be achieved using planar photovoltaic cells or thermal turbines which convert the solar energy into dc electricity. [15]. the energy collector could be a single crystal silicon which receives direct sunlight and is readily available. however, silicon cells are vulnerable to radiation. the collector could as well be gallium arsenide photovoltaic cells which exhibit a level of resistance to both thermal and radiation degradation but is not readily available. 2) means of transmitting power to earth: due to long distances involved, wired transmission is impractical. consequently, wireless power transmission is adopted. usually, lower radio frequency transmission in the microwave spectrum (2.45 ghz) or high frequencies within the optical and infrared range are required. 3) means of receiving power on earth: the means of receiving the transmitted power depends on the transmission mode adopted during the transmission phase [17]. a rectenna (microwave-to-dc converter), which consists of a rectifying circuit and a receiving antenna most suitable for radio frequencies [23], while an array of photovoltaic cells are used for visible and infrared frequency ranges. the sps concept is very involved, and several integrated studies have been undertaken over the years to design produce an optimum template for its operation [6], [13], [19], [24]. the good int. j. anal. appl. 18 (1) (2020) 123 news is that the basic physics of solar power satellites was resolved in the 1960s and 1970s, while the challenges over the years are, basically, engineering and economics [18]. the basic description of sps, earlier shown in fig. 1, can be adapted to two different architectures [19]: 1) geo-based solar power architecture: the satellite can be placed within the geostationary orbit (geo), medium-earth orbit (meo) or low-earth orbit (leo). in geo, the satellite appears to be stationary because the speed of earth’s rotation corresponds to the orbital period of the satellite round the earth. then due to its position, less than 1% of its total time is spent in shadow. if the satellite is placed in the leo, this minimizes the range and brings about reduction in size and weight of the transmitting antenna. however, satellites in leo have a reduced productivity owing to the fact that much of their time is spent in shadow as they orbit the earth. 2) lunar solar power architecture: here, power collector and beam forming equipment are mounted on the moon surface. this architecture, however, is affected by the 14-day lunar night and as such, constant power is not ordinarily provided. in general, three principal geo-based sps concepts (fig. 3) which have been identified and characterized for technical and commercial applications are [13]: 1) microwave wireless power transmission (wpt)/classic power management architecture, involving large discrete structures (e.g., solar array, transmitter, etc.) assembled by a separate facility in space; this concept involves sun-pointed solar energy collection system, a wireless power transmission system which makes use for microwave radio frequency, and ground based rectennas which receive the transmitted power. 2) modular electric/diode array laser wpt/sandwich power management architecture, involving self-assembling solar power-laser-thermal modules of intermediate scale; this can either be electric-laser or solar-pumped laser. the wireless power transmission here involves laser beam generation at the near visible part of the spectrum 3) modular microwave wpt/sandwich power management architecture, involving a large number of very small solar power-microwave-thermal modules that would robotically assemble in orbit. this is a highly modularized architecture which involves light-redirection based energy distribution approach and depends on an integration of int. j. anal. appl. 18 (1) (2020) 124 solar generation, power management and distribution, as well as wireless power transmission subsystems on individual modules so far, the highly modular microwave wpt sandwich sps appears to be the more suitable, offering clear advantages particularly for large-scale commercial base-load power, with fabrication and prototype testing recently performed [25]. the generic functional architecture which can be used to describe any of the sps concepts is as shown in fig. 3, where other systems integral to the sps concepts have been identified such as [12], [15]: 1) earth-to-orbit (eto) transportation; 2) affordable in-space transportation; 3) space assembly, maintenance and servicing; and, 4) ground energy and interface systems; as well as, 5) in-space resources and manufacturing. it is believed that all of these technologies are reasonably near-term and have multiple attractive approaches. however, a great deal of work is needed to bring them to practical fruition. b. challenges associated with sps key areas of concern in the sps concept are in terms of emerging technology, international policy and regulations, health and safety, terrorism, as well as high costs. the aspect of terrorism comes in since high power microwave source with high gain antenna is usually employed in delivering intense burst of energy which can be used as a weapon [26]. the other problems associated with sps find their roots among any of these main areas. for instance, considering how to transmit the energy from the collection point in space to the place where the energy would be used on earth, technology, cost and health factors become critical. fig. 3. generic sps functional architecture int. j. anal. appl. 18 (1) (2020) 125 fig. 4. conceptualized geo-based sps: (a): microwave classic (b): modular electric laser (c): modular sandwich microwave generic sps functional architecture the greatest barrier to the development of sps has been identified as its high launch costs. as a result, a major spacecraft development project can cost many tens, if not hundreds, of millions of dollars, and over a lengthy period of time [27]. such technical challenges greatly influence the economic feasibility of sps. yet, there are other myths bothering on safety for man, animals, high interference, space materials and orbit, machines and the environment [10]. however, in an attempt to delineate the advantages of sps, it is implied that most of the observed challenges can be abated when the promise of bulk clean energy source of future growing energy demand, high transmission and conversion efficiency, ease of transmission, and non-hazardous radiation are factored [28]. iv. recent developments in sps technology in recent times, more world governments have concentrated time and money to study and develop the sps technology. intense efforts, spanning a 40 years period, have been made in the us, canada and europe [28]. china and india both started recent activities, while japan has been undertaking steady technological strides in harnessing the technology up to developmental stage. in the wake of the disaster at the fukushima daiichi nuclear power plant, japan aerospace exploration agency (jaxa), a world-leader in sps systems, now has a technology roadmap that proposes a 1 gw commercial system by 2030 [27]. elsewhere, international conferences, workshops and committees of experts have been sprawling the sps concept in recent times in order to promote international dialogue and coordinate sps efforts. the on-going power symposium of the international astronomical congress (iac), as well as periodic conferences dedicated to sps and wpt have been reported, and also hosted on a dedicated online space solar power library [13], [29]. (a) (b) (c) int. j. anal. appl. 18 (1) (2020) 126 another plus is the recent international assessment developed by the international academy of astronautics (iaa) study [13], which has been serially referenced in this paper. its key findings were: 1) need for new energy technologies such as sps; 2) sps is feasible; 3) sps will soon be economically viable; 4) more research is needed to fine-tune all parts of the technology; 5) low-cost eto transportation will make sps commercially viable; 6) prototype testing-of-concept need to be initiated to boost technology confidence; 7) technological development is currently transient; 8) existing power networks for transmission; and, 9) need for regulatory and policy frameworks. v. conclusion so far, sps technology – not new at all – appears to be receiving wider attention from governments and key stakeholders. to this end, the satellite-based technology is gradually converging most of its trajectories for the development and production of a prototype. besides, the impetus for sps appears to be an underlying drive to address concurrent global challenges of energy demand for a growing world population, in the midst of increasing environmental pollution and certain oil depletion. though bearing in mind the present huge cost of production and other critical factors, sps technology invariably offers great promise to tap directly into the sun’s abundant solar energy for zero-carbon cum sustainable electricity production, and at a significantly greater efficiency than the current earth-based pv systems. the challenge however is that much work still needs to be done in terms of plugging together the technical and economic holes of the sps technology, apparently making it a future source of power generation. acknowledgment authors muncho j. mbunwe & udochukwu b. akuru from university of nigeria, nsukka, nigeria and m. ayaz ahmad from university of tabuk, saudi arabia have been working since a long time in personal joint collaboration. we would like to acknowledge the keen support in financial assistance for this research work to their respective institutes / universities [30-32]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (1) (2020) 127 references [1] glaser, p. e. power from the sun: its future. science, 162 (3856) (1968), 857–861. [2] glaser, p. e. the earth benefits of solar power satellites, the space congress proceeding, 3, (1979), 5-11–5-25. [3] glaser, p. e., maynard, o. e., mackovciak, j., and ralph, e. l, arthur d. little, inc., feasibility study of a satellite solar power station, nasa cr-2357, ntis n74-17784, february 1974. [4] rich, japan’s solar revolution – the sky’s (not) the limit. tofugu.com: posted may 29, 2014. [5] bullis, k., startup to beam power from space, in mit technology review: potential energy. k. bullis (ed.), 2009. [6] mankins, j. c., sps-alpha: the first practical solar power satellite via arbitrarily large phased array (a 2011-2012 nasa niac phase 1 project), nasa innovative advanced concepts program, niac phase 1 final report, 15 september 2012. [7] weitemeyer s, kleinhans d, vogt t, and agert c, integration of renewable energy sources in future power systems: the role of storage, renewable energy, 75(2015), 14-20. [8] fetter, s., space solar power: an idea whose time will never come? phys. soc. 33(1)(2004), 10–11. [9] murphy, t., space-based solar power. available from: http://physics. ucsd.edu/do-themath/2012/03/space-based-solar-power/, 2012. [10] akuru, u.b. and okoro, o.i. a prediction on nigeria’s oil depletion based on hubbert’s model and the need for renewable energy, isrn renewable energy, 2011(2011), article id 285649. [11] ren21, renewables 2010 global status report (paris: ren21 secretariat), 2010, available online: https://www.ren21.net/reports/global-status-report/. [12] akuru, u.b., okoro, o.i. and chikuni, e., impact of renewable energy deployment on climate change in nigeria, journal of energy in southern africa, (26)3, (2015), 125-134. [13] mankins, j. c. and kaya, n. (eds.), space solar power, the first international assessment of space solar power: opportunities, issues and potential pathways forward. internatinal academy of astronautics (iaa), august 2011. [14] international energy outlook 2019. available online at: https://www.eia.gov/outlooks/ieo/. [15] markad c, markad s, and mukhedkar m, wireles power transmission by using solar power satellite, int. j. adv. res. electron. commun. eng. 3(12)(2014), 1793-1796. [16] ji, g., xinbin, h. and li, w. solar power satellite research in china, online journal of space communication, issue no. 16: china sps strategy and schedule, winter 2010. [17] ciotola, m. (ed.), space-based solar energy: a brief review and analysis, national solar power research institute, 1998. [18] mankins, j. c. energy from orbit: solar based solar power, ad astra, national space society, spring 2008. https://www.ren21.net/reports/global-status-report/ https://www.eia.gov/outlooks/ieo/ int. j. anal. appl. 18 (1) (2020) 128 [19] rouge, j. d. space based solar power as an opportunity for strategic security, phase 0 architechture feasibility study, national sceurity space office interim assessment, 10 october, 2007. [20] aleklett, k., jakobsson, m. h. k., lardelli, m., snowden, s. and soderbergh, b. the peak of the oil age – analysisng the world oil production. energy policy, 38(3)( 2010), 1398-1414. [21] fan, w., martin, h., wu, j. and mok, b. space based solar power, http://docshare04.docshare.tips/files/10246/102466914.pdf. [22] nansen, r. h. wireless power transmission: the key to solar power satelittes, ieee aerospace and electronic systems magazine, 11(1)(1996), 33 – 39. [23] mcspadden j. o. and mankins j. c, space solar power programs and microwave wireless power transmission technology, ieee microwave magazine, 3(4)(2002), 46-57. [24] gibbons, j. h., solar power satellites, u.s. government printing office, washington, august 1981. [25] jaffe, p., hodkin, j., harrington, f., person, c., nurnberger, m., nguyen, b., lacava, s., scheiman, d., stewart, g., han, a., hettwer, e., and rhoades d. sandwich module prototype progress for space solar power, acta astronautica, 94(2)(2014), 662-671. [26] pozar d. m, microwave engineering, wiley, 4th edition 2012. [27] sasaki, s., how japan plan to build an orbital solar farm, ieee spectrum, 24(2014), 46-51. [28] mohammed s.s. and ramasamy k. solar powergeneration using sps and wireless power transmission. proc. int. conf. energy environ. 2009, 413-418. [29] https://space.nss.org/space-solar-power-library/ (last accessed 12.12.2019). [30] c. victoria anghel drugarin, vyacheslav v. lyashenko, muncho j. mbunwe, m. ayaz ahmad, pre-processing of images as a source of additional information for image of the natural polymer composites, j. analele universitatii "eftimie murgu" resita. fascicula de inginerie, 25(2) (2018), 11-16. [31] m. ayaz ahmad, mir hashim rasool, jalal h. baker, shafiq ahmad, muncho j. mbunwe, nuclear effect in terms of hurst exponent in 28si-emulsion collisions at 14.6a gev, proc. dae symp. nuclear phys. 63(2018), 956-957. [32] nursabah sarikavakli, syed khalid mustafa, m. ayaz ahmad, vyacheslav lyashenko, muncho j. mbunwe, an interesting approach of bray–liebhafsky (b-l) oscillatory chemical reactions, eur. j. chem. environ. eng. sci. 2(3) (2018), art01. https://space.nss.org/space-solar-power-library/ international journal of analysis and applications volume 17, number 1 (2019), 1-13 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-1 controlled ∗-g-frames and ∗-g-multipliers in hilbert pro-c∗-modules zahra ahmadi moosavi1,∗ and akbar nazari2 1department of mathematics faculty of mathematics and computer shahid bahonar university of kerman, 76169-14111, kerman, iran 2department of mathematics faculty of mathematics and computer shahid bahonar university of kerman, 76169-14111, kerman, iran ∗corresponding author: nazari@uk.ac.ir abstract. a generalization of multiplier, controlled g-frames and g-bessel sequences to ∗-g-frames and ∗-g-bessel sequences in hilbert pro-c∗-modules is presented. it is demonstrated that controlled ∗-g-frames are equivalent to ∗-g-frames in hilbert pro-c∗-modules. 1. introduction frame theory is an application of harmonic analysis. this theory has been rapidly generalized to hilbert spaces and hilbert c∗-modules. in 2005, sun [22] introduced the notion of g-frames as a generalization of frames for bounded operators on hilbert spaces. frank-larson [5] have extended the theory for elements of c∗-algebras and (finitely or countably generated) hilbert c∗-modules have been considered in [1]. it is well known that hilbert c∗-modules are a generalization of hilbert spaces where the inner product takes values in a c∗-algebra rather than in the field of complex numbers. the theory of hilbert c∗-modules received 2018-04-13; accepted 2018-06-22; published 2019-01-04. 2010 mathematics subject classification. 42c15, 46l08. key words and phrases. hilbert pro-c∗-modules; ∗-g-frames; ∗-g-bessel sequences; controlled ∗-g-frames; (c, c′)-controlled ∗-g-frames. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-1 int. j. anal. appl. 17 (1) (2019) 2 has applications in the study of locally compact quantum groups, complete maps between c∗-algebras, noncommutative geometry and kk-theory. not all properties of hilbert spaces hold in hilbert c∗-modules. for instance, the riesz representation theorem for continuous linear functionals on hilbert spaces can not be extended to hilbert c∗-modules [23] and there exist closed subspaces in hilbert c∗-modules that have no orthogonal complement [16]. moreover, as known, every bounded operator on a hilbert space has an adjoint whereas there are bounded operators on hilbert c∗-modules which do not have this property [17]. so, it is to be expected that frames and ∗-frames in hilbert c∗-modules are more complicated than those in hilbert spaces. the properties of g-frames for hilbert c∗-modules have been widely investigated in the literature ( see [1, 5, 12, 25], and the references therein). the paper is organized as follows.in the next section, we give a brief survey of the fundamental definitions and notations of hilbert pro-c∗-modules. section 3 is devoted to investigating ∗-g-frames with a-valued bounds and analyzing their elementary properties. in section 4 we define the concept of controlled ∗-g-frames and we show that a controlled ∗-gframe is equivalent to a ∗-g-frame in hilbert pro-c∗-modules. finally, in section 5 we define multipliers of controlled ∗-g-frame operators in hilbert pro-c∗-modules. 2. preliminaries in this section, we recall some of the basic definitions and properties of pro-c∗-algebras and hilbert modules over them [7, 15, 18]. a pro-c∗-algebra is a complete hausdorff complex topological ∗-algebra a whose topology is determined by its continuous c∗-seminorms in the sense that a net {aλ} converges to 0 iff ρ(aλ) → 0 for any continuous c∗-seminorm ρ on a and we have: (1) ρ(ab) ≤ ρ(a)ρ(b); (2) ρ(a∗a) = ρ(a)2; for all c∗-seminorms ρ on a and a,b ∈a. if the topology of pro-c∗-algebra is determined by only countably many c∗-seminorms, then it is called a σ-c∗-algebra. let a be a unital pro-c∗-algebra with unit 1a and let a ∈ a . then spectrum sp(a) of a ∈ a is the set {λ ∈ c : λ1a − a is not invertible}. if a is not unital, then the spectrum is taken with respect to its unitization ã. if a+ denotes the set of all positive elements of a, then a+ is a closed convex c∗-seminorms on a. we denote by s(a), the set of all continuous c∗-seminorms on a. example 2.1. every c∗-algebra is a pro-c∗-algebra. int. j. anal. appl. 17 (1) (2019) 3 example 2.2. a sub-closed ∗-algebra of a pro-c∗-algebra is a pro-c∗-algebra. proposition 2.1 ( [6]). let a be a unital pro-c∗-algebra with an identity 1a.then for any ρ ∈ s(a), we have: (1) ρ(a) = ρ(a∗) for all a ∈ a; (2) ρ(1a) = 1; (3) if a,b ∈a+ and a ≤ b, then ρ(a) ≤ ρ(b); (4) if 1a ≤ b, then b is invertible and b−1 ≤ 1a; (5) if a,b ∈a+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1; (6) if a,b,c ∈a and a ≤ b then c∗ac ≤ c∗bc; (7) if a,b ∈a+ and a2 ≤ b2, then 0 ≤ a ≤ b. definition 2.1. a pre-hilbert module over pro-c∗-algebra a, is a complex vector space e which is also a left a-module compatible with the complex algebra structure, equipped with an a-valued inner product 〈., .〉 : e ×e →a which is c-and a-linear in its first variable and satisfies the following conditions: (1) 〈x,y〉∗ = 〈y,x〉; (2) 〈x,x〉≥ 0; (3) 〈x,x〉 = 0 iff x = 0; for every x,y ∈ e. we say e is a hilbert a-module (or hilbert pro-c∗-module overa) if e is complete with respect to the topology determined by the family of seminorms ρe(x) = √ ρ(〈x,x〉) x ∈ e,ρ ∈ s(a). let e be a pre-hilbert a-module.by [6], for ρ ∈ s(a) and for all x,y ∈ e, the following cauchybunyakovskii inequality holds: ρ(〈x,y〉)2 ≤ ρ(〈x,x〉)ρ(〈y,y〉). consequently, for each ρ ∈ s(a), we have: ρe(ax) ≤ ρ(a)ρ(x), a ∈a,x ∈ e. let a be a pro-c∗-algebra and e and f be two hilbert a-modules. an a-module map t : e → f is said to bounded if for each ρ ∈ s(a), there is cρ > 0 such that : ρf (tx) ≤ cρ. ρe(x) (x ∈ e), where ρe, respectively ρf , are continuous seminorms on e, respectively f. a bounded a-module map from e to f is called an operators from e to f . we denote the set of all operators from e to f by homa(e,f), and we set homa(e,f) = enda(e) int. j. anal. appl. 17 (1) (2019) 4 proposition 2.2. let t∗ ∈ homa(e,f). we say t is adjointable if there exists an operator t∗ ∈ t ∈ homa(f,e) such that: 〈tx,y〉 = 〈x,t∗y〉 holds for all x ∈ e,y ∈ f. we denote by hom∗a(e,f), the set of all adjointable operator from e to f and end ∗ a(e) = hom ∗ a(e,e) proposition 2.3 ( [6]). let t : e → f and t∗ : f → e be two maps such that the equality 〈tx,y〉 = 〈x,t∗y〉 holds for all x ∈ e, y ∈ f.then t ∈ hom∗a(e,f). it is easy to see that for any ρ ∈ s(a), the map defined by: ρ̂e,f (t) = sup{ρf (t(x) : x ∈ e, ρe(x) ≤ 1}, t ∈ homa(e,f), is a seminorm on homa(e,f). definition 2.2. let e and f be two hilbert modules over pro-c∗-algebra a. then the operator t : e → f is called uniformly bounded (below), if there exists c > 0 such that: ρf (tx) ≤ c ρe(x). (2.1) (c ρe(x) ≤ ρf (tx)) (2.2) the number c in (2.1) is called an upper bound for t and we set : ‖t‖∞ = inf{c : c is an upper bound for t}. clearly, in this case we have: ρ̂(t) ≤‖t‖∞, ∀ρ ∈ s(a). let t be an invertible element in end∗a(e) such that both are uniformly bounded. then by [2, proposition 3.2], for each x ∈ e we have the inequality ‖t−1‖−2∞ 〈x,x〉≤ 〈tx,tx〉≤ ‖t‖ 2 ∞〈x,x〉. (2.3) the following proposition will be used in the next section. proposition 2.4 ( [6]). let t be an uniformly bounded below operator in homa(e,f). then t is closed(range) and injective. int. j. anal. appl. 17 (1) (2019) 5 3. ∗-g-frames in hilbert pro-c∗-modules throughout this section a is a pro-c∗-algebra, u and v are two hilbert a-modules. also {vj}j∈j is a countable sequence of closed submodules of v . definition 3.1. a sequence λ = {λj ∈ hom∗a(u,vj)}j∈j is called a ∗g-frame for u with respect to {vj}j∈j if c〈f,f〉c∗ ≤ ∑ j∈j〈λjf, λjf〉≤ d〈f,f〉d ∗ for all f ∈ u and strictly nonzero elements c,d ∈a. the number c and d are called ∗-g-frame bounds for λ. the ∗-g-frame is called tight if c = d and a parseval if c = d = 1. if in the above we only have the upper bound, then λ is called a ∗-g-bessel sequence. also if for each j ∈ j,vj = v , we call λ a ∗-g-frame for u with respect to v . we mentioned that the set of all g-frames in hilbert pro-c∗-modules are a subset of the family of ∗g-frames. to illustrate this, let λ = {λj}j∈j be a g-frame for u with respect to {vj}j∈j. note that for f ∈ u, ( √ c)1a〈f,f〉( √ c)1a ≤ ∑ j∈j〈λjf, λjf〉( √ d)1a〈f,f〉( √ d)1a therefore, every g-frame for u with real bounds c amd d is a ∗-g-frame for u with a-valued ∗-g-frame bounds ( √ c)1a and ( √ d)1a. example 3.1. let `2(a) be the set of all sequences (an)n∈n of elements of a pro-c∗-algebra a such that the series ∑ i∈n aia ∗ i is convergent in a. then, by [2, example 3.2], ` 2(a) is a hilbert module over a with respect to pointwise operations and inner product defined by: 〈(ai)i∈n, (bi)i∈n〉 = ∑ i∈n aib ∗ i . let a = (ai)i∈n and b = (bi)i∈n in ` 2(a). we define ab = {aibi}i∈n and ρ(a) = √ ρ(〈a,a〉) and a∗ := {ai}i∈n and 〈a,b〉 = ab∗ = ∑ i∈n aib ∗ i . now, let j ∈ j := n and define fj ∈ `2(a) by fj = {f j i }i∈n such that f j i =   1 i 1a i = j; 0 i 6= j, ∀j ∈ n. set λj : ` 2(a) →a by λfj (u) = 〈u,fj〉 for any u ∈ `2(a) . we see that∑ j∈j〈λfj (u), λfj (u)〉≤ 〈u,u〉. thus{λj}j∈j is a ∗-g-bessel sequence . int. j. anal. appl. 17 (1) (2019) 6 definition 3.2. let λ = {λj ∈ end∗a(u,vj)}j∈j be a ∗-g-frame for u with respect to {vj}j∈j with bounds c and d. we define the corresponding ∗-g-frame transform as follows: tλ : u → ⊕ j∈j vj , tλf = {λjf : j ∈ j}, for all f ∈ u. since λ is a ∗-g-frame, hence for each f ∈ u we have: c 〈f,f〉 c∗ ≤ ∑ j∈j〈λjf, λjf〉≤ d 〈f,f〉 d ∗, so tλ is well-defined. also for any ρ ∈ s(a) and f ∈ u the following inequality is obtained: ρ(c)2 ρu (f) ≤ ρ⊕ j vj (tλf) ≤ ρ(d)2 ρu (f). from the above, it follows that the ∗-g-frame transform is an uniformly bounded below operator in enda(u, ⊕ j∈j vj). thus by proposition 2.4, tλ is closed and injective. now, we define the synthesis operator for ∗-g-frame λ as follows: t∗λ : ⊕ j∈j vj → u, t∗λ({yj}j) = ∑ j∈j λ∗j (yj), (3.1) where λ∗j is the adjoint operator of λj. proposition 3.1. the synthesis operator defined by (3.1) is well-defined, uniformly bounded and the adjoint of the transform operator. proof. since λ = {λj : j ∈ j} is a ∗-g-frame for u with respect to {vj}j∈j, there exist c,d ∈a such that for any f ∈ u, c 〈f,f〉 c∗ ≤ ∑ j∈j〈λjf, λjf〉≤ d 〈f,f〉 d ∗. let i be an arbitrary finite subset of j. using the cauchy-bunyakovskii inequality and [24, lemma 2.2], for any ρ ∈ s(a) and (yj)j ∈ ⊕ j∈j vj we have: ρ( ∑ j∈i λ∗j (yj)) = sup{ρ〈 ∑ j∈i λ∗j (yj),f〉 : f ∈ u , ρ(f) ≤ 1} = sup{ρ( ∑ j∈i 〈yj, λjf〉) : f ∈ u , ρ(f) ≤ 1} ≤ sup ρ(f)≤1  ρ(∑ j∈i 〈yj,yj〉)  1/2  ρ(∑ j∈i 〈λjf, λjf〉)  1/2 ≤ sup ρ(f)≤1 ρ(dd∗)1/2ρ(f)(ρ ∑ j∈i 〈yj,yj〉)1/2 ≤  ρ(d) (ρ∑ j∈i 〈yj,yj〉)1/2   . int. j. anal. appl. 17 (1) (2019) 7 now, since the series ∑ j∈j〈yj,yj〉 converges in a, the above inequality shows that ∑ j∈j λ ∗ j (yj) is convergent. hence t∗λ is well-defined. on the other hand, for any f ∈ u and (yj)j ∈ ⊕ j∈j vj, we have: 〈tλ(f), (yj)j〉 = 〈(λjf)j, (yj)j〉 = ∑ j∈j 〈λjf,yj〉 = ∑ j∈j 〈f, λ∗jyj〉 = 〈f, ∑ j∈j λ∗jyj〉 = 〈f,t∗λ(yj)j∈j〉. therefore by proposition 2.2 it follows that the synthesis operator is the adjoint of the transform operator. also, for any ρ ∈ s(a) we have: ρu (t ∗ λ(y)) ≤ ρ(d) ρ⊕j∈j vj (y), y = (yj)j ∈ ⊕ j∈j vj. hence the synthesis operator is uniformly bounded. � let λ = {λj , j ∈ j} be a ∗-g-frame for u with repect to {vj}j∈j. define the corresponding ∗-g-frame operator sλ as follows: sλ = t ∗ λtλ : u → u sλ(f) = ∑ j∈j λ ∗ j λjf. since sλ is a combination of two bounded operators, it is a bounded operator. theorem 3.1. let λ = {λj}j∈j be a ∗-g-frame for u with respect to {vj}j∈j and with bounds c,d. then sλ is an invertible positive operator. also it is a self-adjoint operator such that: ciuc ∗ ≤ sλ ≤ diud∗. (3.2) here iu is the identity function on u. proof. according to the definition of the transform operator, for any f ∈ u we can write: 〈tλ(f),tλ(f)〉 = 〈{λjf}j∈j,{λjf}j∈j〉 = ∑ j∈j〈λjf, λjf〉. since λ is a ∗-g-frame for u with bounds c and d, for each f ∈ u it follows that c〈f,f〉c∗ ≤〈tλ(f),tλ(f)〉≤ d〈f,f〉d∗. on the other hand, 〈sλ(f),f〉 = 〈t∗λtλ(f),f〉 = 〈tλ(f),tλ(f)〉 = 〈f,t ∗ λtλ(f)〉 = 〈f,sλ(f)〉. consequently, sλ is a self-adjoint operator. also, for any f ∈ u, we obtain int. j. anal. appl. 17 (1) (2019) 8 c〈f,f〉c∗ ≤〈sλ(f),f〉≤ d〈f,f〉d∗. it follows that ∗-g-frame operator is positive and (3.2) also holds. moreover, since sλ is one-to-one it follows that sλ is invertible. � according to (3.2) and proposition 2.1 we have the following lemma lemma 3.1. d−1iu (d −1)∗ ≤ s−1λ ≤ c −1iu (c −1)∗. hence the ∗-g-frame operator and its inverse belong to end∗a(u). theorem 3.2. let {λj ∈ end∗a(u,vj)}j∈j and ∑ j∈j〈λjf, λjf〉 converge in the semi-norm for f ∈ u. then λ = {λj}j∈j is a ∗-g-frame for u with respect to {vj}j∈j if and only if there are two strictly nonzero elements c,d ∈a such that for every f ∈ u, ρ(c−1)−1 ρ(〈f,f〉)ρ(c∗−1)−1 ≤ ρ( ∑ j∈j 〈λjf, λjf〉) ≤ ρ(d) ρ(〈f,f〉)ρ(d∗). (3.3) proof. if {λj ∈ end∗a(u,vj)}j∈j is a ∗-g-frame for u with respect to {vj}j∈j, then (〈f,f〉) ≤ c−1( ∑ j∈j 〈λjf, λjf〉)(c∗)−1) and ( ∑ j∈j 〈λjf, λjf〉) ≤ d〈f,f〉d∗. therefore, by proposition 2.1, ρ(c−1)−1 ρ(〈f,f〉)ρ(c∗−1)−1 ≤ ρ( ∑ j∈j 〈λjf, λjf〉) ≤ ρ(d) ρ(〈f,f〉)ρ(d∗). (3.4) for the converse, let (3.3) hold. then we define a linear operator as follows: m : u → ⊕ j∈j vj, m(f) = {λjf}j∈j, ∀f ∈ u, 〈mf,mf〉 = ∑ j∈j 〈λjf, λjf〉, ∀f ∈ u. hence, by (3.3), we have ρu (m(f)) ≤ ρ(d) 1 2 ρu (f) ρ(d ∗) 1 2 . int. j. anal. appl. 17 (1) (2019) 9 this shows that m is uniformly bounded. we write m∗m = k. then 〈m(f),m(f)〉 = 〈m∗m(f),f〉 = 〈k(f),f〉. therefore, k is positive. as, k∗ = (m∗m),k is self-adjoint. on the other hand, 〈k 1 2 f,k 1 2 f〉 = 〈kf,f〉 = ∑ j∈j 〈λjf, λjf〉. now, according to proposition 2.4 and (3.3), k 1 2 is invertible and uniformly bounded; therefore, by [2, proposition 3.2], we have: ‖k− 1 2‖−1∞ 〈f,f〉‖k −1 2‖−1∞ ∗ ≤〈k 1 2 (f),k 1 2 (f)〉≤ ‖k 1 2‖∞〈f,f〉‖k 1 2‖∞ hence {λj}j∈j is a ∗-g-frame. � 4. controlled ∗-g-frames in hilbert pro-c∗-modules in this section, we define the concept of multipliers for ∗-g-bessel sequences and we show that controlled ∗-g-frames are equivalent to ∗-g-frames. let a be a pro-c∗-algebra, u and v be two hilbert a-modules. also, let {vj}j∈j be a countable sequence of closed submodules of v , l(u,v ) and l(u) the collection of all bounded linear operators from u into v and u respectively. gl(u) the set of all bounded operators with a bounded inverse and gl+(u) be the set of positive operators in gl(u). proposition 4.1. let λ = {λj ∈ l(u,vj) : j ∈ j} and θ = {θj ∈ l(u,vj) : j ∈ j} be ∗-g-bessel sequences with bounds bλ and bθ. if for m = {mj}j ⊆ `∞(r), the operator m = mm,λ,θ : u → u m(f) = ∑ j mjλ ∗ jθjf, (4.1) is well-defined, then m is called the ∗-g-multiplier of λ,θ and m. proof. let i be an arbitrary finite subset of j. using the cauchy-bunyakovskii inequality and [24, lemma 2.2], for any ρ ∈ s(a) and f ∈ u we have: ρ( ∑ j∈i mjλ ∗ jθjf) = sup{ρ〈 ∑ j∈i mjλ ∗ jθjf,g〉 : g ∈ u , ρ(g) ≤ 1} = sup{ρ( ∑ j∈i 〈mjθjf, λjg〉) : g ∈ u , ρ(g) ≤ 1} ≤ sup ρ(g)≤1  ρ(∑ j∈i 〈mjθjf,mjθjf〉)  1/2  ρ(∑ j∈i 〈λjg, λjg〉)  1/2 . int. j. anal. appl. 17 (1) (2019) 10 since ∑ j 〈mjθjf,mjθjf〉 = ∑ j mj〈θjf,θjf〉m∗j = ∑ j (ρ(mj)) 2〈θjf,θjf〉 ≤ ‖m‖2∞bθ〈f,f〉b ∗ θ, so by proposition 2.1 we have: ρ( ∑ j〈mjθjf,mjθjf〉) ≤‖m‖ 2 ∞(ρ(f)) 2ρ(bθ) 2. hence we have: ρ( ∑ j∈i mjλ ∗ jθjf) ≤‖m‖∞ ρ(f) ρ(bθ) ρ(bλ) � definition 4.1. let c,c′ ∈ gl+(u). the family λ = {λj ∈ l(u,vj) : j ∈ j} is called a (c,c′)-controlled ∗-g-frame for u with respect to {vj}j∈j, if λ is a ∗-g-bessel sequence and a〈f,f〉a∗ ≤ ∑ j∈j 〈λjcf, λjc′f〉≤ b〈f,f〉b∗, (4.2) for all f ∈ u and strictly nonzero elements a,b ∈a. a,b are called controlled ∗-g-frame bounds. if c′ = i, we call λ = {λj}j a c-controlled ∗-g-frame for u with bounds a,b. if only the second part of the above inequality holds, it is called a (c,c′)-controlled ∗-g-bessel sequence with bound b. lemma 4.1 ( [2]). let x be a hilbert module over c∗-algebra b, s ≥ 0, i.e. this element is positive in c∗-algebra l(u). then for each x ∈ x, 〈sx,x〉≤ ‖s‖〈x,x〉. proposition 4.2. let c ∈ gl+(h). the family λ = {λj ∈ l(u,vj) : j ∈ j} is a ∗-g-frame if and only if λ is a c2controlled ∗-g-frame. proof. let λ be a c2controlled ∗-g-frame with bounds a,b. then a〈f,f〉a∗ ≤ ∑ j∈j 〈λjcf, λjcf〉≤ b〈f,f〉b∗, for f ∈ u. a〈f,f〉a∗ = a〈cc−1f,cc−1f〉a∗ ≤ a‖c‖2〈c−1f,c−1f〉a∗ ≤‖c‖2 ∑ j∈j 〈λjcc−1f,cc−1f〉. int. j. anal. appl. 17 (1) (2019) 11 hence a‖c‖−1〈f,f〉a∗‖c‖−1 ≤ ∑ j∈j 〈λjf, λjf〉. on the other hand for every f ∈ u ∑ j∈j 〈λjf, λjf〉 = ∑ j∈j 〈λjcc−1f,cc−1f〉 ≤ b〈c−1f,c−1f〉b∗ ≤ b‖c−1‖2〈f,f〉b∗. these inequalities yield that λ is a ∗-g-frame with bounds a‖c−1‖,b‖c−1‖. conversely assume that λ is a ∗-g-frame with bounds a′,b′. then for all f ∈ u, a′〈f,f〉a′ ∗ ≤ ∑ j∈j 〈λjf, λjf〉≤ b′〈f,f〉b′ ∗ . so for f ∈ u, ∑ j∈j 〈λjcf, λjcf〉≤ b′〈cf,cf〉b′ ∗ ≤ b′‖c‖2b′ ∗ . for the lower bound, since λ is ∗-g-frame for any f ∈ u, a′〈f,f〉a′ ∗ = a′〈c−1cf,c−1cf〉a′ ∗ ≤ a′‖c−1‖2〈cf,cf〉a′ ∗ ≤‖c−1‖2 ∑ j∈j 〈λjcf, λjcf〉. therefor λ is a c2-controlled ∗-g-frame with bounds a′‖c−1‖,b′‖c−1‖ � 5. multipliers of controlled ∗-g-frames in hilbert pro-c∗-modules in this section, we define the multiplier of a controlled ∗-g-frame for c-controlled ∗-g-frames in hilbert pro-c∗-modules. the definition of general case (c,c′)-controlled ∗-g-frames is similar. lemma 5.1. let c,c′ ∈ gl+(u) and λ = {λj ∈ l(u,vj) : j ∈ j},θ = {θj ∈ l(u,vj) : j ∈ j} be c′2 and c2-controlled ∗-g-bessel sequences for u, respectively. let m = `∞ . then mm,c,θ,λ,c′ : u → u, defined by mm,c,θ,λ,c′f := ∑ j∈j mjcθ ∗ j λjc ′f, is a well-defined bounded operator. int. j. anal. appl. 17 (1) (2019) 12 proof. let λ = {λj ∈ l(u,vj) : j ∈ j},θ = {θj ∈ l(u,vj) : j ∈ j} be c′2 and c2-controlled ∗-g-bessel sequences for u, with bounds b,b′, respectively. for any f,g ∈ u and finite subset i ⊆ j, ρ( ∑ j∈i mjcθ ∗ j λjc ′f) ≤ sup{ρ〈 ∑ j∈i mjcθ ∗ j λjc ′f,g〉 : g ∈ u , ρ(g) ≤ 1} = sup{ρ( ∑ j∈i 〈mjλjc′f,θjc∗g〉) : g ∈ u , ρ(g) ≤ 1} ≤ sup ρ(g)≤1  ρ(∑ j∈i 〈mjλjc′f,mjλjc′f〉)  1/2  ρ(∑ j∈i 〈θjc∗g,θjc∗g〉)  1/2 , since ∑ j 〈mjλjc′f,mjλjc′f〉 = ∑ j mj〈λjc′f, λjc′f〉m∗j = ∑ j (ρ(mj)) 2〈λjc′f, λjc′f〉 ≤ ‖m‖2∞b〈f,f〉b ∗. so by proposition 2.1 we have: ρ( ∑ j 〈mjλjc′f,mjλjc′f〉) = ρ( ∑ j mj〈λjc′f, λjc′f〉m∗j ) ≤‖m‖2 ∞ (ρ(f))2ρ(b)2. hence ρ( ∑ j∈i mjcθ ∗ j λjc ′f) ≤‖m‖∞ ρ(f) ρ(b) ρ(b) ′. this shows that mm,c,θ,λ,c′ is well-defined and ρ(mm,c,θ,λ,c′) ≤‖m‖∞ ρ(b) ρ(b)′. � the above lemma provides a motivation for the following definition. definition 5.1. let c,c′ ∈ gl+(u) and λ = {λj ∈ l(u,vj) : j ∈ j},θ = {θj ∈ l(u,vj) : j ∈ j} be c′2 and c2-controlled ∗-g-bessel sequences for u, respectively. let m = `∞ . the operator mm,c,θ,λ,c′ : u → u, defined by mm,c,θ,λ,c′f := ∑ j∈j mjcθ ∗ j λjc ′f, is called (c,c′)-controlled multiplier operator with symbol m. int. j. anal. appl. 17 (1) (2019) 13 references [1] a. alijani, m. a. dehghan, ∗frames in hilbert -c*-modules, u.p.b. sci. bull. series a, 7(1)5 (2013), 129-140. [2] m.azhini, n. haddadzadeh, fusion frames in hilbert modules over pro-c*-algebras, int. j. ind. math. 5 (2013), no. 2, 109-118. [3] r. j. duffin, and a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341-366. [4] m. frank, d.r. larson, a module frame concept for hilbert c*-modules, in: functional and harmonic analysis of wavelets, san antonio, tx, january, 1990, in: contemp. math, 247, amer. math. soc. providence, ri, 2000, 207-233. [5] m. frank, d.r. larson, frames in hilbert c*-modules and c*-algebras, j. oper. theory 48 (2) (2002) 273-314. [6] n. haddadzadeh,g-frames in hilbert pro-c*-modules, int. j. pure appl. math. 105 (2015), 727-743. [7] a. inoue, locally c*-algebras, mem. fac. sci. kyushu univ. ser. a, 25 (1971), no. 2, 197-235. [8] m. joita, on hilbertmodules over locally c*-algebras. ii, period. math. hung. 51 (1) (2005), 27-36. [9] m. joita, hilbert modules over locally c*-algebras, university of bucharest press, (2006), 150. [10] m. joita,on frames in hilbert modules over pro-c*-algebras, topol. appl. 156 (2008), 83-92. [11] g.g kasparov, hilbert c*-modules, thorem of stinespring and voiculescu, j. operator theory, 4 (1980), 133-150. [12] a. khosravi, b. kosravi, fusion frames and g-frames in hilbert c*-modules, int. j. wavelets multiresolut. inf. process. 6 (3) (2008), 433-446. [13] a. khosravi, m. s. asgari, frames and bases in hilbert modules over locally c*-algebras, int. j. pure appl. math., 14 (2004), no. 2, 169-187. [14] a. khosravi, m. asgari, frames and bases in hilbert modules over locally c*-algebras, indian j. pure appl. math., 14 (2004), no 2, 171-190. [15] e. c. lance, hilbert c*-modules, a toolkit for operator algebraists, london math. soc. lecture note series 210. cambridge univ. press, cambridge, 1995. [16] b. magagajnahosravi, hilbert c*-modules in which all closed submodules are complemented, proc. amer. math. soc, 125 (3) (1997), 849-852. [17] v. m. manuilov, adjointability of operators on hilbert c*-modules, acta math. univ. comenianae, 65 (2) (1996), 161-169. [18] n.c. phillips, inverse limits of c*-algebras, j. operator theory, 19 (1988), 159-195. [19] i. raeburn. s.j. thompson, countably generated hilbert modules, the kasparrov stabilisation theorem, and frames with hilbert modules, proc. amer math. soc. 131 (5) (2003), 1557-1564. [20] m. rashidi-kouchi, a. nazari, on stability of g-frames and g-riesz bases in hilbert c*-modules, int. j. wavelets multiresolut. inf. process., 12 (6) (2014), art. id 1450036. [21] a. rahimi and a. freydooni, controlled g-frames and their g-multipliers in hilbert spaces, an. univ. ovidius constana, ser. mat. 21 (2013), 223-236. [22] w. sun, gframes and g-riesz bases,j. math. anal. appl 322 (2006), 437-452. [23] n. e. wegg olsen, k-theory and c*-algebras, friendly approch, oxford university press, oxford, england, (1993). [24] yu. i. zhuraev, f. sharipov, hilbert modules over locally c*-algebra, arxiv:math. 0011053 v3 [math. oa], (2001). [25] x. xiao. , zeng, some properties of g-frames in hilbert c*-algebra, j. math. anal. appl. 363 (2010), 399-408. 1. introduction 2. preliminaries 3. -g-frames in hilbert pro-c-modules 4. controlled -g-frames in hilbert pro-c-modules 5. multipliers of controlled -g-frames in hilbert pro-c-modules references international journal of analysis and applications volume 16, number 1 (2018), 38-49 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-38 inequalities of fejér type related to generalized convex functions with applications s. mohammadi aslani1, m. rostamian delavar2,∗ and s. m. vaezpour3 1 department of mathematics, karaj branch, islamic azad university, karaj, iran 2department of mathematics, faculty of basic sciences, university of bojnord, bojnord, iran 3department of mathematics, amirkabir university of technology, tehran, iran ∗corresponding author: m.rostamian@ub.ac.ir abstract. this paper deals with some fejér type inequalities related to (η1,η2)-convex functions. in fact the difference between the right and middle part of fejér inequality is estimated without using hölder’s inequality when the absolute value of the derivative of considered function is (η1,η2)-convex. furthermore we give two estimation results when the derivative of considered function is bounded and satisfies a lipschitz condition. 1. introduction and preliminaries the fejér integral inequality for convex functions has been proved in [5]: theorem 1.1. let f : [a,b] → r be a convex function. then f (a + b 2 )∫ b a g(x)dx ≤ ∫ b a f(x)g(x)dx ≤ f(a) + f(b) 2 ∫ b a g(x)dx, (1.1) where g : [a,b] → [0, +∞) is integrable and symmetric to x = a+b 2 ( g(x) = g(a + b−x),∀x ∈ [a,b] ) . received 10th september, 2017; accepted 28th november, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 26a51, 26d15, 52a01. key words and phrases. (η1,η2)-convex function; fejér inequality. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 38 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-38 int. j. anal. appl. 16 (1) (2018) 39 the estimation for difference of the right and middle part in (1.1) is an interesting problem. the following theorem has been proved in [12], that estimates the difference between the right and middle part in (1.1) using hölder’s inequality when the absolute value of the derivative of considered function is convex. theorem 1.2. let f : i◦ ⊂ r → r be a differentiable mapping on i◦, a,b ∈ i◦ with a < b, and w : [a,b] → [0, +∞) be a differentiable mapping and symmetric to a+b 2 . if |f′| is convex on [a,b] , then the following inequality holds: ∣∣∣∣ 1b−a f(a) + f(b)2 ∫ b a w(x)dx− 1 b−a ∫ b a f(x)w(x)dx ∣∣∣∣ ≤ (1.2) 1 2 [∫ 1 0 ( g(x) )p dt ]1 p ( |f′(a)|q + |f′(b)|q 2 )1 q , where g(x) = ∣∣∣∣ ∫ b−(b−a)t a+(b−a)t w(x)dx ∣∣∣∣, for t ∈ [0, 1]. the preinvex functions as a generalization of convex functions was considered by ben-israel and mond in [1] and hanson and mond in [6], but so named by jeyakumar [7]. definition 1.1. [1, 6] a set i ⊆ r is invex with respect to a real bifunction η : i × i → r, if x,y ∈ i,λ ∈ [0, 1] ⇒ y + λη(x,y) ∈ i. also if i is an invex set with respect to η, then a function f : i → r is said to be preinvex if x,y ∈ i and λ ∈ [0, 1] implies f ( y + λη(x,y) ) ≤ λf(x) + (1 −λ)f(y). the following theorem has been proved in [8] which is preinvex version of theorem 1.2. theorem 1.3. let k ⊂ r be an open invex subset with respect to η : k × k → r and a,b ∈ k with a < a + η(b,a). suppose f : k → r is a differentiable mapping on k such that f′ ∈ l([a,a + a + η(b,a)]) and w : [a,a + η(b,a)] → [0, +∞) is an integrable mapping and symmetric to a + 1 2 η(b,a). if |f′|q, q > 1, is preinvex on k, then for every a,b ∈ k with η(b,a) 6= 0 we have the following inequality:∣∣∣∣f(a) + f ( a + η(b,a) ) 2 ∫ a+η(b,a) a w(x)dx− 1 η(b,a) ∫ a+η(b,a) a f(x)w(x)dx ∣∣∣∣ ≤ (1.3) 1 2 (∫ 1 0 gp(t) )1 p ( |f′(a)|q + |f′(b)|q 2 )1 q , where g(t) = ∣∣∣∣ ∫ a+(1−t)η(b,a) a+tη(b,a) w(x)dx ∣∣∣∣, t ∈ [0, 1] and 1p + 1q = 1. int. j. anal. appl. 16 (1) (2018) 40 furthermore the concept of η-convex functions (at the beginning was named by ϕ-convex functions), considered in [4], has been introduced as the following. definition 1.2. consider a convex set i ⊂ r and a bifunction η : f(i) ×f(i) → r. a function f : i → r is called convex with respect to η (briefly η-convex), if f ( λx + (1 −λ)y ) ≤ f(y) + λη ( f(x),f(y) ) , for all x,y ∈ i and λ ∈ [0, 1]. geometrically it says that if a function is η-convex on i, then for any x,y ∈ i, its graph is on or under the path starting from ( y,f(y) ) and ending at ( x,f(y) + η(f(x),f(y) )) . if f(x) should be the end point of the path for every x,y ∈ i, then we have η(x,y) = x−y and the function reduces to a convex one. the following theorem has been proved in [3], where the absolute value of the derivative of considered function is η-convex. theorem 1.4. suppose that f : [a,b] → r is a differentiable function, g : [a,b] → [0, +∞) is a continuous function and symmetric to a+b 2 and |f′| is an η-convex function where η is bounded from above on [a,b]. then ∣∣∣∣f(a) + f(b)2 ∫ b a g(x)dx− ∫ b a f(x)g(x)dx ∣∣∣∣ ≤ (b−a) 4 [ 2 ∣∣f′(b)∣∣ + η(|f′(a)|, |f′(b)|)]∫ 1 0 ∫ 1−t 2 a+ 1+t 2 b 1+t 2 a+ 1−t 2 b g(u)dudt. (1.4) for more results about η-convex functions see [3, 4, 10, 11]. motivated by above works and references therein, we introduce the concept of (η1,η2)-convex functions as a generalization of preinvex and η-convex functions. also we give some fejér type trapezoid inequalities when the absolute value of the derivative of considered function is (η1,η2)-convex but with new face without using of hölder’s inequality. furthermore we obtain two estimation results when the derivative of considered function is bounded and satisfies a lipschitz condition. definition 1.3. let i ⊂ r be an invex set with respect to η1 : i × i → r. consider f : i → r and η2 : f(i) ×f(i) → r. the function f is said to be (η1,η2)-convex if f ( x + λη1(y,x) ) ≤ f(x) + λη2 ( f(y),f(x) ) , for all x,y ∈ i and λ ∈ [0, 1]. note. an (η1,η2)-convex function reduces to (i) an η-convex function if we consider η1(x,y) = x−y for all x,y ∈ i. int. j. anal. appl. 16 (1) (2018) 41 (ii) a preinvex function if we consider η2(x,y) = x−y for all x,y ∈ f(i). (iii) a convex function if satisfies (i) and (ii). we can find an (η1,η2)-convex function which is not convex. example 1.1. consider the function f : [0, +∞) → [0, +∞) by f(x) =   x, 0 ≤ x ≤ 1;1, x > 1. define two bifunction η1 : [0, +∞) × [0, +∞) −→ r and η2 : [0, +∞) × [0, +∞) −→ [0, +∞) by η1(x,y) =   −y, 0 ≤ y ≤ 1;x + y, y > 1, and η2(x,y) =   x + y, x ≤ y;2(x + y), x > y. then f is an (η1,η2)-convex function. but f is not preinvex with respect to η1 and also it is not convex. (consider x = 0, y = 2 and λ > 0). 2. main results in this section without using hölder’s inequality we obtain a trapezoid type inequality related to (1.1). the obtained results are different from (1.2), (1.3) and (1.4) in the face and proof. the following lemma is of importance: lemma 2.1. suppose that i ⊂ r is an invex set with respect to η1. if for a,b ∈ i with η1(b,a) > 0 the function g : [a,a + η1(b,a)] → r is integrable and symmetric to a + 12η1(b,a), then for any 0 ≤ t ≤ 1 2 we have∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds = 2 ∫ 1 2 t g ( a + sη1(b,a) ) ds. (2.1) proof. using the change of variable x = a + sη1(b,a), for 0 ≤ t ≤ 12 we get∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds = (2.2) 1 η1(b,a) [∫ a+η1(b,a) u g(x)dx− ∫ u a g(x)dx ] , where a ≤ u ≤ a + 1 2 η1(b,a). since g is symmetric to a + 1 2 η1(b,a) we have∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x)dx = ∫ a+ 1 2 η1(b,a) a g(x)dx. int. j. anal. appl. 16 (1) (2018) 42 then ∫ a+η1(b,a) u g(x)dx = ∫ a+ 1 2 η1(b,a) u g(x)dx + ∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x)dx = ∫ a+ 1 2 η1(b,a) u g(x)dx + ∫ a+ 1 2 η1(b,a) a g(x)dx. also ∫ a+ 1 2 η1(b,a) a g(x)dx = ∫ u a g(x)dx + ∫ a+ 1 2 η1(b,a) u g(x)dx. so 1 η1(b,a) [∫ a+η1(b,a) u g(x)dx− ∫ u a g(x)dx ] = 2 η1(b,a) ∫ a+ 1 2 η1(b,a) u g(x)dx = 2 ∫ 1 2 t g ( a + sη1(b,a) ) ds. (2.3) using (2.3) in (2.2) we get (2.1). � with the same argument used in the proof of lemma 2.1 we can drive the following lemma. lemma 2.2. suppose that i ⊂ r is an invex set with respect to η1. if for a,b ∈ i with η1(b,a) > 0 the function g : [a,a + η1(b,a)] → r is integrable and symmetric to a + 12η1(b,a), then for any 1 2 ≤ t ≤ 1 we have∫ t 0 g ( a + sη1(b,a) ) ds− ∫ 1 t g ( a + sη1(b,a) ) ds = 2 ∫ t 1 2 g ( a + sη1(b,a) ) ds. (2.4) from lemma 2.1 and lemma 2.2, if g is symmetric nonnegative function, we can obtain two integral inequalities that are useful for our next results. corollary 2.1. suppose that i ⊂ r is an invex set with respect to η1. if for a,b ∈ i with η1(b,a) > 0 the function g : [a,a + η1(b,a)] → [0, +∞) is integrable and symmetric to a + 12η1(b,a), then∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds ≥ 0, 0 ≤ t ≤ 1 2 (2.5) and ∫ t 0 g ( a + sη1(b,a) ) ds− ∫ 1 t g ( a + sη1(b,a) ) ds ≥ 0, 1 2 ≤ t ≤ 1. (2.6) also the following lemma has been proved in [8] which is needed. lemma 2.3. suppose that i◦ ⊂ r is an invex set with respect to η1 and f : i◦ → r is a differentiable mapping on i◦. for any a,b ∈ i◦ with η1(b,a) > 0, if g : [a,a + η1(b,a)] → [0, +∞) is differentiable mapping on i◦ and f′ ∈ l1[a,a + η1(b,a)], then 1 η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) = η1(b,a) 2 ∫ 1 0 p(t)f′ ( a + tη1(b,a) ) dt, int. j. anal. appl. 16 (1) (2018) 43 where p(t) = ∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds. now we are ready to give our main result of the paper which is a trapezoid type inequality related to (1.1) with a new face. theorem 2.1. suppose that i◦ ⊂ r is an invex set with respect to η1 and f : i◦ → r is a differentiable mapping on i◦. for any a,b ∈ i◦ with η1(b,a) > 0, let g : [a,a + η1(b,a)] → [0, +∞) be a differentiable mapping on i◦ and f′ ∈ l1[a,a + η1(b,a)]. if |f′| is a (η1,η2)-convex mapping on [a,a + η1(b,a)], then ∣∣∣∣f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx− ∫ a+η1(b,a) a f(x)g(x)dx ∣∣∣∣ ≤ (2.7) [ 2|f′(a)| + η2 ( |f′(b)|, |f′(a)| )]∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ( a + η1(b,a) −x ) dx. proof. from lemma 2.3, corollary 2.1 and (η1,η2)-convexity of |f′| we have ∣∣∣∣f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx− ∫ a+η1(b,a) a f(x)g(x)dx ∣∣∣∣ = η21(b,a) 2 ∣∣∣∣ ∫ 1 0 [∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds ] f′ ( a + tη1(b,a) ) dt ∣∣∣∣ ≤ η21(b,a) 2 {∫ 1 2 0 ∣∣∣∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds ∣∣∣∣∣f′(a + tη1(b,a))∣∣dt+∫ 1 1 2 ∣∣∣∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds ∣∣∣∣∣f′(a + tη1(b,a))∣∣dt} ≤ η21(b,a) 2 {∫ 1 2 0 (∫ 1 t g ( a + sη1(b,a) ) ds− ∫ t 0 g ( a + sη1(b,a) ) ds ) × [ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dt+∫ 1 1 2 (∫ t 0 g ( a + sη1(b,a) ) ds− ∫ 1 t g ( a + sη1(b,a) ) ds ) × [ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dt } = i. if we change the order of integration in i, then i = η21(b,a) 2 {∫ 1 2 0 ∫ s 0 g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds+ ∫ 1 1 2 ∫ 1 2 0 g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds− ∫ 1 2 0 ∫ 1 2 s g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds+ int. j. anal. appl. 16 (1) (2018) 44 ∫ 1 1 2 ∫ 1 s g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds+ ∫ 1 2 0 ∫ 1 1 2 g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds− ∫ 1 1 2 ∫ s 1 2 g ( a + sη1(b,a) )[ |f′(a)| + tη2 ( |f′(b)|, |f′(a)| )] dtds } . calculating all inner integrals in i we get i = η21(b,a) 2 {∫ 1 2 0 g ( a + sη1(b,a) )( s|f′(a)| + 1 2 s2η2 ( |f′(b)|, |f′(a)| )) ds+ ∫ 1 1 2 g ( a + sη1(b,a) )(1 2 |f′(a)| + 1 8 η2 ( |f′(b)|, |f′(a)| )) ds− ∫ 1 2 0 g ( a + sη1(b,a) )( ( 1 2 −s)|f′(a)| + ( 1 8 − 1 2 s2)η2 ( |f′(b)|, |f′(a)| )) ds+ ∫ 1 1 2 g ( a + sη1(b,a) )( (1 −s)|f′(a)| + ( 1 2 − 1 2 s2)η2 ( |f′(b)|, |f′(a)| )) ds+ ∫ 1 2 0 g ( a + sη1(b,a) )(1 2 |f′(a)| + 3 8 η2 ( |f′(b)|, |f′(a)| )) ds− ∫ 1 1 2 g ( a + sη1(b,a) )( (s− 1 2 )|f′(a)| + ( 1 2 s2 − 1 8 )η2 ( |f′(b)|, |f′(a)| )) ds } . simple form of i can be obtained as the following. i = η21(b,a) 2 {∫ 1 2 0 g ( a + sη1(b,a) )( 2s|f′(a)| + (s2 + 1 4 )η2 ( |f′(b)|, |f′(a)| )) ds+ ∫ 1 1 2 g ( a + sη1(b,a) )( (−2s + 2)|f′(a)| + ( 3 4 −s2)η2 ( |f′(b)|, |f′(a)| )) ds } . if we apply the change of variable x = a + sη1(b,a) in i, we get i = η1(b,a) 2 {∫ a+ 1 2 η1(b,a) a g(x) ([ 2 ( x−a η1(b,a) )] |f′(a)|+ [( x−a η1(b,a) )2 + 1 4 ] η2 ( |f′(b)|, |f′(a)| )) dx+ ∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ([ − 2 ( x−a η1(b,a) ) + 2 ] |f′(a)|+ [3 4 − ( x−a η1(b,a) )2] η2 ( |f′(b)|, |f′(a)| )) dx } . int. j. anal. appl. 16 (1) (2018) 45 on the other hand since g is symmetric to a + 1 2 η1(b,a) then we have ∫ a+ 1 2 η1(b,a) a g(x) ([ 2 ( x−a η1(b,a) )] |f′(a)|+ [( x−a η1(b,a) )2 + 1 4 ] η2 ( |f′(b)|, |f′(a)| )) dx = ∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ([ 2 (a + η1(b,a) −x η1(b,a) )] |f′(a)|+ [(a + η1(b,a) −x η1(b,a) )2 + 1 4 ] η2 ( |f′(b)|, |f′(a)| )) dx. so i = η1(b,a) 2 {∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ([ 2 ( x−a η1(b,a) ) + 2 (a + η1(b,a) −x η1(b,a) )] |f′(a)| + [( x−a η1(b,a) )2 + 1 4 + (a + η1(b,a) −x η1(b,a) )2 + 1 4 ] η2 ( |f′(b)|, |f′(a)| )) dx } = η1(b,a) 2 {∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ( 4 [a + η1(b,a) −x η1(b,a) ] |f′(a)| + 2 [a + η1(b,a) −x η1(b,a) ] η2 ( |f′(b)|, |f′(a)| )) dx } = [ 2|f′(a)| + η2 ( |f′(b)|, |f′(a)| )]∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ( a + η1(b,a) −x ) dx. � corollary 2.2. if in theorem 2.1 we consider η2(x,y) = x−y for all x,y ∈ f(i◦), then ∣∣∣∣f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx− ∫ a+η1(b,a) a f(x)g(x)dx ∣∣∣∣ ≤ (2.8) [ |f′(a)| + |f′(b)| ]∫ a+η1(b,a) a+ 1 2 η1(b,a) g(x) ( a + η1(b,a) −x ) dx. also if we put η1(x,y) = x−y for all x,y ∈ f(i◦) in (2.8) we get ∣∣∣∣f(a) + f(b)2 ∫ b a g(x)dx− ∫ b a f(x)g(x)dx ∣∣∣∣ ≤ [|f′(a)| + |f′(b)|] ∫ b a+b 2 g(x)(b−x)dx. (2.9) furthermore if in (2.9) we set g ≡ 1, then we recapture theorem 2.2 in [2]. ∣∣∣∣f(a) + f(b)2 (b−a) − ∫ b a f(x)dx ∣∣∣∣ ≤ (b−a)28 (|f′(a)| + |f′(b)|). remark 2.1. inequalities (2.8) and (2.9), obtained in corollary 2.2 are new inequalities in the literature. int. j. anal. appl. 16 (1) (2018) 46 3. estimation type results in this section we give two estimation results when the derivative of considered function is bounded and satisfies a lipschitz condition respectively. if the derivative of the considered function is bounded from below and above, then we can drive an estimation type result related to fejér inequality. theorem 3.1. suppose that i◦ ⊂ r is an invex set with respect to η1 and f : i◦ → r is a differentiable mapping on i◦. for any a,b ∈ i◦ with η1(b,a) > 0, let g : [a,a + η1(b,a)] → [0, +∞) be a differentiable mapping on i◦. assume that f′ ∈ l1[a,a + η1(b,a)] and there exist constants m < m such that −∞ < m ≤ f′(x) ≤ m < +∞ for all x ∈ [a,a + η1(b,a)]. then 1 η1(b,a) ∣∣∣∣∣f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx− ∫ a+η1(b,a) a f(x)g(x)dx (3.1) − η1(b,a)(m + m) 4 ∫ 1 0 p(t)dt ∣∣∣∣∣ ≤ η1(b,a)(m −m)4 ∫ 1 0 |p(t)|dt, where p(t) is defined in lemma 2.3. proof. from lemma 2.3 we have 1 η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) = η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) − m + m 2 + m + m 2 ] dt = (m + m)η1(b,a) 4 ∫ 1 0 p(t)dt + η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) − m + m 2 ] dt. so i = 1 η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) − (m + m)η1(b,a) 4 ∫ 1 0 p(t)dt = η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) − m + m 2 ] dt. taking the modulus on i we obtain |i| ≤ η1(b,a) 2 ∫ 1 0 ∣∣∣p(t)∣∣∣∣∣∣f′(a + tη1(b,a))− m + m 2 ∣∣∣dt ≤ (m −m)η1(b,a) 4 ∫ 1 0 |p(t)|dt, since from m ≤ f′ ( a + tη1(b,a) ) ≤ m we have m− m + m 2 ≤ f′ ( a + tη1(b,a) ) − m + m 2 ≤ m − m + m 2 , which implies that ∣∣∣∣f′(a + tη1(b,a))− m + m2 ∣∣∣∣ ≤ m −m2 . int. j. anal. appl. 16 (1) (2018) 47 � corollary 3.1. in theorem 3.1 if we set η1(x,y) = x−y for all x,y ∈ i◦ and g ≡ 1, then∣∣∣∣f(a) + f(b)2 − 1b−a ∫ b a f(x)dx ∣∣∣∣ ≤ m(1 + a− b) + m(1 + b−a)8 . proof. if we consider g ≡ 1, then the relations ||g||∞ = 1 and ∫ 1 0 |p(t)|dt ≤ 1 2 imply that 1 b−a ∣∣∣∣f(a) + f(b)2 (b−a) − ∫ b a f(x)dx ∣∣∣∣ ≤∣∣∣∣m + m4 ∫ 1 0 p(t)dt ∣∣∣∣ + (m −m)(b−a)8 ≤ m + m 8 + (m −m)(b−a) 8 = m(1 + a− b) + m(1 + b−a) 8 . � estimation for difference between the right and middle terms of (1.1) when the derivative of considered function satisfies a lipschitz condition is our next aim. definition 3.1. [9] a function f : [a,b] → r is said to satisfy the lipschitz condition on [a,b] if there is a constant k so that for any two points x,y ∈ [a,b], |f(x) −f(y)| ≤ k|x−y|. theorem 3.2. suppose that i◦ ⊂ r is an invex set with respect to η1 and f : i◦ → r is a differentiable mapping on i◦. for any a,b ∈ i◦ with η1(b,a) > 0, let g : [a,a + η1(b,a)] → [0, +∞) be a differentiable mapping on i◦. assume that f′ is integrable on [a,a + η1(b,a)] and satisfies a lipschitz condition for some k > 0. then ∣∣∣∣∣ 1η1(b,a) ( f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx− ∫ a+η1(b,a) a f(x)g(x)dx ) − η1(b,a) 2 f′ (2a + η1(b,a) 2 )∫ 1 0 p(t)dt ∣∣∣∣∣ ≤ kη1(b,a)2 ∫ 1 0 |t− 1 2 ||p(t)|dt, where p(t) is defined in lemma 2.3. proof. from lemma 2.3 we have 1 η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) = η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) −f′ ( 2a + η1(b,a) 2 ) + f′ ( 2a + η1(b,a) 2 )] dt = η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) −f′ ( 2a + η1(b,a) 2 )] dt+ η1(b,a) 2 f′ ( 2a + η1(b,a) 2 )∫ 1 0 p(t)dt. int. j. anal. appl. 16 (1) (2018) 48 then 1 η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) − η1(b,a) 2 f′ ( 2a + η1(b,a) 2 )∫ 1 0 p(t)dt = η1(b,a) 2 ∫ 1 0 p(t) [ f′ ( a + tη1(b,a) ) −f′ ( 2a + η1(b,a) 2 )] dt. since f′ satisfies a lipschitz condition for k > 0, we have∣∣∣∣f′(a + tη1(b,a))−f′ ( 2a + η1(b,a) 2 )∣∣∣∣ ≤ k ∣∣∣∣a + tη1(b,a) − 2a + η1(b,a)2 ∣∣∣∣ = k ∣∣∣t− 1 2 ∣∣∣η1(b,a). so ∣∣∣∣ 1η1(b,a) (∫ a+η1(b,a) a f(x)g(x)dx− f(a) + f ( a + η1(b,a) ) 2 ∫ a+η1(b,a) a g(x)dx ) − η1(b,a) 2 f′ ( 2a + η1(b,a) 2 )∫ 1 0 p(t)dt ∣∣∣∣ ≤ kη1(b,a)2 ∫ 1 0 ∣∣t− 1 2 ∣∣∣∣p(t)∣∣dt. � corollary 3.2. if in theorem 3.2 we consider g ≡ 1, then∣∣∣∣∣f(a) + f ( a + η1(b,a) ) 2 − 1 η1(b,a) ∫ a+η1(b,a) a f(x)dx ∣∣∣∣∣ ≤ η1(b,a) ( k 12 + 1 4 ∣∣∣f′(2a + η1(b,a) 2 )∣∣∣) . furthermore if we consider η1(x,y) = x−y for all x,y ∈ i◦ we get∣∣∣∣∣f(a) + f(b)2 − 1b−a ∫ b a f(x)dx ∣∣∣∣∣ ≤ (b−a) ( k 12 + 1 4 ∣∣∣f′(a + b 2 )∣∣∣) . 4. acknowledgements the authors are very grateful to professor s. s. dragomir for his valuable suggestions and comments. references [1] a. ben-israel and b. mond, what is invexity?, j. aust. math. soc., ser. b. 28 (1986) 1–9. [2] s. s. dragomir and r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett., 11 (1998) 91–95. [3] m. eshaghi gordji, s. s. dragomir and m. rostamian delavar, an inequality related to η-convex functions (ii), int. j. nonlinear anal. appl., 6(2) (2016) 26–32. [4] m. eshaghi gordji, m. rostamian delavar and m. de la sen, on ϕ-convex functions, j. math. inequal., 10(1) (2016) 173–183. [5] l. fejér, über die fourierreihen, ii, math. naturwise. anz ungar. akad. wiss. 24 (1906) 369–390. int. j. anal. appl. 16 (1) (2018) 49 [6] m.a. hanson and b. mond, convex transformable programming problems and invexity, j. inf. optim. sci., 8 (1987) 201–207. [7] w. jeyakumar, strong and weak invexity in mathematical programming, eur. j. oper. res., 55 (1985) 109–125. [8] m. latif and s. s. dragomir, new inequalities of hermite-hadamard and fejér type preinvexity, j. comput. anal. appl., 19(1) (2015), 725–739. [9] a. w. robert and d. e. varbeg, convex functions, academic press, (1973). [10] m. rostamian delavar and s.s. dragomir, on η-convexity, math. inequal. appl., 20 (2017) 203–216. [11] m. rostamian delavar and m. de la sen, some generalizations of hermite-hadamard type inequalities, springerplus, (2016) 5:1661. [12] m. z. sarikaya, on new hermite hadamard fejér type integral inequalities, stud. univ. babes-bolyai math. 57(3) (2012) 377–386. 1. introduction and preliminaries 2. main results 3. estimation type results 4. acknowledgements references international journal of analysis and applications volume 17, number 2 (2019), 191-207 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-191 solutions of fractional diffusion equations and cattaneo-hristov diffusion model ndolane sene∗ 1laboratoire lmdan, département de mathématiques de la décision, université cheikh anta diop de dakar, faculté des sciences economiques et gestion, bp 5683 dakar fann, senegal ∗corresponding author: ndolanesene@yahoo.fr, ndolane.sene@ucad.edu.sn abstract. the analytical solutions of the fractional diffusion equations in one and two-dimensional spaces have been proposed. the analytical solution of the cattaneo-hristov diffusion model with the particular boundary conditions has been suggested. in general, the numerical methods have been used to solve the fractional diffusion equations and the cattaneo-hristov diffusion model. the laplace and the fourier sine transforms have been used to get the analytical solutions. the analytical solutions of the classical diffusion equations and the cattaneo-hristov diffusion model obtained when the order of the fractional derivative converges to 1 have been recalled. the graphical representations of the analytical solutions of the fractional diffusion equations and the cattaneo-hristov diffusion model have been provided. 1. introduction in fractional calculus, we have many fractional derivatives operators as: the riemann-liouville fractional derivative [34] [36], the caputo fractional derivative [8] [41], the atangana-baleanu fractional derivative [2] [3] [4], the caputo-fabrizio fractional derivative [6] [30], the conformable fractional derivative [42], the generalized fractional derivatives in caputo and riemann-liouville sense [21] [22] [24], and others. fractional calculus has many applications in mechanic, physics and science and engineering. fractional calculus has many applications in the viscoelastic models and the diffusion models. in [12] [13], hristov treats on heat received 2018-12-03; accepted 2019-01-09; published 2019-03-01. 2010 mathematics subject classification. 42a38, 76r50, 26a33 . key words and phrases. fractional diffusion equation; caputo fractional derivative; cattaneo-hristov diffusion model. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 191 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-191 int. j. anal. appl. 17 (2) (2019) 192 diffusion equation in term of the caputo-fabrizio time fractional derivative. in [10], hristov proposes new equations related to the fractional diffusion equations using the atangana-baleanu fractional derivative, see others models in [23]. in [1], alkahtani and atangana discuss the numerical solution of the cattaneo-hristov diffusion equation. in [26] koca et al. propose the numerical solution of the second term of the cattaneohristov diffusion equation. in [27], li et al. have studied the cauchy problem for nonlinear fractional time-space generalized keller-segel equation using the caputo fractional derivative. in [29], yranli et al. devoted to comparing the smoothing performance between the time fractional diffusion equation and the classical diffusion equation using the regulation method, savitzky-golay, and coverer method. in [37], ruan et al. study a simultaneous identification problem of piecewise source term and the fractional order for time-fractional diffusion equation. in [45], zhang and al. propose a discrete form for solving time fractional convection-diffusion equation. in [31], ma et al. study asymptotic of the solutions to the fractional anomalous diffusion equations. several works related to the fractional diffusion equations exist in the literature. the papers [5] [33] [39] [44] treat on fractional diffusion equations. the fractional diffusion equation is obtained when a specific fractional derivative operator replaces the ordinary derivative in the classical diffusion equation. in this paper, we use the caputo fractional derivative. podlubny [35] has introduced the fractional diffusion equation in fractional calculus. we propose the analytical solutions of the fractional diffusion equations in one and two-dimensional spaces. hristov [12] introduced the cattaneo-hristov diffusion equation in fractional calculus. the author [12] opens news problems related to the analytical or approximate solutions of the cattaneo-hristov diffusion equation. koca et al. [26] propose the numerical and analytical solutions of the elastic part of the heat diffusion equation process. alkahtani et al. [1] propose the numerical solution of the complete cattaneo-hristov diffusion equation using the crank-nicholson numerical scheme. hristov [12] proposes an approximate solution of the cattaneo-hristov diffusion equation using the heat-balance integral method (hbim). hristov [12] proposes a double integral-balance method (dim) to get the approximate solution of the cattaneo-hristov diffusion equation. the analytical or approximate solutions of the fractional diffusion equations using hbim and dim were proposed in [14] [15] [16] [17] [18] [19] [20] [32]. the analytical solution of the cattaneo-hristov equation was stated in [26] by koca et al. they give the analytical solution of the elastic part of the heat diffusion equation process. in this paper, we continue the work concerning the analytical solution stated by koca et al. in [26]. in this paper, we propose the analytical solution of the complete cattaneo-hristov diffusion equation of the transient heat equation using an integral method. the integral method uses both the fourier sine transform and the laplace transform. we will notice this integration method will permit to express the analytical solutions of the fractional diffusion equations in the term of the gaussian error function and the mittag-leffler function [9] [40]. the graphical representations of the analytical solutions of some particular fractional diffusion equations are provided. int. j. anal. appl. 17 (2) (2019) 193 the paper is organized as follows: in section 2, we recall preliminary definitions which we will use in this paper. in section 3, we analyze the analytical solutions of the fractional diffusion equation in one-dimensional space. in section 4, we get the analytical solution of the fractional diffusion equation in two-dimensional space. in section 5, we analyze some particular cases graphically. and we finish with section 6 by giving the conclusions and remarks. 2. fractional diffusion equations in this section, we present the fractional differential equations studied in this paper. the problems concern the fractional diffusion equation in one and two-dimensional spaces. the classical diffusion equation defined by the ordinary derivative is popular. many works related to the analytical and the numerical solutions exist. diffusion phenomena, of heat or mass [12] [35] is represented as the following form ∂u(x,t) ∂t = κ2 ∂2u(x,t) ∂x2 (2.1) where κ2 = k ρcp . we add the following informations. • • • k represents the thermal conductivity, • • • ρ represents the specific heat, • • • cp represents the density of the material, • • • u represents the temperature distribution of the material. the fractional diffusion equation is obtained when we replace the ordinary derivative by a fractional derivative operators. the fractional diffusion equation described by the caputo fractional derivative is expressed in one-dimensional space by the following equation [11] [31] [35] dcαu(x,t) = κ 2 ∂ 2u(x,t) ∂x2 (2.2) where dcα represents the caputo fractional derivative operator defined by [28] [40] dcαu(x,t) = 1 γ(1 −α) ∫ t 0 u′(x,s) (t−s)α ds (2.3) all t > 0, α ∈ (0, 1) , and γ(.) denotes the gamma function. κ2 represents the diffusion coefficient for the density of the diffusion material. the boundary conditions considered in this paper are the dirichlet boundary conditions: • • • u(x, 0) = 0 for x > 0, • • • u(0, t) = 1 for t > 0. the fractional diffusion equation exists in two dimensional space. it is expressed as the following form [31] dcαu(x,y,t) = κ 2 { ∂2u(x,y,t) ∂x2 + ∂2u(x,y,t) ∂y2 } (2.4) with the dirichlet boundary conditions defined as int. j. anal. appl. 17 (2) (2019) 194 • • • u(x,y, 0) = 0 for x,y > 0, • • • u(0,y,t) = u(x, 0, t) = 1 for t > 0. the initial boundary conditions play an important role in the integral method. note that, when the boundary conditions change, the form of the analytical solutions changes also. there exist many methods to get the analytical solutions of the fractional diffusion equations: as the laplace transform, as the fourier sine transform [43], as the heat-balance integral method (hbim) [18, 20], as a double integral method (dim) [18, 20], as a multiple integral method (mim) [14]. this paper proposes an integral method consisting of applying both the laplace transform and the fourier sine transform. let recall the laplace transform of the caputo fractional derivative which we will use later [25] [38] l{dcαf(t)} = s αl{f(t)}(s) −sα−1f(0) (2.5) where α ∈ (0, 1). the transformation (2.5 ) is known very useful in the resolution of the fractional differential equations. all solutions obtained in this paper will be rewritten using the mittag-leffler function [9] defined as the following form eα,β (z) = ∞∑ k=0 zk γ(αk + β) . (2.6) where α > 0, β ∈ r and z ∈ c. we obtain the exponential function when α = β = 1 and we obtain the mittag-leffler function with one parameter when β = 1, for more information see in [9]. 3. analytical solution of fractional diffusion equation in one dimensional space in this section, we investigate to find the analytical solution of the fractional diffusion equation in onedimensional space defined as the following form dcαu(x,t) = κ 2 ∂ 2u(x,t) ∂x2 . (3.1) we consider the dirichlet boundary conditions defined as follows: • • • u(x, 0) = 0 for x > 0, • • • u(0, t) = 1 for t > 0. in other words, we assume that the initial temperature of the material is null and the temperature of the plate for all t > 0 is maintained constant u0 = 1. it is essential for our results and the application of our method of resolution. the integral methods as hbim and dim use the finite penetration depth to get the approximate solutions of the fractional diffusion equation (3.1). for more information on these integral methods, see in [18] [20]. in this paper, we adopt the following integral method (see in [43]), described as follows: • • • apply the fourier sine transform, • • • apply the laplace transform, int. j. anal. appl. 17 (2) (2019) 195 figure 1. fractional diffusion model. • • • apply the inverse of laplace transform, • • • apply the inverse of fourier sine transform. this method of resolution seems very useful and practical to get the analytical solution of the fractional diffusion equations. if the temperature of the plate is null, the integral method described above seems no adequate to be applied and it is better to use the classical methods as hbim or dim to solve equation (3.1). we begin the resolution of the fractional differential equation (3.1) by applying the fourier sine transform. multiplying equation (3.1) by 2 π sin wx and integrating it between 0 to ∞, we get that: dcαus(w,t) = κ 2 { 2 π wus(0, t) −w2us(w,t) } dcαus(w,t) = 2κ2w π −κ2w2us(w,t). where us(w,t) denotes the fourier sine transform of u(x,t). rearranging, we obtain the following fractional differential equation defined as dcαus(w,t) + κ 2w2us(w,t) = 2κ2w π . (3.2) the second step of the resolution consists of applying the laplace transform to both sides of equation (3.2), and then we obtain that sαūs(w,s) + κ 2w2ūs(w,s) = 2κ2w πs ūs(w,s) = 2κ2w πs (sα + κ2w2) . (3.3) where ūs(w,s) denotes the laplace transform of us(w,t). the third step of the resolution consists of applying the inverse of the laplace transform to both sides of equation (3.3). to reach our end, we rewrite equation (3.3) as follows: ūs(w,s) = 2 π { 1 s − sα−1 sα + κ2w2 } 1 w . (3.4) int. j. anal. appl. 17 (2) (2019) 196 applying the inverse of laplace transform to both sides of equation (3.4) and using the mittag-leffler function as defined in [9], we get us(w,t) = 2 πw { 1 −eα ( −κ2w2tα )} . (3.5) to get the analytical solution of the fractional diffusion equation (3.1), we apply the inverse of the fourier sine transform to both sides of equation (3.5), and then we obtain the following result u(x,t) = 2 π ∫ ∞ 0 sin wx w { 1 −eα ( −κ2w2tα )} dw = 1 − 2 π ∫ ∞ 0 sin wx w eα ( −κ2w2tα ) dw. (3.6) let now analyze a particular case of the fractional diffusion equation. the classical diffusion equation is obtained when α → 1. to get the analytical solution, we use the laplace transform obtained in equation (3.3). we have the following decomposition ūs(w,s) = 2 π { 1 s − 1 s + κ2w2 } 1 w . (3.7) using the inverse of laplace transform to both sides of equation (3.7), we get the following intermediary solution us(w,t) = 2 πw { 1 − exp ( −κ2w2t )} . (3.8) respecting the procedure of the resolution, we have to apply the inverse of fourier sine transform, and then we obtain the analytical solution of the classical diffusion equation given by u(x,t) = 2 π ∫ ∞ 0 sin wx w { 1 − exp ( −κ2w2t )} dw = 1 − 2 π ∫ ∞ 0 sin wx w exp ( −κ2w2t ) dw = 1 −erf ( x 2κ √ t ) (3.9) where the function erf(.) denotes the gaussian error function. let’s give the behavior of the temperature distribution of the material in some configurations. see in figures 2,3,4 and 5 the behavior of the temperature distribution of the material u in different cases. the figure 2 describes the behavior of the temperature distribution of the material in the diffusion equation (α → 1) when x and t take different values with the diffusion coefficient for the density of the diffusion material fixed to κ2 = 0.85.10−4m2/s (hydrogen ion diffusion coefficient). the figure 3 describes the behavior of the temperature distribution of the material in the diffusion equation (α → 1) when x takes different values and t →∞ and with the diffusion coefficient for the density of the diffusion material fixed to κ2 = 0.85.10−4m2/s. we can observe all the curves decay rapidly. thus the diffusion becomes more than more rapid. the figure 4 describes the behavior of the temperature distribution of the material in the diffusion equation (α → 1) when int. j. anal. appl. 17 (2) (2019) 197 figure 2. surface of the temperature distribution for α → 1, κ2 = 0.85.10−4m2/s figure 3. the temperatures distributions for α → 1, κ2 = 0.85.10−4m2/s and t →∞ successively x = 0.0050, x = 0.0065 and x = 0.0095 and t takes various values with the diffusion coefficient for the density of the diffusion material fixed to κ2 = 0.85.10−4m2/s. we observe when x → 0 then the temperature distribution of the material in the diffusion equation (α → 1) converge to 1. furthermore, we can observe all the curves increase slowly. thus the diffusion is in general very slow. the figure 5 describes the behavior of the temperature distribution of the material in the diffusion equation (α → 1) when successively t = 100, t = 150 and t = 200 and x takes various values with the diffusion coefficient for the density of the diffusion material fixed to κ2 = 0.85.10−4m2/s. we can observe all the curves decay very rapidly. . int. j. anal. appl. 17 (2) (2019) 198 figure 4. the temperatures distributions for α → 1, κ2 = 0.85.10−4m2/s, x = 0.0050; 0.0065; 0.0095 figure 5. the temperatures distributions for α → 1, κ2 = 0.85.10−4m2/s, t = 100; 150; 200 we use a bode plot to interpret the result of this paper graphically. to this end, we use the transfer function given here by the laplace transform. to reach our conclusion, we compute the capacity h(s) = 2κ2 πs (s + κ2w2) using matlab code, we obtain the behavior of the amplitude and the phase of the temperature distribution, see in figure 6. int. j. anal. appl. 17 (2) (2019) 199 figure 6. bode plot of temperature distribution with α → 1, κ = 0.85.10−4m2/s 4. analytical solution of the fractional diffusion equation in two dimensional space in this section, we investigate to find the analytical solution of the fractional diffusion equation in twodimensional space expressed as follows dcαu(x,y,t) = κ 2 { ∂2u(x,y,t) ∂x2 + ∂2u(x,y,t) ∂y2 } (4.1) with dirichlet boundary conditions defined as • • • u(x,y, 0) = 0 for x,y > 0, • • • u(0,y,t) = u(x, 0, t) = 1 for t > 0. we repeat the same reasoning as in section 3. we apply the fourier sine transform. multiplying equation (4.1 ) by 2 π sin wx sin ηy and integrating it between 0 to ∞ successively respecting x and y, we get that: dcαus(w,η,t) = κ 2 { 2(w2 + η2) πwη us(0, t) − (w2 + η2)us(w,η,t) } dcαus(w,η,t) = 2κ2(w2 + η2) πwη −κ2(w2 + η2)us(w,η,t). where us(w,η,t) denotes the fourier sine transform of u(x,y,t). rearranging, we obtain the following fractional differential equation defined as dcαus(w,η,t) + κ 2(w2 + η2)us(w,η,t) = 2κ2(w2 + η2) πwη . (4.2) int. j. anal. appl. 17 (2) (2019) 200 we apply the laplace transform to both sides of equation (4.2). we obtain the following relationships sαūs(w,η,s) + κ 2(w2 + η2)ūs(w,η,t) = 2κ2(w2 + η2) πwηs ūs(w,η,s) = 2κ2(w2 + η2) πwηs (sα + κ2(w2 + η2)) . (4.3) where ūs(w,η,s) denotes the laplace transform of us(w,η,t). to obtain the analytical solution of the fractional diffusion equation (4.1), we rewrite the laplace transform (4.3) as follows ūs(w,η,s) = 2 πwη { 1 s − sα−1 sα + κ2(w2 + η2) } . (4.4) finally, to get the analytical solution of the fractional diffusion equation (4.1), we apply the inverse of laplace transform to both sides of equation (4.4) and the inverse of fourier sine transform on the obtained equation. we get u(x,y,t) = 4 π2 ∫ ∞ 0 sin wx w ∫ ∞ 0 sin ηy y { 1 −eα ( −κ2(w2 + η2)tα )} dηdw. we investigate the analytical solution of the diffusion equation in two-dimensional space obtained when α → 1. to this end, we pick the laplace transform function defined to equation (4.4) when α → 1, defined by ūs(w,η,s) = 2 πwη { 1 s − 1 s + κ2(w2 + η2) } . (4.5) applying the inverse of laplace transform and the inverse of the fourier sine transform, we obtain that u(x,y,t) = 4 π2 ∫ ∞ 0 sin wx w ∫ ∞ 0 sin ηy y { 1 − exp ( −κ2(w2 + η2)t )} dηdw = 1 − 4 π2 ∫ ∞ 0 sin wx w ∫ ∞ 0 sin ηy y exp ( −κ2(w2 + η2)t } dηdw. we use the gaussian error function erf(.), we obtain the following form u(x,y,t) = 1 −erf ( x 2κ √ t ) erf ( y 2κ √ t ) . (4.6) that is the analytical solution of the diffusion equation in two-dimensional space obtained when α → 1. let’s give the behavior of the temperature distribution of the material in the diffusion equation in some configurations. figure 7 describes the behavior of the temperature distribution of the material in the diffusion equation in two-dimensional space when x = y and t takes various values. figure 8 describes the behavior of the temperature distribution of the material in the diffusion equation in two-dimensional space when x = y and t →∞, we observe the gaussian profile of the temperature distribution of the material in the diffusion equation. figure 9 describes the behavior of the temperature distribution of the material in the diffusion equation in two-dimensional space when x = y → 0 and t take various values. we observe all curves increase rapidly. int. j. anal. appl. 17 (2) (2019) 201 figure 7. surface of the temperature distribution of the material in the diffusion equation for α → 1, κ = 0.85.10−4m2/s, x = y figure 8. temperature distribution of the material in the diffusion equation for α → 1, κ = 0.85.10−4m2/s, x = y and t →∞ 5. analytical solution of the cattaneo-hristov diffusion equation hristov in [12] [13], stating with cattaneo constructive relaxation with jeffrey’s kernel proposes a new elastic heat diffusion equation described by the caputo-fabrizio fractional derivative. diffusion phenomena, of heat or mass, are generally explained as a consequence of the conservative law by the relationships [12] ρcp ∂t ∂t = − ∂q ∂x ; q(x,t) = −k ∂t(x,t) ∂x ⇒ ρcp ∂t ∂t = k ∂2t ∂x2 (5.1) where the flux of heat is given by the following relationship q(x,t) = − ∫ t −∞ r(x,t)∇t(x,t−s)ds (5.2) int. j. anal. appl. 17 (2) (2019) 202 figure 9. temperature distribution for α =→ 1, κ2 = 0.85.10−4m2/s, x = y → 0 and t in this case of space independent damping the function r(x,t) it can be represented by the jeffrey kernel r(t) = exp (−(t−s)/τ) where τ designs a finite relaxation term [12] [26]. continuing the constructive equations, the energy balance yields the cattaneo equation defined as the following form [12] ∂t(x,t) ∂t = − k2 τρcp ∫ t 0 exp (−(t−s)/τ) ∂t(x,s) ∂x ds (5.3) with equation (5.3), the energy conservative equation of the internal energy result in the jeffrey type interodifferential equation [12] in the form ∂t(x,t) ∂t = k1 ρcp ∂2t(x,t) ∂x2 + k2 τρcp ∫ t −∞ exp (−(t−s)/τ) ∂2t(x,s) ∂x2 ds. (5.4) finally, using the concept of the caputo-fabrizio fractional derivative recently introduced in [6] and some assumptions, see more details in [6], hristov arrives to the complete cattaneo-hristov diffusion equation [12] [13] expressed as the following form ∂t(x,t) ∂t = a1 ∂2t(x,t) ∂x2 + a2 (1 −α) cf 0 d α t ( ∂2t(x,t) ∂x2 ) . (5.5) where a1 = k1 ρcp and a2 = k2 ρcp with ρ = const, cp = const. the constant k1 and k2 represent successively the effective thermal conductivity and the elastic conductivity. cf0 d α t denotes the caputo-fabrizio fractional derivative, see in [6] and t represents the temperature distribution. the dirichlet boundary conditions considered in this paper is defined in the following form • • • t(x, 0) = 0 for x > 0, • • • t(0, t) = 1 for t > 0. the equation (5.5) is known as the entire cattaneo-hristov equation of transition heat diffusion equation. the cattaneo-hristov diffusion equation allows in-depth investigations of the role of the damping kernel on int. j. anal. appl. 17 (2) (2019) 203 the behavior of the heat diffusion process and the telegraph equation [1]. the second term of the cattaneohristov equation ∂t(x,t) ∂t = a2 (1 −α) cf 0 d α t ( ∂2t(x,t) ∂x2 ) (5.6) is known as the elastic part of the heat diffusion equation process and was subject of investigations done by koca et al. in [26]. in [26] koca et al. propose analytical and numerical solutions of the elastic part of the heat diffusion equation process described by the caputo-fabrizio fractional derivative. in [12], hristov proposes an approximation of the solution using an integral method based on a finite penetration depth. in this section, we investigate to find the analytical solution of the complete cattaneo-hristov diffusion equation (5.5). the boundary conditions considered in this paper are particular cases which we can obtain with cattaneo-hristov model of diffusion. and all results found in this section can be modified when the boundary conditions change. the method of the resolution used in the previous section to get the analytical solution of the fractional diffusion equations in one and two-dimensional spaces doesn’t change. before applying the fourier sine transform and the laplace transform, we recall the laplace transform of the caputo-fabrizio fractional derivative given by l { cf 0 d α t f(t) } = sl{f(t)}(s) −f(0) s + α (1 −s) . (5.7) to get the analytical solution of the complete cattaneo-hristov diffusion equation, we multiply equation (5.5) by 2 π sin wx and integrating it between 0 to ∞; we obtain the following differential equation ∂ts(w,t) ∂t = a1 { 2 π w −w2ts(w,t) } + a2 (1 −α) cf 0 d α t { 2 π w −w2ts(w,t) } = a1 { 2 π w −w2ts(w,t) } −a2w2 (1 −α) cf 0 d α t ts(w,t). (5.8) where ts(w,t) denotes the fourier sine transform of t(x,t). applying the laplace transform to both sides of equation (5.8), we get that t̄s(w,t) = 2a1w (α + (1 −α)s) πs{(1 −α)s2 + (α + (1 −α)(a1w2 + a2w2)) s + a1w2α} . (5.9) where t̄s(w,t) denotes the laplace transform of ts(w,t). let that λ = α 1−α with α 6= 1 and then equation (5.9) can be rewritten as follows t̄s(w,t) = 2a1w (λ + s) πs{s2 + (λ + (a1w2 + a2w2)) s + a1w2λ} . (5.10) the transformation (5.10) is essential in a sense we use it for every specific order. the equation (5.10) can be rewritten as a series, and then we obtain the following relationships t̄s(w,t) = 2 π ∞∑ k=0 (−1)k (a1)kw2k+1λk λs1−(3+k) + s1−(2+k) (s + (λ + (a1w2 + a2))) k+1 . (5.11) int. j. anal. appl. 17 (2) (2019) 204 let that µ = ( λ + (a1w 2 + a2w 2) ) , applying the inverse of laplace transformation and using mittag-leffler functions with three parameters, we get that ts(w,t) = 2 π ∞∑ k=0 (−1)k k! (a1) kw2k+1λk [ λt2k+2e (k) 1,3+k (−µt) + t 2k+1e (k) 1,2+k (−µt) ] . (5.12) finally, we get the analytical solution of the cattaneo-hristov diffusion equation, by applying the inverse of fourier sine transform t(x,t) = 2a1 π ∫ ∞ 0 w sin(wx) ∞∑ k=0 (−1)k k! (a1) kw2kλk × [ λt2k+2e (k) 1,3+k (−µt) + t 2k+1e (k) 1,2+k (−µt) ] dw. (5.13) as in the previous section, we analyze the particular case of the cattaneo-hristov diffusion equation obtained when α → 1. the laplace transform is given using the equation (5.10) by t̄s(w,t) = 2 π 1 w { 1 s − 1 s + a1w2 } . (5.14) applying the inverse of laplace transform to both sides to equation (5.14) and the inverse of fourier sine transform we get t(x,t) = 2 π ∫ ∞ 0 sin(wx) w { 1 − exp(−a1w2t) } dw = 1 − 2 π ∫ ∞ 0 sin(wx) w exp(−a1w2t)dw = 1 −erf ( x 2 √ a1t ) . one can observe this solution is similar to the solution obtained in the classical diffusion equation. thus the surface described by the solution of the particular cattaneo-hristov diffusion equation considered above is identical to the surface represented by the solution of the classical diffusion equation. 6. conclusion the complete cattaneo-hristov equation of the transient heat diffusion equation introduced by hristov was considered in this paper. the problems opened by hristov with this new constructive equation in the fractional diffusion equation are the problem consisting of getting the numerical solutions, the problem consisting of finding the analytical solutions and the problem consisting to get an approximate solutions. hristov proposes an estimate for the solution of the cattaneo-hristov diffusion equation using a finite penetration depth, koca and atangana in their works suggest the analytical and the numerical solutions of the elastic part of the heat diffusion equation process. the numerical solution of the complete cattaneo-hristov equation of the transient heat diffusion equation was considered in recent works done by alkahtani and atangana. this paper proposes the analytical solution of the complete cattaneo-hristov equation of the transient heat diffusion equation. the integral method used in the resolution combines both the fourier sine transform and int. j. anal. appl. 17 (2) (2019) 205 the laplace transform. this paper offers a useful analytical solution of the fractional diffusion equation in two-dimensional space. some special cases of the fractional diffusion equations were discussed and illustrated graphically. references [1] b. s. t. alkahtani and a. atangana. a note on cattaneo-hristov model with non-singular fading memory. therm. sci., 21(1)(2017), 1-7. [2] a. atangana and d. baleanu. new fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. arxiv preprint arxiv:1602.03408, (2016). [3] a. atangana and jf. gomez-aguilar. fractional derivatives with noindex law property: application to chaos and statistics. chaos solitons fractals, 114 (2018), 516-535. [4] a. atangana and i. koca. chaos in a simple nonlinear system with atanganabaleanu derivatives with fractional order. chaos solitons fractals, 89 (2016), 447-454. [5] l. beghin. fractional diffusion-type equations with exponential and logarithmic differential operators. stoc. proc. appl., 128(7)(2018), 2427-2447. [6] m. caputo and m. fabrizio. a new definition of fractional derivative without singular kernel. progr. fract. differ. appl., 1(2)(2015), 1-13. [7] a. c. escamilla, jf. g. aguilar, l. torres, and rf. e. jimnez. a numerical solution for a variable-order reactiondiffusion model by using fractional derivatives with non-local and non-singular kernel. phys. a: stat. mech. appl., 491(2018), 406-424. [8] h. delavari, d. baleanu, and j. sadati. stability analysis of caputo fractional-order nonlinear systems revisited. nonlinear dyn., 67(4) (2012), 2433-2439. [9] e. f. d. goufo. chaotic processes using the two-parameter derivative with non-singular and non-local kernel: basic theory and applications. chaos: an inter. j. nonlinear sci., 26(8) (2016), 084-305. [10] j. hristov. on the atangana-baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation. trends in theory and applications of fractional derivatives with mittag-leffler kernel, springer. 2019. [11] j. hristov. approximate solutions to fractional subdiffusion equations. eur. phys. j. spect. topics, 193(1)(2011), 229-243. [12] j. hristov. transient heat diffusion with a non-singular fading memory: from the cattaneo constitutive equation with jeffrey’s kernel to the caputo-fabrizio time-fractional derivative. therm. sci., 20(2) (2016), 757-762. [13] j. hristov. derivation of the fractional dodson equation and beyond: transient diffusion with a non-singular memory and exponentially fadingout diffusivity. progr. fract. differ. appl, 3(4) (2017), 1-16. [14] j. hristov. multiple integral-balance method basic idea and an example with mullins model of thermal grooving. therm. sci., 21(2017), 1555-1560. [15] j. hristov. the non-linear dodson diffusion equation: approximate solutions and beyond with formalistic fractionalization. math. nat. sci., 1(1) (2017), 1-17. [16] j. hristov. fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. math. model. nat. phenom. 13(1)(2018), 6. [17] j. hristov. integral-balance solution to nonlinear subdiffusion equation. front. fract. calcu., 1(2018), 70. [18] j. hristov. the heat radiation diffusion equation: explicit analytical solutions by improved integral-balance method. therm. sci., 22(2) (2018), 777-788. int. j. anal. appl. 17 (2) (2019) 206 [19] j. hristov. integral balance approach to 1-d space-fractional diffusion models. math. meth. eng., (2019), 111-131, springer. [20] j. hristov. a transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: an improved integralbalance approach. math. meth. eng., (2019), 153-174, springer. [21] f. jarad and t. abdeljawad. a modified laplace transform for certain generalized fractional operators. res. nonlinear anal., (2)(2018), 88-98. [22] f. jarad, e. ugurlu, t. abdeljawad, and dumitru baleanu. on a new class of fractional operators. adv. diff. equa., (1)(2017), 247. [23] h. jordan. steady-state heat conduction in a medium with spatial non-singular fading memory derivation of caputo-fabrizio spacefractional derivative from cattaneo concept with jeffrey’s kernel and analytical solutions. therm. sci., 21(2) (2017), 827-839. [24] u. n. katugampola. a new approach to generalized fractional derivatives. bull. math. anal. appl, 6(4)(2014), 115. [25] a. a. kilbas, m rivero, l rodriguez-germa, and jj trujillo. caputo linear fractional differential equations. ifac proc. 39(11) (2006), 52-57. [26] i. koca and abdon atangana. solutions of cattaneo-hristov model of elastic heat diffusion with caputo-fabrizio and atangana-baleanu fractional derivatives. therm. sci., 21 (2017), 2299-2305. [27] l. li, j. g. liu, and l. wang. cauchy problems for kellersegel type timespace fractional diffusion equation. j. differ. equ., 265(3)(2018), 1044-1096. [28] y. li, y. q. chen, and i. podlubny. mittagleffler stability of fractional order nonlinear dynamic systems. auto., 45(8) (2009), 1965-1969. [29] y. li, f. liu, i. w. turner, and t. li. time-fractional diffusion equation for signal smoothing. appl. math. comp., 326 (2018), 108116. [30] j. losada and j. j. nieto. properties of a new fractional derivative without singular kernel. progr. fract. differ. appl, 1(2)(2015), 87-92. [31] y. ma, f. zhang, and c. li. the asymptotics of the solutions to the anomalous diffusion equations. comput. math. appl., 66(5)(2013), 682-692. [32] t. myers. optimal exponent heat balance and refined integral methods applied to stefan problems. int. j. heat mass transfer, 53(5-6) (2010), 1119-1127. [33] k. m. owolabi and a. atangana. robustness of fractional difference schemes via the caputo subdiffusion-reaction equations. chaos, solitons fractals, 111 (2018), 119-127. [34] i. podlubny. fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (1998), 198. acad. press. [35] i. podlubny. matrix approach to discrete fractional calculus ii: partial fractional differential equations. (2009). [36] s. priyadharsini. stability of fractional neutral and integrodifferential systems. j. fract. calc. appl.,7(1) (2016), 87-102. [37] z. ruan, w. zhang, and zewen wang. simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation. appl. math. comput., 328 (2018), 365-379. [38] k. m. saad, d. baleanu, and a. atangana. new fractional derivatives applied to the kortewegde vries and korteweg-de vries-burgers equations. comput. appl. math., 37 (2018), 52035216. [39] y. salehi, m. t. darvishi, and w. e. schiesser. numerical solution of space fractional diffusion equation by the method of lines and splines. appl. math. comput., 336 (2018), 465-480. [40] n. sene. exponential form for lyapunov function and stability analysis of the fractional differential equations. j. math. comput. sci. 18(4)(2018), 388-397. int. j. anal. appl. 17 (2) (2019) 207 [41] n. sene. lyapunov characterization of the fractional nonlinear systems with exogenous input. fractal fract., (2018), 2(2):17. [42] n. sene. solutions for some conformable differential equations. progr. fract. differ. appl., 4(4)(2018), 493-501. [43] n. sene. stokes first problem for heated flat plate with atangana-baleanu fractional derivative. chaos soli. fract., 117 (2018), 68-75. [44] s. shen, f. liu, and v. v. anh. the analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation. j. comput. appl. math., 345 (2019), 515-534. [45] j. zhang, x. zhang, and b. yang. an approximation scheme for the time fractional convectiondiffusion equation. appl. math. comput., 335(2018), 305-312. 1. introduction 2. fractional diffusion equations 3. analytical solution of fractional diffusion equation in one dimensional space 4. analytical solution of the fractional diffusion equation in two dimensional space 5. analytical solution of the cattaneo-hristov diffusion equation 6. conclusion references international journal of analysis and applications volume 17, number 5 (2019), 850-863 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-850 fixed points of non-smooth functions on finite dimensional ordered banach spaces via clarke generalized jacobian zohari and mardanbeigi∗ 1department of mathematics, science and research branch, islamic azad university, tehran, iran ∗corresponding author: mrmardanbeigi@srbiau.ac.ir abstract. considering lipschitz functions which are not necessarily fréchet differentiable, we obtain a non-smooth version of lakshmikantham’s theorem in finite dimensional ordered banach spaces . we also present an application of the obtained result in dynamical coulomb friction problem. 1. introduction ordered banach spaces are very significant class of vector spaces which are studied widely in theory and applications of mathematics. this class of vector spaces is considered in nonlinear integral equations [2], nonlinear boundary value problems [4], optimal control theory [8], operator equations [20] and etc. on the other hand, an important theory in mathematical analysis is fixed point theory. this theory and its applications in orederd banach spaces have been considered by many researchers. (see [2, 3, 5, 6, 11, 14, 16, 17] and the references therein.) recently, lakshmikantham et al. [14] have proved some fixed point theorems in ordered banach space x for a fréchet differentiable mapping t : x → x. they showed the applications of their results in ode initial value problems and semilinear parabolic initial boundary value problems. vijesh and kumar [19] and mouhadjer and benahmed [16] obtained some generalizations of lakshmikantham’s fixed point theorems. received 2019-05-21; accepted 2019-07-04; published 2019-09-02. 2010 mathematics subject classification. 47h10, 49j52, 46b40. key words and phrases. fixed point; ordered banach spaces; clarke generalized jacobian;coulomb friction. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 850 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-850 int. j. anal. appl. 17 (5) (2019) 851 to obtain fixed points in a class of nonlinear operators in ordered banach spaces, mouhadjer and benahmed introduced a monotone newton-like method in [15], by using lakshmikantham’s fixed point theorems. in this paper, we study these fixed point theorems for lipschitz mappings on finite bancah spaces which are not necessary fréchet differentiable. our main tool is clarke generalized jacobian which was firstly introduced by clarke in [9] for a mapping between two finite dimensional vector spaces. clarke generalized jacobian has heavy calculus rules and this paper, to the best of our knowledge, is the first work which deal to non-smooth fixed point theorem using generalized gradient. since every finite dimensional vector space is isomorphic and homeomorphic to rn for some n, we only focus on mappings f : rn → rn which are not necessary differentiable and we prove some fixed point theorems for lipschitzian ones. finally, to show one of the applications of the obtained results, we consider a non-smooth conic complementary problem and then we investigate the relation between obtained fixed points and the solutions of the problem. this conic complementary problem is arised in the linear discrete coulomb friction problem which studied by acary et al. [1]. the paper is organized as follows. in section 2 preliminaries are given. in section 3 we introduce some new definitions related to clarke generalized jacobian and then we investigate main results. section 4 is devoted to an application of the obtained results. 2. preliminaries let s be a set. denote the convex hull of s (the set of all finite convex combinations of members of s) by co(s). let c ⊆ rn be nonempty. then c is a closed pointed convex cone, when i) c is a closed convex set, ii) for every x ∈ c and every scalar λ ≥ 0, λx ∈ c, iii) c ∩−c = {0}. we have c + c ⊆ c for every convex cone c. using a closed pointed convex cone c ⊆ rn, we can define the following order relation on rn: x ≤ y ⇐⇒ y −x ∈ c. this relation is reflexive, antisymmetric and transitive. if x ≤ y, then i) x + z ≤ y + z, for each z ∈ rn, ii) αx ≤ αy, for all scalar α ≥ 0. for x̄, ȳ ∈ rn such that x̄ ≤ ȳ, the order interval [x̄, ȳ] is defined as [x̄, ȳ] := {z ∈ rn : x̄ ≤ z ≤ ȳ}. a cone, especially used for many applications, is standard cone in finite dimensional euclidean spaces: rn+ := {(x1, . . . ,xn) ∈ r n : xi ≥ 0, i = 1 . . . ,n}. int. j. anal. appl. 17 (5) (2019) 852 definition 2.1. let c ⊆ rn be a closed pointed convex cone. i) c is called normal if there exists a positive constant δ such that for every x,y ∈ rn, 0 ≤ x ≤ y ⇒‖x‖≤ δ‖y‖, ii) c is called regular if every increasing (decreasing) sequence {xn}n∈n which is bounded from above (below) converges. since every finite-dimensional normed space is reflexive, then for closed pointed convex cone c ⊆ rn, normality and regularity are equivalent [12]. definition 2.2. the mapping f : rn → rm is strictly differentiable at x, if there is an m×n-matrix m such that lim y→x,u→0 f(y + u) −f(y) −m(u) ‖u‖ = 0. in this case, m is called strict derivative of f at x. definition 2.3. the mapping f : rn → rm is fréchet differentiable at x, if there is an m×n-matrix m such that lim u→0 f(x + u) −f(x) −m(u) ‖u‖ = 0. in this case, m is called fréchet derivative of f at x. note that, strictly differentiable functions are fréchet differentiable while the converse is not true in general [13]. let the vector-valued function f : rn → rm be locally lipschitz at a given point x, that is, there exists a neighborhood u of x and a positive k such that ‖f(x1) −f(x2)‖≤ k‖x1 −x2‖, ∀x1,x2 ∈ u. by rademacher’s theorem [9], f is differentiable almost everywhere (in the sense of lebesgue measure) on u. definition 2.4. [9, 13] let the vector-valued function f : rn → rm be locally lipschitz at a given point x. the clarke generalized jacobian of f at x, denoted by ∂f(x) is defined as ∂f(x) := co { lim i→∞ ∇f(xi) : xi ∈ ω,xi → x } , where ω is the set of points in u at which f is differentiable at them. note that ∂f(x) is a nonempty convex and compact subset of l(rn,rm), the space of real m×n-matrices. int. j. anal. appl. 17 (5) (2019) 853 3. main results in [14], lakshmikantham et al. considered a mapping t : e → e where e is an ordered banach space and they worked on an order interval [v0,w0] which satisfies some assumptions. fréchet differentiability of t on [v0,w0] is a vital assumption in [14]. one of the lakshmikantham’s abstract fixed point theorems is as follows. theorem 3.1. [14, theorem 2.1] let e be an ordered banach space with regular order cone e+. suppose t : e → e satisfies the following hypotheses: (i) there exist v0,w0 ∈ e such that v0 ≤ tv0, tw0 ≤ w0 and v0 ≤ w0. (ii) the fréchet derivative t ′(u) exists for every u ∈ [v0,w0], and the mapping u 7→ t ′(u)v is increasing on [v0,w0] for all v ∈ e+. (iii) [i −t ′(u)]−1 exists and is a bounded and positive operator for all u ∈ [v0,w0]. then, for n ∈ n, relations vn+1 = tvn + t ′(vn)(vn+1 −vn), wn+1 = twn + t ′(vn)(wn+1 −wn), define an increasing sequence {vn}∞n=0 and a decreasing sequence {wn}∞n=0 which both converge to fixed points of t . these fixed points are equal if (iv) tu1 −tu0 < u1 −u0 whenever v0 ≤ u0 < u1 ≤ w0. assumptions (ii) and (iii) are deeply dependent on fréchet differentiability of t on [v0,w0] , so even for a simple function f(x) = |x| on r we cannot use this theorem (since f is not differentiable at x = 0). in this section, we assume that, the mappings are lipschitz and we impose strictly differentiability only in v̄, the left bound of the order interval [v̄, w̄], and we do not need to assume differentiability in other members of the interval. definition 3.1. let f : rn → rn be locally lipschitz on [v̄, w̄] and let c be a regular convex cone . we say the set-valued u 7→ ∂f(u) is semi-increasing on [v̄, w̄] if for all v,w ∈ [v̄, w̄], with w −v ∈ c, ∂f(w) ◦z ⊆ ∂f(v) ◦z + c, ∀z ∈ c, (3.1) where ◦ denotes the matrix multiplication. remark 3.1. if f : rn → rn is strictly differentiable on [v̄, w̄], then for every x ∈ [v̄, w̄], ∂f(·) is a singleton and lipschitz on [v̄, w̄] [9]. we denote this derivative by f ′(x). if f is semi-increasing on [v̄, w̄] and c = rn+. then, for all v,w ∈ [v̄, w̄], with w −v ∈ rn+ we have, f ′(w) ◦z ⊆ f ′(v) ◦z + rn+, ∀z ∈ r n +. int. j. anal. appl. 17 (5) (2019) 854 thus, for some r ∈ rn+, f ′(w)◦z = f ′(v)◦z+r. this shows that for v ≤ w, f ′(v)◦z ≤ f ′(w)◦z. therefore, u 7→ t ′(u)z is increasing on [v̄, w̄] for all z ∈ rn+. thus, semi-increasing map notion is a generalization of increasing map notion. example 3.1. consider the function f : r2 → r2, defined by f(x,y) = (|x|, |y|) on [(0, 0)t , (1, 1)t ] and c = r2+. where t denotes matrix transposition. then, ∂f(0, 0) =    α 0 0 β   : α,β ∈ [−1, 1]   ∂f(x, 0) =    1 0 0 β   : β ∈ [−1, 1]   0 < x ≤ 1 ∂f(0,y) =    α 0 0 1   : α ∈ [−1, 1]   0 < y ≤ 1 ∂f(x,y) =    1 0 0 1     0 < x,y ≤ 1. if v = (0, 0)t , v ≤ w and z = (z1,z2)t is arbitrary, then figure 1 shows that ∂f(w)◦z ⊆ ∂f(0, 0)◦z+r2+. it is easy to check that in all other cases, (3.1) holds too. thus, f is a semi-increasing map on [(0, 0)t , (1, 1)t ]. figure 1. semi-increasing map in example 3.1 lemma 3.1. let f : rn → rn be locally lipschitz on [v̄, w̄]. if the set-valued u 7→ ∂f(u) is semi-increasing on [v̄, w̄] and v,w ∈ [v̄, w̄], with w −v ∈ c, then ∂f(w) ◦z ⊆ ∂f(v) ◦z −c, ∀z ∈−c. int. j. anal. appl. 17 (5) (2019) 855 proof. let z ∈ −c and ζz ∈ ∂f(w) ◦ z be arbitrary. since −z ∈ c and ζ(−z) ∈ ∂f(w) ◦ (−z) we have ζ(−z) ∈ ∂f(v) ◦ (−z) + c. thus, ζ(−z) = η(−z) + c, for some η ∈ ∂f(v) and c ∈ c. on the other hand, ∂f(u) ⊆ l(rn) for every u ∈ rn, thus, ζ(−z) = −ζz and η(−z) = −ηz and we have ζz = ηz − c. hence, ζz ∈ ∂f(v) ◦z −c, which proves the lemma. � the following lemma plays an important role in the reminder of this paper. lemma 3.2. let f : rn → rn be lipschitz and semi-increasing on an order interval [v̄, w̄]. then for v̄ ≤ v ≤ w ≤ w̄ we have, f(v) −f(w) ∈ ∂f(v) ◦ (v −w) −c. proof. according to the mean value theorem [9, proposition 2.6.5], f(v) −f(w) ∈ co(∂f(co{v,w})) ◦ (v −w). the right-hand side above denotes the convex hull of all points of the form η(v −w), where η ∈ ∂f(u) for some point u ∈ co{v,w}. assume that u = v + t(w −v) where 0 ≤ t ≤ 1. there exist λ1, . . . ,λm ∈ r such that λi ≥ 0 for i ∈{1, . . . ,m} and ∑m i=1 λi = 1 and we have, f(v) −f(w) = ( m∑ i=1 λiζi ) (v −w), (3.2) where ζi ∈ ∂f(u). on the other hand, since v −w ∈−c and u−v ∈ c, by lemma 3.1 we have, ∂f(u) ◦ (v −w) ⊆ ∂f(v) ◦ (v −w) −c. (3.3) now, considering the convexity of ∂f(u), also (3.2) and (3.3), we have f(v) −f(w) ∈ ∂f(v) ◦ (v −w) −c. � in the following, we prove our main results. theorem 3.2. let rn be ordered with regular cone c. assume that the mapping f : rn → rn satisfies the following hypotheses. (i) there exist v̄, w̄ ∈ rn with w̄ − v̄ ∈ c, such that f is lipschitz function on order interval [v̄, w̄], v̄ ≤ f(v̄) and f(w̄) ≤ w̄. (ii) the set-valued u 7→ ∂f(u) is semi-increasing on [v̄, w̄]. (iii) f is strictly differentiable at v̄ with ∂f(v̄) = {ζv̄} such that [i−ζv̄]−1 exists and is a bounded positive operator, that is, [i − ζv̄]−1(c) ⊆ c. then for n ∈ n, relations vn+1 = f(vn) + ζv̄(vn+1 −vn), wn+1 = f(wn) + ζv̄(wn+1 −wn), (3.4) int. j. anal. appl. 17 (5) (2019) 856 with v0 := v̄, define an increasing sequence {vn}n∈n and a decreasing sequence {wn}n∈n which both converge to fixed points of f . these fixed points are equal if (iv) fu1 −fu0 < u1 −u0 whenever v̄ ≤ u0 < u1 ≤ w0. proof. we recall that since f is strictly differentiable at v̄, ∂f(v̄) is a singleton set (see [13, p. 15]). consider v1 = f(v̄) + ζv̄(v1 − v̄). thus, [i − ζv̄]v1 = f(v̄) − ζv̄v̄. this implies v1 = [i − ζv̄]−1(f(v̄) − ζv̄v̄). since [i − ζv̄]−1 is bounded, v1 is well-defined. we show that v̄ ≤ v1 ≤ w̄. the assumption (1) implies: v̄ −v1 ≤ f(v̄) − [f(v̄) + ζv̄(v1 − v̄)] = ζv̄(v̄ −v1). therefore, [i − ζv̄](v̄ −v1) ≤ 0. since, [i − ζv̄]−1 is a positive operator, (v̄ −v1) ≤ [i − ζv̄]−1(0) = 0 and we have v̄ ≤ v1. now, by (i) v1 − w̄ ∈ f(v̄) + ζv̄(v1 − v̄) −f(w̄) −c. and using lemma 3.2 we have, v1 − w̄ ∈ ∂f(v̄) ◦ (v̄ − w̄) + ζv̄(v1 − v̄) −c. therefore, for some c ∈ c we have, v1 − w̄ = ζv̄(v̄ − w̄) + ζv̄(v1 − v̄) − c. (3.5) then (3.5) implies: v1 − w̄ ≤ ζv̄(v1 − w̄). thus [i − ζv̄](v1 − w̄) ≤ 0 and (v1 − w̄) ≤ [i − ζv̄]−1(0) = 0 which implies v1 ≤ w̄. in a similar way we can show that there exists a point w1 such that w1 = f(w̄) + ζv̄(w1 − w̄) and v̄ ≤ w1 ≤ w̄. we claim that v1 ≤ w1. to prove this claim, note that: v1 −w1 = f(v̄) + ζv̄(v1 − v̄) − [f(w̄) + ζv̄(w1 − w̄)]. thus, by lemma 3.2 we have, v1 −w1 ∈ ∂f(v̄) ◦ (v̄ − w̄) + ζv̄(v1 − v̄) − ζv̄(w1 − w̄) −c, similarly, v1 −w1 ≤ ζv̄(v1 −w1) and we have v1 ≤ w1. up to now, we showed that v̄ ≤ v1 ≤ w1 ≤ w̄. we claim for every j ∈ n, v̄ ≤ vj ≤ vj+1 ≤ wj+1 ≤ wj ≤ w̄. since v̄ ≤ v1 and w1 ≤ w̄, we only need to prove vj ≤ vj+1 ≤ wj and vj+1 ≤ wj+1 ≤ wj. we just prove vj ≤ vj+1 and other inequalities are obtained similarly. by lemma 3.2 vj −vj+1 = f(vj−1) + ζv̄(vj −vj−1) − [f(vj) + ζv̄(vj+1 −vj)] ∈ ∂f(vj−1) ◦ (vj−1 −vj) + ζv̄(vj −vj−1) − ζv̄(vj+1 −vj) −c int. j. anal. appl. 17 (5) (2019) 857 since vj−1 − v̄ ∈ c and vj−1 −vj ∈−c, by lemma 3.1 we have, vj −vj+1 ∈ ∂f(v̄) ◦ (vj−1 −vj) + ζv̄(vj −vj−1) − ζv̄(vj+1 −vj) −c. therefore, vj −vj+1 ≤ ζv̄(vj −vj+1), so we have vj −vj+1 ≤ [i −ζv̄]−1(0) = 0 which proves our claim. we found an increasing sequence {vn}n∈n and a decreasing sequence {wn}n∈n such that v̄ ≤ v1 ≤ . . . ≤ vn ≤ wn ≤ . . . ≤ w1 ≤ w̄. since c is a regular cone, {vn}n∈n and {wn}n∈n are convergent. suppose vn → v and wn → w. we will prove vn+1 → f(v) which shows v is a fixed point of f. we have vn+1 −f(v) = f(vn) + ζv̄(vn+1 −vn) −f(v) since f is continuous and vn → v, vn+1 −vn → 0 and f(vn) −f(v) → 0, thus vn+1 → f(v), so f(v) = v. similarly one can show that f(w) = w. if (iv) holds and v < w, then w−v = fw−fv < w−v which is a contradiction. and the uniqueness of the fixed point is proved. � note that under some conditions, the sequences in theorem 3.2, converge quadratically as the following theorem shows. theorem 3.3. let rn be ordered with regular cone c. let the mapping f : rn → rn satisfies the hypotheses of theorem 3.2 and ‖ζv − ζv̄‖≤ l‖u−v‖, ∀ζv ∈ ∂f(v), whenever v̄ ≤ v ≤ u ≤ w̄. then, the sequences {vn}∞n=0 and {wn}∞n=0 converge quadratically to the same fixed point of f . proof. note that, by theorem 3.2, both the sequences {vn}∞n=0 and {wn}∞n=0 converge to the same fixed point u of f. we prove that these sequences converge quadratically. for sequence {vn}∞n=0 we have, vn+1 −u = f(vn) −f(u) + ζv̄(vn+1 −vn) ⊆ ∂f(vn) ◦ (vn −u) + ζv̄(vn+1 −vn) −c. thus, for some ηvn ∈ ∂f(vn) and c ∈ c we have, vn+1 −u = ζvn ◦ (vn −u) + ζv̄(vn+1 −vn) − c = ζvn ◦ (vn −u) + ζv̄(vn+1 −u) − ζv̄(vn −u) − c. int. j. anal. appl. 17 (5) (2019) 858 hence, u−vn+1 = ζvn ◦ (u−vn) − ζv̄(vn+1 −u) + ζv̄(vn −u) + c. this implies [i − ζv̄](u−vn+1) = (ζvn − ζv̄) ◦ (u−vn) + c. since, [i − ζv̄]−1 is a positive operator, we have 0 ≤ u−vn+1 ≤ [i − ζv̄]−1(ζvn − ζv̄) ◦ (u−vn) + c̄, where c̄ = [i −ζv̄]−1(c). on the other hand, the cone c is regular and therefore normal, thus, there exists a positive constant δ such that ‖u−vn+1‖≤ δ‖[i − ζv̄]−1‖‖(ζvn − ζv̄)‖‖u−vn‖ + c̄ ≤ δl‖[i − ζv̄]−1‖‖u−vn‖2 + c̄. for sequence {wn}∞n=0 the claim can be proved by a similar manner. � 4. application in the dynamical coulomb friction problem the dynamical coulomb friction problem in finite dimension with discretized time is associated to the problem of simulating the dynamics of mechanical systems which involve unilateral contact between their parts or with external objects. acary et al. [1] presented a new formulation of this problem. they capture and treat directly the friction model as a parametric quadratic optimization problem with second-order cone constraints coupled with a fixed point equation. in this section, we just use the resulted fixed point theorems obtained in this paper, to investigate problem. firstly, we recall some preliminaries. let x ∈ rd. the normal and tangential components of x with respect to a unit vector e ∈ rd are defined respectively as xn := x te ∈ r, xt := x−xne ∈ rd. for the unit vector e ∈ rd and parameter µ ∈ (0, +∞), the second-order cone ke,µ is defined by ke,µ := {x ∈ rd : ‖xt‖≤ µxn}. if µ = 0 or µ = +∞, we define ke,0 := {x ∈ rd : xt = 0,xn ≥ 0}, ke,∞ := {x ∈ rd : xn ≥ 0}. the dual cone of the second-order cone ke,µ with µ ∈ (0,∞) is defined as k∗e,µ := {s ∈ r d : xts ≥ 0, ∀x ∈ ke,µ} = ke, 1 µ . int. j. anal. appl. 17 (5) (2019) 859 also with the convention 1/0 = ∞ and 1/∞ = 0, (ke,0)∗ = ke,∞ and (ke,∞)∗ = ke,0. consider e1, . . . ,en ∈ rd and µ1, . . . ,µn ∈ [0, +∞]. the associated product cone and its dual cone are respectively l = ke1,µ1 ×···×ken,µn ⊆ rnd , l∗ = k∗e1,µ1 ×···×k ∗ en,µn = ke1, 1 µ1 ×···×ken, 1 µn . set i := { i ∈{1, · · · ,n} : µi 6= 0 } , ni := card i. consider the problem: mv + f = htr ũ = hv + w + es l∗ 3 ũ ⊥ r ∈ l si = ‖ũit‖ for i ∈ i (4.1) with respect to the variable (v,r, ũ,s) ∈ rm × rnd × rnd × rni where ũ := (ũ1, . . . , ũn) ∈ rnd, r := (r1, . . . ,rn) ∈ rnd. moreover, the data of the problem are ei ∈ rd, µi ∈ [0, +∞], m ∈ rm×m, f ∈ rm, h ∈ rnd×m, w ∈ rnd, e ∈ rnd×ni, where matrix m is definite positive and matrix e is constructed by concatenating ni columns ei ∈ rnd, where ei is itself the concatenation of n vectors of rd, all zeros except for the i-th which is µiei. define c(s) := {v ∈ rm : hv + w + es ∈ l∗}, v(s) := arg min v∈c(s) { 1 2 vtmv + ftv } ∈ rm, ũ(s) := hv(s) + w + es ∈ rnd, and f(s) := ( ‖ũit (s)‖ ) i∈i ∈ r ni the following theorem shows the relation between fixed points of f and the solutions of (4.1). theorem 4.1. [1, theorem 3.3] let (v∗,r∗, ũ∗,s∗) solves the problem (4.1), then v∗ = v(s∗) and f(s∗) = s∗. in dimension two , the inverse is also hold. theorem 4.2. [1, theorem 3.7] let the dimension d = 2. then (v∗,r∗, ũ∗,s∗) solves the problem (4.1) if and only if v∗ = v(s∗) and f(s∗) = s∗. int. j. anal. appl. 17 (5) (2019) 860 using these theorems, we can compute a fixed point of f and then check whether the computed fixed point is a solution of the problem (4.1). note that f is a non-smooth function and to compute its fixed points we need to use the non-smooth version fixed point theorem presented in this paper. in the following examples we show the efficiency of the resulted fixed point theorem. example 4.1. let d = 2, n = 3, e1 = e2 = e3 =  0 1  , µ1 = 1, µ2 = 0, µ3 = 2 and m = 2. then e =   0 0 1 0 0 0 0 0 0 0 0 2   . also, i = {1, 3} and ni = 2. for x = ( x1,x2 )t and ei (i = 1, 2, 3), xn = x2 and xt = ( x1, 0 )t . moreover, ke1,µ1 = { x ∈ r2 : |x1| ≤ x2 } , k∗e1,µ1 = ke1,µ1, ke2,µ2 = { x ∈ r2 : x1 = 0,x2 ≥ 0 } , k∗e2,µ2 = { x ∈ r2 : x2 ≥ 0 } ke3,µ3 = { x ∈ r2 : |x1| ≤ 2x2 } k∗e3,µ3 = { x ∈ r2 : |x1| ≤ 1 2 x2 } . these cones and their dual cones are showed in figure 2. figure 2. cones and their dual cones in example 4.1 note that: l = { (α,β,γ)t : α ∈ ke1,µ1,β ∈ ke2,µ2,γ ∈ ke3,µ3 } , l∗ = { (α∗,β∗,γ∗):α∗ ∈ k∗e1,µ1,β ∗ ∈ k∗e2,µ2,γ ∗ ∈ k∗e3,µ3 } . int. j. anal. appl. 17 (5) (2019) 861 assume, m =  1 0 0 1   ,h =   1 0 0 1 1 0 1 0 0 1 1 0   ,w =   1 0 0 1 0 0   ,f =  1 0   ,v =  v1 v2   ,s =  s1 s3   r = (α1,α2,β1,β2,γ1,γ2) t ∈ l, ũ = (α∗1,α ∗ 2,β ∗ 1,β ∗ 2,γ ∗ 1,γ ∗ 2 ) t ∈ l∗. let v ∈ c(s). thus,   α∗1 α∗2 β∗1 β∗2 γ∗1 γ∗2   ∈ l∗ ⇒   1 0 0 1 1 0 1 0 0 1 1 0   ·  v1 v2   +   1 0 0 1 0 0   +   0 0 1 0 0 0 0 0 0 0 0 2   ·  s1 s3   ∈ l∗. therefore,   v1 + 1 v2 + s 1 v1 v1 + 1 v2 v1 + 2s 3   ∈ l∗ ⇒  v1 + 1 v2 + s 1   ∈ k∗e1,µ1,   v1 v1 + 1   ∈ k∗e2,µ2,   v2 v1 + 2s 3   ∈ k∗e3,µ3. hence, |v1 + 1| ≤ v2 + s1,v1 + 1 ≥ 0, |v2| ≤ 1 2 (v1 + 2s 3). so, we have c(s) = { (v1,v2) t ∈ r2 : v1 + 1 ≤ v2 + s1, |v2| ≤ 1 2 (v1 + 2s 3) } . note that c(s) is a closed convex set and j(v) := 1 2 vtmv+ftv is a convex function. therefore, if c(s) 6= ∅, then j(v) has a unique solution. in this example, if s1 ≥ −2s3 + 1, then c(s) 6= ∅. now we must consider the following convex optimization problem: min 1 2 vtmv + ftv = 1 2 (v21 + v 2 2 ) + v1 subject to g1(v) := v1 −v2 −s1 + 1 ≤ 0 g2(v) := |v2|− 1 2 v1 −s3 ≤ 0 int. j. anal. appl. 17 (5) (2019) 862 according to [10, page 150], if v∗ = (v∗1,v ∗ 2 ) t , then there exist λ1,λ2 ≥ 0 such that 0 0   =  v∗1 + 1 v∗2   + λ1   1 −1   + λ2  −12 α   , λigi(v∗) = 0, i = 1, 2 (4.2) where   α = 1 if v∗2 > 0, −1 ≤ α ≤ 1 if v∗2 = 0, α = −1 if v∗2 < 0, then v∗ is the optimal solution. the system (4.2) has a solution if s3 ≤ 1/2. with these condition, v∗1 = 0 and v∗2 = −2s3. thus, v(s) = (0,−2s3). we also have , ũ(s) = hv(s) + w + es =   1 0 0 1 1 0 1 0 0 1 1 0     0 −2s3   +   1 0 0 1 0 0   +   0 0 1 0 0 0 0 0 0 0 0 2   ·  s1 s3   =   1 −2s3 + s1 0 1 −2s3 2s3   and f(s) = (1,−2s3), where s3 ≤ 1 2 . note that from example 3.1, f is a semi-increasing map with c = r2+. set v̄ = (−1,−1)t , w̄ = (2, 0)t . we have ∂f(v̄) =  0 0 0 −2  . with these informations, conditions (i), (ii), (iii) of theorem 3.2 hold. set v0 = (−1,−1)t and vn+1 = (1, 2‖v2n‖) t +  0 0 0 −2   (vn+1 −vn). we have vn = (1, 0) t for every n ∈ n. thus (1, 0)t is a fixed point of f . it is easy to check that the combination of these resulted quantities with r = (0, 0, 0, 0, 0, 1)t , is a solution of system (4.1). references [1] v. acary, f. cadoux, c. lemaréchal, j. malick, a formulation of the linear discrete coulomb friction problem via convex optimization. zamm, z. angew. math. mech. 91 (2011), 155-175. [2] p.r. agarwal, n. hussain, m.a. taoudi, fixed point theorems in ordered banach spaces and applications to nonlinear integral equations, abstr. appl. anal. 2012 (2012), 245872. [3] h. amann, fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. siam rev. 18 (1976), 620-709. [4] h. amann, nonlinear operators in ordered banach spaces and some applications to nonlinear boundary value problems, in: nonlinear operators and the calculus of variations 1976, springer-verlag berlin heidelberg , pp. 1-55. int. j. anal. appl. 17 (5) (2019) 863 [5] h. andrei, p. radu, nonnegative solutions of nonlinear integral equations in ordered banach spaces. fixed point theory, 1 (2004), 65-70. [6] m. berzig, b. samet, positive fixed points for a class of nonlinear operatoes and applications. positivity, 17 (2013), 235-255. [7] s. bonettini, i. loris, f. porta, m. prato, variable metric inexact line-search-based methods for non-smooth optimization. siam j. optim. 26 (2016), 891-921. [8] j. blot, n. hayek, infinite-horizon optimal control in the discrete-time framework. springer-verlag, new york, 2014. [9] f.h. clarke, optimization and non-smooth analysis. society for industrial and applied mathematics, 1990. [10] a. dhara, j. dutta, optimality conditions in convex optimization: a finite-dimensional view. crc press, 2011. [11] t. gnana bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications. nonlinear anal. tma, 65 (2006), 1379-1393. [12] d. guo, y.j. cho, j, zhu, partial ordering methods in nonlinear problems. nova science publishers, new york, 2004. [13] v. jeyakumar, d.t. luc, non-smooth vector functions and continuous optimization. springer, new york, 2008. [14] v. lakshmikantham, s. carl, s. heikkilä, fixed point theorems in ordered banach spaces via quasilinearization. nonlinear anal. tma, 71 (2009), 3448-3458. [15] l. mouhadjer, b. benahmed, a monotone newton-like method for the computation of fixed points, in: le thi h, pham dinh t, nguyen n, editors. modelling, computation and optimization in information systems and management sciences. advances in intelligent systems and computing, vol 359. springer, cham, 2015, pp. 345-356. [16] l. mouhadjer, b. benahmed, fixed point theorem in ordered banach spaces and applications to matrix equations. positivity, 20 (2016), 981-998. [17] j.j. nieto, r. rodŕıguez-lópez, existence and uniqueness of fixed point in partially ordered sets and applications to ordinary diferential equations. acta math. sin. 3 (2007), 2203-2212. [18] w. rudin, functional analysis. mcgraw-hill, inc. 1991. [19] v.a. vijesh, k.h. kumar, wavelet based quasilinearization method for semi-linear parabolic initial boundary value problems. appl. math. comput. 266 (2015), 1163-1176. [20] c.b. zhai, c. yang, c.m. guo, positive solutions of operator equations on ordered banach spaces and applications. comput. math. appl. 56 (2008), 3150-3156. [21] p. zhou, j. du, z. lü, topology optimization of freely vibrating continuum structures based on non-smooth optimization. struct. multidiscip. optim. 56 (2017), 603-618. 1. introduction 2. preliminaries 3. main results 4. application in the dynamical coulomb friction problem references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 132-138 http://www.etamaths.com tripled fixed point results for t-contractions on abstract metric spaces hamidreza rahimi1, calogero vetro2, mujahid abbas3, ghasem soleimani rad4,∗ abstract. in this paper we introduce the notion of t-contraction for tripled fixed points in abstract metric spaces and obtain some tripled fixed point theorems which extend and generalize well-known comparable results in the literature. to support our results, we present an example and an applications to integral equations. 1. introduction and preliminaries in 1922, banach proved his famous fixed point theorem [5]. afterward, many authors considered various definitions of contractive mappings and proved several fixed point theorems, which are extensions and generalizations of banach’s theorem (see, for example, [9, 13, 23]). on the other hand, non-convex analysis has found some applications in optimization theory. fixed point theory in k-metric and k-normed spaces was developed by perov et al. [18], mukhamadijev and stetsenko [17] and others (we refer to a survey by zabrejko [26]). the main idea consists in using an ordered banach space instead of the set of real numbers, as the codomain for a metric. in 2007, huang and zhang [12] reintroduced such as spaces and defined cone metric spaces. then, several fixed point results on cone metric spaces were obtained in [1, 2, 22] and references therein. in 2009, beiranvand et al. [6] defined t-contractions in metric spaces. afterward, some related fixed point theorems were proved in [15]. successively, morales and rajes [16] introduced t-kannan and t-chatterjea contractive mappings in cone metric spaces and studied the existence of fixed points for these mappings. recently, rahimi et al. [19, 21] proved fixed theorems for t-contractions involving two mappings on cone metric spaces. recently, bhaskar and lakshmikantham [8] introduced the concept of coupled fixed point in partially ordered metric spaces, starting a fruitful direction of research followed by many authors, also in the setting of ordered metric and ordered cone metric spaces; see [14, 24] and the references therein. finally, berinde and borcut [7] introduced the notion of tripled fixed point (see also [3, 4]) and obtained results on the existence of tripled fixed points. 2010 mathematics subject classification. 47h10, 46j10, 34a34. key words and phrases. abstract metric space; tripled fixed point; t-contraction; sequentially convergent; subsequentially convergent. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 132 tripled fixed point results for t -contractions 133 in this paper we introduce the notion of t-contraction in tripled fixed point theory and prove some related results on abstract metric spaces. it is worth mentioning that our results do not rely on normality condition on cones involved therein. our theorems extend, unify and generalize well-known results in the literature. following are some definitions and known results needed in the sequel. definition 1.1 ([11, 12]). let e be a real banach space and p a subset of e. then p is called a cone if and only if (a) p is closed, nonempty and p 6= {θ}; (b) a,b ∈ r, a,b ≥ 0, x,y ∈ p implies ax + by ∈ p ; (c) if x ∈ p and −x ∈ p , then x = θ. given a cone p ⊂ e, a partial ordering � with respect to p is defined by x � y ⇐⇒ y −x ∈ p. we shall write x ≺ y to mean x � y and x 6= y. also, we write x � y if and only if y −x ∈ intp , where intp is the interior of p . if intp 6= ∅, the cone p is called solid. a cone p is called normal if there exists a number k > 0 such that, for all x,y ∈ e, we have θ � x � y =⇒‖x‖≤ k‖y‖. the least positive number satisfying the above inequality is called the normal constant of p . definition 1.2. let x be a nonempty set. suppose that a mapping d : x×x → e satisfies the following conditions: (d1) θ � d(x,y) for all x,y ∈ x and d(x,y) = θ if and only if x = y; (d2) d(x,y) = d(y,x) for all x,y ∈ x; (d3) d(x,z) � d(x,y) + d(y,z) for all x,y,z ∈ x. then d is called a cone metric [12] or k-metric [26] on x and (x,d) is called a cone metric space [12] or k-metric space [26]. the concept of k-metric space is more general than that of metric space, in fact each metric space is a k-metric space where x = r and p = [0, +∞). definition 1.3 ([10]). let (x,d) be a k-metric space, {xn} a sequence in x and x ∈ x. then (i) {xn} converges to x if, for every c ∈ e with θ � c there exists an n0 ∈ n such that d(xn,x) � c for all n > n0; (ii) {xn} is called a cauchy sequence if, for every c ∈ e with θ � c there exists an n0 ∈ n such that d(xn,xm) � c for all m,n > n0; (iii) a k-metric space x is said to be complete if every cauchy sequence in x is convergent in x. definition 1.4 ([10]). let (x,d) be a k-metric space, p a solid cone. a mapping t : x → x is said to be: (i) sequentially convergent if the sequence {xn} in x is convergent, whenever {txn} is convergent; (ii) subsequentially convergent if the sequence {xn} has a convergent subsequence, whenever {txn} is convergent; 134 h. rahimi, c. vetro, m. abbas, g. soleimani rad (iii) continuous if for any sequence {xn} in x with lim n→+∞ xn = x implies that lim n→+∞ txn = tx. theorem 1.5 ([19, 20]). let (x,d) be a complete k-metric space, p a solid cone and t : x → x a continuous and one to one mapping. moreover, f : x → x be a mapping satisfying d(tfx,tfy) � α1d(tx,ty) + α2[d(tx,tfx) + d(ty,tfy)] +α3[d(tx,tfy) + d(ty,tfx)], for all x,y ∈ x, where α1,α2,α3 ≥ 0 with α1 + 2α2 + 2α3 < 1. then (i) for each x0 ∈ x, {tfnx0} is a cauchy sequence, (define the iterate sequence {xn} by xn+1 = fn+1x0); (ii) there exists a zx0 ∈ x such that limn→+∞tfnx0 = zx0 ; (iii) if t is subsequentially convergent, then {fnx0} has a convergent subsequence; (iv) there exists a unique wx0 ∈ x such that fwx0 = wx0 ; that is, f has a unique fixed point; (v) if t is sequentially convergent, then, for each x0 ∈ x, the sequence {fnx0} converges to wx0 . definition 1.6 ([25]). an element (x,y,z) ∈ x ×x ×x is called a tripled fixed point of a mapping f : x × x × x → x if x = f(x,y,z), y = f(y,z,x) and z = f(z,x,y). 2. main results definition 2.1. let (x,d) be a k-metric space and t : x → x a mapping. a mapping f : x×x×x → x is said to be a t-contraction, if there exist α,β,γ ≥ 0, with α + β + γ < 1, such that for all x,y,z,x∗,y∗,z∗ ∈ x, we get (1) d(tf(x,y,z),tf(x∗,y∗,z∗)) � αd(tx,tx∗) + βd(ty,ty∗) + γd(tz,tz∗). theorem 2.2. let (x,d) be a complete k-metric space, p a solid cone and f : x × x × x → x a t-contraction, where t : x → x is a continuous and one to one mapping. then (i) {tfn(x0,y0,z0)}, {tfn(y0,z0,x0)} and {tfn(z0,x0,y0)} are cauchy sequences for all x0,y0,z0 ∈ x; (ii) there exist ux0,uy0,uz0 ∈ x such that lim n→+∞ tfn(x0,y0,z0) = ux0 , lim n→+∞ tfn(y0,z0,x0) = uy0, and lim n→+∞ tfn(z0,x0,y0) = uz0 ; (iii) if t is subsequentially convergent, then {tfn(x0,y0,z0)}, {tfn(y0,z0,x0)} and {tfn(z0,x0,y0)} have a convergent subsequence; (iv) there exist uniques wx0,wy0,wz0 ∈ x such that f(wx0,wy0,wz0 ) = wx0, f(wy0,wz0,wx0 ) = wy0, f(wz0,wx0,wy0 ) = wz0 ; that is, f has a unique tripled fixed point; tripled fixed point results for t -contractions 135 (v) if t is sequentially convergent, then, for all x0,y0,z0 ∈ x, the sequence {tfn(x0,y0,z0)} converges to wx0 ∈ x, the sequence {tfn(y0,z0,x0)} converges to wy0 ∈ x and the sequence {tfn(z0,x0,y0)} converges to wz0 ∈ x. proof. let define d : x3 ×x3 → p ; d((x1,y1,z1), (x2,y2,z2)) = d(x1,x2) + d(y1,y2) + d(z1,z2); f : x3 → x3 ; f(x,y,z) = (f(x,y,z),f(y,z,x),f(z,x,y)); t : x3 → x3 ; t(x,y,z) = (tx,ty,tz). then (x3,d) is a complete k-metric space, and t is continuous and one-to-one. it is clear that (x,y,z) ∈ x3 is a tripled fixed point of f if, and only if, it is a fixed point of f. suppose that fn(x,y,z) throughout the text (which is not properly defined) means exactly fn(x,y,z). let k = α + β + γ < 1. therefore, d(tf(x,y,z),tf(x∗,y∗,z∗)) = d(t(f(x,y,z),f(y,z,x),f(z,x,y)),t(f(x∗,y∗,z∗),f(y∗,z∗,x∗),f(z∗,x∗,y∗))) = d((tf(x,y,z),tf(y,z,x),tf(z,x,y)), (tf(x∗,y∗,z∗),tf(y∗,z∗,x∗),tf(z∗,x∗,y∗))) = d(tf(x,y,z),tf(x∗,y∗,z∗)) + d(tf(y,z,x),tf (y∗,z∗,x∗)) + d(tf(z,x,y),tf(z∗,x∗,y∗)) � [αd(tx,tx∗) + βd(ty,ty∗) + d(tz,tz∗)] + [αd(ty,ty∗) + βd(tz,tz∗) + γd(tx,tx∗)] + [αd(tz,tz∗) + βd(tx,tx∗) + d(ty,ty∗)] = (α + β + γ)d(tx,tx∗) + (α + β + γ)d(ty,ty∗) + (α + β + γ)d(tz,tz∗) = k(d(tx,tx∗) + d(ty,ty∗) + d(tz,tz∗)) = kd((tx,ty,tz), (tx∗,ty∗,tz∗)) = kd(t(x,y,z),t(x∗,y∗,z∗)). the proof further follows by theorem 1.5 (taking α1 = k < 1 and α2 = α3 = 0). this completes the proof. � corollary 2.3. let (x,d) be a complete k-metric space, p a solid cone, and t : x → x a continuous and one to one mapping. if f : x ×x ×x → x satisfies (2) d(tf(x,y,z),tf(x∗,y∗,z∗)) � k 3 [d(tx,tx∗) + d(ty,ty∗) + d(tz,tz∗)], for all x,y,z,x∗,y∗,z∗ ∈ x, where k ∈ [0, 1), then the conclusions of theorem 2.2 hold true. proof. the thesis follows easily from theorem 2.2, by putting α = β = γ = k/3 in (1). � corollary 2.4. let (x,d) be a complete k-metric space and p a solid cone. if f : x ×x ×x → x satisfies (3) d(f(x,y,z),f(x∗,y∗,z∗)) � αd(x,x∗) + βd(y,y∗) + γd(z,z∗), for all x,y,z,x∗,y∗,z∗ ∈ x, where α,β,γ ≥ 0 with α + β + γ < 1, then, f has a unique tripled fixed point. 136 h. rahimi, c. vetro, m. abbas, g. soleimani rad proof. the thesis follows easily from theorem 2.2, by putting t = ix, where ix is the identity mapping on x. � corollary 2.5. let (x,d) be a complete k-metric space and p a solid cone. if f : x ×x ×x → x satisfies (4) d(f(x,y,z),f(x∗,y∗,z∗)) � k 3 [d(x,x∗) + d(y,y∗) + d(z,z∗)], for all x,y,z,x∗,y∗,z∗ ∈ x, where k ∈ [0, 1), then f has a unique tripled fixed point. proof. result follows from corollary 2.3, taking t = ix . � example 2.6. let x = [0, 1] and e = c1r[0, 1] endowed with the order induced by p = {φ ∈ e : φ(t) ≥ 0 for t ∈ [0, 1]}. define d : x×x → e by d(x,y)(t) = |x−y|2t, for all x,y ∈ x. clearly, (x,d) is a complete k-metric space with a cone having nonempty interior. next, define the mappings f : x ×x ×x → x and t : x → x by tx = x 2 , for all x ∈ x and f(x,y,z) = x + y + z 6 , for all x,y,z ∈ x. then f satisfies the contractive condition (2) for k = 1/2; that is, d(tf(x,y,z),tf(u,v,w)) � 1 6 [d(tx,tu) + d(ty,tv) + d(tz,tw)], for all x,y,z,u,v,w ∈ x. consequently, corollary 2.3 applies to f, which has a unique tripled fixed point; that is (0, 0, 0). 3. applications let c([0,t],r) be the set of continuous functions defined in [0,t], where t > 0. consider the metric given by d(u,v) = sup t∈[0,t ] |u(t) −v(t)|, for all u,v ∈ r. note that (c([0,t],r),d) is a complete metric space. now, we study the existence and uniqueness of solution to an integral equation, by using corollary 2.5. precisely, we consider the equation (5) x(t) = ∫ t 0 k(t,s)(f(s,x(s)) + g(s,x(s)) + h(s,x(s))) ds + a(t), t ∈ [0,t]. now, we state and prove the following theorem. theorem 3.1. assume that the following conditions hold: (i) k ∈ c([0,t] × [0,t],r) such that sup s,t∈[0,t ] |k(t,s)| = m < 1 t ; (ii) a ∈ c([0,t],r); (iii) f,g,h ∈ c([0,t] ×r,r); (iv) for all xi,yi,zi ∈ c([0,t],r), where i = 1, 2, and t ∈ [0,t] we have |f(t,x1(t)) −f(t,x2(t))| + |g(t,y1(t)) −g(t,y2(t))| + |h(t,z1(t)) −h(t,z2(t))| ≤ 1 3 (|x1(t) −x2(t)| + |y1(t) −y2(t)| + |z1(t) −z2(t)|). tripled fixed point results for t -contractions 137 then, the integral equation (5) has a unique solution. proof. consider the mapping f : c([0,t],r)×c([0,t],r)×c([0,t],r) → c([0,t],r) defined by f(x,y,z)(t) = ∫ t 0 k(t,s)(f(s,x(s)) + g(s,y(s)) + h(s,z(s))) ds + a(t), t ∈ [0,t]. it is easy to show that (x,y,z) is a solution of (5) if and only if (x,y,z) is a tripled fixed point of f. to establish the existence of such a point, we will use corollary 2.5. in fact, by condition (iv), we have easily |f(x1,y1,z1)(t) −f(x2,y2,z2)(t)| ≤ ∫ t 0 |k(t,s)| 1 3 (|x1(s) −x2(s)| + |y1(s) −y2(s)| + |z1(s) −z2(s)|) ds ≤ 1 3 (∫ t 0 |k(t,s)| ds ) (d(x1,x2) + d(y1,y2) + d(z1,z2)), for all xi,yi,zi ∈ c([0,t],r), where i = 1, 2 and t ∈ [0,t]. by (i), it follows that d(f(x1,y1,z1),f(x2,y2,z2)) ≤ mt 3 (d(x1,x2) + d(y1,y2) + d(z1,z2)), for all xi,yi,zi ∈ c([0,t],r), where i = 1, 2. then, condition (4) of corollary 2.5 is satisfied with k = mt < 1 and hence, applying corollary 2.5, we obtain the existence of a unique tripled fixed point of f ; that is, the integral equation (5) has a unique solution. � references [1] m. abbas, g. jungck, common fixed point results for noncommuting mappings without continuity in cone metric spaces, j. math. anal. appl. 341 (2008) 416-420. [2] m. abbas, b.e. rhoades, fixed and periodic point results in cone metric spaces, appl. math. lett. 22 (2009) 511-515. [3] h. aydi, m. abbas, w. sintunavarat, p. kumam, tripled fixed point of w-compatible mappings in abstract metric spaces, fixed point theory appl. 2012, 2012:134. [4] h. aydi, e. karapinar, m. postolache, tripled coincidence point theorems for weak φcontractions in partially ordered metric spaces, fixed point theory appl. (in press). [5] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund. math. j. 3 (1922) 133-181. [6] a. beiranvand, s. moradi, m. omid, h. pazandeh, two fixed point theorems for special mappings, arxiv:0903.1504v1 math fa. (2009). [7] v. berinde, m. borcut, tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, nonlinear anal. 74 (2011) 4889-4897. [8] t. bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal. 65 (2006) 1379-1393. [9] l.b. ćirić, a generalization of banach contraction principle, proc. amer. math. soc. 45 (1974) 267-273. [10] m. filipović, l. paunović, s. radenović, m. rajović, remarks on “cone metric spaces and fixed point theorems of t-kannan and t-chatterjea contractive mappings”, math. comput. modelling 54 (2011) 1467-1472. [11] k. deimling, nonlinear functional analysis, springer-verlag, 1985. [12] l.g. huang, x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 332 (2007) 1467-1475. [13] g. jungck, commuting maps and fixed points, amer. math. monthly 83 (1976) 261-263. [14] v. lakshmikanthama, l. ćirić, coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, nonlinear anal. 70 (2009) 4341-4349. 138 h. rahimi, c. vetro, m. abbas, g. soleimani rad [15] s. moradi, kannan fixed-point theorem on complete metric spaces and generalized metric spaces depended an another function, int. j. math. anal. 5 (47) (2011) 2313-2320. [16] j.r. morales, e. rojas, cone metric spaces and fixed point theorems of t-kannan contractive mappings, int. j. math. anal. 4 (4) (2010) 175-184. [17] e.m. mukhamadiev, v.j. stetsenko, fixed point principle in generalized metric space, izvestija an tadzh. ssr, fiz.-mat.igeol.-chem.nauki. 10 (4) (1969) 8-19 (in russian). [18] a.i. perov, the cauchy problem for systems of ordinary differential equations, approximate methods of solving differential equations, kiev. nauk. dum. (1964) 115-134 (in russian). [19] h. rahimi, b.e. rhoades, s. radenović, g. soleimani rad, fixed and periodic point theorems for t-contractions on cone metric spaces, filomat 27 (5) (2013) 881-888. [20] h. rahimi, g. soleimani rad, fixed point theory in various spaces, lambert academic publishing (lap), germany, 2013. [21] h. rahimi, g. soleimani rad, new fixed and periodic point results on cone metric spaces, journal of linear and topological algebra 1 (1) (2012) 33-40. [22] s. rezapour, r. hamlbarani, some note on the paper cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 345 (2008) 719-724. [23] b.e. rhoades, a comparison of various definition of contractive mappings, trans. amer. math. soc. 266 (1977) 257-290. [24] f. sabetghadam, h.p. masiha, a.h. sanatpour, some coupled fixed point theorems in cone metric space, fixed point theory appl. 2009 (2009), article id 125426, 8 pages. [25] b. samet, c. vetro, coupled fixed point, f-invariant set and fixed point of n-order, ann. funct. anal. 1 (2) (2010) 46-56. [26] p.p. zabrejko, k-metric and k-normed linear spaces: survey, collect. math. 48 (1997) 825859. 1department of mathematics, faculty of science, central tehran branch, islamic azad university, p. o. box 13185/768, tehran, iran 2 dipartimento di matematica e informatica, università degli studi di palermo, via archirafi 34, 90123 palermo, italy 3 department of mathematics and applied mathematics, university of pretoria, lynnwood road, pretoria 0002, south africa 4department of mathematics, faculty of science, central tehran branch, islamic azad university, p. o. box 13185/768, tehran, iran ∗corresponding author international journal of analysis and applications volume 17, number 3 (2019), 464-478 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-464 fractional exponentially m-convex functions and inequalities saima rashid1,2,∗, muhammad aslam noor2 and khalida inyat noor2 1government college university, faisalabad, pakistan 2comsats university islamabad, islamabad, pakistan ∗corresponding author: saimarashid@gcuf.edu.pk abstract. in this article, we introduce a new class of convex functions involving m ∈ [0, 1], which is called exponentially m-convex function. some new hermite-hadamard inequalities for exponentially m-convex functions via reimann-liouville fractional integral are deduced. several special cases are discussed. results proved in this paper may stimulate further research in different areas of pure and applied sciences. 1. introduction convex functions and their variant forms are being used to study a wide class of problems which arises in various branches of pure and applied sciences. this theory provides us a natural, unified and general framework to study a wide class of unrelated problems. for recent applications, generalizations and other aspects of convex functions and their variant forms, see [3–13, 15, 18–21, 24–27, 29–31] and the references therein. an important class of convex functions, which is called exponential convex functions, was introduced and studied by antczak [2], dragomir et al [10] and noor et al [19]. alirezai and mathar [1] have investigated their basic properties along with their potential applications in statistics and information theory, received 2019-02-19; accepted 2019-03-25; published 2019-05-01. 2000 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. convex function; exponential convex function; reimann-liouville fractional integral inequalit; holder’s inequality and power-mean inequality. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 464 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-464 int. j. anal. appl. 17 (3) (2019) 465 see [1, 2, 14]. awan et al [3] and pecaric and jaksetic [26] defined another kind of exponential convex functions and have shown that the class of exponential convex functions unifies various unrelated concepts. it has been shown [17]that the minimum of the differentiable exponentially convex functions on the convex sets can be characterized by an inequality, which is called the exponentially variational inequality. exponentially variational inequalities can be viewed a natural generalization of the variational inequalities, see [32]. for the applications and numerical methods of variational inequalities, see noor [16]. toader [31] defined the m-convexity, an intermediate between the usual convexity and star shaped property. if m = 0, we have the concept of star shaped functions on [a,b]. we would like to emphasize that exponentially convex functions and m-convex functions are two distinct classes of convex functions. it is natural to introduce a new class of convex functions, which unifies these concepts. motivated by these facts, we introduce a new class of convex functions, which is called exponentially m-convex functions. the advantages of fractional calculus have been described and pointed out in the last few decades by many authors. fractional calculus is based on derivatives and integrals of fractional order, fractional differential equations and methods of their solution. the most celebrated inequality has been studied extensively since it is established, is the hermite-hadamard inequality not only established for classical integrals but also for fractional integrals, see [18, 20, 27, 29]. in this paper, we obtain some new hermite-hadamard type inequality for exponentially m-convex functions via riemann-liouville fractional integrals. some special cases are also discussed which can be obtained from results. the ideas and techniques of this paper may motivate further research in this field. 2. preliminaries first of all, we recall the following basic concepts. definition 2.1. [9, 31]. a set k ⊂ r is said to be a m-convex set with respect to a fixed constant m ∈ [0, 1], if (1 − t)a + mtb ∈ k, ∀a,b ∈ k,t ∈ [0, 1]. the m-convex set contains the line segment between points a and mb for every pair of points a and b of k. toader [31] defined the notion of m-convex functions as follows. int. j. anal. appl. 17 (3) (2019) 466 definition 2.2. [31]. a function f : k ⊂ r → r is said to be a m-convex function, where m ∈ [0, 1], if f((1 − t)a + mtb) ≤ (1 − t)f(a) + mtf(b), ∀a,b ∈ k,t ∈ [0, 1]. remark 2.1. clearly, a 1-convex function is a convex function in the ordinary sense. the 0-convex function are the starshaped functions. if we take m = 1, then we recapture the concept of convex functions and if we take t = 1, then f(mb) ≤ mf(b) ∀b ∈ k. this shows that the function f is sub-homogenous. we now consider class of exponentially convex function, which are mainly due to antczak [2], dragomir [8] and noor et al [20], respectively. definition 2.3. [2, 8, 20]. a function f : k ⊂ r −→ r is said to be exponentially convex function,if ef((1−t)a+tb) ≤ [(1 − t)ef(a) + tef(b)], a,b ∈ k, t ∈ [0, 1], (2.1) where f is positive. for the applications of exponentially convex functions in different field of statistics, information theory and mathematical sciences, see [1–3, 17] and the references therein. we would like to point out that u ∈ k is the minimum of the differentiable exponentially convex function f, if and only if, u ∈ k satisfies the inequality 〈f′(u)ef(u),v −u〉≥ 0, ∀v ∈ k, which is called the exponentially variational inequalities. for the more details, see noor and noor [17]. we now introduce a new concept of exponentially m-convex function. definition 2.4. let f : k ⊂ r −→ r is said to be an exponentially m-convex function, where m ∈ (0, 1], if ef((1−t)a+mtb) ≤ [(1 − t)ef(a) + mtef(b)], a,b ∈ k, t ∈ [0, 1]. (2.2) for t = 1 2 , we have ef( a+mb 2 ) ≤ ef(a) + mef(b) 2 , ∀a,b ∈ k. (2.3) the function f is called exponentially jensen m-convex function. we now give the definition of the fractional integral, which is mainly due to [27]. int. j. anal. appl. 17 (3) (2019) 467 definition 2.5. [27]. let α > 0 with n− 1 < α ≤ n, n ∈ n, and 1 < x < v. the leftand right-hand side riemann-liouville fractional integrals of order α of function f are given by jαu+f(x) = 1 γ(α) x∫ u (x− t)α−1f(t)dt, and jαv−f(x) = 1 γ(α) v∫ x (t−x)α−1f(t)dt, where γ(α) is the gamma function. we also made the convention j0 u+ = j0 v− = f(x). we recall the special functions which are known as gamma function, γ(x) = ∞∫ 0 e−ttx−1dt. for appropriate and suitable choice of m, one can obtain several new and known classes of exponentially convex functions as special cases. this shows that the concept of exponentially m-convex function is quite general and unifying one. 3. main results “in this section, we obtain hermite-hadamard type inequalities for exponentially m-convex function via reimann-liouville fractional integral. throughout this section, let i = [a,mb] be an interval in real line. from now onward, we take i = [a,mb], unless otherwise specified.” theorem 3.1. let f : i ⊂ r → r be an exponentially convex function, where m ∈ (0, 1]. if f ∈ l[a,mb], then “ ef( a+mb 2 ) ≤ γ(α + 1) 2(mb−a)α { jα(a)+e f(mb) + mα+1jα(b)−e f( a m ) } ≤ α[ef(a) + m2ef( b m )] + [mef(b) + mef( a m )] α(α + 1) . (3.1) ” int. j. anal. appl. 17 (3) (2019) 468 proof. “let f be an exponentially m-convex function, from the inequality (2.2). then, we have” “ ef( x+my 2 ) ≤ ef(x) + mef(y) 2 , ∀x,y ∈ [a,mb]. ” “substituting x = at + m(1 − t)b and y = (1 − t) a m + mt b m for t ∈ [0, 1]. then” “ 2ef( a+mb 2 ) ≤ [ef(at+m(1−t)b) + mef((1−t) a m +mt b m )]. ” “multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “ 2 α ef( a+mb 2 ) ≤ 1∫ 0 tα−1[ef(at+m(1−t)b) + mef((1−t) a m +mt b m )]dt = 1 (mb−a)α { mb∫ a (mb−u)α−1ef(u)du + mα+1 b∫ a m (v − a m )α−1ef(v)dv } = γ(α) (mb−a)α { jα(a)+e f(mb) + mα+1jα(b)−e f( a m ) } , ” from which, one has “ ef( a+mb 2 ) ≤ γ(α + 1) 2(mb−a)α { jα(a)+e f(mb) + mα+1jα(b)−e f( a m ) } . (3.2) ” on the other hand exponentially m-convexity of f gives “ ef(at+m(1−t)b) + mef((1−t) a m +mt b m ) ≤ t[ef(a) + m2ef( b m ) + (1 − t)[mef(b) + mef( a m )]. ”. multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have “ 1∫ 0 tα−1ef(at+m(1−t)b)dt + m 1∫ 0 tα−1ef((1−t) a m +mt b m )dt ≤ 1∫ 0 tα[ef(a) + m2ef( b m )]dt + 1∫ 0 (tα−1 − tα)[mef(b) + mef( a m )]dt. ” “ γ(α) (mb−a)α { jα(a)+e f(mb) + mα+1jα(b)−e f( a m ) } ≤ ef(a) + m2ef( b m ) α + 1 + mef(b) + mef( a m ) α(α + 1) , ” from which one has “ γ(α + 1) 2(mb−a)α { jα(a)+e f(mb) + mα+1jα(b)−e f( a m ) } ≤ α[ef(a) + m2ef( b m )] + [mef(b) + mef( a m )] α(α + 1) . (3.3) ” “combining inequality (3.2) and inequality (3.3), we get (3.4).” � int. j. anal. appl. 17 (3) (2019) 469 “ corollary 3.1. if we choose m = 1 in theorem 3.1, then we have a new result ef( a+b 2 ) ≤ γ(α + 1) 2(b−a)α { jα(a)+e f(b) + jα(b)−e f(a) } ≤ [ef(a) + ef(b)] α . ” “ corollary 3.2. if we choose m = 1 and α = 1 in theorem 3.1, then we have a new result 2ef( a+b 2 ) ≤ 1 b−a b∫ a ef(x)dx ≤ 2[ef(a) + ef(b)]. ” “ theorem 3.2. let f,g : i ⊂ r → r be an exponentially m-convex function, where m ∈ (0, 1]. if fg ∈ l[a,mb], then γ(α + 1) (mb−a)α { jαmb−e f(a) + jαa+e g(mb) } ≤ ef(a) + meg(b) + α(mef(b) + eg(a)) α(α + 1) . ” proof. let f,g : i ⊂ r → r be an exponentially m-convex function. then “ ef(a(1−t)+mtb) ≤ (1 − t)ef(a) + tmef(b), a,b ∈ [a,mb], t ∈ [0, 1], eg(at+m(1−t)b) ≤ teg(a) + m(1 − t)eg(b), a,b ∈ [a,mb], t ∈ [0, 1]. ” adding both sides of the above inequalities, we have “ ef(a(1−t)+mtb) + eg(at+m(1−t)b) ≤ (1 − t)[ef(a) + meg(b)] + t[mef(b) + eg(a)]. ” “multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “ 1∫ 0 tα−1[ef(a(1−t)+mtb) + eg(at+m(1−t)b)]dt ≤ 1∫ 0 (tα−1 − tα)[ef(a) + meg(b)]dt + 1∫ 0 tα[mef(b) + eg(a)]dt. ” “ γ(α) (mb−a)α { mb∫ a (u−a)α−1ef(u)du + mb∫ a (mb−v)α−1eg(v)dv } ≤ ef(a) + meg(b) α(α + 1) + α(mef(b) + eg(a)) α(α + 1) , int. j. anal. appl. 17 (3) (2019) 470 ” from which, we have “ γ(α + 1) (mb−a)α { jαmb−e f(a) + jαa+e g(mb) } ≤ ef(a) + meg(b) + α(mef(b) + eg(a)) α(α + 1) , ” which is the required result. � “ corollary 3.3. if we choose m = 1 in theorem 3.2, then we have a new result γ(α + 1) (b−a)α { jαb−e f(a) + jαa+e g(b) } ≤ ef(a) + eg(b) + α(ef(b) + eg(a)) α(α + 1) . ” “ corollary 3.4. if we choose m = 1 and α = 1 in theorem 3.2, then we have a new result 1 b−a b∫ a [eg(x) + ef(x)]dx ≤ ef(a) + ef(b) + eg(a) + eg(b) 2 . ” “we now derive hermite-hadamard type inequalities for m-convex functions using reimann-liouville fractional integral.” theorem 3.3. “let α > 0 and f : i ⊂ r → r be an exponentially convex function, where m ∈ (0, 1]. if f ∈ l[a,mb], then” “ ef( a+mb 2 ) ≤ 2α−1γ(α + 1) (mb−a)α [ jα ( a+mb 2 )+ ef(mb) + mα+1jα ( a+mb 2m )− ef( a m ) ] ≤ α 4(α + 1) [ ef(a) −m2ef( a m2 ) ] + m 2 [ ef(b) + me f( a m2 ) ] . ” proof. “let f be an exponentially m-convex function. then, from the inequality (2.3), we have” “ ef( x+my 2 ) ≤ ef(x) + mef(y) 2 , x,y ∈ i. ” “substituting x = t 2 a + m2−t 2 b, y = 2−t 2m a + t 2 b for t ∈ [0, 1]. then” “ 2ef( a+mb 2 ) ≤ ef( t 2 a+m 2−t 2 b) + mef( 2−t 2m a+ t 2 b). int. j. anal. appl. 17 (3) (2019) 471 ” “multiplying both sides of the above inequality with tα−1, and integrating over [0, 1], we have” “ 2 α ef( a+mb 2 ) ≤ 1∫ 0 tα−1ef( t 2 a+m 2−t 2 b)dt + m 1∫ 0 tα−1ef( 2−t 2m a+ t 2 b)dt = 2 a−mb a+mb 2∫ mb ( 2(mb−u) mb−a )α−1 ef(u)du + 2m2 mb−a a+mb 2m∫ a m ( 2(v − a m ) b− a m )α−1 ef(v)dv = 2αγ(α) (mb−a)α [ jα ( a+mb 2 )+ ef(mb) + mα+1jα ( a+mb 2m )− ef( a m ) ] . ” thus “ ef( a+mb 2 ) ≤ 2α−1γ(α + 1) (mb−a)α [ jα ( a+mb 2 )+ ef(mb) + mα+1jα ( a+mb 2m )− ef( a m ) ] = α 2 1∫ 0 tα−1 { ef( t 2 a+m 2−t 2 b) + mef( 2−t 2m a+ t 2 b) } dt ≤ α 2 1∫ 0 tα−1 {[ t 2 ef(a) + m(2 − t) 2 ef(b) ] + m [ m 2 − t 2 e f( a m2 ) + t 2 ef(b) ]} dt = α 4 [ ef(a) −m2ef( a m2 ) ] 1∫ 0 tαdt + mα 2 [ ef(b) + me f( a m2 ) 1∫ 0 tα−1dt ] = α 4(α + 1) [ ef(a) −m2ef( a m2 ) ] + m 2 [ ef(b) + me f( a m2 ) ] , ” which is the required result. � lemma 3.1. “ let α > 0 and f : i ⊂ r → r be a differentiable exponentially m-convex function on the interior i◦ of i, where m ∈ (0, 1]. if |f′| ∈ l[[a,mb], is a m-convex function, then” “ 2α−1γ(α + 1) (mb−a)α [ jα ( a+mb 2 )+ ef(mb) + mα+1jα ( a+mb 2m )− ef( a m ) ] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ] = mb−a 4 [ 1∫ 0 tαef( t 2 a+m 2−t 2 b)f′( t 2 a + m 2 − t 2 b)dt − 1∫ 0 tαef( 2−t 2m a+ t 2 b)f′( 2 − t 2m a + t 2 b)dt ] . (3.4) ” int. j. anal. appl. 17 (3) (2019) 472 proof. it suffices to note that “ mb−a 4 [ 1∫ 0 tαef( t 2 a+m 2−t 2 b)f′( t 2 a + m 2 − t 2 b)dt ] = mb−a 4 [ 2 mb−a ef( a+mb 2 ) − 2α (a−mb) a+mb 2∫ mb ( 2(mb−x) mb−a )α−1 2ef(x)dx a−mb ] = mb−a 4 [ − 2 mb−a ef( a+mb 2 ) + 2α+1γ(α + 1) (mb−a)α+1 jα ( a+mb 2 )− ef(mb) ] . (3.5) ” similarly, one can have “ − mb−a 4 [ 1∫ 0 tαef( 2−t 2m a+ t 2 b)f′( 2 − t 2m a + t 2 b)dt ] = − mb−a 4 [ 2m mb−a ef( a+mb 2m ) − 2α+1γ(α + 1) (mb−a)α+1 jα ( a+mb 2m )+ ef( a m ) ] . (3.6) ” adding (3.5) and (3.6), gives (3.4). � theorem 3.4. “let α > 0 and f : i ⊂ r → r be a differentiable exponentially m-convex function on the interior i◦ of i, where m ∈ (0, 1]. if |f′| ∈ l[[a,mb], is a m-convex function, then” “ ∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4 { 1 4(α + 3) {∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3){∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a m2 ) ∣∣} + m(α + 4) ((α + 2)(α + 3)) { ∆1(a,b) + ∆2( a m2 ,b) }} , ” where “ ∆1(a,b) = ∣∣ef(a)f′(b)∣∣ + ∣∣ef(b)f′(a)∣∣, and ∆2( a m2 ,b) = ∣∣ef( am2 )f′(b)∣∣ + ∣∣ef(b)f′( a m2 ) ∣∣. ” int. j. anal. appl. 17 (3) (2019) 473 proof. “using lemma 3.1and exponentially m-convexity function of f, we have” “∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4 [ 1∫ 0 tα {∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b) ∣∣∣∣ + ∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b) ∣∣∣∣ } dt ] . (3.7) ” using m-convexity of f, we have “∣∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b) ∣∣∣∣∣ + ∣∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b) ∣∣∣∣∣ ≤ {[ t 2 |ef(a)| + m(2 − t) 2 |ef(b)| ][ t 2 |f′(a)| + m(2 − t) 2 |f′(b)| ]} {[ m 2 − t 2 |ef( a m2 )| + t 2 |ef(b)| ][m(2 − t) 2 |f′ a m2 | + t 2 |f′(b)| ]} = t2 4 ∣∣ef(a)f′(a)∣∣ + m2(2 − t)2 4 ∣∣ef(b)f′(b)∣∣ + m(2 − t)t 4 [∣∣ef(a)f′(b)∣∣ + ∣∣ef(b)f′(a)∣∣] + m2(2 − t)2 4 ∣∣ef( am2 )f′( a m2 ) ∣∣ + t2 4 ∣∣ef(b)f′(b)∣∣ + m(2 − t)t 4 [∣∣ef( am2 )f′(b)∣∣ + ∣∣ef(b)f′( a m2 ) ∣∣] = t2 4 {∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(2 − t)2 4 {∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a m2 ) ∣∣} + m(2 − t)t 4 { ∆1(a,b) + ∆2( a m2 ,b) } . (3.8) ” thus “ ∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4 [ 1∫ 0 tα { t2 4 {∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(2 − t)2 4 {∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a m2 ) ∣∣} + m(2 − t)t 4 { ∆1(a,b) + ∆2( a m2 ,b) }} dt ] = mb−a 4 { 1 4(α + 3) {∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3){∣∣ef(b)f′(b)∣∣ + ∣∣ef( am2 )f′( a m2 ) ∣∣} + m(α + 4) ((α + 2)(α + 3)) { ∆1(a,b) + ∆2( a m2 ,b) }} , ” which is the required result. � int. j. anal. appl. 17 (3) (2019) 474 corollary 3.5. [21]. “if we choose m = 1 and α = 1, in theorem 3.4, then” “ ∣∣∣∣ef( a+b2 ) − 1b−a b∫ a ef(x)dx ∣∣∣∣ ≤ b−a 4 [ 7{ ∣∣ef(a)f′(a)∣∣ + ∣∣ef(b)f′(b)∣∣} + 10[∆1(a,b) + ∆2(a,b)] 24 ] . ” theorem 3.5. “let f : i ⊂ r → r be differentiable exponentially m-convex function on the interior i◦ of i, where m ∈ (0, 1]. if |f′| ∈ l[[a,mb], is a m-convex function on i and p−1 + q−1 = 1, where q > 1, then we have” “ ∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4(α + 1) 1 p [{ 1 4(α + 3) |ef(a)f′(a)|q + m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3) |ef(b)f′(b)|q + m(α + 4) (α + 2)(α + 3) ∆3(a,b) }1 q + { m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3) |ef( a m2 ) f′( a m2 )|q + 1 4(α + 3) |ef(b)f′(b)|q + m(α + 4) (α + 2)(α + 3) ∆4( a m2 ,b) }1 q ] , ” “where ∆3(a,b) = |ef(a)f′(a)|q + |ef(b)f′(b)|q and ∆4( a m2 ,b) = |ef( a m2 ) f′( a m2 )|q + |ef(b)f′(b)|q. ” proof. using lemma 3.1 and the power mean inequality, we have “ ∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ int. j. anal. appl. 17 (3) (2019) 475 ” “ ≤ mb−a 4 [ 1∫ 0 tα {∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b) ∣∣∣∣ + ∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b) ∣∣∣∣ } dt ] = mb−a 4 ( 1 (α + 1) )1 p [{ 1∫ 0 tα {∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b) ∣∣∣∣qdt }1 q + { 1∫ 0 tα ∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b) ∣∣∣∣qdt }1 q ] = mb−a 4 ( 1 (α + 1) )1 p [{ 1∫ 0 tα [t2 4 |ef(a)f′(a)|q + m2(2 − t)2 4 |ef(b)f′(b)|q + mt(2 − t) 4 ∆3(a,b) ]}1q + { 1∫ 0 tα [t2 4 |ef(b)f′(b)|q + m2(2 − t)2 4 |ef( a m2 ) f′( a m2 )|q + mt(2 − t) 4 ∆4( a m2 ,b) ]}1q ] = mb−a 4(α + 1) 1 p [{ 1 4(α + 3) |ef(a)f′(a)|q + m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3) |ef(b)f′(b)|q + m(α + 4) (α + 2)(α + 3) ∆3(a,b) }1 q + { m2(α2 + 7α + 14) 4(α + 1)(α + 2)(α + 3) |ef( a m2 ) f′( a m2 )|q + 1 4(α + 3) |ef(b)f′(b)|q + m(α + 4) (α + 2)(α + 3) ∆4( a m2 ,b) }1 q ] , ” which completes the proof. � corollary 3.6. [21]. if we choose m = 1 and α = 1, in theorem 3.5, then ∣∣∣∣ef( a+b2 ) − 1b−a b∫ a ef(x)dx ∣∣∣∣ ≤ b−a 4(2) 1 p [{ 3|ef(a)f′(a)|q + 11|ef(b)f′(b)|q + 20∆3(a,b) 48 }1 q + { 11|ef(a)f′(a)|q + 3|ef(b)f′(b)|q + 20∆4(a,b) 48 }1 q ] , theorem 3.6. let f : i ⊂ r → r be differentiable exponentially m-convex function on the interior i◦ of i, where m ∈ (0, 1]. if |f′| ∈ l[[a,mb], is a m-convex function on i and p−1 + q−1 = 1, where q ≥ 1, then we int. j. anal. appl. 17 (3) (2019) 476 have “ ∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4(pα + 1) 1 p [{ |ef(a)f′(a)|q + 7m2|ef(b)f′(b)|q + 2m∆3(a,b) 12 }1 q + { 7m2|ef( a m2 ) f′( a m2 )|q + |ef(b)f′(b)|q + 2m∆4( am2 ,b) 12 }1 q ] . ” proof. using lemma3.1 and the holder’s inequality, we have “∣∣∣∣2α−1γ(α + 1)(mb−a)α [jα( a+mb2 )+ef(mb) + mα+1jα( a+mb2m )−ef( am )] − 1 2 [ ef( a+mb 2 ) + mef(( a+mb 2m ) ]∣∣∣∣ ≤ mb−a 4 [( 1∫ 0 tpαdt )1 p { 1∫ 0 ∣∣∣∣ef( t2 a+m 2−t2 b)f′( t2a + m2 − t2 b) ∣∣∣∣qdt }1 q + ( 1∫ 0 tpαdt )1 p { 1∫ 0 ∣∣∣∣ef( 2−t2m a+ t2 b)f′( 2 − t2m a + t2b) ∣∣∣∣qdt }1 q ] ” “ ≤ mb−a 4(pα + 1) 1 p [{ 1∫ 0 [t2 4 |ef(a)f′(a)|q + m2(2 − t)2 4 |ef(b)f′(b)|q + mt(2 − t) 4 ∆3(a,b) ] dt }1 q + {[m2(2 − t)2 4 |ef( a m2 ) f′( a m2 )|q + t2 4 |ef(b)f′(b)|q + mt(2 − t) 4 ∆4( a m2 ,b) ]}1q ] = mb−a 4(pα + 1) 1 p [{ |ef(a)f′(a)|q + 7m2|ef(b)f′(b)|q + 2m∆3(a,b) 12 }1 q + { 7m2|ef( a m2 ) f′( a m2 )|q + |ef(b)f′(b)|q + 2m∆4( am2 ,b) 12 }1 q ] , ” which completes the proof. � corollary 3.7. [21]. if we choose m = 1 and α = 1, in theorem 3.6, then “ ∣∣∣∣ef( a+b2 ) − 1b−a b∫ a e f(x) dx ∣∣∣∣ ≤ b−a 4(p + 1) 1 p [{ |ef(a)f′(a)|q + 7|ef(b)f′(b)|q + 2∆3(a, b) 12 }1 q + { 7|ef(a)f′(a)|q + |ef(b)f′(b)|q + 2∆4(a, b) 12 }1 q ] . ” int. j. anal. appl. 17 (3) (2019) 477 4. conclusions in this paper, we have introduced and studied a new class of exponentially convex functions involving the parameter m. we have obtained several new hermite-hadamard inequalities via riemann-liouville fractional integrals. it is shown that previously known results can be obtained as special cases from our results. it is shown that the class of exponentially m-convex functions is quite general, flexible and unifying one. acknowledgments: the authors would like to thank the rector, comsats university islamabad, islamaabd, pakistan for providing excellent research and academic environments. references [1] g. alirezaei and r. mathar, on exponentially concave functions and their impact in information theory, information theory and applications workshop, san diego, california, usa, 2018. [2] t. antczak, (p,r)-invex sets and functions, j. math. anal. appl. 263(2001), 355-379. [3] m. u. awan, m. a. noor and k. i. noor, hermite-hadamard inequalities for exponentiaaly convex functions, appl. math. inform. sci. 2(12)(2018), 405-409. [4] m. k. bakula, m. e. ozdemir and j. pecaric, hadamard type inequalities for m-convex and (α; m)-convex functions, j. inequal. pure appl. math. 9(4)(2008), art. id 96. [5] i. a. baloch and i. iscan, some hermite-hadamard type inequalities for harmonically (s; m)-convex functions in second sense, arxiv:1604.08445v1 [math.ca], 2016. [6] m. k. bacul, j. pecaric and m. ribicic, companion inequalities to jensen’s inequality for m-convex and (α,m) convex functions, j. inequal. pure. appl. math. 7(5)(2006), art. id 194. [7] m. braccamonte, j. gimenez, n. merentes and m. vivas, fejer type inequalities for m-convex functions, publicaciones en ciencias y tecnoloǵıa, 10(1)(2016), 7-11. [8] s. s. dragomir and i. gomm, some hermite-hadamard type inequalities for functions whose exponentials are convex, stud. univ. babes-bolyai math. 60(4)(2015), 527-534. [9] s. s. dragomir and g. toader, some inequalities for m-convex functions, stud. univ. babes-bolyai math. 38 (1993), 21-28. [10] s. s. dragomir, on some new inequalities of hermite-hadamard type for m convex functions, tamkang j. math. 33(1)(2002), 45-55. [11] l. fejéer, uber die fourierreihen, ii, math naturwise. anz ungar. akad. wiss. (24)(1906), 369-390. [12] j. hadamard, etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann. j. math. pure appl. (58)(1893), 171-215. [13] c. y. he, y. wang, b. y.xi and f. qi, hermite-hadamard type inequalities for (α; m)-ha and strongly (α; m)-ha convex functions, j. nonlinear sci. appl. (10)(2017), 205-214. [14] m. mahdavi, exploiting smoothness in statistical learning, sequential prediction, and stochastic optimization. east lansing, mi, usa: michigan state university, (2014). [15] c. p. niculescu and l. e. persson, convex functions and their applications. springer-verlag, new york, (2018). [16] m. a. noor, some developments in general variational inequalities, appl. math. comput. 152(2004), 199-277. [17] m. a. noor and k. i. noor, exponentially convex functions, preprint. int. j. anal. appl. 17 (3) (2019) 478 [18] m. a. noor, k. i. noor and m. u. awan, fractional hermite-hadamard inequalities for convex functions and applications, tbilisi j. math. 8(2)(2015), 103-113. [19] m. a. noor, k. i. noor, and s. rashid, exponential r-convex functions and inequalities, preprint. [20] m. a. noor, k. i. noor and s. rashid, fractal exponential convex functions and inequalities, preprint. [21] s. rashid, m. a. noor and k. i. noor, modified exponential convex functions and inequalities, open acess j. math. theor. phy. 2(2)(2019), 45-51. [22] m. e. ozdemir, m. avci. and h. kavurmaci, hermite-hadamard type inequalities via (α,m)-convexity, j. comput. math. appl. 61(2011), 2614-2620. [23] s. pal and t. k. l. wong, exponentially concave functions and a new information geometry, ann. probab. 46(2)(2018), 1070-1113. [24] j. park, new ostrowski-like type inequalities for differentable (s,m)-convex mappings, int. j. pure appl. math. 78 (8) (2012), 1077-1089. [25] j. e. pecaric, f. proschan and y. l. tong, convex functions, partial orderings and statistical applications, academics press, new york, (1992). [26] j. pecaric and j. jaksetic, exponential onvexity, euler-radau expansions and stolarsky means, rad hrvat. matematicke znanosti, 515(2013), 81-94 [27] i. podlubny, fractional differential equations: mathematics in science and engineering, academic press, san diego, (1999). [28] m. rostamian, s. s. dragomir and m. d. l. sen, estimation type results related to fejér inequality with applications, j. inequal. appl, 2018 (2018), art. id 85. [29] e. set, a. o. akdemir and i. mumcu, the hermite-hadamard type inequality and its extensions for conformable fractional integrals of any order α > 0, preprint. [30] g. toader, some generalizations of the convexity, proc. colloq. approx. opt. cluj-napoca (romania), university of cluj-napoca, 1984, 329-338. [31] g. toader, the order of starlike convex function, bull. appl. comp. math. 85(1998), 347-350. [32] g. stampacchia, formes bilineaires coercivvities sur les ensembles convexes, c. r. acad. sci. paris, 258(1964), 4413-4416. 1. introduction 2. preliminaries 3. main results 4. conclusions references international journal of analysis and applications volume 17, number 2 (2019), 303-310 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-303 approximate solution of fractional integro-differential equations by least squares method d. jabari sabeg1,∗, r. ezzati2 and k. maleknejad2 1department of mathematics, bonab branch, islamic azad university, bonab, iran 2department of mathematics, karaj branch, islamic azad university, karaj, iran ∗corresponding author: davood.jabari@bonabiau.ac.ir abstract. in this paper, least squares approximation method is developed for solving a class of linear fractional integro-differential equations comprising volterra and fredhlom cases. this method is based on a polynomial of degree n to compute an approximate solution of these equations. the convergence analysis of the proposed method is proved. in addition, to show the accuracy and the efficiency of the proposed method, some examples are presented. 1. introduction fractional calculus is a significant branch of mathematics that is used in many fields of science and engineering [2–4]. many researchers have investigated the analytic results on the existence and uniqueness of solutions of the fractional differential equations [5–8]. as we know, for most fractional differential equations, there are not method to obtain analytic solutions, so numerical techniques must be used. during the past years, methods for solving fractional differential equations are developed. additionally, some methods have recently been emerged, such as the adomian decomposition method [10, 11], the operational matrix [12, 13], the collocation method [14, 16], etc. received 2018-10-06; accepted 2018-11-27; published 2019-03-01. 2010 mathematics subject classification. 26a33, 45a05. key words and phrases. fractional differential equation; least squares; approximation method; convergence analysis. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 303 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-303 int. j. anal. appl. 17 (2) (2019) 304 in this paper, by using least squares approximation, a numerical method has been shown to linear fractional integro-differential equations in the following form dαy(t) = p(t)y(t) + f(t) + λ1 ∫ 1 0 k1(t,x)y(x) dx + λ2 ∫ t 0 k2(t,x)y(x) dx t ∈ i = [0, 1] (1.1) with the initial conditions y(k)(0) = dk i = 0, ..,m− 1, m− 1 < α ≤ m, (1.2) where yk(t) stands for the kth-order derivative of y(t) and dα denotes the riemann-liouville fractional derivative of order α. clearly, when λ1 = 0, λ2 = 0, the above equation reduces to a linear fractional differential equation. the rest of the paper is organized as follows: in section 2, we will briefly review some notations and definitions of the fractional calculus theory are used in the paper. in section 3, we introduce the least squares approximation method for solving eq. (1.1), and discuss its convergence. in section 4, we show the efficiency of the proposed method with some numerical examples. section 5, as the final section, presents a conclusion. 2. brief review of fractional calculus in this section, notations and definitions of the fractional calculus theory, which are going to be used in this paper, are presented [1]. 2.1. definition. the riemann-liouville fractional integral operator iα of order α ≥ 0 of a function f(x), is defined as iαf(x) = 1 γ(α) ∫ x 0 (x− t)(α−1)f(t) dt ,α > 0, (2.1) where x > 0 and γ(.) is the euler gamma function. the riemann-liouville fractional derivative of order α will be denoted by dα and defined by dαf(x) = dm dxm (im−αf(x)), (2.2) where m − 1 < α ≤ m,m ∈ n and m is the smallest integer order greater than α. we just mention the following property dαxβ = γ(β + 1) γ(β + 1 −α) xβ−α,β > −1. (2.3) int. j. anal. appl. 17 (2) (2019) 305 3. method of solution in this section, we apply the least squares approximation method for solving eq. (1.1). we define the following operator t(t,y(t)) = dαy(t) −p(t)y(t) −f(t) −λ1 ∫ 1 0 k1(t,x)y(x) dx−λ2 ∫ t 0 k2(t,x)y(x) dx. (3.1) we construct taylor-series expansion for the solution y(t) in eq.(1.1) as y(t) ' yn(t) = n∑ r=0 y(r)(0) r! tr = n∑ r=0 dr r! tr. (3.2) substituting (3.2) into ( 3.1), we have t(t,yn(t)) = d αyn(t) −p(t)yn(t) −f(t) −λ1 ∫ 1 0 k1(t,x)yn(x) dx−λ2 ∫ t 0 k2(t,x)yn(x) dx = dα( n∑ r=0 dr r! tr) −p(t) n∑ r=0 dr r! tr −f(t) − n∑ r=0 dr r! λ1 ∫ 1 0 k1(t,x)x r dx− n∑ r=0 dr r! λ2 ∫ t 0 k2(t,x)x r dx = n∑ r=0 drγ(r + 1) r!γ(r −α + 1) tr−α −p(t) n∑ r=0 dr r! tr −f(t) − n∑ r=0 dr r! λ1 ∫ 1 0 k1(t,x)x r dx− n∑ r=0 dr r! λ2 ∫ t 0 k2(t,x)x r dx = n∑ r=0 drγr(t) −f(t), where γr(t) = γ(r + 1) r!γ(r −α + 1) tr−α −p(t) tr r! − λ1 r! ∫ 1 0 k1(t,x)x r dx− λ2 r! ∫ t 0 k2(t,x)x r dx. let rn(t) := t(t,yn(t)) −t(t,y(t)), t ∈ [0, 1]. remark 3.1. if rn(t) = 0, then y(t) = yn(t); if limn→∞rn(t) = 0, then limn→∞yn(t) = y(t). remark 3.2. for any t ∈ [0, 1], if rn(t) ≡ 0, then yn(t) is an exact solution of eqs. (1.1) and (1.2); if limn→∞rn(t) = 0, then yn(t) converges to the exact solution of eqs. (1.1) and (1.2). let j = j(dm,dm+1, ...,dn) = ∫ 1 0 t2(t,yn(t))dt. (3.3) the problem is to find real constants dm,dm+1, ...,dn such that these constants will minimize j. a necessary condition for the constants dm,dm+1, ...,dn to minimize j is that int. j. anal. appl. 17 (2) (2019) 306 ∂j ∂dj = 0, for each j = m,m + 1, ...,n. by referring (3.3), we get ∂j ∂dj = 2[ n∑ r=0 dr ∫ 1 0 γr(t)γj(t)dt− ∫ 1 0 f(t)γj(t)dt] = 0. (3.4) thus, we have n∑ r=m dr ∫ 1 0 γr(t)γj(t)dt = ∫ 1 0 f(t)γj(t)dt−βj, (3.5) where βj = m−1∑ r=0 dr ∫ 1 0 γr(t)γj(t)dt (3.6) for each j = m,m + 1, ...,n. in order to find yn(t), we have to solve (n−m) a system of linear equations while assuming (n−m) unknowns dr. the system (3.5) can be written in the form: gd = f (3.7) where g =   (γm,γm) (γm,γm+1) . . . (γm,γn) (γm+1,γm) (γm+1,γm+1) . . . (γm+1,γn) ... ... . . . ... (γn,γm) (γn,γm+1) . . . (γn,γn)   , (3.8) d = [dm,dm+1, ...,dn] t , and f = [(γm,f) −βm, (γm+1,f) −βm+1, ..., (γn,f) −βn]t definition 3.3. if eq. (3.7) has a unique solution d, then yn(t) = ∑n r=1 dr r! tr is called an optimal squared approximation solution of eqs. (1.1)-(1.2) defined on a set as span{1, t, t2, ...tn}, t ∈ [0, 1]. remark 3.4. if limn→∞ ∫ 1 0 t2(t,yn(t))dt = 0, then the optimal squared approximation solution yn(t) converges to the exact solution y(t) of eqs. (1.1) and (1.2). we are interested to know that as n → ∞ the optimal squared approximation solution yn(t) will converge to the exact solution y(t) of eqs. (1.1) and (1.2). this conception is proven in theorem 3.5. int. j. anal. appl. 17 (2) (2019) 307 theorem 3.5. suppose y(t), t ∈ [0, 1] is an exact solution and yn(t) is an optimal squared approximation solution of eqs. (1.1) and (1.2). if ∃pn(t) = n∑ r=1 drt r such that ∀t ∈ [0, 1], limn→∞pn(t) = y(t) then lim n→∞ ∫ 1 0 t2(t,yn(t))dt = 0. proof. the proof is similar to proof of theorem 3 in [19]. 4. illustrative examples in this section, we use the presented method in section 3 for solving two examples. example 4.1. for first example, consider the fractional integro-differential equation d0.75y(t) + 1 5 t2ety(t) − ∫ t 0 xety(x) dx = 6t2.25 γ(3.25) , y(0) = 0, where the exact solution is given by y(t) = t3. we applied the presented method with n = 3 for solving this example and achieved the corresponding absolute errors in table 1. table 1: absolute errors for example 1 for n = 3. t = 0 t = 0.2 t = 0.4 t = 0.6 t = 0.8 t = 1.0 0 1.314×10−15 1.045×10−15 1.914×10−16 2.475×10−16 6.717×10−16 example 4.2. consider the equation dαy(t) + 7t2 12 y(t) − ∫ 1 0 txy(x) dx− ∫ t 0 (x + t)y(x) dx = 2t2−α γ(3 −α) − t 4 , y(0) = 0, with the exact solution y(t) = t2. by the presented method in section 3 for n = 2 and different values of α absolute errors are reported in table 2. example 4.3. consider the equation [20] dαy(t) + y(t) = t4 − 1 2 t3 − 3 γ(4 −α) t3−α + 24 γ(5 −α) t4−α, y(0) = 0, 0 ≤ α ≤ 1 whose exact solution is given by y(t) = t4 − 1 2 t3. int. j. anal. appl. 17 (2) (2019) 308 table 2: absolute errors for example 2. t α = 0.2 α = 0.5 α = 0.7 α = 0.9 α = 1.0 0.0 0 0 0 0 0 0.1 0 1.734 × 10−17 1.387 × 10−17 0 0 0.2 0 3.469 × 10−17 2.775 × 10−17 0 0 0.3 0 4.163 × 10−17 4.163 × 10−17 0 0 0.4 0 5.551 × 10−17 5.551 × 10−17 0 0 0.5 0 5.551 × 10−17 5.551 × 10−17 0 0 0.6 0 5.551 × 10−17 5.551 × 10−17 0 0 0.7 0 5.551 × 10−17 1.110 × 10−16 0 0 0.8 0 0 1.110 × 10−16 0 0 0.9 0 1.110 × 10−16 1.110 × 10−16 0 0 1.0 0 0 2.220 × 10−16 0 0 by taking different values of α, we solved the above problem by means of the presented method. the maximum absolute error with the presented method and sct method [20] for n = 4 are compared in table 3. table 3: comparison of maximum absolute error for example 3. α n=4 present method method of [20] 0.01 1.26×10−14 1.2×10−5 0.1 1.41×10−14 1.3×10−4 0.5 5.30 ×10−15 7.8 ×10−4 0.99 1.30 ×10−15 8.6 ×10−4 1 1.14 ×10−15 8.6 ×10−4 example 4.4. consider the equation d 3 2 y(t) + y(t) = 6 γ(2.5) t1.5 + 6 γ(1.5) t0.5 + t3 + t2, y(0) = 0,y′(t) = 0, whose exact solution is given by y(t) = t3 + t2. by applying the technique described in section 3 with m=4, we approximate solution as y(t) = 4∑ r=0 y(r)(0) r! tr = 4∑ r=0 dr r! tr int. j. anal. appl. 17 (2) (2019) 309 here, by using eq. (3.7), we obtain   1.00901 0.422206 0.123762 0.422206 0.19103 0.0586763 0.123762 0.0586763 0.0186265     d2 d3 d4   =   4.55127 1.9906 0.599582   (4.1) finally by solving eq.(4.14), we get d2 = 2,d3 = 6,d4 = 0. thus we can write y(t) = d0 + d1t + d2 t2 2! + d3 t3 3! + d4 t4 4! = t3 + t2, which is the exact solution. 5. conclusion in this paper, we proposed least squares approximation method to solve a class of linear fractional integro-differential equations comprising of fredholm and volterra cases based on a polynomial of degree n. the numerical experiments show that the proposed method can be suitable method for solving these equations. references [1] c. a. monje, y. chen, b. m. vinagre, d. xue, and v. feliu, fractional-order systems and controls, advances in industrial control, springer, london, uk, 2010. [2] s. das, functional fractional calculus for system identification and controls, springer, new york, 2008. [3] i. podlubny, fractional differential equations, academic press inc., san diego, ca, 1999. [4] r.l. magin, fractional calculus in bioengineering, begell house publishers, 2006. [5] m. amairi, m. aoun, s. najar, m.n. abdelkrim, a constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation, appl. math. comput. 217 (2010), 2162-2168. [6] k. diethelm, n.j. ford, multi-order fractional differential equations and their numerical solutions, appl. math. comput. 154 (2004), 621-640. [7] j. deng, l. ma, existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, appl. math. lett. 23 (2010), 676-680. [8] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, elsevier, san diego, 2006. [9] kumar p, agrawal op, numerical scheme for the solution of fractional differential equations of order greater than one, j. comput. nonlinear dynam. 1 (2006), 178-185. [10] s.a.ei-wakil, a. elhanbaly, m. a. abdou, adomian decomposition method for solving fractional nonlinear differential equations, appl. math. comput. 182 (2006), 313-324. [11] momani s, odibat z, numerical approach to differential equations of fractional order, j. comput. appl. math. 207 (2007), 96-110. int. j. anal. appl. 17 (2) (2019) 310 [12] jabari, d. ezzati, r. and maleknejad, k. a new operational matrix for solving two-dimensional nonlinear integral equations of fractional order. cogent math. 4 (2017), art. id 1347017. [13] li y, zhao w, haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, appl. math. comput. 216 (2010), 2276-2285. [14] rawashdeh ea, numerical solution of fractional integro-differential equations by collocation method, appl. math. comput. 176 (2006), 1-6. [15] atanackovic tm, stankovic b, on a numerical scheme for solving differential equations of fractional order, mech. res. commun. 35 (2008), 429-438. [16] pedas arvet, tamme enn, spline collocation methods for linear multi-term fractional differential equations, j. comput. appl. math. 236 (2011), 167-176. [17] odibat z, momani s, numerical methods for nonlinear partial differential equations of fractional order, appl. math. model. 32 (2008), 28-39. [18] wu jl, a wavelet operational method for solving fractional partial differential equations numerically, appl math. comput. 214 (2009), 31-40. [19] q. wang, k. wang, s. chen, least squares approximation method for the solution of volterra-fredholm integral equations, j. comput. appl. math. 272 (2014), 141-147. [20] a.h. bhrawy, m.m. tharwat, a. yildirim duan, a new formula for fractional integrals of chebyshev polynomials: application for solving multi-term fractional differential equations, appl. math. model. 37 (2013), 4245-4252. 1. introduction 2. brief review of fractional calculus 2.1. definition 3. method of solution 4. illustrative examples 5. conclusion references international journal of analysis and applications volume 17, number 6 (2019), 904-916 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-904 on the solutions of a class of fractional hyperbolic integro-differential inclusions aurelian cernea1,2,∗ 1faculty of mathematics and computer science, university of bucharest, academiei 14, 010014 bucharest, romania 2academy of romanian scientists, splaiul independenţei 54, 050094 bucharest, romania ∗corresponding author: acernea@fmil.unibuc.ro abstract. we study a darboux problem associated to a fractional hyperbolic integro-differential inclusion defined by caputo-katugampola fractional derivative and we prove several existence results for this problem. 1. introduction in the last years one may see a strong development of the theory of differential equations and inclusions of fractional order ( [2, 7, 11–13] etc.). the main reason is that fractional differential equations are very useful tools in order to model many physical phenomena. recently, a generalized caputo-katugampola fractional derivative was proposed in [10] by katugampola and afterwards he provided the existence of solutions for fractional differential equations defined by this derivative. this caputo-katugampola fractional derivative extends the well known caputo and caputohadamard fractional derivatives into a single form. even if katugampola fractional integral operator is an erdélyi-kober type operator ( [8]) it is argued ( [10]) that is not possible to derive hadamard equivalence operators from erdélyi-kober type operators. also, in some recent papers [1, 15], several qualitative properties of solutions of fractional differential equations defined by caputo-katugampola derivative were obtained. received 2019-04-04; accepted 2019-04-29; published 2019-11-01. 2010 mathematics subject classification. 34a60, 26a42, 26a33. key words and phrases. fractional derivative; hyperbolic differential inclusion; decomposable set. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 904 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-904 int. j. anal. appl. 17 (6) (2019) 905 the set-valued framework was studied in [5, 6] and existence results are obtained in the situation when the values of the set-valued map are not necesarily convex provided the set-valued map is lipschitz in the state variable. in the present paper we study fractional hyperbolic integro-differential inclusions of the form dα,ρc u(x,y) ∈ f(x,y,u(x,y), (i α,ρ 0 u)(x,y)) a.e. (x,y) ∈ π, (1.1) u(x, 0) = ϕ(x), u(0,y) = ψ(y) (x,y) ∈ π, (1.2) where π = [0,t1] × [0,t2], ϕ(.) : [0,t1] → r, ψ(.) : [0,t2] → r with ϕ(0) = ψ(0), f(., .) : π × r × r → p(r) is a set-valued map, iα,ρ0 is the generalized left-sided mixed integral and d α,ρ c is the mixed caputokatugampola fractional derivative, α = (α1,α2) ∈ [0, 1) × [0, 1) and ρ = (ρ1,ρ2), ρ1,ρ2 > 0. the goal of the present paper is twofold. first, we show that filippov’s ideas ( [9]) can be suitably adapted in order to obtain the existence of a solution of problem 1.1-1.2. we recall that for an ”ordinary” differential inclusion defined by a lipschitzian set-valued map with nonconvex values filippov’s theorem ( [9]) provides the existence of a solution starting from a given ”quasi” solution. at the same time, the result gives an estimate between the ”quasi” solution and the solution of the differential inclusion. secondly, we obtain the existence of solutions continuously depending on a parameter for problem 1.1-1.2. this result is, in fact, a continuous version of our first result. in the proof of this second theorem we essentially use a result of bressan and colombo ( [3]) concerning the existence of continuous selections of lower semicontinuous multifunctions with decomposable values. our second theorem allows to deduce a continuous selection of the solution set of the problem considered. the results in the present paper may be regarded as extensions of the results in [5, 6] to the hyperbolic framework and generalizations of the results in [4] obtained for problems defined by caputo’s derivative to the more general problem 1.1-1.2. the paper is organized as follows: in section 2 we briefly recall some preliminary results that we will use in the sequel and in section 3 we prove the main results of the paper. 2. preliminaries in [10] the following notions were introduced. let ρ > 0. definition 2.1. a) the generalized left-sided fractional integral of order α > 0 of a lebesgue integrable function f : [0,∞) → r is defined by iα,ρf(t) = ρ1−α γ(α) ∫ t 0 (tρ −sρ)α−1sρ−1f(s)ds, int. j. anal. appl. 17 (6) (2019) 906 provided the right-hand side is pointwise defined on (0,∞) and γ(.) is (euler’s) gamma function defined by γ(α) = ∫∞ 0 tα−1e−tdt. b) the generalized fractional derivative, corresponding to the generalized left-sided fractional integral of a function f : [0,∞) → r is defined by dα,ρf(t) = (t1−ρ d dt )n(in−α,ρ)(t) = ρα−n+1 γ(n−α) (t1−ρ d dt )n ∫ t 0 sρ−1f(s) (tρ −sρ)α−n+1 ds if the integral exists and n = [α] + 1. c) the caputo-katugampola generalized fractional derivative is defined by dα,ρc f(t) = (d α,ρ[f(s) − n−1∑ k=0 f(k)(0) k! sk])(t) if n = 1 (i.e., α ∈ [0, 1)), the caputo-katugampola fractional derivative is dα,ρc f(t) = ρα γ(1 −α) ∫ t 0 f′(s) (tρ −sρ)α ds. we note that if ρ = 1, the caputo-katugampola fractional derivative becames the well known caputo fractional derivative. on the other hand, passing to the limit with ρ → 0+, the above definition yields the hadamard fractional derivative. consider i1 = [0,t1], i2 = [0,t2] and π = [0,t1]× [0,t2]. denote by l(π) the σalgebra of the lebesgue measurable subsets of π and by b(r) the family of all borel subsets of r. let c(π, r) be the banach space of all continuous functions from π to r with the norm ||u||c = sup{|u(x,y)|; (x,y) ∈ π} and l1(π, r) be the banach space of functions u(·, ·) : π → r which are integrable, normed by ‖u‖l1 = ∫ t1 0 ∫ t2 0 |u(x,y)|dxdy. let ρ1,ρ2 > 0. the corresponding versions of the above definition in the case of function with two variables are as follows. definition 2.2. a) the generalized left-sided mixed integral of order α = (α1,α2) ∈ [0, 1)× [0, 1) of f(., .) ∈ l1(π, r) is defined by (i α,ρ 0 f)(x,y) = ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1· f(s,t)dsdt. b) the mixed caputo-katugampola fractional-order derivative of order α of f(., .) ∈ l1(π, r) is defined by (dα,ρc f)(x,y) = (i 1−α,ρ 0 ∂2f ∂x∂y )(x,y) = ρ α1 1 ρ α2 2 γ(1−α1)γ(1−α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )−α1 (yρ2 − tρ2 )−α2 ∂ 2f ∂s∂t (s,t)dsdt. in the definition above by 1 −α we mean (1 −α1, 1 −α2) ∈ (0, 1] × (0, 1]. int. j. anal. appl. 17 (6) (2019) 907 definition 2.3. a function u(., .) ∈ c(π, r) is said to be a solution of problem 1.1-1.2 if there exists f(., .) ∈ l1(π, r) such that f(x,y) ∈ f(x,y,u(x,y), (iα,ρ0 u)(x,y)) a.e. (π), (2.1) u(x,y) = ν(x,y) + (i α,ρ 0 f)(x,y), (x,y) ∈ π, (2.2) where ν(x,y) = ϕ(x) + ψ(y) −ϕ(0). the pair (u(., .),f(., .)) is called a trajectory-selection pair of problem 1.1-1.2. the previous definition is justified by the fact that a simple computation shows that u(., .) satisfies dα,ρc u(x,y) ≡ f(x,y), u(x, 0) ≡ ϕ(x), u(0,y) ≡ ψ(y), (x,y) ∈ π if and only if 2.2 is verified. let (x,d) be a metric space. the pompeiu-hausdorff distance of the closed subsets a,b ⊂ x is defined by dh(a,b) = max{d∗(a,b),d∗(b,a)}, d∗(a,b) = sup{d(a,b); a ∈ a}, where d(x,b) = inf{d(x,y); y ∈ b}. with cl(a) we denote the closure of the set a ⊂ x. recall that a subset d ⊂ l1(π, r) is said to be decomposable if for any u(·),v(·) ∈ d and any subset a ∈l(π) one has uχa + vχb ∈ d, where b = i\a. we denote by d the family of all decomposable closed subsets of l1(π, r). let g(., .) : π×rm →p(rn) be a set-valued map. recall that g(., .) is called l(π)⊗b(rm) measurable if for any closed subset c ⊂ rn we have {(x,y,z) ∈ π × rm; f(x,y,z) ∩c} 6= ∅}∈l(π) ⊗b(rm). consider the banach space s := {(ϕ,ψ) ∈ c(i1, r) × c(i2, r); ϕ(0) = ψ(0)} endowed with the norm ||(ϕ,ψ)|| = ||ϕ||c + ||ψ||c and for (ϕ,ψ) ∈ s denote s(ϕ,ψ) the set of all solutions of problem 1.1-1.2. we recall now some results that we are going to use in the next section. lemma 2.1. ( [14]) let g(·, ·) : π → p(rn) be a compact valued measurable multifunction and h(·, ·) : π → rn a measurable function. then there exists a measurable selection g(·, ·) of g(·, ·) such that ‖g(x,y) −h(x,y)‖ = d(h(x,y),g(x,y)), a.e. (π). next (s, d) is a separable metric space and x is a banach space. we recall that a multifunction g(·) : s → p(x) is said to be lower semicontinuous (l.s.c.) if for any closed subset c ⊂ x, the subset {s ∈ s; g(s) ⊂ c} is closed in s. lemma 2.2. ( [3]) let g∗(., .) : π ×s →p(rn) be a closed valued l(π) ⊗b(s) measurable multifunction such that g∗((x,y), .) is l.s.c. for any (x,y) ∈ π. int. j. anal. appl. 17 (6) (2019) 908 then the set-valued map h(.) defined by h(s) = {g ∈ l1(π, rn); g(x,y) ∈ g∗(x,y,s) a.e. (π)} is l.s.c. with nonempty decomposable closed values if and only if there exists a continuous mapping q(.) : s → l1(π, r) such that d(0,g∗(x,y,s)) ≤ q(s)(x,y) a.e. (π), ∀s ∈ s. lemma 2.3. ( [3]) let h(.) : s → d be a l.s.c. set-valued map with closed decomposable values and let f(.) : s → l1(π, rn), p(.) : s → l1(π, r) be continuous such that the multifunction g(.) : s → d defined by g(s) = cl{h ∈ h(s); ||h(x,y) −f(s)(x,y)|| < p(s)(x,y) a.e. (π)} has nonempty values. then g(.) has a continuous selection. 3. the main results in order to obtain an existence result for problem 1.1-1.2 one need the following assumptions on f(., .). hypothesis h1. f(., .) : π × r × r → p(r) is a set-valued map with non-empty, compact values that verifies: i) for all u,v ∈ r, f(., .,u,v) is measurable. ii) there exists k1,k2 > 0 such that for almost all (x,y) ∈ π, dh(f(x,y,u1,v1),f(x,y,u2,v2)) ≤ k1|u1 −u2| + k2|v1 −v2|, ∀u1,v1,u2,v2 ∈ r. in what follows g(., .) ∈ l1(π, r) is given and there exists ξ(., .) ∈ l1(π, r+) with ξ := sup(x,y)∈π(i α,ρ 0 ξ)(x,y) < +∞ which satisfies d(g(x,y),f(x,y,w(x,y), (i α,ρ 0 w)(x,y))) ≤ ξ(x,y) a.e. (π), where w(., .) is a solution of the fractional hyperbolic differential equation dα,ρc w(x,y) = g(x,y) (x,y) ∈ π, (3.1) w(x, 0) = ϕ1(x), w(0,y) = ψ1(y) (x,y) ∈ π, (3.2) with (ϕ1,ψ1) ∈ s. set ν1(x,y) = ϕ1(x) + ψ1(y) −ϕ1(0), (x,y) ∈ π, k3 = t ρ1 1 t ρ2 2 ρ 1−α1 1 ρ 1−α2 2 γ(1+α1)γ(1+α2) and k = k3(k1 + k2k3). int. j. anal. appl. 17 (6) (2019) 909 theorem 3.1. let hypothesis h1 be satisfied, k < 1 and consider g(., .), ξ(., .), w(., .) as above, (ϕ,ψ) ∈ s and ν(x,y) = ϕ(x) + ψ(y) −ϕ(0), (x,y) ∈ π. then there exists (v(., .),f(., .)) a trajectory-selection pair of problem 1.1-1.2 such that |v(x,y) −w(x,y)| ≤ ||ν −ν1||c + ξ 1 −k , ∀(x,y) ∈ π, (3.3) |f(x,y) −g(x,y)| ≤ (k1 + k2k3)(||ν −ν1||c + ξ) 1 −k + ξ(x,y), a.e. (π). (3.4) proof. we define f0(., .) = g(., .), v0(., .) = w(., .). it follows from lemma 2.1 that there exists a measurable function f1(., .) such that f1(x,y) ∈ f(x,y,v0(x,y), (i α,ρ 0 v0)(x,y)) a.e. (π) and for almost all (x,y) ∈ π |f0(x,y) −f1(x,y)| = d(g(x,y),f(x,y,v0(x,y), (i α,ρ 0 v0)(x,y))) ≤ ξ(x,y). define, for (x,y) ∈ π v1(x,y) = ν(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1· tρ2−1f1(s,t)dsdt. since w(x,y) = ν1(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1· tρ2−1f0(s,t)dsdt. one has |v1(x,y) −v0(x,y)| ≤ |ν(x,y) −ν1(x,y)| + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1 (yρ2 − tρ2 )α2−1sρ1−1tρ2−1||f1(s,t) −f0(s,t)||dsdt ≤ ||ν −ν1||c + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2)∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1ξ(s,t)dsdt ≤ ||ν −ν1||c + ξ. from lemma 2.1 we deduce the existence of a measurable function f2(., .) such that f2(x,y) ∈ f(x,y,v1(x,y), (i r 0v1)(x,y)) a.e. (π) and for almost all (x,y) ∈ π |f2(x,y) −f1(x,y)| ≤ d(f1(x,y),f(x,y,v1(x,y), (i α,ρ 0 v1)(x,y))) ≤ dh(f(x,y,v0(x,y), (i α,ρ 0 v0)(x,y)),f(x,y,v1(x,y), (i α,ρ 0 v1)(x,y))) ≤ k1|v1(x,y) −v0(x,y)| + k2|(i α,ρ 0 v0)(x,y) − (i α,ρ 0 v1)(x,y)| ≤ k1(||ν −ν1||c + ξ) + k2 ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1· sρ1−1tρ2−1(||ν −ν1||c + ξ)dsdt = (k1 + k2k3)(||ν −ν1||c + ξ). define, for (x,y) ∈ π v2(x,y) = ν(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1· tρ2−1f2(s,t)dsdt int. j. anal. appl. 17 (6) (2019) 910 and one has |v2(x,y) −v1(x,y)| ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1 sρ1−1tρ2−1|f2(s,t) −f1(s,t)|dsdt ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2− tρ2 )α2−1sρ1−1tρ2−1(k1 + k2k3)(||ν −ν1||c + ξ)dsdt = k(||ν −ν1||c + ξ). assuming that for some p ≥ 2 we have already constructed the sequences (vi(., .)) p i=1, (fi(., .)) p i=1 satisfying |vp(x,y) −vp−1(x,y)| ≤ kp−1(||ν −ν1||c + ξ) (x,y) ∈ π, (3.5) |fp(x,y) −fp−1(x,y)| ≤ (k1 + k2k3)kp−2(||ν −ν1||c + ξ) a.e. (π). (3.6) we apply lemma 2.1 and we find a measurable function fp+1(., .) such that fp+1(x,y) ∈ f(x,y,vp(x,y), (i α,ρ 0 vp)(x,y)) a.e. (π) and for almost all (x,y) ∈ π |fp+1(x,y) −fp(x,y)| ≤ d(fp+1(x,y),f(x,y,vp−1(x,y), (i α,ρ 0 vp−1)(x,y))) ≤ dh(f(x,y,vp(x,y), (i α,ρ 0 vp)(x,y)),f(x,y,vp−1(x,y), (i α,ρ 0 vp−1)(x,y))) ≤ l1|vp(x,y) −vp−1(x,y)| + l2|(i α,ρ 0 vp)(x,y) − (i α,ρ 0 vp−1)(x,y)| ≤ k1[k p−2(||ν −ν1||c + ξ)] + k2k3kp−2(||ν −ν1||c + ξ) = kp−1(||ν −ν1||c +ξ)(k1 + k2k3). define, for (x,y) ∈ π vp+1(x,y) = ν(x,y)+ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1fp+1(s,t)dsdt. (3.7) we have |vp+1(x,y) −vp(x,y)| ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1 sρ1−1tρ2−1|fp+1(s,t) −fp(s,t)|dsdt ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2− tρ2 )α2−1sρ1−1tρ2−1kp−1(||ν −ν1||c + ξ)(k1 + k2k3)dsdt = kp−1(||ν −ν1||c +ξ)k3(k1 + k2k3) = k p(||ν −ν1||c + ξ). taking into account 3.5 we deduce that the sequence (vp(., .))p≥0 is cauchy in c(π, r), so it converges to v(., .) ∈ c(π, r). from 3.6 we infer that the sequence (fp(., .))p≥0 is cauchy in l1(π, r), thus it converges to f(., .) ∈ l1(π, r). using the fact that the values of f(., .) are closed we get that f(x,y) ∈ f(x,y, v(x,y), (i α,ρ 0 v)(x,y)) a.e. (π). int. j. anal. appl. 17 (6) (2019) 911 one may write successively, |ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1fp(s,t)dsdt− ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1f(s,t)dsdt| ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1|fp(s,t)− f(s,t)|dsdt ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −sρ1 )α1−1(yρ2 − tρ2 )α2−1sρ1−1tρ2−1· (k1 + k2k3)|up−1(s,t) −u(s,t)|dsdt ≤ k||up−1(., .) −u(., .)||c. thus, we pass to the limit in 3.2 and we get that v(., .) is a solution of problem 1.1-1.2. at the same time, by adding inequalities 3.5 for any (x,y) ∈ π we have |vp(x,y) −w(x,y)| ≤ |vp(x,y) −vp−1(x,y)| + |vp−1(x,y) −vp−2(x,y)| + . . . + |v2(x,y) −v1(x,y)| + |v1(x,y) −v0(x,y)| ≤ (kp−1 + kp−2 + ... + k + 1)(||ν −ν1||c + ξ) ≤ ||ν−ν1||c+ξ 1−k . (3.8) similarly, by adding inequalities 3.6 for almost all (x,y) ∈ π we have |fp(x,y) −g(x,y)| ≤ |fp(x,y) −fp−1(x,y)| + |fp−1(x,y)− fp−2(x,y)| + . . . + |f2(x,y) −f1(x,y)| + |f1(x,y) −f0(x,y)| ≤ (k1 + k2k3)(k p−2 + ... + k + 1)(||ν −ν1||c + ξ) + ξ(x,y) ≤ (k1 + k2k3) ||ν−ν1||c+ξ 1−k + ξ(x,y). (3.9) finally we pass to the limit with p →∞ in (3.8) and (3.9) and we get (3.3) and (3.4), respectively, which completes the proof. � if in theorem 3.1 we take g = 0, w = 0, ϕ1 = 0, ψ1 = 0 then we obtain the following existence result for solutions of problem 1.1-1.2. corollary 3.1. let hypothesis h1 be satisfied, k < 1 and assume that there exists ξ(., .) ∈ l1(π, r+) with ξ := sup(x,y)∈π(i α,ρ 0 ξ)(x,y) < +∞ such that d(0,f(x,y, 0, 0)) ≤ ξ(x,y) ∀(x,y) ∈ π. then there exists v(., .) ∈ c(π, r) a solution of problem 1.1-1.2 such that |v(x,y)| ≤ ||ν||c + ξ 1 −k , ∀(x,y) ∈ π. next we obtain a continuous version of theorem 3.1. hypothesis h2. i) s is a separable metric space, ϕ(.) → c(i1, r),ψ(.) : s → c(i2, r) and ε(.) : s → (0,∞) are continuous mappings and ϕ(s)(0) ≡ ψ(s)(0). int. j. anal. appl. 17 (6) (2019) 912 ii) there exists the continuous mappings ϕ1(.) → c(i1, r),ψ1(.) : s → c(i2, r) g(.) : s → l1(π, r), ξ(.) : s → l1(π, r) and w(.) : s → c(π, r) such that ϕ1(s)(0) ≡ ψ1(s)(0), (dw(s))α,ρc (x,y) = g(s)(x,y) a.e. (π), ∀s ∈ s, w(s)(x, 0) = ϕ1(s)(x), w(s)(0,y) = ψ1(s)(y) (x,y) ∈ π, ∀s ∈ s, d(g(s)(x,y),f(x,y,w(s)(x,y), (i α,ρ 0 w(s))(x,y))) ≤ ξ(s)(x,y) a.e. (π),∀s ∈ s and the mapping s → ξ(s) := sup(x,y)∈π(i α,ρ 0 ξ(s))(x,y) is continuous. we use next the following notations ν(s)(x,y) = ϕ(s)(x) + ψ(s)(y) − ϕ(s)(0), ν1(s)(x,y) = ϕ1(s)(x) + ψ1(s)(y) −ϕ1(s)(0) (x,y) ∈ π, a(s) = sup(x,y)∈π |ν(s)(x,y) −ν1(s)(x,y)| s ∈ s. theorem 3.2. assume that hypotheses h1 and h2 are satisfied and k < 1. then there exist a continuous mapping v(.) : s → c(π, r) such that for any s ∈ s, v(s)(., .) is a solution of problem 1.1 which satisfies v(s)(x, 0) = ϕ(s)(x), v(s)(0,y) = ψ(s)(y) (x,y) ∈ π,s ∈ s and |v(s)(x,y) −w(s)(x,y)| ≤ a(s) + ε(s) + ξ(s) 1 −k ∀(x,y) ∈ π,∀s ∈ s. proof. we make the following notations v0(., .) = w(., .), ξp(s) := k p−1(a(s) + ε(s) + ξ(s)), p ≥ 1. we consider the set-valued maps g0(.),h0(.) defined, respectively, by g0(s) = {h ∈ l1(π, r); h(x,y) ∈ f(x,y,w(s)(x,y), (i α,ρ 0 w(s))(x,y))a.e.(π)} h0(s) = cl{h ∈ g0(s); |h(x,y) −g(s)(x,y)| < ξ(s)(x,y) + 1 k3 ε(s)}. taking into account that d(g(s)(x,y),f(x,y,w(s)(x,y), (i α,ρ 0 w(s))(x,y)) ≤ ξ(s)(x,y) < ξ(s)(x,y) + 1 k3 ε(s) the set h0(s) is not empty. set f∗0 (x,y,s) = f(x,y,w(s)(x,y), (i α,ρ 0 w(s))(x,y)) and note that d(0,f∗0 (x,y,s)) ≤ |g(s)(x,y)| + ξ(s)(x,y) =: ξ ∗(s)(x,y) and ξ∗(.) : s → l1(i, r) is continuous. applying now lemma 2.2 and lemma 2.3 we obtain the existence of a continuous selection f0 of h0 such that ∀s ∈ s, (x,y) ∈ π, f0(s)(x,y) ∈ f(x,y,w(s)(x,y), (i α,ρ 0 w(s))(x,y)) a.e. (π), ∀s ∈ s, |f0(s)(x,y) −g(s)(x,y)| ≤ ξ0(s)(x,y) = ξ(s)(x,y) + 1 k3 ε(s). int. j. anal. appl. 17 (6) (2019) 913 we define v1(s)(x,y) = ν(s)(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1· zρ1−1tρ2−1f0(s)(z,t)dzdt and one has |v1(s)(x,y) −v0(s)(x,y)| ≤ a(s) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1· (yρ2 − tρ2 )α2−1zρ1−1tρ2−1|f0(s)(z,t) −g(s)(z,t)|dzdt ≤ a(s)+ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1(ξ(s)(z,t) + 1 k3 ε(s))dzdt ≤ a(s) + ξ(s) + ε(s) =: ξ1(s), (x,y) ∈ π,s ∈ s. we construct the sequences of approximations fp(., .) : s → l1(π, r), vp(., .) : s → c(π, r) with the following properties: a) fp(., .) : s → l1(π, r), vp(., .) : s → c(π, r) are continuous, b) fp(s)(x,y) ∈ f(x,y,vp(s)(x,y), (i α,ρ 0 vp(s))(x,y)), a.e. (π), s ∈ s, c) |fp(s)(x,y) −fp−1(s)(x,y)| ≤ (k1 + k2k3)ξp(s), a.e. (π), s ∈ s. d) vp+1(s)(x,y) = ν(s)(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1 · zρ1−1tρ2−1fp(s)(z,t)dzdt, (x,y) ∈ π,s ∈ s. assume that we have already constructed fi(.),vi(.) satisfying a)-c) and define vp+1(.) as in d). from c) and d) one has |vp+1(s)(x,y) −vp(s)(x,y)| ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1· (yρ2 − tρ2 )α2−1zρ1−1tρ2−1|fp(s)(z,t) −fp−1(s)(z,t)|dzdt ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1yρ2 − tρ2 )α2−1zρ1−1tρ2−1(k1+ k2k3)ξp(s)dzdt = k3(k1 + k2k3)ξp(s) = ξp+1(s). (3.10) on the other hand, d(fp(s)(x,y),f(x,y,vp+1(s)(x,y), (i α,ρ 0 vp+1(s))(x,y))) ≤ k1|vp+1(s)(x,y) −vp(s)(x,y)| + k2 ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1· (yρ2 − tρ2 )α2−1zρ1−1tρ2−1|vp+1(s)(z,t) −vp(s)(z,t)|dzdt ≤ (k1 + k2k3)ξp+1(s). (3.11) for any s ∈ s we define the set-valued maps gp+1(s) = {u ∈ l1(π, r); u(x,y) ∈ f(x,y,vp+1(s)(x,y), (i α,ρ 0 vp+1(s))(x,y)) a.e. (π)} and hp+1(s) = cl{u ∈ gp+1(s); |u(x,y) −fp(s)(x,y)| < (k1 + k2k3)ξp+1(s)}. we note that from 3.11 the set hp+1(s) is not empty. set f∗p+1(x,y,s) = f(x,y,vp+1(s)(x,y), (i α,ρ 0 vp+1(s))(x,y)) and note that d(0,f∗p+1(x,y,s)) ≤ |fp(s)(x,y)| + (k1 + k2k3)ξp+1(s) =: ξ ∗ p+1(s)(x,y) int. j. anal. appl. 17 (6) (2019) 914 and ξ∗p+1(.) : s → l1(i, r) is continuous. by lemma 2.2 and lemma 2.3 we obtain the existence of a continuous function fp+1(.) : s → l1(π, r) such that fp+1(s)(x,y) ∈ f(x,y,vp+1(s)(x,y), (i α,ρ 0 vp+1(s))(x,y)) a.e. (π), ∀s ∈ s, |fp+1(s)(x,y) −fp(s)(x,y)| ≤ (k1 + k2k3)ξp+1(s) ∀s ∈ s, (x,y) ∈ π. from 3.10, c) and d) we obtain |vp+1(s)(., .) −vp(s)(., .)|c ≤ ξp+1(s) = kp(a(s) + ε(s) + ξ(s)), (3.12) |fp+1(s)(., .) −fp(s)(., .)|1 ≤ kp−1(k1 + k2k3)t1t2(a(s) + ε(s) + ξ(s)). (3.13) thus, fp(s)(., .), vp(s)(., .) are cauchy sequences in the banach spaces l 1(π, r) and c(π, r), respectively. consider f(.) : s → l1(π, r), v(.) : s → c(π, r) their limits. the function s → a(s) + ε(s) + ξ(s) is continuous, hence locally bounded. therefore 3.13 implies that for every s′ ∈ s the sequence fp(s′)(., .) satisfies the cauchy condition uniformly with respect to s′ on some neighborhood of s. therefore, s → f(s)(., .) is continuous from s into l1(π, r). as before, from 3.12, vp(s)(., .) is cauchy in c(π, r) locally uniformly with respect to s. hence s → v(s)(., .) is continuous from s into c(π, r). at the same time, since vp(s)(., .) converges uniformly to v(s)(., .) and d(fp(s)(x,y),f(x,y,v(s)(x,y), (i α,ρ 0 v(s))(x,y)) ≤ (k1 + k2k3)|vp(s)(x,y) −v(s)(x,y)| a.e. (π), ∀s ∈ s passing to the limit along a subsequence of fp(s)(., .) converging pointwise to f(s)(., .) we obtain f(s)(x,y) ∈ f(x,y,v(s)(x,y), (iα,ρ0 v(s))(x,y)) a.e. (π), ∀s ∈ s. one may write successively, |ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1fp(s)(z,t)dzdt− ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1f(s)(z,t)dzdt| ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1|fp(s)(z,t)− f(s)(z,t)|dzdt ≤ ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1zρ1−1tρ2−1 (k1 + k2k3)|vp−1(s)(z,t) −v(s)(z,t)|dzdt ≤ k||vp−1(s)(., .) −v(s)(., .)||c. so, we pass to the limit in d) and we get ∀(x,y) ∈ π,s ∈ s v(s)(x,y) = ν(s)(x,y) + ρ 1−α1 1 ρ 1−α2 2 γ(α1)γ(α2) ∫ x 0 ∫ y 0 (xρ1 −zρ1 )α1−1(yρ2 − tρ2 )α2−1· zρ1−1tρ2−1f(s)(z,t)dzdt, i.e., v(s)(., .) is the required solution. int. j. anal. appl. 17 (6) (2019) 915 finally, by adding inequalities 3.10 for all p ≥ 1 we get |vp+1(s)(x,y) −w(s)(x,y)| ≤ p+1∑ l=1 ξl(s) ≤ a(s) + ε(s) + ξ(s) 1 −k . (3.14) passing to the limit in 3.14 we obtain the conclusion of the theorem. � theorem 3.2 allows to provide a continuous selection of the solution set of problem 1.1-1.2. hypothesis h3. hypothesis h1 is satisfied, k < 1 and there exists q(., .) ∈ l1(π, r+) with sup(x,y)∈π(i α,ρ 0 q)(x,y) < ∞ such that d(0,f(x,y, 0, 0)) ≤ q(x,y) a.e. (π). corollary 3.2. assume that hypothesis h3 is satisfied. then there exists a function v(., .) : π × s → r such that a) v(., (ξ,η)) ∈s(ξ,η), ∀(ξ,η) ∈ s. b) (ξ,η) → v(., (ξ,η)) is continuous from s into c(π, r). proof. we take s = s, ϕ(µ,η) = µ, ψ(µ,η) = η ∀(µ,η) ∈ s, ε(.) : s → (0,∞) an arbitrary continuous function, g(.) = 0, w(.) = 0, ξ(s)(x,y) ≡ q(x,y) ∀s = (µ,η) ∈ s, (x,y) ∈ π and we apply theorem 3.2 in order to obtain the conclusion of the corollary. � references [1] r. almeida, a. b. malinowski and t. odzijewicz, fractional differential equations with dependence on the caputokatugampola derivative, j. comput. nonlinear dyn. 11 (2016), id 061017. [2] d. băleanu, k. diethelm, e. scalas and j. j. trujillo, fractional calculus models and numerical methods, world scientific, singapore, 2012. [3] a. bressan and g. colombo, extensions and selections of maps with decomposable values, studia math. 90 (1988), 69-86. [4] a. cernea, on an integro-differential inclusion of fractional order, differ. equ. dyn. syst. 3 (2013), 225-236. [5] a. cernea, continuous family of solutions for fractional integro-differential inclusions of caputo-katugampola type, progr. fract. differ. appl. 5 (2019), 37-42. [6] a. cernea, on a fractional integro-differential inclusion of caputo-katugampola type, bull. math. anal. appl. to appear. [7] k. diethelm, the analysis of fractional differential equations, springer, berlin, 2010. [8] a. erdélyi and h. kober, some remarks on hankel transforms, q. j. math. 11 (1940), 212-221. [9] a. f. filippov, classical solutions of differential equations with multivalued right hand side, siam j. control, 5 (1967), 609-621. [10] u. n. katugampola, a new approach to generalized fractional derivative, bull. math. anal. appl. 6 (2014), 1-15. [11] a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [12] k. miller and b. ross, an introduction to the fractional calculus and differential equations, john wiley, new york, 1993. [13] i. podlubny, fractional differential equations, academic press, san diego, 1999. [14] h.d. tuan, on local controllability of hyperbolic inclusions, j. math. syst. estim. control, 4 (1994), 319-339. int. j. anal. appl. 17 (6) (2019) 916 [15] s. zeng, d. băleanu, y. bai and g. wu, fractional differential equations of caputo-katugampola type and numerical solutions, appl. math. comput. 315 (2017), 549-554. 1. introduction 2. preliminaries 3. the main results references international journal of analysis and applications volume 17, number 1 (2019), 33-46 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-33 computing lower bounds of µ-values for a class of rotary electrical machines mutti-ur rehman∗, m. fazeel anwar department of mathematics, sukkur iba university, 65200 sukkur-pakistan ∗corresponding author: mutti.rehman@iba-suk.edu.pk abstract. in this article we present the computations of lower bounds of well-known mathematical quantity in control theory known as structured singular value for a family of structured matrices obtained for a dc motor, that is an electrical machine. the comparison of lower bounds with the well-known matlab function mussv is studied. the structured singular values provide an important tool to synthesize robustness as well as analyze performance and stability of feedback systems. 1. introduction the structured singular value (ssv) [1] is an important and versatile tool in system theory which allows one to address a central problem in control systems i.e. to analyze the stability of a system. the class of structures addressed by the ssv is very generic and covers almost all kinds of parametric perturbations that can be incorporated into the control system via real or complex linear fractional transformations (lfts). we refer the interested readers to see [2 9] and the references there in for more examples and applications of ssv. the exact computations of ssv are np hard [10]. there are however several algorithms available in literature to compute lower and upper bounds for ssv. an upper bound of ssv guaranties the robust stability, while the lower bounds of ssv provides information about instability of linear closed loop systems. received 2018-08-04; accepted 2018-10-06; published 2019-01-04. 2010 mathematics subject classification. 65f15, 34h05. key words and phrases. structured singular value; spectral value set; block diagonal uncertainties; spectral radius; low-rank matrix manifolds, gradient system of odes. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 33 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-33 int. j. anal. appl. 17 (1) (2019) 34 the widely used routine, mussv, available in matlab control toolbox approximates upper bound by means of the diagonal balancing and linear matrix inequalities (lmi) techniques [11, 12]. while on the other hand, lower bound is approximated by generalization of power method [13-14]. in this paper, we present numerical approximations to a lower bound of the ssv associated with mixed real and complex uncertainties. our approach is based on a two level algorithm, inner-outer algorithm, given in [17]. 2. overview of the article section 3 provides the basic framework of proposed problem. we explain how a better approximation of the structured singular values can be obtained by a two level algorithm that is an inner-outer algorithm. the outer algorithm determines the perturbation level while the inner algorithm computes the desired perturbation. the inner algorithm then determines a (local) extremizer of the structured spectral value set. in section 4, we introduce the inner algorithm for the case of pure complex uncertainties. an important characterization of extremizers help us to restrict the problem to a manifold of structured perturbations with normalized and low-rank blocks. a gradient system of odes for finding extremizers on this manifold is established. finally, in section 5, we give a range of numerical experiments to compare lower bounds of ssv obtained with proposed algorithm [17] to those obtained with mussv. 3. framework let c = c and r = r and let cn,n(rn,n) denote the family of complex (real) matrices. for m ∈ cn,n(rn,n) we consider an underlying perturbation set with prescribed block diagonal matrices given below b = {diag(δiii, ∆j) : δi ∈ c(r), ∆j ∈ cn,n(rn,n)}. (3.1) the following definition is given in [1], where i is the n×n identity matrix. definition 3.1. let m ∈ cn,n and ∆ ∈ b is an admissible perturbation. then structured singular value of m is denoted by µb(m) and is defined as follows µb(m) := 1 min{‖∆‖2 : ∆ ∈ b, det(i −m∆) = 0} . (3.2) in above definition 3.2, det(·) represents the determinant of a matrix (i −m∆) while minimum is over an admissible perturbation ∆. for a general set b the ssv become smaller and thus we have an upper bound. the important special case, when the set b only consists of pure complex perturbations, we denote the set by b∗ instead of b. for ∆ ∈ b∗ we have eiθ∆ ∈ b for any θ ∈ r. as a result, this gives us a suitable choice of ∆ ∈ ∆b∗ such that spectral radius attains the maximum value 1 that is ρ(m∆) = 1 iff there is ∆′ ∈ ∆b∗ , which possesses the int. j. anal. appl. 17 (1) (2019) 35 same matrix 2-norm so that m∆′ possess an eigenvalue 1. this in turn implies that the matrix i −m∆′ is singular. this gives us the following alternative definition of the ssv µb∗ (m) :== 1 min{‖∆‖2 : ∆ ∈ b∗, ρ(m∆) = 1} . (3.3) in equ. (3.3), ρ(·) denotes the spectral radius of a matrix m∆. 3.1. overview of the proposed methodology. we need to solve the maximization problem λ(�) = arg max|λ| (3.4) where maximum is over λ for some fixed parameter � > 0. it is clear that µb∗ (m) is the reciprocal of the smallest value of � for which λ(�) = 1. this suggests a two level algorithm. in the inner algorithm, we intend to solve the problem addressed in equ. (3.4). in the outer algorithm we first vary � by using the fast newtons method and this exploits the knowledge of the computation of the exact derivative of the extremizers. we address equ. (3.4) by solving a system of ordinary differential equations. 4. computation of local extremizers in this section we present the inner algorithm for optimization problem addressed in equ. (3.4). we use the following standard eigenvalue perturbation result. lemma 4.1. [15]. consider a family of matrices a : r → cn,n and let λ(t) be an eigenvalue of a(t) for all values of t. the eigenvalue λ(t) tends to converge to the simple eigenvalue λ0 = λ(0), of a0 = a(0) as t → 0. therefor λ(t) is analytic near t = 0 with dλ(t) dt = x∗0 bx0 x∗0 x0 where, x∗0 , x0 are the left and right eigenvectors of a0 = a(0) associated with simple eigenvalue λ0 = λ(0), that is, x∗0 (a0 − λ0i) = 0 and (a0 − λ0i)x∗ = 0 and the matrix b is the time derivative of the matrix valued function a(t). definition 4.2. an admissible perturbation ∆ ∈ b∗ such that ‖∆‖2 ≤ 1 and the matrix (�m∆) for some fixed parameter � > 0 has greatest eigenvalue λmax which maximizes locally the modulus of λ b∗ � (m) is known as a local maximizer. theorem 4.3. let ∆̂ = {diag(δiii, ∆j) : δi ∈ c, ∆j ∈ cmj,mj, ∀i = 1 : s, j = 1 : f} (4.1) with ‖∆̂‖2 = 1 be a local extremizer. the matrix �m∆ possesses the simple eigenvalue λ = |λ|eiθ,θ ∈ r, with x and y being right and left eigenvectors for an eigenvalue λ. the eigenvectors are scaled such that s = eiθy∗x > 0. partitioning x and y according to size and structure of ∆̂ yields x = (xt1 , . . . , x t n, x t n+1, . . . ,x t n+f ) t; int. j. anal. appl. 17 (1) (2019) 36 y = (yt1 , . . . ,y t n , y t n+1, . . . ,y t n+f ) t. (4.2) now we take z = m∗y = (zt1 , . . . , z t n , z t n+1, . . . ,z t n+f ) t and assume the non-degeneracy conditions given by z∗kxk 6= 0 ∀ k = 1, . . . ,n, (4.3) ‖zn+h‖2 · ‖xn+h‖2 6= 0 ∀ h = 1, . . . ,f. (4.4) hence, |δk| = 1 ∀ k = 1, . . . ,s and ‖∆‖2 = 1 ∀h = 1, . . . ,f. theorem 4.4. let ∆̂ = {diag(δiii, ∆j) : δi ∈ c, ∆j ∈ cmj,mj, ∀i = 1 : s, j = 1 : f} (4.5) with ‖∆̂‖2 = 1 be a local extremizer. suppose that λ,x,z are defined and partitioned as in theorem 4.3. assume the non-degeneracy condition of (4.4) and every block possess a singular value which attains the maximum value 1. then the matrix, ∆̂ = {diag(δ1i1, ...,δsis; u1v∗1, ...,ufv ∗ f )} is a local extremizer. remark 4.5. theorem 4.4 allows us to restrict the admissible perturbations in the structured spectral value set given in (3.4) to those with rank-1 blocks. since the frobenius and the matrix 2-norms of a rank-1 matrix are same, this helps us to search for extremizers within the sub-manifold b∗1 = {diag(δ1i1; ∆j) : δi ∈ c, ∆j ∈ c mj,mj,∀i = 1 : s,j = 1 : f}. (4.6) 4.6. approximating extremal points of λb ∗ � (m). in order to approximate the local maxima for λb ∗ � (m) we construct a matrix valued function ∆(t) where ∆(t) ∈ b∗ such that the largest eigenvalue λ of the matrix �m∆(t) achieves the maximum (local) growth. we then derive a gradient system of odes which satisfies the choice of the initial matrix admissible perturbation ∆(t). 4.7. the local optimization problem. we consider the fact that λ = |λ|eiθ is the simple eigenvalue with the corresponding eigenvectors, normalized such that ‖x‖ = ‖y‖ = 1, y∗x = |y∗x|e−iθ. (4.7) as a result of the lemma 4.1, we get the following expression for the change in the largest eigenvalue. d dt |λ|2 = 2|λ| |y∗x| re(z∗∆̇x),z = m∗y (4.8) int. j. anal. appl. 17 (1) (2019) 37 the eigenvectors x and y are defined and normalized as in the theorem 4.3. now, by considering the suitable perturbation ∆ ∈ b∗1 with b∗1 in equ. (4.6), we aim to determine a direction ∆∗ = z that (locally) maximizes the increase of the modulus of λ. this amounts to determining a direction z as given in the following equation z = diag(ω1ir1, . . . ,ωsirn , ω1, . . . , ωf ) (4.9) which is the solution of the following optimization problem z = arg max{re(z∗zx)} subject to re(δiωi) = 0, i = 1 : s, and re〈∆j, ωj〉 = 0, j = 1 : f. (4.10) the linear constraints in maximization problem in equ. (4.10) ensure that z lies in the tangent space of the manifold b∗1 at ∆(t). in particular equ. (4.10) ensures that the norm of each block of the admissible perturbation ∆(t) remain conserved. lemma 4.6. the solution of the maximization problem as discussed in (4.10) is given by z∗ = {diag(ω1ir1, . . . ,ωnirn , ω1, . . . , ωf )}, (4.11) with ωi = νi ( x∗izi −re ( x∗iziδi ) δi ) , i = 1, . . . ,s (4.12) ωj = ζj ( zs+jx ∗ n+j −re〈∆j,zs+jx ∗ s+j〉∆j ) , j = 1, . . . ,f. (4.13) where vi > 0 and ζj > 0. if the right-hand sides are different from zero then z ∈ b∗1 . corollary 4.7. the result of the lemma 4.6 can be written as follows: z∗ = d1pb∗ (zx ∗) −d2∆, (4.14) where pb∗ (·) is the orthogonal projection and d1,d2 ∈ b∗ are diagonal matrices with d1 is positive. 4.8. the gradient system of ordinary differential equations. following lemma 4.6 and corollary 4.7 we consider the following differential equations on the manifold b∗1 . ∆̇ = d1pb∗ (zx ∗) −d2∆. (4.15) where, x(t) is an eigenvector associated to a simple eigenvalue λ(t) of the matrix �m∆(t) for some fixed � > 0. also note that the quantities z(t), d1 and d2 depend on the choice of the matrix valued function that is ∆(t) as well. the differential equ. (4.15) is a gradient system of odes. int. j. anal. appl. 17 (1) (2019) 38 4.9. choice of initial value matrix and �0. in order to compute �0 we choose the initial value matrix ∆0 = dp∆b (wv ∗), (4.16) where d is the positive diagonal matrix such that ∆0 ∈ b∗. as a natural choice for the initialization of the perturbation level, we take �0 as �0 = 1 µ̂b∗ (m) . (4.17) where µ̂b∗ (m) is the upper bound of µ-value approximated by mussv. 5. numerical testing in this section we present various numerical experimentations for both pure and mixed real and complex perturbations. the comparisons of lower bounds of structured singular values for a class of matrices obtained for dc motor are considered. example 1. consider the following five dimensional matrix. m =   −.050 − .113i −.012 − .028i −.216 + .072i −.000 − .000i 6.306 − 2.099i .113 − .050i .028 − .012i −.072 − .216i .000 − .000i 2.099 + 6.306i −.050 − .113i −.012 − .028i −.073 + .072i −.000 − .000i 6.306 − 2.099i −.004 − .008i −.001 − .002i .001 + .005i .142 − .000i −.042 − .153i −.004 − .008i −.001 − .002i −.001 − .002i −.000 − .000i −.042 − .153i   . also, consider the set of block diagonal uncertainties as an input argument. the uncertainty set is taken as b = {diag(δ1i2, ∆1) : δ1 ∈ r, ∆1 ∈ c3,3}. making use of matlab function mussv, we obtain an admissible perturbation set ∆̂, which is given below ∆̂ =   −.149 0 0 0 0 0 −.149 0 0 0 0 0 −.001 − .001i 0 0 0 0 .000 − .0001i 0 0 0 0 −.143 − .044i −.0008 + 0.0033i −.0008 + .0033i   . the 2-norm of admissible perturbation is obtained as 0.1499 while the lower bound of structured singular value is obtained as µlpd(m) = 6.6709 and an upper bound µ u pd(m) = 6.6715. int. j. anal. appl. 17 (1) (2019) 39 applying the algorithm presented in article [17], we obtain the admissible uncertainty �∇̂ with � = 0.1499 and ∇̂ with ∇̂ =   −1 0 0 0 0 0 −1 0 0 0 0 0 −0.0115 − 0.0100i 0.0002 − 0.0002i 0.0002 − 0.0002i 0 0 0.0001 − 0.0003i 0 0 0 0 −0.9541 − 0.2976i −0.0065 + 0.0207i −0.0070 + 0.0206i   . in this case the admissible uncertainty has a unit 2-norm while the obtained lower bound of structured singular value is µlour(m) = 6.6708. example 2. consider the following five dimensional matrix. m =   −.169 − .155i −.042 − .038i −.176 + .137i −.000 − .000i 5.136 − 3.998i .310 − .338i .077 − .084i −.274 − .352i .001 − .001i 7.997 + 10.272i −.169 − .155i −.042 − .038i −.033 + .137i −.000 − .000i 5.136 − 3.998i −.014 − .008i −.003 − .002i .005 + .009i .142 − .000i −.168 − .266i −.014 − .008i −.003 − .002i .005 − .009i .142 − .000i −.168 − .266i   . also, consider the set of block diagonal uncertainties as an input argument. the uncertainty set is taken as b = {diag(δ1i1, ∆1) : δ1 ∈ r, ∆1 ∈ c4,4}. making use of matlab function mussv, we obtain an admissible perturbation set ∆̂, which is given below ∆̂ =   −0.0679 0 0 0 0 0 −.0004 + .0004i −.0002 + .0002i 0 0 0 −.0013 + .0014i −.0007 − .0006i 0 0 0 0 0 0 0 0 .036 − .048i −.024 + .018i −.0007 + .001i −.0007 + .001i   . the 2-norm of admissible perturbation is obtained as 0.1499 while the lower bound of structured singular value is obtained as µlpd(m) = 14.7375 and an upper bound µ u pd(m) = 14.7988. int. j. anal. appl. 17 (1) (2019) 40 applying the algorithm presented in article [17], we obtain the admissible uncertainty �∇̂ with � = 0.0679 and ∇̂ with ∇̂ =   −1 0 0 0 0 0 .005 + .005i −.002 + .002i −.0001 − .0001i −.0001 − .0001i 0 −.018 + .020i −.010 − .009i .0004 − .0005i .0004 − .0005i 0 −.0001 + .0001i 0 0 0 0 .541 + .711i .355 + .270i −.0109 + .018i −.010 + .181i   . in this case the admissible uncertainty has a unit 2-norm while the obtained lower bound of structured singular value is µlour(m) = 14.7375. example 3. consider the following five dimensional matrix. m =   −.283 − .099i −.070 − .024i −.109 + .178i −.001 − .000i 3.195 − 5.200i .299 − .849i .074 − .212i −.534 − .328i .001 − .003i 15.600 + 9.587i −.283 − .099i −.070 − .024i −.033 + .178i −.001 − .000i 3.195 − 5.200i −.021 − .001i −.005 − .0003i .011 + .009i .142 − .000i −.346 − .286i −.021 − .001i −.005 − .0003i .011 + .009i −.0001 − .000 −.346 − .286i   . also, consider the set of block diagonal uncertainties as an input argument. the uncertainty set is taken as b = {diag(δ1i1,δ2i1,δ3i1, ∆1) : δ1,δ2,δ3 ∈ r, ∆1 ∈ c2,2}. making use of matlab function mussv, we obtain an admissible perturbation set ∆̂, which is given below ∆̂ =   −.586 + 1.067i 0 0 0 0 0 −.586 + 1.067i 0 0 0 0 0 −.586 + 1.067i 0 0 0 0 0 .109 .105 0 0 0 −.594 + .631i −.576 + .612i   . the 2-norm of admissible perturbation is obtained as 1.2176 while the lower bound of structured singular value is obtained as µlpd(m) = 0.8213 and an upper bound µ u pd(m) = 0.8213. int. j. anal. appl. 17 (1) (2019) 41 applying the algorithm presented in article [17], we obtain the admissible uncertainty �∇̂ with � = 1.2176 and ∇̂ with ∇̂ =   −.492 + .870 0 0 0 0 0 −.492 + .870 0 0 0 0 0 −.492 + .870 0 0 0 0 0 .087 + .0009i .085 + .001i 0 0 0 −.484 + .520i −.472 + .506i   . in this case the admissible uncertainty has a unit 2-norm while the obtained lower bound of structured singular value is µlour(m) = 0.8213. example 4. consider the following five dimensional matrix. m =   −.326 + .009i −.081 − .002i −.036 + .181i −.001 − .000i 1.065 − 5.282i −.038 − 1.306i −.009 − .326i −.724 − .146i −.0002 − .005i 21.128 + 4.260i −.326 + .009i −.081 + .002i .106 + .181i −.001 1.065 − 5.282i −.020 + .009i −.005 + 0.002i .017 + .006i .142 − .000i −.498 − .196i −.020 + .009i −.005 + .002i .017 + .006i −.0001 − .000i −.498 − 0.196i   . also, consider the set of block diagonal uncertainties as an input argument. the uncertainty set is taken as b = {diag(δ1i1,δ2i1, ∆1) : δ1,δ2,∈ r, ∆1 ∈ c3,3}. making use of matlab function mussv, we obtain an admissible perturbation set ∆̂, which is given below ∆̂ =   −.173 0 0 0 0 0 −.134 0 0 0 0 0 .002 − .005i .0006 + .0002i .0006 + .0002i 0 0 −.0004i 0 0 0 0 .041 + .166i −.014 + .006i −.014 + .006i   . the 2-norm of admissible perturbation is obtained as 0.1734 while the lower bound of structured singular value is obtained as µlpd(m) = 5.7675 and an upper bound µ u pd(m) = 5.7786. int. j. anal. appl. 17 (1) (2019) 42 applying the algorithm presented in article [17], we obtain the admissible uncertainty �∇̂ with � = 2.6959 and ∇̂ with ∇̂ =   −1 0 0 0 0 0 −.911 0 0 0 0 0 .015 − .032i .003 + .0009i .003 + .0009i 0 0 .0002 − .002i .0002 .0002 0 0 .247 + .959i −.083 + .038i −.083 + .038i   . in this case the admissible uncertainty has a unit 2-norm while the obtained lower bound of structured singular value is µlour(m) = 0.3709. in each of following figure, we present the comparison of lower bounds of structured singular values obtained by matlab routine mussv and the algorithm presented in [17], for five dimensional complex matrices. each of these matrix is computed from [16]. 6. conclusion in this article we have considered the problem for the computation of the lower bound of structured singular values for a family of complex matrices obtained in [16]. the computation of structured singular values plays an important role in robust stability and instability in control. the experimental results show the comparison of the lower bounds computed by algorithm [17] when compared to well-known1 matlab function mussv available in matlab control toolbox. 7. nomenclature b family of block diagonal matrices �0 perturbation level ∆0 initial admissible perturbation µ structured singular values references [1] packard, andrew and doyle, john. the complex structured singular value. automatica, 29(1993): 71-109. [2] bernhardsson, bo and rantzer, anders and qiu, li. real perturbation values and real quadratic forms in a complex vector space. linear algebra appl., 270(1998): 131-154. [3] chen, jie and fan, michael kh and nett, carl n. structured singular values with nondiagonal structures. i. characterizations. ieee trans. automatic control, 41(1996): 1507-1511. [4] chen, jie and fan, michael kh and nett, carl n. structured singular values with nondiagonal structures. ii. computation. ieee trans. automatic control, 41(1996): 1511-1516. int. j. anal. appl. 17 (1) (2019) 43 [5] hinrichsen, diederich and pritchard, anthony j. mathematical systems theory i: modelling, state space analysis, stability and robustness. vol. 48. berlin: springer, 2005. [6] karow, michael. µ-values and spectral value sets for linear perturbation classes defined by a scalar product. siam j. matrix anal. appl., 32(2011): 845-865. [7] karow, michael and hinrichsen, diederich and pritchard, anthony j. interconnected systems with uncertain couplings: explicit formulae for mu-values, spectral value sets, and stability radii. siam j. control optim., 45(2006): 856-884. [8] qiu, li and bernhardsson, bo and rantzer, anders and davison, edward j and doyle, jc. a formula for computation of the real stability radius, automatica, 31(1995): 879890. [9] zhou, kemin and doyle, john comstock and glover, keith and others. robust and optimal control. prentice hall new jersey, volume 40: (1996). [10] braatz, richard p and young, peter m and doyle, john c and morari, manfred. computational complexity of µ calculation. ieee trans. automatic control, 39(1994): 1000-1002. [11] young, peter m and newlin, matthew p and doyle, john c. practical computation of the mixed µ problem. american control conference : 2190-2194 (1992) [12] fan, michael kh and tits, andre l and doyle, john c. robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. ieee trans. automatic control, 36(1991): 25-38. [13] young, peter m and doyle, john c and packard, andy and others. theoretical and computational aspects of the structured singular value. syst. control inf., 38(1994): 129-138. [14] packard, andy and fan, michael kh and doyle, john. a power method for the structured singular value. decision and control, 1988., proceedings of the 27th ieee conference on, 2132-2137 (1988). [15] kato, t. perturbation theory for linear operators, classics in mathematics (springer-verlag, berlin, 1995). reprint of the 1980 edition, (1980). [16] fabrizi, andrea and roos, clement and biannic, jean-marc. a detailed comparative analysis of µ lower bound algorithms. european control conference 2014, (2014). [17] rehman, mutti-ur and tabassum, shabana. numerical computation of structured singular values for companion matrices. j. appl. math. phys., 5(2017): 1057-1072. int. j. anal. appl. 17 (1) (2019) 44 figure 1. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. figure 2. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. int. j. anal. appl. 17 (1) (2019) 45 figure 3. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. figure 4. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. int. j. anal. appl. 17 (1) (2019) 46 figure 5. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. figure 6. the comparison of lower bounds of structured singular values for the frequency = 1, 2, 3. 1. introduction 2. overview of the article 3. framework 3.1. overview of the proposed methodology 4. computation of local extremizers 5. numerical testing 6. conclusion 7. nomenclature references international journal of analysis and applications volume 17, number 3 (2019), 361-368 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-361 linear functionals connected with strong double cesaro summability fatih nuray1 and nimet pancaroḡlu akin2,∗ 1department of mathematics, afyon kocatepe university, turkey 2department of mathematics and science education, afyon kocatepe university, turkey ∗corresponding author: npancaroglu@aku.edu.tr abstract. d. borwein characterized linear functionals on the normed linear spaces wp and wp. in this paper we extend his results by presenting definitions for the double strong cesaro mean. using these new notions of strongly p-cesaro summable double sequence and strongly p-cesaro summable bivariate function we present extensions of d. borwein’s results. 1. introduction the first definitions and investigations of the convergence of double sequences are usually atributted to f. pringsheim [12], who studied such sequences and series more than hundred years ago. pringsheim defined what we call the p limit and gave examples of convergence (p convergence) of double sequences with and without the usual convergence of rows and columns. g. h. hardy [4], considered in more details the case of convergence of double sequences where, besides the existence of the p limit, rows and columns converge. f. moricz [6–8] discovered an alternative approach to the hardy convergence, which significantly influenced the whole theory. the following notion of convergence for double sequences was presented by pringsheim in [11]. a double sequence x = {xnm} of real numbers is said to be convergent to l ∈ r in pringsheim’s sense if for any ε > 0, received 2019-01-10; accepted 2019-02-12; published 2019-05-01. 2010 mathematics subject classification. primary 40a05; secondary 40c05. key words and phrases. double sequence; measurable function; bivariate function; cesaro summable function. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 361 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-361 int. j. anal. appl. 17 (3) (2019) 362 there exists nε ∈ n such that |xnm − l| < ε, whenever n,m > nε. in this case we denote such limit as follow: p − lim n,m→∞ xnm = l. a classical notion of sequence space is the following: wp = {x = (xn) : lim n→∞ 1 n n∑ n=1 |xn − `|p = 0}. in [2], d. borwein extended the sequence space wp to the function space wp, the space of real valued functions x, measurable (in the lebesque sense) in the interval (1,∞) for which there is a number ` = `x such that lim t→∞ 1 t ∫ t 1 |x(t) − `|p = 0. by a linear functional we mean one that is real-valued, additive, homogeneous and continuous. it is to be supposed throughout that 1 ≤ p < ∞ and that 1 p + 1 q = 1. 2. main results we begin to the main results with following definitions: definition 2.1. let x = {xnm} be a real double sequence. then the double sequence x is said to be strongly p-cesaro summable to ` if p − lim n,m→∞ 1 nm n∑ n=1 m∑ m=1 |xnm − `|p = 0. the space of all strongly p-cesaro summable double sequences will be denote by w2p. observe that this space is normed by ‖x‖2 = sup n,m≥1 ( 1 nm n∑ n=1 m∑ m=1 |xnm − `|p )1 p . definition 2.2. let x be a real valued bivariate function, measurable (in the lebesque sense) in the (1,∞)× (1,∞). then the bivariate function x is said to be strongly p-cesaro summable to ` if lim t,r→∞ 1 tr ∫ t 1 ∫ r 1 |x(t,r) − `|pdrdt = 0. the space of all strongly p-cesaro summable bivariate functions will be denote by w2p . observe that this space is normed by ‖x‖2 = sup t≥1,r≥1 ( 1 tr ∫ t 1 ∫ r 1 |x(t,r) − `|pdrdt )1 p . given any real double sequence α = {αnm}. we define a double sequence {mnm(α,p)} by int. j. anal. appl. 17 (3) (2019) 363 mnm(α,p) =   sup 2n ≤ v < 2n+1; 2m ≤ u < 2m+1 {|vuαvu|}, if p = 1 ( 1 2n+m ∑2n+1−1 v=2n ∑2m+1−1 u=2m |vuαvu| q )1 q , if p > 1. given any real real valued bivariate function α(t,r) measurable in (1,∞) × (1,∞). we define a double sequence {mnm(α,p)} by mnm(α,p) =   ess.sup 2n ≤ t < 2n+1; 2m ≤ r < 2m+1 {|trα(t,r)|}, if p = 1 ( 1 2n+m ∫ 2n+1 2n ∫ 2m+1 2m |trα(t,r)|q )1 q , if p > 1. theorem 2.1. (i) if f is a linear functional on w2p , then there is a real number a and a real valued bivariate function α, measurable in (1,∞) × (1,∞) such that f(x) = a` + ∫ ∞ 1 ∫ ∞ 1 α(t,r)x(t,r)drdt (2.1) for every x ∈ w2p and ∞∑ n=0 ∞∑ m=0 mnm(α,p) < ∞. (2.2) (ii) if a is a real number and α is a real valued bivariate function, measurable in (1,∞)×(1,∞), satisfying (2.2), then (2.1) defines a linear function on w2p with ‖f‖2 ≤ |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm(α,p) and the integral in (2.1) is absolutely convergent for every x ∈ w2p . proof. let l2p be the linear space of real valued bivariate functions x measurable in (1,∞)×(1,∞) for which∫ ∞ 1 ∫ ∞ 1 |x(t,r)|pdrdt < ∞, with norm ‖x‖l2p = (∫ ∞ 1 ∫ ∞ 1 |x(t,r)|pdrdt )1 p . clearly, if x ∈ l2p, then x ∈ w2p , ` = 0 and ‖x‖2 = ‖x‖w2p ≤‖x‖l2p. consequently the restriction to l 2 p of the given linear functional f on w2p is linear on l 2 p. it follows from standard results that there is a real valued bivariate function α, measurable in (1,∞) × (1,∞), such that f(x) = ∫ ∞ 1 ∫ ∞ 1 α(t,r)x(t,r)drdt (2.3) int. j. anal. appl. 17 (3) (2019) 364 for all x ∈ l2p and either ess.sup {|α(t,r)|} < ∞ if p = 1 1 ≤ t < ∞ 1 ≤ r < ∞ or ∫ ∞ 1 ∫ ∞ 1 |α(t,r)|qdrdt < ∞ if p > 1. to show that α must necessarily satisfy (2.2) we consider the cases p = 1 and p > 1 separately. if p = 1, let mnm = mnm(α, 1). there is a measurable set enm of positive measure |enm| in the (2n, 2n+1)×(2m, 2m+1) such that |trα(t,r)| > mnm − 1 2n+m for all (t,r) ∈ enm. let x(t,r) =   2n+m enm sign(α(t,r)), if (t,r) ∈ enm,n ≤ s,m ≤ u 0, otherwise. then x ∈ l21 and so, by (2.3), ‖f‖2‖x‖2 ≥ f(x) = ∫ ∞ 1 ∫ ∞ 1 x(t,r)α(t,r)drdt = s∑ n=0 u∑ m=0 ∫ ∫ enm 2n+m |enm| |α(t,r)|drdt ≥ 1 4 s∑ n=0 u∑ m=0 1 |enm| ∫ ∫ enm |trα(t,r)|drdt (2.4) ≥ 1 4 s∑ n=0 u∑ m=0 (mnm − 1 2n+m ). furtermore, for 2z ≤ t < 2z+1 ≤ 2s+1, 2h ≤ r < 2h+1 ≤ 2u+1, 1 tr ∫ t 1 ∫ r 1 |x(t,r)|drdt ≤ 1 2z+h ∫ 2z+1 1 ∫ 2h+1 1 |x(t,r)|drdt = 1 2z+h z∑ n=0 h∑ m=0 ∫ ∫ enm x(t,r)|drdt ≤ 1 2z+h z∑ n=0 h∑ m=0 2n+m < 4, and for t > 2s+1, r > 2u+1 1 tr ∫ t 1 ∫ r 1 |x(t,r)|drdt ≤ 1 2s+12u+1 ∫ 2s+1 1 ∫ 2u+1 1 |x(t,r)|drdt < 1. int. j. anal. appl. 17 (3) (2019) 365 hence ‖x‖2 < 4 and so, by (2.4), 4‖f‖2 + 1 4 ∞∑ n=0 ∞∑ m=0 1 2n+m = 4‖f‖2 + 1 ≥ 1 4 ∞∑ n=0 ∞∑ m=0 mnm, which establishes (2.2) in this case. if p > 1, let mnm = mnm(α,p) and let x(t,r) =   (tr)q 2n+m |α(t,r) mnm | q p sign(α(t,r)), if 2n ≤ t < 2n+1 ≤ 2z+1; 2m ≤ r < 2m+1 ≤ 2u+1 and mnm 6= 0 0, otherwise. then x ∈ l2p and so, by (2.3), f(x) = ∫ 2z+1 1 ∫ 2u+1 1 |α(t,r)x(t,r)|drdt = z∑ n=0 u∑ m=0 ∫ 2n+1 2n ∫ 2m+1 2m |α(t,r)x(t,r)|drdt = z∑ n=0 u∑ m=0 mnm. (2.5) furtermore, for 2z ≤ t < 2z+1 ≤ 2s+1, 2h ≤ r < 2h+1 ≤ 2u+1, 1 tr ∫ t 1 ∫ r 1 |x(t,r)|pdrdt ≤ 1 2z+h ∫ 2z+1 1 ∫ 2h+1 1 |x(t,r)|pdrdt = 1 2z+h z∑ n=0 h∑ m=0 ∫ ∫ enm |x(t,r)|pdrdt ≤ 22p 2z+h z∑ n=0 h∑ m=0 2n+m < 22p+2, and for t > 2z+1, r > 2h+1 1 tr ∫ t 1 ∫ r 1 |x(t,r)|pdrdt ≤ 1 2z+12h+1 ∫ 2z+1 1 ∫ 2h+1 1 |x(t,r)|pdrdt < 4p. hence ‖x‖2 < 22+ 2 p and so, by (2.5), ∞∑ n=0 ∞∑ m=0 mnm ≤ 22+ 2 p‖f‖2, which established (2.2) in this case. suppose now p ≥ 1, mnm = mnm(α,p) and x ∈ w2p . then by hölder inequality∫ ∞ 1 ∫ ∞ 1 |α(t,r)x(t,r)|drdt = ∞∑ n=0 ∞∑ m=0 ∫ 2n+1 2n ∫ 2m+1 2m |α(t,r)x(t,r)|drdt ≤ ∞∑ n=0 ∞∑ m=0 mnm ( 2p(1− 1 p )(n+m) ∫ 2n+1 2n ∫ 2m+1 2m ∣∣∣∣x(t,r)tr ∣∣∣∣p drdt )1 p ≤ ∞∑ n=0 ∞∑ m=0 mnm ( 2−(n+m) ∫ 2n+1 2n ∫ 2m+1 2m |x(t,r)|pdrdt )1 p int. j. anal. appl. 17 (3) (2019) 366 ≤ 2 2 p‖x‖2 ∞∑ n=0 ∞∑ m=0 mnm. (2.6) it follows that ∫ ∞ 1 ∫ ∞ 1 |α(t,r)x(t,r)|drdt < ∞ whenever x ∈ w2p , and in particular since the characteristic function of (1,∞) × (1,∞) is in w2p , that∫ ∞ 1 ∫ ∞ 1 |α(t,r)|drdt < ∞. suppose next that x ∈ w2p and ` = `x. let y(t,r) = x(t,r) − ` ynm(t,r) =   y(t,r), if 1 ≤ t ≤ n, 1 ≤ r ≤ m;0, if t ≥ n and r ≥ m. then y ∈ w2p , ynm ∈ l2p and ‖ynm −y‖2 = sup t≥n,r≥m ( 1 tr ∫ t n ∫ r m |x(t,r) − `|p )1 p = o(1) as n,m →∞. but |f(ynm −y)| = |f(ynm) −f(y)| ≤ ‖ynm −y‖2‖f‖2, and so, by (2.3), f(y) = p − lim n,m→∞ f(ynm) = p − lim n,m→∞ ∫ n 1 ∫ m 1 y(t,r)α(t,r)drdt = ∫ ∞ 1 ∫ ∞ 1 x(t,r)α(t,r)drdt− ` ∫ ∞ 1 ∫ ∞ 1 α(t,r)drdt. since both integrals on the right hand side have been shown to be absolutely convergent. taking δ to be characteristic function of (1,∞) × (1,∞) we see that f(x) = f(y + `δ)f(y) + `f(δ) = ∫ ∞ 1 ∫ ∞ 1 x(t,r)α(t,r)drdt + a` where a = f(δ) − ∫∞ 1 ∫∞ 1 α(t,r). this completes the proof of part (i. (ii) it follows from (2.6) that if x ∈ w2p , ` = `x and mnm = mnm(α,p), then |f(x)| = ∣∣∣∣ ∫ ∞ 1 ∫ ∞ 1 x(t,r)α(t,r)drdt + a` ∣∣∣∣ ≤‖x‖22 2p ∞∑ n=0 ∞∑ m=0 mnm + |a`|. (2.7) further, by minkowski’s inequality ( 1 − 1 tr )1 p |`| ≤ ( 1 tr ∫ t 1 ∫ r 1 |x(t,r) − `|pdrdt )1 p + ( 1 tr ∫ t 1 ∫ r 1 |x(t,r)|pdrdt )1 p and the first term on the right hand side is o(1). hence |`| ≤ ‖x‖2 and consequently, by (2.7), |f(x)| ≤ ‖x‖2 ( |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm ) int. j. anal. appl. 17 (3) (2019) 367 for every x ∈ w2p . the additive and homogenous functional f defined by (2.1) is therefore also continuous on w2p and |f(x)| ≤ |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm. finally, by (2.6), the integral in (2.1) is absolutely convergent. thus the proof is completed. � theorem 2.2. (i) if f is a linear functional on w2p, then there is a real number a and a real double sequence α = {αnm} such that f(x) = a` + ∞∑ n=1 ∞∑ m=1 αnmxnm (2.8) for every x = {xnm}∈ w2p and ∞∑ n=0 ∞∑ m=0 mnm(α,p) < ∞. (2.9) (ii) if a is a real number and α = {αnm} is a real double sequence satisfying (2.9), then (2.8) defines a linear function on w2p with ‖f‖2 ≤ |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm(α,p) and the series in (2.8) is absolutely convergent for every x = {xnm}∈ w2p. proof. given any real double sequence x = {xnm}, define a bivariate function x∗ by x∗(t,r) = xnm for n < t ≤ n + 1; m < r ≤ m + 1,n = 1, 2, 3, ...,m = 1, 2, 3, .... it is easily verified that this defines a one to one correspondence between w2p and a linear subspace (w 2 p ) ∗ of w2p such that `x∗ = `x and ‖x∗‖2 ≤‖x‖2 ≤ 2 2 p‖x∗‖2. hence given a linear functional on w2p , the functional f ∗ defined by f∗(x∗) = f(x) is linear on (w2p ) ∗. consequently, by the hahn-banach theorem and theorem2.1, there is a real number a and a real valued bivariate function α∗, integrable over (1,∞) × 1,∞), such that ∞∑ n=0 ∞∑ m=0 mnm(α ∗,p) < ∞ and, for every x ∈ w2p, f(x) = f∗(x∗) = a`x∗ + ∫ ∞ 1 ∫ ∞ 1 α∗(t,r)x∗(t,r)drdt = a`x + ∞∑ n=1 ∞∑ m=1 αnmxnm int. j. anal. appl. 17 (3) (2019) 368 where αnm = ∫n+1 n ∫m+1 m α∗(t,r)drdt. furthermore, for α = {αnm}, ∞∑ n=0 ∞∑ m=0 mnm(α,p) ≤ ∞∑ n=0 ∞∑ m=0 mnm(α ∗,p); and this completes the proof of (i). (ii) if x = {xnm} ∈ w2p mnm = mnm(α,p) and ` = `x then by hölder’s and minkowski’s inequalities, as in the proof of (ii) of theorem2.1, f(x) = a` + ∞∑ n=1 ∞∑ m=1 αnmxnm ≤ |a`| + ∞∑ n=1 ∞∑ m=1 |αnmxnm| ≤ |a`| + 2 2 p‖x‖2 ∞∑ n=0 ∞∑ m=0 mnm ≤‖x‖2 ( |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm ) . the functional f defined by (2.8) is therefore linear on w2p, ‖f‖2 ≤ |a| + 2 2 p ∞∑ n=0 ∞∑ m=0 mnm and the series in (2.8) absolutely convergent. this completes the proof. � references [1] s. banach, theorie des operations lineaires, warsaw, 1932. [2] d. borwein, linear functionals connected with strong cesaro summability, j. london math. soc. 40 (1965), 628-634. [3] t. j. i. a. bromwich, an introduction to the theory of infinite series, second ed., cambridge, 1926 (repr. macmillan, london, 1955). [4] g. h. hardy, on the convergence of certain multiplie series, proc. cambridge philos. soc. 19 (1916-1919), 86-95. [5] f. moricz, tauberian theorems for cesaro summable double sequences, studia math. 110 (1994), 83–96. [6] f. moricz, on the convergence in a restricted sense of multiple series, anal. math. 5(1979), 135–147. [7] f. moricz, some remarks on the notion of regular convergence of multiple series, acta math. hungar. 41 (1983), 161–168. [8] f. moricz, extensions of the spaces c and c0 from single to double sequences, acta math. hungar. 57 (1991), 129–136. [9] mursaleen and osama h. h. edely statistical convergence of double sequences, j. math. anal. appl. 288 (2003), 223–231. [10] r. f. patterson, double sequence core theorems, internat. j. math. and math. sci. 22 (1999), 785–793. [11] a. pringsheim, zur theorie der zweifach unendlichen zahlenfolgen, math. ann. 53 (1900), 289-321. [12] a. pringsheim, elementare theorie der unendlichen doppelreihen, sitzung-berichte. der math.-phys. classe der akad. der wissenschafften zu munchen 27 (1897), 101–152. 1. introduction 2. main results references international journal of analysis and applications volume 16, number 1 (2018), 97-116 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-97 complex neutrosophic subsemigroups and ideals muhammad gulistan1,∗, asghar khan2, amir abdullah1, naveed yaqoob3 1department of mathematics, hazara university, mansehra, pakistan 2department of mathematics, abdul wali khan university mardan, mardan, pakistan 3department of mathematics, college of science al-zulfi, majmaah university, al-zulfi, saudi arabia ∗corresponding author: azhar4set@yahoo.com abstract. in this article we study the idea of complex neutrosophic subsemigroups. we define the cartesian product of complex neutrosophic subsemigroups, give some examples and study some of its related results. we also define complex neutrosophic (left, right, interior) ideal in semigroup. furthermore, we introduce the concept of characteristic function of complex neutrosophic sets, direct product of complex neutrosophic sets and study some results prove on its. 1. introduction in 1965, zadeh, ( [1]) presented the idea of a fuzzy set. atanassov in 1986, ( [2]) initiated the notion of intuitionistic fuzzy set, which is the generalization of a fuzzy set. neutrosophic set was first proposed by smarandache in 1999 ( [5]), which is the generalization of a fuzzy set and intuitionistic fuzzy set. neutrosophic set is characterized by a truth membership function, an indeterminacy membership function and a falsity membership function. it must be noted that there are lots of researchers that worked at complex number and fuzzy sets, for instance buckly ( [6]), nguyen et al. ( [7]) and zhang et al. ( [10]). on the other hand ramot received 19th september, 2017; accepted 5th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 03b52. key words and phrases. complex fuzzy sets; complex neutrosophic sets; fuzzy subsemigroups; complex neutrosophic subsemigroups; complex neutrosophic ideals. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 97 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-97 int. j. anal. appl. 16 (1) (2018) 98 et al. ( [8]) presented a innovative come close to that is entirely unlike from other researchers, wherever they extensive the variety of membership function to unit circle in the complex plane, unlike the others who limited to. further to solve enigma they added an extra terms which is called phase term in translating human language to complex valued functions on physical terms and vice versa (for more information, see ( [8]). abd uazeez et al. in 2012 ( [12]), added the non-membership term to the idea of complex fuzzy set which is known as complex intuitionistic fuzzy sets, the range of values are extended to the unit circle in complex plan for both membership and non-membership functions instead of [0, 1]. in 2016, mumtaz ali et al. ( [14]), more extended the concept of complex fuzzy set, complex intuitionistic fuzzy set, and introduced the concept of complex neutrosophic sets, which is a collection of a complex truth membership function, a complex indeterminacy membership function and a complex falsity membership function. the idea of a fuzzy set in the model of semigroups was first initiated by kuroki in 1979 ( [3]), and defined fuzzy subsemigroups. vildan and halis in 2017 ( [15]), extended the concept of fuzzy subgroups on the base of neutrosophic sets, which is known as neutrosophic subgroups . due to the motivation and inspiration of the above discussion. in this paper we are initiating the study of complex neutrosophic semigroups. this paper introduce the notion of complex neutrosophic subsemigroups and cartesian product of complex neutrosophic subsemigroups with the help of example. we define characteristic function of complex neutrosophic set, direct product of complex neutrosophic sets, complex neutrosophic ideals (left, right, interior) and proved some results. 2. preliminaries here in this part we gathered some basic helping materials. definition 2.1. ( [1]) a function f is defined from a universe x to a closed interval [0, 1] is called a fuzzy set, i.e., a mapping: f : x −→ [0, 1]. definition 2.2. ( [8]) a complex fuzzy set (cfs) c over the universe x, is defined an object having of the form: c = {(x,µc(x)) : x ∈x} where µc(x) = rc(x)·eiωc(x), here the amplitude term rc(x) and phase term ωc(x), are real valued functions, for every x ∈x , the amplitude term µc(x) : x → [0, 1] and phase term ωc(x) lying in the interval [0, 2π]. definition 2.3. ( [13]) let c1 and c2 be any two complex atanassov’s intuitionistic fuzzy sets (caifss) over the universe x, where c1 = {〈 x,rc1 (x) ·e iνc1 (x) ,kc1 (x) ·e iωc1 (x) 〉 : x ∈x } int. j. anal. appl. 16 (1) (2018) 99 and c2 = {〈 x,rc2 (x) ·e iνc2 (x) ,kc2 (x) ·e iωc2 (x) 〉 : x ∈x } . then 1. containment: c1 ⊆c2 ⇔ rc1 (x) ≤ rc2 (x),kc1 (x) ≥ kc2 (x) and νc1 (x) ≤ νc2 (x),ωc1 (x) ≥ ωc2 (x). 2. equal: c1 = c2 ⇔ rc1 (x) = rc2 (x),kc1 (x) = kc2 (x) and νc1 (x) = νc2 (x),ωc1 (x) = ωc2 (x). definition 2.4. ( [14]) let x be a universe of discourse, and x ∈x . a complex neutrosophic set (cns) c in x is characterized by a complex truth membership function ct (x) = pc(x) ·eiµc(x), a complex indeterminacy membership function ci(x) = qc(x)·eiνc(x) and a complex falsity membership function cf (x) = rc(x)·eiωc(x). the values ct (x),ci(x),cf (x) may lies all within the unit circle in the complex plane, where pc(x), qc(x), rc(x) and µc(x), νc(x) ωc(x) are amplitude terms and phase terms, respectively, and where pc(x), qc(x), rc(x) ∈ [0, 1], such that, 0 ≤ pc(x) + qc(x) + rc(x) ≤ 3 and µc(x), νc(x) ωc(x) ∈ [0, 2π]. the complex neutrosophic set can be represented in the form as: c =   〈 x,ct (x) = pc(x) ·eiµc(x),ci(x) = qc(x) ·eiνc(x), cf (x) = rc(x) ·eiωc(x) 〉 : x ∈x   . example 2.1. let x = {x1,x2,x3} be the universe set and c be a complex neutrosophic set which is given by: c =   〈 x1, 0.2e 0.5πi, 0.3e0.6πi, 0.4e0.8πi 〉 , 〈 x2, 0.4e 0.6πi, 0.5e1.3πi, 0.1e0.6πi 〉 ,〈 x3, 0.1e 0.6πi, 0.3e0.9πi, 0.9e0.7πi 〉   . definition 2.5. ( [3]) a fuzzy subset a of a semigroup s is said to be a fuzzy subsemigroup of s if its satisfy the following condition: a(x ·y) ≥a(x) ∧a(y) ∀ x,y ∈s. definition 2.6. ( [15]) let g be any group with multiplication and n be a neutrosophic set on g. then n is said to be a neutrosophic subgroup (nsg) of g, if its satisfy the following conditions: (nsg1): n(x ·y) ≥n(x) ∧n(y), i.e., tn(x ·y) ≥ tn(x) ∧tn(y), in(x ·y) ≥ in(x) ∧ in(y) and fn(x ·y) ≤ fn(x) ∨fn(y). (nsg2): n(x−1) ≥n(x), i.e., tn(x −1) ≥ tn(x), in(x−1) ≥ in(x) and fn(x−1) ≤ fn(x), for all x and y in g. int. j. anal. appl. 16 (1) (2018) 100 lemma 2.1. ( [16]) for a semigroup s, the following conditions are equivalent. (1) s is regular. (2) r∩l = rl for every right ideal r of s and every left ideal l of s. 3. complex neutrosophic subsemigroup note: it should be noted that through out in this part we use a capital letter c to denote a complex neutrosophic set; c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} . definition 3.1. a complex neutrosophic set c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} on a semigroup s is known as a complex neutrosophc subsemigroup (cnsg), if its satisfy the following condition: c(xy) ≥ min{c(x),c(y)} i.e., (i) pc(xy) ·eiµc(xy) ≥ min{pc(x) ·eiµc(x),pc(y) ·eiµc(y)} (ii) qc(xy) ·eiνc(xy) ≥ min{qc(x) ·eiνc(x),qc(y) ·eiνc(y)} (iii) rc(xy) ·eiωc(xy) ≤ max{rc(x) ·eiωc(x),rc(y) ·eiωc(y)}, ∀ x, y ∈s. example 3.1. let s = {1, 2, 3} be a semigroup with the following multiplication table: · 1 2 3 1 1 2 3 2 2 1 3 3 3 3 3 consider a complex neutrosophic set c on s as: c =   〈 1, 0.9e0.7πi, 0.7e0.6πi, 0.5e0.4πi 〉 ,〈 2, 0.8e0.6πi, 0.6e0.5πi, 0.4e0.3πi 〉 ,〈 3, 0.5e0.4πi, 0.4e0.2πi, 0.3e0.2πi 〉   then clearly c is a complex neutrosophic subsemigroup of s. 3.1. cartesian product of complex neutrosophic subsemigroups. definition 3.2. let c1 = {〈 c1t = pc1e iµc1 ,c1i = qc1e iνc1 ,c1f = rc1e iωc1 〉} and c2 = {〈 c2t = pc2e iµc2 ,c2i = qc2e iνc2 ,c2f = rc2e iωc2 〉} int. j. anal. appl. 16 (1) (2018) 101 be any two complex neutrosophic subsemigroups of the semigroups s1 and s2 respectively. then the cartesian product of c1 and c2 denoted by c1 ×c2 is defined as: c1 ×c2 =   〈 (x,y), (c1 ×c2)t (x,y), (c1 ×c2)i(x,y), (c1 ×c2)f (x,y) 〉 / ∀x ∈s1,y ∈s2   where (c1 ×c2)t (x,y) = min{c1t (x),c2t (y)} , (c1 ×c2)i(x,y) = min{c1i(x),c2i(y)} , (c1 ×c2)f (x,y) = max{c1f (x),c2f (y)} , for all x in s1 and y in s2. example 3.2. let s1 = {1, 2, 3} and s2 = {a,b,c} are any two semigroups with the following multiplication tables: · 1 2 3 1 1 2 3 2 2 1 3 3 3 3 3 , · a b c a c b a b b b c c c c c consider c1 =   〈 1, 0.9e0.7πi, 0.7e0.6πi, 0.5e0.4πi 〉 , 〈 2, 0.8e0.6πi, 0.6e0.5πi, 0.4e0.3πi 〉 ,〈 3, 0.5e0.4πi, 0.4e0.2πi, 0.3e0.2πi 〉   and c2 =   〈 a, 0.8e0.7πi, 0.5e0.3πi, 0.4e0.4πi 〉 , 〈 b, 0.6e0.5πi, 0.5e0.4πi, 0.3e0.2πi 〉 ,〈 c, 0.8e0.7πi, 0.7e0.5πi, 0.3e0.2πi 〉   be any two complex neutrosophic subsemigroups of s1 and s2, respectively. now let x = 1 and y = a, then c1 ×c2 = {〈(c1 ×c2)t (1,a), (c1 ×c2)i(1,a), (c1 ×c2)f (1,a)〉 , ...} = {〈min{c1t (1),c2t (a)} , min{c1i(1),c2i(a)} , max{c1f (1), c2f (a)}〉 , ...} = { 〈 min{0.9e0.7πi, 0.8e0.7πi}, min{0.7e0.6πi, 0.5e0.3πi} , max{0.5e0.4πi, 0.4e0.4πi} 〉 , ...} = { 〈 0.8e0.7πi, 0.5e0.3πi, 0.5e0.4πi 〉 , ...}. 4. complex neutrosophic ideals in this section, we define some ideals namely complex neutrosophic (left, right, interior) ideal in semigroup, with the help of examples and study some of its related results. int. j. anal. appl. 16 (1) (2018) 102 4.1. complex neutrosophic left ideal. definition 4.1. a complex neutrosophic set c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} on a semigroup s is known as a complex neutrosophic left ideal of s, if c(xy) ≥c(y) i.e., (i) pc(xy) ·eiµc(xy) ≥ pc(y) ·eiµc(y) (ii) qc(xy) ·eiνc(xy) ≥ qc(y) ·eiνc(y) (iii) rc(xy) ·eiωc(xy) ≤ rc(y) ·eiωc(y), ∀ x, y ∈s. example 4.1. let s = {a,b,c,d} be a semigroup with the following multiplication table: · a b c d a a a a a b a a a a c a a b a d a a b b consider a complex neutrosophic set c on s as: c =   〈 a, 0.9e0.6πi, 0.8e0.5πi, 0.4e0.3πi 〉 , 〈 b, 0.7e0.5πi, 0.6e0.4πi, 0.5e0.4πi 〉 ,〈 c, 0.6e0.4πi, 0.4e0.3πi, 0.7e0.5πi 〉 , 〈 d, 0.5e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉   then c is a complex neutrosophic left ideal of s. 4.2. complex neutrosophic right ideal. definition 4.2. a complex neutrosophic set c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} on a semigroup s is known as a complex neutrosophic right ideal of s, if c(xy) ≥c(x) i.e., (i) pc(xy) ·eiµc(xy) ≥ pc(x) ·eiµc(x) (ii) qc(xy) ·eiνc(xy) ≥ qc(x) ·eiνc(x) (iii) rc(xy) ·eiωc(xy) ≤ rc(x) ·eiωc(x), ∀ x, y ∈s. 4.3. complex neutrosophic ideal. definition 4.3. a complex neutrosophic set c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} on a semigroup s is known as a complex neutrosophic ideal of s, if it is both a complex neutrosophic left ideal and a complex neutrosophic right ideal of s. int. j. anal. appl. 16 (1) (2018) 103 example 4.2. let s = {a,b,c} be a semigroup with the following cayley table: · a b c a a a a b a a a c a a c if we define a complex neutrosophic set c on s as: c =   〈 a, 0.8e0.6πi, 0.6e0.5πi, 0.5e0.4πi 〉 , 〈 b, 0.7e0.6πi, 0.5e0.4πi, 0.6e0.4πi 〉 ,〈 c, 0.7e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉   then obviously c is a complex neutrosophic ideal of s. remark 4.1. every complex neutrosophic left (resp., right) ideal is a complex neutrosophic subsemigroup. but the converse may not be true as seen in the following example. example 4.3. let s = {a,b,c,d} be a semigroup with the following cayley table: · a b c d a a a a a b a a a a c a a a b d a a b c take a complex neutrosophic set c on s as: c =   〈 a, 0.8e0.6πi, 0.6e0.5πi, 0.5e0.4πi 〉 , 〈 b, 0.6e0.6πi, 0.5e0.4πi, 0.6e0.4πi 〉 ,〈 c, 0.8e0.5πi, 0.4e0.3πi, 0.7e0.5πi 〉 , 〈 d, 0.4e0.4πi, 0.3e0.3πi, 0.7e0.5πi 〉   then clearly c is a complex neutrosophic subsemigroup of s. however it is not a complex neutrosophic right ideal of s, because tc (cd) = tc (b) = 0.6e 0.6πi � 0.8e0.5πi = tc (c) . 4.4. complex neutrosophic interior ideal. definition 4.4. a complex neutrosophic set c = {〈 tc = pc ·eiµc,ic = qc ·eiνc,fc = rc ·eiωc 〉} on a semigroup s is known as a complex neutrosophic interior ideal of s, if c(xκy) ≥c(κ) i.e., (i) pc(xκy) ·eiµc(xκy) ≥ pc(κ) ·eiµc(κ) (ii) qc(xκy) ·eiνc(xκy) ≥ qc(κ) ·eiνc(κ) (iii) rc(xκy) ·eiωc(xκy) ≤ rc(κ) ·eiωc(κ), ∀ x, κ, y ∈s. int. j. anal. appl. 16 (1) (2018) 104 example 4.4. let s = {a,b,c,d} be a semigroup with the following multiplication table: · a b c d a a a a a b a a a a c a a b a d a a b b . consider a complex neutrosophic set c on s as: c =   〈 a, 0.7e0.6πi, 0.6e0.4πi, 0.3e0.5πi 〉 , 〈 b, 0, 0.5e0.4πi, 0.5e0.6πi 〉 ,〈 c, 0.5e0.4πi, 0.4e0.3πi, 0.7e0.7πi 〉 , 〈 d, 0, 0.3e0.2πi, 0.7e0.7πi 〉   then c is a complex neutrosophic interior ideal of s. remark 4.2. every complex neutrosophic ideal is a complex neutrosophic interior ideal. but the converse may not be true as seen in the example 4.4. for left tc (dc) = tc (b) = 0 � 0.5e0.4πi = tc (c) right tc (dc) = tc (b) = 0 ≥ 0 = tc (d) . so it is a complex neutrosophic right ideal but not a left ideal. hence c is not a complex neutrosophic ideal. 5. characteristic function of complex neutrosophic set definition 5.1. let h be a non-empty subset over the universe x. then the characteristic complex neutrosophic function of h in x, defined to be a structure: ch = {〈x,tch (x),ich (x),fch (x)〉 : x ∈ h} where tch (x) =   1 ·e i2π if x ∈ h 0 otherwise ich (x) =   1 ·e i2π if x ∈ h 0 otherwise fch (x) =   0 if x ∈ h1 ·ei2π otherwise . definition 5.2. the characteristic function of whole complex neutrosophic set s in semigroup s is defined as; cs = {〈 (1̂tcs , 1 ·e i2π), (1̂ics , 1 ·e i2π), (0̂fcs , 0) 〉 : x ∈s } . int. j. anal. appl. 16 (1) (2018) 105 5.1. direct product of two complex neutrosophic sets. definition 5.3. let c1 = 〈 c1t = pc1e iµc1 ,c1i = qc1e iνc1 ,c1f = rc1e iωc1 〉 and c2 = 〈 c2t = pc2e iµc2 ,c2i = qc2e iνc2 ,c2f = rc2e iωc2 〉 be any two complex neutrosophic sets on s, then the product is define as; c1 ⊗c2 =   〈 x, (pc1 ◦pc2 )(x) ·ei(µc1◦µc2 )(x), (qc1 ◦qc2 )(x) ·ei(νc1◦νc2 )(x), (rc1 ◦rc2 )(x) ·ei(ωc1◦ωc2 )(x) 〉 : x ∈s   where (pc1 ◦pc2 )(x) ·e i(µc1◦µc2 )(x) =   sup x=yκ [ min{pc1 (y)eiµc1 (y),pc2 (κ)eiµc2 (κ)} ] if x = yκ for some y,κ ∈s 0 otherwise (qc1 ◦qc2 )(x) ·e i(νc1◦νc2 )(x) =   sup x=yκ [ min{qc1 (y)eiνc1 (y),qc2 (κ)eiνc2 (κ)} ] if x = yκ for some y,κ ∈s 0 otherwise (rc1 ◦rc2 )(x) ·e i(ωc1◦ωc2 )(x) =   inf x=yκ [ max{rc1 (y)eiωc1 (y),rc2 (κ)eiωc2 (κ)} ] if x = yκ for some y,κ ∈s 1 ·ei2π otherwise for all x in s. proposition 5.1. a complex neutrosophic sets c1,c2 and c3 of a semigroup s, if c1 ⊆ c2, then c1 ⊗c3 ⊆ c2 ⊗c3 and c3 ⊗c1 ⊆c3 ⊗c2. proof: we are proving c1 ⊗c3 ⊆c2 ⊗c3. since c1,c2 and c3 are complex neutrosophic sets of s. let x ∈s. case 1: if x is not expressed as x = yκ, then (c1 ⊗c3)(x) = 〈 0̂, 0̂, 1̂ 〉 and (c2 ⊗c3)(x) = 〈 0̂, 0̂, 1̂ 〉 . clearly, c1 ⊗c3 ⊆c2 ⊗c3. case 2: assume that there exist y,κ ∈s, such that x = yκ. then (pc1 ◦pc3 )(x) ·e i(µc1◦µc3 )(x) = sup x=yκ [ min{pc1 (y)e iµc1 (y),pc3 (κ)e iµc3 (κ)} ] ≤ sup x=yκ [ min{pc2 (y)e iµc2 (y),pc3 (κ)e iµc3 (κ)} ] = (pc2 ◦pc3 )(x) ·e i(µc2◦µc3 )(x). int. j. anal. appl. 16 (1) (2018) 106 similarly, (qc1 ◦qc3 )(x) ·e i(νc1◦νc3 )(x) ≤ (qc2 ◦qc3 )(x) ·e i(νc2◦νc3 )(x). and (rc1 ◦rc3 )(x) ·e i(ωc1◦ωc3 )(x) = inf [ max{rc1 (y)e iωc1 (y),rc3 (κ)e iωc3 (κ)} ] ≥ inf [ max{rc2 (y)e iωc2 (y),rc3 (κ)e iωc3 (κ)} ] = (rc2 ◦rc3 )(x) ·e i(ωc2◦ωc3 )(x). therefore, c1 ⊗c3 ⊆c2 ⊗c3. similarly we can proved c3 ⊗c1 ⊆c3 ⊗c2. � proposition 5.2. let h and k be any subsets of a semigroup s, we have (1) ch ⊗ck = chk ⇒〈tch ◦tck,ich ◦ ick,fch ◦fck〉 = 〈tchk,ichk,fchk〉 . (2) ch ∪ck = ch∪k ⇒〈tch ∪tck,ich ∪ ick,fch ∩fck〉 = 〈tch∪k,ich∪k,fch∩k〉 . (3) ch ∩ck = ch∩k ⇒〈tch ∩tck,ich ∩ ick,fch ∪fck〉 = 〈tch∩k,ich∩k,fch∪k〉 . proof: (1) let α ∈s. if α ∈ hk, then tchk (α) = 1.e i2π, ichk (α) = 1.e i2π and fchk (α) = 0 and α = mn for some m ∈ h and n ∈ k. thus, (tch ◦tck ) (α) = sup α=xy {min{tch (x),tck (y)}} ≥ min{tch (m),tck (n)} = 1.e i2π (ich ◦ ick ) (α) = sup α=xy {min{ich (x),ick (y)}} ≥ min{ich (m),ick (n)} = 1.e i2π and (fch ◦fck ) (α) = inf α=xy {max{fch (x),fck (y)}} ≤ max{fch (m),fck (n)} = 0. it follows that, (tch ◦tck ) (α) = 1.ei2π, (ich ◦ ick ) (α) = 1.ei2π and (fch ◦fck ) (α) = 0. therefore, 〈tch ◦tck,ich ◦ ick,fch ◦fck〉 = 〈tchk,ichk,fchk〉⇒ ch ⊗ck = chk. assume that α /∈ hk, then tchk (α) = 0, ichk (α) = 0 and fchk (α) = 1.e i2π. let y,κ ∈s be such that α = yκ, then we know that y /∈ h or κ /∈ k. assume that y /∈ h, then int. j. anal. appl. 16 (1) (2018) 107 (tch ◦tck ) (α) = sup α=yκ {min{tch (y),tck (κ)}} = sup α=yκ {min{0,tck (κ)}} = 0 = tchk (α) (ich ◦ ick ) (α) = sup α=yκ {min{ich (y),ick (κ)}} = sup α=yκ {min{0,ick (κ)}} = 0 = ichk (α) and (fch ◦fck ) (α) = inf α=yκ {max{fch (y),fck (κ)}} = inf α=yκ { max { 1.ei2π,fck (κ) }} = 1.ei2π = fchk (α). similarly, if κ /∈ k, then (tch ◦tck ) (α) = 0 = tchk (α), (ich ◦ ick ) (α) = 0 = ichk (α) and (fch ◦fck ) (α) = 1.ei2π = fchk (α). therefore ch ⊗ck = chk. proof of (2) and (3) are straightforward. � theorem 5.1. a complex neutrosophic set c on a semigroup s is a complex neutrosophic subsemigroup of s if and only if c⊗c ⊆c. proof: let c be a complex neutrosophic subsemigroup of s, and x ∈s. case 1: if x 6= yκ, for any y,κ ∈s, then obviously c⊗c ⊆c. case 2: if x = yκ, for any y,κ ∈s, then (pc ◦pc)(x) ·ei(µc◦µc)(x) = sup x=yκ [ min{pc(y)eiµc(y),pc(κ)eiµc(κ)} ] ≤ sup x=yκ [ pc(yκ)e iµc(yκ) ] = pc(x) ·eiµc(x). similarly, (qc ◦qc)(x) ·ei(νc◦νc)(x) ≤ qc(x) ·eiνc(x). int. j. anal. appl. 16 (1) (2018) 108 and (rc ◦rc)(x) ·ei(ωc◦ωc)(x) = inf x=yκ [ max{rc(y) ·eiωc(y),rc(κ) ·eiωc(κ)} ] ≥ inf x=yκ [rc(yκ) ·eiωc(yκ)] = rc(x) ·eiωc(x). therefore, c⊗c ⊆c. conversely, suppose c⊗c ⊆c, and assume x = yκ, then pc(yκ) ·eiµc(yκ) ≥ (pc ◦pc)(yκ) ·ei(µc◦µc)(yκ) = sup yκ=yκ [ min{pc(y)eiµc(y),pc(κ)eiµc(κ)} ] = min{pc(y)eiµc(y),pc(κ)eiµc(κ)}. similarly, qc(yκ) ·eiνc(yκ) ≥ min{qc(y)eiνc(y),qc(κ)eiνc(κ)}. and rc(yκ) ·eiωc(yκ) ≤ (rc ◦rc)(yκ) ·ei(ωc◦ωc)(yκ)) = inf yκ=yκ [ max{rc(y)eiωc(y),rc(κ)eiωc(κ)} ] = max{rc(y)eiωc(y),rc(κ)eiωc(κ)}. hence c is a complex neutrosophic subsemigroup of s. � proposition 5.3. a complex neutrosophic set c on a semigroup s, the following are equivalent: (1) c is a complex neutrosophic left ideal of s. (2) s⊗c ⊆c. proof: (1) ⇒ (2) : assume that c is a complex neutrosophic left ideal of s. let x ∈ s, such that (s⊗c)(x) = 〈 0̂, 0̂, 1̂ 〉 , then it is clear s⊗c ⊆c. whenever there exist any two elements y,κ ∈s, such that x = yκ. then (1̂st ◦pc ·e iµc )(x) = sup x=yκ [min{1̂st (y),pc(κ) ·e iµc(κ)}] ≤ sup x=yκ [min{1 ·ei2π,pc(yκ) ·eiµc(yκ)}] = pc(x) ·eiµc(x). similarly, (1̂si ◦qc ·e iνc )(x) ≤ qc(x) ·eiνc(x). int. j. anal. appl. 16 (1) (2018) 109 and (0̂sf ◦rc ·e iω)(x) = inf x=yκ [max{0̂sf (y),rc(κ) ·e iω(κ)}] ≥ inf x=yκ [max{0,rc(yκ) ·eiω(yκ)}] = rc(x) ·eiω(x). therefore, s⊗c ⊆c. conversely, (2) ⇒ (1) : suppose that s⊗c ⊆c. for any elements y,κ of s, let x = yκ. then pc(yκ) ·eiµc(yκ) = pc(x) ·eiµc(x) ≥ (1̂st ◦pc ·e iµc )(x) = sup x=yκ [min{1̂st (y),pc(κ) ·e iµc(κ)}] = pc(κ) ·eiµc(κ). similarly, qc(yκ) ·eiνc(yκ) ≥ qc(κ) ·eiνc(κ). and rc(yκ) ·eiωc(yκ) = rc(x) ·eiωc(x) ≤ (0̂sf ◦rc ·e iωc )(x) = inf x=yκ [max{0̂sf (y),rc(κ) ·e iωc(κ)}] = rc(κ) ·eiωc(κ). hence c is a complex neutrosophic left ideal of s. � proposition 5.4. a complex neutrosophic set c on a semigroup s, the following are equivalent: (1) c is a complex neutrosophic right ideal of s. (2) c⊗s ⊆c. proof: proof is similar to the proposition 5.3. � theorem 5.2. if c is a complex neutrosophic set of a semigroup s, then s⊗c (resp., c⊗s) is a complex neutrosophic left (resp. right) ideal of s. proof: since s⊗(s⊗c) = (s⊗s)⊗c ⊆s⊗c, it follows from proposition 5.3, that s⊗c is a complex neutrosophic left ideal of s. similarly c⊗s is a complex neutrosophic right ideal of s. � theorem 5.3. let s be a left zero subsemigroup of a semigroup s. if c is a complex neutrosophic left ideal of s, then c(x) = c(y) for all x,y ∈ s. int. j. anal. appl. 16 (1) (2018) 110 proof: let x,y ∈ s. then xy = x and yx = y. thus pc(x) ·eiµc(x) = pc(xy) ·eiµc(xy) ≥ pc(y) ·eiµc(y) = pc(yx) ·eiµc(yx) ≥ pc(x) ·eiµc(x). similarly, qc(x) ·eiνc(x) = qc(y) ·eiνc(y). and rc(x) ·eiωc(x) = rc(xy) ·eiωc(xy) ≤ rc(y) ·eiωc(y) = rc(yx) ·eiωc(yx) ≤ rc(x) ·eiωc(x). therefore, c(x) = c(y) for all x,y ∈ s. � theorem 5.4. let s be a right zero subsemigroup of a semigroup s. if c is a complex neutrosophic right ideal of s, then c(x) = c(y) for all x,y ∈ s. proof: proof is similar to the theorem 5.3. � theorem 5.5. let c is a complex neutrosophic left ideal of a semigroup s. if the set of all idempotent elements of s form a left zero subsemigroup of s, then c(x) = c(y) for all idempotent elements x and y of s. proof: let idm(s) be the set of all idempotent elements of s and assume that idm(s) is a left zero subsemigroup of s. for any x,y ∈idm(s), we have xy = x and yx = y. thus pc(x) ·eiµc(x) = pc(xy) ·eiµc(xy) ≥ pc(y) ·eiµc(y) = pc(yx) ·eiµc(yx) ≥ pc(x) ·eiµc(x) = pc(y) ·eiµc(y). similarly, qc(x) ·eiνc(x) = qc(y) ·eiνc(y). and rc(x) ·eiωc(x) = rc(xy) ·eiωc(xy) ≤ rc(y) ·eiωc(y) = rc(yx) ·eiωc(yx) ≤ rc(x) ·eiωc(x) = rc(y) ·eiωc(y). therefore, c(x) = c(y) for all x,y ∈idm(s). � int. j. anal. appl. 16 (1) (2018) 111 theorem 5.6. let c is a complex neutrosophic right ideal of a semigroup s. if the set of all idempotent elements of s form a right zero subsemigroup of s, then c(x) = c(y) for all idempotent elements x and y of s. proof: proof is similar to the theorem 5.5. � proposition 5.5. if s be a semigroup. then the following properties are hold. (1) the intersection of two complex neutrosophic subsemigroups of s is a complex neutrosophic subsemigroup of s. (2) the intersection of two complex neutrosophic left (resp., right) ideals of s is a complex neutrosophic left (resp., right) ideal of s. proof: let c1 = 〈 c1t = pc1 ·e iµc1 ,c1i = qc1 ·e iνc1 ,c1f = rc1 ·e iωc1 〉 and c2 = 〈 c2t = pc2 ·e iµc2 ,c2i = qc2 ·e iνc2 ,c2f = rc2 ·e iωc2 〉 be any two complex neutrosophic subsemigroups of s. let x,y ∈s. then (pc1 ·e iµc1 ∩pc2 ·e iµc2 )(xy) = min{pc1 (xy) ·e iµc1 (xy),pc2 (xy) ·e iµc2 (xy)} ≥ min{min{pc1 (x) ·e iµc1 (x),pc1 (y) ·e iµc1 (y)}, min{pc2 (x) ·e iµc2 (x),pc2 (y) ·e iµc2 (y)}} = min{min{pc1 (x) ·e iµc1 (x),pc2 (x) ·e iµc2 (x)}, min{pc1 (y) ·e iµc1 (y),pc2 (y) ·e iµc2 (y)}} = min{(pc1 ·e iµc1 ∩pc2 ·e iµc2 )(x), (pc1 ·e iµc1 ∩pc2 ·e iµc2 )(y)}. similarly, (qc1 ·e iνc1 ∩qc2 ·e iνc2 )(xy) ≥ min{(qc1 ·e iνc1 ∩qc2 ·e iνc2 )(x), (qc1 ·e iνc1 ∩qc2 ·e iνc2 )(y)}. int. j. anal. appl. 16 (1) (2018) 112 and (rc1 ·e iωc1 ∪rc2 ·e iωc2 )(xy) = max{rc1 (xy) ·e iωc1 (xy),rc2 (xy) ·e iωc2 (xy)} ≤ max{max{rc1 (x) ·e iωc1 (x),rc1 (y) ·e iωc1 (y)}, max{rc2 (x) ·e iωc2 (x),rc2 (y) ·e iωc2 (y)}} = max{max{rc1 (x) ·e iωc1 (x),rc2 (x) ·e iωc2 (x)}, max{rc1 (y) ·e iωc1 (y),rc2 (y) ·e iωc2 (y)}} = max{(rc1 ·e iωc1 ∪rc2 ·e iωc2 )(x), (rc1 ·e iωc1 ∪rc2 ·e iωc2 )(y)}. therefore, c1 ∩c2 is a complex neutrosophic subsemigroup of s. (2) let c1 and c2 be any two complex neutrosophic left ideals of semigroup s, and x,y ∈s. then (pc1 ·e iµc1 ∩pc2 ·e iµc2 )(xy) = min{pc1 (xy) ·e iµc1 (xy),pc2 (xy) ·e iµc2 (xy)} ≥ min{pc1 (y) ·e iµc1 (y),pc2 (y) ·e iµc2 (y)} = (pc1 ·e iµc1 ∩pc2 ·e iµc2 )(y). similarly, (qc1 ·e iνc1 ∩qc2 ·e iνc2 )(xy) ≥ (qc1 ·e iνc1 ∩qc2 ·e iνc2 )(y). and (rc1 ·e iωc1 ∪rc2 ·e iωc2 )(xy) = max{rc1 (xy) ·e iωc1 (xy),rc2 (xy) ·e iωc2 (xy)} ≤ max{rc1 (y) ·e iωc1 (y),rc2 (y) ·e iωc2 (y)} = (rc1 ·e iωc1 ∪rc2 ·e iωc2 )(y). thus c1 ∩c2 is a complex neutrosophic left ideal of semigroup s. the intersection of complex neutrosophic right ideal can be proved in a similar manner. � proposition 5.6. if s be a semigroup. then the following properties are hold. (1) the union of two complex neutrosophic subsemigroups of s is a complex neutrosophic subsemigroup of s. (2) the union of two complex neutrosophic left (resp., right) ideals of s is a complex neutrosophic left (resp., right) ideal of s. proof: let c1 = 〈 c1t = pc1 ·e iµc1 ,c1i = qc1 ·e iνc1 ,c1f = rc1 ·e iωc1 〉 int. j. anal. appl. 16 (1) (2018) 113 and c2 = 〈 c2t = pc2 ·e iµc2 ,c2i = qc2 ·e iνc2 ,c2f = rc2 ·e iωc2 〉 be any two complex neutrosophic subsemigroups of s. let x,y ∈s. then (pc1 ·e iµc1 ∪pc2 ·e iµc2 )(xy) = max{pc1 (xy) ·e iµc1 (xy),pc2 (xy) ·e iµc2 (xy)} ≥ max{min{pc1 (x) ·e iµc1 (x),pc1 (y) ·e iµc1 (y)}, min{pc2 (x) ·e iµc2 (x),pc2 (y) ·e iµc2 (y)}} = pc1 (x) ·e iµc1 (x) ∧pc1 (y) ·e iµc1 (y) ∨ pc2 (x) ·e iµc2 (x) ∧pc2 (y) ·e iµc2 (y) = pc1 (x) ·e iµc1 (x) ∨pc2 (x) ·e iµc2 (x) ∧ pc1 (y) ·e iµc1 (y) ∨pc2 (y) ·e iµc2 (y) = min{(pc1 ·e iµc1 ∪pc2 ·e iµc2 )(x), (pc1 ·e iµc1 ∪pc2 ·e iµc2 )(y)}. similarly, (qc1 ·e iνc1 ∪qc2 ·e iνc2 )(xy) ≥ min{(qc1 ·e iνc1 ∪qc2 ·e iνc2 )(x), (qc1 ·e iνc1 ∪qc2 ·e iνc2 )(y)}. and (rc1 ·e iωc1 ∩rc2 ·e iωc2 )(xy) = min{rc1 (xy) ·e iωc1 (xy),rc2 (xy) ·e iωc2 (xy)} ≤ min{max{rc1 (x) ·e iωc1 (x),rc1 (y) ·e iωc1 (y)}, max{rc2 (x) ·e iωc2 (x),rc2 (y) ·e iωc2 (y)}} = rc1 (x) ·e iωc1 (x) ∨rc1 (y) ·e iωc1 (y) ∧ rc2 (x) ·e iωc2 (x) ∨rc2 (y) ·e iωc2 (y) = rc1 (x) ·e iωc1 (x) ∧rc2 (x) ·e iωc2 (x) ∨ rc1 (y) ·e iωc1 (y) ∧rc2 (y) ·e iωc2 (y) = max{(rc1 ·e iωc1 ∩rc2 ·e iωc2 )(x), (rc1 ·e iωc1 ∩rc2 ·e iωc2 )(y)}. therefore, c1 ∪c2 is a complex neutrosophic subsemigroup of s. int. j. anal. appl. 16 (1) (2018) 114 (2) let c1 and c2 be any two complex neutrosophic left ideals of semigroup s, and x,y ∈s. then (pc1 ·e iµc1 ∪pc2 ·e iµc2 )(xy) = max{pc1 (xy) ·e iµc1 (xy),pc2 (xy) ·e iµc2 (xy)} ≥ max{pc1 (y) ·e iµc1 (y),pc2 (y) ·e iµc2 (y)} = (pc1 ·e iµc1 ∪pc2 ·e iµc2 )(y). similarly, (qc1 ·e iνc1 ∪qc2 ·e iνc2 )(xy) ≥ (qc1 ·e iνc1 ∪qc2 ·e iνc2 )(y). and (rc1 ·e iωc1 ∩rc2 ·e iωc2 )(xy) = min{rc1 (xy) ·e iωc1 (xy),rc2 (xy) ·e iωc2 (xy)} ≤ min{rc1 (y) ·e iωc1 (y),rc2 (y) ·e iωc2 (y)} = (rc1 ·e iωc1 ∩rc2 ·e iωc2 )(y). thus c1 ∪c2 is a complex neutrosophic left ideal of semigroup s. the union of complex neutrosophic right ideal can be proved in a similar manner. � theorem 5.7. if c1 and c2 be a complex neutrosophic right and left ideals of a semigroup s, respectively. then c1 ⊗c2 ⊆c1 ∩c2. proof: let c1 is complex neutrosophic right ideal and c2 is any complex left neutrosophic ideal of s. then by proposition 5.3 and proposition 5.4 we have c1 ⊗c2 ⊆ c1 ⊗s ⊆ c1 and c1 ⊗c2 ⊆ s ⊗c2 ⊆ c2. hence c1 ⊗c2 ⊆c1 ∩c2. � theorem 5.8. if s is regular semigroup, then c1 ⊗c2 = c1 ∩c2 for every complex neutrosophic right ideal c1 = 〈 pc1 ·eiµc1 ,qc1 ·eiνc1 ,rc1 ·eiωc1 〉 and every complex neutrosophic left ideal c2 = 〈 pc2 ·eiµc2 ,qc2 ·eiνc2 ,rc2 ·eiωc2 〉 of s. proof: let α be any element of s. since s is regular, there exist an element x ∈s such that α = αxα. hence we have (pc1 ·e iµc1 ◦pc2 ·e iµc2 )(α) = sup α=yκ {min{pc1 (y) ·e iµc1 (y),pc2 (κ) ·e iµc2 (κ)}} = sup αxα=yκ {min{pc1 (y) ·e iµc1 (y),pc2 (κ) ·e iµc2 (κ)}} ≥ min{pc1 (αx) ·e iµc1 (αx),pc2 (α) ·e iµc2 (α)} ≥ min{pc1 (α) ·e iµc1 (α),pc2 (α) ·e iµc2 (α)} = (pc1 ·e iµc1 ∩pc2 ·e iµc2 )(α). int. j. anal. appl. 16 (1) (2018) 115 similarly, (qc1 ·e iνc1 ◦qc2 ·e iνc2 )(α) ≥ (qc1 ·e iνc1 ∩qc2 ·e iνc2 )(α). and (rc1 ·e iωc1 ◦rc2 ·e iωc2 )(α) = inf α=yκ {max{rc1 (y) ·e iωc1 (y),rc2 (κ) ·e iωc2 (κ)}} = inf αxα=yκ {max{rc1 (y) ·e iωc1 (y),rc2 (κ) ·e iωc2 (κ)}} ≤ max{rc1 (αx) ·e iωc1 (αx),rc2 (α) ·e iωc2 (α)} ≤ max{rc1 (α) ·e iωc1 (α),rc2 (α) ·e iωc2 (α)} = (rc1 ·e iωc1 ∪rc2 ·e iωc2 )(α). so c1 ⊗c2 ⊇c1 ∩c2, and c1 ⊗c2 ⊆c1 ∩c2 is true from theorem 5.7. hence c1 ⊗c2 = c1 ∩c2. � theorem 5.9. for a non-empty subset h of a semigroup s. we have (1) h is a subsemigroup of s if and only if the characteristic complex neutrosophic set ch = 〈tch ,ich ,fch〉 of h in s is a complex neutrosophic subsemigroup of s. (2) h is a left (right) ideal of s if and only if the characteristic complex neutrosophic set ch = 〈tch ,ich ,fch〉 of h in s is a complex neutrosophic left (resp., right) ideal of s. proof: straightforward. � theorem 5.10. for every complex neutrosophic right ideal c1 = 〈tc1,ic1,fc1〉 and every complex neutrosophic left ideal c2 = 〈tc2,ic2,fc2〉 of a semigroup s, if c1 ⊗c2 = c1 ∩c2, then s is regular. proof: assume that c1 ⊗c2 = c1 ∩c2 for every complex neutrosophic right ideal c1 = 〈tc1,ic1,fc1〉 and every complex neutrosophic left ideal c2 = 〈tc2,ic2,fc2〉 of a semigroup s. let r and l be any right and left ideal of s, respectively. in order to see that r∩l⊆rl holds. let α be any element of r∩l, then the characteristic complex neutrosophic sets cr = 〈tcr,icr,fcr〉 and cl = 〈tcl,icl,fcl〉 are a complex neutrosophic right ideal and a complex neutrosophic left ideal of s, respectively, by theorem 5.9. it follows from the hypothesis and proposition 5.2, that is tcrl (α) = (tcr ◦tcl )(α) = (tcr ∩tcl )(α) = tcr∩l (α) = 1.e i2π icrl (α) = (icr ◦ icl )(α) = (icr ∩ icl )(α) = icr∩l (α) = 1.e i2π int. j. anal. appl. 16 (1) (2018) 116 and fcrl (α) = (fcr ◦fcl )(α) = (fcr ∪fcl )(α) = fcr∪l (α) = 0. so that α ∈rl. thus r∩l⊆rl. since the inclusion in the other direction always holds, we obtain that r∩l⊆rl. it follows from lemma 2.1, that s is regular. � references [1] zadeh, l. a. fuzzy sets. inf. control, 8 (1965), 338-353. [2] atanassov, k. t. intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. [3] kuroki, n. fuzzy bi-ideals in semigroups. comment. math. univ. st. pauli, 27 (1979), 17-21. [4] wang, h. et al. single valued neutrosophic sets. proc of 10th int conf on fuzzy theory and technology, salt lake city, utah (2005). [5] smarandache, f. a unifying field in logics. neutrosophy: neutrosophic probability, set and logic, rehoboth; american research press (1999). [6] buckley, j. j. fuzzy complex numbers. fuzzy sets syst. 33 (1989), 333-345. [7] nguyen, h. t. kandel, a. and kreinovich, v. complex fuzzy sets. towards new foundations, ieee. (2000), 7803-5877. [8] ramot, d. milo, r. friedman, m. kandel, a. complex fuzzy sets. ieee trans. fuzzy syst. 10 (2002), 171-186. [9] sveinn, r. j. haskoli islands, verkfraeoideild. complex fuzzy sets. ieee trans. fuzzy syst. 10 (2002), 171 186. [10] zhang, g. dillon, t. s. cai, k. y. ma, j. and lu, j. operation properties and δ-equalities of complex fuzzy sets. int. j. approx. reason. 50 (2009), 1227-1249. [11] jun, y. b. and khan, a. cubic ideals in semigroups. honam math. j. 35 (2013), 607-623. [12] abd ulazeez, m. alkouri, s. and salleh, a. r. complex intuitionistic fuzzy sets. international conference on fundamental and applied sciences, aip conf. proc. 1482 (2012), 464-470. [13] abd ulazeez, m. alkouri, s. and salleh, a. r. complex atanassov’s intuitionistic fuzzy relation. hindawi publishing corporation abstr. appl. anal. 2013 (2013), article id 287382, 18pages. [14] ali, m and smarandache, f. complex neutrosophic set. neural. comput. applic. 28 (2017), 18171834. [15] cetkin, v. and aygun, h. an approach to neutrosophic subgroup and its fundamental properties. j. intell. fuzzy syst. 29 (2015), 1941-1947. [16] iseki, k. a characterization of regular semigroups, proc. japan acad. 32 (1965), 676-677. [17] turksen, i. b. interval-valued fuzzy sets based on normal forms. fuzzy sets syst. 20 (1986), 191-210. 1. introduction 2. preliminaries 3. complex neutrosophic subsemigroup 3.1. cartesian product of complex neutrosophic subsemigroups 4. complex neutrosophic ideals 4.1. complex neutrosophic left ideal 4.2. complex neutrosophic right ideal 4.3. complex neutrosophic ideal 4.4. complex neutrosophic interior ideal 5. characteristic function of complex neutrosophic set 5.1. direct product of two complex neutrosophic sets references international journal of analysis and applications volume 17, number 5 (2019), 734-751 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-734 fixed point theorems for generalized f-contractions and generalized f-suzuki-contractions in complete dislocated sb-metric spaces hamid mehravaran, mahnaz khanehgir∗ and reza allahyari department of mathematics, mashhad branch, islamic azad university, mashhad, iran ∗corresponding author: khanehgir@mshdiau.ac.ir abstract. in this paper, first we describe the notion of dislocated sb-metric space and then we introduce the new notions of generalized f-contraction and generalized f-suzuki-contraction in the setup of dislocated sb-metric spaces. we establish some fixed point theorems involving these contractions in complete dislocated sb-metric spaces. we also furnish some examples to verify the effectiveness and applicability of our results. 1. introduction and preliminaries bakhtin [1] and czerwik [2] introduced b-metric spaces and proved the contraction principle in this framework. in recent times, many authors obtained fixed point results for single-valued or set-valued functions, in the setting of b-metric spaces. in 2012, sedghi et al. [11] introduced the concept of s-metric space by modifying d-metric and g-metric spaces and proved some fixed point theorems for a self-mapping on a complete s-metric space. after that özgür and taş studied some generalizations of the banach contraction principle on s-metric spaces in [8]. they also obtained some fixed point theorems for the rhoades’ contractive condition on s-metric spaces [7]. sedghi et al. [10] introduced the concept of sb-metric space as a generalization of s-metric space and proved some coupled common fixed point theorems in sb-metric space. kishore et al. [4] proved some fixed point received 2019-03-11; accepted 2019-07-24; published 2019-09-02. 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. dislocated metric space; fixed point; generalized f-contraction; generalized f-suzuki-contraction; sb-metric space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 734 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-734 int. j. anal. appl. 17 (5) (2019) 735 theorems for generalized contractive conditions in partially ordered complete sb-metric spaces and gave some applications to integral equations and homotopy theory. on the other hand, wardowski [12] introduced a new contraction, the so-called f-contraction, and obtained a fixed point result as a generalization of the banach contraction principle. thereafter, dung and hang [3] studied the notion of a generalized f-contraction and established certain fixed point theorems for such mappings. recently, piri and kumam [6] extended the fixed point results of [12] by introducing a generalized f-suzuki-contraction in b-metric spaces. motivated by the aforementioned works, in this paper, we first introduce the notion of dislocated sbmetric space and then we describe some fixed point results of [3], [6] by introducing generalized f-contractions and generalized f-suzuki-contractions in dislocated sb-metric spaces. we begin with some basic well-known definitions and results which will be used further on. throughout this paper r, r+, n denote the set of all real numbers, the set of all nonnegative real numbers and the set of all positive integers, respectively. definition 1.1. [11] let x be a nonempty set. an s-metric on x is a function s : x3 → r+ that satisfies the following conditions: (s1) 0 < s(x,y,z) for each x,y,z ∈ x with x 6= y 6= z 6= x, (s2) s(x,y,z) = 0 if and only if x = y = z, (s3) s(x,y,z) ≤ s(x,x,a) + s(y,y,a) + s(z,z,a) for each x,y,z,a ∈ x. then the pair (x,s) is called an s-metric space. definition 1.2. [10] let x be a nonempty set and b ≥ 1 be a given real number. suppose that a mapping sb : x 3 → r+ satisfies: (sb1) 0 < sb(x,y,z) for all x,y,z ∈ x with x 6= y 6= z 6= x, (sb2) sb(x,y,z) = 0 if and only if x = y = z, (sb3) sb(x,y,z) ≤ b ( sb(x,x,a) + sb(y,y,a) + sb(z,z,a) ) for all x,y,z,a ∈ x. then sb is called an sb-metric on x and the pair (x,sb) is called an sb-metric space. definition 1.3. [10] if (x,sb) is an sb-metric space, a sequence {xn} in x is said to be: (1) cauchy sequence if, for each ε > 0, there exists n0 ∈ n such that sb(xn,xn,xm) < ε for all m,n ≥ n0. (2) convergent to a point x ∈ x if, for each ε > 0, there exists a positive integer n0 such that sb(xn,xn,x) < ε or sb(x,x,xn) < ε for all n ≥ n0, and we denote by lim n→∞ xn = x. definition 1.4. [10] an sb-metric space (x,sb) is called complete if every cauchy sequence is convergent in x. int. j. anal. appl. 17 (5) (2019) 736 example 1.1. [9] let x = r. define sb : x3 → r+ by sb(x,y,z) = |x− z| + |y − z| for all x,y,z ∈ x. then (x,sb) is a complete sb-metric space with b = 2. definition 1.5. let (x,sb) be an sb-metric space. then sb is called symmetric if sb(x,x,y) = sb(y,y,x) (1.1) for all x,y ∈ x. it is easy to see that the symmetry condition (1.1) is automatically satisfied by an s-metric [11]. we conclude this section recalling the following fixed point theorems of dung and hang [3] and piri and kumam [6]. for this, we need some preliminaries. definition 1.6. [12] let f be the family of all functions f : (0, +∞) → r such that: (f1) f is strictly increasing, that is for all α,β ∈ (0, +∞) such that α < β, f(α) < f(β), (f2) for each sequence {αn} of positive numbers, lim n→+∞ αn = 0 if and only if lim n→+∞ f(αn) = −∞, (f3) there exists k ∈ (0, 1) such that lim α→0+ αkf(α) = 0. in 2014, piri and kumam [5] described a large class of functions by replacing the condition (f3) in the above definition with the following one: (f3′) f is continuous on (0, +∞). they denote by f the family of all functions f : (0, +∞) → r which satisfy conditions (f1), (f2), and (f3′). example 1.2. (see [5], [13]) the following functions f : (0, +∞) → r are the elements of f. (1) f(α) = − 1√ α , (2) f(α) = −1 α + α, (3) f(α) = 1 1−eα , (4) f(α) = ln α, (5) f(α) = ln α + α. definition 1.7. [3] let (x,d) be a metric space. a mapping t : x → x is said to be a generalized f -contraction on (x,d) if there exist f ∈f and τ > 0 such that, for all x,y ∈ x, d(tx,ty) > 0 ⇒ τ + f ( d(tx,ty) ) ≤ f ( n(x,y) ) , in which n(x,y) = max { d(x,y),d(x,tx),d(y,ty), d(x,ty) + d(y,tx) 2 , d(t2x,x) + d(t2x,ty) 2 ,d(t2x,tx),d(t2x,y),d(t2x,ty) } . int. j. anal. appl. 17 (5) (2019) 737 theorem 1.1. [3] let (x,d) be a complete metric space and let t : x → x be a generalized f -contraction mapping. if t or f is continuous, then t has a unique fixed point x∗ ∈ x and for every x ∈ x the sequence {tnx} converges to x∗. we use fg to denote the set of all functions f : (0, +∞) → r which satisfy conditions (f1) and (f3′) and ψ to denote the set of all functions ψ : r+ → r+ such that ψ is continuous and ψ(t) = 0 if and only if t = 0 (see [6]). definition 1.8. [6] let (x,d) be a b-metric space. a self-mapping t : x → x is said to be a generalized f -suzuki-contraction if there exists f ∈ fg such that, for all x,y ∈ x with x 6= y, 1 2s d(x,tx) < d(x,y) ⇒ f ( s5d(tx,ty) ) ≤ f ( mt (x,y) ) −ψ ( mt (x,y) ) , in which ψ ∈ ψ and mt (x,y) = max { d(x,y),d(t2x,y), d(tx,y) + d(x,ty) 2s , d(t2x,x) + d(t2x,ty) 2s , d(t2x,ty) + d(t2x,tx),d(t2x,ty) + d(tx,x),d(tx,y) + d(y,ty) } . theorem 1.2. [6] let (x,d) be a complete b-metric space and t : x → x be a generalized f -suzukicontraction.then t has a unique fixed point x∗ ∈ x and for every x ∈ x the sequence {tnx} converges to x∗. 2. main results in this section, we first introduce the concept of dislocated sb-metric space and then we demonstrate some fixed point results for generalized f-contractions and generalized f-suzuki-contractions in such spaces. our results are remarkable for two reasons: first dislocated sbmetric is more general, second the contractivity condition involves auxiliary functions form a wider class. definition 2.1. let x be a nonempty set and b ≥ 1 be a given real number. a mapping sb : x3 → r+ is a dislocated sb-metric if, for all x,y,z,a ∈ x, the following conditions are satisfied: (dsb1) sb(x,y,z) = 0 implies x = y = z, (dsb2) sb(x,y,z) ≤ b ( sb(x,x,a) + sb(y,y,a) + sb(z,z,a) ) . a dislocated sb-metric space is a pair (x,sb) such that x is a nonempty set and sb is a dislocated sb-metric on x. in the case that b = 1, sb is denoted by s and it is called dislocated s-metric, and the pair (x,s) is called dislocated s-metric space. definition 2.2. let (x,sb) be a dislocated sb-metric space, {xn} be any sequence in x and x ∈ x. then: (i) the sequence {xn} is said to be a cauchy sequence in (x,sb) if, for each ε > 0, there exists n0 ∈ n such that sb(xn,xn,xm) < ε for each m,n ≥ n0. int. j. anal. appl. 17 (5) (2019) 738 (ii) the sequence {xn} is said to be convergent to x if, for each ε > 0, there exists a positive integer n0 such that sb(x,x,xn) < ε for all n ≥ n0 and we denote it by lim n→∞ xn = x. (iii) (x,sb) is said to be complete if every cauchy sequence is convergent. the following example shows that a dislocated sb-metric need not be a dislocated s-metric. example 2.1. let x = r+, then the mapping sb : x3 → r+ defined by sb(x,y,z) = x + y + 4z is a complete dislocated sb-metric on x with b = 2. however, it is not a dislocated s-metric space. indeed, we have 4 = sb(0, 0, 1) � 2sb(0, 0, 0) + sb(1, 1, 0) = 2. definition 2.3. suppose that (x,sb) is a dislocated sb-metric space. a mapping t : x → x is said to be a generalized f -contraction on (x,sb) if there exist f ∈ f and τ > 0 such that for all x,y ∈ x, sb(tx,tx,ty) > 0 ⇒ τ + f ( b2sb(tx,tx,ty) ) ≤ f ( n(x,y) ) , (2.1) where n(x,y) = max { sb(x,x,y),sb(tx,tx,ty), sb(y,y,tx) 10b8 , sb(x,x,ty) 10b9 , sb(y,y,t 2x) 10b4 } . our first main result is the following. theorem 2.1. let (x,sb) be a complete dislocated sb-metric space and t : x → x be a generalized f -contraction mapping satisfying the following condition: max {sb(y,y,ty) 5b7 + sb(tx,tx,ty) 10b7 , sb(x,x,ty) 10b9 , sb(y,y,t 2x) 10b4 } ≤ sb(tx,tx,ty) for all x,y ∈ x. then t has a unique fixed point v ∈ x. proof. let x0 be an arbitrary point in x and let {xn} be the picard sequence of t based on x0, that is, xn+1 = txn for n = 0, 1, 2, . . . . if there exists n0 ∈ n such that sb(xn0,xn0,xn0+1 ) = 0, then xn0 is a fixed point of t and the existence part of the proof is finished. on the contrary case, assume that sb(xn,xn,xn+1) > 0 for all n ∈ n∪{0}. applying the contractivity condition (2.1), we get f ( b2sb(txn−1,txn−1,txn) ) ≤ f ( n(xn−1,xn) ) − τ. (2.2) using the definition of n(x,y) and the property (dsb2), we obtain that max { sb(xn−1,xn−1,xn),sb(xn,xn,xn+1) } ≤ n(xn−1,xn) (2.3) = max { sb(xn−1,xn−1,xn),sb(xn,xn,txn), sb(xn,xn,txn−1) 10b8 , sb(xn−1,xn−1,txn) 10b9 , sb(xn,xn,txn) 10b4 } ≤ max{sb(xn−1,xn−1,xn),sb(xn,xn,txn), 3sb(xn,xn,xn+1) 10b7 , sb(xn−1,xn−1,txn) 10b9 , sb(xn,xn,txn) 10b4 } = max { sb(xn−1,xn−1,xn),sb(xn,xn,xn+1) } . int. j. anal. appl. 17 (5) (2019) 739 then n(xn−1,xn) = max { sb(xn−1,xn−1,xn),sb(xn,xn,xn+1) } and so (2.2), becomes f ( b2sb(txn−1,txn−1,txn) ) ≤ f ( max { sb(xn−1,xn−1,xn),sb(xn,xn,xn+1) }) − τ. if we assume that max { sb(xn−1,xn−1,xn),sb(txn−1,txn−1,txn) } = sb(txn−1,txn−1,txn) for some n, then we have f ( b2sb(txn−1,txn−1,txn) ) ≤ f ( sb(txn−1,txn−1,txn) ) − τ < f ( sb(txn−1,txn−1,txn) ) . using condition (f1) we conclude that sb(xn,xn,xn+1) < sb(xn,xn,xn+1), which is a contradiction. therefore, for each n ∈ n we have max { sb(xn−1,xn−1,xn),sb(xn,xn,xn+1) } = sb(xn−1,xn−1,xn). applying again (2.2) and condition (f1), we deduce that sb(xn,xn,xn+1) < sb(xn−1,xn−1,xn) for each n. thus { sb(xn,xn,xn+1) } is a nonnegative decreasing sequence of real numbers. then there exists a ≥ 0 such that lim n→+∞ sb(xn,xn,xn+1) = inf n∈n sb(xn,xn,xn+1) = a. we claim that a = 0. to support the claim, let it be untrue and a > 0. then, for any ε > 0, it is possible to find a positive integer m so that sb(xm,xm,txm) < a + ε. so, from (f1), we get f ( sb(xm,xm,txm) ) < f(a + ε). (2.4) it follows from (2.1) that τ + f ( b2sb(txm,txm,t 2xm) ) ≤ f ( n(xm,txm)). (2.5) by a similar argument as (2.3), it yields that n(xm,txm) = max { sb(xm,xm,txm),sb(txm,txm,t 2xm) } . hence (2.5), becomes f ( b2sb(txm,txm,t 2xm) ) ≤ f ( max { sb(xm,xm,txm),sb(txm,txm,t 2xm) }) − τ. (2.6) now if, max{sb(xm,xm,txm),sb(txm,txm,t2xm)} = sb(txm,txm,t2xm) for some m, then (2.6) gives us a contradiction. thus, we infer that max { sb(xm,xm,txm),sb(txm,txm,t 2xm) } = sb(xm,xm,txm), int. j. anal. appl. 17 (5) (2019) 740 and therefore, we have f ( b2sb(txm,txm,t 2xm) ) ≤ f ( sb(xm,xm,txm) ) − τ. it implies that f ( b2sb(t 2xm,t 2xm,t 3xm) ) ≤ f ( sb(txm,txm,t 2xm) ) − τ ≤ f ( b2sb(txm,txm,t 2xm) ) − τ ≤ f ( sb(xm,xm,txm) ) − 2τ. continuing the above process and taking (2.4) into account, we deduce that f ( b2sb(t nxm,t nxm,t n+1xm) ) ≤ f ( sb(t n−1xm,t n−1xm,t nxm) ) − τ ≤ f ( b2sb(t n−1xm,t n−1xm,t nxm) ) − τ ≤ f ( sb(t n−2xm,t n−2xm,t n−1xm) ) − 2τ . . . ≤ f ( sb(xm,xm,txm) ) −nτ < f(a + ε) −nτ, and by passing to the limit as n → +∞ we obtain lim n→+∞ f ( b2sb(t nxm,t nxm.t n+1xm) ) = −∞. this fact together with the condition (f2) implies that lim n→+∞ sb(t nxm,t nxm,t n+1xm) = 0. thus sb(t nxm,t nxm,t n+1xm) < a for n sufficiently large, which is a contradiction with the definition of a. then, lim n→+∞ sb(xn,xn,xn+1) = 0. (2.7) next, we intend to show that the sequence {xn} is a cauchy sequence in x. arguing by contradiction, we assume that there exist ε > 0, and subsequences {xq(n)} and {xp(n)} of {xn} with n < q(n) < p(n) such that sb(xq(n),xq(n),xp(n)) ≥ ε (2.8) for each n ∈ n. further, corresponding to q(n), we can choose p(n) in such a way that it is the smallest integer with q(n) < p(n) satisfying the above inequality, then sb(xq(n),xq(n),xp(n)−1) < ε (2.9) for all n ∈ n. in the light of (2.8) and the condition (2.1), we conclude that f ( b2sb(txq(n)−1,txq(n)−1,txp(n)−1) ) ≤ f ( n(xq(n)−1,xp(n)−1) ) − τ. (2.10) int. j. anal. appl. 17 (5) (2019) 741 by our hypothesis and in view of (dsb2), we get max { sb(xq(n)−1,xq(n)−1,xp(n)−1),sb(txq(n)−1,txq(n)−1,txp(n)−1) } ≤ n(xq(n)−1,xp(n)−1) = max { sb(xq(n)−1,xq(n)−1,xp(n)−1),sb(txq(n)−1,txq(n)−1,txp(n)−1), sb(xp(n)−1,xp(n)−1,txq(n)−1) 10b8 , sb(xq(n)−1,xq(n)−1,txp(n)−1) 10b9 , sb(xp(n)−1,xp(n)−1,txq(n)) 10b4 } ≤ max { sb(xq(n)−1,xq(n)−1,xp(n)−1),sb(txq(n)−1,txq(n)−1,txp(n)−1), sb(xp(n)−1,xp(n)−1,txp(n)−1) 5b7 + sb(txq(n)−1,txq(n)−1,txp(n)−1) 10b7 , sb(xq(n)−1,xq(n)−1,txp(n)−1) 10b9 , sb(xp(n)−1,xp(n)−1,txq(n)) 10b4 } ≤ max { sb(xq(n)−1,xq(n)−1,xp(n)−1),sb(txq(n)−1,txq(n)−1,txp(n)−1) } . it enforces that n(xq(n)−1,xp(n)−1) = max { sb(xq(n)−1,xq(n)−1,xp(n)−1),sb(txq(n)−1,txq(n)−1,txp(n)−1) } . suppose that the maximum on the right-hand side is equal to sb(txq(n)−1,txq(n)−1,txp(n)−1) for some n, then from relation (2.10) together with the condition (f1) we get sb(txq(n)−1,txq(n)−1,txp(n)−1) < sb(txq(n)−1,txq(n)−1,txp(n)−1) which is a contradiction. thus, we find that max { xq(n)−1,xq(n)−1,xp(n)−1),sb(xq(n),xq(n),xp(n)) } = sb(xq(n)−1,xq(n)−1,xp(n)−1) for all n. accordingly, (2.10) becomes f ( b2sb(txq(n)−1,txq(n)−1,txp(n)−1) ) ≤ f ( sb(xq(n)−1,xq(n)−1,xp(n)−1) ) − τ (2.11) and so using (f1) we get sb ( xq(n),xq(n),xp(n) ) < sb(xq(n)−1,xq(n)−1,xp(n)−1). (2.12) regarding to (2.8), (2.12) and employing (dsb2) we observe that ε ≤ sb(xq(n),xq(n),xp(n)) < sb(xq(n)−1,xq(n)−1,xp(n)−1) ≤ 2bsb(xq(n)−1,xq(n)−1,xq(n)) + bsb(xp(n)−1,xp(n)−1,xq(n)) ≤ 2bsb(xq(n)−1,xq(n)−1,xq(n)) + 2b 2sb(xp(n)−1,xp(n)−1,xp(n)−1) +b2sb(xq(n),xq(n),xp(n)−1) ≤ 2bsb(xq(n)−1,xq(n)−1,xq(n)) + 6b 3sb(xp(n)−1,xp(n)−1,xp(n)) +b2sb(xq(n),xq(n),xp(n)−1). combining this result with (2.7) and (2.9) we get ε ≤ lim sup n→+∞ sb(xq(n),xq(n),xp(n)) ≤ lim sup n→+∞ sb(xq(n)−1,xq(n)−1,xp(n)−1) ≤ b 2ε. (2.13) int. j. anal. appl. 17 (5) (2019) 742 in view of (2.13) and (2.11) and applying the conditions (f1) and (f3′), we have f(b2ε) ≤ f ( b2 lim sup n→+∞ sb(xq((n),xq(n),xp(n)) ) ≤ f ( lim sup n→+∞ sb(xq(n)−1,xq(n)−1,xp(n)−1) ) − τ ≤ f(b2ε) − τ. it is a contradiction with τ > 0, and therefore it follows that {xn} is a cauchy sequence in x. by completeness of (x,sb), {xn} converges to some point v ∈ x. then, for each ε > 0, there exists n1 ∈ n such that sb(v,v,xn) < ε, (2.14) for all n ≥ n1. we are going to show that v is a fixed point of t. for this aim, we consider two following cases: case 1. if sb(tv,tv,txn) = 0 for some n ≥ n1, then from (dsb2) we find that sb(tv,tv,v) ≤ 2bsb(tv,tv,txn) + bsb(v,v,txn) ≤ bε. case 2. if sb(tv,tv,txn) > 0 for all n ≥ n1, then using (2.1), we get f ( b2sb(tv,tv,txn) ) ≤ f ( n(v,xn) ) − τ. (2.15) from our assumptions, and using (dsb2), it follows that max { sb(v,v,xn),sb(tv,tv,txn) } ≤ n(v,xn) = max { sb(v,v,xn),sb(tv,tv,txn), sb(xn,xn,tv) 10b8 , sb(v,v,txn) 10b9 , sb(xn,xn,t 2v) 10b4 } ≤ max { sb(v,v,xn),sb(tv,tv,txn), sb(xn,xn,txn) 5b7 + sb(tv,tv,txn) 10b7 , sb(v,v,txn) 10b9 , sb(xn,xn,t 2v) 10b4 } = max { sb(v,v,xn),sb(tv,tv,txn) } . it enforces that n(v,xn) = max { sb(v,v,xn),sb(tv,tv,txn) } . now, if we assume that the maximum on the right-hand side of this equality is equal to sb(tv,tv,txn), then by replacing it in (2.15), we obtain sb(tv,tv,txn) < sb(tv,tv,txn) which is a contradiction. consequently, for each n ∈ n we have max { sb(v,v,xn),sb(tv,tv,txn) } = sb(v,v,xn). hence, (2.15) turns into f ( b2sb(tv,tv,txn) ) ≤ f ( sb(v,v,xn) ) − τ < f ( sb(v,v,xn) ) . employing the condition (f1), we get sb(tv,tv,txn) < sb(v,v,xn). (2.16) int. j. anal. appl. 17 (5) (2019) 743 from (dsb2), (2.16) and (2.14), we deduce that sb(tv,tv,v) ≤ 2bsb(tv,tv,txn1 ) + bsb(v,v,txn1 ) < 3bε. from the arbitrariness of ε in each case, it follows that sb(tv,tv,v) = 0 which implies that tv = v. hence, v is a fixed point of t. finally, we show that t has at most one fixed point. indeed, if v1,v2 ∈ x are two fixed points of t, such that v1 6= v2, then we obtain f ( b2sb(tv1,tv1,tv2) ) ≤ f(n(v1,v2) ) − τ, (2.17) from our hypothesis and by using (dsb2), it follows that sb(v1,v1,v2) ≤ n(v1,v2) ≤ max { sb(v1,v1,v2),sb(tv1,tv1,tv2), sb(v2,v2,tv2) 5b7 + sb(tv1,tv1,tv2) 10b7 , sb(v1,v1,tv2) 10b9 , sb(tv2,tv2,t 2v1) 10b4 } = max { sb(v1,v1,v2),sb(tv1,tv1,tv2) } = sb(v1,v1,v2). then (2.17) becomes f ( b2sb(v1,v1,v2) ) ≤ f(sb(v1,v1,v2) ) − τ. it gives us a contradiction. therefore, v1 = v2 and the fixed point is unique. � now we illustrate our result contained in theorem 2.1 with help of two examples. example 2.2. let (x,sb) be as in example 2.1 and let τ > 0 be an arbitrary fixed number. define the mapping t : x → x by t(x) = e−τ x 8 and take f(α) = ln α + α (α > 0). it is easily verified that n(x,y) = sb(x,x,y) = 2x + 4y. assume that x or y is nonzero, then sb(tx,tx,ty) > 0 and we have τ + f ( b2sb(tx,tx,ty) ) = τ + ln(e−τ (x + 2y)) + e−τ (x + 2y) = ln(x + 2y) + e−τ (x + 2y) ≤ ln(2x + 4y) + 2x + 4y = f ( sb(x,x,y) ) = f ( n(x,y) ) . hence, t is a generalized f -contraction. on the other hand, if we assume that 0 < τ ≤ 0.0250587314, then the following estimate holds: max {sb(y,y,ty) 5b7 + sb(tx,tx,ty) 10b7 , sb(x,x,ty) 10b9 , sb(y,y,t 2x) 10b4 } = max {2y + e−τ y 2 5 × 27 + e−τ ( x 4 + y 2 ) 10 × 27 , 2x + e−τ y 2 10 × 29 , 2y + e−2τ x 16 10 × 24 } ≤ e−τ ( x 4 + y 2 ) = sb(tx,tx,ty). thus all conditions of theorem 2.1 hold and 0 is a unique fixed point of t. int. j. anal. appl. 17 (5) (2019) 744 example 2.3. let x = r, and sb : x3 → r+ be a mapping defined by sb(x,y,z) = x2 2 + y 2 2 + 2z2. then (x,sb) is a complete dislocated sb-metric with b = 2. define the mapping t : x → x by t(x) = x 3 and take f(α) = ln α (α > 0). it is easily checked that n(x,y) = sb(x,x,y) = x2 + 2y2. assume that x or y is nonzero, then sb(tx,tx,ty) > 0 and we have τ + f ( b2sb(tx,tx,ty) ) ≤ f ( n(x,y) ) ⇔ ln( 9 4 ) ≥ τ. also, we observe that max {sb(y,y,ty) 5b7 + sb(tx,tx,ty) 10b7 , sb(x,x,ty) 10b9 , sb(y,y,t 2x) 10b4 } = max {48y2 + 2x2 23040 , 18x2 + 4y2 92160 , 162y2 + 4x2 25920 } ≤ 10240x2 + 20480y2 92160 = sb(tx,tx,ty) for all x,y ∈ x. now, if we assume that 0 < τ ≤ ln( 9 4 ), then all the conditions of theorem 2.1 hold and 0 is a unique fixed point of t . now, we describe the concept of generalized f-suzuki-contraction in the framework of dislocated sb-metric spaces. definition 2.4. let (x,sb) be a dislocated sb-metric space. a mapping t : x → x is said to be a generalized f -suzuki-contraction if there exists f ∈ f such that for all x,y ∈ x 1 2b sb(x,x,tx) < sb(x,x,y) ⇒ f ( 2b3sb(tx,tx,ty) ) ≤ f ( mt (x,y) ) −ψ ( mt (x,y) ) , (2.18) where ψ ∈ ψ and mt (x,y) = max { sb(x,x,y), sb(y,y,ty) 10 , sb(x,x,tx) 10 ,sb(tx,tx,ty), sb(y,y,tx) 18b , sb(tx,tx,t 2x) 2 } . our second main result is the following. theorem 2.2. let (x,sb) be a complete dislocated sb-metric space and t : x → x be a generalized f -suzuki-contraction satisfying the following condition: max {sb(y,y,ty) 10 , sb(x,x,tx) 10 , sb(y,y,ty) 9 + sb(tx,tx,ty) 18 , sb(tx,tx,t 2x) 2 } ≤ sb(tx,tx,ty) for all x,y in x. then t has a unique fixed point in x. proof. let x0 be arbitrary. define xn = txn−1 for each n ∈ n. if there exists n ∈ n such that sb(xn,xn,txn) = 0, then xn = txn and xn becomes a fixed point of t, which completes the proof. therefore, we assume that sb(xn,xn,txn) > 0 for all n ∈ n. taking into account (2.18), we deduce f ( 2b3sb(txn,txn,txn+1) ) ≤ f ( mt (xn,xn+1) ) −ψ ( mt (xn,xn+1) ) . (2.19) int. j. anal. appl. 17 (5) (2019) 745 using (dsb2) we get max { sb(xn,xn,xn+1),sb(xn+1,xn+1,txn+1) } ≤ mt (xn,xn+1) ≤ max { sb(xn,xn,xn+1), sb(xn+1,xn+1,txn+1) 10 , sb(xn,xn,txn) 10 , sb(txn,txn,txn+1), sb(txn,txn,txn+1) 2 , sb(xn+1,xn+1,xn+2) 6 } = max { sb(xn,xn,xn+1),sb(xn+1,xn+1,xn+2) } and combining it with the relation (2.19) we derive f ( 2b3sb(txn,txn,txn+1) ) ≤ f ( max { sb(xn,xn,xn+1),sb(xn+1,xn+1,xn+2 }) −ψ ( max { sb(xn,xn,xn+1),sb(xn+1,xn+1,xn+2 }) . (2.20) if max { sb(xn,xn,xn+1),sb(xn+1,xn+1,xn+2) } = sb(xn+1,xn+1,xn+2), then (2.20) becomes f ( 2b3sb(txn,txn,txn+1) ) ≤ f ( sb(xn+1,xn+1,xn+2) ) −ψ ( s(xn+1,xn+1,xn+2) ) . by the property of ψ and using condition (f1), we obtain 2b3sb(txn,txn,txn+1) < sb(txn,txn,txn+1), which is a contradiction. hence max { sb(xn,xn,xn+1),sb(xn+1,xn+1,xn+2) } = sb(xn,xn,xn+1), then (2.20) becomes f ( 2b3sb(txn,txn,txn+1) ) ≤ f ( sb(xn,xn,xn+1) ) −ψ ( s(xn,xn,xn+1) ) . (2.21) this together with condition (f1) implies that sb(txn,txn,txn+1) < sb(xn,xn,xn+1) for each n ∈ n. then {sb(xn,xn,xn+1)} is a nonnegative decreasing sequence of real numbers. therefore, there exists a ≥ 0 such that lim n→+∞ sb(xn,xn,xn+1) = a. letting n → +∞ in (2.21) and using (f3′) and continuity of ψ, we get f(2b3a) ≤ f(a) −ψ(a). it gives us ψ(a) = 0. by property of ψ we deduce that a = 0. consequently, we have lim n→+∞ sb(xn,xn,xn+1) = 0. (2.22) next, we prove that {xn} is a cauchy sequence in x. if it is not true, then there exist ε > 0 and increasing sequences of natural numbers {p(n)} and {q(n)} such that n < q(n) < p(n), sb(xq(n),xq(n),xp(n)) ≥ ε, sb(xq(n),xq(n),xp(n)−1) < ε (2.23) for all n ∈ n. owing to (2.22), there exists n1 ∈ n such that sb(xq(n),xq(n),txq(n)) < ε (2.24) int. j. anal. appl. 17 (5) (2019) 746 for all n ≥ n1. hence, from (2.23) and (2.24) it follows that 1 2b sb(xq(n),xq(n),txq(n)) < 1 2b ε < sb(xq(n),xq(n),xp(n)) for all n ≥ n1. by using (2.18) we obtain f ( 2b3sb(txq(n),txq(n),txp(n)) ) ≤ f ( mt (xq(n),xp(n)) ) −ψ ( mt (xq(n),xp(n)) ) . (2.25) from our assumptions and regarding (dsb2), we get max { sb(xq(n),xq(n),xp(n)),sb(txq(n),txq(n),txp(n)) } ≤ mt (xq(n),xp(n)) ≤ max { sb(xq(n),xq(n),xp(n)), sb(xp(n),xp(n),xp(n)+1) 10 ,sb(txq(n),txq(n),txp(n)) sb(xq(n),xq(n),xq(n)+1) 10 , sb(xq(n)+1,xq(n)+1,xq(n)+2) 2 , sb(xp(n),xp(n),xp(n)+1) 9 + sb(xq(n)+1,xq(n)+1,xp(n)+1) 18 } ≤ max { sb(xq(n),xq(n),xp(n)),sb(xq(n)+1,xq(n)+1,xp(n)+1) } . then (2.25) becomes f ( 2b3sb(txq(n),txq(n),txp(n)) ) ≤ f ( max { sb(xq(n),xq(n),xp(n)),sb(xq(n)+1,xq(n)+1,xp(n)+1) }) − ψ ( max { sb(xq(n),xq(n),xp(n)),sb(xq(n)+1,xq(n)+1,xp(n)+1) }) . if max { sb(xq(n),xq(n),xp(n)),sb(xq(n)+1,xq(n)+1,xp(n)+1) } = sb(xq(n)+1,xq(n)+1,xp(n)+1) for some n, then we have f ( 2b3sb(txq(n),txq(n),txp(n)) ) ≤ f ( sb(xq(n)+1,xq(n)+1,xp(n)+1) ) − ψ ( sb(xq(n)+1,xq(n)+1,xp(n)+1) ) . obviously, sb(xq(n)+1,xq(n)+1,xp(n)+1) > 0 and by the property of ψ and (f1), we get sb(txq(n),txq(n),txp(n)) < sb(xq(n)+1,xq(n)+1,xp(n)+1), which is a contradiction. duo to this fact, we find that max { sb(xq(n),xq(n),xp(n)),sb(xq(n)+1,xq(n)+1,xp(n)+1) } = sb(xq(n),xq(n),xp(n)) for all n. therefore f ( 2b3sb(txq(n),txq(n),txp(n)) ) ≤ f ( sb(xq(n),xq(n),xp(n)) ) −ψ ( sb(xq(n),xq(n),xp(n)) ) , (2.26) and by (f1), it follows that sb(xq(n)+1,xq(n)+1,xp(n)+1) < sb(xq(n),xq(n),xp(n)). int. j. anal. appl. 17 (5) (2019) 747 in view of (2.23) and (dsb2), we infer that ε ≤ sb(xq(n),xq(n),xp(n)) ≤ 2bsb(xq(n),xq(n),xp(n)−1) + bsb(xp(n),xp(n),xp(n)−1) ≤ 2bsb(xq(n),xq(n),xp(n)−1) + 2b 2sb(xp(n),xp(n),xp(n)) + b 2sb(xp(n)−1,xp(n)−1,xp(n)) ≤ 2bsb(xq(n),xq(n),xp(n)−1) + 6b 3sb(xp(n),xp(n),xp(n)+1) +b2sb(xp(n)−1,xp(n)−1,xp(n)). taking the limit as n → +∞ in the above inequality and regarding (2.22) and (2.23), we deduce that ε ≤ lim n→+∞ sb(xq(n),xq(n),xp(n)) ≤ 2bε. (2.27) on the other hand, we have ε ≤ sb(xq(n),xq(n),xp(n)) ≤ 2bsb(xq(n),xq(n),xq(n)+1) + bsb(xp(n),xp(n),xq(n)+1) ≤ 2bsb(xq(n),xq(n),xq(n)+1) + 2b 2sb(xp(n),xp(n),xp(n)+1) +b2sb(xq(n)+1,xq(n)+1,xp(n)+1). taking the limit supremum as n → +∞ in the above inequality. by using (2.22) we obtain ε b2 ≤ lim sup n→+∞ sb(xq(n)+1,xq(n)+1,xp(n)+1). (2.28) taking the limit supremum as n → +∞ on each side of (2.26) and using conditions (2.27) and (2.28) together with (f1) and (f3′), we deduce that f(2bε) = f(2b3 ε b2 ) ≤ f ( 2b3 lim sup n→+∞ sb(xq(n)+1,sb(xq(n)+1,xp(n)+1) ) ≤ f ( lim sup n→+∞ sb(xq(n),sb(xq(n),xp(n)) ) −ψ ( lim inf n→+∞ sb(xq(n),sb(xq(n),xp(n)) ) ≤ f(2bε) −ψ(ε). it enforces that ψ(ε) = 0, which leads to a contradiction. therefore {xn} is a cauchy sequence in x. since x is a complete dislocated sb-metric space, it follows that there exists v ∈ x in which for each ε > 0, there exists n2 ∈ n such that sb(v,v,xn) < ε (2.29) for all n > n2. now, we prove that v is a fixed point of t. to this end, we show that sb(tv,tv,v) = 0. we consider the following cases: case 1. if sb(v,v,xn) = 0 for sufficiently large n, then v = xn. thus, for sufficiently large n, we can write sb(tv,tv,v) = sb(txn,txn,v) ≤ 2bsb(xn+1,xn+1,xn+2) + bsb(v,v,xn+2). int. j. anal. appl. 17 (5) (2019) 748 letting n → +∞ in the above inequality. from (2.22) and (2.29) we get sb(tv,tv,v) = 0. thus tv = v and v is a fixed point of t . case 2. if there exists n ≥ n2 such that sb(v,v,xn) > 0 and sb(tv,tv,txn) = 0, then from (dsb2) we have sb(tv,tv,v) ≤ 2bsb(tv,tv,txn) + bsb(v,v,xn+1) ≤ bε, which implies that tv = v by virtue of the arbitrariness of ε. case 3. if sb(v,v,xn) > 0 and sb(tv,tv,txn) > 0 for all n ≥ n2, then using (2.18) we obtain f ( 2b3sb(tv,tv,txn) ) ≤ f ( mt (v,xn) ) −ψ ( mt (v,xn) ) . (2.30) thus, by using the hypothesis and taking into account (dsb2), it yields max { sb(v,v,xn),sb(tv,tv,txn) } ≤ mt (v,xn) ≤ max { sb(v,v,xn),sb(tv,tv,txn), sb(xn,xn,txn) 10 sb(v,v,tv) 10 , sb(xn,xn,txn) 9 + sb(tv,tv,txn) 18 , sb(tv,tv,t 2v) 2 } ≤ max { sb(v,v,xn),sb(tv,tv,txn) } . then (2.30) becomes f ( 2b3sb(tv,tv,txn) ) ≤ f ( max { sb(v,v,xn),sb(tv,tv,txn) }) − ψ ( max { sb(v,v,xn),sb(tv,tv,txn) }) . if max { sb(v,v,xn),sb(tv,tv,txn) } = sb(tv,tv,txn), then we have f ( 2b3sb(tv,tv,txn) ) ≤ f ( sb(tv,tv,txn) ) −ψ ( sb(tv,tv,txn) ) . from this it follows that 2b3sb(tv,tv,txn) < sb(tv,tv,txn), which is a contradiction. therefore, max { sb(v,v,xn),sb(tv,tv,txn) } = sb(v,v,xn) and (2.30) becomes f ( 2b3sb(tv,tv,txn) ) ≤ f ( sb(v,v,xn) ) −ψ ( sb(v,v,xn) ) < f ( sb(v,v,xn) ) . thus, from (f1) we get sb(tv,tv,txn) < sb(v,v,xn). (2.31) applying (2.29), (2.31) and (dsb2) we get sb(tv,tv,v) ≤ 2bsb(tv,tv,txn) + bsb(v,v,xn+1) < 3bε for sufficiently large n. it enforces that tv = v by virtue of the arbitrariness of ε. then v is a fixed point of t. int. j. anal. appl. 17 (5) (2019) 749 next, we show the uniqueness. indeed, if v1, v2 are two fixed points of t such that v1 6= v2, then in view of (2.18) we get f ( 2b3sb(tv1,tv1,tv2) ) ≤ f ( mt (v1,v2) ) −ψ ( mt (v1,v2) ) . (2.32) according to our assumptions and by using (dsb2), we find that sb(v1,v1,v2) ≤ mt (v1,v2) ≤ max { sb(v1,v1,v2),sb(tv1,tv1,tv2), sb(v2,v2,tv2) 10 , sb(v1,v1,tv1) 10 , sb(v2,v2,tv2) 9 + sb(tv1,tv1,tv2) 18 } ≤ max { sb(v1,v1,v2),sb(tv1,tv1,tv2) } = sb(v1,v1,v2). then (2.32) becomes f ( 2b3sb(v1,v1,v2) ) ≤ f ( sb(v1,v1,v2) ) −ψ ( sb(v1,v1,v2) ) . from this it follows that 2b3sb(v1,v1,v2) < sb(v1,v1,v2), which is a contradiction. then v1 = v2 and so t has a unique fixed point in x. � example 2.4. let x = {−1, 0, 1}. define the mapping sb : x3 → r+ by sb(x,y,z) =   3 2 , 0 = x = y 6= z = 1 or −1 = x = y 6= z = 1 10 6 , 1 = x = y 6= z 0, x = y = z = −1 or 1 1 5 , otherwise for all x,y,z ∈ x. it is easy to show that (x,sb) is a complete dislocated sb-metric space with b = 32. put f(α) = ln α (α > 0) and ψ(t) = t (t ≥ 0). define t : x → x by t(x) =   0, x = 1 −1, x = −1, 0. note that sb(x,x,y) > 0 and sb(t(x),t(x),t(y)) > 0 if and only if x ∈ {−1, 0}, y = 1 or x = 1, y ∈ {−1, 0}. also, for each x,y ∈ x we have mt (x,y) = sb(x,x,y) and we find that f ( 2b3sb(t(x),t(x),t(y)) ) ≤ f ( sb(x,x,y) ) −ψ ( sb(x,x,y) ) ⇔ ln sb(x,x,y) 2b3sb(t(x),t(x),t(y)) ≥ sb(x,x,y). now, we consider two cases: int. j. anal. appl. 17 (5) (2019) 750 case 2.1. case 1. let x ∈{−1, 0} and y = 1, then sb(x,x,y) = 3 2 , sb(t(x),t(x),t(y)) = sb(−1,−1, 0) = 1 5 , sb(0, 0,t(0)) = 1 5 , sb(−1,−1,t(−1)) = 0. so, we have ln sb(x,x,y) 2b3sb(t(x),t(x),t(y)) = ln 3 2 54 40 = ln 120 108 = 3.0377 ≥ sb(x,x,y) = 3 2 . case 2. let x = 1 and y ∈{−1, 0}, then sb(x,x,y) = 10 6 , sb(t(x),t(x),t(y)) = 1 5 , sb(x,x,t(x)) = 10 6 . so, we have ln sb(x,x,y) 2b3sb(t(x),t(x),t(y)) = ln 10 6 54 40 = ln 400 324 = 3.4369 ≥ sb(x,x,y) = 10 6 . on the other hand, for all x,y ∈ x we have max {sb(y,y,t(y)) 10 , sb(x,x,t(x)) 10 , sb(y,y,t(y)) 9 + sb(t(x),t(x),t(y)) 18 , sb(t(x),t(x),t 2(x)) 2 } ≤ 1 5 = sb(t(x),t(x),t(y)). hence, t is a generalized f -suzuki-contraction which satisfies the assumption of theorem 2.2 and so it has a unique fixed point −1. references [1] i. a. bakhtin, the contraction mapping principle in almost metric spaces, funct. anal. unianowsk gos. ped. inst. 30 (1989), 26–37. [2] s. czerwik, contraction mappings in b-metric spaces, acta math. inform. univ. ostra. 1 (1993), 5–11. [3] n.v. dung and v.l. hang, a fixed point theorem for generalized f-contractions on complete metric spaces, vietnam j. math. 43 (2015), 743–753. [4] g.n.v. kishore, k.p.r. rao, d. panthi, b. srinuvasa rao and s. satyanaraya, some applications via fixed point results in partially ordered sb-metric spaces, fixed point theory appl. 2017 (2017), 10. [5] h. piri and p. kumam, some fixed point theorems concerning f-contraction in complete metric spaces, fixed point theory appl. 2014 (2014), 210. [6] h. piri and p. kumam, fixed point theorems for generalized f-suzuki-contraction mappings in complete b-metric spaces, fixed point theory appl. 2016 (2016), 90 . [7] n.y. özgür and n. taş, some fixed point theorems on s-metric spaces, mat. vesnik 69 (1) (2017), 39-52. [8] n.y. özgür, n. taş and u. celik, new fixed point-circle results on s-metric spaces, bull. math. anal. appl. 9 (2) (2017), 10–23. [9] y. rohená,t. došenović and s. radenović, a note on the paper ”a fixed point theorems in sb-metric spaces”, filomal 31 (11) (2017), 3335–3346. int. j. anal. appl. 17 (5) (2019) 751 [10] sh. sedghi, a. gholidahne, t. došenovic, j. esfahani and s. radenovic, common fixed point of four maps in sb-metric spaces, j. linear topol. alg. 5 (2) (2016), 93–104. [11] sh. sedghi, n. shobe and a. aliouche, a generalization of fixed point theorem in s-metric spaces, mat. vesn. 64 (2012), 258–266. [12] d. wardowski, fixed points of a new type of contractive mappings in complete metric spaces, fixed point theory appl. 2012 (2012), 94. [13] d. wardowski and n.v. dung, fixed points of f-weak contractions on complete metric spaces, demonstr. math. 47 (1) (2014), 146–155. 1. introduction and preliminaries 2. main results references international journal of analysis and applications volume 18, number 1 (2020), 16-32 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-16 mathematical analysis of fractional order co-infection tb and hiv model muhammad farman, muhammad usman, aqeel ahmad∗, m.o. ahmad department of mathematics and statistics, the university of lahore, lahore, pakistan ∗corresponding author: aqeelahmad.740@gmmail.com abstract. a mathematical model of hiv/aids and tb including its co-infections is formulated. we find the equilibrium points and with the help of numerical simulation, we have analyzed that the sub-models of tb, hiv/aids and its co-infections. the caputo and caputo febrizo fractional derivative operator of order α ∈ (0, 1] is employed to obtain the system of fractional differential equations. laplace adomian decomposition method was successfully used for solving the different differential equations.laplace transform is a perfect technique in various field of biological science,engineering,pure and applied mathematics. the latest technique laplace adomian decomposition method is employed on the developed fractional order model for the numerical solutions. finally numerical simulations are also established to investigate the influence of the system parameter on the spread of disease and which show effect of fractional parameter α on our obtained solutions. 1. introduction mathematics plays a key role in many scientific disciplines, primarily as a mathematical modeling tool. mathematics may be thought to play an equally important or even less important role in biology, and mathematics and biology may have something in common. although physics and biology are very different sciences, mathematicians and biologists have developed ”mathematical biology” or ”biomathematics” in the latest disciplines of biological mathematical characterization, as well as research in biology, biomedical and received 2019-10-29; accepted 2019-11-21; published 2020-01-02. 2010 mathematics subject classification. 37c75, 65l07. key words and phrases. dynamical transmission; caputo fractional derivative; simulation; ladm. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 16 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-16 int. j. anal. appl. 18 (1) (2020) 17 biotechnological sciences. the emphasis on biomathematics by the current departmental name is undoubtedly a response to the rapid growth in the use of mathematics in biology in recent years [1], but it is in line with my official duties, including the explicit mention of statistics. the mathematician has taken initiatives, but there is no precedent. as prof. ej williams of melbourne stated in his review of the development of biomathematical models [2], a mathematical model of any phenomenon can only contribute to understanding when it comes to conclusions that can be tested at using experimental or experimental evidence. operational test theory is nothing more than pure mathematics, even though the subject of the discussion is described in biological terms. the use of mathematics in biology is important because biology becomes more quantitative. this new discipline has applications in the fields of biology, biomedicine and biotechnology. the goal of this new discipline is to mathematically represent and model biological problems using various theories and mathematical techniques. mathematical biology has dramatic and practical applications in the fields of biology, biomedicine and biotechnology. the long-term interaction between mathematical methods and biology is explained by the problematic approach: how does the brain work? mathematics plays a role in brain research, from von helmholtz’s [3] early work, which looks for functions similar to energy, the physical and chemical foundations that describe brain dynamics, and freud’s catharsis [4]. norbert wiener’s cybernetics to study biological control mechanisms [5], mathematical research on the nervous system [6] and recent use of mathematics to study consciousness [7, 8]. examine the dynamical transmission and effect of smoking in society [15]. the distinguished features of fractional differential equations are that it outlines memory and transmitted properties of numerous mathematical models. as a fact, that fractional order models are more realistic and practical than the classical integer order models. fractional order derivative produces greater degree of freedom in these models. arbitrary order derivatives are powerful tools for the discretion of the dynamical behavior of various biomaterial and systems [16, 17]. tuberculosis and aids tuberculosis and hiv co-infection refer to people with both hiv infection and latent or active tb. when a person has hiv and tb, each disease accelerates the progression of another disease. in addition to the accelerated progression of hiv infection from latent to active tb, tuberculosis has accelerated the progression of hiv infection [9]. hiv infection and tb bacterial infections are completely different infections. if you are infected with hiv, you will not be infected with tb unless you are in contact with someone who is infected with tb. however, if you live in a country where the prevalence of tb is high, it can happen if you do not realize it. similarly, if you have tb, you will not be infected with hiv unless you engage in activities such as unprotected sex with people already infected with hiv. tuberculosis also occurs earlier than many other opportunistic infections during hiv infection. co-infected people are twice as likely to die as hiv infected people without tb. this is true even when considering antiretroviral treatment (arv) [10]. int. j. anal. appl. 18 (1) (2020) 18 the world health organization (who) estimates that between 2005 and 2012, cooperation activities in tuberculosis prevention and treatment, including tb prevention, antiretroviral therapy and hiv testing, were reinstated at 1.3 million. as a result, these cooperative activities have infected humans by reducing tuberculosis and hiv. in recent years, the number of tb-related deaths among people living with hiv has decreased [11]. overall, the lifetime risk of hiv-negative patients from latent progression to active progression is estimated at about 5-10 percent. for hiv-positive people, the same number represents a 2-year risk [12]. in 2016, approximately 374,000 people with tb and hiv were killed. this is a supplement to the death of only 1.3 million people with tb [13]. it has been reported that people co-infected with hiv and tuberculosis die as a result of hiv infection [14]. the symptoms of active tb and aids make you feel sick. the cough lasts more than 2 weeks, coughing mucus or blood, chest pain. you may also have weakness or fatigue, weight loss, lack of appetite, fever or chills, and night sweats. the link between hiv and tb is opportunistic infection (oi). for people whose immune systems are weakened, infections occur more often or more severely than those with healthy immune systems. treatment with anti-hiv drugs is called antiretroviral therapy (art). antiretroviral therapy protects the immune system and prevents hiv infection from contracting aids. in people co-infected with hiv and tb, antiviral therapy may reduce the risk of developing latent tuberculosis tuberculosis. when you cough or sneeze, the transfer causes tuberculosis and the bacteria of tuberculosis spread through the air. if you are pregnant, you are also more likely to develop tb. age is under 5 years or over 65 years old. drink or inject drugs. do not eat well. once inside the body,tb can be inactive or active. inactive tb is called latent tb. active tb is called tuberculosis. 2. mathematical model firstly split host people represented with t(t) in the form of four sections: susceptible persons w(t), ds(t) dt = bn −βtsit −βhsih −βhtsiht − (ξ + µ)s −εv −ηr (2.1) det dt = βtsit − (µ + χt + αt )et (2.2) deh dt = βtsih − (µ + χh + αh)eh (2.3) deht dt = βhtsiht − (µ + αht )eht + χtet + χheh (2.4) dit dt = αtet − (µ + dt + ψt + γt )it (2.5) dih dt = αheh − (µ + δh + ψh −θ)ih (2.6) int. j. anal. appl. 18 (1) (2020) 19 diht dt = αhteht − (µ + δht + dt −ψt −ψh −θ)ih (2.7) dah dt = δhih − (µ + dh + γh)ah (2.8) daht dt = δhtiht − (µ + dht + γht )aht (2.9) dqt dt = γtit − (µ + dt + σt )qt (2.10) dqh dt = γhah − (µ + dh + σh)qh (2.11) dqht dt = γhtaht − (µ + dht + σht )qht (2.12) dr dt = σtqt + σhqh + σhtqht − (µ + η)r (2.13) dv dt = ξs − (ε + µ)v (2.14) and n(t) = s(t) + et (t) + eh(t) + eht (t) + it (t) + ih(t) + iht (t) + ah(t) + aht (t) + qt (t) +qh(t) + qht (t) + r(t) + v (t) here et (0) = 2000; eh(0) = 2000; eht (0) = 2000; it (0) = 2000; ih(0) = 2000; iht (0) = 2000; ah(0) = 2000; aht (0) = 2000; qt (0) = 2000; qh(0) = 2000; qht (0) = 2000; r(0) = 1000; v (0) = 0; s(0) = 93, 000 (2.15) int. j. anal. appl. 18 (1) (2020) 20 3. model parameters and its formulation : we divided the human population into six classes seiqrv (susceptible-exposed-infected-quarantinedrecovered-vaccinated) and the bird population into three classes sbebib(susceptible-exposed-infected).the state variables and associated parameters of this model are given below. the state variables and associated parameters of co-infected model of hiv and tb s(t)=susceptible humans in time t, et (t)=exposed humans with tb in time t, eh(t)=exposed humans with hiv+ in time t, eht (t)=exposed humans with tb and hiv+ in time t, it (t)=infectious humans with tb in time t, ih(t)=infectious humans with hiv+ in time t, iht (t)=co-infectious humans with tb and hiv+ in time t, ah(t)=aids infected humans in time t aht (t)=aids and tb co-infected humans in time t, qt (t)=quarantined humans who are infected with tb in time t, qh(t)=quarantined humans who are infected with aids in time t, qht (t)=quarantined humans who are co-infected with aids and tb in time t, r(t)=recovered humans in time t v (t)=vaccinated humans in time t, n(t)=total human population in time t, b=birth rate of humans βt =infectivity of tb, βh=infectivity of hiv+ βht =infectivity of both tb and hiv+, η=rate of transmission from recovered to susceptible humans, γt =rate of transmission from tb infected humans to quarantined humans with tb infected, γh=rate of transmission from aids infected humans to quarantined humans with aids infected, γht =rate of transmission from tb and aids co-infected humans to quarantined humans with tb and aids coinfected, αt =rate of transmission from exposed with tb humans to infected tb humans αh=rate of transmission from exposed with hiv+ humans to infected hiv+ humans, αht =rate of transmission from co-exposed with tb and hiv+ humans to co-infected with tb and hiv+ humans, σt =rate of transmission from quarantined humans with tb infected to recovered humans, σh=rate of transmission from quarantined humans with aids infected to recovered humans, σht =rate of transmission from quarantined humans with tb and aids co-infected to recovered humans, ξ=rate of transmission from vaccinated to susceptible humans, ε=rate of transmission from susceptible humans to vaccinated humans, χt =rate of transmission from tb infected humans to coinfected humans with tb and hiv+, χh=rate of transmission from hiv+ infected humans to co-infected humans with tb and hiv+, θ=vertical transmission for hiv+, µ=natural death rate of humans, δh=rate of transmission from hiv+ infected humans to aids infected humans, δht =rate of transmission from tb and hiv+ co-infected humans to tb and aids co-infected humans, dt=death rate due to tb, dh=death rate due to aids, dht =death rate due to tb and aids. parameter values used in mathematical model: int. j. anal. appl. 18 (1) (2020) 21 parameter value parameter value n 1,00,000 χh 0.3 b 0.8 δh 0.6 βt 0.8 δht 0.7 βh 0.85 γt 0.64 βht 0.75 γh 0.3 αt 0.65 γht 0.4 αh 0.7 dt 0.1 αht 0.02 dh 0.5 χt 0.2 dht 0.6 σt 0.8 ψt 0.08 σh 0.4 ψh 0.2 σht 0.1 µ 0.03 ε 0.05 η 0.6 ξ 0.08 θ 0.001 4. qualitative analysis disease free point is (s(t),e(t)t0,e(t)h0,e(t)ht0,i(t)t0,i(t)h0,i(t)ht0,a(t)h0,a(t)ht0, 0,q(t)t0,q(t)h0,q(t)ht0,r(t)0,v (t)0) = (1.438, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5663.7) hence, the endemic point is (se,e(t)te,e(t)he,e(t)hte,i(t)te,i(t)he,i(t)hte,a(t)he,a(t)hte,q(t)te, q(t)he,q(t)hte,r(t)e,v (t)e) = ( (µ + χt + αt )(µ + dt + ψt + γt ) βtαt , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ξbn (1 + �ξ) ) 5. preliminaries definition 1: suppose f ∈h(c,d),d>c,ρ ∈[1,1]: then the newly presented caputo derivative of arbitrary [14] is given by (d ρ t )f(t)= l(ρ) 1−ρ ∫ t c f′(t)exp[−ρt−x 1−ρ]dx where l(ρ) is the normalization of the function satisfying the condition l(0) = l(1) = 1[17]. but, if g /∈h(c,d) under this condition, the derivative is given follows: int. j. anal. appl. 18 (1) (2020) 22 (d ρ t )f(t)=ρ l(ρ) 1−ρ ∫ t c (f(t) −f(x))exp[−ρt−x 1−ρ]dx remark: if σ = 1−ρ ρ ∈ [0, 1], then the above equation assumes the form (d ρ t )f(t) = r(σ) σ ∫ t c (f ′ (t)exp[− t−x σ ]dx,r(0) = r(∞) = 1 morever lim σ→∞ 1 σ exp[− t−x σ ] = ϑ(x− t) its is necessary to have the antiderivative associated with the new fractional derivative. definition 02: let us recall the well known definition of caputo fractional derivative. given c > 0,j ∈ h1(0,c) and 0 < α < 1, the caputo fractional derivative of j of order α is given by cdαj(t) = 1 γ(1 −α) ∫ t 0 (t−s)−α(j ′ (s)ds,t > 0 fractional calculus and, in particular, caputo fractional derivative, finds numerous applications in different areas of science. by changing the kernel (t−s) by the function exp(−α(t−s)/(1−)α) and 1/γ(1 −α)by1/ √ 2π(1 −α2) one obtains the new caputo-fabrizio fractional derivative of order 0 < α < 1. that is cfdαj(t) = 2 −α)m(α) 2(1 −α) ∫ t 0 (exp(− α 1 −α (t−s))j ′ (s)ds where m(α) is a normalization constant depending on α. according to the new definition, it is clear that if j is a constant function, then cdαj(t) = 0 as in the usual caputo derivative. the main difference between old and new definition is that, contrary to the old definition, the new kernel has no singularity for t = s. it is well known that laplace transform plays an important role in the study of ordinary differential equations. in the case of this new fractional definition, it is also known that, for 0 < α < 1 l{cfdαj(t)}s = (2 −α)m(α) 2(s + α(1 −s)) (sl{j(t)}(s) − j(0)) where l{j} denotes the laplace transform of function j. so, it is clear that if we work with caputofabrizio derivative, laplace transform will also be a very useful tool. theorem 1: the closed set d is bounded and positive invariant. proof: dn dt ≤ b −µn so nis bounded above by b µ . hencedn dt < 0whenver(t) > b µ on simplification, we have int. j. anal. appl. 18 (1) (2020) 23 n(t) ≤ n(0)eµt + b µ (1 − eµt) as t → ∞,eµt → 0 and so limt→∞n(t) ≤ bµ thus, d is bounded and positively invariant in r14+ 6. caputo fractional order model converting the epidemic model (2.1 − 2.14) subject to the initial condition (2.15) into fractional order model by using caputo definition, then applying the laplace transform on equation we get l{c0 d α t s(t)} = bnl{1}−βtl{sit}−βhl{sih}−βhtl{siht} − (ξ + µ)l{s}−εl{v}−ηl{r} (6.1) l{c0 d α t e(t)t} = βtl{sit}− (µ + χt + αt )l{et} (6.2) l{c0 d α t e(t)h} = βtl{sih}− (µ + χh + αh)l{eh} (6.3) l{c0 d α t e(t)ht} = βhtl{siht}− (µ + αht )l{eht} + χtl{et} + χhl{eh} (6.4) l{c0 d α t i(t)t} = αtl{et}− (µ + dt + ψt + γt )l{it} (6.5) l{c0 d α t i(t)h} = αhl{eh}− (µ + δh + ψh −θ)l{ih} (6.6) l{c0 d α t i(t)ht} = αhtl{eht}− (µ + δht + dt −ψt −ψh −θ)l{iht} (6.7) l{c0 d α t a(t)h} = δhl{ih}− (µ + dh + γh)l{ah} (6.8) l{c0 d α t a(t)ht} = δhtl{iht}− (µ + dht + γht )l{aht} (6.9) l{c0 d α t q(t)t} = γtl{it}− (µ + dt + σt )l{qt} (6.10) l{c0 d α t q(t)h} = γhl{ah}− (µ + dh + σh)l{qh} (6.11) l{c0 d α t q(t)ht} = γhtl{aht}− (µ + dht + σht )l{qht} (6.12) l{c0 d α t r(t)} = σtl{qt} + σhl{qh} + σhtl{qht}− (µ + η)l{r} (6.13) l{c0 d α t v (t)} = ξl{s}− (ε + µ)l{v} (6.14) int. j. anal. appl. 18 (1) (2020) 24 the series solution is as follows s(t) = 93, 000 − 446, 33, 830 tα α! + 2.1436 × 1012 t2α 2α! + 106, 62400 t3α 3α! 2 α! + 6.016 × 1015 t3α 3α! + ....... et (t) = 2000 + 148, 798, 240 tα α! − 714472000000 t2α 2α! − 6.145 × 1011 t3α 3α! − 25, 600, 000 t3α 3α! 2 α! eh(t) = 2000 + 158097940 tα α! − 7.59164 × 1011 t2α 2α! − 6.529 × 1011 t3α 3α! − 17544, 000 3α! t3α2 α! + ....... eht (t) = 2000 + 139, 500, 900 tα α! − 6.6958 × 1011 t2α 2α! − 5.759 × 1011 t3α 3α! − 63, 480, 000 t3α 3α! 2 α! + ...... it (t) = 2000 − 400 tα α! + 96, 719, 192 t2α 2α! + 83, 200, 000 t3α 3α! + .......... ih(t) = 2000 − 258 tα α! + 110, 668, 794.3 t2α 2α! + 95, 200, 000 3α! t3α + ..... iht (t) = 2000 − 1058 tα α! + 2, 790, 598.842 t2α 2α! + 2, 400, 000 3α! t3α + ......... ah(t) = 2000 − 460 tα α! + 227 t2α 2α! + ...... aht (t) = 2000 − 660 tα α! − 60.8 t2α 2α! + ..... qt (t) = 2000 − 580 tα α! + 283.4 t2α α! + ...... qh(at) = 2000 − 1260 tα α! + 1033.8 t2α 2α! + ..... qht (t) = 2000 − 660 tα α! + 217.8 t2α 2α! + ....... r(t) = 1000 + 1970 tα α! − 2275.1 t2α 2α! + ....... v (t) = 7440 tα α! − 37, 707, 061.6 t2α 2α! − 30, 721, 216 t3α 3α! + ........ 7. fractional order model with caputo febrizio sense the fractional order model of the system (2.1-2.14) by using caputo febrizio derivative is given as. cf 0 d α t s(t) = bn −βtsit −βhsih −βhtsiht − (ξ + µ)s −εv −ηr (7.1) cf 0 d α t e(t)t = βtsit − (µ + χt + αt )et (7.2) cf 0 d α t e(t)h = βtsih − (µ + χh + αh)eh (7.3) cf 0 d α t e(t)ht = βhtsiht − (µ + αht )eht + χtet + χheh (7.4) cf 0 d α t i(t)t = αtet − (µ + dt + ψt + γt )it (7.5) cf 0 d α t i(t)h = αheh − (µ + δh + ψh −θ)ih (7.6) int. j. anal. appl. 18 (1) (2020) 25 cf 0 d α t i(t)ht = αhteht − (µ + δht + dt −ψt −ψh −θ)iht (7.7) cf 0 d α t a(t)h = δhih − (µ + dh + γh)ah (7.8) cf 0 d α t a(t)ht = δhtiht − (µ + dht + γht )aht (7.9) cf 0 d α t q(t)t = γtit − (µ + dt + σt )qt (7.10) cf 0 d α t q(t)h = γhah − (µ + dh + σh)qh (7.11) cf 0 d α t q(t)ht = γhtaht − (µ + dht + σht )qht (7.12) cf 0 d α t r(t) = σtqt + σhqh + σhtqht − (µ + η)r (7.13) cf 0 d α t v (t) = ξs − (ε + µ)v (7.14) the series solution is as follows s(t) = 9300 − 446330830(1 −α) + 2.14239 × 1012(1 −α)2 + 1.84324 × 1012(1 −α)3 − 1182(1 −α) +{446490830α + 4.31477 × 1012α(1 −α) − 1182α2 + 5.52973 × 1012α(1 −α)2 +2.30403435 × 1012α2(1 −α)}t + 2.14258 × 1012α2t2 + 1.53607 × 1011α3t3 et (t) = 2000 + 148798240(1 −α) − 7.14242 × 1011(1 −α)2 − 6.145 × 1011(1 −α)3 +{−1.4284 × 1012α(1 −α) + 148798240α− 1.8435 × 1012α(1 −α)2}t {−7.14306 × 1011α2 − 7.6815 × 1011α2(1 −α)}t2 − 5.1218 × 1010α3t3 eh(t) = 2000 + 1158100000(1 −α) − 7.5892 × 1011(1 −α)2 − 6.5294 × 1011(1 −α)3 +{−1.51783 × 1012α(1 −α) + 158100000α− 1.958 × 1012α(1 −α)2}t {−7.5899 × 1011α2 − 8.16 × 1011α2(1 −α)}t2 − 5.442 × 1010α3t3 eht (t) = 2000 + 139500000(1 −α) − 6.6888 × 1011(1 −α)2 − 5.75 × 1011(1 −α)3 +{−1.33896 × 1012α(1 −α) + 139500000α− 1.727 × 1012α(1 −α)2}t {−3.3444 × 1011α2 − 7.1875 × 1011α2(1 −α)}t2 − 4.792 × 1010α3t3 it (t) == 2000 − 400(1 −α) + 96719196(1 −α)2 + 83200000(1 −α)3 + {193438392α(1 −α) − 400α +249600000α(1 −α)2}t + {48359598α2 + 124800000α2(1 −α)}t2 + 13866666.67α3t3 ih(t) == 2000 − 258(1 −α) + 110670213.9(1 −α)2 + 95200000(1 −α)3 + {221340427.8α(1 −α) − 258α +600000α(1 −α)2}t + {55335106.95α2 + 102935107α2(1 −α)}t2 + 15866666.67α3t3 int. j. anal. appl. 18 (1) (2020) 26 iht (t) == 2000 − 1058(1 −α) + 2790580.842(1 −α)2 + 2400000(1 −α)3 + {5581161.684α(1 −α) −1058α + 7200000α(1 −α)2}t + {1395290421α2 + 3600000α2(1 −α)}t2 + 400000α3t3 a(h)(t) = 2000 − 460(1 −α) + 227(1 −α)2 + {456α(1 −α) − 460α}t + 113.5α2t2 a(ht)(t) = 2000 − 660(1 −α) − 60.8(1 −α)2 −{121.6α(1 −α) + 660α}t− 30.4α2t2 q(t)(t) = 2000 − 580(1 −α) + 283.4(1 −α)2 + {566.8α(1 −α) − 580α}t + 141.7α2t2 q(h)(t) = 2000 − 1260(1 −α) + 1033.8(1 −α)2 + {2067.6α(1 −α) − 1260α}t + 516.9α2t2 q(ht)(t) = 2000 − 660(1 −α) + 217.8(1 −α)2 + {435.6α(1 −α) − 660α}t + 108.9α2t2 r(t) = 1000 + 1970(1 −α) − 2275.5(1 −α)2 + {−4550.5α(1 −α) + 1970α}t− 1137.5α2t2 v (t)) == 7440(1 −α) − 35707061.6(1 −α)2 − 3072192(1 −α)3 + {−7129893.2α(1 −α) + 7440α −92160576α(1 −α)2}t + {−46080800α2 − 17853530.8α2(1 −α)}t2 − 5120202.667α3t3 8. numerical results and discussion the mathematical analysis of seicr epidemic model with nonlinear system of differential equation has been presented. firstly we determined the disease free equilibrium point of fractional order model.for the reliable investigation, evaluation is made for different values of α.hence from figure 1 when we increase the parameter value the rate of susceptible humans will be decrease.from figs.2-4 we observed solution gives better converges to steady state for exposed by increasing the fractional values of α, while from figs.5-9 the compartments converges fastly by putting different values of α.finally in the last fig. the vaccinated humans in time t be increased by decreased the values of α.the laplace adomian decomposition method is an approximate solution in term of infinite power series.hence we can easily observed the steady state solution at fr5actinal order derivative is better, reliable and more effective reasults as compared to ordinary at α = 1. int. j. anal. appl. 18 (1) (2020) 27 time (days) ×10-3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s us ce pt ib le h um an s in ti m e t ×10 6 0 1 2 3 4 5 6 susceptible humans susceptible at α=1 susceptible at α=0.95 susceptible at α=0.90 figure 1. numerical solutions for susceptible humans s(t) in time t at different values of α. time (days) ×10-4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 e t in ti m e t 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 co-infection of hiv and tb α=1 α=0.99 α=0.95 figure 2. numerical solutions for exposed humans with tb et (t) in time t at different values of α. int. j. anal. appl. 18 (1) (2020) 28 time (days) ×10-4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 e h in ti m e t ×10 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 co-infection of hiv and tb α=1 α=0.99 α=0.98 figure 3. numerical solutions for exposed humans with aids eh(t) in time t at different values of α. time (days) ×10-4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 e h t in ti m e t 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 co-infection of hiv and tb α=1 α=0.98 α=0.99 figure 4. numerical solutions for exposed humans with tb and hiv eht (t) in time t at different values of α. int. j. anal. appl. 18 (1) (2020) 29 time (days) ×10-3 0 1 2 3 4 5 6 7 in fe ct io us h um an s w ith h iv a nd t v 0 500 1000 1500 2000 2500 co-infection of hiv and tb co-infected hiv at α=1 co-infected hiv and tb at α=1 co-infected hiv at α=0.9 co-infected hiv and tb at α=0.9 co-infected hiv at α=0.95 co-infected hiv and tb at α=0.95 figure 5. numerical solutions for infectious humans with tb and hiv ih(t) in time t at different values of α. time (days) 0 0.1 0.2 0.3 0.4 0.5 0.6 a h h um an s in ti m e t 1700 1750 1800 1850 1900 1950 2000 co-infection of hiv α=1 α=0.9 α=0.8 figure 6. numerical solutions for aids infected humans ah(t) in time t (year) at different values of α. int. j. anal. appl. 18 (1) (2020) 30 time (days) 0 0.5 1 1.5 2 2.5 a ht h um an s in ti m e t 0 200 400 600 800 1000 1200 1400 1600 1800 2000 co-infection of hiv and tb α=1 α=0.9 α=0.8 figure 7. numerical solutions for aids and tb co-infected humans aht (t) in time t (year) at different values of α. time (days) 0 0.5 1 1.5 q ua ra nt in ed h um an s w ho a re c oin fe ct ed w ith a id s a nd t b in ti m e t 1200 1300 1400 1500 1600 1700 1800 1900 2000 q h infected of hiv and tb with cupoto sence q h infected with aids at α=1 q h co-infected with aids and tb at α=1 q h infected with aids at α=0.9 q h co-infected with aids and tb at α=0.9 q h infected with aids at α=0.8 q h co-infected with aids and tb at α=0.8 figure 8. numerical solutions for quarantined humans who are co-infected with aids and tb qht (t) in time t (year) at different values of α. int. j. anal. appl. 18 (1) (2020) 31 time (days) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 r ec ov er ed h um an s in ti m e t 0 200 400 600 800 1000 1200 1400 1600 1800 2000 co-infection of hiv and tb α=1 α=0.8 α=0.9 figure 9. numerical solutions for recovered humans r(t) in time t at different values of α. time (days) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 v ac ci na te d hu m an s in ti m e t 0 1000 2000 3000 4000 5000 6000 vaccination α=1 α=0.8 α=0.9 figure 10. numerical solutions for vaccinated humans v (t) in time t at different values of α. 9. conclusion we developed a scheme for the solution of co-infection of tb and hiv by using laplace adomian decomposition method. the well known co-infection of tb and hiv model namely susceptible population, exposed population, co-infectious population, quarantined population, recovered population and vaccinated is considered with and without demographic effects. we can solved the detail model of co-infection of tb and hiv with fractional order derivative with caputo sense and caputo fabrizio sense and we use the int. j. anal. appl. 18 (1) (2020) 32 laplace adomian decomposition method to solve the model. it is also observed that to eliminate disease, it is not necessary to vaccinate whole of population. the effect of fractional parameter on our obtained solutions is presented through graphs it is worthy to observe that fractional order derivative show significant changes and memory effect as compared to ordinary derivative. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. t. j. bailey, the mathematical approach to biology and medicine, john wiley, new york, 1967. [2] e. j. williams, the development of biomathematical models. 36th session, international statistical institute, sydney, 1967. [3] h. f. von helmholtz, on the sensations of tone as a physiological basis for the theory of music. (2nd ed.) dover publics, new york, 1954. [4] s. freud, project on a scientific psychology, in the standard edition of the complete psychological works of sigmund freud, vol i, pp 283-346. hogarth press, london, 1966. [5] n. wiener, cybernetics, (2nd ed.) mit press, cambridge, 1961. [6] r. thom, structural stability and morphogenesis, westview press; new ed edition, 2001. [7] g. edelmann, neural darwinism: the theory of neuronal group selection, basic books, new york, 1987. [8] r. penrose, the emperors new mind: concerning computers, minds, and the laws of physics, amer. j. phys. 58(1990), 1214. [9] https://www.unaids.org/en. [10] s. suchindran, is hiv infection a risk factor for multi-drug resistant tuberculosis? a systematic review, plos one, 4 (5) (2009), e5561. [11] unaids, global report: unaids report on the global aids epidemic 2013, world health organization, geneva, switzerland, 2013 [12] implementing the who stop tb strategy: a handbook for national tuberculosis control programmes, geneva, world health organization, 2008, p67. [13] global tuberculosis control 2017, who, geneva, 2017. [14] international classification of diseases (icd), who, geneva, 2010. [15] a. ahmad, m. farman, f. yasin, m. o. ahmad, dynamical transmission and effect of smoking in society, int. j. adv. appl. sci. 5 (2) (2018), 71-75. [16] a. ahmad, m. farman, m.o. ahmad, n. raza, m. abdullah, dynamical behavior of sir epidemic model with non-integer time fractional derivatives: a mathematical analysis, int. j. adv. appl. sci. 5 (1) (2018), 123-129. [17] farah ashraf, aqeel ahmad, muhammad umer saleem, muhammad farman, m. o. ahmad, dynamical behavior of hiv immunology model with non-integer time fractional derivatives, int. j. adv. appl. sci. 5(3) (2018), 39-45. 1. introduction 2. mathematical model 3. model parameters and its formulation 4. qualitative analysis 5. preliminaries 6. caputo fractional order model 7. fractional order model with caputo febrizio sense 8. numerical results and discussion 9. conclusion references international journal of analysis and applications volume 16, number 6 (2018), 904-920 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-904 fixed point results of rational type contractions in b-metric spaces mian bahadur zada1, muhammad sarwar1,∗ and poom kumam2,∗ 1department of mathematics university of malakand, khyber pakhtunkhwa, chakdara dir(l), pakistan 2department of mathematics, faculty of science, king mongkut’s university of technology thonburi (kmutt), 126 pracha uthit road, bang mod, thung khru, bangkok 10140, thailand ∗corresponding authors: sarwar@uom.edu.pk, poom.kum@kmutt.ac.th abstract. the aim of this manuscript is to establish fixed point results satisfying contractive conditions of rational type in the setting of b-metric spaces. the results proved herein are the generalization and extension of some well known results in the existing literature. example is also given in order to illustrate the validity of the presented results. 1. introduction and preliminaries the banach contraction principle [2] is considered to be the pioneering result of the fixed point theory, and plays an important role for solving existence problems in many branches of nonlinear analysis. this principle asserts if (x,d) is a complete metric space and k : x → x satisfies d(kx,ky) ≤ λd(x,y), (1.1) for all x,y ∈ x with λ ∈ [0, 1), then k has a unique fixed point. this principle have been improved and extended by several mathematicians in different directions some of them are as follows: let k be a mapping on a metric space (x,d) and x,y ∈ x, then k is said to be received 2018-06-11; accepted 2018-08-13; published 2018-11-02. 2010 mathematics subject classification. primary 47h10; secondary 54h25. key words and phrases. b–metric spaces; common fixed points; self-maps; cauchy sequence; contractive conditions. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 904 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-904 int. j. anal. appl. 16 (6) (2018) 905 (1) kannan type contraction [10], if there exists a number λ ∈ [ 0, 1 2 ) such that d(kx,ky) ≤ λ[d(x,kx) + d(y,ky)]. (1.2) (2) chatterjee type contraction [4], if there exists a number λ ∈ [ 0, 1 2 ) such that d(kx,ky) ≤ λ[d(x,ky) + d(y,kx)]. (1.3) (3) reich type contraction [12], if there exists a number λ,µ,ν ∈ [0, 1) with λ + µ + ν < 1 such that d(kx,ky) ≤ λd(x,y) + µd(x,kx) + νd(y,ky). (1.4) (4) das and gupta [7] rational type contraction, if there exists a number λ,µ ∈ [0, 1) with λ + µ < 1 such that d(kx,ky) ≤ λd(x,y) + µ d(y,ky)[1 + d(x,kx)] 1 + d(x,y) . (1.5) the contractive conditions on underlying functions play an important role for finding solutions of metric fixed point problems. inspired from the impact of this natural idea, the above contractions have been extended and generalized by several researchers in various spaces such as quasi-metric spaces, cone metric spaces, g-metric spaces, partial metric spaces and vector valued metric spaces etc. following this trend, bakhtin [1] and czerwik [5] generalized metric space with non hausdorff topology called b–metric space to overcome the problem of measurable functions with respect to measure and their convergence. they presented the generalization of the banach contraction principle in b–metric spaces. since then, several papers has been studied by many authors dealing with the existence of fixed point in b–metric spaces (see, [3, 6, 8, 9, 11, 13, 14] and the references therein). the aim of this contribution is to investigate some fixed point results using the concept of the contractive conditions of rational type in the context of b–metric spaces. moreover, an example is given here to illustrate the validity of the obtained results. actually the derived results generalizes the results of [2, 4, 7, 10, 12]. now, we recall some essential notations, definitions and primary results known in the literature. throughout this manuscript, r = set of real numbers, r+ = [0,∞) and n = set of positive integers. definition 1.1. [1, 5] let x be a nonempty set. a function d : x × x → r+ is called a b–metric with coefficient s ≥ 1 if: (1) d(x,y) = 0 ⇔ x = y; (2) d(x,y) = d(y,x) ∀ x,y ∈ x; (3) d(x,y) ≤ s[d(x,y) + d(z,y)] ∀ x,y,z ∈ x. then the pair (x,d) is called a b-metric space. int. j. anal. appl. 16 (6) (2018) 906 remark 1.1. every metric space is b–metric space with s = 1, but in general, a b–metric space need not necessarily be a metric space, as in below example 1.1, (x,d) is b–metric space but not a metric space (see also, examples in [6, 13]). thus, the class of b–metric spaces is larger than the class of metric spaces. example 1.1. let x = r and let the mapping d : x × x → r+ be defined by d(x,y) = |x − y|2 for all x,y ∈ x. then (x,d) is a b–metric space with coefficient s = 2. sintunavarat [14] generalized example 1.1 as: example 1.2. let (x,ρ) be a metric space and p ≥ 1 be a given real number. then d(x,y) = [ρ(x,y)]p is a b–metric on x with parameter s ≤ 2p−1. the following example 1.3 shows that b-metric is not continuous in general (see also, examples in [9, 11]). example 1.3. [8] let x = n∪{∞} and d : x ×x → r be defined by d(m,n) =   0, if m = n,∣∣ 1 m + 1 n ∣∣ , if one of m,n is even and the other is even or ∞, 5, if one of m,n is odd and the other is odd (and m = n) or ∞, 2, otherwise. then, considering all possible cases, it can be checked that (x,d) is a b–metric space with s = 5 2 . however, let xn = 2n for each n ∈ n. then lim n→∞ d(2n,∞) = lim n→∞ 1 2n = 0, that is, xn →∞, but d(xn, 1) = 2 9 5 = d(∞, 1) as n →∞. definition 1.2. [3] let {xn} be a sequence in b–metric space (x,d) and x ∈ x. then (1) {xn} converges to x if and only if for every ε > 0, there exists n(ε) ∈ n, such that d(xn,x) < ε for all n > n(ε) and we write lim n→∞ d(xn,x) = 0 or lim n→∞ xn = x. (2) {xn} is a cauchy sequence if for every ε > 0, there exists n(ε) ∈ n, such that d(xn,xm) < ε for all m,n > n(ε). proposition 1.1. [3] in a b–metric space (x,d), the following assertions hold: • a convergent sequence has a unique limit; • each convergent sequence is cauchy; • a metric space (x,d) is complete if every cauchy sequence is convergent in x. 2. fixed point results in b-metric spaces to present the main results, we need the following lemma. int. j. anal. appl. 16 (6) (2018) 907 lemma 2.1. let (x,d) be a complete b-metric space and l : x → x. let x0 ∈ x and define the sequence {xn} by lxn = xn+1 ∀ n = 0, 1, 2, ... let there exists a mapping λ : x ×x → [0, 1) satisfying λ(lx,y) ≤ λ(x,y) and λ(x,ly) ≤ λ(x,y), for all x,y ∈ x. then λ(xn,y) ≤ λ(x0,y) and λ(x,xn+1) ≤ λ(x,x1) for all x,y ∈ x and n = 0, 1, 2, .... proof. let x,y ∈ x and n = 0, 1, 2, ..., then λ(xn,y) = λ(lxn−1,y) ≤ λ(xn−1,y) = λ(lxn−2,y) ≤ λ(xn−2,y) = ... ≤ λ(x0,y). similarly, λ(x,xn+1) = λ(x,lxn) ≤ λ(x,xn) = λ(x,lxn−1)) = ... ≤ λ(x,x0). � now, we prove the main result. theorem 2.1. let (x,d) be a complete b-metric space and λi : x×x → [0, 1), i = 1, 2, ..., 6. if l : x → x be a self-map such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lx,y) ≤ λi(x,y) and λi(x,ly) ≤ λi(x,y); (ii) d(lx,ly) ≤λ1(x,y)d(x,y) + λ2(x,y) [d(x,ly) + d(y,lx)] s + λ3(x,y)[d(x,lx) + d(y,ly)] + λ4(x,y) d(y,ly)[1 + d(x,lx)] 1 + d(x,y) + λ5(x,y) d(x,ly)d(x,lx) s[1 + d(x,y)] + λ6(x,y) d(x,ly)d(y,lx) s[1 + d(x,y)d(y,lx)] , where λ2(x,y) + λ3(x,y) + λ5(x,y) + s 6∑ i=1 λi(x,y) < 1, with 0 ≤ 6∑ i=1 λi(x,y) < 1. then the mapping l has a unique fixed point in x. proof. let x0 ∈ x and construct a sequence {xn} by the rule lxn = xn+1, ∀ n = 0, 1, 2, ... (2.1) first, we show that {xn} is a cauchy sequence in x. for this, consider d(xn+1,xn+2) = d(lxn,lxn+1), int. j. anal. appl. 16 (6) (2018) 908 by using condition (ii) of theorem 2.1 with x = xn and y = xn+1, we have d(lxn,lxn+1) ≤λ1(xn,xn+1)d(xn,xn+1) + λ2(xn,xn+1) [d(xn,lxn+1) + d(xn+1,lxn)] s + λ3(xn,xn+1)[d(xn,lxn) + d(xn+1,lxn+1)] + λ4(xn,xn+1) d(xn+1,lxn+1)[1 + d(xn,lxn)] 1 + d(xn,xn+1) + λ5(xn,xn+1) d(xn,lxn+1)d(xn,lxn) s[1 + d(xn,xn+1)] + λ6(xn,xn+1) d(xn,lxn+1)d(xn+1,lxn) s[1 + d(xn,xn+1)d(xn+1,lxn)] , using (2.1), we get d(xn+1,xn+2) ≤λ1(xn,xn+1)d(xn,xn+1) + λ2(xn,xn+1) [d(xn,xn+2) + d(xn+1,xn+1)] s + λ3(xn,xn+1)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(xn,xn+1) d(xn+1,xn+2)[1 + d(xn,xn+1)] 1 + d(xn,xn+1) + λ5(xn,xn+1) d(xn,xn+2)d(xn,xn+1) s[1 + d(xn,xn+1)] + λ6(xn,xn+1) d(xn,xn+2)d(xn+1,xn+1) s[1 + d(xn,xn+1)d(xn+1,xn+1)] , with the help of condition (i) of theorem 2.1, we get d(xn+1,xn+2) ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0) [d(xn,xn+2) + d(xn+1,xn+1)] s + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0) d(xn+1,xn+2)[1 + d(xn,xn+1)] 1 + d(xn,xn+1) + λ5(x0,x0) d(xn,xn+2)d(xn,xn+1) s[1 + d(xn,xn+1)] ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0) d(xn,xn+2) s + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0)d(xn+1,xn+2) + λ5(x0,x0) d(xn,xn+2) s . using triangular inequality, we get d(xn+1,xn+2) ≤λ1(x0,x0)d(xn,xn+1) + λ2(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ3(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)] + λ4(x0,x0)d(xn+1,xn+2) + λ5(x0,x0)[d(xn,xn+1) + d(xn+1,xn+2)], int. j. anal. appl. 16 (6) (2018) 909 which implies that d(xn+1,xn+2) ≤ 3∑ i=1 λi(x0,x0) + λ5(x0,x0) 1 − 5∑ i=2 λi(x0,x0) d(xn,xn+1). let h = 3∑ i=1 λi(x0,x0)+λ5(x0,x0) 1− 5∑ i=2 λi(x0,x0) < 1 s · then d(xn+1,xn+2) ≤ hd(xn,xn+1). similarly, d(xn,xn+1) ≤ hd(xn−1,xn). consequently, d(xn+2,xn+1) ≤ hd(xn+1,xn) ≤ h2d(xn,xn−1) ≤ ... ≤ hn+1d(x1,x0). now, for m > n and sh < 1, we have d(xn,xm) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + ... + sm−nd(xm−1,xm), ≤ shnd(x1,x0) + s2hn+1d(x1,x0) + ... + sm−nhm−1d(x1,x0) ≤ [shn + s2hn+1 + ... + sm−nhm−1]d(x1,x0) ≤ shn [ 1 + (sh)1 + (sh)2 + ... + (sh)m−n−1 ] d(x1,x0) ≤ shn 1 −sh d(x1,x0). therefore lim n→∞ d(xn,zm) = 0. hence, {xn} is a cauchy sequence. but x is complete, so there exists t ∈ x such that xn → t as n →∞. next, we show that t is a fixed point of l. for this, assume that lt 6= t, then d(t,lt) 6= 0. now d(t,lt) ≤ d(t,lxn) + d(lxn,lt). (2.2) by applying condition (ii) of theorem 2.1, equation (2.2) become d(t,lt) ≤d(t,lxn) + λ1(xn, t)d(xn, t) + λ2(xn, t) [d(xn,lt) + d(t,lxn)] s + λ3(xn, t)[d(xn,lxn) + d(t,lt)] + λ4(xn, t) d(t,lt)[1 + d(xn,lxn)] 1 + d(xn, t) + λ5(xn, t) d(xn,lt)d(xn,lxn) s[1 + d(xn, t)] + λ6(xn, t) d(xn,lt)d(t,lxn) s[1 + d(xn, t)d(t,lxn)] , int. j. anal. appl. 16 (6) (2018) 910 with the help of equation (2.1) and condition (i) of theorem 2.1, we can write d(t,lt) ≤d(t,xn+1) + λ1(x0, t)d(xn, t) + λ2(x0, t) [d(xn,lt) + d(t,xn+1)] s + λ3(x0, t)[d(xn,xn+1) + d(t,lt)] + λ4(x0, t) d(t,lt)[1 + d(xn,xn+1)] 1 + d(xn, t) + λ5(x0, t) d(xn,lt)d(xn,xn+1) s[1 + d(xn, t)] + λ6(x0, t) d(xn,lt)d(t,xn+1) s[1 + d(xn, t)d(t,xn+1)] . taking limit as n →∞, we get d(t,lt) ≤λ2(x0, t) d(t,lt) s + λ3(x0, t)d(t,lt) + λ4(x0, t)d(t,lt). d(t,lt) ≤ [λ2(z0, t) + sλ3(z0, t) + sλ4(z0, t)] d(t,lt) s . (2.3) but λ2(z0, t) + sλ3(z0, t) + sλ4(z0, t) < 1, so the above inequality (2.3) contradict the fact that d(t,lt) 6= 0. thus lt = t. hence t is a fixed point of l. finally, we have to show that t is a unique fixed point of l. for this, let t∗ 6= t be another fixed point of l. then on putting x = t and y = t∗ in condition (ii) of theorem 2.1, we get d(t,t∗) =d(lt,lt∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) [d(t,lt∗) + d(t∗,lt)] s + λ3(t,t ∗)[d(t,lt) + d(t∗,lt∗)] + λ4(t,t ∗) d(t∗,lt∗)[1 + d(t,lt)] 1 + d(t,t∗) + λ5(t,t ∗) d(t,lt∗)d(t,lt) s[1 + d(t,t∗)] + λ6(t,t ∗) d(t,lt∗)d(t∗,lt) s[1 + d(t,t∗)d(t∗,lt)] ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) [d(t,t∗) + d(t∗, t)] s + λ6(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)d(t∗, t)] , implies that d(t,t∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) 2d(t,t∗) s + λ6(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)d(t∗, t)] ≤[sλ1(t,t∗) + 2λ2(t,t∗) + λ6(t,t∗)] d(t,t∗) s , which is contradiction because sλ1(t,t ∗) + 2λ2(t,t ∗) + λ6(t,t ∗) < 1. hence t is a unique fixed point of l. � from theorem 2.1, we can easily derive the following corollaries and the proofs of which are simple, so we omit it. corollary 2.1. let (x,d) be a complete b-metric space and λi : x ×x → [0, 1), i = 1, 3. if l : x → x be a self-map such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lx,y) ≤ λi(x,y) and λi(x,ly) ≤ λi(x,y); int. j. anal. appl. 16 (6) (2018) 911 (ii) d(lx,ly) ≤ λ3(x,y)[d(x,lx) + d(y,ly)], where 0 ≤ λ3(x,y) < 1s+1 . then the mapping l has a unique fixed point in x. corollary 2.2. let (x,d) be a complete b-metric space and λi : x×x → [0, 1), i = 1, 2, ..., 8. if l : x → x be a self-map such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lx,y) ≤ λi(x,y) and λi(x,ly) ≤ λi(x,y); (ii) d(lx,ly) ≤ λ2(x,y) [d(x,ly) + d(y,lx)] s , where 0 ≤ λ2(x,y) < 1s+1 . then the mapping l has a unique fixed point in x. corollary 2.3. let (x,d) be a complete b-metric space and λi : x ×x → [0, 1), i = 1, 4. if l : x → x be a self-map such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lx,y) ≤ λi(x,y) and λi(x,ly) ≤ λi(x,y); (ii) d(lx,ly) ≤λ1(x,y)d(x,y) + λ4(x,y) d(y,ly)[1 + d(x,lx)] 1 + d(x,y) , where 0 ≤ sλ1(x,y) + λ4(x,y) < 1. then the mapping l has a unique fixed point in x. corollary 2.4. let (x,d) be a complete b-metric space and λi : x ×x → [0, 1), i = 1, 2, 3. if l : x → x be a self-map such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lx,y) ≤ λi(x,y) and λi(x,ly) ≤ λi(x,y); (ii) d(lx,ly) ≤ λ1(x,y)d(x,y) + λ2(x,y)d(x,lx) + λ3(x,y)d(y,ly), where 0 ≤ s[λ1(x,y) + λ2(x,y)] + λ3(x,y) < 1. then the mapping l has a unique fixed point in x. corollary 2.5. let λ6 = 0 and all other conditions of theorem 2.1 are satisfied, then l has a unique fixed point in x. corollary 2.6. let λ5 = λ6 = 0 and all other conditions of theorem 2.1 are satisfied, then l has a unique fixed point in x. corollary 2.7. let λ2 = λ3 = 0 and all other conditions of theorem 2.1 are satisfied, then l has a unique fixed point in x. int. j. anal. appl. 16 (6) (2018) 912 corollary 2.8. let λ4 = λ5 = λ6 = 0 and all other conditions of theorem 2.1 are satisfied, then l has a unique fixed point in x. remark 2.1. (1) in theorem 2.1, if s = 1,λi = 0, for i = 2, 3, 4, 5, 6 and λ1(·) = λ1, we get the banach theorem [2]. (2) in corollary 2.1, if λi = 0, for i = 1, 2, 4, 5, 6, λ3(·) = λ and s = 1, we get the kannan theorem [10]. (3) in corollary 2.2, if λi = 0, for i = 3, 4, 5, 6, λ2(·) = λ and s = 1, we get the chatterjee theorem [4]. (4) in corollary 2.3, if λi = 0, for i = 2, 3, 5, 6, λj(·) = λj for j = 1, 4 and s = 1, we get the result of dass and gupta [7]. (5) in corollary 2.4, if s = 1 and λi(·) = λi for i = 1, 2, 3, we get theorem 3 of [12]. 3. common fixed point results in b-metric spaces for the proof of our next result we use the following lemma. lemma 3.1. let (x,d) be a complete b-metric space and k,l : x → x. let x0 ∈ x and define the sequence {xn} by kx2n = x2n+1 and lx2n+1 = x2n+2 ∀ n = 0, 1, 2, ... let there exists a mapping λ : x ×x → [0, 1) satisfying λ(lkx,y) ≤ λ(x,y) and λ(x,kly) ≤ λ(x,y), for all x,y ∈ x. then λ(x2n,y) ≤ λ(x0,y) and λ(x,x2n+1) ≤ λ(x,x1) for all x,y ∈ x. proof. the proof easily follows from lemma 2.1 . � our next result is proved for a pair of self-maps. theorem 3.1. let (x,d) be a complete b-metric space with s ≥ 1 and λi : x ×x → [0, 1), i = 1, 2, ..., 5. if k,l : x → x be two self-mappings such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lkx,y) ≤ λi(x,y) and λi(x,kly) ≤ λi(x,y); (ii) d(kx,ly) ≤λ1(x,y)d(x,y) + λ2(x,y) d(x,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ3(x,y) d(y,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ4(x,y) d(y,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] + λ5(x,y) d(x,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] ; int. j. anal. appl. 16 (6) (2018) 913 where 5∑ i=2 λi(x,y) + s 5∑ i=1 λi(x,y) + 1 s [λ2(x,y) + λ4(x,y)] < 1, with 0 ≤ 5∑ i=1 λi(x,y) < 1. then k and l have a unique common fixed point in x. proof. let x0 ∈ x and construct a sequence {xn} by the rule kx2n = x2n+1 and lx2n+1 = x2n+2, ∀ n = 0, 1, 2, ... (3.1) first we to show that {xn} is a cauchy sequence in x. for this, consider d(x2k+1,x2k) = d(klx2k−1,lx2k−1). by using condition (ii) of theorem 3.1 with x = lx2k−1 and y = x2k−1, we have d(klx2k−1,lx2k−1) ≤λ1(lx2k−1,x2k−1)d(lx2k−1,x2k−1) +λ2(lx2k−1,x2k−1) d(lx2k−1,klx2k−1)[d(lx2k−1,lx2k−1) + d(x2k−1,lx2k−1)] s[1 + d(lx2k−1,x2k−1)] +λ3(lx2k−1,x2k−1) d(x2k−1,klx2k−1)[d(lx2k−1,lx2k−1) + d(x2k−1,lx2k−1)] s[1 + d(lx2k−1,x2k−1)] +λ4(lx2k−1,x2k−1) d(x2k−1,lx2k−1)[d(lx2k−1,klx2k−1) + d(x2k−1,klx2k−1)] s[1 + d(lx2k−1,x2k−1)] +λ5(lx2k−1,x2k−1) d(lx2k−1,lx2k−1)[d(lx2k−1,klx2k−1) + d(x2k−1,klx2k−1)] s[1 + d(lx2k−1,x2k−1)] ; by equation (3.1), we get d(x2k+1,x2k) ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1)[d(x2k,x2k) + d(x2k−1,x2k)] s[1 + d(x2k,x2k−1)] + λ3(x2k,x2k−1) d(x2k−1,x2k+1)[d(x2k,x2k) + d(x2k−1,x2k)] s[1 + d(x2k,x2k−1)] + λ4(x2k,x2k−1) d(x2k−1,x2k)[d(x2k,x2k+1) + d(x2k−1,x2k+1)] s[1 + d(x2k,x2k−1)] + λ5(x2k,x2k−1) d(x2k,x2k)[d(x2k,x2k+1) + d(x2k−1,x2k+1)] s[1 + d(x2k,x2k−1)] ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1)d(x2k−1,x2k) s[1 + d(x2k,x2k−1)] + λ3(x2k,x2k−1) d(x2k−1,x2k+1)d(x2k−1,x2k) s[1 + d(x2k,x2k−1)] + λ4(x2k,x2k−1) d(x2k,x2k+1) + d(x2k−1,x2k+1) s int. j. anal. appl. 16 (6) (2018) 914 ≤λ1(x2k,x2k−1)d(x2k,x2k−1) + λ2(x2k,x2k−1) d(x2k,x2k+1) s + λ3(x2k,x2k−1) d(x2k−1,x2k+1) s + λ4(x2k,x2k−1) d(x2k,x2k+1) + d(x2k−1,x2k+1) s . from lemma 3.1 and triangular inequality, we can write d(x2k+1,x2k) ≤λ1(x0,x1)d(x2k,x2k−1) + λ2(x0,x1) d(x2k,x2k+1) s + λ3(x0,x1)[d(x2k−1,x2k) + d(x2k,x2k+1)] + λ4(x0,x1) d(x2k,x2k+1) s + λ4(x0,x1)[d(x2k−1,x2k) + d(x2k,x2k+1)]. finally one can get d(x2k+1,x2k) ≤ λ1(x0,x1) + λ3(x0,x1) + λ4(x0,x1) 1 − ( 1 s λ2(x0,x1) + λ3(x0,x1) + (1+s) s λ4(x0,x1) )d(x2k,x2k−1). let h = 5∑ i=1 λi(x0,x1) 1−( (1+s)s λ2(x0,x1)+λ3(x0,x1)+ (1+s) s λ4(x0,x1)+λ5(x0,x1)) < 1 s · then d(x2k+1,x2k) ≤ hd(x2k,x2k−1). (3.2) similarly, consider d(x2k−1,x2k) = d(kx2k−2,lkx2k−2). (3.3) by applying condition (ii) of theorem 3.1 with x = x2k−2 and y = kx2k−2 to equation (3.3) , we get d(kx2k−2,lkx2k−2) ≤λ1(x2k−2,kx2k−2)d(x2k−2,kx2k−2) + λ2(x2k−2,kx2k−2) d(x2k−2,kx2k−2)[d(x2k−2,lkx2k−2) + d(kx2k−2,lkx2k−2)] s[1 + d(x2k−2,kx2k−2)] + λ3(x2k−2,kx2k−2) d(kx2k−2,kx2k−2)[d(x2k−2,lkx2k−2) + d(kx2k−2,lkx2k−2)] s[1 + d(x2k−2,kx2k−2)] + λ4(x2k−2,kx2k−2) d(kx2k−2,lkx2k−2)[d(x2k−2,kx2k−2) + d(kx2k−2,kx2k−2)] s[1 + d(x2k−2,kx2k−2)] + λ5(x2k−2,kx2k−2) d(x2k−2,lkx2k−2)[d(x2k−2,kx2k−2) + d(kx2k−2,kx2k−2)] s[1 + d(x2k−2,kx2k−2)] , int. j. anal. appl. 16 (6) (2018) 915 with the help of (3.1), we get d(x2k−1,x2k) ≤λ1(x2k−2,x2k−1)d(x2k−2,x2k−1) + λ2(x2k−2,x2k−1) d(x2k−2,x2k−1)[d(x2k−2,x2k) + d(x2k−1,x2k)] s[1 + d(x2k−2,x2k−1)] + λ3(x2k−2,x2k−1) d(x2k−1,x2k−1)[d(x2k−2,x2k) + d(x2k−1,x2k)] s[1 + d(x2k−2,x2k−1)] + λ4(x2k−2,x2k−1) d(x2k−1,x2k)[d(x2k−2,x2k−1) + d(x2k−1,x2k−1)] s[1 + d(x2k−2,x2k−1)] + λ5(x2k−2,x2k−1) d(x2k−2,x2k)[d(x2k−2,x2k−1) + d(x2k−1,x2k−1)] s[1 + d(x2k−2,x2k−1)] ≤λ1(x2k−2,x2k−1)d(x2k−2,x2k−1) + λ2(x2k−2,x2k−1) d(x2k−2,x2k) + d(x2k−1,x2k) s + λ4(x2k−2,x2k−1) d(x2k−1,x2k) s + λ5(x2k−2,x2k−1) d(x2k−2,x2k) s . using lemma 3.1, one can get d(x2k−1,x2k) ≤λ1(x0,x1)d(x2k−2,x2k−1) + λ2(x0,x1)[d(x2k−2,x2k−1) + d(x2k−1,x2k)] + λ2(x0,x1) d(x2k−1,x2k) s + λ4(x0,x1) d(x2k−1,x2k) s + λ5(x0,x1)[d(x2k−2,x2k−1) + d(x2k−1,x2k)]. finally, d(x2k−1,x2k) ≤ λ1(x0,x1) + λ2(x0,x1) + λ5(x0,x1) 1 − ( (1+s) s λ2(x0,x1) + 1 s λ4(x0,x1) + λ5(x0,x1) )d(x2k−2,x2k−1). implies that d(x2k−1,x2k) ≤ hd(x2k−2,x2k−1). (3.4) now, from equations (3.2) and (3.4), we have d(x2k+1,x2k) ≤ hd(x2k,x2k−1) ≤ h2d(x2k−1,x2k−2). consequently, we can write d(xn+1,xn) ≤ hd(xn,xn−1) ≤ h2d(xn−1,xn−2) ≤ ... ≤ hnd(x1,x0). now, for m > n and sh < 1, we have int. j. anal. appl. 16 (6) (2018) 916 d(xn,xm) ≤ sd(xn,xn+1) + s2d(xn+1,xn+2) + ... + sm−nd(xm−1,xm), ≤ shnd(x1,x0) + s2hn+1d(x1,x0) + ... + sm−nhm−1d(x1,x0) ≤ [shn + s2hn+1 + ... + sm−nhm−1]d(x1,x0) ≤ shn [ 1 + (sh)1 + (sh)2 + ... + (sh)m−n−1 ] d(x1,x0) ≤ shn 1 −sh d(x1,x0). therefore lim n→∞ d(xn,zm) = 0. hence, {xn} is a cauchy sequence. but x is complete, so there exists t ∈ x such that xn → t as n →∞. next, to show that t is a fixed point of k. for this, consider d(t,kt) ≤ d(t,lx2n+1) + d(lx2n+1,kt). using condition (ii) of theorem 3.1 with x = t and y = x2n+1, we have d(t,kt) ≤d(t,lx2n+1) + λ1(t,x2n+1)d(t,x2n+1) + λ2(t,x2n+1) d(t,kt)[d(t,lx2n+1) + d(x2n+1,lx2n+1)] s[1 + d(t,x2n+1)] + λ3(t,x2n+1) d(x2n+1,kt)[d(t,lx2n+1) + d(x2n+1,lx2n+1)] s[1 + d(t,x2n+1)] + λ4(t,x2n+1) d(x2n+1,lx2n+1)[d(t,kt) + d(x2n+1,kt)] s[1 + d(t,x2n+1)] + λ5(t,x2n+1) d(t,lx2n+1)[d(t,kt) + d(x2n+1,kt)] s[1 + d(t,x2n+1)] . using equation (3.1) and proposition 3.1, we get d(t,kt) ≤d(t,x2n+2) + λ1(t,x1)d(t,x2n+1) + λ2(t,x1) d(t,kt)[d(t,x2n+2) + d(x2n+1,x2n+2)] s[1 + d(t,x2n+1)] + λ3(t,x1) d(x2n+1,kt)[d(t,x2n+2) + d(x2n+1,x2n+2)] s[1 + d(t,x2n+1)] + λ4(t,x1) d(x2n+1,x2n+2)[d(t,kt) + d(x2n+1,kt)] s[1 + d(t,x2n+1)] + λ5(t,x1) d(t,x2n+2)[d(t,kt) + d(x2n+1,kt)] s[1 + d(t,x2n+1)] . taking limit as n →∞, we get d(kt,t) ≤ 0. thus d(kt,t) = 0 implies that kt = t. hence t is a fixed point of k. analogously, using condition (ii) of theorem 3.1 with x = x2n and y = t one can show that t is a fixed point of l. therefore kt = lt = t, that is t is a common fixed point of k and l. int. j. anal. appl. 16 (6) (2018) 917 finally, we prove that t is a unique common fixed point of k and l. for this, suppose that t∗ 6= t be another fixed point of k and l. then putting x = t and y = t∗ in condition (ii) of theorem 3.1, we have d(kt,lt∗) ≤λ1(t,t∗)d(t,t∗) + λ2(t,t∗) d(t,kt)[d(t,lt∗) + d(t∗,lt∗)] s[1 + d(t,t∗)] + λ3(t,t ∗) d(t∗,kt)[d(t,lt∗) + d(t∗,lt∗)] s[1 + d(t,t∗)] + λ4(t,t ∗) d(t∗,lt∗)[d(t,kt) + d(t∗,kt)] s[1 + d(t,t∗)] + λ5(t,t ∗) d(t,lt∗)[d(t,kt) + d(t∗,kt)] s[1 + d(t,t∗)] , which implies that d(t,t∗) ≤λ1(t,t∗)d(t,t∗) + λ3(t,t∗) d(t∗, t)d(t,t∗) s[1 + d(t,t∗)] + λ5(t,t ∗) d(t,t∗)d(t∗, t) s[1 + d(t,t∗)] ≤λ1(t,t∗)d(t,t∗) + λ3(t,t∗) d(t∗, t) s + λ5(t,t ∗) d(t,t∗) s ≤ [sλ1(t,t∗) + λ3(t,t∗) + λ5(t,t∗)] d(t,t∗) s . which is contradiction because sλ1(t,t ∗) + λ3(t,t ∗) + λ5(t,t ∗) < 1, thus d(t∗, t) = 0 and hence t∗ = t. therefore t is a unique common fixed point of k and l. � from theorem 3.1, we can derive the following corollaries and the proof of which is simple, so we omit it. corollary 3.1. if k = l and all other conditions of theorem 3.1 are satisfied, then l has a unique fixed point in x. corollary 3.2. let (x,d) be a complete b-metric space with s ≥ 1 and λi : x × x → [0, 1), i = 1, 2. if k,l : x → x be two self-mappings such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lkx,y) ≤ λi(x,y) and λi(x,kly) ≤ λi(x,y); (ii) d(kx,ly) ≤ λ1(x,y)d(x,y) + λ2(x,y) d(x,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] ; where 0 ≤ sλ1(x,y) + ( s2+s+1 s ) λ2(x,y) < 1. then k and l have a unique common fixed point in x. corollary 3.3. let (x,d) be a complete b-metric space with s ≥ 1 and λi : x × x → [0, 1), i = 1, 4. if k,l : x → x be two self-mappings such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lkx,y) ≤ λi(x,y) and λi(x,kly) ≤ λi(x,y); (ii) d(kx,ly) ≤ λ1(x,y)d(x,y) + λ4(x,y) d(y,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] ; int. j. anal. appl. 16 (6) (2018) 918 where 0 ≤ sλ1(x,y) + ( s2+s+1 s ) λ4(x,y) < 1. then k and l have a unique common fixed point in x. corollary 3.4. let (x,d) be a complete b-metric space with s ≥ 1 and λi : x ×x → [0, 1), i = 1, 2, 4. if k,l : x → x be two self-mappings such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lkx,y) ≤ λi(x,y) and λi(x,kly) ≤ λi(x,y); (ii) d(kx,ly) ≤λ1(x,y)d(x,y) + λ2(x,y) d(x,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ4(x,y) d(y,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] ; where 0 ≤ sλ1(x,y) + ( s2+1 s ) λ2(x,y) + ( s2+s+1 s ) λ4(x,y) < 1. then k and l have a unique common fixed point in x. for the validity of theorem 3.1, we construct the following example. example 3.1. let x = [0, 1] and d : x × x → r+ defined by d(x,y) = (α|x−y|)2 = α2|x − y|2 with α ≥ 8,s = 2. define k,l : x → x by kx = x 4 and lx = x 5 . let λi : x × x → [0, 1), i = 1, 2, ..., 5 are defined as: λ1(x,y) = x 19 + y 21 , λ2(x,y) = x 17 + y 23 , λ3(x,y) = x2 29 + y2 37 , λ4(x,y) = x3 + y3 41 , λ5(x,y) = xy 43 . to check condition (i), we have, since lkx = x 20 and kly = y 20 . then by routine calculation, one can easily obtained that λi(lkx,y) ≤ λi(x,y) and λi(x,kly) ≤ λi(x,y) for all i = 1, 2, ..., 5; to check condition (ii), we have, d(kx,ly) =α2|kx−ly|2 = α2 ∣∣∣x 4 − y 5 ∣∣∣2 ≤ ( x 19 + y 21 ) α2|x−y|2 + ( x 17 + y 23 ) α4 ∣∣x− x 4 ∣∣2 [∣∣x− y 5 ∣∣2 + ∣∣y − y 5 ∣∣2] 2 [1 + α2|x−y|2] + ( x 29 + y 37 ) α4 ∣∣y − x 4 ∣∣2 [∣∣x− y 5 ∣∣2 + ∣∣y − y 5 ∣∣2] 2 [1 + α2|x−y|2] + ( x3 + y3 41 ) α4 ∣∣y − y 5 ∣∣2 [∣∣x− x 4 ∣∣2 + ∣∣y − x 4 ∣∣2] 2 [1 + α2|x−y|2] + (xy 43 ) α4 ∣∣x− y 5 ∣∣2 [∣∣x− x 4 ∣∣2 + ∣∣y − x 4 ∣∣2] 2 [1 + α2|x−y|2] int. j. anal. appl. 16 (6) (2018) 919 d(kx,ly) =λ1(x,y)d(x,y) + λ2(x,y) d(x,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ3(x,y) d(y,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ4(x,y) d(y,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] + λ5(x,y) d(x,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] ; where 5∑ i=2 λi(x,y) + 2 5∑ i=1 λi(x,y) + 1 2 [λ2(x,y) + λ4(x,y)] < 1. thus all the conditions of theorem 3.1 are satisfied, hence k and l has a unique fixed point 0 ∈ x. to state the next result, we need the following lemma the proof of which can be easily obtained from lemma 2.1. lemma 3.2. let (x,d) be a complete b-metric space with s ≥ 1 and k,l : x → x. let x0 ∈ x and define the sequence {xn} by kx2n = x2n+1 and lx2n+1 = x2n+2 ∀ n = 0, 1, 2, ... assume that there exists a mapping λ : x → [0, 1) such that λ(lkx) ≤ λ(x), for all x ∈ x. then λ(x2n) ≤ λ(x0) and λ(x2n+1) ≤ λ(x1) for all x ∈ x and n = 0, 1, 2, .... theorem 3.2. let (x,d) be a complete b-metric space with s ≥ 1 and λi : x → [0, 1), i = 1, 2, ..., 9. if k,l : x → x be two self-mappings such that for all x,y ∈ x the following conditions are satisfied: (i) λi(lkx) ≤ λi(x); (ii) d(kx,ly) ≤λ1(x)d(x) + λ2(x) d(x,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ3(x) d(y,kx)[d(x,ly) + d(y,ly)] s[1 + d(x,y)] + λ4(x) d(y,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] + λ5(x) d(x,ly)[d(x,kx) + d(y,kx)] s[1 + d(x,y)] ; where 5∑ i=2 λi(x) + s 5∑ i=1 λi(x) + 1 s [λ2(x) + λ4(x)] < 1, with 0 ≤ 5∑ i=1 λi(x) < 1. then k and l have a unique common fixed point in x. proof. by using lemma 2.1 and following the same steps as in theorem 3.1 one can easily prove the theorem. � one can deduce corollaries from theorem 3.2 in the same way as derived from theorem 3.1. int. j. anal. appl. 16 (6) (2018) 920 acknowledgements the authors wish to thank the editor and anonymous referees for their comments and suggestions, which helped to improve this paper. references [1] i. a. bakhtin, the contraction mapping principle in quasimetric spaces, funct. anal., unianowsk gos. ped. inst. 30(1989), 26–37. [2] s. banach, sur les oprations dans les ensembles abstraits et leurs applications aux equations integrales, fund. math. 3(1922), 133-181. [3] m. boriceanu, m. bota, a. petrusel, multivalued fractals in b–metric spaces, cent. eur. j. math. 8(2010), 367–377. [4] s. k. chetterjea, fixed point theorems, c. r. acad. bulgara sci. 25(1972), 727–730. [5] s. czerwik, contraction mappings in b–metric spaces, acta math. inform. univ. ostrav. 1(1993), 5–11. [6] s. czerwik, nonlinear set-valued contraction mappings in b–metric spaces, atti sem. mat. fis. univ. modena, 46(1998), 263–276. [7] b. k. dass, s. gupta, an extension of banach contraction principle through rational expression, indian j. pure appl. math. 6(1975), 1455–1458. [8] n. hussain, v. parvaneh, j. r. roshan, z. kadelburg, fixed points of cyclic weakly (ψ,φ,l,a,b)–contractive mappings in ordered b-metric spaces with applications, fixed point theory appl. 2013 (2013), art. id 256. [9] n. hussain, d. dorić, z. kadelburg, s. radenović, suzuki-type fixed point results in metric type spaces, fixed point theory appl. (2012)2012, art. id 126. [10] r. kannan, some results on fixed points, bull. calcutta math. soc. 6(1968), 71–78. [11] p. kumam, n. v. dung, v. t. le hang, some equivalences between cone b–metric spaces and b-metric spaces, abstr. appl. anal. 2013(2013), art. id 573740. [12] s. reich, some remarks concerning contraction mappings, canad. math. bull. 14(1971), 121–124. [13] s. l. singh, b. prasad, some coincidence theorems and stability of iterative procedures, comput. math. appl. 55(2008), 2512–2520. [14] w. sintunavarat, nonlinear integral equations with new admissibility types in b–metric spaces, j. fixed point theory appl. 18(2016), 397416. 1. introduction and preliminaries 2. fixed point results in b-metric spaces 3. common fixed point results in b-metric spaces acknowledgements references international journal of analysis and applications volume 18, number 3 (2020), 513-530 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-513 on the pricing of call-put parities of asian options by reduced differential transform algorithm javed hussain∗, muhammad shoaib khan sukkur iba university, pakistan ∗corresponding author: javed.brohi@iba-suk.edu.pk abstract. the key aim of the paper is to show that how the efficiently the reduced differential transform algorithm (rdta) can be employed to price the exotic financial options. in this paper we have computed the exact solution of the parabolic partial differential equation governing the dynamics of put-call parity in the mathematical theory of asian options, by means of rdta. 1. introduction an option is an agreement that allows the holder to buy (call option) or sell (put option) at a specified future time (expiration or maturity time) an underlying asset at a specified price (strike or exercise price). pricing of an asian option is always complicated due to its path dependents derivatives whose payoffs depend on some form of averaged prices of the underlying asset and no closed form solution exist in general. pricing them efficiently and accurately is very important both in theory and practice. asian option can avoid manipulation of the stock near expiration time. asian options are popular in the financial community as well as in the over-the-counter (otc) market because they are often cheaper than the equivalent classical european options. several analytical approaches have been proposed to address the problem. in [?] a closed form solution of the no-arbitrage price of arithmetic averaged fixed strike price is obtained through the inversion of a laplace transform. received december 12th, 2019; accepted january 14th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 35q91, 91g80. key words and phrases. asian option; put-call parity; reduced differential transform algorithm. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 513 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-513 int. j. anal. appl. 18 (3) (2020) 514 the standard approximation methods based on partial differential equations require some regularity conditions of the solution of the no-arbitrage pde. considering path dependent contingent claims in general setting, the pde is a strongly degenerate parabolic equation in three dimensions (time, the underlying asset price, and the path-dependent variable). in this setting, the needed regularity seemed out of reach. to avoid this difficulty, many authors considered a two-dimensional second-order pde which is obtained from the original one through a change of variable (similarity reduction method) when the contingent claim final payoff has a particular form (see [2, 6, 7, 11, 20, 24, 25]). this method covers a large set of contingent claim contracts, including arithmetic asin options, but not a contingent claim characterized by a general final payoff. the black-scholes pde for the fixed strike arithmetic asian options, considering dividend (see, [17], pp. 277-278) is ∂v ∂t + s −j t ∂v ∂j + σ2 2 s2 ∂2v ∂s2 + (r −q) s ∂v ∂s −rv = 0 (1.1) v (s,j,t) =   (j −k) +, (call option with fixed strike price) (k −j)+, (put option with fixed strike price) . (1.2) v denote the price of an asian option, s is the value of the underlying asset, k the exercise price, t the expiration date, r is the interest rate, q is the continuously paid dividend, σ is the asset volatility, and j = 1 t t∫ 0 sτdτ be the path variable, denote the average of asset price for the period up to t. it is well known that the difference between the prices of european vanilla call and put options is equal to a european forward contract. similarly, we have put-call parity relations for european style asian options [16]. let cfix (s,j,t) and pfix (s,j,t) denote the price of the fixed strike arithmetic averaging asian call option and put option, respectively. their terminal payoff functions are given by cfix (s,j,t) = (j −k)+ = max (j −k, 0) (1.3) pfix (s,j,t) = (k −j)+ = max (k −j, 0) (1.4) let w (s,j,t) denote the difference of cfix and pfix . since both cfix and pfix are governed by the same differential equation [see (1.1)], so does w (s,j,t) .the terminal condition of their difference w (s,j,t) is given by w (s,j,t) = (j −k)+ − (k −j)+ = j −k. int. j. anal. appl. 18 (3) (2020) 515 hence the problem we have to investigate is ∂w ∂t + s −j t ∂w ∂j + σ2 2 s2 ∂2w ∂s2 + (r −q) s ∂w ∂s −rw = 0, (1.5) w |t=t = (j −k) + − (k −j)+ = j −k. in a similar manner, we have other pdes of put-call parities of other types of asian options (floating/fixed and arithmetic/geometric averaging) for the the investigation will see in section 3. in this article, the fundamental objective is to investigate all four versions of pdes of put-call parities of asian options using reduced differential transform method(rdtm) to validate the method. rdtm proposed by keskin [12] and successfully employed to solve many types of linear and nonlinear pdes. rdtm is a reliable semi-analytic method subject to appropriate initial condition. taking into consideration of this method, it is possible to find an exact solution or a closed approximate solution of a differential equation. 2. reduced differential transform method consider a function of two variables u(x,t) and suppose that it can be represented as a product of two single-variable functions, i.e., u(x,t) = f(x)g(t). based on the properties of one dimensional differential transform, the function u(x,t) can be represented as follows: u(x,t) = ( ∞∑ i=0 f(i)xi )( ∞∑ i=0 g(j)tj ) = ∞∑ k=0 uk(x)t k, (2.1) where uk(x) is called t-dimensional spectrum function of u(x,t). the basic definitions of rdtm are introduced as follows (cf. [12–15]): definition 2.1. if function u(x,t) is analytic and differentiated continuously with respect to time t and space x in the domain of interest, then let uk(x) = 1 k! [ ∂k ∂tk u(x,t) ] t=0 (2.2) where the t-dimensional spectrum function uk(x) is the transformed function. in this paper, the lowercase u(x,t) represents the original function, while the uppercase uk(x) stands for the transformed function. definition 2.2. the differential inverse transform of uk(x) is defined as follows: u(x,t) = ∞∑ k=0 uk(x)t k (2.3) then, combining eqs. (2.2) and (2.3) we write u(x,t) = ∞∑ k=0 1 k! [ ∂k ∂tk u(x,t) ] t=0 tk (2.4) int. j. anal. appl. 18 (3) (2020) 516 from the above definitions, it can be found that the concept of the rdtm is derived from the power series expansion. to illustrate the basic concepts of the rdtm, consider the following nonlinear partial differential equation written in an operator form lu(x,t) + ru(x,t) + nu(x,t) = g(x,t), (2.5) with initial condition u(x, 0) = f(x),where l = ∂ ∂t , r is a linear operator which has partial derivatives, nu(x,t) is a nonlinear operator and g(x,t) is an in-homogeneous term. according to the rdtm, we can construct the following iteration formula: (k + 1) uk+1 (x) = gk (x) −ruk (x) −nuk (x) , (2.6) where uk (x) , ruk (x) , nuk (x) and gk (x) are the transformations of the functions lu (x,t) , ru (x,t) , nu (x,t) and g (x,t) respectively. from the initial condition, we write u0 (x) = f (x) . (2.7) substituting (2.7) into (2.6) and by a straightforward iterative calculation, we get the following uk (x) values. then the inverse transformation of the set of values {uk (x)} n k=0 gives approximation solutions as, ũn (x,t) = n∑ k=0 uk (x) t k, (2.8) where n is the order of approximation solution. therefore, the exact solution of problems given by u (x,t) = lim n→∞ ũn (x,t) (2.9) the fundamental mathematical operations performed by rdtm can be readily obtained and are listed in following table. table 1. reduced differential transformation   functional form transformed form u (x,t) uk (x) = 1 k! [ ∂k ∂tk u (x,t) ] t=0 w (x,t) = u (x,t) ± v (x,t) wk (x) = uk (x) ± vk (x) w (x,t) = αu (x,t) wk (x) = αuk (x) , α is a constant w (x,t) = xmtn wk (x) = x mδ (k − n) , the kronecker delta w (x,t) = xmtnu (x,t) wk (x) = x muk−n (x) , when k ≥ n else 0. w (x,t) = u (x,t) v (x,t) wk (x) = k∑ r=0 vr (x) uk−r (x) = k∑ r=0 ur (x) vk−r (x) w (x,t) = ∂ r ∂tr u (x,t) wk (x) = (k + 1) . . . (k + r) uk+r (x) = (k+r)! k! uk+r (x) w (x,t) = ∂ ∂x u (x,t) wk (x) = ∂ ∂x uk (x)   int. j. anal. appl. 18 (3) (2020) 517 3. pricing of four versions of call-put parities of asian options in this section, we will price call-put parities standard pdes of four version of asian options [17], that are: 1) arithmetic average asian option with fixed strike price. 2) geometric average asian option with fixed strike price. 3) arithmetic average asian option with floating strike price. 4) geometric average asian option with floating strike price. by reduced differential transform method to validate the efficiency of the rdtm. 3.1. call-put parity for arithmetic average asian option with fixed strike price. assume that c (s,j,t) and p (s,j,t)denote the valuation of an asian call and put option, respectively. define w (s,j,t) = c (s,j,t) −p (s,j,t) then in {0 ≤ s < ∞, 0 ≤ j < ∞, 0 ≤ t ≤ t} ,w satisfies ∂w ∂t + s −j t ∂w ∂j + σ2 2 s2 ∂2w ∂s2 + (r −q) s ∂w ∂s −rw = 0, (3.1) w |t=t = (j −k) + − (k −j)+ = j −k. by the use of change of variable, ξ = tk − tj s the function w = t s w satisfies the cauchy problem [17] in the domain {ξ ∈ r, 0 ≤ t ≤ t} : ∂w ∂t + σ2 2 ξ2 ∂2w ∂ξ2 − [(r −q) ξ + 1] ∂w ∂ξ −qw = 0, (3.2) w (ξ,t) = −ξ. where w = w (ξ,t) for convenience to apply rdtm, we are assuming that w (ξ,t) = u (ξ,t) ,thus ∂u ∂t + σ2 2 ξ2 ∂2u ∂ξ2 − [(r −q) ξ + 1] ∂u ∂ξ −qu = 0, withu (ξ,t) = −ξ. we take the change of variable τ = t − t, to convert terminal condition into initial condition, so ∂u ∂t = − ∂u ∂τ int. j. anal. appl. 18 (3) (2020) 518 or ∂u ∂τ = σ2 2 ξ2 ∂2u ∂ξ2 − [(r −q) ξ + 1] ∂u ∂ξ −qu, (3.3) withu |τ=0= −ξ. where u = u (ξ,τ) according to the rdtm, we construct the following recurrence relation for the eq. (3.3) (m + 1) um+1 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 um (ξ) − [(r −q) ξ + 1] ∂ ∂ξ um (ξ) −qum (ξ) note: u = u (ξ,τ) → functional form and um = um (ξ) → transformed form. for m = 0 u1 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u0 (ξ) − [(r −q) ξ + 1] ∂ ∂ξ u0 (ξ) −qu0 (ξ) from the initial condition, we write u (ξ, 0) = u0 (ξ) = −ξ so ∂ ∂ξ u0 (ξ) = −1 , ∂2 ∂ξ2 u0 (ξ) = 0 thus u1 (ξ) = 1 + rξ for m = 1 2u2 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u1 (ξ) − [(r −q) ξ + 1] ∂ ∂ξ u1 (ξ) −qu1 (ξ) we get, u2 (ξ) = − 1 2 [ r2ξ + r2 −q2 r −q ] for m = 2 3u3 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u2 (ξ) − [(r −q) ξ + 1] ∂ ∂ξ u2 (ξ) −qu2 (ξ) we have, u3 (ξ) = 1 3! [ r3ξ + r3 −q3 r −q ] by an inductive argument we have following, um (ξ) = (−1)m+1 m! [ rmξ + rm −qm r −q ] , wherem ≥ 0. int. j. anal. appl. 18 (3) (2020) 519 for the solution, differential inverse transform of um (ξ) is defined as below: u (ξ,τ) = ∞∑ m=0 um (ξ) τ m u (ξ,τ) = ∞∑ m=0 (−1)m+1 τm m! [ rmξ + rm −qm r −q ] u (ξ,τ) = ∞∑ m=0 (−1)m+1 τm m! rmξ + ∞∑ m=0 (−1)m+1 τm m! rm −qm r −q u (ξ,τ) = −ξ ∞∑ m=0 (−1)m τmrm m! − 1 r −q ∞∑ m=0 (−1)m τm m! (rm −qm) u (ξ,τ) = −ξ ∞∑ m=0 (−rτ)m m! − 1 r −q [ ∞∑ m=0 (−rτ)m m! − ∞∑ m=0 (−qτ)m m! ] u (ξ,τ) = −ξe−rτ − 1 r −q [ e−rτ −e−qτ ] here ξ = tk − tj s ,u (ξ,t) = w (ξ,t) where τ in terms of t so above equation can be written as w (ξ,t) = − tk − tj s e−r(t−t) − 1 r −q [ e−r(t−t) −e−q(t−t) ] ∵ w (ξ,t) = t s w ⇒ w = s t w (ξ,t) the exact solution, in closed form, is given by c (s,j,t) −p (s,j,t) = w (s,j,t) w (s,j,t) = [ t t j − s (r −q) t −k ] e−r(t−t) − s (r −q) t e−q(t−t) which is same solution as obtained in [17]. consider a six-month call option on stock. if s0 = k = $145,r = 6,q = 3 and σ = 29.5 then graphs of w (s,j,t), for average stock price j ranging from $140 to $150, can be found in figure 1 on next page. 3.2. call-put parity for arithmetic average asian option with floating strike price. consider w (s,j,t) = c (s,j,t) −p (s,j,t) then under appropriate transformations, standard call-put parities satisfies the cauchy problem [17] in the domain {0 ≤ ξ < ∞, 0 ≤ t ≤ t}: ∂w ∂t + σ2 2 ξ2 ∂2w ∂ξ2 + [1 − (r −q) ξ] ∂w ∂ξ −qw = 0, (3.4) w |t=t = 1 − ξ t int. j. anal. appl. 18 (3) (2020) 520 figure 1. plot of w(s,j,t) when s0 = k = $145,r = 6, q = 3 and σ = 29.5 where w = w (ξ,t) taking into consideration w (ξ,t) = u (ξ,t), for the sake of easy utility of rdtm, we have ∂u ∂t + σ2 2 ξ2 ∂2u ∂ξ2 + [1 − (r −q) ξ] ∂u ∂ξ −qu = 0, u (ξ,t) = 1 − ξ t we take the change of variable τ = t − t, to convert terminal condition into initial condition, we have ∂u ∂τ = σ2 2 ξ2 ∂2u ∂ξ2 + [1 − (r −q) ξ] ∂u ∂ξ −qu, (3.5) u |τ=0= 1 − ξ t applying the rdtm to eq. (3.5), we obtain the following recurrence equation (m + 1) um+1 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 um (ξ) + [1 − (r −q) ξ] ∂ ∂ξ um (ξ) −qum (ξ) int. j. anal. appl. 18 (3) (2020) 521 for m = 0 u1 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u0 (ξ) + [1 − (r −q) ξ] ∂ ∂ξ u0 (ξ) −qu0 (ξ) according to the initial condition, we can write u (ξ, 0) = u0 (ξ) = 1 − ξ t so u1 (ξ) = 1 t (rξ − 1) −q for m = 1 2u2 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u1 (ξ) + [1 − (r −q) ξ] ∂ ∂ξ u1 (ξ) −qu1 (ξ) having u2 (ξ) = − 1 2! [ 1 t { r2ξ − r2 −q2 r −q } −q2 ] for m = 2 3u3 (ξ) = σ2 2 ξ2 ∂2 ∂ξ2 u2 (ξ) + [1 − (r −q) ξ] ∂ ∂ξ u2 (ξ) −qu2 (ξ) we will obtained, u3 (ξ) = 1 3! [ 1 t { r3ξ − r3 −q3 r −q } −q3 ] by an inductive argument we have following, um (ξ) = (−1)m+1 m! [ 1 t { rmξ − rm −qm r −q } −qm ] , form ≥ 0 for the solution, inverse differential transform of um (ξ) is defined as below: u (ξ,τ) = ∞∑ m=0 um (ξ) τ m u (ξ,τ) = ∞∑ m=0 (−1)m+1 m! [ 1 t { rmξ − rm −qm r −q } −qm ] τm u (ξ,τ) = − ξ t ∞∑ m=0 (−rτ)m m! + 1 t (r −q) ∞∑ m=0 (−rτ)m − (−qτ)m m! + ∞∑ m=0 (−qτ)m m! u (ξ,τ) = − ξ t e−rτ + 1 t (r −q) ( e−rτ −e−qτ ) + e−qτ u (ξ,τ) = − ξ t e−rτ + 1 t (r −q) e−rτ ( 1 − 1 t (r −q) ) e−qτ here ξ = tj s ,u (ξ,t) = w (ξ,t) where τ in terms of t so above equation can be written as w (ξ,t) = − tj st e−r(t−t) + 1 t (r −q) e−r(t−t) ( 1 − 1 t (r −q) ) e−q(t−t) ∵ w = w s ⇒ w (s,j,t) = sw (ξ,t) int. j. anal. appl. 18 (3) (2020) 522 in closed form, the exact solution is given by c (s,j,t) −p (s,j,t) = − t t je−r(t−t) + s t (r −q) e−r(t−t) ( 1 − 1 t (r −q) ) se−q(t−t) which is same as obtained in [17]. consider a six-month call option on stock. if s0 = k = $145,r = 6,q = 3 and σ = 29.5 then graphs of w (s,j,t), for average stock price j ranging from $140 to $150, can be found in figure 2. figure 2. plot of w(s,j,t) when s0 = k = $145,r = 6, q = 3 and σ = 29.5 3.3. call-put parity for geometric average asian option with fixed strike price. suppose that w (s,j,t) = c (s,j,t) −p (s,j,t) thus in {0 ≤ s < ∞, 0 ≤ j < ∞, 0 ≤ t ≤ t} ,w satisfies the problem: ∂w ∂t + j ln s − ln j t ∂w ∂j + σ2 2 s2 ∂2w ∂s2 + (r −q) s ∂w ∂s −rw = 0, (3.6) w |t=t = (j −k) + − (k −j)+ = j −k. int. j. anal. appl. 18 (3) (2020) 523 by the transformation, ξ = t ln j + (t − t) ln s t in {ξ ∈ r, 0 ≤ t ≤ t} ,w satisfies [17]: ∂w ∂t + σ2 2 ( t − t t )2 ∂2w ∂ξ2 + ( r −q − σ2 2 )( t − t t ) ∂w ∂ξ −rw = 0, (3.7) w |t=t = eξ −k. where w = w (ξ,t) for convenience, we are assuming that w (ξ,t) = u (ξ,t), we get ∂u ∂t + σ2 2 ( t − t t )2 ∂2u ∂ξ2 + ( r −q − σ2 2 )( t − t t ) ∂u ∂ξ −ru = 0, (3.8) u |t=t = eξ −k. we are taking the change of variable τ = t − t, we have ∂u ∂τ = σ2 2 (τ t )2 ∂2u ∂ξ2 + ( r −q − σ2 2 )(τ t ) ∂u ∂ξ −ru, (3.9) u |τ=0= eξ −k. where u = u (ξ,τ) by applying the rdtm, we construct the following iteration formula to eq. (3.9) (m + 1) um+1 (ξ) = σ2 2t2 ∂2 ∂ξ2 um−2 (ξ) + ( r −q − σ 2 2 t ) ∂ ∂ξ um−1 (ξ) −rum (ξ) here we used the property if v (x,t) = xmtnu (x,t) then vk (x) = x muk−n (x) when k ≥ n,else 0, on first and second term. by considering the initial condition, we can write u0 (ξ) = e ξ −k for m = 0 u1 (ξ) = −r ( eξ −k ) for m = 1 u2 (ξ) = 1 2! {( r −q − σ 2 2 t ) eξ + r2 ( eξ −k )} for m = 2 3u3 (ξ) = σ2 2t2 ∂2 ∂ξ2 u0 (ξ) + ( r −q − σ 2 2 t ) ∂ ∂ξ u1 (ξ) −ru2 (ξ) int. j. anal. appl. 18 (3) (2020) 524 we get, u3 (ξ) = 1 3! { σ2 t2 eξ − 3reξ ( r −q − σ 2 2 t ) −r3 ( eξ −k )} for m = 3 4u4 (ξ) = σ2 2t2 ∂2 ∂ξ2 u1 (ξ) + ( r −q − σ 2 2 t ) ∂ ∂ξ u2 (ξ) −ru3 (ξ) substitute u1 (ξ) ,u2 (ξ) and u3 (ξ) in above equation, we get u4 (ξ) = 1 4!  −4 σ 2 t2 reξ + 3 ( r −q − σ 2 2 t )2 eξ + 6 ( r −q − σ 2 2 t ) r2eξ + r4 ( eξ −k ) and so on. for the solution differential inverse transform of um (ξ) is defined as below: u (ξ,τ) = ∞∑ m=0 um (ξ) τ m u (ξ,τ) = eξe  σ2τ3 6t2 + ( r−q− σ 2 2 ) τ2 2t −rτ   −ke−rτ ∵ τ = t − t and from ξ = t ln j + (t − t) ln s t we can write eξ = j t t s (t−t) t so finally, we have the exact solution, in closed form w (s,j,t) =  j t t s (t−t) t e (t−t)  σ2 6 ( t−t t ) 2 + ( r−q− σ 2 2 ) 2 ( t−t t )   −k  e−r(t−t) which is the same result as obtained in [17]. consider a six-month call option on stock. if s0 = k = $145,r = 6,q = 3 and σ = 29.5 then graphs of w (s,j,t), for average stock price j ranging from $140 to $150, can be found in figure 3 on next page. 3.4. call-put parity for geometric average asian option with floating strike price. let w (s,j,t) = c (s,j,t) −p (s,j,t) thus in {0 ≤ s < ∞, 0 ≤ j < ∞, 0 ≤ t ≤ t} ,w satisfies ∂w ∂t + j ln s − ln j t ∂w ∂j + σ2 2 s2 ∂2w ∂s2 + (r −q) s ∂w ∂s −rw = 0, (3.10) w |t=t = (s −j) + − (j −s)+ = s −j. int. j. anal. appl. 18 (3) (2020) 525 figure 3. plot of w(s,j,t) when s0 = k = $145,r = 6, q = 3 and σ = 29.5 under the suitable transformations,the function u satisfies the cauchy problem [17] in the domain{ (x,y) ∈ r2, 0 ≤ t ≤ t } : ∂u ∂t = rw − ( r −q − σ2 2 )[ t − t t ∂u ∂y + ∂u ∂x ] − σ2 2 ∂2u ∂x2 − σ2 2 ( t − t t )2 ∂2u ∂y2 −σ2 ( t − t t ) ∂2u ∂x∂y (3.11) u |t=t = ex −ey. taking the change of variable τ = t − t, we have ∂u ∂τ = σ2 2 (τ t )2 ∂2u ∂y2 + σ2 (τ t ) ∂2u ∂x∂y + σ2 2 ∂2u ∂x2 + ( r −q − σ2 2 )[ τ t ∂u ∂y + ∂u ∂x ] −ru (3.12) u |τ=0= ex −ey. where u = u (x,y,τ) int. j. anal. appl. 18 (3) (2020) 526 using the rdtm to eq. (3.12), we get the following iterative formula (m + 1) um+1 (x,y) = σ2 2t2 ∂2 ∂y2 um−2 (x,y) + σ2 t ∂2 ∂x∂y um−1 (x,y) + σ2 2 ∂2 ∂x2 um (x,y) +( r −q − σ2 2 )[ 1 t ∂ ∂y um−1 (x,y) + ∂ ∂x um (x,y) ] −rum (x,y) with the initial condition u (x,y, 0) = u0 (x,y) = e x −ey for m = 0 u1 (x,y) = 0 + 0 + σ2 2 ∂2 ∂x2 u0 (x,y) + ( r −q − σ2 2 )[ 0 + ∂ ∂x u0 (x,y) ] −ru0 (x,y) thus u1 (x,y) = −qex + rey for m = 1 2u2 (x,y) = 0 + σ2 t ∂2 ∂x∂y u0 (x,y) + σ2 2 ∂2 ∂x2 u1 (x,y) +( r −q − σ2 2 )[ 1 t ∂ ∂y u0 (x,y) + ∂ ∂x u1 (x,y) ] −ru1 (x,y) so u2 (x,y) = 1 2! { q2ex −ey ( r −q − σ 2 2 t + r2 )} for m = 2 3u3 (x,y) = σ2 2t2 ∂2 ∂y2 u0 (x,y) + σ2 t ∂2 ∂x∂y u1 (x,y) + σ2 2 ∂2 ∂x2 u2 (x,y) +( r −q − σ2 2 )[ 1 t ∂ ∂y u1 (x,y) + ∂ ∂x u2 (x,y) ] −ru2 (x,y) so u3 (x,y) = 1 3! { −q3ex −ey ( σ2 t2 − 3 r −q − σ 2 2 t r −r3 )} for m = 3 4u4 (x,y) = σ2 2t2 ∂2 ∂y2 u1 (x,y) + σ2 t ∂2 ∂x∂y u2 (x,y) + σ2 2 ∂2 ∂x2 u3 (x,y) +( r −q − σ2 2 )[ 1 t ∂ ∂y u2 (x,y) + ∂ ∂x u3 (x,y) ] −ru3 (x,y) we have u4 (x,y) = 1 4!  q4ex + 4 σ 2 t2 rey − 6 ( r −q − σ 2 2 t ) r2ey − 3 ( r −q − σ 2 2 t )2 ey −r4ey   int. j. anal. appl. 18 (3) (2020) 527 for the solution, inverse differential transform of um (x,y) is defined as below: u (x,y,τ) = ∞∑ m=0 um (x,y) τ m u (x,y,τ) = exe−qτ −eye  σ2τ3 6t2 + ( r−q− σ 2 2 ) τ2 2t −rτ   ∵ x = ln s ⇒ ex = s and y = t ln j + (t − t) ln s t ⇒ ey = j t t s t−t t also τ = t − t u (x,y,t) = se−q(t−t) −j t t s t−t t e  σ2(t−t)3 6t2 + ( r−q− σ 2 2 ) (t−t)2 2t −r(t−t)   thus we reached to the exact solution, in closed form c (s,j,t) −p (s,j,t) = se−q(t−t) −j t t s t−t t e σ2(t−t)3 6t2 + ( r−q− σ 2 2 ) (t−t)2 2t −r(t−t) which is the same solution as obtained in [17]. consider a six-month call option on stock. if s0 = k = $145,r = 6,q = 3 and σ = 29.5 then graphs of w (s,j,t), for average stock price j ranging from $140 to $150, can be found in figure 4 on next page. int. j. anal. appl. 18 (3) (2020) 528 figure 4. plot of w(s,j,t) when s0 = k = $145,r = 6, q = 3 and σ = 29.5 4. conclusion the reduced differential transform method (rdtm) for put-call parities pdes of asian options has been presented. the solutions obtained by the method are an infinite power series for appropriate initial condition, which can in turn in a closed form, the exact solution. the efficiency of the presented method is validated through all four versions of pdes of asian options put-call parities and found exact solutions same as lishang solutions.we notice that the rdtm technique is highly accurate, rapidly convergent. rdtm is a very easily implementable mathematical tool for the pdes in mathematical finance subject to appropriate initial condition. availability of data and materials: this paper contains no any studies with human participants or animals performed by any of the authors. all the data related to the current study were provided along with this paper. authors’ contributions: both authors contributed equally to the manuscript and typed, read, and approved the final manuscript. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (3) (2020) 529 references [1] a. haghbin, s. hesam, reduced differential transform method for solving seventh order sawada-kotera equations, j. math. computer sci. 5 (1) (2012), 53-59. [2] b. alziary, j. p. drcamps and p. f. koehl, a pde approach to asian options: analytical and numerical evidence, j. bank. finance, 21 (1997), 613-640. [3] birol i̇bi̇ş, application of reduced differential transformation method for solving fourth-order parabolic partial differential equations, j. math. computer sci. 12 (2014), 124-131. [4] b. k. singh, fractional reduced differential transform method for numerical computation of a system of linear and nonlinear fractional pdes, int. j. open probl. compt. math. 9 (3) (2016), 20-38. [5] c. fan, p. li, y. xue, application of differential equations in enzyme kinetics, chem. eng. trans. 71 (2018), 883-888. [6] d.i. cruz and j.m. gonzalez., a different approach for pricing asian options, appl. math. lett. 21 (2008), 303-306. [7] e. barucci, s. polidoro and v. vespri, some results on partial differential equations and asian options, math. models methods appl. sci. 11 (3) (2001), 475-497. [8] h. jafari, h. jassim, s. moshokoa, v. ariyan and f. tchier, reduced differential transform method for partial differential equations within local fractional derivative operators, adv. mech. eng. 8 (4) (2016), 1–6. [9] h. geman and m. yor, bessel processes, asian options, and perpetuities, math. finance, 3 (4) (1993), 349-375. [10] h. rouhparvar, computational technique of linear partial differential equations by reduced differential transform method, int. j. ind. math. 8 (4) (2016), article id ijim-00651. [11] j. n. dewynne and w. t. shaw, differential equations and asymptotic solutions for arithmetic asian options: blackscholes formulae for asian rate calls, eur. j. appl. math. 19 (4) (2008), 353-391. [12] y. keskin, ph.d thesis, selcuk university, 2010. [13] y. keskin, g. oturance, reduced differential transform method for partial differential equations, int. j. nonlinear sci. numer. simul. 10 (6) (2009), 741-749. [14] y. keskin, g. oturance, reduced differential research transform method for fractional partial differential equations, nonlinear sci. lett. a, 1 (1) (2010), 61-72. [15] y. keskin, g. oturance, application of reduced differential transformation method for solving gas dynamics equation, int. j. contemp. math. sci. 5 (22) (2010), 1091-1096. [16] y.k. kwok, mathematical models of financial derivatives, 2nd ed., springer, berlin, 2008. [17] j. lishang, mathematical modeling and methods of option pricing, world scientific, singapore, 2005. [18] mahmoud. r, using the reduced differential transform method to solve nonlinear pdes arises in biology and physics, world appl. sci. j. 23 (8) (2013), 1037-1043. [19] mohammad. t., vineet k., srivastava: a computational modelling of micro strip patch antenna and its solution by rdtm, alex. eng. j. 57 (2018), 1877-1881. [20] l. c. g. rogers and z. shi, the value of an asian option, j. appl. prob. 32 (4) (1995), 1077-1088. [21] v. srivastava, m. awasthi, m. tamsir, rdtm solution of caputo time fractional-order hyperbolic telegraph equation, aip adv. 3 (3) (2013), 032142. [22] s. servi, y. keskin, g. oturanç, reduced differential transform method for improved boussinesq equation, aip conf. proc. 1648 (2015), 370012. [23] t.r. ramesh rao, numerical solution of sine gordon equations through reduced differential transform method, glob. j. pure appl. math. 13 (7) (2017), 3879-3888. int. j. anal. appl. 18 (3) (2020) 530 [24] z. ali, r. rozita and r. hazli, new pricing formula for arithmetic asian options using pde approach, app. math. sci. 5 (77) (2011), 3801-3809. [25] z. ali and r. rozita, solving an asian option pde via the laplace transform, scienceasia 39s (2013), 67-69. 1. introduction 2. reduced differential transform method 3. pricing of four versions of call-put parities of asian options 3.1. call-put parity for arithmetic average asian option with fixed strike price 3.2. call-put parity for arithmetic average asian option with floating strike price 3.3. call-put parity for geometric average asian option with fixed strike price 3.4. call-put parity for geometric average asian option with floating strike price 4. conclusion references int. j. anal. appl. (2022), 20:28 a new ostrowski’s type inequality for quadratic kernel m. m. saleem1, z. ullah2, t. abbas1, m. b. raza1, a. qayyum1,∗ 1institute of southern punjab multan, pakistan 2department of mathematics, division of science and technology, university of education lahore, pakistan ∗corresponding author: atherqayyum@isp.edu.pk abstract. from the past few decades, the integral inequalities have been extensively researched. integral inequalities are applied in innumerable mathematical problems. in current paper, we obtain new versions of ostrowski’s type integral inequalities by implementing proposed general form of 7-step quadratic kernel. applications for cumulative distribution are also provided. 1. introduction in 1938, a ukrainian mathematician a. m. ostrowski [1] discovered the integral inequality. many articles and research books have been dedicated to inequalities and its applications [3]-[7]. in this paper we present 7-step quadratic kernel that further generalize many earlier results contained in [8][12]. several authors have recently addressed the generalization of the ostrowski’s type inequalities. qayyum et.al [9]-[10] applied a 5-step kernel to generalize some ostrowski’s type inequalities. 2. main findings theorem 2.1. let f : [ č, ď ] →r be differentiable on ( č, ď ) , f ′is absolutely continuous on [ č, ď ] and γ ≤ f ′′(ţ) ≤ γ,∀ ţ∈ [ č, ď ] ,then ∀ ÿ ∈ [ č, č+ď 2 ] , we get received: mar. 29, 2022. 2010 mathematics subject classification. 26d10. key words and phrases. ostrowski inequality; numerical integration; quadratic kernel. https://doi.org/10.28924/2291-8639-20-2022-28 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-28 2 int. j. anal. appl. (2022), 20:28 ∣∣∣∣14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) (2.1) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] + f ′(ď) − f ′(č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3} − 1 ď − č ∫ ď č f (ţ) dţ ∣∣∣∣∣ ≤ ω(ÿ)(ď − č)(s −γ) and ∣∣∣∣14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) (2.2) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] + f ′(ď) − f ′(č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3} − 1 ď − č ∫ ď č f (ţ) dţ ∣∣∣∣∣ ≤ ω(ÿ)(ď − č)(γ −s), where s = f ′(ď) − f ′(č) ď − č and ω(ÿ) = 1 768 max {∣∣−65č2 + 231čÿ + 165ďÿ − 101čď −32ď2 − 198ÿ2 ∣∣ ,∣∣−35č2 + 135čÿ + 117ďÿ − 65čď − 26ď2 − 126ÿ2∣∣ , int. j. anal. appl. (2022), 20:28 3∣∣−83č2 + 255čÿ + 141čÿ − 89čď − 26ď2 − 198ÿ2∣∣ ,∣∣127č2 − 297čÿ − 27ďÿ + 43čď − 8ď2 + 162ÿ2∣∣ ,∣∣7č2 − 105čÿ − 219ďÿ + 91čď + 64ď2 + 162ÿ2∣∣ ,∣∣−65č2 + 183čÿ + 69ďÿ − 53čď − 8ď2 − 126ÿ2∣∣ ,∣∣89č2 − 279čÿ − 165ďÿ + 101čď + 32ď2 + 222ÿ2∣∣} . proof. to prove our required results, first of all we introduce a mapping. let f : [ č, ď ] →r be such that f ′ is absolutely continuous on [ č, ď ] . define the kernel k (ÿ,ţ) as: k (ÿ,ţ) =   1 2 (ţ− č)2 ţ ∈ ( č, 3č+ÿ 4 ] 1 2 ( ţ− 7č+ď 8 )2 ţ ∈ ( 3č+ÿ 4 , č+ÿ 2 ] 1 2 ( ţ− 3č+ď 4 )2 ţ ∈ ( č+ÿ 2 , ÿ ] 1 2 ( ţ− č+ď 2 )2 ţ ∈ ( ÿ, č + ď − ÿ ] 1 2 ( ţ− č+3ď 4 )2 ţ ∈ ( č + ď − ÿ, č+2ď−ÿ 2 ] 1 2 ( ţ− č+7ď 8 )2 ţ ∈ ( č+2ď−ÿ 2 , č+4ď−ÿ 4 ] 1 2 ( ţ− ď )2 ţ ∈ ( č+4ď−ÿ 4 , ď ] (2.3) ∀ ÿ ∈ [ č, č+ď 2 ] . then the following identity ∫ ď č k (ÿ,ţ) df ′ (ţ) (2.4) = ∫ ď č f (ţ) dţ− ď − č 4 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3a + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] holds. we know that 1 ď − č ∫ ď č f ′′ (ţ) dţ = f ′(ď) − f ′(č) ď − č (2.5) 4 int. j. anal. appl. (2022), 20:28 1 ď − č ∫ ď č k (ÿ,ţ) dţ = 1 ď − č [ 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 (2.6) − 73 192 ( ÿ − č + ď 2 )3] and implies that 1 ď − č ∫ ď č k (ÿ,ţ) f ′′ (ţ) dţ− 1( ď − č )2 ∫ ď č k (ÿ,ţ) dţ ∫ ď č f ′′ (ţ) dţ (2.7) = 1 ď − č ∫ ď č f (ţ) dţ− 1 4 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] − f ′(ď) − f ′(č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3} . we suppose that rn(ÿ) (2.8) = 1 ď − č ∫ ď č k (ÿ,ţ) f ′′ (ţ) dt − 1( ď − č )2 ∫ ď č k (ÿ,ţ) dt ∫ ď č f ′′ (ţ) dt. if c ∈ r is an arbitray constant, then we have rn(ÿ) (2.9) = 1 ď − č ∫ ď č ( f ′′ (ţ) −c )[ k (ÿ,ţ) − 1 ď − č ∫ ď č k (ÿ, s) ds ] dţ. furthermore, we have |rn(ÿ)| (2.10) ≤ 1 ď − č max ţ∈[č,ď] ∣∣∣∣∣k (ÿ,ţ) − 1ď − č ∫ ď č k (ÿ, s) ds ∣∣∣∣∣ ∫ ď č ∣∣f ′′ (ţ) −c∣∣dţ. int. j. anal. appl. (2022), 20:28 5 now max ∣∣∣∣∣k (ÿ,ţ) − 1ď − č ∫ ď č k (ÿ, s) ds ∣∣∣∣∣ (2.11) = max {∣∣∣∣∣12 ( ÿ − č 4 )2 − β(ÿ) ď − č ∣∣∣∣∣ , ∣∣∣∣∣18 ( ÿ − 3č + ď 4 )2 − β(ÿ) ď − č ∣∣∣∣∣ ,∣∣∣∣∣ 132 ( ÿ − č + ď 2 )2 − β(ÿ) ď − č ∣∣∣∣∣ , ∣∣∣∣∣12 ( ÿ − 3č + ď 4 )2 − β(ÿ) ď − č ∣∣∣∣∣ ,∣∣∣∣∣12 ( ÿ − č + ď 2 )2 − β(ÿ) ď − č ∣∣∣∣∣ , ∣∣∣∣∣18 ( ÿ − č + ď 2 )2 − β(ÿ) ď − č ∣∣∣∣∣ , β(ÿ)ď − č } where β(ÿ) = 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3 . and ω(ÿ) (2.12) = 1 768 max {∣∣−65č2 + 231čÿ + 165ďÿ − 101čď −32ď2 − 198ÿ2 ∣∣ ,∣∣−35č2 + 135čÿ + 117ďÿ − 65čď − 26ď2 − 126ÿ2∣∣ ,∣∣−83č2 + 255čÿ + 141čÿ − 89čď − 26ď2 − 198ÿ2∣∣ ,∣∣127č2 − 297čÿ − 27ďÿ + 43čď − 8ď2 + 162ÿ2∣∣ ,∣∣7č2 − 105čÿ − 219ďÿ + 91čď + 64ď2 + 162ÿ2∣∣ ,∣∣−65č2 + 183čÿ + 69ďÿ − 53čď − 8ď2 − 126ÿ2∣∣ ,∣∣89č2 − 279čÿ − 165ďÿ + 101čď + 32ď2 + 222ÿ2∣∣} . we also have ∫ ď č ∣∣f ′′ (ţ) −γ∣∣dt = (s −γ) (ď − č) (2.13) and ∫ ď č ∣∣f ′′ (ţ) − γ∣∣dţ = (γ −s) (ď − č) . (2.14) so, we attain (2.1) and (2.2) by using (2.5) to (2.14) and taking c = γ and c = γ in (2.10) respectively. corollary 2.1. by replacing ÿ = č in (2.1) and (2.2) , we get∣∣∣∣∣f (č) + f (ď)2 −(ď − č) f ′(ď) − f ′(č) 12 − 1 ď − č ∫ ď č f (ţ)dţ ∣∣∣∣∣ ≤ 1 96 ( ď − č )3 (s −γ) , 6 int. j. anal. appl. (2022), 20:28∣∣∣∣∣f (č) + f (ď)2 −(ď − č) f ′(ď) − f ′(č) 12 − 1 ď − č ∫ ď č f (ţ)dţ ∣∣∣∣∣ ≤ 1 96 ( ď − č )3 (γ −s) . � now some new perturbed ostrowski type inequalities are presented by working with differentiable mapping whose first derivatives are absolutely continuous and the second derivatives belong to f ′′′ ∈ l2 the usual lebesgue spaces which refine and generalize some previous inequalities of this domain. theorem 2.2. let f : [ č, ď ] → r be three times differentiable function on ( č, ď ) . if f ′′′ ∈ l2 [ č, ď ] ,then for all ÿ ∈ [ č, č+ď 2 ] , we have∣∣∣∣14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) (2.15) + 1 2 f ( č + 2b− ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] + f ′(ď) − f ′(č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3} − 1 ď − č ∫ ď č f (ţ) dţ ∣∣∣∣∣ ≤ 1 π ∥∥f ′′′∥∥ 2 [ 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 × ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 1 2 . proof. let rn(x) be defined by (2.7) and (2.8), if we take c = f ′′ ( č+ď 2 ) in (2.9) by applying the cauchy inequality, then |rn(ÿ)| ≤ 1 ď − č ∫ ď č ∣∣∣∣ ( f ′′ (ţ) − f ′′ ( č + ď 2 ))∣∣∣∣ (2.16) × ∣∣∣∣∣k (ÿ,ţ) − 1ď − č ∫ ď č k (ÿ, s) ds ∣∣∣∣∣dţ. int. j. anal. appl. (2022), 20:28 7 ≤ 1 ď − č [∫ ď č ( f ′′ (ţ) − f ′′ ( č + ď 2 ))2 dţ ]1 2 ×  ∫ ď č ( k (ÿ,ţ) − 1 ď − č ∫ ď č k (ÿ, s) ds )2 dţ   1 2 . we apply the diaze-metcalf inequality from [16] to get∫ ď č ( f ′′ (ţ) − f ′′ ( č + ď 2 ))2 dţ ≤ ( ď − č )2 π2 ∥∥f ′′′∥∥2 2 and ∫ ď č ( k (ÿ,ţ) − 1 ď − č ∫ ď č k (ÿ, s) ds )2 dţ (2.17) = ∫ ď č k (ÿ,ţ)2 dţ− 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 = 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 . so, by using the above relations (2.18)-(2.19) , we attain (2.17) . corollary 2.2. by replacing ÿ = č in (2.17) , we get∣∣∣∣∣f (č) + f (ď)2 −(ď − č) f ′(č) + f ′(ď) 12 − 1 ď − č ∫ ď č f (ţ)dţ ∣∣∣∣∣ ≤ 1 π ∥∥f ′′′∥∥ 2 ( ď − č )5 2 1 12 1 √ 5 . � theorem 2.3. let f : [ č, ď ] →r be an absolutely continuous function on ( č, ď ) , with f ′′ ∈ l2 [ č, ď ] . then ∣∣∣∣14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) + 1 2 f ( 3č + ÿ 4 ) (2.18) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ′ ( č + ď − ÿ ) − f ′ (ÿ) + 1 4 f ′ ( č + 2ď − ÿ 2 ) − 1 4 f ′ ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ′ ( č + 4ď − ÿ 4 ) − f ′ ( 3č + ÿ 4 )}] 8 int. j. anal. appl. (2022), 20:28 + f ′(ď) − f ′(č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3} − 1 ď − č ∫ ď č f (ţ) dţ ∣∣∣∣∣ ≤ √ σ(f ′′) ď − č [ 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 1 2 ∀ÿ ∈ [ č, č+ď 2 ] , where σ(f ′′) = ∥∥f ′′∥∥2 2 − ( f ′(ď) − f ′(č) )2 ď − č = ∥∥f ′′∥∥2 2 −s2 ( ď − č ) (2.19) where s = f ′(ď) − f ′(č) ď − č . proof. let rn(x) be defined by (2.7) and (2.8) .if we take c = 1 ď−č ∫ ď č f ′′ (s) ds in (2.9) and applying the cauchy inequality, then we have |rn(ÿ)| ≤ 1 ď − č ∫ ď č ∣∣∣∣∣ ( f ′′ (ţ) − 1 ď − č ∫ ď č f ′′ (s) ds )∣∣∣∣∣ × ∣∣∣∣∣k (ÿ,ţ) − 1ď − č ∫ ď č k (ÿ, s) ds ∣∣∣∣∣dţ. ≤ 1 ď − č  ∫ ď č ( f ′′ (ţ) − 1 ď − č ∫ ď č f ′′ (s) ds )2 dţ   1 2 ×  ∫ ď č ( k (ÿ,ţ) − 1 ď − č ∫ ď č k (ÿ, s) ds )2 dţ   1 2 = √ σ(f ′′) ď − č [ 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 int. j. anal. appl. (2022), 20:28 9 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 1 2 . hence proved (2.21) � corollary 2.3. by replacing ÿ = č in (2.21) we get∣∣∣∣∣f (č) + f (ď)2 −(ď − č) f ′(č) + f ′(ď) 12 − 1 ď − č ∫ ď č f (ţ)dţ ∣∣∣∣∣ ≤ √ σ(f ′′) ( ď − č )3 2 1 12 1 √ 5 . 3. an application to cumulative distribution function consider x is a random variable taking values in the finite interval [ č, ď ] with the probability density function f : [ č, ď ] → [0, 1] and cumulative distributive function f (ÿ) = pr (x ≤ ÿ) = ∫ ÿ č f (ţ)dţ, (3.1) f ( ď ) = pr ( x ≤ ď ) = ∫ ď č f (u)du = 1. (3.2) in this section, we may use the inequalities (2.1)-(2.2), (2.17) and (2.21) to get useful application for cumulative distribution function by using probability density function with smaller error than that which may be obtained by the classical results. theorem 3.1. under the assumption of theorem 2.1, we get the following inequality which holds∣∣∣∣ď −e(x)ď − č − 14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) (3.3) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ( č + ď − ÿ ) − f (ÿ) + 1 4 f ( č + 2ď − ÿ 2 ) − 1 4 f ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ( č + 4ď − ÿ 4 ) − f ( 3č + ÿ 4 )}] − f (ď) − f (č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 ≤ ω(ÿ)(ď − č)(s −γ) 10 int. j. anal. appl. (2022), 20:28 and ∣∣∣∣ď −e(x)ď − č − 14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) (3.4) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ( č + ď − ÿ ) − f (ÿ) + 1 4 f ( č + 2ď − ÿ 2 ) − 1 4 f ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ){ f ( č + 4ď − ÿ 4 ) − f ( 3č + ÿ 4 )}] − f (ď) − f (č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}∣∣∣∣∣ ≤ ω(ÿ)(ď − č)(γ −s) for all ÿ ∈ [ č, č+ď 2 ] . proof. by (3.1) and (3.2) on taking f = f and by applying the fact e(x) = ∫ ď č ţdf (ţ) = ď − ∫ ď č f (ţ)dţ (3.5) we attain (3.3) and (3.4) . � theorem 3.2. by using theorem 2.1, we get the following inequality which holds ∣∣∣∣ď −e(x)ď − č − 14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) (3.6) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ( č + ď − ÿ ) − f (ÿ) + 1 4 f ( č + 2ď − ÿ 2 ) − 1 4 f ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ) × { f ( č + 4ď − ÿ 4 ) − f ( 3č + ÿ 4 )}] int. j. anal. appl. (2022), 20:28 11 − f (ď) − f (č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}∣∣∣∣∣ ≤ 1 π ∥∥f ′′′∥∥ 2 [ 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 1 2 . proof. using (4.36) and by the conditions that we used in above theorem , we get (3.6). � theorem 3.3. with the statement of theorem 2.1, we have the following inequality which holds∣∣∣∣ď −e(x)ď − č − 14 [ f (ÿ) + f ( č + ď − ÿ ) + 1 2 f ( č + ÿ 2 ) (3.7) + 1 2 f ( 3č + ÿ 4 ) + 1 2 f ( č + 2ď − ÿ 2 ) + 1 2 f ( č + 4ď − ÿ 4 ) + ( ÿ − 5č + 3ď 8 ) × { f ( č + ď − ÿ ) − f (ÿ) + 1 4 f ( č + 2ď − ÿ 2 ) − 1 4 f ( č + ÿ 2 )} + 1 8 ( ÿ − 3č + ď 4 ) × { f ( č + 4ď − ÿ 4 ) − f ( 3č + ÿ 4 )}] − f (ď) − f (č)( ď − č )2 { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}∣∣∣∣∣ ≤ √ σ(f ′′) ď − č [ 1 10240 (ÿ − č)5 + 33 320 ( ÿ − 3č + ď 4 )5 − 1057 10240 ( ÿ − č + ď 2 )5 − 1 ď − č { 1 192 (ÿ − č)3 + 3 8 ( ÿ − 3č + ď 4 )3 − 73 192 ( ÿ − č + ď 2 )3}2 1 2 . proof. applying (4.43) and by set of conditions that we used in theorem 3.1, we get (3.7). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. 12 int. j. anal. appl. (2022), 20:28 references [1] a. ostrowski, über die absolutabweichung einer differentienbaren funktionen von ihren integralmittelwert, comment. math. helv. 10 (1938), 226–227. [2] j. amjad, a. qayyum, s. fahad, m. arslan, some new generalized ostrowski type inequalities with new error bounds, innov. j. math. 1 (2022), 30–43. https://doi.org/10.55059/ijm.2022.1.2/23. [3] a. qayyum, s. hussain, a new generalized ostrowski grüss type inequality and applications, appl. math. lett. 25 (2012), 1875–1880. https://doi.org/10.1016/j.aml.2012.02.052. [4] a. qayyum, m. shoaib, a.e. matouk, m.a. latif, on new generalized ostrowski type integral inequalities, abstr. appl. anal. 2014 (2014), 275806. https://doi.org/10.1155/2014/275806. [5] a. qayyum, m. shoaib, m.a. latif, a generalized inequality of ostrowski type for twice differentiable bounded mappings and applications, appl. math. sci. 8 (2014), 1889–1901. https://doi.org/10.12988/ams.2014.4222. [6] a. qayyum, i. faye, m. shoaib, m.a. latif, a generalization of ostrowski type inequality for mappings whose second derivatives belong to l1(a,b) and applications, int. j. pure appl. math. 98 (2015), 169-180. https: //doi.org/10.12732/ijpam.v98i2.1. [7] a. qayyum, m. shoaib, i. faye, some new generalized results on ostrowski type integral inequalities with application, j. comput. anal. appl. 19 (2015), 693-712. https://doi.org/10.48550/arxiv.1505.01520. [8] m.w. alomari, a companion of ostrowski’s inequality for mappings whose first derivatives are bounded and applications numerical integration, kragujevac j. math. 36 (2012), 77-82. [9] a. qayyum, m. shoaib, i. faye, a companion of ostrowski type integral inequality using a 5-step kernel with some applications, filomat. 30 (2016), 3601–3614. https://doi.org/10.2298/fil1613601q. [10] a. qayyum, m. shoaib, i. faye, companion of ostrowski-type inequality based on 5-step quadratic kernel and applications, j. nonlinear sci. appl. 09 (2016), 537–552. https://doi.org/10.22436/jnsa.009.02.19. [11] d.s. mitrinovic, j.e. pecaric, a.m. fink, inequalities involving functions and their integrals and derivatives, springer, dordrecht, 1991. https://doi.org/10.1007/978-94-011-3562-7. https://doi.org/10.55059/ijm.2022.1.2/23 https://doi.org/10.1016/j.aml.2012.02.052 https://doi.org/10.1155/2014/275806 https://doi.org/10.12988/ams.2014.4222 https://doi.org/10.12732/ijpam.v98i2.1 https://doi.org/10.12732/ijpam.v98i2.1 https://doi.org/10.48550/arxiv.1505.01520 https://doi.org/10.2298/fil1613601q https://doi.org/10.22436/jnsa.009.02.19 https://doi.org/10.1007/978-94-011-3562-7 1. introduction 2. main findings 3. an application to cumulative distribution function references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 26-35 http://www.etamaths.com on random coincidence & fixed points for a pair of multi-valued & single-valued mappings pankaj kumar jhade1,∗ and a. s. saluja2 abstract. let (x,d) be a polish space, cb(x) the family all nonempty closed and bounded subsets of x and (ω, σ) be a measurable space. in this paper a pair of hybrid measurable mappings f : ω×x → x and t : ω×x → cb(x), satisfying the inequality (2.1) below are introduced and investigated. it is proved that if x is complete, t(ω, ·), f(ω, ·) are continuous for all ω ∈ ω, t(·,x), f(·,x) are measurable for all x ∈ x and t(ω,ξ(ω)) ⊆ f(ω × x) and f(ω×x) = x for each ω ∈ ω, then there is a measurable mapping ξ : ω → x such that f(ω,ξ(ω)) ∈ t(ω,ξ(ω)) for all ω ∈ ω. 1. introduction random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of random nonlinear operators and their properties and is much needed for the study of various classes of random equations. of course famously random methods have revolutionized the financial markets. random fixed point theorems for random contraction mappings on separable complete metric spaces were first proved by špaček[24] and hanš[7,8]. the survey article by bharucha-reid [1] in 1976 attracted the attention of several mathematicians (see chang and huang[2], hanš[7],[8], špaček[24], huang[10], itoh [11], liu [14], papageorgiou [15],[16], shahzad and hussain [21],shahzad and latif [22], tan and yuan [25]) and give wings to this theory. itoh [11] extended špaček’s and hanš’s theorem to multi-valued contraction mapping . the stochastic version of the wellknown schauder’s fixed point theorem was proved by sehgal and singh [20]. let (x,d) be a metric space and t : x → x a mapping. the class of mappings t satisfying the following contractive conditions: d(tx,ty) ≤ a max{d(x,y),d(x,tx),d(y,ty), d(x,ty) + d(y,tx) 2 } + b max{d(x,tx),d(y,ty)} + c[d(x,ty) + d(y,tx)] (1.1) for all x,y ∈ x, where a,b,c are non-negative real numbers such that b > 0 c > 0 and a + b + 2c = 1, was introduced and investigated by ćirić [3]. ćirić proved that in a complete metric space such mappings have a unique fixed point. this class of mappings was further studied by many authors (ćirić[4],[5], singh and mishra[23], and rhoades et al. [18]).sehgal and singh [20] have generalized ćirić’s 2010 mathematics subject classification. 47h10; 54h25. key words and phrases. separable metric space; random fixed point; random coincidence point ;random multi-function. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 26 on random coincidence & fixed points 27 [4] fixed point theorem to a common fixed point theorem of a pair of mappings and presented some application of such theorems to dynamic programming. in this paper we introduced a new class of nonexpansive type mappings for a pair of multi-valued and single valued mappings which is a stochastic version of ćirić’s [3] fixed point theorem to find the coincidence and fixed points for such class of mappings. 2. preliminaries let (ω, σ) be a measurable space with σ a sigma algebra of subsets of ω and let (x,d) be a metric space. we denote by 2x the family of all subsets of x, by cb(x) the family of all nonempty closed and bounded subsets of x and by h the hausdorff metric on cb(x), induced by the metric d. for any x ∈ x and a ⊆ x, by d(x,a) we denote the distance between xanda, i.e., d(x,a) = inf{d(x,a) : a ∈ a}. a mapping t : ω → 2x is called σ−measurable if for any open subset u of x, t−1(u) = {ω : t(ω)∩u 6= φ}∈ σ.in what follows, when we speak of measurability we will mean σ− measurability. a mapping f : ω × x → x is called a random operator if for any x ∈ x,f(·,x) is measurable. a mapping t : ω×x → cb(x) is called a multi-valued random operator if for every x ∈ x, t(·,x) is measurable.a mapping s : ω → x is called a measurable selector of a measurable multifunction t : ω → 2x if s is measurable and s(ω) ∈ t(ω) for all ω ∈ ω. a measurable mapping ξ : ω → x is called a random fixed point of a random multifunction t : ω ×x → cb(x) if ξ(ω) ∈ t(ω,ξ(ω)) for every ω ∈ ω. a mapping ξ : ω → x is called a random coincidence of t : ω × x → cb(x) and f : ω × x → x if f(ω,ξ(ω)) ∈ t(ω,ξ(ω)) for all ω ∈ ω. the aim of this paper is to prove a stochastic analogue of the ćirić’s [3] fixed point theorem for single valued mappings, extended to a coincidence point theorem for a pair of a random operator f : ω ×x → x and a multi-valued random operator t : ω × x → cb(x), satisfying the following nonexpansive type condition: for each ω ∈ ω, h(t(ω,x),t(ω,y)) ≤ a(ω) max{d(f(ω,x),f(ω,y)),d(f(ω,y),t(ω,y))} + b(ω) max{d(f(ω,x),t(ω,x)),d(f(ω,y),t(ω,y)), d(f(ω,y),t(ω,x))} + c(ω)[d(f(ω,x),t(ω,y)) + d(f(ω,y),t(ω,x))] (2.1) for every x,y ∈ x, where a,b,c : ω → [0, 1) are measurable mappings such that for all ω ∈ ω (2.2) b(ω) > 0 c(ω) > 0 (2.3) a(ω) + b(ω) + 2c(ω) = 1 3. main results now, we give our main results. theorem 3.1. let (x,d) be a complete metric space, (ω, σ) be a measurable space and t : ω ×x → cb(x) & f : ω ×x → x be mappings such that 28 pankaj kumar jhade and a. s. saluja (1) t(ω, ·) and f(ω, ·) are continuous for all ω ∈ ω, (2) t(·,x) and f(·,x) are measurable for all x ∈ x, (3) they satisfy (2.1), where a(ω),b(ω),c(ω) : ω → x satisfy (2.2) and (2.3). if t(ω,ξ(ω)) ⊆ f(ω × x) and f(ω × x) = x for each ω ∈ ω, then there is a measurable mapping ξ : ω → x such that f(ω,ξ(ω)) ∈ t(ω,ξ(ω)) for all ω ∈ ω (i.e. t and f have a random coincidence point). proof. let ψ = {ξ : ω → x} be a family of measurable mappings. define a function g : ω ×x → r+ as follows: g(ω,x) = d(x,t(ω,x)). since x → t(ω,x) is continuous for all ω ∈ ω, we conclude that g(ω, ·) is continuous for all ω ∈ ω. also, since ω → t(ω,x) is measurable for all x ∈ x, we conclude that g(·,x) is measurable(see wagner [26], p 868) for all ω ∈ ω.thus g(ω,x) is the caratheodory function.therefore, if ξ : ω → x is a measurable mapping, then ω → g(ω,ξ(ω)) is also measurable (see [19]). now we shall construct a sequence of measurable mappings {ξn} in ψ and a sequence {f(ω,ξn(ω))} in x as follows.let ξ0 ∈ ψ be arbitrary. then the multifunction g : ω → cb(x) defined by g(ω) = t(ω,ξ0(ω)) is measurable. from the kuratowski-nardzewski [13] selector theorem there is a measurable selector µ1 : ω → x such that µ1(ω) ∈ t(ω,ξ0(ω)) for all ω ∈ ω since µ1(ω) ∈ t(ω,ξ0(ω)) ⊆ x = f(ω × x), let ξ1 ∈ ψ be such that f(ω,ξ1(ω)) = µ1.thus f(ω,ξ1(ω)) ∈ t(ω,ξ0(ω)) for all ω ∈ ω. let k : ω → (1,∞) defined by k(ω) = 1 + b(ω)c(ω) 2 for all ω ∈ ω.then k(ω) is measurable.since k(ω) > 1 and f(ω,ξ1(ω)) is a selector of t(ω,ξ0(ω)), from lemma 2.1 of papageorgiou [15] there is a measurable selector µ2(ω) = f(ω,ξ2(ω)); ξ2 ∈ ψ, such that for all ω ∈ ω: f(ω,ξ2(ω)) ∈ t(ω,ξ1(ω)) and d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) ≤ k(ω)h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) similarly, as f(ω,ξ2(ω)) is a selector of t(ω,ξ1(ω)), there is a measurable selector µ3(ω) = f(ω,ξ3(ω)) of t(ω,ξ2(ω)) ⊆ f(ω ×x) such that d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤ k(ω)h(t(ω,ξ1(ω)),t(ω,ξ2(ω))) continuing in this way we can construct a sequence of measurable mappings µn : ω → x, defined by µn(ω) = f(ω,ξn(ω)); ξn ∈ ψ, such that for all ω ∈ ω: f(ω,ξn+1(ω)) ∈ t(ω,ξn(ω)) and (3.1) d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤ k(ω)h(t(ω,ξn−1(ω)),t(ω,ξn(ω))) on random coincidence & fixed points 29 now from (2.1) h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),d(f(ω,ξ1(ω)),t(ω,ξ1(ω)))} + b(ω) max{d(f(ω,ξ0(ω)),t(ω,ξ0(ω))),d(f(ω,ξ1(ω)),t(ω,ξ1(ω))) ,d(f(ω,ξ1(ω)),t(ω,ξ0(ω)))} + c(ω)[d(f(ω,ξ0(ω)),t(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),t(ω,ξ0(ω)))] (3.2) since f(ω,ξ1(ω)) ∈ t(ω,ξ0(ω)), then d(f(ω,ξ1(ω)),t(ω,ξ0(ω))) = 0 d(f(ω,ξ0(ω)),t(ω,ξ0(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) d(f(ω,ξ1(ω)),t(ω,ξ1(ω))) ≤ h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) thus from (3.2) h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),h(t(ω,ξ0(ω)),t(ω,ξ1(ω)))} + b(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))),h(t(ω,ξ0(ω)),t(ω,ξ1(ω)))} + c(ω)[d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + h(t(ω,ξ0(ω)),t(ω,ξ1(ω)))] (3.3) if we assume that h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) > d(f(ω,ξ0(ω)),f(ω,ξ1(ω))), then from (3.3) and (2.3), we get h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) < a(ω)h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) + b(ω)h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) + 2c(ω)h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) = (a(ω) + b(ω) + 2c(ω))h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) = h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) a contradiction. therefore, we have h(t(ω,ξ0(ω)),t(ω,ξ1(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) since d(f(ω,ξ1(ω)),t(ω,ξ1(ω))) ≤ h(t(ω,ξ0(ω)),t(ω,ξ1(ω))), we have d(f(ω,ξ1(ω)),t(ω,ξ1(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) thus by induction we can show that (3.4) h(t(ω,ξn(ω)),t(ω,ξn+1(ω))) ≤ d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) (3.5) d(f(ω,ξn(ω)),t(ω,ξn(ω))) ≤ d(f(ω,ξn−1(ω)),f(ω,ξn(ω))) for all n ≥ 1 and for all ω ∈ ω from (3.1) and (3.4), we have (3.6) d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤ k(ω)d(f(ω,ξn−1(ω)),f(ω,ξn(ω))) 30 pankaj kumar jhade and a. s. saluja from (3.6), we get d(f(ω,ξ0(ω)),f(ω,ξ2(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) ≤ (1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) (3.7) from (2.1) h(t(ω,ξ0(ω)),t(ω,ξ2(ω))) ≤ a(ω) max{d(f(ω,ξ0(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),t(ω,ξ2(ω)))} + b(ω) max{d(f(ω,ξ0(ω)),t(ω,ξ0(ω))),d(f(ω,ξ2(ω)),t(ω,ξ2(ω))) ,d(f(ω,ξ2(ω)),t(ω,ξ0(ω)))} + c(ω)[d(f(ω,ξ0(ω)),t(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),t(ω,ξ0(ω)))] (3.8) using (3.4), (3.5), (3.6) and (3.7) and by triangle inequality, we get d(f(ω,ξ2(ω)),t(ω,ξ0(ω))) ≤ h(t(ω,ξ1(ω)),t(ω,ξ0(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) d(f(ω,ξ0(ω)),t(ω,ξ2(ω))) ≤ d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),t(ω,ξ2(ω))) ≤ (1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) ≤ (1 + 2k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) now from (3.8), (3.7),(3.6) and (2.3), we have h(t(ω,ξ0(ω)),t(ω,ξ2(ω))) ≤ a(ω)(1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + b(ω)k(ω)d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + 2c(ω)(1 + k(ω))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) = [1 + k(ω)(a(ω) + b(ω) + 2c(ω)) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) = [1 + k(ω) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) as 1 + k(ω) < 2k(ω), we have (3.9) h(t(ω,ξ0(ω)),t(ω,ξ2(ω))) ≤ [2k(ω) − b(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) from (2.3) and (2.1), we have, as f(ω,ξ2(ω)) ∈ t(ω,ξ1(ω)) h(t(ω,ξ1(ω)),t(ω,ξ2(ω))) ≤ a(ω) max{d(f(ω,ξ1(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),t(ω,ξ2(ω)))} + b(ω) max{d(f(ω,ξ1(ω)),t(ω,ξ1(ω))),d(f(ω,ξ2(ω)),t(ω,ξ2(ω))) ,d(f(ω,ξ2(ω)),t(ω,ξ1(ω)))} + c(ω)[d(f(ω,ξ1(ω)),t(ω,ξ2(ω))) + d(f(ω,ξ2(ω)),t(ω,ξ1(ω)))] ≤ [a(ω) + b(ω)] max{d(f(ω,ξ1(ω)),f(ω,ξ2(ω))),d(f(ω,ξ2(ω)),t(ω,ξ2(ω)))} + c(ω)d(f(ω,ξ1(ω)),t(ω,ξ2(ω))) (3.10) on random coincidence & fixed points 31 also by (3.9) since f(ω,ξ1(ω)) ∈ t(ω,ξ0(ω)), we have d(f(ω,ξ1(ω)),t(ω,ξ2(ω))) ≤ h(t(ω,ξ0(ω)),t(ω,ξ2(ω))) ≤ (2k(ω) − b(ω)))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) thus from (3.10) and (3.6), we have h(t(ω,ξ1(ω)),t(ω,ξ2(ω))) ≤ [a(ω) + b(ω)]k(ω)d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) + c(ω)(2k(ω) − b(ω)))d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) = [k(ω)(a(ω) + b(ω) + 2c(ω)) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) implies that (3.11) h(t(ω,ξ1(ω)),t(ω,ξ2(ω))) ≤ [k(ω) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) as a(ω) + b(ω) + 2c(ω) = 1 from (3.1) and (3.11), we have d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤ k(ω)h(t(ω,ξ1(ω)),t(ω,ξ2(ω))) ≤ k(ω)[k(ω) − b(ω)c(ω)]d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) (3.12) as k(ω) = 1 + b(ω)c(ω) 2 , we have k(ω)[k(ω) − b(ω)c(ω)] = ( 1 + b(ω)c(ω) 2 )( 1 + b(ω)c(ω) 2 − b(ω)c(ω) ) = 1 + b2(ω)c2(ω) 4 thus from (3.12) d(f(ω,ξ2(ω)),f(ω,ξ3(ω))) ≤ ( 1 + b2(ω)c2(ω) 4 ) d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) similarly d(f(ω,ξ3(ω)),f(ω,ξ4(ω))) ≤ ( 1 + b2(ω)c2(ω) 4 ) d(f(ω,ξ1(ω)),f(ω,ξ2(ω))) hence by induction d(f(ω,ξn(ω)),f(ω,ξn+1(ω))) ≤ ( 1 + b2(ω)c2(ω) 4 )[ n 2 ] max{d(f(ω,ξ0(ω)),f(ω,ξ1(ω))) ,d(f(ω,ξ1(ω)),f(ω,ξ2(ω)))} (3.13) where [ n 2 ] stands for the greatest integer not exceeding n 2 . also ,since b(ω)c(ω) > 0 for all ω ∈ ω, from (3.13), we have {f(ω,ξn(ω))} is a cauchy sequence in f(ω×x). since f(ω × x) = x is complete, there is a measurable mapping f(ω,ξ(ω)) ∈ f(ω ×x) such that (3.14) lim n→∞ f(ω,ξn(ω)) = f(ω,ξ(ω)) 32 pankaj kumar jhade and a. s. saluja again by triangle inequality and (2.1), we get d(f(ω,ξ(ω)),t(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + d(f(ω,ξn+1(ω)),t(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + h(t(ω,ξn(ω)),t(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + a(ω) max{d(f(ω,ξn(ω)),f(ω,ξ(ω))),d(f(ω,ξ(ω)),t(ω,ξ(ω)))} + b(ω) max{d(f(ω,ξn(ω)),t(ω,ξn(ω))),d(f(ω,ξ(ω)),t(ω,ξ(ω))) ,d(f(ω,ξ(ω)),t(ω,ξn(ω)))} + c(ω)[d(f(ω,ξn(ω)),t(ω,ξ(ω))) + d(f(ω,ξ(ω)),t(ω,ξn(ω)))] thus d(f(ω,ξ(ω)),t(ω,ξ(ω))) ≤ d(f(ω,ξ(ω)),f(ω,ξn+1(ω))) + a(ω) max{d(f(ω,ξn(ω)),f(ω,ξ(ω))),d(f(ω,ξ(ω)),t(ω,ξ(ω)))} + b(ω) max{d(f(ω,ξn(ω)),f(ω,ξn+1(ω))),d(f(ω,ξ(ω)),t(ω,ξ(ω))) ,d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))} + c(ω)[d(f(ω,ξn(ω)),t(ω,ξ(ω))) + d(f(ω,ξ(ω)),f(ω,ξn+1(ω)))] (3.15) taking limit as n →∞, we have d(f(ω,ξ(ω)),t(ω,ξ(ω))) ≤ [a(ω) + b(ω) + c(ω)]d(f(ω,ξ(ω)),t(ω,ξ(ω))) = [1 − c(ω)]d(f(ω,ξ(ω)),t(ω,ξ(ω))) implies that d(f(ω,ξ(ω)),t(ω,ξ(ω))) = 0, as 1−c(ω) < 1 and for ω ∈ ω. hence as t(ω,ξ(ω)) is closed f(ω,ξ(ω)) ∈ t(ω,ξ(ω)), for all ω ∈ ω. � remark 3.2. if in theorem 3.1., f(ω,x) = x for all (ω,x) ∈ ω ×x, then we get the following random fixed point theorem. corollary 3.3. let (x,d) be a separable complete metric space. (ω, σ) be a measurable space and let a mapping t : ω × x → cb(x) be such that t(ω, ·) is continuous for all ω ∈ ω, t(·,x) is measurable for all x ∈ x and h(t(ω,x),t(ω,y)) ≤ a(ω) max{d(x,y),d(x,t(ω,y))} + b(ω) max{d(x,t(ω,x)),d(y,t(ω,y)),d(y,t(ω,x))} + c(ω)[d(x,t(ω,y)) + d(y,t(ω,x))] (3.16) for every x,y ∈ x, where a,b,c : ω → (0, 1) are measurable mappings satisfying (2.2) and (2.3). then there is a measurable mapping ξ : ω → x such that ξ(ω) ∈ t(ω,ξ(ω)) for all ω ∈ ω. corollary 3.4. ([6], corollary 1) let (x,d) be a separable complete metric space. (ω, σ) be a measurable space and let a mapping t : ω ×x → cb(x) be such that t(ω, ·) is continuous for all ω ∈ ω, t(·,x) is measurable for all x ∈ x and h(t(ω,x),t(ω,y)) ≤ a(ω) max{d(x,y),d(x,t(ω,x)),d(y,t(ω,y)) , 1 2 [d(x,t(ω,y)) + d(y,t(ω,x))]} + b(ω) max{d(x,t(ω,x)),d(y,t(ω,y))} + c(ω)[d(x,t(ω,y)) + d(y,t(ω,x))] (3.17) on random coincidence & fixed points 33 for every x,y ∈ x, where a,b,c : ω → (0, 1) are measurable mappings satisfying (2.2) and (2.3). then there is a measurable mapping ξ : ω → x such that ξ(ω) ∈ t(ω,ξ(ω)) for all ω ∈ ω. remark 3.5. the nonexpansive type condition (3.16) includes (3.17) if we set m(x,y) = max{d(x,y),d(x,t(ω,x)),d(y,t(ω,y)), 1 2 [d(x,t(ω,y)) + d(y,t(ω,x))]} for each x,y such that m(x,y) = d(x,y) and a(ω),b(ω),c(ω) : ω → (0, 1) for each x,y such that m(x,y) = max{d(x,t(ω,x)),d(y,t(ω,y))} ,define a(ω) = 0,b(ω) = a(ω) + b(ω),c(ω) = c(ω). for each x,y such that m(x,y) = 1 2 [d(x,t(ω,y)) + d(y,t(ω,x))], define a(ω) = 0,b(ω) = b(ω),c(ω) = a(ω) + 2c(ω). thus corollary (3.3) is an extension of corollary (3.4). finally, we give a simple example in support of theorem 3.1. and corollary 3.3 which shows that these results are actually an improvement of the result of itoh[11]. example 3.6. let (x,d) be any measurable space and k = {0, 1, 2, 4, 6} be the subset of the real line. let the mappings f : ω ×k → k and t : ω ×k → k be defined such that for each ω ∈ ω: f(ω, 0) = 2 f(ω, 1) = 4 f(ω, 2) = 6 f(ω, 4) = 0 f(ω, 6) = 1 t(ω, 0) = 1 t(ω, 1) = 2 t(ω, 2) = 4 t(ω, 4) = 0 t(ω, 6) = 0 then for x = 1 and y = 2, we have d(t(ω, 1),t(ω, 2)) = 4 5 max{‖4 − 6‖,‖6 − 4‖} + 1 20 max{‖4 − 6‖,‖6 − 4‖,‖6 − 2‖} + 1 20 [‖4 − 4‖ + ‖6 − 2‖] = 4 5 .2 + 1 20 .4 + 1 20 .4 = 2 thus, for x = 1 and y = 2, f and t satisfy (2.1) with a(ω) = 4 5 ,b(ω) = 1 20 and c(ω) = 1 20 . it is easy to show that f and t satisfy (2.1) for all x,y ∈ k with the same a(ω),b(ω) and c(ω). also, the rest of the assumptions of theorem 3.1 is satisfied and for ξ(ω) = 4, we have f(ω,ξ(ω)) = 0 = t(ω,ξ(ω)) note that t does not satisfy (3.16) either, as for instance, for x = 2 and y = 4, we have a(ω) max{‖2 − 4‖,‖2 − 0‖} + b(ω) max{‖2 − 4‖,‖4 − 0‖,‖4 − 4‖} +c(ω)[‖2 − 0‖ + ‖4 − 4‖] = 2a(ω) + 4b(ω) + 2c(ω) < 4[a(ω) + b(ω) + 2c(ω)] = 4 = d(t(ω, 2),t(ω, 4)) 34 pankaj kumar jhade and a. s. saluja remark 3.7. our theorem 3.1 generalizes and extends the corresponding fixed point theorems for nonexpansive type single valued mapping of ćirić [3] and rhoades[17]. references [1] a.t bharucha-ried , fixed point theorem in probabilistic analysis, bull. amer. math. soc., 82(1976), 641-645. [2] s.s chang and n.j. huang, on the principle of randomization of fixed points for set valued mappings with applications, norteastern math. j., 7(1991), 486-491. [3] lj. b.ćirić , on some nonexpansive type mappings and fixed points, indian j. pure appl. math., 24(3)(1993), 145-149. [4] lj. b.ćirić, nonexpansive type mappings and a fixed point theorem in convex metric spaces, rend. accad. naz. sci. xl mem. mat., (5) vol.xix (1995), 263-271. [5] lj. b.ćirić, on some mappings in metric spaces and fixed point theorems, acad. roy. belg. bull. cl. sci., (5) t.vi(1995), 81-89. [6] lj. b.ćirić,jeong s. ume and sinǐsa n. ješić, on random coincidence and fixed points for a pair of multi-valued and single-valued mappings, j. ineq. appl., volume 2006(2006), article id 81045, 12 pages. [7] o. hanš, reduzierende zufällige transformationen, czech. math. j. 7(1957), 154-158. [8] o. hanš , random operator equations, proc. 4th berkeley symp. mathematics statistics and probability, vol. ii, part i, pp. 185-202. university of california press, berkeley (1961). [9] c.j. himmelberg , measurable relations. fund. math. 87(1975), 53-72. [10] n.j. huang, a principle of randomization of coincidence points with applications, applied math. lett., 12(1999), 107-113. [11] s. itoh, a random fixed point theorem for multi-valued contraction mapping, pacific j. math., 68(1977), 85-90. [12] t. kubiak , fixed point theorems for contractive type multi-valued mappings, math. japonica, 30(1985), 89-101. [13] k. kuratowski and c. ryll-nardzewski , a general theorem on selectors, bull. acad. polon. sci. ser. sci. math. astronom. phys., 13(1965), 397-403. [14] t.c. liu , random approximations and random fixed points for nonself maps, proc. amer. math. soc., 103(1988), 1129-1135. [15] n.s. papageorgiou , random fixed point theorems for multifunctions, math. japonica, 29(1984), 93-106. [16] n.s. papageorgiou , random fixed point theorems for measurable multifunctions in banach spaces, proc. amer. math. soc., 97(1986),507-514. [17] b.e. rhoades , a generalization of a fixed point theorem of bogin, math. sem. notes, 6(1987), 1-7. [18] b.e. rhoades, s.l. singh and c. kulshrestha , coincidence theorems for some multi-valued mappings, internat. j. math. math. sci., 7(1984), 429-434. [19] r.t. rockafellar , measurable dependence of convex sets and functions in parameters, j. math. anal. appl.,28(1969), 4-25. [20] v.m. sehgal and s.p. singh , on random approximations and a random fixed point theorem for set valued mappings, proc. amer. math. soc., 95(1985), 91-94. [21] n. shahzad and n. hussain , deterministic and random coincidence point results for fnonexpansive maps, j. math. anal. appl., 323 (2006), no. 2, 1038-1046. [22] n. shahzad and a. latif, a random coincidence point theorem, j. math. anal. appl., 245(2000), 633-638. [23] s.l. singh and s.n. mishra , on a ljubomir ćirić’s fixed point theorem for nonexpansive typ maps with applications, indian j. pure appl. math., 33(2002), no. 4, 531-542. [24] a. špaček , zufällige gleichungen, czech math. j., 5(1955), 462-466. [25] k.k. tan,x.z. yuan and n.j. huang , random fixed point theorems and approximations in cones, j. math. anal. appl., 185(1994), 378-390. on random coincidence & fixed points 35 [26] d.h. wagner , survey of measurable selection theorems, siam j. control optim.,15(1977), 859-903. 1department of mathematics, nri institute of information science & technology,bhopal462021 , india 2department of mathematics, j. h. government (pg) college, betul 460001, india ∗corresponding author international journal of analysis and applications volume 17, number 6 (2019), 1034-1051 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-1034 numerical quenching for heat equations with coupled nonlinear boundary flux kouamé béranger edja1,∗, koffi n’guessan2, brou jean-claude koua3 and kidjegbo augustin touré1 1institut national polytechnique houphouët-boigny yamoussoukro, bp 2444, côte d’ivoire 2ufr sed, université alassane ouattara , 01 bp v 18 bouaké 01, côte d’ivoire 3ufr mathématique et informatique, université félix houphouët boigny, côte d’ivoire ∗corresponding author: kouame.edja@inphb.ci abstract. in this paper, we study a numerical approximation of the following problem ut = uxx, vt = vxx, 0 < x < 1, 0 < t < t ; ux(0, t) = u −m(0, t) + v−p(0, t), vx(0, t) = u −q(0, t) + v−n(0, t) and ux(1, t) = vx(1, t) = 0, 0 < t < t, where m, p, q and n are parameters. we prove that the solution of a semidiscrete form of above problem quenches in a finite time only at first node of the mesh. we show that the time derivative of the solution blows up at quenching node. some conditions under which the non-simultaneous or simultaneous quenching occurs for the solution of the semidiscrete problem are obtained. we establish the convergence of the quenching time. finally, some numerical results to illustrate our analysis are given. 1. introduction in this paper, we study the behavior of a semidiscrete approximation of the following heat equations involving nonlinear boundary flux conditions : ut(x,t) = uxx(x,t), vt(x,t) = vxx(x,t), (x,t) ∈ (0, 1) × (0,t), (1.1) received 2019-08-08; accepted 2019-09-23; published 2019-11-01. 2010 mathematics subject classification. 65m06, 65m12, 35k05, 35k55. key words and phrases. numerical quenching; non-simultaneous; heat equation; nonlinear boundary. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1034 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-1034 int. j. anal. appl. 17 (6) (2019) 1035 ux(0, t) = u −m(0, t) + v−p(0, t), vx(0, t) = u −q(0, t) + v−n(0, t), t ∈ (0,t), (1.2) ux(1, t) = 0, vx(1, t) = 0, t ∈ (0,t), (1.3) u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ [0, 1], (1.4) where m,n ≥ 0, p,q > 0, u0 and v0 are positive smooth functions satisfying the compatibility conditions u′0(0) = u −m 0 (0) + v −p 0 (0), v ′ 0(0) = u −q 0 (0) + v −n 0 (0), u ′ 0(1) = 0, v ′ 0(1) = 0, and u ′ 0,v ′ 0 ≥ 0 and u′′0,v′′0 < 0 on (0, 1]. here [0,t) is the maximal time interval such that ∀t ∈ [0,t), inf min 0≤x≤1 {u(x,t),v(x,t)} > 0. we have lim t→t− inf min 0≤x≤1 {u(x,t),v(x,t)} = 0+. the time t can be finite or infinite. if t is finite, then we say that the solution (u,v) quenches in a finite time and t is called the quenching time of (u,v). if t is infinite, then we affirm that the solution (u,v) quenches globally. nonlinear parabolic systems like (1.1)-(1.4) come from chemical reactions, heat transfer, etc, where u and v represent the temperatures of two different materials during heat propagation. the quenching phenomenon of parabolic problems has been the issue of intensive study (see for example [3, 4, 8–10] and the references cited therein), particulary the study of heat equations system with nonlinear boundary conditions has been the subject of investigation of several authors in recent years (see [6, 7, 14, 15, 17] and the references cited therein). in [7] the authors study this problem, they prove that the solution (u,v) quenches in finite time t and the quenching occurs only at the boundary x = 0 for 0 < u0,v0 ≤ 1. they show that • if p < n + 1, there exist initial data such that the non-simultaneous quenching occurs ; • if q ≤ n(m+1) n+1 and p ≥ n + 1 (p ≤ m(n+1) m+1 and q ≥ m + 1), the non-simultaneous quenching occurs for any positive initial data ; • if q ≥ m+1, p ≥ n+1, any quenching must be simultaneous and obtain of results on non-simultaneous quenching rate. moreover, if quenching is simultaneous they found the quenching rate, which depends on the parameter in the flux associated to the other component of the initial data. to the best of our knowledge, no studies have been performed on the numerical approximation of equations (1.1)-(1.4). in this paper, we investigate in the numerical study using a semidiscrete form of (1.1)-(1.4), int. j. anal. appl. 17 (6) (2019) 1036 especially in study of simultaneous and non-simultaneous quenching. for that, we consider a uniform mesh on the interval [0, 1] xi = (i− 1)h, i = 1, . . . ,i, h = 1/(i − 1), uh(t) = (u1(t), . . . ,ui(t)) t , vh(t) = (v1(t), . . . ,vi(t)) t , where ui(t) and vi(t) are the values of the numerical approximation of u and v at the nodes xi at time t. we also denote ϕ1,i and ϕ2,i, respectively, the values of the numerical approximation of u0 and v0 at the nodes xi. by the finite difference method we obtain the following system of odes whose the solution is (uh,vh) : u′i(t) = δ 2ui(t) − bi ( u−mi (t) + v −p i (t) ) , i = 1, . . . ,i, t ∈ (0,th), (1.5) v ′i (t) = δ 2vi(t) − bi ( u −q i (t) + v −n i (t) ) , i = 1, . . . ,i, t ∈ (0,th), (1.6) ui(0) = ϕ1,i vi(0) = ϕ2,i, i = 1, . . . ,i, (1.7) where 0 < ϕ1,i ≤ m, 0 < ϕ2,i ≤ n, i = 1, . . . ,i, δ2ui(t) = ui−1(t) − 2ui(t) + ui+1(t) h2 , 2 ≤ i ≤ i − 1, t ∈ (0,th), δ2u1(t) = 2u2(t) − 2u1(t) h2 , δ2ui(t) = 2ui−1(t) − 2ui(t) h2 , t ∈ (0,th), b1 = 2 h , and bi = 0, i = 2, . . . ,i. here [0,th) is the maximal time interval such that ∀t ∈ [0,th), inf min 1≤i≤i {ui(t),vi(t)} > 0. we have lim t→t− h inf min 1≤i≤i {ui(t),vi(t)} = 0+. the time th can be finite or infinite. if th is finite, then we say that the solution (uh,vh) quenches in a finite time and th is called the semidiscrete quenching time of (uh,vh). if th is infinite, then we affirm that the solution (uh,vh) quenches globally. we show that our semidiscrete scheme reproduces well the conditions for the quenching, quenching set or simultaneous and non-simultaneous quenching of system (1.1)-(1.4). by following, it is also proved that when quenching occurs, the semidiscrete quenching time converges to the theoretical one when the mesh size goes to zero and we give a result on numerical non-simultaneous quenching rate. for previous work on numerical approximations of heat equations with non-linear boundary conditions we refer to [1,2,5,11–13,16] and the references cited therein. the rest of the paper is organized as follows : in the next section, we give int. j. anal. appl. 17 (6) (2019) 1037 some properties concerning our semidiscrete scheme. in section 3, under some conditions, we prove that the solution of the semidiscrete scheme (1.5)-(1.7) quenches in a finite time, we give a result on numerical quenching set. we also show that the time derivative of the solution blows up at quenching node. in section 4 a criterion to identify simultaneous and non-simultaneous quenching is proposed. in section 5, we show the convergence of the semidiscrete scheme and the convergence of the quenching times to the theoretical one when the mesh size goes to zero. finally, in the last section, we give some numerical results to illustrate our analysis. 2. properties of the semidiscrete scheme in this section, we give some auxiliary results for the problem (1.5)-(1.7). definition 2.1. we say that (uh,vh) ∈ ( c1([0,th), r i) )2 is a lower solution of (1.5)-(1.7) if ui ′(t) ≤ δ2ui(t) − bi(ui−m(t) + vi−p(t)), i = 1, . . . ,i, t ∈ (0,th), v ′i (t) ≤ δ 2vi(t) − bi(ui−q(t) + vi−n(t)), i = 1, . . . ,i, t ∈ (0,th), 0 < ui(0) ≤ ϕ1,i, 0 < vi(0) ≤ ϕ2,i, i = 1, . . . ,i, where (uh,vh) is the solution of (1.5)-(1.7). on the other hand, we say that (uh,vh) ∈ ( c1([0,th), r i) )2 is an upper solution of (1.5)-(1.7) if these inequalities are reversed. the following lemma is a discrete form of the maximum principle. lemma 2.1. let eh, ch, αh, βh ∈ (c0([0,th), ri) and uh, vh ∈ c1([0,th), ri) such that u′i(t) − δ 2ui(t) + ei(t)ui(t) + ci(t)vi(t) ≥ 0, i = 1 . . . ,i, t ∈ (0,th), v ′i (t) −δ 2vi(t) + αi(t)ui(t) + βi(t)vi(t) ≥ 0, i = 1 . . . ,i, t ∈ (0,th), ui(0) ≥ 0, vi(0) ≥ 0, i = 1 . . . ,i. then we have ui(t) ≥ 0, vi(t) ≥ 0, i = 1 . . . ,i, t ∈ (0,th). proof. let t0 < th and let (zh(t),wh(t)) = (e λtuh(t),e λtvh(t)) where λ is a real. we find that (zh(t),wh(t)) satisfies the following inequalities : z′i(t) − δ 2zi(t) + (ei(t) −λ)zi(t) + ci(t)wi(t) ≥ 0, i = 1 . . . ,i, t ∈ (0,th), (2.1) w ′i (t) − δ 2wi(t) + αi(t)zi(t) + (βi(t) −λ)wi(t) ≥ 0, i = 1 . . . ,i, t ∈ (0,th), (2.2) int. j. anal. appl. 17 (6) (2019) 1038 zi(0) ≥ 0, wi(0) ≥ 0, i = 1 . . . ,i. (2.3) set m = min { min 1≤i≤i,t∈[0,t0] zi(t), min 1≤i≤i,t∈[0,t0] wi(t) } . since for i ∈ {0, . . . ,i}, zi(t) and wi(t) are continuous functions on a compact, we can assume that m = zi0 (ti0 ) for a certain i0 ∈{0, . . . ,i}. assume m < 0. taking λ negative such that ei0 (ti0 ) −λ > 0 and βi0 (ti0 ) −λ > 0. if ti0 = 0, then zi0 (0) < 0, which contradicts (2.3), hence ti0 6= 0 ; if 1 ≤ i0 ≤ i, we have z′i0 (ti0 ) = limk→0 zi0 (ti0 ) −zi0 (ti0 −k) k ≤ 0. moreover by a straightforward computation we get z′i0 (ti0 ) − δ 2zi0 (ti0 ) + (ei0 (ti0 ) −λ) zi0 (ti0 ) + ci0 (ti0 )wi0 (ti0 ) < 0, but these inequalities contradict (2.1) and the proof is completed. � lemma 2.2. let (uh,vh) and (uh,vh) be lower and upper solutions of (1.5)-(1.7) respectively such that, (uh(0),vh(0)) ≤ (uh(0),vh(0)) then (uh(t),vh(t)) ≤ (uh(t),vh(t)). proof. let us define (zh(t),wh(t)) = (uh(t),vh(t)) − (uh(t),vh(t)). we obtain z′i(t) − δ 2zi(t) −mbi(µi(t))−m−1zi(t) −pbi(νi(t))−p−1wi(t) ≥ 0, i = 1, . . . ,i (2.4) w ′i (t) − δ 2wi(t) −qbi(µi(t))−q−1zi(t) −nbi(νi(t))−n−1wi(t) ≥ 0, i = 1, . . . ,i (2.5) zi(0) ≥ 0, wi(0) ≥ 0, i = 1, . . . ,i (2.6) where µi(t), νi(t) lie, respectively, between ui(t) and ui(t), and between vi(t) and vi(t), for i ∈{1, . . . ,i}. we can rewrite (2.4)-(2.5) as z′i(t) −δ 2zi(t) + ei(t)zi(t) + ci(t)wi(t) ≥ 0, i = 1, . . . ,i, t ∈ (0,th), w ′i (t) −δ 2wi(t) + αi(t)zi(t) + βi(t)wi(t) ≥ 0, i = 1, . . . ,i, t ∈ (0,th), where ei(t) = −mbi(µi(t))−m−1, ci(t) = −pbi(νi(t))−p−1, αi(t) = −qbi(µi(t))−q−1, and βi(t) = −nbi(νi(t))−n−1 i = 1, . . . ,i, ∀t ∈ (0,th). according to lemma 2.1, zi(t) ≥ 0, wi(t) ≥ 0, for i = 1, . . . ,i, ∀t ∈ (0,th) and the proof is completed. � int. j. anal. appl. 17 (6) (2019) 1039 the next lemma gives the properties of the semidiscrete solution. lemma 2.3. let (uh,vh) ∈ ( c1([0,th), r i) )2 be the solution of (1.5)-(1.7) with an initial data (ϕ1,h,ϕ2,h) upper solution such that 0 < ϕ1,i < ϕ1,i+1 ≤ m and 0 < ϕ2,i < ϕ2,i+1 ≤ n for i = 1, . . . ,i − 1. then we have (i) 0 < ui(t) ≤ ϕ1,i ≤ m and 0 < vi(t) ≤ ϕ2,i ≤ n, for i = 1, . . . ,i, t ∈ [0,th); (ii) (ui+1(t),vi+1(t)) > (ui(t),vi(t)), i = 1, . . . ,i − 1, t ∈ (0,th) ; (iii) (u′i(t),v ′ i (t)) ≤ 0, i = 1, . . . ,i, t ∈ (0,th). proof. (i) since (ϕ1,h,ϕ2,h) is an upper solution of (1.5)-(1.7), by the lemma 2.1 and 2.2 we have 0 < ui(t) ≤ m and 0 < vi(t) ≤ n, for i = 1, . . . ,i, t ∈ [0,th). (ii) we argue by contradiction. assume, that t0 the first t > 0, such that (ki,li)(t) = (ui+1−ui,vi+1− vi)(t) > 0, for 1 ≤ i ≤ i − 1, but min{ki0 (t0),li0 (t0)} = 0 for a certain i0 ∈{1, ...,i − 1}. assume that ki0 (t0) = ui0+1(t0) − ui0 (t0) = 0. without lost of generality, we can suppose that i0 is the smallest integer which satisfies the above equality. therefore, by simple computation, (kh,lh) verifies k′h(t) = −a ′kh(t) + b ′u−mh (t) + b ′v −p h (t), l′h(t) = −a ′lh(t) + b ′u −q h (t) + b ′v −nh (t), where a′ = 1 h2   3 −1 0 . . . 0 −1 2 −1 . . . ... 0 . . . . . . . . . 0 ... . . . −1 2 −1 0 . . . 0 −1 3   , b′ =   2 h 0 0 . . . 0 0 0 . . . ... ... . . . . . . . . . ... ... . . . 0 0 0 . . . . . . 0 0   . on the one hand k′i0 (t0) = lim�→0 ki0 (t0) −ki0 (t0 − �) � ≤ 0, and, on the other hand − i∑ j=1 a′i0,jkj(t) + b ′ i0 umi0 (t0) + b ′ i0 v p i0 (t0) > 0. thus we have a contradiction, hence we obtain the desired result. int. j. anal. appl. 17 (6) (2019) 1040 (iii) denote fi(t) = ui(t) −ui(t + ε) and gi(t) = vi(t) −vi(t + ε), for i = 1, . . . ,i, using (i) we obtain fi(0) ≥ 0, gi(0) ≥ 0 for i = 1, . . . ,i. it is not hard to see that f ′i (t) = δ 2fi(t) + mbi(ξi(t)) −m−1fi(t) + pbi(ηi(t)) −p−1gi(t) ≥ 0, g′i(t) = δ 2gi(t) + qbi(ξi(t)) −q−1fi(t) + nbi(ηi(t)) −n−1gi(t) ≥ 0, where ξi(t), ηi(t) lie, respectively, between ui(t + ε) and ui(t) and between vi(t + ε) and vi(t). from lemma 2.1 we get fi(t) ≥ 0 and gi(t) ≥ 0 for i = 1, . . . ,i, t ∈ (0,th). this fact implies the desired result. � 3. quenching and blow-up let (uh,vh) be the solution of (1.5)-(1.7) with 0 < ϕ1,i ≤ m, 0 < ϕ1,i ≤ n for i = 1, . . . ,i. inspired by [4, 7] we prove that (uh,vh) quenches in a finite time and (u ′ h,v ′ h) blows up at quenching node. theorem 3.1. the solution (uh,vh) of (1.5)-(1.7) quenches in a finite time with the only quenching node i =1. proof. integrating (1.5) in time we find ui(t) −ui(0) = ∫ t 0 δ2ui(τ) + bi(u −m i (τ) + v −p i (τ))dτ summing up the above inequality we get i∑ i=1 hui(t) = i∑ i=1 hui(0) + ∫ t 0 ui−1(τ) −ui(τ) h + u2(τ) −u1(τ) h − 2(u−m1 (τ) + v −p 1 (τ))dτ. from (1.5) we have h 2 ui(t) − h 2 ui(0) = ∫ t 0 ui−1(τ) −ui(τ) h dτ, and h 2 u1(t) − h 2 u1(0) = ∫ t 0 u2(τ) −u1(τ) h − (u−m1 (τ) + v −p 1 (τ))dτ. thus h 2 ui(t) + i−1∑ i=2 hui(t) + h 2 u1(t) = h 2 ui(0) + i−1∑ i=2 hui(0) + h 2 u1(0) − ∫ t 0 u−m1 (τ) + v −p 1 (τ)dτ, therefore h 2 ui(t) + i−1∑ i=2 hui(t) + h 2 u1(t) ≤ m − (m−m + n−p)t. int. j. anal. appl. 17 (6) (2019) 1041 proceeding as before, we find that h 2 vi(t) + i−1∑ i=2 hvi(t) + h 2 v1(t) ≤ n − (m−q + n−n)t, which yield a contradiction because uh and vh are positive for all times. then there exists 0 < th < ∞ such that lim t→t− min{u1(t),v1(t)} = 0+. to show i = 1 is the unique quenching node. in everything that follows i ∈ {1, . . . ,i − 1} and t ∈ (0,th). set g(ui(t)) = u −m i (t), f(vi(t)) = v −p i (t), d(ui(t)) = u −q i (t), j(vi(t)) = v −n i (t), and zi(t) = ui+1(t) −ui(t) h −φi(g(ui(t)) + f(vi(t))) (3.1) wi(t) = vi+1(t) −vi(t) h −φi(d(ui(t)) + j(vi(t))) (3.2) where φi, δ 2φi ≥ 0, δ+φi ≤ 0, φi = 0, φ1 = 1, φi(g(ui(0))+f(vi(0))) ≤ δ+ui(0) and φi(d(ui(0))+j(vi(0))) ≤ δ+vi(0). by means of taylor expansions we have δ2(φik(ji(t))) = φik ′(ji(t))δ 2ji(t) + k(ji(t))δ 2φi + k ′(ji(t))δ +φiδ +ji(t) + k ′(ji(t))δ −φiδ −ji(t) +φi (δ+ji(t)) 2 2 k′′(ρi(t)) + φi (δ−ji(t)) 2 2 (k′′(λi(t)), i = 2, . . . ,i − 1, δ2(φ1k(j1(t))) = φ1k ′(j1(t))δ 2j1(t) + k(j1(t))δ 2φ1 + 2k ′(j1(t))δ +φ1δ +j1(t) +φ1(δ +j1(t)) 2k′′(ρ1(t)). if we use the fact that ji, δ +ji(t) and δ 2ji(t) are nonnegative and the hypothesis on φh, we arrive at δ2(φik(ji(t)) ≥ φik′(ji(t))δ2ji(t), i = 1, . . . ,i − 1. (3.3) by using (3.3) we can get z′i(t) − δ 2zi(t) ≥ bi h (g(ui) + f(vi)) + biφig ′(ui) (g(ui) + f(vi)) + biφif ′(ui) (d(ui) + j(vi)) . the above inequalities implies that z′i(t) − δ 2zi(t) + big ′(ui(t))zi(t) + bif ′(vi(t))wi(t) ≥ bi[ 1 h (g(ui(t)) + f(vi(t))) +f′(ui(t))(d(ui(t)) + j(vi(t))) + g ′(ui(t))(g(ui(t)) + f(vi(t))]. we obtain z′i(t) − δ 2zi(t) + big ′(ui(t))zi(t) + bif ′(vi(t))wi(t) ≥ 0, int. j. anal. appl. 17 (6) (2019) 1042 for the parameter h small enough. thus we have z′i(t) −δ 2zi(t) + big ′(ui(t))zi(t) + bif ′(vi(t))wi(t) ≥ 0, w ′i (t) − δ 2wi(t) + bid ′(ui(t))zi(t) + bij ′(vi(t))wi(t) ≥ 0, zi(0) ≥ 0,wi(0) ≥ 0. using the lemma 2.1 we have zi(t) ≥ 0 and wi(t) ≥ 0, for i = 1, . . . ,i − 1 and t ∈ (0,th). this implies that ui+1(t) −ui(t) h ≥ φi(g(ui(t)) + f(vi(t))) ≥ 1 2 ( 1 mm + 1 np ) for i = 1, . . . ,j, with φj = 1 2 , where j ∈{2, . . . ,i − 1}. thus by summing we obtain ui(t) ≥ u1 + (i− 1)h 2 ( 1 mm + 1 np ) ≥ (i− 1)h 2 ( 1 mm + 1 np ) whenever i > 1. the same happens for vh. � theorem 3.2. if lim t→t− h u1(t) = 0 + ( lim t→t− h v1(t) = 0 + ) , then u′h(t) blows up (v ′ h(t) blows up). proof. suppose u′h(t) is bounded. then, there exists a negative constant m such that u ′ h(t) > m. we have i−1∑ i=1 i∑ j=1 h2u′j(t) > i−1∑ i=1 i∑ j=1 h2m. i−1∑ i=1 i∑ j=1 h2m = i−1∑ i=1 ih2m = m 2 + hm 2 . i−1∑ i=1 i∑ j=1 h2u′j(t) = i−1∑ i=2   i∑ j=2 h2u′j(t) + h 2u′1(t)   + h2u′1(t) from (1.5) we arrive at i−1∑ i=1 i∑ j=1 h2u′j(t) = ui(t) −u1(t) − ( v −p 1 (t) + u −m 1 (t) ) + h 2 u′1(t) and lemma 2.3 we arrive at ui(t) −u1(t) − ( v1(t) −p + u1(t) −m) > m 2 + hm. as t → t−h , the left-hand side tends to infinity while the right-side is finite. this contradiction shows that u′h blows up. � int. j. anal. appl. 17 (6) (2019) 1043 4. simultaneous vs. non-simultaneous quenching in this section we consider (uh,vh) the solution of (1.5)-(1.7) with h fixed, and we give some sufficient conditions for the existence of simultaneous and non-simultaneous quenching. theorem 4.1. if uh quenches and vh does not quench in (1.5)-(1.7) then q < m + 1. proof. as vh does not quench, by (1.5) there exists c > 0 such that u′1(t) ≥−cu −m 1 (t), integrating this inequality from t to th, we get u1(t) ≤ c(th − t) 1 m+1 , where c = ((m + 1)c) 1/(m+1) . (4.1) by using (4.1) and (1.6) , we obtain v ′1 (t) ≤ δ 2v1(t) − b1 ( v −n1 (t) + c(th − t) − q m+1 ) . thus v1(th) ≤ c1 −c ∫th 0 (th − t)− q m+1 dt. we can see that this integral diverges if q ≥ m + 1, which is a contradiction and the proof is completed. � corollary 4.1. simultaneous quenching happens if q ≥ m + 1, p ≥ n + 1. lemma 4.1. let (uh,vh) be the solution of (1.5)-(1.7). assume that uh quenches at time th (vh quenches at time th) and δ2ϕ1,i − bi ( ϕ−m1,i + ϕ −p 2,i ) + c ( ϕ−m1,i + ϕ −p 2,i ) ≤ 0, (4.2) δ2ϕ2,i − bi ( ϕ −q 1,i + ϕ −n 2,i ) + c ( ϕ −q 1,i + ϕ −n 2,i ) ≤ 0. (4.3) then there exists a positive constant c such that for t ∈ (0,th) u1(t) m+1 c(m + 1) ≥ th − t ( v1(t) n+1 c(n + 1) ≥ th − t ) , u1(t) ≥ c(th − t) 1 m+1 ( v1(t) ≥ c(th − t) 1 n+1 ) . (4.4) proof. set for i = 1, . . . ,i, t ∈ (0,th), zi(t) = u ′ i(t) + c ( u−mi (t) + v −p i (t) ) and wi(t) = v ′ i (t) + c ( u −q i (t) + v −n i (t) ) . a straightforward calculation gives z′i(t) − δ 2zi(t) + αi(t)zi(t) + βi(t)wi(t) ≤ 0, i = 1, . . . ,i, t ∈ (0,th), w ′i (t) − δ 2wi(t) + ai(t)zi(t) + bi(t)wi(t) ≤ 0, i = 1, . . . ,i, t ∈ (0,th), zi(0) ≤ 0, wi(0) ≤ 0, i = 1, . . . ,i. int. j. anal. appl. 17 (6) (2019) 1044 by virtue of lemma 2.1 zi(t) ≤ 0, wi(t) ≤ 0, i = 1, . . . ,i, t ∈ (0,th). thus we get u′i(t) ≤−cu −m i (t) and v ′ i (t) ≤−cv −n i (t), i = 1, . . . ,i, t ∈ (0,th). (4.5) assume that uh quenches (vh quenches), integrating (4.5) from t to th, we arrive at u1(t) m+1 c(m + 1) ≥ th − t ( v1(t) n+1 c(n + 1) ≥ th − t ) , which implies u1(t) ≥ c(th − t) 1 m+1 ( v1(t) ≥ c(th − t) 1 n+1 ) . � theorem 4.2. if p < n + 1, then there exist initial data such that vh quenches but uh doesn’t. proof. we argue by contradiction. assuming that uh and vh quench simultaneously at time th for any initial data. we have∫ t 0 u′1(s) ds ≥ ∫ th 0 u′1(s) ds = 2 h2 ∫ th 0 u2(s) −u1(s) ds− 2 h ∫ th 0 u−m1 (s) + v −p 1 (s) ds by using the lemma 4.1, we obtain u1(t) ≥ u1(0) + 2 h2 ∫ th 0 u2(s) − u1(s) ds− 2c h ∫ th 0 (th −s)− m m+1 + (th −s)− p n+1 ds as p < n + 1 this integral is converged and u1(t) ≥ c1 −c2t 1 m+1 h −c3t n+1−p n+1 h , with c1, c2, c3 > 0. by summation of (1.6) we observe that − h 2 v ′1 (t) − h 2 v ′i (t) − i−1∑ i=2 hv ′i (t) = u −q 1 (t) + v −n 1 (t), − h 2 v ′1 (t) − h 2 v ′i (t) − i−1∑ i=2 hv ′i (t) ≥ u −q 1 (0) + v −n 1 (0) (4.6) integrate (4.6) from 0 to th, we can obtain vi(0) ( u −q 1 (0) + v −n 1 (0) )−1 ≥ th, then if th is small enough (depending on uh(0) and vh(0)), u1(th) ≥ c0 > 0. we have a contradiction with the hypothesis that uh quenches and the result is obtained as desired. � int. j. anal. appl. 17 (6) (2019) 1045 theorem 4.3. if q ≤ n(m+1) n+1 and p ≥ n + 1 (p ≤ m(n+1) m+1 and q ≥ m + 1) then uh (vh) quenches alone under any positive initial data. proof. assume that there exists initial data such that uh and vh quench simultaneously at time th. without lost of generality, we can suppose that this initial data satisfies (4.2)-(4.3). according to (1.6) v ′1 (t) = δ 2v1(t) − b1(u −q 1 (t) + v −n 1 (t)), v ′1 (t) ≥−b1(u −q 1 (t) + v −n 1 (t)), v1(t) ≤ b1 ∫ th t u −q 1 (s) + v −n 1 (s) ds. from lemma 4.1 we know u1(t) ≥ c(th−t) 1 m+1 , v1(t) ≥ c(th−t) 1 n+1 , moreover q ≤ n(m+1) n+1 . hence there exists c′ > 0 such that v1(t) ≤ c′(th − t) 1 n+1 . let us consider (1.5) u′1(t) = δ 2u1(t) − b1(u−m1 (t) + v −p 1 (t)), u′1(t) ≤ δ 2u1(t) − b1v −p 1 (t), u′1(t) ≤ δ 2u1(t) − b1c′−p(th − t)− p n+1 . integrating both sides from 0 to th, we obtain −u1(0) ≤ c1 − c2 ∫ th 0 (th − t)− p n+1 dt. we can see that the integral diverges if p ≥ n + 1, which is a contradiction. the result is obtained. � remark 4.1. let (uh,vh) be the solution of (1.5)-(1.7) such that the initial data satisfies (4.2)-(4.3). we can see of the lemma 4.1 and the proof of theorem 4.3 that if uh (vh) quenches at time th, then u1(t) ∼ (th − t) 1 m+1 ( v1(t) ∼ (th − t) 1 n+1 ) for t close enough to th. 5. convergence of the semidiscrete quenching time in this section, we study the convergence of the semidiscrete quenching time. now we will show that for each fixed time interval [0,t] where (u,v) is defined, the solution (uh,vh) of (1.5)-(1.7) approximates (u,v) when the mesh parameter h goes to zero. we denote uh(t) = (u(x1, t), . . . ,u(xi, t)) t , vh(t) = (v(x1, t), . . . ,v(xi, t)) t , ‖uh(t)‖∞ = max 1≤i≤i |ui(t)|, ‖uh(t)‖inf = min 1≤i≤i |ui(t)|. theorem 5.1. assume that the problem (1.1)-(1.4) has solution (u,v) ∈ ( c4,1 ([0, 1] × [0,t∗]) )2 and the initial data (ϕ1,h,ϕ2,h) at (1.5)-(1.7) verifies ‖ϕ1,h −uh(0)‖∞ = o(1), ‖ϕ2,h −vh(0)‖∞ = o(1) h → 0. (5.1) int. j. anal. appl. 17 (6) (2019) 1046 then, for h small enough, the semidiscrete problem (1.5)-(1.7) has a unique solution (uh,vh) ∈( c1 ( [0,t∗], ri ))2 such that max t∈[0,t∗] ‖uh(t) −uh(t)‖∞ = o(‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + h2), as h → 0, max t∈[0,t∗] ‖vh(t) −vh(t)‖∞ = o(‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + h2), as h → 0. proof. let σ > 0 be such that (‖u‖∞,‖v‖∞) < σ, t ∈ [0,t∗]. (5.2) then the problem (1.5)-(1.7) has for each h, a unique solution (uh,vh) ∈ ( c1 ( [0,t∗], ri ))2 . let t(h) ≤ t∗ be the greatest value of t > 0 such that max{‖uh(t) −uh(t)‖∞,‖vh(t) −vh(t)‖∞} < 1. (5.3) the relation (5.1) implies t(h) > 0 for h small enough. using the triangle inequality, we obtain ‖uh(t)‖∞ ≤ 1 + σ and, ‖vh(t)‖∞ ≤ 1 + σ for t ∈ (0, t(h)). (5.4) let (e1,h,e2,h)(t) = (uh−uh,vh−vh)(t), ∀t ∈ [0,t∗] be the discretization error. these error functions verify e′1,i(t) = δ 2e1,i(t) + mbi(θi(t)) −m−1e1,i(t) + pbi(θi(t)) −p−1e2,i(t) + o(h 2), e′2,i(t) = δ 2e2,i(t) + qbi(θi(t)) −q−1e1,i(t) + nbi(θi(t)) −n−1e2,i(t) + o(h 2), where θi(t) and θi(t) lie, respectively, between ui(t) and u(xi, t), and between vi(t) and v(xi, t), for i ∈ {1, . . . ,i}. using (5.2) and (5.4), there exist k and l positive constants such that e′1,i(t) ≤ δ 2e1,i(t) + bil|e1,i(t)| + bil|e2,i(t)| + kh2, e′2,i(t) ≤ δ 2e2,i(t) + bil|e1,i(t)| + bil|e2,i(t)| + kh2 let (z,w) ∈ ( c4,1 ([0, 1], [0,t∗]) )2 be such that z(x,t) = ( ‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + qh2 ) e(m+2)t−(1−x) 2 and w = z, ∀(x,t) ∈ [0, 1] × [0,t∗], with m, q positive constants. we can prove by the lemma 2.2 that |e1,i(t)| < z(xi, t), |e2,i(t)| < w(xi, t), 1 ≤ i ≤ i, for t ∈ (0, t(h)). we deduce that ‖uh(t) −uh(t)‖∞ ≤ ( ‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + qh2 ) e(m+2)t, ‖vh(t) −vh(t)‖∞ ≤ ( ‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + qh2 ) e(m+2)t, for t ∈ (0, t(h)). suppose that t∗ > t(h) from (5.3), we obtain 1 = ‖uh(t(h)) −uh(t(h))‖∞ ≤ ( ‖ϕ1,h −uh(0)‖∞ + ‖ϕ2,h −vh(0)‖∞ + qh2 ) e(m+2)t. int. j. anal. appl. 17 (6) (2019) 1047 since the term on the right hand side of the above inequality goes to zero as h tends to zero, we deduce that, 1 ≤ 0, which is impossible. hence we have t(h) = t∗, and the proof is completed. � theorem 5.2. let (u,v) ∈ ( c4,1([0, 1] × [0,t)) )2 be solution of (1.1)-(1.4) with quenches time t and the initial data at (1.5)-(1.7) satisfies (4.2)-(4.3) and (5.1). then the solution (uh,vh) of (1.5)-(1.7) quenches in a finite time th and we have lim h→0 th = t. proof. from theorem 3.1, (uh,vh) quenches in a finite time th. assume that uh quenches. set ε > 0. there exists η > 0 such that y1+m c(m + 1) ≤ ε 2 , 0 ≤ y ≤ η. (5.5) there exists a time t0 ∈ (t − ε 2 ,t) such that 0 < |u(xi, t)| ≤ η2 , for i = 1, . . . ,i, t ∈ [t0,t). setting t1 = t0+t 2 , it is not hard to see that 0 < ‖u(xi, t)‖inf , for t ∈ [0,t1]. from theorem 5.1, it follows that for h sufficiently small ‖uh(t) −uh(t)‖∞ ≤ η 2 . applying the triangle inequality, we get ‖uh(t1)‖inf ≤‖uh(t1) −uh(t1)‖∞ + ‖uh(t1)‖inf ≤ η. since uh quenches, we can deduce from lemma 4.1 and (5.5) that |th −t | ≤ |th −t1| + |t1 −t | ≤ ‖uh(t1)‖1+minf c(m + 1) + ε 2 ≤ ε. the case where vh quenches is analogous. � 6. numerical experiments in this section, we present some numerical approximations to the quenching time of (1.5)-(1.7) for the initial data ϕ1,i = ϕ2,i = 1 + 4 π sin ( π 2 (i− 1)h ) for i = 1, . . . ,i − 1, with different values of m, n, p and q. we also consider the implicit scheme below u (k+1) i −u (k) i ∆tkh = δ2u (k+1) i − bi (( u (k) i )−m + ( v (k) i )−p) , 1 ≤ i ≤ i, v (k+1) i −v (k) i ∆tkh = δ2v (k+1) i − bi (( u (k) i )−q + ( v (k) i )−n) , 1 ≤ i ≤ i, u (0) i = ϕ1,i, v (0) i = ϕ2,i 1 ≤ i ≤ i, where k ≥ 0, ∆tkh = h 2 min { ‖u(k)h ‖ m+1 inf ,‖u (k) h ‖ q+1 inf ,‖v (k) h ‖ p+1 inf ,‖v (k) h ‖ n+1 inf } . int. j. anal. appl. 17 (6) (2019) 1048 definition 6.1. we say that the discrete solution (u (k) h ,v (k) h ) of the implicit scheme quenches in a finite time if lim k→∞ inf{‖u(k)h ‖inf,‖v (k) h ‖inf} = 0 and the series ∑+∞ k=0 ∆t k h converges. the quantity t k h = ∑k−1 j=0 ∆t j h is called the numerical quenching time of the solution (u (k) h ,v (k) h ) and th = ∑+∞ k=0 ∆t k h is called the numerical quenching time of the solution (uh,vh). in tables 1, 2 and 3, in rows, we present the numerical quenching times, the numbers of iterations and the orders of the approximations corresponding to meshes of 16, 32, 64, 128, 256, 512, 1024. we take for the numerical quenching time th = ∑+∞ k=0 ∆t k h which is computed at the first time when ∆t k h = |t k+1 h −t k h| ≤ 10−16. the order(s) of the method is computed from s = log((t4h−t2h)/(t2h−th)) log(2) , where h = 1/(i − 1). table 1. numerical quenching times obtained with the implicit euler method for m = 0.5, p = 1, q = 2, n = 0.5. i th k s 16 0.15390794 34896 32 0.14878519 48454 64 0.14737661 66676 1.86 128 0.14699264 92814 1.87 256 0.14688843 137205 1.88 512 0.14686025 238258 1.89 1024 0.14685267 545941 1.89 table 2. numerical quenching times obtained with the explicit euler method for m = 1, p = 2.5, q = 0.5, n = 1. i th k s 16 0.13630655 168 32 0.13147195 484 64 0.13016571 1540 1.89 128 0.12981187 5384 1.88 256 0.12971578 20063 1.88 512 0.12968969 77515 1.88 1024 0.12968263 305050 1.88 figure 1. on the left, no quenching of uh and on the right, quenching of vh for m = 0.5, p = 1, q = 2, n = 0.5. int. j. anal. appl. 17 (6) (2019) 1049 table 3. numerical quenching times obtained with the implicit euler method for m = 0.3, p = 2, q = 2, n = 0.3. i th k s 16 0.12862938 127 32 0.12271047 372 64 0.12106075 1213 1.84 128 0.12060846 4323 1.87 256 0.12048544 16309 1.88 512 0.12045217 63432 1.89 1024 0.12044321 250457 1.89 figure 2. on the left, quenching of uh and on the right, no quenching of vh for m = 1, p = 2.5, q = 0.5, n = 1. figure 3. on the left, quenching of uh and on the right, quenching of vh for m = 0.3, p = 2, q = 2, n = 0.3. remark 6.1. the various tables of our numerical results show that there is a relationship between the quenching time and flows on the boundaries. if we consider the problem (1.5)-(1.7) in the case where the initial data ϕ2,i = ϕ1,i = 1 + 4 π sin ( π 2 (i− 1)h ) , i = 1, . . . ,i, from figures 1-3, we observe respectively the quenching of (uh,vh). from tables 1-3, we observe the convergence of quenching time th of the solution of (1.5)-(1.7), since the rate of convergence is near 2. this result does not surprise us because of the result int. j. anal. appl. 17 (6) (2019) 1050 established in the previous section. moreover we can see that of the figure 1, vh quenches while uh doesn’t when p < n + 1, of the figure 2, uh quenches while vh doesn’t when q ≤ n(m+1) n+1 and p ≥ n + 1 and of the figure 3, uh and vh quench simultaneously when p ≥ n + 1 and q ≥ m + 1. these numerical results are in fact consistent with the corollary 4.1, theorem 4.2 and theorem 4.3. conclusion in this work, we proposed a semi-discrete scheme, based on finite difference method in space for system of heat equations, coupled by nonlinear boundary flux. the stability and the convergence of semi-discrete scheme are proved respectively. under some conditions the semidiscrete scheme reproduces well the conditions for the quenching, quenching set and simultaneous and non-simultaneous quenching. the analysis in this paper can be extended to more general to some systems of nonlinear parabolic equations. references [1] k.a. adou, k.a. touré, a. coulibaly, numerical study of the blow-up time of positive solutions of semilinear heat equations, far east j. appl. math., 4 (2018), 91-308. [2] t. k. boni, extinction for discretizations of some semilinear parabolic equations, c. r. acad. sci. paris, 333 (2001), 79-800. [3] t. k. boni, h. nachid, d. nabongo, quenching time of semilinear heat equations, miskolc math. notes, 11 (2010), 27-41. [4] c.y. chan, s.i. yuen, parabolic problems with nonlinear absorptions and releases at the boundaries, appl. math. comput., 121 (2001), 203-209. [5] k.b. edja, k.a. touré, b. j.-c. koua, numerical blow-up for a heat equation with nonlinear boundary conditions, j. math. res., 10 (2018), 119-128. [6] r. ferreira, a. de pablo, m.p. llanos, j.d. rossi, incomplete quenching in a system of heat equations coupled at the boundary, j. math. anal. appl., 346 (2008), 1145-154. [7] r.h. ji, c.y. quc, l.d. wang, simultaneous and non-simultaneous quenching for coupled parabolic system, appl. anal., 94 (2015), 233-250. [8] h. kawarada, on solutions of initial-boundary problem for ut = uxx + 1 1−u , publ. rims, kyoto univ., 10 (1975), 729-736. [9] h.a. levine, the quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions, siam j. math. anal., 4 (1983), 1139-1153. [10] h.a. levine, j.t. montgomer the quenching of solutions of some nonlinear parabolic equations, siam j. math. anal., 11 (1980), 842-847. [11] liang k.w, lin ap, tan rce. numerical solution of quenching problems using mesh-dependent variable temporal steps, appl. numer. math., 57 (2007), 791-800. [12] d. nabongo, t.k. boni, quenching for semidiscretizations of a heat equation with a singular boundary condition, asymptotic anal., 59 (2008), 27-38. [13] k.c. n’dri, k.a. touré, g. yoro, numerical blow-up time for a parabolic equation with nonlinear boundary conditions, int. j. numer. methods appl., 17 (2018), 141-160. [14] h. pei, z. li, quenching for a parabolic system with general singular terms, j. nonlinear sci. appl., 7 (2016), 1-10. [15] b. seluk, quenching behavior of a semilinear reaction-diffusion system with singular boundary condition, turk. j. math., 40 (2016), 166-180. int. j. anal. appl. 17 (6) (2019) 1051 [16] m. taha, k. toure, e. mensah, numerical approximation of the blow-up time for a semilinear parabolic equation with nonlinear boundary equation, far east j. appl. math., 60 (2012), 125-167. [17] s.n. zheng, x.f. song, quenching rates for heat equations with coupled nonlinear boundary flux, sci. china ser. a., 51 (2008), 1631-1643. 1. introduction 2. properties of the semidiscrete scheme 3. quenching and blow-up 4. simultaneous vs. non-simultaneous quenching 5. convergence of the semidiscrete quenching time 6. numerical experiments conclusion references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 27-32 http://www.etamaths.com a new chaotic attractor with quadratic exponential nonlinear term from chen’s attractor iftikhar ahmed∗, chunlai mu, and fuchen zhang abstract. in this paper a new three-dimensional chaotic system is proposed, which relies on a nonlinear exponential term and a nonlinear quadratic cross term necessary for folding trajectories. basic dynamical characteristics of the new system are analyzed. compared with the chen system, the equilibrium points of the new system does not contain the origin, and has a greater positive lyapunov index, can produce more complex shaped chaotic attractor. 1. introduction since lorenz found the first chaotic attractor in a three first order autonomous ordinary differential equations (odes) when he studied the atmospheric convection in 1963 [1], many new three dimension (3d) chaotic attractors have been proposed in the last three decades, such as the rossler system [2], the chen system [3], the lü system [4], the liu system [5], and the generalized lorenz system family [6]. new chaotic system can also be achieved by adding or changing the linear/nonlinear term of existing chaotic system. the nonlinear term of system is normally the product of variables at different state. when the system contains nonlinear terms of the exponential function whether there will be chaos phenomenon, yet need research. wei and yang [7] revealed a 3d autonomous chaotic attractor with a nonlinear term in the form of exponential function at the right-hand side in odes as ẋ = ay−ax, ẏ = −by + mxz, ż = n−exy, where the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters was investigated. recently, liang and zhonglin [8] discussed the basic dynamic characteristics of the new chaotic system containing a nonlinear term of exponential function instead of the nonlinear term in lü system. in this paper, a new chaotic system containing a nonlinear term of exponential function instead of the nonlinear term in chen’s system is proposed. the dynamic characteristics and simulation show clearly that proposed system is chaotic same as lorenz chaotic attractor and others, but its topological structure is different from all existing chaotic attractors. 2000 mathematics subject classification. 34c28. key words and phrases. new chaotic attractor, chen’s attractor; quadratic exponential nonlinear term. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 27 28 iftikhar ahmed, chunlai mu, and fuchen zhang this paper is organized as follows. section 2 presents the design of a new chaotic system. section 3 outlines the basic properties of the new system. finally, conclusions are given in section 4. 2. design of a new chaotic system chen’s chaotic system is given by  ẋ = a(y −x) ẏ = (c−a)x + cy −xz ż = xy − bz (1) when a = 35, b = 3, c = 28 ,the lyapunov exponents of system (1) are found to be le1=1.997, le2=0.0002 , le3=-11.9943. the fractal dimension of system (1) is d1=2.1665 . the chaotic attractor is shown in figure 1. system (i) is still in a state of chaos. in system (i), we use exy instead of xy in the third equation, and get a new system (2) :   ẋ = a(y −x), ẏ = (c−a)x + cy −xz, ż = exy − bz, (2) when a = 35, b = 3, c = 28 the lyapunov exponents of system (2) are found to be lei =3.8065, le2=0.010904 , le3=-13.817. there is a positive lyapunov exponent, which is much larger than that of the system (1). the lyapunov dimension of system (2) is: dl = j + 1 |lej+1| j∑ i=1 lei = 2 + le1 + le2 |le3| = 2 + 3.806 + 0.0109 |−13.817| = 2.2762 0 50 100 150 200 250 300 −20 −15 −10 −5 0 5 10 dynamics of lyapunov exponents l 1 =3.8065 l 2 =0.010904 l 3 =−13.817 time l ya p u n o v e xp o n e n ts figure 1. dynamics of lyapunov exponents of system (2) the lyapunov dimensions of the system are fractional. the chaotic attractor is shown in fig. 3. system (2) is also at chaos status. a new chaotic attractor 29 −25 −20 −15 −10 −5 0 5 10 15 20 25 −30 −20 −10 0 10 20 30 x(t) y( t) −25 −20 −15 −10 −5 0 5 10 15 20 25 5 10 15 20 25 30 35 40 45 50 x(t) z( t) −30 −20 −10 0 10 20 30 5 10 15 20 25 30 35 40 45 50 y(t) z( t) figure 2. phase portraits of chaotic attractors of system (i) −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 x(t) y( t) −3 −2 −1 0 1 2 3 0 10 20 30 40 50 60 70 80 90 x(t) z( t) −4 −3 −2 −1 0 1 2 3 4 0 10 20 30 40 50 60 70 80 90 y(t) z( t) figure 3. phase portraits of chaotic attractors of system (2) 30 iftikhar ahmed, chunlai mu, and fuchen zhang 3. basic properties of the new system a. equilibria for equilibrium points we take   a(y −x) = 0, (c−a)x + cy −xz = 0, exy − bz = 0, (3) when a = 35, b = 3, c = 28, system (2) has three equilibrium points: e1(0, 0, 0.333), e2(2.035, 2.035, 21), e3(−2.035,−2.035, 21). the equilibrium point of system (2) does not contain the origin. the jacobian matrix of system (2) is given by j =   −a a 0c−a−z c −x yexy xexy −b  (4) for the equilibrium pointe1 = (0, 0, 0.333) , system (2) has three eigenvalues: λ1 = −3, λ2 = 23.835, λ3 = −30.835. eigenvalues is not all for the positive or negative, according to the routh-hurwitz , e1 = (0, 0, 0.333) is unstable saddle node. for the equilibrium points e2 = (2.035, 2.035, 21) and e3 = (−2.035,−2.035, 21) , system (2) has same eigenvalues: λ1 = −26.053, λ2,3 = 8.026 ± 25.203i. λ1 is a negative real root and λ2,3 are a pair of conjugate roots with positive real part. so e2 = (2.035, 2.035, 21) and e3 = (−2.035,−2.035, 21),are unstable saddle-focus points. b. symmetry and invariance it is easy to see the invariance of system under the coordinate transformation (x,y,z) → (−x,−y,z) i.e., the system has rotation symmetry around the z -axis. −4 −2 0 2 4 −4 −2 0 2 4 0 20 40 60 80 100 x(t) 3d view of new attractor y(t) z( t) figure 4. 3d view of new chaotic system (2) a new chaotic attractor 31 c. dissipativity the three lyapunov exponents and the divergence of the vector field is: 3∑ i=1 lei = ∆v = ∂ẋ ∂x + ∂ẏ ∂y + ∂ż ∂z = −a + c− b = f,(5) where 3∑ i=1 lei denote the three lyapunov exponents of the system. note thatf = −a + c− b = −10 is a negative value, so the system is a dissipative system and an exponential rate is: dv dt = ef = e−10(6) from(6), it can be seen that a volume element v0 is contracted by the flow into a volume element v0e −10t in time t . this means that each volume containing the system trajectory shrinks to zero as t → ∞ at an exponential rate of −10 . therefore, all system orbits are ultimately confined to a specific subset having zero volume and the asymptotic motion settles onto an attractor . d. sensitivity to initial conditions figure 4 shows that the evolution of the chaos trajectories is very sensitive to initial conditions. the initial values of the system are set to [2, 2, 1]t for the solid line and [2.01, 2, 1]t for the dashed line. figure 5. sensitivity of system(2) to initial conditions 4. conclusions a new chaotic system is proposed in this paper, which has exponential term instead of the nonlinear term of the chen’s system. some basic properties of the system have been investigated. compare with chen’s system, the new system has greater chaos interval and much larger lyapunov exponent. its equilibrium point does not contain the origin. even though more important analysis of the system like chaos control, boundedness, and synchronization, will take into account in the future work. 32 iftikhar ahmed, chunlai mu, and fuchen zhang references [1] e. n. lorenz, deterministic non-periodic flow? j.atmos. sci., vol. 20, no. 1, pp. 130-141, 1963. [2] o. e . rossler, an equation for continuous chaos?phys. lett. a, vol. 57, no. 5, pp. 397-399, 1976. [3] g. chen, t. ueta, yet another chaotic attractor? internat. j. bifur. chaos, vol. 9, no. 7, pp. 1465-1457, 1999. [4] j. l? g. chen, a new chaotic attractor conined? internat. j. bifur. chaos, vol. 12, no. 3, pp. 659-662, 2002. [5] c. liu, t. liu, l. liu, k. liu, a new chaotic attractor? chaos, solitons fractals, vol. 22, no. 5, pp. 1031-1038, 2004. [6] s. celikovsky, g. chen, on the generalized lorenz canonical form? chaos, solitons fractals, vol. 26, no. 5, pp. 1271-1276, 2005. [7] z. wei, q. yang, dynamical analysis of a new autonomous 3-d chaotic system only with stable equilibria? nonlinear anal.: rwa, vol. 12, no. 1, pp. 106-118, 2011. [8] z.liang, w. zhonglin, ”design and realization of a new chaotic system”, sensor netw. sec. technol. and prc. commun. syst., 2013 international conference on, 10.1109/sns-pcs.2013.6553844, pp.101-104, 2013. [9] li, x.f., chlouverakis, k.e., xu, d.l.: nonlinear dynamics and circuit realization of a new chaotic flow: a variant of lorenz, chen and lu. nonlinear anal. 10, 2357-2368 (2009) . [10] hahn, w., the stability of motion. springer, new york (1967). college of mathematics and statistics,chongqing university, chongqing 401331, pr china ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 154-163 http://www.etamaths.com index formulas for countably ϕ−set contraction h. salahifard, s. m. vaezpour∗ abstract. in this paper, we study the index formulas for a class of bounded linear operators, namely ϕ−set contractions, acting on a banach space and we discuss some application of this class of operators to the theory of bifurcation points. in particular our results generalize and improve some recent results mentioned in the literature. 1. introduction and preliminary finding necessary and sufficient conditions for the appearance of nontrivial solutions arbitrary close to some points (called bifurcation points) of the trivial branch, with assumption of existence of a known (trivial) branch of solutions for a parametrized family of an equation, is one of the oldest problems of mathematics which have created bifurcation theory. one of the most important role in bifurcation theory is played by index formulas for suitable kind of operators. in recent years, many authors have focused on set-contractive operators and obtained a lot of valuable results (see [12, 7, 10]). nussbaum (1969) [14] developed degree theory for k-set contractive operators (0 ≤ k < 1), which was first introduced by kuratowski at 1930 [11], stuart, toland [18] and amann (1976) [2] established the index formula for k-set contractions and condensing operators. kim (2008)[10] presented an index formula for countably k-set contractive bounded linear operators in a real banach space, by using a degree theory for countably condensing operators. in this paper, we continue to study set-contractive operators and investigate the conditions under which the topological degrees can be defined for a larger class of k-set contraction, namely ϕ−set-contractive operators. moreover, we introduce generalized ϕk−set-contractive operators. it should be noted that this class of operators, as special cases, includes linear bounded operators, nonexpansive operators, completely continuous operators, kset-contractive operators, condensing operators and 1−set contractive operators. correspondingly, we can obtain some new bifurcation theorems of these operators, which improve and extend many famous theorems such as the hetzer’s theorem, nussbaum’s theorem, kim’s theorem, etc. before proceeding to the main results of this paper, we must recall some notations, definitions and theorems we shall need. a function γ : {b ⊂ x : b is bounded} → [0,∞) is said to be a measure of noncompactness on a banach space x, if it satisfies the following conditions: 2010 mathematics subject classification. 47a13, 37g10. key words and phrases. index formula; ϕ-set contractions; fredholm operators; bifurcation points; condensing map. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 154 index formula for ϕ−set contractions 155 (1)(invariance under closure and convex hull): γ( ¯cob) = γ(b), (2)(regularity): γ(b) = 0 if and only if b is relatively compact, (3)(semi-additivity): γ(b1 ∪b2) = max{γ(b1),γ(b2)}, (4)(algebraic semi-additivity): γ(b1 + b2) ≤ γ(b1) + γ(b2) (5)(semi-homogeneity): γ(αb) = |α|γ(b) for all α ∈ r, and (6)(lipschitzianity): |γ(b1)−γ(b2)| ≤ lγρ(b1,b2), where ρ denotes the hausdorff semi-metric, that is, ρ(b1,b2) = inf{� > 0 : b2 ⊂ b1 + �b̄(0, 1),b1 ⊂ b2 + �b̄(0, 1)}. the most important examples of measures of noncompactness are the kuratowski measure of noncompactness (or set measure of noncompactness) α(ω) = inf{r > 0 : xmay be covered by finitely many sets of diameter ≤ r}, and the hausdorff measure of noncompactness (or ball measure of noncompactness) β(ω) = inf{r > 0 : there exists a finite r-net forω in x}. a detailed account of theory and applications of measures of noncompactness may be found in the monographs [1, 3]. note that the kuratowski measure of noncompactness and the hausdorff measure of noncompactness have the above properties; see [1, 19]. a function ϕ : r+ → r+ is said to be a • comparison function if ϕ(0) = 0 and ϕ(t) < t for each t > 0, • semi-comparison function if ϕ(0) = 0 and ϕ(t) < kt for each t > 0 and some k ≥ 0, denote φk = {ϕ : r+ → r+,ϕ(t) < kt for t > 0, ϕ(0) = 0}, for k = 1, we denote φ1 by φ. let k ≥ 1. a continuous operator t : x → x (with respect to γ) is said to be • countably k−set contractive [10]: if γ(t(c)) ≤ kγ(c) for each countable bounded set c ⊆ x. • countably k−set contraction [10]: if γ(t(c)) ≤ kγ(c) for each countable bounded set c ⊆ x and 0 ≤ k < 1. • k−set contraction [18]: if γ(t(c)) ≤ kγ(c) for all bounded sets c ⊂ x. and 1−set contraction, if k = 1 [6]. • countably condensing [10]: if γ(t(c)) < γ(c) for each countable bounded set c ⊂ x with γ(c) > 0. • countably ϕ−set contractive : if γ(t(c)) ≤ ϕ(γ(c)) for some ϕ ∈ φk and each countable bounded set c ⊆ x. •countably ϕ−set contraction [5]: if γ(t(c)) ≤ ϕ(γ(c)) for some ϕ ∈ φ and each countable bounded set c ⊆ x. remark 1.1. clearly, every k−set contractive mapping is a ϕ−set contractive and every bounded linear operator is a k−set contractive mapping by property (6) of measure of noncompactness γ, so every bounded linear operator is a ϕ−set contractive. let t be a bounded linear operator of a banach space x into a banach space y . the null space and the range of t are denoted by n(t) and r(t), respectively. 156 salahifard and vaezpour t is said to be a fredholm operator if the dimension of n(t) is finite and the codimension of r(t) is finite. if the dimension of n(t) is finite and r(t) is closed, then t is said to be a semi-fredholm operator. in this case, i(t) = dimn(t) − codimr(t) is called the index of t . 2. main results in this section, we present a necessary condition for the existence of bifurcation points of u = λgu, where x is a banach space, u ∈ x, λ ∈ r and g : x → x is a countably generalized ϕ−set contraction. this result can be regarded as a generalization of the [9] and [8]. moreover, we extend the index formula for countably ϕ−set contractive, comutable bounded linear operators as follows: theorem 2.1. let t : x → x be a countably ϕ−set contractive bounded linear comutable operator on a real banach space x. if 1 is not an eigenvalue of t , then ind(t, 0) = (−1)ν, where ν is the sum of the multiplicities of the eigenvalues λ > 1 of t . so the results from [10] can be obtained as consequences of our result. in order to show the main theorem above, we shall need some notations, definitions and lemmas to show that the decomposition is possible for countably generalized ϕ−set contractive bounded linear operators. now let us state our main definition which determines an important class of operators including linear bounded operators, nonexpansive operators, completely continuous operators, kset-contractive operators, condensing operators and 1−set contractive operators. definition 2.2. a continuous operator t : ω → x is said to be countably generalized ϕ−set contractive if γ(tn(c)) ≤ ϕ(γ(c)) for some n ∈ n, ϕ ∈ φk, and each countable bounded set c ⊆ x, and is said to be countably generalized ϕ−set contraction if ϕ ∈ φ. the following provides two examples; one is an example of a ϕ−set contraction which is not a k-set contraction for any 0 < k < 1, the other is an example of a generalized ϕ−set contraction which is not a ϕ−set contraction. example 2.3. let x = [0, 1] ∪{2, 3, 4, ...} for the metric, let ρ(x,y) =   |x−y| ifx,y ∈ [0, 1], x + y if one of x,y 6∈ [0, 1]. it is apparent that (x,ρ) is a complete metric space define the mapping t : x → x by tx =   x− 1 2 x2 ifx,y ∈ [0, 1], x− 1 ifx ∈{2, 3, ...}. index formula for ϕ−set contractions 157 then, for x,y ∈ [0, 1] with x?y = t > 0, ρ(x,y) = (x−y)(1 − 1 2 (x + y)) ≤ t(l− 1 2 t) and, if x ∈{2, 3, 4, ...} with x > y, then ρ(x,y) = tx + ty < x− 1 + y = ρ(x,y) − 1. thus, if we define ψ(t) by ψ(t) =   t− 1 2 t2 ift ∈ [0, 1], t− 1 ift > 1. then ψ ∈ φ and t is a ϕ−set contraction because it is easy to prove that every ϕ−contraction mapping is a ϕ−set contraction with respect to the kuratowskii measure of noncompactness. however, as n →∞, ρ(tn, 0)/ρ(n, 0) → 1 so there can be no 0 ≤ k < 1 for which t is a k−set contraction. example 2.4. let x = {1}∪{2n, 3n : n ∈ n} is equipped with the discrete metric and t : x → x be defined by t(x) =   {1} ifx = 1, {3(2n + 1)} ifx = 2n, {1} ifx = 3n and n is odd. if α is the kuratowski measure of noncompactness, then α(t2x) = 0. therefore, t is a generalized ϕ−set contraction. but if c = {2n : n ∈ n}, then α(t(c)) = α(c) = 1 and so t could not be a ϕ−set contraction since ϕ(1) would have to be one. lemma 2.5. let x be a real or complex banach space and t : x → x a bounded linear operator which is countably generalized ϕ−set contraction. let i denote the identity operator in x. then for any relatively compact set m ⊂ x and for any bounded countable set c ⊂ x, m1 = {x ∈ c : x−tx ∈ m} is relatively compact. proof. let m be a relatively compact set in x and c a bounded countable set in x and m1 = {x ∈ c : x − tx ∈ m}. we will show that γ(m1) = 0. suppose that x ∈ m1, so that x = tx + z for some z ∈ m. substituting for x on the right, x = t2x + tz + z, and continuing in this way we find x = tnx + (σn−1j=0 t j)z. if we write m2 = (σ n−1 j=0 t j)(m), the set m2 is relatively compact because it is the continuous image of a relatively compact set. furthermore, the above equality implies that m1 ⊂ tn(m1) +m2, so that γ(m1) ≤ γ(tnm1). since t is generalized countably ϕ−set contraction we have γ(m1) ≤ ϕ(γ(m1)). since ϕ ∈ φ, we have ϕ(γ(m1)) < γ(m1), which yields a contradiction. it follows that γ(m1) = 0 and thus m1 is relatively compact. � 158 salahifard and vaezpour proposition 2.6. let x be a real or complex banach space and t : x → x a bounded linear operator which is countably generalized ϕ−set contraction, then i−t is a semi-fredholm operator. proof. we must show that, the null space of i − t is finite dimensional and the range of i −t is closed in x. but it follows by previous lemma and proposition 2.1 in [10]. � remark 2.7. let x be a real or complex banach space and t : x → x a bounded linear operator which is countably generalized ϕ−set contraction, then i −λt is a semi-fredholm operator for any λ ∈ [0, 1], since γ((λt)n(c)) = λnγ(tn(c)) ≤ λnϕ(γ(c)) ≤ ϕ(γ(c)), so λt is a countably generalized ϕ−set contraction too and by last proposition i −λt is a semi-fredholm operator. proposition 2.8. let x be a real or complex banach space and t : x → x a bounded linear operator which is countably generalized ϕ−set contraction, then i−t is a fredholm operator of index zero. proof. by remark i −λt is a semi-fredholm operator for any λ ∈ [0, 1]. since the index for semi-fredholm operators is constant on its connected components, and {i −λt : λ ∈ [0, 1]} is connected, we have i(i −t) = i(i) = 0. therefore, i −t is a fredholm operator of index zero. � proposition 2.9. let x be a real or complex banach space and t : x → x a bounded linear operator, so by remark 1.1 we have (2.1) γ(tn(c)) ≤ ϕ(γ(c)), for some n ∈ n and some ϕ ∈ φk. let µ ∈ r be such that µn < k−1 for the same n and k in 2.1. then i −µt is a fredholm operator of index zero. proof. since t is a linear operator and satisfying , we have γ((µt)n(c)) = µnγ(tn(c)) ≤ µnϕ(γ(c)) ≤ µnk(γ(c)) = ψ(γ(c)), where ψ(t) = µnkt is belong to φk, thus µt is a countably generalized ϕ−set contraction and therefore i −µt is a fredholm operator of index zero. � let (x, ||.||) be a real banach space and e = r × x a banach space with the norm ||(λ,u)|| = (|λ|2 + ||u||2) 1 2 for (λ,u) ∈ e. consider the following nonlinear equation: (2.2) u = λgu, (λ,u) ∈ e. assume that the operator g : x → x satisfies the following conditions: (h1) gu = lu + hu for all u ∈ x. index formula for ϕ−set contractions 159 (h2) l : x → x is a bounded linear operator (so it satisfies 2.1). (h3) h : x → x is a continuous operator such that ||hu|| ||u|| → 0 as ||u||→ 0. (h4) g : x → x is a generalized ϕ−set contraction. a real number λ is called a characteristic value of l if there exists a nonzero vector u in x such that u = λlu. we call the line {(λ, 0) : λ ∈ r} the set of trivial solutions of 2.2. let s denote the subset of e consisting of all nontrivial solutions of 2.2. a point (µ, 0) is called a bifurcation point of 2.2 if, given any � > 0, there exists an element (λ,u) ∈ s such that |λ−µ|2 + ||u||2 < �2. now we give a necessary condition for the existence of bifurcation points of the above equation, for the case of countably generalized ϕ−set operators, which extends [10]. theorem 2.10. let x be a real banach space and let g : x → x satisfy the hypotheses (h1), (h2) and (h3). suppose that µ ∈ r be such that µn ≤ k−1 and that µ is not a characteristic value of l. then (µ, 0) is not a bifurcation point of 2.2. proof. suppose that µ is not a characteristic value of l. then i −µl is injective. from µn ≤ k−1 it follows by proposition 2.8 that i−µl is fredholm of index zero. hence codimr(i −µl) = dimn(i −µl) = 0 and so r(i −µl) = x. since i −µl is a bijective bounded linear operator on x, the bounded inverse theorem implies that (i − µl)−1 is bounded. assume on the contrary that (µ, 0) is a bifurcation point of 2.2. then there exist a sequence {un} in x \{0} and a sequence {µn} in r such that un = µngun, un 6= 0 and µn 6= µ as n →∞. hence we have ||un|| ≤ ||(i −µl)−1||||(i −µl)un|| = ||(i −µl)−1||||µnhun + (µ−λ)lun|| ≤ ||(i −µl)−1||(|µn|||hun|| + |µn −µ|||l||||un||). therefore 1 ≤ ||(i −µl)−1||(|µn| ||hun|| ||un|| + |µn −µ|||l||). since the right-hand side of the last inequality tends to zero as n → ∞ by (h3), this is a contradiction. we conclude that (µ, 0) is not a bifurcation point of 2.2. � definition 2.11. a bounded linear operator t : x → x on a complex banach space x is said to be commutable if there exists a finite-dimensional linear operator f such that f commutes with t and i −t −f is a one-to-one operator of x onto x. when we say an operataor is finite dimensional, we shall mean its range is finite dimensional and when we say that a linear operator f commutes with t we shall mean (i) the domain of f , d(f), contains the domain of t , (ii) f(x) ∈ d(t) whenever x ∈ d(t), (iii) and tfx = ftx for x ∈ d(t2). browder defined the essential spectrum of a densely defined closed linear operator t on a banach space, in symbols ess(t), to be the set of λ ∈ σ(t) such that at least one of the following conditions holds: (i) r(λi −t), the range of λi −t , is not closed. (ii) λ is a limit point of σ(t). 160 salahifard and vaezpour (iii) ∪∞ν=1n(λi − t)ν is infinite dimensional, where n(λi − t)ν denotes the null space of (λi −t)ν. if t is a densely defined closed linear operator on x, define re(t), the essential spectral radius of t, by re(t) := sup{|λ| : λ ∈ ess(t)}. note that ess(t) is the largest subset of the spectrum σ(t) which remains invariant under perturbations of t by compact operators which commute with t, i.e. ess(t) = {λ : λ ∈ σ(t + c)for every compact operator c such that c(d(t)) ⊂ c, and tcx = ctx for x ∈ d(t2)} now, we may recall another notion of the essential spectrum, introduced by schechter [17], as follows: if one takes the essential spectrum to be the largest subset of the spectrum which remains invariant under arbitrary compact perturbations it yeilds to schechter’s definition. let t be a closed linear operator on a banach space x. the schechter essential spectrum of the operator t is defined by σs(t) = ∩k∈k(x)σ(t + k), where k(x) denote the set of all compact linear operators. it is clear that ess(t) includes properly σs(t) and if we add to σs(t), all limit points of the spectrum, then it will be equivalent to one given by browder, i.e ess(t). the following proposition gives a characterization of the schechter essential spectrum by means of fredholm operators: proposition 2.12. ([[16], theorem 5.4, p. 180]). let x be a banach space and t : x → x be a closed, densely defined linear operator. then λ 6∈ σs(t) if and only if λi −t is a fredholm operator of index zero. lemma 2.13. (nussbaum [13]). let t be an operator on x and r > re(t). then there exists a finite dimensional operator f on x, which commutes with t , such that σ(t + f) ⊂{λ ∈ c : |λ| ≤ r}. lemma 2.14. let t : x → x be a bounded linear operator on a complex banach space x. if re(t) < 1 then t is comutable. proof. by lemma 2.13 there exists a finite dimensional operator f on x, which commutes with t , such that i −t −f is invertible operator of x onto x, so t is commutable. � corollary 2.15. let t : x → x be a densely defined closed linear operator on a complex banach space x which is countably generalized ϕ−set contraction. then re(t) < 1 if and only if 1 is not belong to limit points of spectrum of t . proof. since t is a linear operator satisfying , for |t| ≤ 1, tt is generalized ϕ− set contraction and therefore by 2.8, i − tt is a fredholm operator of index zero, so λi −t for {λ : |λ| ≥ 1} is a fredholm operator of index zero. thus by proposition 2.12, λ 6∈ σs(t) for λ ∈ c with |λ| ≥ 1, and since {λ ∈ c : |λ| = 1} is not a limit point of spectrum of t, therefore σe(t) ⊂{λ ∈ c : |λ| < 1}. index formula for ϕ−set contractions 161 consequently re(t) < 1. � theorem 2.16. let x be a real or complex banach space and t : x → x a bounded linear comutable operator which is countably generalized ϕ−set contraction. then there exist a finite-dimensional subspace n and a closed subspace e of finite codimension such that x = n ⊕ e, n and e are both invariant under t , and (i − tt)|e is a homeomorphism of e onto itself for each t ∈ [0, 1]. proof. since t is a countably generalized ϕ−set contraction, i −t is a fredholm operator of index zero. if c is a closed subspace of x such that i −t|c : c → c is one-to-one, this implies that i −t |c is one-to-one and onto c. let ζ be the complexification of b and let t be the natural extension of t to ζ: t(x + iy) = tx + ity for x,y ∈ c. since t is comutable, there exists a finite-dimensional complex linear operator f such that i − t −f is a one-to-one operator of ζ onto ζ and f commutes with t. now applying theorem 2.7 in [10], the desired result is obtained. � corollary 2.17. let t : x → x be a comutable, countably generalized ϕ−set contractive bounded linear operator. then the sum of the multiplicities of the eigenvalues λ ≥ 1 of t is finite. proof. let λ ≥ 1 be any eigenvalue of t . suppose that x ∈ x is a nonzero vector such that (λi −t)nx = 0 for some positive integer n. let x = z + w, where z ∈ n and w ∈ e, and x = n⊕e is the decomposition described in theorem 2.16. since n and e are invariant under t , we have (λi−t)nz = −(λi−t)nw ∈ n∩e = {0}, and so (λi − t)nw = 0. since (i − λ−1t)|e is one-to-one by theorem 2.16, we have w = 0, and therefore x = z ∈ n. since n is finite dimensional, the conclusion follows. � let ω be a nonempty bounded open set in a banach space x. if t : x → x is a countably γ-condensing operator that has no fixed points on the boundary ∂ω, one may define the degree of i −t on ω as an integer, denoted by deg(i −t, ω, 0) more details of this definition are given in [20]. the above degree has the following basic properties, see [18, theorem 1.3] and [18, corollary 2.1]. lemma 2.18. let ω be a nonempty bounded open set in a banach space x and t : ω̄ → x a countably γ-condensing operator such that t has no fixed points on ∂ω. then the following statements hold: (1) if deg(i −t, ω, 0) = 0, then t has a fixed point in ω. (2) if 0 ∈ ω, then deg(i, ω, 0) = 1. (3) (homotopy invariance) if h : [0, 1] × ω̄ → x is a countably γ-condensing homotopy such that h(t,x) 6= x for all (t,x) ∈ [0, 1] ×∂ω, then deg(i −h(0, .), ω, 0) = deg(i −h(1, .), ω, 0). remark 2.19. if t is a generalized ϕ−set contraction, then tn is a condensing mapping, so we can use the above properties for it. definition 2.20. let ω be an open subset of a banach space x and t : ω → x a ϕ−set contractive operator. if x0 is an isolated fixed point of t , then the index of x0 for t is defined by ind(t,x0) = deg(i −t,b(x0,r), 0); 162 salahifard and vaezpour where b(x0,r) is an open ball in x centered at x0 with radius r so small that t has no fixed points other than x0 in b(x0,r). now, it is time to mention our main theorem which shows that the index formula holds for countably ϕ-set contractive bounded linear operators. proof. of 2.1: since 1 is not an eigenvalue of t , then 0 is the only fixed point of t and ind(t,x0) = deg(i −t,b(x0,r), 0). let x = n ⊕e be the decomposition of x introduced in theorem 2.16. define an operator s : x → x by s = t ◦p , where p denotes the projection onto n. since n is finite dimensional, we obtain that p is compact and so is s. now consider a continuous homotopy h : [0, 1] ×x → x defined by h(t,x) = tsx + (1 − t)tx for (t,x) ∈ [0, 1] ×x. then h is countably γ-condensing on [0, 1] × b̄(0, 1). in fact, for each countable set c ⊂ b̄(0, 1) with γ(c) > 0 we have γ(h([0, 1] ×c)) ≤ γ(co(s(c) ∪t(c))) ≤ max{γ(s(c)),γ(t(c))} ≤ ϕ(γ(c)) < γ(c) because s is compact and t is countably ϕ−set contraction. we claim that h(t,x) 6= x for all (t,x) ∈ [0, 1]×∂b(0, 1). indeed, suppose that h(t0,x0) = x0 for some (t0,x0) ∈ [0, 1] ×∂b(0, 1). let x0 = z + w, where z ∈ n and w ∈ e. then z +w = t0tz + (1−t0)tz + (1−t0)tw. by the invariance of n and e under t , we have z = tz and w = (1−t0)tw. since 1 is not an eigenvalue of t and t is ϕ−set contraction, i−(1−t0)t|e is one-to-one by theorem 2.7, we have z = 0 and w = 0 and hence x0 = 0, which contradicts the assumption that x0 ∈ ∂b(0, 1). lemma 3.1 implies that deg(i −t,b(0, 1), 0) and deg(i −s,b(0, 1), 0) are equal. since s is compact, deg(i −s,b(0, 1), 0) is equal to the lerayschauder degree. using the lerayschauder formula for a compact linear operator (see e.g. [3, theorem 8.10]), we have ind(t, 0) = deg(i −s,b(0, 1), 0) = (−1)ν, where ν is the sum of the multiplicities of the eigenvalues λ > 1 of s. it remains to show that vn = un for every positive integer n, where vn = {(λ,x) : λ > 1,x ∈ x,x 6= 0 and (i −t)nx = 0}; un = {(λ,x) : λ > 1,x ∈ x,x 6= 0 and (i −s)nx = 0}; suppose that (λ,x) ∈ vn and put x = z + w, where z ∈ n and w ∈ e. then (λi−t)nz = −(λi−t)nw ∈ n∩e = {0}. since (i−λ−1t)|e is one-to-one, we have w = 0 and so tx = t(pz) = sx. this shows that vn ⊂ un. now let (λ,x) ∈ un. since s = t on n and s = 0 on e, we have (λi−s)nz = −(λi−s)nw = −λnw = 0, where x = z + w with z ∈ n and w ∈ e. hence w = 0 and sx = tz = tx. this shows that un ⊂ vn. this completes the proof. � index formula for ϕ−set contractions 163 references [1] r.r. akhmerov, m.i. kamenskii, a.s. patapov, a.e. rodkina, b.n. sadovskii. measures of noncompactness and condensing operators, operator theory, advances and applications.55 (1992), birkhuser verlag, basel. [2] h. amann. fixed point equations and nonlinear eigenvalue problems in ordered banach space, siam rev.18 (1976), 620-09. [3] j. appel. measures of noncompactness, condensing operators and fixed points: an appicationoriented survey, fixed point theory. 6 (2005), 157-29. [4] f. e. browder. on the spectral theory of elliptic differential operators. i, math. ann. 142(1961), 22-30. [5] b. c. dhage. condensing mappings and applications to existence theorems for common solution of differential equations, bull. korean math. soc. 36(1999) no. 3, 565-578. [6] l.guozhen. the fixed point index and the fixed point theorems of 1-set contraction mappings, proc. amer. math. soc. 104 (1988), no. 4, 1163-1170. [7] a. granas, j. dugundji. fixed point theory, springer-verlag, new york, berlin, 2003. [8] g. hetzer, v. stallbohm. coincidence degree and rabinowitz’s bifurcation theorem, publications de l’institut mathematique, nouvelle series, tome 20 (1976), 117-129. [9] in-sook kim, sunghui kwon. global bifurcation for generalized k-set contractions, nonlinear analysis. 68 (2008), 3224-231. [10] in-sook kim. index formulas for countably k-set contractive operators, nonlinear analysis.69 (2008), 4182-189. [11] c. kuratowski. sur les espaces complets, fund. math.15 (1930), 301-09. [12] l.s. liu. approximation theorems and fixed point theorems for various class of 1-setcontractive mappings in banach spaces, acta math. sinica.17(2001), 103-12. [13] r. d. nussbaum. the radius of the essential spectrum, duke math. j.38 (1970), 473-78. [14] r.d. nussbaum. the fixed point index and fixed point theorems for k-set contractions, bull. amer. math. soc. 75(1969), 490-95. [15] r.d. nussbaum. asymptotic fixed point theorems for local condensing maps, math. ann. 191 (1971), 181-195. [16] martin schechter. principles of functional analysis, academic press. (1971). [17] martin schechter. on the essential spectrum of an arbitrary operator. i, journal of mathematical analysis and applications.13 (1966), 205-15. [18] c.a. stuart, j.f. toland. the fixed point index of a linear k-set contraction, j. london math. soc.6 (1973), 317-20. [19] j.m. ayerbe toledano, t. dominguez benavides, g. lopez acedo. measures of noncompactness in metric fixed point theory, birkhauser, basel,. (1997). [20] m. vath. fixed point theorems and fixed point index for countably condensing maps, topol. methods nonlinear anal.13 (1999), 341-63. department of mathematics and computer science, amirkabir university of technology, tehran, iran ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 97-112 http://www.etamaths.com common fixed point theorems for four fuzzy mappings animesh gupta1,∗ and neelesh pandey2 abstract. in this paper, we obtain some common fixed point theorems for four fuzzy mappings in complete ordered metric linear spaces. these mappings are assumed to satisfy certain contractive inequality involving functions which are generalizations of altering distance functions. we also note that this fuzzy fixed point result is derivable from a multi-valued fixed point result. 1. introduction in 1965, the theory of fuzzy sets was investigated by zadeh [23]. in 1981, heilpern [11] first introduced the concept of fuzzy contractive mappings and proved a fixed point theorem for these mappings in metric linear spaces. estruch and vidal [10] proved a fixed point theorem for fuzzy contraction mappings in a complete metric spaces which in turn generalized heilpern fixed point theorem. afterwards a number of works appeared in which fixed points of fuzzy mappings satisfying contractive inequalities have been studied (see [9]) a new category of contractive fixed point problems was addressed by m.s. khan et. al [13]. there they introduced altering distance function, which is a control function that alters distance between two points in a metric space. afterwards a number of works have appeared in which altering distances have been used. in references [20] and [21] for example, fixed points of single valued mappings and in [6] fixed points of set valued mappings have been obtained by using altering distance functions. altering distances have been generalized to functions with more than one argument. in [7] a generalization of such functions to a two-variable function and in [8] a generalization to a three-variable function were introduced and applied for obtaining fixed point results in metric spaces. in this paper we introduce a contractive inequality for four fuzzy mappings through a 4-variable generalization of altering distance function and then prove that the two fuzzy mappings defined on a complete ordered metric linear space satisfying such inequality have a common fixed point. we have discussed some specific results, which are obtainable under special choices of the generalized altering distance function. we also show that a more general result in the fixed point theory of multi-valued mappings can be established and the result we obtained for fuzzy mappings can be deduced from the general theorem. 2010 mathematics subject classification. 46s40, 47h10, 54h25. key words and phrases. annihilator, weak annihilator, dominating maps, fuzzy mapping, common fixed point. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 97 98 gupta and pandey 2. preliminaries throughout the rest of the paper unless otherwise stated (x,d) stands for a complete metric space. a fuzzy set in x is a function with domain x and values in [0, 1]. if a is a fuzzy set on x and x ∈ x then the functional value ax is called the grade of membership of x in a. the α−level set of a, denoted by aα, is defined by aα = {x : ax ≥ α}, if α ∈ (0, 1], a0 = {x : ax ≥ 0}, where b denoted the closure of the set b. for any two subsets a and b of x we denote by h(a,b) the hausdroff distance. for any two subsets a and b of x we write δ(a,b) = supα∈a,β∈b d(a,b). definition 2.1. a function ψ : [0, +∞) → [0, +∞) is called an altering distance function if and only if (i) ψ is continuous, (ii) ψ is non-decreasing, (iii) ψ(t) = 0 ⇐⇒ t = 0. choudhury [9] introduced the concept of a generalized altering distance function for three variables. definition 2.2. a function ψ : [0, +∞) × [0, +∞) × [0, +∞) → [0, +∞) is called an altering distance function if and only if (i) ψ is continuous, (ii) ψ is non-decreasing in all three variables, (iii) ψ(x,y,z) = 0 ⇐⇒ x = y = z = 0. rao et al. [18] introduced the concept of a generalized altering distance function for four variables. definition 2.3. a function ψ : [0, +∞)× [0, +∞)× [0, +∞)× [0, +∞) → [0, +∞) is called an altering distance function if and only if (i) ψ is continuous, (ii) ψ is non-decreasing in all three variables, (iii) ψ(x,y,z,w) = 0 ⇐⇒ x = y = z = w = 0. definition 2.4. let (x,d) be a metric space and f,g : x → x. if w = fx = gx, for some x ∈ x, then x is called a coincidence point of f and g, and w is called a coincidence point of f and g. if x = w, then x is a common fixed point of f and g. the pair {f,g} is said to be comparable if and only if limn→+∞d(fgxn,gfxn) = 0, whenever {xn} is a sequence in x such that limn→+∞fxn = limn→+∞gxn = t for some t ∈ x. definition 2.5. let f and g be two self mappings defined on a set x. then f and g are said to be weakly comparable if they commute at every coincidence point. definition 2.6. let x be a nonempty set. then (x,d,�) is called an ordered metric linear space iff (i) (x,d) is a metric linear space, (ii) (x,�) is a partial order. definition 2.7. let (x,�) be a partially ordered set. then x,y ∈ x are comparable if x � y or y � x holds. definition 2.8. let (x,�) be a partially ordered set. a pair (f,g) of self maps of x is said to be weakly increasing if gx � gfx and gx � fgx for all x ∈ x. common fixed point theorems for four fuzzy mappings 99 the notion of partially weakly increasing of pair of mappings is introduced by abbas et al [1]. definition 2.9. let (x,�) be a partially ordered set and f and g be two self maps on x. an ordered pair (f,g) is said to be partially weakly increasing if gx � gfx and gx � fgx for all x ∈ x. note that a pair (f,g) is weakly increasing if and only if ordered pair (f,g) and (g,f) are partially weakly increasing. in the following, an example of an ordered pair (f,g) of self-maps f and g which is partially weakly increasing but not weakly increasing. example 2.10. let x = [0, 1] be endowed with a usual ordering and f,g : x → x be defined by fx = x2 and gx = √ x. clearly, (f,g) is partially weakly increasing but (g,f) is not partially weakly increasing. definition 2.11. let (x,�) be a partially ordered set. a mapping f is called weak annihilator of g if fgx � x for all x ∈ x. example 2.12. let x = [0, 1] be endowed with a usual ordering and f,g : x → x be defined by fx = x2 and gx = x3. thus f is a weak annihilator of g. definition 2.13. let (x,�) be a partially ordered set. a mapping f is called domination if x � fx for each x ∈ x. example 2.14. let x = [0, 1] be endowed with a usual ordering and f : x → x be defined by fx = n √ x. thus f is domination for each x ∈ x. definition 2.15. a subset k of a partially ordered set x is called totally ordered when every two elements of k are comparable. 3. main results now, we proof our main results of this section. theorem 3.1. let (x,d,�) be an ordered complete metric linear space. let t,s,i,j : x → w(x) be four fuzzy mappings satisfying for every pair (x,y) ∈ x ×x such that x and y are comparable, φ1(δ1(sx,ty)) ≤ ψ1 (m(ix,sx)) −ψ2 (m(ix,sx))(3.1) where m(ix,sx) = {d(ix,jy),d1(ix,sx),d1(jy,ty), 1 2 [d1(ix,ty) + d1(jy,sx)]} and ψ1 and ψ2 are generalized altering distance functions (in ψ4) and φ1(x) = ψ1(x,x,x,x). suppose that (i) (i,t) and (j,s) be partially weakly increasing, (ii) t(x) ⊆ i(x) and s(x) ⊆ j(x), (iii) s and t are dominating maps, (iv) t is weak annihilator of i and s is weak annihilator of j, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u. assume either 100 gupta and pandey (a) {s,i} are comparable, s or i is continuous and {t,j} are weakly comparable or (b) {t,j} are comparable, t or j is continuous and {s,i} are weakly comparable . then s,t,i and j have a common fixed point. moreover, the set of common fixed points of s,t,i and j is totally ordered if and only if s,t,i and j have one and only one common fixed point. proof. let x0 ∈ x be an arbitrary point in x. since t(x) ⊆ i(x) and s(x) ⊆ j(x), we can define the sequences {xn} and {yn} in x by {y2n−1} = jx2n−1 ⊂ sx2n−2, {y2n} = ix2n ⊂ tx2n−1,(3.2) for all n ∈ n. by given assumptions x2n−2 � sx2n−2 = jx2n−1 � sjx2n−1 � x2n−1 and x2n−1 � tx2n−1 = ix2n � tix2n � x2n. thus for all n ≥ 1, we have xn � xn+1.(3.3) without loss of generality, we may assume that d(y2n,y2n+1) > 0 ∀n ∈ n.(3.4) if not, then y2n = y2n+1, for some n. putting x = x2n+1 and y = x2n, form (3.3) and the considered contraction (3.1), we have common fixed point theorems for four fuzzy mappings 101 φ1(d(y2n+2,y2n+1)) = φ1(δ1(sx2n+1,tx2n)) ≤ ψ1 (d(ix2n+1,jx2n),d1(ix2n+1,sx2n+1),d1(jx2n,tx2n), 1 2 [d1(ix2n+1,tx2n) + d1(jx2n,sx2n+1)] ) −ψ2 (d(ix2n+1,jx2n),d1(ix2n+1,sx2n+1),d1(jx2n,tx2n), 1 2 [d1(ix2n+1,tx2n) + d1(jx2n,sx2n+1)] ) ≤ ψ1 (d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 [d(y2n+1,y2n+1) + d(y2n,y2n+2)] ) −ψ2 (d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 [d(y2n+1,y2n+1) + d(y2n,y2n+2)] ) (3.5) ≤ ψ1 ( 0,d(y2n+1,y2n+2), 0, 1 2 d(y2n,y2n+2) ) −ψ2 ( 0,d(y2n+1,y2n+2), 0, 1 2 d(y2n,y2n+2) ) .(3.6) using a triangular inequality, we have 1 2 d(y2n,y2n+2) ≤ 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)] ≤ 1 2 d(y2n+1,y2n+2). using this together with a property of the generalized altering function ψ1, we get ψ1 ( 0,d(y2n+1,y2n+2), 0, 1 2 d(y2n,y2n+2) ) ≤ φ1(d(y2n+1,y2n+2). hence, we obtain φ1(d(y2n+1,y2n+2) ≤ φ1(d(y2n+1,y2n+2) −ψ2 ( 0,d(y2n+1,y2n+2), 0, 1 2 d(y2n,y2n+2) ) . this implies that ψ2 ( 0,d(y2n+1,y2n+2), 0, 1 2 d(y2n,y2n+2) ) = 0 which yields that d(y2n,y2n+1) = 0. following the similar arguments, we obtain y2n+2 = y2n+3 and so on. thus {yn} becomes a constant sequence and {y2n} is the common fixed point of i,j,s and t. take for each n, d(y2n,y2n+1) > 0. we claim that 102 gupta and pandey lim n→+∞ d(y2n,y2n+1) = 0.(3.7) by (3.6), we have φ1(d(y2n+2,y2n+1)) = φ1(δ1(sx2n+1,tx2n)) ≤ ψ1 (m(y2n+1,y2n)) −ψ2 (m(y2n+1,y2n))(3.8) where m(y2n+1,y2n) = {d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 d(y2n,y2n+2)}. suppose for some n ∈ n, that d(y2n+2,y2n+1) > d(y2n,y2n+1).(3.9) using (3.9) and a triangular inequality, we have 1 2 d(y2n,y2n+2) ≤ 1 2 [d(y2n,y2n+1) + d(y2n+1,y2n+2)] < d(y2n+1,y2n+2). using this and (3.9) together with a property of the generalized altering distance function ψ1, we get ψ1 (m(y2n+1,y2n)) ≤ φ1(d(y2n+1,y2n+2). where m(y2n+1,y2n) = {d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 d(y2n,y2n+2)}. hence, we obtain φ1(d(y2n+2,y2n+1)) ≤ φ1(d(y2n+2,y2n+1)) −ψ2 (m(y2n+1,y2n)) . where m(y2n+1,y2n) = {d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 d(y2n,y2n+2)}. this implies that ψ2 ( d(y2n+1,y2n),d(y2n+2,y2n+1),d(y2n+1,y2n), 1 2 d(y2n,y2n+2) ) = 0 which yields that d(y2n+1,y2n) = 0. hence, we obtain a contradiction with (3.4). we deduce that d(y2n+1,y2n+2) ≤ d(y2n,y2n+1), ∀n ∈ n.(3.10) similarly, putting x = x2n+1 and y = x2n+2, form (3.3) and the considered contraction (3.1), we have common fixed point theorems for four fuzzy mappings 103 φ1(d(y2n+2,y2n+3)) = φ1(δ1(sx2n+1,tx2n+2)) ≤ ψ1 (d(y2n+1,y2n+2),d(y2n+2,y2n+3),d(y2n+1,y2n+2), 1 2 d(y2n+1,y2n+3) ) −ψ2 (d(y2n+1,y2n+2),d(y2n+2,y2n+3),d(y2n+1,y2n+2), 1 2 d(y2n+1,y2n+3) ) .(3.11) suppose, for some n ∈ n, that d(y2n+2,y2n+3) > d(y2n+1,y2n+2).(3.12) then, by a triangular inequality, we have 1 2 d(y2n+1,y2n+3) ≤ 1 2 [d(y2n+1,y2n+2) + d(y2n+2,y2n+3)] < d(y2n+2,y2n+3). hence, from this, (3.11) and (3.12), we obtain φ1(d(y2n+1,y2n+3)) ≤ φ1(d(y2n+2,y2n+3)) −ψ2 (d(y2n+1,y2n+2),d(y2n+2,y2n+3),d(y2n+1,y2n+2), 1 2 d(y2n+1,y2n+3) ) . this implies that ψ2 ( d(y2n+1,y2n+2),d(y2n+1,y2n+2),d(y2n+2,y2n+3), 1 2 d(y2n+1,y2n+3) ) = 0 which yields that d(y2n+1,y2n+2) = 0. hence, we obtain a contradiction with (3.4). we deduce that d(y2n+1,y2n+2) ≥ d(y2n+2,y2n+3), ∀n ∈ n.(3.13) combining (3.10) and (3.13), we obtain d(y2n,y2n+1) > d(y2n+2,y2n+3), ∀n ∈ n.(3.14) then, {d(y2n+1,y2n+2)} is a nonincreasing sequence of positive real numbers. this implies that there exists r ≥ 0 such that lim n→+∞ d(y2n+1,y2n+2) = r.(3.15) 104 gupta and pandey by (3.8), we have φ1(d(y2n+2,y2n+1)) = φ1(δ1(sx2n+1,tx2n)) ≤ ψ1 (d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 d(y2n,y2n+2) ) −ψ2 (d(y2n+1,y2n),d(y2n+1,y2n+2),d(y2n,y2n+1), 1 2 d(y2n,y2n+2) ) ≤ φ(d(y2n+1,y2n)) −ψ2 (d(y2n+1,y2n),d(y2n+1,y2n+2), d(y2n,y2n+1), 0) .(3.16) letting n → +∞ in (3.16) and using the continuities of φ1 and ψ2, we obtain φ1(r) ≤ φ1(r) −ψ2(r,r,r, 0), which implies that ψ2(r,r,r, 0) = 0 so r = 0. hence lim n→+∞ d(y2n+1,y2n+2) = 0. hence, (3.7) is proved. next, we claim that {yn} is a cauchy sequence. from (3.7), it will be sufficient to prove that {y2n} is a cauchy sequence. we proceed by negation and suppose that {y2n} is not a cauchy sequence. then, there exists ε > 0 for which we can find two sequences of positive integers {m(i)} and {n(i)} such that for all positive integer i, n(i) > m(i) > i, d(ym(i),yn(i)) ≥ ε, d(ym(i),yn(i)−2) < ε.(3.17) from (3.17) and using a triangular inequality, we get ε ≤ d(ym(i),yn(i)) ≤ d(ym(i),yn(i)−2) + d(yn(i)−2,yn(i)−1) + d(yn(i)−1,yn(i)) ≤ ε + d(yn(i)−2,yn(i)−1) + d(yn(i)−1,yn(i)). letting i → +∞ in the above inequality and using (3.7), we obtain lim i→+∞ d(ym(i),yn(i)) = ε.(3.18) again, a triangular inequality gives us |d(yn(i),ym(i)−1) −d(yn(i),ym(i))| ≤ d(ym(i)−1,ym(i)).(3.19) letting i → +∞ in the above inequality and using (3.7) and (3.18), we get lim i→+∞ d(yn(i),ym(i)−1) = ε.(3.20) common fixed point theorems for four fuzzy mappings 105 similarly, we have lim i→+∞ d(yn(i)+1,ym(i)−1) = ε.(3.21) on the other hand, we have d(yn(i),ym(i)) ≤ d(yn(i),yn(i)+1) + d(yn(i)+1,ym(i)) = d(yn(i),yn(i)+1) + d(txn(i),sxm(i)−1). then, from (3.7), (3.18) and the continuity of φ1, we get by letting i → +∞ in the above inequality φ1(ε) ≤ lim i→+∞ d(txn(i),sxm(i)−1).(3.22) now, using the considered contractive condition (3.1) for x = x2m(i)−1 and y = x2n(i), we have φ1(δ1(sx2m(i)−1,tx2n(i))) ≤ ψ1 ( m(x2m(i)−1,tx2n(i)) ) −ψ2 ( m(x2m(i)−1,tx2n(i)) ) m(x2m(i)−1,tx2n(i)) = {d(ix2m(i)−1,jx2n(i)),d1(ix2m(i)−1,sx2m(i)−1), d1(jx2n(i),tx2n(i)), 1 2 [d1(ix2m(i)−1,tx2n(i)) + d1(jx2n(i),sx2m(i)−1)]} φ1(δ1(y2m(i)−1,y2n(i))) ≤ ψ1 ( d(y2m(i)−1,y2n(i)),d(y2m(i)−1,y2m(i)),d(y2n(i),y2n(i)+1), 1 2 [d(y2m(i)−1,y2n(i)+1) + d(y2n(i),y2m(i))] ) −ψ2 ( d(y2m(i)−1,y2n(i)),d(y2m(i)−1,y2m(i)),d(y2n(i),y2n(i)+1), 1 2 [d(y2m(i)−1,y2n(i)+1) + d(y2n(i),y2m(i))] ) . then, from (3.7), (3.20), (3.21) and the continuity of ψ1 and ψ2, we get by letting i → +∞ in the above inequality lim i→+∞ φ1(δ1(sx2m(i)−1,tx2n(i))) ≤ ψ1(ε, 0, 0,ε) −ψ2(ε, 0, 0,ε) ≤ φ1(ε) −ψ2(ε, 0, 0,ε). now, combining (3.1) with the above inequality, we get φ1(ε) ≤ φ1(ε) −ψ2(ε, 0, 0,ε), which implies that ψ2(ε, 0, 0,ε) = 0, that is a contradiction since ε > 0. we deduce that {yn} is a cauchy sequence. 106 gupta and pandey finally, we prove existence of a common fixed point of the four mappings i,j,s and t . since {yn} is a cauchy sequence in complete metric linear space (x,d), there exists a point z ∈ x, such that {y2n} converges to z. therefore, {y2n+1} = jx2n+1 ⊂ sx2n → z, as n → +∞(3.23) and {y2n+2} = ix2n+2 ⊂ tx2n+1 → z, as n → +∞.(3.24) suppose that (a) holds. since {s,i} are comparable, we have lim n→+∞ six2n+2 = lim n→+∞ six2n+2 = iz. also, x2n+1 � tx2n+1 = ix2n+2. now φ1(δ1(six2n+2,tx2n+1)) ≤ ψ1 (d(iix2n+2,jx2n+1),d1(iix2n+2,six2n+2), d1(jx2n+1,tx2n+1), 1 2 [d1(iix2n+2,tx2n+1) + d1(jx2n+1,six2n+2)] ) −ψ2 (d(iix2n+2,jx2n+1),d1(iix2n+2,six2n+2), d1(jx2n+1,tx2n+1), 1 2 [d1(iix2n+2,tx2n+1) + d1(jx2n+1,six2n+2)] ) assume that i is continuous. on passing limit as n → +∞, we obtain φ1(d(iz,z)) ≤ ψ1 (d(iz,z), 0, 0,d(iz,z)) −ψ2 (d(iz,z), 0, 0,d(iz,z)) ≤ φ1(d(iz,z)) −ψ2 (d(iz,z), 0, 0,d(iz,z)) , so ψ2 (d(iz,z), 0, 0,d(iz,z)) = 0, which implies that iz = z.(3.25) now, x2n+1 � tx2n+1 and tx2n+1 → z as n → +∞, so by assumption we have x2n+1 � z and (3.1) becomes φ1(δ1(sz,tx2n+1)) ≤ ψ1 (m(z,x2n+1)) −ψ2 (m(z,x2n+1)) . where m(z,x2n+1) = (d(iz,jx2n+1),d1(iz,sz),d1(jx2n+1,tx2n+1), 1 2 [d1(iz,tx2n+1) + d1(jx2n+1,sz)] ) passing to the limit n → +∞ in the above inequality and using (3.25), common fixed point theorems for four fuzzy mappings 107 φ1(δ1(sz,z)) ≤ ψ1 ( 0,d1(z,sz), 0, 1 2 d1(z,sz) ) −ψ2 ( 0,d1(z,sz), 0, 1 2 d1(z,sz) ) which holds unless ψ2 ( 0,d(z,sz), 0, 1 2 d1(z,sz) ) = 0, so sz = z.(3.26) since s(x) ⊆ j(x), there exists a point w ∈ x such that sz = jw. suppose that tw 6= jw. since z � sz = jw � sjw � w implies z � w. from (3.1), we obtain φ1(δ1(sz,tw)) ≤ ψ1 (d(iz,jw),d1(iz,sz),d1(jw,tw), 1 2 [d1(iz,tw) + d1(jw,sz)] ) −ψ2 (d(iz,jw),d1(iz,sz),d1(jw,tw), 1 2 [d1(iz,tw) + d1(jw,sz)] ) ≤ ψ1 ( 0, 0,d1(jw,tw), 1 2 d1(jw,tw) ) −ψ2 ( 0, 0,d1(jw,tw), 1 2 d1(jw,tw) ) hence jw = tw.(3.27) since t and j are weakly compatible, tz = tsz = tjw = jtw = jsz = jz. thus z is a coincidence point of t and j. now, since x2n � sx2n and sx2n → z as n → +∞, implies that x2n � z, from (3.1) φ1(δ1(sx2n,tz)) ≤ ψ1 (m((x2n,z))) −ψ2 (m((x2n,z))) where m((x2n,z) = (d(ix2n,jz),d1(ix2n,sx2n),d1(jz,tz), 1 2 [d1(ix2n,tz) + d1(jz,sx2n)]) passing to the limit n → +∞ in the above inequality, we have φ1(δ1(z,tz)) ≤ ψ1 (d(z,tz), 0, 0,d(z,tz)) −ψ2 (d(z,tz), 0, 0,d(z,tz)) which gives that 108 gupta and pandey z = tz.(3.28) therefore, sz = tz = iz = jz = z, so z is a common fixed point of i,j,s and t. the proof is similar when s is continuous. similarly, the result follows when (b) holds. now, suppose that set of common fixed points of i,j,s and t is totally ordered. we claim that there is a unique common fixed point of i,j,s and t. assume on contrary that, su = tu = iu = ju = u and sv = tv = iv = jv = v but u 6= v. by supposition, we can replace x = u and y = v in (3.1) to obtain φ1(d(u,v)) ≤ φ1(δ1(su,ty)) ≤ ψ1 (d(iu,jv),d1(iu,su),d1(jv,tv), 1 2 [d1(iu,tv) + d1(jv,su)] ) −ψ2 (d(iu,jv),d1(iu,su),d1(jv,tv), 1 2 [d1(iu,tv) + d1(jv,su)] ) ≤ ψ1 (d(u,v), 0, 0,d(u,v)) −ψ2 (d(u,v), 0, 0,d(u,v)) ≤ φ1(d(u,v)) a contraction, so u = v. conversely, if i,j,s and t have only one common fixed point, then the set of common fixed point of i,j,s and t being singleton is totally ordered. � corollary 3.2. let (x,d,�) be an ordered complete metric linear space. let t,s,i,j : x → w(x) be four fuzzy mappings satisfying for every pair (x,y) ∈ x ×x such that x and y are comparable and there exists a positive lebesgue integrable function u on r+ such that ∫ � 0 u(t)dt > 0 for each � > 0 and that , ∫ φ1(δ1(sx,ty)) 0 u(t)dt ≤ ∫ ψ1(m(x,y)) 0 u(t)dt− ∫ ψ2(m(x,y)) 0 u(t)dt(3.29) where ψ1 and ψ2 are generalized altering distance functions (in ψ4) and φ1(x) = ψ1(x,x,x,x) also m(x,y) = {d(ix,jy),d1(ix,sx),d1(jy,ty), 1 2 [d1(ix,ty) + d1(jy,sx)]} suppose that (i) (i,t) and (j,s) be partially weakly increasing, (ii) t(x) ⊆ i(x) and s(x) ⊆ j(x), (iii) s and t are dominating maps, (iv) t is weak annihilator of i and s is weak annihilator of j, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u. assume either (a) {s,i} are comparable, s or i is continuous and {t,j} are weakly comparable or common fixed point theorems for four fuzzy mappings 109 (b) {t,j} are comparable, t or j is continuous and {s,i} are weakly comparable . then s,t,i and j have a common fixed point. moreover, the set of common fixed points of s,t,i and j is totally ordered if and only if s,t,i and j have one and only one common fixed point. remark 3.3. if we take ψ1(t1, t2, t3, t4) = max{t1, t2, t3, t4} and ψ2(t1, t2, t3, t4) = (1 − k) max{t1, t2, t3, t4}, for k ∈ (0, 1) then φ1(t) = t for all t1, t2, t3, t4 ≥ 0 then the we get following result. corollary 3.4. let (x,d,�) be an ordered complete metric linear space. let t,s,i,j : x → w(x) be four fuzzy mappings satisfying for every pair (x,y) ∈ x ×x such that x and y are comparable, δ1(sx,ty) ≤ k max{d(ix,jy),d1(ix,sx),d1(jy,ty), 1 2 [d1(ix,ty) + d1(jy,sx)]}(3.30) where k ∈ (0, 1) suppose that (i) (i,t) and (j,s) be partially weakly increasing, (ii) t(x) ⊆ i(x) and s(x) ⊆ j(x), (iii) s and t are dominating maps, (iv) t is weak annihilator of i and s is weak annihilator of j, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u. assume either (a) {s,i} are comparable, s or i is continuous and {t,j} are weakly comparable or (b) {t,j} are comparable, t or j is continuous and {s,i} are weakly comparable . then s,t,i and j have a common fixed point. moreover, the set of common fixed points of s,t,i and j is totally ordered if and only if s,t,i and j have one and only one common fixed point. remark 3.5. other results could be derived for other choices of ψ1 and ψ2. corollary 3.6. let (x,d,�) be an ordered complete metric linear space. let t,s,i : x → w(x) be three fuzzy mappings satisfying for every pair (x,y) ∈ x×x such that x and y are comparable, φ1(δ1(sx,ty)) ≤ ψ1 (d(ix,iy),d1(ix,sx),d1(iy,ty), 1 2 [d1(ix,ty) + d1(iy,sx)] ) −ψ2 (d(ix,jy),d1(ix,sx),d1(iy,ty), 1 2 [d1(ix,ty) + d1(iy,sx)] ) (3.31) where ψ1 and ψ2 are generalized altering distance functions (in ψ4) and φ1(x) = ψ1(x,x,x,x). suppose that 110 gupta and pandey (i) (i,t) and (i,s) be partially weakly increasing, (ii) t(x) ⊆ i(x) and s(x) ⊆ i(x), (iii) s and t are dominating maps, (iv) t is weak annihilator of i and s is weak annihilator of i, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u. assume either (a) {s,i} are comparable, s or i is continuous and {t,i} are weakly comparable or (b) {t,i} are comparable, t or i is continuous and {s,i} are weakly comparable . then s,t,i have a common fixed point. moreover, the set of common fixed points of s,t,i is totally ordered if and only if s,t,i have one and only one common fixed point. proof. it follows by taking j = i in theorem 3.1. � corollary 3.7. let (x,d,�) be an ordered complete metric linear space. let t,i,j : x → w(x) be three fuzzy mappings satisfying for every pair (x,y) ∈ x×x such that x and y are comparable, φ1(δ1(tx,ty)) ≤ ψ1 (d(ix,jy),d1(ix,tx),d1(jy,ty), 1 2 [d1(ix,ty) + d1(jy,tx)] ) −ψ2 (d(ix,jy),d1(ix,tx),d1(jy,ty), 1 2 [d1(ix,ty) + d1(jy,tx)] ) (3.32) where ψ1 and ψ2 are generalized altering distance functions (in ψ4) and φ1(x) = ψ1(x,x,x,x). suppose that (i) (i,t) and (j,t) be partially weakly increasing, (ii) t(x) ⊆ i(x) and t(x) ⊆ j(x), (iii) t is dominating maps, (iv) t is weak annihilator of i and j, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u. assume either (a) {t,i} are comparable, t or i is continuous and {t,j} are weakly comparable or (b) {t,j} are comparable, t or j is continuous and {t,i} are weakly comparable . then t,i and j have a common fixed point. moreover, the set of common fixed points of t,i and j is totally ordered if and only if t,i and j have one and only one common fixed point. proof. it follows by taking s = t in theorem 3.1. � common fixed point theorems for four fuzzy mappings 111 corollary 3.8. let (x,d,�) be an ordered complete metric linear space. let t,i : x → w(x) be three fuzzy mappings satisfying for every pair (x,y) ∈ x ×x such that x and y are comparable, φ1(δ1(tx,ty)) ≤ ψ1 (d(ix,iy),d1(ix,tx),d1(iy,ty), 1 2 [d1(ix,ty) + d1(iy,tx)] ) −ψ2 (d(ix,iy),d1(ix,tx),d1(iy,ty), 1 2 [d1(ix,ty) + d1(iy,tx)] ) (3.33) where ψ1 and ψ2 are generalized altering distance functions (in ψ4) and φ1(x) = ψ1(x,x,x,x). suppose that (i) (i,t) be partially weakly increasing, (ii) t(x) ⊆ i(x), (iii) t is dominating maps, (iv) t is weak annihilator of i, (v) if for a nondecreasing sequence {xn} with xn � yn for all n and yn → u implies that xn � u, (vi) {t,i} are comparable, t or i is continuous. then t,i have a common fixed point. moreover, the set of common fixed points of t,i is totally ordered if and only if t,i have one and only one common fixed point. proof. it follows by taking s = t and j = i in theorem 3.1. � 4. acknowledgements the authors are grateful referees for their critical reading of the original manuscript and for their valuable suggestions which improved the quality of this work also we express our sincere thanks to the members of editorial board for consideration this article for publication. references [1] m. abbas, n. talat, s. radenovic, common fixed points of four maps in partially ordered metric spaces, appl. math. lett, 24(2011), 1520-1526. [2] h.m. abu-donia, common fixed points theorems for fuzzy mappings in metric spaces under ψcontraction condition, chaos, solitons & fractals 34 (2007), 538–543. [3] s.c. arora, v. sharma, fixed points for fuzzy mappings, fuzzy sets and systems 110 (2000), 127–130. [4] a. azam, m. arshad, a note on fixed point theorems for fuzzy mappings by p.vijayaraju and m. marudai, fuzzy sets and systems 161 (2010), 1145–1149. [5] a. azam,i. beg ,common fixed points of fuzzy maps, math. comp. modelling 49 (2009), 1331-1336. [6] b. s. choudhury and a. upadhyay, on unique common fixed point for a sequence of multivalued mappings on metric spaces, bulletin of pure and applied science, 19e(2000), 529-533. [7] b. s. choudhury and p. n. dutta, a unified fixed point result in metric spaces involving a two variable function, filomat, 14(2000), 43-48. [8] b. s. choudhury and p. n. dutta, a unified approach to fixed points of self-mappings in metric spaces, (preprint). [9] b.s. choudhury, a common unique fixed point result in metric spaces involving generalized altering distance, math. comm., 10(2005), 105–110. 112 gupta and pandey [10] v.d. estruch and a. vidal, a note on fixed fuzzy points for fuzzy mappings, rend. istit. univ. trieste, 32(2011),39–45. [11] s. heilpern, fuzzy mappings and fixed point theorems, j.math. anal. appl., 83(1981), 566– 569. [12] g. jungck, compatible mappings and common fixed points, internat. j. math. math. sci. 9(1986), 771–779. [13] m. s. khan, m. swaleh and s. sessa, fixed point theorem by altering distances between the points, bull. austral. math. soc., 30(1984), 1-9. [14] b.s. lee, g.m. lee, s.j. cho, d.s. kim, a common fixed point theorem for a pair of fuzzy mappings, fuzzy sets and systems, 98 (1998), 133-136. [15] b.s. lee, s.j. cho, common fixed point theorems for a sequence of fuzzy mappings, internat. j. math & math sciences, 17, no. 3 (1994), 423-428. [16] s.b. nadler jr., multivalued contraction mappings, pacific journal of math., 30 (1969), 475-487. [17] c. v. negoita, d. ralescu, representation theorem for fuzzy concepts, kybernets, 4 (1975), 169-174. [18] k.p.r. rao , g. ravi babu , d. vasu babu, common fixed point theorem through generalized altering distance functions, math. comm. 13(2008), 64-73. [19] a.c.m. ran, m.c.b. reurings, a fixed point theorem in partially ordered sets and some application to matrix equations, proc. amer. math. soc., 132 (2004), 1435-1443. [20] k. p. r. sastry and g. v. r. babu, fixed point theorems in metric spaces by altering distances, bull. cal. math. soc., 90(1998), 175-182. [21] k. p. r. sastry, s. v. r. naidu, g. v. r. babu and g. a. naidu, generalisation of common fixed point theorems for weakly commuting maps by altering distances, tamkang j. math., 31(2000), 243-250. [22] s. sessa, on a weak commutativity condition of mappings in fixed point considerations, publ. inst. math. (beograd) 32 (1982), 149-153. [23] l.a. zadeh, fuzzy sets, informatics and control, 8(1965), 103–112. 1h.no. 93/654, ward no. 2, gandhi chowk, pachmarhi 461881, dist. hoshangabad, (m.p.) india 2department of mathematics, hitkarini college of engineering & technology, jabalpur (m.p.), india ∗corresponding author international journal of analysis and applications volume 18, number 2 (2020), 183-193 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-183 birkhoff normal forms for born-oppenheimer operators nawel latigui1,4, bekkai messirdi2,4,∗ and kaoutar ghomari3,4 1department of mathematics, faculy of exact and applicable sciences, university of oran1 ahmed ben bella, algeria 2high school of electrical engineering and energetics-oran, algeria 3department of mathematics and informatics, enporan, algeria 4laboratory of fundamental and applicable mathematics of oran (lmfao), algeria ∗corresponding author: bmessirdi@yahoo.fr abstract. we describe in this paper a significant spectral reduction method for born-oppenheimer operators with regular potentials, which leads to an adaptable birkhoff normal form theorem for the associated effective hamiltonians. as illustration of the established results, we compute the birkhoff normal form in fermi resonance. 1. introduction for a molecular system with n electrons and n′ nuclei, the hamiltonian, under the born-oppenheimer approximation, can be written as: p(h) = −h2∆x + q(x) , q(x) = −∆y + v (x,y) on l2(rnx × rpy) where n = 3n and p = 3n′ and where h > 0 is a small parameter playing the role of the semi-classical parameter. ∆x (resp. ∆y) is the laplace operator with respect to x (resp. y), x ∈ rn and y ∈ rp, n,n′ ≥ 1, v is the interaction potential between particles. p(h) is called the born-oppenheimer received 2019-10-12; accepted 2019-11-11; published 2020-03-02. 2010 mathematics subject classification. 58k50, 81s10, 47a20. key words and phrases. born-oppenheimer operator; effective hamiltonian; birkhoff normal form; harmonic oscillator; fermi resonance. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 183 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-183 int. j. anal. appl. 18 (2) (2020) 184 hamiltonian. q(x) is the electronic hamiltonian defined on l2(rpy). it is well known that if v (x,y) is smooth verifying suitable assumptions, then z ∈ σ(p(h)) ⇐⇒ z ∈ σ(f(z)) where f(z) is a semiclassical analytic pseudodifferential operator on l2 (rnx) and σ stands for the spectrum, (see e.g. [10], [11], [3]), the main idea, due to born and oppenheimer in [5], is to replace, for fixed x, the operator q(x) by its eigenvalues. this reduction is possible thanks to the pseudodifferential calculus with operator valued symbols. then we are led to consider, the reduced operator (called the effective hamiltonian in the born-oppenheimer approximation): peff (h) = −h2∆x + λ1(x) where λ1(x) is the lowest eigenvalue of q(x), by the minimax principle λ1(x) is simple and analytic if v is sufficiently smooth. motivated by various physical questions we consider the connected problems in the asymptotic h → 0+, note that through standard semiclassical analysis peff (h) can explain the complete spectral picture of p(h) modulo errors in h. we wish to describe the birkhoff normal form near an equilibrium point of p(h). it is well known that a more precise description of the vibrational energies of a molecule is given by the harmonic oscillator, our approach here is to replace q(x) in the born-oppenheimer hamitonian p(h) by its lowest eigenvalue λ1(x) and thus, we are reduced to an effective h-pseudodifferential operator op(eλ), with symbol eλ depending only on (x,ξ). the normal forms in the born-oppenheimer approximation, are introduced here as being those of the schrödinger effective operator peff (h) on l 2 (rnx) . birkhoff normal form is one of the basic tools in quantum and semiclassical mechanics (see e.g. [7], [8]), it has already been used by birkhoff [4] to study some problems of dynamical systems. precisely, the goal of this paper is to analyze the notion of the birkhoff normal form near an equilibrium point and discuss the dynamical consequences for the schrödinger hamiltonian p(h). suppose 0 is a nondegenerate local minimum of λ1(x), by applying semiclassical techniques especially the pseudodifferential calculus with operator valued symbols and the classical quantum formal birkhoff normal form theorem, we show that we can find a canonical transformation putting p(h) as a reasonable perturbation of −∆x + 1 2 〈λ′′1 (0)x,x〉 modulo o(h2). our approach is natural, it consists in computing the normal form of the effective hamiltonian −h2∆x + λ1(x) after reduction of the operator p(h) to peff (h). to our knowledge this is the first attempt to determine the birkoff normal forms for the born-oppenheimer hamiltonians. in section 2, we recall some results on pseudodifferential operators with operator valued symbols. then, we give a representation of the effective hamiltonian and obtain wkb solutions of the hamiltonian p(h). in section 3, we investigate the theorem of birkhoff normal form near an equilibrium point in infinite int. j. anal. appl. 18 (2) (2020) 185 dimension in the born-oppenheimer approximation for p(h) via the effective hamiltonian peff (h). in the fourth section we compute the birkhoff normal form of peff (h) in the fermi resonance. 2. reduction to an effective operator in this section we explain the construction of wkb solutions for the hamiltonian p(h) and several mathematical results concerning the pseudodifferential calculus with operator valued symbols of the bornoppenheimer approximation. for further informations about the pseudodifferential calculus and bkw method we refer the reader to the works of balazard-konlein [2], messirdi [10], baklouti [1] and other authors. 2.1. pseudodifferential calculus with operator valued symbols. for m ∈ r, ω a bounded open subset of rnx and h a complex hilbert space, consider the space of formal power series: sm(ω,h) =   ∞∑ j=0 h−m+j/2sj(x) : sj ∈ c∞(ω,h)   where c∞(ω,h) is the space of c∞-functions mapping ω into h. given ψ ∈ c∞(ω,r) and u a neighborhood of 0 in rnx, we set: ω∗ = {(x,ξ) ∈ ω ×cn : ξ − i∇ψ(x) ∈u} and s0(ω∗,l(h,k)) =   ∞∑ j=0 hjaj(x,ξ) : aj ∈ c∞(ω∗,l(h,k))   where k is hilbert space and l(h,k) is the algebra of all continuous linear operators from h into k. the operator valued functions in s0(ω∗,l(h,k)) are called symbols. for any symbol a = a(x,ξ; h) in s0(ω∗,l(h,k)), by analogy with the action of differential operators on the space e−ψ(x)/hsm(ω,h), one can define an operator op(a) from e−ψ(x)/hsm(ω,h) into e−ψ(x)/hsm(ω,k) by the formula: op(a) ( e−ψ(x)/hs(x,h) ) = e−ψ(x)/h ∑ α∈nn h|α| i|α|α! ∂αξ a(x,i∇ψ(x); h)∂ α y ( s(y,h)eχ(x,y)/h ) y=x (2.1) χ(x,y) = ψ(y) −ψ(x) − (y −x).∇ψ(x) = o ( |x−y|2 ) , s ∈ sm(ω,h). op(a) is called h-pseudodifferential operator with operator valued symbol a(x,ξ; h) = ∞∑ j=0 hjaj(x,ξ). the function a0(x,ξ) (coefficient of h 0) is called principal symbol of the h-pseudodifferential operator op(a). furthermore, such operators verify: eψ(x)/hop(a) ( e−ψ(x)/hs(x,h) ) ∈ sm(ω,h) int. j. anal. appl. 18 (2) (2020) 186 and can be composed using the formula: op(b) ◦op(a) = op(b]a) where a ∈ s0(ω∗,l(h,k)), b ∈ s0(ω∗,l(k,l)) (l is a third hilbert space), the range of op(a) is contained in the domain of op(b), and b]a(x,ξ,h) = ∑ α∈nn h|α| i|α|α! ∂αξ b(x,ξ; h)∂ α xa(x,ξ; h) ∈ s 0(ω∗,l(h,l)). (2.2) this formula makes it possible to inverse asymptotically operators op(a) whose principal symbol a0(x,ξ) is invertible as a linear operator from h into k. 2.2. bkw solutions (scalar case). let us take h = c and recall the following result: theorem 2.1. ( [9]) let a(x,ξ; h) = ∞∑ j=0 hjaj(x,ξ) ∈ s0(ω∗,c) be such that a0(x,ξ) = ξ2 + λ(x) where λ ∈ c∞(ω,r), λ ≥ 0, λ−1(0) = {0} , λ′(0) = 0 and λ′′(0) > 0. let c0 > 0 and n0 be the number of eigenvalues of −∆x + 1 2 〈λ′′(0)x,x〉 in the compact interval [0,c0]. denote by e1, ...,en0 these eigenvalues. then there are formal series: ek(h) = ekh + ∞∑ j=1 ek,jh 1+j/2 and ak(x,h) ∈ smk(ω,c), ek,j, mk ∈ r, k ∈{1, ...,n0} , such that (op(a) −ek(h)) ( e−ψ(x)/hak ) = 0 in e−ψ(x)/hsmk(ω,c) where ψ(x) is the agmon distance associated to the metric λ(x)dx2. the functions e−ψ(x)/hak(x,h) are called the bkw solutions. 2.3. bkw solutions (general case). let v ∈ c∞(ω,l(h2(rpy),l2(rpy))) be ∆-compact: v (x,y) (−∆y + 1) −1 ∈ c∞(ω,l(l2(rpy))) where ω is a bounded open subset of rnx. thus, p(h) is selfadjoint on l 2(rnx×rpy) with domain h2(rnx×rpy) as well as the operator q(x) on l2(rpy) with domain h 2(rpy). denote λ1(x) = inf(σ(q(x))) the lowest energy level (ground state) of operator q(x). suppose that λ1(x) is an isolated eigenvalue of finite multiplicity of q(x), having unique and non-degenerate minimum at 0 : λ1(x) ≥ 0, λ−11 (0) = {0} , λ ′ 1(0) = 0, λ ′′ 1 (0) > 0, (2.3) int. j. anal. appl. 18 (2) (2020) 187 and that λ1(x) is separated from the rest of the spectrum σ(q(x)), i.e., inf x∈rn (inf (σ(q(x))\{λ1(x)})) > 0. (2.4) we also denote by u1(x,y) the first eigenfunction of q(x) associated with λ1(x) and normalized by ‖u1(x,.)‖l2(rpy) = 1 for all x ∈ r n. it can be shown that λ1 ∈ c∞(ω,r) and u1 ∈ c∞(ω,h2(rpy)) (cf. [10]). in particular, the assumption (2.4) implies that the orthogonal projection π(x) on the subspace of l2(rpy) spanned by u1(x,.), x ∈ ω, is c2-regular with respect to x (see [6]). to construct bkw solutions of p(h), the idea here is to use the pseudodifferential calculus with operator valued symbols developed in subsection 2.1. consider, for λ ∈ c, the following symbol: aλ(x; ξ) =   ξ2 + q(x) −λ u1 〈.,u1〉y 0   ∈ s0(ω∗,l(h2(rpy) ⊕c,l2(rpy) ⊕c)), where 〈.,u1〉y is the inner product in l 2(rpy). it follows from the assumptions and (2.1) that: op(aλ) =   p(h) −λ u1 〈.,u1〉y 0   is h-pseudodifferential operator from e−ψ(x)/hsm(ω,h2(rpy)) into e −ψ(x)/hsm(ω,l2(rpy)), with operator valued symbol aλ, where ψ(x) is the agmon distance associated to the metric λ1(x)dx 2. we now describe a method for finding the inverse of op(aλ). using the fact that (∇ψ)2(x) = λ1(x) and the gap assumption (2.4), one can easily show that for |λ| small enough and ξ close enough to i∇ψ(x), re ( π̂(x)q(x)π̂(x) −λ ) > 0 and aλ is invertible with inverse: b0(x,ξ; λ) =   π̂(x) ( ξ2 + π̂(x)q(x)π̂(x) −λ )−1 π̂(x) u1 〈.,u1〉y λ− ξ 2 −λ1(x)   where π̂(x) = 1 − π(x) (see e.g. [3]). in particular, b0(x,ξ; λ) ∈ s0(ω∗,l(l2(rpy) ⊕c,h 2(rpy) ⊕c)). then using the composition formula (2.2), it is easy to construct a symbol: bλ(x,ξ; h) = b0(x,ξ; λ) + hb1(x,ξ; λ) + h 2b2(x,ξ; λ) + ... bλ(x,ξ; h) ∈ s0(ω∗,l(l2(rpy) ⊕c,h 2(rpy) ⊕c)), int. j. anal. appl. 18 (2) (2020) 188 such that aλ]bλ(x,ξ; h) = 1 op(aλ) ◦op(bλ) = i i is the identity operator on e−ψ(x)/hsm(ω,l2(rpy) ⊕c). let us pose: op(bλ) =   e(λ) e+(λ) e−(λ) e∓(λ)   . by lemma 3.1 in [3], we also know that e∓(λ) = op(eλ(x,ξ; λ)) is h-pseudodifferential operator with symbol eλ(x,ξ; λ) ∈ s0(ω∗,c) and its principal symbol is e0(x,ξ; λ) = λ−ξ2 −λ1(x). in particular, f(λ) = λ−e∓(λ) is a scalar h-pseudodifferential operator with principal symbol ξ2 + λ1(x). moreover, we have the following fundamental spectral reduction: λ ∈ σ(p(h)) ⇐⇒ λ ∈ σ(f(λ)). hence, the spectral study of the hamiltonian p(h) on l2(rnx × rpy) is reduced to that of the hpseudodifferential operator f(λ) on l2(rnx) so-called effective hamiltonian of p(h). now use theorem 2.1 with f(λ), |λ| small enough, we find bkw solutions of p(h) as formal series ek(h) = ekh + ∞∑ j=1 ek,jh 1+j/2 and ak ∈ smk(ω,c), such that: (f(ek(h)) −ek(h)) ( e−ψ(x)/hak ) = 0 in the exponentially weighted symbol space e−ψ(x)/hsmk(ω,c). in fact, one can show in many situations that f(λ) = peff (h) + o(h2), which makes it easy to compare (using, for example, the maximum principle) the eigenvalues of p(h) and those of peff (h), and then identify them when h decays to zero fast enough [6]. this reduction will justify in the next section our definition of the normal birkhoff forms of p(h) as those of the effective hamiltonian peff (h). 3. reduction to birkhoff normal form for the effective hamiltonian there exists a very convenient way of constructing a canonical transformation such that we conserve the hamiltonian structure of p(h) by using the birkhoff normal form theorem via the effective hamiltonian peff (h). definition 3.1. we call normal forms of the semi-classical operator p(h), the birkhoff normal forms of the associated effective hamiltonian peff (h). the general philosophy will consist in transforming peff (h) in such a way that the new hamiltonian becomes ĥ2 + λ where ĥ2 is the harmonic oscillator and λ is a reasonable perturbation term who commut int. j. anal. appl. 18 (2) (2020) 189 with ĥ2. we consider here ω = rnx and assume that the hessian matrix λ ′′ 1 (0) is diagonal, let ( 2ν21, ..., 2ν 2 n ) be its eigenvalues, with νj > 0 and ν = (ν1, ...,νn). the rescaling xj → √ νjxj, x = (x1, ...,xn) , transforms p(h) as well as peff (h) into: peff (h) = ĥ2 + γ(x) where ĥ2 is the harmonic oscillator n∑ j=1 νj ( −h2 ∂ 2 ∂x2 j + x2j ) and γ(x) is a smooth function such that γ(x) = o( |x|3) as |x| → 0. in general, γ does not commute with ĥ2, on the other hand we do not have enough information on this perturbation, for that we will use the birkhoff normal form of p(h) which is a transformation of the previous type but more adapted and less restrictive. let: sd (m) =   a (x,ξ; h) : rnx ×rnξ × ]0, 1] −→ c, depends smoothly on x and ξ and for all α ∈ n2n, ∣∣∣∂α(x,ξ)a (x,ξ; h)∣∣∣ ≤ cαhd (1 + |x|2 + |ξ|2)m/2 , cα > 0, uniformly with respect x,ξ and h   where m,d ∈ r. sd (m) is called the semiclassical space of symbols of order d and degree m. for a ∈ sd (m) and u ∈ c∞0 (r2n), we set: (au) (x) = (op~ (a) u) (x) = (2πh) −n ∫ r2n eih −1〈x−x′,ξ〉a ( x + x′ 2 ,ξ; h ) u (x′) dx′dξ. (3.1) a is unbounded linear operator on l2 (rn) with domain c∞0 (r 2n) the space of infinitely differentiable functions on r2n with compact support, a : c∞0 (r 2n) −→ c∞(r2n) is called a semiclassical pseudodifferential operator with h-weyl symbol a of order d and degree m. ψd (m,rn) denotes the set of all semiclassical pseudodifferential operators with symbols in the class sd (m) . different classes of symbols can also be defined, but for our purpose this class is enough. for example, the h-weyl symbol of the harmonic oscillator ĥ2 is the polynomial h2 = n∑ j=1 νj ( ξ2j + x 2 j ) . now, we introduce the space s to be the set of formal series: s =   ∑ α,β∈nn,`∈n tα,β,lx αξβh` : tα,β,l ∈ c for all α,β ∈ nn,` ∈ n   where the degree of xαξβh` is defined by |α| + |β| + 2`, α,β ∈ nn, ` ∈ n, for technical reasons that of h is double-counted. let n ∈ n. let dn be the finite dimensional vector space: dn =   ∑ α,β∈nn,`∈n tα,β,lx αξβh` : tα,β,l ∈ c, α,β ∈ nn,` ∈ n such that |α| + |β| + 2` = n   and on =   ∑ α,β∈nn,`∈n tα,β,lx αξβh` : tα,β,l = 0 if |α| + |β| + 2` < n   . int. j. anal. appl. 18 (2) (2020) 190 note that, for all n ∈ n, dn and on are subspaces of s and s = o0 ⊃o1 ⊃, ..., ⋂ n on = {0} . let 〈., .〉w be the weyl bracket defined on s by: 〈f,g〉w = f̂ĝ − ĝf̂ where f̂ and ĝ are the h-weyl quantizations of symbols f and g, respectively. precisely, 〈ft ,gt〉w = σw (〈f,g〉w ) = σw ( f̂ĝ − ĝf̂ ) where ft and gt are formal taylor series at the origin of f and g in s, respectively and σw denotes the h-weyl symbol. then, 〈., .〉w is antisymmetric satisfying the jacobi identity: 〈〈ft ,g〉w ,ht〉w + 〈〈ht ,ft〉w ,gt〉w + 〈〈gt ,ht〉w ,ft〉w = 0 and the leibniz identity: 〈ft ,gtht〉w = 〈ft ,gt〉w ht + gt 〈ft ,ht〉w . thus, s equipped with the weyl bracket is a lie algebra such that: 〈h,xj〉w = 〈h,ξj〉w = 0 and 〈ξj,xj〉w = −ih, for all j = 1, ...,n x = (x1, ...,xn) and ξ = (ξ1, ...,ξn) ∈ rn. for and any s ∈s, we define a map: ads : s −→s p 7→ ads(p) = 〈s,p〉w which is called the adjoint action. s has a representation on itself, the adjoint representation defined via the map ad. let us consider the important special case of this concept, which is the adjoint action ads for s ∈d2, and especially adh2. let c [z,z,h] be the c-linear space of polynomials spanned by z αzβh` of degree |α|+|β|+2`; α,β ∈ nn,` ∈ n, where z = (x1 + iξ1, ...,xn + iξn) ∈ cn and z is the complex conjugate of z. then, b = { zβzγ : z ∈ cn; β,γ ∈ nn } is a natural basis of c [z,z,h] . the next proposition gives some important properties and results on adh2. proposition 3.1. ( [7]) 1) ih−1adh2 (p) = {h2,p} , where {h2,p} = n∑ j=1 ∂h2 ∂ξj ∂p ∂xj − ∂h2 ∂xj ∂p ∂ξj is the classical poisson bracket. 2) adh2 is diagonal on b, in the sense that adh2 ( zβzγ ) = h〈γ −β,ν〉zβzγ. int. j. anal. appl. 18 (2) (2020) 191 the assumption (2.3) implies that λ1(x) ∈o3, and since h2 + λ1(x) ∈d2, the quantum birkhoff normal form theorem for peff (h) can now be formulated as follows: theorem 3.1. for r ∈o3, there exist s and t in the subspace o3 with real coefficients such that: eih −1ads (h2 + r) = h2 + t where t = t3 + t4 + ... and tj ∈dj commutes with h2 : 〈h2,t〉w = 0. this result is a direct consequence of the birkoff normal form theorem shown for example in the article by ghomari and messirdi [7]. we gain, compared with the bkw constructions developed in the second section, the commutative property for weyl product between the harmonic oscillator and the rest of reduction in birkhoff normal form of the hamiltonian. the birkhoff normal form is a more usable semi-classical reduction involving other interesting spectral properties, especially it conserve the hamiltonian structure and contains enough informations to study the quantum resonances. 4. birkhoff normal form for p(h) in fermi resonance it has been established in the previous sections that p(h) can be reduced to the effective hamiltonian peff (h) modulo o(h2). thus, it is natural to define the birkhoff normal forms of p(h) as those of peff (h) modulo o(h2). let us recall the definitions of the different relations of resonances for the frequencies (ν1, ...,νn) of ĥ2 associated with the eigenvalues of the matrix λ′′1 (0). we say that the frequencies (ν1, ...,νn) are resonant if they are dependent over z, i.e., there exist integers d1, ...,dn ∈ z, not all zero, such that d1ν1 +...+dnνn = 0. the number d = n∑ j=1 |dj| is called the degree of resonance of peff (h). in the particular resonant case where νj = νcdj for all j = 1, ...,n with νc > 0 and d1, ...,dn ∈ n, (νj)j are said to be completely resonant. as an application we study the structure of birkhoff normal form in fermi resonance (νj, 2νj). we compute the birkhoff normal form of peff (h) in the case of 1 : 2 resonance. fermi resonances provide an essential mechanism for intramolecular vibrational energy flow and often dominate the vibrational dynamics in highly excited molecules. first discovered for the co2 molecule, fermi resonances are seen for many molecules. fermi resonance. the harmonic oscillator in fermi resonance is given by: ĥ2 = ( −h2 ∂2 ∂x21 + x21 ) + 2 ( −h2 ∂2 ∂x22 + x22 ) (4.1) int. j. anal. appl. 18 (2) (2020) 192 with symbol h2 = |z1| 2 +2 |z2| 2 where zj = xj+iξj, j = 1, 2. we construct k3 ∈d3 such that 〈h2,k3〉w = 0, so k3 = ∑ 2`+|α|+|β|=3 h`zαzβ with 〈ν,β −α〉 = 0. thus, k3 is generated by the monomials z21z2 and z2z 2 1 and since k3 is real, we can write: k3 = ρre(z 2 1z2), ρ ∈ r. consequently, the birkhoff normal form of the effective hamiltonian peff (h) in the fermi resonance, h2 + w is equal to h2 + k3 + o4, with ρ = 1 2 √ 2 ∂3w ∂x21∂x2 (0) see [8]. the weyl quantization k̂3 of k3 is given by: k̂3 = ρoph ( re(z21z2) ) = ρoph ( x21x2 + 2x1ξ1ξ2 − ξ 2 1x2 ) = ρ [ x21x2 −h 2 ( 2x1 ∂2 ∂x1∂x2 −x2 ∂2 ∂x21 + ∂ ∂x2 )] and finally, ĥ2 + k̂3 = ( −h2 ∂2 ∂x21 + x21 ) + 2 ( −h2 ∂2 ∂x22 + x22 ) (4.2) +ρ [ x21x2 −h 2 ( 2x1 ∂2 ∂x1∂x2 −x2 ∂2 ∂x21 + ∂ ∂x2 )] . acknowledgements: this work is supported by laboratory of fundamental and applicable mathematics of oran (lmfao) and is dedicated to professor bekkai messirdi on the occasion of his 61th birthday. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h. baklouti, asymptotic expansion for the widths of resonances in born-oppenheimer approximation, asymptot. anal. 69(1-2) (2010), 1-29. [2] a. balazard-konlein, asymptotique semi-classique du spectre pour des opérateurs à symbole operatoriel, c. r. acad. sci. paris sér. i math. 301 (1985), 903-906. [3] s. belmohoub and b. messirdi, singular schrödinger operators via grushin problem method, ann. oradea univ. math. fascicola. 24(1) (2017), 83-91. [4] g.d. birkhoff, dynamical systems, ams colloq. publ. 9, ams new york. (1927). [5] m. born and r. oppenheimer, zur quantentheorie der molekeln, ann. physics. 84 (1927), 457-484. [6] j.m. combes and r. seiler, regularity and asymptotic properties of the discrete spectrum of electronic hamiltonians, int. j. quantum chem. 14 (1978), 213-229. int. j. anal. appl. 18 (2) (2020) 193 [7] k. ghomari and b. messirdi, quantum birkhoff-gustavson normal form in some completely resonant cases, j. math. anal. appl. 378 (2011), 306-313. [8] k. ghomari and b. messirdi, hamiltonians spectrum in fermi resonance via the birkhoff-gustavson normal form, int. j. contemp. math. sciences. 4(35) (2009), 1701-1707. [9] b. helffer and j. sjöstrand, multiple wells in the semiclassical limit i, commun. part. diff. equ. 9(4) (1984), 337-408. [10] b. messirdi, asymptotique de born-oppenheimer pour la prédissociation moléculaire (cas de potentiels réguliers), ann. henri poincaré (a). 61 (1992), 255-292. [11] b. messirdi and k. ghomari, resonances of a two-state semiclassical schrödinger hamiltonians, appl. anal. 86(2) (2007), 187-204. 1. introduction 2. reduction to an effective operator 2.1. pseudodifferential calculus with operator valued symbols 2.2. bkw solutions (scalar case) 2.3. bkw solutions (general case) 3. reduction to birkhoff normal form for the effective hamiltonian 4. birkhoff normal form for p(h) in fermi resonance references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 1-8 http://www.etamaths.com a new stability of the s-essential spectrum of multivalued linear operators aymen ammar∗, slim fakhfakh and aref jeribi abstract. we unfold in this paper two main results. in the first, we give the necessary assumptions for three linear relations a, b and s such that σeap,s(a + b) = σeap,s(a) and σeδ,s(a + b) = σeδ,s(a) is true. in the second, considering the fact that the linear relations a, b and s are not precompact or relatively precompact, we can show that σeap,s(a + b) = σeap,s(a) is true. 1. introduction assume that a and s are two bounded operators. accordingly the map p(λ) := λs −a is a linear bundle. in fact many problems of mathematical physics (for example quantum theory, transport theory,...) are meant to shed light on the essential spectra of λs−a. the spectral theory of fredholm linear relations is one case worth mentioning given that this type of operators is unstable under the operation closure inverse and conjugate. but this does not hold for the case of multivalued linear operators. on this account, the investigation of the s-essential spectra of multivalued linear operators seems interesting. historically, in [11] a. jeribi, n. moalla, and s. yengui gave a characterization of the essential spectrum of the operator pencil in order to extend many known results in the literature. in [1] f. abdmouleh, a. ammar, and a. jeribi pursued the study of the s-essential spectra and investigated the s-browder, the s-upper semi-browder, and the s-lower semi-browder essential spectra of bounded linear operators on a banach space x and they introduced the s-riesz projection. moreover, they extended the results of f. abdmouleh and a. jeribi [3] to various types of s-essential spectra. in fact, they gave the characterization of the s-essential spectra of the sum of two bounded linear operators. (see for example [10]). in [4] tereza alvarez, a. ammar, and a. jeribi pursued the study of the s-essential spectra and characterized some s-essential spectra of a closed linear relation in terms of certain linear relations type semi fredholm. in [6] a. ammar characterized some essential spectra of a closed linear relation in terms of certain linear relations type α− and β− atkinson. throughout this work, let x, y and z be three complex normed linear spaces, over k = r or c. a multivalued linear operator (or a linear relation) a from x to y is a mapping from a subspace of x, d(a) := {x ∈ x : ax 6= ∅} called the domain of a, into p(y )\{∅} (collection of non-empty subsets of y) such that a(αx + βy) = αa(x) + βa(y) for all non-zero scalars α,β ∈ c and x,y ∈d(a). if a maps the points of its domain to singletons, then a is said to be a single valued linear operator (or simply an operator). a linear relation is uniquely determined by its graph, g(a), which is defined by g(a) := {(x,y) ∈ x ×y : x ∈d(a) and y ∈ ax}. in this notation, lr(x,y ) denotes the class of all linear relations on x into y . if x = y , we would simply note lr(x,x) := lr(x). the inverse of a is the linear relation a−1 defined by g(a−1) := {(y,x) ∈ y ×x : (x,y) ∈ g(a)}. received 23rd november, 2016; accepted 23rd february, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 47a06. key words and phrases. linear relations; relatively precompact; relatively bounded; s-essential approximate point; s-essential defect. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 ammar, fakhfakh and jeribi the subspace n(a) := a−1(0) is called the null space of a, and a is called injective if n(a) = {0}; i.e., if a−1 is a single valued linear operator. the range of a is the subspace r(a) := a(d(a)), and a is called surjective if r(a) = y . when a is injective and surjective, we say that a is bijective. the quantities α(a) := dim(n(a)) and β(a) := codim(r(a)) = dim(y/r(a)) are called the nullity (or the kernel index) and the deficiency of a, respectively. we also write β(a) := codim(r(a)). the index of a is defined by i(a) := α(a) −β(a) provided that both α(a) and β(a) are not infinite. if α(a) and β(a) are infinite, then a is said to have no index. the set of upper semi-fredholm linear relations from x into y is defined by: φ+(x,y ) := {t ∈cr(x,y ) : r(t) is closed, and α(t) < ∞}, the set of lower semi-fredholm linear relations from x into y is defined by: φ−(x,y ) := {t ∈cr(x,y ) : r(t) is closed, and β(t) < ∞}. if x = y , we would simply note φ+(x,y ) and φ−(x,y ) by respectively φ+(x) and φ−(x). let m be a subspace of x such that m ∩d(a) 6= ∅ and let a ∈lr(x,y ). then, the restriction a|m is the linear relation given by: g(a|m ) := {(m,y) ∈ g(a) : m ∈ m} = g(a) ∩ (m ×y ). for a, b ∈lr(x,y ) and s ∈lr(y,z), the sum a + b and the product sa are the linear relations defined by g(a + b) := {(x,y + z) ∈ x ×y : (x,y) ∈ g(a) and (x,z) ∈ g(b)}, and g(sa) : {(x,z) ∈ x ×z : (x,y) ∈ g(a), (y,z) ∈ g(s) for some y ∈ y} respectively. if λ ∈ k, then λa is defined by: g(λa) := {(x,λy) : (x,y) ∈ g(a)}. if a ∈lr(x) and λ ∈ k, then the linear relation λ−a is given by: g(λ−a) := {(x,y −λx) : (x,y) ∈ g(a)}. we note that ‖ax‖ and ‖a‖ are not real norms. in fact, a nonzero relation can have a zero norm. a is said to be closed if its graph g(a) is a closed subspace of x ×y . the closure of a, denoted by a, is defined in terms of its graph g(a) := g(a). we denote by cr(x,y ) the class of all closed linear relations on x into y . if x = y , we would simply note cr(x,x) := cr(x). if a is an extension of a (that is, a|d(a)), we say that a is closable. let a ∈ lr(x,y ). we say that a is continuous if for each neighbourhood v in r(a), the inverse image a−1(v ) is a neighbourhood in d(a) equivalently ‖a‖ < ∞; open if a−1 is continuous; bounded if d(a) = x and a is continuous; bounded below if it is injective and open; and compact if qaa(bd(a)) is compact in y (bd(a) := {x ∈ d(a) : ‖x‖ ≤ 1}). we denote by kr(x,y ) the class of all compact linear relations on x into y . if x = y , we would simply note kr(x,x) := kr(x). we say that a is precompact if qttbd(t) is totally bounded in y , and strictly singular if there is no infinite dimensional subspace m of d(a) for which a|m is injective, and open. if x is a normed linear space, then x ′ will denote the dual norm of x, i.e., the space of all continuous linear functionals x′ are defined on x with the norm ‖x′‖ = inf{λ : |x′x| ≤ λ‖x‖ for all x ∈ x}. if k ⊂ x and l ⊂ x ′ , we shall adopt the following notations: k⊥ := {x′ ∈ x ′ : x′ = 0 for all x ∈ k}, l> := {x ∈ x : x′ = 0 for all x′ ∈ l}. clearly, k⊥ and l> are closed linear subspaces of x ′ and x respectively. the adjoint of t , t ′, is defined by g(a′) = g(−a−1)⊥ ⊂ y ′ ×x ′ where 〈(y,x), (y′,x′)〉 := 〈x,x′〉 + 〈y,y′〉. this means that stability of the s-essential spectrum of linear relations 3 (y′,x′) ∈ g(a′) if, and only if, y′y −x′x = 0 for all (x,y) ∈ g(t). similarly, we have y′y = x′x for all y ∈ ax, x ∈d(a). hence x′ ∈ a′y if, and only if, y′ax = x′x for all x ∈d(a). let x be a complex banach space and let a ∈ cr(x,y ). suppose that s ∈ lr(x) is a-bounded with a-bounded δ < 1 such that s(0) ⊂ a(0) and d(a) ⊂ d(s). we define the s resolvent set of a by ρs(a) := {λ ∈ c : λs −a is bijective}. in this work, we are concerned with the following s-essential approximate point spectrum of a defined by: σeap,s(a) := ⋂ k∈ka(x) σap,s(a + k). similarly we are concerned with the following s-essential defect spectrum of a defined by: σeδ,s(a) := ⋂ k∈ka(x) σδ,s(a + k), where ka(x) := {k ∈kr(x) : d(a) ⊂d(k) and k(0) ⊂ a(0)}, σap,s(a) := {λ ∈ c : λs −a is not bounded below}, and σδ,s(a) := {λ ∈ c : λs −a is not surjective}. note that if s = i, (the identity operator on x), we recover the usual definition of the essential spectra of a bounded linear operator a. the purpose of this paper is to extend the results in [8] mentioned above to the general case of sessentiel stability in the first place. in the second place, in other hypotheses, we show the stability of s-essential approximate point spectrum. we organize the paper in the following way. section 2 consists in establishing some preliminary results which will be needed in the sequel. the main results of section 3 are lemma 3.1 and lemma 3.2, which give information concerning the equivalence of norm. in section 4, we investigate the stability of the sessential approximate point spectrum and the s-essential defect spectrum of closed and closable linear relations under relatively compact and precompact perturbations on a banach space (see theorem 4.1), and under different hypotheses we find the stability of the s-essential approximate point spectrum (see theorem 4.2). 2. preliminaries the goal of this section consists in establishing some preliminary results which will be needed in the sequel. definition 2.1. [9, definition, iv.3.1] let a ∈lr(x,y ), and let xa denote the vector space d(a) normed by ‖x‖a := ‖x‖ + ‖ax‖, for all x ∈d(a). let ga ∈lr(xa,x) be the identity injection of xa = (d(a),‖.‖a) into x, i.e., d(ga) = xa, ga(x) = x, for all x ∈ xa. definition 2.2. [9, definition, vii.2.1] let a, b ∈ lr(x, y ). b is said to be a-bounded (or bounded relative to a) if d(a) ⊂d(b) and there exist non-negative constants a, and b, such that ‖bx‖≤ a‖x‖+ b‖ax‖ for all x ∈d(a). (2.1) in that case the infimum of all the constant b which satisfies (2.1) is called the a-bound of b. we note that b is a-bounded if, and only if, d(a) ⊂d(b), and bga is bounded. definition 2.3. [9, definition vii.2.1] let a ∈ lr(x,y ). a relation b ∈ lr(x,y ) is said to be a-compact (or compact relative to a) if d(a) ⊂d(b) and bga is compact. b is called a-precompact (or precompact relative to a) if d(a) ⊂d(b) and bga is precompact. lemma 2.1. [7, lemma, 3.1] let s, t ∈lr(x,y ) satisfies s(0) ⊂ t(0) and d(t) ⊂d(s). if s is t -compact, then s is t -bounded. 4 ammar, fakhfakh and jeribi lemma 2.2. [7, lemma, 3.6] let a, b and s ∈lr(x,y ) satisfy b(0)∪s(0) ⊂ a(0). suppose that b is a-bounded with a-bound δ1, s is a-bounded with a-bound δ2, and y is complete. (i) if δ1 + δ2 < 1, and a is closed, then a + b + s is closed. (ii) if δ1 + δ2 < 1 2 , and a + b + s is closed, then a is closed. lemma 2.3. [7, lemma, 4.1] let s ∈ lr(x,y ) and a ∈ f+(x,y ) with dimd(a) = ∞. if s is precompact, then s is strictly singular. if additionally s(0) ⊂ a(0), then a + s ∈f+(x,y ). proposition 2.1. [5, theorem 2.17] let b ∈ lr(x,y ), a ∈ f+(x,y ) with g(b) ⊂ g(a) , and dim d(b) = ∞, then b ∈f+(x,y ). lemma 2.4. [2, lemma 2.3] let x be complete, t ∈cr(x), and k ∈kt (x). (i) if t ∈ φ+(x), then t + k ∈ φ+(x) with i(t + k) = i(t). (ii) if t ∈ φ−(x), then t + k ∈ φ−(x) with i(t + k) = i(t). proposition 2.2. [4, theorem 3.1] let x be complete, a ∈ cr(x) and λ ∈ c. if s ∈ lr(x) is a-bounded with a-bounded δ < 1 such that s(0) ⊂ a(0) and d(a) ⊂d(s), then (i) λ /∈ σeap,s(a) if, and only if, λs −a ∈ φ+(x) and i(λs −a) ≤ 0. (ii) λ /∈ σeδ,s(a) if, and only if, λs −a ∈ φ−(x) and i(λs −a) ≥ 0. to end this section, we present the following proposition suggested by cross in [9]. proposition 2.3. let a, b ∈lr(x,y ) (i) [9, corollary v.2.5] a ∈f+(x,y ) if, and only if, aga ∈f+(xa,y ). (ii) [9, corollary v.2.3] if a is precompact, then a is continuous. (iii) [9, proposition iii.1.5] let d(a) ⊂d(b). if b is continuous, then (a + b)′ = a′ + b′. (iv) [9, proposition v.5.15] let a ∈cr(x,y ). a ∈kr(x,y ) if, and only if, a′ ∈kr(y ′,x′). (v) [9, proposition v.7.5] a ∈f+(x,y ) if, and only if, a′ ∈f−(y ′ ,x ′ ) and a ′ ∈f+(y ′ ,x ′ ) if, and only if, a ∈f−(x,y ). (vi) [9, proposition v.7.8] if dim b(0) < ∞, then a+b−b ∈f+(x,y ) if, and only if, a ∈f+(x,y ). (vii) [9, proposition v.5.27] if a is closable. then a ∈f−(x,y ) if, and only if, aga ∈f−(xa,y ). (viii) [9, proposition v.5.12] let d(a) ⊂ d(b), and let a ∈ f−(x,y ). if b is precompact, then a + b ∈f−(x,y ). 3. main results in [9], book1 claims that ‖a‖ − ‖b‖ ≤ ‖a − b‖ is not in general true. he gives an example (see [9, exercise, ii.1.12]). in the first lemma in this section, we give a necessary and sufficient condition for two linear relations a and b so that the equality of ‖a‖−‖b‖≤‖a−b‖ become justified. lemma 3.1. let a, b ∈lr(x,y ). if b(0) ⊂ a(0) and d(a) ⊂d(b), then (i) ‖ax‖−‖bx‖≤‖ax−bx‖, for x ∈d(a). (ii) ‖ax‖−‖bx‖≤‖ax + bx‖, for x ∈d(a). proof we have for x ∈d(a), by lemma [4, lemma 2.2 (iii)], we get (a−b + b)x = ax, then ‖(a−b + b)x‖ = ‖ax‖, (3.1) (i) using [9, proposition, ii.1.5] and from eqs (3.1), we obtain ‖ax‖ ≤ ‖(a − b)x‖ + ‖bx‖. so ‖ax‖−‖bx‖≤‖ax−bx‖. stability of the s-essential spectrum of linear relations 5 (ii) using [9, proposition, ii.1.5] and from eqs (3.1), we obtain ‖ax‖‖ ≤ ‖(a + b)x‖ + ‖bx‖. so ‖ax‖−‖bx‖≤‖ax + bx‖. lemma 3.2. let a, b, and s ∈lr(x,y ) verifying b(0) ⊂ a(0) and λ ∈ c. if s is a-bounded with a-bound δ1 and b is a-bounded with a-bound δ2 such that δ2 + |λ|δ1 < 1, then ‖.‖a and ‖.‖λs−(a+b) are equivalent. in particular, ‖.‖a and ‖.‖λs−a are equivalent. proof since s is a-bounded with bound δ1 and b is a-bounded with bound δ2, there exist nonnegative constants a, b, a1 and b1 such that, for x ∈ d(a), ‖sx‖ ≤ a‖x‖ + b‖ax‖ and ‖bx‖ ≤ a1‖x‖+b1‖ax‖. so we have −‖bx‖≥−a1‖x‖−b1‖ax‖, thus ‖ax‖−‖bx‖≥−a1‖x‖+ (1−b1)‖ax‖. using lemma 3.1 (ii), we get ‖ax + bx‖≥−a1‖x‖ + (1 − b1)‖ax‖. on the other hand, ‖x‖λs−(a+b) = ‖x‖ + ‖(λs − (a + b))x‖, ≥ ‖x‖ + ‖(a + b))x‖−|λ|‖sx‖, ≥ ‖x‖−a1‖x‖ + (1 − b1)‖ax‖−|λ|‖sx‖, ≥ ‖x‖−a1‖x‖ + (1 − b1)‖ax‖−|λ|a‖x‖− b|λ|‖ax‖, ≥ (1 −a1 −|λ|a)‖x‖ + (1 − b1 −|λ|b)|‖ax‖, ≥ min(1 −a1 −|λ|a, 1 − b1 −|λ|b)|(‖x‖ + ‖ax‖). therefore, ‖x‖λs−(a+b) ≥ k|‖x‖a, with k = min(1 −a1 −|λ|a, 1 − b1 −|λ|b). on the other hand, we obtain ‖x‖λs−(a+b) = ‖x‖ + ‖(λs − (a + b))x‖, ≤ ‖x‖ + ‖ax‖ + ‖bx‖ + |λ|‖sx‖, ≤ ‖x‖ + a1‖x‖ + b1‖ax‖ + ‖ax‖ + |λ|a‖x‖ + b|λ|‖ax‖, ≤ (1 + a1 + |λ|a)‖x‖ + (1 + b1 + |λ|b)|‖ax‖, ≤ max(1 + a1 + |λ|a, 1 + b1 + |λ|b)|(‖x‖ + ‖ax‖). therefore, ‖x‖λs−(a+b) ≤ h|‖x‖a, with h = max(1 + a1 + |λ|a, 1 + b1 + |λ|b). we deduce that ‖.‖a and ‖.‖λs−(a+b) are equivalent. lemma 3.3. let a, b, and s ∈lr(x) and let λ ∈ c. (i) r((λs −a)gb) = r(λs −a). (ii) n((λs −a)gb) = n(λs −a). proof (i) using the fact that gbx = (gb) −1x = x, r(a) = ad(a) and d(ab) = b−1d(a). r((λs −a)gb) = (λs −a)gbd((λs −a)gb), = (λs −a)d((λs −a)gb), = (λs −a)gbd(λs −a), = (λs −a)d(λs −a), = r(λs −a). (ii)n((λs −a)gb) = {x ∈d((λs −a)gb), (λs −a)gb(x) = (λs −a)gb(0)}, = {x ∈d(λs −a), (λs −a)(x) = (λs −a)(0)}, = n(λs −a). proposition 3.1. let x be complete, let a, b, s ∈lr(x) satisfy b(0) ⊂ a(0) and let λ ∈ c. if b is a-precompact, then i(λs −a) = i(λs − (a + b)). 6 ammar, fakhfakh and jeribi proof since b is a-precompact, then bga is precompact, and x is complete. by remark [9, note v.1 p 134] bga is compact. i(λs −a) = i((λs −a)ga), by lemma 3.3, = i((λs −a)ga + bga), by lemma 2.4 (bga is compact), = i((λs − (a + b))ga), = i(λs − (a + b)), by lemma 3.3. lemma 3.4. let a, b ∈lr(x,y ) such that g(a) g(b). we have (i) α(a) ≤ α(b). (ii) β(b) ≤ β(a). (iii) i(a) ≤ i(b). proof (i) we have α(a) := dim(n(a)). then n(a) := {x ∈d(a) : (x, 0) ∈ g(a)}, {x ∈d(a) : (x, 0) ∈ g(b)}, = n(b|d(a)), ⊂ n(b). so, α(a) ≤ α(b). (ii) we have β(a) := codim(r(a)) = dim(y/r(a)). let y ∈ r(a). then, y ∈ ax for all x ∈ d(a). we get by g(a) g(b), y ∈ bx for all x ∈d(a). so, y ∈r(b|d(a)). thus y ∈r(b). we infer that y/r(b) ⊂ y/r(a). then β(b) ≤ β(a). (iii) i(a) := α(a) −β(a) ≤ α(b) −β(b) = i(b). lemma 3.5. let a, b ∈lr(x,y ). if g(a) g(b), then g(a) g(a + b). proof g(a) := {(x,y) ∈ x ×y : x ∈d(a) d(b) and y ∈ ax bx}, {(x,y) ∈ x ×y : x ∈d(a) ∩d(b) and y ∈ ax + bx}, := {(x,y) ∈ x ×y : x ∈d(a + b) and y ∈ (a + b)x}, := g(a + b). 4. stability of σeap,s(.) and σeδ,s(.) in this section, on one level, we study the stability of the s-essential approximate point spectrum and the s-essential defect spectrum of closed and closable linear relations under relatively precompact perturbations on a banach space. on another level, we study the stability of the s-essential approximate point spectrum but under assumptions different from those adopted above. theorem 4.1. let x be complete, a ∈ cr(x), b, s ∈ lr(x) satisfy b(0) ⊂ s(0) ⊂ a(0) and dimd(b) = ∞, and λ ∈ c. if s is a-bounded with a-bound δ1 and b is a-precompact with a-bound δ2 such that δ2 + |λ|δ1 < 1, then σeap,s(a + b) = σeap,s(a), and σeδ,s(a + b) = σeδ,s(a). proof let b be a-precompact, then bga is precompact, and x and xa are complete. by remark [9, note v.1 p 134], we get bga is compact. by lemma 2.1, we get bga is bounded, then b is a-bounded with a-bound δ2. using the fact that s is a-bounded with a-bound δ1 and δ2 + |λ|δ1 < 1 and by applying lemma 2.2 (i), we obtain λs − (a + b) is closed. suppose that λ /∈ σeap,s(a), then by proposition 2.2, λs − a ∈ φ+(x). by proposition 2.3 (i), we get (λs − a)gλs−a ∈ φ+(xa), which gives (λs − a)ga ∈ φ+(xa) by referring to lemma 3.2. since bga is compact and dimd(b) = dimd(bga) = ∞, then using lemma 2.3 it follows that (λs−(a+b))ga ∈ φ+(xa). by lemma 3.2, we obtain (λs−(a+b))gλs−(a+b) ∈ φ+(xa). using stability of the s-essential spectrum of linear relations 7 proposition 2.3 (i), we get λs − (a + b) ∈ φ+(x) and we have i(λs − a) = i(λs − (a + b)) by proposition 3.1, that is λ /∈ σeap,s(a+b) by proposition 2.2. so σeap,s(a+b) ⊆ σeap,s(a). conversely, let λ /∈ σeap,s(a+b). then by proposition 2.2, we have λs−(a+b) ∈ φ+(x). using proposition 2.3 (i), we get (λs−(a+b))gλs−(a+b) ∈ φ+(xa), which gives (λs−(a+b))ga ∈ φ+(xa) by referring to lemma 3.2. since bga is compact, then by lemma 2.3, it follows that (λs−a)ga ∈ φ+(xa). by lemma 3.2, we obtain (λs−a)gλs−a ∈ φ+(xa). using proposition 2.3 (i), we get λs−a ∈ φ+(x). we have i(λs −a) = i(λs − (a + b)) by proposition 3.1, that is λ /∈ σeap,s(a) by proposition 2.2. we infer that σeap,s(a + b) = σeap,s(a). now suppose that λ /∈ σeδ,s(a), then by proposition 2.2, we have λs − a ∈ φ−(x). applying proposition 2.3 (vii), we obtain (λs −a)gλs−a ∈ φ−(xa). using lemma 3.2, we get (λs −a)ga ∈ φ−(xa). since bga is precompact, then by proposition 2.3 (viii), we obtain (λs − (a + b))ga ∈ φ−(xa). resorting to lemma 3.2, we get (λs − (a + b))gλs−(a+b) ∈ φ−(xa). so applying proposition 2.3 (vii), we get (λs − (a + b)) ∈ φ−(x). we have i(λs − a) = i(λs − (a + b)) by proposition 3.1, that is λ /∈ σeδ,s(a + b) by proposition 2.2. then σeδ,s(a + b) ⊂ σeδ,s(a). conversely, let λ /∈ σeδ,s(a + b), then by proposition 2.2, we obtain λs − (a + b) ∈ φ−(x). using proposition 2.3 (vii), we get (λs − (a + b))gλs−(a+b) ∈ φ−(xa). applying lemma 3.2, we get (λs − (a + b))ga ∈ φ−(xa). the latter holds if, and only if, ((λs−(a + b))ga) ′ ∈ φ+(x ′ a) by proposition 2.3 (v). subsequently, using proposition 2.3 (ii) and (iii), we get ((λs − a)ga) ′ + (bga) ′ ∈ φ+(x ′ a). since bga is precompact, then by proposition 2.3 (iv) we have (bga) ′ is precompact. applying lemma 2.4, we have ((λs−a)ga) ′ ∈ φ+(x ′ a). besides using proposition 2.3 (v), we get (λs−a)ga ∈ φ−(xa). so by proposition 2.3 (vii), ((λs −a) ∈ φ−(x). we have i(λs −a) = i(λs − (a + b)) by proposition 3.1. that is λ /∈ σeδ,s(a) by proposition 2.2. we conclude that σeδ,s(a + b) = σeδ,s(a). theorem 4.2. let a ∈ cr(x), b, s ∈ lr(x) and let λ ∈ c. suppose that s is a-bounded with a-bound δ1 and b is a-bounded with a-bound δ2 such that δ2 + |λ|δ1 < 1. if g(b) g(λs) g(a) and dimd(b) = ∞, then (i) σeap,s(a + b) ⊂ σeap,s(a). (ii) if dim b(0) < ∞, then σeap,s(a + b) = σeap,s(a). proof since s is a-bounded with a-bound δ1 and b is a-bounded with a-bound δ2 such that δ2 + |λ|δ1 < 1, then applying lemma 2.2, we obtain λs − (a + b) is closed. (i) suppose that λ /∈ σeap,s(a), then by proposition 2.2, λs −a ∈ φ+(x) and i(λs −a) ≤ 0. since g(b) g(λs) and g(λs) g(a), then applying lemma 3.5, we get g(λs) g(λs −a), and then g(b) g(λs −a). on the one hand, we have g(λs − (a + b)) := {(x,y) ∈ x ×x : (x,y1) ∈ g(λs −a) and (x,y2) ∈ g(b) g(λs −a), where y = y1 + y2}, g(λs −a). on the other hand, dimd(λs − (a + b)) = dim(d(λs − a) ∩d(b)) = dimd(b) = ∞. then by proposition 2.1, we obtain λs − (a + b) ∈ φ+(x). we have g(λs − (a + b)) g(λs −a), using lemma 3.4, we get i(λs − (a + b)) ≤ (λs −a) ≤ 0. so by proposition 2.2, we obtain λ /∈ σeap,s(a + b). then σeap,s(a + b) ⊂ σeap,s(a). 8 ammar, fakhfakh and jeribi (ii) since g(b) g(λs) and g(λs) g(a), then applying lemma 3.5, we get g(λs) g(λs −a) and g(b) g(λs −a). by lemma 3.5, we obtain g(b) g(λs − (a + b)). on the one hand, g(λs −a−b + b) := {(x,y + z) ∈ x ×y : (x,y) ∈ g(λs − (a + b)) and (x,z) ∈ g(b)} {(x,y + z) ∈ x ×y : (x,y) ∈ g(λs − (a + b)) and (x,z) ∈ g(s) g(λs − (a + b))} := g(λs − (a + b)). on the other hand, dimd(λs − a − b + b) = dim(d(λs) ∩d(a) ∩d(b)) = dimd(b) = ∞. let λ /∈ σeap,s(a + b). then by proposition 2.2, we have λs − (a + b) ∈ φ+(x). since λs −a is closed and dim b(0) < 0, then λs−a−b + b is closed and we have g(λs−a−b + b) g(λs−(a + b)), dimd(λs−a−b+b) = ∞, then by proposition 2.1, we get λs−a−b+b ∈ φ+(x). thus by proposition 2.3 (vi), we obtain λs−a ∈ φ+(x). using lemma 3.4, we get i(λs−a) ≤ i(λs−a−b +b) ≤ (λs−(a + b)) ≤ 0. so by proposition2.2, we obtain λ /∈ σeap,s(a). thus, σeap,s(a) ⊂ σeap,s(a + b). we infer that σeap,s(a + b) = σeap,s(a). references [1] f. abdmouleh, a. ammar and a. jeribi, stability of the s-essential spectra on a banach space. math. slovaca 63(2) (2013), 299-320. [2] f. abdmouleh, t. alvarez, a. ammar and a.jeribi, spectral mapping theorem for rakocevic and schmoeger essential spectra of a multivalued linear operator, mediterr. j. math, 12(3) (2015), 10191031. [3] f. abdmouleh and a. jeribi, gustafson, weidman, kato, wolf, schechter, browder, rakocevic and schmoeger essential spectra of the sum of two bounded operators and application to a transport operator, math. nachr. 284(2-3) (2011), 166-176. [4] t. alvarez, a. ammar and a.jeribi, a characterization of some subsets of s-essential spectra of a multivalued linear operator, colloq. math. 135 (2014), 171186. [5] t. alvarez, r.w. cross and d. wilcox, multivalued fredholm type operators with abstract generalised inverses, j. math. anal. appl. 261 (2001), 403-417. [6] a. ammar, a characterization of some subsets of essential spectra of a multivalued linear operator, complex anal. oper. theory 11(1) (2017), 175-196. [7] a. ammar, s. fakhfakh and a. jeribi, relatively bounded perturbation and essential approximate point and defect spectrum of linear relations, prepint (2017). [8] a. ammar, s. fakhfakh and a. jeribi, stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations, j. pseudo-differ. oper. appl. 7(4) (2016), 493-509. [9] r.w. cross, multivalued linear operators, marcel dekker inc. (1998). [10] a. jeribi, spectral theory and applications of linear operators and block operator matrices, springer-verlag. new york (2015). [11] a. jeribi, n. moalla and s. yengui, s-essential spectra and application to an example of transport operators, math. methods appl. sci. 37(16) (2014), 2341-2353. department of mathematics, faculty of sciences of sfax, university of sfax, p.o.box 1171, 3000 sfax, tunisia ∗corresponding author: ammar aymen84@yahoo.fr 1. introduction 2. preliminaries 3. main results 4. stability of eap,s(.) and e,s(.) references int. j. anal. appl. (2022), 20:34 (inf, sup)-hesitant fuzzy ideals of bck/bci-algebras noppakao ratchakhwan1, pongpun julatha1, thiti gaketem2, pannawit khamrot3, rukchart prasertpong4, aiyared iampan2,∗ 1department of mathematics, faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand 2fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 3department of mathematics, faculty of science and agricultural technology, rajamangala university of technology lanna phitsanulok, phitsanulok 65000, thailand 4division of mathematics and statistics, faculty of science and technology, nakhon sawan rajabhat university, nakhon sawan 60000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. in this paper, we introduce the concept of (inf,sup)-hesitant fuzzy ideals, which is a generalization of the concept of interval-valued fuzzy ideals, in bck/bci-algebras and its related properties are investigated. the concept is established in terms of sets, fuzzy sets, negative fuzzy sets, intervalvalued fuzzy sets, pythagorean fuzzy sets, bipolar fuzzy sets and hesitant fuzzy sets. moreover, characterizations of ideals, fuzzy ideals, anti-fuzzy ideals, negative fuzzy ideals, pythagorean fuzzy ideals and bipolar fuzzy ideals of bck/bci-algebras are discussed in terms of (inf,sup)-hesitant fuzzy ideals and interval-valued fuzzy ideals. 1. introduction the concept of fuzzy sets, introduced by zadeh [3], has been widely and successfully applied in many branches: finite state machine, computer science, automata, artificial intelligence, expert, control engineering, robotics and theory of groups, semigroups, bck/bci-algebras and up-algebras. received: may 23, 2022. 2010 mathematics subject classification. 03b52, 03g25, 06f35, 08a72. key words and phrases. bck/bci-algebra; hesitant fuzzy set; (inf, sup)-hesitant fuzzy ideal; interval-valued fuzzy ideal; pythagorean fuzzy ideal; bipolar fuzzy ideal. https://doi.org/10.28924/2291-8639-20-2022-34 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-34 2 int. j. anal. appl. (2022), 20:34 several general, extended and related concepts of fuzzy sets have been introduced and studied such as interval-valued fuzzy sets [4, 5], intuitionistic fuzzy sets [6, 7], pythagorean fuzzy sets [10–12], negative fuzzy sets [13,14], bipolar fuzzy sets [15,16], hesitant fuzzy sets [17,18,20,22] and so forth. bck and bci-algebras are algebraic structures, introduced by imai, iséki and tanaka, that describe fragments of the propositional calculus involving implication known as bck and bci logic (see [29– 31]). in 1991, xi [8] applied the concept of fuzzy sets to bck-algebras. later, a number of authors applied and discussed concept of fuzzy sets and its some general, extended and related concepts to bck/bci-algebras. hong and jun [9] introduced anti-fuzzy ideals of bck-algebras and investigated their some useful properties. subha and dhanalakshmi [12] exposed and studied pythagorean fuzzy ideals of bck-algebras. jun [5] introduced interval-valued fuzzy subalgebras and ideals of bckalgebras, and investigated their related properties and characterizations. lee [16] introduced bipolar fuzzy subalgebras and bipolar fuzzy ideals of bck/bci-algebras, investigated their related properties, and considered equivalent relations on the set of all bipolar fuzzy ideals of bck/bci-algebras. jun and ahn [19] introduced hesitant fuzzy subalgebras and ideals of bck/bci-algebras, and investigated their related properties and important characterizations. muhiuddin et al. [32] introduced hesitant fuzzy translations and hesitant fuzzy extensions of a hesitant fuzzy set on bck/bci-algebras, investigated related properties, and characterized hesitant fuzzy (subalgebras) ideals. studying hesitant fuzzy sets on algebraic structures in the meaning of the infimum or supremum of its images, mosrijai et al. [33] introduced sup-hesitant fuzzy up-subalgebras, up-filters, up-ideals, and strong up-ideals of up-algebras and investigated their related properties. muhiuddin and jun [34] muhiuddin et al. [35] muhiuddin et al. [38], harizavi and jun [37], jun and song [39] and takallo et al. [36] used hesitant fuzzy sets related to the infimum or supremum of their images in study of bck/bci-algebras. jittburus and julatha [24,25], phummee et al. [28], and jittburus et al. [27] used hesitant fuzzy sets related to the infimum or the supremum of their images in study of semigroups. julatha and iampan [21–23,26] used hesitant fuzzy sets related to the infimum or the supremum of their images in study of ternary semigroups and γ-semigroups. as previously stated, it motivated us to study hesitant fuzzy set theory based on ideals of bck/bcialgebras in the meaning of infimum and supremum. on bck/bci-algebras, we introduce (inf, sup)hesitant fuzzy ideals, show that it is a general concept of interval-valued fuzzy ideals, and investigate its related properties. characterizations of (inf, sup)-hesitant fuzzy ideals are established in terms of sets, fuzzy sets, negative fuzzy sets, interval-valued fuzzy sets, pythagorean fuzzy sets, bipolar fuzzy sets and hesitant fuzzy sets. moreover, characterizations of ideals, fuzzy ideals, anti-fuzzy ideals, negative fuzzy ideals, pythagorean fuzzy ideals and bipolar fuzzy ideals of bck/bci-algebras are discussed in terms of (inf, sup)-hesitant fuzzy ideals and interval-valued fuzzy ideals. 2. preliminaries an algebra (x ;�, 0) of type (2, 0) is called a bci-algebra if the followings hold: int. j. anal. appl. (2022), 20:34 3 (i) (∀x,y,z ∈x)(((x �y) � (x �z)) � (z �y) = 0), (ii) (∀x,y ∈x)(((x � (x �y)) �y) = 0), (iii) (∀x ∈x)(x �x = 0), (iv) (∀x,y ∈x)(x �y = 0 = y �x ⇒ x = y). by a bck-algebra we mean a bci-algebra (x ;�, 0) satisfies 0�x = 0 for all x ∈x . for any x,y ∈x , we define x ≤ y by x �y = 0. in a bck/bci-algebra (x ;�, 0), the following hold: (∀x ∈x)(x � 0 = x), (2.1) (∀x,y,z ∈x)((x �y) �z = (x �z) �y). (2.2) a nonempty subset a of a bck/bci-algebra (x ;�, 0) is called an ideal (id) of x if it satisfies the following: 0 ∈a, (2.3) (∀x ∈x)(y ∈a,x �y ∈a⇒ x ∈a). (2.4) we refer the reader to the books [1,2] for further information regarding bck/bci-algebras. in what follows, let x denote a bck/bci-algebra (x ,�, 0) and y denote an arbitrary nonempty set unless otherwise specified. a fuzzy set (fs) [3] in y is an arbitrary function from y into [0, 1]. for fss ζ and ξ in y, we denote ζ ≤ ξ in case that ζ(x) ≤ ξ(x) for all x ∈y. a fs ζ in x is call a fuzzy ideal (fid) [8] of x if it satisfies the following conditions: (∀x ∈x)(ζ(0) ≥ ζ(x)), (2.5) (∀x,y ∈x)(ζ(x) ≥ min{ζ(x �y),ζ(y)}) (2.6) and called an anti-fuzzy ideal (afid) [9] of x if it satisfies the following conditions: (∀x ∈x)(ζ(0) ≤ ζ(x)), (2.7) (∀x,y ∈x)(ζ(x) ≤ max{ζ(x �y),ζ(y)}). (2.8) then ζ is both a fid and an afid of x if and only if it is a constant function. a pythagorean fuzzy set (pfs) [10, 11] on y is an object having the form p = {(x,ζ(x),ξ(x)) |x ∈y} when the functions ζ : y → [0, 1] denote the degree of membership and ξ : y → [0, 1] denote the degree of nonmembership, and 0 ≤ (ζ(x))2 + (ξ(x))2 ≤ 1 for all x ∈ y. for the sake of simplicity, we will use the symbol (ζ,ξ) of the pfs {(x,ζ(x),ξ(x)) | x ∈ y}. for a fs ζ in y, we define a fs ζ 2 by ζ 2 (x) = ζ(x) 2 for all x ∈y. then ( ζ 2 , ξ 2 ) and ( ζ 2 , ζ 2 ) are pfss in y for all fss ζ and ξ in y. thus the concept of pfss is an extension of the concept of fss. a pfs (ζ,ξ) on x is called a pythagorean fuzzy ideal (pfid) [12] of x if ζ is a fid and ξ is an afid of x . a bipolar fuzzy set (bfs) [15] in y is an object having the form b = {(x,ζ(x),ξ(x)) | x ∈ y}, where ζ : y → [−1, 0] is a negative fuzzy set (nfs) in y and ξ : y → [0, 1] is a fs in y. we’ll use 4 int. j. anal. appl. (2022), 20:34 the symbol 〈ζ,ξ〉 for the bfs {(x,ζ(x),ξ(x)) | x ∈y} for the purpose of simplicity. let r be the set of all real numbers. for any element r of r and any function ζ from y into r, define functions r −ζ, r + ζ, rζ and −ζ by: r −ζ : y → r,x 7→ r −ζ(x), (2.9) r + ζ : y → r,x 7→ r + ζ(x) (2.10) rζ : y → r,x 7→ rζ(x) (2.11) −ζ : y → r,x 7→−ζ(x). (2.12) then the followings hold: (1) 〈ζ − 1,ζ〉 is a bfs in y for any fs ζ in y, (2) ( 1+ζ 2 , ξ 2 ) and ( ξ 2 , 1+ζ 2 ) are pfss in y for any bfs 〈ζ,ξ〉 in y, (3) 〈ζ − 1,ξ〉 and 〈ξ− 1,ζ〉 are bfss in y for any pfs (ζ,ξ) in y. thus the concept of bfss is an extension of the concept of fss. a bfs b = 〈ζ,ξ〉 in x is called a bipolar fuzzy ideal (bfid) [16] of x if it satisfies the following conditions: (∀x ∈x)(ζ(0) ≤ ζ(x)), (2.13) (∀x ∈x)(ξ(0) ≥ ξ(x)), (2.14) (∀x,y ∈x)(ζ(x) ≤ max{ζ(x �y),ζ(y)}), (2.15) (∀x,y ∈x)(ξ(x) ≥ min{ξ(x �y),ξ(y)}). (2.16) by a negative fuzzy ideal (nfid) of x we mean a nfs ζ of x satisfies the conditions (2.13) and (2.15). then a bfs 〈ζ,ξ〉 of x is a bfid of x if and only if ζ is a nfid and ξ is a fid of x . by an interval number ă we mean an interval [a−,a+], where 0 ≤ a− ≤ a+ ≤ 1. the set of all interval numbers is denoted by d([0, 1]). for two elements ă = [a−,a+] and b̆ = [b−,b+] in d([0, 1]), define the operations -, =, ≺ and rmin in case of two elements in d([0, 1]) as follows: (1) ă b̆ ⇔ a+ ≤ b+ and a− ≤ b−, (2) ă = b̆ ⇔ a+ = b+ and a− = b−, (3) ă ≺ b̆ ⇔ ă b̆ and ă 6= b̆, (4) rmin{ă, b̆} = [min{a−,b−}, min{a+,b+}]. an interval-valued fuzzy set (ivfs) [4] on y is defined to be a function λ̆ : y → d([0, 1]), where λ̆(x) = [λ̆l(x), λ̆u(x)] for all x ∈y, λ̆l and λ̆u are fss in y such that λ̆l ≤ λ̆u. thus the concept of ivfss is an extension of the concept of fss. an ivfs λ̆ on x is called an interval-valued fuzzy ideal int. j. anal. appl. (2022), 20:34 5 (ivfid) [5] of x if it satisfies: (∀x ∈x)(λ̆(x) λ̆(0)), (2.17) (∀x,y ∈x)(rmin{λ̆(x �y), λ̆(y)}λ̆(x)). (2.18) remark 2.1. an ivfs λ̆ on x is an ivfid of x if and only if λ̆l and λ̆u are fids of x . a hesitant fuzzy set (hfs) [17,18] on y is defined to be a function ω̃ : y → ℘([0, 1]) when ℘([0, 1]) is the set of all subsets of [0, 1]. note that every ivfs on y is a hfs on y. then the concept of hfss is a generalization of the concept of ivfss, and the concept of hfss is an extension of the concept of fss. a hfs ω̃ is a hesitant fuzzy ideal (hfid) [19,20] of x if it satisfies the following: (∀x ∈x)(ω̃(x) ⊆ ω̃(0)), (2.19) (∀x,y ∈x)(ω̃(x �y) ∩ ω̃(y) ⊆ ω̃(x)). (2.20) 3. main results for an element ∇∈ ℘([0, 1]), define inf∇ [24,27] and sup∇ [25,26] by inf∇ = { inf ∇ 0 if ∇ 6= ∅, otherwise, and sup∇ = { sup∇ 0 if ∇ 6= ∅, otherwise. definition 3.1. a hfs ω̃ on x is called an (inf, sup)-hesitant fuzzy ideal ((inf, sup)-hfid) of x if the set [x , ω̃,∇] is an id of x for all ∇ ∈ ℘([0, 1]) when [x , ω̃,∇] := {x ∈ x | inf ω̃(x) ≥ inf∇, sup ω̃(x) ≥ sup∇} is not empty. example 3.1. let x = {0,u,v,w,x} be a bci-algebra [1] with the following cayley table: � 0 u v w x 0 0 0 v w x u u 0 v w x v v v 0 x w w w w x 0 v x x x w v 0 define a hfs ω̃ on x by ω̃(0) = [0.6, 0.8], ω̃(u) = (0.5, 0.7), ω̃(v) = [0.5, 0.6] ∪ {0.7}, ω̃(w) = {0.3, 0.4}, ω̃(z) = (0.3, 0.4). it is routine to verify that ω̃ is an (inf, sup)-hfid of x . 6 int. j. anal. appl. (2022), 20:34 example 3.2. let x = {0,w,x,y,z} be a bck-algebra with the following cayley table: � 0 w x y z 0 0 0 0 0 0 w w 0 0 0 0 x x x 0 0 0 y y x w 0 w z z x w w 0 define a hfs ω̃ on x by ω̃(0) = {0.8, 0.9, 1}, ω̃(w) = (0.6, 0.8], ω̃(x) = ω̃(y) = {0}, ω̃(z) = ∅. it is routine to verify that ω̃ is an (inf, sup)-hfid of x . moreover, we know that ω̃ is not a hfid of x because ω̃(w) * ω̃(0), and ω̃ is not an ivfid of x because it is not an ivfs. for any hfs ω̃ on y, define the fss fω̃ and fω̃ in y by (∀x ∈y)(fω̃(x) = sup ω̃(x)), (3.1) (∀x ∈y)(fω̃(x) = inf ω̃(x)). (3.2) a hfs ϑ̃ on y is called an infimum complement [21, 24] of ω̃ on y if inf ϑ̃(x) = (1 − fω̃)(x) for all x ∈ y and called a supremum complement of ω̃ on y if sup ϑ̃(x) = (1 −fω̃)(x) for all x ∈y. let ic(ω̃) and sc(ω̃) be the set of all infimum complements of ω̃ and the set of all supremum complements of ω̃, respectively. define the hfss ω̃± and ω̃∓ on y by ω̃±(x) = {(1 −fω̃)(x)} and ω̃∓(x) = {(1 −fω̃)(x)} for all x ∈y. then we have ω̃± ∈ ic(ω̃), fω̃± = 1 −fω̃ and ω̃∓ ∈ sc(ω̃), fω̃ ∓ = 1 −fω̃. next, we investigate characterizations of (inf, sup)-hfids of bck/bci-algebras in terms of ids, fids, afids and nfids. lemma 3.1. let ω̃ be a hfs on x . then the followings are equivalent. (1) ω̃ is an (inf, sup)-hfid of x . (2) the set [x , ω̃, ă] is an id of x for all ă ∈d([0, 1]) when [x , ω̃, ă] is not empty. (3) fω̃ and fω̃ are fids of x . (4) f ϑ̃ and f θ̃ are afids of x for all ϑ̃ ∈ ic(ω̃) and θ̃ ∈ sc(ω̃). (5) fω̃± and fω̃ ∓ are afids of x . (6) f ϑ̃ − 1 and f θ̃ − 1 are nfids of x for all ϑ̃ ∈ ic(ω̃) and θ̃ ∈ sc(ω̃). (7) fω̃± − 1 and fω̃ ∓ − 1 are nfids of x . proof. (1) ⇒ (2), (4) ⇒ (5) and (6) ⇒ (7). they are clear. (2) ⇒ (3). let x ∈ x and ă := {t ∈ [0, 1] | inf ω̃(x) ≤ t ≤ sup ω̃(x)}. then ă ∈ d([0, 1]) and x ∈ [x , ω̃, ă]. by the assumption (2), we get [x , ω̃, ă] is an id of x and so 0 ∈ [x , ω̃, ă]. thus sup ω̃(x) = a+ ≤ sup ω̃(0) and inf ω̃(x) = a− ≤ inf ω̃(0), which imply that fω̃(x) ≤ fω̃(0) and fω̃(x) ≤fω̃(0). hence, fω̃ and fω̃ satisfy the condition (2.5). to show that fω̃ and fω̃ satisfy the int. j. anal. appl. (2022), 20:34 7 condition (2.6), let x,y ∈x and b̆ := {t ∈ [0, 1] | min{inf ω̃(y), inf ω̃(x �y)}≤ t ≤ min{sup ω̃(y), sup ω̃(x �y)}}. then b̆ ∈d([0, 1]) and y,x �y ∈ [x , ω̃, b̆]. by the assumption (2), we have x ∈ [x , ω̃, b̆]. thus fω̃(x) = sup ω̃(x) ≥ b+ = min{sup ω̃(y), sup ω̃(x �y)} = min{fω̃(y),fω̃(x �y)}, fω̃(x) = inf ω̃(x) ≥ b− = min{inf ω̃(y), inf ω̃(x �y)} = min{fω̃(y),fω̃(x �y)}. hence, fω̃ and fω̃ satisfy the condition (2.6). therefore, it follows from the conditions (2.5) and (2.6) that fω̃ and fω̃ are fids of x . (3) ⇒ (1). let ∇ be an element of ℘([0, 1]) such that [x , ω̃,∇] 6= ∅. let x ∈ x and y,x � y ∈ [x , ω̃,∇]. then sup ω̃(y) ≥ sup∇, inf ω̃(y) ≥ inf∇, sup ω̃(x�y) ≥ sup∇ and inf ω̃(x�y) ≥ inf∇. by the assumption (3), we have sup ω̃(0) = fω̃(0) ≥fω̃(y) = sup ω̃(y) ≥ sup∇, inf ω̃(0) = fω̃(0) ≥fω̃(y) = inf ω̃(y) ≥ inf∇, sup ω̃(x) = fω̃(x) ≥ min{fω̃(y),fω̃(x �y)} = min{sup ω̃(y), sup ω̃(x �y)}≥ sup∇, and inf ω̃(x) = fω̃(x) ≥ min{fω̃(y),fω̃(x �y)} = min{inf ω̃(y), inf ω̃(x �y)}≥ inf∇. thus 0,x ∈ [x , ω̃,∇]. hence, [x , ω̃,∇] is an id of x . therefore, ω̃ is an (inf, sup)-hfid of x . (3) ⇒ (4). let ϑ̃ ∈ ic(ω̃) and θ̃ ∈ sc(ω̃). by the assumption (3), we obtain that f ϑ̃ and f θ̃ satisfy the conditions (2.5) and (2.6). thus, for all x,y ∈x , we have f θ̃(0) = 1 −fω̃(0) ≤ 1 −fω̃(x) = f θ̃(x), f ϑ̃ (0) = 1 −fω̃(0) ≤ 1 −fω̃(x) = fϑ̃(x), f θ̃(x) = 1 −fω̃(x) ≤ 1 − min{fω̃(y),fω̃(x �y)} = max{1 −fω̃(y), 1 −fω̃(x �y)} 8 int. j. anal. appl. (2022), 20:34 = max{f θ̃(y),f θ̃(x �y)}, f ϑ̃ (x) = 1 −fω̃(x) ≤ 1 − min{fω̃(y),fω̃(x �y)} = max{1 −fω̃(y), 1 −fω̃(x �y)} = max{f ϑ̃ (y),f ϑ̃ (x �y)}. hence, f ϑ̃ and f θ̃ satisfy that conditions (2.7) and (2.8) that they are afids of x . (4) ⇒ (6). let ϑ̃ ∈ ic(ω̃) and θ̃ ∈ sc(ω̃). it is clear that f ϑ̃ − 1 and f θ̃ − 1 are nfss in x . by the assumption (4), we get that f ϑ̃ and f θ̃ satisfy the conditions (2.7) and (2.8). thus, for all x,y ∈x , we get (f θ̃ − 1)(0) = f θ̃(0) − 1 ≤f θ̃(x) − 1 = (f θ̃ − 1)(x), (f ϑ̃ − 1)(0) = f ϑ̃ (0) − 1 ≤f ϑ̃ (x) − 1 = (f ϑ̃ − 1)(x), (f θ̃ − 1)(x) = f θ̃(x) − 1 ≤ max{f θ̃(y),f θ̃(x �y)}− 1 = max{f θ̃(y) − 1,f θ̃(x �y) − 1} = max{(f θ̃ − 1)(y), (f θ̃ − 1)(x �y)}, (f ϑ̃ − 1)(x) = f ϑ̃ (x) − 1 ≤ max{f ϑ̃ (y),f ϑ̃ (x �y)}− 1 = max{f ϑ̃ (y) − 1,f ϑ̃ (x �y) − 1} = max{(f ϑ̃ − 1)(y), (f ϑ̃ − 1)(x �y)}. hence, f ϑ̃ − 1 and f θ̃ − 1 satisfy that conditions (2.13) and (2.15) that they are nfids of x . (5) ⇒ (7). it is similar to prove (4) ⇒ (6). (7) ⇒ (3). let x,y ∈x . since fω̃± − 1 = −fω̃, fω̃ ∓ − 1 = −fω̃ and by the assumption (7), we have −fω̃(0) ≤−fω̃(x), −fω̃(0) ≤−fω̃(x), and −fω̃(x) ≤ max{−fω̃(y),−fω̃(x �y)} = −(min{fω̃(y),fω̃(x �y)}), −fω̃(x) ≤ max{−fω̃(y),−fω̃(x �y)} = −(min{fω̃(y),fω̃(x �y)}). thus fω̃(0) ≥ fω̃(x), fω̃(0) ≥ fω̃(x), fω̃(x) ≥ min{fω̃(y),fω̃(x � y)} and fω̃(x) ≥ min{fω̃(y),fω̃(x � y)}. hence, fω̃ and fω̃ satisfy the conditions (2.5) and (2.6). therefore, fω̃ and fω̃ are fids of x . � proposition 3.1. every ivfid of x is an (inf, sup)-hfid of x . proof. it follows from remark 2.1 and lemma 3.1 � int. j. anal. appl. (2022), 20:34 9 the converse of proposition 3.1 is not generally true, which can see in example 3.2. by proposition 3.1 and example 3.2, we obtain that an (inf, sup)-hfid of a bck/bci-algebra x is a generalization of the concept of an ivfid of x . theorem 3.1. let λ̆ be an ivfs on x . then the followings are equivalent. (1) λ̆ is an ivfid of x . (2) the set [x , λ̆, ă] is an id of x for all ă ∈d([0, 1]) when [x , λ̆, ă] is not empty. (3) λ̆ is an (inf, sup)-hfid of x . proof. it follows from remark 2.1, lemma 3.1 and proposition 3.1. � theorem 3.2. let ω̃ be a hfs on x . the followings are equivalent. (1) ω̃ is an (inf, sup)-hfid of x . (2) λ̆ is an ivfid of x when λ̆ is an ivfs on x such that λ̆l = fω̃ and λ̆u = fω̃. (3) ϑ̃ is an (inf, sup)-hfid of x for all hfs ϑ̃ on x such that f ϑ̃ = fω̃ and fϑ̃ = fω̃. proof. it follows from lemma 3.1 and theorem 3.1. � proposition 3.2. let ω̃ be an (inf, sup)-hfid of x and x,y,z ∈ x such that x � y ≤ z. then fω̃(x) ≥ min{fω̃(y),fω̃(z)} and fω̃(x) ≥ min{fω̃(y),fω̃(z)}. proof. since x �y ≤ z, we have (x �y) �z = 0. thus fω̃(x) ≥ min{fω̃(y),fω̃(x �y)} ≥ min{fω̃(y), min{fω̃(z),fω̃((x �y) �z)}} = min{fω̃(y), min{fω̃(z),fω̃(0)}} = min{fω̃(y),fω̃(z)} and similarly, we hve fω̃(x) ≥ min{fω̃(y),fω̃(z)}. � corollary 3.1. let λ̆ be an ivfid of x and x,y,z ∈x such that x�y ≤ z. then rmin{λ̆(y), λ̆(z)}λ̆(x). proof. it follows from proposition 3.2 and theorem 3.1. � proposition 3.3. let ω̃ be an (inf, sup)-hfid of x and x,y ∈ x such that x ≤ y. then fω̃(x) ≥ fω̃(y) and fω̃(x) ≥fω̃(y). proof. since x ≤ y, we have x �y = 0. then fω̃(x) ≥ min{fω̃(y),fω̃(x �y)} = min{fω̃(y),fω̃(0)} = fω̃(y), fω̃(x) ≥ min{fω̃(y),fω̃(x �y)} = min{fω̃(y),fω̃(0)} = fω̃(y). hence, fω̃(x) ≥fω̃(y) and fω̃(x) ≥fω̃(y). � 10 int. j. anal. appl. (2022), 20:34 corollary 3.2. let λ̆ be an ivfid of x and x,y ∈x such that x ≤ y. then λ̆(y) λ̆(x). proof. it follows from proposition 3.3 and theorem 3.1. � for any subset a of y and ∇, ∆ ∈ ℘([0, 1]), define a map c(a,∇, ∆) [21,23] as follows: c(a,∇, ∆) : y → ℘([0, 1]),x 7→ { ∆ ∇ if x ∈ a, otherwise. we denote c(a) for c(a, [0, 0], [1, 1]) and it is called the characteristic interval-valued fuzzy set of a on x . theorem 3.3. let a be a nonempty subset of x and ∇, ∆ ∈ ℘([0, 1]) such that sup∇ < sup ∆, inf∇ ≤ inf ∆ or sup∇ ≤ sup ∆, inf∇ < inf ∆. then a is an id of x if and only if c(a,∇, ∆) is an (inf, sup)-hfid of x . proof. since a is an id of x , we have 0 ∈ a. then fc(a,∇,∆)(0) = sup ∆ = max{sup ∆, sup∇}≥fc(a,∇,∆)(x), fc(a,∇,∆)(0) = inf ∆ = max{inf ∆, inf∇}≥fc(a,∇,∆)(x) for all x ∈x . thus fc(a,∇,∆) and fc(a,∇,∆) satisfy the condition (2.5). to show that fc(a,∇,∆) and fc(a,∇,∆) satisfy the condition (2.6), let x,y ∈ x . if y /∈ a or x �y /∈ a, then fc(a,∇,∆)(x) ≥ sup∇ = min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)}, fc(a,∇,∆)(x) ≥ inf∇ = min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)}. on the other hand, suppose that y,x �y ∈ a. since a is an id of x , we have x ∈ a. thus fc(a,∇,∆)(x) = sup ∆ = min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)}, fc(a,∇,∆)(x) = inf ∆ = min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)}. hence, fc(a,∇,∆) and fc(a,∇,∆) satisfy the condition (2.6). therefore, fc(a,∇,∆) and fc(a,∇,∆) are ids of x and by lemma 3.1, we obtain that c(a,∇, ∆) is an (inf, sup)-hfid of x . conversely, let x ∈ x and y,x � y ∈ a. then c(a,∇, ∆)(y) = ∆ = c(a,∇, ∆)(x � y). if sup∇ < sup ∆ and inf∇≤ inf ∆, then by lemma 3.1, we have fc(a,∇,∆)(0) ≥fc(a,∇,∆)(x) ≥ min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)} = sup ∆ > sup∇. thus 0,x ∈ a. in the case that sup∇≤ sup ∆ and inf∇ < inf ∆, then by lemma 3.1, we get fc(a,∇,∆)(0) ≥fc(a,∇,∆)(x) ≥ min{fc(a,∇,∆)(y),fc(a,∇,∆)(x �y)} = inf ∆ > inf∇. thus 0,x ∈ a. therefore, a is an id of x . � theorem 3.4. let a be a nonempty subset of x . the followings are equivalent. int. j. anal. appl. (2022), 20:34 11 (1) a is an id of x . (2) c(a,ă, b̆) is an ivfid of x when ă, b̆ ∈d([0, 1]) and ă ≺ b̆. (3) c(a) is an ivfid of x . proof. it follows from theorem 3.3 and theorem 3.1. � for a fs ζ in y and a positive integer n, we define the hfs h(ζ,n) and the ivfs i(ζ,n) on y as follows: h(ζ,n) : y → ℘([0, 1]),x 7→ { ζ 1 + n (x), n + ζ 1 + n (x)} and i(ζ,n) : y →d([0, 1]),x 7→ {t ∈ [0, 1] | ζ 1 + n (x) ≤ t ≤ n + ζ 1 + n (x)}. then the followings are true. (1) suph(ζ,n)(x) = supi(ζ,n)(x), infh(ζ,n)(x) = infi(ζ,n)(x) and h(ζ,n)(x) ⊆ i(ζ,n)(x) for all x ∈y. (2) h(ζ, 1)(x) = {ζ 2 (x), 1+ζ 2 (x)} and i(ζ, 1)(x) = {t ∈ [0, 1] | ζ 2 (x) ≤ t ≤ 1+ζ 2 (x)} for all x ∈y. (3) h(−ζ,n) is a hfs and i(−ζ,n) is an ivfs on y for all nfs ζ in y. next, we use (inf, sup)-hfids and ivfids of bck/bci-algebras to characterize fids in theorem 3.5, afids in theorem 3.6 and nfids in theorem 3.7. theorem 3.5. let ζ be a fs in x . the followings are equivalent. (1) ζ is a fid of x . (2) i(ζ,n) is an ivfid of x for all positive integer n. (3) h(ζ,n) is an (inf, sup)-hfid of x for all positive integer n. (4) ω̃ is an (inf, sup)-hfid of x for all hfs ω̃ on x and positive integer n such that fω̃ = ζ 1+n and fω̃ = n+ζ 1+n . proof. by using theorem 3.2, the conditions (2), (3) and (4) are equivalent. next, we show that (1) and (4) are equivalent. let ω̃ be a hfs on x and n be a positive integer such that fω̃ = ζ 1+n and fω̃ = n+ζ 1+n . by the assumption (1), we have fω̃(0) = ζ(0) 1 + n ≥ ζ(x) 1 + n = fω̃(x), fω̃(0) = n + ζ(0) 1 + n ≥ n + ζ(x) 1 + n = fω̃(x), fω̃(x) = ζ(x) 1 + n ≥ min{ζ(y),ζ(x �y)} 1 + n = min{ ζ(y) 1 + n , ζ(x �y) 1 + n } = min{fω̃(y),fω̃(x �y)}, fω̃(x) = n + ζ(x) 1 + n ≥ n + min{ζ(y),ζ(x �y)} 1 + n = min{ n + ζ(y) 1 + n , n + ζ(x �y) 1 + n } = min{fω̃(y),fω̃(x �y)} 12 int. j. anal. appl. (2022), 20:34 for all x,y ∈ x . hence, fω̃ and fω̃ are fids of x and by using lemma 3.1, we obtain that ω̃ is an (inf, sup)-hfid of x . therefore, (4) is true. conversely, assume that (4) is true. let ω̃ be a hfs on x such that fω̃ = ζ 2 and fω̃ = 1+ζ 2 . by the assumption (4) and lemma 3.1, we obtain that fω̃ = ζ 2 is a fid of x . then for all x,y ∈x , we get ζ(0) = 2(ζ(0) 2 ) ≥ 2(ζ(x) 2 ) = ζ(x) and ζ(x) = 2( ζ(x) 2 ) ≥ 2( min{ζ(y),ζ(x �y)} 2 ) = min{ζ(y),ζ(x �y)}. hence, ζ is an id of x , that is (1) is true. � lemma 3.2. a fs ζ in x is an afid of x if and only if 1 −ζ is a fid of x . proof. assume that ζ is an afid of x . then for all x,y ∈x , we get 1 −ζ(0) ≥ 1 −ζ(x) and 1 −ζ(x) ≥ 1 − max{ζ(y),ζ(x �y)} = min{1 −ζ(y), 1 −ζ(x �y)}. then 1 −ζ is a fid of x . conversely, assume that 1 −ζ is a fid of x . then 1 − (1 −ζ)(0) ≤ 1 − (1 −ζ)(x) and 1 − (1 −ζ)(x) ≤ 1 − min{(1 −ζ)(y), (1 −ζ)(x �y)} = max{1 − (1 −ζ)(y), 1 − (1 −ζ)(x �y)} for all x,y ∈x . since ζ = 1 − (1 −ζ), we obtain that ζ is an afid of x . � theorem 3.6. let ζ be a fs in x . the followings are equivalent. (1) ζ is an afid of x . (2) i(1 −ζ,n) is an ivfid of x for all positive integer n. (3) h(1 −ζ,n) is an (inf, sup)-hfid of x for all positive integer n. (4) ω̃ is an (inf, sup)-hfid of x for all hfs ω̃ on x and positive integer n such that fω̃ = 1−ζ 1+n and fω̃ = 1 + −ζ 1+n . proof. it follows from lemma 3.2 and theorem 3.5. � lemma 3.3. a nfs ζ in x is a nfid of x if and only if −ζ is a fid of x . proof. assume that ζ is a nfid of x . let x,y ∈x . then ζ(0) ≤ ζ(x) and ζ(x) ≤ max{ζ(y),ζ(x � y)}. thus −ζ(0) ≥−ζ(x) and −ζ(x) ≥−(max{ζ(y),ζ(x �y)}) = min{−ζ(y),−ζ(x �y)}. hence, −ζ is a fid of x . int. j. anal. appl. (2022), 20:34 13 conversely, assume that −ζ is a fid of x . then ζ(0) = −(−ζ(0)) ≤−(−ζ(x)) = ζ(x) and ζ(x) = −(−ζ(x)) ≤−(min{−ζ(y),−ζ(x �y)}) = max{−(−ζ(y)),−(−ζ(x �y))} = max{ζ(y),ζ(x �y)} for all x,y ∈x . hence, ζ is a nfid of x . � theorem 3.7. let ζ be a nfs in x . the followings are equivalent. (1) ζ is a nfid of x . (2) i(−ζ,n) is an ivfid of x for all positive integer n. (3) h(−ζ,n) is an (inf, sup)-hfid of x for all positive integer n. (4) ω̃ is an (inf, sup)-hfid of x for all hfs ω̃ on x and positive integer n such that fω̃ = −ζ 1+n and fω̃ = n−ζ 1+n . proof. it follows from lemma 3.3 and theorem 3.5. � for any hfs ω̃ on y and any element ∇ of ℘([0, 1]), define the hfs hω̃∇ on y by hω̃∇(x) = {t ∈∇ | f ω̃± 2 (x) ≤ t ≤ 1+f ω̃ 2 (x)} for all x ∈y. we denote hω̃ for hω̃ [0,1] . then hω̃∇(x) ⊆h ω̃ ∆(x) ⊆h ω̃(x) when x ∈y and ∇⊆ ∆ ⊆ [0, 1]. theorem 3.8. let ω̃ be a hfs on x . the followings are equivalent. (1) ω̃ is an (inf, sup)-hfid of x . (2) hω̃∇ is a hfid of x for all ∇∈ ℘([0, 1]). (3) hω̃ is a hfid of x . proof. (1) ⇒ (2). let x ∈x , ∇∈ ℘([0, 1]) and t ∈hω̃∇(x). then t ∈∇ and f ω̃± 2 (x) ≤ t ≤ 1+f ω̃ 2 (x) . by the assumption (1) and lemma 3.1, we get fω̃±(x) ≥fω̃±(0) and fω̃(x) ≤fω̃(0). thus fω̃± 2 (0) ≤ fω̃± 2 (x) ≤ t ≤ 1 + fω̃ 2 (x) ≤ 1 + fω̃ 2 (0) and so t ∈hω̃(0). hence, hω̃(x) ⊆hω̃(0). therefore, hω̃ satisfies the condition (2.19). to show that hω̃ satisfies the condition (2.20), let x,y ∈ x , ∇ ∈ ℘([0, 1]) and t ∈ hω̃∇(y) ∩ hω̃∇(x �y). then t ∈∇, fω̃± 2 (y) ≤ t ≤ 1+f ω̃ 2 (y) and f ω̃± 2 (x �y) ≤ t ≤ 1+f ω̃ 2 (x �y). 14 int. j. anal. appl. (2022), 20:34 by the assumption (1) and lemma 3.1, we have fω̃±(x) ≤ max{fω̃±(y),fω̃±(x �y)} and fω̃(x) ≥ min{fω̃(y),fω̃(x �y)}. thus fω̃± 2 (x) ≤ max{ fω̃± 2 (y), fω̃± 2 (x �y)} ≤ t ≤ min{ 1 + fω̃ 2 (y), 1 + fω̃ 2 (x �y)} ≤ 1 + fω̃ 2 (x), and so t ∈hω̃∇(x). hence, h ω̃ ∇(y)∩h ω̃ ∇(x�y) ⊆h ω̃ ∇(x). it is showed that h ω̃ ∇ satisfies the condition (2.20). therefore, it follows from the conditions (2.19) and (2.20) that hω̃∇ is a hfid of x for all ∇∈ ℘([0, 1]). (2) ⇒ (3). it is clear. (3) ⇒ (1). let x,y ∈ x . then fω̃± 2 (x), 1+f ω̃ 2 (x) ∈ hω̃(x) and max{fω̃± 2 (y), f ω̃± 2 (x � y)}, min{1+f ω̃ 2 (y), 1+f ω̃ 2 (x � y)} ∈ hω̃(y) ∩ hω̃(x � y). by the assumption (3), we get f ω̃± 2 (x), 1+f ω̃ 2 (x) ∈ hω̃(0) and max{fω̃± 2 (y), f ω̃± 2 (x � y)}, min{1+f ω̃ 2 (y), 1+f ω̃ 2 (x � y)} ∈ hω̃(x). thus f ω̃± 2 (x) ≥ fω̃± 2 (0), 1+f ω̃ 2 (x) ≤ 1+f ω̃ 2 (0), max{fω̃± 2 (y), f ω̃± 2 (x � y)} ≥ fω̃± 2 (x) and min{1+f ω̃ 2 (y), 1+f ω̃ 2 (x � y)} ≤ 1+f ω̃ 2 (x). since fω̃ = 1 − 2( f ω̃± 2 ) and fω̃ = 2( 1+f ω̃ 2 ) − 1, we have fω̃(0) = 2( 1 + fω̃ 2 (0)) − 1 ≥ 2( 1 + fω̃ 2 (x)) − 1 = fω̃(x), fω̃(0) = 1 − 2( fω̃± 2 (0)) ≥ 1 − 2( fω̃± 2 (x)) = fω̃(x), fω̃(x) = 2( 1 + fω̃ 2 (x)) − 1 ≥ 2(min{ 1 + fω̃ 2 (y), 1 + fω̃ 2 (x �y)}) − 1 = min{2( 1 + fω̃ 2 (y)) − 1, 2( 1 + fω̃ 2 (x �y)) − 1} = min{fω̃(y),fω̃(x �y)}, fω̃(x) = 1 − 2( fω̃± 2 (x)) ≥ 1 − 2(max{ fω̃± 2 (y), fω̃± 2 (x �y)}) = min{1 − 2( fω̃± 2 (y)), 1 − 2( fω̃± 2 (x �y))} = min{fω̃(y),fω̃(x �y)}. hence, fω̃ and fω̃ are fids of x and by using lemma 3.1, we obtain that ω̃ is an (inf, sup)-hfid of x . � int. j. anal. appl. (2022), 20:34 15 theorem 3.9. let ω̃ be a hfs on x . the followings are equivalent. (1) ω̃ is an (inf, sup)-hfid of x . (2) (fω̃,f θ̃) is a pfid of x for all θ̃ ∈ sc(ω̃). (3) (fω̃,fω̃ ∓ ) is a pfid of x . (4) (f ω̃ 2 , f ϑ̃ 2 ) is a pfid of x for all ϑ̃ ∈ ic(ω̃). (5) (f ω̃ 2 , f ω̃± 2 ) is a pfid of x . proof. (1) ⇒ (2) and (1) ⇒ (4). they follow from lemma 3.1. (2) ⇒ (3) and (4) ⇒ (5). they are clear. (3) ⇒ (1). by the assumption (3), we obtain that fω̃ is a fid and fω̃ ∓ is an afid of x . since fω̃ = 1 −fω̃ ∓ and lemma 3.2, we get fω̃ is a fid of x . hence, fω̃ and fω̃ are fids of x and by using lemma 3.1, we have that ω̃ is an (inf, sup)-hfid of x . (5) ⇒ (1). it is similar to prove in the case (3) ⇒ (1). � for any pfs p = (ζ,ξ) in y, define the hfs h(p ) on y by h(p )(x) = {t ∈ [0, 1] | 1−ξ 2 (x) ≤ t ≤ 1+ζ 2 (x)} for all x ∈y. theorem 3.10. let p = (ζ,ξ) be a pfs in x . the followings are equivalent. (1) p is a pfid of x . (2) h(p ) is an (inf, sup)-hfid of x . (3) h(p ) is an ivfid of x . proof. it follows from theorem 3.2 and lemmas 3.1 and 3.2. � theorem 3.11. let ω̃ be a hfs on x . the followings are equivalent. (1) ω̃ is an (inf, sup)-hfid of x . (2) 〈f ϑ̃ − 1,fω̃〉 is a bfid of x for all ϑ̃ ∈ ic(ω̃). (3) 〈fω̃± − 1,fω̃〉 is a bfid of x . proof. (1) ⇒ (2). it follows from lemma 3.1. (2) ⇒ (3). it is clear. (3) ⇒ (1). by the assumption (3), we have that fω̃ is a fid and fω̃± − 1 is a nfid of x . since fω̃ = −(fω̃± − 1) and lemma 3.3, we get fω̃ is a fid of x . thus fω̃ and fω̃ are fids of x and by using lemma 3.1, we obtain that ω̃ is an (inf, sup)-hfid of x . � for any bfs b = 〈ζ,ξ〉 on y, define the hfs h〈b〉 on y by h〈b〉(x) = {t ∈ [0, 1] | −ζ 2 (x) ≤ t ≤ 1+ξ 2 (x)} for all x ∈y. theorem 3.12. let b = 〈ζ,ξ〉 be a bfs in x . the followings are equivalent. (1) b is a bfid of x . 16 int. j. anal. appl. (2022), 20:34 (2) h〈b〉 is an (inf, sup)-hfid of x . (3) h〈b〉 is an ivfid of x . proof. it follows from theorem 3.2 and lemmas 3.1 and 3.3. � 4. conclusions in present paper, we have introduced an (inf, sup)-hfid, which is one of genaral concepts of an ivfid, in bck/bci-algebras, and investigated its some important properties. as important study results, characterizations of (inf, sup)-hfids have been discussed by sets, fss, nfss, pfss, hfss, ivfss and bfss. also, we use concepts of (inf, sup)-hfids and ivfids to study characterizations of ids, fids, afids, nfids, pfids and bfids. in our future study of bck/bci-algebras and other algebras, the following objectives considered: • to get more results of hfss in the meaning of the infimum and supremum of its images, • to define neutrosophic sets in bck/bci-algebras and related structures by means of hfss in the meaning of the infimum and supremum of its images, • to define (inf, sup)-type of hfss baded on subalgebras, h-ideals and p-ideals of bck/bcialgebras, • to introduce (inf, sup)-hfids in up-algebras, be-algebras, semigroups and la-semigroups. acknowledgment: this work was supported by (i) university of phayao (up), (ii) thailand science research and innovation (tsri), and (iii) national science, research and innovation fund (nsrf) [grant number: ff65-rim047]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. huang, bci-algebra, science press, beijing, 2006. [2] j. meng, y.b. jun, bck-algebras, kyung moon sa company, seoul, 1994. [3] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [4] l.a. zadeh, the concept of a linguistic variable and its application to approximate reasoning i, inform. sci. 8 (1975), 199-249. https://doi.org/10.1016/0020-0255(75)90036-5. [5] y.b. jun, interval-valued fuzzy subalgebras/ideals in bck-algebras, sci. math. 3 (2000), 435-444. https://www. jams.jp/scm/contents/vol-3-3/3-3-17.pdf. [6] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-0114(86)80034-3. https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/0020-0255(75)90036-5 https://www.jams.jp/scm/contents/vol-3-3/3-3-17.pdf https://www.jams.jp/scm/contents/vol-3-3/3-3-17.pdf https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 int. j. anal. appl. (2022), 20:34 17 [7] t. senapati, m. bhowmik, m. pal, b. davvaz, atanassov’s intuitionistic fuzzy translations of intuitionistic fuzzy subalgebras and ideals in bck/bci-algebras, eurasian math. j. 6 (2015), 96-114. http://mi.mathnet.ru/ emj187. [8] o. xi, fuzzy bck-algebras, math. japon. 36 (1991), 935-942. [9] s.m. hong, y.b. jun, anti fuzzy ideals in bck-algebras, kyungpook math. j. 38 (1998), 145-145. [10] r.r. yager, pythagorean fuzzy subsets, in: 2013 joint ifsa world congress and nafips annual meeting (ifsa/nafips), ieee, edmonton, ab, canada, 2013: pp. 57–61. https://doi.org/10.1109/ifsa-nafips. 2013.6608375. [11] r.r. yager, a.m. abbasov, pythagorean membership grades, complex numbers, and decision making, int. j. intell. syst. 28 (2013), 436–452. https://doi.org/10.1002/int.21584. [12] v.s. subha, p. dhanalakshmi, new type of fuzzy ideals in bck/bci algebras, world sci. news. 153 (2021), 80-92. [13] a. khan, y.b. jun, m. shabir, n-fuzzy quasi-ideals in ordered semigroups, quasigroups relat. syst. 17 (2009), 237-252. [14] y.b. jun, m.s. kang, c.h. park, n-subalgebras in bck/bci-algebras based on point n-structures, int. j. math. math. sci. 2010 (2010), article id 303412. https://doi.org/10.1155/2010/303412. [15] w.r. zhang, bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in: nafips/ifis/nasa ’94. proceedings of the first international joint conference of the north american fuzzy information processing society biannual conference. ieee, san antonio, tx, usa, 1994: pp. 305–309. https://doi.org/10.1109/ijcf.1994.375115. [16] k.j. lee, bipolar fuzzy subalgebras and bipolar fuzzy ideals of bck/bci-algebras, bull. malays. math. sci. soc. 32 (2009), 361-373. [17] v. torra, y. narukawa, on hesitant fuzzy sets and decision, in: 2009 ieee international conference on fuzzy systems, ieee, jeju island, south korea, 2009: pp. 1378–1382. https://doi.org/10.1109/fuzzy.2009.5276884. [18] v. torra, hesitant fuzzy sets, int. j. intell. syst. 25 (2010), 529-539. https://doi.org/10.1002/int.20418. [19] y.b. jun, s.s. ahn, hesitant fuzzy set theory applied to bck/bci-algebras, j. comput. anal. appl. 20 (2016), 635-646. [20] y.b. jun, subalgebras and ideals of bck/bci-algebras in framework of the hesitant intersection, kyungpook math. j. 56 (2016), 371-386. https://doi.org/10.5666/kmj.2016.56.2.371. [21] p. julatha, a. iampan, on inf-hesitant fuzzy γ-ideals of γ-semigroups, adv. fuzzy syst. 2022 (2022), article id 9755894. https://doi.org/10.1155/2022/9755894. [22] p. julatha, a. iampan, a new generalization of hesitant and interval-valued fuzzy ideals of ternary semigroups, int. j. fuzzy log. intell. syst. 21 (2021), 169-175. https://doi.org/10.5391/ijfis.2021.21.2.169. [23] p. julatha, a. iampan, inf-hesitant and (sup, inf)-hesitant fuzzy ideals of ternary semigroups, missouri j. math. sci. (accepted). [24] u. jittburus, p. julatha, inf-hesitant fuzzy interior ideals of semigroups, int. j. math. comput. sci. 17 (2022), 775-783. [25] u. jittburus, p. julatha, new generalizations of hesitant and interval-valued fuzzy ideals of semigroups, adv. math.: sci. j. 10 (2021), 2199-2212. https://doi.org/10.37418/amsj.10.4.34. [26] p. julatha, a. iampan, sup -hesitant fuzzy ideals of γ-semigroups, j. math. comput. sci. 26 (2022), 148-161. http://dx.doi.org/10.22436/jmcs.026.02.05. [27] u. jittburus, p. julatha, a. iampan, inf-hesitant fuzzy ideals of semigroups and their inf-hesitant fuzzy translations, j. discrete math. sci. cryptography (accepted). http://mi.mathnet.ru/emj187 http://mi.mathnet.ru/emj187 https://doi.org/10.1109/ifsa-nafips.2013.6608375 https://doi.org/10.1109/ifsa-nafips.2013.6608375 https://doi.org/10.1002/int.21584 https://doi.org/10.1155/2010/303412 https://doi.org/10.1109/ijcf.1994.375115 https://doi.org/10.1109/fuzzy.2009.5276884 https://doi.org/10.1002/int.20418 https://doi.org/10.5666/kmj.2016.56.2.371 https://doi.org/10.1155/2022/9755894 https://doi.org/10.5391/ijfis.2021.21.2.169 https://doi.org/10.37418/amsj.10.4.34 http://dx.doi.org/10.22436/jmcs.026.02.05 18 int. j. anal. appl. (2022), 20:34 [28] p. phummee, s. papan, c. noyoampaeng, et al. sup-hesitant fuzzy interior ideals of semigroups and their suphesitant fuzzy translations, int. j. innov. comput. inform. control. 18 (2022), 121-132. [29] y. imai, k. iséki, on axiom systems of propositional calculi, proc. japan acad. 42 (1966), 19-21. https://doi. org/10.3792/pja/1195522169. [30] k. iséki, an algebra related with a propositional calculus, proc. japan acad. 42 (1966), 26-29. https://doi. org/10.3792/pja/1195522171. [31] k. iséki, an introduction to the theory of bck-algebras, math. japon. 23 (1978), 1-26. [32] g. muhiuddin, h.s. kim, s.z. song, y.b. jun, hesitant fuzzy translations and extensions of subalgebras and ideals in bck/bci-algebras, j. intell. fuzzy syst. 32 (2017), 43-48. [33] p. mosrijai, a. satirad, a. iampan, new types of hesitant fuzzy sets on up-algebras, math. morav. 22 (2018), 29-39. https://doi.org/10.5937/matmor1802029m. [34] g. muhiuddin, y.b. jun, sup-hesitant fuzzy subalgebras and its translations and extensions, ann. commun. math. 2 (2019), 48-56. [35] g. muhiuddin, h. harizavi, y.b. jun, ideal theory in bck/bci-algebras in the frame of hesitant fuzzy set theory, appl. appl. math. 15 (2020), 337-352. https://digitalcommons.pvamu.edu/aam/vol15/iss1/19. [36] m. mohseni takallo, r.a. borzooei, y.b. jun, sup-hesitant fuzzy p-ideals of bci-algebras, fuzzy inform. eng. 13 (2021), 460–469. https://doi.org/10.1080/16168658.2021.1993668. [37] h. harizavi, y.b. jun, sup-hesitant fuzzy quasi-associative ideals of bci-algebras, filomat 34 (2020), 4189-4197. https://doi.org/10.2298/fil2012189h. [38] g. muhiuddin, a.m. alanazi, m.e. elnair, k.p. shum, inf-hesitant fuzzy subalgebras and ideals in bck/bcialgebras, eur. j. pure appl. math. 13 (2020), 9-18. https://doi.org/10.29020/nybg.ejpam.v13i1.3575. [39] y.b. jun, s.z. song, inf-hesitant fuzzy ideals in bck/bci-algebras, bull. sect. log. 49 (2020), 53-78. https: //doi.org/10.18778/0138-0680.2020.03. https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.5937/matmor1802029m https://digitalcommons.pvamu.edu/aam/vol15/iss1/19 https://doi.org/10.1080/16168658.2021.1993668 https://doi.org/10.2298/fil2012189h https://doi.org/10.29020/nybg.ejpam.v13i1.3575 https://doi.org/10.18778/0138-0680.2020.03 https://doi.org/10.18778/0138-0680.2020.03 1. introduction 2. preliminaries 3. main results 4. conclusions references international journal of analysis and applications volume 18, number 2 (2020), 304-318 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-304 n wave and periodic wave solutions for burgers equations zahia nouri, saida bendaas∗ and houssem eddine kadem département de mathématiques, faculté des sciences, université ferhat abbas, sétif 1. campus el-bez, 19000, sétif, algérie ∗corresponding author: saida bendaas@yahoo.fr abstract. this article concerns the initial boundary value problem for the non linear dissipative burgers equation. our general purpose is to describe the asymptotic behavior of the solution in the cauchy problem with a small parameter ε for this equation and to discuss in particular the cases of the n wave shock and periodic wave shock. we show that the solution of cauchy problem of viscid equation approach the shock type solution for the cauchy problem of the inviscid equation for each case. the results are formulated in classical mathematics and proved with infinitesimal techniques of non standard analysis. 1. introduction burgers equation is the scalar partial differential equation ut + uux = εuxx (1.1) where x ∈ x ⊆ r, t > 0, and u : x ×r+ → r. the parameter ε is typically referred to as the viscosity due to the connection between this equation and the stydy of fluid dynamics. when ε > 0 it is often reffered to the viscous burgers equation, and when ε = 0, it is often reffered to the inviscid burgers equation. burgers equation was proposed as a model of turbulent fluid motion by j.m. burgers in a series of sevral articles. it is one of the most important pdes in the theory of non linear consevation laws. she combining received 2019-05-30; accepted 2019-07-12; published 2020-03-02. 2010 mathematics subject classification. 35l99, 58j45. key words and phrases. non standard analysis; viscid and inviscid burgers equation; n wave shock; periodic wave shock. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 304 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-304 int. j. anal. appl. 18 (2) (2020) 305 both nonlinear propagation effects and diffusive effect. this equation is the approximation for the onedimensional propagation of weak shock waves in a fluid. it can also be used in the description of the variation in vehicle density in highway traffic. burgers introduced the equation to describe the behavior of shock waves, traffic flow and acoustic transmission. this equation plays a relevant role in many different areas of the mathematical physics, specially in fluid mechanics. moreover the simplicity of its formulation, in contrast with the navier-stokes system, makes of the burgers equation a suitable model equation to test different numerical algorithms and results of a varied nature [2], [3]. if the viscous term is null, the remaining equation is hyperbolic this is the inviscid burgers equation ut + uux = 0 (1.2) if the viscous term is dropped from the burgers equation, discontinuities may appear in finite time, even if the initial condition is smooth they give rise to the phenomen of shock waves with important application in physics [10]. this properties make burgers equation a proper model for testing numerical algorithms in flows where severe gradients or shocks are anticipated [1], [16], [18]. recently, kunjan and twinkle [14] used mixture of new integral transform and homotopy perturbation method to find the solution of bugers equation arising in the longitudinal dispersion phenomenon in fluid flow through porous media. discretization methods are well-known techniques for sollving burgers equation. ascher and mclachlan established many methods as multisymplectic box sheme. olayiwola, gbolagade and akinpelu [12] also presented the modified variational iteration method for the numerical solution of generalized burgers-huxley equation. for the boundary value problem, sinai [16] is interested to the initial condition case: null on r−,and brownian on r+. she, aurell and frich [15] with a numerical calculs particularly examine the initial conditions of brownian fractionnair type. to study the initial value problem of burgers equation, classical methods are based on search of solutions of the reduiced problem to deduce existence and asymptotic behaviour of the solutions as ε tends to 0, the passage of the limit is very complicated, but in general the limit exist and it’s a solution for the reduced problem (when ε = 0). for ε small the solution u(x,t) is approximated by this limit [10]. other methods are based on a weak formulation of burgers equation seen as a conservation law satisfied on each of the computational domain called cell or finite volume. stochastic particle method is so used for with different initial conditions. in particular we will take the same initial conditions that we took for the inviscid. this paper completes recent works on the study of boundary value problems of burgers equations for different initial conditions [2][4]. in the presented work, our general purpose is to describe the asymptotic behavior of solutions in boundary value problems with a small parameter ε and to discuss in particular the cases of n wave and periodic wave shocks with new techniques infinitesimal of non-standard analysis. we can conclude that the solution of the cauchy problem of inviscid equation in each case is infinitely close to the solution of he cauchy problem of viscid equation as ε is a parameter positif sufficiently small. we int. j. anal. appl. 18 (2) (2020) 306 introduce the infinitesimal techniques to give a simple formulation for the asymptotic behaviour. it is worth noting that our contribution is an elegant combination of infinitesimal techniques of non-standard analysis and the van den berg method [2], [3], [17]. historically the subject non standard was developed by robinson, reeb, lutz and gose [11]. the nonstandard perturbation theory of differential equations, which is today a well-established tool in asymptotic theory, has its roots in the seventies, when the reebian school (see [11], [17]) introduced the use of nonstandard analysis into the field of perturbed differential equations. our goal in this paper is to generalize these techniques on pde. the paper is organised as follows: section 2 concerns the boundary value problems of inviscid burgers equation, we start with the fitting discontinuous shock then we describe the asymptotic behaviour of solutions for this problem in the n wave and periodic wave cases. section 3 concerns the boundary value problems of viscid burgers equation, it contains basic preliminaries results and deals with our main results about n wave and periodic wave shock cases and its proof, we present it in a non standard form. 2. inviscid burgers equation we will focus first on equation (1.2). specifically, we will deal with the initial value problem  ut + uux = 0 , ∀x ∈ r, t > 0u(ξ, 0) = f(ξ) , t = 0 (2.1) as it as has been suggested previously, although (2.1) seems to be a very innocent problem a priori it hides many unexpected phenomena.this problem does not admit the regular solutions but some weak solutions with certain regularity exist.the burgers equation on the whole line is known to possess traveling waves solutions. using the characteristic method, the solution of the problem (2.1) may be given in a parametric form   u = f(ξ)x = ξ + f(ξ)t (2.2) and shocks must be fitted in such that u = 1 2 (u1 + u2) = 1 2 (f(ξ1) + f(ξ2) (2.3) where f : r → r. is a standard continuous function. ξ1 and ξ2 are the values of ξ on the two sides of the shock [6]. according to (2.2), the solution at time t is obtained from the initial profile u = f(ξ) by translating each point a distance f(ξ)t to the right. the shock cuts out the part corresponding to ξ2 ≥ ξ ≥ ξ1. if the discontinuity line it is a straight line chord betewen the points ξ = ξ1 and ξ = ξ2 on the curve f(ξ). moreover since areas are preserved under the mapping, the equal area property still holds. the chord on int. j. anal. appl. 18 (2) (2020) 307 the f curve cuts off lobes of equal area. the shock determination can then be describe entirely on the fixe f(ξ) curve by drawing all the chords with the equal area propperty can be written analytically as 1 2 {(f(ξ1) + f(ξ2)}(ξ1 − ξ2) = ∫ ξ1 ξ2 f(ξ)dξ (2.4) this is the differential equation for the shock line chord wich verifies the entropic condition such as [6]. since the left hand side is the area under the chord and the right hand side is the area under the f curve. if the shock is at x = s(t) at time t, we also have s(t) = ξ1 + f(ξ1)t (2.5) s(t) = ξ2 + f(ξ2)t (2.6) from (2.5) and (2.6), we have t = ξ1 − ξ2 f(ξ1) −f(ξ2) (2.7) 2.1. single hump. to describe the solutions of the problem (2.1), we assume that the initial condition f verify the following assumptions : (h1): f is equal to a constant u0 outside the range 0 < ξ < l. (h2): f(ξ) > u0 in the range. the theorem bellow gives the assymptotic behavior to the solution behind the shock and at the shock. theorem 2.1. suppose that (h1) and (h2) are satisfied, the solution of the problem (2.1) is given by (2.2) with 0 < ξ < ξ2 and the asymptotic form is u ∼ x t , for u0t < x < u0t + √ 2at (2.8) proof. we consider the problem (2.1) and we suppose that (h1) and (h2) are satisfied. equation (2.4) may be written as 1 2 {(f(ξ1) + f(ξ2) − 2u0}(ξ1 − ξ2) = ∫ l ξ1 (f(ξ) −u0)dξ (2.9) as time goes on ξ1 increase and eventually exceed l. at this stage f(ξ1) = u0 and the shock is moving into the constant region u = u0. the function ξ1(t) can then be eliminated for we have. 1 2 (f(ξ2) −u0) (ξ1 − ξ2) = ∫ l ξ2 (f(ξ) −u0)dξ , t = ξ1 − ξ2 f(ξ2) −u0 (2.10) there for 1 2 (f(ξ2) −u0) 2t = ∫ l ξ2 (f(ξ) −u0)dξ (2.11) int. j. anal. appl. 18 (2) (2020) 308 at this stage the shock position and the value of u just behind the shock are given by  u = f(ξ2)s(t) = ξ2 + f(ξ2)t (2.12) where ξ2 satisfies (2.11). as t is infinitely large we have ξ2 infinitesimal and f(ξ2) approach u0, hence the equation for ξ2(t) takes the limiting form 1 2 (f(ξ2) −u0) 2t ∼ a (2.13) where a = ∫ l 0 (f(ξ) −u0)dξ (2.14) is the area of the hump above the undisturbed value u0.we have ξ2 infinitesimal and f(ξ2) ∼ u0 + √ 2a/t (2.15) therefore the asymptotic formulas for s(t) and u in (2.12) are s(t) ∼ u0t + √ 2at (2.16) u−u0 ∼ √ 2a/t (2.17) at the shock. the shock curve is asymptotically parabolic. the solution behind the shock is given by (2.2) with 0 < ξ < ξ2. since ξ2 is small enouhg as t is small enouhg, all the relevant values of ξ also small enouhg and the asymptotic form is u ∼ x t , for u0t < x < u0t + √ 2at the asymptotic solution and the corresponding (x,t) diagram are shown in figure 1. figure 1. the asymptotic triangular wave � int. j. anal. appl. 18 (2) (2020) 309 figure 2. the profile of the initial condition in the n wave case 2.2. n wave. other problem can be worked out in similar way, one important case is when f(ξ) has a positive and a negative phase about an undisturbed value u0 as in figure 2. there are now two shocks, corresponding to the two compression phases at the front and at the back where f′(ξ) < 0. the families of chords for each are shown in the figure. as t is infinitely large, the pair (ξ2,ξ1) for the front shock approach (0,∞),where as for the rear shock (ξ2,ξ1) approach (−∞, 0). asymptotically the front shock is s ' u0t + √ 2at (2.18) and the jump of u is u−u0 ∼ √ 2a/t (2.19) where a is the area of the f curve above u = u0.the rear shock has x ∼ u0t− √ 2bt (2.20) u−u0 ∼− √ 2b/t (2.21) where b is the area below u = u0.the solution between the shocks is again asymptotically u ∼ x t , u0t− √ 2bt < x < u0t + √ 2at (2.22) the asymptotic form and the (x,t) diagramm are shown in figure 3 and figure 4. figure 3. shock construction for n wave int. j. anal. appl. 18 (2) (2020) 310 figure 4. the asymptotic n wave 2.3. periodic wave. another intersting problem is that of an initial distribution f(ξ) = u0 + a sin 2πξ λ in this case, the shock equations (2.4) simplify considerably for all times t. consider one period 0 < ξ < λ as in figure 5. relations (2.4) becomes (ξ1 − ξ2) sin π λ (ξ1 + ξ2) cos π λ (ξ1 − ξ2) = λ π sin π λ (ξ1 − ξ2) sin π λ (ξ1 + ξ2) (2.23) figure 5. shock construction for a periodic wave and the relevant choice is the trivial one sin π λ (ξ1 + ξ2) = 0 that is ξ1 + ξ2 = λ from the difference and sum of (2.5) and (2.6) we have t = ξ1 − ξ2 2a sin π λ (ξ1 − ξ2) s = u0t + λ 2 respectively. the discontinuity in u at the shock is u2 −u1 = a sin 2πξ1 λ −a sin 2πξ2 λ = 2a sin π λ (ξ1 − ξ2) int. j. anal. appl. 18 (2) (2020) 311 if we introduce ξ1 − ξ2 = λθ π , ξ1 + ξ2 = λ we have t = λ 2πa . θ sin θ s = u0t + λ 2 (2.24) u2 −u1 u0 = 2a u0 sin θ the shock has constant velocity u0 and this result could have been deduced in advance from the symmetry of the problem. the shock starts with zero strength corresponding to θ = 0 at time t = λ/2πa. it reaches a maximum strength of 2a/u0 for θ = π/2, t = λ/4a and decays ultimately with θ approach π, when t is infinitely large u2 −u1 u0 ∼ λ u0t (2.25) it is interesting that the final decay formula does not even depend explicitly on the amplitude a, however the condition for its application is t >> λ/a. for any periodic sinusoidal f(ξ) or not ξ1 − ξ2 → λ as t infinitely large; thence from (2.7) u2 −u1 u0 = f(ξ2) −f(ξ1) u0 ∼ λ u0t (2.26) between successive shocks, the solution for u is linear in x with slope 1/t as before, and the asymptotic form of the entire profile is the sawtooth shown in figure 6. figure 6. asymptotic form of a periodic wave 3. viscid burgers equation in this section we shall present and prove our main results, we discuss the n wave and periodic wave cases in the boundary value problem of viscid burgers equation  ut + uux = εuxx , ∀x ∈ r, t > 0u(ξ, 0) = f(ξ) , t = 0 (3.1) before going further in this cases we need the following proposition and lemma : int. j. anal. appl. 18 (2) (2020) 312 3.1. the cole-hopf transformation. cole and hopf noted the remarkable result [9] that the viscid burgers equation (1.1) may be reduced to the linear heat equation ϕt = εϕxx (3.2) by the non linear transformation u = −2ε[log ϕ]x (3.3) it is again convenient to do the transformation in two steps. firstly are introduced u = ψx so that (1.1) may be integrated to ψt + 1 2 ψ2x = εψxx then we introduce ψ = −2ε[log ϕ] to obtain (3.2). the non linear transformation just eliminates the nonlinear term. the general solution of the heat equation (3.2) is well known and can be handled by a variety of methods. the basic problem considered in section 2 is the initial value problem u = f(ξ) , at t = 0 this is transformed by (3.3) to the initial value problem ϕ = φ(x) = exp { − 1 2ε ∫ x 0 f(η)dη } , t = 0 (3.4) for the heat equation, the solution for ϕ is ϕ = 1 √ 4πεt ∫ +∞ −∞ φ(η) exp { − (x−η)2 4εt } dη (3.5) through (3.3), the solution for u is u(x,t) = ∫ +∞ −∞ x−η t e−g/2εdη∫ +∞ −∞ e −g/2εdη (3.6) where g(η,x,t) = ∫ η 0 f(ν)dν + (x−η)2 2t (3.7) int. j. anal. appl. 18 (2) (2020) 313 3.2. the behavior of solutions as ε small enough. the behavior of the exact solution (3.6) is now considered as ε is small enough. for x, t and f(x) are held fixed as ε is small enough, the dominant contributions to the integrals in (3.6) come from the neighborhood of the stationary points of g. a stationary point is where ∂g ∂η = f(η) − x−η t = 0 let η = ξ(x,t) be such a point that is ξ(x,t) is difined as a solution of f(ξ) − x− ξ t = 0 (3.8) the contribution from the neighborhood of a stationary point η = ξ in an integral is given with the lemma 3.2. lemma 3.1. (the van. den .berg lemma 12). let g be a standard function definied and increasing on [0, +∞[ such that g(v) = avr (1 + δ) for v ' 0 and g(v) > m(v)q. let ϕ be an intern function definied on ]0, +∞[ such that : ϕ(v) = bvs(1 + δ) for v ≈ 0 and such that ∀ d > 0, ∃ standard k and c such that : |ϕ(v)| < k exp(cosh(v)) for v > d. then∫ ∞ 0 ϕ(v) exp(− g(v) 2ε )dv = bγ( (s+1) r ) ra (s+1) r 1 ( 1 2ε ) (s+1) r (3.9) where a and r are positifs standard, m and q are the both positifs. δ is a positif real small enough. b and s are standard, b 6= 0 and s > −1. lemma 3.2. (the nonstandard formula of the method of steepest descents). let ε be a positif rael small enough and let ϕ and g be two standard functions such that : g, is a c2 class function verifies the lemma 3.1, and admits on the ξ point an unique absolute minimum (g′(ξ) = 0 et g′′(ξ) > 0). ϕ(ξ) 6= 0, it is scontinuous on ξ and satisfies the conditions of the lemma 3.1 in the two sens. then∫ +∞ −∞ ϕ(η).e−g/2εdη = ϕ(ξ) √ 4πε√ g′′(ξ) .e−g/2ε(1 + δ) (3.10) δ is a positif real small enough. proof. suppose first that there is only one stationary point ξ(x,t) wich satisfies (3.8) then∫ +∞ −∞ x−η t e−g/2εdη = x−η t √ 4πε√ g′′(ξ) e−g/2ε(1 + δ) (3.11) ∫ +∞ −∞ e−g/2εdη = √ 4πε√ g′′(ξ) e−g/2ε(1 + δ) (3.12) and in (3.6) we have u ' x− ξ t (3.13) int. j. anal. appl. 18 (2) (2020) 314 where ξ(x,t) is defined by (3.8). this asymptotic solution may be rewritten as   u = f(ξ)x = ξ + f(ξ)t it is exactly the solution of (2.1) witch was discussed in section 2. the stationary point ξ(x,t) becomes the characteristic variable. � 3.3. the main results. 3.3.1. n wave. another example, we consider is more easily derived by choosing appropriate solutions for ϕ to satisfy the heat equation and then substituting in (1.4) to obtain u as a rough qualitative guide to the appropriate choice. the profile of u will be some thing like ϕx. to obtain an n wave of u. we choose the source solution of the heat equation for ϕ ϕ = 1 + √ a t .e(−x 2/4εt) (3.14) since ϕ has a δ function behavior as t is infinitesimal, this is a little hard to interpret as an initial value problem on u. however, for any t > 0,it has the form shown in figure 2 with a positive and a negative phase and we may take the profile at any t = t0 to be the initial profile. it should typical of all n wave solutions. theorem 3.1. (i). assume that the initial data f has a profile shown in figure 2, the problem (1.3) admit an unique solution for t > 0 given by u ∼   x t , − √ 2at < x < √ 2at 0, |x| > √ 2at where a = ∫ +∞ −∞ (f(x) −u0) dx. (ii). such solution present n wave chocks, and for ε small enough, this solution is infinitely close to the solution of the inviscid problem (1.4) giveb in section 2. proof. to obtain an n wave for u, we choose the source solution of the heat equation for ϕ given by (3.13). then the corresponding solution for u is u = −2ε ϕx ϕ = x t . √ a t . exp(− x2 4εt ) 1 + √ a t . exp(− x2 4εt ) (3.15) since ϕ has a δ function behavior as t is infinitely small, this is a little hard to interpret as an initial value problem on u. however for any t > 0 it has the form shown in fig.6 with a positive and negative phase int. j. anal. appl. 18 (2) (2020) 315 and we may take the profile at any t = t0 to be the initial profile. it should be typical of all n wave solution. the area under the positive phase of the profile is∫ +∞ 0 udx = −2ε[log ϕ]∞0 = 2ε log[1 + √ a t ] (3.16) the positive phase is infinitely small when t is infinitely large. if the value of (3.14) at the initial time t0 is denoted by a we may introduce a reynolds number r0 = a 2ε = log(1 + √ a t0 ) but as time goes on the effective, reynolds number will be r(t) = 1 2ε ∫ +∞ 0 udx = log(1 + √ a t ) (3.17) and this is infinitely small as t is infinitely large. if r0 >> 1, we may expect the ”inviscid theory” of (2.1) and (2.2) to be a good approximation for some time but as t is infinitely large , r(t) is eventually become dominant. in terms of r0 and t0, a = t0(exp(r0) − 1). hence (3.14) may be written. u = x t .  1 + √ t t0 exp( x2 4εt ) exp(r0) − 1   −1 (3.18) and for r0 >> 1 ( corresponding to t0 << 0),we have exp(r0) − 1 ∼ exp(r0) and exp( x2 4εt ) exp(r0) − 1 ∼ exp( x2 4εt −r0) ∼ exp r0( x2 2at − 1). or u in (3.17) may be approximated by u = x t .  1 + √ t t0 exp[r0( x2 2at − 1)]   −1 (3.19) for x and t . now for fixed t limited and r0 infinitely large we have u ∼   x t , if x2 2at − 1 < 0 , − √ 2at < x < √ 2at 0, if x2 2at − 1 < 0 , |x| > √ 2at this is exactly the inviscid solution. however, for any fixed a and ε we see directely from (3.15) [and it may be verified also from (3.19)] that u ∼ x t . √ a t exp(− x2 4εt ) int. j. anal. appl. 18 (2) (2020) 316 as t is infinitely large. this is the dipole solution of the heat equation. the diffusion dominates the nonlinear term in the final decay. it should be remembered though,that this final period of decay is for extremely large times; the inviscid theory is adequate for most of the interesting range. � 3.3.2. periodic wave. a periodic solution may be obtained by taking for ϕ a distribution of heat sources spaced a distance λ apart. then ϕ = 1 √ 4πεt ∞∑ n=−∞ exp { − (x−nλ)2 4εt } (3.20) theorem 3.2. (i). assume that the initial data f has a profile shown in fig 5.when λ2/4εt >> 1, the problem (3.1) admit an unique solution for t > 0 given by u ∼ x−mλ t , (m− 1/2)λ < x < (m + 1/2)λ (ii). such solution is the periodic wave chock, and for ε small enough, this solution is infinitely close to the solution of the inviscid problem (2.1) given in section 2. proof. to obtain a periodic wave for u, we choose for ϕ a distribution of heat sources spaced a distance λ given by (3.14). then the corresponding solution for u is u = −2ε ϕx ϕ = ∞∑ n=−∞ ( x−nλ t ) exp { − (x−nλ)2 4εt } ∞∑ n=−∞ exp { − (x−nλ)2 4εt } (3.21) for λ2/4εt >>, 1 this implies that √ εt << λ/2, and ∞∑ n=−∞ = 2 ∞∑ n=0 , then for n = m we have (x−nλ)2 4εt = (x−mλ)2 εt << 1 wich gives |x−mλ| << √ εt. and the exponential with the minimum value of (x−nλ)2/4εt will dominate over all the others.therefore the term witch will dominate for (m− 1/2)λ < x < (m + 1/2)λ and (3.15) is approximately u ∼ x−mλ t , for (m− 1/2)λ < x < (m + 1/2)λ this is a sawtooth wave with a periodic set of shocks a distance λ apart, and u jumps from −λ/2t to λ/2t at each shock . the result agrees with he inviscid solution given by (2.20). � corollary 3.1. if λ2 4εt << 1, the initial data can be expanded in a fourier series as ϕ = 1 λ { 1 + 2 ∞∑ n=1 exp ( − 4π2n2 λ2 εt ) cos 2πnx λ } int. j. anal. appl. 18 (2) (2020) 317 and for t > 0, we have u ∼ 8πε λ exp { − 4π2εt λ2 } sin 2πx λ this is a solution of the heat equation . proof. to study the final decay (λ2/4εt) << 1, we may use an alternative form of the solution. the expression (3.14) is periodic in x, and in the interval: −λ/2 < x < λ/2, ϕ → δ(x), as t is infinitely small. the initial condition can be expanded in a fourier series as φ(x) = 1 λ { 1 + 2 ∞∑ n=1 cos 2πnx λ } and the corresponding solution of the heat equation for ϕ is ϕ = 1 λ { 1 + 2 ∞∑ n=1 exp ( − 4π2n2 λ2 εt ) cos 2πnx λ } it may be verified directly that this is the fourier series of .(3.14). in this form u = −2ε ϕx ϕ = 8πε λ ∞∑ n=1 n exp ( − 4π2n2 λ2 εt ) sin 2πnx λ 1 + 2 ∞∑ n=1 exp ( − 4π2n2 λ2 εt ) cos 2πnx λ when εt λ2 >> 1,the term with n = 1 dominate the series and we have u ∼ 8πε λ exp { − 4π2εt λ2 } sin 2πx λ this is a solution of ut = εuxx and the diffusion dominate in the ultimate decay. � 4. conclusion this paper completes recent works on the study of boundary value problems of burgers equations for different initial conditions [2][4]. our general purpose is to describe the asymptotic behavior of solutions in boundary value problem with a small parameter ε and to discuss in particular the n wave shocks and periodic wave shocks cases. the originality of this work consists in introducing new infinitesimal techniques of non-standard analysis. we can conclude that the solution of the cauchy problem of inviscid equation in each case is infinitely close to the solution of he cauchy problem of viscid equation as ε is a parameter positif sufficiently small. we introduce the infinitesimal techniques to give a simple formulation for the asymptotic behavior. it is worth noting that our contribution is an elegant combination of infinitesimal techniques of non standard analysis and the van den berg method [2], [3], [17]. int. j. anal. appl. 18 (2) (2020) 318 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] alexy, samokhin, gradient catastrophes and saw tooth solution for a generalized burgers equation on an interval, j. geom. phys. 85 (2014), 177-184. [2] s. bendaas, the asymptotic behavior of viscid burgers solution in the n wave shock case. a new approach, asian j. math. computer res. 12 (3) (2016), 221-232. [3] s. bendaas, confluence of shocks in burgers equation. a new approach, int. j. differ. equ. appl. 14 (2015), 369-382. [4] s. bendaas, boundary value problems for burgers equations through nonstandard analysis, appl. math. 6 (2015), 10861098. [5] s. bendaas, l’équation de burgers avec un terme dissipatif. une approche non standard, analele universitatii oradea, fasc. math. 15 (2008), 239-252. [6] s. bendaas, quelques applications de l’a.n.s aux e.d.p, thèse de doctorat. université de haute alsace, france, (1994). [7] j. m. burgers, the non linear diffusion equation asymptotic solution and satatistical propblems, reidel, 1974. [8] d. euvrard, résolution numérique des équations aux dérivées partielles, différences finies, éléments finis/masson. paris, milan, barcelone, mexico (1989). [9] e. hopf, the partial differential equation ut + uux = νuxx, commun. pure appl. mech. 3 (1950), 201-230. [10] s. kida, asymptotic properties of burgers turbulence, journal of fluid mechanics, 93(1979), 337-377. [11] r. lutz and m. goze, non standard analysis. a. practical guide with application lectures, notes in math. n 861. springer verlag, berlin (1981). [12] m.o. olayiwola, a.w. gbolagade and f.o. akinpelu, numerical solution of generalized burger’s-huxley equation by modified variational iteration method, j. nigerian assoc. math. phys. 17 (2010), 433-438. [13] a. v. samokhin, evolution of initial data for burgers equation with fixed boundary values, sci. herald mstuca, 194 (2013), 63-70 (in russian). [14] k. shah and t. singh, a solution of the burger’s equation arising in the longitudinal dispersion phenomena in fluid flow through porous media by mixture of new integral transform and homotopy perturbation method, j. geosci. environ. protect. 3 (2015), 24-30. [15] z. s. she, e. aurell and u. frisch, the inviscid burgers equation with initial data of brownien type, commun. math. phys. 148 (1992), 623-641. [16] ya. g. sinai, statistics of shocks in solutions of inviscid burgers equation, commun. math. phys. 148 (1992), 601-621. [17] i. van den berg, non standard asymptotic analysis lectures, notes in math. vol. 1249. springer verlag. [18] a. m. wazwaz, travelling wave solution of generalized forms of burgers, burgers-kdv and burger’s-huxley equations, app. math. comput. 169 (2005), 639-656. 1. introduction 2. inviscid burgers equation 2.1. single hump 2.2. n wave 2.3. periodic wave 3. viscid burgers equation 3.1. the cole-hopf transformation 3.2. the behavior of solutions as 0=x"0122 small enough 3.3. the main results 4. conclusion references int. j. anal. appl. (2022), 20:72 on ωθ̃-µ-open sets in generalized topological spaces fatimah al mahri∗, abdo qahis department of mathematics, college of science and arts, najran university, saudi arabia ∗corresponding author: cahis82@gmail.com abstract. in this paper analogous to [1], we introduce a new class of sets called ωθ̃-µ-open sets in generalized topological spaces which lies strictly between the class of θ̃µ-open sets and the class of ω-µ-open sets. we prove that the collection of ωθ̃-µ-open sets forms a generalized topology. finally, several characterizations and properties of this class have been given. 1. introduction one notion that has received much attention lately is the so-called ω-open sets in a topological space (x,τ) was introduced by hdeib [12], which forms a topology finer than τ. recently, many topological concepts and several interesting results related to this notion have obtained by many authors such as [3], [10], [9], [2]. a collection µ of subsets of a nonempty set x is a generalized topology (gt) if ∅∈ µ and µ is closed under arbitrary unions, this notion was introduced by császár in the sense of [5]. we call the pair (x,µ) a generalized topological space (briefly gts) on x. the elements of µ are called µ-open sets and their complements are called µ-closed sets, see [7], the union of all elements of µ will be denoted by mµ and a gts (x,µ) is said to be strong [7] if x ∈ µ. if a is a subset of a gts (x,µ), then the µ-closure of a, cµ(a), is the intersection of all µ-closed sets containing a and the µ-interior of a, iµ(a), is the union of all µ-open sets contained in a (see [5,7]). it is easy to observe that operators iµ and cµ are idempotent and monotonic a subset a of a gts (x,µ) is µ-open if and only if a = iµ(a), and and iµ(a) = x \cµ(x \a). evidently, a is µ-closed if and only if a = cµ(a), cµ(a) is the smallest µ-closed set containing a, iµ(a) is the largest µ-open set contained in a. over recent years several authors have been working in formulate many topological received: nov. 21, 2022. 2010 mathematics subject classification. 54a05, 54c08. key words and phrases. generalized topology; θ̃µ-open sets; ω-µ-open sets; ωθ̃-µ-open sets; θ̃µ-locally countable; ωθ̃-anti-locally countable. https://doi.org/10.28924/2291-8639-20-2022-72 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-72 2 int. j. anal. appl. (2022), 20:72 concepts to establish new concepts in the structure of gts, see [4], [8], [6] [11], [17], [15], [13] and others. then motivated by the notion of ω-open set in a topological space (x,τ), al ghour and wafa zareer (2016) [1] defined the notions of ω-µ-closed sets and ω-µ-open sets in the structure of gts as follows : a subset a of gts (x,µ) is called ω-µ-closed if it contains all its condensation points. the complement of an ω-µ-closed set is called ω-µ-open. the family of all ω-µ-open subsets of x forms a gt on x, denoted by ωµ. let us now recall some notions defined in [14]. a subset a of gts (x,τ) is said to be θ̃µ-open if and only if for each x ∈ a, there exists u ∈ µ such that x ∈ u ⊆ cµ(u) ∩mµ ⊆ a and the collection of all θ̃µ-open subsets of a gts (x,µ) is denoted by θ̃µ. then θ̃µ is also a gt included in µ. analogous to [1] and by using the notion of θ̃µ-open, we introduce the relatively new notions of ω θ̃ -µ-open as a new class of sets . we present several characterizations, properties, and examples related to the new concepts. in section 2, we use the the notion of θ̃µ-open to introduce ωθ̃-µ-open sets in gts as a new class of sets and we prove that this class lies strictly between the class of θ̃µ-open sets and the class of ω-µ-open sets. moreover, we give some sufficient conditions for the equivalence between the class of ω θ̃ -µ-open sets and the class of ω-µ-open sets. in section 3, several interesting properties of ω θ̃ -µ-open subsets are discussed via the operations of ω θ̃ -interior and ω θ̃ -closure. definition 1.1. [16] a gts (x,µ) is said to be µ-locally indiscrete if every µ-open set in (x,µ) is µ-closed. definition 1.2. [1] a gts (x,µ) is called µ-locally countable if mµ is nonempty and for every point x ∈mµ, there exists a u ∈ µ such that x ∈ u and u is countable. definition 1.3. [14] let (x,µ) be a gts , a ⊆ x and γ θ̃ : p (x) → p (x) be an operation defined as the following: γ θ̃µ (a) = {x ∈ x : cµ(u) ∩mµ ∩a 6= ∅ f or all u ∈ µ,x ∈ u}. theorem 1.1. [1] let (x,µ) be a gts. then mµ = mωµ. theorem 1.2. [1] if (x,µ) is a µ-locally countable gts, then ωµis the discrete topology on mµ. 2. ω θ̃ -µ-open sets we begin this section by introducing the following definition. definition 2.1. let (x,µ) be a gts and a ⊆ x. consider an operation γω θ̃ : p (x) → p (x) defined as the following: γω θ̃ (a) = {x ∈ x : u ∩a is uncountable f or all u ∈ θ̃µ and x ∈ u}. a point x ∈ x is called a int. j. anal. appl. (2022), 20:72 3 θ̃µ-condensation point of a if for all u ∈ θ̃µ such that x ∈ u and u ∩a is uncountable. the set of all θ̃µ-condensation points of a is denoted by γω θ̃ (a). lemma 2.1. let (x,µ) be a gts. the operation γω θ̃ : p (x) → p (x) has the following properties: (1) if a ⊆ b ⊂ x, then γω θ̃ (a) ⊆ γω θ̃ (b) (monotonic property); (2) γω θ̃ (γω θ̃ (a)) ⊆ γω θ̃ (a) for any a ⊆ x (restricting property); (3) if a is any countable subset of x, then γω θ̃ (a) = ∅. proof. (1) let a ⊆ b ⊂ x and x ∈ γω θ̃ (a). then u ∩a is uncountable for each u ∈ θ̃µ and x ∈ u. since a ⊆ b, then u ∩b is uncountable. thus x ∈ γω θ̃ (b) and hence γω θ̃ (a) ⊆ γω θ̃ (b). (2) let x ∈ γω θ̃ (γω θ̃ (a)). then u ∩ γω θ̃ (a) is an uncountable for all u ∈ θ̃µ and x ∈ u. let y ∈ u ⋂ γω θ̃ (a). then y ∈ u and y ∈ γω θ̃ (a) which implies that u ∩a is an uncountable set. hence x ∈ γω θ̃ (a) and therefore γω θ̃ (γω θ̃ (a)) ⊆ γω θ̃ (a). (3) the proof is obvious by definition 2.1. � definition 2.2. let (x,µ) be a gts and a ⊆ x. then a is said to be ω θ̃ -µ-closed if γω θ̃ (a) ⊆ a. the complement of an ω θ̃ -µ-closed set is said to be ω θ̃ -µ-open. the family of all ω θ̃ -µ-open subsets of (x,µ) is denoted by ω θ̃ , where ω θ̃ = {w ⊆ x : γω θ̃ (x\w ) ⊆ x \w}. the following theorem and lemma give a necessary and sufficient condition for ω θ̃ -µ-open sets. theorem 2.1. let (x,µ) be a gts and w ⊆ x. then the following statements are equivalent: (1) w is ω θ̃ -µ-open; (2) if for every x ∈ w there exists a u ∈ θ̃µ such that x ∈ u and u \w is a countable set. proof. (1) ⇒ (2): suppose w is ω θ̃ -µ-open. since x\w is ω θ̃ -µ-closed set, then γω θ̃ (x\w ) ⊆ x\w. this means that for every x ∈ w, x /∈ γω θ̃ (x \w ) and hence there exists a u ∈ θ̃µ such that x ∈ u and u ∩ (x \w ) = u \w is countable. (2) ⇒ (1): let x ∈ w. then by assumption there exists a u ∈ θ̃µ such that x ∈ u and u∩(x\w ) is countable. which implies that x /∈ γω θ̃ (x\w ), γω θ̃ (x\w ) ⊆ x\w and hence x\w is ω θ̃ -µ-closed. therefore w is ω θ̃ -µ-open set. � lemma 2.2. a subset w of a gts (x,µ) is ω θ̃ -µ-open if and only if for every x ∈ w there exists a u ∈ θ̃µ and a countable c ⊆mµ such that x ∈ u \c ⊆ w. proof. necessity. let w be ω θ̃ -µ-open and x ∈ w. by theorem 2.1, there exists u ∈ θ̃µ such that x ∈ u and u\w is countable. let c = u\w. then c is countable, c ⊆mµ and x ∈ u∩(x\c) = u ∩ ( x \ (u ∩x \w ) ) = u ∩w ⊆ w and hence x ∈ u \c ⊆ w. sufficiency. let x ∈ w. from assumption there exists u ∈ θ̃µ and a countable set c ⊆ mµ such that x ∈ u \c ⊆ w. therefore, u \w ⊆ c and u \w is a countable set and this completes the proof. � 4 int. j. anal. appl. (2022), 20:72 theorem 2.2. let (x,µ) be a gts and c ⊆ x. if c is ω θ̃ -µ-closed, then c ⊆ f ∪ b for some ω θ̃ -µ-closed set f and a countable subset b. proof. let c be any ω θ̃ -µ-closed set in (x,µ). then x \c is ω θ̃ -µ-open. by lemma 2.2, for each x ∈ x \ c, there exist a θ̃µ-open set u containing x and a countable subset b ⊆ mµ such that x ∈ u\b ⊆ x\c. thus c ⊆ x\(u\b) = x\(u∩(x\b)) = (x\u)∪b. let f = x\u. then f is ω θ̃ -µ-closed such that c ⊆ f ∪b. � theorem 2.3. let (x,µ) be a gts. then the collection ω θ̃ forms a generalized topology on x. proof. it is clear that ∅ ∈ ω θ̃ . let {wλ : λ ∈ ∆} be a collection of ωθ̃-µ-open subsets of (x,µ) and x ∈ ⋃ λ∈∆ wλ. there exists an λ0 ∈ ∆ such that x ∈ wλ0. since wλ0 is ωθ̃-open set, then by lemma 2.2, there exist u ∈ θ̃µ and a countable set c ⊆ mµ such that x ∈ u \ c ⊆ wλ0 ⊆ ⋃ λ∈∆ wλ. by lemma 2.2, it follows that ⋃ λ∈∆ wλ is ωθ̃-µ-open. hence the collection ωθ̃ is generalized topology on x. � the next theorem obtains that the new class of ω θ̃ -µ-open sets lies strictly between the class of θ̃-µ-open sets and the class of ω-µ-open sets. theorem 2.4. let (x,µ) be a gts. then θ̃µ ⊆ ωθ̃ ⊆ ωµ. proof. to show that θ̃µ ⊆ ωθ̃, let w ∈ θ̃µ and x ∈ w. take u = w and c = ∅. then u ∈ θ̃µ, c ⊆mµ such that x ∈ u \c ⊆ w. therefore, by lemma 2.2, it follows that w ∈ ωθ̃. to show that ω θ̃ ⊆ ωµ, let w ∈ ωθ̃. by theorem 2.1, for each x ∈ w there exists a u ∈ θ̃µ such that x ∈ u and u \w is countable. since θ̃µ ⊆ µ, then u ∈ µ and hence w is ω-µ-open. therefore w ∈ ωµ. � the following diagram follows immediately from the definitions and theorem 2.4. θ̃µ −open =⇒ ωθ̃ −µ−openww� ww� µ−open =⇒ ω −µ−open the converse of these implications need not be true in general as shown by the following examples. example 2.1. consider x = r, a = {4n : n ∈ n} and µ = {∅, [0, 2], [1, 3] ∪ a, [0, 3] ∪ a}. then (x,µ) is a generalized topological space and the family of all θ̃µ-open sets is θ̃µ = {∅, [0, 3] ∪ a}. then [1, 3] ∈ ωµ \ωθ̃, i.e. [1, 3] is ω-µ-open but it is not ωθ̃-µ-open. also, it is easy to check that γω θ̃ (r\ [0, 3]) ⊆r\ [0, 3]. thus [0, 3] ∈ ω θ̃ \ θ̃µ, i.e. [0, 3] is ωθ̃-µ-open but it is not θ̃µ-open example 2.2. let x = {a,b,c,d} with gt µ = {∅,{a,b},{a,c},{a,b,c}}. then {a,c}∈ ω θ̃ \ θ̃µ, i.e. the set {a,c} is ω θ̃ -µ-open but it is not θ̃µ-open. note that the previous examples show that θ̃µ 6= ωθ̃ 6= ωµ in general. int. j. anal. appl. (2022), 20:72 5 remark 2.1. the notions of µ-open and ω θ̃ -µ-open sets are independent of each other. for more clarity in example 2.1, the set [0, 3] is ω θ̃ -µ-open but it is not µ-open and the set [1, 3] ∪a is µ-open but it is not ω θ̃ -µ-open. theorem 2.5. if a gts (x,µ) is a µ-locally indiscrete, then µ ⊆ ω θ̃ . proof. to show that µ ⊆ ω θ̃ , let a ∈ µ and x ∈ a. take u = a. since (x,µ) is µ-locally indiscrete, then cµ(u) = u and we have x ∈ u ⊆ cµ(u)∩mµ ⊆ a. thus a ∈ θ̃µ and by theorem 2.4, θ̃µ ⊆ ωθ̃. therefore a ∈ ω θ̃ . � lemma 2.3. let (x,µ) be a gts. then mµ ∈ θ̃µ. proof. let a = mµ and x ∈ a. then there exists ux ∈ µ such that x ∈ ux. since ux ⊆ cµ(ux ) ⋂ mµ ⊆ a, then a = mµ ∈ θ̃µ. � for a gt µ on a nonempty set x, let mω θ̃ = ⋃ {u ⊆ x : u ∈ ω θ̃ }. thus we have the following theorem. theorem 2.6. let (x,µ) be a gts. then mµ = mω θ̃ proof. by lemma 2.3, mµ ∈ θ̃µ and form theorem 2.4, θ̃µ ⊆ ωθ̃ and hence mµ ⊆ mωθ̃. on the other hand, let x ∈ mω θ̃ . since, mω θ̃ ∈ ω θ̃ , then by lemma 2.2, there exists a u ∈ θ̃µ and a countable set c ⊆mµ such that x ∈ u \c ⊆mω θ̃ . since u ⊆mµ and u is µ-open, it follows that x ∈mµ and hence mω θ̃ ⊆mµ. therefore mµ = mω θ̃ . � by theorem 1.1 and theorem 2.6, we obtain the following corollary corollary 2.1. let (x,µ) be a gts. then mµ = mω θ̃ = mωµ we will denote by (τcoc)x, the cocountable topology on a nonempty set x. theorem 2.7. let (x,µ) be a gts. then (τcoc)u ⊆ ωθ̃ for all u ∈ θ̃µ \{∅}. proof. let u ∈ θ̃µ \ {∅}, w ∈ (τcoc)u and x ∈ w. since w ⊆ u, we have x ∈ u and u \ w = u \ (u ∩v ) for some v ∈ τcoc. now, u \w = u \ (u ∩v ) = u \v . thus u \w is countable set and by theorem 2.1, it follows that w ∈ ω θ̃ . this shows that (τcoc)u ⊆ ωθ̃. � theorem 2.8. for any gts (x,µ), the following statements are equivalent. (1) θ̃µ = ωθ̃. (2) (τcoc)u ⊆ θ̃µ for all u ∈ θ̃µ \{∅}. proof. (1) =⇒ (2): assume that θ̃µ = ωθ̃ and u ∈ θ̃µ \{∅}. then by theorem 2.7, (τcoc)u ⊆ ωθ̃ = θ̃µ. (2) =⇒ (1): suppose that (τcoc)u ⊆ θ̃µ for all u ∈ θ̃µ \{∅}. it is enough to show that ωθ̃ ⊆ θ̃µ. let 6 int. j. anal. appl. (2022), 20:72 w ∈ ω θ̃ and x ∈ w. by lemma 2.2, there exists ux ∈ θ̃µ and a countable set cx ⊆ mµ such that x ∈ ux \cx ⊆ w. thus ux ∩x \cx ∈ (τcoc)ux , where x \cx ∈ τcoc. from assumption ux \cx ∈ (τcoc)ux ⊆ θ̃µ for all x ∈ w, and so ux \cx ∈ θ̃µ. it follows that w = ⋃ {ux \cx : x ∈ w}∈ θ̃µ, and hence θ̃µ = ωθ̃. � proposition 2.1. let (x,µ) be a gts. if θ̃µ is a topology on x, then ωθ̃ is a topology. proof. suppose that θ̃µ is a topology. by theorem 2.3, ωθ̃ is generalized topology. it is enough to show that the collection ω θ̃ is closed under finite intersection. let w, g be ω θ̃ -µ-open sets and x ∈ w ∩ g. then by theorem 2.1, there exist u,v ∈ θ̃µ containing x such that u \ w and v \ g are countable sets. since θ̃µ is a topology, we have x ∈ u ∩ v ∈ θ̃µ. furthermore, (u∩v )\(w ∩g) = (u∩v )∩ [ x\w ∪x\g ] = [(u∩v )\w )]∪[(u∩v )\g)] ⊂ (u\w )∪(v \g). therefore, (u ∩v ) \ (w ∩g) is a countable set and hence w ∩g is ω θ̃ -µ-open. � definition 2.3. let (x,µ) be a gts . then (x,µ) is said to be θ̃µ-locally countable if mµ is nonempty and for every point x ∈mµ, there exists a u ∈ θ̃µ such that x ∈ u and u is countable. the following corollary is a direct result from definition 2.3 and definition 1.2. corollary 2.2. let (x,µ) be a gts. if (x,µ) is θ̃µ-locally countable, then (x,µ) is µ-locally countable. theorem 2.9. if (x,µ) is a θ̃µ-locally countable gts, then ωθ̃ is the discrete topology on mµ. proof. it is enough to show that every singleton subset of mµ is ωθ̃-µ-open. since (x,µ) is θ̃µ-locally countable, then for each x ∈ mµ, there exists a u ∈ θ̃µ such that x ∈ u and u is countable. by theorem 2.7, we have (τcoc)u ⊆ ωθ̃. therefore u \ (u \{x}) = {x}∈ ωθ̃. � the following corollary is a direct result of theorem 2.9. corollary 2.3. let (x,µ) be a strong gts. if (x,µ) is a θ̃µ-locally countable, then ωθ̃ is the discrete topology on x. proposition 2.2. if (x,µ) is a θ̃µ-locally countable gts, then ωθ̃ = ωµ. proof. since (x,µ) is θ̃µ-locally countable, then by theorem 2.9, ωθ̃ is the the discrete topology on mµ. from corollary 2.2 and theorem 1.2, we get ωθ̃ = ωµ. � corollary 2.4. let (x,µ) be a gts. if mµ is a countable nonempty set, then ωθ̃ is the discrete topology on mµ. proof. since mµ is countable nonempty set, then for x ∈ mµ, there exists u ∈ θ̃µ such that u is countable set. thus (x,µ) is θ̃µ-locally countable. from theorem 2.9, we get ωθ̃ is the discrete topology on mµ. � int. j. anal. appl. (2022), 20:72 7 3. further properties of ω θ̃ -µ-open sets definition 3.1. let (x,µ) be a gts and a ⊆ x. a point x ∈ x is called an ω θ̃ -closure point of a if and only if u ∩a 6= ∅ for all u ∈ ω θ̃ and x ∈ u. consider the following operations are defined as follows: (1) γω θ̃ (a) = {x ∈ x : u ∩a 6= ∅, f or all u ∈ ω θ̃ and x ∈ u}; (2) cω θ̃ (a) = ∩{f : a ⊆ f,f is ω θ̃ -µ-closed in x}. lemma 3.1. let (x,µ) be a gts. then cω θ̃ (a) = γω θ̃ (a) for any a ⊆ x. proof. it is enough to show that γω θ̃ (a) is the smallest ω θ̃ -µ-closed set containing a. clearly a ⊆ γω θ̃ (a). further γω θ̃ (a) is ω θ̃ -µ-closed, that is x\γω θ̃ (a) is ω θ̃ -µ-open because for each x ∈ x\γω θ̃ (a) there is ux ∈ ωθ̃ such that x ∈ ux and ux ∩a = ∅. now, for any y ∈ ux implies y ∈ x \γωθ̃ (a) so that x \γω θ̃ (a) = ⋃ x∈x\γω θ̃ (a) ux ∈ ωθ̃. finally if a ⊆ f and f is any ω θ̃ -µ-closed, then x \ f is ω θ̃ -µ-open and (x \ f ) ∩ a = ∅ so that x \f ⊆ x \γω θ̃ (a) and hence γω θ̃ (a) ⊆ f. therefore γω θ̃ (a) is the smallest ω θ̃ -µ-closed set containing a, and by definition 3.1(2), γ ωθ̃ (a) = c ωθ̃ (a). � the proof of the following theorem is straightforward and thus omitted. theorem 3.1. for subsets a,b of gts (x,µ), the following properties hold: (1) if a ⊆ b ⊂ x, then cω θ̃ (a) ⊆ cω θ̃ (b); (2) a ⊆ cω θ̃ (a) for a ⊆ x; (3) cω θ̃ (cω θ̃ (a)) = cω θ̃ (a) for a ⊆ x; (4) a is ω θ̃ -µ-closed if and only if cω θ̃ (a) = a. definition 3.2. let (x,µ) be a gts and a ⊆ x. then we define the following notions: (1) c θ̃µ (a) = ∩{f : a ⊆ f,f is θ̃µ-closed in x}; (2) cωµ(a) = ∩{f : a ⊆ f,f is ω-µ-closed in x}. the proof of the following corollary is straightforward and thus omitted. corollary 3.1. for a subset a of a gts (x,µ), the following properties hold: (1) a is θ̃µ-closed if and only if cθ̃µ(a) = a; (2) a is ω-µ-closed if and only if cωµ(a) = a. lemma 3.2. let (x,µ) be a gts. then γ θ̃µ (a) ⊆ c θ̃µ (a) for any a ⊆ x. proof. let x /∈ c θ̃µ (a). then x ∈ x \c θ̃µ (a) so that there is u ∈ θ̃µ satisfying x ∈ u and u ∩a = ∅. since u ∈ θ̃µ, then there is v ∈ µ such that x ∈ v ⊆ cµ(v ) ∩mµ ⊆ u and cµ(v ) ∩mµ ∩a = ∅, consequently x /∈ γ θ̃ (a). thus we have γ θ̃ (a) ⊆ c θ̃ (a). � 8 int. j. anal. appl. (2022), 20:72 theorem 3.2. let (x,µ) be a gts and a ⊆ x. then the following properties hold: (1) cωµ(a) ⊆ cωθ̃ (a) ⊆ cθ̃µ(a); (2) if a is θ̃µ-closed, then a is ωθ̃-µ-closed; (3) if a is ω θ̃ -µ-closed, then a is ω-µ-closed. proof. (1) to show that cωµ(a) ⊆ cωθ̃ (a), let x /∈ cωθ̃ (a) and so there is a u ∈ ωθ̃ containing x such that u∩a = ∅. from theorem 2.4, we have ω θ̃ ⊆ ωµ, u ∈ ωµ, and hence x /∈ cωµ(a). to show that cω θ̃ (a) ⊆ c θ̃µ (a), let x /∈ c θ̃µ (a) and so there is a u ∈ θ̃µ containing x such that u ∩a = ∅. from theorem 2.4, we have θ̃µ ⊆ ωθ̃, u ∈ ωθ̃, and hence x /∈ cωθ̃ (a). (2) suppose that a is θ̃µ-closed. then by corollary 3.1(1), cθ̃µ(a) = a. thus by (1), cωθ̃ (a) = a and hence a is ω θ̃ -µ-closed. (2) suppose that a is ω θ̃ -µ-closed. then by theorem 3.1(4), cω θ̃ (a) = a . thus by (1), cωµ(a) = a and hence a is ω-µ-closed. � proposition 3.1. let (x,µ) be a θ̃µ-locally countable gts and a ⊆ x. then cωµ(a) = cωθ̃ (a) proof. by theorem 3.2(1), cωµ(a) ⊆ cωθ̃ (a). let x ∈ cωθ̃ (a). then u ∩a 6= ∅ for all u ∈ ωθ̃ and x ∈ u. since (x,µ) is a θ̃µ-locally countable, then by theorem 2.9, ωθ̃ is the discrete topology on mµ and hence ωµ = ωθ̃. which implies that x ∈ cωµ(a) and cωθ̃ (a) ⊆ cωµ(a). hence cωµ(a) = cω θ̃ (a). � theorem 3.3. let (x,µ) be a µ-locally indiscrete gts and let a ⊆ x. then the following properties hold. (1) cµ(a) = cθ̃µ(a); (2) cω θ̃ (a) ⊆ cµ(a); (3) if a is µ-closed in (x,µ), then a is θ̃µ-closed in (x,µ). (4) if a is µ-closed in (x,µ), then a is ω θ̃ -µ-closed in (x,µ). proof. (1) clearly cµ(a) ⊆ cθ̃µ(a). to show that cθ̃µ(a) ⊆ cµ(a), let x /∈ cµ(a). then there exists u ∈ µ such that x ∈ u and u ∩a = ∅. since (x,µ) is a µ-locally indiscrete, cµ(u) = u. it follows that u ⊆ cµ(u) ∩mµ ⊆ u and hence u ∈ θ̃µ. thus x /∈ cθ̃µ(a). (2) since (x,µ) is µ-locally indiscrete. then by theorem 2.5, µ ⊆ ω θ̃ and hence cω θ̃ (a) ⊆ cµ(a). (3) suppose that a is µ-closed in (x,µ), then cµ(a) = a. thus by (1), a = cθ̃µ(a) and hence a is θ̃µ-closed in (x,µ). (4) suppose that a is µ-closed in (x,µ), then cµ(a) = a. thus by (2), a = cω θ̃ (a) and hence a is ω θ̃ -µ-closed in (x,µ). � definition 3.3. a gts (x,µ) is said to be ω θ̃ -anti-locally countable if the intersection of any two ω θ̃ -µ-open sets is either empty or uncountable. the following lemma is used to prove the theorem which is stated below. int. j. anal. appl. (2022), 20:72 9 lemma 3.3. let (x,µ) be ω θ̃ -anti-locally countable and a ⊆ x. if a ∈ ω θ̃ , then c θ̃µ (a) = cω θ̃ (a). proof. suppose that ∅ 6= a ⊆ x and a ∈ ω θ̃ . by theorem 3.2(1), cω θ̃ (a) ⊆ c θ̃µ (a). to show that c θ̃µ (a) ⊆ cω θ̃ (a), let x ∈ c θ̃µ (a) and w ∈ ω θ̃ such that x ∈ w. then by lemma 2.2, there exists u ∈ θ̃µ and a countable set c ⊆ mµ such that x ∈ u \c ⊆ w. since x ∈ u ∩ cθ̃µ(a), u ∩a 6= ∅. choose y ∈ u ∩ a. since a ∈ ω θ̃ , there exists v ∈ θ̃µ and a countable set d ⊆ mµ such that y ∈ v \d ⊆ a. since y ∈ u ∩v and (x,µ) is ω θ̃ -anti-locally countable, then u ∩v is uncountable. thus, (u \c) ∩ (v \d) 6= ∅ and hence a∩w 6= ∅. therefore, x ∈ cω θ̃ (a). � a subset a of gts (x,µ) is said to be θ̃µ-clopen(resp. ωθ̃-µ-clopen) if it is both θ̃µ-open and θ̃µ-closed (resp. ωθ̃-µ-open and ωθ̃-µ-closed). in the following, by using lemma 3.3, we prove the main result in this section. theorem 3.4. let (x,µ) be ω θ̃ -anti-locally countable and a ⊆ x. then, a is θ̃µ-clopen if and only if a is ω θ̃ -µ-clopen. proof. ⇒) suppose that a is θ̃µ-clopen, then a and x \a are θ̃µ-open. since θ̃µ ⊆ ωθ̃, then a and x \a are ω θ̃ -µ-open, and hence a is ω θ̃ -µ-clopen. ⇐) suppose that a is ω θ̃ -µ-clopen. since a and x \a are ω θ̃ -µ-open, the by lemma 3.3, c θ̃µ (a) = cω θ̃ (a) and c θ̃µ (x \a) = cω θ̃ (x \a). since a is ω θ̃ -µ-clopen., then c θ̃µ (a) = cω θ̃ (a) = a and cω θ̃ (x \a) = x \a. therefore, c θ̃µ (a) = a and c θ̃µ (x \a) = x \a and hence a and x \a are θ̃µ-closed sets. this means that a is θ̃µ-clopen. � definition 3.4. let (x,µ) be a gts and a ⊆ x. then, we define the following notions: (1) iω θ̃ (a) = ∪{u ⊆ x : u ⊆ a, u is ω θ̃ -µ-open}; (2) i θ̃ (a) = ∪{u ⊆ x : u ⊆ a, u is θ̃µ-open}; (3) iωµ(a) = ∪{u ⊆ x : u ⊆ a, u is ω-µ-open}. theorem 3.5. for subsets a,b of gts (x,µ), the following properties hold: (1) if a ⊆ b ⊂ x, then iω θ̃ (a) ⊆ iω θ̃ (b); (2) for a ⊆ x, then iω θ̃ (a) ⊆ a; (3) iω θ̃ (iω θ̃ (a)) = iω θ̃ (a) for a ⊆ x; (4) a is ω θ̃ -µ-open if and only if iω θ̃ (a) = a. proof. the proof is obvious � 10 int. j. anal. appl. (2022), 20:72 corollary 3.2. let (x,µ) be a gts and a ⊆ x. then i θ̃µ (a) ⊆ iω θ̃ (a) ⊆ iωµ(a). proof. to show that i θ̃µ (a) ⊆ iω θ̃ (a), let x ∈ i θ̃µ (a). then there is u ∈ θ̃µ such that x ∈ u ⊆ a. by theorem 2.4, u is ω θ̃ -µ-open. thus x ∈ iω θ̃ (a). to show that iω θ̃ (a) ⊆ iωµ(a), let x ∈ iωθ̃ (a). then there is u ∈ ω θ̃ such that x ∈ u ⊆ a. then by theorem 2.4, u is ω-µ-open and hence x ∈ iωµ(a) � theorem 3.6. let (x,µ) be a gts and a ⊆ x. then the following properties hold: (1) cω θ̃ (x \a) = x \ iω θ̃ (a); (2) iω θ̃ (x \a) = x \cω θ̃ (a). proof. (1) let x ∈ cω θ̃ (x \a) and u ∈ ω θ̃ with x ∈ u. since x ∈ cω θ̃ (x \a), u ∩ (x \a) 6= ∅. this implies that x /∈ iω θ̃ (a) and hence x ∈ x \ iω θ̃ (a). conversely, for x ∈ x \ iω θ̃ (a), x /∈ iω θ̃ (a), and then u ∩ (x \a) 6= ∅ for all u ∈ ω θ̃ and x ∈ u which implies x ∈ cω θ̃ (x \a). (2) let x ∈ x \ cω θ̃ (a) if and only if x /∈ cω θ̃ (a) if and only if there is u ∈ ω θ̃ with x ∈ u such that u ∩a = ∅ if and only if x ∈ iω θ̃ (x \a). � 4. conclusion in this paper, we introduced the notion of ω θ̃ -µ-open sets in the sense of generalized topology given in [5]. we have proved that the collection of ω θ̃ -µ-open sets forms a generalized topology on x that lies between the class of θ̃µ-open sets and the class of ω-µ-open sets. the relationships of ωθ̃-µ-open and other well-known generalized open sets are given. several properties of ω θ̃ -µ-open sets which enable us to prove certain of our results are studied and verified. in the upcoming work, we plan to : (1) introduce some concepts in gts using ω θ̃ -µ-open sets such as connectedness, compactness and lindelöfness; (2) introduce continuity and decomposition of continuity via ω θ̃ -µ-open sets. acknowledgements: the authors are grateful to the referees for useful comments and suggestions. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s.a. ghour, w. zareer, omega open sets in generalized topological spaces, j. nonlinear sci. appl. 9 (2016), 3010–3017. https://doi.org/10.22436/jnsa.009.05.93. [2] k. al-zoubi, b. al-nashef, the topology of ω-open subsets, al-manarah j. 9 (2003), 169-179. [3] a. al-omari, m.s. md noorani, regular generalized ω-closed sets, int. j. math. math. sci. 2007 (2007), 16292. https://doi.org/10.1155/2007/16292. [4] a. al-omari, t. noiri, a unified theory of contra-(µ,λ)-continuous functions in generalized topological spaces, acta math. hung. 135 (2012), 31–41. https://doi.org/10.1007/s10474-011-0143-x. [5] á. császár, generalized topology, generalized continuity, acta math. hung. 96 (2002), 351357. https://doi. org/10.1023/a:1019713018007. [6] á. császár, extremally disconnected generalized topologies, ann. univ. sci. budapest. eotvos sect. math. 47 (2004), 91-96. https://doi.org/10.22436/jnsa.009.05.93 https://doi.org/10.1155/2007/16292 https://doi.org/10.1007/s10474-011-0143-x https://doi.org/10.1023/a:1019713018007 https://doi.org/10.1023/a:1019713018007 int. j. anal. appl. (2022), 20:72 11 [7] á. császár, generalized open sets in generalized topologies, acta math. hung. 106 (2005), 53-66. https: //doi.org/10.1007/s10474-005-0005-5. [8] á. császár, product of generalized topologies, acta math. hung. 123 (2009), 127-132. https://doi.org/10. 1007/s10474-008-8074-x. [9] c. carpintero, e. rosas, m. salas, j. sanabria, l. vasquez, generalization of ω-closed sets via operators and ideals, sarajevo j. math. 9 (2013), 293-301. https://doi.org/10.5644/sjm.09.2.13. [10] c. carpintero, n. rajesh, e. rosas, s. saranyasri, on slightly ω-continuous multifunctions, punjab univ. j. math. (lahore), 46 (2014), 51-57. [11] e. korczak-kubiak, a. loranty, r.j. pawlak, baire generalized topological spaces, generalized metric spaces and infinite games, acta math hung. 140 (2013), 203–231. https://doi.org/10.1007/s10474-013-0304-1. [12] h.z. hdeib, ω-closed mappings, rev. colomb. mat. 16 (1982), 65-78. [13] w.k. min, some results on generalized topological spaces and generalized systems, acta math hung. 108 (2005), 171–181. https://doi.org/10.1007/s10474-005-0218-7. [14] w.k. min, remarks on θ̃-open sets in generalized topological spaces, appl. math. lett. 24 (2011) 165–168. https://doi.org/10.1016/j.aml.2010.08.038. [15] v. renukadevi, p. vimaladevi, note on generalized topological spaces with hereditary classes, bol. soc. paran. mat. 32 (2014), 89-97. https://doi.org/10.5269/bspm.v32i1.19401. [16] r. sen, b. roy, iµ∗-open sets in generalized topological spaces, gen. math. 27 (2019), 35-42. [17] z. zhu, w. li, contra continuity on generalized topological spaces, acta math. hung. 138 (2013), 34-43. https://doi.org/10.1007/s10474-012-0215-6. https://doi.org/10.1007/s10474-005-0005-5 https://doi.org/10.1007/s10474-005-0005-5 https://doi.org/10.1007/s10474-008-8074-x https://doi.org/10.1007/s10474-008-8074-x https://doi.org/10.5644/sjm.09.2.13 https://doi.org/10.1007/s10474-013-0304-1 https://doi.org/10.1007/s10474-005-0218-7 https://doi.org/10.1016/j.aml.2010.08.038 https://doi.org/10.5269/bspm.v32i1.19401 https://doi.org/10.1007/s10474-012-0215-6 1. introduction 2. –open sets 3. further properties of –open sets 4. conclusion references international journal of analysis and applications volume 17, number 4 (2019), 620-629 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-620 blow-up, exponential grouth of solution for a nonlinear parabolic equation with p (x)− laplacian amar ouaoua∗ and messaoud maouni laboratory of applied mathematics and history and didactics of mathematics (lamahis) university of 20 august 1955, skikda, algeria ∗corresponding author: a.ouaoua@univ-skikda.dz abstract. in this paper, we consider the following equation ut − div ( |∇u|p(x)−2 ∇u ) + ω |u|m(x)−2 ut = b |u|r(x)−2 u. we prove a finite time blowup result for the solutions in the case ω = 0 and exponential growth in the case ω > 0, with the negative initial energy in the both case. 1. introduction we consider the following boundary problem:  ut − div ( |∇u|p(x)−2 ∇u ) + ω |u|m(x)−2 ut = b |u| r(x)−2 u in ω × (0,t) , u (x,t) = 0, x ∈ ∂ω, t ≥ 0, u (x, 0) = u0 (x) in ω. (1.1) where ω is a bounded domain in rn,n ≥ 1 with smooth boundary ∂ω and b > 0, ω ≥ 0 are constants, p (.) , m (x) and r (.) are given measurable functions on ω satisfying 2 ≤ m1 ≤ m (x) ≤ m2 < p1 ≤ p (x) ≤ p2 < r1 ≤ r (x) ≤ r2 ≤ p∗ (x) . (1.2) received 2019-04-06; accepted 2019-05-07; published 2019-07-01. 2010 mathematics subject classification. 35k55; 35k61; 35k60 . key words and phrases. nonlinear parabolic equation; p (x)− laplacian; blow-up, exponential grouth. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 620 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-620 int. j. anal. appl. 17 (4) (2019) 621 p1 : = ess x∈ω inf(p (x)), p2 := ess x∈ω sup(p (x)), r1 : = ess x∈ω inf(r (x)), r2 := ess x∈ω sup(r (x)), m1 : = ess x∈ω inf(m (x)), m2 := ess x∈ω sup(m (x)), and p∗ (x) =   np(x) esssup x∈ω (n−p(x)) if p2 < n +∞ if p2 ≥ n . we also assume that p (.) , m (.) and r (.) satisfy the log-hölder continuity condition: |q (x) −q (y)| ≤− a log |x−y| , for a.e. x, y ∈ ω, with |x−y| < δ, (1.3) a > 0, 0 < δ < 1. equation (1.1) can be viewed as a generalization of the evolutional p-laplacian equation ut − div ( |∇u|p−2 ∇u ) + ω |u|m−2 ut = b |u| r−2 u, with the constant exponent of nonlinearity p, m, r ∈ (2, ∞) , which appears in various physical contexts. in particular, this equation arises from the mathematical description of the reaction-diffusion/ diffusion, heat transfer, population dynamics processus, and so on (see [11]) and references therein). recently in [1], in the case ω = 0, agaki proved an existence and blow up result for the initial datum u0 ∈ lr(). ôtani [17] studied the existence and the asymptotic behavior of solutions of (1.1) and overcome the difficulties caused by the use of nonmonotone perturbation theory. the quasilinear case, with p 6= 2, requires a strong restriction on the growth of the forcing term |u|r−2u, which is caused by the loss of the elliptic estimate for the p−laplacian operator defined by ∆pu = div(|∇u|p−2∇u) (see [2]). alaoui et al [12] considered the following nonlinear heat equation   ut − div ( |∇u|p(x)−2 ∇u ) = |u|r(x)−2 u + f, in ω × (0,t) , u (x,t) = 0, x ∈ ∂ω × (0,t) , u (x, 0) = u0 (x) in ω. (1.4) where ω is a bounded domain in rn with smooth boundary ∂ω. under suitable conditions on r and p and for f = 0, they showed that any solution with nontrivial initial datum blows up in finite time. in the absence of the diffusion term in equation (1.1) when p (x) = p and r (x) = r proved the existence and plow up results have been established by many authors (see [1 − 3, 9, 14, 17]). int. j. anal. appl. 17 (4) (2019) 622 we should also point out that polat [18] established a blow-up result for the solution with vanishing initial energy of the following initial boundary value problem ut −uxx + |u| m−2 ut = |u| p−2 u. (1.5) where m and p are real constants. in recent years, mush attention has been paid to the study of mathematical models of electro-theological fluids. this models inclode hyperbolic, parapolic or elliptic equations which are nonlinear with respect to the gradient of the thought solution with variable exponents of nonlinearity, (see [4, 5, 10, 15]). our objective in this paper is to study: in the section 3, the blow up of the solutions of the problem (1.1) in the case ω = 0, in the section 4, exponential growth of solution when ω > 0. 2. preliminaries we present in this section some lemmas about the lebesque and sobolev space with variables conponents (see [6 − 8, 12, 13]). let p : ω → [1, + ∞] be a measurable function, where ω is adomain of rn. we define the lebesque space with a variale exponent p (.) by lp(.) (ω) := { v : ω → r : measurable in ω, ap(.) (λv) < +∞, for some λ > 0 } , where ap(.) (v) = ∫ ω |v (x)|p(x) dx. the set lp(.) (ω) equipped with the norm ( luxemburg’s norm) ‖v‖p(.) := inf  λ > 0 : ∫ ω ∣∣∣∣v (x)λ ∣∣∣∣p(x) dx ≤ 1   , lp(.) (ω) is a banach space [13]. we next, define the variable-exponent sobolev space w 1,p(.) (ω) as follows: w 1,p(.) (ω) := { v ∈ lp(.) (ω) such that ∇v exists and |∇v| ∈ lp(.) (ω) } . this is a banach space with respect to the norm ‖v‖w1,p(.)(ω) = ‖v‖p(.) + ‖∇v‖p(.) . furthmore, we set w 1,p(.) (ω) to be the closure of c∞0 (ω) in the space w 1,p(.) 0 (ω). let us note that the space w 1,p(.) (ω) has a differenet definition in the case of variable exponents. however, under condition (1.3) , both definitions are equivalent [13] . the space w−1,p ′ (.) (ω) , dual of w 1,p(.) 0 (ω) , is defined in the same way as the classical sobolev spaces, where 1 p(.) + 1 p ′ (.) = 1. lemma 2.1. (poincaré’s inequality) let ω ⊂ rn be a bounded domain and suppose that p (.) satisfies (1.3) , then ‖v‖p(.) ≤ c‖∇v‖p(.) , for all v ∈ w 1,p(.) 0 (ω) . int. j. anal. appl. 17 (4) (2019) 623 where c > 0 is a constant which depends on p1, p2, and ω only. in particular, ‖∇v‖p(.) define an equivalent norm on w 1,p(.) 0 (ω) . lemma 2.2. if p (.) ∈ c ( ω ) and q : ω → [1, + ∞) is a measurable function such that ess inf x∈ω (p∗ (x) −q (x)) > 0 with p∗ (x) =   np(x) esssup x∈ω (n−p(x)) if p2 < n +∞ if p2 ≥ n. then the embedding w 1,p(.) 0 (ω) ↪→ l q(.) (ω) is continuous and compact. lemma 2.3. ( hölder’s inequality) suppose that p, q, s ≥ 1 are measurable functions defined on ω such that 1 s (y) = 1 p (y) + 1 q (y) , for a.e. y ∈ ω. if u ∈ lp(.) (ω) and v ∈ lq(.) (ω) , then uv ∈ ls(.) (ω) , with ‖uv‖s(.) ≤ 2‖u‖p(.) ‖v‖q(.) . lemma 2.4. if p a measurable function on ω satisfying (1.2) , then we have min { ‖u‖p1 p(.) , ‖u‖p2 p(.) } ≤ ap(.) (u) ≤ max { ‖u‖p1 p(.) , ‖u‖p2 p(.) } , for any u ∈ lp(.) (ω) . 3. blow up in this section, we prove that the solution of equation (1.1) blow up in finite time when ω = 0. we recall that (1.1), becomes   ut − div ( |∇u|p(x)−2 ∇u ) = b |u|r(x)−2 u in ω × (0,t) , u (x,t) = 0, x ∈ ∂ω, t ≥ 0, u (x, 0) = u0 (x) in ω. (3.1) we start with a local existence result for the problem (1.1), which is a direct result of the existence theorem by agaki and ôtani [2]. proposition 3.1. for all u0 ∈ w 1,p(.) 0 (ω), there exists a number t0 ∈ (0,t] such that the problem (1.1) has a solution u on [0,t0] satisfying: u ∈ cw([0,t0]; w 1,p(.) 0 (ω)) ∩c([0,t0], l r(.)(ω)) ∩w 1,2(0,t0; l2(ω)). we define the energy functional associaeted of the problem (1.1) e (t) = ∫ ω 1 p (x) |∇u|p(x) dx− b ∫ ω 1 r (x) |u|r(x) dx. (3.2) int. j. anal. appl. 17 (4) (2019) 624 theorem 3.1. let the assumptions of proposition 1, be satisfied and assume that e (0) < 0. (3.3) then the solution of the problem (3.1) , blow up in finite time. now, we let h (t) := −e (t) , (3.4) and l (t) = 1 2 ∫ ω u2dx. (3.5) to prove our result, we first establesh some lemmas. lemma 3.1. assume that (1.2) and (1.3) , hold and e (0) < 0. then ap(.) (∇u) < bp2 r1 ar(.) (u) , (3.6) and r1 b h (0) < ar(.) (u) . (3.7) proof. we multiply the first equation of (3.1) by ut and integratying over the domain ω, we get d dt  ∫ ω 1 p (x) |∇u|p(x) dx− b ∫ ω 1 r (x) |u|r(x) dx   = −‖ut‖22 , then e ′ (t) = −‖ut‖ 2 2 ≤ 0. (3.8) integrating (3.8) over (0, t) , we obtain e (t) ≤ e (0) < 0. (3.9) by (3.2) and (3.9) , we have ∫ ω 1 p (x) |∇u|p(x) dx < b ∫ ω 1 r (x) |u|r(x) dx, so that ∫ ω 1 p2 |∇u|p(x) dx < ∫ ω b r1 |u|r(x) dx. on the other hand, we have h (t) = − ∫ ω 1 p (x) |∇u|p(x) dx + b ∫ ω 1 r (x) |u|r(x) dx ≤ b ∫ ω 1 r (x) |u|r(x) dx. (3.10) int. j. anal. appl. 17 (4) (2019) 625 then, by (3.10) , (3.4) and (3.9) , we obtain 0 < h (0) < h (t) < b r1 ar(.) (u) . � lemma 3.2. [16] assume that (1.2), (1.3) hold and e (0) < 0. then the solution of (3.1) , satisfies for some c > 0, ar(.) (u) ≥ c‖u‖ r1 r1 . (3.11) proof of theorem 1. we have l ′ (t) = ∫ ω uutdx = ∫ ω u ( div ( |∇u|p(x)−2 ∇u ) + b |u|r(x)−2 u ) dx = −ap(.) (∇u) + bar(.) (u) . (3.12) combining of (3.12) , (3.11) and (3.6) , leads to l ′ (t) ≥ cb ( 1 − p2 r1 ) ‖u‖r1r1 . (3.13) now, we estimate l r1 2 (t) , by the embedding of lr1 (ω) ↪→ l2 (ω) , we get l r1 2 (t) ≤ ( 1 2 ‖u‖2r1 )r1 2 ≤ c‖u‖r1r1 . (3.14) by combining (3.14) and (3.13) , we obtain l ′ (t) ≥ ξl r1 2 (t) . (3.15) a direct integration of (3.15) , then yields l r1 2 −1 (t) ≥ 1 l1− r1 2 (0) − ξt . therefore, l blow up in a time t∗ ≤ 1 l r1 2 −1 (0) . � 4. exponential growth in this section, we prove that the solution of equation (1.1) exponential growth when ω > 0. lemma 4.1. suppose that (1.2) holds and e (0) < 0. then,∫ ω |u|m(x) dx ≤ c ( ‖u‖r1r1 + h (t) ) . (4.1) int. j. anal. appl. 17 (4) (2019) 626 proof. ∫ ω |u|m(x) dx = ∫ ω− |u|m(x) dx + ∫ ω+ |u|m(x) dx, where ω+ = {x ∈ ω / |u (x, t)| ≥ 1} and ω− = {x ∈ ω / |u (x, t)| < 1} . so, we get ∫ ω |u|m(x) dx ≤ c    ∫ ω− |u|r1 dx   m1 r1 +  ∫ ω+ |u|r1 dx   m2 r1   ≤ c ( ‖u‖m1r1 + ‖u‖ m2 r1 ) . exploiting the algebric inequality zv ≤ (z + 1) ≤ ( 1 + 1 a ) (z + a) , ∀z > 0, 0 < v ≤ 1, a ≥ 0, we have ‖u‖m1r1 ≤ c ( ‖u‖r1r1 )m1 r1 ≤ c ( 1 + 1 h (0) )( ‖u‖r1r1 + h (0) ) ≤ c ( ‖u‖r1r1 + h (t) ) . similarly, ‖u‖m2r1 ≤ c ( ‖u‖r1r1 )m2 r1 ≤ c ( 1 + 1 h (0) )( ‖u‖r1r1 + h (0) ) ≤ c ( ‖u‖r1r1 + h (t) ) . this gives ∫ ω |u|m(x) dx ≤ c ( ‖u‖r1r1 + h (t) ) . � theorem 4.1. let the assumptions of proposition 1, be satisfied and assume that (3.3) holds. then the solution of the problem (1.1) , grows exponentially. proof. by the same procedure of the proof the lemma 5, we get e ′ (t) = −‖ut‖ 2 2 −ω ∫ ω |u|m(x)−2 u2t ≤ 0, (4.2) then, we have h ′ (t) = ‖ut‖ 2 2 + ω ∫ ω |u|m(x)−2 u2tdx ≥ 0. (4.3) int. j. anal. appl. 17 (4) (2019) 627 we define g (t) = h (t) + �l (t) . (4.4) for � small to be chosen later. the time derivative of (4.4) , we obtain g ′ (t) = h ′ (t) + � ∫ ω uutdx. by using (1.1) , we get g ′ (t) = h ′ (t) − �ap(.) (∇u) + �bar(.) (u) − �ω ∫ ω |u|m(x)−2 utudx. (4.5) to estimate the last term in the right hand side of (4.5) , by using the following young’s inequality xy ≤ δx2 + δ−1y 2, x, y ≥ 0, δ > 0. ∫ ω |u|m(x)−2 utudx = ∫ ω |u| m(x)−2 2 ut |u| m(x)−2 2 udx ≤ δ ∫ ω |u|m(x)−2 u2tdx + δ −1 ∫ ω |u|m(x) dx. we conclude g ′ (t) ≥ (1 − �δ) ∫ ω |u|m(x)−2 u2tdx + ‖ut‖ 2 2 − �ap(.) (∇u) +�bar(.) (u) − �ωδ −1 ∫ ω |u|m(x) dx. (4.6) then g ′ (t) ≥ (1 − �δ) ∫ ω |u|m(x)−2 u2tdx + ‖ut‖ 2 2 − �ωδ −1 ∫ ω |u|m(x) dx +� (1 −µ) r1h (t) + �bµar(.) (u) + � ( (1 −µ) r1 p2 − 1 ) ap(.) (∇u) , where µ is a constant such that 0 < µ ≤ 1 − p2 r1 . also, by using (3.6) , we obtain g ′ (t) ≥ (1 − �δ) ∫ ω |u|m(x)−2 u2tdx + ‖ut‖ 2 2 − �ωδ −1 ∫ ω |u|m(x) dx +� (1 −µ) r1h (t) + � ( bµ + 1 −µ− p2 r1 ) ar(.) (u) . (4.7) int. j. anal. appl. 17 (4) (2019) 628 then, by lemma 7 and (3.11) , (4.7) becomes g ′ (t) ≥ (1 − �δ) ∫ ω |u|m(x)−2 u2tdx + ‖ut‖ 2 2 − �c ωδ −1 (‖u‖r1r1 + h (t)) +� (1 −µ) r1h (t) + �c ( bµ + 1 −µ− p2 r1 ) ‖u‖r1r1 . (4.8) so that g ′ (t) ≥ (1 − �δ) ∫ ω |u|m(x)−2 u2tdx + ‖ut‖ 2 2 + � ( (1 −µ) r1 − c ωδ−1 ) h (t) +� ( c ( bµ + 1 −µ− p2 r1 ) − c ωδ−1 ) ‖u‖p1p1 . (4.9) so, we chosen δ large sufficient and � small enough for that we can find λ1, λ2 > 0, such that g ′ (t) ≥ λ1h (t) + λ2 ‖u‖ r1 r1 ≥ k1 ( h (t) + ‖u‖r1r1 ) , (4.10) and g (0) = h (0) + �l (0) > 0. similarly in (4.7) , we have ‖u‖22 ≤ c ( h (t) + ‖u‖r1r1 ) . (4.11) on the other hand, by (4.11) , we get g (t) ≤ k2 ( h (t) + ‖u‖r1r1 ) . (4.12) combining with (4.12) and (4.10) , we arrive at g ′ (t) ≥ ηg (t) . (4.13) finally, a simple integration of (4.13) gives g (t) ≥ g (0) eηt, ∀t ≥ 0. (4.14) thus completes the proof. � references [1] g. akagi, local existence of solutions to some degenerate parabolic equation associated with the p-laplacian, j. differential equations 241 (2007), 359–385. [2] g. akagi and m. ôtani, evolutions inclusions governed by subdifferentials in reflexive banach spaces, j. evol. equ. 4 (2004), 519–541. [3] g. akagi and m. ôtani, evolutions inclusions governed by the difference of two subdifferentials in reflexive banach spaces, j. differential equations 209 (2005), 392–415. [4] s.n. antontsev and v. zhikov, higher integrability for parabolic equations of p(x, t)-laplacian type. adv. differ. equ. 10 (2005), 1053-1080. int. j. anal. appl. 17 (4) (2019) 629 [5] y. chen, s. levine and m. rao, variable exponent, linear growth functions in image restoration. siam j. appl. math. 66 (2006), 1383-1406. [6] d. edmunds and j. rakosnik, sobolev embeddings with variable exponent, stud. math. 143 (3) (2000), 267–293. [7] d. edmunds and j. rakosnik sobolev embeddings with variable exponent. ii, math. nachr. 246 (1) (2002), 53–67. [8] x. fan and d. zhao, on the spaces lp(x)(ω) and wm,p(x)(ω), j. math. anal. appl. 263 (2) (2001), 424–446. [9] h. fujita, on the blowing up solutions of the cauchy problem for ut = ∆u+ u 1+α, j. fac. sci. univ. tokyo sect. a.math. 16 (1966), 105–113. [10] y. gao, b. guo and w.gao, weak solutions for a high-order pseudo-parabolic equation with variable exponents. appl. anal. 93 (2) (2014), 322-338. [11] z. jiang, s. zheng, and x. song, blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions, appl. math. lett. 17 (2) (2004), 193–199. [12] a.m. kbiri, s.a. messaoudi. and h.b. khenous, a blow-up result for nonlinear generalized heat equation, comput. math. appl. 68 (12) (2014), 1723–1732. [13] d. lars, p. harjulehto, p. hasto and m. ruzicka, lebesgue and sobolev spaces with variable exponents, in: lecture notes in mathematics, springer, 2011. [14] j. leray and j.l.lions, quelques résultats de visick sur les problémes elliptiques non linéaires pour les méthodes de minty–browder, bull. soc. math. france 93 (1965), 97–107. [15] s.z. lian, w.j. gao, cl. cao and hj. yuan, study of the solutions to a model porous medium equation with variable exponents of nonlinearity. j. math. anal. appl. 342 (2008), 27-38. [16] s.a. messaoudi and a. talahmeh, blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities, math. methods appl. sci. 40 (18) (2017), 6976-6986. [17] m. ôtani, nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, cauchy problems, j. differential equations 46 (1982), 268–299. [18] n. polat, blow up of solution for a nonlinear reaction diffusion equation with multiple nonlinearities, int. j. sci. technol. 2 (2) (2007), 123–128. 1. introduction 2. preliminaries 3. blow up 4. exponential growth references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 216-230 http://www.etamaths.com basic theory for differential equations with unified reimann-liouville and hadamard type fractional derivatives başak karpuz1,∗, umut m. özkan2, tuğba yalçin2 and mustafa k. yildiz2 abstract. in this paper, we extend the definition of the fractional integral and derivative introduced in [appl. math. comput. 218 (2011)] by katugampola, which exhibits nice properties only for numbers whose real parts lie in [0, 1]. we prove some interesting properties of the fractional integrals and derivatives. based on these properties, the following concepts for the new type fractional differential equations are explored: existence and uniqueness of solutions; solutions of autonomous fractional differential equations; dependence on the initial conditions; green’s function; variation of parameters formula. 1. introduction the history of fractional calculus was originated in the seventeenth century, when the half-order derivative was discussed by leibnitz in 1695. since then, this theory became one of the interesting subjects to mathematicians as well as biologists, chemists, economists, engineers and physicists. there are several books written on this subject, for instance [3,9–11,13]. [13] is one of the most comprehensive main tools of the subject, where several types of derivatives (such as riemann-liouville, hadamard, grünwald-letnikov, riesz and caputo) were introduced. derivatives of fractional order are defined by integrals with a fractional order kernel. reimannliouville ([3,9–11,13]) and hadamard ([1,2,7,8,12]) type fractional integrals are two of the most studied forms of fractional integrals. the riemann–liouville fractional integral of order α for a function f is defined by ∫ t s [t−η]α−1 γ(α) f(η)dη for t > s and α > 0, (1.1) which is motivated by the cauchy integral formula∫ t s ∫ η1 s · · · ∫ ηn−1 s f(ηn)dηn · · ·dη2dη1 = ∫ t s [t−η]n−1 γ(n) f(η)dη for t > s and n ∈ n. another one is the hadamard fractional integral introduced in [4], which reads as ∫ t s 1 η1 ∫ η1 s 1 η2 · · · ∫ ηn−1 s f(ηn) ηn dηn · · ·dη2dη1 = 1 γ(n) ∫ t s [ ln ( t η )]n−1 f(η) η dη for t > s and n ∈ n, from which the following fractional integral of f is deduced by∫ t s 1 γ(α) [ ln ( t η )]α−1 f(η) η dη for t > s and α > 0. (1.2) in [5], katugampola unified the reimann-liouville fractional integral and the hadamard fractional integral by ∫ t s [tρ −ηρ]α−1ηρ−1 ρα−1γ(α) f(η)dη for t > s and α > 0, (1.3) received 25th october, 2016; accepted 10th january, 2017; published 1st march, 2017. 2010 mathematics subject classification. primary 26a33; secondary 33e12, 34a08, 34k37. key words and phrases. reimann-liouville fractional calculus; hadamard fractional calculus; existence and uniqueness; dependence on initial conditions; green’s function; variation of parameters formula. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 216 reimann-liouville and hadamard type fractional derivatives 217 where ρ > 0, which we will call as the reimann-liouville-hadamard (rlh) fractional integral. as limρ→0+ tρ−sρ ρ = ln( t s ), we see that (1.3) with ρ = 1 and ρ → 0+ contains (1.1) and (1.2), respectively. this fractional integral has also been extended to fractional derivative in [6], which holds “nicely” for α with re(α) ∈ (0, 1) (see [6, § 3]). motivated by the definition of fractional order derivatives given in [6], we will give a new extended the definition for arbitrary positive numbers. based on this fractional derivative, we will study important properties of the fractional differential equations (fde) defined with this new type of derivatives. the paper covers the following concepts: • existence and uniqueness of solutions to fdes • solutions of autonomous fdes • dependence of solutions on the initial conditions • green’s function for rlh fdes • variation of parameters formula the paper is organized as follows. in § 2, we give the basic definitions and related auxiliary results. § 3 includes the fundamental properties of the fractional integral/derivative, which will be required in the latter sections. in § 4, we will provide existence and uniqueness for solutions of differential equations of the new type of fractional derivative. by using direct substitution technique and the picard iterates, we will consider autonomous type fractional differential equations in § 5. in § 6, we will provide a result on dependence of the initial conditions. § 7 is dedicated to the concept of green’s function and the variation of parameters formula for the new type fractional differential equations. finally, in § 8, we present some directions for future research and make our final discussion to conclude the paper. 2. definitions and auxiliary results let us first introduce the kernel function kαρ : r×r → c, where α ∈ c\z − 0 and ρ ∈ r +, defined by kαρ (t,s) := [tρ −sρ]α−1sρ−1 ρα−1γ(α) for s,t ∈ r. (2.1) we assume for convenience kαρ (t,s) ≡ 0 for α ∈ z − 0 . also, for n ∈ n, k ∈ {1, 2, · · · ,n} and ρ ∈ r +, we let an,k(ρ) :=   [1 − (n− 1)ρ]an−1,1(ρ), k = 1 an−1,k−1(ρ) + [k − (n− 1)ρ]an−1,k(ρ), k = 2, 3, · · · ,n− 1 1, k = n. (2.2) definition 2.1 (cf. [5]). let α ∈ r, ρ ∈ r+ and f : (0,∞) → r. we define the α-order fractional integration of f by [ jαρ f ] (t) :=   ∫ t 0 kαρ (t,η)f(η)dη, α ∈ r\z − 0 f(t), α = 0 (−α)∑ i=1 a(−α),i(ρ) t(−α)ρ−i ( d dt )i f(t), α ∈ z− (2.3) for t > 0. remark 2.1. one can show that an,k(1) = δn,k, where δ is kronecker’s delta. hence, jαρ f = f(−α) for α ∈ z−. example 2.1. for α ∈ r+0 , ν ∈ (−1,∞) and ρ ∈ r +, we have [ jαρ ∗ ρν ] (t) = γ(ν + 1) ραγ(ν + α + 1) tρ(ν+α) for t > 0. 218 karpuz, özkan, yalçin and yildiz the proof is trivial for α = 0. we let α ∈ r+ and compute for t > 0 that [ jα∗ρν ] (t) = ∫ t 0 kαρ (t,η)η ρνdη = ∫ t 0 [tρ −ηρ]α−1ηρ−1 ρα−1γ(α) ηρνdη = tρ(α+ν) ραγ(α) ∫ 1 0 [1 − ζ]α−1ζνdζ = tρ(α+ν) ραγ(α) b(α,ν + 1) = γ(ν + 1) ραγ(α + ν + 1) tρ(α+ν). we proceed by recalling some important properties of the kernel k. lemma 2.1. the following basic properties of the kernel k are true. (i) ∫ t s kαρ (t,η)k β ρ (η,s)dη = k α+β ρ (t,s) for t ≥ s ≥ 0 and α,β ∈ r+. (ii) tρ−1kαρ (t,s) = (−1)αsρ−1kαρ (s,t) for s,t ∈ r and α ∈ c. (iii) ∂ ∂t kα+1ρ (t,s) = tρ−1kαρ (t,s) for s,t ∈ r and α ∈ c\z − 0 . (iv) ∂ ∂s kα+1ρ (t,s) sρ−1 = −kαρ (t,s) for s ∈ r\{0}, t ∈ r and α ∈ c\z − 0 . proof. (i) then, we compute for t ≥ s ≥ 0 that ∫ t s kαρ (t,η)k β ρ (η,s)dη = ∫ t s [tρ −ηρ]α−1ηρ−1 ρα−1γ(α) [ηρ −sρ]β−1sρ−1 ρβ−1γ(β) dη = 1 ρα+β−2γ(α)γ(β) ∫ t s [tρ −ηρ]α−1ηρ−1[ηρ −sρ]β−1sρ−1dη = [tρ −sρ]α+β−1sρ−1 ρα+β−1γ(α)γ(β) ∫ 1 0 [1 − ζ]α−1ζβ−1dζ = [tρ −sρ]α+β−1sρ−1 ρα+β−1γ(α)γ(β) b(α,β) = [tρ −sρ]α+β−1sρ−1 ρα+β−1γ(α + β) = kα+βρ (t,s). (ii) the proof is trivial and thus we omit it here. (iii) for t ≥ s ≥ 0, we have ∂ ∂t kα+1ρ (t,s) = ∂ ∂t [tρ −sρ]αsρ−1 ραγ(α + 1) = αρtρ−1[tρ −sρ]α−1sρ−1 ραγ(α + 1) = tρ−1[tρ −sρ]α−1sρ−1 ρα−1γ(α) = tρ−1kαρ (t,s). (iv) the proof can be given similar to that of (iii). � lemma 2.2. note that for α ∈ r, we have [ jαρ f ] (t) = 1 tρ−1 d dt [jα+1ρ f](t) for t > 0. (2.4) proof. we proceed with the following three distinct cases. reimann-liouville and hadamard type fractional derivatives 219 • let α ∈ r\z−. then, we have for t > 0 that d dt [ jα+1ρ f ] (t) = d dt ∫ t 0 kα+1ρ (t,η)f(η)dη = d dt ∫ t 0 [tρ −ηρ]αηρ−1 ραγ(α + 1) f(η)dη = ∫ t 0 d dt [tρ −ηρ]αηρ−1 ραγ(α + 1) f(η)dη + [tρ − tρ]αηρ−1 ραγ(α + 1) f(t) =αρtρ−1 ∫ t 0 [tρ −ηρ]α−1ηρ−1 ραγ(α + 1) f(η)dη =tρ−1 ∫ t 0 [tρ −ηρ]α−1ηρ−1 ρα−1γ(α) f(η)dη =tρ−1 [ jαρ f ] (t). • let α = −1. then, d dt [j 0ρ f](t) = d dt f(t) = tρ−1 1 tρ−1 d dt f(t) = tρ−1 [ j−1ρ f ] (t) for t > 0. • let α ∈{·· · ,−3,−2}. then, putting n := −α for simplicity, we compute for t > 0 that d dt [ jα+1ρ f ] (t) = d dt n−1∑ i=1 an−1,i t(n−1)ρ−i ( d dt )i f(t) = n−1∑ i=1 d dt an−1,i t(n−1)ρ−i ( d dt )i f(t) = n−1∑ i=1 [ an−1,i t(n−1)ρ−i ( d dt )i+1 − [(n− 1)ρ− i] an−1,i t(n−1)ρ−i+1 ( d dt )i] f(t) =tρ−1 n−1∑ i=1 [ an−1,i tnρ−(i+1) ( d dt )i+1 − [(n− 1)ρ− i] an−1,i tnρ−i ( d dt )i] f(t) =tρ−1 [ n−1∑ i=1 an−1,i tnρ−(i+1) ( d dt )i+1 f(t) − n−1∑ i=1 [(n− 1)ρ− i] an−1,i tnρ−i ( d dt )i f(t) ] =tρ−1 [ n∑ i=2 an−1,i−1 tnρ−i ( d dt )i f(t) − n−1∑ i=1 [(n− 1)ρ− i] an−1,i tnρ−i ( d dt )i f(t) ] =tρ−1 [ an−1,n−1 tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 an−1,i−1 tnρ−i ( d dt )i f(t) − n−1∑ i=2 ( (n− 1)ρ− i )an−1,i tnρ−i ( d dt )i f(t) − [(n− 1)ρ− 1] an−1,1 tnρ−1 d dt f(t) ] =tρ−1 [ an−1,n−1 tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 [ an−1,i−1 − [(n− 1)ρ− i]an−1,i ] 1 tnρ−i ( d dt )i f(t) − [(n− 1)ρ− 1] an−1,1 tnρ−1 d dt f(t) ] =tρ−1 [ an,n tn(ρ−1) ( d dt )n f(t) + n−1∑ i=2 an,i tnρ−i ( d dt )i f(t) + an,1 tnρ−1 d dt f(t) ] =tρ−1 n∑ i=1 an,i tnρ−i ( d dt )i f(t) = tρ−1 [ jαρ f ] (t). the proof is completed by considering the three cases above. � motivated by lemma 2.2, we suggest the following definition for the fractional derivative of a function. 220 karpuz, özkan, yalçin and yildiz definition 2.2. let α ∈ r, ρ ∈ r+ and f : (0,∞) → r. we define the α-order fractional derivative of f iteratedly by [ dαρ f ] (t) :=   [ j−αρ f ] (t), α ∈ r−0 1 tρ−1 d dt [ dα−1ρ f ] (t), α ∈ r+. example 2.2. for α,ν ∈ r+0 and ρ ∈ r +, we have[ dαρ∗ ρν ] (t) = ραγ(ν + 1) γ(ν −α + 1) tρ(ν−α) for t > 0. (2.5) we will prove this by applying induction on n ∈ z+0 for α ∈ [n,n + 1). first, let α ∈ [0, 1), then we have [ dαρ∗ ρν ] (t) = 1 tρ−1 d dt [ dα−1ρ ∗ ρν ] (t) = 1 tρ−1 d dt [ j 1−αρ ∗ ρν ] (t) = 1 tρ−1 d dt γ(ν + 1) ρ1−αγ ( ν + (1 −α) + 1 )tρ(ν+(1−α)) = ραγ(ν + 1) γ(ν −α + 1) tρ(ν−α) for t > 0, where we have applied example 2.1 in the second line above. this proves validity of (2.5) for all α ∈ [0, 1). let n ∈ z+0 , and assume now for all α ∈ [n,n + 1) that (2.5) is true. by definition 2.2, we have for any α ∈ [n + 1,n + 2) that[ dαρ∗ ρν ] (t) = 1 tρ−1 d dt [ dα−1ρ ∗ ρν ] (t) = 1 tρ−1 d dt ρα−1γ(ν + 1) γ ( ν − (α− 1) + 1 )tρ(ν−(α−1)) = ραγ(ν + 1) γ(ν −α + 1) tρ(ν−α) for t > 0, which completes the proof. in the following lemma, we provide a direct form for the definition of the fractional derivative in terms of the coefficients defined in (2.2). lemma 2.3. for α,ρ ∈ r+, we have [ dαρ f ] (t) = dαe∑ i=1 adαe,i(ρ) tdαeρ−i ( d dt )i[ j dαe−αρ f ] (t) for t > 0. proof. we compute that[ dαρ f ] (t) = 1 tρ−1 d dt [ dα−1ρ f ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ dα−2ρ f ] (t) ] = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ dα−dαeρ f ] (t) · · · ]] = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ j dαe−αρ f ] (t) · · · ]] , where we have for a total of dαe usual derivatives above. let us denote g := j dαe−αρ f and use (2.4) repeatedly inside to outside, then [ dαρ f ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ j 0ρ g ] (t) · · · ]] = · · · = 1 tρ−1 d dt [ j−dαe+1ρ g ] (t) = [ j−dαeρ g ] (t), which completes the proof by using (2.3). � reimann-liouville and hadamard type fractional derivatives 221 3. properties of the operators j and d the main result of this section is the following theorem. theorem 3.1. the following properties hold. (i) dαρ = j−αρ for α ∈ r. (ii) jαρ jβρ = jα+βρ for α,β ∈ r + 0 . (iii) dαρdβρ = dα+βρ for α ∈ z + 0 and β ∈ r + 0 or for α,β ∈ r +\z+ with (α + β) 6∈ z+. (iv) dαρjαρ = i for α ∈ r, where i is the identity operator. (v) [ jαρ dαρ f ] (t) = f(t) − dαe∑ i=1 tρ(α−i) ρα−iγ(α− i + 1) [ dα−iρ f ] (0+) for α ∈ r+. proof. (i) for α ∈ r+0 , the proof is similar to that of lemma 2.3, and for α ∈ r −, the proof follows from definition 2.2. (ii) the proof is trivial for α = 0 or β = 0. hence, we consider below the case where α,β ∈ r+. then, [ jαρ j β ρ f ] (t) = ∫ t 0 kαρ (t,η) ∫ η 0 kβρ (η,ζ)f(ζ)dζdη = ∫ t 0 ∫ η 0 kαρ (t,η)k β ρ (η,ζ)f(ζ)dζdη = ∫ t 0 ∫ t ζ kαρ (t,η)k β ρ (η,ζ)f(ζ)dηdζ = ∫ t 0 [∫ t ζ kαρ (t,η)k β ρ (η,ζ)dη ] f(ζ)dζ = ∫ t 0 kα+βρ (t,ζ)f(ζ)dζ = [ jα+βρ f ] (t), where we have applied lemma 2.1 (i) for the last line. (iii) the proof is trivial for α = 0 or β = 0. below, we consider the case where α 6= 0 and β 6= 0. (a) for α ∈ z+ and β ∈ r+, then [ dαρd β ρf ] (t) = 1 tρ−1 d dt [ 1 tρ−1 d dt [ · · · 1 tρ−1 d dt [ dβρf ] (t) · · · ]] = · · · = 1 tρ−1 d dt [ dβ+(α−1)ρ f ] (t) = [ dβ+αρ f ] (t). (b) for α,β ∈ r+\z+ with (α + β) 6∈ z+, then dαρdβρ = dα+βρ as in (ii). (iv) this follows from (i) by using the steps in the proof of (ii) and (iii). (v) performing integration by parts, for α ∈ r+, we obtain [ jα+1ρ d α ρ f ] (t) = [ jα+1ρ 1 ∗ρ−1 [ dα−1ρ f ]′] (t) = ∫ t 0 kα+1ρ (t,η) ηρ−1 [ dα−1ρ f ]′ (η)dη = kα+1ρ (t,η) ηρ−1 [ dα−1ρ f ] (η) ∣∣∣∣η=t η=0+ − ∫ t 0 ∂ ∂η ( kα+1ρ (t,η) ηρ−1 )[ dα−1ρ f ] (η)dη = − [tρ −ηρ]α ραγ(α + 1) [ dα−1ρ f ] (η) ∣∣∣∣η=t η=0+ + ∫ t 0 kαρ (t,η) [ dα−1ρ f ] (η)dη = [ jαρ d α−1 ρ f ] (t) − tρα ραγ(α + 1) [ dα−1ρ f ] (0+) where ∗′ in the first line stands for the usual derivative. using (ii), we get[ j 1ρj α ρ d α ρ f ] (t) = [ j 1ρj α−1 ρ d α−1 ρ f ] (t) − tρα ραγ(α + 1) [ dα−1ρ f ] (0+). 222 karpuz, özkan, yalçin and yildiz an application of d1ρ on both sides yields by using (iv) that[ jαρ d α ρ f ] (t) = [ jα−1ρ d α−1 ρ f ] (t) − tρ(α−1) ρα−1γ(α) [ dα−1ρ f ] (0+). repeating this procedure for a total of dαe times, we get [ jαρ d α ρ f ] (t) = [ jα−dαeρ d α−dαe ρ f ] (t) − dαe∑ i=1 tρ(α−i) ρα−iγ(α− i + 1) [ dα−iρ f ] (0+) (3.1) for all t > 0. by definition 2.1, definition 2.2 and (ii), we have jα−dαeρ d α−dαe ρ = { j 0ρd0ρ = i, α ∈ n jα−dαeρ j dαe−α ρ = j 0ρ = i, α ∈ r+\n, which completes the proof by using this in (3.1). thus, we have justified the validity of each of the properties above, and completed the proof. � 4. existence and uniqueness for rlh fdes let us consider the initial-value problem{[ dαρ y ] (t) = f ( t,y(t) ) for t > 0[ dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (4.1) where α ∈ r+ and y0,y1, · · · ,ydαe−1 ∈ r. suppose that f is defined in a domain ω of a plane (t,y), and define a region r(h,k) ⊂ ω as a set of points (t,y) ∈ ω, which satisfy the inequality∣∣∣∣∣y(t) − dαe∑ i=1 tρ(α−i) ρα−iγ(α− i + 1) ∣∣∣∣∣ ≤ k for all t ∈ (0,h), where h and k are constants. theorem 4.1. let f : ω → r satisfy the lipschitz condition with respect to its second component, i.e., |f(t,y1) −f(t,y2)| ≤ l|y1 −y2| for all (t,y1), (t,y2) ∈ ω, where l ∈ r+, and f be bounded on ω, i.e., |f(t,y)| ≤ m for all (t,y) ∈ ω, where m ∈ r+. further, assume that there exist h,k ∈ r+ such that mhρα ραγ(α + 1) ≤ k. then, there exists a unique and continuous solution of the problem (4.1) in the region r(h,k) ⊂ ω. proof. the method proof based on the ideas in [11, theorem 3.4]. first, consider theorem 3.1 (v) and reduce the problem (4.1) to an equivalent fractional integral equation y(t) = dαe∑ i=1 ydαe−i ρα−iγ(α− i + 1) tρ(α−i) + ∫ t 0 kαρ (t,η)f ( η,y(η) ) dη for t ∈ (0,h]. (4.2) if y satisfies (4.1), then it also satisfies the equation (4.2). on the other hand, if y is a solution of (4.2), then it is satisfies (4.1) initial-value problem. therefore, the equation (4.2) is equivalent to the initial value problem (4.1). now, let us define the sequence of functions {ym}m∈n0 by ym(t) =   dαe∑ i=1 ydαe−i ρα−iγ(α− i + 1) tρ(α−i), m = 0 y0(t) + [ jαρ f ( ∗,ym−1(∗) )] (t), m ∈ n (4.3) reimann-liouville and hadamard type fractional derivatives 223 for t ∈ (0,h]. we will show that limm→∞ym exists and gives the required solution y of the integral equation (4.2). first, it can be shown by induction that ym(t) ∈ r(h,k) for all t ∈ (0,h] and m ∈ n0. indeed, for all t ∈ (0,h] and all m ∈ n0, we obtain |ym(t) −y0(t)| = ∣∣∣∣ ∫ t 0 kαρ (t,η)f ( η,ym−1(η) ) dη ∣∣∣∣ ≤ ∫ t 0 kαρ (t,η) ∣∣f(η,ym−1(η))∣∣dη ≤m ∫ t 0 kαρ (t,η)dη = mtρα ραγ(α + 1) ≤ mhρα ραγ(α + 1) ≤ k and thus |y1(t) −y0(t)| ≤ mhρα ραγ(α + 1) ≤ k for all t ∈ (0,h]. (4.4) let us show by induction that |ym(t) −ym−1(t)| ≤ mlm−1tmρα ρmαγ(mα + 1) for all t ∈ (0,h] and all m ∈ n. (4.5) it follows from (4.4) that (4.5) holds for m = 1. suppose for some m ∈ n that |ym(t) −ym−1(t)| ≤ mlm−1tmρα ρmαγ(mα + 1) for all t ∈ (0,h]. (4.6) then, using (4.3) and (4.6), we have |ym+1(t) −ym(t)| = ∣∣∣∣ ∫ t 0 kαρ (t,η) [ f ( η,ym(η) ) −f ( η,ym−1(η) )] dη ∣∣∣∣ ≤ ∫ t 0 kαρ (t,η) ∣∣f(η,ym(η))−f(η,ym−1(η))∣∣dη ≤l ∫ t 0 kαρ (t,η)|ym(η) −ym−1(η)|dη ≤ mlm ρmαγ(mα + 1) ∫ t 0 kαρ (t,η)η mραdη = mlm ρ(m+1)αt (m+1)ρα γ(α)γ(mα + 1) ∫ 1 0 [1 − ζ]α−1ζmαdζ = mlmt(m+1)ρα ρ(m+1)αγ(α)γ(mα + 1) ∫ 1 0 [1 − ζ]α−1ζmαdζ = mlmt(m+1)ρα ρ(m+1)αγ(α)γ(mα + 1) b(α,mα + 1) = mlmt(m+1)ρα ρ(m+1)αγ ( (m + 1)α + 1 ) for all t ∈ (0,h]. this means that (4.5) is true. let us consider the limiting function y(t) := lim m→∞ ym(t) = y0(t) + ∞∑ j=1 [yj(t) −yj−1(t)] for t ∈ (0,h]. (4.7) according to the estimate (4.5), for t ∈ (0,h], the absolute value of its terms is less than the corresponding terms of the convergent numeric series ∞∑ j=1 |yj(t) −yj−1(t)| ≤ ∞∑ j=1 mlj−1hjρα ρjαγ(jα + 1) = m l ∞∑ j=1 ljhjρα ρjαγ(jα + 1) = m l [ eα,1 ( lhρα ρα ) − 1 ] , 224 karpuz, özkan, yalçin and yildiz where e is the two-parameter mittag-leffler function defined by eα,β(z) := ∞∑ j=0 zj γ(αj + β) for z ∈ c and α,β ∈ c, (4.8) which converges for all values of z (i.e., it is an entire function). this means that the series (4.7) converges uniformly. letting m →∞ in (4.3) and using (4.7), we get y(t) = y0(t) + ∫ t 0 kαρ (t,η)f ( η,y(η) ) dη for all t ∈ (0,h]. therefore, y defined by (4.7) is a solution of (4.2), and thus (4.1). what follows next is to prove the uniqueness of the solution. let us suppose that z is another solution of the equation (4.2), which is continuous in the interval (0,h]. then w(t) := y(t) − z(t) for t ∈ (0,h], then satisfies the equation w(t) = ∫ t 0 kαρ (t,η) [ f ( η,y(η) ) −f ( η,z(η) )] dη (4.9) from which it follows that w(0+) = 0. therefore, w extends continuously to [0,h]. then, |w(t)| ≤ c for all t ∈ (0,h], where c ∈ r+, and we obtain from (4.9) that |w(t)| ≤ cltρα ραγ(α + 1) for all t ∈ (0,h]. repeating this procedure for a total of m ∈ n times, we obtain |w(t)| ≤ clmtmρα ρmαγ(mα + 1) for all t ∈ (0,h]. in the right-hand side, we recognize the general term of the series for the mittag-leffler function eα,1( ltρα ρα ), and therefore lim m→∞ lmtmρα ρmαγ(mα + 1) = 0 for all t ∈ (0,h]. then, we have w(t) ≡ 0 for all t ∈ (0,h], and thus y(t) ≡ z(t) for all t ∈ (0,h]. this ends the proof. � 5. the autonomous equation of rl type let us consider initial-value problem{[ dαρ y ] (t) = λy(t) for t > 0[ dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (5.1) where λ ∈ r. in this case, when compared to (4.1), we have f(t,y) = λy. now, we will introduce two techniques for obtaining the unique solution of (5.1). 5.1. direct substitution. let α ∈ r and β ∈ r and define yα,β(t) := tρβ ρβ eα,β+1 ( λtρα ρα ) for t > 0, where e is the two-parameter mittag-leffler function defined in (4.8). reimann-liouville and hadamard type fractional derivatives 225 then, dαρ y = λy. indeed, we have [ dαρ yα,β ] (t) = [ dαρ ∗ρβ ρβ eα,β+1 ( λ∗ρα ρα )] (t) = [ dαρ ∞∑ j=0 λj∗ρ(αj+β) ραj+βγ(αj + β + 1) ] (t) = ∞∑ j=0 λj [ dαρ∗ρ(αj+β) ] (t) ραj+βγ(αj + β + 1) = ∞∑ j=0 λjtρ(α(j−1)+β) ρα(j−1)+βγ ( α(j − 1) + β + 1 ) = tρ(β−α) ρβ−αγ(β −α + 1) + ∞∑ j=1 λjtρ(α(j−1)+β) ρα(j−1)+βγ ( α(j − 1) + β + 1 ) = tρ(β−α) ρβ−αγ(β −α + 1) + λ tρβ ρβ ∞∑ j=0 λjtραj ραjγ(αj + β + 1) = tρ(β−α) ρβ−αγ(β −α + 1) + λyα,β(t) for all t > 0. here, we see that yα,β solves dαρ y = λy provided that (β−α) is a negative integer. that is, yα,α−i, where i = 1, 2, · · · ,dαe, satisfies dαρ y = λy. moreover, we compute that [ dα−kρ yα,α−i ] (t) = [ dα−kρ ∗ρ(α−i) ρα−i eα,α−i+1 ( λ∗ρα ρα )] (t) = [ dα−kρ ∞∑ j=0 λj∗ρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 )](t) = ∞∑ j=0 λj [ dα−kρ ∗ρ(α(j+1)−i) ] (t) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) = ∞∑ j=0 λjtρ(αj−i+k) ραj−i+kγ(αj − i + k + 1) for t > 0 and k = 1, 2, · · · ,dαe. using the properties of the gamma function and considering the positive powers of t, we find that[ dα−kρ yα,α−i ] (0+) = δi,k for k = 1, 2, · · · ,dαe, where δ is kronecker’s delta. therefore, {yα,α−i} dαe i=1 is the set of normalized fundamental solutions of dαρ y = λy. moreover, the following linear combination of functions y(t) := dαe∑ i=1 ydαe−i tρ(α−i) ρα−i eα,α−i+1 ( λtρα ρα ) for t > 0 forms the solution desired of (5.1). 5.2. the picard iterates. in accordance with the proof of theorem 4.1, let us take ym(t) =   dαe∑ i=1 tρ(α−i) ρα−iγ(α− i + 1) [dα−iρ y](0 +), m = 0 y0(t) + λ [ jαρ ym−1 ] (t), m ∈ n (5.2) for t ∈ (0,h]. we will show by induction that ym(t) = dαe∑ i=1 ydαe−i m∑ j=0 tρ(α(j+1)−i)λj ρ(α(j+1)−i)γ ( α(j + 1) − i + 1 ) (5.3) 226 karpuz, özkan, yalçin and yildiz for all t ∈ (0,h] and m ∈ n0. the claim holds for m = 0 by (5.2). assume for some m ∈ n0 that ym(t) = dαe∑ i=1 ydαe−i m∑ j=0 tρ(α(j+1)−i)λj ρα(j+1)−iγ ( α(j + 1) − i + 1 ) for all t ∈ (0,h], which together with example 2.1 and (5.2) yields ym+1(t) =y0(t) + λ [ jαρ ym ] (t) = y0(t) + λ dαe∑ i=1 ydαe−i m∑ j=0 λj [ jαρ ∗ρ(α(j+1)−i) ] (t) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−iγ(α− i + 1) + dαe∑ i=1 ydαe−i m∑ j=0 λj+1tρ((j+2)α−i) ρ(j+2)α−iγ ( (j + 2)α− i + 1 ) = dαe∑ i=1 ydαe−i m+1∑ j=0 λjtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) for all t ∈ (0,h]. this justifies (5.3). letting m →∞ in (5.3), we obtain the solution of the problem (5.2) as y(t) = lim m→∞ ym(t) = dαe∑ i=1 ydαe−i ∞∑ j=0 λjtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−i eα,α−i+1 ( λtρα ρα ) for t > 0, where e is the two-parameter mittag-leffler function defined in (4.8). 6. dependence on initial conditions let us introduce small changes in the initial conditions of (4.1) and consider{[ dαρ y ] (t) = f ( t,y(t) ) for t > 0[ dα−kρ y ] (0+) = ydαe−k + εdαe−k for k = 1, 2, · · · ,dαe, (6.1) where εdαe−k are arbitrary constants. theorem 6.1. assume that conditions of theorem 4.1 hold. let y and z be respective solutions of the initial value problems (4.1) and (6.1). then, |y(t) −z(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ ρi atρi eα,1−i ( atρα ρα ) for t ∈ (0,h]. proof. in conformity with theorem 4.1, we have y(t) = lim m→∞ ym(t) for t ∈ (0,h], where the sequence of functions {ym}m∈n0 is defined by (4.3) for t ∈ (0,h]. similarly, z(t) = lim m→∞ zm(t) for t ∈ (0,h], where zm(t) =   dαe∑ i=1 tρ(α−i) ρα−iγ(α− i + 1) (ydαe−i + εdαe−i), m = 0 z0(t) + [ jαρ f ( ∗,zm−1(∗) )] (t), m ∈ n (6.2) for t ∈ (0,h]. let us prove by induction that |ym(t) −zm(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 ajtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) (6.3) reimann-liouville and hadamard type fractional derivatives 227 for all t ∈ (0,h] and all m ∈ n0. from (4.3) and (6.2), it directly follows that |y0(t) −z0(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iγ(α− i + 1) for all t ∈ (0,h]. assume now for some m ∈ n that |ym(t) −zm(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 ajtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) (6.4) for all t ∈ (0,h]. then, using (4.3) and (6.2), the lipschitz condition for the function f together with the inequality (6.4), we obtain |ym+1(t) −zm+1(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iγ(α− i + 1) + a ∫ t 0 kαρ (t,η)|ym(η) −zm(η)|dη ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iγ(α− i + 1) + a ∫ t 0 kαρ (t,η) dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 ajηρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 )dη = dαe∑ i=1 ∣∣εdαe−i∣∣ tρ(α−i) ρα−iγ(α− i + 1) + dαe∑ i=1 ∣∣εdαe−i∣∣ m∑ j=0 aj+1tρ((j+2)α−i) ρ(j+2)α−iγ ( (j + 2)α− i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣m+1∑ j=0 ajtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) for all t ∈ (0,h]. this proves (6.3). taking the limit of (6.4) as m →∞, we obtain |y(t) −z(t)| ≤ dαe∑ i=1 ∣∣εdαe−i∣∣ ∞∑ j=0 ajtρ(α(j+1)−i) ρα(j+1)−iγ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣ ρi atρi ∞∑ j=0 aj+1t(j+1)ρα ρα(j+1)γ ( α(j + 1) − i + 1 ) = dαe∑ i=1 ∣∣εdαe−i∣∣ ρi atρi eα,1−i ( atρα ρα ) for all t ∈ (0,h], which completes the proof. � 7. the green’s function for linear equations the objective of this section is to define the green’s function notion for the initial value problem{[ dαρ y ] (t) = p(t)y(t) + f(t) for t > 0[ dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe, (7.1) where p,f : [0,∞) → r are continuous functions, and then present its role in obtaining the solution of the equation. let ∆ := {(t,s) : t > s ≥ 0} and denote by sdαρ f and sjαρ f the fractional derivative and the fractional integral of a function f centered at s ∈ [0,∞), respectively. definition 7.1 (green’s function). let the continuous function gρ : ∆ → r satisfy the following properties. (i) [sdαρgρ(∗,s)](t) = p(t)gρ(t,s) for all (t,s) ∈ ∆. (ii) lim s→t− [sdα−kρ gρ(∗,s)](t) = δk,1 for t > 0 and k = 1, 2, · · · ,dαe. 228 karpuz, özkan, yalçin and yildiz (iii) lim s→t− t→0+ [sdα−kρ gρ(∗,s)](t) = 0 for k = 1, 2, · · · ,dαe− 1. then, gρ is called the green’s function for the initial value problem (7.1). theorem 7.1. let gρ be the green’s function for the initial value problem (7.1). then, y(t) := ∫ t 0 gρ(t,η)ηρ−1f(η)dη for t > 0 (7.2) is the (unique) solution of the initial value problem{[ dαρ y ] (t) = p(t)y(t) + f(t) for t > 0[ dα−kρ y ] (0+) = 0 for k = 1, 2, · · · ,dαe. (7.3) proof. first, we will show that y defined by (7.2) solves the fractional differential equation in (7.3). to this end, we let β = α−dαe + 1, then β ∈ (0, 1]. from (7.2), we have for t > 0 that[ dβρy ] (t) = 1 tρ−1 d dt [ dβ−1ρ y ] (t) = 1 tρ−1 d dt [ j 1−βρ y ] (t) = 1 tρ−1 d dt [∫ t 0 k1−βρ (t,η) ∫ η 0 gρ(η,ζ)ζρ−1f(ζ)dζdη ] = 1 tρ−1 d dt [∫ t 0 ∫ η 0 k1−βρ (t,η)gρ(η,ζ)ζ ρ−1f(ζ)dζdη ] = 1 tρ−1 d dt [∫ t 0 ∫ t ζ k1−βρ (t,η)gρ(η,ζ)ζ ρ−1f(ζ)dηdζ ] = 1 tρ−1 d dt [∫ t 0 [∫ t ζ k1−βρ (t,η)gρ(η,ζ)dη ] ζρ−1f(ζ)dζ ] = 1 tρ−1 d dt [∫ t 0 [ ζj 1−βρ gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ ] = ∫ t 0 1 tρ−1 d dt [ ζj 1−βρ gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + 1 tρ−1 lim ζ→t− [[ ζj 1−βρ gρ(∗,ζ)(t) ] ζρ−1f(ζ) ] , where we have applied the well-known leibnitz rule. using the definition of the fractional derivative, we obtain [ dβρy ] (t) = ∫ t 0 [ ζdβρgρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζdβ−1ρ gρ(∗,ζ) ] (t)f(t) for all t > 0. applying d1 repeatedly for a total of (dαe− 1) times and using definition 7.1 (ii), we find for all t > 0 that[ dαρ y ] (t) = ∫ t 0 [ ζdαρgρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζdα−1ρ gρ(∗,ζ) ] (t)f(t) (7.4) = ∫ t 0 [ ζdαρg(∗,ζ) ] (t)ζρ−1f(ζ)dζ + f(t) =p(t) ∫ t 0 gρ(t,ζ)ζρ−1f(ζ)dζ + f(t) =p(t)y(t) + f(t) for all t > 0 by definition 7.1 (i). thus, y is a solution of the fractional differential equation in (7.3). next, we justify the initial conditions. as in (7.4), we compute that[ dα−kρ y ] (t) = ∫ t 0 [ ζdα−kρ gρ(∗,ζ) ] (t)ζρ−1f(ζ)dζ + lim ζ→t− [ ζdα−k+1ρ gρ(∗,ζ) ] (t)f(t) for all t > 0 and k = 1, 2, · · · ,dαe . by definition 7.1 (iii) and letting t → 0+, we obtain [ dα−kρ y ] (0+) = 0 for k = 1, 2, · · · ,dαe. we have therefore justified that y defined in (7.2) solves the initial value problem (7.3). � reimann-liouville and hadamard type fractional derivatives 229 corollary 7.1. if {hi} dαe i=1 forms the set of normalized fundamental solutions of the homogeneous initial value problem associated with (7.1), i.e.,{ [dαρ hi](t) = p(t)hi(t) for t > 0 [dα−kρ hi](0+) = δi,k for k = 1, 2, · · · ,dαe, and g is the green’s function for the initial value problem (7.1), then the solution of the initial value problem (7.1) is given by y(t) = dαe∑ k=1 ydαe−ihi(t) + ∫ t 0 gρ(t,η)ηρ−1f(η)dη for t > 0. 7.1. representation of solutions for the autonomous equation. in this section, we confine our attention to the linear autonomous initial value problem{[ dαρ y ] (t) = λy(t) + f(t) for t > 0[ dα−kρ y ] (0+) = ydαe−k for k = 1, 2, · · · ,dαe. (7.5) for linear autonomous equations, we can easily verify that gρ(t,s) = gρ ( ρ √ tρ −sρ, 0 ) for all (t,s) ∈ ∆. (7.6) moreover, gρ(∗, 0) is the solution of the associated homogeneous equation{[ dαρ y ] (t) = λy(t) for t > 0[ dα−kρ y ] (0+) = δk,1 for k = 1, 2, · · · ,dαe. due to the discussion made in § 5.1, we see that the green’s function of (7.5) is given by gρ(t,s) = [tρ −sρ]α−1 ρα−1 eα,α ( λ[tρ −sρ]α ρα ) for (t,s) ∈ ∆. further, the variation of parameters formula for the initial value problem (7.5) is given by y(t) = dαe∑ i=1 ydαe−i tρ(α−i) ρα−i eα,α−i+1 ( λtρα ρα ) + ∫ t 0 [tρ −ηρ]α−1 ρα−1 eα,α ( λ[tρ −ηρ]α ρα ) ηρ−1f(η)dη (7.7) for t > 0 (cf. [9, equation (3.1.11)]). 8. final comments the following two examples can be easily verified. example 8.1. for α ∈ r+0 , ν ∈ (−1,∞) and ρ ∈ r +, we have[ sjαρ [∗ ρ −sρ]ν ] (t) = γ(ν + 1) ραγ(ν + α + 1) [tρ −sρ]ν+α for t > s ≥ 0. example 8.2. for α,ν ∈ r+0 and ρ ∈ r +, we have[ sdαρ [∗ ρ −sρ]ν ] (t) = ραγ(ν + 1) γ(ν −α + 1) [tρ −sρ]ν−α for t > s ≥ 0. as our first remark, we would like to say that one can justify (7.6) similar to that in the third part of [11, § 5.1.2]. as an another note, we would like to emphasis that one can justify the variation of parameters formula given in (7.7) by using example 8.2 and applying the picard iterates technique (used in § 5.2) to (7.5). as to some directions for future research, we would like to mention that the study of numerical solutions to fdes and numerical integration techniques would very important. next, we note that the extension of any of the results in this paper to the case fdes with miller-ross type sequential derivatives would be of significant interest too. finally, obtaining the solutions of fdes of the type (7.5) by the laplace transform would also deserve attention for the sake of completeness. 230 karpuz, özkan, yalçin and yildiz references [1] p. l. butzer, a. a. kilbas and j. j. trujillo, compositions of hadamard-type fractional integration operators and the semigroup property, j. math. anal. appl. 269 (2002), no. 2, 387–400. [2] p. l. butzer, a. a. kilbas and j. j. trujillo, fractional calculus in the mellin setting and hadamard-type fractional integrals, j. math. anal. appl. 269 (2002), no. 1, 1–27. [3] k. diethelm, the analysis of fractional differential equations, springer-verlag, berlin, 2010. [4] j. hadamard, essai sur l’étude des fonctions données par leur développement de taylor, journal de mathématiques pures et appliquées 4 (1892), no. 8, 101–186. [5] u. n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218 (2011), no. 3, 860–865. [6] u. n. katugampola, a new approach to generalized fractional derivatives, bull. math. anal. appl. 4 (2014), no. 6, 1–15. [7] a. a. kilbas, hadamard-type fractional calculus, j. korean math. soc. 38 (2001), no. 6, 1191–1204. [8] a. a. kilbas and j. j. trujillo, hadamard-type integrals as g-transforms, integral transforms spec. funct. 14 (2003), no. 5, 413–427. [9] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractional differential equations elsevier science b.v., amsterdam, 2006. [10] k. b. oldham and j. spanier, the fractional calculus theory and applications of differentiation and integration to arbitrary order, academic press, new york, 1974. [11] i. podlubny, fractional differential equations, academic press, new york, 1999. [12] s. pooseh, p. almeida and d. f. m. torres, expansion formulas in terms of integer-order derivatives for the hadamard fractional integral and derivative, numer. funct. anal. optim. 33 (2012), no. 3, 301–319. [13] s. g. samko, a. a. kilbas and o. i. marichev, fractional integrals and derivatives theory and applications, gordon and breach science publishers, yverdon, 1993. 1dokuz eylül university, tınaztepe campus, faculty of science, department of mathematics, buca, 35160 i̇zmir, turkey. 2afyon kocatepe university, ans campus, faculty of science and arts, department of mathematics, 03200 afyonkarahisar, turkey. ∗corresponding author: bkarpuz@gmail.com 1. introduction 2. definitions and auxiliary results 3. properties of the operators j and d 4. existence and uniqueness for rlh fdes 5. the autonomous equation of rl type 5.1. direct substitution 5.2. the picard iterates 6. dependence on initial conditions 7. the green's function for linear equations 7.1. representation of solutions for the autonomous equation 8. final comments references int. j. anal. appl. (2022), 20:50 interval valued intuitionistic fuzzy β-filters on β-algebras kaliyaperumal palanivel1, prakasam muralikrishna2, perumal hemavathi3, ronnason chinram4, pattarawan singavananda5,∗ 1department of mathematics, school of advanced sciences, vellore institute of technology, vellore-632014, india 2pg and research department of mathematics, muthurangam government arts college (autonomus), vellore-632002, india 3saveetha school of engineering, simats, thandalam-602025, india 4division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 5program in mathematics, faculty of science and technology, songkhla rajabhat university, songkhla 90000, thailand ∗corresponding author: pattarawan.pe@skru.ac.th abstract. this study establishes the concept of interval valued intuitionistic fuzzy (invinf) β-filters on β-algebras and a few of its related properties are investigated. some compelling results of interval valued fuzzy β-filters have been examined. further, the notions of products and strong β-filters are also introduced. in addition that, the level set and homomorphism of interval valued intuitionistic fuzzy β-filters are too discussed. furthermore, we enacted that the intersection between two interval valued intuitionistic fuzzy β−filters is again an interval valued intuitionistic fuzzy β-filter. 1. introduction in 2002, neggers et al. [12] proposed the idea of a β-algebra which is an algebraic structure with two operations. the concepts of fuzzy positive implicative and fuzzy associative filters of lattice implication algebras have been initiated in [13,14]. further, the authors in [13,14] have demonstrated that every fuzzy associative filter is a fuzzy associative filter and that every fuzzy positive implicative received: aug. 17, 2022. 2010 mathematics subject classification. 08a72, 03e72. key words and phrases. fuzzy β-filter; intuitionistic fuzzy β-filter; β-algebra; interval valued intuitionistic fuzzy set. https://doi.org/10.28924/2291-8639-20-2022-50 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-50 2 int. j. anal. appl. (2022), 20:50 filter is a fuzzy implicative filter. the equivalent conditions for both fuzzy positive implicative filters and fuzzy associative filters were also provided. xu et al. [20] established the thought of intuitionistic fuzzy implicative filters in lattice implication algebras. jun et al. [8] proposed the concept of fuzzy bcisubalgebras with interval valued membership functions. in 2011, ghorbani [4] proposed the notion of intuitioistic fuzzy filters of residuated lattices. they have illustrated that a residual lattice’s collection of all intuitionistic fuzzy filters is a complete lattice and identified its distributive sublattices. zadeh [21] developed an interval valued fuzzy set was used to extend a fuzzy set (ie. a fuzzy set with an interval valued membership function). an i-v fuzzy set is an interval valued fuzzy set that can be used in various algebraic structures. biswas et al. [2] created fuzzy subgroups with interval membership values in 1994. hoo [9] applied the concepts of filters and ideals in bci-algebras in 1991. in 2015, hemavathi et al. [5–7] discussed interval valued fuzzy β-subalgebras and also applied the concept to intuitionistic fuzzy sets. jun et al. [15] introduced foldness of bipolar fuzzy sets and its application in bck/bcialgebras. takallo et al. [19] discussed the concept of multipolar fuzzy p-ideals of bci-algebras. the concept of multipolar intuitionistic fuzzy hyper bck-ideals in hyper bck-algebras has been developed by seo et al. [16]. borzooei et al. [3] focused on multipolar intuitionistic fuzzy b-algebras. a new perception of cubic multi-polar structures on bck/bci-algebras was approached by al-masarwah et al. [1]. in [10], the authors invented a mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties. muhiuddin et al. [11] applied the theory of linear diophantine fuzzy set into bck/bci-algebras. sujatha et al. [17, 18] introduced fuzzy filters on βalgebras and also developed the concept of intuitiointic fuzzy filters on β-algebras. with all of this in mind, this paper establishes the idea of interval valued fuzzy β-filters on β-algebras and demonstrate few of its intriguing aspects. 2. preliminares this section outlines some of the most important definitions and examples relevant to the study. definition 2.1. a β-algebra is a non-empty set γ with two binary operations + and − and a constant 0 fulfills the following axioms: (1) %δ − 0 = %δ (2) (0 −%δ) + %δ = 0 (3) (%δ − ỳδ) − z̀δ = %δ − (z̀δ + ỳδ) for all %δ, ỳδ, z̀δ ∈ γ. definition 2.2. let f be a mapping from a βalgebra γ to a βalgebra υ , then f is referred as homomorphism, if (1) f (%δ + ỳδ) = f (%) + f (ỳδ) (2) f (%δ − ỳδ) = f (%δ) − f (ỳδ) int. j. anal. appl. (2022), 20:50 3 for all %δ, ỳδ ∈ γ. definition 2.3. a β-subalgebra ξ on a β-algebra γ is referred as β-filter, if (1) %δ 4 ỳδ = %δ + (%δ + ỳδ) ∈ ξ (2) %δ 5 ỳδ = %δ − (%δ − ỳδ) ∈ ξ for all %δ, ỳδ ∈ ξ. definition 2.4. a β-subalgebra ξ on a β-algebra γ is referred as fuzzy β-filter, if (1) �ξ(%δ 4 ỳδ) ≥ min{�ξ(%δ),�ξ(%δ + ỳδ)} and �ξ(%δ 5 ỳδ) ≥ min{�ξ(%δ),�ξ(%δ − ỳδ)} (2) �ξ(ỳδ) ≥ �ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ ξ. 3. interval valued fuzzy β-filters the concept of an interval valued fuzzy (invf) β-filter on a β-subalgebra is introduced in this section. definition 3.1. an invf β-subalgebra ξ on a β-algebra γ is referred as an invf fuzzy β-filter, it satisfies (1) �ξ(%δ 4 ỳδ) ≥ rmin{�ξ(%δ),�ξ(%δ + ỳδ)} and �ξ(%δ 5 ỳδ) ≥ rmin{�ξ(%δ),�ξ(%δ − ỳδ)} (2) �ξ(ỳδ) ≥ �ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ ξ. example 3.1. consider a β-algebra γ = {0,γ1,γ2,γ3} with two binary operations + and − and a constant 0 defined on γ with the cayley’s table: + 0 γ1 γ2 γ3 0 0 0 0 0 γ1 γ1 γ1 γ1 γ1 γ2 0 0 γ2 γ3 γ3 γ3 γ3 γ3 γ3 − 0 γ1 γ2 γ3 0 0 0 0 0 γ1 γ1 γ1 γ1 γ1 γ2 γ2 γ2 γ2 γ2 γ3 γ3 γ3 γ3 γ3 then (γ, +,−, 0) is a β-algebra. thus ξ = {γ1,γ3} is a β-filter on γ. we have ξ is an invf β-subalgebra, with interval membership function �ξ(%δ) =  [0.3, 0.5] : %δ = γ1 [0.4, 0.6] : %δ = γ3 . then it is observed that, ξ is an invf β-filter on γ. 4 int. j. anal. appl. (2022), 20:50 example 3.2. from the example 3.1, ξ is an invf β-subalgebra, define by the membership function �ξ(%δ) =   [0.2, 0.6] : %δ = γ1 [0.1, 0.4] : %δ = γ2 [0.3, 0.5] : %δ = γ3 . then it is observed that, ξ is not an invf β-filter on γ because �ξ(γ2) ≥ �ξ(γ1) ⇒ [0.1, 0.4] � [0.2, 0.6]. lemma 3.1. if ξ1 and ξ2 are two invf β-filters on γ, then so is ξ1 ∩ ξ2. proof. for %δ, ỳδ ∈ γ, �ξ1∩ξ2 (%δ 4 ỳδ) = rmin{�ξ1 (%δ 4 ỳδ),�ξ2 (%δ 4 ỳδ)} ≥ rmin{rmin{�ξ1 (%δ),�ξ1 (%δ + ỳδ)}, rmin{�ξ2 (%δ),�ξ2 (%δ + ỳδ)}} ≥ rmin{rmin{�ξ1 (%δ),�ξ2 (%δ)}, rmin{�ξ1 (%δ + ỳδ),�ξ2 (%δ + ỳδ)}} = rmin{�ξ1∩ξ2 (%δ),�ξ1∩ξ2 (%δ + ỳδ)}. similarly, �ξ1∩ξ2 (%δ 5 ỳδ) ≥ rmin{�ξ1∩ξ2 (%δ),�ξ1∩ξ2 (%δ − ỳδ)}. hence ξ1 ∩ ξ2 is an invf β-filter of γ. � theorem 3.1. every β-filter in invf is also a β-subalgebra in invf. proof. this proof is self-evident, as it follows clearly from the definition of the invf β-filter. every invf β-subalgebra, on the other hand, does not have to be an invf β-filter. � theorem 3.2. for an invf β-filter � of γ, we have �ξ(%δ4 ỳδ) ≥ �ξ(%δ) and �ξ(%δ5 ỳδ) ≥ �ξ(%δ) where %δ ≤ ỳδ. proof. assume that �ξ is an invf β-filter of γ. let %δ, ỳδ ∈ γ. then �ξ(%δ 4 ỳδ) = �ξ(%δ + (%δ + ỳδ)) ≥ rmin{�ξ(%δ),�ξ(%δ + ỳδ)} = rmin{�ξ(%δ), rmin{�ξ(%δ),�ξ(ỳδ)}} (because invf β-filter is an invf β-subalgebra) = rmin{�ξ(%δ),�ξ(%δ)} (because %δ ≤ ỳδ ⇒ �ξ(ỳδ) ≥ �ξ(%δ)) = �ξ(%δ). similarly, �ξ(%δ 5 ỳδ) ≥ �ξ(%δ). � int. j. anal. appl. (2022), 20:50 5 definition 3.2. consider an invf β-filter �ξ of a β-subalgebra γ. for [s1,s2] ∈ d[0, 1], the set �ξ = {%δ ∈ γ : �ξ(%)δ ≥ [s1,s2]} is referred to as a level set of an invf β-filter �ξ of γ. theorem 3.3. an invf subset ξ of a β-algebra γ is an invf β-filter if and only if for any t ∈ d[0, 1] the t−invf level subset ξt = {%δ ∈ γ : ξ(%δ) ≥ t} is either a β-filter or ξt 6= ∅. proof. for an invf level subset of ξ in γ, ξt 6= ∅. then %δ, ỳδ ∈ ξt, ξ(%δ) ≥ t. now, ξ(%δ 4 ỳ) = ξ(%δ + (%δ + ỳδ)) ≥ rmin{ξ(%δ), ξ(%δ + ỳδ)} = rmin{ξ(%δ), rmin{ξ(%δ), ξ(ỳδ)}} = rmin{t,rmin{t,t}} = t. this implies that %δ 4 ỳδ ∈ ξt. similarly, %δ 5 ỳδ ∈ ξt. then ξt is a β-filter of γ. suppose that ξt is a β-filter of γ, on the other hand. for %δ, ỳδ ∈ γ,%δ 4 ỳδ and %δ 5 ỳδ ∈ ξt, this implies that ξ(%δ4 ỳδ) ≥ t and ξ(%δ5 ỳδ) ≥ t ξ(%δ4 ỳδ) = ξ(%δ + (%δ + ỳδ)) ≥ t = rmin{ξ(%δ), ξ(%δ + ỳδ)}. similarly, ξ(%δ 5 ỳδ) ≥ t. this proved ξ is an invf β-filter. � theorem 3.4. consider an onto β-algebra homomorphism f from γ to υ . if ξ2 is a invf β-filter of y , hence its inverse image f−1(ξ2) is again an invf β−filter on γ. proof. let ξ2 be an invf β-filter of y . for any %δ, ỳδ ∈ γ, f−1(�ξ2 (%δ 4 ỳδ)) = f −1(�ξ2 (%δ + (%δ + ỳδ))) = �ξ2 (f (%δ + (%δ + ỳδ))) = �ξ2 (f (%δ)δ + f (%δ + ỳδ)) ≥ rmin{�ξ2 (f (%δ)),�ξ2 (f (%δ + ỳδ))} = rmin{f−1(�ξ2 (%δ)), f −1(�ξ2 (%δ + ỳδ))}. similarly, f−1(�ξ2 (%δ 5 ỳδ)) ≥ rmin{f −1(�ξ2 (%δ)), f −1(�ξ2 (%δ − ỳδ))}. let %δ, ỳδ ∈ γ, so that %δ ≥ ỳδ. subsequently, ξ2 is an invf β-filter, �ξ2 (f (ỳδ)) ≥ �ξ2 (f (%δ)) = f −1(�b(%δ)) such that f−1(�ξ2 (ỳδ)) ≥ f −1(�ξ2 (%δ)). � 3.1. products on invf β-filters on β-algebras. the basic concepts and examples of product on invf β-filters settings are covered in this section. theorem 3.5. an invf β-filter is the cartesian product of any two invf β-filters. 6 int. j. anal. appl. (2022), 20:50 proof. take %δ = (%δ1,%δ2) & ỳδ = (ỳδ1, ỳδ2) ∈ γ ×y & σ = (�ξ1 × �ξ2 ). so σξ1×ξ2 (%δ 4 ỳδ) = �ξ1 ((%δ1,%δ2) 4 (ỳδ1, ỳδ2)) = (�ξ1 × �ξ2 ){((%δ1,%δ2) + ((%δ1,%δ2) + (ỳδ1, ỳδ2)))} ≥ rmin{�ξ1 (%δ1 + (%δ1 + ỳδ1)),�ξ2 (%δ2 + (%δ2 + ỳδ2))} = rmin{rmin{�ξ1 (%δ1),�ξ1 (%δ1 + ỳδ1)}, rmin{�ξ2 (%δ2),�ξ2 (%δ2 + ỳδ2)}} = rmin{rmin{�ξ1 (%δ1),�ξ2 (%δ2)}, rmin{�ξ1 (%δ1 + ỳδ1),�ξ2 (%δ2 + ỳδ2)}} = rmin{σξ1×ξ2 (%δ1,%δ2),σξ1×ξ2 ((%δ1,%δ2) + (ỳδ1, ỳδ2))} = rmin{σξ1×ξ2 (%δ),σξ1×ξ2 (%δ + ỳδ)}. similarly, σξ1×ξ2 (%δ5 ỳδ) ≥ rmin{σξ1×ξ2 (%δ),σξ1×ξ2 (%δ− ỳδ)}. this proved that ξ1 × ξ2 is also an invf β-filter. � theorem 3.6. let γ and υ be two β-algebras. let ξ1t and ξ2s be invf β-filters on γ × y . then(ξ1t × ξ2s) is also a β-filter, if t ≥ s. proof. take %δ = (%δ1,%δ2) & ỳδ = (ỳδ1, ỳδ2) ∈ γ × y & σ = (�ξ1 × �ξ2 ) if t ≥ s. using above theorem, σ(ξ1t × ξ2s)(%δ 4 ỳδ) ≥ s. similarly, σ(ξ1t × ξ2s)(%δ 5 ỳδ) ≥ s. � 3.2. invf strong β-filters. beginning with a description and some examples, this section introduces the notion of an invf strong β-filter on a β-subalgebra. definition 3.3. an invf β-subalgebra ξ of a β-algebra is referred as an invf strong β-filter, if (1) �ξ(%δ 4 ỳδ) = �ξ(%δ 5 ỳδ) (2) �ξ(ỳδ) ≥ �ξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ ξ. example 3.3. for a β-algebra γ = {0,η1,η2,η3} be a with two binary operations + and − constant 0 and defined on γ, we have a cayley’s table + 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 0 0 η1 η3 η3 η3 η3 η3 η3 − 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 η1 η1 η1 η1 η3 η3 η3 η3 η3 then (γ, +,−, 0) is a β-algebra. thus ξ = {η1,η3} is a β-filter on γ. defining the membership function for an invf β-subalgebra ξ as �ξ(%δ) =  [0.2, 0.5] : %δ = η1 [0.3, 0.6] : %δ = η3 . int. j. anal. appl. (2022), 20:50 7 then it is observed that, ξ is an invf strong β-filter on γ. theorem 3.7. every invf strong β-filter is also an invf β-subalgebra. it is not necessary for the converse part of the theorem to be correct. example 3.4. for a β-algebra γ = {0,η1,η2,η3} with two binary operations + and − and constant 0 defined on γ, we have a cayley’s table + 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η3 0 η1 η1 0 η1 η3 η3 η3 η1 η3 η3 − 0 η1 η1 η3 0 0 0 0 0 η1 η1 η1 η1 η1 η1 η1 η1 η1 η1 η3 η3 η3 η3 η3 then (γ, +,−, 0) is a β-algebra. then ξ = {η1,η3} is β-filter on γ. ξ is an invf β-subalgebra, define by the membership function �ξ(%δ) =  [0.4, 0.6] : %δ = η3 [0.3, 0.5] : %δ = η1 . then it is observed that, ξ is not an invf strong β-filter on γ since �ξ(η1 4η3) 6= �ξ(η1 5η3). theorem 3.8. if σ is an invf strong β-filter of γ, then �ξ(%δ 4 ỳδ) ≥ �ξ(ỳδ) where ỳδ ≤ %δ. theorem 3.9. consider be an onto β-algebra homomorphism f from γ to υ . if ξ2 is a invf strong β-filter of υ ,then its inverse image f−1(ξ2) is again an invf strong β-filter on γ. 4. invinf β-filters on β-algebras the concept of interval valued intuitionstic fuzzy (invinf) β-filters on a β-subalgebra is introduced in this section, which starts with the definition. definition 4.1. an invinf β-subalgebra of a β-algebra γ is called as an invinf β-filter on γ, if (1) �ξ(%δ 4 ỳδ) ≥ rmin{�ξ(%δ),�ξ(%δ + ỳδ)} and φξ(%δ 4 ỳδ) ≤ rmax{φξ(%δ),φξ(%δ + ỳδ)} (2) �ξ(%δ 5 ỳδ) ≥ rmin{�ξ(%δ),�ξ(%δ − ỳδ)} and φξ(%δ 5 ỳδ) ≤ rmax{φξ(%δ),φξ(%δ − ỳδ)} (3) �ξ(ỳ) ≥ �ξ(%δ) and φξ(ỳδ) ≤ φξ(%δ) if %δ ≤ ỳδ for all %δ, ỳδ ∈ γ. 8 int. j. anal. appl. (2022), 20:50 example 4.1. consider a β-algebra γ = {0,ρ1,ρ2,ρ3} with two binary operations + and − and a constant 0 defined on γ with the cayley’s table: + 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 0 ρ3 ρ1 ρ2 ρ2 0 ρ2 ρ3 ρ3 ρ3 ρ1 ρ3 ρ3 − 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 ρ2 ρ2 ρ2 ρ2 ρ3 ρ3 ρ3 ρ3 ρ3 now, ξ = {ρ2,ρ3} is a β-filter on γ. defining the membership and non membership function of an invinf β-subalgebra ξ as �ξ(%δ) =  [0.4, 0.6] : %δ = 0,ρ3 [0.3, 0.5] : %δ = ρ1,ρ2 . and φξ(%δ) =  [0.3, 0.4] : %δ = 0,ρ3 [0.4, 0.6] : %δ = ρ1,ρ2 . therefore, ξ is an invinf β-filter on γ. example 4.2. consider a β-algebra γ = {0,ρ1,ρ2,ρ3} with two binary operations + and − and a constant 0 defined on γ with the cayley’s table: + 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 0 ρ1 ρ2 ρ3 ρ3 ρ3 ρ1 ρ2 ρ3 − 0 ρ1 ρ2 ρ3 0 0 0 0 0 ρ1 ρ1 ρ1 ρ1 ρ1 ρ2 ρ2 ρ2 ρ2 ρ2 ρ3 ρ3 ρ3 ρ3 ρ3 now, ξ = {ρ1,ρ2,ρ3} is a β-filter on γ. defining the membership and non membership function of an invinf β-subalgebra ξ as �ξ(%δ) =   [0.2, 0.6] : %δ = 0 [0.4, 0.5] : %δ = ρ1,ρ3 [0.3, 0.7] : %δ = ρ2 and φξ(%δ) =   [0.1, 0.6] : %δ = 0 [0.2, 0.7] : %δ = ρ1 [0.4, 0.5] : %δ = ρ2,ρ3 . this shows that, ξ is not an invinf β-filter on γ because �ξ(ρ3) ≥ �ξ(ρ2) ⇒ [0.4, 0.5] � [0.3, 0.7]. int. j. anal. appl. (2022), 20:50 9 lemma 4.1. if ξ1 and ξ2 be any two invinf β-filters on γ, then ξ1 ∩ ξ2 is also an invinf β-filter of γ. proof. for %δ, ỳδ ∈ γ σξ1∩ξ2 (%δ 4 ỳδ) = rmin{�ξ1 (%δ 4 ỳδ),�ξ2 (%δ 4 ỳδ)} ≥ rmin{rmin{�ξ1 (%δ),�ξ1 (%δ + ỳδ)}, rmin{�ξ2 (%δ),�ξ2 (%δ + ỳδ)}} ≥ rmin{rmin{�ξ1 (%δ),�ξ2 (%δ)}, rmin{�ξ1 (%δ + ỳδ),�ξ2 (%δ + ỳδ)}} = rmin{�ξ1∩ξ2 (%δ),�ξ1∩ξ2 (%δ + ỳδ)}. also, φξ1∩ξ2 (%δ 4 ỳδ) ≤ rmax{φξ1∩ξ2 (%δ),φξ1∩ξ2 (%δ + ỳδ)}. hence, ξ1 ∩ ξ2 is also an invinf β-filter of γ. � lemma 4.2. every invinf β-filter is again an invinf β-subalgebra. proof. the definition of the invinf β-filter leads to this proof. � in general, the converse of the preceding lemma does not seems to be true, as shown by the following example (i.e. every invinf β-subalgebra need not be an invinf β-filter). example 4.3. let γ = {0,ω1,ω2,ω3} be a β-algebra with constant 0 and the cayley’s table: + 0 ω1 ω2 ω3 0 0 0 0 0 ω1 ω1 ω1 ω1 ω1 ω2 ω1 ω1 ω2 0 ω3 ω3 ω3 ω1 ω1 − 0 ω1 ω2 ω3 0 0 0 0 0 ω1 ω1 ω1 ω1 ω1 ω2 ω2 ω2 ω2 ω2 ω3 ω3 ω3 ω3 ω3 now, ξ = {0,ω3} is a β-filter on γ. defining the membership and non membership function of an invinf β-subalgebra ξ as �ξ(%δ) =  [0.3, 0.5] : %δ = 0,ω2 [0.2, 0.4] : %δ = ω1,ω3 . and φξ(%δ) =  [0.3, 0.5] : %δ = 0,ω2 [0.4, 0.6] : %δ = ω1,ω3 . however ξ is not an invinf β-filter on γ because �ξ(ω3) ≥ �ξ(ω1) ⇒ [0.2, 0.4] � [0.3, 0.5]. theorem 4.1. if ξ is an invinf β-filter of γ, then �ξ(%δ4 ỳδ) ≥ �ξ(%δ) and φξ(%δ5 ỳδ) ≤ φξ(%δ) where %δ ≤ ỳδ. 10 int. j. anal. appl. (2022), 20:50 proof. assume that ξ is an invinf β-filter of γ. let %δ, ỳδ ∈ γ. then �ξ(%δ 4 ỳδ) = �ξ(%δ + (%δ + ỳδ)) ≥ rmin{�ξ(%δ),�ξ(%δ + ỳδ)} = rmin{�ξ(%δ), rmin{�ξ(%δ),�ξ(ỳδ)}} = rmin{�ξ(%δ),�ξ(%δ)}∵ %δ ≤ ỳδ ⇒ �ξ(ỳδ) ≤ �ξ(%δ) = �ξ(%δ). similarly, φξ(%δ 5 ỳδ) = φξ(%δ − (%δ − ỳδ)) ≤ rmax{φξ(%δ),φξ(%δ − ỳδ)} = rmax{φξ(%δ), rmax{φξ(%δ),φξ(ỳδ)}} = rmax{φξ(%δ),φξ(%δ)}∵ %δ ≤ ỳδ ⇒ φξ(ỳδ) ≤ φξ(%δ) = φξ(%δ). � definition 4.2. let ξ be an invinf β-filter of a β-subalgebra γ. for s,t ∈ d[0, 1], the set ξs,t = {%δ ∈ γ : �ξ(%δ) ≥ s & φξ(%δ) ≤ t} is referred as a level set of invinf β-filter ξ of γ. theorem 4.2. an invinf subset ξ of a β-algebra γ is an invinf β-filter if and only if for any s,t ∈ d[0, 1] the ξs t−invinf level subset ξs t = {%δ ∈ γ : �ξ(%δ) ≥ s & φξ(%δ) ≤ t} is either a β-filter or ξs,t 6= ∅. proof. consider an invinf level subset of ξ in γ, ξs,t 6= ∅. for any %δ, ỳδ ∈ ξs,t , �ξ(%δ) ≥ s & �ξ(%δ) ≥ s. now �ξ(%δ 4 ỳ) = �ξ(%δ + (%δ + ỳδ)) ≥ rmin{�ξ(%),�ξ(%δ + ỳδ)} = rmin{�ξ(%δ), rmin{�ξ(%δ),�ξ(ỳδ)}} = rmin{s,rmin{s,s}} = s. this implies that %δ 4 ỳδ ∈ ξs t. similarly, �ξ(%δ 5 ỳδ) = rmin{�ξ(%δ),�ξ(%δ − ỳδ)}. analogously, int. j. anal. appl. (2022), 20:50 11 φξ(%δ 4 ỳ) = φξ(%δ + (%δ + ỳδ)) ≤ rmax{φξ(%δ),φξ(%δ + ỳδ)} = rmax{φξ(%δ), rmax{φξ(%δ),φξ(ỳδ)}} = rmax{t,rmax{t,t}} = t. similarly, φξ(%δ5 ỳδ). then φξ(%δ4 ỳδ) ∈ ξs t & φξ(%δ5 ỳ) ∈ ξs t. so, (%δ4 ỳδ) ∈ ξs t & (%δ5 ỳδ) ∈ ξs t. hence ξs t is a β-filter of γ. on the other hand, assume that ξs t is a β-filter of γ. for all %δ, ỳδ ∈ x,%δ4ỳδ and %δ5ỳδ ∈ ξs t. thus �ξ(%δ4 ỳδ) ≥ s and ξ(%δ5 ỳδ) ≥ s. take s = rmin{�ξ(%δ),�ξ(%δ + ỳδ)} for any %δ, ỳδ ∈ x. we have �ξ(%δ4 ỳδ) = �ξ(%δ + (%δ + ỳδ)) ≥ s = rmin{�ξ(%δ),�ξ(%δ + ỳδ)}. similarly, �ξ(%δ5 ỳδ). analogously, for the non membership function. this proves ξ is an invinf β-filter. � theorem 4.3. consider an onto β-algebra homomorphism f from γ to υ . if ξ2 is an invinf β-filter of υ , then its inverse image f−1(ξ2) is also an invinf β-filter on γ. proof. suppose that ξ2 is an invinf β-filter of υ. for any %δ, ỳδ ∈ γ, f−1(�ξ2 (%δ 4 ỳδ)) = f −1(�ξ2 (%δ + (%δ + ỳδ))) = �ξ2 (f (%δ + (%δ + ỳδ))) = �ξ2 (f (%δ) + f (%δ + ỳδ)) ≥ rmin{�ξ2 (f (%δ)),�ξ2 (f (%δ + ỳδ))} = rmin{f−1(�ξ2 (%δ)), f −1(�ξ2 (%δ + ỳδ))}. also, f−1(�ξ2 (%δ 5 ỳδ)) ≥ rmin{f −1(�ξ2 (%δ)), f −1(�ξ2 (%δ − ỳδ))}. analogously, f−1(φξ2 (%δ 4 ỳδ)) = f −1(φξ2 (%δ + (%δ + ỳδ))) = φξ2 (f (%δ + (%δ + ỳδ))) = φξ2 (f (%δ) + f (%δ + ỳδ)) ≤ rmax{φξ2 (f (%δ)),φξ2 (f (%δ + ỳδ))} = rmax{f−1(φξ2 (%δ)), f −1(φξ2 (%δ + ỳδ))}. similarly, f−1(φξ2 (%δ 5 ỳδ)) ≤ rmax{f −1(φξ2 (%δ)), f −1(φξ2 (%δ − ỳδ))}. let %δ, ỳδ ∈ γ, so that %δ ≥ ỳδ. consequently, ξ2 is an invinf β-filter, �ξ2 (f (ỳδ)) ≥ �ξ2 (f (%δ)) = f −1(�ξ2 (%δ)) such that f−1(�ξ2 (ỳδ) ≥ f −1(�ξ2 (%δ)) and φξ2 (f (ỳδ)) ≤ φξ2 (f (%δ)) = f −1(φξ2 (%δ)) such that f −1(φξ2 (ỳδ) ≥ f−1(φξ2 (%δ)). this shows that f −1 is an invinf β-filter on γ. � 12 int. j. anal. appl. (2022), 20:50 5. conclusion in this work, we investigated interval valued filters, the product on interval valued fuzzy βfilters on β-algebras, interval valued fuzzy strong βfilters on β-algebras, and interval valued intuitionistic fuzzy β-filters on β-algebras, as well as their associated outcomes. furthermore, it can be extended to other algebraic structures in the future research works. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. al-masarwah, h. alshehri, algebraic perspective of cubic multi-polar structures on bck/bci-algebras, mathematics. 10 (2022), 1475. https://doi.org/10.3390/math10091475. [2] r. biswas, rosenfeld’s fuzzy subgroups with interval valued membership functions, fuzzy sets syst. 63 (1994), 87-90. https://doi.org/10.1016/0165-0114(94)90148-1. [3] r.a. borzooei, h.s. kim, y.b. jun, s.s. ahn, on multipolar intuitionistic fuzzy b-algebras, mathematics. 8 (2020), 907. https://doi.org/10.3390/math8060907. [4] s. ghorbani, intuitionistic fuzzy filters of residuated lattices, new math. nat. comput. 7 (2011), 499-513. https://doi.org/10.1142/s1793005711002049. [5] p. hemavathi, p. muralikrishna, k. palanivel, a note on interval valued fuzzy β-subalgebras, glob. j. pure appl. math. 11 (2015), 2553-2560. [6] p. hemavathi, p. muralikrishna, k. palanivel, on interval valued intuitionistic fuzzy β-subalgebras, afrika math. 29 (2018), 249-262. https://doi.org/10.1007/s13370-017-0539-z. [7] p. hemavathi, p. muralikrishna, k. palanivel, r. chinram, conceptual interpretation of interval valued t-normed fuzzy β-subalgebra, songklanakarin j. sci. technol. 44 (2022), 339-347. https://doi.org/10.14456/sjst-psu. 2022.48. [8] s.m. hong, y.b. jun, s.j. kim, g.i. kim, fuzzy bci-subalgebras with interval valued membership functions, int. j. math. math. sci. 25 (2001), 135-143. https://doi.org/10.1155/s0161171201005087. [9] c.s. hoo, filters and ideals in bci-algebras, math. japan. 36 (1991), 987-997. https://cir.nii.ac.jp/crid/ 1570009749732647040. [10] p. kaliyaperumal, a. das, a mathematical model for nonlinear optimization which attempts membership functions to address the uncertainties, mathematics 10 (2022), article number 1743 (20 pages). https: //doi.org/10.3390/math10101743. [11] g. muhiuddin, m. al-tahan, a. mahboob, s. hoskova-mayerova, s. al-kaseasbeh, linear diophantine fuzzy set theory applied to bck/bci-algebras, mathematics 10 (2022), article number 2138 (11 pages). https: //doi.org/10.3390/math10122138. [12] j. neggers, k. h. sik, on β-algebras, math. solvaca. 52 (2002), 517-530. http://dml.cz/dmlcz/131570. [13] y.b. jun, s.z. song, on fuzzy implicative filters of lattice implication algebras, j. fuzzy math. 10 (2002), 893-900. [14] y. b. jun, fuzzy positive implicative and fuzzy associative filters of lattice implication algebras, fuzzy sets syst. 121 (2001), 353-357. https://doi.org/10.1016/s0165-0114(00)00030-0. [15] y.b. jun, s.z. song, foldness of bipolar fuzzy sets and its application in bck/bci-algebras, mathematics. 7 (2019), 1036. https://doi.org/10.3390/math7111036. https://doi.org/10.3390/math10091475 https://doi.org/10.1016/0165-0114(94)90148-1 https://doi.org/10.3390/math8060907 https://doi.org/10.1142/s1793005711002049 https://doi.org/10.1007/s13370-017-0539-z https://doi.org/10.14456/sjst-psu.2022.48 https://doi.org/10.14456/sjst-psu.2022.48 https://doi.org/10.1155/s0161171201005087 https://cir.nii.ac.jp/crid/1570009749732647040 https://cir.nii.ac.jp/crid/1570009749732647040 https://doi.org/10.3390/math10101743 https://doi.org/10.3390/math10101743 https://doi.org/10.3390/math10122138 https://doi.org/10.3390/math10122138 http://dml.cz/dmlcz/131570 https://doi.org/10.1016/s0165-0114(00)00030-0 https://doi.org/10.3390/math7111036 int. j. anal. appl. (2022), 20:50 13 [16] y.j. seo, h.s. kim, y.b. jun, s.s. ahn, multipolar intuitionistic fuzzy hyper bck-ideals in hyper bck-algebras, mathematics. 8 (2020), 1373. https://doi.org/10.3390/math8081373. [17] k. sujatha, m. chandramouleeswaran, p. muralikrishna, fuzzy filters on β-algebras, int. j. math. arch. 6 (2015), 162-167. [18] k. sujatha, m. chandramouleeswaran, p. muralikrishna, intuitionstic fuzzy filters on β-algebras, int. j. math. sci. eng. appl. 9 (2015), 117-123. [19] m.m. takallo, s.s. ahn, r.a. borzooei, y.b. jun, multipolar fuzzy p-ideals of bci-algebras, mathematics. 7 (2019), 1094. https://doi.org/10.3390/math7111094. [20] w.t. xu, y. xu, x.d. pan, intuitionistic fuzzy implicative filter in lattice implication algebras, j. jiangnan univ. (nat. sci. ed.) 6 (2009), 736-739. [21] l.a. zadeh, the concept of a linguistic variable and its application to approximate reasoning i, inform. sci. 8 (1975), 199-249. https://doi.org/10.1016/0020-0255(75)90036-5. https://doi.org/10.3390/math8081373 https://doi.org/10.3390/math7111094 https://doi.org/10.1016/0020-0255(75)90036-5 1. introduction 2. preliminares 3. interval valued fuzzy -filters 3.1. products on invf -filters on -algebras 3.2. invf strong -filters 4. invinf -filters on -algebras 5. conclusion references international journal of analysis and applications volume 17, number 1 (2019), 47-63 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-47 nonlinear sequential riemann-liouville and caputo fractional differential equations with nonlocal and integral boundary conditions suphawat asawasamrit1, nawapol phuangthong1, sotiris k. ntouyas2,3 and jessada tariboon1,∗ 1intelligent and nonlinear dynamic innovations research center, department of mathematics, faculty of applied science, king mongkut’s university of technology north bangkok, bangkok 10800, thailand 2department of mathematics, university of ioannina, 451 10 ioannina, greece 3nonlinear analysis and applied mathematics (naam)-research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia ∗corresponding author: jessada.t@sci.kmutnb.ac.th abstract. in this paper, we discuss the existence and uniqueness of solutions for a new class of sequential fractional differential equations of riemann-liouville and caputo types with nonlocal integral boundary conditions, by using standard fixed point theorems. we also demonstrate the application of the obtained results with the aid of examples. received 2018-09-10; accepted 2018-10-24; published 2019-01-04. 2010 mathematics subject classification. 26a33, 34a08, 34b15. key words and phrases. fractional derivatives; fractional integral; boundary value problems; existence; uniqueness; fixed point theorems. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 47 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-47 int. j. anal. appl. 17 (1) (2019) 48 1. introduction fractional differential equations have gained considerable importance due to their widespread applications in various disciplines of social and natural sciences, and engineering. in recent years, there has been a significant development in fractional calculus and fractional differential equations, for instance, see the monographs by kilbas et al. [12], lakshmikantham et al. [14], miller and ross [15], podlubny [16], samko et al. [18], diethelm [9], ahmad et al. [7] and the papers [1, 4–6, 8, 10, 17, 20, 21]. recently in [2] the authors studied a class of nonlinear differential equations with multiple fractional derivatives and caputo type integro-differential boundary conditions  dα[dβx(t) −g(t,x(t))] = f(t,x(t)), t ∈ j := [0,t], x(0) = 0, (dγx)(t) = λ(iδx)(t), (1.1) where dχ is caputo fractional derivative of order χ ∈{α,β,γ}, 0 < α,β,γ < 1, iδ is the riemann-liouville fractional integral of order δ > 0, f,g : j × r → r are given functions and λ 6= γ(β + δ + 1) tγ+δγ(β −γ + 1) . the existence of solutions for the problem (1.1) is established by applying leray-schauder nonlinear alternative [11] and krasnoselskii’s fixed point theorem [13]. the uniqueness result for the problem (1.1) is obtained by means of a celebrated fixed point theorem due to banach. in [3] existence criteria are developed for the solutions of caputo-hadamard type fractional neutral differential equations supplemented with dirichlet boundary conditions  dω[dκx(t) −h(t,x(t))] = f(t,x(t)), t ∈ [1,t], t > 1, x(1) = 0, x(t) = 0, (1.2) where dρ denotes the caputo-hadamard fractional derivatives of order ρ ∈ (0, 1) with ρ ∈ {ω,κ} and f,h : [1,t] ×r → r are appropriate functions. very recently in [19], the authors discussed existence and uniqueness of solutions for two sequential caputo-hadamard and hadamard-caputo fractional differential equations subject to separated boundary conditions as   cdp(hdqx)(t) = f(t,x(t)), t ∈ (a,b), α1x(a) + α2( hdqx)(a) = 0, β1x(b) + β2( hdqx)(b) = 0, (1.3) and   hdq(cdpx)(t) = f(t,x(t)), t ∈ (a,b), α1x(a) + α2( cdpx)(a) = 0, β1x(b) + β2( cdpx)(b) = 0, (1.4) where cdp and hdq are the caputo and hadamard fractional derivatives of orders p and q, respectively, 0 < p,q ≤ 1, f : [a,b] ×r → r is a continuous function, a > 0 and αi,βi ∈ r, i = 1, 2. int. j. anal. appl. 17 (1) (2019) 49 motivated by the above papers, we consider in the present paper the following boundary value problem  rldq[cdrx(t) −g(t,x(t))] = f(t,x(t)), 0 < t < t, x(η) = φ(x), ipx(t) = h(x), (1.5) where rldq, cdr are riemann-liouville and caputo fractional derivatives of orders q,r ∈ (0, 1), respectively, ip is the riemann-liouville fractional integral of order p > 0, f,g : j×r → r are given continuous functions and φ,h : c(j,r) → r are two given functionals. the rest of the paper is arranged as follows. in section 2, we establish basic results that lays the foundation for defining a fixed point problem equivalent to the given problem (1.5). the main results, based on banach’s contraction mapping principle, krasnoselskii’s fixed point theorem and nonlinear alternative of leray-schauder type, are obtained in section 3. examples illustrating the obtained results are also included. 2. preliminaries in this section, we recall some basic concepts of fractional calculus [12, 16] and present known results needed in our forthcoming analysis. definition 2.1. the riemann-liouville fractional derivative of order q for a function f : (0,∞) → r is defined by rldqf(t) = 1 γ(n−q) ( d dt )n ∫ t 0+ (t−s)n−q−1 f(s)ds, q > 0, n = [q] + 1, where [q] denotes the integer part of the real number q, provided the right-hand side is pointwise defined on (0,∞). definition 2.2. the riemann-liouville fractional integral of order q for a function f : (0,∞) → r is defined by iqf(t) = 1 γ(q) ∫ t 0+ (t−s)q−1 f(s)ds, q > 0, provided the right-hand side is pointwise defined on (0,∞). definition 2.3. the caputo derivative of fractional order q for a n-times derivative function f : (0,∞) → r is defined as cdqf(t) = 1 γ(n−q) ∫ t 0+ (t−s)n−q−1 ( d ds )n f(s)ds, q > 0, n = [q] + 1. lemma 2.1. if α + β > 0, then the equation (iαiβu)(t) = (iα+βu)(t), t ∈ j is satisfied for u ∈ l1(j,r). lemma 2.2. let β > α. then the equation (dαiβu)(t) = (iβ−αu)(t), t ∈ j is satisfied for u ∈ c(j,r). lemma 2.3. let n = [α] + 1 if α 6∈ n and n = α if α ∈ n. then the following relations hold: (i) for k ∈{0, 1, 2, . . . ,n− 1}, dαtk = 0; int. j. anal. appl. 17 (1) (2019) 50 (ii) if β > n then dαtβ−1 = γ(β) γ(β −α) tβ−α−1; (iii) iαtβ−1 = γ(β) γ(β + α) tβ+α−1. lemma 2.4. let q > 0. then for y ∈ c(0,t) ∩l(0,t) it holds rliq ( rldqy ) (t) = y(t) + c1t q−1 + c2t q−2 + · · · + cntq−n, where ci ∈ r, i = 1, 2, . . . ,n and n− 1 < q < n. lemma 2.5. let q > 0. then for y ∈ c(0,t) ∩l(0,t) it holds rliq ( cdqy ) (t) = y(t) + c0 + c1t + c2t 2 + · · · + cn−1tn−1, where ci ∈ r, i = 0, 1, 2, . . . ,n− 1 and n = [q] + 1. in the following, for simplicity, we use the notation iq for rliq. lemma 2.6. let p > 0, 0 < q,r ≤ 1, with q + r > 1, λ = γ(q) γ(q + r) tp γ(p + 1) ηq+r−1 − γ(q) γ(p + q + r) tp+q+r−1 6= 0, (2.1) and ĝ,y ∈ c(j,r) and two functionals φ,h : c(j,r) → r. the unique solution of the linear problem  rldq[cdrx(t) − ĝ(t)] = y(t), 0 < t < t, x(η) = φ(x), ipx(t) = h(x), (2.2) is given by x(t) = irĝ(t) + iq+ry(t) + tq+r−1 λ γ(q) γ(q + r) [( φ(x) − irĝ(η) − iq+ry(η) ) tp γ(p + 1) − ( h(x) − ip+rĝ(t) − ip+q+ry(t) )] + 1 λ [ γ(q) γ(q + r) ηq+r−1 ( h(x) − ip+rĝ(t) − ip+q+ry(t) ) − ( φ(x) − irĝ(η) − iq+ry(η) ) γ(q) γ(p + q + r) tp+q+r−1 ] . proof. firstly, we apply the riemann-liouville fractional integral of order q to both sides of equation (2.2), and using lemma 2.4, we have cdrx(t) = ĝ(t) + iqy(t) + c1t q−1, (2.3) int. j. anal. appl. 17 (1) (2019) 51 where a constant c1 ∈ r. after that, using riemann-liouville fractional integral of order r to both sides the above equation and applying lemma 2.5, we get x(t) = irĝ(t) + iq+ry(t) + c1 γ(q) γ(q + r) tq+r−1 + c2, (2.4) where a constant c2 ∈ r. observe that the equation (2.4) is well defined as q + r > 1. using nonlocal boundary condition of problem (2.2) to the above equation, we obtain the linear system c1 γ(q) γ(q + 1) ηq+r−1 + c2 = φ(x) − irĝ(η) − iq+ry(η), c1 γ(q) γ(p + q + 1) tp+q+r−1 + c2 tp γ(p + 1) = h(x) − ip+rĝ(t) − ip+q+ry(t). note that the two functionals φ(x) and h(x) are constants. solving the system of linear equations for constants c1, c2, we have c1 = 1 λ [ tp γ(p + 1) ( φ(x) − irĝ(η) − iq+ry(η) ) − ( h(x) − ip+rĝ(t) − ip+q+ry(t) )] , c2 = 1 λ [ γ(q) γ(q + r) ηq+r−1 ( h(x) − ip+rĝ(t) − ip+q+ry(t) ) − γ(q) γ(p + q + r) tp+q+r−1 ( φ(x) − irĝ(η) − iq+ry(η) )] . substituting two constants c1 and c2 into equation (2.4), we obtain the required solution. the converse follows by direct computation. the proof is completed. � 3. main results let j = [0,t] and c = c(j,r) denotes the banach space of all continuous functions from j to r endowed with the norm defined by ‖x‖ = supt∈j |x(t)|. by lemma 2.6, we define an operator a : c →c by (ax)(t) = irg(s,x(s))(t) + iq+rf(s,x(s))(t) + tq+r−1 λ γ(q) γ(q + r) [( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) tp γ(p + 1) − ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) )] (3.1) + 1 λ [ γ(q) γ(q + r) ηq+r−1 ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) ) − ( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) γ(q) γ(p + q + r) tp+q+r−1 ] , with λ 6= 0. it should be noticed that problem (1.5) can be transformed into a fixed point equation x = ax. int. j. anal. appl. 17 (1) (2019) 52 to accomplish of the study, we will use fixed point theorems to prove that the operator a has fixed points. for the sake of convenience, we define four constants by φ1 = [ tr γ(r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηr γ(r + 1) + tp+r γ(p + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+r γ(p + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηr γ(r + 1) tp+q+r−1 )] , φ2 = [ tq+r γ(q + r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηq+r γ(q + r + 1) + tp+q+r γ(p + q + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+q+r γ(p + q + r + 1) ηq+r−1 + + γ(q) γ(p + q + 1) ηq+r γ(q + r + 1) tp+q+r−1 )] , φ3 = [ γ(q) |λ|γ(q + r) tp+q+r−1 γ(p + 1) + γ(q) |λ|γ(p + q + 1) tp+q+r−1 ] , φ4 = [ γ(q) |λ|γ(q + r) tp+q+r−1 + γ(q) |λ|γ(q + 1) ηq+r−1 ] . the first existence and uniqueness result is obtained by using banach contraction mapping principle. theorem 3.1. let g,f : j × r → r, be continuous functions and φ,h : c(j,r) → r be two functionals satisfying the assumption: (h1) there exist positive constants li, i = 1, 2, 3, 4 such that: |g(t,x) −g(t,y)| ≤ l1|x−y|, |f(t,x) −f(t,y)| ≤ l2|x−y|, t ∈ j,x,y ∈ r, |φ(u) −φ(v)| ≤ l3|u−v| and |h(u) −h(v)| ≤ l4|u−v|, u,v ∈ c(j,r). if the inequality ω1 := l1φ1 + l2φ2 + l3φ3 + l4φ4 < 1, (3.2) holds, then the boundary value problem (1.5) has a unique solution on j. proof. by using the banach’s contraction mapping principle, we shall show that a of a fixed point problem, x = ax, has a unique fixed point which is the unique solution of problem (1.5). to prove the embedding property, we set sup t∈[0,t] |g(t, 0)| = m1 < ∞, sup t∈[0,t] |f(t, 0)| = m2 < ∞, |φ(0)| = m3, |h(0)| = m4, and choose r ≥ m1φ1 + m2φ2 + m3φ3 + m4φ4 1 − ω1 . int. j. anal. appl. 17 (1) (2019) 53 now, we show that abr ⊂ br, where br = {x ∈ c : ‖x‖ ≤ r}. for any x ∈ br, and taking into account assumption (h1), we obtain |ax(t)| ≤ sup t∈[0,t] { irg(s,x(s))(t) + iq+rf(s,x(s))(t) + tq+r−1 λ γ(q) γ(q + r) [( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) tp γ(p + 1) − ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) )] + 1 λ [ γ(q) γ(q + r) ηq+r−1 ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) ) − ( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) γ(q) γ(p + q + r) tp+q+r−1 ]} ≤ ir(|g(s,x(s)) −g(s, 0)| + |g(s, 0)|)(t) +iq+r(|f(s,x(s)) −f(s, 0)| + |f(s, 0)|)(t) + tq+r−1 |λ| γ(q) γ(q + r) [ tp γ(p + 1) ( (|φ(x) −φ(0)| + |φ(0)|) (t) +ir ( |g(s,x(s)) −g(s, 0)| + |g(s, 0)| ) (η) +iq+r ( |f(s,x(s)) −f(s, 0)| + |f(s, 0)| ) (η) ) + ( (|h(x) −h(0)| + |h(0)|) (t) +ip+r ( |g(s,x(s)) −g(s, 0)| + |g(s, 0)| ) (t) +ip+q+r ( |f(s,x(s)) −f(s, 0)| + |f(s, 0)| ) (t) )] + 1 |λ| [ γ(q) γ(q + 1) ηq+r−1 ( (|h(x) −h(0)| + |h(0)|) (t) +ip+r ( |g(s,x(s)) −g(s, 0)| + |g(s, 0)| ) (t) +ip+q+r ( |f(s,x(s)) −f(s, 0)| + |f(s, 0)| ) (t) ) + γ(q) γ(p + q + 1) tp+q+r−1 ( (|φ(x) −φ(0)| + |φ(0)|) (t) +ir ( |g(s,x(s)) −g(s, 0)| + |g(s, 0)| ) (η) +iq+r ( |f(s,x(s)) −f(s, 0)| + |f(s, 0)| ) (η) )] ≤ (l1r + m1) tr γ(r + 1) + (l2r + m2) tq+r γ(q + r + 1) + tq+r−1 |λ| γ(q) γ(q + r) [ tp γ(p + 1) ( (l3r + m3) + (l1r + m1) ηr γ(r + 1) + (l2r + m2) ηq+r γ(q + r + 1) ) + ( (l4r + m4) + (l1r + m1) tp+r γ(p + r + 1) + (l2r + m2) tp+q+r γ(p + q + r + 1) )] int. j. anal. appl. 17 (1) (2019) 54 + 1 |λ| [ γ(q) γ(q + 1) ηq+r−1 ( (l4r + m4) + (l1r + m1) tp+r γ(p + r + 1) + (l2r + m2) tp+q+r γ(p + q + r + 1) ) + γ(q) γ(p + q + r) ( (l3r + m3) + (l1r + m1) ηr γ(r + 1) + (l2r + m2) ηq+r γ(q + r + 1) )] ≤ [ tr γ(r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηr γ(r + 1) + tp+r γ(p + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+r γ(p + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηr γ(r + 1) tp+q+r−1 )] ×(l1r + m1) + [ tq+r γ(q + r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηq+r γ(q + r + 1) + tp+q+r γ(p + q + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+q+r γ(p + q + r + 1) ηq+r−1 + + γ(q) γ(p + q + 1) ηq+r γ(q + r + 1) tp+q+r−1 )] (l2r + m2) + [ γ(q) |λ|γ(q + r) tp+q+r−1 γ(p + 1) + γ(q) |λ|γ(p + q + 1) tp+q+r−1 ] (l3r + m3) + [ γ(q) |λ|γ(q + r) tp+q+r−1 + γ(q) |λ|γ(q + 1) ηq+r−1 ] (l4r + m4) = φ1(l1r + m1) + φ2(l2r + m2) + φ3(l3r + m3) + φ4(l4r + m4) = ω1r + (m1φ1 + m2φ2 + m3φ3 + m4φ4) ≤ r. this mean that ‖ax‖≤ r which yields abr ⊂ br. for all t ∈ [0,t] and for each x,y ∈c, we have |ax(t) −ay(t)| ≤ ir(|g(s,x(s)) −g(s,y(s))|)(t) + iq+r(|f(s,x(s)) −f(s,y(s))|)(t) + tq+r−1 |λ| γ(q) γ(q + r) [ tp γ(p + 1) ( (|φ(x) −φ(y)|) (t) +ir (|g(s,x(s)) −g(s,y(s))|) (η) + iq+r (|f(s,x(s)) −f(s,y(s))|) (η) ) + ( (|h(x) −h(y)|) (t) + ip+r (|g(s,x(s)) −g(s,y(s))|) (t) +ip+q+r (|f(s,x(s)) −f(s,y(s))|) (t) )] + 1 |λ| [ γ(q) γ(q + 1) ηq+r−1 ( (|h(x) −h(y)|) (t) +ip+r (|g(s,x(s)) −g(s,y(s))|) (t) + ip+q+r (|f(s,x(s)) −f(s,y(s))|) (t) ) + γ(q) γ(p + q + 1) tp+q+r−1 ( (|φ(x) −φ(y)|) (t) + ir (|g(s,x(s)) −g(s,x(s))|) (η) int. j. anal. appl. 17 (1) (2019) 55 +iq+r (|f(s,x(s)) −f(s,y(s))|) (η) )] ≤ (l1|x−y|) tr γ(r + 1) + (l2|x−y|) tq+r γ(q + r + 1) + tq+r−1 |λ| γ(q) γ(q + r) [ tp γ(p + 1) ( (l3|x−y|) + (l1|x−y|) ηr γ(r + 1) + (l2|x−y|) ηq+r γ(q + r + 1) ) + ( (l4|x−y|) + (l1|x−y|) tp+r γ(p + r + 1) + (l2|x−y|) tp+q+r γ(p + q + r + 1) )] + 1 |λ| [ γ(q) γ(q + 1) ηq+r−1 ( (l4|x−y|) + (l1|x−y|) tp+r γ(p + r + 1) + (l2|x−y|) tp+q+r γ(p + q + r + 1) ) + γ(q) γ(p + q + r) ( (l3|x−y|) + (l1|x−y|) ηr γ(r + 1) + (l2|x−y|) ηq+r γ(q + r + 1) )] ≤ [ tr γ(r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηr γ(r + 1) + tp+r γ(p + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+r γ(p + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηr γ(r + 1) tp+q+r−1 )] ×(l1|x−y|) + [ tq+r γ(q + r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηq+r γ(q + r + 1) + tp+q+r γ(p + q + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+q+r γ(p + q + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηq+r γ(q + r + 1) tp+q+r−1 )] l2|x−y| + [ γ(q) |λ|γ(q + r) tp+q+r−1 γ(p + 1) + γ(q) |λ|γ(p + q + 1) tp+q+r−1 ] l3|x−y| + [ γ(q) |λ|γ(q + r) tp+q+r−1 + γ(q) |λ|γ(q + 1) ηq+r−1 ] l4|x−y| = φ1l1|x−y|) + φ2l2|x−y|) + φ3l3|x−y|) + φ4l4|x−y|) = ω1|x−y|. the above result implies that ‖ax−ay‖ ≤ ω1‖x−y‖. as ω1 < 1, therefore a is a contraction operator. hence, by the banach contraction mapping principle, we obtain that a has a unique fixed point which is the unique solution of the problem (1.5). the proof is completed. � example 3.1. consider the following nonlinear sequential riemann-liouville and caputo fractional differential equation with nonlocal integral boundary conditions rld 4 5 ( cd 1 2 x(t) − et (t2 + 40) + 20 |x(t)| |x(t)| + 1 ) int. j. anal. appl. 17 (1) (2019) 56 = cos2(2πt) (t + 10) 2 + 50 · ( x2(t) + 2|x(t)| |x(t)| + 1 ) + et, 0 < t < 3, (3.3) x ( 1 2 ) = x2(2) + 2|x(2)| 60(|x(2)| + 1) + 30, i 2 3 x(3) = |x(1)| 25(|x(1)| + 1) . setting constants q = 4/5, r = 1/2, p = 2/3, η = 1/2, t = 3, then we can compute constants as φ1 = 12.42305820, φ2 = 14.24066077, φ3 = 6.845515569, φ4 = 4.835810257. setting functions g(t,x) = et (t2 + 40) + 20 |x| |x| + 1 , f(t,x) = cos2(2πt) (t + 10) 2 + 50 · ( x2 + 2|x| |x| + 1 ) + et φ(x) = x2 + 2|x| 60(|x| + 1) + 30. h(x) = |x| 25(|x| + 1) , so we get |g(t,x) −g(t,y)| ≤ (1/60)|x−y|, |f(t,x) −f(t,y)| ≤ (1/75)|x−y|, |φ(x) −φ(y)| ≤ (1/30)|x−y| and |h(x) − h(y)| ≤ (1/25)|x − y|. therefore the condition (h1) is satisfied with l1 = 1/60, l2 = 1/75, l3 = 1/30 and l4 = 1/25. we can show that ω1 = 0.8185425995 < 1. hence, by theorem 3.1, the boundary value problem (3.3) has a unique solution on [0, 3]. the second existence result will be proved by using the following krasnoselskii’s fixed point theorem. lemma 3.1. (krasnoselskii’s fixed point theorem) [13]. let m be a closed, bounded, convex and nonempty subset of a banach space x. let a, b be the operators such that (a) ax+by ∈ m whenever x,y ∈ m; (b) a is compact and continuous; (c) b is a contraction mapping. then there exists z ∈ m such that z = az +bz. theorem 3.2. assume that g,f : j×r → r, are continuous functions and two functionals φ,h : c(j×r) → r satisfying the assumption (h1). in addition we suppose that: (h2) |g(t,x)| ≤ δ1(t), |f(t,x)| ≤ δ2(t), ∀(t,x) ∈ j ×r and δ1,δ2 ∈ c(j,r+), |φ(u)| ≤ δ3, |h(u)| ≤ δ4, ∀u ∈ c(j ×r) and δ3,δ4 ∈ r+. if the inequality ω2 := l1 ( φ1 − tq γ(r + 1) ) + l2 ( φ2 − tq+r γ(q + r + 1) ) + l3φ3 + l4φ4 < 1, (3.4) then the boundary value problem (1.5) has at least one solution on j. proof. to applied lemma 3.1, we let supt∈j |δ1(t)| = ‖δ1‖, supt∈j |δ2(t)| = ‖δ2‖, and a positive constant r as r ≥‖δ1‖φ1 + ‖δ2‖φ2 + δ3φ3 + δ4φ4. int. j. anal. appl. 17 (1) (2019) 57 define a ball br by br = {x ∈ c : ‖x‖ ≤ r} which is closed, bounded, convex and nonempty subset of a banach space c. in addition, we define the operators p and q on br as (px)(t) = irg(s,x(s))(t) + iq+rf(s,x(s))(t), t ∈ [0,t], (qx)(t) = tq+r−1 λ γ(q) γ(q + r) [( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) tp γ(p + 1) − ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) )] + 1 λ [ γ(q) γ(q + r) ηq+r−1 ( h(x(t)) − ip+rg(s,x(s))(t) − ip+q+rf(s,x(s))(t) ) − ( φ(x(t)) − irg(s,x(s))(η) − iq+rf(s,x(s))(η) ) γ(q) γ(p + q + r) tp+q+r−1 ] , t ∈ [0,t]. obvious that ax = px + qx. to prove that p and q satisfy (a) of lemma 3.1, for x,y ∈ br, we have ‖px + qy‖ ≤ ‖δ1‖ [ tr γ(r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηr γ(r + 1) + tp+r γ(p + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+r γ(p + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηr γ(r + 1) tp+q+r−1 )] +‖δ2‖ [ tq+r γ(q + r + 1) + γ(q) γ(q + r) tq+r−1 |λ| ( tp γ(p + 1) ηq+r γ(q + r + 1) + tp+q+r γ(p + q + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+q+r γ(p + q + r + 1) ηq+r−1 + + γ(q) γ(p + q + 1) ηq+r γ(q + r + 1) tp+q+r−1 )] + δ3 [ γ(q) |λ|γ(q + r) tp+q+r−1 γ(p + 1) + γ(q) |λ|γ(p + q + 1) tp+q+r−1 ] + δ4 [ γ(q) |λ|γ(q + r) tp+q+r−1 + γ(q) |λ|γ(q + 1) ηq+r−1 ] = ‖δ1‖φ1 + ‖δ2‖φ2 + δ3φ3 + δ4φ4 ≤ r. this shows that px + qy ∈ br. the operator q satisfies the condition (c) of lemma 3.1 from assumption (h1) together with (3.4). the final step is to show that the operator p is satisfied condition (b) of lemma 3.1. since the functions f,g are continuous, we get that the operator p is continuous. now we will show that the operator p is compact. int. j. anal. appl. 17 (1) (2019) 58 for any x ∈ br, we obtain ‖px‖≤‖δ1‖ tr γ(q + 1) + ‖δ2‖ tq+r γ(q + r + 1) . therefore, the set p(br) is uniformly bounded. let us let sup(t,x)∈j×br |g(t,x)| = g < ∞ and sup(t,x)∈j×br |f(t,x)| = f < ∞. let t1, t2 ∈ j with t1 < t2. then we have |(px)(t2) − (px)(t1)| ≤ g γ(r) ∣∣∣∣ ∫ t1 0 [ (t2 −s)r−1 − (t1 −s)r−1 ] ds ∣∣∣∣ + gγ(r) ∣∣∣∣ ∫ t2 t1 (t2 −s)r−1ds ∣∣∣∣ + f γ(q + r) ∣∣∣∣ ∫ t1 0 [ (t2 −s)q+r−1 − (t1 −s)q+r−1 ] ds ∣∣∣∣ + f γ(q + r) ∣∣∣∣ ∫ t2 t1 (t2 −s)q+r−1ds ∣∣∣∣ ≤ g γ(r + 1) [|tr2 − t r 2| + 2(t2 − t1) r] + f γ(q + r + 1) [∣∣tq+r2 − tq+r1 ∣∣ + 2(t2 − t1)r] , which is independent of x and tends to zero as t1 → t2. thus, the set p(br) is equicontinuous. hence, by the arzelá-ascoli theorem, the set p(br) is relatively compact. therefore, the operator p is compact which is satisfied condition (b) of lemma 3.1. thus all the assumptions of lemma 3.1 are satisfied. so the boundary value problem (1.5) has at least one solution on j. the proof is completed. � remark 3.1. in the above theorem we can interchange the roles of the operators p and q to obtain a second result replacing (3.4) by the following condition: ω3 := l1 tr γ(r + 1) + l2 tq+r γ(q + r + 1) < 1. (3.5) remark 3.2. since ω2 < ω1 and ω3 < ω1, the condition (3.2) can be relaxed by (3.4) and (3.5). however, the conclusion of both theorems has different mentions between uniqueness and multiplicity of solutions. example 3.2. consider the following nonlinear sequential riemann-liouville and caputo fractional differential equation with nonlocal integral boundary conditions rld 1 2 ( cd 2 3 x(t) − e2t (t2 + 100)2 + 19300 · |x(t)| |x(t)| + 1 ) = cos2(2πt) t2 + 28000 · ( |x(t)| |x(t)| + 1 ) + cos(πt), 0 < t < 4, (3.6) x(2) = |x(3)| 9990(|x(3)| + 1) , i 2 3 x(4) = |x(2)| 9840(|x(2)| + 1) + 35. setting constants q = 1/2, r = 2/3, p = 2/3, η = 2, t = 4, then we can fine that φ1 = 6717.422119, φ2 = 6652.469591, φ3 = 3119.677669, φ4 = 2175.349828. next we set the following functions g(t,x) = e2t (t2 + 100)2 + 19300 · |x| |x| + 1 , int. j. anal. appl. 17 (1) (2019) 59 f(t,x) = cos2(2πt) t2 + 28000 · ( |x| |x| + 1 ) + cos(πt) φ(x) = |x| 9990(|x| + 1) , h(x) = |x| 9840(|x| + 1) + 35. since |g(t,x)−g(t,y)| ≤ (1/29300)|x−y|, |f(t,x)−f(t,y)| ≤ (1/28000)|x−y|, |φ(x)−φ(t,y)| ≤ (1/9990)|x−y| and |h(x) −h(y)| ≤ (1/9840)|x−y|, the condition (h1) fulfilled. it is obvious that |g(t,x)| ≤ e2t 29300 , |f(t,x)| ≤ 1 + cos(πt), |φ(x)| ≤ 1, h(x) ≤ 36. then the condition (h2) is satisfied. in addition we have ω2 = 0.999918 < 1. hence, by theorem 3.2, the boundary value problem (3.6) has at least one solution on [0, 4]. remark 3.3. the problem (3.6) can not be applied by theorem 3.1 since ω1 = 1.000204 > 1. now, our third existence result is based on leray-schauder’s nonlinear alternative. lemma 3.2. (nonlinear alternative for single-valued maps) [11]. let e be a banach space, c be a closed, convex subset of e, u be an open subset of c and 0 ∈ u. suppose that a : u → c is a continuous, compact (that is, a(u) is a relatively compact subset of c) map. then either (i) a has a fixed point in u, or (ii) there is a u ∈ ∂u (the boundary of u in c) and λ ∈ (0, 1) with u = λa(u). theorem 3.3. assume that g,f : j×r → r are continuous functions and two functionals φ,h : c(j×r) → r. in addition we suppose that: (h3) there exist continuous nondecreasing functions ψ1,ψ2 : [0,∞) → (0,∞) and functions p1,p2 ∈ c(j,r+) such that |g(t,x)| ≤ p1(t)ψ1(‖x‖), |f(t,x)| ≤ p2(t)ψ2(‖x‖) for each (t,x) ∈ j ×r; (h4) there exists a constant n > 0 such that n φ1‖p1‖ψ1(n) + φ2‖p2‖ψ2(n) + φ3|φ(n)| + φ4|h(n)| > 1. then the boundary value problem (1.5) has at least one solution on j. proof. let us define a positive number r and let a ball br = {x ∈c : ‖x‖≤ r} be a closed, convex subset of c. next, we will prove that the operator a, defined by (3.1), maps bounded sets (balls) into bounded sets in c. for any t ∈ j and x ∈ br, we have |ax(t)| int. j. anal. appl. 17 (1) (2019) 60 ≤ ir|g(s,x(s))|(t) + iq+r|f(s,x(s))|(t) + tq+r−1 |λ| γ(q) γ(q + r) [( |φ(x(t))| + ir|g(s,x(s))|(η) + iq+r|f(s,x(s))|(η) ) tp γ(p + 1) + ( |h(x(t))| + ip+r|g(s,x(s))|(t) + ip+q+r|f(s,x(s))|(t) )] + 1 |λ| [ γ(q) γ(q + r) ηq+r−1 ( |h(x(t))| + ip+r|g(s,x(s))|(t) + ip+q+r|f(s,x(s))|(t) ) + ( |φ(x(t))| + ir|g(s,x(s))|(η) + iq+r|f(s,x(s))|(η) ) γ(q) γ(p + q + r) tp+q+r−1 ] ≤ ‖p1‖ψ1(‖x‖) tr γ(r + 1) + ‖p2‖ψ2(‖x‖) tq+r γ(q + r + 1) + tq+r−1 |λ| γ(q) γ(q + r) [ tp γ(p + 1) ( |φ(‖x‖)| + (‖p1‖ψ1(‖x‖)) ηr γ(r + 1) + (‖p2‖ψ2(‖x‖)) ηq+r γ(q + r + 1) ) + ( |h(‖x‖)| + (‖p1‖ψ1(‖x‖)) tp+r γ(p + r + 1) + (‖p2‖ψ2(‖x‖)) tp+q+r γ(p + q + r + 1) )] + 1 |λ| [ γ(q) γ(q + 1) ηq+r−1 ( |h(‖x‖)| + (‖p1‖ψ1(‖x‖)) tp+r γ(p + r + 1) + (‖p2‖ψ2(‖x‖)) tp+q+r γ(p + q + r + 1) ) + γ(q) γ(p + q + r) ( |φ(‖x‖)| + (‖p1‖ψ1(‖x‖)) ηr γ(r + 1) + (‖p2‖ψ2(‖x‖)) ηq+r γ(q + r + 1) )] ≤ [ tr γ(r + 1) + γ(q) γ(q + r) tq+r−1 ||λ|| ( tp γ(p + 1) ηr γ(r + 1) + tp+r γ(p + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+r γ(p + r + 1) ηq+r−1 + γ(q) γ(p + q + 1) ηr γ(r + 1) tp+q+r−1 )] ×(‖p1‖ψ1(‖x‖)) + [ tq+r γ(q + r + 1) + γ(q) γ(q + r) tq+r−1 ||λ|| ( tp γ(p + 1) ηq+r γ(q + r + 1) + tp+q+r γ(p + q + r + 1) ) + 1 |λ| ( γ(q) γ(q + 1) tp+q+r γ(p + q + r + 1) ηq+r−1 + + γ(q) γ(p + q + 1) ηq+r γ(q + r + 1) tp+q+r−1 )] ‖p2‖ψ2(‖x‖) + [ γ(q) |λ|γ(q + r) tp+q+r−1 γ(p + 1) + γ(q) |λ|γ(p + q + 1) tp+q+r−1 ] |φ(‖x‖)| + [ γ(q) |λ|γ(q + r) tp+q+r−1 + γ(q) |λ|γ(q + 1) ηq+r−1 ] |h(‖x‖)| = φ1‖p1‖ψ1(‖x‖) + φ2‖p2‖ψ2(‖x‖) + φ3|φ(‖x‖)| + φ4|h(‖x‖)| ≤ φ1‖p1‖ψ1(r) + φ2‖p2‖ψ2(r) + φ3|φ(r)| + φ4|h(r)|. therefore, from the above result, we conclude that ‖ax‖≤ φ1‖p1‖ψ1(r) + φ2‖p2‖ψ2(r) + φ3|φ(r)| + φ4|h(r)|. int. j. anal. appl. 17 (1) (2019) 61 then the set a(br) is uniformly bounded. next, we show that the operator a maps bounded sets into equicontinuous sets of c. let ν1,ν2 ∈ j with ν1 < ν2 and for any x ∈ br, then we have |(ax)(ν2) − (ax)(ν1)| ≤ ir|g(s,x(s))(ν2) −g(s,x(s))(ν1)| + iq+r(|f(s,x(s))(ν2) −f(s,x(s))(ν1)|) + ∣∣∣νq+r−12 −νq+r−11 ∣∣∣ |λ| γ(q) γ(q + r) [ (|φ(x(ν2)) −φ(x(ν1))) tp γ(p + 1) + (|h(x(ν2)) −h(x(ν1))|) ] + 1 |λ| [ γ(q) γ(q + r) ηq+r−1 (|h(x(ν2)) −h(x(ν1))|) + (|φ(x(ν2)) −φ(x(ν1))|) γ(q) γ(p + q + r) tp+q+r−1 ] ≤ ‖p1‖ψ1(r) γ(r + 1) [|tr2 − t r 2| + 2(t2 − t1) r] + ‖p2‖ψ2(r) γ(q + r + 1) [∣∣tq+r2 − tq+r1 ∣∣ + 2(t2 − t1)r] + ∣∣∣νq+r−12 −νq+r−11 ∣∣∣ |λ| γ(q) γ(q + r) [ (|φ(x(ν2)) −φ(x(ν1))|) tp γ(p + 1) + (|h(x(ν2)) −h(x(ν1))|) ] + 1 |λ| [ γ(q) γ(q + r) ηq+r−1 (|h(x(ν2)) −h(x(ν1))|) + (|φ(x(ν2)) −φ(x(ν1))|) γ(q) γ(p + q + r) tp+q+r−1 ] . obviously the right hand side of the above inequality tends to zero independently of x ∈ br as ν1 → ν2, which implies that the set a(br) is equicontinuous. therefore it follows by the arzelá-ascoli theorem that the set a(br) is relative compact. then the operator a is compact. let x(t) be a solution of problem (1.5). then, for t ∈ j and x ∈ br, we have ‖x‖≤ φ1‖p1‖ψ1(‖x‖) + φ2‖p2‖ψ2(‖x‖) + φ3|φ(‖x‖)| + φ4|h(‖x‖)|. consequently, we have ‖x‖ φ1‖p1‖ψ1(‖x‖) + φ2‖p2‖ψ2(‖x‖) + φ3|φ(‖x‖)| + φ4|h(‖x‖)| ≤ 1. let us define a subset of br as u = {x ∈c : ‖x‖ < n}, (3.7) where n is satisfied the condition (h4). note that the operator a : u → c is continuous and completely continuous. from the choice of u, there is no x ∈ ∂u such that x = θax for some θ ∈ (0, 1). then, by nonlinear alternative of leray-schauder type, lemma 3.2, we get that the operator a has a fixed point in u, which is a solution of the boundary value problem (1.5). this completes the proof. � int. j. anal. appl. 17 (1) (2019) 62 example 3.3. consider the following nonlinear sequential riemann-liouville and caputo fractional differential equation with nonlocal integral boundary conditions rld 4 5 ( cd 2 5 x(t) − 2e−t cos2 t 1000 ( |x|5 x4 + 1 + 1 )) = 2 sin4 t 1000 ( x8 |x|7 + 1 + 1 ) , 0 < t < 5, (3.8) x(2) = x(4) 500 , i 3 5 x(5) = x(3) 200 . setting constants q = 4/5, r = 2/5, p = 3/5, η = 2, t = 5, then we get φ1 = 72.200440, φ2 = 129.62057, φ3 = 34.389063 and φ4 = 2.841029. let the following functions g(t,x) = 2e−t cos2 t 1000 ( |x|5 x4 + 1 + 1 ) , f(t,x) = 2 sin4 t 1000 ( x8 |x|7 + 1 + 1 ) , φ(x) = x 500 , h(x) = x 200 . it follows that |g(t,x)| ≤ 2 cos2 t ( |x| + 1 1000 ) and |f(t,x)| ≤ 2 sin4 t ( |x| + 1 1000 ) . hence, we choose p1(t) = 2 cos 2 t, ψ1(|x|) = (|x| + 1)/(1000), p2(t) = 2 sin4 t, ψ2(|x|) = (|x| + 1)/(1000). then there exists a constant n > 0.97645553 satisfying inequality n (72.200440)(2) ( n+1 1000 ) + (129.62057)(2) ( n+1 1000 ) + (34.389063) ∣∣ n 500 ∣∣ + (22.841029) ∣∣ n 200 ∣∣ > 1. thus, by theorem 3.3, the boundary value problem (3.8) has at least one solution on [0, 5]. the following result can be obtained by substituting p1(t),p2(t) ≡ 1 and linear functions ψ1(|x|) = m1|x| + k1 and ψ2(|x|) = m2|x| + k2 in theorem 3.3. corollary 3.1. assume that the continuous functions g,f : j×r → r and two functionals φ,h : c(j×r) → r are satisfied |g(t,x)| ≤ m1|x| + k1, |f(t,x)| ≤ m2|x| + k2 for each (t,x) ∈ j ×r, |φ(x)| ≤ m3|x| + k4, |h(x)| ≤ m4|x| + k4 for each x ∈ c(j,r), where m1,m2,m3,m4 > 0 and k1,k2,k3,k4 ≥ 0. if m1φ1 + m2φ2 + m3φ3 + m4φ4 < 1, then boundary value problem (1.5) has at least one solution on [0,t]. acknowledgements: this research was funded by faculty of applied science, king mongkut’s university of technology north bangkok, thailand. contract no. 6042102. int. j. anal. appl. 17 (1) (2019) 63 references [1] r.p. agarwal, y. zhou, j.r. wang, x. luo, fractional functional differential equations with causal operators in banach spaces, math. comput. modelling 54 (2011), 1440-1452. [2] b. ahmad, s.k. ntouyas, a. alsaedi, m. alnahdi, existence theory for fractional-order neutral boundary value problems, frac. differ. calc. 8 (2018), 111-126. [3] b. ahmad, s.k. ntouyas, a. alsaedi, caputo-type fractional boundary value problems for differential equations and inclusions with multiple fractional derivatives, j. nonlinear funct. anal. 2017 (2017), art. id 52. [4] b. ahmad, s.k. ntouyas, a. alsaedi, new existence results for nonlinear fractional differential equations with three-point integral boundary conditions, adv. difference equ. 2011 (2011) art. id 107384, 11 pp. [5] b. ahmad, s.k. ntouyas, nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions, electron. j. differ. equ. 2012 (2012), no. 98, pp. 1-22. [6] b. ahmad, s.k. ntouyas, a. alsaedi, on fractional differential inclusions with anti-periodic type integral boundary conditions, bound. value probl. 2013 (2013), art. id 82. [7] b. ahmad, a. alsaedi, s.k. ntouyas, j. tariboon, hadamard-type fractional differential equations, inclusions and inequalities. springer, cham, 2017. [8] m. benchohra, j. henderson, s.k. ntouyas, a. ouahab, existence results for fractional order functional differential equations with infinite delay, j. math. anal. appl. 338 (2008), 1340-1350. [9] k. diethelm, the analysis of fractional differential equations, lecture notes in mathematics, springer-verlag berlin heidelberg, 2010. [10] h. ergören, b. ahmad, neutral functional fractional differential inclusions with impulses at variable times, dyn. contin. discrete impuls. syst. ser. b appl. algorithms 24 (2017), 235-246. [11] a. granas, j. dugundji, fixed point theory, springer-verlag, new york, 2003. [12] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations. north-holland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [13] m.a. krasnoselskii, two remarks on the method of successive approximations, uspekhi mat. nauk 10 (1955), 123-127. [14] v. lakshmikantham, s. leela, j.v. devi, theory of fractional dynamic systems, cambridge academic publishers, cambridge, 2009. [15] k. s. miller, b. ross, an introduction to the fractional calculus and differential equations, john wiley, new york, 1993. [16] i. podlubny, fractional differential equations, academic press, san diego, 1999. [17] d. qarout, b. ahmad, a. alsaedi, existence theorems for semi-linear caputo fractional differential equations with nonlocal discrete and integral boundary conditions, fract. calc. appl. anal. 19 (2016), 463-479. [18] s. g. samko, a. a. kilbas, o. i. marichev, fractional integrals and derivatives. theory and applications, gordon and breach, yverdon, 1993. [19] j. tariboon, a. cuntavepanit, s.k. ntouyas, w. nithiarayaphaks, separated boundary value problems of sequential caputo and hadamard fractional differential equations, preprint. [20] c. yu, g. gao, some results on a class of fractional functional differential equations, commun. appl. nonlinear anal. 11 (2004), 67-75. [21] c. yu, g. gao, existence of fractional differential equations, j. math. anal. appl. 310 (2005), 26-29. 1. introduction 2. preliminaries 3. main results acknowledgements: references international journal of analysis and applications volume 17, number 6 (2019), 980-993 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-980 strict solution for a second order differential equation in holder spaces youcef naas and fatima zohra mezeghrani∗ laboratory of mathematics and its applications (lamap), university of oran1, oran 31000, algeria ∗corresponding author: mezeghrani66@yahoo.fr abstract. in this paper we give some results on abstract second order differential elliptic equations of mixed type. the study is performed in holder continuous banach spaces. our main purpose is the study of necessary and sufficient conditions on the data for obtaining existence, uniqueness and maximal regularity properties of the strict solution. the techniques used are based on analytic semigroups theory. 1. introduction . let x be a complex banach space and consider the second order abstract differential problem u′′(x) + au(x) = f(x), x ∈ [0, 1] (1.1) with the boundary conditions u(0) = u0, u ′(1) = u′(0). (1.2) here u0 is a given element in x and a is a closed linear operator of domain d(a) not necessarily dense in x. we assume throughout the paper, the following ellipticity hypothesis: ∀λ > 0,∃(a−λ)−1 ∈l(x) : ||(a−λi)−1||l(x) 6 c 1 + λ (1.3) received 2019-05-23; accepted 2019-07-15; published 2019-11-01. 2010 mathematics subject classification. 34g10, 34k10, 35j25, 47d03. key words and phrases. abstract differential equation, boundary condition, analytic semigroup, holder spaces, square root of operator, strict solution. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 980 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-980 int. j. anal. appl. 17 (6) (2019) 981 our study will treat the existence, uniqueness and regularity of the solution under assumption (1.3). here f belongs to cα([0, 1]; x), 0 < α < 1. let us recall, for the reader’s convenience, that. cα([0, 1]; x) = { f ∈ c([0, 1]; x)/ sup s,t∈[0,1]/t−s6=0 ||f(t) −f(s)||x |t−s|α < +∞ } the main result in this work affirms that, under assumption (1.3), problem (1.1)-(1.2) has a unique strict solution, i.e a function u such that: u ∈ c2([0, 1]; x) ∩c([0, 1]; d(a)) if and only if u0 ∈ d(a) and au0 −f(0) ∈ d(a) and that: u′′,au ∈ cα([0.1]; x) (1.4) if and only if u0 ∈ d(a) and au0 −f(0) ∈ (d(a),x)1−α 2 ,+∞ where for all α ∈]0, 1[, (d(a),x)1−α 2 ,+∞ = { x ∈ x/ sup t>0 ||t α 2 aeatx||x < +∞ } we call property (1.4) the maximal regularity of the solution. we give, then, an explicit representation of the solution using the square root of −a and krein’s method [9] based on order reduction of the equation. all the components of the solution have been analyzed using the sinestrari method [14], the lion’s reiteration theorem [10] and the semigroup techniques, see krein [9]. since the density of d(a) is not assumed, we must carefully use the square root of a, see martinez-sanz [13]. several authors have been interested by equation (1.1) as an abstract problem of elliptic type, i.e under assumption (1.3) with different boundary conditions, where the second member f belongs to two classes of banach spaces, lp and cα([0, 1]; x). different methods have been used; semigroups or dunford integrals. we cite, at first, da prato and grisvard theory on the sum of operators see [5]. in arendt [1] the author proved that problem: u′′(x) + b(x)u′(x) + a(x)u(x) = f(x),x ∈ (0,δ) with u(0) = x ,u′(0) = y, has a unique solution u such that u ∈ w2,p(0,δ; x) ∩ lp(0,δ; d(a)) and u′ ∈ lp(0,δ; d(b)), in the case where d(a) and d(b) embed continuously and densely into x and f belong to lp(0,δ; x). a new approach int. j. anal. appl. 17 (6) (2019) 982 based on the semigroup techniques and fractional powers of operators have been developped by favini et al [6] concerning the complete equation : u′′(x) + 2b(x)u′(x) + a(x)u(x) = f(x), (1.5) under dirichlet boundary conditions. in batty et al [3], the authors studied maximal regularity results for problem (1.5) with boundary conditions: u(0) = u0, u ′(0) = u1. where f belongs to bαpq(0,t; x) and (u0,u1) ∈ (x,db,da)bαpq, 1 6 p 6∞ 1 6 q 6∞ and α ∈ (0, 1 p ). in this paper, we are interested in the resolution of problem (1.1) with mixed boundary conditions (1.3) which make our study difficult especially when the domain of a is not dense in x. we prove existence, uniqueness and maximal regularity of the strict solution. we also give some applications. the plan of this paper is as follows. in section 2 and 3 we will recall some semigroup properties, and give some useful. technical results to analyze the representation of the solution u. section 4 is devoted to the existence, uniqueness and maximal regularity of the strict solution. finaly, section 5 contains an application to a partial differential equation. 2. preliminaries and technical results definition 2.1. we say that ( exq ) x≥0 is a generalized analytic semigroup if q is a linear operator in x, with not dense domain and verifying :  ρ(q) ⊃ sω,δ = { λ ∈ c\{ω} / |arg(λ−ω)| < π 2 + δ } and sup λ∈sω,δ ∥∥(λ−ω) (λi −q)−1∥∥l(x) < +∞, where ω ∈ r and δ ∈ ] 0, π 2 [ . in this case ( exq ) x≥0 is not supposed strongly continuous semigroup (see e. sinestrari [14], a. lunardi [12]). remark 2.1. we fix r > 0, δ0 ∈ ]0,δ[ then ( exq ) x≥0 is defined by exq =   1 2iπ ∫ γ eλx (λi −q)−1 dλ if x > 0 i if x = 0, where γ is the sectoriel boundary curve of sω,δ0\b(ω,r) oriented positively. the following results are valable for all operator q infinitesimal generater of analytic generalized semi group. proposition 2.1. (1) let ϕ ∈ x. then the two following assertions are equivalent : (a) e·qϕ ∈ c([0, 1]; x). (b) ϕ ∈ d(q). int. j. anal. appl. 17 (6) (2019) 983 (2) let θ ∈ ]0, 1[ ,g ∈ cθ([0, 1]; x),ϕ ∈ x. set v(x,ϕ,g,q) = exqϕ + ∫ x 0 e(x−s)qg(s)ds, x ∈ [0, 1] . then the two following assertions are equivalent : (a) v ∈ c1([0, 1]; x) ∩c([0, 1]; d(q)). (b) ϕ ∈ d(q) and g(0) + qϕ ∈ d(q). considering the well known real interpolation space (d(q),x)1−θ,∞ = (x,d(q))θ,∞ proof. (see h. triebel [16] p. 25 et 76). we have also: � theorem 2.1. (1) let θ ∈ ]0, 1[.then the two following assertions are equivalent. (a) e·qϕ ∈ cθ([0, 1]; x). (b) ϕ ∈ (d(q),x)1−θ,∞ . (2) let ϕ ∈ x, θ ∈ ]0, 1[ and g ∈ cθ([0, 1]; x). set v(x) = ∫ x 0 e(x−s)q [g(s) −g(0)] ds, x ∈ [0, 1] , then v ∈ c1,θ([0, 1]; x) ∩cθ([0, 1]; d(q)). (3) let g ∈ c([0, 1]; x) and ϕ ∈ x. set w(x) = exqϕ + ∫ x 0 e(x−s)qg(s)ds, x ∈ [0, 1] , then the two following assertions are equivalent. (a) w ∈ c1,θ([0, 1]; x) ∩cθ([0, 1]; d(q)). (b) g ∈ cθ([0, 1]; x), ϕ ∈ d(q) and g(0) + qϕ ∈ (d(q),x)1−θ,∞ . (4) let g ∈ cθ([0, 1]; x). then q ∫ 1 0 esq (g(s) −g(0)) ds ∈ (d (q)),x)1−θ,∞ . proof. statement 2 is obtained by applying the da prato-grisvard sum theory [5]. statement 3 which improves statement 2 is due to e. sinestrari [14], see also g. da prato [4]. � let g and h be two given x valued functions defined on [0, 1] and θ ∈ ]0, 1[. we write g 'θ h, if g −h ∈ cθ([0, 1]; x). int. j. anal. appl. 17 (6) (2019) 984 proposition 2.2. let h ∈ cθ([0, 1]; x), ϕ ∈ d(q) and set w(x) = exqϕ + ∫ x 0 e(x−s)qh(s)ds, x ∈ [0, 1] ; then qw(·) 'θ e·q (qϕ + h(0)) . proof. it is an easy consequence of theorem 2.1 and proposition 1.2, statement (ii) in sinestrari [14]. � proposition 2.3. assume (1.3). the operator (i −z) has a bounded inverse given by (i −z)−1 = 1 2πi ∫ γ# e2z i −e2z (zi + b)−1dz + i. where γ# is a suitable curve in the complex plane. proof. see lunardi [12]. � we put throughout the paper b = √ −a and z = e−2b remark 2.2. hypothesis (1.3) implies that the operator (− √ −a) generates an analytic semigroup denoted by (e− √ −ax)x>0 on x, see for instance balakrishnan [2]. for u0 ∈ x, consider the follwing abstract function q0 : ]0, 1] −→ x x 7−→ q0(x,b)u0 defined by q0 q0(x,b)u0 = (i −z)−1(i −e−b)−2e−xbu0. we have the following result lemma 2.1. we have: (1) q0(·,b)u0 ∈ c∞(]0, 1] ; d(ak)),k ∈ n (2) ∀x ∈ ]0, 1] ,q′′0 (x,b)u0 + aq0(x,b)u0 = 0, (3) ∃c > 0,∀x ∈ ]0, 1] ,‖q0(x,b)u0‖x ≤ c‖u0‖x . proof. (1) let x > 0,u0 ∈ x. it is not difficult to see that (i −z)−1(i −e−b)−2e−bx = e−bx(i −z)−1(i −e−b)−2, therefore q0(x,b)u0 = e−xb(i −z)−1(i −e−b)−2u0, hence we deduce the first statement using proposition 1.1 in sinestrari [14]. int. j. anal. appl. 17 (6) (2019) 985 (2) for x ∈ ]0, 1], we have: q′0(x,b)u0 = −(i −z) −1(i −e−b)−2e−xbbu0 = −(i −z)−1(i −e−b)−2e−xb √ au0 therefore q′′0 (x,b)u0 = +(i −z) −1(i −e−b)−2e−xbb2u0 = −(i −z)−1(i −e−b)−2e−xbau0 and q′′0 (x,b)u0 + aq0(x,b)u0 = −(i −z) −1(i −e−b)−2e−xbau0 −a(i −z)−1(i −e−b)−2e−xbu0 = −(i −z)−1(i −e−b)−2e−xbau0 − (i −z)−1(i −e−b)−2e−xbau0 = 0 (3) it is well known that there exists m > 0 such that for any x > 0,u0 ∈ x ∥∥e−bxu0∥∥x ≤ m ‖u0‖x (see tanabe [15], page 66, formula (3.27)). thus, ∃c > 0 : ‖q0(x,b)u0‖x = ∥∥(i −z)−1(i −e−b)−2e−xbu0∥∥x ≤ c‖u0‖x . let us specify the behavior of q0(.,b) near 0. � lemma 2.2. (1) let u0 ∈ x. then q0(·, √ −a)u0 ∈ c ([0, 1] ; x) if and only if u0 ∈ d(a) (2) let u0 ∈ d(a). then q0(·, √ −a)u0 ∈ c ([0, 1] ; d(a)) if and only if au0 ∈ d(a) proof. this result is a consequence of commutativity of (i −z)−1 (i−e−b)−2 and a on d(a) ( see sinestrari [14], proposition 1.2, (ii), page 20), we also use the fact that d( √ −a) = d(a), see haase [8], corollary 3.1.11. page 59. � int. j. anal. appl. 17 (6) (2019) 986 for u0 ∈ x, consider the following abstract function : q1 : [0, 1[ −→ x x 7−→ q1(x,b)u0 where q1(x,b)u0 = −(i −z)−1(i −e−b)−2e−(1−x)bu0. we have the following result : lemma 2.3. we have : (1) q1(·, √ −a)u0 ∈ c∞([0, 1[; d(ak)),k ∈ n (2) ∀x ∈ [0, 1[,q′′1 (x, √ −a)u0 + aq1(x, √ −a)u0 = 0, (3) ∃c > 0,∀x ∈ [0, 1[, ∥∥q1(x,√−a)u0∥∥x ≤ c‖u0‖x . proof. it is not difficult to prove this lemma, it is sufficient to replace x by 1 −x. � lemma 2.4. (1) let u0 ∈ x. then q1(·, √ −a)u0 ∈ c ([0, 1] ; x) if and only if u0 ∈ d(a) (2) let u0 ∈ d(a). then q1(·, √ −a)u0 ∈ c ([0, 1] ; d(a)) if and only if au0 ∈ d(a) proof. the proof of this lemma is the same as lemma 2.2 � 3. representation of the solution in this section we assume that (1.3) holds we set : u(1) = u1 let us suppose that problem (1.1)-(1.2) has a strict solution u . then u is the strict solution of the following problem:   u ′′ (x) −b2u(x) = f(x) u(0) = u0 u(1) = u1 (3.1) therefore u is represented by : u(x) = e−xbξ0 + e −(1−x)bξ1 − 1 2 b−1 ∫ x 0 e−(x−s)bf(s)ds− 1 2 b−1 ∫ 1 x e−(s−x)bf(s)ds int. j. anal. appl. 17 (6) (2019) 987 where : ξ0 = (i −z) −1 ( u0 −e−bu1 ) + 1 2 (i −z)−1b−1 (∫ 1 0 e−sbf(s)ds− ∫ 1 0 e−(2−s)bf(s)ds ) ξ1 = (i −z) −1 (−e−bu0 + u1) + 1 2 (i −z)−1b−1 (∫ 1 0 e−(1−s)bf(s)ds− ∫ 1 0 e−(1+s)bf(s)ds ) (see [6]). by derivation we obtain u′(x) = −be−xbξ0 + be−(1−x)bξ1 + 1 2 ∫ x 0 e−(x−s)bf(s)ds− 1 2 ∫ 1 x e−(s−x)bf(s)ds. using that u′(0) = u′(1) we obtain −bξ0 + be−bξ1 − 1 2 ∫ 1 0 e−sbf(s)ds = −be−bξ0 + bξ1 + 1 2 ∫ 1 0 e−(1−s)bf(s)ds. then we deduce : u1= −u0 −b−1(i −e−b)−2 ∫ 1 0 e−sbf(s)ds + b−1(i −e−b)−2 ∫ 1 0 e−(2−s)bf(s)ds +b−1(i −e−b)−2 ∫ 1 0 e−(1+s)bf(s)ds−b−1(i −e−b)−2 ∫ 1 0 e−(1−s)bf(s)ds. therefore u is formally given by : u(x) = +(i −z)−1(i −e−b)−2e−xbu0 − (i −z)−1(i −e−b)−2e−(1−x)bu0 −(i −z)−1(i −e−b)−2e−(4−x)bu0 − (i −z)−1(i −e−b)−2e−(2+x)bu0 +(i −z)−1(i −e−b)−2e−(3+x)bu0 − (i −z)−1(i −e−b)−2e−(1+x)bu0 +(i −z)−1(i −e−b)−2e−(2−x)bu0 + (i −z)−1(i −e−b)−2e−(3−x)bu0 + 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(x+s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(2+x+s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(2+x−s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(4+x−s)bf(s)ds −(i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(1−x+s)bf(s)ds (3.2) +(i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(3−x+s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(2−x+s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(4−x+s)bf(s)ds (3.3) int. j. anal. appl. 17 (6) (2019) 988 − 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(2−x−s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b−1 ∫ 1 0 e−(4−x−s)bf(s)ds − 1 2 b−1 ∫ x 0 e−(x−s)bf(s)ds− 1 2 b−1 ∫ 1 x e−(s−x)bf(s)ds 4. existence, uniqueness and maximal regularity theorem 4.1. let f ∈ cα([0, 1]; x); 0<α<1, we assume that (1.3) holds. then the following assertions are equivalent . (1) problem (1.1)-(1.2) has a unique strict solution u, that is u ∈ c2([0, 1]; x) ∩c([0, 1]; d(a)) (i.e u satisfies (1.1)-(1.2)) (2) u0 ∈ d(a) and au0 −f(0) ∈ d(a) proof. suppose that statement 2 holds, i.e u0 ∈ d(a) and au0 −f(0) ∈ d(a) the solution u of problem (1.1)-(1.2) is given by (3.2), then u′′(x) = −(i −z)−1(i −e−b)−2e−xbau0 + (i −z)−1(i −e−b)−2e−(1−x)bau0 +(i −z)−1(i −e−b)−2e−(4−x)bau0 + (i −z)−1(i −e−b)−2e−(2+x)bau0 −(i −z)−1(i −e−b)−2e−(3+x)bau0 + (i −z)−1(i −e−b)−2e−(1+x)bau0 −(i −z)−1(i −e−b)−2e−(2−x)bau0 − (i −z)−1(i −e−b)−2e−(3−x)bau0 + 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(x+s)bf(s)ds−b ∫ 1 0 e−(2+x+s)bf(s)ds ) + 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(2+x−s)bf(s)ds−b ∫ 1 0 e−(4+x−s)bf(s)ds ) +(i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(3−x+s)bf(s)ds−b ∫ 1 0 e−(1−x+s)bf(s)ds ) + 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(2−x+s)bf(s)ds−b ∫ 1 0 e−(4−x+s)bf(s)ds ) + 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(4−x−s)bf(s)ds−b ∫ 1 0 e−(2−x−s)bf(s)ds ) − 1 2 b ∫ x 0 e−(x−s)bf(s)ds− 1 2 b ∫ 1 x e−(s−x)bf(s)ds +f(x) we write u′′ as int. j. anal. appl. 17 (6) (2019) 989 u ′′ (x) = −(i − z)−1(i − e−b)−2 (au0 − f(0)) e−xb + (i − z)−1(i − e−b)−2 (au0 − f(0)) e−(1−x)b +(i − z)−1(i − e−b)−2 (au0 + f(1)) e−(4−x)b + (i − z)−1(i − e−b)−2e−(2+x)bau0 −(i − z)−1(i − e−b)−2 (au0 − f(0)) e−(3+x)b + (i − z)−1(i − e−b)−2e−(1+x)bau0 −(i − z)−1(i − e−b)−2 (au0 − f(0)) e−(2−x)b − (i − z)−1(i − e−b)−2e−(3−x)bau0 + 1 2 (i − z)−1(i − e−b)−2 ( e −xb b ∫ 1 0 e −sb (f(s) − f(0))ds − b ∫ 1 0 e −(2+x+s)b f(s)ds ) + 1 2 (i − z)−1(i − e−b)−2 ( b ∫ 1 0 e −(2+x−s)b f(s)ds − b ∫ 1 0 e −(4+x−s)b f(s)ds ) +(i − z)−1(i − e−b)−2 ( b ∫ 1 0 e −(3−x+s)b f(s)ds − e−(1−x)bb ∫ 1 0 e −sb (f(s) − f(0))ds ) + 1 2 (i − z)−1(i − e−b)−2 ( b ∫ 1 0 e −(2−x+s)b f(s)ds − b ∫ 1 0 e −(4−x+s)b f(s)ds ) (4.1) + 1 2 (i − z)−1(i − e−b)−2b (∫ 1 0 e −(4−x−s)b f(s)ds − e−(1−x)b ∫ 1 0 e −(1−s)b (f(s) − f(1))ds ) − 1 2 (i − z)−1(i − e−b)−2e−(2−x)bf(1) − 1 2 (i − z)−1(i − e−b)−2e−(5−x)bf(1) − 1 2 (i − z)−1(i − e−b)−2e−(4+x)bf(0) − 3 2 (i − z)−1(i − e−b)−2e−(1+x)bf(0) − 1 2 b ∫ x 0 e −(x−s)b (f(s) − f(x))ds − 1 2 b ∫ 1 x e −(s−x)b (f(s) − f(x))ds + 1 2 (f(x) − f(0)) e−xb + 1 2 (f(x) − f(1)) e−(1−x)b in view of lemmas 2.2 and 2.4 the first and second terms are in c ([0, 1]; x) and all other terms are continuous since f ∈ cα ([0, 1]; x). from which we deduce that u′′ is in c ([0, 1]; x). by the same way, we write: au(x) = +(i −z)−1(i −e−b)−2e−xbau0 − (i −z)−1(i −e−b)−2e−(1−x)bau0 −(i −z)−1(i −e−b)−2e−(4−x)bau0 − (i −z)−1(i −e−b)−2e−(2+x)bau0 +(i −z)−1(i −e−b)−2e−(3+x)bau0 − (i −z)−1(i −e−b)−2e−(1+x)bau0 +(i −z)−1(i −e−b)−2e−(2−x)bau0 + (i −z)−1(i −e−b)−2e−(3−x)bau0 − 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(x+s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(2+x+s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(2+x−s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(4+x−s)bf(s)ds +(i −z)−1(i −e−b)−2b ∫ 1 0 e−(1−x+s)bf(s)ds int. j. anal. appl. 17 (6) (2019) 990 −(i −z)−1(i −e−b)−2b ∫ 1 0 e−(3−x+s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(2−x+s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(4−x+s)bf(s)ds + 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(2−x−s)bf(s)ds − 1 2 (i −z)−1(i −e−b)−2b ∫ 1 0 e−(4−x−s)bf(s)ds + 1 2 b ∫ x 0 e−(x−s)bf(s)ds + 1 2 b ∫ 1 x e−(s−x)bf(s)ds which is equal au(x) = +(i −z)−1(i −e−b)−2 (au0 −f(0)) e−xb − (i −z)−1(i −e−b)−2 (au0 −f(0)) e−(1−x)b −(i −z)−1(i −e−b)−2 (au0 + f(1)) e−(4−x)b − (i −z)−1(i −e−b)−2e−(2+x)bau0 +(i −z)−1(i −e−b)−2 (au0 −f(0)) e−(3+x)b − (i −z)−1(i −e−b)−2e−(1+x)bau0 +(i −z)−1(i −e−b)−2 (au0 −f(0)) e−(2−x)b + (i −z)−1(i −e−b)−2e−(3−x)bau0 − 1 2 (i −z)−1(i −e−b)−2 ( e−xbb ∫ 1 0 e−sb(f(s) −f(0))ds−b ∫ 1 0 e−(2+x+s)bf(s)ds ) − 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(2+x−s)bf(s)ds−b ∫ 1 0 e−(4+x−s)bf(s)ds ) −(i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(3−x+s)bf(s)ds−e−(1−x)bb ∫ 1 0 e−sb(f(s) −f(0))ds ) − 1 2 (i −z)−1(i −e−b)−2 ( b ∫ 1 0 e−(2−x+s)bf(s)ds−b ∫ 1 0 e−(4−x+s)bf(s)ds ) − 1 2 (i −z)−1(i −e−b)−2b (∫ 1 0 e−(4−x−s)bf(s)ds−e−(1−x)b ∫ 1 0 e−(1−s)b(f(s) −f(1))ds ) + 1 2 (i −z)−1(i −e−b)−2e−(2−x)bf(1) + 1 2 (i −z)−1(i −e−b)−2e−(5−x)bf(1) + 1 2 (i −z)−1(i −e−b)−2e−(4+x)bf(0) + 3 2 (i −z)−1(i −e−b)−2e−(1+x)bf(0) + 1 2 b ∫ x 0 e−(x−s)b(f(s) −f(x))ds + 1 2 b ∫ 1 x e−(s−x)b(f(s) −f(x))ds − 1 2 (f(x) −f(0)) e−xb − 1 2 (f(x) −f(1)) e−(1−x)b +f(x) and by the same raisons as u′′, we prove that : au ∈ c([0, 1]; x) and u′′(x) + au(x) = f(x) int. j. anal. appl. 17 (6) (2019) 991 conversely, we suppose that statement 1 holds, i.e u ∈ c2([0, 1]; x) ∩c([0, 1]; d(a)) then u0 = u(0) ∈ d(a), and au0 −f(0) = −u′′(0) ∈ d(a) finally we obtain the following maximal regularity: � theorem 4.2. let f ∈ cα ([0, 1] ; x) , 0 < α < 1, and we suppose that (1.3) holds, then the following assertions are equivalent (1) the unique solution u of problem (1.1)-(1.2) has the maximal regularity property u′′,au ∈ cα([0, 1] ; x) (2) u0 ∈ d(a),au0 −f(0) ∈ (d (a) ,x)1−α 2 ,+∞ proof. assume that there exists a strict solution u of problem (1.1)-(1.2) having the maximal regularity property. from the previous theorem we have u0 ∈ d(a), furthermore, the first and second terms in formula (4.1) are in cα([0, 1]; x) and then e−b· (au0 −f(0)) ∈ cα([0, 1]; x) e(1−·)b (au0 −f(0)) ∈ cα([0, 1]; x) using remark, (f), page 39. in [14], we obtain au0 −f(0) ∈ (d(b),x)1−α,∞ we recall that : (d(b),x)1−α,∞ = (d(a),x)1−α 2 ,∞ . conversely, assume that u0 ∈ d(a),au0 −f(0) ∈ (d (a) ,x)1−α 2 ,+∞ int. j. anal. appl. 17 (6) (2019) 992 using theorem, 1.4. page 361 in [4] we have e− √ −a· (au0 −f(0)) ∈ cα([0, 1]; x) e(1−·) √ −a (au0 −f(0)) ∈ cα([0, 1]; x) ∫ 1 0 be−sb(f(s) −f(0))ds ∈ cα([0, 1]; x) ∫ 1 0 be−(1−s)b(f(s) −f(1))ds ∈ cα([0, 1]; x) and thus u′′, au ∈ cα([0, 1] ; x). � 5. concrete application let x = lp(r) and f ∈ cα([0, 1],l2(r)), 0 < α < 1 and 1 < p < ∞. consider the problem :  ∂2u ∂x2 (x,y) + ∂2u ∂y2 (x,y) = f(x,y) (x,y) ∈ [0, 1] ×r u(0,y) = u0(y) , y ∈ r ∂u ∂x (0,y) = ∂u ∂y (1,y) (5.1) we dfine operator a as follows   d(a) = h2(r) au = u′′ as x is a hilbert space, d(a) is dense in x, moreover (−a) is a self adjoint operator, then d( √ −a) = (d(a),x) 1 2 ,2 = (h 2(r),l2(r)) 1 2 ,2 = h 1(r) see [11]. using fourier transformation, we prove that a verifies (1.3). the following result is a consequence of theorem 4.2 proposition 5.1. problem (5.1) has a unique strict solution u such that u ∈ c2([0, 1]; l2(r)) ∩c([0, 1]; h2(r)) satisfying u′′ ∈ cα([0, 1]; l2(r)) if and only if int. j. anal. appl. 17 (6) (2019) 993 u0 ∈ h2(r) and au0 −f(0) ∈ (h2(r),l2(r))1−α 2 ,∞ = b α 2,∞(r) (the besov space bα2,∞(r) is completely deseribed in grisvard [7] ) acknowledgments. the authors wish to warmly thank the referees for all their useful suggestions. references [1] w. arendt, r. chill, s. fornaro and c.poupaud., lp − maximal regularity for non-autonomous evolution equations, j. differ. equations, 237 (2007), 1-26. [2] a. v. balakrishnan., fractional powers of closed operators and the semigroups generated by them, pac. j. math. 10 (1960), 419-437. [3] c. j. k. batty, r. chill, s. srivastava., maximal regularity in interpolation spaces for second-order cauchy problems, oper. theory: adv. appl. 250 (2015), 49-66. [4] g. da prato., abstract differential equations, maximal regularity, and linearization, proc. symp. pure math. 45 (1986), part 1. [5] g. da prato and p.grisvard., sommes d’opérateurs linéaires et équations différentielles opérationnelles. j. math. pures appl. ix ser. 54 (1975), 305-387. [6] a. favini, r. labbas, s. maingot, h. tanabe and a. yagi., on the solvability and the maximal regularity of complete abstract differential equations of elliptic type, funkcial ekvacioj, 47 (2004), 423-452. [7] p. grisvard., spazi di tracce e applicazioni, rend. mat. (4), vol. 5, série vi (1972), 657-729. [8] m. haase., the functional calculus for sectorial operators, oper. theory adv. appl., vol. 69, birkhuser-verlag, basel (2006). [9] s. g. krein., linear differental equations in banach space, moscou, 1967; english translation: ams, providence, (1971). [10] j.l. lions., théorème de trace et d’interpolation i et ii. annali s.n.s.di pisa, 13, (1959), 389-403 et 14, (1960), 317-331. [11] j. l. lions and j. peetre., sur une classe d’espaces d’interpolation, inst. hautes etudes sci. publ. math., 19 (1964), 5-68. [12] a.lunardi., analytic semigroups and optimal regularity in parabolic problems, birkhâuser, basel, (1995). [13] c. martinez and m. sanz., fractionnal powers of non densely defined operators. annali della scuola normale superiore di pisa-classe di scienze ser. 4, 18 (1991). 443-454. [14] e. sinestrari., on the abstract cauchy problem of parabolic type in space of continuous functions, j. math. anal. appl. 66 (1985), 16-66. [15] j. tanabe., equations of evolution, pitman, london, san francisco, melbourne, (1979). [16] h. triebel., interpolation theory,. function spaces, differential operators, amsterdam, north holland (1978). 1. introduction 2. preliminaries and technical results 3. representation of the solution 4. existence, uniqueness and maximal regularity 5. concrete application references international journal of analysis and applications volume 16, number 2 (2018), 239-253 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-239 new subclass of analytic functions in conical domain associated with ruscheweyh q-differential operator shahid khan1,∗, saqib hussain2, muhammad asad zaighum1, muhammad mumtaz khan3 1department of mathematics, riphah international university islamabad, pakistan 2department of mathematics, comsats institute of information technology, abbottabad, pakistan 3department of public health riphah university of haripur, pakistan ∗corresponding author: shahidmath761@gmail.com abstract. in this paper, we consider a new class of analytic functions which is defined by means of a ruscheweyh q-differential operator. we investigated some new results such as coefficients inequalities and other interesting properties of this class. comparison of new results with those that were obtained in earlier investigation are given as corollaries. 1. introduction let a denote the class of functions f analytic in the open unit disk e = {z : z ∈ c and |z| < 1} and satisfying the normalization condition f (0) = 0 and f′ (0) = 1. received 2017-09-25; accepted 2017-12-07; published 2018-03-07. 2010 mathematics subject classification. primary 05a30, 30c45; secondary 11b65, 47b38. key words and phrases. analytic functions; ruscheweyh q-differential operator; q-derivative operator; conic domains. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 239 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-239 int. j. anal. appl. 16 (2) (2018) 240 thus, the functions in a are represented by the taylor-maclaurin series expansion given by f(z) = z + ∞∑ n=2 anz n, (z ∈ e) . (1.1) let s be the subset of a consisting of the functions that are univalent in e. given functions f, g ∈a, f is said to be subordinate to g in e, denoted by f ≺ g or f (z) ≺ g (z) (z ∈ e) , if there exists a function w ∈p0 where p0 = {w ∈a : w (0) = 0, and |w (z)| < 1 (z ∈ e)} , such that f (z) = g (w (z)) (z ∈ e) . if g is univalent in e, then it follows that f (z) ≺ g (z) (z ∈ e) , ⇒ f (0) = 0 and f (e) ⊂ g (e) . kanas and wísniowska [5, 6] introduced the conic domain ωk,k ≥ 0 as ωk = { u + iv : u > k √ (u− 1)2 + v2 } . we note that ωk is a region in the right half-plane, symmetric with respect to real axis, and contains the point (1, 0). more precisely for k = 0, ω0 is the right half-plane, for 0 < k < 1, ωk is an unbounded region having boundary ∂ωk, a rectangular hyperbola; for k = 1, ω1 is still an unbounded region where ∂ω1 is a parabola, and for k > 1, ωk is a bounded region enclosed by an ellipse. the extremal functions for these conic regions are pk (z) =   1+z 1−z , k = 0, 1 + 2 π2 ( log 1+ √ z 1− √ z )2 , k = 1, 1 1−k2 cosh {( 2 π arccos k ) log 1+ √ z 1− √ z } − k 2 1−k2 , 0 < k < 1, 1 k2−1 sin ( π 2k(κ) ∫ u(z)√ κ 0 dt√ 1−t2 √ 1−κ2t2 ) + k 2 k2−1, k > 1, (1.2) where u(z) = z − √ κ 1 − √ κz (z ∈ e) and κ ∈ (0, 1) is chosen such that k = cosh (πk′(κ)/(4k(κ))). here k(κ) is legender’s complete elliptic integral of first kind and k′(κ) = k( √ 1 −κ2) and k′ (t) is the complementary integral of k (t) for details int. j. anal. appl. 16 (2) (2018) 241 see [1, 5, 6] and more recently [9, 12, 14]. if pk (z) = 1 + l1 (k) z + l2 (k) z 2 + ...,z ∈ e. then it was shown in [6] that for (1.2) one can have l1 (k) =   2a2 1−k2 , 0 ≤ k < 1 8 π2 , k = 1, π2 4k2(t)2(1+t) √ t , k > 1, (1.3) l2 (k) = d (k) l1 (k) , where d (k) =   a2+2 3 , 0 ≤ k < 1 8 π2 , k = 1, (4k(t))2(t2+6t+1)−π2 24k(t)2(1+t) √ t , k > 1, (1.4) with a = 2 π arccos k. furthermore a function p is said to be in the class k −p [a,b] , if and only if p (z) ≺ (a + 1) pk (z) − (a− 1) (b + 1) pk (z) − (b − 1) , k ≥ 0, where pk is defined in (1.2) and −1 ≤ b < a ≤ 1. geometrically, the function p ∈ k − p [a,b] takes all values from the domain ωk[a,b], −1 ≤ b < a ≤ 1, k ≥ 0 which is defined as: ωk[a,b] = { w : < ( (b − 1) w − (a− 1) (b + 1) w − (a + 1) ) > k ∣∣∣∣(b − 1) w − (a− 1)(b + 1) w − (a + 1) − 1 ∣∣∣∣ } , or equivalently ωk[a,b] is a set of numbers w = u + iv such that [( b2 − 1 )( u2 + v2 ) − 2 (ab − 1) u + ( a2 − 1 )]2 > k [( −2 (b + 1) ( u2 + v2 ) + 2 (a + b + 2) u− 2 (a + 1) )2 + 4 (a−b)2 v2 ] . this domain represents the conic type regains for detail see [11]. it can be easily seen that 0 −p [a,b] = p [a,b] introduced in [4] and k −p [1,−1] = p (pk) introduced in [5]. we now recall some basic concept details of the q-calculus which are used in this paper. throughout this paper we assume q to be a fixed number between 0 and 1. for any non-negative integer n, the q-integer number n, [n,q] is defined by: [n,q] = 1 −qn 1 −q = 1 + q + ... + qn−1, [0,q] = 0. (1.5) the q-number shifted factorial is defined by [0,q]! = 1 and [n,q]! = [1,q] [2,q] ... [n,q] . clearly, lim q→1 [n,q] = n and lim q→1 [n,q]! = n!. in general we will denote [t,q] = 1−q t 1−q also for a non-integer number. let f ∈a, and let the q-derivative operator or q-difference operator be defined by ∂qf (z) = f (qz) −f (z) (q − 1) z (z ∈ e) . int. j. anal. appl. 16 (2) (2018) 242 it is easy to observed that for n ∈ n := {1, 2, 3, ...} and z ∈ e ∂qz n = [n,q] zn−1. let the q-generalized pochhammer symbol be defined as [t,q]n = [t,q] [t + 1,q] [t + 2,q] ... [t + n− 1,q] , and for t > 0 let the q-gamma function be defined as γq (t + 1) = [t] γq (t) and γq (1) = 1. the study of opretors play in important role in geomatric functions theory. several diffierential and integral operators were introduced and studied, see for example [2, 3, 13]. kannas et al. define ruscheweyh q-differential operator as follow: definition 1.1. [7] for f ∈a, let the ruscheweyh q-differential operator be defined as follows: rλqf(z) = f(z) ∗fq,λ+1(z), (z ∈ e, λ > −1) (1.6) where fq,λ+1(z) = z + ∞∑ n=2 γq(n + λ) [n− 1,q]!γq(1 + λ) zn, = z + ∞∑ n=2 [λ + 1,q]n−1 [n− 1,q]! zn, = z + ∞∑ n=2 ϕn−1z n. (1.7) where ϕn−1 = γq(n + λ) [n− 1,q]!γq(1 + λ) = [λ + 1,q]n−1 [n− 1,q]! . from (1.6) we obtain that r0qf(z) = f(z), r 1 qf(z) = z∂qf(z) and rmq f(z) = z∂mq (z m−1f(z)) [m,q]! , (m ∈ n). making use of (1.6) and (1.7), the power series of rλqf(z) is given by rλqf(z) = z + ∞∑ n=2 γq(n + λ) [n− 1,q]!γq(1 + λ) anz n = z + ∞∑ n=2 [λ + 1,q]n−1 [n− 1,q]! anz n. (1.8) note that lim q→1 fq,λ+1(z) = z (1 −z)λ+1 int. j. anal. appl. 16 (2) (2018) 243 and lim q→1 rλqf(z) = f(z) ∗ z (1 −z)λ+1 . thus, we can say that ruscheweyh q-differential operator reduces to the differential operator defined by ruscheweyh [16] in the case when q → 1. it is easy to check that z∂ (fq,λ+1(z)) = ( 1 + [λ,q] qλ ) fq,λ+2(z) − [λ,q] qλ fq,λ+1(z). (1.9) making use of (1.6), (1.9) and the properties of hadamard product, we obtain the following equality z∂ ( rλqf(z) ) = ( 1 + [λ,q] qλ ) rλ+1q f(z) − [λ,q] qλ rλqf(z). (1.10) if q → 1, the equality (1.10) implies z ( rλf(z) )′ = (1 + λ) rλ+1f(z) −λrλf(z). which is the well known recurrent formula for ruscheweyh differential operator. using ruscheweyh differential operator various new classes of convex and starlike functions have been defined. now by using ruscheweyh q-differential operator we introduce the following class of functions. definition 1.2. a function f(z) ∈ a is said to be in the class k−usq(λ,a,b,β), k ≥ 0,−1 ≤ b < a ≤ 1, if and only if < ( (b − 1)g (z) − (a− 1) (b + 1)g (z) − (a + 1) ) > k ∣∣∣∣(b − 1)g (z) − (a− 1)(b + 1)g (z) − (a + 1) − 1 ∣∣∣∣ , where g (z) = z∂qr λ qf(z) rλqf(z) + β z2∂2qr λ qf(z) rλqf(z) , or equivalently z∂qr λ qf(z) rλqf(z) + β z2∂2qr λ qf(z) rλqf(z) ∈ k −p[a,b]. (1.11) remark 1.1. it is easily see that lim q→1− k −usq(0,a,b, 0) = k −st (a,b) where k −st (a,b) is a functions class, intrioduced and studied by noor and sarfraz [11]. each of the following lemmas will be needed in our present investigation. lemma 1.1. [15] let h(z) = 1 + ∑∞ n=1 cnz n be subordinate to h(z) = 1 + ∑∞ n=1 cnz n. if h(z) is univalent in e and h(e) is convex, then |cn| ≤ |c1| , n ≥ 1. int. j. anal. appl. 16 (2) (2018) 244 lemma 1.2. ( [8], [10]) if q(z) = 1 + c1z + c2z 2+... is an analytic function with positive real part in e, then ∣∣c2 −vc21∣∣ ≤ 2 max{1, |2v − 1|} . the result is sharp for the functions q(z) = 1 + z2 1 −z2 , or q(z) = 1 + z 1 −z . lemma 1.3. [8] let the function w ∈ e be given by w(z) = c1z + c2z 2 + ... z ∈ e. then for every complex number v, ∣∣c2 −vc21∣∣ ≤ 1 + (|v|− 1) |c1|2 . lemma 1.4. [11] let k ∈ [0,∞) be a fixed and qk(z) = (a + 1)pk(z) − (a− 1) (b + 1)pk(z) − (b − 1) , then qk(z) = 1 + h1(k)z + h2(k)z 2 + ...,z ∈ e. and h1 := h1(k) = a−b 2 l1(k), h2 := h2(k) = a−b 4 {2d(k) − (b + 1)h1}l1(k) where l1(k) and d(k) are defined in (1.3) and (1.4). 2. main results theorem 2.1. a function f ∈a and of the form (1.1) is in the class k−usq(λ,a,b,β), if it satisfies the condition ∞∑ n=2   {2(k + 1){1 − [n,q] −β[n,q][n− 1,q]} + |{(b + 1) [n,q] + β[n,q][n− 1,q] − (a + 1)}|}  ϕn−1 |an| ≤ |b −a| . (2.1) where −1 ≤ b < a ≤ 1, β ≥ 0 and k ≥ 0. int. j. anal. appl. 16 (2) (2018) 245 proof. assume (2.1) is hold, then it suffices to show that  k ∣∣∣∣∣∣ (b−1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a−1) (b+1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a+1) − 1 ∣∣∣∣∣∣ −<  (b−1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a−1) (b+1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a+1) − 1     < 1, we have   k ∣∣∣∣∣∣ (b−1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a−1) (b+1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a+1) − 1 ∣∣∣∣∣∣ −<  (b−1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a−1) (b+1) ( z∂qr λ q f(z) rλq f(z) +β z2∂2qr λ q f(z) rλq f(z) ) −(a+1) − 1     ≤ (k + 1) ∣∣∣∣∣(b − 1) ( z∂qr λ qf(z) + βz 2∂2qr λ qf(z) ) − (a− 1)rλqf(z) (b + 1) ( z∂qrλqf(z) + βz 2∂2qr λ qf(z) ) − (a + 1)rλqf(z) − 1 ∣∣∣∣∣ = 2(k + 1) ∣∣∣∣∣ r λ qf(z) −z∂qrλqf(z) −βz2∂2qrλqf(z) (b + 1) ( z∂qrλqf(z) + βz 2∂2qr λ qf(z) ) − (a + 1)rλqf(z) ∣∣∣∣∣ = 2(k + 1) ∑∞ n=2 (1 − [n,q] −β[n,q][n− 1,q]) ϕn−1anz n (b −a) z + ∑∞ n=2 { (b + 1) [n,q]q + β[n,q][n− 1,q] − (a + 1) } ϕn−1anzn ≤ 2(k + 1) ∑∞ n=2 (1 − [n,q] −β[n,q][n− 1,q]) ϕn−1 |an| |b −a|− ∑∞ n=2 { (b + 1) [n,q]q + β[n,q][n− 1,q] − (a + 1) } ϕn−1 |an| < 1 (by (2.1)) . � when a = 1 − 2α, b = −1,β = 0 with 0 ≤ α < 1, then we have the following known result, proved by kanas and raducanu in [7]. corollary 2.1. a function f ∈ a and of the form (1.1) is in the class k−usq(λ, 1−2α,−1) , if it satisfies the condition ∞∑ n=2 {(k + 1) [n,q] −k −α}ϕn−1 |an| ≤ 1 −α. when q → 1, β = 0,λ = 0, then we have the following known result, proved by noor and sarfraz [11]. corollary 2.2. a function f ∈ a and of the form (1.1 is in the class k −st (a,b), if it satisfies the condition ∞∑ n=2 {2(k + 1)(n− 1) + |n(b + 1) − (a + 1)|} |an| ≤ |b −a| . int. j. anal. appl. 16 (2) (2018) 246 when q → 1, λ = 0,β = 0, a = 1−2α, b = −1 with 0 ≤ α < 1, then we have the following known result, proved by shams et-al. in [18]. corollary 2.3. a function f ∈ a and of the form (1.1) is in the class k −ust (1 − 2α,−1), if it satisfies the condition ∞∑ n=2 {n(k + 1) − (k + α)}|an| ≤ 1 −α, where 0 ≤ α < 1 and k ≥ 0. when λ = 0,β = 0, a = 1 − 2α, b = −1 with 0 ≤ α < 1 and k = 0, then we have the following known result, proved by selverman in [17]. corollary 2.4. a function f ∈ a and of the form (1.1) is in the class 0 −ust (1 − 2α,−1), if it satisfies the condition ∞∑ n=2 {n−α}|an| ≤ 1 −α, 0 ≤ α < 1. theorem 2.2. if f(z) ∈ k −usq(λ,a,b,β) and is of the form (1.1). then |an| ≤ n−2∏ j=0 ( |l1(k)(a−b) − 2[j,q]b| 2 [j + 1,q]{q + β[j + 2,q]}ϕj+1 ) , n ≥ 2, (2.2) where l1(k) is defined by (1.3). proof. let z∂qr λ qf(z) rλqf(z) + β z2∂2qr λ qf(z) rλqf(z) = p(z). (2.3) then p(z) ≺ (a + 1)pk(z) − (a− 1) (b + 1)pk(z) − (b − 1) = [(a + 1)pk(z) − (a− 1)] [(b + 1)pk(z) − (b − 1)] −1 = (a− 1) (b − 1) [ 1 − (a + 1) (a− 1) pk(z) ][ 1 + ∑((b + 1) (b − 1) pk(z) )n] = (a− 1) (b − 1) + ( (a− 1)(b + 1) (b − 1)2 − (a + 1) (b − 1) ) (pk(z)) + ( (a− 1)(b + 1)2 (b − 1)3 − (a + 1)(b + 1) (b − 1)2 ) (pk(z)) 2 + .... by taking pk(z) = 1 + l1(k)z + l2(k)z 2 + ..., after some simplification, we obtain p(z) ≺ ∞∑ n=1 −2(b + 1)n−1 (b − 1)n + { ∞∑ n=1 −2n(a−b)(b + 1)n−1 (b − 1)n+1 } l1(k) + .... int. j. anal. appl. 16 (2) (2018) 247 now we see that the series ∑∞ n=1 −2(b+1)n−1 (b−1)n and ∑∞ n=1 −2n(a−b)(b+1)n−1 (b−1)n+1 are convergent and converge to 1 and a−b 2 respectively. therefore, p(z) ≺ 1 + a−b 2 l1(k)z + .... now if p(z) = 1 + ∑∞ n=1 cnz n, then by lemma 1, we have |cn| ≤ a−b 2 l1(k), n ≥ 1. (2.4) now from (2.3), we have z∂qr λ qf(z) + βz 2∂2qr λ qf(z) = r λ qf(z)p(z), which implies that z + ∞∑ n=2 {[n,q] + β[n,q][n− 1,q]}ϕn−1anzn = ( 1 + ∞∑ n=1 cnz n )( z + ∞∑ n=2 ϕn−1anz n ) . equating coefficients of zn on both sides, we have [n− 1,q]{q + β[n,q]}ϕn−1an = n−1∑ j=1 ϕj−1ajcn−j, a1 = 1. this implies that |an| ≤ 1 [n− 1,q]{q + β[n,q]}ϕn−1 n−1∑ j=1 ϕj−1 |aj| |cn−j| , a1 = 1. using (2.4), we have |an| ≤ (a−b) |l1(k)| 2[n− 1,q]{q + β[n,q]}ϕn−1 n−1∑ j=1 ϕj−1 |aj| , a1 = 1. (2.5) now we prove that (a−b) |l1(k)| 2[n− 1,q]{q + β[n,q]}ϕn−1 n−1∑ j=1 ϕj−1 |aj| ≤ n−2∏ j=0 ( |l1(k)(a−b) − 2[j,q]b| 2 [j + 1,q]{q + β[j + 2,q]} ) . (2.6) for this we use the induction method for n = 2, from (2.5), we have |a2| ≤ (a−b) |l1(k)| 2{q + β[2,q]}ϕ1 . from (2.2), we have |a2| ≤ (a−b) |l1(k)| 2{q + β[2,q]}ϕ1 . for n = 3 from (2.5), we have |a3| ≤ (a−b) |l1(k)| 2[2,q]{q + β[3,q]}ϕ2 {1 + ϕ1a2} ≤ (a−b) |l1(k)| 2[2,q]{q + β[3,q]}ϕ2 { 1 + (a−b) |l1(k)| 2{q + β[2,q]} } . int. j. anal. appl. 16 (2) (2018) 248 from (2.2), we have |a3| ≤ (a−b) |l1(k)| 2{q + β[2,q]}ϕ1 {( |(a−b)l1(k) − 2b| 2 [2,q]{q + β[3,q]}ϕ2 )} ≤ (a−b) |l1(k)| 2{q + β[2,q]}ϕ1 {( (a−b) |l1(k)| + 2 |b| 2 [2,q]{q + β[3,q]}ϕ2 )} ≤ (a−b) |l1(k)| 2 [2,q]{q + β[3,q]}ϕ2 { (a−b) |l1(k)| 2{q + β[2,q]}ϕ1 + 1 {q + β[2,q]}ϕ1 } . let the hypothesis be true for n = m. from (2.4), we have |am| ≤ (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1 n−1∑ j=1 |aj| , a1 = 1 from (2.2), we have |am| ≤ m−2∏ j=0 ( |l1(k)(a−b) − 2[j,q]b| 2 [j + 1,q]{q + β[j + 2,q]}ϕj+1 ) , n ≥ 2 ≤ m−2∏ j=0 ( |l1(k)|(a−b) + 2[j,q] 2 [j + 1,q]{q + β[j + 2,q]}ϕj+1 ) , n ≥ 2. by the induction hypothesis, we have (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1 m−1∑ j=1 ϕj−1 |aj| ≤ m−2∏ j=0 ( |l1(k)|(a−b) + 2[j,q] 2 [j + 1,q]{q + β[j + 2,q]}ϕj+1 ) . (2.7) multiplying both sides by (2.7) (a−b) |l1(k)| + 2[m− 1,q]{q + β[m,q]} 2[m− 1,q]{q + β[m,q]}ϕm−1 , we have m−2∏ j=0 ( |l1(k)|(a−b) + 2[j,q] 2 [j + 1,q]{q + β[j + 1,q]}ϕj+1 ) ≥ { (a−b) |l1(k)| + 2[m− 1,q]{q + β[m,q]} 2[m− 1,q]{q + β[m,q]}ϕm−1 } (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1 m−1∑ j=1 ϕj−1 |aj| = (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1   { (a−b)|l1(k)|+2[m−1,q]{q+β[m,q]} 2[m−1,q]{q+β[m,q]}ϕm−1 ∑m−1 j=1 ϕj−1 |aj| } + ∑m−1 j=1 ϕj−1 |aj|   ≥ (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1  |am| + m−1∑ j=1 ϕj−1 |aj|   = (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1 m∑ j=1 ϕj−1 |aj| . int. j. anal. appl. 16 (2) (2018) 249 that is, (a−b) |l1(k)| 2[m− 1,q]{q + β[m,q]}ϕm−1 m∑ j=1 ϕj−1 |aj| ≤ m−2∏ j=0 ( |l1(k)|(a−b) + 2[j,q] 2 [j + 1,q]{q + β[j + 1,q]}ϕj+1 ) . which shows that inequality (2.7) is true for n = m + 1. hence the required result. � when q → 1, λ = 0 and β = 0, then we have the following known result, proved by noor and sarfraz in [11]. corollary 2.5. a function f ∈ a and of the form (1.1) is in the class k −st [a,b] , if it satisfies the condition |an| ≤ n−2∏ j=0 ( |l1(k)(a−b) − 2jb| 2 (j + 1) ) . when λ = 0,a = 1,b = −1 and β = 0 then we have the following known result, proved by kanas and wisniowska in [6]. corollary 2.6. a function f ∈ a and of the form (1.1) is in the class k −ust [a,b] , if it satisfies the condition |an| ≤ n−2∏ j=0 ( |l1(k) + j| (j + 1) ) . when λ = 0,a = 1−2α, β = 0,b = −1 with 0 ≤ α < 1, then we have the following known result, proved by shams et al. in [18]. corollary 2.7. a function f ∈ a and of the form (1.1) is in the class sd(k,α), if it satisfies the condition |an| ≤ n−2∏ j=0 ( |l1(k)(1 −α) + j| (j + 1) ) . where 0 ≤ α < 1 and k ≥ 0. when λ = 0, β = 0,k = 0, then t1(k) = 2 and we get the following known result, proved in [4] corollary 2.8. a function f ∈ a and of the form (1.1) is in the class s∗[a,b], if it satisfies the condition |an| ≤ n−2∏ j=0 ( |(a−b) − jb| (j + 1) ) , − 1 ≤ b < a ≤ 1. when λ = 0, β = 0, a = 1 − 2α, b = −1 with 0 ≤ α < 1 and k = 0, then we have the following known result, proved by selverman in [17]. corollary 2.9. a function f ∈ a and of the form (1.1) is in the class s∗(α), if it satisfies the condition |an| ≤ n−2∏ j=0 (j − 2α) (n− 1)! , 0 ≤ α < 1. int. j. anal. appl. 16 (2) (2018) 250 theorem 2.3. let −1 ≤ b < a ≤ 1and 0 ≤ k < ∞ be fixed and let f(z) ∈ k −usq(λ,a,b,β) and is of the form (1.1) then for a complex number µ. ∣∣a3 −µa22∣∣ ≤   (a−b)l1(k) 2[2,q]{q+[3,q]β}ϕ2 ∣∣∣{2 + 2d(k)−(1+b)l1(k)2 [2 + 2d(k)−(1+b)l1(k)2 − (a−b) 2{q+β[2,q]}l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )]}∣∣∣ , (µ > δ1) , (a−b)l1(k) 2[2,q]{q+[3,q]β}ϕ2 . (δ1 ≤ µ ≤ δ2) , (a−b)l1(k) 2[2,q]{q+[3,q]β}ϕ2 [ 2d(k)−(1+b)l1(k) 2 (a−b) 2{q+β[2,q]}l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )] (µ < δ2) . (2.8) where δ1 = (ϕ1) 2 ϕ2(a−b)l1(k)   (q + β[2,q]){2 + 2d(k) − (1 + b)l1(k)} +(a−b)l1(k)   , (2.9) δ2 = (ϕ1) 2 ϕ2(a−b)l1(k)   (q + β[2,q]){2d(k) − (1 + b)l1(k) − 2} +(a−b)l1(k)   . (2.10) and l1(k), d(k) are defined in (1.3) and (1.4). proof. if f(z) ∈ k −usq(λ,a,b,β) then it follows that z∂qr λ qf(z) rλqf(z) + β z2∂2qr λ qf(z) rλqf(z) ≺ qk(z) = 1 + a−b 2 l1(k)z + [2d(k) − (1 + b)l1(k)] (a−b) 4 l1(k)z 2 + .... (2.11) now by the definition of subordination there exists a function w analytic in e with w(0) = 0 and |w(z)| < 1 such that z∂qr λ qf(z) rλqf(z) + β z2∂2qr λ qf(z) rλqf(z) = 1 + a−b 2 l1(k)w(z) + [2d(k) − (1 + b)l1(k)] (a−b) 4 l1(k)w 2(z) + .... (2.12) now from lemma 3, equation (2.11) and equation (2.12), we have a2 = (a−b)l1(k) 2{q + β[2,q]}ϕ1 c1, and a3 = (a−b)l1(k) 2 [2,q]{q + β[3,q]}ϕ2 { c2 + { 2d(k) − (1 + b)l1(k) 2 + (a−b) 2{q + β[2,q]} l1(k) } c21 } . int. j. anal. appl. 16 (2) (2018) 251 therefore ∣∣a3 −µa22∣∣ = (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 ∣∣∣∣c2 + { 2d(k) − (1 + b)l1(k) 2 + (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )} c21 ∣∣∣∣∣ . (2.13) this gives ∣∣a3 −µa22∣∣ = (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 ∣∣∣∣c2 − c21 + { 1 + 2d(k) − (1 + b)l1(k) 2 + (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )} c21 ∣∣∣∣∣ . (2.14) suppose that µ > δ1, then using the estimate ∣∣c2 − c21∣∣ ≤ 1 from lemma 3 and the well known estimate |c1| ≤ 1 of the schwarz lemma, we obtain∣∣a3 −µa22∣∣ ≤ (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 ∣∣∣∣ { 2 + 2d(k) − (1 + b)l1(k) 2 − (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )∣∣∣∣∣ . (2.15) the inequality (2.15) is our required assertion (2.8) for µ > δ1. on the other hand if µ < δ2, then (2.13) gives ∣∣a3 −µa22∣∣ ≤ (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 [ |c2| + { 2d(k) − (1 + b)t1(k) 2 + (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )} |c1| 2 ] . applying the estimates |c2| ≤ 1 −|c1| 2 of lemma 3 and |c1| ≤ 1, we have∣∣a3 −µa22∣∣ ≤ (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 [{ 2d(k) − (1 + b)t1(k) 2 + (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )}] . this is the last inequality in (2.8). finally if δ1 < µ < δ2, then∣∣∣∣∣2d(k) − (1 + b)l1(k)2 + (a−b)2{q + β[2,q]}l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )∣∣∣∣∣ ≤ 1. therefore (2.13), yields ∣∣a3 −µa22∣∣ ≤ (a−b)l1(k)2 [2,q]{q + β[3,q]}ϕ2 { |c2| + |c1| 2 } , ≤ (a−b)l1(k) 2 [2,q]{q + β[3,q]}ϕ2 { 1 −|c1| 2 + |c1| 2 } , ≤ (a−b)l1(k) 2 [2,q]{q + β[3,q]}ϕ2 . we get the middle inequality in (2.8). this completes the proof. � int. j. anal. appl. 16 (2) (2018) 252 theorem 2.4. let 0 ≤ k < ∞, −1 ≤ b < a ≤ 1, be fixed and let f(z) ∈ k −usq(λ,a,b,β) and is of the form (1.1) then for a complex number µ. ∣∣a3 −µa22∣∣ ≤ (a−b)l1(k)2[2,q]{q + [3,q]β}ϕ2 max{1, |2v − 1|} , where v is given by (2.17). proof. from (2.13) we have ∣∣a3 −µa22∣∣ = (a−b)l1(k)2[2,q]{q + [3,q]β}ϕ2 ∣∣∣∣c2 − { (1 + b)l1(k) − 2d(k) 2 − (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 )} c21 ∣∣∣∣∣ , = (a−b)l1(k) 2[2,q]{q + [3,q]β}ϕ2 ∣∣c2 −vc21∣∣ (2.16) where v = (1 + b)l1(k) − 2d(k) 2 − (a−b) 2{q + β[2,q]} l1(k) ( 1 −µ ϕ2 (ϕ1) 2 ) . (2.17) applying the lemma 2 on equation (2.16), we obtain the required result. � references [1] n. i. ahiezer, elements of theory of elliptic functions, moscow, 1970. [2] s. hussain, s. khan, m. a. zaighum and m. darus, certain subclass of analytic functions related with conic domains and associated with salagean q-differential operator, aims math. 2(4)(2017), 622-634. [3] s. hussain, s. khan, m. a. zaighum, m. darus and z. shareef, coefficients bounds for certain subclass of biunivalent functions associated with ruscheweyh q-differential operator, j. complex anal. 2017 (2017), article id 2826514. [4] w. janowski, some extremal problems for certain families of analytic functions, ann. polon. math. 28 (1973) 297-326. [5] s. kanas, a. wisniowska, conic regions and k-uniform convexity, j. comput. appl. math, 105 (1999), 327-336. [6] s. kanas, a. wisniowska, conic domains and starlike functions, rev. roumaine math. pures appl. 45 (2000), 647-657. [7] s. kanas, d. raducanu, some class of analytic functions related to conic domains, math. slovaca, 64(5) (2014), 1183-1196. [8] f. r. keogh, e. p. merkes, a coefficient inequality for certain classes of analytic functions, proc. amer. math. soc, 20 (1969), 8-12. [9] n. khan, b. khan, q. z. ahmad and s. ahmad, some convolution properties of multivalent analytic functions, aims math. 2 (2) (2017), 260–268. [10] w. ma, d. minda, a unified treatment of some special classes of univalent functions. in: proc. of the conference on complex analysis (tianjin), 1992 (z. li, f. y. ren, l. yang, s. y. zhang, eds.), conf. proc. lecture notes anal., vol. 1, int. press, massachusetts, 1994, 157-169. [11] k. i. noor, s. n. malik, on coefficient inequalities of functions associated with conic domains, comput. math. appl, 62(2011), 2209-2217. [12] k. i. noor, j. sokol and q. z. ahmad, applications of conic type regions to subclasses of meromorphic univalent functions with respect to symmetric points, rev. r. acad. cienc. exactas fs. nat., ser. a mat. 111 (2017), 947c958. int. j. anal. appl. 16 (2) (2018) 253 [13] k. i. noor, j. sokól and q. z. ahmad, applications of the diffierential operator to a class of meromorphic univalent functions. j. egyptian math. soc. 24 (2) (2016), 181-186. [14] m. nunokawa, s. hussain, n. khan and q. z. ahmad, a subclass of analytic functions related with conic domain, j. clas. anal. 9 (2016), 137-149. [15] w. rogosinski, on the coefficients of subordinate functions, proc. lond. math. soc, 48 (1943), 48-82. [16] s. t. ruscheweyh, new criteria for univalent functions, proc. amer. math. soc, 49 (1975), 109-115. [17] h. selverman, univalent functions with negative coefficients, proc. amer. math. soc, 51 (1975), 109-116. [18] s. shams, s. r. kulkarni, j. m. jahangiri, classes of uniformly starlike and convex functions, int. j. math. math. sci, 55 (2004) 2959-2961. 1. introduction 2. main results references international journal of analysis and applications volume 17, number 3 (2019), 396-405 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-396 existence of time-scale class of three dimensional fractional differential equations rabha w. ibrahim1,∗, and maslina darus2 1ieee:94086547 2centre of modelling and data science, faculty of sciences and technology, universiti kebangsaan malaysia,43600 bangi, selangor, malaysia; maslina@ukm.edu.my ∗corresponding author: rabhaibrahim@yahoo.com abstract. the holomorphic results for fractional differential operator formals have been established. the analytic continuation of these outcomes has been studied for the fractional differential formal   ∂αυ(℘,z) ∂℘α = h(℘,z,υ, ∂υ ∂z , ∂ 2υ ∂z2 ), α ∈ [0, 1) υ(a,z) = ψ(z), in a proximity to z ∈ u, where u is the open unit disk. the benefit of such a problem is that a generalization of two significant problems: the cauchy problem and the diffusion problem. moreover, the analytic solution is given inside the open unit disk, this leads to discuss the solution geometrically. the upper bound of outcomes is determined by suggesting a majorant analytic function in u (for two functions characterized by a power series, a majorant is the summation of a power series with positive coefficients which are not less than the absolute values of the conforming coefficients of the assumed series). this technique is very useful in approximation theory. received 2019-01-13; accepted 2019-02-25; published 2019-05-01. 2010 mathematics subject classification. 34a12,45c30. key words and phrases. fractional calculus; fractional differential equation; fractional operator. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 396 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-396 int. j. anal. appl. 17 (3) (2019) 397 1. introduction time scales calculus [1] cards us to teaching the dynamic equations, which contains both differences and differential equations, both of which are substantial in understanding applications. the dynamical behavior of different classes of fractional operating formals on time scales is presently experiencing active studies. several authors considered the existence and uniqueness solutions for problems involving classical fractional derivative (see [2][10]). holomorphic solution for some complex fractional classes is given in [5][7]. in this work, we use a majorant technique of analytic functions to prove the convergent of outcomes. we generalize some properties by applying the concept of classic fractional derivative formal operator. our construction is furnished by the riemann-liouville fractional operators. definition 1. the riemann-liouville fractional integral formal of the function φ of arbitrary order α > 0 is given by iαa φ(℘) = ∫ ℘ a (℘− τ)α−1 γ(α) φ(τ)dτ. definition 2. the riemann-liouville fractional differential formal of the function φ of arbitrary order α ∈ [0, 1) is given by dαaφ(℘) = d d℘ ∫ ℘ a (℘− τ)−α γ(1 −α) φ(τ)dτ = d d℘ i1−αa φ(℘). definition 3. [8] the majorant formula is given by : σ(χ) = ∑ σiχ i and λ(χ) = ∑ λiχ i, then σ(χ) � λ(χ) if and only if |σi| ≤ |λi| for each i. similarly, if ρ(℘,χ) = ∑ ρik(℘−ε)iχk and θ(t,χ) = ∑ θik(℘−ε)iχk, then ρ(℘,χ) �ε θ(℘,χ) if and only if |ρik| ≤ θik for all i and k. define the family of majorant functions: for each k ∈ n, we set ξ(k)ν (z) = ∞∑ n=0 zn ν(n + 1)k+2 , (|z| < 1,ν ≥ 1). (1.1) clearly that for every k ∈ n,ν ≥ 1, the functional ξ(k)ν converges for all values |z| < 1. further, this functional has some significant majorant correlations as follows: int. j. anal. appl. 17 (3) (2019) 398 proposition 1. the following inequalities hold. (i) ξ(0)ν (z)ξ (0) ν (z) � ξ (0) ν (z); (ii) ξ(0)ν (z) � ξ (1) ν (z) � ξ (2) ν (z) �··· ; (iii) 1 2i+2 ξ(i−1)ν (z) � d dz ξ(k)ν (z) � ξ (k−1) ν (z) and 1 2k+2 ξ(k−2)ν (z) � d2 dz2 ξ(k)ν (z) � ξ (k−2) ν (z); (iv) ξ(k)ν (z)ξ (k) ν (z) · · ·ξ (k) ν (z) � ξ (k) ν (z); (v) 1 1 −εz ξ(k)ν (z) � ci,εξ (k) ν (z), (ε ∈ (0, 1), and ck,ε ∈ (0,∞)); (vi) ξ (k−2) ν (z) (2ν)3(k+2) � dαa ξ (k) ν (z), for sufficient large ν ≥ 1. proof. by employing the formula expansion of ξ (k) ν (z), inequalities (i) and (ii) are achieved. according to the following inequalities: 1 ν2k+2(n + 1)k+1 = n + 1 ν2k+2(n + 1)k+2 = n + 1 ν(2n + 2)k+2 ≤ n + 1 ν(n + 2)k+2 and n + 1 ν(n + 2)k+2 < n + 2 ν(n + 2)k+2 = 1 ν(n + 2)k+1 ≤ 1 ν(n + 1)k+1 , we get (iii). similarly for (iv). to show (v), by arbitrary choice of ε, we assume that εn ≤ ck,ε ν(n + 1)k+2 which leads to 1 1 −εz = ∞∑ n=0 εnzn � ci,εξ(k)ν (z). this implies that for all n 1 1 −εz ξ(k)ν (z) � ci,εξ (k) ν (z). int. j. anal. appl. 17 (3) (2019) 399 finally, by using the relation dαa ξ (k) ν (z) = ∞∑ n=0 γ(n + 1) γ(n + 1 −α)(n + 1)k+2 zn−α, (|z| < 1) we get inequality (vi) for sufficient large ν ≥ 1. � in the same manner of proposition 1, we have the following result: proposition 2. if φ(z) is holomorphic in a proximity of |z| ≤ r0, then φ(z) is majorized by φ(z) � m 1 − ( z r0 )2 � m 1 − (εz r )2 × ξ(k)ν ( z r ) � mck,εξ(k)ν ( z r ), for any 0 < r < εr0. 2. fractional operator formal assume that h(℘,z,υ,v,w), ℘ ∈ j = [a,t] is a holomorphic function in a proximity of the four dim. point (a,b,c,d,e) ∈ j×c4, and suppose that ψ(z) is a holomorphic function in a proximity of z = b achieving ψ(b) = c, ∂ψ ∂z (b) = d and ∂2ψ ∂z2 (b) = e. consider the initial value problem   ∂αυ(℘,z) ∂℘α = h(℘,z,υ, ∂υ ∂z , ∂ 2υ ∂z2 ), υ(a,z) = ψ(z), in a proximity of z = b. (2.1) ( α ∈ [0, 1), z ∈ u, ℘ ∈ j ) eq.(2.1) has cauchy problem when h(℘,z,υ, ∂υ ∂z , ∂2υ ∂z2 ) ≡ θ(℘,z,υ, ∂υ ∂z ) and diffusion problem of fractional order when h(℘,z,υ, ∂υ ∂z , ∂2υ ∂z2 ) ≡ θ(℘,z,υ, ∂2υ ∂z2 ). theorem 3. consider the initial value problem (2.1), then it has a unique holomorphic outcome (υ(℘,z)), in a proximity of (a,b) ∈ j ×c. int. j. anal. appl. 17 (3) (2019) 400 proof. move the point (a,b) into the origin (0, 0) and change the variable as follows: ϕ(℘,z) = υ(℘,z) −ψ(z), where ϕ(℘,z) is the new variable, then we get   ∂αϕ(t,z) ∂℘α = θ(℘,z,ϕ, ∂ϕ ∂z , ∂ 2ϕ ∂z2 ), ϕ(0, 0) = 0, in a proximity of z = 0. (2.2) here, the functional θ(℘,z,ϕ, ∂ϕ ∂z , ∂ 2ϕ ∂z2 ) is holomorphic in a proximity of the origin in i ×c4, ℘ ∈ i = [0, 1]. therefore, it is sufficient to consider (2.2). let the above equation has a unique outcome: ϕ(℘,z) = ∞∑ k=0 ϕk(z) ℘ k, (℘ ∈ i). we show that ϕ(℘,z) converges. let r0 > 0 and ρ > 0 be small enough and the function θ(℘,z,ϕ,v,w) be holomorphic in a proximity of the set s = { (℘,z,ϕ,v,w) ∈ i ×c4; ℘ < τ ≤ 1, |z| ≤ r0, |ϕ| ≤ ρ, |v| ≤ ρ and |w| ≤ ρ } . assume that θ is bounded by m in this domain. since θ is holomorphic, then it has the following construction: θ(℘,z,ϕ,v,w) = ∑ p,q,s,l ap,q,s,l(z)℘ pϕqvswl, ( ℘ ∈ i, (ϕ,v,w) ∈ (c×c×c) ) . in virtue of the cauchy’s inequality and the certainty that the coefficient ap,q,s(z) is holomorphic in a proximity of {z ∈ c; |z| ≤ r0}, implies that ap,q,s,l(z) � m τpρq+s+l 1 1 − ( z r0 )2 . (2.3) in this case, the problem turns to evaluate a function ϑ(℘,z) satisfying the majorant inequalities int. j. anal. appl. 17 (3) (2019) 401   ∂αϑ(℘,z) ∂℘α � ∑ p,q,s,l m τpρq+s+l 1 1−( z r0 )2 ℘pϑq(∂ϑ ∂z )s(∂ 2ϑ ∂z2 )l, ℘ ∈ i ϑ(0, 0) � 0, (2.4) then the function ϑ(℘,z) majorizes the formal solution ϕ(℘,z). assume 0 < r < r0 and define ϑ(℘,z) = lξ(2)ν ( ℘ + ( z r )2 ) , (l > 0 ). (2.5) operating by the fractional differential formal with respect to ℘ we get ∂αϑ(℘,z) ∂℘α = l ∂αξ (2) ν ( ℘ + (z r )2 ) ∂℘α , (l > 0 ). (2.6) then by proposition 1 (vi) we get ∂αϑ(℘,z) ∂℘α � l cν ξ(0)ν ( ℘ + ( z r )2 ) , (2.7) where cν := (2ν) 12. for a constant k0 > 0 again in virtue of proposition 1 (ii) and (iii) we find ∑ p,q,s,l m ρq+s+l 1 1 − ( z r0 )2 1 1 − ℘ τ ϑq( ∂ϑ ∂z )s( ∂2ϑ ∂z2 )l � ∑ p,q,s m ρq+s 1 1 − ( z r0 )2 − ℘ τ {lξ(2)ν ( ℘ + ( z r )2 ) }q ×{ 2r0l r2 ξ(0)ν ( ℘ + ( z r )2 ) }s{ 2r0l r2 ξ(0)ν ( ℘ + ( z r )2 ) }l � ∑ p,q,s m ρq+s 1 1 − ( z r0 )2 − ℘ τ {lξ(0)ν ( ℘ + ( z r )2 ) }q ×{ 2r0l r2 ξ(0)ν ( ℘ + ( z r )2 ) }s{ 2r0l r2 ξ(0)ν ( ℘ + ( z r )2 ) }l � mk0 1 −l/ρ− 4lr0/ρr2 ξ(0)ν ( ℘ + ( z r )2 ) , (2.8) whenever, l ρ + 4r0l r2 < 1. comparing (2.7) and (2.8) with the inequality l cµ ≥ mk0 1 −l/ρ− 4lr0/ρr2 (2.9) then we obtain the majorant inequalities in (2.4) are achieved. note that relation (2.9) holds by choosing a sufficiently small l,r0, such that l ρ + 4r0l r2 < 1. int. j. anal. appl. 17 (3) (2019) 402 hence, ϑ(℘,z) in (2.5) majorizes the formal solution ψ(℘,z). this now implies that ϕ(℘,z) converges in a domain containing {(℘,z) ∈ i ×c; |℘ + (z r )2| < 1}. � 3. continuation outcomes suppose that ω is a proximate of the origin (0, 0) and h(℘,z,υ,v,w), ℘ ∈ i, is a holomorphic function in ω ×cυ ×cv ×cw. consider the following equation: ∂αυ ∂℘α = h(℘,z,υ, ∂υ ∂z , ∂2υ ∂z2 ). (3.1) then we have h(℘,z,υ,v) = ∑ j,p,q aj,p,q(℘,z)υ jvpwq. (3.2) define the following two sets: s0 = {(j,p,q) ∈ n3; aj,p,q(℘,z) 6= 0} and s = {(j,p,q) ∈ s0; j + p + q ≥ 3}. clearly, h is linear if and only if s = ∅; and it is nonlinear otherwise. suppose that h is nonlinear, this implies that s is nonempty with the coefficients formal aj,p,q(℘,z) = ℘ kj,p,qbj,p,q(℘,z), (3.3) where kj,p,q is a non-negative integer and bj,p,q(0,z) = 0. applying (3.1), we get ∂αυ ∂℘α = ∑ j,p,q ℘kj,p,qbj,p,q(℘,z)υ j( ∂υ ∂z )p( ∂2υ ∂z2 )q. (3.4) for κ ∈ r, we have δ(κ) := inf (j,p,q)∈s ( kj,p,q + 1 + κ(j + p + q − 1) ) . (3.5) it is clear that, when κ = 0, we get δ(κ) ≥ 1. also, if κ > sup (j,p,q)∈s ∣∣∣∣ (kj,p,q + 1)j + p + q − 1 ∣∣∣∣ , then δ(κ) is positive. next, we show that υ(℘,z) = o(℘κ) is analytically continued up to a proximate of the origin. int. j. anal. appl. 17 (3) (2019) 403 theorem 4. let υ(℘,z) be a holomorphic solution of (3.4) in ω. then for some κ ∈ r achieving δ(κ) > 0, such that sup z∈c |υ(℘,z)| = o(℘κ), (℘ → 0), and the solution υ(℘,z) can be expanded analytically as a holomorphic outcome of eq. (3.4) up to a approximate of the origin. proof. suppose that υ(℘,z) is a solution for eq. (3.4), which is holomorphic in the domain ω. in addition, we let the expansion (3.2) is valid in the domain ∆ where ∆ := {(℘,z,υ,v,w) : ℘ ≤ 2τ, |z| ≤ 2r, |υ| ≤ ρ, |v| ≤ ρ, |w| ≤ ρ}, such that τ < 1, 2r < 1 and ρ is positive number. let m be a bound of f in ∆. now we suggest the following initial value problem in χ(℘,z) := ∞∑ k=0 χk(z)(℘−ε)k :   ∂αχ(℘,z) ∂℘α = ∑ j,p,q ℘ kj,p,qbj,p,q(℘,z)χ j(∂χ ∂z )p(∂ 2χ ∂z2 )q, ℘ ∈ i χ(ε,z) = υ(ε,z). (3.6) we aim to show that the formal χ(℘,z) converges in a domain including the origin. this leads to υ(℘,z) can be continued analytically by χ(℘,z) up to some neighborhood of the origin. but υ(℘,z) = o(℘κ) as ℘ → 0, this implies that there exists a positive constant c0 > 0 such that |u(ε,z)| ≤ c0ε κ uniformly in z. thus, by proposition 2, for some positive constant c1 > 0, we get υ(ε,z) � c0εκc1ξ(2)ν ( ℘−ε cτ + ( z r )2). (3.7) assume that 0 < κ < 1 (without lose generality). to formulate an inequality satisfying the majorant function, we majorize the expression ℘kj,pbj,p(℘,z) by using ξ (0) ν (z). let λ := ( z r )2 + ℘−ε cτ , then ℘ is majorized by ℘ = ε + (℘−ε) �ε ( ε + 4cτ )( 1 + ℘−ε 4cτ ) �ε ( ε + 4cτ ) ξ(0)ν (λ). (3.8) we proceed to extend the function bj,p,q(℘,z) as follows int. j. anal. appl. 17 (3) (2019) 404 bj,p,q(℘,z) = ∞∑ m=0 b (m) j,p,q(z)℘ m, where each b (m) j,p,q is holomorphic in some domain of {|z| ≤ 2r} and achieves |b(m)j,p,q(z)| ≤ m ρj+p+q(2τ)m+kj,p,q . this estimate poses, b (m) j,p,q(z) � c0c1ξ (0) ν (λ) ρj+p+q(2τ)m+kj,p,q (3.9) where c1 is a positive constant satisfying (3.7). joining relations (3.8) and (3.9) and using proposition 1 (i), we get ℘kj,p,qbj,p,q(℘,z) �ε ∞∑ m=0 [ (ε + 4cτ)ξ(0)ν (λ) ]m+kj,p,q[ c0c1ξ(0)ν (λ) ρj+p+q(2τ)m+kj,p,q ] �ε c0c1 ρj+p+q ξ(0)ν (λ) ∞∑ m=0 (4c)m+kj,p,q. (3.10) putting ε = cτ 2 and 0 < c ≤ 1 and fixing r so that cr < 1/4, 0 < c ≤ 1, we finally get the formal ℘kj,p,qbj,p,q(℘,z) �ε 2c0c1 ρj+p+q ξ(0)ν (λ). therefore, the function w(℘,z) achieves the majorant conclusion   ∂αw ∂℘α �ε ∑ j,p,q 2c0c1 ρj+p+q ξ (0) ν (λ)w j(∂w ∂λ )p(∂ 2w ∂λ2 )q, ℘ ∈ i w(ε,z) �ε εκc0c1 ξ (2) ν (λ) (3.11) is one majorant function for the formal solution χ(℘,z). in the similar manner of the proof of theorem 3 by choosing suitable values for ρ > 0, c > 0, and letting ε = cτ 2 , we have w(℘,z) = εκc0c1ξ (2) ν (λ) achieves the majorant formal given in (3.11). thus, w(℘,z) is holomorphic in a domain involving (0,0); consequently must be true for χ(℘,z). � acknowledgments the authors would like to express their thanks to the reviewers for their important and useful comments to improve the paper. the work here is partially supported by the universiti kebangsaan malaysia grant: gup (geran universiti penyelidikan)-2017-064. int. j. anal. appl. 17 (3) (2019) 405 references [1] m. bohner, a. peterson, dynamic equations on times scale, birkhauser boston, boston, ma, 2001. [2] a. a. kilbas, h. m. srivastava and j.j. trujillo, theory and applications of fractional differential equations. north-holland, mathematics studies, elsevier 2006. [3] j. sabatier, o. p. agrawal, j. a. tenreiro machado, advance in fractional calculus: theoretical developments and applications in physics and engineering, springer, 2007. [4] d. baleanu, b. guvenc ziya, j. a. tenreiro, new trends in nanotechnology and fractional calculus applications, springer, 2009. [5] r. w. ibrahim, existence and uniqueness of holomorphic solutions for fractional cauchy problem, j. math. anal. appl. 380 (2011), 232-240. [6] r. w. ibrahim, on holomorphic solutions for nonlinear singular fractional differential equations, comput. math. appl. 62 (3) (2011), 1084-1090. [7] r. w. ibrahim, generalized ulam-hyers stability for fractional differential equations, int. j. math. 23 (5) (2012), art. id 1250056. [8] r. w. ibrahim, the fractional differential polynomial neural network for approximation of functions, entropy 15 (10) (2013), 4188-4198. [9] r. w. ibrahim, et al., fractional differential texture descriptors based on the machado entropy for image splicing detection. entropy 17 (7) (2015), 4775-4785. [10] r. w. ibrahim and m. darus, analytic study of complex fractional tsallis entropy with applications in cnns, entropy 20 (10) (2018), art. id 722. 1. introduction 2. fractional operator formal 3. continuation outcomes acknowledgments references international journal of analysis and applications volume 16, number 1 (2018), 50-61 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-50 on the limited p-schur property of some operator spaces m.b. dehghani1, s.m. moshtaghioun1,∗ and m. dehghani2 1department of mathematics, yazd university, p. o. box 89195-741, yazd, iran 2department of pure mathematics, faculty of mathematical sciences, university of kashan, p. o. box 87317-53153, kashan, iran ∗corresponding author: moshtagh@yazd.ac.ir abstract. we introduce and study the notion of limited p-schur property (1 ≤ p ≤∞) of banach spaces. also, we establish some necessary and sufficient conditions under which some operator spaces have the limited p-schur property. in particular, we prove that if x and y are two banach spaces such that x contains no copy of `1 and y has the limited p-schur property, then k(x, y ) (the space of all compact operators from x into y ) has the limited p-schur property. 1. introduction a non-empty subset k of a banach space x is said to be limited (resp dunford-pettis (dp)), if for every weak∗-null (resp. weakly null) sequence (x∗n) in the dual space x ∗ of x converges uniformly on k, that is, lim n→∞ sup x∈k |〈x,x∗n〉| = 0 where 〈x,x∗〉 denotes the duality between x ∈ x and x∗ ∈ x∗. in particular, a sequence (xn) ⊂ x is limited if and only if 〈xn,x∗n〉→ 0, for all weak∗-null sequences (x∗n) in x∗. received 11th september, 2017; accepted 27th november, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 47l05; 46b25. key words and phrases. schur property; p-schur property; limited p-schur property; limited p-converging; weakly p-compact. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 50 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-50 int. j. anal. appl. 16 (1) (2018) 51 a subset k of a banach space x is a limited set if and only if for any banach space y , every pointwise convergent sequence (tn) ⊂ l(x,y ) converges uniformly on k, where l(x,y ) denoted the space of all bounded operators from x into y [17, corollary 1.1.2]. it is easily seen that every relatively compact subset of a banach space is limited. but the converse is not true, in general. if every limited subset of banach space x is relatively compact, then x has the gelfand-phillips (gp) property. for example, the classical banach space c0 and `1 have the gp property and every reflexive space and dual space containing no copy of `1 have the same property. a sequence (xn) in banach space x is called weakly p-summable with 1 ≤ p < ∞, if for each x∗ ∈ x∗, the sequence (〈xn,x∗〉) ∈ `p and a sequence (xn) in x is said to be weakly p-convergent to x ∈ x if the sequence (xn − x) ∈ `weakp (x), where `weakp (x) denoted the space of all weakly p-summable sequence in x. also a bounded set k in a banach space is said to be relatively weakly p-compact, 1 ≤ p ≤ ∞ if every sequence in k has a weakly p-convergent subsequence. if the limit point of each weakly p-convergent subsequence is in k, then we call k weakly p-compact set. also, a banach space x is weakly p-compact if the closed unit ball bx of x is a weakly p-compact set. an operator t ∈ l(x,y ) is said to be p-converging if it transfers weakly p-summable sequence into norm null sequences. the class of all p-converging operators from x into y is denoted by cp(x,y ). an operator t ∈ l(x,y ) is limited p-converging if it transfers limited and weakly p-summable sequences into norm null sequences. we denote the space of all limited p-converging operators from x into y by clp(x,y ) [7]. a banach space x has the schur property if every weakly null sequence in x converges in norm. the simplest banach space with the schur property is `1. also a banach space x has the p-schur property (1 ≤ p ≤∞) if every weakly p-summable subset of x is compact. in other words, if 1 ≤ p < ∞, x has the p-schur property if and only if every sequence (xn) ∈ `weakp (x) is a norm null sequence, for example, `p has the 1-schur property. moreover, x has the ∞-schur property if and only if every sequence in cweak0 (x) in norm null where cweak0 (x) containing all weakly null sequences in x. so ∞-schur property coincides with the schur property. also one note that every schur space has the p-schur property [6]. the reader is referred to [2, 11, 14–16] for more information about these concepts. in this note, we study the limited p-schur property of some operator spaces, specially, the space of compact operators. we prove that if x and y are two banach spaces such that x contains no copy of `1 and y has the limited p-schur property, then k(x,y ) has the limited p-schur property. finally, we conclude that if (xα)α∈i are banach spaces and x = (⊕α∈ixα)1 their `1-sum, then the space x has the p-schur property if and only if each factor xα has the same property. int. j. anal. appl. 16 (1) (2018) 52 2. main results recall that the banach space x has the limited p-schur property if every limited weakly p-compact subset of x is relatively compact. more precisely, the banach space x has the limited p-schur property if and only if every limited sequence (xn) ∈ `weakp (x) is norm null. it is easy to see that every banach space with the p-schur property and every banach space with gp property is limited p-schur [7]. moreover, a banach space x has the gp property if and only if every limited weakly null sequence in x is norm null [2, proposition 6.8]. therefore the limited schur (i.e., limited ∞-schur ) property is equivalent to the gp property. also, if a banach space x have the limited p-schur and dp∗p properties, then it has the p-schur property. indeed, a banach space x is said to have the dp∗-property of order p (dp∗p ) if all weakly p-compact sets in x are limited [10]. recall that if m is a closed subspace of l(x,y ), then for arbitrary elements x ∈ x and y∗ ∈ y ∗, the evaluation operators φx : m → y and ψy∗ : m → x∗ on m are defined by φx(t) = tx, ψy∗ (t) = t ∗y∗, (t ∈ m). theorem 2.1. let x and y be two banach spaces such that x is weakly p-compact and y has the p-schur property. then l(x,y ) has the limited p-schur property. proof. suppose that (tn) is a limited weakly p-summable sequence in l(x,y ). we have to prove that (tn) is norm null. we first observe that for every x ∈ x the evaluation operator φx from l(x,y ) to y maps the sequence (tn) to the sequence (tnx). so the latter is also a limited weakly p-summable sequence in y . therefore ‖tnx‖→ 0, since y has the limited p-schur property. now, suppose that (tn) is not norm null. then there is a sequence (xn) in x and ε > 0 such that ‖tnxn‖ > 2ε, for all n ∈ n. since x is weakly p-compact we may assume that there exists x ∈ x such that (xn − x) ∈ `weakp (x). as ‖tnx‖ → 0, we may finally suppose that fn = ‖tnxn −tnx‖ > ε for all n ∈ n. now, choose functional y∗n in by ∗ so that 〈tnxn −tnx,y∗n〉 = fn, and define λn ∈ l(x,y )∗ by 〈t, λn〉 = 〈txn −tx,y∗n〉, for all t ∈ l(x,y ). since |〈t, λn〉| ≤ ‖txn −tx‖ → 0, because (xn −x) ∈ `weakp (x), we see that (λn) is a weak∗-null sequence. but 〈tn, λn〉 = fn > ε > 0 for all n ∈ n. contradicting the assumption that (tn) is limited. � corollary 2.1. let x and y be two banach spaces. if x is reflexive and y has the schur property, then l(x,y ) has the gp property. int. j. anal. appl. 16 (1) (2018) 53 proof. let p = ∞ in theorem 2.1. � corollary 2.2. let x and y be two banach spaces. if x is a weakly p-compact and y ∗ has the p-schur property, then (x⊗̂πy )∗ has the limited p-schur property. proof. it follows easily from the fact that l(x,y ∗) = (x⊗̂πy )∗. � corollary 2.3. let x and y be two banach spaces. if x∗ has the p-schur property and y ∗ is weakly p-compact, then l(x,y ) has the limited p-schur property. proof. the mapping t 7→ t∗ maps l(x,y ) onto a closed subspace of l(y ∗,x∗), which has the limited p-schur property by virtue of theorem 2.1. � in the following theorem we give a necessary and sufficient condition for which a banach space has the limited p-schur property. theorem 2.2. the banach space x has the limited p-schur property if and only if l(x,y ) = clp(x,y ), for every banach space y . proof. suppose that x has the limited p-schur property. if t ∈ l(x,y ) and (xn) ∈ `weakp (x) is a limited sequence, then ‖xn‖→ 0. hence ‖txn‖→ 0. conversely, if y = x, then the identity operator on x is belongs to clp. therefore x has the limited p-scuhr property. � similarly, we can prove that the banach space x has the limited p-schur property if and only if l(y,x) = clp(y,x) for every banach space y . theorem 2.3. if x∗ has the limited p-schur property and y has the schur property, then l(x,y ) has the limited p-schur property. proof. since x∗ has the limited p-schur property, theorem 2.2 implies that each ψy∗ : l(x,y ) → x∗ is limited p-converging. it follows that l(x,y ) has the limited p-schur property. in fact, if l(x,y ) does not have the limited p-schur property, then there exists a limited weakly p-summable sequence (tn) ⊆ l(x,y ) such that ‖tn‖ > ε for all n ∈ n and some ε > 0. choose a sequence xn ∈ bx such that ‖tnxn‖ > ε. on the other hand, ψy∗ is limited p-converging, for all y ∗ ∈ y ∗. therefore ‖t∗ny∗‖ = ‖ψy∗tn‖ → 0. it follows that |〈tnxn,y∗〉| ≤ ‖t∗ny ∗‖‖xn‖→ 0. hence (tnxn) is weakly null and so is norm null. this contradiction shows that l(x,y ) has the limited p-schur property. � int. j. anal. appl. 16 (1) (2018) 54 example 2.1. if x∗ has the limited p-schur property, then `weak1 (x ∗) has the same property. indeed, if one denote `weak ∗ 1 (x ∗) as the space of all sequences (x∗n) ⊂ x∗ such that (〈x,x∗n〉) ∈ `1, for all x ∈ x, then by [5, p. 427], `weak1 (x ∗) = `weak ∗ 1 (x ∗). also, `weak ∗ 1 (x ∗) is isometrically isomorphism to l(x,`1); see e.g., [8, proposition 19.4.3]. since `1 has the schur property, it follows that l(x,`1) = ` weak 1 (x ∗) has the limited p-schur property. if we take p = ∞ in theorem 2.3 we obtain the following result. corollary 2.4. if x∗ has the gp property and y has the schur property, then l(x,y ) has the gp property. theorem 2.4. let x and y be banach spaces. if x has the limited p-schur property and y has the gp property, then the space kw∗ (x ∗,y ) of all compact weak∗-weak continuous operators from x∗ into y has the limited p-schur property. proof. let (tn) be a limited weakly p-summable sequence in kw∗ (x ∗,y ). we have to show that ‖tn‖→ 0. we can choose a sequence (x∗n) in x ∗ such that ‖x∗n‖ = 1 and ‖tnx∗n‖ ≥ 1 2 ‖tn‖ for all n ∈ n. now, we prove that (tnx ∗ n) is weakly null limited sequence in y . fix any y ∗ ∈ y ∗. then for all t ∈ kw∗ (x∗,y ), the operator y∗◦t is a weak∗ continuous linear functional on x∗ so that y∗◦t ∈ x ⊂ x∗∗. thus the operator t 7→ y∗ ◦t from kw∗ (x∗,y ) into x shows that the sequence (y∗ ◦tn) is limited weakly p-summable in x. so ‖y∗ ◦tn‖→ 0 and for each y∗ ∈ y ∗ we have 〈y∗,tnx∗n〉 = 〈y ∗ ◦tn,x∗n〉→ 0 and so (tnx ∗ n) is weakly null. now, assume that (y∗n) is a weak ∗-null sequence in y ∗ and define a sequence (λn) in kw∗ (x ∗,y )∗ by 〈t, λn〉 = 〈tx∗n,y∗n〉. if t ∈ kw∗ (x∗,y ), then t(bx∗ ) is relatively compact and so it is a limited set in y . it follows that lim n→∞ sup x∗∈bx∗ 〈tx∗,y∗n〉 = 0. therefore ‖y∗n ◦ t‖ → 0. thus 〈y∗n ◦ t,x∗n〉 → 0 and so (λn) is weak∗-null in kw∗ (x∗,y )∗. since (tn) is limited, we have 〈tnx∗n,y ∗ n〉 = 〈tn, λn〉→ 0 and so (tnx ∗ n) is limited. finally, the gp property of y yields that ‖tnx∗n‖→ 0 which implies ‖tn‖→ 0. � note that the map t 7→ t∗∗ is an isometric isomorphism from k(x,y ) into kw∗ (x∗,y ). therefore we have the following result. corollary 2.5. let x and y be two banach spaces. if x∗ has the limited p-schur property and y has the gp property, then k(x,y ) has the limited p-schur property. int. j. anal. appl. 16 (1) (2018) 55 since x⊗̂εy may be identified with a closed subspace of kw∗ (x∗,y ) via the isometric embedding x⊗̂εy ↪→ kw∗ (x∗,y ) which is defined by x ⊗ y 7→ θx⊗y, where θx⊗y(x∗) = 〈x,x∗〉y, we have the following corollary. corollary 2.6. if x has the limited p-schur property and y has the gp property, then injective tensor product x⊗̂εy has the limited p-schur property. theorem 2.5. [9, 13] let x and y be two banach spaces and m ⊆ k(x,y ) such that for all x ∈ x, m(x) := {tx : t ∈ m} is relatively compact in y . then under each of the following conditions, m is a relatively compact subset of k(x,y ). (a) x∗∗ has the gp property and for every weak∗-null sequence (x∗∗n ) ⊆ x∗∗, (t∗∗x∗∗n ) is norm null uniformly with respect t ∈ m. (b) x contains no copy of `1 and for every weakly null sequence (xn) ⊆ x, (txn) is norm null uniformly with respect t ∈ m. recall that the operator t ∈ l(x,y ) is said to be limited operator if t(bx) is a limited set in y . the class of all limited operator from x into y is denoted by l(x,y ). on the other hand, t ∈ l(x,y ) if and only if t∗ : y ∗ → x∗ is weak∗-norm sequential continuous cf. [2]. theorem 2.6. let x be a banach space such that x∗ has the gp property. if f is a closed subspace of k(x,y ) and for every x∗∗ ∈ x∗∗, the evaluation operator φx∗∗ on f is limited p-converging, then f has the limited p-schur property. proof. first, observe that the evaluation operator φx∗∗ , as a generalization of φx is denoted by φx∗∗ (t) = t∗∗x∗∗, for all t ∈ m and x∗∗ ∈ x∗∗. let m ⊂ f be a limited weakly p-compact set. since for every x ∈ x, the evaluation map φx is limited pconverging, we conclude that m(x) = {tx : t ∈ m} is relatively compact. since the adjoint of every limited operator is weak∗-norm sequentially continuous, it follows that for every compact operator t ∈ k(x,y ), the operator t∗ is also compact and so is limited. this shows that t∗∗ is weak∗-norm sequentially continuous and therefore for each weak∗-null sequence (x∗∗n ) in x ∗∗, the sequence (t∗∗x∗∗n ) is norm null, that is φx∗∗ is a pointwise norm null sequence of bounded linear operators. hence (φx∗∗n ) converges uniformly on the limited set m [17, corollary 1.1.2]. it follows that lim n→∞ sup t∈m ‖φx∗∗n (t)‖ = 0. then by theorem 2.5 (a) m is relatively compact and so f has the p-schur property. � if one use theorem 2.5 (b) instead of theorem 2.5 (a), we can prove the following theorem. int. j. anal. appl. 16 (1) (2018) 56 theorem 2.7. let x be a banach space containing no copy of `1. if f is a closed subspace of k(x,y ) such that for each x ∈ x, the evaluation operator φx is limited p-converging, then f has the limited p-schur property. recall that a subset h of l(x,y ) is uniformly completely continuous, if for every weakly null sequence (xn) in x, lim n→∞ sup t∈h ‖txn‖ = 0. we remember the following theorem, which has a main role in the proof of the theorem 2.9. theorem 2.8. [13] if x contains no copy of `1, then a subset h ⊆ k(x,y ) is relatively compact if and only if h is uniformly completely continuous and for each x ∈ x, the set φx(h) is relatively compact in y . theorem 2.9. if x contains no copy of `1 and y has the limited p-schur property, then k(x,y ) has the limited p-schur property. proof. if y has the limited p-schur property, then theorem 2.2 shows that each φx : k(x,y ) → y is limited p-converging. now, suppose that h ⊂ k(x,y ) is a limited weakly p-compact set. therefore φx(h) is relatively compact for all x ∈ x. on the other hand, if (xn) is weakly null in x, then complete continuity of each operator t ∈ h implies that ‖φxn (t)‖ = ‖txn‖ → 0. therefore (φxn ) is a norm null sequence at each element t ∈ h and then it is uniformly convergent on the limited set h [17, corolarry 1.1.2]. hence lim n→∞ sup t∈h ‖txn‖ = lim n→∞ sup t∈h ‖φxn (t)‖ = 0. this shows that h is uniformly completely continuous. hence theorem 2.5 (a) shows that h is relatively compact in k(x,y ) and so k(x,y ) has the limited p-schur property. � recall that if 1 ≤ p ≤ ∞, the banach space x has the dunford-pettis property of order p (dpp) if for each banach space y, every weakly compact operator t : x → y is p-converging. for more information about dpp property of banach spaces the reader is referred to [3]. corollary 2.7. if 2 < q < ∞ and 1 q + 1 q∗ = 1, then (`q⊗̂ε`q)∗ and (`q⊗̂π`q)∗ have the limited p-schur property, for all 1 < p < q. proof. since 1 < q∗ < 2 and q∗ < q < ∞, by pitt’s theorem; (see [1, theorem 2.1.4]), every bounded operator t : `q → `q∗ is compact. therefore (`q⊗̂π`q)∗ = l(`q,`q∗ ) = k(`q,`q∗ ) and (`q⊗̂ε`q)∗ = i(`q,`q∗ ) ⊂ k(`q,`q∗ ), where i(`q,`q∗ ) is the space of all integral operators from `q into `q∗ [5, p. 119]. hence it is enough to show that k(`q,`q∗ ) has the limited p-schur property, for all 1 < p < q. in fact, by [3, example 3.3] `q∗ has the dpp property, for all 1 < p < q. it follows from [6, theorem 2.31] that `q∗ has the (limited) int. j. anal. appl. 16 (1) (2018) 57 p-schur property, for all 1 < p < q. on the other hand, `q contains no copy of `1. therefore theorem 2.9 (or corollary 2.5) shows that k(`q,`q∗ ) has the limited p-schur property, for all 1 < p < q. � we also notice that by theorem 2.2, if the closed subspace m of l(x,y ) has the limited p-schur property, then all operators on m, such as evaluation operators, are limited p-converging. therefore the converse of theorem 2.6 is also true. moreover, in the following two theorems 2.11 and 2.12, we will give another sufficient conditions for the limited p-schur property of closed subspace m of some operator spaces with respect to the limited p-converging of evaluation operators. to obtain our next result we need the following well known theorem. theorem 2.10. [9] let x and y be two banach spaces and h be a subset of l(x,y ) such that (1) h(bx) = {tx : t ∈ h,x ∈ bx} is relatively compact. (2) ψy∗ (h) is relatively compact for all y ∗ ∈ y ∗. then h is relatively compact. theorem 2.11. let m be a closed linear subspace of l(x,y ) such that the closed linear span of the set m(x) = {tx : t ∈ m,x ∈ x} has the gp property. if all evaluation operator ψy∗ are limited p-converging, then m has the limited p-schur property. proof. suppose that h is a limited weakly p-compact subset of m. by theorem 2.10, it is enough to show that h(bx) and all ψy∗ (h) are relatively compact in y and x ∗, respectively. for every y∗ ∈ y ∗, the evaluation operator ψy∗ is limited p-converging. therefore ψy∗ (h) is relatively compact. on the other hand, if (y∗n) is a weak ∗-null sequence in y ∗, then the weak∗-norm sequential continuity of the adjoint of eah t ∈ h implies that ‖ψy∗n (t)‖ = ‖t ∗y∗n‖→ 0 as n →∞. therefore (ψy∗n ) converges pointwise on h an so it is converges uniformly on the subset h of m. hence sup{|〈tx,y∗n〉| : t ∈ h,x ∈ bx} = sup{|〈x,t ∗y∗n〉| : t ∈ h,x ∈ bx} = sup t∈h ‖t∗y∗n‖→ 0. thus h(bx) is limited and so is relatively compact. � now, we give a sufficient condition for the limited p-schur property of subspaces of lw∗ (x ∗,y ) of all bounded weak∗-weak continuous operator from x∗ to y . clearly, if t ∈ lw∗ (x∗,y ), then t∗ transfers y ∗ into x. the proof of this theorem is similar to the proof of theorem 3.6 of [6]. so we omit its proof. theorem 2.12. let x and y be banach spaces such that x has the schur property. if m is a closed subspace of lw∗ (x ∗,y ) such that every evaluation operator φx∗ is limited p-converging on m, then m has the limited p–schur property. int. j. anal. appl. 16 (1) (2018) 58 recall that according to [6], a bounded subset k of a banach space x is p-limited if lim n sup x∈k |〈x,x∗n〉| = 0, for every (x∗n) ∈ `weakp (x∗) . a subset k of a dual space x∗ of x is lp-set if lim n sup x∗∈k |〈xn,x∗〉| = 0 for every sequence (xn) ∈ `weakp (x). also, a sequence (x∗n) in x ∗ is an lp-set if and only if limn→∞〈xn,x∗n〉 = 0 for all (xn) ∈ `weakp (x) [7]. it is clear that for every limited subset and every p-limited subset of a dual space is an lp-set. moreover, the following result has been proved in [7]. theorem 2.13. a banach space x is weakly p-compact if and only if every lp-set in x ∗ is relatively compact. theorem 2.14. let x and y be banach spaces. if x contains no copy of `1, y ∗ is weakly p-compact and for every h ∈ l(x,y ∗∗), for every weakly null sequence (xn) ⊂ x, the sequence (hxn) is an lp-set, then k(x,y ) has the gp property and so has the limited p-schur property. proof. let m ⊂ k(x,y ) be a limited set. we have to prove that m is relatively compact. since m(x) = {tx : t ∈ m} is a limited set in y and so is an lp-set, therefore m(x) is a relatively compact set, by theorem 2.13. assume that condition (b) of theorem 2.5 in not verified. so there are a positive number ε, a weakly null sequence (xn) ⊂ x and a sequence (tn) ⊂ m such that for all n ∈ n, ||tnxn|| > ε. now we prove that (tnxn) is weakly null. for every y ∗ ∈ y ∗, the set {t∗ny∗ : n ∈ n} is a dunford-pettis subset of x∗. since (xn) is weakly null, it follows that 〈tnxn,y∗〉 = 〈t∗ny ∗,xn〉→ 0 for every y∗ ∈ y ∗. so the sequence (tnxn) is weakly null. now, we prove that (tnxn) is a p-limited set. suppose that (y ∗ n) ∈ `weakp (y ∗) and h ∈ (x⊗̂πy ∗)∗ = l(x,y ∗∗). as (hxn) is an lp-set in y ∗∗ we have h(xn ⊗ y∗n) = 〈hxn,y∗n〉 → 0 and so (xn ⊗ y∗n) is weakly null in x ⊗π y ∗. since x⊗̂πy ∗ embeds into k(x,y )∗, it follows that (xn ⊗y∗n) is also weakly null in space k(x,y )∗. then it must be that lim n→∞ 〈tnxn,y∗n〉 = lim n→∞ 〈tn,xn ⊗π y∗n〉 = 0, because (tn) is a limited set and so is a dp set. so we have actually proved that (tnxn) is a p-limited set and so lp-set. it follows from theorem 2.13 that it must be a relatively compact set. since it is a weakly null sequence, there is a norm null subsequence and it is a contradiction. � in [18] the authors have been proved that for banach spaces (xα)α∈i, if x = (⊕α∈ixα)1 is their `1-direct sum, then x has the schur property if and only if each factor xα has the same property. here, by a similar idea, we prove that the same condition holds for (limited) p-schur property. int. j. anal. appl. 16 (1) (2018) 59 theorem 2.15. let (xα)α∈i be banach spaces and x = (⊕α∈ixα)1. then the space x has the p-schur property if and only if each xα has the p-schur property. proof. if x = (⊕α∈ixα)1 has the p-schur property, then clearly, every closed subspace of x has the p-schur property. hence each xα has the p-schur property. on the other hand, a straightforward computations shows that a banach space has the p-schur property if and only if all of its closed separable subspaces have the p-schur property. therefore we can assume that each xα is separable and take i = n. hence x = (⊕xk)1 is separable and so has the gp property. if (xn) ∈ `weakp (x), where xn = (bn,k)k∈n, then (bn,k) ∈ `weakp (xk) for all k ∈ n. since xk has the p-schur property, therefore ||bn,k|| → 0 as n → ∞, for all k ∈ n. we have to prove that ||xn|| → 0 or the weakly null sequence (xn) is relatively compact. let {fn}n∈n be a w∗-null sequence in bx∗ . if we show that lim n→∞ 〈xn,fn〉 = 0, then the proof is completed, thanks to the gp property of x. each fn is of the form fn = (an,k)k∈n and for all k ∈ n, an,k w∗−→ 0 in x∗k as n → ∞. to prove that lim n→∞ 〈xn,fn〉 = 0, it is enough to show that sup n ∑ k>m |〈an,k,bn,k〉|→ 0 as m →∞. therefore we have to show that for each ε > 0 there exists m ∈ n such that ∑ k>m |〈an,k,bn,k〉| < ε, (2.1) for all sufficiently large enough n ∈ n. let (2.1) is false. then there is an ε > 0 such that ∑ k>m |〈an,k,bn,k〉| ≥ ε, (2.2) for all m ∈ n and some sufficiently large enough n ∈ n. consider a sequence of positive number, (δk) such that ∑∞ k=1 δk < ε 4 . by the technique given in the proof of main theorem of [18] one can construct two strictly increasing sequences, (nk)k≥1 and (mk)k≥0 such that (1) ∑ j>mk ||bnk,j|| ≤ δk for each k ≥ 1 (2) mk−1∑ j=1 |〈an,j,bnk−1,j〉| ≤ δk for each n ≥ nk (3) ∑ j>mk−1 |〈ank,j,bnk,j〉| ≥ ε. now, let us choose a sequence (λj) such that |λj| = 1, for all j and λj〈ank,j,bnk,j〉 = |〈ank,j,bnk,j〉|, where k ≥ 1 and mk−1 + 1 ≤ j ≤ mk. let h = (hj)j≥1 = (λ1an1,1,λ2an1,2, ...,λm1an1,m1,λm1+1an2,m1+1, ...). int. j. anal. appl. 16 (1) (2018) 60 then ||h|| = sup j≥1 ||hj|| ≤ 1 and 〈h,xnk〉 = ∞∑ j=1 〈hj,bnk,j〉 = k−1∑ i=1 mi∑ j=mi−1+1 λj〈ani,j,bni,j〉 + mk∑ j=mk−1+1 |〈ank,j,bnk,j〉| + ∞∑ j=mk λj〈ank,j,bnk,j〉. with due attention to ||ank,j|| ≤ 1 and inequalities (1), (2) and (3): |〈h,xnk〉| ≥ − k−1∑ i=1 δi + mk∑ j=mk−1+1 |〈ank,j,bnk,j〉|− δk ≥ − k−1∑ i=1 δi + ∑ j>mk−1 |〈ank,j,bnk,j〉|− ∑ j≤mk−1 |〈ank,j,bnk,j〉|− δk ≥ − k−1∑ i=1 δi + ∑ j>mk−1 |〈ank,j,bnk,j〉|− 2δk ≥ ε− k−1∑ i=1 δi − 2δk ≥ ε− 2 ∞∑ i=1 δi > ε 2 . this contradiction shows that (2.1) is true. so lim m→∞ sup n∈n m∑ k=1 |〈an,k,bn,k〉| = 0 therefore lim n→∞ ∑∞ k=1 |〈an,k,bn,k〉| = ∑∞ k=1 limn→∞ |〈an,k,bn,k〉| = 0 since |〈fn,xn〉| ≤ ∑∞ k=1 |〈an,k,bn,k〉| we conclude that lim n→∞ |〈fn,xn〉| = 0 and so ||xn||→ 0. � by a similar technique we have the following theorem. theorem 2.16. suppose that (xα)α∈i are banach spaces and x = (⊕α∈ixα)1. then the space x has the limited p-schur property if and only if each xα has the limited p-schur property. references [1] f. albiac and n.j. kalton, topics in banach space theory, graduate texts in mathematics, 233, springer, new york, 2006. [2] j. bourgain and j. diestel, limited operators and strict consingularity, math. nachr. 119 (1984) 55-58. [3] j. castillo and f. sanchez, dunford-pettis-like properties of continuous vector function spaces, rev. mat. univ. complut. madrid 6 (1993), no. 1, 43-59. [4] d. chen, j. alejandro chvez-domnguez, and li. lei. unconditionally p-converging operators and dunford-pettis property of order p, arxiv preprint arxiv:1607.02161 (2016). [5] a. defant and k. floret, tensor norms and operator ideals, north-holland mathematics studies, 176, north-holland publishing co., amsterdam, 1993. [6] mohammad b. dehghani and s. mohammad moshtaghioun, on the p-schur property of banach spaces, ann. funct. anal. (2017), 14 pages. int. j. anal. appl. 16 (1) (2018) 61 [7] m.b. dehghani and s.m. moshtaghioun, limited p-converging operators and its relation with some geometric properties of banach spaces, (2017), submitted. [8] j. diestel, h. jachowr and a. tonge, absolutely summing operators, cambrigde university press, 1995. [9] g. emmanuele, on relative compactness in k(x, y ), j. math. anal. appl. 379 (2013) 88-90. [10] j. h. fouire and e. d. zeekoei, dp ∗ properties of order p on banach spaces, quaest. math. 37 (2014), no. 3, 349-358. [11] i. ghencia and p. lewis, the dunford-pettis property, the gelfand-phillips property and l-set, colloq. math. 1.6 (2006), 311-324. [12] h. jarchow, locally convex spaces, b.g. teubner, 1981. [13] f. mayoral, compact sets of compact operators in absence of `1, proc. amer. math. soc. 129 (2001), 7982. [14] s.m. moshtaghioun and j. zafarani, weak sequentional convergence in the dual of operator ideas, j. oper. theory 49 (2003), 143-151. [15] a. pelczynski, banach spaces in which every unconditionally converging operator is weakly compact, bull. l’acad. polon. sci. 10 no. 2, (1962), 641-648. [16] r. ryan,the dunford-pettis property and projective tensor products, bull. polish acad. sci. 35 no. 11-12, (1987), 785-792. [17] t. schlumprecht, limited sets in banach spaces, ph. d. dissertation, münchen, (1987). [18] b. tanbay, direct sums and the schur property, turk. j. math. 22 (1999), 349-354. 1. introduction 2. main results references international journal of analysis and applications volume 17, number 4 (2019), 559-577 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-559 normal ruled surfaces of a surface along a curve in euclidean 3-space e3 r. a. abdel-baky1,2,∗ and s. h. nazra3 1deptartment of mathematics, sciences faculty for girls, jeddah university, 21352 jeddah, ksa 2department of mathematics, faculty of science, assiut university, 71516 assiut, egypt 3department of mathematics, umm al-qura university, ksa ∗corresponding author: rbaky@live.com abstract. in this paper, we define a ruled surface normal to a surface along a curve on the surface. then, we analyze the necessary and sufficient condition for that surface to be normal developable. also, we solve the problem when the resulting developable surface is a cylinder, cone or tangent surface. finally, we give some representative examples. 1. introduction in differential geometry, moving frames constitute an important tool while studying curves and surfaces. the most familiar moving frames are the frenet–serret frame along a space curve, and the darboux frame along a surface curve. in euclidean 3-space e3, the darboux frame is constructed by the velocity of the curve and the normal vector of the surface whereas the frenet–serret frame is constructed from the velocity and the acceleration of the curve. expressing the derivatives of these moving frames in terms of the frames themselves includes some real valued functions. these functions are called the curvature and the torsion for received 2019-04-20; accepted 2019-05-16; published 2019-07-01. 2010 mathematics subject classification. 53a04, 53a05, 53a17. key words and phrases. darboux frame; normal ruled surface; orthogonal trajectory. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 559 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-559 int. j. anal. appl. 17 (4) (2019) 560 the frenet–serret frame, and the normal curvature, the geodesic curvature, and the geodesic torsion for the darboux frame (see for example [1-3]). the study of surfaces with a common characteristic curve plays an important role in a diversity of applications. in practical applications, the concept of family of surfaces having a given geodesic curve was first introduced by wang et. al. [4] the basic idea is to regard the wanted surface as an extension from the given characteristic curve , and represent it as a linear combination of the marching-scale functions u(s,t), v(s,t), w(s,t) and the three vector functions t(s), n(s), b(s), which are the unit tangent, the principal normal and the binormal vector of the curve respectively. with the given geodesic curve and isoparametric constraints, they derived the necessary and sufficient conditions for the correct parametric representation of the surface pencil. the extension to ruled and developable surfaces is also outlined. this principal has been used treated extensively in the works (see for example [5-9]). properties of ruled surfaces and their offset surfaces with darboux frame are defined by senturk and yuce [11, 12]. then, the ruled surfaces characteristic properties which are related to the geodesic curvature, the normal curvature and the geodesic torsion are investigated. moreover, some theorems about the integral invariants of the ruled surface and their offset surfaces with darboux frame are given. however, the relevant work on surfaces through characteristic curve on a surface is rare. so, this led us to offer an approach for designing surfaces from a given curve on a surface. for this purpose, we consider a curve on a surface and a ruled surface to the surface along the curve. such a ruled surface is called a normal ruled surface along the curve if it exists. we have a special direction in the darboux frame at each point of the curve which is written as a linear combination of the moving darboux frame. then, the extension to developable surfaces are also outlined. meanwhile, we solved the problem when the resulting developable surface is a cylinder, cone or tangent surface. finally, we illustrate the convenience and efficiency of this approach by some representative examples. 2. preliminaries the ambient space is the euclidean 3-space e3, and for our work we have used [1-3] as general references. a ruled surface in e3 is a surface generated by a straight line l moving along a curve α(s). the various positions of the generating lines are called the rulings of the surface. such a surface, thus, has a parametrization in the ruled form: y(s,v) = α(s) + vx(s), v ∈ r, (2.1) where α(s) is called the base curve, and the line passing through α(s) that is parallel to x(s) is called the ruling of the surface at α(s). the surface y(s,v) is regular if ys×yv 6= 0 for all points, where ys and yv are the partial derivatives of y(s,v) with respect to s and v, respectively. if there exists a common perpendicular to two constructive rulings in the ruled surface, then the foot of the common perpendicular on the main int. j. anal. appl. 17 (4) (2019) 561 rulings is called a central point. the locus of the central point is called striction curve. the parametrization of the striction curve on the ruled surface is given by c(s) = α(s) − < α ′ ,x ′ > ‖x′‖2 x(s) (2.2) we call a point of the ruled surface singular if the partial derivatives ys and yv are linearly dependent. the distribution parameter for a ruled surface describes the winding speed of the tangent planes winding about the ruling. it can be determined by µ(s) = det(α ′ ,x,x ′ ) ‖x′‖2 , (2.3) which only depends on s. it’s well known that y(s,v) is a developable surface if and only if det(α ′ ,x,x ′ ) = 0. the relation between the gaussian curvature k and the distribution parameter is given by the formula: k(s,v) = − µ2 (µ2 + v2) 2 < 0. (2.4) if the ruled surface satisfies y(s,v) = y(s+2π,v) for all s ∈ i , then the ruled surface is called closed. we shall denote the closed normal ruled surface by y-closed ruled surface. a curve which intersects perpendicularly each one of rulings is called an orthogonal trajectory of the ruled surface. it is calculated by < dy,x >=0. (2.5) let α : i ⊆ r → e3 be a unit speed curve; by κ(s) and τ(s) we denote the natural curvature and torsion of α = α(s), respectively. we assume α ′′ (s) 6= 0 for all s ∈ [0,l], since this would give us a straight line. in this paper, α ′ (s) denote the derivative of α with respect to arc length parameter s. for each point of α(s), the set {t(s), n(s), b(s)} is called the serret–frenet frame along α(s), where t(s) = α ′ (s) is the unit tangent, n(s) = α ′′ (s)/ ∥∥∥α′′ (s)∥∥∥ is the unit principal normal, and b(s) = t(s) × n(s) is the unit binormal vector. the arc-length derivative of the serret–frenet frame is governed by the relations:  t ′ (s) n ′ (s) b ′ (s)   =   0 κ(s) 0 −κ(s) 0 τ(s) 0 −τ(s) 0     t(s) n(s) b(s)   , (2.6) let m be a regular surface, and α : i ⊆ r → m is a unit speed curve on m. on the surface, we have the darboux frame {α(s); e1, e2, e3}; e1(s) be the unit tangent vector to α(s), e3 = e3(s) is the surface unit normal restricted to α, and e2= e3×e1 be the unit tangent to the surface m. then, the rotation matrix between serret–frenet frame and darboux frame is  t(s) n(s) b(s)   =   1 0 0 0 cos ϑ sin ϑ 0 −sin ϑ cos ϑ     e1 e2 e3   . (2.7) int. j. anal. appl. 17 (4) (2019) 562 then we have the following darboux formulae:  e ′ 1 e ′ 2 e ′ 3   =   0 κg κn −κg 0 τg −κn −τg 0     e1 e2 e3   = ω(s) ×   e1 e2 e3   , (2.8) where ω(s)=τge1 −κne3 + κge3 is referred to as the darboux vector. here, κn(s) = κ sin ϑ = κn(u) =< α ′′ (s),e3 >, κg(s) = κ cos ϑ = det ( α ′ (s),α ′′ (s),e3(s) ) , τg(s) = τ −ϑ ′ = det ( α ′ (s),e3(s),e ′ 3(s) ) .   (2.9) we call κg = κg(s) a geodesic curvature, κn = κn(s) a normal curvature, and τg = τ + ϑ ′ a geodesic torsion of α(s), respectively. in terms of these quantities, the geodesics, line of curvatures, and asymptotic lines on a smooth surface may be characterized, as loci along which κg = 0, τg = 0, and κn = 0, respectively. the angle of pitch of a closed ruled surface when the set {α(s); e1(s), e2(s), e3(s)} is transposed to the origin 0 by the translation α(s) → 0, it performs there a spherical motion about the fixed center 0. according to the elements of spherical kinematics the motion is an infinitesimal rotation about an instantaneous axis whose direction is given by the darboux vector ω(s). the corresponding steiner vector of the motion is s(s) = ∮ ω = (∮ τg ) e1 − (∮ κn ) e3 + (∮ κg ) e3 the angle of pitch of the y-closed ruled surface is defined by λx =< x,s > . (2.10) thus the steiner vector will be s(s)=λ1e1 −λ2e2 + λ3e3, (2.11) where λi = λi(s) (i=1, 2, 3) are the angle of pitches of the ruled surfaces yi(s,v) = α(s) + vei(s), v ∈ r. (2.12) 2.1. normal ruled surfaces. we now design normal ruled surfaces along the curve α(s) of m as follows: a straight line l in e3 such that it is strictly connected to darboux frame of the unit speed curve α(s) of the given surface m is represented, uniquely with respect to this frame, in the form x(s) = x1(s)e1(s)+x2(s)e2(s)+x3(s)e3(s), x21 + x 2 2 + x 2 3 = 1, x ′ 6= 0,   (2.13) int. j. anal. appl. 17 (4) (2019) 563 where the components xi = xi(s) (i=1, 2, 3) are scalar functions of the arc length parameter s of the base curve α(s). let the set {e1, e2, e3} complete a closed motion along the base curve α(s) ∈ m. then the straight line l generates a closed ruled surface. we can describe the surface by the equation mn : p(s,v) = α(s) + vx(s), v ∈ r. (2.14) to solve our constrained design problem, the normal vector at the point p(s, 0) is (ps ×pv) (s, 0) = x2e3 −x3e2. (2.15) now, it seems natural to pose the following question. under what condition mn is normal ruled surface of the surface m ?. the answer is affirmative and can be stated as follows: mn is normal ruled surface of the surface m if and only if x3 6= 0 and x2 = 0. by substitution: x1(s) =< e,e1>= cos ϕ, x3(s) =< e,e3>= sin ϕ 6= 0, (2.16) so that we have mn : p(s,v) = α(s) + vx(s), v ∈ r, x = cos ϕe1+ sin ϕe3.   (2.17) when we choose different ϕ0, we can get different surfaces. we can control the shape of the surfaces by the value of ϕ(s). we can also calculate that the unit normal vector to the normal ruled surface mn at p(s,v) is: u(s,v) := ps ×pv ‖ps ×pv‖ = − µe2 + ve2 ×x√ µ2 + v2 , (2.18) which is the normal u0 at the point p(s, 0). let φ be the angel between the unit normal vectors u and u0, then tan φ = v µ . (2.19) this result is an expression of the well known chasles theorem: when µ > 0 (µ < 0) the tangent plane turns counterclockwise (clockwise) and the ruled surface is called left-handed (right-handed) ruled surface. as an immediate result we state that: the tangent plane turns evidently through π along a ruling in a non-developable (µ 6= 0) normal ruled surface mn. according to eqs. (2.2), and (2.17) the striction curve is given by: c(s) = α(s) −σ(s)x(s), where σ(s) = ( κn+ϕ ′ ) sin ϕ (κn+ϕ′ ) 2 +(κg cos ϕ−τg sin ϕ)2 .   (2.20) int. j. anal. appl. 17 (4) (2019) 564 also, the distribution parameter is µ(s) = (τg sin ϕ−κg cos ϕ) sin ϕ (κn + ϕ ′ ) 2 + (κg cos ϕ− τg sin ϕ) 2 . (2.21) via eqs. (2.4), and (2.21), the gaussian curvature is k(s,v) = − (τg sin ϕ−κg cos ϕ) 2 sin2 ϕ [1−2v(κg+ϕ′ ) sin ϕ+cos ϕ+v2f(s)] 2 , where f(s) = [ (κg cos ϕ− τg sin ϕ) 2 + ( κn + ϕ ′ )2 cos2 ϕ ] .   (2.22) theorem 1. for the mn-closed ruled surface of m along α(s), the gaussian curvature k(s,v) along a ruling takes the maximum value at the striction point on that ruling. proof. according to eq. (2.22), we have: ∂k(s,v) ∂v = − ∂ ∂v [ (τg sin ϕ−κg cos ϕ) 2 sin2 ϕ [1 − 2v (κg + ϕ ′ ) sin ϕ + cos ϕ + v2f(s)] ] , from which we have ∂k(s,v) ∂v = 0 ⇔ v = ( κn + ϕ ′ ) sin ϕ (κn + ϕ ′ ) 2 + (κg cos ϕ− τg sin ϕ) 2 = σ(s). (2.23) thus, v gives us the maximum of k(s,v) since ∂2k(s,v) ∂v2 |v< 0. thus k(s,v) has its maximum value at the central point on each of the rulings since the central point corresponds to the value v = σ(s). this completes the proof of the theorem. the pitch of a closed ruled surface consider v = v(s) as a function of s with continuous derivatives of a certain order such that the regular curve: γ : β(s) = α(s) + v(s)x(s), (2.24) on the mn-closed ruled surface of m along α(s) is orthogonal to the generator, then we have < β ′ ,x >= 0 ⇒< α ′ ,x > + v ′ =0. the pitch of the mn-closed ruled surface is l(s) = ∮ < β ′ ,x > ds = − ∮ dv = v0 −v1. (2.25) this equation shows that, in general, γ is not a closed curve. after a period, curve γ, starting from the point p0 on the generator intersects the same generator at another point p1 which is generally different from int. j. anal. appl. 17 (4) (2019) 565 p0, i.e. li = p0p1. eq. (2.25) can be also expressed as an integral invariant of the mn-closed ruled surface: by means of eqs. (2.8), and (2.24) we may write β ′ = [ 1 −v ( κn + ϕ ′ ) sin ϕ ] e1 + v [ (κg cos ϕ− τg sin ϕ) e2 + ( κn + ϕ ′ ) cos ϕe3 ] , and therefore eq. (2.25) becomes l(s) = ∮ cos ϕds, (2.26) which is the desired result. also, it can be shown that the tangle of pitch is given by λ(s) = λ1 cos ϕ + λ3 sin ϕ. (2.27) corollary 1. for the mn-closed ruled surface of m along α(s), the short distance between two rulings is the distance measured only on the striction curve which is one of the orthogonal trajectories. proof. fixing two rulings, say for s1 < s2, we compute the length η(v) of an orthogonal trajectory between these two rulings by η(v) = s2∫ s1 ∥∥∥β′∥∥∥ds = √[1 −v (κn + ϕ′ ) sin ϕ]2 + v2f2ds, where f(s) = [ (κg cos ϕ− τg sin ϕ) 2 + ( κn + ϕ ′ )2 cos2 ϕ ] . to find value of v which minimizes η(v), we have to use ∂η(v) ∂v = 0 which gives v = σ(s). this completes the proof. example 1. suppose we are given a parametric curve α(u) = (1 + cos u, sin u, 2 sin u 2 ). after simple computation, we have e1(u) = . α(u)∥∥ .α(u)∥∥ = ( − √ 2 sin u √ 3 + cos u , √ 2 cos u √ 3 + cos u , √ 2 cos u 2√ 3 + cos u ) . then m : x(u,v) = ( 1 + cos u− √ 2v sin u √ 3 + cos u , √ 2v cos u √ 3 + cos u + sin u, √ 2v cos u √ 3 + cos u + 2 sin u 2 ) , which is known as the tangent surface of α(u). it follows that: e3(u) = xu ×xv ‖xu ×xv‖ = ( −3 sin u 2 − sin 3u 2√ 13 + 3 cos u , 2 √ 2 cos ( u 2 )3 √ 13 + 3 cos u , −2 √ 2 √ 13 + 3 cos u ) . int. j. anal. appl. 17 (4) (2019) 566 since .. α(u) = ( −cos u,−sin u,−1 2 sin u 2 ) , we have: κg(u) = det ( . α(u), .. α(u),e3(u) )∥∥ .α(u)∥∥3 = − √ 13 + 3 cos u (3 + cos u) 3 , κn(u) = < .. α(u),e3(u) >∥∥ .α(u)∥∥2 = 0, τg(u) = det ( . α(u),e3(u), . e3(u) )∥∥ .α(u)∥∥2 = 6 cos u 2 13 + 3 cos u , so that α(u) is an asymptotic on m. then we have a normal ruled surface family given by mn : p(u,v) = (1 + cos u, sin u, 2 sin u 2 ) + v (cos ϕe1 + sin ϕe3) , v ∈ r. if we take ϕ(u) = u, then we immediately obtain a member of this family given by mn : p(u,v) = (u cos u,u sin u, 5u) + v (cos ue1 + sin ue3) , v ∈ r. the striction curve is c(u) = α(u) + sin u (κn + 1) x(u) =   1 + cos(u) + sin(u) ( − √ 2 cos(u) sin(u)√ 3+cos(u) − sin(u)(3 sin( u2 )+sin( 3u 2 ))√ 26+6 cos(u) ) , sin(u) + √ 2 sin(u) ( cos(u)2√ 3+cos(u) + 2 cos( u 2 )3 sin(u)√ 13+3 cos(u) ) , 2 sin(u 2 ) + √ 2 sin(u) ( cos( u 2 ) cos(u)√ 3+cos(u) − 2 sin(u)√ 13+3 cos(u) )   , so that, for the curvature functions of mn we have µ(u) = − sin(u) ( cos(u) √ 13+3 cos(u) (3+cos(u))3/2 + 6 cos( u 2 ) sin(u) 13+3 cos(u) )2 1 + ( cos(u) √ 13+3 cos(u) (3+cos(u))3/2 + 6 cos( u 2 ) sin(u) 13+3 cos(u) )2 , k(u,v) = − sin(u)2 ( cos(u) √ 13+3 cos(u) (3+cos(u))3/2 + 6 cos( u 2 ) sin(u) 13+3 cos(u) )2 (1 + cos(u) − 2v sin(u) + v2 (f(u)2))2 , λ(u) = (∫ 2π 0 − √ 13 + 3 cos(u) (3 + cos(u))3/2 du ) sin(u) = (−4.7) sin(u), l(s) = ∫ 2π 0 cos udu = 0. where f(u) = cos(u)2 + ( cos(u) √ 13 + 3 cos(u) (3 + cos(u))3/2 + 6 cos(u 2 ) sin(u) 13 + 3 cos(u) )2 . int. j. anal. appl. 17 (4) (2019) 567 figure 1. the surface m figure 2. the mn ruled surface figure 3. mn ⋃ m with the base curve (red) α(u) and the striction curve (blue) c(u) 2.2. normal developable ruled surface. a developable surface is a special ruled surface which has the same tangent planes along a generator, or to which the tangent planes along a ruling coincide. in view of eq. (2.3), we have: µ(s) = det(α ′ ,x,x ′ ) = 0 ⇔ (κg cos ϕ− τg sin ϕ) sin ϕ = 0. (2.28) we will now investigate this condition in detail: if sin ϕ = 0, then mn can not be normal developable surface (in fact we have imposed ϕ 6= 0). then, according to the assumption of mn being a normal developable ruled surface (mn-developable for short) of m, we have κg cos ϕ− τg sin ϕ = 0, and consequently from eq. (2.17): mn : p(s,v) = α(s) + vx(s), v ∈ r, x(s) = cos ϕe1 + sin ϕe3, cos ϕ = τg√ τ2g +κ 2 g , and sin ϕ = κg√ τ2g +κ 2 g 6= 0.   (2.29) int. j. anal. appl. 17 (4) (2019) 568 theorem 2. the necessary and sufficient condition for mn-developable being a normal along α(s) of m is that there exist a parameter v ∈ r and a function ϕ(s) = tan−1 κg τg , so that mn can be represented by eq. (2.29). as is customary in the literature, there are three types of developable surfaces, the given curve can be classified into three kinds correspondingly. in what follows, we will discuss the relationship between the given curve α(s) ∈ m and its isoparametric developable. in the light of eq. (2.28), we have: < x×x ′ ,α ′ >= 0. (2.30) the first case is when, x×x ′ = 0 ⇔ ( κn + ϕ ′ ) e2 = 0. (2.31) in this case, mn is referred to as a cylindrical surface. since e2 is a nonzero unit vector, then the mndevelopable is a cylindrical surface if and only if ϕ(s) = ϕ0 − s∫ s0 κnds, (2.32) where ϕ0 = ϕ(s0), and s0 is the starting value of the arc length. by similar argument, we can also have the following: x×x ′ 6= 0. (2.33) this implies that the mn-developable is a non-cylindrical surface. therefore, the first derivative of the directrix is α ′ (s) = c ′ (s) + σ(s)x ′ (s) + σ ′ (s)x(s), (2.34) where c ′ is the first derivative of the striction curve, σ(s) is a smooth function. substituting eq. (2.34) into eq. (2.30) gives: < x×x ′ ,c ′ >= 0. (2.35) similarly, there are two possible cases which satisfy eq. (2.35), as presented in the following: the first case is when the first derivative of the striction curve is c ′ = 0. geometrically this condition implies that the striction curve degenerates to a point, and the ruled surface becomes a cone; the striction point of a cone is commonly referred to as the vertex. therefore, the mn-developable is a cone if and only if there exists a fixed point c and a function σ(s) such that: 1 + σ ( κn + ϕ ′ ) sin ϕ−σ ′ cos ϕ = 0, σ ( κn + ϕ ′ ) cos ϕ + σ ′ sin ϕ = 0,   (2.36) int. j. anal. appl. 17 (4) (2019) 569 which imply σ(s) = − sin ϕ κn + ϕ ′ . the second case is when c ′ 6= 0, i.e. σ(s) 6= − sin ϕ κn+ϕ ′ . from eq. (2.35), c ′ is perpendicular to x×x ′ , and, therefore, c ′ is in the plane generated by x and x ′ . the condition for c to be striction curve is therefore equivalent to c ′ and x ′ are perpendicular to each other. therefore, we may conclude that the ruling is parallel to the first derivative of the striction curve, which is also the tangent of the striction curve. this ruled surface is referred too as a tangent ruled surface. so, the mn-developable represented by eq. (2.29) is a tangent surface if and only if there exists a curve c(s) so that: σ(s) 6= − sin ϕ κn + ϕ ′ . so that, for the mn-developable, we have: c(s) = α(s) + sin ϕ κn+ϕ ′ x(s), λ(s) = λ1 cos ϕ + λ3 sin ϕ, l(s) = ∮ τg√ τ2g +κ 2 g ds.   (2.37) theorem 3 (existence and uniqueness). under the above notations there exists a unique mn-developable ruled represented by eq. (2.29). proof. for the existence, we have the mn-developable along α = α(s) represented by eq. (2.29). on the other hand, since mn is a ruled surface, we assume that mn : p(s,v) = α(s) + vζ(s), v ∈ r, with (τg,κn) 6= (0, 0), ζ(s) = ζ1(s)e1+ζ2(s)e2+ζ3(s)e3, ‖ζ(s)‖2 = ζ21 + ζ22 + ζ23 = 1, ζ ′ (s) 6= 0.   (2.38) it can be immediately seen from eqs. (2.8) and (2.38) that mn-ruled is developable if and only if det(α ′ ,ζ,ζ ′ ) = 0 ⇔−ζ3ζ ′ 2 + ζ2ζ ′ 3 − ζ1 (ζ3κg − ζ2κn) + τg ( ζ22 + ζ 2 3 ) = 0. (2.39) on the other hand, since mn is a developable surface which is normal developable surface along α = α(s), we have (ps ×pv) (s,v) = ψ (s,v) e2, (2.40) where ψ = ψ (s,v) is a differentiable function. moreover, the normal vector ps ×pv at the point (s, 0) is (ps ×pv) (s, 0) = −ζ3e2 + ζ2e3. (2.41) int. j. anal. appl. 17 (4) (2019) 570 thus, from eqs. (2.39) and (2.40), one finds that: ζ2 = 0, and ζ3 = −ψ (s, 0) , (2.42) which follows from eq. (2.38) that −ζ1 (ζ3κg) + τg ( ζ23 ) = 0 ζ3 (−ζ1κg + ζ3τg) = 0. if (s, 0) is a regular point (i.e., ψ (s, 0) 6= 0), then ζ3(s) 6= 0. thus, we have ζ1 = τg κg ζ3, with κg 6= 0. (2.43) therefore, we obtain ζ(s) = τg κg ζ3e1+ζ3e3 = ζ3 sin ϕ e(s) , with ϕ 6= 0. (2.44) this means that the direction of ζ(s) is equal to the direction of e(s). if τg 6= 0 (i.e., ϕ 6= π2 ), we have the same result as the above case. on the other hand, suppose that mn-developable has a singular point at (s0, 0). then ψ (s0, 0) = ζ2(s0) = ζ3(s0) = 0, and we have ζ(s0) = ζ1(s0)e1(s0). if the singular point α(s0) is in the closure of the set of points where themn-developable along α(s) is regular, then there exists a point α(s) in any neighborhood of α(s0) such that the uniqueness of the mn-developable holds at α(s). passing to the limit s → s0, uniqueness of the normal developable at s0. suppose that there exists an open interval j ⊆ i such that mn is singular at α(s) for any s ∈ j. then p(s,v) = α(s) + vζ1(s)e1(s) for any s ∈ j. this means that ζ2(s) = ζ3(s) = 0 for s ∈ j. it follows that (ps ×pv) (s,v) = vζ21 (κne2 −κge3) . (2.45) thus the above vector is directed to e2, i.e. ps ×pv ‖ e3(s) if and only if κn 6= 0 and κg = 0 for any s ∈ j. in this case, x(s) = ±e3. this means that uniqueness holds. this ends the proof. example 2. by making use of eqs. (2.29), and example 1, the corresponding surfaces are shown in figures 4, 5 and 6. int. j. anal. appl. 17 (4) (2019) 571 figure 4. the surface m figure 5. the mn developable surface figure 6. mn ⋃ m with the base curve (red) α(u) and the striction curve (blue) c(u) on the other hand, we have assumed that τ2g + κ 2 g 6= 0. if κg = 0, and τg = 0, then lx(s) is undefined, and we have the following theorem: theorem 4. let α : i ⊆ r → m be a unit speed curve on with κg(s) = τg(s) = 0. then, the mndevelopable of m along α(s) is a plane if and only if α is a ruling of mn. proof. in general, the torsion of the curve as a space curve is given by τ(s) = τg + κgκ ′ n −κnκ ′ g κ2g + κ 2 n . if κg(s) = τg(s) = 0, then τ = 0, so that α is a plane curve. moreover, we have e ′ 2 = 0. thus, the mn-developable is a plane along α of m. since α is the intersection of m with the plane mn, α is a ruling. conversely, if α is a ruling of mn-developable, then there exists a plane π such that α(i) = mn ∩m, and e1, e3 ∈ π for any s ∈ i. therefore, π is orthogonal to e2 ∈ π for any s ∈ i. since π is a plane, π is a mn-developable along α ∈mn. this ends the proof. int. j. anal. appl. 17 (4) (2019) 572 as a result the following corollary can be given. corollary 2. let α : i ⊆ r → m be a unit speed curve on with τ2g + κ2g 6= 0. if there are two mndevelopable surfaces along α, then α is a straight line. proof. under the assumption of τ2g + κ 2 g 6= 0 the mn-ruled developable along α ∈mn is unique by theorem 3. if there are two mn-developable surfaces along α with κg(s) = τg(s) = 0, then α is a ruling for these two mn-developable surfaces. if κg(s) = τg(s) = 0 at a point s0 in the closure of the set of points where τ2g + κ 2 g 6= 0, then there exists a point s in any neighborhood of s0 such that the uniqueness of themndevelopable surface holds at s. passing to the limit s → s0; the uniqueness of the mn-developable surface holds at s0. this completes the proof. 2.3. special curves on a surface. in this subsection we consider special curves on a surface. firstly, we consider geodesics on surfaces. let α : i ⊆ r → m be a unit speed curve. since α is a geodesic of m, we have κg = 0, and therefore from eq. (2.9) κn = ±κ, and τg = τ. if we replace κn, τg, and κg in eq. (2.29), we have mn : p(s,v) = α(s) + ve1(s), v ∈ r, which is known as the tangent developable surface of α(s). also, in view of eqs. (2.37), we get: c(s) = α(s), λ(s) = λ1, and l(s) = ∮ ds. in this case l(s) equals the length of the striction curve of mn, and conversely, if l(s) is equal to the striction curve of mn, then mn-developable is a tangential developable. hence the following corollary is true: corollary 3. the mn-developable is a tangential developable if and only if its pitch l is equal to the length of the striction curve. example 3. let m be given as the following parameterization: m : x(s,v) = ( 1 √ 2 cos s, 1 √ 2 sin s, s √ 2 − 2v ) . the darboux frame which belong to this surface are found as: e1(s) = ( − 1√ 2 sin s√ 2 , 1√ 2 cos s√ 2 , 1√ 2 ) , e2(s) = ( − 1√ 2 sin s√ 2 , 1√ 2 cos s√ 2 ,− 1√ 2 ) , e3(s) = (−cos s,−sin s, 0)   moreover, we have κg(s) = 0, κn(s) = 1 √ 2 , and τg(s) = 1 √ 2 . int. j. anal. appl. 17 (4) (2019) 573 thus, the mntangential developable is mn : p(s,v) = ( 1 √ 2 cos s, 1 √ 2 sin s, s √ 2 ) + v cos ( 1 √ 2 ) e1(s). figure 7. the surface m figure 8. the mn tangential surface figure 9. mn ⋃ m with the base curve α(s). secondly, we consider lines of curvature. let α : i ⊆ r → m be a unit speed curve line of curvature with κg 6= 0. since α is a line of curvature, we have τg = 0, and consequently from eqs. (2.29) and (2.37) we obtain mn : p(s,v) = α(s) + ve3(s), v ∈ r, so that c(s) = α(s) + 1 κn e3(s), λ(s) = λ3, and l(s) = 0. as an application of eqs. (2.31) and (2.34) we conclude that: (1)mn is a circular cone if and only if κn is constant (see figures 10, 11, 12). (2)mn is a cylindrical surface if and only if κn = 0 (see figures 13, 14, 15). λ(s) = √ 2π, and l(s) = √ 2π. int. j. anal. appl. 17 (4) (2019) 574 example 4. let m be given as the following parameterization: m : x(s,v) = ( cos s− v √ 2 cos s, sin s− v √ 2 sin s, v √ 2 ) . the darboux frame which belong to this surface are found as: e1(s) = (−sin s, cos s, 0) , e2(s) = ( − 1√ 2 cos s,− 1√ 2 sin s, 1√ 2 ) , e3(s) = ( 1√ 2 cos s, 1√ 2 sin s, 1√ 2 )   also, we have: κg(s) = 1 √ 2 , κn(s) = −1 √ 2 , and τg(s) = 0. the mn-cone is mn : p(s,v) = (cos s, sin s, 0) + v sin ( 1 √ 2 ) e3(s) by eqs. (2.36), we find λ(s) = √ 2π, and l(s) = 0. figure 10. the surface m. figure 11. the mn cone. figure 12. mn ⋃ m with the base curve α(s). thirdly, we consider asymptotic curve. let α : i ⊆ r → m be a unit speed asymptotic curve with τ2g + κ 2 g 6= 0. since α is an asymptotic curve, we have κn = 0, and consequently from eqs. (2.7), (2.8), (2.29) int. j. anal. appl. 17 (4) (2019) 575 and (2.37), we have: mn : p(s,v) = α(s) + vx(s), v ∈ r, x(s) = cos ϕt + sin ϕb, cos ϕ = τ√ τ2+κ2 , and sin ϕ = κ√ τ2+κ2 6= 0,   (2.46) and c(s) = α(s) + sin ϕ ϕ ′ x(s), λ(s) = λ1 cos ϕ + λ3 sin ϕ, l(s) = ∮ τ√ τ2+κ2 ds.   (2.47) then the following corollary can be derived. corollary 4. the necessary and sufficient condition for the mn-developable surface of m along α(s) being as an asymptotic is that there exist a function ϕ(s) = cot−1 τ κ , so that mn can be represented by eqs. (2.46). corollary 4. shows that the property of the constructed mn-developable is completely determined by the given asymptotic, and so is the type of the surface. since there are three types of mn-developable surfaces, the given curve can be classified into three kinds correspondingly. in what follows, we will discuss the relationship between the given asymptotic and its isoparametric mn-developable of m along α(s). suppose the surface constructed by eq. (2.46) is a cylinder, then there is x ′ × x = 0, which results in κτ ′ −τκ ′ =0, namely ( τ κ ) ′ = 0. the ratio of curvature to torsion is a constant, and so the curve α(s) ∈ m is a generalized helix. hence, we can state the following. corollary 5. the mn-developable with α(s) is an asymptotic is a cylinder surface if and only if α(s) is a generalized helix. example 5. we consider asymptotic curve: α(s) = ( 4 5 cos s, 1 − sin s, −3 5 cos s ) , and let m be given as the following parameterization: m : x(s,v) = ( 4 5 cos s− 4 5 v − 3 5 v2, 1 − sin s, −3 5 cos s + 3 5 v − 4 5 v2 ) . after straightforward calculations, we have κg(s) = √ 2 cos s √ 1 + cos 2s , κn(s) = 0, and τg(s) = 0. int. j. anal. appl. 17 (4) (2019) 576 the mn-cylinder is given by mn : p(s,v) =   1 5 cos(s) ( 4 + v ( −4 + √ 2(−3+8v)√ 1+8v2+cos(2s) ) sin(s) ) , 1 −v cos2(s) − sin(s) − 2 √ 2v2 sin2(s)√ 1+8v2+cos(2s) , 1 5 cos(s) ( −3 + v ( 3 − 2 √ 2(2+3v)√ 1+8v2+cos(2s) ) sin(s) )   figure 13. the surface m figure 14. the surface mn figure 15. mn ⋃ m with the base curve α(s). suppose the mn developable is a cone. then, as mentioned above, from eqs. (2.6) and (2.46), we have: c ′ (s) = (1 −σ ′ cos ϕ + σϕ ′ sin ϕ)t− (σ ′ sin ϕ + σϕ ′ cos ϕ)b, which follows that p(s,u) is a circular cone if and only if 0 = (1 −σ ′ cos ϕ + σϕ ′ sin ϕ)t− (σ ′ sin ϕ + σϕ ′ cos ϕ)b. int. j. anal. appl. 17 (4) (2019) 577 we have found, by equating the coefficients of t and b, that: 1 −σ ′ cos ϕ + σϕ ′ sin ϕ = 0 ⇔ 1 − d ds (σ cos ϕ) = 0 ⇔ σ cos ϕ = s + c1, σ ′ sin ϕ + σϕ ′ cos ϕ = 0 ⇔ d ds (σ sin ϕ) = 0 ⇔ σ sin ϕ = c2.   by simple computation, we have τ κ (s) − τ κ (0) = s c , where c is an arbitrary constant. corollary 6. let mn be the normal developable with α(s) expressed by eq. (2.46). then we have the following: (a)mn is a cone if and only if τ κ (s) − τ κ (0) = s c , (b)mn is a tangential developable surface if and only if τ κ (s) − τ κ (0) 6= s c . 3. conclusion in this paper, we study the problem of finding a family of normal ruled surfaces from a given curve on surface. we obtain the parametric representation for a ruled surface family whose members share the same curve as an isoparametric curve. using the darboux frame of the given curve on surface, we present the ruled surface as a linear combination of this frame and analyze the necessary and sufficient condition for that surface to be normal ruled surface. the extension to developable surfaces is also outlined. we illustrate this method by presenting some examples. hopefully these results will lead to a wider usage of surfaces in geometric modeling, garment-manufacture industry, and the manufacturing of products. references [1] b. o’neill. elementary differential geometry. academic press inc., new york 1966. [2] m.p. do carmo. differential geometry of curves and surfaces. englewood cliffs: prentice hall, 1976. [3] g. farin. curves and surfaces for computer aided geometric design, 2nd ed. new york: academic press, 1990. [4] g.j. wang, k. tang, and c.l. tai. parametric representation of a surface pencil with a common spatial geodesic, comput.aided des., 36 (2004), 447–459. [5] e. kasap, f.t. akyildiz, k. orbay. a generalization of surfaces family with common spatial geodesic, appl. math. comput., 201 (2008), 781–789. [6] g. saffak, e. kasap. family of surface with a common null geodesic, internat. j. phys. sci., 4(8) (2009), 428–433. [7] c.y. li, r.h. wang, c.g. zhu. parametric representation of a surface pencil with a common line of curvature, comput. aided des., 43 (9) (2011), 1110–1117. [8] e. bayram, f. guler, e. kasap. parametric representation of a surface pencil with a common asymptotic curve, comput. aided des., 44 (2012), 637–643. [9] c.y. li, r.h. wang, c.g. zhu. an approach for designing a developable surface through a given line of curvature, comput. aided des., 45 (2013) 621–627. [10] g.y. senturk, s. yuce. characteristic properties of the ruled surface with darboux frame. kuwait j. sci. 42 (2) (2015), 14–33. [11] g.y. senturk, s. yuce. bertrand offsets of ruled surfaces with darboux frame, result. math. 72 (3) (2017), 1151–1159. 1. introduction 2. preliminaries 2.1. normal ruled surfaces 2.2. normal developable ruled surface 2.3. special curves on a surface 3. conclusion references international journal of analysis and applications volume 17, number 6 (2019), 928-939 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-928 harmonic analysis on internally gelfand pairs associated to groupoids ibrahima toure∗, kinvi kangni ufr de mathématiques et informatique université félix houphouet-boigny; 22 bp 582 abidjan 22, cote-d’ivoire ∗corresponding author: ibrahima.toure@univ-ufhb.edu.ci; toure@aims.ac.za abstract. let g be a topological locally compact, hausdorff and second countable groupoid with a haar system and k a proper subgroupoid of g with a haar system too. (g, k) is an internally gelfand pair if for any u in the unit space, the algebra of bi-k(u)-invariant functions on g(u) is commutative under convolution. in this work, we give some characterizations of these pairs and extend to this context some classical results of harmonic analysis. 1. introduction the notion of gelfand pair, introduced by i.m.gelfand, has been extensively studied on groups in papers such as [1, 4, 5, 7–10]. it has permitted to extend many results of commutative harmonic analysis to noncommutative case. the notion of groupoid is an extension of the notion of group. in [21, 22], we have extended the notion of gelfand pair from groups to groupoids. in these papers, our analysis is done on a transitive locally compact groupoid, g, and a compact subgroupoid, k. for instance, in [21] we have proved that (g,k) is a gelfand pair if and only if for any u ∈ g(0) the pair of isotropy groups (g(u),k(u)) is a gelfand pair in group sense. thanks to this result, we have extended some results of harmonic analysis from groups to groupoids. but this result is not true in general. for instance, the groupoid algebra is received 2018-11-29; accepted 2019-01-30; published 2019-11-01. 2010 mathematics subject classification. 22a22, 46j10. key words and phrases. groupoids; groupoid representation; internally gelfand pairs. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 928 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-928 int. j. anal. appl. 17 (6) (2019) 929 not necessarily commutative even for abelian groupoids that means groupoids with abelian isotropy groups. nonetheless, there is a ”nice” harmonic analysis on abelian groupoids [3,11,16]. so our purpose is, in order to do harmonic analysis on pairs (g,k) for general locally compact groupoids, to define an alternative notion of gelfand pairs on groupoids taking into account only the isotropy groups. since any compact groupoid is proper and if k is proper then the isotropy groups k(u) are compact, we have giving our definition for proper subgroupoid, k. in fact, for a locally compact groupoid g and a proper subgroupoid k, (g,k) is an internally gelfand pair if for any u in the unit space, the algebra of bi-k(u)-invariant functions on g(u) is commutative under convolution. after notations and setup in the next section, we give in section 3 some characterizations of internally gelfand pairs, in particular we show that a gelfand pair is an internally gelfand pairs. we also study the link between gelfand pairs and internally gelfand pairs. in section 4, we define the notion of g(0)-spherical function associated to internally gelfand pairs and study some properties of these functions. we establish a connection between g(0)-spherical functions and internally irreducible representations introduce by r. bos in his paper [3]. in section 5, we give an extension of bochner theorem. 2. preliminaries we use the notations and setup of this section in the rest of the paper without mentioning. for basic notions on groupoids and haar systems, we refer the reader to [20]. throughout g will be a second countable locally compact hausdorff groupoid with unit space g(0) and left haar system {λu, u ∈ g(0)}. g(2) will denote the set of composable pairs. for x ∈ g, r(x) = xx−1 and d(x) = x−1x are respectively the range and the domain of x. for u,v ∈ g(0), let us put gu = r−1(u), gv = d−1(v), guv = gu ∩gv and for each unit element u, g(u) = {x ∈ g : r(x) = d(x) = u} is the isotropy group at u. the set g′ = {x ∈ g : r(x) = d(x)} is the isotropy group bundle of g. the relation on g(0) defined by: u,v ∈ g(0), u ∼ v iff guv 6= ∅ is an equivalence relation. the equivalence class of u is denoted by [u]g and is called the orbit of u. the graph r = {(r(x),d(x)) : x ∈ g} of this equivalence relation is a groupoid with unit space g(0). the anchor map θ=(r,d) is a continuous homomorphism of g into g(0) ×g(0) with image r. a groupoid is transitive if θ is onto i.e. the range of θ is equal to g(0) ×g(0). otherwise, a groupoid is transitive if it has a single orbit. a groupoid is proper if θ is a proper map. for u ∈ g(0), λu will denote the image of λu by the inverse map and {λu, u ∈ g(0)} is a right haar system on g. let µ be a quasi-invariant measure on g(0) for the haar system {λu,u ∈ g(0)}, ν = ∫ λudµ(u) be the induced measure by µ on g, ν−1 = ∫ λudµ(u) be the inverse of ν, ν2 = ∫ λu ×λudµ(u) be the induced measure by µ on g(2) and ∆ the modular function of µ. there is a decomposition of the left haar system {λu, u ∈ g(0)} for g over r. firstly, there is a measure βuv concentrated on g u v for all (u,v) ∈ r such that βuu is a left haar measure on g(u), and βuv is a translate of βvv i.e. β u v =xβ v v if x ∈ guv . notice that βuv is independent of the choice of x ∈ guv . then, there is a unique borel haar system α={αu : int. j. anal. appl. 17 (6) (2019) 930 u ∈ g(0)} for r with the property that for every u ∈ g(0), we have λu= ∫ βωv dα u(ω,v). cc(g) will denote the space of complex-valued continuous functions on g with compact support, endowed with the inductive limit topology and l1(g,ν) the space of ν− integrable functions on g. in [12], p. hahn defines the following norm on l1(g,ν): ‖ f ‖i = max(‖ f ‖i,r;‖ f ‖i,d) where ‖ f ‖i,r= sup{ ∫ gu |f(x)|dλu(x),u ∈ g(0)}, ‖ f ‖i,d= sup{ ∫ gu |f(x)|dλu(x),u ∈ g(0)} and introduce the following groupoid algebra, i(g,λ,µ) = {f ∈ l1(g,ν) :‖ f ‖i< ∞} under the convolution product defined by: for all f,g ∈ i(g,λ,µ), f ∗g(x) = ∫ gr(x) f(y)g(y−1x)dλr(x)(y). and the involution defined by: for f ∈ i(g,λ,µ), f∗(x) = ∆(x−1)f(x−1) = ∆(x−1)f̌(x). i(g,λ,µ) is a banach ∗-algebra. let k be a proper subgroupoid of g with unit space g(0) and equipped with a haar system {γu, u ∈ g(0)}. as it is explain above, {γu, u ∈ g(0)} has a decomposition {(γuv )(u,v)∈rk, (ρ u)u∈g(0)}, where rk is the graph of the equivalence relation on g(0) seen as unit space of k, such that γu = ∫ γωv dρ u(ω,v). we put i(g\\k) = {f ∈ i(g,λ,µ) : f(kxk′) = f(x)∀x ∈ g,∀k ∈ kr(x),∀k′ ∈ kd(x)}; the space of bi-k− invariant integrable functions which is a banach ∗-subalgebra of i(g,λ,µ). for any f ∈ i(g,λ,µ), let us denote by f\ the bi-k-invariant function defined by: for all x ∈ g, f\(x) = ∫ ∫ f(kxk′)dγr(x)(k)dγ d(x)(k′). if i(g\\k) is commutative for convolution product, we say that (g,k) is a gelfand pair. this notion in groupoids case has been studied by authors in [21, 22]. let h=(hu)u∈g(0) be a hilbert bundle over g(0) and u(h) the unitary groupoid of the bundle h. (π,h) is a unitary continuous representation of g if π is a groupoid morphism of g into u(h) such that for all square integrable sections ξ and η of h, the map x 7→< π(x)ξ(d(x)),η(r(x)) > is continuous. a closed nonzero subbundle m of h (i.e. mu is a nonzero closed subspace of hu for each u ∈ g(0)) is invariant under π if π(x)md(x) ⊂ mr(x), for each x ∈ g. if π admits a non trivial closed invariant subbundle m, it is called reducible. otherwise it is called irreducible. if ξ is a section of h, the subbundle mξ whose leaf at u ∈ g(0) is the closed linear span of the set {π(x)ξ(d(x)) : x ∈ gu} is called the cyclic subbundle generated by ξ. we say that ξ is cyclic if (mξ)u is dense in hu, for each u ∈ g(0). we denote by γµ(h), the hilbert space of square integrable section of h. in [20], j. renault associates to any unitary representation (π,h) a representation l of cc(g) on γµ(h) defined by: (l(f)ξ,η) = ∫ f(x) < π(x)ξ(d(x)),η(r(x)) > dν0(x), int. j. anal. appl. 17 (6) (2019) 931 for all f ∈ cc(g), ξ, η ∈ γµ(h), where ν0=∆ −1 2 ν. l is a bounded non-degenerate ∗-representation of cc(g) where cc(g) is equipped with the norm ‖ · ‖i. we may also define l by: l(f)ξ(u) =∫ gu f(x)π(x)ξ(d(x))∆ −1 2 (x)dλu(x). in [18], the authors extend the notion of positive definite function to groupoids. in fact, a bounded continuous function p : g → c is positive definite if for each u ∈ g(0) and for each f ∈cc(g) we have ∫ ∫ f(x)f̄(y)p(y−1x)dλu(x)dλu(y) ≥ 0. ramsay and walter establish for groupoids the well-known correspondence between positive definite functions and representations. in fact, for any bounded continuous positive definite function p : g → c, there exists a unitary representation π of g on a hilbert bundle h, and a bounded continuous cyclic section ξ of h such that for each x ∈ g, p(x) =< π(x)ξ(d(x)),ξ(r(x)) >. 3. internally gelfand pairs let g be a locally compact, hausdorff and second countable groupoid and k a proper subgroupoid of g. definition 3.1. (g,k) is an internally gelfand pair if for any u ∈ g(0), (g(u),k(u)) is a gelfand pair. the first example is given by (g,g(0)), where g is an abelian groupoid with unit space g(0) which is seen here as a cotrivial groupoid. let us notice that a groupoid g is said abelian if for any u ∈ g(0) the isotropy group g(u) is abelian. the following results give some necessary conditions for internally gelfand pairs. theorem 3.1. let l be a locally compact group acting continuously on a compact space s and let t be a compact subgroup of l acting trivially on s. if (l,t) is a gelfand pair then the pair of transformation groupoids (l ∝ s,t ∝ s) is an internally gelfand pair. proof. we have (t ∝ s)(u) = t ⊂ (l ∝ s)(u) ⊂ l. since (l,t) is a gelfand pair, then ((l ∝ s)(u), (t ∝ s)(u)) is a gelfand pair. (see [1]) � consider the relation on g defined by x ∼ y iff r(x) = r(y) and y−1x ∈ k. it is an equivalence relation and the quotient space g/k, equipped with the quotient topology, is hausdorff and locally compact. the range map r induces a continuous, open surjection p : ġ 7→ r(g) from g/k to g(0) (see [20]). the groupoid g acts on g/k, that is , the map (g,s) 7→ g.s from g∗g/k := {(g,s) ∈ g×g/k : d(g) = p(s)} to g/k, is continuous and satisfy (a) p(g.s) = r(g) (b) (g1g2).s = g1.(g2.s) if (g1,g2) ∈ g(2) and int. j. anal. appl. 17 (6) (2019) 932 (c) p(s).s = s for all s ∈ g/k. for action groupoids, see [14]. let us put g/k ×p g/k = {(ġ1, ġ2) ∈ g/k ×g/k : p(ġ1) = p(ġ2)}. there is an action of g on g/k ×p g/k defined by the relation: g.(s,t) = (g.s,g.t) for g ∈ g and s,t ∈ g/k. theorem 3.2. if for any s,t ∈ g/k, (s,t) ∼ (t,s) then (g,k) is an internally gelfand pair. proof. let u ∈ g(0). for s = uk and x ∈ g(u) we have (s,x−1.s) = x−1(x.s,s) ∼ (x.s,s) ∼ (s,x.s) thus there exists y ∈ gu such that y.s = s and yx−1.s = x.s. the first relation shows that y ∈ k(u) and the second one implies that x−1yx−1.s = s. so we have t ∈ k(u) and x−1yx−1 ∈ k(u). it follows that x−1 ∈ k(u)xk(u) and consequently (g(u),k(u)) is a gelfand pair thanks to proposition i.2 in [8]. . � theorem 3.3. let g be a locally compact, hausdorff and second countable groupoid and k a proper subgroupoid of g. if (g,k) is a gelfand pair then (g,k) is an internally gelfand pair. proof. suppose that (g,k) is a gelfand pair. we consider the map ψ from l1(g(u)\\k(u)) to i(g\\k) defined by f 7→ f where f(x) = ∫ f(kxk′)dγu r(x) (k)dγ d(x) u (k ′) if k d(x) u 6= ∅ and kur(x) 6= ∅ and f(x) = 0 in other case . let’s show that f is actually in i(g\\k). for l ∈ kr(x), l′ ∈ kd(x), we assume first that k d(l′) u 6= ∅ and kur(l) 6= ∅. if k ∈ kd(l ′) u then d(k) = u,r(k) = d(l ′). thus l′k is defined and l′k ∈ kr(l ′) u = k d(x) u that is k d(x) u 6= ∅. we show in the same way that if ku r(l) 6= ∅ then ku r(x) 6= ∅. so, f(lxl′) = ∫ f(klxl′k′)dγur(l)(k)dγ d(l′) u (k ′) = ∫ f(kxlk′)dγud(l)(k)dγ r(l′) u (k ′) = f(x) now, if we assume that k d(l′) u = ∅ or kur(l) = ∅ then f(lxl ′) = 0. if there exists k ∈ kd(x)u then r(k) = d(x) = r(k′) and l′−1k is defined. we have l′−1k ∈ kd(l ′) u . so, k d(x) u = ∅ and f(x) = 0. int. j. anal. appl. 17 (6) (2019) 933 ψ is clearly linear. now for f,h ∈ l1(g(u)\\k(u)), we set f = ψ(f) and h = ψ(h), we have f ∗h(x) = ∫ f(y)h(y−1x)dλr(x)(y) = ∫ f(kyk′)h(ly−1xl′)dγur(y)(k)dγ d(y) u (k ′)dγud(y)(l)dγ d(x) u (l ′)dλr(x)(y) = ∫ f(kyk′)h(ly−1xl′)dγur(x)(k)dγ d(y) u (k ′)dγud(y)(l)dγ d(x) u (l ′)dλd(k)v (y)dµ(v) = ∫ f(yk′)h(ly−1kxl′)dγur(x)(k)dγ d(y) u (k ′)dγud(y)(l)dγ d(x) u (l ′)dλr(k)v (y)dµ(v) = ∫ f(yk′)h(ly−1kxl′)dγur(x)(k)dγ v u(k ′)dγuv (l)dγ d(x) u (l ′)dλur(k′)(y)dµ(v) = ∫ f(y)h(lk′y−1kxl′)dγur(x)(k)dγ v u(k ′)dγuv (l)dγ d(x) u (l ′)dλud(k′)(y)dµ(v) = ∫ f(y)h(y−1kxl′)dγur(x)(k)dγ v u(k ′)dγuv (l)dγ d(x) u (l ′)dλuu(y)dµ(v) = ∫ f(y)h(y−1kxl′)dγur(x)(k)dγ d(x) u (l ′)dλuu(y) = ∫ f ∗h(kxl′)dγur(x)(k)dγ d(x) u (l ′) = ψ(f ∗h) where the line 7 is due to k(u)-biinvariance of h. thus ψ is a morphism of convolution algebras. moreover ψ is injective. in fact, let us notice first that if x ∈ g(u) then f(x) = ∫ f(kxk′)dγuu(k)dγ u u(k ′) = f(x). thus ψ(f) = ψ(h) implies that f = h and in particular, f|g(u) = h|g(u) that is f = h. � the converse is not generally true. for instance if g is abelian, the pair (g,g0) is an internally gelfand pair but not a gelfand pair. for transitive groupoids, the converse is true (see [21]). in the following result, we give a condition for the converse. theorem 3.4. let g be a locally compact hausdorf groupoid with unit space g(0) and let k be a proper subgroupoid of g with unit space g(0) such that (g,k) is an internally gelfand pair. if there exists a unit u ∈ g(0) such that its orbit in k, [u]k, is dense in g(0) then (g,k) is a gelfand pair. proof. since (g,k) is an internally gelfand pair then (g(u),k(u)) is a gelfand pair. it follows thanks to theorem 3.3 of [21] that (g|[u]k,k|[u]k ) is a gelfand pair. so if f,g ∈ i 1(g||k) then f|g|[u]k ∗ g|g|[u]k = g|g|[u]k ∗f|g|[u]k and consequently we have (f∗g)|g|[u]k = (g∗f)|g|[u]k . now it suffices to prove that g|[u]k is dense in g to have f ∗ g = g ∗ f. in fact, since [u]k is dense in g(0), for x ∈ g there exists a sequence {un}n∈n of elements of [u]k which converges to d(x) in g(0). the map d is continuous surjective and open, so thanks to ( [11], proposition i.25, page 20) there exists a subsequence {unj}j∈j of {un}n∈n and a sequence {xj} of g such that d(xj) = unj for any j ∈ j. since an orbit is an invariant subset of g(0) and d(xj) ∈ [u]k then r(xj) are in [u]k. we conclude that {xj} is a sequence of elements of g|[u]k converging to x ∈ g. � int. j. anal. appl. 17 (6) (2019) 934 let h = (hu)u∈g(0) be a continuous hilbert bundle over g(0). π is an internally irreducible representation on h if the restriction of π to g(u) is irreducible for any u ∈ g(0). we denote, as in [3], by irepi(g) the set of equivalence classes of internally irreducible unitary continuous representations of g and by ĝ(u) the set of equivalence classes of irreducible unitary continuous representations of g(u). the map resu : irepi(g) 7→ ĝ(u), designates the restriction map. let us denote by h k(u) u the subspace of k(u)-invariant vectors defined by h k(u) u = {h ∈ hu : π(k)h = h,∀k ∈ k(u)}. theorem 3.5. if (g,k) is an internally gelfand pair then for any internally irreducible unitary representation π on h, dimhk(u)u ≤ 1 for all u ∈ g(0). if for any u ∈ g(0), resu is surjective then the converse holds. proof. since (g(u),k(u)) is a gelfand pair and π|g(u) is irreducible then by classical properties of gelfand pairs dimh k(u) u ≤ 1. for the converse, if πu is a unitary irreducible representation of g(u) then, since resu is surjective, there exists an internally unitary irreducible representation π of g such that πu = π|g(u). so h k(u) u is the space of k(u)− invariant vector corresponding to πu and it follows that (g(u),k(u)) is a gelfand pair. � each h k(u) u is a closed subspace of hu so hk = (h k(u) u )u∈g(0) is a continuous hilbert subbundle of h = (hu)u∈g(0) . we set a = {u ∈ g(0) : h k(u) u 6= {0}} remark 3.1. the set a is an invariant open subset of g. in fact, let us suppose that for x ∈ g, hk(d(x)) d(x) 6= {0}. if ξd(x) is a nonzero vector of h k(d(x)) d(x) we set ηr(x) = ∫ π(k)ξd(x)dγ r(x) d(x) (k). we have ‖ηr(x)‖ = ∫ ∫ < π(k)ξd(x),π(k ′)ξd(x) > dγ r(x) d(x) (k)dγ r(x) d(x) (k′) = ∫ ∫ < π(k′−1k)ξd(x),ξd(x) > dγ r(x) d(x) (k)dγ r(x) d(x) (k′) = ∫ ∫ < ξd(x),ξd(x) > dγ r(x) d(x) (k)dγ r(x) d(x) (k′) = < ξd(x),ξd(x) >= ‖ξd(x)‖ and for k0 ∈ k r(x) r(x) π(k0)ηr(x) = ∫ π(k0k)ξd(x)dγ r(x) d(x) (k) = ∫ π(k)ξd(x)dγ r(x) d(x) (k) = ηr(x) thus ηr(x) ∈ h k(r(x)) r(x) and is nonzero. so h k(r(x)) r(x) 6= {0}. now a is open as the support of a continuous field of hilbert space. we end this section with some examples int. j. anal. appl. 17 (6) (2019) 935 (1) g = a ∝ s a transformation groupoid where a is a locally compact abelian group and s a topological space. let l be a subgroup of a acting continuously and properly on s, so (g = a ∝ s,k = l ∝ s) is an internally gelfand pair. (2) p(m,g,π) a principal fiber. k a compact subgroup of g. (g,k) is a gelfand pair if and only if (p×p g , p×p k ) is an internally gelfand pair. in fact, (p×p g )(m) = g and (p×p k )(m) = k. (3) let g be a proper groupoid with unit space g(0). let us consider a groupoid g̃ = g×(r,d) g = {(x,y) ∈ g×g : (r,d)(x) = (r,d)(y)} with groupoid structure defined in the following way: d(x,y) = d(x) = d(y), r(x,y) = r(x) = r(y); (x,y)(x′,y′) = (xx′,yy′) if d(y) = r(x′) and (x,y)−1 = (x−1,y−1). the set k̃ = {(x,x) : x ∈ g} is a closed subgroupoid of g̃. we have g̃(u) = g(u) × g(u) the cartesian product of g(u) by g(u) and k̃(u) = diag(g(u) × g(u)) = {(x,x) : x ∈ g(u)}. we know (see [7]) that (g(u) × g(u),diag(g(u) × g(u))) is a gelfand pair. so, the pair (g̃,k̃) is an internally gelfand pair. 4. harmonic analysis on pairs (g,k) in this section, (g,k) is an internally gelfand pair. definition 4.1. let ϕ be a bi-k−invariant continuous function on g. ϕ is g(0)-spherical if for any u ∈ g(0), ϕ|g(u), the restriction of ϕ to g(u) is a k(u)spherical function. let’s set that ϕ |g(u)= ϕu and f |g(u)= fu. theorem 4.1. let ϕ be a bi-k−invariant continuous function on g such that ϕu 6= 0 for all u ∈ g(0). then ϕ is g(0)-spherical if and only if for all x,y ∈ g, ∫ ϕ(xky)dγ d(x) r(y) (k) = ϕ(x)ϕ(y) proof. let ϕ be g(0)-spherical. then for any u ∈ g(0) and s,z ∈ g(u) we have ∫ ϕu(skz)dγ u u(k) = ϕ(s)ϕ(z) int. j. anal. appl. 17 (6) (2019) 936 now for x,y ∈ g∫ ϕ(xky)dγ d(x) r(y) (k) = ∫ ϕ((txl)(l−1kk1)(k −1 1 yl1))dγ v r(x)(t)dγ d(x) v (l)dγ r(y) v (k1)dγ d(y) v (l1)dγ d(x) r(y) (k) = ∫ ϕ(txl)(l−1kk1)(k −1 1 yl1)dγ v r(x)(t)dγ d(x) v (l)dγ r(y) v (k1)dγ d(y) v (l1)dγ r(l) r(k1) (k) = ∫ ϕ(txl)k(k−11 yl1)dγ v r(x)(t)dγ d(x) v (l)dγ r(y) v (k1)dγ d(y) v (l1)dγ d(l) d(k1) (k) = ∫ ∫ (ϕ(txl)k(k−11 yl1)dγ v v (k))dγ v r(x)(t)dγ d(x) v (l)dγ r(y) v (k1)dγ d(y) v (l1) = ∫ ϕ(txl)ϕ(k−11 yl1)dγ v r(x)(t)dγ d(x) v (l)dγ r(y) v (k1)dγ d(y) v (l1) = ϕ(x)ϕ(y) for the converse it suffices to write for a fixed u ∈ g(0) the equality for x,y ∈ g(u) and apply the proposition 6.1.5 of [5]. � theorem 4.2. let ϕ be a bi-k-invariant continuous function non identically zero on each g(u). ϕ is g(0)-spherical if and only if for all f ∈ i(g\\k) there exists a continuous map χf on g(0) such that for all u ∈ g(0),ϕu ∗fu = χf (u)ϕu. in particular if there exists a dense orbit [u]k in g(0) then χf is constant on g(0). proof. for any u ∈ g(0), ϕu is a spherical function on g(u), so for all f ∈ i(g\\k) there exists a complex number χf (u) such that ϕu∗fu = χf (u)ϕu. since ϕu is spherical then ϕu(u) = 1 and it follows that χf (u) =∫ g(u) f(x)ϕ(x)dβuu(x). thus the continuity of χf is due to the continuity of the map u 7→ ∫ g(u) f(x)dβuu(x) for any f ∈ cc(g). the converse is trivial. now if u ∼k v then χf (u) = χf (v). in fact, for t ∈ kuv let us consider the map lt from g(v) to g(u) defined by lt(x) = txt −1. lt is a homeomorphism. if β v v is the haar measure on g(v) then it is straightforward to see that the image measure βuu = lt(β v v ) is a haar measure on g(u) and equal to βuu since (g(u),k(u)) being a gelfand pair, g(u) is unimodular. thus χf (u) = ∫ g(u) f(x)ϕ(x−1)dβuu(x) = ∫ g(v) f(txt−1)ϕ(tx−1t−1)dβvv (x) = ∫ g(v) f(x)ϕ(x−1)dβvv (x) = χf (v) so since [u]k is dense in g (0) and χf is continuous then there exists c ∈ c such that χf (w) = c for all w ∈ g(0). � theorem 4.3. let π be an internally irreducible unitary representation on h. ξ a continuous k-invariant section such that || ξ(u) ||= 1 for any u ∈ g(0). then the map ϕ : x 7→ ϕ(x) =< π(x)ξ(d(x)),ξ(r(x)) > is a positive definite g(0)-spherical function. int. j. anal. appl. 17 (6) (2019) 937 the proof is trivial. theorem 4.4. let π be a unitary representation on h admitting a continuous k-invariant section on g(0). if dimh k(u) u = 1 for all u ∈ g(0), then π is internally irreducible. proof. for all u ∈ g(0), π | g(u) is a unitary continuous representation of g(u) on hu. let ξ be a continuous k-invariant section for π on g0. then for all u ∈ g(0) ξ(u) is a k(u)− invariant vector for π | g(u) and since dimh k(u) u = 1 then, thanks to lemme 6.2.3. of [5] (or proposition 2.6 of [8]), π | g(u) is irreducible. � a positive definite function ϕ is said g(0)− elementary if the unitary continuous representation associated to it is internally irreducible. theorem 4.5. let ϕ be a bi-k-invariant, continuous, positive definite function such that ϕ(u) = 1 for all u ∈ g(0). then ϕ is g(0)− spherical if and only if ϕ is g0− elementary. proof. let’s suppose that ϕ is g(0)− spherical and let πϕ be the unitary representation associated to ϕ. we have ϕ(x) =< π(x)ξ(d(x)),ξ(r(x)) > where ξ is a continuous k-invariant section such that || ξ(u) ||= 1 for any u ∈ g(0). let’s set ϕu = ϕ|g(u), the restriction of ϕ to g(u). ϕu is k(u)invariant, continuous, positive definite function such that ϕ(u) = 1. since ϕu is spherical then the representation associated to it, is irreducible and unitarily equivalent to πϕ |g(u). so πϕ is internally irreducible. conversely, ϕ is g(0)− elementary implies that the associated representation is internally irreducible. thus, the positive definite function φ associated to π |g(u) is spherical. but φ = ϕu = ϕ|g(u). so ϕ is g(0)− spherical. � denote by pg(0) the set of positive definite g (0)spherical functions on g and pu the set of positive definite spherical functions on g(u). we know by classical theory (see [5, 8]) that pu equipped with the topology σ(l∞,l1) is locally compact. for u ∈ g(0), let’s consider resu : pg(0) → pu the restriction map. if we equip pg(0) with the coarsest topology making continuous the map resu, then it is locally compact. in this section, we shall suppose that resu is bijective. the choice of the topology of pg(0) makes resu a continuous open bijection and therefore an homeomorphism. we start by given a definition of the fourier transform appropriated to our context. definition 4.2. for a function f ∈ i(g\\k), the fourier transform, noted by f(f), is defined by: f(f)(ϕ) = ∫ g(0) ∫ g(u) f(x)ϕ(x−1)dβuu(x)dµ(u) for all ϕ ∈ pg(0) . we have the following results known in classical case. theorem 4.6. (i) for f,g ∈ i(g\\k), f(f ∗g) = f(f)f(g) (ii) f(f) is continuous on pg(0) and vanishing at infinity (iii) the map f 7→f(f) is a linear transformation. int. j. anal. appl. 17 (6) (2019) 938 proof. (i)for f,g ∈ i(g\\k) f(f ∗g)(ϕ) = ∫ (f ∗g)(x)ϕ(x−1)dβuu(x)dµ(u) = ∫ f(y)g(y−1x)ϕ(x−1)dλu(y)dβuu(x)dµ(u) = ∫ f(y)g(ky−1x)ϕ(x−1)dγud(y)(k)dλ u m(y)dβ u u(x)dµ(m)dµ(u) = ∫ f(yk)g(y−1x)ϕ(x−1)dγum(k)dλ u r(k)(y)dβ u u(x)dµ(m)dµ(u) = ∫ f(yk)g(x)ϕ(x−1y−1)dγum(k)dλ u u(y)dβ u u(x)dµ(m)dµ(u) = ∫ f(k′yk)g(x)ϕ(x−1y−1)dγmu (k ′)dγum(k)dλ d(k′) r(k) (y)dβuu(x)dµ(m)dµ(u) = ∫ f(y)g(x)( ∫ ϕ(x−1ky−1)dγ d(x) r(y) (k))dγmu (k ′)dλ r(k′) d(k) (y)dβuu(x)dµ(m)dµ(u) = ∫ f(y)g(x)ϕ(x−1)ϕ(y−1)dγmu (k ′)dλ r(k′) d(k) (y)dβuu(x)dµ(m)dµ(u) = f(f)f(g) (ii) f(f)(ϕ) = ∫ g(0) φ(u,ϕ)dµ(u) where φ(u,ϕ) = ∫ g(u) f(x)ϕ(x−1)dβuu(x). the map ϕ 7→ φ(u,ϕ) is continuous as the composition of continuous functions ϕ 7→ ϕ|g(u) and f(fu). then, we have |φ(u,ϕ)| ≤ supu∈g(0) ( ∫ g(u) |f(x)|dβuu(x)) ≤ ||f||i. so f(f) is continuous. since f(f)(ϕ) = ∫ g(0) f(fu)(ϕu)dµ(u) and f(fu) vanishing at infinity then f(f) is vanishing at infinity. (iii) the proof is trivial. � theorem 4.7. let φ be a bi-k-invariant continuous positive definite function on g such that φ(u) ≤ 1 for all u ∈ g(0). then there exists a unique measure γ on m0(pg(0) ) such that for all x ∈ g, φ(x) = ∫ p g(0) ω(x)dγ(ω) proof. for any u ∈ g(0), φu the restriction of φ on g(u) is bi-k(u)-invariant continuous positive definite function on g(u). thus thanks to bochner theorem for gelfand pairs, there exists a unique measure θu on m0(pu) such that for all x ∈ g(0), φu(x) = ∫ pu wu(x)dθu(wu). we consider then the measure image γ u of θu by res −1 u . so we obtain a family of measure {γu : u ∈ g(0)} on pg(0) . we put γ = ∫ g(0) γudµ(u). for all x ∈ g we have φ(x) = ∫ φ(kxk′)dγur(x)(k)dγ d(x) u (k ′)dµ(u) = ∫ ωu(kxk ′)dθu(ωu)dγ u r(x)(k)dγ d(x) u (k ′)dµ(u) int. j. anal. appl. 17 (6) (2019) 939 = ∫ resu(ω)(kxk ′)dγ(ω)dγur(x)(k)dγ d(x) u (k ′)dµ(u) = ∫ ω(x)dγu(ω)dµ(u) = ∫ ω(x)dγ(ω) � references [1] m. akouchi, a. bakali, une généralisation des paires de gel’fand, bolletino u.m.i, 7 (1992), 795-822. [2] m. buneci, groupoid algebras for transitive groupoids, math. rep., 5(55) (2003), 9-26. [3] r. bos, groupoids in geometric quantization, phd. thesis, radboud university nijmegen, 2007. [4] j. dieudonné, gelfand pairs and spherical functions, int. j. math. math. sci., 2(2) (1979), 153-162. [5] v. dijk, introduction to harmonic analysis and generalized gelfand pairs, de gruyter stud. math., 36, walter de gruyter and co, berlin, 2009. [6] j. dixmier, les c∗-algèbres et leurs représentations, gauthier-villars, paris, 1964. [7] j. faraut, infinite dimensional spherical analysis, coe lecture note, 10, kyushu university, 2008. [8] j. faraut, analyse harmonique sur les paires de gelfand, les cours du cimpa, 1980. [9] i. f. gelfand, spherical functions on symetrics spaces, dokl. akad. nauk., 70 (1960), 5-8. [10] r. godement, a theory of spherical functions i, trans. amer. math. soc., 73 (1952), 496-556. [11] g. goehle, groupoid crossed products, phd. thesis, dartmouth college, 2009. [12] p. hahn, the regular representations of measure groupoids, trans. amer. math. soc., 242 (1978), 34-72. [13] p. hahn, haar measure for groupoids, trans. amer. math. soc., 242 (1978), 1-33. [14] k. mackenzie, general theory of lie groupoids and lie algebroids, london math. soc. lecture note ser., 213, cambridge university press, 2005. [15] j. f. m {c}clendon, metric families, pac. j. math., 57 (1975), 491-509. [16] p. s. muhly, j. n. renault, d. p. williams, continuous-trace groupoid c*-algebras, iii, trans. math. soc., 348 (1996), 3621-3641. [17] k. j. oty, fourier-stieltjes algebras of r-discrete groupoids, j. oper. theory, 4 (1999), 175-197. [18] a. ramsay, m. e. walter, fourier-stieltjes algebras of locally compact groupoid, j. funct. anal., 148(2) (1997), 314-367. [19] j. renault, the ideal structure of grupoid crossed product c∗-algebras, j. oper. theory, 25 (1991), 3-36. [20] j. renault, a groupoid approach to c∗-algebras, lecture notes in math., 793, springer-verlag, 1980. [21] i. toure, k. kangni, on gelfand pairs associated to transitive groupoids, opuscula math., 33(4) (2013), 751-762. [22] i. toure, k. kangni, on groupoid algebras of biinvariant functions, int. j. math. anal., 6 (43) (2012), 2101-2108. [23] j. j. westman, harmonic analysis on groupoids, pacific j. math., 27(3) (1968), 621-632. 1. introduction 2. preliminaries 3. internally gelfand pairs 4. harmonic analysis on pairs (g,k) references international journal of analysis and applications volume 19, number 2 (2021), 180-192 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-180 radau quadrature for an almost quasi-hermite-fejér-type interpolation in rational spaces shrawan kumar1, neha mathur2, vishnu narayan mishra3,∗, pankaj mathur1,∗ 1department of mathematics and astronomy, university of lucknow, lucknow 2department of mathematics, career convent degree college, lucknow 3department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur, madhya pradesh 484 887, india ∗corresponding authors: vishnunarayanmishra@gmail.com, pankaj mathur14@yahoo.co.in abstract. in this paper, we have studied an almost quasi hermite-fejér-type interpolation in rational spaces. a radau type quadrature formula has also been obtained for the same. 1. introduction hermite fejér and quasi-hermite-fejér-type interpolation processes has been a subject of interest for several mathematicians. in almost all the cases the interpolatory polynomials are considered on the nodes which are the zeros of certain classical orthogonal polynomials. the main idea of the present paper is to construct a rational interpolation process and its corresponding quadrature formula with prescribed nodes based on the chebyshev markov fractions. received november 1st, 2019; accepted march 2nd, 2020; published february 1st, 2021. 2010 mathematics subject classification. primary 05c38, 15a15; secondary 05a15, 15a18. key words and phrases. almost quasi-hermite-fejér-type interpolation; radau-type quadrature; rational space; prescribed poles; chebyshev-markov fractions. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 180 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-180 int. j. anal. appl. 19 (2) (2021) 181 chebyshev and markov introduced rational cosine and sine fractions [3] which generalizes chebyshev polynomials, possesses many similar properties ( [2, 10, 11]) and are called chebychev–markov rational fractions. different aspects of the rational generalization of chebyshev polynomials are discussed in many works ( [1, 12]). in 1962, rusak [9] initiated the study of interpolation processes by means of rational functions on the interval [−1, 1]. the nodes were taken to be the zeros of chebyshev–markov rational fractions. in [6] rational interpolation functions of hermite-fejér-type were constructed [7]. min [4] was the first to consider the rational quasi-hermite-type interpolation. he constructed the interpolatory function and proved its uniform convergence for the continuous functions on the segment with the restriction that the poles of the approximating rational functions should not have limit points on the interval [−1, 1]. recently, based on the ideas of [6] and using method that was different from that of [4], rouba et. al. ( [5], [8]) revisited the rational interpolation functions of hermite-fejér-type. they also proved the uniform convergence of the interpolation process for the function f ∈ c[−1, 1] and obtained explicitly its corresponding lobatto type quadrature formula. in this paper, we have considered an almost quasi-hermite-fejér-type interpolation process on the zeros of the rational chebyshev-markov sine fraction on the semi closed interval (−1, 1], that is, when the interpolatory condition is prescribed only at one of the end points. a radau type quadrature formula corresponding to the interpolation process has also been obtained. 2. preliminaries consider a set of points ak, k = 0, 1, · · · , 2n− 1 which are either real and ak ∈ (−1, 1) or be paired by complex conjugation. also let un(x) be the rational chebyshev-markov sine fraction, un(x) = sin µ2n(x)√ 1 −x2 (2.1) where, µ2n(x) = 1 2 2n−1∑ k=0 arccos x + ak 1 + akx ,(2.2) µ′2n(x) = − λ2n(x)√ 1 −x2 ,(2.3) λ2n(x) = 1 2 2n−1∑ k=0 √ 1 −a2k 1 + akx , n ∈ n.(2.4) also u′n(x) = −cos µ2n(x)λ2n(x) √ 1 −x2 + x sin µ2n(x) (1 −x2)3/2 (2.5) int. j. anal. appl. 19 (2) (2021) 182 and u′n(xk) = − λ2n(xk) (1 −x2k) .(2.6) let r2n−1(a0,a1, · · · ,a2n−1) be a rational space defined as r2n−1(a0,a1, · · · ,a2n−1) := { p2n−1(x)∏2n−1 k=0 (1 + akx) } (2.7) where p2n−1(x) is a polynomial of degree ≤ 2n−1 and {ak}2n−1k=0 are real and belong to [−1, 1] or are paired by complex conjugation. the rational fraction un(x) can be expressed as un(x) = pn−1(x)√ π2n−1k=0 (1 + akx) where pn−1(x) is an algebraic polynomial of degree n−1 with real coefficient. the fraction un(x) has n−1 zeros on the interval (−1, 1) given by, −1 < xn−1 < xn−2 < · · · < x2 < x1 < 1, µ2n(xk) = kπ, k = 1, 2, · · · ,n− 1, where µ2n(x) is given by (2.2). also, the rational function λ2n(x), given by (2.4), can be expressed as λ2n(x) = q2n−1(x)∏2n−1 k=0 (1 + akx) where q2n−1(x) is a polynomial of degree atmost 2n− 1. it has no zeros on [−1, 1]. 3. almost quasi-hermite-fejér-type interpolation let x0 = 1. then for any function f ∈ c(−1, 1] the almost quasi type hermite interpolation function hn(x,f) satisfying the conditions hn(xk,f) = f(xk), k = 0, 1, · · · ,n− 1,(3.1) h′n(xk,f) = yk, k = 1, 2, · · · ,n− 1,(3.2) is given by (3.3) hn(x,f) = n−1∑ k=0 f(xk)ak(x) + n−1∑ k=1 ykbk(x), where yk,k = 1, 2, · · · ,n−1 are arbitrarily given real numbers, {ak(x)}n−1k=0 and {bk(x)} n−1 k=1 are fundamental functions satisfying the conditions  ak(xj) = δkj, j,k = 0, 1, · · · ,n− 1,a′k(xj) = 0, j = 1, 2, · · · ,n− 1,k = 0, 1, · · · ,n− 1(3.4) int. j. anal. appl. 19 (2) (2021) 183 and   bk(xj) = 0, j = 0, 1, · · · ,n− 1,k = 1, 2, · · · ,n− 1,b′k(xj) = δkj, j,k = 1, 2, · · · ,n− 1.(3.5) 4. explicit representation of the fundamental functions lemma 4.1. the fundamental functions {ak(x)}n−1k=0 satisfying the conditions (3.4) can be explicitly represented as for k = 1, 2, · · · ,n− 1 ak(x) = (1 + xk)(1 −x2k)(1 −x){1 − bk(x−xk)}u 2 n(x) λ2n(xk)(x−xk)2λ2n(x) ,(4.1) where bk = 2xk − 1 1 −x2k (4.2) and (4.3) a0(x) = u2n(x) λ2n(x)λ2n(1) . proof. we will show that ak(x), k = 0, 1, · · · ,n− 1 defined by (4.1) and (4.3) satisfy the conditions (3.4) obviously for k = 1, 2, · · · ,n− 1, ak(x0) = 0 and ak(xj) = 0, j = 1, 2, · · · ,n− 1 when j 6= k. for j = k using the l’hospital’s rule, we have lim x→xk ak(x) = (1 −x2k) λ22n(xk) ( lim x→xk sin µ2n(x) (x−xk) )2 = (1 −x2k) λ22n(xk) ( lim x→xk −λ2n(x) cos µ2n(x)√ 1 −x2 )2 = 1. on differentiating (4.1) with respect x we get a′k(x) = (1 + xk)(1 −x2k) λ2n(xk) [ 2{1 − bk(x−xk)} (1 + x)λ2n(x) ( sin µ2n(x) x−xk )( sin µ2n x−xk )′ − bk(1 + x)λ2n(x) + {1 − bk(x−xk)}{(1 + x)λ′2n(x) + λ2n(x)} (1 + x)2λ22n(x) × ( sin µ2n(x) x−xk )2 ] then for j 6= k we have a′k(xj) = 0, j = 1, 2, · · · ,n− 1. for j = k, lim x→xk a′k(x) = (1 −x2k) λ22n(xk) [ 2 lim x→xk (( sin µ2n(x) x−xk )( sin µ2n(x) x−xk )′) − bk(1 + xk)λ2n(xk) + (1 + xk)λ ′ 2n(xk) + λ2n(xk) (1 + xk)λ2n(xk) × ( lim x→xk sin µ2n(x) x−xk )2 ] int. j. anal. appl. 19 (2) (2021) 184 we know that lim x→xk sin µ2n(x) (x−xk) = µ′2n(xk) cos µ2n(xk) = − λ2n(xk)√ 1 −x2k (4.4) and lim x→xk ( sin µ2n(x) x−xk )′ = 1 2 cos µ2n(xk)µ ′′ 2n(xk)(4.5) where µ′′2n(x) = − (1 −x2)λ′2n(x) + xλ2n(x) (1 −x2)3/2 (4.6) then lim x→xk a′k(x) = [ 2xk − 1 (1 −x2k) − bk ] = 0 due to (4.2) which shows that ak(x), k = 1, 2, · · · ,n− 1, given by (4.1), satisfy all the conditions given by (3.4). similarly, for a0(x), given by (4.3), we have that a0(xj) = 0, j = 1, · · · ,n − 1. for j = 0 and using the fact that un(1) = λ2n(1), we have a0(x0) = 1. again by differentiating (4.3) with respect x, we get a′0(xj) = 0, j = 1, 2, · · · ,n − 1. thus a0(x) given by (4.3) satisfy the conditions (3.4) for j = 0, which completes the proof of the lemma. � lemma 4.2. the fundamental functions {bk(x)}n−1k=1 satisfying the conditions (3.5) can be explicitly represented as (4.7) bk(x) = (1 −x)(1 + xk)(1 −x2k)u 2 n(x) λ2n(x)λ2n(xk)(x−xk) . proof. obviously, bk(xj) = 0, k = 1, 2, · · · ,n − 1, j = 0, 1, · · · ,n − 1 and for j 6= k, b′k(xj) = 0, j,k = 1, 2, · · · ,n− 1. for j = k, lim x→xk b′k(x) = (1 −x2k) 2 λ22n(xk) lim x→xk ( u2n(x) (x−xk) ) = (1 −x2k) 2 λ22n(xk) lim x→xk ( 2un(x)u ′ n(x) (x−xk) − ( un(x) x−xk )2) = (1 −x2k) 2 λ22n(xk) (u′n(xk)) 2 = 1, due to (2.6), which proves that bk(x), k = 1, 2, · · · ,n− 1 given by (4.7) satisfy all the conditions given by (3.5). � from lemma 4.1 and lemma 4.2 it follows that hn(f,x) satisfying the conditions (3.1) is an almost quasi hermite interpolation. int. j. anal. appl. 19 (2) (2021) 185 theorem 4.1. the function hn(f,x) is a rational function of degree atmost 2n− 1 that is hn(f,x) ∈r2n−1(a0,a1, · · · ,a2n−1).(4.8) proof. since un ∈pn−1(a0,a1, · · · ,a2n−1), we can express it as un(x) := sn−1(x) (s∗n(x)) 1/2 where s∗n(x) := (x − a0)(x − a1) · · ·(x − a2n−1), sn−1(x) := cn−1(x − x1)(x − x2) · · ·(x − xn−1) and cn−1 depends on n and {ak}2n−1k=0 . so, we have `k(x) = ( s∗n(xk) s∗n(x) )1/2 qk(x), k = 1, 2, · · · ,n− 1,(4.9) where qk(x) := sn−1(x) s′n−1(xk)(x−xk) , k = 1, 2, · · · ,n− 1.(4.10) thus, `k(x) ∈pn−1(a0,a1, · · · ,a2n−1), thus by (3.3), (4.1) and (4.7) we easily find that hn(f,x) = t2n−1(x) q2n−1(x) (4.11) where t2n−1(x) is a polynomial of degree ≤ 2n− 1, which proves the lemma. � let yk = 0, k = 1, 2, · · · ,n− 1 then (3.3) reduces to hn(f,x) = n−1∑ k=1 f(xk)ak(x) + f(1)a0(x)(4.12) which is an almost quasi hermite fejér interpolation function for f ∈ c[−1, 1]. 5. radau-type quadrature formula for a given function f defined on [−1, 1], we define the function (5.1) gn(x,f) = n−1∑ k=0 f(xk)hk(x) where, hk(x) = 1 −x 1 −xk [ 1 − ( u′′n (xk) u ′ n(xk) − 1 (1 −xk) ) (x−xk) ] `2k(x), k = 1, 2, · · · ,n− 1 and h0(x) = u2n(x) u2n(1) . int. j. anal. appl. 19 (2) (2021) 186 we have that gn(f,x) ∈ r2n−1(a0,a1, · · · ,a2n−1). also the rational function gn(f,x) is an almost quasi hermite fejér interpolation function. let (5.2) ak = ∫ 1 −1 1 √ 1 −x2 hk(x)dx, k = 1, 2, · · · ,n− 1 and (5.3) a0 = ∫ 1 −1 1 √ 1 −x2 u2n(x) u2n(1) dx then the radau-type quadrature formula is given by (5.4) ∫ 1 −1 f(x) √ 1 −x2 dx = a0f(1) + n−1∑ k=1 akf(xk). with respect to this quadrature formula, we have the following theorem 5.1. the quadrature formula (5.4) can be expressed as (5.5) ∫ 1 −1 f(x) √ 1 −x2 dx = 2π λ2n(1) f(1) + n−1∑ k=1 π λ2n(xk) f(xk). lemma 5.1. for k = 1, 2, · · · ,n− 1, (5.6) ∫ 1 −1 (1 −x)(x−xk)`2k(x)dx = 0. proof. we have that for k = 1, 2, · · · ,n− 1, `2k(x) = u2n(x) (u′n) 2(xk)(x−xk)2 = (1 −x2k) 2 sin2 µ2n(x) λ22n(xk)(1 −x2)(x−xk)2 .(5.7) also, (5.8) un(1) = lim x→1 sin µ2n(x)√ 1 −x2 = λ2n(1) and (5.9) un(−1) = (−1)n+1λ2n(−1). by these equalities, the left hand side of (5.6) can be represented as (5.10) ik = ∫ 1 −1 sin2 µ2n(x) (1 + x) √ 1 −x2(x−xk) dx consider the transformation (5.11) x = 1 −y2 1 + y2 which gives (5.12) dx = − 4y (1 + y2)2 dy, int. j. anal. appl. 19 (2) (2021) 187 (5.13) x−xk = − 2(y2 −y2k) (1 + y2)(1 + y2k) , (5.14) 1 + x = 2 1 + y2 , (5.15) √ 1 −x2 = 2y 1 + y2 . we know that, (5.16) sin µ2n ( 1 −y2 1 + y2 ) = sin φ2n(y) where sin φ2n(y) is a bernstein sine fraction (5.17) sin φ2n(y) = 1 2i ( χn(y) −χ−1n (y) ) where χn(y) = ∏2n−1 j=0 y−zj y−z̄j and zk are the roots of the equations y 2 + (1 + ak)(1 −ak)−1 = 0, izk > 0, k = 0, 1, · · · , 2n− 1. taking into account the assumptions on the parameters ak, k = 0, 1, · · · , 2n− 1, we have the following: 1) z0 = i, 2) if ak and al are paired by complex conjugation, then the corresponding numbers zk and zl are symmetric with respect to the imaginary axis. besides, the function sin φ2n(y) has zeros at ±yk, yk = √ (1 −xk)/(1 + xk), k = 1, 2, · · · ,n− 1. thus, ik = − 1 + y2k 4 ∫ ∞ −∞ (1 + y2) sin2 φ2n(y) y2 −y2k dy = − 1 + y2k 4 lim z→yk,izk>0 jk(z)(5.18) where (5.19) jk(z) = ∫ ∞ −∞ (1 + y2) sin2 φ2n(y) y2 −z2 dy. from (5.17) we get (5.20) sin2 φ2n(y) = − 1 4 ( χ2n(y) − 2 + χ −2 n (y) ) due to which, we have (5.21) jk(z) = − 1 4 (jk1(z) − 2jk2(z) + jk3(z)) where jk1(z) = ∫ ∞ −∞ (1 + y2)χ2n(y) y2 −z2 dy, jk2(z) = ∫ ∞ −∞ (1 + y2) y2 −z2 dy int. j. anal. appl. 19 (2) (2021) 188 and jk3(z) = ∫ ∞ −∞ (1 + y2)χ−2n (y) y2 −z2 dy. since jk1(z) has only singular point y = z in the upper half plane. thus by the residue theorem we have jk1(z) = 2πi lim y→z (1 + y2)χ2n(y) (y + z) = 1 + z2 z χ2n(z)πi.(5.22) similarly, (5.23) jk3(z) = 1 + z2 z χ−2n (z)πi. also, jk2(z) has only singular point y = z in the upper half plane, therefore by the residue theorem, we have jk2(z) = 2πi lim y→z (1 + y2) (y + z) = 1 + z2 z πi.(5.24) putting the value of jk1(z),jk3(z) and jk2(z) from (5.22), (5.23) and (5.24) respectively in (5.21) we get (5.25) jk(z) = − 1 4 ( 1 + z2 z χ2n(z)πi− 2 1 + z2 z πi + 1 + z2 z χ−2n (z)πi ) which by (5.18) gives (5.26) ik = 1 + y2k 16 lim z→yk,izk>0 ( 1 + z2 z χ2n(z)πi + 1 + z2 z χ−2n (z)πi− 2 1 + z2 z πi ) . since χn(yk) = 1, thus it follows that ik = 0, which completes the proof of the lemma. � lemma 5.2. for k = 1, 2, · · · ,n− 1, (5.27) ak = π λ2n(xk) . proof. due to lemma 5.1 and by putting the value of `2k(x),k = 1, 2, · · · ,n−1, from (5.7) in (5.2), we have (5.28) ak = (1 + xk)(1 −x2k) λ22n(xk) ∫ 1 −1 sin2 µ2n(x) (x−xk)2(1 + x) √ 1 −x2 dx. we find the integrals (5.29) i∗k = ∫ 1 −1 sin2 µ2n(x) (x−xk)2(1 + x) √ 1 −x2 dx, k = 1, 2, · · · ,n− 1, by using the transformation (5.11), (5.12), (5.13), (5.14), (5.15) and (5.16), we have i∗k = (1 + y2k) 2 8 ∫ ∞ −∞ (1 + y2)2 sin2 φ2n(y) (y2 −y2k)2 dy. consider the auxiliary integral j∗k (z) = ∫ ∞ −∞ (1 + y2)2 sin2 φ2n(y) (y2 −z2)2 dy int. j. anal. appl. 19 (2) (2021) 189 then i∗k can be written as i∗k = (1 + y2k) 2 8 lim z→yk,=zk>0 j∗k (z).(5.30) which due to (5.20) can be expressed as i∗k = (1 + y2k) 2 32 lim z→yk,=zk>0 (i∗1k(z) − 2i ∗ 2k(z) + i ∗ 3k(z))(5.31) where, i∗1k(z) = ∫ ∞ −∞ (1 + y2)2 (y2 −z2)2 χ2n(y)dy, i∗2k(z) = ∫ ∞ −∞ (1 + y2)2 (y2 −z2)2 dy and i∗3k(z) = ∫ ∞ −∞ (1 + y2)2 (y2 −z2)2 χ−2n (y)dy. since z0 = i the integrand of i ∗ 1k(z) has only singular point y = z in the upper half plane. thus by the residue theorem, we have i∗1k(z) = 2πi lim y→z d dy ( (1 + y2)2 (y + z)2 χ2n(y) ) = 2πi lim y→z [ χ2n(y) d dy (y2 + 1)2 (y + z)2 + (y2 + 1)2 (y + z)2 d dy χ2n(y) ] .(5.32) since, χn(y) = 2n−1∏ j=0 y −zj y − z̄j which by logarithmic differentiation gives d dy χn(y) = χn(y) 2n−1∑ j=0 zj − z̄j (y −zj)(y − z̄j) . also, d dy ( (y2 + 1)2 (y + z)2 ) = 2y4 + 4y3z + 4yz − 2 (y + z)3 . therefore, i∗1k(z) = 2πiχ 2 n(z) [ 3z4 + 2z2 − 1 4z3 (5.33) + (z2 + 1)2 2z2 2n−1∑ j=0 zj − z̄j (z −zj)(z − z̄j) ] . similarly, i∗3k(z) = 2πiχ −2 n (z) [ 3z4 + 2z2 − 1 4z3 (5.34) + (z2 + 1)2 2z2 2n−1∑ j=0 zj − z̄j (z −zj)(z − z̄j) ] . int. j. anal. appl. 19 (2) (2021) 190 again by residue theorem we have i∗2k(z) = 2πi lim y→z 2y4 + 4y3z + 4yz − 2 (y + z)3 = 2πi ( 3z4 + 2z2 − 1 4z3 ) .(5.35) by (5.33), (5.34), (5.35), (5.31) and (5.30) and taking into account that χ2n(yk) = 1 it follows that i∗k = − πi(1 + y2k) 4 32y2k 2n−1∑ j=0 zj − z̄j (yk −zj)(yk − z̄j) .. since, yk = √ (1 −xk)/(1 + xk) and zk = i √ (1 + ak)/(1 −ak), thus by simple calculation, we have, 2n−1∑ j=0 zj − z̄j (yk −zj)(yk − z̄j) = 2n−1∑ j=0 ( 1 yk −zj − 1 yk − z̄j ) = 2n−1∑ j=0 i √ (1 + aj) √ (1 −aj) 1 + ajxk ( 2 1 + y2k ) = 4iλ2n(xk) (1 + y2k) (5.36) thus i∗k = πλ2n(xk)(1 + y 2 k) 3 8y2k . therefore by (5.28), the lemma follows. � lemma 5.3. for a0, defined by (5.3), we have a0 = ( π λ2n(1) ) .(5.37) proof. by using the transformation (5.11), (5.12), (5.15) and (5.16), we have a0 = 1 4u2n(1) ∫ ∞ −∞ 1 + y2 y2 sin2 φ2n(y)dy which, due to (5.20), can expressed as (5.38) a0 = − 1 16u2n(1) (i1 − 2i2 + i3) where i1 = ∫ ∞ −∞ 1 + y2 y2 χ2n(y)dy, i2 = ∫ ∞ −∞ 1 + y2 y2 dy and i3 = ∫ ∞ −∞ 1 + y2 y2 χ−2n (y)dy. int. j. anal. appl. 19 (2) (2021) 191 since for i1, y = 0 is the only singular point in the upper half plane. thus by the residue theorem, we have i1 = 2πi lim y→0 d dy { (1 + y2)χ2n(y) } (5.39) = 4πi 2n−1∑ j=0 ( 1 z̄j − 1 zj ) . similarly, (5.40) i3 = 4πi 2n−1∑ j=0 ( 1 z̄j − 1 zj ) . the integrand of i2(z) has only singular point y = 0 in the upper half plane. thus by the residue theorem, we have (5.41) i2 = 2πi lim y→0 d dy (1 + y2) = 0. hence using (5.40), (5.40) and (5.41) in (5.38) we get a0 = − πi 2λ22n(1) 2n−1∑ j=0 ( 1 z̄j − 1 zj ) and since 2n−1∑ j=0 ( 1 z̄j − 1 zj ) = − 4 i λ2n(1) hence (5.42) a0 = 2π λ2n(1) which in turn proves the lemma. � by lemma 5.2, lemma 5.3 and (5.4), theorem 5.1 follows. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. borwein and t. erdélyi, polynomials and polynomial inequalities, graduate texts in mathematics 161, springer-verlag, new york (1995). [2] a. l. lukashov, inequalities for the derivatives of rational functions on several intervals, izv. math. 68(3) (2004), 543–565. [3] a. a. markov(1951), izbrannye trudy, teoriya cisel. teoriya veroyatnostei, izdat. akad. nauk sssr, leningrad. [4] g. min, lobatto-type quadrature formula in rational spaces, j. comput. appl. math. 94(1) (1998), 1-12. [5] y. rouba, k. smatrytski and y. dirvuk, rational quasi-hermite-fejer-type interpolation and lobatto-type quadrature formula with chebyshev-markov nodes, jaen j. approx. 7(2) (2015), 291–308 [6] e. a. rovba, interpolation rational operators of fejér and de la valle-poussin type, mat. zametki., 53(2) (1993), 114-121 (in russian, english translation: math. notes. 53 (1993), 195-200. int. j. anal. appl. 19 (2) (2021) 192 [7] e. a. rouba, interpoljacija i rjady furie v ratsionalnoj approksimatsii, grsu, grodno. (2001). [8] y. a. rouba and k. a. smatrytski, rational interpolation in the zeros of chebyshev-markov sine-fractions, dokl. nats. akad. nauk belarusi, 52(5) (2008), 11-15 (in russian). [9] v. n. rusak, interpolation by rational functions with fixed poles, dokl. akad. nauk bssr 6 (1962), 548-550 (in russian). [10] v. n. rusak, on approximations by rational fractions, dokl. akad. nauk bssr 8 (1964), 432-435 (in russian). [11] a. h. turecki, teorija interpolirovanija v zadachakh, izdat “vyssh. skola”, minsk. (1968). [12] j. van deun, electrostatics and ghost poles in near best fixed pole rational interpolation, electron. trans. numer. anal. 26 (2007), 439-452. 1. introduction 2. preliminaries 3. almost quasi-hermite-fejér-type interpolation 4. explicit representation of the fundamental functions 5. radau-type quadrature formula references international journal of analysis and applications volume 16, number 6 (2018), 856-867 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-856 implicit summation formula for 2-variable laguerre-based poly-genocchi polynomials waseem a. khan∗, idrees a. khan and moin ahmad department of mathematics, faculty of science, integral university, lucknow-226026, india ∗corresponding author: waseem08 khan@rediffmail.com abstract. the main object of this paper is to introduce a new class of laguerre-based poly-genocchi polynomials and investigate some properties for these polynomials and related to the stirling numbers of the second kind. we derive summation formulae and general symmetry identities by using different analytical means and applying generating functions. 1. introduction the generalized bernoulli, euler and genocchi polynomials of (real or complex) order α are usually defined by means of the following generating functions (see [1-16]):( t et − 1 )α ext = ∞∑ n=0 b(α)n (x) tn n! , (| t |< 2π; 1α = 1), (1.1) ( 2 et + 1 )α ext = ∞∑ n=0 e(α)n (x) tn n! , (| t |< π; 1α = 1) (1.2) and ( 2t et + 1 )α ext = ∞∑ n=0 g(α)n (x) tn n! , (| t |< π; 1α = 1). (1.3) received 2017-09-19; accepted 2017-12-07; published 2018-11-02. 2010 mathematics subject classification. 33c45, 33c99, 05a10, 05a15. key words and phrases. laguerre polynomials, poly-genocchi polynomials, laguerre-based poly-genocchi polynomials, summation formulae, symmetric identities. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 856 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-856 int. j. anal. appl. 16 (6) (2018) 857 so that obviously bn(x) = b 1 n(x),en(x) = e 1 n(x) and gn(x) = g 1 n(x), (n ∈ n), where n0 = n∪{0}(n = 1, 2, 3, · · ·). the classical polylogarithmic function lik(z) is defined by (see [2], [10]): lik(z) = ∞∑ m=1 zm mk , (k ∈ z). (1.4) the poly-bernoulli numbers and polynomials are defined by following generating functions (see [7], [8], [9]): lik(1 −e−t) et − 1 = ∞∑ n=0 b(k)n tn n! , (1.5) lik(1 −e−t) et − 1 ext = ∞∑ n=0 b(k)n (x) tn n! . (1.6) in the case k = 1 in (1.5) and (1.6), we have b(1)n = bn,b (1) n (x) = bn(x). the poly-genocchi numbers and polynomials are defined by following generating functions (see [14]): 2lik(1 −e−t) et + 1 = ∞∑ n=0 g(k)n tn n! , (1.7) 2lik(1 −e−t) et + 1 ext = ∞∑ n=0 g(k)n (x) tn n! . (1.8) in the case k = 1 in (1.7) and (1.8), we have g(1)n = gn,g (1) n (x) = gn(x). the 2-variable laguerre polynomials (2-vlp) ln(x,y), which is defined by (see [5]): 1 (1 −yt) exp ( −xt 1 −yt ) = ∞∑ n=0 ln(x,y)t n, (| yt |< 1) (1.9) it is equivalently given by (see [6]). exp(yt)c0(xt) = ∞∑ n=0 ln(x,y) tn n! , (1.10) where c0(x) denotes the 0 th order tricomi function. the nth order tricomi functions cn(x) are defined as: cn(x) = ∞∑ r=0 (−1)rxr r!(n + r)! , (n ∈ n0) (1.11) int. j. anal. appl. 16 (6) (2018) 858 with the following generating function: exp ( t− x t ) = ∞∑ n=0 cn(x)t n, (1.12) for t 6= 0 and for all finite x. from (1.9) and (1.10), we get ln(x,y) = n! n∑ s=0 (−1)sxsyn−s (s!)2(n−s)! = ynln(x/y). (1.13) thus, we have ln(x, 0) = (−1)nxn n! , ln(0,y) = y n, ln(x, 1) = ln(x), (1.14) where ln(x) are the classical laguerre polynomials (see [1]). now, we recall here the following definition as follows: the stirling number of the first kind is given by (x)n = x(x− 1) · · ·(x−n + 1) = n∑ l=0 s1(n,l)x l, (n ≥ 0) (1.15) and the stirling number of the second kind is defined by generating function: (et − 1)n = n! ∞∑ l=n s2(l,n) tl l! . (1.16) 2. 2-variable laguerre-based poly-genocchi polynomials let k ∈ z, we inroduce 2-variable laguerre-based poly-genocchi polynomials by the following generating function: 2lik(1 −e−t) et + 1 exp(yt)c0(xt) = ∞∑ n=0 lg (k) n (x,y) tn n! , (2.1) so that lg (k) n (x,y) = n∑ m=0   n m  g(k)n−mlm(x,y). (2.2) when x = y = 0, lg (k) n = g (k) n (0, 0) are called the poly-genocchi numbers. for k = 1 in (2.1), we have 2li1(1 −e−t) et + 1 exp(yt)c0(xt) = ∞∑ n=0 lgn(x,y) tn n! , (2.3) where lgn(x,y) is laguerre-based genocchi polynomials (see [13]). thus, we have lg (k) n (x,y) = lgn(x,y), (n ≥ 0). int. j. anal. appl. 16 (6) (2018) 859 on setting x = 0, (2.1) reduces to the known result of kim et al. [14.,p.eq.(4)4776]: 2lik(1 −e−t) et + 1 exp(yt) = ∞∑ n=0 g(k)n (y) tn n! , (k ∈ z). (2.4) theorem 2.1. the following explicit summation formulae for laguerre-based poly-genocchi polynomials holds true: lg (2) n (x,y) = n∑ m=0   n m   bmm! m + 1 lgn−m(x,y). (2.5) proof. using generating function for laguerre-based poly-genocchi polynomials (2.1), we have ∞∑ n=0 lg (k) n (x,y) tn n! = 2lik(1 −e−t) et + 1 exp(yt)c0(xt) = 2 et + 1 exp(yt)c0(xt) ∫ t 0 1 ez − 1 ∫ t 0 1 ez − 1 · · · 1 ez − 1 ∫ t 0 z ez − 1 dz · · ·dz. in particular k = 2, we have ∞∑ n=0 lg (2) n (x,y) tn n! = 2 et + 1 exp(yt)c0(xt) ∫ t 0 z ez − 1 dz = ( ∞∑ m=0 tmbm m + 1 ) 2t et + 1 exp(yt)c0(xt) = ( ∞∑ m=0 tmbmm! m + 1m! )( ∞∑ n=0 lgn(x,y) tn n! ) . replacing n by n−m in the r.h.s of above equation, we have ∞∑ n=0 lg (2) n (x,y) tn n! = ∞∑ n=0   n∑ m=0   n m   bmm! m + 1 lgn−m(x,y)   tn n! . on equating the coefficients of the like powers of t in both sides, we get (2.5). remark 2.1. on setting x = 0, theorem (2.1) reduces to the known result of kim et al. [14.,p. 4777, theorem (2.1)]. corollary 2.1. for n ≥ 0, we have g(2)n (y) = n∑ m=0   n m   bmm! m + 1 gn−m(y). (2.6) theorem 2.2. for n ≥ 1, the degree of lg (k) n (x,y) is n-1. we have lg (k) n (x,y) n = n−1∑ m=0   n− 1 m   g(k)m+1 m + 1 ln−m−1(x,y). (2.7) proof. from (2.1), we have ∞∑ n=0 lg (k) n (x,y) tn n! = 2lik(1 −e−t) 1 −e−t exp(yt)c0(xt) int. j. anal. appl. 16 (6) (2018) 860 = ( ∞∑ m=0 g(k)m tm m! )( ∞∑ n=0 ln(x,y) tn n! ) replacing n by n−m in above equation and comparing the coefficients of tn, we get lg (k) n (x,y) = n∑ m=0   n m  g(k)m ln−m(x,y), (n ≥ 0). (2.8) from (2.8), we have lg (k) n (x,y) n = n−1∑ m=0   n− 1 m   g(k)m+1 m + 1 ln−m−1(x,y), (n ≥ 1) (2.9) therefore by (2.9), we obtain the result (2.7). remark 2.2. for x = 0, theorem (2.2) reduces to the known result of kim et al. [14.,p. 4778, theorem (2.2)]. corollary 2.2. for n ≥ 1, the degree of g(k)n (x) is n-1. we have g (k) n (y) n = n−1∑ m=0   n− 1 m   g(k)m+1 m + 1 yn−m−1. (2.10) theorem 2.3. for n ≥ 0, we have lg (k) n (x,y) = n∑ p=0 p+1∑ l=1 (−1)l+p+1l!s2(p + 1, l) lk(p + 1)   n p   lgn−p(x,y). (2.11) proof. by using (2.1), we can be written as ∞∑ n=0 lg (k) n (x,y) tn n! = ( lik(1 −e−t) t )( 2t et + 1 exp(yt)c0(xt) ) . (2.12) now 1 t lik(1 −e−t) = 1 t ∞∑ l=1 (1 −e−t)l lk = 1 t ∞∑ l=1 (−1)l lk (1 −e−t)l = 1 t ∞∑ l=1 (−1)l lk l! ∞∑ p=l (−1)ps2(p,l) tp p! = 1 t ∞∑ p=1 p∑ l=1 (−1)l+p lk l!s2(p,l) tp p! = ∞∑ p=0 ( p+1∑ l=1 (−1)l+p+1 lk l! s2(p + 1, l) p + 1 ) tp p! . (2.13) from equations (2.12) and (2.13), we get ∞∑ n=0 lg (k) n (x,y) tn n! = ∞∑ p=0 ( p+1∑ l=1 (−1)l+p+1 lk l! s2(p + 1, l) p + 1 tp p! )( ∞∑ n=0 lgn(x,y) tn n! ) . int. j. anal. appl. 16 (6) (2018) 861 replacing n by n−p in the r.h.s. of above equation and comparing the coefficients of tn in both sides, we arrive at the desired result (2.11). remark 2.3. for x = 0, theorem (2.3) reduces to the known result of kim et al. [14.,p. 4779, theorem (2.3)]. corollary 2.3. for n ≥ 0, we have g(k)n (y) = n∑ p=0 p+1∑ l=1 (−1)l+p+1l!s2(p + 1, l) lk(p + 1)   n p  gn−p(y). (2.14) theorem 2.4. for n ≥ 1, we have lg (k) n (x,y + 1) + lg (k) n (x,y) = 2 n∑ p=1 p∑ l=1 (−1)l+p lk l!s2(p,l)   n p  ln−p(x,y). (2.15) proof. by using definition (2.1), we have ∞∑ n=0 lg (k) n (x,y + 1) tn n! + ∞∑ n=0 lg (k) n (x,y) tn n! = 2lik(1 −e−t) et + 1 exp((y + 1)t)c0(xt) + 2lik(1 −e−t) et + 1 exp(yt)c0(xt) = 2lik(1 −e−t) exp(y)tc0(xt) = ∞∑ p=1 ( 2 p∑ l=1 (−1)l+p lk l!s2(p,l) ) tp p! exp(yt)c0(xt) = ( ∞∑ p=1 ( 2 p∑ l=1 (−1)l+p lk l!s2(p,l) ) tp p! )( ∞∑ n=0 ln(x,y) tn n! ) . replacing n by n−p in the above equation and comparing the coefficients of tn in both sides, we obtain the result (2.15). remark 2.4. taking x = 0, theorem 2.4 reduces to the known result of kim et al. [14.,p. 4780, theorem (2.4)]. corollary 2.4. for n ≥ 1, we have g(k)n (y + 1) + g (k) n (y) = 2 n∑ p=1 p∑ l=1 (−1)l+p lk l!s2(p,l)   n p  yn−p. (2.16) theorem 2.5. for d ∈ n with d ≡ 1(mod2), we have lg (k) n (x,y) = n∑ p=0   n p  dn−p−1 p+1∑ l=0 d−1∑ a=0 (−1)l+p+1l!s2(p + 1, l) lk (−1)algn−p( a + y d ,x). (2.17) int. j. anal. appl. 16 (6) (2018) 862 proof. from equation (2.1), we can be written as ∞∑ n=0 lg (k) n (x,y) tn n! = 2lik(1 −e−t) et + 1 exp(yt)c0(xt) = ( 2lik(1 −e−t) t )( 2t edt + 1 d−1∑ a=0 (−1)a exp((a + y)t)c0(xt) ) = ( ∞∑ p=0 ( p+1∑ l=1 (−1)l+p+1 lk l! s2(p + 1, l) p + 1 ) tp p! )( ∞∑ n=0 dn−1 d−1∑ a=0 (−1)algn( a + y d ,x) tn n! ) . replacing n by n−p in above equation and comparing the coefficient of tn in both sides, we get (2.17). remark 2.5. for x = 0, theorem 2.5 reduces to the known result of kim et al. [14.,p. 4780]. corollary 5. for d ∈ n with d ≡ 1(mod2), we have g(k)n (y) = n∑ p=0   n p  dn−p−1 p+1∑ l=0 d−1∑ a=0 (−1)l+p+1l!s2(p + 1, l) lk (−1)agn−p( a + y d ). (2.18) 3. summation formulae for laguerre-based poly-genocchi polynomials in this section, we establish summation formula for laguerre-based poly-genocchi polynomials by using series techniques method. theorem 3.1. the following implicit summation formulae for laguerre-based poly-genocchi polynomials lg (k) n (x,y) holds true: lg (k) l+p(x,z) = l,p∑ m,n=0   l m     p n   (z −y)m+nlg(k)l+p−m−n(x,y). (3.1) proof. replacing t by t + u and rewrite the generating function (2.1) as 2lik(1 − (e)−(t+u)) et+u + 1 c0(x(t + u)) = e −y(t+u) ∞∑ l,p=0 lg (k) l+p(x,y) tl l! up p! . (3.2) replacing y by z in the above equation and equating the resulting equation to the above equation, we get e(z−y)(t+u) ∞∑ m,l=0 lg (k) l+p(x,y) tl l! up p! = ∞∑ l,p=0 lg (k) l+p(x,z) tl l! up p! . (3.3) on expanding exponential function (3.3) gives ∞∑ n=0 [(z −y)(t + u)]n n! ∞∑ l,p=0 lg (k) l+p(x,y) tl l! up p! = ∞∑ l,p=0 lg (k) l+p(x,z) tl l! up p! (3.4) which on using formula [16, p.52(2)] ∞∑ n=0 f(n) (x + y)n n! = ∞∑ n,m=0 f(n + m) xn n! ym m! , (3.5) int. j. anal. appl. 16 (6) (2018) 863 in the left hand side becomes ∞∑ m,n=0 (z −x)m+ntmun m!n! ∞∑ l,p=0 hg (k) l+p(x,y) tl l! up p! = ∞∑ l,p=0 hg (k) l+p(z,y) tl l! up p! (3.6) now replacing l by l−m, p by p−n and using the lemma [16, p.100(1)] in the left hand side of (3.6), we get ∞∑ m,n=0 ∞∑ l,p=0 (z −x)m+n m!n! lg (k) l+p−m−n(x,y) tl (l−m)! up (p−n)! = ∞∑ l,p=0 lg (k) l+p(x,z) tl l! up p! . (3.7) finally on equating the coefficients of the like powers of t and u in the above equation, we get the required result. remark 3.1. taking l = 0 in assertion (3.1) of theorem 3.1, we deduce the following consequence of theorem 3.1. corollary 3.1. the following summation formula for laguerre-based poly-genocchi polynomials hg (k) n (z,y) holds true: lg (k) p (x,z) = p∑ n=0   p n   (z −y)nlg(k)p−n(x,y). (3.8) remark 3.2. replacing z by z + y in (3.8), we obtain lg (k) p (x,z + y) = p∑ n=0   p n  znlg(k)p−n(x,y). (3.9) theorem 3.2. the following summation formula for laguerre-based poly-genocchi polynomials hg (k) n (z,y) holds true: lg (k) n (x,y + u) = n∑ j=0   n j  ujlg(k)n−j(x,y). (3.10) proof. using (2.1), we can be written as ∞∑ n=0 lg (k) n (x,y + u) tn n! = 2lik(1 −e−t) et + 1 exp((y + u)t)c0(xt) = ( ∞∑ n=0 lg (k) n (x,y) tn n! ) ∞∑ j=0 uj tj j!   now replacing n by n− j and comparing the coefficients of tn in both sides, we obtain (3.10). int. j. anal. appl. 16 (6) (2018) 864 theorem 3.3. the following summation formula for laguerre-based poly-genocchi polynomials hg (k) n (z,y) holds true: lg (k) n (x + w,y + u) = n∑ m=0   n m   lg (k) n−m(x,y)lm(u,w). (3.11) proof. from (2.1) and (1.10), we have 2lik(1 − (e)−t) et + 1 exp((y + u)t)c0((x + w)t) = ( ∞∑ n=0 lg (k) n (x,y) tn n! )( ∞∑ m=0 lm(u,w) tm m! ) . now replacing n by n−m and comparing the coefficients of tn in both sides, we get (3.11). theorem 3.4. the following summation formula for laguerre-based poly-genocchi polynomials lg (k) n (x,y) holds true: lg (k) n (x,y + 1) = n∑ m=0   n m   lg (k) n−m(x,y). (3.12) proof. using definition (2.1), we have ∞∑ n=0 lg (k) n (x,y + 1) tn n! − ∞∑ n=0 lg (k) n (x,y) tn n! = 2lik(1 −e−t) et + 1 exp(yt)c0(xt)(e t − 1) = ( ∞∑ n=0 lg (k) n (x,y) tn n! )( ∞∑ m=0 tm m! ) − ∞∑ n=0 lg (k) n (x,y) tn n! = ∞∑ n=0 n∑ m=0 lg (k) n−m(x,y) tn (n−m)!m! − ∞∑ n=0 lg (k) n (x,y) tn n! . finally, equating the coefficients of the like powers of tn, we get (3.12). 4. identities for 2-variable laguerre-based poly-genocchi polynomials in this section, we derive general symmetry identities for 2-variable laguerre-based poly-genocchi polynomials lg (k) n (x,y) by applying the generating function(2.1). such type of identities have been introduced by several authors (see [11], [12], [13], [15]). theorem 4.1. let a,b > 0 and a 6= b, x,y ∈ r, n ≥ 0, then the following identity holds true: n∑ m=0   n m  an−mbmlg(k)n−m(bx,by)lg(k)m (au,aw) = n∑ m=0   n m  ambn−mlg(k)n−m(ax,ay)lg(k)m (bu,bw). (4.1) int. j. anal. appl. 16 (6) (2018) 865 proof. let g(t) = ( (2lik(1 −e−at)(2lik(1 −e−bt)) (eat + 1)(abt −b−bt) ) exp(ab(y + u)t)c0(abxt)c0(abwt). (4.2) since g(t) is symmetric in a and b and g(t) can written as g(t) = ∞∑ n=0 lg (k) n (bx,by) (at)n n! ∞∑ m=0 lg (k) m (au,aw) (bt)m m! g(t) = ∞∑ n=0   n∑ m=0   n m  an−mbmlg(k)n−m(bx,by)lg(k)m (au,aw)   tn n! . (4.3) similarly, we can show that g(t) = ∞∑ n=0 lg (k) n (ax,ay) (bt)n n! ∞∑ m=0 lg (k) m (bu,bw) (at)m m! g(t) = ∞∑ n=0   n∑ m=0   n m  ambn−mlg(k)n−m(ax,ay)lg(k)m (bu,bw)   tn n! . (4.4) comparing the coefficients of t n n! in (4.3) and (4.4), we arrive at the desired result. remark 4.1. on setting b = 1 in theorem 4.1, we get n∑ m=0   n m  an−mlg(k)n−m(x,y)lg(k)m (au,aw) = n∑ m=0   n m  amlg(k)n−m(ax,ay)lg(k)m (u,w). theorem 4.2. let a,b > 0 and a 6= b, x,y ∈ r and n ≥ 0, then the following identity holds true: n∑ m=0   n m  a−1∑ i=0 b−1∑ j=0 lg (k) n−m ( by + b a i + j,bx ) lg (k) m (au,aw)b man−m = n∑ m=0   n m   b−1∑ i=0 a−1∑ j=0 lg (k) n−m ( ay + a b i + j,ax ) lg (k) m (bu,bw)a mbn−m. (4.5) proof. let g(t) = ( (2lik(1 −e−at))(2lik(1 −e−bt)) (eat + 1)2(ebt + 1)2 ) (eabt + 1)2 exp(ab(y + u)t)c0(abxt)c0(abwt) g(t) = ( 2lik(1 −e−at) eat + 1 ) exp(abyt)c0(abxt) ( eabt + 1 ebt + 1 )( 2lik(1 −e−bt) ebt + 1 ) ×exp(abut)c0(abwt) ( eabt + 1 eat + 1 ) int. j. anal. appl. 16 (6) (2018) 866 = ( 2lik(1 −e−at) (eat + 1 ) exp(abyt)c0(abxt) a−1∑ i=0 (−1)iebti ( 2lik(1 −e−bt) ebt + 1 ) ×exp(abut)c0(abwt) b−1∑ j=0 (−1)jeatj. = ( 2lik(1 −e−t) eat + 1 ) c0(abxt) a−1∑ i=0 b−1∑ j=0 (−1)i+je(by+ b a i+j)at ∞∑ m=0 lg (k) m (au,aw) (bt)m m! = ∞∑ n=0 a−1∑ i=0 b−1∑ j=0 lg (k) n ( by + b a i + j,bx ) (at)n n! ∞∑ m=0 lg (k) m (au,aw) (bt)m (m)! g(t) = ∞∑ n=0   n∑ m=0   n m  a−1∑ i=0 b−1∑ j=0 (−1)i+jlg (k) n−m ( by + b a i + j,bx ) lg (k) m (au,aw)b man−m   tn n! . (4.6) on the other hand g(t) = ∞∑ n=0   n∑ m=0   n m   b−1∑ i=0 a−1∑ j=0 (−1)i+jlg (k) n−m ( ay + a b i + j,ax ) lg (k) m (bu,bw)a mbn−m   tn n! . (4.7) on comparing the coefficients of t n n! in (4.6) and (4.7), we arrive at the desired result (4.5). acknowledgement. all authors would like to thank integral university, lucknow, india, for providing the manuscript number iu/r&d/2017-mcn000240 for the present research work. references [1] l. c. andrews, special functions for engineers and mathematicians, macmillan co. new york, 1985. [2] a. bayad and y. hamahata, polylogarithms and poly-bernoulli polynomials, kyushu. j. math. 65(2011), 15-24. [3] c. h. chang and c. w. ha, on recurrence relation for bernoulli and euler numbers, bull. aust. math. soc. 64(2001), 469-474. [4] g. s. cheon, a note on the bernoulli and euler polynomials, app. math. lett. 16(3)(2003), 365-368. [5] g. dattoli, a. torre, operational methods and two variable laguerre polynomials, atti accad. sci. torino cl. sci. fis. mat. natur. 132(1998) 3-9. [6] g. dattoli, a. torre and a. m. mancho, the generalized laguerre polynomials, the associated bessel functions and applications to propagation problems, radiat. phys. chem. 59(2000), 229-237. [7] h. jolany, m. r. darafsheh, r. e. alikelaye, generalizations of poly-bernoulli numbers and polynomials, int. j. math. comb. 2(2010), 7-14. [8] h. jolany, r. b. corcino, explicit formula for generalization of poly-bernoulli numbers and polynomials with a,b,c parameters, j. class. anal. 6(2015), 119-135. [9] h. jolany, m. aliabadi, r. b. corcino and m. r. darafsheh, a note on multi poly-euler numbers and bernoulli polynomials, gen. math. 20(2-3)(2012), 122-134. [10] m. kaneko, poly-bernoulli numbers, j. thor. nombres bordx. 9 (1997), 221-228. [11] w. a. khan, some properties of the generalized apostol type hermite-based polynomials, kyungpook math. j. 55(2015), 597-614. int. j. anal. appl. 16 (6) (2018) 867 [12] w. a. khan, a note on hermite-based poly-euler and multi poly-euler polynomials, palestine j. math. 6(2017), 204-214. [13] w. a. khan, s. araci, m. acikgoz, a new class of laguerre-based apostol type polynomials, cogent math. 3(2016), art. id 1243839. [14] t. kim, y. s. jang and j. j. seo, a note on poly-genocchi numbers and polynomials, appl. math. sci. 8(2014), 4475-4781. [15] m. a. pathan and w. a. khan, some implicit summation formulas and symmetric identities for the generalized hermiteeuler polynomials, east-west j. math. 16(1) (2014), 92-109. [16] h. m. srivastava and h. l. manocha, a treatise on generating functions ellis horwood limited, new york, 1984. 1. introduction 2. 2-variable laguerre-based poly-genocchi polynomials 3. summation formulae for laguerre-based poly-genocchi polynomials 4. identities for 2-variable laguerre-based poly-genocchi polynomials references international journal of analysis and applications volume 16, number 6 (2018), 842-855 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-842 numerical solution and analysis for acute and chronic hepatitis b muhammad farman, zafar iqbal, aqeel ahmad∗, ali raza and ehsan ul haq department of mathematics and statistics, the university of lahore, lahore, pakistan. ∗corresponding author: aqeelahmad.740@gmmail.com abstract. in this article, we present the transmission dynamic of the acute and chronic hepatitis b epidemic problem to control the spread of hepatitis b in a community. in order to do this, first we present sensitivity analysis of the basic reproduction number r0. we develop a unconditionally convergent nonstandard finite difference scheme by applying mickens approach φ(h) = h + o(h2) instead of h to control the spread of this infection, treatment and vaccination to minimize the number of acute infected, chronically infected with hepatitis b individuals and maximize the number of susceptible and recovered individuals. the stability analysis of the scheme has been developed by theorems which shows the both stable locally and globally. comparison is also made with standard nonstandard finite difference scheme. finally numerical simulations are also established to investigate the influence of the system parameter on the spread of the disease. 1. introduction the scope of mathematics includes mathematical modeling and esoteric mathematics. the flow of work, process, predictions and outcomes can easily be measured with the help of mathematical concepts and theory. therefore, biologists are now extremely dependent on mathematics. mathematical modeling of biological sciences is done by many brilliant scientist [1-3]. the relationship between simple mathematical modeling involves biological system, integer order differential equations that show their dynamics and complex system received 2018-06-26; accepted 2018-09-09; published 2018-11-02. 2010 mathematics subject classification. 37c75, 65l07. key words and phrases. sensitivity analysis; acute hepatitis b; chronic hepatitis b; nsfd. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 842 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-842 int. j. anal. appl. 16 (6) (2018) 843 which describes their changing of structure. the nonlinearity and multi-scale behaviors in mathematical modeling describe the mutual relationship between parameter [4]. in last few decades, many biological models were studied in detail by using classical derivative, few of them in [5,6]. hepatitis b is a potentially life-threatening liver infection caused by the hepatitis b virus. it is a major global health problem. it can cause chronic liver disease and chronic infection and puts people at high risk of death from cirrhosis of the liver and liver cancer [7]. infections of hepatitis b occur only if the virus is able to enter the blood stream and reach the liver. once in the liver, the virus reproduces and releases large numbers of new viruses into the blood stream [8]. this infection has two possible phases: (1) acute and (2) chronic. acute hepatitis b infection lasts less than six months. if the disease is acute, your immune system is usually able to clear the virus from your body, and you should recover completely within a few months. most people who acquire hepatitis b as adults have an acute infection. chronic hepatitis b infection lasts six months or longer. most infants infected with hbv at birth and many children infected between 1 and 6 years of age become chronically infected [7]. about two-thirds of people with chronic hbv infection are chronic carriers. these people do not develop symptoms, even though they harbor the virus and can transmit it to other people. the remaining one-third develop active hepatitis, a disease of the liver that can be very serious. more than 240 million people have chronic liver infections. about 600 000 people die every year due to the acute or chronic consequences of hepatitis b [7,9] hbv can be transmitted from one individual to another individual on different ways, such as transmission of blood, semen and vaginal secretions [20,21,22]. another major transmission of hbv is the unprotected sexual contact, sharing of razors, blades or tooth brushes [3].also the virus transmits from an infected mother to her child during the time of birth. however, hbv cannot be transmitted through water, food, hugging, kissing and causal contact such as in the work place, school, etc. [22]. the mode of transmission of hbv and hiv is the same, but hbv is 50100 times more infectious [25]. hbv infection is a global health problem. according to who about 400million population is infected world wide chronically. in china 93million population are affected due to hbv infections [23,27,28]. vaccine for the prevention of hepatitis b is available in the market that is very effective [24,26]. in the real world phenomena mathematical modeling is one of the powerful tools to describe the dynamical behavior of different diseases [16,17,18,19,29]. mathematical models have been used to help understand the dynamics of viral infections, such as human immunodeficiency virus and hepatitis c infection [11,12]. following these approaches, dynamic models were developed to analyze the changes in hepatitis b virus levels during drug therapy [13,14,15,10]. in this article,we develop a hbv transmission model. the infectious class is divided into two stages, such as acute infectious and chronic infectious stage. thus, the total population is divided into four compartments, s(t) int. j. anal. appl. 16 (6) (2018) 844 susceptible, i1(t) infected with acute hepatitis b, i2(t) infected with chronic hepatitis b and r(t) recovered individuals. in this paper, we investigate the stability and qualitative analysis of acute and chronic hepatitis b model. an unconditionally convergent nonstandard finite difference scheme has been presented to obtain solution of model. the analysis of two different states disease free and endemic equilibrium which means the disease dies out or persist in a population has been made by finding reproductive number. numerical results are presented graphically to show the dynamics of the model. 2. materials and method we used a mathematical model for hbv transmission by extending the work presented in [28].we divide the host population denoted by t(t) into four compartments: susceptible individuals s(t), who are not infective but have the chance to catch the disease; infected i1(t) represents those individuals who are infective with acute hepatitis; i2(t) are those individuals, who are infected with chronic hepatitis and r(t) represents those individuals who have recovered after the infection with a life-time immunity. the flowchart for the transmission of hbv is given in figure 1. figure 1. the flowchart of the model thus, the mathematical model is represented by the following four differentials equations: ds dt = b−αs(t)i2(t) − (µ0 + ν)s(t) (2.1) di1 dt = αs(t)i2(t) − (µ1 + β + γ1)i1(t) (2.2) int. j. anal. appl. 16 (6) (2018) 845 di2 dt = βi1(t) − (µ0 + µ1 + γ2)i2(t) (2.3) dr dt = γ1i1(t) + γ2i2(t) + νs(t) −µ0r(t) (2.4) with initial conditions s(0) ≥ 0,i1(0) ≥ 0,i2(0) ≥,r(0) ≥ 0, here b represents the birth rate, α is the moving rate from susceptible to infected with acute hepatitis b, β is the moving rate from acute stage to infected with chronic hepatitis, γ1 is the recovery rate from acute stage to recovered, γ2 is the recovery rate from chronic stage to recovered compartment, µ0 is the death rate occurring naturally, which is also called natural mortality rate, µ1 is the death rate occurring due to hepatitis b and ν represents hepatitis b vaccination rate. 3. qualitative analysis the model (2.1 − 2.4) is locally asymptotically as well as globally asymptotically stable at disease-free and endemic equilibrium points [29]. for disease-free equilibrium the model (2.1 − 2.4) is both locally and globally stable, if the value of basic reproduction number is less than unity while for the endemic equilibrium the model is stable if the value of the basic reproduction number r0 is greater than unity. model has a disease-free equilibrium, denoted by e0 and defined as, e0 = (s0, 0, 0,r 0), where s0 = b µ0 + ν and r0 = νb µ0(µ0 + ν) . the endemic equilibrium is given by e∗ = (s∗, i∗1 , i ∗ 2 , r ∗), where s∗ = 1 αβ (µ0 + β + γ1)(µ0 + µ1 + γ2) i∗1 = 1 αβ (µ0 + ν)(µ0 + µ1 + γ2)[r0 − 1] i∗2 = 1 α (µ0 + ν)[r0 − 1] r∗ = 1 µ0 [( γ1 αβ (µ0 + ν)(µ0 + µ1 + γ2) + γ2 α (µ0 + ν)[r0 − 1]) + ν αβ (µ0 + β + γ1)(µ0 + µ1 + γ2)] regarding these equilibrium point of the model (2.1−2.4), we have the following results which are proved in [29]. int. j. anal. appl. 16 (6) (2018) 846 3.1. reproductive number. basic reproduction number r0 is defined to be the expected number of secondary infections produced by an index case or the average number of secondary infection arising from a single individual introduced into the susceptible class during its entire infectious period in a totally susceptible population. the basic reproduction number r0 of the model (2.1 − 2.4) in [29] is r0 = αβb (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2) theorem 3.1. if r0 < 1, then the model (2.1−2.4) is locally asymptotically stable at disease-free equilibrium, e0 = ( b µ0+ν , 0, 0, νb µ0(µ0+ν) ), while e0 is unstable saddle point if r0 > 1. theorem 3.2. if r0 ≤ 1, then the model (2.1 − 2.4) is globally asymptotically stable at disease-free equilibrium, e0 = (s0, 0, 0,r0) and unstable otherwise. theorem 3.3. the endemic equilibrium state e1 = (s ∗,i∗1 ,i ∗ 2 ,r ∗) of the model (2.1 − 2.4) is globally asymptotically stable, if r0 > 1, otherwise unstable. prof of these theorems will be given in [29], used in section 4. 3.2. sensitivity analysis of r0: the sensitivity of r0 = αβb (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2) to each of its parameters is ∂r0 ∂α = βb (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2) ≥ 0 ∂r0 ∂β = αb(µ0 + γ1) (µ0 + ν)(µ0 + β + γ1)2(µ0 + µ1 + γ2) ≥ 0 ∂r0 ∂b = αβ (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2) ≥ 0 ∂r0 ∂ν = − αβb (µ0 + ν)2(µ0 + β + γ1)(µ0 + µ1 + γ2) ≤ 0 ∂r0 ∂γ1 = − αβb (µ0 + ν)(µ0 + β + γ1)2(µ0 + µ1 + γ2) ≤ 0 ∂r0 ∂γ2 = − αβb (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2)2 ≤ 0 ∂r0 ∂µ1 = − αβb (µ0 + ν)(µ0 + β + γ1)(µ0 + µ1 + γ2)2 ≤ 0 ∂r0 ∂µ0 = −αβb[(µ0 + ν)(µ0 + β + γ1) + (µ0 + ν)(µ0 + µ1 + γ2) + (µ0 + β + γ1)(µ0 + µ1 + γ2)] (µ0 + ν)2(µ0 + β + γ1)2(µ0 + µ1 + γ2)2 ≤ 0 it can be seen that r0 is most sensitive to change in parameter, here, r0 is increasing with α, b, β, and decreasing with γ1, γ2, ν, µ0, µ1. in other words it found that the sensitivity analysis shows that prevention is better than to control the disease. int. j. anal. appl. 16 (6) (2018) 847 4. nonstandard finite difference (nsfd) scheme a nonstandard finite difference (nsfd) scheme for the system (2.1−2.4) is presented in this section [30]. in recent years, nonstandard finite difference (nsfd) scheme for discrete models have been constructed or tested for a wide range of nonlinear systems of differential equations [31,32,33]. the positivity of the state variables involved in the system is satisfy by proposed method. this property has key role when we solve mathematical models arising in biology because these state variables represent sub-populations which never take negative values. the discretized form of the the system (2.1−2.4) by using nsfd scheme which based on the generalized first order forward method is written as sk+1 −sk h = b−αsk+1ik2 − (µ0 + ν)s k+1 (4.1) sk+1 + hαsk+1ik2 + h(µ0 + ν)s k+1 = sk + bh (4.2) sk+1 = sk + bh 1 + hαik2 + h(µ0 + ν) (4.3) ik+11 − i k 1 h = αik2 s k+1 − (β + µ0 + γ1)ik+11 (4.4) ik+11 + h(β + µ0 + γ1)i k+1 1 = i k 1 + hαi k 2 s k+1 (4.5) ik+11 = ik1 + hαi k 2 s k+1 1 + h(β + µ0 + γ1) (4.6) ik+12 − i k 2 h = βik+11 − (µ1 + µ0 + γ2)i k+1 2 (4.7) ik+12 + h(µ1 + µ0 + γ2)i k+1 2 = i k 2 + hβi k+1 1 (4.8) ik+12 = ik2 + hβi k+1 1 1 + h(µ1 + µ0 + γ2) (4.9) rk+1 −rk h = γ1i k+1 1 + γ2i k+1 2 + νs k+1 −µ0rk+1 (4.10) rk+1(1 + hµ0) = r k + h(γ1i k+1 1 + γ2i k+1 2 + νs k+1) (4.11) rk+1 = rk + h(γ1i k+1 1 + γ2i k+1 2 + νs k+1) 1 + hµ0 (4.12) int. j. anal. appl. 16 (6) (2018) 848 4.1. proposed nsfd scheme. in this section, we design an nsfd scheme [34] that replicates the dynamics of the continuous model (2.1−2.4). let yk = (sk,i1k,i2k,rk)t denoted an approximation of x(tk) where tk = k∆t, with k ∈n , h = ∆t > 0 be a step size then sk+1 −sk φ = b−αsk+1ik2 − (µ0 + ν)s k+1 (4.13) ik+11 − i k 1 φ = αik2 s k+1 − (β + µ0 + γ1)ik+11 (4.14) ik+12 − i k 2 φ = βik+11 − (µ1 + µ0 + γ2)i k+1 2 (4.15) rk+1 −rk φ = γ1i k+1 1 + γ2i k+1 2 + νs k+1 −µ0rk+1 (4.16) which is the new purposed nsfd scheme for the given model, where φ = φ(h) = 1 −e−(β+µ0+γ1)h β + µ0 + γ1 (4.17) the discrete method (4.13 − 4.16) is indeed an nsfd scheme because it is constructed according to mickens rules [33] formalized as follows in [34]. rule 1. the standard denominator h = ∆t of the discrete derivatives is replaced by the complex denominator function in equation (4.17) which satisfies the asymptotic relation φ(h) = h + o(h2) note that the denominator function φ is expected to better capture the dynamics of the continuous model through the presence of the underlying parameters µ0,β,γ1. in fact, exact schemes for a wide range of dynamical systems involve such complex denominator functions [35,36]. rule 2.nonlinear terms in the right-hand side of equation (2.1−2.4) are approximated in a non-local way. for instance, we have i2(tk)s(tk) ' i2ksk+1 instead of i2(tk)s(tk) ' i2ksk 4.2. analysis of the scheme. theorem 4.1. the nsfd scheme (4.13 − 4.16) is a dynamical system on the biological feasible domain k of the continuous model (2.1 − 2.4). proof:first, we prove the positivity of the scheme (4.13 − 4.16). it is easy to show that the nsfd scheme (4.13 − 4.16) takes the explicit form sk+1 = sk + φb 1 + αφik2 + (µ0 + ν)φ int. j. anal. appl. 16 (6) (2018) 849 ik+11 = [1 + αφik2 + (µ0 + ν)φ][i k 1 + αφ(s k + φb)ik2 ] [1 + φ(µ0 + β + γ1)][1 + αφi k 2 + (µ0 + ν)φ] ik+12 = [1 + φ(µ0 + β + γ1)][1 + αφi k 2 + (µ0 + ν)φ](i k 2 + βφi k 1 ) + βαφ 2(sk + φb)ik2 [1 + φ(µ0 + µ1 + γ2)][1 + φ(µ0 + β + γ1)][1 + αφi k 2 + (µ0 + ν)φ] rk+1 = rk.a.b.c.d + φ{γ1(a.d.cik1 + αφ.e) + γ2a(b.c[ik2 + βφik1 ] + αβφ2ik2 .e) + νe.a.b.c} a.b.c.d where a = 1 + µ0φ, b = 1 + φ(µ0 + β + γ1), c = 1 + αφi k 2 + (µ0 + ν)φ d = 1 + φ(µ0 + µ1 + γ2), e = s k + φb thus sk+1 ≥ 0, ik+11 ≥ 0, i k+1 2 ≥ 0, r k+1 ≥ 0 whenever the discrete variables are non-negative at the previous iteration. it remains to prove the positive invariance of k. adding the (4.13) and (4.14),we have [1 + φ(µ0 + ν)]h k+1 = φb + hk − [1 + (µ0 + µ1 + γ1)φ]ik ≤ φb + hk [1 + φ(µ0 + ν)]h k+1 ≤ φb + hk ⇒ hk+1 ≤ b µ0 + ν whenever hk ≤ b µ0 + ν the priori bonds for ik+12 and r k+1 follow the radially from the fact that ik+12 and i k+1 1 and less then or equal hk+1. this complete the proof. theorem 4.2. (1) the disease-free fixed point (resp. the endemic fixed point ) of the nsfd scheme (4.13− 4.16) for the model without recruitment/provision of disease is gas whenever r0 ≤ 1 (resp. whenever r0 > 1). (2) the endemic fixed-point of the nsfd scheme (4.13 − 4.16) for the full model is gas. proof:let yk ∈r4+ be the bounded sequence defined by the nsfd scheme (4.13 − 4.16). we want to prove that yk tends to y ∗, where y ∗ is any of the fixed point. by bolzano weierstrass theorem, there exists a subsequence ynk of yn that converge to some z ∗ as k → +∞. by the assumption made above and the structure of the nsfd scheme (4.13−4.16), y ∗ = z∗ is necessarily either the unique disease-free fixed-point int. j. anal. appl. 16 (6) (2018) 850 e0 (whenever r0 ≤ 1) or the unique endemic fixed-point e∗ or the unique endemic e�, which is las thanks to theorem 4.2. therefore, there exists θ > 0 such that for an initial condition y0 satisfying ‖y 0 −y ∗‖≤ θ we have lim x→+∞ ‖y 0 −y ∗‖ = 0 (4.18) let y 0 be an arbitrary initial condition . as lim x→+∞ ynk = y ∗, there exits a integer k0 such that ‖ynk0 −y ∗‖≤ θ (4.19) in view equation (4.18) and (4.19), we have lim x→+∞,n≥1 ‖ynn −y ∗‖ = lim x→+∞,n≥nk0 ‖ynn −y ∗‖ = 0 (4.20) this prove that y ∗ is gas. table 1. values of physical parameters used in model when r0 < 1 parameter value parameter value n1 100 n2 40 n3 20 n4 5 b 0.4 ν 0.02 β 0.01 µ0 0.03 µ1 0.002 γ1 0.05 γ2 0.06 α 0.005 int. j. anal. appl. 16 (6) (2018) 851 table 2. values of physical parameters used in model when r0 > 1 parameter value parameter value n1 100 n2 40 n3 20 n4 5 b 0.4 ν 0.02 β 0.1 µ0 0.03 µ1 0.04 γ1 0.05 γ2 0.06 α 0.05 4.3. numerical simulations. the mathematical analysis of epidemic model hepatitis b with non-linear incidence has been presented. to observe the effects of the parameters using in this dynamics hepatitis b model (2.1−2.4), conclude several numerical simulations varying the value of parameters given in table 1 and table 2 for r0 < 1 and r0 > 1 respectively. figure 2 and 3 shows the convergence solution for diseases free and endemic equilibria by using nsfd scheme at h = 1. figure 4 and 5 also represent the he convergence solution for diseases free and endemic equilibria by using nsfd scheme at φ = φ(h) + o(h2). the technique create a better impact to control the hepatitis b, it reduces the infected rate and increases the susceptible and recovered population during disease free state as well as in endemic state. 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 disease free equilibrium time,step size h=1 c om pa rm en ta l p op ul at io n susceptible infected with acute hb infected with chronic hb recovered figure 2. numerical solutions for susceptible, acute infected individual, chronic infected individual and recovered population in a time t with step size h = 1 for disease free equilibrium points. int. j. anal. appl. 16 (6) (2018) 852 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 endemic equilibrium time,step size h=1 c om pa rm en ta l p op ul at io n susceptible infected with acute hb infected with chronic hb recovered figure 3. numerical solutions for susceptible, acute infected individual, chronic infected individual and recovered population in a time t with step size h = 1 for endemic equilibrium points. 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 disease free equilibrium time c om pa rm en ta l p op ul at io n susceptible infected with acute hb infected with chronic hb recovered figure 4. numerical solutions for susceptible, acute infected individual, chronic infected individual and recovered population in a time t by using φ = φ(h) with step size h = 1 for disease free equilibrium points. int. j. anal. appl. 16 (6) (2018) 853 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 90 100 endemic equilibrium time c om pa rm en ta l p op ul at io n susceptible infected with acute hb infected with chronic hb recovered figure 5. numerical solutions for susceptible, acute infected individual, chronic infected individual and recovered population in a time t by using φ = φ(h) with step size h = 1 for endemic equilibrium points. 5. conclusion we have considered a mathematical system of equation which describes the hepatitis b disease. the analysis of the system is well established. sufficient conditions for local stability of the dfe point e0 are given in terms of the basic reproduction number r0 of the model, where it is asymptotically stable if r0 < 1. the positive infected equilibrium e ∗ exist when r0 > 1 and sufficient conditions that guarantee the asymptotic stability of the point are given. beside this sensitivity analysis of the parameters involved in threshold parameter r0 is discussed. it is important to note that nonstandard finite difference scheme for mathematical models based on system of differential equations is more powerful approach to compute the convergent solutions for the disease models. the nonstandard finite difference scheme is dynamically consistent, easy to implement and show a good agreement to control the bad impact of hepatitis b for long period of time and to eradicate a death killer factor in the world spread by hepatitis b. finally, we presented the numerical simulation and verified all the analytical results numerically by using nonstandard finite difference scheme to reduce acute as well chronic infected rates for both disease free and endemic equilibria , we are able to control the spreading of hepatitis b in the community. references [1] j. biazar, solution of the epidemic model by adomian decomposition method, appl. math. comput. 173 (2006), 1101-1106. int. j. anal. appl. 16 (6) (2018) 854 [2] s. busenberg, p. driessche, analysis of a disease transmission model in a population with varying size, j. math. biol. 28 (1990), 65-82. [3] a.m.a. el-sayed, s.z. rida and a.a.m. arafa, on the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, int. j. nonlinear sci. 7 (2009), 485-495. [4] o.d. makinde, adomian decomposition approach to a sir epidemic model with constant vaccination strategy, appl. math. comput. 184 (2007), 842-848. [5] a.a.m. arafa, s.z. rida and m. khalil, fractional modeling dynamics of hiv and 4 t-cells during primary infection, nonlinear biomed. phys. 6 (2012), 1-7. [6] c.m. kribs-zaleta, structured models for heterosexual disease transmission, math. biosci. 160 (1999), 83-108. [7] b. buonomo and d. lacitignola, on the dynamics of an seir epidemic model with a convex incidence rate, ricerche mat. 57 (2008), 261-281. [8] who, hepatitis b fact sheet no. 204, the world health organisation, geneva, switzerland, 2013, http://www.who.int/mediacentre/factsheets/fs204/en/. [9] canadian centre for occupational health and safety, hepatitis b, http : //www.ccohs.ca/oshanswers/diseases/hepatitisb.html. [10] a. v. kamyad, r. akbari, a. k heydari and a. heydari mathematical modeling of transmission dynamics and optimal control of vaccination and treatment for hepatitis b virus, comput. math. methods med. 2014 (2014), article id 475451. [11] c.m. stanca, r.m. ruy, w.n. patrick and s.p. alan, modeling the mechanisms of acute hepatitis b virus infection, j. theor. biol. 247 (2007), 23-35. [12] a. perelson, modelling viral and immune system dynamics. nature rev. immunol. 2 (2002), 28-36. [13] a. perelson and r. ribeiro, hepatitis b virus kinetics and mathematical modeling. sem. liv. dis. 24 (2004), 11-15. [14] m. nowak, s. bonhoeffer,a. hill, r. boehme, h. thomas, h. mcdade, viral dynamics in hepatitis b infection. proc. natl acad. sci. usa 93 (1996), 4398-4402. [15] s. lewin, r. ribeiro, t. walters, g. lau, s. bowden, s. locarnini and a. perelson, analysis of hepatitis b viral load decline under potent therapy: complex decay profiles observed. hepatology 34 (2001), 1012-1020. [16] p. colombatto, l. civitano, r. bizzarri, f. oliveri, s. choudhury, r. gieschke, f. bonino and m.r. brunetto, a multiphase model of the dynamics of hbv infection in hbeag-negative patients during pegylated interferon-a2a, lamivudine and combination therapy. antiviral therapy 11 (2006), 197-212. [17] j. wang, j. pang and x. liu,modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, j. biol. dyn. 8 (2014), 99-116. [18] j. wang, r. zhang and t. kuniya, the stability analysis of an sveir model with continuous age-structure in the exposed and infectious classes, j. biol. dyn. 9 (2015), 73-101. [19] g. zaman, y.h. kang and i.h. jung, stability and optimal vaccination of an sir epidemic model, biosystems 93 (2008), 240249. [20] g. zaman, y.h. kang and i.h. jung, optimal treatment of an sir epidemic model with time delay, biosystems 98 (2009), 43-50. [21] d. lavanchy, hepatitis b virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures, j. viral hepat. 11 (2004), 97-107. [22] a.s. lok, e.j heathcote and j.h. hoofnagle, management of hepatitis b, 2000 summary of a workshop, gastroenterology 120 (2001), 18281853. [23] b.j.mcmahon, epidemiology and natural history of hepatitis b, semin. liver dis. 25 (2005), 38. int. j. anal. appl. 16 (6) (2018) 855 [24] m.k. libbus and l.m. phillips, public health management of perinatal hepatitis b virus, public health nurs. 26 (2009), 353-361. [25] j.e. maynard, m.a. kane and s.c. hadler, global control of hepatitis b through vaccination role of hepatitis b vaccine in the expanded programme on immunization, rev. infect. 2 (1989), s574-s578. [26] s. thornley, c. bullen and m. roberts,hepatitis b in a high prevalence newzealand population: a mathematical model applied to infection control policy, j. theor. biol. 254 (2008), 599-603. [27] c.w. shepard, e.p. simard, l. finelli, a.e. fiore and b.p. bell, hepatitis b virus infection epidemiology and vaccination, epidemiol. rev. 28 (2006), 112-125. [28] r. williams, global challenges in liver disease, hepatology 44 (2006), 521-526. [29] t. khan, g. zaman and m. i. chohan, the transmission dynamic and optimal control of acute and chronic hepatitis b, j. biol. dyn. 11(2017), 172-189. [30] r. e. mickens, exact solutions to a finite difference model of a nonlinear reactions advection equation: implications for numerical analysis, numer. methods partial differ. equ. 5 (1989), 313-325. [31] r. e. mickens, applications of nonstandard finite difference schemes, world scientific, singapore (2000). [32] r. anguelov and j.m.s lubuma, nonstandard finite difference method by nonlocal approximations, math. comput. simul. 61 (2003), 465-475. [33] r.e. mickens, nonstandard finite difference models of differential equations, world scientific, singapore (1994). [34] r. anguelov and j.m.s. lubuma, contributions to the mathematics of the nonstandard finite differencemethodandapplications, numer. methods partial differ. equ. 17 (2001), 518-543. [35] j.m.s. lubuma and k.c. patidar, non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, appl. math. comput. 187(2) (2007), 1147-1160. [36] l.w. roeger, exact difference schemes, in a. b. gumel mathematics of continuous and discrete dynamical systems, contemp. math., vol. 618, amer. math. soc., providence, ri, (2014), 147-161. 1. introduction 2. materials and method 3. qualitative analysis 3.1. reproductive number 3.2. sensitivity analysis of r0: 4. nonstandard finite difference (nsfd) scheme 4.1. proposed nsfd scheme 4.2. analysis of the scheme 4.3. numerical simulations 5. conclusion references international journal of analysis and applications volume 17, number 3 (2019), 420-439 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-420 ostrowski type inequality using five step weighted kernel sofian obeidat1,∗, muhammad amer latif1 and ather qayyum2 1department of basic sciences, deanship of preparatory year, university of hai’l, saudi arabia 2department of mathematics, institute of southern punjab multan, pakistan ∗corresponding author: obeidatsofian@gmail.com abstract. the purpose of this paper is to establish weighted version of ostrowski type integral inequalities. the inequalities are obtained by using a newly developed special type of five steps weighted kernel. the introduction of this new kernel gives some new error bounds for various quadrature rules. applications for cumulative distributive functions are considered. 1. introduction the field of inequalities have applications in most of the domains of mathematics. the importance of mathematical inequalities has increased during the past few decades and now it is studied as a separate branch of mathematics. a number of research papers and books have been written on inequalities and their applications (see for instance [9][13]). in many practical problems, it is important to bound one quantity by another quantity. the classical inequalities such as ostrowski’s inequality is very useful for this purpose. ostrowski type inequalities have immediate applications in numerical integration, optimization theory, statistics and integral operator theory. in 1938, ostrowski [8] discovered the following useful integral inequality. received 2019-02-14; accepted 2019-03-07; published 2019-05-01. 2010 mathematics subject classification. primary 26d15, 26d20; secondary 26d99. key words and phrases. ostrowski inequality; numerical integration; composite quadrature rule; cumulative distributive function. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 420 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-420 int. j. anal. appl. 17 (3) (2019) 421 theorem 1.1. let f : [a,b] → r be continuous on [a,b] and differentiable on (a,b) , whose derivative f′ : (a,b) → r is bounded on (a,b) , i.e. ‖f′‖∞ = sup t∈[a,b] |f′ (t)| < ∞ then for all x ∈ [a,b] ∣∣∣∣∣∣ f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤  1 4 + ( x− a+b 2 b−a )2 (b−a)‖f′‖∞ . (1.1) we mention another inequality called grüss inequality [6] which is stated as the integral inequality that establishes a connection between the integral of the product of two functions and the product of their individual integrals. mathematically, it is described as follows: ∣∣∣∣∣∣ 1b−a b∫ a f(x)g(x)dx− 1 b−a b∫ a f(x)dx. 1 b−a b∫ a g(x)dx ∣∣∣∣∣∣ (1.2) ≤ 1 4 (φ −ϕ)(γ −γ), where ϕ ≤ f (x) ≤ φ and γ ≤ g (x) ≤ γ, for all x ∈ [a,b] . the constant 1 4 is sharp in (1.2) . in [4], dragomir and wang combined ostrowski and grüss inequality to give a new inequality which they named ostrowski-grüss type inequalities. in [2], barnett et.al proved some ostrowski type inequality and generalized the trapezoidal inequality. dragomir [3] and liu [5] established some companions of ostrowski type integral inequalities. recently, qayyum et. al. [15] proved some ostrowski type inequalities, they obtained their results by using kernel with five steps. in this paper, we will present the weighted version of the results obtained by qayyum et. al. [15]. throughout the present paper, a weight function (or density function) over some interval [a,b] , where −∞ < a < b < ∞, is a function w : [a,b] −→ [0,∞) with 0 < b∫ a w(t)dt < ∞. int. j. anal. appl. 17 (3) (2019) 422 2. main results definition 2.1. let −∞ < a < b < ∞. let w be a weight function over [a,b] . the 5-step linear kernel with respect to w is denoted by pw, and is defined as follows: pw(x,t) =   t∫ a w (u) du, t ∈ [ a, a+x 2 ] 3 4 t∫ a w (u) du + 1 4 t∫ b w (u) du, t ∈ ( a+x 2 ,x ] 1 2 t∫ a w (u) du + 1 2 t∫ b w (u) du, t ∈ (x,a + b−x] 1 4 t∫ a w (u) du + 3 4 t∫ b w (u) du, t ∈ ( a + b−x, a+2b−x 2 ] t∫ b w (u) du, t ∈ ( a+2b−x 2 ,b ] , (2.1) for x ∈ [ a, a+b 2 ] and t ∈ [0, 1] . the following lemma will be used repeatedly throughout the present paper. lemma 2.1. let −∞ < a < b < ∞. for a weight function w over [a,b] , the identity b∫ a pw(x,t)f ′(t)dt (2.2) = 1 4   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a w (t) f (t) dt, holds for all x ∈ [ a, a+b 2 ] and t ∈ [0, 1] . proof: obvious. lemma 2.2. let −∞ < a < b < ∞, and w be a weight function over [a,b]. if w is symmetric about a+b 2 , then b∫ a pw(x,t)dt = 0. proof: since w is symmetric about a+b 2 , b∫ a tw (t) dt = b∫ a (a + b− t) w (a + b− t) dt = b∫ a (a + b− t) w (t) dt, which implies that b∫ a tw (t) dt = b + a 2 b∫ a w (t) dt. int. j. anal. appl. 17 (3) (2019) 423 using identity (2.2) with f (t) = t, we get that b∫ a pw(x,t)dt = b + a 2 b∫ a w (t) dt− b∫ a tw (t) dt = 0. the converse of lemma 2.2 is not correct in general as shown in the following example. example 2.1. for −∞ < a < b < ∞, let h (x) and w (x) be defined as: h (x) = −2 + 10 3 ( x−a b−a ) , w (x) = 1 9 (h (x)) 2 + 1 27 (h (x)) 3 , for x ∈ [a,b] . clearly, w is not symmetric about a+b 2 . on the other hand, note that b∫ a w (t) dt = 58 729 (b−a) and b∫ a tw (t) dt = 29 729 (b2 −a2), which implies that b∫ a tw (t) dt = a + b 2 b∫ a w (t) dt. thus, b∫ a pw(x,t)dt = 0. now with the help of lemma 2.1, we state and prove some theorems in the following subsections. 2.1. the l1 case. theorem 2.1. let −∞ < a < b < ∞ and f : [a,b] → r be a differentiable function on (a,b). suppose that w is a weight function over [a,b] with b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. if f ′ ∈ l1 [a,b] and d1 ≤ f ′(t) ≤ d2, for all t ∈ [a,b] , where d1, d2 are constants, then the inequality∣∣∣∣∣∣14   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) (2.3) holds for all x ∈ [ a, a+b 2 ] . int. j. anal. appl. 17 (3) (2019) 424 proof. note that for t ∈ [0, 1] and x ∈ [ a, a+b 2 ] , we have a∫ b w (t) dt ≤ pw(x,t) ≤ b∫ a w (t) dt. (2.4) using identity (2.2) with f (t) = t and the fact that b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt, we get that b∫ a pw(x,t)dt = 0, which implies that 1 b−a b∫ a pw(x,t)f ′(t)dt = 1 b−a b∫ a pw(x,t)f ′(t)dt− 1 (b−a)2 b∫ a pw(x,t)dt b∫ a f ′(t)dt applying grüss inequality, we get that∣∣∣∣∣∣ 1b−a b∫ a pw(x,t)f ′(t)dt ∣∣∣∣∣∣ (2.5) = ∣∣∣∣∣∣ 1b−a b∫ a pw(x,t)f ′(t)dt− 1 (b−a)2 b∫ a pw(x,t)dt b∫ a f ′(t)dt ∣∣∣∣∣∣ ≤ 1 4   b∫ a w (t) dt− a∫ b w (t) dt   (d2 −d1) . = 1 2   b∫ a w (t) dt   (d2 −d1) . by lemma 2.1, b∫ a pw(x,t)f ′(t)dt (2.6) = 1 4   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) − b∫ a w (t) f (t) dt. therefore, ∣∣∣∣∣∣14   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) . � int. j. anal. appl. 17 (3) (2019) 425 corollary 2.1. in theorem 2.1, if w is a weight function over [a,b] and is symmetric about a+b 2 , then inequality (2.3) becomes∣∣∣∣∣∣14   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 4   b∫ a w (t) dt   (d2 −d1) (b−a) . (2.7) proof. since w is symmetric about a+b 2 , we have b∫ a pw(x,t)dt = 0, and 1 2 a∫ b w (t) dt ≤ pw(x,t) ≤ 1 2 b∫ a w (t) dt. � we can generalize theorem 2.1 as follows: theorem 2.2. let −∞ < a < b < ∞ and f : [a,b] → r be a differentiable function on (a,b). suppose that w is a weight function over [a,b]. if f ′ ∈ l1 [a,b] and d1 ≤ f ′(t) ≤ d2, for all t ∈ [a,b] , where d1, d2 are constants, then the inequality∣∣∣∣∣∣  1 4   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) −mba (x) n b a   − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) (2.8) holds for all x ∈ [ a, a+b 2 ] , where mba (x) = b∫ a pw(x,t)dt and n b a = f(b)−f(a) b−a . proof. by lemma 2.1, we have 1 b−a b∫ a pw(x,t)f ′(t)dt− 1 (b−a)2 b∫ a pw(x,t)dt b∫ a f ′(t)dt = 1 4 (b−a)   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − f (b) −f (a) (b−a)2 b∫ a pw(x,t)dt− 1 b−a b∫ a w (t) f (t) dt. int. j. anal. appl. 17 (3) (2019) 426 by gruss inequality, we have ∣∣∣∣∣∣ 1b−a b∫ a pw(x,t)f ′(t)dt− 1 (b−a)2 b∫ a pw(x,t)dt b∫ a f ′(t)dt ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) . therefore, ∣∣∣∣∣∣  1 4   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) −mba (x) n b a   − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) . � theorem 2.3. let f : i ⊂ r → r be a differentiable mapping on i0, the interior of the interval i, and let a,b ∈ i with a < b. let w be a weight function over [a,b] with b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. if f ′ ∈ l1 [a,b] with d1 ≤ f ′ (t) ≤ d2 for all t ∈ [a,b], where d1, d2 are constants, then for each x ∈ [ a, a+b 2 ] , we have ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ d2 −d12 b∫ a |pw(x,t)|dt. (2.9) proof. let d = d1+d2 2 using identity (2.2) with f (t) = t, and the fact that b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt, we get that b∫ a pw(x,t)dt = 0, which implies that b∫ a pw(x,t)f ′ (t) dt = b∫ a pw(x,t) (f ′ (t) −d) dt. int. j. anal. appl. 17 (3) (2019) 427 using lemma 2.1, we have b∫ a pw(x,t) (f ′ (t) −d) dt = 1 4   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt. note that, for each t ∈ [a,b] , − ( d2 −d1 2 ) = d1 −d ≤ f ′ (t) −d ≤ d2 −d = d2 −d1 2 which implies that max t∈[a,b] |f ′ (t) −d| ≤ d2 −d1 2 . thus, ∣∣∣∣∣∣ b∫ a pw(x,t) (f ′ (t) −d) dt ∣∣∣∣∣∣ ≤ maxt∈[a,b] |f ′ (t) −d| b∫ a |pw(x,t)|dt ≤ d2 −d1 2 b∫ a |pw(x,t)|dt, which implies that∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ d2 −d12 b∫ a |pw(x,t)|dt. � we can generalize theorem 2.3 as follows: theorem 2.4. let f : i ⊂ r → r be a differentiable mapping on i0, the interior of the interval i, and let a,b ∈ i with a < b. let w be a weight function over [a,b] . if f ′ ∈ l1 [a,b] with d1 ≤ f ′ (t) ≤ d2 for all t ∈ [a,b], where d1, d2 are constants, then for each x ∈ [ a, a+b 2 ] , we have∣∣∣∣∣∣  1 4   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) −mba (x) d   − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ d2 −d12 b∫ a |pw(x,t)|dt. (2.10) where mba = b∫ a pw(x,t)dt and d = d1+d2 2 . int. j. anal. appl. 17 (3) (2019) 428 proof. let d = d1+d2 2 using lemma 2.1, we have b∫ a pw(x,t) (f ′ (t) −d) dt = 1 4   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] −d b∫ a pw(x,t)dt− b∫ a f (t) w (t) dt. using same argument as in the proof of theorem 2.3, we get∣∣∣∣∣∣ b∫ a pw(x,t) (f ′ (t) −d) dt ∣∣∣∣∣∣ ≤ d2 −d12 b∫ a |pw(x,t)|dt, which implies that∣∣∣∣∣∣  1 4   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) −mba (x) d   − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ d2 −d12 b∫ a |pw(x,t)|dt. � theorem 2.5. f : i ⊂ r → r be a differentiable mapping on (a,b) .let w be a weight function over [a,b] with b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. if f ′ ∈ l1 [a,b] with d1 ≤ f ′ (t) ≤ d2 for all t ∈ [a,b], where d1, d2 are constants, then for each x ∈ [ a, a+b 2 ] , we have ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) ( f (b) −f (a) b−a −d1 ) sup t∈[a,b] |pw(x,t)| , (2.11) and ∣∣∣∣∣∣  1 4 b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) ( d2 − f (b) −f (a) b−a ) sup t∈[a,b] |pw(x,t)| . (2.12) int. j. anal. appl. 17 (3) (2019) 429 proof. using identity (2.2) with f (t) = t, and the fact that b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt, we get that b∫ a pw(x,t)dt = 0, which implies that b∫ a pw(x,t)f ′ (t) dt = b∫ a pw(x,t) (f ′ (t) −d) dt, b∫ a f ′ (t) pw(x,t)dt = b∫ a pw(x,t) (f ′ (t) −d1) dt, and b∫ a f ′ (t) pw(x,t)dt = b∫ a pw(x,t) (f ′ (t) −d2) dt. using lemma 2.1 and the triangle inequality we get∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ b∫ a |pw(x,t) (f ′ (t) −d1)|dt, and ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ b∫ a |pw(x,t) (f ′ (t) −d2)|dt. note that b∫ a |pw(x,t)dt (f ′ (t) −d1)|dt ≤ sup t∈[a,b] |pw(x,t)| b∫ a |f ′ (t) −d1|dt and b∫ a |f ′ (t) −d1|dt = b∫ a (f ′ (t) −d1) dt = f (b) −f (b) −d1 (b−a) = (b−a) [ f (b) −f (b) b−a −d1 ] . int. j. anal. appl. 17 (3) (2019) 430 similarly, b∫ a |pw(x,t) (f ′ (t) −d2)|dt ≤ sup t∈[a,b] |pw(x,t)| b∫ a |f ′ (t) −d2|dt. and b∫ a |f ′ (t) −d2|dt = b∫ a d2 −f ′ (t) dt = d2 (b−a) − (f (b) −f (b)) = (b−a) [ d2 − f (b) −f (b) b−a ] . therefore, ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) ( f (b) −f (a) b−a −d1 ) sup t∈[a,b] |pw(x,t)| , and ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) ( d2 − f (b) −f (a) b−a ) sup t∈[a,b] |pw(x,t)| . � 2.2. the l2 case. theorem 2.6. let f : [a,b] → r be an absolutely continuous mapping on (a,b) with f ′ ∈ l2 [a,b] . suppose that w is a weight function over [a,b] with b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. then for each x ∈ [ a, a+b 2 ] , we have ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤‖pw(x,.)‖2 √ ‖f ′‖22 − ( f (b) −f (b) b−a )2 (b−a). (2.13) int. j. anal. appl. 17 (3) (2019) 431 proof. let d = b∫ a f ′ (y) dy. using identity (2.2) with f (t) = t, and the fact that b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt, we get that b∫ a pw(x,t)dt = 0, which implies that b∫ a pw(x,t)f ′ (t) dt = b∫ a pw(x,t)dt (f ′ (t) −d) dt. using lemma 2.1, we have∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ = ∣∣∣∣∣∣ b∫ a (f ′ (t) −d) pw(x,t)dt ∣∣∣∣∣∣ . by cauchy schwartz inequality, ∣∣∣∣∣∣ b∫ a (f ′ (t) −d) pw(x,t)dt ∣∣∣∣∣∣ ≤   b∫ a (f ′ (t) −d)2 dt   1 2   b∫ a (pw(x,t)) 2 dt   1 2 = ‖pw(x,.)‖2   b∫ a (f ′ (t) −d)2 dt   1 2 . since f ′ ∈ l2 [a,b] , b∫ a (f ′ (t) −d)2 dt ≤‖f ′‖22 − ( f (b) −f (b) b−a )2 (b−a) . therefore, ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤‖pw(x,.)‖2 √ ‖f ′‖22 − ( f (b) −f (b) b−a )2 (b−a). � int. j. anal. appl. 17 (3) (2019) 432 theorem 2.7. let f : [a,b] → r be an absolutely continuous mapping on (a,b) with f ′′ ∈ l2 [a,b] . suppose that w is a weight function over [a,b] with b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. then for each x ∈ [ a, a+b 2 ] , we have ∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) π ‖f ′′‖2 ‖pw(x,.)‖2 . (2.14) proof. let d = f ′ ( a + b 2 ) . using identity (2.2) with f (t) = t, and the fact that b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt, we get that b∫ a pw(x,t)dt = 0, which implies that b∫ a pw(x,t)f ′ (t) dt = b∫ a pw(x,t) (f ′ (t) −d) dt. using lemma 2.1, we have∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ = ∣∣∣∣∣∣ b∫ a (f ′ (t) −d) pw(x,t)dt ∣∣∣∣∣∣ ≤   b∫ a (f ′ (t) −d)2 dt   1 2   b∫ a (pw(x,t)) 2 dt   1 2 = ‖pw(x,.)‖2   b∫ a (f ′ (t) −d)2 dt   1 2 . by diaz -metcalf inequality [16] b∫ a (f ′ (t) −d)2 dt ≤ (b−a)2 π2 ‖f ′′‖22 . therefore,∣∣∣∣∣∣14   b∫ a w (t) dt  [f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )] − b∫ a f (t) w (t) dt ∣∣∣∣∣∣ ≤ (b−a) π ‖f ′′‖2 ‖pw(x,.)‖2 . int. j. anal. appl. 17 (3) (2019) 433 � 3. some applications the following lemma will be useful in calculations. lemma 3.1. let −∞ < a < b < ∞ and w be a weight function over [a,b]. if w is symmetric about a+b 2 , then (1) |pw(x,t)| =   t∫ a w (u) du, t ∈ [ a, a+x 2 ] 1 2 ∣∣∣∣∣ t∫a w (u) du− a+b 2∫ t w (u) du ∣∣∣∣∣ , t ∈ (a+x2 ,x] a+b 2∫ t w (u) du, t ∈ ( x, a+b 2 ] t∫ a+b 2 w (u) du, t ∈ ( a+b 2 ,a + b−x ] 1 2 ∣∣∣∣∣a+b−t∫a w (u) du− a+b 2∫ a+b−t w (u) du ∣∣∣∣∣ , t ∈ (a + b−x, a+2b−x2 ] b∫ t w (u) du, t ∈ ( a+2b−x 2 ,b ] . (3.1) (2) b∫ a |pw(x,t)|dt = 2 a+x 2∫ a φ (t) dt + 2 a+b 2∫ x ψ (t) dt + x∫ a+x 2 |φ (t) − ψ (t)|dt and b∫ a |pw(x,t)| 2 dt = 2 a+x 2∫ a [φ (t)] 2 dt + 2 a+b 2∫ x [ψ (t)] 2 dt + 1 2 x∫ a+x 2 [φ (t) − ψ (t)]2 dt, where φ (t) = t∫ a w (u) du and ψ (t) = a+b 2∫ t w (u) du,t ∈ [a,b] . remark 3.1. when w (t) = 1 on [a,b] , where −∞ < a < b < ∞, we get that (1) b∫ a |pw(x,t)|dt = 1 4 (a + b− 2x) (b−x) , for x ∈ [ a, 3a+b 4 ] . (2) b∫ a |pw(x,t)|dt = 1 4 (x−a)2 + 5 4 ( a + b 2 −x )2 + ( 3a + b 4 −x )2 , for x ∈ ( 3a+b 4 ,b ] . int. j. anal. appl. 17 (3) (2019) 434 now we give some applications on the results found. recall that a tagged partition p of a finite interval [a,b] is a finite sequence of numbers a = x0 < x1 < · · · < xn = b, with corresponding values ti ∈ [xi−1,xi], for i = 1, . . . ,n. theorem 3.1. let −∞ < a < b < ∞ and f : [a,b] → r be a differentiable function on (a,b) , and p : a = x0 < x1 < · · · < xn = b be a tagged partition with corresponding values ti ∈ [xi−1, xi−1+xi 2 ], for i = 1, . . . ,n. suppose that w is a weight function over [a,b]. if f ′ ∈ l1 [a,b] and d1 ≤ f ′(t) ≤ d2, for all t ∈ [a,b] , where d1, d2 are constants, then we have the quadrature formula b∫ a w (t) f (t) dt = aw (f,p) + rw (f,p) , where aw (f,p) = n−1∑ i=0 1 4 wxi+1xi ( f (ti+1) + f (xi + xi+1 − ti+1) + f ( xi + ti+1 2 ) +f ( xi + 2xi+1 − ti+1 2 )) −mxi+1xi (ti+1) n xi+1 xi , wxi+1xi = xi+1∫ xi w(t)dt, 0 ≤ i ≤ n− 1, mxi+1xi (ti+1) = xi+1∫ xi pw(x,t)dt, 0 ≤ i ≤ n− 1, nxi+1xi = f (xi+1) −f (xi) xi+1 −xi , 0 ≤ i ≤ n− 1, and the remainder satisfies the inequality |rw (f,p)| ≤ (d2 −d1) 2 n−1∑ i=0 wxi+1xi (xi+1 −xi) . proof. for each 0 ≤ i ≤ n− 1, applying theorem 2.2 on [xi−1,xi] with x = ti+1, we get that∣∣∣∣∣∣ xi+1∫ xi w (t) f (t) dt−  1 4 wxi+1xi   f (ti+1) + f (xi + xi+1 − ti+1) +f ( xi+ti+1 2 ) + f ( xi+2xi+1−ti+1 2 )  −mxi+1xi (ti+1) nxi+1xi   ∣∣∣∣∣∣ ≤ (d2 −d1) 2 wxi+1xi (xi+1 −xi) . using the triangle inequality, we find that∣∣∣∣∣∣ n−1∑ i=0 xi+1∫ xi w (t) f (t) dt−  1 4 wxi+1xi   f (ti+1) + f (xi + xi+1 − ti+1) +f ( xi+ti+1 2 ) + f ( xi+2xi+1−ti+1 2 )  −mxi+1xi (ti+1) nxi+1xi   ∣∣∣∣∣∣ ≤ (d2 −d1) 2 n−1∑ i=0 wxi+1xi (xi+1 −xi) . int. j. anal. appl. 17 (3) (2019) 435 but n−1∑ i=0 xi+1∫ xi w (t) f (t) dt = b∫ a w (t) f (t) dt, which implies that b∫ a w (t) f (t) dt = aw (f,p) + rw (f,p) and |rw (f,p)| ≤ (d2 −d1) 2 n−1∑ i=0 wxi+1xi (xi+1 −xi) . � theorem 3.2. let f : i ⊂ r → r be a differentiable mapping on i0, the interior of the interval i, a,b ∈ i with a < b, and p : a = x0 < x1 < · · · < xn = b be a tagged partition with corresponding values ti ∈ [xi−1, xi−1+xi 2 ], for i = 1, . . . ,n. suppose that w is a weight function over [a,b] . if f ′ ∈ l1 [a,b] with d1 ≤ f ′ (t) ≤ d2 for all t ∈ [a,b], where d1, d2 are constants, then we have the quadrature formula b∫ a w (t) f (t) dt = aw (f,p) + rw (f,p) , where aw (f,p) = n−1∑ i=0 1 4 wxi+1xi ( f (ti+1) + f (xi + xi+1 − ti+1) + f ( xi + ti+1 2 ) +f ( xi + 2xi+1 − ti+1 2 )) −dmxi+1xi (ti+1) , wxi+1xi = xi+1∫ xi w(t)dt, , 0 ≤ i ≤ n− 1, mxi+1xi (ti+1) = xi+1∫ xi pw(x,t)dt, , 0 ≤ i ≤ n− 1, d = d1 + d2 2 , and the remainder satisfies the inequality |rw (f,p)| ≤ (d2 −d1) 2 b∫ a |pw(x,t)|dt. int. j. anal. appl. 17 (3) (2019) 436 proof. for each 0 ≤ i ≤ n− 1, applying theorem 2.4 on [xi−1,xi] with x = ti+1, we get that∣∣∣∣∣∣ xi+1∫ xi w (t) f (t) dt−  1 4 wxi+1xi   f (ti+1) + f (xi + xi+1 − ti+1) +f ( xi+ti+1 2 ) + f ( xi+2xi+1−ti+1 2 )  −dmxi+1xi (ti+1)   ∣∣∣∣∣∣ ≤ (d2 −d1) 2 xi+1∫ xi |pw(x,t)|dt. using the triangle inequality, we find that∣∣∣∣∣∣ n−1∑ i=0 xi+1∫ xi w (t) f (t) dt−  1 4 wxi+1xi   f (ti+1) + f (xi + xi+1 − ti+1) +f ( xi+ti+1 2 ) + f ( xi+2xi+1−ti+1 2 )  −dmxi+1xi (ti+1)   ∣∣∣∣∣∣ ≤ (d2 −d1) 2 n−1∑ i=0 xi+1∫ xi |pw(x,t)|dt. but n−1∑ i=0 xi+1∫ xi w (t) f (t) dt = b∫ a w (t) f (t) dt and n−1∑ i=0 xi+1∫ xi |pw(x,t)|dt = b∫ a |pw(x,t)|dt, which implies that b∫ a w (t) f (t) dt = aw (f,p) + rw (f,p) and |rw (f,p)| ≤ (d2 −d1) 2 b∫ a |pw(x,t)|dt. � before we introduce the next application, recall that if x is a random variable with values in a finite interval [a,b], a < b, and f : [a,b] → [0, 1] is a probability density function, then the cumulative distribution function with respect to f is denoted by f and is defined as: f (x) = ∫ x a f (t) dt for each x ∈ [a,b] . since f satisfies the condition ∫ b a f (x) dx = 1, we find that f (b) = 1, and clearly f (a) = 0. the expectation of x is defined as: e (x) = ∫ b a t df dt dt. int. j. anal. appl. 17 (3) (2019) 437 using integration by parts, we get that e (x) = bf (b) −af (a) − ∫ b a f (t) dt = b− ∫ b a f (t) dt. theorem 3.3. let x be a random variable with values in a finite interval [a,b], a < b, and f : [a,b] → [0, 1] be a probability density function. let w be a differentiable weight function over [a,b] such that w (b) = 1 and b∫ a tw (t) dt = b+a 2 b∫ a w (t) dt. let f be the cumulative distribution function with respect to f . if f ∈ l1 [a,b] and d1 ≤ f(t) ≤ d2, for all t ∈ [a,b] , where d1, d2 are constants, then the inequality∣∣∣∣∣∣14   b∫ a w (t) dt     f (x) + f (a + b−x) +f ( a+x 2 ) + f ( a+2b−x 2 )  − (b−eg) ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) . (3.2) holds for all x ∈ [ a, a+b 2 ] , where eg = b− ∫ b a w (t) f (t) dt. proof. define the function g over [a,b] as follows: g (x) = ∫ x a d dt (wf) dt,x ∈ [a,b] . note that g (a) = 0 and g (b) = ∫ b a d dt (wf) dt = w (b) f (b) −w (a) f (a) = 1. let eg = ∫ b a t d dt (wf) dt. using integration by parts, we get that eg = bw (b) f (b) −aw (a) f (a) − ∫ b a w (t) f (t) dt = b− ∫ b a w (t) f (t) dt, which implies that ∫ b a w (t) f (t) dt = b−eg. int. j. anal. appl. 17 (3) (2019) 438 applying theorem 2.1 on f, we get that∣∣∣∣∣∣14   b∫ a w (t) dt  (f (x) + f (a + b−x) + f (a + x 2 ) + f ( a + 2b−x 2 )) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 12   b∫ a w (t) dt   (d2 −d1) (b−a) , which implies that ∣∣∣∣∣∣14   b∫ a w (t) dt     f (x) + f (a + b−x) +f ( a+x 2 ) + f ( a+2b−x 2 )  − (b−eg) ∣∣∣∣∣∣ ≤ 1 2   b∫ a w (t) dt   (d2 −d1) (b−a) . (3.3) � references [1] m.w. alomari, a companion of ostrowski’s inequality for mappings whose first derivatives are bounded and applications numerical integration, kragujevac j. math. 36 (2012), 77-82. [2] n.s. barnett, s.s. dragomir and i. gomma, a companion for the ostrowski and the generalized trapezoid inequalities, j. math. comput. model. 50 (2009), 179-187. [3] s.s. dragomir, some companions of ostrowski’s inequality for absolutely continuous functions and applications, bull. korean math. soc. 40(2) (2005), 213-230. [4] s.s. dragomir and s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, comput. math. appl. 33(11) (1997), 15-20. [5] z. liu, some companions of an ostrowski type inequality and applications, j. inequal. pure appl. math. 10(2) (2009), 10-12. [6] d.s. mitrinvić, j.e. pecarić and a.m. fink, classical and new inequalities in analysis, kluwer academic publishers, dordrecht, (1993). [7] d.s. mitrinović, j.e. pecarić and a.m. fink, inequalities involving functions and their integrals and derivatives, mathematics and its applications. (east european series), kluwer acadamic publications dordrecht, vol. 53., (1991). [8] a. ostrowski, über die absolutabweichung einer differentienbaren funktionen von ihren integralimittelwert, comment. math. hel. 10 (1938), 226-227. [9] a. qayyum and s. hussain, a new generalized ostrowski grüss type inequality and applications, appl. math. lett. 25 (2012), 1875-1880. [10] a. qayyum, m. shoaib, a.e. matouk and m.a. latif, on new generalized ostrowski type integral inequalities, abstr. appl. anal. 2014 (2014), art. id 275806. [11] a. qayyum, m. shoaib and m. a. latif, a generalized inequality of ostrowski type for twice differentiable bounded mappings and applications, appl. math. sci. 8(38) (2014), 1889-1901. [12] a. qayyum, i. faye, m. shoaib and m.a. latif, a generalization of ostrowski type inequality for mappings whose second derivatives belong to l1(a, b) and applications, int. j. pure appl. math. 98(2) (2015), 169-180. int. j. anal. appl. 17 (3) (2019) 439 [13] a. qayyum, m. shoaib and i. faye, some new generalized results on ostrowski type integral inequalities with application, j. comput. anal. appl. 19(4) (2015), 693-712. [14] a. qayyum, m. shoaib and i. faye, on new weighted ostrowski type inequalities involving integral means over end intervals and application, turk. j. anal. number theory, 3(2) (2015), 61-67. [15] a. qayyum, m. shoaib and i. faye, a companion of ostrowski type integral inequality using a 5-step kernel with some applications, filomat, 30(13) (2016), 3601-3614. [16] n. ujević, new bounds for the first inequality of ostrowski-grüss type and applications, comput. math. appl. 46 (2003), 421-427. 1. introduction 2. main results 2.1. the l1 case. 2.2. the l2 case. 3. some applications references international journal of analysis and applications volume 18, number 1 (2020), 33-49 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-33 an extended s-iteration scheme for g-contractive type mappings in b-metric spaces with graph nilakshi goswami1, nehjamang haokip1, vishnu narayan mishra2,∗ 1department of mathematics, gauhati university, guwahati 781014, india 2department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur 484 887, india ∗corresponding author: vishnunarayanmishra@gmail.com abstract. in this paper, we introduce an extended s-iteration scheme for g-contractive type mappings and prove ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. we also give a numerical example in support of our result and compare the convergence rate between the studied iteration and the modified s-iteration. 1. introduction in 1922, banach gave the proof of a fixed point result, which later on came to be known as the celebrated banach contraction principle. he showed that a contraction mapping t on a complete metric space (x,d) has a unique fixed point. moreover, for an arbitrary point x0 in x, the sequence of picard iterates given by the relation xn = txn−1 n = 1, 2, 3, . . . (1.1) converges to the unique fixed point. in the last few decades, many authors have extended this result by considering a more generalized space, altering the condition of the contraction or by considering different received 2019-05-01; accepted 2019-09-30; published 2020-01-02. 2010 mathematics subject classification. primary 47h10, secondary 54e50. key words and phrases. b-metric space; extended s-iteration scheme; g-contractive type mapping; ∆-convergence; strong convergence. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 33 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-33 int. j. anal. appl. 18 (1) (2020) 34 iteration processes (one may refer to [6]– [8], [9], [10], [13], [14], [19], [21]– [24], [26], [27], [28], [34], [29]– [33] and the references therein). the increasing interest in the study of iteration schemes is accelerated by the advancement in computational mathematics aided by computer programming. we list some of the prominent iteration schemes which are generalizations of (1.1). for x0 ∈ x, the iteration scheme given by xn+1 = (1 −αn)xn + αntxn, n = 0, 1, 2, . . . where {αn}⊂ [0, 1] is called the mann iteration scheme (refer to [20]). for x0 ∈ x, the iteration scheme given by xn+1 = (1 −αn)xn + αntyn yn = (1 −βn)xn + βntxn, n = 0, 1, 2, . . . where {αn} and {βn} are sequences in [0, 1] is called the ishikawa iteration scheme (refer to [15]). in 1976, jungck [16] proved a common fixed point theorem for a pair of mappings s and t satisfying d(tx,ty) ≤ αd(sx,sy) with α ∈ (0, 1) and t(x) ⊂ s(x) in a complete metric space. for x0 in a linear space x, the sequence {sxn} defined by sxn+1 = αnsxn + (1 −αn)txn, n = 0, 1, 2, . . . where {αn} is a sequence in [0, 1] is called the jungck-mann iteration scheme (refer to [39]). if αn = 0, we get the jungck iteration scheme. for x0 ∈ x, the sequence {sxn} defined by sxn+1 = (1 −αn)sxn + αntyn syn = (1 −βn)sxn + βntxn, n = 0, 1, 2, . . . where {αn} and {βn} are sequences in [0, 1] is called the jungck-ishikawa iteration scheme (refer to [25]). in 2007, agarwal et al. [2] introduced the s-iteration scheme and studied its convergence. for a convex subset k of a linear space x and a self mapping t on k, the iterative sequence {xn} of the s-iteration scheme is generated from x1 ∈ k, and is defined by xn+1 = (1 −λn)txn + λntyn, yn = (1 −µn)xn + µntxn, n ∈ n where {λn} and {µn} are real sequences in (0, 1), satisfying ∞∑ n=1 λnµn(1 −µn) = ∞. int. j. anal. appl. 18 (1) (2020) 35 recently, suparatulatorn et al. [40] introduced the modified s-iteration scheme for g-nonexpansive mappings s1 and s2 in banach spaces with graphs. here, for x0 ∈ k, n ≥ 0  xn+1 = (1 −λn)s1xn + λns2yn, yn = (1 −µn)xn + µns1xn (1.2) where {λn} and {µn} are sequences in (0, 1). motivated by [40], in this paper, we consider a convex b-metric space (x,d) with graph and define an extended s-iteration scheme for a triplet of three g-contractive type self mappings on a nonempty closed convex subset k of x. the convergence of this iteration scheme in comparison to the existing modified s-iteration scheme is also discussed with a numerical example. in the following we reproduce the concepts of some of the terms used in this paper. definition 1.1. [5], let x be a non empty set and s ≥ 1 be a given real number. a function d : x×x −→ [0,∞) is called a b-metric if it satisfies the following properties. (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x); and (3) d(x,z) ≤ s[d(x,y) + d(y,z)], for all x,y,z ∈ x. the pair (x,d) is called a b-metric space with coefficient s. in 1970, takahashi [41] introduced the following concept of convex structure in a metric space. definition 1.2. [41] let (x,d) be a metric space. a mapping w : x2 × [0, 1] −→ x satisfying d (z,w(x,y,t)) ≤ td(z,x) + (1 − t)d(z,y) for all x,y,z ∈ x and t ∈ [0, 1] is called a convex structure on x. the above notion of convex structure can as well be extended naturally to b-metric spaces with the condition sd (z,w(x,y,t)) ≤ td(z,x) + (1 − t)d(z,y). (1.3) kirk & ray [17], in 1977, defined a metric space (x,d) to be metrically convex or simply convex if for every distinct elements x and y in x, there exists z in x, distinct from x and y such that d(x,y) = d(x,z) + d(z,y). a natural extension of this notion to b-metric spaces is by the equation d(x,y) = s [d(x,z) + d(z,y)] . int. j. anal. appl. 18 (1) (2020) 36 the mapping w satisfies d(x,y) = s { d (x,w(x,y,t)) + d (x,w(x,y,t)) } for all x,y ∈ x and t ∈ [0, 1] (refer to [41]). this can be seen in proving the following assertion as is done in [1] for metric spaces. if (x,d) is a b-metric space on which a convex structure w is defined, then for all x,y ∈ x and t ∈ [0, 1], sd (x,w(x,y,t)) = td(x,y) and sd (y,w(x,y,t)) = (1 − t)d(x,y). let α = d (x,w(x,y,t)), β = d (y,w(x,y,t)) and γ = d(x,y). then, from (1.3), we get sα ≤ tγ and sβ ≤ (1 − t)γ. now, by the triangle inequality of a b-metric, we have γ ≤ s(α + β) ≤ tγ + (1 − t)γ = γ, that is, γ = s(α + β). now, if sα < tγ, then γ ≤ sα + sβ < γ, which is a contradiction. hence sα = tγ, and consequently, sβ = (1 − t)γ. definition 1.3. [1] a b-metric space (x,d) on which a convex structure w is defined is called a convex b-metric space, denoted by (x,w,d). a subset k of x is called convex if w(x,y,λ) ∈ k whenever x,y ∈ k and λ ∈ [0, 1]. definition 1.4. [38] a convex metric space (x,w,d) is called uniformly convex if for every δ > 0, c > 0 and x,y,z ∈ x, there exists µ > 0 such that d(z,x) ≤ c, d(z,y) ≤ c and d(x,y) ≥ cδ implies d ( z,w ( x,y; 1 2 )) ≤ c(1 −µ) < c. hadamard manifolds and geodesic spaces are nonlinear examples of convex b-metric spaces while uniformly convex banach b-metric spaces and cat(0) spaces are examples of uniformly convex b-metric spaces [11]. definition 1.5. [36] let k be a subset of a b-metric space (x,d) and {xn} be a bounded sequence in x. for x ∈ x, we take r (x,{xn}) = limn→∞ sup d(x,xn). then (1) r ({xn}) = inf {r (x,{xn}) : x ∈ k} is said to be the asymptotic radius of {xn} with respect to k ⊆ x, (2) for any z ∈ k, the set a ({xn}) = {x ∈ x : r (x,{xn}) ≤ r (z,{xn})} is said to be the asymptotic centre of {xn} with respect to k ⊆ x. a sequence {xn} ∆-converges to x if a ({un}) = {x} for every subsequence {un} of {xn}, that is, x is the unique asymptotic centre for every subsequence {un} of {xn}. it is denoted by ∆ − limn→∞xn = x. equivalently, a sequence {xn} in x is said to ∆-converges to a point x ∈ x if lim sup k d ( xnk,x ) ≤ lim sup k d ( xnk,y ) for every subsequence {xnk} of {xn} and every y ∈ x [18]. int. j. anal. appl. 18 (1) (2020) 37 it can be seen that ordinary convergence implies ∆-convergence. however, the converse is not true. example 1.1. † let x be the set of positive rational numbers and define d : x ×x −→ [0,∞) by d(x,y) =   1, x 6= y 0, x = y then (x,d) is a b-metric space. consider the sequence {xn} where xn = 1n , n ∈ n. then evidently, {xn} is not convergent but ∆-convergent to x for every x ∈ x. because, given any x and y in x, we can choose k large enough with xnk 6= x and xnk 6= y so that lim sup k ( d ( xnk,x ) −d ( xnk,y )) ≤ 1 − 1 = 0. definition 1.6. [12] a subset k of a b-metric space (x,d) is said to be chebychev if for every x ∈ x there exists y ∈ k such that d(x,y) < d(k,x) for all k ∈ k and y 6= k. if k is a chebychev subset of a b-metric space x, then the nearest point projection p : x −→ k is defined by sending x to y. as observed in [12], this notion of nearest point projection for chebychev sets is in accordance with that of orthogonal projection onto a subspace of the euclidean space. it was shown in [11] that every closed and convex subset of a uniformly convex b-metric space is chebychev. lemma 1.1. [12] let k be a nonempty, closed and convex subset of a complete uniformly convex metric space (x,w,d). then every bounded sequence {xn} in k has a unique asymptotic centre in k. lemma 1.2. [12] let k be a nonempty, closed and convex subset of a complete uniformly convex metric space (x,w,d). let {xn} be a bounded sequence in k such that a ({xn}) = {y} and r ({xn}) = ρ. if {ym} is another sequence in k such that limm→∞r (ym,{xn}) = ρ, then limm→∞ym = y. lemma 1.3. [12] let (x,w,d) be a uniformly convex metric space and {αn} a sequence in [b,c] ⊂ (0, 1). suppose that the sequences {xn} and {yn} in x are such that lim n→∞ sup d(xn,w) ≤ c, lim n→∞ sup d(yn,w) ≤ c and lim n→∞ sup d (w(xn,yn; αn),w) ≤ c for some w ∈ x and some c ≥ 0. then limn→∞d(xn,yn) = 0. †this example is adapted from https://math.stackexchange.com/a/3370319/445538 int. j. anal. appl. 18 (1) (2020) 38 the proofs of the above lemmas are independent of the property – the triangle inequality of the metric d. thus these results also holds true for the corresponding b-metric spaces. let k be a non-empty subset of a b-metric space (x,d) and ∆ be the diagonal of the cartesian product k ×k, i.e., ∆ = {(x,x) : x ∈ k}. let g be a reflexive digraph (directed graph) with the set v (g) of its vertices coinciding with k, and the set e(g) of its edges containing all loops, i.e., e(g) ⊇ ∆. assuming g has no parallel edges, we identify the graph g with the pair (v (g),e(g)). definition 1.7. [40] the conversion of a graph g is the graph obtained from g by reversing the direction of the edges, denoted by g−1, that is, e(g−1) = {(x,y) ∈ x ×x : (y,x) ∈ g} . a directed graph/digraph g = (v (g),e(g)) is said to be transitive if for every x,y,z ∈ v (g) with (x,y), (y,z) ∈ e(g), we have (x,z) ∈ e(g). definition 1.8. let (x,d) be a b-metric space with coefficient s ≥ 1. a mapping s : x −→ x is said to be a g-contractive type mapping if s is edge-preserving ((x,y) ∈ e(g) implies (sx,sy) ∈ e(g)) and d (sx,sy) ≤ k ( d(x,y) + d(y,sy) ) (1.4) for some k < 1 and for all x,y ∈ x. taking y = sx in the above relation, we get d(sx,s2x) ≤ k ( d(x,sx) + d(sx,s2x) ) that is, d(sx,s2x) ≤ k 1 −k d(x,sx) = k′d(x,sx) for some k′ > 0. if y = w0 ∈ f, then we have d (sx,w0) ≤ kd(x,w0), where f = f(s) is the set of fixed points of s. 2. main results for a b-metric space (x,d) with graph and a non-empty closed convex subset k of x, we introduce the extended s-iteration scheme given below. for x0 ∈ k, int. j. anal. appl. 18 (1) (2020) 39   xn+1 = w (s2s1xn,s3s1yn,αn) , yn = w (s1xn,s3zn,βn) zn = w (s2s1xn,s3s1xn,γn) (2.1) where {αn}, {βn} and {γn} are real sequences in [0, 1] and si : x −→ x is a g-contractive type mapping on k for i = 1, 2, 3. the existence of a fixed point for contractive type mappings given by (1.4) is known from various existing literatures (for example, refer to [11]). in this section, we prove a result on ∆-convergence and strong convergence of the iteration scheme given by (2.1) in a closed convex subset of a uniformly convex b-metric space. let f = ⋂3 i=1 f(si), where f(si) are the sets of fixed points of si. lemma 2.1. let w0 ∈ f be such that (x0,w0), (w0,x0) ∈ e(g). then (xn,w0), (w0,xn), (yn,w0), (w0,yn), (zn,w0), (w0,zn), (xn,yn), (yn,zn) and (xn,xn+1) are in e(g). proof. we will prove by induction. since s1, s2 and s3 are edge-preserving and e(g) is convex, using (2.1) we have (x0,w0) ∈ e(g) =⇒ (w(s2s1x0,s3s1x0; γ0),w0) = (z0,w0) ∈ e(g) and (y0,w0) = (w(s1x0,z0; β0),w0) ∈ e(g). since (x0,w0), (y0,w0) ∈ e(g), we have (x1,w0) = (w(s2s1x0,s3s1y0; α0),w0) ∈ e(g) and therefore, (z1,w0) = (w(s2s1x1,s3s1x1; γ0),w0) ∈ e(g). similarly, (y1,w0) = (w(s1x1,z1; β1),w0) ∈ e(g). now we assume that (xk,w0) ∈ e(g) for some positive integer k. then by the same argument as before, (zk,w0) = (w(s2s1xk,s3s1xk; γk),w0) ∈ e(g), (yk,w0) = (w(s1xk,zk; βk),w0) ∈ e(g) and (xk+1,w0) = (w(s2s1xk,s3s1yk; αk),w0) ∈ e(g). this implies (xk+1,w0) ∈ e(g) which in turn gives (yk+1,w0) and (zk+1,w0) are in e(g). therefore, (xn+1,w0), (yn+1,w0) and (zn+1,w0) are in e(g) for all n ∈ n. in a similar way, we can show that (w0,xn), (w0,yn) and (w0,zn) are in e(g) if (w0,x0) ∈ e(g). int. j. anal. appl. 18 (1) (2020) 40 finally, the transitivity of e(g) implies (xn,yn), (yn,zn) and (xn,xn+1) are also in e(g). � lemma 2.2. if (x,d) is a convex b-metric space and (x0,w0), (w0,x0) ∈ e(g) for arbitrary x0 ∈ x and w0 ∈ f , then d(xn+1,w0) ≤ kd(xn,w0) for all n and hence lim n→∞ d(xn,w0) = 0. proof. if w0 ∈ f, then by lemma 2.1, (xn,w0), (yn,w0), (zn,w0) ∈ e(g). let x′n = s1xn. since s1, s2 and s3 are g-contractive type mappings, we have d (ax,aw0) ≤ k ( d(x,w0) + d(w0,aw0) ) = kd(x,w0) for some k < 1 and a = s1, s2 or s3. now, d(xn+1,w0) = d (w(s2s1xn,s3s1yn,αn),w0) ≤ αnd(s2s1xn,w0) + (1 −αn)d(s3s1yn,w0) ≤ kαnd(s1xn,w0) + k(1 −αn)d(s1yn,w0) = kαnd(s1xn,w0) + k(1 −αn)d (w(s1xn,zn; βn),w0) ≤ kαnd(s1xn,w0) + k(1 −αn)βnd(s1xn,w0) + k(1 −αn)(1 −βn)d(w(s2s1xn,s3s1xn; γn),w0) ≤ k (αn + βn −αnβn) d(s1xn,w0) + k(1 −αn)(1 −βn)( γnd(s2s1xn,w0) + (1 −γn)d(s3s1xn,w0) ) ≤ k (αn + βn −αnβn) d(s1xn,w0) + k(1 −αn)(1 −βn)( γnkd(s1xn,w0) + (1 −γn)kd(s1xn,w0) ) ≤ k ( αn + βn −αnβn + (1 −αn)(1 −βn) ) d(s1xn,w0) = kd(s1xn,w0) ≤ kd(xn,w0) for all n ∈ n. thus the sequence {d(xn,w0)} of positive numbers is monotonically decreasing and hence limn→∞d(xn,w0) exists. in fact, since d(xn+1,w0) ≤ kd(xn,w0) for all n ≥ 0, we have d(xn+1,w0) ≤ knd(x0,w0). this proves the assertion. � lemma 2.3. if x is a convex b-metric space and (x0,w0), (w0,x0) ∈ e(g) for arbitrary x0 ∈ x and w0 ∈ f , then lim n→∞ d(s1xn,xn) = lim n→∞ d(s2xn,xn) = lim n→∞ d(s3xn,xn) = 0. int. j. anal. appl. 18 (1) (2020) 41 proof. from lemma 2.2, we see that lim n→∞ d(xn,w0) = 0. (2.2) then using (1.4) we have d(axn,xn) ≤ sd(axn,w0) + sd(w0,xn) ≤ sk ( d(xn,w0) + d(w0,aw0) ) + sd(xn,w0) ≤ s(k + 1)d(xn,w0) where a = s1, s2 or s3. in the limiting case, we have lim n→∞ d(axn,xn) = 0. � we now prove a result on ∆-convergence in convex b-metric spaces following the method used in [12]. theorem 2.1. let k be a nonempty closed convex subset of a uniformly convex and complete b-metric space x with a continuous convex structure w and, s1,s2,s3 : k −→ k be continuous g-contractive type mappings on k with f 6= ∅. if (x0,w0), (w0,x0) ∈ e(g) for arbitrary x0 ∈ k and some w0 ∈ f , then the sequence {x′n = s1xn} given by (2.1) ∆-converges to an element of f . proof. in lemma 2.2, it is shown that limn→∞d(xn,w0) exists, which in turn shows that the sequence {xn} is bounded. therefore by lemma 1.1, a ({xn}) = {x}. let {vn} be any subsequence of {xn} such that a ({vn}) = {v}. as in theorem 2.4 of [12], we can show that v ∈ k. by lemma 2.3, lim n→∞ d(s1vn,vn) = lim n→∞ d(s2vn,vn) = lim n→∞ d(s3vn,vn) = 0. define um = t mv (t = si, i = 1, 2 or 3) and we observe that d (um,vn) ≤ sd (tmv,tmvn) + m∑ j=1 sjd ( tm−jvn,t m−j+1vn ) ≤ skd ( tm−1v,tm−1vn ) + skd ( tm−1vn,t mvn ) + m∑ j=1 sjd ( tm−jvn,t m−j+1vn ) int. j. anal. appl. 18 (1) (2020) 42 ≤ sk2d ( tm−2v,tm−2vn ) + skd ( tm−1vn,t mvn ) + sk2d ( tm−2vn,t m−1vn ) + m∑ j=1 sjd ( tm−jvn,t m−j+1vn ) ... ≤ skmd (v,vn) + m∑ j=1 (skj + sj)d ( tm−jvn,t m−j+1vn ) ≤ d (v,vn) + d (tvn,vn) m∑ j=1 (skj + sj)k j 1 where k1 = k 1−k > 0. hence, r (um,{vn}) ≤ lim n→∞ sup d (um,vn) ≤ lim n→∞ sup d (v,vn) ≤ r (v,{vn}) , which shows that |r (um,{vn}) −r (v,{vn})| −→ 0 as m →∞. now, from lemma 1.2, limm→∞um = limm→∞t mv = v. k being closed, limm→∞t mv = v ∈ k, and limm→∞tm+1v = tv, which implies tv = v. therefore, by lemma 2.2 (since v ∈ f) limn→∞d (v,xn) exists. now, as in theorem 2.4 of [12], it directly follows that x = v. thus, x is the unique asymptotic centre for any subsequence {vn} of {xn}, showing that {xn} ∆-converges to x. � in [37], shahzad & al-dubiban stated a condition called condition (b) and proved a strong convergence theorem for nonexpansive mappings in banach spaces. we restate the condition in a b-metric setting and prove a strong convergence theorem. the mappings s1,s2,s3 : k −→ k with f = f(s1) ∩f(s2) ∩f(s3) 6= ∅ are said to satisfy condition (b) if there is a non decreasing function f : [0,∞) −→ [0,∞) with f(0) = 0 and f(x) > 0 for all x > 0 such that for all x ∈ k, max{d(s1x,x),d(s2x,x),d(s3x,x)}≥ f (d(x,f)) . theorem 2.2. let k be a nonempty closed convex subset of a uniformly convex and complete b-metric space x with continuous convex structure w and, s1,s2,s3 : k −→ k be g-contractive type mappings on k satisfying f 6= ∅. let (x0,w0), (w0,x0) ∈ e(g) for arbitrary x0 ∈ x and w0 ∈ f . if s1,s2 and s3 satisfy condition (b), then the sequence {xn} given by (2.1) converges strongly to an element of f . int. j. anal. appl. 18 (1) (2020) 43 proof. let w ∈ f. from lemma 2.2, we get that {xn} is a bounded seequence and hence limn→∞d(xn,w) exists. also, we have d(xn+1,w) < d(xn,w) for all n ≥ 1, from which we get that d(xn+1,f) ≤ d(xn,f) for all n ≥ 1. by the same argument as in lemma 2.2, we conclude that limn→∞d(xn,f) exists. by lemma 2.3, we have limn→∞d(sixn,xn) = 0, where i = 1, 2, 3. since s1, s2 and s3 satisfy conddition (b), we get lim n→∞ f (d(xn,f)) = 0 and hence lim n→∞ d(xn,f) = 0. so, there exists a subsequence {xnk} of {xn} and a sequence {wk} in f satisfying d (xnk,wk) ≤ 2 −k. setting nk+1 = nk + j for some j ≥ 1, we have d ( xnk+1,wk ) ≤ d (xnk+j−1,wk) ≤ d (xnk,wk) ≤ 1 2k , using which we get d(wk+1,wk) ≤ s ( d ( wk+1,xnk+1 ) + d ( xnk+1,wk )) ≤ s ( 1 2k+1 + 1 2k ) = s 2k+1 . thus {wk} is a cauchy sequence in f. since f is closed, there exists w∗ ∈ f such that limk→∞wk = w∗. thus, limk→∞xnk = w ∗. as limn→∞d (xn,w ∗) exists and equals 0 by lemma 2.2, the result follows. � 3. numerical example in this section, we present an example with its numerical experiment in support of our results. we also make a comparison of the rate of convergence of the iteration scheme (2.1) to that of the one given in [40]. in 1976, rhoades [35] gave a comparison between two iterations {xn} and {zn}, both converging to a fixed point p of a mapping t : k −→ k by saying {xn} converge faster than {zn} if d(xn,p) ≤ d(zn,p), n ≥ 1, where k is a non-empty closed and convex subset of a complete metric space. int. j. anal. appl. 18 (1) (2020) 44 in numerical analysis, the order of convergence of a real sequence {αn} converging to α is studied using the well known method mentioned below (refer to [4]). let {αn} be a real sequence which converges to α with αn 6= α for all n ∈ n. if lim n→∞ d(αn+1,α) [d(αn,α)] µ = λ for some positive constants λ and µ, then {αn} is said to converge to α of the order µ, with asymptotic error constant λ. for λ < 1, if µ = 1 the sequence is linearly convergent and if µ = 2, the sequence is quadratically convergent. in 2002, using the above method of comparison, berinde [3] compared the rate of convergence between two iteration schemes as given below. let {αn} and {βn} be sequences of positive real numbers converging to α and β, respectively. suppose that lim n→∞ d(αn,α) d(βn,β) = l. (i). if l = 0, then the sequence {αn} is said to converge to α faster than that of the sequence {βn} to {β}. (ii). if 0 < l < ∞, then the sequences {αn} and {βn} are said to have the same rate of convergence. for a nonempty convex subset k of a complete b-metric space x with a self map t : k −→ k, if {xn} and {un} are two iterations both of which converge to a fixed point p of t , then {xn} converges faster than {un} to p if lim n→∞ d(xn,p) d(un,p) = 0. we are now in a position to give an example for our main results and compare the rate of convergence of the studied iteration scheme against the modified s-iteration scheme. in the cases when the b-metric d is induced by the norm ‖.‖x, the mapping w : x2 × [0, 1] −→ x such that w(x,y,t) = (1 − t)x + ty defines a convex structure on x. the iteration (2.1) then takes the following form. for x0 ∈ k,  xn+1 = (1 −αn)s2s1xn + αns3s1yn, yn = (1 −βn)s1xn + βns3zn, zn = (1 −γn)s2s1xn + γns3s1xn (3.1) where {αn}, {βn} and {γn} are real sequences in [0, 1] and si : x −→ x is a g-contractive type mapping on k for i = 1, 2, 3. int. j. anal. appl. 18 (1) (2020) 45 example 3.1. let x = r and k = [0, 2]. let g = (v (g),e(g)) be a directed graph defined by v (g) = k and (x,y) ∈ e(g) if and only if 1 ≤ x,y ≤ 7 4 and x,y ∈ q. we consider the mappings p,q,r : k −→ k given by px = xlog x qx = 1 3 arcsin(x− 1) + 1 and rx = √ x for all x ∈ k. to show that p , q and r are g-contractive type mappings, it is enough to show that they are g-contraction mappings on [0, 2]. now, to show that px = xlog x is a contraction mapping on [1, 2], we note that by mean value theorem, |p(x) −p(y)| |x−y| ≤ c where c = max{p ′x : x ∈ [1, 2]} with p ′x = 2xlog x−1 log x. since p ′(x) < 7 10 for all 1 ≤ x ≤ 2, we see that p is a contraction on [1, 2] and hence a g-contraction mapping. similarly, q′x = 1 3 1√ 1−(x−1)2 ≤ 4 3 √ 7 for all 1 ≤ x ≤ 7 4 and r′x = 1 2 1√ x ≤ 1 2 for all x ≥ 1 implies that q and r are g-contraction mappings on [0, 2]. their common fixed point here being x = 1. consider the real sequences {an}, {bn} and {cn} in [0, 1], where an = n + 1 5n + 3 , bn = n + 4 10n + 7 and cn = n + 2 2n + 3 . let {xn} be a sequence generated by the extended s-iteration (2.1) with x0 = 1.5 and s1 = p , s2 = q, s3 = r and αn = an, βn = bn, γn = cn as defined above. let {un} be a sequence generated by the modified s-iteration (1.2) with u0 = 1.5 and using s1 = p , s2 = q with λn = an and µn = bn. the numerical observations for the error estimates and the rate of convergence for these two iteration schemes are shown in tables 1 & 2 below. int. j. anal. appl. 18 (1) (2020) 46 n modified s-iteration extended s-iteration un |un −un−1| xn |xn −xn−1| 1 1.15489 0.345105 1.04139 0.458612 2 1.02536 0.129532 1.00041 0.0409766 3 1.00201 0.023352 1.00000 0.000411667 4 1.00012 0.00188701 1.00000 4.34451×10−8 5 1.00001 0.000116509 1.00000 4.44089×10−16 6 1.00000 7.01765×10−6 1.00000 0.00000 7 1.00000 4.22157×10−7 1.00000 0.00000 table 1. numerical errors of modified s-iteration and extended s-iteration schemes n modified sextended srate of convergence un xn |un −1| |xn −1| |xn−1||un−1| 1 1.15489 1.04139 0.154895 0.0413883 0.267203 2 1.02536 1.00041 0.025363 0.000411711 0.0162327 3 1.00201 1.00000 0.00201099 4.34451×10−8 0.0000216038 4 1.00012 1.00000 0.000123976 4.44089×10−16 3.58206×10−12 5 1.00001 1.00000 7.46682×10−6 0.00000 0.000000 6 1.00000 1.00000 4.49177×10−7 0.00000 0.000000 7 1.00000 1.00000 2.70198×10−8 0.00000 0.000000 table 2. rate of convergence int. j. anal. appl. 18 (1) (2020) 47 from tables 1 & 2, it is evident that the sequence of iterates {un} and {xn} both converge to 1 ∈ f . we also observe that |xn − 1| ≤ |un − 1| and limn→∞ |xn−1| |un−1| = 0, so the seqeuence of iterates {xn} converges faster than {un}, generated by the modified s-iteration (see figure 1). æ æ æ æ æ à à à à à 0 1 2 3 4 5 0.0 0.1 0.2 0.3 0.4 0.5 n er ro rs à extended sæ modified sfigure 1. comparison of error estimates of the modified s-iteration and the extended s-iteration schemes conclusion in this paper, we have introduced an extended s-iteration scheme for g-contractive type mappings and proved ∆-convergence as well as strong convergence in a nonempty closed and convex subset of a uniformly convex and complete b-metric space with a directed graph. an example is also given to compare the convergence rate between the studied iteration and the modified s-iteration. the iteration considered in this paper may as well be studied for other contractive type mappings and its rate of convergence can be compared with existing iteration schemes. the application of our results for solving constrained optimization problem is also a scope for further study. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (1) (2020) 48 references [1] a. a. abdelhakim, a convexity of functions on convex metric spaces of takahashi and applications, j. egypt. math. soc. 24 (2016), 348–354. [2] r. p. agarwal, d. oregan and d. r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, j. nonlinear convex. anal. 8 (2007), 61–79. [3] v. berinde, iterative approximation of fixed points, editura efemeride, baia mare, 2002. [4] r. l. burden and j. d. faires, numerical analysis, brooks/cole cengage learning, 9th edition, boston, 2010. [5] s. czerwik, contraction mappings in b-metric spaces, acta. math. inform. univ. ostra. 1 (1993), 5–11. [6] d. das and n. goswami, some fixed point theorems on the sum and product of operators in tensor product spaces, int. j. pure appl. math. 109 (2016), no.3, 651–663. [7] d. das, n. goswami and v.n. mishra, some results on fixed point theorems in banach algebras, int. j. anal. appl. 13 (2017), no. 1, 32–40. [8] d. das, n. goswami and v.n. mishra, some results on the projective cone normed tensor product spaces over banach algebras, (online available) bol. soc. paran. mat. (3s.) 38 (2020), no. 1, 197–221. [9] deepmala, a study on fixed point theorems for nonlinear contractions and its applications, ph.d. thesis (2014), pt. ravishankar shukla university, raipur 492 010, chhatisgarh, india. [10] deepmala and h. k. pathak, a study on some problems on existence of solutions for nonlinear functional-integral equations, acta math. sci. 33 b(5) (2013), 1305–1313. [11] h. fukhar-ud-din, existence and approximation of fixed points in convex metric spaces, carpathian j. math. 30 (2014), 175–185. [12] h. fukhar-ud-din, one step iterative scheme for a pair of nonexpansive mappings in a convex metric space, hacet. j. math. stat. 44 (2015), 1023–1031. [13] n. goswami, n. haokip and v. n. mishra, f-contractive type mappings in b-metric spaces and some related fixed point results, fixed point theory and applications 2019 (2019), 13. [14] n. haokip and n. goswami, some fixed point theorems for generalized kannan type mappings in b-metric spaces, proyecciones (antofagasta) 38 (4) (2019), 763-782. [15] s. ishikawa, fixed point by a new iteration method, proc. amer. math. soc. 44 (1974), 147–150. [16] g. jungck, commuting mappings and fixed points, amer. math. monthly 83 (1976), 261–263. [17] w. a. kirk and w. o. ray, a note on lipschitzian mappings in convex metric spaces, canad. math. bull. 20 (1977), 463–466. [18] t. c. lim, remarks on some fixed point theorems, proc. amer. math. soc. 60 (1976), 179 182. [19] x. liu, m. zhou, l.n. mishra, v.n. mishra and b. damjanović, common fixed point theorem of six self-mappings in menger spaces using (clrst ) property, open math. 16 (2018), 1423–1434. [20] w. r. mann, mean value methods in iteration, proc. amer. math. soc. 44 (1953), 506–510. [21] l. n. mishra, h.m. srivastava and m. sen, on existence results for some nonlinear functional-integral equations in banach algebra with applications, int. j. anal. appl., 11 (2016), 1–10. [22] l. n. mishra, k. jyoti, a. rani and vandana, fixed point theorems with digital contractions image processing, nonlinear sci. lett. a, 9 (2018), 104–115. [23] l. n. mishra, s. k. tiwari, v. n. mishra and i. a. khan, unique fixed point theorems for generalized contractive mappings in partial metric spaces, j. funct. spaces, 2015 (2015), article id 960827, 8 pages. int. j. anal. appl. 18 (1) (2020) 49 [24] l. n. mishra, s. k. tiwari and v. n. mishra, fixed point theorems for generalized weakly s-contractive mappings in partial metric spaces, j. appl. anal. comput. 5 (2015), 600–612. [25] m. o. olatinwo and c. o. imoru, some convergence results for the jungck-mann and jungck-ishikawa processes in the class of generalized zamfirescu operators, acta math. univ. comenianae lxxvii (2008), 299–304. [26] h. k. pathak and deepmala, common fixed point theorems for pd-operator pairs under relaxed conditions with applications, j. comput. appl. math. 239 (2013), 103–113. [27] b. patir, n. goswami and l. n. mishra, fixed point theorems in fuzzy metric spaces for mappings with some contractive type conditions, korean j. math. 26 (2018), 307–326. [28] b. patir, n. goswami and v. n. mishra, some results on fixed point theory for a class of generalized nonexpansive mappings, fixed point theory appl. 2018 (2018), 19. [29] t. rasham, a. shoaib, c. park and m. arshad, fixed point results for a pair of multi dominated mappings on a smallest subset in k-sequentially dislocated quasi metric space with application, j. comput. anal. appl., 25 (5) (2018), 975–986. [30] t. rasham, a. shoaib, n. hussain, b. s. alamri and m. arshad, multivalued fixed point results in dislocated b-metric spaces with application to the system of nonlinear integral equations, symmetry, 11 (1) (2019), 40. [31] t. rasham, a. shoaib, b. s. alamri and m. arshad, fixed point results for multivalued contractive mappings endowed with graphic structure, j. math. 2018 (2018) article id 5816364, 8 pages. [32] t. rasham, a. shoaib, b. s. alamri and m. arshad, multivalued fixed point results for new generalized f-dominted contractive mappings on dislocated metric space with application, j. funct. spaces, 2018 (2018), article id 4808764, 12 pages. [33] t. rasham, a. shoaib, b. a. s. alamri, a. asif and m. arshad, fixed point results for α∗-ψ-dominated multivalued contractive mappings endowed with graphic structure, mathematics 7 (2019), 307. [34] a. razani and m. bagherboum, convergence and stability of jungck-type iterative procedures in convex b-metric spaces, fixed point theory appl. 2013 (2013), 331. [35] b. e. rhoades, comments on two fixed point iteration method, j. math. anal. appl. 56 (1976), 741–750. [36] g. s. saluja, strong and ∆-convergence of modified two-step iterations for nearly asymptotically nonexpansive mappings in hyperbolic spaces, int. j. anal. appl. 8 (2015), 39–52. [37] s. shahzad and r. al-dubiban, approximating common fixed points of nonexpansive mappings in banach spaces, georgian math. j. 13 (2006), 529–537. [38] t. shimizu and w. takahashi, fixed points of multivalued mappings in certain convex metric spaces, topol. methods nonlinear anal. 8 (1996), 197–203. [39] s. l. singh, c. bhatnagar and s. n. mishra, stability of jungck-type iterative procedures, int. j. math. math. sci. 19 (2005), 3035–3043. [40] s. suparatulatorn, w. cholamjiak and s. suantai, a modified s-iteration process for g-nonexpansive mappings in banach spaces with graphs, numer. algorithms 77 (2017), 479–490. [41] w. takahashi, a convexity in metric spaces and nonexpansive mappings, kodai math. sem. rep. 22 (1970), 142–149. 1. introduction 2. main results 3. numerical example conclusion references international journal of analysis and applications volume 17, number 2 (2019), 244-259 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-244 hardy-copson type inequalities on time scales for the functions of “n” independent variables m. shahzad ashraf1,∗, khuram ali khan2 and ammara nosheen1 1department of mathematics, the university of lahore (sargodha campus), pakistan 2department of mathematics, university of sargodha, sargodha, pakistan ∗corresponding author: shahzadashraf30@icloud.com abstract. the paper consists of some extensions in hardy and copson type inequalities on time scales. the main results are proved by using induction principle, rules to find derivatives for composition of two functions, hölder’s inequality and fubini’s theorem in time scales settings. the related results and examples are also investigated in seek of applications. 1. introduction advancements in inequalities had started since the end of 19th century, which laid the theocratical foundations of approximation methods. generally it is accepted that the classic work “inequalities [7]”, which appeared in 1934, gave a systematic discipline to a collection of isolated formulas. this work had played a role of a strong stem in continuous growth to modern field of inequalities. many researchers had paid attention to these inequalities since invention of these inequalities. a large number of papers related with extensions, new results, and generalizations can be seen on the topic [5, 8, 10]. copson in [6] extend weighted hardy inequality in the following result: received 2018-08-04; accepted 2018-10-24; published 2019-03-01. 2010 mathematics subject classification. 26d15, 39a13, 34n05. key words and phrases. time scales; hardy-copson inequality; keller’s chain rule. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 244 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-244 int. j. anal. appl. 17 (2) (2019) 245 if % > 1, a(l) = l∑ i=1 a(i)ω(i) ω(i) , then ∞∑ l=1 ω(l)(a(l))% 6 p% ∞∑ l=1 ω(l)a(l)(a(l))%−1 6 p% ∞∑ l=1 ω(l)a%(l). (1.1) in [12] r. p. agarwal et al., proved some dynamic hardy inequalities via time scales calculus. those inequalities as special cases contain some integral and discrete inequalities due to hardy and copson. one of their results is stated as: “consider t is a time scale with a ∈ [0,∞)t, % >1, ω and g be non-negative functions. further assume that ω(ς) = ∫ ς a ω(r)∆r, ω(∞) = ∞, ψ(ς) := ∫ ς a ω(r)g(r)∆r, ∀ ς ∈ [a,∞)t and ∫ ∞ a ( ωσ(ς) ω(ς) )%(%−1) ω(ς)g%(ς)∆ς exists. then ∫ ∞ a ω(ς) (ωσ(ς))% (ψσ(ς))%∆ς 6 % %− 1 ∫ ∞ a ω(ς)g(ς) ω%−1(ς) ψ%−1(ς)∆ς and ∫ ∞ a ω(ς) (ωσ(ς))% (ψσ(ς))%∆ς 6 ( % %− 1 )%∫ ∞ a ( ωσ(ς) ω(ς) )p(p−1) ω(ς)g%(ς)∆ς.” taking into account the worth of inequalities, we are motivated to extend the results of [12] for functions of several variables. 2. preliminaries a closed non-empty subset of real numbers is called a time scale. notation to be used for a time scale is t. r, z, n, n0, [a,b] are examples of time scales whereas rationals, irrationals and (a,b) are not closed, therefore not time scales. few basic concepts related to time scales theory, are as under (see [2–4]): definition 2.1. let t denotes a time scale. for ς ∈ t, the forward jump operator σ : t → t is defined as σ(ς) := inf {z ∈ t; z > ς} , and the backward jump operator ρ : t → t is defined as ρ(ς) := sup{z ∈ t; z < ς} . the point ς ∈ t satisfying σ(ς) > ς is right-scattered whereas the point ς satisfying ρ(ς) < ς is leftscattered. points which are simultaneously left and right scattered are isolated. also, the point ς is called right-dense if ς < sup t and σ(ς) = ς and is called left-dense if ς > inf t and ρ(ς) = ς. the points that are left and right dense simultaneously are dense points. int. j. anal. appl. 17 (2) (2019) 246 definition 2.2. the graininess function µ : t → [0,∞) is defined by, µ(ς) := σ(ς) − ς. definition 2.3. suppose a function g : t → r is satisfying: (a) g is continuous at all right dense ς ∈ t, (b) the left hand limits exist and finite at all left dense ς ∈ t then, g is called right-dense continuous (rd-continuous) in t. the set denoted by crd(t) contains all rdcontinuous functions. remark 2.1. if we exchange the role of left dense points and right dense points in definition of rd-continuous function, we get left dense continuous functions. if function is continuous with respect to both sided dense points, it is continuous function ∀t ∈ t. definition 2.4. let g : t → r and ς ∈ t, if g∆(ς) exists with the condition that, for � > 0, there exists a neighborhood o of ς such that |[g(σ(ς)) −g(s)] −g∆(ς)(σ(ς) −s)| ≤ �|σ(ς) −s|, for all s ∈ o, then g is differentiable at ς and is denoted by g∆(ς). theorem 2.1. assume delta derivatives of g1,g2 : t → r exist at ς ∈ t. then derivative of the product g1g2 : t → r exists at ς ∈ t and satisfies (g1g2) ∆(ς) = g∆1 (ς)g2(ς) + g σ 1 (ς)g ∆ 2 (ς) = g1(ς)g ∆ 2 (ς) + g ∆ 1 (ς)g σ 2 (ς). (2.1) definition 2.5. consider g is rd-continuous function. then for ς0 ∈ t, the function g defined by g := ∫ ς ς0 g(ς)∆ς, for ς ∈ t is called the antiderivative of g. definition 2.6. if α ∈ t, sup t = ∞, and g1 is rd-continuous on the interval [α,∞), then∫ ∞ α g1(ς)∆ς = lim β→∞ ∫ β α g1(ς)∆ς. existence of limit gives convergence of the integral elsewise it diverges. theorem 2.2. if α,β,γ ∈ t, c ∈ r and u1,u2 ∈ crd, then (i) ∫β α uσ1 (ς)u ∆ 2 (ς)∆ς = |u1u2(ς)|βα − ∫β α u∆1 (ς)u2(ς)∆ς, (ii) ∫β α u1(ς)u ∆ 2 (ς)∆ς = |u1u2(ς)|βα − ∫β α u∆1 (ς)u σ 2 (ς)∆ς. these are known as integration by parts formulae. int. j. anal. appl. 17 (2) (2019) 247 theorem 2.3 (chain rule 1). let u2 : r → r be continuous, u2 : t → r is delta differentiable on tκ and suppose that u1 : r → r is continuously differentiable. then there exists c ∈ [ς,σ(ς)] with (u1 ◦u2)∆(ς) = u′1(u2(c))u ∆ 2 (ς). (2.2) theorem 2.4 (chain rule 2). assume u1 : r → r is continuously differentiable and suppose u2 : t → r is delta differentiable. then u1 ◦u2 : t → r is delta differentiable and (u1 ◦u2)∆(ς) = {∫ 1 0 u′1(u2(ς) + hµ(ς)g ∆ 2 (ς))dh } u∆2 (ς) (2.3) holds. chain rule 2 is given by c. pötzsche [11], likewise it also appeared in s. keller’s paper [9]. a simple consequence of theorem 2.4 is given below: (uω1 (ς)) ∆(ς) = ω ∫ 1 0 {huσ1 + (1 −h)u1(ς)} ω−1 dhu∆1 (ς). (2.4) theorem 2.5. (hölder’s inequality) let α,β ∈ t, for rd-continuous functions g1,g2 : [α,β] → r, we have ∫ β α |g1(ς)g2(ς)|∆ς ≤ [∫ β α |g1(ς)|q∆ς ]1 q [∫ β α |g2(ς)|p∆ς ]1 p , (2.5) where p > 1 and q = p/(p− 1). next, we present fubini’s theorem on time scales [1]. theorem 2.6. let t1,t2 be two time scales. suppose s : t1 × t2 → r is integrable with respect to both time scales. define φ(y2) = ∫ t1 s(y1,y2)∆y1 for a.e y2 ∈ λ and ψ(y1) = ∫ t2 s(y1,y2)∆y2 for a.e y1 ∈ t1. then φ and ψ are both differentiable in time scales settings and∫ t1 ∆y1 ∫ t2 s(y1,y2)∆y2 = ∫ t2 ∆y2 ∫ t1 s(y1,y2)∆y1. (2.6) in the sequel, we denote [a,b)t = [a,b) ∩ t, where t be any time scale. also partial derivatives for i ∈ {1, . . . ,n} are denoted by ∂ ∆ςi g(ς1, . . . , ςn) = g ∆i (ς1, . . . , ςn). we also assume that the functions are nonnegative, rd-continuous and the integrals considered are assumed to exist. 3. dynamic hardy & copson-type inequalities via time scales for functions of n independent variables theorem 3.1. let t1, . . . ,tn denote time scales. for p > 1 and i ∈ {1, . . . ,n}, consider ai ∈ [0,∞)ti and g : t1 × . . . × tn → r+. let λi : ti → r+ be such that λi(ςi) := ∫ ςi ai λi(si)∆si and ψ(ς1, . . . , ςn) = int. j. anal. appl. 17 (2) (2019) 248 ∫ ς1 a1 . . . ∫ ςn an ∏n i=1 λi(si)g(s1, . . . ,sn)∆s1 . . . ∆sn. assume λi(∞) = 0 and the delta integrals∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λσii (ςi) [λi(ςi)]p λi(ςi)[g(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn exist. (3.1) then ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) [λσii (ςi)] p [ψ(σ1(ς1), . . . ,σn(ςn))] p∆ς1 . . . ∆ςn ≤ ( p p− 1 )np ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 ( λσii (ςi) λi(ςi) )p(p−1) λi(ςi)g p(ς1, . . . , ςn)∆ς1 . . . ∆ςn, (3.2) where n is any positive integer. proof. we prove the result by using principle of mathematical induction. for n = 1, statement is true by [12, theorem 2.1]. assume for 1 ≤ n ≤ k (3.2) holds. to prove the result for n = k + 1, take left hand side of (3.2) in the following form ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) [λσii (ςi)] p {∫ ∞ ak+1 λk+1(ςk+1) [λ σk+1 k+1 (ςk+1)] p [ψ(σ1(ς1), . . . ,σk+1(ςk+1))] p∆ςk+1 } ∆ς1 . . . ∆ςk. (3.3) consider ik+1 = ∫ ∞ ak+1 λk+1(ςk+1) [λ σk+1 k+1 (ςk+1)] p [ψ(σ1(ς1), . . . ,σk+1(ςk+1))] p∆ςk+1. (3.4) apply integration by parts (theorem 2.2 (i)) on ik+1 w.r.t ςk+1 ∈ [ak+1,∞) to get ik+1 = |uk+1(ςk+1)ψp(σ1(ς1), . . . ,σk(ςk), ςk+1)|∞ak+1 + ∫ ∞ ak+1 −uk+1(ςk+1)[ψp(σ1(ς1), . . . ,σk(ςk), ςk+1)]∆k+1 ∆ςk+1, (3.5) where −uk+1(ςk+1) = ∫ ∞ ςk+1 λk+1(sk+1) [λ σk+1 k+1 (sk+1)] p ∆sk+1 ≤ ( −1 p− 1 )∫ ∞ ςk+1 [λ 1−p k+1(sk+1)] ∆k+1 ∆sk+1. ∴−uk+1(ςk+1) ≤ ( 1 1 −p )( 1 λ p−1 k+1(ςk+1) ) . (3.6) by chain rule (2.4) and for dk+1 ∈ [ςk+1,σk+1(ςk+1)], [ψp(σ1(ς1), . . . ,σk(ςk), ςk+1)] ∆k+1 = pψp−1(σ1(ς1), . . . ,σk(ςk),dk+1)ψ ∆k+1 (σ1(ς1), . . . ,σk(ςk), ςk+1). (3.7) int. j. anal. appl. 17 (2) (2019) 249 since ψ∆k+1 (σ1(ς1), . . . ,σk(ςk), ςk+1) = ∂ ∆ςk+1 ψ(σ1(ς1), . . . ,σk(ςk), ςk+1) = ∫ σ1(ς1) a1 . . . ∫ σk(ςk) ak k∏ i=1 λi(si)× { ∂ ∆ςk+1 ∫ ςk+1 ak+1 λk+1(sk+1)g(s1, . . . ,sk,sk+1)∆sk+1 } ∆s1 . . . ∆sk, (3.8) also ∂ ∆ςk+1 ∫ ςk+1 ak+1 λk+1(sk+1)g(σ1(s1), . . . ,σk(sk),sk+1)∆sk+1 ≥ 0 and σk+1(ςk+1) ≥ dk+1. therefore (3.7) implies ψ∆k+1 (σ1(ς1), . . . ,σk(ςk), ςk+1) = λk+1(ςk+1) ∫ σ1(ς1) a1 . . . ∫ σk(ςk) ak k∏ i=1 λi(si)g(s1, . . . ,sk, ςk+1)∆s1 . . . ∆sk. thus ψ∆k+1 (σ1(ς1), . . . ,σk(ςk), ςk+1) = λk+1(ςk+1)ψk(σ1(ς1), . . . ,σk(ςk), ςk+1), (3.9) where ψk(σ1(ς1), . . . ,σk(ςk), ςk+1) = ∫ σ1(ς1) a1 . . . ∫ σk(ςk) ak k∏ i=1 λi(si)g(s1, . . . ,sk, ςk+1)∆s1 . . . ∆sk. use (3.9) in (3.7) to get, [ψp(σ1(ς1), . . . ,σk(ςk), ςk+1)] ∆k+1 = pψp−1(σ1(ς1), . . . ,σk(ςk),dk+1)λk+1(ςk+1)ψk(σ1(ς1), . . . ,σk(ςk), ςk+1) ≤ pψp−1(σ1(ς1), . . . ,σk(ςk), ςk+1)λk+1(ςk+1)ψk(σ1(ς1), . . . ,σk(ςk), ςk+1). (3.10) ∵ ψ(ς1, ς2, . . . ,ak+1) = 0 and uk+1(∞) = 0, (3.5) becomes ik+1 = ∫ ∞ ak+1 −uk+1(ςk+1)[ψp(σ1(ς1), . . . ,σk(ςk), ςk+1)]∆k+1 ∆ςk+1. (3.11) use (3.6) and (3.10) in (3.11) to get ik+1 ≤ ( p p− 1 )∫ ∞ ak+1 λk+1(ςk+1)ψk(σ1(ς1), . . . ,σk(ςk), ςk+1) λ p−1 k+1(ςk+1) [ψ(σ1(ς1), . . . ,σk+1(ςk+1))] p−1∆ςk+1. (3.12) int. j. anal. appl. 17 (2) (2019) 250 multiply and divide by ( λk+1(ςk+1) [λ σk+1 k+1 (ςk+1)] p )p−1 p on right hand side of (3.12) and then apply hölder’s inequality to get ik+1 ≤ ( p p− 1 )∫ ∞ ak+1  λk+1(ςk+1)λ1−pk+1(ςk+1)ψk(σ1(ς1), . . . ,σk(ςk), ςk+1)[ [λ σk+1 k+1 (ςk+1)] −pλk+1ςk+1 ]p−1 p  p ∆ςk+1   1 p × (∫ ∞ ak+1 λk+1(ςk+1) [λ σk+1 k+1 (ςk+1)] p [ψ(σ1(ς1), . . . ,σk+1(ςk+1))] p∆ςk+1 )p−1 p . (3.13) divide both sides by right most term and take power p on both sides. after simplification, we get ik+1 ≤ ( p p− 1 )p ∫ ∞ ak+1 ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p(p−1) λk+1(ςk+1) ψ p k(σ1(ς1), . . . ,σk(ςk), ςk+1)∆ςk+1. (3.14) substitute (3.14) in (3.3) to get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi) [λσii (ςi)] p [ψ(σ1(ς1), . . . ,σk(ςk),σk+1(ςk+1))] p∆ς1 . . . ∆ςk∆ςk+1 ≤ ( p p− 1 )p ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) [λσii (ςi)] p ×   ∫ ∞ ak+1 ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p(p−1) λk+1(ςk+1)ψ p k(σ1(ς1), . . . ,σk(ςk), ςk+1)∆ςk+1   ∆ς1 . . . ∆ςk. (3.15) use theorem 2.6 “k times” on right hand side of (3.15) to get ∫ ∞ ak+1 ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p(p−1) λk+1(ςk+1) × {∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) [λσii (ςi)] p ψ p k(σ1(ς1), . . . ,σk(ςk), ςk+1)∆ς1 . . . ∆ςk } ∆ςk+1. (3.16) by using induction hypothesis for ψkσ1(ς1), . . . ,σk(ςk), ςk+1) (instead for ψk(σ1(ς1), . . . ,σk(ςk)), ) with fix tk+1 ∈ tk+1, in (3.16) and use fubini’s theorem (theorem 2.6) “k times” to get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi) [λσii (ςi)] p [ψ(σ1(ς1), . . . ,σk+1(ςk+1))] p∆ς1 . . . ∆ςk+1 ≤ ( p p− 1 )(k+1)p ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 ( λσii (ςi) λi(ςi) )p(p−1) λi(ςi)g p(ς1, . . . , ςk+1)∆ς1 . . . ∆ςk+1. thus by principle of mathematical induction, inequality (3.2) holds for all n ∈ z+, which completes the proof. � int. j. anal. appl. 17 (2) (2019) 251 if we assume inf ςi∈[ai,∞)ti λi(ςi) λσii (ςi) = li > 0 ∀ i ∈{1, . . . ,n}, (3.17) in theorem 3.1 (in particular in (3.2)), we obtain the following result. corollary 3.1. for p > 1 and i ∈ {1, . . . ,n}, consider ti be time scales, ai ∈ [0,∞)ti , λi : ti → r+ and g : t1 × . . .×tn → r+. let λi(ςi) and ψ(ς1, . . . , ςn) be defined as in theorem 3.1 such that (3.17) holds and the delta integrals ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λσii (ςi) [λi(ςi)]p λi(ςi)[g(ς1, ς2, . . . , ςn)] p∆ς1∆ς2 . . . ∆ςn exist. then (3.2) takes the form ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) [λσii (ςi)] p [ψ(σ1(ς1), . . . ,σn(ςn))] p∆ς1 . . . ∆ςn ≤ ( p p− 1 )np n∏ i=1 (li) p(1−p) ∫ ∞ a1 . . . ∫ ∞ an λi(ςi)g p(ς1, . . . , ςn)∆ς1 . . . ∆ςn, (3.18) where “n” is any positive integer. remark 3.1. as a special case of corollary 3.1 when ti = r, ∀ i ∈ {1, . . . ,n}, generalization of integral hardy inequality for the functions of n independent variables (note that when ti = r, we have σi(ςi) = ςi and li = 1) takes the form: ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) [λi(ςi)]p (∫ ς1 a1 . . . ∫ ςn an [ n∏ i=1 λi(si)g(s1, . . . ,sn)dsn . . .ds1] p ) dςn . . .dς1 ≤ ( p p− 1 )np ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)g p(ς1, . . . , ςn)dtn . . .dς1. (3.19) as a special case of this inequality when λi(ςi) = 1, ∀ i = {1, . . . ,n}, we have ∫ ∞ a1 . . . ∫ ∞ an 1 [ ∏n i=1(ςi −ai)]p (∫ ς1 a1 . . . ∫ ςn an [g(s1, . . . ,sn)dsn . . .ds1] p ) dςn . . .dς1 ≤ ( p p− 1 )np ∫ ∞ a1 . . . ∫ ∞ an gp(ς1, . . . , ςn)dςn . . .dς1, (3.20) where p > 1. remark 3.2. for p > 1 and ∀ i ∈ {1, 2, . . . ,n}, assume that ti = n with ai = 1 in corollary 3.1. furthermore assume that ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln)g p(l1, . . . , ln) int. j. anal. appl. 17 (2) (2019) 252 is convergent. in this case (3.18) becomes the following discrete copson type inequality for the functions of “n” independent parameters: ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln)(∑l1 s1=1 λ1(s1) . . . ∑ln sn=1 λn(sn) )p ( l1∑ s1=1 . . . ln∑ sn=1 λ1(s1) . . .λn(sn)g(s1, . . . ,sn) )p ≤ ( p p− 1 )kp n∏ i=1 (li) p(1−p) ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln)g p(l1, . . . , ln). (3.21) theorem 3.2. let t1, . . . ,tn denote time scales. for p > 1 and i ∈{1, . . . ,n}, consider ai ∈ [0,∞)ti and g : t1 × . . .×tn → r+. let λi : ti → r+, λi(ςi) := ∫ ςi ai λi(si)∆si, λi(∞) = 0 and the delta integrals∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)[g(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn exist. (3.22) assume for any ςi ∈ [ai,∞)ti , φ(ς1, . . . , ςn) := ∫ ∞ ς1 . . . ∫ ∞ ςn n∏ i=1 ( λi(si) λσii (si) ) g(s1, . . . ,sn)∆s1 . . . ∆sn. (3.23) then ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)[φ(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn ≤ (p)np ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)g p(ς1, . . . , ςn)∆ς1 . . . ∆ςn, (3.24) where n is any positive integer. proof. we prove the result by using principle of mathematical induction. for n = 1, statement is true by [12, theorem 2.5]. assume for 1 ≤ n ≤ k (3.24) holds. to prove result for n = k + 1, take left hand side of (3.24) int the following form ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) {∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk, ςk+1)] p∆ςk+1 } ∆ς1 . . . ∆ςk. (3.25) consider ik+1 = ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1. (3.26) apply integration by parts (theorem 2.2 (i)) on ik+1 w.r.t ςk+1 ∈ [ak+1,∞) to get ik+1 = |φp(ς1, . . . , ςk+1)λk+1(ςk+1)|∞ak+1 + ∫ ∞ ak+1 [− ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] pλ σk+1 k+1 (ςk+1)]∆ςk+1. (3.27) since φ(ς1, . . . , ςk,∞) = 0 and λk+1(ak+1) = 0, (3.27) becomes ik+1 = ∫ ∞ ak+1 [− ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] pλ σk+1 k+1 (ςk+1)]∆ςk+1. (3.28) int. j. anal. appl. 17 (2) (2019) 253 apply chain rule (2.2) to find upper bound of − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] p, for dk+1 ∈ [ςk+1,σk+1(ςk+1)], we get − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] p = −pφp−1(ς1, . . . , ςk,dk+1) ∂ ∆ςk+1 φ(ς1, . . . , ςk, ςk+1). (3.29) also ∂ ∆ςk+1 φ(ς1, . . . , ςk+1) = ∂ ∆ςk+1 ∫ ∞ ς1 . . . ∫ ∞ ςk+1 k+1∏ i=1 ( λi(si) λσii (si) ) g(s1, . . . ,sk+1)∆s1 . . . ∆sk+1 = ∫ ς1 a1 . . . ∫ ςk ak k∏ i=1 ( λi(si) λσii (si) ){ ∂ ∆ςk+1 ∫ ∞ ςk+1 λk+1(sk+1) λ σk+1 k+1 (sk+1) g(s1, . . . ,sk+1)∆sk+1 } ∆s1 . . . ∆sk = ∫ ς1 a1 . . . ∫ ςk ak k∏ i=1 ( λi(si) λσii (si) ){ −λk+1(ςk+1) λ σk+1 k+1 (ςk+1) g(s1, . . . ,sk, ςk+1) } ∆s1 . . . ∆sk = ( −λk+1(ςk+1) λ σk+1 k+1 (ςk+1) )∫ ς1 a1 . . . ∫ ςk ak k∏ i=1 ( λi(si) λσii (si) ) g(s1, . . . ,sk, ςk+1)∆s1 . . . ∆sk. (3.30) since dk+1 ≥ ςk+1, use (3.30) in (3.29) to get, − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] pλ σk+1 k+1 (ςk+1) ≤ pλk+1(ςk+1)φ p−1(ς1, . . . , ςk+1)φk(ς1, . . . , ςk), (3.31) where φk(ς1, . . . , ςk) = ∫ ς1 a1 . . . ∫ ςk ak k∏ i=1 ( λi(si) λσii (si) ) g(s1, . . . ,sk, ςk+1)∆s1 . . . ∆sk. then (3.28) becomes, ik+1 ≤ p ∫ ∞ ak+1 λk+1(ςk+1)φ p−1(ς1, . . . , ςk+1)φk(ς1, . . . , ςk). (3.32) multiply and divide by [λk+1(ςk+1)] p−1 p on r.h.s of (3.32) then apply hölder’s inequality on r.h.s, we get ik+1 ≤ p (∫ ∞ ak+1 λk+1(ςk+1)[φk(ς1, . . . , ςk, ςk+1)] p∆ςk+1 )1 p × (∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 )p−1 p . (3.33) divide both sides by right most term then take power p on both sides and after simplification, we get ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 ≤ pp ∫ ∞ ak+1 λk+1(ςk+1)[φk(ς1, . . . , ςk, ςk+1)] p∆ςk+1. (3.34) int. j. anal. appl. 17 (2) (2019) 254 substitute (3.34) in (3.25), we get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)[φ(ς1, . . . , ςk+1)] p∆ς1 . . . ∆ςk+1 ≤ pp ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) {∫ ∞ ak+1 λk+1(ςk+1)[φk(ς1, . . . , ςk+1)] p∆ςk+1 } ∆ς1 . . . ∆ςk. (3.35) use fubini’s theorem (theorem 2.6) “k times” on right hand side of (3.35), we get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)[φ(ς1, . . . , ςk+1)] p∆ς1 . . . ∆ςk+1 ≤ pp ∫ ∞ ak+1 λk+1(ςk+1) {∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi)φ p k(ς1, . . . , ςk+1)∆ς1 . . . ∆ςk } ∆ςk+1. (3.36) by using induction hypothesis for φk(ς1, . . . , ςk+1) (instead φk(ς1, . . . , ςk+1), with fix tk+1 ∈ tk+1, in (3.16) and again apply theorem 2.6 “k times” on right hand side to get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)[φ(ς1, . . . , ςk+1)] p∆ς1 . . . ∆ςk+1 ≤ p(k+1)p ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)g p(ς1, . . . , ςk+1)∆ς1 . . . ∆ςk+1. (3.37) thus by principle of mathematical induction (3.24) holds for all positive integers n, which completes the proof. � remark 3.3. as a special case of theorem 3.2 when ti = r, ∀ i ∈{1, . . . ,n}, we have following generalization of integral inequality of copson-type for the functions of “n” independent variables (note that when ti = r, we have φ(σ(ς1), . . . ,σ(ςn)) = φ(ς1, . . . , ςn), λσii (ςi) = λi(ςi) and µ(ςi) = 0) and (3.24) takes the form ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) (∫ ∞ ς1 . . . ∫ ∞ ςn n∏ i=1 λi(si) λi(si) g(s1, . . . ,sn)dsn . . .ds1 )p dςn . . .dς1 ≤ pnp ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)g p(ς1, . . . , ςn)dςn . . .dς1. (3.38) as a special case of (3.38) when λi(ςi) = 1, ∀ i ∈{1, . . . ,n}, we have ∫ ∞ a1 . . . ∫ ∞ an (∫ ∞ ς1 . . . ∫ ∞ ςn n∏ i=1 λi(si) si g(s1, . . . ,sn)dsn . . .ds1 )p dςn . . .dς1 ≤ pnp ∫ ∞ a1 . . . ∫ ∞ an gp(ς1, . . . , ςn)dςn . . .dς1, (3.39) where p > 1. int. j. anal. appl. 17 (2) (2019) 255 remark 3.4. for p > 1 and ∀ i ∈ {1, 2, . . . ,n}, assume that ti = n with ai = 1 in theorem 3.2. furthermore assume that ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln)g p(l1, . . . ln) is convergent. in this case (3.24) becomes the following discrete copson type inequality for the functions of “n” independent parameters: ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln) ( ∞∑ s1=l1 . . . ∞∑ sn=ln λ1(s1) . . .λn(sn)∑l1 s1=1 λ1(s1) . . . ∑ln sn=1 λn(sn) g(s1, . . . ,sn) )p ≤ pkp ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln)g p(l1, . . . , ln). (3.40) theorem 3.3. let t1, . . . ,tn denote time scales. for p > 1 and i ∈{1, . . . ,n}, consider ai ∈ [0,∞)ti and g : t1 × . . .×tn → r+. let λi : ti → r+, λi(ςi) := ∫ ςi ai λi(si)∆si, λi(∞) = 0 and the delta integrals∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 ( λi(ςi) λi(ςi) ) [g(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn exist. (3.41) assume for any ςi ∈ [ai,∞)ti φ(ς1, . . . , ςn) := ∫ ∞ ς1 . . . ∫ ∞ ςn n∏ i=1 λi(si) λi(si) g(s1, . . . ,sn)∆s1 . . . ∆sn. (3.42) then ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)[φ(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn ≤ (p)np ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) ( λσii (si) λi(si) )p gp(ς1, . . . , ςn)∆ς1 . . . ∆ςn, (3.43) where n is any positive integer. proof. this result is proved by using principle of mathematical induction. for n = 1, statement is true by [12, theorem 2.8]. assume for 1 ≤ n ≤ k (3.43) holds. to prove the result for n = k + 1, take left hand side of (3.43) in the following form ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) {∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 } ∆ς1 . . . ∆ςk. (3.44) consider ik+1 = ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1. (3.45) use integration by parts theorem 2.2 (i) with φ(ς1, . . . , ςk,∞) = 0 and λk+1(ςk+1) = 0 to get ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 = ∫ ∞ ak+1 − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] pλ σk+1 k+1 (ςk+1)∆ςk+1. (3.46) int. j. anal. appl. 17 (2) (2019) 256 by chain rule (2.4) for dk+1 ∈ [ςk+1, ςk+1], − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] p = −pφp−1(ς1, . . . , ςk,dk+1) ∂ ∆ςk+1 φ(ς1, . . . , ςk+1), (3.47) and ∂ ∆ςk+1 φ(ς1, . . . , ςk+1) = ∫ ∞ ς1 . . . ∫ ∞ ςk k∏ i=1 λi(si) λi(si) { ∂ ∆ςk+1 ∫ ∞ ak+1 λk+1(sk+1) λk+1(sk+1) g(s1, . . . ,sk+1)∆sk+1 } ∆s1 . . . ∆sk. (3.48) also ∂ ∆ςk+1 ∫ ∞ ak+1 λk+1(sk+1) λk+1(sk+1) g(s1, . . . ,sk+1)∆sk+1 = − λk+1(ςk+1) λk+1(ςk+1) g(s1, . . . , ςk+1) ≤ 0 and dk+1 ≥ ςk+1. so after simplifications (3.48) implies ∂ ∆ςk+1 φ(ς1, . . . , ςk+1) = − λk+1(ςk+1) λk+1(ςk+1) ∫ ∞ ς1 . . . ∫ ∞ ςk k∏ i=1 λi(si) λi(si) g(s1, . . . , ςk+1)∆s1 . . . ∆sk. (3.49) substitute (3.49) in (3.47), − ∂ ∆ςk+1 [φ(ς1, . . . , ςk+1)] p = pφp−1(ς1, . . . , ςk,dk+1) λk+1(ςk+1) λk+1(ςk+1) ∫ ∞ ς1 . . . ∫ ∞ ςk k∏ i=1 λi(si) λi(si) g(s1, . . . , ςk+1)∆s1 . . . ∆sk ≤ pφp−1(ς1, . . . , ςk+1) λk+1(ςk+1) λk+1(ςk+1) φk(ς1, . . . , ςk+1) , (3.50) where φk(ς1, . . . , ςk+1) = ∫ ∞ ς1 . . . ∫ ∞ ςk k∏ i=1 λi(si) λi(si) g(s1, . . . , ςk+1)∆s1 . . . ∆sk. put (3.49) in (3.46), ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 ≤ p ∫ ∞ ak+1 λk+1(ςk+1) λ σk+1 k+1 (ςk+1) λk+1(ςk+1) φp−1(ς1, . . . , ςk+1)φk(ς1, . . . , ςk+1)∆ςk+1. (3.51) multiply and divide by [λk+1(ςk+1)] p−1 p on right hand side of (3.51) and apply hölder’s inequality to get ik+1 ≤ p (∫ ∞ ak+1 λk+1(ςk+1) ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p φ p k(ς1, . . . , ςk+1)∆ςk+1 )1 p × (∫ ∞ ak+1 λk+1(ςk+1)φ p(ς1, . . . , ςk+1)∆ςk+1 )p−1 p . (3.52) int. j. anal. appl. 17 (2) (2019) 257 divide both sides by right most term then take power p on both sides and after simplifying, we get ∫ ∞ ak+1 λk+1(ςk+1)[φ(ς1, . . . , ςk+1)] p∆ςk+1 ≤ (p)p ∫ ∞ ak+1 λk+1(ςk+1) ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p φ p k(ς1, . . . , ςk+1)∆ςk+1. (3.53) substitute (3.53) in (3.52), ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)[φ(ς1, . . . , ςk+1)] p∆ς1 . . . ∆ςk+1 ≤ (p)p ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi) {∫ ∞ ak+1 λk+1(ςk+1) ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p φ p k(ς1, . . . , ςk+1)∆ςk+1 } ∆ς1 . . . ∆ςk. (3.54) use theorem 2.6 “k times” on right hand side of (3.54), to get = (p) p ∫ ∞ ak+1 λk+1(ςk+1) ( λ σk+1 k+1 (ςk+1) λk+1(ςk+1) )p { ∫ ∞ a1 . . . ∫ ∞ ak k∏ i=1 λi(ςi)φ p k(ς1, . . . , ςk+1)∆ς1 . . . ∆ςk}∆ςk+1. (3.55) by using induction hypothesis for φk(ς1, . . . , ςk+1) (instead φk(ς1, . . . , ςk+1), with fix tk+1 ∈ tk+1, in (3.55) and again by applying theorem 2.6 on right hand side, we get ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi)[φ(ς1, . . . , ςk+1)] p∆ς1 . . . ∆ςk+1 ≤ (p)(k+1)p ∫ ∞ a1 . . . ∫ ∞ ak+1 k+1∏ i=1 λi(ςi) ( λσii (ςi) λi(ςi) )p gp(ς1, . . . , ςk+1)∆ς1 . . . ∆ςk+1. (3.56) thus by principle of mathematical induction (3.43) holds for all n ∈ z+, which completes the proof. � from theorem 3.3 with condition (3.17), we obtain following result. corollary 3.2. for p > 1 and i ∈ {1, . . . ,n}, consider ti be time scales, ai ∈ [0,∞)ti , λi : ti → r+ and g : t1 × . . .×tn → r+. let λi(ςi) and ψ(ς1, . . . , ςn) be defined as in theorem 3.1 such that (3.17) holds and the delta integrals ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 ( λi(ςi) λi(ςi) ) [g(ς1, . . . , ςn)] p∆ς1 . . . ∆ςn exists. then ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)[φ(ς1, . . . , ςn)] p∆ςn . . . ∆ς1 ≤ pnp n∏ i=1 ( 1 li )p ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)g p(ς1, . . . , ςn)∆ςn . . . ∆ς1. (3.57) int. j. anal. appl. 17 (2) (2019) 258 remark 3.5. as a special case of corollary 3.2 when ti = r, integral inequality of copson-type for the functions of “n” independent variables is ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi) (∫ ∞ ς1 . . . ∫ ∞ ςn n∏ i=1 λi(si) λi(si) g(s1, . . . ,sn)dsn . . .ds1 )p dςn . . .dς1 ≤ pnp ∫ ∞ a1 . . . ∫ ∞ an n∏ i=1 λi(ςi)g p(ς1, . . . , ςn)dςn . . .dς1, (3.58) where p > 1. remark 3.6. assume that ti = n in theorem 3.3, p > 1, ai = 1,∀ i ∈{1, 2, . . . ,n}. furthermore assume that ∑∞ s1=1 . . . ∑∞ sn=1 λ1(s1) . . .λn(sn) λ1(s1+1) λ1(s1) . . . λn(sn+1) λn(sn) gp(s1, . . . ,sn) is convergent. then (3.43) becomes the following discrete inequality of copson’s type for the functions of “n” independent variables: ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1) . . .λn(ln) ( ∞∑ s1=l1 . . . ∞∑ s2=ln λ1(s1) . . .λn(sn) λ1(s1) . . . λn(sn) g(s1, . . . ,sn) )p ≤ pkp ∞∑ l1=1 . . . ∞∑ ln=1 λ1(l1 + 1) . . . λn(ln + 1) λ1(l1) . . . λn(ln) λ1(l1) . . .λn(ln)g p(l1, . . . , ln), where λi(si) = si−1∑ ai=1 λi(si), ∀i = 1, 2, . . . ,n. references [1] m. bohner, a. nosheen, j. peĉarić, a. younus, some dynamic hardy-type inequalities with general kernel, j. math. inequal. 8 (2014), 185-199. [2] m. bohner and s. g. georgiev, multivariable dynamic calculus on time scales, springer intl. publishing, switzerland, 2017. [3] m. bohner and a. peterson, dynamic equations on time scales: an introduction with applications, birkhauser, boston, ma, 2001. [4] m. bohner and a. peterson, advances in dynamic equations on time scales: an introduction with applications, birkhauser, boston, ma, 2003. [5] l. y. chan, some extensions of hardy’s inequality, canad. math. bull. 22 (1979), 165-169. [6] e. t. copson, note on series of positive terms, j. london. math. soc. 2 (1927), 9-12. [7] g. h. hardy, j. e. littlewood and g. pólya, inequalities, 2nd ed., cambridge university press, cambridge, 1952. [8] s. kaijser, l. nikolova, l. e. persson and a. wedestig, hardy-type inequalities via convexity, math. inequal. appl. 8 (2005), 403-417. [9] s. keller, asymotitishces verhalten invarianter faserbündel bei diskretisierung und mittelwertbildung im rahmen der analysis auf zeitskalen, phd thesis, universität augsburg (1999). [10] j. pan, geometric character of domain and the harnack inequality of solution of a singular parabolic equation, math. slovaca, 62 (2012), 721-734. [11] c. pötzsche, chain rule and invariance principle on measure chains, j. comput. appl. math. 141 (1-2) (2002), 249-254. int. j. anal. appl. 17 (2) (2019) 259 [12] s. h. saker, d. o’regan and r. p. agarwal, dynamic inequalities of hardy and copson type on time scales, analysis, 34 (4) (2014), 391-402. 1. introduction 2. preliminaries 3. dynamic hardy & copson-type inequalities via time scales for functions of n independent variables references international journal of analysis and applications volume 18, number 4 (2020), 559-571 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-559 received december 19th, 2019; accepted january 20th, 2020; published may 11th, 2020. 2010 mathematics subject classification. 93a30. key words and phrases. heteroscedasticity; hybridized model; model instability; time series forecasting. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 559 robustifying forecast performance through hybridized arimagarch-type modeling in a discrete-time stochastic series imoh udo moffat 1,*, and emmanuel alphonsus akpan2 1department of statistics, university of uyo, nigeria 2department of mathematical science, abubakar tafawa balewa university, bauchi, nigeria *corresponding author: moffitto2011@gmail.com abstract. the study is aimed at investigating the robustness of forecast performance of a hybridized (arimagarch-type) model over each single component using different periods of horizon to display consistency over time. daily closing share prices were explored from the nigerian stock exchange for first city monument bank and wema bank plc, spanning from january 3, 2006 to december 30, 2016, with a total of 2,713 observations. arima model, garch-type, and hybridized arima-garch-type were considered. the hybridized arima-garch-type was found to produce the best forecast performance in terms of robustness over each single component model and the robustness was found to be consistent over different time horizons for the datasets. the implication is that, it provides an essential remedy to the problem associated with model instability when forecasting a discrete-time stochastic series. 1. introduction forecasting for future observations is one of the objectives of time series application. time series forecasting takes different approaches depending on what it is targeted at. it could be in-sample forecast approach, where the same data used for model formulation are also employed for checking the predictive performance of such model, which is aimed at selecting the best fitted models. also, it could be out-of-sample forecast approach, where the data are https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-559 int. j. anal. appl. 18 (4) (2020) 560 partitioned into two components (training and validation datasets), targeted at achieving forecast accuracy. analytically, the later is advantageous over the former in that it is able to give information about the future values of observed time series, depict forecaster in real time, and strongly discourage model overfitting ([1], [2], [3], [4], [5]). apart from predicting for future observations, forecasting is also a proven and valid tool for selecting a model with best predictive values ([6], [7], [8], [9], [10]). however, in a quest for best predictive performance, model instability has become a subject of discourse and tends to pose a serious threat to time series forecasting accuracy. model instability refers to the tendency for estimated parameters to fluctuate over time. one obvious consequence of model instability is that it leads to wrong choice of model resulting in high variability of prediction error ([11], [12], [13], [14], [15]). one way of overcoming the problem associated with forecast accuracy due to model instability is to account for possible instability in the model ([16], [17], [18]). on the other hand, instead of accounting for fluctuating parameters of a single model in order to achieve improved forecast accuracy, it is a good practice and in line with the recent innovation trend in time series to combat the menace of model instability on forecast accuracy through combined models. this alternative approach involves selecting diverse models and thereafter hybridizing the models to generate forecast, which can depict the provable forecast performance of individual models and at the same time provides robust accuracy. it is evident in the literature that diverse forecast hybridized approaches have supported the assertion that forecasts generated from combined models appeared to be improved and robust over the forecasts obtained by single models ([19, [20], [21], [22], [23], [24], [25]). meanwhile, in nigeria, [7] showed that out-of-sample model selection approach outperformed the in-sample counterpart in describing the characterizations of future observations without necessarily considering the choice of true model by utilizing the advantage of combining both arima and garch-type models to achieve forecast accuracy. [26] looked at possible combination of both arma and arch-type models to form a single model such as arma-arch that could completely capture the linear and non-linear features of financial data. their findings revealed that such combination was sufficient for the time series under study. [27] investigated the carry-over effect of biased estimates of joint arimagarch-type model parameters on forecast accuracy in the presence of outliers and their int. j. anal. appl. 18 (4) (2020) 561 results showed that after adjusting for outliers, marginal improvement on the forecasts was observed. however, these previous studies in the discrete-time stochastic series failed to compare the advantages of the combined arima-garch-type models over its individual components, dwelt on only one time horizon, and failed to predict on different time horizons leading to risky reliance. therefore, this study seeks to bridge this gap by investigating the robustness of forecast performance of arima-garch-type model over each single component using different time horizons to show consistency over time. the motivation for this study is drawn from the fact that arima models are not sufficient for modeling the return series due to the presence of heteroscedasticity, which leads to spurious forecasts. on the other hand, garch-type models are often misspecified with biased parameters and are susceptible to the presence of structural breaks and outliers. thus, these challenges provide the pragmatic reason and form the symbolic platform for adopting the hybridized arima-garch-type model in this study. the remaining aspect of the study is organized as follows: section 2 takes care of materials and methods; section 3 handles discussion of results, while section 4 concludes the study. 2 materials and methods 2.1 the return series the return series, 𝑅𝑡 can be obtained given that,𝑃𝑡, is the price of a unit share at time, t and 𝑃𝑡−1 is the share price at time t−1. 𝑅𝑡 = ∇𝑙𝑛𝑃𝑡 = (1 − 𝐵)𝑙𝑛𝑃𝑡 = 𝑙𝑛𝑃𝑡 − 𝑙𝑛𝑃𝑡−1. (1) here, 𝑅𝑡 in equation (1) is regarded as a transformed series of the share price, 𝑃𝑡, meant to attain stationarity, where both the mean and the variance of the series are stable ([28], [29]), while𝐵is the backshift operator. 2.2 arima models arima model is practically applied to capture the linear dependence in the return series ([30],[28], [26], [31], [32]). however, the fact that the series tends to appear in clusters, which actually results in the violation of assumption of constant variance. also, the linear time series models do not seem to produce accurate out-of-sample forecasts, thus providing a more sensible int. j. anal. appl. 18 (4) (2020) 562 argument for adopting heteroscedastic models ([31], [33]). a typical arima model equation is presented in (2): 𝑅𝑡 = 𝜇𝑡 + 𝑎𝑡 , (2) where 𝜇𝑡 = 𝜑0 + ∑ φjrt−j p j=1 + ∑ θi 𝑞 𝑖=1 𝑎t−i, φ is an autoregressive parameter, and 𝜃 is a moving average parameter. 2.3 garch-type models the generalized autoregressive conditional heteroscedastic (garch-type) models were introduced to account for heteroscedasticity (changing variance) and to overcome the problems associated with violation of assumption of constant variance. the garch-type specification could be symmetric (for example arch) which rely on modeling the conditional variance as a linear function of squared past residuals or asymmetric (for example egarch) which allows for the signs of the innovations (returns) to have impact on the volatility apart from magnitude ([34], [35], [31], [36]). moreover, the generalized autoregressive conditional heteroscedastic (garch-type) models were specified based on the normal distribution for the innovations but could not capture the heavy-tailed property ([34]). therefore, in this study, the student-t distribution is adopted which was traditionally introduced to overcome the weaknesses of the normal distribution in accommodating the heavy-tailed property. 2.3.1 autoregressive conditional heteroscedastic (arch) model arch(q) model as provided in [37] and specified as 𝜎𝑡 2 = 𝜔 + 𝛼1𝑎𝑡−1 2 + ⋯ + 𝛼𝑞 𝑎𝑡−𝑞 2 , (3) where 𝜎𝑡 2 is the conditional variance (heteroscedasticity), 𝜔 is the constant term and𝛼𝑞 is the coefficient of volatility clustering up to order q. 2.3.2 exponential generalized autoregressive conditional heteroscedastic (egarch) model egarch(q,p) model applies the natural logarithm to ensure that the conditional variance is positive and thus overcome the requirement of parameter restrictions ([38]). the egarch (q, p) is defined as, 𝑙𝑛𝜎𝑡 2 = 𝜔 + ∑ 𝛾𝑘 𝑎𝑡−𝑘 𝑟 𝑘=1 + ∑ 𝛼𝑖 (|𝑎𝑡−𝑖 | − 2√𝑣−2𝛤(𝑣+1) 2⁄ (𝑣−1)𝛤(𝑣 2)√𝜋⁄ ) 𝑞 𝑖=1 + ∑ 𝛽𝑗 𝑝 𝑗=1 𝑙𝑛𝜎𝑡−𝑗, 2 (4) 𝛽𝑗is the garch coefficient measuring persistence, and𝛾𝑘is the asymmetric coefficient. int. j. anal. appl. 18 (4) (2020) 563 2.4 hybridized models heteroscedastic models are hybridized of both mean and variance equations. the mean equation represents the arima model as shown in equation (5): 𝑅𝑡 = 𝜑0 + ∑ φjrt−j p j=1 + ∑ θi 𝑞 𝑖=1 𝑎t−i + 𝑎𝑡 , (5) 𝑎𝑡 = 𝜎𝑡 𝑒𝑡 , (6) where 𝑒𝑡 is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero, that is, e(𝑒𝑡) = 0 and variance 1,while 𝑎𝑡 in (6) is the standardized residual term that follows arch(q) and egarch(q,p) models in (3) and (4), respectively. putting it differently, equation (6) provides the link between the arima and the garch-type models. 2.5 model evaluation criteria the methods of forecast evaluation based on forecast error include mean squared error (mse), and mean absolute error (mae). these criteria measure forecast accuracy. these measures are employed in this study because of their popularity. the measures are computed as follows: mse = 1 𝑛 ∑ ℮𝑖 2𝑛 𝑖 =1 (7) mae = 1 𝑛 ∑ |℮𝑖 | 𝑛 𝑖=1 (8) where ℮𝑖 is the forecast error and n is the number of forecast error. 3. results and discussion the data are divided into two groups; training and testing which is purposed at checking the forecasting performance. the training data set ranging from january 3, 2006 to november 24, 2016, consisting of 2690 was used for model formulation while the testing data set was used for evaluating the forecasting performance at 8, 16 and 23 horizons. 3.1 plot analysis the plots of share prices in figures 1-2 showed a fluctuating movement away from the common means indicating the presence of nonstationarity. however, figures 3-4 represent the return series, which are the stationary, because they are clustered around the common means, indicating the presence of changing variance. int. j. anal. appl. 18 (4) (2020) 564 figure 1: share price series of first city monument bank source: data analysis figure 2: share price series of wema bank source: data analysis figure 3: return series of first city monument bank source: data analysis 0 5 10 15 20 25 2006 2008 2010 2012 2014 2016 fc mb 0 2 4 6 8 10 12 14 16 2006 2008 2010 2012 2014 2016 we ma ba nk sh ar es -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2006 2008 2010 2012 2014 2016 ld _f cm b int. j. anal. appl. 18 (4) (2020) 565 figure 4: return series of wema bank source: data analysis 3.3 evaluation of forecast performance considering the return series of first city monument bank, arima(0,1,1) model was found to be adequate in capturing the linear dependence in the data. on the other hand, arch(2)-t model was adequate in handling the heteroscedasticity in the data. however, combining the two models resulted in an arima(0,1,1)-arch(2)-t model could jointly express both the linear and nonlinear properties of the series. since our aim is to assess the forecast performance of the combined model in comparison to the individual component models at different horizons, the mse and mad were explored as the forecast performance evaluation measures and their values at different horizons are shown in table i. table i: evaluation of forecast performance for first city monument bank horizon 8 16 23 model mse mae mse mae mse mae arima arima(0,1,1) 3.267𝑒 −4 0.01379 4.792𝑒 −4 0.01681 4.626𝑒 −4 0.01718 garchtype arch(2)-t 3.333𝑒 −4 0.01438 4.778𝑒 −4 0.01711 4.500𝑒 −4 0.01717 arimagarchtype arima(0,1,1)arch(2)-t 2.327𝑒 −4 0.01330 4.667𝑒 −4 0.01602 4.300𝑒 −4 0.01710 source: data analysis -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 2006 2008 2010 2012 2014 2016 ld_ we ma ba nk sh ar es int. j. anal. appl. 18 (4) (2020) 566 to show the improvement of forecast performance of the joint arima(0,1,1)-arch(2)-t model over the arima(0,1,1) model, from table ii, it was found that arima(0,1,1)-arch(2)-t model outperformed arima(0,1,1) model by; 0.0094% and 0.049% as indicated by mse and mae, respectively at horizon 8; 0.00125% and 0.079% as shown by mse and mae, respectively at horizon 16; 0.00326% and 0.008% as indicated by mse and mae, respectively at horizon 23. table ii: percentage improvement of forecasting performance between arima and arimagarch-type models for first city monument bank horizon 8 16 23 model mse mae mse mae mse mae arima arima(0,0,1) 3.267𝑒 −4 0.01379 4.792𝑒 −4 0.01681 4.626𝑒 −4 0.01718 arimagarchtype arima(0,0,1)arch(2)-t 2.327𝑒 −4 0.01330 4.667𝑒 −4 0.01602 4.300𝑒 −4 0.01710 percentage difference 0.94𝑒 −2 0.049 0.125𝑒 −2 0.079 0.326𝑒 −2 0.008 source: data analysis from table iii, it was found that arima(0,1,1)-arch(2)-t model also outperformed arch(2)-t model by 0.01006% and 11.1% (as indicated by the respective values of mse and mae) at horizon 8; 0.00111% and 0.109% (as indicated by the respective values of mse and mae) at horizon 16; while 0.002% and 0.007% (as indicated by the respective values of mse and mae) at horizon 23. table iii: percentage improvement of forecasting performance between garch and arima-garch-type models for first city monument bank horizon 8 16 23 model mse mae mse mae mse mae garchtype arch(2)-t 3.333𝑒 −4 0.01438 4.778𝑒 −4 0.01711 4.5𝑒 −4 0.01717 arimagarchtype arima(0,0,1)arch(2)-t 2.327𝑒 −4 0.01330 4.667𝑒 −4 0.01602 4.3𝑒 −4 0.01710 percentage difference 1.006𝑒 −2 11.1 0.111𝑒 −2 0.109 0.2𝑒 −2 0.007 source: data analysis int. j. anal. appl. 18 (4) (2020) 567 for the return series of wema bank, arima(2,1,1) model was successfully adequate in handling the linear dependence in the data. egarch(1,1)-t model, on the other hand, was adequate in expressing the heteroscedasticity in the data. combining the two models resulted in an arima(2,1,1)-egarch(1,1)-t model, which was able to jointly capture both the linear and nonlinear properties of the series. assessing the forecast performance of the combined model in comparison to the individual component models at different time horizons, the mse and mad were explored as the forecast performance evaluation measures and their values at different horizons are shown in table iv. table iv: evaluation of forecast performance for wema bank horizon 8 16 23 model mse mae mse mae mse mae arima arima(2,1,1) 7.690𝑒 −4 0.02354 7.821𝑒 −4 0.02352 8.848𝑒 −4 0.02226 garchtype egarch(1,1)-t 7.559𝑒 −4 0.02296 7.753𝑒 −4 0.02323 8.803𝑒 −4 0.02206 arimagarchtype arima(2,1,1)egarch(1,1)-t 7.446𝑒 −4 0.02148 7.503𝑒 −4 0.02131 8.750𝑒 −4 0.02202 source: data analysis from table v, it was found that arima(2,1,1)-egarch(1,1)-t model outperformed arima(2,1,1) model by 0.00244% and 0.206% (as indicated by the respective values of mse and mae) at horizon 8; 0.00318% and 0.221% (as indicated by the respective values of mse and mae) at horizon 16; while 0.00098% and 0.024% (as indicated by the respective values of mse and mae) at horizon 23. table v: percentage improvement of forecasting performance between arima and arimagarch-type models for wema bank horizon 8 16 23 model mse mae mse mae mse mae arima arima(2,0,1) 7.690𝑒 −4 0.02354 7.821𝑒 −4 0.02352 8.848𝑒 −4 0.02226 arimagarchtype arima(2,0,1)egarch(1,1)-t 7.446𝑒 −4 0.02148 7.503𝑒 −4 0.02131 8.750𝑒 −4 0.02202 percentage difference 0.244𝑒 −2 0.206 0.318𝑒 −2 0.221 0.098𝑒 −2 0.024 source: data analysis int. j. anal. appl. 18 (4) (2020) 568 from table vi, it was found that arima(2,1,1)-egarch(1,1)-t model outperformed egarch(1,1)-t model by 0.00113% and 0.148% (as indicated by the respective values of mse and mae) at horizon 8; 0.0025% and 0.192% (as shown by the respective values of mse and mae) at horizon 16; while 0.00053% and 0.004% (as indicated by the respective values of mse and mae) at horizon 23. table vi: percentage improvement of forecasting performance between arima and arima-garch-type models for wema bank horizon 8 16 23 model mse mae mse mae mse mae garchtype egarch(1,1)-t 7.559𝑒 −4 0.02296 7.753𝑒 −4 0.02323 8.803𝑒 −4 0.02206 arimagarchtype arima(2,0,1)egarch(1,1)-t 7.446𝑒 −4 0.02148 7.503𝑒 −4 0.02131 8.750𝑒 −4 0.02202 percentage difference 0.113𝑒 −2 0.148 0.25𝑒 −2 0.192 0.053𝑒 −2 0.004 source: data analysis so far, arima(0,1,1)-arch(2)-t model appeared to be more robust than each of arima(0,1,1) and arch(2)-t models for the return series of first city monument bank, while arima(2,1,1)-egarch(1,1)-t model seemed to be more robust than each of arima(2,1,1) and egarch(1,1)-t models for the return series of wema bank by producing the least forecast errors (as measured by mse and mae at 8, 16 and 23 time horizons), while the percentage difference between each of the component models and the combined model provides the quantity of improvement measured. these findings are in tandem with the studies of [19], [20], [21], [22], [23], [24], [25]. evidently, the study has provided the needed improvement on the work of [27] by showing that the robustness of forecast performance of arima-garch-type model over each component using different horizon periods is consistent over time. 4. conclusion in summary, our findings revealed that the forecast performances of the combined models are better and more robust than those of individual components. actually, the robustness of their performances is consistent over different time horizons, which is a clear indication of an insignificant variability of the prediction error. this is particularly, a remedy to int. j. anal. appl. 18 (4) (2020) 569 model instability, which is a process that results in high variability of prediction error. conversely, it is recommended that model instability should be accounted for in each of the component models and their forecast performances compared to that of the combined model in order to assess whether they all result in near robustness. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. carriero, a. b. galvao and g. kapetanios, a comprehensive evaluation of macroeconomic forecasting method, int. j. forecast. 35(2019), 1636-1657. [2] p. gorgi, s. j. koopman and m. li, forecasting economic time series using score-driven dynamic models with mixed-data sampling, int. j. forecast. 35(2019), 1735-1747. [3] e. granziera and t. sekhposyan, predicting relative forecasting performance: an empirical investigation, int. j. forecast. 35(2019), 1636-1657. [4] a. s. gabriel, evaluating the forecasting performance of garch models: evidence from romania, precedia-soc. behav. sci. 62(2012), 1006-1010. [5] h. leeb, evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process, bernoulli. 14(2008), 661-690. [6] e. gulay and h. emec, the stock returns volatility based on the garch(1,1) model: the superiority of the truncated standard normal distribution in forecasting volatility, iran. econ. rev. 23(2019), 87-108. [7] i. u. moffat and e. a. akpan, selection of heteroscedastic models: a time series forecasting approach, appl. math. 10(2019), 333-348. [8] a. graefe, k. c. green and j. s. armstrong, accuracy gains from conservative forecasting: tests using variations of 19 econometric models to predict 154 elections in 10 countries, plos one 14(2019), e0209850. [9] j. ding, v. tarokh and g. gang, model selection techniquesan overview, 1eee signal proc. mag. 21(2018), 1-21. [10] m. pilatowska, information and prediction criteria in selecting the forecasting model, dyn. econ. model. 11(2011), 21-40 int. j. anal. appl. 18 (4) (2020) 570 [11] i. georgiev, d. i. harvey, s. j. leybourne and a. m. r. taylor, testing for parameter instability in predictive regression model, j. econ. 204(2018), 101-118. [12] s. chen, c. cui and j. zhang, on testing for structural break of coefficients in factor-augmented regression models, econ. lett. 161(2017), 141-145. [13] h. lee, the effect of level shift in the unconditional variance on predicting conditional volatility, j. econ. theory econometrics, 26(2015), 36-56. [14] s. farhani, tests of parameters instability: theoretical study and empirical analysis on two types of models (arma model and market model), int. j. econ. financial issues. 2(2012), 246-266. [15] h. zou and y. yang, combining time series models for forecasting, int. j. forecast. 20(2004), 69-84. [16] h. hong, n. chen, f. o’brien and j. ryan, stock return predictability and model instability: evidence from mainland china and hong kong, q. rev. econ. finance. 68(2018), 132-142. [17] d. pettenuzzo and a. timmermann, forecasting macreconomic variability under model instability, j. bus. econ. stat. 35(2017), 183-201. [18] r. giacomini and b. rossi, forecast comparisons in unstable environments, j. appl. econometrics. 25(2010), 595-620. [19] b. fazelabdolabadi, a hybrid bayesian-network proposition for forecasting the crude oil price, financial innov. 5(2019), 30. [20] e. spiliotis, f. fetropoulos and v. assimakopoulos, improving the forecasting performance of temporal hierarchies, plos one 14(2019), e0223422. [21] d. ardia, k. bluteau, k. boudt and l. catania, forecasting risk with markov-switching garch models: a large-scale performance study, int. j. forecast. 34(2018), 733-747. [22] s. di sanzo, a markov switching long memory model of crude oil price return volatility, energy econ. 74(2018), 351-359. [23] y. runfang, d. jiangze and l. xiaotao, improved forecast ability of oil market volatility based on combined markov switching and garch-class model, inform. technol. quant. manage. 122(2017), 415-422. [24] s. gunay, markov regime switching generalized autoregressive conditional heteroscedastic model and volatility modeling for oil returns, int. j. energy econ. policy. 5(2015), 979-985. [25] g.p. zhang, time series forecasting using a hybrid arima and neural network model, neurocomputing. 50(2003), 159-175. [26] e.a. akpan, i. u. moffat and n. b. ekpo, arma-arch modeling of the returns of first bank of nigeria, eur. sci. j. 12(2016), 257-266. int. j. anal. appl. 18 (4) (2020) 571 [27] e. a. akpan, k. e. lasis, a. adamu and h. b. rann, evaluation of forecasts performance of arimagarch-type models in the light of outliers, world sci. news. 119(2019), 68-84. [28] i.u. moffat and e. a. akpan, white noise analysis: a measure of time series model adequacy, appl. math. 10(2019), 989-1003. [29] e.a. akpan and i. u. moffat, modeling the effects of outliers on the estimation of linear stochastic time series model, int. j. anal. appl. 17(2019), 530-547. [30] e. a. akpan and i. u. moffat, detection and modeling of asymmetric garch effects in a discretetime series. int. j. stat. probab. 6(2017), 111-119. [31] r. s. tsay, analysis of financial time series. (3rded.). new york: john wiley & sons inc., (2010), 97140. [32] r. s. tsay, time series and forecasting: brief history and future research, j. amer. stat. assoc. 95(2000), 638-643. [33] p.h. franses and d. vandijk,non-linear time series models in empirical finance. (2nded.). new york, cambridge university press, (2003), 135-147. [34] i.u. moffat and e. a. akpan, modeling heteroscedasticity of discrete-time series in the face of excess kurtosis, glob. j. sci. front. res., f, math. decision sci. 18(2018), 21-32. [35] c. francq and j. zakoian, garch models: structure, statistical inference and financial applications. (1sted.). chichester, john wiley &sons ltd, (2010) 19-220. [36] r.f. engle and v. k.ng, measuring and testing the impact of news on volatility, j. finance. 48(1993), 1749-1778. [37] r.f. engle, autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflations, econometrica. 50(1982), 987-1007. [38] d.b. nelson, conditional heteroscedasticity of asset returns. a new approach. econometrica, 59(1991), 347-370. international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 139-143 http://www.etamaths.com estimation of comparative growth properties of entire and meromorphic functions in terms of their relative order arkojyoti biswas abstract. in this paper we discuss some comparative growth properties of entire and meromorphic functions on the basis of their relative order which improve some earlier results. 1. introduction, definitions and notations. let f be a non constant entire function in the open complex plane c and mf (r) = max{|f(z)| : |z| = r}.then mf (r) is strictly increasing,its inverse m−1f : (|f(0)| ,∞) → (0,∞) exists and is such that lim r→∞ m−1f (r) = ∞. two entire functions f and g are said to be asymptotically equivalent if there exists l (0 < l < ∞) such that mf (r) mg(r) → l as r →∞ and in that case we write f ∼ g.clearly if f ∼ g then g ∼ f. the order and lower order of an entire function are defined in the following way: definition 1. the order ρf and lower order λf of an entire function f are defined as follows: ρf = lim sup r→∞ log log mf (r) log r and λf = lim inf r→∞ log log mf (r) log r . the function f is said to be of regular growth if ρf = λf. the notion of order of an entire function was much improved by the introduction of the relative order of two entire functions.in this connection bernal [1] gave the following definition. 2010 mathematics subject classification. 30d30, 30d35. key words and phrases. entire and meromorphic functions; growth; relative order; asymptotic properties. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 139 140 arkojyoti biswas definition 2. [1]let f and g be two entire functions.the relative order ρg(f) of f with respect to g is defined as follows: ρg(f) = inf{µ > 0 : mf (r) < mg(rµ) for all sufficiently large values of r} = lim sup r→∞ log m−1g mf (r) log r . if we take g (z) as exp z then we see that ρg(f) = ρf and this shows that the relative order generalised the concept of the order of an entire function. similarly the relative lower order λg(f) is defined as λg(f) = lim inf r→∞ log m−1g mf (r) log r . for the case of a meromorphic function this generalisation was due to lahiri and banerjee [5].they introduced the notion of relative order ρg(f) of f with respect to g where f is meromorphic as follows: ρg(f) = inf{µ > 0 : tf (r) < tg(rµ) for all sufficiently large values of r} = lim sup r→∞ log t−1g tf (r) log r . similarly the relative lower order λg(f) of f with respect to g is defined by λg(f) = lim inf r→∞ log t−1g tf (r) log r . in a recent paper datta and biswas [2] studied some growth properties of entire functions using relative order.in this paper we discuss some comparative growth properties of entire and meromorphic functions in terms of their relative order which improves some results of datta and biswas [2]. 2. lemmas. in this section we present two lemmas which will be needed in the sequel. lemma 1. if g1 and g2 be two entire functions with property (a) such that g1 ∼ g2.if f be meromorphic then ρg1 (f) = ρg2 (f). lemma 1 follows from theorem 5 {cf.[3]} on putting l(r) ≡ 1. lemma 2 ([4]). if f,g be two meromorphic function and g is of regular growth.then ρg(f) = ρf ρg . 3. theorems. in this section we present the main results of our paper. theorem 1. let f be meromorphic and g,h be two entire functions with non zero finite orders.then lim inf r→∞ log t−1g tf (r) log t−1h tf (r) ≤ ρg(f) ρh(f) ≤ lim sup r→∞ log t−1g tf (r) log t−1h tf (r) . estimation of comparative growth properties 141 proof. from the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (1) log t−1g tf (r) < (ρg(f) + ε) log r. also for a sequence of values of r tending to infinity we get that (2) log t−1g tf (r) > (ρg(f) −ε) log r. again for arbitrary ε (> 0) and for all sufficiently large values of r we obtain that (3) log t−1h tf (r) < (ρh(f) + ε) log r and for a sequence of values of r tending to infinity we get that (4) log t−1h tf (r) > (ρh(f) −ε) log r. now from (1) and (4) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1h tf (r) < (ρg(f) + ε) (ρh(f) −ε) . as ε(> 0) is arbitrary it follows that (5) lim inf r→∞ log t−1g tf (r) log t−1h tf (r) ≤ ρg(f) ρh(f) . also from (2) and (3) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1h tf (r) > (ρg(f) −ε) (ρh(f) + ε) . as ε(> 0) is arbitrary it follows that (6) lim sup r→∞ log t−1g tf (r) log t−1h tf (r) ≥ ρg(f) ρh(f) . thus from (5) and (6) theorem 1 follows.this completes the proof. � corollary 1. if g and h are of regular growths then using lemma 2 we get from theorem 1 that lim inf r→∞ log t−1g tf (r) log t−1h tf (r) ≤ ρh ρg ≤ lim sup r→∞ log t−1g tf (r) log t−1h tf (r) . corollary 2. if g and h are of regular growths and g ∼ h then using lemma 1 we get from theorem 1 that lim inf r→∞ log t−1g tf (r) log t−1h tf (r) ≤ 1 ≤ lim sup r→∞ log t−1g tf (r) log t−1h tf (r) . remark 1. the converse of corollary 2 is not always true which is evident from the following example. example 1. let g (z) = exp z and h (z) = exp(2z) so that mg(r) = e r and mh(r) = e 2r .now mg(r) mh(r) → 0 as r →∞ and so g1 � g2 .also tg(r) = r π and th(r) = r π 142 arkojyoti biswas and therefore t−1g (r) = πr and t −1 h (r) = π 2 r. but lim r→∞ log t−1g tf (r) log t−1h tf (r) = lim r→∞ log πtf (r) log π 2 tf (r) = 1. theorem 2. let f,h be meromorphic and g be entire functions with non zero finite order.then lim inf r→∞ log t−1g tf (r) log t−1g th(r) ≤ ρg(f) ρg(h) ≤ lim sup r→∞ log t−1g tf (r) log t−1g th(r) . proof. from the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (7) log t−1g th(r) < (ρg(h) + ε) log r and for a sequence of values of r tending to infinity we get that (8) log t−1g th(r) > (ρg(h) −ε) log r. now from (1) and (8) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1g th(r) < (ρg(f) + ε) (ρg(h) −ε) . as ε(> 0) is arbitrary it follows that (9) lim inf r→∞ log t−1g tf (r) log t−1g th(r) ≤ ρg(f) ρg(h) . also from (2) and (7) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1g th(r) > (ρg(f) −ε) (ρg(h) + ε) . as ε(> 0) is arbitrary it follows that (10) lim sup r→∞ log t−1g tf (r) log t−1g th(r) ≥ ρg(f) ρg(h) . from (9) and (10) we obtain theorem 2.this completes the proof. � corollary 3. if g and h are of regular growths then using lemma 2 we get from theorem 2 that lim inf r→∞ log t−1g tf (r) log t−1g th(r) ≤ ρf ρh ≤ lim sup r→∞ log t−1g tf (r) log t−1g th(r) . theorem 3. let f, h be meromorphic and g,k be entire functions with non zero finite orders.then lim inf r→∞ log t−1g tf (r) log t−1k th(r) ≤ ρg(f) ρk(h) ≤ lim sup r→∞ log t−1g tf (r) log t−1k th(r) . estimation of comparative growth properties 143 proof. from the definition of relative order we get for arbitrary ε (> 0) and for all sufficiently large values of r that (11) log t−1k th(r) < (ρk(h) + ε) log r. also for a sequence of values of r tending to infinity we get that (12) log t−1k th(r) > (ρk(h) −ε) log r. now from (1) and (12) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1k th(r) < (ρg(f) + ε) (ρk(h) −ε) . as ε(> 0) is arbitrary it follows that (13) lim inf r→∞ log t−1g tf (r) log t−1k th(r) ≤ ρg(f) ρk(h) . also from (2) and (11) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1k th(r) > (ρg(f) −ε) (ρk(h) + ε) . as ε(> 0) is arbitrary it follows that (14) lim sup r→∞ log t−1g tf (r) log t−1k th(r) ≥ ρg(f) ρk(h) . from (13) and (14) we obtain theorem 3.this completes the proof. � references [1] l.bernal: orden relativo de crecimiento de functiones enteras, collectanea mathematica, 39(1988), 209-229. [2] s.k.datta and t.biswas: relative order of composite entire functions and some related growth properties, bull. cal. math. soc., 102 (2010), 259-266. [3] s.k.datta and a.biswas: estimation of relative order of entire and meromorphic functions in terms of slowly changing functions, int.j.contemp.math.sciences, 6 (2011), 1175-1186. [4] s.k.datta and a.biswas: a note on relative order of entire and meromorphic functions,international j. of math. sci. & engg. appls. 6 (2012), 413-421. [5] b.k. lahiri and d.banerjee: relative order of entire and meromorphic functions, proc.nat.acad.sci.india, 69 (1999), 339-354. ranaghat yusuf institution, p.o.-ranaghat, dist-nadia, pin-741201,west bengal, india international journal of analysis and applications volume 16, number 5 (2018), 712-732 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-712 simultaneous determination of distance between sets by multivalued kannan type coupling binayak s. choudhury1, pranati maity2, nikhilesh metiya3 and mihai postolache4,5,6,∗ 1department of mathematics, indian institute of engineering science and technology, shibpur, howrah 711103, west bengal, india 2department of mathematics, national institute of technology, rourkela, rourkela 769008, odisha, india 3department of mathematics, sovarani memorial college, jagatballavpur, howrah 711408, west bengal, india 4department of general education, china medical university, taichung 40402, taiwan 5gh. mihoc−c. iacob institute of mathematical statistics and applied mathematics of the romanian academy, bucharest 050711, romania 6department of mathematics & computer science, university politehnica of bucharest, 313 splaiul independenţei, bucharest 060042, romania ∗corresponding author: mi.postolache@mail.cmuh.org.tw; mihai@mathem.pub.ro received 2018-04-22; accepted 2018-06-23; published 2018-09-05. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. multivalued coupling; coulpled best proximity point; metric spaces; uniformly convex banach space. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 712 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-712 int. j. anal. appl. 16 (5) (2018) 713 abstract. in this paper we define a multivalued kannan type coupling between two subsets of a metric space and use it to obtain the distance between the two subsets through the determination of two pairs of points simultaneously. the problem is a multivalued coupled proximity point problem which falls under the general category of global optimization and is approached from the standpoint of fixed point theory. we use uc-property which is a geometric property that holds automatically for appropriate pairs of subsets of uniformly convex banach spaces and is adapted to metric spaces by certain postulations. the main results are illustrated with examples. corresponding results are obtained in banach spaces. the work is in the domain of setvalued analysis. 1. introduction the purpose of the present work is to find algorithmically the distance between two sets through the determination of two pairs of points simultaneously for which we use a multivalued coupling between the two sets. we define the coupling here. we assume that the multivalued coupling satisfies a kannan-type inequality in which δ-distance is used. it is a new approach to the global optimization problem known as best proximity point problem which seeks to determine the distance between two sets with the help of mappings defined between them. the problem is treated as that of finding an optimal approximate solution of a fixed point equation. it was introduced in the work of eldered [15] and has been considered by several researchers subsequently. singlevalued mappings were originally used in these problems. later multivalued functions were also utilized. some instances of works on best proximity point problems using single valued mappings are [1, 6–8, 11, 13, 20, 25, 26] while use of multivalued functions in these problems appear in works like [2, 12, 18, 24]. here we have a different approach to the problem mentioned above in which we use a coupled mapping between two sets. essentially we find an optimal approximate solution to coupled fixed point equations which allows us to determine simultaneously the required distance through two different pairs of points. moreover we use multivalued coupling. the coupling is assumed to satisfy an inequality which is of kannan type. kannan type inequalities are different from banach’s contraction inequality and has very different properties including the possibility that the mappings satisfying these inequalities may be discontinuous. these inequalities first appeared in the work of kannan [21, 22] and has been discussed in a large number of subsequent works [10, 14, 16, 19, 23]. 2. mathematical preliminaries throughout this paper (x, d) stands for a general metric space. let a and b be two subsets of x. a pair of points (a,b) with a ∈ a and b ∈ b is said to be a best proximity pair if d(a, b) = d(a, b) where d(a, b) = inf{d(x, y) : x ∈ a and y ∈ b}. if t : a −→ b is a non-self mapping for which there exists z ∈ a such that d(z, tz) = d(a, b), that is, where (z, tz) is a best proximity pair, we say that z is a int. j. anal. appl. 16 (5) (2018) 714 best proximity point of t and the problem of finding such a point is known as the best proximity point problem. it is a global optimization problem in that it seeks to find a point z ∈ a which minimizes the value of d(z, tz) over z ∈ a subject to the condition that the minimum is d(a, b). on the other hand it is an extension of the fixed point problem and reduces to that problem in the special case where a∩b 6= φ. the following are the concepts from setvalued analysis which we shall use in this paper. let (x, d) be a metric space. let n(x) := the collection of all non-empty subsets of x; b(x) := the collection of all non-empty bounded subsets of x and cb(x) := the collection of all non-empty closed and bounded subsets of x. now for a, b ∈ b(x), the functions d and δ are defined as follows. d(a, b) = inf { d(a, b) : a ∈ a, b ∈ b } and δ(a, b) = sup { d(a, b) : a ∈ a, b ∈ b } . d(a, b) is sometimes denoted as d(a, b) or dist(a, b). if a = {a}, then we write d(a, b) = d(a, b) and δ(a, b) = δ(a, b). also in addition, if b = {b}, then d(a, b) = d(a, b) and δ(a, b) = d(a, b). for a, b, c ∈ b(x), the definition of δ(a, b) yields the following: δ(a, b) = δ(b, a); δ(a, b) ≤ δ(a, c) + δ(c, b); δ(a, b) = 0 iff a = b = {a}; δ(a, a) = diam a [17]. the δ-distance has all the properties of a metric except one. it has been used in works like [4, 5, 9]. we use this concept in our theorem. let a and b be two non-empty subsets of a metric space (x, d) and t : a → cb(b) a multivalued mapping. a point x∗ ∈ a is called a best proximity point of t if d(x∗, tx∗) = d(a, b) [2]. this is a natural generalization of its singlevalued counterpart described in the above. a fixed point x of a multivalued mapping t is given by the following inclusion relation x ∈ tx. now there need not be a fixed point of the multivalued mapping in general. here the task in the best proximity point problem is to find a global minima of the function x 7→ d(x, tx) by constraining an approximate solution of the inclusion relation x ∈ tx to satisfy d(x, tx) = d(a, b). here we take another approach to the above problem. for that purpose we define singlevalued and multivalued coupling functions. definition 2.1. let a and b be two non-empty subsets of a metric space (x, d). a mapping f : x×x −→ x is said to be a coupling with respect to a and b if f(x, y) ∈ b for (x, y) ∈ a×b and f(x, y) ∈ a for (x, y) ∈ b ×a. definition 2.2. let a and b be two non-empty subsets of a metric space (x, d). a multivalued mapping f : x × x −→ n(x) is said to be a multivalued coupling with respect to a and b if f(x, y) ⊆ b for (x, y) ∈ a×b and f(x, y) ⊆ a for (x, y) ∈ b ×a. int. j. anal. appl. 16 (5) (2018) 715 definition 2.3 (property uc [26]). let a and b be two non-empty subsets of a metric space (x, d). then the pair (a, b) is said to satisfy the property uc if for sequences {xn} and {x′n} in a and {yn} in b, lim n−→∞ d(xn, yn) = d(a, b) and lim n−→∞ d(x′n, yn) = d(a, b) =⇒ lim n−→∞ d(xn, x ′ n) = 0. lemma 2.1 ( [26]). let a and b be two non-empty subsets of a metric space (x, d). assume that the pair (a,b) has the property uc. let {xn} and {yn} be sequences in a and b respectively such that either of the following holds: lim m−→∞ sup n≥m d(xm, yn) = d(a, b) or lim n−→∞ sup m≥n d(xm, yn) = d(a, b). then {xn} is a cauchy sequence. our purpose here is to realize the minimum distance between two sets a and b through a coupled best proximity point for singlevalued and multivalued coupling which we define below. definition 2.4. let a and b be two non-empty subsets of a metric space (x, d) and f : x ×x −→ x be a coupling with respect to a and b. an element (x, y) ∈ a×b is called a coupled best proximity point of f if d(x, f(x, y)) = d(a, b) and d(y, f(y, x)) = d(a, b). definition 2.5. let a and b be two non-empty subsets of a metric space (x, d) and f : x×x −→ n(x) be a multivalued coupling with respect to a and b. an element (x, y) ∈ a × b is called a coupled best proximity point of f if d(x, f(x, y)) = d(a, b) and d(y, f(y, x)) = d(a, b). it is noted that proximity points for coupled mapping have been defined in [20]. our definition is for both singlevalued and multivalued couplings which is different from the above mentioned concept. also the present work is in a different course from that of [20]. here the problem of finding coupled best proximity points for multivalued couplings is viewed as that of finding simultaneous optimal approximate solutions of the coupled fixed point inclusions x ∈ f(x, y) and y ∈ f(y, x) for a multivalued coupling f in the set a×b such that the solution satisfies d(x, f(x, y)) = d(y, f(y, x)) = d(a, b). in general, the exact solutions may not exist. this is surely the case where d(a, b) 6= 0 which is of interest here. from another viewpoint, it is the problem of simultaneous minimization of d(x, f(x, y)) and d(y, f(y, x)) for x ∈ a, y ∈ b such that the minimum values at the point of optimality are the global minimum d(a,b). it is to be noted that the pairs (x, f(x, y)) and (y, f(y, x)) are in general different. thus the distance between the two sets is obtained through two different pairs of points, that is, in the process we obtain two best proximity pairs. also the coupled best proximity point may not be unique. we discuss an application of our main result in uniformly convex banach spaces. int. j. anal. appl. 16 (5) (2018) 716 definition 2.6 ( [3]). a banach space x is said to be uniformly convex if for every � satisfying 0 < � ≤ 2 there corresponds a δ(�) > 0 such that the conditions hold for all x, y ∈ x, ||x|| = ||y|| = 1 and ||x−y|| ≥ � =⇒|| x + y 2 || ≤ 1 − δ(�). we use the following results in our application. lemma 2.2 ( [15]). let a be a non-empty closed and convex subset and b be a non-empty closed subset of a uniformly convex banach space. let {xn} and {zn} be sequences in a and {yn} be a sequence in b such that (i) ||zn −yn|| −→ d(a, b) as n −→ ∞ and (ii) for every � > 0 there exists n0 such that for all m > n ≥ n0, ||xm −yn|| ≤ d(a, b) + �. then for every � > 0 there exists n1 such that for all m > n ≥ n1, ||xm −zn|| ≤ �. lemma 2.3 ( [15]). let a be a non-empty closed and convex subset and b be a non-empty closed subset of a uniformly convex banach space. let {xn} and {zn} be sequences in a and {yn} be a sequence in b such that (i) ||xn −yn|| −→ d(a, b) as n −→∞ and (ii) ||zn −yn|| −→ d(a, b) as n −→∞. then ||xn −zn|| −→ 0 as n −→∞. here in our theorem we have established the existence of a coupled proximity points of a multivalued coupling in a metric space (x, d) which satisfies certain coupled inequality. the theorems are illustrated with examples. 3. main results theorem 3.1. let a and b be two non-empty closed subsets of a complete metric space (x, d) and f : x ×x −→ b(x) be a coupling with respect to a and b. suppose there exists k ∈ [0, 1 2 ) such that for x, v ∈ a and y, u ∈ b, the following inequality holds δ(f(x, y), f(u, v)) ≤ k [ d(x, f(x, y)) + d(u, f(u, v)) ] + (1 − 2k) d(a, b) (3.1) then there exist two sequences {xn} and {yn} in a and b respectively such that lim n−→∞ d(xn, yn+1) = d(a, b) and lim n−→∞ d(yn, xn+1) = d(a, b). further, if {xn} and {yn} are cauchy sequences, then f has a coupled best proximity point. proof. starting with x0 ∈ a and y0 ∈ b, we construct two sequences {xn} and {yn} respectively in a and b as follows xn+1 ∈ f(yn, xn) and yn+1 ∈ f(xn, yn) for all n ≥ 0. (3.2) int. j. anal. appl. 16 (5) (2018) 717 by (3.1) and (3.2), we have d(x1, y2) ≤ δ(f(y0, x0), f(x1, y1)) = δ(f(x1, y1), f(y0, x0)) ≤ k [ d(x1, f(x1, y1)) + d(y0, f(y0, x0)) ] + (1 − 2k) d(a, b) ≤ k [ d(x1, y2) + d(y0, x1) ] + (1 − 2k) d(a, b), which implies that d(x1, y2) ≤ k 1 −k d(y0, x1) + 1 − 2k 1 −k d(a, b). put t = k 1 −k . then 0 ≤ t < 1 and the above inequality becomes d(x1, y2) ≤ t d(y0, x1) + (1 − t) d(a, b). (3.3) again, using (3.1) and (3.2), we have d(y1, x2) ≤ δ(f(x0, y0), f(y1, x1)) ≤ k [ d(x0, f(x0, y0)) + d(y1, f(y1, x1)) ] + (1 − 2k) d(a, b) ≤ k [ d(x0, y1) + d(y1, x2) ] + (1 − 2k) d(a, b), which implies that d(y1, x2) ≤ k 1 −k d(x0, y1) + 1 − 2k 1 −k d(a, b) = t d(x0, y1) + (1 − t) d(a, b). (3.4) using (3.1) and (3.2), we have d(x2, y3) ≤ δ(f(y1, x1), f(x2, y2)) = δ(f(x2, y2), f(y1, x1)) ≤ k [ d(x2, f(x2, y2)) + d(y1, f(y1, x1)) ] + (1 − 2k) d(a, b) ≤ k [ d(x2, y3) + d(y1, x2) ] + (1 − 2k) d(a, b), that is, d(x2, y3) ≤ k 1 −k d(y1, x2) + ( 1 − k 1 −k ) d(a, b) = t d(y1, x2) + (1 − t) d(a, b) ≤ t [ t d(x0, y1) + (1 − t) d(a, b) ] + (1 − t) d(a, b) [ by (3.4) ] = t2 d(x0, y1) + (1 − t2) d(a, b). (3.5) int. j. anal. appl. 16 (5) (2018) 718 using (3.1) and (3.2), we have d(y2, x3) ≤ δ(f(x1, y1), f(y2, x2)) ≤ k [ d(x1, f(x1, y1)) + d(y2, f(y2, x2)) ] + (1 − 2k) d(a, b) ≤ k [ d(x1, y2) + d(y2, x3) ] + (1 − 2k) d(a, b), that is, d(y2, x3) ≤ k 1 −k d(x1, y2) + ( 1 − k 1 −k ) d(a, b) = t d(x1, y2) + (1 − t) d(a, b) ≤ t [ t d(y0, x1) + (1 − t) d(a, b) ] + (1 − t) d(a, b) [ by (3.3) ] = t2 d(y0, x1) + (1 − t2) d(a, b). (3.6) again, using (3.1) and (3.2), we have d(x3, y4) ≤ δ(f(y2, x2), f(x3, y3)) = δ(f(x3, y3), f(y2, x2)) ≤ k [ d(x3, f(x3, y3)) + d(y2, f(y2, x2)) ] + (1 − 2k) d(a, b) ≤ k [ d(x3, y4) + d(y2, x3) ] + (1 − 2k) d(a, b), that is, d(x3, y4) ≤ k 1 −k d(y2, x3) + ( 1 − k 1 −k ) d(a, b) = t d(y2, x3) + (1 − t) d(a, b) ≤ t [ t2 d(y0, x1) + (1 − t2) d(a, b) ] + (1 − t) d(a, b) [ by (3.6) ] = t3 d(y0, x1) + (1 − t3) d(a, b). (3.7) and d(y3, x4) ≤ δ(f(x2, y2), f(y3, x3)) ≤ k [ d(x2, f(x2, y2)) + d(y3, f(y3, x3)) ] + (1 − 2k) d(a, b) ≤ k[d(x2, y3) + d(y3, x4)] + (1 − 2k) d(a, b), int. j. anal. appl. 16 (5) (2018) 719 that is, d(y3, x4) ≤ k 1 −k d(x2, y3) + ( 1 − k 1 −k ) d(a, b) = t d(x2, y3) + (1 − t) d(a, b) ≤ t [ t2 d(x0, y1) + (1 − t2) d(a, b) ] + (1 − t) d(a, b) [ by (3.5) ] = t3 d(x0, y1) + (1 − t3) d(a, b). (3.8) let m be any positive integer. assume that if n is odd and n ≤ m, then d(xn, yn+1) ≤ tn d(y0, x1) + (1 − tn) d(a, b) (3.9) and d(yn, xn+1) ≤ tn d(x0, y1) + (1 − tn) d(a, b). (3.10) assume that if n is is even and n ≤ m, then d(xn, yn+1) ≤ tn d(x0, y1) + (1 − tn) d(a, b) (3.11) and d(yn, xn+1) ≤ tn d(y0, x1) + (1 − tn) d(a, b). (3.12) let m be even. then d(xm+1,ym+2) ≤ δ(f(ym, xm), f(xm+1, ym+1)) = δ(f(xm+1, ym+1), f(ym, xm)) ≤ k [ d(xm+1, f(xm+1, ym+1) + d(ym, f(ym, xm))) ] + (1 − 2k) d(a, b) ≤ k [ d(xm+1, ym+2) + d(ym, xm+1)] + (1 − 2k) d(a, b), that is, d(xm+1, ym+2) ≤ k 1 −k d(ym, xm+1) + (1 − k 1 −k )d(a, b) ≤ t [ tmd(y0, x1) + (1 − tm) d(a, b) ] + (1 − t) d(a, b) (by (3.12)) ≤ tm+1d(y0, x1) + (1 − tm+1) d(a, b). and d(ym+1,xm+2) ≤ δ(f(xm, ym), f(ym+1, xm+1)) ≤ k [ d(xm, f(xm, ym)) + d(ym+1, f(ym+1, xm+1)) ] + (1 − 2k) d(a, b) ≤ k [ d(xm, ym+1) + d(ym+1, xm+2) ] + (1 − 2k) d(a, b), int. j. anal. appl. 16 (5) (2018) 720 that is, d(ym+1, xm+2) ≤ k 1 −k d(xm, ym+1) + (1 − k 1 −k ) d(a, b) ≤ t [ tmd(x0, y1) + (1 − tm) d(a, b) ] + (1 − t) d(a, b) (by (3.11)) ≤ tm+1 d(x0, y1) + (1 − tm+1) d(a, b). let m be odd. then d(xm+1, ym+2) ≤ δ(f(ym, xm), f(xm+1, ym+1)) = δ(f(xm+1, ym+1), f(ym, xm)) ≤ k [ d(xm+1, f(xm+1, ym+1)) + d(ym, f(ym, xm)) ] + (1 − 2k) d(a, b) ≤ k [ d(xm+1, ym+2) + d(ym, xm+1) ] + (1 − 2k) d(a, b), that is, d(xm+1, ym+2) ≤ k 1 −k d(ym, xm+1) + (1 − k 1 −k ) d(a, b) ≤ t [tmd(x0, y1) + (1 − tm) d(a, b)] + (1 − t) d(a, b) (by (3.10)) ≤ tm+1d(x0, y1) + (1 − tm+1) d(a, b). and d(ym+1, xm+2) ≤ δ(f(xm, ym), f(ym+1, xm+1)) ≤ k [ d(xm, f(xm, ym)) + d(ym+1, f(ym+1, xm+1)) ] + (1 − 2k) d(a, b) ≤ k [ d(xm, ym+1) + d(ym+1, xm+2) ] + (1 − 2k) d(a, b), that is, d(ym+1,xm+2) ≤ k 1 −k d(xm, ym+1) + (1 − k 1 −k ) d(a, b) ≤ t [tmd(y0, x1) + (1 − tm) d(a, b)] + (1 − t) d(a, b) (by(3.9)) ≤ tm+1 d(y0, x1) + (1 − tm+1) d(a, b). thus (3.9)-(3.12) are valid for m + 1. by induction method, (3.9) and (3.10) are valid for all odd integers n and (3.11) and (3.12) are valid for all even integers n. therefore, we conclude that for all integers n ≥ 0, we have d(x2n+1, y2n+2) ≤ t2n+1d(y0, x1) + (1 − t2n+1) d(a, b) (3.13) and d(x2n, y2n+1) ≤ t2n d(x0, y1) + (1 − t2n) d(a, b). (3.14) int. j. anal. appl. 16 (5) (2018) 721 also, for all integer n ≥ 0, we conclude that d(y2n+1, x2n+2) ≤ t2n+1 d(x0, y1) + (1 − t2n+1) d(a, b) (3.15) and d(y2n, x2n+1) ≤ t2n d(y0, x1) + (1 − t2n) d(a, b). (3.16) since {xn} and {yn} sequences respectively in a and b and 0 ≤ t < 1, taking n −→ ∞ in (3.13), (3.14), (3.15) and (3.16), we have lim n−→∞ d(x2n, y2n+1) = lim n−→∞ d(x2n+1, y2n+2) = d(a, b) (3.17) and lim n−→∞ d(y2n, x2n+1) = lim n−→∞ d(y2n+1, x2n+2) = d(a, b). (3.18) now, (3.17) and (3.18) respectively implies that lim n−→∞ d(xn, yn+1) = d(a, b) and lim n−→∞ d(yn, xn+1) = d(a, b). (3.19) further, suppose that {xn} and {yn} are cauchy sequences. here x is complete and a and b are closed subsets of x. since {xn} and {yn} are sequences respectively in a and b, there exist x ∈ a and y ∈ b such that xn −→ x and yn −→ y as n −→∞. (3.20) by (3.19) and (3.20), we have lim n−→∞ d(xn, yn+1) = d(x, y) = d(a, b) (3.21) and lim n−→∞ d(yn, xn+1) = d(y, x) = d(x, y) = d(a, b). (3.22) since f is coupling with respect to a and b, and x ∈ a, y ∈ b, we have f(x, y) ⊆ b. then we have d(a, b) ≤ d(x, f(x, y)). (3.23) using (3.1) and (3.2), we obtain d(x, f(x, y)) ≤ δ(x, f(x, y)) ≤ d(x, xn+1) + δ(xn+1, f(x, y)) ≤ d(x, xn+1) + δ(f(yn, xn), f(x, y)) = d(x, xn+1) + δ(f(x, y), f(yn, xn)) ≤ d(x, xn+1) + k [ d(x, f(x, y)) + d(yn, f(yn, xn)) ] + (1 − 2k) d(a, b) ≤ d(x, xn+1) + k [ d(x, f(x, y)) + d(yn, xn+1) ] + (1 − 2k) d(a, b), int. j. anal. appl. 16 (5) (2018) 722 which implies that d(x, f(x, y)) ≤ 1 1 −k d(x, xn+1) + k 1 −k d(yn, xn+1) + ( 1 − k 1 −k ) d(a, b). taking n −→∞ in the above inequality, using (3.20) and (3.22), we have d(x, f(x, y)) ≤ k 1 −k d(a, b) + ( 1 − k 1 −k ) d(a, b) = d(a, b). (3.24) combining (3.23) and (3.24), we have d(a, b) ≤ d(x, f(x, y)) ≤ d(a, b), which implies d(x, f(x, y)) = d(a, b). similarly, we can prove that d(y, f(y, x)) = d(a, b). hence we have d(x, f(x, y)) = d(a, b) and d(y, f(y, x)) = d(a, b), that is, (x, y) is a coupled best proximity point of f. � example 3.1. let x = c[0, π] with chebyshev metric or sup metric. let a = {fr ∈ x : 0 ≤ r ≤ 1} and b = {gr ∈ x : 0 ≤ r ≤ 1}, where fr(x) = 1 −r sin x and gr(x) = r sin x− 1, for x ∈ [0, π]. let f : x ×x −→ b(x), f(u, v) =   {g1}, if u ∈ a and v ∈ b {f1}, if u ∈ b and v ∈ a {u + v}, otherwise. let k ∈ [0, 1 2 ) be arbitrary. it is verified that all the conditions of the theorem 3.1 are satisfied and (f1, g1) is a coupled best proximity point of f . in the next theorem we show that in case when the pair (a, b) satisfies the property uc, the sequences {xn} and {yn} constructed in theorem 3.1 actually converge to the coupled best proximity point of the kannan type coupling. theorem 3.2. let (x, d) be a complete metric space and (a, b) be a pair of non-empty closed subsets of x satisfying the property uc. let f : x × x −→ b(x) be a coupling with respect to a and b such that (3.1) is satisfied. then there exist two sequences {xn} and {yn} in a and b respectively such that lim n−→∞ d(xn, yn+1) = d(a, b) and lim n−→∞ d(yn, xn+1) = d(a, b). also, f has a coupled best proximity point. proof. we take the same sequences {xn} and {yn} as in theorem 3.1. then, lim n−→∞ d(xn, yn+1) = d(a, b) and lim n−→∞ d(yn, xn+1) = d(a, b) follows from theorem 3.1. so we have lim n−→∞ d(x2n, y2n+1) = d(a, b) and lim n−→∞ d(y2n+1, x2n+2) = d(x2n+2, y2n+1) = d(a, b). using the property uc, we have lim n−→∞ d(x2n, x2n+2) = 0, (3.25) int. j. anal. appl. 16 (5) (2018) 723 again, lim n−→∞ d(y2n, x2n+1) = d(a, b) and lim n−→∞ d(x2n+1, y2n+2) = d(y2n+2, x2n+1) = d(a, b). using the property uc, we have lim n−→∞ d(y2n, y2n+2) = 0. (3.26) from (3.25) and (3.26), we have that lim n−→∞ d(x2n, x2n+2) = lim n−→∞ d(y2n, y2n+2) = 0. (3.27) next we prove that {x2n} and {y2n} are cauchy sequences. we first establish that given � > 0 we can find a positive integer n such that for all m, n > n, max { d(x2m, y2n), d(y2m, x2n) } < d(a, b) + �. (3.28) if (3.28) is not valid, then there exists an � > 0 for which we can find two sequences of positive integers {2m(p)} and {2n(p)} such that for all positive integers p, 2n(p) > 2m(p) > p, max { d(x2m(p), y2n(p)), d(y2m(p), x2n(p)) } ≥ d(a, b) + � and max { d(x2m(p), y2n(p)−2), d(y2m(p), x2n(p)−2) } < d(a, b) + �. then d(a, b) + � ≤ max { d(x2m(p), y2n(p)), d(y2m(p), x2n(p)) } ≤ max { d(x2m(p), y2n(p)−2) + d(y2n(p)−2, y2n(p)), d(y2m(p), x2n(p)−2) + d(x2n(p)−2, x2n(p)) } ≤ max { d(x2m(p), y2n(p)−2), d(y2m(p), x2n(p)−2) } + max { d(y2n(p)−2, y2n(p)), d(x2n(p)−2, x2n(p)) } < d(a, b) + � + max { d(y2n(p)−2, y2n(p)), d(x2n(p)−2, x2n(p)) } . taking p −→∞ and using (3.27), we obtain lim p−→∞ max { d(x2m(p), y2n(p)), d(y2m(p), x2n(p)) } = d(a, b) + �. (3.29) int. j. anal. appl. 16 (5) (2018) 724 again, max { d(x2m(p), y2n(p)), d(y2m(p), x2n(p)) } ≤ max { δ(f(y2m(p)−1, x2m(p)−1), f(x2n(p)−1, y2n(p)−1)), δ(f(x2m(p)−1, y2m(p)−1), f(y2n(p)−1, x2n(p)−1)) } = max { δ(f(x2n(p)−1, y2n(p)−1), f(y2m(p)−1, x2m(p)−1)), δ(f(x2m(p)−1, y2m(p)−1), f(y2n(p)−1, x2n(p)−1)) } ≤ max { k [ d(x2n(p)−1,f(x2n(p)−1,y2n(p)−1)) + d(y2m(p)−1,f(y2m(p)−1,x2m(p)−1)) ] + (1 − 2k) d(a, b), k [ d(x2m(p)−1,f(x2m(p)−1,y2m(p)−1)) + d(y2n(p)−1,f(y2n(p)−1,x2n(p)−1)) ] + (1 − 2k) d(a, b) } ≤ max { k [ d(x2n(p)−1, y2n(p)) + d(y2m(p)−1, x2m(p)) ] + (1 − 2k) d(a, b), k [ d(x2m(p)−1, y2m(p)) + d(y2n(p)−1, x2n(p)) ] + (1 − 2k) d(a, b) } . taking p →∞, using (3.19) and (3.29), we get d(a, b) + � ≤ d(a, b), which is a contradiction. therefore, (3.28) holds. since {xn} and {yn} are sequences in a and b respectively, we have d(a, b) ≤ d(x2m, y2n) and d(a, b) ≤ d(y2m, x2n) for all m, n, and hence d(a, b) ≤ max { d(x2m, y2n), d(y2m, x2n) } for all m, n. then from (3.28), we can write that given any � > 0 there exists an integer n such that for all m, n > n d(a, b) − � < d(a, b) ≤ max { d(x2m, y2n), d(y2m, x2n) } < d(a, b) + �, which implies that for all m, n > n d(a, b) − � < d(a, b) ≤ d(x2m, y2n) < d(a, b) + �, and d(a, b) − � < d(a, b) ≤ d(y2m, x2n) < d(a, b) + �. therefore, lim m,n−→∞ d(x2m, y2n) = d(a, b) and lim m,n−→∞ d(y2m, x2n) = d(a, b). int. j. anal. appl. 16 (5) (2018) 725 from the above double limits we conclude that lim n−→∞ sup m≥n d(x2m, y2n) = d(a, b) or lim m−→∞ sup n≥m d(x2m, y2n) = d(a, b) and lim n−→∞ sup m≥n d(y2m, x2n) = d(a, b) or lim m−→∞ sup n≥m d(y2m, x2n) = d(a, b). thus, by lemma 2.1 it follows that {x2n} and {y2n} are a cauchy sequences in a and b respectively. as a and b are closed subsets of the complete metric space x, there exist u ∈ a and v ∈ b such that x2n −→ u and y2n −→ v as n −→∞. (3.30) since f is coupling with respect to a and b, and u ∈ a, v ∈ b, we have f(u, v) ⊆ b. then we have d(a, b) ≤ d(u, f(u, v)). (3.31) now d(u, f(u, v)) ≤ δ(u, f(u, v)) ≤ d(u, x2n) + δ(x2n, f(u, v)) ≤ d(u, x2n) + δ(f(y2n−1, x2n−1), f(u, v)) = d(u, x2n) + δ(f(u, v), f(y2n−1, x2n−1)) ≤ d(u, x2n) + k [ d(u, f(u, v)) + d(y2n−1, f(y2n−1, x2n−1)) ] + (1 − 2k) d(a, b) ≤ d(u, x2n) + k [ d(u, f(u, v)) + d(y2n−1, x2n) ] + (1 − 2k) d(a, b), which implies that d(u, f(u, v)) ≤ 1 1 −k d(u, x2n) + k 1 −k d(y2n−1, x2n) + ( 1 − k 1 −k ) d(a, b). taking n −→∞ in the above inequality, using (3.30) and the first part of the theorem, we have d(u, f(u, v)) ≤ k 1 −k d(a, b) + ( 1 − k 1 −k ) d(a, b) = d(a, b). (3.32) combining (3.31) and (3.32), we have d(a, b) ≤ d(u, f(u, v)) ≤ d(a, b), which implies d(u, f(u, v)) = d(a, b). similarly, we can prove that d(v, f(v, u)) = d(a, b). hence we have d(u, f(u, v)) = d(a, b) and d(v, f(v, u)) = d(a, b), that is, (u, v) is a coupled best proximity point of f. � int. j. anal. appl. 16 (5) (2018) 726 example 3.2. let x = r2 with the metric defined as d((x, y), (u, v)) = |x−u|+|y−v| for (x, y), (u, v) ∈ x. let a = {0}× [0, 1] and b = {1}× [0, 1]. let f : x ×x −→ b(x) be defined as f(x, y) =   {(1, 1)}, if x = (0,α) ∈ a and y = (1,β) ∈ b {(0, 1)}, if x = (1,β) ∈ b and y = (0,α) ∈ a {(u, v) :| u |≤ x2 + y2 and | v |≤ x2 + y2}, otherwise. let k ∈ [0, 1 2 ) be arbitrary. then all the conditions of theorem 3.2 are satisfied and ( (0, 1), (1, 1) ) is a coupled best proximity point of f. 4. application to uniformly convex banach spaces we now discuss the implications of our result in previous section in uniformly convex banach spaces for the case of single valued couplings. in this section d(a, b) = inf { ||a − b|| : a ∈ a, b ∈ b } and the notation d(a, b) is also denoted as d(a, b) or dist(a, b). theorem 4.1. let a and b be two non-empty closed subsets of a banach space x. let f : x ×x −→ x be a coupling with respect to a and b satisfying the following inequality: ||f(x, y) −f(u, v)|| ≤ k [ ||x−f(x, y)|| + ||u−f(u, v)|| ] + (1 − 2k) d(a, b) (4.1) where x, v ∈ a, y, u ∈ b and k ∈ [0, 1 2 ). then for arbitrary (x0, y0) ∈ a×b, the sequences {xn} and {yn} constructed as xn+1 = f(yn, xn) and yn+1 = f(xn, yn) for all n ≥ 0, (4.2) satisfy lim n−→∞ ||xn −yn+1|| = d(a, b) and lim n−→∞ ||yn −xn+1|| = d(a, b). (4.3) further, if {xn} and {yn} are cauchy sequences, then f has a coupled best proximity point. proof. {x}∈ b(x) for every x ∈ x. define a mapping t : x×x −→ b(x) as t(x, y) = { f(x, y) } , for x, y ∈ x. then all the conditions of the theorem reduce to the conditions of theorem 3.1 and hence by application of theorem 3.1, f has a coupled best proximity point. � lemma 4.1. let f : x ×x −→ x, where x is a normed linear space, be a coupling with respect to two subsets a and b of x. suppose there exists k ∈ [0, 1 2 ) such that (4.1) is satisfied for all x, v ∈ a and y, u ∈ b. then for arbitrary (x0, y0) ∈ a × b, the sequences {xn} and {yn} constructed following (4.2) satisfy lim n−→∞ ||xn −yn|| = d(a, b). proof. by theorem 4.1, it can be proved that the sequences {xn} and {yn} satisfy (4.3), that is, lim n−→∞ ||xn −yn+1|| = d(a, b) and lim n−→∞ ||yn −xn+1|| = d(a, b). int. j. anal. appl. 16 (5) (2018) 727 by (4.1) and (4.2), we have for n ≥ 1, ||xn −yn|| = ||f(yn−1, xn−1) −f(xn−1, yn−1)|| = ||f(xn−1, yn−1) −f(yn−1, xn−1)|| ≤ k [ ||xn−1 −f(xn−1, yn−1)|| + ||yn−1 −f(yn−1, xn−1)|| ] + (1 − 2k) d(a, b) = k [ ||xn−1 −yn|| + ||yn−1 −xn|| ] + (1 − 2k) d(a, b). (4.4) since {xn} and {yn} are sequences in a and b respectively, we have d(a, b) ≤ ||xn −yn|| for all n ≥ 1. (4.5) it follows from (4.4) and (4.5) that d(a, b) ≤ ||xn −yn|| ≤ k [ ||yn−1 −xn|| + ||xn−1 −yn|| ] + (1 − 2k) d(a, b). taking n −→∞ in the above inequality using (4.3), we have lim n−→∞ ||xn −yn|| = d(a, b). � remark 4.1. the above lemma is also generally valid in metric spaces. it is not necessary to assume x to be a normed linear space. we use this result only in the theorem for uniformly convex banach spaces. for this reason we assume x to be a normed linear space in particular. theorem 4.2. let a and b be two non-empty closed and convex subsets of a uniformly convex banach space x. let f : x ×x −→ x be a coupling with respect to a and b. suppose there exists k ∈ [0, 1 2 ) such that (4.1) is satisfied for all x, v ∈ a and y, u ∈ b. then f has a coupled best proximity point. proof. let x0 ∈ a and y0 ∈ b. we construct two sequences {xn} and {yn} respectively in a and b which satisfy (4.2), that is, xn+1 = f(yn, xn) and yn+1 = f(xn, yn) for all n ≥ 0. by theorem 4.1, we have (4.3). now by (4.3), we have lim n−→∞ ||xn −yn+1|| = d(a, b) and lim n−→∞ ||yn+1 −xn+2|| = ||xn+2 −yn+1|| = d(a, b). by application of lemma 2.3, we have lim n−→∞ ||xn −xn+2|| = 0, (4.6) again, by (4.3), we have lim n−→∞ ||yn −xn+1|| = d(a, b) and lim n−→∞ ||xn+1 −yn+2|| = ||yn+2 −xn+1|| = d(a, b). by application of lemma 2.3, we have lim n−→∞ ||yn −yn+2|| = 0. (4.7) int. j. anal. appl. 16 (5) (2018) 728 next we prove that {x2n}, {y2n}, {x2n+1} and {y2n+1} are cauchy sequences. in the following we consider the case of the sequences {x2n} and {y2n}. from (4.6) and (4.7), we have that lim n−→∞ ||x2n −x2n+2|| = lim n−→∞ ||y2n −y2n+2|| = 0. (4.8) we first establish that given � > 0 we can find a positive integer n such that for all m, n > n, max { ||x2m −y2n||, ||y2m −x2n|| } < d(a, b) + �. (4.9) by lemma 4.1, we have lim n−→∞ ||x2n −y2n|| = d(a, b) (4.10) if (4.9) is not valid, then there exists an � > 0 for which we can find two sequences of positive integers {2m(p)} and {2n(p)} such that for all positive integers p, 2n(p) > 2m(p) > p, max { ||x2m(p) −y2n(p)||, ||y2m(p) −x2n(p)|| } ≥ d(a, b) + � and max { ||x2m(p) −y2n(p)−2||, ||y2m(p) −x2n(p)−2|| } < d(a, b) + �. then d(a, b) + � ≤ max { ||x2m(p) −y2n(p)||, ||y2m(p) −x2n(p)|| } ≤ max { ||x2m(p) −y2n(p)−2|| + ||y2n(p)−2 −y2n(p)||, ||y2m(p) −x2n(p)−2|| + ||x2n(p)−2 −x2n(p)|| } ≤ max { ||x2m(p) −y2n(p)−2||, ||y2m(p) −x2n(p)−2|| } + max { ||y2n(p)−2 −y2n(p)||, ||x2n(p)−2 −x2n(p)|| } < d(a, b) + � + max { ||y2n(p)−2 −y2n(p)||, ||x2n(p)−2 −x2n(p)|| } . taking p −→∞ and using (4.8), we obtain lim p−→∞ max { ||x2m(p) −y2n(p)||, ||y2m(p) −x2n(p)|| } = d(a, b) + �. (4.11) int. j. anal. appl. 16 (5) (2018) 729 again, for all p, max { ||x2m(p)+2 −y2n(p)+2||, ||y2m(p)+2 −x2n(p)+2|| } ≤ max { ||x2m(p)+2 −x2m(p)|| + ||x2m(p) −y2n(p)|| + ||y2n(p) −y2n(p)+2||, ||y2m(p)+2 −y2m(p)|| + ||y2m(p) −x2n(p)|| + ||x2n(p) −x2n(p)+2|| } ≤ max { ||x2m(p) −y2n(p)||, ||x2n(p) −y2m(p)|| } + max { ||x2m(p)+2 −x2m(p)|| + ||y2n(p) −y2n(p)+2||, ||y2m(p)+2 −y2m(p)|| + ||x2n(p) −x2n(p)+2|| } and max { ||x2m(p) −y2n(p)||, ||x2n(p) −y2m(p)|| } ≤ max { ||x2m(p) −x2m(p)+2|| + ||x2m(p)+2 −y2n(p)+2|| + ||y2n(p)+2 −y2n(p)||, ||x2n(p) −x2n(p)+2|| + ||x2n(p)+2 −y2m(p)+2|| + ||y2m(p)+2 −y2m(p)|| } ≤ max { ||x2m(p)+2 −y2n(p)+2||, ||x2n(p)+2 −y2m(p)+2|| } + max { ||x2m(p) −x2m(p)+2|| + ||y2n(p)+2 −y2n(p)||, ||x2n(p) −x2n(p)+2|| + ||y2m(p)+2 −y2m(p)|| } . taking p −→∞ in the above two inequalities, and using (4.8) and (4.11), we conclude that lim p−→∞ max { ||x2m(p)+2 −y2n(p)+2||, ||y2m(p)+2 −x2n(p)+2|| } = d(a, b) + �. (4.12) again max { ||x2m(p)+2 −y2n(p)+2||, ||y2m(p)+2 −x2n(p)+2|| } = max { ||f(y2m(p)+1, x2m(p)+1) −f(x2n(p)+1, y2n(p)+1)||, ||f(x2m(p)+1, y2m(p)+1) −f(y2n(p)+1, x2n(p)+1)|| } = max { ||f(x2n(p)+1, y2n(p)+1) −f(y2m(p)+1, x2m(p)+1)||, ||f(x2m(p)+1, y2m(p)+1) −f(y2n(p)+1, x2n(p)+1)|| } ≤ max { k [ ||x2n(p)+1 −y2n(p)+2|| + ||y2m(p)+1 −x2m(p)+2|| ] + (1 − 2k) d(a, b), k [ ||x2m(p)+1 −y2m(p)+2|| + ||y2n(p)+1 −x2n(p)+2|| ] + (1 − 2k) d(a, b) } . taking p → ∞, using (4.3) and (4.12), we get d(a, b) + � ≤ d(a, b), which is a contradiction. therefore (4.9) holds. int. j. anal. appl. 16 (5) (2018) 730 since {xn} and {yn} are sequences in a and b respectively, we have d(a, b) ≤ ||x2m −y2n|| and d(a, b) ≤ ||y2m −x2n|| for all m, n, and hence d(a, b) ≤ max { ||x2m −y2n||, ||y2m −x2n|| } for all m, n. then from (4.9), we can write that given any � > 0 there exists an integer n such that for all m, n > n d(a, b) − � < d(a, b) ≤ max { ||x2m −y2n||, ||y2m −x2n|| } < d(a, b) + �, which implies that for all m, n > n d(a, b) − � < d(a, b) ≤ ||x2m −y2n|| < d(a, b) + �, and d(a, b) − � < d(a, b) ≤ ||y2m −x2n|| < d(a, b) + �. therefore, lim m,n−→∞ ||x2m −y2n|| = d(a, b) and lim m,n−→∞ ||y2m −x2n|| = d(a, b). (4.13) by (4.10), (4.13) and lemma 2.3, we have lim m,n−→∞ ||x2m −x2n|| = 0 and lim m,n−→∞ ||y2m −y2n|| = 0, (4.14) which implies that {x2n} and {y2n} are a cauchy sequences in a and b respectively. the sets a and b being closed, there exist u ∈ a and v ∈ b such that x2n −→ u and y2n −→ v as n −→∞. (4.15) since f is coupling with respect to a and b, and u ∈ a, v ∈ b, we have f(u, v) ∈ b. then we have d(a, b) ≤ ||u−f(u, v)||. (4.16) now ||u−f(u, v)|| ≤ ||u−x2n|| + ||x2n −f(u, v)|| ≤ ||u−x2n|| + ||f(y2n−1, x2n−1) −f(u, v)|| = ||u−x2n|| + ||f(u, v) −f(y2n−1, x2n−1)|| ≤ ||u−x2n|| + k [ ||u−f(u, v)|| + ||y2n−1 −f(y2n−1, x2n−1)|| ] + (1 − 2k) d(a, b) ≤ ||u−x2n|| + k [ ||u−f(u, v)|| + ||y2n−1 −x2n|| ] + (1 − 2k) d(a, b), which implies that ||u−f(u, v)|| ≤ 1 1 −k ||u−x2n|| + k 1 −k ||y2n−1 −x2n|| + ( 1 − k 1 −k ) d(a, b). int. j. anal. appl. 16 (5) (2018) 731 taking n −→∞ in the above inequality, using (4.3) and (4.15), we have ||u−f(u, v)|| ≤ k 1 −k d(a, b) + ( 1 − k 1 −k ) d(a, b) = d(a, b). (4.17) combining (4.16) and (4.17), we have d(a, b) ≤ ||u−f(u, v)|| ≤ d(a, b), which implies ||u − f(u, v)|| = d(a, b). similarly, we can prove that ||v − f(v, u)|| = d(a, b). hence (u, v) is a coupled best proximity point of f . � conclusion. coupled fixed point results are known for their applications. here we have presented an instance of an application where the concept of coupled fixed points has been utilized in the solution of a minimization problem. the approaches adapted here is new to the category of problems considered. it appears that the present method can be further applied to similar problems through other types of couplings satisfying different inequalities. this can be treated as an open problem. acknowledgement: the work is supported by the science and engineering research board, government of india, under research project no. pdf/2016/000353. the support is gratefully acknowledged. references [1] a. abkar, m. gabeleh, best proximity points for cyclic mappings in ordered metric spaces, j. optim. theory appl. 151 (2011), 418-424. [2] a. abkar, m. gabeleh, the existence of best proximity points for multivalued non-self-mappings, rev. r. acad. cienc. exactas fs. nat., ser. a mat., racsam, 107 (2013), 319-325. [3] r. p. agarwal, d. o‘regan, d. r. sahu, fixed point theory for lipschitzian-type mappings with applications, springer, new york, 2009. [4] i. altun, d. turkoglu, some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation, filomat 22 (2008), 13-21. [5] m. a. ahmed, common fixed point theorems for weakly compatible mappings, rocky mountain j. math. 33 (2003), 1189-1203. [6] c. d. bari, t. suzuki, c. vetro, best proximity points for cyclic meir keeler contractions, nonlinear anal. 69 (2008), 3790-3794. [7] a. bejenaru, a. pitea, fixed point and best proximity point theorems in partial metric spaces, j. math. anal. 7(4) (2016), 25-44. [8] b. s. choudhury, p. maity, p. konar, a global optimality result using nonself mappings, opsearch 51(2) (2014), 312-320. [9] b. s. choudhury, n. metiya, p. maity, coincidence point results of multivalued weak c-contractions on metric spaces with a partial order, j. nonlinear sci. appl. 6 (2013), 7-17. [10] b. s. choudhury, p. maity, cyclic coupled fixed point result using kannan type contractions, j. operators 2014 (2014), art. id 876749, 5 pages. [11] b. s. choudhury, p. maity, best proximity point results in generalized metric spaces, vietnam j. math. 44 (2016), 339-349. int. j. anal. appl. 16 (5) (2018) 732 [12] b. s. choudhury, p. maity, n. metiya, best proximity point results in setvalued analysis, nonlinear anal. modelling control. 21 (2016), 293-305. [13] b. s. choudhury, p. maity, k. sadarangani, a best proximity point theorem using discontinuous functions, j. convex anal. 24 (2017), 41-53. [14] b. damjanovic, d. dragan, multivalued generalizations of the kannan fixed point theorem, filomat 25(1) (2011), 125-131. [15] a. a. eldred, p. veeramani, existence and convergence of best proximity points, j. math. anal. appl. 323 (2006), 10011006. [16] e. karapinar, i. m. erhan, best proximity point on different type contractions, appl. math. info. sci. 5(3) (2011), 558-569. [17] b. fisher, common fixed points of mappings and setvalued mappings, rostock math. colloq. 18 (1981), 69-77. [18] m. gabeleh, best proximity points: global minimization of multivalued non-self mappings, optim. lett. 8 (2014), 11011112. [19] j. gornicki, fixed point theorems for kannan type mappings, j. fixed point theory appl. 19 (3) (2017), 2145-2152. [20] a. ilchev, b. zlatanov, error estimates for approximation of coupled best proximity points for cyclic contractive maps, appl. math. comput. 290 (2016), 412-425. [21] r. kannan, some result on fixed points, bull. calcutta math. soc. 60 (1968), 71-76. [22] r. kannan, some result on fixed points-ii, amer. math. monthly 76 (1969), 405-408. [23] m. petric, best proximity point theorems for weak cyclic kannan contraction, filomat 25(1) (2011), 145-154. [24] v. pragadeeswarar, m. marudai, p. kumam, best proximity point theorems for multivalued mappings on partially ordered metric spaces, nonlinear sci. appl. 9 (2016), 1911-1921. [25] w. shatanawi, a. pitea, best proximity point and best proximity coupled point in a complete metric space with (p)property, filomat 29(1) (2015), 63-74. [26] t. suzuki, m. kikkawa, c. vetro, the existence of best proximity points in metric spaces with the property uc, nonlinear anal. 71 (2009), 2918-2926. 1. introduction 2. mathematical preliminaries 3. main results 4. application to uniformly convex banach spaces references international journal of analysis and applications volume 17, number 5 (2019), 803-808 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-803 generalization of bateman polynomials asad ali∗, muhammad zafar iqbal, bilal anwer, ather mehmood department of mathematics and statistics, university of agriculture faisalabad, pakistan ∗corresponding author:: mrasadali5100@gmail.com abstract. in this paper, generalize the bateman polynomials in terms of generalized hypergeometric function of the type pfp. establish different forms of extended polynomials such as series expansion, generating functions and recurrence relations. 1. introduction bateman polynomials are the family of fn orthogonal polynomials. many of researchers generalized the classical results on the bateman polynomials. a large dedicated literature, numbers of relevant properties, extensions, generalizations and applications of bateman polynomials are available in [1], [2], [4], [7], [10] and [11]. the bateman polynomials fn(x) generated by ∞∑ n=0 fn(x)t n = (1 − t)−1ψ ( −4xt (1 − t)2 ) , (1.1) have the following classical properties. fn(x) = ∞∑ k=0 (−n)k(1 + n)kγkxk ( 1 2 )k(1)k , (1.2) fn(x) = 2f2(−n, 1 + n; 1, 1; x), (1.3) xf ′ n(x) − nfn(x) = −nfn−1(x) − xf ′ n−1(x), n ≥ 1, (1.4) received 2019-04-08; accepted 2019-05-13; published 2019-09-02. 2010 mathematics subject classification. 26c05, 65q30. key words and phrases. bateman polynomials; generating functions; recurrence relations. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 803 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-803 int. j. anal. appl. 17 (5) (2019) 804 xf ′ n(x) − nfn(x) = − n−1∑ k=0 fk(x) − 2x n−1∑ k=0 f ′ k(x), n ≥ 1, (1.5) xf ′ n(x) − nfn(x) = n−1∑ k=0 (−1)n−k(1 + 2k)fk(x), n ≥ 1. (1.6) 2. main results in this section we determine generalized properties of classical bateman polynomials, series expansion, generating function and recurrence relations. for this let ψ(u) have a formal powerseries expansion ψ(u) = ∞∑ n=0 γnu n, (2.1) define a polynomials f (α) n (x) by ∞∑ n=0 f(α)n (x) n = (1 − t)−1−αψ ( −ppxt qq(1 − t)p ) . (2.2) where p ≥ 2, q = p − 1 and α is any non-negative real parameter. theorem 2.1. if n is non-negative integer then, f(α)n (x) = (1 + α)n n! n∑ k=0 (−n)k( 1+α+nq )k( 2+α+n q )k....( q+α+n q )kx kγk ( 1+α p )k( 2+α p )k....( p+α p )k . (2.3) proof: from (2.1) and (2.2) ∞∑ n=0 f(α)n (x)t n = ∞∑ k=0 (−1)k(p)pkxktkγk (q)qk(1 − t)1+α+pk , by using (1), pp 58 of [1] ∞∑ n=0 f(α)n (x)t n = ∞∑ n=0 ∞∑ k=0 (−1)k(p)pk(1 + α)n+pkγkxktn+k (q)qkn!(1 + α)pk , by using lemma 11, pp 57 of [1] ∞∑ n=0 f(α)n (x)t n = ∞∑ n=0 n∑ k=0 (−1)k(p)pk(x)k(1 + α)n+qkγktn (q)qk(n − k)!(1 + α)pk , int. j. anal. appl. 17 (5) (2019) 805 = ∞∑ n=0 n∑ k=0 (1 + α)n n! (−1)kn!(p)pk(x)k(1 + α + n)qkγktn (q)qk(n − k)!(1 + α)pk , equating the coefficients of tn, we obtain (2.3). theorem 2.2. if n ≥ 1, then f(α)n (x) = (1 + α)n n! pfp  −n, 1 + α + n q , 2 + α + n q ... q + α + n q ; 1, 1...1︸ ︷︷ ︸ p−times ; x   . (2.4) proof: if we choose γk = ( 1+α p )k( 2+α p )k....( p+α p )k (k!)p+1 . in (2.3) then our yield is (2.4). theorem 2.3. if n is non-negative integer then, xf ′(α) n (x) − nf (α) n (x) = −(α + n)f (α) n−1(x) − qxf ′(α) n−1(x). (2.5) proof: in order to derive (2.5), consider f = ∞∑ n=0 f(α)n (x)t n = (1 − t)−1−αψ(v). where, v = −ppxt qq(1 − t)p . differentiate with respect to x fx = ∞∑ n=0 f ′(α) n (x)t n = (1 − t)−1−αψ ′ (v) −ppt qq(1 − t)p , int. j. anal. appl. 17 (5) (2019) 806 differentiate with respect to t ft = ∞∑ n=0 f(α)n (x)nt n−1 = (1 + α)(1 − t)−2−αψ(v) − (1 − t)−1−αψ ′ (v) ∂v ∂t , where, ∂v ∂t = −ppx(1 + qt) qq(1 − t)p+1 . ft = ∞∑ n=0 f(α)n (x)nt n−1 = (1 + α)(1 − t)−2−αψ(v) − x pp(1 − t)−2−α−p(1 + qt) qq ψ ′ (v), therefore f satisfies the partial differential equation x(1 + qt)fx − t(1 − t)ft + (1 + α)tf = 0. x(1 + qt) ∞∑ n=0 f ′(α) n (x)t n − t(1 − t ∞∑ n=0 f(α)n (x)nt n−1) + (1 + α)t ∞∑ n=0 f(α)n (x)t n = 0, ∞∑ n=0 [xf ′(α) n (x) − nf (α) n (x)]t n = − ∞∑ n=0 (1 + α + n)f(α)n (x)t n+1 − qx ∞∑ n=0 f ′(α) n (x)t n+1, = − ∞∑ n=1 (α + n)f (α) n−1(x)t n − qx ∞∑ n=1 f ′(α) n−1(x)t n, which leads to (2.5). theorem 2.4. if n is non-negative integer then, xf ′(α) n (x) − nf (α) n (x) = −(1 + α) n−1∑ k=0 f (α) k (x) − px n−1∑ k=0 f ′(α) k (x). (2.6) proof: int. j. anal. appl. 17 (5) (2019) 807 f also satisfies the partial differential equation xfx − xtfx + pxtfx − tft + t2ft + (1 + α)tf = 0. xfx − tft = − (1 + α)t 1 − t f − pxt 1 − t fx. x ∞∑ n=0 f ′(α) n (x)t n − t ∞∑ n=0 f(α)n (x)nt n−1 = −(1 + α) ∞∑ n=0 tn+1 ∞∑ k=0 f (α) k (x)t k − px ∞∑ n=0 tn+1 ∞∑ k=0 f ′(α) k (x)t k, ∞∑ n=0 [xf ′(α) n (x) − nf (α) n (x)]t n = −(1 + α) ∞∑ n=0 ∞∑ k=0 f (α) k (x)t n+k+1 − px ∞∑ n=0 ∞∑ k=0 f ′(α) k (x)t n+k+1 = − ∞∑ n=1 [(1 + α) n−1∑ k=0 f (α) k (x) − px n−1∑ k=0 f ′(α) k (x)]t n, which leads to (2.6). theorem 2.5. if n is non-negative integer then, xf ′(α) n (x) − nf (α) n (x) = n−1∑ k=0 (−q)n−k(1 + α + pk)f(α)k (x). (2.7) proof: f satisfies the partial differential equation xfx + qxtfx − tft − qt2ft + pt2ft + (1 + α)tf = 0. xfx − tft = − (1 + α)t 1 + qt f − pt2 1 + qt ft, ∞∑ n=0 [xf ′(α) n (x) − nf (α) n (x)]t n = −(1 + α) ∞∑ n=0 ∞∑ k=0 (−q)nf(α)k (x)t n+k+1 − p ∞∑ n=0 ∞∑ k=0 (−q)nf(α)k (x)kt n+k+1, int. j. anal. appl. 17 (5) (2019) 808 = ∞∑ n=1 n−1∑ k=0 (−q)n−k(1 + α + pk)f(α)k (x)t n, which gives (2.7). for α = 0 and p = 2 the equations (2.2) to (2.7) reduces to (1.1) to (1.6). theorem 2.6. if n ≥ 1, then the polynomials f(α)n (x) also satisfying the following property ∞∑ n=0 f(α)n (x)t n = (1 − t)−1−αpfp  1 + α + n p , 2 + α + n p ... q + α + n p ; 1, 1...1︸ ︷︷ ︸ p−times ; −ppxt qq   . (2.8) acknowledgments the authors express their sincere gratitude to dr. ghulam farid for useful discussions and invaluable advice. references [1] e. d. rainville, special functions, the macmillan company, new york, 1960. [2] g. andrews, r. askey and r. roy, special functions, cambridge university press, 2004. [3] g. andrews, r. askey and r. roy. special functions, cambridge university press, 1999. [4] s. b. trickovic and m. s. stankovic, on the orthogonality of classical orthogonal polynomials, integral transforms spec. funct., 14(2003), 129-138. [5] r. diaz and e. pariguan, on hypergeometric functions and pochhammer k-symbol, divulg. mat., 15 (2007), 179-192. [6] k. y. chen and h. m. srivastava, a limit relationship between laguerre and hermite polynomials. integral transforms spec. funct., 16(2005), 75 80. [7] e. h. doha, h. m. ahmed and s. i. el-soubhy, explicit formulae for the coefficients of integrated expansions of laguerre and hermite polynomials and their integrals, integral transforms spec. funct., 20 (2009), 491 503. [8] i. krasikov and a. zarkh, equioscillatory property of the laguerre polynomials, j. approx. theory, 162(2010), 2021 2047. [9] s. alam and a. k. chongdar, on generating functions of modified laguerre polynomials, rev. real academia de ciencias. zaragoza, 62(2007), 91 98. [10] m. a. khan and a. k. shukla, on some generalized sister celines polynomials, czechoslovak math. j., 49(3) (1999), 527-545. [11] h. m. srivastava and p. w. karlsson, multiple gaussian hypergeometric series, halstedpress (ellis horwood limited, chichester), john wiley and sons, new york, chichester, brisbane and toronto, 1985. 1. introduction 2. main results references international journal of analysis and applications volume 18, number 2 (2020), 319-331 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-319 generalizations of minkowski and beckenbach–dresher inequalities and functionals on time scales rabia bibi1,∗, anees ur rahman2 and muhammad shahzad2 1department of mathematics, abbottabad university of science and technology, havelian, abbottabad, pakistan 2department of mathematics, hazara university, mansehra, pakistan ∗corresponding author: emaorr@gmail.email abstract. we generalize integral forms of the minkowski inequality and beckenbach–dresher inequality on time scales. also, we investigate a converse of minkowski’s inequality and several functionals arising from the minkowski inequality and the beckenbach–dresher inequality. 1. introduction and preliminaries a time scale t is an arbitrary nonempty closed subset of the real numbers. the theory of time scales was introduced by stefan hilger [7] in order to unify the theory of difference equations and the theory of differential equations. for an introduction to the theory of dynamic equations on time scales, we refer to [3, 8]. martin bohner and gusein sh. guseinov [4, 5] defined the multiple riemann and multiple lebesgue integration on time scales and compared the lebesgue ∆-integral with the riemann ∆-integral. let n ∈ n be fixed. for each i ∈{1, . . . ,n}, let ti denote a time scale and λn = t1 × . . .×tn = {t = (t1, . . . , tn) : ti ∈ ti, 1 ≤ i ≤ n} received 2019-08-18; accepted 2019-09-24; published 2020-03-02. 2010 mathematics subject classification. primary 26d15; secondary 26a51, 34n05. key words and phrases. minkowski inequality, beckenbach–dresher inequality, time scales integrals. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 319 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-319 int. j. anal. appl. 18 (2) (2020) 320 an n-dimensional time scale. let µ∆ be the σ-additive lebesgue ∆-measure on λ n and f be the family of ∆-measurable subsets of λn. let e ∈ f and (e,f,µ∆) be a time scale measure space. then for a ∆-measurable function f : e → r, the corresponding ∆-integral of f over e will be denoted according to [5, (3.18)] by∫ e f(t1, . . . , tn)∆1t1 . . . ∆ntn, ∫ e f(t)∆t, ∫ e fdµ∆, or ∫ e f(t)dµ∆(t). by [5, section 3], all theorems of the general lebesgue integration theory, including the lebesgue dominated convergence theorem, hold also for lebesgue ∆-integrals on λn. here we state fubini’s theorem for integrals on time scales. it is used in the proofs of our main results. theorem 1.1 (fubini’s theorem). let (x,m,µ∆) and (y,l,ν∆) be two finite-dimensional time scale measure spaces. if f : x ×y → r is a ∆-integrable function and if we define the functions ϕ(y) = ∫ x f(x,y)dµ∆(x) for a.e. y ∈ y and ψ(x) = ∫ y f(x,y)dν∆(y) for a.e. x ∈ x, then ϕ is ∆-integrable on y and ψ is ∆-integrable on x and∫ x dµ∆(x) ∫ y f(x,y)dν∆(y) = ∫ y dν∆(y) ∫ x f(x,y)dµ∆(x). (1.1) hölder’s inequality and minkowski’s inequality and their converses for multiple integration on time scales were investigated in [1]. these inequalities hold for both riemann integrals and lebesgue integrals on time scales. for completeness, let us recall these inequalities from [1]. theorem 1.2 (hölder’s inequality [1, theorem 6.2]). for p 6= 1, define q = p/(p − 1). let (e,f,µ∆) be a time scale measure space. assume w, f, g are nonnegative functions such that wfp, wgq, wfg are ∆-integrable on e. if p > 1, then∫ e w(t)f(t)g(t)dµ∆(t) ≤ (∫ e w(t)fp(t)dµ∆(t) )1/p × (∫ e w(t)gq(t)dµ∆(t) )1/q . (1.2) if 0 < p < 1 and ∫ e wgqdµ∆ > 0, or if p < 0 and ∫ e wfpdµ∆ > 0, then (1.2) is reversed. theorem 1.3 (minkowski’s inequality [1, theorem 7.2]). let (e,f,µ∆) be a time scale measure space. for p ∈ r, assume w, f, g, are nonnegative functions such that wfp, wgp, w(f + g)p are ∆-integrable on e. if p ≥ 1, then (∫ e w(t)(f(t) + g(t))pdµ∆(t) )1 p ≤ (∫ e w(t)fp(t)dµ∆(t) )1/p + (∫ e w(t)gp(t)dµ∆(t) )1/p . (1.3) if 0 < p < 1 or p < 0, then (1.3) is reversed provided each of the two terms on the right-hand side is positive. int. j. anal. appl. 18 (2) (2020) 321 theorem 1.4 (converse of hölder’s inequality [1, theorem 11.3]). for p 6= 1, define q = p/(p − 1). let (e,f,µ∆) be a time scale measure space. assume w, f, g are nonnegative functions such that wfp, wgq, wfg are ∆-integrable on e. suppose 0 < m ≤ f(t)g−q/p(t) ≤ m for all t ∈ e. if p > 1, then ∫ e w(t)f(t)g(t)dµ∆(t) ≥ k(p,m,m) (∫ e w(t)fp(t)dµ∆(t) )1/p × (∫ e w(t)gq(t)dµ∆(t) )1/q , (1.4) where k(p,m,m) = |p|1/p|q|1/q (m −m)1/p|mmp −mmp|1/q |mp −mp| . (1.5) if 0 < p < 1 or p < 0, then (1.4) is reversed provided either ∫ e wgqdµ∆ > 0 or ∫ e wfpdµ∆ > 0. in [2] bibi et al., obtain integral forms of minkowski’s and beckenbach–dresher inequality on time scales. in this paper we generalize these inequalities and investigate functional obtained from our new inequalities. 2. minkowski inequalities let ul(x1,x2, . . . ,xl), vm(x1,x2, . . . ,xm), gk(x1,x2, . . . ,xk), are real valued functions of l,m, and k variables, respectively. let (x,m,µ∆) and (y,l,ν∆) be time scale measure spaces. then, throughout in the following sections, we use the following notations: ul = ul(x) = ul(u1(x),u2(x), . . . ,ul(x)), (2.1) vm = vm(y) = vm(v1(y),v2(y), . . . ,vm(y)), fk = fk(x,y) = fk(f1(x,y),f2(x,y), . . . ,fk(x,y)), where {ui(x)}li=1, {vi(y)} m i=1, {fi(x,y)} k i=1, are defined on x, y , and x ×y , respectively. in the sequel, we assume that all occurring integrals are finite. theorem 2.1 (integral minkowski inequality). if p ≥ 1, then [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p ≤ ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) (2.2) holds provided all integrals in (2.2) exists. if 0 < p < 1 and∫ x (∫ y fkvmdν∆ )p uldµ∆ > 0, ∫ y fkvmdν∆ > 0 (2.3) holds, then (2.2) is reversed. if p < 0 and (2.3) and∫ x f p kuldµ∆ > 0, (2.4) hold, then (2.2) is reversed as well. int. j. anal. appl. 18 (2) (2020) 322 proof. let p ≥ 1. put h(x) = ∫ y fk(x,y)vm(y)dν∆(y). now, by using fubini’s theorem (theorem 1.1) and hölder’s inequality (theorem 1.2) on time scales, we have ∫ x hp(x)ul(x)dµ∆(x) = ∫ x h(x)hp−1(x)ul(x)dµ∆(x) = ∫ x (∫ y fk(x,y)vm(y)dν∆(y) ) hp−1(x)ul(x)dµ∆(x) = ∫ y (∫ x fk(x,y)h p−1(x)ul(x)dµ∆(x) ) vm(y)dν∆(y) ≤ ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p (∫ x hp(x)ul(x)dµ∆(x) )p−1 p vm(y)dν∆(y) = ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) (∫ x hp(x)ul(x)dµ∆(x) )p−1 p and hence (∫ x hp(x)ul(x)dµ∆(x) )1 p ≤ ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y). for p < 0 and 0 < p < 1, the corresponding results can be obtained similarly. � theorem 2.2 (converse of integral minkowski inequality). suppose 0 < m ≤ fk(x,y)∫ y fk(x,y)vm(y)dν∆(y) ≤ m for all x ∈ x, y ∈ y. if p ≥ 1, then [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p ≥ k(p,m,m) ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) (2.5) provided all integrals in (2.5) exist, where k(p,m,m) is defined by (1.5). if 0 < p < 1 and (2.3) holds, then (2.5) is reversed. if p < 0 and (2.3) and (2.4) hold, then (2.5) is reversed as well. proof. let p ≥ 1. put h(x) = ∫ y fk(x,y)vm(y)dν∆(y). int. j. anal. appl. 18 (2) (2020) 323 then by using fubini’s theorem (theorem 1.1) and the converse hölder inequality (theorem 1.4) on time scales, we get ∫ x hp(x)ul(x)dµ∆(x) = ∫ x (∫ y fk(x,y)vm(y)dν∆(y) ) hp−1(x)ul(x)dµ∆(x) = ∫ y (∫ x fk(x,y)h p−1(x)ul(x)dµ∆(x) ) vm(y)dν∆(y) ≥k(p,m,m) ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1/p × (∫ x hp(x)ul(x)dµ∆(x) )p−1 p vm(y)dν∆(y). dividing both sides by (∫ x hp(x)ul(x)dµ∆(x) )p−1 p , we obtain (2.5). for 0 < p < 1 and p < 0, the corresponding results can be obtained similarly. � now we define the rth power mean m[r](fk,µ∆) of the function fk with respect to the measure µ∆ by m[r](fk,µ∆) =   (∫ x frk (x,y)ul(x)dµ∆(x)∫ x ul(x)dµ∆(x) )1 r if r 6= 0, exp (∫ x log fk(x,y)ul(x)dµ∆(x)∫ x ul(x)dµ∆(x) ) if r = 0, (2.6) where ∫ x uldµ∆ > 0. corollary 2.1. let 0 < s ≤ r. then m[r](m[s](fk, dν∆), dµ∆) ≥ k (r s ,m,m ) m[s](m[r](fk, dµ∆), dν∆). proof. by putting p = r/s and replacing fk by f s k in (2.5), raising to the power of 1 s and dividing by (∫ x ul(x)dµ∆(x) )1 r (∫ y vm(y)dν∆(y) )1 s , we get the above result. � 3. minkowski functionals in this section, we will consider some functionals which arise from the minkowski inequality. similar results (but not for time scales measure spaces) can be found in [9]. let fk and vm be fixed functions satisfying the assumptions of theorem 2.1. let us consider the functional m1 defined by m1(ul) = [∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p − ∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x), int. j. anal. appl. 18 (2) (2020) 324 where ul is a nonnegative function on x such that all occurring integrals exist. also, if we fix the functions fk and ul, then we can consider the functional m2(vm) = ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) − [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p , where vm is a nonnegative function on y such that all occurring integrals exist. remark 3.1. (i) it is obvious that m1 and m2 are positive homogeneous, i.e., m1(aul) = am1(ul), and m2(avm) = am2(vm), for any a > 0. (ii) if p ≥ 1 or p < 0, then m1(ul) ≥ 0, and if 0 < p < 1, then m1(ul) ≤ 0. (iii) if p ≥ 1, then m2(vm) ≥ 0, and if p < 1 and p 6= 0, then m2(vm) ≤ 0. theorem 3.1. (i) if p ≥ 1 or p < 0, then m1 is superadditive. if 0 < p < 1, then m1 is subadditive. (ii) if p ≥ 1, then m2 is superadditive. if p < 1 and p 6= 0, then m2 is subadditive. (iii) suppose ul1 and ul2 are nonnegative functions such that ul2 ≥ ul1. if p ≥ 1 or p < 0, then 0 ≤ m1(ul1) ≤ m1(ul2), (3.1) and if 0 < p < 1, then (3.1) is reversed. (iv) suppose vm1 and vm2 are nonnegative functions such that vm2 ≥ vm1. if p ≥ 1, then 0 ≤ m2(vm1) ≤ m2(vm2), (3.2) and if p < 1 and p 6= 0, then (3.2) is reversed. proof. first we show (i). we have m1(ul1 + ul2) −m1(ul1) −m1(ul2) = [∫ y (∫ x fp(x,y)(ul1 + ul2)(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p − ∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p (ul1 + ul2)(x)dµ∆(x) − [∫ y (∫ x f p k (x,y)ul1(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p + ∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul1(x)dµ∆(x) − [∫ y (∫ x f p k (x,y)ul2(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p int. j. anal. appl. 18 (2) (2020) 325 + ∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul2(x)dµ∆(x) = [∫ y (∫ x f p k (x,y)(ul1 + ul2)(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p − [∫ y (∫ x f p k (x,y)ul1(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p − [∫ y (∫ x f p k (x,y)ul2(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p . using the minkowski inequality (1.3) for integrals (theorem 1.3) with p replaced by 1/p, we have m1(ul1 + ul2) −m1(ul1) −m1(ul2)   ≥ 0 if p ≥ 1 or p < 0,≤ 0 if 0 < p ≤ 1. (3.3) so, m1 is superadditive for p ≥ 1 or p < 0, and it is subadditive for 0 < p ≤ 1. the proof of (ii) is similar: after a simple calculation, we have m2(vm1 + vm2) −m2(vm1) −m2(vm2) = [∫ x (∫ y fk(x,y)vm1(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p + [∫ x (∫ y fk(x,y)vm2(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p − [∫ x (∫ y fk(x,y)(vm1 + vm2)(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p . using the minkowski inequality (2.2) for integrals (theorem 2.1), we have that this is nonnegative for p ≥ 1 and nonpositive for p < 1 and p 6= 0. now we show (iii). if p ≥ 1 or p < 0, then using superadditivity and positivity of m1, ul2 ≥ ul1 implies m1(ul2) = m1(ul1 + (ul2 −ul1)) ≥ m1(ul1) + m1(ul2 −ul1) ≥ m1(ul1), and the proof of (3.1) is established. if 0 < p < 1, then using subadditivity and negativity of m1, ul2 ≥ ul1 implies m1(ul2) ≤ m1(ul1) + m1(ul2 −ul1) ≤ m1(ul1). the proof of (iv) is similar. � remark 3.2. put x,y ⊆ n, then for fixed fk and ul, the function m2 has the form m2(vm1) = ∑ j∈y vm1(j) (∑ i∈x ul(i)fk(i,j) p )1/p −  ∑ i∈x ul(i)  ∑ j∈y vm1(j)fk(i,j)  p  1/p , int. j. anal. appl. 18 (2) (2020) 326 where f(i,j) = fk(i,j) ≥ 0. if p ≥ 1, then the mapping m2 is superadditive, and vm2(j) ≥ vm1(j) for all j ∈ y implies 0 ≤ ∑ j∈y vm1(j) (∑ i∈x ul(i)fk(i,j) p )1/p −  ∑ i∈x ul(i)  ∑ j∈y vm1(j)fk(i,j)  p  1/p ≤ ∑ j∈y vm2(j) (∑ i∈x ul(i)fk(i,j) p )1/p −  ∑ i∈x ul(i)  ∑ j∈y vm2(j)fk(i,j)  p  1/p provided all occurring sums are finite. corollary 3.1. (i) suppose ul1 and ul2 are nonnegative functions such that cul2 ≥ ul1 ≥ cul2, where c,c ≥ 0. if p ≥ 1 or p < 0, then cm1(ul2) ≤ m1(ul1) ≤ cm1(ul2), and if 0 < p < 1, then the above inequality is reversed. (ii) suppose vm1 and vm2 are nonnegative functions such that cvm2 ≥ vm1 ≥ cvm2, where c,c ≥ 0. if p ≥ 1, then cm2(vm2) ≤ m2(vm1) ≤ cm2(vm2), and if p < 1 and p 6= 0, then the above inequality is reversed. corollary 3.2. if vm1 and vm2 are nonnegative functions such that vm2 ≥ vm1, then m[0] (∫ y fk(x,y)vm1(y)dν∆(y),µ∆ ) − ∫ y m[0](fk,µ∆)vm1(y)dν∆(y) ≤ m[0] (∫ y fk(x,y)vm2(y)dν∆(y),µ∆ ) − ∫ y m[0](fk,µ∆)vm2(y)dν∆(y), (3.4) where m[0](fk,µ∆) is defined in (2.6). the next result gives another property of m1, but a similar result can also be stated for m2. theorem 3.2. let ϕ : [0,∞) → [0,∞) be a concave function. suppose ul1 and ul2 are nonnegative functions such that ϕ◦ul1, ϕ◦ul2, ϕ◦ (αul1 + (1 −α)ul2) are ∆-integrable for α ∈ [0, 1]. if p ≥ 1, then m1(ϕ◦ (αul1 + (1 −α)ul2)) ≥ αm1(ϕ◦ul1) + (1 −α)m1(ϕ◦ul2), and if 0 < p < 1, then the above inequality is reversed. int. j. anal. appl. 18 (2) (2020) 327 proof. we show this only for p ≥ 1 as the other case follows similarly. since ϕ is concave, we have ϕ(αul1 + (1 −α)ul2)) ≥ αϕ(ul1) + (1 −α)ϕ(ul2). now, from (3.1) and (3.3), we have m1(ϕ◦ (αul1 + (1 −α)ul2)) ≥ m1(α(ϕ◦ul1) + (1 −α)(ϕ◦ul2)) ≥ m1(α(ϕ◦ul1)) + m1((1 −α)(ϕ◦ul2)) ≥ αm1(ϕ◦ul1) + (1 −α)m1(ϕ◦ul2), and the proof is established. � let fk, ul and vm be fixed functions satisfying the assumptions of theorem 2.1. let us define functionals m3 and m4 by m3(a) = [∫ y (∫ a f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) ]p − ∫ a (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) and m4(b) = ∫ b (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y)− [∫ x (∫ b fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]1 p , where a ⊆ x and b ⊆ y . the following theorem establishes superadditivity and monotonicity of the mappings m3 and m4. theorem 3.3. (i) suppose a1,a2 ⊆ x and a1 ∩a2 = ∅. if p ≥ 1 or p < 0, then m3(a1 ∪a2) ≥ m3(a1) + m3(a2), and if 0 < p < 1, then the above inequality is reversed. (ii) suppose a1,a2 ⊆ x and a1 ⊆ a2. if p ≥ 1 or p < 0, then m3(a1) ≤ m3(a2), and if 0 < p < 1, then the above inequality is reversed. (iii) suppose b1,b2 ⊆ y and b1 ∩b2 = ∅. if p ≥ 1, then m4(b1 ∪b2) ≥ m4(b1) + m4(b2), and if p < 1 and p 6= 0, then the above inequality is reversed. (iv) suppose b1,b2 ⊆ y and b1 ⊆ b2. if p ≥ 1, then m4(b1) ≤ m4(b2), and if p < 1 and p 6= 0, then the above inequality is reversed. int. j. anal. appl. 18 (2) (2020) 328 the proof of theorem 3.3 is omitted as it is similar to the proof of theorem 3.1. remark 3.3. for p ≥ 1, if sm is a subset of y with m elements and if sm ⊇ sm−1 ⊇ . . . ⊇ s2, then we have m4(sm) ≥ m4(sm−1) ≥ . . . ≥ m4(s2) ≥ 0 and m4(sm) ≥ max{m4(s2) : s2 is any subset of sm with 2 elements}. 4. beckenbach–dresher inequalities let ul, vm, fk be defined as in (4.1). let fn(x1,x2, . . . ,xn), gt(x1,x2, . . . ,xt) are real valued functions of n, and t variables, respectively. let (x,m,µ∆), (x,m,λ∆) and (y,l,ν∆) be time scale measure spaces. then, throughout in the following sections, we use the following notations: wn = wn(x) = wn(w1(x),w2(x), . . . ,wn(x)), (4.1) gt = gt(x,y) = gt(g1(x,y),g2(x,y), . . . ,gt(x,y)), where ul and wn are nonnegative functions on x, vm is a nonnegative function on y , fk is a nonnegative function on x ×y with respect to the measure (µ∆ ×ν∆), and gt is a nonnegative function on x ×y with respect to the measure (λ∆ ×ν∆). in the sequel, we assume that all occurring integrals are finite. theorem 4.1. if s ≥ 1, q ≤ 1 ≤ p, and q 6= 0 (4.2) or s < 0, p ≤ 1 ≤ q, and p 6= 0, (4.3) then [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q ≤ ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )s p(∫ x gqt (x,y)wn(x)dλ∆(x) )s−1 q vm(y)dν∆(y) (4.4) provided all occurring integrals in (4.4) exist. if 0 < s ≤ 1, p ≥ 1, q ≤ 1, and q 6= 0, (4.5) then (4.4) is reversed. int. j. anal. appl. 18 (2) (2020) 329 proof. assume (4.2) or (4.3). by using the integral minkowski inequality (2.2) and hölder’s inequality (1.2), we have [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q ≤ [∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )1 p vm(y)dν∆(y) ]s [∫ y (∫ x gqt (x,y)wn(x)dλ∆(x) )1 q vm(y)dν∆(y) ]s−1 =  ∫ y ((∫ x f p k (x,y)ul(x)dµ∆(x) )s p )1 s vm(y)dν∆(y)  s ×  ∫ y ((∫ x gqt (x,y)wn(x)dλ∆(x) )1−s q ) 1 1−s vm(y)dν∆(y)   1−s ≤ ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )s p (∫ x gqt (x,y)wn(x)dλ∆(x) )1−s q vm(y)dν∆(y). if (4.5) holds, then the reversed inequality in (4.4) can be proved in a similar way. � 5. beckenbach–dresher functionals let fk, gt, ul, wn be fixed functions satisfying the assumptions of theorem 4.1. we define the beckenbach–dresher functional bd(vm) by bd(vm) = ∫ y (∫ x f p k (x,y)ul(x)dµ∆(x) )s p(∫ x gqt (x,y)wn(x)dλ∆(x) )s−1 q vm(y)dν∆(y) − [∫ x (∫ y fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q , where we suppose that all occurring integrals exist. theorem 5.1. if (4.2) or (4.3) holds, then bd(vm1 + vm2) ≥ bd(vm1) + bd(vm2). (5.1) if vm2 ≥ vm1, then bd(vm1) ≤ bd(vm2). (5.2) if c,c ≥ 0 and cvm2 ≥ vm1 ≥ cvm2, then cbd(vm2) ≥ bd(vm1) ≥ cbd(vm1). (5.3) if (4.5) holds, then (5.1), (5.2) and (5.3) are reversed. int. j. anal. appl. 18 (2) (2020) 330 proof. assume (4.2) or (4.3). then we have bd(vm1 + vm2) −bd(vm1) −bd(vm2) = [∫ x (∫ y fk(x,y)vm1(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm1(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q + [∫ x (∫ y fk(x,y)vm2(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm2(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q − [∫ x (∫ y fk(x,y)vm1(y)dν∆(y) + ∫ y fk(x,y)vm2(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ y gt(x,y)vm1(y)dν∆(y) + ∫ y gt(x,y)vm2(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q ≥ 0, where in the last inequality we used (4.4) from theorem 4.1. using theorem 4.1 again, vm2 ≥ vm1 implies bd(vm2) = bd(vm1 + (vm2 −vm1)) ≥ bd(vm1) + bd(vm2 −vm1) ≥ bd(vm1). the proof of (5.3) is similar. if (4.5) holds, then the reversed inequalities of (5.1), (5.2) and (5.3) can be proved in a similar way. � let fk, gt, ul, vm, wn be fixed functions. we define a functional bd1 by bd1(a) = ∫ a (∫ x f p k (x,y)ul(x)dµ∆(x) )s p(∫ x gqt (x,y)wn(x)dλ∆(x) )s−1 q vm(y)dν∆(y) − [∫ x (∫ a fk(x,y)vm(y)dν∆(y) )p ul(x)dµ∆(x) ]s p[∫ x (∫ a gt(x,y)vm(y)dν∆(y) )q wn(x)dλ∆(x)]s−1q , where a ⊆ y . for bd1, the following result holds. theorem 5.2. (i) suppose a1,a2 ⊆ y and a1 ∩a2 = ∅. if (4.2) or (4.3) holds, then bd1(a1 ∪a2) ≥ bd1(a1) + bd1(a2), and if (4.5) holds, then the above inequality is reversed. (ii) suppose a1,a2 ⊆ y and a1 ⊆ a2. if (4.2) or (4.3) holds, then bd1(a1) ≤ bd1(a2), and if (4.5) holds, then the above inequality is reversed. the proof of theorem 5.2 is omitted as it is similar to the proof of theorem 5.1. int. j. anal. appl. 18 (2) (2020) 331 remark 5.1. if sk ⊆ x has k elements and if sm ⊇ sm−1 ⊇ . . . ⊇ s2, then (4.2) or (4.3) implies bd1(sm) ≥ bd1(sm−1) ≥ ···≥ bd1(s2) ≥ 0 and bd1(sm) ≥ max{bd1(s2) : s2 is any subset of sm with 2 elements}, while (4.5) implies the reversed inequalities with max replaced by min. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. anwar, r. bibi, m. bohner, and j. pečarić, integral inequalities on time scales via the theory of isotonic linear functionals, abstr. appl. anal. 2011(2011), art. id 483595. [2] r. bibi, m. bohner, j. pečarić, and s. varošanec, minkowski and beckenbach-dresher inequalities and functionals on time scales, j. math. inequal. appl. 2013(2013), 299–312. [3] m. bohner and a. peterson, dynamic equations on time scales: an introduction with applications, birkhäuser, boston, 2001. [4] m. bohner and g. sh. guseinov, multiple integration on time scales, dynam. systems appl. 14 (2005), 579–606. [5] m. bohner and g. sh. guseinov, multiple lebesgue integration on time scales, adv. difference equ. 2006 (2006), art. id 26391. [6] b. guljaš, c. e. m. pearce, and j. pečarić, some generalizations of the beckenbach–dresher inequality, houston j. math. 22 (1996), 629–638. [7] s. hilger, ein maßkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, ph. d. thesis, universität würzburg, 1988. [8] s. hilger, analysis on measure chains — a unified approach to continuous and discrete calculus, results math. 18 (1990), 18–56. [9] b. ivanković, j. pečarić, and s. varošanec, properties of mappings related to the minkowski inequality, mediterranean j. math. 8 (2011), 543–551. [10] s. varošanec, a generalized beckenbach–dresher inequality and related results, banach j. math. anal. 4 (2010), 13–20. 1. introduction and preliminaries 2. minkowski inequalities 3. minkowski functionals 4. beckenbach–dresher inequalities 5. beckenbach–dresher functionals references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 178-194 http://www.etamaths.com growth and complex oscillation of linear differential equations with meromorphic coefficients of [p,q] −ϕ order rabab bouabdelli and benharrat belaïdi∗ abstract. this paper is devoted to considering the growth of solutions of complex higher order linear differential equations with meromorphic coefficients under some assumptions for [p, q] − ϕ order and we obtain some results which improve and extend some previous results of h. hu and x. m. zheng; x. shen, j. tu and h. y. xu and others. 1. introduction and main results throughout this paper, a meromorphic function will means meromorphic in the whole complex plane. in this paper, we assume that readers are familiar with the fundamental results and standard notations of the nevanlinna’s theory of meromorphic functions (see [9, 18]). consider for n ≥ 2 the linear differential equations (1.1) f(n) + an−1f (n−1) + · · · + a1f′ + a0f = 0, (1.2) f(n) + an−1f (n−1) + · · · + a1f′ + a0f = f, where a0, · · · ,an−1,f are meromorphic functions. in [11, 12], juneja, kapoor and bajpai investigated some properties of entire functions of [p,q]-order and obtained some results concerning their growth. in [16], in order to maintain accordance with general definitions of the entire function f of iterated p−order [13, 14], liu-tushi gave a minor modification of the original definition of the [p,q]-order given in [11, 12]. by this new concept of [p,q]-order, the [p,q]-order of solutions of complex linear differential equations (1.1) and (1.2) was investigated in the unit disc and in the complex plane (see e.g. [2, 3, 4, 15, 16]). in [6] , i. chyzhykov, j. heittokangas and j. rättyä introduced the definition of ϕ−order of a meromorphic function in the unit disc as follows. definition 1.1 ([6]) let ϕ : [0, 1) → (0, +∞) be a non-decreasing unbounded function, the ϕ−order of f in the unit disc is defined by σ (f,ϕ) = lim sup r→+∞ log+ t (r,f) log ϕ (r) , 2010 mathematics subject classification. 34m10, 30d35. key words and phrases. meromorphic functions, [p, q] − ϕ order, [p, q] − ϕ type, [p, q] − ϕ exponent of convergence, differential equation. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 178 growth and complex oscillation of linear differential equations 179 and where in the following, t (r,f) is the characteristic function of nevanlinna. on the basic of definition 1.1, recently in [17] , x. shen, j. tu and h. y. xu introduced the new concept of [p,q] − ϕ order of meromorphic functions in the complex plane to study the growth and zeros of second order linear differential equations. for all r ∈ r, we define exp1 r := er and expp+1 r := exp ( expp r ) , p ∈ n. we also define for all r sufficiently large log1 r := log r and logp+1 r := log ( logp r ) , p ∈ n. moreover, we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r. definition 1.2 [17] let ϕ : [0, +∞) → (0, +∞) be a non-decreasing unbounded function, and p,q be positive integers and satisfy p ≥ q ≥ 1. then the [p,q] − ϕ order and [p,q]−ϕ lower order of a meromorphic function f are respectively defined by σ[p,q] (f,ϕ) = lim sup r→+∞ logp t (r,f) logq ϕ (r) , µ[p,q] (f,ϕ) = lim inf r→+∞ logp t (r,f) logq ϕ (r) . definition 1.3 let f be a meromorphic function satisfying 0 < σ[p,q] (f,ϕ) = σ < ∞. then the [p,q] −ϕ type of f (z) is defined by τ[p,q] (f,ϕ) = lim sup r→+∞ logp−1 t (r,f)[ logq−1 ϕ (r) ]σ . definition 1.4 let p,q be integers such that p ≥ q ≥ 1. let f be a meromorphic function satisfying 0 < µ[p,q] (f,ϕ) = µ < ∞. then the lower [p,q] −ϕ type of f is defined by τ[p,q] (f,ϕ) = lim inf r→+∞ logp−1 t (r,f)[ logq−1 ϕ (r) ]µ . definition 1.5 ([17]) let f be a meromorphic function. then, the [p,q] − ϕ exponent of convergence of zero-sequence (distinct zero-sequence) of f is defined by λ[p,q] (f,ϕ) = lim sup r→+∞ logp n ( r, 1 f ) logq ϕ (r) , λ̄[p,q] (f,ϕ) = lim sup r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) . and the lower exponent of distinct zero-sequence of f is defined by λ̄[p,q] (f,ϕ) = lim inf r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) . remark 1.1. if ϕ (r) = r in the definitions 1.2-1.5, then we obtain the standard definitions of the [p,q]−order, [p,q]−type and [p,q]−exponent of convergence. 180 bouabdelli and belaïdi remark 1.2 [17] throughout this paper, we assume that ϕ : [0, +∞) → (0, +∞) is a non-decreasing unbounded function and always satisfies the following two conditions : (i) lim r→+∞ logp+1 r logq ϕ(r) = 0. (ii) lim r→+∞ logq ϕ(αr) logq ϕ(r) = 1 for some α > 1. from remark 1.2, we can obtain the following proposition. proposition 1.1 suppose that ϕ (r) satisfies the condition (i) − (ii) . a) if f (z) is a meromorphic function, then λ[p,q] (f,ϕ) = lim sup r→+∞ logp n ( r, 1 f ) logq ϕ (r) = lim sup r→+∞ logp n ( r, 1 f ) logq ϕ (r) , λ̄[p,q] (f,ϕ) = lim sup r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) = lim sup r→+∞ logp n ( r, 1 f ) logq ϕ (r) . b) if f (z) is a meromorphic function, then λ̄[p,q] (f,ϕ) = lim inf r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) = lim inf r→+∞ logp n ( r, 1 f ) logq ϕ (r) . proof. we prove only b), for the proof of a) see [17] . we have n ( r, 1 f ) = r∫ 0 n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) log r. it follows that for r > r0 > 1 n ( r, 1 f ) −n ( r0, 1 f ) = r∫ 0 n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) log r −  r0∫ 0 n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) log r0   = r∫ r0 n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) (log r − log r0) (1.3) = r∫ r0 n̄ ( t, 1 f ) t dt ≤ n̄ ( r, 1 f ) log r r0 . then by (1.3) and lim r→+∞ logp+1 r logq ϕ(r) = 0, we obtain lim inf r→+∞ logp n ( r, 1 f ) logq ϕ (r) growth and complex oscillation of linear differential equations 181 (1.4) ≤ max  lim infr→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) , lim sup r→+∞ logp+1 r logq ϕ (r)   = lim infr→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) . on the other hand, since α > 1, we have for r > 1 n ( αr, 1 f ) = αr∫ 0 n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) log αr ≥ αr∫ r n̄ ( t, 1 f ) − n̄ ( 0, 1 f ) t dt + n̄ ( 0, 1 f ) log αr ≥ ( n̄ ( r, 1 f ) − n̄ ( 0, 1 f )) log α + n̄ ( 0, 1 f ) log αr (1.5) = n̄ ( r, 1 f ) log α + n̄ ( 0, 1 f ) log r ≥ n̄ ( r, 1 f ) log α. by (1.5) and lim r→+∞ logq ϕ(αr) logq ϕ(r) = 1, we get lim inf r→+∞ logp n ( αr, 1 f ) logq ϕ (αr) ≥ lim inf r→+∞  logp n̄ ( r, 1 f ) logq ϕ (r) . logq ϕ (r) logq ϕ (αr)   ≥ lim inf r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) .lim inf r→+∞ logq ϕ (r) logq ϕ (αr) (1.6) = lim inf r→+∞ logp n̄ ( r, 1 f ) logq ϕ (r) . by (1.4) and (1.6), it is easy to see that conclusion of b) holds. many authors have investigated complex oscillation properties of (1.1) and obtained many results when the coefficients in (1.1) are entire or meromorphic functions under some assumptions of [p,q]−order. recently, hu and zheng investigated the growth of solutions of (1.1) and obtained the following results. theorem a ([10]) let p,q be integers such that p ≥ q > 1 or p > q = 1, and let a0, · · · ,an−1 be meromorphic functions. assume that λ[p,q] ( 1 a0 ) < µ[p,q] (a0) < ∞, and that max { σ[p,q] (aj) ,j = 1, · · · ,n− 1 } ≤ µ[p,q] (a0) and max { τ[p,q] (aj) : σ[p,q] (aj) = µ[p,q] (a0) ,j 6= 0 } < τ[p,q] (a0) = τ. if f (6≡ 0) is a meromorphic solution of (1.1) satisfying n (r,f) n (r,f) < expp+1 { b logq r } ( b ≤ µ[p,q] (a0) ) , then we have λ̄[p+1,q] (f −ψ) = µ[p+1,q] (f) = µ[p,q] (a0) ≤ σ[p,q] (a0) = σ[p+1,q] (f) = λ̄[p+1,q] (f −ψ) , 182 bouabdelli and belaïdi where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ) < µ[p,q] (a0) . theorem b ([10]) let p,q be integers such that p ≥ q > 1 or p > q = 1, and let a0, · · · ,an−1 be meromorphic functions. assume that λ[p,q] ( 1 a0 ) < µ[p,q] (a0) < ∞, and that max { σ[p,q] (aj) ,j = 1, · · · ,n− 1 } ≤ µ[p,q] (a0) and lim sup r→+∞ n−1∑ j=1 m(r,aj) m(r,a0) < 1. if f (6≡ 0) is a meromorphic solution of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq r } ( b ≤ µ[p,q] (a0) ) , then we have λ̄[p+1,q] (f −ψ) = µ[p+1,q] (f) = µ[p,q] (a0) ≤ σ[p,q] (a0) = σ[p+1,q] (f) = λ̄[p+1,q] (f −ψ) , where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ) < µ[p,q] (a0) . for the case that the dominant coefficient a0 is replaced by an arbitrary coefficient as (s ∈{1, · · · ,n− 1}), they obtained the following. theorem c ([10]) let p,q be integers such that p ≥ q ≥ 1, and let a0, · · · ,an−1 be meromorphic functions. suppose that there exists one as (0 ≤ s ≤ n − 1) with λ[p,q] ( 1 as ) < µ[p,q] (as) < ∞ and that max{σ[p,q] (aj) , j 6= s} ≤ µ[p,q] (as) and max{τ[p,q] (aj) : σ[p,q] (aj) = µ[p,q] (as) , j 6= s} < τ[p,q] (as) = τ. then every transcendental meromorphic solution f ( 6≡ 0) of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq r } (b ≤ µ[p,q] (as)) satisfies µ[p+1,q] (f) ≤ µ[p,q] (as) ≤ µ[p,q] (f) and σ[p+1,q] (f) ≤ σ[p,q] (as) ≤ σ[p,q] (f) . moreover, every non-transcendental meromorphic solution f of (1.1) is a polynomial with degree deg (f) ≤ s− 1. the main purpose of this paper is to make use of the concept of meromorphic functions of [p,q] −ϕ-order to improve the results above. theorem 1.1 let p,q be integers such that p ≥ q > 1 or p > q = 1, and let a0, · · · ,an−1 be meromorphic functions. assume that λ[p,q] ( 1 a0 ,ϕ ) < µ[p,q] (a0,ϕ) < ∞, and that max{σ[p,q] (aj,ϕ) ,j = 1, · · · ,n− 1} ≤ µ[p,q] (a0,ϕ) and max{τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) , j 6= 0} < τ[p,q] (a0,ϕ) = τ, and where ϕ satisfies the conditions lim r→+∞ logp+1 r logq ϕ(r) = 0 and lim r→+∞ logq−1 ϕ(αr) logq−1 ϕ(r) = 1 for some α > 1. if f ( 6≡ 0) is a meromorphic solution of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq ϕ (r) } ( b ≤ µ[p,q] (a0,ϕ) ) , then we have λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) ≤ σ[p,q] (a0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) , where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ,ϕ) < µ[p,q] (a0,ϕ) . theorem 1.2 let p,q be integers such that p ≥ q > 1 or p > q = 1, and let a0, · · · ,an−1 be meromorphic functions. assume that λ[p,q] ( 1 a0 ,ϕ ) < µ[p,q] (a0,ϕ) < ∞, and that max{σ[p,q] (aj,ϕ) ,j = growth and complex oscillation of linear differential equations 183 1, · · · ,n−1}≤ µ[p,q] (a0,ϕ) and lim sup r→+∞ n−1∑ j=1 m(r,aj) m(r,a0) < 1, and where ϕ satisfies the conditions (i) − (ii) of the remark 1.2. if f (6≡ 0) is a meromorphic solution of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq ϕ (r) } ( b ≤ µ[p,q] (a0,ϕ) ) , then we have λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) ≤ σ[p,q] (a0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) , where ψ (z) (6≡ 0) is a meromorphic function with σ[p+1,q] (ψ,ϕ) < µ[p,q] (a0,ϕ) . theorem 1.3 let p,q be integers such that p ≥ q ≥ 1, and let a0, · · · ,an−1 be meromorphic functions. suppose that there exists one as (0 ≤ s ≤ n− 1) with λ[p,q] ( 1 as ,ϕ ) < µ[p,q] (as,ϕ) < ∞ and that max { σ[p,q] (aj,ϕ) ,j 6= s } ≤ µ[p,q] (as,ϕ) and max { τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (as,ϕ) , j 6= s } < τ [p,q] (as,ϕ) = τ, and where ϕ satisfies the conditions (i) − (ii) of the remark 1.2. then every transcendental meromorphic solution f (6≡ 0) of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq ϕ (r) } ( b ≤ µ[p,q] (as,ϕ) ) satisfies µ[p+1,q] (f,ϕ) ≤ µ[p,q] (as,ϕ) ≤ µ[p,q] (f,ϕ) and σ[p+1,q] (f,ϕ) ≤ σ[p,q] (as,ϕ) ≤ σ[p,q] (f,ϕ) . moreover, every non-transcendental meromorphic solution f (z) of (1.1) is a polynomial with degree deg (f) ≤ s− 1. remark 1.3. if we put ϕ (r) = r in the theorems 1.1, 1.2, 1.3, then we obtain theorems a, b, c. 2. auxiliary lemmas we need the following lemmas to obtain our results. lemma 2.1 ([5]) let f be a meromorphic solution of (1.1) assuming that not all coefficients aj (z) are constants. given a real constant γ > 1, and denoting t (r) = n−1∑ j=0 t (r,aj) , we have log m (r,f) < t (r){(log r) log t (r)}γ , if p = 0 and log m (r,f) < r2p+γ−1t (r){log t (r)}γ , if p > 0, outside of an exceptional set ep with ∫ ep tp−1dt < +∞. remark 2.1. especially, if p = 0, then the exceptional set e0 has finite logarithmic measure ∫ e0 dt t = mle0. lemma 2.2 ([1] , [8]) let g : [0, +∞) → r, h : [0, +∞) → r be monotone increasing functions. if (i) g (r) ≤ h (r) outside of an exceptional set of finite linear measure, or (ii) g (r) ≤ h (r) , r /∈ e1 ∪(0, 1], where e1 ⊂ [1,∞) is a set of finite logarithmic measure, then for any β > 1, there exists r0 = r0 (β) > 0 such that g (r) ≤ h (βr) for all r > r0. 184 bouabdelli and belaïdi lemma 2.3 ([9]) let f be a transcendental meromorphic function and n ≥ 1 be an integer. then m ( r, f(n) f ) = o (log (rt (r,f))) outside of a possible exceptional set e2 of r of finite linear measure, and if f is of finite order of growth, then m ( r, f(n) f ) = o (log r) . lemma 2.4 let p,q be integers such that p ≥ q ≥ 1, and let f be a meromorphic function satisfying µ[p,q] (f,ϕ) = µ < ∞ ( σ[p,q] (f,ϕ) = σ < ∞ ) , where ϕ (r) only satisfies lim r→+∞ logq ϕ(αr) logq ϕ(r) = 1 for some α > 1. then there exists a set e3 ⊂ (1,∞) of infinite logarithmic measure such that for all r ∈ e3, we have µ = lim r→+∞ r∈e3 logp t (r,f) logq ϕ (r) ,  σ = lim r→+∞ r∈e3 logp t (r,f) logq ϕ (r)   and for any given ε > 0 and sufficiently large r ∈ e3 t (r,f) < expp { (µ + ε) logq ϕ (r) } ( t (r,f) > expp { (σ −ε) logq ϕ (r) }) . proof. we prove only the first assumption, for the second we use the same proof. by the definition 1.2, there exists an increasing sequence {rn} ∞ n=1 tending to ∞ satisfying ( 1 + 1 n+1 ) rn < rn+1 and µ = µ[p,q] (f,ϕ) = lim rn→∞ logp t (rn,f) logq ϕ (rn) . then for any given ε > 0, there exists an n1 such that for n ≥ n1 and any r ∈[ rn, ( 1 + 1 n ) rn ] , we have logp t (r,f) logq ϕ (r) ≤ logp t (( 1 + 1 n ) rn,f ) logq ϕ (( 1 + 1 n ) rn ) logq ϕ((1 + 1n)rn) logq ϕ (rn) . when q ≥ 1, we have logq ϕ((1+ 1 n )rn) logq ϕ(rn) → 1 (n → +∞). let e3 = ∞⋃ n=n1 [ rn, (1 + 1 n )rn ] , for any given ε > 0 and all r ∈ e3, we have lim r→+∞ r∈e3 logp t (r,f) logq ϕ (r) ≤ lim rn→∞ logp t (( 1 + 1 n ) rn,f ) logq ϕ (( 1 + 1 n ) rn ) = µ[p,q] (f,ϕ) , where mle3 = ∞∑ n=n1 (1+ 1n )rn∫ rn dt t = ∞∑ n=n1 log ( 1 + 1 n ) = ∞. on the other hand, we have lim r→+∞ r∈e3 logp t (r,f) logq ϕ (r) ≥ lim inf r→+∞ logp t (r,f) logq ϕ (r) = µ[p,q] (f,ϕ) . growth and complex oscillation of linear differential equations 185 therefore, lim r→+∞ r∈e3 logp t (r,f) logq ϕ (r) = µ[p,q] (f,ϕ) and for any given ε > 0 and sufficiently large r ∈ e3 t (r,f) < expp { (µ + ε) logq ϕ (r) } . lemma 2.5 let f1,f2 be meromorphic functions of [p,q] − ϕ order satisfying σ[p,q] (f1,ϕ) > σ[p,q] (f2,ϕ) , where ϕ (r) only satisfies lim r→+∞ logq ϕ(αr) logq ϕ(r) = 1 for some α > 1. then there exists a set e4 ⊂ (1, +∞) having infinite logarithmic measure such that for all r ∈ e4, we have lim r→+∞ t (r,f2) t (r,f1) = 0. proof. set σ1 = σ[p,q] (f1,ϕ) , σ2 = σ[p,q] (f2,ϕ) (σ1 > σ2) . by lemma 2.4, there exists a set e4 ⊂ (1, +∞) having infinite logarithmic measure such that for any given 0 < ε < σ1−σ2 2 and all sufficiently large r ∈ e4 t (r,f1) > expp { (σ1 −ε) logq ϕ (r) } and for all sufficiently large r t (r,f2) < expp { (σ2 + ε) logq ϕ (r) } . from this we can get t (r,f2) t (r,f1) < expp { (σ2 + ε) logq ϕ (r) } expp { (σ1 −ε) logq ϕ (r) } = 1 exp { expp−1 { (σ1 −ε) logq ϕ (r) } − expp−1 { (σ2 + ε) logq ϕ (r) }}, r ∈ e4. since 0 < ε < σ1−σ2 2 , then we have lim r→+∞ t (r,f2) t (r,f1) = 0, r ∈ e4. remark 2.2 if µ[p,q] (f1,ϕ) > µ[p,q] (f2,ϕ) , then we get the same result. lemma 2.6 let p,q be integers such that p ≥ q ≥ 1, and let a0, · · · ,an−1, f ( 6≡ 0) be meromorphic functions. if f is a meromorphic solution of (1.2) satisfying max { σ[p,q] (f,ϕ) ,σ[p,q] (aj,ϕ) ,j = 0, · · · ,n− 1 } < µ[p,q] (f,ϕ) , then we have λ̄[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = µ[p,q] (f,ϕ) , where ϕ satisfies the conditions (i) − (ii) of remark 1.2. proof. by (1.2) , we get (2.1) 1 f = 1 f ( f(n) f + an−1 f(n−1) f + · · · + a1 f′ f + a0 ) . 186 bouabdelli and belaïdi it is easy to see that if f has a zero at z0 of order α (α > n) , and a0, · · · ,an−1 are analytic at z0, then f must have a zero at z0 of order α−n. hence (2.2) n ( r, 1 f ) ≤ nn ( r, 1 f ) + n ( r, 1 f ) + n−1∑ j=0 n (r,aj) . by the lemma 2.3 and (2.1), we have (2.3) m ( r, 1 f ) ≤ m ( r, 1 f ) + n−1∑ j=0 m (r,aj) + o (log t (r,f) + log r) (r /∈ e2) , where e2 ⊂ (1, +∞) is a set of r of finite linear measure. by (2.2) and (2.3) , we get t (r,f) = t ( r, 1 f ) + o (1) ≤ nn ( r, 1 f ) + t (r,f) (2.4) + n−1∑ j=0 t (r,aj) + o{log (rt (r,f))} (r /∈ e2) . since max{σ[p,q] (f,ϕ) , σ[p,q] (aj,ϕ) , j = 0, · · · ,n− 1} < µ[p,q] (f,ϕ) , then (2.5) max { t (r,f) t (r,f) , t (r,aj) t (r,f) ( j = 0, · · · ,n− 1) } → 0, r → +∞. also, for all sufficiently large r, we have (2.6) log (t (r,f)) = o{t (r,f)} . by (2.4) − (2.6) , for all |z| = r /∈ e2, we have (2.7) (1 −o (1)) t (r,f) ≤ nn ( r, 1 f ) + o (log r) . by definition 1.2, proposition 1.1, lemma 2.2 and (2.7) , we get (2.8) µ[p,q] (f,ϕ) ≤ λ̄[p,q] (f,ϕ) . since µ[p,q] (f,ϕ) ≥ λ[p,q] (f,ϕ) ≥ λ̄[p,q] (f,ϕ) , then by (2.8) , we have λ̄[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = µ[p,q] (f,ϕ) . using the same method above, lemma 2.5 and lemma 2.2 we can prove the following lemma. lemma 2.7 let p,q be integers such that p ≥ q ≥ 1, and let a0, · · · ,an−1, f ( 6≡ 0) be meromorphic functions. if f is a meromorphic solution of (1.2) satisfying max { σ[p,q] (f,ϕ) ,σ[p,q] (aj,ϕ) ,j = 0, · · · ,n− 1 } < σ[p,q] (f,ϕ) < +∞, then we have λ[p,q] (f,ϕ) = λ[p,q] (f,ϕ) = σ[p,q] (f,ϕ) , where ϕ satisfies the conditions (i) − (ii) of remark 1.2. lemma 2.8 let p,q be integers such that p ≥ q ≥ 1 and let a0, · · · ,an−1 be meromorphic functions such that max { σ[p,q] (aj,ϕ) : j 6= s } ≤ µ[p,q] (as,ϕ) < ∞, where ϕ satisfies the conditions (i)−(ii) of remark 1.2. if f ( 6≡ 0) is a meromorphic growth and complex oscillation of linear differential equations 187 solution of (1.1) satisfying n(r,f) n(r,f) < expp+1 { b logq ϕ (r) } ( b ≤ µ[p,q] (as,ϕ) ) , then we have µ[p+1,q] (f,ϕ) ≤ µ[p,q] (as,ϕ) . proof. by (1.1), we know that the poles of f can only occur at the poles of a0, · · · ,an−1. by n(r,f) n(r,f) < expp+1 { b logq ϕ (r) } ( b ≤ µ[p,q] (as,ϕ) ) , we have n (r,f) < expp+1 { b logq ϕ (r) } n (r,f) ≤ expp+1 { b logq ϕ (r) }n−1∑ j=0 n (r,aj) (2.9) ≤ expp+1 { b logq ϕ (r) }n−1∑ j=0 t (r,aj) . then by (2.9) , we have (2.10) t (r,f) ≤ m (r,f) + expp+1 { b logq ϕ (r) }n−1∑ j=0 t (r,aj) . by lemma 2.4, there exists a set e3 of infinite logarithmic measure such that for any given ε > 0 and sufficiently large r ∈ e3, we have (2.11) t (r,as) ≤ expp {( µ[p,q] (as,ϕ) + ε ) logq ϕ (r) } . since max { σ[p,q] (aj,ϕ) : j 6= s } ≤ µ[p,q] (as,ϕ) , for the above ε > 0 and sufficiently large r, we have (2.12) t (r,aj) ≤ expp {( µ[p,q] (as,ϕ) + ε ) logq ϕ (r) } , j 6= s. by (2.11) , (2.12), lemma 1.1 and remark 1.2, there exists a set e0 of r of finite logarithmic measure such that for sufficiently large r ∈ e3�e0 m (r,f) ≤ exp   n−1∑ j=0 t (r,aj)  (log r) log  n−1∑ j=0 t (r,aj)    γ   (2.13) ≤ expp+1 {( µ[p,q] (as,ϕ) + 2ε ) logq ϕ (r) } . from (2.10) and (2.13) , we get lim inf r→+∞ logp+1 t (r,f) logq ϕ (r) ≤ lim inf r→+∞ r∈e3�e0 logp+1 t (r,f) logq ϕ (r) ≤ µ[p,q] (as,ϕ) + 3ε. since ε > 0 is arbitrary, we have µ[p+1,q] (f,ϕ) ≤ µ[p,q] (as,ϕ) . lemma 2.9 let p,q be integers such that p ≥ q > 1 or p > q = 1 and let a0, · · · ,an−1 be meromorphic functions. assume that λ[p,q] ( 1 a0 ,ϕ ) < µ[p,q] (a0,ϕ) and that max { σ[p,q] (aj,ϕ) : j = 1, · · · ,n− 1 } ≤ µ[p,q] (a0,ϕ) = µ, 0 < µ < ∞, and max { τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) ,j 6= 0 } < τ [p,q] (a0,ϕ) = τ, 0 < τ < ∞, where ϕ satisfies the conditions (i) − (ii) of remark 1.2. if f ( 6≡ 0) is a meromorphic solution of (1.1) , then we have µ[p+1,q] (f,ϕ) ≥ µ[p,q] (a0,ϕ) . 188 bouabdelli and belaïdi proof. suppose that f (6≡ 0) is a meromorphic solution of (1.1) . by (1.1) , we obtain (2.14) −a0 = f(n) f + an−1 f(n−1) f + · · · + a1 f′ f . by λ[p,q] ( 1 a0 ,ϕ ) < µ[p,q] (a0,ϕ) , we have n (r,a0) = o (t (r,a0)) , r → +∞. then by (2.14) , we get (2.15) t (r,a0) = m (r,a0) + n (r,a0) ≤ n−1∑ j=1 m (r,aj) + n−1∑ j=1 m ( r, f(j) f ) + o (t (r,a0)) . hence, by (2.15) and lemma 2.3 that (2.16) t (r,a0) ≤ o  n−1∑ j=1 m (r,aj) + log (rt (r,f))   , for sufficiently large r → +∞, r /∈ e2, where e2 is a set of r of finite linear measure. set b = max{σ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) < µ[p,q] (a0,ϕ) = µ, j = 1, · · · ,n− 1}. if σ[p,q] (aj,ϕ) < µ[p,q] (a0,ϕ) = µ, then for any ε (0 < 2ε < µ− b) and all r → +∞, we have m (r,aj) ≤ t (r,aj) ≤ expp { (b + ε) logq ϕ (r) } (2.17) < expp { (µ−ε) logq ϕ (r) } = expp−1 {( logq−1 ϕ (r) )µ−ε} . set τ1 = max { τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) , j 6= 0 } , then τ1 < τ. if σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) , τ[p,q] (aj,ϕ) ≤ τ1 < τ, then for r → +∞ and any ε (0 < 2ε < τ − τ1) , we have (2.18) m (r,aj) ≤ t (r,aj) < expp−1 { (τ1 + ε) ( logq−1 ϕ (r) )µ} . by the definition of the lower [p,q] −ϕ type, for r → +∞, we have (2.19) t (r,a0) > expp−1 { (τ −ε) ( logq−1 ϕ (r) )µ} . when p ≥ q > 1 or p > q = 1, we have for r → +∞ expp−1 { (τ1 + ε) ( logq−1 ϕ (r) )µ} = o ( expp−1 { (τ −ε) ( logq−1 ϕ (r) )µ}) . by substituting (2.17) − (2.19) into (2.16) , we obtain (2.20) expp−1 { (τ − 2ε) ( logq−1 ϕ (r) )µ} ≤ o (log (rt (r,f))) , r /∈ e2,r → +∞. then by (2.20) , remark 1.2 and lemma 2.2, we have µ[p+1,q] (f,ϕ) ≥ µ[p,q] (a0,ϕ) . lemma 2.10 let p,q be integers such that p ≥ q ≥ 1 and let f be a meromorphic function with 0 < σ[p,q] (f,ϕ) < ∞, where ϕ (r) only satisfies lim r→+∞ logq−1 ϕ(αr) logq−1 ϕ(r) = 1 for some α > 1. then for every ε > 0, there exists a set e5 ⊂ (1,∞) of infinite logarithmic measure such that τ[p,q] (f,ϕ) = lim r→+∞ r∈e5 logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) . growth and complex oscillation of linear differential equations 189 proof. by the definition of the [p,q] − ϕ type, there exists a sequence {rn} ∞ n=1 tending to ∞ satisfying ( 1 + 1 n ) rn < rn+1, and τ[p,q] (f,ϕ) = lim rn→∞ logp−1 t (rn,f)( logq−1 ϕ (rn) )σ[p,q](f,ϕ) . then for any given ε > 0, there exists an n1 such that for n ≥ n1 and any r ∈[ rn, ( 1 + 1 n ) rn ] , we have logp−1 t (rn,f)( logq−1 ϕ (rn) )σ[p,q](f,ϕ) ( logq−1 ϕ (rn) logq−1 ϕ [( 1 + 1 n ) rn ])σ[p,q](f,ϕ) ≤ logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) . when q ≥ 1, we have logq−1 ϕ(rn) logq−1 ϕ[(1+ 1 n )rn] → 1, rn →∞. set e5 = ∞⋃ n=n1 [ rn, ( 1 + 1 n ) rn ] . then, we have lim r→+∞ r∈e5 logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) ≥ limrn→∞ logp−1 t (rn,f)(logq−1 ϕ (rn))σ[p,q](f,ϕ) = τ[p,q] (f,ϕ) and ∫ e5 dr r = ∞∑ n=n1 (1+ 1n )rn∫ rn dt t = ∞∑ n=n1 log ( 1 + 1 n ) = ∞. therefore, by the evident fact that lim r→+∞ r∈e5 logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) ≤ lim supr→+∞ r∈e5 logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) = τ[p,q] (f,ϕ) , we have τ[p,q] (f,ϕ) = lim r→+∞ r∈e5 logp−1 t (r,f)( logq−1 ϕ (r) )σ[p,q](f,ϕ) . the proof of the following two lemmas is essentially the same as in the corresponding results for the usual order and lower order. for details, see chapter 2 of the book by goldberg-ostrovskii [7] and chapter 1 of the book by yang-yi [18]. so, we omit the proofs. lemma 2.11 let p ≥ q ≥ 1 be integers, and let f and g be non-constant meromorphic functions of [p,q] −ϕ order. then we have ρ[p,q] (f + g,ϕ) ≤ max { ρ[p,q] (f,ϕ) ,ρ[p,q] (g,ϕ) } and ρ[p,q] (fg,ϕ) ≤ max { ρ[p,q] (f,ϕ) ,ρ[p,q] (g,ϕ) } . furthermore, if ρ[p,q] (f,ϕ) > ρ[p,q] (g,ϕ) , then we obtain ρ[p,q] (f + g,ϕ) = ρ[p,q] (fg,ϕ) = ρ[p,q] (f,ϕ) . 190 bouabdelli and belaïdi lemma 2.12 let p ≥ q ≥ 1 be integers, and let f and g be non-constant meromorphic functions with ρ[p,q] (f,ϕ) as [p,q]−ϕ order of f and µ[p,q] (g,ϕ) as lower [p,q] −ϕ order of g. then we have µ[p,q] (f + g,ϕ) ≤ max { ρ[p,q] (f,ϕ) ,µ[p,q] (g,ϕ) } and µ[p,q] (fg,ϕ) ≤ max { ρ[p,q] (f,ϕ) ,µ[p,q] (g,ϕ) } . furthermore, if µ[p,q] (g,ϕ) > ρ[p,q] (f,ϕ) , then we obtain µ[p,q] (f + g,ϕ) = µ[p,q] (fg,ϕ) = µ[p,q] (g,ϕ) . 3. proof of theorems proof of theorem 1.1 by lemma 1.1 and (2.10) , we have as in lemma 2.8 t (r,f) ≤ expp+1 {( σ[p,q] (a0,ϕ) + 3ε ) logq ϕ (r) } , for any ε > 0 and r /∈ e0,r → +∞, where e0 is a set of r of finite logarithmic measure. by lemma 2.2, we get σ[p+1,q] (f,ϕ) ≤ σ[p,q] (a0,ϕ) . set d = max{σ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) < σ[p,q] (a0,ϕ) ,j = 1, · · · ,n−1}. if σ[p,q] (aj,ϕ) < µ[p,q] (a0,ϕ) ≤ σ[p,q] (a0,ϕ) or σ[p,q] (aj,ϕ) ≤ µ[p,q] (a0,ϕ) < σ[p,q] (a0,ϕ) , then for any given ε ( 0 < 2ε < σ[p,q] (a0,ϕ) −d ) and sufficiently large r, we have (3.1) t (r,aj) ≤ expp { (d + ε) logq ϕ (r) } = expp−1 {( logq−1 ϕ (r) )d+ε} . set τ1 = max{τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) ,j 6= 0}. if σ[p,q] (aj,ϕ) = µ[p,q] (a0,ϕ) = σ[p,q] (a0,ϕ) , then we have τ1 < τ ≤ τ[p,q] (a0,ϕ) . therefore (3.2) t (r,aj) ≤ expp−1 { (τ1 + ε) ( logq−1 ϕ (r) )σ[p,q](a0,ϕ)} , holds for any r → +∞ and any given ε ( 0 < 2ε < τ[p,q] (a0,ϕ) − τ1 ) . by the definition of the [p,q] −ϕ type and lemma 2.10, and sufficiently large r ∈ e5, where e5 is a set of r of infinite logarithmic measure, we have (3.3) t (r,a0) > expp−1 {( τ[p,q] (a0,ϕ) −ε )( logq−1 ϕ (r) )σ[p,q](a0,ϕ)} . then by (2.16) and (3.1) − (3.3) , for all sufficiently large r, r ∈ e5�e2 and the above ε, we obtain (3.4) expp−1 {( τ[p,q] (a0,ϕ) − 2ε )( logq−1 ϕ (r) )σ[p,q](a0,ϕ)} ≤ o (log rt (r,f)) , where e2 is a set of r of finite linear measure. then, we have σ[p+1,q] (f,ϕ) ≥ σ[p,q] (a0,ϕ) . thus, we have σ[p+1,q] (f,ϕ) = σ[p,q] (a0,ϕ) . by lemmas 2.8 and 2.9, we have µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) . now we need to prove λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) and λ̄[p+1,q] (f −ψ,ϕ) = σ[p+1,q] (f,ϕ) . setting g = f − ψ, since σ[p+1,q] (ψ,ϕ) < µ[p,q] (a0,ϕ) , then by lemmas 2.11 and 2.12 we have σ[p+1,q] (g,ϕ) = σ[p+1,q] (f,ϕ) = σ[p,q] (a0,ϕ) , and µ[p+1,q] (g,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) , λ̄[p+1,q] (g,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) growth and complex oscillation of linear differential equations 191 and λ̄[p+1,q] (g,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) . by substituting f = g + ψ,f′ = g′ + ψ′, · · · ,f(n) = g(n) + ψ(n) into (1.1) , we get (3.5) g(n) + an−1g (n−1) + · · · + a0g = − [ ψ(n) + an−1ψ (n−1) + · · · + a0ψ ] . if f = ψ(n)+an−1ψ (n−1)+· · ·+a0ψ ≡ 0, then by lemma 2.9, we have µ[p+1,q] (ψ,ϕ) ≥ µ[p,q] (a0,ϕ) , which is a contradiction. hence f (z) 6≡ 0. since f (z) 6≡ 0 and σ[p+1,q] (f,ϕ) ≤ σ[p+1,q] (ψ,ϕ) < µ[p,q] (a0,ϕ) = µ[p+1,q] (f,ϕ) = µ[p+1,q] (g,ϕ) ≤ σ[p+1,q] (g,ϕ) = σ[p+1,q] (f,ϕ) , then by lemma 2.7 and (3.5) , we have λ̄[p+1,q] (g,ϕ) = λ[p+1,q] (g,ϕ) = σ[p+1,q] (g,ϕ) = σ[p,q] (a0,ϕ) , i.e., λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) = σ[p+1,q] (f,ϕ) = σ[p,q] (a0,ϕ) . by lemma 2.6 and (3.5) , we have λ̄[p+1,q] (g,ϕ) = µ[p+1,q] (g,ϕ) , i.e., λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) . therefore λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) ≤ σ[p,q] (a0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) . the proof of the theorem is complete. proof of theorem 1.2 by the first part of the proof of theorem 1.1, we can get σ[p+1,q] (f,ϕ) ≤ σ[p,q] (a0,ϕ) . by (3.6) lim sup r→+∞ n−1∑ j=1 m (r,aj) m (r,a0) < 1 we have for r → +∞ (3.7) n−1∑ j=1 m (r,aj) < δm (r,a0) , where δ ∈ (0, 1) . by λ[p,q] ( 1 a0 ,ϕ ) < µ[p,q] (a0,ϕ) , we have n (r,a0) = o (t (r,a0)) , r → +∞. by (2.15) and (3.7), for r → +∞,r /∈ e2, we obtain (3.8) t (r,a0) = m (r,a0) + n (r,a0) ≤ δt (r,a0) + o (log rt (r,f)) + o (t (r,a0)) , where e2 is a set of r of finite linear measure. by lemma 2.2 and (3.8) , we have σ[p+1,q] (f,ϕ) ≥ σ[p,q] (a0,ϕ) . then we have σ[p+1,q] (f,ϕ) = σ[p,q] (a0,ϕ) . by (3.8) and lemma 2.2, we have µ[p+1,q] (f,ϕ) ≥ µ[p,q] (a0,ϕ) . by lemma 2.8, we have µ[p+1,q] (f,ϕ) ≤ µ[p,q] (a0,ϕ) , then we get µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) . by using the similar proof of theorem 1.1, we can get λ̄[p+1,q] (f −ψ,ϕ) = µ[p+1,q] (f,ϕ) = µ[p,q] (a0,ϕ) ≤ σ[p,q] (a0,ϕ) = σ[p+1,q] (f,ϕ) = λ̄[p+1,q] (f −ψ,ϕ) = λ[p+1,q] (f −ψ,ϕ) . the proof of the theorem is complete. proof of theorem 1.3 suppose that f is rational solution of (1.1) . if f is either a rational function with a pole of multiplicity n ≥ 1 at z0 or a polynomial with degree 192 bouabdelli and belaïdi deg (f) ≥ s, then f(s) (z) 6≡ 0. if max{σ[p,q] (aj,ϕ) ,j 6= s} < µ[p,q] (as,ϕ) = µ, then we have µ[p,q] (0,ϕ) = µ[p,q] ( f(n) + an−1f (n−1) + · · · + a0f,ϕ ) = µ[p,q] (as,ϕ) = µ > 0, which is a contradiction. set τ1 = max{τ[p,q] (aj,ϕ) : σ[p,q] (aj,ϕ) = µ[p,q] (as,ϕ) ,j 6= s}. if σ[p,q] (aj,ϕ) = µ[p,q] (as,ϕ) , τ[p,q] (aj,ϕ) ≤ τ1 < τ, then we may choose constants δ1,δ2 such that τ1 < δ1 < δ2 < τ. for sufficiently large r, we have (3.9) m (r,aj) ≤ t (r,aj) ≤ expp−1 { δ1 ( logq−1 ϕ (r) )µ} . if σ[p,q] (aj,ϕ) < µ[p,q] (as,ϕ) , then for sufficiently large r and any given ε ( 0 < 2ε < µ[p,q] (as,ϕ) −σ[p,q] (aj,ϕ) ) , we obtain (3.10) m (r,aj) ≤ t (r,aj) ≤ expp {( σ[p,q] (aj,ϕ) + ε ) logq ϕ (r) } . under the assumption that λ[p,q] ( 1 as ,ϕ ) < µ[p,q] (as,ϕ) , for sufficiently large r, we have (3.11) n (r,as) = o (t (r,as)) . by the definition of the lower [p,q] −ϕ type, for sufficiently large r, we get (3.12) t (r,as) ≥ expp−1 { δ2 ( logq−1 ϕ (r) )µ} . by (1.1) , we have (3.13) t (r,as) ≤ n (r,as) + ∑ j 6=s m (r,aj) + o (log r) , for sufficiently large r. hence, by substituting (3.9) , (3.10) and (3.11) into (3.13) we have the contradiction. therefore, if f is a non-transcendental meromorphic solution, then it must be a polynomial with degree deg (f) ≤ s− 1. now, we assume that f is a transcendental meromorphic solution of (1.1) . by (1.1) , we have (3.14) −as = f f(s) [ f(n) f + · · · + as+1 f(s+1) f + as−1 f(s−1) f + · · · + a0 ] . noting that m ( r, f f(s) ) ≤ t (r,f) + t ( r, 1 f(s) ) = t (r,f) + t ( r,f(s) ) + o (1) ≤ t (r,f) + (s + 1) t (r,f) + o (t (r,f)) + o (1) (3.15) = (s + 2) t (r,f) + o (t (r,f)) + o (1) . by lemma 2.3, (3.14) and (3.15) , we obtain t (r,as) = m (r,as) + n (r,as) (3.16) ≤ n (r,as) + ∑ j 6=s m (r,aj) + (s + 3) t (r,f) + o (log (rt (r,f))) , growth and complex oscillation of linear differential equations 193 for sufficiently large r /∈ e2, where e2 is a set of r of finite linear measure. then by (3.9)−(3.12) , (3.16) and lemma 2.2, we can get µ[p,q] (f,ϕ) ≥ µ[p,q] (as,ϕ) and σ[p,q] (f,ϕ) ≥ σ[p,q] (as,ϕ) . by lemma 1.1 and (2.10) , we have (3.17) t (r,f) ≤ expp+1 {( σ[p,q] (as,ϕ) + 3ε ) logq ϕ (r) } , for any ε > 0, and r /∈ e0, r → +∞, where e0 is a set of r of linear logarithmic measure. then by (3.17) and lemma 2.2, we have σ[p+1,q] (f,ϕ) ≤ σ[p,q] (as,ϕ) . by lemma 2.8, we obtain µ[p+1,q] (f,ϕ) ≤ µ[p,q] (as,ϕ) . then we get σ[p+1,q] (f,ϕ) ≤ σ[p,q] (as,ϕ) ≤ σ[p,q] (f,ϕ) and µ[p+1,q] (f,ϕ) ≤ µ[p,q] (as,ϕ) ≤ µ[p,q] (f,ϕ) . the proof of the theorem is complete. acknowledgements. this paper is supported by university of mostaganem (umab) (cnepru project code b02220120024). references [1] s. bank, general theorem concerning the growth of solutions of first-order algebraic differential equations, compositio math. 25 (1972), 61–70. [2] b. beläıdi, growth of solutions of linear differential equations in the unit disc, bull. math. analy. appl., 3 (2011), no. 1, 14–26. [3] b. beläıdi, growth of solutions to linear differential equations with analytic coefficients of [p, q]-order in the unit disc, electron. j. differential equations 2011, no. 156, 1–11. [4] b. beläıdi, on the [p, q]-order of analytic solutions of linear differential equations in the unit disc, novi sad j. math. 42 (2012), no. 1, 117–129. [5] y. m. chiang and h. k. hayman, estimates on the growth of meromorphic solutions of linear differential equations, comment. math. helv. 79 (2004), no. 3, 451–470. [6] i. chyzhykov, j. heittokangas and j. rättyä, finiteness of ϕ−order of solutions of linear differential equations in the unit disc, j. anal. math. 109 (2009), 163–198. [7] a. goldberg and i. ostrovskii, value distribution of meromorphic functions, transl. math. monogr., vol. 236, amer. math. soc., providence ri, 2008. [8] g. g. gundersen, finite order solutions of second order linear differential equations, trans. amer. math. soc. 305 (1988), no. 1, 415-429. [9] w. k. hayman, meromorphic functions, oxford mathematical monographs clarendon press, oxford 1964. [10] h. hu and x. m. zheng, growth of solutions of linear differential equations with meromorphic coefficients of [p, q] −order, math. commun. 19(2014), 29-42. [11] o. p. juneja, g. p. kapoor and s. k. bajpai, on the [p, q]-order and lower [p, q]-order of an entire function, j. reine angew. math. 282 (1976), 53–67. [12] o. p. juneja, g. p. kapoor and s. k. bajpai, on the [p, q]-type and lower [p, q]-type of an entire function, j. reine angew. math. 290 (1977), 180–190. [13] l. kinnunen, linear differential equations with solutions of finite iterated order, southeast asian bull. math. 22 (1998), no. 4, 385–405. [14] i. laine, nevanlinna theory and complex differential equations, de gruyter studies in mathematics, 15. walter de gruyter & co., berlin-new york, 1993. [15] l. m. li and t. b. cao, solutions for linear differential equations with meromorphic coefficients of [p, q]-order in the plane, electron. j. differential equations 2012 (2012), no. 195, 1–15. [16] j. liu, j. tu and l. z. shi, linear differential equations with entire coefficients of [p, q]-order in the complex plane, j. math. anal. appl. 372 (2010), 55–67. [17] x. shen, j. tu and h. y. xu, complex oscillation of a second-order linear differential equation with entire coefficients of [p, q] − ϕ order, adv. difference equ. 2014 (2014), article id 200, 14 pages. [18] c. c. yang and h. x. yi, uniqueness theory of meromorphic functions, mathematics and its applications, 557. kluwer academic publishers group, dordrecht, 2003. 194 bouabdelli and belaïdi department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 42-51 http://www.etamaths.com oscillation criteria for second-order nonlinear functional dynamic equations with damping on time scales emi̇ne tuǧla and fatma serap topal∗ abstract. in this paper, we study oscillatory behavior of second-order dynamic equations with damping under some assumptions on time scales. new theorems extend and improve the results in the literature. illustrative examples are given. 1. introduction during the past decades, the questions regarding the study of oscillatory properties of differential equations with damping or distributed deviating arguments have become an important area of research due to the fact that such equations arise in many real life problems. in 1988, hilger introduced the theory of time scales in his ph.d. thesis [1] in order to unify continuous and discrete analysis; see also [4]. preliminaries about time scale calculus can be found in [2, 3] and omitted here. there has been much research achievement about the oscillation of dynamic equations on time scales in the last few years; see the papers [5-8, 10,11, 13-16, 18-20] and the references therein. in [9], chen et al. investigated the oscillation of a second-order nonlinear dynamic equation with positive and negative coefficients of the form (r(t)x∆(t))4 + p(t)f(x(ξ(t))) −q(t)f(x(δ(t))) = 0. in [17], s. enel concerned with the oscillatory behavior of all solutions of nonlinear second order damped dynamic equation (r(t)ψ(x∆(t)))∆ + p(t)ψ(x∆(t)) + q(t)f(xσ(t)) = 0. in [12], erbe et al. studied the oscillatory behavior of the solutions of the second order nonlinear functional dynamic equation (a(t)(x∆(t))γ)∆ + n∑ i=0 pi(t)φαi(x(gi(t))) = 0, on an arbitrary time scale t. in this study, we are concerned with the oscillation of solutions of second order dynamic equations with damping terms of the following form (r(t)g(x(t),x∆(t)))∆ +p(t)g(x(t),x∆(t)) +q1(t)f1(x(τ1(t))) +q2(t)f2(x(τ2(t))) = 0 (1.1) on a time scale t such that inf t = t0 and sup t = ∞. this paper is organized as follows. in this section we give some assumptions and lemmas that we need through our work. in section 2, we establish some new sufficient conditions for oscillation of (1.1). finally, in section 3, we present some examples to illustrate our results. now, we mention some definitions and lemmas from calculus on time scales which can be found in [2-3]. lemma 1.1. assume that g : t → r is strictly increasing and that t̃ := g(t) = {g(t) : t ∈ t} is a time scale. if f : t → r is an rd-continuous functions, g is differentiable with rd-continuous derivative, received 10th january, 2017; accepted 17th march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 4k11; 39a10; 39a99. key words and phrases. nonlinear dynamic equations; damping equations; oscillation solutions; time scales. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 42 oscillation for nonlinear functional dynamic equations 43 and a,b ∈ t, then ∫ b a f(t)g4(t)4t = ∫ g(b) g(a) (f ◦g−1)(s)4̃s, where g−1 is the inverse function of g and 4̃ denotes the derivative on t̃. lemma 1.2. every rd-continuous function has an antiderivative. in particular if t0 ∈ t, then f defined by f(t) := ∫ t t0 f(τ)4τ for t ∈ t is an antiderivative of f. lemma 1.3. assume that f : t → r is strictly increasing and that t̃ := f(t) = {f(t) : t ∈ t} is a time scale. let g : t̃ → r. if f4(t) and g4̃(f(t)) exist for t ∈ tκ, then (g ◦f)4 = (g4̃ ◦f)f4, where 4̃ denotes the derivative on t̃. definition 1.1. a function p : t → r is said to be regressive provided 1 + µ(t)p(t) 6= 0 for all t ∈ tκ, where µ(t) = σ(t) − t. the set of all regressive rd-continuous functions p : t → r is denoted by r. let p ∈r for all t ∈ t. the exponential function on t is defined by ep(t,s) = exp (∫ t s ζµ(r)(p(r))∆r ) where ζµ(s) is the cylinder transformation given by ζµ(r)(p(r)) := { 1 µ(r) log(1 + µ(r)p(r)), if µ(r) > 0; p(r), if µ(r) = 0. the exponential function y(t) = ep(t,s) is the solution to the initial value problem y ∆ = p(t)y, y(s) = 1. other properties of the exponential function are given in the following lemma [3, theorem 1.39]. lemma 1.4. let p,q ∈r. then i. e0(t,s) ≡ 1 and ep(t,t) ≡ 1; ii. ep(σ(t),s) = (1 + µ(t)p(t))ep(t,s); iii. 1 ep(t,s) = e (t,s) where, p(t) = − p(t) 1+µ(t)p(t) ; iv. ep(t,s) = 1 ep(s,t) = e p(s,t); v. ep(t,s)ep(s,r) = ep(t,r); vi. ep(t,s)eq(t,s) = ep⊕q(t,s); vii. ep(t,s) eq(t,s) = ep q(t,s); viii. ( 1 ep(.,s) )∆ = − p(t) eσp (.,s) . throughout this paper we assume that the followings: (c1) t0 ∈ t and [t0,∞)t = {t ∈ t : t ≥ t0}, (c2) r ∈ crd([t0,∞)t, (0,∞)) and ∫∞ t0 1 r(t) ∆t = ∞, (c3) p,q1,q2 ∈ crd([t0,∞)t, [0,∞)) (c4) τ1,τ2 ∈ crd(t,t), lim t→∞ τ1(t) = lim t→∞ τ2(t) = ∞, τ2 has inverse function τ−12 ∈ crd(t,t), v := τ−12 ◦τ1 ∈ crd(t,t), τ ∆ 1 ,v ∆ ∈ crd([t0,∞)t, (0,∞)), τ1(t),v(t) ≤ t for t ∈ [t0,∞)t, τ1([t0,∞)t) = [τ1(t0),∞)t, v([t0,∞)t) = [v(t0),∞)t, where τ1([t0,∞)t) = {τ1(t) : t ∈ [t0,∞)t} and v([t0,∞)t) = {v(t) : t ∈ [t0,∞)t}, (c5) f1,f2 ∈ c(r,r), there exist positive constants l1,l2,m such that f1(u) u ≥ l1, 0 < f2(u) u ≤ l2 and | f2(u) |≤ m for u 6= 0 and q1(t) e−p r (σ(t),t0) l1 −q2(v(t))l2v∆(t) > 0 for t ∈ [t0,∞)t, (c6) g ∈ c(r × r,r), there exist positive constants l3 such that g(u,v) v ≤ l3 and vg(u,v) > 44 tuǧla and topal 0 for v 6= 0, (c7) ∫ ∞ t [ 1 r(s) ∫ s v(s) q2(u)∆u ] ∆s < ∞ for every sufficiently large t ∈ t, (c8) − p(t) r(t) is positively regressive, which means 1 −µ(t)p(t) r(t) > 0. the following lemma has an important role to prove our main results. lemma 1.5. assume that (c1)−(c7) hold. furthermore, suppose that x is a positive solution of (1.1) on [t0,∞)t, then ( r(t)g(x(t),x∆(t)) e−p r (t,t0) )∆ < 0 and x∆(t) > 0 on [t0,∞)t. proof. easily we get( r(t)g(x(t),x∆(t)) e−p r (t,t0) )∆ = (r(t)g(x(t),x∆(t)))∆e−p r (t,t0) − (e−pr (t,t0)) ∆r(t)g(x(t),x∆(t)) e−p r (t,t0)e−pr (σ(t),t0) = (r(t)g(x(t),x∆(t)))∆ + p(t)g(x(t),x∆(t)) e−p r (σ(t),t0) = −q1(t)f1(x(τ1(t))) −q2(t)f2(x(τ2(t))) e−p r (σ(t),t0) < 0. this implies that r(t)g(x(t),x∆(t)) e−p r (t,t0) is decreasing. we claim that x∆(t) > 0 on [t0,∞)t. if not, then there is t ≥ t1 such that r(t)g(x(t),x∆(t)) e−p r (t,t0) ≤ r(t1)g(x(t1),x ∆(t1)) e−p r (t1,t0) := a < 0. from (c6), we get x∆(t) ≤ a l3 e−p r (t,t0) r(t) . integrating from t1 to t and using decreasing of e−p r (.,t0), we have x(t) −x(t1) ≤ ae−p r (t1,t0) l3 ∫ t t1 1 r(s) ∆s. so x(t) ≤−∞. this implies that x(t) is eventually negative which is a contradiction. hence, x∆(t) > 0 on [t0,∞)t. � 2. main results in this section, we’ll obtain some new oscillation criteria of second-order dynamic equation (1.1) with damping by using the generalized riccati transformation and the inequality technique. theorem 2.1. assume that (c1) − (c8) hold. furthermore, suppose that there exists a positive function α ∈ crd([t0,∞)t,r) such that for every sufficiently large t, lim sup t→∞ ∫ t t ([ q1(s)l1 e−p r (σ(s),t0) −q2(v(s))l2v∆(s) ] α(s) − (α∆+ (−s))2(r(τ1(s)))l3 4α(s)τ∆1 (s) ) ∆s = ∞, (2.1) where α∆+ (s) = max{α∆(s), 0}. then every solution of (1.1) is oscillatory. proof. assume that x is a nonoscillatory solution of (1.1). without loss of generality, we may assume x is an eventually positive solution of (1.1). that is, there exists t1 ∈ t for t ≥ t1 and x(t) > 0. we defined the function z by z(t) = ∫ t t1 g(x(s),x∆(s)) e−p r (s,t0) ∆s + ∫ ∞ t 1 r(s) ∫ s v(s) q2(u)f2(x(τ2(u)))∆u∆s ≥ ∫ t t1 g(x(s),x∆(s)) e−p r (s,t0) ∆s ≥ 0. oscillation for nonlinear functional dynamic equations 45 thus, we get z∆(t) = (∫ t t1 g(x(s),x∆(s)) e−p r (s,t0) ∆s )∆ + (∫ ∞ t 1 r(s) ∫ s v(s) q2(u)f2(x(τ2(u)))∆u∆s )∆ = g(x(t),x∆(t)) e−p r (t,t0) − 1 r(t) ∫ t v(t) q2(u)f2(x(τ2(u)))∆u and r(t)z∆(t) = r(t)g(x(t),x∆(t)) e−p r (t,t0) − ∫ t v(t) q2(u)f2(x(τ2(u)))∆u = r(t)g(x(t),x∆(t)) e−p r (t,t0) − ∫ t v(t1) q2(u)f2(x(τ2(u)))∆u + ∫ v(t) v(t1) q2(u)f2(x(τ2(u)))∆u. making substitution s = v(u), we have∫ t t1 q2(v(u))f2(x(τ1(u)))v ∆(u)∆u = ∫ v(t) v(t1) q2(s)f2(x(τ1(v −1(s))))∆̃s = ∫ v(t) v(t1) q2(s)f2(x(τ2(s)))∆̃s for t ∈ [t1,∞)t. (2.2) according to condition v([t0,∞)t) = [v(t0),∞)t in (c4), we get that the derivative ∆ on t is equal to the derivative 4̃ on t̃ := v([t0,∞)t) in (2.2). hence, we conclude∫ t t1 q2(v(u))f2(x(τ1(u)))v ∆(u)∆u = ∫ v(t) v(t1) q2(s)f2(x(τ2(s)))∆s for t ∈ [t1,∞)t. thus, for t ∈ [t1,∞)t, we get r(t)z∆(t) = r(t)g(x(t),x∆(t)) e−p r (t,t0) − ∫ t v(t1) q2(u)f2(x(τ2(u)))∆u + ∫ t t1 q2(v(u))f2(x(τ1(u)))v ∆(u)∆u (r(t)(z∆(t))∆ = ( r(t)g(x(t),x∆(t)) e−p r (t,t0) )∆ −q2(t)f2(x(τ2(t))) + q2(v(t))f2(x(τ1(t)))v∆(t) = (r(t)g(x(t),x∆(t)))∆e−p r (t,t0) − (e−pr (t,t0)) ∆r(t)g(x(t),x∆(t)) e−p r (t,t0)e−pr (σ(t),t0) −q2(t)f2(x(τ2(t))) +q2(v(t))f2(x(τ1(t)))v ∆(t) = (r(t)g(x(t),x∆(t)))∆ + p(t)g(x(t),x∆(t)) e−p r (σ(t),t0) −q2(t)f2(x(τ2(t))) +q2(v(t))f2(x(τ1(t)))v ∆(t) = −q1(t)f1(x(τ1(t))) −q2(t)f2(x(τ2(t))) e−p r (σ(t),t0) −q2(t)f2(x(τ2(t))) +q2(v(t))f2(x(τ1(t)))v ∆(t) ≤ −q1(t)l1x(τ1(t)) e−p r (σ(t),t0) + q2(v(t))l2x(τ1(t))v ∆(t) = − [ q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v∆(t) ] x(τ1(t)) = −q(t)x(τ1(t)) < 0, where q(t) = q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v∆(t). 46 tuǧla and topal thus, there exists t2 ∈ [t1,∞)t such that r(t)z∆(t) strictly decreasing on [t2,∞)t and either eventually positive or eventually negative. since r(t) > 0 for t ∈ [t0,∞)t, z∆(t) is also either eventually positive or eventually negative. we claim z∆(t) > 0 for t ∈ [t2,∞)t. (2.3) assume that (2.3) does not hold, then there exists tξ ∈ [t2,∞)t such that z∆(tξ) < 0. since r(t)z∆(t) is strictly decreasing on [t2,∞)t, it is clear that r(t)z∆(t) ≤ r(tξ)z∆(tξ) = −c < 0 for t ∈ [tξ,∞)t. thus, we obtain z∆(t) ≤ −c 1r(t) for t ∈ [tξ,∞)t. by integrating both sides of the last inequality from tξ to t, we get z(t) −z(tξ) ≤−c ∫ t tξ 1 r(s) ∆(s) for t ∈ [tξ,∞)t. noticing (c2), we have lim t→∞ z(t) = −∞. this contradicts z(t) ≥ 0. therefore, (2.3) holds. thus, we have z∆(t) > 0 on [t2,∞)t. define the function w by generalized riccati substitution w(t) := α(t) r(t)z∆(t) x(τ1(t)) . there exist t3 ∈ [t2,∞)t such that w(t) > 0 for t ∈ [t3,∞)t. easily, we get w∆ = (rz∆)∆ α x◦ τ1 + (rz∆)σ ( α x◦ τ1 )∆ = (rz∆)∆ α x◦ τ1 + (rz∆)σ ( α∆ (x◦ τ1)σ − (x◦ τ1)∆α x◦ τ1(x◦ τ1)σ ) ≤ (rz∆)∆ α x◦ τ1 + α∆+ (rz∆)σ (x◦ τ1)σ −α (rz∆)σ (x◦ τ1)σ (x◦ τ1)∆ x◦ τ1 = (rz∆)∆ α x◦ τ1 + α∆+ wσ ασ −α wσ ασ (x◦ τ1)∆ x◦ τ1 , where α∆+ (s) = max{α∆(s), 0}. thus, we have w∆ ≤−qα + α∆+ wσ ασ −α wσ ασ (x◦ τ1)∆ x◦ τ1 . from the chain rule, we know that (x◦ τ1)∆ = (x∆̃ ◦ τ1)τ∆1 . according to condition τ1([t0,∞)t) = [τ1(t0),∞)t in (c4), we get that the derivative ∆ on t is equal to the derivative 4̃ on t̃ := τ1([t0,∞)t). so, we have w∆ ≤−qα + α∆+ wσ ασ −α wσ ασ (x∆ ◦ τ1)τ∆1 x◦ τ1 . also z∆(t) = g(x(t),x∆(t)) − 1 r(t) ∫ t v(t) q2(u)f2(x(τ2(u)))∆u ≤ g(x(t),x∆(t)) ≤ l3x∆(t), implies that −x∆(t) ≤− z∆(t) l3 and so w∆ ≤−qα + α∆+ wσ ασ −α wσ ασ (z∆ ◦ τ1) x◦ τ1 τ∆1 l3 . oscillation for nonlinear functional dynamic equations 47 since τ1(t) ≤ t ≤ σ(t) and r(t)z∆(t) is strictly decreasing on [t2,∞)t, we get (r ◦ τ1)(z∆ ◦ τ1) ≥ (rz∆)σ and (z∆ ◦ τ1) ≥ (rz∆)σ (r ◦ τ1) . thus, we get w∆ ≤ −qα + α∆+ wσ ασ − α l3 wσ ασ (rz∆)σ (r ◦ τ1) τ∆1 x◦ τ1 = −qα + α∆+ wσ ασ − α l3 ( wσ ασ )2 (x◦ τ1)σ x◦ τ1 τ∆1 r ◦ τ1 . (2.4) from (c4) we see that τ1(t) is strictly increasing on [t0,∞)t. since t ≤ σ(t), we have τ1(t) ≤ τσ1 (t). since x∆(t) > 0, we get x ◦ τ1(t) ≤ x ◦ τσ1 (t). hence, from (2.4) there exist a sufficiently large t4 ∈ [t3,∞)t such that w∆ ≤ −qα + α∆+ wσ ασ − α l3 ( wσ ασ )2 τ∆1 r ◦ τ1 = −qα− [ α∆+ 2 √ (r ◦ τ1)l3 ατ∆1 − wσ ασ √ ατ∆1 (r ◦ τ1)l3 ]2 + (α∆+ ) 2(r ◦ τ1)l3 4ατ∆1 ≤ −qα + (α∆+ ) 2(r ◦ τ1)l3 4ατ∆1 . (2.5) integrating both sides of the last inequality from t4 to t, we get w(t) −w(t4) ≤− ∫ t t4 [ (q(s)α(s)) − (α∆+ (s)) 2(r(τ1(s)))l3 4α(s)τ∆1 (s) ] ∆s since w(t) > 0 for t ∈ [t3,∞)t we have∫ t t4 [ (q(s)α(s)) − (α∆+ (s)) 2(r(τ1(s)))l3 4α(s)τ∆1 (s) ] ∆s ≤ w(t4) −w(t) ≤ w(t4) and lim sup t→∞ ∫ t t4 [ (q(s)α(s)) − (α∆+ (s)) 2(r(τ1(s)))l3 4α(s)τ∆1 (s) ] ∆s ≤ w(t4) < ∞, which is a contradiction to (2.1). the proof is completed. � theorem 2.2. assume that (c1) − (c8) hold. let h be an rd-continuous function defined as follows: h : dt ≡{(t,s) : t ≥ s ≥ t0, t,s ∈ [t0,∞)t}→ r, such that h(t,t) = 0, for t ≥ t0, h(t,s) > 0, for t > s ≥ t0, and h has a nonpositive rd-continuous delta partial derivative h∆s with respect to the second variable and lim sup t→∞ 1 h(t,t) ∫ t t h(t,s) ( q(s)α(s) − (α∆+ (−s))2(r(τ1(s)))l3 4α(s)τ∆1 (s) ) ∆s = ∞ (2.6) for every sufficiently large t , where q(t) = q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v∆(t) and α∆+ (s) = max{α∆(s), 0}, then all solutions of (1.1) are oscillatory. 48 tuǧla and topal proof. assume that x is a nonoscillatory solution of (1.1). without loss of generality, we may assume x is an eventually positive solution of (1.1). proceeding as in the proof of the theorem 2.1, we get w∆(t) ≤−q(t)α(t) + (α∆+ ) 2(t)(r ◦ τ1(t))l3 4α(t)τ∆1 (t) . multiplying by h(t,s) and then integrating from t4 to t, we obtain∫ t t4 h(t,s)w∆(s)4s ≤ ∫ t t4 h(t,s) ( −q(s)α(s) + (α∆+ ) 2(s)(r ◦ τ1(s))l3 4α(s)τ∆1 (s) ) ∆s. since ∫ t t4 h(t,s)w∆(s)∆s = h(t,s)w(s) |s=ts=t4 − ∫ t t4 h∆s (t,s)w σ(s)∆s, we get −h(t,t4)w(t4) ≤ ∫ t t4 h(t,s) ( −q(s)α(s) + (α∆+ ) 2(s)(r ◦ τ1(s))l3 4α(s)τ∆1 (s) ) ∆s. thus, we have∫ t t4 h(t,s) ( q(s)α(s) − (α∆+ ) 2(s)(r ◦ τ1(s))l3 4α(s)τ∆1 (s) ) ∆s ≤ h(t,t4)w(t4) and so 1 h(t,t4) ∫ t t4 h(t,s) ( q(s)α(s) − (α∆+ ) 2(s)(r ◦ τ1(s))l3 4α(s)τ∆1 (s) ) ∆s ≤ w(t4) < ∞, which contradicts with (2.6). this completes the proof. � theorem 2.3. assume that (c1)−(c8) hold. let h be an rd-continuous function defined as follows: h : dt ≡{(t,s) : t ≥ s ≥ t0, t,s ∈ [t0,∞)t}→ r, such that h(t,t) = 0, for t ≥ t0, h(t,s) > 0, for t > s ≥ t0, and h has an rd-continuous ∆−partial derivative h∆s on dt with respect to the second variable. let h : dt → r be an rd-continuous function satisfying h∆s (t,s) + h(t,s) α∆+ (s) ασ(s) = h(t,s) ασ(s) √ h(t,s), (t,s) ∈ dt and lim sup t→∞ 1 h(t,t) ∫ t t ( h(t,s)q(s)α(s) − [h(t,s)]2(r ◦ τ1)(s)l3 4α(s)τ∆1 (s) ) ∆s = ∞ (2.7) for every sufficiently large t , where q(t) = q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v∆(t) and α∆+ (s) = max{α∆(s), 0}, then all the solutions of (1.1) are oscillatory. proof. assume that x is a nonoscillatory solution of (1.1). without loss of generality, we may assume x is an eventually positive solution of (1.1). proceeding as in the proof of the theorem 2.1, we have (2.5). multiplying (2.5) by h(t,s) and then integrating from t4 to t, we obtain∫ t t4 h(t,s)q(s)α(s)∆s ≤ − ∫ t t4 h(t,s)w∆(s)∆s + ∫ t t4 h(t,s) α∆+ (s) ασ(s) wσ(s)∆s − ∫ t t4 h(t,s) α(s)τ∆1 (s) (ασ(s))2(r ◦ τ1)(s)l3 [wσ(s)]2∆s. thus, using ∫ t t4 h(t,s)w∆(s)∆s = [h(t,s)w(s)]s=ts=t4 − ∫ t t4 h∆s (t,s)w σ(s)∆s, oscillation for nonlinear functional dynamic equations 49 we have ∫ t t4 h(t,s)q(s)α(s)∆s ≤ h(t,t4)w(t4) + ∫ t t4 ([ h∆s (t,s) + h(t,s) α∆+ (s) ασ(s) ] wσ(s) −h(t,s) α(s)τ∆1 (s) (ασ(s))2(r ◦ τ1)(s)l3 [wσ(s)]2 ) ∆s ≤ h(t,t4)w(t4) + ∫ t t4 ( h(t,s) ασ(s) √ h(t,s)wσ(s) −h(t,s) α(s)τ∆1 (s) (ασ(s))2(r ◦ τ1)(s)l3 [wσ(s)]2 ) ∆s = h(t,t4)w(t4) + ∫ t t4 ( [h(t,s)]2(r ◦ τ1)(s)l3 4α(s)τ∆1 (s) − [ h(t,s) 2ασ(s) √ (ασ(s))2(r ◦ τ1)(s)l3 α(s)τ∆1 (s) − √ h(t,s) α(s)τ∆1 (s) (ασ(s))2(r ◦ τ1)(s)l3 wσ(s) ]2 ∆s ≤ h(t,t4)w(t4) + ∫ t t4 [h(t,s)]2(r ◦ τ1)(s)l3 4α(s)τ∆1 (s) ∆s. so, we get 1 h(t,t4) ∫ t t4 ( h(t,s)q(s)α(s) − [h(t,s)]2(r ◦ τ1)(s)l3 4α(s)τ∆1 (s) ) ∆s ≤ w(t4) < ∞, which is a contradiction to (2.7). the proof is completed. � 3. examples example 3.1. let t = r. consider the equation ( 1 t x′(t) 2 + sin2(x(t)) )′ + t 1 3 x′(t) 2 + sin2(x(t)) + 1 t2 x(t 1 5 − 3)(x(t 1 5 − 3)2 + 4) + 10 t21 x(t2 − 3) x2(t2 − 3) + 1 = 0, (3.1) for t ≥ t0 := 4 here r(t) = 1 t ,p(t) = t 1 3 ,q1(t) = 1 t2 ,q2(t) = 10 t21 ,τ1(t) = t 1 5 − 3,τ2(t) = t2 − 3, g(x(t),x′(t)) = x′(t) 2+sin2(x(t)) , f1(u) = u(u 2 + 4) and f2(u) = u u2+1 , f1(u) u = u2 + 4 ≥ 4 := l1 and f2(u) u = 1 u2+1 ≤ 1 := l2 for u 6= 0, | f2(u) |≤ 12 := m, g(x(t),x′(t)) x′(t) ≤ 1 2 := l3. thus, we obtain ∫∞ t0 1 r(t) dt = ∫∞ t0 tdt = ∞,v(t) = t 1 10 < t for t ∈ [4,∞) and 1 − µ(t)p(t) r(t) = 1 > 0 for t ∈ [4,∞). also, we get q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v′(t) = 4e(t−4) 4 3 t2 − 1 t3 = 4te(t−4) 4 3 − 1 t3 > 0 for t ∈ [4,∞) and∫ ∞ t [ 1 r(s) ∫ s v(s) q2(u)du ] ds = ∫ ∞ t [ s ∫ s s 1 10 10 u21 du ] ds = ∫ ∞ t s17 − 1 2s19 ds < ∞ for t ∈ [4,∞). hence, we have∫ ∞ t ([ q1(s)l1 e−p r (σ(s),t0) −q2(v(s))l2v′(s) ] α(s) − (α′+(s)) 2(r(τ1(s)))l3 4α(s)τ∆1 (s) ) ds = ∫ ∞ t ( 4se(s−4) 4 3 − 1 s3 s3 − 3s2 1 s 1 5 −3 1 2 4s3 1 5s 4 5 ) ds 50 tuǧla and topal = ∫ ∞ t ( 4se(s−4) 4 3 − 1 − 15 8(s 2 5 − 3s 1 5 ) ) ds = ∫ ∞ t ( 32(s 7 5 − 3s 6 5 )e(s−4) 4 3 − 8(s 2 5 − 3s 1 5 ) − 15 8(s 2 5 − 3s 1 5 ) ) ds = ∞. therefore, according to theorem 2.1, every solution of (3.1) is oscillatory on [4,∞). example 3.2. let t = 2n0 . consider the equation( tx∆(t) )∆ + 1 t2 x∆(t) + t + 1 2 x( t 2 )(x2( t 2 ) + 2) + 1 2t2 x(2t) 2 + x2(2t) = 0, (3.2) for t ∈ [t0,∞)t, t ≥ t0 := 2 here, r(t) = t, p(t) = 1 t2 , q1(t) = t+1 2 , q2(t) = 1 2t2 , τ1(t) = t 2 < t, τ2(t) = 2t, g(x(t),x ∆(t)) = x∆(t), f1(u) = u(u 2 + 2) and f2(u) = u u2+2 , f1(u) u = u2 + 2 ≥ 2 := l1 and f2(u) u = 1 u2+2 ≤ 1 2 := l2 for u 6= 0, | f2(u) |≤ 12 := m, g(x(t),x∆(t)) x∆(t) = 1, l3 = 1. therefore, we obtain ∫∞ t0 1 r(t) ∆t = ∫∞ 2 1 t ∆t = ∞, v(t) = t 4 < t,v∆(t) = 1 4 for t ∈ [2,∞)t and 1 −µ(t)p(t) r(t) = 1 − t 1 t3 = 1 − 1 t2 > 0 for t ∈ [2,∞)t. also, we get q1(t)l1 e−p r (σ(t),t0) −q2(v(t))l2v∆(t) > q1(t)l1 −q2(v(t))l2v∆(t) = t3 + t2 − 1 t2 > 0 for t ∈ [2,∞)t and∫ ∞ t [ 1 r(s) ∫ s v(s) q2(u)∆u ] ∆s = ∫ ∞ t [ 1 s ∫ s s 4 1 2u2 ∆u ] ∆s = ∫ ∞ t [ 1 s ( −1 u )∣∣∣∣s s 4 ∆u ] ∆s = ∫ ∞ t 3 s2 ∆s < ∞. hence we have∫ ∞ t ([ q1(s)l1 e−p r (σ(s),t0) −q2(v(s))l2v∆(s) ] α(s) − (α∆+ (s)) 2(r(τ1(s)))l3 4α(s)τ∆1 (s) ) ∆s = ∫ ∞ t ([ s3 + s2 − 1 s2 ] s− s 2 4s1 2 ) ∆s = ∫ ∞ t ( s3 + s2 − 1 s − 1 4 ) ∆s = ∞. thus, according to theorem 2.1, every solution of (3.2) is oscillatory on [2,∞)t. references [1] s. hilger, analysis on measure chains-a unified approach to continuous and discrete calculus, results math. 18 (1990) 18-56. [2] m. bohner, a. peterson, dynamic equations on time scales, an introduction with applications, birkhäuser, boston, 2001. [3] m. bohner and a. peterson (eds), advances in dynamic equations on time scales, birkhäuser, boston, 2003. [4] r.p. agarwal, m. bohner, d. o’regan, a. peterson, dynamic equations on time scales: a survey, j. comput. and appl. math. 141 (1-2)(2002) 1-26. [5] r.p. agarwal, m. bohner, s.h. saker, oscillation of second order delay dynamic equations, can. appl. math. q. 13 (2005) 1-18. [6] h.a. agwa, a.m.m. khodier, h.a. hassan, oscillation of second-order nonlinear delay dynamic equations on time scales, int. j. differ. equ. 2011 (2011) article id 863801. [7] m. bohner, some oscillation criteria for first order delay dynamic equations, far east j. appl. math. 18 (3)(2005) 289-304. [8] m. bohner, s. h. saker, oscillation of second order nonlinear dynamic equations on time scales, rocky mountain j. math. 34 (4) (2004) 1239-1254. [9] d.x. chen, p.x. qu, y.h. lan, oscillation of second-order nonlinear dynamic equations with positive and negative coefficients, adv. differ. equ. 2013 (2013) art. id 168. [10] l. erbe, a. peterson, s.h. saker, oscillation criteria for second-order nonlinear dynamic equations on time scales, j. lond. math. soc. 67 (3) (2003) 701-714. [11] l. erbe, a. peterson, s.h. saker, oscillation criteria for second-order nonlinear delay dynamic equations, j. math. anal. appl. 333 (1) (2007) 505-522. oscillation for nonlinear functional dynamic equations 51 [12] l. erbe, t. hassan, a. peterson, oscillation of second order functional dynamic equations, int. j. differ. equ. 5 (2) (2010) 175-193. [13] t.s. hassan, oscillation criteria for half-linear dynamic equations on time scales, j. math. anal. appl. 345 (1) (2008) 176-185. [14] s.h. saker, oscillation criteria of second-order half-linear dynamic equations on time scales, j. comput. appl. math. 177 (2) (2005) 375-387. [15] s. sun, z. han, c. zhang, oscillation of second-order delay dynamic equations on time scales, j. appl. math. comput. 30 (1-2) (2009) 459-468. [16] y. s. ahiner, oscillation of second-order delay differential equations on time scales, nonlinear anal., theory, methods appl. 63 (5-7) (2005) 1073-1080. [17] m.t. s. enel, oscillation theorems for the second order damped nonlinear dynamic equation on time scales, contemp. anal. appl. math. 1 (2) (2013) 167-180. [18] q. zhang, oscillation of second-order half-linear delay dynamic equations with damping on time scales, j. comput. appl. math. 235 (5) (2011) 1180-1188. [19] q. zhang, l. gao, oscillation of second-order nonlinear delay dynamic equations with damping on time scales, j. appl. math. comput. 37 (1-2) (2011) 145-158. [20] q. zhang, l. gao, l. wang, oscillation of second-order nonlinear delay dynamic equations on time scales, comput. math. appl. 61 (8) (2011) 2342-2348. department of mathematics, ege university,35100 bornova, izmir-turkey ∗corresponding author: f.serap.topal@ege.edu.tr 1. introduction 2. main results 3. examples references international journal of analysis and applications volume 17, number 6 (2019), 940-957 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-940 nonlinear coupled fractional order systems with integro-multistrip-multipoint boundary conditions bashir ahmad1,∗, ahmed alsaedi2, sotiris k. ntouyas1,2 1nonlinear analysis and applied mathematics (naam)-research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia 2department of mathematics, university of ioannina, 451 10 ioannina, greece ∗corresponding author: bashirahmad qau@yahoo.com abstract. we study the existence and uniqueness of solutions for a nonlinear system of coupled fractional differential equations equipped with nonlocal coupled integro-multistrip-multipoint boundary conditions. our results are new in the sense that the given boundary conditions connect the values of the known functions over the given domain with the ones described on different sub-segments and different nonlocal positions within the given domain. we make use of banach contraction mapping principle, leray-schauder alternative and krasnoselskii fixed point theorem to prove the desired results for the problem at hand. an example illustrating the existence and uniqueness result is also presented. 1. introduction fractional calculus is found to be more practical and effective than the classical calculus in the mathematical modeling of several real world phenomena. the topic of fractional differential equations has evolved as an important and significant area of investigation in view of its numerous applications in viscoelasticity, electroanalytical chemistry, and many physical problems [1][4]. in recent years, many works have been devoted to the study of the mathematical aspects of fractional order differential equations. many advanced received 2019-03-22; accepted 2019-04-10; published 2019-11-01. 2010 mathematics subject classification. 26a33, 34b15. key words and phrases. boundary value problem; fractional derivative; fractional integral; fixed point theorem. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 940 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-940 int. j. anal. appl. 17 (6) (2019) 941 and efficient methods have been applied to develop the existence theory for fractional differential equations. one of the powerful tools for developing the existence crireria for solutions to such equations is based on the fixed point theory. many authors applied fixed point theorems to establish the existence theory for nonlinear fractional differential equations; for example, see [5][13] and the references cited therein. on the other hand, the coupled systems of nonlinear fractional differential equations also received considerable attention. such systems appear in various disciplines of applied nature, for instance, see [14, 15]. the tools of the fixed point theory also played a key role in developing the existence theory for the coupled systems of fractional differential equations [18][20]. in [21], the authors investigated a coupled system of nonlinear fractional differential equations:  cdαx(t) = f(t,x(t),y(t),dγy(t)), t ∈ [0,t], 1 < α ≤ 2, 0 < γ < 1, cdβy(t) = g(t,x(t),dδx(t),y(t)), t ∈ [0,t], 1 < β ≤ 2, 0 < δ < 1, equipped with coupled nonlocal and integral boundary conditions of the form:  x(0) = h(y), ∫ t 0 y(s)ds = µ1x(η), y(0) = φ(x), ∫ t 0 x(s)ds = µ2y(ξ), η,ξ ∈ (0,t), where cdi denote the caputo fractional derivatives of order i, i = α,β,γ,δ, f,g : [0,t] × r × r × r → r, h,φ : c([0,t],r) → r are given continuous functions, and µ1,µ2 are real constants. in [22], the existence of solutions for the following boundary value problem of coupled system of nonlinear fractional differential equations was discussed:  cdqx(t) = f(t,x(t),y(t),c dσ1y(t)), 1 < q ≤ 2, 0 < σ1 < 1, t ∈ [0, 1], cdpy(t) = g(t,x(t),c dσ2x(t),y(t)), 1 < p ≤ 2, 0 < σ2 < 1, t ∈ [0, 1],   x(0) = ψ1(y), x(1) = a1 ∫ ξ 0 y(s)ds + b1 m−2∑ i=1 αiy(ηi), y(0) = ψ2(x), y(1) = a2 ∫ ξ 0 x(s)ds + b2 m−2∑ i=1 βix(ηi), 0 < ξ < η1 < η2 < · · · < ηm−2 < 1, m ≥ 3, where cdj (j = p,q,σ1,σ2) denote the caputo fractional derivative of order j, f,g : [0, 1] × r × r × r → r, ψ1,ψ2 : c([0, 1],r) → r are given appropriate functions, a1,a2,b1 and b2 are real constants and αi, βi, i = 1, 2, . . . ,m− 2, are positive real constants. in this paper, we are concerned with existence of solutions for a nonlinear system of coupled fractional differential equations: dσx(t) = f(t,x(t),y(t)), n− 1 < σ < n, n ≥ 3, t ∈ j := [0, 1], dφx(t) = g(t,x(t),y(t)), m− 1 < φ < m, m ≥ 3, t ∈ j := [0, 1], (1.1) int. j. anal. appl. 17 (6) (2019) 942 subject to integro-multistrip-multipoint boundary conditions: x( î )(0) = 0, î = 0, 1, 2, . . . ,n− 2,∫ 1 0 x(s)ds = p∑ i=2 βi−1 ∫ ηi ηi−1 y(s)ds + q∑ j=1 γj y(ρj), βi−1 > 0,γj > 0, y( ĵ )(0) = 0, ĵ = 0, 1, 2, . . . ,m− 2,∫ 1 0 y(s)ds = µ∑ i=2 β′i−1 ∫ θi θi−1 x(s)ds + λ∑ j=1 γ′j x(ζj), β ′ i−1 > 0,γ ′ j > 0, (1.2) where dσ,dφ are the standard riemann-liouville fractional derivatives of order σ and φ respectively, f,g : j × r × r → r are continuous functions and 0 < η1 < η2 < ... < ηp < ρ1 < ρ2 < ... < ρq < 1, 0 < θ1 < θ2 < ... < θµ < ζ1 < ζ2 < ... < ζλ < 1 with p,q,µ,λ ∈ n. the rest of the paper is organized as follows. in section 2, we recall some basic definitions of fractional calculus and present an auxiliary lemma, which plays a pivotal role in obtaining the main results presented in section 3. we also discuss an example for illustration of the existence-uniqueness result. 2. preliminaries first of all, we recall some basic definitions of fractional calculus [2]. definition 2.1. the fractional integral of order r with the lower limit zero for a function f is defined as irf(t) = 1 γ(r) ∫ t 0 f(s) (t−s)1−r ds, t > 0, r > 0, provided the right hand-side is point-wise defined on [0,∞), where γ(·) is the gamma function, which is defined by γ(r) = ∫∞ 0 tr−1e−tdt. definition 2.2. the riemann-liouville fractional derivative of order r > 0, n − 1 < r < n, n ∈ n, is defined as dr0+f(t) = 1 γ(n−r) ( d dt )n ∫ t 0 (t−s)n−r−1f(s)ds, where the function f has absolutely continuous derivatives upto order (n− 1). the following lemma is of great importance in the proof of our main results. lemma 2.1. let h,k ∈ c(j,r) and ω = 1 σφ − λ1λ2 6= 0, where λ1 = 1 φ p∑ i=2 βi−1(η φ i −η φ i−1) + q∑ j=1 γjρ φ−1 j , int. j. anal. appl. 17 (6) (2019) 943 λ2 = 1 σ µ∑ i=2 β′i−1(θ σ i −θ σ i−1) + λ∑ j=1 γ′jζ σ−1 j . then the solution of the linear fractional differential system dσx(t) = h(t), n− 1 < σ < n, t ∈ j := [0, 1], dφy(t) = k(t), m− 1 < φ < m, t ∈ j := [0, 1], (2.1) supplemented with the boundary conditions (1.2) is equivalent to the system of integral equations x(t) = 1 γ(σ) ∫ t 0 (t−s)σ−1h(s)ds + tσ−1 ω { 1 φ ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1k(s)dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1k(s)ds− 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1h(s)dsdt ) (2.2) +λ1 ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1h(τ)dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1h(s)ds− 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1k(s)dsdt )} , and y(t) = 1 γ(φ) ∫ t 0 (t−s)φ−1k(s)ds + tφ−1 ω { 1 σ ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1h(s)dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1h(s)ds− 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1h(s)dsdt ) (2.3) +λ2 ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1k(τ)dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1k(s)ds− 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1h(s)dsdt )} . proof. as argued in [2], the general solution of the equations dσx(t) = h(t), n− 1 < σ < n and dφy(t) = k(t), m− 1 < φ < m, can be written as x(t) = b1t σ−1 + b2t σ−2 + · · · + bntσ−n + 1 γ(σ) ∫ t 0 (t−s)σ−1h(s)ds, (2.4) y(t) = d1t φ−1 + d2t φ−2 + · · · + dmtφ−m + 1 γ(φ) ∫ t 0 (t−s)φ−1k(s)ds, (2.5) where bi, i = 1, 2, . . . ,n, and dj,j = 1, . . . ,m are arbitrary constants. using the conditions x ( î )(0) = 0, î = 0, 1, 2, . . . ,n − 2, and y( ĵ )(0) = 0, ĵ = 0, 1, 2, . . . ,m − 2, we find that b2 = b3 = · · · = bn = 0 and d1 = d2 = . . . = dm = 0. thus (2.4) and (2.5) become x(t) = b1t σ−1 + 1 γ(σ) ∫ t 0 (t−s)σ−1g(s)ds, (2.6) y(t) = d1t φ−1 + 1 γ(φ) ∫ t 0 (t−s)φ−1g(s)ds. (2.7) int. j. anal. appl. 17 (6) (2019) 944 using the conditions ∫ 1 0 x(s)ds = ∑p i=2 βi−1 ∫ηi ηi−1 y(s)ds + ∑q j=1 γjy(ρj) and∫ 1 0 y(s)ds = ∑µ i=2 β ′ i−1 ∫ θi θi−1 x(s)ds + ∑λ j=1 γ ′ jx(ζj) in (2.6) and (2.7), we get 1 σ b1 − λ1d1 = p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1k(τ)dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1k(s)ds− 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1h(s)dsdt, −λ2b1 + 1 φ d1 = µ∑ i=2 β′i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1h(τ)dτds + λ∑ j=1 γ′j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1h(s)ds− 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1k(s)dsdt. solving the above system for b1 and d1, we find that b1 = 1 ω ( 1 φ [ p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1k(τ)dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1k(s)ds− 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1h(s)dsdt ] +λ1 [ µ∑ i=2 β′i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1h(τ)dτds + λ∑ j=1 γ′j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1h(s)ds− 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1k(s)dsdt ]) , and d1 = 1 ω ( 1 σ [ µ∑ i=2 β′i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1h(τ)dτds + λ∑ j=1 γ′j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1h(s)ds− 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1k(s)dsdt ] +λ2 [ p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1k(τ)dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1k(s)ds− 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1h(s)dsdt ]) . inserting the above values of b1 and d1 in (2.6) and (2.7) leads to the solutions (2.2) and (2.3). the converse follows by direct computation. the proof is completed. � 3. main results let us introduce the space x = {x(t)|x(t) ∈ c([0, 1],r)} endowed with the norm ‖x‖ = sup{|x(t)|, t ∈ [0, 1]}. obviously (x,‖ · ‖) is a banach space. then the product space (x × x,‖(x,y)‖) is also a banach space equipped with norm ‖(x,y)‖ = ‖x‖ + ‖y‖. int. j. anal. appl. 17 (6) (2019) 945 in view of lemma 2.1, we define an operator t : x ×x → x ×x by t(x,y)(t) =   t1(x,y)(t) t2(x,y)(t)   , where t1(x,y)(t) = 1 γ(σ) ∫ t 0 (t−s)σ−1f(s,x(s),y(s))ds + tσ−1 ω { 1 φ ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1g(s,x(s),y(s))dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1g(s,x(s),y(s))ds (3.1) − 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1f(s,x(s),y(s))dsdt ) +λ1 ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1f(τ,x(τ),y(τ))dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1f(s,x(s),y(s))ds − 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1g(s,x(s),y(s))dsdt )} , and t2(x,y)(t) = 1 γ(φ) ∫ t 0 (t−s)φ−1g(s,x(s),y(s))ds + tφ−1 ω { 1 σ ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1f(τ,x(τ),y(τ))dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1f(s,x(s),y(s))ds (3.2) − 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1g(s,x(s),y(s))dsdt ) +λ2 ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1g(τ,x(τ),y(τ))dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1g(s,x(s),y(s))ds − 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1f(s,x(s),y(s))dsdt )} . for the sake of computational convenience, we define q1 = 1 γ(σ + 1) + 1 |ω| [ 1 φ 1 γ(σ + 2) + |λ1| ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) + λ∑ j=1 |γ′j| ζσj γ(σ + 1) )] , (3.3) q2 = 1 |ω| [ 1 φ ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) ) + |λ1| 1 γ(φ + 2) ] , (3.4) int. j. anal. appl. 17 (6) (2019) 946 q3 = 1 |ω| [ 1 σ ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) + λ∑ j=1 |γ′j| ζσj γ(σ + 1) ) + |λ2| 1 γ(σ + 2) ] , (3.5) q4 = 1 γ(φ + 1) + 1 |ω| [ 1 σ 1 γ(φ + 2) + |λ2| ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) )] . (3.6) in the first result, we prove the existence and uniqueness of solutions for the system (1.1)-(1.2) via banach contraction mapping principle. theorem 3.1. assume that: (h1): f,g : [0, 1] × r × r → r are continuous functions and there exist positive constants `1 and `2 such that for all t ∈ [0, 1] and xi,yi ∈ r, i = 1, 2, we have |f(t,x1,x2) −f(t,y1,y2)| ≤ `1(|x1 −y1| + |x2 −y2|), |g(t,x1,x2) −g(t,y1,y2)| ≤ `2(|x1 −y1| + |x2 −y2|). if (q1 + q3)`1 + (q2 + q4)`2 < 1, where qi, i = 1, 2, 3, 4 are given by (3.3)-(3.6), then the system (1.1)-(1.2) has a unique solution on [0, 1]. proof. define supt∈[0,1] f(t, 0, 0) = n1 < ∞ and supt∈[0,1] g(t, 0, 0) = n2 < ∞ and r > 0 such that r > (q1 + q3)n1 + (q2 + q4)n2 1 − (q1 + q3)`1 − (q2 + q4)`2 . we show that tbr ⊂ br, where br = {(x,y) ∈ x ×x : ‖(x,y)‖≤ r}. by the assumption (h1), for (u,v) ∈ br, t ∈ [0, 1], we have |f(t,x(t),y(t))| ≤ |f(t,x(t),y(t)) −f(t, 0, 0)| + |f(t, 0, 0)| ≤ `1(|x(t)| + |y(t)|) + n1 ≤ `1(‖x‖ + ‖y‖) + n1 ≤ `1r + n1, and |g(t,x(t),y(t))| ≤ `2(‖x‖ + ‖y‖) + n2 ≤ `2r + n2, which lead to |t1(x,y)(t)| ≤ 1 γ(σ) ∫ t 0 (t−s)σ−1(`1r + n1)ds + 1 |ω| { 1 φ ( p∑ i=2 |βi−1| 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1(`2r + n2)dτds int. j. anal. appl. 17 (6) (2019) 947 + q∑ j=1 |γj| 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1(`2r + n2)ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1(`1r + n1)dsdt ) +|λ1| ( µ∑ i=2 |β′i−1| 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1(`1r + n1)dτds + λ∑ j=1 |γ′j| 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1(`1r + n1)ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1(`2r + n2)dsdt )} = (q1`1 + q2`2)r + q1n1 + q2n2. hence ‖t1(x,y)‖≤ (q1`1 + q2`2)r + q1n1 + q2n2. in a similar manner, one can find that ‖t2(x,y)‖≤ (q3`1 + q4`2)r + q3n1 + q4n2. consequently, we have ‖t(x,y)‖≤ [(q1 + q3)`1 + (q2 + q4)`2]r + (q1 + q3)n1 + (q2 + q4)n2 ≤ r. now, for (x2,y2), (x1,y1) ∈ x ×x, and for any t ∈ [0, 1], we get |t1(x2,y2)(t) −t1(x1,y1)(t)| ≤ 1 γ(σ) ∫ t 0 (t−s)σ−1`1(‖x2 −x1‖ + ‖y2 −y1‖)ds + 1 |ω| { 1 φ ( p∑ i=2 |βi−1| 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1`2(‖x2 −x1‖ + ‖y2 −y1‖)dτds + q∑ j=1 |γj| 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1`2(‖x2 −x1‖ + ‖y2 −y1‖)ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1`1(‖x2 −x1‖ + ‖y2 −y1‖)dsdt ) +|λ1| ( µ∑ i=2 |β′i−1| 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1`1(‖x2 −x1‖ + ‖y2 −y1‖)dτds + λ∑ j=1 |γ′j| 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1`1(‖x2 −x1‖ + ‖y2 −y1‖)ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1`2(‖x2 −x1‖ + ‖y2 −y1‖)dsdt )} ≤ (q1`1 + q2`2)(‖x2 −x1‖ + ‖y2 −y1‖), int. j. anal. appl. 17 (6) (2019) 948 which implies that ‖t1(x2,y2) −t1(x1,y1)‖≤ (q1`1 + q2`2)(‖x2 −x1‖ + ‖y2 −y1‖). (3.7) similarly, we obtain ‖t2(x2,y2)(t) −t2(x1,y1)‖≤ (q3`1 + q4`2)(‖x2 −x1‖ + ‖y2 −y1‖). (3.8) thus it follows from (3.7) and (3.8) that ‖t(x2,y2) −t(x1,y1)‖≤ [(q1 + q3)`1 + (q2 + q4)`2](‖x2 −x1‖ + ‖y2 −y1‖). since (q1 + q3)`1 + (q2 + q4)`2 < 1, therefore, t is a contraction. so, by banach fixed point theorem, the operator t has a unique fixed point, which corresponds to a unique solution of problem (1.1)-(1.2). this completes the proof. � in the following theorem, we prove the existence of solutions for the system (1.1)-(1.2) by means of leray-schauder alternative. lemma 3.1. (leray-schauder alternative) ( [23] p. 4.) let f : e → e be a completely continuous operator (i.e., a map that restricted to any bounded set in e is compact). let e(f) = {x ∈ e : x = λf(x) for some 0 < λ < 1}. then either the set e(f) is unbounded, or f has at least one fixed point. theorem 3.2. assume that: (h3): f,g : [0, 1] × r × r → r are continuous functions and there exist real constants ki,γi ≥ 0, (i = 0, 1, 2) and k0 > 0,γ0 > 0 such that ∀xi ∈ r (i = 1, 2), |f(t,x1,x2)| ≤ k0 + k1|x1| + k2|x2|, |g(t,x1,x2)| ≤ γ0 + γ1|x1| + γ2|x2|. if (q1 + q3)k1 + (q2 + q4)γ1 < 1 and (q1 + q3)k2 + (q2 + q4)γ2 < 1, where mi (i = 1, 2, 3, 4) are given by (3.3)-(3.6), then there exists at least one solution to the system (1.1)(1.2) on [0, 1]. proof. first we show that the operator t : x × x → x × x is completely continuous. by continuity of functions f and g, it is easy to show that the operator t is continuous. let ω ⊂ x ×x be bounded. then there exist positive constants l1 and l2 such that |f(t,x(t),y(t))| ≤ l1, |g(t,x(t),y(t))| ≤ l2, ∀(x,y) ∈ ω. int. j. anal. appl. 17 (6) (2019) 949 then, for any (x,y) ∈ ω, we have |t1(x,y)(t)| ≤ 1 γ(σ) ∫ t 0 (t−s)σ−1l1ds + 1 |ω| { 1 φ ( p∑ i=2 |βi−1| 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1l2dτds + q∑ j=1 |γj| 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1l2ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1l1dsdt ) +|λ1| ( µ∑ i=2 |β′i−1| 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1l1dτds + λ∑ j=1 |γ′j| 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1l1ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1l2dsdt )} ≤ q1l1 + q2l2, which implies that ‖t1(x,y)‖≤ q1l1 + q2l2. in a similar manner, we can get ‖t2(x,y)‖≤ q3l1 + q4l2. thus, it follows from the above inequalities that the operator t is uniformly bounded, since ‖t(x,y)|| ≤ (q1 + q3)l1 + (q2 + q4)l2. next, we show that t is equicontinuous. let t1, t2 ∈ [0, 1] with t1 < t2. then we have |t1(x(t2),y(t2)) −t1(x(t1),y(t1))| ≤ l1 { 1 γ(σ) ∫ t1 0 [(t2 −s)σ−1 − (t1 −s)σ−1]ds + 1 γ(σ) ∫ t2 t1 (t2 −s)σ−1ds } + |tσ−12 − t σ−1 1 | |ω| {[ 1 φ 1 γ(σ + 2) + |λ1| ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) )] l1 + [ 1 φ ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) ) + |λ1| 1 γ(φ + 2) ] l2 } ≤ l1 γ(σ + 1) [2(t2 − t1)σ + |tσ2 − t σ 1 |] + |tσ−12 − t σ−1 1 | |ω| {[ 1 φ 1 γ(σ + 2) + |λ1| ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) )] l1 + [ 1 φ ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) ) + |λ1| 1 γ(φ + 2) ] l2 } . analogously, we can obtain |t2(x(t2),y(t2)) −t2(x(t1),y(t1))| int. j. anal. appl. 17 (6) (2019) 950 ≤ l2 γ(φ + 1) [2(t2 − t1)φ + |t2 − t φ 1 |] + |tφ−12 − t φ−1 1 | |ω| {[ 1 σ ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) + λ∑ j=1 |γ′j| ζσj γ(σ + 1) ) + |λ2| 1 γ(σ + 2) ] l1 + [ 1 σ 1 γ(φ + 2) + |λ2| ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) )] l2 } . from the preceding inequalities, we deduce that the operator t(x,y) is equicontinuous, and thus the operator t(x,y) is completely continuous. finally, it will be verified that the set e = {(x,y) ∈ x ×x|(x,y) = λt(x,y), 0 ≤ λ ≤ 1} is bounded. let (x,y) ∈e, with (x,y) = λt(x,y). for any t ∈ [0, 1], we have x(t) = λt1(x,y)(t), y(t) = λt2(x,y)(t). then |x(t)| ≤ q1(k0 + k1|x| + k2|y|) + q2(γ0 + γ1|x| + γ2|y|) = q1k0 + q2γ0 + (q1k1 + q2γ1)|x| + (q1k2 + q2γ2)|y|, and |y(t)| ≤ q3(k0 + k1|x| + k2|y|) + q4(γ0 + γ1|x| + γ2|y|) = q3k0 + q4γ0 + (q3k1 + q4γ1)|x| + (q3k2 + q4γ2)|y|. hence we have ‖x‖≤ q1k0 + q2γ0 + (q1k1 + q2γ1)‖x‖ + (q1k2 + q2γ2)‖y|| and ‖y‖≤ q3k0 + q4γ0 + (q3k1 + q4γ1)‖x‖ + (q3k2 + q4γ2)‖y‖, which imply that ‖x‖ + ‖y‖ ≤ (q1 + q3)k0 + (q2 + q4)γ0 + [(q1 + q3)k1 + (q2 + q4)γ1]‖x‖ +[(q1 + q3)k2 + (q2 + q4)γ2)]‖y‖. consequently, ‖(x,y)‖≤ (q1 + q3)k0 + (q2 + q4)γ0 m0 , where m0 = min{1 − [(q1 + q3)k1 + (q2 + q4)γ1], 1 − [(q1 + q3)k2 + (q2 + q4)γ2)]}, which proves that e is bounded. thus, by lemma 3.1, the operator t has at least one fixed point. hence the system (1.1)-(1.2) has at least one solution. the proof is complete. � our last result is based on krasnoselskii fixed point theorem [24]. int. j. anal. appl. 17 (6) (2019) 951 lemma 3.2. (krasnoselskii) let m be a closed, bounded, convex and nonempty subset of a banach space x. let a,b be operators mapping m to x such that (a) ax + by ∈ m where x,y ∈ m; (b) a is compact and continuous; (c) b is a contraction mapping. then there exists z ∈ m such that z = az + bz. theorem 3.3. assume that f,g : [0, 1] × r → r are continuous functions satisfying assumption (h1) in theorem 3.1. in addition we suppose that there exist two positive constants l1,l2 such that for all t ∈ [0, 1] and xi,yi ∈ r, i = 1, 2, |f(t,x1,x2)| ≤ l1 and |g(t,x1,x2)| ≤ l2. (3.9) if `1 γ(σ + 1) + `2 γ(φ + 1) < 1, (3.10) then the problem (1.1)-(1.2) has at least one solution on [0, 1]. proof. in order to verify the hypotheses of lemma 3.2, we decompose the operator t into four operators t1,1,t1,2,t2,1 and t2,2 on bδ = {(x,y) ∈ x ×y : ‖(x,y)‖≤ δ} as follows: t1,1(x,y)(t) = tσ−1 ω { 1 φ ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1g(s,x(s),y(s))dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1g(s,x(s),y(s))ds − 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1f(s,x(s),y(s))dsdt ) +λ1 ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1f(τ,x(τ),y(τ))dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1f(s,x(s),y(s))ds − 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1g(s,x(s),y(s))dsdt )} , t1,2(x,y)(t) = 1 γ(σ) ∫ t 0 (t−s)σ−1f(s,x(s),y(s))ds, and t2,1(x,y)(t) = tφ−1 ω { 1 σ ( µ∑ i=2 β ′ i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1f(τ,x(τ),y(τ))dτds + λ∑ j=1 γ ′ j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1f(s,x(s),y(s))ds − 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1g(s,x(s),y(s))dsdt ) int. j. anal. appl. 17 (6) (2019) 952 +λ2 ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1g(τ,x(τ),y(τ))dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1g(s,x(s),y(s))ds − 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1f(s,x(s),y(s))dsdt )} , t2,2(x,y)(t) = 1 γ(φ) ∫ t 0 (t−s)φ−1g(s,x(s),y(s))ds. notice that t1(x,y)(t) = t1,1(x,y)(t) + t1,2(x,y)(t) and t2(x,y)(t) = t2,1(x,y)(t) + t2,2(x,y)(t) on bδ and that the ball bδ is a closed, bounded and convex subset of the banach space x × x. let us select δ ≥ max{q1l1 + q2l2,q3l1 + q4l2} and show that tbδ ⊂ bδ for verifying condition (a) of lemma 3.2. setting x = (x1,x2),y = (y1,y2) ∈ bδ, and using condition (3.9), we obtain |t1,1(x,y)(t) + t1,2(x,y)(t)| ≤ 1 γ(σ) ∫ t 0 (t−s)σ−1l1ds + 1 |ω| { 1 φ ( p∑ i=2 |βi−1| 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1l2dτds + q∑ j=1 |γj| 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1l2ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1l1dsdt ) +|λ1| ( µ∑ i=2 |β′i−1| 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1l1dτds + λ∑ j=1 |γ′j| 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1l1ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1l2dsdt )} = q1l1 + q2l2 ≤ δ. likewise, we can find that |t2,1(x̂, ŷ)(t) + t2,2(x̂, ŷ)(t)| ≤ q3l1 + q4l2 ≤ δ. clearly the above two inequalities lead to the fact that t1(x,y) + t2(x̂, ŷ) ∈ bδ. now we establish that the operator (t1,2,t2,2) is a contraction satisfying condition (c) of lemma 3.2. for (x1,y1), (x2,y2) ∈ bδ, we have |t1,2(x1,y1)(t) −t1,2(x2,y2)(t)| = 1 γ(σ) ∫ t 0 (t−s)σ−1|f(s,x1(s),y1(s)) −f(s,x2(s),y2(s))|ds ≤ `1 γ(σ + 1) (‖x1 −y1‖ + ‖x2 −y2‖), (3.11) and |t2,2(x1,y1)(t) −t2,2(x2,y2)(t)| int. j. anal. appl. 17 (6) (2019) 953 = 1 γ(φ) ∫ t 0 (t−s)φ−1|g(s,x1(s),y1(s)) −g(s,x2(s),y2(s))|ds ≤ `2 γ(φ + 1) (‖x1 −y1‖ + ‖x2 −y2‖). (3.12) it follows from (3.11) and (3.12) that ‖(t1,2,t2,2)(x1,y1) − (t1,2,t2,2)(x2,y2)‖ ≤ [ `1 γ(σ + 1) + `2 γ(φ + 1) ] (‖x1 −y1‖ + ‖x2 −y2‖), which is a contraction by (3.10). therefore, the condition (c) of lemma 3.2 is satisfied. next we will show that the operator (t1,1,t2,1) satisfies the condition (b) of lemma 3.2. by applying the continuity of the functions f,g on [0, 1]×r×r, we can conclude that the operator (t1,1,t2,1) is continuous. for each (x,y) ∈ bδ, we have |t1,1(x,y)(t)| ≤ 1 |ω| { 1 φ ( p∑ i=2 |βi−1| 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1l2dτds + q∑ j=1 |γj| 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1l2ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1l1dsdt ) +|λ1| ( µ∑ i=2 |β′i−1| 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1l1dτds + λ∑ j=1 |γ′j| 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1l1ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1l2dsdt )} = p∗, and |t2,1(x,y)(t)| ≤ tφ−1 |ω| { 1 σ ( µ∑ i=2 β′i−1 1 γ(σ) ∫ θi θi−1 ∫ s 0 (s− τ)σ−1l1dτds + λ∑ j=1 γ′j 1 γ(σ) ∫ ζj 0 (ζj −s)σ−1l1ds + 1 γ(φ) ∫ 1 0 ∫ t 0 (t−s)φ−1l2dsdt ) +λ2 ( p∑ i=2 βi−1 1 γ(φ) ∫ ηi ηi−1 ∫ s 0 (s− τ)φ−1l2dτds + q∑ j=1 γj 1 γ(φ) ∫ ρj 0 (ρj −s)φ−1l2ds + 1 γ(σ) ∫ 1 0 ∫ t 0 (t−s)σ−1l1dsdt )} = q∗, int. j. anal. appl. 17 (6) (2019) 954 which lead to the fact that ‖(t1,1,t2,1)(x,y)‖≤ p∗ + q∗. thus the set (t1,1,t2,1)bδ is uniformly bounded. in the next step, we will show that the set (t1,1,t2,1)bδ is equicontinuous. for t1, t2 ∈ [0, 1] with t1 < t2 and for any (x,y) ∈ bδ, we obtain |t1,1(x,y)(t2) −t1,1(x,y)(t1)| ≤ |tσ−12 − t σ−1 1 | |ω| {[ 1 φ 1 γ(σ + 2) + |λ1| ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) )] l1 + [ 1 φ ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) ) + |λ1| 1 γ(φ + 2) ] l2 } ≤ + |tσ−12 − t σ−1 1 | |ω| {[ 1 φ 1 γ(σ + 2) + |λ1| ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) )] l1 + [ 1 φ ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) ) + |λ1| 1 γ(φ + 2) ] l2 } . analogously, we can obtain |t2,1(x,y)(t2) −t2,1(x,y)(t1)| ≤ |tφ−12 − t φ−1 1 | |ω| {[ 1 σ ( µ∑ i=2 |β′i−1| θσ+1i −θ σ+1 i−1 γ(σ + 2) + λ∑ j=1 |γ′j| ζσj γ(σ + 1) ) + |λ2| 1 γ(σ + 2) ] l1 + [ 1 σ 1 γ(φ + 2) + |λ2| ( p∑ i=2 |βi−1| η φ+1 i −η φ+1 i−1 γ(φ + 2) + q∑ j=1 |γj| ρ φ j γ(φ + 1) )] l2 } . thus |(t1,1,t2,1)(x,y)(t2) − (t1,1,t2,1)(x,y)(t1)| tends to zero as t1 → t2 independent of (x,y) ∈ bδ. therefore the set (t1,1,t2,1)bδ is equicontinuous. thus it follows by the arzelá-ascoli theorem that the operator (t1,1,t2,1) is compact on bδ. by the conclusion of lemma 3.2, we deduce that the problem (1.1)(1.2) has at least one solution on [0, 1]. this completes the proof. � example 3.1. consider the following system of fractional boundary value problems  cd3/2x(t) = 1 4(t + 2)2 |x(t)| 1 + |x(t)| + 1 + 1 32 sin2 y(t) + 1 √ t2 + 1 , t ∈ [0, 1], cd5/2y(t) = 1 32π sin(2πx(t)) + |y(t)| 16(1 + |y(t)|) + 1 2 , t ∈ [0, 1], ∫ 1 0 x(s)ds = 5 2 ∫ 1/6 1/7 y(s)ds + 4 ∫ 1/5 1/6 y(s)ds + 3 2 y (1 4 ) + 2y (1 3 ) , ∫ 1 0 y(s)ds = 2 ∫ 1/2 1/3 x(s)ds + 4 ∫ 2/3 1/2 x(s)ds + x (1 3 ) + 1 2 x (2 3 ) . (3.13) here σ = 3/2,φ = 3/2,β1 = 5/2,β2 = 4,η1 = 1/7,η2 = 1/6,η3 = 1/5,γ1 = 3/2,γ2 = 2,ρ1 = 1/4,ρ2 = 1/3,β′1 = 2,β ′ 2 = 4,γ ′ 1 = 1,γ ′ 2 = 1/2,θ1 = 1/3,θ2 = 1/2,θ3 = 2/3,ζ1 = 1/3,ζ2 = 2/3, f(t,x,y) = 1 4(t + 2)2 |x| 1 + |x| + 1 + 1 32 sin2 y, and g(t,x,y) = 1 32π sin(2πx) + |y| 16(1 + |y|) + 1 2 . with the given data, we find int. j. anal. appl. 17 (6) (2019) 955 that q1 ≈ 1.618712, q2 ≈ 0.096199, q3 ≈ 1.276613, q4 ≈ 0.199029. note that |f(t,x1,x2) −f(t,y1,y2)| ≤ 1 16 |x1−x2|+ 1 16 |y1−y2|, |g(t,x1,x2)−g(t,y1,y2)| ≤ 1 16 |x1−x2|+ 1 16 |y1−y2|, and (q1 +q3)`1 +(q2 +q4)`2 ≈ 0.217861 < 1. thus all the conditions of theorem 3.1 are satisfied and consequently, its conclusion applies to the problem (3.13). 4. conclusions in this paper we introduced and solved a new boundary value problem consisting of nonlinear coupled fractional differential equations and nonlocal coupled integro-multistrip-multipoint boundary conditions. assuming different conditions on the nonlinear functions involved in the given problem, we have presented the criteria ensuring the existence of solutions for the problem at hand by applying banach contraction mapping principle, leray-schauder alternative and krasnoselskii fixed point theorem. the obtained results are of quite general nature and lead to several interesting special cases (new results) by fixing the values of the parameters involved in the problem appropriately. for example, if we take all βi−1 = 0, i = 2, . . . ,p in the results of this paper, we obtain the ones associated with the boundary conditions of the form: x ( î ) (0) = 0, ∫ 1 0 x(s)ds = q∑ j=1 γj y(ρj), î = 0, 1, 2, . . . ,n− 2, y ( ĵ ) (0) = 0, ∫ 1 0 y(s)ds = µ∑ i=2 β ′ i−1 ∫ θi θi−1 x(s)ds + λ∑ j=1 γ ′ j x(ζj), ĵ = 0, 1, 2, . . . ,m− 2. letting all β′i−1 = 0, i = 2, . . . ,p, our results correspond to the ones with the boundary conditions: x ( î ) (0) = 0, ∫ 1 0 x(s)ds = p∑ i=2 βi−1 ∫ ηi ηi−1 y(s)ds + q∑ j=1 γj y(ρj), î = 0, 1, 2, . . . ,n− 2, y ( ĵ ) (0) = 0, ∫ 1 0 y(s)ds = λ∑ j=1 γ ′ j x(ζj), ĵ = 0, 1, 2, . . . ,m− 2. in case all βi−1 = 0, i = 2, . . . ,p and β ′ i−1 = 0, i = 2, . . . ,µ, the results of this paper lead to the ones for the integro-multipoint boundary conditions: x ( î ) (0) = 0, ∫ 1 0 x(s)ds = q∑ j=1 γj y(ρj); y ( ĵ ) (0) = 0, ∫ 1 0 y(s)ds = λ∑ j=1 γ ′ j x(ζj), where î = 0, 1, 2, . . . ,n− 2 and ĵ = 0, 1, 2, . . . ,m− 2. fixing all γj = 0,j = 1, . . . ,q and γ ′ j = 0, j = 1, . . . ,λ in the results of this paper, we get the ones associated with coupled integro-multistrip conditions of the form: x( î )(0) = 0, ∫ 1 0 x(s)ds = p∑ i=2 βi−1 ∫ ηi ηi−1 y(s)ds, î = 0, 1, 2, . . . ,n− 2, y( ĵ )(0) = 0, ∫ 1 0 y(s)ds = µ∑ i=2 β′i−1 ∫ θi θi−1 x(s)ds, ĵ = 0, 1, 2, . . . ,m− 2. int. j. anal. appl. 17 (6) (2019) 956 references [1] i. podlubny, fractional differential equations, academic press, san diego, 1999. [2] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies, 204, elsevier science b.v., amsterdam, 2006. [3] k.s. miller, b. ross, an introduction to the fractional calculus and fractional differential equations, wiley and sons, new york, 1993 [4] j. sabatier, o.p. agrawal, j.a.t. machado (eds.), advances in fractional calculus: theoretical developments and applications in physics and engineering, springer, dordrecht, 2007. [5] s. liang, j. zhang, existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, math. comput. modelling 54 (2011) 1334-1346. [6] c. goodrich, existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, comput. math. appl. 61 (2011), 191-202. [7] j.r. wang, y. zhou, m. feckan, on recent developments in the theory of boundary value problems for impulsive fractional differential equations, comput. math. appl. 64 (2012), 3008-3020. [8] b. ahmad, on nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order, results math. 63 (2013), 183-194. [9] c. zhai, l. xu, properties of positive solutions to a class of four-point boundary value problem of caputo fractional differential equations with a parameter, commun. nonlinear sci. numer. simul. 19 (2014), 2820-2827. [10] j. henderson, n. kosmatov, eigenvalue comparison for fractional boundary value problems with the caputo derivative, fract. calc. appl. anal. 17 (2014), 872-880. [11] y. ding, z. wei, j. xu, d. o’regan, extremal solutions for nonlinear fractional boundary value problems with p-laplacian, j. comput. appl. math. 288 (2015), 151-158. [12] h. wang, existence of solutions for fractional anti-periodic bvp, results math. 68 (2015), 227-245. [13] b. ahmad, s.k. ntouyas, some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions, rev. r. acad. cienc. exactas fis. nat. ser. a math. racsam 110 (2016), 159-172. [14] m. faieghi, s. kuntanapreeda, h. delavari, d. baleanu, lmi-based stabilization of a class of fractional-order chaotic systems, nonlinear dynam. 72 (2013), 301-309. [15] f. zhang, g. chen c. li, j. kurths, chaos synchronization in fractional differential systems, phil. trans. r. soc. a 371 (2013), 20120155. [16] b. ahmad, s.k. ntouyas, a. alsaedi, fractional differential equations and inclusions with nonlocal generalized riemannliouville integral boundary conditions, int. j. anal. appl. 13 (2017), 231-247. [17] s. asawasamrit, n. phuangthong, s.k. ntouyas, j. tariboon, nonlinear sequential riemann-liouville and caputo fractional differential equations with nonlocal and integral boundary conditions, int. j. anal. appl. 17 (2019), 47-63. [18] c. s. goodrich, existence of a positive solution to systems of differential equations of fractional order, comput. math. appl. 62 (2011), 1251-1268. [19] s.k. ntouyas, m. obaid, a coupled system of fractional differential equations with nonlocal integral boundary conditions, adv. difference equ. 2012 (2012), art. id 130. [20] b. ahmad, s.k. ntouyas, existence results for a coupled system of caputo type sequential fractional differential equations with nonlocal integral boundary conditions, appl. math. comput. 266 (2015), 615-622. int. j. anal. appl. 17 (6) (2019) 957 [21] b. ahmad, s.k. ntouyas, a. alsaedi, on a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, chaos solitons & fractals 83 (2016), 234-241. [22] a. alsaedi, s.k. ntouyas, d. garout, b. ahmad, coupled fractional-order systems with nonlocal coupled integral and discrete boundary conditions, bull. malays. math. sci. soc. 42 (2019), 241-266. [23] a. granas, j. dugundji, fixed point theory, springer-verlag, new york, 2005. [24] m.a. krasnoselskii, two remarks on the method of successive approximations, uspekhi mat. nauk 10 (1955), 123-127. 1. introduction 2. preliminaries 3. main results 4. conclusions references int. j. anal. appl. (2023), 21:9 received: nov. 22, 2022. 2020 mathematics subject classification. 03e72, 54a40. key words and phrases. topology; fuzzy sets; separation properties; covering properties fuzzy paracompactness. https://doi.org/10.28924/2291-8639-21-2023-9 © 2023 the author(s) issn: 2291-8639 1 regularity and paracompactness in fuzzy topological spaces francisco gallego lupiáñez* dpt. mathematics, universidad complutense, madrid (spain) *corresponding author: fg_lupianez@mat.ucm.es abstract. in this paper we obtain two characterizations of regular fuzzy topological spaces using luo's and abd el-monsef and others' paracompact fuzzy topological spaces. 1. introduction in this paper we obtain two characterizations of fuzzy regularity as a fuzzy covering property. indeed, we show that one can characterize fuzzy regularity as a paracompact-type fuzzy property in luo's and abd el-monsef and others' both senses. 2. definitions and main results definition 1 [1] let  be a set in a fts ( ),x  and let (   )0,1 , 0,1 ;r s  we define ( ){ }[ ] :r x x x r    = ( ){ }( ) :s x x x s     =  r r r    = definition 2 [1] let a be a family of sets and be a set in a fts ( ),x  . we say that a is locally finite (resp. ∗-locally finite) in for each point e in , there exists ( )q e  such that     https://doi.org/10.28924/2291-8639-21-2023-9 2 int. j. anal. appl. (2023), 21:9 is quasi-coincident (resp. intersects) with at most a finite number of sets of a ; we often omit the word "in  " when x = . definition 3 [1] a family of sets a is called a q -cover of a set μ if for each ,( )x supp  there exist a a such that  and are quasi-coincident at x . let ( 0,1 .r  a is called a r − q cover of if a is a q -cover of .r    definition 4 [1] let ( 0,1 r  , be a set in a fts . we say that  is r -paracompact (resp. * r -paracompact) if for each -open q -cover of there exists an open refinement of it which is both locally finite (resp.∗-locally finite) in and a r q -cover of . the fuzzy set is called s-paracompact (resp. s*-paracompact) if for every ( 0,1r  , is r -paracompact (resp. *r -paracompact). definition 5 [2] a family of fuzzy sets u is called an l -cover of a fuzzy set if .      u definition 6 [2] let be a fuzzy set in a fts . we say that  is fuzzy paracompact (resp. ∗-fuzzy paracompact) if for each open l -cover b of and for each ( 0,1 ,  there exists an open refinement *b of b which is both locally finite (resp. ∗-locally finite) in and l -cover of  − . we say that a fts is fuzzy paracompact (resp. ∗-fuzzy paracompact) if each constant set in x is fuzzy paracompact (resp. ∗-fuzzy paracompact). theorem 1 let a fuzzy hausdorff fts (in any of wuyts and lowen’s definitions that are good extensions of hausdorffness). then is fuzzy regular if and only if for each ( 0,1 ,r  for each r -open q-cover of and for each fuzzy point x  of x there exists an open refinement of it which is both ∗-locally finite in x  and a r -q-cover of . proof (⇒) for each ( 0,1 ,r  let u be a r -open q-cover of ( ),x  , and x be a fuzzy point of x . then, we have that the family of crisp sets ( )1 { } r u u − u∣ , is an open cover of  ( ),x  , which is hausdorff and regular ([3], [4]). then ([5], [6], [7]), it has an open refinement [ ] x v    ( ),x  r        ( ),x    ( ),x  ( ),x  ( ),x  ( ),x  ( ),x  3 int. j. anal. appl. (2023), 21:9 which is a cover of x , and is locally finite in x . for each v x v we have an v u u with ( )1 ( ) . v r v u −  let .{ }v vu vx x =  w v∣ then, x w , is both an open refinement of u and a r q− -cover of ( ),x  , and also is ∗-locally finite in x  , indeed, because x v is locally finite in x, we have an open neigborhood g of x that g intersects with only finite number of members of x v . then ( ) g q x    intersects with only a finite number of members of x w . (⇐) let u [ ] be an open cover of ( ],[ )x  ; then  u u u∣ is an open q-cover of 1 , x and, for each ,x x it has an open refinement r x v which is a q-cover of 1 x and also locally finite in 1 r x − . let  (1 ) | rr xv v− =w v ; then [ ]w is both a refinement of u and a cover of ( ],[ )x  . also, w is locally finite in x . indeed: we take 1 1 ( x ) r o q −  such that 1 o , is quasicoincident with only a finite number of members 1 ,..., n v v , of r x v . let 1 ( ) ( ) r oo = , then [ ].x o   for each v rx v , if (1 )r o v −  , we have a crisp point y x , such that ( ) ( ) ( ) ( )1 1, 1 , 1,o y r v y r o y v y  − +  then 1o qv and  1,..., nv v v . hence the neighborhood o of x intersects with only a finite number of members ( ) ( ) (1 ) (1 )1 ,..., . rnr v v − − w theorem 2. let ( ),x  be a fuzzy hausdorff fts (in any of wuyts and lowen’s definitions that are good extensions of hausdorffness [3]). then ( ),x  is fuzzy regular if and only if for each r i , and for each open l -cover b of r , for each ( 0,1 ,  and for each fuzzy point x of x , there exists an open refinement b * of it which is both ∗-locally finite in x  and l -cover of .r − proof (⇒) for each r i , and for each open l -cover b of r , for each ( 0,1 ,  and for each fuzzy point x  of x , we have that the family of crisp sets ( ){ ,1] } [ ]¹(g gr  −=  − u b∣ is an open cover of ( ],[ )x  which is hausdorff and regular ([3, 4]). then ([5],[6],[7]), it has an open refinement * [ ] x u which is a cover of x and is locally finite in x . for each *xv u , there 4 int. j. anal. appl. (2023), 21:9 exists ,( ]¹ 1( ) v g r  − − u , such that (( ])¹ ,1 v v g r  −  − . so  *|v v xg v   b* = u is refinement of b . then, there exists * x v u such that x v and ( ) .vg x r  − so, ,( )( ) v v g x r   − and  *| .v v xg v r    − u since *xu is locally finite in x , there exists  a  with x a , such that intersects with at most a finite number of members of * x u . then, there exists a   such that a x q   and a  intersects with a finite number of fuzzy sets of b * . (⇐) let [ ]u be an open cover of x and x x , then { } u u= b u∣ is an open l -cover of ( ),x  and for each, r i is { } u ru   u∣ . for each ( 0,1 ,  there exists an open refinement *b of b which is both locally finite in x  and l -cover of r − . this implies that 1 { } rg g −  b*∣ for all 1 .  let  1(( ,1])g r g−= −  b *u* , then [ ]u* is an open refinement of u (indeed, for each 1(( ,1])g r − − u * there exists g v u such that ( ( )¹ ,1 )gg r v − −  . since   1g g r    − b * , then u * is an open refinement of u . and, since x x there exists ( )a q x which intersects with only 1,.., .ng g b * since ( ) 1,a x +  we have ( ) 1a x  − , then ( ( )  ¹ 1 ,1 .x a  − −  if ( ( ) ( ( ) ¹ 1 ,1 ¹ ,1a g r − −−  −   , there exists some point z, such that ( ) 1a z  − and ( )g z r  − , so a g   . then, if the neigborhood ( ( )¹ 1 ,1a − − of x intersects with infinite members of u * , a intersects with infinite members of *b .thus u * is locally finite in x . this yields that the hausdorff topological space ( ],[ )x  is regular ([5], [6], [7]) and ( ),x  is fuzzy regular ([4]). 3. discussion in this paper, fuzzy regularity is characterized as a fuzzy covering property. future research could obtain characterization of other fuzzy separation properties as fuzzy covering properties. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. 5 int. j. anal. appl. (2023), 21:9 references [1] m.k. luo, paracompactness in fuzzy topological spaces, j. math. anal. appl. 130 (1988), 55–77. https://doi.org/10.1016/0022-247x(88)90386-1. [2] m.e.a. el-monsef, f.m. zeyada, s.n. el-deeb, i.m. hanafy, good extensions of paracompactness, math. japonica. 37 (1992), 195-200. [3] p. wuyts, r. lowen, on separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces, and fuzzy uniform spaces, j. math. anal. appl. 93 (1983), 27–41. https://doi.org/10.1016/0022247x(83)90217-2. [4] y.m. liu, m.k. luo, fuzzy topology, world scientific publishing, singapore, 1997. [5] j. abdelhay, caracterizacao dos espacos topologicos regulares e normais por meio de coberturas, gazeta de matematica (lisboa) 9 (1948), 8–9. (in portuguese). [6] j.m. boyte, point (countable) paracompactness, j. aust. math. soc. 15 (1973), 138–144. https://doi.org/10.1017/s144678870001288x. [7] j. chew, regularity as a relaxation of paracompactness, amer. math. mon. 79 (1972), 630–632. https://doi.org/10.1080/00029890.1972.11993101. https://doi.org/10.1016/0022-247x(88)90386-1 https://doi.org/10.1016/0022-247x(83)90217-2 https://doi.org/10.1016/0022-247x(83)90217-2 https://doi.org/10.1017/s144678870001288x https://doi.org/10.1080/00029890.1972.11993101 international journal of analysis and applications volume 17, number 6 (2019), 1019-1033 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-1019 new results on the conformable fractional sumudu transform: theories and applications zeyad al-zhour1,∗, fatimah alrawajeh2, nouf al-mutairi2 and raed alkhasawneh3 1department of basic engineering sciences, college of engineering, imam abdulrahman bin faisal university, p.o. box 1982, dammam, saudi arabia 2department of mathematics, college of science, imam abdulrahman bin faisal university, p.o. box 1982, dammam, saudi arabia 3department of general courses, college of applied studies and community service, imam abdulrahman bin faisal university, p.o. box 1982, dammam, saudi arabia ∗corresponding author: zalzhour@iau.edu.sa; zeyad1968@yahoo.com abstract. in this paper, we generalize the formula of sumudu transform to the conformable fractional order and some interesting and important rules of this transform and conformable fractional laplace transform are derived and discussed. moreover, we present the general analytical solution of a singular and nonlinear conformable fractional poissonboltzmann differential equation based on the conformable fractional sumudu transform. also, our proposed method is applied successfully for obtaining the general solutions of some linear and nonhomogeneous conformable fractional differential equations. finally, the results show that our proposed method is an efficient and can be applied for finding the general solutions of the all cases realted to the conformable fractional differential equations. received 2019-06-28; accepted 2019-08-07; published 2019-11-01. 2010 mathematics subject classification. 26a33, 44a05, 44a10. key words and phrases. conformable fractional operator; sumudu transform; laplace transform; poisson-boltzmann differential equation; adomain decomposition method. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1019 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-1019 int. j. anal. appl. 17 (6) (2019) 1020 1. introduction and preliminaries various types of fractional derivatives were introduced such as: riemann-liouville, caputo, modified riemann-liouville, hadamard, grunwald, letnikov and riesz operators. the most two popular definitions of fractional derivatives of order α > 0 are ([1], [2], [3], [4], [5], [6], [7]): (i) riemann-liouville fractional derivative: rdαf(x) = 1 γ(n−α) dn dxn ∫ x 0 (x− t)n−α−1f(t)dt. (1.1) (ii) caputo fractional derivative: cdαf(x) = 1 γ(n−α) ∫ x 0 (x− t)n−α−1 dn dxn f(t)dt. (1.2) here γ(.) denotes to the gamma function. in the last few decades, fractional differentiation has been used applied scientists for solving several fractional differential equations and they proved that the fractional calculus is very useful in several fields of applications with some restrictions such as: physics (quantum mechanics and thermodynamics), chemistry, biology, economics, engineering, signal and image processing and control theory ([2], [3], [4], [5], [6], [7], [8], [9], [10]). very recently, the discrepancies between known definitions can be solved in simple way by presenting a new fractional definition which is called the ”conformable fractional derivative ” and defined for a given function f : [0,∞) → r of fractional (ordinary) order α > 0 by khalil et el. [11] as follows: dnαf(x) = lim ε→0 f[α]−1(x + εx[α]−α) −f[α]−1(x) ε , n− 1 < α ≤ n, x > 0, (1.3) where [α] is the smallest integer number greater than or equal α and n ∈ n. as a special case, if 0 < α ≤ 1, then we have: dαf(x) = lim ε→0 f(x + εx1−α) −f(x) ε . (1.4) this definition is very easy for calculating derivatives and solving fractional differential equations compared with other fractional definitions such as the definitions of liouville-reimman and caputo fractional derivatives. it has received a lot of attention and many applications have been remodeled using this new definition ([12], [13], [14], [15], [16], [17], [18], [19]). moreover, it has many interesting advantages which make its easier and flexible more than the definitions of other fractional derivatives ([12], [13], [14], [15], [16], [17], [18], [19]). some of these adavantages are: (i) it satisfies the all concepts of ordinary calculus such as: quotient, product and chain rules, rolle’s theorem and mean-value theorem. (ii) a non-differentiable function can be αdifferentiable in the conformable sense. (iii) it can be easily extended to generalize many integral transforms such as: laplace, mellin, natural and sumudu transforms. int. j. anal. appl. 17 (6) (2019) 1021 also, the conformable fractional integral has been also defined of order α > 0 by: iαf(x) = ∫ x 0 f(t) tα−1dt. (1.5) in fact, if f(x) is an n -differentiable function at x > 0 and α ∈ (0, 1], n ∈ n, then [11]: (i) dαf(x) = x1−α d dx f(x), (1.6) (ii) dαiαf(x) = f(x). (1.7) in the literature there are several works on the theory and applications of integral transforms such as the laplace and sumudu transforms ([20], [21], [22], [23], [24], [25], [26], [27], [28]) that are widely used in physics, electric circuit theory, astronomy, as well as engineering and sciences. sumudu transform was introduced by watugala [20 ] and defined over the following set of functions: a = { f(x) : ∃ m,τ1,τ2 > 0, |f(x)| < me |x| τj , if x ∈ (−1)j × [0,∞),j = 1, 2 } , (1.8) as follows: s[f(x)] = f(u) = ∫ ∞ 0 e−xf(ux)dx, u ∈ (−τ1,τ2), (1.9) or equivalently: s[f(x)] = f(u) = 1 u ∫ ∞ 0 e− x u f(x)dx, u > 0. (1.10) this transform has many interesting advantages over other integral transforms especially the ”unity” feature which could provide convergence when solvimg differential equations and also used to solve problems without resorting to a new frequency domain while for example, laplace transform must satisfy the dirichlet condition which is f(x) must be piecewise continuous which means that it must be single valued but can have a finite number of finite isolated discontinuues for x > 0. also this transform possesses many interesting advantages which make its visualization easier and some of these advantages can be found in ([20], [21], [22], [23], [24], [25], [26], [27], [28]). there are several methods and techniques were applied for obtaining the analytical and numerical solution of such nonlinear and singular fractional differential equations such as the fractional laplace transform method [19], generalized kudryashov method [29], adomian decomposition method ([30], [31]) and modified kudryashov method [32]. note that the fractional order differential equations are now span a half-century or more and play a crucial role in several theoretical and applied sciences such as, but certainly not limited to, theoretical biology and ecology, solid state physics, viscoelasticity, fiber optics, signal processing and electric int. j. anal. appl. 17 (6) (2019) 1022 control theory, stochastic based finance and thermodynamics ([19], 20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]). finally, the theory of thermal explosions originally proposed by frank-kamenetzky ([33], [34]) and revisited by barenblatt et al. [ 35] has been required the solution of the nonlinear poisson-boltzmann differential equation which is given and solved numerically by chambré [36]. the ordinary poisson-boltzmann differential equation as follows ([36], [37]): d2y dx2 + β x dy dx = ey (1.11) this equation is also a very useful in many settings, whether it be to understand physiological interfaces, polymer science, electron interactions in a semiconductor; in the critical theory of gravitation and combustion or explosion. and to describe the distribution of the electric potential in solution in the direction normal to a charged surface and from this equation, many other equations can been derived with a number of different assumptions. in this paper, we extend the definition of sumudu transform to fractional order and derive a list of interesting rules and properties of this extension which including the rules of the conformable fractional laplace transform. also, a very nice relationship between conformable fractional sumudu and laplace transforms is derived and proved. moreover, we give two important and attractive applications for conformable fractional sumudu transform. these applications are: firstly, we apply the conforrmable fractional sumudu transform together with adomain decomposition method for presenting the general analytical solution of a singular and nonlinear conformable fractional poisson-boltzmann differential equation. secondly, we also apply the conformable fractional sumudu transform to find the general solutions of some linear and nonhomogeneous conformable fractional differential equations. finally, the results show that our proposed method is an efficient method and applied successfully to find the general solutions of the all cases (singular, linear, nonlinear, homogeneous and nonhomogeneous) realted to the conformable fractional differential equations. 2. conformable fractional laplace and sumudu transforms in this section, we introduce (with proofs) a list of important basic rules and properties for the conformable fractional laplace and sumudu transforms involving the nice relationship between of these transforms which are playing a central role in the solutions of conformable fractional differential equations. definition 2.1.: let f : [0,∞) → r be a given function and 0 < α ≤ 1. then the conformable fractional laplace transform of f is defined as: lα{f(x)} = fα(s) = ∫ ∞ 0 e−s xα α f(x)xα−1dx, (2.1) provided the integral exists. int. j. anal. appl. 17 (6) (2019) 1023 theorem 2.1.: let f : [0,∞) → r be a given function and 0 < α ≤ 1. then lα{dαf(x)} = sfα(s) −f(0), s > 0. (2.2) proof. by using definition 2.1 and integration by parts, we have: lα{dαf(x)} = ∫ ∞ 0 e−s xα α dαf(x)xα−1dx = ∫ ∞ 0 e−s xα α x1−αf′(x)xα−1dx = ∫ ∞ 0 e−s xα α f′(x)dx = [ e−s xα α f(x) ]∞ 0 − ∫ ∞ 0 f(x) ( − s α α xα−1 ) e−s xα α dx = lim c→∞ [ e−s xα α f(x) ]c 0 + s ∫ ∞ 0 f(x) xα−1e−s xα α dx = sfα(s) −f(0), which completes the proof of theorem 2.1. � theorem 2.2 .: let f : [0,∞) → r be a function. then fα(s) = l { f(αx) 1 α } (s). (2.3) proof. by using definition 2.1 and letting v = x α α , we have: fα(s) = ∫ ∞ 0 e−s xα α f(x)xα−1dx = ∫ ∞ 0 e−svf(αv) 1 α dv = l { f(αv) 1 α } , which completes the proof of theorem 2.2. � theorem 2.3.: let c, a, p ∈ r and 0 < α ≤ 1. then (i) lα{c}(s) = c s , s > 0. (2.4) (ii) lα{xp}(s) = α p α γ(1 + p α ) s1+ p α , s > 0. (2.5) (iii) lα { e axα α } (s) = 1 s−a , s > a. (2.6) (iv) lα { sin( axα α ) } (s) = a s2 + a2 , s > 0. (2.7) (v) lα { cos( axα α ) } (s) = s s2 + a2 , s > 0. (2.8) (vi) lα { sinh( axα α ) } (s) = a s2 −a2 , s > |a| . (2.9) (vii) lα { cosh( axα α ) } (s) = s s2 −a2 , s > |a| . (2.10) proof. follows by applying definition 2.1. � int. j. anal. appl. 17 (6) (2019) 1024 theorem 2.4. : let f and g : [0,∞) → r andl let λ, µ, a ∈ r and 0 < α ≤ 1. then (i) lα{λf(x) + µg(x)} = λfα(s) + µgα(s),s > 0, (2.11) (ii) lα { e−a xα α f(x) } = fα(s + a), s > |a| , (2.12) (iii) lα{iαf(x)} = fα(s) s ,s > 0, (2.13) (iv) lα { xnα αn f(x) } = (−1)n dn dsn fα(s),s > 0, (2.14) (v) lα{(f ∗g)(x)} = fα(s)gα(s), s > 0, (2.15) where f ∗g is the convolution product of f and g. proof. (i) straightforward (ii) by using thereom 2.2, we get: lα { e−a xα α f(x) } = l { e −a α (αx) α. 1 α f(αx) 1 α } = l { e−axf(αx) 1 α } = l { f(αx) 1 α } |s→s+a = fα(s + a). (iii) by using theorem 2.1, we have: lα{dαiαf(x)} = slα{iαf(x)}− iαf(0). since iαf(0) = 0, then we obtain: fα(s) = slα{iαf(x)} lα{iαf(x)} = fα(s) s . (vi) by using theorem 2.2 , we obtain: lα { xnα αn f(x) } = l { (αx)nα. 1 α αn f(αx) 1 α } = l { xnf(αx) 1 α } = (−1)n dn dsn fα(s). (v) by using theorem 2.2, we get: lα{(f ∗g)(x)} = l { (f ∗g)(αx) 1 α } = l { f(αx) 1 α } l { g(αx) 1 α } = fα(s) gα(s). which completes the proof of theorem 2.4. � int. j. anal. appl. 17 (6) (2019) 1025 definition 2.2.: over the following set of functions: aα = { f(x) : ∃ m,τ1,τ2 > 0, |f(x)| < me |xα| ατj , if xα ∈ (−1)j × [0,∞),j = 1, 2 } , (2.16) then the conformable fractional sumudu transform of f can be generalized by: sα[f(x)] = fα(u) = 1 u ∫ ∞ 0 e −xα αu f(x)dαx, (2.17) where dαx = x α−1dx , 0 < α ≤ 1 and provided the integral exists. the relationship between the conformable fractional sumudu and conformable fractional laplace transforms is given as in the next result. theorem 2.5.: let f : [0,∞) → r be a given function and 0 < α ≤ 1.then sα[f(x)] = fα( 1u) u . (2.18) proof. applying definition 2.2, we get: sα[f(x)] = 1 u ∫ ∞ 0 e− xα αu f(x)dαx. letting v = x α α ⇒ dv = xα−1dx, then we have: sα[f(x)] = 1 u ∫ ∞ 0 e− v u f(αv) 1 α dv = 1 u l { f(αv) 1 α } s→ 1 u = fα( 1u) u , which completes the proof of theorem 2.5. � theorem 2.6. : let f : [0,∞) → r be a given function and 0 < α ≤ 1.then sα[dαf(x)] = fα(u) u − f(0) u . (2.19) proof. using theorems 2.5 and 2.1, we get: sα[dαf(x)] = lα{dαf(x)}s→ 1 u u = [sfα(s) −f(0)]s→ 1 u u = fα( 1u) u2 − f(0) u = fα(u) u − f(0) u , which completes the proof of theorem 2.6. � theorem 2.7.: let f : [0,∞) → r be an n-differentiable function and 0 < α ≤ 1.then sα[dnαf(x)] = sα[f(x)] un − f(0) un , 0 < α ≤ 1 and n ∈ n. (2.20) proof. follows by using the induction process on n and theorem 2.6. � int. j. anal. appl. 17 (6) (2019) 1026 now we introduce the basic rules of conformable fractional sumudu transform for some certain functions as in the next result. theorem 2.8.: let a, c ∈ r and 0 < α ≤ 1.then we have: (i) sα[c] = c. (2.21) (ii) sα [ ea xα α ] = 1 1 −au , u > 1 a . (2.22) (iii) sα [ sin(a xα α ) ] = au 1 + a2u2 , u > 1 |a| . (2.23) (iv) sα [ cos(a xα α ) ] = 1 1 + a2u2 , u > 1 |a| . (2.24) (v) sα [ sinh(a xα α ) ] = au 1 −a2u2 , u > 1 |a| . (2.25) (vi) sα [ cosh(a xα α ) ] = 1 1 −a2u2 , u > 1 |a| . (2.26) (vii) sα [ xnα αn ] = γ(n + 1)un, u > 0. (2.27) proof. by using theorems 2.5 and 2.2, we have: (i) sα[c] = lα{c}s→ 1 u u = { c s } s→ 1 u u = c. (ii) sα [ ea xα α ] = lα { ea xα α } s→ 1 u u = l{eax}s→ 1 u u = { 1 s−a } s→ 1 u u = 1 1 −au . (iii) sα [ sin(a xα α ) ] = lα { sin(ax α α ) } s→ 1 u u = l{sin(ax)}s→ 1 u u = { a s2+a2 } s→ 1 u u = au 1 + a2u2 . similarly we can prove (iv) and then can easy to prove (v) and (vi) based on (iii) and (iv) of theorem 2.8. (vii) sα [ xnα αn ] = lα { xnα αn } s→ 1 u u = γ(n + 1)un+1 u = γ(n + 1)un. which completes the proof of theorem 2.8. � int. j. anal. appl. 17 (6) (2019) 1027 theorem 2.9.: let f and g : [0,∞) → r given functiins and let λ, µ ∈ r and 0 < α ≤ 1.then we have: (i) linearly property: sα{λf(x) + µg(x)} = λfα(u) + µgα(u), (2.28) (ii) shifting property: sα [ e−a xα α f(x) ] = fα( 1u + a) u . (2.29) (iii) integral property: sα [iαf(x)] = fα( 1 u ). (2.30) (iv) convolution property: sα [(f ∗g)(x)] = ufα(u)gα(u). (2.31) (v) power product property: sα [ xnα αn f(x) ] = 1 u [ (−1)n dn dsn fα(s) ] s→ 1 u . (2.32) proof. (i) straightforward by using definition 2.2. (ii) by applying theorems 2.5 and 2.2, we have: sα [ e−a xα α f(x) ] = lα { e−a xα α f(x) } s→ 1 u u = l { e−axf(αx) 1 α } s→ 1 u u = fα( 1u + a) u . (iii) by using theorem 2.5 and eq.(2.13), we have: sα [iαf(x)] = lα{iαf(x)}s→ 1 u u = fα( 1 u ). (iv) by applying theorem 2.5 and eq. (2.15), we have: sα [(f ∗g)(x)] = lα{f ∗g}s→ 1 u u = [fα(s)gα(s)]s→ 1 u u = fα( 1u)gα( 1 u ) u = ufα(u)gα(u). (v) by using theorem 2.5 and eq.(2.3), we have: sα [ xnα αn f(x) ] = lα { xnα αn f(x) } s→ 1 u u = 1 u l { xnf(αx) 1 α } s→ 1 u = 1 u [ (−1)n dn dsn fα(s) ] s→ 1 u . which completes the proof of theorem 2.9. � int. j. anal. appl. 17 (6) (2019) 1028 theorem 2.10.: let y = f(x) be an n-differentiable function and 0 < α ≤ 1. then sα [ xα α dnαy ] = u d du (u sα[dnαy]) , n = 1, 2, ... (2.33) proof. let dαx = x α−1dx , then we have: sα[dnαy] = 1 u ∫ ∞ 0 e− xα αu dnαy dαx d du (u sα[dnαy]) = ∫ ∞ 0 d du e− xα αu dnαy dαx = 1 u2 ∫ ∞ 0 e− xα αu xα α dnαy dαx = 1 u ∫ ∞ 0 1 u e− xα αu xα α dnαy dαx = 1 u sα [ xα α dnαy ] . which concludes the result as in eq. (2.33). � 3. analytical solution of a singular and nonlinear conformable fractional poisson-boltzmann differential equation in this section, we apply the conformable fractional sumudu transform togethor with adomain decomposition method to present the general analytical solution of the following singular and nonlinear conformable fractional poisson boltzmann differential equation: d2αy + β xα dαy = ey, y(0) = 0, (3.1) where 0 < α ≤ 1 and β > 0. to solve this problem: multiply eq. (3.1) by x α α and take sαof both sides, then by applying theorems 2.10 and 2.6, we get: u d du ( yα(u) u ) + γ ( yα(u) u ) = sα ( xα α ey ) , (3.2) where γ = β α . since u 6= 0 and integrating eq. (3.2) with respect to z, we get: yα(u) u = ∫ u 0 1 z sα [ xα α ey ] dz −γ ∫ u 0 yα(z) z2 dz (3.3) yα(u) = u ∫ u 0 1 z sα [ xα α ey ] dz −uγ ∫ u 0 yα(z) z2 dz. (3.4) suppose the solution y(x) and non-linear function ey by the following infinite series: y(x) = ∞∑ n=0 yn(x) , e y = ∞∑ n=0 an(x), (3.5) int. j. anal. appl. 17 (6) (2019) 1029 where an is the adomain polynomial of e y. note that: a◦ = e y◦, a1 = y1e y◦, (3.6) a2 = y2e y◦ + 1 2! y21e y◦, ... substituting eq. (3.5) into eq.(3.4), we have: sα [ ∞∑ n=0 yn(x) ] = u ∫ u 0 1 z sα [ xα α an ] dz −uγ ∫ u 0 sα[ ∑∞ n=0 yn(x)] z2 dz. (3.7) by taking s−1α of both sides of eq (3.7), we obtain: ∞∑ n=0 yn(x) = s−1α [ u ∫ u 0 1 z sα[ xα α an]dz −uγ ∫ u 0 sα[ ∑∞ n=0 yn(x)] z2 dz ] . thus, the general solution of eq.(3.1) is given by: y(x) = s−1α [ u ∫ u 0 1 z sα[ xα α an]dz −uγ ∫ u 0 sα[ ∑∞ n=0 yn(x)] z2 dz ] , (3.8) where y◦ = 0 yn+1 = s−1α [ u ∫ u 0 1 z sα[ xα α an]dz −uγ ∫ u 0 sα[yn(x)] z2 dz ] . (3.9) now, (i) for n = 0, then: y1 = s−1α [ u ∫ u 0 1 z sα[ xα α a◦]dz −uγ ∫ u 0 sα[y◦] z2 dz ] = s−1α [ u ∫ u 0 1 z sα[ xα α ]dz ] = 1 2! x2α α2 . (3.10) (ii) for n = 1, then: y2 = s−1α [ u ∫ u 0 1 z sα[ xα α a1]dz −uγ ∫ u 0 sα[y1] z2 dz ] = s−1α [ u ∫ u 0 1 γ(3)z sα[ x3α α3 ]dz −uγ ∫ u 0 sα[x 2α α2 ] γ(3)z2 dz ] = s−1α [ γ(4) 3γ(3) u4 −γu2 ] = 1 4! x4α α4 − γ 2! x2α α2 . (3.11) int. j. anal. appl. 17 (6) (2019) 1030 (iii) for n = 2, then y3 = s−1α [ u ∫ u 0 1 z sα [ xα α a2 ] dz −uγ ∫ u 0 sα[y2] z2 dz ] = s−1α [ u ∫ u 0 1 z sα [ xα α y2e y◦ + 1 2! xα α y21e y◦ ] dz −uγ ∫ u 0 sα[y2] z2 dz ] = γ(6) 5γ(7) x6α α6 − γ(4) 6γ(5) γ x4α α4 + γ(6) 40γ(7) x6α α6 − γ2 2! x2α α2 − 1 3γ(5) x4α α4 . (3.12) now by above discussion, then it is easy to get the power series solution of eq. (3.1) as follows: y(x) = y◦ + y1 + y2 + y3 + ....... = 1 2! x2α α2 + 1 4! x4α α4 − γ 2! x2α α2 + ..... ((3.13)) 4. analytical solutions of some linear and nonhomogeneous conformable fractional differential equations in this section, we apply the conformable fractional sumudu transform to present the general analytical solutions of some linear and nonhomogeneous conformable fractional differential equations as in the following problems. problem 4.1.: consider the following linear and nonhomogeneous conformable fractional differential equation: d3αy + dαy = xα α , y(0) = 0. (4.1) by applying the conformable fractional sumudu transform sα of both sides of eq. (4.1) and using theorem 2.7, then we get: yα(u) u3 + yα(u) u = γ(2)u, which implies that: yα(u) [ 1 + u2 u3 ] = u. thus, yα(u) = u4 1 + u2 . (4.2) by taking s−1α of both sides of eq. (4.2), then we obtain the general solution of eq. (4.1) as follows: y(x) = s−1α [ u2 − 1 + 1 u2 + 1 ] = x2α 2α2 + cos xα α − 1. (4.3) int. j. anal. appl. 17 (6) (2019) 1031 problem 4.2.: consider the following linear and nonhomogeneous conformable fractional differential equation: d2αy + y = e 2xα α , y(0) = 1. (4.4) by applying sα of both sides of eq. (4.4) and using theorems 2.7 and 2.8, we obtain: yα(u) u2 − 1 u2 + yα(u) = 1 1 − 2u , which implies that: yα(u) [ 1 + u2 u2 ] = 1 1 − 2u + 1 u2 . (4.5) by taking s−1α of both sides of eq. (4.5), then we obtain the general solution of eq. (4.4) as follows: y(x) = s−1α [ u2 (1 + u2)(1 − 2u) ] + s−1α [ 1 1 + u2 ] = s−1α [ au + b 1 + u2 ] + s−1α [ c 1 − 2u ] + s−1α [ 1 1 + u2 ] = −2 5 s−1α [ u 1 + u2 ] − 1 5 s−1α [ 1 1 + u2 ] + 1 5 s−1α [ 1 1 − 2u ] + s−1α [ 1 1 + u2 ] = − 2 5 sin xα α + 4 5 cos xα α + 1 5 e2 xα α . (4.6) problem 4.3.: consider the following linear and nonhomogeneous conformable fractional differential equation: d2αy −y = cos( xα α ), y(0) = 0. (4.7) by applying sα of both sides of eq. (4.7) and using theorems 2.7 and 2.8, we obtain: yα(u) u2 −yα(u) = 1 1 + u2 , which implies that: yα(u) = u2 (1 −u2)(1 + u2) . (4.8) by taking s−1α of both sides of eq. (4.8), then we obtain the general solution of eq. (4.7) as follows: y(x) = s−1α [ u2 (1 −u2)(1 + u2) ] = s−1α [ au + b 1 −u2 ] + s−1α [ du + c 1 + u2 ] = 1 2 s−1α 1 1 −u2 − 1 2 s−1α [ 1 1 + u2 ] = 1 2 cosh xα α − 1 2 cos xα α . (4.9) int. j. anal. appl. 17 (6) (2019) 1032 5. conclusion we have developed the conformable fractional sumudu transform for presenting the general analytical solution of the singular and nonlinear conformable fractional poisson -boltzmann equation and also for presenting the general solutions of some linear and nonhomogeneuos conformable fractional differential equations. in our opinion, it is worth to extend some other conformable fractional transforms such as: the conformable fractional natural and conformable fractional mellin trabsforns and using them in many application and comparisons. how to use conformable fractional sumudu and laplace transforns for solving some other nonlinear and singular conformable fractional differential equations such as conformable fractional laneemden and conformable fractional van der pol oscillator differential equations still need further researches. references [1] m. herzallah, notes on some fractional calculus operators and their properties, j. frac. calc. appl., 5 (35) (19) (2014), 1-10. [2] m. caputo, linear model of dissipation whose q is almost frequency independent, geophys. j. int., 13 (5) (1967), 529-539. [3] k. s. miller and b. ross , an introduction to the fractional calculus and fractional differential equations, wiley and sons, ny, usa, 1993. [4] i. pondlubny, fractional differential equations:an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, academic press, usa, 1998. [5] a. el-ajou, o. abu arqub, z. al-zhour and s. momani, new results on fractional power series: theories and applications, entropy, 15 (2013), 5305-5323. [6] a. kilicman and z. al-zhour, kronecker operational matrices for fractional calculus and some applications, appl. math. comput., 187 (2007), 250-265. [7] v. daftardar-gejji, fractional calculus theory and applications, narosa publishing house, 2014. [8] j. leszczynski, an introduction of fractional mechanics, czestochowa university of technology, 2011. [9] a. atangana, s. demiray and h. bulut, modelling the non-linear wave motion within the scope of the fractional calculus, abstr. appl. anal., 2014 (2014), art. id 481657, 7 pages. [10] g. adomian, solving frontier problems of physics the decomposition method, kluwer academic publishers, 1994. [11] r. khalil, m. horani, a. yousef and m. sababheh, a new definition of fractional derivative, j. comput. appl. math., 264 (2014), 65-70. [12] t. abdeljawad, on conformable fractional calculus, j. comput. appl. math., 279 (2015), 57-66. [13] m. abu hammad and r. khalil, conformable fractional heat differential equation, int. j. pure appl. math., 94(2) (2014), 215-221. [14] z. ayati, j. biazar and m. ilei, general solution of bernoulli and riccati fractional differential equations based on conformable fractional derivative, int. j. appl. math. res., 6(2) (2017), 49-51. [15] u. ghosh, m. sarkar and d. shantanu, solution of linear fractional non-homogeneous differential equations with jumarie fractional derivative and evaluation of particular integral, amer. j. math. anal., 3(3) (2015), 54-64. [16] l. wang and j. fu, non-noether symmetries of hamiltonian systems with conformable fractional derivatives, chin. phys. b, 25(1) (2016), 4501. int. j. anal. appl. 17 (6) (2019) 1033 [17] h. guebbai and m. ghiat, new conformable fractional derivative definition for positive and increasing functions and its generalization, adv. dyn. syst. appl., 11 (2) (2016), 105-111. [18] a kareem, conformable fractional derivatives and it is applications for solving fractional differential equations, iosr j. math., 13 (2017), 81-87. [19] a. h. khader, the conformable laplace transform of the fractional chebyshev and legendre polynnomials, msc.thesis, zarqa university, 2017. [20] g. k. watugala, sumudu transform: a new integral transform to solve differential equations and control engineering problems, int. j. math. educ. sci. technol., 24 (1) (1993), 35-43. [21] m. a. asiru, sumudu transform and solution of integral equations of convolution type, int. j. math. educ. sci., 32 (6)(2001), 906-910. [22] s. demiray, h. bulut and f. bin muhammad, sumudu transform method for analytical solutions of fractional type ordinary differential equations, math. problems eng., 6 ( 2015), 131-690. [23] v. g. gupta, b. shrama and a. kilicman, a note on fractional sumudu transform, j. appl. math., 2010 (2010), art. id 154189, 9 pages. [24] m. a asiru, further properties of the sumudu transform and its applications, int. j. math. educ. sci. technol., 33 (3) (2002), 441–449. [25] m. a asiru, classroom note: application of the sumudu transform to discrete dynamic systems, int. j. math. educ. sci. technol., 34 (6) (2003), 944–949 [26] a. kılıcman, h. eltayeb and p. r. agarwal, on sumudu transform and system of differential equations, abstr. appl. anal., 2010 (2010), art. id 598702, 11 pages. [27] kılıcman a. and gadain h., on the applications of laplace and sumudu transforms, j. franklin inst., 347 (5) (2010) 848-862. [28] j. vashi and m. g.timol, laplace and sumudu transforms and their application, int. j. innov. sci., eng. technol., 3 (8) (2016), 538-542. [29] s. demiray, y. pandir and h. bulut, generalized kudryashov method for time-fractional differential equations, abstr. appl. anal., 2014 (2014), art. id 901540, 13 pages. [30] r. mittal and r. nigam, solution of fractional integraldifferential equations by a domian decomposition method, int. j. appl. math. mech., 4 (2) (2008), 87-94. [31] g. adomian, differential equation with singular coefficients, j. appl. math. comput., 47 (1992), 179-184. [32] a.tandogan, y. pandir and y. gurefe, solutions of the nonlinear differential equations by use of modified kudryashov method, turk. j. math computer sci., 2013 (2013), art. id 20130021, 7 pages. [33] d. a. frank-kamenetsky, dokl. akad. nauk ussr 18, 411, 1938. [34] d. a. frank-kamenetsky, diffusion and heat transfer in chemical kinetics, 2nd ed., plenum, new york, 1969. [35] g. i. barenblatt, j. b. bell and w. y. crutchfield, the thermal explosion revisited, proc. natl. acad. sci., 95 (1998), 13384-13386. [36] p. l. chambré, on the solution of the poisson-boltzmann equation with application to the theory of thermal explosions, j. chem. phys., 20 (1952), 1795-1797. [37] l. samaj and e. trizac, poisson-boltzmann thermodynamics of counterions confined by curved hard walls, phys. rev., 93 (2016), 012601. 1. introduction and preliminaries 2. conformable fractional laplace and sumudu transforms 3. analytical solution of a singular and nonlinear conformable fractional poisson-boltzmann differential equation 4. analytical solutions of some linear and nonhomogeneous conformable fractional differential equations 5. conclusion references int. j. anal. appl. (2022), 20:32 a plancherel theorem on a noncommutative hypergroup brou kouakou germain1,∗, ibrahima toure2, kinvi kangni2 1université de man, côte d’ivoire 2université félix houphouet boigny, côte d’ivoire ∗corresponding author: germain.brou@univ-man.edu.ci, broukouakou320@yahoo.fr abstract. let g be a locally compact hypergroup and let k be a compact sub-hypergroup of g. (g,k) is a gelfand pair if mc(g//k), the algebra of measures with compact support on the double coset g//k, is commutative for the convolution. in this paper, assuming that (g,k) is a gelfand pair, we define and study a fourier transform on g and then establish a plancherel theorem for the pair (g,k). 1. introduction hypergroups generalize locally compact groups. they appear when the banach space of all bounded radon measures on a locally compact space carries a convolution having all properties of a group convolution apart from the fact that the convolution of two point measures is a probability measure with compact support and not necessarily a point measure. the intention was to unify harmonic analysis on duals of compact groups, double coset spaces g//h (h a compact subgroup of a locally compact group g), and commutative convolution algebras associated with product linearization formulas of special functions. the notion of hypergroup has been sufficiently studied (see for example [2,4,6,7]). harmonic analysis and probability theory on commutative hypergroups are well developed meanwhile where many results from group theory remain valid (see [1]). when g is a commutative hypergroup, the convolution algebra mc(g) consisting of measures with compact support on g is commutative. the typical example of commutative hypergroup is the double coset g//k when g is a locally compact group, k is a compact subgroup of g such that (g,k) is a gelfand pair. in [4], r. i. jewett has shown the existence of a positive measure called plancherel measure on the dual space ĝ of a commutative hypergroup g. when the hypergroup g is not commutative, it is possible to involve a received: may 24, 2022. 2010 mathematics subject classification. 43a62, 43a22, 20n20. key words and phrases. hypergroups; gelfand pair; probability measure; plancherel theorem; multiplicative function. https://doi.org/10.28924/2291-8639-20-2022-32 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-32 2 int. j. anal. appl. (2022), 20:32 compact sub-hypergroup k of g leading to a commutative subalgebra of mc(g). in fact, if k is a compact sub-hypergroup of a hypergroup g, the pair (g,k) is said to be a gelfand pair if mc(g//k) the convolution algebra of measures with compact support on g//k is commutative. the notion of gelfand pairs for hypergroups is well-known (see [3, 8, 9]). the goal of this paper is to extend jewett work’s by obtaining a plancherel theorem over gelfand pair associated with non-commutative hypergroup. in the next section, we give notations and setup useful for the remainder of this paper. in section 3, we introduce first the notion of k-multiplicative functions and obtain some of their characterizations. thanks to these results, we establish a one to one correspondence between the space of k-multiplicative functions and the dual space of g. then, we define a fourier tranform on mb(g), the algebra of bounded measures on g and on k(g), the algebra of continuous functions on g with compact support. finally, using the fact that g//k is a commutative hypergroup, we prove that there exists a nonnegative measure (the plancherel measure) on the dual space of g. 2. notations and preliminaries we use the notations and setup of this section in the rest of the paper without mentioning. let g be a locally compact space. we denote by: c(g) (resp. m(g)) the space of continuous complex valued functions (resp. the space of radon measures) on g, cb(g) (resp. mb(g)) the space of bounded continuous functions (resp. the space of bounded radon measures) on g, k(g) (resp. mc(g)) the space of continuous functions (resp. the space of radon measures) with compact support on g, c0(g) the space of elements in c(g) which are zero at infinity, c(g) the space of compact sub-space of g, δx the point measure at x ∈ g, spt(f ) the support of the function f . let us notice that the topology on m(g) is the cône topology [4] and the topology on c(g) is the topology of michael [5]. definition 2.1. g is said to be a hypergroup if the following assumptions are satisfied. (h1) there is a binary operator ∗ named convolution on mb(g) under which mb(g) is an associative algebra such that: i) the mapping (µ,ν) 7−→ µ∗ν is continuous from mb(g)×mb(g) in mb(g). ii) ∀x,y ∈ g, δx ∗δy is a measure of probability with compact support. iii) the mapping: (x,y) 7−→ supp(δx ∗δy) is continuous from g ×g in c(g). (h2) there is a unique element e (called neutral element) in g such that δx ∗ δe = δe ∗ δx = δx,∀x ∈ g. int. j. anal. appl. (2022), 20:32 3 (h3) there is an involutive homeomorphism: x 7−→ x from g in g, named involution, such that: i) (δx ∗ δy)− = δy ∗ δx,∀x,y ∈ g with µ−(f ) = µ(f−) where f−(x) = f (x),∀f ∈ c(g) and µ ∈ m(g). ii) ∀x,y,z ∈ g, z ∈ supp(δx ∗δy) if and only if x ∈ supp(δz ∗δy). the hypergroup g is commutative if δx ∗δy = δy ∗δx,∀x,y ∈ g. for x,y ∈ g, x ∗y is the support of δx ∗δy and for f ∈ c(g), f (x ∗y)≡ (δx ∗δy)(f )= ∫ g f (z)d(δx ∗δy)(z). the convolution of two measures µ,ν in mb(g) is defined by: ∀f ∈ c(g) (µ∗ν)(f )= ∫ g ∫ g (δx ∗δy)(f )dµ(x)dν(y)= ∫ g ∫ g f (x ∗y)dµ(x)dν(y), for µ in mb(g), µ∗ =(µ)−. so mb(g) is a *-banach algebra. definition 2.2. h ⊂ g is a sub-hypergroup of g if the following conditions are satisfied. (1) h is non empty and closed in g, (2) ∀x ∈ h,x ∈ h, (3) ∀x,y ∈ h, supp(δx ∗δy)⊂ h. let us now consider a hypergroup g provided with a left haar measure µg and k a compact subhypergroup of g with a normalized haar measure ωk. let us put mµg(g) the space of measures in mb(g) which are absolutely continuous with respect to µg. mµg(g) is a closed self-adjoint ideal in mb(g). for x ∈ g, the double coset of x with respect to k is k∗{x}∗k = {k1 ∗x ∗k2;k1,k2 ∈ k}. we write simply kxk for a double coset and recall that kxk = ⋃ k1,k2∈k supp(δk1∗δx ∗δk2). all double coset form a partition of g and the quotient topology with respect to the corresponding equivalence relation equips the double cosets space g//k with a locally topology ( [1], page 53). the natural mapping pk : g −→ g//k defined by: pk(x) = kxk ,x ∈ g is an open surjective continuous mapping. a function f ∈ c(g) is said to be invariant by k or k − invariant if f (k1 ∗x ∗k2)= f (x) for all x ∈ g and for all k1,k2 ∈ k. we denote by c\(g), (resp. k\(g)) the space of continuous functions (resp. continuous functions with compact support) which are k−invariant. for f ∈ c\(g), one defines the function f̃ on g//k by f̃ (kxk)= f (x) ∀x ∈ g. f̃ is well defined and it is continuous on g//k. conversely, for all continuous function ϕ on g//k, the function f = ϕ◦pk ∈ c\(g). one has the obvious consequence that the mapping f 7−→ f̃ sets up a topological isomorphism between the topological vector spaces c\(g) and c(g//k) (see [8,9]). so, for any f in c\(g), f = f̃ ◦pk. otherwise, we consider the k-projection f 7−→ f \ (by identifying f \ and f̃ \) from c(g) into c(g//k) where for x ∈ g,f \(x) = ∫ k ∫ k f (k1 ∗ x ∗ k2)dωk(k1)dωk(k2). if f ∈ k(g), then f \ ∈ k(g//k). for a measure µ ∈ m(g), one defines µ \ by µ \ (f )= µ(f \) for f ∈k(g). µ is said to be k−invariant if µ \ = µ and we denote by m\ (g) the set of all those measures. considering these properties, one 4 int. j. anal. appl. (2022), 20:32 defines a hypergroup operation on g//k by: δkxk ∗δkyk(f̃ )= ∫ k f (x∗k∗y)dωk(k) (see [2, p. 12] ). this defines uniquely the convolution (kxk) ∗ (kyk) on g//k. the involution is defined by: kxk = kxk and the neutral element is k. let us put m = ∫ g δkxkdµg(x), m is a left haar measure on g//k. we say that (g,k) is a gelfand pair if the convolution algebra mc(g//k) is commutative. mc(g//k) is topologically isomorphic to m \ c (g). considering the convolution product on k(g), k(g) is a convolution algebra and k\(g) is a subalgebra. thus (g,k) is a gelfand pair if and only if k\(g) is commutative ( [3], theorem 3.2.2). 3. plancherel theorem let g be a locally compact hypergroup and let k be a compact sub-hypergroup of g. in this section, we assume that (g,k) is a gelfand pair. 3.1. k-multiplicative functions. let us put g\ b the space of continuous, bounded function φ on g such that: (i) φ is kinvariant, (ii) φ(e)=1, (ii) ∫ k φ(x ∗k ∗y)dwk(k)= φ(x)φ(y) ∀x,y ∈ g. let ĝ be the sub-space of g\ b containing the elements φ in g\ b such that φ(x)= φ(x) ∀x ∈ g. ĝ is called the dual space of the hypergroup g. remark 3.1. (1) if φ ∈ ĝ, then φ− ∈ ĝ. (2) equipped with the topology of uniform convergence on compacta, ĝ is a locally compact hausdorff space. (3) in general, ĝ is not a hypergroup. definition 3.2. a complex-valued function χ on g will be called a multiplicative (resp. kmultiplicative) function if χ is continuous and not identically zero, and has the property that: χ(x ∗y)= χ(x)χ(y) (resp. ∫ k χ(x ∗k ∗y)dwk(k)= χ(x)χ(y)) ∀x,y ∈ g. a multiplicative (resp. k-multiplicative) function on mb(g) is a continuous complex-valued function f not identically zero on m\ b (g), and has the property that: f(µ∗ν)= f(µ)f(ν) (resp. f(µ∗wk ∗ν)= f(µ)f(ν)) ∀µ,ν ∈ mb(g). for χ ∈ cb(g), not identically zero, let put fχ(µ)= ∫ g χdµ for µ ∈ mb(g). int. j. anal. appl. (2022), 20:32 5 proposition 3.3. let f be a k-multiplicative function on mb(g), then: i) f is multiplicative on m\ b (g). ii) f(wk)= f(δe)=1. iii) ∀µ ∈ mb(g), f(µ\)= f(µ) iv) ∀k ∈ k, f(δk)=1. proof. i) just remember that µ∗wk = µ,∀µ ∈ m \ b (g). ii) let ν ∈ m\ b (g) such that f(ν) 6=0. f(ν)= f(ν ∗wk)= f(ν)f(wk) =⇒ f(wk)=1. f(ν)= f(ν ∗wk ∗δe)= f(ν)f(δe) =⇒ f(δe)=1. iii) let µ ∈ mb(g). since µ\ = wk ∗µ∗wk, we have f(µ\) = f(wk ∗µ∗wk) = f(wk ∗µ∗wk ∗wk) = f(wk ∗µ) = f(δe ∗wk ∗µ) = f(µ). iv) if k ∈ k, δ\ k = wk. using (ii) and (iii), we have f(δk)=1. � proposition 3.4. let φ ∈ g\ b . i) fφ is a bounded linear k-multiplicative function on mb(g). ii) fφ is not identically zero on m \ µg(g). proof. i) that is clear that fφ is linear and bounded. let µ,ν ∈ mb(g). we have fφ(µ∗wk ∗ν)= ∫ g ∫ k ∫ g φ(x ∗k ∗y)dµ(x)dwk(k)dν(y) = ∫ g φ(x)dµ(x) ∫ g φ(x)dν(y) = fφ(µ)fφ(ν). morever, fφ(wk)= ∫ k φ(k)dwk(k)=1 6=0. ii) if µ ∈ mµg(g), then µ \ = wk ∗µ∗wk ∈ m \ µg(g). let f ∈k(g) with spt(f )⊂ k such that∫ g f (x)dug(x)=1. f \µg ∈ m \ µg(g) and fφ(f \µg)= fφ(f µg) = ∫ g φ(x)f (x)dug(x) = ∫ k f (x)dug(x) =1 6=0. 6 int. j. anal. appl. (2022), 20:32 � theorem 3.5. 1) let e be a multiplicative linear function on m\µg(g) not identically zero. there exists a unique k-multiplicative linear function f on mb(g) such that f = e on m \ µg(g). 2) let f be a bounded linear k-multiplicative function on mb(g) not identically zero on m \ µg(g).there exists a unique function φ in g \ b such that f = fφ. proof. 1) let ν ∈ m\µg(g) such that e(ν) 6=0 and put f(µ)= e(µ\ ∗ν) e(ν) , f or µ ∈ mb(g) f is well defined since mµg(g) is an ideal in mb(g). let us first see that f is multiplicative on m\ b (g). for µ and µ′ in m\ b (g), we have f(µ∗µ′)= e(µ∗µ′ ∗ν) e(ν) = e(ν ∗µ∗µ′ ∗ν) e(ν)2 = e(ν ∗µ) e(ν) e(µ′ ∗ν) e(ν) = e(ν ∗µ∗ν) e(ν)2 f(µ′) = f(µ)f(µ′). moreover f(wk)= e(wk ∗ν) e(ν) = e(ν) e(ν) =1. so for µ and µ′ in mb(g), we have f(µ∗wk ∗µ′)= f(wk ∗ (wk ∗µ∗wk)∗ (wk ∗µ′ ∗wk)∗wk) = f((wk ∗µ∗wk)∗ (wk ∗µ′ ∗wk)) = f(µ\ ∗µ′\) = f(µ)f(µ′). the uniqueness stems from proposition 3.3. 2) let f be a bounded linear k-multiplicative function on mb(g). let ν ∈ m \ µg(g) such that f(ν) 6=0. if µ1,µ2 ∈ mb(g) then |f(µ1)−f(µ2)|= ∣∣∣f(µ\1)−f(µ\2)∣∣∣ = ∣∣∣f(µ\1 ∗ν)−f(µ\2 ∗ν)∣∣∣ |f(ν)| = ∣∣f((µ1 ∗ν −µ2 ∗ν)\)∣∣ f(ν) ≤ ‖f‖ f(ν) ‖µ1 ∗ν −µ2 ∗ν‖ . int. j. anal. appl. (2022), 20:32 7 thus f is positive-continuous by ( [4], theorem 5.6b). by ( [4], theorem 2.2d) there exists a bounded continuous function h on g such that f(µ)= ∫ g h(x)dµ(x). so φ = h. � 3.2. fourier transform on mb(g). definition 3.6. let µ ∈ mb(g), the fourier transform of µ is the map µ̂ : ĝ −→ c defined by: µ̂(φ)= ∫ g φ(x)dµ(x). proposition 3.7. i) for µ ∈ mb(g), µ̂ ∈ cb(ĝ). ii) for µ ∈ mb(g), µ̂ = µ̂ \ . iii) for µ ∈ mµg(g), µ̂ ∈ c0(ĝ). iv) if µ ∈ m\ b (g) and ν ∈ mb(g), then µ̂∗ν = µ̂ν̂. proof. i) we can see that, µ̂(φ)= µ(φ) ∀φ ∈ ĝ. ii) for φ ∈ ĝ, we have µ̂(φ)= fφ−(µ). so µ̂ \ (φ)= fφ−(µ \ )= fφ−(µ)= µ̂(φ). iii) this comes from theorem 3.5 and ( [4], theorem 6.3g) iv) let φ belongs to ĝ, we have µ̂∗ν(φ)= ∫ g φ−(x)dµ∗ν(x) = ∫ g ∫ g φ−(x ∗y)dµ(x)dν(y) = ∫ g [∫ g ( ∫ k ∫ k φ−(k1 ∗x ∗k2 ∗y)dωk(k1)dωk(k2))dµ(x) ] dν(y) = ∫ g [∫ g ( ∫ k ( ∫ k φ−((k1 ∗x)∗k2 ∗y)dωk(k2))dωk(k1))dµ(x) ] dν(y) = ∫ g φ−(y) [∫ g ( ∫ k φ−(k1 ∗x)dωk(k1))dµ(x) ] dν(y) = ∫ g φ−(y) [∫ g (φ−(x)dµ(x) ] dν(y) = ∫ g φ−(x)dµ(x) ∫ g φ−(y)dν(y) = µ̂(φ)ν̂(φ). � remark 3.8. by the definition, the mapping µ 7−→ µ̂ from mb(g) to cb(ĝ) is continuous. 3.3. fourier transform on g. definition 3.9. let f ∈ k\(g), the fourier transform of f is the map f̂ : ĝ −→ c defined by: f̂ (φ)= ∫ g φ(x)f (x)dug(x) 8 int. j. anal. appl. (2022), 20:32 proposition 3.10. i) for f ∈k(g), f̂ \ = f̂ µg ∈ c0(ĝ). ii) if f ∈k\(g) and g ∈k(g), then ̂f ∗g = f̂ ĝ\. proof. i) for any f in k(g), we have f̂ \(φ)= ∫ g φ−(x)( ∫ k ∫ k f (k1 ∗x ∗k2)dωk(k1)dωk(k2))dug(x) = ∫ g f (x)( ∫ k ∫ k φ−(k1 ∗x ∗k2)dωk(k1)dωk(k2))dug(x) = ∫ g φ(x)f (x)dug(x)= f̂ µg(φ) ∀φ ∈ ĝ since f µg ∈ mµg(g), then f̂ µg ∈ c0(ĝ). ii) let f ∈k\(g) and g ∈k(g). for φ ∈ ĝ, we have ̂f ∗g(φ)= ∫ g φ−(x)f ∗g(x)dµg(x) = ∫ g φ−(x)( ∫ g f (x ∗y)g(y)dµg(y))dµg(x) = ∫ g g(y)( ∫ g φ−(x ∗y)f (x)dµg(x))dµg(y) = ∫ g g(y) ∫ k ∫ k ∫ g φ−(k1 ∗x ∗k2 ∗y)f (x)dµg(x)dωk(k1)dωk(k2)dµg(y) = ∫ g g(y)φ−(y)dµg(y) ∫ g f (x) ∫ k φ−(k1 ∗x)dωk(k1)dµg(x) = ∫ g φ−(y)g(y)dµg(y) ∫ g φ−(x)f (x)dωk(k1)dµg(x) = f̂ (φ)ĝ(φ). � we therefore extend the spherical fourier transform to all k(g) with f̂ = f̂ \ for any f ∈k(g) and to l1(g,µg) and l 2(g,µg). we have the following result. theorem 3.11. there exists a unique nonnegative measure π on ĝ such that∫ g |f (x)|2dµg(x)= ∫ ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ) for all f in l1(g,µg)∩l2(g,µg). the space { f̂ : f ∈k(g) } is dense in l2(ĝ,π). proof. considering the space ĝ//k defined by [4], φ̃ ∈ ĝ//k if and only if φ = φ̃◦pk ∈ ĝ. let ϕ̃ belongs to cb(ĝ//k). let us consider ϕ : ĝ −→c defined by: ϕ(φ)= ϕ̃(φ̃). int. j. anal. appl. (2022), 20:32 9 ϕ ∈ cb(ĝ) and the mapping cb(ĝ//k) −→ cb(ĝ) ϕ̃ 7−→ ϕ is a linear bijection, specificaly ϕ ∈ k(ĝ) ⇐⇒ ϕ̃ ∈ k(ĝ//k). by ( [4], theorem. 7.3i), there exist a unique nonnegative measure π̃ on ĝ//k such that ∫ g//k ∣∣∣f̃ (kxk)∣∣∣2dm(kxk) =∫ ĝ//k ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) for f̃ ∈ l1(g//k,m)∩ l2(g//k,m). let us consider the mapping π defined by π(ϕ) = π̃(ϕ̃) for ϕ ∈ k(ĝ).π is a measure on ĝ. since π̃ is nonnegative, then π is nonnegative. otherwise, note that ˜̂ f = ̂̃ f for f ∈ k\(g). indeed since f ∈ k\(g) then f̃ ∈ k(g//k) and f̂ ∈ cb(ĝ). so ̂̃ f and ˜̂ f belong to cb(ĝ//k). for φ̃ ∈ ĝ//k, we have ̂̃ f (φ̃)= ∫ g//k φ̃(kxk)f̃ (kxk)dm(kxk) = ∫ g//k φ̃−(kxk)f̃ (kxk)dm(kxk) = ∫ g φ−(x)f (x)dug(x) = f̂ (φ)= ˜̂ f (φ̃) let f ∈k\(g). we have ∫ ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ)= ∫ ĝ//k ∣∣∣∣˜̂f (φ̃) ∣∣∣∣2dπ̃(φ̃) = ∫ ĝ//k ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) = ∫ g//k ∣∣∣f̃ (kxk)∣∣∣2dm(kxk) = ∫ g |f (x)|2dµg(x). as f̂ = f̂ \ ∀f ∈k(g) and g unimodular, we deduce that ∫ ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ)= ∫g |f (x)|2dµg(x) ∀f ∈ k(g). by the continuity of the fourier transform and by application of the dominated convergence theorem, we conclude that ∫ g |f (x)|2dµg(x) = ∫ ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ) for any f belongs to l1(g,µg)∩ l2(g,µg). let π ′ a nonnegative measure on ĝ such that ∫ g |f (x)|2dµg(x)= ∫ ĝ ∣∣∣f̂ (φ)∣∣∣2dπ′(φ) for all f in l1(g,µg)∩l2(g,µg). as above but in reverse order π′ defines a nonnegative measure π̃′ on ĝ//k such that ∫ g//k ∣∣∣f̃ (kxk)∣∣∣2dm(kxk) = ∫ ĝ//k ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) for f̃ ∈ l1(g//k,m)∩ l2(g//k,m). that is π̃′ = π̃ seen the uniqueness of π̃, so π = π′. let us put f(k(g)) = 10 int. j. anal. appl. (2022), 20:32{ f̂ ; f ∈k(g } . let ϕ ∈k(ĝ) such that 〈 f̂ ,ϕ 〉 = ∫ ĝ f̂ (φ)ϕ(φ)dπ(φ)=0 ∀f ∈k\(g). we have〈 f̂ ,ϕ 〉 =0 ∀f ∈k\(g) =⇒ ∫ ĝ f̂ (φ)ϕ(φ)dπ(φ)=0 ∀f ∈k\(g) =⇒ ∫ ĝ ˜̂ f (φ̃)ϕ̃(φ̃)dπ̃(φ̃)=0 ∀f ∈k\(g) =⇒ 〈˜̂ f , ϕ̃ 〉 =0 ∀f ∈k(g) =⇒ 〈̂̃ f , ϕ̃ 〉 =0 ∀f̃ ∈k(g//k) =⇒ ϕ̃ =0 since f(k(g//k)) is dense in l2(ĝ//k,π̃) =⇒ ϕ =0. so (f(k(g)))⊥ ∩k(ĝ) = {0}. since k(ĝ) is dense in l2(ĝ,π), then (f(k(g)))⊥ = {0} and f(k(g)) is dense in l2(ĝ,π). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] w.r. bloom, h. heyer, harmonic analysis of probability measures on hypergroups, de gruyter studies in mathematics 20, walter de gruyter, berlin, 1995. [2] c.f. dunkl, the measure algebra of a locally compact hypergroup, trans. amer. math. soc. 179 (1973), 331–348. https://doi.org/10.1090/s0002-9947-1973-0320635-2. [3] b.k. germain, k. kinvi, on gelfand pair over hypergroups, far east j. math. 132 (2021), 63–76. https://doi. org/10.17654/ms132010063. [4] r.i. jewett, spaces with an abstract convolution of measures, adv. math. 18 (1975), 1–101. https://doi.org/ 10.1016/0001-8708(75)90002-x. [5] l. nachbin, on the finite dimensionality of every irreducible unitary representation of a compact group, proc. amer. math. soc. 12 (1961), 11-12. https://doi.org/10.1090/s0002-9939-1961-0123197-5. [6] k.a. ross, centers of hypergroups, trans. amer. math. soc. 243 (1978), 251–269. https://doi.org/10.1090/ s0002-9947-1978-0493161-2. [7] r. spector, mesures invariantes sur les hypergroupes, trans. amer. math. soc. 239 (1978), 147–165. https: //doi.org/10.1090/s0002-9947-1978-0463806-1. [8] l. székelyhidi, spherical spectral synthesis on hypergroups, acta math. hungar. 163 (2020), 247–275. https: //doi.org/10.1007/s10474-020-01068-9. [9] k. vati, gelfand pairs over hypergroup joins, acta math. hungar. 160 (2019), 101–108. https://doi.org/10. 1007/s10474-019-00946-1. https://doi.org/10.1090/s0002-9947-1973-0320635-2 https://doi.org/10.17654/ms132010063 https://doi.org/10.17654/ms132010063 https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1090/s0002-9939-1961-0123197-5 https://doi.org/10.1090/s0002-9947-1978-0493161-2 https://doi.org/10.1090/s0002-9947-1978-0493161-2 https://doi.org/10.1090/s0002-9947-1978-0463806-1 https://doi.org/10.1090/s0002-9947-1978-0463806-1 https://doi.org/10.1007/s10474-020-01068-9 https://doi.org/10.1007/s10474-020-01068-9 https://doi.org/10.1007/s10474-019-00946-1 https://doi.org/10.1007/s10474-019-00946-1 1. introduction 2. notations and preliminaries 3. plancherel theorem 3.1. k-multiplicative functions. 3.2. fourier transform on mb(g) 3.3. fourier transform on g references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 170-177 http://www.etamaths.com on stability of convolution of janowski functions khalida inayat noor and humayoun shahid∗ abstract. in this paper, the classes s∗ [a, b] and c [a, b] are discussed in terms of dual sets. using duality, various geometric properties of mentioned class are analyzed. problem of neighborhood as well as stability of convolution of s∗ [a, b] and c [a, b] are studied. some of our results generalize previously known results. 1. introduction let a be the class of all functions of the form f (z) = z + ∞∑ n=2 anz n, (1.1) which are analytic in open unit disc e = {z ∈ c : |z| < 1}. let s ⊂a be the class of functions which are univalent and also s∗ (α) and c (α) be the well known subclasses of s which, respectively consist of starlike and convex functions of order α. if f (z) and g (z) are analytic in e, we say that f (z) is subordinate to g (z) , written as f ≺ g or f (z) ≺ g (z) if there exist a schwarz function w (z) which is analytic in e with w (0) = 0 and |w (z)| < 1 where z ∈ e, such that f (z) = g (w (z)) , z ∈ e. also, if g ∈ s, then f (z) ≺ g (z) if and only if f (0) = g (0) and f (e) ⊂ g (e) . a number of subclasses of analytic functions were introduced using subordination. in 1973, janowski [2] introduced the class p [a,b] which is defined as p [a,b] = { p (z) : p (z) ≺ 1 + az 1 + bz } , where −1 ≤ b < a ≤ 1. geometrically, p (e) is contained in the open disc centered on the real axis having diameter end points 1−a 1−b and 1+a 1+b with centered at 1−ab 1−b2 . for specific values of a and b we obtain many known subclasses of p [a,b] . some specific cases include (i). p [1,−1] = p, the class of caratheodory functions. (ii). p [1 − 2α,−1] = p (α) , the class of caratheodory functions of order α. (iii). p (z) ∈ p [α, 0] satisfies the condition |p (z) − 1| < α, see [4]. using p [a,b] , janowski [2] introduced s∗ [a,b] and c [a,b] which are defined as s∗ [a,b] = { f ∈a : zf′ (z) f (z) ≺ 1 + az 1 + bz , z ∈ e } and c [a,b] = { f ∈a : (zf′ (z)) ′ f′ (z) ≺ 1 + az 1 + bz , z ∈ e } , where −1 ≤ b < a ≤ 1. we note that alexander relation holds between s∗ [a,b] and c [a,b]. received 27th october, 2016; accepted 9th january, 2017; published 1st march, 2017. 2010 mathematics subject classification. 30c45. key words and phrases. convex; univalent functions; convolution; coefficent result; neighbourhood; stability of convolution; janowski functions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 170 stability of convolution of janowski functions 171 the convolution (hadamard) of two functions f (z) given by (1.1) and g (z) = z+ ∞∑ n=2 bnz n is defined as (f ∗g) (z) = (g ∗f) (z) = z + ∞∑ n=2 anbnz n. let v ⊂a the dual set v ∗ (see [6]) is defined as following v ∗ = { g ∈a : (f ∗g) (z) z 6= 0, ∀f ∈ a, z ∈ e } . (1.2) silverman et al. [8] proved that s∗ [a,b] = g∗, where g∗ represents the dual set of g defined in (1.2) and g is given by g = { g ∈a : g (z) = z −lz2 (1 −z)2 } , (1.3) where l = e −iθ+a a−b and θ ∈ [0, 2π]. using the alexander type relation, c [a,b] = h ∗ where h = { h ∈a : h (z) = z + (1 − 2l) z2 (1 −z)3 } , (1.4) where l is same as given in (1.3) and −1 ≤ b < a ≤ 1. for f ∈a and is of form (1.1) and δ ≥ 0, the nδ neighborhood of function f is defined as following (see [7]). nδ (f) = { g (z) = z + ∞∑ n=2 bnz n ∈ a : ∞∑ n=2 n |bn −an| ≤ δ } . ruscheweyh proved many inclusion results of nδ (f) especially n1 4 (f) ⊂ s∗ for all f ∈ c. for x, y ⊂ a. the convolution is called stable univalent if there exist δ > 0 such that nδ (f) ∗ nδ (g) ⊂ s, where f ∈ x and g ∈ y. the constant δ is defined as δ (x ∗y,z) = sup{δ : nδ (f) ∗nδ (g) ⊂ z} . (1.5) in the current paper, we estimate the coefficient bounds of functions given in (1.3) and (1.4). using these estimates we discuss some interesting properties of nδ (f) for different classes and inclusion properties of nδ (f) . 2. preliminaries to prove our main results, we need the following lemmas. lemma 2.1. [8]. let −1 ≤ b < a ≤ 1 and θ ∈ [0, 2π] . then g∗ = s∗ [a,b] , where g = { g ∈ a : g (z) = z − e −iθ+a a−b z 2 (1 −z)2 } . lemma 2.2. [8]. let −1 ≤ b < a ≤ 1 and θ ∈ [0, 2π] . then h∗ = c [a,b] , where h =  h ∈ a : h (z) = z + ( 1 − 2e −iθ+a a−b ) z2 (1 −z)3   . (2.1) lemma 2.3. [5]. let ψ be convex and g be starlike in e. then, for f analytic in e with f (0) = 1, ψ∗fg ψ∗g is contained in the convex hull of f (e) . 172 noor and shahid 3. main results theorem 3.1. let −1 ≤ b < a ≤ 1, then for h (z) = z + ∞∑ n=2 cnz n ∈ g,∣∣∣∣n (1 + b) − (a + 1)a−b ∣∣∣∣ ≤ |cn| ≤ n (1 −b) + a− 1a−b , proof. for h ∈ g, the coefficients can be written as cn = n(1 −l) + l, where l = e −iθ+a a−b and θ ∈ [0, 2π] . to find the maximum value of |cn (θ)| where θ varies from 0 to 2π, consider |cn (θ)| 2 = (nb −a)2 + (n− 1)2 + 2 (n− 1) (nb −a) cos θ (a−b)2 = φ (θ) , φ (θ) attains its maximum value at θ = π as φ′ (π) = −2(n−1)(nb−a) (a−b)2 sin π = 0 and φ ′′ (π) = −2(n−1)(nb−a) (a−b)2 cos π < 0 as nb − a < 0.the maximum value of φ (θ) is φ (π) = ( n(b−1)+1−a a−b )2 , we note that φ (θ) ≤ φ (π) for all θ ∈ [0, 2π] . substituting the value of φ (π) we obtain |cn| ≤ n (1 −b) + a− 1 a−b . now again consider φ (θ) and we note that φ (z) has its minimum at θ = 0 and φ (0) = ( n(b+1)−(a+1) a−b )2 . thus we obtain |cn| ≥ ∣∣∣∣n (b + 1) − (a + 1)a−b ∣∣∣∣ . this completes the proof. � applying the alexander type relation between set g and h we obtain following corollary 3.1. let −1 ≤ b < a ≤ 1, then for h (z) = z + ∞∑ n=2 cnz n ∈ h,∣∣∣∣n [n (b + 1) − (a + 1)]a−b ∣∣∣∣ ≤ |cn| ≤ n [n (1 −b) + a− 1]a−b . corollary 3.2. let −1 ≤ b < a ≤ 1 and let f (z) = z + λzn, n ≥ 2. then f ∈ s∗ [a,b] if and only if |λ| ≤ a−b n (1 −b) + a− 1 . (3.1) proof. let f (z) = z + λzn where λ is given in inequality (3.1) and then for g ∈ g, consider∣∣∣∣(f ∗g) (z)z ∣∣∣∣ ≥ 1 −|λ| |cn|zn−1, z ∈ e. now using theorem 3.1 and value of λ given in (3.1), we obtain∣∣∣∣(f ∗g) (z)z ∣∣∣∣ > 0, z ∈ e. hence f ∈ s∗ [a,b] . conversely, now consider f (z) = z + λzn ∈ s∗ [a,b] and let g (z) = z + ∞∑ n=2 n(1−b)+a−1 a−b z n and (f ∗g) (z) z = 1 + λ n (1 −b) + a− 1 a−b zn−1 6= 0. if |λ| > a−b n(1−b)+a−1 , then there exist ξ ∈ e such that (f ∗g) (ξ) ξ = 0. which is a contradiction, hence |λ| ≤ a−b n(1−b)+a−1. � stability of convolution of janowski functions 173 corollary 3.3. let −1 ≤ b < a ≤ 1 and let f (z) = z + λzn, n ≥ 2. then f ∈ c [a,b] if and only if |λ| ≤ a−b n [n (1 −b) + a− 1] . using the coefficient bounds of functions in set g, we now give alternate method to prove the theorem given in [1]. corollary 3.4. let −1 ≤ b < a ≤−1 and let f is of the form (1.1) and satisfy ∞∑ n=2 [n (1 −b) + a− 1] |an| ≤ a−b, then f ∈ s∗ [a,b] . proof. let f (z) = z + ∞∑ n=2 anz n and g (z) = z + ∞∑ n=2 n(1−b)+a−1 a−b z n, consider (f ∗g) (z) z = 1 + ∞∑ n=2 n (1 −b) + a− 1 a−b anz n−1, z ∈ e. it is known from lemma 2.1 that f ∈ s∗ [a,b] if and only if (f∗g)(z) z 6= 0. now∣∣∣∣(f ∗g) (z)z ∣∣∣∣ ≥ 1 − ∞∑ n=2 n (1 −b) + a− 1 a−b |an| |z| n−1 > 0, which gives us the required condition. � we now consider two specific functions fα (z) = f (z) + αz 1 + α (3.2) and fn,α (z) = f (z) + α n zn (n ≥ 2) . (3.3) here α is a non zero complex number also we note that if f (z) ∈a, then both fα (z) and fn,α (z) ∈a. the geometric properties of these functions are studied by various authors (see [3]). using these two functions, we study the geometric properties of nδ (f) for classes of s ∗ [a,b] and c [a,b]. we first discuss the relation between f (z) and fα (z) in the following lemma. lemma 3.1. let −1 ≤ b < a ≤ 1, f ∈ a and δ > 0 and let for for all α ∈ c, fα ∈ s∗ [a,b] (or c [a,b]), then f ∈ s∗ [a,b] (or c [a,b]) furthermore for all g ∈ g (or h)∣∣∣∣(f ∗g) (z)z ∣∣∣∣ > δ, where |α| < δ and z ∈ e. proof. since fα ∈ s∗ [a,b] then by lemma 2.1, we know that for all g ∈ g, (fα ∗g) (z) z 6= 0, z ∈ e. using (3.2) and simplifying, we obtain (f ∗g) (z) z 6= −α, for all α. thus we obtain ∣∣∣∣(f ∗g) (z)z ∣∣∣∣ > δ. using lemma 2.1, we obtain that f ∈ s∗ [a,b] . this completes the proof. � applying the similar method, we have the following result. 174 noor and shahid lemma 3.2. let −1 ≤ b < a ≤ 1, f ∈ a and δ > 0 and let for for all α, fn,α ∈ s∗ [a,b] , then for all h ∈ g ∣∣∣∣(f ∗h) (z)zcn ∣∣∣∣ > δn, where |α| < δ and z ∈ e. using theorem 3.1 in lemma 3.1, we obtain the following. corollary 3.5. let −1 ≤ b < a ≤ 1, f ∈ a and δ > 0 and let for for all α, fn,α ∈ s∗ [a,b] , then for all h ∈ g ∣∣∣∣(f ∗h) (z)z ∣∣∣∣ > δn ∣∣∣∣n (b + 1) − (a + 1)a−b ∣∣∣∣ , where |α| < δ and z ∈ e. we now prove the following theorem 3.2. let −1 ≤ b < a ≤ 1 and δ > 0 if for all α, fα ∈ s∗ [a,b] then nδ1 (f) ⊂ s∗ [a,b] where δ1 = δ (a−b) 1 −b . proof. let g ∈ nδ1 (f) and g (z) = z + ∞∑ n=2 bnz n. to prove that g ∈ s∗ [a,b] , it is enough to show that (g ∗h) (z) z 6= 0, where h ∈ g and z ∈ e. consider ∣∣∣∣(g ∗h) (z)z ∣∣∣∣ = ∣∣∣∣(f ∗h) (z)z + ((g −f) ∗h) (z)z ∣∣∣∣ ≥ ∣∣∣∣(f ∗h) (z)z ∣∣∣∣− ∣∣∣∣((g −f) ∗h) (z)z ∣∣∣∣ . using lemma 3.1 and series representations of f (z) , g (z) and h (z) , we obtain∣∣∣∣(g ∗h) (z)z ∣∣∣∣ > δ − ∞∑ n=2 (n (1 −b) − (1 −a)) |bn −an| a−b . (3.4) since ∞∑ n=2 (n (1 −b) − (1 −a)) |bn −an| a−b ≤ 1 −b a−b ∞∑ n=2 n |bn −an| ≤ 1 −b a−b δ1. (3.5) using (3.4) in (3.5), we obtain ∣∣∣∣(g ∗h) (z)z ∣∣∣∣ > δ − 1 −ba−bδ1 > 0. hence δ1 = δ (a−b) 1 −b . this completes the proof. � theorem 3.3. let −1 ≤ b < a ≤ 1. f ∈ c [a,b] , then fα ∈ s∗ [a,b] for |α| < 14. proof. let f (z) = z + ∞∑ n=2 anz n. then fα (z) = f (z) + αz (1 + α) = (f (z) ∗ψ (z)) , z ∈ e. stability of convolution of janowski functions 175 here ψ (z) = z − α 1+α z2 1 −z . using the properties of convolution we obtain f (z) ∗ψ (z) = zf′ (z) ∗ ( ψ (z) ∗ log ( 1 1 −z )) . since f ∈ c [a,b], zf′ ∈ s∗ [a,b] , also if |α| < 1 4 , ψ ∈ s∗. applying the convolution we obtain ψ (z) ∗ log ( 1 1 −z ) = z∫ 0 ψ (t) t dt. (3.6) using the alexander relation in (3.6), we obtain ψ (z)∗log ( 1 1−z ) ∈ c. using lemma 2.3 one can prove that c ∗s∗ [a,b] ⊂ s∗ [a,b] , hence fα (z) = zf ′ (z) ∗ ( ψ (z) ∗ log ( 1 1 −z )) ∈ s∗ [a,b] , |α| < 1 4 . this completes the proof. � we now prove the following. theorem 3.4. let −1 ≤ b < a ≤−1. if f ∈ c [a,b] , then nδ (f) ⊂ s∗ [a,b] where δ = a−b4(1−b). proof. if f ∈ c [a,b] , then by theorem 3.3 fα ∈ s∗ [a,b] for |α| < 14. choosing δ = 1 4 and applying theorem 3.2, we obtain our required result. � for specific values of a and b we have the following corollary 3.6. [7]. if f ∈ c [1,−1] = c, then nδ (f) ⊂ s∗ where δ = 14. corollary 3.7. if f ∈ c [1 − 2β,−1] = c (β) , then nδ (f) ⊂ s∗ (β) where δ = 1−β4 and 0 ≤ β < 1. we now prove the stability of convolution given in (1.5) for different classes of nδ (f) . in the next theorem i represent the identity function i (z) = z. theorem 3.5. let −1 ≤ b < a ≤ 1. the following relation holds δ (i ∗ i,c [a,b]) ≥ √ a−b 1 −b (3.7) δ (i ∗ i,s∗ [a,b]) ≥ √ 2(a−b) 1 −b (3.8) δ (c [a,b] ∗c,c [a,b]) = 0 (3.9) δ (s∗ [a,b] ∗c,c [a,b]) = 0 (3.10) δ (c [a,b] ∗c,s∗ [a,b]) ≥ √ 4 + (a−b)2 2 (1 −b)2 − 2 = δ0. (3.11) proof. let f,g ∈ nδ (i) , then applying definition of nδ (f) , we obtain ∞∑ n=2 n |an| ≤ δ and ∞∑ n=2 n |bn| ≤ δ. consider ∞∑ n=2 n (n (1 −b) − 1 + a) |an| |bn| a−b ≤ 1 −b a−b ∞∑ n=2 n2 |an| |bn| ≤ 1 −b a−b δ2. now for h ∈ h,∣∣∣∣((f ∗g) ∗h) (z)z ∣∣∣∣ ≥ ∞∑ n=2 n (n (1 −b) − 1 + a) |an| |bn| a−b − 1 ≥ 1 −b a−b δ2 − 1 > 0. 176 noor and shahid using value of δ given in (3.7), we obtain our first inequality. similarly consider f, g ∈ nδ (i) and consider ∞∑ n=2 (n (1 −b) − 1 + a) |an| |bn| a−b ≤ 1 −b a−b ∞∑ n=2 n2 |an| |bn| ≤ 1 −b 2 (a−b) δ2, thus we obtain∣∣∣∣((f ∗g) ∗h) (z)z ∣∣∣∣ ≥ ∞∑ n=2 (n (1 −b) − 1 + a) |an| |bn| a−b − 1 ≥ 1 −b 2(a−b) δ2 − 1 > 0. which gives us inequality in (3.8). to prove (3.9), consider f (z) = z + ( a−b 2(2(1−b)−1+a) ) z2 ∈ c [a,b]and g (z) = g0 (z) + δ2z 2 ∈ c, where g0 (z) = z 1−z . taking the convolution of f and g, we get (f ∗g) (z) = z + ( a−b 2 (2 (1 −b) − 1 + a) + δ (a−b) 4 (2 (1 −b) − 1 + a) ) z2, applying corollary 3.3 with n = 2, (f ∗g) (z) ∈ c [a,b] if and only if, δ = 0. to prove (3.10), we are applying the same method with f (z) = z + ( a−b (2(1−b)−1+a) ) z2 and g (z) = g0 (z) + δ 2 z2. for relation given in (3.11), consider f0 ∈ c [a,b] and g0 ∈ c and f ∈ nδ (f0) and g ∈ nδ (g0), then for h ∈ g ∣∣∣∣(f ∗g ∗h) (z)z ∣∣∣∣ ≥ ∣∣∣∣(f0 ∗g0 ∗h) (z)z ∣∣∣∣− ∣∣∣∣(f0 ∗ (g −g0) ∗h) (z)z ∣∣∣∣ − ∣∣∣∣(g0 ∗ (f −f0) ∗h) (z)z ∣∣∣∣ (3.12) − ∣∣∣∣((f −f0) ∗ (g −g0) ∗h) (z)z ∣∣∣∣ . applying lemma 2.3 one can prove f0 ∗g0 ∈ s∗ [a,b] and using theorem 3.4, we obtain∣∣∣∣(f ∗g ∗h) (z)z ∣∣∣∣ > a−b4 (1 −b). (3.13) if f0 (z) = z + ∞∑ n=2 a0nz n and g0 (z) = z + ∞∑ n=2 b0nz n and we know that f0 (z) ∈ c [a,b] ⊂ c therefore |a0n| ≤ 1 and |b0n| ≤ 1. now∣∣∣∣(f0 ∗ (g −g0) ∗h) (z)z ∣∣∣∣ ≤ ∞∑ n=2 |a0n| |bn − b0n| |n (1 −b) − 1 + a| a−b ≤ 1 −b a−b δ. (3.14) using definition of nδ (f) , we know that n |an −a0n| ≤ δ or |an −a0n| ≤ δ2 as n ≥ 2. now consider∣∣∣∣((f −f0) ∗ (g −g0) ∗h) (z)z ∣∣∣∣ ≤ ∞∑ n=2 |an −a0n| |bn − b0n| |n (1 −b) − 1 + a| a−b ≤ (1 −b) δ2 2 (a−b) . (3.15) using (3.13), (3.14) and (3.15) in (3.12), we obtain∣∣∣∣(f ∗g ∗h) (z)z ∣∣∣∣ ≥ a−b4 (1 −b) − 2 (1 −b)a−b δ − (1 −b) δ 2 2 (a−b) > 0. solving for δ we obtain relation given in (3.11) which is non negative when δ ≤ δ0. this completes the proof. � stability of convolution of janowski functions 177 references [1] o. p. ahuja, families of analytic functions related to ruscheweyh derivatives and subordinate to convex functions, yokohama math. j., 41(1993), 39-50. [2] w. janowski, some extremal problems for certain families of analytic functions, i. ann. polon. math., 28(1973), 298-326. [3] s. kanas, stability of convolution and dual sets for the class of k−uniformly convex and k−starlike functions, zeszyty naukowe politechniki rzeszowskiej matematyka, 22(1998), 51-64. [4] s. ponnusamy and v. singh, convolution properties of some classes of analytic functions, j. math. sci., 89(1998), 1008-1020. [5] s. ruscheweyh, t. sheil-small, hadamard products of schlicht functions and the pólya-schoenberg conjecture, comment. math. helv., 48(1973), 119-135. [6] s. ruscheweyh, duality for hadamard products with applications to extremal problems for functions regular in the unit disc, trans. amer. math. soc., 210(1975), 63-74. [7] s. ruscheweyh, neighborhoods of univalent functions, proc. amer. math. soc., 81(1981), 521-527. [8] h. silverman and e. m. silvia, subclasses of starlike functions subordinate to convex functions, canad. j. math., 37(1985), 48-61. department of mathematics comsats institute of information technology park road, islamabad, pakistan ∗corresponding author: shahid humayoun@yahoo.com 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications volume 18, number 2 (2020), 172-182 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-172 a note on generalized indexed product summability a. mishra1, b. p. padhy1,∗, b. k. majhi2 and u. k. misra3 1dept. of mathematics, kiit, deemed to be university, bhubaneswar, odisha, india 2dept. of mathematics, centurion university of technology and management, odisha, india 3dept. of mathematics, national institute of science and technology, berhampur, odisha, india ∗corresponding author: birupakhya.padhyfma@kiit.ac.in abstract. in the past, many researchers like szasz, rajgopal, parameswaran, ramanujan, das, sulaiman, have established results on products of two summability methods. in the present article, we have established a result on generalized indexed product summability which not only generalizes the result of misra et al [2] and paikray et al [3] but also the result of sulaiman [7]. 1. introduction if we look back to the history, it is found that, in 1952, szasz [8] published some results on products of summability methods. subsequently, rajgopal [5] in 1954, parameswaran [4] in 1957, ramanujan [6] in 1958 etc. published some more results on products of summability methods. later das [1] in 1969 proved a result on absolute product summability. in 2008, sulaiman [7] published a result on indexed product summability of an infinite series. the result of sulaiman was then extended by paikray et al.[3] in 2010 and misra et al [2] in 2011. let ∑ an be an infinite series with the sum of partial sums {sn}. let {pn} be a sequence of positive real received 2019-11-02; accepted 2019-12-02; published 2020-03-02. 2010 mathematics subject classification. 40d25. key words and phrases. |(r,qn)(r,pn)|k-product summability; |(n,qn)(n,pn)|k-product summability; |(n,qn)(n,pn),αn|k-product summability; |(n,qn)(n,pn),αn; δ|k-product summability; |(n,qn)(n,pn),αn,δ,µ|k-product summability. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 172 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-172 int. j. anal. appl. 18 (2) (2020) 173 constants such that pn = p0 + p1 + p2 + ... + pn →∞ as n →∞ (p−i = p−i = 0). (1.1) the sequence-to-sequence transformation tn = 1 n n∑ ν=0 pνsν (1.2) defines the (r,pn) transform of {sn} generated by {pn}. the series ∑ an is said to be summable |r,pn|k,k ≥ 1, if ∞∑ n=1 nk−1|tn − tn−1|k < ∞. (1.3) similarly, the sequence-to-sequence transformation tn = 1 n n∑ ν=0 pn−νsν (1.4) defines the (n,pn) transform of {sn} generated by {pn}. let {τn} be the sequence of (n,qn) transform of the (n,pn) transform of {sn}, generated by the sequence {qn} and {pn} respectively.that is τn = 1 qn n∑ r=0 qn−r 1 pr r∑ ν=0 pr−νsν then the series ∑ an is said to be summable |(n,qn)(n,pn)|k,k ≥ 1, if ∞∑ n=1 nk−1|τn − τn−1|k < ∞, (1.5) and the series ∑ an is said to be summable |(n,qn)(n,pn),δ|k,k ≥ 1, 1 ≥ δk ≥ 0 if ∞∑ n=1 nδk+k−1|τn − τn−1|k < ∞. (1.6) similarly, if {αn} is a sequence of positive numbers, then the series ∑ an is said to be summable |(n,qn)(n,pn),αn|k,k ≥ 1, if ∞∑ n=1 αn k−1|τn − τn−1|k < ∞, (1.7) and the series ∑ an is summable |(n,qn)(n,pn),αn; δ|k,k ≥ 1, 1 ≥ δk ≥ 0, if ∞∑ n=1 αn δk+k−1|τn − τn−1|k < ∞. (1.8) for, µ a real number, the series ∑ an is summable |(n,qn)(n,pn),αn,δ,µ|k,k ≥ 1, 1 ≥ δk ≥ 0, if ∞∑ n=1 αn µ(δk+k−1)|τn − τn−1|k < ∞. (1.9) int. j. anal. appl. 18 (2) (2020) 174 we assume through out this paper that qn = q0 +q1 +...+qn →∞ as n →∞ and pn = p0 +p1 +...+pn →∞ as n →∞. 2. known theorems in 2008, sulaiman [7] has proved the following theorem. theorem 2.1. let k ≥ 1 and {λn} be a sequence of constants. let us define fν = n∑ r=ν qr pr , fν = n∑ r=ν prfr (2.1) let pnqn = o(pn) such that ∞∑ n=ν+1 nk−1qn k qn kqn−1 = o ( (νqν) k−1 qk−1ν ) . (2.2) then the sufficient condition for the implication ∑ an is summable |r,rn|k ⇒ ∑ anλn is summable |(r,qn)(r,pn)|k are |λν|fν = o (qν) , (2.3) |λν| = o (qν) , (2.4) pνrν|λν| = o (qν) , (2.5) pνqνrν|λν| = o (qνqν−1rν) , (2.6) pnqnrn|λn| = o (pnqnrn) , (2.7) rν−1|∆λν|fν−1 = o (qνrν) , (2.8) and rν−1|∆λν| = o (qνrν) , (2.9) where rn = r1 + r2 + ... + rn. subsequently paikray et al [3] generalized the above theorem by replacing the (r,pn) summability by a summability. he proved: theorem 2.2. let k ≥ 1 and {λn} be a sequence of constants. let us define fν = n∑ r=ν qrarν, fν = n∑ r=ν fr (2.10) int. j. anal. appl. 18 (2) (2020) 175 then the sufficient condition for the implication ∑ an is summable |r,rn|k ⇒ ∑ anλn is summable |(r,qn)(a)|k are m+1∑ n=ν+1 nk−1qn k qn kqn−1 = o ( 1 λν k ) , (2.11) ( n∑ r=ν qr k k−1 ) = o(qν), (2.12) ( n∑ r=ν akr,ν ) = o ( νk−1 ) , (2.13) rν = o (rν) , (2.14) qn qn = o (1) , (2.15) qnλnan,n qn−1 = o (1) , (2.16) (∆λν) k qνk−1 = o ( νk−1 ) , (2.17) ∆λν λν = o (1) , (2.18) and λν k qνk−1 = o ( νk−1 ) , (2.19) where rn = r1 + r2 + ... + rn. in 2011, misra et al [2], generalize the above theorems and proved the following theorem. theorem 2.3. for the sequences of real constants {pn} and {qn} and the sequence of positive numbers {αn}, we define fν = n∑ i=ν qn−ipi−ν pi and fν = n∑ i=ν fi (2.20) let qn = o (qnpn) (2.21) and m+1∑ n=ν+1 {f(αn)}k(αn)k−1qnk qn kqn−1 = o ( (νqν) k−1 qkν ) as m →∞. (2.22) int. j. anal. appl. 18 (2) (2020) 176 then for any sequence {rn} and {λn}, the sufficient conditions for the implication ∑ an is summable |r,rn|k ⇒ ∑ anλn is summable |(n,qn)(n,pn),αn; f|k,k ≥ 1, are |λν|fν = o (qν) , (2.23) |λn| = o (qn) , (2.24) rνfν|λν| = o (qνrν) , (2.25) qnrnfn|λn| = o (qnqn−1rn) , (2.26) rν−1fν+1|∆λν| = o (qνrν) , (2.27) rν−1|∆λν| = o (qνrν) , (2.28) qnrn|λn| = o (qnqn−1rn) , (2.29) ∞∑ n=1 nk−1|tn|k = o(1), (2.30) and ∞∑ n=2 {f(αn)}k(αn)k−1|tn|k = o(1), (2.31) where rn = r1 + r2 + ... + rn. in what follows, we established a theorem on generalized product summability of the infinite series ∑ anλn in the following form: 3. main theorem theorem 3.1. for ’µ’ a real number, the sequences of real constants {pn} and {qn} and the sequence of positive numbers {αn}, we define fν = n∑ i=ν qn−ipi−ν pi and fν = n∑ i=ν fi (3.1) let qn = o (qnpn) (3.2) and ∞∑ n=ν+1 αn µ(kδ+k−1)qn k qn kqn−1 = o ( (νqν) k−1 qkν ) as m →∞. (3.3) int. j. anal. appl. 18 (2) (2020) 177 then for any sequence {rn} and {λn}, the sufficient conditions for the implication ∑ an is summable |r,rn|k ⇒ ∑ anλn is summable |(n,qn)(n,pn),αn,δ,µ|k,k ≥ 1, are |λν|fν = o (qν) , (3.4) |λn| = o (qn) , (3.5) rνfν|λν| = o (qνrν) , (3.6) qnrnfn|λn|αnµδ = o (qnqn−1rn) , (3.7) rν−1fν+1|∆λν| = o (qνrν) , (3.8) rν−1|∆λν| = o (qνrν) , (3.9) qnrn|λn|αnµδ = o (qnqn−1rn) , (3.10) ∞∑ n=1 nk−1|tn|k = o(1), (3.11) and ∞∑ n=2 (αn) µ(k−1)|tn|k = o(1), (3.12) where rn = r1 + r2 + ... + rn. 4. proof of theorem 3.1 let {tn ′ } be the (r,rn) transform of the series ∑ an. then tn ′ = 1 r n∑ ν=0 rνsν tn = tn ′ − t ′ n−1 = rn rnrn−1 n∑ ν=1 rν−1aν let {sn} be the sequence of partial sums of the series ∑ anλn and {τn} be the sequence of (n,qn)(n,pn)transform of the series ∑ anλn. then τn = 1 qn n∑ r=0 qn−r 1 pr r∑ ν=0 pr−νsν = 1 qn n∑ ν=0 sν n∑ r=ν qn−νpr−ν pr = 1 qn n∑ ν=0 fνsν (4.1) int. j. anal. appl. 18 (2) (2020) 178 hence tn = τn − τn−1 = 1 qn n∑ ν=0 fνsν − 1 qn−1 n−1∑ ν=0 fνsν = − qn qnqn−1 n∑ ν=0 fνsν + fnsn qn−1 = − qn qnqn−1 n∑ r=0 fr r∑ ν=0 aνλν + fn qn−1 n∑ ν=0 aνλν = − qn qnqn−1 n∑ r=0 arλr r∑ ν=0 fν + fn qn−1 n∑ ν=0 aνλν (4.2) = − qn qnqn−1 n∑ ν=1 rν−1aν ( λν rν−1 n∑ r=ν fr ) + q0p0 pnqn−1 n∑ ν=1 rν−1aν ( λν rν−1 ) = − qn qnqn−1 [ n−1∑ ν=1 ( ν∑ r=1 rr−1ar ) ∆ ( λν rν−1 n∑ r=ν fr ) + ( n∑ ν=1 rν−1aν ) λn rn−1 fn ] + q0p0 pnqn−1 [ n−1∑ ν=1 ( ν∑ r=1 rr−1ar ) ∆ ( λν rν−1 ) + ( n∑ ν=1 rν−1aν ) λn rn−1 ] = − qn qnqn−1 [ n−1∑ ν=1 { λνfνtν + rν−1 rν fνλνtν + rν−1 rν (∆λν) fν+1tν } + rn rn λnfntn ] + q0p0 pnqn−1 [ n−1∑ ν=1 { λνtν + rν−1 rν (∆λν) tν } + rn rn λntn ] = 7∑ i=1 tn,i,say. (4.3) in order to prove this theorem, using (4.3) and minokowski’s inequality, it is sufficient to show that ∞∑ n=1 αn µ(δk+k−1)|tn,i|k < ∞ for i = 1, 2, 3, 4, 5, 6, 7. on applying holder’s inequality, we have m+1∑ n=2 αn µ(δk+k−1)|tn,1|k = m+1∑ n=2 αn µ(δk+k−1)| qn qnqn−1 n−1∑ ν=1 λνfνtν|k int. j. anal. appl. 18 (2) (2020) 179 ≤ m+1∑ n=2 αn µ(δk+k−1) qn k qn kqn−1 n−1∑ ν=1 |λν|kfνk|tν|k qνk−1 ( 1 qn−1 n−1∑ ν=1 qν )k−1 = o(1) m∑ ν=1 1 qνk−1 |λν|kfνk|tν|k m+1∑ n=ν+1 αn µ(δk+k−1)qn k qn kqn−1 = o(1) m∑ ν=1 1 qνk−1 |λν|kfνk|tν|k (νqν) k−1 qν k , using (3.2) = o(1) m∑ ν=1 νk−1|tν|k ( |λν|fν qν )k = o(1) m∑ ν=1 νk−1|tν|k using (3.4) = o(1) as m →∞. next m+1∑ n=2 αn µ(δk+k−1)|tn,2|k = m+1∑ n=2 αn µ(δk+k−1)| qn qnqn−1 n−1∑ ν=1 rν−1 rν fνλνtν|k ≤ m+1∑ n=2 αn µ(δk+k−1) qn k qn kqn−1 n−1∑ ν=1 rν kfν k|λν|k|tν|k qνk−1rνk ( 1 qn−1 n−1∑ ν=1 qν )k−1 = o(1) m∑ ν=1 rν kfν k|λν|k|tν|k qνk−1rνk m+1∑ n=ν+1 αn µ(δk+k−1)qn k qn kqn−1 = o(1) m∑ ν=1 νk−1|tν|k ( rνfν|λν| rνqν )k = o(1) m∑ ν=1 νk−1|tν|k using (3.6) = o(1) as m →∞. further m+1∑ n=2 αn µ(δk+k−1)|tn,3|k = m+1∑ n=2 αn µ(δk+k−1)| qn qnqn−1 n−1∑ ν=1 rν−1 rν fν+1(∆λν)tν|k ≤ m+1∑ n=2 αn µ(δk+k−1) qn k qn kqn−1 n−1∑ ν=1 (rν−1) k (fν+1) k|∆λν|k|tν|k qνk−1rνk ( 1 qn−1 n−1∑ ν=1 qν )k−1 int. j. anal. appl. 18 (2) (2020) 180 = o(1) m∑ ν=1 (rν−1) k (fν+1) k|∆λν|k|tν|k qνk−1rνk m+1∑ n=ν+1 αn µ(δk+k−1)qn k qn kqn−1 using (3.3) = o(1) m∑ ν=1 νk−1|tν|k ( rν−1fν+1|∆λν| rνqν )k = o(1) m∑ ν=1 νk−1|tν|k using (3.7) = o(1) as m →∞. again, m+1∑ n=2 αn µ(δk+k−1)|tn,4|k = m+1∑ n=2 αn µ(δk+k−1)| qn qnqn−1 rnλnfntn rn |k ≤ m+1∑ n=2 αn µ(δk+k−1)|tn|k ( qnrnfn|λn| qnqn−1rn )k = m+1∑ n=2 αn µ(k−1)|tn|k ( qnrnfn|λn|αnµδ qnqn−1rn )k = o(1) m+1∑ n=2 αn µ(k−1)|tn|k, using (3.7) = o(1) as m →∞. next, m+1∑ n=2 αn µ(δk+k−1)|tn,5|k = m+1∑ n=2 αn µ(δk+k−1)| p0q0 pnqn−1 n−1∑ ν=1 λνtν|k ≤ o(1) m+1∑ n=2 αn µ(δk+k−1) 1 pn kqn−1 n−1∑ ν=1 |λν|k|tν|k qνk−1 ( 1 qn−1 n−1∑ ν=1 qν )k−1 = o(1) m∑ ν=1 |λν|k|tν|k qνk−1 m+1∑ n=ν+1 αn µ(δk+k−1) pn kqn−1 = o(1) m∑ ν=1 |λν|k|tν|k qνk−1 m+1∑ n=ν+1 αn µ(δk+k−1)qn k qn kqn−1 using (3.2) = o(1) m∑ ν=1 νk|tν|k ( |λν| qν )k = o(1) m∑ ν=1 νk|tν|k using (3.6) = o(1) as m →∞. int. j. anal. appl. 18 (2) (2020) 181 again, m+1∑ n=2 αn µ(δk+k−1)|tn,6|k = m+1∑ n=2 αn µ(δk+k−1)| p0q0 pnqn−1 n−1∑ ν=1 rν−1 rν (∆λν)tν|k ≤ o(1) m+1∑ n=2 αn µ(δk+k−1) 1 pn kqn−1 n−1∑ ν=1 (rν−1) k|∆λν|k|tν|k rνkqνk−1 ( 1 qn−1 n−1∑ ν=1 qν )k−1 = o(1) m∑ ν=1 (rν−1) k|∆λν|k|tν|k rνkqνk−1 m+1∑ n=ν+1 αn µ(δk+k−1) pn kqn−1 = o(1) m∑ ν=1 νk−1|tν|k ( rν−1|∆λν| rνqν )k = o(1) m∑ ν=1 νk−1|tν|k using (3.9) = o(1) as m →∞. finally, m+1∑ n=2 αn µ(δk+k−1)|tn,7|k = m+1∑ n=2 αn µ(δk+k−1)| p0q0 pnqn−1 rn rn λntn|k = o(1) m+1∑ n=2 αn µ(δk+k−1)|tn|k ( rn|λn| pnqn−1rn )k = o(1) m+1∑ n=2 αn µ(δk+k−1)|tn|k ( qnrn|λn| qnqn−1rn )k = o(1) m+1∑ n=2 αn µ(k−1)|tn|k ( qnrn|λn|αnµδ qnqn−1rn )k = o(1) m+1∑ n=2 αn µ(k−1)|tn|k, using (3.10) = o(1) as m →∞. this completes the proof of the theorem. int. j. anal. appl. 18 (2) (2020) 182 5. conclusion for µ = 1, the summability method |(n,qn)(n,pn),αn,δ,µ|k reduces to the summability method |(n,qn)(n,pn),αn,δ|k. for, f(αn) = (αn)δ and δ ≥ 0, |(n,qn)(n,pn),αn,δ; f|k summability reduces to |(n,qn)(n,pn),αn,δ|k summability. again, for δ = 0, |(n,qn)(n,pn),αn,δ|k summability reduces to |(n,qn)(n,pn),αn|ksummability and for αn = n, |(n,qn)(n,pn),αn|ksummability reduces to |(n,qn)(n,pn)|k-summability. when pn = 1 = qn, |(n,qn)(n,pn)|k-summability is same as |(r,qn)(r,pn)|k-summability. also, |(r,qn)(r,pn)|k-summability reduces to |(r,qn)(a)|k-summability when (r,pn)-summability is replaced by asummability. from the above results and discussions, we are in a conclusion that our results are more generalized and in particular generalizes the results of sulaiman [7], paikray et al [3] and misra et al [2]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] das, g., tauberian theorems for absolute norlund summability, proc. lond. math. soc. 19 (2) (1969), 357-384. [2] misra, m., padhy, b.p., buxi, s.k. and misra, u.k., on indexed product summability of an infinite series, j. appl. math. bioinform. 1 (2) (2011), 147-157. [3] paikray, s.k., misra, u.k. and sahoo, n.c., product summability of an infinite series, int. j. computer math. sci. 1 (7) (2010), 853-863. [4] parameswaran, m.r., some product theorems in summability, math. z. 68 (1957), 19-26. [5] rajgopal, c.t., theorems on product of two summability methods with applications, j. indian math. soc. 18 (1) (1954), 88-105. [6] ramanujan, m.s., on products of summability methods, math. z. 69 (1) (1958), 423-428. [7] sulaiman, w.t., a note on product summability of an infinite series, int. j. math. sci. 2008 (2008), article id 372604. [8] szasz, o., on products of summability methods, proc. amer. math. soc. 3 (2) (1952), 257-263. 1. introduction 2. known theorems 3. main theorem 4. proof of theorem 3.1 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 68-82 http://www.etamaths.com characterization of biorthogonal multiwavelet packets with arbitrary dilation matrix firdous a. shah1 and r. abass2,∗ abstract. in this paper, we investigate the characterization of biorthogonal multiwavelet packets associated with arbitrary matrix dilations and particularly of orthonormal multiwavelet packets by means of basic equations in fourier domain. 1. introduction it is well-known that the classical orthonormal wavelet bases have poor frequency localization. for example, if the wavelet ψ is band limited, then the measure of the supp of ψ̂j,k is 2 j-times that of supp ψ̂. to overcome this disadvantage, coifman etal. [8] constructed univariate orthogonal wavelet packets. the fundamental idea of wavelet packet analysis is to construct a library of orthonormal bases for l2(r), which can be searched in real time for the best expansion with respect to a given application. well known daubechies orthogonal wavelets are a special case of wavelet packets. chui and li [6] generalized the concept of orthogonal wavelet packets to the case of non-orthogonal wavelet packets so that they can be can be employed to the spline wavelets and so on. the introduction of biorthogonal wavelet packets attributes to cohen and daubechies [7]. they have also shown that all the wavelet packets, constructed in this way, are not led to riesz bases for l2(r). shen [18] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets. other notable generalizations are the wavelet packets related to the walsh polynomials on r+ [13,14,16], higher dimensional wavelet packets with arbitrary dilation matrix [9], the orthogonal version of vector-valued wavelet packets [5] and the m-band framelet packets [17]. on the other hand, multiwavelets are natural extension and generalization of traditional wavelets. they have received considerable attention from the wavelet research communities both in the theory as well as in applications. they can be seen as vector valued-wavelets that satisfy conditions in which matrices are involved rather than scalars as in the wavelet case. multiwavelets can own symmetry, orthogonality, short support and high order vanishing moments, however traditional wavelets can not possess all these properties at the same time (see [10]). as far as the characterization of multiwavelets is concerned, calogero studied the characterization of all multiwavelets associated with general expanding maps of rn in [4]. 2010 mathematics subject classification. 42c40; 42c15; 65t60. key words and phrases. multiwavelet; multiresolution analysis; scaling function; multiwavelet packet; matrix dilation; fourier transform. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 68 characterization of biorthogonal multiwavelet 69 the calogero’s work was further extended by bownik [2], taking into consideration the dilation matrices which preserves the standard lattice zn in terms of affine systems. in the same year, another characterization of orthonormal multiwavelets was given by rzeszotnik [11] for expanding dilations that preserves the lattice zn. however, bownik [3] has presented a new approach to characterize all orthonormal multiwavelets by means of basic equations in the fourier domain. recently, yang and cheng [20] have generalized the concept of wavelet packets to the case of multiwavelet packets associated with a dilation factor a which were more flexible in applications. subsequently, behera [1] extended the results of yang and cheng to the multivariate multiwavelet packets associated with a dilation matrix a. he proved lemmas on the so-called splitting trick and several theorems concerning the fourier transform of the multiwavelet packets and the construction of multiwavelet packets to show that their translates form an orthonormal basis of l2(rd). later on, sun and li [19] have given the construction and properties of generalized orthogonal multiwavelet packets based on the results discussed in [20]. orthogonal wavelet packets have many desired properties such as compact support, good frequency localization and vanishing moments. however, there is no continuous symmetry which is a much desired property in imaging the compression and signal processing. to achieve symmetry, several generalizations of scalar orthogonal wavelet packets have been investigated in literature. the biorthogonal wavelet packets achieve symmetry where the orthogonality is replaced by the biorthogonality. the characterization of multiwavelet packets associated with the dilation matrix a on general lattices has been studied by the author in [12, 15]. in this paper, we further investigate the characterization of biorthogonal multiwavelet packets associated with arbitrary matrix dilations and particularly of orthonormal multiwavelet packets by means of basic equations in fourier domain. we have structured the article as follows. in section 2, we state some basic preliminaries, notations and definitions including the definition of multiresoltion analysis associated with arbitrary dilation matrix a and the corresponding multiwavelet packets. in section 3, we establish our main results concerning with the characterization of biorthogonal multiwavelet packets on rd. 2. notations and preliminaries throughout, this paper, we use the following notations. let r and c be all real and complex numbers, respectively. z and z+ denote all integers and all non-negative integers, respectively. zd and rd denote the set of all d-tuples integers and d-tuples of reals, respectively. assume that we have an expansive dilation matrix a, i.e., all eigenvalues λ of a satisfy |λ| > 1 and preserves the lattice γ. let a = |deta|,a∗ = transpose of a and b be a d×d non-singular matrix. also, if a is expanding so is a∗. considering zd as an additive group, we see that azd is a normal subgroup of zd so we can form the cosets of azd in zd. it is well known fact that the number of distinct cosets of azd in zd is equal to a = |deta| (see[21]). with a and b defined as above, we consider 70 shah and abass λ(a,b) = { α ∈ rd : ∃ (j,m) ∈ z×b∗−1(zd) : α = a∗−jm } , (2.1) and ia,b(α) = { (j,m) ∈ z×b∗−1(zd) : α = a∗−jm } . (2.2) the set λ(a,b) is thought of as the set of all a-adic vectors relative to the lattice b∗−1(zd), i. e., the set of representatives of the equivalence classes of z×b∗−1(zd) with respect to the equivalence relation defined by (j,m) ∼ (j′,m′) if and only if α = a∗−jm = a∗−j ′ . further, the set ia,b(α) is the set of points of z×b∗−1(zd) in the equivalence class of α ∈ λ(a,b). since it is a well known fact that for every dilation matrix a, there exists a hermitian norm ‖·‖∗ in rd, and constants λmax ≥ λmin > 1, such that if b denotes the unit ball in the new norm, centered at the origin, then b ⊂ λminb ⊂ a∗(b) ⊂ λmaxb. for each k ∈ z, we define hk as hk = a ∗k(b), 2h0 ⊂ hη, |b| = 1. where η be the smallest integer. then, the quasi-distance ρ on rd induced by the dilation a∗ is given by ρ(ξ,ζ) = { |deta|j if ξ − ζ ∈ hj+1 \hj 0 if ξ = ζ. furthermore, it is easy to verify that the hardy-littlewood maximal operator mhlf(ζ) = sup k∈z 1 |hk| ∫ ζ+hk ∣∣f(ξ)∣∣dξ is bounded from l1 to l1-weak norm and lim k→−∞ 1 |hk| ∫ ξ+hk f(ξ) dξ = f(ξ), for a.e. ξ ∈ rd. (2.3) definition 2.1. a countable family {fα}α∈a of elements in a separable hilbert space h is a frame if there exist constants a,b, 0 < a ≤ b < ∞ satisfying a ∥∥f∥∥2 2 ≤ ∑ α∈a ∣∣〈f,fα〉∣∣2 ≤ b∥∥f∥∥22 (2.4) for all f ∈ h. the constants a and b independent of f for which (2.4) holds are called frame bounds. a frame is a tight frame if a and b can be chosen so that a = b and is a normalized tight frame if a = b = 1. if only the right hand side inequality holds in (2.4), we say that {fα}α∈a is a bessel squence with constant b. characterization of biorthogonal multiwavelet 71 lemma 2.2 [3]. two families {fα : α ∈a} and { f̃α : α ∈a } constitute a biorthogonal pair if and only if they are bessel sequences and satisfy p(f,g) = ∑ α∈a 〈 f,fα 〉〈 f̃α,g 〉 = 〈 f,g 〉 for all f,g in a dense subset d of h, where p(f,g) is a bi-linear functional on h×h. using polarization identity along with the definition 2.1 implies that p(f,f) = ∥∥f∥∥2, for all f ∈ l2(rd), (2.5) which is equivalent to p(f,g) = 〈f,g〉, for f ∈d. we recall the notion of higher dimensional multiresolution analysis associated with multiplicity l and orthogonal multiwavelets of l2 ( rd ) (see [1]). definition 2.3. a sequence {vj}j∈z of closed subspaces of l 2 ( rd ) is called a multiresolution analysis (mra) of l2 ( rd ) of multiplicity l associated with the dilation matrix a if the following conditions are satisfied: (i) vj ⊂ vj+1 for all j ∈ z; (ii) ⋃ j∈zvj is dense in l 2 ( rd ) and ⋂ j∈zvj = {0} ; (iii) f ∈ vj if and only if f(a·) ∈ vj+1 for all j ∈ z; (iv) there exist l-functions φ = {ϕ1,ϕ2, . . . ,ϕl} ∈ v0, such that the system of functions {ϕ`(x−k)} l `=1,k∈zd, forms an orthonormal basis for subspace v0. the l-functions whose existence is asserted in (iv) are called scaling functions of the given mra. given a multiresolution analysis {vj}j∈z, we define another sequence {wj}j∈z of closed subspaces of l 2 ( rd ) by wj = vj+1 vj,j ∈ z. these subspaces inherit the scaling property of {vj}, namely f ∈ wj if and only if f(a·) ∈ wj+1. (2.6) further, they are mutually orthogonal, and we have the following orthogonal decompositions: l2 ( rd ) = ⊕ j∈z wj = v0 ⊕ (⊕ j≥0 wj ) . (2.7) a set of functions {ψr` : 1 ≤ ` ≤ l, 1 ≤ r ≤ a− 1} in l 2 ( rd ) is said to be a set of basic multiwavelets associated with the mra of multiplicity l if the collection{ ψr` (.−k) : 1 ≤ r ≤ a− 1, 1 ≤ ` ≤ l, k ∈ z d } 72 shah and abass forms an orthonormal basis for w0. now, in view of (2.6) and (2.7), it is clear that if {ψr` : 1 ≤ ` ≤ l, 1 ≤ r ≤ a− 1} is a basic set of multiwavelets, then{ aj/2ψr` (a j.−k) : j ∈ z, k ∈ zd, 1 ≤ ` ≤ l, 1 ≤ r ≤ a− 1 } forms an orthonormal basis for l2 ( rd ) (see [1]). for any n ∈ z+, we define the basic multiwavelet packets ωn` ; 1 ≤ ` ≤ l recursively as follows. we denote ω0` = ϕ`, 1 ≤ ` ≤ l, the scaling functions and ωr` = ψ r ` ,r ∈ z +, 1 ≤ ` ≤ l as the possible candidates for basic multiwavelets. then, define ωs+ar` (x) = l∑ j=1 ∑ k∈zd hs`jk a 1/2 ωr` (ax−k), 0 ≤ s ≤ a− 1, 1 ≤ ` ≤ l (2.8) where ( hs`jk ) is a unitary matrix (see [1]). taking fourier transform on both sides of (2.8), we obtain ( ωs+ar` )∧ (ξ) = l∑ j=1 hs`j(b −1ξ) ( ωr` )∧ (b−1ξ). (2.9) note that (2.8) defines ωn` for every non-negative integer n and every ` such that 1 ≤ ` ≤ l. the set of functions {ωn` : n ∈ z +, 1 ≤ ` ≤ l} as defined above are called the basic multiwavelet packets corresponding to the mra {vj}j∈z of l 2(rd) of multiplicity l associated with matrix a. definition 2.4. let {ωn` : n ∈ z +, 1 ≤ ` ≤ l} be the basic multiwavelet packets. the collection p = { |deta|j/2ωn` (a.− k) : 1 ≤ ` ≤ l,j ∈ z,k ∈ z d } is called the general multiwavelet packets associated with mra {vj : j ∈ z} of l2 ( rd ) of multiplicity l over matrix dilation a. corresponding to some orthonormal scaling vector φ = ω0` , the family of multiwavelet packets ωn` defines a family of subspaces of l 2(rd) as follows: unj = span { aj/2ωn` (a jx−k) : k ∈ zd, 1 ≤ ` ≤ l } ; j ∈ z,n ∈ z+. (2.10) observe that u0j = vj, u 1 j = wj = a−1⊕ r=1 urj , j ∈ z characterization of biorthogonal multiwavelet 73 so that the orthogonal decomposition vj+1 = vj ⊕wj, can be written as u0j+1 = a−1⊕ r=0 urj . (2.11) a generalization of this result for other values of n = 1, 2, . . . can be written as unj+1 = a−1⊕ r=0 uan+rj , j ∈ z. (2.12) the following proposition is proved in [1]. proposition 2.5. if j ≥ 0, then wj = a−1⊕ r=0 urj = a2−1⊕ r=a urj−1 = · · · = at+1−1⊕ r=at urj−t = aj+1−1⊕ r=aj ur0 where unj is defined in (2.10). using this decomposition, we get the multiwavelet packets decomposition of subspaces wj, j ≥ 0. similar to the orthogonal multiwavelet packets, the biorthogonal multiwavelet packets associated with the biorthogonal scaling vector φ̃ are given by ω̃s+ar` (x) = l∑ j=1 ∑ k∈zd h̃s`jk a 1/2 ω̃r` (ax−k), 0 ≤ s ≤ a− 1, 1 ≤ ` ≤ l. (2.13) implementation of fourier transform of (2.13) yields ( ω̃s+ar` )∧ (ξ) = l∑ j=1 h̃s`j(b −1ξ) ( ω̃r` )∧ (b−1ξ). (2.14) let ωn` be general multiwavelet packets associated with the dilation matrix a. then, we consider the system f(a,b) = { ωn`,j,k : j ∈ z,k ∈ z d,` = 1, . . . ,l,aj ≤ n < aj+1 } (2.15) where ωn`,j,k(x) = |deta| j/2 ωn` ( ajx−bk ) . similarly, for the biorthogonal multiwavelet packets, we have f̃(a,b) = { ω̃n`,j,k : j ∈ z,k ∈ z d,` = 1, . . . ,l,aj ≤ n < aj+1 } (2.16) where ω̃n`,j,k(x) = |deta| j/2 ω̃n` ( ajx−bk ) . 74 shah and abass the bi-linear functional p(f,g) associated to the multiwavelet packets systems f(a,b) and f̃(a,b) is given by p(f,g) = aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ∑ k∈zd 〈 f,ωn`,j,k 〉〈 ω̃n`,j,k,g 〉 . (2.16) we will also consider the set d as a dense subset of l2 ( rd ) defined by d = { f ∈ l2 ( rd ) : f̂ ∈ l∞(rd), f̂ has compact support in rd \{0} } . 3. characterization of biorthogonal multiwavelet packets in this section, we prove our main results concerning the characterization of biorthogonal multiwavelet packets associated with arbitrary matrix dilations by means of the fourier transform. theorem 3.1. suppose {ωn` : n ∈ z +,` = 1, . . . ,l} and {ω̃n` : n ∈ z +,` = 1, . . . ,l} are the basic multiwavelet packets associated with a pair of biorthogonal scaling functions φ and φ̃ such that the following functions are locally integrable: aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ∣∣ω̂n` (a∗jξ)∣∣2, a j+1−1∑ n=aj l∑ `=1 ∑ j∈z ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 . (3.1) then, the bi-linear functional p(f,g) converges absolutely for all f,g ∈ d. moreover, the multiwavelet packets ωn` and ω̃ n ` satisfy: 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α) ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗j(ξ + a∗−jm) ) = δα,0, (3.2) for a.e. ξ ∈ rd and for all α ∈ λ(a,b), if and only if p(f,g) = 〈f,g〉, for all f,g ∈ d. proof. first of all we prove that p(f,g) is absolutely convergent. for this, fix j ∈ z and let gj = aj+1−1∑ n=aj l∑ `=1 ∑ k∈zd 〈f,ωn`,j,k〉〈ω̃ n `,j,k,f〉. (3.3) implementation of parseval’s identity gives 〈 f,ωn`,j,k 〉 = |deta|j/2 ∫ rd f̂ ( a∗jζ ) ω̂n` (ζ) e 2πib(k)·ζ dζ, characterization of biorthogonal multiwavelet 75 and 〈 ω̃n`,j,k,f 〉 = |deta|j/2 ∫ rd f̂ ( a∗jζ ) ˆ̃ωn` (ξ) e −2πib(k)·ξ dξ. let f n `,j(ξ) = ∑ s∈zd f̂ ( a∗j(ξ + b∗−1s) ) ω̂n` ( ξ + b∗−1s ) . then, by virtue of fourier inversion formula for the function f n `,j ◦b ∗−1, we obtain f n `,j(ξ) = ∑ k∈zd { |detb| ∫ b∗−1([0,1]d) f n `,j(ζ) e 2πibk·ζ dζ } e−2πibk·ξ = |detb| ∑ k∈zd {∫ rd f̂ ( a∗jζ ) ω̂n` (ζ) e 2πibk·ζ dζ } e−2πibk·ξ. thus, gj as defined in (3.3) can be written as gj = |deta|j |detb| aj+1−1∑ n=aj l∑ `=1 ∫ rd f̂ ( a∗jξ ) ˆ̃ωn` (ξ) f n `,j(ξ) dξ = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∫ rd f̂(ξ) ˆ̃ωn` ( a∗−jξ ) f n `,j ( a∗−jξ ) dξ = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∫ rd f̂(ξ) ˆ̃ωn` ( a∗−jξ ) {∑ s∈zd f̂ ( ξ + a∗jb∗−1s ) ω̂n` ( a∗−jξ + b∗−1s )} dξ. now, in order to show that the convergence of ∑ j∈z gj is absolute and unconditional, it is sufficient to prove that the following two series are absolutely convergent: aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ∫ rd f̂(ξ) ˆ̃ωn` ( a∗−jξ ) f̂(ξ) ω̂n` ( a∗−jξ ) dξ, and aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ∫ rd f̂(ξ) ˆ̃ωn` ( a∗−jξ ) ∑ s∈zd\{0} f̂ ( ξ + a∗jb∗−1s ) ω̂n` ( a∗−jξ + b∗−1s )dξ. 76 shah and abass from our assumptions on the basic multiwavelet packets ωn` and ω̃ n ` , it is clear that the first of these series converges absolutely. moreover, we have 2 ∣∣∣ˆ̃ωn` (a∗−jξ) ω̂n` (a∗−jξ + b∗−1s)∣∣∣ ≤ ∣∣∣ˆ̃ωn` (a∗−jξ)∣∣∣2 + ∣∣∣ω̂n` (a∗−jξ + b∗−1s)∣∣∣2. further, it is easy to verify that the convergence of the second series follows from the convergence of: aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ∑ s∈zd\{0} ∫ rd ∣∣∣f̂(ξ)∣∣∣ ∣∣∣f̂(ξ + a∗jb∗−1s)∣∣∣∣∣∣ˆ̃ωn` (a∗−jξ)∣∣∣2 dξ = ∫ rd aj+1−1∑ n=aj l∑ `=1  ∑ j∈z ∑ s∈zd\{0} |deta|j ∣∣∣f̂(a∗jξ)∣∣∣∣∣∣f̂(a∗jξ + a∗jb∗−1s)∣∣∣   ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ, and from the convergence of a similar series, with ω̃n` replaced by ω n ` . but as s 6= 0, therefore there exists j ∈ z such that f̂ ( a∗jξ ) f̂ ( a∗jξ + a∗jb∗−1s ) = 0, for all j ≥ j. on the other hand, for each fixed j ∈ z, and ξ ∈ rd, the number of s ∈ zd, for which the above product is nonzero, is less than or equal to c|deta|−j for some constant c. thus, we have ∑ j∈z ∑ s∈zd\{0} |deta|j ∣∣∣f̂(a∗jξ)∣∣∣∣∣∣f̂(a∗jξ + a∗jb∗−1s)∣∣∣ ≤ c ∑ j≤j ∥∥∥f̂∥∥∥2 ∞ χf ( a∗jξ ) , where f is compact in rd\{0} . observe that if b′ < |a∗jξ| < b, there exists k > 0, which does not depend on ξ, such that the number of j for which this is nonzero is less than k for every ξ. hence, the above sum can be estimated from above by ck‖f̂‖2∞ and it proves the convergence of second sum. hence, we can rearrange the series for p(f,g) to obtain p(f,f) = ∑ α∈λ(a,b) ∫ rd f̂(ξ)f̂(ξ + α) ×   1|detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α) ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗j(ξ + α) )dξ. therefore, it is enough to show that if p(f,g) = 〈f,g〉 for all f,g ∈ d, then the second condition follows. for this, we write p(f,g) = m(f,g) + r(f,g), characterization of biorthogonal multiwavelet 77 with m(f,g) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∫ rd ĝ(ξ)f̂(ξ)  ∑ j∈z ˆ̃ωn` ( a∗jξ ) ω̂n` (a ∗jξ)   dξ, and r(f,g) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ α∈λ(a,b)\{0} ∫ rd ĝ(ξ)f̂(ξ + α) ∑ (j,m)∈ia,b(α) ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗j(ξ + α) ) dξ. now, let us fix, ξ0 ∈ rd \{0} ,k ∈ z, and consider f = g = f1, where f1 is defined by f̂1(ξ) = 1 |hk|1/2 χhk(ξ). then, m(f1,f1) = 1 |detb||hk| aj+1−1∑ n=aj l∑ `=1 ∫ hk ∑ j∈z ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗jξ ) dξ, and ∣∣r(f1,f1)∣∣ ≤ 1|detb||hk| aj+1−1∑ n=aj l∑ `=1 ∑ α∈λ(a,b) α 6=0 ∑ (j,m)∈ia,b(α) × ∫ hk∩(α+hk) ∣∣∣ˆ̃ωn` (a∗jξ) ω̂n` (a∗j(ξ + α))∣∣∣dξ ≤ 1 |detb||hk|   aj+1−1∑ n=aj l∑ `=1 ∑ α∈λ(a,b) α6=0 ∑ (j,m)∈ia,b(α) ∫ hk∩(α+hk) ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj l∑ `=1 ∑ α∈λ(a,b) α6=0 ∑ (j,m)∈ia,b(α) ∫ hk∩(α+hk) ∣∣ω̂n` (a∗j(ξ + α))∣∣2dξ   1/2 . to estimate r(f1,f1), we observe that if α 6∈ hk+η, then hk ∩ (α + hk) = ∅. therefore, we may assume that α ∈ hk+η and α 6= 0. also, if (j,m) ∈ ia,b(α), then m ∈ a∗j(hk+η) ∩b∗−1(zd) and j ≥−k + c1, 78 shah and abass where c1 is the largest integer such that hk+η ∩b∗−1(zd) = {0} . therefore, under these observations, we have ∣∣r(f1,f1)∣∣ ≤ 1|detb||hk|   aj+1−1∑ n=aj l∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈a∗j(hk+η)∩b ∗−1(zd) ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj l∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈a∗j(hk+η)∩b ∗−1(zd) ∫ ξ0+hk ∣∣ω̂n` (a∗jξ)∣∣2dξ   1/2 . now, in order to estimate the first factor in the above product, we observe that 1 |hk| aj+1−1∑ n=aj l∑ `=1 ∑ j≥−k+c1 ∑ m 6=0 m∈a∗j(hk+η)∩b∗−1(zd) ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ ≤ |deta|−k aj+1−1∑ n=aj l∑ `=1 ∑ j≥−k+c1 c|deta|j+k|deta|−j ∫ a∗j(ξ0+hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ ≤ c ∑ j≥−k+c1 ∫ a∗j(ξ0+hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ. here, we have used the fact that the number of points of the lattice b∗−1(zd), different from the origin and contained in the set a∗j(hk+η) = hj+k+η, is smaller than a constant multiple of the volume of this set. similar estimate holds for the second factor. since the sets a∗j(ξ0+hk), j ∈ z, are pairwise disjoint for sufficiently large |k|, so we may conclude that r(f1,f1) → 0, as k →−∞ by the lebesgue dominated convergence theorem. therefore, we have 1 = lim k→−∞ 1 |detb| aj+1−1∑ n=aj l∑ `=1 1 |hk| ∫ ξ0+hk ∑ j∈z ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗jξ ) dξ = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ j∈z ˆ̃ωn` ( a∗jξ0 ) ω̂n` (a ∗jξ0), which proves our claim for α = 0. this also shows that m(f,g) = 〈f,g〉, and thus r(f,g) = 0, for f,g ∈d. now, we choose α0 ∈ λ(a,b) \{0}, and write r(f,g) = r1(f,g) + r2(f,g), characterization of biorthogonal multiwavelet 79 where r1(f,g) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∫ rd ĝ(ξ)f̂(ξ+α0) ∑ (j,m)∈ia,b(α0) ˆ̃ωn` (a ∗jξ) ω̂n` ( a∗j(ξ + α0) ) dξ, and r2(f,g) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ α∈λ(a,b) α 6=0,α0 ∫ rd ĝ(ξ)f̂(ξ+α) ∑ (j,m)∈ia,b(α) ˆ̃ωn` ( a∗jξ ) ω̂n ` ( a∗j(ξ + α) ) dξ. let ξ0 ∈ rd \{0} be a lebesgue point of differentiability for the functions aj+1−1∑ n=aj l∑ `=1 ∞∑ j=j ∣∣ω̂n` (a∗jξ)∣∣2 and a j+1−1∑ n=aj l∑ `=1 ∞∑ j=j ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 , j ∈ z. then, for given k ∈ z, we define f2 and g2 as follows: f̂2(ξ + α0) = 1 |hk|1/2 χξ0+hk(ξ), ĝ2(ξ + α0) = 1 |hk|1/2 χξ0+hk(ξ). using equation (2.3), we obtain lim k→−∞ r1(f2,g2) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α0) ˆ̃ωn` ( a∗jξ0 ) ω̂n` ( a∗j(ξ0 + α0) ) . to estimate r2(f2,g2), we observe that ĝ2(ξ)f̂2(ξ + α) 6≡ 0 is only possible when α ∈ α0 + hk+η. since α = (a∗)−jm ∈ λ(a,b) \{0,α0} , there exists j0 ∈ z such that (a∗)−jm 6∈ α0 + hη for any m ∈ b∗−1(zd)\{0} and j ≤ j0. thus, r2(f2,g2) can be re-written as r2(f2,g2) = 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∞∑ j=j1 ∑ m 6=0, a∗−jm−α0∈hk+η ∫ rd ĝ2(ξ)f̂2(ξ + α) ˆ̃ω n ` ( a∗jξ ) ω̂n ` ( a∗j(ξ + α) ) dξ + 1 |detb| aj+1−1∑ n=aj l∑ `=1 j1∑ j=j0 ∑ m 6=0, a∗−jm−α0∈hk+η ∫ rd ĝ2(ξ)f̂2(ξ + α) ˆ̃ω n ` ( a∗jξ ) ω̂n ` ( a∗j(ξ + α) ) dξ = r2,1(f2,g2) + r2,2(f2,g2), where j1 ∈ z. since r2,2(f2,g2) is now a finite sum, and the number of m’s satisfying the condition a∗−jm−α0 ∈ hk+η ⊂ hη may now be estimated independently of k ≤ 0, we have limk→−∞r2,2(f2,g2) = 0 by lebesgue dominated convergence theorem. to estimate r2,1(f2,g2), we will show that for every ε > 0, there exists 80 shah and abass j1 ∈ z such that |r2,1(f2,g2)| ≤ ε for sufficiently large |k|. in fact, as in the case of r(f1,g1), we have r2,1(f2,g2) ≤ 1 |detb||hk|   aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∑ m 6=0, a∗−jm−α0∈hk+η ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ   1/2 ×   aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∑ m 6=0, a∗−jm−α0∈hk+η ∫ ξ0+hk ∣∣ω̂n` (a∗jξ)∣∣2 dξ   1/2 . therefore, it is enough to estimate just one of these factors, namely: 1 |hk| aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∑ m 6=0, a∗−jm−α0∈hk+η ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ ≤ 1 |hk| aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ( 1 + c|deta|k+j+η )∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ = aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 1 |hk| ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ + a j+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∫ (a∗)j(ξ0+hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ. here, we have used the fact that the number of points of the lattice b∗−1(zd) that are contained in the set a∗j(α0 + hk+η) = (a ∗)jα0 + hj+k+η, is smaller than one plus a constant multiple of the volume of this set. let j1 ∈ z be such that aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∣∣∣ˆ̃ωn` (a∗jξ0)∣∣∣2 < ε/2. then, by our choice of ξ0 and equation (2.3), we have lim k→−∞ sup aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 1 |hk| ∫ ξ0+hk ∣∣∣ˆ̃ωn` (a∗jξ)∣∣∣2 dξ < ε/2. therefore, by virtue of lebesgue dominated convergence theorem, we get lim k→−∞ sup aj+1−1∑ n=aj l∑ `=1 ∑ j≥j1 ∫ a∗j(ξ0+hk) ∣∣∣ˆ̃ωn` (ξ)∣∣∣2 dξ = 0. since the sets a∗j(α0 + hk+η), j ∈ z, are pairwise disjoint for sufficiently large |k|, therefore, for every ε > 0, there exist j1 such that lim k→−∞ sup ∣∣r2,1(f2,g2)∣∣ ≤ ε. characterization of biorthogonal multiwavelet 81 combining these observations with the fact that ε is arbitrary, we obtain 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α0) ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗j(ξ + α0) ) = 0, for all α0 ∈ λ(a,b) \{0}. an immediate consequence of the above theorem is the following: corollary 3.2. let {ωn` : n ∈ z +,` = 1, . . . ,l} be the basic multiwavelet packets associated with the scaling vector φ. then 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α) ω̂n` ( a∗jξ ) ω̂n` ( a∗j(ξ + a∗−jm) ) = δα,0, (3.4) for a.e. ξ ∈ rd and for all α ∈ λ(a,b), if and only if the system f(a,b) given by (2.15) is a normalized tight frame for l2 ( rd ) . theorem 3.3. if {ωn` : n ∈ z +,` = 1, . . . ,l} and {ω̃n` : n ∈ z +,` = 1, . . . ,l} are bessel families and have the property that the functions in (3.1) are locally integrable. then, they are biorthogonal if and only if 1 |detb| aj+1−1∑ n=aj l∑ `=1 ∑ (j,m)∈ia,b(α) ˆ̃ωn` ( a∗jξ ) ω̂n` ( a∗j(ξ + a∗−jm) ) = δα,0, (3.5) for a.e. ξ ∈ rd and for all α ∈ λ(a,b). moreover, if ωn` = ω̃ n ` and ‖ω n ` ‖2 = ‖ω̃n` ‖2 = 1 for n ∈ z +,` = 1, . . . ,l. then, the system f(a,b) forms an orthonormal basis for l2 ( rd ) . proof. the proof of this theorem follows from (2.2) and theorem 3.1. references [1] b. behera, multiwavelet packets and frame packets of l2 ( rd ) , proc. indian acad. sci. 111(4) (2001), 439-463. [2] m. bownik, a characterization of affine dual frames of l2 ( rn ) , j. appl. comput. harmon. anal. 8 (2000), 203-221. [3] m. bownik, on characterization of multiwavelets in l2(rn), proc. amer. math. soc. 129(11) (2001), 3265-3274. [4] a. calogero, a characterization of wavelets on general lattices, j. geom. anal. 10 (2000), 597-622. [5] q. chen and z. chang, a study on compactly supported orthogonal vector valued wavelets and wavelet packets, chaos. solit. fract. 31 (2007), 10241034. [6] c. chui and c. li, non-orthogonal wavelet packets, siam j. math. anal. 24(3) (1993), 712-738. 82 shah and abass [7] a. cohen and i. daubechies, on the instability of arbitrary biorthogonal wavelet packets, siam j. math. anal. 24(5) (1993), 1340-1354. [8] r. coifman, y. meyer, s. quake and m.v. wickerhauser, signal processing and compression with wavelet packets, technical report, yale university (1990). [9] j. han and z. cheng, on the splitting trick and wavelets packets with arbitrary dilation matrix of l2(rs), chaos. solit. fract. 40 (2009), 130137. [10] f. keinert, wavelets and multiwavelets, chapman & hall, crc, 2004. [11] z. rzeszotnik, calderón’s condition and wavelets, collect. math. 52 (2001), 181-191. [12] f.a. shah, a characterization of multiwavelet packets on general lattices, int. j. non-linear anal. appl. 6 (1) (2015), 69-84. [13] f.a. shah, construction of wavelet packets on p-adic field, int. j. wavelets multiresolut. inf. process. 7(5) (2009), 553-565. [14] f.a. shah, biorthogonal p-wavelet packets related to the walsh polynomials, j. classical anal. 2 (2013), 135-146. [15] f.a. shah and k. ahmad, characterization of multiwavelet packets in l2 ( rd ) , jordan j. math. statist. 3(3) (2010), 159-180. [16] f.a. shah and l. debnath, p-wavelet frame packets on a half-line using the walsh-fourier transform, integ. transf. spec. funct. 22(12) (2011), 907-917. [17] f.a. shah and l. debnath, explicit construction of m-band tight framelet packets, analysis. 32(4) (2012), 281-294. [18] z. shen, non-tensor product wavelet packets in l2(rs), siam j. math. anal. 26(4) (1995), 1061-1074. [19] l. sun and g. li, generalized orthogonal multiwavelet packets, chaos. solit. fract. 42 (2009), 2420-2424. [20] s. yang and z. cheng, a-scale multiple orthogonal wavelet packets, appl. math. china series. 13 (2000), 61-65. [21] p. wojtaszczyk, a mathematical introduction to wavelets, cambridge university press, cambridge, 1997. 1department of mathematics, university of kashmir, south campus, anantnag-192101, jammu and kashmir, india 2department of mathematical sciences, bgsb university, rajouri-185234 , jammu and kashmir, india ∗corresponding author int. j. anal. appl. (2022), 20:48 fuzzy ideals and fuzzy filters on topologies generated by fuzzy relations kheir saadaoui1,∗, soheyb milles2, lemnaouar zedam3 1laboratory lmpa, department of mathematics, university of m’sila, algeria 2laboratory lmpa, department of mathematics and informatics, university center of barika, algeria 3laboratory lmpa, department of mathematics, university of m’sila, algeria ∗corresponding author: kheir.saadaoui@univ-msila.dz abstract. recently, mishra and srivastava have introduced and studied the notion of fuzzy topology generated by fuzzy relation and several properties were proved. in this paper, we mainly investigate the lattice structure of fuzzy open sets in this topology, and show its various properties and characteristics. additionally, we extend to this lattice the notions of fuzzy ideal and fuzzy filter. for each of these notions, we fully characterize them in terms of this lattice meet and join operations. 1. introduction the notions of ideals and filters have studied in many algebraic structures (e.g., semi-groups, rings, mv-algebras, lattices, et cetera) and used as tools to investigate properties, representations and characterizations of these algebraic structures [16], [7], [13], [14]. in addition to their theoretical roles, they have used in some areas of applied mathematics, especially, in topology and its analysis approaches. ideals and filters are appeared to provide very general contexts to unify the various notions of sequences convergence and limit in arbitrary topological spaces, and to express completeness and compactness in metric spaces [3], [17]. in the fuzzy setting and its extensions, several authors introduced and investigated the concepts of fuzzy ideals and fuzzy filters in different structures. the first approach considered fuzzy ideal and fuzzy filter as fuzzy sets on crisp structures, like on lattices or on residuated lattices [5], [9], [15]. the received: aug. 17, 2022. 2010 mathematics subject classification. 03b52, 03g10, 06b10. key words and phrases. fuzzy set; lattice; ideal; filter; topology. https://doi.org/10.28924/2291-8639-20-2022-48 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-48 2 int. j. anal. appl. (2022), 20:48 second approach proposed similar notions on (intuitionistic) fuzzy structures [2], [10], [12]. the third approach considered neutrosophic ideal and neutrosophic filter as neutrosophic sets [1], [20]. the present study is motivated by the work of mishra and srivastava [11] that have considered the notion of fuzzy topology generated by a fuzzy relation. more specifically, we deepen the study of a lattice structure of fuzzy open sets on this topology, and providing its various characteristics and properties. we pay particular attention to the notion of fuzzy ideal (resp. fuzzy filter) on this topology generated by a fuzzy relation. furthermore, we provide a characterization of these notion of fuzzy ideal (resp. fuzzy filter) based on the meet and the join operations of the introduced lattice. this paper is organized as follows. in section 2, we recall some basic concepts related to fuzzy sets, fuzzy relations and fuzzy topology. in section 3, we provide the lattice structure of fuzzy open sets in a fuzzy topology generated by a fuzzy relation, and we show its various properties and characteristics. in section 4, we introduce the notions of fuzzy ideal (resp. fuzzy filter) on the lattice of fuzzy open sets, and some basic properties are given. finally, some conclusions and future research in section 5 are presented. 2. basic concepts this section contains the basic definitions and properties of fuzzy sets, fuzzy topology and some related notions that will be needed throughout this paper. 2.1. fuzzy sets. in this subsection, we recall some basic concepts of fuzzy sets. let x be a universe, a fuzzy subset a = {〈x,µa(x)〉 | x ∈ x} of x defined by zadeh in 1965 [18] is characterized by a membership function µa : x → [0,1], where µa(x) is interpreted as the degree of membership of the element x in the fuzzy subset a for each x ∈ x. for fuzzy sets, several operations are defined. here we present only those which are related to the present paper. let a = {〈x,µa(x)〉 | x ∈ x} and b = {〈x,µb(x)〉 | x ∈ x} be two fuzzy subsets on x, then (i) a ⊆ b if µa(x)≤ µb(x), for any x ∈ x; (ii) a = b if µa(x)= µb(x), for any x ∈ x; (iii) a∩b = {〈x,µa(x)∧µb(x)〉 | x ∈ x}; (iv) a∪b = {〈x,µa(x)∨µb(x)〉 | x ∈ x}; (v) a = {〈x,1−µa(x)〉 | x ∈ x}; (vi) supp(a)= {x ∈ x | µa(x) > 0}; (vii) ker(a)= {x ∈ x | µa(x)=1}. in the sequel, we need the following definition of level set (which is also often called α-cuts) of a fuzzy set. int. j. anal. appl. (2022), 20:48 3 definition 2.1. [6] let a be a fuzzy set on a nonempty set x. the α-cut of a is the crisp subset aα = {x ∈ x | µa(x)≥ α}, for any α ∈ [0,1]. 2.2. fuzzy relations. zadeh [19] introduced the concept of fuzzy relation as a natural generalization of crisp relation. definition 2.2. [19] a fuzzy binary relation (a fuzzy relation, for short) from a nonempty set x to a nonempty set y is a fuzzy subset in x ×y , i.e., is an expression r given by r = {〈(x,y),µr(x,y)〉 | (x,y)∈ x ×y} , where µr : x ×y → [0,1] for any (x,y) ∈ x × y . the value µr(x,y) is called the degree of relation between x and y under the fuzzy relation r. next, we need to recall the following definitions [19]. let r and p are two fuzzy relations from a nonempty set x to a nonempty set y . r is said to be contained in p or we say that p contains r, denoted by r ⊆ p , if for all (x,y)∈ x×y it holds that µr(x,y)≤ µp(x,y). the transpose (inverse) rt of r is the fuzzy relation from the nonempty set y to the nonempty set x defined by rt = {〈(x,y),µrt(x,y)〉 | (x,y)∈ x ×y}, where µrt(x,y)= µr(y,x) for any (x,y)∈ x ×y. the intersection of two fuzzy relations r and p from a nonempty set x to a nonempty set y is defined as r∩p = {〈(x,y),µr∩p(x,y)〉}, where µr∩p(x,y)=min(µr(x,y),µp(x,y)) for any (x,y)∈ x ×y . the union of two fuzzy relations r and p from a nonempty set x to a nonempty set y is defined as r∪p = {〈(x,y),µr∪p(x,y)〉}, where µr∪p(x,y)=max(µr(x,y),µp(x,y)) for any (x,y)∈ x ×y . in general, if a is a set of fuzzy relations from a nonempty set x to a nonempty set y , then⋂ r∈a r = {〈(x,y),µ∩r∈ar(x,y)〉}, 4 int. j. anal. appl. (2022), 20:48 where µ∩r∈ar(x,y)= infr∈a µr(x,y) for any (x,y)∈ x ×y ;⋃ r∈a r = {〈(x,y),µ∪r∈ar(x,y)〉}, where µ∪r∈ar(x,y)= supr∈a µr(x,y) for any (x,y)∈ x ×y . 2.3. fuzzy topology. definition 2.3. [4] [fuzzy topology] a fuzzy topology (ft, for short) on a nonempty set x is a family τ of fuzzy sets on x which satisfies the following axioms: (i) ∅,x ∈ τ ; (ii) g1 ∩g2 ∈ τ for any g1,g2 ∈ τ; (iii) ⋃ gi ∈ τ for any {gi : i ∈ j}⊆ τ. in this case, the pair (x,τ) is called a fuzzy topological space (fts, for short) and any fs in τ is known as a fuzzy open set (fos, for short) in x. the complement of a fuzzy open set is called a fuzzy closed set (fcs, for short) in x. example 2.1. let x = {x1,x2,x3} and a1,a2,a3 ∈ fs(x) such that a1 = {〈x1,0.4〉,〈x2,0.7〉,〈x3,0.1〉}, a2 = {〈x1,0.3〉,〈x2,0.6〉,〈x3,0.2〉}, a3 = {〈x1,0.2〉,〈x2,0.5〉,〈x3,0.2〉}. then, τ = {∅,x,a1,a2,a3} is a fuzzy topology on x. 2.4. fuzzy topology generated by a fuzzy relation. the notion of fuzzy topology generated by a fuzzy relation was previously proposed by mishra and srivastava [11]. definition 2.4. [11] let x be a nonempty crisp set and r = {〈(x,y),µr(x,y)〉 | x,y ∈ x} a fuzzy relation on x. then for any x ∈ x, the fuzzy sets lx and rx defined by: µlx(y)= µr(y,x), for any y ∈ x, µrx(y)= µr(x,y), for any y ∈ x, are called respectively the lower and the upper contour of x. we denote by τ1, the fuzzy topology generated by the set of all lower contours and τ2, the fuzzy topology generated by the set of all upper contours. consequently, we denote by τr, the fuzzy topology generated by s the set of all lower and upper contours and it’s called the fuzzy topology generated by r. definition 2.5. let r be a fuzzy relation on the set x and τr is the fuzzy topology generated by r and let u1, u2 are two fuzzy open sets on τr. the u1 is said to be contained in u2 (in symbols, u1 ⊆ u2) if µu1(xi)≤ µu2(xi) for any xi ∈ x. in this case, we also say that u1 is smaller than u2. int. j. anal. appl. (2022), 20:48 5 example 2.2. let x = {x,y} and r be a fuzzy relation on x given by: µr(., .) x y x 0.5 0.7 y 0.4 0.6 then lx, ly, rx and ry are the fuzzy sets on x given by : lx = {〈x,0.5〉;〈y,0.4〉}, ly = {〈x,0.7〉;〈y,0.6〉}, rx = {〈x,0.5〉;〈y,0.7〉}, ry = {〈x,0.4〉;〈y,0.6〉}. notice that ry ⊆ rx and ry ⊆ ly. the fuzzy topology τr generated by the fuzzy topology generated by r is the fuzzy topology generated by s = {lx,ly} ∪ {rx,ry}, i.e., τr = {∅,x,lx,ly,rx,ry,lx ∩ry,ly ∩rx,lx ∪ry,ly ∪rx}, where lx ∩ry = {〈x,0.4〉;〈y,0.4〉}, ly ∩rx = {〈x,0.5〉;〈y,0.6〉}, lx ∪ry = {〈x,0.5〉;〈y,0.6〉}, and ly ∪rx = {〈x,0.7〉;〈y,0.7〉}. example 2.3. let x = {x,y,z} and r be a fuzzy relation on x given by: µr(., .) x y z x 1 0.5 0 y 0 1 0.8 z 0.7 0 1 lx, ly, lz, rx, ry and rz are the fuzzy sets on x given by : lx = {〈x,1〉;〈y,0〉;〈z,0.7〉} ; ly = {〈x,0.5〉;〈y,1〉;〈z,0〉} ; lz = {〈x,0〉;〈y,0.8〉;〈z,1〉} ; rx = {〈x,1〉;〈y,0.5〉;〈z,0〉} ; ry = {〈x,0〉;〈y,1〉;〈z,0.8〉} ; rz = {〈x,0.7〉;〈y,0〉;〈z,1〉}. the fuzzy topology τr is generated by s = {lx,ly,lz}∪{rx,ry,rz}. thus, τr = { ∅,x,lx,ly,lz,rx,ry,rz,{〈x,0.5〉;〈y,0〉;〈z,0〉},{〈x,0〉;〈y,0〉;〈z,0.7〉}, {〈x,1〉;〈y,0〉;〈z,0〉},{〈x,0.7〉;〈y,0〉;〈z,0.7〉},{〈x,0〉;〈y,0.8〉;〈z,0〉},{〈x,0.5〉;〈y,0.5〉;〈z,0〉}, 6 int. j. anal. appl. (2022), 20:48 {〈x,0〉;〈y,1〉;〈z,0〉},{〈x,0〉;〈y,0.8〉;〈z,0.8〉},{〈x,0〉;〈y,0〉;〈z,1〉},{〈x,0〉;〈y,0.5〉;〈z,0〉}, {〈x,0〉;〈y,0〉;〈z,0.8〉},{〈x,0.7〉;〈y,0〉;〈z,0〉},{〈x,1〉;〈y,1〉;〈z,0.7〉},{〈x,1〉;〈y,0.8〉;〈z,1〉}, {〈x,1〉;〈y,0.5〉;〈z,0.7〉},{〈x,1〉;〈y,1〉;〈z,0.8〉},{〈x,1〉;〈y,0〉;〈z,1〉},{〈x,0.5〉;〈y,1〉;〈z,1〉}, {〈x,1〉;〈y,1〉;〈z,0〉},{〈x,0.5〉;〈y,1〉;〈z,0.8〉},{〈x,0.7〉;〈y,1〉;〈z,1〉},{〈x,0〉;〈y,1〉;〈z,1〉}, {〈x,0.7〉;〈y,0.8〉;〈z,1〉},{〈x,1〉;〈y,1〉;〈z,0.8〉},{〈x,1〉;〈y,0.5〉;〈z,1〉} } . 3. the lattice of fuzzy open sets on fuzzy topology generated by a fuzzy relation in this section, we mainly investigate the lattice of all fuzzy open sets on a given fuzzy topology generated by a fuzzy relation. the following theorem provides the lattice structure of fuzzy open sets on a fuzzy topology generated by a fuzzy relation. theorem 3.1. let x be a nonempty set, r a fuzzy relation on x and τr be a fuzzy topology generated r. then the family l = {ui | ui is a fuzzy opn set on τr}, is a lattice. proof. the fact that τr is a fuzzy topology guarantees that the union and intersection of two fuzzy open sets are also fuzzy open sets. hence, (l,⊆) is a lattice. � remark 3.1. one can easily see that (i) (l,∩,∪,∅,x) is a boolean algebra. (ii) if x is a finite set, then the lattice (l,⊆) is complete. 4. fuzzy ideals and filters on the lattice of fuzzy open sets in this section, we introduce the notions of fuzzy ideal (resp. fuzzy filter) on the lattice of fuzzy open sets, and a characterization in terms of this lattice meet and join operations is given. throughout this section, l denotes the lattice of fuzzy open sets of τr a fuzzy topology generated by a fuzzy relation r on a nonempty set x. 4.1. definitions. definition 4.1. a fuzzy set i on l is called a fuzzy ideal if for all a,b ∈ l the following conditions hold: (i) µi(a∪b)≥ µi(a)∧µi(b), (ii) µi(a∩b)≥ µi(a)∨µi(b). definition 4.2. a fuzzy set f on l is called a fuzzy filter if for all a,b ∈ l the following conditions hold: int. j. anal. appl. (2022), 20:48 7 (i) µf(a∪b)≥ µf(a)∨µf(b), (ii) µf(a∩b)≥ µf(a)∧µf(b). the following proposition expresses the relationship between a fuzzy ideal and a fuzzy filter on a lattice of fuzzy open sets. its proof is straightforward. proposition 4.1. let ld be the order-dual lattice of l and a be fuzzy set on (l). then it holds that a is a fuzzy ideal on l if and only if a is a fuzzy filter on ld, and conversely. we need also the following result. proposition 4.2. let a and b are two fuzzy sets on l, then it holds that (i) if a and b are two fuzzy ideals on l, then a∩b is a fuzzy ideal on l; (ii) if a and b are two fuzzy filters on l, then a∩b is a fuzzy filter on l. 4.2. basic characterization of fuzzy ideals and filters on a lattice of fuzzy open sets. in this subsection, we provide interesting characterization of fuzzy ideals (resp. filters) on the lattice of fuzzy open sets in terms of its meet and its join operations. theorem 4.1. i is a fuzzy ideal on l if and only if the following condition is satisfied: µi(a∪b)= µi(a)∧µi(b), for any a,b ∈ l . proof. suppose that i is a fuzzy ideal on l, then for any a,b ∈ l it holds that µi(a ∪ b) ≥ µi(a)∧µi(b). since a v a∪b and b v a∪b, it follows from definition 4.1 (ii) that µi(a)= µi(a∩ (a∪b))≥ µi(a)∨µi(a∪b)≥ µi(a∪b) . in the same manner, µi(b) ≥ µi(a∪b) . hence, µi(a)∧µi(b) ≥ µi(a∪b). thus, µi(a∪b) = µi(a)∧µi(b) . conversely, suppose that µi(a∪b)= µi(a)∧µi(b) for any a,b ∈ l. then it is easy to see that µi(a∪b) ≥ µi(a)∧µi(b) for any a,b ∈ l. next, we will show that µi(a∩b) ≥ µi(a)∨µi(b) for any a,b ∈ l. let a,b ∈ l, since a ∪ (a ∩ b) = a and b ∪ (a ∩ b) = b then it holds that µi(a ∪ (a ∩ b)) = µi(a) and µi(b ∪ (a ∩ b)) = µi(b). from hypothesis it follows that µi(a) ∧ µi(a ∩ b) = µi(a) and µi(b) ∧ µi(a ∩ b) = µi(b). hence, µi(a ∩ b) ≥ µi(a) and µi(a∩b)≥ µi(b). thus, µi(a∩b)≥ µi(a)∨µi(b), for any a,b ∈ l. therefore, i is a fuzzy ideal on l. � in the same line, the following theorem provides a characterization of fuzzy filters on the lattice of fuzzy open sets in terms of its meet operation. theorem 4.2. f is a fuzzy filter on l if and only if the following condition is satisfied: µf(a∩b)= µf(a)∧µf(b), for any a,b ∈ l . 8 int. j. anal. appl. (2022), 20:48 proof. the proof is a direct application of proposition 4.1 and theorem 4.1. � as corollaries of the above theorems, we obtain the following interesting properties of fuzzy ideals and fuzzy filters on a lattice of fuzzy open sets. corollary 4.1. let i be a fuzzy ideal on l and a,b ∈ l. if a v b, then µi(a) ≥ µi(b), (i.e., the mapping µi is antitone). corollary 4.2. let f be a fuzzy filter on l and a,b ∈ l. if a v b, then µf(a)≤ µf(b), (i.e., the mapping µf is monotone). 5. conclusion and future work in this article, we have studied properties of lattices on fuzzy topology generated by fuzzy relation, and provided their various characteristics. we have introduced and studied the notions of fuzzy ideal (resp. fuzzy filter) on the lattice of fuzzy open sets and we have discussed some its basic properties. we anticipate that these notions of fuzzy ideals (resp. fuzzy filters) will facilitate the study and the representations of the different kinds of fuzzy lattices. due to the usefulness of these notions, we think it makes sense to study some kinds of fuzzy ideals (resp. fuzzy filters) on fuzzy topology generated by fuzzy relation. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. bennoui, l. zedam, s. milles, several types of single-valued neutrosophic ideals and filters on a lattice, twms j. app. eng. math. in press. [2] s. boudaoud, l. zedam, s. milles, principal intuitionistic fuzzy ideals and filters on a lattice, discuss. math.: gen. algebra appl. 40 (2020), 75–88. https://doi.org/10.7151/dmgaa.1325. [3] n. bourbaki, topologie générale, springer-verlag, berlin heidelberg, 2007. [4] c.l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182–190. https://doi.org/10.1016/ 0022-247x(68)90057-7. [5] b. davvaz, o. kazanci, a new kind of fuzzy sublattice (ideal, filter) of a lattice, int. j. fuzzy syst. 13 (2011), 55–63. [6] j.a. goguen, the logic of inexact concepts, synthese, 19 (1969), 325–373. https://www.jstor.org/stable/ 20114646. [7] b.a. davey, h.a. priestley, introduction to lattices and order, second edition, cambridge university press, cambridge, 2002. [8] j.l. kelley, general topology, van nostrand, princeton, new jersey, 1955. [9] y. liu, m. zheng, characterizations of fuzzy ideals in coresiduated lattices, adv. math. phys. 2016 (2016), 6423735. https://doi.org/10.1155/2016/6423735. https://doi.org/10.7151/dmgaa.1325 https://doi.org/10.1016/0022-247x(68)90057-7 https://doi.org/10.1016/0022-247x(68)90057-7 https://www.jstor.org/stable/20114646 https://www.jstor.org/stable/20114646 https://doi.org/10.1155/2016/6423735 int. j. anal. appl. (2022), 20:48 9 [10] s. milles, e. rak, l. zedam, intuitionistic fuzzy complete lattices, in: k.t. atanassov, o. castillo, j. kacprzyk, m. krawczak, p. melin, s. sotirov, e. sotirova, e. szmidt, g. de tré, s. zadrożny (eds.), novel developments in uncertainty representation and processing, springer international publishing, cham, 2016: pp. 149–160. https: //doi.org/10.1007/978-3-319-26211-6_13. [11] s. mishra, r. srivastava, fuzzy topologies generated by fuzzy relations, soft comput. 22 (2016), 373–385. https://doi.org/10.1007/s00500-016-2458-6. [12] s. milles, l. zedam, e. rak, characterizations of intuitionistic fuzzy ideals and filters based on lattice operations, j. fuzzy set valued anal. 2017 (2017), 143–159. https://doi.org/10.5899/2017/jfsva-00399. [13] b.s. schröder, ordered sets, birkhauser, boston, usa, 2002. [14] m.h. stone, the theory of representations of boolean algebras, trans. amer. math. soc. 40 (1936), 37–111. https://doi.org/10.2307/1989664. [15] m. tonga, maximality on fuzzy filters of lattices, afr. mat. 22 (2011), 105–114. https://doi.org/10.1007/ s13370-011-0009-y. [16] b. van gasse, g. deschrijver, c. cornelis, e.e. kerre, filters of residuated lattices and triangle algebras, inform. sci. 180 (2010), 3006–3020. https://doi.org/10.1016/j.ins.2010.04.010. [17] s. willard, general topology, addison-wesley publishing company, reading, massachusetts, 1970. [18] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 331–352. [19] l.a. zadeh, similarity relations and fuzzy orderings, inform. sci. 3 (1971), 177–200. https://doi.org/10.1016/ s0020-0255(71)80005-1. [20] l. zedam, s. milles, a. bennoui, ideals and filters on a lattice in neutrosophic setting, appl. appl. math. 16 (2021), 1140–1154. https://doi.org/10.1007/978-3-319-26211-6_13 https://doi.org/10.1007/978-3-319-26211-6_13 https://doi.org/10.1007/s00500-016-2458-6 https://doi.org/10.5899/2017/jfsva-00399 https://doi.org/10.2307/1989664 https://doi.org/10.1007/s13370-011-0009-y https://doi.org/10.1007/s13370-011-0009-y https://doi.org/10.1016/j.ins.2010.04.010 https://doi.org/10.1016/s0020-0255(71)80005-1 https://doi.org/10.1016/s0020-0255(71)80005-1 1. introduction 2. basic concepts 2.1. fuzzy sets 2.2. fuzzy relations 2.3. fuzzy topology 2.4. fuzzy topology generated by a fuzzy relation 3. the lattice of fuzzy open sets on fuzzy topology generated by a fuzzy relation 4. fuzzy ideals and filters on the lattice of fuzzy open sets 4.1. definitions 4.2. basic characterization of fuzzy ideals and filters on a lattice of fuzzy open sets 5. conclusion and future work references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 9-19 http://www.etamaths.com some integral inequalities for local fractional integrals m. zeki sarikaya1,∗, samet erden2 and hüseyin budak1 abstract. in this paper, firstly we extend some generalization of the hermite-hadamard inequality and bullen inequality to generalized convex functions. then, we give some important integral inequalities related to these inequalities. 1. introduction definition 1.1 (convex function). the function f : [a,b] ⊂ r → r, is said to be convex if the following inequality holds f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) for all x,y ∈ [a,b] and t ∈ [0, 1] . we say that f is concave if (−f) is convex. the classical hermite-hadamard inequality which was first published in [8] gives us an estimate of the mean value of a convex function f : i → r, f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 (1.1) in [1], bullen proved the following inequality which is known as bullen’s inequality for convex function. let f : i ⊂ r → r be a convex function on the interval i of real numbers and a,b ∈ i with a < b. the inequality 1 b−a ∫ b a f(x)dx ≤ 1 2 [ f ( a + b 2 ) + f(a) + f(b) 2 ] . an account the history of this inequality can be found in [3]. surveys on various generalizations and developments can be found in [12] and [2]. recently in [5], the author established this inequality for twice differentiable functions. in the case where f is convex then there exists an estimation better than (1.1). in [6], farissi gave the refinement of the inequality (1.1) as follows: theorem 1.1. assume that f : i → r is a convex function on i. then for all λ ∈ [0, 1], we have f ( a + b 2 ) ≤ l (λ) ≤ 1 b−a b∫ a f (x) dx ≤ l (λ) ≤ f (a) + f (b) 2 , where l (λ) := λf ( λb + (2 −λ) a 2 ) + (1 −λ) f ( (1 + λ) b + (1 −λ) a 2 ) and l (λ) := 1 2 (f (λb + (1 −λ) a) + λf (a) + (1 −λ) f (b)) . for more information recent developments to above inequalities, please refer to [2][7], [9][11], [14] and so on. received 1st december, 2016; accepted 9th february, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 26d15, 26a51, 52a40, 52a41. key words and phrases. hermite-hadamard inequality; local fractional integral; fractal space; generalized convex function. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 9 10 sarikaya, erden and budak 2. preliminaries recall the set rα of real line numbers and use the gao-yang-kang’s idea to describe the definition of the local fractional derivative and local fractional integral, see [15, 16] and so on. recently, the theory of yang’s fractional sets [15] was introduced as follows. for 0 < α ≤ 1, we have the following α-type set of element sets: zα : the α-type set of integer is defined as the set {0α,±1α,±2α, ...,±nα, ...} . qα : the α-type set of the rational numbers is defined as the set {mα = ( p q )α : p,q ∈ z, q 6= 0}. jα : the α-type set of the irrational numbers is defined as the set {mα 6= ( p q )α : p,q ∈ z, q 6= 0}. rα : the α-type set of the real line numbers is defined as the set rα = qα ∪jα. if aα,bα and cα belongs the set rα of real line numbers, then (1) aα + bα and aαbα belongs the set rα; (2) aα + bα = bα + aα = (a + b) α = (b + a) α ; (3) aα + (bα + cα) = (a + b) α + cα; (4) aαbα = bαaα = (ab) α = (ba) α ; (5) aα (bαcα) = (aαbα) cα; (6) aα (bα + cα) = aαbα + aαcα; (7) aα + 0α = 0α + aα = aα and aα1α = 1αaα = aα. the definition of the local fractional derivative and local fractional integral can be given as follows. definition 2.1. [15] a non-differentiable function f : r → rα, x → f(x) is called to be local fractional continuous at x0, if for any ε > 0, there exists δ > 0, such that |f(x) −f(x0)| < εα holds for |x−x0| < δ, where ε,δ ∈ r. if f(x) is local continuous on the interval (a,b) , we denote f(x) ∈ cα(a,b). definition 2.2. [15] the local fractional derivative of f(x) of order α at x = x0 is defined by f(α)(x0) = dαf(x) dxα ∣∣∣∣ x=x0 = lim x→x0 ∆α (f(x) −f(x0)) (x−x0) α , where ∆α (f(x) −f(x0)) =̃γ(α + 1) (f(x) −f(x0)) . if there exists f(k+1)α(x) = k+1 times︷ ︸︸ ︷ dαx ...d α xf(x) for any x ∈ i ⊆ r, then we denoted f ∈ d(k+1)α(i), where k = 0, 1, 2, ... definition 2.3. [15] let f(x) ∈ cα [a,b] . then the local fractional integral is defined by, ai α b f(x) = 1 γ(α + 1) b∫ a f(t)(dt)α = 1 γ(α + 1) lim ∆t→0 n−1∑ j=0 f(tj)(∆tj) α, with ∆tj = tj+1 − tj and ∆t = max{∆t1, ∆t2, ..., ∆tn−1} , where [tj, tj+1] , j = 0, ...,n − 1 and a = t0 < t1 < ... < tn−1 < tn = b is partition of interval [a,b] . here, it follows that ai α b f(x) = 0 if a = b and ai α b f(x) = −bi α a f(x) if a < b. if for any x ∈ [a,b] , there exists ai α x f(x), then we denoted by f(x) ∈ iαx [a,b] . definition 2.4 (generalized convex function). [15] let f : i ⊆ r → rα. for any x1,x2 ∈ i and λ ∈ [0, 1] , if the following inequality f(λx1 + (1 −λ)x2) ≤ λαf(x1) + (1 −λ)αf(x2) holds, then f is called a generalized convex function on i. here are two basic examples of generalized convex functions: (1) f(x) = xαp, x ≥ 0, p > 1; (2) f(x) = eα(x α), x ∈ r where eα(xα) = ∞∑ k=0 xαk γ(1+kα) is the mittag-lrffer function. some integral inequalities for local fractional integrals 11 theorem 2.1. [13] let f ∈ dα(i), then the following conditions are equivalent a) f is a generalized convex function on i b) f(α) is an increasing function on i c) for any x1,x2 ∈ i, f(x2) −f(x1) ≥ f(α)(x1) γ (1 + α) (x2 −x1) α . corollary 2.1. [13] let f ∈ d2α(a,b). then f is a generalized convex function ( or a generalized concave function) if and only if f(2α)(x) ≥ 0 ( or f(2α)(x) ≤ 0 ) for all x ∈ (a,b) . lemma 2.1. [15] (1) (local fractional integration is anti-differentiation) suppose that f(x) = g(α)(x) ∈ cα [a,b] , then we have ai α b f(x) = g(b) −g(a). (2) (local fractional integration by parts) suppose that f(x),g(x) ∈ dα [a,b] and f(α)(x), g(α)(x) ∈ cα [a,b] , then we have ai α b f(x)g (α)(x) = f(x)g(x)|ba −a i α b f (α)(x)g(x). lemma 2.2. [15] we have i) dαxkα dxα = γ(1 + kα) γ(1 + (k − 1) α) x(k−1)α; ii) 1 γ(α + 1) b∫ a xkα(dx)α = γ(1 + kα) γ(1 + (k + 1) α) ( b(k+1)α −a(k+1)α ) , k ∈ r. lemma 2.3 (generalized hölder’s inequality). [15] let f,g ∈ cα [a,b] , p,q > 1 with 1p + 1 q = 1, then 1 γ(α + 1) b∫ a |f(x)g(x)|(dx)α ≤   1 γ(α + 1) b∫ a |f(x)|p (dx)α   1 p   1 γ(α + 1) b∫ a |g(x)|q (dx)α   1 q . in [13], mo et al. proved the following generalized hermite-hadamard inequality for generalized convex function: theorem 2.2 (generalized hermite-hadamard inequality). let f(x) ∈ i(α)x [a,b] be a generalized convex function on [a,b] with a < b. then f ( a + b 2 ) ≤ γ (1 + α) (b−a)α a iαb f(x) ≤ f (a) + f (b) 2α . (2.1) the aim of this paper is to extend the generalized hermite-hadamard inequalities and generalized bullen inequalities to generalized convex functions. 3. main results theorem 3.1 (generalized hermite–hadamard-type inequality). let f(x) ∈ i(α)x [a,b] be a generalized convex function on [a,b] with a < b. then f ( a + b 2 ) ≤ h (λ) ≤ γ (1 + α) (b−a)α a iαb f(x) ≤ h (λ) ≤ f (a) + f (b) 2α , (3.1) where h (λ) := λαf ( λb + (2 −λ) a 2 ) + (1 −λ)α f ( (1 + λ) b + (1 −λ) a 2 ) and h (λ) := 1 2α [f (λb + (1 −λ) a) + λαf (a) + (1 −λ)α f (b)] . 12 sarikaya, erden and budak proof. let f be a generalized convex. then, applying (2.1) on the subinterval [a,λb + (1 −λ) a], with λ 6= 0, we have f ( λb + (2 −λ) a 2 ) (3.2) ≤ 1 λα (b−a)α λb+(1−λ)a∫ a f (t) (dt) α ≤ f (a) + f (λb + (1 −λ) a) 2α . applying (2.1) again on [λb + (1 −λ) a,b], with λ 6= 1, we get f ( (1 + λ) b + (1 −λ) a 2 ) (3.3) ≤ 1 (1 −λ)α (b−a)α b∫ λb+(1−λ)a f (t) (dt) α ≤ f (λb + (1 −λ) a) + f (b) 2α . multiplying (3.2) by λα, (3.3) by (1 −λ)α, and adding the resulting inequalities, we get: h (λ) ≤ γ (1 + α) (b−a)α a iαb f(x) ≤ h (λ) (3.4) where h (λ) and h (λ) are defined as in theorem 3.1. using the fact that f is a generalized convex function, we obtain f ( a + b 2 ) (3.5) = f ( λ λb + (2 −λ) a 2 + (1 −λ) (1 + λ) b + (1 −λ) a 2 ) ≤ λαf ( λv + (2 −λ) a 2 ) + (1 −λ)α f ( (1 + λ) b + (1 −λ) a 2 ) ≤ λα 2α [f (λb + (1 −λ) a) + f (a)] + (1 −λ)α 2α [f (b) + f (λb + (1 −λ) a)] = 1 2α [f (λb + (1 −λ) a) + λαf (a) + (1 −λ)α f (b)] ≤ f (a) + f (b) 2α . then by (3.4) and (3.5), we get (3.1). � theorem 3.2. let g(x) ∈ d2α [a,b] such that there exist constants m,m ∈ rα so that m ≤ g(2α) (x) ≤ m for x ∈ [a,b]. then m (bα + aαbα + aα) γ (1 + 3α) − m γ (1 + 2α) ( a2α + b2α 2α ) (3.6) ≤ γ (1 + α) (b−a)α a iαb g(x) −g ( a + b 2 ) ≤ m γ (1 + 2α) ( a2α + b2α 2α ) − m (bα + aαbα + aα) γ (1 + 3α) . some integral inequalities for local fractional integrals 13 and m γ (1 + 2α) ( a2α + b2α 2α ) − m (bα + aαbα + aα) γ (1 + 3α) (3.7) ≤ g(a) + g(b) 2α − γ (1 + α) (b−a)α a iαb g(x) ≤ m (bα + aαbα + aα) γ (1 + 3α) − m γ (1 + 2α) ( a2α + b2α 2α ) . proof. let f(x) = g(x) − m γ(1+2α) x2α, then f(2α) (x) = g(2α) (x) − m ≥ 0, which shows that f is generalized convex on (a,b). appliying ineqaulity (2.1) for f , then we have g ( a + b 2 ) − m γ (1 + 2α) ( a + b 2 )2α = f ( a + b 2 ) ≤ γ (1 + α) (b−a)α a iαb f(x) = 1 (b−a)α ∫ b a [ g(x) − m γ (1 + 2α) x2α ] (dx) α = γ (1 + α) (b−a)α a iαb g(x) − 1 (b−a)α m γ (1 + 2α) γ (1 + 2α) γ (1 + 3α) ( b3α −a3α ) . this implies that m (bα + aαbα + aα) γ (1 + 3α) − m γ (1 + 2α) ( a + b 2 )2α ≤ γ (1 + α) (b−a)α a iαb g(x) −g ( a + b 2 ) which proves the first inequality in (3.6). to prove the second part of (3.6), we apply the same argument for the generalized convex mapping f(x) = m γ(1+2α) x2α −g(x); x ∈ [a,b]. by applying the second part of the generalized hermite-hadamard inequality (2.1) for the mapping f(x) = g(x) − m γ(1+2α) x2α as follows g(a) + g(b) 2α − m γ (1 + 2α) ( a2α + b2α 2α ) = f(a) + f(b) 2α ≥ γ (1 + α) (b−a)α a iαb f(x) = 1 (b−a)α ∫ b a [ g(x) − m γ (1 + 2α) x2α ] (dx) α = γ (1 + α) (b−a)α a iαb g(x) − 1 (b−a)α m γ (1 + 2α) γ (1 + 2α) γ (1 + 3α) ( b3α −a3α ) . 14 sarikaya, erden and budak this is equivalent to m γ (1 + 2α) ( a2α + b2α 2α ) − m (bα + aαbα + aα) γ (1 + 3α) ≤ g(a) + g(b) 2α − γ (1 + α) (b−a)α a iαb g(x) which proves the rest part of (3.7). the second part is established in a similar manner; and we omit the details which completes the proof. � we prove the following inequality which is generalized bullen inequality for generalized convex function. theorem 3.3 (generalized bullen inequality). let f(x) ∈ i(α)x [a,b] be a generalized convex function on [a,b] with a < b. then we have the inequality γ (1 + α) (b−a)α a iαb f(x) ≤ 1 2α [ f ( a + b 2 ) + f (a) + f (b) 2α ] . (3.8) proof. using the theorem 2.2, we find that 2αγ (1 + α) (b−a)α 1 γ (1 + α) b∫ a f (x) (dx) α = 2αγ (1 + α) (b−a)α   1 γ (1 + α) a+b 2∫ a f (x) (dx) α + 1 γ (1 + α) b∫ a+b 2 f (x) (dx) α   ≤ f ( a+b 2 ) + f (a) 2α + f (b) + f ( a+b 2 ) 2α = f ( a + b 2 ) + f (a) + f (b) 2α . hence, the proof is completed. � theorem 3.4. let i ⊆ r be an interval, f : i0 ⊆ r → rα (i0 is the interior of i) such that f ∈ d2α(i0) and f(α) ∈ cα [a,b] for a,b ∈ i0 with a < b. then, for all x ∈ [a,b] , we have the following identity 1 2α (b−a)α (γ (1 + α))2 b∫ a ( x− a + b 2 )α p(x)f(2α) (x) (dx) α (3.9) = 1 2α [ f ( a + b 2 ) + f (a) + f (b) 2α ] − γ (1 + α) (b−a)α a iαb f(x) where p(x) =   (a−x)α , [ a, a+b 2 ) (b−x)α , [ a+b 2 ,b ] . some integral inequalities for local fractional integrals 15 proof. using the local fractional integration by parts, we have 1 γ (1 + α) b∫ a ( x− a + b 2 )α p(x)f(2α) (x) (dx) α = 1 γ (1 + α) a+b 2∫ a ( x− a + b 2 )α (a−x)α f(2α) (x) (dx)α + 1 γ (1 + α) b∫ a+b 2 ( x− a + b 2 )α (b−x)α f(2α) (x) (dx)α = ( x− a + b 2 )α (a−x)α f(α) (x) ∣∣∣∣ a+b 2 a − γ (1 + α) γ (1 + α) a+b 2∫ a ( 3a + b 2 − 2x )α f(α) (x) (dx) α + ( x− a + b 2 )α (b−x)α f(α) (x) ∣∣∣∣b a+b 2 − γ (1 + α) γ (1 + α) b∫ a+b 2 ( a + 3b 2 − 2x )α f(α) (x) (dx) α . using the local fractional integration by parts again, we find that 1 γ (1 + α) b∫ a ( x− a + b 2 )α p(x)f(2α) (x) (dx) α = γ (1 + α) (b−a)α f ( a + b 2 ) + γ (1 + α) (b−a)α f (a) + f (b) 2α − 2α (γ (1 + α)) 2 γ (1 + α) b∫ a f (x) (dx) α . if we devide the resulting equality with 2αγ (1 + α) (b−a)α, then we complete the proof. � theorem 3.5. suppose that the assumptions of theorem 3.4 are satisfied, then we have the following inequality ∣∣∣∣ 12α [ f ( a + b 2 ) + f (a) + f (b) 2α ] − γ (1 + α) (b−a)α a iαb f(x) ∣∣∣∣ ≤ (b−a)(1+ 1 p )α 8αγ (1 + α) (b(p + 1,p + 1)) 1 p ∥∥∥f(2α) (x)∥∥∥ q where, p,q > 1, 1 p + 1 q = 1, ∥∥f(2α)∥∥ q is defined by ∥∥∥f(2α)∥∥∥ q =   1 γ (1 + α) b∫ a ∣∣∣f(2α)(x)∣∣∣q (dx)α   1 q 16 sarikaya, erden and budak and b (x,y) is defined by b (x,y) = 1 γ (1 + α) 1∫ 0 t(x−1)α (1 − t)(y−1)α (dt)α . proof. taking madulus in (3.9) and using generalized hölder inequality, we have∣∣∣∣ 12α [ f ( a + b 2 ) + f (a) + f (b) 2α ] − γ (1 + α) (b−a)α a iαb f(x) ∣∣∣∣ (3.10) ≤ 1 2α (b−a)α (γ (1 + α))2 b∫ a ∣∣∣∣x− a + b2 ∣∣∣∣α |p(x)| ∣∣∣f(2α) (x)∣∣∣ (dx)α ≤ 1 2α (b−a)α γ (1 + α)   1 γ (1 + α) b∫ a ∣∣∣f(2α) (x)∣∣∣q (dx)α   1 q ×   1 γ (1 + α) b∫ a ∣∣∣∣x− a + b2 ∣∣∣∣pα |p(x)|p (dx)α   1 p = ∥∥f(2α)∥∥ q 2α (b−a)α γ (1 + α)   1 γ (1 + α) a+b 2∫ a ( a + b 2 −x )pα (x−a)pα (dx)α + 1 γ (1 + α) b∫ a+b 2 ( x− a + b 2 )pα (b−x)pα (dx)α   1 p = ∥∥f(2α)∥∥ q 2α (b−a)α γ (1 + α) (k1 + k2) 1 p . for calculating integral k1, using changing variable with x = (1 − t)a + ta+b2 , we obtain k1 = 1 γ (1 + α) a+b 2∫ a ( a + b 2 −x )pα (x−a)pα (dx)α (3.11) = ( b−a 2 )(2p+1)α 1 γ (1 + α) 1∫ 0 (1 − t)pαtpα (dt)α = ( b−a 2 )(2p+1)α b(p + 1,p + 1). similarliy, using changing variable with x = (1 − t)a+b 2 + tb, we have k2 = 1 γ (1 + α) b∫ a+b 2 ( x− a + b 2 )pα (b−x)pα (dx)α (3.12) = ( b−a 2 )(2p+1)α b(p + 1,p + 1) some integral inequalities for local fractional integrals 17 putting (3.11) and (3.12) in (3.10), we obtain∣∣∣∣ 12α [ f ( a + b 2 ) + f (a) + f (b) 2α ] − γ (1 + α) (b−a)α a iαb f(x) ∣∣∣∣ ≤ ∥∥f(2α)∥∥ q 2α (b−a)α γ (1 + α) ( 2α (b−a)(2p+1)α 2(2p+1)α b(p + 1,p + 1) )1 p = (b−a)(1+ 1 p )α 8αγ (1 + α) (b(p + 1,p + 1)) 1 p ∥∥∥f(2α)∥∥∥ q which completes the proof. � theorem 3.6. the assumptions of theorem 3.4 are satisfied. if the mapping ϕ(x) =   (a−x)α ( x− a+b 2 )α f(2α) (x) , [ a, a+b 2 ) (b−x)α ( x− a+b 2 )α f(2α) (x) , [ a+b 2 ,b ] . is a generalized convex, then we have the inequality (b−a)2α 64α (γ (1 + α)) 2 [ f(2α) ( 3a + b 4 ) + f(2α) ( a + 3b 4 )] ≤ 1 2α [ f ( a + b 2 ) + f (a) + f (b) 2α ] − γ (1 + α) (b−a)α a iαb f(x) ≤ (b−a)2α 128α (γ (1 + α)) 2 [ f(2α) ( 3a + b 4 ) + f(2α) ( a + 3b 4 )] . proof. applying the first inequality (2.1) for the mapping ϕ, we get γ (1 + α) (b−a)α 2α γ (1 + α) a+b 2∫ a ϕ (x) (dx) α (3.13) ≥ ϕ ( 3a + b 4 ) = (b−a)2α 16α f(2α) ( 3a + b 4 ) and γ (1 + α) (b−a)α 2α γ (1 + α) b∫ a+b 2 ϕ (x) (dx) α (3.14) ≥ ϕ ( a + 3b 4 ) = (b−a)2α 16α f(2α) ( a + 3b 4 ) . applying the inequality (3.8) for the mapping ϕ, we have γ (1 + α) (b−a)α 2α γ (1 + α) a+b 2∫ a ϕ (x) (dx) α (3.15) ≤ 1 2α [ ϕ ( 3a + b 4 ) + ϕ (a) + ϕ ( a+b 2 ) 2α ] = (b−a)2α 32α f(2α) ( 3a + b 4 ) 18 sarikaya, erden and budak and γ (1 + α) (b−a)α 2α γ (1 + α) b∫ a+b 2 ϕ (x) (dx) α (3.16) ≤ 1 2α [ ϕ ( a + 3b 4 ) + ϕ ( a+b 2 ) + ϕ (b) 2α ] = (b−a)2α 32α f(2α) ( a + 3b 4 ) . adding the inequalities (3.13)-(3.16) and from theorem 3.4, we write (b−a)2α 16α [ f(2α) ( 3a + b 4 ) + f(2α) ( a + 3b 4 )] ≤ γ (1 + α) (b−a)α 2α γ (1 + α) b∫ a+b 2 ϕ (x) (dx) α = 4α (γ (1 + α)) 2 [ 1 2α ( f ( a + b 2 ) + f (a) + f (b) 2α ) − γ (1 + α) (b−a)α a iαb f(x) ] ≤ (b−a)2α 32α [ f(2α) ( 3a + b 4 ) + f(2α) ( a + 3b 4 )] . if we devide the resulting inequality with 4α (γ (1 + α)) 2 , then we complete the proof. � references [1] p. s. bullen, error estimates for some elementary quadrature rules, univ. beograd. publ. elektrotehn. fak. ser. mat. fiz. (1978) 602-633, (1979) 97-103. [2] s. s. dragomir and r.p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (5) (1998), 91-95. [3] s. s. dragomir and c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, rgmia monographs, victoria university, 2000. [4] s. s. dragomir and r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (5) (1998), 91-95. [5] a. el farissi, z. latreuch and b. belaidi, hadamard-type inequalities for twice diffrentiable functions, rgmia research report collection, 12 (1) (2009), art. id 6. [6] a. el farissi, simple proof and refinement of hermite-hadamard inequality, j. math. inequal. 4 (3) (2010), 365-369. [7] x. gao, a note on the hermite-hadamard inequality, jmi jour. math. ineq..4 (4) (2010), 587-591. [8] j. hadamard, etude sur les proprietes des fonctions entieres et en particulier d’une fonction consideree par riemann, j. math. pures appl. 58 (1893), 171-215. [9] u. s. kirmaci and m. e. ozdemir, on some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, appl. math comput. 153 (2004), 361–368. [10] u. s. kirmaci and r. dikici, on some hermite-hadamard type inequalities for twice differentiable mappings and applications, tamkang j. math. 44 (1) (2013), 41–51. [11] u. s. kirmaci, inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, appl. math. comput. 147 (2004) 137–146. [12] d. s. mitrinovic, j. e. pečarič, and a. m. fink, classical and new inequalities in analysis, ser. math. appl. (east european ser.). dordrecht: kluwer academic publishers group, vol. 61, 1993. [13] h. mo, x. sui and d. yu, generalized convex functions on fractal sets and two related inequalities, abstr. appl. anal. 2014 (2014), art. id 636751, 7 pages. [14] m. z. sarikaya and h. yaldiz, on the hadamard’s type inequalities for l-lipschitzian mapping, konuralp j. math. 1 (2) (2013), 33-40. [15] x. j. yang, advanced local fractional calculus and its applications, world science publisher, new york, 2012. [16] j. yang, d. baleanu and x. j. yang, analysis of fractal wave equations by local fractional fourier series method, adv. math. phys. 2013 (2013), art. id 632309. some integral inequalities for local fractional integrals 19 1department of mathematics, faculty of science and arts, düzce university, konuralp campus, düzceturkey 2department of mathematics, faculty of science, bartın university, bartin-turkey ∗corresponding author: sarikayamz@gmail.com 1. introduction 2. preliminaries 3. main results references int. j. anal. appl. (2023), 21:48 on weakly s-2-absorbing submodules govindarajulu narayanan sudharshana∗ department of mathematics, annamalai university, chidambaram 608001, tamil nadu, india ∗corresponding author: sudharshanasss3@gmail.com abstract. let r be a commutative ring with identity and let m be a unitary r-module. in this paper, we introduce the notion of weakly s-2-absorbing submodule. suppose that s is a multiplicatively closed subset of r. a submodule p of m with (p :r m)∩s = ∅ is said to be a weakly s-2-absorbing submodule if there exists an element s ∈s such that whenever a, b ∈r and m ∈m with 0 6= abm ∈p , then sab ∈ (p :m) or sam ∈p or sbm ∈p . we give the characterizations, properties and examples of weakly s-2-absorbing submodules. 1. introduction throughout this paper, r denotes a commutative ring with non zero identity and m is an r module. prime ideals and submodules have vital role in ring and module theory. of course a proper submodule p of m is called prime if am ∈ p for a ∈ r and m ∈ m implies a ∈ (p :r m) or m ∈ p where (p :r m) = {r ∈ r : rm ⊆ p}. several generalizations of these concepts have been studied extensively by many authors [9], [13], [6], [16], [3], [11], [14], [5]. in 2007, atani and farzalipour introduced the concept of weakly prime submodules as a generalization of prime submodules. a proper submodule p of m is defined as weakly prime if for a ∈ r and m ∈ m, whenever for 0 6= am ∈ p implies a ∈ (p :r m) or m ∈ p as in [5]. a new kind of generalization of prime submodule has been introduced and studied by sengelen sevim et. al. in 2019 in [14]. for a multiplicatively closed subset s of r, that is, s satisfies the following conditions: (i) 1 ∈ s and (ii) s1s2 ∈ s for each s1, s2 ∈ s, a proper submodule p of an r-module m with (p :r m) ∩s = ∅ is called an s-prime submodule if there exists s ∈ s such that for a ∈ r and m ∈ m, if am ∈ p then either sa ∈ (p :r m) or sm ∈ p. in particular an ideal i of r is called received: apr. 5, 2022. 2010 mathematics subject classification. 06f25. key words and phrases. weakly s-prime; s-2-absorbing submodule; weakly s-2-absorbing submodule. https://doi.org/10.28924/2291-8639-21-2023-48 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-48 2 int. j. anal. appl. (2023), 21:48 as s-prime ideal if i is an s-prime submodule of an r-module r, [10]. after that, the concept of weakly s-prime submodule was introduced as a generalization of s-prime submodules in [11]. here, for a multiplicatively closed subset s of r, they called a submodule p of an r-module m with (p :r m) ∩s = ∅ a weakly s-prime submodule if there exists s ∈ s such that for a ∈ r and m ∈ m, if 0 6= am ∈ p then either sa ∈ (p :r m) or sm ∈ p. in particular, a proper ideal i of r disjoint with s is said to be weakly s-prime if there exists s ∈ s such that for a, b ∈ r and 0 6= ab ∈ i then either sa ∈ i or sb ∈ i [3]. one of the important generalizations of prime submodule is the concept of 2-absorbing submodule. in 2011, darani and soheilnia [6] introduced the concepts of 2-absorbing and weakly 2-absorbing submodules of modules over commutative rings with identities. a proper submodule p of a module m over a commutative ring r with identity is said be a 2-absorbing submodule (weakly 2-absorbing submodule) of m if whenever a, b ∈ r and m ∈ m with abm ∈ p (0 6= abm ∈ p ), then abm ⊆ p or am ∈ p or bm ∈ p . predictably, a proper ideal i of r is 2-absorbing ideal if and only if i is a 2-absorbing submodule of r-module r. recently, the concept of s-2-absorbing submodules was introduced in [16] which is a generalization of s-prime submodules and 2-absorbing submodules. a submodule p of m is said to be an s-2absorbing submodule if (p :r m) ∩s = ∅ and there exists a fixed s ∈ s such that for a, b ∈ r and m ∈ m, if abm ∈ p then either sab ∈ (p :r m) or sam ∈ p or sbm ∈ p. in particular, an ideal i of r is an s-2-absorbing ideal if i is an s-2-absorbing submodule of r-module r. our objective in this paper is to define and study the concept of weakly s-2-absorbing submodule as an extension of the above concepts. a submodule p of m is said to be a weakly s-2-absorbing submodule if (p :r m)∩s = ∅ and there exists an element s ∈ s such that for a, b ∈ r and m ∈ m, if 0 6= abm ∈ p then either sab ∈ (p :r m) or sam ∈ p or sbm ∈ p. in this case, we say that p is associated to s. in particular, an ideal i of r is a weakly s-2-absorbing ideal if i is a weakly s-2-absorbing submodule of r-module r. some characterizations of weakly s-2-absorbing submodules are obtained. besides, we investigate relationships between s-2-absorbing submodule and weakly s-2-absorbing submodule and also between weakly s-prime and weakly s-2-absorbing submodules of modules over commutative rings. 2. characterizations of weakly s-2-absorbing submodules we start with the definitions and relationships of the main concepts of the paper. definition 2.1. let s be a multiplicatively closed subset of r. a submodule p of an r-module m is called a weakly s-2-absorbing submodule if (p :r m) ∩s = ∅ and there exists an element s ∈ s such that, whenever a, b ∈ r and m ∈ m, 0 6= abm ∈ p implies sab ∈ (p : m) or sam ∈ p or sbm ∈ p. in this case, we say that p is associated to s. in particular, an ideal i of r is a weakly s-2-absorbing ideal if i is a weakly s-2-absorbing submodule of r-module r int. j. anal. appl. (2023), 21:48 3 example 2.1. consider the z-module m = z ×z6 and let p = 2z× < 3̄ >. then p is a weakly s-2-absorbing submodule of m where s = {2n : n ∈ n∪{0}}. indeed, let (0, 0̄) 6= r1r2(r ′,m) ∈ p for r1,r2,r ′ ∈ z and m ∈ z6 such that 2r1r2 /∈ (p : m) = 6z. then r1r2m ∈< 3̄ > with r1, r2 /∈ 3z and so m ∈< 3̄ > also r ′ ∈ 2z. thus, 2r1(r ′,m) ∈ p as needed. example 2.2. consider the submodule p =< 6 > of the z-module z and the multiplicatively closed subset s = {5n : n ∈n∪{0}}. then p is a weakly s-2-absorbing submodule. it is clear that every s-2-absorbing submodule is a weakly s-2-absorbing submodule. since the zero submodule is (by definition) a weakly s-2-absorbing submodule of any r-module, hence the converse is not true in general and the following example shows this. example 2.3. consider r = z, m = z/30z, p = 0 and s = z −{0}. then 2.3(5 + 30z) = 0 ∈ p while 1.2.3 /∈ (p : m), 1.2(5 + 30z) /∈ p and 1.3(5 + 30z) /∈ p. therefore p is not s-2-absorbing while it is weakly s-2-absorbing. every weakly 2-absorbing submodule p of an r-module m satisfying (p : m) ∩s = ∅ is a weakly s-2-absorbing submodule of m and the two concepts coincide if s ⊆ u(r) where u(r) denotes the set of units in r. the following example shows that the converse need not be true. example 2.4. suppose that m = z ×z is an r = z ×z-module and p = pz ×{0} is a submodule of m where p is prime. then p is weakly s-2-absorbing submodule of m where s = z −{0}×{0}. indeed, let (0, 0) 6= (r1, r2)(r3, r4)(m1,m2) ∈ p for (r1, r2), (r3, r4) ∈ z ×z and (m1,m2) ∈ m such that s(r1, r2)(r3, r4) /∈ (p : m) = 0. then either r1 or r3 or m1 must be p and either r2 or r4 or m2 must be 0. thus s(p,r2)(m1,m2) ∈ p as needed. on the other hand, p is not a weakly 2-absorbing submodule since (0, 0) 6= (p, 1)(1, 0)(1, 1) ∈ p but neither (p, 1)(1, 0) ∈ (p : m) nor (p, 1)(1, 1) ∈ p nor (1, 0)(1, 1) ∈ p. hence p is not weakly 2-absorbing. lemma 2.1. let s be a multiplicatively closed subset of r and p be a submodule of m. if p is weakly s-prime, then there exists an element s ∈ s of p such that 0 6= abm ∈ p for all a, b ∈ r and m ∈ m implies sbm ⊆ p whenever sam /∈ p . proof. let a, b ∈ r and m ∈ m. assume that 0 6= abm ∈ p. then 0 6= b(am) ∈ p. since p is weakly s-prime, there exists s ∈ s of p such that sb ∈ (p : m) or sam ∈ p. hence if sam /∈ p, then we get sbm ⊆ p. proposition 2.1. let s be a multiplicatively closed subset of r and p be a submodule of m. if p is weakly s-prime, then it is weakly s-2-absorbing. proof. let a, b ∈ r and m ∈ m be such that 0 6= abm ∈ p . since p is weakly s-prime, there exists s ∈ s of p such that sa ∈ (p : m) or sbm ∈ p. if sbm ∈ p , then we are done. suppose 4 int. j. anal. appl. (2023), 21:48 sbm /∈ p , then by lemma2.1, we get sam ⊆ p and consequently sabm ⊆ p. hence p is weakly s-2-absorbing. the converse of the previous proposition need not be true, is illustrated in the following example. example 2.5. suppose that m = z × z is an r = z × z-module and p = 2z × {0} is a submodule of m. then p is weakly s-2-absorbing where s = (2z + 1) × {0}. indeed, let (0, 0) 6= (r1, r2)(r3, r4)(m1,m2) ∈ p for (r1, r2), (r3, r4) ∈ z × z and (m1,m2) ∈ m such that s(r1, r2)(r3, r4) /∈ (p : m) = 0. then either r1 or r3 or m1 must be in 2z. without loss of generality, assume that r1 ∈ 2z. then s(r1, r2)(m1,m2) ∈ 2z ×{0} as needed. on the other hand, we have (0, 0) 6= (2, 0)(1, 1) ∈ p. now neither s(2, 0) ∈ (p : m) nor s(1, 1) ∈ p. hence p is not weakly s-prime. let r be a ring and s ⊆ r a multiplicatively closed subset of r. the saturation s∗ of s is defined as s∗={r ∈ r: r 1 is a unit of s−1r }. note that s∗ is a multiplicatively closed subset containing s. proposition 2.2. if m is an r-module and s is a mltiplicatively closed subset of r. then the following statements hold. (i) suppose that s1 ⊆ s2 are multiplicatively closed subsets of r. if p is a weakly s1-2-absorbing submodule and (p : m) ∩s2 = ∅, then p is a weakly s2-2-absorbing submodule. (ii) a submodule p of m is a weakly s-2-absorbing submodule if and only if it is a weakly s∗-2absorbing submodule. (iii) if p is a weakly s-2-absorbing submodule of m, then s−1p is a weakly 2-absorbing submodule of s−1m. proof. (i): it is clear. (ii):let p be weakly s-2-absorbing. suppose (p : m)∩s∗ 6= ∅. then we have t ∈ (p : m)∩s∗ and this implies that t 1 .a s = 1 for some a ∈ r and s ∈ s as t 1 is a unit of s−1r. thus ta = s ∈ s implies ta ∈ s and so (p : m) ∩s 6= ∅ which is a contradiction. hence (p : m) ∩s∗ = ∅. by (i), p is a weakly s∗-2-absorbing submodule as s ⊆ s∗. conversely, let a, b ∈ r and m ∈ m such that 0 6= abm ∈ p. since p is weakly s∗-2-absorbing, there exists s” ∈ s∗ of p such that s”ab ∈ (p : m) or s”am ∈ p or s”bm ∈ p. since s” ∈ s∗, we have s” 1 .t s = 1 for some t ∈ r, s ∈ s. then s”t = s ∈ s and so s”t ∈ s. then sab ∈ (p : m) or sam ∈ p or sbm ∈ p. thus p is weakly s-2-absorbing. (iii) let a s1 , b s2 ∈ s−1r and m s3 ∈ s−1m be such that 0m s 6= a s1 b s2 m s3 ∈ s−1p. then we get 0m 6= sabm ∈ p for some s ∈ s. by assumption, there exists s4 ∈ s of p such that s4(sa)b ∈ (p : m) or s4(sa)m ∈ p or s4bm ∈ p. then as1 b s2 = s4s s4s ab s1s2 ∈ s−1(p : m) ⊆ (s−1p : s−1m) or a s1 m s3 = s4s s4s am s1s3 ∈ s−1p or b s2 m s3 = s4 s4 bm s2s3 ∈ s−1p . hence s−1p is weakly 2-absorbing submodule of s−1m. the converse of (iii) in the above proposition need not be true is shown by the following example. int. j. anal. appl. (2023), 21:48 5 example 2.6. consider the z-module m = q3 and s = z −{0}. let p = {(r1, r2, 0) : r1, r2 ∈ z}. note that (p : m) = 0 and (p : m) ∩s = ∅. if a = 2, b = 3 and m = (1 2 , 1 3 , 0), then (0, 0, 0) 6= 2.3(1 2 , 1 3 , 0) = (3, 2, 0) ∈ p. if we take s = 5 ∈ s, then clearly 5.2.3 /∈ (p : m), 5.2(1 2 , 1 3 , 0) /∈ p, 5.3(1 2 , 1 3 , 0) /∈ p. thus p is not weakly s-2-absorbing. from the fact that s−1m is a vectorspace over the field s−1z that is q and the proper subspace s−1p is 2-absorbing [16], we have s−1p is a weakly 2-absorbing submodule by [6]. proposition 2.3. let s be a multiplicatively closed subset of r and m be an r-module. then the intersection of two weakly s-prime submodule is a weakly s-2-absorbing submodule. proof. let p1, p2 be two weakly s-prime submodules of m and p = p1 ∩ p2. let a, b ∈ r and m ∈ m be such that 0 6= abm ∈ p. since p1 is weakly s-prime and 0 6= a(bm) ∈ p1, there exists s1 ∈ s of p1 such that s1a ∈ (p1 : m) or s1bm ∈ p1. again as p2 is weakly s-prime and 0 6= bam ∈ p2 there exists s2 ∈ s of p2 such that s2b ∈ (p2 : m) or s2am ∈ p2. now consider the following four cases. case 1: s1a ∈ (p1 : m) and s1bm /∈ p1 s2b ∈ (p2 : m) and s2am /∈ p2. now, put s = s1s2 ∈ s. then sab ∈ (p1 : m) and sab ∈ (p2 : m) and so sabm ⊆ p1 ∩ p2 = p. hence sab ∈ (p : m). case 2: s1a ∈ (p1 : m) and s1bm /∈ p1 s2am ∈ p2 and s2b /∈ (p2 : m). then s1am ∈ s1am ⊆ p1 and s2am ∈ p2 implies that sam ∈ p where s = s1s2 ∈ s. case 3: s1bm ∈ p1 and s1a /∈ (p1 : m) s2am /∈ p2 and s2b ∈ (p2 : m) then clearly sbm ∈ p where s = s1s2 ∈ s. case 4: s1bm ∈ p1 and s1a /∈ (p1 : m) s2am ∈ p2 and s2b /∈ (p2 : m) as p1 is weakly s-prime and 0 6= abm ∈ p1 and also s1am /∈ p1 gives that s1bm ⊆ p1 by lemma 2.1. for the same reason, we get s2am ⊆ p2. then clearly sab ∈ (p : m) where s = s1s2 ∈ s. hence p is weakly s-2-absorbing. the following result provides some condition under which a weakly s-2-absorbing submodule is s-2absorbing. theorem 2.1. let s be a multiplicatively closed subset of r and p be a weakly s-2-absorbing submodule of m. if p is not s-2-absorbing, then (p : m)2p = 0. proof. by our assmption, there exists s ∈ s of p such that, whenever x, y ∈ r and m ∈ m, 0 6= xym ∈ p implies sxy ∈ (p : m) or sxm ∈ p or sym ∈ p. suppose (p : m)2p 6= 0, we claim that p is s-2-absorbing. let a, b ∈ r and m ∈ m be such that abm ∈ p. if abm 6= 0, then 6 int. j. anal. appl. (2023), 21:48 sab ∈ (p : m) or sam ∈ p or sbm ∈ p. so assume that abm = 0. now, first we assume that abp 6= 0. then abp0 6= 0 for some p0 ∈ p implies 0 6= abp0 = ab(m + p0) ∈ p. then sab ∈ (p : m) or sa(m + p0) ∈ p or sb(m + p0) ∈ p by our assumption. hence sab ∈ (p : m) or sam ∈ p or sbm ∈ p. hence we may assume that abp = 0. if a(p : m)m 6= 0, then aq0m 6= 0 for some q0 ∈ (p : m). then 0 6= aq0m = a(b+q0)m ∈ p. then, we get sa(b+q0) ∈ (p : m) or sam ∈ p or s(b+q0)m ∈ p. hence sab ∈ (p : m) or sam ∈ p or sbm ∈ p. so we can assume that a(p : m)m = 0. in the same manner, we can assume that b(p : m)m = 0. since (p : m)2p 6= 0, there exists x0, y0 ∈ (p : m) and m0 ∈ p with x0y0m0 6= 0. if ay0m0 6= 0, then 0 6= ay0m0 = a(b+y0)(m+m0) ∈ p since abm = 0, abm0 ∈ abp = 0 and ay0m = amy0 ∈ am(p : m) = 0. hence, by our assumption sa(b + y0) ∈ (p : m) or sa(m + m0) ∈ p or s(b + y0)(m + m0) ∈ p and so sab ∈ (p : m) or sam ∈ p or sbm ∈ p. so we can assume that ay0m0 = 0. in the same manner, we can assume that x0y0m = 0 and x0bm0 = 0. since x0y0m0 6= 0, we have 0 6= x0y0m0 = (a + x0)(b + y0)(m + m0) ∈ p since abm = 0, abm0 ∈ abp = 0 and ay0m = amy0 ∈ am(p : m) = 0. then, s(a + x0)(b + y0) ∈ (p : m) or s(a + x0)(m + m0) ∈ p or s(b + y0)(m + m0) ∈ p. hence sab ∈ (p : m) or sam ∈ p or sbm ∈ p. hence p is s-2-absorbing. recall that an r-module m is said to be a multiplication module if for each submodule n of m, n = im for some ideal i of r. if n1, n2 are two submodules of m, then n1 = am and n2 = bm for some ideals a, b of r. the product of n1 and n2 is defined as n1n2 = abm [4]. also note that this product is independent of the presentations of submodules n1 and n2 of m [4, theorem 3.4]. a submodule n of an r-module m is called a nilpotent submodule if (n : m)kn = 0 for some positive integer k [1]. corollary 2.1. let s be a multiplicatively closed subset of r and p be a submodule of m. assume that p is a weakly s-2-absorbing submodule of m that is not s-2-absorbing, then 1, p is nilpotent. 2, if m is a multiplication module, then p3 = 0. proof. 1. immediate from the definition of nilpotent submodule and by theorem 2.1. 2. by theorem 2.1, (p : m)2p = 0. then (p : m)3m = (p : m)2(p : m)m = 0. thus p3 = 0. if n is a proper submodule of a non-zero r-module m. then the m-radical of n, denoted by m-radn is defined as the intersection of all prime submodules of m containing n [12], [8]. if a is an ideal of the ring r then the m-radical of a (considered as a submodule of the r-module r) is denoted by √ a and consists of all elements r of r such that rn ∈ a for some positive integer n [8]. also it is shown in [8, theorem 2.12] that if n is a proper submodule of a multiplication r-module m, then m-radn = ( √ (n : m))m. int. j. anal. appl. (2023), 21:48 7 proposition 2.4. assume that m is a faithful multiplication r-module, s is a multiplicatively closed subset of r and p is a submodule of m. let p be a weakly s-2-absorbing submodule of m. if p is not s-2-absorbing, then p ⊆ m-rad0. proof. suppose p is not s-2-absorbing. by theorem 2.1, (p : m)2p = 0. since (p : m)2(p : m)m ⊆ (p : m)2p, we have (p : m)3 ⊆ ((p : m)2p : m) = (0 : m) = 0. let a ∈ (p : m), then a3 = 0 and so a ∈ √ 0. thus (p : m) ⊆ √ 0. hence p = (p : m)m ⊆ √ 0m = m-rad0. proposition 2.5. if s is a multiplicatively closed subset of r and p is a submodule of a cyclic faithful r-module m, then p is a weakly s-2-absorbing submodule of m if and only if (p : m) is a weakly s-2-absorbing ideal of r. proof. let p be a weakly s-2-absorbing submodule of m. assume that m = rm for some m ∈ m and let 0 6= abc ∈ (p : m) for some a, b, c ∈ r. then abcm ∈ p. if abcm 6= 0, then their exists an element s ∈ s of p such that sab ∈ (p : m) or sacm ∈ p or sbcm ∈ p. if sab ∈ (p : m), then we are done. if sacm ∈ p, then sac ∈ (p : m) = (p : m) as m is cyclic. likewise, if sbcm ∈ p, then sbc ∈ (p : m). then, assume that abcm = 0, we get abc ∈ (0 : m) = (0 : m). as m is faithful, we have abc = 0, a contradiction. hence (p : m) is a weakly s-2-absorbing ideal of r. conversely, let 0 6= abm′ ∈ p for some a, b ∈ r and m′ ∈ m. then m′ = cm for some c ∈ r and we get 0 6= abcm ∈ p. this implies abc ∈ (p : m) = (p : m). if abc 6= 0, then there exists an element s′ ∈ s of (p : m) such that s′ab ∈ (p : m) or s′bc ∈ (p : m) or s′ac ∈ (p : m). if s′ab ∈ (p : m), then we are done. if s′bc ∈ (p : m), then s′bc ∈ (p : m) and so s′bm′ ∈ p. likewise if s′ac ∈ (p : m), then s′am′ ∈ p . now, assume that abc = 0, then abcm = 0.m = 0, a contradiction. hence p is weakly s-2-absorbing. proposition 2.6. if s is a multiplicatively closed subset of r and p is a submodule of a cyclic rmodule m, then p is an s-2-absorbing submodule of m if and only if (p : m) is an s-2-absorbing ideal of r. after recalling the concepts of triple-zero in various papers like [9], [7], we give the following result which is an analogue of [9, theorem 3.10]. theorem 2.2. let s be a multiplicatively closed subset of r and let p be a weakly s-2-absorbing submodule of m. if a, b ∈ r, m ∈ m with abm = 0 and sab /∈ (p : m), sam /∈ p , sbm /∈ p for any s ∈ s, then (1) abp = a(p : m)m = b(p : m)m = 0 (2) a(p : m)p = b(p : m)p = (p : m)2m = 0 proof. (1). if abp 6= 0, then for some p ∈ p, abp 6= 0. since 0 6= abp = ab(m + p) ∈ p, then by assumption there exists s ∈ s of p such that sab ∈ (p : m) or sa(m + p) ∈ p or sb(m + p) ∈ p. 8 int. j. anal. appl. (2023), 21:48 hence sab ∈ (p : m) or sam ∈ p or sbm ∈ p , which is not possible by our assumption. hence abp = 0. if a(p : m)m 6= 0, then for some r ∈ (p : m), arm 6= 0. since 0 6= arm = a(r + b)m ∈ p, then there exists s ∈ s of p such that sa(r + b) ∈ (p : m) or sam ∈ p or s(r + b)m ∈ p. that is sab ∈ (p : m) or sam ∈ p or sbm ∈ p, which is not possible by our assumption. thus a(p : m)m = 0. the similar argument prove that b(p : m)m = 0. (2). assume that a(p : m)p 6= 0. then for some r ∈ (p : m), p ∈ p, 0 6= arp ∈ p. as 0 6= arp = a(b + r)(m + p). by (1), we get 0 6= a(b + r)(m + p) ∈ p, then there exists s ∈ s of p such that sa(b + r) ∈ (p : m) or sa(m + p) ∈ p or s(b + r)(m + p) ∈ p. hence sab ∈ (p : m) or sam ∈ p or sbm ∈ p, a contradiction by our assumption. hence a(p : m)p = 0. now, if (p : m)2m 6= 0, then for some r1, r2 ∈ (p : m), 0 6= r1r2m ∈ p. since by (1), 0 6= r1r2m = (a+r1)(b+r2)m ∈ p, then there exists s ∈ s of p such that s(a+r1)(b+r2) ∈ (p : m) or s(a + r1)m ∈ p or s(b + r2)m ∈ p and so sab ∈ (p : m) or sam ∈ p or sbm ∈ p, a contradiction by our assumption. hence (p : m)2m = 0. we recall that if n is a submodule of an r-module m and a is an ideal of r, then the residual of n by a is the set (n :m a) = {m ∈ m : am ⊆ n}. it is clear that (n :m a) is a submodule of m containing n. more generally, for any subset b ⊆ r, (n :m b) is a submodule of m containing n. proposition 2.7. let s be a multiplicatively closed subset of r. for a submodule p of an r-module m with (p : m) ∩s = ∅, the following assertions are equivalent. (1) p is a weakly s-2-absorbing submodule of m. (2) for any a, b ∈ r, there exists s ∈ s such that, if sabm * p, then (p : ab) = (0 : ab) or (p : ab) ⊆ (p : sa) or (p : ab) ⊆ (p : sb) (3) for any a, b ∈ r and for any submodule k of m, there exists s ∈ s such that, if 0 6= abk ⊆ p then sab ∈ (p : m) or sak ⊆ p or sbk ⊆ p. proof. (1) =⇒ (2) let a, b ∈ r. let m ∈ (p : ab). if abm = 0, then clearly m ∈ (0 : ab). if abm 6= 0, that is if 0 6= abm ∈ p, then by (1), there exist s ∈ s of p such that sab ∈ (p : m) or sam ∈ p or sbm ∈ p. clearly, if sabm * p, we conclude that either sam ∈ p or sbm ∈ p. as (0 : ab) ⊆ (p : ab), we get (p : ab) = (0 : ab) or (p : ab) ⊆ (p : sa) or (p : ab) ⊆ (p : sb). (2) =⇒ (3) let a, b ∈ r and k be a submodule of m such that 0 6= abk ⊆ p and, for the element s ∈ s of (2), we have to claim that sab ∈ (p : m) or sak ⊆ p or sbk ⊆ p. if sab ∈ (p : m), then there is nothing to prove. suppose sab /∈ (p : m). as abk ⊆ p, we have k ⊆ (p : ab) and by (2), we have k ⊆ (0 : ab) or k ⊆ (p : sa) or k ⊆ (p : sb). if k ⊆ (0 : ab), then abk = 0, a contradiction. if k ⊆ (p : sa), then sak ⊆ p as required. (3) =⇒ (1) let a, b ∈ r and m ∈ m with 0 6= abm ∈ p. clearly ab < m >⊆ p . if int. j. anal. appl. (2023), 21:48 9 ab < m > 6= 0, by (3), sab ∈ (p : m) or sam ∈ sa < m >⊆ p or sbm ∈ sb < m >⊆ p. if ab < m >= 0, then abm ∈ ab < m >= 0, a contradiction. theorem 2.3. let s be a multiplicatively closed subset of r and p be a submodule of an r-module m. if p is a weakly s-2-absorbing submodule of m. then (1) there exists an s ∈ s such that for any a, b ∈ r, if abk ⊆ p and 0 6= 2abk for some submodule k of m, then sab ∈ (p : m) or sak ⊆ p or sbk ⊆ p . (2) there exists an s ∈ s such that for an ideal i of r and a submodule k of m, if aik ⊆ p and 0 6= 4aik, where a ∈ r, then sai ∈ (p : m) or sak ⊆ p or sik ⊆ p . (3) there exists an s ∈ s such that for all ideals i, j of r and submodule k of m, if 0 6= ijk ⊆ p and 0 6= 8(ij + (i + j)(p : m))(k + p ), then sij ⊆ (p : m) or sik ⊆ p or sjk ⊆ p. in particular this holds if the group (m, +) has no elements of order 2. proof. (1) by our assmption, there exists s ∈ s of p such that, whenever x, y ∈ r and m ∈ m, 0 6= xym ∈ p implies sxy ∈ (p : m) or sxm ∈ p or sym ∈ p. let a, b ∈ r such that abk ⊆ p and 0 6= 2abk for some submodule k of m. now, we will show that sab ∈ (p : m) or sak ⊆ p or sbk ⊆ p . suppose sab /∈ (p : m). then proving that sak ⊆ p or sbk ⊆ p is enough. let k be an arbitrary element of k. as abk ∈ abk ⊆ p , if abk 6= 0, then sab ∈ (p : m) or sak ∈ p or sbk ∈ p. thus we have k ∈ (p : sa) or k ∈ (p : sb) since sab /∈ (p : m). hence sak ⊆ p or sbk ⊆ p. if abk = 0. since 0 6= 2abk, for some k1 ∈ k, we get 0 6= 2abk1 and clearly 0 6= abk1 ∈ p. then we get sak1 ∈ p or sbk1 ∈ p since sab /∈ (p : m). put k2 = k + k1 and so 0 6= abk2 ∈ p. then sak2 ∈ p or sbk2 ∈ p since sab /∈ (p : m). this leads to the following cases. case 1: sak1 ∈ p and sbk1 ∈ p since sak2 ∈ p or sbk2 ∈ p, we have sak ∈ p or sbk ∈ p . thus sak ∈ p or sbk ∈ p. case 2: sak1 ∈ p and sbk1 /∈ p suppose sak /∈ p and sbk /∈ p. then sak2 = sak1 + sak /∈ p and so sbk2 ∈ p. hence sa(k2 + k1) /∈ p and similarly sb(k2 + k1) /∈ p. as p is weakly s-2-absorbing and sab /∈ (p : m), hence ab(k2 + k1) = 0. but ab(k2 + k1) = ab(k1 + k + k1) = 2abk1, a contradiction as 2abk1 6= 0. thus sak ∈ p or sbk ∈ p and so sak ⊆ p or sbk ⊆ p. case 3: sak1 /∈ p and sbk1 ∈ p the proof is same as that of case 2. (2) by our assmption, there exists s ∈ s of p such that, whenever x, y ∈ r and m ∈ m, 0 6= xym ∈ p implies sxy ∈ (p : m) or sxm ∈ p or sym ∈ p. let i be an ideal of r and k be a submodule of m such that aik ⊆ p and 0 6= 4aik, where a ∈ r. we have to prove that sai ∈ (p : m) or sak ⊆ p or sik ⊆ p. suppose sai * (p : m), for some i ∈ i we have sai /∈ (p : m). let us first prove that there exists b ∈ i such that 0 6= 4abk and sab /∈ (p : m). since 0 6= 4aik, for some i′ ∈ i, 0 6= 4ai′k. suppose sai′ /∈ (p : m) or 0 6= 4aik, if we put b = i′, we get sab /∈ (p : m) and 0 6= 4abk and if we put b = i, we get 0 6= 4abk and 10 int. j. anal. appl. (2023), 21:48 sab /∈ (p : m). from the above, clearly by putting b = i′ or b = i, we get the result. hence assume that sai′ ∈ (p : m) and 4aik = 0. hence 0 6= 4a(i + i′)k ⊆ p and sa(i + i′) /∈ (p : m). thus we find b ∈ i such that 0 6= 4abk and sab /∈ (p : m). as 0 6= 4abk, we get 0 6= 2abk and by (1), since abk ⊆ aik ⊆ p and sab /∈ (p : m), we get sak ⊆ p or sbk ⊆ p. if sak ⊆ p, there we are done. thus assume that sak * p and so sbk ⊆ p. now to exhibit that sai ∈ (p : m) or sik ⊆ p. let i” ∈ i. if 2ai”k 6= 0, then by (1), sai” ∈ (p : m) or si”k ⊆ p since sak * p. thus we get i” ∈ ((p : m) : sa) or i” ∈ (p : sk). therefore i ⊆ ((p : m) : sa) or i ⊆ (p : sk). then we are done. if 2ai”k = 0, then clearly 0 6= 2a(b + i”)k and a(b + i”)k ⊆ p, by (1) sa(b + i”) ∈ (p : m) or s(b + i”)k ⊆ p since sak * p, (b + i”) ∈ (p : sk) or (b + i”) ∈ ((p : m) : sa). (i): if (b + i”) ∈ (p : sk), then si”k ⊆ p as sbk ⊆ p . hence i” ∈ (p : sk). (ii): now assume (b + i”) ∈ ((p : m) : sa) and (b + i”) /∈ (p : sk). consider 0 6= 4abk = 2a(b+i” +b)k and a(b+i” +b)k ⊆ p . by (1), sa(b+i” +b) ∈ (p : m) or s(b+i” +b)k ⊆ p since sak * p . as sab /∈ (p : m), we have sa(b + i” + b) /∈ (p : m). then we have s(b + i” + b)k ⊆ p. since (b + i”) /∈ (p : sk), we have s(b + i” + b)k * p. therefore (b + i”) ∈ (p : sk). since sbk ⊆ p, we have si”k ⊆ p and so i” ∈ (p : sk). consequently i ⊆ ((p : m) : sa) or i ⊆ (p : sk) and hence as sai * (p : m), we get sik ⊆ p . (3) let i, j be the ideals of r and k be a submodule of m such that 0 6= ijk ⊆ p and 0 6= 8(ij + (i + j)(p : m))(k + p ). since 0 6= 8(ij + (i + j)(p : m))(k + p ) = 8ijk + 8i(p : m)k + 8j(p : m)k + 8ijp + 8i(p : m)p + 8j(p : m)p . as a result, one of the types listed below has been satisfied. type 1: 0 6= 8ijk. then for some j ∈ j, 0 6= 8jik and so 0 6= 4jik. as jik ⊆ p , by (2), there exists s ∈ s such that sji ⊆ (p : m) or sik ⊆ p or sjk ⊆ p. if sik ⊆ p, then we are done and so assume that sik * p that is sji ⊆ (p : m) or sjk ⊆ p. we claim that sij ⊆ (p : m) or sjk ⊆ p. let j′ ∈ j be an arbitrary element. if 0 6= 4j′ik, by (2), sj′i ⊆ (p : m) or sj′k ⊆ p since sik * p .then j′ ∈ ((p : m) : si) or j′ ∈ (p : sk). hence we get the result. now let 4j′ik = 0. as 0 6= 4(j + j′)ik ⊆ p , by (2), s(j + j′)i ⊆ (p : m) or s(j + j′)k ⊆ p since sik * p. hence we get s(j + j′)i ⊆ (p : m) or s(j + j′)k ⊆ p. thereby we get the four cases. case 1: sji ⊆ (p : m) and s(j + j′)i ⊆ (p : m). hence we get sj′i ⊆ (p : m), that is sij ⊆ (p : m) case 2: sjk ⊆ p and s(j + j′)k ⊆ p hence we get sj′k ⊆ p, that is sjk ⊆ p case 3: sji ⊆ (p : m) and sjk * p. s(j + j′)k ⊆ p and s(j + j′)i * (p : m). this can be represented as j ∈ ((p : m) : si) and j /∈ (p : sk), j + j′ ∈ (p : sk) and int. j. anal. appl. (2023), 21:48 11 j + j′ /∈ ((p : m) : si). hence j + j′ + j /∈ ((p : m) : si) and j + j′ + j /∈ (p : sk). now consider 0 6= 8jik = 4(j + j′ + j)ik and by (2), s(j + j′ + j)i ⊆ (p : m) or s(j + j′ + j)k ⊆ p since sik * p. hence we get j + j′ + j ∈ ((p : m) : si) or j + j′ + j ∈ (p : sk) and this is not possible. therefore, since j ∈ ((p : m) : si) or j ∈ (p : sk) and j + j′ ∈ (p : sk) or j + j′ ∈ ((p : m) : si), there must be any one of the following holds. (i) j ∈ (p : sk) and j + j′ ∈ (p : sk) and j + j′ /∈ ((p : m) : si), then j′ ∈ (p : sk). (ii) j ∈ ((p : m) : si) and j /∈ (p : sk) and j + j′ ∈ ((p : m) : si), then j′ ∈ ((p : m) : si). case 4: s(j + j′)i ⊆ (p : m) and s(j + j′)k * p sjk ⊆ p and sji * (p : m). similar to the above case, we have j′ ∈ ((p : m) : si) or j′ ∈ (p : sk). thus sij ⊆ (p : m) or sjk ⊆ p. type 2: if 0 6= 8ijp and 8ijk = 0, then 0 6= 8ij(k + p ) ⊆ p and by type 1, sij ⊆ (p : m) or sj(k + p ) ⊆ p or si(k + p ) ⊆ p and so sij ⊆ (p : m) or sjk ⊆ p or sik ⊆ p . type 3: if 0 6= 8j(p : m)k and 8ijk = 0, then 0 6= 8j(p : m)k = 8j(i + (p : m))k and so by type 1, sj(i + (p : m)) ⊆ (p : m) or sjk ⊆ p or s(i + (p : m))k ⊆ p . hence sij ⊆ (p : m) or sjk ⊆ p or sik ⊆ p. likewise if 0 6= 8i(p : m)k, we get the result. type 4: if 0 6= 8j(p : m)p and 8ijk = 8ijp = 8j(p : m)k = 8i(p : m)k = 0. then 0 6= 8j(p : m)p = 8j(i + (p : m))(k + p ) and by type 1, sj(i + (p : m)) ⊆ (p : m) or sj(k + p ) ⊆ p or s(i + (p : m))(k + p ) ⊆ p. hence sij ⊆ (p : m) or sjk ⊆ p or sik ⊆ p. likewise if 0 6= 8i(p : m)p, we have the result. to prove the particular case, let (m, +) be a group having no subgroups of order 2. we have to show that 0 6= 8ijk. if this happens, we get the result by type 1. suppose 8ijk = 0. let 0 6= a ∈ ijk. as 8a = 0, so the group (m, +) has a subgroup of order 2, 4 or 8, which is a contradiction. corollary 2.2. let s be a multiplicatively closed subset of r and i be a weakly s-2-absorbing ideal of r. (1) there exists s ∈ s such that for any a, b ∈ r and for any ideal a of r, if aba ⊆ i and 0 6= 2aba, then sab ∈ i or saa ⊆ i or sba ⊆ i. (2) there exists s ∈ s such that for any a ∈ r, ideals a, b of r, if aab ⊆ i and 0 6= 4aab, then saa ⊆ i or sab ⊆ i or sab ⊆ i. (3) there exists s ∈ s such that for any ideals a, b, c of r, if 0 6= abc ⊆ i and 0 6= 8(ab(c + i) + ac(b + i) + bc(a + i) + ai(b + c) + bi(a + c) + ci(a + b) + i2(a + b + c)), then sab ⊆ i or sbc ⊆ i or sac ⊆ i. in particular, this happens when the group (r, +) has no elements of order 2. proposition 2.8. let φ: m1 → m2 be a module homomorphism where m1 and m2 are r-modules and s be a multiplicatively closed subset of r. then the following holds. 1. if φ is a monomorphism and k is a weakly s-2-absorbing submodule of m2 with (φ−1(k) : 12 int. j. anal. appl. (2023), 21:48 m1) ∩s = ∅, then φ−1(k) is a weakly s-2-absorbing submodule of m1. 2. if φ is an epimorphism and p is a weakly s-2-absorbing submodule of m1 containing kerφ, then φ(p ) is a weakly s-2-absorbing submodule of m2. proof. 1. let a, b ∈ r and m1 ∈ m1 be such that 0 6= abm1 ∈ φ−1(k). then 0 6= φ(abm1) = abφ(m1) ∈ k as φ is a monomorphism. since k is weakly s-2-absorbing, there exists s ∈ s such that sab ∈ (k : m2) or saφ(m1) ∈ k or sbφ(m1) ∈ k. if sab ∈ (k : m2), then sab ∈ (φ−1(k) : m1) since (k : m2) ⊆ (φ−1(k) : m1) and if saφ(m1) ∈ k or sbφ(m1) ∈ k, we have φ(sam1) ∈ k implies sam1 ∈ φ−1(k) or φ(sbm1) ∈ k implies sbm1 ∈ φ−1(k). hence φ−1(k) is a weakly s-2-absorbing submodule of m1. 2. first observe that (φ(p ) : m2) ∩ s = ∅. indeed, assume that s′ ∈ (φ(p ) : m2) ∩ s. then φ(s′m1) = s′φ(m1) = s′m2 ⊆ φ(p ) and so s′m1 ⊆ p as kerφ ⊆ p. this shows that s′ ∈ (p : m1) and so (p : m1) ∩ s 6= ∅, a contradiction occurs since p is a weakly s-2-absorbing submodule of m1. now, let a, b ∈ r and m2 ∈ m2 be such that 0 6= abm2 ∈ φ(p ). as we can write m2 = φ(m1) for some m1 ∈ m1 and so 0 6= abm2 = ab(φ(m1)) = φ(abm1) ∈ φ(p ). since kerφ ⊆ p, we have 0 6= abm1 ∈ p. then there exists s ∈ s such that sab ∈ (p : m1) or sam1 ∈ p or sbm1 ∈ p . consequently we get sab ∈ (φ(p ) : m2) or φ(sam1) = saφ(m1) = sam2 ∈ φ(p ) or φ(sbm1) = sbφ(m1) = sbm2 ∈ φ(p ). hence φ(p ) is weakly s-2-absorbing submodule of m2. corollary 2.3. let s be a multiplicatively closed subset of r. p1 and p2 are two submodules of m with p2 ⊆ p1. 1. if k is a weakly s-2-absorbing submodule of m with (k : p1) ∩s = ∅, then k ∩p1 is a weakly s-2-absorbing submodule of p1. 2. if p1 is a weakly s-2-absorbing submodule of m, then p1/p2 is a weakly s-2-absorbing submodule of m/p2. 3. if p1/p2 is a weakly s-2-absorbing submodule of m/p2 and p2 is a weakly s-2-absorbing submodule of m, then p1 is a weakly s-2-absorbing submodule of m. proof. 1. consider the injection i : p1 → m defined by i(p1) = p1 for all p1 ∈ p1. we have to show that (i−1(k) : p1) ∩ s = ∅. indeed, if s ∈ (i−1(k) : p1) ∩ s, then sp1 ⊆ i−1(k). as i−1(k) = k∩p1, we have sp1 ⊆ k∩p1 ⊆ k and so s ∈ (k : p1)∩s, a contradiction as k is weakly s-2-absorbing. thus by proposition 2.8(1), we conclude the result. 2. consider the canonical epimorphism π : m → m/p2 defined by π(m) = m + p2. then π(p1) = p1/p2 is a weakly s-2-absorbing submodule of m/p2 by proposition 2.8(2). 3. let a, b ∈ r and m ∈ m be such that 0 6= abm ∈ p1. then ab(m + p2) = abm + p2 ∈ p1/p2. if ab(m + p2) 6= p2, then there exists s1 ∈ s of p1/p2 implies s1ab ∈ (p1/p2 : m/p2) or s1a(m + p2) ∈ p1/p2 or s1b(m + p2) ∈ p1/p2. hence s1ab ∈ (p1 : m) or s1am ∈ p1 or s1bm ∈ p1. if abm ∈ p2, then by assumption, there exists s2 ∈ s of p2 such that s2ab ∈ (p2 : m) ⊆ (p1 : m) or int. j. anal. appl. (2023), 21:48 13 s2am ∈ p2 ⊆ p1 or s2bm ∈ p2 ⊆ p1. it follows that p1 is a weakly s-2-absorbing submodule of m associated with s = s1s2 ∈ s. we need to recall the following lemma for the next result. lemma 2.2. [2] for an ideal iof a ring r and a submodule n of a finitely generated faithful multiplication r-module m, the following hold. 1. (in :r m) = i(n :r m). 2. if i is finitely generated faithful multiplication, then (a) (in :m i) = n. (b) whenever n ⊆ im, then (jn :m i) = j(n :m i) for any ideal j of r. proposition 2.9. let i be a finitely generated faithful multiplication ideal of a ring r, s be a multiplicatively closed subset of r and p be a submodule of a finitely generated faithful multiplication cyclic r-module m. 1. if ip is a weakly s-2-absorbing submodule of m and (p : m) ∩s = ∅, then either i is a weakly s-2-absorbing ideal of r or p is a weakly s-2-absorbing submodule of m. 2. p is a weakly s-2-absorbing submodule of im if and only if (p :m i) is a weakly s-2-absorbing submodule of m. proof. (1) suppose p = m, we get i = i(p :r m) = (ip :r m) by lemma 2.2. since ip is a weakly s-2-absorbing submodule of m, by proposition 2.5, i is a weakly s-2-absorbing ideal of r. now, suppose p is a proper submodule of m. by lemma 2.2, (ip :m i) = p and so (p : m) = ((ip :m i) :r m) = (i(p :r m) :m i). let a, b ∈ r and m ∈ m be such that 0 6= abm ∈ p. since i is faithful, then (0 :m i) = annr(i)m = 0 [2], and so 0 6= abim ⊆ ip . by proposition 2.7, there exists s ∈ s of ip such that sab ∈ (ip : m) or saim ⊆ ip or sbim ⊆ ip . if sab ∈ (ip : m), then sab ∈ (p : m). if saim ⊆ ip, then sam ∈ (ip : i) = p. likewise if sbim ⊆ ip, then sbm ∈ p. hence p is a weakly s-2-absorbing submodule of m. (2) suppose p is a weakly s-2-absorbing submodule of im. then (p :r im) ∩ s = ((p :m i) :r m) ∩ s = ∅. let a, b ∈ r and m ∈ m be such that 0 6= abm ∈ (p :m i). if abim = 0, then abm ∈ (0 :m i) = annr(i)m = 0, a contradiction. hence 0 6= abim ⊆ p. by proposition 2.7, there exists s ∈ s of p such that sab ∈ (p : im) or saim ⊆ p or sbim ⊆ p. if sab ∈ (p :r im), then sab ∈ ((p :m i) :r m). if saim ⊆ p, then sam ∈ (p : i) and similarly if sbim ⊆ p , we get sbm ∈ (p : i) as required. conversely, suppose (p :m i) is a weakly s-2-absorbing submodule of m. then clearly ((p :m i) :r m) ∩s = (p :r im) ∩s = ∅. let a, b ∈ r and x ∈ im be such that 0 6= abx ∈ p. clearly ab < x >⊆ p. since x ∈ im, by lemma 2.2, ab(< x >:m i) = (ab < x >:m i) ⊆ (p :m i). if ab(< x >:m i) = 0, then since abx ∈ (abix :m i) and ix ⊆ im, by lemma 2.2, we have abx ∈ ab(ix :m i) ⊆ ab(< x >:m i) = 0, a contradiction. so we have 0 6= ab(< x >:m i) ⊆ 14 int. j. anal. appl. (2023), 21:48 (p :m i). by proposition 2.7, there exists s′ ∈ s of (p :m i) such that s′ab ∈ ((p :m i) :r m) or s′a(< x >:m i) ⊆ (p :m i) or s′b(< x >:m i) ⊆ (p :m i). if s′ab ∈ ((p :m i) :r m), then s′ab ∈ (p :r im). if s′a(< x >:m i) ⊆ (p :m i), then is′a(< x >:m i) ⊆ p. since s′ax ∈ s′a < x >= s′a(i < x >:m i) = s′ai(< x >:m i) ⊆ p by lemma 2.2. likewise if s′b(< x >:m i) ⊆ (p :m i), then s′bx ∈ p . hence p is a weakly s-2-absorbing submodule of im. acknowledgment: the author thanks prof. dr. p. dheena, professor, department of mathematics, annamalai university, for suggesting the problem and going through the proof. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.m. ali, idempotent and nilpotent submodules of multiplication modules, commun. algebra. 36 (2008), 4620–4642. https://doi.org/10.1080/00927870802186805. [2] m.m. ali, residual submodules of multiplication modules, beitr. algebra geom. 46 (2005), 405-422. [3] f.a.a. almahdi, e.m. bouba, m. tamekkante, on weakly s-prime ideals of commutative rings, an. univ. ovidius const. ser. mat. 29 (2021), 173–186. https://doi.org/10.2478/auom-2021-0024. [4] r. ameri, on the prime submodules of multiplication modules, int. j. math. math. sci. 2003 (2003), 1715–1724. https://doi.org/10.1155/s0161171203202180. [5] s.e. atani, f. farzalipour, on weakly prime submodules, tamkang j. math. 38 (2007), 247–252. https://doi. org/10.5556/j.tkjm.38.2007.77. [6] a.y. darani, f. soheilnia, 2-absorbing and weakly 2-absorbing submodules, thai j. math. 9 (2011), 577-584. [7] a.y. darani, f. soheilnia, u. tekir, g. ulucak, on weakly 2-absorbing primary submodules of modules over commutative rings, j. korean math. soc. 54 (2017), 1505–1519. https://doi.org/10.4134/jkms.j160544. [8] z.a. el-bast, p.p. smith, multiplication modules, commun. algebra. 16 (1988), 755-779. https://doi.org/10. 1080/00927878808823601. [9] n.j. groenewald, on weakly prime and weakly 2-absorbing modules over noncommutative rings, kyungpook math. j. 61 (2021), 33–48. https://doi.org/10.5666/kmj.2021.61.1.33. [10] a. hamed, a. malek, s-prime ideals of a commutative ring, beitr. algebra geom. 61 (2019), 533–542. https: //doi.org/10.1007/s13366-019-00476-5. [11] h.a. khashan, e.y. celikel, on weakly s-prime submodules, (2021). https://doi.org/10.48550/arxiv.2110. 14639. [12] r.l. mccasland, m.e. moore, on radicals of submodules of finitely generated modules, can. math. bull. 29 (1986), 37–39. https://doi.org/10.4153/cmb-1986-006-7. [13] s. moradi, a. azizi, weakly 2-absorbing submodules of modules, turk. j. math. 40 (2016), 350-364. [14] e. sengelen sevim, t. arabaci, u. tekir, on s-prime submodles, turkish journal of mathematics 43.2(2019),10361046. [15] p.f. smith, some remarks on multiplication modules, arch. math. 50 (1988), 223–235. https://doi.org/10. 1007/bf01187738. [16] g. ulucak, ü. tekir, s. koç, on s-2-absorbing submodules and vn-regular modules, an. univ. ovidius const. ser. mat. 28 (2020), 239–257. https://doi.org/10.2478/auom-2020-0030. https://doi.org/10.1080/00927870802186805 https://doi.org/10.2478/auom-2021-0024 https://doi.org/10.1155/s0161171203202180 https://doi.org/10.5556/j.tkjm.38.2007.77 https://doi.org/10.5556/j.tkjm.38.2007.77 https://doi.org/10.4134/jkms.j160544 https://doi.org/10.1080/00927878808823601 https://doi.org/10.1080/00927878808823601 https://doi.org/10.5666/kmj.2021.61.1.33 https://doi.org/10.1007/s13366-019-00476-5 https://doi.org/10.1007/s13366-019-00476-5 https://doi.org/10.48550/arxiv.2110.14639 https://doi.org/10.48550/arxiv.2110.14639 https://doi.org/10.4153/cmb-1986-006-7 https://doi.org/10.1007/bf01187738 https://doi.org/10.1007/bf01187738 https://doi.org/10.2478/auom-2020-0030 1. introduction 2. characterizations of weakly s-2-absorbing submodules references international journal of analysis and applications volume 18, number 2 (2020), 254-261 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-254 certain subfamily of harmonic functions related to sălăgean q-differential operator sh. najafzadeh1, deborah olufunmilayo makinde2,∗ 1department of mathematics, payame noor university, p. o. box: 19395–3697, tehran, iran 2department of mathematics, obafemi awolowo university, 220005, ile-ife, osun state, nigeria ∗corresponding author: funmideb@yahoo.com abstract. the theory of q–calculus operators are applied in many areas of sciences such as complex analysis. in this paper we apply sălăgean q–differential operator to harmonic functions and introduce sharp coefficient bounds, extreme points, distortion inequalities and convexity results. 1. introduction we state some notations regarding to q–calculus used in this article, see [1, 4] and [6]. for 0 < q < 1 and positive integer n, the q–integer number is denoted by [n]q and introduced by: [n]q = 1 −qn 1 −q = 1 + q + q2 + . . . + qn−1. (1.1) we can easily conclude that: lim q→1− [n]q = n. if f(z) be analytic in this open unit disk u = {z ∈ c : |z| < 1} and normalized by f(0) = f′(0) − 1 = 0, then the q–difference operator of q–calculus operated on f given by: ∂qf(z) = f(z) −f(qz) z(1 −q) , (1.2) where lim q→1− ∂qf(z) = f ′(z), for example see [2, 5] and [8]. received 2019-04-24; accepted 2019-05-23; published 2020-03-02. 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. q–calculus; harmonic; univalent; sălăgean operator; convex set. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 254 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-254 int. j. anal. appl. 18 (2) (2020) 255 for f(z) = z + ∑∞ k=2akz k, the sălăgean q–differential operator is defined by: s0qf(z) = f(z) s1qf(z) = z∂qf(z) = f(z) −f(qz) (1 −q) ... smq f(z) = z∂q ( sm−1q f(z) ) = f(z) ∗ ( z + ∑∞ k=2[k] m q z k ) = z + ∞∑ k=2 [k]mq akz k, (1.3) where m is a positive integer and “∗” is the familiar hadamard product or convolution of two analytic functions. since lim q→1− smq (z) = z + ∞∑ k=2 kmakz k, is the famous sălăgean operator [9], so the operator smq is called sălăgean q–differential operator. let sh denote the class of functions: f = h + g (1.4) which are harmonic, univalent and sense-preserving in u and normalized by f(0) = f′(0) − 1 = 0, where h and g are analytic in u take the form: h(z) = z + ∞∑ k=2 akz k and g(z) = ∞∑ k=1 bkz k, (0 6 b1 < 1). (1.5) also, we call h and g analytic part and co-analytic part of f respectively, see [3]. hence f ∈sh is of the type: f(z) = z + ∞∑ k=2 akz k + ∞∑ k=1 bkzk (1.6) now, we consider the sălăgean q–differential operator of harmonic functions f = h + g, by: smq f(z) = s m q h(z) + (−1) msmq g(z), (1.7) where smq is defined by (1.3) and h and g are of the type (1.5). for more details see [7]. we denote by s∗h the family of functions of the type (1.4) where: h(z) = z − ∞∑ k=2 |ak|zk , g(z) = ∞∑ k=1 |bk|zk, |b1| < 1. (1.8) for a > 0, 0 6 b,c 6 1, 0 6 d < 1 and γ ∈ r let s∗ h(γ) (a,b,c,d) denote the class of functions in s∗h of the type (1.5) such that: re { (1 −a)(1 −b) s0qf(z) z + (a + b) ( smq f(z) )′ z′ −ceiγ ( smq f(z) )′′ z′′ + ( ceiγ −ab )} > d, (1.9) int. j. anal. appl. 18 (2) (2020) 256 where z′ = ∂ ∂θ (z) = iz, ( smq f(z) )′ = ∂ ∂θ ( smq f(re iθ) ) = iz ( smq h )′ − iz(smq g)′ (1.10) z′′ = ∂2 ∂θ2 (z) = −z, ( smq f(z) )′′ = ∂2 ∂θ2 ( smq f(re iθ) ) = −z ( smq h )′ −z2(smq h)′′ −z(smq g)′ −z2(smq g)′′. (1.11) we further denote by s∗ h(γ) (a,b,c,d) the subclass of s h(γ) (a,b,c,d) consisting of harmonic functions f = h + g so that h and g are of the form (1.8) and satisfying (1.9). 2. main results in our first theorem, we introduce a sufficient coefficient condition for functions in s h(γ) (a,b,c,d) and then we show that this condition is also necessary for f(z) ∈s∗ h(γ) (a,b,c,d). theorem 2.1. suppose f = h + g, where h and g be given by (1.5) and: ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak|+ ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk|6 1 −d. (2.1) then f(z) ∈s h(γ) (a,b,c,d). proof. in view of the fact that: “ re{w}> 0 ⇐⇒|w + 1 −d|> |w − 1 −d|, ” and letting: w = (1 −a)(1 −b) smq f(z) z + (a + b) ( smq f(z) )′ z′ −ceiγ ( smq f(z) )′′ z′′ + ( ceiθ −ab ) , it is enough to show that: |w + 1 −d|− |w − 1 −d|> 0. but by using (1.10) and (1.11) we have: |w + 1 −d| = ∣∣∣∣∣(1 −a)(1 −b) ( 1 + ∞∑ k=2 ak[k] m q z k−1 + ∞∑ k=1 bk[k] m q (z) k−1 ) + (a + b) ( 1 + ∞∑ k=2 kak[k] m q z k−1 − ∞∑ k=1 kbk[k] m q (z) k−1 ) −ceiγ ( 1 + ∞∑ k=2 kak[k] m q z k−1 + ∞∑ k=2 k(k − 1)ak[k]mq z k−1 + ∞∑ k=1 kbk[k] m q (z) k−1 + ∞∑ k=1 k(k − 1)bk[k]mq (z) k−1 ) int. j. anal. appl. 18 (2) (2020) 257 + ceiγ −ab + 1 −d ∣∣∣∣∣ 6 2 −d − ∞∑ k=1 ∣∣1 + (a + b)(k − 1) + ab −ck2[k]mq ∣∣ |ak| ∣∣∣∣zkz ∣∣∣∣ − ∞∑ k=1 ∣∣1 − (a + b)(k − 1) + ab −ck2∣∣ [k]mq |bk| ∣∣∣∣zkz ∣∣∣∣ , and |w − 1 −d|6 d + ∞∑ k=2 ∣∣(a + b)(k − 1) + ab −ck2∣∣ [k]mq |ak| ∣∣∣∣zkz ∣∣∣∣ + ∞∑ k=1 ∣∣1 − (a + b)(k − 1) + ab −ck2∣∣ [k]mq |bk| ∣∣∣∣zkz ∣∣∣∣ . so by using (2.1), we get: |w + 1 −d|− |w − 1 −d|> 2 [ 1 −d − ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak|− ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk| ] > 0. � remark 2.1. the coefficient bound (2.1) is sharpt for the function: f(z) = z + ∞∑ k=2 xk |(a + b)k − (1 −a−b + ab) −ck2| [k]mq zk + ∞∑ k=1 yk |(a + b)k − (1 −a−b + ab) −ck2| [k]mq (z)k, where 1 1 −d ( ∞∑ k=2 |xk| + ∞∑ k=1 |yk| ) = 1. theorem 2.2. let f = h + g ∈s∗h. then f(z) ∈s ∗ h(γ) (a,b,c,d) if and only if: ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak| + ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk|6 1 −d. (2.2) proof. from theorem 2.1, and since s∗ h(γ) (a,b,c,d) ⊂ s h(γ) (a,b,c,d), we conclude the “if” part. for the “only if” part, suppose that f(z) ∈s∗ h(γ) (a,b,c,d). thus for z = reiθ ∈ u, we have: re { (1 −a)(1 −b) smq f(z) z + (a + b) ( smq f(z) )′ z′ −ceiγ ( smq f(z) )′′ z′′ + ceiγ + ceiγ −ab } = re { (1 −a)(1 −b) ( 1 + ∞∑ k=2 ak[k] m q z k−1 + ∞∑ k=1 bk[k] m q (z) k−1 ) int. j. anal. appl. 18 (2) (2020) 258 + (a + b) ( 1 + ∞∑ k=2 kak[k] m q z k−1 − ∞∑ k=1 kbk[k] m q (z) k−1 ) −ceiγ ( 1 + ∞∑ k=2 kak[k] m q z k−1 + ∞∑ k=2 k(k − 1)ak[k]mq z k−1 + ∞∑ k=1 kbk[k] m q (z) k−1 + ∞∑ k=1 k(k − 1)bk[k]mq (z) k−1 ) + ceiγ −ab } > 1 − ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak|rk−1 − ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk|rk−1 > d. the above inequality holds for all z = reiθ ∈ u. so if z = r → 1, we obtain the required result (2.2). now the proof is complete. � 3. geometric properties of s∗ h(γ) (a,b,c,d) in this section, we first introduce extreme points of s∗ h(γ) (a,b,c,d) and then we obtain the distortion bounds for f ∈s∗ h(γ) (a,b,c,d). finally we show that the class s∗ h(γ) (a,b,c,d) is a convex set. theorem 3.1. f = h + g ∈s∗ h(γ) (a,b,c,d) if and only if it can be expressed: f(z) = x1z + ∞∑ k=2 xkhk(z) + ∞∑ k=1 ykgk(z), (z ∈ u), (3.1) where hk(z) = z − 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq zk, (k = 2, 3, . . .), (3.2) gk(z) = 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq (z)k, (k = 1, 2, . . .), (3.3) x1 > 0, y1 > 0, x1 + ∑∞ k=2xk + ∑∞ k=1yk = 1, xk > 0 and yk > 0 for k = 2, 3, . . .. proof. if f is given by (3.1), then: f(z) = z − ∞∑ k=2 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq xkz k + ∞∑ k=1 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq yk(z) k. since by (2.2), we have: ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq × ( 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq ) |xk| + ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq × × ( 1 −d |(a + b)k − (1 −a−b + ab) −ck2| [k]mq ) |yk| int. j. anal. appl. 18 (2) (2020) 259 = (1 −d) ( ∞∑ k=2 |xk| + ∞∑ k=1 |yk| ) = (1 −d)(1 −x1) 6 1 −d. so f(z) ∈s∗ h(γ) (a,b,c,d). conversely, suppose f(z) ∈s∗ h(γ) (a,b,c,d). by putting: x1 = 1 − ( ∞∑ k=2 xk + ∞∑ k=1 yk ) , where xk = ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq 1 −d |ak|, yk = ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq 1 −d |bk|, we conclude the required representation (3.1), so the proof is complete. � theorem 3.2. if f(z) ∈s∗ h(γ) (a,b,c,d), |z| = r < 1, then: |f(z)|> (1 −|b1|)r − 1 [2]mq ( 1 −d (a + b) + (1 + ab) − 4c − 2(a + b) − (1 + ab) −c (a + b) + (1 + ab) − 4c |b1| ) r2, (3.4) and |f(z)|6 (1 −|b1|)r + 1 [2]mq ( 1 −d (a + b) + (1 + ab) − 4c − 2(a + b) − (1 + ab) −c (a + b) + (1 + ab) − 4c |b1| ) r2. (3.5) proof. suppose f(z) ∈s∗ h(γ) (a,b,c,d), then by (2.2), we have: |f(z)| = ∣∣∣∣∣z − ∞∑ k=2 |ak|zk + ∞∑ k=1 |bk|(z)k ∣∣∣∣∣ = ∣∣∣∣∣z + |b1|(z) − ∞∑ k=2 ( |ak|zk −|bk|(z)k )∣∣∣∣∣ > r −|b1|r − 1 −d (a + b) + (1 + ab) − 4c [ ∞∑ k=2 ((a + b) + (1 + ab) − 4c 1 −d |ak| + (a + b) + (1 + ab) − 4c 1 −d |bk| ) rk ] > (1 −|b1|)r − 1 −d (a + b) + (1 + ab) − 4c [ ∞∑ k=2 ((a + b)(k − 1) + (1 + ab) − 4c 1 −d |ak| + (a + b)(k − 1) − (1 + ab) − 4c 1 −d |bk| ) rk ] > (1 −|b1|)r − 1 −d (a + b) + (1 + ab) − 4c ( 1 − 2(a + b) − (1 + ab) −c 1 −d |b1| ) r2 = (1 −|b1|)r − 1 [2]mq ( 1 −d (a + b) + (1 + ab) − 4c − 2(a + b) − (1 + ab) −c (a + b) + (1 + ab) − 4c |b1| ) r2. relation (3.5) can be proved by using the similar statements. so the proof is complete. � int. j. anal. appl. 18 (2) (2020) 260 theorem 3.3. if fj(z), j = 1, 2, . . ., belongs to s∗h(γ)(a,b,c,d), then the function f(z) = ∑∞ j=1λjfj(z) is also in s∗ h(γ) (a,b,c,d), where fj(z) defined by: fj(z) = z − ∞∑ k=2 ak,jz k + ∞∑ k=1 bk,j(z) k, ( j = 1, 2, . . . , ∑∞ j=1λj = 1 ) . in the other worlds, s∗ h(γ) (a,b,c,d), is a convex set. proof. since fj(z) ∈s∗h(γ)(a,b,c,d), so by (2.2), we get: ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak| ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk,j|6 1 −d, (j = 1, 2, . . .). also f(z) = ∞∑ j=1 λjfj(z) = z − ∞∑ k=k ( ∞∑ j=1 λjak,j ) zk + ∞∑ k=1 ( ∞∑ j=1 λjbk,j ) (z)k. now, according to theorem 2.2, we have: ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq ∣∣∣∣∣∣ ∞∑ j=1 λjak,j ∣∣∣∣∣∣ + ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq ∣∣∣∣∣∣ ∞∑ j=1 λjbk,j ∣∣∣∣∣∣ = ∞∑ j=1 ( ∞∑ k=2 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |ak,j| + ∞∑ k=1 ∣∣(a + b)k − (1 −a−b + ab) −ck2∣∣ [k]mq |bk,j| ) λj 6 (1 −d) ∞∑ j=1 λj = 1 −d. thus, f(z) ∈s∗ h(γ) (a,b,c,d). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. agrawal. coefficient estimates for some classes of functions associated with q–function theory. bull. australian math. soc. 95(3)(2017), 446–456. [2] g. e. andrews, r. askey, and r. roy. encyclopedia of mathematics and its applications. special functions, 71, 1999. [3] j. clunie and t. sheil-small. harmonic univalent functions. ann. acad. sci. fenn. ser. a. i. math. vol. 9, 1984, 3-25. [4] g. gasper and m. rahman. basic hypergeometric series, volume 35. cambridge university press cambridge, uk, 1990. vol. 35 encycl. math. appl. [5] m. govindaraj and s. sivasubramanian. on a class of analytic functions related to conic domains involving q–calculus. anal. math. 43(3)(2017), 475–487. [6] f. jackson. q–difference equations. amer. j. math. 32(4)(1910), 305–314. int. j. anal. appl. 18 (2) (2020) 261 [7] j. m. jahangiri. harmonic univalent functions defined by q–calculus operators. arxiv preprint arxiv:1806.08407, 2018. [8] s. d. purohit and r. k. raina. certain subclasses of analytic functions associated with fractional q–calculus operators. math. scand. 109(2011), 5570. [9] g. s. sălăgean. subclasses of univalent functions. in complex analysisfifth romanian-finnish seminar, springer, 1983, 362–372. 1. introduction 2. main results 3. geometric properties of references international journal of analysis and applications volume 17, number 1 (2019), 122-131 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-122 generalized convex function and associated petrović’s inequality a. ur. rehman1, g. farid1 and vishnu narayan mishra2,3,∗ 1comsats university islamabad, attock campus, kamra road, attock 43600, pakistan 2department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur, madhya pradesh 484 887, india 3l. 1627 awadh puri colony beniganj, phase iii, opposite industrial training institute (i.t.i.), ayodhya 224 001, uttar pradesh, india ∗corresponding author: vishnunarayanmishra@gmail.com abstract. in this paper, petrović’s inequality is generalized for h−convex functions, when h is supermultiplicative function. it is noted that the case for h−convex functions does not lead the particular cases for p−function, godunova-levin functions, s−godunova-levin functions and s−convex functions due to the conditions imposed on h. to cover the case, when h is submultiplicative, petrović’s inequality is generalized for h−concave functions. 1. introduction let [c,d] be an interval containing (0, 1) and h : [c,d] → r be a non-negative function. a function f : [a,b] → r is said to be an h−convex, if f is non-negative and for all x,y ∈ [a,b], α ∈ (0, 1), one has f(αx + (1 −α)y) ≤ h(α)f(x) + h(1 −α)f(y). (1.1) received 2018-09-12; accepted 2018-11-09; published 2019-01-04. 2010 mathematics subject classification. 52a41. key words and phrases. petrović’s inequality; convex functions; h−convex functions; concave functions; h−concave functions. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 122 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-122 int. j. anal. appl. 17 (1) (2019) 123 if above inequality is reversed, then f is said to be h-concave. the h−convex function was introduced by s. varošanec in [1]. the important thing about these function is that it generalized many other generalization of convex function like s−convex functions, godunova-levin functions, s−godunova-levin functions and p−functions given in [1–3]. remark 1.1. particular value of h in inequality (1.1) gives us the following results: i. h(α) = α gives the convex functions. ii. h(α) = 1 gives the p−functions. iii. h(α) = αs and α ∈ (0, 1) gives the s−convex functions of second sense. iv. h(α) = 1 α and α ∈ (0, 1) gives the godunova-levin functions. v. h(α) = 1 αs and α ∈ (0, 1) gives the s−godunova-levin functions of second sense. in case of h−concavity, following results are valid: vi. h(α) = 1 gives the reverse p−functions. vii. h(α) = αs and α ∈ (0, 1) gives the s−concave functions of second sense. viii. h(α) = 1 α gives the reverse godunova-levin functions. ix. h(α) = 1 αs gives the reverse s−godunova-levin functions of second sense. in [6] (also see [7, p. 154]), m. petrović proved the following result, which is known as petrović’s inequality in the literature. theorem 1.1. suppose that (x1, ...,xn) and (p1, ...,pn) be non-negative n-tuples such that ∑n k=1 pkxk ≥ xi for i = 1, ...,n and ∑n k=1 pkxk ∈ [0,a]. if f is a convex function on [0,a], then the inequality n∑ k=1 pkf(xk) ≤ f ( n∑ k=1 pkxk ) + ( n∑ k=1 pk − 1 ) f(0) (1.2) is valid. in recent years, h−convex functions are considered in literature by many researchers and mathematicians, for example, see [1, 3, 5, 8, 9] and references there in. many authors worked on petrović’s inequality by giving results related to it, for example see [6, 10–12] and it has been generalized for m−convex functions by m. bakula et.al. in [13]. in [14], petrović’s inequality was generalized on coordinates by using the definition of convex functions on coordinates. in this paper, petrović’s inequality is generalized for h−convex functions, in the case, when h is supermultiplicative function. in case, when h is submultiplicative, petrović’s inequality is generalized for h−concave functions. also the results has been generalized on coordinates in the plane. int. j. anal. appl. 17 (1) (2019) 124 2. generalized petrović’s inequality for h-convex function a function h : [c,d] → r is said to be a submultiplicative function if h(xy) ≤ h(x)h(y), (2.1) for all x,y ∈ [c,d]. if the above inequality is reversed, then h is said to be supermultiplicative function. if equality holds in the above inequality, then h is said to be multiplicative function. here we state important lemma, which is very helpful in proving petrović’s inequality for h−convex functions. this lemma is generalization of result given in [7, page 152]. lemma 2.1. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. also let h : [0,a] → r be a positive function and f : [0,a] → r be a function. if f(x) h(x−c) is increasing for x > c on [0,a], then n∑ k=1 pkf(xk) ≤ n∑ k=1 pkh(xk − c) h ( n∑ k=1 pkxk − c )f ( n∑ k=1 pkxk ) . (2.2) proof. since ∑n k=1 pkxk ≥ xj > c for all j = 1, ...,n and f(x) h(x−c) is increasing on [0,a], f ( n∑ k=1 pkxk ) h ( n∑ k=1 pkxk − c ) ≥ f(xj) h(xj − c) , that is, h ( n∑ k=1 pkxk − c ) f(xj) ≤ h(xj − c)f ( n∑ k=1 pkxk ) . multiplying above inequality by pj and taking sum for j = 1, ...,n, one has h ( n∑ k=1 pkxk − c ) n∑ j=1 pjf(xj) ≤ n∑ j=1 pjh(xj − c)f ( n∑ k=1 pkxk ) . this is equivalent to the required result. � the following theorem consists of the result for generalized petrović’s inequality for h−convex functions. theorem 2.1. let (x1, ...,xn) be non-negative n-tuples and (p1, ...,pn) be positive n-tuples such that n∑ k=1 pkxk ∈ [0,a] and n∑ k=1 pkxk ≥ xj ≥ c for j = 1, ...,n and c ∈ [0,a]. (2.3) also let h : [0,a] → r+ be a supermultiplicative function such that h(α) + h(1 −α) ≤ 1, for all α ∈ (0, 1). (2.4) if f : [0,a] → r be an h−convex function on [0,a], then int. j. anal. appl. 17 (1) (2019) 125 n∑ j=1 pjf(xj) ≤ n∑ j=1 pjh(xj − c) h ( n∑ k=1 pkxk − c )f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − n∑ j=1 pjh(xj − c) h ( n∑ k=1 pkxk − c )  f(c). (2.5) proof. suppose f is h−convex and ph(x) = f(x) −f(c) h(x− c) . we take y > x > c and x = αy + (1 −α)c, then ph(x) = f(αy + (1 −α)c) −f(c) h(αy + (1 −α)c− c) ≤ h(α)f(y) + [h(1 −α) − 1] f(c) h(α(y − c)) . using the fact that h is supermultiplicative, one has ph(x) ≤ h(α)f(y) + [h(1 −α) − 1] f(c) h(α)h(y − c)) since h(1 −α) − 1 ≤−h(α), this implies ph(x) ≤ f(y) h(y − c) − f(c) h(y − c) = ph(y). as we have proved if f is h−convex, then f(x)−f(c) h(x−c) is increasing for x > c so substituting f(x) by f(x)−f(c) in lemma 2.1, one has n∑ j=1 pj (f(xj) −f(c)) ≤ n∑ j=1 pjh(xj − c) h ( n∑ k=1 pkxk − c ) [f ( n∑ k=1 pkxk ) −f(c) ] . the above inequality leads to the required result. � the following theorem is a simple consequence of the above theorem just by taking c = 0. it can be considered as petrović’s inequality for h−convex functions. theorem 2.2. let the conditions given in theorem 2.1 are valid. if f : [0,a] → r be an h−convex function on [0,a], then n∑ j=1 pjf(xj) ≤ n∑ j=1 pjh(xj) h ( n∑ k=1 pkxk )f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − n∑ j=1 pjh(xj) h ( n∑ k=1 pkxk )  f(0). (2.6) from theorem 2.1, one can get a generalization of petrović’s inequality. int. j. anal. appl. 17 (1) (2019) 126 theorem 2.3. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be a convex function on [0,a], then n∑ j=1 pjf(xj) ≤ n∑ j=1 pj(xj − c)( n∑ k=1 pkxk − c )f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − n∑ j=1 pj(xj − c)( n∑ k=1 pkxk − c )  f(c). (2.7) proof. let us consider h(x) = x, then clearly h is supermultiplicative and condition (2.4) is valid. taking this value of h in theorem 2.1 leads us to required result. � remark 2.1. taking h(x) = x in theorem 2.2 or c = 0 in theorem 2.3 leads to theorem 1.1. 3. generalized petrović’s inequality for h-concave function in the previous section, one can see that the condition on function h given in (2.4) restrict us to give petrović’s type inequalities for particular cases of h−convex functions given in remark 1.1. if we consider reverse inequality in (2.4), then it covers those particular cases but instead of h−convex function, we have h−concave function. lemma 3.1. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi for i = 1, ...,n and ∑n k=1 pkxk ∈ [0,a]. also let h : [0,a] → r be a positive function and f : [0,a] → r be a function. if f(x) h(x−c) is decreasing for x > c on [0,a], then reverse of inequality (2.2) is valid. proof. the proof is similar to the lemma 2.1. � in the following theorem, reverse of (2.5) has been concluded. the notable thing is the requirements of submultiplicity and reverse of (2.4) for function h along with h-concavity of the function f. theorem 3.1. let (x1, ...,xn) be non-negative n-tuples and (p1, ...,pn) be positive n-tuples and the conditions given in (2.3) are valid. also let h : [0,a] → r+ be a submultiplicative function such that h(α) + h(1 −α) ≥ 1, for all α ∈ (0, 1). (3.1) if f : [0,a] → r be an h−concave function on [0,a], then reverse of (2.5) is valied, that is, n∑ j=1 pjf(xj) ≥ n∑ j=1 pjh(xj − c) h ( n∑ k=1 pkxk − c )f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − n∑ j=1 pjh(xj − c) h ( n∑ k=1 pkxk − c )  f(c). (3.2) int. j. anal. appl. 17 (1) (2019) 127 proof. first we will prove that f(x)−f(c) h(x−c) is decreasing for x > c when f is h−concave function. for this purpose consider ph(x) = f(x) −f(c) h(x− c) . we take y > x > c and x = αy + (1 −α)c, then ph(x) = f(αy + (1 −α)c) −f(c) h(αy + (1 −α)c− c) ≥ h(α)f(y) + [h(1 −α) − 1] f(c) h(α(y − c)) . using the fact that h is submultiplicative, so we have ph(x) ≥ h(α)f(y) + [h(1 −α) − 1] f(c) h(α)h(y − c)) since h(1 −α) − 1 ≥−h(α), this implies ph(x) ≥ f(y) h(y − c) − f(c) h(y − c) = ph(y). this proves that f(x)−f(c) h(x−c) is decreasing in [0,a] for x > c. now substituting f(x) by f(x) −f(c) in lemma 3.1 give us the required result. � in the following theorem, we give petrović’s inequality for h−concave functions. it is simple consequence of the previous theorem by just taking c = 0. theorem 3.2. let the conditions given in theorem 3.1 are valid. if f : [0,a] → r be an h−concave function on [0,a], then the reverse of (2.6) is valid. in the following theorem, we give the generalized petrović’s inequality for concave functions. theorem 3.3. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be concave function on [0,a], then the reverse of (2.7) is valid. proof. if we take h(x) = x in (3.2), we get the required result. � remark 3.1. by taking h(x) = x in theorem 3.2 or c = 0 in theorem 3.3, one can get the reverse of inequality (1.2) in the case when f is concave function. in the following theorem, we give the petrović’s type inequality for reverse p−functions. theorem 3.4. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi for i = 1, ...,n and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be a reverse p−function on [0,a], then n∑ j=1 pjf(xj) ≥ n∑ j=1 pjf ( n∑ k=1 pkxk ) (3.3) int. j. anal. appl. 17 (1) (2019) 128 proof. if we take h(x) = 1, then it fulfils the condition of theorem 3.1 and follows the required result. � in the following theorem, we give the generalized petrović’s type inequality for reverse godunova-levin functions. theorem 3.5. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be a reverse godunova-levin function on [0,a], then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk − c ) n∑ j=1 pj xj − c f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk − c ) n∑ j=1 pj xj − c  f(c). (3.4) proof. consider h(x) = 1 x , then h(α) + h(1 −α) = 1 α + 1 1 −α > 1 for all α ∈ (0, 1). using above value of h in theorem 3.1 gives the required result. � the following theorem is a simple consequence of the previous theorem. it is worth stating as petrović’s type inequality for reverse godunova-levin functions. theorem 3.6. let the conditions given in theorem 3.1 are valid. if f : [0,a] → r be a reverse godunovalevin function on [0,a], then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk ) n∑ j=1 pj xj f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk ) n∑ j=1 pj xj  f(0). (3.5) proof. putting c = 0 in theorem 3.6 leads to required result. � before giving two important theorems, let us consider h(h) = h(α) + h(1 −α) − 1,α ∈ (0, 1), then for different values of h, that is, for αs and 1 αs , we take g1(α) := h(α s) = αs + (1 −α)s − 1 and g2(α) := h ( 1 αs ) = 1 αs + 1 (1 −α)s − 1, where s ∈ (0, 1). int. j. anal. appl. 17 (1) (2019) 129 figure 1. graph of g1 at different value of s. one can see that g1 is positive for α ∈ (0, 1) and at different value of s. the line at bottom is at s = 1, the next curve is for s = 1 2 and so on. figure 2. graph of g2 at different value of s. one can see that g2 is positive for α ∈ (0, 1) and at different value of s. the curve at top is at s = 1, the below one is for s = 1 2 and so on. from figures 1 and 2, one can see that g1 and g2 are positive, therefore h(α) = α s and h(α) = 1 αs for α,s ∈ (0, 1) satisfied the conditions of theorem 3.1, but these functions does not satisfy the conditions of theorem 2.1. hence the above two particular values of h in theorem 3.1 leads us the following two theorems. theorem 3.7. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be a reverse s-godunova-levin int. j. anal. appl. 17 (1) (2019) 130 function on [0,a]. then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk − c )s n∑ j=1 pj (xj − c)s f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk − c )s n∑ j=1 pj (xj − c)s  f(c). (3.6) theorem 3.8. suppose that (x1, ...,xn) and (p1, ...,pn) be two non-negative n-tuples such that ∑n k=1 pkxk ≥ xi > c for i = 1, ...,n, c ∈ [0,a] and ∑n k=1 pkxk ∈ [0,a]. if f : [0,a] → r be a s-concave function on [0,a], then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk − c )s n∑ j=1 pj (xj − c)s f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk − c )s n∑ j=1 pj (xj − c)s  f(c). (3.7) if we take c = 0, then we have following petrović’s type inequalities. theorem 3.9. let the conditions given in theorem 1.1 are valid. if f : [0,a] → r be a reverse s-godunovalevin function on [0,a], then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk )s n∑ j=1 pj (xj)s f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk )s n∑ j=1 pj (xj)s  f(c). (3.8) theorem 3.10. let the conditions given in theorem 1.1 are valid. if f : [0,a] → r be a s-concave function on [0,a], then n∑ j=1 pjf(xj) ≥ ( n∑ k=1 pkxk )s n∑ j=1 pj (xj)s f ( n∑ k=1 pkxk ) +   n∑ j=1 pj − ( n∑ k=1 pkxk )s n∑ j=1 pj (xj)s  f(0). (3.9) proof. put c = 0 in theorem 3.8, one has the required result. � 4. concluding remarks this paper generalized the petrović’s inequality for h−convex (h−concave) functions. it has been noted that under certain conditions on h, theorem 2.1 provide the generalization of petrović’s inequality for h−convex functions, but this generally does not leads to godunova-levin functions, p−functions, s−godunovalevin functions and s−convex functions. theorem 3.1 give the petrović’s inequality for h−concave under certain condition on h. interesting, it give reverse of those particular cases for which theorem 2.1 fails. int. j. anal. appl. 17 (1) (2019) 131 it is an open problem to find such generalization of petrović’s inequality for h−convex functions with some suitable conditions on h, which lead to all particular cases of h−convex functions specially mentioned in remark 1.1. acknowledgement the authors are very grateful to the editor and reviewers for their careful and meticulous reading of the paper. the research work of the 1st author is supported by higher education commission of pakistan under nrpu 2016, project no. 5421. references [1] s. varošanec, on h−convexity, j. math. anal. appl. 326(1) (2007), 303–311. [2] s. s. dragomir, j. e. pečarić and l. e. persson , some inequalities of hadamard inequality, soochow j. math. 21(3) (1995), 335–341. [3] a. házy, bernstein-doetsch type results for h-convex functions, math. inequal. appl. 14(3) (2011), 499–508. [4] s. s dragomir, on hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, taiwanese j. math. 5(4) (2001), 775–788. [5] m. alomari, m. a. latif, on hadmard-type inequalities for h-convex functions on the co-ordinates, int. j. math. anal. 3(33) (2009), 1645–1656. [6] m. petrović’s, sur une fonctionnelle, publ. math. univ. belgrade 1 (1932), 146–149. [7] j. e. pečarić, f. proschan and y. l. tong, convex functions, partial orderings and statistical applications, academic press, new york, 1991. [8] m. bombardelli, s. varoanec, properties of h-convex functions related to the hermitehadamardfejr inequalities, comput. math. appl. 58(9) (2009), 1869–1877. [9] a. olbryś, on separation by h-convex function, tatra mt. math. publ. 62(1) (2015), 105–111. [10] s. butt, j. pečarić and a. u. rehman, exponential convexity of petrović and related functional, j. inequal. appl. 2011(1) (2011), 16 pp. [11] j. pečarić and v. čuljak, inequality of petrović and giaccardi for convex function of higher order, southeast asian bull. math. 26(1) (2003), 57–61. [12] j. e. pečarić, on the petrović inequality for convex functions, glasnik matematicki 18(38) (1983), 77–85. [13] m. k. bakula, j. pečarić and m. ribičić, companion inequalities to jensen’s inequality for m-convex and (α,m)-convex functions, j. inequal. pure appl. math. 7(5) (2006), 32 pp. [14] a. u. rehman, m. mudessir, h. t. fazal and g. farid, petrović’s inequality on coordinates and related results, cogent math. 3(1) (2016), 11 pp. [15] j. pečarić and j. peric, improvements of the giaccardi and the petrovic inequality and related stolarsky type means, an. univ. craiova, ser. mat. inf. 39(1) (2012), 65–75. 1. introduction 2. generalized petrovic's inequality for h-convex function 3. generalized petrovic's inequality for h-concave function 4. concluding remarks acknowledgement references international journal of analysis and applications volume 17, number 6 (2019), 917-927 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-917 shehu transform and applications to caputo-fractional differential equations rachid belgacem1, dumitru baleanu2,3,∗, ahmed bokhari1 1department of mathematics, faculty of exact sciences and informatics, hassiba benbouali university of chlef, algeria 2department of mathematics, faculty of arts and sciences, cankaya university, tr-06530 ankara, turkey 3institute of space science, r-077125 măgurle-bucharest, romania ∗corresponding author: dumitru@cankaya.edu.tr abstract. in this manuscript we establish the expressions of the shehu transform for fractional riemannliouville and caputo operators. with the help of this new integral transform we solve higher order fractional differential equations in the caputo sense. three illustrative examples are discussed to show our approach. 1. introduction one of the most effective methods to solve differential equations is to use integrals transformation. the main advantage of this method is that it transforms the differential problem to an algebraic problem. we recall that the laplace’s transformation which is widely used to solve differential and integral equations. the sumudu transform was first defined in 1993 by watugala who used it to solve engineering control problems [17]. received 2019-05-24; accepted 2019-07-11; published 2019-11-01. 2010 mathematics subject classification. 26a33, 65r10, 44a20 . key words and phrases. shehu transform; caputo derivative; mittag-leffler function. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 917 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-917 int. j. anal. appl. 17 (6) (2019) 918 the shehu transform was introduced recently by shehu maitama and weidong zhao [16] and it is a generalization of the laplace and the sumudu integral transforms. the authors have used it to solve ordinary and partial differential equations [16]. the shehu transform is obtained over the set a by [16] : a = { f (t) : ∃n,η1,η2 > 0, |f (t)| < n exp ( |t| ηi ) , if t ∈ (−1)i × [0,∞) } . (1.1) by h [f (t)] = v (s,u) = ∫ ∞ 0 exp ( − st u ) f (t) dt. (1.2) obviously, the shehu transform is linear as the laplace and sumudu transformations. theorem 1.1. [16] if the function f(n)(t) is the nth derivative of the function f(t) ∈ a, then its shehu transform is defined by h [ f(n)(t) ] = (u s )−n v (s,u) − n−1∑ k=0 (u s )(k+1)−n f(k) (0) , n ≥ 1. (1.3) some properties of shehu transform are given in [16]. for our results we need some other definitions and some properties. definition 1.1. a generalization of the exponential function is given by [10] eα (z) = ∞∑ k=0 zk γ (αk + 1) , α ∈ c, re (α) > 0. (1.4) a generalization of mittag-leffler function eα (z) is defined as follows [18]: eα,β (z) = ∞∑ k=0 zk γ (αk + β) , α,β ∈ c, re (α) , re (β) > 0. (1.5) a generalization of mittag-leffler function eα,β (z) of (1.5) is introduced by prabhakar [14], as follows: e γ α,β (z) = ∞∑ n=0 γk γ (αk + β) zk k! , α,β,γ ∈ c, re (α) , re (β) > 0, re (γ) > 0. (1.6) where γk denotes the familiar pochhammer symbol. lemma 1.1. [6] in the complex plane c, for any re (α) , re (β) > 0, re (γ) > 0 and ω ∈ c. s ( tγ−1e γ α,β (ωt α) ) = uβ−1 (1 −ωuα)−γ . (1.7) corollary 1.1. sumudu transform of mittag-leffer function eα,β (z) = ∑∞ n=0 zn γ(nα+β) , α,β ∈ c, re (α) , re (β) > 0 exists and given by s ( tγ−1eα,β ( ωtβ )) = uγ−1 ( 1 −ωuβ )−1 . (1.8) int. j. anal. appl. 17 (6) (2019) 919 definition 1.2. let f ∈ l1 (a,b) . if α ≥ 0, then left sided riemann–liouville fractional integral of order α is defined by [1, 12, 13] iα0+f (t) = 1 γ (α) t∫ 0 (t− τ)α−1 f (τ) dτ = 1 γ (α) tα−1 ∗f (t) , α > 0, t > 0, (1.9) i0f (t) = f (t) . definition 1.3. let f ∈ l1 (a,b) , and m−1 < α ≤ m. the caputo fractional derivative of order α (α > 0) is defined as [1, 5, 12] cdα0+f (t) =   1 γ(m−α) t∫ 0 (t− τ)m−α−1 f(m) (τ) dτ, m− 1 < α ≤ m. ∂m ∂tm f (t) if α = m. remark 1.1. [13] under the terms of the previous definition, we have cdα0+f (t) = 1 γ (m−α) tm−α−1 ∗f(m) (t) . (1.10) lemma 1.2. [9, lemma 2.22 p.96] if f(t) ∈ acn [a,b] or f(t) ∈ cn [a,b] , then ( iαc0+ d α 0+ ) f(t) = f(t) − n−1∑ k=0 f(k) (0) k! tk. (1.11) as the next theorem shows, the shehu transform is closely connected with the sumudu transform, theorem 1.2. [3] let f(t) ∈ a with sumudu transform g (u). then the shehu transform v (s,u) of f(t) is given by v (s,u) = u s g (u s ) . (1.12) lemma 1.3. [15, p. 140-141] if f ∈ l1 (a,b) for any b > a and of exponential order. then iα0+f (t) also of exponential order. 2. main result in this section, we present some results on the transformation of shehu, as a complementary result of what can be seen in [16]. theorem 2.1. let a ∈ c∗ and let f (at) ∈ a. if v (s,u) denote the shehu transform of f. then h (f (at)) = 1 a v (s,au) . int. j. anal. appl. 17 (6) (2019) 920 proof. using the definition of shehu transform eq.(1.1), we get h (f (at)) = ∫ ∞ 0 exp ( − s u t ) f (at) dt. if we set τ = at (t = τ/a), then h (f (at)) = 1 a ∫ ∞ 0 exp ( − s au τ ) f (τ) dτ = 1 a v (s,au) . � theorem 2.2. let a ∈ c∗ and let f (t) ∈ a with shehu transform v (s,u) . then h ( eatf (t) ) = v (s−au,u) . proof. using eq.(1.2), we have h ( eatf (t) ) = ∫ ∞ 0 exp ( at− s u t ) f (t) dt = ∫ ∞ 0 exp ( − s−au u t ) f (t) dt. by setting s′ = s−au, we get h ( eatf (t) ) = ∫ ∞ 0 exp ( − s′ u t ) f (t) dt = v (s′,u) = v (s−au,u). � theorem 2.3. for x > 0, the shehu transform of tx−1 is v (s,u) = γ (x) (u s )x . (2.1) proof. for x > 0, the gamma function is defined by γ (x) = ∫ ∞ 0 τx−1e−τdτ. if we set τ = s u t ( t = u s τ ) , then we have γ (x) = ∫ ∞ 0 (s u t )x−1 e− s u t s u dt = (s u )x ∫ ∞ 0 tx−1e− s u tdt = (s u )x h ( tx−1 ) . then, h ( tx−1 ) = γ (x) ( u s )x . � int. j. anal. appl. 17 (6) (2019) 921 lemma 2.1. in the complex plane c, for any re (α) , re (β) > 0, re (γ) > 0 and ω ∈ c. shehu transform of e γ α,β (ωt α) is given by h ( tβ−1e γ α,β (ωt α) ) = (u s )β ( 1 −ω (u s )α)−γ . (2.2) proof. using eqs.(1.7), (1.12), we get h ( tβ−1e γ α,β (ωt α) ) = (u s )(u s )β−1 ( 1 −ω (u s )α)−γ = (u s )β ( 1 −ω (u s )α)−γ . � corollary 2.1. shehu transform of mittag-leffler function eα,β (z) = ∑∞ k=0 zn γ(αk+β) , α,β ∈ c, re (α) , re (β) > 0 exists and given by h ( tβ−1eα,β (ωt α) ) = (u s )β ( 1 −ω (u s )α)−1 . (2.3) proof. using eq.(2.2) and since eα,β (z) = e 1 α,β (z) , we get the desired result. � the next theorem shows the shehu transform convolution theorem. theorem 2.4. let f(t) and g(t) be in a, having shehu transforms v (s,u) and w (s,u), respectively. then the shehu transform of the convolution of f and g (f ∗g) (t) = ∫ ∞ 0 f (t) g (t− τ) dτ, (2.4) is given by h ((f ∗g) (t)) = v (s,u) w (s,u) . (2.5) proof. first, recall that the sumudu transform of f ∗g is given by [2] s ((f ∗g) (t)) = uf(u)g(u). (2.6) where f(u) and g(u), are the sumudu transforms of f(t) and g(t) respectively. now, since, by the relation (1.12), h [v (t) ∗w (t)] = u s s [v (t) ∗w (t)] = (u s )2 f (u s ) g (u s ) = (u s ) f (u s ) × (u s ) g (u s ) = v (s,u) w (s,u) . � int. j. anal. appl. 17 (6) (2019) 922 theorem 2.5. let f satisfy the conditions of lemma 1.3. then the shehu transform of iαt f (t) exists and given by h (iαt f (t)) = (u s )α v (s,u) . (2.7) proof. since by equation eq.(1.9) above, iα 0+ f (t) = 1 γ(α) tα−1 ∗ f (t) , then by theorems 2.3 and theorem 2.4, we have, h (iαt f (t)) = 1 γ (α) h ( tα−1 ) h (f (t)) = 1 γ (α) γ (α) (u s )α v (s,u) = (u s )α v (s,u) . � theorem 2.6. if f ∈ acn (a,b) for any b > a and of exponential order. then h ( cdα0 f (t) ) = (s u )α v (s,u) − n−1∑ k=0 (s u )α−(k+1) f(k) (0) . (2.8) proof. since ( iαc 0+ dα0+ ) f(t) = f(t) − ∑n−1 k=0 f(k)(0) k! tk, by lemma 1.2, we have h (( iαc0+ d α 0+ ) f(t) ) = h ( f(t) − n−1∑ k=0 f(k) (0) k! tk ) , thus (u s )α h ( cdα0+f(t) ) = v (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) , finally, we get h ( cdα0+f(t) ) = (u s )−α v (s,u) − n−1∑ k=0 (u s )k+1−α f(k) (0) . by other method, we can use eq. (1.12), and that [8] s ( cdα0 f (t) ) = u−α ( g (u) − n−1∑ k=0 ukf(k) (0) ) . in fact, h ( cdα0+f(t) ) = (u s )(u s )−α ( g (u s ) − n−1∑ k=0 (u s )k f(k) (0) ) = (u s )1−α (s u v (s,u) − n−1∑ k=0 (u s )k f(k) (0) ) = (u s )−α ( v (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) ) . int. j. anal. appl. 17 (6) (2019) 923 by anothor method, we have by remark 1.1, cdα 0+ f (t) = 1 γ(n−α)t n−α−1 ∗f(n) (t) , n− 1 < α ≤ n, then by using theorems 2.3 and theorem 2.4, we obtain h ( cdα0+f(t) ) = h ( 1 γ (n−α) tn−α−1 ∗f(n) (t) ) = 1 γ (n−α) h ( tn−α−1 ) h ( f(n) (t) ) = 1 γ (n−α) γ (n−α) (u s )n−α [(s u )n v (s,u) − n−1∑ k=0 (s u )n−(k+1) f(k) (0) ] = (u s )−α [ v (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) ] . � 3. applications we take into consideration a general linear ordinary differential equation with fractional order as follows: cdα0+y(t) = n∑ i=1 biy (i) (t) + g (t) , n− 1 < α ≤ n (3.1) subject to the initial condition y(i) (0) = ai, i = 0, ...,n− 1, (3.2) where ai,bj ∈ r, g (t) ∈ a. when we get shehu transform of (3.1) taking into consideration (1.3) and (2.8), we obtain shehu transform of (3.1) as follows h ( cdα0+y(t) ) = h ( n∑ i=1 biy (i) (t) + g (t) ) , by the linearity of shehu transform, we have h ( cdα0+y(t) ) = n∑ i=0 bih ( y(i) (t) ) + h (g (t)) , = b0y (t) + n∑ i=1 bih ( y(i) (t) ) + h (g (t)) using eqs.(1.3), (2.8), we obtain (u s )−α v (s,u) − n−1∑ k=0 (u s )k+1−α y(k) (0) = b0v (s,u) + n∑ i=1 bi [(u s )−i v (s,u) − i−1∑ k=0 (u s )k+1−i y(k) (0) ] + h (g (t)) int. j. anal. appl. 17 (6) (2019) 924 (u s )−α v (s,u) − n∑ i=0 bi (u s )−i v (s,u) = n−1∑ k=0 ak (u s )k+1−α − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + h (g (t)) v (s,u) = ((u s )−α − n∑ i=0 bi (u s )−i)−1 (n−1∑ k=0 ak (u s )k+1−α (3.3) − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + h (g (t)) ) . operating the inverse shehu transform on both sides of eq. (3.3), we get the solution of eq. (3.1) as follows: y (t) = h−1  ((u s )−α − n∑ i=0 bi (u s )−i)−1 (n−1∑ k=0 ak (u s )k+1−α (3.4) − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + h (g (t)) )] . example 3.1. when n = 1,b0 = −1 and b1 = g (t) = 0 , we obtain [11] cdαy (t) + y (t) = 0, 0 < α ≤ 1, t > 0, (3.5) with initial condition y (0) = 1. (3.6) substituting n,b0,b1 and g in (3.4), we get : y (t) = h−1  ((u s )−α − 1∑ i=0 bi (u s )−i)−1 (u s )1−α . y (t) = h−1 [(u s )( 1 − (−1) (u s )α)−1] thus, by eq.(2.3), we have v (s,u) = h (eα (−tα)) . (3.7) when we get the inverse sumudu transform of (3.7) , we find exact solution of eq.(3.5) as follows: y (t) = eα (−tα) . example 3.2. below we give the following particular example, which where debate in the literature and here important application in several world problems. consider the bagley-torvik equation [7] d2y (t) +c d3/2y (t) + y (t) = t + 1, (3.8) int. j. anal. appl. 17 (6) (2019) 925 with the initial conditions y (0) = y′ (0) = 1. (3.9) in this case, we have n = 2,b0 = b2 = −1,b1 = 0 and g (t) = t + 1. applying eq.(3.4), we get y (t) = h−1  ((u s )−3/2 − 2∑ i=0 bi (u s )−i)−1 ( 1∑ k=0 ak (u s )k+1−3/2 (3.10) − 2∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + u s + (u s )2)] . then, y (t) = h−1 (( u s )−1/2 + ( u s )1/2 + ( u s )−1 + 1 + u s + ( u s )2( u s )−3/2 + ( u s )−2 + 1 ) = h−1 (( u s )−1 + ( u s )−1/2 + u s( u s )−2 + ( u s )−3/2 + 1 + 1 + ( u s )1/2 + ( u s )2( u s )−2 + ( u s )−3/2 + 1 ) = h−1 ( u s + (u s )2) . (3.11) taking the inverse shehu transform of eq. (3.11), yields y (t) = t + 1, which is the exact solution. example 3.3. consider the following homogeneous fractional ordinary differential equation : [4] cd 1 2 y (t) + y (t) = t2 + γ (3) γ ( 5 2 )t32 , t > 0, (3.12) with initial condition y (0) = 0. (3.13) in order to find exact solution of (3.12), we apply eq.(3.4) , for n = 1, b0 = −1,b1 = 0 and g (t) = t2 + γ(3) γ( 52 ) t 3 2 , we obtain y (t) = h−1 ((u s )−1 2 + 1 )−1 ( 2 (u s )3 + γ (3) (u s )5 2 ) = h−1  2 (us)3 + γ (3) (us)52( u s )−1 2 + 1   = h−1 ( 2 (u s )3 ) . when we take the inverse shehu transform of 2 ( u s )3 , we get the analytical solution of eq.(3.12) y (t) = t2. int. j. anal. appl. 17 (6) (2019) 926 4. conclusions in the field of fractional calculus finding a new integral transform for solving the ordinary od partial fractional differential equations it is always useful. in this manuscript, the newly suggested shehu integral transform was applied to solve hgher order fractional differential equations with caputo derivative. we show the efficiency and and high accuracy of the suggested integral transform. references [1] d. baleanu, k. diethelm, e. scalas, and j. j. trujillo, fractional calculus: models and numerical methods, vol. 3 of series on complexity, nonlinearity and chaos, world scientific publishing, boston, mass, usa, 2012. [2] f. b. m. belgacem, a. a. karaballi, and s. l. kalla, analytical investigations of the sumudu transform and applications to integral production equations, math. probl. eng. 2003 (2003), article id 439059. [3] a. bokhari, r. belgacem, d. baleanu, application of shehu transform to atangana-baleanu derivatives, j. math. computer sci. (in press). [4] h. bulut, h. m. baskonus, and f. b. m. belgacem, the analytical solution of some fractional ordinary differential equations by the sumudu transform method, abstr. appl. anal. 2013 (2013), article id 203875. [5] m. caputo, linear model of dissipation whose q is almost frequency independent-ii, geoph. j. royal astron. soc. 13 (5) (1967), 529–539. [6] v. g. gupta and b. sharma, application of sumudu transform in reaction-diffusion systems and nonlinear waves, appl. math. sci., 4 (9) (2010), 435-446. [7] h. jafari, s. das, h. tajadodi, solving a multi-order fractional di erential equation using homotopy analysis method, j. king saud univ. sci. 23 (2011) 151-155. [8] q. d. katatbeh, f. b. m. belgacem, applications of the sumudu transform to fractional differential equations, nonlinear stud. 18 (1) (2011), 99-112. [9] a. a. a. kilbas, h. m. srivastava, j. j. trujillo; theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [10] m.g. mittag-leffler, sur la nouvelle fonction e(x). comptes rendus acad. sci. paris (ser. ii) 137 (1903), 554–558. [11] z. m. odibat and s. momani, an algorithm for the numerical solution of differential equations of fractional order, j. appl. math. inform. 26 (1-2) (2008), 15–27. [12] k.b. oldham, j. and spanier, the fractional calculus:theory and applications of differentiation and integration to arbitrary order, academic press, new york and london, 1974. [13] i. podlubny, fractional differantial equations, academic press, san diego, calif, usa, 1999. [14] t.r. prabhakar, a singular integral equation with generalized mittag-leffler function in the kernel, yokohama math. j. 19 (1971), 7–15 [15] s. g. samko , a. a. kilbas and o.i. marichev, fractional integrals and derivatives, yverdon-les-bains, switzerland: gordon and breach science publishers, yverdon, 1993. [16] s. maitama and w. zhao, new integral transform: shehu transform a generalization of sumudu and laplace transform for solving differential equations, int. j. anal. appl. 17 (2) (2019), 167-190. [17] g. k. watugala, sumudu transform: a new integral transform to solve differential equations and control engineering problems, int. j. math. educ. sci. technol. 24 (1) (1993), 35–43. int. j. anal. appl. 17 (6) (2019) 927 [18] wiman, a.: ueber den fundamentalsatz in der theorie der functionen e(x). acta math. 29 (1905), 191–201. 1. introduction 2. main result 3. applications 4. conclusions references international journal of analysis and applications volume 18, number 5 (2020), 784-798 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-784 sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualifications izhar ahmad1,∗, krishna kummari2 and s. al-homidan1 1department of mathematics and statistics, king fahd university of petroleum and minerals, dhahran 31261, saudi arabia 2department of mathematics, gitam-hyderabad campus, hyderabad-502329, india ∗corresponding author: drizhar@kfupm.edu.sa abstract. in this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. sufficient optimality conditions for a lu optimal solution are derived under generalized convexity assumptions. moreover, appropriate duality results are discussed for a mond-weir type dual problem. in addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem. 1. introduction due to the mathematical challenges and important roles in various fields, mathematical programs with vanishing constraints have attracted many mathematicians in the past decade. mathematical programming problem with vanishing constraints is a constrained optimization problem and it is closely related to the mathematical programs with equilibrium constraints, see for example [9, 10, 14]. this problem was first studied by achtziger and kanzow in [2] and this serves as a model for many problems from topology and structural optimization (see [2, 5]). for mathematical programming problems with vanishing constraints, received october 14th, 2019; accepted november 13th, 2019; published july 17th, 2020. 2010 mathematics subject classification. 26a51, 49j35, 90c32. key words and phrases. interval-valued optimization problem; vanishing constraints; constraint qualifications; generalized convexity; sufficiency; duality. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 784 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-784 int. j. anal. appl. 18 (5) (2020) 785 it is well known that the usual nonlinear programming constraint qualifications such as slater constraint qualification, mangasarian-fromovitz constraint qualification, cottle constraint qualification and linear independence constraint qualification do not hold (see [12]), while mishra et al. [12] proved that the standard generalized guignard constraint qualification holds in many situations, and some sufficient conditions are presented in [12]. the guignard constraint qualification (gcq) was introduced by guignard [8] and is one of the weakest among the most prominent constraint qualifications such as the slater constraint qualification [19], abadie constraint qualification [1], mangasarian-fromovitz constraint qualification [11], cottle constraint qualification [7] and linear independence constraint qualification etc. for more information and inter-relation between these constraint qualifications one can see the survey papers [15, 22]. in recent years, a number of approaches have been developed to deal with interval-valued optimization problems. in [24, 25], wu derived karush-kuhn-tucker type optimality conditions for a optimization problem with an interval-valued objective function. further, the karushkuhn-tucker type necessary optimality conditions for a optimization problem in which objective and constraints functions are assumed to be interval valued were investigated by singh et al. [17]. however, optimality conditions for an interval-valued multiobjective programming with generalized differentiable functions (viz. gh-differentiable functions) are discussed in [18]. bhurjee and panda [6] provided an overview of an interval-valued optimization problem by developing a methodology to study the efficient solution for an interval-valued optimization problem. for more details related to interval-valued optimization problems, we refer to the papers (see, for example [3,13,16,20,23,26]). to the author’s knowledge, there are no results for an interval-valued mathematical programming problem with vanishing constraints in the literature. therefore, this paper focuses on an interval-valued mathematical programming problem with vanishing constraints to explore the sufficient optimality conditions and mond-weir type duality results. the rest of the article is organized as follows: some background material and preliminary definitions are provided in section 2. the sufficient optimality conditions for a lu optimal solution for considered problem under generalized convexity assumptions are given in section 3. in section 4, weak, strong and strict converse duality theorems are discussed for a mond-weir type dual model. finally, section 5 is devoted to the conclusion. 2. preliminaries for a nonempty subset q of rn, we use the notations clq and clcoq to denote the closure of q and closure of the convex hull of q, respectively. let θ be the set of all closed and bounded intervals in r. let θ1 = [τ l,τu ], θ2 = [ρ l,ρu ] ∈ θ, then we have (i ) θ1 + θ2 = {τ + ρ | τ ∈ θ1 and ρ ∈ θ2} = [τl + ρl,τu + ρu ], int. j. anal. appl. 18 (5) (2020) 786 (ii ) −θ1 = {−τ | τ ∈ θ1} = [−τu,−τl], (iii ) θ1 − θ2 = θ1 + (−θ2) = [τl −ρu,τu −ρl], (iv ) k + θ1 = {k + τ | τ ∈ θ1} = [k + τl,k + τu ], (v ) kθ1 = {kτ | τ ∈ θ1} =   [kτl,kτu ], if k ≥ 0, [kτu,kτl], if k < 0, where k is a real number. for θ1 = [τ l,τu ] and θ2 = [ρ l,ρu ], the order relation ≤lu is defined as follows: (i) θ1 ≤lu θ2 if and only if τl ≤ ρl and τu ≤ ρu . (ii) θ1 <lu θ2 if and only if θ1 ≤lu θ2 and θ1 6= θ2. it is obvious that, θ1 <lu θ2 if and only if τl < ρl and τu < ρu, or, τl ≤ ρl and τu < ρu, or, τl < ρl and τu ≤ ρu. an interval-valued function ψ : rn → θ can be written as ψ(x) = [ψl(x), ψu (x)], where ψl(x), ψu (x) are real-valued functions satisfying the condition ψl(x) ≤ ψu (x) and θ be the set of all closed and bounded intervals in r. in the present analysis, we consider the following inter-valued optimization problem with vanishing constraints: (ivvc) min x∈f ψ(x) = [ψl(x), ψu (x)] subject to ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) ≥ 0, ∀i = 1, 2, ...,r, φi(x)`i(x) ≤ 0, ∀i = 1, 2, ...,r, where ψ : rn → θ is an interval-valued function and ψl, ψu , ϕi,ζi,`i, φi : rn → r are assumed to be continuously differentiable functions. the feasible region is given by f = {x ∈ rn|ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) ≥ 0, ∀i = 1, 2, ...,r, φi(x)`i(x) ≤ 0, ∀i = 1, 2, ...,r}. int. j. anal. appl. 18 (5) (2020) 787 definition 2.1 (sun and wang [21]). a point ā ∈ f is said to be a lu optimal solution to (ivvc), if there exists no x0 ∈ f such that ψ(x0) <lu ψ(ā). let x∗ ∈ f be any feasible solution of the (ivvc). the following index sets will be used in the sequel. λϕ = {i ∈{1, 2, ...,p}|ϕi(x∗) = 0}, λζ = {1, 2, ...,q}, λ+ = {i ∈{1, 2, ...,r}|`i(x∗) > 0}, λ0 = {i ∈{1, 2, ...,r}|`i(x∗) = 0}. furthermore, the index set λ+ can be divided into the following subsets λ+0 = {i ∈{1, 2, ...,r}|`i(x∗) > 0, φi(x∗) = 0}, λ+− = {i ∈{1, 2, ...,r}|`i(x∗) > 0, φi(x∗) < 0}. similarly, the index set λ0 can be partitioned in the following way λ0+ = {i ∈{1, 2, ...,r}|`i(x∗) = 0, φi(x∗) > 0}, λ00 = {i ∈{1, 2, ...,r}|`i(x∗) = 0, φi(x∗) = 0}, λ0− = {i ∈{1, 2, ...,r}|`i(x∗) = 0, φi(x∗) < 0}. also, for x∗ ∈ f, we define the sets qk, q k , k = l,u and q as follows: qk = { x ∈ rn|ψi(x) ≤ ψi(x∗),∀i = l,u, i 6= k, ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) ≥ 0, ∀i = 1, 2, ...,r, φi(x)`i(x) ≤ 0, ∀i = 1, 2, ...,r } . q k = { x ∈ rn|ψi(x) ≤ ψi(x∗),∀i = l,u, i 6= k, ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) = 0, φi(x) ≥ 0, ∀i ∈ λ0+, φi(x) ≤ 0,`i(x) ≥ 0, ∀i ∈ λ0− ∪ λ00 ∪ λ+0 ∪ λ+− } . and q = { x ∈ rn|ψl(x) ≤ ψl(x∗), ψu (x) ≤ ψu (x∗), int. j. anal. appl. 18 (5) (2020) 788 ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) = 0, φi(x) ≥ 0, ∀i ∈ λ0+, φi(x) ≤ 0,`i(x) ≥ 0, ∀i ∈ λ0− ∪ λ00 ∪ λ+0 ∪ λ+− } . the linearizing cone q k , k = l,u at x∗ ∈ f is given by l(q k ; x∗) = { δ ∈ rn|∇ψi(x∗) t δ ≤ 0,∀i = l,u, i 6= k, ∇ϕi(x∗)tδ ≤ 0, ∀i ∈ λϕ, ∇ζi(x∗)tδ = 0, ∀i ∈ λζ, ∇`i(x∗)tδ = 0, ∀i ∈ λ0+, ∇`i(x∗)tδ ≥ 0, ∀i ∈ λ00 ∪ λ0−, ∇φi(x∗)tδ ≤ 0, ∀i ∈ λ+0 ∪ λ00 } . and the symbol t denotes the transpose of a matrix. the linearizing cone to q at x∗ ∈ q, given by. l(q; x∗) = l(q l ; x∗) ∩l(q u ; x∗). definition 2.2. the tangent cone to q at x∗ ∈ clq is defined by t(x∗) = { δ ∈ rn|∃{xn}⊆ f,{tn} ↓ 0 : xn → x∗ and xn −x∗ tn → δ } the modified guignard constraint qualification was introduced by mishra et al. ( [12], definition 6.14) for a mathematical programming problem with vanishing constraints. from this perspective, we define the modified guignard constraint qualification (ivvc-gcq) for an interval-valued optimization problem (ivvc) as follows. definition 2.3. the modified guignard constraint qualification (ivvc-gcq) is said to holds at x∗ ∈ f, if l(q; x∗) ⊆ clcot(ql; x∗) ∩ clcot(qu ; x∗). mishra et al. [12] proved the karush-kuhn-tucker type necessary optimality conditions for a multiobjective optimization problem with vanishing constraints under modified guignard constraint qualification. along the lines of mishra et al. ( [12] theorem 6.4), if we set m = 2, we acquire the following karush-kuhntucker type necessary optimality conditions for (ivvc) as follow: int. j. anal. appl. 18 (5) (2020) 789 theorem 2.1. let x∗ ∈ f be a lu optimal solution of (ivvc) such that (ivvc-gcq) holds at x∗. then there exist 0 < λl,λl ∈ r, µi ∈ r+, i = 1, 2, ...,p, γi ∈ r,i = 1, 2, ...,q and η`i,η φ i ∈ r,i = 1, 2, ...,r such that λl∇ψl(x∗) + λu∇ψu (x∗) + p∑ i=1 µi∇ϕi(x∗) + q∑ i=1 γi∇ζi(x∗) − r∑ i=1 η`i∇`i(x ∗) + r∑ i=1 ηφi ∇φi(x ∗) = 0, (2.1) ϕi(x ∗) ≤ 0, µiϕi(x∗) = 0, ∀i = 1, 2, ...,p, (2.2) ζi(x ∗) = 0, ∀i = 1, 2, ...,q, (2.3) η`i = 0, i ∈ λ+, η ` i ≥ 0, i ∈ λ00 ∪ λ0−, η ` i free, i ∈ λ0+, (2.4) ηφi = 0, i ∈ λ+− ∪ λ0− ∪ λ0+, η φ i ≥ 0, i ∈ λ+0 ∪ λ00, (2.5) η`i`i(x ∗) = 0,ηφi φi(x ∗) = 0,∀i = 1, 2, ...,r. (2.6) we define the following index sets which will be useful to prove the sufficient optimality conditions and duality results. λ+ϕ = {i ∈{1, 2, ...,p}|µi > 0}, λ + ζ = {i ∈ λζ|γi > 0}, λ−ζ = {i ∈ λζ|γi < 0}, λ + + = {i ∈ λ+|η ` i > 0} λ+0 = {i ∈ λ0|η ` i > 0}, λ − 0 = {i ∈ λ0|η ` i < 0}, λ−0+ = {i ∈ λ0+|η φ i < 0}, λ − 00 = {i ∈ λ00|η φ i < 0}, λ−+0 = {i ∈ λ+0|η φ i < 0}, λ + 00 = {i ∈ λ00|η φ i > 0}, λ−00 = {i ∈ λ00|η φ i < 0}, λ + +0 = {i ∈ λ+0|η φ i > 0}, λ−+0 = {i ∈ λ+0|η φ i < 0}, λ + 0− = {i ∈ λ0−|η φ i > 0}, λ++− = {i ∈ λ+−|η φ i > 0}. we now turn our attention to define some well-known concepts of convexity and generalized convexity for a real valued differentiable function (see, for example, [4]). int. j. anal. appl. 18 (5) (2020) 790 definition 2.4. let ω : x ⊆ rn → r be a continuously differentiable function. then, ω is said to be a (strictly) convex at (x 6= x∗ ∈ x) x∗ ∈ x if for any x ∈ x, we have ω(x) − ω(x∗)(>) ≥ (x−x∗)t∇ω(x∗). definition 2.5. let ω : x ⊆ rn → r be a continuously differentiable function. then, ω is said to be a quasiconvex at x∗ ∈ x if for any x ∈ x, we have ω(x) ≤ ω(x∗) ⇒ (x−x∗)t∇ω(x∗) ≤ 0, equivalently (x−x∗)t∇ω(x∗) > 0 ⇒ ω(x) > ω(x∗). definition 2.6. let ω : x ⊆ rn → r be a continuously differentiable function. then, ω is said to be a (strictly) pseudoconvex at x∗ ∈ x if for any x ∈ x, we have (x−x∗)t∇ω(x∗) ≥ 0 ⇒ ω(x)(>) ≥ ω(x∗), equivalently ω(x)(≤) < ω(x∗) ⇒ (x−x∗)t∇ω(x∗) < 0. 3. sufficient optimality conditions in this section, we establish sufficient optimality conditions for the problem (ivvc) using the concept of generalized convexity. theorem 3.1 (sufficient optimality conditions). let x̃ ∈ f and there exist 0 < λl,λu ∈ r, µi ∈ r+, i = 1, 2, ...,p, γi ∈ r,i = 1, 2, ...,q and η`i, η φ i ∈ r,i = 1, 2, ...,r such that λl∇ψl(x̃) + λu∇ψu (x̃) + p∑ i=1 µi∇ϕi(x̃) + q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηφi ∇φi(x̃) = 0, (3.1) ϕi(x̃) ≤ 0, µiϕi(x̃) = 0, ∀i = 1, 2, ...,p, (3.2) ζi(x̃) = 0, ∀i = 1, 2, ...,q, (3.3) η`i = 0, i ∈ λ+, η ` i ≥ 0, i ∈ λ00 ∪ λ0−, η ` i free, i ∈ λ0+, (3.4) ηφi = 0, i ∈ λ+− ∪ λ0− ∪ λ0+, η φ i ≥ 0, i ∈ λ+0 ∪ λ00, (3.5) int. j. anal. appl. 18 (5) (2020) 791 η`i`i(x̃) = 0,η φ i φi(x̃) = 0,∀i = 1, 2, ...,r. (3.6) further, assume that λlψl(.) + λu ψu (.) is pseudoconvex at x̃ on f and that p∑ i=1 µiϕi(.), ζi(.)(i ∈ λ+ζ ), −ζi(.)(i ∈ λ−ζ ), −`i(.)(i ∈ λ + + ∪λ + 0 ), `i(.)(i ∈ λ − 0 ), −φi(.)(i ∈ λ − 0+ ∪λ − 00 ∪λ − +0), φi(.)(i ∈ λ + 00 ∪λ + 0−∪λ + +0 ∪ λ++−) are quasiconvex at x̃ on f. then x̃ is a lu optimal solution of the problem (ivvc). proof. suppose contrary to the result that x̃ is not a lu optimal solution to the problem (ivvc), then by definition 2.1 there exists x0 ∈ f such that ψ(x0) <lu ψ(x̃). that is,   ψl(x0) < ψ l(x̃) ψu (x0) < ψ u (x̃) , or   ψl(x0) ≤ ψl(x̃) ψu (x0) < ψ u (x̃) , or   ψl(x0) < ψ l(x̃) ψu (x0) ≤ ψu (x̃) . since λl > 0, λu > 0, therefore the above inequalities yield λlψl(x0) + λ u ψu (x0) < λ lψl(x̃) + λu ψu (x̃), which by pseudoconvexity of λlψl(.) + λu ψu (.) at x̃ on f, we obtain (x0 − x̃)t [ λl∇ψl(x̃) + λu∇ψu (x̃) ] < 0. (3.7) for x0 ∈ f, µi ∈ r+, i = 1, 2, ...,p, we have µiϕi(x0) ≤ 0, i = 1, 2, ...,p, which in view of (3.2) implies that p∑ i=1 µiϕi(x0) ≤ p∑ i=1 µiϕi(x̃), which by quasiconvexity of p∑ i=1 µiϕi(.) at x̃ on f, we get (x0 − x̃)t p∑ i=1 µi∇ϕi(x̃) ≤ 0. (3.8) by similar arguments, we have (x0 − x̃)t∇ζi(x̃) ≤ 0,∀i ∈ λ+ζ , −(x0 − x̃)t∇ζi(x̃) ≤ 0,∀i ∈ λ−ζ , −(x0 − x̃)t∇`i(x̃) ≤ 0,∀i ∈ λ++ ∪ λ + 0 , (x0 − x̃)t∇`i(x̃) ≤ 0,∀i ∈ λ−0 , −(x0 − x̃)t∇φi(x̃) ≤ 0,∀i ∈ λ−0+ ∪ λ − 00 ∪ λ − +0, (x0 − x̃)t∇φi(x̃) ≤ 0,∀i ∈ λ+00 ∪ λ + 0− ∪ λ + +0 ∪ λ + +−, int. j. anal. appl. 18 (5) (2020) 792 which by the definition of index sets one has (x0 − x̃)t [ q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηφi ∇φi(x̃) ] ≤ 0. (3.9) on adding (3.7), (3.8) and (3.9), we get (x0 − x̃)t [ λl∇ψl(x̃) + λu∇ψu (x̃) + p∑ i=1 µi∇ϕi(x̃) + q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηφi ∇φi(x̃) ] < 0, which contradicts (3.1). this completes the proof of this theorem. � now, we verify the sufficient optimality conditions by the following example. example 3.1. consider the following interval-valued optimization problem: (ivvc-1) min x∈f1 ψ(x) = [ψl(x), ψu (x)] = [x + x3,x5] subject to `1(x) = 1 + x 3 ≥ 0, φ1(x)`1(x) = x(1 + x 3) ≤ 0, which is the form of (ivvc) with n = 1, p = q = 0 and r = 1. the feasible region of (ivvc-1) is f1 = {x ∈ r|`1(x) ≥ 0, φ1(x)`1(x) ≤ 0}. note that x̃ = 0 is a feasible solution of (ivvc-1) and it can be easily observe that there exist 0 < λl,λu ∈ r, η`1, and η φ 1 ∈ r such that the relations (3.1)-(3.6) hold for the problem (ivvc-1). also, it is not difficult to see that λlψl(.) + λu ψu (.) is pseudoconvex at x̃ on f1 and `1(x), φ1(x) are quasiconvex at x̃ on f1. since all the assumptions of theorem 3.1 are satisfied, then x̃ = 0 is a lu optimal solution of the problem (ivvc-1). 4. mond-weir type duality we present the following mond-weir type dual for (ivvc). (imwdvc) max ψ(y) = [ ψl(y), ψu (y) ] subject to λl∇ψl(y) + λu∇ψu (y) + p∑ i=1 µi∇ϕi(y) + q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηφi ∇φi(y) = 0, (4.1) int. j. anal. appl. 18 (5) (2020) 793 µi ≥ 0,µiϕi(y) ≥ 0, ∀i = 1, 2, ...,p, (4.2) γi ∈ r,γiζi(y) = 0, ∀i = 1, 2, ...,q, (4.3) η`i ≥ 0, ∀i ∈ λ+,η ` i ∈ r,∀i ∈ λ0, (4.4) −η`i`i(y) ≥ 0, ∀i = 1, 2, ...,r, (4.5) 0 < λl,λu ∈ r,ηφi ≤ 0, ∀i ∈ λ0+, η φ i ≥ 0, ∀i ∈ (λ0− ∪ λ+−), (4.6) ηφi ∈ r, ∀i ∈ (λ00 ∪ λ+0), η φ i φi(y) ≥ 0, ∀i = 1, 2, ...,r. (4.7) we denote by w1 the set of all feasible solutions of the problem (imwdvc) and let prw1 = {y ∈ rn|(y,λl,λu,µ,γ,η`,ηφ) ∈ w1} be the projection of the set w1 on rn. now, we prove duality results between problems (ivvc) and (imwdvc) under certain generalized convexity assumptions imposed on the involved functions. theorem 4.1 (weak duality). let x ∈ f and (y,λl,λu,µ,γ,η`,ηφ) ∈ w1. further, assume that λlψl(.)+ λu ψu (.) is pseudoconvex at y on f ∪ prw1 and that p∑ i=1 µiϕi(.), ζi(.)(i ∈ λ+ζ ), −ζi(.)(i ∈ λ − ζ ), −`i(.)(i ∈ λ++ ∪λ + 0 ), `i(.)(i ∈ λ − 0 ), −φi(.)(i ∈ λ − 0+ ∪λ − 00 ∪λ − +0), φi(.)(i ∈ λ + 00 ∪λ + 0−∪λ + +0 ∪λ + +−) are quasiconvex at y on f∪ prw1, then ψ(x) ≥lu ψ(y). proof. suppose, contrary to the result, that ψ(x) <lu ψ(y). that is,   ψl(x) < ψl(y) ψu (x) < ψu (y) , or   ψl(x) ≤ ψl(y) ψu (x) < ψu (y) , or   ψl(x) < ψl(y) ψu (x) ≤ ψu (y) . since λl > 0, λu > 0, therefore the above inequalities yield λlψl(x) + λu ψu (x) < λlψl(y) + λu ψu (y), which by pseudoconvexity of λlψl(.) + λu ψu (.) at y on f∪ prw1, we obtain (x−y)t [ λl∇ψl(y) + λu∇ψu (y) ] < 0. (4.8) int. j. anal. appl. 18 (5) (2020) 794 for x ∈ f, µi ≥ 0, i = 1, 2, ...,p, we have µiϕi(x) ≤ 0, i = 1, 2, ...,p, which in view of (4.2) implies that p∑ i=1 µiϕi(x) ≤ p∑ i=1 µiϕi(y), which by quasiconvexity of p∑ i=1 µiϕi(.) at y on f∪ prw1, we get (x−y)t p∑ i=1 µi∇ϕi(y) ≤ 0. (4.9) by similar arguments, we have (x−y)t∇ζi(y) ≤ 0,∀i ∈ λ+ζ , −(x−y)t∇ζi(y) ≤ 0,∀i ∈ λ−ζ , −(x−y)t∇`i(y) ≤ 0,∀i ∈ λ++ ∪ λ + 0 , (x−y)t∇`i(y) ≤ 0,∀i ∈ λ−0 , −(x−y)t∇φi(y) ≤ 0,∀i ∈ λ−0+ ∪ λ − 00 ∪ λ − +0, (x−y)t∇φi(y) ≤ 0,∀i ∈ λ+00 ∪ λ + 0− ∪ λ + +0 ∪ λ + +−, which by the definition of index sets one has (x−y)t [ q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηφi ∇φi(y) ] ≤ 0. (4.10) on adding (4.8), (4.9) and (4.10), we get (x−y)t [ λl∇ψl(y) + λu∇ψu (y) + p∑ i=1 µi∇ϕi(y) + q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηφi ∇φi(y) ] < 0, which contradicts (4.1). this completes the proof of this theorem. � now, we verify the weak duality theorem by the following example. example 4.1. consider the following interval-valued optimization problem: (ivvc-2) min x∈f2 ψ(x) = [ψl(x), ψu (x)] = [x + x3,x5] subject to `1(x) = x 3 ≥ 0, φ1(x)`1(x) = (−2 + x)x3 ≤ 0, which is the form of (ivvc) with n = 1, p = q = 0 and r = 1. the feasible region of (ivvc-2) is f2 = {x ∈ r|`1(x) ≥ 0, φ1(x)`1(x) ≤ 0}. int. j. anal. appl. 18 (5) (2020) 795 for any feasible x ∈ f2, the corresponding mond-weir type dual problem for the primal problem (ivvc-2) is given by (imwdvc-1) max ψ(y) = [ ψl(y), ψu (y) ] = [y + y3,y5] subject to λl∇ψl(y) + λu∇ψu (y) −η`1∇`1(y) + η φ 1 ∇φ1(y) = λl + 3λly2 + 5λuy4 − 3η`1y 2 + ηφ1 = 0, η`1 ≥ 0, if 1 ∈ λ+, η ` 1 ∈ r, if i ∈ λ0, ηφ1 ≤ 0, if 1 ∈ λ0+, η φ 1 ≥ 0, if 1 ∈ (λ0− ∪ λ+−), η φ 1 ∈ r, if i ∈ (λ00 ∪ λ+0), 0 < λl,λu ∈ r, −η`1`1(y) ≥ 0, η φ 1 φ1(y) ≥ 0. let w2 be the set of all feasible solutions of the problem (imwdvc-1) and note that, (y,λl,λu,η`1,η φ 1 ) = (0, 1 2 , 1 2 , 1,−1 2 ) is a feasible solution for (imwdvc-1). furthermore, it is not difficult to see that λlψl(.) + λu ψu (.) is pseudoconvex at y on f2 ∪ prw2 and `1(.), φ1(.) are quasiconvex at y on f1 ∪ prw2. also, for the feasible solutions x = 1 for (ivvc-2) and (y,λl,λu,η`1,η φ 1 ) = (0, 1 2 , 1 2 , 1,−1 2 ) for (imwdvc1), we observe that ψ(x) >lu ψ(y). hence the weak duality theorem 4.1 is verified. theorem 4.2 (strong duality). let x̃ be a lu optimal solution to the problem (ivvc) and the generalized guignard constraint qualification (ivvc-gcq) is satisfied at x̃. then there exist λ̃u > 0, λ̃l > 0, µ̃ ∈ rp+, γ ∈ rq, η` ∈ rr, and ηφ ∈ rr such that (x̃,λl,λu,µ,γ,η`,ηφ) is a feasible solution for (imwdvc) and the two objective values are same. further, if all the assumptions of the theorem 4.1 are fulfilled, then the point (x̃,λl,λu,µ,γ,η`,ηφ) is a lu optimal solution of (imwdvc). proof. by assumption x̃ is a lu optimal solution for (ivvc) and the generalized guignard constraint qualification (ivvc-gcq) is satisfied at this point, then by theorem 2.1, there exist λ̃u > 0, λ̃l > 0, µ̃ ∈ rp+, γ ∈ rq, η` ∈ rr, and ηφ ∈ rr such that the conditions (2.1)-(2.6) are satisfied. thus, (x̃,λl,λu,µ,γ,η`,ηφ) is feasible in (imwdvc), moreover, the corresponding objective values of (ivvc) and (imwdvc) are equal. further, if (x̃,λl,λu,µ,γ,η`,ηφ) is not a lu optimal solution to (imwdvc), then there exists a feasible solution (ỹ,λl,λu,µ,γ,η`,ηφ) for (imwdvc), such that the following inequality is satisfied ψ(x̃) <lu ψ(ỹ). int. j. anal. appl. 18 (5) (2020) 796 this is a contradiction to the theorem 4.1. hence (x̃,λl,λu,µ,γ,η`,ηφ) is a lu optimal solution to (imwdvc). � theorem 4.3 (strict converse duality). let x̃ ∈ f and (ỹ, λ̃l, λ̃u, µ̃, γ̃, η̃`, η̃φ) ∈ w1 such that λ̃lψl(x̃) + λ̃u ψu (x̃) ≤ λ̃lψl(ỹ) + λ̃u ψu (ỹ). (4.11) further, assume that λ̃lψl(.) + λ̃u ψu (.) is strictly pseudoconvex at ỹ on f ∪ prw1 and that p∑ i=1 µ̃iϕi(.), ζi(.)(i ∈ λ+ζ ), −ζi(.)(i ∈ λ − ζ ), −`i(.)(i ∈ λ + + ∪ λ + 0 ), `i(.)(i ∈ λ − 0 ), −φi(.)(i ∈ λ − 0+ ∪ λ − 00 ∪ λ − +0), φi(.)(i ∈ λ+00 ∪ λ + 0− ∪ λ + +0 ∪ λ + +−) are quasiconvex at ỹ on f∪ prw1, then x̃ = ỹ. proof. suppose, contrary to the result, that x̃ 6= ỹ. since x̃ and (ỹ, λ̃l, λ̃u, µ̃, γ̃, η̃`, η̃φ) are feasible solutions in problems (ivvc) and (imwdvc), respectively, then p∑ i=1 µ̃iϕi(x̃) ≤ p∑ i=1 µ̃iϕi(ỹ), which by quasiconvexity of ∑p i=1 µ̃iϕi(.) at ỹ on f∪ prw1, we get (x̃− ỹ)t p∑ i=1 µ̃i∇ϕi(ỹ) ≤ 0. (4.12) by similar arguments as in theorem 4.1, we have (x̃− ỹ)t∇ζi(ỹ) ≤ 0,∀i ∈ λ+ζ , −(x̃− ỹ)t∇ζi(ỹ) ≤ 0,∀i ∈ λ−ζ , −(x̃− ỹ)t∇`i(ỹ) ≤ 0,∀i ∈ λ++ ∪ λ + 0 , (x̃− ỹ)t∇`i(ỹ) ≤ 0,∀i ∈ λ−0 , −(x̃− ỹ)t∇φi(ỹ) ≤ 0,∀i ∈ λ−0+ ∪ λ − 00 ∪ λ − +0, (x̃− ỹ)t∇φi(ỹ) ≤ 0,∀i ∈ λ+00 ∪ λ + 0− ∪ λ + +0 ∪ λ + +−, which by the definition of index sets one has (x̃− ỹ)t [ q∑ i=1 γ̃i∇ζi(ỹ) − r∑ i=1 η̃`i∇`i(ỹ) + r∑ i=1 η̃φi ∇φi(ỹ) ] ≤ 0. (4.13) on adding (4.12) and (4.13), we get (x̃− ỹ)t [ p∑ i=1 µ̃i∇ϕi(ỹ) + q∑ i=1 γ̃i∇ζi(ỹ) − r∑ i=1 η̃`i∇`i(ỹ) + r∑ i=1 η̃φi ∇φi(ỹ) ] ≤ 0, which together with (4.1), implies that (x̃− ỹ)t [ λl∇ψl(y) + λu∇ψu (y) ] ≥ 0. int. j. anal. appl. 18 (5) (2020) 797 in view of strict pseudoconvexity of λ̃lψl(.) + λ̃u ψu (.) at ỹ on f∪ prw1, the above inequality gives λ̃lψl(x̃) + λ̃u ψu (x̃) > λ̃lψl(ỹ) + λ̃u ψu (ỹ), which contradicts (4.11). this completes the proof of this theorem. � 5. conclusion in this paper, we have derived sufficient optimality conditions for an interesting class of interval-valued optimization problems with vanishing constraints under generalized convexity assumptions. furthermore, weak, strong and strict converse duality results for a mond-weir type dual model have been established. it would be interesting to see whether the results derived in this paper hold for a non-differentiable multiple interval-valued objective programming problems with vanishing constraints. we shall investigate it in our forthcoming papers. acknowledgements: this research is financially supported by the king fahd university of petroleum and minerals, saudi arabia, under the internal project no. in171012. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. m. abadie, on the kuhn-tucker theorem, nonlinear programming, j. abadie ed., john wiley, new york, 1967, pp. 21-36. [2] w. achtziger and c. kanzow, mathematical programs with vanishing constraints: optimality conditions and constraint qualifications, math. program. 114 (2008), 69-99. [3] y. an, g. ye, d. zhao and w. liu, hermite-hadamard type inequalities for interval (h1, h2)-convex functions, mathematics, 7 (2019), art. id 436. [4] m. s. bazaraa, h.d. sherali and c.m. shetty, nonlinear programming. theory and algorithms. 2nd edition, john wiley & sons, hoboken, 1993. [5] m. p. bendsøe and o. sigmund, topology optimization-theory, methods and applications, 2nd ed., springer, heidelberg, germany, 2003. [6] a. k. bhurjee and g. panda, efficient solution of interval optimization problem, math. method. oper. res. 76 (2012), 273-288. [7] r. w. cottle, a theorem of fritz john in mathematical programming, rand memorandum rm-3858-pr, rand corporation, 1963. [8] m. guignard, generalized kuhn-tucker conditions for mathematical programming problems in a banach space, siam j. control, 7 (1969), 232-241. [9] a. khare and t. nath, enhanced fritz john stationarity, new constraint qualifications and local error bound for mathematical programs with vanishing constraints, j. math. anal. appl. 472 (2019), 1042-1077. int. j. anal. appl. 18 (5) (2020) 798 [10] z.-q. luo, j.-s. pang and d. ralph, mathematical programs with equilibrium constraints, cambridge university press, cambridge, 1997. [11] o. l. mangasarian and s. fromovitz, the fritz john necessary optimality condition in the presence of equality and inequality constraints, j. math. anal. appl. 17 (1967), 37-47. [12] s. k. mishra, v. singh, v. laha and r. n. mohapatra, on constraint qualifications for multiobjective optimization problems with vanishing constraints, optimization methods, theory and applications, springer berlin heidelberg (2015), 95-135. [13] r. osuna-gómez, b. hernández-jiménez, y. chalco-cano and g. ruiz-garzón, new efficiency conditions for multiobjective interval-valued programming problems, inform. sci. 420 (2017), 235-248. [14] j. v. outrata, m. kočvara and j. zowe, nonsmooth approach to optimization problems with equilibrium constraints, nonconvex optimization and its applications, kluwer, dordrecht, 1998. [15] d. w. peterson, a review of constraint qualifications in finite-dimensional spaces, siam rev. 15 (1973), 639-654. [16] a. sadeghi, m. saraj and n. m. amiri, efficient solutions of interval programming problems with inexact parameters and second order cone constraints, mathematics, 6 (2018), art id 270. [17] d. singh, b. a. dar and a. goyal, kkt optimality conditions for interval valued optimization problems, j. nonlinear anal. optim. 5 (2014), 91-103. [18] d. singh, b. a. dar and d. s. kim, kkt optimality conditions in interval-valued multiobjective programming with generalized differentiable functions, european j. oper. res. 254 (2016), 29-39. [19] m. slater, lagrange multipliers revisited: a contribution to nonlinear programming, cowles commission discussion paper, mathematics, 403, 1950. [20] i. m. stancu-minasian, stochastic programming with multiple objective functions, d reidei publishing company, bordrecht, 1984. [21] y. sun and l. wang, optimality conditions and duality in nondifferentiable intervalvalued programming, j. ind. manage. optim. 9 (2013), 131-142. [22] z. wang and s. c. fang, on constraint qualifications: motivation, design and inter-relations, j. ind. manage. optim. 9 (2013), 983-1001. [23] l. wang, g. yang, h. xiao, q. sun, j. ge, interval optimization for structural dynamic responses of an artillery system under uncertainty, eng. optim. 52 (2020), 343–366. [24] h. c. wu, on interval valued nonlinear programming problems, j. math. anal. appl. 338 (2008), 299-316. [25] h. c. wu, the karush kuhn tuker optimality conditions in multiobjective programming problems with interval-valued objective functions, european j. oper. res. 196 (2009), 49-60. [26] h. c. wu, solving the interval-valued optimization problems based on the concept of null set, j. ind. manage. optim. 14 (3) (2018), 1157-1178. 1. introduction 2. preliminaries 3. sufficient optimality conditions 4. mond-weir type duality 5. conclusion references international journal of analysis and applications volume 17, number 5 (2019), 711-721 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-711 generalized statistical convergence of double sequences in paranormed spaces abdullah alotaibi1,2, alaa mohammed aljahili2 and s. a. mohiuddine1,∗ 1operator theory and applications research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia 2department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia ∗corresponding author: mohiuddine@gmail.com abstract. we introduce the notion of (λ,µ)-statistical convergence of double sequences in a setting of paranormed space and prove that every convergent sequence is (λ,µ)-statistically convergent but not conversely by supporting an illustrative example. we also define the notions of (λ,µ)-statistical cauchy and strongly (λ,µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ,µ)statistical convergence. 1. introduction and preliminaries the notion of statistical convergence, which is an extension of the idea of common convergence, was first appeared, under the name of almost convergence, in the first edition of the celebrated monograph of zygmund [32]. this idea was introduced by fast [11] as follows: the sequence x = (xk) is statistically convergent to ` if for every ε > 0, limn n −1|{k ≤ n : |xk − `| ≥ ε}| = 0. some basic properties of statistical convergence were studied by schoenberg [30], s̆alát [31] and connor [9]. an interesting notion of statistically received 2019-01-17; accepted 2019-02-21; published 2019-09-02. 2010 mathematics subject classification. 40a05, 40g05. key words and phrases. double sequence; statistical convergence; statistical cauchy; strong p-cesàro summability; paranormed space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 711 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-711 int. j. anal. appl. 17 (5) (2019) 712 cauchy sequence was first defined by fridy [12] and he also showed that it is equivalent to statistical convergence. thereafter, this notion turned out to be one of the most active areas of research in summability theory. the statistical convergence was studied in various setup such as topological hausdorff groups [8], normed spaces [15], locally convex hausdorff topological spaces [16], paranormed spaces [2], random 2-normed spaces [19] and many others. mursaleen [24] presented a generalization of statistical convergence with the help of non-decreasing sequence λ = (λk) such that λk+1 ≤ λk + 1 and λ1 = 0 of positive numbers tending to ∞ and called it λ-statistical convergence. we also refer to the recent work in [1, 3, 4, 6, 7, 10, 14, 17, 18, 21, 22] for some applications of convergence methods to approximation theorems. pringsheim [29] extended the notion of usual convergence from single sequences of real numbers to double sequences as follows: a double sequence x = (xjk) has a pringsheim limit ξ (convergent to ξ in pringsheim’s sense), in symbols, we shall write (p) lim x = ξ, provided that given an � > 0 there exists an n ∈ n such that |xjk − ξ| < � whenever j,k > n. also, x = (xjk) is bounded, denoted by l∞, if ‖x‖ = sup j,k |xjk| < ∞. it is well known that every convergent single sequence is bounded but this fact need not be true for double sequences. statistical convergence extended to double sequences by mursaleen and edely [26] with the help of two dimensional analogue of natural density of subsets of n × n and further it was defined and studied in intuitionistic fuzzy normed spaces, locally solid riesz spaces and paranormed spaces by mursaleen and mohiuddine [27, 28], mohiuddine et al. [20] and arani et al. [5], respectively. also, we refer to [13, 23]. let k ⊂ n×n. then, the double natural density of k is defined by δ2(k) = (p) lim m,n |k(m,n)| mn provided that the limit exists, where |k(m,n)| be the numbers of (j,k) in k such that j ≤ m and k ≤ n. x = (xjk) is said to be statistically convergent to ξ if for each � > 0, (p) lim m,n (mn)−1|{(j,k),j ≤ m,k ≤ n : |xjk − ξ| ≥ �}| = 0. mursaleen et al. [25] defined and studied the notion of (λ,µ)-statistical convergence for double sequences where λ = λm and µ = µn are two non-decreasing sequences of positive real numbers each tending to ∞ such that λ1 = 0, λm+1 ≤ λm + 1 and µ1 = 0, µn+1 ≤ µn + 1 for all m,n. the (λ,µ)-density of the set k ⊆ n×n is given by δλ,µ(k) = (p) lim m,n 1 λmµn |{m−λm + 1 ≤ j ≤ m,n−µn + 1 ≤ k ≤ n : (j,k) ∈ k}| int. j. anal. appl. 17 (5) (2019) 713 provided that the limit exists. we remark that λm = m and µn = n, the (λ,µ)-density reduces to the natural double density. a double sequence x = (xjk) is (λ,µ)-statistically convergent to ξ if for every � > 0, (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : |xjk − ξ| ≥ �}| = 0 where im = [m−λm + 1,m] and jn = [n−µn + 1,n]. if x is a linear space and g : x → r such that (i) x = 0 ⇒ g(x) = 0, (ii) g(x + y) ≤ g(x) + g(y), (iii) g(−x) = g(x) and (iv) if tk → t (k → ∞) and xk → x (k → ∞) in the sense that g(xk −x) → 0 (k → ∞) for scalars tk, t and the vectors xk,x ∈ x, then tkxk → tx (k → ∞) in the sense that g(tkxk − tx) → 0 (k →∞), then g is said to be a paranorm on x and the pair (x,g) is called a paranormed space. note that if g(x) = 0 implies x = 0, then paranorm g is called a total paranorm on x and the pair (x,g) is called a total paranormed space. it is to further note that each seminorm (norm) on x is a paranorm (total) but not conversely. 2. (λ,µ)-statistical convergence in paranormed spaces in this section, we introduce the notion of convergence and (λ,µ)-statistical convergence in the framework of paranormed space and prove various interesting results and display an illustrative example is support of our result. definition 2.1. let (x,g) be a paranormed space. we say that a double sequence x = (xjk) is convergent, shortly, g2-convergent, to ξ in (x,g) if for each � > 0, there is k0 ∈ n such that g(xjk − ξ) < � whenever j,k ≥ k0. in symbols, one writes g2lim x = ξ, and ξ is called the g2-limit of x. definition 2.2. let (x,g) be a paranormed space. we say that a double sequence x = (xjk) is (λ,µ)statistically convergent, shortly, g(sλ,µ)-convergent, to ξ in (x,g), if for each � > 0, the set {(j,k) ∈ n×n : g(xjk − ξ) ≥ �} has (λ,µ)-density zero, equivalently, one writes (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : g(xjk − ξ) ≥ �}| = 0. in this case, one writes g(sλ,µ)lim x = ξ. if we choose λm = m and µn = n then the notion of g(sλ,µ)convergence is reduced to statistically convergence for double sequence in (x,g) due to arani et al. [5]. we denote this by g(s2)-convergence and write g(s2)lim x = ξ. theorem 2.3. if a double sequence x = (xjk) is g(sλ,µ)-convergent then g(sλ,µ)-limit is unique. proof. assume that g(sλ,µ)lim x = ξ ′ and g(sλ,µ)lim x = ξ ′′. let � > 0 be given. we now define the following two sets b′(�) = {(j,k) ∈ n×n : g(xjk−ξ′) ≥ �/2} and b′′(�) = {(j,k) ∈ n×n : g(xjk−ξ′′) ≥ �/2}. int. j. anal. appl. 17 (5) (2019) 714 since g(sλ,µ)lim x = ξ ′, one obtains δλ,µ(b ′(�)) = (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : g(xjk − ξ′) ≥ �/2}| = 0. similarly, the assumption g(sλ,µ)lim x = ξ ′′ gives δλ,µ(b ′′(�)) = 0. now, let b(�) = b′(�) ∪ b′′(�). then δλ,µ(b(�)) = 0 and so δλ,µ(b c(�)) = 1 since bc(�) is a nonempty set. now if (j,k) ∈ n×n\b(�), then g(ξ′ − ξ′′) ≤ g(xjk − ξ′) + g(xjk − ξ′′) < �/2 + �/2 = �. since � > 0 is arbitrary, one obtains g(ξ′ − ξ′′) = 0 which yields ξ′ = ξ′′. � theorem 2.4. if a double sequence x = (xjk) is g2-convergent to ξ, then it is g(sλ,µ)-convergent to the same limit. proof. assume that (xjk) is g2-convergent to ξ, that is, g2lim x = ξ. let � > 0 be given. then, there is n ∈ n such that g(xjk − ξ) < � for all j,k ≥ n. since, the set k(�) = {(j,k) ∈ n × n : g(xjk − ξ) ≥ �} is contained in n×n, hence δλ,µ(k(�)) = 0, that is, (xjk) is g(sλ,µ)-convergent to ξ. example 2.5. the present example proves that the converse of last theorem is not true in general. let x = `(1/jk) =  x = (xjk) : ∞∑ j=1 ∞∑ k=1 | xjk |1/jk< ∞   with the paranorm g(x) = ∞∑ j=1 ∞∑ k=1 | xjk |1/jk . let us define x = (xjk) by xjk =   jk for m− [ √ λm] + 1 ≤ j ≤ m,n− [ √ µn] + 1 ≤ k ≤ n, 0 otherwise. for 0 < � < 1, one writes b(�) := {(j,k) ∈ n×n : g(xjk) ≥ �}. it is easy to see that g(xjk) =   (jk) 1/jk for m− [ √ λm] + 1 ≤ j ≤ m,n− [ √ µn] + 1 ≤ k ≤ n, 0 otherwise; and hence we obtain (p) lim jk g(xjk) :=   1 for m− [ √ λm] + 1 ≤ j ≤ m,n− [ √ µn] + 1 ≤ k ≤ n, 0 otherwise. int. j. anal. appl. 17 (5) (2019) 715 we obtain that (xjk) is not convergent in (x,g) (g2lim x does not exist) but δλ,µ(b(�)) = 0, that is, g(sλ,µ)lim x = 0. hereby, we conclude that the converse of above theorem 2.4 need not be true in general. � the proof of the following theorem is straightforward and hence omitted. theorem 2.6. let (x,g) be a paranormed space and assume that g(sλ,µ)lim x ′ = ξ′ and g(sλ,µ)lim x ′′ = ξ′′. then (i) g(sλ,µ)lim(x ′ ±x′′) = ξ′ ± ξ′′, (ii) g(sλ,µ)lim αx ′ = cξ′ for c ∈ r. theorem 2.7. let (x,g) be a paranormed space. then, a double sequence x = (xjk) is (λ,µ)-statistically convergent to ξ in (x,g) if and only if there exists a set b = {(jm,kn) : j1 < j2 < ... < jm < ...; k1 < k2 < ... < kn < ...}⊆ n×n with δ(b) = 1 such that glimm,n xjmkn = ξ. proof. suppose that x = (xjk) is g(sλ,µ)-convergent to ξ, that is, g(sλ,µ)lim x = ξ. for s = 1, 2, ..., one writes b(s) = { (m,n) ∈ n×n : g(xjmkn − ξ) ≥ 1 s } and d(s) = { (m,n) ∈ n×n : g(xjmkn − ξ) < 1 s } . then δλ,µ(b(s)) = 0, d(1) ⊃ d(2) ⊃ ... ⊃ d(i) ⊃ d(i + 1) ⊃ ..., (1) and δλ,µ(d(s)) = 1 (s = 1, 2, ...). (2) we need to prove that (xjmkn ) is g2-convergent to ξ for (m,n) ∈ d(s). let us assume, on contrary, that (xjmkn ) is not g2-convergent to ξ. consequently, there is � > 0 such that g(xkn − ξ) ≥ � for infinitely many terms. let us write d(�) = {(m,n) ∈ n×n : g(xjmkn − ξ) < �} and � > 1 s , s ∈ n. then δλ,µ(d(�)) = 0, and by (1), d(s) ⊂ d(�). hence δλ,µ(d(s)) = 0, which contradicts (2) and therefore (xjmkn ) is g2-convergent to ξ. int. j. anal. appl. 17 (5) (2019) 716 conversely, let us assume there exists a set b = {(jm,kn) : j1 < j2 < ... < jm < ...; k1 < k2 < k3 < ... < kn < ...} ⊆ n × n with δ(b) = 1 such that glimn→∞xjmkn = ξ. then there is a positive integer n such that g(xmn − ξ) < � for m,n > n. put b(�) = {(m,n) ∈ n×n : g(xmn − ξ) ≥ �} and b′ = {(jn+1,kn+1), (jn+2,kn+2), ...}. then δλ,µ(b′) = 1 and b(�) ⊆ n × n −b′ which implies that δλ,µ(b(�)) = 0. hence g(sλ,µ)lim x = ξ. � we are now defining the notion of (λ,µ)-statistically cauchy double sequence in a paranormed space and prove that it is equivalent to the notion of (λ,µ)-statistically convergence double sequence. definition 2.8. let (x,g) be a paranormed space. we say that x = (xjk) is (λ,µ)-statistically cauchy double sequence in (x,g), denoted by g(sλ,µ)-cauchy, if for every � > 0 there exist m,n ∈ n such that, for all j,m ≥ m, k,n ≥ n, we have (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : g(xjk −xmn) ≥ �}| = 0. theorem 2.9. let x = (xjk) be a double sequence in a complete paranormed space (x,g). then, x is g(sλ,µ)-convergent iff it is g(sλ,µ)-cauchy. proof. assume that g(sλ,µ)lim x = ξ. then, the set g(�) = {(j,k) ∈ n×n : g(xjk − ξ) ≥ �/2} has (λ,µ)-density zero which yields δλ,µ(g c(�)) = δλ,µ({(j,k) ∈ n×n : g(xjk − ξ) < �}) = 1. suppose (m,n) ∈ gc(�). therefore, g(xmn − ξ) < �/2. now, let h(�) := {(j,k) ∈ n×n : g(xjk −xmn) ≥ �}. we need to show that h(�) ⊂ g(�). let (j,k) ∈ h(�). then g(xjk −xmn) ≥ � and hence g(xjk − ξ) ≥ �/2, i.e., (j,k) ∈ g(�). otherwise, if g(xjk − ξ) < ε then � ≤ g(xjk −xmn) ≤ g(xjk − ξ) + g(xmn − ξ) < � 2 + � 2 = �, which is not possible. thus h(�) ⊂ g(�) and hence δλ,µ(h(�)) = δλ,µ ({(j,k) ∈ n×n : g(xjk −xmn) ≥ �}) = 0. therefore (xjk) is (λ,µ)-statistically cauchy in (x,g). int. j. anal. appl. 17 (5) (2019) 717 conversely, assume that x = (xjk) is g(sλ,µ)-cauchy but not g(sλ,µ)-convergent. then there exist m,n ∈ n such that for all j,m ≥ m, k,n ≥ n, the set a(�) = {(j,k) ∈ n×n : g(xjk −xmn) ≥ �}, has (λ,µ)-density zero, that is, δλ,µ(a(�) = 0 and δλ,µ(e(�)) = 0, where e(�) = { (j,k) ∈ n×n : g(xjk − ξ) < � 2 } , that is, (p) lim m,n 1 λmµn ∣∣∣{(j,k),j ∈ im,k ∈ jn : g(xjk − ξ) < � 2 }∣∣∣ = 0, which yields δλ,µ(e c(�)) = 1. since g(xjk−xmn) ≤ 2g(xjk−ξ) < �, if g(xjk−ξ) < �/2. thus, δλ,µ(ac(�)) = 0 and so δλ,µ(a(�) = 1. this is a contradiction to our assumption that x is g(sλ,µ)-cauchy. hence x is g(sλ,µ)convergent. 3. strong summability for double sequences in (x,g) we give the idea of strong (λ,µ)p-summability in the setting of paranormed space (x,g) and obtain its relation with g(sλ,µ)-convergence. definition 3.1. let (x,g) be a paranormed space and let p be a positive real number. the double sequence x = (xjk) is said to be strongly (λ,µ)p-summable to the limit ξ in (x,g), denoted by xjk −→ ξ[vλ,µ,g]p, if (p) lim m,n 1 λmµn ∑ j∈im ∑ k∈jn (g (xjk − ξ)) p = 0 (0 < p < ∞). theorem 3.2. one has the following: (i) if 0 < p < ∞ and xjk −→ ξ[vλ,µ,g]p, then x = (xjk) is g(sλ,µ)-convergent to ξ. (ii) if x = (xjk) ∈l∞ and g(sλ,µ)-convergent to ξ then xjk −→ ξ[vλ,µ,g]p, where p ∈ (0,∞). proof. (i) assume that xjk −→ ξ[vλ,µ,g]p. then, as m,n →∞, one obtains 0 ←− 1 λmµn ∑ j∈im ∑ k∈jn (g(xjk − ξ))p ≥ 1 λmµn ∑ j∈im (g(xk−ξ))p≥� ∑ k∈jn (g(xk−ξ))p≥� (g(xk − ξ))p ≥ �p λmµn |f(�)|, where f(�) = {j ∈ im,k ∈ jn : (g(xjk − ξ))p ≥ �}. that is, (p) limm,n→∞ 1λmµn |f(�)| = 0 and so δλ,µ(f(�)) = (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : (g(xjk − ξ))p ≥ �}| = 0. thus x = (xjk) is g(sλ,µ)-convergent to ξ. int. j. anal. appl. 17 (5) (2019) 718 (ii) assume that a double sequence x = (xjk) is bounded and g(sλ,µ)-convergent to ξ. let � > 0 be given. then, we have δλ,µ(f(�)) = 0. since x ∈l∞, there is an m > 0 such that g(xjk − ξ) ≤ m. we have 1 λmµn ∑ j∈im ∑ k∈jn (g(xjk − ξ))p = 1 λmµn ∑ j∈im (j,k)/∈f(�) ∑ k∈jn (j,k)/∈f(�) (g(xjk − ξ))p + 1 λmµn ∑ j∈im (j,k)∈f(�) ∑ k∈jn (j,k)∈f(�) (g(xjk − ξ))p. if we take (j,k) /∈ f(�) then 1 λmµn ∑ j∈im (j,k)/∈f(�) ∑ k∈jn (j,k)/∈f(�) (g(xjk − ξ))p < �. on the other and, if (j,k) ∈ f(�), we have 1 λmµn ∑ j∈im (j,k)∈f(�) ∑ k∈jn (j,k)∈f(�) (g(xjk − ξ))p ≤ (sup g(xjk − ξ)) 1 λmµn (|{j ∈ im,k ∈ jn : (g(xjk − ξ))p ≥ �}|) ≤ m λmµn |{j ∈ im,k ∈ jn : (g(xjk − ξ))p ≥ �}|. we see that the right of above inequality tends to zero as m,n → ∞, since δλ,µ(f(�)) = 0. hence, we conclude that xk −→ ξ[vλ,µ,g]p. � remark 3.3. if we choose λm = m and µn = n, then strong (λ,µ)p-summablity in a paranormed space is reduced to the notion of strong p-cesàro summablity for double sequences in the same setup, denoted by [c1,1,g]p. then, we have the following corollary from theorem 3.3. corollary 3.4. let (x,g) be a paranormed space. (i) if p ∈ (0,∞) and xjk −→ ξ[c1,1,g]p, then g(s2)lim x = ξ. (ii) if x = (xjk) ∈l∞ and g(s2)lim x = ξ then xjk −→ ξ[c1,1,g]p (p ∈ (0,∞)). theorem 3.5. if a double x = (xjk) is strongly (λ,µ)p-summable or (λ,µ)-statistically convergent to ξ in (x,g), then there is a convergent double sequence y = (yjk) and a (λ,µ)-statistically null double sequence z = (zjk) such that y = (yjk) is convergent to ξ in pringsheim’s sense, x = y + z and (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : zjk 6= 0}| = 0. (3) moreover, if a double sequence x = (xjk) is bounded, then both (yjk) and (zjk) are bounded. int. j. anal. appl. 17 (5) (2019) 719 proof. it is clear from theorem 3.2 that if a double sequence xjk −→ ξ[vλ,µ,g]p, then it is (λ,µ)-statistically convergent to ξ. let us take s(0) = 0 and choose a strictly increasing sequence s(1) < s(2) < s(3) < ... of positive integers such that 1 λmµn |{(j,k),j ∈ im,k ∈ jn : |xjk − ξ| ≥ l}| < l−1, for m,n > s(l). we are defining y = (yjk) and z = (zjk) as follows: choose zjk = 0 and yjk = xjk if s(0) < j,k < s(1). suppose l ≥ 1 and s(l) < j,k < s(l + 1). we now set yjk =   xjk, zjk = 0 for |xjk − ξ| < l −1, ξ, zjk = xjk − ξ for |xjk − ξ| ≥ l−1. clearly, x = y + z and double sequences y and z are bounded if a double sequence x is bounded. we have to show that y = (yjk) is convergent to ξ in the pringsheim’s sense. for given � > 0, let us choose l such that � > 1/l. we can see that for j,k > s(l), one obtains |yjk − ξ| < � (since |yjk − ξ| = |xjk − ξ| < �) if |xjk − ξ| < l−1 and |yjk − ξ| = |ξ − ξ| = 0 if |xjk − ξ| > l−1. if follows that (yjk) is convergent to ξ in the pringsheim’s sense. it remains to prove that (3) holds. it is enough to prove that if δ > 0 and l ∈ n such that 1/l < δ, then |{(j,k),j ∈ im,k ∈ jn : zjk 6= 0}| < δ ∀ m,n > s(l). as we have seen from the construction that if s(l) < j,k ≤ s(l + 1) then zjk = 0 only if |xjk − ξ| > 1/l. it follows that if s(r) < j,k ≤ s(r + 1), then {(j,k),j ∈ im,k ∈ jn : zjk 6= 0}⊆{(j,k),j ∈ im,k ∈ jn : |xjk − ξ| > 1/r}. consequently, if s(r) < j,k ≤ s(r + 1) and r > l, one obtains 1 λmµn |{(j,k),j ∈ im,k ∈ jn : zjk 6= 0}| ⊆ 1 λmµn |{(j,k),j ∈ im,k ∈ jn : |xjk − ξ| > 1/r}| < 1/r < 1/l < δ. thus, we have the following (p) lim m,n 1 λmµn |{(j,k),j ∈ im,k ∈ jn : zjk 6= 0}| = 0. int. j. anal. appl. 17 (5) (2019) 720 references [1] t. acar and s. a. mohiuddine, statistical (c, 1)(e, 1) summability and korovkin’s theorem, filomat 30(2) (2016), 387-393. [2] a. alotaibi and a. m. alroqi, statistical convergence in a paranormed space, j. inequal. appl. 2012 (2012), 39. [3] a. alotaibi, m. mursaleen, s. a. mohiuddine, korovkin type approximation theorems for σ-convergence of double sequences, j. nonlinear convex anal. 16(1) (2015), 183-192. [4] m. a. alghamdi and m. mursaleen, λ-statistical convergence in paranormed space, abstr. appl. anal. 2013 (2013), article id 264520. [5] f. a. arani, m. e. gordji and s. talebi, statistical convergence of double sequence in paranormed spaces, j. math. comput. sci. 10 (2014), 47-53. [6] c. belen and s. a. mohiuddine, generalized weighted statistical convergence and application, appl. math. comput. 219 (2013), 9821-9826. [7] n. l. braha, h. m. srivastava and s. a. mohiuddine, a korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, appl. math. comput. 228 (2014), 162-169. [8] h. çakalli, on statistical convergence in topological groups, pure appl. math. sci. 43 (1996), 27-31. [9] j. s. connor, the statistical and strong p-cesàro convergence of sequences, analysis 8 (1988), 47-63. [10] o. h. h. edely, s. a. mohiuddine and a. k. noman, korovkin type approximation theorems obtained through generalized statistical convergence, appl. math. lett. 23 (2010), 1382-1387. [11] h. fast, sur la convergence statistique, coll. math. 2 (1951), 241-244. [12] j. a. fridy, on statistical convergence, analysis 5(4) (1985), 301-313. [13] b. hazarika and v. kumar, on asymptotically double lacunary statistical equivalent sequences in ideal context, j. inequal. appl. 2013 (2013), 543. [14] u. kadak and s. a. mohiuddine, generalized statistically almost convergence based on the difference operator which includes the (p,q)-gamma function and related approximation theorems, results math. 73 (2018), 9. [15] e. kolk, the statistical convergence in banach spaces, tartu ul. toime. 928 (1991), 41-52. [16] i. j. maddox, statistical convergence in a locally convex space, math. camb. phil. soc. 104 (1988), 141-145. [17] s. a. mohiuddine, statistical weighted a-summability with application to korovkin’s type approximation theorem, j. inequal. appl. 2016 (2016), 101. [18] s. a. mohiuddine and b. a. s. alamri, generalization of equi-statistical convergence via weighted lacunary sequence with associated korovkin and voronovskaya type approximation theorems, rev. r. acad. cienc. exactas fs. nat., ser. a mat., racsam 113 (3) (2019), 1955-1973. [19] s. a. mohiuddine and m. aiyub, lacunary statistical convergence in random 2-normed spaces, appl. math. inf. sci. 6(3) (2012), 581-585. [20] s. a. mohiuddine, a. alotaibi and m. mursaleen, statistical convergence of double sequences in locally solid riesz spaces, abstr. appl. anal. 2012 (2012), 719729. [21] s. a. mohiuddine, a. asiri and b. hazarika, weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, int. j. gen. syst. 48(5) (2019), 492-506. [22] s. a. mohiuddine and b. hazarika, some classes of ideal convergent sequences and generalized difference matrix operator, filomat 31(6) (2017), 1827-1834 [23] s. a. mohiuddine, b. hazarika and a. alotaibi, on statistical convergence of double sequences of fuzzy valued functions, j. intell. fuzzy syst. 32 (2017), 4331-4342. int. j. anal. appl. 17 (5) (2019) 721 [24] m. mursaleen, λ-statistical convergence, math. slovaca 50 (2000), 111-115. [25] m. mursaleen, c. çakan, s. a. mohiuddine and e. savaş, generalized statistical convergence and statistical core of double sequences, acta math. sin. 26(11) (2010), 2131-2144. [26] m. mursaleen and o. h. h. edely, statistical convergence of double sequences, j. math. anal. appl. 288 (2003), 223-231. [27] m. mursaleen and s. a. mohiuddine, statistical convergence of double sequences in intuitionistic fuzzy normed spaces, chaos solitons fractals 41(5) (2009), 2414-2421. [28] m. mursaleen and s. a. mohiuddine, on lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, j. comput. appl. math. 233 (2009), 142-149. [29] a. pringsheim, zur ttheorie der zweifach unendlichen zahlenfolgen, math. ann. 53 (1900), 289-321. [30] i. j. schoenberg, the integrability of certain functions and related summability methods, amer. math. monthly 66 (1959), 361-375. [31] t. s̆alát, on statistically convergent sequences of real numbers, math. slovaca 30 (1980), 139-150. [32] a. zygmund, trigonometrical series, vol. 5(1935) of monografýas de matemáticas, warszawa-lwow. 1. introduction and preliminaries 2. (,)-statistical convergence in paranormed spaces 3. strong summability for double sequences in (x,g) references international journal of analysis and applications volume 16, number 1 (2018), 137-148 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-137 impulsive diffusion equation on time scales tuba gulsen, shaida saber mawlood sian, emrah yilmaz∗ and hikmet koyunbakan department of mathematics, faculty of science, firat university, 23119, elazig, turkey ∗corresponding author: emrah231983@gmail.com abstract. application of boundary value problems (bvp’s) on an arbitrary time scale t is a fairly new and important subject in mathematics. in this study, we deal with an eigenvalue problem for impulsive diffusion equation with boundary conditions on t . we generalize some noteworthy results about spectral theory of classical diffusion equation into t . also, some eigenfunction estimates of the impulsive diffusion eigenvalue problem are established on t. 1. introduction time scale theory was first considered by stefan hilger [1] in 1988 in his doctoral dissertation under supervision of bernard aulbach [2] to unify the two approaches of dynamic modelling: difference and differential equations. however, similar ideas have been used before and go back at least to the introduction of the rieamann-stieltjes integral which unifies sums and integrals. more specifically, t is an arbitrary, non-empty, closed subset of r. many results related to differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be totally different in nature. because of these reasons, the theory of dynamic equations is an active area of research. the time scale calculus can be applied to any fields in which dynamic processes are described by discrete or continuous time models. so, the calculus of time scale has various applications involving non-continuous domains like modeling of certain bug populations, chemical reactions, phytoremediation of metals, wound healing, maximization problems in received 3rd october, 2017; accepted 8th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 34n05, 34l40, 34l05. key words and phrases. time scales; impulsive diffusion equation; spectral theory. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 137 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-137 int. j. anal. appl. 16 (1) (2018) 138 economics and traffic problems. in recent years, several authors have obtained many important results about different topics on time scales (see [3–7]). although there are many studies in the literature on time scales, very little work has been done about bvp’s. the first studies on these type problems for linear ∆− differential equations on t were fulfilled by chyan, davis, henderson and yin [8] in 1998 and agarwal, bohner and wong [9] in 1999. in [8], the theory of positive operators according to a cone in a banach space is applied to eigenvalue problems related to the second order linear ∆−differential equations on t to prove existence of a smallest positive eigenvalue and then a theorem proved to compare the smallest positive eigenvalue for two problems of that type. in [9], an oscillation theorem is given for sturm-liouville (sl) eigenvalue problem on t with separated boundary conditions and rayleigh’s principle is studied. in 2002, agarwal, bohner and o’regan [10] presented some new existence results for time scale bvp’s on infinite intervals. guseinov [11] investigated some eigenfunction expansions for the simple sl eigenvalue problem on t in 2007. in that paper, the existence of the eigenvalues and eigenfunctions is proved and mean square convergent and uniformly convergent expansions in eigenfunctions are established by guseinov. later, huseynov and bairamov generalized the results of guseinov to the more general eigenvalue problem in 2009 [12]. zhang and ma investigated solvability of sl problems on t at resonance in 2010 [13]. guo zhang and sun [14] gave a variational approach for sl bvp’s on t in 2011. erbe, mert and peterson [15] derived formulas for finding two linearly independent solutions of the sl dynamic equation in 2012. yilmaz, koyunbakan and ic gave some substantial results for diffusion equation on t in 2015 [16]. yaslan [17] proved the existence of positive solutions for second-order impulsive bvp’s on t in 2016. allahverdiev, eryilmaz and tuna [18] considered dissipative sl operators with a spectral parameter under the boundary condition on bounded time scales in 2017. gulsen and yilmaz [19] explained some properties of dirac system on time scales in 2017. as can be seen from the literature, the studies about spectral theory on time scales have focused on sl equation and dirac system. we notice that there isn’t any research about impulsive diffusion equation on time scales according to our survey. hence, this study will be the first related to spectral theory of impulsive diffusion equation on t. our study will give a lead to mathematicians to solve some direct and inverse spectral problems for any types of operators on t. to give basic results, we need to remind some fundamental concepts on time scales. forward and backward jump operators at t ∈ t, for t < sup t are defined as σ(t) = inf{s ∈ t : s > t}, ρ(t) = sup{s ∈ t : s < t}, respectively (supplemented by inf φ = sup t and sup φ = inf t where φ indicates the empty set). these operators are well-defined and map t into t. at the same time, t is said to be left dense, left scattered, right dense and right scattered if ρ(t) = t,ρ(t) < t,σ(t) = t and σ(t) > t, respectively. the distance from an int. j. anal. appl. 16 (1) (2018) 139 arbitrary element t ∈ t to the closest element on the right is called graininess of t and is determined by µ(t) = σ(t) − t. a closed interval on t is defined by [a,b]t = {t ∈ t : a ≤ t ≤ b}, where a and b are fixed points of t with a < b. for standard notions and notations related to time scales theory, we refer to the reference [3]. we also need to explain tk along with the set t to define ∆− derivative of a function. if t has left scattered maximum m, then tk = t−{m}. otherwise, we put tk = t [3]. assume y : t → r is a function and t ∈ tk. then, we define y∆(t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood u = (t−δ,t + δ) ∩t of t for some δ > 0 such that ∣∣[y(σ(t)) −y(s)] −y∆(t) [σ(t) −s]∣∣ ≤ ε |σ(t) −s| , for all s ∈ u. we call y∆(t) as the delta (∆) or hilger derivative of y at t. the function y : t → r is called rd−continuous if it is continuous at right dense points on t and its left sided limits exist (finite) at left dense points on t. the set of all rd−continuous functions on t is denoted by crd(t). the set of all functions that are ∆−differentiable and whose ∆−derivative is rd−continuous on t is denoted by c1rd(t). however, y is regulated if its right sided limits exist (finite) at all right dense points on t and its left sided limits exist (finite) at all left dense points on t. let y be a regulated function on t. the indefinite ∆−integral of y is denoted by ∫ y(t)∆t = y (t) + c, where c is an arbitrary constant and y is pre-antiderivative of y. finally, definite ∆− integral of y is defined by ∫ s r y(t)∆t = y (s) −y (r), for all r,s ∈ t. here, we want to state some major spectral properties of impulsive diffusion eigenvalue problem of the form ly = −y∆∆(t) + {q(t) + 2λp(t)}yσ(t) = λ2$(t)yσ(t), t ∈ [ρ(a),b] ∩t, (1.1) with the boundary conditions y∆(ρ(a)) = 0, (1.2) αy(b) + βy∆(b) = 0, (1.3) int. j. anal. appl. 16 (1) (2018) 140 where λ is a spectral parameter and $(t) =   1, ρ(a) < t < b 2 ϑ2, b 2 < t < b , 0 < ϑ 6= 1. we suppose that p,q : [ρ(a),b] ∩ t → r are continuous functions; a,b ∈ t with a < b, yσ = y(σ) and( α2 + β2 ) 6= 0. y(t) is known as eigenfunction of the problem (1.1)-(1.3). by taking t = r in (1.1), we obtain below classical impulsive diffusion equation (or quadratic pencil of schrödinger equation) −y′′(t) + {q(t) + 2λp(t)}y(t) = λ2$(t)y(t). (1.4) (1.4) is reduced to sturm-liouville equation when p(t) = 0 and $(t) = 1 [20]. now, we give some information related to the physical meaning and historical development of diffusion equation in classical spectral theory. the problem of defining the interactions among colliding particles is an attractive issue in mathematical physics. in many cases, a definition can be carried out through a well known theoretical model. particularly, one is interested in collisions of two spinless particles, and it is assumed that the s−wave binding energies and s−wave scattering matrix are completely known from collision experiments. s−wave schrödinger equation with a radial static potential v can be defined by y′′ + [e −v (x)] y = 0, x ≥ 0, (1.5) where v depends on energy and has the following energy dependence v (x,e) = u(x) + 2 √ eq(x). (1.6) u(x) and q(x) are complex-valued functions. (1.5) reduces to the klein-gordon s−wave equation with the static potential q(x), for a particle of zero mass and the energy √ e with the supplementary condition u(x) = −q2(x) [21]. the klein-gordon equation is an attempt to marry special relativity and quantum mechanics. it is derived from einstein’s energy equation where the energy and momentum terms are replaced with quantum mechanical operators. the klein-gordon equation describes particles of spin−0 hence it describes the higgs boson. however, it is not positive definite because it is a second order differential equation and when considering its interpretation from within a quantum mechanical framework we run into trouble namely negative probability densities, which are not allowed in quantum mechanics [22]. jaulent and jean [21] explained the actual background of diffusion operator in 1972. maksudov and guseinov [23] considered the spectral theory of diffusion operator in 1979. then, gasymov and guseinov [24] determined the diffusion operator from spectral data in 1981. fragela [25] studied on this operator with integral boundary conditions in 1984. maksudov and guseinov [26] gave some solutions of the inverse scattering problem for diffusion operator on whole axis in 1986. bairamov, çakar and celebi [27] obtained some results for diffusion operator with spectral singularities, disctrete spectrum and principal functions int. j. anal. appl. 16 (1) (2018) 141 in 1997. moreover, inverse nodal problem was first considered by koyunbakan [28] in 2006 for diffusion operator. koyunbakan [29] obtained uniqueness results and reconstruction formulas in 2007 with panakhov. koyunbakan and yilmaz [30] expressed reconstruction formulas for the k−th derivatives of the first potential function q in diffusion equation by using nodal points in 2008. yang [31] extended their results to the second potential function p in 2010. yang and zettl [32] solved half-inverse diffusion problem in 2012. we can replicate these examples. there are many other researches on diffusion operator in literature by many authors (see [33–42]). this study is collocated as follows: in section 2, we prove some basic theorems for impulsive diffusion equation on t. using some methods, we get asymptotic estimate of eigenfunction for the problem (1.1)-(1.3) in section 3. in section 4, we give a conclusion to summarize our study. 2. some spectral properties of impulsive diffusion equation on time scales it is well known that the problem (1.1)-(1.3) has only real, simple eigenvalues and its eigenfunctions are orthogonal when t = r [24]. the below results will generalize this basic consequences to an arbitrary time scale. theorem 2.1. the eigenvalues of the problem (1.1)-(1.3) are all simple. proof. let λ ∈ r be spectral parameter and y1(t), y2(t) be eigenfunctions of (1.1)-(1.3). then, we have ly1 = −y∆∆1 (t) + {q(t) + 2λp(t)}y σ 1 (t) = λ 2$(t)yσ1 (t), ly2 = −y∆∆2 (t) + {q(t) + 2λp(t)}y σ 2 (t) = λ 2$(t)yσ2 (t). multiplying these equations by yσ2 and y σ 1 respectively and substracting, we get ( y1y ∆ 2 −y2y ∆ 1 )∆ = 0. (2.1) by taking ∆−integral of (2.1) on [ρ(a),b], we conclude w [y1,y2] ∣∣∣bρ(a) = c. using the conditions (1.2), (1.3), one can easily obtain that c = 0. then( y1 y2 )∆ = 0. the last equality means that y1 = ky2 (k constant) and it implies y1 and y2 are linearly dependent eigenfunctions of (1.1)-(1.3). this completes the proof. theorem 2.2. all eigenvalues of the problem (1.1)-(1.3) are all real. int. j. anal. appl. 16 (1) (2018) 142 proof. let λ0 be a complex eigenvalue and y(t) be an eigenfunction of the problem (1.1)-(1.3) related to λ0. then, we obtain { y∆y −y∆y }∆ = y∆∆yσ −y∆∆yσ = { q(t) + 2λ0p(t) −λ20$(t) } yσyσ − { q(t) + 2λ0p(t) −λ 2 0$(t) } yσyσ = ( λ 2 0 −λ 2 0 ) $(t)yσyσ − 2 ( λ0 −λ0 ) p(t)yσyσ = ( λ 2 0 −λ 2 0 ) $(t) |yσ(t)|2 − 2 ( λ0 −λ0 ) p(t) |yσ(t)|2 . if we take ∆−integral of the last equality from ρ(a) to b, we get ( λ 2 0 −λ 2 0 ) b∫ ρ(a) $(t) |yσ(t)|2 ∆t− 2 ( λ0 −λ0 ) b∫ ρ(a) p(t) |yσ(t)|2 ∆t = y∆(b)y(b) −y∆(b)y(b) −y∆(ρ(a))y(ρ(a)) + y∆(ρ(a))y(ρ(a)) = 0, by considering the conditions (1.2), (1.3). so, we have ( λ0 + λ0 ) b∫ ρ(a) $(t) |yσ(t)|2 ∆t− 2 b∫ ρ(a) p(t) |yσ(t)|2 ∆t = 0 ⇒ yσ(t) = 0 , where ( λ0 + λ0 ) $(t) − 2p(t) > 0 for λ0 6= λ0. this is a contradiction. hence, eigenvalues of the problem (1.1)-(1.3) are all real. theorem 2.3. let y1,y2 ∈ c1rd(t) be the eigenfunctions of the problem (1.1)-(1.3). then, a) (ly1) y σ 2 − (ly2) yσ1 = w ∆(y1,y2) on [ρ(a),b] ∩t. b) < ly1,y σ 2 > − < ly2,yσ1 >= w(y1,y2)(b) −w(y1,y2)(ρ(a)), where w(y1,y2) = y1y ∆ 2 −y2y∆1 is the wronskian of the functions y1 and y2. proof. a) definition of the wronskian and product rule for ∆−derivative give following result: w ∆(y1,y2) = y ∆∆ 2 y σ 1 −y ∆∆ 1 y σ 2 = −yσ1 ( −y∆∆2 + (q(t) + 2λp(t)) y σ 2 ) + yσ2 ( −y∆∆1 + (q(t) + 2λp(t)) y σ 1 ) = (ly1) y σ 2 − (ly2) y σ 1 . int. j. anal. appl. 16 (1) (2018) 143 b) by using definition of wronskian and inner product on the set of so-called rd−continuous functions, we have < ly1,y σ 2 > − < ly2,y σ 1 >= b∫ ρ(a) [(ly1) y σ 2 − (ly2) y σ 1 ] ∆t = b∫ ρ(a) {( −y∆∆1 + (q(t) + 2λp(t)) y σ 1 ) yσ2 − ( −y∆∆2 + (q(t) + 2λp(t)) y σ 2 ) yσ1 } ∆t = − b∫ ρ(a) { y∆∆1 y σ 2 −y ∆∆ 2 y σ 1 } ∆t = b∫ ρ(a) {w(y1,y2)} ∆ ∆t = w(y1,y2)(b) −w(y1,y2)(ρ(a)). hence, this completes the proof. in classical spectral theory, these equalities are known as lagrange’s identity and green’s formula, respectively. theorem 2.4. the equality y (t,λ) ∂ ∂λ y∆ (t,λ) −y∆ (t,λ) ∂ ∂λ y (t,λ) = −2λ t∫ ρ(a) $(τ) (yσ (τ,λ)) 2 ∆τ + 2 t∫ ρ(a) p(τ) (yσ (τ,λ)) 2 ∆τ, holds for all t ∈ [ρ(a),b] ∩t and λ ∈ r. proof. let γ, λ ∈ r with γ 6= λ. then, { y (t,γ) y∆ (t,λ) −y∆ (t,γ) y (t,λ) }∆ = yσ (t,γ) y∆∆ (t,λ) −y∆∆ (t,γ) yσ (t,λ) = yσ (t,γ) { q(t) + 2λp(t) −λ2$(t) } yσ (t,λ) −yσ (t,λ) { q(t) + 2γp(t) −γ2$(t) } yσ (t,γ) = ( γ2 −λ2 ) $(t)yσ(t,λ)yσ(t,γ) − 2 (γ −λ) p(t)yσ(t,λ)yσ(t,γ). dividing both sides of above equality by λ−γ and taking limit as γ → λ, we have lim γ→λ { y (t,γ) y∆ (t,λ) −y∆ (t,γ) y (t,λ) }∆ λ−γ = − lim γ→λ {(γ + λ) $(t)yσ(t,λ)yσ(t,γ) − 2p(t)yσ(t,λ)yσ(t,γ)} ⇒ { y (t,λ) ∂ ∂λ y∆ (t,λ) −y∆ (t,λ) ∂ ∂λ y (t,λ) }∆ = −2λ$(t) (yσ (t,λ))2 + 2p(t) (yσ (t,λ))2 . int. j. anal. appl. 16 (1) (2018) 144 by taking ∆−integral of the last equality from ρ(a) to t, we get t∫ ρ(a) { y (τ,λ) ∂ ∂λ y∆ (τ,λ) −y∆ (τ,λ) ∂ ∂λ y (τ,λ) }∆ ∆τ = −2λ t∫ ρ(a) $(τ) (uσ (τ,λ)) 2 ∆τ + 2 t∫ ρ(a) p(τ) (uσ (τ,λ)) 2 ∆τ. since y (ρ(a),λ) = 1 and y∆ (ρ(a),λ) = 0, it yields ∂ ∂λ y (ρ(a),λ) = 0 and ∂ ∂λ y∆ (ρ(a),λ) = 0. finally, after some computations, we obtain y (t,λ) ∂ ∂λ y∆ (t,λ) −y∆ (t,λ) ∂ ∂λ y (t,λ) = −2λ t∫ ρ(a) $(τ) (uσ (τ,λ)) 2 ∆τ + 2 t∫ ρ(a) p(τ) (uσ (τ,λ)) 2 ∆τ. so, the proof is complete. theorem 2.5. the eigenfunctions y1(t,λ1) and y2(t,λ2) of the problem (1.1)-(1.3) corresponding to distinct eigenvalues λ1 and λ2 are orthogonal , i.e (λ1 + λ2) b∫ ρ(a) $(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t− 2 b∫ ρ(a) p(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t = 0. proof. let us use following equality { y∆1 (t,λ1)y2(t,λ2) −y ∆ 2 (t,λ2)y1(t,λ1) }∆ = {( λ22 −λ 2 1 ) $(t) − 2 (λ2 −λ1) p(t) } yσ1 (t,λ1)y σ 2 (t,λ2). taking ∆−integral of the last equality from ρ(a) to b, we get b∫ ρ(a) { y∆1 (t,λ1)y2(t,λ2) −y ∆ 2 (t,λ2)y1(t,λ1) }∆ ∆t = ( λ22 −λ 2 1 ) b∫ ρ(a) $(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t − 2 (λ2 −λ1) b∫ ρ(a) p(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t = y∆1 (b,λ1)y2(b,λ2) −y ∆ 2 (b,λ2)y1(b,λ1) −y∆1 (ρ(a),λ1)y2(ρ(a),λ2) + y ∆ 2 (ρ(a),λ2)y1(ρ(a),λ1) = 0, and (λ1 + λ2) b∫ ρ(a) $(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t− 2 b∫ ρ(a) p(t)yσ1 (t,λ1)y σ 2 (t,λ2)∆t = 0, for λ1 6= λ2. then, it shows that the eigenfunctions y1(t,λ1) and y2(t,λ2) corresponding to distinct eigenvalues are always orthogonal. int. j. anal. appl. 16 (1) (2018) 145 3. main results in this section, we acquire asymptotic estimate for the eigenfunction y(t,λ) of the problem (1.1)-(1.3) on t whose all points are right dense. theorem 3.1. the eigenfunction y(t,λ) and its ∆−derivative of the problem (1.1)-(1.3) have the following asymptotic estimates; y(t,λ) = cosλ (t,ρ(a)) − 1 λ t∫ ρ(a) (q(s) + 2λp(s)) y(s,λ) sinλ(t,s)∆s, and y∆(t,λ) = −λ sinλ (t,ρ(a)) − t∫ ρ(a) (q(s) + 2λp(s)) y(s,λ) cosλ(t,s)∆s, where t ∈ [ρ(a),b] ∩t. proof. by equation (1.1), we can write 1 λ t∫ ρ(a) (q(s) + 2λp(s)) y(s,λ) sinλ(t,s)∆s = 1 λ t∫ ρ(a) sinλ(t,s) ( y∆∆(s,λ) + λ2$(s)y(s,λ) ) ∆s (3.1) = 1 λ t∫ ρ(a) sinλ(t,s)y ∆∆(s,λ)∆s + λ t∫ ρ(a) sinλ(t,s)$(s)y(s,λ)∆s. after using integration by parts twice for the first integral in (3.1), we get 1 λ t∫ ρ(a) sinλ(t,s)y ∆∆(s,λ)∆s = 1 λ ( sinλ(t,s)y ∆(s,λ) )∣∣∣tρ(a) − 1λ t∫ ρ(a) λ cosλ(t,s)y ∆(s,λ)∆s = 1 λ sinλ(t,t)y ∆(t,λ) − 1 λ sinλ(t,ρ(a))y ∆(ρ(a),λ) − (cosλ(t,s)y(s,λ)) ∣∣∣tρ(a) + t∫ ρ(a) −λ sinλ(t,s)y(s,λ)∆s = − 1 λ sinλ(t,ρ(a))y ∆(ρ(a),λ) −y(t,λ) + cosλ(t,ρ(a))y(ρ(a),λ) −λ t∫ ρ(a) sinλ(t,s)y(s,λ)∆s, and by the conditions (1.2) and (1.3), we get 1 λ t∫ ρ(a) sinλ(t,s)y ∆∆(s,λ)∆s = −y(t,λ) + cosλ(t,ρ(a)) −λ t∫ ρ(a) sinλ(t,s)y(s,λ)∆s. (3.2) therefore, considering (3.1) and (3.2) we have int. j. anal. appl. 16 (1) (2018) 146 y(t,λ) = cosλ (t,ρ(a)) − 1 λ t∫ ρ(a) (q(s) + 2λp(s)) y(s,λ) sinλ(t,s)∆s, (3.3) where $(t) = 1. differentiating (3.3) with respect to t, we obtain y∆(t,λ) = −λ sinλ (t,ρ(a)) − 1 λ sinλ (σ(t), t) (q(t) + 2λp(t)) y(t,λ) − 1 λ t∫ ρ(a) w(s)y(s,λ)λ cosλ (t,ρ(a)) ∆s, or since sinλ (σ(t), t) = sinλ (t,t) = 0, we have y∆(t,λ) = −λ sinλ (t,ρ(a)) − t∫ ρ(a) (q(s) + 2λp(s)) y(s,λ) cosλ(t,s)∆s. (3.4) this completes proof. 4. conclusion impulsive type diffusion eigenvalue problems appear in many fields of mathematical physics as quantum field theory, relativistics physics. because of this importance, we notice that obtaining the solutions of these type problems on time scales will attract attention. in fact, our main goal is to obtain the asymptotic estimates for eigenvalues of this problem on t. during the next period, we want to study ambarzumyan’s theorem and inverse problem for different type of operators on t. we believe that this study will be an important step for this purpose. references [1] s. hilger, ein maßkettenkalkül vnit anwendung auf zentruvnsvnannigfaltigkeiten, [ph.d. thesis], universitüt würzburg, 1988. [2] b. aulbach, s. hilger, a unified approach to continuous and discrete dynamics, qualitative theory of differential equations, szeged (1988); colloq. math. soc. jános bolyai, north-holland, amsterdam, 53 (1990), 37–56. [3] m. bohner and a. peterson, dynamic equations on time scales: an introduction with applications, boston (ma), birkhäuser, boston inc, 2001. [4] f. m. atici, g. sh. guseinov, on green’s functions and positive solutions for boundary value problems on time scales, j. comput. appl. math. 141 (2002), 75–99. [5] e. bairamov, y. aygar and t. koprubasi, the spectrum of eigenparameter-dependent discrete sturm–liouville equations, j. comput. appl. math. 235 (16) (2011), 4519-4523. [6] y. aygar and m. bohner, on the spectrum of eigenparameter-dependent quantum difference equations, appl. math. inf. sci. 9 (4) (2015), 1725-1729. [7] g. sh. guseinov, self-adjoint boundary value problems on time scales and symmetric green’s functions, turk. j. math. 29 (2005), 365–380. [8] c. j. chyan, j. m. davis, j. henderson, w. k. c. yin, eigenvalue comparisons for differential equations on a measure chain, electron. j. diff. equ. 35 (1998), 1-7. int. j. anal. appl. 16 (1) (2018) 147 [9] r. p. agarwal, m. bohner and p. j. y. wong, sturm–liouville eigenvalue problems on time scales, appl. math. comput. 99 (1999), 153–166. [10] r. p. agarwal, m. bohner, d. o’regan, time scale boundary value problems on infinite intervals, j. comput. appl. math. 141 (2002), 27–34. [11] g. sh. guseinov, eigenfunction expansions for a sturm–liouville problem on time scales, int. j. difference equ. 2 (2007), 93–104. [12] a. huseynov and e. bairamov, on expansions in eigenfunctions for second order dynamic equations on time scales, nonlinear dyn. syst. theory 9 (2009), 77-88. [13] y. zhang and l. ma, solvability of sturm-liouville problems on time scales at resonance, j. comput. appl. math. 233 (2010), 1785-1797. [14] q. g. zhang, h. r. sun, variational approach for sturm-liouville boundary value problems on time scales, j. appl. math. comput. 36 (1-2) (2011), 219–232. [15] l. erbe, r. mert and a. peterson, spectral parameter power series for sturm–liouville equations on time scales, appl. math. comput. 218 (2012), 7671-7678. [16] e. yilmaz, h. koyunbakan and u. ic, some spectral properties of diffusion equation on time scales. contemp. anal. appl. math. 3 (2015), 238-246. [17] i̇. yaslan, existence of positive solutions for second-order impulsive boundary value problems on time scales, mediterr. j. math. 13 (4) (2016), 1613–1624. [18] b. p. allahverdiev, a. eryilmaz and h. tuna, dissipative sturm-liouville operators with a spectral parameter in the boundary condition on bounded time scales, electron. j. diff. equ. 95 (2017), 1–13. [19] t. gulsen and e. yilmaz, spectral theory of dirac system on time scales, appl. anal. 96 (2017), 2684-2694. [20] y. cakmak and s. isık, half inverse problem for the impulsive diffusion operator with discontinuous coefficient, filomat 30 (1) (2016), 157-168. [21] m. jaulent and c. jean, the inverse s-wave scattering problem for a class of potentials depending on energy, commun. math. phys. 28 (3) (1972), 177-220. [22] a. wazwaz, partial differential equations: methods and applications, balkema publishers, leiden, 2002. [23] f. g. maksudov, m. m. guseinov, a quadratic pencil of operators in the presence of a continuous spectrum, (russian) dokl. akad. nauk azerbaidzhan ssr 35 (1) (1979), 9–13. [24] m. g. gasymov and g. sh. guseinov, determination of a diffusion operator from the spectral data, dokl. akad. nauk azerbaijan sssr 37 (2) (1981), 19-23. [25] a. fragela, quadratic pencils of differential operators with integral boundary conditions, (russian) differential equations and their applications (russian), 50–52, moskov. gos. univ., moscow, 1984. [26] f. g. maksudov, g. sh. guseinov, on the solution of the inverse scattering problem for the quadratic bundle of the one-dimensional schrödinger operators on the whole axis, (russian) dokl. akad. nauk sssr 289 (1) (1986), 42–46. [27] e. bairamov, ö. çakar and a. o. çelebi, quadratic pencil of schrödinger operators with spectral singularities: discrete spectrum and principal functions, j. math. anal. appl. 216 (1) (1997), 303-320. [28] h. koyunbakan, a new inverse problem for the diffusion operator, appl. math. lett. 19 (10) (2006), 995-999. [29] h. koyunbakan, e. s. panakhov, a uniqueness theorem for inverse nodal problem, inverse probl. sci. eng. 5 (6) (2007), 517-524. [30] h. koyunbakan and e. yilmaz, reconstruction of the potential function and its derivatives for the diffusion operator, z. nat. forsch. a: phys. sci. 63 (3-4) (2008), 127-130. int. j. anal. appl. 16 (1) (2018) 148 [31] c. f. yang, reconstruction of the diffusion operator from nodal data, z. nat. forsch. a: phys. sci. 65 (1-2) (2010), 100-106. [32] c. f. yang and a. zettl, half inverse problems for quadratic pencils of sturm-liouville operators, taiwanese j. math. 16 (5) (2012), 1829-1846. [33] r. hryniv and n. pronska, inverse spectral problems for energy-dependent sturm-liouville equations, inverse probl. 28 (8) (2012), 085008. [34] s. a. buterin and c. t. shieh, inverse nodal problem for differential pencils, appl. math. lett. 22 (2009), 1240-1247. [35] a. b. yakhshimuratov, o. r. allaberganov, the inverse problem for a quadratic pencil of sturm-liouville operators with a periodic potential on a half-axis, (russian) uzbek. mat. zh. 3 (2006), 96–107. [36] i. m. guseinov and i. m. nabiev, the inverse spectral problem for pencils of differential operators, sb. math. 198 (11) (2007), 1579-1598. [37] h. koyunbakan, inverse problem for a quadratic pencil of sturm-liouville operator, j. math. anal. appl. 378 (2) (2011), 549–554. [38] y. p. wang, the inverse problem for differential pencils with eigenparameter dependent boundary conditions from interior spectral data, appl. math. lett. 25 (7) (2012), 1061-1067. [39] r. kh. amirov, a. nabiev, inverse problems for the quadratic pencil of the sturm-liouville equations with impulse, abstr. appl. anal. 2013 (2013), art. id 361989, 10 pp [40] l. k. sharma, p. v. luhanga and s. chimidza, potentials for the klein-gordon and dirac equations, chiang mai j. sci. 38 (4) (2011), 514-526. [41] k. chadan, d. colton, l. paivarinta and w. rundell, an introduction to inverse scattering and inverse spectral problems, siam, philadelphia, pa, 1997. [42] a. d. orujov, on the spectrum of the quadratic pencil of differential operators with periodic coefficients on the semi-axis, bound. value probl. 2015 (2015), art. id 117, 16 pp. 1. introduction 2. some spectral properties of impulsive diffusion equation on time scales 3. main results 4. conclusion references int. j. anal. appl. (2022), 20:29 applications of spherical fuzzy sets in ternary semigroups wasitthirawat krailoet1, ronnason chinram1,∗ montakarn petapirak1, aiyared iampan2 1division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 2department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand ∗corresponding author: ronnason.c@psu.ac.th abstract. in this paper, we introduce the notions of spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups by using the concepts of ternary subsemigroups and ideals in ternary semigroups. we investigate their properties. moreover, we study roughness of spherical fuzzy ideals in ternary semigroups. 1. introduction the theory of ternary algebraic system was investigated by lehmer [8] in 1932, but earlier such structures were studied by kasner [5] who gave the idea of n-ary algebras. furthermore, the ideal theory in ternary semigroups was established by sioson [12]. in 1965, the notion of fuzzy sets was initiated by zadeh [14]. the fuzzy set is an extension of classical sets and represented by using a generalization of the indicator of classical sets that is called a membership function. later, the concept of fuzzy set was applied to study in many algebraic structures. in 1981, kuroki [6] provided some properties of fuzzy ideals. in 2013, iampan [4] gave the definition and characterized the properties of ideal extensions in ternary semigroups. after the introduction of ordinary fuzzy sets, the concept of rough sets was given by pawlak [10] in 1982 which is defined depending on some equivalence relation on a universal finite set. the combination of theories of fuzzy sets and rough sets has been discussed in many research papers through all the years until 1990, when dubois and prade [3] proposed the notion of received: may 10, 2022. 2010 mathematics subject classification. 03e72. key words and phrases. ideals; fuzzy ideals; rough fuzzy sets; spherical fuzzy sets; rough ideals; rough spherical fuzzy ideals; ternary semigroups. https://doi.org/10.28924/2291-8639-20-2022-29 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-29 2 int. j. anal. appl. (2022), 20:29 rough fuzzy sets. in 2009, petchkhaew and chinram [11] studied fuzzy, rough and rough fuzzy ternary subsemigroups (left ideals, right ideals, lateral ideals, ideals) of ternary semigroups. later, in 2012, kar and sarkar [7] focused on studying fuzzy ideals of ternary semigroups and their related properties. in 2016, wang and zhan [13] established the rough semigroups and the rough fuzzy semigroups based on fuzzy ideals. in 2019, ashraf et al. [1] introduced the notion of spherical fuzzy set, which is a generalization of the picture fuzzy sets, intuitionistic fuzzy sets and pythagorean fuzzy sets when the degree of abstinence is involved, as it provides enlargement of the space of degrees of truthfulness (membership), abstinence (hesitancy) and falseness (non-membership). recently, in 2020, chinram and panityakul [2] introduced rough pythagorean fuzzy ideals in ternary semigroups and gave some remarkable properties. our aim of this paper is to study spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups by using the concepts of ternary subsemigroups and ideals in ternary semigroups. moreover, we study roughness of spherical fuzzy sets and spherical fuzzy ideals in ternary semigroups. 2. preliminaries in this section, we shall recall some basic definitions that will be used in this paper. 2.1. ternary semigroups. a non-empty set t together with a ternary operation, called ternary multiplication, denoted by juxtaposition, is said to be a ternary semigroup if (abc)de = a(bcd)e = ab(cde) for all a,b,c,d,e ∈ t. for any three non-empty subsets a,b and c of a ternary semigroup t , a product abc is defined by abc = {abc | a ∈ a,b ∈ b and c ∈ c}. example 2.1. (1) the following example (banach’s example) shows that a ternary semigroup does not necessarily reduce an ordinary semigroup. let t = {−i,0, i} be a ternary semigroup under ternary multiplication over c. we obtain that t is not a semigroup under multiplication over c. (2) let z− be the set of all negative integers. then z− is a ternary semigroup under ternary multiplication over z. we obtain that z− is not a semigroup under multiplication over z. (3) the set of all odd permutation is a ternary semigroup under ternary composition. it is not a semigroup under composition. a non-empty subset s of a ternary semigroup t is called a ternary subsemigroup of t if s3 ⊆ s. let i be a non-empty subset of a ternary semigroup t . then i is called a left ideal of t if tti ⊆ i, a lateral ideal of t if tit ⊆ i and a right ideal of t if itt ⊆ i. a non-empty subset i of a ternary semigroup t is called an ideal of t if i is a left ideal, a lateral ideal and a right ideal of t. an ideal i of a ternary semigroup t is called a proper ideal if i 6= t . int. j. anal. appl. (2022), 20:29 3 2.2. fuzzy sets. a fuzzy subset of a set s is a function s × s → [0,1]. let f and g be any two fuzzy subsets of any set s. (1) f ⊆ g if f (a)≤ g(a) for all a ∈ s. (2) (f ∩g)(a)=min{f (a),g(a)} for all a ∈ s. (3) (f ∪g)(a)=max{f (a),g(a)} for all a ∈ s. a fuzzy subset f of a ternary semigroup t is called a fuzzy ternary subsemigroup of t if f (xyz)≥min{f (x), f (y), f (z)} for all x,y,z ∈ t. a fuzzy subset f of t is called a fuzzy left ideal of t if f (xyz) ≥ f (z) for all x,y,z ∈ t , a fuzzy lateral ideal of t if f (xyz) ≥ f (y) for all x,y,z ∈ t and a fuzzy right ideal of t if f (xyz) ≥ f (x) for all x,y,z ∈ t . a fuzzy subset f of a ternary semigroup t is called a fuzzy ideal of t if it is a fuzzy left ideal, a fuzzy lateral ideal and a fuzzy right ideal of t, i.e., f (xyz)≥max{f (x), f (y), f (z)} for all x,y,z ∈ t. for any three fuzzy sets f1, f2 and f3 of a ternary semigroup t. the product f1◦ f2◦ f3 of f1, f2 and f3 is defined by (f1◦ f2◦ f3)(y)=  supy=y1y2y3 min{f1(y1), f2(y2), f3(y3)} if y ∈ t 3, 0 otherwise. it is obvious that the product f1◦ f2◦ f3 of fuzzy subsets f1, f2 and f3 of a ternary semigroup t is also a fuzzy subset of t . let f(t) be the set of all fuzzy subsets of a ternary semigroup t. then f(t) is a ternary semigroup under this product. 2.3. spherical fuzzy sets. let s be a universal set. a spherical fuzzy set on s s := {< x,µs(x),ηs(x),νs(x) >| x ∈ s} where µs : s → [0,1], ηs : s → [0,1] and νs : s → [0,1] represent the degree of membership, the degree of hesitancy and the degree of non-membership of x ∈ s with the condition 0 ≤ (µs(x))2 + (ηs(x)) 2 +(νs(x)) 2 ≤ 1. we may also denote a spherical fuzzy set s by s =(µs,ηs,νs). example 2.2. let f be any fuzzy subset of a set s. let µs : s → [0,1], ηs : s → [0,1] and νs : s → [0,1] be defined by µs(x)= f (x),ηs(x)=0 and νs(x)=1− f (x). then s := {< x,µs(x),ηs(x),νs(x) >| x ∈ s} is a spherical fuzzy set on s. let s1 = (µs1,ηs1,νs1) and s2 = (µs2,ηs2,νs2) be any two spherical fuzzy set of a universal set s. we say that s1 ⊆s2 if and only if µs1(x)≤ µs2(x),ηs1(x)≤ ηs1(x) and νs1(x)≥ νs1(x) for all x ∈ s. 4 int. j. anal. appl. (2022), 20:29 3. main results 3.1. spherical fuzzy ideals in ternary semigroups. we define spherical fuzzy ternary subsemigroups and spherical fuzzy ideals in ternary semigroups as follows: definition 3.1. a spherical fuzzy set s =(µs,ηs,νs) on a ternary semigroup t is called a spherical fuzzy ternary subsemigroup of t if, for all a,b,c ∈ t (1) µs(abc)≥min{µs(a),µs(b),µs(c)}, (2) ηs(abc)≥min{ηs(a),ηs(b),ηs(c)}, (3) νs(abc)≤max{νs(a),νs(b),νs(c)}. definition 3.2. a spherical fuzzy set s =(µs,ηs,νs) on a ternary semigroup t is called (1) a spherical fuzzy left ideal of t if for all a,b,c ∈ t , µs(abc)≥ µs(c), ηs(abc)≥ ηs(c) and νs(abc)≤ νs(c), (2) a spherical fuzzy lateral ideal of t if for all a,b,c ∈ t, µs(abc)≥ µs(b), ηs(abc)≥ ηs(b) and νs(abc)≤ νs(b), (3) a spherical fuzzy right ideal of t if for all a,b,c ∈ t, µs(abc)≥ µs(a), ηs(abc)≥ ηs(a) and νs(abc)≤ νs(a), (4) a spherical fuzzy ideal of t if for all a,b,c ∈ t , µs(abc)≥max{µs(a),µs(b),µs(c)}, ηs(abc)≥max{ηs(a),ηs(b),ηs(c)} and νs(abc)≤min{νs(a),νs(b),νs(c)}. next, we define the product of three spherical fuzzy sets. definition 3.3. let s1, s2 and s3 be any three spherical fuzzy sets on a ternary semigroup t. the product s1◦s2◦s3 of s1, s2 and s3 is defined by s1◦s2◦s3 =((µs1◦µs2◦µs3),(ηs1◦ηs2◦ηs3),(νs1◦νs2◦νs3)) where (µs1◦µs2◦µs3)(x)=   sup x=abc min{µs1(a),µs2(b),µs3(c)}, if x ∈ t 3; 0, otherwise, (ηs1◦ηs2◦ηs3)(x)=   sup x=abc min{ηs1(a),ηs2(b),ηs3(c)}, if x ∈ t 3; 0, otherwise, int. j. anal. appl. (2022), 20:29 5 and (νs1◦νs2◦νs3)(x)=   inf x=abc max{νs1(a),νs2(b),νs3(c)}, if x ∈ t 3; 1, otherwise. theorem 3.1. let s =(µs,ηs,νs) be a spherical fuzzy set on a ternary semigroup t . then s is a spherical fuzzy ternary subsemigroup of t if and only if s◦s◦s ⊆s. proof. assume that s is a spherical fuzzy ternary subsemigroup of t . let x ∈ t . if x /∈ t3, we obtain that (µs◦µs◦µs)(x)=0≤ µs(x), (ηs◦ηs◦ηs)(x)=0≤ ηs(x) and (νs◦νs◦νs)(x)=1≥ νs(x). now, assume that x ∈ t3, we obtain that (µs◦µs◦µs)(x)= sup x=abc min{µs(a),µs(b),µs(c)}≤ sup x=abc µs(abc)= µs(x), (ηs◦ηs◦ηs)(x)= sup x=abc min{ηs(a),ηs(b),ηs(c)}≤ sup x=abc ηs(abc)= ηs(x) and (νs◦νs◦νs)(x)= inf x=abc max{νs(a),νs(b),νs(c)}≥ inf x=abc νs(abc)= νs(x). hence s◦s◦s ⊆s. conversely, let a,b,c ∈ t . µs(abc) ≥ (µs◦µs◦µs)(abc) = sup abc=x1x2x3 min{µs(x1),µs(x2),µs(x3)} ≥ min{µs(a),µs(b),µs(c)}, ηs(abc) ≥ (ηs◦ηs◦ηs)(abc) = sup abc=x1x2x3 min{ηs(x1),ηs(x2),ηs(x3)} ≥ min{ηs(a),ηs(b),ηs(c)} and νs(abc) ≤ (νs◦νs◦νs)(abc) = inf abc=x1x2x3 max{νs(x1),νs(x2),νs(x3)} ≤ max{νs(a),νs(b),νs(c)}. this implies that s is a spherical fuzzy ternary subsemigroup of t . � 6 int. j. anal. appl. (2022), 20:29 let t := (µt ,ηt ,νt ) be a spherical fuzzy set on a ternary semigroup t defined by µt (x)=1 and ηt (x)= νt (x)=0 for all x ∈ t . the following theorem holds. theorem 3.2. let s = (µs,ηs,νs) be a spherical fuzzy set on a ternary semigroup t. if s is a spherical fuzzy left ideal of t , then t ◦t ◦s ⊆s. proof. assume that s is a spherical fuzzy left ideal of t. if x /∈ t3, we obtain that (µt ◦µt ◦µs)(x)=0≤ µs(x), (ηt ◦ηt ◦ηs)(x)=0≤ ηs(x) and (νt ◦νt ◦νs)(x)=1≥ νs(x). now, assume that x ∈ t3, we obtain that (µt ◦µt ◦µs)(x)= sup x=abc min{µt (a),µt (b),µs(c)}= sup x=abc µs(c)≤ µs(x), (ηt ◦ηt ◦ηs)(x)= sup x=abc min{ηt (a),ηt (b),ηs(c)}=0≤ ηs(x) and (νt ◦νt ◦νs)(x)= inf x=abc max{νt (a),νt (b),νs(c)}= inf x=abc νs(c)≥ νs(x). hence t ◦t ◦s ⊆s. � theorem 3.3. let s = (µs,ηs,νs) be a spherical fuzzy set on a ternary semigroup t . if s is a spherical fuzzy lateral ideal of t , then t ◦s◦t ⊆s. proof. the proof is similar to that of theorem 3.2. � theorem 3.4. let s = (µs,ηs,νs) be a spherical fuzzy set on a ternary semigroup t . if s is a spherical fuzzy right ideal of t , then s◦t ◦t ⊆s. proof. the proof is similar to that of theorem 3.2. � 3.2. rough spherical fuzzy sets in ternary semigroups. the aims of this subsection is to connect rough set theory and spherical fuzzy sets of ternary semigroups. definition 3.4. an equivalence relation ρ on a ternary semigroup t is called a congruence if for all x1,x2,x3,y1,y2,y3 ∈ t (x1,y1),(x2,y2),(x3,y3)∈ ρ ⇒ (x1x2x3,y1y2y3)∈ ρ. the congruence class of x ∈ t is denoted by [x]ρ. a congruence ρ on t is called complete if [y1]ρ[y2]ρ[y3]ρ = [y1y2y3]ρ for all y1,y2,y3 ∈ t. int. j. anal. appl. (2022), 20:29 7 definition 3.5. let ρ be a congruence on a ternary semigroup t and s =(µs,ηs,νs) be the spherical fuzzy set on a ternary semigroup t . (1) the lower approximation is defined as app(s)= {< y,µs(y),ηs(y),νs(y) >| y ∈ t}, where µs(y) = inf y ′∈[y]ρ µs(y ′), ηs(y) = inf y ′∈[y]ρ ηs(y ′) and νs(y) = sup y ′∈[y]ρ νs(y ′) with the condition that 0≤ (µs(y))2 +(ηs(y))2 +(νs(y))2 ≤ 1. (2) the upper approximation is defined as app(s)= {< y,µs(y),ηs(y),νs(y) >| y ∈ t}, where µs(y) = sup y ′∈[y]ρ µs(y ′), ηs(y) = sup y ′∈[y]ρ ηs(y ′) and νs(y) = inf y ′∈[y]ρ νs(y ′) with the condition that 0≤ (µs(y))2 +(ηs(y))2 +(νs(y))2 ≤ 1. (3) the rough spherical fuzzy set of t is defined by app(s)= (app(s),app(s)). theorem 3.5. let ρ be a congruence on a ternary semigroup t and s1 = (µs1,ηs1,νs1) and s2 = (µs2,ηs2,νs2) be any two spherical fuzzy sets on t. the following statements hold. (1) if s1 ⊆s2, then app(s1)⊆ app(s2) and app(s1)⊆ app(s2). (2) app(s1 ∩s2)⊆ app(s1)∩app(s2). (3) app(s1 ∪s2)= app(s1)∪app(s2). (4) app(s1 ∩s2)= app(s1)∩app(s2). (5) app(s1)∪app(s2)⊆ app(s1 ∪s2). proof. (1) assume that s1 ⊆s2. then µs1 ≤ µs2, ηs2 ≤ ηs1 and νs2 ≥ νs1. thus for all y ∈ t, we have µs1(y)= sup y ′∈[y]ρ µs1(y ′)≤ sup y ′∈[y]ρ µs2(y ′)= µs2(y), ηs1(y)= sup y ′∈[y]ρ ηs1(y ′)≤ sup y ′∈[y]ρ ηs2(y ′)= ηs2(y) and νs1(y)= inf y ′∈[y]ρ νs1(y ′)≥ inf y ′∈[y]ρ νs2(y ′)= νs2(y). this implies that app(s1)⊆ app(s2). similarly, we have app(s1)⊆ app(s2). (2) since s1 ∩s2 ⊆s1 and s1 ∩s2 ⊆s2, app(s1 ∩s2)⊆ app(s1)∩app(s2) by (1). (3) note that app(s1)∪app(s2)= (µs1 ∪µs2,ηs1 ∩ηs2,νs1 ∩νs2) 8 int. j. anal. appl. (2022), 20:29 and app(s1 ∪s2)= (µs1∪s2,ηs1∪s2,νs1∪s2). let y ∈ t . then (µs1 ∪µs2)(y)=max{µs1(y),µs2(y)} =max{ sup y ′∈[y]ρ µs1(y ′), sup y ′∈[y]ρ µs2(y ′)} = sup y ′∈[y]ρ max{µs1(y ′),µs2(y ′)} = sup y ′∈[y]ρ µs1∪s2(y ′) = µs1∪s2(y), (ηs1 ∪ηs2)(y)=max{ηs1(y),ηs2(y)} =max{ sup y ′∈[y]ρ ηs1(y ′), sup y ′∈[y]ρ ηs2(y ′)} = sup y ′∈[y]ρ max{ηs1(y ′),ηs2(y ′)} = sup y ′∈[y]ρ ηs1∪s2(y ′) = ηs1∪s2(y) and (νs1 ∩νs2)(y)=min{νs1(y),νs2(y)} =min{ inf y ′∈[y]ρ νs1(y ′), inf y ′∈[y]ρ νs2(y ′)} = inf y ′∈[y]ρ min{νs1(y ′),νs2(y ′)} = inf y ′∈[y]ρ νs1∪s2(y ′) = νs1∪s2(y). (4) note that app(s1)∩app(s2)= (µs1 ∩µs2,ηs1 ∩ηs2,νs1 ∪νs2) and app(s1 ∩s2)= (µs1∩s2,ηs1∩s2,νs1∩s2). int. j. anal. appl. (2022), 20:29 9 let y ∈ t . then (µs1 ∩µs2)(y)=min{µs1(y),µs2(y)} =min{ inf y ′∈[y]ρ µs1(y ′), inf y ′∈[y]ρ µs2(y ′)} = inf y ′∈[y]ρ min{µs1(y ′),µs2(y ′)} = inf y ′∈[y]ρ µs1∩s2(y ′) = µs1∩s2(y), (ηs1 ∩ηs2)(y)=min{ηs1(y),ηs2(y)} =min{ inf y ′∈[y]ρ ηs1(y ′), inf y ′∈[y]ρ ηs2(y ′)} = inf y ′∈[y]ρ min{ηs1(y ′),ηs2(y ′)} = inf y ′∈[y]ρ ηs1∩s2(y ′) = ηs1∩s2(y) and (νs1 ∪νs2)(y)=max{νs1(y),νs2(y)} =max{ sup y ′∈[y]ρ νs1(y ′), sup y ′∈[y]ρ νs2(y ′)} = sup y ′∈[y]ρ max{νs1(y ′),νs2(y ′)} = sup y ′∈[y]ρ νs1∩s2(y ′) = νs1∩s2(y). (5) since s1 ⊆s1 ∪s2 and s2 ⊆s1 ∪s2,app(s1)∪app(s2)⊆ app(s1 ∪s2) by (1). � theorem 3.6. let ρ be a congruence relation on a ternary semigroup t and s be a spherical fuzzy set on t . then app(s) is also a spherical fuzzy set on t. 10 int. j. anal. appl. (2022), 20:29 proof. let y ∈ t . then (µs(y)) 2 +(ηs(y)) 2 +(νs(y)) 2 =( inf y ′∈[y]ρ µs(y ′))2 +( inf y ′∈[y]ρ ηs(y ′))2 +( sup y ′∈[y]ρ νs(y ′))2 = inf y ′∈[y]ρ (µs(y ′))2 + inf y ′∈[y]ρ (ηs(y ′))2 + sup y ′∈[y]ρ (νs(y ′))2 ≤ inf y ′∈[y]ρ (µs(y ′))2 + inf y ′∈[y]ρ (ηs(y ′))2 + sup y ′∈[y]ρ (1− (µs(y ′))2 − (ηs(y ′))2) ≤ inf y ′∈[y]ρ (µs(y ′))2 + inf y ′∈[y]ρ (ηs(y ′))2 +1− inf y ′∈[y]ρ (µs(y ′))2 − inf y ′∈[y]ρ (ηs(y ′))2 =1. this implies that 0 ≤ (µs(y))2 +(ηs(y))2 +(νs(y))2 ≤ 1. therefore, app(s) is a spherical fuzzy set on t. � let s be a spherical fuzzy set on a ternary semigroup t . note that app(s) need not be a spherical fuzzy set on t , as can be seen in the following example. example 3.1. let t = {i,−i} be the ternary semigroup under the ternary multiplication, ρ = t ×t and s be a spherical fuzzy set on t defined by µs(i)=1,ηs(i)=0,νs(i)=0 and µs(−i)=0,ηs(−i)=1,νs(−i)=0. then µs(i)= µs(−i)=1,ηs(i)= ηs(−i)=1,νs(i)= νs(−i)=0. in this example, we have that app(s) is not a spherical fuzzy set on t . 3.3. rough spherical fuzzy ideals in ternary semigroups. the aims of this subsection is to connect rough set theory and spherical fuzzy ideals of ternary semigroups. theorem 3.7. let ρ be a complete congruence relation on a ternary semigroup t . if s is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of t , then app(s) is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of t. proof. let y1,y2,y3 ∈ t. µs(y1y2y3)= inf y∈[y1y2y3]ρ µs(y) = inf y∈[y1]ρ[y2]ρ[y3]ρ µs(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ µs(abc) ≥ inf abc∈[y1]ρ[y2]ρ[y3]ρ µs(c)= inf c∈[y3] µs(c)= µs(y3), int. j. anal. appl. (2022), 20:29 11 ηs(y1y2y3)= inf y∈[y1y2y3]ρ ηs(y) = inf y∈[y1]ρ[y2]ρ[y3]ρ ηs(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ ηs(abc) ≥ inf abc∈[y1]ρ[y2]ρ[y3]ρ ηs(c)= inf c∈[y3] ηs(c)= ηs(y3) and νs(y1y2y3)= sup y∈[y1y2y3]ρ νs(y) = sup y∈[y1]ρ[y2]ρ[y3]ρ νs(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ νs(abc) ≤ sup abc∈[y1]ρ[y2]ρ[y3]ρ νs(c)= sup c∈[y3] νs(c)= νs(y3). this implies that µs(y1y2y3)≥ µs(y3), ηs(y1y2y3)≥ ηs(y3) and νs(y1y2y3)≤ νs(y3). then app(s) is a spherical fuzzy left ideal of t. the proofs of other cases are similar. � corollary 3.1. let ρ be a complete congruence relation on a ternary semigroup t . if s is a spherical fuzzy ideal of t, then app(s) is a spherical fuzzy ideal of t . proof. this follows from theorem 3.7. � theorem 3.8. let ρ be a congruence relation on a ternary semigroup t. if s is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of t and app(s) is a spherical fuzzy set of t , then app(s) is a spherical fuzzy left ideal [spherical fuzzy lateral ideal, spherical fuzzy right ideal] of t . proof. let y1,y2,y3 ∈ t. µs(y1y2y3)= sup y∈[y1y2y3]ρ µs(y) ≥ sup y∈[y1]ρ[y2]ρ[y3]ρ µs(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ µs(abc) ≥ sup abc∈[y1]ρ[y2]ρ[y3]ρ µs(c)= sup c∈[y3]ρ µs(c)= µs(y3), ηs(y1y2y3)= sup y∈[y1y2y3]ρ ηs(y) ≥ sup y∈[y1]ρ[y2]ρ[y3]ρ ηs(y)= sup abc∈[y1]ρ[y2]ρ[y3]ρ ηs(abc) ≥ sup abc∈[y1]ρ[y2]ρ[y3]ρ ηs(c)= sup c∈[y3]ρ ηs(c)= ηs(y3) 12 int. j. anal. appl. (2022), 20:29 and νs(y1y2y3)= inf y∈[y1y2y3]ρ νs(y) ≤ inf y∈[y1]ρ[y2]ρ[y3]ρ νs(y)= inf abc∈[y1]ρ[y2]ρ[y3]ρ νs(abc) ≤ inf c∈[y3]ρ[y2]ρ[y3]ρ νs(c)= inf c∈[y3]ρ νs(c)= νs(y3). this implies that µs(y1y2y3)≥ µs(y3), ηs(y1y2y3)≥ ηs(y3) and νs(y1y2y3)≤ νs(y3). then app(s) is a spherical fuzzy left ideal of t. the proofs of other cases are similar. � corollary 3.2. let ρ be a congruence relation on a ternary semigroup t. if s is a spherical fuzzy ideal of t and app(s) is a spherical fuzzy set of t , then app(s) is a spherical fuzzy ideal of t. proof. this follows from theorem 3.8. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. ashraf, s. abdullah, t. mahmood, f. gahni, t. mahmood, spherical fuzzy sets and their applications in multi-attribute decision making problems, j. intell. fuzzy syst. 36 (2019), 2829–2844. https://doi.org/10. 3233/jifs-172009 [2] r. chinram, t. panityakul, rough pythagorean fuzzy ideals in ternary semigroups, j. math. computer sci. 20 (2020), 302–312. https://doi.org/10.22436/jmcs.020.04.04 [3] d. dubois, h. prade, rough fuzzy sets and fuzzy rough sets, int. j. gen. syst. 17 (1990), 191–209. https: //doi.org/10.1080/03081079008935107. [4] a. iampan, some properties of ideal extensions in ternary semigroups, iran. j. math. sci. inform. 8 (2013), 67–74. https://doi.org/10.7508/ijmsi.2013.01.007. [5] e. kasner, an extension of the group concept, bull. amer. math. soc. 10 (1904), 290-291 [6] n. kuroki, on fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy sets syst. 5 (1981), 203–215. https: //doi.org/10.1016/0165-0114(81)90018-x. [7] s. kar, p. sarkar, fuzzy ideals of ternary semigroups, fuzzy inf. eng. 4 (2012), 181–193. https://doi.org/10. 1007/s12543-012-0110-4 [8] d.h. lehmer, a ternary analogue of abelian groups, amer. j. math. 54 (1932), 329–338. https://doi.org/10. 2307/2370997. [9] j. los, on the extending of models (i), fundam. math. 42 (1955), 38–54. https://bibliotekanauki.pl/ articles/1383224. [10] z. pawlak, rough sets, int. j. computer inf. sci. 11 (1982), 341–356. https://doi.org/10.1007/bf01001956. [11] p. petchkhaew, r. chinram, fuzzy, rough and rough fuzzy ideals in ternary semigroups, int. j. pure appl. math. 56 (2009), 21–36. [12] f.m. siosan, ideal theory in ternary semigroups, math. japon. 10 (1965), 63–84. [13] q. wang, j. zhan, rough semigroups and rough fuzzy semigroups based on fuzzy ideals, open math. 14 (2016), 1114–1121. https://doi.org/10.1515/math-2016-0102. https://doi.org/10.3233/jifs-172009 https://doi.org/10.3233/jifs-172009 https://doi.org/10.22436/jmcs.020.04.04 https://doi.org/10.1080/03081079008935107 https://doi.org/10.1080/03081079008935107 https://doi.org/10.7508/ijmsi.2013.01.007 https://doi.org/10.1016/0165-0114(81)90018-x https://doi.org/10.1016/0165-0114(81)90018-x https://doi.org/10.1007/s12543-012-0110-4 https://doi.org/10.1007/s12543-012-0110-4 https://doi.org/10.2307/2370997 https://doi.org/10.2307/2370997 https://bibliotekanauki.pl/articles/1383224 https://bibliotekanauki.pl/articles/1383224 https://doi.org/10.1007/bf01001956 https://doi.org/10.1515/math-2016-0102 int. j. anal. appl. (2022), 20:29 13 [14] l. a. zadeh, fuzzy sets, inf. control, 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65)90241-x. [15] s. zeng, a. hussain, t. mahmood, m. i. ali, s. ashraf, m. munir, covering-based spherical fuzzy rough set model hybrid with topsis for multi-attribute decision-making, symmetry 11 (2019), 547. https://doi.org/ 10.3390/sym11040547. https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.3390/sym11040547 https://doi.org/10.3390/sym11040547 1. introduction 2. preliminaries 2.1. ternary semigroups 2.2. fuzzy sets 2.3. spherical fuzzy sets 3. main results 3.1. spherical fuzzy ideals in ternary semigroups 3.2. rough spherical fuzzy sets in ternary semigroups 3.3. rough spherical fuzzy ideals in ternary semigroups references international journal of analysis and applications volume 16, number 1 (2018), 62-74 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-62 some results on controlled k−frames in hilbert spaces m. nouri1,2,∗, a. rahimi2 and sh. najafzadeh2 1department of mathematics, payame noor university, p.o.box 19395-3697 tehran, iran 2department of mathematics, university of maragheh, maragheh, iran ∗corresponding author: mohammadnoori562@yahoo.com abstract. controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator. also k-frames have been introduced to study atomic systems with respect to bounded linear operator. in this paper, the notion of controlled k-frames will be studied and it will be shown that controlled k-frames are equivalent to k-frames under some mild conditions. finally, the stability of controlled k-bessel sequences under perturbation will be discussed with some examples. 1. introduction frames in hilbert spaces were first proposed by duffin and schaeffer to deal with nonharmonic fourier series in 1952 [9] and widely studied from 1986 by daubechies et al. [10]. now, frames play an important role not only in the theoretics also in many kinds of applications and have been widely applied in signal processing [13], sampling [11,12], coding and communications [19], filter bank theory [3], system modeling [8] and so on. over the years, various extentions of the frame theory have been investigated and proposed, such as the fusion frames [5, 6] to deal with hierarchical data processing, g-frames [20] by sun to deal with all existing frames as united object, oblique dual frames [11] by elder to deal with sampling reconstructions, and etc. received 11th september, 2017; accepted 29th november, 2017; published 3rd january, 2018. 2000 mathematics subject classification. primary 42c40; secondary 41a58, 47a58. key words and phrases. bessel sequence; controlled frame; frame; k-frame; perturbation. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 62 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-62 int. j. anal. appl. 16 (1) (2018) 63 the notion of k-frames were recently introduced by l. gǎvruta to study the atomic systems with respect to a bounded linear operator k in hilbert spaces. k-frames are more general than ordinary frames in sense that the lower frame bound only holds for the elements in the range of the k, where k is a bounded linear operator in a separable hilbert space h. recent addition to these generalized frames are the controlled frames [1]. controlled frames have been introduced to improve the numerical efficiency of iterative algorithms for inverting the frame operator on abstract hilbert spaces, however, they were used earlier just as a tool for spherical wavelets [2]. the main advantage of these frames lies in the fact that they retain all the advantages of standard frames but additionally they give a generalized way to check the frame condition while offering a numerical advantage in the sense of preconditioning. recent developments in this direction can be found in [14–18] and the references therein. in this paper, the concept of controlled k-frame will be defined and it will be shown that any controlled k-frame is equivalent to a k-frame. finally, we will discuss the stability of controlled k-bessel sequences under perturbation. throughout this paper, h is a separable hilbert space, b(h) is the family of all bounded linear operators on h and k ∈ b(h). gl(h) denotes the set of all bounded linear operators which have bounded inverses and gl+(h) denotes the set of all positive operators in gl(h). the paper is organized as follows: section 2 contains some preliminary result. in section 3, we define the concept of controlled k-frame and we will show that controlled k-frames are equivalent to k-frames. in section 4, we discuss the stability of a more general perturbation for controlled k-bessel sequence. in section 5, we will examine with some examples the perturbation of controlled k-bessel sequences. 2. preliminaries and notations in this section, some necessary definitions and theorems are presented. definition 2.1. a bounded operator t ∈ b(h) is called positive (respectively, non-negative), if 〈tf,f〉 > 0 for all f 6= 0 (respectively, 〈tf,f〉≥ 0 for all f). every non-negative operator is clearly self-adjoint. if a ∈ b(h) is non-negative, then there exists a unique non-negative operator b such that b2 = a. furthermore, b commutes with every operator that commutes with a. this will be denoted by b = a 1 2 . let b+(h) be the set of positive operators on h. for self-adjoint operators t1 and t2, the notation t1 ≤ t2 or t2 −t1 ≥ 0 means 〈t1f,f〉≤ 〈t2f,f〉 ,∀f ∈ h. int. j. anal. appl. 16 (1) (2018) 64 the following result is needed in the sequel, but straightforward to prove: proposition 2.1. [1] let t : h → h be a linear operator. then the following conditions are equivalent: a. there exist m > 0 and m < ∞, such that mi ≤ t ≤ mi, b. t is positive and there exist m > 0 and m < ∞, such that m‖f‖2 ≤ ‖t 1 2 f‖2 ≤ m‖f‖2 for all f ∈ h, c. t is positive and t 1 2 ∈ gl(h), d. there exists a self-adjoint operator a ∈ gl(h), such that a2 = t, e. t ∈ gl+(h), f. there exist constants m > 0 and m < ∞ and operator c ∈ gl+(h), such that m′c ≤ t ≤ m′c, g. for every c ∈ gl+(h), there exist constants m > 0 and m < ∞, such that m′c ≤ t ≤ m′c. it is well-known that all bounded operators u on a hilbert space h are not invertible: an operator u needs to be injective and surjective in order to be invertible. for doing this, one can use right-inverse operator. the following lemma shows that if an operator u has closed range, there exists a right-inverse operator u† in the following sense: lemma 2.1. [7] let h1 and h2 be hilbert spaces and suppose that u : h2 → h1 is a bounded operator with closed range ru . then there exists a bounded operator u † : h1 → h2 which uu†x = x, ∀x ∈ ru. the operator u† in the lemma 2.3 is called the pseudo-inverse of u. in the literature, one will often see the pseudo-inverse of an operator u with closed range defined as the unique operator u† satisfying that nu† = r ⊥ u, ru† = n ⊥ u , uu †x = x, ∀x ∈ ru. definition 2.2. a sequence {fi}i∈i in h is called a frame for h, if there exist constants 0 < a ≤ b < ∞ such that a‖f‖2 ≤ ∑ i∈i |〈f,fi〉|2 ≤ b‖f‖2 , ∀f ∈ h. if a = b, then {fi}i∈i is called a tight frame and if a = b = 1, it is called a parseval frame. if only the right inequality of the above inequality holds,{fn}n∈i is called a bessel sequence. int. j. anal. appl. 16 (1) (2018) 65 remark 2.1. the frame operator sf = ∑ i∈i〈f,fi〉fi associated with a frame {fi}i∈i is a bounded, invertible and positive operator on h. this provides the reconstruction formulas f = s−1sf = ∑ i∈i 〈f,fi〉s−1fi = ∑ i∈i 〈f,s−1fi〉fi, ∀f ∈ h. furthermore, ai ≤ s ≤ bi and b−1i ≤ s−1 ≤ a−1i. definition 2.3. let c ∈ gl(h). a frame controlled by the operator c or c-controlled frame is a family of vectors {fi}i∈i in h, such that there exist constants 0 < mc ≤ mc < ∞, verifying mc‖f‖2 ≤ ∑ i∈i 〈f,fi〉〈cfi,f〉≤ mc‖f‖2 , ∀f ∈ h. the controlled frame operator s is defined by sf = ∑ i∈i 〈f,fi〉cfi, ∀f ∈ h. definition 2.4. let k ∈ b(h). a sequence {fn}∞n=1 ⊂ h is called a k-frame for h, if there exist constants a,b > 0 such that a‖k∗f‖2 ≤ ∞∑ n=1 |〈f,fn〉|2 ≤ b‖f‖2, ∀f ∈ h. (2.1) we call a and b the lower and upper frame bounds for k-frame, respectively. if only the right inequality of the above inequality holds, {fn}∞n=1 ⊂ h is called a k-bessel sequence. remark 2.2. if k = i, then k-frame are just the ordinary frame. remark 2.3. in the following, we will assume that r(k) is closed, since this can assume that the pseudoinverse k† of k exists. because of the higher generality of k-frames, some properties of ordinary frames can not hold for kframes, such as the frame operator of a k-frame is not an isomorphism. for more differences between k-frames and ordinary frames, we refer to [21]. definition 2.5. let k ∈ b(h). a sequence {fn}∞n=1 ⊂ h is called an atomic system for k, if the following conditions are satisfied: (1) {fn}∞n=1 is a bessel sequence. (2) for any x ∈ h, there exists ax = {an}∈ l2 such that kx = ∞∑ n=1 anfn where ‖ax‖l2 ≤ c‖x‖, c is positive constant. int. j. anal. appl. 16 (1) (2018) 66 suppose that {fn}∞n=1 is a k-frame for h. obviously it is a bessel sequence, so we can define the following operator t : l2 → h, ta = ∞∑ n=1 anfn, a = {an}∈ l2, it follows that t∗ : h → l2 t∗f = {〈f,fn〉}∞n=1, ∀f ∈ h. let s = tt∗, we obtain sf = ∞∑ n=1 〈f,fn〉fn, ∀f ∈ h. we call t, t∗ and s the synthesis operator, analysis operator and frame operator for k-frame {fn}∞n=1, respectively. the following theorem gives a characterization of k-frames in hilbert spaces. proposition 2.2. let {fn}∞n=1 be a bessel sequence in h. then {fn}∞n=1 is a k-frame for h, if and only if there exists a > 0 such that s ≥ akk∗, where s is the frame operator for {fn}∞n=1. proof. the sequence {fn}∞n=1 is a k-frame for h with frame bounds a,b and frame operator s if and only if a‖k∗f‖2 ≤ ∞∑ k=1 |〈f,fn〉|2 = 〈sf,f〉≤ b‖f‖2 , ∀f ∈ h, (2.2) that is 〈akk∗f,f〉≤ 〈sf,f〉≤ 〈bf,f〉 , ∀f ∈ h. so the conclusion holds. � remark 2.4. frame operator of a k-frames is not invertible on h in general, but we can show that it is invertible on the subspace r(k) ⊂ h. in fact, since r(k) is closed, there exists a pseudo-inverse k† of k, such that kk†f = f , ∀f ∈ r(k) , namely kk†|r(k) = ir(k), so we have i∗r(k) = (k †|r(k))∗k∗. hence for any f ∈ r(k), we obtain ‖f‖ = ‖(k†|r(k))∗k∗f‖≤‖k†‖.‖k∗f‖, that is, ‖k∗f‖2 ≥‖k†‖−2‖f‖2. combined with (2.2), we have 〈sf,f〉≥ a‖k∗f‖2 ≥ a‖k†‖−2‖f‖2 , ∀f ∈ r(k). (2.3) int. j. anal. appl. 16 (1) (2018) 67 so, from the definition of k-frame we have a‖k†‖−2‖f‖≤‖sf‖≤ b‖f‖ , ∀f ∈ r(k), (2.4) which implies that s : r(k) → s(r(k)) is a homeomorphism. furthermore, we have b−1‖f‖≤‖s−1f‖≤ a−1‖k†‖2‖f‖ , ∀f ∈ s(r(k)). 3. controlled k-frames controlled frames for spherical wavelets were introduced in [2] to get a numerically more efficient approximation algorithm and the related theory. for general frames, it was developed in [1]. for getting a numerical solution of a linear system of equations ax = b, we can solve the system of equations pax = pb, where p is a suitable preconditioning matrix. it was the main motivation for introducing controlled frames in [2]. controlled frames extended to g-frames in [17] and for fusion frames in [15]. in this section, the concept of controlled frames and controlled bessel sequences will be extended to k-frames and it will be shown that controlled k-frames are equivalent k-frames. definition 3.1. let c ∈ gl+(h) and let ck = kc. the family {fn}∞n=1 is called c-controlled k-frame for h, if {fn}∞n=1 is a k-bessel sequence and there exist constants a > 0 and b < ∞ such that a‖c 1 2 k∗f‖2 ≤ ∞∑ n=1 〈f,fn〉〈f,cfn〉≤ b‖f‖2 , ∀f ∈ h. the constants a and b are called c-controlled k-frame bounds. if c = i, the c-controlled k-frame {fn}∞n=1 is a k-frame for h with bounds a and b. if the second part of the above inequality holds, it called c-controlled k-bessel sequence with bound b. definition 3.2. let c ∈ gl+(h). a sequence {fn}∞n=1 ∈ h is a c-controlled bessel sequence for h if and only if the operator lc : h → h , lcf = ∞∑ n=1 〈f,fn〉cfn, ∀f ∈ h. is well defined and there exists constant b < ∞ such that ∞∑ n=1 〈f,fn〉〈f,cfn〉≤ b‖f‖2 , ∀f ∈ h. definition 3.3. the operator lc : h → h and lcf = ∑∞ n=1〈f,fn〉cfn where f ∈ h is called the c-controlled bessel sequence operator, also lcf = csf. the following lemma characterizes c-controlled k-frames in term of their operators. lemma 3.1. let {fn}∞n=1 be a c-controlled k-frame in h, for c ∈ gl+(h). then ai‖c 1 2 k†‖2 ≤ lc ≤ bi. int. j. anal. appl. 16 (1) (2018) 68 proof. suppose that {fn}∞n=1 is a c-controlled k-frame with bounds a and b. then a‖c 1 2 k∗f‖2 ≤ ∞∑ n=1 〈f,fn〉〈f,cfn〉≤ b‖f‖2 , ∀f ∈ h. for f ∈ h a‖c 1 2 k∗f‖2 ≤〈f,lcf〉≤ b‖f‖2 i.e. a‖c 1 2 k∗‖2i ≤ lc ≤ bi. � the following proposition shows that for evaluation a family {fn}∞n=1 ⊂ h to be a controlled k-frame it is sufficient to check just a simple operator inequality. proposition 3.1. let {fn}∞n=1 be a bessel sequence in h and c ∈ gl+(h). then {fn}∞n=1 is a c-controlled k-frame for h if and only if there exists a > 0 such that cs ≥ cakk∗. proof. the sequence {fn}∞n=1 is a controlled k-frame for h with frame bounds a,b and frame operator s if and only if a‖c 1 2 k∗f‖2 ≤ ∞∑ n=1 〈f,fn〉〈f,cfn〉≤ b‖f‖2 , ∀f ∈ h. that is, 〈cakk∗f,f〉≤ 〈csf,f〉≤ 〈bf,f〉, ∀f ∈ h. � the following proposition shows that any controlled k-frame is a k-frame. proposition 3.2. let {fn}∞n=1 be a c-controlled k-frame and c ∈ gl+(h). then {fn}∞n=1 is a k-frame for h. proof. suppose that {fn}∞n=1 is a controlled k-frame with bounds a and b. then for any f ∈ h, a‖k∗f‖2 = a‖c− 1 2 c 1 2 k∗f‖2 ≤ a‖c 1 2‖2‖c− 1 2 k∗f‖2 ≤ ‖c 1 2‖2 ∞∑ n=1 〈f,fn〉〈f,c0fn〉 = ‖c 1 2‖2 ∞∑ n=1 |〈f,fn〉|2. hence for f ∈ h, a‖c 1 2‖−2‖k∗f‖2 ≤ ∞∑ n=1 |〈f,fn〉|2 int. j. anal. appl. 16 (1) (2018) 69 on the other hand for every f ∈ h, ∞∑ n=1 |〈f,fn〉|2 = 〈f,sf〉 = 〈f,c−1csf〉 = 〈(c−1cs) 1 2 f, (c−1cs) 1 2 f〉 = ‖(c−1cs) 1 2 f‖2 ≤ ‖c− 1 2‖2‖(cs) 1 2 f‖2 = ‖c− 1 2‖2〈f,csf〉 ≤ ‖c− 1 2‖2b‖f‖2. these inequalities yields that {fn}∞n=1 is a k-frame with bounds a‖c 1 2‖−2 and b‖c− 1 2‖2. � the following proposition show that any k-frame is a controlled k-frame under some conditions. proposition 3.3. let c ∈ gl+(h) be a self adjoint and kc = ck, if {fn}∞n=1 is k-frame for h, then {fn}∞n=1 is a c-controlled k-frame for h. proof. suppose that {fn}∞n=1 be a k-frame with bounds a′ and b′. then for all f ∈ h, a′‖k∗f‖2 ≤ ∞∑ n=1 |〈f,fn〉|2 ≤ b′‖f‖2. a′‖c 1 2 k∗f‖2 = a′‖k∗c 1 2 f‖2 ≤ ∞∑ n=1 〈c 1 2 f,fn〉〈c 1 2 f,fn〉 = 〈c 1 2 f, ∞∑ n=1 〈fn,c 1 2 f〉fn〉 = 〈c 1 2 f,c 1 2 sf〉 = 〈f,csf〉. hence a′‖c 1 2 k∗f‖2 ≤〈f,csf〉 for every f ∈ h. on the other hand for every f ∈ h, |〈f,csf〉|2 = |〈c∗f,sf〉|2 = |〈cf,sf〉|2 ≤‖cf‖2‖sf‖2 ≤‖c‖2‖f‖2b‖f‖2. hence, a′‖c 1 2 k∗f‖2 ≤〈f,csf〉≤ b′‖c‖‖f‖2. therefore {fn}∞n=1 is a c-controlled k-frame with bounds a′ and b′‖c‖. � int. j. anal. appl. 16 (1) (2018) 70 4. perturbation for controlled k-bessel sequences one of the most important problems in the studying of frames and its applications specially on wavelet and gabor systems is the invariance of these systems under perturbation. at the first, the problem of perturbation studied by paley and wiener for bases and then extended to frames. there are many versions of perturbation of frames in hilbert spaces, banach space, hilbert c∗-modules and etc. in the last decade, several authors have generalized the paley-wiener perturbation theorem to the perturbation of frames in hilbert spaces. the most general result of these was the following obtained by casazza and christensen [4]. in this section, we mainly give an important on stability of perturbation for k-frames.to do this, we have to introduce tree lemmas below first. lemma 4.1. [7] a sequence {fn}∞n=1 ⊂ h is a bessel sequence with bound b in h, if and only if the operator t : l2 → h, ta = ∑ anfn is welldefined and bounded operator with ‖t‖≤ √ b. lemma 4.2. [7] if {fn}∞n=1 is an ordinary frame for h, then {kfn}∞n=1 is a k−frames for h. lemma 4.3. [7] let t1 ∈ l(x,y ) and let t2 : x → y be linear. if there exist two constants λ1,λ2 ∈ [0, 1] such that ‖t1x−t2x‖≤ λ1‖t1x‖ + λ2‖t2x‖ , ∀x ∈ x then t2 ∈ l(x,y ). moreover, if t1 is invertible on x, then t2 is also invertible on x, and we have 1 −λ1 1 + λ2 ‖t1x‖≤‖t2x‖≤ 1 + λ1 1 −λ2 ‖t1x‖ , ∀x ∈ x and 1 −λ2 1 + λ1 . 1 ‖t1‖ ‖y‖≤‖t−12 y‖≤ 1 + λ2 1 −λ1 ‖t−11 ‖‖y‖ , ∀y ∈ y. theorem 4.1. [4] let {xj}j∈j be a frame for a hilbert space h with frame bounds c and d. assume that {yj}j∈j is a sequence of h and that there exist λ1,λ2,µ > 0 such that max{λ1 + µ√ c ,λ2} < 1. suppose one of the following conditions holds for any finite scalar sequence {cj} and every x ∈ h. then {yj}j∈j is also a frame for h; (1) ( ∑ j∈j |〈x,xj −yj〉| 2) 1 2 ≤ λ1( ∑ j∈j |〈x,xj〉| 2) 1 2 + λ2( ∑ j∈j |〈x,yj〉| 2) 1 2 + µ‖x‖, (2) ‖ ∑n i=1 cj(xj −yj)‖≤ λ1‖ ∑n i=1 cjxj‖ + λ2‖ ∑n i=1 cjyj‖ + µ( ∑n i=1 |cj| 2) 1 2 . moreover, if {xj}j∈j is a riesz basis for h and {yj}j∈j satisfies (2), then {yj}j∈j is also a riesz basis for h. the perturbation theorem investigated by x. xiao, y. zhu, l. gǎvruta to k-frames [21]: int. j. anal. appl. 16 (1) (2018) 71 theorem 4.2. suppose that {fn}∞n=1 is a k-frame for h, and α,β ∈ [0,∞], such that max{α+γ √ a−1‖k+‖,β} < 1. if {gn}∞n=1 ⊂ h and satisfy, ‖ n∑ k=1 ck(fk −gk)‖≤ α‖ n∑ k=1 ckfk‖ + β‖ n∑ k=1 ckgk‖ + γ( n∑ k=1 |ck|2) 1 2 , for any ci (i ∈ n), then {gn}∞n=1 is a pq(r(k))k-frame for h, with frame bounds [ √ a‖k+‖−1(1 −α) −γ]2 (1 + β)2‖k‖2 , [ √ b(1 + α) + γ]2 (1 −β)2 , where pq(r(k)) is a orthogonal projection operator for h to q(r(k)), q = ut ∗, t,u are synthesis operator for {fn}∞n=1 and {gn}∞n=1 respectively. motivating the above theorems, we prove perturbation for controlled k-bessel sequences. theorem 4.3. suppose that {fn}+∞n=1 ⊂ h is a c-controlled k-frame for h, with frame bounds a, b, and α,β,γ ∈ [0,∞), such that max{α + γ √ a−1‖k+‖,β} < 1. if {gn}+∞n=1 ⊂ h and satisfy, ‖ n∑ k=1 ckfk − n∑ k=1 ckgk‖ ≤ α‖c‖‖ n∑ k=1 ckfk‖ + β‖ n∑ k=1 ckgk‖ +γ( n∑ k=1 |ck|2) 1 2 , (4.1) for any ci, then {gn}+∞n=1 is a controlled k-bessel sequence for h with bound ( (1 + α‖c‖) √ b + γ 1 −β )2, where t,u are the synthesis operator for {fn}+∞n=1 and {gn} +∞ n=1, respectively. proof. let {fn}+∞n=1 be a frame for h, so by lemma 4.1, the frame operator t is bounded and ‖t‖≤ √ b. the condition (4.1) implies that for all finite sequences {ck}, ‖ n∑ k=1 ckgk‖ = ‖− n∑ k=1 ck(fk −gk) + n∑ k=1 ckfk‖≤‖− n∑ k=1 ck(fk −gk)‖ + ‖ n∑ k=1 ckfk‖ ≤ (1 + α‖c‖)‖ n∑ k=1 ckfk‖ + β‖ n∑ k=1 ckgk‖ + γ( n∑ k=1 |ck|2) 1 2 . this calculation actually holds for all {ck}+∞k=1 ∈ l 2(n). to see this, at the first we have to prove that∑∞ k=1 ckgk is convergent for any given {ck} +∞ k=1 ∈ l 2(n). given n,m ∈ n with n > m, ‖ n∑ k=1 ckfk − m∑ k=1 ckfk‖ = ‖ n∑ k=m+1 ckfk‖ int. j. anal. appl. 16 (1) (2018) 72 ≤ (1 + α‖c‖)‖ n∑ k=m+1 ckfk‖ + β‖ n∑ k=m+1 ckgk‖ + γ( n∑ k=m+1 |ck|2) 1 2 . since {ck}∞k=1 ∈ l 2(n) and ∑∞ k=1 ckfk is convergent, this implies that { ∑n k=1 ckfk} ∞ n=1 is a cauchy sequence in h and therefore convergent, thus the pre-frame operator u is well defined on l2(n); it follows that for all {ck}∞k=1 ∈ l 2(n), ‖ ∞∑ k=1 ckgk‖≤ (1 + α‖c‖)‖ ∞∑ k=1 ckfk‖ + β‖ ∞∑ k=1 ckgk‖ + γ( ∞∑ k=1 |ck|2) 1 2 , (4.2) in terms of the operator t,u, (4.3) states that ‖u{ck}∞k=1‖≤ (1 + α‖c‖)‖t{ck} ∞ k=1‖ + β‖u{ck} ∞ k=1‖ + γ( ∞∑ k=1 |ck|2) 1 2 ≤ ((1 + α‖c‖) √ b + γ)( ∞∑ k=1 |ck|2) 1 2 + β‖u{ck}∞k=1‖ , ∀{ck} ∞ k=1 ∈ l 2(n). so ‖u{ck}∞k=1‖≤ ((1 + α‖c‖) √ b + γ) 1 −β ( ∞∑ k=1 |ck|2) 1 2 . (4.3) via lemma 4.1, this estimation shows that {gk}∞k=1 is a bessel sequence with bound ((1 + α‖c‖) √ b + γ)2 (1 −β)2 . � 5. examples in this section, we give some examples that examines the stability of the controlled k-bessel sequences under perturbation. example 5.1. suppose that h = c3, {gn}3n=1 = {e1,e2,e3}, where e1 =   1 0 0   , e2 =   0 1 0   , e3 =   0 0 1   . now define k ∈ l(h) as follows k : h → h, ke1 = e1, ke2 = e1, ke3 = e2. obviously, {gn}3n=1 is an ordinary frame for h. by lemma 4.2, we know that {fn}3n=1 = {kgn}3n=1 is a k-frame for h. by proposition 3.3, we can show that {fn}+∞n=1 is a controlled k-frame for h. example 5.2. let h = c3,{ei}3i=1 be the orthonormal basis for h. now define k ∈ l(h) as follows k : h → h, ke1 = 2e1, ke2 = 2e1, ke3 = 6e2. int. j. anal. appl. 16 (1) (2018) 73 let {fi}3i=1 = {2e1, 2e1, 6e2}, we know {fi} 3 i=1 is a k−frame for h by lemma 4.2. we can take k + as follows k+e1 = e1 + e2 4 , k+e2 = e3 6 , k+e3 = 0. (5.1) it’s easy the calculate the adjoint of k as follows k+e1 = 2e1 + 2e2, k +e2 = 6e3, k +e3 = 0. since {fi}3i=1 is a k−frame for h, by the definition of k−frame we can get 0 < a ≤ 1, so let me take a = 1. from (5.1) we can obtain ‖k+‖ = 1 2 √ 2 . let α = 0.95,β = 0.96,γ = 0.001, it’s easy the check that max{α + γ √ a−1‖k+‖,β} < 1 holds. now, take g1 = e1,g2 = e1,g3 = 5e2, for any ci(i = 1, 2, 3), we have∥∥∥∥∥ 3∑ k=1 ck(fk −gk) ∥∥∥∥∥ = √ (c1 + c2)2 + c 2 3, γ ( n∑ i=1 c2k )1 2 = 0.001 √ c21 + c 2 2 + c 2 3 ≥ 0, α ∥∥∥∥∥ n∑ k=1 ckfk ∥∥∥∥∥ = 0.95 √ 4(c1 + c2)2 + 36c 2 3 = √ 3.16(c1 + c2)2 + 32.49c 2 3,∥∥∥∥∥ n∑ k=1 ckgk ∥∥∥∥∥ = √ 3.6864(c1 + c2)2 + 23.04c 2 3 ≥ 0. it is trivial to check that (4.1) holds. moreover, we can show that {gi}3i=1 = {e1,e1, 5e2} is a pq(r(k))k−frame for h. in fact, by the definitions of t,u we get qf = ut∗f = 3∑ k=1 〈f,fk〉gk,f ∈ h. for any f ∈ h, suppose that f = c1e1 + c2e2 + c3e3, then qf = 3∑ k=1 〈c1e1 + c2e2 + c3e3,fk〉gk = 4c1e1 + 30c2e2, which implies that r(q) = span{e1,e2}. by the definition of k we can calculate ‖k‖ = 6, now we leave the reader to verify that (4.3) holds. 6. conclusion in this article, controlled k-frames is first defined. then, we examined the conditions that controlled k-frames are equivalent to k-frames (under certain conditions). at the end, the stability of the controlled k-bessel sequences were checked under perturbation. references [1] p. balazs, j. p. antoine, a. grybos, wighted and controlled frames, int. j. wavelets multiresolut. inf. process. 8(1) (2010) 109-132. [2] i. bogdanova, p. vandergheynst, j .p . antoine, l. jacques, m. morvidone, stereographic wavelet frames on the sphere, appl. comput. harmon. anal. (19) (2005) 223-252. int. j. anal. appl. 16 (1) (2018) 74 [3] h. bolcskei, f. hlawatsch , h. g. feichtinger,frame theoretic analysis of oversampled filter banks, ieee trans. signal process. 46 (1998) 3256-3268. [4] p. casazza, o. christensen, perturbation of operators and applications to frame theory, j. fourier anal. appl. 3 (1997) 543-557. [5] p. g. casazza, g. kutyniok, frames of subspaces. wavelets, frames and operator theory, college park, md, contemp. math., vol.345. american mathematical society, providence,(2004) 87-113. [6] p.g. casazza, g. li .s. kutyniok, fusion frames and distributed processing, appl. comput. harmon. anal.25 (2008) 114-132. [7] o. christensen, an introduction to frames and riesz bases, birkhäuser, boston 2003. [8] n. e. duday ward, j. r. partington, a construction of rational wavelets and frames in hardy-sobolev space with applications to system modelling. siam.j.control optim.36 (1998) 654-679. [9] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. math. soc.72 (1952) 341-366. [10] i. daubechies, a. grossmann, y. meyer, painless non orthogonal expansions, j. math. phys. 27(1986) 1271-1283. [11] y. c. eldar, sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, j. fourier. anal. appl. 9(1) (2003) 77-96. [12] y. c. eldar, t. werther, general framework for consistent sampling in hilbert spaces, int. j. walvelets multi. inf. process. 3(3) (2005) 347-359. [13] p. j. s. g. ferreira, mathematics for multimedia signal processing ii: discrete finite frames and signal reconstruction, in: byrnes, j.s. (ed.) signals processing for multimedia, pp. 35-54.ios press, amsterdam (1999). [14] d. hua and y. huang, controlled k-g-frames in hilbert spaces, results. math. (2016) doi 10.1007/s00025-016-0613-0. [15] a. khosravi and k. musazadeh, controlled fusion frames, methods funct. anal. topol. 18(3) (2012), 256-265. [16] k. musazadeh and h. khandani, some results on controlled frames in hilbert spaces, acta math. sci. 36b(3) (2016), 655-665. [17] a. rahimi, a. fereydooni, controlled g-frames and their g-multipliers in hilbert spaces , an. tiin. univ. ovidius constana, 2(12) (2013), 223-236. [18] m. rashidi-kouchi and a. rahimi, on controlled frames in hilbert c∗-modules, int. j. wavelets multiresolut. inf. process. 15(4) (2017), art. id 1750038. [19] t. strohmer, r. jr. heath, grass manian frames with applications to coding and communications, appl. comput. harmon. anal. 14 (2003), 257275. [20] w. sun, g-frames and g-riesz bases, j. math. anal. 322 (2006) 437-452. [21] x. xiao, y. zhu, l. gǎvruta, some properties of k-frames in hilbert spaces , results. math. 63 (2013), 1243-1255. 1. introduction 2. preliminaries and notations 3. controlled k-frames 4. perturbation for controlled k-bessel sequences 5. examples 6. conclusion references international journal of analysis and applications volume 17, number 4 (2019), 652-658 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-652 univalent functions formulated by the salagean-difference operator rabha w. ibrahim1,∗, and maslina darus2 1ieee:94086547 2centre of modelling and data science, faculty of sciences and technology, universiti kebangsaan malaysia,43600 bangi, selangor, malaysia email address: maslina@ukm.edu.my ∗corresponding author: rabhaibrahim@yahoo.com abstract. we present a class of univalent functions tm(κ,α) formulated by a new differential-difference operator in the open unit disk. the operator is a generalization of the well known salagean’s differential operator. based on this operator, we define a generalized class of bounded turning functions. inequalities, extreme points of tm(κ,α), some convolution properties of functions fitting to tm(κ,α), and other properties are discussed. 1. introduction let λ be the class of analytic function formulated by f(z) = z + ∞∑ n=2 anz n, z ∈ u = {z : |z| < 1}. received 2019-03-02; accepted 2019-04-01; published 2019-07-01. 2010 mathematics subject classification. 30c45. key words and phrases. fractional calculus; fractional differential equation; fractional operator. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 652 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-652 int. j. anal. appl. 17 (4) (2019) 653 we symbolize by t(α) the subclass of λ for which <{f′(z)} > α in u. for a function f ∈ λ, we present the following difference operator d0κf(z) = f(z) d1κf(z) = zf ′(z) + κ 2 (f(z) −f(−z) − 2z) , κ ∈ r ... dmκ f(z) = dκ(d m−1 κ f(z)) = z + ∞∑ n=2 [n + κ 2 (1 + (−1)n+1)]m anzn. (1.1) it is clear that when κ = 0, we have the salagean’s differential operator [1]. we call dmκ the salageandifference operator. moreover, dmκ is a modified dunkl operator of complex variables [2] and for recent work [3]. dunkl operator describes a major generalization of partial derivatives and realizes the commutative law in rn. in geometry, it attains the reflexive relation, which is plotting the space into itself as a set of fixed points. example 1. (see figs 1 and 2) • let f(z) = z/(1 −z) then d11f(z) = z + 2z 2 + 4z3 + 4z4 + 6z5 + 6z6 + ... • let f(z) = z/(1 −z)2 then d11f(z) = z + 4z 2 + 12z3 + 16z4 + 30z5 + 36z6 + ... we proceed to define a generalized class of bounded turning utilizing the the salagean-difference operator. let tm(κ,α) denote the class of functions f ∈ λ which achieve the condition <{(dmκ f(z)) ′} > α, 0 ≤ α ≤ 1, z ∈ u, m = 0, 1, 2, ... . clearly, t0(κ,α) = t(α) (the bounded turning class of order α). the hadamard product or convolution of two power series is denoted by (∗) achieving f(z) ∗h(z) = ( z + ∞∑ n=2 anz n ) ∗ ( z + ∞∑ n=2 ηnz n ) = z + ∞∑ n=2 anηnz n. (1.2) the aim of this effort is to present several important properties of the class tm(κ,α). for this purpose, we need the following auxiliary preliminaries. int. j. anal. appl. 17 (4) (2019) 654 figure 1. d11(z/(1 −z)) figure 2. d11(z/(1 −z)2) lemma 1. let {an}∞n=0 be a convex null sequence (a0 − a1 ≥ a1 − a2, ... ≥ 0). then the function ρ(z) = a0/2 + ∑∞ n=1 an z n, is analytic and <ρ(z) > 0 in u. lemma 2. if ρ(z) is analytic in u, ρ(0) = 1 and <ρ(z) > 1/2,z ∈ u, then for any function % analytic in u, the function ρ∗% assigns its credits in the convex hull of %(u). int. j. anal. appl. 17 (4) (2019) 655 lemma 3. [4] for all z ∈ u the sum < (∑ n=2 zn−1 n + 1 ) > − 1 3 . there are different techniques of studying the class of bounded turning functions, such as using partial sums or applying jack lemma [5][7]. 2. results in this section, we illustrate our results. theorem 4. tm+1(κ,α) ⊂ tm(κ,α). proof. let f ∈ tm+1(κ,α), then we have <{1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m+1 anzn−1} > α. dividing the last inequality by 1 −α and adding +1 we obtain the inequality <{1 + 1 2(1 −α) ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m+1 anzn−1} > 1 2 . by employing the definition of the convolution, we have the construction (dmκ f(z)) ′ = 1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m anzn−1 = ( 1 + 1 2(1 −α) ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m+1 anzn−1 ) ∗ ( 1 + 2(1 −α) ∞∑ n=2 zn−1 n + κ 2 (1 + (−1)n+1) ) . in view of lemma 1, with a0 = 1 and an = 1/(n + κ 2 (1 + (−1)n+1),n = 1, 2, ..., we have < ( 1 + 2(1 −α) ∞∑ n=2 zn−1 n + κ 2 (1 + (−1)n+1) ) > α. in virtue of lemma 2, we arrive at the required result. � theorem 5. tm+1(κ,α) ⊂ tm(κ,β), β ≤ α, 0 ≤ κ ≤ 1/2. proof. let f ∈ tm+1(κ,α) then we have <{1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m+1 anzn−1} > α. int. j. anal. appl. 17 (4) (2019) 656 also, we have the convolution (dmκ f(z)) ′ = 1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m anzn−1 = ( 1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m+1 anzn−1 ) ∗ ( 1 + ∞∑ n=2 zn−1 n + κ 2 (1 + (−1)n+1) ) . it is clear that n + κ 2 (1 + (−1)n+1) ≤ n + 2κ ≤ n + 1, 0 ≤ κ ≤ 1/2. by applying lemma 3 on the second term of the above convolution, we obtain < ( 1 + ∞∑ n=2 zn−1 n + κ 2 (1 + (−1)n+1) ) > 2/3. thus, we attain that <(dmκ f(z)) ′ > 2 3 α. by considering β := 2 3 α ≤ α, α ∈ [0, 1], we attain the requested result. � theorem 6. let f ∈ tm(κ,α) and h ∈ c, the set of convex univalent functions (c ⊂ λ ). then f ∗ h ∈ tm(κ,α). proof. by the marx-strohhacker theorem [8], if h is convex univalent in u, then <{ h(z) z } > 1/2. utilizing convolution properties, we obtain <(dmκ (f ∗h)(z)) ′ = < (h(z) z ∗dmκ f(z) ′ ) . but <(dmκ f(z)′) > α; thus, in view of lemma 2, we have the desire conclusion. � theorem 7. let f,h ∈ tm(κ,α). then f ∗h ∈ tm(κ,β), where β := κ(2α + 1) + 4α− 1 2(κ + 1) , 0 ≤ κ ≤ 1. int. j. anal. appl. 17 (4) (2019) 657 proof. define a function h ∈ λ as follows: h(z) = z + ∞∑ n=2 ϑnz n, z ∈ u. since h ∈ tm(κ,α) then <{1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m ϑnzn−1} > α. let ϕ0 = 1, and in general, we have ϕn = κ + 1 [(n + 1)(n + κ 2 (1 + (−1)n+2) + 1)]m , n ≥ 1, 0 ≤ κ ≤ 1, m = 1, 2, .... obviously, the sequence {ϕn}∞n=0 is a convex null sequence. therefore, by lemma 1, we conclude that <{1 + ∞∑ n=2 κ + 1 [(n + 1)(n + κ 2 (1 + (−1)n+2) + 1)]m zn−1} > 1 2 . now the convolution ( 1 + ∞∑ n=2 n[n + κ 2 (1 + (−1)n+1)]m ϑnzn−1 ) ∗ ( 1 + ∞∑ n=2 κ + 1 [(n)(n + κ 2 (1 + (−1)n+1))]m zn−1 ) = 1 + ∞∑ n=2 (κ + 1) ϑnz n−1 satisfies the real <{1 + ∞∑ n=2 (κ + 1) ϑnz n−1zn−1} > α. in other words, we have <{ h(z) z } = <{1 + ∞∑ n=2 ϑnz n−1} > κ + α α + 1 . thus, <{ h(z) z } = <{1 + ∞∑ n=2 ϑnz n−1 − 2α + κ− 1 2(κ + 1) } > 1 2 . but f,h ∈ tm(κ,α), this implies that <{ (h(z) z − 2α + κ− 1 2(κ + 1) ) ∗dmκ (f)(z)) ′} > α. consequently, we conclude that int. j. anal. appl. 17 (4) (2019) 658 <{ (h(z) z ) ∗dmκ (f)(z)) ′} > κ(2α + 1) + 4α− 1 2(κ + 1) := β. thus, by lemma 2 and the fact <(dmκ (f ∗h)(z)) ′ = < (h(z) z ∗dmκ f(z) ′ ) , we realize the requested result. � note that some applications of the dunkl operator in a complex domain can be found in [9]. acknowledgments the work here is partially supported by the universiti kebangsaan malaysia grant: gup ( geran universiti penyelidikan)-2017-064. references [1] g.s. salagean, subclasses of univalent functions, complex analysis-fifth romanian-finnish seminar, part 1 (bucharest, 1981), lecture notes in math., vol. 1013, springer, berlin, (1983), 362–372. [2] c.f. dunkl, differential-difference operators associated with reflections groups, trans. amer. math. soc. 311 (1989), 167– 183. [3] r. w. ibrahim, new classes of analytic functions determined by a modified differential-difference operator in a complex domain, karbala int. j. modern sci. 3(1) (2017), 53–58. [4] j.m. jahangiri, k. farahmand, partial sums of functions of bounded turning, j. inequal. pure appl. math. 4(4) (2003), article 79. [5] m. darus m, r.w ibrahim, partial sums of analytic functions of bounded turning with applications, comput. appl. math. 29 (1) 2010), 81–88. [6] r.w ibrahim, m. darus, extremal bounds for functions of bounded turning, int. math. forum. 6 (33) (2011), 1623–1630. [7] r.w ibrahim, upper bound for functions of bounded turning. math. commun. 17(2) (2012), 461–468. [8] s.s. miller, and p. t. mocanu. differential subordinations: theory and applications. crc press, 2000. [9] r.w ibrahim, optimality and duality defined by the concept of tempered fractional univex functions in multi-objective optimization. int. j. anal. appl. 15 (1) (2017), 75–85. 1. introduction 2. results acknowledgments references international journal of analysis and applications volume 17, number 1 (2019), 14-25 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-14 on i-asymptotically lacunary statistical equivalence of functions on amenable semigroups ömer ki̇şi̇∗, burak çakal department of mathematics, faculty of science, bartın university, 74100, bartın, turkey ∗corresponding author: okisi@bartin.edu.tr abstract. in this study we define the notions of asymptotically paper, we introduce the concept of iasymptotically statistical equivalent and i-asymptotically lacunary statistical equivalent functions defined on discrete countable amenable semigroups. in addition to these definitions, we give some inclusion theorems. 1. introduction fast [5] presented an interesting generalization of the usual sequential limit which he called statistical convergence for number sequences. schoenberg [24] established some basic properties of statistical convergence and also studied the concept as a summability method. using lacunary sequences fridy and orhan defined lacunary statistical convergence in [6]. also, in another study, they gave the relationships between the lacunary statistical convergence and the cesàro summability. after their definition, freedman et al. [7] established the connection between the strongly cesàro summable sequences and the strongly lacunary summable sequences. the concept of i-convergence of real sequences is a generalization of statistical convergence which is based on the structure of the ideal i of subsets of the set of natural numbers. p. kostyrko et al. [8] introduced the concept of i-convergence of sequences in a metric space and studied some properties of this convergence. received 2018-07-04; accepted 2018-09-07; published 2019-01-04. 2010 mathematics subject classification. 40a05, 40c05. key words and phrases. folner sequence; amenable group; equivalent functions; statistical convergence; lacunary sequences; i−convergence. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 14 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-14 int. j. anal. appl. 17 (1) (2019) 15 recently, the idea of statistical convergence and lacunary convergence was further extended by das et al. [2] to i-statistical convergence, i-lacunary statistical convergence. in 1993, mursaleen [15] defined λ-statistical convergence by using the λ sequence. let λ denote the set of all non-decreasing sequences λ = (λn) of positive numbers tending to ∞ such that λn+1 ≤ λn +1 and λ1 = 1. asymptotic equivalence of sequences was introduced by pobyvanets [18]. marouf’s work [9] was extension of pobyvanets’s work. in 2003, patterson [16] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. in [17] asymptotically lacunary statistical equivalent which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences was studied. also in [20], i-asymptotically statistical equivalent and i-asymptotically lacunary statistical equivalent sequences were examined. let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold, and w(g) and m(g) denote the spaces of all real valued functions and all bounded real functions f on g respectively. m(g) is a banach space with the supremum norm ‖f‖∞ = sup{|f(g)| : g ∈ g}. nomika [23] showed that, if g is countable amenable group, there exists a sequence {sn} of finite subsets of g such that (i) g = ∪∞i=1sn, (ii) sn ⊂ sn+1, n = 1, 2, 3, ..., (iii) limn→∞ |sng−∩sn| |sn| = 1, limn→∞ |gsn−∩sn| |sn| = 1 for all g ∈ g. here |a| denotes the number of elements in the finite set a. any sequence of finite subsets of g satisfying (i), (ii) and (iii) is called a folner sequence for g. amenable semigroups were studied by [1]. the concept of summability in amenable semigroups was introduced in [13], [14]. in [3], douglas extended the notion of arithmetic mean to amenable semigroups and obtained a characterization for almost convergence in amenable semigroups. in [21], the notions of convergence and statistical convergence, statistical limit point and statistical cluster point of functions on discrete countable amenable semigroups were introduced. nuray f. rhoades b.e. [22] defined the notions of asymptotically, statistically, almost statistically and strong almost asymptotically equivalent functions defined on discrete countable amenable semigroups. in addition to these definitions, they gave some inclusion theorems. also, they proved that the strong almost asymptotically equivalence of the functions f(g) and h(g) defined on discrete countable amenable semigroups does not depend on the particular choice of the folner sequence. in [12], the concepts of σ-uniform density of subsets a of the set n of positive integers and corresponding iσ-convergence of functions defined on discrete countable amenable semigroups were introduced. furthermore, for any folner sequence inclusion relations between iσ-convergence and invariant convergence also iσ-convergence and [vσ]p-convergence were given. they introduced the concept of iσ-statistical convergence int. j. anal. appl. 17 (1) (2019) 16 and iσ-lacunary statistical convergence of functions defined on discrete countable amenable semigroups. in addition to these definitions, they gave some inclusion theorems. also, they made a new approach to the notions of [v,λ]-summability, σ-convergence and λ-statistical convergence of folner sequences by using ideals and introduced new notions, namely, iσ-[v,λ]-summability, iσ-λ-statistical convergence of folner sequences. kişi and güler [11] introduced the concepts of sσ-asymptotically equivalent, sσ,λ-asymptotically equivalent, σ-asymptotically lacunary statistical equivalent and strong (σ,θ)-asymptotically equivalent functions defined on discrete countable amenable semigroups. the purpose of the study [10] was to extend the notions of i-convergence, i-limit superior and i-limit inferior, i-cluster point and i-limit point to functions defined on discrete countable amenable semigroups. also, a new approach to the notions of [v,λ]-summability and λ-statistical convergence by using ideals and introduce new notions, namely, i-[v,λ]-summability and i-λ-statistical convergence to functions was defined on discrete countable amenable semigroups. this study presents the notion of i-asymptotically lacunary statistical equivalence which is a natural combination of i-asymptotically equivalence, lacunary statistical equivalence for functions defined on discrete countable amenable semigroups. we introduce new concepts, and establish certain inclusion theorems. 2. definitions and notations definition 2.1. [21] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (g) is said to be convergent to s, for any folner sequence {sn} for g, if for each ε > 0 there exists k0 ∈ n such that |f (g) −s| < ε for all m > k0 and g ∈ g \ sm. definition 2.2. [21] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w(g) is said to be strongly summable to s, for any folner sequence {sn} for g, if lim n→∞ 1 |sn| ∑ g∈sn |f (g) −s| = 0, where |sn| denotes the cardinality of the set sn. definition 2.3. [21] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w(g) is said to be statistically convergent to s, for any folner sequence {sn} for g, if for every ε > 0, lim n→∞ 1 |sn| |{g ∈ sn : |f (g) −s| ≥ ε}| = 0 the set of all statistically convergent functions will be denoted s(g). int. j. anal. appl. 17 (1) (2019) 17 definition 2.4. [22] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. two nonnegative functions f,h ∈ w(g) are said to be asymptotically equivalent, for any folner sequence {sn} for g, if for every ε > 0 there exists k0 ∈ n such that∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ < ε for all m > k0 and g ∈ g\sm. it will be denoted by f ∼ h. definition 2.5. [22] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. two nonnegative functions f,h ∈ w(g) are said to be strong asymptotically equivalent, for any folner sequence {sn} for g, if lim n→∞ 1 |sn| ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ = 0 it will be denoted by f ∼w h. definition 2.6. [22] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. two nonnegative functions f,h ∈ w(g) are said to be statistically equivalent, for any folner sequence {sn} for g, if for every ε > 0, lim n→∞ 1 |sn| |{g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε}| = 0 definition 2.7. [10] let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (g) is said to be i-convergent to s for any folner sequence {sn} for g, if for every ε > 0; {g ∈ sn : |f (g) −s| ≥ ε}∈i; i.e., |f (g) −s| < ε a.a.g. the set of all i-convergent sequences will be denoted by i (g). 3. main results definition 3.1. let g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (g) is said to be i-statistically convergent to s, for any folner sequence {sn} for g, if for each ε > 0 and δ > 0,{ n ∈ n : 1 |sn| |{g ∈ sn : |f (g) −s| ≥ ε}|≥ δ } ∈i. the set of all i-statistically convergent folner sequences will be denoted by si (g). definition 3.2. let θ be lacunary sequence and g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w(g) is said to be lacunary statistically convergent to s, for any folner sequence {sn} for g, if for every ε > 0, lim r→∞ 1 hr |{g ∈ sn : |f (g) −s| ≥ ε}| = 0. int. j. anal. appl. 17 (1) (2019) 18 the set of all lacunary statistically convergent folner sequences will be denoted sθ(g). definition 3.3. let θ be lacunary sequence, i ⊆ 2n be an admissible ideal in n and g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (g) is said to be i-lacunary statistically convergent to s, for any folner sequence {sn} for g, if for each ε > 0 and δ > 0,{ r ∈ n : 1 hr |{g ∈ sn : |f (g) −s| ≥ ε}|≥ δ } ∈i. the set of all i-lacunary statistically convergent sequences will be denoted by sθi (g). definition 3.4. let θ be lacunary sequence, i ⊆ 2n be an admissible ideal in n and g be a discrete countable amenable semigroup with identity in which both right and left cancelation laws hold. f ∈ w (g) is said to be strongly i-lacunary convergent to s or nθi (g)-convergent to s, for any folner sequence {sn} for g, if for each ε > 0,  r ∈ n : 1hr ∑ g∈sn |f (g) −s| ≥ ε   ∈i. the set of all strongly i-lacunary convergent sequences will be denoted by nθi (g). definition 3.5. two nonnegative functions f,h ∈ w(g) are said to be asymptotically i-equivalent, for any folner sequence {sn} for g, if every ε > 0,{ g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε } ∈i. it will be denoted by f ∼i(g) h. definition 3.6. two nonnegative functions f,h ∈ w(g) are said to be i-asymptotically statistical equivalent, for any folner sequence {sn} for g, if for each ε > 0 and δ > 0, the following set{ n ∈ n : 1 |sn| ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ ≥ δ } belongs to i. it will be denoted by f ∼si(g) h. for i = ifin, i-asymptotically statistical equivalence coincides with asymptotically statistical equivalence for any folner sequence {sn} for g. definition 3.7. two nonnegative functions f,h ∈ w(g) are said to be i-asymptotically lacunary statistical equivalent, for any folner sequence {sn} for g, if for every ε > 0 and δ > 0, the following set{ r ∈ n : 1 hr ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ ≥ δ } belongs to i. it will be denoted by f ∼s θ i(g) h. int. j. anal. appl. 17 (1) (2019) 19 for i = ifin, i-asymptotically lacunary statistical equivalence coincides with asymptotically lacunary statistical equivalence for any folner sequence {sn} for g. definition 3.8. two nonnegative functions f,h ∈ w(g) are said to be strongly i-asymptotically lacunary equivalent, for any folner sequence {sn} for g, if for every ε > 0 r ∈ n : 1hr ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε   ∈i. it will be denoted by f ∼n θ i(g) h. definition 3.9. two nonnegative functions f,h ∈ w(g) are said to be strongly λi (g)-asymptotically equivalent, for any folner sequence {sn} for g, if for every ε > 0 n ∈ n : 1λn ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε   ∈i. it will be denoted by f ∼vλi (g) h. definition 3.10. two nonnegative functions f,h ∈ w(g) are said to be i-asymptotically λ-statistical equivalent provided that for every ε, δ > 0,{ n ∈ n : 1 λn ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ ≥ δ } ∈i. it will be denoted by f ∼sλi (g) h. theorem 3.1. let λ ∈ λ and i is an admissible ideal in n. if f ∼vλi (g) h then f ∼sλi (g) h. proof. assume that f ∼vλi (g) h and ε > 0. then, ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ∑ g∈sn&|f(g)h(g)−1|≥ε ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε. ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ and so, 1 ε.λn ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ 1λn ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ . then for any δ > 0, { n ∈ n : 1 λn ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ} ⊆ { n ∈ n : 1 λn ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ εδ } . since right hand belongs to i then left hand also belongs to i and this completes the proof. � int. j. anal. appl. 17 (1) (2019) 20 theorem 3.2. let λ ∈ λ and i is an admissible ideal in n. if f,h ∈ m (g) are bounded functions and f ∼sλi (g) h then f ∼vλi (g) h. proof. let f,h ∈ m (g) are bounded functions and f ∼sλi (g) h. then, there is an m such that∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≤ m for all k. for each ε > 0, 1 λn ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ = 1λn ∑g∈sn&|f(g)h(g)−1|≥ε ∣∣∣f(g)h(g) − 1∣∣∣ + 1 λn ∑ g∈sn&|f(g)h(g)−1|<ε ∣∣∣f(g)h(g) − 1∣∣∣ ≤ m 1 λn ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ + ε 2 then, { n ∈ n : 1 λn ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε } ⊆ { n ∈ n : 1 λn ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ ≥ ε 2m } ∈i. therefore f ∼vλi (g) h. � theorem 3.3. if lim inf λn |sn| > 0 then f ∼si(g) h implies f ∼sλi (g) h. proof. assume that lim inf λn |sn| > 0 there exists a δ > 0 such that λn |sn| ≥ δ for sufficiently large n. for given ε > 0 we have, 1 |sn| { k ≤ |sn| : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε } ⊇ 1 |sn| { k ∈ in : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε } , where in = [n−λn + 1,n]. therefore, 1 |sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ 1|sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ λn |sn| . 1 λn ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ. 1 λn ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ then for any η > 0 we get{ n ∈ n : 1 λn ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ η } ⊆ { n ∈ n : 1|sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ ηδ} ∈i, and this completes the proof. � theorem 3.4. if λ = (λn) ∈ ∆ be such that limn→∞ λn |sn| = 1, then f ∼sλi (g) h is subset of f ∼si(g) h. int. j. anal. appl. 17 (1) (2019) 21 proof. let δ > 0 be given. since limn→∞ λn |sn| = 1, we can choose m ∈ n such that ∣∣∣∣ λn|sn| − 1 ∣∣∣∣ < δ2 , for all n ≥ m. now observe that, for ε > 0 1 |sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ = 1|sn| ∣∣∣{k ≤ |sn|−λn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + 1 |sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≤ |sn|−λn |sn| + 1 |sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≤ 1 − ( 1 − δ 2 ) + 1 |sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ = δ 2 + 1 |sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ , for all n ≥ m. hence,{ g ∈ sn : 1 |sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ } ⊂ { g ∈ sn : 1 |sn| ∣∣∣{k ∈ in : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ2 } ∪{1, 2, ...,m} if f ∼sλi (g) h, then the set on the right hand side belongs to i and so the set on the left hand side also belongs to i. this shows that f ∼si(g) h. � definition 3.11. two nonnegative functions f,h ∈ w(g) are said to be strongly cesáro i-asymptotically equivalent, provided that for every ε > 0, g ∈ sn : 1|sn| ∑ 1≤k≤|sn| ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε   ∈i. it will be denoted by f ∼[c1(i)](g) h. theorem 3.5. if f ∼vλi (g) h, then f ∼[c1(i)](g) h. proof. assume that f ∼vλi (g) h and ε > 0. then, 1 |sn| ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ = 1|sn| |sn|−λn∑ k=1 ∣∣∣f(g)h(g) − 1∣∣∣ + 1|sn| ∑ k∈in,g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≤ 1 λn |sn|−λn∑ k=1 ∣∣∣f(g)h(g) − 1∣∣∣ + 1λn ∑k∈in,g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≤ 2 λn ∑ k∈in,g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ int. j. anal. appl. 17 (1) (2019) 22 and so,  g ∈ sn : 1|sn| ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε   ⊆  g ∈ sn : 1λn ∑ k∈in,g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε2   belongs to i. hence f ∼[c1(i)](g) h. � theorem 3.6. let λ ∈ λ and i is an admissible ideal in n. if f,h ∈ m (g) are bounded and f ∼sλi (g) h then f ∼c1(i)(g) h. proof. suppose that f,h ∈ m (g) and f ∼sλi (g) h. since f,h ∈ m (g), they are bounded, assume that∣∣∣f(g)h(g) − 1∣∣∣ < m for all g. let ε > 0 be given and set ln = { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε2 } . for each ε > 0, 1 λn ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ = 1λn ∑g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ + 1λn ∑g∈sn\ln ∣∣∣f(g)h(g) − 1∣∣∣ ≤ m 1 λn ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + ε2 . then, { n ∈ n : 1 λn ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε } ⊆ { n ∈ n : 1 λn ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ ≥ ε 2m } ∈i. therefore, f ∼vλi (g) h. � theorem 3.7. if f ∼c1(i)(g) h, then, f ∼s(g) h. proof. let f ∼c1(i)(g) h, and ε > 0 given. then, 1 |sn| ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ 1|sn| ∑ g∈sn&|f(g)h(g)−1|≥ε ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε. ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ and so 1 ε. |sn| ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ 1|sn| ∣∣∣∣ { k ≤ |sn| : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ . so for a given δ > 0, { n ∈ n : 1|sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ} ⊆ { n ∈ n : 1|sn| ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ εδ } ∈i. therefore f ∼s(g) h. � int. j. anal. appl. 17 (1) (2019) 23 theorem 3.8. let f, h ∈ m (g). if f ∼s(g) h then f ∼c1(i)(g) h. proof. suppose that f, h ∈ m (g) and f ∼sλ(g) h. then, there is a m such that ∣∣∣f(g)h(g) − 1∣∣∣ ≤ m for all k. given ε > 0, we have 1 |sn| ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ = 1|sn| ∑ g∈sn&|f(g)h(g)−1|≥ε ∣∣∣f(g)h(g) − 1∣∣∣ + 1|sn| ∑ g∈sn&|f(g)h(g)−1|<ε ∣∣∣f(g)h(g) − 1∣∣∣ ≤ 1|sn|.m ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + 1|sn|.ε ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ < ε}∣∣∣ ≤ m |sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ + ε. then, for any δ > 0,{ n ∈ n : 1|sn| ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ δ } ⊆ { n ∈ n : 1|sn| ∣∣∣{k ≤ |sn| : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δm } ∈i. therefore f ∼c1(i)(g) h. � theorem 3.9. let f,h ∈ w(g) two nonnegative functions. then, (a) if f ∼n θ i(g) h, then f ∼s θ i(g) h, (b) if f,h ∈ m (g) and f ∼s θ i(g) h, then f ∼n θ i(g) h. proof. (a). let ε > 0 and f ∼n θ i(g) h. then, we can write∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε. ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ and so 1 εhr ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ 1hr ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε }∣∣∣∣ . then, for any δ > 0 { r ∈ n : 1 hr ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε}∣∣∣ ≥ δ} ⊆ { r ∈ n : 1 hr ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε.δ } ∈i. int. j. anal. appl. 17 (1) (2019) 24 hence, we have f ∼s θ i(g) h. (b) suppose that f,h ∈ m (g) and f ∼s θ i(g) h. since f,h ∈ m (g), they are bounded, assume that∣∣∣f(g)h(g) − 1∣∣∣ < m for all g. 1 hr ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ = 1hr ∑g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ + 1hr ∑g∈sn\ln ∣∣∣f(g)h(g) − 1∣∣∣ ≤ m hr ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ + ε 2 . and define the sets d1 =  r ∈ n : 1hr ∑ g∈sn ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε   and d2 = { r ∈ n : 1 hr ∣∣∣∣ { g ∈ sn : ∣∣∣∣f (g)h (g) − 1 ∣∣∣∣ ≥ ε2 }∣∣∣∣ ≥ ε2m } . if r /∈ d2, then 1hr ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ < ε 2m . also we can get 1 hr ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≤ mhr ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ + ε 2 < ε 2 + ε 2 = ε. thus, r /∈ d1. consequently, we have{ r ∈ n : 1 hr ∑ g∈sn ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε } ⊆ { r ∈ n : 1 hr ∣∣∣{g ∈ sn : ∣∣∣f(g)h(g) − 1∣∣∣ ≥ ε2 }∣∣∣ ≥ ε 2m } ∈i. therefore, f ∼n θ i(g) h. this completes the proof. � references [1] m. day, amenable semigroups, illinois j. math. 1 (1957), 509–544. [2] p. das, e. savaş and s. kr. ghosal, on generalized of certain summability methods using ideals, appl. math. letter, 36 (2011), 1509-1514. [3] s.a. douglass, on a concept of summability in amenable semigroups, math. scand. 28 (1968), 96-102. [4] s.a. douglass, summing sequences for amenable semigroups, michigan math. j. 20 (1973), 169-179. [5] h. fast, sur la convergence statistique, colloq. math. 2 (1951), 241–244. [6] j. a fridy and orhan c., lacunary statistical convergence, pacific j. math., 160(1) (1993), 43-51. [7] a.r. freedman, j.j. sember and m. raphael, some cesàro type summability spaces, proc. lond. math. soc., 37 (1978), 508-520. [8] p. kostyrko, t. salát and w. wilezyński, i-convergence, real anal. exchange, 26(2) (2000), 669-686. [9] m. marouf, asymptotic equivalence and summability, internat. j. math. math. sci., 16(4) (1993), 755-762. int. j. anal. appl. 17 (1) (2019) 25 [10] ö. kişi and e. güler, a generalized statistical convergence via ideals defined by folner sequence on amenable semigroup, in prooceding of 4th international conference on analysis and its applications, kırşehir, turkey (2018), 104-110. [11] ö. kişi and e. güler, σ-asymptotically lacunary statistical equivalent functions on amenable semigroups, far east j. appl. math., 97(6) (2017), 275-287. [12] ö. kişi and b. çakal, on iσ-convergence of folner sequence on amenable semigroups, ntmsci 6(2) (2018), 222-235. [13] p.f. mah, summability in amenable semigroups, trans. amer. math. soc. 156 (1971), 391–403. [14] p.f. mah, matrix summability in amenable semigroups, proc. amer. math. soc. 36 (1972), 414–420. [15] m. mursaleen, λ-statistical convergence, math. slovaca, 50(1) (2000), 111-115. [16] r.f. patterson, on asymptotically statistically equivalent sequences, demostratio math., 36(1) (2003), 149-153. [17] r.f. patterson and e. savaş, on asymptotically lacunary statistical equivalent sequences, thai j. math., 4(2) (2006), 267-272. [18] i.p. pobyvanets, asymptotic equivalence of some linear transformations defined by a nonnegative matrix and reduced to generalized equivalence in the sense of cesàro and abel, mat. fiz. no. 28 (1980), 83–87. [19] e. savaş and p. das, a generalized statistical convergence via ideals, appl. math. lett., 24 (2011), 826–830. [20] e. savaş, on i-asymptotically lacunary statistical equivalent sequences, adv. difference equ., 2013 (2013), art. id 111. [21] f. nuray and b.e. rhoades, some kinds of convergence defined by folner sequences, analysis, 31(4) (2011), 381–390. [22] f. nuray and b.e. rhoades, asymptotically and statistically equivalent functions defined on amenable semigroups, thai j. math., 11(2) (2013), 303–311. [23] i. nomika, folner’s conditions for amenable semigroups, math. scand., 15 (1964), 18–28. [24] i.j. schoenberg, the integrability of certain functions and related summability methods, amer. math. monthly, 66 (1959), 361-375. 1. introduction 2. definitions and notations 3. main results references international journal of analysis and applications volume 17, number 5 (2019), 821-837 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-821 properties of operations for fuzzy soft sets over fully up-semigroups akarachai satirad and aiyared iampan∗ department of mathematics, school of science, university of phayao, phayao 56000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. the aim of this manuscript is to apply distributivity laws of several fuzzy sets for any fuzzy sets and study distributivity laws with any fuzzy soft sets. we investigate properties of some operations for fuzzy soft sets over fully up-semigroups and their interrelation with respect to different operations such as “(restricted) union”, “(extended) intersection”, “and”, and “or”. 1. introduction and preliminaries several researches introduced a new class of algebras related to logical algebras and semigroups such as: in 1993, jun et al. [7] introduced the notion of bci-semigroups. in 2018, iampan [6] introduced the notion of fully up-semigroups. in 1999, to solve complicated problems in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as the probability theory, the theory of (intuitionistic) fuzzy sets, the theory of vague sets, the theory of interval mathematics, and the theory of rough sets. however, all of these theories have their own difficulties which are pointed out in [11]. in 2001, maji et al. [10] introduced the concept of fuzzy soft sets as a generalization of the standard soft sets, and presented an application of fuzzy soft sets in a decision making received 2019-06-02; accepted 2019-07-11; published 2019-09-02. 2010 mathematics subject classification. 03g25; 08a72. key words and phrases. fully up-semigroup; fuzzy soft set; (restricted) union; (extended) intersection; and; or. this work was supported by the unit of excellence, university of phayao. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 821 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-821 int. j. anal. appl. 17 (5) (2019) 822 problem. in 2010, jun et al. [8] applied fuzzy soft set for dealing with several kinds of theories in bck/bcialgebras. the notions of fuzzy soft bck/bci-algebras, (closed) fuzzy soft ideals and fuzzy soft p-ideals are introduced, and related properties are investigated. in 2013, rehman et al. [13] studied some operations of fuzzy soft sets and give fundamental properties of fuzzy soft sets. they discuss properties of fuzzy soft sets and their interrelation with respect to different operations such as union, intersection, restricted union and extended intersection. then, they illustrate properties of and and or operations by giving counter examples. also we prove that certain de morgan’s laws hold in fuzzy soft set theory with respect to different operations on fuzzy soft sets. in 2019, satirad and iampan [16] introduced ten types of fuzzy soft sets over fully up-semigroups, and investigate the algebraic properties of fuzzy soft sets under the operations of (extended) intersection and (restricted) union. before we begin our study, we will give the definition of a up-algebra. definition 1.1. [5] an algebra a = (a, ·, 0) of type (2, 0) is called a up-algebra where a is a nonempty set, · is a binary operation on a, and 0 is a fixed element of a (i.e., a nullary operation) if it satisfies the following axioms: (up-1): (∀x, y, z ∈ a)((y ·z) · ((x ·y) · (x ·z)) = 0), (up-2): (∀x ∈ a)(0 ·x = x), (up-3): (∀x ∈ a)(x · 0 = 0), and (up-4): (∀x, y ∈ a)(x ·y = 0, y ·x = 0 ⇒ x = y). from [5], we know that the notion of up-algebras is a generalization of ku-algebras (see [12]). on a up-algebra a = (a, ·, 0), we define a binary relation ≤ on a [5] as follows: (∀x, y ∈ a)(x ≤ y ⇔ x ·y = 0). example 1.1. [18] let x be a universal set and let ω ∈p(x) where p(x) means the power set of x. let pω(x) = {a ∈p(x) | ω ⊆ a}. define a binary operation · on pω(x) by putting a·b = b∩(ac∪ω) for all a, b ∈pω(x) where ac means the complement of a subset a. then (pω(x), ·, ω) is a up-algebra and we shall call it the generalized power up-algebra of type 1 with respect to ω. let pω(x) = {a ∈p(x) | a ⊆ ω}. define a binary operation ∗ on pω(x) by putting a ∗ b = b ∪ (ac ∩ ω) for all a, b ∈ pω(x). then (pω(x),∗, ω) is a up-algebra and we shall call it the generalized power up-algebra of type 2 with respect to ω. in particular, (p(x), ·,∅) is a up-algebra and we shall call it the power up-algebra of type 1, and (p(x),∗, x) is a up-algebra and we shall call it the power up-algebra of type 2. int. j. anal. appl. 17 (5) (2019) 823 example 1.2. [3] let n be the set of all natural numbers with two binary operations ◦ and • defined by (∀x, y ∈ n)  x◦y =   y if x < y,0 otherwise   and (∀x, y ∈ n)  x•y =   y if x > y or x = 0,0 otherwise   . then (n,◦, 0) and (n,•, 0) are up-algebras. for more examples of up-algebras, see [2, 6, 17, 18]. in a up-algebra a = (a, ·, 0), the following assertions are valid (see [5, 6]). (∀x ∈ a)(x ·x = 0), (1.1) (∀x, y, z ∈ a)(x ·y = 0, y ·z = 0 ⇒ x ·z = 0), (1.2) (∀x, y, z ∈ a)(x ·y = 0 ⇒ (z ·x) · (z ·y) = 0), (1.3) (∀x, y, z ∈ a)(x ·y = 0 ⇒ (y ·z) · (x ·z) = 0), (1.4) (∀x, y ∈ a)(x · (y ·x) = 0), (1.5) (∀x, y ∈ a)((y ·x) ·x = 0 ⇔ x = y ·x), (1.6) (∀x, y ∈ a)(x · (y ·y) = 0), (1.7) (∀a, x, y, z ∈ a)((x · (y ·z)) · (x · ((a ·y) · (a ·z))) = 0), (1.8) (∀a, x, y, z ∈ a)((((a ·x) · (a ·y)) ·z) · ((x ·y) ·z) = 0), (1.9) (∀x, y, z ∈ a)(((x ·y) ·z) · (y ·z) = 0), (1.10) (∀x, y, z ∈ a)(x ·y = 0 ⇒ x · (z ·y) = 0), (1.11) (∀x, y, z ∈ a)(((x ·y) ·z) · (x · (y ·z)) = 0), and (1.12) (∀a, x, y, z ∈ a)(((x ·y) ·z) · (y · (a ·z)) = 0). (1.13) definition 1.2. [6] let a be a nonempty set, · and ∗ are binary operations on a, and 0 is a fixed element of a (i.e., a nullary operation). an algebra a = (a, ·,∗, 0) of type (2, 2, 0) in which (a, ·, 0) is a up-algebra and (a,∗) is a semigroup is called a fully up-semigroup (in short, an f-up-semigroup) if the operation “∗” is distributive (on both sides) over the operation “·”. int. j. anal. appl. 17 (5) (2019) 824 definition 1.3. [20] a fuzzy set f in a nonempty set u (or a fuzzy subset of u) is described by its membership function ff. to every point x ∈ u, this function associates a real number ff(x) in the interval [0, 1]. the number ff(x) is interpreted for the point as a degree of belonging x to the fuzzy set f, that is, f := {(x, ff(x)) | x ∈ u}. we say that a fuzzy set f in u is constant if its membership function ff is constant. rosenfeld [14] introduced the notion of fuzzy subsemigroups (resp., fuzzy ideals) of semigroups as follows: definition 1.4. a fuzzy set f in a semigroup a = (a,∗) is called (1) a fuzzy subsemigroup of a if (∀x, y ∈ a)(ff(x∗y) ≥ min{ff(x), ff(y)}). (2) a fuzzy ideal of a if (∀x, y ∈ a)(ff(x∗y) ≥ max{ff(x), ff(y)}). clearly, a fuzzy ideal is a fuzzy subsemigroup. definition 1.5. [9] let {fi}i∈i be a nonempty family of fuzzy sets in a nonempty set u where i is an arbitrary index set. the intersection of fi, denoted by ⋂ i∈i fi, is described by its membership function f⋂ i∈i fi which defined as follows: (∀x ∈ u)(f⋂ i∈i fi (x) = inf{ffi(x)}i∈i ). the union of fi, denoted by ⋃ i∈i fi, is described by its membership function f ⋃ i∈i fi which defined as follows: (∀x ∈ u)(f⋃ i∈i fi (x) = sup{ffi(x)}i∈i ). theorem 1.1. let fi and f be fuzzy sets in a nonempty set x where i is a nonempty set. then the following properties hold: (1) f ∩ ( ⋃ i∈i fi) = ⋃ i∈i (f ∩ fi), (2) ( ⋃ i∈i fi) ∩ f = ⋃ i∈i (fi ∩ f), (3) f ∪ ( ⋂ i∈i fi) = ⋂ i∈i (f ∪ fi), and (4) ( ⋂ i∈i fi) ∪ f = ⋂ i∈i (fi ∪ f). proof. let x ∈ x. (1) first, we investigate left hand side of the equality. assume that ⋃ i∈i fi = f ∪. then f ∩ ( ⋃ i∈i fi) = f ∩ f ∪. also, ff∩f∪ (x) = min{ff(x), ff∪ (x)} = min{ff(x), f⋃i∈i fi (x)} = min{ff(x), sup{ffi(x)}i∈i}. int. j. anal. appl. 17 (5) (2019) 825 consider the right hand side of the equality. assume that f ∩ fi = f∩i for all i ∈ i. then f⋃ i∈i f ∩ i (x) = sup{ff∩ i (x)}i∈i = sup{ff∩fi (x)}i∈i = sup{min{ff(x), ffi (x)}}i∈i. it is clear that min{ff(x), sup{ffi(x)}i∈i} = sup{min{ff(x), ffi (x)}}i∈i . therefore, f∩( ⋃ i∈i fi) = ⋃ i∈i (f∩ fi). (2) by using techniques as in (1), then (2) can be derived. (3) first, we investigate left hand side of the equality. assume that ⋂ i∈i fi = f ∩. then f ∪ ( ⋂ i∈i fi) = f ∪ f∩. also, ff∪f∩ (x) = max{ff(x), ff∩ (x)} = max{ff(x), f⋂ i∈i fi (x)} = max{ff(x), inf{ffi(x)}i∈i}. consider the right hand side of the equality. assume that f ∪ fi = f∪i for all i ∈ i. then f⋂ i∈i f ∪ i (x) = inf{ff∪ i (x)}i∈i = inf{ff∪fi (x)}i∈i = inf{max{ff(x), ffi (x)}}i∈i. it is clear that max{ff(x), inf{ffi(x)}i∈i} = inf{max{ff(x), ffi (x)}}i∈i . therefore, f∪( ⋂ i∈i fi) = ⋂ i∈i (f∪ fi). (4) by using techniques as in (3), then (4) can be derived. � somjanta et al. [19], guntasow et al. [4], and satirad and iampan [16] introduced the notion of fuzzy up-subalgebras (resp., fuzzy near up-filters, fuzzy up-filters, fuzzy up-ideals, fuzzy strongly up-ideals) of up-algebras as follows: definition 1.6. a fuzzy set f in a up-algebra a = (a, ·, 0) is called (1) a fuzzy up-subalgebra of a if (∀x, y ∈ a)(ff(x ·y) ≥ min{ff(x), ff(y)}). (2) a fuzzy near up-filter of a if (i) (∀x ∈ a)(ff(0) ≥ ff(x)), and (ii) (∀x, y ∈ a)(ff(x ·y) ≥ ff(y)). (3) a fuzzy up-filter of a if (i) (∀x ∈ a)(ff(0) ≥ ff(x)), and int. j. anal. appl. 17 (5) (2019) 826 (ii) (∀x, y ∈ a)(ff(y) ≥ min{ff(x ·y), ff(x)}). (4) a fuzzy up-ideal of a if (i) (∀x ∈ a)(ff(0) ≥ ff(x)), and (ii) (∀x, y, z ∈ a)(ff(x ·z) ≥ min{ff(x · (y ·z)), ff(y)}). (5) a fuzzy strongly up-ideal of a if (i) (∀x ∈ a)ff(0) ≥ ff(x), and (ii) (∀x, y, z ∈ a)(ff(x) ≥ min{ff((z ·y) · (z ·x)), ff(y)}). we know that the notion of fuzzy up-subalgebras is a generalization of fuzzy near up-filters, the notion of fuzzy near up-filters is a generalization of fuzzy up-filters, the notion of fuzzy up-filters is a generalization of fuzzy up-ideals, and the notion of fuzzy up-ideals is a generalization of fuzzy strongly up-ideals. moreover, fuzzy strongly up-ideals and constant fuzzy sets coincide in up-algebras. satirad and iampan [15,16] introduced the notion of fuzzy ups-subalgebras (resp., fuzzy upi-subalgebras, fuzzy near ups-filters, fuzzy near upi-filters, fuzzy ups-filters, fuzzy upi-filters, fuzzy ups-ideals, fuzzy upiideals, fuzzy strongly ups-ideals, fuzzy strongly upi-ideals) of f-up-semigroups as follows: definition 1.7. a fuzzy set f in an f-up-semigroup a = (a, ·,∗, 0) is called (1) a fuzzy ups-subalgebra of a if f is a fuzzy up-subalgebra of (a, ·, 0) and a fuzzy subsemigroup of (a,∗). (2) a fuzzy upi-subalgebra of a if f is a fuzzy up-subalgebra of (a, ·, 0) and a fuzzy ideal of (a,∗). (3) a fuzzy near ups-filter of a if f is a fuzzy near up-filter of (a, ·, 0) and a fuzzy subsemigroup of (a,∗). (4) a fuzzy near upi-filter of a if f is a fuzzy near up-filter of (a, ·, 0) and a fuzzy ideal of (a,∗). (5) a fuzzy ups-filter of a if f is a fuzzy up-filter of (a, ·, 0) and a fuzzy subsemigroup of (a,∗). (6) a fuzzy upi-filter of a if f is a fuzzy up-filter of (a, ·, 0) and a fuzzy ideal of (a,∗). (7) a fuzzy ups-ideal of a if f is a fuzzy up-ideal of (a, ·, 0) and a fuzzy subsemigroup of (a,∗). (8) a fuzzy upi-ideal of a if f is a fuzzy up-ideal of (a, ·, 0) and a fuzzy ideal of (a,∗). (9) a fuzzy strongly ups-ideal of a if f is a fuzzy strongly up-ideal of (a, ·, 0) and a fuzzy subsemigroup of (a,∗). (10) a fuzzy strongly upi-ideal of a if f is a fuzzy strongly up-ideal of (a, ·, 0) and a fuzzy ideal of (a,∗). theorem 1.2. [15, 16] the intersection of any nonempty family of fuzzy ups-subalgebras (resp., fuzzy upi-subalgebras, fuzzy near ups-filters, fuzzy near upi-filters, fuzzy ups-filters, fuzzy upi-filters, fuzzy upsideals, fuzzy upi-ideals, fuzzy strongly ups-ideals, fuzzy strongly upi-ideals) of an f-up-semigroup is also int. j. anal. appl. 17 (5) (2019) 827 a fuzzy ups-subalgebra (resp., fuzzy upi-subalgebra, fuzzy near ups-filter, fuzzy near upi-filter, fuzzy upsfilter, fuzzy upi-filter, fuzzy ups-ideal, fuzzy upi-ideal, fuzzy strongly ups-ideal, fuzzy strongly upi-ideal). theorem 1.3. [15, 16] the union of any nonempty family of fuzzy near upi-filters (resp., fuzzy strongly ups-ideals, fuzzy strongly upi-ideals) of an f-up-semigroup is also a fuzzy near upi-filter (resp., fuzzy strongly ups-ideal, fuzzy strongly upi-ideal). 2. fuzzy soft sets over fully up-semigroups from now on, we shall let a be an f-up-semigroup a = (a, ·,∗, 0) and p be a set of parameters. let f(a) denotes the set of all fuzzy sets in a. a subset e of p is called a set of statistics. definition 2.1. let e ⊆ p . a pair (f̃, e) is called a fuzzy soft set over a if f̃ is a mapping given by f̃ : e → f(a), that is, a fuzzy soft set is a statistic family of fuzzy sets in a. in general, for every e ∈ e, f̃[e] := {(x, f f̃[e] (x)) | x ∈ a} is a fuzzy set in a and it is called a fuzzy value set of statistic e. definition 2.2. let (f̃, e1) and (g̃, e2) be two fuzzy soft sets over a common universe u. the union [10] of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1) ∪ (g̃, e2) = (h̃, e) satisfying the following conditions: (i) e = e1 ∪e2 and (ii) for all e ∈ e, h̃[e] =   f̃[e] if e ∈ e1 \e2 g̃[e] if e ∈ e2 \e1 f̃[e] ∪ g̃[e] if e ∈ e1 ∩e2. the restricted union [13] of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1)d(g̃, e2) = (h̃, e) satisfying the following conditions: (i) e = e1 ∩e2 6= ∅ and (ii) h̃[e] = f̃[e] ∪ g̃[e] for all e ∈ e. definition 2.3. [10] let (f̃, e1) and (g̃, e2) be two fuzzy soft sets over a common universe u. the or of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1) ∨ (g̃, e2) = (h̃, e) satisfying the following conditions: (i) e = e1 ×e2 and (ii) h̃[e1, e2] = f̃[e1] ∪ g̃[e2] for all (e1, e2) ∈ e. definition 2.4. let (f̃, e1) and (g̃, e2) be two fuzzy soft sets over a common universe u. the extended intersection [13] of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1)∩(g̃, e2) = (h̃, e) satisfying the following conditions: int. j. anal. appl. 17 (5) (2019) 828 (i) e = e1 ∪e2 and (ii) for all e ∈ e, h̃[e] =   f̃[e] if e ∈ e1 \e2 g̃[e] if e ∈ e2 \e1 f̃[e] ∩ g̃[e] if e ∈ e1 ∩e2. the intersection [1] of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1) e (g̃, e2) = (h̃, e) satisfying the following conditions: (i) e = e1 ∩e2 6= ∅ and (ii) h̃[e] = f̃[e] ∩ g̃[e] for all e ∈ e. definition 2.5. [10] let (f̃, e1) and (g̃, e2) be two fuzzy soft sets over a common universe u. the and of (f̃, e1) and (g̃, e2) is defined to be the fuzzy soft set (f̃, e1) ∧ (g̃, e2) = (h̃, e) satisfying the following conditions: (i) e = e1 ×e2 and (ii) h̃[e1, e2] = f̃[e1] ∩ g̃[e2] for all (e1, e2) ∈ e. definition 2.6. a fuzzy soft set (f̃, e) over a is called a fuzzy soft ups-subalgebra based on e ∈ e (we shortly call an e-fuzzy soft ups-subalgebra) of a if a fuzzy set f̃[e] in a is a fuzzy ups-subalgebra of a. if (f̃, e) is an e-fuzzy soft ups-subalgebra of a for all e ∈ e, we say that (f̃, e) is a fuzzy soft ups-subalgebra of a. we can call fuzzy soft sets that fuzzy soft upi-subalgebras (fuzzy soft near ups-filters, fuzzy soft near upi-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals, fuzzy soft strongly ups-ideals, and fuzzy soft strongly upi-ideals ) based on a statistic or fuzzy soft upi-subalgebras (fuzzy soft near ups-filters, fuzzy soft near upi-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals, fuzzy soft strongly ups-ideals, and fuzzy soft strongly upi-ideals ) of a if fuzzy soft sets satisfy statement in definition 2.6. we will introduce the notions of the restricted union, the union, the intersection, the extended intersection, the and, and the or of any fuzzy soft sets and apply to f-up-semigroups. definition 2.7. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the restricted union of (f̃i, ei) is defined to be the fuzzy soft set di∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ⋂ i∈i ei 6= ∅ and (ii) f̃[e] = ⋃ i∈i f̃i[e] for all e ∈ e. int. j. anal. appl. 17 (5) (2019) 829 theorem 2.1. the restricted union of family of fuzzy soft near upi-filters of a is also a fuzzy soft near upi-filter. proof. let (f̃i, ei) be a fuzzy soft near upi-filters of a for all i ∈ i. assume that di∈i (f̃i, ei) = (f̃, e) be the restricted union of (f̃i, ei) for all i ∈ i. then e = ⋂ i∈i ei 6= ∅. let e ∈ e. by theorem 1.3, we have f̃[e] = ⋃ i∈i f̃i[e] is a fuzzy near upi-filter of a. therefore, (f̃, e) is an e-fuzzy soft near upi-filter of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft near upi-filter of a. � in the same way as theorem 2.1, we can use theorem 1.3 to prove that the restricted union of family of fuzzy soft strongly ups-ideals (resp., fuzzy soft strongly upi-ideals) of a is also a fuzzy soft strongly ups-ideal (resp., fuzzy soft strongly upi-ideal). definition 2.8. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the union of (f̃i, ei) is defined to be the fuzzy soft set ⋃ i∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ⋃ i∈i ei and (ii) f̃[e] = ⋃ j∈j f̃j[e] for all e ∈ e with e ∈ ⋂ j∈j ej − ⋃ k∈i−j ek where ∅ 6= j ⊆ i. theorem 2.2. the union of family of fuzzy soft near upi-filters of a is also a fuzzy soft near upi-filter. proof. let (f̃i, ei) be a fuzzy soft near upi-filters of a for all i ∈ i. assume that ⋂ i∈i (f̃i, ei) = (f̃, e) be the union of (f̃i, ei) for all i ∈ i. then e = ⋃ i∈i ei. let e ∈ e. case 1: |j| = |i|. by theorem 2.1, we have f̃[e] = ⋂ i∈i f̃i[e] is a fuzzy near upi-filter of a. case 2: |j| = 1, that is, j is a singleton set. then f̃[e] = ⋂ j∈{j} f̃j[e] = f̃j[e] is a fuzzy near upi-filter of a. case 3: 1 < |j| < |i|. then f̃[e] = ⋂ j∈j f̃j[e]. since e ∈ ej for all j ∈ j and e /∈ ek for some k ∈ i − j and by same case 1, we have f̃[e] is a fuzzy near upi-filter of a. therefore, (f̃, e) is an e-fuzzy soft near upi-filter of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft near upi-filter of a. � in the same way as theorem 2.2, we can prove that the union of family of fuzzy soft strongly ups-ideals (resp., fuzzy soft strongly upi-ideals) of a is also a fuzzy soft strongly ups-ideal (resp., fuzzy soft strongly upi-ideal). in [16], we show that the union of two fuzzy soft ups-subalgebras (resp., fuzzy soft upi-subalgebras, fuzzy soft near ups-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals) of a is not fuzzy soft ups-subalgebra (resp., fuzzy soft upi-subalgebra, fuzzy soft near ups-filter, fuzzy soft ups-filter, fuzzy soft upi-filter, fuzzy soft ups-ideal, fuzzy soft upi-ideal). int. j. anal. appl. 17 (5) (2019) 830 definition 2.9. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the intersection of (f̃i, ei) is defined to be the fuzzy soft setei∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ⋂ i∈i ei 6= ∅ and (ii) f̃[e] = ⋂ i∈i f̃i[e] for all e ∈ e. theorem 2.3. the intersection of family of fuzzy soft ups-subalgebras of a is also a fuzzy soft upssubalgebra. proof. let (f̃i, ei) be a fuzzy soft ups-subalgebras of a for all i ∈ i. assume that ei∈i (f̃i, ei) = (f̃, e) is the intersection of (f̃i, ei) for all i ∈ i. then e = ⋂ i∈i ei 6= ∅. let e ∈ e. by theorem 1.2, we have f̃[e] = ⋂ i∈i f̃i[e] is a fuzzy ups-subalgebra of a. therefore, (f̃, e) is an e-fuzzy soft ups-subalgebra of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft ups-subalgebra of a. � in the same way as theorem 2.3, we can use theorem 1.2 to prove that the intersection of family of fuzzy soft upi-subalgebras (resp., fuzzy soft near ups-filters, fuzzy soft near upi-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals, fuzzy soft strongly ups-ideals, fuzzy soft strongly upi-ideals) of a is also a fuzzy soft upi-subalgebra (resp., fuzzy soft near ups-filter, fuzzy soft near upi-filter, fuzzy soft ups-filter, fuzzy soft upi-filter, fuzzy soft ups-ideal, fuzzy soft upi-ideal, fuzzy soft strongly ups-ideal, fuzzy soft strongly upi-ideal). definition 2.10. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the extended intersection of (f̃i, ei) is defined to be the fuzzy soft set⋂ i∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ⋃ i∈i ei and (ii) f̃[e] = ⋂ j∈j f̃j[e] for all e ∈ e with e ∈ ⋂ j∈j ej − ⋃ k∈i−j ek where ∅ 6= j ⊆ i. theorem 2.4. the extended intersection of family of fuzzy soft ups-subalgebras of a is also a fuzzy soft ups-subalgebra. proof. let (f̃i, ei) be a fuzzy soft ups-subalgebras of a for all i ∈ i. assume that ⋂ i∈i (f̃i, ei) = (f̃, e) is the extended intersection of (f̃i, ei) for all i ∈ i. then e = ⋃ i∈i ei. let e ∈ e. case 1: |j| = |i|. by theorem 2.3, we have f̃[e] = ⋂ i∈i f̃i[e] is a fuzzy ups-subalgebra of a. case 2: |j| = 1, that is, j is a singleton set. then f̃[e] = ⋂ j∈{j} f̃j[e] = f̃j[e] is a fuzzy ups-subalgebra of a. case 3: 1 < |j| < |i|. then f̃[e] = ⋂ j∈j f̃j[e]. since e ∈ ej for all j ∈ j and e /∈ ek for some k ∈ i − j and by same case 1, we have f̃[e] is a fuzzy ups-subalgebra of a. int. j. anal. appl. 17 (5) (2019) 831 therefore, (f̃, e) is an e-fuzzy soft ups-subalgebra of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft ups-subalgebra of a. � in the same way as theorem 2.4, we can prove that the extended intersection of family of fuzzy soft upisubalgebras (resp., fuzzy soft near ups-filters, fuzzy soft near upi-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals, fuzzy soft strongly ups-ideals, fuzzy soft strongly upi-ideals) of a is also a fuzzy soft upi-subalgebra (resp., fuzzy soft near ups-filter, fuzzy soft near upifilter, fuzzy soft ups-filter, fuzzy soft upi-filter, fuzzy soft ups-ideal, fuzzy soft upi-ideal, fuzzy soft strongly ups-ideal, fuzzy soft strongly upi-ideal). definition 2.11. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the and of (f̃i, ei) is defined to be the fuzzy soft set ∧ i∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ∏ i∈i ei and (ii) f̃[(ei)i∈i ] = ⋂ i∈i f̃i[ei] for all (ei)i∈i ∈ e. theorem 2.5. the and of family of fuzzy soft ups-subalgebras of a is also a fuzzy soft ups-subalgebra. proof. let (f̃i, ei) be a fuzzy soft ups-subalgebras of a for all i ∈ i. by means of definition 2.11, we assume that ∧ i∈i (f̃i, ei) = (f̃, e) such that e = ∏ i∈i ei and f̃[(ei)i∈i ] = ⋂ i∈i f̃i[ei] for all (ei)i∈i ∈ e. assume that e = (ei)i∈i ∈ e and let x, y ∈ a. then f f̃[e] (x ·y) = f⋂ i∈i f̃i[ei] (x ·y) = inf{f f̃i[ei] (x ·y)}i∈i ≥ inf{min{f f̃i[ei] (x), f f̃i[ei] (y)}}i∈i = min{inf{f f̃i[ei] (x)}i∈i, inf{ff̃i[ei](y)}i∈i} = min{f⋂ i∈i f̃i[ei] (x), f⋂ i∈i f̃i[ei] (y)} = min{f f̃[e] (x), f f̃[e] (y)}, and f f̃[e] (x∗y) = f⋂ i∈i f̃i[ei] (x∗y) = inf{f f̃i[ei] (x∗y)}i∈i ≥ inf{min{f f̃i[ei] (x), f f̃i[ei] (y)}}i∈i = min{inf{f f̃i[ei] (x)}i∈i, inf{ff̃i[ei](y)}i∈i} = min{f⋂ i∈i f̃i[ei] (x), f⋂ i∈i f̃i[ei] (y)} = min{f f̃[e] (x), f f̃[e] (y)}. int. j. anal. appl. 17 (5) (2019) 832 therefore, f̃[e] is a fuzzy ups-subalgebra of a, that is, (f̃, e) is an e-fuzzy soft ups-subalgebra of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft ups-subalgebra of a. � in the same way as theorem 2.5, we can use theorem 1.2 to prove that the and of family of fuzzy soft upi-subalgebras (resp., fuzzy soft near ups-filters, fuzzy soft near upi-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals, fuzzy soft strongly ups-ideals, fuzzy soft strongly upi-ideals) of a is also a fuzzy soft upi-subalgebra (resp., fuzzy soft near ups-filter, fuzzy soft near upi-filter, fuzzy soft ups-filter, fuzzy soft upi-filter, fuzzy soft ups-ideal, fuzzy soft upi-ideal, fuzzy soft strongly ups-ideal, fuzzy soft strongly upi-ideal). definition 2.12. let {(f̃i, ei) | i ∈ i} be a nonempty family of fuzzy soft sets over a common universe u where i is an arbitrary index set. the or of (f̃i, ei) is defined to be the fuzzy soft set ∨ i∈i (f̃i, ei) = (f̃, e) satisfying the following conditions: (i) e = ∏ i∈i ei and (ii) f̃[(ei)i∈i ] = ⋃ i∈i f̃i[ei] for all (ei)i∈i ∈ e. theorem 2.6. the or of family of fuzzy soft near upi-filters of a is also a fuzzy soft near upi-filter. proof. let (f̃i, ei) be a fuzzy soft near upi-filters of a for all i ∈ i. by means of definition 2.12, we assume that ∨ i∈i (f̃i, ei) = (f̃, e) such that e = ∏ i∈i ei and f̃[(ei)i∈i ] = ⋃ i∈i f̃i[ei] for all (ei)i∈i ∈ e. assume that e = (ei)i∈i ∈ e and let x, y ∈ a. then f f̃[e] (0) = f⋃ i∈i f̃i[ei] (0) = sup{f f̃i[ei] (0)}i∈i ≥ sup{f f̃i[ei] (x)}i∈i = f⋃ i∈i f̃i[ei] (x) = f f̃[e] (x), f f̃[e] (x ·y) = f⋃ i∈i f̃i[ei] (x ·y) = sup{f f̃i[ei] (x ·y)}i∈i ≥ sup{f f̃i[ei] (y)}i∈i = f⋃ i∈i f̃i[ei] (y) = f f̃[e] (y), and int. j. anal. appl. 17 (5) (2019) 833 f f̃[e] (x∗y) = f⋃ i∈i f̃i[ei] (x∗y) = sup{f f̃i[ei] (x∗y)}i∈i ≥ sup{max{f f̃i[ei] (x), f f̃i[ei] (y)}}i∈i = max{sup{f f̃i[ei] (x)}i∈i, sup{ff̃i[ei](y)}i∈i} = max{f⋂ i∈i f̃i[ei] (x), f⋂ i∈i f̃i[ei] (y)} = max{f f̃[e] (x), f f̃[e] (y)}. therefore, f̃[e] is a fuzzy near upi-filter of a, that is, (f̃, e) is an e-fuzzy soft near upi-filter of a. but since e is an arbitrary statistic of e, we have (f̃, e) is a fuzzy soft near upi-filter of a. � in the same way as theorem 2.6, we can use theorem 1.3 to prove that the or of family of fuzzy soft strongly ups-ideals (resp., fuzzy soft strongly upi-ideals) of a is also a fuzzy soft strongly ups-ideal (resp., fuzzy soft strongly upi-ideal). the following example shows that the or of two fuzzy soft ups-subalgebras of a is not fuzzy soft upssubalgebra. example 2.1. let a be the set of four series of the iphone, that is, a = {5, 6, 7, x}. define two binary operations · and ∗ on a as the following cayley tables: · x 7 6 5 x x 7 6 5 7 x x 6 5 6 x 7 x 5 5 x 7 6 x ∗ x 7 6 5 x x x x x 7 x x x x 6 x x x 7 5 x x 7 x then a = (a, ·,∗, x) is an f-up-semigroup. let (f̃1, e1) and (f̃2, e2) be two fuzzy soft sets over a where e1 := {price, beauty, specifications}and e2 := {price, stability} with f̃1[price], f̃1[beauty], f̃1[specifications], f̃2[price], and f̃2[stability] are fuzzy sets in a defined as follows: f̃1 x 7 6 5 price 0.9 0.7 0.9 0.2 beauty 1 0.8 0.3 0.2 specifications 0.6 0.5 0.3 0.4 f̃2 x 7 6 5 price 0.9 0.3 0.2 0.8 stability 0.7 0.2 0.5 0.2 int. j. anal. appl. 17 (5) (2019) 834 then (f̃1, e1) and (f̃2, e2) are two fuzzy soft ups-subalgebras of a. since (price, price) ∈ e1 ×e2, we have (f f̃1[price]∪f̃2[price] )(5 ∗ 6) = (f f̃1[price]∪f̃2[price] )(7) = 0.7 � 0.8 = min{0.8, 0.9} = min{(f f̃1[price]∪f̃2[price] )(5), (f f̃1[price]∪f̃2[price] )(6)}. thus f̃1[price]∪f̃2[price] is not a fuzzy ups-subalgebra of a, that is, (f̃1, e1)∪(f̃2, e2) is not a (price, price)fuzzy soft ups-subalgebra of a. hence, (f̃1, e1)∪(f̃2, e2) is not a fuzzy soft ups-subalgebra of a. moreover, (f̃1, e1) ∨ (f̃2, e2) is not a fuzzy soft ups-subalgebra of a. we can apply those examples in [16] to check that the or of two fuzzy soft upi-subalgebras (resp., fuzzy soft near ups-filters, fuzzy soft ups-filters, fuzzy soft upi-filters, fuzzy soft ups-ideals, fuzzy soft upi-ideals) of a is not fuzzy soft upi-subalgebra (resp., fuzzy soft near ups-filter, fuzzy soft ups-filter, fuzzy soft upi-filter, fuzzy soft ups-ideal, fuzzy soft upi-ideal). we prove that certain distributive laws hold in fuzzy soft set theory with respect to the restricted union, the union, the intersection, and the extended intersection on any fuzzy soft sets. theorem 2.7. let (f̃i, ei) and (f̃, e) be fuzzy soft sets over a common universe u where i is a nonempty set. then the following properties hold: (1) (f̃, e) e ( ⋃ i∈i (f̃i, ei)) = ⋃ i∈i ((f̃, e) e (f̃i, ei)), (2) ( ⋃ i∈i (f̃i, ei)) e (f̃, e) = ⋃ i∈i ((f̃i, ei) e (f̃, e)), (3) (f̃, e) d ( ⋂ i∈i (f̃i, ei)) = ⋂ i∈i ((f̃, e) d (f̃i, ei)), (4) ( ⋂ i∈i (f̃i, ei)) d (f̃, e) = (f̃i, ei)) d ⋂ i∈i ((f̃, e), (5) (f̃, e) ∩ (di∈i (f̃i, ei)) =di∈i ((f̃, e) ∩ (f̃i, ei)), (6) (di∈i (f̃i, ei)) ∩ (f̃, e) =di∈i ((f̃i, ei) ∩ (f̃, e)), (7) (f̃, e) ∪ (ei∈i (f̃i, ei)) =ei∈i ((f̃, e) ∪ (f̃i, ei)), (8) (ei∈i (f̃i, ei)) ∪ (f̃, e) =ei∈i ((f̃i, ei) ∪ (f̃, e)), (9) (f̃, e) e (di∈i (f̃i, ei)) =di∈i ((f̃, e) e (f̃i, ei)), (10) (di∈i (f̃i, ei)) e (f̃, e) =di∈i ((f̃i, ei) e (f̃, e)), (11) (f̃, e) d (ei∈i (f̃i, ei)) =ei∈i ((f̃, e) d (f̃i, ei)), and (12) (ei∈i (f̃i, ei)) d (f̃, e) =ei∈i ((f̃i, ei) d (f̃, e)). proof. (1) first, we investigate left hand side of the equality. suppose that ⋃ i∈i (f̃i, ei) = (g̃, e u ) is the union of (f̃i, ei) for all i ∈ i. then eu = ⋃ i∈i ei and for any e ∈ e u , g̃[e] = ⋃ j∈j f̃j[e] with int. j. anal. appl. 17 (5) (2019) 835 e ∈ ⋂ j∈j ej− ⋃ k∈i−j ek where ∅ 6= j ⊆ i. thus (f̃, e)e( ⋃ i∈i (f̃i, ei)) = (f̃, e)e(g̃, e u ) = (h̃, eui ). for any e ∈ eui = e∩eu 6= ∅, h̃[e] = f̃[e]∩g̃[e] where e∩eu = e∩( ⋃ i∈i ei) = ⋃ i∈i (e∩ei). by considering g̃ as piecewise defined function, we have h̃[e] = f̃[e]∩( ⋃ j∈j f̃j[e]) with e ∈ ⋂ j∈j (e∩ej)− ⋃ k∈i−j (e∩ek) where ∅ 6= j ⊆ i. consider the right hand side of the equality. suppose that (f̃, e) e (f̃i, ei) = (̃ii, e i i ) is the intersection of (f̃, e) and (f̃i, ei) for all i ∈ i. then eii = e ∩ ei 6= ∅ and for any e ∈ e i i , ĩi[e] = f̃[e] ∩ f̃i[e]. now, ⋃ i∈i ((f̃, e) e (f̃i, ei)) = ⋃ i∈i (̃ii, e i i ) = (j̃, e iu ), where eiu = ⋃ i∈i e i i = ⋃ i∈i (e ∩ ei). for any e ∈ eiu , j̃[e] = ⋃ j∈j ĩj[e] with e ∈ ⋂ j∈j e i j − ⋃ k∈i−j e i k where ∅ 6= j ⊆ i. considering ĩi as piecewise functions for all i ∈ i, we have j̃[e] = ⋃ j∈j (f̃[e] ∩ f̃j[e]) with e ∈ ⋂ j∈j (e ∩ ej) − ⋃ k∈i−j (e ∩ ek) where ∅ 6= j ⊆ i. by theorem 1.1 (1), it is clear that h̃ and j̃ are same set-valued mapping. hence, (f̃, e) e ( ⋃ i∈i (f̃i, ei)) = ⋃ i∈i ((f̃, e) e (f̃i, ei)). (2) by using techniques as in (1) and by theorem 1.1 (2), then (2) can be derived. (3) by using techniques as in (1) and by theorem 1.1 (3), then (3) can be derived. (4) by using techniques as in (1) and by theorem 1.1 (4), then (4) can be derived. (5) first, we investigate left hand side of the equality. suppose that di∈i (f̃i, ei) = (g̃, eru ) is the restricted union of (f̃i, ei) for all i ∈ i. then eru = ⋂ i∈i ei 6= ∅ and for any e ∈ e ru , g̃[e] = ⋃ i∈i f̃i[e]. thus (f̃, e) ∩ (di∈i (f̃i, ei)) = (f̃, e) ∩ (g̃, eru ) = (h̃, eruei ). for any e ∈ eruei = e ∪eru , we have h̃[e] =   f̃[e] if e ∈ e \eru g̃[e] if e ∈ eru \e f̃[e] ∩ g̃[e] if e ∈ e ∩eru. by taking into account the definition of g̃ along with h̃, we can write h̃[e] =   f̃[e] if e ∈ e \ ( ⋂ i∈i ei)⋃ i∈i f̃i[e] if e ∈ ( ⋂ i∈i ei) \e f̃[e] ∩ ( ⋃ i∈i f̃i[e]) if e ∈ e ∩ ( ⋂ i∈i ei). consider the right hand side of the equality. suppose that (f̃, e) ∩ (f̃i, ei) = (̃ii, eeii ) is the extended intersection of (f̃, e) and (f̃i, ei) for all i ∈ i. then for any e ∈ eeii = e ∪ei, we have ĩi[e] =   f̃[e] if e ∈ e \ei f̃i[e] if e ∈ ei \e f̃[e] ∩ f̃i[e] if e ∈ e ∩ei. now, di∈i ((f̃, e) ∩ (f̃i, ei)) = di∈i (̃ii, eeii ) = (j̃, eeiru ) where eeiru = ⋂ i∈i e i i = ⋂ i∈i (e ∪ ei) = e∪( ⋂ i∈i ei) 6= ∅. for any e ∈ e eiru , j̃[e] = ⋃ i∈i ĩi[e]. by taking into account the properties of operations int. j. anal. appl. 17 (5) (2019) 836 in set theory and considering ĩi as piecewise defined functions for all i ∈ i, we have j̃[e] =   ⋃ i∈i f̃[e] if e ∈ e \ ( ⋂ i∈i ei)⋃ i∈i f̃i[e] if e ∈ ( ⋂ i∈i ei) \e⋃ i∈i (f̃[e] ∩ f̃i[e]) if e ∈ e ∩ ( ⋂ i∈i ei). and so j̃[e] =   f̃[e] if e ∈ e \ ( ⋂ i∈i ei)⋃ i∈i f̃i[e] if e ∈ ( ⋂ i∈i ei) \e⋃ i∈i (f̃[e] ∩ f̃i[e]) if e ∈ e ∩ ( ⋂ i∈i ei). by theorem 1.1 (1), it is clear that h̃ and j̃ are same set-valued mapping. hence, (f̃, e)∩(di∈i (f̃i, ei)) = di∈i ((f̃, e) ∩ (f̃i, ei)). (6) by using techniques as in (5) and by theorem 1.1 (2), then (6) can be derived. (7) by using techniques as in (5) and by theorem 1.1 (3), then (7) can be derived. (8) by using techniques as in (5) and by theorem 1.1 (4), then (8) can be derived. (9) first, we investigate left hand side of the equality. suppose that di∈i (f̃i, ei) = (g̃, eru ) is the restricted union of (f̃i, ei) for all i ∈ i. then eru = ⋂ i∈i ei 6= ∅ and for any e ∈ e ru , g̃[e] = ⋃ i∈i f̃i[e]. thus (f̃, e) e (di∈i (f̃i, ei)) = (f̃, e) e (g̃, eru ) = (h̃, erui ). for any e ∈ erui = e ∩ eru = e ∩ ( ⋂ i∈i ei) 6= ∅, we have h̃[e] = f̃[e] ∩ g̃[e] = f̃[e] ∩ ( ⋃ i∈i f̃i[e]). consider the right hand side of the equality. suppose that (f̃, e) e (f̃i, ei) = (̃ii, e i i ) is the intersection of (f̃, e) and (f̃i, ei) for all i ∈ i. then eii = e ∩ ei 6= ∅ and for any e ∈ e i i , ĩi[e] = f̃[e] ∩ f̃i[e]. now, di∈i ((f̃, e) e (f̃i, ei)) = di∈i (̃ii, eii ) = (j̃, eiru ), where eiru = ⋂ i∈i e i i = ⋂ i∈i (e ∩ ei) 6= ∅. for any e ∈ eiru , j̃[e] = ⋃ j∈j ĩj[e] = ⋃ j∈j (f̃[e] ∩ f̃i[e]). since ⋂ i∈i (e ∩ ei) = e ∩ ( ⋂ i∈i ei), we have eiru = erui . by theorem 1.1 (1), it is clear that h̃ and j̃ are same set-valued mapping. hence, (f̃, e) e (di∈i (f̃i, ei)) = di∈i ((f̃, e) e (f̃i, ei)). (10) by using techniques as in (9) and by theorem 1.1 (2), then (10) can be derived. (11) by using techniques as in (9) and by theorem 1.1 (3), then (11) can be derived. (12) by using techniques as in (9) and by theorem 1.1 (4), then (12) can be derived. � acknowledgment the authors would also like to thank the anonymous referee for giving many helpful suggestion on the revision of present paper. references [1] b. ahmad and a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), article id 586507. [2] m. a. ansari, a. haidar, and a. n. a. koam, on a graph associated to up-algebras, math. comput. appl. 23 (2018), no. 4, 61. int. j. anal. appl. 17 (5) (2019) 837 [3] n. dokkhamdang, a. kesorn, and a. iampan, generalized fuzzy sets in up-algebras, ann. fuzzy math. inform. 16 (2018), no. 2, 171–190. [4] t. guntasow, s. sajak, a. jomkham, and a. iampan, fuzzy translations of a fuzzy set in up-algebras, j. indones. math. soc. 23 (2017), no. 2, 1–19. [5] a. iampan, a new branch of the logical algebra: up-algebras, j. algebra relat. top. 5 (2017), no. 1, 35–54. [6] a. iampan, introducing fully up-semigroups, discuss. math., gen. algebra appl. 38 (2018), no. 2, 297–306. [7] y. b. jun, s. m. hong, and e. h. roh, bci-semigroups, honam math. j. 15 (1993), no. 1, 59–64. [8] y. b. jun, k. j. lee, and c. h. park, fuzzy soft set theory applied to bck/bci-algebras, comput. math. appl. 59 (2010), 3180–3192. [9] k. h. lee, first course on fuzzy theory and applications, springer-verlag berlin heidelberg, republic of south korea, 2005. [10] p.k. maji, r. biswas, and a.r. roy, fuzzy soft sets, j. fuzzy math. 9 (2001), no. 3, 589–602. [11] d. molodtsov, soft set theory-first results, comput. math. appl. 37 (1999), 19–31. [12] c. prabpayak and u. leerawat, on ideals and congruences in ku-algebras, sci. magna 5 (2009), no. 1, 54–57. [13] a. rehman, s. abdullah, m. aslam, and m. s. kamran, a study on fuzzy soft set and its operations, ann. fuzzy math. inform. 6 (2013), no. 2, 339–362. [14] a. rosenfeld, fuuzy groups, j. math, anal. appl. 35 (1971), 512–517. [15] a. satirad and a. iampan, fuzzy sets in fully up-semigroups, manuscript accepted for publication in ital. j. pure appl. math., july 2018. [16] a. satirad and a. iampan, fuzzy soft sets over fully up-semigroups, eur. j. pure appl. math. 12 (2019), no. 2, 294–331. [17] a. satirad, p. mosrijai, and a. iampan, formulas for finding up-algebras, int. j. math. comput. sci. 14 (2019), no. 2, 403–409. [18] a. satirad and a. iampan, generalized power up-algebras, int. j. math. comput. sci. 14 (2019), no. 1, 17–25. [19] j. somjanta, n. thuekaew, p. kumpeangkeaw, and a. iampan, fuzzy sets in up-algebras, ann. fuzzy math. inform. 12 (2016), no. 6, 739–756. [20] l. a. zadeh, fuzzy sets, inf. control, 8 (1965), 338–353. 1. introduction and preliminaries 2. fuzzy soft sets over fully up-semigroups acknowledgment references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 114-124 doi: 10.28924/2291-8639-15-2017-114 evolutes of hyperbolic dual spherical curve in dual lorentzian 3-space rashad a. abdel-baky1,2,∗ abstract. based on the e. study’s map, we study a timelike ruled surface as a curve on the hyperbolic dual unit sphere in dual lorentzian 3-space d31. then, as applications of the singularity theory of smooth functions, we define the notation of evolutes for timelike ruled surfaces and establish the relationships between their geometric invariants. finally, an example of application is introduced and explained in detail. 1. introduction despite its long history, the theory of surface is still one of the most important interesting topics in differential geometry and it is being study by many mathematicians until now. among the surfaces, a ruled surface has been drawing attention to scientists as well as mathematicians because of its various application such as the study of design problems in spatial mechanisms and physics, kinematics and computer aided design (cad). there exists a vast literature on the subject including several monographs, for example [1-7]. rather unexpectedly dual numbers have been applied to study the motion of a line space; they seem even be the most appropriate apparatus for this end. in screw and dual number algebra, the e. study’s map concludes: the set of all oriented lines in euclidean 3-space e3 is in one-toone correspondence with set of points of the dual unit sphere in the dual 3space d3. more details on the necessary basic concepts about the dual elements and the one-to-one correspondence between ruled surfaces and one-parameter dual spherical motions can be found in [7-10]. if we take the minkowski 3-space e31 instead of e 3 the e. study’s map can be stated as follows: the timelike and spacelike dual unit vectors of hyperbolic and lorentzian dual unit spheres h2+ and s 2 1 at the lorentzian 3-space d 3 1 are in one-to-one correspondence with the directed timelike and spacelike lines of the space of lorentzian lines e31, respectively [12]. then a differentiable curve on h 2 + corresponds to a timelike ruled surface at e31. similarly the timelike (resp. spacelike) curve on s 2 1 corresponds to any spacelike (resp. timelike) ruled surface at e31. then, the study of ruled surfaces in the minkowski 3-space is more interesting than the euclidean case. one of the main techniques for applying the singularity theory to euclidean differential geometry is to consider the distance squared function and the height function on a submanifold of e3 [13, 14]. there are some articles concerning singularities of surfaces and classical geometric invariants of space curves for several kinds of geometry [13-23]. in these articles the corresponding functions depend on each geometry. in this paper, we consider the lorentzian dual distance function on a dual curve in h2+. as an application of singularity theory to the lorentzian dual height function, we detect the hyperbolic dual evolute and classify singularities of it. by the main result we showed that the hyperbolic dual evolute can be defined in the case when the dual geodesic curvature σ 6= ±1. then, applying to study’s map, we established the relationships between singularities of these subjects and geometric invariants of timelike ruled surface which are deeply related to the order of contact with evolutes. finally, an example illustrates the application of the obtained formulae was introduced. received 20th september, 2017; accepted 21st october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 53a04, 53a05, 53a17. key words and phrases. keyword1; blaschke frame; evolute of the dual spherical curve; singularity. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 114 evolutes of hyperbolic dual spherical curve 115 2. basic concepts in this section we list some notions, formulas and conclusions for the theory of dual numbers and dual lorentzian vectors (see for instance refs. [1–4, 8-11]). let r3 denote the vector space with its usual vector structure. we denote (x1,x2,x3) the coordinates of a vector with respect to the canonical basis of r3. the three-dimensional minkowski 3-space is the metric space e31 = (r 3,<,>), where the metric <,> is < x, y >= −x1y1 + x2y2 + x3y3, x = (x1,x2,x3) , y = (y1,y2,y3) , (2.1) which is called the lorentzian metric. for any two vectors x = (x1,x2,x3) and y = (y1,y2,y3) of e31, the lorentzian vector product is defined by x × y = (−(x2y3 −x3y2), (x1y3 −x3y1), (x1y2 −x2y1)) . (2.2) a vector x ∈ e31 is said to be spacelike if < x, x >>0 or x = 0, timelike if < x, x ><0 and lightlike or null if < x, x >=0 and x 6= 0. a timelike or light-like vector in e31 is said to be causal. we point out that the null vector x = 0 is considered of spacelike type although it satisfies < x, x >=0. for x ∈e31 the norm is defined by ‖x‖ = √ |< x, x >|, then the vector x is called a spacelike unit vector if < x, x >=1 and a timelike unit vector if < x, x >= −1. similarly, a regular curve in e31 can locally be spacelike, timelike or null (lightlike), if all of its velocity vectors are spacelike, timelike or null (lightlike), respectively. the angle between two vectors in minkowski 3-space e31 is defined by [8-11]: definition 1 i) spacelike angle: let x and y be spacelike vectors in e31 that span a spacelike vector subspace; then we have |< x, y >| ≤ ‖x‖‖y‖, and hence, there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖cos θ. this number is called the spacelike angle between the vectors x and y. ii) central angle: let x and y be spacelike vectors in e31 that span a timelike vector subspace; then we have |< x, y >| > ‖x‖‖y‖, and hence, there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖cosh θ. this number is called the central angle between the vectors x and y. iii) lorentzian timelike angle: let x be spacelike vector and y be timelike vector in e31. then there is a unique real number θ ≥ 0 such that < x, y >= ‖x‖‖y‖sinh θ. this number is called the lorentzian timelike angle between the vectors x and y. a ruled surface m in e31 is a surface generated by a straight line l moving along a curve c(s). the various positions of the generating lines are called the rulings of the surface. such a surface, thus, has a parametrization in the ruled form [1-6]: m : y(s,v) = c(s) + vx(s), s ∈ i, v ∈ r, (2.3) such that < x,x>=σ(±1), < x ′ ,x ′ >=η(±1), < c ′ , x ′ >= 0; ′ = d ds . in this case the curve c = c(s) is the striction curve, and the parameter s is the arc length of the non-null spherical curve x = x(s). let now excluding x is constant or null or x ′ null. as usual blaschke frame relative to x(s) will be defined as the frame of which this line and the central normal t(s) = x ′ (s) to the ruled surface at the central point are two edges. the third edge g(s)= x×t is the central tangent to the ruled surface m. the frame { x = x(s), t(s) = x ′ , g(s) = x × t} is called blaschke frame. then, we have x × t = g, x×g =σt, t×g= −ηx, < g, g >= −ση. (2.4) therefore, the following blaschke formulae hold:  x ′ t ′ g ′   =   0 1 0−ση 0 γ 0 σγ 0     xt g   , (2.5) where γ(s) = det(x ′′ , x ′ , x) is the geodesic curvature function of the non-null spherical x(s). in terms of the blaschke frame {x, t , g } with signs σ, η, −ση, the striction curve c can be reconstructed from c ′ (s) = σγx −σηµg. (2.6) 116 abdel-baky the functions γ(s), γ(s) and µ(s) are called the curvature functions or construction parameters of the ruled surface. the geometrical meanings of these invariants are explained as follows: γ is the geodesic curvature of the spherical image curve x = x(s); γ describes the angle between the tangent of the striction curve and the ruling of the surface; and µ is the distribution parameter of the ruled surface at the ruling x. note that m is a timelike surface when (σ,η) = (±1, 1). in fact (σ,η) = (−1,−1) is impossible because of the causal character. 2.1. the e. study’s map. the set of dual numbers is d = {a = a + εa∗ | a, a∗ ∈ r, (2.7) where ε 6= 0 is called the dual operator with the algebraic property of ε2 = 0. sums and products of dual numbers are well defined using the dual operator. analogously, for all pairs (x, x∗) ∈ e31 ×e31 the set d31 = {x = x + εx ∗, ε 6= 0, ε2 = 0}, (2.8) together with the lorentzian inner product < x, y >=< x, y > + ε(< y, x∗ > + < y∗, x >), (2.9) forms the dual lorentzian 3-space d31. thereby a point x = (x1,x2,x3) t has dual coordinates xi = (xi + εx ∗ i ) ∈ d. the norm is defined by < x, x > 1 2 := ‖x‖ = ‖x‖(1+ε < x, x∗ > ‖x‖2 ). (2.10) the hyperbolic and lorentzian dual unit spheres, respectively, are: h2+ = {x ∈d 3 1 | −x 2 1 + x 2 2 + x 2 3 = −1, x1 > 0}, (2.11) and s21 = {x ∈d 3 1 | −x 2 1 + x 2 2 + x 2 3 = 1}. (2.12) this yields f1×f2= f3, f2×f3= −f1, f3×f1= f2, (2.13) where f1, f2, and f3, are the base at the origin point 0 (0, 0, 0) of the dual lorentzian 3-space d31. via this, the e. study’s map can be stated as follows: the dual unit spheres are shaped as a pair of conjugate hyperboloids. the common asymptotic cone represents the set of null lines, the ring shaped hyperboloid represents the set of spacelike lines, and the oval shaped hyperboloid forms the set of timelike lines, opposite points of each hyperboloid represent the pair of opposite vectors on a line (see fig. 1). figure 1 evolutes of hyperbolic dual spherical curve 117 3. timelike ruled surface as a dual curve we use in this section the notations of the preceding section. the e. study’s map allows us to rewrite eq. (2.3) by the dual vector function as: m : x(s) = x(s) + εc(s)×x(s) = x(s) + εx∗(s), (3.1) where x∗ is the moment of x about the origin in e31. the representation of directed lines in e 3 1 by dual unit vectors brings about several advantages and from now on we do not distinguish between directed lines and their representing dual unit vectors. therefore, the dual arc length ds = ds + εds∗ of the dual curve x(s) ∈ h2+ or s21 is: s(s) = s∫ s0 √ |< x′, x′ >|ds. (3.2) then the dual parameter is determined such that ∥∥dx ds ∥∥ = 1. so, we have ds ds = 1 |< x′, x′ >| . (3.3) therefore, we derive the dual lorentzian form of the blaschke frame equations in exactly the same way as in eq. (2.5): d ds   xt g   =   0 1 0−ση 0 σ 0 σς 0     xt g   , (3.4) where σ = σ(s) is the dual geodesic curvature function of x(s) ∈ h2+ or s21. under the assumption that σ 6= ±1, we define the dual evolute of x(s) ∈ h2+ or s21 as follows: b(s)= σx + g√ |σ2 − 1| . (3.5) we remark that b(s) is located in h2+ if and only if σ 2 > 1, otherwise it is in s21. therefore, we consider a pseudo dual circle on h2+ or s 2 1 is described by the equation s(r, b0) = { x(s) ∈ h2+ or s 2 1 | < x, b0 >= r(s) } , (3.6) where r = ρ + ερ∗ is a dual spherical radius of curvature, and b0 is a fixed dual unit vector which determines the pseudo dual circle’s center. then, we have the following proposition. proposition 1. let x : i ⊆ d → h2+ or s21 be a unit speed dual curve with σ2 6= ±1. then dς ds = 0 iff b0= ±b. under this condition, x(s) ∈ h2+ or s21 is a part of pseudo dual circle whose center is b. proof. for the first differential of b we get: db ds = ∓ σ ′ |< x′, x′ >| (√ |σ2 − 1| )3 2 ( σ ′ x + σg ) . (3.7) then b0= ±b iff σ ′ (s) = 0. under this condition we put r = σ√ |σ2−1| with σ2 6= ±1. so x(s) ∈ h2+ or s21 is a part of pseudo dual circle whose center is b. through the reminder of this work we will study a non-developable timelike ruled surface characterized by (σ,η) = (−1, 1). therefore, we have x × t = g, x × g = − t, t × g = − x, < g, g >=1, (3.8) and under the assumption that σ2 > 1, we also have: b(s)= σx + g √ σ2 − 1 , with < b, b >= − 1. (3.9) 118 abdel-baky in terms of the blaschke frame, we can show that: ds = √ < x ′ , x ′ >ds = √ < x ′ , x ′ > + 2ε < x ′ , x∗ ′ >ds = √ 1 + 2ε < t, c ′ × x + c × t >ds = (1 + εµ) ds, (3.10) which imply ds ds = 1 −εµ. (3.11) then, we determine σ = σ(s) by dg ds = (1 −εµ) { dg ds + ε [ dc ds × g + c× dg ds ]} this expression is further expanded using eqs. (2.5) and (2.6) to yield dg ds = (1 −εµ){−γt + ε [(−γx + µg) × g −γc × t]} = (1 −εµ) [−γt + ε (γt −γt∗)] = (1 −εµ) (−γ + εγ) t = [−γ + ε (γ + γµ)] t. (3.12) comparing eqs. (3.4) and (3.12) we see that σ is defined in terms of γ, µ and γ as: σ = γ −ε (γ + γµ) . (3.13) similar to the books in [14, 15], a dual point b0 of h2+ will be said to be a bk evolute of the dual curve x(s) in h2+ at s ∈ r if, for all i such that 1 ≤ i ≤ k, < b0, xi(s) >= 0, but < b0, xk+1(s) > 6= 0. here xi denotes the i-th derivatives of x with respect to the dual arc length of x(s) in h2+. for the first evolute b of x(s), we have < b, x ′ >= ± < b, t >= 0, and < b, x ′′ >= ± < b, x+σg >= 0. so, b is at least a b2 evolute of x(s) ∈ h2+. 3.1. height dual functions. let x : i ⊆ d → h2+ be a dual curve x(s) in h2+ with σ2 > 1. we now define a smooth dual function ht : i × h2+ → d, by ht (s, b0) =< b0, x >. we call ht a hyperbolic timelike height dual function on x(s) in h2+. we use the notation he(s) = h t (s, b0) for any fixed b0 of h2+. proposition 2. let x : i ⊆ d → h2+ be a dual curve x(s) in h2+ with σ2 > 1. then the following holds: 1he will be invariant in the first approximation iff b0 ∈ sp{x,g}, that is, h ′ e = 0 ⇔< x ′ , b0>=0 ⇔< t, b0>=0 ⇔ b0=a1x+a2g; (3.14) for some dual numbers a1,a2 ∈ d, and a21 −a22 = −1. 2he will be invariant in the second approximation iff b0 is b2 evolute of x(s) ∈ h2+, that is, h ′ e = h ′′ e = 0 ⇔ b0= ± b, and σ 2 > 1. (3.15) 3he will be invariant in the third approximation iff b0 is b3 evolute of x(s) ∈ h2+, that is, h ′ e = h ′′ e = h ′′′ e = 0 ⇔ b0= ±b, σ 2 > 1, and σ ′ 6= 0. (3.16) 4he will be invariant in the fourth approximation iff b0 is b4 evolute of x(s) ∈ h2+, that is, h ′ e = h ′′ e = h ′′′ e = h (iv) e = 0 ⇔ b0= ±b, σ 2 > 1, σ ′ = 0, and σ ′′ 6= 0. (3.17) proof. for the first differential of he we get: h ′ e =< x ′ , b0>. (3.18) so, we get: h ′ e = 0 ⇔< t, b0>=0 ⇔ b0=a1x+a2g; (3.19) evolutes of hyperbolic dual spherical curve 119 for some dual numbers a1,a2 ∈ d, and a21 −a22 = −1, the result is clear. 2differentiation of eq. (3.18) leads to: h ′′ e =< x ′′ , b0>= < x + σg, b0> . (3.20) by using eq. (3.19),we have: h ′ e = h ′′ e = 0 ⇔< x ′ , b0>= < x ′′ , b0>=0 ⇔ b0= ± x ′ × x ′′∥∥x′ × x′′∥∥ = ±b. (3.21) 3differentiation of eq. (3.20) leads to: h ′′′ e =< x ′′′ , b0 >= ( 1 − σ2 ) < t, b0> +σ ′ < g, b0> (3.22) hence, we have: h ′ e = h ′′ e = h ′′′ e = 0 ⇔ b0= ±b, σ 2 > 1, and σ ′ 6= 0. (3.23) 4by the similar arguments, we can also have: h ′ e = h ′′ e = h ′′′ e = h (iv) e = 0 ⇔ b0= ±b, σ 2 > 1, σ ′ = 0, and σ ′′ 6= 0. (3.24) the proof is completed. according to the above proposition, we have: (a) the osculating circle s(r, b0) of x(s) in h2+ is determined by the equations < b0, x >=r, < x ′, b0 >= 0,< x ′′ , b0 >= 0, (3.25) which are obtained from the condition that the osculating circle should have contact of at least third order at x(s0) iff σ ′ 6= 0. then, as in the euclidean 3-space, the first and last two equations, respectively, determine the osculating timelike line congruence of the trajectory of the line x and its axis b0 [12]. (b) the osculating circle s(r, b0) and the dual curve x(s) in h2+ have at least fourth order at x(s0) iff σ ′ = 0, and σ ′′ 6= 0. in this way, considering the evolutes of a general timelike ruled surface we can get a sequence of evolutes b2, b3, ...,bn. the properties and the relationship between these evolutes and their involute timelike surfaces are very interesting problems. for example, it is easy to see that when b0=±b, and σ ′ 6= 0, m is traced during a lorentzian screw motion about b0, by the line x located at r =const. relative to b0. 4. unfoldings of dual functions of one variable in this section we will use the same technique on the singularity theory for families of dual smooth functions. detailed descriptions are found in the books [11, 12]. let f: (d×dr, (s0, x0)) → d be a dual smooth function, and f(s) = fx0 , fx0 (s) = f(s, x0). then f is called an r-parameter dual unfolding of f(s). we say that f(s) has ak singularity at s0 if f(p)(s0) = 0 for all 1 ≤ p ≤ k, and f(p+1)(s0) 6= 0. we also say that f(s) has a≥k singularity at s0 if f(p)(s0) = 0 for all 1 ≤ p ≤ k. let the (k − 1)-jet of the partial derivative ∂f ∂xi at s0 be j (k−1) ( ∂f ∂xi (s, x0) ) = σk−1j=1 ljis j (without the dual constant term), for i = 1, ...,r. then f(s, x) is called a (p) versal dual unfolding iff the (k − 1)×r dual matrix of coefficients (lji) has rank (k − 1). (this certainly requires k−1 ≤ r, so the smallest value of r is k − 1). we now state important sets about the unfoldings relative to the above notations. the singular dual set of f(s, x) is the set sf = { x ∈h2+| there exists s with ∂f ∂s = 0 at (s, x) } . (4.1) the bifurcation dual set bf of f is the set [11, 12]: bf = { x ∈h2+| there exists s with ∂f ∂s = ∂2f ∂s2 = 0 at (s, x) } . (4.2) 120 abdel-baky then similar to [11], we state the following theorem: theorem 1. let f: d×dr, ((s0, x0)) → d be a dual r-parameter unfolding of f(s), which has the ak singularity (k ≥ 1) at s0. suppose that f is a (p) versal dual unfolding then: (1) if k = 2, then bf is locally diffepmorphic to {0}×dr−1; (2) if k = 3, then bf is locally diffepmorphic to c̃×dr−2, where c̃ = {(x1,x2)| x21 = x32} is the ordinary cusp. for the dual curve x(s) ∈ h2+, with σ2 > 1, and he(s) = ht (s, b0), the bifurcation dual set of ht is given as follows: bht = { x ∈h2+| b = ± σx + g √ σ2 − 1 , σ2 > 1 } . (4.3) hence, we have the following fundamental proposition: proposition 3. for the dual unit speed curve x(s)= (x1(s),x2(s),x3(s)) on h2+, with σ(s0) 6= 0 and σ2(s0) 6= ±1. if the he(s) = ht (s, b0) has the ak-singularity (k = 2, 3) at s0 ∈ d, then ht is the (p) versal dual unfolding of he0 (s0). proof. since b0= (z1,z2,z3) ∈ h2+, −z21 + z22 + z23 = −1, z1 > 0. z1,z2, and z3 can’t be all zero. without loss of generality, suppose z3 6= 0. then by z3 = √ −1 + z21 −z22 , we have ht (s, b0) = −z1x1(s) + z2x2(s) + √ −1 + z21 −z22x3(s). (4.4) so ∂ht ∂z1 = ( −x1(s) + z1x3(s)√ −1+z21−z 2 2 ) , ∂ht ∂z2 = ( x2(s) − z2x3(s)√ −1+z21−z 2 2 ) .   (4.5) we also have ∂ ∂s ∂ht ∂z1 = ( −x ′ 1(s) + z1x ′ 3(s)√ −1+z21−z 2 2 ) , ∂ ∂s ∂ht ∂z2 = ( x ′ 2(s) − z2x ′ 3(s)√ −1+z21−z 2 2 ) ,   (4.6) and ∂2 ∂s2 ∂ht ∂z1 = ( −x ′′ 1 (s) + z1x ′′ 3 (s)√ −1+z21−z 2 2 ) , ∂2 ∂s2 ∂ht ∂z2 = ( x ′′ 2 (s) − z2x ′′ 3 (s)√ −1+z21−z 2 2 ) .   (4.7) let b0= (z10,z20,z30) ∈ h2+, and assume z30 6= 0, then j1 ( ∂ht ∂z1 (s, b0) ) = ( −x ′ 1(s0) + z1x ′ 3(s0) z30 ) s, j1 ( ∂ht ∂z2 (s, b0) ) = ( x ′ 2(s0) − z2x ′ 3(s0) z30 ) s,   (4.8) and j2 ( ∂ht ∂z1 (s, b0) ) = ( −x ′ 1(s0) + z1x ′ 3(s0) z30 ) s + 1 2 ( −x ′′ 1 (s0) + z1x ′′ 3 (s0) z30 ) s2, j2 ( ∂ht ∂z2 (s, b0) ) = ( x ′ 2(s0) − z2x ′ 3(s0) z30 ) s + 1 2 ( x ′′ 2 (s0) − z2x ′′ 3 (s0) z30 ) s2.   (4.9) (i) if he0 (s0) has the a2-singularity at s0 ∈ d, then h ′ e0 (s0) = 0. so the (2 − 1) × 2 dual matrix of coefficients (lji) is: a = ( −x ′ 1(s0) + z1x ′ 3(s0) z30 x ′ 2(s0) − z2x ′ 3(s0) z30 ) ; (4.10) evolutes of hyperbolic dual spherical curve 121 suppose that the rank of the matrix a is zero, then we have: x ′ 1(s0) = z1x ′ 3(s0) z30 , x ′ 2(s0) = z2x ′ 3(s0) z30 . (4.11) since ∥∥∥x′(s0)∥∥∥ = ‖t(s0)‖ = 1, we have x′3(s0) 6= 0, so that we have the contradiction as follows: 0 = < ( x ′ 1(s0),x ′ 2(s0),x ′ 3(s0) ) , (z1,z2,z30) > (4.12) = −x ′ 1(s0)z1 + x ′ 2(s0)z2 + x ′ 3(s0)z30 = − z21x ′ 3(s0) z30 + z22x ′ 3(s0) z30 + x ′ 3(s0)z30 = −x ′ 3(s0) z30 6= 0. therefore rank (a) = 1, and ht is the (p) versal dual unfolding of he at s0. (ii) if he0 (s0) has the a3-singularity at s0 ∈ d, then h ′ e0 (s0) = h ′′ e0 (s0) = 0, and by proposition 1: b(s0)= ± ( σx + g √ σ2 − 1 ) (s0), (4.13) where σ2(s0) 6= ±1, σ ′ (s0) = 0, and σ ′′ (s0) 6= 0. so the (3 − 1) × 2 dual matrix of the coefficients (lji) is b = ( l11 l12 l21 l22 ) =   −x ′ 1 + z1x ′ 3 z30 x ′ 2 − z2x ′ 3√ −1+z21−z 2 2 −x ′′ 1 + z1x ′′ 3 z30 x ′′ 2 − z2x ′′ 3 z30   . (4.14) for the purpose, we also require the 2 × 2 matrix b to be non-singular, which always does. in fact, the determinate of this matrix at s0 is det (b) = 1 z30 ∣∣∣∣∣∣∣ −x ′ 1 x ′ 2 x ′ 3 −x ′′ 1 x ′′ 2 x ′′ 3 z10 z20 z30 ∣∣∣∣∣∣∣ (4.15) = 1 z30 < x ′ ×x ′′ , e0 > = ± 1 z30 < x ′ ×x ′′ , ( σx + g √ σ2 − 1 ) > (4.16) since x ′ = t, we have x ′′ = x+σg. substituting these relations to the above equality, we have det (b) = ∓ 1 z30 σ(s0)√ σ2(s0) − 1 6= 0. (4.17) this means that rank (b) = 2. this completes the proof. theorem 2. let x(s) be a dual unit speed curve on h2+, then the dual spherical evolute of x(s) is: (1) diffepmorphic to a timelike oriented line if σ ′ (s0) 6= 0; (2) diffepmorphic to the cusp c̃ at s0 ∈ d if σ ′ (s0) = 0, and σ ′′ (s0) 6= 0. proof. for the proof of assertion (1), from eq. (3.9) we have db ds := b ′ = ∓ σ ′(√ σ2 − 1 )3 2 ( σ ′ x + σg ) . (4.18) therefore b is locally diffepmorphic to a timelike oriented line if σ ′ (s0) 6= 0. for the assertion (2), from proposition 2, and theorem 1, the bifurcation set bht at b0=± ( σx+g√ σ2−1 ) (s0) is locally diffepmorphic to the ordinary cusp c̃ in h2+ if σ ′ (s0) = 0, and σ ′′ (s0) 6= 0. 122 abdel-baky example 1. the dual coordinates xi = (xi + εx ∗ i ) of an arbitrary point x of the dual hyperbolic unit sphere h2+, centered at the origin, may be expressed as: m : x = (cosh θ, sinh θ cos φ, sinh θ sin φ) , (4.19) where θ = ϑ + εϑ∗, and φ = ϕ + εϕ∗ are dual hyperbolic and spacelike angles with ϑ∗, ϑ ∈ r, and 0 ≤ ϕ ≤ 2π, respectively. moreover, let us consider x = x(t), t ∈ r corresponding to a timelike ruled surface m .therefore, we have:  xt g   =   cosh θ sinh θ cos φ sinh θ sin φ0 −sin φ cos φ sinh θ cosh θ cos φ cosh θ sin φ     f1f2 f3   . (4.20) thus, we get the blaschke equations: d dt   xt g   =   0 dφdt sinh θ dθdtdφ dt sinh θ 0 −dφ dt cosh θ dθ dt dφ dt cosh θ 0     xt g   , (4.21) from which we obtain ds = √( dφ dt sinh θ )2 + ( dθ dt )2 dt, σ(t) = dφ dt cosh θ√ ( dφdt sinh θ) 2 +( dθdt ) 2 .   (4.22) now, let us take ϑ = c1(real const.), and ϑ ∗ = c2(real const.). in this case, we have µ := ds ∗ ds = ( dϕ∗ dt /dϕ dt ) + ϑ∗ coth ϑ, γ = coth ϑ, −γ = ( dϕ∗ dt /dϕ dt ) coth ϑ + ϑ∗,   (4.23) and the dual evolute b is given as follows: b(t) = σx − g √ σ2 − 1 = f1. (4.24) according to theorem 2, we have that the evolute of x = x(t) in h2+ is locally diffeomorphic to a timelike oriented line. moreover, let y(y1,y2,y3) denote the position vector of an arbitrary point of m. then, considering eqs. (3.1), and (4.19) we find: y(ϕ,v) =   ϕ∗ + v cosh ϑϑ∗ sin ϕ + v sinh ϑ cos ϕ −ϑ∗ cos ϕ + v sinh ϑ sin ϕ   , v ∈ r. (4.25) from eq. (4.25) we have: m : y22 ϑ∗2 + y23 ϑ∗2 − y 21 (ϑ∗ coth ϑ) 2 = 1, (4.26) where y1 = y1 − ϕ∗. since ϑ, and ϑ∗ are two-independent parameters, we can say that m is, in generally, a family of lorentzian one-sheeted hyperboloids with two parameters, so it represents a quadratic timelike line congruence. the intersection of this timelike congruence and the corresponding spacelike planes y1 −ϕ∗ = 0 is the one-parameter family of euclidean circles y22 + y23 = ϑ ∗2. if we take ϕ∗(t) = hϕ, and ϕ(t) = ϕ then we immediately obtain a member of this line congruence as shown in fig. 2. ; ϕ ∈ [0, 2π], v ∈ [−3, 3], ϑ = π 2 , ϑ∗ = h = 1. (4.27) 5. conclusion mathematical techniques used the e. study’s map have been shown to be suitable for study dual hyperbolic invariants as applications of the singularity theory of smooth dual functions. hopefully these results will lead to a wider usage of the geometric properties of the timelike ruled surfaces an analogue of the problem addressed in this paper may be consider for x(s) ∈ s21 in the dual lorentzian 3-space d31. we will study this problem in the future. evolutes of hyperbolic dual spherical curve 123 figure 2 references [1] o. bottema, and b. roth. theoretical kinematics, north-holland press, new york 1979. [2] a. karger, and j. novak. space kinematics and lie groups, gordon and breach science publishers, new york 1985. [3] j m. mc carthy j m. an introduction to theoretical kinematics, london: the mit press 1990. [4] h. pottman, and j. wallner. computational line geometry, springer-verlag, berlin, heidelberg 2001. [5] r.a. abdel-baky, and f.r. al-solamy. a new geometrical approach to one-parameter spatial motion, j. eng. math. 60 (2008). 149–172. [6] r.a. abdel-baky, and r.a al-ghefari. on the one-parameter dual spherical motions, comput. aided geom. des. 28 (2011), 23–37. [7] r.a al-ghefari, and r.a. abdel-baky. kinematic geometry of a line trajectory in spatial motion, j. mech. sci. tech. 29 (9) (2015), 3597-3608. [8] b. o’neil. semi-riemannian geometry geometry, with applications to relativity, academic press, new york, 1983. [9] w. sodsiri. ruled linear weingarten surfaces in minkowski 3-space, soochow j. math., 29(4) (2003), 435-443. [10] w. kuhnel. differential geometry (2nd edition), amer. math. soc., 2006. [11] r. lopez. differential geometry of curves and surfaces in lorentz-minkowski space, arxiv.org/abs/0810.3351v1 2008. [12] y. yayli, a. caliskan, and h. h. u̧gurlu, the e. study maps of circles on dual hyperbolic and lorentzian unit spheres h20 and s 2 0, math. proc. r. ir. acad. 102a (2002), no. 1, 37-47. [13] m. onder, and hh ugurlu. frenet frames and invariants of timelike ruled surfaces, ain shams eng. j. 4 (2013), 507–513. [14] bruce, j. w.; giblin, p. j.: curves and singularities. 2nd. ed. cambridge univ. press, cambridge 1992. [15] ir. porteous. geometric differentiation for the intelligence of curves and surfaces, second edition, cambridge university press, cambridge, 2001. [16] s. izumiya and n. takeuchi, geometry of ruled surfaces, applicable math., in the golden age, (2003), 305–338. [17] s. izumiya and n. takeuchi. special curves and ruled surfaces, beitrage zur algebra und geometrie contributions to algebra and geometry, 44 (2003), no. 1, 203-212. [18] s. izumiya and a. takiyama, a time-like surface in minkowski 3-space which contains pseudocircles, proc. edinburgh math. soc. (2) 40, 1 (1997), 27-136. [19] s. izumiya, d-h. pe. and t. sano. the lightcone gauss map and lightcone developable of a spacelike curve in minkowski 3-space, j. glasgow math. j. 42 (2000), 75-89. [20] s. izumiya, d. pei and t. sano, singularities of hyperbolic gauss maps, proc. london math. soc., 86 (2003), 485–512. [21] s. izumiya and m. c. romero fuster, the horospherical gauss-bonnet type theorem in hyperbolic space, j. math. soc. japan 58 (2006), 965–984. [22] s. izumiya, k. saji and n. takeuchi, circular surfaces, adv. geom. 7 (2007), 295–313. [23] s. izumiya, legendrian dualities and spacelike hypersurfaces in the lightcone, mosc. math. j. 9 (2009), 325–357. 1department of mathematics, sciences faculty for girls, king abdulaziz university, p.o. box 126300, jeddah 21352, saudi arabia 124 abdel-baky 2department of mathematics, faculty of science, university of assiut, assiut 71516, egypt ∗corresponding author: rbaky@live.com 1. introduction 2. basic concepts 2.1. the e. study's map 3. timelike ruled surface as a dual curve 3.1. height dual functions 4. unfoldings of dual functions of one variable 5. conclusion references international journal of analysis and applications volume 17, number 3 (2019), 342-360 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-342 pata-type fixed point results in bν(s)-metric spaces fangyuan dong, peisheng ji∗ and xiaohui wang school of mathematics and statistics, qingdao university, shangdong 266071, p. r. china ∗corresponding author: jipeish 1@sina.com abstract. the aim of this is to study fixed point theorems in bν (s)-metric spaces under the pata-type conditions. as consequences, we establish common fixed point results of pata-type for two maps in bν (s)metric spaces. 1. introduction the banach contraction principle introduced by banach [6] is one of the most important results in mathematical analysis. it is the most widely applied fixed point result in many branches of mathematics and generalized in many different directions. some generalizations of the notion of a metric space have been proposed by some authors, such as, rectangular metric spaces, semi metric spaces, pseudo metric spaces, probabilistic metric spaces, fuzzy metric spaces, quasi metric spaces, quasi semi metric spaces, d metric spaces, and cone metric spaces (see [1, 2, 7, 20, 24, 26, 29–33, 36]). the other direction of investigation is concerned with generalizations of contractive condition (see [3, 10, 11, 18, 19] and others in literature). one of the interesting recent results of this kind was obtained by v. pata in [23]. several scholars have already used pata-type conditions to obtain new fixed point results (see [5, 9, 14–17]). v. pata obtained the following interesting refinement of the classical banach contraction principle. received 2018-12-01; accepted 2019-01-30; published 2019-05-01. 2010 mathematics subject classification. 54h25; 47h10. key words and phrases. fixed point; bν (s)-metric space; pata-type contractive mapping; triangular α-admissible mapping. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 342 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-342 int. j. anal. appl. 17 (3) (2019) 343 theorem 1.1. [23] let (x,d) be a metric space, f : x → x, let λ ≥ 0,η ≥ 1 and β ∈ [0,η] be fixed constants and ψ : [0, 1] → [0,∞) be an increasing function, vanishing with continuity at 0. if the inequality d(fx,fy) ≤ (1 −ε)d(x,y) + λεηψ(ε)[1 + ||x|| + ||y||]β is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. here, ||x|| = d(x,x0) for a chosen point x0 ∈ x. it was also shown by an example that the previous theorem is real generalization of banach’s result. more results of this kind were subsequently obtained by various authors. b-metric spaces were firstly used by i.a. bakhtin and s. czerwik. definition 1.1. [4,8] let x be a nonempty set, s ≥ 1 be a given real number. a mapping d : x×x → [0,∞) is called a b-metric with parameter s if for all x,y ∈ x the following holds: (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x) for all x,y ∈ x ; (3) d(x,y ≤ s[d(x,z) + d(z,y)] for all x,y,z ∈ x (b-triangular inequality). then the pair (x,d) is called a b-metric space. remark 1.1. in general, b-metric might not be continuous functions (see example in [4, 8]). definition 1.2. [7] let x be a nonempty set. let d : x × x → [0,∞) be a mapping such that for all x,y ∈ x and distinct points u,v ∈ x, each distinct from x and y: (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x); (3) d(x,y ≤ d(x,u) + d(u,v) + d(v,y) (rectangular inequality). then d is called a generalized metric and the pair (x,d) is called generalized metric space (or shortly gms). remark 1.2. obviously, each metric space is a generalized metric space, but the converse is not true. moreover, sarma et al. [32] and samet [31] presented examples showing that generalized metric spaces might not be hausdorff and, again, that generalized metric might be discontinuous. also, suzuki showed in [35] that, in general, generalized metric spaces do not have a compatible topology. as a combination of b-metric and generalized metric spaces, b-rectangular metric spaces were introduced and used in [12, 22, 28]. definition 1.3. [12] let x be a nonempty set and s ≥ 1 be a fixed real number.. let d : x ×x → [0,∞) be a mapping such that for all x,y ∈ x and distinct points u,v ∈ x, each distinct from x and y: (1) d(x,y) = 0 if and only if x = y; int. j. anal. appl. 17 (3) (2019) 344 (2) d(x,y) = d(y,x); (3) d(x,y ≤ s[d(x,u) + d(u,v) + d(v,y)] (b-rectangular inequality). then d is called a b-rectangular metric and the pair (x,d) is called b-rectangular metric space with parameter s. in 2017, z.d. mitrovic and s. radenovic [21] introduced the concept of bµ(s)-metric space as follows. definition 1.4. [21] let x be a nonempty set. let d : x×x → [0,∞) be a mapping and let ν ∈ n, s ≥ 1. then (x,d) is said to be a bν(s)-metric space if for all x,y ∈ x and for all distinct points u1,u2, · · · ,uν ∈ x, each of them different from x and y, the following hold: (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x); (3) d(x,y ≤ s[d(x,u1) + d(u1,u2) + · · · + d(uν,y)] (bν(s)-metric inequality). they note that: b1(1)-metric space is usual metric space, b1(s)-metric space is b-metric space with coefficient of bakhtin and czerwik, b2(1)-metric space is generalized metric space, b2(s)-metric space is rectangular b-metric space with coefficient s of george et al., bν(1)-metric space is ν-generalized metric space of branciari. definition 1.5. [21] let (x,d) be a bν(s)-metric space, {xn} be a sequence in x and x ∈ x. then (i) the sequence {xn} is said to be convergent in (x,d) and converges to x, if for every ε > 0 there exists n0 ∈ n such that d(xn,x) < ε for all n > n0 and this fact represented by limn→∞xn = x or xn →∞ as n →∞. (ii) the sequence {xn} is said to be cauchy sequence in (x,d) if for every ε > 0 there exists positive integer n(ε) such that d(xm,xn) < ε for all m,n > n(ε). (iii) (x,d) is said to be a complete bµ(s)-metric space if for every cauchy sequence in x converges to some x. and they proved the following theorem: theorem 1.2. [21] let (x,d) be a complete bν(s)-metric space and suppose that t : x → x be a selfmapping satisfying: d(tx,ty) ≤ λd(x,y) for all x,y ∈ x, where λ ∈ [0, 1). then t has a unique fixed point. int. j. anal. appl. 17 (3) (2019) 345 definition 1.6. [34] let x be a nonempty set, t : x → x and α : x ×x → [0,∞). we say that t is a triangular α-admissible mapping if (1) α(x,y) ≥ 1 implies α(tx,ty) ≥ 1, for x,y ∈ x; (2) α(x,z) ≥ 1, α(z,y) ≥ 1 implies α(x,y) ≥ 1, for all x,y,z ∈ x. lemma 1.1. [34] let t is a triangular α-admissible mapping. assume that there exists x0 ∈ x such that α(x0,tx0) ≥ 1. define sequence {xn} by xn = tnx0. then α(xm,xn) ≥ 1 for all m,n ∈ n with m < n. the following lemmas will be used for proving our main results. lemma 1.2. let (x,d) be a bν(s)-metric space and let {xn} be a sequence in x with distinct elements ( xn 6= xm for n 6= m). suppose that d(xn,xn+p) tends to 0 as n →∞ for all p = 1, 2, · · · ,ν, and xn → x as n →∞. then 1 s d(x,y) ≤ lim inf n→∞ d(xn,y) ≤ lim sup n→∞ d(xn,y) ≤ sd(x,y), for all y ∈ x with y 6= x. proof. since {xn} be a sequence in x with distinct elements, we can assume that xn is different from x and y for all n ∈ n. by the bν(s)-metric inequality, we have d(x,y) ≤ s[d(x,xn+ν−1) + d(xn+ν−1,xn+ν−2) + · · · + d(xn+1,xn) + d(xn,y)], d(xn,y) ≤ s[d(xn,xn+ν−1) + d(xn+ν−1,xn+ν−2) + · · · + d(xn+1,x) + d(x,y)]. since d(xn,xn+p) tends to 0 as n → ∞ for all p = 1, 2, · · · ,ν, and xn → x as n → ∞, taking lim infn→∞ on the both sides of the first inequality and taking lim supn→∞ on the both sides of the second inequality, it follows that 1 s d(x,y) ≤ lim inf n→∞ d(xn,y) ≤ lim sup n→∞ d(xn,y) ≤ sd(x,y). lemma 1.3. let (x,d) be a bν(s)-metric space and let {xn} be a sequence in x with distinct elements ( xn 6= xm for n 6= m). suppose that d(xn,xn+p) tends to 0 as n →∞ for all p = 1, 2, · · · ,ν and {xn} is not a cauchy sequence. then there exist � > 0 and two sequence {mk} and {nk} of positive integers such that nk > mk + ν, mk ≥ k and � ≤ lim inf k→∞ d(xnk,xmk) ≤ lim sup n→∞ d(xnk,xmk) ≤ s�, � s ≤ lim inf k→∞ d(xnk−1,xmk−1) ≤ lim sup n→∞ d(xnk−1,xmk−1) ≤ s�. int. j. anal. appl. 17 (3) (2019) 346 proof. since {xn} is not a cauchy sequence, there exists � > 0 for which we can choose two subsequences {xmk} and {xnk} of {xn} such that nk is the smallest index for which nk > mk ≥ k and d(xmk,xnk) ≥ �. (1.1) this means that d(xmk,xmk+1) < �,d(xmk,xmk+2) < �, · · · ,d(xmk,xnk−1) < �. (1.2) since limn→∞d(xn,xn+p) = 0 for all p = 1, 2, · · · ,ν, we can assume that nk > mk + ν. using (1.1), (1.2) and bν(s)-metric inequality, we have ε ≤ d(xmk,xnk) ≤ s[d(xmk,xnk−ν) + d(xnk−ν,xnk−ν+1) + · · · + d(xnk−2,xnk−1) + d(xnk−1,xnk)] ≤ s[� + d(xnk−ν,xnk−ν+1) + · · · + d(xnk−2,xnk−1) + d(xnk−1,xnk)]. since limn→∞d(xn,xn+1) = 0, we get � ≤ lim inf k→∞ d(xnk,xmk) ≤ lim sup k→∞ d(xnk,xmk) ≤ s�. using (1.1) and bν(s)-metric inequality, we have � ≤ d(xmk,xnk) ≤s[d(xmk,xmk−1) + d(xmk−1,xnk−1) + d(xnk−1,xnk−2) + · · · + d(xnk−ν+1,xnk)]. since limn→∞d(xn,xn+p) = 0 for all p = 1, 2, · · · ,ν, we get � ≤ s lim inf k→∞ d(xnk−1,xmk−1), that is � s ≤ lim inf k→∞ d(xmk−1,xnk−1). using (1.2) and bν(s)-metric inequality, we have d(xmk−1,xnk−1) ≤ s[d(xmk−1,xmk) + d(xmk,xnk−ν) + d(xnk−ν,xnk−ν+1) + · · · + d(xnk−3,xnk−2) + d(xnk−2,xnk−1)] ≤ s[d(xmk−1,xmk) + � + d(xnk−ν,xnk−ν+1) + · · · + d(xnk−3,xnk−2) + d(xnk−2,xnk−1)]. by taking the upper limit as k →∞ in the above inequality, since limn→∞d(xn,xn+1) = 0, we get lim sup k→∞ d(xmk−1,xnk−1) ≤ s�. int. j. anal. appl. 17 (3) (2019) 347 thus � s ≤ lim inf k→∞ d(xmk−1,xnk−1) ≤ lim sup k→∞ d(xmk−1,xnk−1) ≤ sε. � lemma 1.4. let {an} and {bn} be two sequences of nonnegative numbers. if lim n→∞ bn = 0, lim n→∞ max{an,bn} = a, then limn→∞an = a. 2. main results throughout the paper, f(t) denotes the set of fixed points of the mapping t . for a given bν(s)-metric space (x,d) and a fixed x0 ∈ x, we will denote ||x|| = d(x,x0) for x ∈ x. we denote by ψ the family of all functions ψ : [0, 1] → [0,∞) which is an increasing function, continuous at 0, with ψ(0) = 0. theorem 2.1. let (x,d) be a complete bν(s)-metric space with s ≥ 1, t : x → x and α : x ×x → [0,∞) a given function. suppose that following conditions are satisfied: (1) t is a triangular α-admissible mapping; (2) there exist λ ≥ 0,η ≥ 1, β ∈ [0,η] and ψ ∈ ψ such that for every ε ∈ [0, 1] and for all x,y ∈ x with α(x,y) ≥ 1 and d(tx,ty) > 0, sd(tx,ty) <(1 −ε)m(x,y) + λεηψ(ε)[1 + ||x|| + ||y|| + ||tx|| + ||ty||]β (2.1) where m(x,y) = max{d(x,y),d(x,tx),d(y,ty), d(x,tx)d(y,ty) 1 + d(x,y) }; (3) there exists x0 ∈ x such that α(x0,tx0) ≥ 1; (4) if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n ∈ n and xn → x as n → ∞, then there exists a subsequence {xn(k)} of {xn} such that α(xn(k),x) ≥ 1 for all k ∈ n. then t has a fixed point u and {tnx0} converges to u. further, if all x,y ∈ f(t), we have α(x,y) ≥ 1, then t has a unique fixed point in x. int. j. anal. appl. 17 (3) (2019) 348 proof. let x0 ∈ x satisfies α(x0,tx0) ≥ 1. we construct the sequence {xn} in x by xn = txn−1 = tnx0 for n ∈ n. if xn = xn+1 for some n ∈ n, then xn is a fixed point of t . consequently, we suppose that xn 6= xn+1 for all n ∈ n. since t is a triangular α-admissible mapping, by lemma 1.1, we have α(xn,xm) ≥ 1, for all n,m ∈ n with n < m. (2.2) step i. we will show that the sequence {d(xn,xn+1)} is decreasing. indeed, putting ε = 0, x = xn, y = xn+1 in (2.1), we obtain sd(xn,xn+1) = sd(txn−1,txn) < m(xn−1,xn), (2.3) where m(xn−1,xn) = max{d(xn−1,xn),d(xn−1,txn−1),d(xn,txn), d(xn−1,txn−1)d(xn,txn) 1 + d(xn−1,xn) } = max{d(xn−1,xn),d(xn−1,xn),d(xn,xn+1), d(xn−1,xn)d(xn,xn+1) 1 + d(xn−1,xn) } = max{d(xn−1,xn),d(xn,xn+1)}. (2.4) combining (2.3) and (2.4), we have sd(xn,xn+1) < max{d(xn−1,xn),d(xn,xn+1)}. hence d(xn,xn+1) < 1 s d(xn−1,xn) (2.5) for all n ∈ n. thus the sequence {d(xn,xn+1)} is decreasing. step ii. we will prove that xn 6= xm for all n 6= m. suppose that xn = xm for some n > m, so we have xn+1 = txn = txm = xm+1. by (2.5), we have d(xn,xn+1) < d(xn−1,xn) < · · · < d(xm,xm+1) = d(xn,xn+1) a contradiction. thus xn 6= xm for all n 6= m. step iii. we will show that for p = 1, 2, · · · ,ν, the sequence {d(xn,xn+p)} is bounded. indeed, since α(xn,xn+p) ≥ 1 and d(xn,xn+p) > 0, putting ε = 0, x = xn, y = xn+p in (2.1), we obtain sd(xn,xn+p) = sd(txn−1,txn+p−1) < m(xn−1,xn+p−1), (2.6) int. j. anal. appl. 17 (3) (2019) 349 where m(xn−1,xn+p−1) = max{d(xn−1,xn+p−1),d(xn−1,txn−1),d(xn+p−1,txn+p−1), d(xn−1,txn−1)d(xn+p−1,txn+p−1) 1 + d(xn−1,xn+p−1) } = max{d(xn−1,xn+p−1),d(xn−1,xn),d(xn+p−1,xn+p), d(xn−1,xn)d(xn+p−1,xn+p) 1 + d(xn−1,xn+p−1) } ≤ max{d(xn−1,xn+p−1),d(xn−1,xn),d(xn−1,xn)2}. combining (2.6), we have sd(xn,xn+p) < max{d(xn−1,xn+p−1),d(xn−1,xn),d(xn−1,xn)2}. taking an = d(xn,xn+p) and bn = d(xn,xn+1), since s ≥ 1, we have san < max{an−1,bn−1,b2n−1}. since sbn < bn−1 ≤ max{an−1,bn−1,b2n−1} and sb2n < b2n−1 ≤ max{an−1,bn−1,b2n−1}, we have max{an,bn,b2n} < 1 s max{an−1,bn−1,b2n−1} (2.7) for all n ∈ n. thus the sequence {max{an,bn,b2n}}n∈n is decreasing. thus k = sup{d(xn,xn+p),d(xn,xn+1),d(xn,xn+1)2 : n = 1, 2, · · · ; p = 1, 2, · · · ,ν} < ∞. (2.8) step iv. we will prove that the sequence cn = d(xn,x0) is bounded. using (2.5), we deduce the following estimate cn = d(xn,x0) ≤ s[d(x0,xν−1) + d(xν−1,xν−2) + · · · + d(x2,x1) + d(x1,xn+1) + d(xn+1,xn)] ≤ s[cν−1 + (ν − 1)c0] + sd(tx0,txn). therefore, we infer from (2.1) that cn ≤(1 −ε)m(x0,xn) + λεηψ(ε)[1 + ||xn|| + ||x1|| + ||xn+1||]β + s[cν−1 + (ν − 1)c0], (2.9) int. j. anal. appl. 17 (3) (2019) 350 where m(x0,xn) = max{d(x0,xn),d(x0,tx0),d(xn,txn), d(x0,tx0)d(xn,txn) 1 + d(x0,xn) } = max{d(x0,xn),d(x0,x1),d(xn,xn+1), d(x0,x1)d(xn,xn+1) 1 + d(x0,xn) } ≤ max{cn,c1,c21}. (2.10) combining (2.9) and (2.10), as β ≤ η we have cn ≤(1 −ε)[max{cn,c1,c21}] + λεηψ(ε)[1 + cn + c1 + cn+1] η + s[cν−1 + (ν − 1)c0]. (2.11) suppose that the sequence cn = d(xn,x0) is not bounded. then there is a subsequence {cni} satisfying that cni ≥ max{1,c1,c21, 1 + νk} for all i ∈ n and cni →∞. using (2.8), we have cn+1 = d(xn+1,x0) ≤ s[d(x0,xn) + d(xn,xn+1) + · · · + d(xn+ν−2,xn+ν−1) + d(xn+ν−1,xn+1)] ≤ sd(x0,xn) + sνk = scn + sνk. thus, for all i ∈ n, (2.11) implies that cni ≤ (1 −ε)cni + λε ηψ(ε)[1 + (1 + s)cni + sνk] η + s[cν−1 + (ν − 1)c0] ≤ (1 −ε)cni + λε ηψ(ε)(3scni) η + s[cν−1 + (ν − 1)c0]. thus we have cni ≤ (1 −ε)cni + aε ηψ(ε)cηni + b for some a,b > 0. hence εcni ≤ aε ηψ(ε)cηni + b. now, as in [23], the choice ε = εi = (1 + b)/cni leads to the contradiction 1 ≤ a(1 + b)ηψ(εi) → 0. hence the sequence cn = d(xn,x0) is bounded. step v. for p = 1, 2, · · · ,ν, limn→∞d(xn,xn+p) = 0. int. j. anal. appl. 17 (3) (2019) 351 for all ε ∈ (0, 1] and for x = xn,y = xn+p we have sd(xn,xn+p) = sd(txn−1,txn+p−1) < (1 −ε)m(xn−1,xn+p−1) + λεηψ(ε)[1 + ||xn|| + ||xn+p−1|| + ||xn+p||]β ≤ (1 −ε)m(xn−1,xn+p−1) + bεηψ(ε), b > 0 (2.12) where m(xn−1,xn+p−1) = max{d(xn−1,xn+p−1),d(xn−1,txn−1),d(xn+p−1,txn+p−1), d(xn−1,txn−1)d(xn+p−1,txn+p−1) 1 + d(xn−1,xn+p−1) } = max{d(xn−1,xn+p−1),d(xn−1,xn),d(xn+p−1,xn+p), d(xn−1,xn)d(xn+p−1,xn+p) 1 + d(xn−1,xn+p−1) } ≤ max{d(xn−1,xn+p−1),d(xn−1,xn),d(xn−1,xn)2}. (2.13) for p = 1, using (2.13) and (2.5), the inequality (2.12) implies that sd(xn,xn+1) = sd(txn−1,txn) ≤ (1 −ε)m(xn−1,xn) + bεηψ(ε) ≤ (1 −ε)d(xn−1,xn) + bεηψ(ε). (2.14) by (2.5), the sequence {d(xn,xn+1)} is converges. if limn→∞d(xn,xn+1) = d∗ > 0, it follows from (2.14) that d∗ ≤ bψ(ε), that is d∗ = 0. a contradiction. thus limn→∞d(xn,xn+1) = 0. in the following, we assume that d(xn,xn+1) < 1 for all n ∈ n. thus d(xn,xn+1) 2 < d(xn,xn+1) (2.15) for all n ∈ n. fixed p ≥ 2. (2.12), (2.13) and (2.15) imply that sd(xn,xn+p) ≤ (1 −ε) max{d(xn−1,xn+p−1),d(xn−1,xn)} + bεηψ(ε), that is, using the notations in step iii, an ≤ san ≤ (1 −ε) max{an−1,bn−1} + bεηψ(ε). (2.16) int. j. anal. appl. 17 (3) (2019) 352 from step iii, we see that max{an,bn} is decreasing. since limn→∞ bn = 0, by lemma 1.4, we have lim n→∞ an = lim n→∞ max{an,bn} = t. if t > 0, taking the limit as n →∞ on both sides of (2.16), we have t ≤ bψ(ε) for all ε ∈ (0, 1], that is t = 0. a contradiction. step vi. we will show that {xn} is cauchy sequence in x. suppose, to the contrary, that is, {xn} is not a cauchy sequence. by step v and lemma 1.3, there exist δ > 0 and two sequence {mk} and {nk} of positive integers such that nk > mk + ν, mk ≥ k and δ ≤ lim inf k→∞ d(xnk,xmk) ≤ lim sup n→∞ d(xnk,xmk) ≤ sδ, δ s ≤ lim inf k→∞ d(xnk−1,xmk−1) ≤ lim sup n→∞ d(xnk−1,xmk−1) ≤ sδ. (2.17) now putting x = xmk−1,y = xnk−1 in (2.1), since d(xnk,xmk) > 0 and α(xnk,xmk) ≥ 1, we obtain sδ ≤ sd(xmk,xnk) = sd(txmk−1,txnk−1) < (1 −ε)m(xmk−1,xnk−1) + bεψ(ε), (2.18) where m(xmk−1,xnk−1) = max{d(xmk−1,xnk−1),d(xmk−1,txmk−1),d(xnk−1,txnk−1), d(xmk−1,txmk−1)d(xnk−1,txnk−1) 1 + d(xmk−1,xnk−1) } = max{d(xmk−1,xnk−1),d(xmk−1,xmk),d(xnk−1,xnk), d(xmk−1,xmk)d(xnk−1,xnk) 1 + d(xmk−1,xnk−1) }. (2.19) using step v and (2.19), we have lim sup k→∞ m(xmk−1,xnk−1) ≤ sδ. taking the limit of supermum as k →∞ in (2.18), sδ ≤ bψ(ε), that is, δ = 0, a contradiction. hence {xn} is cauchy sequence in x. since (x,d) is complete, there exists u ∈ x such that lim n→∞ d(xn,u) = 0. int. j. anal. appl. 17 (3) (2019) 353 step vii. we show that u is a fixed point of t. using theorem 2.1 (iv), there exists a subsequence {xnk} of {xn} such that α(xnk,u) ≥ 1 for all k ∈ n. suppose that u 6= tu, so d(u,tu) > 0. since {xn} is a sequence with distinct elements, we can assume that xn 6= tu for all n ∈ n. putting x = xnk,y = u in (2.1), we get d(u,tu) ≤ s[d(u,xnk+ν) + d(xnk+ν−1,xnk+ν−2) + · · · + d(xnk+2,xnk+1) + d(xnk+1,tu)] < s[d(u,xnk+ν) + d(xnk+ν−1,xnk+ν−2) + · · · + d(xnk+2,xnk+1)]+ (1 −ε)m(xnk,u) + bεψ(ε), (2.20) where m(xnk,u) = max{d(xnk,u),d(xnk,txnk),d(u,tu), d(xnk,txnk)d(u,tu) 1 + d(xnk,u) } = max{d(xnk,u),d(xnk,xnk+),d(u,tu), d(xnk,xnk+1)d(u,tu) 1 + d(xnk,u) }. (2.21) using step v and (2.21), we have lim k→∞ m(xnk,u) = d(u,tu). taking the limit as k →∞ in (2.20), using step v, we have d(u,tu) ≤ (1 −ε)d(u,tu) + bεψ(ε), from which we have d(u,tu) ≤ bψ(ε), that is d(u,tu) = 0, a contradiction. thus u = tu. step viii. finally, we prove that the fixed point of t is unique. suppose that u,v are two fixed points of t such that u 6= v. then by the hypothesis, α(u,v) ≥ 1. hence, from (2.1) with ε = 0, x = u and y = v we have sd(u,v) = sd(tu,tv) < m(u,v) where m(u,v) = max{d(u,v),d(u,tu),d(v,tv), d(u,tu)d(v,tv) 1 + d(u,v) } = d(u,v). thus d(u,v) < d(u,v), a contradiction. � note. in theorem 2.1, if s > 1, the inequality (2.1) can be replaced by sd(tx,ty) ≤(1 −ε)m(x,y)) + λεηψ(ε)[1 + ||x|| + ||y|| + ||tx|| + ||ty||]β. int. j. anal. appl. 17 (3) (2019) 354 moreover, if ν ≥ 2, we can give another method to prove that {xn} is a cauchy sequence. in fact, from (2.7) and (2.8), we have max{an,bn,b2n}≤ k sn for all n ∈ n. using bµ(s)-inequality, for all n,p ∈ n, we have d(xn,xn+pν) ≤ s[d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xn+ν−1,xn+ν)] + s2[d(xn+ν,xn+ν+1) + d(xn+ν+1,xn+ν+2) + · · · + d(xn+2ν−1,xn+2ν)] · · · + sp[d(xn+(p−1)ν,xn+(p−1)ν+1) + d(xn+(p−1)ν+1,xn+(p−1)ν+2) + · · · + d(xn+pν−1,xn+pν)] ≤ s[ k sn + k sn+1 + · · · + k sn+ν−1 ] + s2[ k sn+ν + k sn+ν+1 + · · · + k sn+2ν−1 ] + · · · + sp[ k sn+(p−1)ν + k sn+(p−1)ν+1 + · · · + k sn+pν−1 ] ≤ sν k sn 1 − 1 sν−1 , d(xn,xn+pν+1) ≤ s[d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xn+ν−1,xn+ν)] + s2[d(xn+ν,xn+ν+1) + d(xn+ν+1,xn+ν+2) + · · · + d(xn+2ν−1,xn+2ν)] · · · + sp[d(xn+(p−1)ν,xn+(p−1)ν+1) + d(xn+(p−1)ν+1,xn+(p−1)ν+2) + · · · + d(xn+pν−1,xn+pν+1)] ≤ sν k sn 1 − 1 sν−1 , · · · d(xn,xn+pν+ν−1) ≤ s[d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xn+ν−1,xn+ν)] + s2[d(xn+ν,xn+ν+1) + d(xn+ν+1,xn+ν+2) + · · · + d(xn+2ν−1,xn+2ν)] · · · int. j. anal. appl. 17 (3) (2019) 355 + sp[d(xn+(p−1)ν,xn+(p−1)ν+1) + d(xn+(p−1)ν+1,xn+(p−1)ν+2) + · · · + d(xn+pν−1,xn+pν+ν−1)] ≤ sν k sn 1 − 1 sν−1 , this implies d(xn,xn+m) ≤ sν k sn 1 − 1 sν−1 . thus {xn} is a cauchy sequence. corollary 2.1. let (x,�,d) be a partially ordered and complete bν(s)-metric space with s ≥ 1. suppose that following conditions are satisfied: (1) t is a increasing mapping with respect �, that is tx � ty if x � y; (2) there exist λ ≥ 0,l ≥ 0,η ≥ 1, β ∈ [0,η] and ψ ∈ ψ such that for every ε ∈ [0, 1] and for all x,y ∈ x with x � y and d(tx,ty) > 0, sd(tx,ty) ≤(1 −ε)m(x,y) + λεηψ(ε)[1 + ||x|| + ||y|| + ||tx|| + ||ty||]β where m(x,y) = max{d(x,y),d(x,tx),d(y,ty), d(x,tx)d(y,ty) 1 + d(x,y) }; (3) there exists x0 ∈ x such that x0 � tx0;. (4) xn � x for all n ∈ n whenever {xn} is nondecreasing sequence in x such that xn → x ∈ x. then t has a fixed point u and {tnx0} converges to u. further, if all x,y ∈ f(t), x and y are comparable, then t has a unique fixed point in x. proof. define α : x ×x → [0,∞) as α(x,y) =   1, if x � y, 0, otherwise. clearly, by theorem 2.1, t has a fixed point. � int. j. anal. appl. 17 (3) (2019) 356 3. common fixed point results in this section, we prove some common fixed point results for two self-mappings. following [27], we introduce the notion of f −α-admissible mapping. definition 3.1. let x be a non-empty set. and let t,f : x −→ x and α : x×x −→ [0,∞). the mapping t is f −α-admissible if, for all x,y ∈ x such that α(fx,fy) ≥ 1, we have α(tx,ty) ≥ 1. clearly, if f is the identity mapping, then t is α-admissible. theorem 3.1. let (x,d) be a complete bν(s)-metric space with s ≥ 1 and α : x ×x → [0,∞) be a given function. let t,f : x → x be two mappings. suppose that following conditions are satisfied:: (1) t is an f −α-admissible mapping; (2) there exist λ ≥ 0,η ≥ 1, β ∈ [0,η] and ψ ∈ ψ such that for every ε ∈ [0, 1] and for all x,y ∈ x with α(fx,fy) ≥ 1 and d(tx,ty) > 0, sd(tx,ty) ≤(1 −ε)m(fx,fy) + λεηψ(ε)[1 + ||fx|| + ||fy|| + ||tx|| + ||ty||]β (4.1) where m(fx,fy) = max{d(fx,fy),d(fx,tx),d(fy,ty), d(fx,tx)d(fy,ty) 1 + d(fx,fy) }; (3) there exists x0 ∈ x such that α(fx0,tx0) ≥ 1; (4) if {yn} is a sequence in x such that α(yn,yn+1) ≥ 1 for all n ∈ n and yn → y as n → ∞, then there exists a subsequence {yn(k)} of {yn} such that α(yn(k),y) ≥ 1 for all k ∈ n. then t and f have a point of coincidence in x. moreover, if t and f are weakly compatible, then t and f have a common fixed point. further, if all points of coincidence of f and t , we have α(fx,fy) ≥ 1, then t and f have a unique point of coincidence in x.. before we prove this theorem, we introduce the following lemma. lemma 3.1. [13] let x be a non-empty set and let f : x → x be a self-mapping. then there exists a subset e of x such that fe = fx and f|e is injective. int. j. anal. appl. 17 (3) (2019) 357 proof. by lemma 3.1, there exists e ⊆ x such that fe = fx and f : e −→ x is one-to-one. now, define a map h : f(e) −→ f(e) by h(f(x)) = tx. since f is one-to-one on e, h is well defined. note that for all fx,fy ∈ f(e) with α(fx,fy) ≥ 1,d(h(fx),h(fy)) > 0, then (4.1) can be rewrite as sd(h(fx),h(fy)) ≤(1 −ε)m(fx,fy) + λεηψ(ε)[1 + ||fx|| + ||fy|| + ||h(fx)|| + ||h(fy)||]β, m(fx,fy) = max{d(fx,fy),d(fx,h(fx)),d(fy,h(fy)), d(fx,h(fx))d(fy,h(fy)) 1 + d(fx,fy) }; thus for all x′,y′ ∈ f(e) with α(x′y′) ≥ 1 and d(hx′,hy′) > 0, we have sd(hx′,hy′) ≤(1 −ε)m(x′,y′) + λεηψ(ε)[1 + ||x′|| + ||y′|| + ||hx′|| + ||hy′||]β where m(x′,y′) = max{d(x′,y′),d(x′,hx′),d(y′,hy′), d(x′,hx′)d(y′,hy′) 1 + d(x′,y′) }. since f(e) = f(x) is complete, by using theorem 2.1, there exists x0 ∈ e such that h(fx0) = fx0. hence t and f have a point of coincidence in x. it is clear that t and f have a common fixed point whenever t and f are weakly compatible. � in 2017, m. rangamma and p. m. reddy [25] established a unique common fixed point theorem for t-contraction of two self mappings on generalized cone b-metric spaces with solid cone. in the following theorem, a unique common fixed point theorem for t-contraction of two self mappings on bν(s)-metric spaces is established. theorem 3.2. let (x,d) be a complete bν(s)-metric space and α : x × x → [0,∞) be a given function. let t,f : x → x be a mappings. suppose that t is one to one and t(x) is a complete subspace of x, and the following conditions are satisfied: (1) if α(tx,ty) ≥ 1 then α(tfx,tfy) ≥ 1, and α(x,y) ≥ 1,α(y,z) ≥ 1 implies α(x,z) ≥ 1, x,y,z ∈ x; (2) there exist λ ≥ 0,η ≥ 1, β ∈ [0,η] and ψ ∈ ψ such that for every ε ∈ [0, 1] and for all x,y ∈ x with α(tx,ty) ≥ 1 and d(tfx,tfy) > 0, sd(tfx,tfy) ≤(1 −ε)m(tx,ty) + λεηψ(ε)[1 + ||tx|| + ||ty|| + ||tfx|| + ||tfy||]β (4.3) where m(tx,ty) = max{d(tx,ty),d(tx,tfx),d(ty,tfy), d(tx,tfx)d(ty,tfy) 1 + d(tx,ty) }; int. j. anal. appl. 17 (3) (2019) 358 (3) there exists x0 ∈ x such that α(tx0,tfx0) ≥ 1; (4) if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n ∈ n and xn → x as n → ∞, then there exists a subsequence {xn(k)} of {xn} such that α(xn(k),x) ≥ 1 for all k ∈ n. then f has a fixed point in x. further, if all x,y ∈ f(f), we have α(tx,ty) ≥ 1 then f has a unique fixed point in x. moreover, if f and t are commuting at the fixed point of f, then f and t have a unique common fixed point in x. proof. since t is one to one, the conditions (i) and (ii) can be restated as (i′) if α(tx,ty) ≥ 1 then α(tft−1tx,tft−1ty) ≥ 1, and α(x,y) ≥ 1,α(y,z) ≥ 1 implies α(x,z) ≥ 1, x,y,z ∈ x. (ii′) if α(tx,ty) ≥ 1 and d(tft−1tx,tft−1ty) > 0 implies sd(tft−1tx,tft−1ty) ≤ (1 −ε)m(tx,ty) + λεηψ(ε)[1 + ||tx|| + ||ty|| + ||tft−1tx|| + ||tft−1ty||]β where m(tx,ty) = max{d(tx,ty),d(tx,tft−1tx),d(ty,tft−1ty), d(tx,tft−1tx)d(ty,tft−1ty) 1 + d(tx,ty) }. let f′ = tft−1. then we have (i′′) f′ is a triangular α-admissible mapping in tx. (ii′′) for all x′,y′ ∈ tx, if α(x′,y′) ≥ 1 and d(f′x,f′y) > 0 implies sd(f′x′,f′y′) ≤(1 −ε)m(x′,y′) + λεηψ(ε)[1 + ||x′|| + ||y′|| + ||f′x′|| + ||f′y′||]β where m(x′,y′) = max{d(x′,y′),d(x′,f′x′),d(y′,f′y′), d(x′,f′x′)d(y′,f′y) 1 + d(x′,y′) }. then, by theorem 2.1, there exist x′ = tx ∈ tx such that f′tx = tx, that is tfx = tx. since t is one to one, we get fx = x. if x ∈ f(f), then tfx = tx and tft−1tx = tx, which mains that tx is a fixed point of f′. thus if for all x,y ∈ f(f), α(tx,ty) ≥ 1, then, by theorem 2.1, f′ has a unique fixed point. it follows that f has a unique fixed point. moreover, if f and t are commuting at the unique fixed point int. j. anal. appl. 17 (3) (2019) 359 x of f, then tx = tfx = ftx, i.e., tx is also a fixed point of f. since f has unique fixed point, we have tx = x, i.e., x is also the fixed point of t . so f and t have a unique common fixed point in x. � references [1] m. abbas, n. saleem and m. de la sen, optimal coincidence point results in partially ordered non-archimedean fuzzy metric spaces, fixed point theory appl. 2016(2016), art. id 44. [2] j. ahmad, m. arshad and c. vetro, on a theorem of khan in a generalized metric space, int. j. anal. 2013(2013), art. id 852727. [3] arslan hojat ansari and anchalee kaewcharoen, c-class functions and fixed point theorems for generalized α-η-ψ-ϕ-fcontraction type mappings in α-η-complete metric spaces, j. nonlinear sci. appl. 9(2016), 4177-4190. [4] i.a. bakhtin, the contraction mapping principle in quasi-metric space [in russian], funk.an. ulianowsk gos. ped. inst. 30(1989), 26-37. [5] s. balasubramanian, a pata-type fixed point theorem, math. sci. 8(2014), 65-69. [6] s. banach, sur les operations dans les ensembles abstraits et leur application aux equations integrals, fundam. math. 3(1922), 133-181. [7] a. branciari, a fixed point theorem of banach-caccioppoli type on a class of generalized metric spaces, publ. math. debrecen 57(2000), 31-37. [8] s. czerwik, contraction mappings in b-metric spaces, acta math. inf. univ. ostrav. 1(1993), 5-11. [9] m. eshaghi, s.mohseni, m.r. delavar, m. de la sen, g. h. kim and a. arian, pata contractions and coupled type fixed points, fixed point theory appl. 2014(2014), art. id 130. [10] k. fan, extensions of two fixed point theorems of f. e. browder, math. z. 112(1969), 234-240. [11] b. fisher, on a theorem of khan, riv. math. univ. parma. 4(1978), 135-137. [12] r. george, s. radenovic, k.p. reshma and s. shukla, rectangular b-metric spaces and contraction principle, j. nonlinear sci. appl. 8(2016), 1005–1013. [13] r. h. haghi, s. rezapour and n. shahzad, some fixed point generalizations are not real generalizations, nonlinear anal., model. control 74(2011), 1799-1803. [14] z. kadelburg and s. radenovic, fixed point and tripled fixed point theorems under pata-type conditions in ordered metric space, int. j. anal. appl. 6(2014),113-122. [15] z. kadelburg and s. radenovic, fixed point theorems for pata-type maps in ordered metric space, j. egypt. math. soc. 24(2016), 77-82. [16] z. kadelburg and s. radenovic, a note on pata-type cyclic contractions, sarajevo j. math. 11(2015), 235-245. [17] z. kadelburg and s. radenovic, pata-type common fixed point results in b-metric and b-rectangular metric spaces, j. nonlinear sci. appl. 8(2015), 944-954. [18] m. s. khan, m. swaleh and s. sessa, fixed point theorems by altering distances between the points, bull. aust. math. soc. 30(1984), 1-9. [19] w. a. kirk and n. shahzad, generalized metrics and caristis theorem, fixed point theory appl. 2013(2013), art. id 129. [20] ma zhenhua, jiang lining and sun hongkai, c*-algebra-valued metric spaces and related fixed point theorems, fixed point theory appl. 2014(2014), art. id 206. [21] z.d. mitrovic and s. radenovic, the banach and reich contractions in bν (s)-metric spaces, j. fixed point theory appl. 19(2017), 3087-3905. int. j. anal. appl. 17 (3) (2019) 360 [22] d.d. mitrovic, and s. radenovic, a common fixed point theorem of jungck in rectangular b-metric spaces, acta math. hungar. 153(2017), 401-407. [23] v. pata, a fixed point theorem in metric spaces, j. fixed point theory appl. 10(2011), 299-305. [24] h. piri, s. rahrovi and p. kumam, khan type fixed point theorems in a generalized metric space, j. math. computer sci. 16(2016), 211-217. [25] m. rangamma and p. m. reddy, a common fixed point theorem for t-contractions on generalized cone b-metric spaces, commun. korean math. soc. 32(2017), 65-74. [26] z. raza, n. saleem and m. abbas, optimal coincidence points of proximal quasi-contraction mappings in non-archimedean fuzzy metric spaces, j. nonlinear sci. appl. 9(2016), 3787-3801. [27] v. l. rosa and p. vetro, common fixed points for α − ψ − φ-contractionseneralized in g metric spaces, nonlinear anal., model. control 19(2014), 43-54. [28] j. r. roshan, vahid parvaneh, z. kadelburg and n. hussain, new fixed point results in b-rectangular metric spaces, nonlinear anal., model. control 21(2006),614-634. [29] n. saleem, m. abbas and z. raza, optimal coincidence best approximation solution in non-archimedean fuzzy metric spaces, iran. j. fuzzy syst. 13(2016), 113-124. [30] n. saleem, b. ali, m. abbas and z. raza, fixed points of suzuki type generalized multivalued mappings in fuzzy metric spaces with applications, fixed point theory appl. 2015(2015), art. id 36. [31] b. samet, discussion on ’a fixed point theorem of banach-caccioppoli type on a class of generalized metric spaces’ by branciari, publ. math. debrecen, 76(2010), 493-494. [32] i. r. sarma, j. m. rao and s. s. rao, contractions over generalized metric spaces, j. nonlinear sci. appl. 2(2009), 180-182. [33] shen congcong, jiang lining and ma zhenhua, c*-algebra-valued g-metric spaces and related fixedpoint theorems, j. function spaces 2018(2018), article id 3257189. [34] vahid parvaneh, nawab hussain and zoran kadelburg, generalized wardowski type fixed point theorems via α-admissable fg-contractions in b-metric spaces, acta math. sci. 5(2016), 1445-1456. [35] t.suzuki, generalized metric spaces do not have the compatible topology, abstr. appl. anal. 2014(2014), art. id 458098. [36] wang liguang, liu bo and bai ran, stability of a mixed type functional equation on multi-banach spaces, a fixed point approach, fixed point theory appl. 2010(2010), art. id 283827. 1. introduction 2. main results 3. common fixed point results references international journal of analysis and applications volume 17, number 1 (2019), 76-104 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-76 diminishing choquet hesitant 2-tuple linguistic aggregation operator for multiple attributes group decision making ismat beg1∗, raja noshad jamil2 and tabasam rashid2 1lahore school of economics, lahore-53200, pakistan 2university of management and technology, lahore-54770, pakistan ∗corresponding author: ibeg@lahoreschool.edu.pk abstract. in this article, we develop a diminishing hesitant 2-tuple averaging operator (dh2ta) for hesitant 2-tuple linguistic arguments. dh2ta work in the way that it aggregate all hesitant 2-tuple linguistic elements and during the aggregation process it also controls the hesitation in translation of the resultant aggregated linguistic term. we develop a scalar product for hesitant 2-tuple linguistic elements and based on the scalar product a weighted diminishing hesitant 2-tuple averaging operator (dwh2ta) is introduced. moreover, combining choquet integral with hesitant 2-tuple linguistic information, the diminishing chouqet hesitant 2-tuple average operator (dch2ta) is defined. the proposed operators higher reflect the correlations among the elements. after investigating the properties of these operators, a multiple attribute decision making method based on dch2ta operator is proposed. finally, an example is given to illustrate the significance and usefulness of proposed method. 1. introduction different procedures wherein problems that manage indefinite and vague data mostly involves the vulnerability of their definition structures. utilizing numerical modelling to represent such indeterminate data would not be reliably adequate. in these conditions wherein the vulnerability would not be of probabilistic received 2018-10-17; accepted 2018-11-26; published 2019-01-04. 2010 mathematics subject classification. 91b06; 94d05; 90b50; 91b10; 46s40; 03e72. key words and phrases. hesitant 2-tuple model; aggregation operator; choquet integral; multiple attribute group decision making; supply chain management. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 76 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-76 int. j. anal. appl. 17 (1) (2019) 77 nature and the capacities are unclear, it is hard to give numerical specific information. usually the decision makers that take an interest in this kind of issues utilize linguistic descriptors to particular their evaluation and identified with the uncertain potential they have concerning the issues [38,40]. therefore, the utilization of linguistic demonstration in problems managing non-probabilistic instability shows up justification and has made effective result in particular fields, for example: situation realization [35], decision models [6, 9, 49, 64], information retrieval [25, 30], risk evaluation [18], engineering analysis [39, 40], sensory evaluation [10, 36], performance appraisal [1, 2], data mining [27], social alternative [19] and waste management [51]. these accomplishments have not been possible without systems to complete the improvement of computing with words (cw) [34] that implies the utilization of linguistic knowledge. the accompanying algorithm indicates how these translations to be functional. algorithm 1.1. step 1. input data in the form of linguistic terms or 2-tuple linguistic terms step 2. translation into equivalent numeric value step 3. manipulation step 4. retranslation into linguistic terms / 2-tuple linguistic terms accordingly step 5. output data these ideas for cw have an edge on probability theory [26, 33], the uncertainty models, in these problems are alternatively involving the imprecision and vagueness of the linguistic descriptors. for this reason other tools as fuzzy logic [71] and the fuzzy linguistic process [72] used specific computational models for cw, for instance: • the linguistic computational model created on membership functions such as [13, 37] these models based on the fuzzy linguistic approach and makes the computations instantly on the membership function of the linguistic terms by way of utilizing the extension principle [16, 31]. • foundation of the linguistic symbolic computational models are on ordinal scales [65]. these models represent the understanding in keeping with the fuzzy linguistic technique and makes use of the ordered structure of the linguistic term set to achieve symbolic computations in such ordered linguistic scales. equivalent tactics founded on this mode of computing has been discussed in [14, 62]. it notable that this mannequin has been frequently applied to decision making practices due to its easy adaptation and effortlessness for decision makers [65]. linguistic models seek after the computational plan showed by means of yager in [66, 68] can be seen in general algorithm 1.1, that features out the significance of the interpretation and translation approaches in cw and likewise mendel and wu [42] highlighted similar techniques in computing with perceptions . in the article author discussed that firstly, taking data linguistically and translates into a computing int. j. anal. appl. 17 (1) (2019) 78 tool for manipulative structure. in the second stage, incorporates taking the outcomes from the control, computing device, arrange and change them into linguistic information as an approach to be reasonable by method for individuals that is without uncertainty, one of the essential desire of cw [42]. these linguistic computational units present a most important weak point, in view that they carried out the translation step as an approximation method to precise the outcome in the usual expression area (initial term set) scary a lack of accuracy [23]. to obstruct such inaccuracy in the translation step was once offered the 2-tuple linguistic computational model [22, 41]. it is a typical mannequin that broadens the utilization of records adjusting the fuzzy linguistic strategy representation with including a parameter with essential linguistic representation as an approach to show signs of improvement exactness of the linguistic calculations after the re-interpretation step, holding the cw plan stated in algorithm 1.1 and the work out the capacity of the result. in recent times, numerous aggregation operators have been produced for the 2-tuple linguistic model to assess diverse decision making issues [59]. herrera and mart́ınez [22] proposed the 2-tuple arithmetic weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended 2-tuple weighted averaging operator. xu et al. [63] developed the extended geometric mean operator, the extended arithmetic averaging operator, the extended ordered weighted averaging operator and the extended ordered weighted geometric operator . jiang and fan [28], proposed the 2-tuple ordered weighted averaging operator and the 2-tuple ordered weighted geometric operator. the extended 2-tuple ordered weighted averaging operator was proposed in [73]. the extended 2-tuple weighted geometric operator and the extended 2-tuple ordered weighted geometric operator have been developed in [60]. herrera et al. [24] proposed an unbalanced linguistic computational model that helpful for calculating the 2-tuple fuzzy linguistic computational model to achieve processes of evaluating words for unbalanced term sets in an accurate mode without loss of information. furthermore, dong et al. [15] proposed a consistency improving model which preserves the utmost original knowledge and preferences in the process of improving consistency and it also guarantees that the elements in the optimal adjusted unbalanced linguistic preference relation are all simple unbalanced linguistic terms. aggregation operators examined over, the attributes are thought to be autonomous of each other, which are differentiated by an independent axiom [43, 44, 57]. but in the real decision making practice, the characteristics of the problem are often dependent or correlated to each other. choquet integral [11] is one of the valuable instrument to build up the model an issue, which utilize the properties as between reliance or connection to each other. choquet integral has examined and connected all the basic properties of the decision making problems [32, 50, 67, 69]. yager [67] proposed the induced choquet ordered averaging operator to aggregate a group of real arguments while in [69], yager combined the intuitionistic fuzzy sets with choquet integral. the intuitionistic fuzzy choquet integral operator obtained in [9]. tan and chen [53] developed int. j. anal. appl. 17 (1) (2019) 79 the induced choquet ordered averaging operator. xu [64] proposed the intuitionistic fuzzy correlated averaging operator, the intuitionistic fuzzy correlated geometric operator, the interval-valued intuitionistic fuzzy correlated averaging operator and the interval-valued intuitionistic fuzzy correlated geometric operator to aggregate the intuitionistic fuzzy information and the interval-valued intuitionistic fuzzy information. beg and rashid [4] used choquet integral, for selection of bike when the criteria include interactions among each others. yang and zhiping [70] proposed, 2-tuple correlated averaging operator, the 2-tuple correlated geometric operator and the generalized 2-tuple correlated averaging operator combined with choquet integral. joshi et al. [29] developed novel hesitant probabilistic fuzzy linguistic ordered weighted averaging and hesitant probabilistic fuzzy linguistic ordered weighted geometric aggregation operators for ill structured and complex decision making problems. torra [55] discussed that hesitant fuzzy set can deal with the conditions where the assessment of a selection under each and every criterion is represented by several feasible values, not by a margin of error, or some probability distribution on the possible values. for instance, decision maker gives the membership value of x into a, and so they wish to assign 0.23, 0.26 and 0.31, which is a hesitant fuzzy element {0.23, 0.26, 0.31} rather than the interval between 0.23 and 0.31. use these qualities of the hesitant fuzzy set, beg and rashid [5] proposed hesitant 2-tuple linguistic information to take care of marginal error. as we observed in known literature for hesitant sets, the aggregation operators produced more hesitation during aggregation process with respect to given hesitant elements [74]. due to this we develop new operational laws for hesitant 2-tuple. these operational laws reduce the hesitation during aggregate process. we used these operational laws to develop a diminishing hesitant 2-tuple averaging operator (dh2ta) for hesitant 2-tuple linguistic arguments. dh2ta worked in two ways, firstly it aggregated all hesitant 2-tuple linguistic elements and secondly it also reduces the hesitation in resultant 2-tuple linguistic term. in this article, to check applicability of our method we apply it to supply chain management area. in today globalization era, a suitable supplier selection is a core issue of supply chain management that effect the overall performance as without efficient suppliers it is impossible to produce low-cost and high quality products [3, 56]. especially for organizations that spend a high level of their business income on parts and material supplies and whose material costs is a large part of aggregate costs. an organized and transparent approach regarding the choice of supplier is essential for these organizations. supplier selection is a procedure by which suppliers are assessed, evaluated, and then selected to become a part of the company’s supply chain [8]. to overcome the supply chain risk, reduced the production cost, optimize inventory levels and the end profitability are major targets of supply chain management [12, 21]. there exist some well known method for supplier section for instant, matrix approach [20], vendor performance matrix approach [52], vendor profile analysis [54], analytic hierarchy process (ahp) [47, 48] and multiple objective programming (mop) [17]. int. j. anal. appl. 17 (1) (2019) 80 in this paper, we use the notion of hesitant 2-tuple linguistic information which was proposed by beg and rashid [5] to develop a diminishing hesitant 2-tuple averaging operator (dh2ta) for hesitant 2-tuple linguistic arguments. the rest of the paper is structured as follows: some basic concepts are presented to understand our proposal in section 2. in section 3 we propose some definition which is ranking the hesitant 2tuple linguistic information in section 4, we define diminishing hesitant 2-tuple averaging (dh2ta) operator, and discussed some properties of dh2ta. in section 5, we merge choquet integral with the operator dh2ta and developed a new operator diminishing choquet hesitant 2-tuple average operator (dch2ta) and also discussed different properties of dch2ta. the multiple attribute decision making method based on dch2ta is proposed in section 6. in section 7, a numerical example is given to illustrate the developed approach and to demonstrate its feasibility and practicality. concluding remarks are given in last section. 2. hesitant fuzzy sets some important preliminary concepts are given in this section to understand our proposed aggregation operators. hesitant fuzzy set was defined by torra [55] to match the vagueness of real life, when some one is hesitant about membership value. definition 2.1. [55] for a reference set x. the hesitant fuzzy set on x is defined by function that will give a subset of [0, 1] when applied to x. to be easily understood, xia et al. [61] expressed the hfs by a mathematical symbol: e = {< x,he(x) > |x ∈ x}, where he(x) is set of values form [0, 1], known as the possible membership degrees of x to set e. also h = he(x) is called hesitant fuzzy element (hfe). to find order between two hfes, xia et al. [61] defined score function as follow: definition 2.2. [61] let e be a hfe and h ∈ e then score function ”s” of e is s (e) = 1 n(e) n(e)∑ i=1 hi where n(e) be total number of elements in e. let e1 and e2 be two hfes then, if s(e1) < s(e2) then e1 ≺ e2 and if s(e1) = s(e2) then e1 ≈ e2 int. j. anal. appl. 17 (1) (2019) 81 let e, e1 and e2 be elements of a hesitant fuzzy set a then following basic operations are introduced by xia et al. [61]: (1) eα = ∪h∈e{hα} , α > 0. (2) αe = ∪h∈e{1 − (1 −h) α} ,α > 0. (3) h1 ⊕h2 = ∪h1∈e1,h2∈e2 {h1 + h2 −h1h2} . (4) h1 ⊗h2 = ∪h1∈e1,h2∈e2 {h1h2} . next we study concise review of 2-tuple linguistic information and some important basic concepts which are necessary to develop the aggregation operator for hesitant 2-tuple linguistic information. assume that l = {li | i = 2n+ 1,∀n ∈ n} where n be the set of natural number and li be representation of a possible value for linguistic variable. the set l have the following properties by [22]: p 1. the set l must be ordered: li ≥ lj if i ≥ j, p 2. the maximum of any two linguistic terms is max(li, lj) = li if li ≥ lj, p 3. the minimum of any two linguistic terms is min(li, lj) = li if li ≤ lj. the cardinality of the set l must be low enough that is not to impose unnecessary precision for users and it should be rich enough to allow discrimination of the performance of the individual criteria in the limited number of ranking. psychologist [45] recommend the use of 7 ± 2 labels. due to this point of view, a linguistic term set, l with seven labels can be defined as follows: l = {l0 = extremely unattractive (eu), l1 = fairly unattractive (fu), l2 = unattractive (u), l3 = normal (n), l4 = attractive (a), l5 = fairly attractive (fa), l6 = extremely attractive (ea)}. in the literature different models have been recommended for processing of linguistic information. in this paper, we have implemented 2-tuple linguistic representation model, which is based on symbolic translation [22]. symbolic translation is defined as follow: definition 2.3. [22] let l = {l0, l1, ..., lg} be the set of linguistic terms, δi ∈ [0, g] for any i ∈{0, 1, ...,g}, j = round(δi) and ςj = δi − j =⇒ ςj ∈ [−0.5, 0.5), then ςj is called the value of the symbolic translation. where round(δi) is the usual round operation on label index of set l. definition 2.4. [22] let l = {l0, l1, l2, ..., lg} be the set of linguistic terms set and δi be the number representing the aggregation result of symbolic operation. the function 4 used to obtain the 2-tuple linguistic information equivalent to δi is defined as: 4 : [0, g] −→ l× [−0.5, 0.5), 4(δi) = (lj, ςj) with   lj j = round(δi)ςj = δi − j ςj ∈ [−0.5, 0.5) int. j. anal. appl. 17 (1) (2019) 82 figure 1. structure of different 2-tuple linguistic elements inverse function of 4 is always exist and denoted by 4−1. 4−1 : l× [−0.5, 0.5) −→ [0, g], 4−1 (lj, ςj) = ςj + j = δi example 2.1. suppose we have different 2 − tuple elements x1 = (nothing, 0.19), x2 = (low, 0.43), x3 = (medium,−0.22) and x4 = (v ery high,−0.19) then the structure of these elements is described in figure 1. definition 2.5. [22] let (li, ςi) and (lj, ςj) be two 2-tuple linguistic elements, then order between them is according to an ordinary lexicographic order: (1) if i < j then (li, ςi) < (lj, ςj) , (2) if i = j then • if ςi < ςj then (li, ςi) < (lj, ςj) • if ςi = ςj then (li, ςi) = (lj, ςj) definition 2.6. [58] a fuzzy measure α on the set x is a set function α : p(x) → [0, 1] satisfying the following conditions: (1) α(∅) = 0, α(x) = 1; (2) if b ⊆ c ⇒ α(b) ≤ α(c), ∀ b,c ⊆ x; (3) α(b ∪c) = α(b) + α(c) + λα(b)α(c) ∀ b,c ⊆ x and b ∩c = ∅, where λ ∈ (−1, +∞). int. j. anal. appl. 17 (1) (2019) 83 by parameter λ the interaction between criteria can be represented. n⋃ i=1 xi = x for a finite set x. the λ− fuzzy measure α satisfied the following equation α (x) = α ( n⋃ i=1 xi ) =   1 λ { n∏ i=1 (1 + λα(xi)) − 1 } if λ 6= 0 n∑ i=1 α(xi) λ = 0 (2.1) where xi ∩xj = ∅ for all i,j = 1, 2, ...,n and i 6= j. the number α(xi) for a subset with a single element {xi} is called a fuzzy density. α (a) =   1 λ { n∏ i=1 (1 + λα(xi)) − 1 } if λ 6= 0 n∑ i=1 α(xi) λ = 0 (2.2) based on above equation, the value of λ can be find from the following equation, if α(x) = 1 then, 1 = 1 λ { n∏ i=1 (1 + λα(xi)) − 1 } (2.3) in the above definition, if λ = 0, then the third condition reduces to the axiom of the additive measure i.e. α(b ∪c) = α(b) + α(c) ∀b,c ⊆ x and b ∩c = ∅ if the elements of b are independent, then α(b) = ∑ xi∈b α(xi) ∀b ⊆ x. 3. hesitant 2-tuple linguistic information hesitant 2-tuple linguistic information model is introduced by beg and rashid [5] to manage the conditions in which information described is in linguistic term and decision maker has some hesitation to decide its possible linguistic translations. let x be a universe of discourse and l = {l0, l1, l2, ..., lg} be the linguistic term set then a hesitant 2-tuple linguistic term set in x is an expression e = {(x,h(x)) : x ∈ x} , where h(x) = (li, ςi,j) be the hesitant linguistic information by mean of 2-tuple, ςi,j is finite subset of [−0.5, 0.5) which represent the possible translations of li and j be the cardinality of ςi,j and i ∈{0, 1, 2, ...,g} definition 3.1. let h(x) = (li, ςi,j) be a h2tle then score function s of h(x) is s (h(x)) = 1 j ∑ γ∈ςi,j γ where j is the cardinality of ςi,j to find order between two h2tle we use the score function defined in definition 3.1 definition 3.2. let h1(x) = (li, ςi,j) and h2(x) = (lk, ςk,p) be two h2tles, then order between them is according to an ordinary lexicographic order: int. j. anal. appl. 17 (1) (2019) 84 (1) if i < k then h1(x) ≺ h2(x), (2) if i = k and • s(h1(x)) < s(h2(x)) then h1(x) ≺ h2(x) • s(h1(x)) = s(h2(x)) then h1(x) = h2(x) definition 3.3. let ( lip, ςip,jp ) be k h2tles. if k ∈ n,ip ∈ {0, 1, 2, ...,g} and jp be cardinality of ςip,jp, then minkp=1 ( lip, ςip,jp ) is defined as follow: (1) if all ( lip, ςip,jp ) are different due to different ip then, let min k p=1(ip) = i ∈ {0, 1, 2, ...,g}. if ςi,j be represent the translations of li and j be the cardinality of ςi,j, then by definition 3.2 k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (2) if ip = i but ςip,jp are different for each ( lip, ςip,jp ) . if ςi,j is represent the translations of lip = li with s ( lip, ςip,jp ) = minkp=1 ( 1 jp ∑ γ∈ςip,jp γ ) then k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (3) if all ( lip, ςip,jp ) are equal, such that ip = i and ςip,jp = ςi,j, then by definition 3.2 k min p=1 ( lip, ςip,jp ) = (li, ςi,j) (4) if all ( lip, ςip,jp ) h2tles are equal by definition 3.2 but still there is some possibility exist that ςip,jp are different, but score are same then min k p=1 ( lip, ςip,jp ) is one with maximum hesitation. definition 3.4. let ( lip, ςip,jp ) be k h2tles. if k ∈ n, ip ∈ {0, 1, 2, ...,g} and jp be cardinality of ςip,jp, then maxkp=1 ( lip, ςip,jp ) is defined as follow: (1) if all ( lip, ςip,jp ) are different due to different ip then, let max k p=1(ip) = i ∈ {0, 1, 2, ...,g}. if ςi,j be represent the translations of li and j be the cardinality of ςi,j, then by definition 3.2 k max p=1 ( lip, ςip,jp ) = (li, ςi,j) (2) if ip = i but ςip,jp are different for each ( lip, ςip,jp ) . if ςi,j is represent the translations of lip = li with s ( lip, ςip,jp ) = maxkp=1 ( 1 jp ∑ γ∈ςip,jp γ ) then k max p=1 ( lip, ςip,jp ) = (li, ςi,j) (3) if all ( lip, ςip,jp ) are equal, such that ip = i and ςip,jp = ςi,j, then by definition 3.2 k max p=1 ( lip, ςip,jp ) = (li, ςi,j) int. j. anal. appl. 17 (1) (2019) 85 (4) if all ( lip, ςip,jp ) h2tles are equal by definition 3.2 but still there is some possibility exist that ςip,jp are different, but score are same then max k p=1 ( lip, ςip,jp ) is one with minimum hesitation. definition 3.5. if (lk1, ςk1,j1 ) is a hesitant 2-tuple linguistic element, g be the upper limit of the linguistic term set and λ ≥ 0 is any scalar. then the scalar product for h2tle is defined as follows: λ (lk1, ςk1,j1 ) = (λlk1,λςk1,j1 ) = (li, ςi,j) (li, ςi,j) is calculated as follow let β1 = { µ|µ = ((g + 1) ( 1 − ( 1 − g + γk1,j1 g + 1 )λ) ∀γk1,j1 ∈ ςk1,j1 } , then β2 = ⋃ {θ1|θ1 = round(µ), ∀µ ∈ β1}, i = ( ∑ θ1∈β2 θ1 + k1 ) |β2| + 1 , |β2| be cardinality of β2ςip,jp = {θ2|θ2 = different(µ−θ1)}, jp be cardinality of each ςip,jp ςi,j = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = k⋃ p=1 ςip,jp, while ς1 = |η|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 min {( ςip,jp )})       and ς2 = |η|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       , where round (∗) is usual round operation. 4. diminishing hesitant 2-tuple averaging operator beg and rashid [5] discussed a model which is characterized by a linguistic term and its possible symbolic translations. this model is more suitable for dealing with fuzziness and uncertainty than the 2-tuple linguistic arguments. in this section, we defined an operator for the hesitant 2-tuple linguistic elements to handle the situation, where experts face some hesitation to present its possible linguistic translations. definition 4.1. if h1 = (li1, ςi1,j1 ) ,h2 = (li2, ςi2,j2 ) , ...,hk = (lik, ςik,jk) are k, 2-tuple hesitant linguistic terms where jp is the cardinality of ςip,jp, then diminishing hesitant 2-tuple averaging operator (dh2ta) is int. j. anal. appl. 17 (1) (2019) 86 defined as dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) where i = round ( i1 + i2 + ... + ik k ) and ςi,j = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = k⋃ p=1 ςip,jpwe have ς1 = |η|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       and ς2 = |η|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp )}) , max ( k⋃ p=1 { min ( ςip,jp )})       where round (∗) be the round function and |η| be the cardinality of η. example 4.1. let h1 = (l2,{−0.3,−0.25,−0.1, 0.0, 0.2}), h2 = (l3,{−0.2,−0.1, 0.1 , 0.2, 0.25}) and h3 = (l3,{0.1, 0.23, 0.3}) be 2-tuple hesitant linguistic terms, then, dh2ta(h1,h2,h3) = (l3,{0.1, 0.2}). theorem 4.1. let (li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk) be k 2-tuple hesitant linguistic terms, where j1,j2, ...,jk be the cardinality of ςi1,j1, ςi2,j2, ..., ςik,jk respectively, i1,i2, ..., ik ∈{0, 1, 2, ...,m} if all k 2-tuple hesitant linguistic terms are equal i.e. li1 = li2 = ... = lik = li and also ςi1,j1 = ςi2,j2 = ... = ςik,jk = ςi,j then, dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) . proof. as li1 = li2 = ... = lik = li and ςi1,j1 = ςi2,j2 = ... = ςik,jk = ςi,j therefore,( i1 + i2 + ... + ik k ) = ( i + i + ... + i k ) = i. and ςi,j = k⋃ p=1 ςip,jp. (4.1) let ε1 = min ( k⋃ p=1 { max ( ςip,jp )}) , ε2 = max ( k⋃ p=1 { min ( ςip,jp )}) , r1 = max (ε1,ε2) and r2 = min (ε1,ε2) (4.2) int. j. anal. appl. 17 (1) (2019) 87 from equation 4.1 and 4.2, ς1 = k⋃ p=1 {min (rp,r1)} = ςi,j and ς2 = k⋃ p=1 {max (rp,r2)} = ςi,j =⇒ ςi,j = ς1 ∩ ς2 where rp ∈ ςi,j hence dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) . � theorem 4.2. let (li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk) be k, 2-tuple hesitant linguistic terms where jp be the cardinality of ςip,jp for ip = 0, 1, 2, ...,g, jp = 1, 2, ...,n and p = 1, 2, ...,k, then min((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ max ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) . proof. as each 2-tuple hesitant linguistic term ( lip, ςip,jp ) consist of two parts such that lip be the hesitant linguistic information by mean of 2-tuple and ςip,jp is a finite subset of [−0.5, 0.5) which represent the possible translations of lip, where jp be the cardinality of ςip,jp and ip ∈{0, 1, 2, ...,g},then case 1. if all ( lip, ςip,jp ) are different due to different ip then, by definition 4.1 dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) i = round ( i1 + i2 + ... + ik k ) clearly k min p=1 (ip) ≤ i ≤ k max p=1 (ip) therefore by definitions 3.2, 3.3,3.4 and 4.1 we have min((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) (4.3) ≤ dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ max ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) case 2. if ip = i for all ( lip, ςip,jp ) but ςip,jp are different for each p = 1, 2, 3...,k, then, round ( i1 + i2 + ... + ik k ) = round ( i + i + ... + i k ) = i (4.4) let dh2ta((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ςi,j), min((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς min i,j ), and max((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς max i,j ). consider s (li, ςi,j) , s ( li, ς min i,j ) and s ( li, ς max i,j ) are scores of (li, ςi,j) , ( li, ς min i,j ) and ( li, ς max i,j ) respectively. then, by definition of 3.2 and 4.1, int. j. anal. appl. 17 (1) (2019) 88 case 2.1. let max(ς min i,j ) ≤ min(ς max i,j ) =⇒ max(ς min i,j ) ≤ r ≤ min(ς max i,j )∀r ∈ ςi,j =⇒ s ( li, ς min i,j ) ≤ s (li, ςi,j) ≤ s ( li, ς max i,j ) . therefore,( li, ς min i,j ) ≤ dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) (4.5) case 2.2. let max(ς min i,j ) ≥ min(ς max i,j ) =⇒ min(ς max i,j ) ≤ r ≤ max(ς min i,j )∀r ∈ ςi,j =⇒ s ( li, ς min i,j ) ≤ s (li, ςi,j) ≤ s ( li, ς max i,j ) . therefore,( li, ς min i,j ) ≤ dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) (4.6) case 2.3. let min(ς min i,j ) ≤ max(ς max i,j ) ≤ max(ς min i,j ) =⇒ ςi,j = ς max i,j =⇒ s ( li, ς max i,j ) = s (li, ςi,j). therefore, dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = ( li, ς max i,j ) (4.7) equations 4.3,4.4,4.5,4.6 and 4.7 provide the required result. � theorem 4.3. if ( lip, ςip,jp ) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) for ip, i ′ p ∈{0, 1, 2, ...,g} , jp,j ′ p ∈{1, 2, ...,n} and p = 1, 2, ...,k, then dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ dh2ta (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) proof. given that ( lip, ςip,jp ) ≤ ( l ′ ip , ς ′ ip,jp ) for all p = 1, 2, ...,k, case 1. if, ∀p = 1, 2, ...,k, ip < i ′ p =⇒ (∑k p=1 ip k ) < (∑k p=1 i ′ p k ) =⇒ dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) (4.8) < dh2ta (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) case 2. if ip = i ′ p which implies that (∑k p=1 ip k ) = (∑k p=1 i ′ p k ) = i. let dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) , dh2ta((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′ i,j), min((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς min i,j ), min((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′min i,j ), max((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) = (li, ς max i,j ) and max((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )) = (l ′ i, ς ′ max i,j ) int. j. anal. appl. 17 (1) (2019) 89 therefore by theorem 4.2, ( li, ς min i,j ) ≤ dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( li, ς max i,j ) , (4.9) and ( l ′ i, ς ′min i,j ) ≤ dh2ta (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) ≤ ( l ′ i, ς ′ max i,j ) . (4.10) as, ( li, ς min i,j ) ≤ ( l ′ i, ς ′ min i,j ) , ( li, ς max i,j ) ≤ ( l ′ i, ς ′ max i,j ) and( lip, ςip,jp ) ≤ ( l ′ ip , ς ′ ip,jp ) , therefore s ( li, ς min i,j ) ≤ s ( l ′ i, ς ′ min i,j ) , s ( li, ς max i,j ) ≤ s ( l ′ i, ς ′ max i,j ) and s ( lip, ςip,jp ) ≤ s ( l ′ ip , ς ′ ip,jp ) for each p. therefore by theorem 4.2 dh2ta((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ dh2ta((l ′ i1 , ς ′ i1,j1 ), (l ′ i2 , ς ′ i2,j2 ), ..., (l ′ ik , ς ′ ik,jk )), which is required result. � theorem 4.4. let ( li′p , ςi′p,jp ) be a permutation of p hesitant 2-tuples linguistic elements of ( lip, ςip,jp ) ,where ip, i ′ p ∈{0, 1, 2, ...,g} , jp,j ′ p = {1, 2, ...,n} and p = 1, 2, ...,k then, dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = dh2ta (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) proof. let us consider (σ(1),σ(2), ...,σ(k)) be permutation of (1, 2, ...,k) such that( lip, ςip,jp ) σ(1) ≤ ( lip, ςip,jp ) σ(2) ≤ ... ≤ ( lip, ςip,jp ) σ(k) then, ( lip, ςip,jp ) σ(p) = ( li′p , ςi′p,jp ) σ(p) ∀p = 1, 2, ...,k therefore, (ip)σ(p) = ( i ′ p ) σ(p) =⇒ (∑k p=1 ip k ) = (∑k p=1 i ′ p k ) ∀p = 1, 2, ...,k (4.11) also, s ( (lip, ςip,jp)σ(p) ) = s (( li′p , ςi′p,jp ) σ(p) ) ∀p = 1, 2, ...,k (4.12) from equation 4.11 and 4.12 dh2ta ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = dh2ta (( l ′ i1 , ς ′ i1,j1 ) , ( l ′ i2 , ς ′ i2,j2 ) , ..., ( l ′ ik , ς ′ ik,jk )) . � int. j. anal. appl. 17 (1) (2019) 90 5. hesitant 2-tuple linguistic information aggregation operators based on the choquet integral in this section, we develop diminishing choquet hesitant 2-tuple average operator (dch2ta) by selecting choquet integral to find the weights for dwh2ta. we also discussed different properties of dch2ta. definition 5.1. let h1 = (li1, ςi1,j1 ) , h2 = (li2, ςi2,j2 ) , ...,hk = (lik, ςik,jk) be k, 2-tuple hesitant linguistic terms where jp be the cardinality of ςip,jp for any finite natural number p. x be the set of attributes and α be the fuzzy measure on x, then diminishing choquet hesitant 2-tuple average operator (dch2ta) is defined as follow: dch2taα (h1,h2, ...,hk) (5.1) = dh2ta ({( α ( hσ(p) ) −α ( hσ(p−1) )) (li, ςi,j)σ(p) |p = 1, 2, ...,k }) , here (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k) such that (li, ςi,j)σ(1) ≥ (li, ςi,j)σ(2) ≥ ... ≥ (li, ςi,j)σ(k), xσ(p) is the attribute corresponding to (li, ςi,j)σ(p) and hσ(p) = { xσ(l)|l ≤ p } for p ≥ 1, hσ(0) = ∅. theorem 5.1. let (li1, ςi1,j1 ) = (li2, ςi2,j2 ) = ... = (lik, ςik,jk) be all equal k, hesitant 2-tuples linguistic elements such that i1 = i2 = ... = ik and ςi1,j1 = ςi2,j2 = ... = ςik,jk. if ip ∈ {0, 1, 2, ...,g} , jp ∈ {1, 2, ...,n} and p = 1, 2, ...,k, x is the set of attributes and α be the fuzzy measure on x, then diminishing choquet hesitant 2-tuple average operator (dch2ta) is always dch2taα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) , where (li, ςi,j) = wp ( lip, ςip,jp ) p , wp = α ( hσ(p) ) −α ( hσ(p−1) ) = 1 k and (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k). proof. given that, ( lip, ςip,jp ) σ(p) = ( lip, ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k, therefore( lip ) σ(p) = ( lip ) σ(p−1) also ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k by definition 3.5 scalar product for wp−1 and wp are a1 =   (µ) σ(p) |(µ) σ(p) = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )wp) |∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   and b1 =   (µ) σ(p−1) |(µ) σ(p−1) = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p−1) g+1 )wp−1) |∀ ( γip,jp ) σ(p−1) ∈ ( ςip,jp ) σ(p−1)   int. j. anal. appl. 17 (1) (2019) 91 this implies that, a1 = b1 as wp−1 = wp and ( lip, ςip,jp ) σ(p) = ( lip, ςip,jp ) σ(p−1) ∀p = 1, 2, 3, ...,k. (5.2) let, a2 = ⋃{ θ1 σ(p) |θ1 σ(p) = round((µ) σ(p) ), ∀(µ) σ(p) ∈ a1 } and b2 = ⋃{ θ1 σ(p−1) |θ1 σ(p−1) = round((µ) σ(p−1) ), ∀(µ) σ(p−1) ∈ b1 } . by equation 5.2 a2 = b2. (5.3) as (ip)σ(p) = (ip)σ(p−1) therefore by equation 5.3,  ∑ θ1 σ(p) ∈a2 θ1 σ(p) + (ip)σ(p)   (|a2| + 1) =   ∑ θ1 σ(p−1) ∈b2 θ1 σ(p−1) + (ip)σ(p−1)   (|b2| + 1) = i and ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) for all p, =⇒ k⋃ p=1 ( ςip,jp ) σ(p) = k⋃ p=1 ( ςip,jp ) σ(p−1) = η1(say) (5.4) where, ( ςip,jp ) σ(p) = { θ2 σ(p) |θ2 σ(p) = different((µ) σ(p) −θ1 σ(p) ) } and ( ςip,jp ) σ(p−1) = { θ2 σ(p−1) |θ2 σ(p−1) = different((µ) σ(p−1) −θ1 σ(p−1) ) } . as ( ςip,jp ) σ(p) = ( ςip,jp ) σ(p−1) for all p, =⇒ max ( ςip,jp ) σ(p) = max ( ςip,jp ) σ(p−1) for all p therefore min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) = min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) and max ( k⋃ p=1 { min ( ςip,jp ) σ(p) }) = max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) }) . int. j. anal. appl. 17 (1) (2019) 92 by equation 5.4 let rq ∈ η1 then, (ς1) σ(p) = |η1|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p) })       (ς1) σ(p−1) = |η1|⋃ q=1   min  rq, max   min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) })       and (ς2) σ(p) = |η1|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp ) σ(p) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p) })       (ς2) σ(p−1) = |η1|⋃ q=1   max  rq, min   min ( k⋃ p=1 { max ( ςip,jp ) σ(p−1) }) , max ( k⋃ p=1 { min ( ςip,jp ) σ(p−1) })       by equations 5.2,5.3 and 5.4 we have, (ς1) σ(p) = (ς1) σ(p−1) and (ς2) σ(p) = (ς2) σ(p−1) therefore, (ς1) σ(p) ∩ (ς2) σ(p) = (ς1) σ(p−1) ∩ (ς2) σ(p−1) = ςi,j. (5.5) given that wp = α ( hσ(p) ) −α ( hσ(p−1) ) = 1 k for all p, therefore by definition 3.5 and equation 5.5 we have, wp ( lip, ςip,jp ) p = wp−1 ( lip, ςip,jp ) p−1 = (li, ςi,j) (say) therefore by theorem 4.1 we have required result, dch2taα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) = (li, ςi,j) � theorem 5.2. let ( lip, ςip,jp ) be k, hesitant 2-tuples linguistic elements, if ip ∈ {0, 1, 2, ...,g} , jp ∈ {1, 2, ...,n} and p = 1, 2, ...,k. x be the set of attributes and α be the fuzzy measure on x, then for any g use as upper limit of the linguistic term set, then diminishing choquet hesitant 2-tuple average operator (dch2ta) must satisfied, ( lip, min ( ςip,jp )) σ(k) ≤ dch2taα (h1,h2, ...,hk) ≤ ( lip, max(ςip,jp) ) σ(1) where (σ(1),σ(2), ...,σ(k)) be the permutation of (1, 2, ...,k) such that (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1). int. j. anal. appl. 17 (1) (2019) 93 proof. as, (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1) also ( lip, min ( ςip,jp )) σ(k) ≤ ( lip, ςip,jp ) σ(k) and ( lip, ςip,jp ) σ(1) ≤ ( lip, max ( ςip,jp )) σ(1) . therefore ( lip, min ( ςip,jp )) σ(k) ≤ (lip, ςip,jp)σ(k) ≤ (lip, ςip,jp)σ(k−1) ≤ ... ≤ (lip, ςip,jp)σ(1) ≤ ( lip, max ( ςip,jp )) σ(1) . because, 0 ≤ α ( hσ(p) ) −α ( hσ(p−1) ) ≤ 1 ∀p = 1, 2, 3, ...,k therefore by definition 3.5 we have, let τp = α ( hσ(p) ) −α ( hσ(p−1) )( lip, ςip,jp ) ∀p = 1, 2, ...,k τp =   µ2|µ2 = (g + 1) ( 1 − ( 1 − g+(γip,jp) g+1 )α(hσ(p))−α(hσ(p−1))) ∀ ( γip,jp ) ∈ ( ςip,jp ) where p = 1, 2, ...k   and βp = ⋃ {θ2|θ2 = round(µ2),∀µ2 ∈ τp} =⇒ λp = ( ∑ θ2∈βp θ2 + ip ) |βp| + 1 , |βp| be cardinality of βp. clearly, (ip)σ(k) ≤ round (∑ λp k ) = i ′ ≤ (ip)σ(1) . (5.6) let li′ be the linguistic term of dch2taα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) . by definition 5.1 we observe that translation of li′ , ( say ςi′,j′ ) may truncate the extreme values of ( ςip,jp ) σ(k) and ( ςip,jp ) σ(1) i.e. it must satisfied the following condition, min ( (ςip,jp)σ(k) ) ≤ γ ′ ip,jp ≤ max ( (ςip,jp)σ(1) ) ∀γ ′ ip,jp ∈ ςi′,j′ (5.7) therefore, from equation 5.6 and 5.7 we have, ( lip, min ( ςip,jp )) σ(k) ≤ dch2taα ((li1, ςi1,j1 ) , (li2, ςi2,j2 ) , ..., (lik, ςik,jk)) ≤ ( lip, max(ςip,jp) ) σ(1) � int. j. anal. appl. 17 (1) (2019) 94 theorem 5.3. let ( lip, ςip,jp ) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) for all p = 1, 2, ...,k if for ip, i ′ p ∈ {0, 1, 2, ...,g} , jp,j ′ p ∈ {1, 2, ...,n} . let x be the set of attributes and α be the fuzzy measure on x, then, dch2taα((li1, ςi1,j1 ), (li2, ςi2,j2 ), ..., (lik, ςik,jk)) ≤ dch2taα((l ′ i ′ 1 , ς ′ i ′ 1,j ′ 1 ), (l ′ i2 , ς i ′ 2,j ′ 2 ), ..., (l ′ i ′ k , ς i ′ k ,j ′ k )) proof. if lip = l ′ i ′ p then order of ( lip, ςip,jp ) and ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) depend on possible translations of lip and l ′ i ′ p . as,( lip, ςip,jp ) σ(p) ≤ ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) σ(p) ∀p = 1, 2, 3, ...,k. therefore, ∑ γip,jp∈ ( ςip,jp ) σ(p) γip,jp n ≤ ∑ γ ′ i ′ p ,j ′ p ∈ ( ς ′ i ′ p ,j ′ p ) σ(p) γ ′ i ′ p,j ′ p n ∀p = 1, 2, 3, ...,k (5.8) we also know that 0 ≤ ( α ( hσ(p) ) −α ( hσ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k also 0 ≤ ( α ( h ′ σ(p) ) −α ( h ′ σ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k implies that β1 = α ( hσ(p) ) −α ( hσ(p−1) )( lip, ςip,jp ) σ(p) =   µ|µ = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )α(hσ(p))−α(hσ(p−1))) ∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   let β2 = ⋃ {θ1|θ1 = round(µ),∀µ ∈ β1} =⇒ (i)σ(p) = ( ∑ θ1∈β2 θ1 + (ip)σ(p) ) |β2| + 1 , |β2| be cardinality of β2 (5.9) β ′ 1 = α ( hσ(p) ) −α ( hσ(p−1) )( lip, ςip,jp ) =   µ ′ |µ ′ = (g + 1)  1 −  1 − g+ ( γ ′ i ′ p,j ′ p ) σ(p) g+1   α ( h ′ σ(p) ) −α ( h ′ σ(p−1) )  ∀ ( γ ′ i ′ p,j ′ p ) σ(p) ∈ ( ς ′ i ′ p,j ′ p ) σ(p)   int. j. anal. appl. 17 (1) (2019) 95 let β ′ 2 = ⋃{ θ ′ 1|θ ′ 1 = round(µ ′ ),∀µ ′ ∈ β ′ 1 } =⇒ ( i ′ ) σ(p) = ( ∑ θ ′ 1∈β ′ 2 θ ′ 1 + ( i ′ p ) σ(p) ) |β′2| + 1 , |β ′ 2| be cardinality of β ′ 2 (5.10) from equations 5.8, 5.9 and 5.10 we have, dch2taα (h1,h2, ...,hk) ≤ dch2taα ( h ′ 1,h ′ 2, ...,h ′ k ) if ip < i ′ p then obviously from equation 5.9 and 5.10 we have (i)σ(p) ≤ ( i ′ ) σ(p) =⇒ dch2taα (h1,h2, ...,hk) ≤ dch2taα ( h ′ 1,h ′ 2, ...,h ′ k ) � theorem 5.4. let h ′ p = ( l ′ i ′ p , ς ′ i ′ p,jp ) p be a permutation of p hesitant 2-tuples linguistic elements of hp =( lip, ςip,jp ) p . where ip = 0, 1, 2, ...,g, jp = 1, 2, ...,n and p = 1, 2, ...,k. x be the set of attributes and α be the fuzzy measure on x, then, dch2taα (h1,h2, ...,hk) = dch2taα ( h ′ 1,h ′ 2, ...,h ′ k ) proof. let us consider (σ(1),σ(2), ...,σ(k)) be permutation of (1, 2, ...,k) such that( lip, ςip,jp ) σ(1) ≤ ( lip, ςip,jp ) σ(2) ≤ ... ≤ ( lip, ςip,jp ) σ(k) then, ( lip, ςip,jp ) σ(p) = ( l ′ i ′ p , ς ′ i ′ p,j ′ p ) σ(p) such that ip = i ′ p and ςip,jp = ς ′ i ′ p,j ′ p ∀p = 1, 2, 3, ...,k (5.11) we also know that 0 ≤ ( α ( hσ(p) ) −α ( hσ(p−1) )) ≤ 1 ∀p = 1, 2, 3, ...,k implies that β1 = α ( hσ(p) ) −α ( hσ(p−1) )( lip, ςip,jp ) σ(k) =   µ|µ = (g + 1) ( 1 − ( 1 − g+(γip,jp)σ(p) g+1 )α(hσ(p))−α(hσ(p−1))) ∀ ( γip,jp ) σ(p) ∈ ( ςip,jp ) σ(p)   . let β2 = ⋃ {θ1|θ1 = round(µ),∀µ ∈ β1} =⇒ (i)σ(p) = ( ∑ θ1∈β2 θ1 + (ip)σ(k) ) |β2| + 1 , |β2| be cardinality of β2 (5.12) int. j. anal. appl. 17 (1) (2019) 96 β ′ 1 = α ( hσ(p) ) −α ( hσ(p−1) )( lip, ςip,jp ) =   µ ′ |µ ′ = (g + 1)  1 −  1 − g+ ( γ ′ i ′ p,j ′ p ) σ(p) g+1   α ( h ′ σ(p) ) −α ( h ′ σ(p−1) )  ∀ ( γ ′ i ′ p,j ′ p ) σ(p) ∈ ( ς ′ i ′ p,j ′ p ) σ(p)   . let β ′ 2 = ⋃{ θ ′ 1|θ ′ 1 = round(µ ′ ),∀µ ′ ∈ β ′ 1 } =⇒ ( i ′ ) σ(p) = ( ∑ θ ′ 1∈β ′ 2 θ ′ 1 + ( i ′ p ) σ(p) ) |β′2| + 1 , |β ′ 2| be cardinality of β ′ 2. (5.13) from equations 5.11, 5.12 and 5.13 β1 = β ′ 1 and β2 = β ′ 2∀p = 1, 2, 3, ...,k =⇒ dch2taα (h1,h2, ...,hk) = ch2tnα ( h ′ 1,h ′ 2, ...,h ′ k ) which is required proof. � 6. an application of dch2ta operators to multiple attribute decision making in this section dch2ta operator is applied to multiple attribute decision making problems based on hesitant 2-tuple linguistic information. firstly, we developed a decision making method for utilization of dch2ta operator. let d = {d1,d2, ...,dr} be the set of ”r” decision makers, x = {x1,x2, ...,xm} be the set of alternatives and y = {y1,y2, ...,yn} be the set of attributes. step 1. the decision makers developed the decision matrices mp = [( l p ijk, ς p )] m×n , where ( l p ijk, ς p ) be the hesitant evaluation of the alternatives xi determined by the decision makers dp based on attributes yj, where i = 1, 2, ...,m, j = 1, 2, ...,n, and p = 1, 2, ..,r, where ς p ⊂ [−0.5, 0.5) and k ∈{0, 1, 2, ...g}. step 2. find the matrix magg = [ dh2ta ( l p ijk, ς p )] m×n , where, dh2ta ( l p ijk, ς p ) is an aggregate value of ( l p ijk, ς p ) (i = 1, 2, ...,m,j = 1, 2, ...,n) for all decision maker’s evaluation as follow: dh2ta ( (l1ijk1, ς 1), (l2ijk2, ς 2), ..., (lnijkn, ς n) ) = (lk, ςk) where k = round ( k1 + k2 + ... + kn n ) and ςk = {x|x ∈ ς1 ∩ ς2 } for all rq ∈ η = n⋃ j=1 ςjwe have int. j. anal. appl. 17 (1) (2019) 97 ς1 = |η|⋃ q=1   min  rq, max   min ( n⋃ j=1 {max (ςp)} ) , max ( n⋃ j=1 min{(ςp)} )       and ς2 = |η|⋃ q=1   max  rq, min   min ( n⋃ j=1 {max (ςp)} ) , max ( j⋃ j=1 {min (ςp)} )       where round (∗) be the round function and |η| be the cardinality of η. step 3. confirm the fuzzy measures of attributes sets of b. we use the dch2ta operator define in definition 5.1 to aggregate the values to find overall values (l, ς)i (i = 1, 2, ...,m) of alternatives ai. (l, ς)i = dch2taα ((li1, ςi1), (li2, ςi2), ..., (lin, ςin)) = dh2ta ( wi1(liσ(1), ςiσ(1)),wi2(liσ(2), ςiσ(2)), ...,win(liσ(n), ςiσ(n)) ) where (σ(1),σ(2), ...,σ(n)) be the permutation of (1, 2, ...,n) such that (liσ(1), ςiσ(1)) ≥ (liσ(2), ςiσ(2)) ≥ ... ≥ (liσ(n), ςiσ(n)) and wij = α(hiσ(j)) −α(hiσ(j−1)) is the set of attributes corresponding to (liσ(1), ςiσ(1)),(liσ(2), ςiσ(2)),..., (liσ(n), ςiσ(n)). step 4. rank these aggregate values (l, ς)i (i = 1, 2, ...,m) in descending order according to the rule in definition 3.2 and select (l, ς)i with the largest value. 7. illustrative example in order to demonstrate the significance of our newly proposed method, we consider an example where mr. robert, a food chain owner, wants to hire a supplier for raw food material for his chain. to save hedge risks, a three member committee (decision makers), d = {d1,d2,d3} has been created to select the most suitable supplier. decision makers short listed five potential suppliers after initial analysis for supplier’s capabilities. let s = {s1, s2, s3, s4, s5} be the set of short listed suppliers. during the supplier selection process, decision maker decide to consider the following set of attributes for judgments y = {y1(price), y2(quality), y3(delivery time), y4(financial status of the company)}. in numerous practical group decision making problems in supply chain management, the contractor selection or determination of an accomplice for an endeavor in the field of production network administration, military framework effectiveness assessment, etc. decision makers normally need to give their preferences over alternatives. as preference information given by decision makers is normally imprecise. it might be due int. j. anal. appl. 17 (1) (2019) 98 to hesitations, uncertainty or vagueness about preferences as a decision should be made under time pressure and lack of information or knowledge processing capacities especially when financial condition turns out to be more complex. the best choice for decision maker is to handle data in hesitant 2-tuple elements due to effectiveness of them in these particular situations. consider that decision makers evaluate the alternatives with respect to the attributes in 2-tuple linguistic arguments to form decision matrices mp where p = {1, 2, 3}. step 1. develop decision matrices mp = [( l p ijk, ς p )] 5×4 , ςp ⊂ [−0.5, 0.5) m1 =   (m,{−0.3, 0.0, 0.2}) (g,{0.45, 0.32, 0.2}) (p,{0.2, 0.3}) (p,{−0.3, 0.1}) (p,{0.0, 0.2, 0.1}) (m,{−0.48,−0.2, 0.0}) (m,{−0.45, 0.1}) (g,{−0.2, 0.1, 0.2}) (g,{−0.3, 0.1, 0.2}) (m,{−0.0, 0.2}) (v g,{−0.2.0.0, 0.4}) (p,{−0.3, 0.1, 0.2}) (v g,{−0.1, 0.0, 0.2}) (p,{0.0, 0.2, 0.4}) (p,{−0.5,−0.3}) (m,{−0.45,−0.25}) (eg,{−0.4,−0.3, 0.1}) (p,{−0.1, 0.2, 0.3}) (v p,{−0.45,−0.2}) (g,{−0.4,−0.1, 0.0})   m2 =   (p,{−0.3,−0.1}) (v g,{−0.1, 0.0, 0.1}) (v p,{−0.2, 0.3}) (m,{0.1, 0.2, 0.4}) (v p,{0.4}) (p,{0.2, 0.3}) (g,{0.3, 0.4}) (v g,{−0.1,−0.45,−0.2}) (m,{0.0, 0.3}) (p,{−0.1, 0.2}) (g,{0.1, 0.3}) (v p,{−0.3,−0.2, 0.0}) (eg,{0.2, 0.4}) (m,{−0.4, 0.3}) (p,{0.2, 0.4}) (g,{0.1, 0.3, 0.4}) (g,{−0.2, 0.1}) (m,{−0.2, 0.15}) (p,{−0.1, 0.2}) (v g,{−0.1, 0.3})   m3 =   (g,{−0.5, 0.1, 0.2}) (v g,{0.2, 0.3}) (m,{0.1, 0.2}) (v p,{0.0, 0.1, 0.2}) (m,{−0.4,−0.1}) (p,{0.0, 0.2, 0.4}) (v g,{−0.3,−0.2}) (m,{−0.2,−0.1, 0.0}) (p,{−0.2, 0.0, 0.1}) (v g,{−0.05, 0.2}) (g,{0.0, 0.1, 0.25}) (v p,{−0.3,−0.2, 0.0}) (g,{−0.3,−0.1, 0.0}) (g,{0.0, 0.25, 0.45}) (p,{0.1, 0.2, 0.3}) (m,{−0.1, 0.2, 0.3}) (m,{−0.1, 0.1, 0.3}) (p,{−0.2,−0.1, 0.0}) (m,{0.1, 0.4, 0.45}) (eg,{−0.05, 0.25})   step 2 use the dh2ta operator to aggregate value of ( l p ijk, ς p ) (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4, p = 1, 2, 3 and k ∈{0, 1, 2, ..., 6}) for all decision maker’s evaluation as follow: magg =  (m,{−0.3,−0.1}) (v g,{−0.1, 0.1, 0.2}) (p,{0.2}) (p,{0.1}) (p,{−0.1, 0.0, 0.1, 0.2, 0.4}) (p,{0.0, 0.2}) (g,{−0.2, 0.1, 0.3}) (g,{−0.2,−0.1}) (m,{0.1}) (m,{−0.05, 0.2}) (g,{0.1, 0.25}) (v p,{−0.3,−0.2, 0.0}) (v g,{0.0, 0.2}) (m,{0.0, 0.2, 0.25, 0.3}) (p,{−0.3, 0.1, 0.2}) (m,{−0.25,−0.1, 0.1}) (g,{−0.1, 0.1}) (p,{−0.1, 0.0) (p,{−0.2,−0.1, 0.1}) (v g,{−0.05, 0.0})   int. j. anal. appl. 17 (1) (2019) 99 step 3. to find the fuzzy measures for attributes of y = {y1(price), y2(quality), y3(delivery time), y4(financial status of company)} and parameter λ. let α(y1) = 0.3, α(y2) = 0.25 ,α(y3) = 0.15 and α(y4) = 0.29. then by equation 2.3, λ = 0.00277 and by equation 2.2, α(y1,y2) = 0.5502, α(y1,y3) = 0.4501, α(y1,y4) = 0.5902, α(y2,y3) = 0.4001, α(y2,y4) = 0.5402, α(y3,y4) = 0.4401, α(y1,y2,y3) = 0.7004, α(y1,y2,y4) = 0.8406, α(y1,y3,y4) = 0.7405, α(y2,y3,y4) = 0.6904, α(y1,y2,y3,y4) = 1. to find dch2ta aggregate value for the following elements, firstly we use wij = α(hiσ(j))−α(hiσ(j−1)) weight for each element. (l1σ(1), ς1σ(1)) = (v g,{−0.1, 0.1, 0.2}) (l1σ(2), ς1σ(2)) = (m,{−0.3,−0.1}) (l1σ(3), ς1σ(3)) = (p,{0.2}) (l1σ(4), ς1σ(4)) = (p,{0.1}) as h1σ(1) = {y2}, h1σ(2) = {y1,y2} and h1σ(3) = {y1,y2,y3}, h1σ(4) = {y1,y2,y3,y4} we can get w11 = 0.25, w12 = 0.3002 and w13 = 0.1502, w14 = 0.2996. (l, ς)1 = dch2taα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) = (m,{−0.0700,−0.0537,−0.0163, 0.2138}) s((l, ς)1) = 0.0184 similarly, find the values of (l, ς)2, (l, ς)3, (l, ς)4 and (l, ς)5 are (l, ς)2 = dch2taα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)2 = (m,{−0.1967,−0.146,−0.0921}) s ((l, ς)2) = −0.1449 (l, ς)3 = dch2taα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)3 = (m,{−0.0674,−0.0072, 0.0924, 0.2177}) s ((l, ς)3) = 0.0589 (l, ς)4 = dch2taα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)4 = (m,{0.0462, 0.0654, 0.0955}) and s ((l, ς)4) = 0.0690 int. j. anal. appl. 17 (1) (2019) 100 and (l, ς)5 = dch2taα ( (l1σ(1), ς1σ(1)), (l1σ(2), ς1σ(2)), (l1σ(3), ς1σ(3)), (l1σ(4), ς1σ(4)) ) (l, ς)5 = (m,{−0.3002,−0.2846,−0.2118,−0.0477,−0.0380,−0.0163}) s ((l, ς)5) = −0.1498 as by definition 3.2 (l, ς)4 > (l, ς)3 > (l, ς)1 > (l, ς)2 > (l, ς)5, hence s4 � s3 � s1 � s2 � s5. therefore, the most suitable supplier’s option is s4, second, third and four position suppliers are s3,s1 and s2 respectively, while the worst suppliers option is s5 . 8. discussion and conclusion herrera and mart́ınez [22], discussed a symbolic model and name it 2-tuple linguistic representation model. the 2-tuple linguistic model use words toward processing without loss of any information. in their proposed 2-tuple model, the linguistic term sets were consistent and symmetrically distributed. in view of the herrera and mart́ınez [22], the following models have been considered afterward: • wang and hao model [59], • herrera et al. model [24], • numerical scale model [15]. in each of these models, linguistic term sets examined consistently and symmetrically scattered. moreover, the symbolic proportions over linguistic terms are precise qualities, and just a single linguistic term set is considered for translation of these qualities. but these models did not address where hesitation occurs between the translation of arguments. beg and rashid [5] introduced the concept of hesitant 2-tuple linguistic model to merge herrera and mart́ınez’s [22], 2-tuple linguistic model with torra’s [55], hesitant fuzzy set. hesitant 2-tuple linguistic model is very helpful for the situation where decision maker may hesitant to pick a possible value of translation for a linguistic term as it will not cause any loss of information in the process. beg and rashid [5] used hesitant 2-tuple linguistic model for the situation where the attributes in the decision making problem are evaluated by hesitant 2-tuple linguistic arguments and they used topsis technique to illustrate hesitant 2-tuple linguistic model’s efficiency and feasibility in real-world decision making applications. as topsis technique use maximum and minimum distance or similarity from all terms provide the best option accordingly. some time the resultant value did not reflect the true picture and fail to find the best result over the argument. particularly, where we have an interrelation between the arguments. choquet integral [11] is the best choice where interrelationship is required. in today globalization era, choice of a suitable supplier for the business in the sense of supply chain management has become a key strategic consideration. but due to natural human hesitation, incomplete supplier information and performances and market uncertainty, a supplier selection process has become more int. j. anal. appl. 17 (1) (2019) 101 complicated. due to this, it is difficult for decision makers to express their conclusion on the suppliers with exact and crisp values and the evaluations are often expressed in linguistic terms. in such circumstances fuzzy set theory is a very appropriate tool to deal with this kind of problems. in this paper, we have observed a situation that the attributes within the selection for decision making problems are interactive or interdependent and analyze the values in the form of 2 tuple hesitant linguistic arguments. by utilizing the choquet integral, we have developed dh2ta and dch2ta aggregation operators. the properties of both operators are studied, such as commutativity, boundedness and monotonicity. we proved that dh2ta operator is an idempotent operator. we also utilized dch2ta operator to the more than one attribute group decision making problems for hesitant 2-tuple linguistic understanding and proposed a method for group decision making problems. an illustrative example has been given to demonstrate the proposed decision making approach. we observe that dch2ta is suitable for conditions where decision making problems are interdependent. the operator dch2ta has the properties to reduce hesitation in aggregated value of hesitant 2-tuple linguistic elements. in real decision making problem, there involve the interrelationships between the arguments. often bonfeeroni mean operators (bm) [7] and muirhead mean operators (mm) [46] used as the tools where interrelationships between arguments exist. as we observed that diminishing operational laws have the ability to reduce hesitation in resultant argument. in future, we will use this capability of diminishing operational laws and will purpose bm and mm for hesitant 2-tuple linguistic model. conflict of interest. the authors declare that they have no conflict of interest. references [1] r. d. andrés, m. espinilla and l. mart́ınez, an extended hierarchical linguistic model for managing integral evaluation, int. j. comput. intell. syst. 3 (4) (2010), 486–500. [2] r. d. andrés, j.l. garćıa-lapresta and l. mart́ınez, a multi-granular linguistic model for management decision-making in performance appraisal, soft computing, 14 (1) (2010), 21–34. [3] c. araz and i. ozkarahan, supplier evaluation and management system for strategic sourcing based on a new multicriteria sorting procedure, int. j. production economics, 106 (2) (2007), 585–606. [4] i. beg and t. rashid, multi-criteria of bike purchasing using fuzzy choquet integral, j. fuzzy math. 22 (3)(2014), 677-694. [5] i. beg and t. rashid, hesitant 2-tuple linguistic information in multiple attributes group decision making, j. intell. fuzzy syst. 30 (2016), 109–116. [6] i. beg and t. rashid, modelling uncertainties in multi-criteria decision making using distance measure and topsis for hesitant fuzzy sets, j. artif. intell. soft comput. res. 7(2)(2017), 103-109. [7] c. bonferroni, sulle medie multiple di potenze, bolletino matematica italiana, 5 (1950), 267–270. [8] c. t. chen, c. t. lin, and s. f. huang, a fuzzy approach for supplier evaluation and selection in supply chain management, int. j. production economics, 102 (2) (2006), 289–301. [9] c.t. chen, p. f. pai and w. z. hung, an integrated methodology using linguistic promethee and maximum deviation method for third-party logistics supplier selection, int. j. comput. intell. syst. 3 (4) (2010), 438–451. int. j. anal. appl. 17 (1) (2019) 102 [10] y. chen, x. zeng m. happiette, p. bruniaux, r. ng and w. yu, optimisation of garment design using fuzzy logic and sensory evaluation techniques, eng. appl. artif. intell. 22 (2) (2009), 272–282. [11] g. choquet, theory of capacities. ann. inst. fourier, 5 (1953), 131–295. [12] s. y. chou and y. h. chang, a decision support system for supplier selection based on a strategy-aligned fuzzy smart approach, expert syst. appl. 34 (4) (2008), 2241–2253. [13] r. degani and g. bortolan, the problem of linguistic approximation in clinical decision making, int. j. approx. reason. 2 (1988), 143–162. [14] m. delgado, j.l. verdegay and m.a. vila, on aggregation operations of linguistic labels, int. j. intell. syst. 8 (3) (1993), 351–370. [15] y.c. dong, c.c. li and f. herrera, an optimization-based approach to adjusting the unbalanced linguistic preference relations to obtain a required consistency level, inf. sci. 292 (2015), 27–38. [16] d. dubois and h. prade, fuzzy sets and systems: theory and applications, kluwer academic, new york, 1980. [17] c. x. feng, j. wang, and j. s. wang, an optimization model for concurrent selection of tolerances and suppliers, comput. ind. eng. 40 (1-2) (2001), 15–33. [18] n. fenton and w. wang, risk and confidence analysis for fuzzy multicriteria decision making, knowl.based syst. 19 (6) (2006), 430–437. [19] j.l. garćıa-lapresta, b. llamazares and m. mart́ınez-panero, a social choice analysis of the borda rule in a general linguistic framework, int. j. comput. intell. syst. 3 (4) (2010), 501–513. [20] r. e. gregory, source selection: a matrix approach, j. purchas. mater. manag. 22 (2) (1986), 24–29. [21] s. h. ha and r. krishnan, a hybrid approach to supplier selection for the maintenance of a competitive supply chain, expert syst. appl. 34 (2) (2008), 1303–1311. [22] f. herrera and l. mart́ınez, a 2-tuple fuzzy linguistic representation model for computing with words, ieee trans. fuzzy syst. 8 (2000), 746–752. [23] f. herrera and l. mart́ınez, the 2-tuple linguistic computational model advantages of its linguistic description, accuracy and consistency, int. j. uncertain. fuzziness knowl.-based syst. 9 (2001), 33–48. [24] f. herrera, e. herrera-viedma and l. mart́ınez, a fuzzy linguistic methodology to deal with unbalanced linguistic term sets, ieee trans. fuzzy syst. 16(2) (2008), 354–370. [25] e. herrera-viedma, a.g. lópez-herrera, m. luque and c. porcel, a fuzzy linguistic irs model based on a 2-tuple fuzzy linguistic approach, int. j. uncertain. fuzziness knowl.-based syst. 15 (2) (2007), 225–250. [26] v.n. huynh and y. nakamori, a satisfactory-oriented approach to multi-expert decision-making under linguistic assessments, ieee trans. syst. man cybern. 35 (2) (2005), 184–196. [27] h. ishibuchi, t. nakashima and m. nii, classification and modeling with linguistic information granules: advanced approaches to linguistic data mining, springer, berlin, 2004. [28] y.p. jiang and z.p. fan, property analysis of the aggregation operators for 2-tuple linguistic information, control decision, 18(6) (2003), 754–757. [29] d.k. joshi, i. beg and s. kumar, hesitant probabilistic fuzzy linguistic sets with applications in multi-criteria group decision making problems, mathematics 6(4)(2018), article id 47. [30] a. khalid and i. beg, incomplete hesitant fuzzy preference relations in group decision making, int. j. fuzzy syst. 19(3)(2017), 637-645. [31] g.j. klir and b. yuan, fuzzy sets and fuzzy logic: theory and applications, prentice-hall ptr, 1995. int. j. anal. appl. 17 (1) (2019) 103 [32] c. labreuche and m. grabisch, generalized choquet-like aggregation functions for handling bipolar scales, eur. j. oper. res. 172 (3) (2006), 931–955. [33] j. lawry, a framework for linguistic modelling, artif. intell. 155 (1–2) (2004), 1–39. [34] c. c. li, y. donga, f. herrera, e. h. viedm and l. mart́ınez, personalized individual semantics in computing with words for supporting linguistic group decision making. an application on consensus reaching, inf. fusion, 33 (2017), 29-40. [35] j. lu, g. zhang and f. wu, team situation awareness using web-based fuzzy group decision support systems, int. j. comput. intell. syst. 1 (1) (2008), 51–60. [36] j. lu, y. zhu, x. zeng, l. koehl, j. ma and g. zhang, a linguistic multi-criteria group decision support system for fabric hand evaluation, fuzzy optim. decis. mak. 8 (4) (2009), 395–413. [37] o. martin and g.j. klir, on the problem of retranslation in computing with perceptions, int. j. general syst. 35 (6) (2006), 655–674. [38] l. mart́ınez, d. ruan, f. herrera, e. herrera-viedma and p.p. wang, linguistic decision making: tools and applications, inf. sci. 179 (14) (2009), 2297–2298. [39] l. mart́ınez, j. liu, d. ruan and j.b. yang, dealing with heterogeneous information in engineering evaluation processes, inf. sci. 177 (7) (2007), 1533–1542. [40] l. mart́ınez, j. liu, j. b. yang and f. herrera, a multigranular hierarchical linguistic model for design evaluation based on safety and cost analysis, int. j. intell. syst. 20 (12) (2005), 1161–1194. [41] l. mart́ınez, r. m. rodŕıguez and f. herrera, 2-tuple linguistic model computing with words in decision making, springer (2015). [42] j.m. mendel and d. wu, perceptual computing: aiding people in making subjective judgments, ieee-wiley, (2010). [43] r. mesiar and a. kolesarova, on the fuzzy set theory and aggregation functions: histor and some recent advances, iran. j. fuzzy syst. in press. [44] w.r.w. mohd and l. abdullah, aggregation methods in group decision making: a decade survey, informatica 41 (2017), 71–86. [45] g.a. miller, the magical number seven, plus or minus two: some limits on our capacity of processing information, psychol. rev. 63 (1956), 81–97. [46] r. f. muirhead, some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, proc. edinburgh math. soc. 21(3) (1902), 144–162. [47] r. narasimhan, an analytic approach to supplier selection, j. purchas. supply manag. 1 (1983), 27–32. [48] r. l. nydick and r. p. hill, using the analytic hierarchy process to structure the supplier selection procedure, int. j. purchas. mat. manag. 28 (2) (1992), 31–36. [49] l. pérez-domı́nguez, l. rodŕıguez-picón, a. alvarado-iniesta, d.l. cruz and z.s. xu, moora under pythagorean fuzzy set for multiple criteria decision making, complexity, 2018 (2008), article id 2602376, 10 pages [50] i. saad, s. hammadi, m. benrejeb and p. borne, choquet integral for criteria aggregation in the flexible job-shop scheduling problems, math. comput. simul. 76 (2008), 447–462. [51] h. shi, h. c. liu, p. li, x. and g. xu, an integrated decision making approach for assessing healthcare waste treatment technologies from a multiple stakeholder, waste manag. 59 (2017), 508-517. [52] w. r. soukup, supplier selection strategies, j. purchas. mat. manag. 23 (3) (1987), 7–12. [53] c. q. tan and x. h. chen, intuitionistic fuzzy choquet integral operator for multi-criteira decision making, expert syst. appl. 37 (2010), 149–157. [54] thompson, vendor prole analysis, j. purchas. mat. manag. 26 (1) (1990), 11–18. int. j. anal. appl. 17 (1) (2019) 104 [55] v. torra, hesitant fuzzy sets, int. j. intell. syst. 25 (2010), 529–539. [56] r. j. vokurka, j. choobineh, and l. vadi, a prototype expert system for the evaluation and selection of potential suppliers, int. j. oper. prod. manag. 16 (12) (1996), 106–127. [57] p. wakker, additive representations of preferences, kluwer academic publishers, (1999). [58] z. wang and g. klir, fuzzy measure theory, new york: plenum press, (1992). [59] j.h. wang and j.y. hao, a new version of 2-tuple fuzzy linguistic representation model for computing with words, ieee trans. fuzzy syst. 14(3) (2006), 435–445. [60] g.w. wei, method for two-tuple linguistic group decision making based on the et-wg and et-owg operators, expert syst. appl. 37 (2010), 7895–7900. [61] m.m. xia, z.s. xu and b. zhu, geometric bonferroni means with their application in multi-criteria decision making, knowl. based syst. 40 (2013), 88-100. [62] z.s. xu, a method based on linguistic aggregation operators for group decision making with linguistic preference relations, inf. sci. 166(1–4) (2004), 19–30. [63] z.s. xu, eowa and eowg operators for aggregating linguistic labels based on linguistic preference relations, int. j. uncertain. fuzziness knowl.-based syst. 12 (2004), 791–810. [64] z. xu, s. shang, w. qian and w. shu, a method for fuzzy risk analysis based on the new similarity of trapezoidal fuzzy numbers, expert syst. appl. 37 (3) (2010), 1920–1927. [65] r.r. yager, a new methodology for ordinal multi objective decisions based on fuzzy sets, decision sci. 12 (1981), 589–600. [66] r.r. yager, computing with words and information/intelligent systems 2: applications, chapter approximate reasoning as a basis for computing with words, physica verlag, (1999), 50–77. [67] r.r. yager, induced aggregation operators, fuzzy sets syst. 137 (2003), 59–69. [68] r.r. yager, on the retranslation process in zadeh’s paradigm of computing with words, ieee transactions on systems, man, and cybernetics – part b: cybernetics, 34 (2) (2004), 1184–1195. [69] r.r. yager, owa aggregation of intuitionistic fuzzy sets, int. j. general syst. 38 (6) (2009), 617–641. [70] w. yang and z. chen, new aggregation operators based on the choquet integral and 2-tuple linguistic information, expert syst. appl. 39 (2012), 2662–2668 [71] l. zadeh, fuzzy sets, inf. control, 8 (1965), 338–353. [72] l. zadeh, the concept of a linguistic variable and its application to approximate reasoning, part iii, inf. sci. 9 (1) (1975), 43–80. [73] y. zhang and z.p. fan, an approach to linguistic multiple attribute decision-making with linguistic information based on elowa operator, syst. eng. 24(12) (2006), 324–339. [74] b. zhu, z. s. xu and m. m. xia, hesitant fuzzy geometric bonferroni means, inf. sci. 205 (2012), 72–85. 1. introduction 2. hesitant fuzzy sets 3. hesitant 2-tuple linguistic information 4. diminishing hesitant 2-tuple averaging operator 5. hesitant 2-tuple linguistic information aggregation operators based on the choquet integral 6. an application of dch2ta operators to multiple attribute decision making 7. illustrative example 8. discussion and conclusion references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 164-169 http://www.etamaths.com bounds of certain dynamic inequalities on time scales deepak b. pachpatte abstract. in this paper we study explicit bounds of certain dynamic integral inequalities on time scales. these estimates give the bounds on unknown functions which can be used in studying the qualitative aspects of certain dynamic equations. using these inequalities we prove the uniqueness of some partial integro-differential equations on time scales. 1. introduction in 1989 german mathematician stefan hilger [4] initiated the study of time scale in his ph.d dissertation. dynamic inequalities on time scales has applications in various fields. during past few years many authors have studied various types of dynamic equations and inequalities on time scales, its properties and applications [1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. motivated by need for the diverse applications and to widen the scope of such inequalities in this paper we obtain some explicit bounds of certain dynamic inequalities on time scales. 2. main results let r denotes the real number z the set of integers and t denotes the arbitrary time scales and it = i ∩ t, i = [t0,∞). let crd be the set of all rd continuous function. we assume here basic understaing of time scale calculus. the research monograph [2, 3] gives the basic information on time scales calculus. now we give here our main results. theorem 2.1 let p(t,s),q(t,s) ∈ crd(it × it,r+) and be nondecreasing for t ∈ it for each s ∈ it and (2.1) u (t) ≤ c + t∫ t0 p (t,τ)u (τ) ∆τ + α∫ t0 q (t,τ)u (τ) ∆τ, for t ∈ it where c ≥ 0 is a constant. if (2.2) k (t) = α∫ t0 q (t,τ) ep (τ,s)∆τ < 1, 2010 mathematics subject classification. 26d10, 26e70, 34n05. key words and phrases. explicit bounds; integral inequality; dynamic equations; time scales. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 164 bounds of certain dynamic inequalities on time scales 165 then (2.3) u (t) ≤ c 1 −k (t) ep (t,τ) . proof. for fixed x, t0 ≤ x ≤ α then for t0 ≤ t ≤ x we have (2.4) u (t) ≤ c + t∫ t0 p (x,τ)u (τ) ∆τ + α∫ t0 q (x,τ)u (τ) ∆τ. define a function w(t,x), t0 ≤ t ≤ x by right hand side of (2.4) we get (2.5) u(t) ≤ w(t,x), t0 ≤ t ≤ x, (2.6) w (t0,x) = c + α∫ t0 q (x,τ)u (τ) ∆τ, and (2.7) w∆ (t,x) = p (x,t) u (t) ≤ p (x,t) w (t) , for t0 ≤ t. by taking t = s and integrating it with respect to s from t0 to x we have (2.8) w (x,x) ≤ w (t0,x) ep (t0,x) . since x is arbitrary in (2.5) and (2.8) x replaced by t and u(t) ≤ w(t,t) we get (2.9) u (t) ≤ w (t0, t) ep (t0,s) , where (2.10) w (t0, t) = c + α∫ t0 q (t,τ)u (τ) ∆τ. using (2.9) on the right hand side of (2.10) and by (2.2) we have (2.11) w (t0, t) ≤ c 1 −k (t) . using (2.11) in (2.9) we get the result theorem 2.2 let f(t,τ),g(t,τ),h(t,τ) ∈ crd(it × it,r+), f(t,τ),g(t,τ) are nondecreasing in t for each τ ∈ it and if (2.12) u (t) ≤ c + t∫ t0 f (t,τ)  u (τ) + τ∫ t0 h (τ,s) u (s) ∆s   ∆τ + α∫ t0 g (t,τ)u (τ) ∆τ. for t ∈ it. then (2.13) u (t) ≤ c 1 −k1 (t) ef (t0,τ) , where (2.14) k1 (t) = τ∫ t0 g (t,τ)ef (t0,τ) ∆τ < 1, 166 pachpatte (2.15) f (t,τ) = f (t,τ)  1 + τ∫ t0 h (τ,ξ) ∆ξ   . proof. let c > 0 and for any fix x ∈ it then for t0 ≤ t ≤ x. from (2.13) we have (2.16) u (t) ≤ c+ t∫ t0 f (x,τ)  u (τ) + τ∫ t0 h (τ,s) u (s) ∆s   ∆τ + α∫ t0 g (x,τ)u (τ) ∆τ. define a function w(t,x), t ∈ [t0,x], u(t) ≤ w(t,x),w(t,x) > 0 (2.17) w (t0,x) = c + α∫ t0 g (x,τ)u (τ) ∆τ, and w∆ (t,x) = f (x,t)  u (τ) + t∫ t0 h (t,s) u (s) ∆s   ≤ f (x,t)  w (t) + t∫ t0 h (t,s) w (s,x) ∆s   .(2.18) from (2.17) and since w(t,x) is nondecreasing in t we have (2.19) w∆ (t,x) w (t,x) ≤ f (x,t)  1 + t∫ t0 h (t,s) ∆s   , for t ∈ [t0,x]. now taking t = ξ and integrating with respect to ξ from t0 to x we get (2.20) w (x,x) ≤ w (t0,x) ef (t0,τ) since x is arbitrary with x replaced by t we have for t ∈ it, (2.21) w (t,t) ≤ w (t0, t) ef (t0,τ) , and (2.22) w (t0, t) = c + α∫ t0 g (t,τ)u (τ) ∆τ, for t ∈ it. since u(t) ≤ w(t,t) we get from (2.21) (2.23) u (t) ≤ w (t0, t) ef (t0,τ) . now from (2.23), (2.22) and from (2.14) we have (2.24) w (t0, t) ≤ c 1 −k1 (t) . using (2.24) in (2.25) we get (2.25) u (t) ≤ c 1 −k1 (t) ef (t0,τ) . bounds of certain dynamic inequalities on time scales 167 theorem 2.3 let a,b,c ∈ crd(it,r+) and (2.26) u (t) ≤ c + t∫ t0 a (τ)  u (τ) + τ∫ t0 b (s) u (s) ∆s + α∫ t0 d (s)u (s) ∆s   ∆τ, (2.27) z = α∫ t0 d (s)ea+b (t0,s) ∆s < 1, then (2.28) u(t) ≤ c 1 −z ea+b (t0,τ) , for t ∈ it. proof. now define a function w(t) by right hand side of (2.26) then w(t0) = c,u(t) ≤ w(t) and w∆ (t) = a (t)  u (t) + t∫ t0 b (s) u (s) ∆s + α∫ t0 d (s)u (s) ∆s   ≤ a (t)  w (t) + t∫ t0 b (s) w (s) ∆s + α∫ t0 d (s)w (s) ∆s   ,(2.29) for t ∈ it . now define a function v(t) by (2.30) v(t) = w (t) + t∫ t0 b (s) w (s) ∆s + α∫ t0 d (s)w (s) ∆s, then w(t) ≤ v(t), w∆ (t) ≤ a(t)v(t) (2.31) v (t0) = c + α∫ t0 d (s)w (s) ∆s, and v∆ (t) = w∆ (t) + b (t) w (t) ≤ a (t) v (t) + b (t) w (t) ≤ [a(t) + b(t)] w (t) .(2.32) we get (2.33) v (t) ≤ v (t0) ea+b ((t0, t)) , for t ∈ it. using (2.33) in w(t) ≤ v(t) we have (2.34) w (t) ≤ v (t0) ea+b(t0, t). now from (2.33), (2.31) and from (2.27) we have (2.35) v (t0) ≤ c 1 −z , using (2.34) in (2.33) and we have u(t) ≤ w(t) we get (2.28). 168 pachpatte 3. applications now we consider the following dynamic equation y (t) = a (t) + t∫ t0 g  t,τ,y (τ) , τ∫ t0 b (τ,s,y (τ)) ∆s   ∆τ + α∫ t0 d (t,τ,x (τ))∆τ(3.1) for t ∈ it where y(t) is unknown function and α ∈ crd(it,r+), b,d ∈ crd(it × rn,rn), g ∈ crd(it ×rn ×rn,rn). now we give the application of theorem 2.2 for studying certain properties of solution of equation (3.1). theorem 3.1 suppose that the function a,b,d,g as in (3.1) satisfy the conditions (3.2) |a (t)| ≤ c, (3.3) |b (t,τ,y)| ≤ h (t,τ) |y| , (3.4) |d (t,τ,y)| ≤ g (t,τ) |y| , (3.5) |g (t,τ,y,x)| ≤ f (t,τ) (|y| + |x|) , where f(t,τ),g(t,τ),h(t,τ) and c are as given in theorem 2.2. let k1(t) be as in (2.14). if y(t), t ∈ it is a solution of (3.1) then (3.6) |y (t)| ≤ c 1 −k1 (t) ef (τ,t0) , for t ∈ it where f is defined by (2.15). proof. since y(t) is solution of equation (3.1) and using (3.2) − (3.5) we get |y (t)| ≤ c + t∫ t0 f (t,τ)  |y (x)| + τ∫ t0 h (τ,s) |y (τ)|∆s   ∆τ + α∫ t0 g (t,τ) |y (τ)|∆τ(3.7) now an application of theorem 2.2 to (3.7) we get (3.6). references [1] s. andras and a. meszaros, wendroff type inequalities on time scales via picard operators, math. inequal. appl.,17,1(2013),159-174. [2] m. bohner and a. peterson, dynamic equations on time scales, birkhauser boston/berlin, (2001). [3] m. bohner and a. peterson, advances in dynamic equations on time scales, birkhauser boston/berlin, (2003). [4] s. hilger, analysis on measure chain-a unified approch to continuous and discrete calculus, results. math., 18:18-56, 1990. [5] w. n. li, some pachpatte type inequalities on time scales, comput. math. appl. vol. 57, iss. 2, 2009, p.275-282 bounds of certain dynamic inequalities on time scales 169 [6] l. li, m. han, some new dynamic opial type inequalities and applications for second order integro-differential dynamic equations on time scales, appl. math. comput.,vol. 232, 2014, p. 542-547 [7] f. meng, j. shao, some new volterra fredholm type dynamic integral inequalities on time scales, appl. math. comput., vol. 223, 2013, p. 444-451. [8] d. b. pachpatte, explicit estimates on integral inequalities with time scale, j. inequal. pure. appl. math., vol. 7, issue 4, artivle 143, 2006. [9] d. b. pachpatte, integral inequalitys for partial dynamic equations on time scales, electron. j. differential equations,vol. 2012 (2012), no. 50, 1c7. [10] d. b. pachpatte, properties of solutions to nonlinear dynamic integral equations on time scales, electron. j. differential equations,vol. 2008(2008). no. 130. pp.1-8. [11] s.h. saker, applications of opial inequalities on time scales on dynamic equations with damping terms, math. comput. modelling, vol. 58, iss. 11?12, 2013, p.1777-1790 [12] y. sun, t. hassan, some nonlinear dynamic integral inequalities on time scales, appl. math. comput., vol 220, 2013, p. 221-225. [13] a. tuna, s. kutukcu, some integral inequalities on time scales, appl. math. mech. 2008, 29(1):23?29. [14] l. yin and f. qi, some integral inequalities on time scales, results. math. 64 (2013), 371?381. department of mathematics, dr. b.a.m. university, aurangabad, maharashtra 431004, india international journal of analysis and applications volume 17, number 4 (2019), 530-547 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-530 received 2019-05-01; accepted 2019-06-10; published 2019-07-01. 2010 mathematics subject classification. 82b31. key words and phrases. arima; outliers; return series; time series. ©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 530 modeling the effects of outliers on the estimation of linear stochastic time series model emmanuel alphonsus akpan1,* and imoh udo moffat2 1department of mathematical science, abubakar tafawa balewa university, bauchi, nigeria 2department of mathematics and statistics, university of uyo, nigeria *corresponding author: eubong44@gmail.com abstract. this study investigates the effects of outliers on the estimates of arima model parameters with particular attention given to the performance of two outlier detection and modeling methods targeted at achieving more accurate estimates of the parameters. the two methods considered are: an iterative outlier detection aimed at obtaining the joint estimates of model parameters and outlier effects, and an iterative outlier detection with the effects of outliers removed to obtain an outlier free series, after which a successful arima model is entertained. we explored the daily closing share price returns of fidelity bank, union bank of nigeria, and unity bank from 03/01/2006 to 24/11/2016, with each series consisting of 2690 observations from the nigerian stock exchange. arima (1, 1, 0) models were selected based on the minimum values of akaike information criteria which fitted well to the outlier contaminated series of the respective banks. our findings revealed that arima (1, 1, 0) models which fitted adequately to the outlier free series outperformed those of the parameter-outlier effects jointestimated model. furthermore, we discovered that outliers biased the estimates of the model parameters by reducing the estimated values of the parameters. the implication is that, in order to achieve more accurate estimates of arima model parameters, it is needful to account for the presence of significant outliers and preference should be given to the approach of cleaning the series of outliers before subsequent entertainment of adequate linear time series models. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-530 int. j. anal. appl. 17 (4) (2019) 531 1. introduction outliers are common characterizations of every time series. in general, outliers are extreme observations that deviate from the overall pattern of the sample. statistically, outliers are those observations whose standard deviations are greater than 3 in absolute value, which is the value of kurtosis occupied by the normal distribution. however, the effects of outliers on the linear time series models cannot be overemphasized; such effects range from false inference, introduction of biases in the model parameters, model misspecification and misleading confidence interval ([1], [2], [3], [4]). by efficiency, we mean the goodness of an estimator of a model which can be measured by variance, that is, a model with the smallest variance is considered to be superior as regarding efficiency. to reiterate the need for efficiency of the estimates of model parameters by considering the presence of outliers, this study applied two outlier identification and modeling methods. the first is the modified iterative method proposed by [5], which involves the joint estimation of the model parameters and the magnitude of outlier effects. the second is the modified iterative method proposed by [6], which involves identification of outliers sequentially by searching for most relevant anomaly, estimating its effect and removing it from the data. the estimation of the model parameters is again done on the outlier corrected series, and further iteration of the process is carried out until no significant perturbation is found. actually, the motivation for this study is derived from the fact that previous studies such as [7], [8], [9], [10] failed to consider outliers while modeling returns series in nigeria. thus, this gap in knowledge is fully addressed in our work. this work is further organized as follows: section 2 takes care of materials and methods; section 3 handles the results and discussion while section 4 treats the conclusion. 2. materials and methods 2.1 return series the returns series (𝑅𝑡) can be obtained given that 𝑃𝑡 is the price of a unit shares at time t and 𝑃𝑡−1 is the price of shares at time t−1. thus 𝑅𝑡 = ∇𝑙𝑛𝑃𝑡 = (1 − 𝐵)𝑙𝑛𝑃𝑡 = 𝑙𝑛 𝑃𝑡 − 𝑙𝑛 𝑃𝑡−1 (1) in equation (1), 𝑅𝑡 is regarded as a transformed series of the price (𝑃𝑡) of shares meant to attain stationarity such that both the mean and the variance of the series are stable [11] while 𝐵 is the backshift operator. int. j. anal. appl. 17 (4) (2019) 532 2.2 autoregressive integrated moving average (arima) model [3] considered the extension of arma model to deal with homogenous non-stationary time series in which 𝑋𝑡 , is non-stationary but its 𝑑 𝑡ℎ difference is a stationary arma model. denoting the 𝑑𝑡ℎ difference of 𝑋𝑡 by 𝜑(𝐵) = 𝜙(𝐵)∇𝑑 𝑋𝑡 = 𝜃(𝐵) 𝑡 (2) where 𝜑(𝐵) is the nonstationary autoregressive operator such that d of the roots of 𝜑(𝐵) = 0 are unity and the remainder lie outside the unit circle while 𝜙(𝐵) is a stationary autoregressive operator. it should be noted that in equation (2), the presence of outliers is not taken into consideration. 2.3 joint model of arima and outlier-effects 𝑅𝑡 = ∑ φjrt−j p j=1 + ∑ θi 𝑞 𝑖=1 𝑎t−i + 𝑎𝑡 + ∑ 𝜔𝑗 𝑘 𝑗=1 𝑉𝑗(b)𝐼𝑡 (𝑇) , (3) where 𝑉𝑗(b) = 1 for an ao, and 𝑉𝑗(b) = 𝜃(𝐵) 𝜑(𝐵) for an io at t = 𝑇𝑗, 𝑉𝑗(b) = (1 – 𝐵) −1 for a ls, 𝑉𝑗(b) = (1 – 𝛿 𝐵)−1 for an tc, and 𝜔 is the size of the outlier. for more details on the types of outliers and estimation of their effects, see [1], [12], [3], [4], [5], [13]. 2.4 arima model for outlier-adjusted return series 𝑅𝑡 − ∑ φjrt−j p j=1 − ∑ θi 𝑞 𝑖=1 𝑎t−i − ∑ 𝜔𝑗 𝑘 𝑗=1 𝑉𝑗(b)𝐼𝑡 (𝑇) = 𝑎𝑡, (4) where 𝑎𝑡 is the outlier free series. meanwhile, equations (3) and (4) represent major modifications on equation (2) to account for the presence of outliers. 2.5 outliers in time series generally, in time series, four types of outliers are identified and they are as follows: additive outlier, innovative outlier, level shift outlier and temporary outlier [12]. 2.5.1 additive outlier (ao) a time series 𝑌1, …, 𝑌𝑇 affected by the presence of an additive outlier at t = t is given by 𝑌𝑡 = { 𝑋𝑡 , 𝑡 ≠ 𝑇 𝑋𝑡 + 𝜔, 𝑡 = 𝑇 = 𝑋𝑡 + 𝜔𝐼𝑡 (𝑇) = 𝜃(𝐵) 𝜑(𝐵) 𝑎𝑡 + 𝜔𝐼𝑡 (𝑇) (5) for t = 1, …,t, where 𝐼𝑡 (𝑇) = { 1 , 𝑡 = 𝑇, 0, 𝑡 ≠ 𝑇, is the indicator variable representing the presence or absence of an outlier at time t, 𝑋𝑡 follows an arima model, 𝜔 is an outlier size. hence, an additive outlier affects only a single observation (see also [1], [12], [3], [4]). 2.5.2 innovative outlier (io) a time series𝑌1, …, 𝑌𝑇 affected by the presence of an innovative outlier at t = t is given by int. j. anal. appl. 17 (4) (2019) 533 𝑌𝑡 = 𝑋𝑡 + 𝜃(𝐵) 𝜑(𝐵) 𝜔𝐼𝑡 (𝑇) = 𝜃(𝐵) 𝜑(𝐵) (𝑎𝑡 + 𝜔𝐼𝑡 (𝑇) ) (6) hence, an innovative outlier affects all observations 𝑌𝑡 , 𝑌𝑡+1 ,…, beyond time t through the memory of the system described by 𝜓(b) = 𝜃(𝐵) 𝜑(𝐵) , such that 𝑌𝑡 = 𝑋𝑡 + ψ(b)𝜔𝐼𝑡 (𝑇) . meanwhile, according to [12], the innovation of a time series 𝑌1, …, 𝑌𝑇 is affected by 𝑌𝑡 = 𝑒𝑡 + 𝜔𝐼𝑡 (𝑇) (5) where 𝑒𝑡are the innovations of the uncontaminated series 𝑋𝑡. 2.6.3 level shift (ls) a time series 𝑌1, …, 𝑌𝑇 affected by the presence of a level shift at t = t is given by 𝑌𝑡 = 𝑋𝑡 + 𝜔𝑆𝑡 (𝑇) (6) where 𝑆𝑡 (𝑇) = (1 − 𝐵)−1𝐼𝑡 (𝑇) . note that level shift affects all the observation of the series after t = t. hence, according to [12], level shift serially affects the innovations as follows: 𝑎𝑡 = 𝑒𝑡 + π(b)𝜔𝑆𝑡 (𝑇) (7) where 𝜋(𝐵) = (1 − 𝜋1𝐵 − 𝜋2𝐵 2 − ⋯ ) 2.7.4 temporary change (tc) a time series 𝑌1, …, 𝑌𝑇 affected by the presence of a temporary change at t = t is given by 𝑌𝑡 = 𝑋𝑡 + 1 1−𝛿𝐵 𝜔𝐼𝑡 (𝑇) (8) where 𝛿 is an exponential decay parameter such that 0 < 𝛿 < 1. if 𝛿 tends to 0, the temporary change reduces to an additive outlier, whereas if 𝛿 tends to 1, the temporary change reduces to a level shift. the temporary change affects the innovations as follows: 𝑎𝑡 = 𝑒𝑡 + 𝜋(𝐵) 1−𝛿𝐵 𝜔𝐼𝑡 (𝑇) (9) if 𝜋(𝐵) is close to 1 − 𝛿𝐵, the effect of temporary change on the innovations is very close to the effect of an innovative outlier. otherwise, the temporary change can affect several observations with a decreasing effect after t = t [12]. 3. results and discussion 3.1 time plots inspecting the plots in figures 1-3, it is obvious that they are characterized by upward and downward movements away from the common mean, which clearly indicates the existence of nonstationarity. int. j. anal. appl. 17 (4) (2019) 534 figure 1: price series of fidelity bank shares figure 2: price series of union bank shares figure 3: price series of unity bank shares also, the plots in figures 4 6 indicate that the returns series cluster around the mean which implies that the series are stationary. 0 2 4 6 8 10 12 14 2006 2008 2010 2012 2014 2016 fi db an k 0 5 10 15 20 25 30 35 40 45 50 55 2006 2008 2010 2012 2014 2016 ub ns ha re s 0 1 2 3 4 5 6 7 8 9 10 2006 2008 2010 2012 2014 2016 un it yb an ks ha re s int. j. anal. appl. 17 (4) (2019) 535 figure 4: return series of fidelity bank figure 5: return series of union bank figure 6: return series of unity bank -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 2006 2008 2010 2012 2014 2016 ld_ fi db an k -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2006 2008 2010 2012 2014 2016 ld _u bn sh ar es -0.5 0 0.5 1 1.5 2 2.5 2006 2008 2010 2012 2014 2016 ld _u ni ty ba nk sh ar es int. j. anal. appl. 17 (4) (2019) 536 3.2 linear time series modeling return series of fidelity bank from figures 7 and 8, both acf and pacf indicate that mixed model is possible. the following models, arima(1, 1, 0), arima(0, 1, 1), arima(1, 1, 1), arima(1, 1, 2) and arima (2, 1, 1) are entertained tentatively. figure 7: acf of return series of fidelity bank figure 8: pacf of return series of fidelity bank from table i, arima(1, 1, 1) model has the smallest aic but one of its parameters is not significant. secondly, arima(1, 1, 2) model has the second smallest aic yet its parameters are not significant. hence, arima(1, 1, 0) model is selected based on the ground that its parameter is significant and has the nearest minimum aic. table i: arima models for return series of fidelity bank model parameter akaike information criteria (aic) log likelihood 𝝋𝟏 𝝋𝟐 𝛉𝟏 𝛉𝟐 arima(1,1,0) 0.1606∗∗∗ −11562.17 5783.09 arima(0,1,1) 0.1494∗∗∗ −11559.28 5780.64 arima(1,1,1) 0.2569∗∗∗ − 0.0986 −11563.16 5783.58 arima(1,1,2) −0.0498 0.2071 0.0628 −11562.88 5784.44 arima(2,1,1) −0.0721 0.0619 0.2288 −11561.98 5783.99 *** significance at 5% level int. j. anal. appl. 17 (4) (2019) 537 furthermore, evidence from ljung-box q-statistics shows that arima(1, 1, 0) model is adequate at 5% level of significance given the q-statistics at lags 1, 4, 8, and 24 given by q(1) = 0.0376, q(4) = 5.4261, q(8) = 9.8001, and q(24) = 23.379 with the respective p-values of 0.8462, 0.2463, 0.2793, and 0.4975. 3.3 identification of outliers in the residual series of arima(1, 1, 0) model fitted to the return series of fidelity bank considering the critical value, c = 4, and based on the condition that n ≥ 450, we identified sixteen (16) different outliers that have contaminated the residual series of arima(1, 1, 0) model, as indicated in table ii. they are: two (2) innovation outliers (io), five (5) additive outliers, and nine (9) temporary change outliers. table ii: types of outliers identified in the residual series of arima(1, 1, 0) model fitted to the return series of fidelity bank type observation index location estimate t-statistic io 1555 26/04/2012 -0.09798041 -4.198390 ao 1789 08/04/2013 -0.10865950 -4.715609 ao 1841 21/06/2013 -0.10673597 -4.632131 ao 2539 15/04/2016 -0.17301613 -7.508560 ao 2042 11/04/2014 -0.30477209 -13.226510 tc 827 18/05/2009 0.07540548 4.049288 tc 847 16/06/2009 -0.07692527 -4.130901 tc 859 02/07/2009 -0.07537282 -4.047534 tc 1665 04/10/2012 0.08360953 4.489847 tc 1724 01/02/2013 0.07510564 4.033187 tc 2263 05/03/2015 0.07816849 4.197662 tc 2280 30/03/2015 0.09555288 5.131207 io 2292 17/04/2015 -0.09220965 -4.046644 ao 2043 14/04/2014 0.24193892 10.598998 tc 691 27/10/2008 -0.06641433 -4.025161 tc 950 11/11/2009 0.06557061 4.004060 int. j. anal. appl. 17 (4) (2019) 538 to account for the effect of outliers, the method of joint estimation of the parameter of arima (1, 1, 0) model with outliers identified in table ii is performed as indicated in table iii. comparing the values of aic = −11922.67 and log likelihood = 5979.34 of the joint model of arima(1, 1, 0) with outliers effects with that of arima (1, 1, 0) model having aic = −11562.17 and log likelihood = 5783.09, it is obvious that the joint model of arima (1, 1, 0) with outliers effects has a lower aic and a higher log likelihood value, thus making it a better model than the arima (1, 1, 0) model where the influence of outliers is not taken into consideration. table iii: joint model of arima (1, 1, 0) and outlier-effects fitted to return series fidelity bank estimate std. error z value pr(>|z|) ar1 0.171530 0.019142 8.9607 < 2.2e−16 ∗∗∗ io1555 -0.084464 0.024209 -3.4890 0.0004849∗∗∗ ao1789 -0.109211 0.025841 -4.2262 2.376e−05 ∗∗∗ ao1841 -0.107286 0.025841 -4.1518 3.299e−05 ∗∗∗ ao2042 -0.273962 0.026198 -10.4573 < 2.2e−16 ∗∗∗ ao2539 -0.173178 0.025830 -6.7045 2.022e−11 ∗∗∗ tc827 0.075179 0.021072 3.5677 0.0003601∗∗∗ tc847 -0.076153 0.021068 -3.6147 0.0003007 ∗∗∗ tc859 -0.074623 0.021069 -3.5418 0.0003974 ∗∗∗ tc1665 0.083147 0.021082 3.9439 8.016e−05 ∗∗∗ tc1724 0.074547 0.021090 3.5348 0.0004081∗∗∗ tc2263 0.078614 0.021097 3.7264 0.0001943∗∗∗ tc2280 0.095246 0.021087 4.5168 6.277e-06∗∗∗ io2292 -0.071450 0.024232 -2.9486 0.0031921∗∗ ao2043 0.197831 0.026196 7.5520 4.286e-14∗∗∗ tc691 -0.070928 0.021078 -3.3651 0.0007653∗∗∗ tc950 0.071171 0.021068 3.3782 0.0007295∗∗∗ 3.4 building arima(1, 1, 0) model for outlier-adjusted return series of fidelity bank here, the second method is applied which is the removal of the outliers effects to obtain an outlier-adjusted series. then, arima(1, 1, 0) model fitted well to the outlier-adjusted series with its parameter significant at 5% level [see table iv] and is found to be adequate given the int. j. anal. appl. 17 (4) (2019) 539 q-statistics at lags 1, 4, 8, and 24 given by q(1) = 0.0003, q(4) = 4.2007, q(8) = 13.92, and q(24) = 29.649 with the corresponding p-values of 0.99, 0.38, 0.09, and 0.20. table iv: arima (1, 1, 0) model for outlier-adjusted return series of fidelity bank model parameter (𝝋) akaike information criteria log likelihood arima (1, 1, 0) 0.1715∗∗∗ −11954.67 5979.34 *** significance at 5% level arima (1, 1, 0) model with the least aic = −11954.67 appears to be better than that of the joint model of arima (1, 1, 0) with outliers effects. on comparing the estimates of arima(1, 1, 0) model fitted to the outlier contaminated series with the arima(1, 1, 0) model when adjusted for outliers using the two proposed methods, it is found that the estimates of both the joint arima(1, 1, 0) model with outliers effects and the arima(1, 1, 0) model fitted to the outlier adjusted series are the same. however, the later tends to outperform the former on the basis of smallest information criteria. of paramount interest is the discovery that outliers introduced substantial bias in the estimate of arima (1, 1, 0) model by 0.0109 as shown in table v. again, the modified iterative method produced a model with smallest variance as indicated in table v, hence, adjudged the most efficient method. table v: effect of outliers on estimate of arima(1, 1, 0) model for return series of fidelity bank model arima (1,1,0) (for outlier contaminated) joint arima (1,1,0) with outliers effects arima (1,1,0) (for outlier adjusted) bias introduced parameter (𝝋𝟏) 0.1606 0.1715 0.1715 −0.0109 aic −11562.18 -11922.67 −11954.67 standard error 0.0190 0.0191 0.0189 variance 0.000795 0.000691 0.000687 log likelihood 5783.09 5979.34 5979.34 int. j. anal. appl. 17 (4) (2019) 540 3.5 linear time series modeling of return series of union bank from figures 9 and 10, both acf and pacf indicate that the following mixed model could be entertained tentatively: arima(1, 1, 0), arima(0, 1, 1) and arima(1, 1, 1). figure 9: acf of return series of union bank figure 10: pacf of return series of union bank from table vi, arima (1, 1, 0) model is selected based on the ground that its parameter is significant and has the minimum aic. table vi: arima models for return series of union bank model parameter akaike information criteria (aic) log likelihood 𝝋𝟏 𝛉𝟏 arima (1,1,0) 0.1014∗∗∗ −9132.26 4567.13 arima (0,1,1) 0.0963∗∗∗ −9130.87 4566.43 arima (1,1,1) 0.2455 − 0.1453 −9131.12 4567.56 *** significance at 5% level furthermore, evidence from ljung-box q-statistics shows that arima(1, 1, 0) model is adequate at 5% level of significance given the q-statistics at lags 1, 4, 8 and 24 given by q(1) = int. j. anal. appl. 17 (4) (2019) 541 0.0133, q(4) = 2.3753, q(8) = 4.318 and q(24) = 7.9309 with the corresponding p-values of 0.9082, 0.6671, 0.8274, and 0.9991. 3.6 identification of outliers in the residual series of arima(1, 1, 0) model fitted to the return series of union bank here, we consider the critical value, c = 5 given that c = 4 was not sufficient for computing weights of outliers and about nineteen (19) different outliers are identified to have contaminated the residual series of arima(1, 1, 0) model, four (4) innovation outliers (io), eight (8) additive outliers and seven (7) temporary change outliers, as shown in table vii. table vii: types of outliers identified in the residual series of arima(1, 1, 0) model fitted to the return series of union bank type observation index location estimate t-statistic io 458 16/11/2007 -0.20259320 -9.867965 io 1472 23/12/2011 -0.22031597 -10.731210 io 1831 07/06/2013 0.10533493 5.130683 io 1843 25/06/2013 0.10590627 5.158511 ao 150 15/08/2006 -0.13856541 -6.783874 ao 705 14/11/2008 -0.20086454 -9.833910 ao 1471 22/12/2011 1.67935140 82.217553 ao 1830 06/06/2013 -0.11483241 -5.621956 ao 1842 24/06/2013 -0.10581300 -5.180384 ao 1984 21/01/2014 -0.10648119 -5.213098 ao 1994 04/02/2014 0.16239480 7.950512 tc 691 27/10/2008 -0.08071046 -5.129738 tc 901 31/08/2009 -0.08274861 -5.259278 tc 1470 22/12/2011 0.53378545 33.925958 tc 1523 09/03/2012 -0.08218825 -5.223663 tc 1541 04/04/2012 0.07869209 5.001456 tc 1824 28/05/2013 0.11353246 7.215815 tc 2534 08/04/2016 -0.08059290 -5.122266 ao 1748 06/02/2013 -0.11923771 -5.160464 int. j. anal. appl. 17 (4) (2019) 542 again, applying the first method as indicated in table viii, it is found that the values of aic = −11560.27 and log likelihood = 5800.13 for the joint model of arima(1, 1, 0) with outliers effects when compared to that of arima (1, 1, 0) model with aic = −9132.26 and log likelihood = 4567.13 are respectively smaller and higher, making the former a better model than the later. table viii: joint model of arima (1, 1, 0) and outliers effects fitted to return series of union bank estimate std. error z value pr(>|z|) ar1 0.265411 0.018828 14.0965 < 2.2e−16 ∗∗∗ io458 -0.176690 0.024975 -7.0747 1.497e−12 ∗∗∗ io1472 -0.045126 0.026449 -1.7061 0.0879825 . io1831 0.049676 0.025686 1.9340 0.0531185 . io1843 0.049638 0.025664 1.9341 0.0530983 . ao150 -0.152926 0.027115 -5.6399 1.701e−08 ∗∗∗ ao705 -0.209666 0.027091 -7.7393 9.999e−15 ∗∗∗ ao1471 1.676966 0.029599 56.6554 < 2.2e−16 ∗∗∗ ao1830 -0.122852 0.027860 -4.4096 1.036e−05 ∗∗∗ ao1842 -0.094486 0.027815 -3.3969 0.0006816 ∗∗∗ ao1984 -0.118687 0.027104 -4.3790 1.192e−05 ∗∗∗ ao1994 0.169260 0.027084 6.2495 4.117e−10 ∗∗∗ tc691 -0.072536 0.023951 -3.0285 0.0024576 ** tc901 -0.076155 0.023946 -3.1803 0.0014712 ** tc1470 -0.004099 0.025804 -0.1589 0.8737844 tc1523 -0.075499 0.023947 -3.1528 0.0016173 ** tc1541 0.079253 0.023929 3.3120 0.0009264 *** tc1824 0.106075 0.024035 4.4134 1.018e−05 ∗∗∗ tc2534 -0.084954 0.023936 -3.5492 0.0003865 *** ao1748 -0.110790 0.027141 -4.0821 4.463e−05 ∗∗∗ 3.7 building arima (1, 1, 0) model for outlier-adjusted return series of union bank using the second method, which is removing the effects of the outliers and afterward, arima(1, 1, 0) model is fitted to the outlier-adjusted series with its parameter significant at 5% level [table ix], it is found to be adequate at 5% level of significance given the q-statistics at int. j. anal. appl. 17 (4) (2019) 543 lags 1, 14, 18, and 24 having q(1) = 0.0030, q(14) = 19.228, q(18) = 24.611 and q(24) = 27.717 with the corresponding p-values of 0.956, 0.1564, 0.136, and 0.2722. table ix: arima (1,1,0) model for outlier adjusted return series of union bank model parameter (𝝋) akaike information criteria log likelihood arima (1,1,0) 0.2654∗∗∗ −11598.27 5800.13 *** significance at 5% level arima (1, 1, 0) model fitted to the outlier adjusted series with least aic = −11598.27 is found to be a better model than that of the joint estimation of arima (1, 1, 0) with outliers effect, and that of arima (1, 1, 0) model without outliers effect. again, the effects of outliers on the estimate of arima(1, 1, 0) model fitted to the return series of union bank is similar to that of the fidelity bank although the estimate of the model is reduced by 0.164 and the modified iterative method is also adjudged superior in term of efficiency given that it produced a model with minimum variance as shown in table x. table x: effect of outliers on estimate of arima (1, 1, 0) model for return series of union bank model arima (1,1,0) (for outlier contaminated) joint arima (1,1,0) and outlier effect arima (1,1,0) (for outlier adjusted) bias introduced parameter 0.1014 0.2654 0.2654 −0.164 aic -9130.26 -11560.27 -11598.27 standard error 0.0192 0.0188 0.0186 variance 0.001963 0.000785 0.000784 log-likelihood 4567.13 5800.13 5800.13 3.8 linear time series modeling of return series of unity bank again, using the same procedures as in the first two banks, arima(1, 1, 0) model is found to be adequate for the return series of the unity bank. however, about thirty three (33) different outliers are identified to have contaminated the residuals series of arima(1,1,0) model, two (2) innovation outliers (io), six (6) additive outliers, fifteen (15) temporary change and ten (10) level shift at c = 5 as shown in table xi and the joint estimation of the parameter of arima(1, 1, 0) model and outliers effects is shown in table xii. int. j. anal. appl. 17 (4) (2019) 544 table xi: types of outliers identified in the residual series of arima (1, 1, 0) model fitted to the return series of unity bank type observation index location estimate t-statistic io 2293 20/04/2015 -0.180979695 -7.444781 ao 248 10/01/2007 1.098612289 45.331893 ao 1906 24/09/2013 -0.200532990 -8.274566 ao 2292 17/04/2015 2.302585093 95.011264 tc 247 09/01/2007 0.365004553 19.903211 tc 1736 18/01/2013 0.107790532 5.877674 tc 1745 01/02/2013 0.112419182 6.130068 tc 1753 13/02/2013 -0.118923561 -6.484743 tc 1762 26/02/2013 0.091895380 5.010932 tc 2291 16/04/2015 0.758297010 41.348923 tc 2298 27/04/2015 -0.142415961 -7.765752 tc 2304 06/04/2015 -0.098876918 -5.391626 tc 2446 30/11/2015 -0.093629262 -5.105479 tc 2458 16/12/2015 0.112419118 6.130064 tc 2460 18/12/2015 0.104980801 5.724463 tc 2467 04/01/2016 -0.106045605 -5.782525 tc 2469 06/01/2016 -0.119002493 -6.489047 io 1905 23/09/2013 0.127354627 5.132279 ao 1904 20/09/2013 -0.163097022 -6.592937 ls 243 29/12/2006 -0.003141767 -5.772181 ls 251 15/01/2007 -0.002771753 -5.084049 ls 347 11/06/2007 -0.002837928 -5.102009 ls 520 19/06/2008 -0.003114010 -5.387808 ls 598 13/06/2008 -0.003027395 -5.143002 ls 613 04/07/2008 -0.003035068 -5.137530 ls 631 30/07/2008 -0.002988789 -5.037234 ls 635 05/08/2008 -0.003001055 -5.052994 ls 2286 09/04/2015 -0.008202488 -6.130741 tc 1901 17/09/2013 0.097493034 5.208012 tc 2477 18/01/2016 0.096324642 5.145597 ls 607 26/06/2008 0.022239276 28.623733 ao 1336 09/06/2011 -0.128187865 -5.110079 ao 1872 05/08/2013 -0.144642972 -5.766045 int. j. anal. appl. 17 (4) (2019) 545 table xii: joint model of arima (1, 1, 0) and outliers effect fitted to return series of unity bank estimate std. error z value pr(>|z|) ar1 0.22870458 0.01895846 12.0635 < 2.2e−16 ∗∗∗ io2293 0.00020692 0.02847462 0.0073 0.9942018 ao248 1.09943444 0.03077300 35.7272 < 2.2e−16 ∗∗∗ ao1906 -0.19318151 0.03242140 -5.9585 2.546e−09 ∗∗∗ ao2292 2.30319017 0.03189738 72.2062 < 2.2e−16 ∗∗∗ tc247 -0.00386092 0.03081394 -0.1253 0.9002878 tc1736 0.09934947 0.02489870 3.9901 6.603e−05 ∗∗∗ tc1745 0.11091559 0.02491948 4.4510 8.549e−06 ∗∗∗ tc1753 -0.12798775 0.02489883 -5.1403 2.743e−07 ∗∗∗ tc1762 0.10385821 0.02488517 4.1735 3.000e−05 ∗∗∗ tc2291 0.00423615 0.02733850 0.1550 0.8768592 tc2298 -0.12316459 0.02511656 -4.9037 9.404e−07 ∗∗∗ tc2304 -0.07679364 0.02506410 -3.0639 0.0021848∗∗ tc2446 -0.08214679 0.02500377 -3.2854 0.0010185∗∗ tc2458 0.09014905 0.02690186 3.3510 0.0008051∗∗∗ tc2460 0.07420557 0.02694465 2.7540 0.0058872∗∗ tc2467 -0.07462796 0.02693742 -2.7704 0.0055984∗∗ tc2469 -0.08905130 0.02691343 -3.3088 0.0009370∗∗∗ io1905 0.08527383 0.03084516 2.7646 0.0056997∗∗ ao1904 -0.19483838 0.03033146 -6.4236 1.331e−10∗∗∗ ls243 0.00112671 0.01580103 0.0713 0.9431541 ls251 0.00098712 0.01609793 0.0613 0.9511048 ls347 -0.00156994 0.00489397 -0.3208 0.7483691 ls520 -0.00603173 0.00523178 -1.1529 0.2489502 ls598 -0.02950821 0.01304279 -2.2624 0.0236717∗ ls613 -0.05489807 0.01706044 -3.2179 0.0012915∗∗ ls631 0.00951673 0.01947408 0.4887 0.6250634 ls635 -0.00408405 0.01769983 -0.2307 0.8175170 ls2286 -0.00206496 0.00221715 -0.9314 0.3516679 tc1901 0.11851208 0.02564415 4.6214 3.811e−06 ∗∗∗ tc2477 0.10312584 0.02499954 4.1251 3.706e−05 ∗∗∗ ls607 0.08270945 0.01887366 4.3823 1.174e−05 ∗∗∗ ao1336 -0.12237989 0.02901086 -4.2184 2.460e−05 ∗∗∗ ao1872 -0.12246391 0.02910831 -4.2072 2.586e−05 ∗∗∗ int. j. anal. appl. 17 (4) (2019) 546 the effects of outliers on the estimate of arima (1, 1, 0) model fitted to the return series of unity bank is similar to those of the first two banks only that the estimate of the model is reduced by 0.1501, as shown in table xiii. table xiii: effect of outliers on estimate of arima (1, 1, 0) model for return series unity bank model arima (1,1,0) (for outlier contaminated) joint arima (1,1,0) and outliers effects arima (1,1,0) (for outlier adjusted) bias introduced parameter 0.0786 0.2287 0.2287 −0.1501 aic −7588.08 −11206.23 −11272.23 standard error 0.0192 0.0190 0.0188 variance 0.00348 0.000885 0.000884 log likelihood 3795.04 5638.12 5638.12 4. conclusion in all, it is discovered that outliers introduced substantial biases in the estimates of the arima models of the returns series considered and the two methods employed are sufficient and adequate in handling outliers in such time series. meanwhile, to ensure efficiency of the estimated parameters of linear models, it is needful and commendable to account for the presence of outliers with preference given to modified iterative method. furthermore, the fact that volatility clustering exist in the return series calls for entertainment and modeling of heteroscedasticity in future studies. references [1] i. u. moffat and e. a. akpan, identification and modeling of outliers in a discrete-time stochastic series, amer. j. theor. appl. stat. 6 (2017), 191-197. [2] r. s. tsay, analysis of financial time series. (3rd ed.). new york: john wiley & sons inc., (2010), 97-140. [3] g.e.p. box, g. m. jenkins and g.c. reinsel, time series analysis: forecasting and control. (3rd ed.). new jersey: wiley and sons, (2008), 5-22. [4] c. chen and l. m. liu, joint estimation of model parameters and outlier effects in time series. j. amer. stat. assoc. 8 (1993), 84-297. int. j. anal. appl. 17 (4) (2019) 547 [5] w. w. s. wei, time series analysis univariate and multivariate methods. (2nd ed.). new york: adison westley, (2006), 33-59. [6] f. battaglia and l. orfei, outlier detection and estimation in nonlinear time series, j. time seri. anal. 26 (2002), 108-120. [7] a. b. abdullahi and h. r. bakari, modeling the nigeria stock market (shares)evidence from time series analysis, int. j. eng. sci. 3 (2014), 1-12. [8] e. j. ekpenyong and u. p. udoudo, short-term forecasting of nigeria inflation rates using seasonal arima model, sci. j. appl. math. stat. 5 (2016), 101-107. [9] o. m. olayiwola, a. a. amalare and s. o. adebesin, prediction of returns on all-share index of nigeria stock exchange, pac. j. sci. technol. 17 (2016), 114-119. [10] o. s. ajao, o. s. obafemi, and f. a. bolarinwa, modeling dollar-naira exchange rate in nigeria, nigerian stat. soc. 1 (2017), 191-198. [11] e. a. akpan and i. u. moffat, detection and modeling of asymmetric garch effects in a discrete-time series, int. j. stat. probab. 6 (2017), 111-119. [12] m. j. sanchez and d. pena, the identification of multiple outliers in arima models, commun. stat. theory methods, 32(6) (2013), 1265-128. [13] i. chang, g. c. tiao and c. chen, estimation of time series parameters in the presence of outliers, technometrics, 30 (1988), 193-204. international journal of analysis and applications issn 2291-8639 volume 2, number 2 (2013), 111-123 http://www.etamaths.com complex oscillation of solutions and their derivatives of non-homogenous linear differential equations in the unit disc zinelaâbidine latreuch and benharrat belaïdi∗ abstract. in this paper, we study the complex oscillation of solutions and their derivatives of the differential equation f′′ + a (z) f′ + b (z) f = f (z) , where a (z) , b (z) ( 6≡ 0) and f (z) (6≡ 0) are meromorphic functions of finite iterated p-order in the unit disc ∆ = {z : |z| < 1}. 1. introduction and main results throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the nevanlinna’s value distribution theory on the complex plane and in the unit disc ∆ = {z : |z| < 1} (see [11] , [12] , [15] , [16] , [18]). we need to give some definitions and discussions. firstly, let us give two definitions about the degree of small growth order of functions in ∆ as polynomials on the complex plane c. there are many types of definitions of small growth order of functions in ∆ (see [9, 10]) . definition 1.1 [9, 10] let f be a meromorphic function in ∆, and d (f) = lim sup r→1− t (r,f) log 1 1−r = b. 2010 mathematics subject classification. 34m10, 30d35. key words and phrases. linear differential equations, meromorphic functions, iterated p−exponent of convergence of the sequence of zeros, unit disc. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 111 112 latreuch and belaïdi if b < ∞, then we say that f is of finite b degree (or is non-admissible). if b = ∞, then we say that f is of infinite degree (or is admissible), both defined by characteristic function t(r,f). definition 1.2 [9, 10] let f be an analytic function in ∆, and dm (f) = lim sup r→1− log+ m (r,f) log 1 1−r = a < ∞ (or a = ∞) . then we say that f is a function of finite a degree (or of infinite degree) defined by maximum modulus function m(r,f) = max |z|=r |f (z)| . moreover, for f ⊂ [0, 1) , the upper and lower densities of f are defined by dens∆f = lim sup r→1− m (f ∩ [0,r)) m ([0,r)) , dens∆f = lim inf r→1− m (f ∩ [0,r)) m ([0,r)) respectively, where m (g) = ∫ g dt 1−t for g ⊂ [0, 1) . now we give the definitions of iterated order and growth index to classify generally the functions of fast growth in ∆ as those in c, see [4] , [14] , [15] . let us define inductively, for r ∈ [0, 1) , exp1 r = er and expp+1 r = exp(expp r), p ∈ n. we also define for all r sufficiently large in (0, 1) , log1 r = log r and logp+1 r = log(logp r),p ∈ n. moreover, we denote by exp0 r = r, log0 r = r, exp−1 r = log1 r, log−1 r = exp1 r. definition 1.3 [5] the iterated p-order of a meromorphic function f in ∆ is defined by ρp (f) = lim sup r→1− log+p t (r,f) log 1 1−r (p ≥ 1) . for an analytic function f in ∆, we also define ρm,p (f) = lim sup r→1− log+p+1 m (r,f) log 1 1−r (p ≥ 1) . complex oscillation 113 remark 1.1 it follows by m. tsuji in [18] that if f is an analytic function in ∆, then ρ1 (f) ≤ ρm,1 (f) ≤ ρ1 (f) + 1. however, it follows by proposition 2.2.2 in [15] that ρm,p (f) = ρp (f) (p ≥ 2) . definition 1.4 [5] the growth index of the iterated order of a meromorphic function f(z) in ∆ is defined by i (f) =   0, if f is non-admissible, min{p ∈ n,ρp (f) < ∞} , if f is admissible, ∞, if ρp (f) = ∞ for all p ∈ n. for an analytic functionf in ∆, we also define im (f) =   0, if f is non-admissible, min{p ∈ n,ρm,p (f) < ∞} , if f is admissible, ∞, if ρm,p (f) = ∞ for all p ∈ n. definition 1.5 [6, 7] let f be a meromorphic function in ∆. then the iterated p−exponent of convergence of the sequence of zeros of f (z) is defined by λp (f) = lim sup r→1− log+p n ( r, 1 f ) log 1 1−r , where n ( r, 1 f ) is the counting function of zeros of f (z) in {z ∈ c : |z| < r}. similarly, the iterated p-exponent of convergence of the sequence of distinct zeros of f (z) is defined by λp (f) = lim sup r→1− log+p n ( r, 1 f ) log 1 1−r , where n ( r, 1 f ) is the counting function of distinct zeros of f (z) in {z ∈ c : |z| < r}. 114 latreuch and belaïdi definition 1.6 [8] the growth index of the iterated convergence exponent of the sequence of zeros of f(z) in ∆ is defined by iλ (f) =   0, if n ( r, 1 f ) = o ( log 1 1−r ) , min{p ∈ n,λp (f) < ∞} , if some p ∈ n with λp (f) < ∞, ∞, if λp (f) = ∞ for all p ∈ n. the complex oscillation theory of solutions of linear differential equations in the complex plane c was started by bank and laine in 1982 ([1]). after their wellknown work, many important results have been obtained on the growth and the complex oscillation theory of solutions of linear differential equations in the unit disc ∆ = {z : |z| < 1} , (see [2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 16, 20]) . recently, the second author (see, [2]) extended some results of [6, 20] to the case of higher order linear differential equations with analytic coefficients. he investigated the relation between solutions and their derivatives of the differential equation (1.1) f(k) + a (z) f = 0 and analytic functions of finite iterated p-order, and obtained the following results: theorem a [2] let h be a set of complex numbers satisfying dens∆ {|z| : z ∈ h ⊂ ∆} > 0, and let a (z) 6≡ 0 be an analytic function in ∆ such that ρm,p (a) = σ < ∞ and for real number α > 0, we have for all ε > 0 sufficiently small, |a (z)| ≥ expp { α ( 1 1 −|z| )σ−ε} as |z|→ 1− for z ∈ h. if ϕ (z) is an analytic function in ∆ such that ϕ(k−j) (z) 6≡ 0 (j = 0, · · · ,k) with finite iterated p−order ρp (ϕ) < ∞, then every solution f 6≡ 0 of (1.1) , satisfies λp ( f(j) −ϕ ) = λp ( f(j) −ϕ ) = ρp (f) = ∞ (j = 0, · · · ,k) , λp+1 ( f(j) −ϕ ) = λp+1 ( f(j) −ϕ ) = ρp+1 (f) = ρm,p (a) (j = 0, · · · ,k) . complex oscillation 115 theorem b [2] let h be a set of complex numbers satisfying dens∆ {|z| : z ∈ h ⊂ ∆} > 0, and let a (z) 6≡ 0 be an analytic function in ∆ such that ρp (a) = σ < ∞ and for real number α > 0, we have for all ε > 0 sufficiently small, |a (z)| ≥ expp−1 { α ( 1 1 −|z| )σ−ε} as |z|→ 1− for z ∈ h. if ϕ (z) is an analytic function in ∆ such that ϕ(k−j) (z) 6≡ 0 (j = 0, · · · ,k) with finite iterated p−order ρp (ϕ) < ∞, then every solution f 6≡ 0 of (1.1) , satisfies λp ( f(j) −ϕ ) = λp ( f(j) −ϕ ) = ρp (f) = ∞ (j = 0, · · · ,k) , σ ≤ λp+1 ( f(j) −ϕ ) = λp+1 ( f(j) −ϕ ) = ρp+1 (f) ≤ ρm,p (a) (j = 0, · · · ,k) . in this paper we consider the oscillation problem of solutions and their derivatives of second order non-homogenous linear differential equation (1.2) f′′ + a (z) f′ + b (z) f = f (z) , where a (z) , b (z) 6≡ 0 and f (z) 6≡ 0 are meromorphic functions of finite iterated p-order in ∆. it is a natural to ask what about the exponent of convergence of zeros of f(j) (z) (j = 0, 1, 2, · · ·) , where f is a solution of (1.2) . for some related papers in the complex plane on the usual order see, [17, 19] . the main purpose of this paper is give an answer to this question. before we state our results we need to define the following notations (1.3) aj (z) = aj−1 (z) − b′j−1 (z) bj−1 (z) for j = 1, 2, 3, · · · , (1.4) bj (z) = a ′ j−1 (z) −aj−1 (z) b′j−1 (z) bj−1 (z) + bj−1 (z) for j = 1, 2, 3, · · · and (1.5) fj (z) = f ′ j−1 (z) −fj−1 (z) b′j−1 (z) bj−1 (z) for j = 1, 2, 3, · · · , where a0 (z) = a (z) , b0 (z) = b (z) and f0 (z) = f (z) . we obtain the following results. 116 latreuch and belaïdi theorem 1.1 let a (z) , b (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions of finite iterated p−order in ∆ such that bj (z) 6≡ 0 and fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) . if f is a meromorphic solution in ∆ of (1.2) with ρp (f) = ∞ and ρp+1 (f) = ρ, then f satisfies λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ (j = 0, 1, 2, · · ·) and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ (j = 0, 1, 2, · · ·) . theorem 1.2 let a (z) , b (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions in ∆ with finite iterated p−order such that bj (z) 6≡ 0 and fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) . if f is a meromorphic solution in ∆ of (1.2) with ρp (f) > max{ρp (a) ,ρp (b) ,ρp (f)} , then λp ( f(j) ) = λp ( f(j) ) = ρp (f) (j = 0, 1, 2, · · ·) . remark 1.2 in theorems 1.1, 1.2, the conditions bj (z) 6≡ 0 and fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) are necessary. for example f (z) = exp ( 1 1−z )2 − 1 satisfies (1.2) where a (z) = −3 1−z , b (z) = − 4 (1−z)6 , f (z) = 4 (1−z)6 and ρ1 (f) = 1 > max{ρ1 (a) ,ρ1 (b) ,ρ1 (f)} = 0. on the other hand, we have a1 = a− b′ b = − 9 1 −z , b1 = a ′ −a b′ b + b = 15 (1 −z)2 − 4 (1 −z)6 , f1 = f ′ −f b′ b ≡ 0, and λ1 (f) = 1 > λ1 (f ′) = 0. here, we give some sufficient conditions on the coefficients which guarantee bj (z) 6≡ 0 and fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) , and we obtain: theorem 1.3 let a (z) , b (z) 6≡ 0 and f (z) 6≡ 0 be analytic functions in ∆ with finite iterated p−order such that β = ρp (b) > max{ρp (a) ,ρp (f)} . then all nontrivial solutions of (1.2) satisfy ρp (b) ≤ λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) ≤ ρm,p (b) (j = 0, 1, 2, · · ·) complex oscillation 117 with at most one possible exceptional solution f0 such that ρp+1 (f0) < ρp (b) . in the next, we set σp (f) = lim sup r→1− logp m (r,f) log 1 1−r . theorem 1.4 let a (z) , b (z) 6≡ 0 and f (z) 6≡ 0 be meromorphic functions in ∆ with finite iterated p-order such that σp (b) > max{σp (a) ,σp (f)} . if f is a meromorphic solution in ∆ of (1.2) with ρp (f) = ∞ and ρp+1 (f) = ρ, then f satisfies λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ (j = 0, 1, 2, · · ·) and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ (j = 0, 1, 2, · · ·) . 2. some lemmas lemma 2.1 [2] let f be a meromorphic function in the unit disc for which i (f) = p ≥ 1 and ρp (f) = β < ∞, and let k ∈ n. then for any ε > 0, m ( r, f(k) f ) = o ( expp−2 ( 1 1 −r )β+ε) for all r outside a set e1 ⊂ [0, 1) with ∫ e1 dr 1−r < ∞. lemma 2.2 [7] let a0,a1, · · · ,ak−1,f 6≡ 0 be meromorphic functions in ∆, and let f be a meromorphic solution of the differential equation (2.1) f(k) + ak−1 (z) f (k−1) + · · · + a0 (z) f = f (z) such that i (f) = p (0 < p < ∞) . if either max{i (aj) (j = 0, 1, · · · ,k − 1) , i (f)} < p or max{ρp (aj) (j = 0, 1, · · · ,k − 1) ,ρp (f)} < ρp (f) , 118 latreuch and belaïdi then iλ (f) = iλ (f) = i (f) = p and λp (f) = λp (f) = ρp (f) . using the same arguments as in the proof of lemma 2.2 (see, the proof of lemma 2.5 in [7]), we easily obtain the following lemma. lemma 2.3 let a0, a1, · · · , ak−1, f 6≡ 0 be finite iterated p−order meromorphic functions in the unit disc ∆. if f is a meromorphic solution with ρp (f) = ∞ and ρp+1 (f) = ρ < ∞ of equation (2.1) , then λp (f) = λp (f) = ρp (f) = ∞ and λp+1 (f) = λp+1 (f) = ρp+1 (f) = ρ. lemma 2.4 [7] let p ∈ n, and assume that the coefficients a0, · · · ,ak−1 and f 6≡ 0 are analytic in ∆ and ρp (aj) < ρp (a0) for all j = 1, · · · ,k − 1. let αm := max{ρm,p (aj) : j = 0, · · · ,k − 1} . (i) if ρm,p+1 (f) > αm, then all solutions f of (2.1) satisfy ρm,p+1 (f) = ρm,p+1 (f) . (ii) if ρm,p+1 (f) < αm, then all solutions f of (2.1) satisfy ρp (a0) ≤ ρm,p+1 (f) ≤ αm, with at most one exeption f0 satisfying ρm,p+1 (f0) < ρp (a0) . (iii) if ρm,p+1 (f) < ρp (a0) , then all solutions f of (2.1) satisfy ρp (a0) ≤ λp+1 (f) = λp+1 (f) = ρm,p+1 (f) ≤ αm, with at most one exception f0 satisfying ρm,p+1 (f0) < ρp (a0) . 3. proof of theorems proof of theorem 1.1. for the proof, we use the principle of mathematical induction. since b (z) 6≡ 0 and f (z) 6≡ 0, then by using lemma 2.3 we have λp (f) = λp (f) = ρp (f) = ∞ and λp+1 (f) = λp+1 (f) = ρp+1(f) = ρ. complex oscillation 119 dividing both sides of (1.2) by b, we obtain (3.1) 1 b f′′ + a b f′ + f = f b . differentiating both sides of equation (3.1) , we have (3.2) 1 b f(3) + (( 1 b )′ + a b ) f′′ + (( a b )′ + 1 ) f′ = ( f b )′ . multiplying now (3.2) by b, we get (3.3) f(3) + a1f ′′ + b1f ′ = f1, where a1 = a− b′ b , b1 = a ′ −a b′ b + b and f1 = f ′ −f b′ b . since b1 6≡ 0 and f1 6≡ 0 are meromorphic functions with finite iterated p-order, then by using lemma 2.3 we obtain λp (f ′) = λp (f ′) = ρp (f) = ∞ and λp+1 (f ′) = λp+1 (f ′) = ρp+1(f) = ρ. dividing now both sides of (3.3) by b1, we obtain (3.4) 1 b1 f(3) + a1 b1 f′′ + f′ = f1 b1 . differentiating both sides of equation (3.4) and multplying by b1, we get (3.5) f(4) + a2f (3) + b2f ′′ = f2, where a2,b2 6≡ 0 and f2 6≡ 0 are meromorphic functions defined in (1.3) − (1.5) . by using lemma 2.3 we obtain λp (f ′′) = λp (f ′′) = ρp (f) = ∞ and λp+1 (f ′′) = λp+1 (f ′′) = ρp+1 (f) = ρ. 120 latreuch and belaïdi suppose now that (3.6) λp ( f(k) ) = λp ( f(k) ) = ρp (f) = ∞, λp+1 ( f(k) ) = λp+1 ( f(k) ) = ρp+1 (f) = ρ for all k = 0, 1, 2, · · · ,j − 1, and we prove that (3.6) is true for k = j. by the same procedure as before, we can obtain f(j+2) + ajf (j+1) + bjf (j) = fj, where aj,bj 6≡ 0 and fj 6≡ 0 are meromorphic functions defined in (1.3) − (1.5) . by using lemma 2.3 we obtain λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ. the proof of theorem 1.1 is complete. proof of theorem 1.2. by a similar reasoning as theorem 1.1 and by using lemma 2.2, we obtain λp ( f(j) ) = λp ( f(j) ) = ρp (f) (j = 0, 1, 2, · · ·) . proof of theorem 1.3. by lemma 2.4 (iii), all nontrivial solutions of (1.2) satisfy ρp (b) ≤ λp+1 (f) = λp+1 (f) = ρp+1 (f) ≤ ρm,p (b) with at most one exceptional solution f0 such that ρp (b) > ρp+1 (f0). by using (1.3) and lemma 2.1 we have m (r,aj) ≤ m (r,aj−1) + o ( expp−2 ( 1 1 −r )β+ε) (β = ρp (bj−1)) outside a set e1 ⊂ [0, 1) with ∫ e1 dr 1−r < ∞, for all j = 1, 2, 3, · · · , which we can write as (3.7) m (r,aj) ≤ m (r,a) + o ( expp−2 ( 1 1 −r )β+ε) (j = 1, 2, 3, · · ·) . on the other hand, we have from (1.4) complex oscillation 121 bj = aj−1 ( a′j−1 aj−1 − b′j−1 bj−1 ) + bj−1 = aj−1 ( a′j−1 aj−1 − b′j−1 bj−1 ) + aj−2 ( a′j−2 aj−2 − b′j−2 bj−2 ) + bj−2 (3.8) = j−1∑ k=0 ak ( a′k ak − b′k bk ) + b. now we prove that bj 6≡ 0 for all j = 1, 2, 3, · · · . for that we suppose there exists j ∈ n such that bj = 0. by (3.7) and (3.8) we have t (r,b) = m (r,b) ≤ j−1∑ k=0 m (r,ak) + o ( expp−2 ( 1 1 −r )β+ε) ≤ jm (r,a) + o ( expp−2 ( 1 1 −r )β+ε) (3.9) = jt (r,a) + o ( expp−2 ( 1 1 −r )β+ε) which implies the contradiction ρp (b) ≤ ρp (a) . hence bj 6≡ 0 for all j = 1, 2, 3, · · · . suppose now there exists j ∈ n such that fj = 0. then, from (1.5) f ′j−1 (z) −fj−1 (z) b′j−1 (z) bj−1 (z) = 0 which implies (3.10) fj−1 (z) = cbj−1 (z) , where c ∈ c∗. by (3.8) and (3.10) we have (3.11) 1 c fj−1 = j−2∑ k=0 ak ( a′k ak − b′k bk ) + b. on the other hand, we have from (1.5) (3.12) m (r,fj) ≤ m (r,f) + o ( expp−2 ( 1 1 −r )β+ε) (j = 1, 2, 3, · · ·) . 122 latreuch and belaïdi by (3.11) , (3.12) and lemma 2.1, we have t (r,b) = m (r,b) ≤ j−2∑ k=0 m (r,ak) + m (r,fj−1) + o ( expp−2 ( 1 1 −r )β+ε) ≤ (j − 1) m (r,a) + m (r,f) + o ( expp−2 ( 1 1 −r )β+ε) = (j − 1) t (r,a) + t (r,f) + o ( expp−2 ( 1 1 −r )β+ε) which implies the contradiction ρp (b) ≤ max{ρp (a) ,ρp (f)} . since bj 6≡ 0, fj 6≡ 0 (j = 1, 2, 3, · · ·) , then by applying theorem 1.1 and lemma 2.4 (iii) we have ρp (b) ≤ λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) ≤ ρm,p (b) (j = 0, 1, 2, · · ·) with at most one exceptional solution f0 such that ρp (b) > ρp+1 (f0) . 4. proof of theorem 1.4 using the same reasoning as theorem 1.1, we obtain theorem 1.4. references [1] s. bank and i. laine, on the oscillation theory of f′′ + a (z) f = 0. where a is entire, trans. amer. math. soc. 273 (1982), no. 1, 351–363. [2] b. beläıdi, oscillation of fast growing solutions of linear differential equations in the unit disc, acta univ. sapientiae math. 2 (2010), no. 1, 25–38. [3] b. beläıdi, a. el farissi, fixed points and iterated order of differential polynomial generated by solutions of linear differential equations in the unit disc, j. adv. res. pure math. 3 (2011), no. 1, 161–172. [4] l. g. bernal, on growth k-order of solutions of a complex homogeneous linear differential equation, proc. amer. math. soc. 101 (1987), no. 2, 317–322. [5] t. b. cao and h. x. yi, the growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, j. math. anal. appl. 319 (2006), no. 1, 278–294. [6] t. b. cao, the growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, j. math. anal. appl. 352 (2009), no. 2, 739-748. [7] t. b. cao and z. s. deng, solutions of non-homogeneous linear differential equations in the unit disc, ann. polo. math. 97(2010), no. 1, 51-61. [8] t. b. cao, c. x. zhu, k. liu, on the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc, j. math. anal. appl. 374 (2011), no. 1, 272–281. complex oscillation 123 [9] z. x. chen and k. h. shon, the growth of solutions of differential equations with coefficients of small growth in the disc, j. math. anal. appl. 297 (2004), no. 1, 285–304. [10] i. e. chyzhykov, g. g. gundersen and j. heittokangas, linear differential equations and logarithmic derivative estimates, proc. london math. soc. (3) 86 (2003), no. 3, 735–754. [11] w. k. hayman, meromorphic functions, oxford mathematical monographs clarendon press, oxford, 1964. [12] j. heittokangas, on complex differential equations in the unit disc, ann. acad. sci. fenn. math. diss. 122 (2000), 1-54. [13] j. heittokangas, r. korhonen and j. rättyä, fast growing solutions of linear differential equations in the unit disc, results math. 49 (2006), no. 3-4, 265–278. [14] l. kinnunen, linear differential equations with solutions of finite iterated order, southeast asian bull. math. 22 (1998), no. 4, 385-405. [15] i. laine, nevanlinna theory and complex differential equations, de gruyter studies in mathematics, 15. walter de gruyter & co., berlin-new york, 1993. [16] i. laine, complex differential equations, handbook of differential equations: ordinary differential equations. vol. iv, 269–363, handb. differ. equ., elsevier/north-holland, amsterdam, 2008. [17] z. latreuch and b. beläıdi, on the zeros of solutions and their derivatives of second order non-homogenous linear differential equations, submitted. [18] m. tsuji, potential theory in modern function theory, chelsea, new york, (1975), reprint of the 1959 edition. [19] j. tu, h. y. xu and c. y. zhang, on the zeros of solutions of any order of derivative of second order linear differential equations taking small functions, electron. j. qual. theory differ. equ. 2011, no. 23, 1-17. [20] g. zhang, a. chen, fixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc, electron j. qual. theory differ. equ., 2009, no. 48, 1-9. department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem-(algeria) ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 195-204 http://www.etamaths.com fixed point theorem of modified s-iteration process for ciric quasi contractive operator in cat(0) spaces g. s. saluja abstract. the aim of this paper is to study the strong convergence of modified s-iteration process for ciric quasi contractive operator in the framework of cat (0) spaces. also we give an application of our result with supporting example. our result improves and extends some corresponding previous result from the existing literature (see, e.g., [3, 29] and many others). 1. introduction and preliminaries a metric space x is a cat(0) space if it is geodesically connected and if every geodesic triangle in x is at least as ’thin’ as its comparison triangle in the euclidean plane. it is well known that any complete, simply connected riemannian manifold having nonpositive sectional curvature is a cat(0) space. other examples include pre-hilbert spaces (see [6]), r-trees (see [22]), euclidean buildings (see [7]), the complex hilbert ball with a hyperbolic metric (see [15]), and many others. for a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to bridson and haefliger [6]. fixed point theory in a cat(0) space was first studied by kirk (see [23, 24]). he showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete cat(0) space always has a fixed point. since, then the fixed point theory for single-valued and multi-valued mappings in cat(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [2], [9], [12]-[14], [16], [20]-[21], [25]-[26], [28], [30]-[31] and references therein). it is worth mentioning that the results in cat(0) spaces can be applied to any cat(k) space with k ≤ 0 since any cat(k) space is a cat(k′) space for every k′ ≥ k (see,e.g., [6]). let (x,d) be a metric space. a geodesic path joining x ∈ x to y ∈ x (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ r to x such that c(0) = x, c(l) = y and d(c(t),c(t′)) = |t − t′| for all t,t′ ∈ [0, l]. in particular, c is an isometry, and d(x,y) = l. the image α of c is called a geodesic (or metric) segment joining x and y. we say x is (i) a geodesic space if any two points of x are joined by a geodesic and (ii) a uniquely geodesic if there is exactly 2010 mathematics subject classification. 54h25, 54e40. key words and phrases. ciric quasi contractive operator; modified s-iteration process; fixed point; strong convergence; cat(0) space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 195 196 saluja one geodesic joining x and y for each x,y ∈ x, which we will denoted by [x,y], called the segment joining x to y. a geodesic triangle 4(x1,x2,x3) in a geodesic metric space (x,d) consists of three points in x (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). a comparison triangle for geodesic triangle 4(x1,x2,x3) in (x,d) is a triangle 4(x1,x2,x3) := 4(x1,x2,x3) in r2 such that dr2 (xi,xj) = d(xi,xj) for i,j ∈{1, 2, 3}. such a triangle always exists (see [6]). cat(0) space a geodesic metric space is said to be a cat(0) space if all geodesic triangles of appropriate size satisfy the following cat(0) comparison axiom. let 4 be a geodesic triangle in x, and let 4 ⊂ r2 be a comparison triangle for 4. then 4 is said to satisfy the cat(0) inequality if for all x,y ∈ 4 and all comparison points x,y ∈4, d(x,y) ≤ dr2 (x,y).(1.1) complete cat(0) spaces are often called hadamard spaces (see [19]). if x,y1,y2 are points of a cat(0) space and y0 is the midpoint of the segment [y1,y2] which we will denote by (y1 ⊕y2)/2, then the cat(0) inequality implies d2 ( x, y1 ⊕y2 2 ) ≤ 1 2 d2(x,y1) + 1 2 d2(x,y2) − 1 4 d2(y1,y2).(1.2) the inequality (1.2) is the (cn) inequality of bruhat and tits [8]. the above inequality was extended in [13] as d2(z,αx⊕ (1 −α)y) ≤ αd2(z,x) + (1 −α)d2(z,y) −α(1 −α)d2(x,y)(1.3) for any α ∈ [0, 1] and x,y,z ∈ x. let us recall that a geodesic metric space is a cat(0) space if and only if it satisfies the (cn) inequality (see [[6], p.163]). moreover, if x is a cat(0) metric space and x,y ∈ x, then for any α ∈ [0, 1], there exists a unique point αx⊕ (1 −α)y ∈ [x,y] such that d(z,αx⊕ (1 −α)y) ≤ αd(z,x) + (1 −α)d(z,y),(1.4) for any z ∈ x and [x,y] = {αx⊕ (1 −α)y : α ∈ [0, 1]}. a subset c of a cat(0) space x is convex if for any x,y ∈ c, we have [x,y] ⊂ c. the mann iteration process [27] is defined by the sequence {xn}, (1.5) { x1 ∈ c, xn+1 = (1 −αn)xn + αntxn, n ≥ 1, where {αn} is a sequence in (0,1). fixed point theorem of modified s-iteration process 197 further, the ishikawa iteration process [17] is defined by the sequence {xn}, (1.6)   x1 ∈ c, xn+1 = (1 −αn)xn + αntyn, yn = (1 −βn)xn + βntxn, n ≥ 1, where {αn} and {βn} are the sequences in (0,1). this iteration process reduces to the mann iteration process when βn = 0 for all n ≥ 1. in 2007, agarwal, o’regan and sahu [1] introduced the s-iteration process in a banach space, (1.7)   x1 ∈ c, xn+1 = (1 −αn)txn + αntyn, yn = (1 −βn)xn + βntxn, n ≥ 1, where {αn} and {βn} are the sequences in (0,1). note that (1.3) is independent of (1.2) (and hence (1.1)). they showed that their process is independent of those of mann and ishikawa and converges faster than both of these (see [[1], proposition 3.1]). we now modify (1.7) in a cat(0) space as follows. let c be a nonempty closed convex subset of a complete cat(0) space x and t : c → c be a mapping. suppose that {xn} is a sequence generated iteratively by (1.8)   x1 ∈ c, xn+1 = (1 −αn)txn ⊕αntyn, yn = (1 −βn)xn ⊕βntxn, n ≥ 1, where and throughout the paper {αn}, {βn} are the sequences such that 0 ≤ αn, βn ≤ 1 for all n ≥ 1. we recall the following definitions in a metric space (x,d). a mapping t : x → x is called an a-contraction if d(tx,ty) ≤ ad(x,y) for all x, y ∈ x,(1.9) where a ∈ (0, 1). the mapping t is called kannan mapping [18] if there exists b ∈ (0, 1 2 ) such that d(tx,ty) ≤ b [d(x,tx) + d(y,ty)] for all x, y ∈ x.(1.10) the mapping t is called chatterjea mapping [10] if there exists c ∈ (0, 1 2 ) such that d(tx,ty) ≤ c [d(x,ty) + d(y,tx)] for all x, y ∈ x.(1.11) in 1972, zamfirescu [32] obtained the following interesting fixed point theorem. 198 saluja theorem z. let (x,d) be a complete metric space and t : x → x a mapping for which there exists the real number a, b and c satisfying a ∈ (0, 1), b, c ∈ (0, 1 2 ) such that for any pair x, y ∈ x, at least one of the following conditions holds: (z1) d(tx,ty) ≤ ad(x,y), (z2) d(tx,ty) ≤ b [d(x,tx) + d(y,ty)], (z3) d(tx,ty) ≤ c [d(x,ty) + d(y,tx)]. then t has a unique fixed point p and the picard iteration {xn}∞n=0 defined by xn+1 = txn, n = 0, 1, 2, . . . converges to p for any arbitrary but fixed x0 ∈ x. the conditions (z1) − (z3) can be written in the following equivalent form d(tx,ty) ≤ h max { d(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty) + d(y,tx) 2 } ,(1.12) ∀x, y ∈ x and 0 < h < 1, has been obtained by ciric [11] in 1974. a mapping satisfying (1.12) is called ciric quasi-contraction. it is obvious that each of the conditions (z1) − (z3) implies (1.12). an operator t satisfying the contractive conditions (z1) − (z3) in the theorem z is called z-operator. in 2000, berinde [3] introduced a new class of operators on a normed space e satisfying ‖tx−ty‖≤ δ‖x−y‖ + l‖tx−x‖ , (∗) for any x, y ∈ e, 0 ≤ δ < 1 and l ≥ 0. he proved that this class is wider than the class of zamfirescu operators and used the mann iteration process to approximate fixed points of this class of operators in a normed space given in the form of following theorem. theorem b. let c be a nonempty closed convex subset of a normed space e. let t : c → c be an operator satisfying (∗). let {xn}∞n=0 be defined by: for x1 = x ∈ c, the sequence {xn}∞n=0 given by xn+1 = (1 − bn)xn + bntxn, n ≥ 0, where {bn} is a sequence in [0,1]. if f(t) 6= ∅ and ∑∞ n=1 bn = ∞, then {xn} ∞ n=0 converges strongly to the unique fixed point of t . in this paper, inspired and motivated by [1, 32], we study s-iteration process and establish strong convergence theorem to approximate the fixed point for ciric fixed point theorem of modified s-iteration process 199 quasi contractive operator in the framework of cat(0) spaces. we need the following useful lemmas to prove our main result in this paper. lemma 1.1. (see [28]) let x be a cat(0) space. (i) for x, y ∈ x and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = td(x, y) and d(y, z) = (1 − t) d(x, y). (a) we use the notation (1 − t)x⊕ ty for the unique point z satisfying (a). (ii) for x, y ∈ x and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z) ≤ (1 − t)d(x,z) + td(y,z). lemma 1.2. (see [4]) let {pn}∞n=0, {qn}∞n=0, {rn}∞n=0 be sequences of nonnegative numbers satisfying the following condition: pn+1 ≤ (1 −sn)pn + qn + rn, ∀n ≥ 0, where {sn}∞n=0 ⊂ [0, 1]. if ∑∞ n=0 sn = ∞, limn→∞ qn = o(sn) and ∑∞ n=0 rn < ∞, then limn→∞ pn = 0. 2. strong convergence theorem in cat(0) space in this section, we establish strong convergence result of modified s-iteration process to approximate a fixed point for ciric quasi contractive operator in the framework of cat(0) spaces. theorem 2.1. let c be a nonempty closed convex subset of a complete cat(0) space x and let t : c → c be an operator satisfying the condition (1.12). let {xn} be defined by the iteration scheme (1.8). if ∑∞ n=0 αnβn = ∞, then {xn} converges strongly to the unique fixed point of t . proof. by theorem z, we know that t has a unique fixed point in c, say u. consider x, y ∈ c. since t is a operator satisfying (1.12), then if d(tx,ty) ≤ h 2 [d(x,tx) + d(y,ty)] ≤ h 2 [d(x,tx) + d(y,x) + d(x,tx) + d(tx,ty)], implies ( 1 − h 2 ) d(tx,ty) ≤ h 2 d(x,y) + hd(x,tx), which yields (using the fact that 0 < h < 1) d(tx,ty) ≤ ( h/2 1 −h/2 ) d(x,y) + ( h 1 −h/2 ) d(x,tx).(2.1) 200 saluja if d(tx,ty) ≤ h 2 [d(x,ty) + d(y,tx)] ≤ h 2 [d(x,tx) + d(tx,ty) + d(y,x) + d(x,tx)], implies ( 1 − h 2 ) d(tx,ty) ≤ h 2 d(x,y) + hd(x,tx), which also yields (using the fact that 0 < h < 1) d(tx,ty) ≤ ( h/2 1 −h/2 ) d(x,y) + ( h 1 −h/2 ) d(x,tx).(2.2) denote δ = max { h, h/2 1 −h/2 } = h, l = max { h 1 −h/2 , h 1 −h/2 } = h 1 −h/2 . thus, in all cases, d(tx,ty) ≤ δ d(x,y) + ld(x,tx) = hd(x,y) + ( h 1 −h/2 ) d(x,tx).(2.3) holds for all x, y ∈ c. also from (1.12) with y = u = tu, we have d(tx,u) ≤ h max { d(x,u), d(x,tx) 2 , d(x,u) + d(u,tx) 2 } ≤ h max { d(x,u), d(x,tx) 2 , d(x,u) + d(u,tx) 2 } ≤ h max { d(x,u), d(x,u) + d(u,tx) 2 , d(x,u) + d(u,tx) 2 } . (2.4) since for non-negative real numbers a and b, we have a + b 2 ≤ max{a, b}.(2.5) using (2.5) in (2.4), we have d(tx,u) ≤ hd(x,u).(2.6) now (2.6) gives d(txn,u) ≤ hd(xn,u)(2.7) and d(tyn,u) ≤ hd(yn,u).(2.8) fixed point theorem of modified s-iteration process 201 using (1.8), (2.8) and lemma 1.1(ii), we have d(yn,u) = d((1 −βn)xn ⊕βntxn,u) ≤ (1 −βn)d(xn,u) + βnd(txn,u) ≤ (1 −βn)d(xn,u) + hβnd(xn,u) = [1 − (1 −h)βn]d(xn,u).(2.9) now using (1.8), (2.7), (2.9) and lemma 1.1(ii), we have d(xn+1,u) = d((1 −αn)txn ⊕ααntyn,u) ≤ (1 −αn)d(txn,u) + αnd(tyn,u) ≤ (1 −αn)hd(xn,u) + hαnd(yn,u) ≤ (1 −αn)hd(xn,u) + hαn[1 − (1 −h)βn]d(xn,u) = [1 − (1 −h)αnβn]hd(xn,u) ≤ [1 − (1 −h)αnβn]d(xn,u) = [1 −an]d(xn,u),(2.10) where an = (1 −h)αnβn. since 0 < h < 1; αn, βn ∈ [0, 1] and by assumption of the theorem ∑∞ n=0 αnβn = ∞, it follows that ∑∞ n=0 an = ∞. hence, by lemma 1.2, we get that limn→∞ d(xn,u) = 0. therefore {xn} converges strongly to a fixed point of t . to show uniqueness of the fixed point u, assume that u1, u2 ∈ f(t) and u1 6= u2. applying (1.12) and using the fact that 0 < h < 1, we obtain d(u1,u2) = d(tu1,tu2) ≤ h max { d(u1,u2), d(u1,tu1) + d(u2,tu2) 2 , d(u1,tu2) + d(u2,tu1) 2 } = h max { d(u1,u2), d(u1,u1) + d(u2,u2) 2 , d(u1,u2) + d(u2,u1) 2 } = h max { d(u1,u2), 0, d(u1,u2) } ≤ hd(u1,u2) < d(u1,u2), since 0 < h < 1, which is a contradiction. therefore u1 = u2. thus {xn} converges strongly to the unique fixed point of t . this completes the proof. the contraction condition (1.9) makes t continuous function on x while this is not the case with contractive conditions (1.10), (1.11) and (2.3). the contractive conditions (1.10) and (1.11) both included in the class of zamfirescu operators and so their convergence theorems for modified s-iteration process 202 saluja are obtained in theorem 2.1 in the setting of cat(0) space. remark 2.1. our result extends the corresponding result of [29] to the case of modified s-iteration process and from uniformly convex banach space to the setting of cat(0) spaces. remark 2.2. theorem 2.1 also extends theorem b to the case of modified siteration process and from normed space to the setting of cat(0) spaces. 3. application to contraction of integral type theorem 3.1. let c be a nonempty closed convex subset of a complete cat(0) space x and let t : c → c be an operator satisfying the following condition: ∫ d(t x,t y) 0 µ(t)dt ≤ h ∫ max{d(x,y), d(x,t x)+d(y,t y) 2 , d(x,t y)+d(y,t x) 2 } 0 µ(t)dt (3.1) for all x, y ∈ x and 0 < h < 1, where µ: [0, +∞) → [0, +∞) is a lebesgueintegrable mapping which is summable (i.e. with finite integral) on each compact subset of [0, +∞), nonnegative, and such that for each ε > 0, ∫ ε 0 µ(t)dt > 0. let {xn} be defined by the iteration process (1.8). if ∑∞ n=0 αnβn = ∞, then {xn} converges strongly to the unique fixed point of t . proof. the proof of theorem 3.1 follows from theorem 2.1 by taking µ(t) = 1 over [0, +∞) since the contractive condition of integral type transforms into a general contractive condition (1.12) not involving integrals. this completes the proof. example 3.1. let x = {0, 1, 2, 3, 4} and d be the usual metric of reals. let t : x → x be given by { tx = 3, if x = 0 = 2, otherwise. again let µ: [0, +∞) → [0, +∞) be given by µ(t) = 1 for all t ∈ [0, +∞). then µ: [0, +∞) → [0, +∞) is a lebesgue-integrable mapping which is summable (i.e. with finite integral) on each compact subset of [0, +∞), nonnegative, and such that for each ε > 0, ∫ ε 0 µ(t)dt > 0. let us take x = 0, y = 1. then from condition (3.1), we have 1 = ∫ d(t x,t y) 0 µ(t)dt ≤ h ∫ max{d(x,y), d(x,t x)+d(y,t y) 2 , d(x,t y)+d(y,t x) 2 } 0 µ(t)dt = h max{1, 2, 2} which implies h ≥ 1 2 . now if we take 0 < h < 1, then condition (3.1) is satisfied and 2 is of course a unique fixed point of t. the following corollary is a special case of theorem 3.1. fixed point theorem of modified s-iteration process 203 corollary 3.1. let c be a nonempty closed convex subset of a complete cat(0) space x and let t : c → c be an operator satisfying the following condition:∫ d(t x,t y) 0 µ(t)dt ≤ h ∫ d(x,y) 0 µ(t)dt(3.2) for all x, y ∈ x and h ∈ (0, 1), where µ: [0, +∞) → [0, +∞) is a lebesgueintegrable mapping which is summable (i.e. with finite integral) on each compact subset of [0, +∞), nonnegative, and such that for each ε > 0, ∫ ε 0 µ(t)dt > 0. let {xn} be defined by the iteration process (1.8). if ∑∞ n=0 αnβn = ∞, then {xn} converges strongly to the unique fixed point of t . condition (3.2) is called branciari [5] contractive condition of integral type. putting µ(t) = 1 in the condition (3.2), we get banach contraction condition. proof of corollary 3.1. the proof of corollary 3.1 immediately follows from theorem 2.1 by taking µ(t) = 1 over [0, +∞) and max { d(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty) + d(y,tx) 2 } = d(x,y) since the contractive condition of integral type transforms into a general contractive condition (1.9) not involving integrals. this completes the proof. example 3.2. let x be the real line with the usual metric d and suppose c = [0, 1]. define t : c → c by tx = x+1 2 for all x,y ∈ c. obviously t is selfmapping with a unique fixed point 1. again let µ: [0, +∞) → [0, +∞) be given by µ(t) = 1 for all t ∈ [0, +∞). then µ: [0, +∞) → [0, +∞) is a lebesgue-integrable mapping which is summable (i.e. with finite integral) on each compact subset of [0, +∞), nonnegative, and such that for each ε > 0, ∫ ε 0 µ(t)dt > 0. if x,y ∈ [0, 1], then we have d(tx,ty) = ∣∣∣x−y 2 ∣∣∣. let us take x = 0, y = 1. then from condition (3.2), we have 1 2 = ∫ d(t x,t y) 0 µ(t)dt ≤ h.1 = h ∫ d(x,y) 0 µ(t)dt which implies h ≥ 1 2 . now if we take 0 < h < 1, then condition (3.2) is satisfied and 1 is of course a unique fixed point of t. references [1] r.p. agarwal, d. o’regan, d.r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, j. nonlinear convex anal. 8(1) (2007), 61-79. [2] a. abkar and m. eslamian, common fixed point results in cat(0) spaces, nonlinear anal.: theory, method and applications, 74 (2011), no.5, 1835-1840. [3] v. berinde, iterative approximation of fixed points, baia mare: efemeride, 2000. [4] v. berinde, iterative approximation of fixed points, springer-verlag, berlin heidelberg, 2007. [5] a. branciari, a fixed point theorem for mappings satisfying a general contractive condition of integral type, int. j. math. math. sci. 29 (2002), 531-536. 204 saluja [6] m.r. bridson and a. haefliger, metric spaces of non-positive curvature, 319 of grundlehren der mathematischen wissenschaften, springer, berlin, germany, 1999. [7] k.s. brown, buildings, springer, new york, ny, usa, 1989. [8] f. bruhat and j. tits, ”groups reductifs sur un corps local”, institut des hautes etudes scientifiques. publications mathematiques, 41 (1972), 5-251. [9] p. chaoha and a. phon-on, a note on fixed point sets in cat(0) spaces, j. math. anal. appl. 320 (2006), no.2, 983-987. [10] s.k. chatterjee, fixed point theorems compactes, rend. acad. bulgare sci. 25 (1972), 727730. [11] l.b. ciric, a generalization of banach principle, proc. amer. math. soc. 45 (1974), 727-730. [12] s. dhompongsa, a. kaewkho and b. panyanak, lim’s theorems for multivalued mappings in cat(0) spaces, j. math. anal. appl. 312 (2005), no.2, 478-487. [13] s. dhompongsa and b. panyanak, on 4-convergence theorem in cat(0) spaces, comput. math. appl. 56 (2008), no.10, 2572-2579. [14] r. espinola and a. fernandez-leon, cat(k)-spaces, weak convergence and fixed point, j. math. anal. appl. 353 (2009), no.1, 410-427. [15] k. goebel and s. reich, uniform convexity, hyperbolic geometry, and nonexpansive mappings, 83 of monograph and textbooks in pure and applied mathematics, marcel dekker inc., new york, ny, usa, 1984. [16] n. hussain and m.a. khamsi, on asymptotic pointwise contractions in metric spaces, nonlinear anal.: theory, method and applications, 71 (2009), no.10, 4423-4429. [17] s. ishikawa, fixed points by a new iteration method, proc. amer. math. soc. 44 (1974), 147-150. [18] r. kannan, some results on fixed point theorems, bull. calcutta math. soc. 10 (1968), 71-76. [19] m.a. khamsi and w.a. kirk, an introduction to metric spaces and fixed point theory, pure appl. math, wiley-interscience, new york, ny, usa, 2001. [20] s.h. khan and m. abbas, strong and 4-convergence of some iterative schemes in cat(0) spaces, comput. math. appl. 61 (2011), no.1, 109-116. [21] a.r. khan, m.a. khamsi and h. fukhar-ud-din, strong convergence of a general iteration scheme in cat(0) spaces, nonlinear anal.: theory, method and applications, 74 (2011), no.3, 783-791. [22] w.a. kirk, fixed point theory in cat(0) spaces and r-trees, fixed point and applications, 2004 (2004), no.4, 309-316. [23] w.a. kirk, geodesic geometry and fixed point theory, in seminar of mathematical analysis (malaga/seville, 2002/2003), 64 of coleccion abierta, 195-225, university of seville secretary of publications, seville, spain, 2003. [24] w.a. kirk, geodesic geometry and fixed point theory ii, in international conference on fixed point theory and applications, 113-142, yokohama publishers, yokohama, japan, 2004. [25] w. laowang and b. panyanak, strong and ∆-convergence theorems for multivalued mappings in cat(0) spaces, j. inequal. appl. 2009 (2009), article id 730132, 16 pages. [26] l. leustean, a quadratic rate of asymptotic regularity for cat(0)-spaces, j. math. anal. appl. 325 (2007), no. 1, 386-399. [27] w.r. mann, mean value methods in iteration, proc. amer. math. soc. 4 (1953), 506-510. [28] y. niwongsa and b. panyanak, noor iterations for asymptotically nonexpansive mappings in cat(0) spaces, int. j. math. anal. 4 (2010), no.13, 645-656. [29] b.e. rhoades, fixed point iteration using infinite matrices, trans. amer. math. soc. 196 (1974), 161-176. [30] s. saejung, halpern’s iteration in cat(0) spaces, fixed point theory and applications, 2010 (2010), article id 471781, 13 pages. [31] n. shahzad, fixed point results for multimaps in cat(0) spaces, topology and its applications, 156 (2009), no.5, 997-1001. [32] t. zamfirescu, fixed point theorems in metric space, arch. math. (basel), 23 (1972), 292-298. department of mathematics, govt. nagarjuna, p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications volume 17, number 6 (2019), 974-979 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-974 on properties of certain analytic multiplier transform of complex order deborah olufunmilayo makinde1,∗, shahram najafzadeh2 1department of mathematics, obafemi awolowo university, ile-ife, nigeria 2department of mathematics, payame noor university, p. o. box: 19395–3697, tehran, iran ∗corresponding author: funmideb@yahoo.com abstract. the focus of this paper is to investigate the subclasses s∗c(γ,µ,α,λ; b), ts∗c(γ,µ,α,λ; b) = t ∩ s∗c(γ,µ,α,λ; b) and obtain the coefficient bounds as well as establishing its relationship with certain existing results in the literature. 1. introduction let a be the class of normalized analytic functions f in the open unit disc u = {z ∈ c : |z| < 1} with f(0) = f′(0) = 0 and of the form f(z) = z + ∞∑ n=2 anz n, an ∈ c, (1.1) and s the class of all functions in a that are univalent in u. also, the subclass of functions in a that are of the form f(z) = z − ∞∑ n=2 anz n, an ≥ 0, (1.2) is denoted by t and the subclasses s∗(α), c(γ) are given respectively by s∗(α) = { f ∈ s : re ( zf′(z) f(z) ) > γ z ∈ u } (1.3) received 2019-07-05; accepted 2019-08-12; published 2019-11-01. 2010 mathematics subject classification. 30c45. key words and phrases. analyticity; univalent; linear transformation; coefficient bounds. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 974 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-974 int. j. anal. appl. 17 (6) (2019) 975 c(α) = { f ∈ s : re ( 1 + zf′′(z) f′(z) ) > γ z ∈ u,≥ γ < 1 } . (1.4) moreover, the class ts∗(γ) denoted by t ∩ s∗(γ) which is the subclass of function f ∈ t such that f is starlike of order γ and respectively, tc(γ) is the class of function f ∈ t such that f is convex of order γ. an interesting unification of the classes s∗(α) and c(γ) denoted by s∗c(γ,β) which satisfies the condition re { zf′(z) + βz2f′(z) βzf′(z) + (1 −β)f(z) } > γ 0 ≥ γ < 1,z ∈ u. (1.5) has been extensively studied by different researchers, for example, see [6] and [1,2,3]. the special cases for β = 0, 1 are given by s∗(γ) and c((γ)) respectively. furthermore, the class ts∗c(γ,β) which is the subclass of function f ∈ t such that f belongs the class s∗c(γ,β), was studied by altintas et al. and other researchers. for details see [ 3, 5, 6 ]. using the unification in (5), nizami mustafa [6] introduced and investigated the class s∗c(γ,β; τ) and ts∗c(γ,β; τ), 0 ≤ α < 1; β ∈ [0, 1]; τ ∈ c which he defined as follows a function f ∈ s given by (1.1) is said to belong to the class s∗c(γ,β; τ) if the following condition is satisfied re { 1 + 1 τ [ zf′(z) + βz2f′(z) βzf′(z) + (1 −β)f(z) − 1 ]} > γ 0 ≥ γ < 1; β ∈ [0, 1]; τ ∈ c −{0},z ∈ u. (1.6) meanwhile, the author in [4] defined a linear transformation dmα,λf by dmα,λf(z) = z + ∞∑ n=2 α ( 1 + λ(n + α− 2) 1 + λ(α− 1) )m anz n, 0 ≤ λ ≤ 1; α ≥ 1; m ∈ n∪ 0 (1.7) motivated by the work of mustafa in [6], we study the effect of the application of the linear operator dmα,λf on the unification of the classes of the functions s∗c(γ,β; τ). now, we define the class s∗c(γ,α,λ; b) to be class of functions f ∈ s which satisfies the condition re { 1 + 1 b [ z(dmα,λf) ′(z) + µz2(dmα,λf) ′′(z) µz(dmα,λf) ′(z) + (1 −µ)(dmα,λf)(z) − 1 ]} > γ, 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪0 (1.8) also, we denote by dt the subclass of the class of functions in (7) which is of the form dmα,λf(z) = z − ∞∑ n=2 α ( 1 + λ(n + α− 2) 1 + λ(α− 1) )m anz n, 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪ 0 (1.9) and denote by ts∗c(γ,µ,α,λ; b) = t ∩s∗c(γ,µ,α,λ; b) which is the class of functions f in (1.9) such that f belong to the class s∗c(γ,µ,α,λ; b) = t ∩s∗c(γ,µ,α,λ; b). in this paper, we investigate the subclasses s∗c(γ,µ,α,λ; b) and ts∗c(γ,µ,α,λ; b) = t ∩s∗c(γ,µ,α,λ; b) int. j. anal. appl. 17 (6) (2019) 976 2. coeffiecient bounds for the classes s∗cλα(γ,µ; b) and ts ∗cλα(γ,µ; b) theorem 2.1. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ,α,λ; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪ 0 if ∞∑ n=2 [ α ( 1 + λ(n + α− 2) 1 + λ(α− 1) )m [1 + µ(n− 1)][n + |b|(1 −γ) − 1] ] |an| ≤ |b|(1 −γ) the result is sharp for the function dmα,λf(z) = z + |b|(1 −γ)(1 + λ(α− 1))m α[1 + µ(n− 1)][n + |b|(1 −γ)](1 + λ(n + α− 2))m zn n ≥ 2 proof. by (1.8), f belong to the class s∗c(γ,µ,α,λ; b) if re { 1 + 1 b [ z(dmα,λf) ′(z) + µz2(dmα,λf) ′′(z) µz(dmα,λf) ′(z) + (1 −µ)(dmα,λf)(z) − 1 ]} > γ it suffices to show that: ∣∣∣∣∣1b [ z(dmα,λf) ′(z) + µz2(dmα,λf) ′′(z) µz(dmα,λf) ′(z) + (1 −µ)(dmα,λf)(z) − 1 ]∣∣∣∣∣ < 1 −γ (2.1) simple computation in (2.1), using (1.7), we have:∣∣∣∣∣1b [ z(dmα,λf) ′(z) + µz2(dmα,λf) ′′(z) µz(dmα,λf) ′(z) + (1 −µ)(dmα,λf)(z) − 1 ]∣∣∣∣∣ = ∣∣∣∣∣∣1b   z + ∑∞n=2 nα ( 1+λ(n+α−2) 1+λ(α−1) )m anz n + µ ∑∞ n=2 n(n− 1)α ( 1+λ(n+α−2) 1+λ(α−1) )m anz n µz + ∑∞ n=2 µnα ( 1+λ(n+α−2) 1+λ(α−1) )m anzn + (1 −µ) ( z + ∑∞ n=2 α ( 1+λ(n+α−2) 1+λ(α−1) )m anzn ) − 1   ∣∣∣∣∣∣ = ∣∣∣∣∣∣1b  z + ∑∞n=2 nα[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m anz n z + ∑∞ n=2 α(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m anzn − 1   ∣∣∣∣∣∣ ≤ 1 b  ∑∞n=2 α(n− 1)[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m |an| 1 − ∑∞ n=2 α(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m |an|   which is bounded by 1 −γ if∑∞ n=2 α(n− 1)[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m |an| ≤ |b|(1 −γ)1 − ∑∞ n=2 α(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m |an| which is equivalent to∑∞ n=2 [ α(n− 1)[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m + α|b|(1 −γ)(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m] |an| ≤ |b|(1 −γ) which implies that ∞∑ n=2 [ α ( 1 + λ(n + α− 2) 1 + λ(α− 1) )m [1 + µ(n− 1)][n + |b|(1 −γ) − 1] ] |an| ≤ |b|(1 −γ) (2.2) int. j. anal. appl. 17 (6) (2019) 977 thus, (2.1) is satisfied if (2.2) is satisfied. � corollary 2.1. let f be as defined in (1) and the function dmα,λf belongs to the class s ∗c(γ,µ,α,λ; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪ 0. then |an| ≤ |b|(1 −γ)(1 + λ(α− 1))m α[1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n + α− 2))m corollary 2.2. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ, 1,λ,m; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 [(1 + λ(n− 1))m [1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.3) the result is sharp for the function dmα,λf(z) = z + |b|(1 −γ) [1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n− 1))m zn, n ≥ 2 corollary 2.3. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ, 1,λ, 1; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 [(1 + λ(n− 1)) [1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.4) the result is sharp for the function dmα,λf(z) = z + |b|(1 −γ) [1 + µ(n− 1)][n + |b|(1 −γ) − 1](1 + λ(n− 1)) zn, n ≥ 2 corollary 2.4. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ, 1, 1, 1; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 [n[1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.5) the result is sharp for the function dmα,λf(z) = z + |b|(1 −γ) n[1 + µ(n− 1)][n + |b|(1 −γ) − 1] zn, n ≥ 2 corollary 2.5. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ, 1, 0, 1; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 [[1 + µ(n− 1)][n + |b|(1 −γ) − 1]] |an| ≤ |b|(1 −γ) (2.6) the result is sharp for the function dmα,λf(z) = z + |b|(1 −γ) [1 + µ(n− 1)][n + |b|(1 −γ)−] zn, n ≥ 2 this result agrees with the theorem 2.1 in [6]. int. j. anal. appl. 17 (6) (2019) 978 corollary 2.6. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ, 0, 1,λ, 0; 1), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 [[1 + µ(n− 1)][n−γ]] |an| ≤ 1 −γ (2.7) the result is sharp for the function dmα,λf(z) = z + 1 −γ [1 + µ(n− 1)][n−γ] zn, n ≥ 2 this result agrees with the corollary 2.1 in [6]. corollary 2.7. let f be as defined in (1.1). then the function dmα,λf belongs to the class s ∗c(γ,µ, 1,λ, 0; 1), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; m ∈ n∪ 0 if ∞∑ n=2 (n−γ)|an| ≤ 1 −γ (2.8) the result is sharp for the function dmα,λf(z) = z + 1 −γ n−γ zn, n ≥ 2 this result agrees with the corollary 2.2 in [6]. theorem 2.2. let f ∈ dt . then the function dmα,λf belongs to the class dts ∗c(γ,µ,α,λ; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪ 0 if and only if ∞∑ n=2 α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)] ( x y )m |an| ≤ |b|(1 −γ) proof. we shall prove only the necessity part of the theorem as the sufficiency proof is similar to the proof of theorem 1. let f be as defined in (1.1) and dmα,λf belongs to the class ts ∗c(γ,µ,α,λ; b), 0 ≥ γ < 1,z ∈ u; 0 ≤ λ,µ ≤ 1; α ≥ 1; m ∈ n∪ 0; b ∈ r−{0}, we have re { 1 + 1 b [ z(dmα,λf) ′(z) + µz2(dmα,λf) ′′(z) µz(dmα,λf) ′(z) + (1 −µ)(dmα,λf)(z) − 1 ]} > γ (2.9) using (1.7) in (2.9) and by algebraic simplification, we have re   − ∑∞ n=2 α(n− 1)[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m anz n b { z − ∑∞ n=2 α(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m anzn }   ≥ γ − 1 choosing z to be real and z −→ 1, we have − ∑∞ n=2 α(n− 1)[1 + µ(n− 1)] ( 1+λ(n+α−2) 1+λ(α−1) )m an b { 1 − ∑∞ n=2 α(1 + µ(n− 1)) ( 1+λ(n+α−2) 1+λ(α−1) )m an } ≥ γ − 1 (2.10) int. j. anal. appl. 17 (6) (2019) 979 b ∈ r−{0} implies that b could be greater or less than zero. let b > 0 in (19), we have − ∞∑ n=2 α(n− 1)[1 + µ(n− 1)] ( x y )m an ≥ (γ − 1)b { 1 − ∞∑ n=2 α(1 + µ(n− 1)) ( x y )m an } (2.11) where x = 1 + λ(n + α− 2) and y = 1 + λ(α− 1) from (20), we have ∞∑ n=2 α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)] ( x y )m |an| ≤ b(1 −γ) (2.12) now suppose b < 0, which implies that b = −|b| and substituting b = −|b| in (19), we have∑∞ n=2 α(n− 1)[1 + µ(n− 1)] ( x y )m an |b| { 1 − ∑∞ n=2 α(1 + µ(n− 1)) ( x y )m an } ≥ (2.13) ∑∞ n=2 α(n− 1)[1 + µ(n− 1)] ( x y )m |an| ≥ (γ − 1)|b| { 1 − ∑∞ n=2 α(1 + µ(n− 1)) ( x y )m an } which implies ∞∑ n=2 α(n− 1)[1 + µ(n− 1)][n + b(1 −γ)] ( 1 + λ(n + α− 2) 1 + λ(α− 1) )m |an| ≥−b(1 −γ) (2.14) from (21) and (23), the proof of the necessity is completed. � references [1] altintas, o., on a subclass of certain starlike functions with negative coefficient. math. japon., 36 (1991), 489-495. [2] altintas, o., irmak, h. and srivastava, h. m., fractional calculus and certain starlke functions with negative coefficients. comput. math. appl., 30 (1995), no. 2, 9-16. [3] altintas, o., özkan, ö. and srivastava, h. m., neighborhoods of a certain family of multivalent functions with negative coefficients. comput. math. appl., 47 (2004), 16671672. [4] makinde, d.o., a new multiplier differential operator. adv. math., sci. j., 7 (2018), no.2, 109 -114. [5] irmak, h., lee, s. h. and cho, n. e., some multivalently starlike functions with negative coefficients and their subclasses defined by using a differential operator. kyungpook math. j., 37 (1997), 43-51. [6] mustafa, n., the various properties of certain subclasses of analytic functions of complex order. arxiv:1704.04980 [math.cv], 2017. 1. introduction 2. coeffiecient bounds for the classes s*c(,;b) and ts*c(,;b) references international journal of analysis and applications volume 16, number 5 (2018), 702-711 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-702 c-algebrability of pathological sets of product integrable functions fatemeh farmanesh, ali farokhinia∗ department of mathematics, shiraz branch, islamic azad university, shiraz, iran ∗corresponding author: alifarokinia@gmail.com abstract. in this paper we investigate linear algebraic structures in the set of product integrable matrixvalued functions and find c-generated algebras in l([a, b], rn×n)\l∗([a, b], rn×n) and d([a, b], rn×n)\l([a, b], rn×n). 1. introduction if x is a vector space, a subset m of x is called lineable if m ∪{0} contains an infinite dimensional vector space. if x is a linear algebra and m ⊆ x, one calls m a κ-algebrable set if m ∪{0} contains a κ-generated algebra, that is, an algebra which has a minimal system of generators of cardinality κ. these notions were coined by v.i. guariy [1, 9] and then became a criterion for measuring how much large linear algebraic structures could be found in a set of functions with weird properties (see [2, 6–8]). another criterion is the concept of strong algebrability introduced by glab and bartoszewicz in [5]. let κ be a cardinal number and x be a linear commutative algebra. a subset m of x is called strongly κ-algebrable if m ∪{0} contains a κ-generated algebra isomorphic to a free algebra. in this paper we seek a linear algebraic structures in the spaces of product integrable function. the notion of product integral has been introduced by vito volterra about the end of the 19th century, who studied received 2018-04-26; accepted 2018-08-02; published 2018-09-05. 2000 mathematics subject classification. 47a16, 47l10. key words and phrases. algebrable; lebesgue integrable; product integrable. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 702 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-702 int. j. anal. appl. 16 (5) (2018) 703 linear systems of differential equations w ′(t) = a(t)w(t), t ∈ [a,b] w(a) = i, where i is the identity matrix, a : [a,b] → rn×n is a given continuous function and w : [a,b] → rn×n is the unknown function (see [17]). later, ludwig schlesinger introduced the definition of the riemann product integral as follows: given a tagged partition of an interval [a,b], which is a collection of point-interval pairs d = (ξi, [ti−1, ti]) m i=1, where a = t0 ≤ t1 ≤ ... ≤ tm = b and ξi ∈ [ti−1, ti] for every i ∈{1, 2, ...,m}. we refer to t0, t1, ..., tm as the division points of d, while ξ1,ξ2, ...,ξm are the tags of d. remark 1.1. if we replace ξi ∈ [ti−1, ti] by ξi ∈ [a,b], then the collection d is called a free tagged partition. given a function δ : [a,b] → r+ (called a gauge on [a,b]), a free tagged partition is called δ-fine if [ti−1, ti] ⊂ (ξi −δ(ξi),ξi + δ(ξi)) , i = {1, 2, ...,m}. now consider a matrix function a : [a,b] → rn×n with entries {aij} n i,j=1 . put ∆ti = ti − ti−1 , i = 1, 2, ...,m , υ(d) = max ∆ti 1≤i≤m , and define p(a,d) = m∏ i=1 (i + a(ξi)∆ti) = (i + a(ξ1)∆t1)(i + a(ξ2)∆t2)....(i + a(ξm)∆tm). in case the limit limυ(d)→0 p(a,d) exists, it is called the riemann product integral of the function a on the interval [a,b] and is denoted by the symbol (i + a(t)dt) b∏ a . in this paper r([a,b],rn×n) denotes the set of all riemann product integrable functions. utilizing step functions schlesinger generalized this definition and introduced the lebesgue product integral (see [11, 12, 16]). let us recall some facts that will be needed: 1. a function a : [a,b] → rn×n is called a step function if there exists numbers a = t0 < t1 < ... < tm = b such that a is constant function on every interval (tk−1, tk), k = 1, 2, ...,m. 2. for a ∈ rn×n we will use the operator norm ‖a‖ = sup{‖ax‖ : ‖x‖≤ 1} , where ‖ax‖ and ‖x‖ denote the euclidean norms of vectors ax, x ∈ rn. 3. a sequence of functions {ak : [a,b] → rn×n}k∈n is called uniformly bounded if there exists a number m ∈ r such that ‖ak(x)‖≤ m for all k ∈ n and all x ∈ [a,b]. int. j. anal. appl. 16 (5) (2018) 704 theorem 1.1. [16, lemma 3.5.4 and theorem 3.5.5] let ak : [a,b] → rn×n, k ∈ n, be a uniformly bounded sequence of step functions such that limk→∞ak(x) = a(x) a.e. on [a,b]. then lim k→∞ ‖ak −a‖1 = lim k→∞ b∫ a ‖ak(x) −a(x)‖dx = 0, and the limit lim k→∞ (i + ak(x)dx) b∏ a exists and is independent of the choice of the sequence {ak}. definition 1.2. [16, definiton 3.5.6] consider the function a : [a,b] → rn×n. assume there exists a uniformly bounded sequence of step functions ak : [a,b] → rn×n such that limk→∞ak(x) = a(x) a.e. on [a,b], then the function a is called lebesgue product integrable and we define (i + a(x)dx) b∏ a = lim k→∞ (i + ak(x)dx) b∏ a . the symbole l∗([a,b],rn×n) denotes the set of all lebesgue product integrable functions. it is easy to show that l∗([a,b],rn×n) = {a : [a,b] → rn×n : a is measurable and bounded}. let us recall that a function a : [a,b] → rn×n is called bochner intagrable if there is a sequence of simple functions ak : [a,b] → rn×n, k ∈ n such that lim k→∞ ak(t) = a(t) a.e. on [a,b] and lim k→∞ ‖ak −a‖1 = 0. thus by theorem 1.1 and definition 1.2, each a ∈ l∗([a,b],rn×n) is bochner intagrable. after that schlesinger extended the definition of l∗([a,b],rn×n) to all matrix functions with lebesgue integrable (not necessarily bounded) entries and used the next symbole: l([a,b],rn×n) = {a : [a,b] → rn×n : (l) b∫ a ‖a(x)‖dx < ∞}. the symbole (l) estands for the lebesgue integral. taking account of theorem 1.1 it is natural to state the following definition. definition 1.3. [16, definiton 3.8.1] a function a : [a,b] → rn×n is called product integrable if there exists a sequence of step functions {ak} such that lim k→∞ ‖ak −a‖1 = 0. we define (i + a(x)dx) b∏ a = lim k→∞ (i + ak(x)dx) b∏ a int. j. anal. appl. 16 (5) (2018) 705 remark 1.2. since step functions belong to the complete space l([a,b],rn×n), every product integrable function also belongs to l([a,b],rn×n). moreover, step functions form a dense subset in this space, and hence (i + a(x)dx) b∏ a exists if and only if a ∈ l([a,b],rn×n), i.e., the lebesgue integral ∫ b a ‖a(t)‖dt is finite. concerning the above definitions of product integral we have the following chain of strict inclusions: r([a,b],rn×n) ⊂ l∗([a,b],rn×n) ⊂ l([a,b],rn×n). 2. the exponential function and the product integral recall that for every a ∈ rn×n the matrix exponential is defined by ea = ∞∑ k=0 ak k! . theorem 2.1. [16, theorem 3.2.2] consider a riemann integrable function a : [a,b] → rn×n. then lim υ(d)→0 m∏ k=1 ea(ξk)∆tk = lim υ(d)→0 m∏ k=1 (i + a(ξk)∆tk) = (i + a(t)dt) b∏ a , where partitions are as in introduction. remark 2.1. if a ∈ l∗([a,b],rn×n) and {ak} ∞ k=1 is a uniformly bounded sequence of step functions in l∗([a,b],rn×n) such that ak → a a.e. on [a,b], then by [16, theorem 3.6.3] we have (i+a(x)dx) b∏ a = lim k→∞ (i+ ak(x)dx) b∏ a . now every function ak is associated with a partition dk : a = t k 0 < t k 1 < ... < t k m(k) = b such that ak(x) = a k j , x ∈ (t k j−1, t k j ), and lim k→∞ υ(dk) = 0. so by the definition of lebesgue product integrable functions, (i + a(x)dx) b∏ a = lim k→∞ (i + ak(x)dx) b∏ a = lim k→∞ m(k)∏ j=1 exp(akj ∆t k j ). moreover schlesinger in [16, p. 485-486] proved the product integral might be also calculated as (i + a(x)dx) b∏ a = lim k→∞ m(k)∏ j=1 (i + (akj ∆t k j ). we remark that each a ∈ l∗([a,b],rn×n) is bochner integrable and hence the product integrals b∏ a exp(a(t)dt and b∏ a (i + a(t)dt) exist and equal to each other; see [13, theorem 14, theorem 16]. thus according to the previous discussion, theorem 2.1 holds for all a ∈ l∗([a,b],rn×n). int. j. anal. appl. 16 (5) (2018) 706 now cosider a function a ∈ l([a,b],rn×n). by the definition 1.3 there exists a sequence of step functions {ak} ∞ k=1 such that lim k→∞ ‖ak −a‖1 = 0 and (i + a(t)dt) b∏ a = lim k→∞ (i + ak(t)dt) b∏ a . thus theorem 2.1 does also hold for a ∈ l([a,b],rn×n). so we can state the next theorem. theorem 2.2. let a : [a,b] → rn×n be a matrix function and a ∈ l([a,b],rn×n), then exp◦a is product integrable. 3. lebesgue product integrable functions the next definition and theorem provide important tools for proving the existence of infinitely generated algebras in the family of real or complex functions. definition 3.1 ( [3]). we say that a function f : r → r is an exponential-like function (of rank m) whenever f is given by f(x) = m∑ i=1 aie bix for some distinct nonzero real numbers b1,b2, ...,bm and some nonzero real numbers a1,a2, ...,am. theorem 3.2 ( [3,4]). let f ⊂ r[0,1] and assume that there exists a function f ∈f such that fof ∈f\{0} for every exponential-like function f : r → r. then f is strongly c-algebrable. more exactly, if h ⊂ r is a set of cardinality c and linearly independent over the rationals q, then exp◦(rf), r ∈ h, are free generators of an algebra contained in f ∪{0}. note that in all proofs we apply theorem 3.2 theorem 3.3. the set of riemann real valued integrable functions is strongly c-algebrable. proof. volterra in [17] showed that the riemann integrable functions are product integrable, thus by theorem 2.1 and theorem 3.2 the proof follows. � theorem 3.4. the set of real valued lebesgue integrable functions is strongly c-algebrable. proof. schlesinger in [12, 16] showed the product integrability of lebesgue integrable functions. so by theorem 2.2 and theorem 3.2, the proof is complete. � theorem 3.5. the set l([a,b],rn×n)\l∗([a,b],rn×n) is strongly c-algebrable. proof. let a : [0, 1] → rn×n be given by a(x) = (aij(x))ni,j=1 such that for each i,j = 1, 2, ..,n, aij(x) =   1√ x x ∈ (0, 1] 0 x = 0 . int. j. anal. appl. 16 (5) (2018) 707 so for some y ∈ rn×1 and ‖y‖≤ 1, a(x)y =   a11 . . . a1n ... . . . ... an1 · · · ann     y1 ... yn   =   a11y1 + a12y2 + · · · + a1nyn ... an1y1 + an2y2 + · · · + annyn   , ‖a(x)y‖≥ √ n x (y1 + · · · + yn) 2 ≥ 1 x , x ∈ (0, 1]. thus a is not bounded and so a and exp◦ (a) are not in l∗([0, 1],rn×n). now let am : [0, 1] → rn×n be given by am(x) = ( (m) bij (x)) n i,j=1 such that for each i,j = 1, 2, ..,n, (m) bij (x) =   0 x ∈ [0, 1 m ] 1√ x x ∈ ( 1 m , 1] . given an arbitrary i and j, and note that for m ≥ 2, (m) bij (x) is lebesgue integrable on [0, 1]. since lim m→∞ (m) bij (x) = aij(x) for each x ∈ [0, 1], so by the monotone convergence theorem aij(x) is lebesgue integrable. thus a and exp◦(a) are in l([0, 1],rn×n) so f ◦(a) is in l([0, 1],rn×n), for every exponentiallike function f, and the proof is complete by theorem 3.2. � theorem 3.6. the set of l([a,b],rn×n)\r([a,b],rn×n) is c-algebrable. proof. since l([a,b],rn×n)\l∗([a,b],rn×n) ⊆ l([a,b],rn×n)\r([a,b],rn×n), the preceding theorem implies that l([a,b],rn×n)\r([a,b],rn×n) is c-algebrable. � 4. product integrability of denjoy integrable matrix-valued functions the following definition generalizes the concept of denjoy product integration. definition 4.1. consider the function a : [a,b] → rn×n and let [c,d] ⊂ [a,b]. the oscilation of a on the interval [c,d] is the number osc(a, [c,d]) = sup{‖a(ξ1) −a(ξ2)‖ : ξ1,ξ2 ∈ [c,d]} . the abbreviations ac, bv and acg stand for “absolutely continuous”, “bounded variations” and “generalized absolutely continiuous”, respectively. definition 4.2. let a : [a,b] → rn×n and e ∈ [a,b]. 1. the strong variation of f on e is defined by v∗(f,e) = sup { n∑ i=1 osc(f, [ci,di]) } , where the supremum is taken over all finite collections {[ci,di] : 1 ≤ i ≤ n} of non-overlapping intervals that have endpoints in e. int. j. anal. appl. 16 (5) (2018) 708 2. the function f is of bounded variation in the restricted sense on e (briefely a is bv∗ on e ) if v∗(f,e) is finite. 3. the function a is absolutely continuous in the restricted sense on e (briefely a is ac∗ on e ) if for each ε > 0, there exists δ > 0 such that n∑ i=1 osc(a, [ci,di]) < ε, whenevere {[ci,di] : 1 ≤ i ≤ n} is a finite collection of non-overlapping intervals that have endpoints in e and satisfy n∑ i=1 (di − ci) < δ. 4. the function a is generalized absolutely continuous in the restricted sense on e (briefely a is acg∗ on e ) if a ∣∣ e is continuous on e and e can be written as a countable union sets on each of which a is ac∗. note that in general, v (f,e) ≤ v∗(f,e) and hence a is bv (ac,bv g,acg) on e if it is bv∗(ac∗,bv g∗,acg∗) on e. definition 4.3. the function a : [a,b] → rn×n is denjoy integrable on [a,b] if there exists an acg∗ function a : [a,b] → rn×n such that a′ = a a.e. on [a,b]. theorem 4.4. [15, theorem 6.2] let f : [a,b] → rn×n and e ⊆ [a,b]. (1) if f is ac(acg,ac∗,acg∗) on e, then f is bv (bv g,bv∗,bv g∗) on e. (2) if f is bv∗ on e, then f is bv∗ on e. (3) suppose that e is closed with a,b ∈ e and let g be the linear extension of f to [a,b]. if f is bv (ac) on e, then g is bv (ac) on [a,b]. remark 4.1. let p be a perfect set. a perfect portion of p is a set of the form p∩[c,d] where p∩(c,d) 6= ∅, c,d ∈ p, and p ∩ [c,d] is a perfect set. theorem 4.5. [15, theorem 6.10] suppose that f : [a,b] → rn×n is acg(acg∗) on [a,b] and let e ⊂ [a,b] be a perfect set. then there is a perfect portion e ∩ [c,d] of e such that f is ac(ac∗) on e ∩ [c,d]. ( note that in this case, each subinterval of [a,b] contains an interval on which the function f is ac(ac∗). the endpoints of all the intervals on which f is ac(ac∗) form a dence set in [a,b] ). we recall that the next lemma and proposition are mentioned in [15] as exercises. lemma 4.1. let f : [a,b] → rn×n, and e be a closed set with bounds a and b, and let [a,b] − e = ∞⋃ n=1 (an,bn). suppose that g is the linear extension of f from e to [a,b] and c ∈ e. then g(x)−g(c) x−c is between f(an)−f(c) an−c and f(bn)−f(c) bn−c for each x ∈ (an,bn). in particular, if c is two-sided limit point of e and f is differentiable at c, then g is differentiable at c and g′(c) = f ′(c). int. j. anal. appl. 16 (5) (2018) 709 proof. first we note that g = f on e and g is linear on each of the intervals contiguous to e. for each x ∈ [an,bn], we have g(x) = f(bn) −f(an) bn −an (x−an) + f(an), and hence an easy calculation completes the proof. � proposition 4.1. suppose that a : [a,b] → rn×n is denjoy integrable on [a,b]. then [a,b] = ∪∞n=1en where each en is closed and a is lebesgue integrable on each en. proof. by the hypothesis, there exists an acg∗ function a : [a,b] → rn×n such that a′ = a a.e. on [a,b], and we can write [a,b] = ∪∞n=1en, where a is ac∗ on each en. by theorem 4.4 we can assume that each en is closed. then by theorem 4.5 there exists a perfect portion en ∩ [c,d] of en for n ∈ n, such that a is ac∗ on en∩[c,d]. let g : [c,d] → rn×n be the linear extension of a ∣∣ en∩[c,d] to [c,d]. by part 3 of theorem 4.4, g is ac on [c,d]. so the function g′ exists a.e. and is lebesgue integrable on [c,d]. but by lemma 4.1 a′ = g′ = a a.e. on en ∩ [c,d], so the function a is lebesgue integrable. � theorem 4.6. let a : [a,b] → rn×n be denjoy integrable on [a,b], then it is product integrable. proof. let d([a,b],rn×n) be endowed by the norm ‖a‖ = (d) b∫ a ‖a(t)‖dt, where (d) stands for the denjoy integral. by proposition 4.1 there exists subsets en such that [a,b] = ∪∞n=1en where for each n ∈ n, en is non-overlapping, closed and a is lebesgue integrable on en. let an be the restriction of a to en for each n ∈ n. then each an is lebesgue integrable and so product integrable and hence for each an there exists a sequense of step functions {ank} ∞ k=1 such that ank : en → r n×n and lim k→∞ ‖ank −an‖en = limk→∞ ∫ en ‖ank (x) −an(x)‖dx = 0 for each n, put an = infen and bn = supen, so both an,bn are in en. thus for each en there exist t0, t1, ..., tn such that t0 = an ≤ t1 ≤ ... ≤ tn = bn, and ank is constant on (tk−1, tk) for k = 1, . . . ,n. now let {bk} ∞ k=1 be a sequence of step functions on [a,b] such that [a,b] = ∞⋃ n=1 en and bk = ank on each en. then by dominated convergence theorem we have the followings: lim k→∞ ‖bk −a‖1 = lim k→∞ b∫ a ‖bk(x) −a(x)‖dx = lim k→∞ ∞∑ n=1 ∫ en ‖ank (x) −an(x)‖dx = 0, int. j. anal. appl. 16 (5) (2018) 710 i.e., bk converges to a also in the norm of space d([a,b],rn×n) and hence by [16, theorem 3.5.5] lim k→∞ (i + bk(x))dx b∏ a exists. so the proof is complete. � 5. c-algebrability of the set of denjoy product integrable in this section, some pathological properties (more precisely algebrability) of sets of product integrable functions contained in d([a,b],rn×n)\l([a,b],rn×n) are investigated. first we note that a matrix a = {aij} n i,j=1 is called regular if it has a nonzero determinant. definition 5.1. a function a : [a,b] → rn×n is called perron product integrable if there is a regular matrix b ∈ rn×n such that for every ε > 0 there is a function δ : [a,b] → (0,∞) such that ‖b −p(a,d)‖ < ε for every δ-fine partition d of [a,b]. theorem 5.2. consider the function a : [a,b] → rn×n in d([a,b],rn×n). then b∏ a ea(t)dt = (i + a(t)dt) b∏ a . proof. by [10, theorem 2.12] and [15, theorem 11.2], the proof is clear. � corollary 5.1. if a : [a,b] → rn×n is product integrable function, then exp ◦ (a) is product integrable function. theorem 5.3. the set of product integrable functions is strongly c-algebrable. proof. by corollary 5.1 and theorem 3.2 the proof follows. � proposition 5.1. [15, theorem 7.11] suppose that f : [a,b] → r is denjoy integrable on each subinterval [c,d] ⊆ (a,b). if d∫ c f converges to a finite limit as c → a+ and d → b−, then f is denjoy integrable on [a,b] and b∫ a f = lim c→a+ d→b− d∫ c f. theorem 5.4. the set of d([a,b],rn×n)\l ([a,b],rn×n) is strongly c-algebrable. proof. let ∞∑ n=1 cn be a nonabsolutely convergent series of real numbers and let in = ( 2−n, 2−n+1 ) , n ∈ n. define the function a : [0, 1] → rn×n by a(x) = (aij(x))ni,j=1, such that for each i,j = 1, 2, ...,n (aij(x)) =   2ncn x ∈ in 0 otherwise. note that 1∫ 0 |aij(x)|dx = ∞∑ n=1 ∫ in |aij(x)|dx = ∞∑ n=1 ∣∣2−ncn2n∣∣ = ∞∑ n=1 |cn| = ∞. int. j. anal. appl. 16 (5) (2018) 711 hence neither aij nor a is lebesgue integrable on [0, 1]. now we are going to show that a is denjoy integrable on [0, 1]. for each 0 < α < 1 both of functions aij and a are bounded on [α, 1], so they are lebesgue integrable on [α, 1]. let b(x) = 1∫ x aij for each x ∈ (0, 1]. the function b is linear on each in. it follows that b(x) is between b(2−n) and b(2n) for each x ∈ in. now b(2−n) = n∑ k=1 ck and lim n→∞ b (2−n) = ∞∑ k=1 ck. therefore lim x→0+ b (x) = ∞∑ n=1 cn and according to proposition 5.1, aij is denjoy integrable on [0, 1] for each i,j = 1, 2, ...,n. thus for each aij(x) there exists an acg∗ function fij such that f ′ ij(x) = aij(x) a.e. on x ∈ [0, 1]. now put f(x) = (fij(x))ni,j=1 for each x ∈ [0, 1]. so f ′(x) = (f′ij(x)) n i,j=1 = (aij(x)) n i,j=1 = a(x) a.e. on [0,1]. hence a is denjoy integrable on x ∈ [0, 1]. one can see easily that exp◦aij is denjoy integrable and so is exp◦a. thus by theorem 3.2 the proof is complete. � references [1] r. aron, v. i. gurariy and j. b. seoane-sepúlveda, lineability and spaceability of sets of functions on r, proc. amer. math. soc. 133 (3) (2005), 795-803. [2] r. aron and j. b. seoane-sepúlveda, algebrability of the set of everywhere surjective functions on c, bull. belg. math. soc. simon stevin 14 (1) (2007), 25-31. [3] m. balcerzak, a. bartoszewicz and m. filipczac, nonseparable spaceability and strong algebrability of sets of continuous singular functions, j. math. anal. appl. 407 (2) (2013), 263-269. [4] a. bartoszewicz, m. bieniea, m. filipczac and s. glab, strong c-algebrability of strong sierpinski-zygmund, smooth nowhere analytic and other sets of functions, j. math. anal. appl. 412 (2) (2015), 620-630. [5] a. bartoszewicz and s. glab, strong c-algebrability of sets of sequences and functions, proc. amer. math. soc. 141 (2013), 827-835. [6] a. farokhinia, algebrability of space of quasi-everywhere surjective functions. b. math. anal. appl. 6 (6) (2014), 38-43. [7] a. farokhinia, lineability of denjoy integrable function, j. math. ext. 11 (1) (2017), 57-65. [8] a. farokhinia, lineability of space of quasi-everywhere surjective functions. j. math. ext. 6 (3) (2013), 45-51. [9] v. i. gurariy and l. qurta, on lineability of sets of continuous functions, j. math. anal. appl. (1) (2004), 62-72. [10] j. jarńık and j. kurzweil, a general form of the product integral and linear ordinary, czech. math. j. 37 (4) (1987), 642-659. [11] f. r. riesz, sur lintegrale de lebesgue. acta mathematica 42 (1919), 191-205. [12] l. schlesinger, neue grandlagen fur einen infinittesimalkul der matrizen, mathematische zeitschrift 33 (1931), 33-61. [13] a. slav́ık and s̆. schwabic, henstock-kurzweil and mcshane product integrals; descriptive definations, czech. math. j. 58 (133) (2008), 241-269. [14] s̆. schwabic, bochner product integration, math. bohem. 119 (1994), 305-335. [15] r. gordon, the integrals of lebesgue, denjoy, perron and henstock, american mathematical society, 1994. [16] a. slav́ık, product integration, its history and applications, matfyzpress, prague, 2007. [17] v. volterra and b. hostinsky, operations infinitesimales lineaires, gauthier-villars, paris, 1938. adison-wesley. publishing company 1979. 1. introduction 2. the exponential function and the product integral 3. lebesgue product integrable functions 4. product integrability of denjoy integrable matrix-valued functions 5. c-algebrability of the set of denjoy product integrable references int. j. anal. appl. (2022), 20:73 unusual nonpolynomial van der pol oscillator equations with exact harmonic and isochronous solutions kolawolé kêgnidé damien adjaï, marcellin nonti, jean akande, marc delphin monsia∗ department of physics, university of abomey-calavi, abomey-calavi, 01.bp.526, cotonou, benin ∗corresponding author: monsiadelphin@yahoo.fr abstract. we do not know van der pol-type equations with nonlinear restoring force having explicitly an exact periodic solution. we present, for the first time, nonpolynomial van der pol oscillator equations that do not satisfy the classical existence theorems. we exhibit their exact harmonic and isochronous solutions and prove the existence of limit cycles by using averaging theory. we also present first integrals and exact solutions of polynomial van der pol-duffing equations to show that they do not have any limit cycle. additionally, we prove that the damped duffing-type equations are equivalent to the conservative duffing equations exhibiting nonoscillatory solutions. 1. introduction the lienard equation: ẍ + f (x)ẋ +g(x)=0, (1.1) where overdot is the derivative with respect to time and f (x) and g(x) are functions of x, is one of the most important autonomous second-order differential equations. this importance results in the fact that it often occurs in mathematical modelling of physical and engineering systems. consequently, equation (1.1) has been widely investigated in the literature. a celebrated equation of this type is the van der pol oscillator [1–3]: ẍ +β(x2 −1)ẋ +x =0, (1.2) where β � 0, mentioned having a unique limit cycle only in the light of qualitative theory of differential equations and existence theorems [1–3] since it has no known exact and general solutions. thus, received: nov. 12, 2022. 2010 mathematics subject classification. 34a05, 34c07, 34c25, 34c29, 65l06. key words and phrases. van der pol-duffing equation, nonpolynomial van der pol-type oscillator, damped duffing equation, first integrals, exact harmonic and limit cycle solutions, existence theorem. https://doi.org/10.28924/2291-8639-20-2022-73 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-73 2 int. j. anal. appl. (2022), 20:73 the result known currently under the name of the lienard-levinson-smith theorem [1–3] has been intensively applied to investigate the existence of limit cycles for the equations of type (1.1). in this way the van der pol-duffing equation [4] is: ẍ +β(x2 −1)ẋ +γx +αx3 =0, (1.3) where α, β and γ are constants and has been considered for a long time as a self-excited oscillator. however, similar to the van der pol equation, equation (1.3) is not integrable in general. recently, udwadia and cho [5] succeeded in showing that equation (1.3) in the form: ẍ +β(x2 −1)ẋ −3α ( 1+ 3α β3 ) x +αx3 =0, (1.4) does not have any limit cycle by explicitly calculating its general solution. in [6], the author established a first integral of equations of type (1.3) but in terms of special functions, namely, in terms of hypergeometric functions. monsia et al. [4], for the first time, successfully derived the time-independent first integral of a more general form of equation (1.4) in terms of elementary functions such that it became possible to obtain the lagrangian of equations of type (1.3). the above explicitly shows that the van der pol equation with polynomial nonlinear restoring force can have no limit cycle. moreover, it is worth mentioning that the exact and general solution depicted by udwadia and cho [5] is not periodic. recently, akande et al. [7] successfully showed that the lienard equation: ẍ − q1x√ q22 −x2 ẋ =0, (1.5) admits exact harmonic and isochronous periodic solutions, while this equation does not satisfy the classical existence theorems. this fact is also observed for the lienard equations: ẍ ±q3 √ q4q5 −q6q7x2ẋ +q4q7x +q4q7x2 −q3q4q5 =0, (1.6) studied by akplogan et al. [8], where the parameters qi, i = 1, ....,7 are arbitrary constants. in this situation, let us consider the following equation with van der pol damping called van der pol-type equation: ẍ +β(x2 −a)ẋ +γx +λ(c1 −c2x2)p =0, (1.7) where β � 0, γ � 0, a � 0, λ, c1, c2, and p are constants. when λ = 0, equation (1.7) reduces to the van der pol equation (1.2) for γ = 1. equation (1.7) generalizes the equation given in [ [1] p.146] as an exercise for c1 = 0, and p = 1. if c1 = 0 and p = 3 2 , equation (1.7) becomes the van der pol-duffing equation (1.3) mentioned above. for c2 = 0, equation (1.7) takes the form of the biased van der pol equation, which appears in [ [9] p.287]. to the best of our knowledge, such a van der pol equation with nonpolynomial restoring force has not been previously investigated in the literature. from this perspective, the question is to ask if equation (1.7) can exhibit harmonic and limit cycle oscillations. thus, the objective is to show the nonexistence of limit cycles for van der pol equations with polynomial restoring force and to prove the existence of the harmonic and limit cycle int. j. anal. appl. (2022), 20:73 3 solutions of equation (1.7). to that end, we first present some results for the van der pol equations with polynomial restoring force (section 2) and second, explicitly prove the existence of the exact harmonic and limit cycle solutions of equation (1.7) (section 3). finally, the results are compared to numerical solutions using the fourth-order runge-kutta algorithm, and we give a conclusion for the work. 2. van der pol equations with polynomial restoring force in this part, we present time-independent first integrals allowing the determination of lagrangian and exact solutions of van der pol-type equations with polynomial restoring forces. 2.1. van der pol-duffing equations. consider the equation: ẍ − (k1x2 +3k2)ẋ +2k22x −2k 2 1x 5 =0, (2.1) where k1 and k2 are arbitrary parameters. a first integral of this equation can be written as follows: ẋ = k2x +k1x 3. (2.2) putting k1 =−µ and 3k2 = k, one can, from equation (2.1), arrive at: ẍ +(µx2 −k)ẋ + 2k2 9 x −2µ2x5 =0, (2.3) which is a quintic duffing-van der pol equation. consider the equation: ẍ +(3k1x 2 −2k2)ẋ +k22x −k1k2x 3 =0, (2.4) which denotes the cubic duffing-van der pol equation. the first integral of equation (2.4) has the form: ẋ = k2x −k1x3. (2.5) the cubic-quintic duffing-van der pol equation: ẍ +k1(x 2 −1−2k1)ẋ +2k31x +2k 2 1(1−k1)x 3 −2k21x 5 =0, (2.6) has the time-independent first integral: k1x 5 = k1x 3 −x2ẋ. (2.7) the first integral of the equation: ẍ −k2(1−x2)ẋ −k1(k1 +k2)x + k1k2 3 x3 −k1k3 =0, (2.8) can be written as follows: ẋ =(k1 +k2)x − k2 3 x3 +k3, (2.9) where k3 is an arbitrary parameter. 4 int. j. anal. appl. (2022), 20:73 consider the equation: ẍ −k1(1−x2)ẋ +k2(k1 −k2)x − k1k2 3 x3 =0. (2.10) its first integral can be read: ẋ =(k1 −k2)x − k1 3 x3. (2.11) 2.2. generalized van der pol-type equation. now, consider the more general van der pol-type equation with polynomial restoring force: ẍ + ( k2 k3 `+ k1 k3 x1−` ) ẋ + (`−1)k21 k23 x3−2` + 2(`−1)k1k2 k23 x2−` + (`−1)k22 k23 x =0. (2.12) one can verify that equation (2.12) has the time-independent first integral: ẋ =− 1 k3 ( k1x 2−` +k2x ) , (2.13) where ` is an arbitrary parameter. it suffices to put ` = (2−n) into equation (2.12) to recover the general form considered in [6]. the author derived the first integral of this type of equation (2.12) in terms of hypergeometric functions. equation (2.12) with ` = (2− n) has also been investigated by chandrasekar et al. [10]. the first integrals derived by these authors [10] using the so-called generalized extended prelle-singer method are functions of time. the general solution of equation (2.12), as can be verified, using the first integral (2.13), becomes: x(t)= [ 1 k2 ( −k1 +e −k2 k3 (`−1)(t+k) )] 1 `−1 , (2.14) where ` 6= 1, and k is an integration constant. an interesting case of equation (2.12) consists of putting ` =−1 to obtain: ẍ + ( k1 k3 x2 − k2 k3 ) ẋ − 2k21 k23 x5 − 4k1k2 k23 x3 − 2k22 k23 x =0. (2.15) the van der pol-duffing equation (2.15) is equivalent to the conservative cubic-quintic duffing equation: ẍ − 3k21 k23 x5 − 4k1k2 k23 x3 − k22 k23 x =0, (2.16) obtained by using the first integral (2.13), where ` = −1. in this regard, the general solution of equation (2.15) or (2.16), taking into consideration equation (2.14), can read: x(t)= (k2) 1 2[ e 2k2 k3 (t+k) −k1 ]1 2 . (2.17) int. j. anal. appl. (2022), 20:73 5 this result shows that the widely studied cubic-quintic duffing equation is (2.12) in fact a pseudooscillator. from equation (2.13), the more general van der pol-type equation with polynomial nonlinear restoring force becomes equivalent to the generalized duffing equation: ẍ − k1k2 k23 (3− `)x2−` − k21 k23 (2− `)x3−2` − k22 k23 x =0. (2.18) conservative equation (2.18) also has the general solution (2.14). it is worth noting that the conservative cubic-quintic duffing equation (2.16) is equivalent by using the first integral (2.13) to the dissipative lienard equation: ẍ + 4k2 k3 ẋ − 3k21 k23 x5 + 3k22 k23 x =0, (2.19) with the general solution (2.17). equation (2.19) is also known as the damped-quintic duffing equation.the general solution (2.17) of equation (2.19) contradicts the results of existence of the bounded periodic solutions exhibited in [11]. the above shows that the van der pol-type equations with polynomial restoring force do not have any limit cycle. that being so, we can establish the exact general harmonic solution and limit cycle of equation (1.7). 3. exact harmonic and limit cycle solutions 3.1. exact harmonic and isochronous solutions. in this part, we look for equation (1.7), an exact harmonic solution: x(t)= acos(wt +ϕ), (3.1) where a, w and ϕ are arbitrary parameters. thus, substituting equation (3.1) into equation (1.7) yields, after a few algebraic calculations, the results are c1 = a = a2, γ = w2, c2 =1, λ =±βw, and p = 3 2 . the arbitrary constant ϕ can be determined by using initial conditions. from these integrability conditions, the desired van der pol-type equation (1.7) with nonlinear restoring force takes the form: ẍ +β(x2 −a2)ẋ +w2x ±βw(a2 −x2) 3 2 =0. (3.2) comparing equation (3.2) with lienard equation (1.1) yields f (x) = β(x2 − a2) and g(x) = w2x ±βw(a2 −x2) 3 2 . hence, for g(x)= w2x −βw(a2 −x2) 3 2 , equation (3.2) is written: ẍ +β(x2 −a2)ẋ +w2x −βw(a2 −x2) 3 2 =0, (3.3) which admits the exact and general solution (3.1). for g(x)= w2x +βw(a2−x2) 3 2 , equation (3.2) becomes: ẍ +β(x2 −a2)ẋ +w2x +βw(a2 −x2) 3 2 =0, (3.4) and has the exact and general solution: x(t)= asin(wt +ϕ1), (3.5) 6 int. j. anal. appl. (2022), 20:73 where ϕ1 is an arbitrary constant that can be calculated by the application of initial conditions. 3.2. existence theorem analysis. as seen, the functions f (x) and g(x) do not satisfy the conditions required by usual theorems for the existence of a centre at the origin [1–3,12]. for example, according to theorem 11.3 of [ [1] p.390], the origin is a centre for the lienard equation (1.1) when f (x) and g(x) are odd, and g(0) = 0. however, the previous expression of f (x) is not odd but rather even. the previous formulas of g(x) are not odd, and g(0) 6=0 while the general solutions (3.1) and (3.5) show that the origin is an isochronous centre for equations (3.3) and (3.4). 3.3. phase plane analysis. consider equation (3.3). then, the equivalent dynamical system can be written as follows:   ẋ = y ẏ =−β(x2 −a2)y −w2x +βw(a2 −x2) 3 2 . (3.6) the equilibrium points are given by y =0, and: βw(a2 −x2) 3 2 −w2x =0. (3.7) one can easily observe that x = 0 does not satisfy equation (3.7). therefore, for the qualitative theory of differential equations, equation (3.3) cannot have the origin as an equilibrium point, which contradicts the exact and general harmonic solutions (3.1). according to equation (3.7), x = 0, when: βwa3 =0. (3.8) equation (3.8) holds only, as w 6=0, and a 6=0, when β =0. in this case, equation (3.3) reduces to the linear harmonic oscillator equation: ẍ +w2x =0. (3.9) the previous analysis also holds for equation (3.4). the above shows that the classical theorems for the existence of isochronous centres clearly exclude a number of lienard equations, as seen in several previous papers [7,8]. 3.4. application of averaging method for equation (3.3). in this part, we investigate the existence of a limit cycles for equation (3.3) using the averaging method. equation (3.3) can be written as follows: ẍ +w2x +β [ (x2 −a2)ẋ −w(a2 −x2) 3 2 ] =0. (3.10) the equation (3.10) has the form: ẍ +w2x = εf(ẋ,x), (3.11) int. j. anal. appl. (2022), 20:73 7 where f(ẋ,x) = −(x2 − a2)ẋ + w(a2 − x2) 3 2 and β = ε, such that ε is small parameter, that is, 0 < ε � 1, as required in the application of the averaging method. now, we seek for equation (3.11) the solution of the form: x(t)= r(t)cos ( wt +φ(t) ) , (3.12) under the initial conditions: x(0)= a, ẋ(0)=0, (3.13) such that: ẋ(t)=−wr(t)sin ( wt +φ(t) ) , (3.14) and ṙ(t)cos ( wt +φ(t) ) − r(t)φ̇(t)sin ( wt +φ(t) ) =0. (3.15) hence, knowing that: (a2 −x2) 3 2 =(a2 − r2(t) 2 ) [ 1− 3r2(t) 4(a2 − r 2(t) 2 ) cos(2wt +2φ(t)) ] , (3.16) one can, after a little algebraic manipulation, obtain: f(ẋ,x)= w [ a2 − r2(t) 2 ]3 2 − 3 4 w [ a2 − r2(t) 2 ]1 2 r2(t)cos(2wt +2φ(t))+ 1 4 wr3(t)sin(3wt +3φ(t))+wr(t) [ r2(t) 4 −a2 ] sin(wt +φ(t)). (3.17) from: ṙ(t)=− ε 2πw ∫ 2π 0 f [ r(t)cos(wt +φ(t)),−wr(t)sin(wt +φ(t)) ] sin(wt +φ(t))d(wt +φ(t)), (3.18) and φ̇(t)=− ε 2πwr(t) ∫ 2π 0 f [ r(t)cos(wt+φ(t)),−wr(t)sin(wt+φ(t)) ] cos(wt+φ(t))d(wt+φ(t)). (3.19) after a few algebra, it results in: ṙ(t)= εr(t) 8 [ 4a2 − r2(t) ] (3.20) integrating, after separation of variables, yields: r(t)= 2a[ 1+e−εa 2(t+k) ]1 2 , (3.21) 8 int. j. anal. appl. (2022), 20:73 where k is a constant of integration. using the initial conditions (3.13), the following is obtained: r(t)= 2a[ 1+3e−εa 2t ]1 2 , (3.22) where e−εa 2k =3. now, from equation (3.19): φ̇(t)=0, that is: φ(t)= ϕ0, (3.23) where ϕ0 is a constant. in this context, the desired solution (3.12) takes the form: x(t)= 2a[ 1+3e−εa 2t ]1 2 cos(wt +ϕ0). (3.24) when a = w =1, and ϕ0 =0, the solution (3.23) becomes: x(t)= 2 √ 1+3e−εt cos(t). (3.25) it is worth mentioning that the formula (3.25) is the solution obtained for the van der pol oscillator equation: ẍ +ε(x2 −1)ẋ +x =0, (3.26) by strogatz [ [9], p.225] using the averaging method. 4. numerical applications to do so, it is first necessary to determine the constants ϕ and ϕ1 from the initial conditions. 4.1. solution (3.1) in terms of x0 and v0. from the general initial conditions x(0)= x0 and ẋ(0)= v0, one can obtain, using the general solution (3.1), the system of algebraic equations: x0 = acosϕ, v0 =−wasinϕ, (4.1) which yields the constant: ϕ = arctan ( − v0 wx0 ) . (4.2) using this expression, the general solution (3.1) takes the form: x(t)= acos [ wt −arctan ( v0 wx0 )] . (4.3) figure 1 shows the graphical comparison of the result (4.3) in the circles line with the solution obtained by numerical integration of equation (3.3) in solid line where a = 1, w = 1, β = 0.01, x0 =1, and v0 =0.001. int. j. anal. appl. (2022), 20:73 9 figure 1. comparison of solution (4.3) to the numerical solution of equation (3.3).typical values are a =1, w =1, β =0.01, x0 =1, and ϑ =0.001. 4.2. solution (3.12) in terms of x0 and v0. under the general initial conditions that x(0)= x0 and ẋ(0)= v0, solution (3.5) leads to the system of algebraic equations: x0 = asinϕ1, v0 = wacosϕ1, (4.4) such that the constant ϕ1 is defined as: ϕ1 = arccotan ( v0 wx0 ) . (4.5) from this, the general solution (3.5) can be rewritten in the form: x(t)= asin [ wt +arccot ( v0 wx0 )] . (4.6) the graphical comparison of this solution (4.6) in the circles line with the result obtained by numerical integration of equation (3.4) is depicted in figure 2, where a = 1, w = 1, β = 0.01, x0 =1, and v0 =0.001. figure 2. comparison of solution (4.6) to the numerical solution of equation (3.4).typical values are a =1, w =1, β =0.01, x0 =1, and ϑ =0.001. 10 int. j. anal. appl. (2022), 20:73 5. conclusion we have presented exceptional nonpolynomial van der pol oscillator equations in this paper. we have exhibited their exact harmonic and limit cycle solutions while they do not satisfy classical theorems for the existence of at least one periodic solution. we have proven that the van der pol-duffing-type equations are equivalent to the conservative duffing equations so that they could not admit limit cycle oscillations. additionally, we have shown that the damped quintic duffing equations are equivalent to the conservative cubic-quintic duffing equations. it was for the first time such results have been obtained in the literature. authors’ contributions: all authors have equal contributions in this paper. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.w. jordan, p. smith, nonlinear ordinary differential equations, 4th ed, oxford university press, new york, 2007 [2] r.e. mickens, oscillations in planar dynamic systems, world scientific, singapore, 1996. https://doi.org/10. 1142/2778. [3] m. cioni, g. villari, an extension of dragilev?s theorem for the existence of periodic solutions of the liénard equation, nonlinear anal.: theory meth. appl. 127 (2015), 55-70. https://doi.org/10.1016/j.na.2015.06.026. [4] m. d. monsia, j. akande, k.k.d. adjaï, on the non-existence of limit cycles of the duffing-van der pol type equations, (2021). https://doi.org/10.6084/m9.figshare.14547501.v1. [5] f.e. udwadia, h. cho, first integrals and solutions of duffing-van der pol type equations, j. appl. mech. 81 (2013), 034501. https://doi.org/10.1115/1.4024673. [6] t. stachowiak, hypergeometric first integrals of the duffing and van der pol oscillators, j. differ. equ. 266 (2019), 5895-5911. https://doi.org/10.1016/j.jde.2018.10.049. [7] j. akande, k.k.d. adjaï, a.b. yessoufou, et al. exact and sinusoidal periodic solutions of lienard equation without restoring force, (2021). https://doi.org/10.6084/m9.figshare.14546019. [8] a.r.o. akplogan, k.k.d. adjaï, j. akande, et al. modified van der pol-helmohltz oscillator equation with exact harmonic solutions, (2021). https://doi.org/10.21203/rs.3.rs-730159/v1. [9] s.h. strogatz, nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, westview press, cambridge, 2001. [10] v.k. chandrasekar, s.n. pandey, m. senthilvelan, et al. a simple and unified approach to identify integrable nonlinear oscillators and systems, j. math. phys. 47 (2006), 023508. https://doi.org/10.1063/1.2171520. [11] e.e. obinwanne, a.r. okeke, on application of lyapunov and yoshizawa’s theorems on stability, asymptotic stability, boundaries and periodicity of solution of duffing’s equation, asian j. appl. sci. 2 (2014), 970-975. [12] m. sabatini, on the period function of liénard systems, j. differ. equ. 152 (1999), 467-487. https://doi.org/ 10.1006/jdeq.1998.3520. https://doi.org/10.1142/2778 https://doi.org/10.1142/2778 https://doi.org/10.1016/j.na.2015.06.026 https://doi.org/10.6084/m9.figshare.14547501.v1 https://doi.org/10.1115/1.4024673 https://doi.org/10.1016/j.jde.2018.10.049 https://doi.org/10.6084/m9.figshare.14546019 https://doi.org/10.21203/rs.3.rs-730159/v1 https://doi.org/10.1063/1.2171520 https://doi.org/10.1006/jdeq.1998.3520 https://doi.org/10.1006/jdeq.1998.3520 1. introduction 2. van der pol equations with polynomial restoring force 2.1. van der pol-duffing equations 2.2. generalized van der pol-type equation 3. exact harmonic and limit cycle solutions 3.1. exact harmonic and isochronous solutions 3.2. existence theorem analysis 3.3. phase plane analysis 3.4. application of averaging method for equation (3.3) 4. numerical applications 4.1. solution (3.1) in terms of x0 and v0. 4.2. solution (3.12) in terms of x0 and v0. 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 52-63 http://www.etamaths.com on generalized inequalities of hermite-hadamard type for convex functions çetin yildiz1,∗, m. emin özdemir2 abstract. in this paper, new integral inequalities of hermite-hadamard type are developed for n−times differentiable convex functions. also a parallel development is made base on concavity. 1. introduction a function f : [a, b] ⊂ r → r is said to be convex if whenever x, y ∈ [a, b] and t ∈ [0, 1], the following inequality holds: f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y). we say that f is concave if (−f) is convex. this definition has its origins in jensen’s results from [6] and has opened up the most extended, useful and multi-disciplinary domain of mathematics, namely, convex analysis. convex curves and convex bodies have appeared in mathematical literature since antiquity and there are many important results related to them. on november 22, 1881, hermite (1822-1901) sent a letter to the journal mathesis. this letter was published in mathesis 3 (1883, p: 82) and in this letter an inequality presented which is well-known in the literature as hermite-hadamard integral inequality: f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 (1.1) where f : i ⊆ r → r is a convex function on the interval i of a real numbers and a, b ∈ i with a < b. if the function f is concave, the inequality in (1.1) is reversed. the inequalities (1.1) have become an important cornerstone in mathematical anlysis and optimization. many uses of these inequalities have been discovered in a variety of settings. moreover , many inequalities of special means can be obtained for a particular choice of the function f. due to the rich geometrical significance of hermite-hadamard’s inequality, there is growing literature providing its new proofs, extensions, refinements and generalizations, see for example ( [4], [7][11], [13], [14][19]) and the references therein. in 2000, cerone et. al. (see [3]) proved the following generalization for n−times differentiable functions. theorem 1.1. let f : [a, b] → r be a mapping such that the derivative f(n−1) (n ≥ 1) is absolutely continuous on [a, b]. then∫ b a f(t)dt = n−1∑ k=0 1 (k + 1)! [ (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k)(b) ] + 1 n! ∫ b a (x− t)nf(n)(t)dt, for all x ∈ [a, b]. received 10th january, 2017; accepted 22nd march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 26d15, 26d10. key words and phrases. hermite-hadamard inequality; hölder inequality; convex functions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 52 convex functions 53 for other recent results concerning the n−times differentiable functions see [1][3], [5], [7], [12], [17], [18] where further references are given. in [8], kavurmacı et. al. obtained the following theorems. theorem 1.2. let f : i ⊆ r → r be a differentiable mapping on i◦ such that f′ ∈ l[a, b], where a, b ∈ i with a < b. if |f′| is convex on [a, b], then the following inequality holds:∣∣∣∣∣(x−a)f(a) + (b−x)f (b)(b−a) − 1b−a ∫ b a f(u)du ∣∣∣∣∣ (1.2) ≤ (x−a)2 (b−a) [ 2|f′(a)| + |f′(x)| 6 ] + (b−x)2 (b−a) [ |f′(x)| + 2|f′(b)| 6 ] for each x ∈ [a, b]. theorem 1.3. let f : i ⊆ r → r be a differentiable mapping on i◦ such that f′ ∈ l[a, b], where a, b ∈ i with a < b. if |f′| p p−1 is convex on [a, b] and for some fixed q > 1, then the following inequality holds: ∣∣∣∣∣(x−a)f(a) + (b−x)f (b)(b−a) − 1b−a ∫ b a f(u)du ∣∣∣∣∣ (1.3) ≤ ( 1 p + 1 )1 p ( 1 2 )1 q × [ (x−a)2 [|f′(a)|q + |f′(x)|q] 1 q + (b−x)2 [|f′(x)|q + |f′(b)|q] 1 q b−a ] and q = p p−1. theorem 1.4. let f : i ⊆ r → r be a differentiable mapping on i◦ such that f′ ∈ l[a, b], where a, b ∈ i with a < b. if |f′|q is convex on [a, b] and for some fixed q ≥ 1, then the following inequality holds: ∣∣∣∣∣(x−a)f(a) + (b−x)f (b)(b−a) − 1b−a ∫ b a f(u)du ∣∣∣∣∣ (1.4) ≤ 1 2 ( 1 3 )1 q × [ (x−a)2 [|f′(x)|q + 2|f′(a)|q] 1 q + (b−x)2 [|f′(x)|q + 2|f′(b)|q] 1 q b−a ] for each x ∈ [a, b]. the main purpose of the present paper is to establish several new inequalities for n− times differantiable mappings that are connected with the celebrated hermite-hadamard integral inequality. 2. main results lemma 2.1. let f : [a, b] → r be n-times differentiable functions. if f(n) ∈ l[a, b], then∫ b a f(t)dt = n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! (2.1) +(−1)n (x−a)n+1 n! ∫ 1 0 (t− 1)nf(n)(tx + (1 − t)a)dt +(−1)n (b−x)n+1 n! ∫ 1 0 (1 − t)nf(n)(tx + (1 − t)b)dt where x ∈ [a, b] and n natural number, n ≥ 1. 54 yildiz and özdemir proof. the proof is by mathematical induction. the case n = 1 is [ [8], lemma 1]. assume that (2.1) holds for ”n” and let us prove it for ”n + 1”. that is, we have to prove the equality ∫ b a f(t)dt = n∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! (2.2) +(−1)n+1 (x−a)n+2 (n + 1)! ∫ 1 0 (t− 1)n+1f(n+1)(tx + (1 − t)a)dt +(−1)n+1 (b−x)n+2 (n + 1)! ∫ 1 0 (1 − t)n+1f(n+1)(tx + (1 − t)b)dt where x ∈ [a, b]. then, we can write i = (x−a)n+2 (n + 1)! ∫ 1 0 (t− 1)n+1f(n+1)(tx + (1 − t)a)dt + (b−x)n+2 (n + 1)! ∫ 1 0 (1 − t)n+1f(n+1)(tx + (1 − t)b)dt and integrating by parts gives i = (x−a)n+2 (n + 1)! { (t− 1)n+1 f(n)(tx + (1 − t)a) x−a ∣∣∣∣1 0 − m + 1 x−a ∫ 1 0 (t− 1)nf(n)(tx + (1 − t)a)dt } + (b−x)n+2 (n + 1)! { (1 − t)n+1 f(n)(tx + (1 − t)b) x− b ∣∣∣∣1 0 + m + 1 x− b ∫ 1 0 (1 − t)nf(n)(tx + (1 − t)b)dt } = (−1)n+2 (x−a)n+1 (n + 1)! f(n)(a) − (x−a)n+1 n! ∫ 1 0 (t− 1)nf(n)(tx + (1 − t)a)dt + (b−x)n+1 (n + 1)! f(n)(b) − (b−x)n+1 n! ∫ 1 0 (1 − t)nf(n)(tx + (1 − t)b)dt. now, using the mathematical induction hypothesis, we get (2.3) 1 (−1)n ∫ b a f(t)dt = 1 (−1)n n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! +(−1)n+2 (x−a)n+1 (n + 1)! f(n)(a) + (b−x)n+1 (n + 1)! f(n)(b) − i. convex functions 55 multiplying the both sides of (2.3) by (−1)n, we obtain ∫ b a f(t)dt = n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! + (x−a)n+1 (n + 1)! f(n)(a) + (−1)n (b−x)n+1 (n + 1)! f(n)(b) −(−1)n { (x−a)n+2 (n + 1)! ∫ 1 0 (t− 1)n+1f(n+1)(tx + (1 − t)a)dt + (b−x)n+2 (n + 1)! ∫ 1 0 (1 − t)n+1f(n+1)(tx + (1 − t)b)dt } = n∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! +(−1)n+1 (x−a)n+2 (n + 1)! ∫ 1 0 (t− 1)n+1f(n+1)(tx + (1 − t)a)dt +(−1)n+1 (b−x)n+2 (n + 1)! ∫ 1 0 (1 − t)n+1f(n+1)(tx + (1 − t)b)dt. thus, the identity (2.2) and the lemma is proved. � theorem 2.1. let f : i ⊂ r → r be n−times differentiable function, a, b ∈ i and a < b. if f(n) ∈ l[a, b] and ∣∣f(n)∣∣ (n ≥ 1) is convex on [a, b], then we have ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.4) ≤ (x−a)n+1 (n + 2)! { (n + 1) ∣∣∣f(n)(a)∣∣∣ + ∣∣∣f(n)(x)∣∣∣} + (b−x)n+1 (n + 2)! {∣∣∣f(n)(x)∣∣∣ + (n + 1) ∣∣∣f(n)(b)∣∣∣} where x ∈ [a, b]. proof. from lemma 2.1 and using the properties of modulus, we can write ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! ∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)a)∣∣∣dt + (b−x)n+1 n! ∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)b)∣∣∣dt. 56 yildiz and özdemir since ∣∣f(n)∣∣ is convex on [a, b], it follows that∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! ∫ 1 0 (1 − t)n [ t ∣∣∣f(n)(x)∣∣∣ + (1 − t) ∣∣∣f(n)(a)∣∣∣]dt + (b−x)n+1 n! ∫ 1 0 (1 − t)n [ t ∣∣∣f(n)(x)∣∣∣ + (1 − t) ∣∣∣f(n)(b)∣∣∣]dt = (x−a)n+1 (n + 2)! { (n + 1) ∣∣∣f(n)(a)∣∣∣ + ∣∣∣f(n)(x)∣∣∣} + (b−x)n+1 (n + 2)! {∣∣∣f(n)(x)∣∣∣ + (n + 1) ∣∣∣f(n)(b)∣∣∣} . this completes the proof. � remark 2.1. in the inequality (2.4), if we choose n = 1, then we have the inequality (1.2). corollary 2.1. in the inequality (2.4), if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we have∣∣∣∣∣f(a) + f(b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a)2 192 { 3 |f′′(a)| + 2 ∣∣∣∣f′′ ( a + b 2 )∣∣∣∣ + 3 |f′′(b)| } ≤ (b−a)2 48 {|f′′(a)| + |f′′(b)|} . theorem 2.2. let f : i ⊂ r → r be n−times differentiable function, x ∈ [a, b] and a < b. if f(n) ∈ l[a, b] and ∣∣f(n)∣∣q (n ≥ 1) is convex on [a, b], then we the following inequality:∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.5) ≤ ( 1 np + 1 )1 p  (x−a) n+1 n! [∣∣f(n)(a)∣∣q + ∣∣f(n)(x)∣∣q 2 ]1 q + (b−x)n+1 n! [∣∣f(n)(x)∣∣q + ∣∣f(n)(b)∣∣q 2 ]1 q   where 1 p + 1 q = 1. proof. using lemma 2.1 and hölder integral inequality, we obtain∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! (∫ 1 0 (1 − t)npdt )1 p (∫ 1 0 ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt)1q + (b−x)n+1 n! (∫ 1 0 (1 − t)npdt )1 p (∫ 1 0 ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt)1q . convex functions 57 since ∣∣f(n)∣∣q is convex on [a, b], then ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! ( 1 np + 1 )1 p (∫ 1 0 [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(a)∣∣∣q]dt)1q + (b−x)n+1 n! ( 1 np + 1 )1 p (∫ 1 0 [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(b)∣∣∣q]dt)1q = ( 1 np + 1 )1 p  (x−a) n+1 n! [∣∣f(n)(a)∣∣q + ∣∣f(n)(x)∣∣q 2 ]1 q + (b−x)n+1 n! [∣∣f(n)(x)∣∣q + ∣∣f(n)(b)∣∣q 2 ]1 q   which completes the proof. � remark 2.2. in theorem 2.2, if we choose n = 1, then we have the inequality (1.3). corollary 2.2. in theorem 2.2, if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we obtain ∣∣∣∣∣f(a) + f(b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a)2 16 ( 1 2p + 1 )1 p ×   [ |f′′(a)|q + ∣∣f′′(a+b 2 )∣∣q 2 ]1 q + [∣∣f′′(a+b 2 )∣∣q + |f′′(b)|q 2 ]1 q   . theorem 2.3. let f : i ⊂ r → r be n−times differentiable function and a < b. if f(n) ∈ l[a, b] and∣∣f(n)∣∣q is convex on [a, b], then we get ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.6) ≤ ( q − 1 nq + q −p− 1 )1−1 q × { (x−a)n+1 n! [ 1 p + 2 ∣∣∣f(n)(a)∣∣∣q + 1 (p + 1)(p + 2) ∣∣∣f(n)(x)∣∣∣q]1q + (b−x)n+1 n! [ 1 (p + 1)(p + 2) ∣∣∣f(n)(x)∣∣∣q + 1 p + 2 ∣∣∣f(n)(b)∣∣∣q]1q } where 1 p + 1 q = 1 and x ∈ [a, b]. 58 yildiz and özdemir proof. from lemma 2.1 and using the properties of modulus, we get ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! ∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)a)∣∣∣dt + (b−x)n+1 n! ∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)b)∣∣∣dt = (x−a)n+1 n! ∫ 1 0 (1 − t)n(1 − t) p q (1 − t) p q ∣∣∣f(n)(tx + (1 − t)a)∣∣∣dt + (b−x)n+1 n! ∫ 1 0 (1 − t)n(1 − t) p q (1 − t) p q ∣∣∣f(n)(tx + (1 − t)b)∣∣∣dt. using the hölder integral inequality, we can write ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n!  ∫ 1 0 [ (1 − t)n (1 − t) p q ] q q−1 dt  1− 1 q (∫ 1 0 (1 − t)p ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt)1q + (b−x)n+1 n!  ∫ 1 0 [ (1 − t)n (1 − t) p q ] q q−1 dt  1− 1 q (∫ 1 0 (1 − t)p ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt)1q . since ∣∣f(n)∣∣q is convex on [a, b], then ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! (∫ 1 0 (1 − t) nq−p q−1 dt )1−1 q (∫ 1 0 (1 − t)p [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(a)∣∣∣q]dt)1q + (b−x)n+1 n! (∫ 1 0 (1 − t) nq−p q−1 dt )1−1 q (∫ 1 0 (1 − t)p [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(b)∣∣∣q]dt)1q = ( q − 1 nq + q −p− 1 )1−1 q × { (x−a)n+1 n! [ 1 p + 2 ∣∣∣f(n)(a)∣∣∣q + 1 (p + 1)(p + 2) ∣∣∣f(n)(x)∣∣∣q]1q + (b−x)n+1 n! [ 1 (p + 1)(p + 2) ∣∣∣f(n)(x)∣∣∣q + 1 p + 2 ∣∣∣f(n)(b)∣∣∣q]1q } which completes the proof of the theorem. � convex functions 59 corollary 2.3. in theorem 2.3, if we choose n = 1, we have∣∣∣∣∣(x−a)f(a) + (b−x)f (b)(b−a) − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ (2.7) ≤ ( q − 1 2q −p− 1 )1−1 q × { (x−a)2 (b−a) [ 1 p + 2 |f′(a)|q + 1 (p + 1)(p + 2) |f′(x)|q ]1 q + (b−x)2 (b−a) [ 1 (p + 1)(p + 2) |f′(x)|q + 1 p + 2 |f′(b)|q ]1 q } . corollary 2.4. in the inequality (2.7), if we choose x = a+b 2 , then we obtain∣∣∣∣∣f(a) + f (b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ b−a 4 ( q − 1 2q −p− 1 )1−1 q × {[ 1 p + 2 |f′(a)|q + 1 (p + 1)(p + 2) ∣∣∣∣f′ ( a + b 2 )∣∣∣∣q ]1 q + [ 1 (p + 1)(p + 2) ∣∣∣∣f′ ( a + b 2 )∣∣∣∣q + 1p + 2 |f′(b)|q ]1 q } . corollary 2.5. in theorem 2.3, if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we have∣∣∣∣∣f(a) + f (b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a)2 16 ( q − 1 3q −p− 1 )1−1 q × {[ 1 p + 2 |f′′(a)|q + 1 (p + 1)(p + 2) ∣∣∣∣f′′ ( a + b 2 )∣∣∣∣q ]1 q + [ 1 (p + 1)(p + 2) ∣∣∣∣f′′ ( a + b 2 )∣∣∣∣q + 1p + 2 |f′′(b)|q ]1 q } . theorem 2.4. for n ≥ 1, let f : i ⊂ r → r be n−times differentiable function and a < b. if f(n) ∈ l[a, b] and ∣∣f(n)∣∣q is convex on [a, b], for q ≥ 1, then the following inequality holds: ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.8) ≤ (x−a)n+1 (n + 1)! [ (n + 1) ∣∣f(n)(a)∣∣q + ∣∣f(n)(x)∣∣q (n + 2) ]1 q + (b−x)n+1 (n + 1)! [∣∣f(n)(x)∣∣q + (n + 1) ∣∣f(n)(b)∣∣q (n + 2) ]1 q . 60 yildiz and özdemir proof. from lemma 2.1 and using the well known power-mean integral inequality, we have ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! (∫ 1 0 (1 − t)ndt )1−1 q (∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt)1q + (b−x)n+1 n! (∫ 1 0 (1 − t)ndt )1−1 q (∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt)1q . since ∣∣f(n)∣∣q is convex on [a, b], for q ≥ 1, then we obtain ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 n! ( 1 n + 1 )1−1 q (∫ 1 0 (1 − t)n [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(a)∣∣∣q]dt)1q + (b−x)n+1 n! ( 1 n + 1 )1−1 q (∫ 1 0 (1 − t)n [ t ∣∣∣f(n)(x)∣∣∣q + (1 − t) ∣∣∣f(n)(b)∣∣∣q]dt)1q = (x−a)n+1 (n + 1)! [ (n + 1) ∣∣f(n)(a)∣∣q + ∣∣f(n)(x)∣∣q (n + 2) ]1 q + (b−x)n+1 (n + 1)! [∣∣f(n)(x)∣∣q + (n + 1) ∣∣f(n)(b)∣∣q (n + 2) ]1 q . hence, the proof of the theorem is completed. � remark 2.3. in theorem 2.4, if we choose n = 1, we obtain the inequality (1.4). corollary 2.6. in the inequality (2.8) if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we get ∣∣∣∣∣f(a) + f (b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a)2 48   [ 3 |f′′(a)|q + ∣∣f′′(a+b 2 )∣∣q 4 ]1 q + [∣∣f′′(a+b 2 )∣∣q + 3 |f′′(b)|q 4 ]1 q   . theorem 2.5. let f : i ⊂ r → r be n−times differentiable function and a < b. if f(n) ∈ l[a, b] and∣∣f(n)∣∣q is concave on [a, b], then we obtain ∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.9) ≤ ( 1 np + 1 )1 p { (x−a)n+1 n! ∣∣∣∣f(n) ( a + x 2 )∣∣∣∣ + (b−x)n+1n! ∣∣∣∣f(n) ( x + b 2 )∣∣∣∣ } where 1 p + 1 q = 1. convex functions 61 proof. from lemma 2.1 and hölder integral inequality, we obtain∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.10) ≤ (x−a)n+1 n! {(∫ 1 0 (1 − t)npdt )1 p (∫ 1 0 ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt)1q + (∫ 1 0 (1 − t)npdt )1 p (∫ 1 0 ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt)1q } . since ∣∣f(n)∣∣q is concave on [a, b], we can write the following inequalities via jensen inequality: (2.11)∫ 1 0 ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt = ∫ 1 0 t0 ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt ≤ (∫ 1 0 t0dt )∣∣∣∣∣f(n) (∫ 1 0 (tx + (1 − t)a)dt∫ 1 0 t0dt )∣∣∣∣∣ q = ∣∣∣∣f(n) ( a + x 2 )∣∣∣∣q and similarly ∫ 1 0 ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt ≤ ∣∣∣∣f(n) ( x + b 2 )∣∣∣∣q . (2.12) thus, if we use (2.11) and (2.12) in the inequality (2.10), we obtain the inequality of (2.9). this completes the proof. � corollary 2.7. in the inequality (2.9) if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we obtain∣∣∣∣∣f(a) + f (b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a)2 16 ( 1 2p + 1 )1 p {∣∣∣∣f′′ ( 3a + b 4 )∣∣∣∣ + ∣∣∣∣f′′ ( a + 3b 4 )∣∣∣∣ } . theorem 2.6. let f : i ⊂ r → r be n−times differentiable function and a < b. if f(n) ∈ l[a, b] and∣∣f(n)∣∣q is concave on [a, b], for q ≥ 1, then the following inequality holds: (2.13)∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ ≤ (x−a)n+1 (n + 1)! ∣∣∣∣f(n) ( (n + 1)a + x n + 2 )∣∣∣∣ + (b−x)n+1(n + 1)! ∣∣∣∣f(n) ( x + (n + 1)b n + 2 )∣∣∣∣ . proof. from lemma 2.1 and using the well known power-mean inequality, we have∣∣∣∣∣ ∫ b a f(t)dt− n−1∑ k=0 (x−a)k+1f(k)(a) + (−1)k(b−x)k+1f(k) (b) (k + 1)! ∣∣∣∣∣ (2.14) ≤ (x−a)n+1 n! (∫ 1 0 (1 − t)ndt )1−1 q (∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt)1q + (b−x)n+1 n! (∫ 1 0 (1 − t)ndt )1−1 q (∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt)1q . 62 yildiz and özdemir using the jensen inequality, we can write (2.15)∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)a)∣∣∣q dt ≤ (∫ 1 0 (1 − t)ndt )∣∣∣∣∣f(n) (∫ 1 0 (1 − t)n(tx + (1 − t)a)dt∫ 1 0 (1 − t)ndt )∣∣∣∣∣ q = ( 1 n + 1 )∣∣∣∣f(n) ( (n + 1)a + x n + 2 )∣∣∣∣q and similarly ∫ 1 0 (1 − t)n ∣∣∣f(n)(tx + (1 − t)b)∣∣∣q dt ≤ ( 1 n + 1 )∣∣∣∣f(n) ( x + (n + 1)b n + 2 )∣∣∣∣q . (2.16) thus, if we use (2.15) and (2.16) in the inequality (2.14), we obtain the inequality of (2.13). the proof of the theorem is completed. � corollary 2.8. in theorem 2.6, if we choose n = 2, x = a+b 2 and f′(x) = f′(a + b − x) (that is, f′ symmetric function), then we have∣∣∣∣∣f(a) + f (b)2 − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (b−a) 2 48 {∣∣∣∣f′′ ( 7a + b 8 )∣∣∣∣ + ∣∣∣∣f′′ ( a + 7b 8 )∣∣∣∣ } . references [1] s.-p. bai, s.-h. wang and f. qi, some hermite-hadamard type inequalities for n-time differentiable (α,m)-convex functions, j. inequal. appl. 2012 (2012), article id 267. [2] p. cerone, s.s. dragomir and j. roumeliotis, some ostrowski type inequalities for n-time differentiable mappings and applications, demonstr. math. 32 (4) (1999), 697-712. [3] p. cerone, s.s. dragomir and j. roumeliotis and j. šunde, a new generalization of the trapezoid formula for n-time differentiable mappings and applications, demonstr. math. 33 (4) (2000), 719-736. [4] s.s. dragomir and c.e.m. pearce, selected topics on hermite-hadamard inequalities and applications, rgmia monographs, victoria university, 2000. online:[http://www.staxo.vu.edu.au/rgmia/monographs/hermite hadamard.html]. [5] d.-y. hwang, some inequalities for n-time differentiable mappings and applications, kyung. math. j. 43 (2003), 335-343. [6] j. l. w. v. jensen, on konvexe funktioner og uligheder mellem middlvaerdier, nyt. tidsskr. math. b., 16 (1905), 49-69. [7] w.-d. jiang, d.-w. niu, y. hua and f. qi, generalizations of hermite-hadamard inequality to n-time differentiable function which are s-convex in the second sense, analysis (munich), 32 (2012), 209-220. [8] h. kavurmaci, m. avci, m.e. özdemir, new inequalities of hermite–hadamard type for convex functions with applications, j. inequal. appl. 2011 (2011), article id 86. [9] u.s. kırmacı, inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, appl. math. comput., 147 (2004), 137–146. [10] u.s. kırmacı, m.k. bakula, m.e. özdemir and j. pećarić, hadamard-type inequalities for s-convex functions, appl. math. comput., 193 (2007), 26-35. [11] m.e. özdemir and u.s. kırmacı, two new theorem on mappings uniformly continuous and convex with applications to quadrature rules and means, appl. math. comput. 143 (2003), 269-274. [12] m.e. özdemir, ç. yıldız, new inequalities for n-time differentiable functions, arxiv:1402.4959v1. [13] m.e. özdemir, ç. yıldız, new inequalities for hermite-hadamard and simpson type with applications, tamkang j. math. 44 (2) (2013) 209-216. [14] a. saglam, m.z sarıkaya and h. yıldırım, some new inequalities of hermite-hadamard’s type, kyung. math. j. 50 (2010), 399-410. [15] m.z. sarıkaya and n. aktan, on the generalization some integral inequalities and their applications, math. comput. modelling, 54 (2011), 2175-2182. [16] e. set, m.e. özdemir and s.s. dragomir, on hadamard-type inequalities involving several kinds of convexity, j. inequal. appl. 2010 (2010) article id 286845. [17] ç. yıldız, new inequalities of the hermite-hadamard type for n-time differentiable functions which are quasiconvex, j. math. inequal. (10) (3) (2016), 703-711. [18] s.h. wang, b.-y. xi and f. qi, some new inequalities of hermite-hadamard type for n-time differentiable functions which are m-convex, analysis (munich), 32 (2012), 247-262. convex functions 63 [19] b.-y. xi and f. qi, some integral inequalities of hermite-hadamard type for convex functions with applications to means, j. funct. spaces appl., 2012 (2012), article id 980438. 1atatürk university, k. k. education faculty, department of mathematics, 25240, campus, erzurum, turkey 2uludağ university, education faculty, department of mathematics, 16059, bursa, turkey ∗corresponding author: cetin@atauni.edu.tr 1. introduction 2. main results references int. j. anal. appl. (2022), 20:54 direct solution of black-scholes-merton european put option model on dividend yield with modified-log payoff function s.e. fadugba1,∗, a.a. adeniji2, m.c. kekana2, j.t. okunlola3, o. faweya4 1department of mathematics, ekiti state university, ado ekiti, 360001, nigeria 2department of mathematics, tshwane university of technology, pretoria, south africa 3department of mathematical and physical sciences, afe babalola university, ado ekiti, nigeria 4department of statistics, ekiti state university, ado ekiti, 360001, nigeria ∗corresponding author: sunday.fadugba@eksu.edu.ng abstract. this paper proposes a framework based on the celebrated transform of mellin type (mt) for the direct solution of the black-scholes-merton european put option model (bsmepom) on dividend yield (dy) with modified-log payoff function (mlpf) under the geometric brownian motion. the focal goal of this paper is to use mt to obtain a valuation formula for the european put option (epo) which pays a dy with mlpf. by means of the mt and its inversion formula, the price of epo on dy was expressed in terms of integral equation. the valuation formula of epo was obtained with the help of the convolution property of mt and final time condition. mt was tested on an illustrative example in order to measure its performance, effectiveness and suitability. the mlpf was compared with other existing payoff functions. hence, the effect of dy on the pricing of epo with mlpf was also investigated. 1. introduction option valuation has become extremely popular in computational finance. this popularity has been displayed as one of the key major areas in derivative security. in other words, option valuation has contributed greatly to the financial markets. there is a massive growth in trading activities on derivatives globally from the inception of the black-scholes pricing formula [1,2]. it is noteworthy to say that the black-scholes models for linear payoff function has been used by many researchers and as well as become one of the utmost areas in financial markets over the last few decades. immediately received: aug. 17, 2022. 2010 mathematics subject classification. 91b24, 91b28, 91g20. key words and phrases. convolution property; dividend yield; final time condition; mellin transform. https://doi.org/10.28924/2291-8639-20-2022-54 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-54 2 int. j. anal. appl. (2022), 20:54 after the huge success recorded by the black-scholes model for vanilla option flavours, several other valuation formula were developed for options pricing with different payoff functions such as mellin transform, binomial model, finite difference method, monte carlo method, e.t.c; see [3] – [6]. for mathematical framework, some implementations of transform methods of different types in financial markets; see [7]– [15]. ghevariya [16] solved the classical black-scholes european put option model for modified-log payoff function with the help of the mt. in this paper, a direct solution of bsmepom via the celebrated transform of mellin type is proposed in the sense of dy and mlpf. the remaining part of the paper is listed as follows, section 2 captures the brief concepts of mt. a new result that captures the governing model for epo on dy with mlpf and the solution of bsmepom on dy with mlpf is stated and proved in section 3. an illustrative example on the application of mt to epo is captured by section 4. section 5 is the concluding part of the paper. 2. mellin transform this section captures some definitions of terms based on the framework of the mellin transform. 2.1. definitions of terms. definition 2.1. let f (x) be a locally lebesgue integrable function. the mellin transform of f (x) is defined as m[f (x),ω] := f̃ (ω)= ∫ ∞ 0 f (x)xω−1dx (2.1) the mellin transform variable ω is a complex number, ω = <(.)+ i=(.), where <(.) is the real part, i is the imaginary unit and =(.) is the imaginary part. definition 2.2. if f (x) is an integrable function with fundamental strips (a,b), then if c is such that a < c < b and {f̃ (ω) : ω = c + it} is integrable, the inverse mellin transform is defined as m−1[f̃ (ω)]= f (x)= 1 2πi ∫ c+i∞ c−i∞ f̃ (ω)x−ωdω (2.2) remark 2.1. for more details on the condition that ensures the existence of mt; see [17]. remark 2.2. the fundamental operational properties of the mellin transforms such as scaling, shifting, derivatives, integrals, convolution, multiplicative convolution and parseval’s formula are well detailed in [9,12,17,18]. 3. the solution of bsmepom on dy with mlpf ghevariya derived bsm formula on non-dividend yield for ml-payoff function [16]. in this section, black-scholes-merton formula on dividend yield with mlpf is derived via the mt in the following result. int. j. anal. appl. (2022), 20:54 3 theorem 3.1. consider the bsmepom on dy with mlpf of the form ∂pe(st,t) ∂t +(r −q)st ∂pe(st,t) ∂st + (σst) 2 2 ∂2pe(st,t) ∂(st) 2 − rpe(st,t)=0 (3.1) subject to the boundary conditions lim st→∞ pe(st,t)=0 on [0,t) (3.2) lim st→0 pe(st,t)= k er(t−t) on [0,t) (3.3) and mlpf pe(st ,t)= [ st ln ( k st )]+ on [0,∞) (3.4) where pe(st,t), t, t, st, k, σ, r and q are the price of epo, current time, time to expiry, underlying asset price, strike price, volatility, risk-free interest rate and dy, respectively, then the valuation formula for bsmepom on dy with mlpf is given by pe(st,t)= ste −qτ [σ√τη(d)− (d1 +d2τ)n(−d)] (3.5) with d1 = ln ( st k ) , (3.6) d2 = ( r −q + σ2 2 ) , (3.7) d = d1 +d2τ σ √ τ , (3.8) τ = t − t,η(κ)= 1 √ 2π e− κ2 2 ,n(κ)= ∫ κ −∞ η(κ)dκ. (3.9) proof. taking the mt of (3.1) and using its linearity, independence of time derivatives and shifting properties and rearranging terms, one obtains ∂p̃e(ω,t) ∂t =− σ2 2 (ω2 +ω(1−α1)−α2)p̃e(ω,t) (3.10) where α1 = 2(r −q) σ2 ,α2 = 2r σ2 solving (3.1), yields p̃e(ω,t)= m(ω)e −1 2 σ2(ω2+ω(1−α1)−α2)t (3.11) but m(ω)=m(pe(st ,t),ω)e 1 2 σ2(ω2+(1−α1)ω−α2)t (3.12) which is equivalent to m(ω)= g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)t (3.13) 4 int. j. anal. appl. (2022), 20:54 substituting (3.13) into (3.11), yields p̃e(ω,t)= g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)τ (3.14) with τ = t − t. by means of (2.2), (3.14) yields pe(st,t)= 1 2πi ∫ c+i∞ c−i∞ g̃(ω)e 1 2 σ2(ω2+(1−α1)ω−α2)τs−ωt dω (3.15) which is the integral equation for governing equation (3.1). let ξ(st)= 1 2πi ∫ c+i∞ c−i∞ e σ2 2 (ω2+ω(1−α1)−α2)s−ωt dω (3.16) using the fact that e 1 2 σ2(ω2+(1−α1)ω−α2)τ = e−β1(β 2 1+α2)+β1(ω+β2) 2 (3.17) where β1 = σ2τ 2 ,β2 = 1−α1 2 (3.18) thus ξ(st)= e−β1(β 2 1+α2) 2πi ∫ c+i∞ c−i∞ eβ1(ω+β2) 2 s−ωt dω (3.19) using the transformation given by [19]. eφω 2 = 1 2 √ π ∫ ∞ 0 1 √ φ exp ( −(lnst)2 4φ ) (st) ω−1dst, <(φ)≥ 0 (3.20) yields ξ(st)= e −β1(β21+α2) sβ2 σ √ 2πτ e −1 2 ( ln(s) σ √ τ )2 (3.21) similarly, ξ ( st v ) = e−β1(β 2 1+α2) (st v )β2 σ √ 2πτ e −1 2   ln ( st v ) σ √ τ  2 (3.22) using the terminal condition (3.4), then g(st)=m−1(g̃(ω))= [ st ln ( k st )]+ (3.23) also g(v)= [ v ln ( k v )]+ (3.24) with the help of the convolution property of mt, (3.15) becomes pe(st,t)= ∫ ∞ 0 g(v)ξ ( st v ) 1 v dv (3.25) substituting (3.22) and (3.24) into (3.25), one gets pe(st,t)= ∫ ∞ 0 [ v ln ( k v )]+ e−β1(β 2 1+α2) (st v )β2 σ √ 2πτ e −1 2   ln ( st v ) σ √ τ  2 1 v dv (3.26) int. j. anal. appl. (2022), 20:54 5 pe(st,t)= e −β1(β21+α2) s β2 t σ √ 2πτ ∫ k 0 [ v ln ( k v )] e −1 2   ln ( st v ) σ √ τ  2 1 v dv (3.27) simplifying further, yields pe(st,t)= e −β1(β21+α2) s β2 t σ √ 2πτ ∫ k 0 ln(k) 1 vβ2 e −1 2   ln ( st v ) σ √ τ  2 dv −e−β1(β 2 1+α2) s β2 t σ √ 2πτ ∫ k 0 ln(v) 1 vβ2 e −1 2   ln ( st v ) σ √ τ  2 dv (3.28) pe(st,t)= e −β1(β21+α2) s β2 t σ √ τ [ln(k)g1 −g2] (3.29) where g1 = 1 √ 2π ∫ k 0 1 vβ2 e −1 2   ln ( st v ) σ √ τ  2 dv (3.30) g2 = 1 √ 2π ∫ k 0 ln(v) vβ2 e −1 2   ln ( st v ) σ √ τ  2 dv (3.31) let y = ln ( s v ) σ √ τ (3.32) thus g2 = σ √ τs −β2+1 t e β1(β2−1)2[σ √ τj1 − ln(st)j2] (3.33) where j1 = 1 √ 2π ∫ ln( stv ) σ √ τ ∞ ye− 1 2 (y−σ √ τ(β2−1))2dy (3.34) j2 = 1 √ 2π ∫ ln( stv ) σ √ τ ∞ e− 1 2 (y−σ √ τ(β2−1))2dy (3.35) taking t = y −σ √ τ(β2 −1),d = ln ( st k ) −σ2τ(β2 −1) σ √ τ = ln ( st k ) + ( r −q + σ 2 2 ) τ σ √ τ (3.36) equations (3.34) and (3.35) become j1 =−[η(d)+σ √ τ(β2 −1)n(−d)] (3.37) and j2 =−n(−d) (3.38) 6 int. j. anal. appl. (2022), 20:54 respectively. substituting (3.37) and (3.38) into (3.33), yields g2 =−σ √ τs −β2+1 t e β1(β2−1)2[σ √ τη(d)+(σ2τ(β2 −1)− ln(st))n(−d)] (3.39) similarly, g1 = σ √ τs −β2+1 t e β1(β2−1)2[n(−d)] (3.40) using (3.39), (3.40), the values of α2, β1, β2 and (3.29), the result follows. hence, this completes the proof � 4. numerical example consider the valuation of the epo on a dy with mlpf via the mt using the following parameters s,k,r,σ,q,t in table 1. table 1. the parameters parameters values s in dollars 100 k in dollars 100, 110, 120, 130, 140, 150 r 8% σ 0.5 q 0,5%,20%,60%,100% t in months 6 the results obtained are displayed in tables 2 and 3. table 2. the comparative study of mlpf, log payoff [20] and linear payoff [1] with q =0 k mlpf log payoff [20] linear payoff [1] 100 0 0 0 110 9.5310 0.0953 10 120 18.2322 0.1823 20 130 26.2364 0.2624 30 140 33.6472 0.3365 40 150 40.5465 0.4055 50 int. j. anal. appl. (2022), 20:54 7 figure 1. the plots of table 2. table 3. the effect of dy on the price of epo with mlpf k/q 0 0.05 0.2 0.6 1 100 9.5684 10.3063 12.6497 19.3538 25.5118 110 13.7482 14.6425 17.3986 24.6869 30.6874 120 18.4538 19.4684 22.5140 30.0284 35.6212 130 23.5158 24.6103 27.8197 35.2583 40.2814 140 28.7839 29.9193 33.1776 40.3074 44.6658 150 34.1365 35.2780 38.4899 45.1414 48.7877 figure 2. physical interpretation of the effect of dividend yield on the price of epo using table 3. 8 int. j. anal. appl. (2022), 20:54 5. conclusion a direct solution of bsmepom via the celebrated transform of mellin type in the sense of dy and mlpf has been proposed in this paper. the mt has the ability of handling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as laplace and fourier types. the integral equation for the representation of the price of epo with dy was obtained. the closed form approximation formula for epo was also obtained via mt with the help of its convolution property and final time condition. moreover, the mt was tested on some parameters to show its performance, effectiveness, and suitability. from table 2, it is clearly seen that the mlpf used in this present paper performed better than the log payoff function used in [20] and also was found to be very close to the linear payoff function of plain vanilla [1]. it is observed from table 3, that the holder is more beneficial to enter into a european put option. in other words, however, the benefits of these cash flows are given to the holder of a put option. table 3 shows that increase in dy leads to increase in the prices of the epo with mlpf. the effect of dy is captured in figure 2. hence, from the results displayed in figures 1 and 2, it can be concluded that mt is suitable for the valuation of epo on mlpf with dy due to its capacity power of solving bsmepom directly in terms of market price. acknowledgements: the authors wish to thank tshwane university of technology for their financial support and the department of higher education and training, south africa. author contributions: the authors contributed, read and approved the final manuscript. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] f. black, m. scholes, the pricing of options and corporate liabilities, j. political econ. 81 (1973), 637–654. https://doi.org/10.1086/260062. [2] s.e. fadugba, c.r. nwozo, valuation of european call options via the fast fourier transform and the improved mellin transform, j. math. finance. 06 (2016), 338–359. https://doi.org/10.4236/jmf.2016.62028. [3] r.c. merton, option pricing when underlying stock returns are discontinuous, j. financial econ. 3 (1976), 125-144. https://doi.org/10.1016/0304-405x(76)90022-2. [4] m.j. brennan, e.s. schwartz, finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, j. financial quant. anal. 13 (1978), 461-474. https://doi.org/10.2307/2330152. [5] j.c. cox, s.a. ross, m. rubinstein, option pricing: a simplified approach, j. financial econ. 7 (1979), 229–263. https://doi.org/10.1016/0304-405x(79)90015-1. [6] p. boyle, m. broadie, p. glasserman, monte carlo methods for security pricing, j. econ. dyn. control. 21 (1997), 1267–1321. https://doi.org/10.1016/s0165-1889(97)00028-6. [7] f.s. emmanuel, the mellin transforms method as an alternative analytic solution for the valuation of geometric asian option, appl. comput. math. 3 (2014), 1-7, https://doi.org/10.11648/j.acm.s.2014030601.11. [8] d.j. manuge, p.t. kim, a fast fourier transform method for mellin-type option pricing, (2014). https://doi. org/10.48550/arxiv.1403.3756. https://doi.org/10.1086/260062 https://doi.org/10.4236/jmf.2016.62028 https://doi.org/10.1016/0304-405x(76)90022-2 https://doi.org/10.2307/2330152 https://doi.org/10.1016/0304-405x(79)90015-1 https://doi.org/10.1016/s0165-1889(97)00028-6 https://doi.org/10.11648/j.acm.s.2014030601.11 https://doi.org/10.48550/arxiv.1403.3756 https://doi.org/10.48550/arxiv.1403.3756 int. j. anal. appl. (2022), 20:54 9 [9] c.r. nwozo, s.e. fadugba, mellin transform method for the valuation of some vanilla power options with nondividend yield, int. j. pure appl. math. 96 (2014), 79-104, https://doi.org/10.12732/ijpam.v96i1.7. [10] c.r. nwozo, s.e. fadugba, performance measure of laplace transforms for pricing path dependent options, int. j. pure appl. math. 94 (2014), 75-197. https://doi.org/10.12732/ijpam.v94i2.5. [11] s.e. fadugba, c.r. nwozo, mellin transform method for the valuation of the american power put option with nondividend and dividend yields, j. math. finance. 05 (2015), 249–272. https://doi.org/10.4236/jmf.2015.53023. [12] s.e. fadugba, solution of fractional order equations in the domain of the mellin transform, j. nigerian soc. phys. sci. 1 (2019), 138–142. https://doi.org/10.46481/jnsps.2019.31. [13] s.e. fadugba, c.r. nwozo, closed-form solution for the critical stock price and the price of perpetual american call options via the improved mellin transforms, int. j. financial markets derivat. 6 (2018), 269-286. https: //doi.org/10.1504/ijfmd.2018.097489. [14] s.e. fadugba, c.t. nwozo, perpetual american power put options with non-dividend yield in the domain of mellin transforms, palestine j. math. 9 (2020), 371-385. [15] s.e. fadugba, laplace transform for the solution of fractional black-scholes partial differential equation for the american put options with non-dividend yield, int. j. stat. econ. 20 (2019), 10-17. [16] s.j. ghevariya, bsm european option pricing formula for ml-payoff function with mellin transform, int. j. math. appl. 6 (2018), 34-36. [17] p. flajolet, x. gourdon, p. dumas, mellin transforms and asymptotics: harmonic sums, theor. computer sci. 144 (1995), 3–58. https://doi.org/10.1016/0304-3975(95)00002-e. [18] a.h. zemanian, generalized integral transformation, dover publications, new york, 1987. [19] a. erdélyi, w. magnus, f. oberhettinger, et al. tables of integral transforms, vol. 1-2, first edition, mcgraw-hill, new york, 1954. [20] p. wilmott, paul wilmott on quantitative finance, second edition, john wiley & sons, hoboken, 2006. https://doi.org/10.12732/ijpam.v96i1.7 https://doi.org/10.12732/ijpam.v94i2.5 https://doi.org/10.4236/jmf.2015.53023 https://doi.org/10.46481/jnsps.2019.31 https://doi.org/10.1504/ijfmd.2018.097489 https://doi.org/10.1504/ijfmd.2018.097489 https://doi.org/10.1016/0304-3975(95)00002-e 1. introduction 2. mellin transform 2.1. definitions of terms 3. the solution of bsmepom on dy with mlpf 4. numerical example 5. conclusion references international journal of analysis and applications volume 17, number 4 (2019), 503-516 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-503 stability and convergence analysis of smoking impact in society with algorithm aspects aqeel ahmad∗, maryam shahid, muhammad farman, m.o. ahmad department of mathematics and statistics, the university of lahore, lahore, pakistan ∗corresponding author: aqeelahmad.740@gmail.com abstract. in this manuscript, an epidemic model employed the dynamics of drugs usage among adults. among smokers, often the desire to quit smoking arises. a large number of smokers attempt to quit, but only a few of them are successful. a non-linear mathematical model is employed to study and assess the dynamics of smoking and its impact on public health in a community. we prove the essential properties, bounded, positivity and well-posed, also local and global stability analysis has been made for the epidemic model. the sensitivity analysis of the model is provided by threshold or reproductive number as well as analyzed qualitatively. we develop an unconditionally convergent nonstandard finite difference scheme by applying mickens approach φ(h) = h + o(h2) instead of h to control the spread of bad impact in society. finally numerical simulations are also established to investigate the influence of the system parameters on the spread of the smoking impact in society. 1. introduction the scope of mathematics includes mathematical modeling and esoteric mathematics. the flow of work, process, predictions and outcomes can easily be measured with the help of mathematical concepts and theories. therefore, biologists are now extremely dependent on mathematics. mathematical modeling of biological sciences has been conducted [1–3]. the relationship between simple mathematical modeling involves biological system, integer order differential equations that show their dynamics and complex system which describes their changing of structure. the nonlinearity and multi-scale behaviors in mathematical modeling received 2019-04-11; accepted 2019-05-14; published 2019-07-01. 2010 mathematics subject classification. 37m05,92b05. key words and phrases. stability analysis; qualitative analysis; convergence analysis; sensitivity analysis. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 503 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-503 int. j. anal. appl. 17 (4) (2019) 504 describe the mutual relationship between parameters [4]. in last few decades, many biological models were studied in detail by using classical derivative [5–8]. smoking is the large problem in the entire world. despite overwhelming facts about smoking, it is still a very bad habit which is widely spread and accepted socially. smoking is a bad habit in which a substance is burned and the resulting smoke breathed to taste and absorbed into the bloodstream [14]. tobacco pandemic is one of the largest public medicinal threats to the world has ever faced, as it puts to death up to half of its users. smoking kills about six million people each year of whom more than five million are ex-smokers and smokers and over 500,000 are nonsmokers revealed to second-hand smoke. tobacco users who pass away prematurely deprive their families of earnings, lift the cost of fitness care and through them in deep financial crisis. the world health organization forecasts that by 2030, ten million persons will pass away every year due to tobacco associated illnesses. numerous mathematical models for smoking have been developed in the last few years [9, 13]. epidemiology is related with the spread of diseases in population and the major factors that effects the transmission of diseases [15]. the subject of smoking is one of the interesting areas in study of epidemiology. there is a lot of work that has been done on the smoking epidemics models [16–18]. mathematical models are established to understand the dynamics of the contagious diseases. these models are divided into compartments with population are called compartmental models. the assumptions of transfer data from one to another compartment depends upon the nature and time rate. the major idea in these models is that the people will start to live healthy in a community. diseases may effect the healthy peoples but effected one could become healthy again in the community [19–22]. different numerical technique can be employed for epidemic model to analyse the data for control strategy. different technique converges conditionally and diverge for large step size for epidemic model but nsfd scheme converges unconditionally. in this paper, we investigate the stability and qualitative analysis of the smoking model. an unconditionally convergent nonstandard finite difference scheme has been presented to obtain solution of smoking model. the analysis of two different states disease free and endemic equilibrium which means the disease dies out or persist in a population has been made by finding reproductive number. numerical results are presented graphically to show the dynamics of the model. 2. materials and method the concept of mathematical modeling has been prolonged to define the stability and qualitative features of giving up smoking models from 2000 [26–28]. many researchers considered various smoking models like [23], in which they examined numerous classes of smokers (potential smokers p(t), smokers s(t), and quit smokers q(t). then [24] modified the proposed model and introduced a new class named as chain smokers. later on, zeb et al. [25] studied a smoking model in which they argued local and global stability of the model and int. j. anal. appl. 17 (4) (2019) 505 its general solutions in which the collaboration between occasional and potential smokers occurs. in [26] the author presented a modified smoking model in which the dynamic behavior has been discussed numerically. the proposed smoking model [8] in the form of system of differential equation is given as: dp dt = bn −β1lp − (d1 + µ)p + τq (2.1) dl dt = β1lp −β2ls − (d2 + µ)l (2.2) ds dt = β2ls − (γ + d3 + µ)s (2.3) dq dt = γs − (τ + d4 + µ)q, (2.4) dn dt = (b−µ)n − (d1p + d2l + d3s + d4q) (2.5) with initial conditions p(0) = n1,l(0) = n2,s(0) = n3,q(0) = n4,n(0) = n5, where p, l, s, q and n represents the smoking status of time dependent sub-compartments b, µ, γ and τ are the parameter involved β1 and β2 are the transmission coefficient d1, d2, d3 and d4 are the death rates of the sub-compartments of the smoking model. 3. stability of the model in this section, we find the basic reproduction number and stability of the model. we prove that our model is locally and globally stable for disease-free-equilibrium. since all our model parameters are positive or non-negative, it is important to show that all state variables remain positive or non-negative for all positive initial conditions for 0. from our model equation, we have dn dt = (b−µ)n − (d1p + d2l + d3s + d4q) ≤ b−µ)n the closed set is d = (p,l,q,s,n ∈ r5+ : n ≤ b µ ) theorem 1: the closed set d is bounded and positive invariant. proof: dn dt ≤ bn −µn, so n is bounded above by bn −µn, hence dn dt < 0 and t > b µ . on simplification we have, n(t) ≤ n(0)e−µt + b µ (1 −e−µt). as t →∞, e−µ → 0, so limt→∞n(t) ≤ bµ thus, d is bounded and positively invariant in r5+ theorem 2: assume that model (2.1 − 2.4) has global solution corresponding to non-negative initial int. j. anal. appl. 17 (4) (2019) 506 condition then solution is non negative all times. . proof:assume p(0) ≥ 0, l(0) ≥ 0,q(0) ≥ 0, equation can be written as dp dt = bn − (β1l + d1 + µ)p dp dt = bn −dp this linear first order differential equation in p which has solution p = p(0)exp( ∫ t 0 −a(p)dp + exp ∫ t 0 −a(p)dt× 0 ∫ t 0 πexp( ∫ u 0 −a(w)dw))du ≥ 0 hence p ≥ 0 for all t ≥ 0, regarding the non-negativity of the remaining variables, we consider the subsystem dl dt = β1lp −β2ls − (d2 + µ)l (3.1) ds dt = β2ls − (γ + d3 + µ)s (3.2) dq dt = γs − (τ + d4 + µ)q, (3.3) this can be written in matrix form dy dt = my (t) + b(t) (3.4) where y (t) =   l(t) s(t) q(t)  , m =   −β1 − (d2 + µ) 0 0 0 −(γ + d3 + µ) 0 0 γ −(τ + d4 + µ)   and b(t) =   0 0 γ   we note that m is a metzler matrix (i.e with non-negative of t-diagonal entries) in opinion of the before now recognized non-negatively of p. thus an equation (3.4) is a monotone system [46]. therefore, r3+ invariant under the flow system (3.4). this complete the proof of the theorem. 4. qualitative analysis by substituting the values of parameters in given system of differential equations and taking rate of change with respect to time zero, we get bn −β1lp − (d1 + µ)p + τq = 0, (4.1) β1lp −β2ls − (d2 + µ)l = 0, (4.2) int. j. anal. appl. 17 (4) (2019) 507 β2ls − (γ + d3 + µ)s = 0, (4.3) γs − (τ + d4 + µ)q = 0, (4.4) (b−µ)n − (d1p + d2l + d3s + d4q) = 0, (4.5) by solving equations. (4.1 − 4.5), we get the disease free equilibrium point e0 = (p,l,s,q) i.e e0 = (0, 0, 0, 0), which is trivial solutions. if β1 < d2+µ then the disease free equilibrium point e1 = (p0,l0,s0,q0) i.e e1 = (1, 0, 0, 0) solving the system of equations (4.1 − 4.5), we get e∗0 = (p ∗,l∗,s∗,q∗), where p∗ = µ + d2 β1 + β1d2(µ + γ + d3) − (b−µ)(µ + d2)(−β2(µ + d1 − bd1) + β1(µ + γ + d3)) β1[β1γτ + (b−µ)(−β2(µ + d1 + bd1) + β1(µ + γ + d3 + d4 + d3(µ + τ + d4))] l∗ = µ + γ + d3 β2 s∗ = β1d2(µ + γ + d3) − (b−µ)(µ + d2)(−β2(µ + d1 − bd1) + β1(µ + γ + d3)) β2[β1γτ + (b−µ)(−β2(µ + d1 + bd1) + β1(µ + γ + d3)) + β1d4 + β1d3(µ + τ + d4)] q∗ = γ[β1d2(µ + γ + d3) − (b−µ)(µ + d2)(−β2(µ + d1 − bd1) + β1(µ + γ + d3))] β2(µ + τ + d4)[β1γτ + (b−µ)(−β2(µ + d1 + bd1) + β1(µ + γ + d3 + d4 + d3(µ + τ + d4))] theorem 3: e0 of the given system is locally asymptotically stable if re(λ) < 0. proof: λ can be evaluated from the relation |j −λi| = 0. consider the jacobian matrix again as j =   −β1l− (d1 + µ) −β1p 0 τ β1l β1p −β2s − (d2 + µ) −β2l 0 0 β2s β2l− (γ + d3 + µ) 0 0 0 γ −τ −d4 −µ   j =   −d1 −µ 0 0 τ 0 −(d2 + µ) 0 0 0 0 −γ −d3 −µ 0 0 0 γ −τ −d4 −µ   int. j. anal. appl. 17 (4) (2019) 508 the eigne values of the above matrix according to the equilibrium point e0 = (0, 0, 0, 0) are −(d1 + µ), −(d2 + µ), −(γ + d3 + µ), −(τ + d4 + µ) with negative real parts represents that the given system is locally asymptotically stable. also the system is stable for the point e1 = (1, 0, 0, 0). theorem 4: the the disease free equilibrium e0 = (1, 0, 0, 0, 0) of subsystem (2.1 − 2.4) is globally asymptotically stable (gas) if r0 < 1 . proof:let p0 = 1 and consider the following combination of linear function and voltera type lyapunove function m0 = m0(p,l,s,q) = p −p0ln(p) + l + b1s + b2q using the function that p0 = 1, lie derivative of m0 in the direction of vector field given by the right hand side of equation (2.1 − 2.4) is dm0 dt = dp dt (1 − 1 p ) + dl dt + b1 ds dt + b2 dq dt dm0 dt = (bn −β1lp − (d1 + µ)p + τq)(1 − 1 p ) + β1lp −β2ls − (d2 + µ)l + b1(β2ls −(γ + d3 + µ))s + b2(γs − (τ + d4 + µ))q dm0 dt = − bn p (1 + τq−p) + (β1 + (b1 − 1)β2s − (d2 + µ))l + (b2γ − b1(γ + d3 + µ))s +(τ − b2(τ + d4 + µ))q choose b1, b2 b2γ − b1(γ + d3 + µ) = 0 and τ − b2(τ + d4 + µ) = 0 we get b2 = τ τ + d4 + µ b1 = τγ (τ + d4 + µ)(γ + d3 + µ) with in this mind dm0 dt becomes dm0 dt = − bn p (1 + τq−p) + (β1 + ( τγ (τ + d4 + µ)(γ + d3 + µ) − 1)β2s − (d2 + µ))l dm0 dt = − bn p (1 + τq−p) + (β1(τ + d4 + µ)(γ + d3 + µ) + (τγ − (τ + d4 + µ)(γ + d3 + µ))β2s −(d2 + µ)(τ + d4 + µ)(γ + d3 + µ)) ≤ 0 since it is easy to see that the largest invariant subset contained in the set. ε = {(p,l,s,q)�k0/ dm0 dt = 0} int. j. anal. appl. 17 (4) (2019) 509 5. reproductive number consider the jacobian matrix as j =   −β1l− (d1 + µ) −β1p 0 τ β1l β1p −β2s − (d2 + µ) −β2l 0 0 β2s β2l− (γ + d3 + µ) 0 0 0 γ −(τ + d4 + µ)   . since the jacobian matrix is j = f −v then the matrix f and v can be written as f =   0 −β1 0 0 0 β1 0 0 0 0 0 0 0 0 0 0   , v =   β1l + (d1 + µ) β1p 0 −τ −β1l −β1p + β2s + (d2 + µ) β2l 0 0 −β2s −β2l + (γ + d3 + µ) 0 0 0 −γ (τ + d4 + µ)   . we know that k = fv −1 and using the relation |k −λi| = 0 for the eigen value λ , we get λ = β1 µ + d2 , which shows the reproductive number r0 = β1 µ+d2 . by substituting the values of parameters, we get r0 = 0.020408 < 1. since reproductive number r0 < 1, so the constructed system is in disease free state. 6. sensitivity analysis of r0 the sensitivity of r0 = β1 µ+d2 , to each of its parameters is ∂r0 ∂β1 = 1 µ + d2 ≥ 0 ∂r0 ∂µ = − β1 (µ + d2)2 ≤ 0 ∂r0 ∂d2 = − β1 (µ + d2)2 ≤ 0 int. j. anal. appl. 17 (4) (2019) 510 it can be seen that r0 is most sensitive to change in parameter, here, r0 is increasing with β1 and decreasing with µ, d2. in other words it found that the sensitivity analysis shows that prevention is better than to control the disease. table 1. values of physical parameters used in smoking model parameter value parameter value n1 40 n2 40 n3 60 n4 80 n5 200 d1 0.33 d2 0.44 d3 0.55 d4 0.66 β1 0.001 β2 0.001 µ 0.05 b 0.1 γ 0.99 τ 0.2 7. nonstandard finite difference (nsfd) scheme a nonstandard finite difference (nsfd) scheme for the system (2.1−2.5) is presented in this section [29]. in recent years, nonstandard finite difference (nsfd) scheme for discrete models have been constructed or tested for a wide range of nonlinear systems of differential equations [30–32]. the positivity of the state variables involved in the system is satisfy by proposed method. this property has key role when we solve mathematical models arising in biology because these state variables represent sub-populations which never take negative values. the discretized form of the the system (2.1−2.5) by using nsfd scheme which based on the generalized first order forward method is written as pk+1 −pk φ = bnk −β1lkpk+1 − (d1 + µ)pk+1 + τqk (7.1) pk+1 + β1l kpk+1φ + φ(d1 + µ)p k+1 = pk + bφnk + τφqk (7.2) pk+1 = pk + bφnk + τφqk 1 + β1lkφ + φ(d1 + µ) (7.3) lk+1 −lk φ = β1l kpk+1 −β2lk+1sk − (d2 + µ)lk+1 (7.4) lk+1 + φβ2l k+1sk + φ(d2 + µ)l k+1 = lk + β1l kpk+1 (7.5) lk+1 = lk + β1l kpk+1 1 + φβ2sk + φ(d2 + µ) (7.6) int. j. anal. appl. 17 (4) (2019) 511 sk+1 −sk φ = β2l k+1sk −sk+1(γ + d3 + µ) (7.7) sk+1 = β2l k+1skφ−φsk+1(γ + d3 + µ) (7.8) sk+1(1 + φ(γ + d3 + µ)) = β2l k+1skφ + sk (7.9) sk+1 = β2l k+1skφ + sk 1 + φ(γ + d3 + µ) (7.10) qk+1 −qk = φγsk+1 −φ(τ + d4 + µ)qk+1 (7.11) qk+1(1 + φ(τ + d4 + µ)) = φγs k+1 + qk (7.12) qk+1 = φγsk+1 + qk 1 + φ(τ + d4 + µ) (7.13) which is the purposed nsfd scheme for the given model, where φ = φ(h) = 1 −e−(d3+µ+γ)h d3 + µ + γ (7.14) the discrete method given in (22, 25, 29, 32) is indeed an nsfd scheme because it is constructed according to mickens rules [32] formalized as follows in [33]. rule 1. the standard denominator h = ∆t of the discrete derivatives is replaced by the complex denominator function in equation (33) which satisfies the asymptotic relation φ(h) = h + o(h2) note that the denominator function φ is expected to better capture the dynamics of the continuous model through the presence of the underlying parameters d3,µ,γ. in fact, exact schemes for a wide range of dynamical systems involve such complex denominator functions [34, 35]. rule 2. nonlinear terms in the right-hand side of equation (2.1−2.5) are approximated in a non-local way. for instance, we have ltkptk ' lkpk+1 instead of ltkstk ' lkpk 8. analysis of the scheme theorem 5: the nsfd scheme (22, 25, 29, 32) is a dynamical system on the biological feasible domain k of the continuous model (2.1 − 2.5). proof: first, we prove the positivity of the scheme (22, 25, 29, 32). it is easy to show that the nsfd scheme (22, 25, 29, 32) takes the explicit form pk+1 = pk + bφnk + τφqk 1 + β1lkφ + φ(d1 + µ) int. j. anal. appl. 17 (4) (2019) 512 lk+1 = lk(1 + β1l kφ + φ(d1 + µ)) + (β1l k)(pk + bφnk + τφqk) (1 + φβ2sk + φ(d2 + µ))(1 + β1lkφ + φ(d1 + µ)) sk+1 = β2s kφa + skb (1 + φ(γ + d3 + µ))b qk+1 = φγ(β2s kφa + skb) + qk(1 + φ(γ + d3 + µ))b (1 + φ(τ + d4 + µ))(1 + φ(γ + d3 + µ))b where a = lk(1+β1l kφ+φ(d1+µ))+(β1l k)(pk+bφnk+τφqk), b = (1+φβ2s k+φ(d2+µ))(1+β1l kφ+φ(d1+µ)) thus pk+1 ≥ 0, lk+1 ≥ 0, sk+1 ≥ 0, qk+1 ≥ 0 whenever the discrete variables are non-negative at the previous iteration. it remains to prove the positive invariance of k. adding the (22, 25) and (29) we have [1 + φ(d1 + µ)]h k+1 = φbn + hk − [1 + (d2 + µ)φ]lk+1 − [1 + φ(γ + d3 + µ)]sk+1 ≤ φbn + hk [1 + φ(µ + d1)]h k+1 ≤ φbn + hk ⇒ hk+1 ≤ bn d1 + µ whenever hk ≤ bn µ + d1 the priori bonds for qk+1 and nk+1 follow the radially from the fact that lk+1 and sk+1 are less then or equal to hk+1. this complete the proof. 9. numerical simulations the mathematical analysis of smoking epidemic model with non-linear incidence has been presented. firstly, we investigate the basic reproduction number r0 for the system (2.1−2.5) which completely characterized the stability of the disease free and endemic equilibrium. we observed that, if r0 < 1, the disease free state at e0 and e1 is locally stable. to observe the effects of the parameters using in this dynamics smoking model (2.1−2.5), conclude several numerical simulations varying the value of parameters. these simulations reveals that a change in time and step size h affects the dynamics of the epidemic as shown in figures 1 and figures 2. by applying the mickens approach, we use φ = φ(h) instead of step size h in figure 2 and figure 4. comparison is made by highlighting the point in each graph which shows that the smokers reduces within 20 weeks when we used φ. its interpretation for a longer period reduces the infected individuals in the health system. when initial condition changes to p(0) = 40,l(0) = 80,s(0) = 120,q(0) = 160,n(0) = 400, the convergence to disease free equilibrium point remain consistent as shown in figure 3 and figures 4. int. j. anal. appl. 17 (4) (2019) 513 time (weeks), step size h=1 0 2 4 6 8 10 12 14 16 18 20 c o m p a rm e n ta l p o p u la tio n 0 10 20 30 40 50 60 70 80 disease free equilibrium potential smokers light smokers smokers quit smokers x: 20 y: 3.747e-07 figure 1. numerical solutions for potential smokers, light smokers, smokers and quit smokers in a time t (weeks) with step size h = 1 time (weeks),for φ =φ(h) with h=1 0 2 4 6 8 10 12 14 16 18 20 c o m p a rm e n ta l p o p u la tio n 0 10 20 30 40 50 60 70 80 disease free equilibrium potential smokers light smokers smokers quit smokers x: 19.53 y: 8.774e-09 figure 2. numerical solutions for potential smokers, light smokers, smokers and quit smokers in a time t (weeks)for φ = φ(h) with h = 1 int. j. anal. appl. 17 (4) (2019) 514 time (weeks), step size h=1 0 2 4 6 8 10 12 14 16 18 20 c o m p a rm e n ta l p o p u la tio n 0 20 40 60 80 100 120 140 160 disease free equilibrium potential smokers light smokers smokers quit smokers x: 20 y: 9.083e-07 figure 3. numerical solutions for potential smokers, light smokers, smokers and quit smokers in a time t (weeks) with step size h = 1 with different initial conditions time (weeks), for φ =φ(h) with h=1 0 2 4 6 8 10 12 14 16 18 20 c o m p a rm e n ta l p o p u la tio n 0 20 40 60 80 100 120 140 160 disease free equilibrium potential smokers light smokers smokers quit smokers x: 19.53 y: 2.35e-08 figure 4. numerical solutions for potential smokers, light smokers, smokers and quit smokers in a time t (weeks) for φ = φ(h) with h = 1 with different initial conditions int. j. anal. appl. 17 (4) (2019) 515 10. conclusion it is an important to note that nonstandard finite difference scheme for mathematical models based on system of differential equations is more powerful approach to compute the convergent solution. the constructed unconditionally convergent nonstandard finite difference (nsfd) scheme for smoking model preserve the positivity of all values of h (step size) which shows that the developed scheme is stable. the nonstandard finite difference scheme is dynamically consistent, easy to implement and shows a good agreement to analyze the bad impact of smoking for long period of time and represents their dynamical behavior graphically. threshold condition shows most sensitive effect regarding their parameters. we prove the essential properties, bounded, positivity and well-posed, also local and global stability analysis has been made to analyze the smoking effects in the community. numerical simulations are carried out to check the actual behavior of the model. references [1] j. biazar, solution of the epidemic model by adomian decomposition method, appl. math. comput. 173 (2006), 1101-1106. [2] s. busenberg and p. driessche, analysis of a disease transmission model in a population with varying size, j. math. biol. 28 (1990), 65-82. [3] a.m.a. el-sayed, s.z. rida and a.a.m. arafa, on the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, int. j. nonlinear sci. 7 (2009), 485-495. [4] a.a.m. arafa, s.z. rida and m. khalil, fractional modeling dynamics of hiv and 4 t-cells during primary infection, nonlinear biomed. phys. 6 (2012), 1-7. [5] c.m. kribs-zaleta, structured models for heterosexual disease transmission, math. biosci. 160 (1999), 83-108. [6] b. buonomo and d. lacitignola, on the dynamics of an seir epidemic model with a convex incidence rate, ricerche mat. 57 (2008), 261-281. [7] x. liu and c. wang, bifurcation of a predator-prey model with disease in the prey, nonlinear dyn. 62 (2010), 841-850. [8] f. haq, k. shah, g.u rahman and m. shahzad. numerical solution of fractional order smoking model via laplace adomian decomposition method, alex. eng. j. 57 (2018), 1061-1069. [9] c. chavez and b. song; dynamical models of tuberculosis and their applications; math. biosci. eng. 1 (2004), 361-404. [10] a. mcneill, m. raw, j. whybrow and p. bailey; national strategy for smoking cessation treatment in england; addiction 100 (s.2) (2005), 1-11. [11] r.p. sargent, r.m. shepard and s.a. glantz; admission for myocardial infarction associated with public smoking bun; br.med. j. 1 (2004), 328-977. [12] y.m. terry-mcelrath, m.a. wakefield, s. emery, h. saffer, g.m. szczypka and p. o. malley p; state antitobacco advertising and smoking outcomes by gender and race/ethnicity; ethnicity and health 12 (2007), 339-362. [13] r. ullah, m. khan, g. zaman, s. islam, m.a. khan, s. jan and t. gul, dynamical featurers of mathemtical model on smoking, j. appl. environ. biol. sci., 6 (2016), 92-96. [14] http://en.wikipedia.org/wiki/smoking (3th october, 2016). [15] https://en.wikipedia.org/wiki/epidemiology (17th november, 2016). int. j. anal. appl. 17 (4) (2019) 516 [16] c. castillo-garsow, g. jordan-salivia, and a. rodriguez herrera, mathematical models for the dynamics of tobacco use, recovery, and relapse, technical report series bu-1505m, cornell university, ithaca, ny, usa, (1997). [17] o. sharomi and a. b. gumel, curtailing smoking dynamics: a mathematical modeling approach, appl. math. comput. 195 (2008), 475-499. [18] g. zaman, qualitative behavior of giving up smoking model; bull. malaysian math. sci. soc. 2 (2011), 403-415. [19] s.a. matintu, smoking as epedemic: modeling and simulation study, american j. appl. math. 5 (2017), 31-38. [20] a. ahmad, m. farman, f. yasin and m. o. ahmad, dynamical transmission and effect of smoking in society, int. j. adv. appl. sci. 5(2) (2018), 71-75 [21] f. ashraf, a. ahmad, m. u. saleem, m. farman and m.o. ahmad, dynamical behavior of hiv immunology model with non-integer time fractional derivatives, int. j. adv. appl. sci. 5(3) (2018), 39-45, . [22] a. ahmad, m. farman, m. o ahmad, n. raza and abdullah, dynamical behavior of sir epidemic model with non-integer time fractional derivatives: a mathematical analysis, int. j. adv. appl. sci. 5(1) (2018), 123-129. [23] j.b. swartz, use of a multistage model to predict time trends in smoking induced lung cancer, j. epidemiol. commun. healt. 46 (1992), 11-31. [24] f. brauer and c. castillo-cha vez, mathematical models in population biology and epidemiology, springer, (2001). [25] a. zeb, g. zaman, v.s. erturk, b. alzalg, f. yousafzai and m. khan, approximating a giving up smoking dynamic on adolescent nicotine dependence in fractional order, plos one, 11 (2016), 10-15. [26] g. zaman, optimal campaign in the smoking dynamics, comput. math. method. med. 2011 (2011), article id 163834. [27] g. zaman, qualitative behavior of giving up smoking models, bull. malay. math. sci. soc. 34 (2011), 403-415. [28] v. suat erturk, g. zamanb and s. momanic, a numeric analytic method for approximating a giving up smoking model containing fractional derivatives, comput. math. appl. 64 (2012), 3065-3074. [29] r. e. mickens, exact solutions to a finite difference model of a nonlinear reactions advection equation: implications for numerical analysis, numer. methods partial differ. equations, 5 (1989), 313-325. [30] r. e.mickens, applications of nonstandard finite difference schemes, world scientific, singaporen (2000). [31] r. anguelov and j. m.-s. lubuma, nonstandard finite difference method by nonlocal approximations, math. comput. simul. 61 (2003), 465-475. [32] r. e. mickens, nonstandard finite difference models of differential equations, world scientific, singapore (1994). [33] r. anguelov and j.m.-s. lubuma, contributions to the mathematics of the nonstandard nite dierencemethodandapplications, numer. methods partial differ. equations, 17 (2001), 518-543. [34] j.m.-s. lubuma and k.c. patidar, non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, appl. math. comput. 187(2) (2007), 1147-1160. [35] l.w. roeger, exact difference schemes, in a. b. gumel mathematics of continuous and discrete dynamical systems, contemp. math. 618 (2014), 147-161. 1. introduction 2. materials and method 3. stability of the model 4. qualitative analysis 5. reproductive number 6. sensitivity analysis of r0 7. nonstandard finite difference (nsfd) scheme 8. analysis of the scheme 9. numerical simulations 10. conclusion references international journal of analysis and applications volume 17, number 6 (2019), 958-973 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-958 on generalized k-uniformly close-to-convex functions of janowski type afis saliu1,2,∗ 1department of mathematics, comsats university, chak shahzad, islamabad, 44000, pakistan 2department of mathematics, gombe state university, gombe state, nigeria ∗corresponding author: afissaliu@gsu.edu.ng abstract. this work is concerned with the class of analytic functions that maps open unit disk onto conic domains. necessary condition, arc length, growth rate of coefficients, radius problems and property of some integral transformation under the class are examined. 1. introduction let a be the class of functions f(z) whose series representation is given by f(z) = z + ∞∑ n=2 anz n, (1.1) and regular in an open unit disk ∆ = {z : |z| < 1}. let s denotes the class of univalent functions in ∆ and c(ρ),s∗(ρ) and k(ρ), 0 ≤ ρ < 1 be the well known subclasses of s which consist of convex, starlike and close-to-convex functions of order ρ respectively. c(0) ≡ c,s∗(0) ≡ s∗ and k(0) ≡ k are the classes of convex, starlike and close-to-convex functions in ∆ respectively. a function f ∈ a is subordinate to g ∈ a (written as f(z) ≺ g(z)) if there exists a function w(z) with |w(z)| < 1 and w(0) = 0 such that f(z) = g(w(z)). in addition, if g(z) is univalent in ∆, then f(0) = g(0) and f(∆) ⊂ g(∆) [4]. 2010 mathematics subject classification. 30c45, 30c50, 30c55. key words and phrases. analytic functions; janowski functions; conic domains. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 958 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-958 int. j. anal. appl. 17 (6) (2019) 959 let h be the class of functions p(z) = 1 + ∞∑ n=1 cnz n that are regular in ∆ with p(0) = 1. then p ∈ p [a,b], −1 ≤ b < a ≤ 1 if and only if p (z) ≺ 1+az 1+bz , or equivalently p (z) = (a + 1)h(z) − (a− 1) (b + 1)h(z) − (b − 1) , where h ∈ p[1,−1] = p, the class of functions with positive real part. this class of functions was first considered and study extensively by janowski [6]. kanas and wisniowska [7,8] introduced the class of k-uniformly convex functions and k-uniformly starlike function denoted by k − ucv and k − ust respectively, which were defined subject to the conic domain ωk,k ≥ 0, given by ωk = {u + iv : u > k √ (u− 1)2 + v2 } (1.2) this domain represents the right half plane for k = 0, the right branch of hyperbola for 0 < k < 1 and an ellipse when k > 1. the function pk(z) plays an extremal role for all functions that maps ∆ onto ωk and it is given by pk(z) =   1+z 1−z , k = 0, 1 + 2 π2 ( log 1+ √ z 1− √ z )2 , k = 1, 1 + 2 1−k2 sinh 2 [ ( 2 π arccos k) arctan √ z ] , 0 < k < 1, 1 + 1 1−k2 sin [ π 2r(t) u(z)√ t∫ 0 1√ 1−x2 √ 1−(tx)2 dx ] + 1 k2−1, k > 1, (1.3) where u(z) = z− √ t 1− √ tz , t ∈ (0, 1),z ∈ ∆ and t is chosen such that k = cosh(πr ′(t) 4r(t) ), r(t) is legendre’s complete elliptic integral of the first kind and r′(t) is the complementary integral of r(t) [7]. we denote by p(pk), the class of functions that are subordinate to pk(z). ronning [20] proved that for p ∈ p(pk), there exists a function h ∈p such that p(z) = hγ(z) and γ is given as : γ = 2 π arctan ( 1 k ) (1.4) it was also shown in [9] that for pk(z) = 1 + δkz + · · · ∈ p(pk), δk =   8(arccos k)2 π2(1−k2) , 0 ≤ k < 1, 8 π2 , k = 1, π2 4(k2−1) √ t(1+t)r2(t) , k > 1. (1.5) very recently, k.i. noor [18] extended the domain ωk to that of janowski type, ωk[a,b], − 1 ≤ b < a ≤ 1 and defined it as ωk[a,b] = { w: re ( (b − 1)h(z) − (a− 1) (b + 1)h(z) − (a + 1 ) > k ∣∣∣∣(b − 1)h(z) − (a− 1)(b + 1)h(z) − (a + 1) − 1 ∣∣∣∣ } (1.6) int. j. anal. appl. 17 (6) (2019) 960 or alternatively, ωk[a,b] = { u + iv : [(b2 − 1)(u2 + v2) − 2(ab − 1)u + (a2 − 1)]2 >k2[(−2(b + 1)(u2 + v2) + 2(a + b + 2)u− 2(a + 1))2 + 4(a−b)2v2] } . (1.7) geometrically, the effect of ωk[a,b] on ωk was described in [18]. we denote by k−p [a,b], −1 ≤ b < a ≤ 1, the class of functions that map ∆ onto ωk[a,b]. equivalently, we say p ∈ k −p [a,b] if and only if p(z) ≺ (a + 1)pk(z) − (a− 1) (b + 1)pk(z) − (b − 1) , k ≥ 0, −1 ≤ b < a ≤ 1, (1.8) where pk is given by (1.3). also, it is worthy to note that p ∈ k − p [a,b] ⊂ p(β1) which implies that p(z) = (1 −β1)h1(z) + β1, (see [18]) where h1 ∈p and β1 is given by β1 = 2k + 1 −a 2k + 1 −b (1.9) we extend the class k −p [a,b] as follows : definition 1.1. let p(z) ∈h. then p ∈ k −pµ[a,b] if and only if for µ ≥ 2,k ≥ 0,−1 ≤ b < a ≤ 1, we have p(z) = µ + 2 4 p1(z) − µ− 2 4 p2(z), p1,p2 ∈ k −p [a,b]. (1.10) definition 1.2. let p(z) ∈h. then p ∈ k −pµ[a,b,α] if and only if for µ ≥ 2,k ≥ 0, − 1 ≤ b < a ≤ 1,α ∈ [0, 1), we have p(z) = (1 −α)h(z) + α, h(z) ∈ k −pµ[a,b]. (1.11) for k = 0,a = 1,b = −1,α = 0, we have the class pµ introduced and studied in [22]. also, when µ = 2,α = 0, we get the class k − p [a,b], which was first considered by k.i. noor in [18]. the class k −pµ[1 − 2γ∗,−1, 0], γ∗ ∈ [0, 1) is the same as the class k −pµ(γ∗) studied in [17]. we now define the following classes of functions. definition 1.3. let f ∈ a. then f ∈ k −urµ[a,b,α], k ≥ 0, µ ≥ 2, α ∈ [0, 1),−1 ≤ b < a ≤ 1, if and only if zf′(z) f(z) ∈ k −pµ[a,b,α]. definition 1.4. let f ∈ a. then f ∈ k −uvµ[a,b,α], k ≥ 0, µ ≥ 2, α ∈ [0, 1),−1 ≤ b < a ≤ 1, if and only if (zf′(z))′ f′(z) ∈ k −pµ[a,b,α]. it is obvious to note that f ∈ k −uvµ[a,b,α] ⇔ zf′(z) ∈ k −urµ[a,b,α]. also, 0 −uvµ[1,−1, 0] = vµ, is the class of functions of bounded boundary rotation(see [1], [22]) int. j. anal. appl. 17 (6) (2019) 961 definition 1.5. let f ∈a. then f ∈ k−tµ[a,b,c,d,α], k ≥ 0, µ ≥ 2, α ∈ [0, 1),−1 ≤ b < a ≤ 1,−1 ≤ d < c ≤ 1 if and only if there exists g ∈ k −urµ[c,d,α] such that zf′(z) g(z) ∈ k −p [a,b] or equivalently as f′(z) g′(z) ∈ k −p [a,b] where g ∈ k −uvµ[c,d,α]. we note the following special cases. i 0 −t2[a,b,c,d, 0] = k[a,b,c,d], is the class of functions studied by silvia [23]. ii 0 −t2[1,−1, 1,−1, 0] = k, is the class of close to convex functions examined by kaplan [10]. iii k −t2[a,b,c,d, 0] = k −uk[a,b,c,d], the class considered in [12] iv 1−t2[1,−1, 1,−1, 0] = ucc, is the class of uniformly close to convex functions explored by kumar and ramesha [11]. v 1−t2[1−2ρ,−1, 1,−1, 0] = ucc(ρ) is the class of uniformly close to convex of order ρ, −1 ≤ ρ < 1 that was taken into account in [24]. vi 0 −tµ[1 − 2ρ,−1, 1 − 2ρ,−1, 0] = tµ(ρ),ρ ∈ [0, 1), is the class of functions that was studied by k.i. noor in [16] vii 0 − tµ[1,−1, 1,−1, 0] = tµ, is the class of generalized close-to-convex functions introduced and studied in [15]. let g(a,b,c; z) = γ(c) γ(a)γ(c−a) ∫ 1 0 ua−1(1 −u)c−a−1(1 −zu)−bdu, (1.12) where rea > 0,re(c−a) > 0, γ denotes the gamma function and g(a,b,c; z) is hypergeometric function. unless if otherwise stated, we lay down once and for all that k ≥ 0, µ ≥ 2, α ∈ [0, 1),−1 ≤ b < a ≤ 1,−1 ≤ d < c ≤ 1. 2. some preliminary lemmas we need the following lemmas. lemma 2.1. [1] every function f ∈vµ is a close to convex function of order µ2 − 1. lemma 2.2. [13] let p ∈ p [a,b]. then |arg p(z)| ≤ sin−1 (a−b)r 1 −abr2 . (2.1) int. j. anal. appl. 17 (6) (2019) 962 lemma 2.3. [5] let p(z) ∈h with rep (z) > 0,z = reiθ(0 < r < 1). then 2π∫ 0 |p(reiθ)|λdθ < c(λ) 1 (1 −r)λ−1 , (2.2) where c(λ) is a constant and λ > 1. lemma 2.4. [1] let g ∈vµ. then for µ > 3 there exists s1 ∈s∗(0) and p ∈p such that zg′(z) = s1(z)(p(z)) µ 2 −1 lemma 2.5. [19] let q(z) be analytic in ∆ with q(0) = 1. if % ≥ 1, rec ≥ 0, 0 ≤ θ1 < θ2 ≤ 2π, z = reiθ, then θ2∫ θ1 re { q(z) + %zq′(z) c% + q(z) } dθ > −σπ, (0 < σ ≤ 1), implies θ2∫ θ1 req(z)dθ > −σπ. 3. main result in this section, we present our main work. theorem 3.1. let f ∈ k −uvµ[c,d,α]. then f′(z) = (φ′(z))1−β, where φ ∈ vµ, β = α + (1 −α)β1 and β1 is given by (1.9). proof. let f ∈ k −uvµ[c,d,α]. then (zf′(z))′ f′(z) = (1 −α) [ µ + 2 4 p1(z) − µ− 2 4 p2(z) ] + α, p1,p2 ∈ k −p [c,d] ⊂p(β2). where β2 = 2k+1−c 2k+1−d . therefore, (zf′(z))′ f′(z) = µ + 2 4 h1(z) − µ− 2 4 h2(z), h1,h2 ∈p(β), where β = (1 −α)β2 + α = 2k + 1 −c + α(c −d) 2k + 1 −d . (3.1) thus, there exist f1,f2 ∈ c(β) such that (zf′(z))′ f′(z) = µ + 2 4 (zf′1(z)) ′ f′1(z) − µ− 2 4 (zf′2(z)) ′ f′2(z) int. j. anal. appl. 17 (6) (2019) 963 integrating and using the result due to brannan [1], we have f′(z) = (f′1(z)) ( µ+2 4 ) (f′2(z)) ( µ−2 4 ) =ϕ′(z), ϕ ∈vµ(β) =(φ′(z))1−β, φ ∈vµ. � theorem 3.2. let f ∈ k −tµ[a,b,c,d,α]. then for µ > 2, 0 ≤ θ1 < θ2 ≤ 2π,z = reiθ, θ2∫ θ1 re (zf′(z))′ f′(z) dθ > −π [ (1 −α) ( c −d 2k + 1 −d ) ( µ 2 − 1) + γ ] , where γ is given by (1.4). proof. let f′(z) = g′(z)h(z), (3.2) where g ∈ k −uvµ[c,d,α], h ∈ k −p [a,b]. from theorem 3.1, we write zg′(z) = zβ(zφ′(z))1−β. by logarithmic differentiation and using lemma 2.1, it follows that θ2∫ θ1 (zg′(z))′ g′(z) dθ =β(θ2 −θ1) + (1 −β) θ2∫ θ1 (zφ′(z))′ φ′(z) dθ >−π(1 −β)( µ 2 − 1). (3.3) since h ∈ k − p [a,b], there exists h1 ∈ p [a,b] such that h(z) = (h1(z))γ, where γ is given by (1.4). therefore, f′(z) = g′(z)(h1(z)) γ and thus, θ2∫ θ1 re (reiθf′(reiθ))′ f′(reiθ) dθ = θ2∫ θ1 re (reiθg′(reiθ))′ g′(reiθ) dθ + γ θ2∫ θ1 re reiθh′1(re iθ) h1(reiθ) dθ. (3.4) but θ2∫ θ1 re reiθh′1(re iθ) h1(reiθ) dθ = arg h1(re iθ2 ) − arg h1(reiθ1 ), which implies max h1∈p[a,b] ∣∣∣∣∣ θ2∫ θ1 re reiθh′1(re iθ) h1(reiθ) dθ ∣∣∣∣∣ ≤ maxh1∈p[a,b] |arg h1(reiθ2 )| + maxh1∈p[a,b] |arg h1(reiθ1 )| . (3.5) int. j. anal. appl. 17 (6) (2019) 964 using lemma 2.2 in (3.5), we get max h1∈p[a,b] ∣∣∣∣∣ θ2∫ θ1 re reiθh′1(re iθ) h1(reiθ) dθ ∣∣∣∣∣ ≤ π − 2 cos−1 (a−b)r1 −abr2 . this means that θ2∫ θ1 re reiθh′1(re iθ) h1(reiθ) dθ ≥−π. (3.6) using (3.3) and (3.6) in (3.4), we have the result and this completes the proof. � corollary 3.1. [14] let f ∈ k −tµ[1,−1, 1,−1, 0]. then for µ > 2, 0 ≤ θ1 < θ2 ≤ 2π,z = reiθ, θ2∫ θ1 re (zf′(z))′ f′(z) dθ > −π ( µ− 2 2(k + 1) + γ ) . corollary 3.2. let f ∈ 1 −tµ[1,−1, 1,−1, 0]. then for µ > 2, 0 ≤ θ1 < θ2 ≤ 2π,z = reiθ, θ2∫ θ1 re (zf′(z))′ f′(z) dθ > −π µ 4 . corollary 3.3. [15] let f ∈ 0 −tµ[1,−1, 1,−1, 0]. then for 0 ≤ θ1 < θ2 ≤ 2π,z = reiθ, θ2∫ θ1 re (zf′(z))′ f′(z) dθ > −π µ 2 . using goodman result in [3], we have the following. corollary 3.4. the function f ∈ k −tµ[a,b,c,d,α] is univalent in ∆ for µ < 2 ( 1−γ 1−α )( 2k+1−d c−d ) + 2. remark 3.1. let f ∈ k−tµ[a,b,c,d,α]. then setting zf′(z) f(z) = p(z) in theorem 3.2 and applying lemma 2.5, we have θ2∫ θ1 re zf′(z) f(z) dθ > −π [ (1 −α) ( c −d 2k + 1 −d ) ( µ 2 − 1) + γ ] . theorem 3.3. let f ∈ k −tµ[a,b,c,d,α]. then |arg f′(z)| ≤ (1 −β)µ sin−1 r + γ sin−1 (a−b)r 1 −abr2 , where γ is given by (1.4). proof. let f′(z) = (φ′(z))1−β(h(z))γ, φ ∈vµ,h ∈ p [a,b], int. j. anal. appl. 17 (6) (2019) 965 where β is given by (3.1). therefore, |arg f′(z)| ≤ (1 −β)|arg(φ′(z))||arg(h(z))γ|. since it is well known in [21], that for φ ∈ vµ, |arg(φ′(z))| ≤ µ sin−1 r. using this results and lemma 2.2, the proof is complete. � theorem 3.4. let p(z) = 1 + ∞∑ n=1 cnz n ∈ k −pµ[a,b,α]. then for n ≥ 1, |cn| ≤ µ(1 −α)(a−b)|δk| 4 , where δk is given by (1.5). proof. since p ∈ k −pµ[a,b,α], then we write p(z) = µ + 2 4 [(1 −α)h1(z) + α] − µ− 2 4 [(1 −α)h2(z) + α], h1,h2 ∈ k −p [a,b]. (3.7) let h1(z) = 1 + ∞∑ n=1 anz n and h2(z) = 1 + ∞∑ n=1 bnz n. then from (3.7), we have 1 + ∞∑ n=1 cnz n = 1 + ∞∑ n=1 [ µ + 2 4 an − µ− 2 4 bn ] zn. comparing the coefficient of zn, we obtain |cn| ≤ (1 −α) [ µ + 2 4 |an| + µ− 2 4 |bn| ] . it has been established in [18] that for h(z) = 1 + ∞∑ n=1 dnz n ∈ k −p [a,b], |dn| ≤ (a−b)|δk| 2 , n ≥ 1 and δk is given by (1.5). using this result, it follows that |cn| ≤ µ(1 −α)(a−b)|δk| 4 and this completes the proof. � corollary 3.5. let p(z) = 1 + ∞∑ n=1 cnz n ∈ 1 −p2[1,−1, 0]. then for n ≥ 1, |cn| ≤ 8 π2 corollary 3.6. [13] let p(z) = 1 + ∞∑ n=1 cnz n ∈ 0 −p2[a,b, 0]. then for n ≥ 1, |cn| ≤ a−b corollary 3.7. [4] let p(z) = 1 + ∞∑ n=1 cnz n ∈ 0 −p2[1,−1, 0]. then for n ≥ 1, |cn| ≤ 2. int. j. anal. appl. 17 (6) (2019) 966 theorem 3.5. let p(z) = 1 + ∞∑ n=1 cnz n ∈ k −pµ[a,b,α]. then for z = reiθ(0 < r < 1), 1 2π 2π∫ 0 |p(z)|2dθ ≤ 1 + [( µ(1−α)(a−b)|δk| 4 )2 − 1 ] r2 1 −r2 . proof. using perseval’s identity, we have 1 2π 2π∫ 0 |p(z)|2dθ = 1 + ∞∑ n=1 |cn|2r2n. applying theorem 3.4, we get 1 2π 2π∫ 0 |p(z)|2dθ ≤1 + [ µ(1 −α)(a−b)|δk| 4 ]2 ∞∑ n=1 r2n = 1 + [( µ(1−α)(a−b)|δk| 4 )2 − 1 ] r2 1 −r2 . � corollary 3.8. for p ∈ 0 −p2[a,b, 0] = p [a,b], 1 2π 2π∫ 0 |p(z)|2dθ ≤ 1 + [(a−b)2 − 1]r2 1 −r2 . corollary 3.9. [16] for p ∈ 0 −p2[1 − 2γ,−1, 0] = p(γ), 1 2π 2π∫ 0 |p(z)|2dθ ≤ 1 + [µ2(1 −γ)2 − 1]r2 1 −r2 . theorem 3.6. [arc length problem] let f ∈ k −tµ[a,b,c,d,α]. then for µ > 2−γ1−β + 2, lr(f) ≤ c(γ,β,a,b) ( 1 1 −r )(1−β)( µ 2 +1)+γ−1 (r → 1), where c(γ,β,a,b) ( 1 1−r ) is a constant that only depends on γ,β,a,b, where γ and β are respectively given by (1.4) and (3.1). proof. for f ∈ k −tµ[a,b,c,d,α], and application of the result due to brannan [1], we have zf′(z) = zβ ( s1(z)(p1(z)) µ 2 −1 )1−β (h1(z)) γ, (3.8) where s1 ∈s∗,p1 ∈p and h1 ∈ p [a,b]. therefore, for z = reiθ (r < 1), lr(f) = 2π∫ 0 |zf′(z)|dθ int. j. anal. appl. 17 (6) (2019) 967 using (3.8), distortion theorem for starlike function s1(z), hölder’s inequality, corollary 3.8 and lemma 2.3, it follows that lr(f) ≤ 2π (1 −r)2(1−β) ( 1 2π ∫ 2π 0 |p1(z)| (1−β)(µ−2) 2−γ dθ )2−γ 2 ( 1 2π ∫ 2π 0 |h1(z)|2 dθ )γ 2 ≤ 2π (1 −r)2(1−β) ( 1 + [(a−b)2 − 1]r2 1 −r2 )γ 2 ( 1 2π ∫ 2π 0 |p1(z)| (1−β)(µ−2) 2−γ dθ )2−γ 2 ≤c(γ,β,a,b,α) ( 1 1 −r )(1−β)( µ 2 +1)+γ−1 (r → 1), where c(γ,β,a,b,α) = [2π(a−b)2] γ 2 (c(λ)) 2−γ 2 and λ = (1 −β)(µ− 2) 2 −γ > 1. � theorem 3.7. let f ∈ k −tµ[a,b,c,d,α]. then for µ > 2−γ1−β + 2, an = o(1)n (1−β)( µ 2 +1)+γ−2 (n →∞), where o(1) is a constant depending on γ,β,a and b, and γ, β are respectively given by (1.4) and (3.1). proof. by cauchy theorem, we have for z = reiθ n|an| ≤ 1 2πrn ∫ 2π 0 |zf′(z)|dθ = 1 2πrn lr(f). now, by applying theorem 3.6 and setting r = 1 − 1 n as n →∞, we have the result. � theorem 3.8. let f ∈ k −tµ[a,b,c,d,α]. then for µ > 2−γ1−β + 2,∣∣|an+1|− |an| ∣∣ ≤ d(γ,β,µ,a,b)nη−1, where η = 2(γ−β−1)+µ(1−β)−2 2 and d(γ,β,µ,a,b) is a constant that depends only on γ,β,µ,a,b, and γ, β are given by (1.4) and (3.1) respectively. proof. for f ∈ k −tµ[a,b,c,d,α], we write zf′(z) = zβ ( s1(z)(p1(z)) µ 2 −1 )1−β (h1(z)) γ, where s1 ∈s∗,p1 ∈p and h1 ∈ p [a,b]. let z1 be complex number with |z1| = r. then by cauchy theorem,∣∣∣∣z1(n + 1)|an+1|−n|an| ∣∣∣∣ ≤ 12πrn+1 ∫ 2π 0 |z1 −z| ∣∣∣∣zβ (s1(z)(p1(z)) µ2 −1)1−β (h1(z))γ ∣∣∣∣. int. j. anal. appl. 17 (6) (2019) 968 since s1 ∈ s∗, then by a result due to golusin [2], |(z1 −z)s1| ≤ 2r2 1 −r2 , we have ∣∣∣∣z1(n + 1)|an+1|−n|an| ∣∣∣∣ ≤ 2rβ+22πrn+1(1 −r2) ∫ 2π 0 ∣∣∣∣(s1(z))−β(p1(z))( µ2 −1)(1−β)(h1(z))γ ∣∣∣∣. now, using distortion theorem for starlike function s1(z), holder’s inequality, corollary 3.8 and lemma 2.3, we have ∣∣∣∣z1(n + 1)|an+1|−n|an| ∣∣∣∣ ≤ 2rn+1(1 −r)1−2β ( 1 2π ∫ 2π 0 |p1(z)| (1−β)(µ−2) 2−γ dθ )2−γ 2 × ( 1 2π ∫ 2π 0 |h1(z)|2 dθ )γ 2 ≤ 2 rn+1(1 −r)1−2β ( 1 + [(a−b)2 − 1]r2 1 −r2 )γ 2 × (∫ 2π 0 |p1(z)| (1−β)(µ−2) 2−γ dθ )2−γ 2 ≤ d1(γ,β,µ,a,b) rn+1 ( 1 1 −r )2γ−2β+µ(1−β)−2 2 , where d1 is a constant. setting r = 1 − 1n and z1 = n n+1 , the proof is completed. � corollary 3.10. for f ∈ k −t2[a,b,c,d, 0], ∣∣∣∣|an+1|− |an| ∣∣∣∣ ≤ d1(γ,β,a,b)n(γ−2β)−1, where d1(γ,β,a,b) is a constant that depends on γ,β,a,b. corollary 3.11. [15] for f ∈ 0 −tµ[1,−1, 1,−1, 0], ∣∣∣∣|an+1|− |an| ∣∣∣∣ ≤ d1(µ)nµ2 −1, where d1(γ,β,a,b) is a constant that depends only on µ. theorem 3.9. f ∈ k −tµ[a,b,c,d,α]. maps the disk |z| < r∗ onto a convex domain, where r∗ is the smallest positive real number of the equation 1 −λ1r + λ2r2 + λ3r3 + λ4r4 = 0 (3.9) int. j. anal. appl. 17 (6) (2019) 969 where λ1 =a + b + (1 −β)µ + γ(a−b), λ2 =1 + ab + (1 −β)µ(a + b), λ3 =(2β − 1)(a + b) − (1 −β)µab + γ(a−b), λ4 =ab(1 − 2β), and γ, β are respectively given by (1.4) and (3.1). proof. we set zf′(z) = zβ (zφ′(z)) 1−β (h(z))γ, where φ ∈vµ and h ∈ p[a,b]. by logarithmic differentiation, we have re (zf′(z))′ f(z) = β + (1 −β)re (zφ′(z))′ φ(z) + γre zh′(z) h(z) . for h ∈ p [a,b] and φ ∈vµ , it is known in [6] and [22] respectively that re zh′(z) h(z) ≥− (a−b)r (1 −ar)(1 −b)r and re (zφ′(z))′ φ(z) ≥ r2 −µr + 1 1 −r2 . using these results, we have that re (zf′(z))′ f(z) ≥ [ β(1 −r2)(1 −ar)(1 −br) + (1 −β)(r2 −µr + 1)(1 −ar)(1 −br) −γ(a−b)r(1 −r2) ]/ (1 −r2)(1 −ar)(1 −br). if we let t(r) = β(1 −r2)(1 −ar)(1 −br) + (1 −β)(r2 −µr + 1)(1 −ar)(1 −br) −γ(a−b)r(1 −r2). then t(1) = (1 −β)(2 −µ)(1 −a)(1 −b) ≤ 0, since µ ≥ 2 and t(0) = 1 > 0. thus, r ∈ (0, 1). hence, the theorem is proved. � corollary 3.12. let f ∈ 0 −t2[1,−1, 1,−1, 0]. then f ∈ c for |z| < 2 − √ 3. int. j. anal. appl. 17 (6) (2019) 970 corollary 3.13. let f ∈ 0 −tµ[1,−1, 1,−1, 0]. then f(z) maps the disc |z | < 1 2 [ µ + 2 − √ µ2 + 4µ ] onto a convex domain using the well-known distortion theorems for φ ∈vµ and h1 ∈p, we prove the following. theorem 3.10. let f ∈ k −tµ[1,−1,c,d,α]. then a2(a,b,c,r1) ≤ |f(z) | ≤ a1(a,b,c,r1), where a1(a,b,c,r1) = 2b−1 a [ ra1 g(a,b,c,−r1) −g(a,b,c,−1) ] and a2(a,b,c,r1) = 2b−1 a [ g(a,b,c,−1) −r−a1 g(a,b,c,−r −1 1 ) ] , where a =( µ 2 − 1)(1 −β)γ + 1, b =2β, c =( µ 2 − 1)(1 −β)γ + 2, and, γ and β are given by (1.4) and (3.1) respectively. proof. let f′(z) = (φ′(z))1−β(h1(z)) γ, φ ∈vµ, h1 ∈p. then |f(z) | ≤ ∫ |z | 0 |φ(t) |1−β|h1(t) |γ dt ≤ ∫ |z | 0 (1 + t)( µ 2 −1)(1−β) (1 − t)( µ 2 +1)(1−β) (1 + t)γ (1 − t)γ dt = ∫ |z | 0 ( 1 + t 1 − t )( µ 2 +1)(1−β) (1 + t)γ (1 − t)2(1−β)+γ .dt let ξ = 1+t 1−t, then 1 − t = 2 1+ξ ,dt = 2 (1+ξ)2 dξ, (1 + t) = 2ξ 1+ξ . therefore, |f(z) | ≤22β−1 ∫ 1+|z | 1−|z | 1 ξ( µ 2 −1)(1−β)+γ(1 + ξ)−2βdξ =2b−1(i1 − i2), (3.10) int. j. anal. appl. 17 (6) (2019) 971 where i2 = ∫ 1 0 ξa−1(1 − ξ)c−a−1(1 − (1ξ))−bdξ = 1 a g(a,b,c,−1). (3.11) to calculate i1, let ξ = r1u and r1 = 1+r 1−r . then i1 = ∫ 1 0 (r1u) ( µ 2 −1)(1−β)+γ(1 + r1u) −2βr1du =ra1 ∫ 1 0 ua−1(1 + r1u) −bdu = ra1 a g(a,b,c,−r1). (3.12) using (3.10), (3.11) and (3.12), the upper bound is obvious. now, we proceed to calculate the lower bound. let dr denotes the radius of the largest schlicht disk centered at the origin contained in the image of |z | < r under the function f(z). then there exists a point z0 with |z0 | = r, such that |f(z) | = dr. the ray from 0 to f(z) lies entirely in the image of ∆ and the inverse image of this ray is a curve γ in |z | < r. thus, dr =|f(z0) | = ∫ γ |f′(z) ||dr | = ∫ |z | 0 |φ′(ρ) |1−β|h1(ρ) |γdρ ≥ ∫ |z | 0 ( 1 −ρ 1 + ρ )( µ 2 11)(1−β) (1 −ρ)γ (1 + ρ)2(1−β)+γ dρ. let ν = 1−ρ 1+ρ . then dρ = −2 (1+ν)2 dν and 1 −ρ = 2ν 1+ν . going through the same process as it has been done for the upper bound, we easily obtain the lower bound and this completes the proof. � theorem 3.11. let f,g ∈ k−tµ[a,b,c,d,α], η1,c1,δ and v be positively real, η1 < 1, c1 > η1, η1 = v +δ. then for the function f(z) defined by [f(z)]η1 = c1z η1−c1 ∫ z 0 t(c1−δ−v)−1(f(t))δ(g(t))vdt, (3.13) ∫ θ2 θ1 re zf ′(z) f(z) > −π [ (1 −α) ( c −d 2k + 1 −d )( µ 2 − 1 ) + γ ] , where 0 ≤ θ1 < θ2 ≤ 2π. int. j. anal. appl. 17 (6) (2019) 972 proof. it has been shown in [17] that the integral transformation (3.13) is a well defined analytic function in ∆. let h(z) = zf′(z) f(z) . then, differentiating (3.13), we obtain f(z)η1 [(c1 −η1) + η1h(z)] = c1f(z)δg(z)v. differentiating logarithmically and with some simple computations, we have h(z) + 1 η zh′(z) (c1−η1) η1 + h(z) = δ η1 zf′(z) f(z) + v η1 zg′(z) g(z) . now, since f,g ∈ k −tµ[a,b,c,d,α], then application of remark 3.1 gives∫ θ2 θ1 re { h(z) + 1 η zh′(z) (c1−η1) η1 + h(z) } > −π [ (1 −α) ( c −d 2k + 1 −d )( µ 2 − 1 ) + γ ] . on using lemma 2.5 with % = 1 η , c = c1 −η, we complete the proof. � theorem 3.12. let f,g ∈ k −tµ[a,b,c,d,α] and −1 < ρ ≤ 0, 0 < α1 ≤ 1. then for the function j(z) defined by j(z) = [ (ρ + 1 α1 )z 1− 1 α1 ∫ z 0 t 1 α1 −2 f(t)ρg(t)dt ] 1 1+ρ , (3.14) ∫ θ2 θ1 re zj′(z) j(z) > −π [ (1 −α) ( c −d 2k + 1 −d )( µ 2 − 1 ) + γ ] , where 0 ≤ θ1 < θ2 ≤ 2π. proof. set zj′(z) j(z) = g(z). then differentiating (3.14) logarithmically and with some simple calculations, we get ( 1 α1 − 1 ) + (ρ + 1)g(z) = z 1 α1 −1 f(z)ρg(z)∫ z 0 t 1 α1 −2 f(t)ρg(t)dt . this implies ( ρ + 1 α1 ) f(z)ρg(z) = (( 1 α1 − 1 ) + (ρ + 1)g(z) ) j(z)ρ+1. differentiating logarithmically once again, we have h(z) + 1 1+ρ zh′(z) ( 1 α1 −1) 1+ρ + h(z) = ρ 1 + ρ zf′(z) f(z) + 1 1 + ρ zg′(z) g(z) . applying lemma 2.5 with % = 1 1+ρ , c = 1 α1 − 1, it follows that ∫ θ2 θ1 re zj′(z) j(z) > −π [ (1 −α) ( c −d 2k + 1 −d )( µ 2 − 1 ) + γ ] . � int. j. anal. appl. 17 (6) (2019) 973 references [1] d.a. brannan, on functions of bounded boundary rotation, proc. edln. math. soc. 2(1968/69), 330–347. [2] g. golusin, on distortion theorems and coefficients of univalent functions, mat. sb. 19(1946), 183-203 . [3] a.w. goodman, on close-to-convex functions of higher order, ann. univ. sci. budapest, eötöus sect. math., 25(1972), 17-30 [4] a.w. goodman, univalent functions, vols. i& ii, polygonal publishing house, washinton, 1983. [5] w.k. hyman, on functions with positive real part, j. london math. soc. 36(1961), 34-48. [6] w. janowski, some extremal problems for certain families of analytic functions, i, ann. polon. math. 28 (1973), 297-326. [7] s. kanas and a. wisniowska, conic regions and k-uniform convexity, j. comput. appl. math. 105(1999), 327-336. [8] s. kanas and a. wisniowska, conic domains and starlike functions, rev. roum. math. pures appl. 45(2000), 647-657. [9] s. kanas, coefficient estimates in subclasses of the caratheodory class related to conical domain, acta math. univ. comenian, 74(2005), no. 2, 149-161. [10] w. kaplan, close-to-convex schilcht functions, mich. math. j. 1 (1952), 169-185. [11] s. kumar and c. ramesha, subordination properties of uniformly convex and uniformly close to convex functions, j. ramanujan math. soc. 9 (1994), no. 2, 203-214. [12] m. shahi, a. muhammad and n.m. sarfraz, janoswski type close-to-convex functions associated with conic regions, j. ineq. appl. 2017,1(2017), 259. [13] b.s. mehrok, a subclass of close-to-convex functions, int. j. math. anal, 4,(2010), no. 27, 1319-1327. [14] k.i. noor and m.a. noor, higher order close-to-convex functions related with conic domain, appl. math. inf. sci. 8(2014), no. 5, 2455. [15] k.i. noor, on a generalization of close-to-convexity, internat. j. math. math. sci. 6(1983), no. 2, 327-334. [16] k.i. noor, on subclasses of close-to-convex functions of higher order, internat. j. math. math. sci. 15(1992), no. 2, 279-290. [17] k.i. noor, on a generalization of uniformly convex and related functions, comput. math. appl. 61(2011), 117-125. [18] k.i. noor and s.n. malik, on coefficient inequalities of functions associated with conic domains, comput. math. appl. 62(2011), 2209-2217. [19] k.i. noor and m.a. salim, on some classes of analytic functions, j. inequal. pure appl. math. 5(2004), no. 4, 98. [20] f. ronning, uniform convex functions and a corresponding class of starlike functions, proc. amer. math. soc. 118(1993), 189-196. [21] b. pinchuk, a variational method for functions of bounded boundary rotation, trans. amer. math. soc. 138(1969) 107-113. [22] b. pinchuk, functions with bounded boundary rotation, isr. j. math. 10 (1971), 716. [23] e.m. silvia, subclasses of close-to-convex functions, int. j. math. math. sci. 3(1983), 449-458. [24] k.g. subramanian, t.v. sudharsan and h. silverman , on uniforly close-to-convex functions and uniformly quasiconvex functions, int.j. math. math. sci.48(2003), 3053-3058. [25] d.k. thomas, on starlike and close-to-convex univalent functions, j. lond. math. soc. 42 (1967), 427-435. 1. introduction 2. some preliminary lemmas 3. main result references international journal of analysis and applications volume 18, number 1 (2020), 63-70 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-63 random common fixed point theorems for two pairs of nonlinear contractive maps in polish spaces kanayo stella eke∗, hudson akewe and jimevwo godwin oghonyon department of mathematics, covenant university, canaanland, km 10 idiroko road, p. m. b. 1023, ota, ogun state, nigeria ∗corresponding author: kanayo.eke@covenantuniversity.edu.ng abstract. this research work proves the random common fixed point theorem for two pairs of random weakly compatible mappings fulfilling certain generalized random nonlinear contractive conditions in polish spaces. an example is given to support the validity of our results. our results generalize and extend some works in literature. 1. introduction the random fixed point theory introduced in 1950 by prague school of probabilistic plays very important role in the theory of random integral, random differential equations and other areas of applied mathematics. some classical fixed point theorems in different abstract spaces are proved in the context of random fixed point theory (see; akewe et al.[1] , rashwan and albaqeri [2], hans [3] and nieto et al. [4]). the common fixed point of two pairs of weakly compatible mappings satisfying certain contractive conditions in g-partial metric spaces without assuming the continuity of any of the maps involved was proved by eke and akinlabi [8]. the random common fixed point of two pairs of random subsequentially continuous mappings with compatibility of type (e) satisfying certain generalized contractive conditions in polish spaces ( separable metric space) was established by rashwan and hammed [9]. in this paper, we prove the random version of received 2019-01-18; accepted 2019-02-25; published 2020-01-02. 2010 mathematics subject classification. 47h10. key words and phrases. polish space; random operators; random weakly compatible maps; random coincidence point; random common fixed point. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 63 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-63 int. j. anal. appl. 18 (1) (2020) 64 the result of eke and akinlabi [8] in the context of polish spaces by using the contractive maps of rashwan and hammed [9]. our results are extension and an improvement on some related results in the literature. 2. preliminaries let (ω,φ) be a measurable space, a a separable metric space and σn : ω → a a measurable sequence. an operator f : ω ×a → a is random operator, if for every x ∈ a, the mapping f(.,x) : ω → a is measurable. a measurable mapping ω : ω → a is a random fixed point of a random operator f : ω × a → a if f(v,ω(v)) = ω(v) for each v ∈ ω, (details of these definitions can be found in beg and abbas [5], choudhury and ray [6] and choudhury and upadhyah [7]). the following theorems are the results of eke and akinlabi [8] and rashwan and hammed [9] respectively. theorem 1.1 [8]: let b, d, e and f be self-maps of a g-partial metric space a satisfying b(a) ⊂ f(a), d(a) ⊂ e(a) and gp(ba,ba,db) ≤ hua,a,b(b,d,e,f), and gp(ba,db,db) ≤ hua,b,b(b,d,e,f), where h ∈ (0, 1) and ua,a,b(b,d,e,f) ∈ {gp(ea,ea,fb),gp(ba,ba,ea),gp(db,db,fb), gp(ba,ba,fb) + gp(db,db,ea) 2 } (1.1) and ua,b,b(b,d,e,f) ∈ {gp(ea,fb,fb),gp(ba,ea,ea),gp(db,fb,fb), gp(ba,fb,fb) + gp(db,ea,ea) 2 } (1.2) for all a,b ∈ a. if one of b(a), d(a), e(a) or f(a) is a complete subspace of a, then {b,e} and {d,f} have a unique point of coincidence in x. moreover if {b,e} and {d,f} are weakly compatible, then b, d, e and f have a unique common fixed point. theorem 1.2 [9]: let a be a polish space and b,d,e,f : ω × a → a are four random mappings satisfy d(b(v,a),f(v,b)) ≤ φ(max{d(e(v,a),f(v,a)),d(e(v,a),f(v,a)),d(f(v,b),d(v,b)), d(e(v,a),d(v,b))+d(f (v,y),b(v,x)) 2 }), int. j. anal. appl. 18 (1) (2020) 65 for all a,b ∈ a, and v ∈ ω, where φ : [0,∞) → [0,∞) is contractive modulus and non decreasing such that φ(0) = 0. if the two pairs {b,e} and {d,f} are weakly random subsequentially continuous and random compatible of type (e). then b,d,e and f have a unique common random fixed point in a. 3. main results in this section we present our results as follow: theorem 2.1 : let a be a polish space and b,d,e,f, : φ × a → a are two pairs random operators fulfilling b(v,a) ⊂ f(v,a), d(v,a) ⊂ e(v,a) and d(b(v,a),d(v,b)) ≤ ω(max{d(e(v,a),f(v,b)),d(e(v,a),b(v,a)),d(f(v,b),d(v,b)), d(e(v,a),d(v,b))+d(f (v,b),b(v,a)) 2 }), (2.1) for every a,b ∈ a, and v ∈ φ where ω : [0,∞) → [0,∞) is a comparison function. if one of b(v, a), d(v, a), e(v, a) or f(v, a) is a complete subspaces of a, then {b,e} and {d,f} have a unique random point of coincidence in a. additionally, if {b,e} and {d,f} are random weakly compatible, then b,d,e and f have a unique random common fixed point in a. proof: let a0(v) ∈ a be an arbitrary random variable in a. consider a1(v) ∈ a such that b(v,a0) = f(v,a1), since b(a) ⊂ f(a). suppose d(a) ⊂ e(a) then there is a2(v) ∈ a such that d(v,a1) = e(v,a2). consequently, two sequences an(v) and bn(v) in a can be generated such that; b(v,a2k(v)) = f(v,a2k+1(v)) = b2k+1(v) d(v,a2k+1(v)) = e(v,a2k+2(v)) = b2k+2(v). for a given k ∈ n and employing (2.1) we get d(bk(v),bk+1(v)) = d(b(v,ak−1(v)),d(v,ak(v))) ≤ ω(max{d(e(v,ak−1(v)),f(v,ak(v))), d(e(v,ak−1(v)),b(v,ak−1(v))),d(f(v,ak(v)),d(v,ak(v))), d(e(v,ak−1(v)),d(v,ak(v))) + d(f(v,ak(v)),b(v,ak−1(v))) 2 }) = ω(max{d(bk−1(v),bk(v)),d(bk−1(v),bk(v)),d(bk(v),bk+1(v)), d(bk−1(v),bk+1(v)) + d(bk(v),bk(v)) 2 }) ≤ ω(max{d(bk−1(v),bk(v)),d(bk(v),bk+1(v)), d(bk−1(v),bk(v)) + d(bk(v),bk+1(v)) 2 }) ≤ ω(d(bk−1(v),bk(v))) continuing the process and by induction we have d(bk(v),bk+1(v)) ≤ ωkd(b0(v),b1(v)). (2.2) int. j. anal. appl. 18 (1) (2020) 66 for n > m and using the triangle inequality we have d(bm(v),bn(v)) ≤ d(bm(v),bm+1(v)) + d(bm+1(v),bm+2(v)) + d(bm+2(v),bm+3(v)) + · · · + d(bn−1(v),bn(v)) ≤ ωmd(b0(v),b1(v)) + ωm+1d(b0(v),b1(v)) + ωm+2d(b0(v),b1(v)) + · · · + ωm+n−1d(b0(v),b1(v)) < (ωm + ωm+1 + ωm+2 + · · · + ωm+n−1)d(b0(v),b1(v)) ≤ ωm 1 − ωm d(b0(v),b1(v)). with the condition of ω we have that {bk(v)} is a cauchy sequence in a. if e(v, a) is a complete subspace of a, then there is b0 ∈ a such that e(v,ak(v)) = bk(v) converges to ω. likewise, e(v,ak(v)) = d(v,ak+1(v)) = bk → ω and f(v,ak−1(v)) = b(v,ak−2(v)) = bk−1 → ω as k →∞. if there is z in a such that e(v,z(v)) = ω(v). then we prove that b(v,z(v)) = ω(v). otherwise, we show that b(v,z(v)) 6= ω(v). d(b(v,z(v)),ω(v)) ≤ d(b(v,z(v)),d(v,ak(v))) + d(d(v,ak(v),ω(v)) ≤ ω(max{d(e(v,z(v)),f(v,ak(v))),d(e(v,z(v)),b(v,z(v))), d(f(v,ak(v)),d(v,ak(v))), d(e(v,z(v)),d(v,ak(v))) + d(f(v,ak(v)),b(v,z(v))) 2 }) +d(d(v,ak(v),ω(v)). as k →∞ we have d(b(v,z(v)),ω(v)) ≤ ω(max{d(ω(v),ω(v)),d(ω(v),b(v,z(v))),d(ω(v),ω(v)), d(ω(v),ω(v)) + d(ω(v),b(v,z(v))) 2 }) ≤ ω(d(b(v,z(v)),ω(v))) < d(b(v,z(v)),ω(v)), a contradiction, hence we have d(b(v,z(v)) = ω(v). this shows that b(v,z(v)) = e(v,z(v)) = ω(v). since ω(v) ∈ b(v,a) ⊂ f(v,a), then there is a u(v) ∈ a such that f(v,u(v)) = ω(v). we claim that d(v,u(v)) = ω(v). on the other hand, we assume that d(v,u(v)) 6= ω(v). int. j. anal. appl. 18 (1) (2020) 67 so using (1) we obtain, d(ω(v),d(v,u(v))) ≤ d(ω(v),b(v,ak(v))) + (b(v,ak(v)),d(v,u(v))) ≤ ω(max{d(e(v,ak(v)),f(v,u(v))),d(e(v,u(v)),b(v,ak(v))), d(d(v,u(v)),f(v,u(v))), d(e(v,ak(v)),d(v,u(v))) + d(f(v,u(v)),b(v,ak(v))) 2 }) +d(b(v,ak(v),ω(v)). as k →∞ we obtain d(ω(v),d(v,u(v))) ≤ ω(max{d(ω(v),d(v,u(v))),d(d(v,u(v)),ω(v)), d(d(v,u(v)),ω(v)) 2 }) ≤ ω(d(ω(v),d(v,u(v))) < d(ω(v),d(v,u(v)). this is a contradiction according to the condition of ω. therefore d(v,u(v)) = ω(v) . this shows that {b,e} and {d,f} have a common point of coincidence. consider {b,e} and {d,f} being random weakly compatible, then b(v,ω(v)) = b(v,e(v,z(v))) = e(v,b(v,z(v))) = e(v,ω(v)) = ω1(v) (say) and d(v,ω(v)) = d(v,f(v,u(v))) = f(v,d(v,u(v))) = f(v,ω(v)) = ω2(v) (say). now we prove that the points of coincidence are unique. d(ω1(v),ω2(v)) = d(b(v,ω1(v)),f(v,ω2(v)) ≤ ω(max{d(e(v,ω1(v)),f(v,ω2(v))),d(e(v,ω1(v)),b(v,ω1(v))), d(f(v,ω2(v)),d(v,ω2(v))), d(e(v,ω1(v))),d(v,ω2(v))) + d(f(v,ω2(v)),b(v,ω1(v))) 2 }) ≤ ω(max{d(ω1(v),ω2(v)),d(ω1(v),ω1(v)),d(ω2(v),ω2(v)), d(ω1(v),ω2(v)) + d(ω2(v),ω1(v)) 2 }) ≤ ω(d(ω1(v),ω2(v))) < d(ω1(v),ω2(v)). this shows that ω1(v) = ω2(v) by the property of ω. therefore b(v,ω(v)) = e(v,ω(v)) = d(v,ω(v)) = f(v,ω(v)) now, we prove that ω(v) is the common fixed point of b,d,e and f in a. we claim that ω(v) = d(v,ω(v)). int. j. anal. appl. 18 (1) (2020) 68 suppose ω(v) 6= d(v,ω(v)) then using (2.1) we have, d(ω(v),d(v,ω(v))) = d(b(v,ω(v)),d(v,ω(v))) ≤ ω(max{d(e(v,ω(v)),f(v,ω(v))),d(e(v,ω(v)),b(v,ω(v))), d(f(v,ω(v)),d(v,ω(v))), d(e(v,ω(v))),d(v,ω(v))) + d(f(v,ω(v)),b(v,ω(v))) 2 }) ≤ ω(max{d(ω(v),d(v,ω(v))),d(d(v,ω(v)),ω(v)), d(d(v,ω(v)),ω(v)) 2 }) ≤ ω(d(ω(v),d(v,ω(v))) < d(ω(v),d(v,ω(v)) this contradict our assumption that ω(v) 6= d(v,ω(v)). hence ω(v) = d(v,ω(v)). thus ω(v) is the random common fixed point of b,d,e and f. for the uniqueness of the random common fixed point of b,d,e and f, we assume a different random common fixed point of b,d,e and f say ω′(v) such that ω(v) = ω′(v) .on the other hand, let ω(v) 6= ω′(v) and using (2.1) we get d(ω(v),ω′(v)) = d(b(v,ω(v)),d(v,ω′(v))) ≤ ω(max{d(e(v,ω(v)),f(v,ω′(v))),d(e(v,ω(v)),b(v,ω(v))), d(f(v,ω′(v)),d(v,ω′(v))), d(e(v,ω(v))),d(v,ω′(v))) + d(f(v,ω′(v)),b(v,ω(v))) 2 }) ≤ ω(max{d(ω(v),ω′(v)),d(ω(v),ω(v)),d(ω′(v),ω′(v)), d(ω(v),ω′(v)) + d(ω′(v),ω(v)) 2 }) ≤ ω(d(ω(v),ω′(v)) < d(ω(v),ω′(v)), a contradiction, hence ω(v) = ω′(v). remark : theorem 2.1 gives an independent version of the result of rashwan and hammed [9](theorem 9) because the result is proved without the assumption of weakly random subsequential continuity and compatibility of type (e). theorem 2.1 proves the random version of the result of eke and akinlabi[8] (theorem 2.1) with general contractive mappings in the context of polish space. if ω(t) = k(t) in theorem 2.1 then we obtain the following corollary. corollary 2.2 : let a be a polish space and b,d,e,f, : φ × a → a are two pairs random operators fulfilling d(b(v,a),d(v,b)) ≤ int. j. anal. appl. 18 (1) (2020) 69 k(max{d(e(v,a),f(v,b)),d(e(v,a),b(v,a)),d(f(v,b),d(v,b)), d(e(v,a),d(v,b))+d(f (v,b),b(v,a)) 2 }), for every a,b ∈ a, and v ∈ φ where k ∈ [0, 1). if one of b(v, a), d(v, a), e(v, a) or f(v, a) is a complete subspaces of a, then {b,e} and {d,f} have a unique random point of coincidence in a. additionally, if {b,e} and {d,f} are random weakly compatible, then b,d,e and f have a unique random common fixed point in a. example: let x = r with the usual metric d and φ = [0, 1]. define ω : (0,∞) → [0,∞) by ω(t) = t. let the random operator b,d,e,f : φ×a → a be defined by e(v,a(v)) = f(v,a(v)) =   0, if a(v) ≤ 1 2, otherwise (3.1) b(v,a(v)) = d(v,a(v)) =   0, if a(v) ≤ 1 1 2 , otherwise (3.2) the pairs {b,e} and {d,f} are weakly compatible, b(v,a) ⊂ f(v,a), and d(v,a) ⊂ e(v,a). the contractive condition is satisfied and the unique random common fixed point of b,d,e and f is 0. conclusion: the random common fixed point of two pairs of generalized random nonlinear contractive mappings employing the property of random weakly compatible mappings is proved in the context of polish space. further work can be done using different contractive mappings in this space. we provided an example to support our result. acknowledgement: the authors are grateful to covenant university for supporting this research financially. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h. akewe , k. s. eke and v. olisama. on the equivalance of stochastic fixed point iterations for generalized ϕcontractive -like operators. int. j. anal. 2018(2018), article id 9576137. [2] r. a. rashwan and d. m. albaqeri. a common fixed point theorem and application to random integral equations. int. j. appl. math. res. 3(1)(2014), 71-80. [3] o. hans. random operator equations. proceedings of the fourth berkeley symposium on mathematical statistics and probability ii, part i, (1961), 85-202. [4] j. j. nieto, a. ouahab and r. rodriguez-lopez. random fixed point theorems in partially ordered metric spaces. fixed point theory appl. 2016(2016), art. id 98. [5] i. beg and m. abbas. iterative procedures for solution of random equations in banach spaces, j. math. anal. appl. 315(2006), 181-201. int. j. anal. appl. 18 (1) (2020) 70 [6] b. s. choudhury and m. ray. convergence of an iteration leading to a solution of a random operator equation, j. appl. math. stoch. anal. 12(1999), 161-168. [7] b. s. choudhury and a. upadhyay. an iteration leading to random solutions and fixed points of operators, soochow j. math. 25(1999), 395-400. [8] k. s. eke and g. o. akinlabi. common fixed point theorems for four maps in g-partial metric spaces. amer. j. appl. sci. 14(3)(2017), 372-380. [9] r. a. rashwan and h. a. hammad. random common fixed point theorem for random weakly subsequentially continuous generalized contractions with application. int. j. pure appl. math. 109(4)(2016), 813-826. 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 20-26 http://www.etamaths.com algebraic hyper-structures associated to nash equilibrium point and applications a. delavar khalafi and b. davvaz∗ abstract. in this paper, we generalize some concepts of the game theory such as nash equilibrium point, saddle point and existence theorems on hyper-structures. based on new definitions and theorems, we obtain some important results in the game theory. a few suitable examples have been given for better understanding. 1. introduction and preliminaries algebraic hyperstructures are suitable generalizations of classical algebraic structures. in a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. more exactly, if h is a non-empty set and p∗(h) is the set of all non-empty subsets of h, then we consider maps of the following type: fi : h×h −→p∗(h), where i ∈ {1, 2, . . . ,n} and n is a positive integer. the maps fi are called (binary) hyperoperations. for all x,y of h, fi(x,y) is called the (binary) hyperproduct of x and y. an algebraic system (h,f1, . . . ,fn) is called a (binary) hyperstructure. usually, n = 1 or n = 2. under certain conditions, imposed to the maps fi, we obtain the so-called semihypergroups, hypergroups, hyperrings or hyperfields. sometimes, external hyperoperations are considered, which are maps of the following type: h : r×h −→p∗(h), where r 6= h. usually, r is endowed with a ring or a hyperring structure. several books have been written on this topic, see [1, 2, 6, 13]. hyperstructure theory both extends some well-known group results and introduce new topics leading us to a wide variety of applications, as well as to a broadening of the investigation fields, for example see [4, 5, 8, 10–12]. a recent book on hyperstructures [2] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. another book [6] is devoted especially to the study of hyperring theory. several kinds of hyperrings are introduced and analyzed. the volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. the theory of suitable modified hyperstructures can serve as a mathematical background in the field of quantum communication systems. optimization theory is the study of the extremal values of a function: its minima and maxima. in mathematics, optimization refers to choosing the best element from some set of available alternatives. nonlinear programming deals with the problem of optimizing an objective function in the presence of some constraints. in [8], we generalized the optimization theory on algebraic hyperstructures. the game theory is another framework which has been generalized of optimization theory. the famous mathematician von numan has been proposed his important game theory in 1928. game theory is an important branch of applied mathematics in which decision maker chooses his strategy with regards to strategies of other players. in this theory any player tries to choose his best strategy for obtaining maximum pay off function. the methods and applications in the game theory are very different. in [9], bileaner two person nonzero sum game has been considered. in the mathematical domain the extensions of the previous works are very popular. our contribution in this paper is the extensions of some previous concepts in game theory on the hyperstructures and have considered several new examples that we can’t solve them via usual game theory. it means that we have obtained a new widespread in the game theory. 2010 mathematics subject classification. 20n20, 91a99. key words and phrases. algebraic hyperstructure; optimization; game theory; nash equilibrium point. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 20 algebraic hyper-structures associated to nash equilibrium point and applications 21 2. game theory in this paper, we address a hyper-structures as follows: ? : h ×h → h ⊗h ⊆p∗(h), (2.1) · : f ×h → h, + : h ×h → h, where h 6= ∅, ? is a commutative hyperoperation such that ?(h×h) = h⊗h, · and + are commutative binary operations and f is a filed. henceforth, let f = r. convex and concave functions play an important role in almost all branches of mathematics as well as other areas of science and engineering. convex and concave functions have many special and important properties. in this paper we use some these properties in game theory . let define p∗(x) = {x ? y ∈ h ⊗h : x, y ∈ x} for all non-empty subset x in h. now, we formulate the n-person game theory problem on hyper-structures as follows: define the function fi : p ∗(x1) ×···×p∗(xn) → r, i = 1, · · · ,n, where xi is any non-empty subset of h and fi and xi are a pay off function and a strategy set of i-th player, respectively. let w = x1 ×···×xn. the next definition plies an important role in the following discussions. definition 2.1. the n-tuple (x1 ?y1, · · · ,xn ?yn) ∈ p∗(x1)×···×p∗(xn) is called nash equilibrium point, if the following inequalities hold. f1(x1 ? y1, · · · ,xn ? yn) 6 f1(x1 ? y1, · · · ,xn ? yn), (2.2) for all x1 ? y1 ∈ p∗(x1), ... fn(x1 ? y1, · · · ,xn ? yn) 6 fn(x1 ? y1, · · · ,xn ? yn), for all xn ? yn ∈ p∗(xn). henceforth, for simplicity let consider n = 2. as a special case we consider the following situation. if we have f1(x1 ? y1,x2 ? y2) + f2(x1 ? y1,x2 ? y2) = 0, for all (x1 ?y1,x1 ?y1) ∈ p∗(x1)×p∗(x2). thus, the nash equilibrium point satisfies in the following inequalities, f1(x1 ? y1,x2 ? y2) 6 f1(x1 ? y1,x2 ? y2) 6 f1(x1 ? y1,x2 ? y2), (2.3) for all (x1 ? y1,x2 ? y2) ∈ p∗(x1) ×p∗(x2), or equivalently f2(x1 ? y1,x2 ? y2) 6 f2(x1 ? y1,x2 ? y2) 6 f2(x1 ? y1,x2 ? y2), (2.4) for all (x1 ? y1,x2 ? y2) ∈ p∗(x1) ×p∗(x2). the next definition plays an important role in the game theory. definition 2.2. the pair (x1 ? y1,x2 ? y2) ∈ p∗(x1) ×p∗(x2) is called the saddle point, if satisfies the first inequalities (3) or second inequalities (4). suppose that v = inf x2,y2∈x2 sup x1,y1∈x1 f(x1 ? y1,x2 ? y2) and v = sup x1,y1∈x1 inf x2,y2∈x2 f(x1 ? y1,x2 ? y2). clearly, we have v ≤ v. definition 2.3. the strategy x1, y1 ∈ x1 is called max-min if v = inf x2,y2∈x2 f(x1 ? y1,x2 ? y2), and similarly, the strategy x2 ? y2 ∈ x2 is called min-max, if v = sup x1,y1∈x1 f(x1 ? y1,x2 ? y2). the following theorem gives us a necessary and sufficient condition that guaranties the existence of saddle point. 22 delavar khalafi and davvaz theorem 2.1. suppose that the pay-off function f(x1 ? y1,x2 ? y2) on x1 ×x2 is given. there is a saddle point (x1 ? y1,x2 ? y2) if and only if sup x1,y1∈x1 inf x2,y2∈x2 f(x1 ? y1,x2 ? y2) = inf x2,y2∈x2 sup x1,y1∈x1 f(x1 ? y1,x2 ? y2). (2.5) in addition x1 ? y1 ∈ p∗(x1) and x2 ? y2 ∈ p∗(x2) are max-min and min-max, respectively. proof. suppose that (x1 ? y1,x2 ? y2) is a saddle point. we have v ≤ sup x1,y1∈x1 f(x1 ? y1,x2 ? y2) = f(x1 ? y1,x2 ? y2) = inf x2,y2∈x2 f(x1 ? y1,x2 ? y2) ≤ v, so the equation (5) is held. now, suppose that we have the equation (5) and x1 ? y1 and x2 ? y2 are max-min and min-max strategies, respectively. then, f(x1 ? y1,x2 ? y2) ≤ sup x1,y1∈x1 f(x1 ? y1,x2 ? y2) = v = v = inf x2,y2∈x2 f(x1 ? y1,x2 ? y2) ≤ f(x1 ? y1,x2 ? y2). this completes the proof. � the saddle point does not always exist. the following example denotes such a situation. example 2.1. suppose that h = r and x1 = x2 = [0, 1]. we define f : p∗([0, 1]) −→ r and ? respectively as follows: x ? y = x, f(x1 ? y1,x2 ? y2) = 3x 2 1 − 5x1x2 + 3x 2 2. clearly, m(x1 ? y1) = min x2,y2∈x2 f(x1 ? y1,x2 ? y2) = 11x21 12 . then, we have: v = max x1,y1∈x1 11x21 12 = 11 12 , x1 = 1. similarly, n(x2 ? y2) = maxx1,y1∈x1 f(x1 ? y1,x2 ? y2) = max{3x22, 3x22 − 5x2 + 3}, x2 = 3 5 . therefore, we have v = min x2,y2∈x2 n(x2 ? y2) = 27 25 , v < v. now, we obtain max-min and min-max strategies. as min x2,y2∈x2 f(1 ? y1,x2 ? y2) = min x2∈x2 3 − 5x2 + 3x22 = 11 12 = v, max x1,y1∈x1 f(x1 ? y1, 3 5 ? y2) = max x1∈x1 3x21 − 3x1 + 27 25 = 27 25 = v. therefore, {1 ? y1 : y1 ∈ [0, 1]} and {35 ? y2 : y2 ∈ [0, 1]} are max-min and min-max strategy set, respectively. according to theorem (2.5), (1 ? y1, 3 5 ? y2) is not a saddle point. the above example shows that we must generalize the previous saddle point definition. definition 2.4. let � > 0. the pair (x�1 ? y � 1,x � 2 ? y � 2) ∈ p∗(x1) ×p∗(x2) is called �-saddle point of f(x1 ? y1,x2 ? y2) on x1 ×x2, if f(x1 ? y1,x � 2 ? y � 2) − � ≤ f(x � 1 ? y � 1,x � 2 ? y � 2) ≤ f(x � 1 ? y � 1,x2 ? y2) + � (2.6) for all x1, y1 ∈ x1, x2, y2 ∈ x2. lemma 2.1. let x1 ?y1, x2 ?y2 are max-min and min-max strategies, respectively and � = v−v ≥ 0. then, (x1 ? y1, x2 ? y2) is an �−saddle point of f(x1 ? y1,x2 ? y2) on x1 ×x2. proof. if v = v, so x1 ? y1, x2 ? y2 is a saddle point. let � = v −v > 0. clearly, x1 ? y1, x2 ? y2 is an �−saddle point. � using the above definition, we consider the following example. example 2.2. in example 2, let � = v −v. as f(x1 ? y1, 3 5 ? y2) − � = 3x21 − 3x1 + 27 25 − ( 27 25 − 11 12 ) ≤ f(1 ? y1, 35 ? y2) = 27 25 ≤ f(1 ? y1,x2 ? y2) + � = 3 − 5x2 + 3x22 + ( 27 25 − 11 12 ), algebraic hyper-structures associated to nash equilibrium point and applications 23 for all x1, y1 ∈ x1, x2, y2 ∈ x2, so (x�1 ? y�1,x�2 ? y�2) = (1 ? y1, 3 5 ? y2), is a �−saddle point of f(x1 ? y1,x2 ? y2). similar to saddle point, we can generalize the concepts of min-max and max-min as follows: definition 2.5. let � > 0. the strategies x�1 ? y � 1 and x � 2 ? y � 2 are �− max-min and �−min-max, if inf x2,y2∈x2 f(x�1 ? y � 1,x2 ? y2) ≥ v − � and sup x1,y1∈x1 f(x1 ? y1,x � 2 ? y � 2) ≤ v + �. in the remaining part, we consider the some new topology characters of hyper-structures [7] and their applications in game theory. let h be a metric space, x, y be compact subsets of h and y (x1 ? y1) = arg min x2, y2∈x2 f(x1 ? y1,x2 ? y2) = {x̂2 ? ŷ2|x̂2, ŷ2 ∈ x2, f(x1 ? y1, x̂2 ? ŷ2) = min x2, y2∈x2 f(x1 ? y1,x2 ? y2)}. theorem 2.2. suppose that the pay off function f(x1?y1,x2?y2) is a continuous function on x1×x2 and p∗(x1), p ∗(x2) are compact sets in h ⊗ h. then, the function g(x1 ? y1) = min x2, y2∈x2 f(x1 ? y1,x2 ? y2) on x1 is a continuous function. proof. let {xk1} and {yk1} be two sequences in x1 such that convergence to x1 and y1, respectively. by considering g(xk1 ? y k 1 ), we can prove that it converges to g(x1 ? y1). in contradiction, there are subsequences {xkl1 }, {y kl 1 } in x1 such that lim l→∞ g(x kl 1 ? y kl 1 ) 6= g(x1 ? y1). choosing sequence {xkl2 ?y kl 2 ∈ y (x kl 1 ?y kl 1 )}, based on compactness of x2, we have lim l→∞ x kl 2 ?y kl 2 = x2 ?y2. we must show that x2 ? y2 ∈ y (x1 ? y1). by definition xkl2 ? y kl 2 , we have g(x kl 1 ? y kl 1 ) = f(x kl 1 ? y kl 1 ,x kl 2 ? y kl 2 ) ≤ f(x kl 1 ? y kl 1 ,x2 ? y2), for all x2, y2 ∈ x2. in the above inequality, when l →∞, we conclude that f(x1 ? y1,x2 ? y2) ≤ f(x1 ? y1,x2 ? y2), for all x2, y2 ∈ x2. therefore, x2 ? y2 ∈ y (x1 ? y1), that is lim l→∞ g(x kl 1 ? y kl 1 ) = lim l→∞ f(x kl 1 ? y kl 1 ,x kl 2 ? y kl 2 ) = f(x1 ? y1,x2 ? y2) = g(x1 ? y1). � under what conditions is the function y (x1 ? y1) continuous? we assert the sufficient condition, that guaranties the continuous function y (x1 ? y1). theorem 2.3. suppose that the conditions of previous theorem have been satisfied and for any x1, y1 ∈ x1, y (x1 ? y1) = {x2 ? y2} be a singleton. then, y (x1 ? y1) is a continuous function on x1. proof. suppose that the function y (x1 ?y1) isn’t continuous in x1 ?y1 ∈ p∗(x1), so there is a sequence {xk1 ? yk1} in p ∗(x1), such that it converges {x1 ? y1}, but {xk1 ? yk1} = {x k 2 ? y k 2} does not convergent to y (x1 ?y1) = {x2 ?y2}. therefore, there is an �−neighborhood n?� (y (x1 ?y1)) ⊂ p∗(x2) that does not contain infinite number of elements {xk2 ? yk2}. rely on compactness of p∗(x2) −n?� (y (x1 ? y1)), we have subsequence {y (xkl1 ?y kl 1 )} = {x kl 2 ?y kl 2 }⊂ p ∗(x2) −n?� (y (x1 ?y1)), such that it convergent to x′2 ? y ′ 2 6= x2 ? y2. according to the previous theorem, we conclude that x′2 ? y′2 = y (x1 ? y1), which it contradicts to singleton. � definition 2.6. let f : p∗(x) → r, where x is non-empty convex subset in h. the function f is called a convex function on p∗(x) if f([λx1 + (1 −λ)x2] ? [λy1 + (1 −λ)y2]) ≤ λf(x1 ? y1) + (1 −λ)f(x2 ? y2) for each x1,x2,y1,y2 ∈ x, x1 ? y1, x2 ? y2 ∈ p∗(x) and for all 0 ≤ λ ≤ 1. the function is called strictly convex on p∗(x) if the inequality is satisfied as a strict inequality for each distinct x1 ? y1, x2 ? y2 ∈ p∗(x) and 0 < λ < 1. the function f is called concave (strictly concave) on x if −f is convex (strictly convex) on x. 24 delavar khalafi and davvaz the following function is an example of convex function [8]. example 2.3. let h = r+ ×r+. suppose that zmin = min{x1,x2,y1,y2}, zmax = max{x1,x2,y1,y2}, (x1,y1) ? (x2,y2) = [zmin, zmax] × [zmin, zmax] ⊆ r+ ×r+ and f : h⊗h → r is defined by f((x1,y1)?(x2,y2)) = zmax−zmin, for all (x1,y1), (x2,y2) ∈ x, where x is any non-empty convex subset in h. suppose that ((x̄1, ȳ1), (x̄2, ȳ2), z̄) and ((x̂1, ŷ1), (x̂2, ŷ2), ẑ) in epi?f and x̄λ = λx̄1 + (1 −λ)x̄2, ȳλ = λȳ1 + (1 −λ)ȳ2, x̂λ = λx̂1 + (1 −λ)x̂2, ŷλ = λŷ1 + (1 −λ)ŷ2. one can show that max{x̄λ, ȳλ, x̂λ, ŷλ}− min{x̄λ, ȳλ, x̂λ, ŷλ}≤ λz̄ + (1 −λ)ẑ. that is, ((x̄λ, ȳλ), (x̂λ, ŷλ), λz̄ + (1 −λ)ẑ) ∈ epi?f. therefore, epi?f is a convex set, so we conclude that the function f is also a convex function. theorem 2.4. let h = rn, x1, x2 be convex and compact subsets of h and f(x1 ? y1,x2 ? y2) is a continuous function on x1 × x2. suppose that for any x2 ? y2 ∈ p∗(x2), f(x1 ? y1,x2 ? y2) be a concave function with respect to x1 ?y1 ∈ p∗(x1) and for any x1 ?y1 ∈ p∗(x1) it is a convex function with respect to x2 ? y2 ∈ p∗(x2). then, the function f(x1 ? y1,x2 ? y2) has a saddle point. proof. : at first , we consider the special case that the function f(x1 ?y1,x2 ?y2) be a strictly convex w.r.t x2 ? y2 ∈ p∗(x2). then, for any x1 ? y1 ∈ p∗(x1), the function f(x1 ? y1,x2 ? y2) obtains its unique minimum on x2 in y (x1 ? y1). according to previous theorems, we conclude g(x1 ? y1) and y (x1 ? y1) are continuous function on x1. suppose that the function g(x1 ? y1) obtains its minimum on x1 in x1 ? y1. we can show that (x1 ? y1,y (x1 ? y1)) is a saddle point of f(x1 ? y1,x2 ? y2) on x1×x2. let x1 ?y1 ∈ p∗(x1) be arbitrary, 0 < λ < 1 and y = y ([(1−λ)x1 +λx1]?[(1−λ)y1 +λy1]). according to concavity of the function f(x1 ? y1,x2 ? y2) with respect to x1 ? y1 ∈ p∗(x1), we have (1 −λ)g(x1 ? y1) + λf(x1 ? y1,y ([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1])) ≤ (1 −λ)f(x1 ? y1,y ([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1])) +λf(x1 ? y1,y ([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1])) ≤ f([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1],y ([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1])) = g([(1 −λ)x1 + λx1] ? [(1 −λ)y1 + λy1]) ≤ g(x1 ? y1). therefore, λf(x1 ?y1,y ([(1−λ)x1 + λx1] ? [(1−λ)y1 + λy1])) ≤ λg(x1 ?y1). divided by λ and it tends to zero, we have the following inequalities: f(x1 ? y1,y (x1 ? y1)) ≤ g(x1 ? y1) = f(x1 ? y1,y (x1 ? y1)) ≤ f(x1 ? y1,x2 ? y2), for all x1, y1 ∈ x1 and x2, y2 ∈ x2. in general we consider the following perturbed function f�(x1 ?y1,x2 ?y2) = f(x1 ?y1,x2 ?y2) + �h(x2 ?y2) where h(x2 ?y2) is a continuous and strictly convex function on x2. clearly f�(x1 ?y1,x2 ?y2) is a continuous, concave with respect to x1 ?y1 and strictly convex with respect to x2 ? y2 function. using previous discussion, we have f�(x1 ? y1,x � 2 ? y � 2) ≤ f�(x � 1 ? y � 1,x � 2 ? y � 2) ≤ f�(x � 1 ? y � 1,x2 ? y2), (2.7) for all x1, y1 ∈ x1 and x2, y2 ∈ x2. let � = �k in the inequalities (7) and �k → 0+. because of compactness of x1 and x2, we conclude that x � 1 ? y � 1 → x1 ? y1 and x�2 ? y�2 → x2 ? y2. therefore, (x1 ? y1,x2 ? y2) is a saddle point of the function f(x1 ? y1,x2 ? y2). � 3. applications in this section, we consider some examples in game theory and explain our theory in the previous section. algebraic hyper-structures associated to nash equilibrium point and applications 25 3.1. examples. the following examples show that, how we can use nash equilibrium point in practice. first of all, we show that the nash equilibrium point gives a generalization of usual optimization problem. example 3.1. since the optimization problem is a special case of the game theory with one player, so we obtain the optimization model as follows: f1(x1 ? y1) 6 f1(x1 ? y1), for all x1 ? y1 ∈ p∗(x1). this means that x1 ? y1 ∈ argmin{f(x1 ? y1) : x1 ? y1 ∈ p∗(x1)}. example 3.2. let h = {f ∈ c|f : r → r} and x1, x2 ⊆ h, where x1 and x2 are such continuous functions that have left and right inverse. we define the following function. f1(g,h) = ∫ dg ⋂ dh [(g −h)(x)]2dx, where g and h are the left and right inverse of an arbitrary function and dg and dh are the domains of g and h, respectively. let f2(g,h) = −f1(g,h). clearly (g,g) = (f,f) is a saddle point, i.e., the left and right inverse are the same. 3.2. numerical examples. example 3.3. in example 3, we define x1 = {(x, 0) : 0 ≤ x ≤ m}, x2 = {(0,y) : 0 ≤ y ≤ n}. clearly, x1, x2 are compact and convex subsets of h. according to example 3, the function h(x1?y1) = zmax−zmin = max{x11,x21} and e(x2?y2) = zmax−zmin = max{y12,y22}, where x1 = (x11, 0), y1 = (x 2 1, 0), x2 = (0,y 1 2), y2 = (0,y 2 2), x1, y1 ∈ x1, x2, y2 ∈ x2 are convex functions. let f(x1 ? y1, x2 ? y2) = e(x2 ? y2) −h(x1 ? y1). we can show that the function f(x1 ? y1,x2 ? y2) is a continuous, concave function with respect to x1?y1) ∈ p∗(x1) and convex function with respect to x2?y2) ∈ p∗(x2). therefore, the all assumptions of theorem (2.11) has been held. clearly, {(x1 ? y1,x2 ? y2) : max{x11,x21} = max{y12,y22}} is the set of saddle points of the function f(x1 ? y1,x2 ? y2) on x1 ×x2. example 3.4. let define h = r and x1 = x2 = h, x ? y = [min{x,y},max{x,y}], f(x1 ? y1,x2 ? y2) = (x 2 2 + y 2 2) − (x 2 1 + y 2 1), for all x1, y1 ∈ x1 and x2, y2 ∈ x2. in this example, x1, x2 are not compact subset of h. we choose x1 = y1 = x2 = y2 = 0. then, x1 ? y1 = {0}, x2 ? y2 = {0} and f({0},{0}) = 0. therefore, (x1 ? y1,x2 ? y2) = ({0},{0}) is a saddle point, that is f(x1 ? y1,x2 ? y2) = −(x21 + y 2 1) ≤ f(x1 ? y1,x2 ? y2) = 0 ≤ f(x1 ? y1,x2 ? y2) = (x 2 2 + y 2 2), for all x1, y1 ∈ x1 and x2, y2 ∈ x2. thus, theorem (2.11) is a sufficient condition. references [1] p. corsini, prolegomena of hypergroup theory, second edition, aviani editore, 1993. [2] p. corsini and v. leoreanu, applications of hyperstructure theory, advances in mathematics, kluwer academic publishers, dordrecht, 2003. [3] i. cristea and m. stefanescu, hypergroups and n-ary relations, eur. j. comb. 31 (2010), 780-789. [4] b. davvaz, a. dehghan nezhad and a. benvidi, chemical hyperalgebra: dismutation reactions, match commun. math. comput. chem. 67 (2012) 55-63. [5] b. davvaz, a. dehghan nezhad and a. benvidi, chain reactions as experimental examples of ternary algebraic hyperstructures, match commun. math. comput. chem. 65 (2011) 491-499. [6] b. davvaz and v. leoreanu-fotea, hyperring theory and applications, international academic press, usa, 2007. [7] a. dehghan nezhad and b. davvaz, universal hyperdynamical systems, bull. korean math. soc. 47 (2010), 513-526. [8] a. delavar khalafi and b.davvaz, algebraic hyper-structures associated to convex analysis and applications, filomat 26 (2012) 55-65. [9] a. delavarkhalafi solution methods for equilibrium points in bileaner non-zero sum games, ph.d. thesis (2002) (in russian). [10] m. ghadiri, b. davvaz and r. nekouian, hv-semigroup structure on f2-offspring of a gene pool, int. j. biomath. 5 (2012), art. id 1250011. [11] s. hošková and j. chvalina, discrete transformation hypergroups and transformation hypergroups with phase tolerance space, discre. math. 308 (2008), 4133-4143. 26 delavar khalafi and davvaz [12] v. leoreanu-fotea, b. davvaz, fuzzy hyperrings, fuzzy sets and syst. 160 (2009), 2366-2378. [13] t. vougiouklis, hyperstructures and their representations, hadronic press, florida, 1994. department of mathematics, yazd university, yazd, iran ∗corresponding author: davvaz@yazd.ac.ir 1. introduction and preliminaries 2. game theory 3. applications 3.1. examples 3.2. numerical examples references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 170-177 http://www.etamaths.com hankel determinant for a class of analytic functions related with lemniscate of bernoulli ashok kumar sahoo1 and jagannath patel2,∗ abstract. the object of the present investigation is to solve fekete-szegö problem and determine the sharp upper bound to the second hankel determinant for a new class r̃ of analytic functions in the unit disk. 1. introduction and preliminaries let a be the class of functions f of the form (1.1) f(z) = z + ∞∑ n=2 anz n which are analytic in the open unit disk u = {z ∈ c : |z| < 1}. a function f ∈ a is said to be starlike of order ρ and convex of order ρ, if and only if re{zf′(z)/f(z)} > ρ and re{(1 + zf′′(z))/f′(z)} > ρ for 0 ≤ ρ < 1 and z ∈u. by usual notations, we write these classes of functions by s ?(ρ) and k (ρ), respectively. we denote s ?(0) = s ? and k (0) = k , the familiar subclasses of starlike and convex functions in u. further, we say that a function f ∈ a is in the class r(ρ), if it satisfies the inequality: (1.2) re{f′(z)} > ρ (z ∈u) we note that r(ρ) is a subclass of close-to-convex functions order ρ(0 ≤ ρ < 1) in u. we write r(0) = r, the familiar class functions in a whose derivatives have a positive real part in u. a function f is said to be subordinate to a function g, written as f ≺ g, if there exists a schwarz function w with w(0) = 0 and |w(z)| < 1 such that f(z) = g(w(z)),z ∈ u. in particular, if g is univalent in u, then f(0) = g(0) and f(u) ⊂ g(u). let p denote the class of analytic functions φ normalized by (1.3) φ(z) = 1 + p1z + p2z 2 + · · · (z ∈u) such that re{φ(z)} > 0 in u. 2010 mathematics subject classification. 30c45. key words and phrases. analytic function; subordination; fekete-szegö problem; hankel determinant. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 170 analytic functions related with lemniscate of bernoulli 171 definition. a function f ∈ a is said to be in the class r̃, if it satisfies the condition (1.4) ∣∣∣(f′(z))2 − 1∣∣∣ < 1 (z ∈u). it follows from (1.4) and the definition of subordination that a function f ∈ r̃ satisfies the following subordination relation (1.5) f′(z) ≺ √ 1 + z (z ∈u). to bring out the geometrical significance of the class r̃, we set h(z) = √ 1 + z, z ∈u and note that ω = h(eiθ) = √ 1 + eiθ (0 ≤ θ ≤ 2π). which yields ω2 − 1 = eiθ or |ω2 − 1| = 1. letting ω = u + iv, we deduce that (u2 + v2)2 = 2(u2 −v2). thus, h(u) is the region bounded by the right half of the lemniscate of bernoulli given by { u + iv ∈ c : (u2 + v2)2 = 2(u2 −v2) } , which implies that the derivative of functions in r̃ have a positive real part and hence univalent in u [1]. noonan and thomas [12] defined the q-th hankel determinant of the function f, given by (1.1) by (1.6) hq(n) = ∣∣∣∣∣∣∣∣∣ an an+1 · · · an+q−1 an+1 an+2 · · · an+q ... ... ... ... an+q−1 an+q · · · an+2q−2 ∣∣∣∣∣∣∣∣∣ (a1 = 1,n,q ∈ n). the determinant given in (1.6) has been studied by several authors with the subject of inquiry ranging from the rate of growth of hq(n) (as n →∞) [13] to the determination of precise bounds with specific values of n and q for certain subclasses of analytic functions in the unit disc u. for n = 1,q = 2,a1 = 1 and n = q = 2, the hankel determinant simplifies to h2(1) = |a3 −a22| and h2(2) = |a2a4 −a23|. we refer to h2(2) as the second hankel determinant. it is known [1] that if the function f, given by (1.1) is analytic and univalent in u, then the sharp inequality h2(1) = |a3 −a22| ≤ 1 holds. for a family f of functions in a of the form (1.1), the more general problem of finding the sharp upper bounds for the functionals |a3−µa22| (µ ∈ r or µ ∈ c) is popularly known as fekete-szegö problem for the class f. the fekete-szegö problem for the known classes of univalent functions, starlike functions, convex functions and closeto-convex functions has been completely settled ([2], [5], [6], [7]). recently, janteng et al. [3, 4] have obtained the sharp upper bounds to the second hankel determinant h2(2) for the family r. for initial work on the class r one may refer to the paper by macgregor [11]. in our present investigation, by following the techniques devised by libera and zlotkiewicz [8, 9], we solve the fekete-szegö problem and also determine the sharp upper bound to the second hankel determinant h2(1) for the class r̃. to establish our main results, we shall need the followings lemmas. 172 sahoo and patel lemma 1.1. let the function φ, given by (1.3) be a member of the class p. then (1.7) |pk| ≤ 2 (k ≥ 1) and (1.8) ∣∣p2 −ν p21∣∣ ≤ 2 max{1, |2ν − 1|}. the estimate (1.7) is sharp for the function ϕ(z) = (1+z)/(1−z),z ∈u, whereas the estimate (1.8) is sharp for the functions given by ϕ and ψ(z) = (1+z2)/(1−z2), z ∈ u. we note that the estimate (1.7) is contained in [1] and the estimate (1.8) is obtained in [10]. lemma 1.2 ([9],see also [8]). if the function φ, given by (1.3) belongs to the class p, then (1.9) p2 = 1 2 { p21 + (4 −p 2 1)x } and (1.10) p3 = 1 4 { p31 + 2(4 −p 2 1)p1x− (4 −p 2 1)p1x 2 + 2(4 −p21)(1 −|x| 2)z } for some complex numbers x,z satisfying |x| ≤ 1 and |z| ≤ 1. 2. main results now, we determine an upper bound for the fekete-szegö problem of the class r̃. theorem 2.1. if the function f, given by (1.1) belongs to the class r̃, then for any µ ∈ c (2.1) |a3 −µa22| ≤ 1 6 max { 1, |2 + 3 µ| 8 } . the estimate in (2.1) is sharp. proof. from (1.5), it follows that (2.2) f′(z) = √ 1 + w(z) (z ∈u), where w is analytic and satisfies the condition w(0) = 0 and |w(z)| < 1 in u. setting (2.3) χ(z) = 1 + w(z) 1 −w(z) = 1 + p1z + p2z 2 + · · · (z ∈u), we see that χ ∈ p. from (2.3), we get (2.4) w(z) = χ(z) − 1 χ(z) + 1 (z ∈u) so that by (2.2) and (2.4), we get (2.5) f′(z) = ( 2χ(z) 1 + χ(z) )1 2 (z ∈u). analytic functions related with lemniscate of bernoulli 173 now, by substituting the series expansion of χ from (2.3) in (2.5), it is easily seen that ( 2χ(z) 1 + χ(z) )1 2 = 1 + 1 4 p1z + ( 1 4 p2 − 5 32 p21 ) z2 + ( 1 4 p3 − 5 16 p1p2 + 13 128 p31 ) z3 + · · · .(2.6) differentiating the series expansion of f given by (1.1) with respect to z and comparing the coefficients of z,z2 and z3 in (2.6), we deduce that a2 = 1 8 p1(2.7) a3 = 1 12 ( p2 − 5 8 p21 ) (2.8) a4 = 1 16 ( p3 − 5 4 p1p2 + 13 32 p31 ) .(2.9) thus, by using (2.7) and (2.8), we get (2.10) ∣∣a3 −µa22∣∣ = 112 ∣∣∣∣p2 − 116 (10 + 3µ)p21 ∣∣∣∣ the expression in (2.10) with the aid of (1.8) yields the required estimate (2.1). the estimate in (2.1) is sharp for the function f0 ∈ a defined by (2.11) f′0(z) = {√ 1 + z2, |2 + 3 µ| ≤ 8 √ 1 + z, |2 + 3 µ| > 8. this completes the proof of theorem 2.1. � letting µ = 0(or µ = 1 respectively) in theorem 2.1, we get corollary 2.1. if the function f, given by (1.1) belongs to the class r̃, then |a3| ≤ 1 6 and |a3 −a22| ≤ 1 6 .(2.12) the estimates in (2.12) are sharp for the function f0 ∈ a defined by (2.13) f′0(z) = √ 1 + z2 (z ∈u). if µ ∈ r, then theorem 2.1 reduces to corollary 2.2. let µ ∈ r. if the function f, given by (1.1) belongs to the class r̃, then (2.14) ∣∣a3 −µa22∣∣ ≤   − 2 + 3µ 48 , µ ≤− 10 3 1 6 , − 10 3 ≤ µ ≤ 2 2 + 3µ 48 , µ > 2. the estimates in (2.14) are sharp. 174 sahoo and patel proof. first, we assume that µ < −10/3. then, (2+3µ)/8 < −1 so that |2+3µ|/8 > 1. hence by using (2.1), we get (2.15) |a3 −µa22| ≤ |2 + 3µ| 48 = − 2 + 3µ 48 . next, if −10/3 ≤ µ ≤ 2, then |2 + 3µ| ≤ 1 so that (2.16) |a3 −µa22| ≤ 1 6 again by the use of (2.1). finally, if µ > 2, then (2 + 3µ)/8 > 1. thus, by (2.1) (2.17) |a3 −µa22| ≤ 2 + 3µ 48 . the estimates are sharp for the function f1 defined in u by f′1(z) = √ 1 + z, for µ < −10/3 or µ > 2, and for the function f0 given by (2.13) in the case −10/3 ≤ µ ≤ 2. � in the following theorem, we find the sharp upper bound to the second hankel determinant for the class r̃. theorem 2.2. let the function f, given by (1.1) be a member of the family r̃. then (2.18) ∣∣a2a4 −a23∣∣ ≤ 136. the estimate in (2.18) is sharp. proof. from (2.7), (2.8) and (2.9), we have∣∣a2a4 −a23∣∣ = ∣∣∣∣ 1128 ( p1p3 − 5 4 p21p2 + 13 32 p41 ) − 1 144 ( p22 − 5 4 p21p2 + 25 64 p41 )∣∣∣∣ = 1 16 ∣∣∣∣18p1p3 − 5288p21p2 − 19p22 + 172304p41 ∣∣∣∣ .(2.19) since the function χ, given by (2.3) and the function χ(eiθz) (θ ∈ r) are in the class p simultaneously, we assume without loss of generality that p1 > 0. for convenience of notation, we write p1 = p (0 ≤ p ≤ 2). now, by using lemma 2.2 in (2.19), we get∣∣a2a4 −a23∣∣ = 1 16 ∣∣∣∣ ( 1 32 p4 + 1 16 (4 −p2)p2x− 1 32 (4 −p2)p2x2 + 1 16 (4 −p2)p(1 −|x|2)z ) − ( 5 576 p4 + 5 576 (4 −p2)p2x ) − ( 1 36 p4 + 1 18 (4 −p2)p2x + 1 36 (4 −p2)2x2 ) + 17 2304 p4 ∣∣∣∣ = 1 16 ∣∣∣∣ 52304p4 − 1576 (4 −p2)p2x− 1288{8(4 −p2) + 9p2}(4 −p2)x2 + 1 16 (4 −p2)p(1 −|x|2)z ∣∣∣∣ (2.20) analytic functions related with lemniscate of bernoulli 175 for some x (|x| ≤ 1) and for some z (|z| ≤ 1). applying the triangle inequality in (2.20) and replacing |x| by y in the resulting equation, we get∣∣a2a4 −a23∣∣ ≤ 116 { 5 2304 p4 + 1 576 (4 −p2)p2y + 1 288 (4 −p2)(2 −p)(16 −p)y2 + 1 16 (4 −p2)p } = g(p,y) (0 ≤ p ≤ 2, 0 ≤ y ≤ 1) (say).(2.21) we next maximize the function g(p,y) on the closed rectangle [0, 2]× [0, 1]. differentiating the function g, given in (2.21) with respect to y, we deduce that (2.22) ∂g ∂y = 1 9216 (4 −p2)p2 + 1 2304 (4 −p2)(2 −p)(16 −p)y > 0 for 0 < p < 2 and 0 < y < 1. thus, in view of (2.22), the function g(p,y) cannot have a maximum in the interior on the closed rectangle [0, 2]× [0, 1]. therefore, for fixed p ∈ [0, 2] (2.23) max 0≤y≤1 g(p,y) = g(p, 1) = f(p) (say), where f(p) = 1 16 { 5 2304 p4 + 1 576 (4 −p2)p2 + 1 288 (4 −p2)(2 −p)(16 −p) + 1 16 (4 −p2)p } (0 ≤ p ≤ 2).(2.24) on differentiating the function f , given by (2.24) followed by a simple calculation yields f ′(p) = − 1 9216 (7p2 + 104)p < 0 which implies that the function f is a decreasing function of p so that max0≤p≤2 f(p) occurs at p = 0. thus, the upper bound in (2.21) corresponds to p = 0 and y = 1 from which we get the required estimate (2.18). equality holds in (2.18) for the function f0 ∈ a , given by (2.13) and the proof of theorem 2.2 is thus completed. � next, we determine the upper bound for the fourth coefficient of functions belonging to the class r̃. theorem 2.3. if the function f, given by (1.1) belongs to the class r̃, then (2.25) |a4| ≤ 1 8 and the estimate is sharp. proof. using lemma 1.1 in (2.9) and following the lines of proof of theorem 1.2, we deduce that |a4| ≤ 1 32 { p3 16 + (4 −p2)p 2 y + (4 −p2)p 2 y2 + (4 −p2)(1 −y2) } = 1 32 { p3 16 + (4 −p2)p 2 t + (4 −p2)(p− 2) 2 t2 + (4 −p2) } = g(p,t) (say),(2.26) 176 sahoo and patel where p ∈ [0, 2] and y ∈ [0, 1]. we next maximize the function g(p,y) on the closed rectangle [0, 2]×[0, 1]. suppose that the maximum of g occurs at the interior point of [0, 2] × [0, 1]. differentiating the function g with respect to y, we get ∂g ∂y = 1 128 (4 −p2){p + 4(p− 2)y}. for y ∈ (0, 1) and fixed p ∈ (0, 2), it is easily seen that ∂g ∂y > 0, which shows that g is a decreasing function of y contradicting our assumption. therefore, (2.27) max{g(p,y)}0≤y≤1 = g(p, 0) = 1 32 { p3 16 + (4 −p2) } = f(p) (say). from (2.27), we have f ′(p) = 1 32 { 3 16 p2 − 2p) } and f ′′(p) = 1 32 { 3 8 p− 2) } < 0 for p = 0. this implies that f attains its maximum at p = 0. hence, we get the required result. the estimate in (2.25) is sharp for the function f ∈ a , defined by f′(z) = √ 1 + z3 (z ∈u). � references [1] p.l. duren, univalent functions, grundlehren der mathematischen wissenschaften, 259, springer-verlag, new york, usa (1983). [2] m. fekete and g. szegö, eine bemerkung über ungerede schlichte funktionen, j. london math. soc., 8 (1933), 85-89. [3] a. janteng, s.a. halim and m. darus, coefficient inequality for a function whose derivative has a positive real part, j. inequal. pure appl. math., 7 (2006), article id 50. [4] a. janteng, s.a. halim and m. darus, estimate on the second hankel functional for functions whose derivative has a positive real part, j. quality measurement and analysis, 4 (2008), 189-195. [5] f.r. keogh and e.p. merkes, a coefficient inequality for certain classes of analytic functions, proc. amer. math. soc., 20 (1969), 8-12. [6] w. koepf, on the fekete-szegö problem for close-to-convex functions-ii, arch. math.(basel), 49 (1987), 420-433. [7] w. koepf, on the fekete-szegö problem for close-to-convex functions, proc. amer. math. soc., 101 (1987), 89-95. [8] r.j. libera and e.j. zlotkiewicz, early coefficient of the inverse of a regular convex function, proc. amer. math. soc. 85 (2) (1982), 225-230. [9] r.j. libera and e.j. zlotkiewicz, coefficient bounds for the inverse of a function with derivative in p, proc. amer. math. soc. 87 (2) (1983), 251-257. [10] w. c. ma and d. minda, a unified treatment of some special classes of univalent functions, proceedings of the conference on complex analysis (tianjin, 1992), z. li, f. ren, l. yang and s. zhang (eds.), int. press, cambridge, ma,(1994) 157-169. [11] t.h. macgregor, functions whose derivative have a positive real part. trans. amer. math. soc. 104(3) (1962), 532-537. [12] j.w. noonan and d.k. thomas, on the second hankel determinant of areally mean p-valent functions, trans. amer. math. soc., 223 (1976), 337-346. [13] k. i. noor, hankel determinant problem for the class of functions with bounded boundary rotation, rev. roum. math. pures et appl., 28 (1983), no. 8, 731 739. analytic functions related with lemniscate of bernoulli 177 1department of mathematics, veer surendra sai university of technology, sidhi vihar, burla-768 018, india 2department of mathematics, utkal university, vani vihar, bhubaneswar-751004, india ∗corresponding author int. j. anal. appl. (2022), 20:31 a brief note on qs/bp/boi-algebras aiyared iampan1,∗, pongpun julatha2 and nareupanat lekkoksung3 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2department of mathematics, faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand 3division of mathematics, faculty of engineering, rajamangala university of technology isan, khon kaen campus, khon kaen 40000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. in this paper, the three types of the logical algebra of interest are qs-algebras, boi-algebras, and bp-algebras, which are continuously studied by algebraists. since the terms for each of the above logical algebras appear to be independent of each other, it may be understood that they are different. therefore, the main purpose of this brief note is to show that qs-algebras, boi-algebras, and bpalgebras satisfying (bm) as one. 1. introduction and preliminaries several algebras with one binary and one nullary operations were proposed to develop an algebraic replica of implication reduction in classical or non-classical propositional logics. in 1966, imai and iséki [4,5] introduced bck-algebras and bci-algebras, which were important algebras and also inspired the creation of new types of algebra. it is known that the class of bck-algebras is a proper subclass of the class of bci-algebras. in 1999, ahn and kim [2] introduced the concept of qs-algebras (see definition 1.1) and thereafter in 2013 ahn and han [1] introduced the concept of bp-algebras (see definition 1.2). in 2019, el-gendy [3] introduced the concept of boi-algebras (see definition 1.3). the three types of the logical algebra of interest in this paper are qs-algebras, boi-algebras, and bp-algebras, which are continuously studied by algebraists. since the terms for each of the above received: may 16, 2022. 2010 mathematics subject classification. 06f35, 03g25, 03b05. key words and phrases. qs-algebra; boi-algebra; bp-algebra. https://doi.org/10.28924/2291-8639-20-2022-31 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-31 2 int. j. anal. appl. (2022), 20:31 logical algebras appear to be independent of each other, it may be understood that they are different. therefore, the main purpose of this brief note is to show that qs-algebras, boi-algebras, and bpalgebras satisfying (bm) as one. before we begin the study, let’s review the definitions of qs-algebras, bp-algebras, and boi-algebras according to definitions 1.1, 1.2, and 1.3, respectively. definition 1.1. [2] a qs-algebra is a non-empty set x with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈x)(x ∗x =0), (qs1) (∀x ∈x)(x ∗0= x), (qs2) (∀x,y,z ∈x)((x ∗y)∗z =(x ∗z)∗y), (qs3) (∀x,y,z ∈x)((z ∗x)∗ (z ∗y)= y ∗x). (bm) example 1.1. [2] let x = {0,1,2} with the following cayley table: ∗ 0 1 2 0 0 0 0 1 1 0 0 2 2 0 0 then x is a qs-algebra. ahn and kim proved the following proposition (see [2]). proposition 1.1. if x =(x;∗,0) is a qs-algebra, then (∀x,y ∈x)((x ∗ (x ∗y))∗y =0), (1.1) (∀x,y,z ∈x)(((x ∗z)∗ (y ∗z))∗ (x ∗y)=0), (1.2) (∀x ∈x)(0∗ (0∗ (0∗x))=0∗x). (1.3) definition 1.2. [1] a bp-algebra is a non-empty set x with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈x)(x ∗x =0), (qs1) (∀x,y ∈x)(x ∗ (x ∗y)= y), (bp1) (∀x,y,z ∈x)((x ∗z)∗ (y ∗z)= x ∗y). (bp2) int. j. anal. appl. (2022), 20:31 3 example 1.2. [1] let x = {0,1,2,3} with the following cayley table: ∗ 0 1 2 3 0 0 1 2 3 1 1 0 3 2 2 2 3 0 1 3 3 2 1 0 then x is a bp-algebra. ahn and han proved the following proposition (see [1]). proposition 1.2. if x =(x;∗,0) is a bp-algebra, then (∀x ∈x)(0∗ (0∗x)= x), (1.4) (∀x,y ∈x)(0∗ (x ∗y)= y ∗x), (1.5) (∀x ∈x)(x ∗0= x). (qs2) definition 1.3. [3] a boi-algebra is a non-empty set x with a constant 0 and a binary operation ∗ satisfying axioms: (∀x ∈x)(x ∗x =0), (qs1) (∀x,y ∈x)(x ∗ (x ∗y)= y), (bp1) (∀x,y,z ∈x)((x ∗y)∗z =(x ∗z)∗y). (qs3) example 1.3. [3] let x = {0,1,2} with the following cayley table: ∗ 0 1 2 0 0 2 1 1 1 0 2 2 2 1 0 then x is a boi-algebra. el-gendy proved the following proposition (see [3]). 4 int. j. anal. appl. (2022), 20:31 proposition 1.3. if x =(x;∗,0) is a boi-algebra, then (∀x ∈x)(x ∗0= x), (qs2) (∀x,y ∈x)(0∗ (x ∗y)= y ∗x), (1.6) (∀x,y,z ∈x)((z ∗x)∗ (z ∗y)= y ∗x), (bm) (∀x,y,z ∈x)((x ∗z)∗ (y ∗z)= x ∗y), (bp2) (∀x,y ∈x)((x ∗y)∗ (0∗y)= x), (1.7) (∀x,y ∈x)(x ∗ (x ∗ (x ∗y))= x ∗y), (1.8) (∀x,y ∈x)((x ∗y)∗x =0∗y), (1.9) (∀x,y,z ∈x)((x ∗y)∗z =(x ∗y)∗ (0∗z)). (1.10) 2. results in this section, we will link that qs-algebras and boi-algebras are the same, and boi-algebras and bp-algebras satisfying (bm) are the same. theorem 2.1. every qs-algebra is a boi-algebra. proof. it only needs to show (bp1). replacing x by 0 and z by x in (bm) and by (qs2), we obtain (bp1). � theorem 2.2. every boi-algebra is a qs-algebra. proof. it only needs to show (qs2) and (bm). it is immediately obtained by (qs2) and (bm). � from theorems 2.1 and 2.2, we get the following theorem. theorem 2.3. qs-algebras and boi-algebras are the same. theorem 2.4. every boi-algebra is a bp-algebra. proof. it only needs to show (bp2). it is immediately obtained by (bp2). � theorem 2.5. every bp-algebra satisfying (bm) is a boi-algebra. proof. it only needs to show (qs3). let x = (x;∗,0) be a bp-algebra satisfying (bm). let x,y,z ∈x. then (x ∗y)∗z =(x ∗y)∗ (x ∗ (x ∗z)) (by (bp1)) =(x ∗z)∗y, (by (bm)) so (qs3) is satisfied. hence, x is a boi-algebra. � from theorems 2.4 and 2.5, we get the following theorem. int. j. anal. appl. (2022), 20:31 5 theorem 2.6. boi-algebras and bp-algebras satisfying (bm) are the same. from theorems 2.3 and 2.6, we get the following corollary. corollary 2.1. qs-algebras, boi-algebras, and bp-algebras satisfying (bm) are the same. we can summarize the diagram as follows. acknowledgment the authors wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s.s. ahn, j.s. han, on bp-algebras, hacettepe j. math. stat. 42 (2013), 551–557. [2] s.s. ahn, h.s. kim, on qs-algebras, j. chungcheong math. soc. 12 (1999), 33–41. [3] o.r. el-gendy, on boi-algebras, int. j. math. comput. appl. res. 9 (2019), 13–28. [4] y. imai, k. iséki, on axiom systems of propositional calculi. xiv, proc. japan acad. 42 (1966), 19–22. https: //doi.org/10.3792/pja/1195522169. [5] k. iséki, an algebra related with a propositional calculus, proc. japan acad. 42 (1966), 26–29. https://doi. org/10.3792/pja/1195522171. https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522171 https://doi.org/10.3792/pja/1195522171 1. introduction and preliminaries 2. results acknowledgment references international journal of analysis and applications volume 18, number 3 (2020), 356-365 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-356 some results about a boundary value problem on mixed convection m. boulekbache1,∗, m. aïboudi1 and k. boudjema djeffal2 1département de mathématiques, faculté des sciences exactes et appliquées. université oran 1 ahmed ben bella, laboratoire de recherche d’analyse mathématique et applications"l.a.m.a", oran, algérie 2département de mathématiques, faculté des sciences exactes et informatique. université de hassiba benbouali, chlef, algérie ∗corresponding author: medbouke@gmail.com abstract. the purpose of this paper is to study the autonomous third order non linear differential equation f′′′ + ff′′ + g(f′) = 0 on [0, +∞[ with g(x) = βx(x − 1) and β > 1, subject to the boundary conditions f(0) = a ∈ r, f′(0) = b < 0 and f′(t) → λ ∈ {0, 1} as t → +∞. this problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as mixed convection in porous medium or flow adjacent to a stretching wall. our goal, here is to investigate by a direct approach this boundary value problem as completely as possible, say study existence or non-existence and uniqueness solutions and the sign of this solutions according to the value of the real parameter β. 1. introduction in fluid mechanics, the problems are usually governed by systems of partial differential equations. in modeling of boundary layer, this is sometimes possible, and in some cases, the system of partial differential received november 3rd, 2019; accepted november 27th, 2019; published may 1st, 2020. 2010 mathematics subject classification. 34b15, 34c11, 76d10. key words and phrases. third order non linear differential equation; boundary value problem; mixed convection; concave solution; convex solution; convex-concave solution. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 356 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-356 int. j. anal. appl. 18 (3) (2020) 357 equations reduces to a systems involving a third order differential equation of the form f′′′ + ff′′ + g(f′) = 0, (1.1) where the function g : r → r is assumed to be locally lipschitz. if g(x) = 0, the equation is the blasius equation (1907) see [6], [15]. the case g(x) = β(x2 − 1) was first given by falkner and skan (1931) see [13]. the case g(x) = βx2, this case occurs in the study of free convection (1966) see [3], [5], [7], [9], [12]. and for g(x) = βx(x − 1) is the mixed convection (2003) see [1], [2], [4], [8], [10], [11], [14], [16]. in this paper is to investigate this last case with β > 1. we consider the equation f′′′ + ff′′ + βf′(f′ −1) = 0 (1.2) and we associate to equation (1.2) the boundary value problem:  f′′′ + ff′′ + βf′(f′ −1) = 0 f(0) = a, a ∈ r f′(0) = b < 0 f′(t) −→ λ as t −→ +∞ (pβ;a,b,λ) where λ ∈ {0,1} and β > 1. this problem arises in the study of mixed convection boundary layer near a semi-infinite vertical plate embedded in a saturated porous medium, with a prescribed power law of the distance from the leading edge for the temperature. the parameter β is a temperature power-law profile and b is the mixed convection parameter, namely b = ra pe − 1, with ra the rayleigh number and pe the péclet number. the case where a ≥ 0, b ≥ 0, β > 0 and λ ∈ {0,1} was treated by aïboudi and al.see [1], and for a ∈ r, b ≤ 0, 0 < β < 1 see [2], and the results obtained generalize the ones of [11]. in [8], brighi and hoernel established some results about the existence and uniqueness of convex and concave solution of (pβ;a,b,1) where −2 < β < 0 and b > 0. these results can be recovered from [10], where the general equation f′′′ + ff′′ + g(f′) = 0 is studied. in [16], some theoretical results can be found about the problem (pβ;0,b,1) with −2 < β < 0, and b < 0. in [14] and [16], the method used by the authors allows them to prove the existence of a convex solution for the case a = 0 and seems difficult to generalize for a 6= 0. the problem (pβ;a,b,λ) with β = 0 is the well known blasius problem. in the following, we note by fc a solution of the problem to the initial values below and by [0,tc) the right maximal interval of its existence:  f′′′ + ff′′ + βf′(f′ −1) = 0 f(0) = a f′(0) = b f′′(0) = c (pβ;a,b,c) int. j. anal. appl. 18 (3) (2020) 358 to solve the boundary value problem (pβ;a,b,λ) we will use the shooting method, which consists of finding the values of a reel parameter c for which the solution of (1.2) satisfying the initial conditions. 2. on blasius equation in this section, we recall some results about subsolutions and supersolutions of the blasius equation. recall that the so-called blasius equation is the third order ordinary differential equation f′′′ + ff′′ = 0 i.e eq.(1.1) with g = 0. let us notice that, for any τ ∈ r, the function hτ : t 7→ 3t−τ is a solution of blasius equation on each (−∞,τ) and (τ,+∞). let i ⊂ r be an interval and f : i −→ r be a function. definition 2.1. we say that f is a subsolution (resp. a supersolution) of the blasius equation f′′′+ff′′ = 0 if f is of class c3 and if f′′′ + ff′′ ≤ 0 on i (resp.f′′′ + ff′′ ≥ 0 on i). proposition 2.1. let t0 ∈ r. there does not exist no positive concave subsolution of the blasius equation on the interval [t0,+∞). proof. see [10], proposition 2.11. � proposition 2.2. let t0 ∈ r. there does not exist no positive convex supersolution of the blasius equation on the interval [t0,+∞). proof. see [10], proposition 2.5. � 3. preliminary results proposition 3.1. let f be a solution of the equation (1.2) on some maximal interval i = (t−,t+) and β > 1. 1. if f is any anti-derivative of f on i, then (f′′ef )′ = −βf′(f′ −1)ef . 2. assume that t+ = +∞ and that f′(t) −→ λ ∈ r as t → +∞. if moreover f is of constant sign at infinity, then f′′(t) −→ 0 as t → +∞. 3. if t+ = +∞ and if f′(t) −→ λ ∈ r as t → +∞, then λ = 0 or λ = 1. 4. if t+ < +∞, then f′′ and f′ are unbounded near t+. 5. if there exists a point t0 ∈ i satisfying f′′(t0) = 0 and f′(t0) = µ, where µ = 0 or 1 then for all t ∈ i, we have f(t) = µ(t− t0) + f(t0). 6. if f′(t) → 0 as t → +∞, then f(t) does not tend to −∞ or +∞ as t → +∞. proof. the first item follows immediately from equation (1.2). for the proof of items 2-5, see [3], and item 6 see [1]. � int. j. anal. appl. 18 (3) (2020) 359 proposition 3.2. let us suppose that f be a solution of equation (1.2) on the maximal interval i = (t−,t+) (1) let h1 = f′′ + f(f′ −1) then h′1 = (1−β)f′(f′ −1), for all t ∈ i; (2) let h2 = 3f′′2 + βf′2(2f′ −3) then h′2 = −6ff′′2, for all t ∈ i ; (3) let h3 = 2ff′′ −f′2 + (2f′ −β)f2 then h′3 = 2(2−β)ff′2, for all t ∈ i; (4) let h4 = f′′ + ff′ then h′4 = (1−β)f′2 + βf′, for all t ∈ i; (5) let h5 = f′ + 12f 2 then h′5 = h4 = f ′′ + ff′, for all t ∈ i. proof. this statements follows immediately from equation (1.2). � 4. the boundary value problem (pβ;a,b,λ) let the boundary value problem (pβ;a,b,λ), we are interested here in a concave, convex and convex-concave solutions of a problem (pβ;a,b,λ) and there sign. we used shooting method to find these solutions, this method consists of finding the values of a parameter c ∈ r for which the solution of (pβ;a,b,c) satisfying the initial conditions f′(0) = a,f′(0) = b and f′′(0) = c, exists up to infinity and is such that f′(t) → λ as t → +∞. define the following sets: c0 = {c ≤ 0 : f′′c ≤ 0 on [0,tc )}, c1 = {c > 0 : f′c ≤ 0 and f ′′ c ≥ 0 on [0,tc)}, c2 = {c > 0 : ∃tc ∈ [0,tc),∃εc > 0 s.t f′c < 0 on (0, tc), f′c > 0 on (tc, tc + εc) and f ′′ c > 0 on (0, tc + εc)}, c3 = {c > 0 : ∃sc ∈ [0,tc),∃εc > 0 s.t f′′c > 0 on (0,sc), f′′c < 0 on (sc,sc + εc) and f ′ c < 0 on (0,sc + εc)}. int. j. anal. appl. 18 (3) (2020) 360 remark 4.1. it is easy to prove that c0, c1, c2 and c3 are disjoint nonempty open subsets of r, c0 = ]−∞,0] and c1 ∪c2 ∪c3 =]0,+∞[ (see appendix a of [10] with g(x) = βx(x−1) and β > 0). lemma 4.1. let β > 0. if c ∈ c0, then tc < +∞. moreover, fc is concave solution, decreasing and f′c(t)→−∞ as t→tc. proof. if c ∈ c0, the result follows from proposition 3.1 item 1, we have f′′c (t) < 0 and f′′c (t) < 0 for all t ∈ [0,+∞), then fc is a no positive concave subsolution of the blasius equation on [0,+∞) if a < 0, and on [t0,+∞) such that fc(t0) = 0 if a > 0, with f′c(t)→ −∞ as t→tc. if we assume that tc = +∞, this leads to a contradiction with proposition 3.1, then tc < +∞. � lemma 4.2. let β > 0. then fc is a convex solution of the boundary value problem (pβ,a,b,0) if and only if c ∈ c1. proof. see appendix a of [10] with g(x) = βx(x−1) and β > 0. � lemma 4.3. let β > 0. if c ∈ c3, then tc < +∞. moreover, fc is convex-concave, decreasing and f′c(t) →−∞ as t → tc. proof. see [2], lemma 5.3. � remark 4.2. from proposition 3.1 items 1,3 and 5, if c ∈ c2, then there are only three possibilities for the solution of the initial value problem (pβ;a,b,c): (1) fc is convex and f′c(t) → +∞ as t → tc (with tc ≤ +∞); (2) there exists a point t0 ∈ [0,tc) such that f′′c (t0) = 0 and f′c(t0) > 1; (3) fc is a convex solution of (pβ;a,b,1). the next proposition shows that the case (1) cannot hold. proposition 4.1. let β > 0. there does not exit c ≥ 0, such that fc is convex and f′c(t) → +∞ as t → tc on its right maximal interval of existence [0,tc). proof. assume that fc is convex on its right maximal interval of existence [0,tc) and f′c(t) → +∞ as t → tc. there exist t0 ∈ [0,t0), which the function h2 is decreasing for t > t0, this is a contradiction as t →tc. � proposition 4.2. let β > 1. if there exist t0 ∈ [0,tc) such that f′c(t0) = 0 and f′′c (t0) < 0, then for all t > t0, f′′c (t) < 0 and f ′ c(t) 6= 0. proof. let fc is convex on its right maximal interval of existence [0,tc), suppose there exist t1 > t0 such that f′c(t1) = 0, hence the function h4 is decreasing on [t0, t1], therefore h4(t0) > h4(t1), we have f ′′ c (t0) > f ′′ c (t1), which yields a contradiction. � int. j. anal. appl. 18 (3) (2020) 361 5. the a < 0 case proposition 5.1. let β > 1, the boundary value problem (pβ;a,b,1) has no convex solution. proof. let fc is convex on maximal interval of existence [0,tc), such that f′c(t) → 1 as t → tc, then there exist t0 ∈ [0,tc), such that f′c(t0) = 0, the function h1 is creasing for all t > t0, therefore h1(t) > h1(t0) for t > t0, we have f′′c (t) −f′′c (t0)>−fc(t) (f′c(t)−1) > 0, we obtain a contradiction for t large enough because f′′c (t) −→ 0 and fc(t) > 0. � proposition 5.2. the boundary value problem (pβ;a,b,0) has no negative convex-concave solution. proof. let fc is convex-concave on maximal interval of existence [0,tc), such that f′c(t) → 0 as t → tc, then there exist t0 ∈ [0,tc) such that f′c(t0) = 0, the function h2 is creasing for all t > t0, we have 3f′′2c (t0) < h2(t) for all t > t0, h2(t) → 0 as t → +∞, a contradiction. � remark 5.1. if the boundary value problem (pβ;a,b,0) has a convex-concave solution, then this solution changes the sign. lemma 5.1. if c ∈ c1, then there exist c∗ such that 0 < c < c∗, tc = +∞, and the solution fc is negative on [0,+∞). proof. let fc is solution on maximal interval of existence [0,tc), if c ∈ c1, then tc = +∞, the function h2 is creasing on [0,tc), it follows that 3c2 + βb2(2b− 3) < 0, we obtain c < −b √ β(3−2b) 3 , and the solution fc is negative because a < 0 and f′c < 0. � lemma 5.2. if c ∈ c3, then there exist c∗ such that 0 < c < c∗, tc < +∞ and the solution fc is negative on [0,tc). proof. if c ∈ c3, then f′c →−∞, and tc < +∞, other results same proof that lemma 5.1. � remark 5.2. it follows from lemma 5.1 and lemma 5.2, there exist c∗ > 0 such that c > c∗, c2 6= ∅ and here the solution fc is convex-concave. lemma 5.3. let 1 < β < 2, if c ∈ c2 and fc is a no positive solution on maximal interval of existence [0,tc), then for all t ∈ [0,tc) we have fc(t) ≤ max { a, b√ β } , tc < +∞ and f′c(t) →−∞ as t → tc. proof. let c ∈ c2 and fc is a no positive solution on maximal interval of existence [0,tc). from the proposition 3.1, 4.2 and 5.2, there exist t0 ∈ [0,tc) such that f′c(t0) = 0. moreover, the function h3 is decreasing on [0,tc), we have h3(0) > h3(t0), it follows that, −b2 > 2ac − b2 + (2b − β)a2 > 2fc(t0)f′′c (t0) − βf2c (t0) > −βf2c (t0), we get fc(t0) < b√ β for all t ∈ [0,tc), the conclusion follows from that, for all t ∈ [0,tc ) , if a < b√β we have fc(t) ≤ fc(t0) and, if a > b√ β we have fc(t) ≤ a with tc < +∞, and f′c →−∞ as t → tc. � int. j. anal. appl. 18 (3) (2020) 362 lemma 5.4. if c ∈ c2, and if there exist t1 ∈[0,tc) such that f′′c (t1) = 0 and fc(t1) < 0, then f′c(t1) > 3 2 . proof. if c ∈ c2, there exist t0 ∈ [0,tc) such that f′c(t0) = 0, fc(t0) < 0, and there exist t1 > t0 such that f′′c (t1) = 0, we suppose fc(t1) < 0 and f ′ c(t1) < 3 2 , the function h2 is creasing on [0, t1), we have 3f′′c (t0) < βf ′ c 2(t1)(2f ′ c(t1)−3), we obtain a contradiction. � remark 5.3. thanks to the previous lemma, if we have f′c(t1) < 3 2 and fc is convex-concave solution on maximal interval of existence [0,tc), then fc changes the sign. lemma 5.5. for 1 < β < 2 and b < −1, if c ∈ c2 and if there exist t0 ∈ [0,tc) such that fc(t0) = 0, then f′c(t0) > 1. proof. let fc is convex-concave solution on maximal interval of existence [0,tc), 1 < β < 2 and b < −1, if there exist t0 ∈ [0,tc) such that fc(t0) = 0, the function h3 is decreasing on [0, t0), we have h3(0) > h3(t0), therefore −b2 > −f′2c (t0), and we obtain f′c(t0) > −b > 1. � lemma 5.6. if c ∈ c2 and there exist t1 ∈ [0,tc) such that f′′c (t1) = 0 and if fc(t1) < 0, then f′c(t1) > −β 1−β . proof. if c ∈ c2, there exist t0 ∈ [0,tc) such that f′c(t0) = 0 and f′′c (t0) > 0, there exist t1 > t0 such that f′′c (t0) = 0, we suppose f ′ c(t1) < −β 1−β , the function h4 is creasing on [t0, t1], we have f ′′ c (t0) < fc(t1)f ′ c(t1), this is a contradiction. � theorem 5.1. let β > 1, a < 0 and b < 0. (1) the boundary value problem (pβ;a,b,0) has as least one negative convex solution on [0,+∞). (2) the boundary value problem (pβ;a,b,1) has no convex solution on [0,+∞). (3) the boundary value problem (pβ;a,b,+∞) has no convex solution on [0,tc). proof. the first result follows from remark 4.1 and lemma 4.2, the second result follows from proposition 5.1 and the third result follows from proposition 4.1. � 6. the a > 0 case let a,b ∈ r with b < 0 and a > 0. we assume β > 1, and fc be a solution of the initial value problem (pβ;a,b,c) on the right maximal interval of existence [0,tc), c > 0. before that, and in order to complete the study, let us divide the sets c2 and c3 into the following two int. j. anal. appl. 18 (3) (2020) 363 subsets: c2.1 = {c ∈ c2; f′c > 0 on [tc,tc )}, c2.2 = {c ∈ c2; ∃sc > tc s.t f′c > 0 on [tc,sc) and f′c(sc) = 0}, c3.1 = {c ∈ c3; fc(sc) < 0}, c3.2 = {c ∈ c3; fc(sc) > 0}. proposition 6.1. if c ∈ c1 ∪c2 ∪c3.1, then c > −ab proof. if c ∈ c1, tc = +∞, f′c(t) → 0 as t → +∞, the function h4 is decreasing on [0,+∞), we have c + ab > 0, if c ∈ c2 ∪ c3.1, there exist t0 ∈ [0,tc) such that f′c(t0) = 0 or fc(t0) = 0, we have c + ab ≥ f′′c (t0) > 0. � remark 6.1. if c ≤−ab then c ∈ c3.2 and tc < +∞. thus c3.2 6= ∅ and the convex part of the solution fc is positive. proposition 6.2. if c ∈ c1 ∪c2.1 and b > −12a 2, then tc = +∞ and the solution fc is positive. proof. let fc solution of the initial value problem (pβ;a,b,c) on the right maximal interval of existence [0,tc), c > 0, if c ∈ c1 ∪ c2.1, thanks to propositions 3.1 and 4.1 it follows that tc = +∞, no we suppose there exist t0 ∈ [0,tc) such that fc(t0) = 0, the function h4 is decreasing for all t > 0, we have h4(t0) = f′′c (t0), therefore h5 is creasing on [0, t0), we obtain b + 12a 2 < f′c(t0) < 0, this is a contradiction. � remark 6.2. if c ∈ c2.2 and b > −12a 2, the solution fc is positive on [0, t0), t0 is the point such that t0 > sc with fc(t0) = 0 and sc be as in definition of c2.2. lemma 6.1. let β > 1 and −1 2 a2 < b < 0. if fc be solution of the initial value problem (pβ;a,b,c), on the right maximal interval of existence [0,tc) and if there exist t0 ∈ [0,tc) such that fc(t0) = 0 and f′c(t0) < 0 then f′′c (t0) < 0. proof. for contradiction, let us that t0 ∈ [0,tc) with fc(t0) = 0 and f′c(t0) < 0, the function h4 is decreasing on [0, t0) and h4(t0)=f′′c (t0) > 0 then for all t∈ [0, t0), h4 > 0, and h5 is creasing on [0, t0), we have b + 1 2 a2 < f′c(t0) < 0, this is a contradiction. � proposition 6.3. let 1 < β < 2, b > −1 2 a2 and c ∈ c2.2. for all t ∈ [0,tc ) , one has fc(t) < √ b2+(β−2b)a2 β . proof. let c ∈ c2.2 and sc be as in the definition of c2.2, the function h3 is creasing on [0,sc), we have: −b2 + (2b−β)a2 < 2ac− b2 + (2b−β)a2 < 2fc(sc)f′′c (sc)−βf2c (sc) < −βf2c (sc), int. j. anal. appl. 18 (3) (2020) 364 which implies that fc(sc) < √ b2+(β−2b)a2 β . from the proposition 4.2, the conclusion follows from that , for all t ∈ [0,tc ) , we have fc(t) ≤ fc(sc). � lemma 6.2. if c ∈ c1 ∪c2.1 and b > −12a 2. then tc = +∞ and there exist c∗ > 0 such that c > c∗. proof. let c ∈ c1 ∪ c2.1, and b > −12a 2. by the definition of c1 and c2.1, thanks to proposition 6.2, we have tc = +∞ and f′c is bounded. otherwise the function h2 is decreasing for t > 0, we obtain 3c2 + βb2(2b−3) > 0, which implies that c > −b √ β(3−2b) 3 . � remark 6.3. there exist c∗ > 0, if c < c∗, then there exist t0 ∈ [0,tc) such that fc(t0) = 0, f′c(t0) < 0 and f′′c (t0) < 0 say that c ∈ c2.2 ∪c3.2, since if c ∈ c2.1 then tc = +∞. let us divide the set c2.1 into the following two subsets: c2.1.1 = {c ∈ c2.1; f′c(t) → 0 as t → +∞}, c2.1.2 = {c ∈ c2.1; f′c(t) → 1 as t → +∞}. proposition 6.4. let 1 < β < 2, if c ∈ c1 ∪c3 ∪c2.2 ∪c2.1.1. then there exist c∗ > 0 such that c < c∗. proof. let fc solution of the initial value problem (pβ;a,b,c) on the right maximal interval of existence [0,tc), either there exist t0 ∈ [0,tc) such that fc(t0) = 0 or f′c(t0) = 0 if tc < +∞, and if tc = +∞, we have f′c(t) → 0 as t → +∞, from proposition 3.1 item 6, it follows that the function h3 is creasing on [0, t0) or [0,+∞), we get 2ac− b2 + (2b−β)a2 < 0, which implies that c < b 2+(β−2b)a2 2a . � remark 6.4. from proposition 6.4 there exist c∗ > 0, such that for c ≥ c∗, then c ∈ c2.1.2. thus c2.1.2 6= ∅. corollary 6.1. if 1 < β < 2, a > 0, b < 0 and b > −1 2 a2, then the problem (pβ;a,b,1) has as least one positive convex or positive convex-concave solution on [0,+∞). proof. this follows immediately from remark 6.4, lemma 6.2 and proposition 6.3. � theorem 6.1. let β > 1, a > 0 and b < 0. (1) the boundary value problem (pβ;a,b,0) has as least one convex solution on [0,+∞) if in addition we have b > −1 2 a2 it will be no negative convex solution. (2) the boundary value problem (pβ;a,b,−∞) has infinity convex-concave solutions on the maximal interval of existence [0,tc) with tc < +∞, if in addition we have b > −12a 2 the convex part of these solutions will be no negative. (3) the boundary value problem (pβ;a,b,+∞) has no convex solution on [0,tc). int. j. anal. appl. 18 (3) (2020) 365 proof. the first result follows from remark 4.1, lemma 4.2 and proposition 6.2 , the second result follows from proposition 3.1, proposition 4.2 and remark 6.1, and the third result follows from proposition 4.1. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. aiboudi, i. bensari-khellil, b. brighi, similarity solutions of mixed convection boundary-layer flows in a porous medium. differ. equ. appl. 9(1)(2017), 69-85. [2] m. aiboudi, k. boudjema djeffal, b. brighi, on the convex and convex-concave solutions of opposing mixed convection boundary layer flow in a porous medium, abstr. appl. anal. 2018 (2018), id 4340204. [3] m. aiboudi, b. brighi, on the solutions of a boundary value problem arising in free convection with prescribed heat flux, arch. math. 93 (2009), 165-174. [4] e.h. aly, l. elliott, d.b. ingham, mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. eur. j. mech. b, fluids, 22 (2003), 529-543. [5] z. belhachemi, b. brighi, k. taous, on a family of differential equations for boundary layer approximations in porous media, eur. j. appl. math. 12(4) (2001), 513-528. [6] b. brighi, a. fruchard, t. sari, on the blasius problem, adv. differ. equ. 13 (5-6)(2008), 509-600. [7] b.brighi, j.-d. hoernel, on general similarity boundary layer equation, acta math. univ. comenian. 77 (2008), 9-22. [8] b. brighi, j.-d. hoernel, on the concave and convex solutions of mixed convection boundary layer approximation in a porous medium. appl. math. lett. 19 (1) (2006), 69-74. [9] b.brighi, sur un problème aux limites associé à l’équation différentielle f′′′ + ff′′ + 2f′′ = 0, ann. sci. math. québec, 33 (1) (2012), 355-391. [10] b. brighi, the equation f′′′ + ff′′ + g(f′) = 0 and the associated boundary value problems, results math. 61 (3-4) (2012), 355-391. [11] m. guedda, multiple solutions of mixed convection boundary-layer approximations in a porous medium. appl. math. lett. 19 (1) (2006), 63-68. [12] m. guedda, nonuniqueness of solutions to differential equations for boundary layer approximations in porous media, c. r. mecanique, 330 (2002), 279-283. [13] s. p. hastings and w. c. troy, oscillating solutions of the falkner-skan equations for positive β, j. differ. equ. 71(1) (1988), 123-144. [14] g.c. yang, an extension result of the opposing mixed convection problem arising in boundary layer theory. appl. math. lett. 38 (1) (2014), 180-185. [15] g.c.yang, an upper bound on the critical value β∗ involved in the blasius problem, j. inequal. appl. 2010 (2010), id 960365. [16] g.c. yang, l. zhang, l.f.dang, existence and nonexistence of solutions on opposing mixed convection problems in boundary layer theory. eur. j. mech. b, fluids, 43 (2014), 148-153. 1. introduction 2. on blasius equation 3. preliminary results 4. the boundary value problem (p;a,b,) 5. the a<0 case 6. the a>0 case references international journal of analysis and applications volume 17, number 1 (2019), 26-32 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-26 on nonclassical impulsive ordinary differential equations with nonlocal conditions s. a. bishop∗, m. c. agarana and j. g. oghonyon department of mathematics, covenant university, ota, ogun state, nigeria ∗corresponding author: sheila.bishop@covenantuniversity.edu.ng abstract. results on mild solutions of nonclassical differential equations with impulsive and nonlocal conditions are extended to a case when the nonlocal conditions are necessarily non lipschitz and non compact. 1. introduction we study the following quantum stochastic differential equation (qsde) with impulsive nonlocal conditions introduced in [1]; dz(t) = a(t)z(t) + e(t, (z(t))d∧π (t) + f(t,z(t))daf (t) +g(t,z(t))da+g (t) + h(t,z(t))dt, almost all t ∈ i,t 6= tk,k = 1, ...,m (1) ∆z(tk) = jk(z(t − k )),k = 1, ...,m z(t0) = z0 + g(z), t ∈ [0,t] where (i) a is a family of semigroup defined in [1] received 2018-07-04; accepted 2018-09-12; published 2019-01-04. 2010 mathematics subject classification. 35a24. key words and phrases. nonclassical ordinary differential equations (nodes); non-compact; nonlocal conditions; impulse effect; non lipschitz; stochastic processes. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 26 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-26 int. j. anal. appl. 17 (1) (2019) 27 (ii) e, f, g, h are stochastic processes. (iii) jk ∈ c(b̃, b̃),k = 1, 2, ...,m and ∆z(tk) is the difference between z(t+k ) and z(t−k ). (v) g : b̃ → pc(i,sesq(id⊗ie)) is a nonlocal condition that is not lipschitz and not compact. (vi) z ∈ b̃ is a stochastic process and η,ξ ∈ id⊗ie is arbitrary. problems with nonlocal conditions have been an area of interest, mostly because of the advantage they have over initial value problems. existence of solution of nonlocal problems for different types of differential equations were extensively discussed in the literature by using various methods (see [1, 2, 3-14] and the references therein). the motivation for this study, is that nonlocal problems occur naturally when modeling physical problems. in [2], impulsive quantum stochastic differential equations (iqsde) with initial value conditions were studied. the multivalued maps are lower respectively upper semicontinuous. in [1], existence results for eq.(1) with nonlocal conditions that are completely continuous were established. we showed that the function g which constitute the nonlocal condition is compact and lipschitz continuous. several interesting results on nonlocal impulsive differential equations satisfying some lipschitz and compactness conditions have been established in [6-9]. in this study, existence of solution of eq. (1) is established with nonlocal conditions that are not necessarily lipschitz and compact. we adopt the most suitable fixed point method to establish this result. impulsive qsdes have found applications in quantum continuous measurements, especially when the mean number of photons up to time ti is momentary giving rise to impulses on the counting stochastic processes associated with the observables x(ti). see [1, 2] and the references therein. 2. preliminaries the definitions of the following spaces l2loc(b̃)mvs, b̃, pc(i, b̃), pc ′(i, b̃), pc(i,sesq(id⊗ie)) and pc′(i,sesq(id⊗ie)) are adopted from [1, 2]. the spaces b̃ and pc(i,sesq(id⊗ie)) denote the locally convex and banach spaces respectively. the hausdorff distance, ρ(a,b) is defined as: ρ(a,b) = max(δ(a,b),δ(b,a)),a,b ∈ clos(c) and d(z,b) = infy∈b|z −y|, δ(a,b) = supz∈ad(z,b) where x ∈ c is as defined in [1] and ρ is a metric. definition 1. a stochastic process z ∈ pc(i,ã) is called a solution of eq. (1) if it satisfies the integral int. j. anal. appl. 17 (1) (2019) 28 equation z(t) = s(t)[z0 + g(z)] + ∫ t 0 s(t−s)(e(s, (z(s))dλπ(s) + f(s,z(s))daf (s) +g(s,z(s))da+g (s) + h(s,z(s))ds) + ∑ 0<tk<t s(t− tk)jk(z(tk)) (2) for t ∈ [0,t]. to establish a mild solution with non lipschitz, non compact and nonlocal condition, we must assume that z(t) near zero does not affect the nonlocal condition g(z). we use the equivalent form of eq. (1) given by, d dt 〈η,z(t)ξ〉 = p(t,z(t))(η,ξ). (3) remark. equation (3) is also known as nonclassical ordinary differential equation. see [1] and the references therein. next, we state the following assumptions: (h1) the map p in eq. (3) is continuous and well defined in [1]. hence, there exists a function k p ηξ : [0,t] → ir+ so that ‖p(t,y) −p(t,z)‖ηξ ≤ k p ηξ(t)‖y −z‖ηξ , t ∈ [0,t], y,z ∈ b̃. (h2) jk ∈ c(b̃, b̃) and t(.) are compact operators (h3) for each z0 ∈ b̃, we have constants hηξ > 0 and m > 0 so that ‖s(t)‖ηξ ≤ m,t ≥ 0 and m ( ‖z0‖ηξ + sup ϕ∈hh ‖g(ϕ)‖ηξ + k p ηξ(t) sup s,t∈[0,t] ‖p(s,ϕ(s))‖ηξ + sup ϕ∈zh m∑ k=1 ‖jk(ϕ(tk))‖ηξ ) ≤ hηξ where hh := { ϕ ∈ pc([0,t],ã) : ‖(ϕ(t))‖ηξ ≤ hηξ, t ∈ [0,t] } (h4) g : pc([0,t],ã) →ã is continuous and constitute the nonlocal condition. also g : hh → bd let δ depend on hηξ ∈ (0, t1) and g(ϕ) = g(φ), ϕ,φ ∈ hh where ϕ(s) = φ(s),s ∈ [δ,t] and bd denote a bounded set. int. j. anal. appl. 17 (1) (2019) 29 3. main result theorem 1. let conditions (h1)-(h4) hold. then for z0 ∈ b̃, problem (1) has at least a solution. proof. let δ ∈ (0, t1), define h(δ) and hh(δ) as ; h(δ) := pc([δ,t], b̃) for functions in pc([0,t], b̃) on [δ,t] and hh(δ) := {ϕ ∈ h(δ) : ‖ϕ(t)‖ηξ ≤ hηξ, t ∈ [δ,t]} . let z ∈ hh(δ) be fixed. then define a map γz on hh by γz(ϕ)(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉 + ∫ t 0 s(t−s)p(s, (ϕ(s))(η,ξ)ds + ∑ 0<tk<t s(t− tk)jk(u(tk)), t ∈ [0,t], z̃ ∈ hh(δ) where z̃(t) = z(t), if t ∈ [δ,t] and z̃(t) = z(δ), if t ∈ [0,δ] this shows that γz is a continuous mapping from hh to hh. also, we have ‖γnz (ϕ)(t) − γ n z (φ)(t)‖ηξ = ∣∣∣∣ ∫ t 0 s(t−s)(p(s,ϕ(s))(η,ξ) −p(s,φ(s))(η,ξ)ds) ∣∣∣∣ ≤ m ∫ t 0 ‖p(s,ϕ(s)) −p(s,φ(s))‖ηξds ≤ m ∫ t 0 kpηξ(s)‖ϕ(s) −φ(s)‖ηξds (3) since the map s → sups∈[0,t] ‖ϕ(s) −φ(s)‖ηξ is continuous, we let rηξ = sup s∈[0,t] ‖ϕ(s) −φ(s)‖ηξ and nηξ(t) = ∫ t 0 kpηξ(s)ds. then from (3), we get ‖γnz (ϕ)(t) − γ n z (φ)(t)‖ηξ ≤ rηξ(mnηξ(t)) n n! ,n = 1, 2, .... (4) int. j. anal. appl. 17 (1) (2019) 30 this can also be proved by induction, see [7] in [1]. so we get ‖γmz (ϕ)(t) − γ m z (φ)(t)‖ηξ = ∥∥∥∥∥ n∑ m=k+1 (γmz (ϕ)(t) − γ m z (φ)(t)) ∥∥∥∥∥ ηξ ≤ n∑ m=k+1 ‖(γmz (ϕ)(t) − γ m z (φ)(t))‖ηξ ≤ n∑ m=k+1 rηξ(mnηξ(t)) m m! where t ∈ [0,t], ϕ,φ ∈ hh,m = 1, 2, ... now by considering large m, we see that γmz is a contraction operator on hh and hence γz has a unique fixed point given by ϕz(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉 + ∫ t 0 s(t−s)p(s,ϕz(s))(η,ξ)ds + ∑ 0<tk<t s(t− tk)jk(z(tk)) ∈ hh, t ∈ [0,t] from the above, define a map υ : hh(δ) → hh(δ) by υ(z)(t)(η,ξ) = ϕz(t)(η,ξ) = 〈η, [z0 + g(z̃)]ξ〉 + ∫ t 0 s(t−s)p(s,ϕz(s))(η,ξ)ds ≤ ∑ 0<tk<t s(t− tk)jk(z(tk)), t ∈ [δ,t] (5) the next following steps show that the map υ has a fixed point: by applying lebesgue dominated convergence theorem and arzela-ascoli theorem we show that; (1) the operator υ is continuous; (2) it maps bd sets into bd sets in pc(i,sesq(id⊗ie)); (3) υ maps bd sets into equicontinuous sets in pc(i,sesq(id⊗ie)) and by applying the schauder-tychonov’s fixed point theorem we get a z∗ ∈ hh(δ), which is a fixed point. by replacing υ(z) with z, g(z̃) with g(z̃∗) in (5), we get z(t)(η,ξ) = 〈η, [z0 + g(z̃∗)]ξ〉 + ∫ t 0 s(t−s)p(s,z(s))(η,ξ)ds ≤ ∑ 0<tk<t s(t− tk)jk(z∗(tk)), t ∈ [0,t] (6) int. j. anal. appl. 17 (1) (2019) 31 but g(z̃∗) = g(z̃), z̃∗(tk) = z̃(tk) and by the definition of the map υ, z̃∗(t) = υ(z̃∗)(t) = ϕz̃∗ (t) = z̃(t). so that by (6), we conclude that z(t) is the required solution of eq. (1) which is the desired result. conclusion using the equivalent form of qsde (1) given by eq. (3) and having satisfied the conditions of the appropriate fixed point theorem, we conclude that a fixed point exists and it is a solution of the problem. conflict of interest the authors declare that there is no conflict of interests. references [1] s. a. bishop, e. o. ayoola, j. g. oghonyon, existence of mild solution of impulsive quantum stochastic differential equation with nonlocal conditions. anal. math. phys. 7 (3) (2017), 255-265. [2] s. a. bishop and p. e. oguntunde, existence of solutions of impulsive quantum stochastic differential inclusion, j. eng. appl. sci. 10 (7) (2015), 181-185. [3] t. caraballo, m. a. hammami, l. mchiri, practical exponential stability of impulsive stochastic functional differential equations. syst. control lett. 109 (2017), 43-48. [4] t. cardinali, p. rubbioni, nonlinear anal., theory methods appl. 75 (2) (2012), 871-879. [5] m. a. dialloa, k. ezzinbib, a. sene, impulsive integro-differential equations with nonlocal conditions in banach spaces. trans. a. razmadze math. inst. 171 (2017), 304-315. [6] q.dong, g. li, existence of solutions for semilinear differential equations with nonlocal conditions in banach spaces. electron j. qual. theory diff. equ. 47 (2009), art. id 47. [7] z.b. fan, g. li, existence results for semilinear differential equations with nonlocal and impulsive conditions. j. funct. anal. 258 (2010), 1709-1727. [8] g. li, existence results for semilinear differential equations with nonlocal conditions and impulsive conditions. j. funct. anal. 258(2010), 1709-1727. [9] l. zhu , q. dong and g. li, impulsive differential equations with nonlocal conditions in general banach spaces. advances in diff. equ. 2012 (2012), art. id 10. [10] a. g. okeke, s a. bishop,s. h. khan iterative approximation of fixed point of multivalued ρ-quasi-nonexpansive mappings in modular function spaces with applications, j. funct. spaces 2018 (2018), art. id 1785702. [11] o. bolojan-nica, g. infante, r. precupa, existence results for systems with coupled nonlocal initial conditions. nonlinear anal., theory methods appl. 94 (2014), 231-242 [12] s. ji and g. li, a unified approach to nonlocal impulsive differential equations with the measure of noncompactness. adv. diff. equ. 2012 (2012), art. id 182. int. j. anal. appl. 17 (1) (2019) 32 [13] x.m. xue, semilinear nonlocal problems without the assumptions of compactness in banach spaces. anal. appl. 8 (2010), 211-225. [14] x.m, xue, nonlocal nonlinear differential equations with a measure of noncompactness in banach spaces.nonlinear anal., theory methods appl. 70 (2009), 2593-2601. 1. introduction 2. preliminaries 3. main result references international journal of analysis and applications volume 17, number 4 (2019), 674-685 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-674 application of srivastava-attiya operator to the generalization of mocanu functions khalida inayat noor and shujaat ali shah∗ comsats university islamabad, pakistan ∗corresponding author: shahglike@yahoo.com abstract. in this paper we introduce certain subclasses of analytic functions by applying srivastava-attiya operator. our main purpose is to derive inclusion results by using concept of conic domain and subordination techniques. we also deduce some new as well as well-known results from our investigations. 1. introduction let χ denotes the class of analytic functions f(z) in the open unit disk f = {z : |z| < 1} such that f(z) = z + ∞∑ n=2 anz n. (1.1) subordination of two functions f and g is denoted by f ≺ g and defined as f(z) = g(w(z)), where w(z) is schwarz function in f. let s, s∗ and c denotes the subclasses of χ of univalent functions, starlike functions and convex functions respectively. for 0 ≤ δ < 1, s∗(δ) and c(δ) are the subclasses of s of functions f satisfies; zf′(z) f(z) ≺ 1 + (1 − 2δ) z 1 −z , (z ∈ f) , (1.2) received 2019-04-24; accepted 2019-05-27; published 2019-07-01. 2010 mathematics subject classification. 30c45, 30c55. key words and phrases. srivastava-attiya operator; mocanu functions; conic domains. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 674 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-674 int. j. anal. appl. 17 (4) (2019) 675 (zf′(z)) ′ f′(z) ≺ 1 + (1 − 2δ) z 1 −z , (z ∈ f) , (1.3) respectively. mocanu [13] introduced the class mα of α−convex functions f ∈ s satisfies; ( (1 −α) zf′(z) f(z) + α (zf′(z)) ′ f′(z) ) ≺ 1 + z 1 −z , (1.4) where α ∈ [0, 1], f(z) z f′(z) 6= 0. and z ∈ f. we see that m0 = s∗ and m1 = c. this class is vastly studied by several authors. see [4, 15, 17–19]. for k ∈ [0,∞) , kanas and wisniowska [8, 9] introduced the classes k − ucv of k-uniformly convex functions and k − st of k-starlike functions. the analytic conditions for these classes are given [6–9] as; k −ucv = { f ∈ s : re ( 1 + zf′′(z) f′(z) ) > k ∣∣∣∣zf′′(z)f′(z) ∣∣∣∣ } , (z ∈ f) . (1.5) k −st = { f ∈ s : re ( zf′(z) f(z) ) > k ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ } , (z ∈ f) . (1.6) we can rewrite the above relations easily as; re (p(z)) > k |p(z) − 1| , (1.7) where p(z) = 1 + zf′′(z) f′(z) or p(z) = zf′(z) f(z) . it is clear that p(f) is conic domain defined as; ωk = {w ∈ c : re (w) > k |w − 1|} , (1.8) or ωk = { u + iv : u > k √ (u− 1)2 + v2 } , (0 ≤ k < ∞) . (1.9) these conic domains are being studied by several authors. see [2, 6, 14, 16]. sokol and nonukawa [23] introduced the class defined as; mn = { f ∈ s:re ( 1 + zf′′(z) f′(z) ) > ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ } , (z ∈ f) . (1.10) it is obvius that mn ⊂ c. recently s. sivasubramanian et al. [22] extend the sokol and nonukawa’s work in terms of conic domains. they introduced a new class k −mn of functions f ∈ s such that re ( 1 + zf′′(z) f′(z) ) > k ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ , (z ∈ f) . (1.11) in motivation of the work [23], a. rasheed et al. [21], introduced an interesting class k −umα (0 ≤ α ≤ 1) of functions f ∈ s such that int. j. anal. appl. 17 (4) (2019) 676 re [ (1 −α) zf′(z) f(z) + α (zf′(z)) ′ f′(z) ] > k ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ , (z ∈ f) . (1.12) obviously, we can see k −um1 = k −mn and 1 −um1 = mn. we recall a hurwitz-lerch zeta function φ(s,b; z) [25] defined by φ(s,a; z) = ∞∑ n=2 zn (n + b) s , (1.13) ( b � c\z−0 ; s � c when |z| < 1; re(s) > 1 when |z| = 1), where c and z−0 denotes the set of complex numbers and the set of negative integers respectively. srivastava and attiya [24] introduced the linear operator js,b : χ → χ defined in terms of the convolution ( or hadamard product ), by js,bf(z) = gs,b(z) ∗f(z), (1.14) where gs,b(z) = (1 + b) s [φ(s,b; z) − bs] , (1.15) with z � f, b � c\z−0 and s � c. therefore, using (1.13) to (1.15), we have js,bf(z) = z + ∞∑ n=2 ( 1 + b n + b )s anz n, (1.16) where z � f, b � c\z−0 and s � c. the srivastava-attiya operator generalizes the integral operators introduced by alexandar [1], libera [10], bernardi [3] and jung et al. [5]. in 2007, raducanu and srivastava [20] introduced and studied the class s∗s,b(δ) of functions f � χ satisfies js,bf(z) � s ∗(δ). now by using concepts of conic domains and srivastava-attiya integral operator, we introduce new classes as following. definition 1.1. let k ∈ [0,∞) and α,β ∈ [0, 1]. then f ∈ k −um(α,β) if and only if re [ (1 −α) zf′(z) f(z) + α (zf′(z)) ′ f′(z) ] > k ∣∣∣∣(1 −β)zf′(z)f(z) + β (zf ′(z)) ′ f′(z) − 1 ∣∣∣∣ , (z ∈ f) . some of the special cases are given below and we refer to [8, 9, 21–23]. special cases: (i) for β = 0, the class k −um(α,β) reduces to the class k −umα. see [21]. (ii) for α = 1 and β = 0, the class k −um(α,β) reduces to the class k −mn. see [22]. (iii) for α = 1, β = 0 and k = 1, the class k −um(α,β) reduces to the class mn. see [23]. (iv) for α = 1 and β = 1 , the class k −um(α,β) reduces to the class k −ucv . see [9]. int. j. anal. appl. 17 (4) (2019) 677 (v) for α = 0 and β = 0 , the class k −um(α,β) reduces to the class k −st. see [8]. definition 1.2. let α,β � [0, 1], k � [0,∞), b � c\z−0 and s � c. then f � k −umsb (α,β) if and only if js,bf(z) � k −um(α,β). clearly, for s = 0 the classes k −umsb (α,β) and k −um(α,β) coincides. 2. preliminaries lemma 2.1. [12] let ~ be an analytic function on f except for at most one pole on ∂f and univalent on f, ℘ be an analytic function in f with ℘(0) = ~(0) and ℘(z) 6= ℘(0), z ∈ f. if ℘ is not subordinate to ~, then there exist points z0 ∈ f, ξ0 ∈ ∂f and ε ≥ 1 for which ℘ (|z| < |z0|) ⊂ ~ (f) , ℘(z0) = ~(ξ0), z0℘′(z0) = εξ0℘′(ξ0). lemma 2.2. [6] if f ∈ s∗(α) for some α ∈ [ 1 2 , 0 ) , then re ( f(z) z ) > 1 3 − 2α . lemma 2.3. [6] if re (√ f′(z) ) > α for some α ∈ [ 1 2 , 0 ] , then re ( f(z) z ) > 2α2 + 1 3 . 3. main results theorem 3.1. let k ∈ [0,∞) and α,β ∈ [0, 1]. also, let p be a function analytic in the unit disk such that p(0) = 1. if re [ p(z) + α zp′(z) p(z) ] −k ∣∣∣∣p(z) − 1 + βzp′(z)p(z) ∣∣∣∣ > 0, then p(z) ≺ 1 + (1 − 2γ)z 1 −z := h(z), where γ = γ(k,α,β) is given by γ(k,α,β) = 1 4 [√ (α− 2k + βk)2 (1 + k) 2 + 8 (α + βk) (1 + k) − (α− 2k + βk) (1 + k) ] . (3.1) proof. we may assume that γ ≥ 1 2 since the condition re ( p(z) + zp′(z) p(z) ) > 0 implies at least re (p(z)) > 1 2 . (see [11]). suppose now, on the contrary that p ⊀ h. then, by lemma 2.1, there exist z0 ∈ f, ξ0 ∈ ∂f and m ≥ 1 such that int. j. anal. appl. 17 (4) (2019) 678 p(z0) = γ + ix, z0p ′(z0) = my, where y ≤− (1 −γ)2 + x2 2 (1 −γ) , (x,y ∈ r) . using these relations, we have re [ p(z0) + α z0p ′(z0) p(z0) ] −k ∣∣∣∣p(z0) − 1 + βz0p′(z0)p(z0) ∣∣∣∣ > 0, or 0 < re [ p(z0) + α z0p ′(z0) p(z0) ] −k ∣∣∣∣p(z0) − 1 + βz0p′(z0)p(z0) ∣∣∣∣ = re [ γ + ix + α my γ + ix ] −k ∣∣∣∣γ + ix− 1 + β myγ + ix ∣∣∣∣ = γ + αmyγ γ2 + x2 −k ∣∣∣∣∣(γ + ix) 2 − (γ + ix) + βmy γ + ix ∣∣∣∣∣ ≤ γ − αγ 2(1 −γ) ( (1 −γ)2 + x2 γ2 + x2 ) −k √ (x + y x2) 2 + tx2 γ2 + x2 = r(x), where x = (2γ+β)(1−γ) 2 , y = 2(1−γ)+β 2(1−γ) and t = (2γ − 1) 2 . the function r(x) is even in regard of x. now we have to show that r(x) has maximum value at x = 0 when α,β ∈ [0, 1] and γ ∈ [ 1 2 , 1 ) . we can easily check r′(x) = −x   αγ (2γ − 1) (1 −γ) (γ2 + x2) −k  2y (x + y x2) + t − 2 √ (x + y x2) 2 + tx2 γ2 + x2     . then r′(x) = 0, if and only if, x = 0. since α,β ∈ [0, 1] and γ ∈ [ 1 2 , 1 ) . so one can see r′′(x) = − [ α (2γ − 1) γ(1 −γ) − k 2 { (2(1 −γ) + β) (2γ + β) + 2 (2γ − 1)2 }] < 0. thus r(x) has maximum value at x = 0, that is r(x) ≤ r(0) = γ − αγ(1 −γ) 2γ − k(1 −γ) (2γ + β) 2γ = 0, for γ = γ(k,α,β) is given by (3.1), which contradicts the assumption. hence p(z) ≺ 1 + (1 − 2γ)z 1 −z := h(z), where γ = γ(k,α,β) is given by (3.1). � theorem 3.2. let α,β � [0, 1], k � [0,∞), b � c\z−0 and s � c. then k −umsb (α,β) ⊂ s ∗ s,b(γ), where γ = γ(k,α,β) is given by (3.1). int. j. anal. appl. 17 (4) (2019) 679 proof. let f ∈ k −umsb (α,β). then, by definition 1.2, js,bf(z) � k −um(α,β), that is re [ (1 −α) z (js,bf(z)) ′ js,bf(z) + α ( z (js,bf(z)) ′)′ (js,bf(z)) ′ ] > k ∣∣∣∣∣(1 −β)z (js,bf(z)) ′ js,bf(z) + β ( z (js,bf(z)) ′)′ (js,bf(z)) ′ − 1 ∣∣∣∣∣ , (z ∈ f) . putting p(z) = z(js,bf(z)) ′ js,bf(z) , we have re [ p(z) + α zp′(z) p(z) ] −k ∣∣∣∣p(z) − 1 + βzp′(z)p(z) ∣∣∣∣ > 0. our required result follows easily by using theorem 3.1. � when s = 0, then we have the following new result for class k −um(α,β) theorem 3.3. let k ∈ [0,∞) and α,β ∈ [0, 1]. then k −um(α,β) ⊂ s∗(γ), where γ = γ(k,α,β) is given by (3.1). the proof is straight forward by putting p(z) = zf′(z) f(z) and using theorem 3.1. when k = 0, then we have the following result for a class 0−um(α,β) = mα, introduced by mocanu [13]. corollary 3.1. let f ∈ mα. then f ∈ s∗(γ), where γ (α) = −α + √ α2 + 8α 4 . (3.2) when β = 0, then we have the following result, proved in [21]. corollary 3.2. let f ∈ k −um(α, 0) = k −umα. then f ∈ s∗(γ), where γ (α,k) = (2ϑ−η) + √ (2ϑ−η)2 + 8η 4 , (3.3) where ϑ = k k+1 , η = α+k k+1 . when α = 1, β = 0, then we have the following result, proved in [22]. corollary 3.3. let f ∈ k −um(1, 0) = k −mn. then f ∈ s∗(γ), where γ(k) = 1 4   √( 1 − 2k 1 + k )2 + 8 (1 + k) − ( 1 − 2k 1 + k ) . (3.4) when α = 1, β = 0, and k = 1, then we have the following result, proved in [22]. corollary 3.4. let f ∈ 1 −um(1, 0) = mn. then f ∈ s∗(γ), where γ ' 0.6403. when α = β = 1, then we have the following result, proved in [6]. int. j. anal. appl. 17 (4) (2019) 680 corollary 3.5. let f ∈ k −um(1, 1) = k −ucv . then f ∈ s∗(γ), where γ(k) = 1 4   √( 1 −k 1 + k )2 + 8 − ( 1 −k 1 + k ) . (3.5) theorem 3.4. let f ∈ k −umsb (α,β). then js,bf(z) z ≺ 1 + (1 − 2η)z 1 −z , where η = 1 3−2γ and γ = γ(k,α,β) is given by (3.1). proof. let f ∈ k −umsb (α,β). then by theorem 3.2 we have z (js,bf(z)) ′ js,bf(z) ≺ 1 + (1 − 2γ)z 1 −z , where γ = γ(k,α,β) is given by (3.1). using lemma 2.2, we get js,bf(z) z ≺ 1 + (1 − 2η)z 1 −z , where η = 1 3−2γ . � when s = 0, then one can prove the following result by using theorem 3.3 together with lemma 2.2. theorem 3.5. let f ∈ k −um(α,β). then f(z) z ≺ 1 + (1 − 2η)z 1 −z , where η = 1 3−2γ and γ = γ(k,α,β) is given by (3.1). when α = 1, β = 0, then we have the following result, proved in [22]. corollary 3.6. let f ∈ k −um(1, 0) = k −mn. then f(z) z ≺ 1 + (1 − 2η)z 1 −z , where η = 1 3−2γ and γ = γ(k) is given by (3.4). when α = 1, β = 0 and k = 1, then we have the following result, proved in [22]. corollary 3.7. let f ∈ 1 −um(1, 0) = mn. then f(z) z ≺ 1 + (1 − 2η)z 1 −z , where η ' 0.58159. when α = β = 1, then we have the following result, proved in [6]. corollary 3.8. let f ∈ k −um(1, 1) = k −ucv . then f(z) z ≺ 1 + (1 − 2η)z 1 −z , where η = 1 3−2γ and γ = γ(k) is given by (3.5). int. j. anal. appl. 17 (4) (2019) 681 when α = β = k = 1, then we have the following result, proved in [6]. corollary 3.9. let f ∈ 1 −um(1, 1) = 1 −ucv . then re ( f(z) z ) > 0.6289. theorem 3.6. let α,β � [0, 1], k � [0,∞), b � c\z−0 and s � c. if re [√ (js,bf(z)) ′ + α z (js,bf(z)) ′′ 2 (js,bf(z)) ′ ] > k ∣∣∣∣(js,bf(z))′ + βz (js,bf(z))′′2 (js,bf(z))′ − 1 ∣∣∣∣ , then √ (js,bf(z)) ′ ≺ 1 + (1 − 2γ)z 1 −z ⇒ js,bf(z) z ≺ 1 + (1 − 2η)z 1 −z where η = 2γ 2+1 3 and γ = γ(k,α,β) is given by (3.1). proof. if we put p(z) = √ (js,bf(z)) ′ , then zp′(z) p(z) = z (js,bf(z)) ′′ 2 (js,bf(z)) ′ . the proof follows easily by using theorem 3.1 along with lemma 2.3. � we can deduce the following result from theorem 3.6 by choosing s = 0. theorem 3.7. let k ∈ [0,∞) and α,β ∈ [0, 1]. if re [√ f′(z) + α zf′′(z) 2f′(z) ] > k ∣∣∣∣√f′(z) + βzf′′(z)2f′(z) − 1 ∣∣∣∣ , then √ f′(z) ≺ 1 + (1 − 2γ)z 1 −z ⇒ f(z) z ≺ 1 + (1 − 2η)z 1 −z where η = 2γ 2+1 3 and γ = γ(k,α,β) is given by (3.1). when α = 1, β = 0, then we have the following result, proved in [22]. corollary 3.10. if re [√ f′(z) + zf′′(z) 2f′(z) ] > k ∣∣∣√f′(z) − 1∣∣∣ , then re (√ f′(z) ) > γ ⇒ re ( f(z) z ) > η, where η = 2γ 2+1 3 and γ = γ(k) is given by (3.4). when k = 1, then we have the following result, proved in [22]. int. j. anal. appl. 17 (4) (2019) 682 corollary 3.11. if re [√ f′(z) + zf′′(z) 2f′(z) ] > ∣∣∣√f′(z) − 1∣∣∣ , then re (√ f′(z) ) > γ ' 0.64 ⇒ re ( f(z) z ) > η ' 0.60. for k = 0, we have the following result, refer to [22]. corollary 3.12. if re [√ f′(z) + zf′′(z) 2f′(z) ] > 0, then re (√ f′(z) ) > γ ' 0.64 ⇒ re ( f(z) z ) > η ' 0.60. theorem 3.8. let α,β � [0, 1], k � [0,∞), b � c\z−0 and s � c. if re [ js,bf(z) z + α ( z (js,bf(z)) ′ js,bf(z) − 1 )] > k ∣∣∣∣js,bf(z)z + β ( z (js,bf(z)) ′ js,bf(z) − 1 ) − 1 ∣∣∣∣ , then js,bf(z) z ≺ 1 + (1 − 2γ)z 1 −z , where γ = γ(k,α,β) is given by (3.1). the proof follows easily by substituting p(z) = js,bf(z) z in theorem 3.1. for s = 0, we can easily deduce the following result. theorem 3.9. let k ∈ [0,∞) and α,β ∈ [0, 1]. if re [ f(z) z + α ( zf′(z) f(z) − 1 )] > k ∣∣∣∣f(z)z + β ( zf′(z) f(z) − 1 ) − 1 ∣∣∣∣ , then f(z) z ≺ 1 + (1 − 2γ)z 1 −z , where γ = γ(k,α,β) is given by (3.1). when α = 1, β = 0, then we have the following result, proved in [22]. corollary 3.13. if re [ zf′(z) f(z) + f(z) z − 1 ] > k ∣∣∣∣f(z)z − 1 ∣∣∣∣ ⇒ f(z)z ≺ 1 + (1 − 2γ)z1 −z where γ = γ(k) is given by (3.4). when α = 1, β = 0 and k = 1, then we have the following result, proved in [22]. int. j. anal. appl. 17 (4) (2019) 683 corollary 3.14. if re [ zf′(z) f(z) + f(z) z − 1 ] > ∣∣∣∣f(z)z − 1 ∣∣∣∣ ⇒ re ( f(z) z ) > γ ' 0.64. when α = 1, β = 0 and k = 0, then we have following result. corollary 3.15. if re [ zf′(z) f(z) + f(z) z − 1 ] > 0 ⇒ re ( f(z) z ) > 1 2 . if we substitute p(z) = (js,bf(z)) ′ in theorem 3.1, then we have the following result. theorem 3.10. let α,β � [0, 1], k � [0,∞), b � c\z−0 and s � c. if re [ (js,bf(z)) ′ + α z (js,bf(z)) ′′ (js,bf(z)) ′ ] > k ∣∣∣∣(js,bf(z))′ + βz (js,bf(z))′′(js,bf(z))′ − 1 ∣∣∣∣ , then re ( (js,bf(z)) ′) > γ where γ = γ(k,α,β) is given by (3.1). for s = 0, we have the following result. theorem 3.11. let k ∈ [0,∞) and α,β ∈ [0, 1]. if re [ f′(z) + α zf′′(z) f′(z) ] > k ∣∣∣∣f′(z) + βzf′′(z)f′(z) − 1 ∣∣∣∣ , then re (f′(z)) > γ where γ = γ(k,α,β) is given by (3.1). when α = 1, β = 0, then we have the following result, proved in [22]. corollary 3.16. if re [ f′(z) + zf′′(z) f′(z) ] > k |f′(z) − 1|⇒ re (f′(z)) > γ ' 0.64. when α = 1, β = k = 0, then we have the following result. corollary 3.17. if re [ f′(z) + zf′′(z) f′(z) ] > 0, then re (f′(z)) > 1 2 . int. j. anal. appl. 17 (4) (2019) 684 references [1] j.w. alexander, functions which map the interior of the unit circle upon simple regions, ann. math. (ser. 2). 17 (1915), 12-22. [2] h.a. al-kharsani and a. sofo, subordination results on harmonic k-uniformly convex mappings and related classes, comput. math. appl. 59 (2010), 3718-3726. [3] s.d. bernardi, convex and starlike univalent functions, trans. amer. mat. soc. 135 (1969), 429-446. [4] j. dziok, classes of functions associated with bounded mocanu variation, j. inequal. appl. 2013 (2013), art. id. 349. [5] i.b. jung, y.c. kim, and h.m. srivastava, the hardy space of analytic functions associated with certain one-parameter families of integral operators, j. math. anal. appl. 176 (1993), 138-147. [6] s. kanas, subordinations for domains bounded by conic sections, bull. belg. math. soc. simon stevin. 15 (2008), 589-598. [7] s. kanas, techniques of the differential subordination for domains bounded by conic sections, int. j. math. math. sci. 38 (2003), 2389-2400. [8] s. kanas and a. wisniowska, conic domain and starlike functions, rev. roumaine math. pures appl. 45 (2000), 647-657. [9] s. kanas and a. wisniowska, conic regions and k-uniform convexity, j. comput. math. 105 (1999), 327-336. [10] r.j. libera, some classes of regular univalent functions, proc. amer. math. soc. 16 (1965), 755-758. [11] s.s. miller and p.t. mocanu, differential subordinations and applications, marcel dekker, inc. new york-basel. 2000. [12] s.s. miller and p.t. mocanu, differential subordinations and univalent functions, michigan math. j. 28 (1981), 157-171. [13] p.t. mocanu, une propriete de convexite generlise dans la theorie de la representation conforme, mathematica (cluj). 11 (1969), 127-133. [14] k.i. noor, on generalization of uniformly convex and related functions, comput. math. appl. 61 (2011), 117-125. [15] k.i. noor and s. hussain, on certain analytic functions associated with ruscheweyh derivatives and bounded mocanu variation, j. math. anal. appl. 340 (2008), 1145-1152. [16] k.i. noor and s.n. malik, on coefficient inequalities of functions associated by conic domains, comput. math. appl. 62 (2011), 2209-2217. [17] k.i. noor and s.n. malik, on generalized bounded mocanu variation associated with conic domain, math. comput. modell. 55 (2012), 844-852. [18] k.i. noor and a. muhammad, on analytic functions with generalized bounded mocanu variation, appl. math. comput. 196 (2008), 802-811. [19] k.i. noor and w. ul-haq, on some implication type results involving generalized bounded mocanu variations, comput. math. appl. 63 (2012), 1456-1461. [20] d. răducanu and h.m. srivastava, a new class of analytic functions defined by means of convolution operator involving hurwitz-lerch zeta function, integral transforms spec. funct. 18 (2007), 933-943. [21] a. rasheed, s. hussain, m.a. zaighum and z. shareef, analytic functions related with mocanu class, int. j. anal. appl. 16 (2018), 783-792. [22] s. sivasubramanian, m. govindaraj and k. piejko, on certain class of univalent functions with conic domains involving sokol-nunokawa class, u.p.b. sci. bull. series a. 80 (2018), 123-134. [23] j. sokol and m. nunokawa, on some class of convex functions, c. r. math. acad. sci. paris. 353 (2015), 427-431. [24] h.m. srivastava and a.a. attiya, an integral operator associated with the hurwitz-lerch zeta function and differential subordination. integral transforms spec. funct. 18 (2007), 207–216. int. j. anal. appl. 17 (4) (2019) 685 [25] h.m. srivastava and j. choi, series associated with the zeta and related functions, dordrecht, boston, london, kluwer academic publishers, 2001. 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications volume 17, number 5 (2019), 793-802 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-793 hermite-hadamard type inequalities for m-convex and (α,m)-convex stochastic processes serap özcan∗ department of mathematics, faculty of arts and sciences, kırklareli university, 39100 kırklareli, turkey ∗corresponding author: serapozcann@yahoo.com abstract. in this paper, the concepts of m-convex and (α,m)-convex stochastic processes are introduced. several new inequalities of hermite-hadamard type for differentiable m-convex and (α,m)-convex stochastic processes are established. the results obtained in this work are the generalizations of the known results. 1. introduction stochastic convexity and its applications is of great importance in statistics and probability, because it provides numerical approximations for existing probabilistic quantities. in 1980, nikodem [10] defined convex stochastic processes and investigated their properties. in 1988, shaked et al. [16] defined stochastic convexity and gave its applications. in 1992, skowronski [17] introduced some new types of convex stochastic processes and obtained some further results on these processes. in 2012, kotrys [6] extended classical hermite-hadamard inequality to convex stochastic processes. in recent years, there have been many studies on the above mentioned processes. for recent generalizations and improvements on convex stochastic processes, please refer to [4][8], [11][15], [19]. received 2019-05-24; accepted 2019-06-14; published 2019-09-02. 2010 mathematics subject classification. 26d15, 26a51, 60g99. key words and phrases. convex stochastic process; m-convex stochastic process; (α,m)-convex stochastic process; meansquare integral; hermite-hadamard type inequality. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 793 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-793 int. j. anal. appl. 17 (5) (2019) 794 2. preliminaries let (ω,κ,p) be a probability space. a function x : ω → r is called a random variable if it is κmeasurable. let i ⊂ r be an interval. then, a function x : i × ω → r is called a stochastic process if for every t ∈ i the function x (t, ·) is a random variable. let p − lim and e [x (t, ·)] denote the limit in probability and the expectation value of random variable x (t, ·), respectively. then, a stochastic process x : i × ω → r is called (1) continuous in probability in the interval i, if p − lim t→t0 x (t, ·) = x (t0, ·) for all t0 ∈ i. (2) mean square continuous in the interval i, if lim t→t0 e [ (x (t, ·) −x (t0, ·)) 2 ] = 0 for all t0 ∈ i. (3) mean-square differentiable at a point t ∈ i if there is a random variable x′ (t, ·) : i × ω → r such that x′ (t, ·) = p − lim t→t0 x (t, ·) −x (t0, ·) t− t0 . let x : i × ω → r be a stochastic process with e [ (x (t, ·))2 ] < ∞ for all t ∈ i. let u = t0 < t1 < t2 < ... < tn = b be a partition of [u,v] if the identity lim n→∞ e [ ( ∑ x (θk) (tk − tk−1) −y ) 2 ] = 0 holds for all normal sequences of partitions of the interval [u,v] and for all θk ∈ [tk−1, tk], k = 1, 2, ...,n. then, we can write y (·) = ∫ v u x (t, ·) dt (a.e.). the assumption of the mean-square continuity of the stochastic process x is enough for the mean-square integral to exist. definition 2.1. [10] the stochastic process x : i × ω → r is convex if for all λ ∈ [0, 1] and u,v ∈ i the inequality x (λu + (1 −λ) v, ·) ≤ λx (u, ·) + (1 −λ) x (v, ·) (a.e.) (2.1) is satisfied. if the inequality (2.1) is assumed only for λ = 1 2 , then the stochastic process x is called jensenconvex or 1 2 -convex. in [6], kotrys defined convex stochastic processes as following: int. j. anal. appl. 17 (5) (2019) 795 theorem 2.1. let x : i × ω → r be a jensen-convex stochastic process and mean-square continuous in the interval i. then the following inequality holds for all u,v ∈ i, u < v. x ( u + v 2 , · ) ≤ 1 v −u ∫ v u x (t, ·) dt ≤ x (u, ·) + x (v, ·) 2 (a.e.). (2.2) definition 2.2. [18] let m ∈ [0, 1]. the function f : [0,c] → r, c > 0, is said to be m-convex, if f (tx + m (1 − t) y) ≤ tf (x) + m (1 − t) f (y) is satisfied for every x,y ∈ [0,c] and t ∈ [0, 1]. definition 2.3. [9] let α,m ∈ [0, 1]. the function f : [0,c] → r, c > 0, is said to be (α,m)-convex, if f (tx + m (1 − t) y) ≤ tαf (x) + m (1 − tα) f (y) is satisfied for every x,y ∈ [0,c] and t ∈ [0, 1]. for further information about m-convex and (α,m)-convex functions, please refer to [1], [2], [5], [13]. theorem 2.2. [3] let a,b ∈ r with a < b and let f : [a,b] → r be a differentiable function on (a,b). if |f′| is convex on [a,b], then∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) (|f ′ (a)| + |f′ (b)|) 8 . theorem 2.3. [3] let a,b ∈ r with a < b and let f : [a,b] → r be a differentiable function on (a,b). suppose p ∈ r with p > 1. if |f′|q is convex on [a,b] for q ∈ r with q > 1, then∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ b−a2 (p + 1) 1p [ |f′ (a)|q + |f′ (b)|q 2 ]1 q , where 1 p + 1 q = 1. 3. main results in order to establish our main results we give the following definitions and lemma: definition 3.1. the stochastic process x : [a,b] × ω → r is said to be m-convex where m ∈ [0, 1], if x (λu + m (1 −λ) v, ·) ≤ λx (u, ·) + m (1 −λ) x (v, ·) holds for all u,v ∈ [a,b] and λ ∈ [0, 1]. definition 3.2. the stochastic process x : [a,b]×ω → r is said to be (α,m)-convex where (α,m) ∈ [0, 1]2, if x (λu + m (1 −λ) v, ·) ≤ λαx (u, ·) + m (1 −λα) x (v, ·) holds for all u,v ∈ [a,b] and λ ∈ [0, 1]. int. j. anal. appl. 17 (5) (2019) 796 lemma 3.1. [11] let x : i◦ ⊆ r × ω → r be a mean-square differentiable stochastic process on i◦ and u,v ∈ i◦ with u < v. if x′ is mean-square integrable on [u,v], then the following inequality holds almost everywhere: x (u, ·) + x (v, ·) 2 − 1 v −u ∫ v u x (t, ·) dt = v −u 2 ∫ 1 0 (1 − 2λ) x′ (λu + (1 −λ) v, ·) dλ. now we obtain results for stochastic processes whose derivatives absolute values raise to some certain power are m-convex and (α,m)-convex. theorem 3.1. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′| is m-convex stochastic process on [u,v] for m ∈ (0, 1], then the following inequality holds almost everywhere:∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u8 [ |x′ (u, ·)| + m ∣∣∣x′( v m , · )∣∣∣] . proof. from lemma 3.1, we obtain∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 ∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|dλ. since |x′| is m-convex stochastic process on [u,v] for all u,v ∈ i, λ ∈ [0, 1] and m ∈ (0, 1], we have |x′ ((λu + (1 −λ) v) , ·)| = ∣∣∣x′(λu + m (1 −λ) v m , · )∣∣∣ ≤ λ |x′ (u, ·)| + m (1 −λ) ∣∣∣x′( v m , · )∣∣∣ . hence we have ∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 [ |x′ (u, ·)| ∫ 1 0 |1 − 2λ|(1 −λ) dλ + m ∣∣∣x′( v m , · )∣∣∣∫ 1 0 |1 − 2λ|λdλ ] . since ∫ 1 0 |1 − 2λ|(1 −λ) dλ = ∫ 1 0 |1 − 2λ|λdλ = 1 4 , we obtain the desired result. � remark 3.1. for m = 1, theorem 3.1 becomes to theorem 5 in [11]. int. j. anal. appl. 17 (5) (2019) 797 theorem 3.2. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′|q is m-convex stochastic process on [u,v] for q > 1 and 1 p + 1 q = 1, then the following inequality holds almost everywhere: ∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (p + 1) 1 p [ |x′ (u, ·)|q + m ∣∣x′( v m , · )∣∣q 2 ]1 q . (3.1) proof. by lemma 3.1 and using well known hölder’s inequality, we have∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (∫ 1 0 |1 − 2λ|p dλ )1 p (∫ 1 0 |x′ (λu + (1 −λ) v, ·)|dλ )1 q . (3.2) since |x′|q is m-convex stochastic process on [u,v] for all u,v ∈ i with u < v, λ ∈ [0, 1] and m ∈ (0, 1], we have |x′ (λu + (1 −λ) v, ·)|q ≤ λ |x′ (u·)|q + m (1 −λ) ∣∣∣x′( v m , · )∣∣∣q . thus we obtain ∫ 1 0 |x′ (λu + (1 −λ) v, ·)|q dλ ≤ ∫ 1 0 [ λ |x′ (u, ·)|q + m (1 −λ) ∣∣∣x′( v m , · )∣∣∣q]dλ = 1 2 |x′ (u, ·)|q + m 2 ∣∣∣x′( v m , · )∣∣∣q . (3.3) moreover, since ∫ 1 0 |1 − 2λ|p dλ = ∫ 1/2 0 (1 − 2λ)p dλ + ∫ 1 1/2 (2λ− 1)p dλ = 1 p + 1 , (3.4) utilizing inequalities (3.3) and (3.4) in (3.2), we get the inequality (3.1). � remark 3.2. for m = 1, theorem 3.2 becomes to corollary 6 in [11]. theorem 3.3. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′|q is m-convex stochastic process on [u,v] for m ∈ (0, 1], q ≥ 1, then the following inequality holds almost everywhere :∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 4 [ |x′ (u, ·)|q + m ∣∣x′( v m , · )∣∣q 2 ]1 q . (3.5) int. j. anal. appl. 17 (5) (2019) 798 proof. for q = 1, the proof is the same as that of theorem 3.1. suppose that q > 1. from lemma 3.1 and using well known power-mean inequality, we have∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (∫ 1 0 |1 − 2λ|dλ )1−1 q (∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|q dλ )1 q . (3.6) using m-convexity of the stochastic process |x′|q on [u,v] in the second integral on the right side of the inequality (3.6), we have∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|q dλ ≤ ∫ 1 0 |1 − 2λ| [ λ |x′ (u, ·)|q + m (1 −λ) ∣∣∣x′( v m , · )∣∣∣q]dλ = |x′ (u, ·)|q ∫ 1 0 λ |1 − 2λ|dλ + m ∣∣∣x′( v m , · )∣∣∣q ∫ 1 0 (1 −λ) |1 − 2λ|dλ = 1 4 |x′ (u, ·)|q + m 4 ∣∣∣x′( v m , · )∣∣∣q . a usage of the last inequality in (3.6) gives the desired result. � remark 3.3. for q = 1, the inequality (3.5) reduces to the inequality proved in theorem 3.1. if q = p p−1 (p > 1), then one has 4 p > p + 1 and so 1 4 < 1 2(p+1) 1 p . this shows that the inequality (3.5) is better than the one given by (3.1) in theorem 3.2. now we establish our results for (α,m)-convex stochastic processes. theorem 3.4. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′| is (α,m)-convex stochastic process on [u,v] for m ∈ (0, 1], q ≥ 1, then the following inequality holds almost everywhere:∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 [ m1 |x′ (u, ·)| + mm2 ∣∣∣x′( v m , · )∣∣∣] (3.7) where m1 = 1 + α2α 2α (1 + α) (2 + α) , (3.8) m2 = 1 2 −m1. (3.9) int. j. anal. appl. 17 (5) (2019) 799 proof. from lemma 3.1, we have ∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 ∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|dλ. (3.10) since |x′| is (α,m)-convex stochastic process on [u,v] for all u,v ∈ i with u < v, (α,m) ∈ (0, 1]2 and λ ∈ [0, 1], we have ∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|dλ ≤ |x′ (u, ·)| ∫ 1 0 |1 − 2λ|λαdλ + m ∣∣∣x′( v m , · )∣∣∣∫ 1 0 |1 − 2λ|(1 −λα) dλ = m1 |x′ (u, ·)| + m ( 1 2 −m1 )∣∣∣x′( v m , · )∣∣∣ (3.11) where ∫ 1 0 |1 − 2λ|λαdλ = 1 + α2α 2α (1 + α) (2 + α) = m1, and ∫ 1 0 |1 − 2λ|(1 −λα) dλ = 1 2 − 1 + α2α 2α (1 + α) (2 + α) = 1 2 −m1 = m2. using the inequality (3.11) in the inequality (3.10), we get the required result. � remark 3.4. for (α,m) = (1, 1), theorem 3.4 becomes to theorem 5 in [11]. theorem 3.5. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′|q is (α,m)-convex stochastic process on [u,v] for (α,m) ∈ (0, 1]2, q ≥ 1, then the following inequality holds almost everywhere:∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (p + 1) 1 p [ α |x′ (u, ·)|q + m ∣∣x′( v m , · )∣∣q 1 + α ]1 q (3.12) where 1 p + 1 q = 1. proof. using lemma 3.1 and hölder’s inequality, we have∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (∫ 1 0 |1 − 2λ|p dλ )1 p (∫ 1 0 |x′ (λu + (1 −λ) v, ·)|q dλ )1 q . (3.13) by (α,m)-convexity of the stochastic processes |x′|q on [u,v], we have for every λ ∈ [0, 1] |x′ ((λu + (1 −λ) v) , ·)|q ≤ λα |x′ (u, ·)|q + m(1 −λα) ∣∣∣x′( v m , · )∣∣∣q int. j. anal. appl. 17 (5) (2019) 800 for (α,m) ∈ (0, 1]2. hence ∫ 1 0 |x′ (λu + (1 −λ) v, ·)| ≤ |x′ (u, ·)|q ∫ 1 0 λαdλ + m ∣∣∣x′( v m , · )∣∣∣q ∫ 1 0 (1 −λα) dλ = 1 1 + α |x′ (u, ·)|q + mα 1 + α ∣∣∣x′( v m , · )∣∣∣q . utilizing of the above inequality in (3.13) and the fact ∫ 1 0 |1 − 2λ|p dλ = 1 p + 1 completes the proof. � remark 3.5. for (α,m) = (1, 1), theorem 3.5 becomes to corollary 6 in [11]. theorem 3.6. suppose b∗ > 0. let x : i ⊂ [a,b∗] × ω → r be a differentiable stochastic process on i◦ and let x′ be mean-square integrable on [u,v] where u,v ∈ i with u < v. if |x′|q is (α,m)-convex stochastic process on [u,v] for (α,m) ∈ (0, 1]2, q ≥ 1, then the following inequality holds almost everywhere: ∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 ( 1 2 )1−1 q [ m1 |x′ (u, ·)| q + mm2 ∣∣∣x′( v m , · )∣∣∣q]1q , where m1 = 1 + α2α 2α (1 + α) (2 + α) , m2 = 1 2 −m1. proof. for q = 1, the proof is similar to that of theorem 3.4. now suppose that q > 1. using lemma 3.1 and power-mean inequality, we have ∣∣∣∣x (u, ·) + x (v, ·)2 − 1v −u ∫ v u x (t, ·) dt ∣∣∣∣ ≤ v −u 2 (∫ 1 0 |1 − 2λ|dλ )1−1 q (∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|q dλ )1 q . (3.14) int. j. anal. appl. 17 (5) (2019) 801 since |x′|q is (α,m)-convex stochastic process on [u,v] for every λ ∈ [0, 1] and (α,m) ∈ (0, 1]2, we have∫ 1 0 |1 − 2λ| |x′ (λu + (1 −λ) v, ·)|q dλ ≤ ∫ 1 0 |1 − 2λ| [ λα |x′ (u, ·)|q + m (1 −λα) ∣∣∣x′( v m , · )∣∣∣q]dλ = |x′ (u, ·)|q ∫ 1 0 |1 − 2λ|λαdλ + ∣∣∣x′( v m , · )∣∣∣q ∫ 1 0 |1 − 2λ|(q −λα) dλ = m1 |x′ (u, ·)| q + m2 ∣∣∣x′( v m , · )∣∣∣q . (3.15) using the inequality (3.15) in the inequality (3.14) we get the desired result. � references [1] m. k. bakula, m. e. özdemir and j. pec̆arić, hadamard type inequalities for m-convex and (α,m)-convex functions, j. inequal. pure appl. math., 9(4) (2008), article 96. [2] m. k. bakula, j. pec̆arić and m. ribićić, companion inequalities to jensen’s inequality for m-convex and (α,m)-convex functions, j. inequal. pure appl. math., 7(5) (2006), article 194. [3] s. s. dragomir and r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and trapezoidal formula, appl. math. lett., 11(5) (1998), 91–95. [4] l. gonzalez, n. merentes and m. valera-lopez, some estimates on the hermite-hadamard inequality through convex and quasi-convex stochastic processes, math. eterna, 5(5) (2015), 745–767. [5] i̇. i̇şcan, h. kadakal and m. kadakal, some new integral inequalities for functions whose nth derivatives in absolute value are (α,m)-convex functions, new trends math. sci., 5(2) (2017), 180–185. [6] d. kotrys, hermite-hadamard inequality for convex stochastic processes, aequationes math., 83 (2012), 143–151. [7] d. kotrys, remarks on strongly convex stochastic processes, aequationes math., 86 (2013), 91–98. [8] l. li and z. hao, on hermite-hadamard inequality for h-convex stochastic processes, aequationes math., 91 (2017), 909–920. [9] v. g. miheşan, a generalization of the convexity, seminer on functional equations, approximation and convexity, clujnapoca, romania, 1993. [10] k. nikodem, on convex stochastic processes, aequationes math., 20 (1980), 184–197. [11] n. okur, i̇. i̇şcan and e. yuksek dizdar, hermite-hadamard type inequalities for p-convex stochastic processes, int. j. optim. control, theor. appl., 9(2) (2019), 148–153. [12] m. z. sarıkaya, h. yaldız and h. budak, some integral inequalities for convex stochastic processes, acta math. univ. comenianae, 85 (2016), 155–164. [13] e. set, m sardari, m. e. özdemir and j. rooin, on generalizations of the hadamard inequality for (α,m)-convex functions, kyungpook math. j., 52 (2012), 307–317. [14] e. set, m. tomar and s. maden, hermite-hadamard type inequalities for s-convex stochastic processes in the second sense, turk. j. anal. numb. theory, 2(6) (2016), 202–207. [15] e. set, m. z. sarıkaya and m. tomar, hermite-hadamard type inequalities for coordinates convex stochastic processes, math. aeterna, 5(2) (2015), 363–382. [16] m. shaked and j. g. shanthikumar, stochastic convexity and its applications, adv. appl. probab., 20 (1988), 427–446. [17] a. skowronski, on some properties of j-convex stochastic processes, aequationes math., 44 (1992), 249–258. int. j. anal. appl. 17 (5) (2019) 802 [18] g. toader, some generalizations of the convexity, proc. colloq. approx. optim., univ. cluj-napoca, cluj-napoca, romania, (1985), 329–338. [19] m. tomar, e. set and s. maden, hermite-hadamard type inequalities for log-convex stochastic processes, j. new theory, 2 (2015), 23–32. 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications volume 18, number 2 (2020), 212-242 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-212 received 2019-12-30; accepted 2020-01-27; published 2020-03-02. 2010 mathematics subject classification. 91b42. key words and phrases. consumers’ attitude, repurchase intention, functional foods. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 212 attitudes and repurchase intention of consumers towards functional foods in ho chi minh city, vietnam nhu-ty nguyen1,2,* 1school of business, international university, vietnam 2vietnam national university hcmc; quarter 6, linh trung ward, thu duc district, hcmc, vietnam *corresponding author: nhutynguyen@gmail.com; nhutynguyen@hcmiu.edu.vn abstract.although the demand of functional foods is increasing rapidly, and vietnam is considered as a much potential market for the development of functional food market. however, there are many unsolved problems remaining in functional foods market, and these problems have never ceased to draw public attention and provoke debates. thus, the purpose of this research is to investigate the factors affecting to consumers’ attitude and how consumers’ attitude affecting to consumer’s repurchase intention. the research will be conducted based on the questionnaires’ results collect from people who have used functional foods in ho chi minh city, vietnam. questionnaires will be spread to consumers by hands and online survey. there are 260 responses are valid. the results show that there are four factors that positively affect to consumers’ attitude. these are (1) consumers’ knowledge, (2) perceived role, (3) trust, (4) subjective norms. moreover, the findings also show that when people have positive feelings about products, they will increase their intention to products. research also finds that trust is a direct relationship with consumers’ repurchases intention. from this conclusion, the study gives some recommendations to increase consumers’ attitude towards functional foods as well as the repurchase intention. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-212 int. j. anal. appl. 18 (2) (2020) 213 introduction 1. research problems although the demand of functional foods is increasing rapidly and vietnam is considered as a much potential market for the development of functional food market [1]. however, there are many unsolved problems remains in functional foods market and these problems have never ceased to draw public attention and provoke debates. the biggest problem for vietnamese company is building trust in consumers toward functional foods (according to dantri website in 2017). in 2000, the number of vietnamese people using functional food is only about 500,000, and they mostly live in big cities such as ho chi minh city and ha noi. however, in 2017, this number rapidly increased up to 20 million people (above 21% of population). in which, only 20% people use domestic functional food product and the rest use foreign made products from south korea, us, japan, etc. (according to vnexpress in 2018). the domestic products are dominated by foreign products. the first reason for this situation is that consumers in developing countries like vietnam are often in favor of foreign goods rather than domestic goods ([2]; [3]; [4];[5]). another reason is that advertising of functional foods exaggerates than the benefits for health they really are, and it breaks the trust of vietnamese customer when they don’t receive the benefits as advertisements. moreover, the reason also comes from the low level of knowledge and the unconsciousness about functional foods. this led to the serious outcomes such as functional foods and dangers behind advertising, overusing functional food being in danger on recent newspapers (according to datviet in 2018). besides, vietnamese are often affected by advice from people around them such as their relatives or their friends and make decisions.secondly, with the rapidly increasing of functional food demand, although vietnamese authorities have formulated policies for developing domestic products and tightly controlling the quality of functional food in vietnam, there exists the large quantity of fake and poor good quality products based on the incoherence of legal systems and policies. as a result, functional foods market lose consumers’ trust towards products ([6]; [7]) therefore, understanding these problems,this research is carried out with the purpose of understanding profoundly into consumers’ behavior by examining the influence of perception factors to consumers’ attitude and how consumers’ attitude affecting to their repurchase intention as well as investigatingthe incentives behind their repurchase intention. moreover, this research also recommends some solutions based on the results of research. it can int. j. anal. appl. 18 (2) (2020) 214 be practical for vietnamese enterprises and authorities to develop functional food market and attract customers. the research will be conducted in ho chi minh city market and general functional foods. 2. research objectives the purposes of this study are proposed as follow: ✓ examine the influential factors on consumer attitude towards functional food in a rapid growing market ✓ explore the impact of attitude on repurchase intention ✓ to understand the problems and challenges, propose solutions for improving consumers’ attitude and repurchase intention, increasing competitiveness of vietnamese functional food products and supporting vietnamese authorities in developing and managing the market 3. research questions with these purposes, the research are raised to achieve the research purposes ✓ what kind of factors influence consumers’ attitude toward functional foods? ✓ how does consumers’ attitude impact on consumers’ repurchase intention toward functional foods product? ✓ what are the recommendations to enhance the consumers’ attitude and repurchase intention toward the products? literature review 1. conceptual model review this study is carried out to understand about consumers’ behaviors and how consumers attitude affecting to consumers’ repurchase intentionby adapting a model form the research” consumer attitude and purchase intention towards organic food – a case study in china” conducted by mingyan yang, sarah al-shaaban and tram b. nguyen. the research was undertaken in 2014 [8] in order to explore the factors affecting consumers’ attitude and the impact of consumers’ attitude on consumer’s purchase intention towards organic food. int. j. anal. appl. 18 (2) (2020) 215 figure 2.1 research model ofmingyan yang, sarah al-shaaban and tram b. nguyen by adapting the theory of planned behavior [9] which illustrated that “behavioral intention refers to an individual’s subjective likelihood of performing a certain behavior” and “intention can be related to attitude in some extent”, the model they studied including five factors “health consciousness”, “consumer knowledge”, “environmental concern”, “personal norms” and subjective norms”, the mediating factor “attitude” and the dependent variable “purchase intention”. the research examined that consumer knowledge; personal norms and health consciousness are the key important factors that influence to consumer’s attitude and purchase intention. moreover, the findings also found that consumers have positive attitude towards functional foods will be more likely to purchase them ([2]; [10]; [1]). there are many researchers in vietnam have conducted study to consumer’s acceptance and consumers’ purchase intention; however; there has not been any study about consumers’ repurchase intention and their attitude toward functional food, especially in ho chi minh city ([11]; [12]; [13]; [14]). besides, consumers’ attitude can be significantly different from countries to countries. therefore, the study will based on this model and adjust to make it applicable in vietnam market. 2. research concepts 2.1 consumers’ repurchase intention the concept of repurchase intention means the future intention to use good or service ([15]). according to jackson’s points of view ([16]), “repurchase intention” like “consumer behavioral intention” that measure the tendency towards using rate of customers about such as continuing, int. j. anal. appl. 18 (2) (2020) 216 increasing or decreasing the amount of good or services. the decision of reselecting the same goods and services are explained by the good experiences about products such as perceived product quality, price or trust they have been through. from consumer perspective, consumer attitude and commitment to purchase a certain product is the consequence of repurchase intention ([1]). in this research, repurchase intention towards functional foods is the likelihood that consumers will continue to buy functional foods in ho chi minh city. 2.2 consumers’ attitude consumers’ attitude describes the consumers’ feeling which is negative or positive towards an object and activities ([17]). according to ([18]), attitude is defined as the emotion of people in evaluating an object or activities. according to ([19]), consumers’ attitude can make consumption behavior happen such as how frequently people go shop food. people who have more positive attitude have a tendency to buy more products ([20]). besides basing on experiences, consumers’ attitude also comes from the expectations and beliefs about products ([9]; [20]; [21]). in the previous researches, theory of planned behavior (tpb) is often applied to examine the consumers’ attitude ([21]; [20]). h6: consumers’ attitude has positive relationship to consumers’ repurchase intention 2.3 consumer knowledge the biggest difference between developing functional foods and normal foods is the requirement of scientific evidences; therefore, the research process needs experts from many fields in order to have these evidences ([22]). in a quality research 1999, ifcf pointed out that knowledge and trust are the key motivation for buying and using or not using functional foods in diet. the higher levels of knowledge consumers have towards functional foods, the higher cognitive capacity and exact performance they have ([23]). consumers can gain knowledge from various channels such as social media, social networks, health experts and advertisements. when they have high knowledge about functional foods, they can easily make right decisions to choose or refuse product. h1: there is a positive relationship between consumer knowledge and consumers’ attitude int. j. anal. appl. 18 (2) (2020) 217 2.4 price ares et al. (2010) [24] conclude that high prices may affect consumers' intention to buy in two ways: reducing the intention to buy due to spending too much on the product, or positively affect purchasing intention because it creates better perception about product quality. the research of annunziata and vecchio (2012) [25] demonstrates that different groups of customers have different ways of evaluate the importance of prices. there are many people think that functional foods with higher price must go with higher quality. with these consumers, they have high income and are willing to pay for healthier alternatives regardless of price. on the other hands, verbeke (2005) [20] also indicates that one of the main barriers for consumers’ attitude and intention relates to very high price. several consumers prefer good quality functional foods with affordable price. they will consider wasting or saving before buying functional foods. childs (1997) [26] concludes that price and perceived price are better predicted factors (compared to trust) for the purchase habit of functional foods in the future. this leads the suggestion for this study to focus on perceived price as a determinant of acceptance and purchase. h2: there is a positive relationship between price and consumers’ attitude 2.5 subjective norms subjective norms are defined as the perceived social pressure on to perform or not perform actions ([9]). subjective norms are decisively factor that is widely applied to understand about human behavior ([27]). it has been said that “although many people think of themselves as individuals, people, most of the time, look to others for guidance in how to behave” and they may feel “unwanted emotions” when not act like others ([27]). in marketing sector, many researchers claim that subjective norms are the key factor determine consumers’ purchasing intention and ([28], [29]). according to shepherd, (1999) [30], choosing foods is influenced by many factors including subjective norms factors. the previous researches proved the relationship between subjective norms (foods other people claim that i should eat) and the amount of food using ([31]). according to aarts and dijksterhuis, (2003) [21], a special feature of these standard influences is that its impact seems unconscious, and people often deny that they are influenced by others. h3: there is a positive relationship between subjective norms and consumers’ attitude int. j. anal. appl. 18 (2) (2020) 218 2.6 trust as cited in carmina and carlos (2011, p.283), moorman et al. (1993) [32] defined: “trust as a willingness to rely on an exchange partner in whom one has confidence.” because of the awareness of the roles of foods for health, consumers believe functional foods play an important role for protecting and improving people’s health ([28]). citizens in developed countries increasingly have more responsibility for their health and they claim that food and diet have relationship with people’s health ([33]). according to the research of niva (2007) [34], consumers are more health conscious, and they are willing to change their eating habits to have a better health. if the perceived role of functional foods for health mainly includes general aspects about recognizing about the important of health, controlling and orientation behaviors, trust in the context of functional foods definitely relates to perceived benefits of functional foods for health ([20]). therefore, the hypothesis is proposed as follow: h4: there is a positive relationship between trust and consumers’ attitude 2.7 perceived role perceiving roles of functional foods for health plays an important part in affecting functional foods acceptance. in the previous research, there are various concepts about trust and roles of functional foods and these concepts come from the effect of themselves on health ([35]), have consciousness about health requirements ([36]). in the research “functional food choices: impacts of trust and health beliefs” ([31]). the author states that both trust and health perceived are the key factors affecting to consumer’ attitude. h5: there is a positive relationship between perceived role and consumers’ attitude int. j. anal. appl. 18 (2) (2020) 219 proposed research model research methodology 1. research design this research is conducted to test the feasible factors affecting to consumers’ attitude towards functional foods and the relationship between attitude and repurchase intention. it will be based on the previous researches and theoretical frameworks in order to develop six hypotheses. this is the reason why deductive research method will be appropriate to be implemented in this case. moreover, empirical data is collected by using survey questionnaire with the purpose of testing the existing theories and modifying mode. from this point of view, deductive approach completely satisfies this requirement. with the purpose of collecting, analyzing and generalizing data, quantitative research will be implemented along with deductive research. specifically, the quantitative approach is appropriate in this circumstance because the research intends to reach a large number of consumers in ho chi minh city by survey questionnaires. furthermore, the main objective of int. j. anal. appl. 18 (2) (2020) 220 this research is to test the hypothesis and theoretical model based on the previous study. therefore, quantitative method is fulfilled the requirements. 2. sample of research the conceptual model has 7 variables, in which 5 variables are independent and 2 variables are dependent. according to ([37]), the applicable sample size should be at least 150 and 200. with the limited financial ability and time period, the research will carry out 260 surveys 3. data collection method in this research, primary data will be collected by giving online and offline survey to consumers from 10th april 2019 to 24th april 2019. with this method, it is the best way to reach a large amount of consumers using functional foods. in specified, questionnaires will be directly delivered to each consumer by hands at public place or by social media channel such as facebook, zalo and so on. 4. questionnaire design the surveys are spread out among consumers in ho chi minh city by hands or by social media channel such as facebook, zalo and so on. there are two parts in the questionnaire; the first is designed to collect demographic information including job, age, gender, income. it will help researcher understand about the target object. the left part is created based on the factors affecting to consumers’ attitude, consumer’s attitude and repurchase intention. the questionnaires have 26 question designed from the theoretical concepts. the demographic part of questionnaires is designed in ordinal, nominal scale. the second part which is designed in order to measure the theoretical concept based on likert measurement scale where 1 represents for totally disagree and 5 stands for totally agree. all questions are closed question, easy to understand and designed in vietnamese language. respondents chose one of the existing alternatives without expressing new opinion. with this method, it will ensure the related between data and conceptual model 5. data analysis method data will be analyzed by using spss and amos software. analysis method is chosen according to study objectives; therefore, this research will conduct the following analysis o descriptive statistics o reliability test o exploratory factor analysis (efa) o confirmatory factor analysis (cfa) o structural equation modeling (sem) int. j. anal. appl. 18 (2) (2020) 221 60.800% 39.200% gender female male 27.700% 31.900% 26.200% 14.200% age from 20 to under 35 from 35 to under 50 over 50 under 20 research results and finding discussion 1. demographic analysis objects percentage discussion gender from the percentage of respondents, it can be seen that the number of female using functional foods outweighs the number of male and occupies 61.97% of total. those numbers can demonstrate that female have more demand for functional foods than male because female extremely care about health care foods than male and they will make more purchasing decisions for functional foods. age the highest percentage of respondents by age is from the age of 35 to under 50 occupying 32.5 percent of total and followed by the age of from 20 to under 35 (27.2%). the percentage of the age “over 50” is approximately equal the percentage of the age from 20 to under 35(26.4%). finally, the lowest percentage (14%) is of consumers who age “under 20”. these percentages imply that functional foods are widely used at any age; however, middle age who has a stable income and consciousness for their health is the main consumers for functional products. therefore, people in the middle age will be potential target customer that business should consider. int. j. anal. appl. 18 (2) (2020) 222 36% 23% 27% 14% income from 10 to 20 million vnd from 5 to 10 million vnd over 20 million vnd under 5 milliion vnd 29% 5% 36% 2% 14% 14% occupation labor office staff other business manager student income the highest percentage of people who use functional foods have high income from 10-20 million and over 20 million. people have the income of “5 million to 10 million” as well as under 5 million is also occupied rather high percentage. the reason is that there has a huge amount of functional foods in market, so price level of functional foods also varies from low to high price to compete with other brands. occupation it can be seen that office staff occupies a highest percentage of 69.8%. they have stable income and may be suffered by many health problems because sitting for a long periods of timeevery day. therefore, this may the reason why they use functional foods to protect their health and avoid health disease. next are the retirees who are suffered from many health diseases because of aging. labor is the group has lowest percentage and take a very small respondents of 5.3% int. j. anal. appl. 18 (2) (2020) 223 2. reliability test – cronbach’s alpha and exploratory factor analysis (efa) scale mean if item deleted scale variance if item deleted corrected itemtotal correlation cronbach's alpha if item deleted cronbach's alpha 0.859 sun1 my relatives often use functional foods in their diet 10.29 5.930 .771 .793 sun2 my relatives claim that using functional foods brings benefits for health 10.21 6.654 .636 .853 sun3 my friends often use functional foods in their diet 10.53 6.806 .698 .823 sun4 my relatives often use functional foods 10.61 7.543 .767 .812 cronbach's alpha .782 rop1 functional foods play an important role for my health 11.18 6.553 .606 .729 rop2 i feel to have control over for my health 11.17 6.046 .507 .772 rop3 i feel to use more functional foods which bring more benefits for my health as compared to 5 years ago 11.10 5.449 .627 .708 rop4 functional foods can help me improve my emotion 11.24 5.510 .637 .703 cronbach's alpha .827 prk1 i clearly understand about positive effects of functional foods on my health 10.12 7.685 .733 .746 prk2 i highly appreciate my knowledge about functional foods 10.22 8.844 .582 .812 prk3 i know using functional foods can be likely help me improve my health 10.15 7.526 .678 .770 prk4 i know using functional foods can have side – effects 10.12 7.542 .631 .794 cronbach's alpha .835 pri1 functional foods are not so expensive compared to the enormous benefits they brings for my health 10.40 9.570 .674 .788 int. j. anal. appl. 18 (2) (2020) 224 pri2 i think that functional foods’ price which is higher than normal foods is appropriate with their benefits 10.24 8.763 .641 .810 pri3 i always compare the price of functional foods in different stores before i buy them 10.35 8.992 .814 .727 pri4 the high price of function foods is not the main reason for me to not buy it 10.23 10.813 .563 .834 cronbach's alpha .841 ati1 i like using functional foods in my diet 7.40 3.623 .781 .714 ati2 functional foods are suitable with my health protection demand 7.37 3.424 .677 .812 ati3 i will be glad to suggest other people using functional foods 7.49 3.664 .670 .814 cronbach's alpha .784 rei1 i experienced functional foods , and i intend to continue to use it 10.92 5.962 .566 .744 rei2 i intend to use more functional foods in the future 11.09 5.567 .603 .725 rei3 i plan to buy more types of functional foods 11.08 5.580 .618 .718 rei4 i will buy functional foods regularly 11.05 5.326 .582 .739 cronbach's alpha .846 tru1 functional foods are likely to have positive impacts on my health 14.37 9.933 .728 .798 tru2 functional foods help me take my personal health on my hands 14.59 9.717 .719 .798 tru3 functional foods are a convenient way of meeting daily nutrient requirements that i never meet with conventional diet 14.48 9.077 .667 .813 tru4 i can improve my personal health by using functional foods 14.37 9.693 .632 .821 tru6 safety levels of functional foods are tested carefully 14.56 10.463 .546 .842 after conducting reliability test, all independent variables meet the standard requirements of reliability will be retested by efa to test the correlation andamong factors. from the results of table below, it can be seen that the value of kmo is 0.824 (≥ 0.05) and the significance of bartlett’s test is 0.000 (< 0.05) demonstrates that extracting variables is acceptable andthere correlationbetween observed variables int. j. anal. appl. 18 (2) (2020) 225 table 4.20 kmo and bartlett's test kaiser-meyer-olkin measure of sampling adequacy. .833 bartlett's test of sphericity approx. chi-square 3512.001 df 378 sig. .000 total variance explained factor initial eigenvalues extraction sums of squared loadings rotation sums of squared loadingsa total % of variance cumulative % total % of variance cumulative % total 1 7.308 26.099 26.099 6.899 24.639 24.639 4.417 2 2.687 9.596 35.695 2.316 8.273 32.911 4.023 3 2.431 8.682 44.377 2.043 7.296 40.207 2.601 4 2.111 7.540 51.917 1.675 5.983 46.190 4.176 5 1.691 6.039 57.956 1.276 4.556 50.747 4.199 6 1.537 5.488 63.444 1.106 3.952 54.698 3.181 7 1.049 3.745 67.190 .729 2.603 57.302 4.716 8 .883 3.152 70.342 9 .778 2.780 73.122 10 .694 2.480 75.602 11 .671 2.396 77.998 12 .628 2.244 80.242 13 .580 2.070 82.311 14 .520 1.859 84.170 15 .471 1.681 85.851 16 .463 1.655 87.507 17 .426 1.522 89.028 18 .411 1.467 90.495 19 .381 1.361 91.856 20 .361 1.290 93.146 21 .330 1.177 94.323 22 .299 1.067 95.390 23 .289 1.034 96.423 24 .243 .869 97.292 25 .225 .804 98.096 26 .211 .755 98.850 27 .174 .621 99.471 28 .148 .529 100.000 extraction method: principal axis factoring. a. when factors are correlated, sums of squared loadings cannot be added to obtain a total variance. int. j. anal. appl. 18 (2) (2020) 226 table 4.21 pattern matrixa factor 1 2 3 4 5 6 7 tru1 .896 tru2 .866 tru4 .643 tru3 .624 tru6 .552 sun4 .897 sun3 .839 sun1 .773 sun2 .642 pri3 .957 pri1 .723 pri2 .702 pri4 .644 prk1 .896 prk3 .756 prk4 .704 prk2 .606 rei2 .804 rei3 .761 rei4 .601 rei1 .522 rop3 .765 rop1 .753 rop4 .737 rop2 .509 ati1 .988 ati3 .627 ati2 .600 extraction method: principal axis factoring. rotation method: promax with kaiser normalization. a. rotation converged in 6 iterations. from the total variance explained table above, 6 factors are extracted with the cumulative (%) of 57.302 percent. this illustrates that they can explicate around 57 percent of data variability. these observed variables have loading factors are greater than 0.5 which fulfill the conditions of convergent and discriminated validity. 3. cfa test after conducting efa, cfa is the next step which is employed to examine the measurement model before testing the relationship between factors by using the simultaneous equation model. int. j. anal. appl. 18 (2) (2020) 227 in specific, cfa is aimed to test the consistence degree of constructional measures and the researchers’ understanding of that construct. in cfa test, p-value, convergent validity, discriminant validity and reliability should be taken into account to verify the conformity of model figure 4.21 cfa test int. j. anal. appl. 18 (2) (2020) 228 testing the convergence and the reliability estimate ave cr tru1 <--trust .841 0.539 0.851 tru2 <--trust .836 tru4 <--trust .656 tru3 <--trust .720 tru6 <--trust .582 sun4 <--subjective_norms .885 0.628 0.869 sun3 <--subjective_norms .855 sun1 <--subjective_norms .768 sun2 <--subjective_norms .639 pri3 <--price .940 0.586 0.847 pri1 price .717 pri2 price .710 pri4 price .665 prk1 product_knowledge .832 0.555 0.832 prk3 product_knowledge .754 prk4 product_knowledge .712 prk2 product_knowledge .672 rei2 repurchase_intention .867 0.551 0.827 rei3 <--repurchase_intention .848 rei4 <--repurchase_intention .627 rei1 <--repurchase_intention .584 rop3 <--perceived_role .806 0.514 0.807 rop1 <--perceived_role .717 rop4 <--perceived_role .739 rop2 <--perceived_role .588 ati1 <--attitude .868 0.654 0.850 ati2 <--attitude .790 ati3 <--attitude .765 int. j. anal. appl. 18 (2) (2020) 229 measure results evaluation chi – square/df (cmin/df) 1.670 great cfi 0.934 great gfi 0.874 acceptable agfi 0.845 great rmsea(root mean squared error of approximation) 0.051 acceptable pclose 0.416 great tli 0.925 great from the result of model fit, all the indices is rather good. however, with the purpose achieving the best value of model fitness, the modification indices need to be considered to examine new whether covariance should be draw between two observed variables. the new covariance will be drawn when the modification indices are highest. m.i. par change e20 <--> e21 73.363 .429 e18 <--> e19 50.109 .297 as we can see in the modification indices above, there is a very high modification index between e20 and e21 as well as e8 and e9. therefore, the new covariance should be drawn between these errors. after drawing the new covariance, the model fit results is improved significantly. the gfi value increases to 0.894, the rmsea decrease to 0.037(<0.05) and the chi-square also decreases to 1.345. these indices are sufficient to satisfy with the conditions in the table above. in conclusion, the model fit of cfa is highly adaptable the data int. j. anal. appl. 18 (2) (2020) 230 int. j. anal. appl. 18 (2) (2020) 231 4. structural equation modeling (sem) sem is conducted with the purpose of testing the hypothesis relationships in the research model regression weights: (group number 1 default model) int. j. anal. appl. 18 (2) (2020) 232 . estimate s.e. c.r. p attitude <--trust .330 .073 4.512 *** attitude <--subjective_norms .449 .077 5.860 *** attitude <--price .063 .041 1.533 .125 attitude <--product_knowledge .158 .064 2.447 .014 attitude <--perceived_role .181 .067 2.718 .007 repurchasei ntention <--attitude .376 .060 6.231 *** from the table, the p-value of price and attitude is 0.131, which is greater than 0.05, so the relationship between price and attitude is insignificant. the rest factors which are subjective norms, trust, product knowledge and perceived role are significantly related to consumers’ attitude with the p value of less than 0.05. in addition, the relationship between consumer’s attitude and consumers’ repurchase intention is also significant with the p-value of *** (less than 0.05). therefore, there only has the hypothesis h2 will be rejected from the model int. j. anal. appl. 18 (2) (2020) 233 estimate s.e. c.r. p label attitude <--trust .330 .074 4.450 *** attitue <--subjective_norms .414 .075 5.556 *** attitude <--product_knowledge .170 .065 2.612 .009 attitude <--perceived_role .189 .068 2.798 .005 repurchase_intention <--attitude .374 .060 6.215 *** int. j. anal. appl. 18 (2) (2020) 234 standardized regression weights: (group number 1 default model) estimate attitude <--trust .292 attitude <--subjective_norms .347 attitude <--product_knowledge .185 attitude <--perceived_role .186 repurchase_intention <--attitude .597 as the figure above, it can be seen that all indices are satisfied with the sem criteria such as chisquare/df = 1.467 (< 2), gfi = 0.901 (>0.9), tli = 0.954 (>0.9), cfi= 0.960 (>0.9) and rmsea = 0.042 (< 0.08). the measurement model is totally fit the data. bootstrap testing parameter se se-se mean bias se-bias attitude <--trust .088 .002 .334 .004 .003 attitude <--subjective_norms .084 .002 .417 .003 .003 attitude <--product_knowledge .075 .002 .168 -.002 .002 attitude <--perceived_role .076 .002 .189 .000 .002 repurchase_intention <--attitude .083 .002 .369 -.005 .003 parameter cr attitude <--trust 1.333333 attitude <--subjective_norms 1 attitude <--product_knowledge -1 attitude <--perceived_role 0 repurchase_intention <--attitude -1.66667 int. j. anal. appl. 18 (2) (2020) 235 the cr value is less than 1.96, the research model is trustworthy hypothesis checking statements results h1 there is a positive relationship between consumer knowledge and consumers’ attitude accept h2 there is a positive relationship between price and consumers’ attitude reject h3 there is a positive relationship between subjective norms and consumers’ attitude accept h4 there is a positive relationship between trust and consumers’ attitude accept h5 there is a positive relationship between perceived role and consumers’ attitude accept h6 consumers’ attitude has positive relationship to consumers’ repurchase intention accept consumer knowledge subjective norms perceived role trust consumers’ attitude consumers’ repurchase intention int. j. anal. appl. 18 (2) (2020) 236 conclusions and recommendations 1. discussion of the findings the results of this research aimed to respond to research’s question put in the first chapter. the findings will be discussed and compared with the previous studies which are stated in the literature review part as following what kind of factors influence consumers’ attitude toward functional foods? in the research, there are five factors are proposed for the impacts on consumer’s attitude. however, after the analysis process, the results are shown as follow: (1) subjective norms: in this research, subjective norms positively influence to consumers’ attitude towards functional foods and it matches with the theory of reason action ([38]). this theory claims that purchase intention is the outcomes of consumers’ attitude towards a product or service and there is a close link between consumer’s attitude and subjective norms. it is easy to understand in asia culture, especially vietnamese culture because asian consumers often make decisions based on the opinion of other people and response to what society expected to them more than choosing products based on their individual preferences. therefore, people will have positive attitude towards functional foods when people around them such as relatives and friends highly appreciate the benefits of functional foods and recommends them use (2) trust: in this research, trust has profound influence on consumers’ attitude and it is totally satisfied with the exploration of previous research such as the “conceptual model of consumers’ willingness to eat functional food” ([39]). this paper states that trust relates to benefit perceived from using functional foods, and it is the strongest factors affecting to consumers’ willingness to buy these foods. however, it is inconsistent with the study “functional food choices: impacts of trust and health beliefs ([31]) about the level of influence on consumers’ attitude. in this study, authors emphasize that perceived role is carried heavier weight than trust. however, in this research trust are weighted more heavily than perceived role among the factors affecting to consumers’ attitude. the reason is that there are many functional foods are produced in vietnam have ingredients from natural ingredients, so the effects of these functional foods may need more time to exert effects on consumers’ health. although these natural products need longer time to exert effects, they are totally safe for people. therefore, with people who prefer int. j. anal. appl. 18 (2) (2020) 237 natural –origin ingredients will have positive attitude based on their trust towards these products. (3) consumer knowledge: this factor is proposed based on the study “consumer acceptance of functional foods: socio demographic, cognitive and attitudinal determinants. ([20]). in this study, the authors conclude that there is a difference in knowledge about functional foods across gender and education, and knowledge positively impacts on consumers’ acceptance of functional foods. the results of this research are also in line with the investigation of verbeke about the influence level of knowledge on consumers’ acceptance. the more knowledge about functional foods people have, the more positive attitude they are. (4) perceived role: from the analysis process, this study examines that role perceived significantly impacts on consumers’ attitude, which is similar to the results of the research “functional food choices: impacts of trust and health beliefs ([31]). the author states that both trust and health perceived are the key factors affecting to consumer’ attitude. the paper consumers’ acceptance of functional foods in ho chi minh city ([37]) also complete agree with the important role of functional foods on consumers’ acceptance in ho chi minh city (5) price: price is not perceived as the influential factors towards functional foods in this research. it appears to contradict the study “consumer acceptance of functional foods: socio demographic, cognitive and attitudinal determinants” of verbeke (2005) [20]. this study indicates that one of the main barriers for consumers’ attitude and intention relates to very high price. the reason is that there have large number of variety functional foods in the market, and functional foods’ prices are much competitive with each other and affordable for all segmentations. another reason is that thanks to the development of technology, functional foods producers can produce high quality products with low cost, so it is not difficult to find a kind of quality functional foods without charging a high price. moreover, other factors such as subjective norms, trust, consumer knowledge and perceived role have a great impact on consumers’ attitude and dominate the factor of price. when people think about a kind of functional foods, they will consider about these factors and make decision rather than its price. therefore,price of functional foods doesn’t play an important role in consumers’ attitude. how does consumers’ attitude impact on consumers’ repurchase intention toward functional foods product? int. j. anal. appl. 18 (2) (2020) 238 consumers’ attitude greatly impact on consumers’ behavior based on the theory of planned behavior ([9]). from the research process, the author states that the consumers’ behaviors towards a product are the consequence of consumers’ attitude to this product. therefore, this research based on this theory andinvestigate the relationship between consumers’ attitude and repurchase intention. there are many previous research examine the relationship between consumers’ attitude and purchase intention. for instance, the research “consumer attitude and purchase intention towards organic food – a case study in china” conducted by mingyan yang, sarah al-shaaban and tram b. nguyen (2014) [8]” examined that attitude have positive influence on consumer’s purchase intention towards functional foods. the results of this study are in line with the previous research; however, this research explores the impacts of attitude on repurchase intention. moreover,with the purpose of making it appropriate with vietnamese market and functional foods market, some factors that affecting to consumers’ attitude are changed. recommendations build consumers’ trust by completing the legal framework for manufacturing and trading functional foods. nowadays, one of the biggest challenges of the functional food industry is the lack of a clearly legal framework for production and trading, quality standards as well as how to manage and punish illegal activities. more and more negative phenomena with increasing levels are happening in the functional food market such as inaccurate or exaggerated information in the packaging, products that have not been examined and recognized by authorities, over advertising, counterfeit goods. this leads to lose consumers’ trust towards functional foods. consequently, this situation has seriously affected to the development of potential functional foods market in vietnam. therefore, businesses operating in this area need to coordinate with the authorities to develop legal system, standards, regulations and management regulations to protect their rights and benefits of businesses, step by step creating a solid foundation to maintain and rebuild consumers’ trust. this is not only a right but also an obligation of businesses to supply safe and beneficial products for consumers enhance consumer awareness by emphasizing the message "functional food is a nutritious food ". not only experts like doctors but also consumers are increasingly aware of the close relationship between nutrition and health status ([29]). in order to meet the necessary nutrient demand, as well as provide nutrients to enhance health and prevent disease, besides normal foods, int. j. anal. appl. 18 (2) (2020) 239 functional foods can be a good choice for bringing convenience and efficiency to consumers, especially those who are busy and pressured. therefore, businesses need to send a clear message in their communication plan that functional food is a kind of food with added, enhanced nutrients to provide essential nutrients, strengthen resistance, and reduce the risk of disease. developing long-term plans aimed at introducing functional food to improve consumer knowledge. knowledge of functional foods also has a significant impact on consumer acceptance. the more consumers understand about functional components as well as the effects of those components on health, the more they are likely to use functional foods. therefore, businesses need to develop specific plans to spread the knowledge of functional foods to consumers. the implementation of the plan needs to be flexible, diverse, regular, easy to understand, attractive, abundant and also need to coordinate with other ways to improve knowledge of consumers by organizing workshop, tv programs with the participants of doctors or nutritionist, contest about nutrition. 5.4 limitations the first limitation of the research is the research sample. the questionnaires mainly are spread in district 1 and district 3 and there are only 260 valid responses. therefore, the research’s results can’t reflect the entire population in ho chi minh city the second is about the factors including in the research. besides factors from the studying, there may have many other factors that can impact on consumers’ attitude and repurchase intention. because of the limitation about finance and time period, the research can’t examine all factors. acknowledgements the author would like to thank ms. nguyen ngoc tuong vy from school of business, international university – vietnam national university, hcmc for her editorial assistance. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (2) (2020) 240 references [1] nguyen, n. t. optimizing factors for accuracy of forecasting models in food processing industry: a context of cacao manufacturers in vietnam. ind. eng. manage. syst. 18(4)(2019), 808-824. [2] nguyen, n. t., & tran, t. t. raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl. 31(11)(2019), 7963-7974. [3] nguyen, n. t., & nguyen, l. x. t. applying dea model to measure the efficiency of hospitality sector: the case of vietnam. int. j. anal. appl. 17(6) (2019), 994-1018. [4] nguyen, n. t., & tran, t. t. mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dyn. nat. soc. 2015 (2015), art. id 858157. [5] nguyen, n. t., & tran, t. t. facilitating an advanced product layout to prioritize hot lots in 450 mm wafer foundry in the semiconductor industry. int. j. adv. appl. sci. 3(6)(2016), 14-23. [6] nguyen, n. t., & tran, t. t. a study of the strategic alliance for vietnam domestic pharmaceutical industry: a dynamic integration of a hybrid dea and gm (1, 1) approach. j.grey syst. 30 (4) (2018), 134-151. [7] nguyen, n. t., & tran, t. t. a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9)(2018), 73-81. [8] yang, m., al-shaaban, s., & nguyen, t. b. consumer attitude and purchase intention towards organic food : a quantitative study of china (dissertation) (2014). retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-34944 [9] ajzen i., the theory of planned behavior, org. behavior human decision proc. 50 (1991), 179-211. [10] nguyen, n. t., tran, t. t., wang, c. n., & nguyen, n. t. optimization of strategic alliances by integrating dea and grey model. j. grey syst. 27(1)(2015), 38-56. [11] tran, t. t. evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci. 3(12)(2016), 5-20. [12] tran, t. t. an empirical research on selecting the targeted suppliers and purchasing process of supermarket. int. j. adv. appl. sci. 4(4) (2017), 96-109. [13] tran, t. t. forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci. 4(9)(2017), 86-95. [14] wang, c. n., nguyen, n. t., & tran, t. t. integrated dea models and grey system theory to evaluate past-to-future performance: a case of indian electricity industry.sci. world j. 2015(2015).art. id 638710. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-34944 int. j. anal. appl. 18 (2) (2020) 241 [15] fornell, c., a national customer satisfaction barometer: the swedish experience. j. of market., 56(1) (1992), 6-21. [16] jackson, s. e., & schuler, r. s., a meta-analysis and conceptual critique of research on role ambiguity and role conflict in work settings. org. behavior human decision proc., 36(1) (1985), 16-78. [17] ferrell, o. c., & pride, w. m., marketing: concepts and strategies. houghton mifflin (1991). [18] eagly, a. h., & chaiken, s., the advantages of an inclusive definition of attitude. social cognition, 25(5) (2007), 582-602. [19] siegrist, m, sampfli, n and kastenholz, h. consumers’ willingness to buy functional foods.the influence of carrier, benefit and trust.appetite,51(2008), 526-529. [20] verbeke, w. consumer acceptance of functional foods: socio-demographic, cognitive and attitudinal determinants. food qual. prefer.16 (1) (2005), 45-57. [21] aarts, h and dijksterhuis, a., the silence of the library: environment, situational norm,and social behaviour. j. person. soc. psychol.84 (1) (2003), 18-28. [22] fogliano, v and vitaglione, p. functional foods: planning and development. mol.nutritionfood res.49 (3) (2005), 256-262. [23] siró, i, kapolna, e, kapolna, b and lugasi, a.functional food.product development,marketing and consumer acceptance a review. appetite, 51 (3) (2008), 456-467. [24] ares, g., besio, m., giménez, a., & deliza, r., relationship between involvement and functional milk desserts intention to purchase. influence on attitude towards packaging characteristics. appetite, 55(2) (2010), 298-304. [25] annunziata, a., & vecchio, r., consumer perception of functional foods: a conjoint analysis with probiotics. food quality and preference, 28(1) (2013), 348-355. [26] childs, n. m., foods that help prevent disease: consumer attitudes and public policy implications. j. of cons. marketing, 14(6) (1997), pp. 433-447. https://doi.org/10.1108/07363769710186015 [27] perkins, n. d., post-translational modifications regulating the activity and function of the nuclear factor kappa b pathway. oncogene, 25(51) (2006), 6717-6730. int. j. anal. appl. 18 (2) (2020) 242 [28] promotosh, b., & sajedul, i., young consumers’ purchase intentions of buying green products. a study based on the theory of planned behavior. umea school of business, spring semester (2011). [29] menrad, k. market and marketing of functional food in europe.j. food eng.56 (2) (2003), 181-188. [30] shepherd, r.social determinants of food choice.proceedings of the nutritionsociety, 58(04)(1999), 807-812. [31] ding, y., veeman, m. m., & adamowicz, w. l. functional food choices: impacts of trust and health control beliefs on canadian consumers’ choices of canola oil. food policy, 52(2015), 92-98. [32] moorman, c., deshpande, r., & zaltman, g. (1993). factors affecting trust in market research relationships. journal of marketing, 57(1), 81-101. [33] urala, n and lähteenmäki, l. attitudes behind consumer’s willingness to use functional foods.food qual. prefer. 15(2004),793–803. [34] niva, m., ‘all foods affect health’: understandings of functional foods and healthy eating among health-oriented finns. appetite, 48(3) (2007), 384-393. [35] hilliam, m. functional foods: the western consumer view point. nutrition rev.54 (11) (1996), 189194. [36] bech-larsen, t., & grunert, k. g., the perceived healthiness of functional foods: a conjoint study of danish, finnish and american consumers' perception of functional foods. appetite, 40(1) (2003), 9-14. [37] nguyen, n.t. performance evaluation in strategic alliances: a case of vietnamese construction industry. glob. j. flex. syst. manage. 21(1) (2020), 85-99. [38] lähteenmäki, l. consumers’ changing attitudes towards functional foods. food qual. prefer.18 (1) (2007), 1-12. [39] babicz-zielinska, e., & jezewska-zychowicz, m., conceptual model of consumer’s willingness to eat functional foods. roczniki państwowego zakładu higieny, 68(1) (2017). int. j. anal. appl. (2023), 21:11 on global existence of the fractional reaction-diffusion system’s solution iqbal m. batiha1,2,∗, nabila barrouk3, adel ouannas4, waseem g. alshanti1 1department of mathematics, al-zaytoonah university of jordan, amman 11733, jordan 2nonlinear dynamics research center (ndrc), ajman university, ajman, uae 3department of mathematics and informatics, mohamed cherif messaadia university, souk ahras, algeria 4department of mathematics and computer science, university of larbi ben m’hidi, oum el bouaghi, algeria ∗corresponding author: i.batiha@zuj.edu.jo abstract. the purpose of this paper is to prove the global existence of solution for one of most significant fractional partial differential system called the fractional reaction-diffusion system. this will be carried out by combining the compact semigroup methods with some l1-estimate methods. our investigation can be applied to a wide class of fractional partial differential equations even if they contain nonlinear terms in their constructions. 1. introduction in this paper, we intend to study the following nonlinear parabolic system:{ ∂u ∂t −d1 (−∆)α u = f (u,v) , in ]0, +∞[ × ω, ∂v ∂t −d2 (−∆)β v = g (u,v) , in ]0, +∞[ × ω, (1.1) subject to the following boundary conditions: ∂u ∂η = ∂v ∂η = 0, or u = v = 0, in ]0, +∞[ ×∂ω, (1.2) and the initial data: u (0, ·) = u0 (·) , v (0, ·) = v0 (·) , in ω, (1.3) received: nov. 25, 2022. 2020 mathematics subject classification. 35a01, 35k57, 34a08. key words and phrases. semigroup methods; fractional reaction-diffusion systems; local mild solution; global solution. https://doi.org/10.28924/2291-8639-21-2023-11 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-11 2 int. j. anal. appl. (2023), 21:11 where ω is a regular and bounded domain of rn, (n ≥ 1), with its boundary ∂ω, u = u (t,x), v = v (t,x) are two real-valued functions such that x ∈ ω and t > 0, and where (−∆)δ is a non local operator that accounts for the anomalous diffusion [1,2] so that 0 < δ < 1, (δ = α or β), and d1, d2 are two constants of diffusion assumed to be nonnegative, whereas f and g are two functions in which they "enough regular". it should be furthermore mentioned that the functions u (0, ·) and v (0, ·) are assumed to be continuous and nonnegative. besides, the local existence of the solution (u,v) in times is classical, and moreover it is not negative if u0 and v0 are so. it is worth mentioned that system (1.1)-(1.3) arises in many field of applied science such as physics, chemistry and various biological processes including population dynamics and others, see [3] and references therein. in this regard and in order to make system (1.1)-(1.3) more reality, we assume that the following hypothesis: • the initial data u0 and v0 are nonnegative functions such that: u0,v0 ∈ l1 (ω) . (1.4) • the two functions f and g are a quasi-positives functions, i.e., f (0,v) ≥ 0, g (u, 0) ≥ 0, ∀u,v ≥ 0. (1.5) • it exists a nonnegative constant c independent of (ξ1,ξ2) such that: f (ξ1,ξ2) + g (ξ1,ξ2) ≤ c (ξ1 + ξ2) ,∀(ξ1,ξ2) ∈r2+. (1.6) • in addition, we have: f (ξ1,ξ2) ≤ c (ξ1 + ξ2) ,∀(ξ1,ξ2) ∈r2+. (1.7) the main question we want to address here is the existence of global solution for system (1.1)(1.3). in fact, the subject of the global existence of fractional reaction-diffusion systems has received a lot of attention in the last decades and several outstanding results have been proved by some of the major experts in the field, see [4–10]. in the same context, replacing the anomalous diffusion operator by the standard laplacian operator (−∆) was firstly studied in one-dimensional space. this notion has been investigated by many authors by considering certain special forms of the nonlinear terms f and g. in particular, alikakos showed in [11] the existence of global bounded solutions whenever f (u,v) = −g (u,v) = −uvσ, for 1 < σ < n + 2 n . the extension of this result for σ > 1 is studied later by masuda [12]. following that, haraux and youkana generalized the result of masuda via the functional of lyapunov in reference [13]. actually, they performed their generalization by putting f (u,v) = g (u,v) = −uψ (v), where ψ is a nonlinear function satisfying the condition: lim v→+∞ [log (1 + ψ (v))] v = 0. in the same regard, barabanova generalized the result of haraux and youkana in [14] by concerning with the global existence of nonnegative solutions of a reaction-diffusion equation with exponential int. j. anal. appl. (2023), 21:11 3 nonlinearity. lately, it has been shown that there is also another very powerful method relying on compact semigroups could be used for examining the global existence of solutions for a reactiondiffusion equation [15–19]. for a better understanding, we send the reader to the works of moumeni and barrouk [20,21]. later on, a more general model was studied by haraux and kirane [22]. they took different diffusion coefficients for the two equations and for the general nonlinear terms. they proved the existence of global bounded solutions and investigated their asymptotic behavior. equally, hnaien et al. proved in reference [23] the existence of a local mild solution, global existence solution and asymptotic behavior of solutions for the system (1.1)-(1.3) when f (u,v) = −λuv and g (u,v) = λuv −µv. the remainder of this paper is organized as follows. in section 2, we present some definitions and preliminaries. in section 3, we provide some results related to the compactness of a proposed operator. in section 4, we prove the existence of a local mild solution, positivity and global existence of solution for particular system. finally, the global existence of solutions for system (1.1)-(1.3) are studied in section 5, followed by section 6 that abbreviates the work. 2. preliminaries in this part, some preliminaries and overview of the local existence and global existence of solution for fractional reaction-diffusion system are illustrated. this will pave the way to introduce our findings later on. definition 2.1. let f (u,v) ∈ x, where x is a banach space. the function f is locally lipschitz if for all t1 ≥ 0 and all constant k > 0, there exist a constant l (k,t1) > 0 such that: ‖f (u1,v1) −f (u2,v2)‖≤ l |(u1,v1) − (u2,v2)| , is satisfied ∀(u1,v1) , (u2,v2) ∈ r×r with |(u1,v1)| ≤ k, |(u2,v2)| ≤ k and t ∈ [0,t1] such that t > 0. lemma 2.1. let a be m-dissipative operator in the banach space x and s (t) be a semigroup engendered by a. let f be a function locally lipschitz. then, for all u0 ∈ x, there exists tmax = t (u0) such that the system:   u ∈ c ([0,t ] ,d (a)) ∩c1 ([0,t ] ,x) , du dt −au = f (u (s)) , u (0) = u0 (2.1) admits a unique local solution u verifying u (t) = s (t) u0 + ∫ t 0 s (t − s) f (u (s)) ds, ∀t ∈ [0,tmax] . next, some further preliminaries associated with the existence of global solution for the fractional reaction-diffusion system (1.1)-(1.3) will be recalled. 4 int. j. anal. appl. (2023), 21:11 theorem 2.1. [5] consider the following classical boundary-eigenvalue system for the fractional power of the laplacian in ω with homogeneous neumann boundary condition:  (−∆)α ϕk = λαk ϕk, in ω, ∂ϕk ∂η = 0, on ω, where ω is an open bounded domain in rn and d ( (−∆)α ) = { u ∈ l2 (ω) , ∂u ∂η = 0, ∥∥(−∆)α u∥∥ l2(ω) < +∞ } , such that: ∥∥(−∆)α u∥∥2 l2(ω) = ∑+∞ k=1 |λαk 〈u,ϕk〉| 2 . then this system has a countable system of eigenvalues of the laplacian operator in l2 (ω) with homogeneous neumann boundary condition in which 0 < c ≤ λ1 < λ2 < ... < λj < ... so that λj →∞ as j →∞, and ϕk is the corresponding eigenvectors for k = 1, 2, . . . , +∞. thus, based on what we have mentioned above, we can infer that, for u ∈ d ( (−∆)α ) , we have: (−∆)α u = ∑+∞ k=1 λαk 〈u,ϕk〉ϕk. in addition, with using integration by parts, we can have the following formula:∫ ω u (x) (−∆)α v (x) dx = ∫ ω v (x) (−∆)α u (x) dx, (2.2) for u,v ∈ d ( (−∆)α ) . lemma 2.2. let θ ∈ c∞0 (q) such that θ ≥ 0. then, there is a nonnegative function φ ∈ c 1,2 (q) that represents a solution of the system:  −φt −d∆φ = θ on q, φ (t,x) = 0 on (0,t ) ×∂ω, φ (t,x) = 0 on ω. (2.3) actually, in accordance with ladyzenskaya and solonnikov in [24], we observe that system (2.3) possesses a unique nonnegative solution. moreover, for all q ∈ ]1, +∞[, we note that there exists a nonnegative constant c independent of θ such that: ‖φ‖lp(q) ≤ c ‖θ‖lq(q) . besides, for all ω0 ∈ l1 (ω) and h ∈ l1 (q), we can have the following equalities:∫ q (s (t) ω0 (x)) θdxdt = ∫ ω ω0 (x) φ (0,x) dx, (2.4) and ∫ q (∫ t 0 s (t − s) h (s,x) ds ) θdxdt = ∫ q h (s,x) φ (s,x) dxds. (2.5) int. j. anal. appl. (2023), 21:11 5 3. compactness of operator in this section, we will provide a result connected with a compactness of operator l that define the solution of system (2.1) in the case where the initial value equals zero (u (0) = 0), i.e., l (f ) (t) = u (t) = ∫ t 0 s (t − s) f (u (s)) ds, ∀t ∈ [0,t ] . from this point of view, we will first recall the dunford–pettis theorem, which can be found with its proof in [25]. this will help us to derive our first result in this work. theorem 3.1 (dunford–pettis). let z be a bounded set in l1 (ω). then z has compact closure in the weak topology σ ( l1,l∞ ) if and only if z is equiintegrable, i.e., (a) { ∀ε > 0,∃δ > 0, such that ∫ a |f | < ε, ∀a ⊂ ω, measurable with |a| < δ, ∀f ∈z } , (b) { ∀ε > 0, ∃ω ⊂ ω, measurable with |ω| < ∞,such that ∫ ω\ω |f | < ε, ∀f ∈z } . theorem 3.2. if for all t > 0, the operator s (t) is compact, then l is compact in l1 ([0,t ] ,x). proof. the proof of this result consists of two steps. step 1: we show that s (λ) l : f → s (λ) l (f ) is compact in l1 ([0,t ] ,x), i.e., we show that the set {s (λ) l (f ) (t) : ‖f‖1 ≤ 1} is relatively compact in l 1 ([0,t ] ,x) , ∀t ∈ [0,t ]. to this aim, we notice that due to s (t) is compact, then the operator t → s (t) is continuous over ]0, +∞[ in l(x). therefore, we have: ∀ε > 0,∀δ > 0, ∃η > 0. ∀0 ≤ h ≤ η, ∀t ≥ δ, ‖s (t + h) −s (t)‖l(x) ≤ ε. now, if one chooses λ = δ, we obtain: s (λ)u (t + h) −s (λ) u (t) = ∫ t+h 0 s (λ + t + h− s) f (u (s)) ds − ∫ t 0 s (λ + t − s) f (u (s)) ds = ∫ t+h t s (λ + t + h− s) f (u (s)) ds + ∫ t 0 (s (λ + t + h− s) −s (λ + t − s)) f (u (s)) ds, for 0 ≤ t ≤ t −h. consequently, based on the inequality: ‖s (λ) u (t + h) −s (λ) u (t)‖x ≤ ∫ t+h t ‖f (u (s))‖x ds + ε ∫ t 0 ‖f (u (s))‖x ds, we can define v (t) by: v (t) = { u (t) if 0 ≤ t ≤ t, 0 otherwise . therefore, we have: ‖s (λ) v (t + h) −s (λ) v (t)‖1 ≤ (h + εt )‖f (u (s))‖1 , 6 int. j. anal. appl. (2023), 21:11 which implies that {s (λ) v : ‖f‖1 ≤ 1} is equiintegrable. hence, we infer {s (λ) l (f ) (t) : ‖f‖1 ≤ 1} is relatively compact in l 1 ([0,t ] ,x), and so s (λ) l is compact. step 2: we show that s (λ) l converges towards l when λ goes towards 0 in l1 ([0,t ] ,x). for this purpose, we observe: s (λ) u (t) −u (t) = ∫ t 0 s (λ + t − s) f (u (s)) ds − ∫ t 0 s (t − s) f (u (s)) ds. so, for t ≥ δ, we can have: ‖s (λ) u (t) −u (t)‖≤ ∫ t δ ‖s (λ + s) −s (s)‖l(x) ‖f (u (s))‖ds + 2 ∫ t t−δ ‖f (u (s))‖ds. immediately, if we choose 0 < λ < η, we get: ‖s (λ) u (t) −u (t)‖≤ ε ∫ t δ ‖f (u (s))‖ds + 2 ∫ t t−δ ‖f (u (s))‖ds. besides, for 0 ≤ t < δ, we can have: ‖s (λ) u (t) −u (t)‖≤ 2 ∫ t 0 ‖f (u (s))‖ds. as f ∈ l1 (0,t,x), we gain: ‖s (λ) u (t) −u (t)‖≤ (εt + 2δ)‖f (u (s))‖1 . thus, as λ → 0, then s (λ) u → u in l1 ([0,t ] ,x), where the operator l is a uniform limit with compact linear operator between two banach spaces, which confirms that l is compact in l1 ([0,t ] ,x). � remark 3.1. the semigroup s (t) generated by the operator d (−∆)δ is compact in l1 (ω). 4. study of a particular system this section is divided into three subsections so that the first one aims to deals with the local existence of solution for a first-order system derived from system (1.1)-(1.3), then the positivity of such solution will be discussed, followed by exploring the global existence of the solution of the derived system. thus, in order to achieve this objective, we first convert system (1.1)-(1.3) to an abstract first-order system in the banach space x = l1 (ω)×l1 (ω). to this aim, we define the functions un0 and vn0 by: un0 = min (u0,n) , and vn0 = min (v0,n) , for all n > 0. it is clear that un0 and vn0 verify (1.4), i.e., un0,vn0 ∈ l 1 (ω) and un0 ≥ 0, vn0 ≥ 0. int. j. anal. appl. (2023), 21:11 7 thus, based on the previous assumptions, we can formulate the first-order system derived from system (1.1)-(1.3) as:   ∂wn ∂t −awn = f (wn) in [0,t [ × ω, ∂wn ∂η = 0, or wn = 0 in [0,t [ ×∂ω, wn (0, ·) = wn0 (·) in ω. (4.1) 4.1. local existence of solution for system (4.1). in this subsection, we intend to discuss the local existence of solution for system (4.1). in this connection, we let wn = (un,vn), wn0 = (un0,vn0 ) and f = (f ,g). besides, we suppose a is an operator defined as: a = ( d1 (−∆)α 0 0 d2 (−∆)β ) , where d (a) := { wn ∈ l1 (ω) ×l1 (ω) : ( (−∆)α un, (−∆)β vn ) ∈ l1 (ω) ×l1 (ω) } . in view of the above assumptions, system (4.1) can be returned to the shape of system (2.1). thus, if (un,vn) is a solution of system (4.1), then it verifies the following integral equations:  un (t) = s1 (t) un0 + ∫ t 0 s1 (t − s) f (un (s) ,vn (s)) ds, vn (t) = s2 (t) vn0 + ∫ t 0 s2 (t − s) g (un (s) ,vn (s)) ds, (4.2) where s1 (t) and s2 (t) are the semigroups of contractions in l1 (ω) generated by the operator d1 (−∆)α and d2 (−∆)β. theorem 4.1. there exists tm > 0 such that (un,vn) is a local solution of (4.1), for all t ∈ [0,tm]. proof. due to s1 (t) and s2 (t) are semigroups of contraction and as f is locally lipschitz for 0 ≤ un0,vn0 ≤ n, then ∃tm > 0 such that (un,vn) is a local solution of system (4.1) on [0,tm]. � theorem 4.2. let un0,vn0 ∈ l 1 (ω), then there exists a maximal time tmax > 0 and a unique mild solution (un,vn) ∈ c ( [0,tmax) ,l 1 (ω) ×l1 (ω) ) of system (4.1) subject to either tmax = +∞, or tmax < +∞ and lim t→tmax (‖un (t)‖∞ + ‖vn (t)‖∞) = +∞. proof. for arbitrary t > 0, we define the banach space as: et := { (un,vn) ∈ c ( [0,t ] ,l1 (ω) ×l1 (ω) ) : ‖(un,vn)‖≤ 2‖(un0,vn0 )‖ = r } , where ‖·‖∞ := ‖·‖l∞(ω) and ‖·‖ is the norm of et defined by: ‖(un,vn)‖ := ‖un‖l∞([0,t ],l∞(ω)) + ‖vn‖l∞([0,t ],l∞(ω)) . 8 int. j. anal. appl. (2023), 21:11 next, for every (un,vn) ∈ et , we define ψ (un,vn) := (ψ1 (un,vn) , ψ2 (un,vn)) as: ψ1 (un,vn) = s1 (t) un0 + ∫ t 0 s1 (t − s) f (un (s) ,vn (s)) ds, ψ2 (un,vn) = s2 (t) vn0 + ∫ t 0 s2 (t − s) g (un (s) ,vn (s)) ds, for t ∈ [0,t ]. now, we will prove the local existence of solution for the considered system by the banach fixed point theorem. to this aim, we let ψ : et → et and (un,vn) ∈ et . this leads to infer the inequality: ‖ψ1 (un,vn)‖∞ ≤ ‖un0‖∞ + c (‖un‖∞ + ‖vn‖∞) t ≤ ‖un0‖∞ + c ( ‖un0‖∞ + ‖vn0‖∞ ) t, (by maximum principle). similarly, we have: ‖ψ2 (un,vn)‖∞ ≤‖vn0‖∞ + c ( ‖un0‖∞ + ‖vn0‖∞ ) t. this immediately implies: ‖ψ (un,vn)‖ ≤ ‖un0‖∞ + ‖vn0‖∞ + 2c ( ‖un0‖∞ + ‖vn0‖∞ ) t ≤ 2 ( ‖un0‖∞ + ‖vn0‖∞ ) , by choosing t such that t ≤ ‖un0‖∞ + ‖vn0‖∞ cr . therefore, we gain ψ (un,vn) ∈ et for t ≤ ‖un0‖∞+‖vn0‖∞ cr . now, to complete the proof, we need to show that ψ is a contraction map. in this regard, we have: ‖ψ1 (un,vn) − ψ1 (ũn, ṽn)‖∞ ≤ l ∫ t 0 ‖(un,vn) − (ũn, ṽn)‖∞dτ ≤ lt (‖ṽn −vn‖∞ + ‖ũn −un‖∞) , for (un,vn) , (ũn, ṽn) ∈ et . similarly, we obtain: ‖ψ2 (un,vn) − ψ2 (ũn, ṽn)‖∞ ≤ lt (‖ṽn −vn‖∞ + ‖ũn −un‖∞) . actually, the above two estimates imply: ‖ψ (un,vn) − ψ (ũn, ṽn)‖∞ ≤ 2lt (‖ṽn −vn‖∞ + ‖ũn −un‖∞) ≤ 1 2 ‖(un,vn) − (ũn, ṽn)‖ , where t ≤ max ( ‖un0‖∞+‖vn0‖∞ cr , 1 4l ) . this exactly shows the contraction result. hence, by the banach fixed point theorem, system (4.1) admits a unique mild solution (un,vn) ∈ et . in general, this solution can be extended on a maximal interval [0,tmax) where tmax := sup{t > 0 : (un,vn) is a solution to (4.1) in et} . � int. j. anal. appl. (2023), 21:11 9 however, with the aim of showing the global existence of solution for the system at hand, we need the fact that such solution should be positive, and this what we aim to address in the next subsection. 4.2. positivity of solution for system (4.1). in what follow, we intend to prove the positivity of solution for system (4.1). this would help us to discuss the global existence of such solution. in this respect, we introduce next another result. lemma 4.1. let (un,vn) be the solution of system (4.1) such that: un0 (x) ≥ 0, vn0 (x) ≥ 0, x ∈ ω. then un (t,x) ≥ 0 and vn (t,x) ≥ 0, ∀(t,x) ∈ ]0,t [ × ω. proof. let ūn (t,x) = 0 in ]0,t [ × ω =⇒ ∂ūn ∂t = 0 and (−∆)α ūn = 0. then, according to the hypothesis (1.5), we can have: ∂un ∂t −d1 (−∆)α un − f (un,vn) = 0 ≥ ∂ūn ∂t −d1 (−∆)α ūn − f (ūn,vn) , and un (0,x) = un0 (x) ≥ 0 = ūn (0,x) . hence, by the comparison theorem, we obtain: un (t,x) ≥ ūn (t,x) , which implies un (t,x) ≥ 0. in a similar manner, we can gain vn (t,x) ≥ 0, and this completes the proof. � 4.3. global existence of solution for system (4.1). to prove the global existence of the solution of system (4.1) for all nonnegative t, it is enough, according to routh [26], to find an estimate of the solution for all t ≥ 0 in l1 (ω). in this regard, we introduce the next lemma. lemma 4.2. let (un,vn) be the solution of system (4.1), then there exists m (t), which depends only on t, such that for all 0 ≤ t ≤ tm, we have: ‖un + vn‖l1(ω) ≤ m (t) . based on this estimate, we confirm that the solution (un,vn) given by theorem 4.1 is a global solution. proof. first of all, it should be noted that we can write system (4.1) in the form:  ∂un ∂t −d1 (−∆)α un = f (un,vn) , in [0,t [ × ω, ∂vn ∂t −d2 (−∆)β vn = g (un,vn) , in [0,t [ × ω, ∂un ∂η = ∂vn ∂η = 0, or un = vn = 0, in [0,t [ ×∂ω, un (0,x) = un0 (x) , vn (0,x) = vn0 (x) , in ω. (4.3) 10 int. j. anal. appl. (2023), 21:11 with the use of the first and second equations of system (4.3), we can obtain: ∂ ∂t (un + vn) −d1 (−∆)α un −d2 (−∆)β vn = f (un,vn) + g (un,vn) . by taking into account assumption (1.6), we have: ∂ ∂t (un + vn) −d1 (−∆)α un −d2 (−∆)β vn ≤ c (un + vn) . now, let us integrate the above inequality over ω, and then use the integration by parts performed on formula (2.2) to get ∫ ω (−∆)α un (x) dx = 0 and ∫ ω (−∆)β vn (x) dx = 0. this would gives: ∫ ω ∂ ∂t (un + vn) dx ≤ c ∫ ω (un + vn) dx or ∂ ∂t ∫ ω (un + vn) dx∫ ω (un + vn) dx ≤ c. by integrate the above inequality over [0,t], we get: ln ∫ ω (un + vn) dx ∣∣∣∣t 0 ≤ ct or ln ∫ ω (un + vn) dx∫ ω (un0 + vn0 ) dx ≤ ct, which implies: ∫ ω (un + vn) dx∫ ω (un0 + vn0 ) dx ≤ exp (ct) , i.e., ⇒ ∫ ω (un + vn) dx ≤ exp (ct) ∫ ω (un0 + vn0 ) dx ⇒ ∫ ω (un + vn) dx ≤ exp (ct) ∫ ω (u0 + v0) dx, as if un0 ≤ u0, vn0 ≤ v0. now, let us assume that m (t) = exp (ct)‖u0 + v0‖l1(ω). then, due to un and vn are positives, we gain: ‖un + vn‖l1(ω) ≤ m (t) , 0 ≤ t ≤ tm, which completes the proof. � in the following content, we provide a further result that aims to show the existence of the solution’s estimate (un,vn) for system (4.1) in l1 (q). lemma 4.3. for any solution (un,vn) of system (4.1), there is a constant k (t), which depends only on t, such that: ‖un + vn‖l1(q) ≤ k (t)‖u0 + v0‖l1(ω) . int. j. anal. appl. (2023), 21:11 11 proof. in order to prove this result, we multiply the first equation of system (4.2) by θ, and then integrate the result over q. accordingly, by using (2.4) and (2.5), we obtain:∫ q unθdxdt = ∫ q s1 (t) un0 (x) θdxdt + ∫ q (∫ t 0 s1 (t − s) f (un,vn) ds ) θdxdt = ∫ ω un0 (x) φ (0,x) dx + ∫ q f (un,vn) φ (s,x) dxds. moreover, we find:∫ q vnθdxdt = ∫ ω vn0 (x) φ (0,x) dx + ∫ q g (un,vn) φ (s,x) dxds. this yields:∫ q (un + vn) θdxdt = ∫ ω (un0 (x) + vn0 (x)) φ (0,x) dx + ∫ q (f (un,vn) + g (un,vn)) φ (s,x) dxds ≤ ∫ ω (u0 (x) + v0 (x)) φ (0,x) dx + ∫ q c (un + vn) φ (s,x) dxds. consequently, using holder inequality implies:∫ q (un + vn) θdxdt ≤ ‖u0 + v0‖l1(ω) .‖φ (0,x)‖l∞(q) + c‖un + vn‖l1(q) .‖φ‖l∞(q) ≤ ( ‖u0 + v0‖l1(ω) + c‖un + vn‖l1(q) ) .‖φ‖l∞(q) ≤ max (1,c) ( ‖u0 + v0‖l1(ω) + ‖un + vn‖l1(q) ) .‖φ‖l∞(q) ≤ k1 (t) ( ‖u0 + v0‖l1(ω) + ‖un + vn‖l1(q) ) .‖θ‖l∞(q) , where k1 (t) ≥ max (c,cc). now, since θ is arbitrary in c∞0 (q), then we have: ‖un + vn‖l1(q) ≤ k1 () ( ‖u0 + v0‖l1(ω) + ‖un + vn‖l1(q) ) . thus, by taking k (t) = k1(t) 1−k1(t) , we obtain: ‖un + vn‖l1(q) ≤ k (t)‖u0 + v0‖l1(ω) , which finishes the proof. � 5. global existence of solution for system (1.1)-(1.3) in this section, we will provide one of the main results of this work. in particular, with the help of using the four assumptions (1.4)-(1.7), we will explore the global existence of solution for system (1.1)-(1.3). 12 int. j. anal. appl. (2023), 21:11 theorem 5.1. suppose that the hypotheses (1.4)-(1.7) are satisfied. then there exists a solution (u,v) of system (1.1)-(1.3) of the form:  u (t) = s1 (t) u0 + ∫ t 0 s1 (t − s) f (u (s) ,v (s)) ds, ∀t ∈ [0,t [ , v (t) = s2 (t) v0 + ∫ t 0 s2 (t − s) g (u (s) ,v (s)) ds, ∀t ∈ [0,t [ , (5.1) where u,v ∈ c ( [0, +∞[ ,l1 (ω) ) , f (u,v) ,g (u,v) ∈ l1 (q) such that q = (0,t ) ×ω for all t > 0, and where s1 (t) and s2 (t) are the semigroups of contractions in l1 (ω) generated by d1 (−∆)α and d2 (−∆)β. proof. to prove this result, we define the operator l by: l : (w0,h) → sd (t) w0 + ∫ t 0 sd (t − s) h (s) ds, where sd (t) is the semigroup of contraction generated by the operator d (−∆) δ. according to theorem 3.2 and as sd (t) is compact, then the operator l is an adding of two compact operators in l1 (q), and so l is compact in l1 (q) ×l1 (q). therefore, there exists a subsequence ( unj,vnj ) of (un,vn) such that ( unj,vnj ) converges towards (u,v) in l1 (q) × l1 (q). let us now show that( unj,vnj ) is a solution of system (4.2), i.e.,  unj (t,x) = s1 (t) un0 + ∫ t 0 s1 (t − s) f ( unj (s) ,vnj (s) ) ds, vnj (t,x) = s2 (t) vn0 + ∫ t 0 s2 (t − s) g ( unj (s) ,vnj (s) ) ds, (5.2) that is, it is enough to show that (u,v) verifies (5.1). in this regard, it should be clearly noted that if j → +∞, then we gain un0 → u0 and vn0 → v0, and so f ( unj,vnj ) → f (u,v) , g ( unj,vnj ) → g (u,v) a.e. (5.3) thus to show that (u,v) verifies (5.1), it remains to show that: f ( unj,vnj ) → f (u,v) , g ( unj,vnj ) → g (u,v) , in l1 (q) when j → +∞. to this aim, we integrate the equations of system (4.1) over q coupled with take (2.2) into account to obtain: −d1 ∫ q (−∆)α unjdxdt = 0, −d2 ∫ q (−∆)β vnjdxdt = 0. consequently, we have: ∫ ω unjdx − ∫ ω un0dx = ∫ q f ( unj,vnj ) dxdt,∫ ω vnjdx − ∫ ω vn0dx = ∫ q g ( unj,vnj ) dxdt, int. j. anal. appl. (2023), 21:11 13 such that: − ∫ q f ( unj,vnj ) dxdt ≤ ∫ ω u0dx, (5.4) and − ∫ q g ( unj,vnj ) dxdt ≤ ∫ ω v0dx. (5.5) now, let us assume: nn = c ( unj + vnj ) − f ( unj,vnj ) , mn = c ( unj + vnj ) − f ( unj,vnj ) −g ( unj,vnj ) . as a result, it is clear that, according to the two assumptions (1.6) and (1.7) that can be respectively used in of (5.4) and (5.5), the terms nn and mn are positives. this would lead to the following assertions: ∫ q nndxdt ≤ c ∫ q ( unj + vnj ) dxdt + ∫ ω u0dx,∫ q mndxdt ≤ c ∫ q ( unj + vnj ) dxdt + ∫ ω (u0 + v0) dx. consequently, lemma 4.3 implies:∫ q nndxdt < +∞, ∫ q mndxdt < +∞, which immediately gives:∫ q ∣∣f (unj,vnj)∣∣dxdt ≤ c ∫ q ( unj + vnj ) dxdt + ∫ q nndxdt < +∞, and ∫ q ∣∣g(unj,vnj)∣∣dxdt ≤ c ∫ q ( unj + vnj ) dxdt + ∫ q mndxdt < +∞. now, we assume hn = nn + c ( unj + vnj ) and ψn = mn + c ( unj + vnj ) . clearly, one can observe that hn and ψn are positives in l1 (q), and∣∣f (unj,vnj)∣∣ ≤ hn a.e, and ∣∣g(unj,vnj)∣∣ ≤ ψn a.e. let us now combine this result with (5.3), and then apply the convergence theorem dominated by lebesgue to obtain: f ( unj,vnj ) → f (u,v) g ( unj,vnj ) → g (u,v) in l1 (q) . by passing in the limit j → +∞ of (5.2) in l1 (q), we obtain:  u (t) = s1 (t) u0 + ∫ t 0 s1 (t − s) f (u (s) ,v (s)) ds, v (t) = s2 (t) v0 + ∫ t 0 s2 (t − s) g (u (s) ,v (s)) ds. hence, (u,v) satisfies (5.1), and consequently (u,v) is the solution of system (1.1)-(1.3). � 14 int. j. anal. appl. (2023), 21:11 6. conclusions in this paper, the global existence of solution for the fractional reaction-diffusion system has been discussed and proved as well. the compact semigroup methods and some l1-estimates have been utilized for this purpose. several theoretical results have been consequently inferred and derived. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. ilic, f. liu, i. turner, v. anh, numerical approximation of a fractional-in-space diffusion equation (ii) – with nonhomogeneous boundary conditions, fract. calc. appl. anal. 9 (2006), 333-349. http://eudml.org/doc/ 11286. [2] g. karch, nonlinear evolution equations with anomalous diffusion, in: qualitative properties of solutions to partial differential equations. jindrich necas center for mathematical modelling lecture notes, vol. 5 (matfyzpress, prague, 2009), pp. 25–68. [3] r.a. fisher, the wave of advance of advantageous genes, ann. eugenics. 7 (1937), 355–369. https://doi.org/ 10.1111/j.1469-1809.1937.tb02153.x. [4] s. bonafede, d. schmitt, "triangular" reaction–diffusion systems with integrable initial data, nonlinear anal.: theory methods appl. 33 (1998), 785–801. https://doi.org/10.1016/s0362-546x(98)00042-x. [5] t. diagana, some remarks on some strongly coupled reaction-diffusion equations, (2003). https://doi.org/ 10.48550/arxiv.math/0305152. [6] s. kouachi, a. youkana, global existence for a class of reaction-diffusion systems, bull. polish. acad. sci. math. 49 (2001), 1-6. [7] t.e. oussaeif, b. antara, a. ouannas, et al. existence and uniqueness of the solution for an inverse problem of a fractional diffusion equation with integral condition, j. funct. spaces. 2022 (2022), 7667370. https: //doi.org/10.1155/2022/7667370. [8] a. ouannas, f. mesdoui, s. momani, et al. synchronization of fitzhugh-nagumo reaction-diffusion systems via one-dimensional linear control law, arch. control sci. 31 (2021), 333–345. https://doi.org/10.24425/acs. 2021.137421. [9] i.m. batiha, a. ouannas, r. albadarneh, et al. existence and uniqueness of solutions for generalized sturm–liouville and langevin equations via caputo–hadamard fractional-order operator, eng. comput. 39 (2022), 2581–2603. https://doi.org/10.1108/ec-07-2021-0393. [10] z. chebana, t.e. oussaeif, a. ouannas, et al. solvability of dirichlet problem for a fractional partial differential equation by using energy inequality and faedo-galerkin method, innov. j. math. 1 (2022), 34–44. https://doi. org/10.55059/ijm.2022.1.1/4. [11] n.d. alikakos, lp-bounds of solutions of reaction-diffusion equations, commun. part. differ. equ. 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113. [12] k. masuda, on the global existence and asymptotic behavior of solutions of reaction-diffusion equations, hokkaido math. j. 12 (1983), 360-370. https://doi.org/10.14492/hokmj/1470081012. [13] a. haraux, a. youkana, on a result of k. masuda concerning reaction-diffusion equations, tohoku math. j. (2). 40 (1988), 159-163. https://doi.org/10.2748/tmj/1178228084. [14] a. barabanova, on the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity, proc. amer. math. soc. 122 (1994), 827-831. http://eudml.org/doc/11286 http://eudml.org/doc/11286 https://doi.org/10.1111/j.1469-1809.1937.tb02153.x https://doi.org/10.1111/j.1469-1809.1937.tb02153.x https://doi.org/10.1016/s0362-546x(98)00042-x https://doi.org/10.48550/arxiv.math/0305152 https://doi.org/10.48550/arxiv.math/0305152 https://doi.org/10.1155/2022/7667370 https://doi.org/10.1155/2022/7667370 https://doi.org/10.24425/acs.2021.137421 https://doi.org/10.24425/acs.2021.137421 https://doi.org/10.1108/ec-07-2021-0393 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.1080/03605307908820113 https://doi.org/10.14492/hokmj/1470081012 https://doi.org/10.2748/tmj/1178228084 int. j. anal. appl. (2023), 21:11 15 [15] t.e. oussaeif, b. antara, a. ouannas, et al. existence and uniqueness of the solution for an inverse problem of a fractional diffusion equation with integral condition, j. funct. spaces. 2022 (2022), 7667370. https: //doi.org/10.1155/2022/7667370. [16] i.m. batiha, z. chebana, t.e. oussaeif, et al. on a weak solution of a fractional-order temporal equation, math. stat. 10 (2022), 1116–1120. https://doi.org/10.13189/ms.2022.100522. [17] n. anakira, z. chebana, t.-e. oussaeif, i.m. batiha, a. ouannas, a study of a weak solution of a diffusion problem for a temporal fractional differential equation, nonlinear funct. anal. appl. 27 (2022), 679–689. https: //doi.org/10.22771/nfaa.2022.27.03.14. [18] i.m. batiha, solvability of the solution of superlinear hyperbolic dirichlet problem, int. j. anal. appl. 20 (2022), 62. https://doi.org/10.28924/2291-8639-20-2022-62. [19] m. bezziou, z. dahmani, i. jebril, et al. solvability for a differential system of duffing type via caputo-hadamard approach, appl. math. inform. sci. 11 (2022), 341–352. https://doi.org/10.18576/amis/160222. [20] a. moumeni, n. barrouk, existence of global solutions for systems of reaction-diffusion with compact result, int. j. pure appl. math. 102 (2015), 169-186. https://doi.org/10.12732/ijpam.v102i2.1. [21] a. moumeni, n. barrouk, triangular reaction diffusion systems with compact result, glob. j. pure appl. math. 11 (2015), 4729-4747. [22] a. haraux, m. kirane, estimations c1 pour des problèmes paraboliques semi-lineaires, ann. fac. sci. toulouse math. 5 (1983), 265-280. http://www.numdam.org/item?id=afst_1983_5_5_3-4_265_0. [23] d. hnaien, f. kellil, r. lassoued, asymptotic behavior of global solutions of an anomalous diffusion system, j. math. anal. appl. 421 (2015), 1519–1530. https://doi.org/10.1016/j.jmaa.2014.07.083. [24] o.a. ladyzenskaya, v.a. solonnikov, n.n. uralceva, linear and quasilinear equations of parabolic type, american mathematical society, providence, ri (1968). [25] n. dunford, j.t. schwartz, linear operators, 3 volume set, interscience, (1972). [26] f. rothe, global existence of reaction-diffusion systems, lecture notes in mathematics, 1072, springer, berlin, (1984). https://doi.org/10.1155/2022/7667370 https://doi.org/10.1155/2022/7667370 https://doi.org/10.13189/ms.2022.100522 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.28924/2291-8639-20-2022-62 https://doi.org/10.18576/amis/160222 https://doi.org/10.12732/ijpam.v102i2.1 http://www.numdam.org/item?id=afst_1983_5_5_3-4_265_0 https://doi.org/10.1016/j.jmaa.2014.07.083 1. introduction 2. preliminaries 3. compactness of operator 4. study of a particular system 4.1. local existence of solution for system (4.1) 4.2. positivity of solution for system (4.1) 4.3. global existence of solution for system (4.1) 5. global existence of solution for system (1.1)-(1.3) 6. conclusions references international journal of analysis and applications volume 17, number 1 (2019), 64-75 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-64 l2-uncertainty principle for the weinstein-multiplier operators ahmed saoudi1,2,∗ and imen ali kallel1 1northern border university, college of science, p.o. box 1231, arar 91431, saudi arabia 2tunis el manar university, faculty of science of tunis, campus universitaire, 2092, tunisia ∗corresponding author: ahmed.saoudi@ipeim.rnu.tn abstract. the aim of this paper is establish the heisenberg-pauli-weyl uncertainty principle and donhostark’s uncertainty principle for the weinstein l2-multiplier operators. 1. introduction the weinstein operator ∆dw,α defined on r d+1 + = r d × (0,∞), by ∆dw,α = d+1∑ j=1 ∂2 ∂x2j + 2α + 1 xd+1 ∂ ∂xd+1 = ∆d + lα, α > −1/2, where ∆d is the laplacian operator for the d first variables and lα is the bessel operator for the last variable defined on (0,∞) by lαu = ∂2u ∂x2d+1 + 2α + 1 xd+1 ∂u ∂xd+1 . the weinstein operator ∆dw,α has several applications in pure and applied mathematics, especially in fluid mechanics [4]. received 2018-09-14; accepted 2018-10-26; published 2019-01-04. 2010 mathematics subject classification. 43a32, 44a15. key words and phrases. weinstein operator; l2-multiplier operators; heisenberg-pauli-weyl uncertainty principle; donhostark’s uncertainty principle. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 64 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-64 int. j. anal. appl. 17 (1) (2019) 65 the weinstein transform generalizing the usual fourier transform, is given for ϕ ∈ l1α(r d+1 + ) and λ ∈ rd+1+ , by fw,α(ϕ)(λ) = ∫ rd+1 + ϕ(x)λdα(x,λ)dµα(x), where dµα(x) is the measure on rd+1+ = r d × (0,∞) and λdα is the weinstein kernel given respectively later by (2.1) and (2.4). let m be a function in l2α(r d+1 + ) and let σ be a positive real number. the weinstein l 2-multiplier operators is defined for smooth functions ϕ on rd+1+ , in [14] as tw,m,σϕ(x) := f−1w,α (mσfw,α(ϕ)) (x), x ∈ r d+1 + , (1.1) where the function mσ is given by mσ(x) = m(σx). these operators are a generalization of the multiplier operators tm associated with a bounded function m and given by tm(ϕ) = f−1(mf(ϕ)), where f(ϕ) denotes the ordinary fourier transform on rn. these operators gained the interest of several mathematicians and they were generalized in many settings in [1, 3, 6, 13, 14, 16–18]. in this work we are interested the l2 uncertainty principles for the weinstein multiplier operators. the uncertainty principles play an important role in harmonic analysis. these principles state that a function ϕ and its fourier transform f(ϕ) cannot be simultaneously sharply localized. many aspects of such principles are studied for several fourier transforms. many uncertainty principles have already been proved for the weinstein transform fw,α, namely by n. ben salem, a. r. nasr [2] and mejjaoli h. and salhi m. [9]. the authors have established in [9] the heisenberg-pauli-weyl inequality for the weinstein transform, by showing that, for every ϕ in l2α(r d+1 + ) ‖ϕ‖α,2 ≤ 2 2α + d + 2 ‖|x|ϕ‖α,2‖|y|fw,α(ϕ)‖α,2. (1.2) in the present paper we are interested in proving an analogue of heisenberg-pauli-weyl uncertainty principle for the operators tw,m,σ. more precisely, we will show, for ϕ ∈ l2α(r d+1 + ) ‖ϕ‖α,2 ≤ 2‖|y|fw,α(ϕ)‖α,2 2α + d + 2 (∫ rd+1 + ∫ ∞ 0 |x|2|tw,m,σϕ(x)|2α,2 dσ σ dµα(x) )1 2 , provided m be a function in l2α(r d+1 + ) satisfying the admissibility condition∫ ∞ 0 |mσ(x)| dσ σ = 1, a.e. x ∈ rd+1+ . (1.3) moreover, for β,δ ∈ [1,∞) and ε ∈ r, such that βε = (1 −ε)δ, we will show ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βtw,m,σϕ∥∥εα,2 ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 . int. j. anal. appl. 17 (1) (2019) 66 using the techniques of donoho and stark [5], we show uncertainty principle of concentration type for the l2 theory. let ϕ be a function in l2α(r d+1 + ) and m ∈ l1α(r d+1 + ) ∩l2α(r d+1 + ) satisfying the admissibility condition (1.3). if ϕ is �-concentrated on ω and tw,m,σϕ is ν-concentrated on σ, then ‖m‖α,1 (µα(ω)) 1 2 (∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) )1 2 ≥ 1 − (� + ν), where θα is the measure on (0,∞) ×rd+1+ given by dθα(σ,x) := (dσ/σ)dαµ(x). this paper is organized as follows. in section 2, we recall some basic harmonic analysis results related with the weinstein operator ∆dw,α and we introduce preliminary facts that will be used later. in section 3, we establish heisenberg-pauli-weyl uncertainty principle for the operators tw,m,σ. the last section of this paper is devoted to donoho-stark’s uncertainty principle for the weinstein l2multiplier operators. 2. harmonic analysis associated with the weinstein operator in this section, we shall collect some results and definitions from the theory of the harmonic analysis associated with the weinstein operator ∆dw,α. main references are [10–12]. in the following we denote by • rd+1+ = rd × (0,∞). • x = (x1, ...,xd,xd+1) = (x′,xd+1). • −x = (−x′,xd+1). • c∗(rd+1), the space of continuous functions on rd+1, even with respect to the last variable. • s∗(rd+1), the space of the c∞ functions, even with respect to the last variable, and rapidly decreasing together with their derivatives. • lpα(r d+1 + ), 1 ≤ p ≤∞, the space of measurable functions f on r d+1 + such that ‖f‖α,p = (∫ rd+1 + |f(x)|p dµα(x) )1/p < ∞, p ∈ [1,∞), ‖f‖α,∞ = ess sup x∈rd+1 + |f(x)| < ∞, where dµα(x) = x2α+1d+1 (2π)d22αγ2(α + 1) dx. (2.1) • aα(rd+1) = { ϕ ∈ l1α(r d+1 + ); fw,αϕ ∈ l1α(r d+1 + ) } the wiener algebra space. we consider the weinstein operator ∆dw,α defined on r d+1 + by ∆dw,α = d+1∑ j=1 ∂2 ∂x2j + 2α + 1 xd+1 ∂ ∂xd+1 = ∆d + lα, α > −1/2, (2.2) int. j. anal. appl. 17 (1) (2019) 67 where ∆d is the laplacian operator for the d first variables and lα is the bessel operator for the last variable defined on (0,∞) by lαu = ∂2u ∂x2d+1 + 2α + 1 xd+1 ∂u ∂xd+1 . the weinstein operator ∆dw,α have remarkable applications in diffrerent branches of mathematics. for instance, they play a role in fluid mechanics [4]. 2.1. the eigenfunction of the weinstein operator. for all λ = (λ1, ...,λd+1) ∈ cd+1, the system ∂2u ∂x2j (x) = −λ2ju(x), if 1 ≤ j ≤ d lαu(x) = −λ2d+1u(x), u(0) = 1, ∂u ∂xd+1 (0) = 0, ∂u ∂xj (0) = −iλj, if 1 ≤ j ≤ d (2.3) has a unique solution denoted by λdα(λ,.), and given by λdα(λ,x) = e −i<x′,λ′>jα(xd+1λd+1) (2.4) where x = (x′,xd+1), λ = (λ ′,λd+1) and jα is is the normalized bessel function of index α defined by jα(x) = γ(α + 1) ∞∑ k=0 (−1)kx2k 2kk!γ(α + k + 1) . the function (λ,x) 7→ λdα(λ,x) has a unique extension to cd+1×cd+1, and satisfied the following properties. proposition 2.1. i). for all (λ,x) ∈ cd+1 ×cd+1 we have λdα(λ,x) = λ d α(x,λ). (2.5) ii). for all (λ,x) ∈ cd+1 ×cd+1 we have λdα(λ,−x) = λ d α(−λ,x). (2.6) iii). for all (λ,x) ∈ cd+1 ×cd+1 we get λdα(λ, 0) = 1. (2.7) vi). for all ν ∈ nd+1, x ∈ rd+1 and λ ∈ cd+1 we have ∣∣dνλλdα(λ,x)∣∣ ≤‖x‖|ν|e‖x‖‖=λ‖ (2.8) where dνλ = ∂ ν/(∂λν11 ...∂λ νd+1 d+1 ) and |ν| = ν1 + ... + νd+1. in particular, for all (λ,x) ∈ r d+1 ×rd+1, we have ∣∣λdα(λ,x)∣∣ ≤ 1. (2.9) int. j. anal. appl. 17 (1) (2019) 68 2.2. the weinstein transform. definition 2.1. the weinstein transform is given for ϕ ∈ l1α(r d+1 + ) by fw,α(ϕ)(λ) = ∫ rd+1 + ϕ(x)λdα(λ,x)dµα(x), λ ∈ r d+1 + , (2.10) where µα is the measure on rd+1+ given by the relation (2.1). some basic properties of this transform are as follows. for the proofs, we refer [11, 12]. proposition 2.2. (1) for all ϕ ∈ l1α(r d+1 + ), the function fw,α(ϕ) is continuous on r d+1 + and we have ‖fw,αϕ‖α,∞ ≤‖ϕ‖α,1 . (2.11) (2) the weinstein transform is a topological isomorphism from s∗(rd+1+ ) onto itself. the inverse transform is given by f−1w,αϕ(λ) = fw,αϕ(−λ), for all λ ∈ r d+1 + . (2.12) (3) parseval formula: for all ϕ,φ ∈s∗(rd+1+ ), we have∫ rd+1 + ϕ(x)φ(x)dµα(x) = ∫ rd+1 + fw,α(ϕ)(x)fw,α(φ)(x)dµα(x). (2.13) (4) plancherel formula: for all ϕ ∈s∗(rd+1+ ), we have ‖fw,αϕ‖α,2 = ‖ϕ‖α,2 . (2.14) (5) plancherel theorem: the weinstein transform fw,α extends uniquely to an isometric isomorphism on l2α(r d+1 + ). (6) inversion formula: let ϕ ∈ l1α(r d+1 + ) such that fw,αϕ ∈ l1α(r d+1 + ), then we have ϕ(λ) = ∫ rd+1 + fw,αϕ(x)λdα(−λ,x)dµα(x), a.e. λ ∈ r d+1 + . (2.15) 2.3. the translation operator associated with the weinstein operator. definition 2.2. the translation operator ταx , x ∈ r d+1 + associated with the weinstein operator ∆ d w,α, is defined for a continuous function ϕ on rd+1+ which is even with respect to the last variable and for all y ∈ rd+1+ by ταx ϕ(y) = cα ∫ π 0 ϕ ( x′ + y′, √ x2d+1 + y 2 d+1 + 2xd+1yd+1 cos θ ) (sin θ) 2α dθ, with cα = γ(α + 1) √ πγ(α + 1/2) . int. j. anal. appl. 17 (1) (2019) 69 by using the weinstein kernel, we can also define a generalized translation, for a function ϕ ∈ s∗(rd+1) and y ∈ rd+1+ the generalized translation ταx ϕ is defined by the following relation fw,α(ταx ϕ)(y) = λ d α(x,y)fw,α(ϕ)(y). (2.16) the following proposition summarizes some properties of the weinstein translation operator. proposition 2.3. the translation operator ταx , x ∈ r d+1 + satisfies the following properties. i). for ϕ ∈ c∗(rd+1), we have for all x,y ∈ rd+1+ ταx ϕ(y) = τ α y ϕ(x) and τ α 0 ϕ = ϕ. ii). let ϕ ∈ lpα(r d+1 + ), 1 ≤ p ≤∞ and x ∈ r d+1 + . then τ α x ϕ belongs to l p α(r d+1 + ) and we have ‖ταx ϕ‖α,p ≤‖ϕ‖α,p . (2.17) note that the aα(rd+1+ ) is contained in the intersection of l1α(r d+1 + ) and l ∞ α (r d+1 + ) and hence is a subspace of l2α(r d+1 + ). for ϕ ∈aα(r d+1 + ) we have ταx ϕ(y) = cα,d ∫ rd+1 + λdα(x,z)λ d α(−y,z)fw,αϕ(z)dµα(z). (2.18) by using the generalized translation, we define the generalized convolution product ϕ∗w ψ of the functions ϕ, ψ ∈ l1α(r d+1 + ) as follows ϕ∗w ψ(x) = ∫ rd+1 + ταx ϕ(−y)ψ(y)dµα(y). (2.19) this convolution is commutative and associative, and it satisfies the following properties. proposition 2.4. i) for all ϕ,ψ ∈ l1α(r d+1 + ), (resp. ϕ,ψ ∈ s∗(r d+1 + )), then ϕ ∗w ψ ∈ l1α(r d+1 + ), (resp. ϕ∗w ψ ∈s∗(rd+1+ )) and we have fw,α(ϕ∗w ψ) = fw,α(ϕ)fw,α(ψ). (2.20) ii) let p,q,r ∈ [1,∞], such that 1 p + 1 q − 1 r = 1. then for all ϕ ∈ lpα(r d+1 + ) and ψ ∈ lqα(r d+1 + ) the function ϕ∗w ψ belongs to lrα(r d+1 + ) and we have ‖ϕ∗w ψ‖α,r ≤‖ϕ‖α,p‖ψ‖α,q . (2.21) iii) let ϕ,ψ ∈ l2α(r d+1 + ). then ϕ∗w ψ = f−1w,α (fw,α(ϕ)fw,α(ψ)) . (2.22) iv) let ϕ,ψ ∈ l2α(r d+1 + ). then ϕ ∗w ψ belongs to l2α(r d+1 + ) if and only if fw,α(ϕ)fw,α(ψ) belongs to l2α(r d+1 + ) and we have fw,α(ϕ∗w ψ) = fw,α(ϕ)fw,α(ψ). (2.23) int. j. anal. appl. 17 (1) (2019) 70 v) let ϕ,ψ ∈ l2α(r d+1 + ). then ‖ϕ∗w ψ)‖α,2 = ‖fw,α(ϕ)fw,α(ψ)‖α,2, (2.24) where both sides are finite or infinite. 3. heisenberg-pauli-weyl uncertainty principle in this section we establish heisenberg-pauli-weyl uncertainty principle for the operator tw,m,σ. theorem 3.1. let m be a function in l2α(r d+1 + ) satisfying the admissibility condition (1.3). then, for ϕ ∈ l2α(r d+1 + ), we have ‖ϕ‖α,2 ≤ 2‖|y|fw,α(ϕ)‖α,2 2α + d + 2 (∫ rd+1 + ∫ ∞ 0 |x|2|tw,m,σϕ(x)|2α,2 dσ σ dµα(x) )1 2 . (3.1) proof. let ϕ ∈ l2α(r d+1 + ). the inequality (3.1) holds if ‖|y|fw,α(ϕ)‖α,2 = +∞ or ∫ rd+1 + ∫ ∞ 0 |x|2|tw,m,σϕ(x)|2α,2 dσ σ dµα(x) = +∞. let us now assume that ‖|y|fw,α(ϕ)‖α,2 + ∫ rd+1 + ∫ ∞ 0 |x|2|tw,m,σϕ(x)|2α,2 dσ σ dµα(x) < +∞. inequality (1.2) leads to ∫ rd+1 + |tw,m,σϕ(x)|2α,2dµα(x) < (∫ rd+1 + |x|2|tw,m,σϕ(x)|2α,2dµα(x) )1 2 × (∫ rd+1 + |y|2|fw,α(tw,m,σϕ(.))(y)|2α,2dµα(y) )1 2 . integrating with respect to dσ/σ, we get ‖tw,m,σϕ)‖2α,2 < ∫ ∞ 0 (∫ rd+1 + |x|2|tw,m,σϕ(x)|2α,2dµα(x) )1 2 × (∫ rd+1 + |y|2|fw,α(tw,m,σϕ(.))(y)|2α,2dµα(y) )1 2 dσ σ . from [14, theorem 2.3] and schwarz’s inequality, we obtain int. j. anal. appl. 17 (1) (2019) 71 ‖ϕ‖2α,2 < (∫ ∞ 0 ∫ rd+1 + |x|2|tw,m,σϕ(x)|2α,2dµα(x) dσ σ )1 2 × (∫ ∞ 0 ∫ rd+1 + |y|2|fw,α(tw,m,σϕ(.))(y)|2α,2dµα(y) dσ σ )1 2 . from (1.1), fubini-tonnelli’s theorem and the admissibility condition (1.3), we have ∫ ∞ 0 ∫ rd+1 + |y|2|fw,α(tw,m,σϕ(.))(y)|2α,2dµα(y) dσ σ = ∫ ∞ 0 ∫ rd+1 + |y|2|mσ(y)|2|fw,α(ϕ)(y)|2α,2dµα(y) dσ σ = ∫ rd+1 + |y|2|fw,α(ϕ)(y)|2α,2dµα(y). this gives the result and completes the proof of the theorem. � theorem 3.2. let m be a function in l2α(r d+1 + ) satisfying the admissibility condition (1.3) and β,δ ∈ [1,∞). let ε ∈ r, such that βε = (1 −ε)δ then, for ϕ ∈ l2α(r d+1 + ), we have ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βtw,m,σϕ∥∥εα,2 ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 . (3.2) proof. let ϕ ∈ l2α(r d+1 + ). the inequality (3.1) holds if ∥∥|x|βtw,m,σϕ∥∥εα,2 = +∞ or ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 = +∞. let us now assume that ϕ ∈ l2α(r d+1 + ) with ϕ 6= 0 such that ∥∥|x|βtw,m,σϕ∥∥εα,2 + ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 < +∞, therefore, for all δ > 1, we have ∥∥|x|βtw,m,σϕ∥∥ 1βα,2 ‖tw,m,σϕ‖ 1β′α,2 = ∥∥∥|x|2|tw,m,σϕ|2β ∥∥∥12α,β ∥∥∥|tw,m,σϕ| 2β′ ∥∥∥12 α,β′ , with β′ = β β−1. applying the hölder’s inequality, we get ‖|x|tw,m,σϕ‖α,2 ≤ ∥∥|x|βtw,m,σϕ∥∥ 1βα,2 ‖tw,m,σϕ‖ 1β′α,2 . according to [14, theorem 2.3], we have for all β ≥ 1 ‖|x|tw,m,σϕ‖α,2 ≤ ∥∥|x|βtw,m,σϕ∥∥ 1βα,2 ‖ϕ‖ 1β′α,2 , (3.3) int. j. anal. appl. 17 (1) (2019) 72 with equality if β = 1. in the same manner, for all δ ≥ and using plancherel formula (2.14), we get ‖|y|fw,α(ϕ)‖α,2 ≤ ∥∥|y|δfw,α(ϕ)∥∥1δα,2 ‖ϕ‖ 1δ′α,2 , (3.4) with equality if δ = 1. by using the fact that βε = (1−ε)δ and according to inequalities (3.3) and (3.4), we have  ‖|x|tw,m,σϕ‖α,2 ‖|y|fw,α(ϕ)‖α,2 ‖ϕ‖ 1 β′ + 1 δ′ α,2  βδ ≤ ∥∥|x|βtw,m,σϕ∥∥εα,2 ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 , with equality if β = δ = 1. next by theorem 3.1, we obtain ‖ϕ‖α,2 ≤ ( 2 2α + d + 2 )βε ∥∥|x|βtw,m,σϕ∥∥εα,2 ∥∥|y|δfw,α(ϕ)∥∥1−εα,2 , which completes the proof of the theorem. � 4. donoho-stark’s uncertainty principle definition 4.1. (i) let ω be a measurable subset of rd+1+ , we say that the function ϕ ∈ l2α(r d+1 + ) is �-concentrated on ω, if ‖ϕ−χωϕ‖α,2 ≤ �‖ϕ‖α,2, (4.1) where χω is the indicator function of the set ω. (ii) let σ be a measurable subset of (0,∞) × rd+1+ and let ϕ ∈ l2α(r d+1 + ). we say that tw,m,σϕ is ν-concentrated on σ, if ‖tw,m,σϕ−χσtw,m,σϕ‖2,α ≤ ν‖tw,m,σ‖2,α, (4.2) where χς is the indicator function of the set σ. we need the following lemma for the proof of donoho-stark’s uncertainty principle. lemma 4.1. let m,ϕ ∈ l1α(r d+1 + ) ∩ l2α(r d+1 + ). then the operators tw,m,σ satisfy the following integral representation. tw,m,σ = 1 σ2α+d+2 ∫ rd+1 + ψα(x,y)ϕ(y)dµα(y), (σ,x) ∈ (0,∞) ×rd+1+ , where ψα(x,y) = ∫ rd+1 + mσ(z)λ d α(λ,x)λ d α(λ,−y)dµα(z). proof. the result follows from the definition of the weinstein l2-multiplier operators (1.1) and the inversion formula of the weinstein transform (2.12) using fubini-tonnelli’s theorem. � int. j. anal. appl. 17 (1) (2019) 73 theorem 4.1. let ϕ be a function in l2α(r d+1 + ) and m ∈ l1α(r d+1 + )∩l2α(r d+1 + ) satisfying the admissibility condition (1.3). if ϕ is �-concentrated on ω and tw,m,σϕ is ν-concentrated on σ, then ‖m‖α,1 (µα(ω)) 1 2 (∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) )1 2 ≥ 1 − (� + ν), where θα is the measure on (0,∞) ×rd+1+ given by dθα(σ,x) := (dσ/σ)dαµ(x). proof. let ϕ be a function in l2α(r d+1 + ). assume that 0 < µα(ω) < ∞ and∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) < ∞. according to [14, theorem 2.3] and inequalities (4.1)-(4.2), we get ‖tw,m,σϕ−χσtw,m,σ(χωϕ)‖2,α ≤ ‖tw,m,σϕ−χσtw,m,σϕ‖2,α +‖tw,m,σϕ−χσtw,m,σ(ϕ−χωϕ)‖2,α ≤ ‖tw,m,σϕ−χσtw,m,σ(ϕ−χωϕ)‖2,α +ν‖tw,m,σϕ‖2,α ≤ (� + ν)‖ϕ‖2,α. by triangle inequality it follows that ‖tw,m,σϕ‖2,α ≤ ‖tw,m,σϕ−χσtw,m,σ(χωϕ)‖2,α + ‖χσtw,m,σ(χωϕ)‖2,α ≤ (� + ν)‖ϕ‖2,α + ‖χσtw,m,σ(χωϕ)‖2,α. (4.3) on the other hand, we have ‖χσtw,m,σ(χωϕ)‖2,α = (∫ ∫ σ |tw,m,σ(χωϕ)(x)|2dθα(σ,x) )1 2 and moreover m,χωϕ ∈ l1α(r d+1 + ) ∩l2α(r d+1 + ), then by lemma 4.1, we obtain |tw,m,σ(χωϕ)(x)| ≤ 1 σ2α+d+2 ‖m‖1,α‖ϕ‖2,α(µα(ω)) 1 2 . therefore, thus ‖χσtw,m,σ(χωϕ)‖2,α ≤ ‖m‖1,α‖ϕ‖2,α(µα(ω)) 1 2 × (∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) )1 2 . hence, according to last inequality and (4.3) ‖tw,m,σ(ϕ)‖2,α ≤ ‖m‖1,α‖ϕ‖2,α(µα(ω)) 1 2 × (∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) )1 2 + (� + ν)‖ϕ‖2,α. int. j. anal. appl. 17 (1) (2019) 74 applying plancherel formula [14, theorem 2.3], we obtain ‖m‖α,1 (µα(ω)) 1 2 (∫ ∫ σ 1 σ2(2α+d+2) dθα(σ,x) )1 2 ≥ 1 − (� + ν), which completes the proof of the theorem. � corollary 4.1. if σ = {(σ,x) ∈ (0,∞) ×rd+1+ : σ ≥ %} for some % > 0, one assumes that ρ = max { 1/σ : (σ,x) ∈ σ for some x ∈ rd+1+ } . then by the previous theorem, we deduce that ρ2α+d+2‖m‖α,1 (µα(ω)) 1 2 (θα(σ)) 1 2 ≥ 1 − (� + ν). acknowledgement 4.1. the authors gratefully acknowledge the approval and the support of this research study by the grant no.7385-sci-2017-1-8-f from the deanship of scientific research at northern boder university, arar, k.s.a. references [1] j.p. anker, lp fourier multipliers on riemannian symmetric spaces of the noncompact type, ann. math. (2). 132(3) (1990), 597-628. [2] n. ben salem and ar. nasr, heisenberg-type inequalities for the weinstein operator, integral transforms spec. funct. 26(9) (2015), 700-718. [3] j. j. betancor, ó. ciaurri and j. l. varona, the multiplier of the interval [−1, 1] for the dunkl transform on the real line, j. funct. anal. 242(1) (2007), 327-336. [4] m. brelot, equation de weinstein et potentiels de marcel riesz, semin. theor. potent., paris, no. 3, lect. notes math. 3 (1978), 18-38. [5] d.l.donoho and p.b. stark, uncertainty principles and signal recovery, siam j. appl. math. 49(3) (1989), 906931. [6] j. gosselin and k. stempak, a weak-type estimate for fourier-bessel multipliers, proc. amer. math. soc. 106(3) (1989), 655-662. [7] g. kimeldorf and g. wahba, some results on tchebycheffian spline functions and stochastic processes, j. math. anal. appl. 33(1) (1971), 82-95. [8] t. matsuura, s. saitoh and d. trong, approximate and analytical inversion formulas in heat conduction on multidimensional spaces, j. inverse ill-posed probl. 13(5) (2005), 479-493. [9] h. mejjaoli and m. salhi, uncertainty principles for the weinstein transform, czech. math. j. 61 (2011), 941-974. [10] h. ben mohamed and b. ghribi, weinstein-sobolev spaces of exponential type and applications, acta math. sin., engl. ser. 29 (3) (2013), 591-608. [11] z. ben nahia and n. ben salem, on a mean value property associated with the weinstein operator, potential theory icpt ’94. proceedings of the international conference, kouty, czech republic, berlin: de gruyter (1996), 243-253. [12] z. ben nahia and n. ben salem, spherical harmonics and applications associated with the weinstein operator, potential theory icpt ’94. proceedings of the international conference, kouty, czech republic, berlin: de gruyter (1996), 233-241. [13] a. nowak and k. stempak, relating transplantation and multipliers for dunkl and hankel transforms, math. nachr. 281(11) (2008), 1604-1611. int. j. anal. appl. 17 (1) (2019) 75 [14] a. saoudi, calderón’s reproducing formulas for the weinstein l2-multiplier operators, arxiv:1801.08939 [math.ap]. [15] s.saitoh, approximate real inversion formulas of the gaussian convolution, appl. anal. 83(7) (2004), 727-733. [16] f. soltani, lp-fourier multipliers for the dunkl operator on the real line, j. funct. anal. 209(1) (2004), 16-35. [17] f. soltani, multiplier operators and extremal functions related to the dual dunkl-sonine operator, acta math. sci., ser. b, engl. ed. 33(2) (2013), 430-442. [18] f. soltani, dunkl multiplier operators on a class of reproducing kernel hilbert spaces. j. math. res. appl. 36(6) (2016), 689-702. 1. introduction 2. harmonic analysis associated with the weinstein operator 2.1. the eigenfunction of the weinstein operator 2.2. the weinstein transform 2.3. the translation operator associated with the weinstein operator 3. heisenberg-pauli-weyl uncertainty principle 4. donoho-stark's uncertainty principle references international journal of analysis and applications volume 18, number 1 (2020), 1-15 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1 on the equiform differential geometry of aw(k)-type curves in pseudo-galilean 3-space m. khalifa saad1,2,∗ and h. s. abdel-aziz2 1department of mathematics, faculty of science, islamic university of madinah, 170 madinah, ksa 2department of mathematics, faculty of science, sohag university, 82524 sohag, egypt ∗corresponding author: mohammed.khalifa@iu.edu.sa abstract. the aim of this paper is to study aw(k)-type (1 ≤ k ≤ 3) curves according to the equiform differential geometry of the pseudo-galilean space g13. we give some geometric properties of aw(k) and weak aw(k)-type curves. moreover, we give some relations between the equiform curvatures of these curves. finally, examples of some special curves are given and plotted to support our main results. 1. introduction the geometry of space is associated with mathematical group. the idea of invariance of geometry under transformation group may imply that, on some spacetimes of maximum symmetry there should be a principle of relativity which requires the invariance of physical laws without gravity under transformations among inertial systems [1]. the theory of curves and the curves of constant curvature in the equiform differential geometry of the isotropic spaces i13 , i 2 3 and the galilean space g3 are described in [2] and [3], respectively. the pseudo-galilean space is one of the real cayley-klein spaces. it has projective signature (0, 0, +,−) according to [2]. the absolute of the pseudo-galilean space is an ordered triple {w,f,i} where w is the ideal plane, f a line in w and i is the fixed hyperbolic involution of the points of f. in [4], from the differential received 2019-10-24; accepted 2019-11-20; published 2020-01-02. 2010 mathematics subject classification. 53a04, 53a35, 53c40. key words and phrases. aw(k)-type curves; spacelike and timelike curves; general helix; equiform geometry; pseudo-galilean space. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1 int. j. anal. appl. 18 (1) (2020) 2 geometric point of view, k. arslan and a. west defined the notion of aw(k)-type submanifolds. since then, many works have been done related to aw(k)-type submanifolds (see, for example, [5–10]). in [9], özgür and gezgin studied a bertrand curve of aw(k)-type and furthermore, they showed that there is no such bertrand curve of aw(1) and aw(3)-types if and only if it is a right circular helix. in addition, they studied weak aw(2)-type and aw(3)-type conical geodesic curves in euclidean 3-space e3. besides, in 3-dimensional galilean space and lorentz space, the curves of aw(k)-type were investigated in [6, 8]. in [7], the authors gave curvature conditions and characterizations related to aw(k)-type curves in en and in [10], the authors investigated curves of aw(k)-type in the 3-dimensional null cone. this paper is organized as follows. in section 2, the basic notions and properties of a pseudo-galilean geometry are reviewed. in section 3, properties of the equiform geometry of the pseudo-galilean space g13 are given. section 4 contains a study of aw(k)-type equiform frenet curves. finally, some examples of special curves in g13 are included in section 5. 2. basic concepts in this section, we recall some basic notions from pseudo-galilean geometry [11,12]. in the inhomogeneous affine coordinates for points and vectors (point pairs) the similarity group h8 of g 1 3 has the following form x̄ = a + b.x, ȳ = c + d.x + r. cosh θ.y + r. sinh θ.z, z̄ = e + f.x + r. sinh θ.y + r. cosh θ.z, (2.1) where a,b,c,d,e,f,r and θ are real numbers. particularly, for b = r = 1, the group (2.1) becomes the group b6 ⊂ h8 of isometries (proper motions) of the pseudo-galilean space g13. the motion group leaves invariant the absolute figure and defines the other invariants of this geometry. it has the following form x̄ = a + x, ȳ = c + d.x + cosh θ.y + sinh θ.z, z̄ = e + f.x + sinh θ.y + cosh θ.z. (2.2) according to the motion group in the pseudo-galilean space, there are non-isotropic vectors a(a1,a2,a3) (for which holds a1 6= 0) and four types of isotropic vectors: spacelike (a1 = 0, a22 − a23 > 0), timelike (a1 = 0, a 2 2 −a23 < 0) and two types of lightlike vectors (a1 = 0,a2 = ±a3). the scalar product of two vectors u = (u1,u2,u3) and v = (v1,v2,v3) in g 1 3 is defined by 〈u,v〉 =   u1v1, if u1 6= 0 or v1 6= 0,u2v2 −u3v3 if u1 = 0 and v1 = 0. int. j. anal. appl. 18 (1) (2020) 3 we introduce a pseudo-galilean cross product in the following way u×g13 v = ∣∣∣∣∣∣∣∣∣ 0 −j k u1 u2 u3 v1 v2 v3 ∣∣∣∣∣∣∣∣∣ , where j = (0, 1, 0) and k = (0, 0, 1) are unit spacelike and timelike vectors, respectively. let us recall basic facts about curves in g13, that were introduced in [13–15]. a curve γ(s) = (x(s),y(s),z(s)) is called an admissible curve if it has no inflection points (γ̇× γ̈ 6= 0) and no isotropic tangents (ẋ 6= 0) or normals whose projections on the absolute plane would be lightlike vectors (ẏ 6= ±ż). an admissible curve in g13 is an analogue of a regular curve in euclidean space [12]. for an admissible curve γ(s) : i ⊆ r → g13, the curvature κ(s) and torsion τ(s) are defined by κ(s) = √ |ÿ(s)2 − z̈(s)2| (ẋ(s))2 , τ(s) = ÿ(s) ... z (s) − ... y (s)z̈(s) |ẋ(s)|5 ·κ2(s) , (2.3) expressed in components. hence, for an admissible curve γ : i ⊆ r → g13 parameterized by the arc length s with differential form ds = dx is given by γ(x) = (x,y(x),z(x)). (2.4) the formulas (2.3) have the following form κ(x) = √ |y′′ (x)2 −z′′ (x)2|, τ(x) = y ′′ (x)z ′′′ (x) −y ′′′ (x)z ′′ (x) κ2(x) . (2.5) the associated trihedron is given by e1 = γ ′(x) = (1,y ′ (x),z ′ (x)), e2 = 1 κ(x) γ ′′ (x) = 1 κ(x) (0,y ′′ (x),z ′′ (x)), e3 = 1 κ(x) (0,�z ′′ (x),�y ′′ (x)), (2.6) where � = +1 or � = −1, chosen by criterion det(e1,e2,e3) = 1, that means∣∣∣y′′ (x)2 −z′′ (x)2∣∣∣ = �(y′′ (x)2 −z′′ (x)2). the curve γ given by (2.4) is timelike (resp. spacelike) if e2(s) is a spacelike (resp. timelike) vector. the principal normal vector or simply normal is spacelike if � = +1 and timelike if � = −1. for derivatives of the tangent e1, normal e2 and binormal e3 vector fields, the following frenet formulas in g 1 3 hold: e′1(x) = κ(x)e2(x), e′2(x) = τ(x)e3(x), e′3(x) = τ(x)e2(x). (2.7) int. j. anal. appl. 18 (1) (2020) 4 3. frenet equations according to the equiform geometry of g13 this section contains some important facts about equiform geometry. the equiform differential geometry of curves in the pseudo-galilean space g13 has been described in [11]. in the equiform geometry a few specific terms will be introduced. so, let γ(s) : i → g13 be an admissible curve in the pseudo-galilean space g13, the equiform parameter of γ is defined by σ := ∫ 1 ρ ds = ∫ κds, where ρ = 1 κ is the radius of curvature of the curve γ. then, we have ds dσ = ρ. (3.1) let h be a homothety with center at origin and the coefficient µ. if we put γ̄ = h(γ), then it follows s̄ = µs and ρ̄ = µρ, where s̄ is the arc-length parameter of γ̄ and ρ̄ is the radius of curvature of this curve. therefore, σ is an equiform invariant parameter of γ (see [11]). notation 3.1. the functions κ and τ are not invariants of the homothety group, then from (2.3) it follows that κ̄ = 1 µ κ and τ̄ = 1 µ τ. now we define the frenet formulas of the curve γ with respect to its equiform invariant parameter σ in g13. the vector t = dγ dσ , is called a tangent vector of the curve γ. from (2.6) and (3.1), we get t = dγ ds ds dσ = ρ · dγ ds = ρ · e1. (3.2) also, the principal normal and the binormal vectors are respectively, given by n = ρ · e2, b = ρ · e3. (3.3) it is easy to show that {t, n, b} is an equiform invariant frame of γ. on the other hand, the derivatives of these vectors with respect to σ are given by  t n b   ′ =   ρ̇ 1 0 0 ρ̇ ρτ 0 ρτ ρ̇     t n b   . (3.4) the functions k : i → r defined by k = ρ̇ is called the equiform curvature of the curve γ and t : i → r defined by t = ρτ = τ κ is called the equiform torsion of this curve. in the light of this, the formulas (3.4) int. j. anal. appl. 18 (1) (2020) 5 analogous to the frenet formulas in the equiform geometry of the pseudo-galilean space g13 can be written as   t n b   ′ =   k 1 0 0 k t 0 t k     t n b   . (3.5) the equiform parameter σ = ∫ κ(s)ds for closed curves is called the total curvature, and it plays an important role in global differential geometry of euclidean space. also, the function τ κ has been already known as a conical curvature and it also has interesting geometric interpretation. notation 3.2. let γ : i → g13 be a frenet curve in the equiform geometry of g13, the following statements are true ( for more details, see [11, 13] ): (1) if γ(s) is an isotropic logarithmic spiral in g13. then, k =const. 6= 0 and t = 0, (2) if γ(s) is a circular helix in g13. then, k =0 and t =const. 6= 0, (3) if γ(s) is an isotropic circle in g13. then, k =0 and t = 0. 4. aw(k)-type curves in the equiform geometry of g13 let γ(s) : i → g13 be a curve in the equiform geometry of the pseudo-galilean space g13. the curve γ is called a frenet curve of osculating order l if its derivatives: γ′(s),γ′′(s),γ′′′(s), ...,γ(l)(s), are linearly dependent and γ′(s),γ′′(s),γ′′′(s), ...,γ(l+1)(s), are no longer linearly independent for all s ∈ i. to each frenet curve of order 3, one can associate an orthonormal 3-frame {t, n, b} along γ, such that γ′(s) = 1 ρ t, called the equiform frenet frame (eqs. (3.5)). now, we consider equiform frenet curves of osculating order 3 in g13 and discuss some important results. let γ(s) : i → g13 be a frenet curve in the equiform geometry of the pseudo-galilean space. by the use of frenet formulas (3.5), we obtain the higher order derivatives of γ as follows γ′(s) = dγ dσ dσ ds = 1 ρ t, γ′′(s) = 1 ρ2 n, γ′′′(s) = 1 ρ3 (−kn+t b) , γ′′′′(s) = 1 ρ4 [(2k2+t 2 −k′)n + (t ′ − 3kt )b]. int. j. anal. appl. 18 (1) (2020) 6 notation 4.1. let us write q1 = 1 ρ2 n, (4.1) q2 = 1 ρ3 (−kn+t b) , (4.2) q3 = 1 ρ4 [(2k2+t 2 −k′)n + (t ′ − 3kt )b]. (4.3) notation 4.2. γ′(s),γ′′(s),γ′′′(s) and γ′′′′(s) are linearly dependent if and only if q1,q2 and q3 are linearly dependent. definition 4.1. [5] frenet curves (of osculating order 3) in the equiform geometry of the pseudo-galilean space g13 are called curves of type: (1) equiform aw(1) if they satisfy q3 = 0, (2) equiform aw(2) if they satisfy ‖q2‖ 2 q3 = 〈q3,q2〉q2, (3) equiform aw(3) if they satisfy ‖q1‖ 2 q3 = 〈q3,q1〉q1, (4) weak equiform aw(2) if they satisfy q3 = 〈q3,q∗2〉q ∗ 2, (4.4) (5) weak equiform aw(3) if they satisfy q3 = 〈q3,q∗1〉q ∗ 1, (4.5) where q∗1 = q1 ‖q1‖ , q∗2 = q2 −〈q2,q∗1〉q∗1 ‖q2 −〈q2,q∗1〉q∗1‖ . (4.6) proposition 4.1. let γ : i → g13 be a frenet curve (of osculating order 3) in the equiform geometry of the pseudo-galilean space g13, therefore (i) γ is of type weak equiform aw(2) if and only if 2k2 + t 2 −k′ = 0, (4.7) (ii) γ is of type weak equiform aw(3) if and only if t ′ − 3kt (s) = 0. (4.8) proof. using definition 4.1 and notation 4.1, the proof will be obvious. � int. j. anal. appl. 18 (1) (2020) 7 theorem 4.1. let γ : i → g13 be a frenet curve (of osculating order 3) in the equiform geometry of the pseudo-galilean space g13. then γ is of type equiform aw(1) if and only if −k′ + 2k2 + t 2 = 0, 3kt −t ′ = 0. (4.9) proof. since γ is of type equiform aw(1), then from (4.3), we obtain 1 ρ4 [(2k2+t 2(s) −k′)n + (t ′ − 3kt )b] = 0. as we know, the vectors n and b are linearly independent, so we can write 2k2+t 2 −k′ = 0 and t ′ − 3kt = 0. the converse statement is straightforward and therefore, the proof is completed. � theorem 4.2. let γ : i → g13 be a frenet curve (of osculating order 3) in the equiform geometry of the pseudo-galilean space g13. then γ is of type equiform aw(2) if k2t −kt ′ + tk′ −t 3 = 0. (4.10) proof. assuming that γ is a frenet curve in the equiform geometry of g13 , then from (4.2) and (4.3), one can write q2 = a11n + a12b, q3 = a21n + a22b, where a11,a12, a21 and a22 are differentiable functions. since q2 and q3 are linearly dependent, hence coefficients determinant equals zero, that is ∣∣∣∣∣∣ a11 a12 a21 a22 ∣∣∣∣∣∣ = 0, (4.11) where a11 = −1 ρ3 k, a12 = 1 ρ3 t , a21 = 1 ρ4 [−k′ + 2k2 + t 2], a22 = 1 ρ4 [−3kt + t ′]. (4.12) from (4.11) and (4.12), we obtain (4.10). � int. j. anal. appl. 18 (1) (2020) 8 theorem 4.3. let γ : i → g13 be a frenet curve (of osculating order 3) in the equiform geometry of g13. then γ is of equiform aw(3)-type if t ′ − 3kt = 0. (4.13) proof. using definition 4.1 and eqs. (4.1) and (4.3), we obtain (4.13). � 5. computational examples we consider some examples (timelike and spacelike curves [11, 12]) which characterize equiform general (circular) helices with respect to the frenet frame {t, n, b} in the equiform geometry of g13 which satisfy some conditions of equiform curvatures (i)k = k(s),t = t (s) (ii)k =const. 6= 0,t =const. 6= 0 (iii)k =const. 6= 0,t =0. example 5.1. consider the equiform timelike general helix r : i −→ g13,i ⊆ r which parameterized by the arc length s with differential form ds = dx is given by r(x) = (x,y(x),z(x)), where x(s) = s, y(s) = e−as (a2 − b2)2 (( a2 + b2 ) cosh (bs) + 2ab sinh (bs) ) , z(s) = e−as (a2 − b2)2 ( 2ab cosh (bs) + ( a2 + b2 ) sinh (bs) ) ; a,b ∈ r−{0} . the corresponding derivatives of r are as follows r′ = ( 1, −e−as (a2 − b2) (a cosh (bs) + b sinh (bs)) , e−as (b2 −a2) (b cosh (bs) + a sinh (bs)) ) , r′′ = ( 0,e−as cosh (bs) ,e−as sinh (bs) ) , r′′′ = ( 0,e−as (−a cosh (bs) + b sinh (bs)) ,e−as (b cosh (bs) −a sinh (bs)) ) . the tangent vector of r has the form e1 = (x ′,y′,z′) = ( 1, −e−as (a2 − b2) (a cosh (bs) + b sinh (bs)) , e−as (b2 −a2) (b cosh (bs) + a sinh (bs)) ) , and the two normals (normal and binormal) of the curve are, respectively e2 = (0, cosh (bs) , sinh (bs)) , e3 = (0, sinh (bs) , cosh (bs)) ; det[e1, e2, e3] = 1. int. j. anal. appl. 18 (1) (2020) 9 therefore, the curvature and torsion of r are respectively, given by κ = e−as, τ = b . from the equiform frenet formulas, we can express the vector fields t, n, b as follows t = ( eas, −1 (a2 − b2) (a cosh (bs) + b sinh (bs)) , 1 (b2 −a2) (b cosh (bs) + a sinh (bs)) ) , n = (0,eas cosh (bs) ,eas sinh (bs)) , b = (0,eas sinh (bs) ,eas cosh (bs)) , respectively. in the light of this, the equiform curvatures are given by k = aeas,t = −beas. figure 1. equiform timelike general helix with k = 5e5s,t = −2e5s. example 5.2. let r : i −→ g13,i ⊆ r be the equiform spacelike general helix, and it is given by r(x) = (x,y(x),z(x)), int. j. anal. appl. 18 (1) (2020) 10 where x(s) = s, y(s) = e−as (a2 − b2)2 ( 2ab cosh (bs) + ( a2 + b2 ) sinh (bs) ) , z(s) = e−as (a2 − b2)2 (( a2 + b2 ) cosh (bs) + 2ab sinh (bs) ) ; a,b ∈ r−{0} . for the coordinate functions of r, we have r′ = ( 1, e−as (b2 −a2) (b cosh (bs) + a sinh (bs)) , −e−as (a2 − b2) (a cosh (bs) + b sinh (bs)) ) , r′′ = ( 0,e−as sinh (bs) ,e−as cosh (bs) ) , r′′′ = ( 0,e−as (b cosh (bs) −a sinh (bs)) ,e−as (b sinh (bs) −a cosh (bs)) ) . also, the associated trihedron is given by e1 = ( 1, e−as (b2 −a2) (b cosh (bs) + a sinh (bs)) , −e−as (a2 − b2) (a cosh (bs) + b sinh (bs)) ) , e2 = (0, sinh (bs) , cosh (bs)) , e3 = (0,−cosh (bs) ,−sinh (bs)) . the curvature and torsion of this curve are κ = e−as, τ = −b . furthermore, the tangent, normal and binormal vector fields in the equiform geometry of g13 are obtained as follows t = ( eas, 1 (b2 −a2) (b cosh (bs) + a sinh (bs)) , −1 (a2 − b2) (a cosh (bs) + b sinh (bs)) ) , n = (0,eas sinh (bs) ,eas cosh (bs)) , b = (0,−eas cosh (bs) ,−eas sinh (bs)) , respectively. the equiform curvatures of r are k = aeas,t = −beas. example 5.3. consider the equiform timelike circular helix r : i −→ g13,i ⊆ r is given by r(x) = (x,y(x),z(x)), int. j. anal. appl. 18 (1) (2020) 11 figure 2. equiform spacelike general helix with k = 5e5s,t = −2e5s. where x(s) = s, y(s) = a3s b (b2 −a2) ( b sinh ( b a ln(as) ) −a cosh ( b a ln(as) )) , z(s) = a3s b (b2 −a2) ( b cosh ( b a ln(as) ) −a sinh ( b a ln(as) )) ; a,b ∈ r−{0} . for this curve, the equiform vector fields are obtained as follows t = ( s a , as b cosh ( b a ln(as) ) , as b sinh ( b a ln(as) )) , n = ( 0, s a sinh ( b a ln(as) ) , s a cosh ( b a ln(as) )) , b = ( 0, s a cosh ( b a ln(as) ) , s a sinh ( b a ln(as) )) , respectively. it follows that k = 1 a ,t = −b a2 . int. j. anal. appl. 18 (1) (2020) 12 figure 3. equiform timelike circular helix with k = 1 2 ,t = −5 4 . example 5.4. let the equiform spacelike circular helix r : i −→ g13,i ⊆ r be r(x) = (x,y(x),z(x)), where x(s) = s, y(s) = a3s b (b2 −a2) ( b cosh ( b a ln(as) ) −a sinh ( b a ln(as) )) , z(s) = a3s b (b2 −a2) ( b sinh ( b a ln(as) ) −a cosh ( b a ln(as) )) ; a,b ∈ r−{0} . here, the equiform differential vectors respectively, are as follows t = ( s a , as b sinh ( b a ln(as) ) , as b cosh ( b a ln(as) )) , n = ( 0, s a cosh ( b a ln(as) ) , s a sinh ( b a ln(as) )) , b = ( 0,− s a sinh ( b a ln(as) ) ,− s a cosh ( b a ln(as) )) . equiform curvature and equiform torsion are calculated as follows k = 1 a ,t = b a2 . int. j. anal. appl. 18 (1) (2020) 13 figure 4. equiform spacelike circular helix with k = 1 3 ,t = 4 9 . example 5.5. let r : i −→ g13,i ⊆ r be a equiform timelike isotropic logarithmic spiral which parameterized by the arc length s with differential form ds = dx, and is given by r(x) = (x,y(x), 0), where x(s) = s, y(s) = as + b a2 (ln(as + b) − 1) , z(s) = 0; a,b ∈ r−{0} . for this curve, we get r′ = ( 1, ln(as + b) a , 0 ) , r′′ = ( 0, 1 as + b , 0 ) , r′′′ = ( 0, −a (as + b) 2 , 0 ) , int. j. anal. appl. 18 (1) (2020) 14 and e1 = ( 1, ln(as + b) a , 0 ) , e2 = (0, 1, 0) , e3 = (0, 0, 1) ; κ = 1 as + b , τ = 0. in this case, equiform frenet vectors and equiform curvatures are as follows t = ( as + b, (as + b) ln(as + b) a , 0 ) , n = (0,as + b, 0) , b = (0, 0,as + b) , k = a,t = 0. respectively. figure 5. equiform timelike isotropic logarithmic spiral with k = 2,t = 0. from aforementioned calculations, according to (proposition 4.2 and theorems 4.1 − 4.3), the first four examples are not characterize curves of equiform aw(1), weak equiform aw(2) or weak equiform aw(3)types. on the other hand, the last example shows that the curve is of equiform aw(2) and aw(3)-types and it is not of equiform aw(1)-type. also, this curve is of weak equiform aw(2) and not of weak equiform aw(3)-types. int. j. anal. appl. 18 (1) (2020) 15 6. conclusion in this paper, we have considered some special curves of equiform aw(k)-type of the pseudo-galilean 3space. also, using the equiform curvature conditions of these curves, the necessary and sufficient conditions for them to be equiform aw(k) and weak equiform aw(k)-types are obtained. furthermore, some examples to support our main results are given and plotted. acknowledgment this research was supported by islamic university of madinah. we would like to thank our colleagues from deanship of scientific research who provided insight and expertise that greatly assisted the research. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] i. yaglom, a simple non-euclidean geometry and its physical basis, springer-verlag, in new york, 1979. [2] b. j. pavković, equiform geometry of curves in the isotropic spaces i13 and i 2 3 , rad jazu, 1986, 39-44. [3] b. j. pavković and i. kamenarović, the equiform differential geometry of curves in the galilean space g3, glasnik mat. 22 (42) (1987), 449-457. [4] k. arslan and a. west, product submanifolds with pointwise 3-planar normal sections, glasgow math. j. 37 (1) (1995), 73-81. [5] k. arslan and c. özgür, curves and surfaces of aw(k) -type, geometry and topology of submanifolds ix, world scientific, 1999, 21-26. [6] m. külahci, m. bektas and m. ergüt, on harmonic curvatures of null curves of the aw(k)-type in lorentzian space, z. naturforsch. a, 63 (5-6) (2008), 248-252. [7] m. külahci and m. ergüt, bertrand curves of aw(k)-type in lorentzian space, nonlinear anal., theory methods appl. 70 (2009), 1725-1731. [8] m. külahci, a.o. öğrenmiş and m. ergüt, new characterizations of curves in the galilean space g3, int. j. phys. math. sci. 1 (2010), 49-57. [9] c. özgür and f. gezgin, on some curves of aw(k)-type, differ. geom. dyn. syst. 7 (2005), 74-80. [10] d. w. yoon, general helices of aw(k)-type in the lie group, j. appl. math. 2012 (2012), article id 535123. [11] z. erjavec and b. divjak, the equiform differential geometry of curves in the pseudo-galilean space, math. commun. 13 (2008), 321-332. [12] z. erjavec, on generalization of helices in the galilean and the pseudo-galilean space, j. math. res. 6 (3) (2014), 39-50. [13] b. divjak, the general solution of the frenet’s system of differential equations for curves in the pseudo-galilean space g13, math. commun. 2 (1997), 143-147. [14] b. divjak, geometrija pseudogalilejevih prostora, ph. d. thesis, university of zagreb, 1997. [15] b. divjak, curves in pseudo-galilean geometry, ann. univ. sci. budapest. sect. math. 41 (1998), 117-128. 1. introduction 2. basic concepts 3. frenet equations according to the equiform geometry of g31 4. aw(k)-type curves in the equiform geometry of g31 5. computational examples 6. conclusion acknowledgment references international journal of analysis and applications volume 16, number 1 (2018), 75-82 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-75 new subfamily of meromorphic convex functions in circular domain involving q-operator bakhtiar ahmad∗ and muhammad arif department of mathematics, abdul wali khan university mardan, kp, pakistan ∗corresponding author: pirbakhtiarbacha@gmail.com abstract. the main object of the present paper is to investigate a number of useful properties such as sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for a new subclass of meromorphic convex functions, which are defined here by means of a newly defined q-linear differential operator. 1. introduction and definitions quantum calculus (q-calculus), which is the study of classical calculus without the notion of limits, attracted the researchers because of its applications in various branches of mathematics, physics and various other branches of science, for details see [6, 7].the q-analogue of derivative and integral operators were introduced by jackson [13, 14] along with some applications of q-calculus. later on aral and gupta [5–7] introduced the q-baskakov durrmeyer operator by using q-beta function while the author’s in [4, 8, 9] discussed the qgeneralization of complex operators known as q-picard and q-gauss-weierstrass singular integral operators. kanas and răducanu [15] gave the q-analogue of ruscheweyh differential operator using the concepts of convolution and then studied some of its properties. more applications of this operator can be seen in the paper [3]. received 13th september, 2017; accepted 4th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. meromorphic functions; janowski functions; q-differential operator. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 75 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-75 int. j. anal. appl. 16 (1) (2018) 76 in this paper a q-differential operator for meromorphic functions using convolution is defined. we use this operator to define and study some properties of a family of meromorphic convex functions associated with circular domain. let a denote the family of all meromorphic functions f that are analytic in the punctured disc d = {z ∈ c : 0 < |z| < 1} and satisfying the normalization f(z) = 1 zp + ∞∑ k=1 ak+pz k+p, (z ∈ d) . (1.1) also let ms∗ (α) and mk(α) denote the well known families of meromorphic starlike and meromorphic convex functions of order α (0 ≤ α < 1) respectively. for f and g be two meromorphic functions that are analytic in d and have the form (1.1), then convolution of these functions can be defined by f(z) ∗g(z) = 1 zp + ∞∑ k=1 ak+pbk+pz k+p, (z ∈ d) . for 0 < q < 1, the q-derivative of a function f is defined by ∂qf(z) = f (qz) −f(z) z (q − 1) , (z 6= 0, q 6= 1) . (1.2) simple calculations yields that for n ∈ n := {1, 2, 3, . . .} and z ∈ d ∂q { ∞∑ n=1 anz n } = ∞∑ n=1 [n,q] anz n−1, (1.3) where [n,q] = 1 −qn 1 −q = 1 + n∑ l=1 ql, [0,q] = 0. for any non-negative integer n the q-number shift factorial is defined by [n,q]! =   1, n = 0,[1,q] [2,q] [3,q] · · · [n,q] , n ∈ n. also the q-generalized pochhammer symbol for x ∈ r is given by [x,q]n =   1, n = 0,[x,q][x + 1,q] . . . [x + n− 1,q], n ∈ n, and for x > 0, let q-gamma function is defined as γq (x + 1) = [x,q] γq (t) and γq (1) = 1. we now define a function φp (q,µ; z) = 1 zp + ∞∑ n=1 λn+p z n+p, (µ > −1, z ∈ d), (1.4) int. j. anal. appl. 16 (1) (2018) 77 with λn+p = γq (µ + n + p + 1) γq (µ + 1) [n + p,q]! = [µ + 1,q]n+p [n + p,q]! . it is quite clear that the series defined in (1.4) is convergent absolutely in d. using the function φ (q,µ; z) and definition of q-derivative along with the idea of convolutions, we now define the differential operator lµ+p−1q : ap → ap by lµ+p−1q f (z) = φp (q,µ; z) ∗f(z) = 1 zp + ∞∑ n=1 λn+panz n+p , (µ > −1, z ∈ d) . (1.5) also for more details on the q-analogue of differential operators see the work [1, 2, 17]. motivated from the work studied in [10, 12, 18–20], we now define a subfamily mc∗q (p,µ,a,b) of ap by using the operator lµq as follows; definition 1.1. let −1 ≤ b < a ≤ 1 and 0 < q < 1. then a function f ∈ ap is in the class mc∗q (p,µ,a,b) , if it satisfies −qp∂q ( z∂qlµ+p−1q f (z) ) [p,q] ∂ql µ+p−1 q f (z) ≺ 1 + az 1 + bz . (1.6) where the notation ”≺” denotes the familiar subordinations. equivalently, a function f ∈ ap is in the class mc∗q (p,µ,a,b) , if and only if∣∣∣∣∣∣ qp∂q ( z∂qlµ+p−1q f (z) ) + [p,q] ∂qlµ+p−1q f (z) a [p,q] ∂ql µ+p−1 q f (z) + bqp∂q ( z∂ql µ+p−1 q f (z) ) ∣∣∣∣∣∣ < 1. (1.7) 2. the main results and their consequences theorem 2.1. let f ∈ ap be of the form (1.1) and satisfy the inequality ∞∑ n=1 qp[n+p,q][µ+1,q]n+p [n+p,q]! (qp[n + p,q] (1 + b) + (1 + a) [p,q]) |an+p| ≤ [p,q] 2 (a−b) . (2.1) then the function f ∈mc∗q (p,µ,a,b) . proof. to show f ∈ mc∗q (p,µ,a,b) , we only need to prove the inequality (1.7). for this using (1.5), and then with the help of (1.2) and (1.3) we have∣∣∣∣∣∣ qp∂q ( z∂qlµ+p−1q f (z) ) + [p,q] ∂qlµ+p−1q f (z) a [p,q] ∂ql µ+p−1 q f (z) + bqp∂q ( z∂ql µ+p−1 q f (z) ) ∣∣∣∣∣∣ int. j. anal. appl. 16 (1) (2018) 78 = ∣∣∣∣∣∣ qp ( [p,q]2 q2pzp+1 + ∑∞ n=1 λn+p[n+p,q] 2an+pz n+p−1 ) +[p,q] ( − [p,q] qpzp+1 + ∑∞ n=1 λn+p[n+p,q]an+pz n+p−1 ) a[p,q] ( − [p,q] qpzp+1 + ∑∞ n=1 λn+p[n+p,q]an+pzn+p−1 ) +bqp ( [p,q]2 q2pzp+1 + ∑∞ n=1 λn+p[n+p,q] 2an+pzn+p−1 ) ∣∣∣∣∣∣ = ∣∣∣∣∣ ∑∞ n=1 λn+p[n+p,q](q p[n+p,q]+[p,q])an+pz n+p−1 −(a−b)[p,q] 2 qpzp+1 + ∑∞ n=1 λn+p[n+p,q](a[p,q]+bq p[n+p,q])an+pzn+p−1 ∣∣∣∣∣ = ∣∣∣ ∑∞n=1 qpλn+p[n+p,q](qp[n+p,q]+[p,q])an+pzn+2p (a−b)[p,q]2+ ∑∞ n=1 qpλn+p[n+p,q](a[p,q]+bqp[n+p,q])an+pzn+2p ∣∣∣ ≤ ∑∞ n=1 qpλn+p[n+p,q](q p[n+p,q]+[p,q])|an+p| (a−b)[p,q]2− ∑∞ n=1 qpλn+p[n+p,q](a[p,q]+bqp[n+p,q])|an+p| < 1, where we have used the inequality (2.1) and this completes the proof. � theorem 2.2. let f ∈mc∗q (p,µ,a,b) and has the form (1.1) . then for |z| = r 1 rp − τ1rp ≤ |f(z)| ≤ 1 rp + τ1r p, where τ1 = (a−b) [p,q]! [p,q]2 qp[µ + 1,q]p+1((1 + a) [p,q] + qp[p + 1,q] (1 + b)) . proof. consider |f(z)| = ∣∣∣∣∣ 1zp + ∞∑ n=1 an+p z n+p ∣∣∣∣∣ , ≤ 1 |zp| + ∞∑ n=1 |an+p| |z| n+p = 1 rp + ∞∑ n=1 |an+p| rn+p as |z| = r < 1 so rn+p < rp and |f(z)| ≤ 1 rp + rp ∞∑ n=1 |an+p| (2.2) similarly |f(z)| ≥ 1 rp −rp ∞∑ n=1 |an+p| (2.3) since (2.1) implies that ∞∑ n=1 qp[n+p,q][µ+1,q]n+p [n+p,q]! (qp[n + p,q] (1 + b) + (1 + a) [p,q]) |an+p| ≤ [p,q] 2 (a−b) . but qp [µ+1,q]p+1 [p,q]! ((1 + a) [p,q] + qp[p + 1,q] (1 + b)) ∞∑ n=1 |an+p| ≤ ∞∑ n=1 qp [µ+1,q]n+p [n+p,q]! ((1 + a) [p,q] + qp[n + p,q] (1 + b)) |an+p| . hence qp [µ+1,q]p+1 [p,q]! ((1 + a) [p,q] + qp[p + 1,q] (1 + b)) ∞∑ n=1 |an+p| ≤ [p,q] 2 (a−b) , int. j. anal. appl. 16 (1) (2018) 79 which gives ∞∑ n=1 |an+p| ≤ [p,q]2(a−b)[p,q]! qp((1+a)[p,q]+qp[p+1,q](1+b))[µ+1,q]p+1 now by putting this value in (2.2) and (2.3) we get the required result. � theorem 2.3. let f ∈mc∗q (p,µ,a,b) and has the form (1.1) . then for |z| = r [p,q]m qmp+ζrm+p − τ2rp ≤ ∣∣∂mq f(z)∣∣ ≤ [p,q]mqmp+ζrm+p + τ2rp. where τ2 = [p,q] 2 (a−b) [p,q]! ((1 + a) [p,q] + qp[p + 1,q] (1 + b)) and ζ = m∑ n=1 n. proof. by the virtue of (1.2) and (1.3) , we can write ∂mq f(z) = (−1)m [p,q]m qmp+ζzp+m + ∞∑ n=1 [n + p− (m− 1) ,q]m+1ap+nzp+n−m. since |z| = r < 1 so rp+n−m ≤ rp for m ≤ n hence ∣∣∂mq f(z)∣∣ ≤ [p,q]mqmp+ζrm+p + rp ∞∑ n=1 [n + p− (m− 1) ,q]m+1 |ap+n| , (2.4) and similarly ∣∣∂mq f(z)∣∣ ≥ [p,q]mqmp+ζrm+p −rp ∞∑ n=1 [n + p− (m− 1) ,q]m+1 |ap+n| . (2.5) now by using (2.1) and the following inequality qp ((1+a)[p,q]+qp[p+1,q](1+b)) [p,q]! ∞∑ n=1 [µ + p,q]p+n |ap+n| ≤ ∞∑ n=1 qp [µ+p,q]n+p [n+p,q]! ((1 + a) [p,q] + qp[p + n,q] (1 + b)) |an| , we have ∞∑ n=1 [µ + p,q]n+p |an+p| ≤ (a−b) [p,q]2[p,q]! qp ((1 + a) [p,q] + qp[p + 1,q] (1 + b)) , but certainly ∞∑ n=1 [n + p− (m− 1) ,q]m+1 |ap+n| ≤ ∞∑ n=1 [µ + p,q]n+p |an+p| , which implies ∞∑ n=1 [n + p− (m− 1) ,q]m+1 |ap+n| ≤ (a−b) [p,q]2[p,q]! qp ((1 + a) [p,q] + qp[p + 1,q] (1 + b)) . finally, using this in (2.4) and (2.5) we obtain the required result. � theorem 2.4. let f ∈mc∗q (p,µ,a,b) . then f ∈mcp (α) for |z| < r1, where r1 = ( p (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n) (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! ) 1 n+2p . int. j. anal. appl. 16 (1) (2018) 80 proof. let f ∈mc∗q (p,µ,a,b). to prove f ∈mcp (α) , we only need to show∣∣∣∣ zf′′(z) + (p + 1) f′(z)zf′′(z) + (1 + 2α−p) f′(z) ∣∣∣∣ ≤ 1. using (1.1) along with some simple computation yields ∞∑ n=1 (p + n) (n + p + α) p (p−α) |an+p| |z| n+2p ≤ 1. (2.6) from (2.1) , we can easily obtain that ∞∑ n=1 qp[p+n,q][µ+p,q]n+p [n+p,q]! ( ((1+a)[p,q]+qp[p+n,q](1+b)) (a−b)[p,q]2 ) |an+p| < 1. now inequality (2.6) will be true, if the following holds ∞∑ n=1 (p+n)(n+p+α) p(p−α) |an+p| |z| n+2p < ∞∑ n=1 qp[µ+p,q]n+p [n+p−1,q]! ( ((1+a)[p,q]+qp[p+n,q](1+b)) (a−b)[p,q]2 ) |an+p| , which implies that |z|n+2p < p (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n) (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! , and so |z| < ( p (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n) (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! ) 1 n+2p = r1, we get the required condition. � theorem 2.5. let f ∈mc∗q (p,µ,a,b). then f ∈ms ∗ p (α) for |z| < r2, where r2 = ( (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! ) 1 n+2p . proof. we know that f ∈ms∗p (α) , if and only if∣∣∣∣ zf′(z) + pf(z)zf′(z) − (p− 2α)f(z) ∣∣∣∣ ≤ 1. using (1.1) and upon simplification yields ∞∑ n=1 ( n + p + α p−α ) |an+p| |z| n+2p ≤ 1. (2.7) now from (2.1) we can easily obtain ∞∑ n=1 qp[µ+p,q]n+p [n+p−1,q]! ( ((1+a)[p,q]+qp[p+n,q](1+b)) (a−b)[p,q]2 ) |an+p| < 1. for inequality (2.7) to be true it will be enough if ∞∑ n=1 ( n+p+α p−α ) |an+p| |z| n+2p < ∞∑ n=1 qp[µ+p,q]n+p [n+p−1,q]! ( ((1+a)[p,q]+qp[p+n,q](1+b)) (a−b)[p,q]2 ) |an+p| . int. j. anal. appl. 16 (1) (2018) 81 this gives |z|n+2p < (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! , and hence |z| < ( (p−α) qp ((1 + a) [p,q] + qp[p + n,q] (1 + b)) [µ + p,q]n+p (p + n + α) (a−b) [p,q]2 [n + p− 1,q]! ) 1 n+2p = r2. thus we obtain the required result. � references [1] i. aldawish and m. darus, starlikness of q-differential operator involving quantum calculus, korean j. math., 22(4)(2014), 699 − 709. [2] h. aldweby and m. darus, a subclass of harmonic univalent functions associated with q-analogue of dziok-srivastava operator, isrn math. anal., 2013(2013), article id 382312, 6 pages. [3] h. aldweby and m. darus, some subordination results on q-analogue of ruscheweyh differential operator, abstr. appl. anal., 2014(2014), article id 958563, 6 pages. [4] a. aral, on the generalized picard and gauss weierstrass singular integrals, j. compu. anal. appl., 8(3)(2006), 249 − 261. [5] a. aral and v. gupta, generalized q-baskakov operators, math. slovaca, 61(4)(2011), 619 − 634. [6] a. aral and v. gupta, on the durrmeyer type modification of the q-baskakov type operators, non-linear anal. theory, methods and appl., 72(3 − 4)(2010), 1171 − 1180. [7] a. aral and v. gupta, on q-baskakov type operators, demonstratio mathematica, 42(1)(2009), 109 − 122. [8] g. a. anastassiu and s. g. gal, geometric and approximation properties of generalized singular integrals, j. korean math. soci., 23(2)(2006), 425 − 443. [9] g. a. anastassiu and s. g. gal, geometric and approximation properties of some singular integrals in the unit disk, j. inequ. appl., 2006(2016), article id 17231, 19 pages. [10] j. dziok, g. murugusundaramoorthy and j. soko l, on certain class of meromorphic functions with positive coefcients, acta mathematica scientia b, 32(4)(2012), 1 − 16. [11] m. r. ganigi and b. a. uralegaddi, new criteria for meromorphic univalent functions, bull. math. soc. sci. math. roumanie (n.s.), 33(81)(1989), 9 − 13. [12] a. huda and m. darus, integral operator defined by q-analogue of liu-srivastava operator, studia univ. babes-bolyai ser. math. 58(4)(2013), 529 − 537. [13] f. h. jackson, on q-definite integrals, the quarterly j. pure appl. math., 41(1910), 193 − 203. [14] f. h. jackson, on q-functions and a certain difference operator, trans. royal soc. edinburgh, 46(2)(1909), 253 − 281. [15] s. kanas and d. răducanu, some class of analytic functions related to conic domains, math. slovaca 64(5)(2014), 1183 − 1196. [16] m. s. liu, on a subclass of p-valent close to convex functions of type α and order β, j. math. study, 30 (1) (1997), 102−104 (chinese) [17] a. mohammed and m. darus, a generalized operator involving the q-hypergeometric function, mat. vesnik, 65(4)(2013), 454 − 465. [18] c. pommerenke, on meromorphic starlike functions, pac. j. math. 13(1963), 221 − 235. int. j. anal. appl. 16 (1) (2018) 82 [19] t. m. seoudy and m. k. aouf, coefficient estimates of new classes of q-starlike and q-convex functions of complex order, j. math. inequal. 10(1)(2016), 135 − 145. [20] b. a. uralegaddi and c. somanatha, certain diferential operators for meromorphic functions, houston j. math. 17(2)(1991), 279 − 284. 1. introduction and definitions 2. the main results and their consequences references international journal of analysis and applications volume 17, number 4 (2019), 578-585 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-578 improved link prediction using pca ankita1,∗, nanhay singh2 1university school of information communication and technology, guru gobind singh indraprastha university, new delhi, india 2ambedkar institute of advanced communication technologies and research, guru gobind singh indraprastha university, new delhi, india ∗corresponding author: ankita.it.13@gmail.com abstract. link prediction is known as a challenging problem in the area of online social media. earlier, learning model for link prediction task has been proposed by many researchers. but the classification of imbalanced and high dimensional data is an interesting and challenging problem in machine learning due to presence of unbalanced and redundant or correlated data which break down the classification performance. in this paper, we have balanced the data and used principle component analysis (pca) to reduce the correlated data and improved the performance of link prediction model. experiment is carried out on social network data set and the use of pca method has improved the performance in classification of links. 1. introduction link prediction problem: online social media is a structure of users, where nodes are the users or entity and edges represent collaboration, interaction or association between users. link prediction is known as a fundamental problem in the online social network where task is to predict links in the near future. it can be defined as a given social network graph consisting of nodes as users and links between them at time t, predict new links between users at time t’(where t is less than t’) as shown in figure 1. as new relationship received 2019-03-04; accepted 2019-04-05; published 2019-07-01. 2010 mathematics subject classification. 91d30. key words and phrases. feature; reduction; link; learning; prediction . c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 578 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-578 int. j. anal. appl. 17 (4) (2019) 579 keeps on adding in social network and old relations may be deleted, this is considered as the challenging problem. figure 1. link preditcion example there are many different approaches proposed by researchers in the past and applied in different areas. for a co-authorship network, using link prediction, future association between authors can be predicting. similarly in online social network, the friend connections between the users can be identified or predict. in the area of spam mail detection, link prediction model can be used to detect anomalies in the emails [1]. the link prediction techniques also have been applied in disease prediction by folino and pizzuti [2]. they have applied link prediction to predict the diseases. the diseases are represented as the nodes and the incidence of the diseases in a patient is represented by an edge. using link prediction, a score between two diseases is identified and using link score, probability of co-occurrence of the diseases is identified. link prediction model has also been applied in recommender system [3, 4], where new product can be recommended to users based on user preferences or rating. due to wide range of application in different areas, link prediction problem has attracted more attention for the researchers. the link prediction methods can be classified as similarity based methods or learning based methods. in the case of similarity based model, similarity score is assigned on the edge of the network by using similarity metrics and used to predict links between two nodes. in [5] [6], similarity based model has been used by authors to predicting links in the graph network. in learning based model the link prediction is considered as the classification problem and in [7], supervised learning model used to predict links in the social network graph. most of the existing work have used the static nature of the network and have used the supervised learning algorithm using topological features of the network without using any dimension reduction techniques to improve the feature set. in this paper, we propose a supervised learning algorithm where features vector created using network information and to improve the performance of learning based model we have balance the data and used principle component analysis (pca) for dimension reduction. principle component analysis is a method to convert the attributes of a dataset into uncorrelated datasets. the components of this new dataset are known as principle components (pcs). pca is used to reduce the high dimensional data into low dimension datasets having high variance. highly correlated attributes in a dataset can degrade the performance in prediction model. int. j. anal. appl. 17 (4) (2019) 580 the contribution of this paper: 1) we proposed a learning based model using features based model. 2) we have balanced the data using oversampling and used dimension reduction technique, pca to improve the prediction accuracy. 3) we have used karate club dataset in our experiment to show effectiveness of our approach and used accuracy, recall, precision and f-measure for performance evaluation. here, paper is ordered as: section 2 – the related work; section 3 – proposed methodology for solving link prediction problem; section 4 – experimental results and discussion; section 5 – conclusion of the work. 2. related work liben-nowell and kleinberg [8] have described link prediction problem, where vertex in the graph represents a person in online social network and edges between vertexes represent association or interaction between them. the problem of link prediction can be considered as a supervised learning model where we use the link information of training dataset to train the learning model. from the trained model, prediction of links can be made. many researchers have used learning based algorithm to predict links in online social network. hasan [9] identified set of features and shows effectiveness of features in link prediction method. they have compared different classes of supervised learning algorithm in terms of their prediction accuracy. in [10] authors have considered weighted network using supervised learning model. in their work, comparison of the link prediction model in co authorship network has been done which shows better results in supervised learning model comparatively to unsupervised model. similarly in [11] authors have used classification model in the healthcare domain where model can predict future association among physician and shows good results in experiment. as common neighbour is one of simplest similarity measures between two nodes, the authors in [12] have used common neighbours between nodes and proposed, a probabilistic model using the naive bayes classifier where by identifying different roles of common neighbours and by giving different weights to them , the proposed model outperform in the experiment results. there exists no. of classification model for supervised learning such as svm, k nearest neighbour, decision tree. in [9] authors have used the co-authorship dataset and have used bagging and support vector machine for learning model. also the regression model is used in [13] for the link prediction. for predicting links in the network, this problem also being solved using feature based link prediction model, where any popular supervised classification techniques can be employ [14] and the major challenge is to select the set of features.in this feature based learning model, topological features of the graph is considered mostly by the researchers [15]. there are many topology based features like node based, path based features which being considered for feature based learning based model. though many work has included topological features, but to the best of our knowledge, lesser have given importance to centrality based features and have used pca on these features. in our proposed methodology we have included centrality features in the feature vector and further int. j. anal. appl. 17 (4) (2019) 581 used dimension reduction technique, pca (principal component analysis) and oversampling of dataset to improve the link prediction performance. 3. methodology the proposed approach is using dimension reduction technique which reduce the high dimension features to low dimension features. the proposed methodology for feature based link prediction using pca is represented in the figure 2 and steps are explained as follows: 3.1. steps in proposed approach: step 1: extract features from the dataset: as our proposed approach is based on feature based learning model, the selection of features is an important task. in our proposed method, following are the components of features vector: 1) common neighbours(cn) :common neighbours metric is based on the thought that if two nodes i and j have many common neighbour nodes, the probability to have link in the future is more. 2) resource allocation index (rai): it is motivated by a resource allocation process and measures how much resource is transmitted between a and b.. therefore, the similarity of node a and node b can be defined as the sum of the inverse of the degree of each of the common neighbor between a and b. 3) betweeness centrality (bc): betweeness centrality, determines of the degree to which a given node is in the shortest paths between the other nodes in the graph. a node (a) has high betweeness if the shortest paths between many pairs of the other nodes in the graph pass through that node (a). 4) closeness centrality (clc): it identifies the most significant nodes in the network. for node a, clc is defined as the ratio of the total number of nodes(n) in the graph minus one to the sum of the shortest distances of the node a to every other node in the graph. 5) degree centrality(dc) : degree centrality is defined as the no. of connections of the node in a network graph g(n,e) .the degree centrality is normalized by dividing is by the maximum degree in graph and defined as: dc(a)(degree centrality of node a)=k(a)/(n-1) k(a) is the degree of the node a and (n-1) maximum degree in a graph. step 2: balance the data using oversampling technique: by using the extracted features of the network, the links between the nodes will be identifies a f link or nf link. before performing classification we have checked for any missing values and balance the data using oversampling of the data. for the balancing of the dataset oversampling or under sampling can be done. for our work we have employed oversampling of the data , by which the the resultant dataset have become balanced. step 3: apply dimension reduction technique, principal component analysis (pca) on the resultant feature vector which provides relevant and uncorrelated dataset. the pca is a procedure that converts the correlated attributes into linearly uncorrelated attributes. int. j. anal. appl. 17 (4) (2019) 582 step 4: perform classification using principal components (pcs) step 5: evaluate the model using accuracy, recall, precision and f-measure. figure 2. proposed approach for feature based link prediction using pca 3.2. evaluation metrics that we have used in our experiment: 1. accuracy: it is the performance metrics to measure the accuracy of the model. as defines in [16], it is the ratio of total no. of correct predictions to the total no. of samples or predictions accuracy= correct prediction (true positive+ true negative)/total no. of samples 2. precision: precision talks about how precise and accurate model is out of total predictive positive.it is define as no. of true positive samples divided by total no. of predictive positive samples. precision: true positive samples/ total predictive positive samples 3. recall: recall metrics is the ratio of true positive samples to the total actual positive samples. recall : true positive samples/ total actual positive samples 4. f-measure: f -measure is the function of precision and recall and it is defined as: f-measure = 2 * (precision * recall) / (precision + recall) 4. experiment results and discussion we evaluate our model on karate club dataset; the network graph of the dataset is shown in the figure 3. in the zachary karate club dataset, nodes are 34 and edges are 78. the node is represented as members of club and the relationship or link between the club members is represented by f link and those members who don’t have any link is represented by nf link. in our proposed approach we have applied dimension reduction technique on feature vector which gives us principal components. using these principle components we have int. j. anal. appl. 17 (4) (2019) 583 figure 3. network graph of karate club dataset, nodes=34 and f links=78 table 1. accuracy using pca and without using pca classifier/parameter for evaluation knn decision tree naive bayes svm using pca 0.9343 0.8216 0.7559 0.8779 without using pca 0.8967 0.7887 0.7512 0.8545 applied the classification using k nearest neighbour, naive bayes (nb), svm (support vector machine) and decision tree. in table i and figure 4, the accuracy of proposed approach using pca has been compared with classification of links without using pca. the results are better when we have applied the dimension reduction technique. figure 4. accuracy using pca and without using pca other parameters that we have used for evaluation of our proposed approach is precision, recall and f-measure. the results of classification of links with pca and without using pca are shown in table ii and figure 5. f-measure which is balanced of recall and precision is better using pca than without using pca in all the cases. int. j. anal. appl. 17 (4) (2019) 584 table 2. recall, precision, f-measure using pca and without using pca classifier/parameter for evaluation knn decision tree naive bayes svm recall using pca 0.88 0.85 0.81 0.86 recall without pca 0.82 0.85 0.84 0.79 precision using pca 1 0.77 0.66 0.92 precision without pca 1 0.7 0.61 0.95 fmeasure using pca 0.93 0.80 0.72 0.88 f-measure without pca 0.90 0.76 0.70 0.86 figure 5. f-score, recall and precision 5. conclusion dimension reduction technique allows to reduce the high dimension data to low dimension data and gives the uncorrelated dataset. using one of the most popular techniques of dimension reduction i.e. pca (principle component analysis), we have proposed a feature based learning model. in our approach we have balance the data using oversampling and then applied the principal component analysis on the extracted features which is included common neighbour, resource allocation index and centrality based features (degree centrality, closeness centrality and betweeness centrality). the results depict the performance (accuracy and f-measure) of classification based link prediction model is improved after applying pca. references [1] huang, z, dajun zeng, d. a link prediction approach to anomalous email detection. in: ieee international conference on systems, man and cybernetics, san diego, ca, 2006, 1131?1136. [2] folino, f. and pizzuti, c., link prediction approaches for disease networks. in international conference on information technology in bio-and medical informatics. springer, berlin, heidelberg, 2012, 99-108. [3] esslimani, i., brun, a. and boyer, a., densifying a behavioral recommender system by social networks link prediction methods. social network anal. mining, 1(3)(2011), 159-172. [4] chen, h., li, x. and huang, z., link prediction approach to collaborative filtering. in proceedings of the 5th acm/ieeecs joint conference on digital libraries (jcdl’05), ieee, 2005, 141-142. . int. j. anal. appl. 17 (4) (2019) 585 [5] lü, l., jin, c.h. and zhou, t., similarity index based on local paths for link prediction of complex networks. phys. rev. e, 80(4)(2009), 046122. [6] liu, w. and lü, l., link prediction based on local random walk. europhys. lett. 89(5)(2010), 58007. [7] benchettara, n., kanawati, r. and rouveirol, c., supervised machine learning applied to link prediction in bipartite social networks. in 2010 international conference on advances in social networks analysis and mining. ieee 2010, 326-330. [8] liben-nowell, david, and kleinberg, jon. the link prediction problem for social networks. j. amer. soc. inf. sci. technol. 58(7)(2007), 1019-1031 [9] al hasan, m., chaoji, v., salem, s. and zaki, m., april. link prediction using supervised learning. in sdm06: workshop on link analysis, counter-terrorism and security, 2006. [10] de sa, h.r. and prudencio, r.b., supervised link prediction in weighted networks. in the 2011 international joint conference on neural networks, ieee, 2011, 2281-2288. [11] almansoori, w., gao, s., jarada, t.n., elsheikh, a.m., murshed, a.n., jida, j., alhajj, r. and rokne, j., link prediction and classification in social networks and its application in healthcare and systems biology. network modeling analysis in health informatics and bioinformatics, 1(1-2)(2012), 27-36. [12] liu, z., zhang, q.m., lü, l. and zhou, t., link prediction in complex networks: a local nave bayes model. europhys. lett. 96(4)(2011), 48007. [13] o’madadhain, j., hutchins, j. and smyth, p., prediction and ranking algorithms for event-based network data. acm sigkdd explorations newsletter, 7(2)(2005), 23-30. [14] al hasan, m. and zaki, m.j., a survey of link prediction in social networks. in social network data analytics. springer, boston, ma. 2011. 243-275. [15] kashima, h. and abe, n., a parameterized probabilistic model of network evolution for supervised link prediction. in sixth international conference on data mining (icdm’06). ieee. 2006, 340-349. [16] fawcett, t., an introduction to roc analysis. pattern recognition letters, 27(8)(2006), 861-874. [17] liu, h., hu, z., haddadi, h. and tian, h., hidden link prediction based on node centrality and weak ties. europhys. lett. 101(1)(2013), 18004. [18] freeman, l.c., centrality in social networks conceptual clarification. social networks, 1(3)(1978), 215-239. [19] sabidussi g., the centrality of a graph, psychometrika 31(4)(1966), 581-603. [20] yao, l., wang, l., pan, l. and yao, k., link prediction based on common-neighbors for dynamic social network. proc. computer sci. 83(2016), 82-89. 1. introduction 2. related work 3. methodology 3.1. steps in proposed approach: 3.2. evaluation metrics that we have used in our experiment: 4. experiment results and discussion 5. conclusion references international journal of analysis and applications volume 16, number 2 (2018), 209-221 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-209 on generalized local property of |a; δ|k-summability of factored fourier series b. b. jena1, vandana2,∗, s. k. paikray1 and u. k. misra3 1department of mathematics, veer surendra sai university of technology, burla 768018, odisha, india 2department of management studies, indian institute of technology, madras, tamil nadu-600 036, india 3mathematics, national institute of science and technology, pallur hills, golanthara 761008, odisha, india ∗corresponding author: vdrai1988@gmail.com abstract. the convergence of fourier series of a function at a point depends upon the behaviour of the function in the neighborhood of that point and it leads to the local property of fourier series. in the proposed paper a new result on local property of |a; δ|k-summability of factored fourier series has been established that generalizes a theorem of sarigöl [13] (see [m. a. sariögol, on local property of |a|k-summability of factored fourier series, j. math. anal. appl. 188 (1994), 118-127]) on local property of |a|k-summability of factored fourier series. 1. introduction and motivation suppose ∑ an be a given infinite series with sequence of partial sum (sn) and let a = (anv) be a lower triangular matrix with nonzero diagonal entries. then a defines the sequence-to-sequence transformation received 2017-09-21; accepted 2017-12-07; published 2018-03-07. 2010 mathematics subject classification. 40f05; 40d25. key words and phrases. fourier series; lower triangular matrix; |a; δ|k-summability; local property. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 209 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-209 int. j. anal. appl. 16 (2) (2018) 210 from the sequence s = (sn) to a(s) = (an(s)), with an(s) = n∑ v=0 anvsv. (1.1) a series ∑ an is summable |a|k (k ≥ 1) if, (see [13]) ∞∑ n=1 |ann|1−k|an(s) −an−1(s)|k < ∞, (1.2) and the series ∑ an is summable |a; δ|k (k ≥ 1) if, (see [6]) ∞∑ n=1 |ann|1−k−δk|an(s) −an−1(s)|k < ∞. (1.3) let us consider two lower triangular matrices ā and â associated with a as follows: ānv = n∑ r=v anr, (n,v = 0, 1, 2, ..., ) and ânv = ānv − ān−1,v. (n = 1, 2, 3, ..., ). in special case, when a = (n̄ ,pn) then |a,δ|k-summability reduces to |n̄ ,pn; δ|k-summability and for k = 1, (|n̄ ,pn; δ|) is equivalent to |r,pn; δ|-summability (see [2]). also, if we take a = (c,α) with (α > −1), then |a,δ|k-summability becomes |c,α, (α− 1)(1 − 1/k)δ|k in flett’s notation. furthermore, for double absolute factorable summability matrix (see [11]). we use the notations ∆cn = cn − cn+1 and ∆̄cn,v = cnv − cn−1,v, c−1,0 = 0, (n,v = 0, 1, 2, ..., ). a sequence (λn) is called a convex sequence if, ∆2(λn) ≥ 0 for every n ∈ z+, where int. j. anal. appl. 16 (2) (2018) 211 ∆2(λn) = ∆(λn) − ∆(λn+1) and ∆(λn) = λn −λn+1. let f(t) ∈ l(−π,π) be a 2π periodic function. without loss of generality let us consider that a0 = 0 in the fourier series expansion of f(t) that is, ∫ π −π f(t)dt = 0. (1.4) thus the fourier series expansion of f(t) becomes: f(t) = ∞∑ n=1 (an cos nt + bn sin nt) = ∞∑ n=1 an(t). (1.5) it is well known that the convergence of the fourier series at t = x is a local property of f [16] (i.e., it depends only on the behavior of f in an arbitrarily small neighborhood of x) and hence the summability of the fourier series at t = x by any regular linear summability method is also a local property of f. moreover, as regards to the approximation of fourier series of functions see the recent results [9], [10] and [5]. 2. preliminaries dealing with riesz summability and local property of fourier series, mohanty [12] has established that |r, log(n), 1|-summability of a factored fourier series ∑ an log(n + 1) (2.1) of a function f(t) at any point t = x is a local property of the generating function of f(t) but the summability |c, 1| of this series is not. subsequently, replacing the series (2.1) by ∑ an(t) (log log(n + 1))δ (δ > 1). (2.2) matsumoto [7] as obtained a new result on local property of |r,pn, 1|-summability. generalizing the above result bhatt [1] proved the following theorem: theorem 2.1. suppose (λn) is a convex sequence such that ∑ λn n is convergent, then the |r, log(n), 1|summability of a factored fourier series ∑ an(t)λn log(n) at any point t = x is a local property of f(t). int. j. anal. appl. 16 (2) (2018) 212 by replacing the factor λnlog(n) in a most general form, mishra [8] has proved the following theorem. theorem 2.2. suppose (pn) be a sequence satisfying following conditions: pn = o(npn), pn∆pn = o(pnpn+1). then the |n̄ ,pn|-summability of a factored fourier series ∞∑ n=1 an(t)λnpn(npn)−1 (2.3) at any point t = x is a local property of f(t), where (λn) is a convex sequence. replacing |n̄ ,pn|-summability in mishra’s result, bor [3] proved a more general form on |n̄ ,pn|ksummability method. quite recently, bor [4] introduced the following result on |n̄ ,pn|k-summability of a factored fourier series at any point t = x as a local property of f(t) under more appropriate conditions then those given in the theorem. theorem 2.3. let the positive sequence (pn) and a sequence (λn) be such that ∆xn = o(n −1); ∞∑ n=1 1 n {|λn|k + |λn+1|k}xk−1n 5 ∞; ∞∑ n=1 (xkn + 1)|∆λn| 5 ∞, where xn = (npn) −1pn. then the |n ,pn|k-summability of a factored fourier series ∞∑ n=1 λnxnan(t) at any point t = x is a local property of f(t). later sarigöl (see [13]) has proved the following int. j. anal. appl. 16 (2) (2018) 213 theorem 2.4. suppose that a = (anv) is a positive normal matrix satisfying an−1,v = anv, (n 5 v + 1) ān,0 = 1 (n = 0, 1, 2, ..., ) n−1∑ v=1 avvân,v−1 = o(ann), ∆xn = o(n −1), where xn = 1 (nann) . if a sequence (λn) satisfying following conditions ∞∑ n=1 n−1{|λkn| + |λn+1| k}xkn−1 5 ∞, ∞∑ n=1 (xkn + 1)|∆λn| 5 ∞. then the |a|k-summability of a factored fourier series ∞∑ n=1 λnxnan(t) at any point t = x is a local property of f(t). again to improve upon and generalize theorem 2.4, sulaiman [14] has proved the following theorem for a normal matrix. theorem 2.5. let a = (anv) is a normal matrix satisfying |ân,v+1| ≤ |ann|, ∞∑ n=v+1 |ân,v+1| ≤∞, n−1∑ v=1 |avv||ân,v+1| = o(|ann|), ∆xn = o( 1 n ), int. j. anal. appl. 16 (2) (2018) 214 where xn = 1 (nann) . if a sequence (λn) satisfying the following conditions ∞∑ n=1 n−1{|λkn| + |λn+1| k}xkn−1 5 ∞, ∞∑ n=1 xn|∆λn| 5 ∞. then the |a|k-summability of a factored fourier series ∞∑ n=1 λnxnan(t) at any point t = x is a local property of f(t). 3. main result in the present paper, we have established a new result on local property of |a,δ|k-summability of factored fourier series ∞∑ n=1 λnxnan(t) in the form of a theorem as follows. theorem 3.1. suppose a = (anv) is a positive normal matrix such that an−1,v ≥ an,v (n 5 v + 1); (3.1) ān,0 = 1 (n = 0, 1, ..., ); (3.2) n−1∑ v=1 avvân,v−1 = o(ann); (3.3) m+1∑ n=v+1 ân,v+1a −δk nn = o(v δk); (3.4) m+1∑ n=v+1 a−δknn |∆̄anv| = o(v δk), ; (3.5) ∆xn = o(n −1), (3.6) where xn = 1 (nann) . if a sequence (λn) satisfying the following conditions ∞∑ n=1 n−1{|λ|k + |λn+1|k}xknn δk 5 ∞; (3.7) ∞∑ n=1 (xkn + 1)|∆λn|n δk 5 ∞. (3.8) int. j. anal. appl. 16 (2) (2018) 215 then the |a,δ|k-summability of a factored fourier series ∞∑ n=1 λnxnan(t) at any point t = x is a local property of f(t). remark 3.1. the element ânv = 0 for each n,v. in fact, it is easily seen from the positiveness of the matrix, (3.1) and (3.2), that â00 = 1, ânv = ān0 − āv−1,0 + v−1∑ j=0 (an−1,j −anj) = v−1∑ j=0 (an−1,j −anj) = 0 (1 5 v 5 n) and equal to zero otherwise. in order to prove the above theorem we need the a lemma as follows. lemma 3.1. suppose that the matrix a and the sequence (λn) satisfy the conditions of the theorem, and that (sn) is bounded. then factored fourier series ∞∑ n=1 λnxnan(t) is summable to |a,δ|k (k = 1, δ = 0). proof. let (tn) be an a− transformation of n∑ i=1 λixian(t), then tn = n∑ i=0 anisi = n∑ i=1 ani i∑ v=1 λvxv = n∑ v=1 λvxv n∑ i=v ani = n∑ v=1 ānvλnxv ∆̄tn = tn −tn−1 = n∑ v=1 (ānv − ān−1,v)λvxv = n∑ v=1 ânvλvxv ∆̄tn = n−1∑ v=1 (ânvλvxv)sv + annλnxnsn but, ∆(ânvλvxv) = λvxv∆ânv + ∆(λvxv)ân,v+1 = λvxv∆̄anv + (xv∆λv + ∆xvλv+1)ân,v+1. int. j. anal. appl. 16 (2) (2018) 216 ∆̄tn = n−1∑ v=1 ân,v+1xv∆λvsv + n−1∑ v=1 ân,v+1λv+1∆xvsv + n−1∑ v=1 ∆̄anvλvxvsv + annλnxnsn = tn,1 + tn,2 + tn,3 + tn,4, (say). to complete the proof, it is sufficient to show that by using minkowski’s inequality ∞∑ n=1 a1−k−δknn |tn,m| k < ∞ (m = 1, 2, 3, 4). using hölder inequality and (3.1), (3.2), (3.8), let i1 = m+1∑ n=2 a1−k−δknn |tn,1| k 5 m+1∑ n=2 a1−k−δknn { n−1∑ v=1 ân,v+1xv|∆λv||sv| }k = o(1) m+1∑ n=2 a1−k−δknn { n−1∑ v=1 ân,v+1xv|∆λv| }k = o(1) m+1∑ n=2 a−δknn n−1∑ v=1 ân,v+1x k v |∆λv| { (ann) −1 n−1∑ v=1 ân,v+1|∆λv| }k−1 . since, ân,v+1 = n∑ r=v+1 (anr −an−1,r) = n∑ r=0 (an−1,r −an,r) 5 n−1∑ r=0 (an−1,r −anr) = ān−1,0 − ān0 + ann = ann. ⇒ n−1∑ v=1 ân,v+1|∆λv| 5 ann n−1∑ v=1 |∆λv| = o(ann). int. j. anal. appl. 16 (2) (2018) 217 i1 = o(1) m+1∑ n=2 a−δknn n−1∑ v=1 ân,v+1x k v |∆λv| = o(1) m∑ v=1 xkv |∆λv| m+1∑ n=v+1 ân,v+1a −δk nn = o(1) m∑ v=1 xkv |∆λv|v δk = o(1). using hölder inequality, and (3.3), (3.4), (3.6), (3.7), i2 = m+1∑ n=2 a1−k−δknn |tn,2| k 5 m+1∑ n=2 a1−k−δknn { n−1∑ v=1 ân,v+1|λv+1||∆xv||sv| }k = o(1) m+1∑ n=2 a1−k−δknn { n−1∑ v=1 ân,v+1|λv+1|avvxv }k = o(1) m+1∑ n=2 (ann) −δk n−1∑ v=1 ân,v+1|λv+1|kavvxkv { (ann) −1 n−1∑ v=1 avvân,v+1 }k−1 = o(1) m+1∑ n=2 (ann) −δk n−1∑ v=1 ân,v+1|λv+1|kavvxkv = o(1) m∑ v=1 avvx k v |λv+1| k m+1∑ n=v+1 a−δknn ân,v+1 = o(1) m∑ v=1 avvx k v |λv+1| kvδk = o(1) m∑ v=1 1 v xkv |λv+1| kvδk = o(1). int. j. anal. appl. 16 (2) (2018) 218 using hölder inequality, and (3.1), (3.2), i3 = m+1∑ n=2 a1−k−δknn |tn,3| k 5 m+1∑ n=2 a1−k−δknn { n−1∑ v=1 |∆̄anv||λv|xv|sv| }k = o(1) m+1∑ n=2 a1−k−δknn { n−1∑ v=1 |∆̄anv||λv|xv }k = o(1) m+1∑ n=2 a−δknn n−1∑ v=1 |∆̄anv||λv|kxkv { (ann) −1 n−1∑ v=1 |∆̄anv| }k−1 . we know n−1∑ v=1 |∆̄anv| = n−1∑ v=1 (an−1,v −anv) = ān−1,0 − ān,0 + an0 −an−1,0 + ann = an0 −an−1,0 + ann ≤ ann. i3 = o(1) m+1∑ n=2 a−δknn n−1∑ v=1 |∆̄anv||λv|kxkv = o(1) m∑ v=1 |λv|kxkv m+1∑ n=v+1 a−δknn |∆̄anv| = o(1) m∑ v=1 |λv|kxkvv δkavv = o(1). int. j. anal. appl. 16 (2) (2018) 219 finally, using (3.7), i4 = ∞∑ n=1 a1−k−δknn |tn,4| 5 ∞∑ n=1 a1−k−δknn {ann|λn|xn|sn|} k = o(1) ∞∑ n=1 a1−k−δknn {ann|λn|xn} k = o(1) ∞∑ n=1 (ann) −δkann|λ|kxkn = o(1) ∞∑ n=1 (ann) −δk|λ|kxkn 1 n = o(1). thus the proof of the above lemma is established. proof of the theorem 3.1. since the convergence of the fourier series at a point is a local property of its generating function f(t), the theorem follows by formula from chapter ii of the book (see details [17]) and from the above lemma 3.1. applications. now we apply the theorem to the weighted mean in which a = (anv) is defined as anv = pvp −1 n , when (0 5 v 5 n) where pn = p0 + p1 + ... + pn; therefore, it is well known that ānv = p −1 n (pn −pv−1) and ân,v+1 = (pnpn−1) −1pnpv. one can now easily verify that taking δ = 0 the conditions of the theorem reduce to those of theorem 2.3. we may now ask weather there are some examples (other then weighted mean methods) of matrices a that satisfy the hypotheses of the theorem. for this, apply the theorem to the cesàro method of order α with (0 5 α 5 1) in which a is given by [15] anv = aα−1n−v aαn . int. j. anal. appl. 16 (2) (2018) 220 it is well known that ānv = aαn−v aαn and ânv = vaα−1n−v naαn . it is now seen that by taking account of aαn ≈ nα γ(α+1) conditions (3.1)-(3.8) are satisfied. therefore the above theorem is same as the following result. corollary 3.1. let k ≥ 1 and 0 ≤ α ≤ 1. if (λn) a convex sequence satisfying following conditions: ∞∑ n=1 nαk−α−k{|λ|k + |λn+1|k}nδk 5 ∞, ∞∑ n=1 |∆λn|nδk 5 ∞. then the |c,α, (α− 1)(1 − 1 k )δ|k summability of a factored fourier series ∞∑ n=1 λnxnan(t) with xn = a α n at any point t = x is a local property of the generating function f(t). 4. conclusion the result obtained here is more general in the sense that, by substituting δ = 0, the |a; δ|k-summability reduces to |a|k-summability. acknowledgment the authors would like to express their heartfelt thanks to the editors and anonymous referees for their most valuable comments and constructive suggestions which leads to the significant improvement of the earlier version of the manuscript. references [1] s. n. bhatt, an aspect of local property of |r,log, 1| summability of the factored fourier series, proc. natl. inst. india 26 (1960), 69-73. [2] h. bor, a note on local property of factored fourier series, j. non. anal. 64 (2006), 513-517. [3] h. bor, local property of |n̄ ,pn|k-summability of factored fourier series, bull. inst. math. acad. sinica 17 (1989), 165-170. [4] h. bor, on the local property of |n̄ ,pn|k-summability of factored fourier series, j. math. anal. 163 (1992), 220-226. [5] deepmala, piscoran laurian-ioan, approximation of signals (functions) belonging to certain lipschitz classes by almost riesz means of its fourier series, j. inequal. appl. 2016 (2016), art. id 163. [6] t. m. fleet, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc. 37 (1957), 113-141. [7] k. matsumoto, local property of the summability |r,pn, 1|, tohoku math. j 8 (1956), 114-124. int. j. anal. appl. 16 (2) (2018) 221 [8] k. n. mishra, multipliers for |n̄ ,pn|-summability of fourier series, bull.inst. math. acad. sinica 14 (1984), 431-438. [9] v. n. mishra, some problems on approximations of functions in banach spaces, ph.d. thesis (2007), indian institute of technology, roorkee 247 667, uttarakhand, india. [10] v. n. mishra, l. n. mishra, trigonometric approximation of signals (functions) in lp(p ≥ 1)norm, int. j. contem. math. sci. 7 (2012), 909-918. [11] v. n. mishra, s. k. paikray, p. palo, p. n. samanta, m. misra, u. k. misra, on double absolute factorable matrix summability, tbilisi math. j. 10 (2017), 29-44. [12] r. mohanty, on the summability |r, logw, 1| of fourier series, j. london math. soc. 25 (1950), 67-72. [13] m. a. sariögol, on local property of |a|k-summability of factored fourier series, j. math. anal. appl. 188 (1994), 118-127. [14] w. t. sulaiman, on local property of absolute weighted mean summability of fourier series, bull. math. anal. appl. 4 (2011), 163-168. [15] d. s yu and p. zhou, a new class of matrices and its applications to absolute summability factor theorems, math. comput. model. 57 (2013), 401-412. [16] e. c. tichmarsh, the theory of functions, oxford university press, london, (1961). [17] a. zygmund, trigonometric series, vol. i, cambridge univ. press, cambridge, (1959). 1. introduction and motivation 2. preliminaries 3. main result 4. conclusion acknowledgment references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 144-160 http://www.etamaths.com harmonic beta-preinvex functions and inequalities muhammad aslam noor1,2,∗, khalida inayat noor2 and sabah iftikhar2 abstract. in this paper, we introduce and study a new class of harmonic convex functions which is called harmonic beta-preinvex functions. we establish some estimates, involving the euler beta function and the hypergeometric function of the integral ∫ a+η(b,a) a (x−a)p(a+η(b,a)−x)qf(x)dx for the class of functions whose certain powers of the absolute value are generalized harmonic preinvex function. some special cases are also discussed. results obtained in this paper can be viewed as significant contribution in this fascinating and dynamic field. 1. introduction theory of convex functions had not only stimulated new and deep results in many branches of mathematical and engineering sciences, but also provided us a unified and general frame work to study a wide class of unrelated problems. for recent applications, generalizations and other aspects of convex functions , see [1, 8, 10, 22, 23, 25–28, 36] and the references therein. hanson [6] introduced the concept of invex functions. note that the invex functions are not convex functions. ben-israel and mond [2] introduced the concept of invex sets and peinvex functions. they proved that a every differentiable preinvex function is an function, but the converse is not true. however, noor and noor [14] have established the equivalence between the invex functions and differentiable preinvex functions under some certain conditions. noor [13] proved that the minimum of the differentiable preinvex function on the invex sets can be characterized by a class of variational inequalities, called the variational-like inequalities. noor [15–17] also established some hermite-hadamard type integral inequalities for preinvex functions and log-preinvex functions. in fact, it has been shown that a function f is a preinvex function, if and only if, it satisfies the hermite-hadamard type integral inequalities. this result can be viewed an analogous to the convex functions. these results proved to be the starting point to derive various integral inequalities for different classes of preinvex functions and their variant forms. for recent developments, see [15–21, 29] and the references therein. tunc et. al. [36] introduced beta-convex functions and established some inequalities. noor et al [29] introduced and investigated the beta-preinvex functions. they derived several integral estimates for the betapreinvex functions. for more details, [15–20, 29] and references therein. the class of harmonic convex function was introduced by anderson [1] and iscan [8]. it is natural to unify these different concepts. motivated by these facts, noor et al. [23] introduced and investigated another class of harmonic convex functions, which is called harmonic preinvex functions and can be viewed as significant generalization of both the harmonic convex functions and preinvex functions. noor et al. [30] also considered the harmonic beta-convex functions and derived some estimates for the integrals inequalities. motivated and inspired by the on going research in the convexity theory, we introduce and consider a new class of harmonic convex functions. this new of convex functions is called the harmonic betapreinvex function. it is shown that several new classes of harmonic convex functions and harmonic received 8th november, 2016; accepted 23rd january, 2017; published 1st march, 2017. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. harmonic preinvex functions; beta function; hypergeometric function; hermite-hadamard type inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 144 harmonic beta-preinvex functions 145 preinvex functions can be obtained as special cases. we obtain some new hermite-hadamard type inequalities and estimates for the class of functions whose certain powers of the absolute value are harmonic beta-preinvex involving the euler beta function and the hypergeometric functions. results obtained in this paper continue to hold for the various classes of convex functions. we also mention that the main results of this paper can be derived via the generalized harmonic beta-preinvex functions involving an arbitrary non-negative function. the ideas and techniques of this paper may be starting point for further research. 2. preliminaries let i be a nonempty closed set in r\{0} . let f : i = [a,a+η(b,a)] ⊆ r\{0}−→ r be a continuous function and η(·, ·) : i×i −→ r be a continuous bifunction. in this section, we recall the following new and known concepts. definition 2.1. [23]. a set i ⊆ r\{0} is said to be a harmonic invex set with respect to the bifunction η(·, ·), if x(x + η(y,x)) x + (1 − t)η(y,x) ∈ i, ∀x,y ∈ i,t ∈ [0, 1]. if η(y,x) = y − x, then the harmonic invex set reduces to harmonic convex set. clearly, every harmonic convex set is invex set but the converse is not true. we now introduce a new class of harmonic convex functions, which is called the harmonic beta-preinvex functions. this class unifies the concept of harmonic beta-convex and beta-preinvex functions. definition 2.2. a function f : i ⊆ r\{0}→ r is said to be harmonic beta-preinvex function, where p,q > −1, if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y), ∀x,y ∈ i,t ∈ (0, 1). (2.1) if t = 1 2 , then, we have f ( 2x(x + η(y,x)) 2x + η(y,x) ) ≤ f(x) + f(y) 2p+q (2.2) which is called the harmonic jensen beta-preinvex function. we now discuss some important special cases of harmonic beta-preinvex functions, which include some new and known ones. i). if p = 1 and q = 0, then definition 4.1, reduces to the definition of classical harmonic preinvex functions. definition 2.3. [23]. a function f : i ⊆ r \{0}−→ r is said to be harmonic preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ i,t ∈ [0, 1]. ii). if p = −1 and q = 0, then definition 4.1, reduces to the definition of godunova-levin harmonic preinvex functions. definition 2.4. a function f : i ⊆ r \ {0} −→ r is said to be godunova-levin harmonic preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ f(x) 1 − t + f(y) t , ∀x,y ∈ i,t ∈ (0, 1). iii). if p = 0 and q = 0, then definition 4.1, reduces to the definition of harmonic p-preinvex functions. 146 noor, noor and iftikhar definition 2.5. a function f : i ⊆ r \ {0} −→ r is said to be harmonic p -preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ f(x) + f(y), ∀x,y ∈ i,t ∈ [0, 1]. iv). if p = 1 and q = 1, then definition 4.1, reduces to the definition of harmonic tgs-preinvex functions. definition 2.6. a function f : i ⊆ r \ {0} −→ r is said to be harmonic tgs-preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ t(1 − t)[f(x) + f(y)], ∀x,y ∈ i,t ∈ [0, 1]. v). if p = s and q = 0, then definition 4.1, reduces to the definition of harmonic s-preinvex functions. definition 2.7. a function f : i ⊆ r \ {0} −→ r is said to be harmonic s-preinvex function with respect to η(·, ·), where s ∈ [−1, 1], if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ (1 − t)sf(x) + tsf(y), ∀x,y ∈ i,t ∈ (0, 1). vi). if p = −s and q = 0, then definition 4.1, reduces to the definition of godunova-levin harmonic s-preinvex functions. definition 2.8. a function f : i ⊆ r \{0} −→ r is said to be godunova-levin harmonic s-preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ (1 − t)−sf(x) + t−sf(y), ∀x,y ∈ i,t ∈ (0, 1). vii). if p = 1 2 and q = −1 2 , then definition 4.1, reduces to the definition of generalized harmonic mt-preinvex functions. definition 2.9. a function f : i ⊆ r \ {0} −→ r is said to be generalized harmonic mt -preinvex function with respect to η(·, ·), if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ √ 1 − t √ t f(x) + √ t √ 1 − t f(y), ∀x,y ∈ i,t ∈ (0, 1). definition 2.10. a function f : i ⊆ r \{0} −→ r is said to be harmonic log-beta-preinvex function on i, where p,q > −1, if f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q, ∀x,y ∈ i,t ∈ (0, 1). it follows that log f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ (1 − t)ptq log f(x) + tp(1 − t)q log f(y). from definition 2.10, we have f ( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y). this shows that, harmonic log-beta-preinvex function implies harmonic beta-preinvex function, but the converse is not true. if η(y,x) = y −x, then definition 2.2 reduces to: harmonic beta-preinvex functions 147 definition 2.11. a function f : i ⊆ r\{0}→ r is said to be harmonic beta-convex function, where p,q > −1, if f ( xy tx + (1 − t)y ) ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y), ∀x,y ∈ i,t ∈ (0, 1). (2.3) if t = 1 2 , then, we have f ( 2xy x + y ) ≤ f(x) + f(y) 2p+q (2.4) which is called the harmonic jensen beta-convex function. these classes of harmonic beta-convex functions were introduced and studied by noor et al. [30]. for appropriate and suitable choices of p,q > −1 and the bifunction η(., .), one can obtain several new and known classes of harmonic convex functions from definition 4.1, and definition 2.10. this shows that harmonic beta-preinvex functions are quite general and unifying ones. definition 2.12. a function f : [a,a + η(b,a)] ⊂ r\{0}−→ r is said to be harmonic symmetric with respect to 2a(a+η(b,a)) 2a+η(b,a) , if f(x) = f ( a(a + η(b,a))x (2a + η(b,a))x−a(a + η(b,a)) ) ∀x ∈ [a,a + η(b,a)]. we recall the following special functions which are known as beta function and hypergeometric function respectively. β(x,y) = γ(x)γ(y) γ(x + y) = ∫ 1 0 tx−1(1 − t)y−1dt, x,y > 0, 2f1[a,b; c,z] = 1 β(b,c− b) ∫ 1 0 tb−1(1 − t)c−b−1(1 −zt)−adt, c > b > 0, |z| < 1. we also need the following assumption regarding the bifucntion η(., .), which plays an important role in the derivation of the main results of this paper. condition c [11]. let i ⊆ r be an invex set with respect to bifunction η(·, ·) : i × i → r. for any x,y ∈ i and any t ∈ [0, 1], we have η(y,y + tη(x,y)) = −tη(x,y) η(x,y + tη(x,y)) = (1 − t)η(x,y). 3. main results in this section, we derive hermite-hadamard inequalities for harmonic beta-preinvex function. theorem 3.1. let f : i = [a,a + η(b,a)] ⊆ r \ {0} −→ r be harmonic beta-preinvex function with a < a + η(b,a). if f ∈ l[a,a + η(b,a)], then 2p+q−1f ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x) x2 dx ≤ γ(p + 1)γ(q + 1) γ(p + q + 2) [f(a) + f(b)] proof. let f be harmonic beta-preinvex function. then, taking x = a(a+η(b,a)) a+(1−t)η(b,a) and y = a(a+η(b,a)) a+tη(b,a) in (4.4), and using condition c, we have f ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 2p+q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] = 1 2p+q [∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt + ∫ 1 0 f ( a(a + η(b,a)) a + tη(b,a) ) dt ] 148 noor, noor and iftikhar this implies 2p+q−1f ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x) x2 dx now consider a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x) x2 dx = ∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt ≤ f(a) ∫ 1 0 (1 − t)ptqdt + f(b) ∫ 1 0 tp(1 − t)qdt = [f(a) + f(b)]β(p + 1,q + 1), which is the required result. � theorem 3.2. let f,g : i ⊂ r\{0}−→ r be harmonic beta-preinvex functions. if f,g ∈ l[a,a+η(b,a)], then a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g ( a(a+η(b,a))x (2a+η(b,a))x−a(a+η(b,a)) ) x2 dx ≤ γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) = f(a)g(a) + f(b)g(b) (3.1) n(a,b) = f(a)g(b) + f(b)g(a) (3.2) proof. let f,g be harmonic beta-preinvex functions. then f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b) (3.3) g ( a(a + η(b,a)) a + tη(b,a) ) ≤ (1 − t)ptqg(a) + tp(1 − t)qg(b). (3.4) from (3.3) and (3.4), we have f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)] (3.5) integrating both sides of (3.5), we obtain∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) dt ≤ ∫ 1 0 [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)]dt = [f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt + [f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt = m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) thus a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g ( a(a+η(b,a))x (2a+η(b,a))x−a(a+η(b,a)) ) x2 dx ≤ m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) ≤ γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), which is the required result. � harmonic beta-preinvex functions 149 if g ( a(a+η(b,a))x (2a+η(b,a))x−a(a+η(b,a)) ) = g(x) in theorem 3.2, then it reduces to the following result. corollary 3.1. let f,g : i ⊂ r \ {0} −→ r be harmonic beta-preinvex functions. if f,g ∈ l[a,a + η(b,a)], then a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx ≤ γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) m(a,b) + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. theorem 3.3. let f,g : i ⊂ r\{0}−→ r be harmonic beta-preinvex functions. if fg ∈ l[a,a+η(b,a)], then 22(p+q)−1f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) − a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx ≤ γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. proof. let f be harmonic beta-preinvex function. then taking x = a(a+η(b,a)) a+(1−t)η(b,a) and y = a(a+η(b,a)) a+tη(b,a) in (4.4) and using condition c, we have f ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 2p+q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] , g ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 2p+q [ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + g ( a(a + η(b,a)) a + tη(b,a) )] . consider f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 22p+2q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] [ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + g ( a(a + η(b,a)) a + tη(b,a) )] ≤ 1 22p+2q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) +f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) +[(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b) ] +[tp(1 − t)qf(a) + (1 − t)ptqf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b) ]] . 150 noor, noor and iftikhar integrating over [0, 1], we have ∫ 1 0 f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) dt ≤ 1 22p+2q [∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt + ∫ 1 0 f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] = 1 22p+2q [∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt + ∫ 1 0 f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) dt +2m(a,b)β(p + q + 1,p + q + 1) + 2n(a,b)β(2p + 1, 2q + 1) ] = 1 22p+2q−1 [ a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dt +m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) ] . this completes the proof. � theorem 3.4. let f,g : i ⊂ r\{0}−→ r be harmonic beta-preinvex functions. if fg ∈ l[a,a+η(b,a)], then a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a µ(x) f(a)g(x) + f(b)g(x) x2 dx + a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a µ(x) g(a)f(x) + g(b)f(x) x2 dx ≤ γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) m(a,b) + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) n(a,b) + a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx, where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively and µ(x) = ( ap(a + η(b,a)q((a + η(b,a)) −x)p(x−a)q xp+qη(b,a)p+q ) proof. let f, g be harmonic beta-preinvex functions. then, we have f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b), g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) ≤ (1 − t)ptqg(a) + tp(1 − t)qg(b). harmonic beta-preinvex functions 151 now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ r) and x1 < x2, x3 < x4, we have f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) [(1 − t)ptqg(a) + tp(1 − t)qg(b)] +g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) [(1 − t)ptqf(a) + tp(1 − t)qf(b)] ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b)] +f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) = [f(a)g(a) + f(b)g(b)]t2p(1 − t)2q + [f(a)g(b) + f(b)g(a)]tp+q(1 − t)p+q +f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) integrating over [0, 1], we have g(a) ∫ 1 0 (1 − t)ptqf ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt +g(b) ∫ 1 0 tp(1 − t)qf ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt +f(a) ∫ 1 0 (1 − t)ptqg ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt +f(b) ∫ 1 0 tp(1 − t)qg ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt ≤ [f(a)g(a) + f(b)g(b)] ∫ 1 0 t2p(1 − t)2qdt +[f(a)g(b) + f(b)g(a)] ∫ 1 0 tp+q(1 − t)p+qdt + ∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt this implies a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a µ(x) f(a)g(x) + f(b)g(x) x2 dx + a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a µ(x) g(a)f(x) + g(b)f(x) x2 dx ≤ m(a,b)β(2p + 1, 2q + 1) + n(a,b)β(p + q + 1,p + q + 1) + a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx, which is the required result. � 152 noor, noor and iftikhar theorem 3.5. let f,g : i ⊂ r\{0}−→ r be harmonic beta-preinvex functions. if fg ∈ l[a,a+η(b,a)], then f ( 2a(a + η(b,a)) 2a + η(b,a) ) a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a g(x) x2 dx +g ( 2a(a + η(b,a)) 2a + η(b,a) ) a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x) x2 dx ≤ 1 2p+q [ a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b) ] +2p+q−1f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) , where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. proof. let f, g be harmonic beta-preinvex function. then taking x = a(a+η(b,a)) a+(1−t)η(b,a) and y = a(a+η(b,a)) a+tη(b,a) in (4.4) and using condition c, we have f ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 2p+q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] , g ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 2p+q [ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + g ( a(a + η(b,a)) a + tη(b,a) )] . now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ r) and x1 < x2, x3 < x4, we have 1 2p+q f ( 2a(a + η(b,a)) 2a + η(b,a) )[ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + g ( a(a + η(b,a)) a + tη(b,a) )] + 1 2p+q g ( 2a(a + η(b,a)) 2a + η(b,a) )[ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] ≤ 1 22p+2q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] [ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) +g ( a(a + η(b,a)) a + tη(b,a) )] + f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) ≤ 1 22p+2q [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) +f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) +2[f(a)g(a) + f(b)g(b)]tp+q(1 − t)p+q +2[f(a)g(b) + f(b)g(a)]t2p(1 − t)2q ] +f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) harmonic beta-preinvex functions 153 integrating over [0, 1], we have 1 2p+q f ( 2a(a + η(b,a)) 2a + η(b,a) )∫ 1 0 [ g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + g ( a(a + η(b,a)) a + tη(b,a) )] dt + 1 2p+q g ( 2a(a + η(b,a)) 2a + η(b,a) )∫ 1 0 [ f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) + f ( a(a + η(b,a)) a + tη(b,a) )] dt ≤ 1 22p+2q [∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt + ∫ 1 0 f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] + ∫ 1 0 f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) dt = 1 22p+2q [∫ 1 0 f ( a(a + η(b,a)) a + (1 − t)η(b,a) ) g ( a(a + η(b,a)) a + (1 − t)η(b,a) ) dt + ∫ 1 0 f ( a(a + η(b,a)) a + tη(b,a) ) g ( a(a + η(b,a)) a + tη(b,a) ) dt +2m(a,b)β(p + q + 1,p + q + 1) + 2n(a,b)β(2p + 1, 2q + 1) ] +f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) from the above inequality, it follows that f ( 2a(a + η(b,a)) 2a + η(b,a) ) a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a g(x) x2 dx +g ( 2a(a + η(b,a)) 2a + η(b,a) ) a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x) x2 dx ≤ 1 2p+q [ a(a + η(b,a)) η(b,a) ∫ a+η(b,a) a f(x)g(x) x2 dx + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b) ] +2p+q−1f ( 2a(a + η(b,a)) 2a + η(b,a) ) g ( 2a(a + η(b,a)) 2a + η(b,a) ) , which is the requires result. � remark 3.1. if η(b,a) = b−a, then we obtain the new integral inequalities for the class of harmonic beta-convex functions. 4. integral inequalities we need the following lemma in order to obtain new integral inequalities related to harmonic beta-preinvex function. 154 noor, noor and iftikhar lemma 4.1. if f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r is a function such that f ∈ l[a,a + η(b,a)], then the following equality holds for some fixed α,β > 0.∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t f ( a(a + η(b,a)) at ) dt, where at = a + (1 − t)η(b,a). theorem 4.1. let f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a + η(b,a)] and |f| is harmonic beta-preinvex function on [a,a + η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) , where ϕ1(t; a,b) = ∫ 1 0 tα+q(1 − t)β+p a α+β+2 t dt = β(α + q + 1,β + p + 1) bα+β+2 2f1[α + β + 2,α + q + 1; α + β + p + q + 2; 1 − a b ] (4.1) ϕ2(t; a,b) = ∫ 1 0 tα+p(1 − t)β+q a α+β+2 t dt = β(α + p + 1,β + q + 1) bα+β+2 2f1[α + β + 2,α + p + 1; α + β + p + q + 2; 1 − a b ] (4.2) proof. using lemma 4.1 and harmonic beta-preinvexity of |f|, we have∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣dt ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t { (1 − t)ptq|f(a)| +tp(1 − t)q|f(b)| } dt = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( |f(a)| ∫ 1 0 tα+q(1 − t)β+p a α+β+2 t dt +|f(b)| ∫ 1 0 tα+p(1 − t)β+q a α+β+2 t dt ) = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) . this completes the proof. � theorem 4.2. let f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a+η(b,a)] and |f|λ is harmonic beta-preinvex function on [a,a+η(b,a)] and α,β > 0, harmonic beta-preinvex functions 155 λ ≥ 1, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , where ϕ1(t; a,b), ϕ2(t; a,b) are given by (4.1) and (4.2) respectively, and ϕ3(t; a,b) = ∫ 1 0 tα(1 − t)β a α+β+2 t dt = β(α + 1,β + 1) bα+β+2 2f1[α + β + 2,α + 1; α + β + 2; 1 − a b ] proof. using lemma 4.1, harmonic beta-preinvexity of |f|λ and power mean inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣dt ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tα(1 − t)β a α+β+2 t dt )1− 1 λ (∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣λdt ) 1 λ ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tα(1 − t)β a α+β+2 t dt )1− 1 λ (∫ 1 0 tα(1 − t)β a α+β+2 t { (1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ } dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tα(1 − t)β a α+β+2 t dt )1− 1 λ ( |f(a)|λ ∫ 1 0 tα+q(1 − t)β+p a α+β+2 t dt + |f(b)|λ ∫ 1 0 tα+p(1 − t)β+q a α+β+2 t dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , which the the required result. � theorem 4.3. let f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a+η(b,a)] and |f|λ is harmonic beta-preinvex function on [a,a+η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ , 156 noor, noor and iftikhar where 1 λ + 1 µ = 1 and ϕ4(t; a,b) = ∫ 1 0 tαµ(1 − t)βµ a (α+β+2)µ t dt = β(αµ + 1,βµ + 1) b(α+β+2)µ 2f1[(α + β + 2)µ,αµ + 1; (α + β)µ + 2; 1 − a b ]. proof. using lemma 4.1, harmonic beta-preinvexity of |f|λ and the holder’s integral inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣dt ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµ a (α+β+2)µ t dt ) 1 µ (∫ 1 0 ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣λdt ) 1 λ ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµ a (α+β+2)µ t dt ) 1 µ (∫ 1 0 [(1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ]dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ . this completes the proof. � theorem 4.4. let f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a+η(b,a)] and |f|λ is harmonic beta-preinvex function on [a,a+η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a)β 1 µ (pµ + 1,qµ + 1)( |f(a)|λϕ5(t; a,b) + |f(b)|λϕ6(t; a,b) ) 1 λ , where 1 λ + 1 µ = 1 and ϕ5(t; a,b) = ∫ 1 0 tq(1 − t)p a (α+β+2)λ t dt = β(q + 1,p + 1) b(α+β+2)λ 2f1[(α + β + 2)λ,q + 1; p + q + 2; 1 − a b ] ϕ6(t; a,b) = ∫ 1 0 tp(1 − t)q a (α+β+2)λ t dt = β(p + 1,q + 1) b(α+β+2)λ 2f1[(α + β + 2)λ,p + 1; p + q + 2; 1 − a b ]. harmonic beta-preinvex functions 157 proof. using lemma 4.1, harmonic beta-preinvexity of |f|λ and the holder integral inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣dt ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 1 a (α+β+2)λ t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣λdt ) 1 λ ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 1 a (α+β+2)λ t { (1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ } dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ ( |f(a)|λ ∫ 1 0 tq(1 − t)p a (α+β+2)λ t dt + |f(b)|λ ∫ 1 0 tp(1 − t)q a (α+β+2)λ t dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a)β 1 µ (pµ + 1,qµ + 1)( |f(a)|λϕ5(t; a,b) + |f(b)|λϕ6(t; a,b) ) 1 λ . this completes the proof. � theorem 4.5. let f : i = [a,a + η(b,a)] ⊆ r\{0}−→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a+η(b,a)] and |f|λ is harmonic beta-preinvex function on [a,a+η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a)(ϕ7(t; a,b)) 1 µ( |f(a)|λβ(αλ + q + 1,βλ + p + 1) +|f(b)|λβ(αλ + p + 1,βλ + q + 1) ) 1 λ , where 1 λ + 1 µ = 1 and ϕ7(t; a,b) = ∫ 1 0 1 a (α+β+2)µ t dt = 2f1[(α + β + 2)µ, 1; 2; 1 − ab ] b(α+β+2)µ . 158 noor, noor and iftikhar proof. using lemma 4.1, harmonic beta-preinvexity of |f|λ and the holder’s integral inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β a α+β+2 t ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣dt ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 1 a (α+β+2)µ t dt ) 1 µ (∫ 1 0 tαλ(1 − t)βλ ∣∣∣∣f ( a(a + η(b,a)) at )∣∣∣∣λdt ) 1 λ ≤ aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 1 a (α+β+2)µ t dt ) 1 µ (∫ 1 0 tαλ(1 − t)βλ[(1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ]dt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) (∫ 1 0 1 a (α+β+2)µ t dt ) 1 µ ( |f(a)|λ ∫ 1 0 tαλ+q(1 − t)βλ+pdt + |f(b)|λ ∫ 1 0 tαλ+p(1 − t)βλ+qdt ) 1 λ = aα+1(a + η(b,a))β+1ηα+β+1(b,a) ( ϕ7(t; a,b) ) 1 µ( |f(a)|λβ(αλ + q + 1,βλ + p + 1) + |f(b)|λβ(αλ + p + 1,βλ + q + 1) ) 1 λ , this completes the proof. � remarks results obtained in this paper can be the extended for generalized harmonic beta-preinvex functions with appropriate and suitable modifications. to be more precise, we introduce the generalized harmonic beta-preinvex functions as: definition 4.1. a function f : i ⊆ r \ {0} → r is said to be generalized harmonic beta-preinvex function, if there exists a non-negative arbitrary function h such that f( x(x + η(y,x)) x + (1 − t)η(y,x) ) ≤ h(1 − t)f(x) + h(t)f(y), ∀x,y ∈ i,t ∈ (0, 1). (4.3) if t = 1 2 , then, we have f ( 2x(x + η(y,x)) 2x + η(y,x) ) ≤ h( 1 2 ){f(x) + f(y)}, ∀x,y ∈ i, (4.4) which is called the generalized harmonic jensen beta-preinvex function. we note that, if h(t) = (1 − t)ptq, p,q > −1, then the generalized harmonic beta-preinvex functions reduce to the harmonic beta-preinvex functions as defined in definition 2.2. for appropriate choice of the bifunctions η(., .) and the non-negative function h(.), one can obtain several new classes of convex, harmonic convex, harmonic preinvex functions and their variant forms. these class of convex functions are the subject of our future work. harmonic beta-preinvex functions 159 acknowledgements authors are pleased to acknowledge the support of distinguished scientist fellowship program (dsfp), king saud university, riyadh, saudi arabia. the authors would like to thank dr. s. m. junaid zaidi(h.i.,s.i.), rector, comsats institute of information technology, pakistan, for providing excellent research and academic environment. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] a. ben-isreal and b. mond. what is invexity? j. austral. math. soc. ser. b, 28(1)(1986), 1-9. [3] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publisher, dordrechet, holland, (2002). [4] g. cristescu, improved integral inequalities for product of convex functions, j. inequal. pure appl. math., 6(2)(2005) art. id 35. [5] j. hadamard, etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann. j. math. pure appl., 58(1893), 171-215. [6] m. a. hanson. on sufficiency of the kuhn-tucker conditions, j. math. anal. appl. 80(1981), 545-550. [7] c. hermite, sur deux limites d’une integrale definie. mathesis, 3(1883), 82. [8] i, iscan, hermite-hadamard type inequalities for harmonically convex functions. hacett, j. math. stats., 43(6)(2014), 935-942. [9] i, iscan, m. aydin and s. dikmenoglu, new integral inequalities via harmonically convex functions, math. stat., 3(5)(2015), 134-140. [10] m. liu, new integral inequalities involving beta function via p-convexity, miskol. math. notes, 15(2)(2014), 585591. [11] s. r. mohan and s. k. neogy, on ivex sets and preinvex functions, j. math. anal. appl. 189(1995), 901-908. [12] c. p. niculescu and l. e. persson, convex functions and their applications, springer-verlag, new york, (2006). [13] m. a. noor, variational-like inequalities, optimization, 30(1994), 323-330. [14] m. a. noor and k. i. noor, some characterizations of strongly preinvex functions, j. math. anal. appl. 316(2006), 697-706. [15] m. a. noor, hermite-hadamard integral inequalities for log-preinvex functions, j. math. anal. approx. theory, 2(2007), 126-131. [16] m. a. noor, hadamard integral inequalities for product of two preinvex functions, nonl. anal. forum, 14(2009), 167-173. [17] on hadamard integral inequalities involving two log-preinvex functions, j. inequal. pure appl. math. 8(1)(2007), 1-14. [18] m. a. noor, k. i. noor and m. u. awan, fractional hermite-hadamrd inequalities for for two kinds of s-preinvex functions, nonlinear sci. lett. a. 8(1)(2017), 11-24. [19] m. a. noor, k. i. noor and m. u. awan, some quantum integral inequalities via preinvex functions, appl. math. comput. 269(2015), 242-251. [20] m. a. noor, k. i. noor and s. khan, some inequalities for geometrically preinvex functions, proc. jungjeon math. soc. 19(1)(2016), 67-81. [21] m. a. noor, s. khan and k. i. noor, integral inequalities for geometrically log-preinvex functions, appl. math. inform. sci. lett. 4(3)(2016), 103-110. [22] m. a. noor, k. i. noor, m. u. awan and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b. sci. bull. serai a, 77(1)(2015), 5-16. [23] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic preinvex functions, saussurea, 6(2)(2016), 34-53. [24] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions, twms j. pure appl. math. 7(1) (2016), 3-19 [25] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic nonconvex functions, magnt res. report. 4(1)(2016), 24-40. [26] m. a. noor, k. i. noor and s. iftikhar, some newtons’s type inequalities for harmonic convex functions. j. adv. math. stud., 9(1)(2016), 7-16. [27] m. a. noor, k. i. noor, s. iftikhar and k. al-bany, inequalities for mt-harmonic convex functions, j. adv. math. stud., 9(2)(2016), 194-207. [28] m. a. noor, k. i. noor and s. iftikhar, integral inequalities of hermite-hadamard type for harmonic (h,s)-convex functions, int. j. anal. appl., 11(1)(2016), 61-69. [29] m. a. noor, k. i. noor and s. iftikhar, some integral inequalities for beta-preinvex functions, inter. j. anal. appl. 13(1)(2017), 41-53. [30] m. a. noor, k. i. noor and s. iftikhat, harmonic beta-convex functions and integral inequalities, preint(2017). [31] m. a. noor, k. i. noor, s. iftikhar and f. safdar, integral inequalities for relative harmonic (s,η)-convex functions, appl. math. comput. sci. 1(1)(2016), 27-34. [32] b. g. pachpatte, on some inequalities for convex functions. rgmia res. rep. coll., 6(e)(2003), 1-8. 160 noor, noor and iftikhar [33] m. e. ozdemir, e. set, and m. alomari, integral inequalities via several kinds of convexity, creat. math. inform., 20(1)(2011), 62-73. [34] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [35] d. d. stancu, g. coman, and p. blaga, analiza numericǎ si teoria aproximǎ arii. clujnapoca: presa universitara clujeanǎ, ii(2002). [36] m. tunc, u. sanal and e. gov, some hermite-hadamard inequalities for beta-convex and its fractional applications, new trends math. sci. 3(4)(2015), 18-33. 1department of mathematics, king saud university, riyadh, saudi arabia 2department of mathematics, comsats institute of information technology, park road, islamabad, pakistan ∗corresponding author: noormaslam@ksu.edu.sa, noormaslam@gmail.com 1. introduction 2. preliminaries 3. main results 4. integral inequalities remarks acknowledgements references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 64-68 http://www.etamaths.com some new ostrowski type inequalities via fractional integrals ghulam farid∗ abstract. we have found a new version of well known ostrowski inequality in a very simple and antique way via riemann-liouville fractional integrals. also some related results have been derived. 1. introduction ostrowski type inequalities via riemann-liouville fractional integrals ostrowski inequality. in 1938, the following celebrated inequality was established by ostrowski [11]. theorem 1.1. let f : i −→ r where i is an interval in r, be a mapping differentiable in i◦, the interior of i and a,b ∈ i◦, a < b. if |f′(t)| ≤ m, for all t ∈ [a,b], then we have∣∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ [ 1 4 + (x− a+b 2 )2 (b−a)2 ] (b−a)m, (1.1) for x ∈ [a,b]. it is well known as ostrowski inequality and its consideration by a lot of mathematicians reflects importance and motivation. in fact ostrowski inequality plays a vital role while studying the error bounds of different numerical quadrature rules for example mid point’s, trapezoidal’s, simpson’s and other generalized riemann type. it also motivated the researchers to find its refinements, generalizations, extensions and their applications (see, [1–5, 12] and references therein). riemann-liouville fractional integral operators. fractional calculus deals with the study of integral and differential operators of non-integral order. many mathematicians like liouville, riemann and weyl made major contributions to the theory of fractional calculus. the study on the fractional calculus continued with contributions from fourier, abel, lacroix, leibniz, grunwald and letnikov, (for details see, [6, 8, 10]). riemann-liouville fractional integral operator is the first formulation of an integral operator of non-integral order. definition 1.1. [14] let f ∈ l1[a,b]. then the riemann-liouville fractional integrals of f of order α > 0 with a ≥ 0 is defined by iαa+f(x) = 1 γ(α) ∫ x a (x− t)α−1f(t)dt, x > a and iαb−f(x) = 1 γ(α) ∫ b x (t−x)α−1f(t)dt, x < b. in fact these formulations of fractional integral operators have been established due to letnikov [9], sonin [13] and then by laurent [7]. received 12th january, 2017; accepted 17th march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 26b15, 26a33, 26a24. key words and phrases. ostrowski inequality; fractional integrals. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 64 some new ostrowski type inequalities via fractional integrals 65 fractional ostrowski type inequalities. remaining within the assumptions of ostrowski inequality following more general inequality is observed. theorem 1.2. under the assumptions of theorem 1.1 we have∣∣∣f(x) ((b−x)β + (x−a)α)−(γ(β + 1)iβb−f(x) + γ(α + 1)iαa+f(x))∣∣∣ (1.2) ≤ m ( β β + 1 (b−x)β+1 + α α + 1 (x−a)α+1 ) , x ∈ [a,b] where α,β > 0. proof. for t ∈ [a,x],α > 0 we have (x− t)α ≤ (x−a)α. (1.3) under given condition on f′ and by (1.3) we have∫ x a (m −f′(t))(x− t)αdt ≤ (x−a)α ∫ x a (m −f′(t))dt and ∫ x a (m + f′(t))(x− t)αdt ≤ (x−a)α ∫ x a (m + f′(t))dt. integrating and simplifying the calculations we obtain the following inequalities f(x)(x−a)α − γ(α + 1)iαa+f(x) ≤ mα α + 1 (x−a)α+1 and γ(α + 1)iαa+f(x) −f(x)(x−a) α ≤ mα α + 1 (x−a)α+1. above inequalities result the following inequality |f(x)(x−a)α − γ(α + 1)iαa+f(x)| ≤ mα α + 1 (x−a)α+1. (1.4) now on the other hand for t ∈ [x,b],β > 0 we have (t−x)β ≤ (b−x)β. (1.5) under given condition on f′ and by (1.5) we have∫ b x (m −f′(t))(t−x)βdt ≤ (b−x)β ∫ b x (m −f′(t))dt and ∫ b x (m + f′(t))(t−x)βdt ≤ (b−x)β ∫ b x (m + f′(t))dt. integrating and simplifying the calculations we obtain the following inequalities f(x)(b−x)β − γ(β + 1)iβ b− f(x) ≤ mβ β + 1 (b−x)β+1 and γ(β + 1)i β b− f(x) −f(x)(b−x)β ≤ mβ β + 1 (b−x)β+1. above inequalities result the following inequality∣∣∣f(x)(b−x)β − γ(β + 1)iβb−f(x)∣∣∣ ≤ mββ + 1 (b−x)β+1. (1.6) by adding (1.4) and (1.6) we get (1.2). � the following more general result for a differentiable function which is bounded below as well as bounded above holds. 66 farid theorem 1.3. let f : i −→ r where i is an interval in r, be a mapping differentiable in i◦, the interior of i and a,b ∈ i◦, a < b. if m < f′(t) ≤ m for all t ∈ [a,b], then we have( (x−a)α − (b−x)β ) f(x) − ( γ(α + 1)iαa+f(x) − γ(β + 1)i β b− f(x) ) ≤ mα α + 1 (x−a)α+1 − mβ β + 1 (b−x)β+1, x ∈ [a,b] and ( (b−x)β − (x−a)α ) f(x) + ( γ(α + 1)iαa+f(x) − γ(β + 1)i β b− f(x) ) ≤ mβ β + 1 (b−x)β+1 − mα α + 1 (x−a)α+1, x ∈ [a,b], where α,β > 0 proof. proof is on the same lines just after comparing conditions on derivative of f, of the proof of theorem 1.2, let we left it for the reader. � in the following we have obtained a related result to fractional ostrowski inequality (1.2). theorem 1.4. under the assumptions of theorem 1.1 we have∣∣∣((b−x)βf(b) + (x−a)αf(a))−(γ(β + 1)iβx+f(b) + γ(α + 1)iαx−f(a))∣∣∣ (1.7) ≤ m ( β β + 1 (b−x)β+1 + α α + 1 (x−a)α+1 ) , x ∈ [a,b] where α,β > 0. proof. for t ∈ [a,x],α > 0 we have (t−a)α ≤ (x−a)α. (1.8) under given condition on f′ and by (1.8) we have∫ x a (m −f′(t))(t−a)αdt ≤ (x−a)α ∫ x a (m −f′(t))dt and ∫ x a (m + f′(t))(t−a)αdt ≤ (x−a)α ∫ x a (m + f′(t))dt. integrating and simplifying the calculations we obtain the following inequalities γ(α + 1)iαx−f(a) −f(a)(x−a) α ≤ mα α + 1 (x−a)α+1 and f(a)(x−a)α − γ(α + 1)iαx−f(a) ≤ mα α + 1 (x−a)α+1. above inequalities result the following inequality |f(a)(x−a)α − γ(α + 1)iαx−f(a)| ≤ mα α + 1 (x−a)α+1. (1.9) now on the other hand for t ∈ [x,b],β > 0 we have (b− t)β ≤ (b−x)β. (1.10) under given condition on f′ and by (1.10) we have∫ b x (m −f′(t))(b− t)βdt ≤ (b−x)β ∫ b x (m −f′(t))dt and ∫ b x (m + f′(t))(b− t)βdt ≤ (b−x)β ∫ b x (m + f′(t))dt. integrating and simplifying the calculations we obtain the following inequalities f(b)(b−x)β − γ(β + 1)iβ x+ f(b) ≤ mβ β + 1 (b−x)β+1 some new ostrowski type inequalities via fractional integrals 67 and γ(β + 1)i β x+ f(b) −f(b)(b−x)β ≤ mβ β + 1 (b−x)β+1. above inequalities result the following inequality∣∣∣f(b)(b−x)β − γ(β + 1)iβx+f(b)∣∣∣ ≤ mββ + 1 (b−x)β+1. (1.11) by adding (1.9) and (1.11) we get (1.7). � some implications. following implications have been observed. corollary 1.1. if β takes value α in (1.2), then we leads the following fractional ostrowski inequality |f(x) ((b−x)α + (x−a)α) − γ(α + 1) (iαb−f(x) + i α a+f(x))| ≤ m α α + 1 ( (b−x)α+1 + (x−a)α+1 ) , x ∈ [a,b], where α > 0. corollary 1.2. if β and α simultaneously take value 1, then we lead to the ostrowski inequality (1.1). corollary 1.3. if β takes value α in theorem 1.4, then we lead to the following inequality |((b−x)αf(b) + (x−a)αf(a)) − γ(α + 1) (iαx+f(b) + i α x−f(a))| ≤ mα α + 1 ( (b−x)α+1 + (x−a)α+1 ) , x ∈ [a,b], where α > 0. remark 1.1. following the steps of the proof of theorem 1.2 line by line with α = β = 1, an alternative proof of the ostrowski inequality is followed (see, [5]). remark 1.2. if m is replaced with −m in theorem 1.3, then with some rearrangements one can get theorem 1.2. remark 1.3. a more general form of theorem 1.4 like theorem 1.3 for a differentiable function which is bounded below as well as bounded above holds which we leave for reader. acknowledgement this research is supported by the higher education commission of pakistan. references [1] p. cerone, s. s. dragomir, midpoint-type rules from an inequalities point of view, handbook of analyticcomputational methods in applied mathematics, editor: g. anastassiou, crc press, new york, 2000. [2] s. s. dragomir, ostrowski-type inequalities for lebesgue integral: a survey of recent results, aust. j. math. anal. appl., 14 (1) (2017), 1-287. [3] s. s. dragomir, t. m. rassias, (eds.) ostrowski-type inequalities and applications in numerical integration, kluwer academic publishers, dordrecht, boston, london, 2002. [4] s. s. dragomir, s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and some numerical quadrature rules, comput. math. appl., 33 (1997), 15-20. [5] g. farid, straightforward proofs of ostrowski inequality and some related results, int. j. anal. 2016 (2016), article id 3918483. [6] a. a. kilbas, h. m. srivastava, j. j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, 204, elsevier, new york-london, 2006. [7] h. laurent, sur le calcul des derivees a indicies quelconques, nouv. annales de mathematiques., 3 (3) (1884), 240-252. [8] m. lazarević, advanced topics on applications of fractional calculus on control problems, system stability and modeling, wseas press, 2014. [9] a. v. letnikov, theory of differentiation with arbitray pointer (russian), matem. sbornik., 3 (1868), 1-66. [10] k. miller, b. ross, an introduction to the fractional calculus and fractional differential equations, john, wiley and sons inc, new york, 1993. [11] a. ostrowski, über die absolutabweichung einer dierentierbaren funktion von ihren integralmittelwert, comment. math. helv., 10 (1938), 226–227. [12] x. qiaoling, z. jian, l. wenjun, a new generalization of ostrowski-type inequality involving functions of two independent variables, comput. math. appl., 60 (2010), 2219–2224. 68 farid [13] n. y. sonin, on differentiation with arbitray index, moscow matem. sbornik., 6 (1) (1869), 1-38. [14] z. tomovski, r. hiller, h. m. srivastava, fractional and operational calculus with generalized fractional derivative operators and mittag-leffler function, integral transforms spec. funct., 21 (11) (2010), 797-814. comsats institute of information technology, department of mathematics attock campus, attock, pakistan ∗corresponding author: faridphdsms@hotmail.com 1. introduction ostrowski type inequalities via riemann-liouville fractional integrals ostrowski inequality riemann-liouville fractional integral operators fractional ostrowski type inequalities some implications acknowledgement references international journal of analysis and applications volume 17, number 5 (2019), 879-891 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-879 computing structured singular values for sturm-liouville problems mutti-ur rehman1,∗, ghulam abbas1 and arshad mehmood2 1department of mathematics, sukkur iba university, sukkur-sindh 65200 pakistan 2department of mathematics, the university of lahore, gujrat campus, gujrat-punjab 50700 pakistan ∗corresponding author: mutti.rehman@iba-suk.edu.pk abstract. in this article we present numerical computation of pseudo-spectra and the bounds of structured singular values (ssv) for a family of matrices obtained while considering matrix representation of sturmliouville (s-l) problems with eigenparameter-dependent boundary conditions. the low rank ode’s based technique is used for the approximation of the bounds of ssv. the lower bounds of ssv discuss the instability analysis of linear system in system theory. the numerical experimentation show the comparison of bounds of ssv computed by low rank ode’s technique with the well-known matlab routine mussv available in matlab control toolbox. 1. introduction the spectrum of a matrix sturm-liouville (s-l) problem was characterized by f.v.atkinson in terms of the spectrum of a class of unitary matrices in his famous book titled discrete and continuous boundary value problems. the atkinson’s team had developed a prototype fortran code in order to numerically compute the spectrum of s-l problems [1]. the spectrum of a regular, self-adjoint s-l problem is not bounded and therefore as a result is infinite. the spectrum of a regular, self-adjoint s-l problem is bounded and finite when the coefficients of s-l problem satisfies the external imposed conditions. for a positive integer n, exactly n eigenvalues are computed for a class of s-l problems [2]. the finite spectrum to s-l problem while received 2019-06-12; accepted 2019-07-17; published 2019-09-02. 2010 mathematics subject classification. 15a18, 15a03, 80m50. key words and phrases. eigen values; singular values; singular vectors; low rank ode’s. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 879 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-879 int. j. anal. appl. 17 (5) (2019) 880 considering the transmission conditions are studied in [3] which shows the equivalent matrix representation. the matrix representation of s-l problems for eigen parameter dependent boundary conditions are studied in [4–6]. a s-l problem is one of the form −(py′)′ + q y = λwy on an open interval i = (a,b) with −∞ < a < b < ∞ having the eigen parameter dependent boundary conditions of the form aηθ(a) + bηθ(b) = 0, θ = (y py ′)t with aη =  ηα′1 −α1 −ηα′2 + α2 0 0   ,bη =   0 0 ηβ′1 + β1 −ηβ′2 −β2   , where αi, α ′ i, βi, β ′ i ∈ r, ∀i = 1 : 2, such that ξ1 = α ′ 1α2 −α1α ′ 2 6= 0, ξ2 = β ′ 1β2 −β1β ′ 2 6= 0. the parameter η is the spectral parameter while the coefficients satisfying the minimal conditions r = 1 p ,q,w ∈ l(i,c) where l(i,c) is the complex valued function which are lebesgue integrable on the interval i. the s-l problem as considered above is said to be of atkinson type if for n > 1 there exists a = a0 < b0 < · · · < an < bn = b with r = 1 p = 0 on [ak,bk], wk = ∫ bk ak wdw 6= 0, ∀k = 1 : n and q = w = 0 on [bk−1,ak], rk = ∫ bk ak rdr 6= 0, ∀k = 1 : n. the s-l problem with the eigen parameter dependent boundary conditions is equivalent to matrix eigenvalue problem of the form (p + q) u = ηwu where u = [v0,u0,u1, ...,un,vn+1] is an eigenvector. this finally lead us to the following eigenvalue problem where we aim to approximate eigenvalues ηi, singular values σi and structured singular values µi of the matrix (p + q)w−1 with (p + q) u = ηwu, in turns this implies that ( (p + q)w−1 −ηi ) u = 0. the singular value decomposition (svd) is one of the important tool in the modern days numerical analysis and specially in numerical linear algebra. the applications of svd and it’s basic theory are studied in [7]. the main idea of svd is based upon the factorization of the given matrix a ∈ rm,n with m ≥ n. the int. j. anal. appl. 17 (5) (2019) 881 svd tool splits up matrix a into further three matrices u, σ and v t. the matrix σ is the one look like as a diagonal matrix and having the non-negative quantities σi along it’s main diagonal. the quantities σi are known as the singular values of a. the singular values are the positive square roots of the spectrum of matrix ata rather than a. the columns vectors of u are known as the left singular vectors of a while the orthonormal eigen vectors of aat. on the other hand, the column vectors of the orthogonal matrix v acts as the right singular vectors of a and orthonormal eigen vectors for ata. both singular values and singular vectors are relatively insensitive to the perturbations across the elements of the matrix under consideration. these quantities are also insensitive to finite precision error [8]. the singular values are well-conditioned with respect to an accuracy [9]. golub-kahan-reinsch singular value decomposition algorithm [10] is the standard numerical algorithm used for the approximation of the singular values of a matrix. hestenes algorithm [11] and kogbetliantz algorithm [12] acts as the parallel algorithms for the computation of the singular values σi. the algorithm by golub-kahan-reinsch is computationally very efficient on the sequential machine but however it’s not much attractive on the parallel processor [13]. the structured singular values (ssv) is the generalization of the singular values of a square, rectangular matrix a ∈ km,n with k = r,c. ssv was first introduced by j.c.doyle [14]. the ssv tool widely used in control, system theory to investigate both stability and instability of feedback systems. for applications we refer [15]. unfortunately, the computation of an exact value of ssv is not possible and appear as an np-hard problem [17] which allows to develop numerical methods in order to approximate bounds of ssv. the lower bounds of ssv are approximated by using power method [16] while [18] is used to approximate it’s upper bounds. the lower bounds of ssv provide sufficient information about the instability of closed loop system while upper bounds is used to study the stability of feedback system in linear control theory. 1.1. preliminaries. definition 1.1. the spectrum of a square a complex valued matrix m ∈ cn,n is defined as λ(m) = {λ ∈ c : |(λi −m) | = 0}. definition 1.2. the pseudospectrum of a complex matrix m ∈ cn,n with a small positive real parameter � > 0 is defined as λ�(m) = {λ ∈ c : |(λi −m) −1 | ≥ 1 � }. definition 1.3. for a small positive parameter � ≥ 0. a number λ belongs to epsilon-pseudo-spectrum of an operator a, denoted by λ�(a) and satisfies the following equivalent conditions (i) λ ∈ λ(a + e) for some perturbation e having ‖e‖≤ �; int. j. anal. appl. 17 (5) (2019) 882 (ii) ∃ u ∈ cn,n having ‖u‖ = 1 such that ‖au−λu‖≤ �; (iii) λ ∈ ρ(a) and ‖(λi −a)−1 ‖≥ 1 � or λ ∈ λ(a) where ρ(a) denotes the spectral radius of the matrix a. definition 1.4. the pseudospectrum of a complex matrix m ∈ cn,n with a small positive real parameter � > 0 is defined as λ�(m) = {λ ∈ c : |(λi −m) −1 | ≥ 1 � }. definition 1.5. unstructured uncertainty b or stuctured uncertainty b is stable transfer matrix or structured stable transfer matrix having the form. b = {diag(δiii; ∆j) : δi ∈ c, ∆j ∈ cmj,mj,∀i = 1 : s,j = 1 : f}. definition 1.6. for a given n-dimensional square matrix m ∈ cn,n and underlying perturbation set b, the µ-value is defined as µb(m) = 1 min{‖∆‖2 : ∆ ∈ b,det(i −m∆) = 0} . unless no such ∆ cause (i −m∆) to be singular for which µb(m) = 0. 1.2. reformulation of µ-values. in this section we reformulate the µ-values on the basis of structured spectral value sets. the key idea for the reformulation of the structured singular values is to shift the largest eigenvalue of the matrix valued function i − m∆(t) such that for λmax = 1 the new eigenvalue η = 0 as η = 1−λmax and it achieve the maximum value to be one when λmax = 0. on the basis of this mathematical construction, the reformulation of structured singular values is given as below. definition 1.7. for a given m ∈ cn,n and perturbation level � > 0, the structured spectral value set is denoted by λb� (m) and is defined as λb� (m) = {λ ∈ λ(�m∆), ∆ ∈ b,‖∆‖2 ≤ 1}, where λ(�m∆) denotes the spectrum of the matrix valued function (�m∆), and is simply a disk centered at origin 0. definition 1.8. the structured epsilon spectral value set for a given m ∈ cn,n and � ≥ 0, is defined as σb� (m) = {η : 1 −λ : λ ∈ λ b � (m)}. definition 1.9. for a given m ∈ cn,n and an underlying perturbation set b the µ-value is defined as µb(m) = 1 arg min�>0 { max|λ| = 1,λ ∈ λb� (m) }. int. j. anal. appl. 17 (5) (2019) 883 2. pseudo-spectrum in this section we present the pseudospectra for matrices under consideration to whom the goal is to approximate structured singular values. for this purpose we make use of the software package eigtool [19]. eigtool is routinely used for plotting unstructured pseudospectra of the matrices under consideration. in figures 1-4, we show the computation of the pseudospectra of a different matrices as taken in examples 1-4. we also show the absolute values and the real part of the eigenvalues computed for each matrix. the spectrum of the eigenvalues in 3-dimensional space is also shown by making use of eigtool. let a be an n-dimensional matrix and let λ(a) denotes the set of all the eigenvalues of the matrix a. let ‖a‖ denotes the matrix-norm of the matrix a induced by an inner product space 〈·, ·〉. the computation of the pseudo-spectra of an operator is very straightforward but at the same its very costly too. the boundaries associated with the pseudo-spectrum are nothing but just the level curves of the resolvent corresponding to operator a, that is, ‖(λi −a)−1 ‖. the computations of the level curves involves the computation of the numerical values of λ at the grid point in the complex planes and then to compute the desired contour plots. for the computation of the �-pseudo-spectrum, the computation of an admissible perturbation e such that ‖e‖ = � for the perturbed matrix a + e. for the computation of the �-pseudo-spectrum the determination of the sets l� and u� is essential such that l� ≤ λ� ≤ u�. here, l�(a) = {λ ∈ ρ(a) : b(λ) ≥ 1�}∪ λ(a) acts as a lower bound of the �-pseudo-spectrum with � ≥ 0. for an upper bounds of the pseudo-spectrum, u�(a) = {λ ∈ ρ(a) : b(λ) ≥‖(λi −a) −1 ‖} for all λ ∈ ρ(a). for a complete detail we refer [20] and the reference therein. dim = 3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 (a) pseudospectrum of 3-dim real valued matrix figure 1. matlab interface for computing pseudospectrum. the graphical representation show the pseudospectrum of the 3-dimensional real valued matrix (example 1) int. j. anal. appl. 17 (5) (2019) 884 dim = 4 −2 0 2 4 6 8 −4 −3 −2 −1 0 1 2 3 4 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 (a) pseudospectrum of 4-dim real valued matrix figure 2. matlab interface for computing pseudospectrum. the graphical representation show the pseudospectrum of the 4-dimensional real valued matrix (example 2) dim = 5 −4 −2 0 2 4 6 8 −5 −4 −3 −2 −1 0 1 2 3 4 5 −1.1999 −1.0999 −0.9999 −0.8999 −0.7999 −0.6999 −0.5999 −0.4999 −0.3999 −0.2999 −0.1999 −0.0999 0.0001 (a) pseudospectrum of 5-dim real valued matrix figure 3. matlab interface for computing pseudospectrum. the graphical representation show the pseudospectrum of the 5-dimensional real valued matrix (example 3) int. j. anal. appl. 17 (5) (2019) 885 dim = 6 −5 0 5 10 15 −6 −4 −2 0 2 4 6 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 (a) pseudospectrum of 6-dim real valued matrix figure 4. matlab interface for computing pseudospectrum. the graphical representation show the pseudospectrum of the 6-dimensional real valued matrix (example 4) 3. proposed methodology in order to solve the maximization problem discussed in definition 3.3, we make use of numerical method based upon low-rank ordinary differential equations technique. the numerical method is mainly composed of two-level algorithm, that is, inner-algorithm and outer-algorithm. in the inner-algorithm the main objective is to first construct then solve a gradient system of ordinary differential equations. on the other hand in the outer-algorithm we vary the perturbation level � > 0 by means of fast newton iteration. the outeralgorithm computes an exact derivative of an extremizer say ∆(�) for ∆ ∈ b and � > 0. a complete detail of numerical method under consideration is given in [15]. next, we discuss the computation of an extremizer. for this purpose, we first approximate the derivative of an eigenvalue matrix λ(p) of a smooth matrix family say a(p) for some fixed parameter p. 3.1. approximation of an extremizers. a matrix valued function ∆ ∈ b having the largest singular value bounded above by 1 and the matrix valued function (i − �m∆) having a smallest eigenvalue which minimizes the modulus of structured spectral value set ∑b � (m) is known as an extremizer. the following theorem computes extremizer for a chosen smallest complex number belonging to the set ∑b � (m). theorem 5.1. for a perturbation ∆ ∈ b having the block diagonal structure ∆ = {diag(δ1i1, . . .δs′ is′ ,δs′ +1is′ +1, . . .δsis; ∆1, . . . , ∆f}, int. j. anal. appl. 17 (5) (2019) 886 with ‖∆‖2 = 1, acts as a local extremizer of structured spectral value set. for a simple smallest eigenvalue λ = |λ|eιθ,θ ∈ r of matrix valued function (i − �m∆) having right and left evectors x and y scaled as s = eιθy∗x and let z = m∗y. the non-degeneracy conditions z∗kxk 6= 0, ∀ = 1 : s ′ re(z∗kxk) 6= 0, ∀ = 1 : s ′ + 1 : s and ||zs+h||.||xs+h|| 6= 0, ∀h = 1 : f, holds. then magnitude of each complex scalar δi∀i = 1 : s appears to be exactly equal to 1 while each full block possesses a unit 2-norm. 3.2. gradiant system of ode’s. the gradient system of odes for an admissible perturbation ∆ ∈ b to approximate a local extremizer of smallest eigenvalue λ = |λ|�iθ, is obtained as, δ̇i = νi(x ∗ izi −re(x ∗ iziδ̄i)δi); i = 1 : s ′ δ̇l = sign(re(z ∗ l xl)ψ(−1,1)(δl); l = s ′ + 1 : s ∆̇j = νj(zs+jx ∗ s+j −re〈∆j; zs+jx ∗ s+j〉); j = 1 : f, where δi ∈ c, ∀i = 1 : s ′ , δl ∈ r for l = s ′ + 1 and ψ(−1,1), the characteristic function. for more discussion in the construction of gradient system of odes in above equations, we refer to [15]. 3.3. outer-algorithm. in outer-algorithm the main aim is to vary � > 0, the perturbation level by means of fast newton’s itaration. in turn 1 � will provide us the approximation of lower bound of µ-value. we make use of fast newton’s iteration in order to solve a problem |λ(�)| = 1, (3.1) in eq. (5.1), � > 0. in order to solve eq. (5.1), we need to compute d d� (|λ(�)|) , the derivative. the following theorem (5.2) help us to compute d d� (|λ(�)|), when |λ(�)| is simple and ∆(0), λ(0) are assumed to remains smooth in the neighboring region of perturbation level � > 0 theorem 5.2 consider matrix valued function ∆ ∈ b. let x and y as a function of perturbation level � > 0 acts as right and left eigenvectors of matrix valued function (�m∆). consider the scaling of these vector accordingly of theorem (5.2). let z = m∗y and assume that non-degenracy conclusions as discussed in theorem (5.2) yields then, d d� (|λ(�)|) = 1 |y(�∗)x(�)| s∑ i=1 |zi(�)∗xi(�)| + f∑ j=1 ||zs+j(�)||.||ys+j(�)|| > 0. int. j. anal. appl. 17 (5) (2019) 887 3.4. choice of suitable initial value matrix and initial perturbation level. for a suitable choice of the initial value matrix ∆0 and an initial perturbation level �0, we refer to [15]. 4. numerical experimentation example 1. consider a three dimensional real valued matrix m = (p + q)w−1 taken from [21]. m =   −1 −2 0 1 2 2 0 −1 2   . we take the underlying perturbation as θb = {diag(δ1i1, ∆1) : δ1 ∈ r, ∆1 ∈ c2,2}. the well-known matlab routine mussv approximates the bounds of ssv as follows along with the required perturbation ∆̂ as ∆̂ =   −0.2877 0 0 0 0.2124 0.0358 0 0.1880 0.0316   . we compute the matrix 2-norm of ∆̂, that is, ‖∆̂‖2 = 0.2877. the mussv routine computes an upper bound µ upper pd = 3.4762 meanwhile a same lower bound is computed µ lower pd = 3.4762. algorithm [15] computes the lower bounds of ssv as follows while the admissible perturbation �∗∆∗ is obtained as ∆∗ =   −1 0 0 0 0.7384 0.1243 0 0.6537 0.1100   . the perturbation level is computed as �∗ = 2 and an admissible perturbation possesses a unit 2.norm, that is, ‖∆∗‖2 = 1. the lower bound of ssv is obtained as µlowernew = 3.4762. figure. 9 shows the numerical approximations of both lower and an upper bound of ssv. the graphical interpretation shows that in various cases the obtained results for the lower and upper bounds of ssv via low rank ode’s and matlab routine mussv are similar. in some cases it’s clear that obtained results via mussv for the lower bounds of ssv dominates than those obtained by low rank ode’s technique. int. j. anal. appl. 17 (5) (2019) 888 0 0.5 1 1.5 2 2.5 3 3.5 4 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 frequency(rad/sec) u p p e r/ l o w e r b o u n d s b y m u ss v upper bounds by mussv lower bounds by mussv lower bounds by nalgo figure 5. comparison of the bounds of ssv approximated by matlab function mussv and nalgo for a 3-dim real matrix valued function at frequencies w = 1 : 5. example 2. consider a four dimensional real valued matrix m = (p + q)w−1 taken from [21]. m =   −1 −6 0 0 1 6 1.3333 0 0 4 2 1.5 0 0 0.6667 1   . figure. 10 shows the numerical approximations of both lower and an upper bound of ssv. the graphical interpretation shows that in various cases the obtained results for the lower and upper bounds of ssv via low rank ode’s and matlab routine mussv are similar. in some cases it’s clear that obtained results via mussv for the lower bounds of ssv dominates than those obtained by low rank ode’s technique. 0 0.5 1 1.5 2 2.5 3 3.5 4 6 6.5 7 7.5 8 8.5 9 9.5 10 frequency(rad/sec) u p p e r/ l o w e r b o u n d s o f s s v upper bounds by mussv lower bounds by mussv lower bounds by nalgo figure 6. comparison of the bounds of ssv approximated by matlab function mussv and nalgo for a 4-dim real matrix valued function at frequencies w = 1 : 5. int. j. anal. appl. 17 (5) (2019) 889 example 3. consider a five dimensional real valued matrix m = (p + q)w−1 taken from [21]. m =   −1 −6 0 0 0 1 6 1.3333 0 0 0 4 0 −1.5 0 0 0 2.6667 −1 0.5 0 0 0 0.5 1   . figure. 11 shows the numerical approximations of both lower and an upper bound of ssv. the graphical interpretation shows that in various cases the obtained results for the lower and upper bounds of ssv via low rank ode’s and matlab routine mussv are similar. in some cases it’s clear that obtained results via mussv for the lower bounds of ssv dominates than those obtained by low rank ode’s technique. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 10 frequency(rad/sec) u p p e r/ l o w e r b o u n d s o f s s v upper bounds by mussv lower bounds by mussv lower bounds by nalgo figure 7. comparison of the bounds of ssv approximated by matlab function mussv and nalgo for a 5-dim real matrix valued function at frequencies w = 1 : 5. example 4. consider a six dimensional real valued matrix m = (p + q)w−1 taken from [21]. m =   −1 −6 0 0 0 0 1 6 1.3333 0 0 0 0 4 0 −1.5 0 0 0 −6 −0.6667 2.5 0 0 0 0 0 −2 12 0.5 0 0 0 0 2 1   . figure. 12 shows the numerical approximations of both lower and an upper bound of ssv. the graphical interpretation shows that in various cases the obtained results for the lower and upper bounds of ssv via int. j. anal. appl. 17 (5) (2019) 890 low rank ode’s and matlab routine mussv are similar. in some cases it’s clear that obtained results via mussv for the lower bounds of ssv dominates than those obtained by low rank ode’s technique. 0 0.5 1 1.5 2 2.5 3 3.5 4 6 7 8 9 10 11 12 13 frequency(rad/sec) u p p e r/ l o w e r b o u n d s o f s s v upper bounds by mussv lower bounds by mussv lower bounds by nalgo figure 8. comparison of the bounds of ssv approximated by matlab function mussv and nalgo for a 6-dim real matrix valued function at frequencies w = 1 : 5. 5. conclusion. in this article we have presented numerical computation of pseudo-spectra and the bounds of structured singular values (ssv) for a family of matrices obtained while considering the matrix representation of sturm-liouville (s-l) problems with eigenparameter-dependent boundary conditions. the numerical experimentation shows that: • in some cases the lower bounds of ssv obtained by low rank ode’s based technique are sharper than the one approximated by matlab routine mussv. • the matlab routine mussv is very fast compare to low rank ode’s based technique. • the matlab routine mussv additionally approximate an upper bounds of ssv which is not possible while making use of low rank ode’s based technique. references [1] atkinson, fv and krall, am and leaf, gk and zettl, a. on the numerical computation of eigenvalues of matrix sturmliouville problems with matrix coefficients. argonne national laboratory reports, darien, 1987. [2] kong, q and wu, h and zettl, a. sturm–liouville problems with finite spectrum. j. math. anal. appl., 263 (2001) 748–762. [3] ao, ji-jun and sun, jiong and zhang, mao-zhu. the finite spectrum of sturm–liouville problems with transmission conditions. appl. math. comput., 218 (2001), 1166–1173. [4] kong, qingkai and volkmer, hans and zettl, anton. matrix representations of sturm–liouville problems with finite spectrum. results math., 54 (2009), 103–116. int. j. anal. appl. 17 (5) (2019) 891 [5] ao, ji-jun and sun, jiong and zhang, mao-zhu. matrix representations of sturm–liouville problems with transmission conditions. computers math. appl., 63 (2012), 1335–1348. [6] ao, ji-jun and sun, jiong and zhang, mao-zhu. the finite spectrum of sturm–liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. results math., 63 (2013), 1057–1070. [7] vandewalle, joos and de moor, bart. a variety of applications of singular value decomposition in identification and signal processing. svd signal proc. algorithms appl. architect amsterdam, 1988, 43–91. [8] wilkinson, james hardy. the algebraic eigenvalue problem. oxford clarendon, vol. 662, 1965. [9] klema, virginia and laub, alan. the singular value decomposition: its computation and some applications. ieee trans. automatic control, 25 (1980), 164–176. [10] golub, gene and kahan, william. calculating the singular values and pseudo-inverse of a matrix. j. soc. ind. appl. math., ser. b, numer. anal., 2 (1965), 205–224. [11] hestenes, magnus r. inversion of matrices by biorthogonalization and related results. j. soc. ind. appl. math., 6 (1958), 51–90. [12] kogbetliantz, eg. solution of linear equations by diagonalization of coefficients matrix. q. appl. math., 13 (1955), 123–132. [13] luk, franklin t. computing the singular value decomposition on the illiac iv. cornell university, year. 1980. [14] doyle, john. analysis of feedback systems with structured uncertainties. iee proc., part d , 129 (1982), 242–250. [15] guglielmi, nicola and rehman, mutti-ur and kressner, daniel. a novel iterative method to approximate structured singular values. siam j. matrix anal. appl., 38 (2017), 361–386. [16] packard, andy and fan, michael kh and doyle, john. a power method for the structured singular value. proc. 27th ieee conf. decision control, 1988, 2132–2137. [17] braatz, richard p and young, peter m and doyle, john c and morari, manfred. computational complexity of µ calculation. ieee trans. automatic control 39 (1994), 1000-1002. [18] fan, michael kh and tits, andré l and doyle, john c. robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. ieee trans. automatic control 39 (1994), 25-38. [19] wright, thomas g and trefethen, ln. eigtool. software available at http://www.comlab.ox.ac.uk/pseudospectra/eigtool, 2002. [20] reddy, satish c and schmid, peter j and henningson, dan s. pseudospectra of the orr–sommerfeld operator. siam j. appl. math., 53 (1993), 15–47. [21] ao, ji-jun and sun, jiong. matrix representations of sturm–liouville problems with eigenparameter-dependent boundary conditions. linear algebra appl., 438 (2013), 2359–2365. 1. introduction 1.1. preliminaries 1.2. reformulation of -values 2. pseudo-spectrum 3. proposed methodology 3.1. approximation of an extremizers 3.2. gradiant system of ode's 3.3. outer-algorithm 3.4. choice of suitable initial value matrix and initial perturbation level. 4. numerical experimentation 5. conclusion. references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 144-153 http://www.etamaths.com approximating fixed points of generalized nonexpansive mappings in banach spaces bapurao c. dhage abstract. in this paper, we prove a fixed point theorem for the selfmaps of a closed convex and bounded subset of the banach space satisfying a generalized nonexpansive type condition. some results concerning the approximations of fixed points with krasnoselskii and mann type iterations are also proved under suitable conditions. our results include the well-known result of kannan (1968) and bose and mukherjee (1981) as the special cases with a different and constructive method. 1. introduction let (x,d) be a metric space. then banach contraction principle states that if x is complete and f : x → x satisfies the condition (1.1) d(fx,fy) ≤ αd(x,y) for all x,y ∈ x and 0 ≤ α < 1, then f has a unique fixed point. the mapping f satisfying the condition (1.1) is called contraction and when α = 1, f is called nonexpansive. the nonexansive mappings have been studied by kirk and goebel [6] for fixed points. bogin [1] considered a class of generalized nonexpansive mappings characterized by the inequality (1.2) d(fx,fy) ≤ ad(x,y) + b[d(x,fx) + d(y,fy)] + b[d(x,fy) + d(y,fx)] for all x,y ∈ x, where a,b,c are nonnegative real numbers satisfying (1.3) a + 2b + 2c = 1 for the study of fixed points. recently ciric [3] generalized the above class of mappings (1.2)-(1.3) to a wider class mappings characterized by the inequality d(fx,fy) ≤ a max { d(x,y),d(x,fx),d(y,fy), 1 2 [d(x,fy) + d(y,fx)] } + b max{d(x,fx),d(y,fy)} + c[d(x,fy) + d(y,fx)](1.4) for all x,y ∈ x, where the real numbers a,b,c ≥ 0 satisfy the condition (1.5) a + b + 2c = 1. 2010 mathematics subject classification. 47h10. key words and phrases. banach space; nonexpansive mappings; fixed points. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 144 generalized nonexpansive mappings in banach spaces 145 similarly, the study of nonexansive mappings in banach spaces has been made extensively by several authors. bose and mukherjee [2] studied the class of generalized nonexansive mappings for the study of fixed points characterized by the inequality (1.6) ‖fx−fy‖≤ a‖x−y‖ + b [‖x−fx‖ + ‖y −fy‖] + c [‖x−fx‖ + ‖y −fy‖] for all x,y ∈ x, where a,b,c are nonnegative real numbers, a > 0 satisfying the condition (1.7) 3a + 2b + 4c = 1. the aim of the present note is to generalize the above class of mappings (1.6)(1.7) and prove a couple of fixed point theorems under a generalized contraction condition with a different method which in turn generalize fixed point theorems of bose and mukherjee [2] as the special cases. 2. generalized nonexpansive mappings given a non-empty, closed, convex and bounded subset c of the banach space x, consider the class of nonexpansive type mappings f : c → c characterized by the inequality ‖fx−fy‖≤ a max { ‖x−y‖,‖x−fx‖,‖y −fy‖, 1 2 [‖x−fy‖ + ‖y −fx‖] } + b [ ‖x−fx‖ + ‖y −fy‖ ] + c max { ‖x−fy‖,‖y −fx‖ } (2.1) for all x,y ∈ x, where the real numbers a,b,c ≥ 0 satisfy the inequality (2.2) a + b + c ≤ 1 2 . the generalized nonexpansive mappings characterized by the inequalities (2.1) and (2.2) have been considered in dhage [4] in the setting of a metric space for fixed points and are different from the class of ciric’s mappings characterized by the inequalities (1.6) and (1.7). in this section we prove a couple of results concerning the existence of fixed point for the class of generalized nonexpansive mappings (2.1) and (2.2) in a banach space via a scheme of krasnoselskii type iterations. theorem 2.1. let c be a non-empty, closed, convex and bounded subset of the normed linear space x and let f : c → c be a mapping satisfying the inequality (2.1) and (2.2) with a > 0. if the sequence {xn} defined by (2.3) xn+1 = (1 − t)xn + tfxn, n = 0, 1, 2, ...; for some t ∈ (0, 1) and for some x = x0 ∈ c converges to u, then u is a unique fixed point of f. 146 dhage proof. by (2.1), one gets ‖xn+1 −fu‖≤ (1 − t)‖xn −fu‖ + t‖fxn −fu‖ ≤ (1 − t)‖xn −fu‖ + a { ‖xn −u‖,‖xn −fx‖,‖u−fu‖, 1 2 [‖xn −fu‖ + ‖u−fxn‖] } + b [‖xn −fxn + ‖u−fu‖ + ‖] + c max{‖xn −fu‖,‖u−fxn‖}.(2.4) now, xn+1 = (1 − t)xn + tfxn, and so we have (xn+1 −xn) = −t(xn −fxn). this further implies that ‖xn+1 −xn‖ = t‖xn −fxn‖−→ 0 as n →∞. taking the limit as n →∞ in (2.4), we obtain ‖u−fu‖≤ (1 − t)‖u−fu‖ + ta max { 0, 0,‖u−fu‖, 1 2 ‖u−fu‖ } + tb [0 + ‖u−fu‖] + tc max{‖u−fu‖, 0} ≤ [(1 − t) + ta + tb + tc]‖u−fu‖ ≤ (1 − t + a + b + c)‖u−fu‖. since a + b + c < 1, we may choose t ∈ (0, 1) such that t > a + b + c. then from the above inequality, we obtain so u = fu. to prove uniqueness, let v(6= u) be another fixed point of f. then by (2.1), ‖u−v‖ = ‖fu−fv‖ ≤ a max { ‖u−v‖,‖u−fu‖,‖v −fv‖, 1 2 [‖u−fv‖ + ‖v −fu‖] } + b [‖u−fu‖ + ‖v −fv‖] + c max{‖u−fv‖,‖v −fu‖} = (a + c)‖u−v)‖ which is a contradiction. hence u = v and the proof of the theorem is complete. � theorem 2.2. let c be a non-empty, closed, convex and bounded subset of a banach space x. if f : c → c satisfies the inequalities (2.1) and (2.2) with a > 0, b > 0, then f has a unique fixed point. proof. let x = x0 ∈ c be arbitrary and consider the sequence {xn} defined by (2.3). then, we have x1 −x2 = (1 − t)(x0 −x1) + t(fx0 −fx2). generalized nonexpansive mappings in banach spaces 147 then, by (2.1), we obtain ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + t‖fx0 −fx1‖ ≤ (1 − t)‖x0 −x1‖ + ta max { ‖x0 −x1‖,‖x0 −fx0‖,‖x1 −fx1‖, 1 2 [‖x0 −x1‖ + ‖x1 −x2‖] } + tb [‖x0 −fx0‖ + ‖x1 −fx1‖] + tc max{‖x0 −fx1‖,‖x1 −fx0‖}.(2.5) now, x1 = (1 − t)x0 + tfx0, and so we have ⇒ x1 −x0 = −t(x0 −fx0). this further implies that t‖x0 −fx0‖ = ‖x0 −x1‖. again, x2 = (1 − t)x1 + tfx1, and so we have x2 −x1 = −t(x1 −fx1) which again implies that t‖x1 −fx1‖ = ‖x1 −x2‖. similarly, (x0 −fx1) = (x0 −x1) + (x1 −fx1), implies t(x0 −fx1) = t(x0 −x1) + t(x1 −x2)), and t‖x0 −fx1‖≤ t‖x0 −x1‖ + t‖x1 −x2‖. again, x1 −fx0 = x1 −x0 + x0 −fx0 = (x1 −x2) + (x0 −fx0), which gives t(x1 −fx0) = t(x1 −x0) + t(x0 −f(x0) = (1 − t)(x0 −x1), or, t‖x1 −fx0‖ = (1 − t)‖x0 −x1‖. 148 dhage substituting the above values in (2.5), ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a max { ‖x0 −x1‖,‖x0 −x1‖,‖x1 −x2‖, 1 2 [ [ (1 − t)‖x0 −x1‖ + t‖x0 −x1‖ + ‖x1 −x2‖ ]} + b [ ‖x0 −x1‖ + ‖x1 −x2‖ ] + c max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } = (1 − t)‖x0 −x1‖ + a max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} + b [ ‖x0 −x1‖ + ‖x1 −x2‖ ] + c max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } .(2.6) now there are three cases: case i: suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} = ‖x0 −x1‖ and max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } = ‖x0 −x1‖ for t > 1 2 . then from (2.6), (1 − b)‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + (a + b)‖x0 −x1‖ + ct‖x0 −x1‖ + c‖x1 −x2‖. therefore, ‖x1 −x2‖≤ ( (1 − t) + a + b + ct 1 − b− c ) ‖x0 −x1‖ ≤ ( (1 − t) + a + b + c 1 − b− c ) ‖x0 −x1‖ = α1‖x0 −x1‖ case ii: suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} = ‖x1 −x2‖. then, ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a‖x1 −x2‖ + b‖x0 −x1‖ + b‖x1 −x2‖ + ct‖x0 −x1‖ + c‖x1 −x2‖ ≤ ( 1 − t + b + c 1 −a− b− c ) ‖x0 −x1‖ ≤ α2‖x0 −x1‖ [t > a + 2b + 2c] generalized nonexpansive mappings in banach spaces 149 case iii: suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖+‖x1 −x2‖ ]} = 1 2 [‖x1 −x2‖+‖x0 −x1‖]. then, ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a 2 ‖x0 −x1‖ + a 2 ‖x1 −x2‖ + b‖x0 −x1‖ + b‖x1 −x2‖ + c‖x0 −x1‖ + c‖x1 −x2‖ ≤ ( 1 − t + a 2 + b + c 1 − a 2 − b− c ) ‖x0 −x1‖ ≤ α3‖x0 −x1‖ [t > a + 2b + 2c]. let α = max{α1,α2,α3}, then in all above three cases we obtain ‖x1 −x2‖≤ α‖x0 −x1‖. therefore, ‖xn −xn+1‖≤ n+p∑ i=n ‖xi −xi+1‖ ≤ αn 1 −α ‖x0 −x1‖ −→ 0 as n →∞. this shows that {xn} is a cauchy is a sequence in c. since c is a closed subset of a complete space, it is complete. hence {xn} is convergent and converse to a point u ∈ c. the rest of the proof is similar to theorem 2.1 and so we omit the details. � corollary 2.1. let c be a non-empty, closed, convex and bounded subset of the normed linear space x and let f : c → c. suppose that there exists a positive integer r such that f satisfies the contraction condition ‖frx−fry‖≤ a max { ‖x−y‖,‖x−frx‖,‖y −fry‖, 1 2 [‖x−fry‖ + ‖y −frx‖] } + b [ ‖x−frx‖ + ‖y −fry‖ ] + c max { ‖x−fry‖,‖y −frx‖ } (2.7) for all x,y ∈ c, where the real numbers a,b,c ≥ 0, a > 0, satisfy the inequality (2.8) a + b + c ≤ 1 2 . if the sequence {xn} defined by (2.9) xn+1 = (1 − t)xn + tfrxn, n = 0, 1, 2, ...; for some t ∈ (0, 1) and for some x = x0 ∈ c converges to u, then u is a unique fixed point of f. proof. by theorem 2.1 above, the mapping fr has a unique fixed point, say p ∈ c. then we have fr(p) = p. therefore, fr(fp) = fr+1(p) = f(fr(p)) = fp showing that fp is again a fixed point of fr. by uniqueness of p, we get fp = p. thus, f has 150 dhage a unique fixed point p in c and the sequence of iterations given by (2.9) converges to p. the proof of the theorem is complete. � in the following section we prove that the mann iterations of the mapping f in a uniformly convex banach space satisfying (2.1) and (2.2). 3. convergence of mann iterations the following definitions is well-known in the literature. definition 3.1. a self mapping f of a convex subset c of a banach space x is said to be quasi-nonexpansive provided f has a fixed point and if p is a fixed point of f, then ‖fx−p‖≤‖x−p‖ for all x ∈ c. in a uniformly banach space, senter and dotson, jr., have conditions under which the sequence of mann types of iterates of a quasi-nonexpansive mapping converges to a fixed point of the mapping in question. we denote by f(f) the set of all fixed points of f in c. condition i: let c be a convex subset of a uniformly convex banach space x. a mapping f : c → c is said to satisfy condition i if there is a nondecreasing function β : [0,∞) → [0,∞) with β(0) = 0, f(r) > 0 for r ∈ (0,∞) satisfying ‖x− fx‖ > β(d(x,f(f))) for all x ∈ c, where β(d(x,f(f))) = inf{ ‖x−p‖ : p ∈ f(f)}. condition i: let c be a convex subset of a uniformly convex banach space x. a mapping f : c → c is said to satisfy condition i if there is a real number α > 0 such that ‖x−fx‖≥ αd(x,f(f)) for all x ∈ c. it can be easily shown that a mapping which satisfies condition ii also satisfies condition i. now, we state a key theorem of senter and dotson [9] which is used in what follows. before going to the theorem we define the mann iterations of the mapping f on a subset c of the banach space x. let x1 ∈ c be arbitrary and let m(x1, tn,f) be a sequence {xn} defined by xn+1 = (1 − tn)xn + tnf(xn), where tn ∈ [β,γ], 0 < β < γ < 1 and n ∈ n. theorem 3.1 (senter and dotson [9]). let x be a uniformly convex banach space, c a closed, convex and bounded subset of x and let f be a nonexpansive mapping of c into itself. if f satisfies condition i, then for arbitrary x1 ∈ c, the sequence m(x1, tn,f) converges to a member of f(f). below we prove a result concerning the convergence of the sequence of mann iterations to the fixed point of generalized nonexpansive mappings in a uniformly banach space. theorem 3.2. let c be a closed, convex and bounded subset of a uniformly banach space x and let f : c → c be a generalized nonexpansive mapping satisfying the inequalities (2.1) and (2.2). then f has a unique fixed point p and for arbitrary x1 ∈ c, the sequence m(x1, tn,f) of mann iterations converges to p. proof. by theorem 2.1, f has a unique fixed point p in c. we show that the sequence m(x1, tn,f) of mann iterations converges to p for arbitrary x1 ∈ c. this will be achieved in the following two steps: generalized nonexpansive mappings in banach spaces 151 step i: f is quasi-nonexpansive on c. we first show that f is a quasi-nonexpansive mapping on c into itself. assume the contrary, that is, ‖fx−p‖ > ‖x−p‖ for some x ∈ c. then by (2.1), we have ‖fx−p‖ = ‖fx−fp‖ ≤ a max{‖x−p‖,‖x−fx‖,‖p−fp‖, 1 2 [ ‖x−fp‖ + ‖p−fx‖]} + b [ ‖x−fx‖ + ‖p−fp‖ ] + c max{‖x−fp‖,‖p−fx‖} = a max{‖x−p‖,‖x−fx‖, 1 2 [ ‖x−p‖ + ‖fx−p‖ ] } + b‖x−fx‖ + c max{‖x−p‖,‖fx−p‖} ≤ a max{‖x−p‖,‖x−fx‖,‖fx−p‖} + b‖x−fx‖ + c‖fx−p‖ ≤ a max{‖x−fx‖,‖fx−p‖} + b‖x−fx‖ + c‖fx−p‖.(3.1) now there are two cases: case i: suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. then from (3.1), we obtain ‖fx−p‖≤ (a + b + c)‖fx−p‖ which is a contradiction, since a + b = c ≤ 1 2 . case ii: suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. then from (3.1), we obtain ‖fx−p‖≤ (a + b + c)‖x−fx‖ ≤ (a + b + c)[‖x−p‖ + ‖fx−p‖] = (a + b + c)‖x−p‖ + (a + b + c)‖fx−p‖ which further implies that ‖fx−p‖≤ [ a + b + c 1 − (a + b + c) ] ‖x−p‖ which is a contradiction, since a + b + c 1 − (a + b + c) ≤ 1. thus, in both the cases, we obtain a contradiction. therefore, we conclude that ‖fx−p‖≤‖x−p‖ for all x ∈ c and consequently f is quasi-nonexpansive on c. step i: f satisfies condition ii on c. let x ∈ c be arbitrary. then, (3.2) ‖x−p‖≤‖x−fx‖ + ‖fx−p‖. 152 dhage now, by (2.1), ‖fx−p‖ = ‖fx−fp‖ ≤ a max{‖x−p‖,‖x−fx‖,‖p−fp‖, 1 2 [ ‖x−fp‖ + ‖p−fx‖]} + b [ ‖x−fx‖ + ‖p−fp‖ ] + c max{‖x−fp‖,‖p−fx‖} = a max{‖x−p‖,‖x−fx‖} + b‖x−fx‖ + c‖x−p‖.(3.3) now there are two cases: case i: suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. then from (3.1), we obtain ‖fx−p‖≤ (a + c)‖x−p‖ + b‖x−fx‖. substituting above value in (3.2), we obtain ‖x−fx‖≥ 1 3 ‖x−p‖ = 1 3 d(x,f(f)). case ii: suppose that max{‖x−fx‖,‖x−p‖} = ‖x−fx‖. then from (3.1), we obtain ‖fx−p‖≤ (a + b)‖x−fx‖ + c‖x−p‖. substituting above value in (3.2), we obtain ‖x−fx‖≥ 1 3 ‖x−p‖ = 1 3 d(x,f(f)). thus, f satisfies condition ii with α = 1 3 . consequently f satisfies condition i and by an application of theorem 3.1, for arbitrary x1 ∈ c, the sequence m(x1, tn,f) of mann iterations of f converges to p. this completes the proof. � corollary 3.1. let c be a closed, convex and bounded subset of a uniformly banach space x and let f : c → c. suppose that there exists a positive integer r such that f satisfies the generalized contraction condition (2.7) and (2.8). then f has a unique fixed point p and for arbitrary x1 ∈ c, the sequence m(x1, tn,fr) of mann iterations converges to p. proof. by theorem 2.1, the mapping f has a unique fixed point p in c which is also a unique fixed point of fr. now the desired conclusion follows by a direct application of theorem 3.2. � as a consequence of theorem 3.2, we obtain the following fixed point theorem of bose and mukherjee [2] as a corollary. corollary 3.2. let c be a closed, convex and bounded subset of a uniformly banach space x and let f : c → c be a generalized nonexpansive mapping satisfying the inequalities (1.6) and (1.7). then f has a unique fixed point p and for arbitrary x1 ∈ c, the sequence m(x1, tn,f) of mann iterations converges to p. generalized nonexpansive mappings in banach spaces 153 references [1] j. bogin, a generalization of a fixed point theorem of goebel, kirk and shimi, canad math. bull. 19 (1976),7-12. [2] r. k. bose, r. n. mukherjee, approximating fixed points if some mappings, proc. amer. math. soc. 82 (1981), 603-606. [3] lj. b. ciric, on some nonexpansive type mapppings and fixed points, indian j. pure appl. math., 24 (1993), 145-149. [4] b. c. dhage, generalized nonexpansive mappings and fixed points in metric spaces, (submitted) [5] w. c. dotson jr., fixed points of quasi-nonexpansive mappings, j. austral. math. soc. 13 (1972), 167-170. [6] k. goebel, w. a. kirk, topics in metric fixed point theory, cambridge univ. press 1990. [7] r. kannan, some reswults on fixed points, bull. cal. math. soc. 60 (1968), 71-76. [8] w. r. mann, mean value methods in iterations, proc. amer. math. soc. 4 (1953), 506-510. [9] h. f. senter, w. g. dotson, approximating fixed points if nonexpansive mappings, proc. amer. math. soc. 44 (1974), 375-379. kasubai, gurukul colony, ahmedpur-413 515, dist: latur, maharashtra, india international journal of analysis and applications volume 16, number 2 (2018), 149-161 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-149 l-dunford–pettis and almost l-dunford–pettis sets in dual banach lattices halimeh ardakani1∗ and manijeh salimi2 1department of mathematics, payame noor university, iran 2department of mathematics, farhangian university, iran ∗corresponding author: halimeh ardakani@yahoo.com abstract. following the concept of l–limited sets in dual banach spaces introduced by salimi and moshtaghioun, we introduce the concepts of l–dunford–pettis and almost l–dunford–pettis sets in dual banach lattices and then by a class of operators on banach lattices, so called disjoint dunford–pettis completely continuous operators, we characterize banach lattices in which almost l–dunford–pettis subsets of their dual, coincide with l–dunford–pettis sets. 1. introduction a subset a of a banach space x is called limited (resp. dunford–pettis (dp)), if every weak∗ null (resp. weak null) sequence (x∗n) in x ∗ converges uniformly on a, that is lim n→∞ sup a∈a |〈a,x∗n〉| = 0. also if a ⊆ x∗ and every weak null sequence (xn) in x converges uniformly on a, we say that a is an l–set. every relatively compact subset of e is dp. if every dp subset of a banach space x is relatively compact, then x has the relatively compact dp property (abb. dprcp). for example, dual banach spaces with the weak radon-nikodym property (see [11], in short wrnp) and schur spaces (i.e., weak and norm received 2017-10-15; accepted 2017-12-16; published 2018-03-07. 2010 mathematics subject classification. primary 46a40; secondary 46b40, 46b42. key words and phrases. dunford–pettis set; relatively compact dunford–pettis property; dunford–pettis completely continuous operator. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 149 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-149 int. j. anal. appl. 16 (2) (2018) 150 convergence of sequences in x coincide) have the dprcp [6]. also we recall that a banach space x has the dprcp if and only if every dp and weakly null sequence (xn) in x is norm null. recently, the authors in [14], introduced the class of dunford–pettis completely continuous (abb. dpcc) operators on banach spaces. in fact, a bounded linear operator t : x → y between two banach spaces is dpcc if it carries dp and weakly null sequences in x to norm null ones in y . the class of all dpcc operators from x to y is denoted by dpcc(x,y ). in this article, by the definition of l–limited sets in [12] in dual banach spaces, we introduce the concepts of l–dp and almost l–dp sets in banach lattices and then we obtain banach lattices in which two classes of sets coincide. finally by introducing the concept of disjoint dp completely continuous (abb. dpdcc) operators between banach lattices and positive dprcp, we obtain some characterizations of them and then the relation between the positive dprcp of e and dp dcc operators on e is treated. the class of all dpdcc operators from e to y is denoted by dpdcc(e,y ). here, we remember some definitions and terminologies from banach lattice theory. it is evident that if e is a banach lattice, then its dual e∗, endowed with the dual norm and pointwise order, is also a banach lattice. the norm ‖.‖ of a banach lattice e is order continuous if for each generalized net (xα) such that xα ↓ 0 in e, (xα) converges to 0 for the norm ‖.‖, where the notation xα ↓ 0 means that the net (xα) is decreasing, its infimum exists and inf(xα) = 0. a banach lattice e is said to be σ–dedekind complete if every countable subset of e that is bounded above has a supremum. a subset a of e is called solid if |x| ≤ |y| for some y ∈ a implies that x ∈ a and the solid hull of a is the smallest solid set containing a and is exactly the set sol(a) = {y ∈ e : |y| ≤ |x|, for some x ∈ a}. throughout this article, x and y denote the arbitrary banach spaces and x∗ refers to the dual of the banach space x. we use lw∗(x ∗,y ) for banach spaces of all bounded weak∗-weak continuous operators from x∗ to y . also e and f denote arbitrary banach lattices and e+ = {x ∈ e : x ≥ 0} refers to the positive cone of the banach lattice e. be is the closed unit ball of e. if x is an element of a banach lattice e, then absolute value of x is denoted by |x|. if a,b belong to e and a ≤ b, the interval [a,b] is the set of all x ∈ e such that a ≤ x ≤ b. a subset of a banach lattice is called order bounded if it is contained in an order interval. a linear mapping t from e into f is called order bounded if it carries order bounded subsets of e into order bounded sets. we recall from [10] that, an element x belonging to a riesz space e is discrete, if x > 0 and |y| ≤ x implies y = tx for some real number t. if every order interval [0,y] in e contains a discrete element, then e is said to be a discrete riesz space. the lattice operations in the banach lattice e are weakly sequentially continuous if for every weakly null sequence (xn) in e, |xn|→ 0 for σ(e,e∗). we refer the reader for undefined terminologies, to the classical references [1], [2], [10]. int. j. anal. appl. 16 (2) (2018) 151 2. (l)-dunford–pettis sets in banach lattices following the introducing of the concept l–limited sets in [12], we define l–dp sets and we give some properties of them in banach spaces and specially in banach lattices. a norm bounded subset b of a dual banach space x∗ is said to be an l–limited set if every weakly null and limited sequence (xn) of x converges uniformly to zero on the set b, that is supf∈b |f(xn)|→ 0. definition 2.1. a norm bounded subset b of a dual banach space x∗ is said to be an l–dp set if every weakly null and dp sequence (xn) of x converges uniformly to zero on the set b, that is supf∈b |f(xn)|→ 0. it is clear that every l-set set in x∗ is l–dp and every subset of an l–dp set is the same. also, it is evident that every l–dp set is weak∗ bounded and so is bounded. similar to [12, theorem 2.2], we obtain: (a) absolutely closed convex hull of an l–dp set is an l–dp set, (b) relatively weakly compact subsets of dual banach spaces are l–dp set, (c) every weak∗ null sequence in dual banach space is an l–dp set. note that the converse of assertion (b) in general, is false. in fact, the following theorem 2.1, shows that the closed unit ball of `∞ is an l–dp set. theorem 2.1. a banach space x has the dprcp iff every bounded subset of x ∗ is an l–dp set. proof. since the banach space x has the dprcp iff every dp and weakly null sequence (xn) in x is norm null [14], the proof is clear. � the following theorem 2.2, gives a necessary and sufficient condition for banach spaces that l–sets and l–dp sets in its dual coincide. we recall that an operator t : x → y between two banach spaces is completely continuous, if t carries weakly null sequences in x to norm null ones, and the class of completely continuous operators is denoted by cc(x,y ). theorem 2.2. a banach space x has the dp property iff each l–dp set in x∗ is an l–set. proof. suppose x has the dp property. since every weakly null sequence in x is dp so every l–dp set in x∗ is an l-set. conversely, it is enough to show that for each banach space y , cc(x,y ) = dpcc(x,y ) [14, theorem 1.5]. if t : x → y is dpcc, it is clear that t∗(by∗) is an l–dp set. so by hypothesis, it is an l–set and we know that the operator t : x → y is completely continuous iff t∗(by∗) is an l–set. � corollary 2.1. a banach space with the dprcp has the dp property if and only if it has the schur property. proof. it is clear that the banach space x has the schur property if and only if every bounded subset of x∗ is l–set. now, if x has the dp property and dprcp, then by theorem 2.1, unit ball x ∗ is l–dp and so it is an l–set. the converse of the assertion is also clear. � int. j. anal. appl. 16 (2) (2018) 152 theorem 2.3. let a be an l–dp subset of a dual banach lattice e∗ and e has the weakly sequentially continuous lattice operations. then |a| = {|a| : a ∈ a} is an l– dp set. proof. we show that every weakly null and dp sequence (xn) in e converges uniformly on |a|, that is, limn→∞ supx∗∈a |〈xn, |x∗|〉| = 0. from [10, lemma 1.4.4], 〈|xn|, |x∗|〉 = max{〈zn,x∗〉 : |zn| ≤ |xn|} for all n. so, there exists zn ∈ e, such that |zn| ≤ |xn| and 〈|xn|, |x∗|〉 = 〈zn,x∗〉. since e has the weakly sequentially continuous lattice operations, the sequences (|xn|) and so (zn) are weakly null. since the set a is l–dp, supx∗∈a |〈zn,x∗〉| → 0. from supx∗∈a|〈xn, |x∗|〉| ≤ supx∗∈a〈|xn|, |x∗|〉, we have supx∗∈a |〈xn, |x∗|〉|→ 0 and then the set |a| is l–dp. � definition 2.2. a banach space x has the l–dp property, if every l–dp subset of x∗ is relatively weakly compact. theorem 2.4. for a banach space x, the following are equivalent: (a) x has the l–dp property, (b) for each banach space y , dpcc(e,y ) = w(e,y ), (c) dpcc(x,`∞) = w(x,`∞). proof. (a) ⇒ (b). suppose that x has the l-dp property and t : x → y is dpcc. thus t∗(by∗) is an l–dp set. so by hypothesis, it is relatively weakly compact and t is a weakly compact operator. (b) ⇒ (c). it is obvious. (c) ⇒ (a). if x does not have the l–dp property, there exists an l–dp subset a of x∗ that is not relatively weakly compact. so there is a sequence (xn) ⊂ a with no weakly convergent subsequence. now we show that the operator t : x → `∞ by tx = (〈x,x∗n〉) , x ∈ x is dpcc, but it is not weakly compact. as (x∗n) ⊂ a is an l–dp set, for every weakly null and dp sequence (xm) in x we have ‖txm‖ = sup n |〈xm,x∗n〉|→ 0 thus t is a dpcc operator. it is easy to see that t∗(e∗n) = x ∗ n , n ∈ n thus t∗ is not a weakly compact operator and neithe is t . this finishes the proof. � the classical banach lattices `p, where 1 ≤ p < ∞ and schur spaces are discrete kb-space and so they have the dprcp [3]. the following corollary shows that the classical banach lattices `p, where 1 < p < ∞ have the l–dp property. int. j. anal. appl. 16 (2) (2018) 153 corollary 2.2. a banach space with the dprcp has the l–dp property if and only if it is reflexive. proof. if a banach space x has the dprcp, then by [14], the identity operator on x is dpcc and so is weakly compact, thanks to the l–dp property of x. hence x is reflexive. � recall that a banach space x is said to have the reciprocal dp property (abb. rdp) if every completely continuous operator on x is weakly compact [7]. theorem 2.5. if a banach space x has the l–dp property, then it has the rdp. proof. for arbitrary banach space y , let t : x → y be a completely continuous operator. thus it is dpcc and so by theorem 2.4, t is weakly compact. hence x has the rdp property. � in the following, we show that the l–dp property is carried by every complemented subspace. theorem 2.6. if a banach space x has the l–dp property, then every complemented subspace of x has the l–dp property. proof. consider a complemented subspace y of x and a projection map p : x → y . suppose p : y → `∞ is a dpcc operator, so tp : x → `∞ is also dpcc. since x has the l–dp property, by theorem 2.4, tp is weakly compact. hence t is weakly compact. � the following evident proposition gives a characterization of the l–dp property property by l–dp setes. proposition 2.1. let x be a banach space. then the following are equivalent: (a) x has the l–dp property, (b) every l–dp sequence in x∗ is relatively weakly compact. theorem 2.7. let e be a banach lattice with the l–dp property. then for each f ∈ (e∗)+, [−f,f] is an l–dp set. proof. it is evident that every l-set in e∗ is an l–dp set. if e is a banach lattice e with the l–dp property, then every l–set in e∗ is relatively weakly compact and so by [?, theorem 3.1], e∗ has an order continuous norm. hence by [2, theorem 4.9], for each f ∈ (e∗)+, [−f,f] is relatively weakly compact and so it is an l–dp set. � in the rest of this section by using some techniques to those in [4], we investigate additional properties of l–dp sets. proposition 2.2. let x be a banach space and b be a bounded subset of x∗. then the following are equivalent: int. j. anal. appl. 16 (2) (2018) 154 (a) b is an l–dp set, (b) for each sequence (fn) in b, fn(xn) → 0, for every weakly null and dp sequence (xn) of x. proof. (a) ⇒ (b). this is from the inequality |fn(xn)| ≤ supf∈b |f(xn)| for each sequence (fn) in b and for every weakly null and dp sequence (xn) of x. (b) ⇒ (a). assume that b is not an (l) dp set in x∗. then there exsits an � > 0 and a weakly null and dp sequence (xn) in x such that supf∈b |f(xn)| > � for all n. this implies the existence of a sequence fn in b such that |fn(xn)| > �, for all n. � as in the previous proposition 2.2, we can easily conclude that, for a norm bounded sequence (fn) of x ∗, the subset {fn : n ∈ n} is an l–dp set iff fn(xn) → 0, for every weakly null and dp sequence (xn) of x. proposition 2.3. let t be an operator from a banach space x into a banach lattice e and f ∈ (e∗)+. then the following are equivalent: (a) t∗[−f,f] is an l–dp set, (b) for every weakly null and dp sequence (xn) of x, f(|t(xn)|) → 0. proof. it follows immediately from the equality f(|t(xn)|) = supg∈t∗[−f,f] |g(xn)|. � by taking t = ide in proposition 2.3, for each f ∈ (e∗)+, [−f,f] is an l–dp set iff for every weakly null and dp sequence (xn) of e, (|xn|) is weakly null. the next main result, gives an equivalent condition to t∗(b) be an l–dp set, where b is a norm bounded solid subset of e∗ and t is an operator from a banach space x into a banach lattice e. recall that a sequence (xn) in a banach lattice e is (pairwise) disjoint, if for each i 6= j, |xi|∧ |xj| = 0. theorem 2.8. let t be an operator from a banach space x into a banach lattice e and b be a norm bounded solid subset of e∗. then the following are equivalent: (a) t∗(b) is an l–dp set in x∗, (b) t∗[−f,f] and {t∗fn : n ∈ n} are l–dp sets, for each f ∈ b+ and for each norm bounded disjoint sequence (fn) ∈ b+. proof. the proof is similar to [4, theorem 2.7]. � by taking t = ide in theorem 2.8, we obtain a norm bounded solid subset b of e ∗ is an l–dp set iff [−f,f] and {fn : n ∈ n} are l–dp sets, for each f ∈ b+ and for each disjoint sequence (fn) ∈ b+. the next result characterizes dpcc operators by l–dp sets. theorem 2.9. for an operator t from a banach space x into a banach lattice e, the following are equivalent: int. j. anal. appl. 16 (2) (2018) 155 (a) t is dpcc, (b) t∗(be∗) is an l– dp set, where be∗ is the closed unit ball of e ∗, (c) t∗[−f,f] and {t∗fn : n ∈ n} are l–dp sets, for each f ∈ (be∗)+ and for each norm bounded disjoint sequence (fn) ∈ (be∗)+, (d) |t(xn)|→ 0 for σ(e,e∗) and fn(txn) → 0, for every weakly null and dp sequence (xn) in x and for each disjoint sequence (fn) in (be∗) +. proof. (a) ⇔ (b). by the equality supf∈t∗(be∗) |f(xn)| = ‖txn‖e, t ∗(be∗) is an l–dp set in x ∗, if and only if, t is a dpcc operator. by theorem 2.8, the statements (b) and (c) are equivalent and the equivalence (c) ⇔ (d) is a direct consequence of proposition 2.3. � 3. almost l-dp sets in banach lattices in this section we introduce a new class of sets and operators. definition 3.1. let e be a banach lattice and x be a banach space. then (a) a norm bounded subset b of a dual banach lattice e∗ is said to be an almost l–dp set if every disjoint weakly null and dp sequence (xn) of e converges uniformly to zero on the set b, that is supf∈b |f(xn)|→ 0. (b) an operator t from a banach lattice e into a banach space x is a disjoint dp completely continuous (abb. dpdcc) operator if the sequence (‖txn‖) converges to zero for every disjoint weakly null and dp sequence in e. note that every l–dp set of a dual banach lattice, is an almost l–dp set, but the converse is false, in general. in fact for many banach lattices e with the positive dprcp and without the dprcp, the closed unit ball of the dual banach lattice e∗ is an almost l–dp set, but it is not l–dp set. as an example, the closed unit ball b`∞ of `∞ is an almost l–dp set in `∞, but the closed unit ball b(`∞)∗ is not an almost l–dp set in (`∞) ∗. in the following, we give a useful chracterization of almost l-dp sets, that is proved by the method of proposition 2.2. as we mentioned at the end of the previous section, we use some techniques to those in [4]. proposition 3.1. let e be a banach lattice and b be a norm bounded set in e∗. then the following are equivalent: (a) b is an almost l–dp set, (b) for each sequence (fn) in b, fn(xn) → 0, for every disjoint weakly null and dp sequence (xn) of e. int. j. anal. appl. 16 (2) (2018) 156 in particular, we obtain: proposition 3.2. let e be a banach lattice and (fn) be a norm bounded sequence in e ∗. then the following are equivalent: (a) the subset {fn : n ∈ n} is an almost l–dp set, (b) fn(xn) → 0, for every disjoint weakly null and dp sequence (xn) of e. similar to [4], [−f,f] is an almost l–dp set in e∗, for each f ∈ (e∗)+. also for an order bounded operator from a banach lattice e into a banach lattice f , t∗([−f,f]) is an almost l–dp set, for each f ∈ (f∗)+. theorem 3.1. let t be an order bounded operator from a banach lattice e into a banach lattice f and b be a norm bounded solid subset of f∗. then the following are equivalent: (a) t∗(b) is an almost l–dp set in e∗, (b) {t∗fn : n ∈ n} is an almost l–dp set, for each f ∈ b+ and for each disjoint sequence (fn) in b+. (c) fn(txn) → 0, for every disjoint weakly null and dp sequence (xn) of e+ and for each disjoint sequence (fn) in b +. proof. the proof is the same as the proof of theorem 2.9. � by taking t = ide in theorem 3.1, we obtain a norm bounded solid subset b of e ∗ is an almost l–dp set iff {fn : n ∈ n} is an almost l–dp set for each disjoint sequence (fn) in b+. the next result characterizes the class of dpdcc operators by almost l– dp sets. theorem 3.2. for an order bounded operator t from a banach lattice e into another banach lattice f , the following are equivalent: (a) t is dpdcc, (b) t∗(bf∗) is an almost l– dp set, where bf∗ is the closed unit ball of f ∗, (c) {t∗(fn) : n ∈ n} is an almost l–dp set for each disjoint sequence (fn) in (bf∗)+, (d) fn(t(xn)) → 0, for every disjoint weakly null and dp sequence (xn) of e+ and for each disjoint sequence (fn) in (bf∗) +. proof. (a) ⇔ (b). by the equality, supf∈t∗(bf∗) |f(xn)| = ‖txn‖f , for every sequence (xn) in e, it follows easily that, t∗(bf∗) is an almost l-limited set in e ∗ if and only if t is dpdcc. by theorem 3.1, the statements (b) and (c) are equivalent and the equivalence (c) ⇔ (d) is a direct consequence of proposition 3.2. � int. j. anal. appl. 16 (2) (2018) 157 now the concept of positive dprcp in banach lattices is introduced and banach lattices with the positive dprcp is characterized. next we give some properties of dp dcc operators from an arbitrary banach lattice e to another f , related to the positive dprcp of the banach lattice e. definition 3.2. a banach lattice e has the positive dprcp if each weakly null and dp sequence with the positive terms in e is norm null. it is clear that the dprcp implies the positive dprcp, but the converse is false, in general. for example, l1[0, 1] has the positive dprcp without the dprcp. theorem 3.3. for a banach lattice e, the following are equivalent: (a) e has the positive dprcp, (b) every weakly null and disjoint dp sequence in e converges to zero in norm. proof. (a) ⇒ (b). let (xn) be a weakly null and disjoint dp sequence in e. from [15, proposition 1.3], the sequence (|xn|) is weakly null and by [8, lemma 3.7], it is dp in e. from (a), the sequence (|xn|) and so (xn) converges to zero in norm. (b) ⇒ (a). suppose that infn‖xn‖ = c > 0 for some weakly null and dp sequence (xn) ⊂ e+. putting yn = c −1xn and using [9, corollary 5] we find a subsequence (ynk ), a constant d > 0, and a disjoint sequence (zk) of e + such that 0 < zk ≤ ynk and ‖zk‖≥ d. it is clear that disjoint dp sequence (zk) tends weakly to zero, but ‖zk‖≥ d. this fact contradicts the assumption. � theorem 3.4. a banach lattice e has the positive dprcp iff every bounded set in e ∗ is an almost l–dp set. proof. from theorem 3.3, a banach lattice e has the positive dprcp iff every disjoint weakly null and dp sequence in e is norm null. � theorem 3.5. let e be a banach lattice. then the following are equivalent: (a) e has the positive dprcp, (b) for each banach space y , dpdcc(e,y ) = l(e,y ), (c) dpdcc(e,`∞) = l(e,`∞). proof. (a) ⇒ (b). if e has the positive dprcp and (xn) is a weakly null and disjoint dp sequence in e, then by theorem 3.3, (xn) is norm null and for each bounded operator t on e, ‖txn‖ → 0; that is, dpdcc(e,f) = l(e,f). (b) ⇒ (c). it is obvious. (c) ⇒ (a). if e does not have the positive dprcp, then by theorem 3.3, there exists a weakly null and int. j. anal. appl. 16 (2) (2018) 158 disjoint dp sequence (xn) in e such that ‖xn‖ = 1, for all n. choose a normalized sequence (x∗n) in e∗ such that |〈xn,x∗n〉| = 1, for all n, and define the operator t : e → `∞ by tx = (〈x,x∗n〉) , x ∈ e. but t is not dpdcc, since the sequence (xn) is weakly null and disjoint dp and ‖txn‖≥ 1, for all n. � in the following theorem 3.6, we show that the positive dprcp and the dprcp, coincide in the class of discrete banach lattices. let us start with the following lemma. lemma 3.1. c0 dose not have the positive dprcp. proof. it is enough to remember that c0 dose not have the positive schur property and use the fact that every weakly null sequence in c0 is dp. by [13], a banach lattice has the positive schur property, whenever 0 ≤ xn → 0 weakly implies ‖xn‖→ 0 � now we are able to formulate the following equivalence condition. theorem 3.6. let e be a discrete banach lattice. then e has the positive dprcp, if and only if, it has the dprcp. proof. since the positive dprcp is inherited by closed riesz subspaces and c0 does not have the positive dprcp, then e does not contain any order copy of c0. according to [10, corollary 2.4.12], e is kb space, and so it possesses the dprcp by [?]. � corollary 3.1. the dual banach lattice c(k)∗ has the positive dprcp, where k is a compact hausdorff space. proof. for each positive and weakly null sequence (fn) in c(k) ∗, ‖fn‖ = fn(1k) → 0, where 1k denotes the constant function 1 on k. that is c(k)∗ has the positive dprcp. on the other hands from [2], the banach lattice c(k)∗ is discrete and by theorem 3.6, it has the dprcp. � theorem 3.7. let t : e → x from a banach lattice e be an operator. then the following are equivalent: (a) t is dpdcc, (b) the sequence (‖txn‖) converges to zero for every weakly null and dp sequence in e+, (c) the sequence (‖txn‖) converges to zero for every disjoint weakly null and dp sequence in e+. proof. the proof is similar to [5, theorem 2.2]. � let m ⊂ l(x,y ) be a banach lattice. if m has the positive dprcp, then by theorem 3.5 all of the evaluation operators φx and ψy∗ are dp dcc operators, where φx(t) = tx and ψy∗(t) = t ∗y∗ for x ∈ x, int. j. anal. appl. 16 (2) (2018) 159 y∗ ∈ y ∗ and t ∈m. now, we show that the dpaccness of evaluation operators is a sufficient condition for the positive dprcp of their domain. theorem 3.8. let y has the schur property and m⊂ l(x,y ) be a banach lattice. if for every y∗ ∈ y ∗, the evaluation operator ψy∗ on m is dpdcc, then m has the positive dprcp. proof. if m does not have the positive dprcp, by theorem 3.3, there exists a weakly null and disjoint dp sequence (tn) in m and some � > 0 such that ‖tn‖ > �, for all n. so there exists a sequence (xn) in bx such that ‖tn(xn)‖ > �, for all n. on the other hand, the evaluation operator ψy∗ on m is dpdcc for all y∗ ∈ y ∗ and so ‖t∗n(y∗)‖ = ‖ψy∗(tn)‖→ 0. hence |〈tnxn,y∗〉| ≤ ‖t∗n(y∗)‖→ 0. so the sequence (tnxn) is weakly null and it is norm null by the schur property, a fact that is impossible. � theorem 3.9. let x has the schur property and m ⊂ lw∗(x∗,y ) be a banach lattice. if for every x∗ ∈ x∗, the evaluation operator φx∗ on m is dpdcc, then m has the positive dprcp. proof. if m does not have the positive dprcp, by theorem 3.3, there exists a weakly null and disjoint dp sequence (tn) in m and some � > 0 such that ‖tn‖ > �, for all n. on the other hand, the evaluation operator φx∗ on m is dpdcc for all x∗ ∈ x∗ and so ‖tn(x∗)‖ = ‖φx∗(tn)‖ → 0. since ‖t∗n‖ > �, there exists a sequence (y∗n) in by∗ such that ‖t∗ny∗n‖ > �, for all n. but the schur property of x shows that the weakly null sequence (t∗ny ∗ n) is norm null, which is a contradiction. � two final theorems of this section, are a relationship between order weakly compact and m-weakly compact operators with a dpdcc operator. recall that a continuous operator t : e → x from a banach lattice e to a banach space x is order weakly compact if and only if ‖txn‖→ 0 for every disjoint order bounded sequence (xn) in e [2, theorem 5.57]. theorem 3.10. evere dpdcc operator on a banach lattice e is order weakly compact. proof. let (xn) be an order bounded disjoint sequence of e. it follows from [2] and [?] that (xn) is a weakly null and dp sequence. since t is dpdcc then, ‖txn‖→ 0. hence t is order weakly compact. � an operator t : e → x from a banach lattice to a banach space is said to be m-weakly compact if ‖txn‖→ 0 holds for every norm bounded disjoint sequence (xn) in e [10]. in [14], the authors proved that each dpcc operator from a banach lattice e to a banach space x is m-weakly compact when e∗ has an order continuous norm and e has the dp∗ property (that is, every relatively weakly compact set in e is limited). in fact, we have a similar conclusion about dpdcc operators. theorem 3.11. let e be a banach lattice and x be a banach space. if e∗ has an order continuous norm and e has the dp property, then each dpdcc operator t : e → x is m-weakly compact. int. j. anal. appl. 16 (2) (2018) 160 proof. let t : m → x be a dpdcc operator and let (xn) be a bounded disjoint sequence in e. it follows from [10, corollary 2.9] that (xn) is weakly null and so it is dp by the dp property of e. by our hypothesis on t , we have ‖txn‖→ 0 and then t is m-weakly compact. � 4. almost l–dp sets which are l–dp sets as we noted in the beginning of section 3, every l–dp set in the dual banach lattice e∗, is an almost l–dp set, but the converse is false in general. in this section we characterize banach lattices in which the class of almost l–dp sets and that of l–dp sets coincide in their dual. theorem 4.1. for a banach lattice e, the following are equivalent: (a) each almost l–dp set in e∗ is an l–dp set, (b) for each banach space y , dpdcc(e,y ) = dpcc(e,y ), (c) dpdcc(e,`∞) = dpcc(e,`∞). proof. (a) ⇒ (b). let t : e → y be an operator. by the equality sup f∈t∗(by∗) |f(xn)| = ‖txn‖y , for every sequence (xn) in e, it follows easily that, t ∗(by∗) is an almost l–dp (respectively, l–dp) set in e∗, if and only if, t is a dpdcc (respectively, dpcc) operator. now, let t be a dpdcc operator. then t∗(by∗) is an almost l–dp set in e ∗ and from the hypothesis (a), it is an l–dp set in e∗. hence t is a dpcc operator. (b) ⇒ (c). it is clear. (c) ⇒ (a). let b be an almost l–dp set in e∗. to prove that b is an l–dp set, it sufficies to show that fn(xn) → 0 for each sequence (fn) in b and for every weakly null and dp sequence (xn) in e (see proposition 2.2). consider the operator s : e → `∞ defined by s(x) = (fn(x))∞n=1, for each x ∈ e. as b is almost l–dp, s is a dpdcc operator. in fact, for every weakly null and disjoint dp sequence (zi) in e, we have ‖szi‖∞ = ‖fn(zi)∞n=0‖∞ ≤ sup f∈b |f(zi)|→ 0, as i → ∞. it follows that s is a dpdcc operator and so from our hypothesis, s is dpcc. so ‖sxn‖∞ → 0 and the desired conclusion follows from the inequality |fn(xn)| ≤ ‖sxn‖∞ for each n. � we recall that, an operator t from a banach space x into a banach lattice e is said to be semicompact if for each � > 0 there exists some u ∈ e+ satisfying t(bx) ⊂ [−u,u] + �be. according to [4, theorem 4.3], each operator t : e → x is dpdcc, whenever its adjoint t∗ : x∗ → e∗ is semicompact. int. j. anal. appl. 16 (2) (2018) 161 at the end of this section, it should be noted that the adjoint of a dpdcc operator is not necessary dpdcc and vice versa. for example, the identity operator on the banach lattice `1 is dp dcc (because `1 has the dprcp, [14]) but its adjoint, id`∞ : `∞ → `∞, is not dpdcc. in fact, if en = (0, 0, ..., 1, 0, ...) with n’th entry equals to 1 and all others zero, then (en) is an order bounded disjoint sequence of `∞. hence (en) is weakly null and by [?], it is dp, but ‖id`∞(en)‖ = ‖en‖∞ = 1 for all n. also the identity operator on `∞ is not dpdcc but its adjoint is dpdcc, because (`∞) ∗ has dprcp. also by theorem 3.2, b(`∞)∗ is not an almost l–dp set in (`∞) ∗, as noted that in the begining of section 3. references [1] c. d. aliprantis and o. burkishaw, locally solid riesz spaces, academic press, new york, london, 1978. [2] c. d. aliprantis and o. burkishaw, positive operators, academic press, new york, london, 1978. [3] b. aqzzouz and k. bouras, dunford-pettis sets in banach lattices, acta math. univ. comenianae, 81 (2012), 185–196. [4] b. aqzzouz and k. bouras, l–sets and almost l– sets in banach lattices, quaest. math., 36 (2013), 107–118. [5] b. aqzzouz and a. elbour, some characterizations of almost dunford–pettis operators and applications, j. positivity 15 (2011), 369–380. [6] g. emmanuele, banach spaces in which dunford-pettis sets are relatively compact, arch. math., 58 (1992), 477–485. [7] g. emmanuele, the reciprocal dunford-pettis property and projective tensor products, math. proc. cambridge philos. soc., 109 (1992), 161–166. [8] k.e. fahri, n. machrafi and m. moussa, banach lattices with the positive dunford-pettis relatively compact property, extracta math., 80 (2015), 161–179. [9] g. groenewegen and p. meyer-nieberg, an elementary and unified approach to disjoint sequence theorems, indag. math., 48 (1986), 313–317. [10] p. meyernieberg, banach lattices, universitext, springer– verlag, berlin, 1991. [11] k. musial, the weak radon-nikodym property in banach spaces, studia math., 64 (1979), 151–173. [12] m. salimi and s. m. moshtaghioun, a new class of banach spaces and its relation with some geometric properties of bancah spaces, abstr. appl. anal., id 212957, 2012. [13] j. a. sanchez, positive schur property in banach lattices, extraccta math., 7 (1992), 161-163. [14] y. wen, ji. chen, characterizations of banach spaces with relatively compact dunford-pettis sets, adv. math., to appear. [15] w. wnuk, on the dual positive schur property in banach lattices, j. positivity, 17 (2013), 759–773. 1. introduction 2. (l)-dunford–pettis sets in banach lattices 3. almost l-dp sets in banach lattices 4. almost l–dp sets which are l–dp sets references international journal of analysis and applications volume 17, number 4 (2019), 596-619 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-596 multilinear bmo estimates for the commutators of multilinear fractional maximal and integral operators on the product generalized morrey spaces feri̇t gürbüz hakkari university, faculty of education, department of mathematics education, hakkari 30000, turkey corresponding author: feritgurbuz@hakkari.edu.tr abstract. in this paper, we establish multilinear bmo estimates for commutators of multilinear fractional maximal and integral operators both on product generalized morrey spaces and product generalized vanishing morrey spaces, respectively. similar results are still valid for commutators of multilinear maximal and singular integral operators. 1. introduction and main results the classical morrey spaces lp,λ have been introduced by morrey in [21] to study the local behavior of solutions of second order elliptic partial differential equations(pdes). in recent years there has been an explosion of interest in the study of the boundedness of operators on morrey-type spaces. it has been obtained that many properties of solutions to pdes are concerned with the boundedness of some operators on morreytype spaces. morrey spaces can complement the boundedness properties of operators that lebesgue spaces can not handle. morrey spaces which we have been handling are called classical morrey spaces(see [21]). but, classical morrey spaces are not totally enough to describe the boundedness properties. to this end, we need to generalize parameters p and q, among others p, but this issue will exceed the scope of the article, so we pass this part. though we do not consider the direct applications of morrey spaces to pdes, morrey received 2019-03-25; accepted 2019-04-22; published 2019-07-01. 2010 mathematics subject classification. 42b20, 42b25, 42b35. key words and phrases. multi-sublinear fractional maximal operator; multilinear fractional integral operator; multilinear commutator; generalized morrey space; generalized vanishing morrey space; multilinear bmo space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 596 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-596 int. j. anal. appl. 17 (4) (2019) 597 spaces can be applied to pdes. applications to the second order elliptic partial differential equations can be found in [10] and [26]. we will say that a function f ∈ lp,λ = lp,λ (rn) if sup x∈rn,r>0  r−λ ∫ b(x,r) |f (y)|p dy   1/p < ∞. (1.1) here, 1 < p < ∞ and 0 < λ < n and the quantity of (1.1) is the (p,λ)-morrey norm, denoted by ‖f‖lp,λ. in recent years, more and more researches focus on function spaces based on morrey spaces to fill in some gaps in the theory of morrey type spaces (see, for example, [11–14, 16, 18, 23, 25, 29]). moreover, these spaces are useful in harmonic analysis and pdes. but, this topic exceeds the scope of this paper. thus, we omit the details here. on the other hand, the study of the operators of harmonic analysis in vanishing morrey space, in fact has been almost not touched. a version of the classical morrey space lp,λ(rn) where it is possible to approximate by ”nice” functions is the so called vanishing morrey space v lp,λ(rn) has been introduced by vitanza in [27] and has been applied there to obtain a regularity result for elliptic pdes. this is a subspace of functions in lp,λ(rn), which satisfies the condition lim r→0 sup x∈rn 0<t<r  t−λ ∫ b(x,t) |f (y)|p dy   1/p = 0, where 1 < p < ∞ and 0 < λ < n for brevity, so that v lp,λ(rn) =  f ∈ lp,λ(rn) : limr→0 supx∈rn 0<t<r t− λ p‖f‖lp(b(x,t)) = 0   . for the properties and applications of vanishing morrey spaces, see also [1]. after studying morrey spaces in detail, researchers have passed to the concept of generalized morrey spaces. firstly, motivated by the work of [21], mizuhara [19] introduced generalized morrey spaces mp,ϕ. then, the generalized morrey spaces mp,ϕ with normalized norm is defined as follows: definition 1.1. (generalized morrey space) let ϕ(x,r) be a positive measurable function on rn×(0,∞). if 0 < p < ∞, then the generalized morrey space mp,ϕ ≡ mp,ϕ(rn) is defined by{ f ∈ llocp (r n) : ‖f‖mp,ϕ = sup x∈rn,r>0 ϕ(x,r)−1|b(x,r)|− 1 p‖f‖lp(b(x,r)) < ∞ } . obviously, the above definition recover the definition of lp,λ(rn) if we choose ϕ(x,r) = r λ−n p , that is lp,λ (rn) = mp,ϕ (rn) | ϕ(x,r)=r λ−n p . int. j. anal. appl. 17 (4) (2019) 598 everywhere in the sequel we assume that inf x∈rn,r>0 ϕ(x,r) > 0 which makes the above spaces non-trivial, since the spaces of bounded functions are contained in these spaces. we point out that ϕ(x,r) is a measurable nonnegative function and no monotonicity type condition is imposed on these spaces. in [12], [16], [18], [19] and [25], the boundedness of the maximal operator and calderón-zygmund operator on the generalized morrey spaces mp,ϕ has been obtained, respectively. for brevity, in the sequel we use the notations mp,ϕ (f; x,r) := |b(x,r)|− 1 p ‖f‖lp(b(x,r)) ϕ(x,r) , and mwp,ϕ (f; x,r) := |b(x,r)|− 1 p ‖f‖wlp(b(x,r)) ϕ(x,r) . in this paper, extending the definition of vanishing morrey spaces [27], we introduce generalized vanishing morrey spaces v mp,ϕ(rn) with normalized norm in the following form. definition 1.2. (generalized vanishing morrey space) the generalized vanishing morrey space v mp,ϕ(rn) is defined by { f ∈ mp,ϕ(rn) : lim r→0 sup x∈rn mp,ϕ (f; x,r) = 0 } . everywhere in the sequel we assume that lim r→0 1 inf x∈rn ϕ(x,r) = 0, (1.2) and sup 0<r<∞ 1 inf x∈rn ϕ(x,r) < ∞, (1.3) which make the spaces v mp,ϕ(rn) non-trivial, because bounded functions with compact support belong to this space. the spaces v mp,ϕ(rn) are banach spaces with respect to the norm ‖f‖v mp,ϕ ≡‖f‖mp,ϕ = sup x∈rn,r>0 mp,ϕ (f; x,r) . the spaces v mp,ϕ(rn) are also closed subspaces of the banach spaces mp,ϕ(rn), which may be shown by standard means. furthermore, we have the following embeddings: v mp,ϕ ⊂ mp,ϕ, ‖f‖mp,ϕ ≤‖f‖v mp,ϕ. on the other hand, it is well known that, for the purpose of researching non-smoothness partial differential equation, mathematicians pay more attention to the singular integrals. moreover, the fractional type operators and their weighted boundedness theory play important roles in harmonic analysis and int. j. anal. appl. 17 (4) (2019) 599 other fields, and the multilinear operators arise in numerous situations involving product-like operations, see [2, 3, 5–8, 14, 17, 20, 24] for instance. first of all, we recall some basic properties and notations used in this paper. let rn be the n-dimensional euclidean space of points x = (x1, ...,xn) with norm |x| = (∑n i=1x 2 i )1/2 and corresponding m-fold product spaces (m ∈ n) be (rn)m = rn ×···×rn. let b = b(x,r) denotes open ball centered at x of radius r for x ∈ rn and r > 0 and bc(x,r) its complement. also |b(x,r)| is the lebesgue measure of the ball b(x,r) and |b(x,r)| = vnrn, where vn = |b(0, 1)|. we also denote by −→y = (y1, . . . ,ym), d−→y = dy1 . . .dym, and by −→ f the m-tuple (f1, ...,fm), m, n the nonnegative integers with n ≥ 2, m ≥ 1. let −→ f ∈ llocp1 (r n) ×···×llocpm (r n). then multi-sublinear fractional maximal operator m (m) α is defined by m(m)α (−→ f ) (x) = sup t>0 |b (x,t)| α n   m∏ i=1 1 |b (x,t)| ∫ b(x,t) |fi (yi)|  d−→y , 0 ≤ α < mn. from definition, if α = 0 then m (m) α is the multi-sublinear maximal operator m (m) and also; in the case of m = 1, m (m) α is the classical fractional maximal operator mα. the theory of multilinear calderón-zygmund singular integral operators, originated from the works of coifman and meyer’s [4], plays an important role in harmonic analysis. its study has been attracting a lot of attention in the last few decades. a systematic analysis of many basic properties of such multilinear singular integral operators can be found in the articles by coifman-meyer [4], grafakos-torres [7–9], chen et al. [2], fu et al. [5]. let t(m) (m ∈ n) be a multilinear operator initially defined on the m-fold product of schwartz spaces and taking values into the space of tempered distributions, t(m) : s (rn) ×···×s (rn) → s (rn) . following [7], recall that the m(multi)-linear calderón-zygmund operator t(m) (m ∈ n) for test vector −→ f = (f1, . . . ,fm) is defined by t(m) (−→ f ) (x) = ∫ (rn)m k (x,y1, . . . ,ym) { m∏ i=1 fi (yi) } dy1 · · ·dym, x /∈ m⋂ i=1 suppfi, where k is an m-calderón-zygmund kernel which is a locally integrable function defined off the diagonal y0 = y1 = · · · = ym on (rn) m+1 satisfying the following size estimate: |k (x,y1, . . . ,ym)| ≤ c |(x−y1, . . . ,x−ym)| mn , for some c > 0 and some smoothness estimates, see [7–9] for details. the result of grafakos and torres [7,9] shows that the multilinear calderón-zygmund operator is bounded on the product of lebesgue spaces. int. j. anal. appl. 17 (4) (2019) 600 theorem 1.1. [7,9] let t(m) (m ∈ n) be an m-linear calderón-zygmund operator. then, for any numbers 1 ≤ p1, . . . ,pm,p < ∞ with 1p = 1 p1 +· · ·+ 1 pm , t(m) can be extended to a bounded operator from lp1×···×lpm into lp, and bounded from l1 ×···×l1 into l 1 m ,∞. on the other hand, the multilinear fractional type operators are natural generalization of linear ones. their earliest version was originated on the work of grafakos [6] in 1992, in which he studied the multilinear maximal function and multilinear fractional integral defined by m(m)α (−→ f ) (x) = sup t>0 1 rn−α ∫ |y|<r ∣∣∣∣∣ m∏ i=1 fi (x−θiy) ∣∣∣∣∣dy, and i(m)α (−→ f ) (x) = ∫ rn 1 |y|n−α m∏ i=1 fi (x−θiy) dy, where θi (i = 1, . . . ,m) are fixed distinct are nonzero real numbers and 0 < α < n. we note that, if we simply take m = 1 and θi = 1, then mα and iα are just the operators studied by muckenhoupt and wheeden in [22]. in this paper we deal with another kind of multilinear operator which was defined by kenig and stein [17] for −→ f = (f1, . . . ,fm), which is called multilinear fractional integral operator as follows i(m)α (−→ f ) (x) = ∫ (rn)m 1 |(x−y1, . . . ,x−ym)| mn−α { m∏ i=1 fi (yi) } d−→y , whose kernel is |k (x,y1, . . . ,ym)| = |(x−y1, . . . ,x−ym)| −mn+α , 0 < α < mn, where f1, . . . ,fm : rn → r are measurable and |(x−y1, . . . ,x−ym)| = √ m∑ i=1 |x−yi| 2 . they [17] proved that i (m) α (m ∈ n) is of strong type (lp1 ×lp2 ×···×lpm,lq) and weak type (lp1 ×lp2 ×···×lpm,lq,∞). if we take m = 1, i (m) α (m ∈ n) is the classical fractional integral operator iα. moreover, their main result (theorem 1 in [17]) is the multi-version of well-known hardy-littlewood-sobolev inequality. later, weighted inequalities for the multilinear fractional integral operators have been established by moen [20] and chen-xue [3], respectively. yu and tao [29] have also obtained the boundedness of the operators i (m) α , t (m) and m(m) (m ∈ n) on the product generalized morrey spaces, respectively. now, we will give some properties related to the space of functions of bounded mean oscillation, bmo, which play a great role in the proofs of our main results, introduced by john and nirenberg [15] in 1961. this space has become extremely important in various areas of analysis including harmonic analysis, pdes and function theory. bmo-spaces are also of interest since, in the scale of lebesgue spaces, they may be considered and appropriate substitute for l∞. appropriate in the sense that are spaces preserved by a wide class of important operators such as the hardy-littlewood maximal function, the hilbert transform and which can be used as an end point in interpolating lp spaces. int. j. anal. appl. 17 (4) (2019) 601 let us recall the definition of the space of bmo(rn). definition 1.3. [12, 16] the space bmo(rn) of functions of bounded mean oscillation consists of locally summable functions with finite semi-norm ‖b‖∗ ≡‖b‖bmo = sup x∈rn,r>0 1 |b(x,r)| ∫ b(x,r) |b(y) − bb(x,r)|dy < ∞, (1.4) where bb(x,r) is the mean value of the function b on the ball b(x,r). the fact that precisely the mean value bb(x,r) figures in (1.4) is inessential and one gets an equivalent seminorm if bb(x,r) is replaced by an arbitrary constant c : ‖b‖∗ ≈ sup r>0 inf c∈c 1 |b (x,r)| ∫ b(x,r) |b (y) − c|dy. each bounded function b ∈ bmo. moreover, bmo contains unbounded functions, in fact log|x| belongs to bmo but is not bounded, so l∞(rn) ⊂ bmo(rn). in 1961 john and nirenberg [15] established the following deep property of functions from bmo. theorem 1.2. [15] if b ∈ bmo(rn) and b (x,r) is a ball, then ∣∣{x ∈ b (x,r) : |b(x) − bb(x,r)| > ξ}∣∣ ≤ |b (x,r) |exp (− ξ c‖b‖∗ ) , ξ > 0, where c depends only on the dimension n. by theorem 1.2, we can get the following results. corollary 1.1. [12, 16] let b ∈ bmo(rn). then, for any q > 1, ‖b‖∗ ≈ sup x∈rn,r>0   1|b(x,r)| ∫ b(x,r) |b(y) − bb(x,r)|pdy   1 p (1.5) is valid. corollary 1.2. [12, 16] let b ∈ bmo(rn). then there is a constant c > 0 such that ∣∣bb(x,r) − bb(x,t)∣∣ ≤ c‖b‖∗(1 + ln t r ) for 0 < 2r < t, (1.6) and for any q > 1, it is easy to see that ‖b− (b)b‖lq(b) ≤ cr n q ‖b‖∗ ( 1 + ln t r ) , (1.7) where c is independent of b, x, r and t. now inspired by definition 1.3, we can give the definition of multilinear bmo (= bmo). indeed in this paper we will consider a multilinear version (= multilinear bmo or bmo) of the bmo. int. j. anal. appl. 17 (4) (2019) 602 definition 1.4. we say that −→ b = (b1, . . . ,bm) ∈ bmo if ∥∥∥−→b ∥∥∥ bmo = sup x∈rn,r>0 m∏ i=1 1 |b(x,r)| ∫ b(x,r) ∣∣∣bi (yi) − (bi)b(x,r)∣∣∣dyi < ∞, where (bi)b(x,r) = 1 |b(x,r)| ∫ b(x,r) bi(yi)dyi. remark 1.1. notice that (bmo) m is contained in bmo and we have ∥∥∥−→b ∥∥∥ bmo ≤ m∏ i=1 ‖bi‖∗ , so (bmo) m ⊂ bmo is valid. we now make some conventions. throughout this paper, we use the symbol a . b to denote that there exists a positive consant c such that a ≤ cb. if a . b and b . a, we then write a ≈ b and say that a and b are equivalent. for a fixed p ∈ [1,∞), p′ denotes the dual or conjugate exponent of p, namely, p′ = p p−1 and we use the convention 1 ′ = ∞ and ∞′ = 1. remark 1.2. let 0 < α < mn and 1 < pi < ∞ with 1p = m∑ i=1 1 pi , 1 qi = 1 pi − α mn , 1 q = m∑ i=1 1 qi = 1 p − α n and −→ b = (b1, . . . ,bm) ∈ (bmo) m for i = 1, . . . ,m. then, from corollary 1.2, it is easy to see that m∏ i=1 ‖bi − (bi)b‖lqi(b) ≤ c m∏ i=1 |b(x,r)| 1 qi ‖bi‖∗ ( 1 + ln t r ) , (1.8) and m∏ i=1 ‖bi − (bi)b‖lqi(2b) ≤ m∏ i=1 ( ‖bi − (bi)2b‖lqi(2b) + ‖(bi)b − (bi)2b‖lqi(2b) ) . m∏ i=1 |b(x,r)| 1 qi ‖bi‖∗ ( 1 + ln t r ) . (1.9) on the other hand, xu [28] has established the boundedness of the commutators generated by m-linear calderón-zygmund singular integrals and rbmo functions with nonhomogeneity on the product of lebesgue space. inspired by [2, 3, 7, 9, 24, 28], commutators t (m) −→ b generated by m-linear calderón-zygmund operators t(m) and bounded mean oscillation functions −→ b = (b1, . . . ,bm) is given by t (m) −→ b (−→ f ) (x) = ∫ (rn)m k (x,y1, . . . ,ym) [ m∏ i=1 [bi (x) − bi (yi)] fi (yi) ] d−→y , int. j. anal. appl. 17 (4) (2019) 603 where k (x,y1, . . . ,ym) is a m-linear calderón-zygmund kernel, bi ∈ (bmo) i (rn) for i = 1, . . . ,m. note that tb is the special case of t (m) −→ b with taking m = 1. similarly, let bi (i = 1, . . . ,m) be a locally integrable functions on rn, then the commutators generated by m-linear fractional integral operators and −→ b = (b1, . . . ,bm) is given by i (m) α, −→ b (−→ f ) (x) = ∫ (rn)m 1 |(x−y1, . . . ,x−ym)| mn−α [ m∏ i=1 [bi (x) − bi (yi)] fi (yi) ] d−→y , where 0 < α < mn, and fi (i = 1, . . . ,m) are suitable functions. the commutators of a class of multi-sublinear maximal operators corresponding to t (m) −→ b and i (m) α, −→ b (m ∈ n) above are, respectively, defined by m (m) −→ b (−→ f ) (x) = sup t>0   m∏ i=1 1 |b (x,t)| ∫ b(x,t) [|bi (x) − bi (yi)|] |fi (yi)|  d−→y , and m (m) α, −→ b (−→ f ) (x) = sup t>0 |b (x,t)| α n   m∏ i=1 1 |b (x,t)| ∫ b(x,t) [|bi (x) − bi (yi)|] |fi (yi)|  d−→y , 0 ≤ α < mn. the following result is known. lemma 1.1. [24] (strong bounds of i (m) −→ b ,α ) let 0 < αi < n, 1 < p1, . . . ,pm < ∞, 1p = m∑ i=1 1 pi , α = m∑ i=1 αi and 1 q = 1 p − α n . then there is c > 0 independent of −→ f and −→ b such that ∥∥∥i(m)−→ b ,α (−→ f )∥∥∥ lq(rn) ≤ c m∏ i=1 ‖bi‖∗‖fi‖lpi(rn) . using the idea in the proof of lemma 3.2 in [13], we can obtain the following corollary 1.3: corollary 1.3. (strong bounds of m (m) α, −→ b ) under the assumptions of lemma 1.1, the operator m (m) α, −→ b is bounded from lp1 (r n) ×···lpm(rn) to lq(rn). moreover, we have ∥∥∥m(m) α, −→ b (−→ f )∥∥∥ lq(rn) ≤ c m∏ i=1 ‖bi‖∗‖fi‖lpi(rn) . proof. set ĩ (m) −→ b ,α (|f|) (x) = ∫ (rn)m 1 |(x−y1, . . . ,x−ym)| mn−α [ m∏ i=1 [|bi (x) − bi (yi)|] |fi (yi)| ] d−→y 0 < α < mn. int. j. anal. appl. 17 (4) (2019) 604 it is easy to see that lemma 1.1 holds for ĩ (m) −→ b ,α . on the other hand, for any t > 0, we have ĩ (m) −→ b ,α (|f|) (x) ≥ ∫ (b(x,t))m 1 |(x−y1, . . . ,x−ym)| mn−α [ m∏ i=1 [|bi (x) − bi (yi)|] |fi (yi)| ] d−→y ≥ 1 tmn−α ∫ b(x,t) [ m∏ i=1 [|bi (x) − bi (yi)|] |fi (yi)| ] d−→y . taking supremum over t > 0 in the above inequality, we get m (m) α, −→ b (−→ f ) (x) ≤ c−1n,αĩ (m) −→ b ,α (|f|) (x) cn,α = |b (0, 1)| mn−α n . (1.10) � as a simple corollary of lemma 1.1 and corollary 1.3, we can obtain the following result. corollary 1.4. (strong bounds of t (m) −→ b and m (m) −→ b ) let 1 < p1, . . . ,pm < ∞ and 0 < p < ∞ with 1p = m∑ i=1 1 pi . then there is c > 0 independent of −→ f and −→ b such that ∥∥∥t(m)−→ b (−→ f )∥∥∥ lp(rn) ≤ c m∏ i=1 ‖bi‖∗‖fi‖lpi(rn) , ∥∥∥m(m)−→ b (−→ f )∥∥∥ lp(rn) ≤ c m∏ i=1 ‖bi‖∗‖fi‖lpi(rn) . the purpose of this paper is to consider the mapping properties on mp1,ϕ1 ×···×mpm,ϕm and v mp1,ϕ1 × ···×v mpm,ϕm for the commutators of multilinear fractional maximal and integral operators, respectively. similar results still hold for commutators of multilinear maximal and singular integral operators. commutators of multilinear fractional maximal and integral operators on product generalized morrey spaces have not also been studied so far and this paper seems to be the first in this direction. now, let us state the main results of this paper. theorem 1.3. let 0 < α < mn and 1 ≤ pi < mnα with 1 p = m∑ i=1 1 pi , 1 q = m∑ i=1 1 pi + m∑ i=1 1 qi − α n and −→ b ∈ (bmo)m (rn) for i = 1, . . . ,m. let functions ϕ,ϕi : rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies the condition ∞∫ r ( 1 + ln t r )m essinft<τ<∞ m∏ i=1 ϕi(x,τ)τ n p t n  1q− m∑ i=1 1 qi  +1 dt ≤ cϕ (x,r) , (1.11) where c does not depend on x ∈ rn and r > 0. int. j. anal. appl. 17 (4) (2019) 605 then, i (m) α, −→ b and m (m) α, −→ b (m ∈ n) are bounded operators from product space mp1,ϕ1 ×···×mpm,ϕm to mq,ϕ. moreover, we have ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ mq,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖mpi,ϕi , (1.12) ∥∥∥m(m) α, −→ b (−→ f )∥∥∥ mq,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖mpi,ϕi . (1.13) corollary 1.5. let 1 < pi < ∞ and 0 < p < ∞ with 1p = m∑ i=1 1 pi and −→ b ∈ (bmo)m (rn) for i = 1, . . . ,m. let functions ϕ,ϕi : rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies the condition ∞∫ r ( 1 + ln t r )m essinft<τ<∞ m∏ i=1 ϕi(x,τ)τ n p t n p +1 dt ≤ cϕ (x,r) , where c does not depend on x ∈ rn and r > 0. then, t (m) −→ b and m (m) −→ b (m ∈ n) are bounded operators from product space mp1,ϕ1 ×···×mpm,ϕm to mp,ϕ. moreover, we have ∥∥∥t(m)−→ b (−→ f )∥∥∥ mp,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖mpi,ϕi , ∥∥∥m(m)−→ b (−→ f )∥∥∥ mp,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖mpi,ϕi . our another main result is the following. theorem 1.4. let 0 < α < mn and 1 ≤ pi < mnα with 1 p = m∑ i=1 1 pi , 1 q = m∑ i=1 1 pi + m∑ i=1 1 qi − α n and −→ b ∈ (bmo)m (rn) for i = 1, . . . ,m. let functions ϕ,ϕi : rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies conditions (1.2)-(1.3) and ∞∫ r ( 1 + ln t r )m m∏ i=1 ϕi(x,t) t n p t n  1q− m∑ i=1 1 qi  +1 dt ≤ c0ϕ (x,r) , (1.14) where c0 does not depend on x ∈ rn and r > 0, lim r→0 ln 1 r inf x∈rn ϕ(x,r) = 0 (1.15) and cδ := ∞∫ δ (1 + ln |t|)m sup x∈rn m∏ i=1 ϕi(x,t) t n p t n  1q− m∑ i=1 1 qi  +1 dt < ∞ (1.16) for every δ > 0. int. j. anal. appl. 17 (4) (2019) 606 then, i (m) α, −→ b and m (m) α, −→ b (m ∈ n) are bounded operators from product space v mp1,ϕ1 ×···×v mpm,ϕm to v mq,ϕ. moreover, we have∥∥∥i(m) α, −→ b (−→ f )∥∥∥ v mq,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖v mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖v mpi,ϕi , (1.17) ∥∥∥m(m) α, −→ b (−→ f )∥∥∥ v mq,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖v mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖v mpi,ϕi . (1.18) corollary 1.6. let 1 < pi < ∞ and 0 < p < ∞ with 1p = m∑ i=1 1 pi and −→ b ∈ (bmo)m (rn) for i = 1, . . . ,m. let functions ϕ,ϕi : rn×(0,∞) → (0,∞) (i = 1, . . . ,m) and (ϕ1, . . . ,ϕm,ϕ) satisfies conditions (1.2)-(1.3) and ∞∫ r ( 1 + ln t r )m m∏ i=1 ϕi(x,t) t n p t n p +1 dt ≤ c0ϕ (x,r) , where c0 does not depend on x ∈ rn and r > 0, lim r→0 ln 1 r inf x∈rn ϕ(x,r) = 0 and cδ := ∞∫ δ (1 + ln |t|)m sup x∈rn m∏ i=1 ϕi(x,t) t n p t n p +1 dt < ∞ for every δ > 0. then, t (m) −→ b and m (m) −→ b (m ∈ n) are bounded operators from product space v mp1,ϕ1 ×···×v mpm,ϕm to v mp,ϕ. moreover, we have∥∥∥t(m)−→ b (−→ f )∥∥∥ v mp,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖v mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖v mpi,ϕi , ∥∥∥m(m)−→ b (−→ f )∥∥∥ v mp,ϕ . ∥∥∥−→b ∥∥∥ bmo ‖fi‖v mpi,ϕi . m∏ i=1 ‖bi‖∗‖fi‖v mpi,ϕi . the article is organized as follows. a key lemma is given and proved in section 2. section 3 will be devoted to the proofs of the theorems (theorems 1.3 and 1.4) stated above. 2. a key lemma in order to prove the main results (theorems 1.3 and 1.4), we need the following lemma. lemma 2.1. let x0 ∈ rn, 0 < α < mn and 1 ≤ pi < mnα with 1 p = m∑ i=1 1 pi , 1 q = m∑ i=1 1 pi + m∑ i=1 1 qi − α n and −→ b ∈ (bmo)m (rn) for i = 1, . . . ,m. then the inequality ‖i(m) α, −→ b (−→ f ) ‖lq(b(x0,r)) . m∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n  1q− m∑ i=1 1 qi  +1 (2.1) int. j. anal. appl. 17 (4) (2019) 607 holds for any ball b(x0,r) and for all −→ f ∈ llocp1 (r n) ×···×llocpm (r n). proof. in order to simplify the proof, we consider only the situation when m = 2. actually, a similar procedure works for all m ∈ n. thus, without loss of generality, it is sufficient to show that the conclusion holds for i (2) α, −→ b (−→ f ) = i (2) α,(b1,b2) (f1,f2). we just consider the case pi > 1 for i = 1, 2. for any x0 ∈ rn, set b = b (x0,r) for the ball centered at x0 and of radius r and 2b = b (x0, 2r). indeed, we also decompose fi as fi (yi) = fi (yi) χ2b + fi (yi) χ(2b)c for i = 1, 2. and, we write f1 = f 0 1 + f ∞ 1 and f2 = f 0 2 + f ∞ 2 , where f 0 i = fiχ2b, f ∞ i = fiχ(2b)c, for i = 1, 2. thus, we have ∥∥∥i(2)α,(b1,b2) (f1,f2)∥∥∥lq(b(x0,r)) ≤ ∥∥∥i(2)α,(b1,b2) (f01 ,f02)∥∥∥lq(b(x0,r)) + ∥∥∥i(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥lq(b(x0,r)) + ∥∥∥i(2)α,(b1,b2) (f∞1 ,f02)∥∥∥lq(b(x0,r)) + ∥∥∥i(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥lq(b(x0,r)) = f1 + f2 + f3 + f4. firstly, we use the boundedness of i (2) α,(b1,b2) from lp1 ×lp2 into lq (see lemma 1.1) to estimate f1, and we obtain f1 = ∥∥∥i(2)α,(b1,b2) (f01 ,f02)∥∥∥lq(b(x0,r)) . 2∏ i=1 ‖bi‖∗‖fi‖lpi(2b) . r n q 2∏ i=1 ‖bi‖∗‖fi‖lpi(2b) ∞∫ 2r dt t n q +1 . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n q +1 . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . secondly, for f2 = ∥∥∥i(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥lq(b(x0,r)), we decompose it into four parts as follows: f2 . ∥∥∥[(b1 −{b1}b)] [(b2 −{b2}b)] i(2)α (f01 ,f∞2 )∥∥∥ lq(b(x0,r)) + ∥∥∥[(b1 −{b1}b)] i(2)α [f01 , (b2 −{b2}b) f∞2 ]∥∥∥ lq(b(x0,r)) + ∥∥∥[(b2 −{b2}b)] i(2)α [(b1 −{b1}b) f01 ,f∞2 ]∥∥∥ lq(b(x0,r)) + ∥∥∥i(2)α [(b1 −{b1}b) f01 , (b2 −{b2}b) f∞2 ]∥∥∥ lq(b(x0,r)) ≡ f21 + f22 + f23 + f24. int. j. anal. appl. 17 (4) (2019) 608 let 1 < p1,p2 < 2n α , such that 1 p = 1 p1 + 1 p2 , 1 q = 1 p − α n , 1 r = 1 q1 + 1 q2 and 1 q = 1 r + 1 q . then, using hölder’s inequality and by (1.8) we have f21 = ∥∥∥(b1 − (b1)b) (b2 − (b2)b) i(2)α (f01 ,f∞2 )∥∥∥ lq(b(x0,r)) . ‖(b1 − (b1)b) (b2 − (b2)b)‖lr(b(x0,r)) ∥∥∥i(2)α (f01 ,f∞2 )∥∥∥ lq(b(x0,r)) . ‖b1 − (b1)b‖lq1 (b(x0,r)) ‖b2 − (b2)b‖lq2 (b(x0,r)) ×r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n q +1 . 2∏ i=1 ‖bi‖∗|b(x0,r)| 1 q1 + 1 q2 r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n q +1 . 2∏ i=1 ‖bi‖∗r n ( 1 q1 + 1 q2 ) r n ( 1 p1 + 1 p2 −α n ) ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) t n ( 1 q1 + 1 q2 ) dt t n q +1 = 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 , where in the second inequality we have used the following fact: it is clear that |(x0 −y1, x0 −y2)| 2n−α ≥ |x0 −y2| 2n−α . by hölder’s inequality, we have ∣∣∣i(2)α (f01 ,f∞2 ) (x)∣∣∣ . ∫ rn ∫ rn ∣∣f01 (y1)∣∣ |f∞2 (y2)| |(x−y1,x−y2)| 2n−αdy1dy2 . ∫ 2b |f1 (y1)|dy1 ∫ (2b)c |f2 (y2)| |x0 −y2| 2n−αdy2 ≈ ∫ 2b |f1 (y1)|dy1 ∫ (2b)c |f2 (y2)| ∞∫ |x0−y2| dt t2n−α+1 dy2 . ‖f1‖lp1 (2b) |2b| 1− 1 p1 ∞∫ 2r ‖f2‖lp2 (b(x0,t)) |b (x0, t)| 1− 1 p2 dt t2n−α+1 . ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n q +1 , where 1 p = 1 p1 + 1 p2 . thus, the inequality ∥∥∥i(2)α (f01 ,f∞2 )∥∥∥ lq(b(x0,r)) . r n q ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n q +1 is valid. int. j. anal. appl. 17 (4) (2019) 609 on the other hand, for the estimates used in f22, f23, we have to prove the below inequality: ∣∣∣i(2)α [f01 , (b2 (·) − (b2)b) f∞2 ] (x)∣∣∣ . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 p1 + 1 p2 ) +1−α . (2.2) to estimate f22, the following inequality ∣∣∣i(2)α [f01 , (b2 (·) − (b2)b) f∞2 ] (x)∣∣∣ . ∫ 2b |f1 (y1)|dy1 ∫ (2b)c |b2 (y2) − (b2)b| |f2 (y2)| |x0 −y2| 2n−α dy2 is satisfied. it’s obvious that ∫ 2b |f1 (y1)|dy1 . ‖f1‖lp1 (2b) |2b| 1− 1 p1 , (2.3) and using hölder’s inequality and by (1.6) and (1.7) we have ∫ (2b)c |b2 (y2) − (b2)b| |f2 (y2)| |x0 −y2| 2n−α dy2 . ∫ (2b)c ∣∣∣b2 (y2) − (b2)b(x0,r)∣∣∣ |f2 (y2)|   ∞∫ |x0−y2| dt t2n−α+1  dy2 . ∞∫ 2r ∥∥∥b2 (y2) − (b2)b(x0,t)∥∥∥lq2 (b(x0,t)) ‖f2‖lp2 (b(x0,t)) |b (x0, t)|1− ( 1 p2 + 1 q2 ) dt t2n−α+1 + ∞∫ 2r ∣∣∣(b2)b(x0,t) − (b2)b(x0,r)∣∣∣‖f2‖lp2 (b(x0,t)) |b (x0, t)|1− 1p2 dtt2n−α+1 . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 |b (x0, t)| 1 q2 ‖f2‖lp2 (b(x0,t)) |b (x0, t)| 1− ( 1 p2 + 1 q2 ) dt t2n−α+1 + ‖b2‖∗ ∞∫ 2r ( 1 + ln t r ) |b (x0, t)|‖f2‖lp2 (b(x0,t)) |b (x0, t)| 1− 1 p2 dt t2n−α+1 . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 ‖f2‖lp2 (b(x0,t)) dt t n ( 1+ 1 p2 ) +1−α . (2.4) hence, by (2.3) and (2.4), it follows that: ∣∣∣i(2)α [f01 , (b2 (·) − (b2)b) f∞2 ] (x)∣∣∣ . ‖b2‖∗‖f1‖lp1 (2b) |2b| 1− 1 p1 ∞∫ 2r ( 1 + ln t r )2 ‖f2‖lp2 (b(x0,t)) dt t n ( 1+ 1 p2 ) +1−α . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 p1 + 1 p2 ) +1−α . int. j. anal. appl. 17 (4) (2019) 610 this completes the proof of inequality (2.2). thus, let 1 < τ < ∞, such that 1 q = 1 q1 + 1 τ . then, using hölder’s inequality and from (2.2) and (1.7), we get f22 = ∥∥∥[(b1 −{b1}b)] i(2)α [f01 , (b2 −{b2}b) f∞2 ]∥∥∥ lq(b(x0,r)) . ‖b1 − (b1)b‖lq1 (b) ∥∥∥i(2)α [f01 , (b2 − (b2)b) f∞2 ]∥∥∥ lτ(b) . 2∏ i=1 ‖bi‖∗ |b (x0,r)| 1 q1 + 1 τ × ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 p1 + 1 p2 ) +1−α . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . similarly, f23 has the same estimate above, here we omit the details, thus the inequality f23 = ∥∥∥[(b2 −{b2}b)] i(2)α [(b1 −{b1}b) f01 ,f∞2 ]∥∥∥ lq(b(x0,r)) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 is valid. now we turn to estimate f24. similar to (2.2), we have to prove the following estimate for f24: ∣∣∣i(2)α [(b1 − (b1)b) f01 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ ≤ 2∏ i=1 ‖bi‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . (2.5) firstly, the following inequality ∣∣∣i(2)α [(b1 − (b1)b) f01 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ . ∫ 2b |b1 (y1) − (b1)b| |f1 (y1)|dy1 ∫ (2b)c |b2 (y2) − (b2)b| |f2 (y2)| |x0 −y2| 2n−α dy2 is valid. it’s obvious that from hölder’s inequality and (1.7) ∫ 2b |b1 (y1) − (b1)b| |f1 (y1)|dy1 . ‖b1‖∗ |b (x0,r)| 1− 1 p1 ‖f1‖lp1 (2b) . (2.6) then, by (2.4) and (2.6) we have ∣∣∣i(2)α [(b1 − (b1)b) f01 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ ≤ 2∏ i=1 ‖bi‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . int. j. anal. appl. 17 (4) (2019) 611 this completes the proof of inequality (2.5). therefore, by (2.5) we deduce that f24 = ∥∥∥i(2)α [(b1 − (b1)b) f01 , (b2 − (b2)b) f∞2 ]∥∥∥ lq(b) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . considering estimates f21, f22, f23, f24 together, we get the desired conclusion f2 = ∥∥∥i(2)α,(b1,b2) (f01 ,f∞2 )∥∥∥lq(b(x0,r)) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . similar to f2, we can also get the estimates for f3, f3 = ∥∥∥i(2)α,(b1,b2) (f∞1 ,f02)∥∥∥lq(b(x0,r)) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . at last, we consider the last term f4 = ∥∥∥i(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥lq(b(x0,r)). we split f4 in the following way: f4 . f41 + f42 + f43 + f44, where f41 = ∥∥∥(b1 − (b1)b) (b2 − (b2)b) i(2)α (f∞1 ,f∞2 )∥∥∥ lq(b) , f42 = ∥∥∥(b1 − (b1)b) i(2)α [f∞1 , (b2 − (b2)b) f∞2 ]∥∥∥ lq(b) , f43 = ∥∥∥(b2 − (b2)b) i(2)α [(b1 − (b1)b) f∞1 ,f∞2 ]∥∥∥ lq(b) , f44 = ∥∥∥i(2)α [(b1 − (b1)b) f∞1 , (b2 − (b2)b) f∞2 ]∥∥∥ lq(b) . now, let us estimate f41, f42, f43, f44 respectively. for the term f41, let 1 < τ < ∞, such that 1q = ( 1 q1 + 1 q2 ) + 1 τ , 1 τ = 1 p1 + 1 p2 − α n . then, by hölder’s inequality and (1.8), we get f41 = ∥∥∥(b1 − (b1)b) (b2 − (b2)b) i(2)α (f∞1 ,f∞2 )∥∥∥ lq(b) . ‖b1 − (b1)b‖lq1 (b) ‖b2 − (b2)b‖lq2 (b) ∥∥∥i(2)α (f∞1 ,f∞2 )∥∥∥ lτ(b) . 2∏ i=1 ‖bi‖∗ |b (x0,r)| 1 q1 + 1 q2 r n τ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n τ +1 . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 , where in the second inequality we have used the following fact: int. j. anal. appl. 17 (4) (2019) 612 noting that |(x0 −y1, x0 −y2)| 2n−α ≥ |x0 −y1| n−α 2 |x0 −y2| n−α 2 and by hölder’s inequality, we get ∣∣∣i(2)α (f∞1 ,f∞2 ) (x)∣∣∣ . ∫ rn ∫ rn ∣∣f1 (y1) χ(2b)c∣∣ ∣∣f2 (y2) χ(b)c∣∣ |(x0 −y1,x0 −y2)| 2n−α dy1dy2 . ∫ (2b)c ∫ (2b)c |f1 (y1)| |f2 (y2)| |x0 −y1| n−α 2 |x0 −y2| n−α 2 dy1dy2 . ∞∑ j=1 2∏ i=1 ∫ 2j+1b\2jb |fi (yi)| |x0 −yi| n−α 2 dyi . ∞∑ j=1 2∏ i=1 ( 2jr )−n+α 2 ∫ 2j+1b |fi (yi)|dyi . ∞∑ j=1 ( 2jr )−2n+α 2∏ i=1 ‖fi‖lpi(2j+1b) ∣∣2j+1b∣∣1− 1pi . ∞∑ j=1 2j+2r∫ 2j+1r ( 2j+1r )−2n+α−1 2∏ i=1 ‖fi‖lpi(2j+1b) ∣∣2j+1b∣∣1− 1pi dt . ∞∑ j=1 2j+2r∫ 2j+1r 2∏ i=1 ‖fi‖lpi(b(x0,t)) |b (x0, t)| 1− 1 pi dt t2n+1−α . ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) |b (x0, t)| 2− ( 1 p1 + 1 p2 ) dt t2n+1−α . ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n τ +1 . moreover, for p1, p2 ∈ [1,∞) the inequality ∥∥∥i(2)α (f∞1 ,f∞2 )∥∥∥ lτ(b(x0,r)) . r n τ ∞∫ 2r 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n τ +1 (2.7) is valid. for the terms f42, f43, similar to the estimates used for (2.2), we have to prove the following inequality: ∣∣∣i(2)α [f∞1 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 p1 + 1 p2 ) +1−α . (2.8) int. j. anal. appl. 17 (4) (2019) 613 noting that |(x0 −y1, x0 −y2)| 2n−α ≥ |x0 −y1| n−α 2 |x0 −y2| n−α 2 , we get ∣∣∣i(2)α [f∞1 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ . ∫ rn ∫ rn |b2 (y2) − (b2)b| ∣∣f1 (y1) χ(2b)c∣∣ ∣∣f2 (y2) χ(2b)c∣∣ |(x0 −y1,x0 −y2)| 2n−α dy1dy2 . ∫ (2b)c ∫ (2b)c |b2 (y2) − (b2)b| |f1 (y1)| |f2 (y2)| |x0 −y1| n−α 2 |x0 −y2| n−α 2 dy1dy2 . ∞∑ j=1 ∫ 2j+1b\2jb |f1 (y1)| |x0 −y1| n−α 2 dy1 ∫ 2j+1b\2jb |b2 (y2) − (b2)b| |f2 (y2)| |x0 −y2| n−α 2 dy2 . ∞∑ j=1 ( 2jr )−2n+α ∫ 2j+1b |f1 (y1)|dy1 ∫ 2j+1b |b2 (y2) − (b2)b| |f2 (y2)|dy2. on the other hand, it’s obvious that ∫ 2j+1b |f1 (y1)|dy1 ≤‖f1‖lp1 (2j+1b) ∣∣2j+1b∣∣1− 1p1 , (2.9) and using hölder’s inequality and by (1.6) and (1.7) ∫ 2j+1b |b2 (y2) − (b2)b| |f2 (y2)|dy2 ≤‖b2 − (b2)2j+1b‖lq2 (2j+1b) ‖f2‖lp2 (2j+1b) ∣∣2j+1b∣∣1−( 1p2 + 1q2 ) + |(b2)2j+1b − (b2)b|‖f2‖lp2 (2j+1b) ∣∣2j+1b∣∣1− 1p2 . ‖b2‖∗ ∣∣2j+1b∣∣ 1q2 (1 + ln 2j+1r r ) ‖f2‖lp2 (2j+1b) ∣∣2j+1b∣∣1−( 1p2 + 1q2 ) + ‖b2‖∗ ( 1 + ln 2j+1r r )∣∣2j+1b∣∣‖f2‖lp2 (2j+1b) ∣∣2j+1b∣∣1− 1p2 . ‖b2‖∗ ( 1 + ln 2j+1r r )2 ∣∣2j+1b∣∣1− 1p2 ‖f2‖lp2 (2j+1b) . (2.10) hence, by (2.9) and (2.10), it follows that: ∣∣∣i(2)α [f∞1 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ . ∞∑ j=1 ( 2jr )−2n+α ∫ 2j+1b |f1 (y1)|dy1 ∫ 2j+1b |b2 (y2) − (b2)b| |f2 (y2)|dy2 . ‖b2‖∗ ∞∑ j=1 ( 2jr )−2n+α ( 1 + ln 2j+1r r )2 ∣∣2j+1b∣∣2−( 1p1 + 1p2 ) 2∏ i=1 ‖fi‖lpi(2j+1b) int. j. anal. appl. 17 (4) (2019) 614 . ‖b2‖∗ ∞∑ j=1 2j+2r∫ 2j+1r ( 2j+1r )−2n+α−1 ( 1 + ln 2j+1r r )2 ∣∣2j+1b∣∣2−( 1p1 + 1p2 ) 2∏ i=1 ‖fi‖lpi(2j+1b) dt . ‖b2‖∗ ∞∑ j=1 2j+2r∫ 2j+1r ( 1 + ln 2j+1r r )2 ∣∣2j+1b∣∣2−( 1p1 + 1p2 ) 2∏ i=1 ‖fi‖lpi(2j+1b) dt t2n−α+1 . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 |b (x0, t)| 2− ( 1 p1 + 1 p2 ) 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t2n−α+1 . ‖b2‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 p1 + 1 p2 ) +1−α . this completes the proof of (2.8). now we turn to estimate f42. let 1 < τ < ∞, such that 1q = 1 q1 + 1 τ . then, by hölder’s inequality, (1.7) and (2.8), we obtain f42 = ∥∥∥(b1 − (b1)b) i(2)α [f∞1 , (b2 − (b2)b) f∞2 ]∥∥∥ lq(b) . ‖(b1 − (b1)b)‖lq1 (b) ∥∥∥i(2)α [f∞1 , (b2 − (b2)b) f∞2 ]∥∥∥ lτ(b) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . similarly, f43 has the same estimate above, here we omit the details, thus the inequality f43 = ∥∥∥(b2 − (b2)b) i(2)α [(b1 − (b1)b) f∞1 ,f∞2 ]∥∥∥ lq(b) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 is valid. finally, to estimate f44, similar to the estimate of (2.8), we have ∣∣∣i(2)α [(b1 − (b2)b) f∞1 , (b2 − (b2)b) f∞2 ] (x)∣∣∣ . ∞∑ j=1 ( 2jr )−2n+α ∫ 2j+1b |b1 (y1) − (b1)b| |f1 (y1)|dy1 ∫ 2j+1b |b2 (y2) − (b2)b| |f2 (y2)|dy2 . 2∏ i=1 ‖bi‖∗ ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . int. j. anal. appl. 17 (4) (2019) 615 thus, we have f44 = ∥∥∥i(2)α [(b1 − (b1)b) f∞1 , (b2 − (b2)b) f∞2 ]∥∥∥ lq(b) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . by the estimates of f4j above, where j = 1,2,3,4, we know that f4 = ∥∥∥i(2)α,(b1,b2) (f∞1 ,f∞2 )∥∥∥lq(b(x0,r)) . 2∏ i=1 ‖bi‖∗r n q ∞∫ 2r ( 1 + ln t r )2 2∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n ( 1 q − ( 1 q1 + 1 q2 )) +1 . consequently, combining all the estimates for f1, f2, f3, f4, we complete the proof of lemma 2.1. � 3. proofs of the main results now we are ready to return to the proofs of theorems 1.3 and 1.4. 3.1. proof of theorem 1.3. proof. to prove theorem 1.3, we will use the following relationship between essential supremum and essential infimum ( essinf x∈e f (x) )−1 = esssup x∈e 1 f (x) , (3.1) where f is any real-valued nonnegative function and measurable on e (see [30], page 143). indeed, we consider (1.12) firstly. since −→ f ∈ mp1,ϕ1 × ··· × mpm,ϕm, by (3.1) and the non-decreasing, with respect to t, of the norm m∏ i=1 ‖fi‖lpi(b(x,t)), we get m∏ i=1 ‖fi‖lpi(b(x,t)) essinf 0<t<τ<∞ m∏ i=1 ϕi(x,τ)τ n p ≤ esssup 0<t<τ<∞ m∏ i=1 ‖fi‖lpi(b(x,t)) m∏ i=1 ϕi(x,τ)τ n p ≤ esssup 0<τ<∞,x∈rn m∏ i=1 ‖fi‖lpi(b(x,t)) m∏ i=1 ϕi(x,τ)τ n p ≤ m∏ i=1 ‖fi‖mpi,ϕi . (3.2) int. j. anal. appl. 17 (4) (2019) 616 for 1 < p1, . . . ,pm < ∞, since (ϕ1, . . . ,ϕm,ϕ) satisfies (1.11) and by (3.2), we have ∞∫ r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x,t)) dt t n  1q− m∑ i=1 1 qi  +1 ≤ ∞∫ r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x,t)) essinf t<τ<∞ m∏ i=1 ϕi(x,τ)τ n p essinf t<τ<∞ m∏ i=1 ϕi(x,τ)τ n p t n  1q− m∑ i=1 1 qi  +1 dt ≤ c m∏ i=1 ‖fi‖mpi,ϕi ∞∫ r ( 1 + ln t r )m essinft<τ<∞ m∏ i=1 ϕi(x,τ)τ n p t n  1q− m∑ i=1 1 qi  +1 dt ≤ c m∏ i=1 ‖fi‖mpi,ϕi ϕ(x,r). (3.3) then by (2.1) and (3.3), we get ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ mq,ϕ = sup x∈rn,r>0 ϕ (x,r) −1 |b(x,r)|− 1 q ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ lq(b(x,r)) . m∏ i=1 ‖bi‖∗ sup x∈rn,r>0 ϕ (x0,r) −1 ∞∫ r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n  1q− m∑ i=1 1 qi  +1 . m∏ i=1 ‖bi‖∗‖fi‖mpi,ϕi . thus we obtain (1.12). the conclusion of (1.13) is a direct consequence of (1.10) and (1.12). indeed, from the process proving (1.12), it is easy to see that the conclusions of (1.12) also hold for ĩ (m) −→ b ,α . combining this with (1.10), we can immediately obtain (1.13), which completes the proof. � 3.2. proof of theorem 1.4. proof. since the inequalities (1.17) and (1.18) hold by theorem 1.3, we only have to prove the implication lim r→0 sup x∈rn r− n p m∏ i=1 ‖fi‖lpi(b(x,r)) m∏ i=1 ϕi(x,r) = 0 implies lim r→0 sup x∈rn r− n q ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ lq(b(x,r)) ϕ(x,r) = 0. (3.4) to show that sup x∈rn r− n q ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ lq(b(x,r)) ϕ(x,r) < ε for small r, int. j. anal. appl. 17 (4) (2019) 617 we use the estimate (2.1): sup x∈rn r− n q ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ lq(b(x,r)) ϕ(x,r) . sup x∈rn m∏ i=1 ‖bi‖∗ ϕ(x,r) ∞∫ r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x,t)) dt t n  1q− m∑ i=1 1 qi  +1 . we take r < δ0, where δ0 is small enough and split the integration: r− n q ∥∥∥i(m) α, −→ b (−→ f )∥∥∥ lq(b(x,r)) ϕ(x,r) ≤ c [iδ0 (x,r) + jδ0 (x,r)] , (3.5) where δ0 > 0 (we may take δ0 < 1), and iδ0 (x,r) := m∏ i=1 ‖bi‖∗ ϕ(x,r) δ0∫ r ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n  1q− m∑ i=1 1 qi  +1 , and jδ0 (x,r) := m∏ i=1 ‖bi‖∗ ϕ(x,r) ∞∫ δ0 ( 1 + ln t r )m m∏ i=1 ‖fi‖lpi(b(x0,t)) dt t n  1q− m∑ i=1 1 qi  +1 , and r < δ0. now we can choose any fixed δ0 > 0 such that sup x∈rn t− n p m∏ i=1 ‖fi‖lpi(b(x,t)) m∏ i=1 ϕi(x,t) < ε 2cc0 , t ≤ δ0, where c and c0 are constants from (1.14) and (3.5), which is possible since −→ f ∈ v mp1,ϕ1 ×···×v mpm,ϕm. this allows to estimate the first term uniformly in r ∈ (0,δ0): m∏ i=1 ‖bi‖∗ sup x∈rn ciδ0 (x,r) < ε 2 , 0 < r < δ0 by (1.14). for the second term, writing 1 + ln t r ≤ 1 + |ln t|+ ln 1 r , by the choice of r sufficiently small because of the conditions (1.15) we obtain jδ0 (x,r) ≤ cδ0 + c̃δ0 ln 1 r ϕ(x,r) m∏ i=1 ‖bi‖∗‖fi‖v mpi,ϕi , where cδ0 is the constant from (1.16) with δ = δ0 and c̃δ0 is a similar constant with omitted logarithmic factor in the integrand. then, by (1.15) we can choose r small enough such that sup x∈rn jδ0 (x,r) < ε 2 , which completes the proof of (3.4). int. j. anal. appl. 17 (4) (2019) 618 for m (m) α, −→ b , we can also use the same method to obtain the desired result, but we omit the details. therefore, the proof of theorem 1.4 is completed. � acknowledgement: this study has been given as the plenary talk by the author at the “the international workshop on mathematical methods in engineering” (mme-2017), çankaya university, ankara, turkey, 27-29 april 2017. references [1] x. n. cao and d. x. chen, the boundedness of toeplitz-type operators on vanishing morrey spaces, anal. theory appl., 27 (2011), 309-319. [2] d. x. chen, j. chen and s. mao, weighted lp estimates for maximal commutators of multilinear singular integrals, chin. ann. math., 34b(6) (2013), 885-902. [3] x. chen and q. y. xue, weighted estimates for a class of multilinear fractional type operators, j. math. anal. appl., 362 (2010), 355-373. [4] r. r. coifman and y. meyer, on commutators of singular integrals and bilinear singular integrals, trans. amer. math. soc., 212 (1975), 315-331. [5] z. w. fu, y. lin and s. z. lu, λ-central bmo estimates for commutators of singular integral operators with rough kernel, acta math. sin. (engl. ser.), 24 (2008), 373-386. [6] l. grafakos, on multilinear fractional integrals, studia. math., 102 (1992), 49-56. [7] l. grafakos and r. h. torres, multilinear calderón-zygmund theory, adv. math., 165 (2002), 124-164. [8] l. grafakos and r. h. torres, maximal operator and weighted norm inequalities for multilinear singular integrals, indiana univ. math. j., 51 (2002), 1261-1276. [9] l. grafakos and r. h. torres, on multilinear singular integrals of calderón-zygmund type, in: proceedings of the 6th international conference on harmonic analysis and partial differential equations (el escorial), in: publ. mat., vol. extra, 2002, 57-91. [10] m. giaquinta, multiple integrals in the calculus of variations and non-linear elliptic systems. princeton, new jersey: princeton univ. press, 1983. [11] f. gürbüz, weighted morrey and weighted fractional sobolev-morrey spaces estimates for a large class of pseudodifferential operators with smooth symbols, j. pseudo-differ. oper. appl., 7(4) (2016), 595-607. [12] f. gürbüz, sublinear operators with rough kernel generated by calderón-zygmund operators and their commutators on generalized morrey spaces, math. notes, 101(3-4) (2017), 429-442. [13] f. gürbüz, some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted morrey spaces, canad. math. bull., 60(1) (2017), 131-145. [14] f. gürbüz, multi-sublinear operators generated by multilinear fractional integral operators and local campanato space estimates for commutators on the product generalized local morrey spaces, adv. math. (china), 47(6) (2018), 855-880. [15] f. john and l. nirenberg, on functions of bounded mean oscillation, commun. pure appl. math., 14 (1961), 415-426. [16] t. karaman, boundedness of some classes of sublinear operators on generalized weighted morrey spaces and some applications [ph.d. thesis], ankara university, ankara, turkey, 2012 (in turkish). [17] c. e. kenig and e. m. stein, multilinear estimates and fractional integration, math. res. lett., 6 (1999), 1-15. int. j. anal. appl. 17 (4) (2019) 619 [18] y. lin, strongly singular calderón-zygmund operator and commutator on morrey type spaces, acta math. sin. (engl. ser.), 23(11) (2007), 2097-2110. [19] t. mizuhara, boundedness of some classical operators on generalized morrey spaces, harmonic analysis (s. igari, editor), icm 90 satellite proceedings, springer verlag, tokyo (1991), 183-189. [20] k. moen, weighted inequalities for multilinear fractional integral operators, collect. math., 60 (2009), 213-238. [21] c. b. morrey, on the solutions of quasi-linear elliptic partial differential equations, trans. amer. math. soc., 43 (1938), 126-166. [22] b. muckenhoupt and r. l. wheeden, weighted norm inequalities for fractional integrals, trans. amer. math. soc., 192 (1974), 261-274. [23] d. k. palagachev and l. g. softova, singular integral operators, morrey spaces and fine regularity of solutions to pde’s, potential anal., 20 (2004), 237-263. [24] z. y. si and s. z. lu, weighted estimates for iterated commutators of multilinear fractional operators, acta math. sin. (engl. ser.), 28(9) (2012), 1769-1778. [25] l. g. softova, singular integrals and commutators in generalized morrey spaces, acta math. sin. (engl. ser.), 22(3) (2006), 757-766. [26] m. e. taylor, tools for pde: pseudodifferential operators, paradifferential operators, and layer potentials, volume 81 of math. surveys and monogr. ams, providence, r.i., 2000. [27] c. vitanza, functions with vanishing morrey norm and elliptic partial differential equations, in: proceedings of methods of real analysis and partial differential equations, capri, pp. 147-150. springer (1990). [28] j. xu, boundedness in lebesgue spaces for commutators of multilinear singular integrals and rbmo functions with non-doubling measures, sci. china (series a), 50 (2007), 361-376. [29] x. yu and x. x. tao, boundedness of multilinear operators on generalized morrey spaces, appl. math. j. chinese univ., 29(2) (2014), 127-138. [30] r. l. wheeden and a. zygmund, measure and integral: an introduction to real analysis, vol. 43 of pure and applied mathematics, marcel dekker, new york, ny, usa, 1977. 1. introduction and main results 2. a key lemma 3. proofs of the main results 3.1. proof of theorem 1.3. 3.2. proof of theorem 1.4. references international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 123-131 http://www.etamaths.com absolute monotonicity of functions related to estimates of first eigenvalue of laplace operator on riemannian manifolds feng qi1,2,∗ and miao-miao zheng1 abstract. the authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of laplace operator on riemannian manifolds. 1. background and mail results in [38, 39], j. q. zhong and h. c. yang obtained that the first eigenvalue λ1 of laplace operator on a compact riemannian monifold m with non-negative ricci curvature satisfies (1.1) λ1 ≥ π2 d2 , where d denotes the diameter of m. the inequality (1.1) improves corresponding results in [11, 12]. for proving the inequality (1.1), the authors introduced in [38, lemma 4] and [39, lemma 4] the function (1.2) ψ(θ) =   4 π (θ + sin θ cos θ) − 2 sin θ cos2 θ , θ ∈ ( − π 2 , π 2 ) ±1, θ = ± π 2 and obtained that the function y(θ) = ψ(θ) satisfies ψ′(θ) ≥ 0, the differential equation (1.3) y(θ) − sin θ + y′ sin θ cos θ − 1 2 y′′(θ) cos2 θ = 0, and the inequality (1.4) 0 ≤ ψ′(θ) cos θ ≤ 2 ( 4 π − 1 ) on [ −π 2 , π 2 ] . these results were ever employed in [37, p. 348, lemma 4]. in [8, p. 3], it was pointed out that ψ′(θ) ≥ 0 and |ψ(θ)| ≤ 1 on [ −π 2 , π 2 ] . for more information, please refer to [18, lemma 4], [23, lemma 1 and proposition 7], [26, lemma 4], and [27, proposition 3]. 2010 mathematics subject classification. primary 26a48, 33b10; secondary 11b68, 34a05, 44a10, 58c40. key words and phrases. absolutely monotonic function; completely monotonic function; trigonometric function; laplace operator; eigenvalue. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 123 124 f. qi and m.-m. zheng let m be a m-dimensional compact riemannian manifold with boundary ∂m, the inner radius of m be ρ, the ricci curvature of m be not less than −r, and the mean curvature of ∂m be not less than −h0, where r and h0 are positive scalars. theorem 3 in [35, p. 331] reads that the first eigenvalue µ1 of m under dirichlet boundary condition satisfies (1.5) µ1 ≥ π2 4ρ2 − 1 2 r− 2 3 (m− 1)h0 π ρ . for proving the inequality (1.5), the author considered the functions (1.6) p(θ) =   2 cos2 θ ∫ π/2 θ t cos2 t d t, θ ∈ ( − π 2 , π 2 ) 0, θ = ± π 2 and (1.7) φ(θ) =   1 cos2 θ ∫ π/2 θ cos2 t d t, θ ∈ ( − π 2 , π 2 ) 0, θ = π 2 and obtained in [35, pp. 338–340] that p′(θ) ≤ 0 and φ′(θ) ≤ 0 on [ 0, π 2 ] , that (1.8) ∫ π/2 0 p(θ) d θ = π 2 , ∫ π/2 0 φ(θ) d θ = 1 2 , and that the function z(θ) = 1 + αp(θ) + βφ(θ) satisfies z ( π 2 ) = 1 and (1.9) z(θ) = 1 + α cos2 θ −z′(θ) cos θ sin θ + 1 2 z′′(θ) cos2 θ, θ ∈ [ 0, π 2 ] . in [18, propositions 11 and 12], [23, propositions 2, 3, and 5], and [27, propositions 1 and 2], it was obtained that the function y (θ) = p(θ) satisfies the differential equation (1.10) y ′′(θ) cos2 θ − 2y ′(θ) sin θ cos θ − 2y (θ) + 2 cos2 θ = 0 and the inequalities (1.11) p′(θ) sin θ ≤ 0, |p′(θ) cos θ| ≤ 8 3 , p(θ) ≤ π2 8 − 1 2 for θ ∈ [ − π 2 , π 2 ] . in [9], it was established that the function p(θ) cos θ is increasing on[ 0, π 2 ] , that the function p′(θ) is decreasing, and that (1.12) π2 8 − 1 2 ≤ p(θ) cos θ ≤ π 3 , p(θ) ≤ 1 5 + cos2 θ on [ −π 2 , π 2 ] . see also [34, p. 699]. in [13, theorem 1.1], it was obtained that the first positive eigenvalue λ of laplace operator on a closed n-dimensional riemannian manifold with ricci curvature ric(m) ≥ (n− 1)k > 0 has the lower bound (1.13) λ ≥ 1 2 (n− 1)k + π2 4r2 , where r is the largest interior radius of the nodal domains of eigenfunctions of the eigenvalue λ. for verifying the above conclusion, the author considered in [13, absolute and complete monotonicity of some functions 125 lemma 3.1] the function ξ(t) = −2p(t) and obtained some conclusions on ξ(t), which may be reformulated as follows. (1) for t ∈ ( −π 2 , π 2 ) , the function ξ(t) meets 1 2 ξ′′(t) cos2 t− ξ′(t) cos t sin t− ξ(t) = 2 cos2 t,(1.14) ξ′(t) cos t− 2ξ(t) sin t = 4t cos t;(1.15) (2) for t ∈ [ −π 2 , π 2 ] , (1.16) 1 − π2 4 = ξ(0) ≤ ξ(t) ≤ ξ ( ± π 2 ) = 0 and ∫ π/2 0 ξ(t) d t = − π 2 ; (3) the derivative ξ′(t) is increasing on [ −π 2 , π 2 ] , (1.17) ξ′ ( ± π 2 ) = ± 2π 3 , and ξ′(t)  < 0, t ∈ ( − π 2 , 0 ) , > 0, t ∈ ( 0, π 2 ) ; (4) for t ∈ [ −π 2 , π 2 ] , (1.18) 2 ( 3 − π2 4 ) ≤ ξ′(t) t ≤ 4 3 , and for t ∈ ( 0, π 2 ) , (1.19) [ ξ′(t) t ]′ > 0; (5) for t ∈ [ −π 2 , π 2 ] , (1.20) ξ′′(t) > 0, ξ′′ ( ± π 2 ) = 2, and ξ′′(0) = 2 ( 3 − π2 4 ) ; (6) for t ∈ [ −π 2 , π 2 ] , (1.21) ξ′′′ ( π 2 ) = 8π 15 , ξ′′′(t)  < 0, t ∈ ( − π 2 , 0 ) , > 0, t ∈ ( 0, π 2 ) . by calculus, it is easy to see that ψ(θ) =   2 π [2θ + sin(2θ) −π sin θ] sec2 θ, θ ∈ ( − π 2 , π 2 ) , ±1, θ = ± π 2 , (1.22) p(θ) =   ( π2 8 − 1 2 θ2 ) sec2 θ −θ tan θ − 1 2 , θ ∈ ( − π 2 , π 2 ) , 0, θ = ± π 2 , (1.23) and (1.24) φ(θ) =  − 1 4 [2θ + sin(2θ) −π] sec2 θ, θ ∈ ( − π 2 , π 2 ) , 0, θ = π 2 . see also [28, pp. 6–7]. for more information, please read [4, 5, 10, 17, 24, 25, 30, 36] and closely related references therein. 126 f. qi and m.-m. zheng a function f is said to be completely monotonic on an interval i if it has derivatives of all orders on i and satisfies (1.25) 0 ≤ (−1)k−1f(k−1)(x) < ∞ for x ∈ i and k ∈ n, where f(0)(x) means f(x) and n stands for the set of all positive integers. see [14, chapter xiii], [31, chapter 1], or [33, chapter iv]. the class of completely monotonic functions may be characterized by the famous hausdorff-bernstein-widder theorem [33, p. 161, theorem 12b]: a necessary and sufficient condition that f(x) should be completely monotonic for 0 < x < ∞ is that (1.26) f(x) = ∫ ∞ 0 e−xt d α(t), where α(t) is non-decreasing and the above integral converges for 0 < x < ∞. recall from [14, chapter xiii] or [33, chapter iv] that a function f is said to be absolutely monotonic on an interval i if it has derivatives of all orders and (1.27) f(k−1)(t) ≥ 0 for t ∈ i and k ∈ n. theorem 12c in [33, p. 162] states that a necessary and sufficient condition that f(x) should be absolutely monotonic in −∞ < x < 0 is that (1.28) f(x) = ∫ ∞ 0 ext d α(t), where α(t) is non-decreasing and the integral converges for −∞ < x < 0. for more information on completely and absolutely monotonic functions, please refer to [6, 7, 19, 20, 21, 22, 29] and closely related references therein. in this paper, we will prove the following absolute and complete monotonicity of functions related to estimates of first eigenvalue of laplace operator on riemannian manifolds. theorem 1.1. the functions ψ(θ) and 8 π −2−ψ′(θ) cos θ are absolutely monotonic on ( 0, π 2 ) . theorem 1.2. the function −p′(θ) is absolutely monotonic on ( 0, π 2 ) . theorem 1.3. the function φ(θ) is completely monotonic on ( −π 2 , π 2 ) . 2. proofs of theroems 1.1 to 1.3 proof of theorem 1.1. the function ψ(θ) may be rewritten as ψ(θ) = 4 π tan θ + 4 π θ sec2 θ − 2 tan θ sec θ = 4 π tan θ + 4 π θ(tan θ)′ − 2(sec θ)′ = 4 π (θ tan θ)′ − 2(sec θ)′. it is well known [1, p. 75, 4.3.67 and 4.3.69] that the tangent tan x and the secant sec x can be expanded into power series (2.1) tan z = ∞∑ n=1 (−1)n−122n(22n − 1)b2n z2n−1 (2n)! absolute and complete monotonicity of some functions 127 and (2.2) sec z = ∞∑ n=0 (−1)ne2n z2n (2n)! for |z| < π 2 , where bn for n ≥ 0 are bernoulli numbers which may be defined by the power series expansion (2.3) z ez − 1 = ∞∑ n=0 bn zn n! = 1 − z 2 + ∞∑ k=1 b2k z2k (2k)! , |z| < 2π and en for n ≥ 0 stand for euler numbers which are integers and may be defined by (2.4) 2ez e2z + 1 = ∞∑ n=0 en n! zn = ∞∑ n=0 e2n z2n (2n)! , |z| < π, see [1, p. 804, 23.1.1 and 23.1.2] or [32, p. 3, (1.1) and p. 15]. consequently, ψ(θ) = 4 π ∞∑ n=1 (−1)n−122n(22n − 1)b2n (2n− 1)! θ2n−1 − 2 ∞∑ n=1 (−1)n (2n− 1)! e2nθ 2n−1 = 2 ∞∑ n=1 1 (2n− 1)! (−1)n−1 [ 2 π 22n(22n − 1)b2n + e2n ] θ2n−1. in [1, p. 805, 23.1.15], it was listed that (2.5) 4n+1(2n)! π2n+1 > (−1)ne2n > 1 1 + 3−1−2n 4n+1(2n)! π2n+1 , n ∈{0}∪n. in [2], it was obtained that the double inequality (2.6) 2(2n)! (2π)2n 1 1 − 2α−2n ≤ (−1)n−1b2n ≤ 2(2n)! (2π)2n 1 1 − 2β−2n holds for n ∈ n if and only if α ≤ 0 and β ≥ 2 + ln(1−6/π 2) ln 2 = 0.649 . . . . as a result, (−1)n−1 [ 2 π 22n(22n − 1)b2n + e2n ] > 2 π 22n(22n − 1) 2(2n)! (2π)2n 1 1 − 2−2n − 4n+1(2n)! π2n+1 = 0. this implies that the function ψ(θ) is absolutely monotonic on [ 0, π 2 ] . direct calculation and utilization of (2.1) and (2.2) yield 8 π − 2 −ψ′(θ) cos θ = 4 sec2 θ − 8 π (θ tan θ sec θ + sec θ) − 4 + 8 π = 4(tan θ)′ − 8 π [θ(sec θ)′ + sec θ] − 4 + 8 π = 4 ∞∑ n=1 (−1)n−1(2n− 1)22n(22n − 1)b2n (2n)! θ2n−2 − 8 π [ ∞∑ n=1 (−1)n (2n− 1)! e2nθ 2n + ∞∑ n=0 (−1)n (2n)! e2nθ 2n ] − 4 + 8 π 128 f. qi and m.-m. zheng = 4 ∞∑ n=1 2n + 1 (2n)! [ 22(n+1)(22(n+1) − 1)(−1)nb2(n+1) 2(n + 1)(2n + 1) − 2 π (−1)ne2n ] θ2n. employing the inequalities (2.5) and (2.6) reveals 22(n+1)(22(n+1) − 1)(−1)nb2(n+1) 2(n + 1)(2n + 1) − 2 π (−1)ne2n > 22(n+1)(22(n+1) − 1) 2(n + 1)(2n + 1) 2(2n + 2)! (2π)2n+2 1 1 − 2−2n−2 − 2 π 4n+1(2n)! π2n+1 = 0. this means that the function 8 π −2−ψ′(θ) cos θ is absolutely monotonic on [ 0, π 2 ] . the proof of theorem 1.1 is complete. � proof of theorem 1.2. straightforward computation and utilization of (2.1) yield −p′(θ) = 1 2 [θ2(tan θ)′]′ + (θ tan θ)′ − π2 8 (tan θ)′′ = 1 2 ∞∑ n=1 (−1)n−122n(22n − 1)b2n θ2n−1 (2n− 2)! + ∞∑ n=1 (−1)n−122n(22n − 1)b2n θ2n−1 (2n− 1)! − π2 8 ∞∑ n=2 (−1)n−1(2n− 1)(2n− 2)22n(22n − 1)b2n θ2n−3 (2n)! = ∞∑ n=1 22n−1 [ (2n + 1)(22n − 1)(−1)n−1b2n − π2 2(n + 1) (22n+2 − 1)(−1)nb2n+2 ] θ2n−1 (2n− 1)! . accordingly, to prove the absolute monotonicity of the function −p′(θ), it suffices to show the inequality (2.7) |b2n+2| |b2n| = (−1)nb2n+2 (−1)n−1b2n ≤ 22n − 1 22n+2 − 1 2(n + 1)(2n + 1) π2 , n ∈ n. in [32, p. 5, (1.14)], it was listed that (2.8) b2n = (−1)n+12(2n)! (2π)2n ∞∑ m=1 1 m2n , n ∈ n. then (2.9) (−1)nb2n+2 (−1)n−1b2n = 2(n + 1)(2n + 1) π2 1 4 ∑∞ m=1 1 m2n+2∑∞ m=1 1 m2n , n ∈ n. hence, to prove the inequality (2.7), it is sufficient to verify 1 4 ∑∞ m=1 1 m2n+2∑∞ m=1 1 m2n ≤ 22n − 1 22n+2 − 1 , n ∈ n, absolute and complete monotonicity of some functions 129 which may be rearranged as( 1 − 1 22n+2 ) ∞∑ m=1 1 m2n+2 ≤ ( 1 − 1 22n ) ∞∑ m=1 1 m2n , n ∈ n. this inequality is a special case of lemma 2.1 in [3, 40], which may be slightly modified as follows: the sequence( 1 − 1 2n ) ∞∑ m=1 1 mn = ∞∑ m=1 1 mn − ∞∑ m=1 1 (2m)n = ∞∑ m=1 1 (2m− 1)n , n ≥ 2 is decreasing in n. the proof of theorem 1.2 is complete. � remark 2.1. for more information on the inequality (2.7), please refer to [15, 16] and closely related references therein. proof of theorem 1.3. by definition, it is easy to see that a function f(x) is completely monotonic in (a,b) if and only if f(−x) is absolutely monotonic in (−b,−a). see [33, p. 145, definition 2c]. hence, it is sufficient to prove that the function φ(−θ) is absolutely monotonic on ( −π 2 , π 2 ) . it is easy to see that φ(−θ) = 1 4 [2θ + sin(2θ) + π] sec2 θ = 1 4 [2θ(tan θ)′ + 2 tan θ + π(tan θ)′] = 1 4 [2(θ tan θ)′ + π(tan θ)′]. utilization of (2.1) leads to φ(−θ) = 1 4 [ 2 ∞∑ n=1 22n(22n − 1)(−1)n−1b2n θ2n−1 (2n− 1)! + π ∞∑ n=1 (2n− 1)22n(22n − 1)(−1)n−1b2n θ2n−2 (2n)! ] . since (−1)n−1b2n > 0 for all n ∈ n, all the coefficients of θk for k ≥ 0 in the power series expansion of φ(−θ) are positive. therefore, the function φ(−θ) is absolutely monotonic on ( −π 2 , π 2 ) . the proof of theorem 1.3 is complete. � references [1] m. abramowitz and i. a. stegun (eds), handbook of mathematical functions with formulas, graphs, and mathematical tables, national bureau of standards, applied mathematics series 55, 9th printing, washington, 1970. [2] h. alzer, sharp bounds for the bernoulli numbers, arch. math. (basel) 74 (2000), 207-211. [3] h.-f. ge, new sharp bounds for the bernoulli numbers and refinement of becker-stark inequalities, j. appl. math. 2012 (2012), article id 137507, 7 pages. [4] b.-n. guo, q.-m. luo, and f. qi, monotonicity results and inequalities for the inverse hyperbolic sine function, j. inequal. appl. 2013 (2013), article id 536, 6 pages. [5] b.-n. guo and f. qi, sharpening and generalizations of shafer-fink’s double inequality for the arc sine function, filomat 27 (2013), no. 2, 261–265. [6] b.-n. guo and f. qi, a property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 72 (2010), no. 2, 21–30. 130 f. qi and m.-m. zheng [7] b.-n. guo and f. qi, on the degree of the weighted geometric mean as a complete bernstein function, afr. mat. 26 (2015), in press. [8] f.-b. hang and x.-d. wang, a remark on zhong-yang’s eigenvalue estimate. int. math. res. not. imrn 2007, no. 18, article id rnm064, 9 pages. [9] q.-d. hao and b.-n. guo, a method of finding extremums of composite functions of trigonometric functions, kuàng yè (mining) (1993), no. 4, 80–83. (chinese) [10] z.-h. huo, f. qi, and b.-n. guo, laplace operator ∆ and its representations, zhèngzhōu fǎngzh̄i gōngxúeyùan xúebào (journal of zhengzhou textile institute) 4 (1993), no. 2, 52–57. (chinese) [11] p. li. poincaré inequalities on riemannian manifolds, seminar on differential geometry (ann. of math. stud. 102), 73–83, princeton university press, 1982. [12] p. li and s. t. yau, estimates of eigenvalues of a compact riemannian manifold, geometry of the laplace operator (proc. sympos. pure math., univ. hawaii, honolulu, hawaii, 1979), 205–239, proc. sympos. pure math., xxxvi, amer. math. soc., providence, r.i., 1980. [13] j. ling, the first eigenvalue of a closed manifold with positive ricci curvature, proc. amer. math. soc. 134 (2006), no. 10, 3071–3079. [14] d. s. mitrinović, j. e. pečarić, and a. m. fink, classical and new inequalities in analysis, kluwer academic publishers, dordrecht-boston-london, 1993. [15] f. qi, a double inequality for ratios of bernoulli numbers, researchgate dataset, available online at http://dx.doi.org/10.13140/2.1.2367.2962. [16] f. qi, a double inequality for ratios of bernoulli numbers, rgmia res. rep. coll. 17 (2014), article 103, 4 pages. [17] f. qi, an estimate of the gap of the first two eigenvalues in the schrödinger operator, jiāozuò kuàngyè xuéyuàn xuébào (journal of jiaozuo mining institute) 12 (1993), no. 2, 108–112. [18] f. qi, estimates of the gap of two eigenvalues in the schrödinger operator and the first eigenvalue of laplace operator, thesis supervised by professor yi-pei chen and submitted for the master degree of sceince in mathematics at xiamen university by feng qi in april 1989. (chinese) [19] f. qi, integral representations and complete monotonicity related to the remainder of burnside’s formula for the gamma function, j. comput. appl. math. 268 (2014), 155–167. [20] f. qi, properties of modified bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, math. inequal. appl. (2015), in press. [21] f. qi and c. berg, complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified bessel function, mediterr. j. math. 10 (2013), no. 4, 1685–1696. [22] f. qi, p. cerone, and s. s. dragomir, complete monotonicity of a function involving the divided difference of psi functions, bull. aust. math. soc. 88 (2013), no. 2, 309–319. [23] f. qi and b.-n. guo, lower bound of the first eigenvalue for the laplace operator on compact riemannian manifold, chinese quart. j. math. 8 (1993), no. 2, 40–49. [24] f. qi and b.-n. guo, sharpening and generalizations of shafer’s inequality for the arc sine function, integral transforms spec. funct. 23 (2012), no. 2, 129–134. [25] f. qi, b.-n. guo, and r.-q. cui, estimates of the upper bound of the difference of two arbitrary neighboring eigenvalues of the schrödinger operator, j. math. (wuhan) 16 (1996), no. 1, 81–86. (chinese) [26] f. qi, b.-n. guo, and q.-d. hao, estimate of the lower bound for the gap between the first two eigenvalues of laplace operator, kuàng yè (mining) (1994), no. 2, 86–93. (chinese) [27] f. qi, h.-c. li, b.-n. guo and q.-m. luo, inequalities and estimates of the eigenvalue for laplace operator, jiāozuò kuàngyè xuéyuàn xuébào (journal of jiaozuo mining institute) 13 (1994), no. 3, 89–95. (chinese) [28] f. qi, d.-w. niu, and b.-n. guo, refinements, generalizations, and applications of jordan’s inequality and related problems, j. inequal. appl. 2009 (2009), article id 271923, 52 pages. [29] f. qi and s.-h. wang, complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified bessel functions, glob. j. math. anal. 2 (2014), no. 3, 91–97. [30] f. qi, l.-q. yu, and q.-m. luo, estimates for the upper bounds of the first eigenvalue on submanifolds, chinese quart. j. math. 9 (1994), no. 2, 40–43. [31] r. l. schilling, r. song, and z. vondraček, bernstein functions—theory and applications, 2nd ed., de gruyter studies in mathematics 37, walter de gruyter, berlin, germany, 2012. absolute and complete monotonicity of some functions 131 [32] n. m. temme, special functions: an introduction to classical functions of mathematical physics, wiley 1996. [33] d. v. widder, the laplace transform, princeton university press, princeton, 1946. [34] h. c. yang, estimate of the first eigenvalue of laplace operator on riemannian manifolds whose ricci curvature has a negative lower bound, sci. sinica ser. a 32 (1989), no. 7, 689–700. (chinese) [35] h. c. yang, estimates of the first eigenvalue of compact riemannian manifolds with boundary with dirichlet boundary conditions acta math. sinica 34 (1991), no. 3, 329–342. (chinese) [36] d.-g. yang, lower bound estimates of the first eigenvalue for compact manifolds with positive ricci curvature, pacific j. math. 190 (1999), no. 2, 383–398. [37] q. h. yu and j. q. zhong, lower bounds of the gap between the first and second eigenvalues of the schrödinger operator. trans. amer. math. soc. 294 (1986), no. 1, 341–349. [38] j. q. zhong and h. c. yang, estimates of the first eigenvalue of laplace operator on compact riemannian manifolds, sci. sinica ser. a 26 (1983), no. 9, 812–820. (chinese) [39] j. q. zhong and h. c. yang, on the estimate of the first eigenvalue of a compact riemannian manifold, sci. sinica ser. a 27 (1984), no. 12, 1265–1273. [40] l. zhu and j.-k. hua, sharpening the becker-stark inequalities, j. inequal. appl. 2010 (2010), article id 931275, 4 pages. 1department of mathematics, college of science, tianjin polytechnic university, tianjin city, 300387, china 2college of mathematics, inner mongolia university for nationalities, tongliao city, inner mongolia autonomous region, 028043, china ∗corresponding author international journal of analysis and applications volume 17, number 2 (2019), 260-274 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-260 dynamical behavior of fractional order svir epidemic model aqeel ahmad∗, nouman javeed, muhammad farman, m.o. ahmad, amna hafeez and ali raza department of mathematics and statistics, the university of lahore, lahore, pakistan ∗corresponding author: aqeelahmad.740@gmmail.com abstract. in this paper, we proposed a fractional order svir epidemic model is incorporated to investigate its dynamical behavior in random environments. s, v, i, r are known as variables and these variables represent the number of susceptible, vaccinated, infected and recovered cells from viruses in the body. the caputo fractional derivative operator of order α � (0,1] is employed to obtain the system of fractional differential equations. the basic reproductive number is derived for a general viral production rate which determines the local stability of the infection free equilibrium. the stability and sensitivity analysis of fractional order has been made and verify the non-negative unique solution. the solution of the time fractional model has been procured by employing laplace adomian decomposition method (ladm) and the accuracy of the scheme is presented by convergence analysis. finally numerical solutions are also established to investigate the influence of system parameter on the spread of disease and which show the effect of fractional parameter on our obtained solution. 1. introduction it is evident that science subjects such as physics and chemistry are associated with mathematics. however, biology is the subject which is not usually associated with mathematics. but with the help of technology, relevant research suggests that there are quantifiable aspects of life science as well which can be measured with the help of mathematics. mathematics plays a very crucial role in understanding and exploring the natural world in this regard. the combination of mathematics and biology gave birth to new field that is received 2018-10-17; accepted 2018-11-27; published 2019-03-01. 2010 mathematics subject classification. 37c75,65l07. key words and phrases. stability analysis; dynamical transmission; caputo fractional derivative; ladm. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 260 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-260 int. j. anal. appl. 17 (2) (2019) 261 mathematical biology. mathematical biology is defined as a field of research and inquiry that investigates and explores mathematical representations of biological systems. the scope of mathematics includes mathematical modeling and esoteric mathematics. the flow of work, process, predictions and outcomes can easily be measured with the help of mathematical concepts and theory. therefore, biologists are now extremely dependent on mathematics. mathematical modeling of biological sciences is done by many brilliant scientist [1-3]. the relationship between simple mathematical modeling involves biological system, integer order differential equations that show their dynamics and complex system which describes their changing of structure. the nonlinearity and multi-scale behaviors in mathematical modeling describe the mutual relationship between parameters [4]. in last few decades, many biological models were studied in detail by using classical derivatives, few of them are [5-9]. mathematics has always beneficial and play important role in the rising science. mathematics and physical sciences each had important effects on the development of each other. mathematics has been playing a significant role in the progress of life and social sciences, and these sciences has started to influence the development of mathematics. biomedical science is one of the most important fields of science and expected well growing future. biology is concern with the subject of mathematician for improvement. the mostly use of mathematics in biology become more qualitative to quantitative. mathematical biology is the study of bio informatics and its relevant fields. the earlier stages of mathematical biology were dominated by physical systems, often involving specific mathematical models of bio systems and there components or compartments. mathematical modeling and analysis is use to recognize biology and biomedical system[10,11]. examine the dynamical transmission and effect of smoking in society [18]. the distinguished features of fractional differential equations are that it outlines memory and transmitted properties of numerous mathematical models. as a fact, that fractional order models are more realistic and practical than the classical integer order models. fractional order derivative produces greater degree of freedom in these models. arbitrary order derivatives are powerful tools for the discretion of the dynamical behavior of various biomaterial and systems [19, 20]. it is expected that biomedical science will be the foremost field of research in future. for the development of the health of their subject, mathematicians should get involved in the field of biology. there are lots of examples that suggest how mathematics has benefited from physics. it is clear that if mathematicians do not take genuine interest in the field of biosciences they will miss the opportunity to be a part of the most vital and interesting scientific enterprises of all times. mathematical biology is an emerging, well-recognized subject. moreover, in my opinion it is considered the most exciting modern application of mathematics. the burgeoning use of mathematics in biology is likely to happen since the field of biology became more significant. the intricacy of the biological sciences makes integrative involvement crucial. for the mathematician, biology has opened up new avenues and windows for further research. similarly for the int. j. anal. appl. 17 (2) (2019) 262 biologist, mathematical modelling offers another research tool proportionate with new powerful laboratory proficiency. however, it will be fruitful only if it is used relevantly recognizing its limitations. one of the important roles mathematics plays in the field of biology is the formation and creation of mathematical models. these equations, models or formulas can anticipate or illustrate natural happenings, for example behavioral patterns of an organism and measurement of population with the passage of time. as far as mathematical models are concerned it makes easier for scientist to analyze and describe a measurable phenomenon. it has been observed that most of the fields of medicine are also dependent upon mathematical models. mathematical models helps the scientist especially in the frequencies of gene expression and the rates of spreading of diseases. for the sake of argument lets suppose that a scientist is interested in studying butterfly migrations. the biologist simply entrees into that field and count a sample population of a specific area. after that, he multiplies his sample numbers by the total geographical range to find out a population estimate. he will then return to his lab and review and read other researcher’s reports and literature on butterflies. he will observe their migration pattern over the span. he will further employ the technique of vector calculations to predict and for see their future path. finally, he will investigate previous years’ data on the butterfly numbers and location to formulate a likely error margin for his prediction. at every step of this process, the biologist has to focus on mathematics to gauge, predict, analyze and understand natural phenomena. 2. mathematical model in this chapter, we proposed a fractional order svir epidemic model is incorporated to investigate its dynamical behavior in random environments. s, v, i, r are known as variables and these variables represent the number of susceptible, vaccinated,infected and recovered cells from viruses in the body. the caputo fractional derivative operator of order α � (0,1] is employed to obtain the system of fractional differential equations. the basic reproductive number is derived for a general viral production rate which determines the local stability of the infection free equilibrium. the stability and sensitivity analysis of fractional order has been made and verify the non-negative unique solution.the solution of the time fractional model has been procured by employing laplace adomian decomposition method (ladm) and the accuracy of the scheme is presented by convergence analysis. finally numerical solutions are also established to investigate the influence of system parameter on the spread of disease and which show the effect of fractional parameter on our obtained solution. thus, the mathematical model is represented by the following four differentials equations: this model is the basic compartmental model in epidemiology. since then,many compartmental models such as sir or sirs models have been established to predict the various properties of the diseases spreading.hence vaccination can also be considered by adding some compartment naturally into the basic epidemic models int. j. anal. appl. 17 (2) (2019) 263 for certain diseases. in formulation of svir epidamic model, here we uses four different variables here s represents the compartment of susceptible cells (i.e., the compartment of those individuals which are not infected an able to catch the disease), v compartment represents the vaccinated cells, i compartment represents the already infected cells and r represents the recovered cells from viruses in the body. this model has the following form: ds dt = µ−µs −βsi −αs, (2.1) dv dt = αs −β1v i −γ1v −µv, (2.2) di dt = βsi + β1v i −γi −µi, (2.3) dr dt = γ1v + γi −µr. (2.4) here s(t),i(t) and r(t) denote the densities of susceptible, infected and recovered individuals,respectively. v(t) denotes the density of vaccinees who have begun the vaccination prodess.from [12] it follows that all the parameters in the model are positive and have the following features, µ is the recruitment rate and natural death rate of the population, β denotes the transmission coefficient between compartments s and i, α is the rate at which susceptible individuals are moved into the vaccination process, β1 denotes the disease transmission rate when the vaccinees contact with infected individuals before obtaining immunity, γ1 is the average rate for the vaccinees move into recovered population, γ is the recovery rate of infected individuals. we assume thatβ1 is less than β because the vaccinating may have some partial immunity during the process. the fractional order extension for the ordinary differential equations of the model have been first studied in [13,14]. the new system of differential equation is represented by the fractional system of differential equations (fdes) is given as follows. dα1 (s) = µ−µs −βsi −αi (2.5) dα2 (v ) = αs −β1v i −λ1v −µv (2.6) dα3 (i) = βsi + β1v i −γi −µi (2.7) dα4 (r) = λ1v + λi −µr (2.8) with the initial conditions s(0) = n1 = 0.4,v (0) = n2 = 0.4,i(0) = n3 = 0.2,r(0) = n4 = 0.1 (2.9) int. j. anal. appl. 17 (2) (2019) 264 3. qualitative analysis to evaluate the equilibrium point ,we take dα1 (s) = 0, dα2 (v ) = 0, dα3 (i) = 0, dα4 (r) = 0 above model becomes µ−µs −βsi −αs = 0 (3.1) αs −β1v i −γ1v −µv = 0 (3.2) βsi + β1v i −γi −µi = 0 (3.3) γ1v + γi −µr = 0 (3.4) after simplifications (2.1),(2.2),(2.3) and (2.4) we get s = µ α+µ , v = αµ (α+µ)(µ+γ1) , i = 0, r = αγ1 (α+µ)(µ+γ1) after parametric values µ = 1, β = 15, α = 15, β1 = 10, γ = 2, γ1 = 4 we get (s,v,i,r)=(0.0625, 0.1875, 0, 0.75) 3.1. reproductive number. consider the jacobian matrix to find the reproductive number by using next generation technique j =   −µ−βi −α 0 −βs 0 α β1i −γ1 −µ β1v 0 βi β1i βs + β1v −γ −µ 0 0 γ1 γ −µ   . (3.5) we take j = f v, where f =   0 0 0 0 0 β1i β1v 0 0 β1i β1v 0 0 0 0 0   . (3.6) int. j. anal. appl. 17 (2) (2019) 265 and v =   µ + βi + α 0 +βs 0 −α γ1 + µ 0 0 −βi 0 −βs + γ + µ 0 0 −γ1 −γ µ   . (3.7) by using the relation |k −λi|=0, where k = fv −1 and s = µ α+µ , v = αµ (α+µ)(µ+γ1) and i=0 we get eigen value λ = αµβ1(µ + γ1) α(γ + µ) + µ(−β + γ + µ) which represents the reproductive number r0 = αµβ1(µ + γ1) α(γ + µ) + µ(−β + γ + µ) which is r0 = 1.2 > 1 4. stability analysis theorem 4.1. the endemic equilibrium state e1 = (s,v,i,r) of the model (1.1)-(1.4) is locally asymptotically stable, if r0 > 1, otherwise unstable. to prove this theorem, we have to show that the re(λ) < 0. consider the jacobian matrix as j =   −µ−βi −α 0 −βs 0 α −β1i −γ1 −µ −β1v 0 βi β1i βs + β1v −γ −µ 0 0 γ1 γ −µ   (4.1) int. j. anal. appl. 17 (2) (2019) 266 take i=0 (as infected component), we get j0 =   −µ−α 0 −βs 0 α −γ1 −µ −β1v 0 0 0 βs + β1v −γ −µ 0 0 γ1 γ −µ   (4.2) by using the relation |j0 −λi|=0 we get the result λ1 = −α−µ < 0 λ2 = −µ < 0 λ3 = −µ−γ1 < 0 λ4 = −γ −µ + βµα+µ + β1αµ (α+µ)(µ+γ1) < 0 all the real values of λ are negative. so,our system for endemic equilibrium is locally asymptotically stable. 5. preliminaries in this section, we give some fundamental results and definitions from fractional calculus. for detailed over view of the topic readers are referred to [15,16]. definition 4.1 the riemann-liouville fractional integration of order α ∈ (0, 1) of the function f ∈ l1([0,t],r) is defined as as iα0+f(t) = 1 γ(α) ∫ t 0 (t−s)α−1f(s)ds. the riemann-liouville derivative has certain disadvantages such that the fractional derivative of a constant is not zero. therefore, we will make use of caputo’s definition owing to its convenience for initial conditions of the fractional differential equations. definition 4.2 the definitions of laplace transform of caputo’s derivative is written as l{cdαy(t)} = sαh(s) − σn−1k=0s α−i−1y(k)(0), n− 1 < α ≤ n; ‘ n ∈ n. for arbitrary ci ∈ r, i = 0, 1, 2, ...,n− 1, where n = [α] + 1 and [α] shows the integer part of α. 6. the laplace-adomian decomposition method consider the fractional-order epidemic model (1.5)-(1.8) subject to the initial condition (1.9). the nonlinear term in this model is si and v i for φ1 = φ2 = φ3 = φ4 = φ5 = 1 the fractional order model converts int. j. anal. appl. 17 (2) (2019) 267 to the classical epidemic model. by using laplace transform on the system given in equations (1.5)-(1.8), we get l{dα1t s} = µl{1}−µl{s}−βl{si}−αl{s} l{dα2t v} = αl{s}−β1l{v i}−λ1l{v}−µl{v} l{dα3t i} = βl{si}−β1l{v i}−γl{i}−µl{i} l{dα4t r} = λ1l{v}−λl{i}−µl{r} by the defination laplace transform, we get sα1 l{s}−sα1−1s(0) = µl{1}−µl{s}−βl{si}−αl{s} sα2 l{v}−sα2−1v (0) = αl{s}−β1l{v i}−λ1l{v}−µl{v} sα3 l{i}−sα3−1i(0) = βl{si}−β1l{v i}−γl{i}−µl{i} sα4 l{r}−sα4−1r(0) = λ1l{v}−λl{i}−µl{r} by using the initial condition l{s} = n1 s + µ sα1+1 − µ sα1 l{s}− β sα1 l{si}− α sα1 l{s} (6.1) l{v} = n2 s + α sα2 l{s}− β1 sα2 l{v i}− λ1 sα2 l{v}− µ sα2 l{v} (6.2) l{i} = n3 s + β sα3 l{si}− β1 sα3 l{v i}− γ sα3 l{i}− µ sα3 l{i} (6.3) l{r} = n4 s + λ1 sα4 l{v} + λ sα4 l{i}− µ sα4 l{r} (6.4) it should be assumed that method gives the solution as an infinite series s = ∞∑ k=0 sk,v = ∞∑ k=0 vk,i = ∞∑ k=0 ik,r = ∞∑ k=0 rk (6.5) the nonlinearity si and v ican be written as i = ∞∑ k=0 ak,v i = ∞∑ k=0 bk (6.6) int. j. anal. appl. 17 (2) (2019) 268 where ak and bk is called the adomian polynomials given as ak = 1 k! dk dλk [ k∑ j=0 λjsj k∑ j=0 λjij]|λ=0 (6.7) bk = 1 k! dk dλk [ k∑ j=0 λjvj k∑ j=0 λjij]|λ=0 (6.8) from equations 3.1, 3.2, 3.3 and 3.4, we have the followings results l{s0} = n1 s + µ sα1+1 , l{v0} = n2 s , l{i0} = n3 s , l{r0} = n4 s (6.9) similarly, we have l{s1} = − µ sα1 l{s0}− β sα1 l{a0}− α sα1 l{s0} (6.10) . . . l{sk+1} = − µ sα1 l{sk}− β sα1 l{ak}− α sα1 l{sk} (6.11) l{v1} = α sα2 l{s0}− β1 sα2 l{b0}− λ1 sα2 l{v0}− µ sα2 l{v0} (6.12) . . . l{vk+1} = α sα2 l{sk}− β1 sα2 l{bk}− λ1 sα2 l{vk}− µ sα2 l{vk} (6.13) l{i1} = β sα3 l{a0} + β1 sα3 l{b0}− γ sα3 l{i0}− µ sα3 l{i0} (6.14) . . . l{ik+1} = β sα3 l{ak} + β1 sα3 l{bk}− γ sα3 l{ik}− µ sα3 l{ik} (6.15) int. j. anal. appl. 17 (2) (2019) 269 l{r1} = λ1 sα4 l{v0} + λ sα4 l{i0}− µ sα4 l{r0} (6.16) . . . l{rk+1} = λ1 sα4 l{vk} + λ sα4 l{ik}− µ sα4 l{rk} (6.17) the purpose of the work is to analysis the mathematical behavior of the solution s(t),v (t),i(t),r(t) for the different values of φ. by applying the inverse laplace transform to both sides of the equation (3.9), we get the values of s0,v0,i0,r0 and used for further process. putting the values of s0,v0,i0,r0 and a0,b0 into the equations (3.12-3.16) and get the values of s1,v1,i1,r1. similarly we find the remaining term s2,s3,s4, ...., v2,v3,v4, .... , i2,i3,i4, ...., r2,r3,r4, .... and in the same manners. we have the series solution in the form s(t) = s0 + s1 + s2 + s3 + s4 + .... (6.18) v (t) = v0 + v1 + v2 + v3 + v4 + .... (6.19) i(t) = i0 + i1 + i2 + i3 + i4 + ... (6.20) r(t) = r0 + r1 + r2 + r3 + r4 + .... (6.21) the laplace adomian decomposition method is an analytical approximate solution in terms of infinite power series. for numerical results, we used the followings table values of parameters are considered from table 1. values of physical parameters used in svir model when r0 > 1 parameter value parameter value α 15 β 15 γ 2 µ 1 β1 10 γ1 4 by applying laplace inverse property on equation (3.18-3.21)as describes above, we can find the solution of fractional model in series form is written as s(t) = 0.4 − 6.6 tα1 α1! + 125.4 t2α1 2α1! + 361 t3α1 3α1! − 9 tα1+α3 (α1 + α3)! − 18 t2α1+α3 (2α1 + α3)! , −21 (α1 + α3)!t 2α1+α3 α1α3(2α1 + α3)! − 45 (2α1 + α3)!t 3α1+α3 α1!(α1 + α3)!(3α1 + α3)! + ..... (6.22) int. j. anal. appl. 17 (2) (2019) 270 v (t) = 0.4 + 3.2 tα2 α2! − 99 tα1+α2 (α1 + α2)! − 285 t2α1+α2 (2α1 + α2)! − 6 tα2+α3 (α2 + α3)! , −12 tα1+α2+α3 (α1 + α2 + α3)! − 22.4 t2α2 2α2! − 105 tα1+2α2 (α1 + 2α2)! + ...... (6.23) i(t) = 0.2 + 1.5 tα3 α3! + 3 tα1+α3 (α1 + α3)! + 91.8 t2α3 2α3! + 21 tα1+2α3 (α1 + 2α3)! + 21 (α1 + α3)!t α1+2α3 α1!α3!(α1 + 2α3)! , +45 (2α1 + α3)!t α1+2α3 α1!(α1 + α3)!(α1 + 2α3)! − 22.8 tα1+α3 (α1 + α3)! − 57 t2α1+α3 (2α1 + α3)! + 6.4 tα2+α3 (α2 + α3)! , +30 tα1+α2+α3 (α1 + α2 + α3)! + ...... (6.24) r(t) = 0.1 + 1.9 tα4 α4! + 12.8 tα2+α4 (α2 + α4)! + 60 tα1+α2+α4 (α1 + α2 + α4)! + 2.28 tα3+α4 (α3 + α4)! , +6 tα1+α3+α4 (α1 + α3 + α4)! − 1.9 t2α4 2α4! + ..... (6.25) 7. numerical results and discussion the mathematical analysis of svir epidemic model with nonlinear system of fractional differential equation has been presented. the numerical results of susceptible, vaccinated, infected and recovered population for different values of α are established in tables 2-5 by using ladm. for the reliable investigation, evaluation is made for different values of α and can be seen from figure 1-4. we observe that fractional order svir epidamic model has more degree of freedom as compared to ordinary derivatives. by taking non-integer values of fractional parameter, remarkable responses of the compartments of the proposed model are obtained. for different values of α solution converges to steady state and gives the better convergence by decreasing the fractional values of α. int. j. anal. appl. 17 (2) (2019) 271 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (years) 0 10 20 30 40 50 60 s us pe ct ab le s (t) endemic equilibrium 1 = 2 = 3 = 0.9 1 = 2 = 3 =0.70 1 = 2 = 3 = 0.66 figure 1. numerical solutions for susceptible s(t) population in a time t (months) at different values of α. 0 0.005 0.01 0.015 0.02 0.025 0.03 time (years) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 v ac ci na tio n v( t) endemic equilibrium 1 = 2 = 3 = 0.8 1 = 2 = 3 = 0.775 1 = 2 = 3 = 0.75 figure 2. numerical solutions for vaccination v (t) population in a time t (months) at different values of α. int. j. anal. appl. 17 (2) (2019) 272 0 2 4 6 8 10 12 14 16 18 time (years) 0 50 100 150 200 250 300 i nf ec te d i(t ) endemic equilibrium 1 =0.519, 2 =0.25, 3 = 0.1 1 =0.519, 2 =0.2, 3 = 0.1 1 =0.519, 2 =0.17, 3 = 0.1 figure 3. numerical solutions for infected i(t) population in a time t (months) at different values of α. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 time (years) 0 0.5 1 1.5 2 2.5 r ec ov er ed r (t) endemic equlibrium 1 = 2 = 3 = 4 = 1 1 = 2 = 3 = 4 = 0.9 1 = 2 = 3 = 4 = 0.85 figure 4. numerical solutions for recovered r(t) population in a time t (months) at different values of α. 8. conclusion it is important to note that laplace adomian decomposition method for mathematical models based on system of fractional order differential equations is more powerful approach to compute the convergent solutions.we developed a scheme for analytical solution of epidemic fractional svir model by using laplace int. j. anal. appl. 17 (2) (2019) 273 adomian decomposition method. the well-known epidemic model namely susceptible-vaccination-infectedrecovered (svir) is considered with and without demographic effects. the model represents population dynamics during the disease as a set of non-linear coupled ordinary differential equations. there is no exact solution available in the literature for this model up to the best of authors knowledge. it is observed that the infection rate and reproductive numbers play a key role for an epidemic to occur and the epidemic can be controlled by vaccination. it is also observed that to eliminate the disease, it is not necessary to vaccinate whole of the population.the effect of fractional parameter on our obtained solutions is presented through tables and graphs. it is worthy to observe that fractional derivatives show significant changes and memory effects as compared to ordinary derivatives. finally, we estimated the parameter that characterize the behavior of disease and present numerical simulations. the disease free equilibrium state is carried out by presenting re(λ) < 0, i.e. the population is viable. references [1] j. biazar, solution of the epidemic model by adomian decomposition method, appl. math. comput. 173 (2006), 1101-1106. [2] s. busenberg, p. driessche, analysis of a disease transmission model in a population with varying size, j. math. biol. 28 (1990), 65-82. [3] ae a.m.a. el-sayed, s.z. rida, a.a.m. arafa, on the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber, int. j. nonlinear sci. 7 (2009), 485-495. [4] o.d. makinde, adomian decomposition approach to a sir epidemic model with constant vaccination strategy, appl. math. comput. 184 (2007), 842-848. [5] a.a.m. arafa, s.z. rida, m. khalil, fractional modeling dynamics of hiv and 4 t-cells during primary infection, nonlinear biomed. phys. 6 (2012), 1-7. [6] c.m. kribs-zaleta, structured models for heterosexual disease transmission, math. biosci. 160 (1999), 83-108. [7] b. buonomo, d. lacitignola, on the dynamics of an seir epidemic model with a convex incidence rate, ricerche mat. 57 (2008), 261-281. [8] x. liu, c. wang, bifurcation of a predator-prey model with disease in the prey, nonlinear dyn. 62 (2010), 841-850. [9] fazal haq, kamal shah, ghaus ur rahman, muhammad shahzad. numerical solution of fractional order smoking model via laplace adomian decomposition method, alex. eng. j. 57 (2) (2018), 1061-1069 [10] j.d murray: mathematical biology i. an introduction,usa ,springer , verlag berlin heidelberg. (2002). [11] a. c. guyton and j. e. hall, text book of medical physiology, elsevier saunders, st. louis, edition 11 972 (2012). [12] erturk, vs, zaman, g, momani, s: a numeric analytic method for approximating a giving up smoking model containing fractional derivatives. comput. math. appl. 64 (2012), 3068-3074. [13] zeb, a, chohan, i, zaman, g: the homotopy analysis method for approximating of giving up smoking model in fractional order. appl. math. 3 (2012), 914-919. [14] alkhudhari, z, al-sheikh, s, al-tuwairqi, s: global dynamics of a mathematical model on smoking. appl. math. 2014 (2014), article id 847075. [15] r. magin, fractional calculus in bioengineering, begell house publishers, 2004. [16] j. biazar, solution of the epidemic model by adomian decomposition method, appl. math. comput. 173 (2006), 1101-1106. int. j. anal. appl. 17 (2) (2019) 274 [17] t. khan, g. zaman and m. i. chohan, the transmission dynamic and optimal control of acute and chronic hepatitis b, j. biol. dyn. 11 (1) (2017), 172-189. [18] aqeel ahmad, muhammad farman, faisal yasin, m. o. ahmad, dynamical transmission and effect of smoking in society, int. j. adv. appl. sci. 5(2) (2018), 71-75 [19] aqeel ahmad, muhammad farman, m. o. ahmad, nauman raza, m. abdullah, dynamical behavior of sir epidemic model with non-integer time fractional derivatives: a mathematical analysis, int. j. adv. appl. sci. 5(1) (2018), 123-129 [20] farah ashraf, aqeel ahmad, muhammad umer saleem, muhammad farman, m. o. ahmad, dynamical behavior of hiv immunology model with non-integer time fractional derivatives, int. j. adv. appl. sci. 5(3) (2018), 39-45 1. introduction 2. mathematical model 3. qualitative analysis 3.1. reproductive number 4. stability analysis 5. preliminaries 6. the laplace-adomian decomposition method 7. numerical results and discussion 8. conclusion references international journal of analysis and applications volume 16, number 5 (2018), 689-701 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-689 some properties of analytic functions associated with conic type regions khalida inayat noor1, nazar khan1,2, maslina darus3,∗, qazi zahoor ahmad2 and bilal khan2 1department of mathematics, comsats institute of information technology, park road, islamabad, pakistan 2department of mathematics, abbottabad university of science and technology, abbottabad, pakistan 3school of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, 43600 bangi, selangor, malaysia ∗corresponding author: maslina@ukm.edu.my abstract. the main purpose of this investigation is to define new subclasses of analytic functions with respect to symmetrical points. these functions map the open unit disk onto certain conic regions in the right half plane. we consider various corollaries and consequences of our main results. we also point out relevant connections to some of the earlier known developments. 1. introduction and definitions let h denote the class of functions analytic in the unit disk e = {z : |z| < 1}. let a denote the class of analytic functions in the open unit disk e and satisfying the following conditions f(0) = f′ (0) − 1 = 0. received 2018-05-09; accepted 2018-07-10; published 2018-09-05. 2010 mathematics subject classification. primary 05a30, 30c45; secondary 11b65, 47b38. key words and phrases. analytic functions; symmetric points; conic type regions. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 689 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-689 int. j. anal. appl. 16 (5) (2018) 690 therefore, for f ∈a, one has f(z) = z + ∞∑ n=2 anz n (∀z ∈ e) . (1.1) also let s be the subclass of a which consists of univalent functions in e. moreover the class of starlike functions in e will be denoted by s∗, which consists of normalized functions f ∈a that satisfy the following inequality: < ( zf′ (z) f (z) ) > 0 (∀ z ∈ e) . (1.2) similarly the class c of convex functions in e consists of normalized functions f ∈ a that satisfy the following inequality: < ( (zf′ (z)) ′ f′ (z) ) > 0 (∀ z ∈ e) . (1.3) for two functions f and g, analytic in e, we say that f is subordinate to g, denoted by f(z) ≺ g(z) or f ≺ g, if there exists a schwarz function w which is analytic in e with w (0) = 0 and |w(z)| ≤ |z| such that f(z) = g (w(z)) . furthermore if the function g is univalent in e, then one can find that f(z) ≺ g(z) ⇐⇒ 0 = g (0) and f (e) ⊆ g (e) . we next denote by p the class of analytic functions p, which are normalized by p (z) = 1 + ∞∑ n=1 pnz n (1.4) such that <(p (z)) > 0. definition 1.1. a function f ∈a is said to belongs to the class s∗s , if and only if zf′(z) f(z) −f(−z) ≺ 1 + z 1 −z (∀ z ∈ e) . the class s∗s , of starlike functions with respect to symmetrical points, was introduced by sakaguchi in 1959, ( see [23]). remark 1.1. for function f ∈ a the idea of alexander’s theorem [7] was used by das and singh [6] for defining the class cs of convex functions with respect to symmetrical points, in the following way: f(z) ∈cs ⇐⇒ zf′(z) ∈s∗s . int. j. anal. appl. 16 (5) (2018) 691 definition 1.2. a given function h with h (0) = 1 is said to belong to the class p [a,b] if and only if h (z) ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1. the analytic functions class p [a,b] was introduced by janowski [9], who showed that h (z) ∈p [a,b] if and only if there exist a function p ∈p such that h (z) = (a + 1) p (z) − (a− 1) (b + 1) p (z) − (b − 1) , −1 ≤ b < a ≤ 1. geometrically a function h belongs to p [a,b] if and only if it maps the open unit disk e onto the disk defined by the domain ω [a,b] = { t ∈ c : ∣∣∣∣t− 1 −ab1 −b2 ∣∣∣∣ < a−b1 −b2 } . historically speaking, the conic domain ωk, k ≥ 0, was first introduced by kanas and wísniowska (see [11] and [12]) as ωk = { u + iv : u > k √ (u− 1)2 + v2 } . moreover for fixed k this domain represents the right half plane (k = 0), a parabola (k = 1), the right branch of hyperbola (0 < k < 1) and an ellipse (k > 1), see also [17], [18] and recently [21]. indeed the extremal functions for these conic regions are pk(z) =   1 1−k2 cosh {( 2 π arccos k ) log 1+ √ z 1− √ z } − k 2 1−k2 (0 ≤ k < 1) 1 + 2 π2 ( log 1+ √ z 1− √ z )2 (k = 1) 1 k2−1 sin ( π 2k(κ) ∫ u(z)√ κ 0 dt√ 1−t2 √ 1−κ2t2 ) + k 2 k2−1 (k > 1) , (1.5) where u(z) = z − √ κ 1 − √ κz (∀ z ∈ e) and κ ∈ (0, 1) is chosen such that k = cosh (πk′(κ)/(4k(κ))). here k(κ) is legendre’s complete elliptic integral of first kind and k′(κ) = k( √ 1 −κ2) i.e. k′ (t) is the complementary integral of k (t), [1], [2]. assume that pk(z) = 1 + t1(k)z + t2(k)z 2 + . . . (∀ z ∈ e) . int. j. anal. appl. 16 (5) (2018) 692 then it was shown in [13] that for (1.5) one can have t1 := t1(k) =   2a2 1−k2 (0 ≤ k < 1) 8 π2 (k = 1) π2 4ķ2(t)2(1+t) √ t (k > 1) , (1.6) t2 := t2(k) = d(k)t1(k), where d(k) =   a2+2 3 (0 ≤ k < 1) 8 π2 (k = 1) (4k(κ))2(t2+6t+1)−π2 24k(κ)2(1+t) √ t (k > 1) (1.7) with a = 2 π arccos k. noor et al. [16] combine the concepts of janowski functions and the conic regions and define the following: definition 1.3. a function h ∈h is said to be in the class k-p [a,b], if and only if h(z) ≺ (a + 1)pk(z) − (a− 1) (b + 1))pk(z) − (b − 1) k ≥ 0, (1.8) where pk(z) is defined by (1.5) and −1 ≤ b < a ≤ 1. geometrically, each function h ∈ k-p[a,b] takes all values in the domain ωk[a,b],−1 ≤ b < a ≤ 1,k ≥ 0 which is defined as ωk[a,b] = { w : < ( (b − 1)w − (a− 1) (b + 1))w − (a + 1) ) > k ∣∣∣∣ (b − 1)w − (a− 1)(b + 1))w − (a + 1) − 1 ∣∣∣∣ } , or equivalently ωk[a,b] is a set of numbers w = u + iv such that[( b2 − 1 )( u2 + v2 ) − 2 (ab − 1) u + ( a2 − 1 )]2 > k [( −2(b + 1)) ( u2 + v2 ) + 2 (a + b + 2) u− 2(a + 1) )2 + 4(a−b)2v2 ] . this domain represents the conic type regions for detail (see [16]). one can observe that 0-p [a,b] = p [a,b] , introduced by janowski (see [9]) and k-p [1,−1] = p (pk) , introduced by kanas and wísniowska (see [11]). int. j. anal. appl. 16 (5) (2018) 693 in the recent years, several interesting subclasses of analytic functions have been introduced and investigated, see for example [3], [4], [5], [19] and [22]. motivated and inspired by the recent research going on and from the above mentioned work, we now introduce some new subclasses of analytic functions as following: definition 1.4. a function f ∈ s is said to be in the class k-uss [a,b] , k ≥ 0, −1 ≤ b < a ≤ 1, if and only if 2zf′(z) f(z) −f(−z) ≺ (a + 1)pk(z) − (a− 1) (b + 1))pk(z) − (b − 1) (∀ z ∈ e) . (1.9) remark 1.2. first of all, it is easily seen that 0-uss [1,−1] = s∗s , the class of starlike functions with respect to symmetric points introduced and studied by sakaguchi (see [23]). secondly, we have 0-uss [a,b] = s∗s [a,b] , the class of janowski starlike functions with respect to symmetric points introduced by goel and mehrok in 1982 (see [8]). thirdly, we have k-uss [1,−1] = k-sts, introduced and studied by noor (see [20]). definition 1.5. a function f ∈ s is said to be in the class k-ucs [a,b] , k ≥ 0,−1 ≤ b < a ≤ 1, if and only if 2 (zf′(z)) ′ (f(z) −f(−z))′ ≺ (a + 1)pk(z) − (a− 1) (b + 1))pk(z) − (b − 1) (∀ z ∈ e) . remark 1.3. from definiton 1.5 it is readily observe that 0-ucs [1,−1] = cs, the class of convex functions with respect to symmetric points introduced and studied by das and singh, (see [6]). secondly we have 0-ucs [a,b] = c∗s [a,b] , the class of janowski convex functions with respect to symmetric points introduced by janteng and halim in 2008 (see [10]) . and finally k-ucs [1,−1] = k-ucvs, introduced and studied by noor (see [20]). int. j. anal. appl. 16 (5) (2018) 694 2. a set of lemmas each of the following lemmas will be needed in our present investigation. lemma 2.1. [14] if a function w ∈h is of the form w(z) = c1z + c2z 2 + . . . and |w(z)| ≤ |z| (∀ z ∈ e) , (2.1) then for every complex number s, we have ∣∣c2 −sc21∣∣ ≤ 1 + (|s|− 1) ∣∣c21∣∣ . lemma 2.2. let k ∈ [0,∞) be a fixed and qk(z) = (a + 1)pk(z) − (a− 1) (b + 1))pk(z) − (b − 1) . then qk(z) = 1 + h1(k)z + h2(k)z 2 + . . . (∀ z ∈ e) (2.2) and h1 := h1(k) = a−b 2 t1(k) (2.3) h2 := h2(k) = (a−b)t1(k) 4 {2d(k) − (b + 1)t1(k)} (2.4) where t1(k) and d(k) are defined by (1.6) and (1.7). proof. we have (a + 1)pk(z) − (a− 1) = {(b + 1)pk(z) − (b − 1)} { 1 + h1z + h2z 2 + . . . } . therefore, we obtain 2 + (a + 1) { t1z + t2z 2 + . . . } = [ 2 + (b + 1) { t1z + t2z 2 + . . . }][ 1 + h1z + h2z 2 + . . . ] (2.5) comparing the coefficients at z gives (a + 1)t1 = (b + 1)t1 + 2h1, so we obtain the first equality (2.3). similarly, comparing the coefficients at z2 gives (a + 1)t2 = 2h2 + (b + 1)h1t1 + (b + 1)t2, so we have (a−b)t2 − (b + 1)h1t1 = 2h2. int. j. anal. appl. 16 (5) (2018) 695 applying (2.3) gives h2 = (a−b) 2 t2 − (a−b)(b + 1) 4 t 21 = (a−b) 2 d(k)t1 − (a−b)(b + 1) 4 t 21 = (a−b)t1 4 {2d(k) − (b + 1)t1} . so, we obtain the second equality (2.4). this completes the proof. � 3. main results in this section, we will prove our main results. throughout our discussion, we assume that −1 ≤ b < a ≤ 1 and k ≥ 0. theorem 3.1. let f ∈ k-uss [a,b]. then the function ϕ(z) = 1 2 (f(z) −f(−z)) , (3.1) belongs to k-us[a,b] in e, where k-us[a,b] is the class of janowski starlike functions g(z) ∈a such that zg′(z) g(z) ∈ k-p[a,b]. proof. taking logarithmic differentiation of (3.1), we have zϕ′(z) ϕ(z) = z (f(z)) ′ + z (f(−z))′ (f(z) −f(−z)) . then we find after some simplification that zϕ′(z) ϕ(z) = 1 2 [ 2z (f(z)) ′ (f(z) −f(−z)) + 2z (f(−z))′ (f(−z) −f(z)) ] = 1 2 [p1(z) + p2(z)] , p1,p2 ∈ k-p [a,b] (∀ z ∈ e) . moreover one can find that k-p [a,b] is a convex set ( see [16]), it follows that zϕ′(z) ϕ(z) ∈ k-p [a,b] and thus ϕ(z) ∈ k-us [a,b]. � remark 3.1. the above theorem shows that the class k-uss [a,b] is a subclass of the class of close-toconvex functions. int. j. anal. appl. 16 (5) (2018) 696 theorem 3.2. let 0 ≤ k < ∞ be fixed. assume that a function qk defined in lemma 2.2, has the form (2.2). if the function h(z) = 1 + b1z + b2z 2 + . . . is a member of the function class k-p [a,b], then for −∞ < u < ∞, ∣∣b2 −ub21∣∣ ≤   a−b 2 t1(k) { u (a−b) 2 t1(k) − 12 {[2d(k) − (b + 1)]t1(k)} } (u > α1) a−b 2 t1(k), (α1 ≤ u ≤ α2) a−b 2 t1(k) { 1 2 {[2d(k) − (b + 1)]t1(k)) } −u(a−b) 2 t1(k) (u < α2) , (3.2) where α1 = [2 + 2d(k) − (b + 1)]t1(k) (a−b)t1(k) , α2 = [2d(k) − (b + 1)]t1(k) − 2 (a−b)t1(k) , and t1, d(k) are defined by (1.6) and (1.7). proof. if f ∈ k-p [a,b] then it follows that h(z) ≺ qk(z) = 1 + a−b 2 t1(k)z + (a−b) [2d(k) − (b + 1)] t1(k) 4 t1(k)z 2 + . . . (∀z ∈ e) . (3.3) now by the definition of subordination there exists a function w analytic in e with w (0) = 0 and |w(z)| < 1 such that w(z) = c1z + c2z2 + · · · and h(z) = 1 + a−b 2 t1(k)w(z) + (a−b) [2d(k) − (b + 1)] t1(k) 4 t1(k)w 2(z) + . . . . (3.4) now from (2.1), (3.3) and (3.4), we have b1 = a−b 2 t1(k)c1, b2 = a−b 2 t1(k) { c2 + [2d(k) − (b + 1)] t1(k) 2 c21 } . therefore, we obtain b2 −ub21 = a−b 2 t1(k) { c2 + { [2d(k) − (b + 1)] t1(k) 2 −u a−b 2 t1(k) } c21 } . (3.5) this gives ∣∣b2 −ub21∣∣ = a−b2 t1(k) ∣∣∣∣c2 − c21 + { 1 + [2d(k) − (b + 1)] t1(k) 2 −u a−b 2 t1(k) } c21 ∣∣∣∣ . int. j. anal. appl. 16 (5) (2018) 697 suppose that u > α1, then using the estimate ∣∣c2 − c21∣∣ ≤ 1 from lemma 2.1 and the well known estimate |c1| ≤ 1 of the schwarz lemma, we obtain∣∣b2 −ub21∣∣ ≤ a−b2 t1(k) { u (a−b) 2 t1(k) − (2d(k) − (b + 1))t1(k)) 2 } . this is the first inequality in (3.2). on the other hand if u < α2, then (3.5) gives∣∣b2 −ub21∣∣ ≤ a−b2 t1(k) { |c2| + { (2d(k) − (b + 1))t1(k)) 2 −u (a−b) 2 t1(k) } |c1| 2 } . applying the estimates |c2| ≤ 1 −|c1| 2 of lemma 2.1 and |c1| ≤ 1, we have∣∣b2 −ub21∣∣ ≤ a−b2 t1(k) { 1 + { (2d(k) − (b + 1))t1(k)) 2 −u (a−b) 2 t1(k) − 1 } |c1| 2 } ≤ a−b 2 t1(k) { (2d(k) − (b + 1))t1(k)) 2 −u (a−b) 2 t1(k) } . this is the last inequality in (3.2). finally if α1 < u < α2, then∣∣∣∣(2d(k) − (b + 1))t1(k))2 −u(a−b)2 t1(k) ∣∣∣∣ ≤ 1. therefore (3.5), yields ∣∣b2 −ub21∣∣ ≤ a−b2 t1(k) { |c2| + |c1| 2 } ≤ a−b 2 t1(k) { 1 −|c1| 2 + |c1| 2 } = a−b 2 t1(k). we get the middle inequality in (3.2). this completes the proof. � remark 3.2. in above theorem if we set a = 1 and b = −1 we have the result given in [15]. theorem 3.3. let the function f given by (2.1) be in the class k-uss [a,b]. then ∣∣µa22 −a3∣∣ ≤ 12   a−b 2 t1(k) { µ(a−b) 4 t1(k) − (2d(k)−(b+1))t1(k)) 2 } (u > δ1) a−b 2 t1(k) (δ1 ≤ u ≤ δ2) a−b 2 t1(k) { (2d(k)−(b+1))t1(k)) 2 − µ(a−b) 4 t1(k) } (u < δ2) , where δ1 = 2 (2 + 2d(k) − (b + 1))t1(k)) (a−b)t1(k) , δ1 = 2 (2d(k) − (b + 1))t1(k) − 2) (a−b)t1(k) . and t1, d(k) are defined by (1.6) and (1.7). int. j. anal. appl. 16 (5) (2018) 698 proof. by definition of the class k-uss [a,b], there exists a function h ∈s, represented by h(z) = 1 + b1z + b2z 2 + . . . and subordinate to qk, where qk is given by (2.2), such that 2zf′(z) f(z) −f(−z) = h(z) (∀ z ∈ e) . substituting the corresponding series expansions and by equating coefficients of z and z2, we obtain a2 = 1 2 b1, a3 = 1 2 b2. therefore ∣∣µa22 −a3∣∣ ≤ 12 ∣∣∣∣µb212 − b2 ∣∣∣∣ . an application of theorem 3.2, with u = µ 2 , we obtain the result asserted by theorem 3.3. � theorem 3.4. a function f ∈ k-uss [a,b], if and only if 1 z { f(z) ∗ [ z −mz2 (1 −z)2 (1 + z) ]} 6= 0 (∀ z ∈ e) , (0 < θ < 2π) , (3.6) where m = (a + b + 2)pk(e iθ) − (a + b − 2) (a−b)pk(eiθ) + (a−b) . (3.7) proof. if f ∈ k-uss [a,b], then we have f(z) := 2zf′(z) f(z) −f(−z) ≺ (a + 1)pk(z) − (a− 1) (b + 1)pk(z) − (b − 1) (∀ z ∈ e) . (3.8) for 0 ≤ k ≤ 1 the function pk(z) has a pole at z = 1 and the curve pk(eiθ), θ ∈ (0, 2π), is the imaginary axis, a hyperbola or an ellipse. for k > 1 the function pk(z) is analytic on the unit disk. in each of the cases, if f(z) ∈ k −uss [a,b], then f(|z| < 1) lies on the right with respect this curve, or 2zf′(z) f(z) −f(−z) 6= (a + 1)pk(e iθ) − (a− 1) (b + 1)pk(eiθ) − (b − 1) (∀ z ∈ e and 0 < θ < 2π) . a simple computations gives 1 z   zf ′(z) [ (b + 1)pk(e iθ) − (b − 1) ] − 1 2 [f(z) −f(−z)]×[ (a + 1)pk(e iθ) − (a− 1) + (b + 1)pk(eiθ) − (b − 1) ]   6= 0, (3.9) for z ∈ e, θ ∈ (0, 2π). using the convolution properties f(z) ∗ z (1 −z)2 = zf′(z) and f(z) ∗ z 1 −z2 = 1 2 [f(z) −f(−z)] (∀ z ∈ e) , we have that 1 z   f(z) ∗ [ z[(b+1)pk(eiθ)−(b−1)] (1−z)2 − z[(a+1)pk(eiθ)−(a−1)+(b+1)pk(eiθ)−(b−1)] 1−z2 ]   6= 0. int. j. anal. appl. 16 (5) (2018) 699 hence it follows that 1 z  f(z) ∗ z − (b+1)pk(e iθ)−(b−1)+(a+1)pk(eiθ)−(a−1) (b+1)pk(eiθ)−(b−1)−(a+1)pk(eiθ)−(a−1) z2 (1 −z)2(1 + z)   6= 0 (3.10) for z ∈ e, θ ∈ (0, 2π), which is the required conditions (3.6) and (3.7). conversely, suppose that the condition (3.6) holds. therefore we have 2zf′(z) f(z) −f(−z) 6= (a + 1)pk(e iθ) − (a− 1) (b + 1)pk(eiθ) − (b − 1) (∀ z ∈ e) . (3.11) suppose that h(z) = (a + 1)pk(z) − (a− 1) (b + 1))pk(z) − (b − 1) (∀ z ∈ e) . now from relation (3.11), it is clear that h (∂e) ∩f (e) = ∅. therefore the simply connected domain f (e) is contained in a connected component of c\h (∂e). the univalence of the function h together with the fact h (0) = h (0) = 1 shows that f ≺ h which shows that f ∈ k-uss [a,b]. � in its special case when k = 0, theorem 3.4 yields the following known result. corollary 3.1. for λ = 0,−1 ≤ b < a ≤ 1. a function f ∈ k-usλs [a,b] , if and only if 1 z  f(z) ∗  z + [(b+a+2)pk(eiθ)−(b+a−2)](b−a)(pk(eiθ)−1) z2 (1 −z)2 (1 + z)     6= 0 (∀ z ∈ e and 0 ≤ θ < 2π) . if, in theorem 3.4, we set −b = 1 = a and k = 0, we obtain the following result. corollary 3.2. a function f ∈ 0-uss [1,−1], if and only if 1 z { f(z) ∗ [ z ( 1 −ze−iθ ) (1 −z)2 (1 + z) ]} 6= 0 (∀ z ∈ e) , 0 < θ < 2π. theorem 3.5. if f ∈s, then f ∈ k-ucs [a,b], if and only if 1 z { f(z) ∗ 1 + 2z3 + [m − 3] z2 − 3mz4 z (1 −z)3 (1 + z)2 } 6= 0 (∀ z ∈ e) , 0 < θ < 2π, where m is given by (3.7). int. j. anal. appl. 16 (5) (2018) 700 proof. let g(z) = z + mz2 (1 −z)2 (1 + z) , then zg′(z) = z + mz4 + (m + 2) z3 + (2m + 1) z2 (1 −z)3 (1 + z)2 . now using the alexander type relation between k-uss [a,b] and k-ucs [a,b], the identity zf′(z) ∗g(z) = f(z) ∗zg′(z), and theorem 3.4, we obtain the required result. � 4. acknowledgements the work here is supported by ukm grant: gup-2017-064. conflict of interest: there is no conflict of intrests between the authors, financial or whatsoever. declaration: all authors agreed with the contents of the manuscript. references [1] n. i. ahiezer, elements of theory of elliptic functions, moscow, 1970. [2] g. d. anderson, m. k. vamanamurthy and m. k. vourinen, conformal invariants, inequalities and quasiconformal maps, wiley-interscience, 1997. [3] m. al-kaseasbeh and m. darus, inclusion and convolution properties of a certain class of analytic functions, eurasian math. j. 8 (4) (2017), 11-17. [4] m. caglar, h. ohan and e. deniz, majorization for certain subclass of analytic functions involving the generalized noor integral operator, filomat, 27 (1) (2013), 143-148. [5] m. darus and s. owa, new subclasses concerning some analytic and univalent functions, chinese j. math. (2017), article id 4674782, 4 pages. [6] r. n. das and p. singh, radius of convexity for certain subclass of close-to-convex functions, j. indian math soc. 41 (1977), 363-369. [7] p. l. duren, univalent functions, grundlehren der mathematischen wissenschaften, band 259, springer-verlag, new york, berlin, heidelberg and tokyo, 1983. [8] m. r. goel and b. s. mehrok, a subclass of univalent functions, houston j. math. 8 (1982), 343-357. [9] w. janowski, some extremal problem for certain families of analytic functions i, ann. polon. math. 28 (1973), 298-326. [10] a. janteng and s. a. halim, a subclass of convex functions with respect to symmetric points, proceedings of the 16th national symposium on science mathematical, 2008. [11] s. kanas and a. wísniowska, conic regions and k-uniform convexity, j. comput. appl. math. 105 (1999), 327-336. [12] s. kanas and a. wísniowska, conic domains and starlike functions, rev. roumaine math. pures appl. 45 (2000), 647-657. [13] s. kanas, coefficient estimate in subclasses of the caratheodary class related to conic domains, acta math. univ. comenianae lxxiv. 2 (2005), 149-161. int. j. anal. appl. 16 (5) (2018) 701 [14] f. r. keogh and e. p. merkes, a coefficient inequality for certain classes of analytic functions, proc. amer. math. soc. 20 (1969), 8-12. [15] a. k. mishra and p. gochhayat, a coefficient inequality for a sublclass of the caratheodory functions defined using conical domains, comput. math. appl. 61 (2011), 2816-2820. [16] k. i. noor and s. n. malik, on coefficient inequalities of functions associated with conic domains, comput. math. appl. 62 (2011), 2209-2217. [17] k. i. noor, on a generalization of uniformly convex and related functions, comput. math. appl. 61 (2011), 117-125. [18] k. i. noor, m. arif and m. w. ul-haq, on k-uniformly close-to-convex functions of complex order, appl. math. comput. 215 (2009), 629-635. [19] k. i. noor, n. khan and m. a. noor, on generalized spiral-like analytic functions, filomat, 28 (7) (2014), 1493-1503. [20] k. i. noor, on uniformly univalent functions with respect to symmetrical points, j. math. ineq. 2014 (2014), 1-14. [21] k. i. noor, q. z. ahmad and m. a. noor, on some subclasses of analytic functions defined by fractional derivative in the conic regions, appl. math. inf., sci. 9 (2) (2015), page 819. [22] m. obradovic and p. ponnusanny, radius of univalence of certain class of analytic functions, filomat, 27 (2013), 1085-1090. [23] k. sakaguchi, on the theory of univalent mapping, j. math. soc. japan, 11 (1959), 72-80. [24] h. m. srivastava, t. n. shanmugam, c. ramachandran and s. sivassurbramanian, a new subclass of k-uniformly convex functions with negative coefficients, j. inequal. pure, appl. math. 8 (43) (2007), art. id 43. 1. introduction and definitions 2. a set of lemmas 3. main results 4. acknowledgements references international journal of analysis and applications volume 17, number 4 (2019), 548-558 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-548 fractional integral inequalities of grüss type via generalized mittag-leffler function g. farid1, a. u. rehman1, vishnu narayan mishra2,3,∗ and s. mehmood4 1comsats university islamabad, attock campus, kamra road, atoock 43600, pakistan 2department of mathematics, indira gandhi national tribal university, lalpur, amarkantak 484 887, madhya pradesh, india 3l. 1627 awadh puri colony, beniganj, phase-iii, opposite industrial training institute (i.t.i.), ayodhya-224 001, uttar pradesh, india 4gbps sherani, hazro attock, pakistan ∗corresponding author: vishnunarayanmishra@gmail.com abstract. we use generalized fractional integral operator containing the generalized mittag-leffler function to establish some new integral inequalities of grüss type. a cluster of fractional integral inequalities have been identified by setting particular values to parameters involved in the mittag-leffler special function. presented results contain several fractional integral inequalities which reflects their importance. received 2018-12-10; accepted 2019-01-16; published 2019-07-01. 2010 mathematics subject classification. 26a33, 26d10, 33e12. key words and phrases. grüss inequality; generalized fractional integral operator; mittag-leffler function. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 548 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-548 int. j. anal. appl. 17 (4) (2019) 549 1. introduction in 1935, grüss [5] proved the following inequality 1 a2 −a1 ∫ a2 a1 f1(t)f2(t)dt− ( 1 a2 −a1 ∫ a2 a1 f1(t)dt )( 1 a2 −a1 ∫ a2 a1 f2(t)dt ) (1.1) ≤ (m −m)(n −n) 4 , where f and g are two integrable functions on [a,b] and satisfying the following conditions m ≤ f1(x) ≤ m, n ≤ f2(x) ≤ n m,m,n,n ∈ r, x ∈ [a,b]. in the literature inequality (1.1) is well known as the grüss inequality. inequality (1.1) remains in the focus of researchers especially working in the field of mathematical analysis. a lot of authors are working on (1.1) and have produced important results for different kinds of functions. in recent years, many important and fascinating grüss type inequalities have been established (see for example [7, 9, 15]). our interest in this paper is to give some generalized fractional integral inequalities of grüss type by use of generalized fractional integral operators due to the mittag-leffler function. in the following we define an extended generalized mittag-leffler function e γ,δ,k,c µ,α,l (t; p) as fallows: definition 1.1. [3] let µ,α,l,γ,c ∈ c, <(µ),<(α),<(l) > 0, <(c) > <(γ) > 0 with p ≥ 0, δ > 0 and 0 < k ≤ δ + <(µ). then the extended generalized mittag-leffler function eγ,δ,k,cµ,α,l (t; p) is defined by e γ,δ,k,c µ,α,l (t; p) = ∞∑ n=0 βp(γ + nk,c−γ) β(γ,c−γ) (c)nk γ(µn + α) tn (l)nδ , (1.2) here (c)nk denotes the generalized pochhammer symbol (c)nk = γ(c + nk) γ(c) , bp is an extension of the beta function bp(x,y) = ∫ 1 0 tx−1(1 − t)y−1e− p t(1−t) dt (<(x),<(y),<(p) > 0) . the corresponding generalized fractional integral operator � γ,δ,k,c µ,α,l,ω,af is defined as fallows: definition 1.2. [3] let ω,µ,α,l,γ,c ∈ c, <(µ),<(α),<(l) > 0, <(c) > <(γ) > 0 with p ≥ 0, δ > 0 and 0 < k ≤ δ + <(µ). let f ∈ l1[a,b] and x ∈ [a,b]. then the generalized fractional integral operator � γ,δ,k,c µ,α,l,ω,af is defined by: ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) = ∫ x a (x− t)α−1eγ,δ,k,cµ,α,l (ω(x− t) µ; p)f(t)dt. (1.3) int. j. anal. appl. 17 (4) (2019) 550 from generalized fractional integral operator we have ( � γ,δ,k,c µ,α,l,ω,a1 ) (x; p) = ∫ x a (x− t)α−1eγ,δ,k,cµ,α,l (w(x− t) µ; p)dt = ∫ x a (x− t)α−1 ∞∑ n=0 bp(γ + nk,c−γ) b(γ,c−γ) (c)nk γ(µn + α) wn(x− t)µn (l)nδ dt = ∞∑ n=0 bp(γ + nk,c−γ) b(γ,c−γ) (c)nk γ(µn + α) wn (l)nδ ∫ x a (x− t)µn+α−1dt = (x−a)α ∞∑ n=0 bp(γ + nk,c−γ) b(γ,c−γ) (c)nk γ(µn + α) wn (l)nδ (x−a)µn 1 µn + α . hence ( � γ,δ,k,c µ,α,l,ω,a1 ) (x; p) = (x−a)α eγ,δ,k,cµ,α+1,l(w(x−a) µ; p). we use the following notation in our results cα(x; p) = ( � γ,δ,k,c µ,α,l,ω,a1 ) (x; p). (1.4) integral operators are very useful in solving integral as well as differential equations. several types of integral operators have been studied by the mathematicians (see for example [1, 2, 4, 6, 8, 11–14]). in this paper at first some generalized fractional integral inequalities and their particular cases are established. then a generalized fractional korkine’s identity is proved. at the end grüss fractional integral inequality via generalized fractional integral operator have been obtained. the presented inequality contained several versions of grüss inequality in fractional calculus. 2. main results first we prove the following fractional inequality. theorem 2.1. let f,ψ1,ψ2 ∈ l1[a,b] such that ψ1(x) ≤ f(x) ≤ ψ2(x) ∀x ∈ [a,b]. (2.1) then for extended generalized fractional integral operator (1.3) we have the following inequality: ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) (2.2) ≥ ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) proof. from (2.1) we have (ψ2(u) −f(u)) (f(v) −ψ1(v)) ≥ 0 ∀u,v ∈ [a,b]. (2.3) int. j. anal. appl. 17 (4) (2019) 551 this gives the following inequality: ψ2(u)f(v) + ψ1(v)f(u) ≥ ψ1(v)ψ2(u) + f(u)f(v). (2.4) multiplying (2.4) by (x − u)α−1eγ,δ,k,cµ,α,l (ω(x − u) µ; p) on both sides and integrating with respect to u over [a,x], the following inequality is obtained:∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)ψ2(u)f(v)du (2.5) + ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)ψ1(v)f(u)du ≥ ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)ψ1(v)ψ2(u)du + ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)f(u)f(v)du. using the definition 1.2 we get f(v) ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) + ψ1(v) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) (2.6) ≥ ψ1(v) ( � γ,δ,k,c µ,α,l,ω,aψ2(u) ) (x; p) + f(v) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p). now multiplying (2.6) by (x − v)β−1eγ,δ,k,cµ,β,l (ω(x − v) µ; p) on both sides and integrating with respect to v over [a,x], the following inequality is obtained: ( � γ,δ,k,c µ,β,l,ω,aψ2 ) (x; p) ∫ x a (x−v)β−1eγ,δ,k,cµ,β,l (ω(x−v) µ; p)f(v)dv (2.7) + ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ∫ x a (x−v)β−1eγ,δ,k,cµ,β,l (ω(x−v) µ; p)ψ1(v)dv ≥ ( � γ,δ,k,c µ,β,l,ω,aψ2 ) (x; p) ∫ x a (x−v)β−1eγ,δ,k,cµ,β,l (ω(x−v) µ; p)ψ1(v)dv + ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ∫ x a (x−v)β−1eγ,δ,k,cµ,β,l (ω(x−v) µ; p)f(v)dv. by using the definition 1.2 and then after simple calculation we get the required inequality (2.2). � a particular case is given as follows. corollary 2.1. let f ∈ l1[a,b] and m1,m2 be two real numbers such that m1 ≤ f(x) ≤ m2 ∀x ∈ [a,b]. then we have m2cα(x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) + m1 ( � γ,δ,k,c µ,α,l,ω,af ) (x; p)cβ(x; p) ≥ m1m2cα(x; p)cβ(x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p). (2.8) int. j. anal. appl. 17 (4) (2019) 552 proof. proof follows on the same lines as the proof of theorem 2.1 just use ψ1(x) = m1 and ψ2(x) = m2 as constant functions. � some more inequalities are given in the next result. theorem 2.2. let f,ψ1,ψ2 ∈ l1[a,b] such that (2.1) holds. also let g ∈ l1[a,b] and there exist φ1 and φ2 such that φ1(x) ≤ g(x) ≤ φ2(x) ∀x ∈ [a,b]. (2.9) then for extended generalized fractional integral (1.3) we have the following inequalities: (i) ( � γ,δ,k,c µ,β,l,ω,aφ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) + ( � γ,δ,k,c µ,β,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) (2.10) ≥ ( � γ,δ,k,c µ,β,l,ω,aφ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p), (ii) ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) + ( � γ,δ,k,c µ,β,l,ω,aφ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ≥ ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p), (iii) ( � γ,δ,k,c µ,β,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p) + ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ≥ ( � γ,δ,k,c µ,β,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p), (iv) ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aφ1 ) (x; p) + ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ≥ ( � γ,δ,k,c µ,β,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,aφ1 ) (x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p), proof. (i) from (2.1) and (2.9) we have (ψ2(u) −f(u))(g(v) −φ1(v)) ≥ 0, (2.11) that gives ψ2(u)g(v) + φ1(v)f(u) ≥ ψ1(v)ψ2(u) + f(u)g(v). (2.12) multiplying (2.12) by (x − u)α−1eγ,δ,k,cµ,α,l (ω(x − u) µ; p)(x − v)β−1eγ,δ,k,cµ,β,l (ω(x − v) µ; p) on both sides and integrating with respect to u and v over [a,x] then by using definition 1.2 we get (i). to prove (ii) − (iv), we use the following inequalities instead of (2.11) respectively (ii) (φ2(u) −g(u))(f(v) −ψ1(v)) ≥ 0, (iii) (ψ2(u) −f(u))(g(v) −φ2(v)) ≤ 0, (iv) (ψ1(u) −f(u))(g(v) −φ1(v)) ≤ 0, then on the same lines as done to obtain (i) one can get inequalities (ii) − (iv). � special cases are stated as follows. int. j. anal. appl. 17 (4) (2019) 553 corollary 2.2. let f,g ∈ l1[a,b]. also let m1,m2,n1 and n2 be real constants such that m1 ≤ f(x) ≤ m2, n1 ≤ g(x) ≤ n2, ∀x ∈ [a,b]. then we have (i) n1cβ(x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) + m2cα(x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p) ≥ n1m2cβ(x; p)cα(x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p), (ii) m1cβ(x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) + n2cα(x; p) ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ≥ m1n2cβ(x; p)cα(x; p) + ( � γ,δ,k,c µ,β,l,ω,af ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p), (iii) m2cα(x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p) + n2cβ(x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ≥ m2n2cβ(x; p)cα(x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p), (iv) m1cα(x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p) + n1cβ(x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ≥ m1n1cβ(x; p)cα(x; p) + ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,β,l,ω,ag ) (x; p). proof. proof follows on the same lines as the proof of theorem 2.1 just use ψ1(x) = m1,ψ2(x) = m2,φ1(x) = n1 and φ2(x) = n2 as constant functions. � next we give the korkine’s identity which is used in the next result. theorem 2.3. let f,ψ1,ψ2 ∈ l1[a,b] such that (2.1) holds. then for extended generalized fractional integral (1.3) we have the following equality: cα(x; p) ( � γ,δ,k,c µ,α,l,ω,af 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ]2 (2.13) = [( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ] × [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ] −cα(x; p) [( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ] × [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ] + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ1f ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ2f ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) −cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ1ψ2 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p). int. j. anal. appl. 17 (4) (2019) 554 proof. for any u,v ∈ [a,b] we have (ψ2(v) −f(v))(f(u) −ψ1(u)) + (ψ2(u) −f(u))(f(v) −ψ1(v)) (2.14) − (ψ2(u) −f(u))(f(u) −ψ1(u)) − (ψ2(v) −f(v))(f(v) −ψ1(v)) = f2(u) + f2(v) − 2f(u)f(v) + ψ2(v)f(u) + ψ1(u)f(v) −ψ1(u)ψ2(v) + ψ2(u)f(v) + ψ1(v)f(u) −ψ1(v)ψ2(v) −ψ2(u)f(u) + ψ1(u)ψ2(u) −ψ1(u)f(u) −ψ2(v)f(v) + ψ1(v)ψ2(v) −ψ1(v)f(v). now multiplying (2.14) by (x − u)α−1eγ,δ,k,cµ,α,l (ω(x − u) µ; p)(x − v)α−1eγ,δ,k,cµ,α,l (ω(x − v) µ; p) on both sides and integrating with respect to u and v over [a,x] then by using definition 1.2 we get the required identity (2.13). � the last result is the generalized fractional grüss inequality. theorem 2.4. let f and g be a two functions such that f,g ∈ l1[a,b]. also let ψ1,ψ2,φ1 and φ2 be four integrable functions satisfying (2.1) and (2.9). then for extended generalized fractional integral (1.3) we have the following inequality: ∣∣∣cα(x; p) (�γ,δ,k,cµ,α,l,ω,afg) (x; p) −(�γ,δ,k,cµ,α,l,ω,af) (x; p) (�γ,δ,k,cµ,α,l,ω,ag) (x; p)∣∣∣ (2.15) ≤ √ g(f,ψ1,ψ2)g(g,φ1,φ2), where g(u,v,w) = [( � γ,δ,k,c µ,α,l,ω,aw ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,au ) (x; p) ] × [( � γ,δ,k,c µ,α,l,ω,au ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,av ) (x; p) ] + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,avu ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,av ) (x; p) ( � γ,δ,k,c µ,α,l,ω,au ) (x; p) + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,awu ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aw ) (x; p) ( � γ,δ,k,c µ,α,l,ω,au ) (x; p) −cα(x; p) ( � γ,δ,k,c µ,α,l,ω,avw ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,av ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aw ) (x; p). proof. since f and g are two integrable functions we have [f(u) −f(v)] [g(u) −g(v)] (2.16) = f(u)g(u) + f(v)g(v) −f(u)g(v) −f(v)g(u). int. j. anal. appl. 17 (4) (2019) 555 multiplying (2.16) by 1 2 (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) and integrating with respect to u and v over [a,x], the following inequality is obtained:( 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) (2.17) × [f(u) −f(v)] [g(u) −g(v)] dudv) = ( � γ,δ,k,c µ,α,l,ω,afg ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p). now by using cauchy-schwarz inequality we have( 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) (2.18) × [f(u) −f(v)] [g(u) −g(v)] dudv)2 ≤ 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) × [f(u) −f(v)]2 dudv × 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) × [g(u) −g(v)]2 dudv. from (2.18) one can have 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) (2.19) × [f(u) −f(v)]2 dudv = cα(x; p) ( � γ,δ,k,c µ,α,l,ω,af 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ]2 . similarly, 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) (2.20) × [g(u) −g(v)]2 dudv = cα(x; p) ( � γ,δ,k,c µ,α,l,ω,ag 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ]2 . using (2.19) and (2.20) in (2.18) we have( 1 2 ∫ x a ∫ x a (x−u)α−1eγ,δ,k,cµ,α,l (ω(x−u) µ; p)(x−v)α−1eγ,δ,k,cµ,α,l (ω(x−v) µ; p) (2.21) × [f(u) −f(v)] [g(u) −g(v)] dudv)2 ≤ cα(x; p) ( � γ,δ,k,c µ,α,l,ω,af 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ]2 ×cα(x; p) ( � γ,δ,k,c µ,α,l,ω,ag 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ]2 . int. j. anal. appl. 17 (4) (2019) 556 now combining (2.17) and (2.21) we have (( � γ,δ,k,c µ,α,l,ω,afg ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) )2 (2.22) ≤ cα(x; p) ( � γ,δ,k,c µ,α,l,ω,af 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ]2 ×cα(x; p) ( � γ,δ,k,c µ,α,l,ω,ag 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ]2 . since (ψ2(x) −f(x))(f(x) −ψ1(x)) ≥ 0 and (φ2(x) −g(x))(g(x) −φ1(x)) ≥ 0, therefore cα(x; p) ( � γ,δ,k,c µ,α,l,ω,a(ψ2 −f)(f −ψ1) ) (x; p) ≥ 0, and cα(x; p) ( � γ,δ,k,c µ,α,l,ω,a(φ2 −g)(g −φ1) ) (x; p) ≥ 0. by theorem 2.3 we have cα(x; p) ( � γ,δ,k,c µ,α,l,ω,af 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ]2 (2.23) ≤ [( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) ] × [( � γ,δ,k,c µ,α,l,ω,af ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ] + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ1f ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ2f ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,af ) (x; p) −cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aψ1ψ2 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,aψ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aψ2 ) (x; p). = g(f,ψ1,ψ2). int. j. anal. appl. 17 (4) (2019) 557 similarly cα(x; p) ( � γ,δ,k,c µ,α,l,ω,ag 2 ) (x; p) − [( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ]2 (2.24) ≤ [( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) ] [( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aφ1 ) (x; p) ] + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aφ1f ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aφ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) + cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aφ2g ) (x; p) − ( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,ag ) (x; p) −cα(x; p) ( � γ,δ,k,c µ,α,l,ω,aφ1φ2 ) (x; p) + ( � γ,δ,k,c µ,α,l,ω,aφ1 ) (x; p) ( � γ,δ,k,c µ,α,l,ω,aφ2 ) (x; p). = g(g,φ1,φ2). combining (2.23),(2.24) with (2.22), we get the required inequality (2.15). � concluding remarks since the extended generalized fractional integral operator contains itself several known fractional integral operators for particular values of involved parameters. for example selecting p = 0, fractional integral inequalities for fractional integral operators defined by salim and faraj in [12], selecting l = δ = 1, fractional integral inequalities for fractional integral operators defined by rahman et al. in [11], selecting p = 0 and l = δ = 1, fractional integral inequalities for fractional integral operators defined by shukla and prajapati in [13] and see also [14], selecting p = 0 and l = δ = k = 1, fractional integral inequalities for fractional integral operators defined by prabhakar in [10], selecting p = ω = 0 fractional integral inequalities for riemann-liouville fractional integrals. therefore the presented results contain all such results for these particular fractional integral operators as special cases. acknowledgement this work is supported under nrpu 2016, project no. 5421 by the higher education commission of pakistan. references [1] a. akkurt, z. kaçar and h. yildirim, generalized fractional integral inequalities for continuous random variables, j. probab. stat. 2015(2015), art. id 958980. [2] a. akkurt, s. kilinç and h. yildirim, grüss type inequalities involving the generalized gauss hypergeometric functions, int. j. pure appl. math. 106(4) 2016, 1103-1114. [3] m. andrić, g. farid and j. pečarić, a further extension of mittag-leffler function, fract. calc. appl. anal., 21(5) (2018), 1377-1395. [4] a. s. balakishiyev, e. a. gadjieva, f. gürbüz and a. serbetci, boundedness of some sublinear operators and their commutators on generalized local morrey spaces, complex var. elliptic equ. 63(11) (2018), 1620-1641. int. j. anal. appl. 17 (4) (2019) 558 [5] g. grüss, über das maximum des absolten betrages von 1 b−a ∫ b a f(x)g(x)dx − 1 (b−a)2 ∫ b a f(x)dx ∫ b a f(x)dx, math. z. 39 (1935), 215-226. [6] f. gürbüz, some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted morrey spaces, canad. math. bull. 60(1) (2017), 131-145. [7] e. kaçar and h. yildirim, grüss type integral inequalities for generalized riemann-liouville fractional integrals, int. j. pure appl. math. 101(1) (2015), 55-70. [8] v. n. mishra, k. khatri, l. n. mishra, deepmala, inverse result in simultaneous approximation by baskakov-durrmeyerstancu operators, j. inequal. appl. 2013 (2013), art. id 586. [9] x. li, r. n. mohapatra and r. s. rodriguez, grüss-type inequalities, j. math. anal. appl. 267 (2002), 434-443. [10] t. r. prabhakar, a singular integral equation with a generalized mittag-leffler function in the kernel, yokohama math. j. 19 (1971), 7-15. [11] g. rahman, d. baleanu, m. a. qurashi, s. d. purohit, s. mubeen and m. arshad, the extended mittag-leffler function via fractional calculus, j. nonlinear sci. appl. 10 (2017), 4244-4253. [12] t. o. salim and a. w. faraj, a generalization of mittag-leffler function and integral operator associated with integral calculus, j. frac. calc. appl., 3(5) (2012), 1-13. [13] a. k. shukla and j. c. prajapati, on a generalization of mittag-leffler function and its properties, j. math. anal. appl. 336 (2007), 797-811. [14] h. m. srivastava and z. tomovski, fractional calculus with an integral operator containing generalized mittag-leffler function in the kernal, appl. math. comput. 211(1) (2009), 198-210. [15] g. wang, p. agarwal and m. chand, certain grüss type inequalities involving the generalized fractional integral operator, j. inequal. appl. 2014 (2014), art. id 147. 1. introduction 2. main results concluding remarks acknowledgement references int. j. anal. appl. (2022), 20:25 interval-valued intuitionistic fuzzy subalgebras/ideals of hilbert algebras aiyared iampan1,∗, v. vijaya bharathi2, m. vanishree3, n. rajesh3 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2no. 42, cauvery nagar east, thanjavur-613005, tamilnadu, india 3department of mathematics, rajah serfoji government college (affliated to bharathidasan university), thanjavur-613005, tamilnadu, india ∗corresponding author: aiyared.ia@up.ac.th abstract. in this paper, the concept of interval-valued intuitionistic fuzzy sets to subalgebras and ideals of hilbert algebras is introduced. the inverse image of interval-valued intuitionistic fuzzy subalgebras and interval-valued intuitionistic fuzzy ideals of hilbert algebras is studied and some related properties are investigated. equivalence relations on interval-valued intuitionistic fuzzy ideals are discussed. 1. introduction the concept of fuzzy sets was proposed by zadeh [15]. the theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. after the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. the integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1, 3, 4]. the idea of intuitionistic fuzzy sets suggested by atanassov [2] is one of the extensions of fuzzy sets with better applicability. applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multi-criteria decision-making [6–8]. the concept of hilbert algebras was introduced in early 50-ties by henkin and skolem for some investigations of implication in intuitionistic and other non-classical logics. in 60-ties, these algebras were studied especially by horn and diego from algebraic point of view. diego proved received: apr. 8, 2022. 2010 mathematics subject classification. 20n05, 94d05, 03e72. key words and phrases. hilbert algebra; interval-valued intuitionistic fuzzy subalgebra; interval-valued intuitionistic fuzzy ideal. https://doi.org/10.28924/2291-8639-20-2022-25 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-25 2 int. j. anal. appl. (2022), 20:25 (cf. [11]) that hilbert algebras form a variety which is locally finite. hilbert algebras were treated by busneag (cf. [9], [10]) and jun (cf. [13]) and some of their filters forming deductive systems were recognized. dudek (cf. [12]) considered the fuzzification of subalgebras and deductive systems in hilbert algebras. in this paper, the concept of interval-valued intuitionistic fuzzy sets to subalgebras and ideals of hilbert algebras is introduced. the inverse image of interval-valued intuitionistic fuzzy subalgebras and interval-valued intuitionistic fuzzy ideals of hilbert algebras is studied and some related properties are investigated. equivalence relations on interval-valued intuitionistic fuzzy ideals are discussed. 2. preliminaries before we begin the study, let’s review the definition of hilbert algebras, which was defined by diego [11] in 1966. definition 2.1. a hilbert algebra is a triplet x = (x, ·, 1), where h is a nonempty set, · is a binary operation, and 1 is a fixed element of x such that the following axioms hold: (1) (∀x,y ∈ x)(x · (y ·x) = 1), (2) (∀x,y,z ∈ x)((x · (y ·z)) · ((x ·y) · (x ·z)) = 1), (3) (∀x,y ∈ x)(x ·y = 1,y ·x = 1 ⇒ x = y). the following result was proved in [12]. lemma 2.1. let x = (x, ·, 1) be a hilbert algebra. then (1) (∀x ∈ x)(x ·x = 1), (2) (∀x ∈ x)(1 ·x = x), (3) (∀x ∈ x)(x · 1 = 1), (4) (∀x,y,z ∈ x)(x · (y ·z) = y · (x ·z)). in a hilbert algebra x = (x, ·, 1), the binary relation ≤ is defined by (∀x,y ∈ x)(x ≤ y ⇔ x ·y = 1), which is a partial order on x with 1 as the largest element. definition 2.2. [14] a nonempty subset i of a hilbert algebra x = (x, ·, 1) is called an ideal of x if (1) 1 ∈ i, (2) (∀x ∈ x,∀y ∈ i)(x ·y ∈ i), (3) (∀x ∈ x,∀y1,y2 ∈ i)((y1 · (y2 ·x)) ·x ∈ i). a fuzzy set [15] in a nonempty set x is defined to be a function µ : x → [0, 1], where [0, 1] is the unit closed interval of real numbers. int. j. anal. appl. (2022), 20:25 3 definition 2.3. [12] a fuzzy set µ in a hilbert algebra x = (x, ·, 1) is said to be a fuzzy subalgebra of x if the following condition holds: (∀x,y ∈ x)(µ(x ·y) ≥ min{µ(x),µ(y)}). definition 2.4. [5] a fuzzy set µ in a hilbert algebra x = (x, ·, 1) is said to be a fuzzy ideal of x if the following conditions hold: (1) (∀x ∈ x)(µ(1) ≥ µ(x)), (2) (∀x,y ∈ x)(µ(x ·y) ≥ µ(y)), (3) (∀x,y1,y2 ∈ x)(µ((y1 · (y2 ·x)) ·x) ≥ min{µ(y1),µ(y2)}). definition 2.5. [2] let x be a nonempty set. an intuitionistic fuzzy set a in x is defined to be a structure a := {(x,µa(x),γa(x)) | x ∈ x}, (2.1) where µa : x → [0, 1] is the degree of membership of x to [0, 1] and γa : x → [0, 1] is the degree of non-membership of x to [0, 1] such that (∀x ∈ x)(0 ≤ µa(x) + γa(x) ≤ 1), and the intuitionistic fuzzy set a in (2.1) is simply denoted by a = (µa,γa). let d[0, 1] be the set of all closed subintervals of [0, 1]. consider two elements d1,d2 ∈ d[0, 1]. if d1 = [a1,b1] and d2 = [a2,b2], then rmin{d1,d2} = [min{a1,a2}, min{b1,b2}] and rmax{d1,d2} = [max{a1,a2}, max{b1,b2}]. if di = [ai,bi] ∈ d[0, 1] for i = 1, 2, . . ., then we define rsupi{di} = [sup i {ai}, sup i {bi}]. similarly, we define rinfi{di} = [inf i {ai}, inf i {bi}]. now we call d1 ≥ d2 if and only if a1 ≥ a2 and b1 ≤ b2. similarly, the relations d1 ≤ d2 and d1 = d2 are defined. definition 2.6. an interval-valued intuitionistic fuzzy (ivif) set a in x is an object having the form a = {(x,µa(x),γa(x)) | x ∈ x}, where µa : x → d[0, 1] and γa : x → d[0, 1]. the intervals µa(x) and γa(x) denote the intervals of the degree of belongingness and non-belongingness of the element 4 int. j. anal. appl. (2022), 20:25 x to the set d[0, 1], where µa(x) = [µla(x),µ u a(x)] and γa(x) = [γ l a(x),γ u a(x)] for all x ∈ x with the following condition: (∀x ∈ x)(0 ≤ µua(x) + γ u a(x) ≤ 1). for the sake of simplicity, we shall use the symbol a = (µa,γa) for the ivif set a = {(x,µa(x),γa(x)) | x ∈ x}. also note that µa(x) = [1 − µua(x), 1 − µ l a(x)] and γa(x) = [1 −γua(x), 1 −γ l a(x)] for all x ∈ x, where [µa(x),γa(x)] represents the complement of x in a. 3. ivif subalgebras of hilbert algebras definition 3.1. an ivif set a = (µa,γa) in a hilbert algebra x is called an ivif subalgebra of x if (∀x,y ∈ x) ( µa(x ·y) ≥ rmin{µa(x),µa(y)} γa(x ·y) ≤ rmax{γa(x),γa(y)} ) . (3.1) example 3.1. let x = {1,x,y,z, 0} with the following cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 then x is a hilbert algebra. we define an ivif set a = (µa,γa) as follows: µa(a) = { [0.5, 0.6] if a ∈{1,x,y,z} [0.1, 0.2] if a = 0 and γa(a) = { [0.3, 0.4] if a ∈{1,x,y,z} [0.4, 0.5] if a = 0. then a is an ivif subalgebra of x. proposition 3.1. every ivif subalgebra a = (µa,γa) of a hilbert algebra x satisfies µa(1) ≥ µa(x) and γa(1) ≤ γa(x) for all x ∈ x, where µa(1) and γa(1) are the upper bound and lower bound of µa(x) and γa(x), respectively. proof. for any x ∈ x, we have µa(1) = µa(x ·x) ≥ rmin{µa(x),µa(x)} = rmin{[µla(x),µ u a(x)], [µ l a(x),µ u a(x)]} = [µla(x),µ u a(x)] = µa(x) int. j. anal. appl. (2022), 20:25 5 and γa(1) = γa(x ·x) ≤ rmax{γa(x),γa(x)} = rmax{[γla(x),γ u a(x)], [γ l a(x),γ u a(x)]} = [γla(x),γ u a(x)] = γa(x). � proposition 3.2. if an ivif set a = (µa,γa) in a hilbert algebra x is an ivif subalgebra, then (∀x ∈ x) ( µa(1 ·x) ≥ µa(x) γa(1 ·x) ≤ γa(x) ) . (3.2) proof. for any x ∈ x, we have µa(1 ·x) ≥ rmin{µa(1),µa(x)} = rmin{µa(x ·x),µa(x)} ≥ rmin{rmin{µa(x),µa(x)},µa(x)} = µa(x) and γa(1 ·x) ≤ rmax{γa(1),γa(x)} = rmax{γa(x ·x),γa(x)} ≤ rmax{rmax{γa(x),γa(x)},γa(x)} = γa(x). � theorem 3.1. an ivif set a = (µa,γa) = ([µla,µ u a], [γ l a,γ u a]) in a hilbert algebra x is an ivif subalgebra of x if and only if µla,µ u a,γ l a, and γ u a are fuzzy subalgebras of x. proof. let µla and µ u a be fuzzy subalgebras of x and x,y ∈ x. then µ l a(x ·y) ≥ min{µ l a(x),µ l a(y)} and µua(x ·y) ≤ min{µ u a(x),µ u a(y)}. now, µa(x ·y) = [µla(x ·y),µ u a(x ·y)] ≥ [min{µla(x),µ l a(y)}, min{µ u a(x),µ u a(y)}] = rmin{[µla, (x),µ u a(x)], [µ l a(y),µ u a(y)]} = rmin{µa(x),µa(y)}. again, let γla and γ u a be fuzzy subalgebras of x and x,y ∈ x. then γ l a(x · y) ≤ max{γ l a(x),γ l a(y)} and γua(x ·y) ≤ max{γ u a(x),γ u a(y)}. now, γa(x ·y) = [γla(x ·y),γ u a(x ·y)] ≤ [max{γla(x),γ l a(y)}, max{γ u a(x),γ u a(y)}] = rmax{[γla, (x),γ u a(x)], [γ l a(y),γ u a(y)]} = rmax{γa(x),γa(y)}. 6 int. j. anal. appl. (2022), 20:25 hence, a = {[µla,µ u a], [γ l a,γ u a]} is an ivif subalgebra of x. conversely, assume that a is an ivif subalgebra of x. for any x,y ∈ x, [µla(x ·y),µ u a(x ·y)] = µa(x ·y) ≥ rmin{µa(x),µa(y)} = rmin{[µla(x),µ u a(x)], [µ l a(y),µ u a(y)]} = [min{µla(x),µ l a(y)}, min{µ u a(x),µ u a(y)}] and [γla(x ·y),γ u a(x ·y)] = γa(x ·y) ≤ rmax{γa(x),γa(y)} = rmax{[γla(x),γ u a(x)], [γ l a(y),γ u a(y)]} = [max{γla(x),γ l a(y)}, max{γ u a(x),γ u a(y)}]. thus µla(x ·y) ≥ min{µ l a(x),µ l a(y)},µ u a(x ·y) ≥ min{µ u a(x),µ u a(y)},γ l a(x ·y) ≤ max{γ l a(x),γ l a(y)}, and γua(x ·y) ≤ max{γ u a(x),γ u a(y)}. therefore, µ l a,µ u a,γ l a, and γ u a are fuzzy subalgebras of x. � theorem 3.2. if a = (µa,γa) and b = (µb,γb) are two ivif subalgebras of a hilbert algebra x, then a∩b = (µa∩b,γa∪b) is an ivif subalgebra of x. proof. let x,y ∈ x. since a and b are ivif subalgebras of x, by theorem 3.1, we have µa∩b(x ·y) = [µla∩b(x ·y),µ u a∩b(x ·y)] = [min{µla(x ·y),µ l b(x ·y)}, min{µ u a(x ·y),µ u b(x ·y)}] ≥ [min{µla∩b(x),µ l a∩b(y)}, min{µ u a∩b(x),µ u a∩b(y)}] = rmin{µa∩b(x),µa∩b(y)} and γa∪b(x ·y) = [γla∪b(x ·y),γ u a∪b(x ·y)] = [max{γla(x ·y),γ l b(x ·y)}, max{γ u a(x ·y),γ u b(x ·y)}] ≤ [max{γla∪b(x),γ l a∪b(y)}, max{γ u a∪b(x),γ u a∪b(y)}] = rmax{γa∪b(x),γa∪b(y)}. hence, a∩b = (µa∩b,γa∪b) is an ivif subalgebra of x. � corollary 3.1. let {ai | i = 1, 2, 3, · · ·} be a family of ivif subalgebras of a hilbert algebra x. then ∞ ∩ i=1 ai is also an ivif subalgebra of x, where ∞ ∩ i=1 ai = {(x, rminµai (x), rmaxγai (x)) | x ∈ x}. for any elements x and y of a hilbert algebra x, let n∏ x ·y denotes the expression x ·(· · ·(x ·(x ·y))), where x occurred n times. theorem 3.3. let a = (µa,γa) be an ivif subalgebra of a hilbert algebra x and let n ∈n. then (1) µa ( n∏ x ·x ) ≥ µa(x) for any odd number n, (2) γa ( n∏ x ·x ) ≤ γa(x) for any odd number n, int. j. anal. appl. (2022), 20:25 7 (3) µa ( n∏ x ·x ) = µa(x) for any even number n, (4) γa ( n∏ x ·x ) = γa(x) for any even number n. proof. let x ∈ x and assume that n is odd. then n = 2p− 1 for some positive integer p. we prove the theorem by induction. now, µa(x ·x) = µa(1) ≥ µa(x) and γa(x ·x) = γa(1) ≤ γa(x). suppose that µa ( 2p−1∏ x ·x ) ≥ µa(x) and γa ( 2p−1∏ x ·x ) ≤ γa(x). then by assumption, µa ( 2(p+1)−1∏ x ·x ) = µa ( 2p+1∏ x ·x ) = µa ( 2p−1∏ x · (x · (x ·x)) ) = µa ( 2p−1∏ x ·x ) ≥ µa(x) and γa ( 2(p+1)−1∏ x ·x ) = γa ( 2p+1∏ x ·x ) = γa ( 2p−1∏ x · (x · (x ·x)) ) = γa ( 2p−1∏ x ·x ) ≤ γa(x), which proves (1) and (2). proofs are similar for the cases (3) and (4). � definition 3.2. let a = (µa,γa) be an ivif set defined in a hilbert algebra x. the ivif sets ⊕a and ⊗a are defined as ⊕a = {(x,µa(x),µa(x)) | x ∈ x} and ⊗a = {(x,γa(x),γa(x)) | x ∈ x}. theorem 3.4. if a = (µa,γa) is an ivif subalgebra of a hilbert algebra x, then ⊕a and ⊗a both are ivif subalgebras. proof. let x,y ∈ x. then µa(x ·y) = [1, 1] −µa(x ·y) ≤ [1, 1] − rmin{µa(x),µa(y)} = rmax{1 − µa(x), 1 − µa(y)} = rmax{µa(x),µa(y)}. hence, ⊕a is an ivif subalgebra of x. let x,y ∈ x. then γa(x · y) = [1, 1] −γa(x · y) ≥ [1, 1] − rmax{γa(x),γa(y)} = rmin{1 −γa(x), 1 −γa(y)} = rmin{γa(x),γa(y)}. hence, ⊗a is also an ivif subalgebra of x. � the sets {x ∈ x | µa(x) = µa(1)} and {x ∈ x | γa(x) = γa(1)} are denoted by µ1a and γ 1 a, respectively. these two sets are also subalgebra of a hilbert algebra x. theorem 3.5. let a = (µa,γa) be an ivif subalgebra of a hilbert algebra x, then the sets µ1a and γ1a are subalgebras of x. 8 int. j. anal. appl. (2022), 20:25 proof. let x,y ∈ µ1a. then µa(x) = µa(1) = µa(y) and so µa(x ·y) ≤ rmin{µa(x),µa(y)} = µa(1). by using proposition 3.1, we have µa(x ·y) = µa(1); hence, x ·y ∈ µ1a. again, let x,y ∈ γ 1 a. then γa(x) = γa(1) = γa(y) and so, γa(x · y) ≤ rmax{γa(x),γa(y)} = γa(1). again, by proposition 3.1, we have γa(x ·y) = γa(1); hence, x ·y ∈ γ1a. therefore, the sets µ 1 a and γ 1 a are subalgebras of x. � theorem 3.6. let b be a nonempty subset of a hilbert algebra x and a = (µa,γa) be an ivif set in x defined by µa(x) = { [α1,α2] if x ∈ b [β1,β2] otherwise and γa(x) = { [θ1,θ2] if x ∈ b [δ1,δ2] otherwise for all [α1,α2], [β1,β2], [θ1,θ2], [δ1,δ2] ∈ d[0, 1] with [α1,α2] ≥ [β1,β2] and [θ1,θ2] ≤ [δ1,δ2] and α2 +θ2 ≤ 1 and β2 +δ2 ≤ 1. then a is an ivif subalgebra of x if and only if b is a subalgebra of x. moreover, µ1a = b = γ 1 a. proof. let a be an ivif subalgebra of x. let x,y ∈ b. then µa(x ·y) ≥ rmin{µa(x),µa(y)} = rmin{[α1,α2], [α1,α2]} = [α1,α2] and γa(x ·y) ≤ rmax{γa(x),γa(y)} = rmax{[α1,α2], [α1,α2]} = [α1,α2]. so x ·y ∈ b. hence, b is a subalgebra of x. conversely, suppose that b is a subalgebra of x. let x,y ∈ x. consider two cases: case (i): if x,y ∈ b, then x ·y ∈ b. thus µa(x ·y) = [α1,α2] = rmin{µa(x),µa(y)} and γa(x ·y) = [θ1,θ2] = rmax{γa(x),γa(y)}. case (ii): if x /∈ b or y /∈ b, then µa(x · y) ≥ [β1,β2] = rmin{µa(x),µa(y)} and γa(x · y) ≤ [θ1,θ2] = rmax{γa(x),γa(y)}. hence, a is an ivif subalgebra of x. now, µ1a = {x ∈ x | µa(x) = µa(1)} = {x ∈ x | µa(x) = [α1,α2]} = b and γ1a = {x ∈ x | γa(x) = γa(1)} = {x ∈ x | γa(x) = [θ1,θ2]} = b. � definition 3.3. let a = (µa,γa) be an ivif subalgebra of a hilbert algebra x. for [s1,s2], [t1,t2] ∈ d[0, 1], the sets u(µa : [s1,s2]) = {x ∈ x | µa(x) ≥ [s1,s2]} is called an upper [s1,s2]-level of a and l(γa : [t1,t2]) = {x ∈ x | γa(x) ≤ [t1,t2]} is called a lower [t1,t2]-level of a. theorem 3.7. if a = (µa,γa) is an ivif subalgebra of a hilbert algebra x, then the upper [s1,s2]-level and lower [t1,t2]-level of a are subalgebras of x. int. j. anal. appl. (2022), 20:25 9 proof. let x,y ∈ u(µa : [s1,s2]). then µa(x) ≤ [s1,s2] and µa(y) ≤ [s1,s2]. it follows that µa(x · y) ≤ rmin{µa(x),µa(y)} ≤ [s1,s2] so that x · y ∈ u(µa : [s1,s2]). hence, u(µa : [s1,s2]) is a subalgebra of x. let x,y ∈ l(γa : [t1,t2]). then γa(x) ≤ [t1,t2] and γa(y) ≤ [t1,t2]. thus γa(x · y) ≤ rmax{γa(x),γa(y)} ≤ [t1,t2] so that x · y ∈ l(γa : [t1,t2]). hence, l(γa : [t1,t2]) is a subalgebra of x. � theorem 3.8. let a = (µa,γa) be an ivif set in a hilbert algebra x such that the sets u(µa : [s1,s2]) and l(γa : [t1,t2]) are subalgebras of x for every [s1,s2], [t1,t2] ∈ d[0, 1]. then a is an ivif subalgebra of x. proof. let [s1,s2], [t1,t2] ∈ d[0, 1] be such that u(µa : [s1,s2]) and l(γa : [t1,t2]) are subalgebras of x. in contrary, let x0,y0 ∈ x be such that µa(x0 · y0) < rmin{µa(x0),µa(y0)}. let µa(x0) = [θ1,θ2],µa(y0) = [θ3,θ4], and µa(x0·y0) = [s1,s2]. then [s1,s2] < rmin{[θ1,θ2], [θ3,θ4]} = [min{θ1,θ3}, min{θ2,θ4}]. so, s1 < min{θ1,θ3} and s2 < min{θ2,θ4}. consider, [ρ1,ρ2] = 1 2 [µa(x0 ·y0) + rmin{µa(x0),µa(y0)}] = 1 2 [[s1,s2] + [min{θ1,θ3}, min{θ2,θ4}]] = [ 1 2 (s1 + min{θ1,θ3}), 12 (s2 + min{θ2,θ4})]. therefore, min{θ1,θ3} > ρ1 = 12 (s1 +min{θ1,θ3}) > s1 and min{θ2,θ4} > ρ2 = 1 2 (s2 +min{θ2,θ4}) > s2. hence, [min{θ1,θ3}, min{θ2,θ4}] > [ρ1,ρ2] > [s1,s2], so that x0 ·y0 /∈ u(µa : [s1,s2]), which is a contradiction because µa(x0) = [θ1,θ2] ≥ [min{θ1,θ3}, min{θ2,θ4}] > [ρ1,ρ2] and µa(y0) = [θ3,θ4] ≥ [min{θ1,θ3}, min{θ2,θ4}] > [ρ1,ρ2]. this implies that x0 · y0 ∈ u(µa : [s1,s2]). thus µa(x · y) ≤ rmin{µa(x),µa(y)} for all x,y ∈ x. again, in contrary, let x0,y0 ∈ x be such that γa(x0 · y0) > rmax{γa(x0),γa(y0)}. let γa(x0) = [η1,η2],γa(y0) = [η3,η4], and γa(x0 · y0) = [t1,t2]. then [t1,t2] > rmax{[η1,η2], [η3,η4]} = [max{η1,η3}, max{η2,η4}]. so t1 > max{η1,η3} and t2 > max{η2,η4}. let us consider, [β1,β2] = 1 2 [γa(x0 ·y0) + rmax{γa(x0),γa(y0)}] = 1 2 [[t1,t2] + [max{η1,η3}, max{η2,η4}] = [ 1 2 (t1 + max{η1,η3}), 12 (t2 + max{η2,η4})]. therefore, max{η1,η3} < β1 = 12 [(t1 + max{η1,η3})] < t1 and max{η2,η4} < β2 = 1 2 [(t2 + max{η2,η4})] < t2. hence, [max{η1,η3}, max{η2,η4}] < [β1,β2] < [t1,t2] so that x0 · y0 /∈ l(γa : [t1,t2]), which is a contradiction because γa(x0) = [η1,η2] ≤ [max{η1,η3}, max{η2,η4}] > [β1,β2] 10 int. j. anal. appl. (2022), 20:25 and γa(y0) = [η3,η4] ≥ [max{η1,η3}, max{η2,η4}] > [β1,β2]. this implies that x0 · y0 ∈ l(γa : [t1,t2]). thus γa(x · y) ≥ rmax{γa(x),γa(y)} for all x,y ∈ x. therefore, a is an ivif subalgebra of x. � theorem 3.9. any subalgebra of a hilbert algebra x can be realized as both the upper [s1,s2]-level and lower [t1,t2]-level of some ivif subalgebra of x. proof. let b be an ivif subalgebra of x and a be an ivif set on x defined by µa(x) ={ [α1,α2] if x ∈ b [0, 0] otherwise and γa(x) = { [β1,β2] if x ∈ b [1, 1] otherwise for all [α1,α2], [β1,β2] ∈ d[0, 1] and α2 + β2 ≤ 1. we consider the following cases: case (i) if x,y ∈ b, then µa(x) = [α1,α2],γa(x) = [β1,β2], and µa(y) = [α1,α2], γa(y) = [β1,β2]. thus µa(x ·y) = [α1,α2] = rmin{[α1,α2], [α1,α2]} = rmin{µa(x),µa(y)} and γa(x ·y) = [β1,β2] = rmax{[β1,β2], [β1,β2]} = rmax{γa(x),γa(y)}. case (ii) if x ∈ b and y /∈ b, then µa(x) = [α1,α2],γa(x) = [β1,β2], and µa(y) = [0, 0], γa(y) = [1, 1]. thus µa(x ·y) ≥ [0, 0] = rmin{[α1,α2], [0, 0]} = rmin{µa(x),µa(y)} and γa(x ·y) ≥ [1, 1] = rmax{[β1,β2], [1, 1]} = rmax{γa(x),γa(y)}. case (iii) if x /∈ b and y ∈ b, then µa(x) = [0, 0],γa(x) = [1, 1],µa(y) = [α1,α2], and γa(y) = [β1,β2]. thus µa(x ·y) ≥ [0, 0] = rmin{[0, 0], [α1,α2]} = rmin{µa(x),µa(y)} and γa(x ·y) ≤ [1, 1] = rmax{[1, 1], [β1,β2]} = rmax{γa(x),γa(y)}. case (iv) if x /∈ b and y /∈ b, then µa(x) = [0, 0],γa(x) = [1, 1],µa(y) = [0, 0], and γa(y) = [1, 1]. thus µa(x ·y) ≤ [0, 0] = rmin{[0, 0], [0, 0]} = rmin{µa(x),µa(y)} and γa(x ·y) ≥ [1, 1] = rmax{[1, 1], [1, 1]} = rmax{γa(x),γa(y)}. therefore, a is an ivif subalgebra of x. � int. j. anal. appl. (2022), 20:25 11 theorem 3.10. let b be a subset of a hilbert algebra x and a be an ivif set in x defined by µa(x) ={ [α1,α2] if x ∈ b [0, 0] otherwise and γa(x) = { [β1,β2] if x ∈ b [1, 1] otherwise for all [α1,α2], [β1,β2] ∈ d[0, 1] and α2 + β2 ≤ 1. if a is realized as a lower level subalgebra and an upper level subalgebra of some ivif subalgebra of x, then b is a subalgebra of x. proof. let a be an ivif subalgebra of x and x,y ∈ b. then µa(x) = [α1,α2] = µa(y) and γa(x) = [β1,β2] = γa(y). thus µa(x ·y) ≤ rmin{µa(x),µa(y)} = rmin{[α1,α2], [α1,α2]} = [α1,α2] and γa(x ·y) ≥ rmax{γa(x),γa(y)} = rmax{[β1,β2], [β1,β2]} = [β1,β2], which imply that x ·y ∈ b. hence, b is a subalgebra of x. � 4. ivif ideals of hilbert algebras definition 4.1. an ivif set a = (µa,γa) in a hilbert algebra x is said to be an ivif ideal of x if the following conditions are hold: (∀x ∈ x) ( µa(1) ≥ µa(x) γa(1) ≤ γa(x) ) , (4.1) (∀x,y ∈ x) ( µa(x ·y) ≥ µa(y) γa(x ·y) ≤ γa(y) ) , (4.2) (∀x,y1,y2 ∈ x) ( µa((y1 · (y2 ·x)) ·x) ≥ rmin{µa(y1),µa(y2)} γa((y1 · (y2 ·x)) ·x) ≤ rmax{γa(y1),γa(y2)} ) . (4.3) example 4.1. let x = {1,x,y,z, 0} with the following cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 then x is a hilbert algebra. we define an ivif set a = (µa,γa) as follows: µa(x) = { [0.5, 0.6] if x ∈{1,x,y,z} [0.1, 0.2] if x = 0 and γa(x) = { [0.3, 0.4] if x ∈{1,x,y,z} [0.4, 0.5] if x = 0. 12 int. j. anal. appl. (2022), 20:25 then a is an ivif ideal of x. proposition 4.1. if a = (µa,γa) is an ivif ideal of a hilbert algebra x, then (∀x,y ∈ x) ( µa((y ·x) ·x) ≥ µa(y) γa((y ·x) ·x) ≤ γa(y) ) . (4.4) proof. putting y1 = y and y2 = 1 in (4.3), we have µa((y ·x) ·x) ≥ rmin{µa(y),µa(1)} = µa(y) and γa((y ·x) ·x) ≤ rmax{γa(y),γa(1)} = γa(y). � lemma 4.1. if a = (µa,γa) is an ivif ideal of a hilbert algebra x, then (∀x,y ∈ x) ( x ≤ y ⇒ { µa(x) ≤ µa(y) γa(x) ≥ γa(y) ) . (4.5) proof. let x,y ∈ x be such that x ≤ y. then x ·y = 1 and so µa(y) = µa(1 ·y) = µa(((x ·y) · (x ·y)) ·y) ≥ rmin{µa(x ·y),µa(x)} ≥ rmin{µa(1),µa(x)} = µa(x). thus γa(y) = γa(1 ·y) = γa(((x ·y) · (x ·y)) ·y) ≤ rmax{γa(x ·y),γa(x)} ≤ rmax{γa(1),γa(x)} = γa(x). � theorem 4.1. every ivif ideal of a hilbert algebra x is an ivif subalgebra of x. proof. let a = (µa,γa) be an ivif ideal of x. since y ≤ x ·y for all x,y ∈ x, it follows from lemma 4.1 that µa(y) ≥ µa(x ·y),γa(y) ≤ γa(x ·y). it follows from (4.2) that µa(x ·y) ≥ µa(y) ≥ rmin{µa(x ·y),µa(x)} ≥ rmin{µa(x),µa(y)} int. j. anal. appl. (2022), 20:25 13 and γa(x ·y) ≤ γa(y) ≤ rmax{γa(x ·y),γa(x)} ≤ rmax{γa(x),γa(y)}. hence, a is an ivif subalgebra of x. � theorem 4.2. an ivif set a = (µa,γa) = {[µla,µ u a], [γ l a,γ u a]} in a hilbert algebra x is an ivif ideal of x if and only if µla,µ u a,γ l a, and γ u a are fuzzy ideals of x. proof. since µla(1) ≥ µ l a(x),µ u a(1) ≥ µ u a(x),γ l a(1) ≤ γ l a(x), and γ u a(1) ≤ γ u a(x), we have µa(1) ≥ µa(x) and γa(1) ≤ γa(x). let x,y ∈ x. then µa(x ·y) = [µla(x ·y),µ u a(x ·y)] ≥ [µ l a(y),µ u a(y)] = µa(y) and γa(x ·y) = [γla(x ·y),γ u a(x ·y)] ≤ [γ l a(y),γ u a(y)] = γa(y). let x,y1,y2 ∈ x. then µa((y1 · (y2 ·x)) ·x) = [µla((y1 · (y2 ·x)) ·x),µ u a((y1 · (y2 ·x)) ·x)] ≥ [min{µla(y1),µ l a(y2)}, min{µ u a(y1),µ u a(y2)}] = rmin{[µla(y1),µ u a(y1)], [µ l a(y2),µ u a(y2)]} = rmin{µa(y1),µa(y2)} and γa((y1 · (y2 ·x)) ·x) = [γla((y1 · (y2 ·x)) ·x),γ u a((y1 · (y2 ·x)) ·x)] ≤ [max{γla(y1),γ l a(y2)}, max{γ u a(y1),γ u a(y2)}] = rmax{[γla(y1),γ u a(y1)], [γ l a(y2),γ u a(y2)]} = rmax{γa(y1),γa(y2)}. hence, a = {[µla,µ u a], [γ l a,γ u a]} is an ivif ideal of x. conversely, assume that a is an ivif ideal of x. let x ∈ x. then [µla(1),µ u a(1)] = µa(1) ≥ µa(x) = [µ l a(x),µ u a(x)]; hence, µ l a(1) ≥ µ l a(x) and γ l a(1) ≤ γ l a(x). let x,y ∈ x. then [µ l a(x · y),µua(x ·y)] = µa(x ·y) ≥ µa(y) = [µ l a(y),µ u a(y)]; hence, µ l a(x ·y) ≥ µ l a(y) and µ u a(x ·y) ≥ µ u a(y). also, [γla(x · y),γ u a(x · y)] = γa(x · y) ≤ γa(y) = [γ l a(y),γ u a(y)]; hence, γ l a(x · y) ≤ γ l a(y) and γua(x ·y) ≤ γ u a(y). let x,y1,y2 ∈ x. then µla((y1 · (y2 ·x)) ·x),µ u a((y1 · (y2 ·x)) ·x)] = µa((y1 · (y2 ·x)) ·x) ≥ rmin{µa(y1),µa(y2)} = rmin{[µla(y1),µ u a(y1)], [µ l a(y2),µ u a(y2)]} = [min{µla(y1),µ l a(y2)}, min{µ u a(y1),µ u a(y2)}]. 14 int. j. anal. appl. (2022), 20:25 hence, µla((y1 ·(y2 ·x)) ·x) ≥ min{µ l a(y1),µ l a(y2)} and µ u a((y1 ·(y2 ·x)) ·x) ≥ min{µ u a(y1),µ u a(y2)}. also, [γla((y1 · (y2 ·x)) ·x),γ u a((y1 · (y2 ·x)) ·x)] = γa((y1 · (y2 ·x)) ·x) ≤ rmax{γa(y1),γa(y2)} = rmax{[γla(y1),γ u a(y1)], [γ l a(y2),γ u a(y2)]} = [max{γla(y1),γ l a(y2)}, max{γ u a(y1),γ u a(y2)}]. hence, γla((y1 ·(y2 ·x)) ·x) ≤ max{γ l a(y1),γ l a(y2)} and γ u a((y1 ·(y2 ·x)) ·x) ≤ max{γ u a(y1),γ u a(y2)}. therefore, µla,µ u a,γ l a, and γ u a are fuzzy ideals of x. � proposition 4.2. if a = (µa,γa) and b = (µb,γb) are ivif ideals of a hilbert algebra x, then a∩b is an ivif ideal of x. proof. let a = (µa,γa) and b = (µb,γb) be ivif ideals of x. let x ∈ x. then µa∩b(1) = [µ l a∩b(1),µ u a∩b(1)] = [min{µla(1),µ l b(1)}, min{µ u a(1),µ u b(1)}] ≥ [min{µla(x),µ l b(x)}, min{µ u a(x),µ u b(x)}] = [µla∩b(x),µ u a∩b(x)] = µa∩b(x) and γa∪b(1) = [γ l a∪b(1),γ u a∪b(1)] = [max{γla(1),γ l b(1)}, max{γ u a(1),γ u b(1)}] ≤ [max{γla(x),γ l b(x)}, max{γ u a(x),γ u b(x)}] = [γla∪b(x),γ u a∪b(x)] = γa∪b(x). let x,y ∈ x. then µa∩b(x ·y) = [µla∩b(x ·y),µ u a∩b(x ·y)] = [min{µla(x ·y),µ l b(x ·y)}, min{µ u a(x ·y),µ u b(x ·y)}] ≥ [min{µla(y),µ l b(y)}, min{µ u a(y),µ u b(y)}] = [µla∩b(y),µ u a∩b(y)] = µa∩b(y) and γa∪b(x ·y) = [γla∪b(x ·y),γ u a∪b(x ·y)] = [max{γla(x ·y),γ l b(x ·y)}, max{γ u a(x ·y),γ u b(x ·y)}] ≤ [max{γla(y),γ l b(y)}, max{γ u a(y),γ u b(y)}] = [γla∪b(y),γ u a∪b(y)] = γa∪b(y). int. j. anal. appl. (2022), 20:25 15 let x,y1,y2 ∈ x. then µa∩b((y1 · (y2 ·x)) ·x) = [µla∩b((y1 · (y2 ·x)) ·x),µ u a∩b((y1 · (y2 ·x)) ·x)] = [ min{µla((y1 · (y2 ·x)) ·x),µ l b((y1 · (y2 ·x)) ·x)}, min{µua((y1 · (y2 ·x)) ·x),µ u b((y1 · (y2 ·x)) ·x)} ] ≥ [ min{min{µla(y1),µ l a(y2)}, min{µ l b(y1),µ l b(y2)}}, min{min{µua(y1),µ u a(y2)}, min{µ u b(y1),µ u b(y2)}} ] = [ min{min{µla(y1),µ l b(y1)}, min{µ l a(y2),µ l b(y2)}}, min{min{µua(y1),µ u b(y2)}, min{µ u a(y2),µ u b(y2)}} ] = [min{µla∩b(y1),µ l a∩b(y2)}, min{µ l a∩b(y1),µ l a∩b(y2)}] = rmin{µa∩b(y1),µa∩b(y2)} and γa∪b((y1 · (y2 ·x)) ·x) = [γla∪b((y1 · (y2 ·x)) ·x),γ u a∪b((y1 · (y2 ·x)) ·x)] = [ max{γla((y1 · (y2 ·x)) ·x),γ l b((y1 · (y2 ·x)) ·x)}, max{γua((y1 · (y2 ·x)) ·x),γ u b((y1 · (y2 ·x)) ·x)} ] ≤ [ max{min{γla(y1),γ l a(y2)}, min{γ l b(y1),γ l b(y2)}}, max{min{γua(y1),γ u a(y2)}, min{γ u b(y1),γ u b(y2)}} ] = [ max{min{γla(y1),γ l b(y1)}, min{γ l a(y2),γ l b(y2)}}, max{min{γua(y1),γ u b(y2)}, min{γ u a(y2),γ u b(y2)}} ] = [max{γla∩b(y1),γ l a∩b(y2)}, max{γ l a∩b(y1),γ l a∩b(y2)}] = rmax{γa∩b(y1),γa∩b(y2)}. hence, a∩b is an ivif ideal of x. � corollary 4.1. if {ai = (µai,γai ) | i ∈ ∆} is a family of ivif ideals of a hilbert algebra x, then ⋂ i∈∆ ai is an ivif ideal of x. corollary 4.2. if a = (µa,γa) is an ivif ideal of a hilbert algebra x, then a is also an ivif ideal of x. theorem 4.3. if a = (µa,γa) is an ivif ideal of a hilbert algebra x, then ⊕a and ⊗a are both ivif ideals. proof. assume that a = (µa,γa) is an ivif ideal of x. let x ∈ x. then µa(1) = 1 − µa(1) ≤ 1 − µa(x) ≤ µa(x). let x,y ∈ x. then µa(x · y) = 1 − µa(x · y) ≤ 1 − µa(y) ≤ µa(y). let 16 int. j. anal. appl. (2022), 20:25 x,y1,y2 ∈ x. then µa((y1 · (y2 ·x)) ·x) = 1 −µa((y1 · (y2 ·x)) ·x) ≤ 1 − rmin{µa(y1),µa(y2)} = rmax{1 −µa(y1), 1 −µa(y2)} = rmax{µa(y1),µa(y2)}. hence, ⊕a is an ivif ideal of x. let x ∈ x. then γa(1) = 1 − γa(1) ≥ 1 − γa(x) ≥ γa(x). let x,y ∈ x. then γa(x · y) = 1 −γa(x ·y) ≥ 1 −γa(y) ≥ γa(y). let x,y1,y2 ∈ x. then γa((y1 · (y2 ·x)) ·x) = 1 −γa((y1 · (y2 ·x)) ·x) ≥ 1 − rmax{γa(y1),γa(y2)} = rmin{1 −γa(y1), 1 −γa(y2)} = rmin{γa(y1),γa(y2)}. hence, ⊗a is an ivif ideal of x. � theorem 4.4. an ivif set a = (µa,γa) is an ivif ideal of a hilbert algebra x if and only if for every [s1,s2], [t1,t2] ∈ d[0, 1], the sets u(µa : [t1,t2]) and l(g, [s1,s2]) are either empty or ideals of x. proof. let a = (µa,γa) be an ivif ideal of x and let [s1,s2], [t1,t2] ∈ d[0, 1] be such that u(µa : [t1,t2]) and l(γa : [s1,s2]) are nonempty sets of x. it is clear that 1 ∈ u(µa : [t1,t2])∩l(γa : [s1,s2]) since µa(1) ≥ [t1,t2] and γa(1) ≤ [s1,s2]. let x ∈ x and y ∈ u(µa : [t1,t2]). then µa(y) ≥ [t1,t2]. it follows that µa(x · y) ≥ µa(y) ≥ [t1,t2] so that x · y ∈ u(µa : [t1,t2]). let x ∈ x and y1,y2 ∈ u(µa : [t1,t2]). then µa(y1) ≥ [t1,t2] and µa(y2) ≥ [t1,t2]. hence, µa((y1 · (y2 ·x)) ·x) ≥ min{µa(y1),µa(y2)} ≥ [t1,t2] so that (y1 · (y2 · x)) · x ∈ u(µa : [t1,t2]). hence, u(µa : [t1,t2]) is an ideal of x. let x ∈ x and y ∈ l(γa : [s1,s2]). then γa(y) ≤ [s1,s2]. it follows that γa(x·y) ≤ γa(y) ≤ [s1,s2] so that x·y ∈ l(γa : [s1,s2]). let x ∈ x and y1,y2 ∈ l(γa : [s1,s2]). then γa(y1) ≤ [s1,s2] and γa(y2) ≤ [s1,s2]. hence, γa((y1 · (y2 ·x)) ·x) ≤ max{γa(y1),γa(y2)}≤ [s1,s2] so that (y1 · (y2 ·x)) ·x ∈ l(γa : [s1,s2]). hence, l(γa : [s1,s2]) is an ideal of x. assume now that every nonempty sets u(µa : [t1,t2]) and l(γa : [s1,s2]) are ideals of x. if µa(1) ≥ µa(x) is not true for all x ∈ x, then there exists x0 ∈ x such that µa(1) < µa(x0). but in this case for [s1,s2] = 1 2 (µa(1) + µa(x0)). then x0 ∈ u(µa : [s1,s2]), that is u(µa : [s1,s2]) 6= ∅. since by the assumption, u(µa : [s1,s2]) is an ideal of x, then µa(1) ≥ [s1,s2], which is impossible. hence, µa(1) ≥ µa(x). if γa(1) ≤ γa(x) is not true, then there exists y0 ∈ x such that γa(1) < γa(y0). but in this case for [s′0,s ′′ 0 ] = 1 2 (γa(1) + γa(y0)). then y0 ∈ l(γa : [s′0,s ′′ 0 ]), that is l(γa : [s ′ 0,s ′′ 0 ]) 6= ∅. since by the assumption, l(γa : [s′0,s ′′ 0 ]) is an ideal of x, then γa(1) ≤ [s ′ 0,s ′′ 0 ], which is impossible. hence, γa(1) ≤ γa(x). if µa(x · y) ≥ µa(y) is not true for all x,y ∈ x, then there exist x0,y0 ∈ x such that µa(x0 · y0) < µa(y0). let [t1,t2] = 12 (µa(x0 · y0) + µa(y0)). then t ∈ [0, 1] and µa(x0 · y0) < t < µa(y0), which prove that y0 ∈ u(µa : t). since u(µa : t) is an ideal of x, int. j. anal. appl. (2022), 20:25 17 x0 · y0 ∈ u(µa : t). hence, µa(x0 · y0) ≥ t, a contradiction. thus µa(x · y) ≥ µa(y) is true for all x,y ∈ x. if γa(x · y) ≤ γa(y) is not true for all x,y ∈ x, then there exist x0,y0 ∈ x such that γa(x0 ·y0) > γa(y0). let [t′0,t ′′ 0 ] = 1 2 (γa(x0 ·y0) + γa(y0)). then [t′0,t ′′ 0 ] ∈ d[0, 1] and γa(x0 ·y0) > [t′0,t ′′ 0 ] > γa(y0), which prove that y0 ∈ l(γa : [t ′ 0,t ′′ 0 ]). since l(γa : [t ′ 0,t ′′ 0 ]) is an ideal of x, x0·y0 ∈ l(γa : [t′0,t ′′ 0 ]). hence, γa(x0·y0) ≤ [t ′ 0,t ′′ 0 ], a contradiction. thus γa(x·y) ≤ γa(y) is true for all x,y ∈ x. suppose that µa((y1·(y2·x))·x) ≥ rmin{µa(y1),µa(y2)} is not true for all x,y1,y2 ∈ x. then there exist u0,v0,x0 ∈ x such that µa((u0 · (v0 · x0))) · x0) < rmin{µa(u0),µa(v0)}. taking [p′,p′′] = 1 2 (µa((u0 ·(v0 ·x0)))·x0) + rmin{µa(u0),µa(v0)}). then µa((u0 ·(v0 ·x0))·x0) < [p′,p′′] < rmin{µa(u0),µa(v0)}, which prove that u0,v0 ∈ u(µa : [p′,p′′]). since u(µa : p[p′,p′′]) is an ideal of x, (u0·(v0·x0))x0 ∈ u(µa : [p′,p′′]), a contradiction. thus µa((y1·(y2·x))·x) ≥ rmin{µa(y1),µa(y2)} is true for all x,y1,y2 ∈ x. suppose that γa((y1·(y2·x))·x) ≤ rmax{γa(y1),γa(y2)} is not true for all x,y1,y2 ∈ x. then there exist u0,v0,x0 ∈ x such that γa((u0 ·(v0 ·x0)·x0) > rmax{γa(u0),γa(v0)}. taking [p′0,p ′′ 0 ] = 1 2 (γa((u0 · (v0 ·x0)) ·x0) + rmax{γa(u0),γa(v0)}). then γa((u0 · (v0 ·x0)) ·x0) > [p′0,p ′′ 0 ] > rmax{γa(u0),γa(v0)}, which prove that u0,v0 ∈ l(γa : [p ′ 0,p ′′ 0 ]). since l(γa : [p ′ 0,p ′′ 0 ]) is an ideal of x, (u0 · (v0 · x0)) · x0 ∈ l(γa : [p′0,p ′′ 0 ]), a contradiction. thus γa((y1 · (y2 · x)) · x) ≤ rmax{γa(y1),γa(y2)} is true for all x,y1,y2 ∈ x. hence, a is an ivif ideal of x. � 5. product of ivif subalgebras/ideals in hilbert algebras definition 5.1. let a = (µa,γa) and b = (µb,γb) be ivif sets in hilbert algebras x and y , respectively. the cartesian product a×b = {((x,y), (µa×µb)(x,y), (γa×γb)(x,y)) | x ∈ x,y ∈ y} defined by (µa ×µb)(x,y) = rmin{µa(x),µb(y)} and (γa ×γb)(x,y) = rmax{γa(x),γb(y)}, where µa ×µb : x ×y → d[0, 1] and γa ×γb : x ×y → d[0, 1] for all x ∈ x and y ∈ y . remark 5.1. let x and y be hilbert algebras. we define the binary operation · on x × y by (x,y) · (u,v) = (x ·u,y ·v) for every (x,y), (u,v) ∈ x ×y , then clearly (x ×y, ·, (1, 1)) is a hilbert algebra. proposition 5.1. if a = (µa,γa) and b = (µb,γb) are ivif subalgebras of hilbert algebras x and y , respectively, then the cartesian product a×b is also an ivif subalgebra of x ×y . 18 int. j. anal. appl. (2022), 20:25 proof. let (x1,y1), (x2,y2) ∈ x ×y . then (µa ×µb)((x1,y1) · (x2,y2)) = (µa ×µb)((x1 ·x2), (y1 ·y2)) = rmin{µa(x1 ·x2),µb(y1 ·y2)} ≥ rmin{rmin{µa(x1),µa(x2)}, rmin{µb(y1),µb(y2)}} = rmin{rmin{µa(x1),µb(y1)}, rmin{µa(x2),µb(y2)}} = rmin{(µa ×µb)(x1,y1), (µa ×µb)(x2,y2)} and (γa ×γb)((x1,y1) · (x2,y2)) = (γa ×γb)((x1 ·x2), (y1 ·y2)) = rmin{γa(x1 ·x2),γb(y1 ·y2)} ≤ rmin{rmax{γa(x1),γa(x2)}, rmax{γb(y1),γb(y2)}} = rmax{rmin{γa(x1),γb(y1)}, rmin{γa(x2),γb(y2)}} = rmax{(µa ×µb)(x1,y1), (µa ×µb)(x2,y2)}. hence, a×b is an ivif subalgebra of x ×y . � lemma 5.1. if a = (µa,γa) and b = (µb,γb) are two ivif subalgebras of hilbert algebras x and y , respectively, then ⊕(a×b) = (µa ×µb,µa ×µb) is an ivif subalgebra of x ×y . proof. it is sufficient to prove only the part of µa ×µb. let (x1,y1), (x2,y2) ∈ x ×y . then (µa ×µb)((x1,y1) · (x2,y2)) = (µa ×µb)((x1 ·x2), (y1 ·y2)) = rmax{µa(x1 ·x2),µb(y1 ·y2)} = rmax{1 −µa(x1 ·x2), 1 −µb(y1 ·y2)} ≤ rmax{1 − rmin{µa(x1),µa(x2)}, 1 − rmin{µb(y1),µb(y2)}} = rmax{rmax{1 −µa(x1), 1 −µb(y1)}, rmax{1 −µa(x2), 1 −µb(y2)}} = rmax{rmax{µa(x1),µb(y1)}, rmax{µa(x2),µb(y2)}} = rmax{(µa ×µb)(x1,y1), (µa ×µb)(x2,y2)}. hence, ⊕(a×b) is an ivif subalgebra of x ×y . � lemma 5.2. if a = (µa,γa) and b = (µb,γb) are two ivif subalgebras of hilbert algebras x and y , respectively, then ⊗(a×b) = (γa ×γb,γa ×γb) is an ivif subalgebra of x ×y . int. j. anal. appl. (2022), 20:25 19 proof. it is sufficient to prove only the part of γa ×γb. let (x1,y1), (x2,y2) ∈ x ×y . then (γa ×γb)((x1,y1) · (x2,y2)) = (γa ×γb)((x1 ·x2), (y1 ·y2)) = rmin{γa(x1 ·x2),γb(y1 ·y2)} = rmin{1 −γa(x1 ·x2), 1 −γb(y1 ·y2)} ≥ rmin{1 − rmax{γa(x1),γa(x2)}, 1 − rmax{γb(y1),γb(y2)}} = rmin{rmin{1 −γa(x1), 1 −γb(y1)}, rmin{1 −γa(x2), 1 −γb(y2)}} = rmin{rmin{γa(x1),γb(y1)}, rmin{γa(x2),γb(y2)}} = rmin{(γa ×γb)(x1,y1), (γa ×γb)(x2,y2)}. hence, ⊗(a×b) is an ivif subalgebra of x ×y . � theorem 5.1. the ivif sets a = (µa,γa) and b = (µb,γb) are ivif subalgebras of hilbert algebras x and y , respectively if and only if ⊕(a×b) and ⊗(a×b) are ivif subalgebras of x ×y . proof. it follows from lemmas 5.1 and 5.2. � proposition 5.2. if a = (µa,γa) and b = (µb,γb) are two ivif ideals of hilbert algebras x and y , respectively, then the cartesian product a×b is also an ivif ideal of x ×y . proof. let (x,y) ∈ x ×y . then (µa ×µb)(1, 1) = rmin{µa(1),µb(1)} ≥ rmin{µa(x),µb(y)} = (µa ×µb)(x,y) and (γa ×γb)(1, 1) = rmax{γa(1),γb(1)} ≤ rmax{γa(x),γb(y)}} = (γa ×γb)(x,y). let (x1,x2), (y1,y2) ∈ x ×y . then (µa ×µb)((x1,x2) · (y1,y2)) = (µa ×µb)((x1 ·y1), (x2 ·y2)) = rmin{µa(x1 ·y1),µb(x2 ·y2)} ≥ rmin{µa(y1),µb(y2)} = (µa ×µb)(y1,y2) and (γa ×γb)((x1,x2) · (y1,y2)) = (γa ×γb)((x1 ·y1), (x2 ·y2)) = rmax{γa(x1 ·y1),γb(x2 ·y2)} ≤ rmax{γa(y1),γb(y2)} = (γa ×γb)(y1,y2). 20 int. j. anal. appl. (2022), 20:25 let (x1,y1), (x2,y2), (x3,y3) ∈ x ×y . then (µa ×µb)(((x2,y2) · ((x3,y3) · (x1,y1))) · (x1,y1)) = (µa ×µb)(((x2 · (x3 ·x1)) ·x1)), (y2 · (y3 ·y1)) ·y1)) = rmin{µa((x2 · (x3 ·x1)) ·x1),µb((y2 · (y3 ·y1)) ·y1)} ≥ rmin{rmin{µa(x2),µa(x3)}, rmin{µb(y2),µb(y3)}} = rmin{rmin{µa(x2),µb(y2)}, rmin{µa(x3),µb(y3)}} = rmin{(µa ×µb)(x2,y2), (µa ×µb)(x3,y3)} and (γa ×γb)(((x2,y2) · ((x3,y3) · (x1,y1))) · (x1,y1)) = (γa ×γb)(((x2 · (x3 ·x1)) ·x1)), (y2 · (y3 ·y1)) ·y1)) = rmax{γa((x2 · (x3 ·x1)) ·x1),γb((y2 · (y3 ·y1)) ·y1)} ≤ rmax{rmax{γa(x2),γa(x3)}, rmax{γb(y2),γb(y3)}} = rmax{rmax{γa(x2),γb(y2)}, rmax{γa(x3),γb(y3)}} = rmax{(γa ×γb)(x2,y2), (γa ×γb)(x3,y3)}. hence, a×b is an ivif ideal of x ×y . � lemma 5.3. if a = (µa,γa) and b = (µb,γb) are two ivif ideals of hilbert algebras x and y , respectively, then ⊕(a×b) = (µa ×µb,µa ×µb) is an ivif ideal of x ×y . proof. it is sufficient to prove only the part of µa ×µb. let (x,y) ∈ x ×y . then (µa ×µb)(1, 1) = rmax{µa(1),µb(1)} = rmax{1 −µa(1), 1 −µb(1)} ≤ rmax{1 −µa(x), 1 −µb(y)} = (µa ×µb)(x,y). let (x1,x2), (y1,y2) ∈ x ×y . then (µa ×µb)((x1,x2) · (y1,y2)) = (µa ×µb)((x1 ·y1), (x2 ·y2)) = rmax{µa(x1 ·y1),µb(x2 ·y2)} = rmax{1 −µa(x1 ·y1), 1 −µb(x2 ·y2)} ≤ rmax{1 −µa(y1), 1 −µb(y2)} = rmax{µa(y1),µb(y2)} = (µa ×µb)(y1,y2). int. j. anal. appl. (2022), 20:25 21 let (x1,y1), (x2,y2), (x3,y3) ∈ x ×y . then (µa ×µb)(((x2,y2) · ((x3,y3) · (x1,y1))) · (x1,y1)) = (µa ×µb)(((x2 · (x3 ·x1)) ·x1)), (y2 · (y3 ·y1)) ·y1)) = rmax{µa((x2 · (x3 ·x1)) ·x1),µa((y2 · (y3 ·y1)) ·y1)} = rmax{1 −µa((x2 · (x3 ·x1)) ·x1), 1 −µa((y2 · (y3 ·y1)) ·y1)} ≤ rmax{1 − rmin{µa(x2),µa(x3)}, 1 − rmin{µb(y2),µb(y3)}} = rmax{rmax{1 −µa(x2), 1 −µb(y2)}, rmax{1 −µa(x3), 1 −µb(y3)}} = rmax{rmax{µa(x2),µb(y2)}, rmax{µa(x3),µb(y3)}} = rmax{(µa ×µb)(x2,y2), (µa ×µb)(x3,y3)}. hence, ⊕(a×b) is an ivif ideal of x ×y . � lemma 5.4. if a = (µa,γa) and b = (µb,γb) are two ivif ideals of hilbert algebras x and y , respectively, then ⊗(a×b) = (γa ×γb,γa ×γb) is an ivif ideal of x ×y . proof. it is sufficient to prove only the part of γa ×γb. let (x,y) ∈ x ×y . then (γa ×γb)(1, 1) = rmin{γa(1),γb(1)} = rmin{1 −γa(1), 1 −γb(1)} ≥ rmin{1 −γa(x), 1 −γb(y)} = (γa ×γb)(x,y). let (x1,x2), (y1,y2) ∈ x ×y . then (γa ×γb)((x1,x2) · (y1,y2)) = (γa ×γb)((x1 ·y1), (x2 ·y2)) = rmin{γa(x1 ·y1),γb(x2 ·y2)} = rmin{1 −γa(x1 ·y1), 1 −γb(x2 ·y2)} ≥ rmin{1 −γa(y1), 1 −γb(y2)} = rmin{γa(y1),γb(y2)} = (γa ×γb)(y1,y2). let (x1,y1), (x2,y2), (x3,y3) ∈ x ×y . then (γa ×γb)(((x2,y2) · ((x3,y3) · (x1,y1))) · (x1,y1)) = (γa ×γb)(((x2 · (x3 ·x1)) ·x1), ((y2 · (y3 ·y1)) ·y1)) = rmin{γa((x2 · (x3 ·x1)) ·x1),γa((y2 · (y3 ·y1)) ·y1)} = rmin{1 −γa((x2 · (x3 ·x1)) ·x1), 1 −γa((y2 · (y3 ·y1)) ·y1)} ≥ rmin{1 − rmax{γa(x2),γa(x3)}, 1 − rmax{γb(y2),γb(y3)}} = rmin{rmin{1 −γa(x2), 1 −γb(y2)}, rmin{1 −γa(x3), 1 −γb(y3)}} = rmin{rmin{γa(x2),γb(y2)}, rmin{γa(x3),γb(y3)}} = rmin{(γa ×γb)(x2,y2), (γa ×γb)(x3,y3)}. hence, ⊗(a×b) is an ivif subalgebra of x ×y . � 22 int. j. anal. appl. (2022), 20:25 theorem 5.2. the ivif sets a = (µa,γa) and b = (µb,γb) are ivif ideals of hilbert algebras x and y , respectively if and only if ⊕(a×b) and ⊗(a×b) are ivif ideals of x ×y . proof. it follows from lemmas 5.3 and 5.4. � theorem 5.3. let a = (µa,γa) and b = (µb,γb) be any two ivif sets in hilbert algebras x and y , respectively. if a × b is an ivif subalgebra of x × y , then nonempty upper [s1,s2]-level cut u(µa × µb : [s1,s2]) and nonempty lower [t1,t2]-level cut l(γa × γb : [t1,t2]) are subalgebras of x ×y for all [s1,s2], [t1,t2] ∈ d[0, 1]. proof. it follows from theorem 3.7. � theorem 5.4. let a = (µa,γa) and b = (µb,γb) be any two ivif sets in hilbert algebras x and y , respectively. if a × b is an ivif ideal of x × y , then nonempty upper [s1,s2]-level cut u(µa×µb : [s1,s2]) and nonempty lower [t1,t2]-level cut l(γa×γb : [t1,t2]) are ideals of x×y for all [s1,s2], [t1,t2] ∈ d[0, 1]. proof. it follows from theorem 4.4. � a mapping f : x → y of hilbert algebras is called a homomorphism if f (x · y) = f (x) · f (y) for all x,y ∈ x. note that if f : x → y is a homomorphism of hilbert algebras, then f (1) = 1. let f : x → y be a homomorphism of hilbert algebras. for any ivif set a = (µa,γa) in y , we define an ivif set f−1(a) = (µf−1(a),γf−1(a)) in x by µf−1(a)(x) = µa(f (x)) and γf−1(a)(x) = γa(f (x)) ∀x ∈ x. proposition 5.3. let f : x → y be a homomorphism of a hilbert algebra x into a hilbert algebra y and a = (µa,γa) an ivif subalgebra of y . then the inverse image f−1(a) of a is an ivif subalgebra of x. proof. let x,y ∈ x. then µf−1(a)(x ·y) = µa(f (x ·y)) = µa(f (x) · f (y)) ≥ rmin{µa(f (x)),µa(f (y))} = rmin{µf−1(a)(x),µf−1(a)(y)} and γf−1(a)(x ·y) = γa(f (x ·y)) = γa(f (x) · f (y)) ≤ rmax{γa(f (x)),γa(f (y))} = rmax{γf−1(a)(x),γf−1(a)(y)}. hence, f−1(a) of a is an ivif subalgebra of x. � int. j. anal. appl. (2022), 20:25 23 theorem 5.5. let f : x → y be a homomorphism of a hilbert algebra x into a hilbert algebra y and a = (µa,γa) be an ivif ideal of y . then the inverse image f−1(a) = (µf−1(a),γf−1(a)) is an ivif ideal of x. proof. since f is a homomorphism of x into y , then f (1) = 1 ∈ y and, by the assumption, µa(f (1)) = µa(1) ≥ µa(y) for every y ∈ y . in particular, µa(f (1)) ≥ µa(f (x)) for all x ∈ x. hence, µf−1(a)(1) ≥ µf−1(a)(x) for all x ∈ x. also, γa(f (1)) = γa(1) ≤ γa(y) for every y ∈ y . in particular, γb(f (1)) ≤ γb(f (x)) for all x ∈ x. hence, γf−1(a)(1) ≤ γf−1(a)(x) for all x ∈ x, which proves (4.1). now let x,y ∈ x. then, by the assumption, µf−1(a)(x ·y) = µa(f (x ·y)) = µa(f (x) · f (y)) ≥ µa(f (y)) = µf−1(a)(y) and γf−1(a)(x ·y) = γa(f (x ·y)) = γa(f (x) · f (y)) ≤ γa(f (y)) = γf−1(a)(y), which proves (4.2). let x,y1,y2 ∈ x. then by assumption, µf−1(a)((y1 · (y2 ·x)) ·x) = µa(f (y1 · (y2 ·x) ·x)) = µa(f (y1) · (f (y2 ·x)) · f (x)) = µa(f (y1 · (y2 ·x)) · f (x)) = µa(f (y1 · (y2 ·x)) ·x) ≥ rmin{µa(f (y1)),µa(f (y2))} = rmin{µf−1(a)(y1),µf−1(a)(y2)} and γf−1(a)((y1 · (y2 ·x)) ·x) = γa(f (y1 · (y2 ·x) ·x)) = γa(f (y1) · (f (y2 ·x)) · f (x)) = γa(f (y1 · (y2 ·x)) · f (x)) = γa(f (y1 · (y2 ·x)) ·x)) ≤ rmax{γa(f (y1)),γa(f (y2))} = rmax{γf−1(a)(y1),γf−1(a)(y2)}, which proves (4.3). hence, f−1(a) is an ivif ideal of x. � 6. equivalence relations on ivif subalgebras/ideals let i (h) be the family of all ivif ideals of a hilbert algebra x and let t = [t1,t2] ∈ d[0, 1]. define binary relations ut and lt on i (h) as follows: (a,b) ∈ ut ⇔ u(µa : t) = u(µb : t), (a,b) ∈ lt ⇔ l(γa : t) = l(γb : t), respectively, for a = (µa,γa) and b = (µb,γb) in i (h). then clearly ut and lt are equivalence relations on i (h). for any a = (µa,γa) ∈ i (h), let [a]ut (resp., [a]lt) denote the equivalence class 24 int. j. anal. appl. (2022), 20:25 of a modulo ut (resp., lt), and denote by i (h)/ut (resp., i (h)/lt) the system of all equivalence classes modulo ut (resp., lt); so i (h)/ut = {[a]ut | a = (µa,γa) ∈ i (h)}, respectively, i (h)/lt = {[a]lt | a = (µa,γa) ∈ i (h)}. now let i(h) denote the family of all ideals of x and let t = [t1,t2] ∈ d[0, 1]. define maps ft and gt from i (h) to i(h) ∪{∅} by ft(a) = u(µa : t) and gt(a) = l(γa : t), respectively, for all a = (µa,γa) ∈ i (h). then ft and gt are clearly well defined. theorem 6.1. for any t = [t1,t2] ∈ d[0, 1], the maps ft and gt are surjective from i (h) to i(h) ∪{∅}. proof. let t = [t1,t2] ∈ d[0, 1]. note that 0 = (0,1) is in i (h), where 0 and 1 are ivif sets in x defined by 0(x) = [0, 0] and 1(x) = [1, 1] for all x ∈ x. obviously, ft(0) = u(0,t) = u([0, 0] : [t1,t2]) = ∅ = l([1, 1] : [t1,t2]) = l(1 : t) = gt(0). let g( 6= ∅) ∈ i(h). for g = (χg,χg) ∈ i (h), we have ft(g) = u(χg : t) = g and gt(g) = l(χg; t) = g. hence, ft and gt are surjective. � theorem 6.2. the quotient sets i (h)/ut and i (h)/lt are equipotent to i(h) ∪{∅} for every t = [t1,t2] ∈ d[0, 1]. proof. for t = [t1,t2] ∈ d[0, 1], let f ∗t (resp., g∗t ) be a map from i (h)/ut (resp., i (h)/lt) to i(h) ∪{∅} defined by f ∗t ([a]ut ) = ft(a) (resp., g∗t ([a]lt ) = gt(a)) for all a = (µa,γa) ∈ i (h). if u(µa : t) = u(µb : t) and l(γa : t) = l(γb : t) for a = (µa,γa) and b = (µb,γb) ∈ i (h), then (a,b) ∈ ut and (a,b) ∈ lt; hence, [a]ut = [b]ut and [a]lt = [b]lt. therefore, the maps f ∗t and g∗t are injective. now let g( 6= ∅) ∈ i(h). for g = (χg,χg) ∈ i (h), we have f ∗t ([g]ut = ft(g) = u(χg : t) = g, g∗t ([g]lt = gt(g) = l(χg,t) = g. finally, for 0 = (0,1) ∈ i (h), we get f ∗t ([0]ut = ft(0) = u(0,t) = ∅, g∗t ([0]lt = gt(0) = l(1,t) = ∅. this shows that f ∗t and g ∗ t are surjective. � for any t = [t1,t2] ∈ d[0, 1], we define another relation rt on i (h) as follows: (a,b) ∈ rt ⇔ u(µa : t) ∩l(γa : t) = u(µb : t) ∩l(γb : t) for any a = (µa,γa),b = (µb,γb) ∈ i (h). then the relation rt is also an equivalence relation on i (h). int. j. anal. appl. (2022), 20:25 25 theorem 6.3. for any t = [t1,t2] ∈ d[0, 1], the map ϕt : i (h) → i(h) ∪ {∅} is defined by ϕt(a) = ft(a) ∩gt(a) for each a = (µa,γa) ∈ i (h) as surjective. proof. let t = [t1,t2] ∈ d[0, 1]. for 0 = (0,1) ∈ i (h), ϕt(0) = ft(0) ∩gt(0) = u(0,t) ∩l(0,t) = ∅. for any h ∈ i (h), there exists h = (χh,χh) ∈ i (h) such that ϕt(h) = ft(h) ∩gt(h) = u(χh : t) ∩l(χh,t) = h. hence, ϕt is surjective. � theorem 6.4. for any t = [t1,t2] ∈ d[0, 1], the quotient set i (h)/rt is equipotent to i(h) ∪{∅}. proof. let t = [t1,t2] ∈ d[0, 1] and let ϕ∗t : i (h)/rt → i(h)∪{∅} be a map defined by ϕ∗t([a]rt ) = ϕt(a) for all [a]rt ∈ i (h)/rt. if ϕ∗t([a]rt ) = ϕ∗t([b]rt ) for any [a]rt, [b]rt ∈ i (h)/rt, then ft(a) ∩ gt(a) = ft(b) ∩ gt(b), that is, u(µa : t) ∩ l(γa : t) = u(µb : t) ∩ l(γb : t); hence, (a,b) ∈ rt. it follows that [a]rt = [b]rt so that ϕ∗t is injective. for 0 = (0,1) ∈ i (h), ϕ∗t([0]rt ) = ϕt(0) = ft(0) ∩gt(0) = u(0,t) ∩l(1,t) = ∅. if h ∈ i (h), then for h = (χh,χh) ∈ i (h), we have ϕ∗t([h]rt ) = ϕt(h) = ft(h) ∩gt(h) = u(χh : t) ∩l(χh,t) = h. hence, ϕ∗t is surjective, this completes the proof. � the same type of results are also true for ivif subalgebras. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. ahmad, a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [2] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87–96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [3] m. atef, m.i. ali, t.m. al-shami, fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, comput. appl. math. 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x. [4] n. caǧman, s. enginoǧlu, f. citak, fuzzy soft set theory and its application, iran. j. fuzzy syst. 8 (2011), 137-147. [5] w.a. dudek, y.b. jun, on fuzzy ideals in hilbert algebra, novi sad j. math. 29 (1999), 193-207. [6] h. garg, s. singh, a novel triangular interval type-2 intuitionistic fuzzy sets and their aggregation operators, iran. j. fuzzy syst. 15 (2018), 69-93. https://doi.org/10.22111/ijfs.2018.4159. https://doi.org/10.1155/2009/586507 https://doi.org/10.1155/2009/586507 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1007/s40314-021-01501-x https://doi.org/10.22111/ijfs.2018.4159 26 int. j. anal. appl. (2022), 20:25 [7] h. garg, k. kumar, an advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, soft comput. 22 (2018), 4959–4970. https: //doi.org/10.1007/s00500-018-3202-1. [8] h. garg, k. kumar, distance measures for connection number sets based on set pair analysis and its applications to decision-making process, appl intell. 48 (2018), 3346–3359. https://doi.org/10.1007/s10489-018-1152-z. [9] d. busneag, a note on deductive systems of a hilbert algebra, kobe j. math. 2 (1985), 29-35. [10] d. busneag, hilbert algebras of fractions and maximal hilbert algebras of quotients, kobe j. math. 5 (1988), 161-172. [11] a. diego, sur les algébres de hilbert, collection de logique math. ser. a (ed. hermann, paris) 21 (1966), 1-52. [12] w.a. dudek, on fuzzification in hilbert algebras, contrib. gen. algebra. 11 (1999), 77-83. [13] y.b. jun, deductive systems of hilbert algebras, math. japon. 43 (1996), 51-54. [14] i. chajda, r. halas, congruences and ideals in hilbert algebras, kyungpook math. j. 39 (1999), 429-429. [15] l.a. zadeh, fuzzy sets, inf. control. 8 (1965), 338-353. https://doi.org/10.1007/s00500-018-3202-1 https://doi.org/10.1007/s00500-018-3202-1 https://doi.org/10.1007/s10489-018-1152-z 1. introduction 2. preliminaries 3. ivif subalgebras of hilbert algebras 4. ivif ideals of hilbert algebras 5. product of ivif subalgebras/ideals in hilbert algebras 6. equivalence relations on ivif subalgebras/ideals references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 27-33 http://www.etamaths.com existence of solutions for a certain boundary value problem associated to a fourth order differential inclusion aurelian cernea1,2,∗ abstract. existence of solutions for a fourth order differential inclusion with cantilever boundary conditions is investigated. new results are obtained when the right hand side has convex or non convex values. 1. introduction fourth order differential equations are often used in engineering and physical problems. boundary value problems associated to fourth order differential equations appear in elasticity theory describing stationary states of the deflection of an elastic beam. the same equation can describe the ”effect of the shear” when investigating transverse vibrations. as a consequence there was an intensive development of the study of such problems. in the single valued case several results concerning existence, localization and multiplicity of solutions may be found in [3], [4], [5], [8], [10], [12] etc.. this paper is devoted to the following boundary value problem x(4) ∈ f(t,x), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0, (1.1) where f(., .) : [0, 1] ×rn →p(rn) is a set-valued map. the aim of our paper is to consider the more general framework of set-valued problems and to present three existence results for problem 1.1. our results are obtained under several hypotheses concerning the regularity of the set-valued map f and are based on a nonlinear alternative of lerayschauder type, on bressan-colombo selection theorem for lower semicontinuous set-valued maps with decomposable values and on kuratowski and ryll-nardzewski selection theorem. we mention that the methods used are rather known in the theory of differential inclusions, however their exposition in the framework of problem 1.1 is new. the paper is organized as follows: in section 2 we recall some preliminary facts that we need in the sequel, in section 3 we prove our results using fixed point techniques and in section 4 we provide a filippov type existence result. 2. preliminaries in this section we sum up some basic facts that we are going to use later. let (x,d) be a metric space with the corresponding norm |.| and let i ⊂ r be a compact interval. denote by l(i) the σ-algebra of all lebesgue measurable subsets of i, by p(x) the family of all nonempty subsets of x and by b(x) the family of all borel subsets of x. if a ⊂ i then χa(.) : i → {0, 1} denotes the characteristic function of a. for any subset a ⊂ x we denote by a the closure of a. recall that the pompeiu-hausdorff distance of the closed subsets a,b ⊂ x is defined by dh (a,b) = max{d∗(a,b),d∗(b,a)}, d∗(a,b) = sup{d(a,b); a ∈ a}, where d(x,b) = infy∈b d(x,y). as usual, we denote by c(i,x) the banach space of all continuous functions x(.) : i → x endowed with the norm |x(.)|c = supt∈i|x(t)| and by l1(i,x) the banach space of all (bochner) integrable functions x(.) : i → x endowed with the norm |x(.)|1 = ∫ i |x(t)|dt. received 4th december, 2016; accepted 6th march, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 34a60, 34b18. key words and phrases. differential inclusion; fixed point; decomposable set. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 27 28 cernea a subset d ⊂ l1(i,x) is said to be decomposable if for any u(·),v(·) ∈ d and any subset a ∈l(i) one has uχa + vχb ∈ d, where b = i\a. consider t : x → p(x) a set-valued map. a point x ∈ x is called a fixed point for t(.) if x ∈ t(x). t(.) is said to be bounded on bounded sets if t(b) := ∪x∈bt(x) is a bounded subset of x for all bounded sets b in x. t(.) is said to be compact if t(b) is relatively compact for any bounded sets b in x. t(.) is said to be totally compact if t(x) is a compact subset of x. t(.) is said to be upper semicontinuous if for any x0 ∈ x, t(x0) is a nonempty closed subset of x and if for each open set d of x containing t(x0) there exists an open neighborhood v0 of x0 such that t(v0) ⊂ d. let e a banach space, y ⊂ e a nonempty closed subset and t(.) : y → p(e) a multifunction with nonempty closed values. t(.) is said to be lower semicontinuous if for any open subset d ⊂ e, the set {y ∈ y ; t(y) ∩d 6= ∅} is open. t(.) is called completely continuous if it is upper semicontinuous and totally compact on x. it is well known that a compact set-valued map t(.) with nonempty compact values is upper semicontinuous if and only if t(.) has a closed graph. we recall the following nonlinear alternative of leray-schauder type proved in [11] and its consequences. theorem 2.1. let d and d be the open and closed subsets in a normed linear space x such that 0 ∈ d and let t : d → p(x) be a completely continuous set-valued map with compact convex values. then either i) the inclusion x ∈ t(x) has a solution, or ii) there exists x ∈ ∂d (the boundary of d) such that λx ∈ t(x) for some λ > 1. corollary 2.1. let br(0) and br(0) be the open and closed balls in a normed linear space x centered at the origin and of radius r and let t : br(0) → p(x) be a completely continuous set-valued map with compact convex values. then either i) the inclusion x ∈ t(x) has a solution, or ii) there exists x ∈ x with |x| = r and λx ∈ t(x) for some λ > 1. corollary 2.2. let br(0) and br(0) be the open and closed balls in a normed linear space x centered at the origin and of radius r and let t : br(0) → x be a completely continuous single valued map with compact convex values. then either i) the equation x = t(x) has a solution, or ii) there exists x ∈ x with |x| = r and x = λt(x) for some λ < 1. if f(., .) : i × x → p(x) is a set-valued map with compact values we define sf : c(i,x) → p(l1(i,x)) by sf (x) := {f ∈ l1(i,x); f(t) ∈ f(t,x(t)) a.e. (i)}. we say that f(., .) is of lower semicontinuous type if sf (.) is lower semicontinuous with nonempty closed and decomposable values. the next result is proved in [2]. theorem 2.2. let s be a separable metric space and g(.) : s →p(l1(i,x)) be a lower semicontinuous set-valued map with closed decomposable values. then g(.) has a continuous selection (i.e., there exists a continuous mapping g(.) : s → l1(i,x) such that g(s) ∈ g(s) ∀s ∈ s). a set-valued map g : i →p(x) with nonempty compact convex values is said to be measurable if for any x ∈ x the function t → d(x,g(t)) is measurable. a set-valued map f(., .) : i × x → p(x) is said to be carathéodory if t → f(t,x) is measurable for any x ∈ x and x → f(t,x) is upper semicontinuous for almost all t ∈ i. moreover, f(., .) is said to be l1-carathéodory if for any l > 0 there exists hl(.) ∈ l1(i,r) such that sup{|v|; v ∈ f(t,x)}≤ hl(t) a.e. (i), ∀x ∈ bl(0). theorem 2.3. let x be a banach space, let f(., .) : i ×x →p(x) be a l1-carathéodory set-valued map with sf (x) 6= ∅ for all x(.) ∈ c(i,x) and let γ : l1(i,x) → c(i,x) be a linear continuous mapping. then the set-valued map γ ◦sf : c(i,x) →p(c(i,x)) defined by (γ ◦sf )(x) = γ(sf (x)) a fourth order differential inclusion 29 has compact convex values and has a closed graph in c(i,x) ×c(i,x). the proof of theorem above may be found in [9], note that if dimx < ∞, and f(., .) is as in theorem 2.5, then sf (x) 6= ∅ for any x(.) ∈ c(i,x) (e.g., [9]). we recall also a selection result ( [1]) which is a version of the celebrated kuratowski and ryllnardzewski selection theorem. lemma 2.1. consider x a separable banach space, b is the closed unit ball in x, h : i →p(x) is a set-valued map with nonempty closed values and g : i → x,l : i → r+ are measurable functions. if h(t) ∩ (g(t) + l(t)b) 6= ∅ a.e.(i), then the set-valued map t → h(t) ∩ (g(t) + l(t)b) has a measurable selection. in what follows i = [0, 1], and v = {x ∈ w4,1(i,r); x(0) = x′(0) = x′′(1) = x′′′(1) = 0} with the norm ||x||v = ||x(4)||1. by a solution of problem (1.1) we mean a function x(.) ∈ v for which there exists a function f(.) ∈ l1(i,r) with f(t) ∈ f(t,x(t)), a.e. (i) such that x(4)(t) = f(t) a.e. (i). the next technical result is proved in [3]. lemma 2.2. if f(.) : [0, 1] → r is an integrable function, then the solution of the boundary value problem x(4) = f(t), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0 is given by x(t) = ∫ 1 0 g(t,s)f(s)ds, where g(t,s) := { s2 6 (3t−s), if 0 ≤ s < t ≤ 1, t2 6 (3s− t), if 0 ≤ t < s ≤ 1 obviously, |g(t,s)| ≤ 1 2 ∀t.s ∈ i. 3. existence via fixed points we are able now to present the two existence results for problem (1.1) using fixed point techniques. we consider first the case when f(., .) is convex valued. hypothesis h1. i) f(., .) : i×r →p(r) has nonempty compact convex values and is carathéodory. ii) there exist ϕ(.) ∈ l1(i,r) with ϕ(t) > 0 a.e. (i) and there exists a nondecreasing function ψ : [0,∞) → (0,∞) such that sup{|v|; v ∈ f(t,x)}≤ ϕ(t)ψ(|x|) a.e. (i), ∀x ∈ r. theorem 3.1. assume that hypothesis h1 is satisfied and there exists r > 0 such that r > 1 2 |ϕ|1ψ(r). (3.1) then problem 1.1 has at least one solution x(.) such that |x(.)|c < r. proof. let x = w4,1(i,r) and consider r > 0 as in 3.1. it is obvious that the existence of solutions to problem 1.1 reduces to the existence of the solutions of the integral inclusion x(t) ∈ ∫ 1 0 g(t,s)f(s,x(s))ds, t ∈ i. (3.2) consider the set-valued map t : br(0) →p(w4,1(i,r)) defined by t(x) := {v(.) ∈ w4,1(i,r); v(t) := ∫ 1 0 g(t,s)f(s)ds, f ∈ sf (x)}. (3.3) we show that t(.) satisfies the hypotheses of corollary 2.1. first, we show that t(x) ⊂ w4,1(i,r) is convex for any x ∈ w4,1(i,r). if vi ∈ t(x) then there exist fi ∈ sf (x) such that for any t ∈ i one has vi(t) = ∫ 1 0 g(t,s)fi(s)ds, i = 1, 2. 30 cernea let 0 ≤ α ≤ 1. then for any t ∈ i we have (αv1 + (1 −α)v2)(t) = ∫ 1 0 g(t,s)[αf1(s) + (1 −α)f2(s)]ds. the values of f(., .) are convex, thus sf (x) is a convex set and hence αf1 + (1 −α)f2 ∈ t(x). secondly, we show that t(.) is bounded on bounded sets of w4,1(i,r). let b ⊂ w4,1(i,r) be a bounded set. then there exist m > 0 such that |x|c ≤ m ∀x ∈ b. if v ∈ t(x) there exists f ∈ sf (x) such that v(t) = ∫ 1 0 g(t,s)f(s)ds. one may write for any t ∈ i |v(t)| ≤ ∫ 1 0 |g(t,s)|.|f(s)|ds ≤ ∫ 1 0 |g(t,s)|ϕ(s)ψ(|x(t)|)ds and therefore |v|c ≤ 12|ϕ|1ψ(m) ∀v ∈ t(x), i.e., t(b) is bounded. we show next that t(.) maps bounded sets into equi-continuous sets. let b ⊂ w4,1(i,r) be a bounded set as before and v ∈ t(x) for some x ∈ b. there exists f ∈ sf (x) such that v(t) = ∫ 1 0 g(t,s)f(s)ds. then for any t,τ ∈ i we have |v(t) −v(τ)| ≤ | ∫ 1 0 g(t,s)f(s)ds− ∫ 1 0 g(τ,s)f(s)ds| ≤ ∫ 1 0 |g(t,s) −g(τ,s)|.|f(s)|ds ≤ ∫ 1 0 |g(t,s) −g(τ,s)|ϕ(s)ψ(m)ds. it follows that |v(t) −v(τ)|→ 0 as t → τ. therefore, t(b) is an equi-continuous set in w4,1(i,r). we apply now arzela-ascoli’s theorem we deduce that t(.) is completely continuous on w4,1(i,r). in the next step of the proof we prove that t(.) has a closed graph. let xn ∈ w4,1(i,r) be a sequence such that xn → x∗ and vn ∈ t(xn) ∀n ∈ n such that vn → v∗. we prove that v∗ ∈ t(x∗). since vn ∈ t(xn), there exists fn ∈ sf (xn) such that vn(t) = ∫ 1 0 g(t,s)fn(s)ds. define γ : l 1(i,r) → w4,1(i,r) by (γ(f))(t) := ∫ 1 0 g(t,s)f(s)ds. one has maxt∈i |vn(t) −v∗(t)| = |vn(.) −v∗(.)|c → 0 as n →∞ we apply theorem 2.3 to find that γ ◦ sf has closed graph and from the definition of γ we get vn ∈ γ ◦ sf (xn). since xn → x∗, vn → v∗ it follows the existence of f∗ ∈ sf (x∗) such that v∗(t) = ∫ 1 0 g(t,s)f∗(s)ds. therefore, t(.) is upper semicontinuous and compact on br(0). we apply corollary 2.1 to deduce that either i) the inclusion x ∈ t(x) has a solution in br(0), or ii) there exists x ∈ x with |x|c = r and λx ∈ t(x) for some λ > 1. assume that ii) is true. with the same arguments as in the second step of our proof we get r = |x(.)|c ≤ 12|ϕ|1ψ(r) which contradicts 3.1. hence only i) is valid and theorem is proved. � we consider now the case when f(., .) is not necessarily convex valued. our existence result in this case is based on the leray-schauder alternative for single valued maps and on bressan colombo selection theorem. hypothesis h2. i) f(., .) : i × r → p(r) has compact values, f(., .) is l(i) ⊗b(r) measurable and x → f(t,x) is lower semicontinuous for almost all t ∈ i. ii) there exist ϕ(.) ∈ l1(i,r) with ϕ(t) > 0 a.e. (i) and there exists a nondecreasing function ψ : [0,∞) → (0,∞) such that sup{|v|; v ∈ f(t,x)}≤ ϕ(t)ψ(|x|) a.e. (i), ∀x ∈ r. theorem 3.2. assume that hypothesis h2 is satisfied and there exists r > 0 such that condition 3.1 is satisfied. then problem 1.1 has at least one solution on i. proof. we note first that if hypothesis h2 is satisfied then f(., .) is of lower semicontinuous type (e.g., [7]). therefore, we apply theorem 2.2 with s = w4,1(i,r) and g(.) = sf (.) to deduce that there a fourth order differential inclusion 31 exists a continuous mapping f(.) : w4,1(i,r) → l1(i,r) such that f(x) ∈ sf (x) ∀x ∈ w4,1(i,r). we consider the corresponding problem x(t) = ∫ 1 0 g(t,s)f(x(s))ds, t ∈ i (3.4) in the space x = w4,1(i,r). it is clear that if x(.) ∈ w4,1(i,r) is a solution of the problem (3.4) then x(.) is a solution to problem 1.1. let r > 0 that satisfies condition 3.1 and define the set-valued map t : br(0) →p(w4,1(i,r)) by (t(x))(t) := ∫ 1 0 g(t,s)f(x(s))ds. obviously, the integral equation 3.4 is equivalent with the operator equation x(t) = (t(x))(t), t ∈ i. it remains to show that t(.) satisfies the hypotheses of corollary 2.2. we show that t(.) is continuous on br(0). from hypotheses h2 ii) we have |f(x(t))| ≤ ϕ(t)ψ(|x(t)|) a.e. (i) for all x(.) ∈ w4,1(i,r). let xn,x ∈ br(0) such that xn → x. then |f(xn(t))| ≤ ϕ(t)ψ(r) a.e. (i). from lebesgue’s dominated convergence theorem and the continuity of f(.) we obtain, for all t ∈ i lim n→∞ (t(xn))(t) = ∫ 1 0 g(t,s)f(xn(s))ds = ∫ 1 0 g(t,s)f(x(s))ds = (t(x))(t) i.e., t(.) is continuous on br(0). repeating the arguments in the proof of theorem 3.1 with corresponding modifications it follows that t(.) is compact on br(0). we apply corollary 2.2 and we find that either i) the equation x = t(x) has a solution in br(0), or ii) there exists x ∈ x with |x|c = r and x = λt(x) for some λ < 1. as in the proof of theorem 3.1 if the statement ii) holds true, then we obtain a contradiction to 3.1. thus only the statement i) is true and problem 1.1 has a solution x(.) ∈ w4,1(i,r) with |x(.)|c < r � 4. a filippov type existence result in this section we consider the, even, more general problem x(4) ∈ f(t,x,v (x)(t)), a.e. ([0, 1]), x(0) = x′(0) = x′′(1) = x′′′(1) = 0, (4.1) where f : [0, 1] × r × r → p(r) is a set-valued map, v : c([0, 1],r) → c([0, 1],r) is a nonlinear volterra integral operator defined by v (x)(t) = ∫ t 0 k(t,s,x(s))ds with k(., ., .) : [0, 1] × r × r → r a given function. we show that filippov’s ideas ( [6]) can be suitably adapted in order to obtain the existence of solutions for problem 4.1. in order to prove our results we need the following hypotheses. hypothesis h3. i) f(., .) : i ×r×r →p(r) has nonempty closed values and is l(i) ⊗b(r×r) measurable. ii) there exists l(.) ∈ l1(i, (0,∞)) such that, for almost all t ∈ i,f(t, ., .) is l(t)-lipschitz in the sense that dh (f(t,x1,y1),f(t,x2,y2)) ≤ l(t)(|x1 −x2| + |y1 −y2|) ∀ x1,x2,y1,y2 ∈ r. iii) k(., ., .) : i ×r×r → r is a function such that ∀x ∈ r, (t,s) → k(t,s,x) is measurable. iv) |k(t,s,x) −k(t,s,y)| ≤ l(t)|x−y| a.e. (t,s) ∈ i × i, ∀x,y ∈ r. we use next the following notations m(t) := l(t)(1 + ∫ t 0 l(u)du), t ∈ i, m0 = ∫ 1 0 m(t)dt. 32 cernea theorem 4.1. assume that hypothesis h3 is satisfied and m0 < 2. let y(.) ∈ c(i,r) be such that y(0) = y′(0) = y′′(1) = y′′′(1) = 0 and there exists p(.) ∈ l1(i,r+) with d(y(4)(t),f(t,y(t),v (y)(t))) ≤ p(t) a.e. (i). then there exists x(.) a solution of problem 4.1 satisfying for all t ∈ i |x(t) −y(t)| ≤ 1 2 −m0 ∫ 1 0 p(t)dt. proof. the set-valued map t → f(t,y(t),v (y)(t)) is measurable with closed values and f(t,y(t), v (y)(t)) ∩{y(4)(t) + p(t)[−1, 1]} 6= ∅ a.e. (i). it follows from lemma 2.1 that there exists a measurable selection f1(t) ∈ f(t,y(t),v (y)(t)) a.e. (i) such that |f1(t) −y(4)(t)| ≤ p(t) a.e. (i) (4.2) define x1(t) = ∫ 1 0 g(t,s)f1(s)ds and one has |x1(t) −y(t)| ≤ 12 ∫ 1 0 p(t)dt. we claim that it is enough to construct the sequences xn(.) ∈ c(i,r), fn(.) ∈ l1(i,r), n ≥ 1 with the following properties xn(t) = ∫ 1 0 g(t,s)fn(s)ds, t ∈ i, (4.3) fn(t) ∈ f(t,xn−1(t),v (xn−1)(t)) a.e. (i), (4.4) |fn+1(t) −fn(t)| ≤ l(t)(|xn(t) −xn−1(t)| + ∫ t 0 l(s)|xn(s) −xn−1(s)|ds) a.e. (i) (4.5) if this construction is realized then from 4.2-4.5 we have for almost all t ∈ i |xn+1(t) −xn(t)| ≤ 1 2 ( m0 2 )n ∫ 1 0 p(t)dt ∀n ∈ n. indeed, assume that the last inequality is true for n− 1 and we prove it for n. one has |xn+1(t) −xn(t)| ≤ ∫ 1 0 |g(t,t1)|.|fn+1(t1) −fn(t1)|dt1 ≤ 1 2 ∫ 1 0 l(t1)[|xn(t1) −xn−1(t1)| + ∫ t1 0 l(s)|xn(s) −xn−1(s)|ds]dt1 ≤ 1 2∫ 1 0 l(t1)(1 + ∫ t1 0 l(s)ds)dt1.( 1 2 )nmn−10 ∫ 1 0 p(t)dt = 1 2 ( m0 2 )n ∫ 1 0 p(t)dt therefore {xn(.)} is a cauchy sequence in the banach space c(i,r), hence converging uniformly to some x(.) ∈ c(i,r). therefore, by 4.5, for almost all t ∈ i, the sequence {fn(t)} is cauchy in r. let f(.) be the pointwise limit of fn(.). moreover, one has |xn(t) −y(t)| ≤ |x1(t) −y(t)| + ∑n−1 i=1 |xi+1(t) −xi(t)| ≤ 1 2 ∫ 1 0 p(t)dt + ∑n−1 i=1 ( 1 2 ∫ 1 0 p(t)dt)( m0 2 )i = 1 2 ∫ 1 0 p(t)dt 1−m0 2 . (4.6) on the other hand, from 4.2, 4.5 and 4.6 we obtain for almost all t ∈ i |fn(t) −y(4)(t)| ≤ n−1∑ i=1 |fi+1(t) −fi(t)| + |f1(t) −d q cy(t)| ≤ l(t) ∫ 1 0 p(t)dt 2 −m0 + p(t). hence the sequence fn(.) is integrably bounded and therefore f(.) ∈ l1(i,r). using lebesgue’s dominated convergence theorem and taking the limit in 4.2, 4.4 we deduce that x(.) is a solution of 1.1. finally, passing to the limit in 4.6 we obtained the desired estimate on x(.). it remains to construct the sequences xn(.),fn(.) with the properties in 4.2-4.5. the construction will be done by induction. since the first step is already realized, assume that for some n ≥ 1 we already constructed xn(.) ∈ c(i,r) and fn(.) ∈ l1(i,r), n = 1, 2, ...n satisfying 4.2, 4.5 for n = 1, 2, ...n and 4.4 for n = 1, 2, ...n − 1. the set-valued map t → f(t,xn (t),v (xn )(t)) is measurable. moreover, the map a fourth order differential inclusion 33 t → l(t)(|xn (t) − xn−1(t)| + ∫ t 0 l(s)|xn (s) − xn−1(s)|ds) is measurable. by the lipschitzianity of f(t, .) we have that for almost all t ∈ i f(t,xn (t)) ∩{fn (t) + l(t)(|xn (t) −xn−1(t)| + ∫ t 0 l(s)|xn (s) −xn−1(s)|ds)[−1, 1]} 6= ∅. lemma 2.1 yields that there exist a measurable selection fn+1(.) of f(.,xn (.),v (xn )(.)) such that for almost all t ∈ i |fn+1(t) −fn (t)| ≤ l(t)(|xn (t) −xn−1(t)| + ∫ t 0 l(s)|xn (s) −xn−1(s)|ds). we define xn+1(.) as in 4.2 with n = n + 1. thus fn+1(.) satisfies 4.4 and 4.5 and the proof is complete. � the assumptions in theorem 4.1 are satisfied, in particular, for y(.) = 0 and therefore with p(.) = l(.). we obtain the following consequence of theorem 4.1. corollary 4.1. assume that hypothesis h3 is satisfied, m0 < 2 and d(0,f(t, 0,v (0)(t)) ≤ l(t) a.e. (i). then there exists x(.) a solution of problem 4.1 satisfying for all t ∈ i, |x(t)| ≤ 1 2−m0 ∫ 1 0 l(t)dt. if f does not depend on the last variable, hypothesis h3 becames hypothesis h4. i) f(., .) : i×r →p(r) has nonempty closed values and is l(i)⊗b(r) measurable. ii) there exists l(.) ∈ l1(i, (0,∞)) such that, for almost all t ∈ i, f(t, .) is l(t)-lipschitz in the sense that dh (f(t,x1),f(t,x2)) ≤ l(t)|x1 −x2| ∀ x1,x2 ∈ r. denote l0 = ∫ 1 0 l(t)dt. corollary 4.2. assume that hypothesis h4 is satisfied, m0 < 2 and d(0,f(t, 0) ≤ l(t) a.e. (i). then there exists x(.) a solution of problem 1.1 satisfying for all t ∈ i |x(t)| ≤ l0 2 −l0 . references [1] j.p. aubin and h. frankowska, set-valued analysis, birkhäuser, basel, 1990. [2] a. bressan and g. colombo, extensions and selections of maps with decomposable values, studia math. 90 (1988), 69-86. [3] a. cabada, r. precup, l. saavedra and s.a. tersian, multiple positive solutions to a fourth-order boundary value problem, electronic j. differ. equ. 2016 (2016), art. 254. [4] j.a. cid, d. franco and f. minhos, positive fixed points and fourth-order equations, bull. lond. math. soc. 41 (2009), 72-78. [5] r. enguica and l. sanchez, existence and localization of solutions for fourth-order boundary value problems, electron. j. differ. equ. 2007 (2007), art. 127. [6] a.f. filippov, classical solutions of differential equations with multivalued right hand side, siam j. control 5 (1967), 609-621. [7] m. frignon and a. granas, théorèmes d’existence pour les inclusions différentielles sans convexité, c. r. acad. sci. paris, ser. i 310 (1990), 819-822. [8] j.r. graef, l. kong, q. kong and b. yang, positive solutions to a fourth order boundary value problem, results math. 59 (2011), 141-155. [9] a. lasota and z. opial, an application of the kakutani-ky-fan theorem in the theory of ordinary differential equations, bull. acad. polon. sci. math., astronom. physiques 13 (1965), 781-786. [10] f. li, q. zhang and z. liang, existence and multiplicity of solutions of a kind of fourth-order boundary value problem, nonlinear anal. 62 (2005), 803-816. [11] d. o’ regan, fixed point theory for closed multifunctions, arch. math. (brno) 34 (1998), 191-197. [12] l. yang, h. chen and x. yang, the multiplicity of solutions for fourth-order equations generated from a boundary condition, appl. math. lett. 24 (2011), 1599-1603. 1faculty of mathematics and computer science, university of bucharest, academiei 14, 010014 bucharest, romania 2academy of romanian scientists, splaiul independenţei 54, 050094 bucharest, romania ∗acernea@fmil.unibuc.ro 1. introduction 2. preliminaries 3. existence via fixed points 4. a filippov type existence result references international journal of analysis and applications volume 17, number 5 (2019), 892-903 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-892 hadamard and fejér-hadamard type inequalities for convex and relative convex functions via an extended generalized mittag-leffler function ghulam farid1, vishnu narayan mishra2,3,∗, sajid mehmood4 1department of mathematics, comsats university islamabad, attock campus, pakistan 2department of mathematics, indira gandhi national tribal university, lalpur, amarkantak-484 887, madhya pradesh, india 3l. 1627 awadh puri colony, beniganj, phase-iii, opposite industrial training institute (i.t.i.), ayodhya-224 001, uttar pradesh, india 4govt boys primary school sherani, hazro, attock, pakistan ∗corresponding author: vishnunarayanmishra@gmail.com abstract. in this paper, we will prove the hadamard and the fejér-hadamard type integral inequalities for convex and relative convex functions due to an extended generalized mittag-leffler function. these results contain several fractional integral inequalities for the well known fractional integral operators. 1. introduction convex functions are very useful in the field of mathematical inequalities. definition 1.1. let i be an interval of real numbers. then a function f : i → r is said to be convex function, if for all x,y ∈ i and 0 ≤ λ ≤ 1, the following inequality holds: f(xλ + (1 −λ)y) ≤ λf(x) + (1 −λ)f(y). received 2019-01-22; accepted 2019-04-09; published 2019-09-02. 2010 mathematics subject classification. 26b25, 26a51, 26a33, 33e12. key words and phrases. convex functions; hadamad inequality; fejér-hadamard inequality; mittag-leffler function; fractional integral operators. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 892 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-892 int. j. anal. appl. 17 (5) (2019) 893 convex functions are equivalently defined by the following inequality which is well known as the hadamard inequality: f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 where f : i → r is a convex function on i and a,b ∈ i,a < b. following definitions are given in [8]. definition 1.2. let tg be a set of real numbers. this set tg is said to be relative convex with respect to an arbitrary function g : r → r, if (1 − t)x + tg(y) ∈ tg where x,y ∈ r such that x,g(y) ∈ tg, 0 ≤ t ≤ 1. definition 1.3. a function f : tg → r is said to be relative convex, if there exists an arbitrary function g : r → r, such that f((1 − t)x + tg(y)) ≤ (1 − t)f(x) + tf(g(y)), holds, where x,y ∈ r such that x,g(y) ∈ tg, 0 ≤ t ≤ 1. noor et al. proved the following hadamard type integral inequality in [8] for relative convex functions via riemann-liouville fractional integral operators. theorem 1.1. let f be a positive relative convex function and integralable on [a,g(b)]. then for α > 0, the following inequalities hold: f ( a + g(b) 2 ) ≤ γ(α + 1) 2(g(b) −a)α [iαa+f(g(b)) + i α b− f(a)] ≤ f(a) + f(g(b)) 2 . now we define the extended generalized mittag-leffler function e γ,δ,k,c µ,α,l (.; p) as follows: definition 1.4. [2] let µ,α,l,γ,c ∈ c, <(µ),<(α),<(l) > 0, <(c) > <(γ) > 0 with p ≥ 0, δ > 0 and 0 < k ≤ δ + <(µ). then the extended generalized mittag-leffler function eγ,δ,k,cµ,α,l (t; p) is defined by: e γ,δ,k,c µ,α,l (t; p) = ∞∑ n=0 βp(γ + nk,c−γ) β(γ,c−γ) (c)nk γ(µn + α) tn (l)nδ , (1.1) where βp is the generalized beta function defined by: βp(x,y) = ∫ 1 0 tx−1(1 − t)y−1e− p t(1−t) dt and (c)nk is the pochhammer symbol defined as (c)nk = γ(c+nk) γ(c) . in [2], properties of the generalized mittag-leffler function are discussed and it is given that e γ,δ,k,c µ,α,l (t; p) is absolutely convergent for k < δ + <(µ). let s be the sum of series of absolute terms of the mittag-leffler int. j. anal. appl. 17 (5) (2019) 894 function e γ,δ,k,c µ,α,l (t; p), then we have ∣∣∣eγ,δ,k,cµ,α,l (t; p)∣∣∣ ≤ s. we use this property of mittag-leffler function to prove the results of this paper. remark 1.2. mittag-leffler function (1.1) is the generalization of following functions: (i) by setting p = 0, it reduces to the salim-faraj function e γ,δ,k,c µ,α,l (t) defined in [12]. (ii) by setting l = δ = 1, it reduces to the function eγ,k,cµ,α (t; p) defined by rahman et al. in [11]. (iii) by setting p = 0 and l = δ = 1, it reduces to the shukla-prajapati function eγ,kµ,α(t) defined in [13] (see also [14]). (iv) by setting p = 0 and l = δ = k = 1, it reduces to the prabhakar function eγµ,α(t) defined in [10]. the corresponding left and right sided generalized fractional integral operators � γ,δ,k,c µ,α,l,ω,a+ and � γ,δ,k,c µ,α,l,ω,b− are defined as follows: definition 1.5. [2] let ω,µ,α,l,γ,c ∈ c, <(µ),<(α),<(l) > 0, <(c) > <(γ) > 0 with p ≥ 0, δ > 0 and 0 < k ≤ δ + <(µ). let f ∈ l1[a,b] and x ∈ [a,b]. then the generalized fractional integral operators � γ,δ,k,c µ,α,l,ω,a+ f and � γ,δ,k,c µ,α,l,ω,b− f are defined by:( � γ,δ,k,c µ,α,l,ω,a+ f ) (x; p) = ∫ x a (x− t)α−1eγ,δ,k,cµ,α,l (ω(x− t) µ; p)f(t)dt, (1.2) and ( � γ,δ,k,c µ,α,l,ω,b− f ) (x; p) = ∫ b x (t−x)α−1eγ,δ,k,cµ,α,l (ω(t−x) µ; p)f(t)dt. (1.3) remark 1.3. operators in (1.2) and (1.3) are the generalization of the following fractional integral operators: (i) by setting p = 0, they reduce to the fractional integral operators defined by salim-faraj in [12]. (ii) by setting l = δ = 1, they reduce to the fractional integral operators defined by rahman et al. in [11]. (iii) by setting p = 0 and l = δ = 1, they reduce to the fractional integral operators defined by srivastavatomovski in [14]. (iv) by setting p = 0 and l = δ = k = 1, they reduce to the fractional integral operators defined by prabhakar in [10]. (v) by setting p = ω = 0, they reduces to the riemann-liouville fractional integrals. in [5] the hadamard and the fejér-hadamard inequalities for convex functions via generalized fractional integral operator containing the mittag-leffler function defined in [12] have been proved. in [1, 6, 8], the hadamard and the fejér-hadamard type inequalities for convex and relative convex functions via riemann-liouville fractional integral operators and extended generalized fractional integral operators have been proved. in this paper, we give fractional integral inequalities of the hadamard and the fejérhadamard type for convex and relative convex functions by using the extended generalized mittag-leffler function. we also produce the results which are given in [1, 6, 8] by setting particular values of parameters. int. j. anal. appl. 17 (5) (2019) 895 2. main results following lemmas are useful to establish the main results. lemma 2.1. let f : [a,b] → r be a function such that f ∈ l1[a,b] and symmetric about a+b2 with a < b. then for extended generalized fractional integral operators (1.2) and (1.3), the following equality holds: ( � γ,δ,k,c µ,α,l,ω,a+ f ) (b; p) = ( � γ,δ,k,c µ,α,l,ω,b− f ) (a; p) (2.1) = ( � γ,δ,k,c µ,α,l,ω,a+ f ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− f ) (a; p) 2 . proof. using symmetricity of f we have f(a + b− t) = f(t), therefore by (1.2) of definition 1.5, we have ( � γ,δ,k,c µ,α,l,ω,a+ f ) (b; p) = ∫ b a (b− t)α−1eγ,δ,k,cµ,α,l (ω(b− t) µ; p)f(t)dt, (2.2) putting t = a + b− t in above, we get ( � γ,δ,k,c µ,α,l,ω,a+ f ) (b; p) = ∫ b a (t−a)α−1eγ,δ,k,cµ,α,l (ω(t−a) µ; p)f(t)dt. by using definition 1.5, we get ( � γ,δ,k,c µ,α,l,ω,a+ f ) (b; p) = ( � γ,δ,k,c µ,α,l,ω,b− f ) (a; p). (2.3) therefore we get (2.1). � lemma 2.2. let f : [a,b] → r be a function such that f′ ∈ l1[a,b] with a < b. if g : [a,b] → r is integrable and symmetric about a+b 2 , then for extended generalized fractional integral operators (1.2) and (1.3), the following equality holds: ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] (2.4) − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ] = ∫ b a [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ] f′(t)dt. int. j. anal. appl. 17 (5) (2019) 896 proof. one can note that ∫ b a [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ] f′(t)dt = ∫ b a [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ] f′(t)dt + ∫ b a [ − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ] f′(t)dt. by simple calculation, we get ∫ b a [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ] f′(t)dt = f(b) (∫ b a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ) − ∫ b a ( (b− t)α−1eγ,δ,k,cµ,α,l (ω(b− t) µ; p ) fg(t)dt. by using definition 1.5, we get f(b) ( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) − ( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p). now by using lemma 2.1, we have ∫ b a [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ] f′(t)dt (2.5) = f(b) 2 [( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − ( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) and ∫ b a [ − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ] f′(t)dt (2.6) = f(a) 2 [( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p). by adding (2.6) and (2.5), we get (2.4). � in the following we give integral inequality of the hadamard type. theorem 2.3. let f : [a,b] → r be a differentiable function such that f′ ∈ l1[a,b] with a < b. if |f′| is convex on [a,b] and g : i → r is continuous and symmetric function about a+b 2 , then for extended generalized int. j. anal. appl. 17 (5) (2019) 897 fractional integral operators (1.2) and (1.3), the following inequality holds:∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ ‖ g ‖∞ s(b−a)α+1 α(α + 1) ( 1 − 1 2α ) [|f′(a) + f′(b)|], for k < δ + <(µ) and ‖ g ‖∞= sup t∈[a,b] |g(t)|. proof. by using lemma 2.2, we have∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] (2.7) − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ ∫ b a ∣∣∣∣ [∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ]∣∣∣∣∣ |f′(t)|dt. since |f′| is convex, so we have |f′(t)| ≤ b− t b−a |f′(a)| + t−a b−a |f′(b)| (2.8) where t ∈ [a,b]. from symmetricity of g, we have∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds = ∫ a+b−t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(a + b−s)ds = ∫ a+b−t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds. this gives ∣∣∣∣ ∫ t a (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds (2.9) − ∫ b t (s−a)α−1eγ,δ,k,cµ,α,l (ω(s−a) µ; p)g(s)ds ∣∣∣∣∣ = ∣∣∣∣∣ ∫ a+b−t t (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ∣∣∣∣∣ ≤   ∫a+b−t t |(b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)|ds,t ∈ [a, a+b 2 ]∫ t a+b−t |(b−s) α−1e γ,δ,k,c µ,α,l (ω(b−s) µ; p)g(s)|ds,t ∈ [a+b 2 ,b]. int. j. anal. appl. 17 (5) (2019) 898 from (2.7), (2.8), (2.9) and absolute convergence of mittag-leffler function, we get∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] (2.10) − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ ∫ a+b 2 a (∫ a+b−t a |(b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)|ds )( b− t b−a |f′(a)| + t−a b−a |f′(b)| ) dt + ∫ b a+b 2 (∫ t a+b−t |(b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)|ds ) × ( b− t b−a |f′(a)| + t−a b−a |f′(b)| ) dt ≤ ‖ g ‖∞ s α(b−a) [∫ a+b 2 a ((b− t)α − (t−a)α(b− t)|f′(a)|)dt + ∫ a+b 2 a ((b− t)α − (t−a)α(t−a)|f′(b)|)dt + ∫ b a+b 2 ((t−a)α − (b− t)α(b− t)|f′(a)|)dt + ∫ b a+b 2 ((t−a)α − (b− t)α(t−a)|f′(b)|)dt ] . as we have ∫ a+b 2 a ((b− t)α − (t−a)α) (b− t)dt = (b−a)α+2 α + 1 ( α + 1 α + 2 − 1 2α+1 ) and ∫ a+b 2 a ((b− t)α − (t−a)α) (t−a)dt = (b−a)α+2 α + 1 ( 1 α + 2 − 1 2α+1 ) . by using the values of above integrals in (2.10), we have∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ ‖ g ‖∞ s α(b−a) (b−a)α+2 α + 1 [( α + 1 α + 2 − 1 2α+1 ) + ( 1 α + 2 − 1 2α+1 )] [|f′(a)| + |f′(b)|] = ‖ g ‖∞ s α(α + 1) (b−a)α+1 ( 1 − 1 2α ) [|f′(a)| + |f′(b)|]. � remark 2.4. (i) if we put p = 0 in theorem 2.3, then we get [1, theorem 2.3]. (ii) if we put ω = p = 0 in theorem 2.3, then we get [6, theorem 2.6]. int. j. anal. appl. 17 (5) (2019) 899 theorem 2.5. let f : [a,b] → r be a differentiable function such that f′ ∈ l1[a,b] with a < b. if |f′|q, q ≥ 1 is convex on [a,b] and g : i → r is continuous and symmetric function about a+b 2 , then for extended generalized fractional integral operators (1.2) and (1.3), the following inequality holds: ∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] (2.11) − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ 2 ‖ g ‖∞ s(b−a)α+1 α(α + 1) ( 1 − 1 2α ) (|f′(a)|q + |f′(b)|q) 1 q , for k < δ + <(µ) and ‖ g ‖∞= sup t∈[a,b] |g(t)|. proof. by using lemma 2.2, power mean inequality, the inequality (2.9) takes the following form: ∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] (2.12) − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤ [∫ b a ∣∣∣∣∣ ∫ a+b−t t (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ∣∣∣∣∣dt ]1−1 q [∫ b a ∣∣∣∣∣ ∫ a+b−t t (b−s)α−1eγ,δ,k,cµ,α,l (ω(b−s) µ; p)g(s)ds ∣∣∣∣∣ |f′(t)|qdt ]1 q . using absolute convergence of mittag-leffler function and ‖ g ‖∞= sup t∈[a,b] |g(t)|, we have ∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤‖ g ‖ 1−1 q ∞ s 1−1 q [∫ a+b 2 a (∫ a+b−t t (b−s)α−1ds ) dt + ∫ b a+b 2 (∫ t a+b−t (b−s)α−1ds ) dt ]1−1 q ×‖ g ‖ 1 q ∞ s 1 q [∫ a+b 2 a (∫ a+b−t t (b−s)α−1ds ) |f′(t)|qdt + ∫ b a+b 2 (∫ t a+b−t (b−s)α−1ds ) |f′(t)|qdt ]1 q . int. j. anal. appl. 17 (5) (2019) 900 by some calculation, we have∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤‖ g ‖∞ s [ (b−a)α+1 α(α + 1) ( 1 − 1 2α ) + (b−a)α+1 α(α + 1) ( 1 − 1 2α )]1−1 q × [∫ a+b 2 a ((b− t)α − (t−a)α) |f′(t)|qdt + ∫ b a+b 2 ((b− t)α − (t−a)α) |f′(t)|qdt ]1 q . since |f′|q is convex, so we have |f′(t)|q ≤ b− t b−a |f′(a)|q + t−a b−a |f′(b)|q. (2.13) hence ∣∣∣∣ ( f(a) + f(b) 2 )[( � γ,δ,k,c µ,α,l,ω,a+ g ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− g ) (a; p) ] − [( � γ,δ,k,c µ,α,l,ω,a+ fg ) (b; p) + ( � γ,δ,k,c µ,α,l,ω,b− fg ) (a; p) ]∣∣∣ ≤‖ g ‖∞ s [ 2 (b−a)α+1 α(α + 1) ( 1 − 1 2α )]1−1 q × [∫ a+b 2 a ((b− t)α − (t−a)α) ( b− t b−a |f′(a)|q + t−a b−a |f′(b)|q ) dt + ∫ b a+b 2 ((b− t)α − (t−a)α) ( b− t b−a |f′(a)|q + t−a b−a |f′(b)|q ) dt ]1 q . from it (2.11) can be obtained. � remark 2.6. (i) if we put p = 0 in theorem 2.5, then we get [1, theorem 2.5]. (ii) if we put ω = p = 0 in theorem 2.5, then we get [6, theorem 2.8]. in the following we give the hadamard inequality for relative convex functions via generalized fractional integral operators. theorem 2.7. let f : [a,g(b)] → r be a function such that f ∈ l1[a,g(b)] with a < b. if f is relative convex on [a,g(b)], then for extended generalized fractional integral operators (1.2) and (1.3), the following inequalities hold: f ( a + g(b) 2 )( � γ,δ,k,c µ,α,l,ω′,a+ 1 ) (g(b); p) (2.14) ≤ 1 2 [( � γ,δ,k,c µ,α,l,ω′,a+ f ) (g(b); p) + ( � γ,δ,k,c µ,α,l,ω′,g(b)− f ) (a; p) ] ≤ f(a) + f(g(b)) 2 ( � γ,δ,k,c µ,α,l,ω′,g(b)− 1 ) (a; p), int. j. anal. appl. 17 (5) (2019) 901 where ω′ = ω (g(b)−a)µ . proof. since f is relative convex, so we have f ( a + g(b) 2 ) = f [( 1 2 (ta + (1 − t)g(b) ) + ( 1 − 1 2 ) ((1 − t)a + tg(b)) ] (2.15) ≤ 1 2 f (ta + (1 − t)g(b)) + 1 2 f ((1 − t)a + tg(b)) . multiplying (2.15) by 2tα−1e γ,δ,k,c µ,α,l (ωt µ; p) on both sides and then integrating over [0, 1], we have 2f ( a + g(b) 2 )∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)dt (2.16) ≤ ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)f (ta + (1 − t)g(b)) dt + ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)f ((1 − t)a + tg(b)) dt. putting x = ta + (1 − t)g(b) and y = (1 − t)a + tg(b) in above, we have 2f ( a + g(b) 2 )∫ a g(b) ( g(b) −x g(b) −a )α−1 e γ,δ,k,c µ,α,l ( ω ( g(b) −x g(b) −a )µ ; p )( −dx g(b) −a ) (2.17) ≤ ∫ a g(b) ( g(b) −x g(b) −a )α−1 e γ,δ,k,c µ,α,l ( ω ( g(b) −x g(b) −a )µ ; p ) f(x) ( −dx g(b) −a ) + ∫ g(b) a ( y −a g(b) −a )α−1 e γ,δ,k,c µ,α,l ( ω ( y −a g(b) −a )µ ; p ) f(y) ( dy g(b) −a ) . by using definition 1.5, we get 2f ( a + g(b) 2 )( � γ,δ,k,c µ,α,l,ω′,a+ 1 ) (g(b); p) (2.18) ≤ [( � γ,δ,k,c µ,α,l,ω′,a+ f ) (g(b); p) + ( � γ,δ,k,c µ,α,l,ω′,g(b)− f ) (a; p) ] . again by using the relative convexity of f, we have f(ta + (1 − t)g(b)) + f((1 − t)a + tg(b)) ≤ tf(a) + (1 − t)f(g(b)) + (1 − t)f(a) + tf(g(b)). (2.19) multiplying (2.19) by tα−1e γ,δ,k,c µ,α,l (ωt µ; p) on both sides and then integrating over [0, 1], we have ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)f(ta + (1 − t)g(b))dt + ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)f((1 − t)a + tg(b))dt ≤ ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)tf(a) + (1 − t)f(g(b))dt + ∫ 1 0 tα−1e γ,δ,k,c µ,α,l (ωt µ; p)(1 − t)f(a) + tf(g(b))dt. int. j. anal. appl. 17 (5) (2019) 902 putting x = ta + (1 − t)g(b) and y = (1 − t)a + tg(b) in above and then using definition 1.5, we get[( � γ,δ,k,c µ,α,l,ω′,a+ f ) (g(b); p) + ( � γ,δ,k,c µ,α,l,ω′,g(b)− f ) (a; p) ] (2.20) ≤ [f(a) + f(g(b))] ( � γ,δ,k,c µ,α,l,ω′,g(b)− 1 ) (a; p). combining it with (2.18), (2.14) is obtained. � remark 2.8. (i) if we put p = 0 in theorem 2.7, then we get [1, theorem 2.8]. (ii) if we put ω = p = 0 and k = 1 in theorem 2.7, then we get theorem 1.1. in the upcoming theorem we give the generalization of previous result. theorem 2.9. let f : [g(a),g(b)] → r be a function such that f ∈ l1[g(a),g(b)] with a < b. if f is relative convex on [g(a),g(b)], then for extended generalized fractional integral operators (1.2) and (1.3), the following inequalities hold: f ( g(a) + g(b) 2 )( � γ,δ,k,c µ,α,l,ω′,g(a)+ 1 ) (g(b); p) ≤ 1 2 [( � γ,δ,k,c µ,α,l,ω′,g(a)+ f ) (g(b); p) + ( � γ,δ,k µ,α,l,ω′,g(b)− f ) (a; p) ] ≤ f(g(a)) + f(g(b)) 2 ( � γ,δ,k,c µ,α,l,ω′,g(b)− 1 ) (g(a); p), where ω′ = ω (g(b)−g(a))µ . proof. proof of this theorem is on the same lines of the proof of theorem 2.7. � remark 2.10. (i) if we put p = 0 in theorem 2.9, then we get [1, theorem 2.10]. (ii) if we put ω = p = 0 in theorem 2.9, then we get [4, corollary 1]. acknowledgement the research work of ghulam farid is supported by higher education commission of pakistan under nrpu 2016, project no. 5421. references [1] g. abbas, g. farid, some integral inequalities of the hadamard and the fejér hadamard type via generalized fractional integral operator, j. nonlinear anal. optim. 8 (2018), 12pp. [2] m. andrić, g. farid, j. pečarić, a further extension of mittag-leffler function, fract. calc. appl. anal. 21(5) (2018), 1377-1395. [3] d. i. duca, l. lupa, saddle points for vector valued functios: existance, necessary and sufficient theorems, j. glob. optim. 53 (2012), 431–440. [4] g. farid, a. u. rehman, m. zahra, on hadamard inequalities for relative convex functions via fractional integrals, nonlinear anal. forum 21(1) (2016), 77-86. int. j. anal. appl. 17 (5) (2019) 903 [5] g. farid, hadamard and fejér-hadamard inequalities for generalized fractional integrals involving special functions, konuralp j. math. 4(1) (2016), 108–113. [6] i. iscan, hermite hadamard fejér type inequalities for convex functions via fractional integrals, stud. univ. babes-bolyai math. 60(3) (2015), 355–366. [7] m. a. noor, differential non-convex functions and general variational inequalities, appl. math. comput. 199(2) (2008), 623–630. [8] m. a. noor, k. i. noor, m. u. awan, generalized convexity and integral inequalities, appl. math. inf. sci. 9(1) (2015), 233-243. [9] j. pečarić, f. proschan, y. l. tong, convex funtions, partial orderings and statistical applications, academic press, new york, 1992. [10] t. r. prabhakar, a singular integral equation with a generalized mittag-leffler function in the kernel, yokohama math. j. 19 (1971), 7-15. [11] g. rahman, d. baleanu, m. a. qurashi, s. d. purohit, s. mubeen, m. arshad, the extended mittag-leffler function via fractional calculus, j. nonlinear sci. appl. 10 (2017), 4244-4253. [12] t. o. salim, a. w. faraj, a generalization of mittag-leffler function and integral operator associated with integral calculus, j. frac. calc. appl. 3(5) (2012), 1–13. [13] a. k. shukla, j. c. prajapati, on a generalization of mittag-leffler function and its properties, j. math. anal. appl. 336 (2007), 797-811. [14] h. m. srivastava, z. tomovski, fractional calculus with an integral operator containing generalized mittagleffler function in the kernel, appl. math. comput. 211(1) (2009), 198–210. 1. introduction 2. main results acknowledgement references n f  0 ,1,1, 2 , 3, 5, ... 0 0f  1 1f  1 2 , 2 k k k f p f q f k w i t h      0 1 ,f a f b  , , &p q a b , , &p q a b 1 , 2p q a b    1 2 0 1 2 2 2 , 2 k k k v v v f o r k w i t h v v          0k k v  1 1 1 2 1 2 2 k k k v          1 2 0 1 2 , 2 0 , 1 k k k j j j k w i t h j j        1 2 1 2 k k k j        1 2 0 1 2 , 2 2 , 1 k k k j j j k w i t h j j        1 2 k k k j     1 2 &  2 2 0x x   k v k j 2 1 2 2 1 4 1 1 2 , 0 & 0 k k p k k p p v j v v w h e r e k p             2 1 2 1 2 1 2 11 2 2 2 1 1 2 1 2 2 k p k p k k k p k v j                            4 2 4 2 2 1 1 2 1 2 1 2 1 2 1 2 2 2 k p k p k p p                          2 1 1 2 4 1 1 2 1 2 2 p p k k p v                  2 1 4 1 1 2 k k p p v v       2 2 1 4 1 ( ) 0 , : k k k i i f p t h e n v j v     1 2 1 2 1 4 2 ( ) 1 , : 4 k k k k i i i f p t h e n v j v        1 2 2 2 1 4 3 ( ) 2 , : 4 k k k k i i i i f p t h e n v j v        1 2 2 1 4 2 2 4 , 0 & 0 k k p k k p p v j j j w h e r e k p          1 2 2 2 4 2 2 4 , 0 & 0 k k p k k p p v j v v w h e r e k p           2 1 2 2 2 4 2 ( ) 0 , : 2 k k k k i i f p t h e n v j v       2 1 2 2 4 3 ( ) 1 , : k k k i i i f p t h e n v j v      2 3 2 2 2 2 4 4 ( ) 2 , : 2 k k k k i i i i f p t h e n v j v          1 2 2 2 4 2 1 2 4 , 0 & 0 k k p k k p p v j j j w h e r e k p           2 2 4 4 , 0 & 0 k k p k k p p v j v v w h e r e k p       2 1 2 2 4 ( ) 0 , : 2 k k k k i i f p t h e n v j v     2 1 2 1 2 4 1 ( ) 1 , : 2 k k k k i i i f p t h e n v j v         2 1 2 2 2 4 2 ( ) 2 , : 3 2 k k k k iii i f p th e n v j v        2 2 4 1 12 4 , 0 & 0 k k p k k p p v j j j w h e r e k p         2 1 2 2 1 4 1 1 2 , 0 & 0 k k p k k p p v j v v w h e r e k p            2 2 1 4 1 ( ) 0 , : k k k i i f p t h e n v j v     2 1 2 1 2 1 4 ( ) 1 , : 2 k k k k i i i f p t h e n v j v       2 2 2 1 4 1 ( ) 2 , : 4 k k k k i i i i f p t h e n v j v         2 1 2 2 1 4 2 2 2 , 0 & 0 k k p k k p p v j j j w h e r e k p           2 1 2 2 1 4 1 1 2 , 0 & 0 k k p k k p p v j v v w h e r e k p           2 2 1 4 1 ( ) 0 , : 4 k k k k i i f p t h e n v j v      2 1 2 1 4 2 ( ) 1 , : 4 k k k k i i i f p t h e n v j v       2 2 2 1 4 3 ( ) 2 , : k k k i i i i f p t h e n v j v        2 1 2 2 1 4 2 2 2 , 0 & 0 k k p k k p p v j j j w h e r e k p          2 2 4 4 , 0 & 0 k k p k k p p v j v v w h e r e k p        2 1 2 2 4 ( ) 0 , : 2 k k k k i i f p t h e n v j v     2 1 2 4 1 ( ) 1 , : k k k i i i f p t h e n v j v     2 2 2 4 2 ( ) 2 , : 4 k k k k i i i i f p t h e n v j v       2 2 4 1 12 4 , 0 & 0 k k p k k p p v j j j w h e r e k p         2 1 2 2 4 2 2 , 0 & 0 k k k p k p p v j v v w h e r e k p         2 1 2 2 4 ( ) 0 , : 2 k k k k i i f p t h e n v j v     2 2 1 4 1 ( ) 1 , : k k k i i i f p t h e n v j v     2 3 2 2 2 4 2 ( ) 2 , : 2 k k k k i i i i f p t h e n v j v         2 1 2 2 4 1 1 2 2 , 0 & 0 k k k p k p p v j j j w h e r e k p          international journal of analysis and applications volume 17, number 5 (2019), 809-820 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-809 bifurcation and stability analysis of a discrete time sir epidemic model with vaccination özlem ak gümüş1,∗, a. george maria selvam2 and d. abraham vianny2 1adıyaman university, faculty of arts and sciences, department of mathematics, 02040, adiyaman, turkey 2sacred heart college (autonomous), department of mathematics, tirupattur-635 601, vellore dt., tamil nadu, india ∗corresponding author: akgumus@adiyaman.edu.tr abstract. in this paper, we study the qualitative behavior of a discrete-time epidemic model with vaccination. analysis of the model shows forth that the disease free equilibrium (dfe) point is asymptotically stable if the basic reproduction number r0 is less than one, while the endemic equilibrium (ee) point is asymptotically stable if the basic reproduction number r0 is greater than one. the results are reinforced with numerical simulations and enhanced with graphical representations like time trajectories, phase portraits and bifurcation diagrams for different sets of parameter values. 1. introduction mathematical models defining biological events has an important place in the study of population dynamics. most of the biological occurrences in nature are illustrated by discrete time, which point to, that there are particular time instants at which the basic events in the system can occur, and it is not essential that at these discrete time instants only a exclusive event happens. the most realistic approach to non-overlapping generations like fish or insect populations, is created with discrete time system ( [6], [9], [15], [16], [17], [18]). received 2019-05-26; accepted 2019-07-08; published 2019-09-02. 2010 mathematics subject classification. 39a10, 39a28, 39a30. key words and phrases. difference equations; epidemic model; bifurcation; stability theory. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 809 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-809 int. j. anal. appl. 17 (5) (2019) 810 one of the other famous examples identified by these systems are epidemic models [10]. modeling an outbreak that is progressing through the population allows us to examine the consequences of ways of preventing or controlling the disease. although the work of kermack and mckendrick have the basic foundation for epidemics, the first attempt in explaining, predicting or modeling of epidemics dates back to over a century is made by hamer (1906), ross (1911). these early models operate on the principle where individuals can be classified by their epidemiological status which are susceptible to the infection, infected and recovered (immune) ( [12], [13], [14]). discrete time models are more suitable than continuous time models to examine infectious diseases due to many reasons. statistical data on diseases are collected at a specific time. in this case, the appropriate model defining the disease will be the discrete time model [19]. on the other hand, the studies on discrete time models obtained from the continuous time model by using nonstandard discretization technique are more suitable to avoid mathematical complexity with regularity of solutions ( [20], [21]). furthermore, although the continuous-time logistic equation has only equilibrium dynamics, the well known discrete logistic equation which is discrete counterpart of it exhibits period doubling bifurcation to chaos ( [22], [23]). 2. sir epidemic model with vaccination the mathematical modelling of infectious diseases has a significant role in the studies of dynamical system. because studies on the dynamics of these models help us to control diseases like swine flu, bird flue and aids. sir models are suitable to define the transmission of infectious diseases with lifelong immunity such as chicken pox, measles, smallpox, mumps and sars. the sir model is one of the simplest and most fundamental of all epidemiological models and in these models with a single epidemic, births and deaths are ignored, and so, only two transitions are possible: infection (moving individuals from the susceptible to the infected class) and recovery (moving individuals from the infected to the recovered class). the assumptions in this model are that the per capita rate that a given susceptible individual becomes infected is proportional to the prevalence of infection in the population and that infected individuals recovers at a constant rate. the fundamental parameter that governs the behavior of the epidemic is the basic reproductive ratio, r0 which is defined as the average number of secondary cases produced by a single infectious individual in a totally susceptible population. vaccination can be included in a epidemic model by assuming a proportion of susceptible individuals are vaccinated during each time interval ( [1], [3], [4], [5]). vaccination operates by reducing the pool of susceptible individuals, and when this is reduced sufficiently, an infectious disease cannot spread within the population. most importantly, it is not necessary to vaccinate everyone to prevent an epidemic, immunizing someone not only protects that person but confers some protection to the population in general [11]. int. j. anal. appl. 17 (5) (2019) 811 3. the discrete time system the author of [2] has presented the dynamics of the sir epidemic model which is as follows: st+1 =st − a n itst + β(rt + it) it+1 = a n itst + (1 −β −γ)it rt+1 =(1 −β)rt + γit + pst (3.1) where a > 0, 0 < β < 1 and 0 < γ < 1. in this paper, we focus on the dynamics of a sir epidemic model by including vaccination to the model as presented in [2]. the general sir epidemic model is of the following form [1]: st+1 =(1 −p)st − a n itst + β(rt + it) it+1 = a n itst + (1 −β −γ)it rt+1 =(1 −β)rt + γit + pst such that the initial conditions s0,i0 and r0 which are positive real numbers with (s0 +i0 +r0 = n). here 0 < p + a < 1 and 0 < β + γ < 1. also, β is the probability of birth, γ is the probability of recovery, p is the proportion of vaccinated, a is the contact rate and n is the total population size. moreover, we have the following equivalent two dimensional system using the relation st + it + rt = n. st+1 =(1 −p)st − a n itst + β(n −st) it+1 = a n itst + (1 −β −γ)it (3.2) where p, a, β and γ have positive values. 4. stability of equilibrium points and numerical simulations in this section, we consider the discrete-time system (3.2). foremost, we discuss the existence of equilibrium points for (3.2), and then study the stability of the equilibrium points by using the characteristic polynomial or the eigenvalues of the jacobian matrix evaluated at each of the fixed points. lemma 4.1. [7] let q(x) = x2 + bx + c. suppose that q(1) > 0, x1 and x2 are two roots of q(x) = 0. then (i) |x1| < 1 and |x2| < 1 if and only if q(−1) > 0 and c < 1; (ii) |x1| < 1 and |x2| > 1 (or |x1| > 1 and |x2| < 1) if and only if q(−1) < 0; (iii) |x1| > 1 and |x2| > 1 if and only if q(−1) > 0 and c > 1; (iv) x1 = −1 and |x2| 6= 1 if and only if q(−1) = 0 and b 6= 0, 2; (v) x1 and x2 are complex and |x1| = |x2| = 1 if and only if b2 − 4c < 0 and c = 1. int. j. anal. appl. 17 (5) (2019) 812 lemma 4.2. [8]the characteristic polynomial q(x) = x2 + bx + c where b=-(trace of the jacobian matrix) and c= determinant of the jacobian matrix has all its roots inside the unit open disk (|x| < 1) if and only if (i) q(1) > 0 and q(−1) > 0. (ii) d+1 = 1 + c > 0 and d−1 = 1 −c > 0 now, we will investigate the equilibrium points and then analyze the stability of these equilibrium points. for analyzing the local stability of equilibrium points (s∗,i∗), we give the following theorems. theorem 4.1. the model (3.2) has two equilibrium points, p0 = ( βn β+p , 0 ) and p1 = ( (β+γ)n a , n(aβ−(β+γ)(p+β)) a(β+γ) ) . proof. when we examine the following equilibrium points (s∗,i∗) of the model (3.2), we easily obtain the equilibrium points of the model (3.2) by using st = st+1 = s ∗ and it = it+1 = i ∗ : s∗ = (1 −p)s∗ − a n i∗s∗ + β(n −s∗) i∗ = a n i∗s∗ + (1 −β −γ)i∗ � theorem 4.2. suppose that p + β < 1. the disease-free equilibrium (dfe) point p0 = ( βn β+p , 0 ) of the system (3.2) is locally asymptotically stable (las) if aβ (β + p)(β + γ) < 1. (4.1) proof. by considering (3.2), we can get the jacobian matrix evaluated p0 as jp0 =   1 −p−β −aβ(β+p) 0 aβ (β+p) + (1 −β −γ)   . the eigenvalues of this matrix are x1 = 1 −p−β, x2 = aβ (p + β) + (1 −γ −β). if β + p < 1, then it is easy to see that x1 = 1 −p−β < 1, and also since β + γ < 1, x2 > 0 is always true. consequently, if |x2| = aβ(β+p) + (1 −β −γ) < 1, then we get aβ (β+p)(β+γ) < 1. � corollary 4.1. the basic reproductive ratio r0 is referred as aβ (β+p)(β+γ) . this ratio is a threshold parameter. if r0 < 1, then there exists that the dfe point is las. we consider the initial conditions (s(0),i(0)) = (70, 30) for numerical study. int. j. anal. appl. 17 (5) (2019) 813 example 4.1. (a) for the dfe point, we assume the parameter values as n = 100, p = 0.0005, a = 0.1, β = 0.02, γ = 0.2. the eigenvalues are |x1| = 0.9795 < 1, |x2| = 0.8776 < 1 and r0 = 0.4435 < 1 then the dfe point p0 = (97.561, 0) of the model (3.2) is las (see figure-1). (b)we take the parameter values as n = 100,p = 0.05,a = 1.7,β = 1.7,γ = 0.1. the eigenvalues are |x1| = 0.7500 < 1, |x2| = 0.8514 < 1 and r0 = 0.9175 < 1 then the dfe point p0 = (97.1429, 0) of the model (3.2) is las (see figure-2). note that trace jp0 > 0. figure 1. time plots and phase portrait of dfe point p0 with stability r0 < 1 figure 2. time plots and phase portrait of dfe point p0 with stability r0 < 1 theorem 4.3. if 1 < r0 < 2 (p+β) , then the endemic equilibrium (ee) point p1 = ( (β+γ)n a , n(aβ−(β+γ)(p+β)) a(β+γ) ) of the system (3.2) is las. int. j. anal. appl. 17 (5) (2019) 814 proof. by considering (3.2), we can write the jacobian matrix evaluated at p1 as jp1 =   1 − βaβ+γ −β −γ βa β+γ − (p + β) 1   , (4.2) if it is organized as relating to r0, we find jp1 =   1 − (β + p)r0 −aβ(β+p)r0 (p + β)(r0 − 1) 1   , the characteristic polynomial of the jacobian matrix at jp1 is as follows: q(x) = x2 − (2 − (β + p)r0)x + 1 − (β + p)r0 + aβ ( 1 − 1 r0 ) . (4.3) for the stability of the ee point of (3.2), we get 0 < aβ ( 1 − 1 r0 ) < r0(p + β), or equivalently 1 < r0 < βa (β + γ)2(p + β) + 1. (4.4) from lemma 4.2. note that aβ β + γ < 2 (4.5) is always provided. equivalently, we have r0 < 2 (p + β) . (4.6) if (4.4) and (4.6) are compared, then we get 2 (p + β) < βa (β + γ)2(p + β) + 1. (4.7) thus the proof is completed. � corollary 4.2. if 1 < r0 < 2 (p+β) , then q(1) > 0, q(−1) > 0 and c < 1 is always confirmed such that 0 < p + a < 1 and 0 < β + γ < 1. proof. from (4.2), the characteristic polynomial is as follows: q(x) = x2 + ( aβ β + γ − 2 ) x + 1 − aβ β + γ + aβ − (β + γ)(β + p). (4.8) obviously, q(1) > 0 is always true, since q(1) > 0, r0 > 1. also, we obtain q(−1) = 4 + aβ − (β + γ)(β + p) − 2aβ β + γ (4.9) d−1 = 1 −c = aβ β + γ −aβ + (β + γ)(β + p) (4.10) and d+1 = 1 + c = 2 − aβ β + γ + aβ − (β + γ)(β + p). (4.11) int. j. anal. appl. 17 (5) (2019) 815 here, we take c = q(0) = 1 − aβ β + γ + aβ − (β + γ)(β + p) (4.12) such that r0 > 1. note that whenever q(−1) > 0, d+1 > 0 is always true. by considering (4.10), we can write, aβ −aβ(β + γ) + (β + γ)2(β + p) > 0 such that β + γ < 1. it clear that aβ −aβ(β + γ) > 0. so 1 −c > 0 is always provided. similarly, we can write by considering (4.9). (β + γ)[4 + aβ − (β + γ)(β + p)] − 2aβ > 0 such that aβ − (β + γ)(β + p) > 0. then, we get q(−1) > 0. from (4.5) and by the positive state of the ee point of (3.2), the result is clear. � example 4.2. for the ee point, we take the parameter values as (a) n = 100, p = 0.0005, a = 0.6, β = 0.025, γ = 0.3. applying the conditions, we get q(1) = 0.0068 > 0, q(−1) = 3.9144 > 0, c = 0.9606 < 1 and r0 = 1.81 > 1 and thus the ee point p1 = (54.1667, 3.4423) of the model (3.2) is las (see figure-3). (b) we take the parameter values as n = 100, p = 0.05, a = 4.2, β = 1.6, γ = 0.1 and applying the conditions we see that q(1) = 3.9150 > 0, q(−1) = 0.0092 > 0, c = 0.9621 < 1 and r0 = 2.3957 > 1 and so the ee point p1 = (40.4762, 54.8319) of the model (3.2) is las (see figure-4). figure 3. time plots and phase portrait of ee point p1 with stability r0 > 1 5. bifurcation in this section, we give the bifurcation diagrams of the susceptible and infected populations of the model (3.2). the bifurcation diagrams are considered for four cases: case (i): fixing parameters n = 100, β = 0.8, p = 0.0005, γ = 0.1 and varying a. the bifurcation diagrams of model (3.2) are plotted with contact rate a ∈ (3.0, 4.15) as the bifurcation int. j. anal. appl. 17 (5) (2019) 816 figure 4. time plots of ee point p1 with stability r0 > 1 parameter and the system undergoes periodic doubling or flip bifurcation. when a ∈ (3.0, 3.36) there appears stability. in the range a ∈ (3.36, 3.8) periodic-2 orbits, for a ∈ (3.8, 3.9) periodic-4 orbits and a ∈ (3.9, 3.95) periodic-8 orbits occur, leading to chaos for a ∈ (3.95, 4.15). local amplifications corresponding to figure (5) for a ∈ [3.75, 4.15] can be seen in figure(6). figure 5. bifurcation diagrams for susceptible and infected populations with a ∈ (3.0, 4.15) case (ii): fixing parameters n = 100, p = 0.0005, β = 0.8, a = 3.5 and varying γ. the bifurcation diagrams of model (3.2) are plotted with recovery rate γ ∈ (0, 0.4), as the bifurcation parameter. when γ ∈ (0, 0.023) there appears chaos. in the range γ ∈ (0.023, 0.03) periodic-8 orbits, for γ ∈ (0.03, 0.05) periodic-4 orbits, for γ ∈ (0.05, 0.12) periodic-2 orbits occur which is called as periodic half bifurcation. finally for the range γ ∈ (0.12, 0.4) there appears stability (see figure-7). case (iii): fixing parameters n = 100, β = 0.8, a = 3.5, γ = 0.1 and varying p. the bifurcation diagrams of model (3.2) are plotted in the particular range of p ∈ (0, 0.2), with proportion int. j. anal. appl. 17 (5) (2019) 817 figure 6. local amplification corresponding to figure (5) for a ∈ (3.75, 4.15) figure 7. bifurcation diagrams for susceptible and infected populations with γ ∈ (0, 0.4) vaccinated rate as the bifurcation parameter. when p ∈ (0, 0.02) there appears chaos. in the range p ∈ (0.02, 0.056) there appears stability, in the range p ∈ (0.056, 0.127) there appears periodic-2 orbits, in the range p ∈ (0.127, 0.15) periodic-4 orbits, in the range p ∈ (0.15, 0.16) periodic-8 orbits and in the range p ∈ (0.16, 0.185) there is chaos (see figure-8). case (iv): fixing parameters n = 100, p = 0.0005, a = 4.2, γ = 0.1 and varying β. the bifurcation diagrams of the model (3.2) are plotted in the particular range of β ∈ (1.0, 2.5), with birth rate as the bifurcation parameter. local amplification corresponding to figure (9) for β ∈ (1.5, 2.5) can be seen in figure (10). int. j. anal. appl. 17 (5) (2019) 818 figure 8. bifurcation diagrams for susceptible and infected populations with p ∈ (0, 0.2) figure 9. bifurcation diagrams for susceptible and infected population with β ∈ (1.0, 2.5) 6. conclusion in this paper, we consider an discrete time sir epidemic model with vaccination and obtained the conditions for the existence of the equilibrium points and discussed the stability of the system at dfe and ee points. also the numerical examples ascertain the theoretical findings. time plots and phase portraits are presented for the susceptible and infected populations for biological feasible parameters. bifurcation diagrams and local amplifications of the same are presented. the discrete model exhibits varied and rich dynamical behavior. estimates on r0 have been obtained to determine the emergence of diseases such as measles, chickenpox and smallpox [24]. we present the dynamics of the model with the effect of vaccine ( [1], [2]). in example 4.1-(a) and in example 4.2-(a), we observe that the diseases free equilibrium is local asymptotic stable since int. j. anal. appl. 17 (5) (2019) 819 figure 10. local amplification corresponding to figure (9) for β ∈ (1.5, 2.5) r0 < 1 (see figure-1) and the endemic equilibrium point is local asymptotic stable since r0 > 1 (see figure3) by taking p = 0.0005 and n = 100. example 4.1-(b) shows that there is a decrease in the number of susceptible persons even if the vaccination rate increases when the rate of recovery decreases and the rate of contact increases (see figure-2). if the rate of contact increases further, example 4.2-(b) demonstrates an increase in the number of diseases (see figure-4). figure 5 points the bifurcation diagrams for susceptible and infected populations with changing values of a. in figure 6, we exhibit the local amplification corresponding to figure 5. figure 7 shows the bifurcation diagrams for susceptible and infected populations with changing values γ. figure 8 displays the bifurcation diagrams for susceptible and infected populations with changing values p. lastly, for the particular range of β, local amplification corresponding to figure 9 which shows bifurcation diagrams are presented in figure 10. consequently, the lower contact rate of a has an effect of reducing the disease ( [1], [24]). also increasing of the rate of vaccination has a reinforcing effect on the reduction of the disease as other parameters remain constant. references [1] l.j.s. allen, an introduction to mathematical biology, pearson, new jersey, 2007. [2] q. din, qualitative behavior of a discrete sir epidemic model, int. j. biomath. 9 (6) (2016), 1650092. [3] a. george maria selvam and d. abraham vianny, behavior of a discrete fractional order sir epidemic model, int. j. eng. technol. 7 (2018), 675-680. [4] a. george maria selvam, d. abraham vianny and mary jacintha, stability in a fractional order sir epidemic model of childhood diseases with discretization, iop conf. ser., j. phys., conf. ser., 1139 (2018), 012009. [5] a.george maria selvam and d.abraham vianny, discrete fractional order sir epidemic model of childhood diseases with constant vaccination and its stability, int. j. techn. innov. modern eng. sci., 4 (11) (2018), 405-410. [6] e.a. grove and g. ladas, periodicities in nonlinear difference equations, chapman & hall/crc press, boca raton, 2004. int. j. anal. appl. 17 (5) (2019) 820 [7] x. liu, d. xiao, complex dynamic behaviors of a discrete-time predator-prey system. chaos solution fract., 32 (2007), 80-94. [8] x. liu, c. mou, w. niu, d. wang, stability analysis for discrete biological models using algebraic methods. math. comput sci. 5 (2011), 247-262. [9] r m. may, simple mathematical models with very complicated dynamics, nature, 261 (1976), 459-467. [10] h. sedaghat, nonlinear difference equations: theory with applications to social science models, kluwer academic publishers, dordrecht, netherlands, 2003. [11] m. j. keeling and l. danon, mathematical modelling of infectious diseases, br. med. bull. 92 (1) (2009), 33-42. [12] kermack, w. o. mckendrick, a. g, a contribution to the mathematical theory of epidemics, proc. royal soc. ser a, 115 (772) (1927), 700-721. [13] w. hamer, ii. epidemic disease in england. lancet, 1 (1906), 733-739. [14] r. ross, the prevention of malaria. 2nd ea. john murray, london, 1911. [15] o. ak gumus, global and local stability analysis in a nonlinear discretetime population model, adv. difference equ. 2014 (2014), 299. [16] h merdan, o. ak gumus, stability analysis of a general discrete-time population model involving delay and allee effects. appl. math. comput. 219 (2012), 1821-1832. [17] q din, öa gümüş and h khalil, neimark-sacker bifurcation and chaotic behavior of a modified host-parasitoid model, z. naturforsch., a, 72 (1) (2017), 25-37. [18] h. merdan, ö. ak gümüş and g. karahisarli, global stability analysis of a general scalar difference equation, discontinuity, nonlinear. complex. 7 (3) (2018), 225-232. [19] i. longili, the generalized discrete-time epidemic model with immunity: a synthesis, math. biosci. 82(1986), 19-41. [20] s. jang, s. elaydi, difference equations from discretization of a continuous epidemic model with immigration of infectives, can. appl. math. q. 11 (2003), 93-105. [21] x. ma, y. zhou and h. cao, global stability of the endemic equilibrium of a discrete sir epidemic model, adv. difference equ. 2013 (2013), 42. [22] l. allen, an introduction to deterministic models in biology, prentice-hall, 2004. [23] s. elaydi, discrete chaos, chapman and hall/crc, 2007. [24] may, r. m., parasitic infections as regulators of animal populations. amer. scientist, 71 (1983), 36-45. 1. introduction 2. sir epidemic model with vaccination 3. the discrete time system 4. stability of equilibrium points and numerical simulations 5. bifurcation 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 201-215 http://www.etamaths.com coupled best proximity point theorem in metric spaces animesh gupta1,∗, s.s. rajput2 and p.s. kaurav2 abstract. the purpose of this article is to generalized the result of w. sintunavarat and p. kumam [29]. we also give an example in support of our theorem for which result of w. sintunavarat and p. kumam [29] is not true. moreover we establish the existence and convergence theorems of coupled best proximity points in metric spaces, we apply this results in a uniformly convex banach space. contents this article is organized in the following order. section 1 : in this section we give some basic concepts of the best proximity point theorems also we give some previous known results which are used to prove of our main result. section 2 : in this section we study the existence and convergence of coupled best proximity points for cyclic contraction pair. we also give an example in support of our theorem. section 3 : in this section, we give the new coupled fixed point theorem for a cyclic contraction pair. we also give an example in support of our theorem. section 4 : in this section authors would like to express their sincere thanks to the editorial board and referees. 1. introduction and preliminaries fixed point theory is one of the most useful tools in analysis. the first result of fixed point theorem is given by banach s. [4] by the general setting of complete metric space using which is known as banach contraction principle. this principle has been generalized by many researchers in many ways like by [2], [9], [10], [24], [33], [34], [40] and so on. one of the important thing in [4] is banach contraction principle is true for self mapping. in case of non self mapping (say t) the mapping does not has a fixed point. then the researchers find an element x such that d(x,tx) is minimum or near to zero for a given problem which implies that x and tx are very closed says close proximity to each other. due to this problem the theory of fixed point is converted into the theory of best proximity point. on the other words, proximity 2000 mathematics subject classification. 47h10,54h25,46j10, 46j15. key words and phrases. coupled fixed point, coupled common fixed point, coupled best proximity point, mixed monotone. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 201 202 gupta, rajput and kaurav theory is a generalization of fixed point theory. basically the proximity theory is useful tool to find proximity point when the given mapping is non self. let a and b be two non empty subsets of x such that t : a → b then a point x ∈ a for which d(x,tx) = d(a,b) is called a best proximity point of t . it should be noted that best proximity point reduced to fixed point when the mapping t is self mapping that is a = b. in 1969, fan [12] presented a classical result for best approximation theorem which as follows, theorem 1 ([12]). if a is a nonempty convex subset of a hausdorff locally convex topological vector space b and t : a → b is continuous mapping, then there exists an element x ∈ a such that d(x,tx) = d(tx,a). afterword a number of authors have derived extensions of fan’s theorem and the best approximation theorem in many directions such as prolla [26], sehgal and singh [27, 28], wlodarczyk and plebaniak [43, 44, 45, 46], vetrivel et al. [42], eldred and veramani [11], mongkolkeha and kumam [25] and basha and veeramani [5, 6, 7, 8] (see also [3, 15, 16, 17, 18, 19, 20, 21] and reference therein.) beside this, bhaskar and lakshmikantham [13] introduced the notion of mixed monotone mapping and proved some coupled fixed point theorems for mapping satisfying mixed monotone property. after the result of [13] there are lots of work presented by many authors such as [1], [14], [30], [31] (see also reference therein.) the concept of coupled best proximity point theorem is introduced by w. sintunavarat and p. kumam [29] and proved coupled best proximity theorem for cyclic contraction. our purpose of this article is to generalized the result of [29] also we give an example in support of our main theorem. first we recall some basic definitions and examples that are related to the main results of this article. throughout this article we denote by n the set of all positive integers and by r the set of all real numbers. for nonempty subsets a and b of a metric space (x,d), we let (1.1) d(a,b) = inf{d(x,y) : x ∈ a and y ∈ b} stands for the distance between a and b. a banach spaces x is said to be (1) strictly convex if the following implication holds for all x,y ∈ x: ‖x‖ = ‖y‖ = 1 and x 6= y =⇒‖x+y 2 ‖ < 1. (2) uniformly convex if for each � with 0� ≤ 2, there exists δ > 0 such that thee following implication holds for all x,y ∈ x: ‖x‖≤ 1,‖y‖≤ 1 and ‖x−y‖≥ � =⇒‖x+y 2 ‖ < 1 − δ. it is easily to see that a uniformly convex banach space x is strictly but the converges is not true. definition 2 ([41]). let a and b be nonempty subsets of a metric space (x,d). the ordered pair (a,b) satisfies the property uc if the following holds: coupled best proximity point theorem in metric spaces 203 if {xn} and {zn} are sequences in a and {yn} is a sequence in b such that d(xn,yn) → d(a,b) and d(zn,yn) → d(a,b), then d(xn,zn) → 0. example 3. let a and b be nonempty subsets of a metric space (x,d). the following are examples of a pair of nonempty subsets (a,b) satisfying the property uc. (1) every pair of nonempty subsets a,b of a metric space (x,d) such that d(a,b) = 0. (2) every pair of nonempty subsets a,b of a uniformly convex banach space x such that a is convex. (3) every pair of nonempty subsets a,b of a strictly convex banach space which a is convex and relatively compact and the closure of b is weakly compact. definition 4 ([39]). let a and b be nonempty subsets of a metric space (x,d). the ordered pair (a,b) satisfies the property uc∗ if (a,b) has property uc and the following condition holds: if {xn} and {zn} are sequences in a and {yn} is a sequence in b satisfying: (1) d(zn,yn) → d(a,b) (2) for every � > 0 there exists n ∈ n such that d(xm,yn) ≤ d(a,b) + � for all m > n ≥ n. then for every � > 0 there exists n1 ∈ n such that d(xm,zn) ≤ d(a,b) + � for all m > n ≥ n1. example 5 ([39]). let a and b be nonempty subsets of a metric space (x,d). the following are examples of a pair of nonempty subsets (a,b) satisfying the property uc∗. (1) every pair of nonempty subsets a,b of a metric space (x,d) such that d(a,b) = 0. (2) every pair of nonempty closed subsets a,b of a uniformly convex banach space x such that a is convex. definition 6. let a and b be nonempty subsets of a metric space (x,d) and t : a → b be a mapping. a point x ∈ a is said to be a best proximity point of t if it satisfies the condition that d(x,tx) = d(a,b). it can be observed that a best proximity point reduces to a fixed point if the underlying mapping is a self mapping. definition 7 ([13]). let a be a nonempty subset of a metric space x and f : a×a → a. a point (x,y) ∈ a×a is called a coupled fixed point of f if x = f(x,y), y = f(y,x). 204 gupta, rajput and kaurav 2. coupled best proximity point theorems in this section we study the existence and convergence of coupled best proximity points for cyclic contraction pair. definition 8. let a and b be nonempty subsets of a metric space x and f : a×a → b. an ordered coupled (x,y) ∈ a×a is called a coupled best proximity point of f if, d(x,f(x,y)) = d(y,f(y,x)) = d(a,b). it is easy to see that if a = b in definition 8, then a coupled best proximity point reduces to a coupled fixed point. next,w. sintunavarat and p. kumam [29] introduce the notion of a cyclic contraction for two mappings which as follows. definition 9. let a and b be nonempty subsets of a metric space x, f : a×a → b and g : b ×b → a. the ordered pair (f,g) is said to be a cyclic contraction if there exists a non-negative number α < 1 such that d(f(x,y),g(u,v)) ≤ α 2 [d(x,u) + d(y,v)] + (1 −α)d(a,b) for all (x,y) ∈ a×a and (u,v) ∈ b ×b. now we introduced the following notion of cyclic contraction for two mappings which is generalization of [29] as follows. definition 10. let a and b be nonempty subsets of a metric space x, f : a×a → b and g : b ×b → a. the ordered pair (f,g) is said to be a cyclic contraction if there exists a non-negative number p + q < 1 such that d(f(x,y),g(u,v)) ≤ [pd(x,u) + qd(y,v)] + (1 − (p + q))d(a,b) for all (x,y) ∈ a×a and (u,v) ∈ b ×b. note that if (f,g) is a cyclic contraction, then (g,f) is also a cyclic contraction. also if we take p = q = α 2 in definition 10 then we get definition 9. following example show that definition 10 is generalization of definition 9. example 11. let x = r with the usual metric d(x,y) =| x−y | also a = [6, 12] and b = [−12,−6]. it easy to see that d(a,b) = 12. define f : a×a → b and g : b ×b → a by f(x,y) = −3x− 2y − 6 6 and g(x,y) = −3x− 2y + 6 6 . for arbitrary (x,y) ∈ a×a, (u,v) ∈ b ×b and fixed k = 1 2 , l = 1 3 , we get d(f(x,y),g(u,v)) = ∣∣∣∣−3x− 2y − 66 − −3u− 2v + 66 ∣∣∣∣ ≤ 3 | x−u | +2 | y −v | 6 + 2 = kd(x,u) + ld(y,v) + (1 − (k + l))d(a,b). this implies that (f,g) is a cyclic contraction with p = 1 2 and q = 1 3 . coupled best proximity point theorem in metric spaces 205 the following lemma plays an important role in our main results. lemma 12. let a and b be nonempty subsets of a metric space x, f : a×a → b and g : b ×b → a and (f,g) be a cyclic contraction. if (x0,y0) ∈ a×a and we define xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n ∪ {0}, then d(xn,xn+1) → d(a,b), d(xn+1,xn+2) → d(a,b), d(yn,yn+1) → d(a,b) and d(yn+1,yn+2) → d(a,b). proof. for each n ∈ n, we have d(xn,xn+1) = d(f(xn,yn),g(xn−1,yn−1)) ≤ pd(xn,xn−1) + qd(yn,yn−1) + (1 − (p + q))d(a,b) similarly we have d(yn,yn+1) = d(f(yn,xn),g(yn−1,xn−1)) ≤ pd(yn,yn−1) + qd(xn,xn−1) + (1 − (p + q))d(a,b) therefore, by letting dn = d(xn,xn+1) + d(yn,yn+1) by adding above inequality we have dn ≤ (p + q)dn−1 + 2(1 − (p + q))d(a,b) similarly we can show that dn−1 ≤ (p + q)dn−2 + 2(1 − (p + q))d(a,b) consequently we have d1 ≤ (p + q)d0 + 2(1 − (p + q))d(a,b) if d0 = 0 then (x0,y0) is a coupled best proximity point of f and g. now let d0 > 0 for each n ≥ m we have d(xn,xm) ≤ d(xn,xn−1) + d(xn−1,xn−2) + ......... + d(xm+1,xm) d(yn,ym) ≤ d(yn,yn−1) + d(yn−1,yn−2) + ......... + d(ym+1,ym) which gives d(xn,xm) + d(yn,ym) ≤ dn−1 + dn−2 + dn−3....... + dm dn ≤ (p + q)nd0 + 2n(1 − (p + q)n)d(a,b) taking n →∞ we have d(xn,xn+1) + d(yn,yn+1) → d(a,b) 206 gupta, rajput and kaurav implies that d(xn,xn+1) → d(a,b) d(yn,yn+1) → d(a,b) for all n ∈ n. by similar argument, we also have d(xn+1,xn+2) → d(a,b), d(yn+1,yn+2) → d(a,b). � lemma 13. let a and b be nonempty subsets of a metric space x such that (a,b) and (b,a) have a property uc, f : a×a → b and g : b ×b → a and let the ordered pair (f,g) is a cyclic contraction. if (x0,y0) ∈ a×a and define xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n ∪{0}, then for � > 0, there exists a positive integer n0 such that for all m > n ≥ n0 (2.1) pd(xm,xn+1) + qd(ym,yn+1) < d(a,b) + �. proof. by lemma 12, we have d(xn,xn+1) → d(a,b), d(xn+1,xn+2) → d(a,b), d(yn,yn+1) → d(a,b), d(yn+1,yn+2) → d(a,b). since (a,b) has a property uc, we get d(xn,xn+2) → 0. a similar argument shows that d(yn,yn+2) → 0. as (b,a) has a property uc, we also have d(xn+1,xn+3) → 0, d(yn+1,yn+3) → 0. suppose that (2.1) does not hold. then there exists �′ > 0 such that for all k ∈ n, there is mk > nk ≥ k satisfying pd(xmk,xnk+1) + qd(ymk,ynk+1) ≥ d(a,b) + � ′. further, corresponding to nk, we can choose mk in such a way that it is the smallest integer with mk > nk and satisfying above relation. then pd(xmk−2,xnk+1) + qd(ymk−2,ynk+1) < d(a,b) + � ′. therefore, we get d(a,b) + �′ ≤ pd(xmk,xnk+1) + qd(ymk,ynk+1) ≤ p[d(xmk,xmk−2) + d(xmk−2,xnk+1)] +q[d(ymk,ymk−2) + d(ymk−2,ynk+1)] < pd(xmk,xmk−2) + qd(ymk,ymk−2)] + d(a,b) + � ′. letting k →∞, we obtain to see that pd(xmk,xnk+1) + qd(ymk,ynk+1) → d(a,b) + � ′. coupled best proximity point theorem in metric spaces 207 by using the triangle inequality, we get pd(xmk,xnk+1) + qd(ymk,ynk+1) ≤ p[d(xmk,xmk+2) + d(xmk+2,xnk+3) + d(xnk+3,xnk+1) +q[d(ymk,ymk+2) + d(ymk+2,ynk+3) + d(ynk+3,ynk+1)] = p[d(xmk,xmk+2) + d(g(xmk+1,ymk+1),f(xnk+2,ynk+2)) + d(xnk+3,xnk+1)] +q[d(ymk,ymk+2) + d(g(ymk+1,xmk+1),f(ynk+2,xnk+2)) + d(ynk+3,ynk+1)] ≤ p[d(xmk,xmk+2) + pd(xmk+1,xnk+2) + qd(ymk+1,ynk+2) +(1 − (p + q))d(a,b) + d(xnk+3,xnk+1)] +q[d(ymk,ymk+2) + pd(ymk+1,ynk+2) + qd(xmk+1,xnk+2) + (1 − (p + q))d(a,b) + d(ynk+3,ynk+1) ≤ (p + q)[d(xmk,xmk+2) + d(xnk+3,xnk+1) + d(ymk,ymk+2) + d(ynk+3,ynk+1)] +(p + q)2[d(xmk+1,xnk+2) + d(ymk+1,ynk+2)] + (1 − (p + q) 2)d(a,b). taking k →∞, we get d(a,b) + �′ ≤ (p + q)2[d(a,b) + �′] + (1 − (p + q)2)d(a,b) = d(a,b) + (p + q)2�′ which contradicts. therefore, we can conclude that (2.1) holds. � lemma 14. let a and b be nonempty subsets of a metric space x, (a,b) and (b,a) satisfy the property uc∗. let f : a×a → b, g : b ×b → a and (f,g) be a cyclic contraction. if (x0,y0) ∈ a×a and define xn+1 = f(xn,yn) yn+1 = f(yn,xn) and xn+2 = g(xn+1,yn+1) yn+2 = g(yn+1,xn+1) for all n ∈ n∪{0}, then {xn}, {yn}, {xn+1} and {yn+1} are cauchy sequences. proof. by lemma 12, we have d(xn,xn+1) → d(a,b) and d(xn+1,xn+2) → d(a,b). since (a,b) satisfies property uc, we get d(xn,xn+2) → 0. similarly, we also have d(xn+1,xn+3) → 0 because (b,a) satisfies property uc. we now show that for every � > 0 there exists n ∈ n such that (2.2) d(xm,xn+1) ≤ d(a,b) + � for all m > n ≥ n suppose (2.2) not hold, then there exists � > 0 such that for all k ∈ n there exists mk > nk ≥ k such that (2.3) d(xmk,xnk+1) > d(a,b) + �. further, corresponding to nk, we can choose mk in such a way that it is the smallest integer with mk > nk and satisfying above relation. now we have d(a,b) + � < d(xmk,xnk+1) ≤ d(xmk,xmk−2) + d(xmk−2,xnk+1) ≤ d(x2mk,x2mk−2) + d(a,b) + �. 208 gupta, rajput and kaurav taking k →∞, we have d(x2mk,x2nk+1) → d(a,b) +�. by lemma 13, there exists n ∈ n such that (2.4) pd(xmk,xnk+1) + qd(ymk,ynk+1) < d(a,b) + � for all m > n ≥ n. by using the triangle inequality, we get d(a,b) + � < d(xmk,xnk+1) ≤ d(xmk,xmk+2) + d(xmk+2,xnk+3) + d(xnk+3,xnk+1) = d(xmk,xmk+2) + d(g(xmk+1,ymk+1),f(xnk+2,ynk+2)) +d(xnk+3,xnk+1) ≤ d(xmk,xmk+2) + [pd(xmk+1,xnk+2) + qd(ymk+1,ynk+2)] +(1 − (p + q))d(a,b) + d(xnk+3,xnk+1) = p[d(f(xmk,ymk ),g(xnk+1,ynk+1))] + q[d(f(ymk,xmk ),g(ynk+1,xnk+1))] +(1 − (p + q))d(a,b) + d(xmk,xmk+2) + d(xnk+3,xnk+1) ≤ p [ p[d(xmk,xnk+1) + qd(ymk,ynk+1) + (1 − (p + q))d(a,b)] ] +q [ [pd(ymk,ynk+1) + qd(xmk,xnk+1) + (1 − (p + q))d(a,b)] ] +(1 − (p + q))d(a,b) + d(xmk,xmk+2) + d(xnk+3,xnk+1) = (p + q)2[d(xmk,xnk+1) + d(ymk,ynk+1)] +(1 − (p + q)2)d(a,b) + d(xmk,xmk+2) + d(xnk+3,xnk+1) < (p + q)2(d(a,b) + �) + (1 − (p + q)2)d(a,b) + d(xmk,xmk+2) + d(xnk+3,xnk+1) = (p + q)2� + d(a,b) + d(xmk,xmk+2) + d(xnk+3,xnk+1). taking k →∞, we get d(a,b) + � ≤ d(a,b) + (p + q)2� which contradicts. therefore, condition (2.2) holds. since (2.2) holds and d(xn,xn+1) → d(a,b), by using property uc∗ of (a,b), we have {xn} is a cauchy sequence. in similar way, we can prove that {yn},{xn+1} and {yn+1} are cauchy sequences. � here we state the main results of this article in the existence and convergence of coupled best proximity points for cyclic contraction pairs on nonempty subsets of metric spaces satisfying the property uc∗. theorem 15. let a and b be nonempty closed subsets of a metric space x such that (a,b) and (b,a) have a property uc∗, f : a×a → b and g : b ×b → a and let the ordered pair (f,g) is a cyclic contraction. if (x0,y0) ∈ a×a and define xn+1 = f(xn,yn) yn+1 = f(yn,xn) and xn+2 = g(xn+1,yn+1) yn+2 = g(yn+1,xn+1) coupled best proximity point theorem in metric spaces 209 for all n ∈ n ∪{0}. then f has a coupled best proximity point (r,s) ∈ a3 and g has a coupled best proximity point (p′,q′,r′) ∈ b3. moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. furthermore, if r = s and r′ = s′, then d(r,r′) + d(s,s′) = 2d(a,b). proof. by lemma 12, we get d(xn,xn+1) → d(a,b). using lemma 14, we have {xn} and {yn} are cauchy sequences. thus, there exists r,s ∈ a such that xn → r, yn → s. we obtain that (2.5) d(a,b) ≤ d(r,xn−1) ≤ d(r,xn) + d(xn,xn−1). letting n →∞ in (2.5), we have d(r,xn−1) → d(a,b). by a similar argument we also have d(s,yn−1) → d(a,b). it follows that d(xn,f(r,s)) = d(g(xn−1,yn−1),f(r,s)) ≤ pd(xn−1,p) + qd(yn−1,q) + (1 − (p + q))d(a,b). taking n →∞, we get d(p,f(p,q,r)) = d(a,b). similarly, we can prove that d(s,f(s,r)) = d(a,b) therefore, we have (r,s) is a coupled best proximity point of f. in similar way, we can prove that there exists r′,s′ ∈ b such that xn+1 → r′ and yn+1 → s′. moreover, we have d(r′,g(r′,s′)) = d(a,b), and d(s′,f(s′,r′)) = d(a,b) and so (r′,s′) is a coupled best proximity point of g. finally, we assume that r = s and r′ = s′ and then we show that d(r,r′) + d(s,s′) = 2d(a,b). for all n ∈ n, we obtain that d(xn,xn+1) = d(g(xn−1,yn−1),f(xn,yn)) ≤ pd(xn−1,xn) + qd(yn−1,yn) + (1 − (p + q))d(a,b). letting n →∞, we have (2.6) d(r,r′) ≤ pd(r,r′) + d(s,s′) + (1 − (p + q))d(a,b). for all n ∈ n, we have d(yn,yn+1) = d(g(yn−1,xn−1),f(yn,xn)) ≤ pd(yn−1,yn) + qd(xn−1,xn) + (1 − (p + q))d(a,b). letting n →∞, we have d(s,s′) ≤ pd(s,s′) + d(r,r′) + (1 − (p + q))d(a,b). similarly we can write, it follows from (2.6)and (2.7) that d(r,r′) + d(s,s′) ≤ pd(r,r′) + qd(s,s′) + 2(1 − (p + q))d(a,b) 210 gupta, rajput and kaurav which implies that (2.7) d(r,r′) + d(s,s′) ≤ 2d(a,b). since d(a,b) ≤ d(r,r′) and d(a,b) ≤ d(s,s′), we have (2.8) d(r,r′) + d(s,s′) ≥ 2d(a,b). from (2.7) and (2.8), we get (2.9) d(r,r′) + d(s,s′) = 2d(a,b). this complete the proof. � note that every pair of nonempty closed subsets a,b of a uniformly convex banach space x such that a is convex satisfies the property uc. therefore, we obtain the following corollary. corollary 16. let a and b be nonempty closed convex subsets of a uniformly convex banach space x, f : a×a → b and g : b ×b → a and let the ordered pair (f,g) be a cyclic contraction. let (x0,y0) ∈ a×a and define xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n ∪{0}. then f has a coupled best proximity point (r,s) ∈ a × a and g has a coupled best proximity point (r′,s′) ∈ b × b. moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. furthermore, if r = s and r′ = s′, then d(r,r′) + d(s,s′) = 2d(a,b). next, we give some illustrative example of corollary 16. example 17. consider uniformly convex banach space x = r with the usual norm. let a = [1, 2] and b = [−1,−2].thus d(a,b) = 2. define f : a×a → b and g : b ×b → a by f(x,y) = −2x− 3y − 1 6 and g(x,y) = −2x− 3y + 1 6 . for arbitrary (x,y) ∈ a×a and (u,v) ∈ b ×b and fixed p = 1 3 and q = 1 2 we get d(f(x,y),g(u,v)) = ∣∣∣∣−x−y − 16 − −u−v + 16 ∣∣∣∣ ≤ 2|x−u| + 3|y −v| 6 + 1 3 = 1 3 d(x,u) + 1 2 d(y,v) + (1 − (p + q))d(a,b) this implies that (f,g) is a cyclic contraction with α = 1 2 . since a and b are closed convex, we have (a,b) and (b,a) satisfy the property uc∗. therefore, all hypothesis of corollary 16 hold. so f has a coupled best proximity point and g has a coupled best proximity point. we note that a point (1, 1) ∈ a × a is a unique coupled best proximity point theorem in metric spaces 211 coupled best proximity point of f and a point (−1,−1, ) ∈ b×b is a unique coupled best proximity point of g. furthermore, we get d(1,−1) + d(1,−1) = 4 = 2d(a,b). next, we give the coupled best proximity point result in compact subsets of metric spaces. theorem 18. let a and b be nonempty compact subsets of a metric space x, f : a × a → b and g : b × b → a and let the ordered pair (f,g) be a cyclic contraction. let (x0,y0) ∈ a×a and define xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n ∪{0}. then f has a coupled best proximity point (r,s) ∈ a × a and g has a coupled best proximity point (r′,s′) ∈ b × b. moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. furthermore, if r = s and r′ = s′, then d(p,p′) + d(q,q′) + d(r,r′) = 2d(a,b). proof. since x0,y0 ∈ a and xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n∪{0}, we have xn,yn ∈ a and xn+1,yn+1 ∈ a for all n ∈ n∪{0}. as a is compact, the sequences {xn} and {yn} have convergent subsequences {xnk} and {ynk} respectively, such that xnk → r ∈ a, ynk → s ∈ a. now, we have (2.10) d(a,b) ≤ d(r,xnk−1) ≤ d(r,xnk ) + d(xnk,xnk−1) by lemma 12, we have d(xnk,xnk−1) → d(a,b). taking k → ∞ in (2.10), we get d(r,xnk−1) → d(a,b). by a similar argument we observe that d(s,xnk−1) → d(a,b). note that d(a,b) ≤ d(xnk,f(r,s)) = d(g(xnk−1,ynk−1),f(r,s)) ≤ pd(xnk−1,r) + qd(ynk−1,s) + (1 − (p + q))d(a,b). taking k →∞, we get d(r,f(r,s)) = d(a,b). similarly, we can prove that d(s,f(s,r)) = d(a,b). thus f has a coupled best proximity (r,s) ∈ a × a. in similar way, since b is compact, we can also prove that g has a coupled best proximity point (r′,s′) ∈ b ×b. for d(r,r′) + d(s,s′) = 2d(a,b) similar to the final step of the proof of theorem 15. this complete the proof. � 212 gupta, rajput and kaurav 3. coupled fixed point theorems in this section, we give the new coupled fixed point theorem for a cyclic contraction pair. theorem 19. let a and b be nonempty closed subsets of a metric space x, f : a × a → b and g : b × b → a and let the ordered pair (f,g) be a cyclic contraction. let (x0,y0) ∈ a×a and define xn+1 = f(xn,yn), xn+2 = g(xn+1,yn+1) yn+1 = f(yn,xn), yn+2 = g(yn+1,xn+1) for all n ∈ n∪{0}. if d(a,b) = 0, then f has a coupled fixed point (r,s) ∈ a×a and g has a coupled fixed point (r′,s′) ∈ b×b. moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. furthermore, if r = r′ and s = s′, then f and g have a common coupled fixed point in (a∩b)2. proof. since d(a,b) = 0, we get (a,b) and (b,a) satisfy the property uc. therefore, by theorem 15, claim that f has a coupled best proximity point (r,s) ∈ a×a that is (3.1) d(r,f(r,s)) = d(s,f(s,r)) = d(a,b) and g has a coupled best proximity point (r′,s′) ∈ b ×b that is (3.2) d(r′,g(r′,s′)) = d(s′,g(s′,r′)) = d(a,b). from (3.1) and d(a,b) = 0 , we conclude that r = f(r,s), s = f(s,r). that is (r,s) is a coupled fixed point of f . it follows from (3.2) and d(a,b) = 0, we get r′ = g(r′,s′), and s′ = g(s′,r′) that is (r′,s′) is a coupled fixed point of g. next, we assume that r = r′ and s = s′ and then we show that f and g have a unique common coupled fixed point in (a∩b)2. from theorem 15, we get (3.3) d(r,r′) + d(s,s′) = 2d(a,b). since d(a,b) = 0, we get d(r,r′) + d(s,s′) = 0 which implies that r = r′ and s = s′. therefore, we conclude that (r,s) ∈ (a∩b)2 is common coupled fixed point of f and g. � example 20. consider x = r with the usual metric, a = [−2, 0] and b = [0, 2]. define f : a×a → b and g : b ×b → a by f(x,y) = − 2x + 3y 6 and g(u,v) = − 2u + 3v 6 . coupled best proximity point theorem in metric spaces 213 then d(a,b) = 0 and (f,g) is a cyclic contraction with p = 1 3 and q = 1 2 . indeed, for arbitrary (x,y) ∈ a×a and (u,v) ∈ b ×b, we have d(f(x,y),g(u,v)) = ∣∣∣∣−2x + 3y6 + 2u + 3v6 ∣∣∣∣ ≤ 1 6 (2 | x−u | +3 | y −v |) ≤ pd(x,u) + qd(y,v) + (1 − (p + q))d(a,b). therefore, all hypothesis of theorem 19 hold. so f and g have a common coupled fixed point and this point is (0, 0) ∈ (a∩b)2. if we take a = b in theorem 19, then we get the following results. corollary 21. let a be a nonempty closed subset of a complete metric space x, f : a × a → a and g : a × a → a and let the ordered pair (f,g) be a cyclic contraction. let (x0,y0) ∈ a×a and define xn+1 = f(xn,yn) yn+1 = f(yn,xn) and xn+2 = g(xn+1,yn+1) yn+2 = g(yn+1,xn+1) for all n ∈ n ∪{0}. then f has a coupled fixed point (r,s) ∈ a×a and g has a coupled fixed point (r′,s′) ∈ b ×b. moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. furthermore, if r = r′ and s = s′, then f and g have a common coupled fixed point in a×a. we take f = g in corollary 21, then we get the following results corollary 22. let a be nonempty closed subsets of a complete metric space x, f : a×a → a and d(f(x,y),f(u,v)) ≤ pd(x,u) + qd(y,v) for all (x,y), (u,v) ∈ a×a. then f has a coupled fixed point (r,s) ∈ a×a. 4. acknowledgements the authors thank the editor and the referees for their useful comments and suggestions for improving the quality of this research. references [1] m. abbas, w. sintunavarat, p. kumam, coupled fixed point in partially ordered g metric spaces. fixed point theory appl. 2012, 31 (2012) [2] a. d. arvanitakis, a proof of the generalized banach contraction conjecture. proc. am. math. soc. 131(12), 3647–3656 (2003) [3] r. p. agarwal, m. a. alghamdi, n. shahzad, fixed point theory for cyclic generalized contractions in partial metric spaces, fixed point theory appl., 2012, 2012:40 [4] s. banach, sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales, fund. math. 3 (1922) 133-181. 214 gupta, rajput and kaurav [5] s. s. basha, best proximity point theorems generalizing the contraction principle. nonlinear anal. 74, 5844–5850 (2011) [6] s. s. basha, p. veeramani, best approximations and best proximity pairs. acta. sci. math. (szeged) 63, 289–300 (1997) [7] s. s. basha, p. veeramani, best proximity pair theorems for multifunctions with open fibres. j. approx. theory 103, 119–129 (2000) [8] s. s. basha, p. veeramani, d.v. pai, best proximity pair theorems. indian j. pure appl. math. 32, 1237–1246 (2001) [9] d. w. boyd, j. s. w. wong, on nonlinear contractions. proc. am. math. soc. 20, 458–464 (1969) [10] b. s. choudhury, k. p. das, a new contraction principle in menger spaces. acta math. sin. 24(8), 1379–1386 (2008) [11] a. a. eldred, p. veeramani, existence and convergence of best proximity points. j. math. anal. appl. 323, 1001–1006 (2006) [12] k. fan, extensions of two fixed point theorems of f. e. browder. math. z. 112, 234–240 (1969) [13] t. gnana-bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications. nonlinear anal. tma 65, 1379–1393 (2006) [14] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications. nonlinear anal., theory methods appl. 11 (1987) 623–632. [15] e. karapınar, fixed point theory for cyclic weak φ-contraction, appl. math. lett., 24 (2011) 822?825. [16] e. karapinar, k. sadarangani, fixed point theory for cyclic (phi,psi) contractions fixed point theory appl., 2011:69 ,(2011) [17] e. karapinar, i.m. erhan, a.y. ulus, fixed point theorem for cyclic maps on partial metric spaces, appl. math. inf. sci. 6 (2012), no:1, 239-244. [18] e. karapinar, i.m. erhan, cyclic contractions and fixed point theorems, filomat, 26 (2012),no:4, 777-782 [19] e. karapinar, best proximity points of cyclic mappings, appl. math. lett., 25 (2012), 1761–1766. [20] s. karpagam and s. agrawal: best proximity points theorems for cyclic meir-keeler contraction maps, nonlinear anal.,74,(2011) 1040–1046. [21] w. a. kirk, p. s. srinavasan and p. veeramani: fixed points for mapping satisfying cylical contractive conditions, fixed point theory, 4(2003), 79-89. [22] g. s. r. kosuru and p. veeramani, cyclic contractions and best proximity pair theorems, arxiv:1012.1434v2 [math.fa] 29 may 2011, 14 pages. [23] j. merryfield, b. rothschild, j.d.jr. stein, an application of ramsey’s theorem to the banach contraction principle. proc. am. math. soc. 130(4), 927–33 (2002) [24] c. mongkolkeha, w. sintunavarat, p. kumam,: fixed point theorems for contraction mappings in modular metric spaces. fixed point theory appl. 2011, 93 (2011) [25] c. mongkolkeha, p. kumam, best proximity point theorems for generalized cyclic contractions in ordered metric spaces. j. opt. theory appl. (2012) (in press), http://dx.doi.org/10.1007/s10957-012-9991-y [26] j.p. prolla, fixed point theorems for set valued mappings and existence of best approximations. numer. funct. anal. optim. 5, 449–455 (1983) [27] v. m. sehgal, s. p. singh, a generalization to multifunctions of fan’s best approximation theorem. proc. am. math. soc. 102, 534–537 (1988) [28] v. m. sehgal, s. p. singh, a theorem on best approximations. numer. funct. anal. optim.10, 181–184 (1989) [29] w. sintunavarat, p. kumam, coupled best proxitmity point theorem in metric spaces. fixed point theory appl.2012/1/93 (2012). [30] w. sintunavarat, y.j. cho, p. kumam, coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. fixed point theory appl. 2011, 81 (2011) [31] w. sintunavarat, y.j. cho, p. kumam, coupled fixed point theorems for weak contraction mapping under f-invariant set. abstr. appl. anal. 2012, 15 (article id 324874) (2012) [32] w. sintunavarat, p. kumam,: weak condition for generalized multi-valued (f,α,β)-weak contraction mappings. appl. math. lett. 24, 460–465 (2011) coupled best proximity point theorem in metric spaces 215 [33] w. sintunavarat, p. kumam,: gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. j. inequal. appl. 2011, 3 (2011) [34] w. sintunavarat, y.j. cho, kumam, p: common fixed point theorems for c-distance in ordered cone metric spaces. comput. math. appl. 62, 1969–1978 (2011) [35] w. sintunavarat, p. kumam,: common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces. j. appl. math. 2011, 14 (article id 637958) (2011) [36] w. sintunavarat, p. kumam,: common fixed point theorems for hybrid generalized multivalued contraction mappings. appl. math. lett. 25, 52–57 (2012) [37] w. sintunavarat, p. kumam,: common fixed point theorems for generalized operator classes and invariant approximations. j. inequal. appl. 2011, 67 (2011) [38] w. sintunavarat, p. kumam,: generalized common fixed point theorems in complex valued metric spaces and applications. j. inequal. appl. 2012, 84 (2012) [39] w. sintunavarat, p. kumam, coupled best proximity point theorem in metric spaces. fixed point theory appl. 2012, 93 (2012) [40] t. suzuki,: a generalized banach contraction principle that characterizes metric completeness. proc. am. math. soc. 136(5), 1861–1869 (2008) [41] t. suzuki, m. kikkawa, c. vetro, the existence of best proximity points in metric spaces with the property uc, nonlinear analysis: theory, methods & applications 71 (7?), 2918-2926 (2009) [42] v. vetrivel, p. veeramani, p. bhattacharyya, some extensions of fan’s best approximation theorem. numer. funct. anal. optim. 13, 397–402 (1992) [43] k. wlodarczyk, r. plebaniak, a. banach, best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. nonlinear anal. 70(9), 3332–3342 (2009) [44] k. wlodarczyk, r. plebaniak, a. banach, erratum to: best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. nonlinear anal. 71, 3583–3586 (2009) [45] k. wlodarczyk, r. plebaniak, a. banach, best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. nonlinear anal. 70, 3332–3341 (2009) [46] k. wlodarczyk, r. plebaniak, c. obczynski, convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. nonlinear anal. 72, 794–805 (2010) 1department of mathematics, vidhyapeeth institute of science & technology, bhopal india 2department of mathematics, govt. p.g. college, gadarwara, distnarsingpur (m.p.) india ∗corresponding author international journal of analysis and applications volume 18, number 2 (2020), 243-253 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-243 the analytical solution of telegraph equation of space-fractional order derivative by the aboodh transform method mohamed elarbi benattia1,∗, kacem belghaba2 1high school of economic, laboratory of mathematics and its applications (lamap), oran, algeria 2university of oran 1, laboratory of mathematics and its applications (lamap), oran, algeria ∗corresponding author: mohamed.benattia74@yahoo.com abstract. in this article, an analytical solution based on the series expansion method is proposed to solve the telegraph equation of space fractional order (tesfo), namely the aboodh transformation method (atm) subjected to the appropriate initial condition. using atm, it is possible to find exact solution or a closed approximate solution of a differential equation. finally, several numerical examples are given to illustrate the accuracy and stability of this method. 1. introduction in the last few decades, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation processes and so on [1]. suspension flows are traditionally modeled by parabolic partial differential equations. sometimes they can be better modeled by hyperbolic equations such as the telegraph equation, which have parabolic asymptotic. in particular the experimental data described in [1] seem to be better modeled by the telegraph equation than by the heat equation. the telegraph equation is used in signal analysis for transmission and propagation of electrical signals and also used modeling reaction diffusion. the different type solutions of the fractional telegraph equations have been received 2019-12-22; accepted 2020-01-20; published 2020-03-02. 2010 mathematics subject classification. 65r10, 26a33. key words and phrases. aboodh transform; fractional differential equation; caputo fractional derivative; telegraph equation. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 243 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-243 int. j. anal. appl. 18 (2) (2020) 244 discussed by momani [2] by using decomposition method, yildirim [3] by homotopy perturbation method. our concern in this work is to consider the space-fractional telegraph equations as dαxu(x, t) = aut + utt + bu(x, t) + g(x, t), 0 < x < 1 where t ≥ 0, 0 < α ≤ 2, a, b are given constants, g(x, t) is given function. the main objective of this paper is to introduce a new analytical and approximate solution of spatial fractional telegraphic equations using the aboodh transformation method(atm), where in [5] authors proposed a sumudu transformation method (stm) which is used to solve this equation. 2. preliminary 2.1. fundamental properties of fractional calculus. in this section we give definitions and some basic results. definition 2.1. an aboodh transform is defined for functions of exponential order. we consider functions in the set f defined by; f = { f(t) : ∣∣f(t)∣∣ < me−vt, if t ∈[0;∞[, m,k1,k2 > 0; k1 ≤ v ≤ k2} (2.1) for a given function in the set f , m must be finite number and k1, k2 may be finite or infinite with variable v define as k1 ≤ v ≤ k2. then, the aboodh transform denoted by the operator a(:) is defined by the integral equation: t (v) = a [ f(t) ] = 1 v ∞∫ 0 f(t)e−vtdt, t ≥ 0, k1 ≤ v ≤ k2. (2.2) for a given function in the set f, m must be finite number and k1, k2 may be finite or infinite with variable v define as k1 ≤ v ≤ k2. then, the aboodh transform denoted by the operator a(:) is defined by the integral equation: t (v) = a [ f(t) ] = 1 v ∞∫ 0 f(t)e−vtdt, t ≥ 0, k1 ≤ v ≤ k2. (2.3) standard aboodh transform for some special functions found are given below in table (2.1). int. j. anal. appl. 18 (2) (2020) 245 f(t) t (v) = a [ f(t) ] 1 1 v2 t 1 v3 tn, n ≥ 1 n! vn+2 eat 1 v2 −av sin(at) 1 v(v2 + a2) cos(at) 1 v2 + a2 sinh(at) 1 v(v2 −a2) tcosh(at) 1 v2 −a2 table(2.1): aboodh transform of some functions. definition 2.2. the riemann-liouville fractional integral of order α ∈ r+ is defined as d−αf (t) = iαf(t) = 1 γ (α) t∫ 0 f(x) (t−x)1−α dx, 0 < α ≤ 1 (2.4) i0f(t) = f(t) properties of the operator iα can be found in for α,β > 0, and γ > −1, we have: iαiβf(t) = iβiαf(t) = iα+βf(t) iαtγ = γ (γ + 1) γ (α + γ + 1) tα+γ definition 2.3. the caputo fractional derivative (cfd) operator dαt of order α is dαt f(t) = i n−αdnf(t) = 1 γ (n−α) t∫ 0 f(n)(x) (t−x)1+α−n dx, x > 0 (2.5) for n− 1 < α ≤ n, n ∈ n, t > 0. definition 2.4. the mittage leffler function eα (z) with α > 0, is definite by the following series: eα (z) = zα γ (nα + 1) , z ∈ c (2.6) where n ∈ z+, α ∈ r+. definition 2.5. the aboodh transform a [dαxf(x)] of the fractional derivative using the caputo idea of the function is given by: a [dαxf(x)] ] = vαt (v) − n−1∑ k=0 f(k)(0) v2−α+k (2.7) int. j. anal. appl. 18 (2) (2020) 246 it is easy to understand that: a [dαt f(x; t)] ] = vαa [f(x; t)] ] − n−1∑ k=0 f(k)(x; 0) v2−α+k , n− 1 < α ≤ n, (2.8) remark 2.1. the aboodh transform is linear, i.e., if α and β are any constants and f(t) and g(t) are functions defined over the set f above, then a [ αf(t) ±βg(t) ] = αa [ f(t) ] ±βa [ g(t) ] . 3. procedure solution using atm for solving linear tesfo we consider the following linear tesfo of the form: dαxu(x, t) = aut + utt + bu(x, t) + g(x, t), 0 < x < 1, (3.1) t ≥ 0, 0 < α ≤ 2 where g(x, t) is the source term and a ,b are constants. with initial condition ∂(r)u(0, t) ∂xr = u(r)(0, t) ∣∣∣ t=0 = fr(t), r = 0, 1, 2, .......,n− 1. (3.2) now applying the at into eq(3.1) we have: a [dαxu(x, t)] = a [aut + utt + bu(x, t)] + a [g(x, t)] (3.3) substituting eq(2.8) into eq(3.3) we get: vαa [u(x; t)] ] − m−1∑ k=0 u(k)(0; t) v2−α+k = a [aut + utt + bu(x, t)] + a [g(x, t)] (3.4) a [u(x; t)] = m−1∑ k=0 fk(t) v2+k + v−αa [aut + utt + bu(x, t)] + v −αa [g(x, t)] (3.5) so, according to aboodh decomposition method (adm) we can obtain the solution result u(x, t) as u(x, t) = ∞∑ n=0 un(x, t) (3.6) now, substituting eq(3.6) into eq(3.5) gives a [ ∞∑ n=0 un(x, t) ] = m−1∑ k=0 fk(t) v2+k + v −α a [ a ( ∞∑ n=0 un(x, t) ) t + ( ∞∑ n=0 un(x, t) ) tt + b ∞∑ n=0 un(x, t) ] + v −α a [g(x, t)] (3.7) from eq(3.7) we can define all the coefficients of un+1(x, t) so we get the zero coefficients u0(x, t) as: a [u0(x, t)] = m−1∑ k=0 fk(t) v2+k int. j. anal. appl. 18 (2) (2020) 247 the first component u1(x, t) as: a [u1(x, t)] = v −αa [a (u0(x, t))t + (u0(x, t))tt + bu0(x, t) + g(x, t)] finally the remaining coefficients of un+1(x, t)can be find in a way like each coefficients is found by using the coming before components. a [un+1(x, t)] = v −αa [a (un(x, t))t + (un(x, t))tt + bun(x, t) + g(x, t)] , n ≥ 0. applying the aboodh inverse to the above equations yields the following: u0(x, t) = a −1 [ m−1∑ k=0 fk(t) v2+k ] u1(x, t) = a −1 [v−αa [a (u0(x, t))t + (u0(x, t))tt + bu0(x, t) + g(x, t)]] ... un+1(x, t) = a −1 [v−αa [a (un(x, t))t + (un(x, t))tt + bun(x, t) + g(x, t)]] so that, the as un(x, t) is given as: un(x, t) = n−1∑ j=0 uj(x, t) (3.8) such that lim n→∞ un(x, t) = u(x, t) (3.9) 4. illustrative examples in this section we shall test two examples using the atm to solve the tesfo and the solutions we got it by using the present procedure will be comparing with original es. example 4.1. consider the following homogeneous tesfo dαxu(x, t) = utt + ut + u, x, t ≥ 0, 0 < α ≤ 2, (4.1) with initial conditions   u(0, t) = e−t, t ≥ 0 ux(0, t) = e −t, t ≥ 0 (4.2) we appling the at with (2.8) into (4.1) and (4.2) we get: vαa [u(x; t)] − 1∑ k=0 u(k)(0; t) v2−α+k = a [u(x, t)tt + u(x, t)t + u(x, t)] (4.3) so, we have a [u(x; t)] = e−t v2 + e−t v3 + v−αa [u(x, t)tt + u(x, t)t + u(x, t)] (4.4) int. j. anal. appl. 18 (2) (2020) 248 so, according to adm we can obtain the solution result u(x, t) as u(x, t) = ∞∑ n=0 un(x, t) substituting (3.6) into (4.4) gives a [ ∞∑ n=0 un(x, t) ] = e−t v2 + e−t v3 + v−αa [( ∞∑ n=0 un(x, t) ) )tt + ( ∞∑ n=0 un(x, t) ) )t + ( ∞∑ n=0 un(x, t) )] (4.5) according to equation (4.5), we can calculate the terms un+1(x, t) so, we get the coefficients of u0(x, t) as a [u0(x; t)] = e−t v2 + e−t v3 (4.6) so, we can use the aboodh inverse in (4.6), we get u0(x; t) = a −1 [ e−t v2 + e−t v3 ] = e−t + xe−t and in the same way we calculate the coefficients of u1(x, t) a [u1(x, t)] = v −αa [(u0(x, t)) tt + (u0(x, t)) t + (u0(x, t))] (4.7) also, we have u1(x, t) = a −1 [v−αa [(u0(x, t)) tt + (u0(x, t)) t + (u0(x, t))]] (4.8)   u1(x, t) = a −1 [v−αa [e−t + xe−t]] = a−1 [ e−t vα + e−t vα+3 ] = e−ta−1 [ 1 vα+2 + 1 vα+3 ] = e−t ( xα γ(α + 1) + xα+1 γ(α + 2) ) (4.9) we can find the coefficients of un(x, t) with the recurente relation as follows un+1(x, t) = a −1 [v−αa [(un(x, t)) tt + (un(x, t)) t + (un(x, t))]] , ∀n ≥ 0 (4.10) also, we have u2(x, t) = e −t ( x2α γ(2α + 1) + x2α+1 γ(2α + 2) ) ... ... un(x, t) = e −t ( xnα γ(nα + 1) + xnα+1 γ(nα + 2) ) (4.11) int. j. anal. appl. 18 (2) (2020) 249 finally, we obtain the approximate solution un(x, t) = e −t ( 1 + x + xα γ(α + 1) + xα+1 γ(α + 2) + x2α γ(2α + 1) + x2α+1 γ(2α + 2) + .................. ) (4.12) if we put α = 1 in (4.12), we can conclude the exact solution u(x, t) = e−t ( 1 + x + x + x2 2! + x2 2! + x3 3! + x3 3! + .................. ) = 2e−t+x −e−t (a) (1.1) (b) (1.2) (c) (1.3) figure 1. comparison between (1.1) the exact solution for α = 1 and (1.2), (1.3) the approximative solutions using 4-term of the atm for α = 1.7 and α = 1.9 respectively. int. j. anal. appl. 18 (2) (2020) 250 example 4.2. we consider a linear telegraph equation described by dαxu(x, t) = utt + 2ut + u x, t ≥ 0, 0 < α ≤ 2, (4.13) with initial conditions   u(0, t) = e−3t, t ≥ 0 ux(0, t) = 2e −3t, t ≥ 0 (4.14) we appling the at with (2.8) into (4.13) and (4.14) we get: vαa [u(x; t)] − 1∑ k=0 u(k)(0; t) v2−α+k = a [u(x, t)tt + 2u(x, t)t + u(x, t)] (4.15) so, we have a [u(x; t)] = e−3t v2 + 2 e−3t v3 + v−αa [u(x, t)tt + 2u(x, t)t + u(x, t)] (4.16) so, according to adm we can obtain the solution result u(x; t) as u(x, t) = ∞∑ n=0 un(x, t) (4.17) substituting (3.6) into (4.16) gives a [ ∞∑ n=0 un(x, t) ] = e−3t v2 + 2 e−3t v3 + v −α a [( ∞∑ n=0 un(x, t) ) )tt + 2 ( ∞∑ n=0 un(x, t) ) )t + ( ∞∑ n=0 un(x, t) )] (4.18) according to equation (4.18), we can calculate the terms un+1(x, t). so, we get the coefficients of u0(x, t) as a [u0(x; t)] = e−3t v2 + 2 e−3t v3 (4.19) we use the aboodh inverse in (4.19), we obtain u0(x; t) = e −3t + 2xe−3t and in the same way we calculate the coefficients of u1(x, t) a [u1(x, t)] = v −αa [(u0(x, t)) tt + 2 (u0(x, t)) t + (u0(x, t))] (4.20) also, we have u1(x, t) = a −1 [v−αa [(u0(x, t)) tt + 2 (u0(x, t)) t + (u0(x, t))]] (4.21) int. j. anal. appl. 18 (2) (2020) 251   u1(x, t) = a −1 [ v−αa [ 4e−3t + 8xe−3t ]] = e−3ta−1 [ 4 vα+2 + 8 vα+3 ] = 4e−3t ( xα γ(α + 1) + 2xα+1 γ(α + 2) ) (4.22) we can find the coefficients of un(x, t) with the recurrence relation as follows un+1(x, t) = a −1 [v−αa [(un(x, t)) tt + 2 (un(x, t)) t + (un(x, t))]] (4.23) also, we have   u2(x, t) = a −1 [v−αa [(u1(x, t)) tt + 2 (u1(x, t)) t + (u1(x, t))]] = 4a−1 [ v−αa [ e−3t ( 4xα γ(α + 1) + 8xα+1 γ(α + 2) )]] = 4e−3ta−1 [ 4 v2α+2 + 8 v2α+3 ] = 16e−3t ( x2α γ(2α + 1) + 2x2α+1 γ(2α + 2) ) (4.24) we can give the general solution as follow un(x, t) = 4 ne−3t ( xnα γ(nα + 1) + 2xnα+1 γ(nα + 2) ) (4.25) finally, we obtain the approximate solution un(x, t) = e −3t ( 1 + 2x + 4xα γ(α + 1) + 8xα+1 γ(α + 2) + 16x2α γ(2α + 1) + 32x2α+1 γ(2α + 2) + ....... 4nxnα γ(nα + 1) + 2 4nxnα+1 γ(nα + 2) + ..... ) (4.26) if we put α = 2 in (4.26), we can conclude the exact solution u(x, t) = e−3t ( 1 + 2x + 4x2 2! + 8x3 3! + 16x4 4! + ..... ) = e−3te2x = e−3t+2x (4.27) int. j. anal. appl. 18 (2) (2020) 252 (a) (2.1) (b) (2.2) (c) (2.3) figure 2. comparison between (2.1) the exact solution for α = 2 and (2.2), (2.3) the approximative solutions using 4-term of the atm for α = 1.7 and α = 1.9 respectively. conclusion. the application of atm was extended successfully for solving the tesfo. the atm was clearly very efficient and powerful technique in finding the approximative solution of the proposed equations. in order to check the effectiveness of the introduced procedure, two numerical examples are tested, by comparing the approximative solution with the exact solution. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (2) (2020) 253 references [1] s. kumar. a new analytical modelling for fractional telegraph equation via laplace transform. appl. math. model. 38 (13) (2014), 3154-3163. [2] s. momani. analytic and approximate solutions of space and time fractional telegraph equations. appl. math. comput., 170 (2005), 1126-1134. [3] a. yildirim. he’s homotopy perturbation method for solving the spaceand time-fractional telegraph equations. int. j. comput. math. 87 (13) (2010), 2998-3006. [4] m. elarbi benattia, k. belghaba. numerical solution for solving fractional differential equations using shifted chebyshev wavelet. gen. lett. math. 3 (2) (2017), 101-110. [5] mohammed g. s. al-safi, wurood r. abd al-hussein, ayad ghazi naser al-shammari. a new approximate solution for the telegraph equation of space-fractional order derivative by using sumudu method. iraqi j. sci. 59 (3a) (2018), 1301-1311. [6] mohammed al-safi, g.s., farah,l.j. and muna, s.a. numerical solution for telegraph equation of space fractional order using legendre wavelets spectral tau algorithm, aust. j. basic appl. sci. 10 (12) (2016), 383-391. [7] gupta, v.g., sharma, b. and kilicman, a. a note on fractional sumudu transform. journal of applied mathematics, 2010 (2010), article id 154189. [8] sunil, k. 2013. a new analytical modelling for fractional telegraph equation via laplace transform. appl. math. model. 38 (13) (2014), 3154-3163. [9] abdelbagy a. alshikh, mohand m. abdelrahim mahgob. a comparative study between laplace transform and two new integrals “elzaki” transform and “aboodh” transform. pure appl. math. j. 5 (5) (2016), 145-150. 1. introduction 2. preliminary 2.1. fundamental properties of fractional calculus 3. procedure solution using atm for solving linear tesfo 4. illustrative examples conclusion references international journal of analysis and applications volume 18, number 2 (2020), 149-160 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-149 ideals on generalized topological spaces fahad alsharari∗ department of mathematics, college of science and human studies, hotat sudair, majmaah university, majmaah 11952, saudi arabia ∗corresponding author: f.alsharari@mu.edu.sa abstract. in this paper, we define the g-closure operator and investigate some of its crucial properties. we also introduce and study the concept of ψg-classes and generalized compatibly of generalized topology with ideal. this work is generalization of [4]. 1. introduction the idea of ”idealizing” of a topological space can be found in some classical texts of kuratowski ( [14], [15]) and vaidyanathaswamy [18]. some early applications of ideal topological spaces can be found in various branches of mathematics, like a generalization of cantor-bendixson theorem by freud [10], or in measure theory by scheinberg [17]. in 1990 jankovic̀ and hamlett [13] wrote a paper in which they, among their results, included many other results in this area using modern notation, and logically and systematically arranging them. this paper rekindled the interest in this topic, resulting in many generalizations of the ideal topological space and many generalizations of the notion of open sets, like in papers of jafari and rajesh [11], and manoharan and thangavelu [16]. in 1966 velicko [19] introduced the notions of θ-open and θ-closed sets, and also a θ-closure, examining h-closed spaces in terms of an arbitrary filterbase. a space x is called h-closed if every open cover of x has a finite subfamily whose closures cover x. it turned out that θ-open sets completely correspond to the received 2019-11-22; accepted 2020-01-13; published 2020-03-02. 2010 mathematics subject classification. 54a05. key words and phrases. generalized local function; g-closure operator; ψg-classes; generalized compatibly of generalized topology with ideal. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 149 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-149 int. j. anal. appl. 18 (2) (2020) 150 already known notion of θ-continuity, introduced in 1943 by fomin [9]. in 1975 dickman and porter [8] continued the study of h-closed spaces using θ-closed sets proving that an h-closed space is not a countable union of nowhere dense θ-closed sets. in 1980, jankovic̀ [12] proved that a space is hausdorff if and only if every compact set is θ-closed. recent applications of θ-open sets can be found in the paper of caldas, jafari and latif [6], and in the paper of cammaroto, catalioto, pansera and tsaban [7]. in [5], al-omari and noiri introduced the local closure function as a generalization of the θ-closure and the local function in an ideal topological space. they proved some basic properties for the local closure function, and also introduced two new topologies obtained from the original one using the local closure function. afterwards, many properties in topological spaces have been explored by various researchers ( [20], [21], [22], [23], [24], [25]) ideal topological space is a beautiful mixture of topology and geometry. a generalized topology (briefly, gt) [1], µ on a nonempty set x is a collection of subsets of x such that φ ∈ µ and µ is closed under arbitrary unions. elements of µ will be called µ-open sets, and a subset a of (x,µ) will be called µ-closed if (x \a) is µ-open. the pair (x,µ) will be called a generalized topological space (briefly, gts) ( [2], [3]). by a space x or (x,µ), we will always mean a gts. clearly, every topological space is a gts. if u is a subset of a space (x,µ), then the µ-closure clµ(u) of µ is the intersection of all µ-closed sets containing u and the µ-interior intµ(u) of u is the union of all µ-open sets contained in µ [2]. let (x,τ) be a topological space, for each u ∈ p(x) and x ∈ x. then, u is called generalized open neighborhood of x if x ∈ u with u is µ-open set. that is ng(x) = {u is µ-open: x ∈ u}. 2. ideal generalized local function in this section we introduce and study the concept of ideal generalized topological spaces. we also investigate some of its properties. definition 2.1. an ideal generalized topological space is a generalized topological space (x,µ) with an ideal i on x and is denoted by (x,µ,i). for a subset a⊂ x, ag(i,µ) = {x ∈ x : u ∩a /∈ i for each u ∈ ng(x)}. clearly, every ideal topological space is ideal generalized topological space. lemma 2.1. for a⊆ (x,µ,i) we have, clµ(ag) = ag. proof. one implication is immediate. conversely, we show that clµ(ag) ⊆ag. let x ∈ clµ(ag). then for each µ-open set g containing x, g ∩ag 6= φ. thus, x ∈ ag(i,µ) which implies, for each generalized open neighborhood u of x, u ∩a /∈ i. therefore, (g ∪ u) ∩a /∈ i, implies x ∈ag(i,µ). hence, clµ(ag) ⊆ag. which completes the proof � int. j. anal. appl. 18 (2) (2020) 151 corollary 2.1. let (x,µ,i) be an ideal generalized topological space. then, ag(φ,µ) = clµ(a). proof. follows from lemma 2.1 � theorem 2.1. let (x,µ,i) be an ideal generalized topological space and a,b ⊆ x. then: (1) ag is µ-closed set, (2) if i1 ⊆ i2, then ag(i1) ⊇ag(i2), (3) if a⊆b, then ag ⊆bg, (4) ag ⊆ clµ(a), (5) (ag)g ⊆ag, (6) ag ∪bg = (a∪b)g, (7) ag −bg = (a−b)g, (8) if b ∈ i, then (a∪b)g = (a−b)g = ag, (9) if b ∈ i, then (x −b)g = xg, (10) if u ∈ µ, then (u ∩ag) ⊆ (u ∩a)g . proof. (1) if x /∈ag, then for some µ-open set u, we have x ∈ u and u ∩a∈ i. this implies u ⊆ x −ag, which means that x −ag is µ-open set. thus, ag is µ-closed set. (2) it is clear. (3) let x /∈ bg. then, there exists u ∈ ng(x) containing x such that u ∩ b ∈ i. this implies u ∩a⊆ u ∩b ∈ i. hence, x /∈ag . thus, ag ⊆bg. (4) follows directly from (1). int. j. anal. appl. 18 (2) (2020) 152 (5) since ag ⊆ clµ(a), (ag)g ⊆ clµ(ag). by (1), (ag)g ⊆ clµ(ag) = ag. (6) one implication is immediate from (2), that is (ag ∪bg) ⊂ (a∪b)g. to prove the reveres inclusion. let x /∈ (ag ∪bg), then x /∈ ag or x /∈ bg . then there exists u1 ∈ ng(x) and u2 ∈ ng(x) such that u1 ∩a ∈ i and u2 ∩b ∈ i. since i is hereditary and additive, then (u1 ∩ u2) ∩ (a∪b) ∈ i. thus, x /∈ (a∪b)g . hence, (ag ∪bg) ⊃ (a∪b)g. (7) since a = (a∩b) ∪ (a−b) for any a,b ⊆ x. then by (2), we have ag −bg = [(a∩b) ∪ (a−b)]g −bg = (a−b)g. (8) by using (6) and the fact that, if b ∈ i, then bg = φ. (9) obvious. (10) let x ∈ u ∩ag. then x ∈ u and x ∈ ag. this implies that there exists g ∈ ng(x) such that g∩a∈ i. since, x ∈ u ∩g, u ∩ (g∩a) ∈ i. hence, x ∈ (u ∩a)g . � theorem 2.2. let (x,µ,i) be an ideal generalized topological space and {ai}i∈j be a family of subsets of x. then: (1) ( ⋃ (ai)g : i ∈ j) ⊆ ( ⋃ ai : i ∈ j)g, (2) ( ⋂ ai : i ∈ j)g ⊆ ( ⋂ (ai)g : i ∈ j), proof. (1) since ai ⊆ ⋃ ai, for each i ∈ j, by theorem 2.1(2), we have (ai)g ⊆ ( ⋃ ai)g, for each i ∈ j. this implies ( ⋃ (ai)g : i ∈ j) ⊆ ( ⋃ ai : i ∈ j)g. (2) since ⋂ ai ⊆ai, ( ⋂ ai)g ⊆ (ai)g , for each i ∈ j. thus, ( ⋂ ai : i ∈ j)g ⊆ ( ⋂ (ai)g : i ∈ j). � now, we define the g-closure operator, denoted by cl?µ for a generalized topology µ ?(i) finer than τ as follows: cl?µ(a) = a∪ag for every a⊆ x. we will occasionally write ag or ag(i) for ag(µ,i) and it will cause no ambiguity. we will denote by int?µ(a) and cl?µ(a) the interior and closure of a⊆ (x,µ,i), respectively, with respect to µ?. int. j. anal. appl. 18 (2) (2020) 153 theorem 2.3. the class β(µ,i) = {u −e : u ∈ µ,e ∈ i} is a base for a generalized topology. proof. for every β1,β2 ∈ β, we have β1 = u1 −e1 and β2 = u2 −e2 where u1,u2 ∈ µ and e1,e2 ∈ i. then β1 ∩β2 = (u1 −e1) ∩ (u2 −e2) = (u1 ∩x −e1) ∩ (u2 ∩x −e2) = (u1 ∩u2) ∩ (x −e1 ∩x −e2) = (u1 ∩u2) ∩x − (e1 ∪e2) = (u1 ∩u2) − (e1 ∪e2) ∈ β the generalized topology which have β(µ,i) as a base is called ?-generalized topology and is denoted by µ?. � theorem 2.4. let (x,µ,i) be an ideal generalized topological space. then for each a,b ∈ (x,µ,i) we have: (1) if a⊆b, then cl?g(a) ⊆ cl?g(b). (2) cl?µ(cl ? µ(a)) ⊆ cl?µ(a). (3) cl?µ(a∪b) = cl?µ(a) ∪cl?µ(b). (4) a⊆ cl?µ(a) ⊆ clµ(a). (5) if i = φ, then, ag = clµ(a) ⊆ cl(a). (6) if i = px, then, ag = φ and a = clµ(a). proof. we shall verify only the statements (2) and the remainder of this theorem can be proved similarly (2) form theorem 2.1(5) we have cl?µ(cl ? µ(a)) = cl ? µ(a) ∪ (cl ? µ(a)µ = (a∪aµ) ∪ (a∪aµ)µ = (a∪aµ) = cl?µ(a). � theorem 2.5. if i1 and i2 are two ideals on (x,µ) such that i1 ⊆ i2, then: (1) ag(i1) ⊇ag(i2). int. j. anal. appl. 18 (2) (2020) 154 (2) µ?(i1) ⊆ µ?(i2). proof. (1) since a is a generalized open local function of i1 at x, it must also be a generalized open local function of i2 at x (since every i1 is i2). hence, ag(i1) ⊇ag(i2). (2) since i1 ⊆ i2 and by theorem 2.1(2), aµ(i1) ⊇ ag(i2). this implies that cl?µ(a)(i2,µ) ⊆ cl?µ(a)(i1,µ)). therefore, µ?(i1) ⊆ µ?(i2). � theorem 2.6. if (x,µ,i) is an ideal generalized topological space and a⊆ x. then ag−(ag)g ⊆ (a−ag)g. proof. let x ∈ ag − (ag)g. then x ∈ ag ∩ (x − (ag)g). thus, x ∈ ag that is, there exists u ∈ ng(x) such that u ∩a /∈ i. hence, for each u ∈ ng(x), u ∩ (a−ag) /∈ i, which implies x ∈ (a−ag) and this completes the proof. � theorem 2.7. let (x,µ) be a generalized topological space with i1 and i2 two ideals on x and a ⊆ x. then: (1)ag(i1 ∩ i2,µ) = ag(i1,µ) ∪ag(i2,µ). (2) ag(i1 ∪ i2,µ) = ag(i1,µ?(i2)) ∩ag(i2,µ?(i1)). proof. (1) since i1 ∩ i2 ⊆ i1, then from theorem 2.1 (2) we have ag(i1,µ) ⊆ ag(i1 ∩ i2,µ). similarly, ag(i2,µ) ⊆ag(i1 ∩ i2,µ). hence, ag(i1,µ) ∪ag(i2,µ) ⊆ag(i1 ∩ i2,µ) (2.1) to prove the reverse inclusion, let x /∈ag(i1,µ)∪ag(i2,µ), then x /∈ag(i1,µ) and x /∈ag(i2,µ), implies that there exists u1 ∈ ng(x) such that u1 ∩a∈ i1. again, x /∈ag(i2,µ), implies there exists u2 ∈ ng(x) such that u2 ∩a∈ i2.. therefore, (u1 ∩u2) ∩a∈ (i1 ∩ i2). hence, x /∈ag(i1 ∩ i2,µ), implies that ag(i1 ∩ i2,µ) ⊆ag(i1,µ) ∪ag(i2,µ). (2.2) therefore, equations (2.1) and (2.2) establish the result. (2) assume that x /∈ ag(i1 ∪ i2,µ), then there exists u ∈ ng(x) such that u ∩a ∈ i1 ∪ i2. let e ∈ i1 and h∈ i2 such that u ∩a = e∪h, because of the heredity of i, we may assume e∪h = φ. thus we have u ∩a−e = h and u ∩a−h = e. thus, (u −e) ∩a = h ∈ i1 and (u −h) ∩a = e ∈ i2. therefore, x /∈ag(i2,µ?(i1)) or x /∈ag(i1,µ?(i2)) and hence, ag(i1,µ?(i2)) ∩ag(i2,µ?(i1)) ⊆ag(i1 ∪ i2,µ). (2.3) int. j. anal. appl. 18 (2) (2020) 155 now assume that x /∈ ag(i1,µ?(i2)). this implies that there exist u ∈ ng(x) and h ∈ i2 such that (u−h)∩a∈ i1. we may assume, because of the heredity of i2, that h⊆a. put e = (u−h)∩a and we have u ∩a = e∪h∈ i1 ∪i2. thus, x /∈ag(i1 ∪i2,µ). thus we have shown that ag(i1 ∪i2,µ) ⊆ag(i1,µ?(i2)). similarly, we have that ag(i1 ∪ i2,µ) ⊆ag(i2,µ?(i1)). therefore, ag(i1 ∪ i2,µ) ⊆ag(i1,µ?(i2)) ∩ag(i2,µ?(i1)). (2.4) from (2.3) and (2.4) the result is establised. � remark 2.1. put i1 = i2 in the above theorem, the following corollary answers about the relationship between µ? and [µ?]?. corollary 2.2. let (x,µ) be a generalized topological space with i an ideal on x and a⊆ x. then: (1) ag(i1,µ) = ag(i1,µ?). (2) µ? = [µ?]? definition 2.2. a subset a of the space (x,µ,i) is said to be µ-closed set iff ag ⊆ a. equivalently, a is said to be µ?-closed iff cl?µ(a) = a. theorem 2.8. the following statements are equivalent for a subset a of a space (x,µ,i). (1) a∈ µ? . (2) a is µ?-closed set. (3) (x −a)g ⊆ (x −a). (4) a⊆ (x −a)g . proof. it is clear. � 3. ψg-classes definition 3.1. if (x,µ,i) is ideal generalized topological spaces, we define an operator ψg(µ,i) : px → µ as follows: for every a⊆ x, ψg(µ,i)(a) = {x: there exists u ∈ ng(x) such that u −a∈ i}. equivalently, ψg(µ,i)(a) = x − (x −a)g for each a⊆ x. we denote ψg(µ,i) by ψg when no ambiguity. lemma 3.1. for a⊆ (x,µ,i) we have: (1) if i = φ, then ψg(a) = intµ(a). (2) if i = px, then ψg(a) = x. int. j. anal. appl. 18 (2) (2020) 156 proof. obvious. � theorem 3.1. for a⊆ (x,µ,i) we have: : (1) ψg(a) = ∪{u ∈ µ : u −a∈ i}. (2) if u ∈ µ, then ψg(u) = ∪{m∈ µ : (m−u) ∪ (u −m) ∈ i}. proof. (1) follows immediately form definition 3.1. (2) ) if we denote ∪{m ∈ µ : (m − a) ∪ (u − m) ∈ i} by ψg′ (a)1. by heredity of i we have {m∈ µ : (m−a) ∪ (u −m) ∈ i}⊆{m∈ µ : (m−a) ∈ i} and hence by (1) we have for every a⊆ x ψg′ (a) ⊆ ψg(a). (3.1) now assume that u ∈ µ and x ∈ ψg(u). then there exists m ∈ µ such that x ∈ (m− u) ∈ i. let n = m∪u . then, n ∈ µ and x ∈ (n −u) ∪ (u −n) = (m −u) ∪φ = (m−u) ∈ i. thus, x ∈ ψg′ (u). hence ψg(u) ⊆ ψg′ (u) (3.2) from (3.1) and (3.2) we have ψg(u) = ψg′ (u), for every u ∈ µ. � theorem 3.2. let a⊆ (x,µ,i) be a generalized topological space. (1) if a⊆ x, then ψg(a) is µ-open set. (2) if a⊆b, then ψg(a) ⊆ ψg(b). (3) if a,b ∈ px, then ψg(a∩b) = ψg(a) ∩ ψg(b). (4) if u ∈ µ, then u ∈ ψg(u). (5) if a⊆ x, then ψg(a) ⊆ ψg(ψg(a)). (6) if a⊆ x, then ψg(a) = ψg(ψg(a)) iff (x −a)g = ((x −a)g)g. (7) if a⊆ i, then ψg(a) = (x −ag). (8) if a⊆ x and e ⊆ i, then ψg(a−e) = ψg(a). (9) if a⊆ x and e ⊆ i, then ψg(a∪e) = ψg(a). (10) if (a−b) ∪ (b−a) ∈ i, then ψg(a) = ψg(b). proof. (1) follows from theorem 3.2(1). int. j. anal. appl. 18 (2) (2020) 157 (2) since ψg(µ,i)(a) = x − (x −a)g ⊆ x − (x −b)g = ψg(µ,i)(b). (3) one implication is immediate, i.e. ψg(a∩b) ⊆ ψg(a) ∩ ψg(b). conversely, let x ∈ ψg(a) ∩ ψg(b), then x ∈ ψg(a) and x ∈ ψg(b) from definition 3.1, there exists u1,u2 ∈ ng(x) such that u1 −a ∈ i and u2 −b ∈ i. let u3 = u1 ∩ u2 and we have u3 −a ∈ i and u3 −b ∈ i (by heredity of i), thus u3 − (a∩b) = (u3 −a) ∪ (u3 −b) ∈ i (by additivity of i) and hence x ∈ ψg(a∩b) which implies ψg(a) ∩ ψg(b) ⊆ ψg(a∩b) and this completes the proof. (4) if u ∈ µ, then x −u is µ-cloed set which implies (x −u)g ⊆ (x −u) (theorem 2.8) and hence by definition 3.1, we have u ⊆ x − (x −u)g = ψg(u). (5) since ψg(a) is µ-open set, then by (4) we have the result. (6) since ψg(µ,i)(a) = x − (x −a)g and by hypothesis, ψg(ψg(a)) = ψg[x − (x −a)g] = x − [x − (x − (x −a)g)]g = x − [(x −a)g]g = x − [(x −a)g]g = x − (x −a)g = ψg(a) (7) follows from definition 3.1 and theorem 2.1(9). (8) from theorem 2.1(9) and definition 3.1, we have, ψg(a−e) = x − [x − (a−e)]g = x − [x − (a∩x −e)]g = x − [(x −a) ∪e]g = x − [(x −a)]g = ψg(a). (9) from theorem 2.1(9) and definition 3.1, we have, ψg(a∪e) = x − [x − (a∪e)]g = x − [(x −a) ∪e]g = x − [(x −a)]g = ψg(a). (10) follows immediately from (8) and (9). � corollary 3.1. let (x,µ,i) be an ideal generalized topological space for every a ⊆ x and e ∈ i, then, ψg(a−e) = ψg(a) = ψg(a∪e). proof. follows immediately from theorem 3.2 (8) and 3.2 (9). � theorem 3.3. let (x,µ,i) be an ideal generalized topological space, then µ? = {a⊆ x : a⊆ ψg(a)}. int. j. anal. appl. 18 (2) (2020) 158 proof. let δ = {a⊆ x : a⊆ ψg(a)}. firstly, we show that δ is a generalized topology. observe that φ ⊆ ψg(φ) and ⊆ ψg(x). now, if a,b ⊆ δ, then by theorem 3.2(3) we have a∩b ⊆ ψg(a∩b) = ψg(a)∩ψg(b), this implies that a∩b ∈ δ. if {ai}i∈γ ⊆ δ, then ai ⊆ ψg(∪ai) for every i ∈ γ. this implies ∪ai ⊆ ψg(∪ai) and we have shown that δ is a topology. secondly, we show that µ? = δ. now if u ∈ µ? and x ∈ u , there exists v ∈ µ? and e ∈ i such that x ∈ v −e ⊆ u . clearly, v − u ∈ e,. so that v − u ∈ i by heredity and hence x ∈ ψg(u). thus, u ∈ ψg(u) and we have shown µ? ⊆ δ. now, let a ∈ δ we have by definition 3.1 that a ∈ ψg(a), implies a ⊆ x − [(x −a)]g, which implies (x −a)g ⊆ (x −a), which also implies x −a is µ?-open and hence a∈ µ?. thus, µ? = δ. � 4. some forms of ?-compatible definition 4.1. if (x,µ,i) is an ideal generalized topological space. then i is said to be ?-compatible with µ denoted by i ∼ µ, if for every a ⊆ x and for every x ∈ a, there exists u ⊆ ag(x) such that u ∩a ∈ i, then a∈ i. theorem 4.1. let (x,µ,i) be an ideal generalized topological space. if i ∼ µ, then the base β(µ,i) for µ?(i) is a generalized topology and hence β(µ,i) = µ?(i) and all µ-open in µ?(i) are of simple form, i.e., µ?(i) = {u −m : u ∈ µ}. proof. follows immediately from the definition. � theorem 4.2. if i ∼ µ, then the following statements are equivalent for a subset a of a space (x,µ,i). (1) a∈ i. (2) ag = φ. (3) a∩ag = φ. proof. (1) ⇒ (2): let a ∈ i. then from hypothesis for every x ∈ x and every u ∈ ng(x), we have u ∩a∈ i. thus x /∈ag which means ag = φ. (2) ⇒ (3): obvious. (3) ⇒ (1): a /∈ i. then according to ?-compatible of µ with i, there is an x ∈ a such that for every u ∈ ng(x), u ∩a /∈ i, so x ∈ag which means a∩ag 6= φ, which is a contradiction. � int. j. anal. appl. 18 (2) (2020) 159 theorem 4.3. let (x,µ,i) be an ideal generalized topological space. then i ∼ µ. iff ψg(a) −a ∈ i, for every a⊆ x. proof. necessity let i ∼ µ and a ⊆ x. we observe that x ∈ ψg(a) −a iff x /∈ a and x /∈ (x −a)g iff x /∈ a and there exists u ∈ ng(x), such that u −a ∈ i, iff there exists x ∈ u ∈ ng(x), such that x ∈ (u −a) ∈ i. now for each x ∈ ψg(a)−a and u ∈ ng(x) and u ∩(ψg(a)−a) ∈ i, by heredity , since i ∼ µ, then (ψg(a) −a) ∈ i. sufficiency: let a ⊆ x, and let for every x ∈ a there exists u ∈ ng(x) such that (u ∩ a) ∈ i and now we prove that a ∈ i. we observe that ψg(x −a) − (x −a) = {x : ∃u ∈ ng(x) such that x ∈{[u−(x−a)] ∈ i}}. this implies ψg(x−a)−(x−a) = {x : ∃u ∈ ng(x) such that x ∈ (u∩a) ∈ i}. thus we have a⊆ ψg(x −a) − (x −a) ∈ i and hence a∈ i by the heredity of i. � theorem 4.4. let (x,µ,i) be an ideal generalized topological space with i ∼ µ. then (1)(a−ag) ∈ i, for every a⊆ x. (2) (ag)g = ag , for every a⊆ x. proof. assume that x ∈ (a − a)g , then for every u ∈ ng(x), we have (a − ag) ∩ u /∈ i or (u ∩a) −ag /∈ i. this implies (a∩u) /∈ i and so x ∈ ag . thus, x ∈ a−ag from which we conclude that (a−ag) ∩ (a−ag)g = φ, from theorem 4.2 we have a−ag ∈ i. (2) let i ∼ µ. then for every a⊆ x, a−ag ∈ i. this implies a−ag = φ, by using theorem 2.1(7), we have ag − (ag)g ⊆ (a−ag)g = φ. hence, ag ⊆ (ag)g. however, by theorem 2.1(4) we have ag ⊆ (ag)g. thus, ag = (ag)g . � theorem 4.5. let (x,µ,i) be an ideal generalized topological space with i ∼ µ. then ψg(ψg(a)) = ψg(a) for every a⊆ x. proof. from theorem 3.2(5) we have ψg(a) ⊆ ψg(ψg(a)). since i ∼ µ, then from corollary 3.1, ψg(a) = a∪e for some e ∈ i and hence by theorem 3.2(9) ψg(ψg(a)) = ψg(a∪e) = ψg(a). � acknowledgement: the author would like to express their sincere thanks to the referees for their useful suggestions and comments. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (2) (2020) 160 references [1] a. császár, generalized topology, generalized continuity, acta math. hungar. 96 (2002), 351-357. [2] a. császár, generalized open sets in generalized topologies, acta math. hun gar. 106 (12) (2005), 53-66. [3] a. császár, further remarks on the formula for -interior, acta math. hungar. 113 (2006), 325-332. [4] a. császár, modification of generalized topologies via hereditary classes, acta math. hungar. 115 (2007), 29-36. [5] a. al-omari and t. noiri, local closure functions in ideal topological spaces, novi sad j. math. 43(2) (2013), 139–149. [6] m. caldas, s. jafari and r.m. latif, sobriety via θ-open sets, an. s¸ tiint .̧ univ. al. i. cuza ias i̧. mat. (n.s.), 56(1) (2010), 163–167. [7] f. cammaroto, a. catalioto, b.a. pansera and b. tsaban, on the cardinality of the θ-closed hull of sets, topol. appl. 160(18) (2013), 2371–2378. [8] r.f. dickman jr and j.r. porter, θ-closed subsets of hausdorff spaces, pac. j. math. 59(2) (1975), 407–415. [9] s. fomin, extensions of topological spaces, ann. math. 44 (3) (1943), 471–480. [10] g. freud, ein beitrag zu dem satze von cantor und bendixson, acta math. acad. sci. hungar. 9 (1958), 333–336. [11] s. jafari and n. rajesh, generalized closed sets with respect to an ideal, eur. j. pure appl. math. 4(2) (2011), 147–151. [12] d.s. janković, on some separation axioms and θ-closure, mat. vesnik, 4 (17) (32) (72) (1980), 439-450. [13] d. janković and t.r. hamlett, new topologies from old via ideals. amer. math. mon. 97(4) (1990), 295–310. [14] k. kuratowski, topologie i, warszawa, 1933. [15] k. kuratowski, topology, vol. i, academic press, new york, london, 1966. [16] r. manoharan and p. thangavelu, some new sets and topologies in ideal topological spaces, chin. j. math. 2013 (2013), article id 973608. [17] s. scheinberg, topologies which generate a complete measure algebra, adv. math. 7 (1971), 231–239. [18] r. vaidyanathaswamy, the localisation theory in set-topology. proc. indian acad. sci., sect. a. 20 (1944), 51–61. [19] n.v. velicko, h-closed topological spaces. mat. sb. (n.s.) 70 (112) (1966) 98–112 (in russian); in: american mathematical society translations, vol. 78, american mathematical society, providence, ri, (1969), 103–118 [20] f. alsharari and y. m. saber, gθ ?τj τi -fuzzy closure operator, new math. nat. comput. in press. https://doi.org/10.1142/s1793005720500088 [21] a.m. zahran, s.e. abbas, s.a. abd el-baki and y.m. saber, decomposition of fuzzy continuity and fuzzy ideal continuity via fuzzy idealization, chaos solitons fractals, 42 (2009), 3064–3077. [22] y.m. saber and m.a. abdel-sattar, ideals on fuzzy topological spaces, appl. math. sci. 8 (2014), 1667 1691. [23] y.m. saber and f. alsharari, generalized fuzzy ideal closed sets on fuzzy topological spaces in s̆ostak sense, int. j. fuzzy logic intell. syst. 18 (3) (2018), 161-166. [24] f. alsharari and y.m. saber, separation axioms on fuzzy ideal topological spaces in s̆ostak’s sense, int. j. adv. appl. sci. 7 (2) (2020), 78 – 84. [25] y.m. saber, f. alsharari and f. smarandache, on single-valued neutrosophic ideals in s̆ostak sense, symmetry, 12 (2) (2020), 193. 1. introduction 2. ideal generalized local function 3. g-classes 4. some forms of -compatible references international journal of analysis and applications volume 18, number 1 (2020), 104-116 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-104 generating sets and a structure of the wreath product of groups with non-faithful group action ruslan v. skuratovskii1,2,∗ 1faculty of mathematics and computer science, ntuu, kpi named by ”i. sikorskiy”, pr. pobedy 37, kiev 03056, ukraine 2faculty of computer science and applied mathematics, iapm, frometovskaiy 2, kiev 03039, ukraine ∗corresponding author: r.skuratovskii@kpi.ua, ruslan@unicyb.kiev.ua, ruslcomp@mail.ru abstract. given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. we generalize the result presented in the book of j. meldrum [11] also the results of a. woryna [4]. the quotient group of the restricted and unrestricted wreath product by its commutator is found. the generic sets of commutator of wreath product were investigated. the structure of wreath product with non-faithful group action is investigated. we strengthen the results from the author [17, 19] and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. we generalise the results of meldrum j. [11] about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group a does not have to act faithfully). the commutator of such a group, its minimal generating set and the center of such products has been investigated here. the minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found. received 2019-10-03; accepted 2019-11-20; published 2020-01-02. 2010 mathematics subject classification. 20b27, 20e08, 20b22. key words and phrases. wreath product of group; minimal generating set of commutator subgroup of wreath product of groups; center of non regular wreath product; spherically homogeneous rooted tree; semidirect product. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 104 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-104 int. j. anal. appl. 18 (1) (2020) 105 1. introduction the form of commutator presentation [11] has been given here in the form of wreath recursion [10] and additionally, its commutator width has been studied. the results about commutators’s structure given in [11] were improved. lucchini a. [6] previously investigated a case of the generating set of cn−1p og, where g denotes a finite n-generated group, p is a prime which does not divide the order |g| and cp denotes the cyclic group of order p. the results of lucchini a. [6] tell us that the wreath product cn−1p o g is also n-generated. we firstly consider the active group g which is cyclic and then generalize this wreath product for both iterated wreath products and for the direct product of iterated wreath products of cyclic groups. it should be noted that to some extent a similar question for iterated wreath product was studied was studied by bondarenko i [3]. one of the goal of our research is to study the center and commutator subgroup of wreath product with non-faithful action of active group on the set. also as the goal of our paper is the minimal generating set and upper bound of minimal size of the generating set of the commutator subgroup of such class of group. the structure of center and quotient group by its commutator subgroup for a of such non-regular wreath product were still not investigated. 2. prelimenaries let g be a group. we denote by d(g) the minimal number of generators of the group g [3, 6]. the commutator width of g [14], denoted cw(g), is defined to be the least integer n, such that every element of g′ is a product of at most n commutators if such an integer exists, and otherwise is cw(g) = ∞. the estimations of the upper bound of generating set of commutator subgroup were given by [14]. the property of commutator widths for groups and elements has proven to be important and in particular, its connections with stable commutator length and bounded cohomology has become significant. meldrum j. [11] briefly considered one form of commutators of the wreath product a o b. in order to obtain a more detailed description of this form, we take into account the commutator width (cw(g)) as presented in work of muranov a. [12]. the form of commutator presentation [11] has been given here in the form of wreath recursion [10] and additionally, its commutator width has been studied. the subtree of x∗ (or t) which is induced by the set of vertices ∪ki=0x i is denoted by x[k] (or tk). denote the restriction of the action of an automorphism g ∈ autx∗ to the subtree x[l] by g(v)|x[l] . it should be noted that a restriction g(v)|x[1] is called the vertex permutation (v.p) of g in a vertex v. int. j. anal. appl. 18 (1) (2020) 106 3. minimal generating set of direct product of wreath products of cyclic groups this work strengthens previous results by the author [17] and will additionally consider a new class of groups. this class is precisely the wreath-cyclic groups and will be denoted by =. let g ∈=, then this class is constructed by formula: g = ( n0 o j0=0 ckj0 ) × ( n1 o j1=0 ckj1 ) ×···× ( nl o jl=0 ckjl ), 1 ≤ kji < ∞,ni < ∞, where the orders of cij are denoted by ij. it should be noted that at the end of this product, a semidirect product could arise with a given homomorphism φ, which is defined by a free action on the set z. in other words, one would obtain a group of the form ( k∏ i=1 gi )n nφz. note that the last group here is isomorphic to one of the fundamental orbital groups of (f) of the morse function f. namely, we have π0 (s,f |∂m ) [21]. consider now the group h = n o j=1 cij , whose orders ij for all cij are mutually coprime for all j > 1 and whose number of cyclic factors in the wreath product is finite. we will call such group h wreath-cyclic. note that the multiplication rule of automorphisms g, h which are presented in the form of wreath recursion [13] g = (g(1),g(2), . . . ,g(d))σg, h = (h(1),h(2), . . . ,h(d))σh, is given precisely by the formula: g ·h = (g1hσg(1),g2hσg(2), . . . ,gdhσg(d))σgσh. in the general case, if an active group is not cyclic, then the cycle decomposition of an n-tuple for automorphism sections will induce the corresponding decomposition of the σg. if σ is v.p of automorphism g at vij and all the vertex permutations below vij are trivial, then we do not distinguish σ from the section gvij of g which is defined by it. that is to say, we can write gvij = σ = (vij)g as proposed by bartholdi l., grigorchuk r. and šuni z. [1]. we now make use of both rooted and directed automorphisms as introduced by bartholdi l., grigorchuk r. and šuni z. [1]. recall that we denote a truncated tree by t. definition 3.1. an automorphism of t is said to be rooted if all of its vertex permutations corresponding to non-empty words are trivial. let l = x1x2x3 · · · be an infinite ray in t. definition 3.2. the automorphism g of t is said to be directed along the infinite ray l if all vertex permutations along l and all vertex permutations corresponding to vertices whose distance to the ray l is at least two are trivial. in such case, we say that l is the spine of g (as exemplified in figure 1). it should be noted that because we consider only truncated trees and truncated automorphisms here and for convenience, we will say rooted automorphism instead of truncated rooted automorphism. int. j. anal. appl. 18 (1) (2020) 107 we reformulate and generalize the result of a. woryna [4] about a minimal generating set of iterated wreath product. also we make the statement more general after this theorem. …1,1 1,2 1,k … 2,1 2,2 2,n … 2,n+1 2,n+2 2,2n 3,nm+1 3,nm+2 … 2,k(n-1)+1 2,kn … 3,nm+m+1 3,nm+2m … 4,nml+ml+1 … 4, nml+ml+l ø …1,1 1,2 1,k ø … 2,12,2 2,n … 2,kn … 2,n+1 2,2n fig. 1. directed automorphism fig. 2. rooted automorhism theorem 3.1. if orders of cyclic groups cni, cnj are mutually coprime i 6= j, then the group g = ci1 o ci2 o · · · ocim admits two generators, namely β0, β1. proof. construct the generators of n o j=0 cij as a rooted automorphism β0 (figure 2) and a directed automorphism β1 [1] along a path l (figure 1) on a rooted labeled truncated tree tx. we consider the group g = ci1 oci2 o· · ·ocim. construct the generating set of ci1 oci2 o· · ·ocim, where the active group is on the left. denote by lcm1 = lcm(i2, i3, . . . , im) the least common multiplier of the orders by i2, i3, . . . , im. in a similar fashion, we denote lcmk = lcm(i1, i2, . . . , ik−1, ik+1, . . . , im) similarly. we utilise a presentation of those wreath product elements from a tableaux of kaloujnine l. [9] which has the form σ = [a1,a2(x), a3 (x1,x2) , . . .]. additionally, we use a subgroup of tableau with length n which has the form σ(n) = [a1,a2(x1), . . .an(x1, . . . ,xn)] . the tableaux which has first n trivial coordinates was denoted in [20] by (n)σ = [e, . . . ,e,αn+1(x1, . . . ,xn),αn+1(x1, . . . ,xn+1), . . .] . int. j. anal. appl. 18 (1) (2020) 108 the canonical set of generators for the wreath product of cp o · · · o cp o cp was used by dmitruk y. and sushchanskii v. [7] and additionally utilized by the author [16]. this set has form σ′1 = [π1, e, e, . . . ,e] ,σ ′ 2 = [e1, π2, e, . . . ,e] , . . . ,σ ′ n = [e1, e, . . . ,e,πn] . (3.1) we split such a tableau into sections with respect to (3.1), where the i-th section corresponds to portrait of α at i-th level. the first section corresponds to an active group and the crown of wreath product g, the second section is separated with a semicolon to a base of the wreath product. the sections of the base of wreath product are divided into parts by semicolon and these parts correspond to groups cij which form the base of wreath product. the l-th section of of a tableau presentation of automorphism β1 corresponds to portrait of automorphism β1 on level x l. the portrait of automorphisms β1 on level x l is characterised by the sequence (e, . . . ,e,πl,e, . . . ,e), where coordinate πl is the vertex number of unique non trivial v.p on x l, the sequence has i0i1 . . . il−1 coordinates. therefore, our first generator has the form β0 = [π1,e,e, . . . ,e], which is the rooted automorphism. the second generator has the form β1 =  e; π2, e,e, . . . ,e︸ ︷︷ ︸ i1 ; i1i2︷ ︸︸ ︷ e,e, . . . ,e︸ ︷︷ ︸ i2 ,π3,e, . . . ,e; i2i3+i3︷ ︸︸ ︷ e, . . . ,e,π4,e, . . . ,e︸ ︷︷ ︸ i1i2i3 ; e, . . . ,e   , it should be noted that after the last (fourth) semicolon (or in other words before π5) there are i2i3i4 + i3i4 + i4 trivial coordinates. there are i2i3i4i5 + i3i4i5 + i4i5 + i5 trivial coordinates before π6 (or in other words after the fifth semicolon but before π6). in a section after k − 1 semicolon the coordinate of a nontrivial element πk is calculated in a similar way. we know from [20] that β1is generator of (2)g, i.e. 2-base of g. recall that (k)g calls k -th base of g. the subgroup (k)g is a subgroup of all tableaux of form (k)u with u ∈ g. let cn = 〈πn〉 and set σ1 = β0. we have to show that our generating set {β0,β1} generates the whole canonical generating set. for this, we obtain the second new generator σ2 in form of the tableau σlcm22 = β lcm2 1 =  e; πlcm22 , e,e, . . . ,e︸ ︷︷ ︸ i1 ; e,e, . . . ,e︸ ︷︷ ︸ i1i2 ; e,e, . . . ,e︸ ︷︷ ︸ i1i2i3 ; e, . . . , e   . because ord(π1) = i1 and (i1, lcm1) = 1, we find that the element π lcm1 1 is generator of ci1 since ord(π1) = ord(π lcm1 1 ). we obtain that σ2 = ( βlcm21 )lcm−12 (modi2) , which corresponds to generator σ2 of canonical generating set (3.1). observe that b3 = σ −1 1 β1 is generator of (3)g, i.e. it is precisely a 3-base of g. int. j. anal. appl. 18 (1) (2020) 109 it is known [20] that the generator σ2 precisely generates the group that is isomorphic to the group [u]2 for all 2-nd coordinate tableaux. from the same principle, one can obtain that σ3 = β lcm3 1 =  e; e, e,e, . . . ,e︸ ︷︷ ︸ i1 ; i1i2︷ ︸︸ ︷ e,e, . . . ,e︸ ︷︷ ︸ i2 ,πlcm33 ,e, . . . ,e; i2i3︷ ︸︸ ︷ e, . . . ,e,e,e, . . . ,e︸ ︷︷ ︸ i1i2i3 ; e . . . e   . this generator σ3 generates the group which is isomorphic to the group of all (2i1 +2)-th coordinate tableaux, which is precisely [u]2i1+2 [20]. making use of the same principle allows us to express all the σi from our canonical generating set. note that if it were a self-similar group, then it would be more useful to present it in terms of wreath recursion, as the set where β0 is the rooted automorphism. given a permutational representation of cij we can present our group by wreath recursion. we present β1 by wreath recursion as β1 = (π2,β2,e,e, . . . ,e). it would be written in form σlcm21 = β1 lcm2 = (π2 lcm2,β2 lcm2,e,e, . . . ,e) = (π2 lcm(2),e,e, . . . ,e), since ord(π2) = i2 and (i2, lcm2) = 1 then the element π lcm2 2 is generator of ci2 too, because ord(π2) = ord(π lcm2 2 ). we then obtain the second generator σ2 of canonical generating set by exponentiation( βlcm21 )lcm−12 (modi2) = (π2,e, . . . ,e). since we have obtained σ2 = (π2,e, . . . ,e), we can express σ −1 2 =( π−12 ,e, . . . ,e ) , where π2 is a state of σ2. consider an alternative recursive constructed generating set which consists of nested automorphism β1 states which are β2, β3,. . . ,βm and the automorphism β0. the state β2 is expressed as follows σ −1 2 β1 = (e,β2,e, . . . ,e). it should be noted that a second generator of a recursive generating set could be constructed in an other way, namely β′2 = β1 i2 = (π2 i2,β2 i2,e,e, . . . ,e) = (e,β2 i2, . . . ,e,e), where β2 is the state in a vertex of the second level x 2. we can then express the next state β2 of β1 by multiplying σ −1 2 β1 = (e,β2,e, . . . ,e). therefore, by a recursive approach, we obtain β2 = (π3,β3,e, . . . ,e) and analogously we obtain β lcm3 2 = σ lcm3 3 = (π lcm3 3 ,e, . . . ,e). similarly, we obtain β lcmk k−1 = σ lcmk k = ( π lcmk k ,e, . . . ,e ) and σk = ( β lcmk k−1 )lcm−1 k (mod ik) = (πk,e, . . . ,e). the k-th generator of the recursive generating set can therefore be expressed as σ−1k βk−1 = (e,βk,e, . . . ,e). the last generator of our generating set has another structure, namely σm = (πm,e, . . . ,e) which concludes the proof. � let n o j=0 cij be generated by β0 and β1 and m o l=0 ckl = 〈α0,α1〉. denote an order of g by |g|. int. j. anal. appl. 18 (1) (2020) 110 theorem 3.2. if (|α0|, |β0|) = 1 and (|α1|, |β1|) = 1 or (|α0|, |β1|) = 1 and (|α1|, |β0|) = 1, then there exists a generating set of 2 elements for the wreath-cyclic group g = ( n o j=0 cij ) × ( m o l=0 ckl), where ij are orders of cij . proof. the generators α1 and β1 are directed automorphisms, α0, β0 are rooted automorphisms [1]. the structure of tableaux are described above in theorem 1. in case (|α0|, |β0|) = 1 are mutually coprime and (|α1|, |β1|) = 1 are mutually coprime, then we group generator α0 and β0 in vector that is first generator of direct product ( n o j=0 cij ) × ( m o l=0 ckl). therefore, the first generator of g has form (α0, β0) and the second generator has form of vector (β1, α1 ). the generator α1 has a similar structure. in order to express the generator σ2 of the canonical set (3.1) from 〈α0,β1〉 we change the exponent from β1 to lcm2. analogously, we obtain σk = β1 lcmk which concludes the proof. � 4. center and commutator subgroup of wreath product their minimal generating sets let us find upper bound of generators number for g′. let a be a group and b a permutation group, i.e. a group a acting upon a set x, where the active group a can act not faithfully. consider the set of all pairs {(a,f),f : x → h,a ∈a}. we define a product on this set as {(a1,f1)(a2,f2) := (a1a2,f1fa12 )}, where fa21 (x) = f1(a2(x)). theorem 4.1. if w = (a,x) o (b,y ), where |x| = n, |y | = m and active group a acts on x transitively, then d (w ′) ≤ (n− 1)d(b) + d(b′) + d(a′). proof. the generators of w ′ in form of tableaux [2]: a′i = (ai; e,e,e, . . . ,e), t1 = (e; hj1,e,e, . . . ,cj1 ), . . ., tk = (e; e,e,e, . . . ,hjk,e, . . . ,cjk), tl = (e; e,e,e, . . . ,hjl,cjl), where hj,cjl ∈ sb, b = 〈sb〉, ai ∈ sa, a = 〈sa〉. note that, on a each coordinate of tableau, that presents a commutator of [a; h1, . . . ,hn] and [b; g1, . . . ,gn], a, b ∈ a,hi,gj ∈ b can be product of form a1a2a−11 a −1 2 ∈ a ′ and higa(i)h −1 ab(i) g−1 aba−1(i) ∈ b, according to corollary 4.9 [11]. this products should satisfy the following condition: n∏ i∈x higa(i)h −1 ab(i) g−1 aba−1(i) ∈b′. (4.1) that is to say that the product of coordinates of wreath product base is an element of commutator b′. as it was described above it is subdirect product of b×b×···×b︸ ︷︷ ︸ n with the additional condition (4.1). this int. j. anal. appl. 18 (1) (2020) 111 is the case because not all element of the subdirect product are independent because the elements must be chosen in such a way that (4.1) holds. we may rearrange the factors in the product in the following way: n∏ i=1 higa(i)h −1 ab(i) g−1 aba−1(i) = ( n∏ i=1 higih −1 i g −1 i )[g,h] ∈b ′. where [g,h] is a commutator in case cw(b) = 1. we express this element from b′ as commutator [g,h] if cw(b) = 1. in the general case, we would have cw(b)∏ j=1 [gj,hj] instead of this element. this commutator are formed as product of commutators of rearranged elements of n∏ i=1 higa(i)h −1 ab(i) g−1 aba−1(i) . therefore, we have a subdirect product of n the copies of the group b which has been equipped by condition (4.1). the multiplier cw(b)∏ j=1 [gj,hj] from b′, which has at least d(b′) generators n∏ i=1 higa(i)h −1 ab(i) g−1 aba−1(i) = ( n∏ i=1 higih −1 i g −1 i ) cw(b)∏ j=1 [gj,hj] ∈b′. since ( n∏ i=1 higih −1 i g −1 i ) = e and the product cw(b)∏ j=1 [gj,hj] belongs to b′, then condition (4.1) holds. the assertion of a theorem on a recursive principle is easily generalized on multiple wreath product of groups. thus minimal total amount consists of at least d (b′) generators for n − 1 factors of group b, d (b′) generators for the dependent factor from b′ and d(a) generators of the group a. it should be noted that not all the elements of commutator subgroup, that has structure of the subdirect product, are independent by (4.1), at least one of them must be chosen carefully such that would be (4.1) satisfied. this implies the estimation d (w ′) ≤ (n− 1)d(b) + d(b′). thus minimal total amount consists of at least d (b′) generators for n − 1 factors of group b, d (b′) generators for the dependent factor from b′ and d(a) generators of the group a which concludes the proof. � we shall consider special case when a passive group (b,y ) of w is a perfect group. since we obtain a direct product of n − 1 the copies of the group b then according to corollary 3.2. of wiegold j. [22] d (bn) ≤ d(b) + n− 1 [22]. more exact upper bound give us theorem a. [22], which use s a the size of the smallest simple image of g. therefore, in this case our upper bound has the form d (w ′) ≤ clogsn + d(b′) + d(a′). now we consider non-regular wreath product, where active group can be both as infinite as finite and consider a center of such group. we generalize a result of meldrum j. [11] because we consider not only the permutation wreath product groups, but the group a does not have to act on the set x faithfully, hence int. j. anal. appl. 18 (1) (2020) 112 (a,x) o b is not regular wreath product, where b is a passive group. recall that an action is said to be faithful if for every g ∈ g, there exists x from g-space x such that xg 6= x. let x = {x1,x2, . . . ,xn} be a-space. if an non faithfully action by conjugation determines a shift of copies of b from direct product bn then we have not standard wreath product (a,x) ob that is semidirect product of a and ∏ xi∈x b that is anϕ(b) n and the following proposition holds. let k = ker(a,x) that is subgroup of a that acts on x as a pointwise stabilizer, that is kernel of action of a on x. denote by z(4̃(b)) the subgroup of diagonal [5] fun(x,z(b)) of functions f : x → z(b) which are constant on each orbit of action of a on x for unrestricted wreath product, and denote by z(4(bn)) the subgroup of diagonal fun(x,z(bn)) of functions with the same property for restricted wreath product, where n is number of non-trivial coordinates in base of wreath product. proposition 4.1. a center of the group (a,x) ob is direct product of normal closure of center of diagonal of z(bn) i.e. (e ×z(4(bn))), trivial an element, and intersection of (k) ×e with z(a). in other words, z((a,x) ob) = 〈(1; h,h,. . . ,h︸ ︷︷ ︸ n ), e, z(k,x) o e〉' 〈z(a) ∩k) ×z(4(bn)〉, where h ∈ z(b), |x| = n. for restricted wreath product with n non-trivial coordinate: z((a,x) ob) = 〈(1; . . . ,h,h, . . . ,h, . . .), e, z(k,x) o e〉' (z(a) ∩k) ×z(4(bn)). in case of unrestricted wreath product we have: z((a,x) ob) = 〈(1; . . . ,h−1,h0,h1, . . . ,hi,hi+1, . . . , ), e, z(k,x) o e〉' (z(a) ∩k) ×z(4̃(b)). proof. the elements of center subgroup have to satisfy the condition: f : x → b such is constant on each orbit oj of action a on x i.e. f(x) = bi for any x ∈ oj. also every bx: bx ∈ z(b). indeed the elements of form (1; h,h,. . . ,h︸ ︷︷ ︸ n ) will not be changed by action of conjugation of any element from a because any permutation elements coordinate of diagonal of bn does not change it. also h commutes with any element of base of (a,x) ob because h from centre of b. since the action is defined by shift on finite set x, |x| = n is not faithfully, then its kernel k 6= e which confirms the proposition. also elements of subgroup (a,x)oe) belongs to z((a,x) ob) iff it acts trivial on x. � this is generalization of theorem 4.2 from the book [11] because action of a is not faithfully. example 4.1. if a = z then a centre z((z,x) ob) = 〈(1; h,h,. . . ,h︸ ︷︷ ︸ n ), e, nzne : h ∈ z(4(bn))〉. since the action defined by shift on finite set x is not faithfully, and its kernel is isomorphic to nz because cyclic shift on n coordinates is invariant on x. int. j. anal. appl. 18 (1) (2020) 113 generating set for commutator subgroup (zn ozm) ′ , where zn, zm have presentation in additive form, is the following: h1 = (0; 1, 0, . . . ,m− 1) , h2 = (0; 0, 1, 0, . . . ,m− 1) , ... hn−1 = (0; 0, . . . , 1,m− 1) . thus, it consist of n tableaux of form hi = (hi1, . . . ,him) and relations for coordinate of any tableau hi, i ∈ {1, . . . ,n− 1} is hi1 + · · · + hin−1 ≡ 0(mod m). according to theorem 3, for wreath product of abelian groups presented in multiplicative form, this relation has the form n∏ i=1 hifiπah −1 iπaπb f−1i πaπbπ −1 a [h,f] = n∏ i=1 (hifiπah −1 iπaπb f−1i πaπbπ −1 a i+2∏ j=i+1 [ hj,fjπa ] ) = e. example 4.2. if g = zn ozm is standard wreath product, then d(g′) = n− 1. let g = z ox z and g = a ox b be a restricted wreath product, where only n non-trivial elements in coordinates of base of wreath product which are indexed by elements from x, in degenerated case | x |= n. z acts on x by left shift. also a acts transitively from left. remark 4.1. the quotient group of a restricted wreath products g = z ox z by a commutator subgroup is isomorphic to z×z. in previous conditions if g = aox b then, g/g′ = a/a′×b/b′. if g = zn ozm, where (m, n) = 1, then d(g/g′) = 1. if g = zoz is an unrestricted regular wreath product then g/g′ ' z×e ' z. proof. consider the element of g = a ox b, where a can be z which acts on x by left shift, then elements of commutator subgroup has form: [e; . . . ,h−n, . . . ,h0,h1, . . . ,hn, . . . , ], where hi ∈ b. according to corollary 4.9 [11] the commutator of elements h = [a; h1, . . . ,hn], g = [b; g1, . . . ,gn], g,h ∈ g satisfies the condition (4.1), which for case where b is abelian such: n∏ i=1 higa(i)h −1 ab(i) g−1 aba−1(i) = e, where gi, hi are non trivial coordinates from base of group, a, b ∈ a, gi,hj ∈ b. the commutator with the shifted coordinate higa(i)h−1ab(i)g −1 aba−1(i) appears within the i-th coordinate position due to action of a. according to corollary 4.9 [11] the set of elements satisfying condition (4.1) forms a commutator. also the equivalent condition can be formulated: n∏ i=1 higih −1 i g −1 i ∈b ′, (4.2) int. j. anal. appl. 18 (1) (2020) 114 therefore, if b is abelian an element h of g belongs to g′ iff h satisfy a condition: n∏ i=1 hi = e. for unrestricted wreath product to show that all base of wreath product is in the commutator subgroup we choose an element [e; . . . ,h−1,h0,h1, . . .], where hi is variable, and form a commutator which is an arbitrary element [e; . . . ,g−1,g0,g1, . . .] of wreath product base: [e; . . . ,h−1,h0,h1, . . .][σ; e,e, . . . ,e][e; . . . ,h −1 −1,h −1 0 ,h −1 1 , . . .][σ −1; e,e, . . . ,e] = = [e; . . . ,g−1,g0,g1, . . .]. for convenience we present z in additive form. then to previous equality holds the following equations have to be satisfied: h0 − h1 = g0,h1 − h2 = 0,h2 − h3 = 0, .... it implies that h1 = h0 − 1, h2 = h1, h3 = h2, ... hi + 1 = hi. therefore hi = 0, i ≥ 1. from other side we have h−1 − h0 = g0,h−1 − h−2 = 0,h−2 − h−3 = 0, .... so h−i = g0, for all i < 0. that is impossible in the restricted case but possible in the unrestricted. as a corollary g/g′ ' z ×z for restricted case. thus, for unrestricted case all base of g is in g′ as a corollary g/g′ ' z ×e. thus, this group is a subdirect product of b ×b ×···×b︸ ︷︷ ︸ n with the additional condition (4.2) where, because for any element of the subgroup of coordinates there exists a surjective homomorphism acting upon b, we can conclude that g′ must be a subdirect product. the commutator subgroup is the kernel of homomorphism ϕ : g � g/g′. more precisely, g = (z,x) o (z,y ) � g/g′ ' z/z′ ×z/z′ = z×z. in case g = a o b the kerϕ has the same structure, the homomorphism ϕ maps those elements of bn, as base of g, which satisfy n∏ i=1 hi = e, i.e. the elements of b ′ in e of the group g/g′. thus, kerϕ = g′. to show that the properties of injectivity and surjectivity hold for this homomorphism, we chose the elements from g which have the form [e; e, . . .e,h,e, . . . ,e] that can be generator in canonical form of generating set of wreath product (3.1), where h /∈ g′, corresponding to a a specimen from the quotient group b/b′. also we chose independently, an element of the form [a; e, . . . ,e, . . . ,e] corresponding to a specimen of the quotient group a/a′. therefore, we must have a one-to-one correspondence between g/g′ and a/a′×b/b′. in this case, we obtain g/g′ ' [ a/a′ ×b/b′ ] . the basic property of homomorphism for generators in canonical form (3.1) is obviously accomplished. in the scenario when the action of z upon the n elements from the set is isomorphic to the action of zn elements on the set or the action of the zn elements on itself. in case g = z oz we have g/g′ ' [z×z]. for the group g = zn ozm the same is true with g/g′ ' [zn ×zm] and dependently of fact of (m,n) = 1 or not can admits one or two generators. � let f : m → r now be a c∞ morse function. let d(m) be a group of diffeomorphisms which preserve the morse function [21] f on m (möbius). consider a group h of automorphisms of critical sets xi on m which are induced by the action of diffeomorphisms h of a group d(m) which preserve the morse function f. int. j. anal. appl. 18 (1) (2020) 115 in other words, the h here are from the stabilizer s (f) /d(m). we note that the generators with stabilizers with the right action by diffeomorphisms π0s(f|xi,∂xi) are τi. the generators of the cyclic group z which define a shift are ρ. since the group action is continuous, this implies that the ρ can realize only cyclic shifts, else one would change the domains of of simple connectedness xi (critical sets) order. the group h ' zn(z)n = 〈ρ, τ〉 with defined above homomorphism in autzn has two generators and non trivial relations [18]〈 ρ,τ1, . . . ,τn ∣∣ρτi( mod n)ρ−1 = τi+1( mod n) , τiτj = τjτi, i, j ≤ n〉 . corollary 4.1. a center of the group h = znϕ(z) n is a normal closure of sets: diagonal of zn, trivial an element and subgroup that is kernel of action by conjugation of elements of zn that is (〈ρ2n〉 ' 2nz). in other words, z(h) = 〈(1; h,h,. . . ,h︸ ︷︷ ︸ n ), e, 2nz n e ' 2nz×z.〉, where h, g ∈ z. proof. since the action is defined by conjugation and relation ρ2nτiρ −2n = τi holds then the element (ρ 2n,e) commutates with every (e,τi). the stabilizer of such an action over the z-space x = {x1,x2, . . . ,x2n} is the subgroup 2nz. so subgroup stabilize all xi of z-space m. other words subgroup 〈ρ2n〉 belongs to kernel of action φ. besides the element (1; h,h,. . . ,h︸ ︷︷ ︸ n ) will not be changed by action of conjugation of any element from h because any permutation elements coordinate of diagonal of zn does not change it. thus, z(h) ' 2nz×z. � corollary 4.2. the centre of a group of the form znφ(b) n ' (z,x) o b generates, by normal closure of: center of diagonal of bn, trivial an element, and nz o x e. 5. conclusion the minimal generating set for wreath-cyclic groups have been constructed. the investigation of structure of wreath product that described in book of meldrum [11] was generalized on case of non-faithful group action of an active group. the center of wreath product, where active group action is non-faithfully. new estimations of the upper bound of generating set of commutator subgroup was obtained. acknowledgement: we thanks to antonenko alexandr for a graphical support, also we are grateful to samoilovych i. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (1) (2020) 116 references [1] laurent bartholdi, rostislav i grigorchuk, and zoran šuni. branch groups. in handbook of algebra, volume 3, elsevier, 2003, 989–1112. [2] agnieszka bier and vitaliy sushchansky. kaluzhnin’s representations of sylow p-subgroups of automorphism groups of p-adic rooted trees. algebra discr. math., 19 (2015), 19-38. [3] ievgen v bondarenko. finite generation of iterated wreath products. arch. math., 95 (4) (2010), 301–308. [4] a. woryna, the rank and generating set for iterated wreath products of cyclic groups, commun. algebra, 39 (7) (2011), 2622-2631. [5] john dixon and brian mortimer. permutation groups, springer science & business media, volume 163, 1996. [6] andrea lucchini. generating wreath products and their augmentation ideals. rend. semin. mat. univ. padova, 98 (1997), 67–87. [7] yu v dmitruk and vi sushchanskii. structure of sylow 2-subgroups of the alternating groups and normalizers of sylow subgroups in the symmetric and alternating groups. ukr. math. j., 33 (1981), 235–241. [8] i. martin isaacs. commutators and the commutator subgroup. amer. math. mon., 9 (1977), 720–722. [9] léo kaloujnine. sur les p-groupes de sylow du groupe symétrique du degré pm. c. r. acad. sci., 221 (1945), 222–224. [10] yaroslav lavrenyuk. on the finite state automorphism group of a rooted tree. algebra discr. math., 2002 (2002), 79–87. [11] john dp meldrum. wreath products of groups and semigroups, volume 74. crc press, 1995. [12] alexey muranov. finitely generated infinite simple groups of infinite commutator width. int. j. algebra comput., 17 (2007), 607–659. [13] volodymyr nekrashevych. self-similar groups, mathematical surveys and monographs, amer. math. soc., new york, volume 117, 2005. [14] nikolay nikolov. on the commutator width of perfect groups. bull. lond. math. soc., 36 (2004), 30–36. [15] vladimir sharko. smooth and topological equivalence of functions on surfaces. ukr. math. j., 55 (2003), 832–846. [16] ruslan skuratovskii. corepresentation of a sylow p-subgroup of a group s n. cybern. syst. anal., 45(1) (2009), 25–37. [17] ruslan skuratovskii. minimal generating sets for wreath products of cyclic groups, groups of automorphisms of ribe graph and fundumental groups of some morse functions orbits. in algebra, topology and analysis (summer school), (2016), 121–123. [18] ruslan skuratovskii. minimal generating set and a structure of the wreath product of groups, and the fundamental group of the orbit morse function. arxiv:1901.00061 [math.gr], (2019), 1-14. [19] ruslan skuratovskii. minimal generating sets of cyclic groups wreath product (in russian). in international conference, mal’tsev meetting, (2018), 118. [20] vitaly ivanovich sushchansky. normal structure of the isometric metric group spaces of p-adic integers. algebraic structures and their application. kiev, (1988), 113–121. [21] sergiy maksymenko. deformations of functions on surfaces by isotopic to the identity diffeomorphisms. arxiv:1311.3347 [math.gt], (2013). [22] james wiegold. growth sequences of finite groups. j. aust. math. soc., 17(2)(1974), 133–141. 1. introduction 2. prelimenaries 3. minimal generating set of direct product of wreath products of cyclic groups 4. center and commutator subgroup of wreath product their minimal generating sets 5. conclusion references international journal of analysis and applications volume 16, number 1 (2018), 16-24 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-16 generalized rough cesàro and lacunary statistical triple difference sequence spaces in probability of fractional order defined by musielak-orlicz function a. esi1 and n. subramanian2,∗ 1department of mathematics,adiyaman university,02040,adiyaman, turkey 2department of mathematics, sastra university, thanjavur-613 401, india ∗corresponding author: nsmaths@gmail.com abstract. we generalized the concepts in probability of rough cesàro and lacunary statistical by introducing the difference operator ∆αγ of fractional order, where α is a proper fraction and γ = (γmnk) is any fixed sequence of nonzero real or complex numbers. we study some properties of this operator involving lacunary sequence θ and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. the main focus of the present paper is to generalized rough cesàro and lacunary statistical of triple difference sequence spaces and investigate their topological structures as well as some inclusion concerning the operator ∆αγ . 1. introduction a triple sequence (real or complex) can be defined as a function x : n×n×n → r (c) , where n,r and c denote the set of natural numbers, real numbers and complex numbers respectively. the different types of notions of triple sequence was introduced and investigated at the initial by sahiner et al. [10,11], esi et received 31st august, 2017; accepted 28th november, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 40f05, 40j05, 40g05. key words and phrases. analytic sequence; musielak-orlicz function; triple sequences; chi sequence; cesàro summable; lacunary statistical. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 16 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-16 int. j. anal. appl. 16 (1) (2018) 17 al. [1-3], datta et al. [4],subramanian et al. [12], debnath et al. [5] and many others. a triple sequence x = (xmnk) is said to be triple analytic if supm,n,k |xmnk| 1 m+n+k < ∞. the space of all triple analytic sequences are usually denoted by λ3. a triple sequence x = (xmnk) is called triple gai sequence if ((m + n + k)! |xmnk|) 1 m+n+k → 0 as m,n,k →∞. the notion of difference sequence spaces (for single sequences) was introduced by kizmaz [6] as follows z (∆) = {x = (xk) ∈ w : (∆xk) ∈ z} for z = c,c0 and `∞, where ∆xk = xk −xk+1 for all k ∈ n. the difference triple sequence space was introduced by debnath et al. (see [5]) and is defined as ∆xmnk = xmnk −xm,n+1,k −xm,n,k+1 + xm,n+1,k+1 −xm+1,n,k + xm+1,n+1,k + xm+1,n,k+1 −xm+1,n+1,k+1 and ∆0xmnk = 〈xmnk〉 . 2. some new difference triple sequence spaces with fractional order let γ (α) denote the euler gamma function of a real number α. using the definition γ (α) with α /∈ {0,−1,−2,−3, · · ·} cab be expressed as an improper integral as follows: γ (α) = ∫∞ 0 e−xxα−1dx, where α is a positive proper fraction. we have defined the generalized fractional triple sequence spaces of difference operator ∆αγ (xmnk) = ∞∑ u=0 ∞∑ v=0 ∞∑ w=0 (−1)u+v+w γ (α + 1) (u + v + w)!γ (α− (u + v + w) + 1) xm+u,n+v,k+w. (2.1) in particular, we have (i) ∆ 1 2 (xmnk) = xmnk − 116xm+1,n+1,k+1 −··· . (ii) ∆− 1 2 (xmnk) = xmnk + 5 16 xm+1,n+1,k+1 + · · · . (iii) ∆ 2 3 (xmnk) = xmnk − 481xm+1,n+1,k+1 − ··· . now we determine the new classes of triple difference sequence spaces ∆αγ (x) as follows: ∆αγ (x) = { x : (xmnk) ∈ w3 : ( ∆αγx ) ∈ x } , (2.2) where ∆αγ (xmnk) = ∑∞ u=0 ∑∞ v=0 ∑∞ w=0 (−1)u+v+wγ(α+1) (u+v+w)!γ(α−(u+v+w)+1)xm+u,n+v,k+w and x ∈ χ3∆f (x) = χ 3 f ( ∆αγxmnk ) = µmnk ( ∆αγx ) = [ fmnk (( (m + n + k)! ∣∣∆αγ∣∣) 1m+n+k , 0̄)] . proposition 2.1. (i) for a proper fraction α, ∆α : w ×w ×w → w ×w ×w defined by equation of (2.1) is a linear operator. int. j. anal. appl. 16 (1) (2018) 18 (ii) for α,β > 0, ∆α ( ∆β (xmnk) ) = ∆α+β (xmnk) and ∆ α (∆−α (xmnk)) = xmnk. proof: omitted. proposition 2.2. for a proper fraction α and f be an musielak-orlicz function, if χ3f (x) is a linear space, then χ 3∆αγ f (x) is also a linear space. proof: omitted 3. definitions and preliminaries throughout the article w3,χ3 (∆) , λ3 (∆) denote the spaces of all, triple gai difference sequence spaces and triple analytic difference sequence spaces respectively. subramanian et al. (see [12]) introduced by a triple entire sequence spaces, triple analytic sequences spaces and triple gai sequence spaces. the triple sequence spaces of χ3 (∆) , λ3 (∆) are defined as follows: χ3 (∆) = { x ∈ w3 : ((m + n + k)! |∆xmnk|) 1/m+n+k → 0 asm,n,k →∞ } , λ3 (∆) = { x ∈ w3 : supm,n,k |∆xmnk| 1/m+n+k < ∞ } . definition 3.1. an orlicz function ([see [7]) is a function m : [0,∞) → [0,∞) which is continuous, nondecreasing and convex with m (0) = 0, m (x) > 0, for x > 0 and m (x) → ∞ as x → ∞. if convexity of orlicz function m is replaced by m (x + y) ≤ m (x) + m (y) , then this function is called modulus function. lindenstrauss and tzafriri ([8]) used the idea of orlicz function to construct orlicz sequence space. a sequence g = (gmn) defined by gmn (v) = sup{|v|u− (fmnk) (u) : u ≥ 0} ,m,n,k = 1, 2, · · · is called the complementary function of a musielak-orlicz function f. for a given musielak-orlicz function f, [see [9] ] the musielak-orlicz sequence space tf is defined as follows tf = { x ∈ w3 : if (|xmnk|) 1/m+n+k → 0 asm,n,k →∞ } , where if is a convex modular defined by if (x) = ∑∞ m=1 ∑∞ n=1 ∑∞ k=1 fmnk (|xmnk|) 1/m+n+k ,x = (xmnk) ∈ tf. we consider tf equipped with the luxemburg metric d (x,y) = ∑∞ m=1 ∑∞ n=1 ∑∞ k=1 fmnk ( |xmnk|1/m+n+k mnk ) is an exteneded real number. definition 3.2. let α be a proper fraction. a triple difference sequence spaces of ∆αγx = ( ∆αγxmnk ) is said to be ∆αγ strong cesàro summable to 0̄ if int. j. anal. appl. 16 (1) (2018) 19 limuvw→∞ 1 uvw ∑u m=1 ∑v n=1 ∑w k=1 ∣∣∆αγxmnk, 0̄∣∣ = 0. in this we write ∆αγxmnk →[c,1,1,1] ∆αγxmnk. the set of all ∆αγ strong cesàro summable triple sequence spaces is denoted by [c, 1, 1, 1]. definition 3.3. let α be a proper fraction and β be a nonnegative real number. a triple difference sequence spaces of ∆αγx = ( ∆αγxmnk ) is said to be ∆αγ rough strong cesàro summable in probability to a random variable ∆αγx : w ×w ×w → r×r×r with respect to the roughness of degree β if for each � > 0, limuvw→∞ 1 uvw ∑u m=1 ∑v n=1 ∑w k=1 p (∣∣∆αγxmnk, 0̄∣∣ ≥ β + �) = 0. in this case we write ∆αγxmnk →[c,1,1,1]p∆β ∆αγxmnk. the class of all β∆ α γ− strong cesàro summable triple sequence spaces of random variables in proability and it will be denoted by β [c, 1, 1, 1] p∆ . 4. rough cesàro summable of triple of ∆αγ in this section by using the operator ∆αγ , we introduce some new triple difference sequence spaces of rough cesàro summable involving lacunary sequences θ and arbitrary sequence p = (prst) of strictly positive real numbers. if α be a proper fraction and β be nonnegative real number. a triple difference sequence spaces of ∆αγx = ( ∆αγxmnk ) is said to be ∆αγ− rough strong cesàro summable in probability to a random variable ∆αγx : w ×w ×w → r×r×r with respect to the roughness of degree β if for each � > 0 then define the triple difference sequence spaces as follows: (i) c ( ∆αγ ,p ) θ = ∑∞ r=1 ∑∞ s=1 ∑∞ t=1 p ( fmnk [∣∣∣ 1hrst ∑(mnk)∈irst ∆αγx∣∣∣prst] ≥ β + �) < ∞. in this case we write c ( ∆αγ ,p ) θ →[c,1,1,1] p∆ β c ( ∆αγ ,p ) θ . the class of all βc ( ∆αγ ,p ) θ − rough strong cesàro summable triple sequence spaces of random variables in probability and it will be denoted by β [c, 1, 1, 1] p∆ . (ii) c [ ∆αγ ,p ] θ = ∑∞ r=1 ∑∞ s=1 ∑∞ t=1 p ( 1 hrst ∑ (mnk)∈irst fmnk [∣∣∆αγx∣∣prst] ≥ β + �) < ∞. in this case we write c [ ∆αγ ,p ] θ →[c,1,1,1] p∆ β c [ ∆αγ ,p ] θ . the class of all βc [ ∆αγ ,p ] θ − rough strong cesàro summable triple sequence spaces of random variables in probability. (iii) cλ ( ∆αγ ,p ) θ = p ( fmnk [∣∣∣ 1hrst ∑(mnk)∈irst ∆αγx∣∣∣prst] ≥ β + �) < ∞. in this case we write cλ (∆αγ ,p)θ →[c,1,1,1]p∆β cλ ( ∆αγ ,p ) θ . the class of all βcλ ( ∆αγ ,p ) θ − rough strong cesàro summable triple sequence spaces of random variables in probability. (iv) cλ [ ∆αγ ,p ] θ = 1 hrst ∑ (mnk)∈irst p ( fmnk [∣∣∆αγx∣∣prst] ≥ β + �) < ∞. in this case we write cλ [∆αγ ,p]θ →[c,1,1,1]p∆β cλ [ ∆αγ ,p ] θ . the class of all βcλ [ ∆αγ ,p ] θ − rough strong cesàro summable triple sequence spaces of random variables in probability. (v) n ( ∆αγ ,p ) θ = limrst→∞ 1 hrst ∑ (mnk)∈irst p ( fmnk [∣∣∆αγx, 0̄∣∣prst] ≥ β + �) = 0. in this case we write n (∆αγ ,p)θ →[c,1,1,1]p∆β int. j. anal. appl. 16 (1) (2018) 20 n ( ∆αγ ,p ) θ . the class of all βn ( ∆αγ ,p ) θ − rough strong cesàro summable triple sequence spaces of random variables in probability. theorem 4.1. if α be a proper fraction, β be nonnegative real number, f be an musielak-orlicz function and (prst) is a triple difference analytic sequence then the sequence spaces c ( ∆αγ ,p ) θ , c [ ∆αγ ,p ] θ , cλ ( ∆αγ ,p ) θ , cλ [ ∆αγ ,p ] θ and n ( ∆αγ ,p ) θ are linear spaces. proof: because the linearity may be proved in a similar way for each of the sets of triple sequences, hence it is omitted. theorem 4.2. if α be a proper fraction, β be nonnegative real number, f be an musielak-orlicz function and (prst) , for all r,s,t ∈ n, then the triple difference sequence spaces c [ ∆αγ ,p ] θ is a bk-space with the luxemburg metric is defined by d (x,y)1 = ∑∞ u=0 ∑∞ v=0 ∑∞ w=0 fmnk [ γuvwxuvw uvw ] + limuvw→∞ 1 uvw ∑∞ r=1 ∑∞ s=1 ∑∞ t=1 fmnk [ p ( 1 hrst ∑ (m,n,k)∈irst ∣∣∆αγx∣∣p) ≥ β + �]1/p , 1 ≤ p. also if prst = 1 for all (r,s,t) ∈ n, then the triple difference spaces cλ [ ∆αγ ,p ] θ and n ( ∆αγ ,p ) θ are bkspaces with the luxemburg metric is defined by d (x,y)2 = ∑∞ u=0 ∑∞ v=0 ∑∞ w=0 fmnk [ γuvwxuvw uvw ] + limuvw→∞ 1 uvw 1 hrst ∑ (m,n,k)∈irst fmnk [ p (∣∣∆αγx∣∣) ≥ β + �] . proof: we give the proof for the space cλ [ ∆αγ ,p ] θ and that of others followed by using similar techniques. suppose (xn) is a cauchy sequence in cλ [ ∆αγ ,p ] θ , where xn = (xij`) n and xm = ( xmij` ) are two elements in cλ [ ∆αγ ,p ] θ . then there exists a positive integer n0 (�) such that |xn −xm| → 0 as m,n → ∞. for all m,n ≥ n0 (�) and for each i,j,` ∈ n. therefore  x11uvw x 12 uvw ... ... x21uvw x 22 uvw ... ... . . .   and   ∆αγx 11 ij` ∆ α γx 12 ij` ... ... ∆αγx 21 ij` ∆ α γx 22 ij` ... ... . . .   are cauchy sequences in complex field c and cλ [ ∆αγ ,p ] θ respectively. by using the completness of c and cλ [ ∆αγ ,p ] θ we have that they are convergent and suppose that xnij` → xij` in c and ( ∆αγx n ij` ) → yij` in cλ [ ∆αγ ,p ] θ for each i,j,` ∈ n as n → ∞. then we can find a triple sequence space of (xij`) such that yij` = ∆ α γxij` for i,j,` ∈ n. these xsij` can be interpreted as xij` = 1 γij` ∑i−m u=1 ∑j−n v=1 ∑`−k w=1 ∆ α γyuvw = 1 γij` ∑i u=1 ∑j v=1 ∑` w=1 ∆ α γyu−m,v−n,w−k, (y1−m,1−n,1−k = y2−m,2−n,2−k = · · · = y000 = 0) . for sufficiently large (i,j,`) ; that is, int. j. anal. appl. 16 (1) (2018) 21 ( ∆αγx n ) =   ∆αγx 11 ij` ∆ α γx 12 ij` ... ... ∆αγx 21 ij` ∆ α γx 22 ij` ... ... . . .   converges to ( ∆αγxij` ) for each i,j,` ∈ n as n →∞. thus |xm −x|2 → 0 as m →∞. since cλ [ ∆αγ ,p ] θ is a banach luxemburg metric with continuous coordinates, that is |xn −x|2 → 0 implies ∣∣∣xnij` −xij`∣∣∣ → 0 for each i,j,` ∈ n as n →∞, this shows that cλ [ ∆αγ ,p ] θ is a bk-space. theorem 4.3. if α be a proper fraction, β be nonnegative real number, f be an musielak-orlicz function and (prst) , for all r,s,t ∈ n, then the triple difference sequence space c ( ∆αγ ,p ) θ is a bk-space with the luxemburg metric is defined by d (x,y)3 = ∑∞ u=0 ∑∞ v=0 ∑∞ w=0 fmnk [ γuvwxuvw uvw ] + limuvw→∞ 1 uvw ∑∞ r=1 ∑∞ s=1 ∑∞ t=1 fmnk [ p (∣∣∣ 1hrst ∑(m,n,k)∈irst ∆αγx∣∣∣p) ≥ β + �]1/p , 1 ≤ p. also if prst = 1 for all (r,s,t) ∈ n, then the triple difference spaces cλ ( ∆αγ ,p ) θ is a bk-spaces with the luxemburg metric is defined by d (x,y)4 = ∑∞ u=0 ∑∞ v=0 ∑∞ w=0 fmnk [ γuvwxuvw uvw ] + limuvw→∞ 1 uvw fmnk [ p (∣∣∣ 1hrst ∑(m,n,k)∈irst ∆αγx∣∣∣) ≥ β + �] . proof: the proof follows from theorem 4.2. now, we can present the following theorem, determining some inclusion relations with out proof, since it is a routine verification. theorem 4.4. let α,ξ be two positive proper fractions α > ξ > 0 and β be two nonnegative real number, f be an musielak-orlicz function and (prst) = p , for each r,s,t ∈ n be given.then the following inclusions are satisfied: (i) c ( ∆ξγ,p ) θ ⊂ c ( ∆αγ ,p ) θ , (ii) c [ ∆ξγ,p ] θ ⊂ c [ ∆αγ ,p ] θ , (i) c ( ∆αγ ,p ) θ ⊂ c ( ∆αγ ,q ) θ , 0 < p < q. 5. rough lacunary statistical convergence of triple of ∆αγ in this section by using the operator ∆αγ , we introduce some new triple difference sequence spaces involving rough lacunary statistical sequences spaces and arbitrary sequence p = (prst) of strictly positive real numbers. int. j. anal. appl. 16 (1) (2018) 22 definition 5.1. the triple sequence θi,`,j = {(mi,n`,kj)} is called triple lacunary if there exist three increasing sequences of integers such that m0 = 0,hi = mi −mr−1 →∞ as i →∞ and n0 = 0,h` = n` −n`−1 →∞ as ` →∞. k0 = 0,hj = kj −kj−1 →∞ as j →∞. let mi,`,j = min`kj,hi,`,j = hih`hj, and θi,`,j is determine by ii,`,j = {(m,n,k) : mi−1 < m < mi andn`−1 < n ≤ n` andkj−1 < k ≤ kj} ,qi = mimi−1 ,q` = n` n`−1 ,qj = kj kj−1 . definition 5.2. let α be a proper fraction, f be an musielak-orlicz function and θ = {mrnskt}(rst)∈n ⋃ 0 be the triple difference lacunary sequence spaces of ( ∆αγxmnk ) is said to be ∆αγ− lacunary statistically convergent to a number 0̄ if for any � > 0, limrst→∞ 1 hrst ∣∣{(m,n,k) ∈ irst : fmnk [∣∣∆αγxmnk, 0̄∣∣] ≥ �}∣∣ = 0 , where ir,s,t = {(m,n,k) : mr−1 < m < mr andns−1 < n ≤ ns andkt−1 < k ≤ kt} ,qr = mrmr−1 ,qs = ns ns−1 ,qt = kt kt−1 . in this case write ∆αγx →sθ ∆αγx. definition 5.3. if α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and θ = {mrnskt}(rst)∈n ⋃ 0 be the triple difference sequence spaces of lacunary. a number x is said to be ∆αγ −nθ− convergent to a real number 0̄ if for every � > 0, limrst→∞ 1 hrst ∑ m∈ir ∑ n∈is ∑ k∈it fmnk [∣∣∆αγxmnk, 0̄∣∣] = 0. in this case we write ∆αγxmnk →nθ 0̄. definition 5.4. let α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers. a triple difference sequence spaces of random variables is said to be ∆αγ− rough lacunary statistically convergent in probability to ∆αγx : w ×w × w → r×r×r with respect to the roughness of degree β if for any �,δ > 0, limrst→∞ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ = 0 and we write ∆αγxmnk →spβ 0̄. it will be denoted by βspθ . definition 5.5. let α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers. a triple difference sequence spaces of random variables is said to be ∆αγ− rough nθ− convergent in probability to ∆αγx : w ×w ×w → r×r×r with respect to the roughness of degree β if for any � > 0, limrst→∞ 1 hrst ∑ m∈ir ∑ n∈is ∑ k∈it∣∣{p ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �)}∣∣ = 0, and we write ∆αγxmnk →npθβ ∆αγx. the class of all β −nθ− convergent triple difference sequence spaces of random variables in probability will be denoted by βnpθ . definition 5.6. let α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers. a triple difference sequence spaces of random variables is said to be ∆αγ− rough lacunary statistically cauchy if there exists a number n = n (�) int. j. anal. appl. 16 (1) (2018) 23 in probability to ∆αγx : w ×w ×w → r×r×r with respect to the roughness of degree β if for any �,δ > 0, limrst→∞ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk −xn )∣∣)]prst ≥ β + �) ≥ δ}∣∣ = 0. theorem 5.1. let α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers, 0 < p < ∞. (i) if (xmnk) → ( n ( ∆αγ ,p ) θ ) for prst = p then (xmnk) → ( ∆αγ (sθ) ) . (ii) if x ∈ ( ∆αγ (sθ) ) , then (xmnk) → ( n ( ∆αγ ,p ) θ ) . proof: let x = (xmnk) ∈ ( n ( ∆αγ ,p ) θ ) and � > 0, ∣∣{p ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �)}∣∣ = 0. we have 1 hrst ∑ (mnk)∈irst ∣∣{p ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �)}∣∣ ≥ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣(β+�δ )p . so we observe by passing to limit as r,s,t →∞, limrst→∞ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ ≤( δ α+� )p p ( limrst→∞ 1 hrst ∑ (m,n,k)∈irst ∣∣∆αγxmnk∣∣p) = 0. which implies that xmnk → (∆αγ (sθ)) . suppose that x ∈ ∆αγ ( λ3 ) and (xmnk) → ( ∆αγ (s) ) . then it is obvious that ( ∆αγx ) ∈ λ3 and 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �) ≥ δ}∣∣ → 0 as r,s,t → ∞. let � > 0 be given and there exists u0v0w0 ∈ n such that∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]prst ≥ β + �2) ≥ δ2}∣∣ ≤ �2(d(∆αγx,y))λ3 + δ2, where ∑∞ u=1 ∑∞ v=1 ∑∞ w=1 |γuvwxuvw| = 0, for all r ≥ u0,s ≥ v0, t ≥ w0. further more, we can write∣∣∆αγxmnk∣∣ ≤ d(∆αγxmnk,y)∆αγ ≤ d(∆αγx,y)λ3 = d (x,y)∆αγx. for r,s,t ≥ u0,v0,w0. 1 hrst ∑ (mnk)∈irst p ([ fmnk (∣∣∆αγxmnk∣∣)]p) = 1hrst p (∑(mnk)∈irst [fmnk (∣∣∆αγxmnk∣∣)]p) + 1 hrst p (∑ (mnk)/∈irst [ fmnk (∣∣∆αγxmnk∣∣)]p) < 1hrst p ( hrst ( � 2 + δ 2 ) + hrst � d(x,y) p ∆αγx 2 d(x,y) p ∆αγx + δ 2 ) = � + δ. hence (xmnk) → ( n ( ∆αγ ,p ) θ ) . corollary 5.1. if α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers then the following statements are hold: (i) s ⋂ λ3 ⊂ ∆αγ (sθ) ⋂ ∆αγ ( λ3 ) , (ii) ∆αγ (sθ) ⋂ ∆αγ ( λ3 ) = ∆αγ ( w3p ) . theorem 5.2. let α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers. if x = (xmnk) is a ∆ α γ− triple difference rough lacunary statistically convergent sequence, then x is a ∆αγ− triple difference rough lacunary statistically cauchy sequence. proof: assume that (xmnk) → ( ∆αγ (sθ) ) and �,δ > 0. then 1 δ ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �2)}∣∣ for almost all m,n,k and if we select n, then 1 δ ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxn∣∣)]prst ≥ β + �2)}∣∣ holds. now, we have∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk −xn )∣∣)]prst)}∣∣ ≤ 1 δ ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxmnk∣∣)]prst ≥ β + �2)}∣∣ + int. j. anal. appl. 16 (1) (2018) 24 1 δ ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxn∣∣)]prst ≥ β + �2)}∣∣ < 1δ (β + �) = �, for almost m,n,k. hence (xmnk) is a ∆αγ− rough lacunary statistically cauchy. theorem 5.3. if α be a proper fraction, β be nonnegative real number,f be an musielak-orlicz function and arbitary sequence p = (prst) of strictly positive real numbers and 0 < p < ∞, then n ( ∆αγ ,p ) θ ⊂ ∆αγ (sθ) . proof: suppose that x = (xmnk) ∈ n ( ∆αγ ,p ) θ and∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxmnk∣∣)]p ≥ β + �)}∣∣. therefore we have 1 hrst ∑ (mnk)∈irst p ([ fmnk (∣∣∆αγxmnk∣∣)]p) ≥ 1hrst ∑(mnk)∈irst (β + �)p ≥ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγxmnk∣∣)]p ≥ β + �)}∣∣ (β + �)p . so we observe by passing to limit as r,s,t →∞, limrst→∞ 1 hrst ∣∣{(m,n,k) ∈ irst : p ([fmnk (∣∣∆αγ (xmnk)∣∣)]p ≥ β + �) ≥ δ}∣∣ < 1 (β+�)p ( p ( limrst→∞ 1 hrst ∑ (m,n,k)∈irst [ fmnk (∣∣∆αγ (xmnk)∣∣)]p)) = 0 implies that x ∈ ∆αγ (sθ) . hence n ( ∆αγ ,p ) θ ⊂ ∆αγ (sθ) . competing interests: the authors declare that there is not any conflict of interests regarding the publication of this manuscript. references [1] a. esi , on some triple almost lacunary sequence spaces defined by orlicz functions, res. rev., discr. math. struct., 1(2) (2014), 16-25. [2] a. esi and m. necdet catalbas,almost convergence of triple sequences, glob. j. math. anal., 2(1) (2014), 6-10. [3] a. esi and e. savas, on lacunary statistically convergent triple sequences in probabilistic normed space,appl. math. inf. sci., 9 (5) (2015), 2529-2534. [4] a. j. datta a. esi and b.c. tripathy,statistically convergent triple sequence spaces defined by orlicz function , j. math. anal., 4(2) (2013), 16-22. [5] s. debnath, b. sarma and b.c. das ,some generalized triple sequence spaces of real numbers , j. nonlinear anal. optim., 6(1) (2015), 71-79. [6] h. kizmaz , on certain sequence spaces, can. math. bull., 24(2) (1981), 169-176. [7] p.k. kamthan and m. gupta, sequence spaces and series, lecture notes, pure and applied mathematics, 65 marcel dekker, inc. new york , 1981. [8] j. lindenstrauss and l. tzafriri, on orlicz sequence spaces, israel j. math., 10 (1971), 379-390. [9] j. musielak, orlicz spaces,lectures notes in math.,1034, springer-verlag, 1983. [10] a. sahiner, m. gurdal and f.k. duden, triple sequences and their statistical convergence, selcuk j. appl. math. , 8 no. (2)(2007), 49-55. [11] a. sahiner, b.c. tripathy , some i related properties of triple sequences, selcuk j. appl. math. , 9(2)(2008), 9-18. [12] n. subramanian and a. esi, the generalized tripled difference of χ3 sequence spaces, glob. j. math. anal., 3(2) (2015), 5460. 1. introduction 2. some new difference triple sequence spaces with fractional order 3. definitions and preliminaries 4. rough cesro summable of triple of 5. rough lacunary statistical convergence of triple of references international journal of analysis and applications volume 17, number 6 (2019), 994-1018 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-994 received 2019-05-06; accepted 2019-07-03; published 2019-11-01. 2010 mathematics subject classification. 91b02. key words and phrases. hospitality; evaluation; forecasting; dea; gm(1,1). ©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 994 applying dea model to measure the efficiency of hospitality sector: the case of vietnam nhu-ty nguyen*, linh-xuan thi nguyen school of business, international university – vietnam national university hcmc; quarter 6, linh trung ward, thu duc district, hcmc, vietnam; e-mail: nhutynguyen@hcmiu.edu.vn (n.-t., nguyen) *corresponding author: nhutynguyen@hcmiu.edu.vn abstract. tourism industry is one of the world’s largest industries with a global economic contribution of over 7.6 trillion dollars in 2016 which provides an equal or even surpasses the business volume of oil exports, and food and beverage.as the current climate of the globe, vietnam’s tourism in general, hospitality in particular has attracted investment from not only domestic enterprises but many international hospitality corporations which create a fierce competitive than ever.identifying inefficient activities and providing improvement in whole process is crucial. the present research aims to study and evaluate the performance of vietnam hospitality industry through 20 chosen companies that qualify criteria of data envelopment analysis (dea) model and malmquist productivity index. it would be a useful tool in benchmarking the efficient firms and inefficient ones operating in the industry and help the former to improve their efficiency. the researcher uses 5 input variables (cost of good sales; sales expense; operation expense; fixed assets and owner equity) and 2 output variables (revenues and profit after tax).dmu1 and dmu8 face with huge fluctuation in efficiency which acquires the management board to review and improve their operation process to ensure the sustainable development of the firm in current competitive market. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-994 mailto:nhutynguyen@hcmiu.edu.vn int. j. anal. appl. 17 (6) (2019) 995 1. introduction nowadays, tourism industry is one of the world’s largest industries with a global economic contribution of over 7.6 trillion dollars in 2016 which provides an equal or even surpass the business volume of oil exports, and food and beverage. tourism has significant grown and various diversification to be one of the fastest growing industry. the development of national economy and encircles a growing number of new destinations encouraged the development of modern tourism. as the current climate of the globe, vietnam’s tourism in general, hospitality in particular has attracted investment from not only domestic enterprises but many international hospitality corporations which create a fierce competitive than ever. hospitality’s benefit will go toward the development of tourist and the competition can be expected. global hospitalities like six sense, sheraton, nikko, ect. has joined to the competition. hotel managers agree that enhancing their performance can be their advantages and those advantages can be identified with the competitive benchmarking ([1]). as the nature of hotel service is simultaneous and perishable, how to manage the customer demands and service capacity will affect in the profitability of company ([2]). how to efficiently operate in comparation with industrial benchmark and rivals is crucial in this industry. identifying inefficient activities and providing improvement in whole process is crucial. the present research aims to study and evaluate the performance of vietnam hospitality industry through 20 chosen companies that qualify criteria of data envelopment analysis (dea) model and malquist productivity index. it would be a useful tool in benchmarking the efficient firms and inefficient ones operating in the industry and help the former to improve their efficiency the objectives of the research are summarized as following: b. to evaluate the efficiency of 20 listed firms in vietnam hospitality industry using the data for the past 5 years from 2013 to 2017 c. to forecast the performance of those 20 dmus in the next 5 years from 2018 to 2022 and then use the forecasted data an input to evaluate their performances d. to compare the efficiency of the past and the future between the set of data 20132017 and the set of data 2018-2022 in order to evaluate the past-to-future performance. int. j. anal. appl. 17 (6) (2019) 996 2. literature review the grey system theory, introduced by professor deng julong in 1988 whichis well known as beneficial tool to take uncertain system with “small sample and poor information” as the research object. moreover, it is a useful method to forecast and has been applied successfully in many fields and present satisfaction results. recently, the application of the systems has be successfully employed in calculating of input and output values of organizations, agriculture, meteorological, forestry and disaster predictions. the method is not complicated and be used easily. the small set of data can be used rather than a large number of data. therefore, it is practically to apply the method in the paper. data envelopment analysis (dea) is popular in new researchers which is suggested firstly in 1978 bycharnes, cooper and rodes. dea is a nonparametric method in which the researcher is allowed to use multiple inputs and outputs. the inputs and outputs are combined, the relative efficiency for a whole organization or parts of an organization (or called decisionmaking units [dmus]) is calculated. in a sample of dmus, the best performing ones is identified. an efficiency frontier is, the dmus on the frontier are efficient (best practice) and the ones that are below efficiency frontier are inefficient. the efficiency index will have values from 0 to 1 (or 0 to 100%). the result 1 implied the efficient unit and vice versa the result less than 1 represents the inefficient. the dmus in dea in the sample have similar activities so that a similar group of inputs and outputs can be identified. moreover, the dmus have to operate in a similar environment ([3]). different from the traditional accounting method, dea model has the benefits that it is proper to compare the relative performance between multiple performance measures ([4]). there are two approaches in dea: (1) the input-oriented approach and (2) the out-put oriented approach. the input-oriented approach argues minimization of input for the given outputs. meanwhile, the output-oriented approach is the maximization of outputs for the given inputs. table 2.1 indicates several previous studies that applied dea model in evaluating the performance and efficiency of companies in hospitality industry in different countries and some similar industries as well. int. j. anal. appl. 17 (6) (2019) 997 table 2. 1 table of input and output author input output johns, howcroft, and drake (1997) [5] available room nights, labourtimes , food and beverage costs and utilities cost number of room night sold, reserved cost, food and beverage revenue hwang and chang (2003) [6] number of full-time employees, number of rooms, food and beverage department areas, operating expenses revenue of each divisions: room, food and beverage; and other revenue chiang, tsai, and wang (2004) [7] number of hotel rooms, food and beverage department areas, employees, total cost yielding index, food and beverage revenue, other revenue barros (2005) [8] number of full-time employees, cost of labour, rooms, hotel areas, property’s book value, operation expenses, external expense sales revenue, number of guests, and number of night stay barros and santos (2006) [9] number of full-time employees, cost of labour, capital sales revenue, added value, earnings önüt and soner (2006) [10] number of full-time employees, consumption of electricity, water and liquefied occupancy rate, sales revenue, number of guests chen (2007) [11] cost of labour , cost of f&b, cost of materials total revenue davutyan (2007) [12] number of available beds, number of fulltime employees, operating cost beds sold to return customers divided by number of available beds, beds sold min et al. (2008) [13] sales expense, total labour cost, operation expenses and non-operating expenses revenue of each divisions: room, food and beverage; and other revenue barros et al. (2009) [14] number of full-time employees, physical capital sales revenue, added value neves and lourenco (2009) [15] cost of goods and services, current assets, net fixed assets, andowner equity, revenues and earnings (ebitda) perrigot et al. (2009) [16] hotel establish ages, number of labour, number of rooms. number of hotel openings during the year, franchising contract: royalties in percentage, chain ranking sales revenue, room revenues, other revenues and occupancy rate int. j. anal. appl. 17 (6) (2019) 998 yu and lee (2009) [17] full-time employees in each department: the room service department, the f&b department, number of hotel rooms, floor area in the f&b service department; cost for each service sector, shared input revenue of each divisions: room, food and beverage; and other revenue chen, hu, and liao (2010) [18] number of hotel rooms; number of employees; floor area in the f&b service department revenue of each divisions: room, food and beverage; and other revenue hsieh and lin (2010) [19] room expenses, number of employees of the room department, food and beverage cost, employees of food and beverage department; area of rooms, catering floors revenue of room service, food and beverage service hsieh et al. (2010) [20] number of hotel rooms; number of employees, facilities expenses, operation expenses occupancy rate, sales revenue assaf and magnini (2011) [21] number of hotel rooms; number of employees, operational costs occupancy rate, sales revenue avkiran (2011) [22] number of full-time staff, permanent parttime staff, number of room sales revenue and a double room cost chen (2011) [23] number of employees, area of floors, guest rooms, operation expenses, and depreciation expenses occupancy rate, number of guests and guest satisfaction index, room revenue, other revenue yen and othman (2011) [24] number of full-time employees, cost of labour, rooms, hotel areas, property’s book value, operation expenses, external expense no nights occupied, number of guests; occupancy rate, revenue of room service, food and beverage service honma and hu (2012) [25] number employees, number of temporary staff, number of seats in restaurants and bars, number of rooms sales revenue manasakis et al. (2013) [26] no. of employees, number employees, operation expenses sales revenues and total number of spent nights katarina poldrugovac, metkatekavcic& sandra jankovic (2016) [27] expenses of each division: room, energy, f&b, labor and other sales revenue and occupancy rate int. j. anal. appl. 17 (6) (2019) 999 3. methodologies 3.1. collecting dmus the research was only conducted 20 companies whose financial reports are audited by reliable institutions and collected from vietnam stock exchanges market or company’s official website from 2013 to 2017 and denoted from dmu1 to dmu20 as the order in the table 3.1. in addition, the financial result of vietnam hospitality companies from 2013 to 2017 are also generated in the bellow table: 4. table 3. 2 decision making unit number code company name denote 1 dmu1 ben thanh tourist and service jsc btv 2 dmu2 dong a hotel corp jsc dah 3 dmu3 dic tourist and trading jsc dcd 4 dmu4 lang son exim tourist jsc dxl 5 dmu5 commercial and service join stock jsc investment power ein 6 dmu6 hoi an tourist service co hot 7 dmu7 post hotel jsc nph 8 dmu8 ninh van bay travel real estale jsc nvt 9 dmu9 ocean hotel & service jsc och 10 dmu10 petroleum phuongdong tourism jsc pdc 11 dmu11 sai gon hotel jsc sgh 12 dmu12 thuy ta jsc ttj 13 dmu13 vung tau intourco resort vir 14 dmu14 thanh cong tourist & service jsc vng 15 dmu15 corporation tourist of baria vung tau vtg 16 dmu16 dak lak tourist jsc. dld 17 dmu17 my tra tourist & service co mtc 18 dmu18 the national oil service jsc of vietnam oscvn 19 dmu19 dong nai tourist co dnt 20 dmu20 kim lien tourist co kimlien int. j. anal. appl. 17 (6) (2019) 1000 3.2 dea malmquist productivity index one of the standard approaches for measuring productivity that applied in many researches overtime is the malmquist productivity index, especially when nonparametric specifications are applied to micro data. malmquist productivity index was first proposed by caves, christensen and diewert (1982) and then further modified later by färe, grosskopf, lidgren and roos in 1995. malmquist productivity index (mpi) is a tool for measurement of productivity changes of a dmu over periods of time. it is defined as the product of “catch-up” and “frontier-shift” terms. the catch-up term is the degree of efforts that the dmu attained for improving its efficiency, while the frontier-shift term reflects the change in the efficient frontiers surrounding the dmu between the two time periods 1 and 2. dmu0 at periods 1 and 2 is denoted by (𝑥0 1 , 𝑦0 1) and (𝑥0 2 , 𝑦0 2). the efficiency score of dmu (𝑥0, 𝑦𝑜 ) 𝑡1 is measured by the technological frontier t2: 𝑑𝑡2 ((𝑥0, 𝑦0) 𝑡1 )(𝑡1 = 1,2 𝑎𝑛𝑑 𝑡2 = 1,2) c stands for the efficiency change (catchup effect) and is determined by the following formula: 𝐶 = 𝑑2((𝑥0, 𝑦𝑜 ) 2) 𝑑1((𝑥0, 𝑦𝑜 ) 1) the technological change (frontier-shift effect) denoted by f has the formula: 𝐹 = [ 𝑑1((𝑥0, 𝑦𝑜 ) 1) 𝑑2((𝑥0, 𝑦𝑜) 1) . 𝑑1((𝑥0, 𝑦𝑜 ) 2) 𝑑2((𝑥0, 𝑦𝑜 ) 2) ] 1/2 malmquist productivity index (mpi) is the product of c and f, that is, mpi = (catch-up) x (frontier-shift) or 𝑀𝑃𝐼 = [ 𝑑1((𝑥0, 𝑦𝑜 ) 2) 𝑑1((𝑥0, 𝑦𝑜 ) 1) . 𝑑2((𝑥0, 𝑦𝑜) 2) 𝑑2((𝑥0, 𝑦𝑜) 1) ] 1/2 if the malmquist productivity index (mpi) is greater than 1 (mpi>1), it indicates progress in relative efficiency from period 1 to period 2. productivity remains unchanged if mpi equal to 1 (mpi=1) and it demonstrate a regress when mpi is less than 1 (mpi<1). 3.3 establishing inputs and outputs together with the financial report of hotel in vietnam, there are five inputs chosen are cost of good sales, sales expenses, operation expenses (including electricity and labor cost), fixed assets and equity. the reasons for this range choosing as bellow: cost of good sales is direct cost that contribute to goods and service of hospitality firm. int. j. anal. appl. 17 (6) (2019) 1001 fixed asset is a long-term tangible piece of property that a firm owns and uses in its operations to generate income. a fixed asset is bought for production or supply of goods or services, for rental to third party or for use in the organization. it has a physical form and is reported on the balance sheet as property, plant and equipment (pp&e). in this research which related to hospitality, fixed asset is considered to be important due to the information it provides. sales expenses are used for selling activities like incentive, marketing expense. in a hotel, marketing department is indirect factor to raise the sales revenue. operation expenses includes energy expenses, labor cost which is direct factor which create service to provide for customer. in hotel industry, labor or quality of human resource plays a significant role to generate profit. equity or owner’s equity presents the owner’s fund for business in various operation. two chosen outputs factors are considered as sales and profit after tax (pat). sales are the transactions between parties where the buyers receive goods (tangible or intangible), services and/or assets in exchange for money. it can also refer to an agreement between the buyer and seller of the selected good or service. profit after tax is defined as the net amount earned by a business after all taxation related expenses have been deducted. the profit after tax is often assessment of what a business is really earning and hence can use in its operations than its total revenues. in the previous research, occupancy rate is a non-financial output of equation but it is not objectively provided by vietnamese hotel company, so we cannot put it in the output variables. 4. empirical result and analysia 4.1. empirical result a forecast inputs/ outputs of next 5 years from 2019 to 2022 will be generated by gm (1,1) model. a sample is presented to illustrate the procedure of gm (1,1) forecasting applied in the research. the researcher takes factor of sales revenue of ben thanh tourist and service jsc. (btv) in period of time from 2018 to 2022 to demonstrate the calculation process and other variables are calculated in the same way. the researcher use the gm(1,1) model for trying to forecast the variance of primitive series as follows. int. j. anal. appl. 17 (6) (2019) 1002 first, the primitive series is created: 𝑋(0) = (455,117 ; 601,329 ; 597,653 ; 672,090 ; 814,010) secondly, perform the accumulated generating operation (ago): 𝑋(1) = (455,117; 1,056,446; 1,654,099; 2,326,189; 3,140,199) 𝑥(1)(1) = 𝑥(0)(1) = 455,117 𝑥(1)(2) = 𝑥(0)(1) + 𝑥(0)(2) = 1,056,446 𝑥(1)(3) = 𝑥(0)(1) + 𝑥(0)(2) + 𝑥(0)(3) = 1,654,099 𝑥(1)(4) = 𝑥(0)(1) + 𝑥(0)(2) + 𝑥(0)(3) + 𝑥(0)(4) = 2,326,189 𝑥(1)(5) = 𝑥(0)(1) + 𝑥(0)(2) + 𝑥(0)(3) + 𝑥(0)(4) + 𝑥(0)(5) = 3,140,199 third, create the different equations of gm(1,1) to find 𝑋(1) series, and the following mean obtained by the mean equation is 𝑧(1)(2) = 1 2 (455,117 + 1,056,446) = 755,781.5 𝑧(1)(3) = 1 2 (1,056,446 + 1,654,099) = 1,355,272.5 𝑧(1)(4) = 1 2 (1,654,099 + 2,326,189) = 1,990,144 𝑧(1)(5) = 1 2 (2,326,189 + 3,140,199) = 2,733,194 fourth, solve the equations. to find a and b, the primitive series values are substituted into grey differential equation to obtain 601,329 + 𝑎 𝑥 755,781.5 = 𝑏 597,653 + 𝑎 𝑥 1,355,272.5 = 𝑏 672,090 + 𝑎 𝑥 1,990,144 = 𝑏 814,010 + 𝑎 𝑥 2,733,194 = 𝑏 then, convert the linear equations into the form of a matrix let 𝐵 = [ −755,781.5 1 −1,355,272.5 1 −1,990,144 1 −2,733,194 1 ] , 𝜃 = [ 𝑎 𝑏 ], 𝑦𝑁 = [ 601,329 597,653 672,090 814,010 ]. and then use the least square method to find a and b: [ 𝑎 𝑏 ] = 𝜃 = (𝐵𝑇 𝑦𝑁 ) = [ −0.110619605 482,266.0641 ]. int. j. anal. appl. 17 (6) (2019) 1003 use the two coefficients a and b to generate the whitening equation of the differential equation: 𝑑𝑥(1) 𝑑𝑡 − 0.110619605 × 𝑥(1) = 482,266.0641. find the prediction model from 𝑋(1)(𝑘 + 1) = (𝑋(0)(1) − 𝑏 𝑎 ) 𝑒 −𝑎𝑘 + 𝑏 𝑎 𝑋(1)(𝑘 + 1) = (455,117 − 482,266.0641 −0.110619605 ) 𝑒 0.110619605 + 482,266.0641 −0.110619605 = (4,814,796.860)𝑒 0.110619605 − 4,359,679.806 substitute different values of k into the equation: 𝑘 = 0 𝑋(1)(1) = 455,117 𝑘 = 1 𝑋(1)(2) = 1,018,303 𝑘 = 2 𝑋(1)(3) = 1,647,365 𝑘 = 3 𝑋(1)(4) = 2,350,009 𝑘 = 4 𝑋(1)(5) = 3,134,841 𝑘 = 5 𝑋(1)(6) = 4,011,475 𝑘 = 6 𝑋(1)(7) = 4,990,648 𝑘 = 7 𝑋(1)(8) = 6,084,356 𝑘 = 8 𝑋(1)(9) = 7,305,994 𝑘 = 9 𝑋(1)(10) = 8,670,527 derive the predicted value of the original series according to the accumulated generating operation and obtain 𝑥(0)(1) = 𝑥(1)(1) = 455,117 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2013 𝑥(0)(2) = 𝑥(1)(2) − 𝑥(1)(1) = 563,186 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2014 𝑥(0)(3) = 𝑥(1)(3) − 𝑥(1)(2) = 629,062 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2015 𝑥(0)(4) = 𝑥(1)(4) − 𝑥(1)(3) = 702,643 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2016 𝑥(0)(5) = 𝑥(1)(5) − 𝑥(1)(4) = 784,831 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2017 𝑥(0)(6) = 𝑥(1)(6) − 𝑥(1)(5) = 876,633 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2018 𝑥(0)(7) = 𝑥(1)(7) − 𝑥(1)(6) = 979,173 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2019 𝑥(0)(8) = 𝑥(1)(8) − 𝑥(1)(7) = 1,093,707 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2020 𝑥(0)(9) = 𝑥(1)(9) − 𝑥(1)(8) = 1,221,638 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2021 𝑥(0)(10) = 𝑥 (1)(10) − 𝑥(1)(9) = 1,364,533 − 𝑓𝑜𝑟 𝑦𝑒𝑎𝑟 2022 int. j. anal. appl. 17 (6) (2019) 1004 the other input and output factors’ forecasting results will be carried out same as the above process. the results of all dmus from 2018 to 2022 could be acquired and the detailed numbers are generated in the table 4.1 to 4.5 respectively: table 4.1. financial data of decision-making units 2018 denote i (cogs) i (sales expenses) i (operatio n expense) i (equity capital) i (fixed asset) o (revenue) o (profit after tax dmu1 32,908,616 32,908,616 32,908,616 32,908,616 32,908,616 32,908,616 32,908,616 dmu2 13,478,159 47,923 209,550 26,711,591 41,077,064 17,825,572 2,639,877 dmu3 76,576,376 802,055 132,735 1,357,945 5,424,210 548,536 (2,160) dmu4 483,545 55,537 132,735 14,095,615 582,156 17,757,563 418,969 dmu5 9,162,200 61,518 600,414 13,824,261 705,966 8,950,675 155,308 dmu6 6,060,193 1,181,491 1,029,223 4,872,856 3,765,479 8,046,307 335,272 dmu7 42,334 4,466 12,848 1,236,428 1,948 34,135 15,871 dmu8 4,788,941 992,087 3,287,557 38,014,824 8,803,001 9,747,591 4,601,632 dmu9 28,121,728 6,317,735 12,613,506 52,338,060 58,484,432 53,715,440 (7,762,659) dmu10 2,887,402 17,091 222,029 6,591,606 5,915,483 3,456,240 394,069 dmu11 1,193,309 314,599 10,513,538 1,820,995 2,393,822 992,115 dmu12 2,441,583 1,720,458 115,510 5,709,437 501,977 4,605,586 274,796 dmu13 1,763,348 692,579 3,960,691 2,839,775 2,782,695 348,412 dmu14 37,453,279 382,746 1,269,673 8,568,309 9,799,291 12,423,095 345,689 dmu15 3,334,348 1,390,279 2,425,122 7,386,611 5,098,755 6,958,109 9,656 dmu16 2,654,787 38,652 360,443 3,260,395 3,260,395 7,124,050 (28,570) dmu17 1,329,496 130,241 263,803 2,339,723 2,035,763 1,850,575 37,359 dmu18 9,886,471 1,978,471 626,407 3,988,739 1,634,898 12,610,975 983,349 dmu19 9,886,471 1,978,471 2,007,085 4,381,002 1,634,898 12,610,975 531,090 dmu20 6,381,260 23,930 329,038 1,655,251 555,370 6,381,260 291,194 http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm int. j. anal. appl. 17 (6) (2019) 1005 table 4.2. financial data of decision-making units 2019 denote i (cogs) i (sales expenses) i (operatio n expense) i (equity capital) i (fixed asset) o (revenue) o (profit after tax dmu1 37,098,046 1,890,424 2,615,960 11,466,654 2,850,215 42,572,760 981,061 dmu2 21,412,310 53,533 229,447 39,762,448 57,698,449 27,611,697 4,412,162 dmu3 68,177,527 725,295 124,253 1,333,963 5,484,919 486,002 (2,272) dmu4 432,150 53,533 124,253 14,329,849 649,092 19,072,378 492,506 dmu5 12,158,274 62,712 605,491 14,007,484 678,952 9,033,212 169,854 dmu6 6,229,626 1,851,392 1,252,261 4,905,668 3,618,884 8,493,387 278,994 dmu7 46,613 4,359 12,831 1,306,662 656 32,394 16,405 dmu8 4,991,351 976,854 3,225,616 38,288,765 6,531,108 10,084,064 7,736,101 dmu9 31,027,610 6,744,248 13,857,607 52,755,069 62,891,546 61,426,778 (8,089,791) dmu10 2,964,553 11,267 190,777 6,789,405 5,928,333 3,541,795 609,336 dmu11 1,321,688 317,410 15,698,135 1,689,007 2,854,516 1,945,718 dmu12 2,363,252 1,829,433 118,395 8,232,072 441,592 4,616,801 272,509 dmu13 1,797,353 847,900 3,994,845 2,770,284 2,859,928 327,333 dmu14 98,443,453 506,905 1,456,691 9,160,646 10,956,611 16,671,791 498,003 dmu15 3,300,936 1,449,366 2,653,246 7,303,900 5,231,296 7,135,472 9,271 dmu16 2,627,426 38,730 372,926 3,180,629 3,180,629 6,972,281 (16,456) dmu17 1,402,484 138,824 309,527 2,343,045 2,024,305 2,015,901 42,898 dmu18 12,235,304 2,016,607 847,652 4,199,560 1,530,489 14,777,243 1,220,173 dmu19 12,235,304 2,016,607 2,054,724 4,691,514 1,530,489 14,777,243 590,032 dmu20 6,711,212 24,138 218,970 1,385,341 459,497 6,711,212 241,473 http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm int. j. anal. appl. 17 (6) (2019) 1006 table 4.3. financial data of decision-making units 2020 denote i (cogs) i (sales expenses) i (operatio n expense) i (equity capital) i (fixed asset) o (revenue) o (profit after tax dmu1 41,820,811 1,927,365 2,826,492 11,535,781 2,705,067 47,552,493 972,973 dmu2 34,017,037 59,799 251,234 59,189,743 81,045,496 42,770,341 7,374,271 dmu3 60,699,858 655,882 116,312 1,310,405 5,546,309 430,596 (2,390) dmu4 386,217 51,601 116,312 14,567,975 723,724 20,484,545 578,951 dmu5 16,134,074 63,931 610,612 14,193,136 652,972 9,116,510 185,761 dmu6 6,403,796 2,901,122 1,523,632 4,938,702 3,477,997 8,965,309 232,162 dmu7 51,325 4,254 12,813 1,380,886 221 30,741 16,957 dmu8 5,202,316 961,855 3,164,843 38,564,681 4,845,548 10,432,152 13,005,660 dmu9 34,233,763 7,199,555 15,224,416 53,175,401 67,630,760 70,245,150 (8,430,709) dmu10 3,043,766 7,427 163,924 6,993,140 5,941,210 3,629,468 942,196 dmu11 1,463,879 320,246 23,439,442 1,566,584 3,403,871 3,815,907 dmu12 2,287,434 1,945,311 121,351 11,869,299 388,472 4,628,044 270,242 dmu13 1,832,013 1,038,055 4,029,293 2,702,493 2,939,305 307,529 dmu14 258,752,069 671,340 1,671,255 9,793,932 12,250,612 22,373,540 717,427 dmu15 3,267,857 1,510,964 2,902,829 7,222,116 5,367,283 7,317,356 8,901 dmu16 2,600,346 38,808 385,840 3,102,814 3,102,814 6,823,745 (9,479) dmu17 1,479,478 147,973 363,177 2,346,372 2,012,912 2,195,997 49,259 dmu18 15,142,176 2,055,478 1,147,038 4,421,524 1,432,748 17,315,624 1,514,033 dmu19 15,142,176 2,055,478 2,103,495 5,024,033 1,432,748 17,315,624 655,516 dmu20 7,058,225 24,348 145,722 1,159,444 380,174 7,058,225 200,241 http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm int. j. anal. appl. 17 (6) (2019) 1007 table 4.4. financial data of decision making units 2021 denote i (cogs) i (sales expenses) i (operatio n expense) i (equity capital) i (fixed asset) o (revenue) o (profit after tax dmu1 47,144,808 1,965,028 3,053,969 11,605,324 2,567,311 53,114,705 964,952 dmu2 54,041,754 66,798 275,089 88,108,905 113,839,670 66,250,982 12,324,995 dmu3 54,042,336 593,112 108,879 1,287,264 5,608,385 381,507 (2,514) dmu4 345,167 49,738 108,879 14,810,058 806,937 22,001,272 680,569 dmu5 21,409,977 65,172 615,775 14,381,248 627,985 9,200,576 203,158 dmu6 6,582,836 4,546,045 1,853,810 4,971,958 3,342,594 9,463,452 193,192 dmu7 56,514 4,151 12,796 1,459,326 74 29,173 17,528 dmu8 5,422,198 947,087 3,105,214 38,842,585 3,595,001 10,792,256 21,864,655 dmu9 37,771,215 7,685,600 16,726,037 53,599,081 72,727,100 80,329,479 (8,785,994) dmu10 3,125,096 4,896 140,851 7,202,988 5,954,115 3,719,311 1,456,888 dmu11 1,621,367 323,107 34,998,261 1,453,036 4,058,950 7,483,687 dmu12 2,214,048 2,068,528 124,381 17,113,585 341,741 4,639,314 267,993 dmu13 1,867,341 1,270,854 4,064,038 2,636,362 3,020,885 288,923 dmu14 680,112,603 889,116 1,917,424 10,470,997 13,697,439 30,025,286 1,033,533 dmu15 3,235,111 1,575,181 3,175,889 7,141,247 5,506,805 7,503,876 8,545 dmu16 2,573,546 38,887 399,202 3,026,902 3,026,902 6,678,373 (5,460) dmu17 1,560,700 157,725 426,126 2,349,704 2,001,582 2,392,183 56,563 dmu18 18,739,663 2,095,099 1,552,167 4,655,220 1,341,249 20,290,038 1,878,664 dmu19 18,739,663 2,095,099 2,153,422 5,380,121 1,341,249 20,290,038 728,268 dmu20 7,423,180 24,560 96,976 970,383 314,545 7,423,180 166,050 http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm int. j. anal. appl. 17 (6) (2019) 1008 table 4.5. financial data of decision making units 2022 denote i (cogs) i (sales expenses) i (operatio n expense) i (equity capital) i (fixed asset) o (revenue) o (profit after tax dmu1 53,146,575 2,003,427 3,299,752 11,675,287 2,436,571 59,327,529 956,998 dmu2 85,854,367 74,616 301,210 131,157,506 159,903,649 102,622,345 20,599,393 dmu3 48,115,006 536,350 101,921 1,264,531 5,671,156 338,014 (2,644) dmu4 308,480 47,944 101,921 15,056,164 899,718 23,630,302 800,023 dmu5 28,411,119 66,438 620,983 14,571,854 603,955 9,285,418 222,185 dmu6 6,766,881 7,123,632 2,255,540 5,005,438 3,212,462 9,989,274 160,763 dmu7 62,226 4,051 12,779 1,542,221 25 27,684 18,118 dmu8 5,651,373 932,545 3,046,709 39,122,491 2,667,197 11,164,790 36,758,085 dmu9 41,674,200 8,204,458 18,375,766 54,026,138 78,207,476 91,861,505 (9,156,251) dmu10 3,208,598 3,228 121,025 7,419,133 5,967,049 3,811,379 2,252,739 dmu11 1,795,798 325,994 52,257,143 1,347,717 4,840,100 14,676,872 dmu12 2,143,017 2,199,551 127,487 24,674,985 300,631 4,650,612 265,763 dmu13 1,903,351 1,555,863 4,099,082 2,571,848 3,104,729 271,442 dmu14 1,787,630,73 8 1,177,536 2,199,853 11,194,869 15,315,139 40,293,928 1,488,917 dmu15 3,202,692 1,642,127 3,474,635 7,061,284 5,649,953 7,695,151 8,204 dmu16 2,547,022 38,966 413,027 2,952,848 2,952,848 6,536,099 (3,145) dmu17 1,646,380 168,119 499,985 2,353,041 1,990,317 2,605,895 64,950 dmu18 23,191,844 2,135,483 2,100,385 4,901,267 1,255,593 23,775,387 2,331,112 dmu19 23,191,844 2,135,483 2,204,535 5,761,447 1,255,593 23,775,387 809,093 dmu20 7,807,007 24,774 64,536 812,150 260,245 7,807,007 137,697 because the smallest value is – 9,156,251 usd which is forecast value of factor profit after tax of dmu9, all values will be scale up usd10,000,000 for carrying the dea model. http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/nvt-ctcp-bat-dong-san-du-lich-ninh-van-bay.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm http://finance.vietstock.vn/och-ctcp-khach-san-va-dich-vu-dai-duong.htm int. j. anal. appl. 17 (6) (2019) 1009 4.2 forecasting accuracy the predicting data of dmus for next 5 years from 2018 to 2022 generated by gm (1,1) model is used as inputs for dea – malmquist to study the performance of them in future. since the study mentioned in previous chapter, forecasting data should be accuracy to ensure that result of future data and upcoming analysis. therefore, mape (mean absolut percentage error) is employed to calculate forecasting error between the two sets of data. the results are summarized in the following table: table 4. 1 mape 2013-2017 denote dmu average mape (%) btv dmu1 5.16% dah dmu2 11.30% dcd dmu3 15.06% dxl dmu4 9.50% ein dmu5 10.40% hot dmu6 5.65% nph dmu7 10.76% nvt dmu8 13.43% och dmu9 10.49% pdc dmu10 12.46% sgh dmu11 9.06% ttj dmu12 5.71% vir dmu13 2.65% vng dmu14 12.68% vtg dmu15 1.72% dld dmu16 2.05% mtc dmu17 2.46% oscvn dmu18 2.96% dnt dmu19 1.78% kimlien dmu20 9.75% average mape 7.75% the result of average mape is only 7.75 which proves that g(1,1) model is qualified to apply to forecast the future value of dmus in this research. int. j. anal. appl. 17 (6) (2019) 1010 4.3 pearson correlation the pearson correlation is conducted to confirm the relationship between input and output. pearson test confirms that the lower correlation implies the less correlated and higher correlation implies the closer correlated between two variables. the correlation value is always from -1 to 1. the closer to -1 and 1 is correlation, the more perfect is linear relationship formed. tables from 4.6 to 4.15 confirm that the correlations comply well with the earlier condition of the dea model as their correlation coefficients show strong positive associations. hence, it proves that the input and output are chosen appropriately. and there is no elimination of any variable. table 4. 6. correlation coefficient 2013 cogs sales expenses operation expense equity capital fixed asset revenue profit after tax cogs 1.0000 0.2600 0.0118 0.0402 0.0911 0.0198 0.1703 sales expenses 0.2600 1.0000 0.7352 0.8324 0.7104 0.8123 0.8735 operation expense 0.0118 0.7352 1.0000 0.8156 0.7452 0.8539 0.8434 equity capital 0.0402 0.8324 0.8156 1.0000 0.8850 0.8381 0.8921 fixed asset 0.0911 0.7104 0.7452 0.8850 1.0000 0.6227 0.7515 revenue 0.0198 0.8123 0.8539 0.8381 0.6227 1.0000 0.9156 profit after tax 0.1703 0.8735 0.8434 0.8921 0.7515 0.9156 1.0000 int. j. anal. appl. 17 (6) (2019) 1011 table 4. 7. correlation coefficient 2014 cogs sales expenses operation expense equity capital fixed asset revenue profit after tax cogs 1.0000 0.2343 0.0904 -0.0025 0.1016 0.0505 -0.1021 sales expenses 0.2343 1.0000 0.8323 0.6870 0.6726 0.8217 -0.7957 operation expense 0.0904 0.8323 1.0000 0.8062 0.8456 0.7739 -0.9894 equity capital -0.0025 0.6870 0.8062 1.0000 0.8333 0.7424 -0.7460 fixed asset 0.1016 0.6726 0.8456 0.8333 1.0000 0.6132 -0.8230 revenue 0.0505 0.8217 0.7739 0.7424 0.6132 1.0000 -0.7203 profit after tax -0.1021 -0.7957 -0.9894 -0.7460 -0.8230 -0.7203 1.0000 table 4. 8. correlation coefficient 2015 cogs sales expenses operation expense equity capital fixed asset revenue profit after tax cogs 1.0000 0.2107 -0.0014 -0.0277 0.0483 0.0105 0.0769 sales expenses 0.2107 1.0000 0.8204 0.6763 0.6628 0.6466 0.1385 peration expense -0.0014 0.8204 1.0000 0.9400 0.8725 0.6454 -0.2134 equity capital -0.0277 0.6763 0.9400 1.0000 0.8499 0.6655 -0.3462 fixed asset 0.0483 0.6628 0.8725 0.8499 1.0000 0.4731 -0.1748 revenue 0.0105 0.6466 0.6454 0.6655 0.4731 1.0000 0.2546 profit after tax 0.0769 0.1385 -0.2134 -0.3462 -0.1748 0.2546 1.0000 int. j. anal. appl. 17 (6) (2019) 1012 table 4. 9. correlation coefficient 2016 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 0.2970 0.1834 0.0597 0.1903 0.1770 -0.2140 sales expenses 0.2970 1.0000 0.8827 0.6186 0.6957 0.7956 -0.7954 peration expense 0.1834 0.8827 1.0000 0.8139 0.8506 0.8320 -0.8946 equity capital 0.0597 0.6186 0.8139 1.0000 0.8293 0.7135 -0.6195 fixed asset 0.1903 0.6957 0.8506 0.8293 1.0000 0.6900 -0.7823 revenue 0.1770 0.7956 0.8320 0.7135 0.6900 1.0000 -0.6654 profit after tax -0.2140 -0.7954 -0.8946 -0.6195 -0.7823 -0.6654 1.0000 table 4. 10. correlation coefficient 2017 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 0.2738 0.0782 0.1658 0.2289 0.2283 -0.0705 sales expenses 0.2738 1.0000 0.5854 0.6972 0.7026 0.7681 0.0229 peration expense 0.0782 0.5854 1.0000 0.6625 0.6444 0.5513 0.7644 equity capital 0.1658 0.6972 0.6625 1.0000 0.9313 0.8728 0.1421 fixed asset 0.2289 0.7026 0.6444 0.9313 1.0000 0.7571 0.1400 revenue 0.2283 0.7681 0.5513 0.8728 0.7571 1.0000 0.0368 profit after tax -0.0705 0.0229 0.7644 0.1421 0.1400 0.0368 1.0000 int. j. anal. appl. 17 (6) (2019) 1013 table 4.11. correlation coefficient 2018 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 0.2980 0.3054 0.1616 0.3360 0.2457 0.1812 sales expenses 0.2980 1.0000 0.9777 0.4537 0.4661 0.5485 0.8969 peration expense 0.3054 0.9777 1.0000 0.5956 0.5894 0.6744 0.8136 equity capital 0.1616 0.4537 0.5956 1.0000 0.8376 0.8329 0.2441 fixed asset 0.3360 0.4661 0.5894 0.8376 1.0000 0.8638 0.1923 revenue 0.2457 0.5485 0.6744 0.8329 0.8638 1.0000 0.2097 profit after tax 0.1812 0.8969 0.8136 0.2441 0.1923 0.2097 1.0000 table 4. 12. correlation coefficient 2019 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 0.1606 0.1695 0.0740 0.2340 0.2927 -0.1422 sales expenses 0.1606 1.0000 0.9058 0.5273 0.5455 0.7524 -0.6269 peration expense 0.1695 0.9058 1.0000 0.6972 0.6597 0.7924 -0.6064 equity capital 0.0740 0.5273 0.6972 1.0000 0.8241 0.6996 -0.0339 fixed asset 0.2340 0.5455 0.6597 0.8241 1.0000 0.7172 -0.3054 revenue 0.2927 0.7524 0.7924 0.6996 0.7172 1.0000 -0.4332 profit after tax -0.1422 -0.6269 -0.6064 -0.0339 -0.3054 -0.4332 1.0000 int. j. anal. appl. 17 (6) (2019) 1014 table 4. 13. correlation coefficient 2020 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 0.0376 0.0861 0.0304 0.1510 0.2398 -0.0629 sales expenses 0.0376 1.0000 0.8935 0.3862 0.4328 0.6811 -0.5096 peration expense 0.0861 0.8935 1.0000 0.5470 0.5512 0.7493 -0.4604 equity capital 0.0304 0.3862 0.5470 1.0000 0.8510 0.6913 0.2533 fixed asset 0.1510 0.4328 0.5512 0.8510 1.0000 0.7468 -0.0658 revenue 0.2398 0.6811 0.7493 0.6913 0.7468 1.0000 -0.2555 profit after tax -0.0629 -0.5096 -0.4604 0.2533 -0.0658 -0.2555 1.0000 table 4. 14. correlation coefficient 2021 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 -0.0101 0.0399 -0.0058 0.0874 0.2049 -0.0409 sales expenses -0.0101 1.0000 0.8530 0.2162 0.3065 0.5597 -0.3953 peration expense 0.0399 0.8530 1.0000 0.3708 0.4389 0.6768 -0.3315 equity capital -0.0058 0.2162 0.3708 1.0000 0.8758 0.6976 0.4465 fixed asset 0.0874 0.3065 0.4389 0.8758 1.0000 0.7912 0.1196 revenue 0.2049 0.5597 0.6768 0.6976 0.7912 1.0000 -0.0761 profit after tax -0.0409 -0.3953 -0.3315 0.4465 0.1196 -0.0761 1.0000 int. j. anal. appl. 17 (6) (2019) 1015 table 4. 15. correlation coefficient 2022 cogs sales expenses peration expense equity capital ixed asset revenue profit after tax cogs 1.0000 -0.0164 0.0227 -0.0297 0.0480 0.1919 -0.0448 sales expenses -0.0164 1.0000 0.7701 0.0648 0.1816 0.3946 -0.3049 peration expense 0.0227 0.7701 1.0000 0.2107 0.3329 0.5756 -0.2408 equity capital -0.0297 0.0648 0.2107 1.0000 0.8907 0.7249 0.5323 fixed asset 0.0480 0.1816 0.3329 0.8907 1.0000 0.8434 0.2355 revenue 0.1919 0.3946 0.5756 0.7249 0.8434 1.0000 0.0708 profit after tax -0.0448 -0.3049 -0.2408 0.5323 0.2355 0.0708 1.0000 5. conclusion in conclusion, the research employs dea model and malmquist production index model to study and evaluate performance of past-to-future performance of 20-listed company in vietnam hospitality industry. at beginning, researcher collects data of 20 qualified listed companies (decision making unit) with 2 sets of data: the original set of data and the future set of data. the original set of data is collected on vietstock.vn, cophieu68.com and website of companies where their financial reports are audited by reliable institution. the researcher uses 5 input variables (cost of good sales; sales expense; operation expense; fixed assets and owner equity) and 2 output variables (revenues and profit after tax). the association level of those variables is examined by pearson correlation and the result confirms the close correlated relationship among variables which is qualified to the requirement of dea model. then dea-malmquist is applied first time to analysis the original data set (past data from 2013 to 2017) in order to evaluate performance of dmus in the past. next, the future data set is generated by employed gm (1,1). the error of forecasting model is calculated by mean absolute percentage error (mape). the average mape of all dmus is 7.75% which is in acceptant range (less than 10%). the forecast data will be input for dea-malmquist model to evaluate performance of dmus in next 5 year from 2018 to 2022. int. j. anal. appl. 17 (6) (2019) 1016 in the first-time employing dea model with the past data, the average score of all dmus is 1.053 score. the most efficient company is dmu1 (average score is 1.053), however, it inconsistently performs during studying period. in the first two period of time from 2013 to 2015, dmu1 (btv) has the highest score 1.051 and 1.140 respectively. in the other hand, dmu8 (nvt) is the least efficient firm from 2014 to 2015. ninh van tourist experienced loss in period time from 2014 to 2015 due to their huge investment to real estate investment and development, but the real estate market was in deep crisis at that time. together with the down trend of the world economy, ninh van tourist made consecutive losses. interestingly, period of year 2015 to 2016 experiences a converse trend where dmu1 is the lease efficient firm with 0.878 score and dum8 is the most one with 1.209 score. the down trend of dmu1 is since the event of oil rig haiyang shi you 981 in southeast sea between vietnam and china in july 2015 cause the dramatically reduce in chinese visitors. ben thanh tourist should diversify their target customers avoid the similar event rather than mostly relying visitors from any one country. sai gon hotel jsc (dmu11) as its mpi score has tendency to reduce and be the least efficiency in period 2016 to 2017 which requests its management board to improve as soon as possible in order not to be kick out of the market.although the overall trend of hospitality industry is table, dmu1 and dmu8 face with huge fluctuation in efficiency which acquire the management board to review and improve their operation process to ensure the sustainable development of the firm in current competitive market. according to the result of second time dea – malmquist production index, the future performance of all dmus is more consistent efficiency than the past ones. dmu1 seems to have outstanding performance from the past up to future data recorded, however, the declining tendency occurs in the last intervals. it raises the attention to management board of ben thanh tourist and service jsc to review its performance and operation frequently to ensure their advantages in hospitality industry. dmu9 (och) and dmu14 (vng) are potential to perform better in future. the rest dmus are stable within score interval from 0.969 to 1.054. references [1] min, h., & min, h., benchmarking the quality of hotel services: managerial perspectives, int. j. qual. reliab. manage. 14 (1997), 582–597. int. j. anal. appl. 17 (6) (2019) 1017 [2] chen, c. t., hu, j. l., & liao, j. j., tourists’ nationalities and the cost efficiency of international tourist hotels in taiwan, afr. j. business manage. 4 (2010), 3440–3446. [3] dyson, r. g., allen, r., camanho, a. s., podinovski, v. v., sarrico, c. s., & shale, e. a., pitfalls and protocols in dea, eur. j. oper. res. 132 (2001), 245–259. [4] hsieh, l. f., wang, l. h., huang, y. c., & chen, a., an efficiency and effectiveness model for international tourist hotels in taiwan, service ind. j. 30 (2010), 2183–2199. [5] johns, n., howcroft, b., & drake, l., the use of data envelopment analysis to monitor hotel productivity, progr. tourism hosp. res. 3 (1997), 119–127. [6] hwang, s. n., & chang, t. y., using data envelopment analysis to measure hotel managerial efficiency change in taiwan, tourism manage. 24 (2003), 357–369. [7] chiang, w. e., tsai, m. h., & wang, l. s. m., a dea evaluation of taipei hotels, ann. tourism res. 31 (2004), 712–715. [8] barros, c. p., measuring efficiency in the hotel sector, ann. tourism res. 32 (2005), 456–477. [9] barros, c. a. p., & santos, c. a., the measurement of efficiency in portuguese hotels using data envelopment analysis, j. hosp. tourism res. 30 (2006), 378–400. [10] onut s and s soner, “energy efficiency assessment for the antalya region hotels in turkey, energy buildings, 38 (8) (2006), 964-997. [11] chen, c. f., applying the stochastic frontier approach to measure hotel managerial efficiency in taiwan, tourism manage. 28 (2007), 696–702. [12] davutyan, nurhan, measuring the quality of hospitality at antalya, int. j. tourism res. 9 (2007), 51 – 57. [13] min, h., min, h., &joo, s. j., a data envelopment analysis-based balanced scorecard for measuring the comparative efficiency of korean luxury hotels, int. j. qual. reliab. manage. 25 (2008), 349–365. [14] barros, c. p., peypoch, n., &solonandrasana, b., efficiency and productivity growth in hotel industry, int. j. tourism res. 11 (2009), 389–402. [15] perrigot, r., cliquet, g., & piot-lepetit, i., plural form chain and efficiency: insights from the french hotel chains and the dea methodology, eur. manage. j. 27 (2009), 268–280. [16] tourist, m. n.,statistic figure, retrieved from ministry national administration of tourist: http://vietnamtourism.gov.vn/index.php/items/13461, (2018, march). http://vietnamtourism.gov.vn/index.php/items/13461 int. j. anal. appl. 17 (6) (2019) 1018 [17] yen, f. l., & othman, m., data envelopment analysis to measure efficiency of hotels in malaysia, segi rev. 4 (2011), 25–36. [18] chen, c. t., hu, j. l., & liao, j. j., tourists’ nationalities and the cost efficiency of international tourist hotels in taiwan, afr. j. business manage. 4 (2010), 3440–3446. [19] hsieh, l. f., & lin, l. h., a performance evaluation model for international tourist hotels in taiwan – an application of the relational network dea, int. j. hosp. manage. 29 (2010), 14–24. [20] hsieh, l. f., wang, l. h., huang, y. c., & chen, a., an efficiency and effectiveness model for international tourist hotels in taiwan, service ind. j. 30 (2010), 2183–2199. [21] assaf, a., &knežević, c. l., the performance of the slovenian hotel industry: evaluation post-privatisation, int. j. tourism res. 12 (2011), 462–471. [22] avkiran, necmi, applications of data envelopment analysis in the service sector, 2011, 10.1007/978-1-4419-6151-8_15 [23] wang, f. c., hung, w. t., & shang, j. k., measuring pure managerial efficiency of international tourist hotels in taiwan, service ind. j. 26 (2006), 59–71. [24] yen, f. l., & othman, m., data envelopment analysis to measure efficiency of hotels in malaysia, segi rev. 4 (2011), 25–36. [25] honma, s., & hu, j. l., analyzing japanese hotel efficiency, tourism hosp. res. 12 (2012), 155–167. [26] manasakis, c., apostolakis, a., &datseris, g., using data envelopment analysis to measure hotel efficiency in crete, int. j. contemp. hosp. manage. 25 (2013), 510–535. [27] katarina poldrugovaca, m. t., efficiency in the hotel industry: an empirical examination of the most influential factors, economic research-ekonomska istraživanja, (2016). international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 50-58 http://www.etamaths.com new integral inequalities in quantum calculus kamel brahim1,∗, sabrina taf2 and bochra nefzi1 abstract. in this paper, we study the q-analogue of klamkin-mclenaghan’s and grueb-reinboldt’s inequalities then we use the riemann-liouville fractional q-integral to get some new integral results. 1. introduction let us consider (1.1) t(f,g; a,b) = 1 b−a ∫ b a f(x)g(x)dx− ( 1 b−a ∫ b a f(x)dx )( 1 b−a ∫ b a g(x)dx ) where f and g are two integrable functions on [a,b], and tq(f,g; a,b) = 1 b−a ∫ b a f(x)g(x)dqx− ( 1 b−a ∫ b a f(x)dqx )( 1 b−a ∫ b a g(x)dqx ) where f and g are two functions defined on [a,b]q. the well-known grüss integral inequality can be stated as follows (see [10, 15]): (1.2) |t(f,g; a,b)| ≤ 1 4 (m −m)(n −n), provided that f and g are two integrable functions on [a,b] such that (1.3) 0 < m ≤ f(x) ≤ m < ∞, 0 < n ≤ g(x) ≤ n < ∞, ,x ∈ [a,b]. the constant 1 4 is best possible. gauchman gave the q-integral grüss inequality as follows (see [8]): |tq(f,g; a,b)| ≤ 1 4 (m −m)(n −n), in [12], the authors proved the following klamkin-mclenaghan inequality (1.4) n∑ k=1 wka 2 k n∑ k=1 wkb 2 k − ( n∑ k=1 wkakbk )2 ≤ (√ m n − √ m n )2 n∑ k=1 wkakbk n∑ k=1 wkb 2 k, 2010 mathematics subject classification. 35a23. key words and phrases. grüss inequality, fractional q-integral, klamkin-mclenaghan inequality, grueb-reinboldt inequality. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 50 new integral inequalities in quantum calculus 51 where 0 < m n ≤ ak bk ≤ m n < ∞, wk > 0, k = 1, . . . ,n. in [9], the authors proved the following grueb-reinboldt inequality (1.5) n∑ k=1 wka 2 k n∑ k=1 wkb 2 k ≤ (mn + mn)2 4nmnm ( n∑ k=1 wkakbk )2 in [6], dragomir and diamond proved that (1.6) |t(f,g; a,b)| ≤ 1 4 · (m −m)(n −n) √ mmnn · 1 b−a ∫ b a f(x)dx · 1 b−a ∫ b a g(x)dx, and (1.7) |t(f,g; a,b)| ≤ (√ m − √ m )(√ n − √ n )√ 1 b−a ∫ b a f(x)dx · 1 b−a ∫ b a g(x)dx. in recent years, many researches have studies (1.1) and number of inequalities appeared in literature (see [1, 2, 3, 4, 5, 14, 18]). the main objective of this paper is to establish some new q-fractional integral inequalities of klamkin-mclenaghan and grueb-reinboldt type. this paper is organized as follows: in section 2, we present some preliminary results and notation. in section 3, we state the q-analogue of klamkin-mclenaghan and grueb-reinboldt inequalities, then we establish some new q-fractional integral inequalities. 2. basic definitions for the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper (see [7, 13, 16]). we write for a,b ∈ c and q ∈ (0, 1), (a; q)∞ = ∞∏ k=0 (1 −aqk), (a− b)(α) = aα ( b a ; q)∞ (qα b a ; q)∞ . the q-jackson integral from 0 to a is defined by (see [11]) (2.1) ∫ a 0 f(x)dqx = (1 −q)a ∞∑ n=0 f(aqn)qn, provided the sum converges absolutely. the q-jackson integral in a generic interval [a,b] is given by (see [11]) (2.2) ∫ b a f(x)dqx = ∫ b 0 f(x)dqx− ∫ a 0 f(x)dqx. in the case a = bqn, we can write (2.3) ∫ b a f(x)dqx = (1 −q)b n−1∑ k=0 f(bqk)qk. 52 brahim, taf and nefzi the fractional q-integral of the riemann-liouville type is (see [16])( jαq f ) (x) = 1 γq(α) ∫ x 0 (x−qt)(α−1)f(t)dqt; α > 0 = xα γq(α) (1 −q) ∞∑ n=0 (1 −qn+1)(α−1)f(xqn)qn.(2.4) where γq(α) = 1 1 −q ∫ 1 0 ( u 1 −q )α−1 eq(qu)dqu, and eq(t) = ∞∏ k=0 (1 −qkt). the q-fractional integration has the following semi-group property for α,β ∈ r+ (jβq j α q f)(x) = (j α+β q f)(x). for the expression (2.4), when f(x) = xλ, we get another expression that will be used later: jαq (x λ) = γq(λ + 1) γ(α + λ + 1) xα+λ. finally, for b > 0 and a = bqn,n = 1, 2, ...,∞, we write (2.5) [a,b]q = {bqk : 0 ≤ k ≤ n}, [0,b]q = {bqk : k ∈ n}. 3. main results theorem 1. let f and g be two functions defined on [a,b]q satisfying the condition 0 < m ≤ f(t) ≤ m < ∞, 0 < n ≤ g(t) ≤ n < ∞, t ∈ [a,b]q(3.1) then one has the inequality (3.2) |tq(f,g; a,b)| ≤ (√ m− √ m )(√ n− √ n )√ 1 b−a ∫ b a f(x)dqx · 1 b−a ∫ b a g(x)dqx. the following lemma is used to prove theorem 1: lemma 1. let h and l are two functions defined on [a,b]q such that 0 < m1 ≤ h(t) ≤ m1 < ∞, 0 < n1 ≤ l(t) ≤ n1 < ∞, t ∈ [a,b]q.(3.3) then, we have (3.4)∫ b a h2(x)dqx ∫ b a l2(x)dqx − (∫ b a h(x)l(x)dqx )2 ≤ (√ m1 n1 − √ m1 n1 )2 ∫ b a h(x)l(x)dqx ∫ b a l2(x)dqx proof. from the condition (3.8), we have m1 √ qk ≤ h(bqk) √ qk ≤ m1 √ qk, and n1 √ qk ≤ l(bqk) √ qk ≤ n1 √ qk. new integral inequalities in quantum calculus 53 using the klamkin-mclenaghan inequality (1.4), we obtain n−1∑ k=0 h2(bqk)qk n−1∑ k=0 l2(bqk)qk − ( n−1∑ k=0 h(bqk)l(bqk)qk )2 ≤ (√ m1 n1 − √ m1 n1 )2 n−1∑ k=0 h(bqk)l(bqk)qk n−1∑ k=0 l2(bqk)qk. from (2.3), we get∫ b a h2(x)dqx ∫ b a l2(x)dqx − (∫ b a h(x)l(x)dqx )2 ≤ (√ m1 n1 − √ m1 n1 )2 ∫ b a h(x)l(x)dqx ∫ b a l2(x)dqx lemma 1 is thus proved. � proof of theorem 1: using the cauchy-schwartz inequality for double integrals, we have |tq(f,g; a,b)| = ∣∣∣∣∣ 12(b−a)2 ∫ b a ∫ b a (f(x) −f(y))(g(x) −g(y))dqxdqy ∣∣∣∣∣ ≤ 1 2(b−a)2 [∫ b a ∫ b a (f(x) −f(y))2dqxdqy · ∫ b a ∫ b a (g(x) −g(y))2dqxdqy ]1 2 = 1 2(b−a)2 [ 4 [ (b−a) ∫ b a f2(x)dqx− (∫ b a f(x)dqx )2] × [ (b−a) ∫ b a g2(x)dqx− (∫ b a g(x)dqx )2]]1 2 = [ 1 b−a ∫ b a f2(x)dqx− ( 1 b−a ∫ b a f(x)dqx )2]1 2 × [ 1 b−a ∫ b a g2(x)dqx− ( 1 b−a ∫ b a g(x)dqx )2]1 2 .(3.5) by lemma 1, we get 1 b−a ∫ b a f2(x)dqx − ( 1 b−a ∫ b a f(x)dqx )2 ≤ (√ m − √ m )2 1 b−a ∫ b a f(x)dqx(3.6) and 1 b−a ∫ b a g2(x)dqx − ( 1 b−a ∫ b a g(x)dqx )2 ≤ (√ n − √ n )2 1 b−a ∫ b a g(x)dqx(3.7) 54 brahim, taf and nefzi from (3.5), (3.6) and (3.7), we deduce the desired inequality (3.9). theorem 1 is thus proved. corollary 1. let f and g be two functions defined on [0,b]q satisfying the condition 0 < m ≤ f(t) ≤ m < ∞, 0 < n ≤ g(t) ≤ n < ∞, t ∈ [0,b]q(3.8) then one has the inequality (3.9) |tq(f,g; 0,b)| ≤ 1 b (√ m − √ m )(√ n − √ n )√∫ b 0 f(x)dqx · ∫ b 0 g(x)dqx. proof. by taking a = bqn in the previous theorem and by tending n to ∞ we obtain the result. � theorem 2. let f and g be two functions defined on [a,b]q satisfying the condition (3.1). then one has the inequality (3.10) |tq(f,g; a,b)| ≤ (m −m)(n −n) 4 √ nmnm · 1 b−a ∫ b a f(x)dqx · 1 b−a ∫ b a g(x)dqx lemma 2. [17] let f and g are two functions defined on [a,b]q satisfying the condition (3.8). then we have (3.11) ∫ b a f2(x)dqx ∫ b a g2(x)dqx ≤ (mn + mn)2 4nmnm (∫ b a f(x)g(x)dqx )2 proof of theorem 2: using lemma 2, we get (b−a) ∫ b a f2(x)dqx ≤ (m + m)2 4mm (∫ b a f(x)dqx )2 , hence (3.12) 1 b−a ∫ b a f2(x)dqx− ( 1 b−a ∫ b a f(x)dqx )2 ≤ (m −m)2 4mm ( 1 b−a ∫ b a f(x)dqx )2 . similarly, we have (3.13) 1 b−a ∫ b a g2(x)dqx− ( 1 b−a ∫ b a g(x)dqx )2 ≤ (n −n)2 4nn ( 1 b−a ∫ b a g(x)dqx )2 . from (3.5), (3.12) and (3.13), we deduce the desired inequality (3.14). theorem 2 is thus proved. corollary 2. let f and g be two functions defined on [0,b]q satisfying the condition (3.8). then one has the inequality (3.14) |tq(f,g; 0,b)| ≤ (m −m)(n −n) 4b2 √ nmnm · ∫ b 0 f(x)dqx · ∫ b 0 g(x)dqx theorem 3. let f and g be two positive functions defined on [0,∞) satisfying the condition (3.15) 0 < m ≤ f(τ) ≤ m < ∞, 0 < n ≤ g(τ) ≤ n < ∞, τ ∈ [0, t], t > 0. new integral inequalities in quantum calculus 55 then for all α > 0, b > 0 we have (3.16) ∣∣∣∣ bαγq(α + 1)jαq (f(b)g(b)) −jαq f(b)jαq g(b) ∣∣∣∣ ≤ (√ m − √ m )(√ n − √ n ) bα γq(α + 1) √ jαq f(b)j α q g(b) the following lemma is used to prove theorem 3: lemma 3. let h and l be two positive functions on [0,∞) such that (3.17) 0 < m1 ≤ h(τ) ≤ m1 < ∞, 0 < n1 ≤ l(τ) ≤ n1 < ∞, τ ∈ [0, t], t > 0. then for all α > 0, b > 0, we have (3.18) jαq h 2(b)jαq l 2(b) − ( jαq (h(b)l(b)) )2 ≤ (√ m1 n1 − √ m1 n1 )2 jαq (h(b)l(b))j α q l 2(b). proof. using the klamkin-mclenaghan inequality (1.4), we obtain ∞∑ k=0 h2(bqk)(1 −qk+1)(α−1)qk ∞∑ k=0 l2(bqk)(1 −qk+1)(α−1)qk − ( ∞∑ k=0 h(bqk)l(bqk)(1 −qk+1)(α−1)qk )2 ≤ (√ m1 n1 − √ m1 n1 )2 ∞∑ k=0 h(bqk)l(bqk)(1 −qk+1)(α−1)qk ∞∑ k=0 l2(bqk)(1 −qk+1)(α−1)qk. from (2.4), we obtain jαq h 2(b)jαq l 2(b) − ( jαq (h(b)l(b)) )2 ≤ (√ m1 n1 − √ m1 n1 )2 jαq (h(b)l(b))j α q l 2(b). lemma 3 is thus proved. � proof of theorem 3: define (3.19) q(τ,ρ) = (f(τ) −f(ρ))(g(τ) −g(ρ)) multiplying (3.19) by (b−qτ)(α−1)(b−qρ)(α−1) γ2q(α) and double integrating with respect to τ and ρ from 0 to b, we get 1 γ2q(α) ∫ b 0 ∫ b 0 (b−qτ)(α−1)(b−qρ)(α−1)q(τ,ρ)dqτdqρ(3.20) = 2 bα γq(α + 1) jαq (f(b)g(b)) − 2j α q f(b)j α q g(b). 56 brahim, taf and nefzi on the other hand, using the cauchy-schwartz inequality we get∣∣∣∣∣ 1γ2q(α) ∫ b 0 ∫ b 0 (b−qτ)(α−1)(b−qρ)(α−1)q(τ,ρ)dqτdqρ ∣∣∣∣∣ ≤ [ 1 γ2q(α) ∫ b 0 ∫ b 0 (b−qτ)(α−1)(b−qρ)(α−1)(f(τ) −f(ρ))2dqτdqρ ]1 2 × [ 1 γ2q(α) ∫ b 0 ∫ b 0 (b−qτ)(α−1)(b−qρ)(α−1)(g(τ) −g(ρ))2dqτdqρ ]1 2 . then, ∣∣∣∣∣ 1γ2q(α) ∫ b 0 ∫ b 0 (b−qτ)(α−1)(b−qρ)(α−1)q(τ,ρ)dqτdqρ ∣∣∣∣∣(3.21) ≤ 2 [ bα γq(α + 1) jαq f 2(b) − ( jαq f(b) )2]12 ×[ bα γq(α + 1) jαq g 2(b) − ( jαq g(b) )2]12 using lemma 3, we get (3.22) bα γq(α + 1) jαq f 2(b) − ( jαq f(b) )2 ≤ (√m −√m)2 bα γq(α + 1) ( jαq f(b) ) and (3.23) bα γq(α + 1) jαq g 2(b) − ( jαq g(b) )2 ≤ (√n −√n)2 bα γq(α + 1) ( jαq g(b) ) . using (3.20), (3.22) and (3.23), we deduce the desired inequality (3.16). theorem 3 is thus proved. theorem 4. let f and g be two positive functions defined on [0,∞) satisfying the condition (3.15). then for all α > 0, b > 0, we have (3.24)∣∣∣∣ bαγq(α + 1)jαq (f(b)g(b)) −jαq f(b)jαq g(b) ∣∣∣∣ ≤ (m −m)(n −n) 4 √ mmnn jαq f(b)j α q g(b). to prove theorem 4 we need the following result: lemma 4. let h and l be two positive functions on [0,∞) such that the condition (3.17). then for all α > 0, b > 0, we have (3.25) jαq h 2(b)jαq l 2(b) ≤ (mn + mn)2 4nmnm (jαq h(b)l(b)) 2 proof. using grueb-reinboldt inequality (1.5), we obtain( ∞∑ k=0 h2(bqk)(1 −qk+1)α−1qk )( ∞∑ k=0 l2(bqk)(1 −qk+1)α−1qk ) ≤ (mn + mn)2 4nmnm ( ∞∑ k=0 h(bqk)l(bqk)(1 −qk+1)α−1qk )2 (3.26) new integral inequalities in quantum calculus 57 which implies that (3.27) (jαq h 2(b))(jαq l 2(b)) (jαq h(b)l(b)) 2 ≤ (mn + mn)2 4nmnm lemma 4 is thus proved. � proof of theorem 4: using lemma 4, we get bα γq(α + 1) jαq f 2(b) ≤ (m + m)2 4mm (jαq f(b)) 2 thus, (3.28) bα γq(α + 1) jαq f 2(b) − (jαq f(b)) 2 ≤ (m −m)2 4mm (jαq f(b)) 2 by the same, we obtain (3.29) bα γq(α + 1) jαq g 2(b) − (jαq g(b)) 2 ≤ (n −n)2 4nn (jαq g(b)) 2 using (3.20), (3.28) and (3.29), we deduce the desired inequality (3.24). theorem 4 is thus proved. references [1] anber. a and dahmani. z., new integral results using pólya-szegö inequalities, acta et comm. univ. tari de math. 17 (2) (2013), 171-178. [2] brahim. k and taf. s., on some fractional q-integral inequalities, malaya journal of matematik 3(1)(2013), 21-26. [3] brahim. k and taf. s., some fracional intehral inequalities in quantum calculus, journal of fractional calculus and application. 4(2)(2013), 245-250. [4] dahmani. z., tabahrit. l and taf.s., new generalisation of grüss inequality using riemannliouville fractional integrals, bull. math. anal. appl. 2(2010), 93-99. [5] dragomir. s.s., some integral inequalities of grüss type, indian j.pure apll. math. 31(2000),397-415. [6] dragomir. s.s. and diamond. n.t., integral inequalities of grüss type via pólya-szegö and shisha-mond results,east asian math. j.19 (2003), 27-39. [7] gasper. g and rahman. r., basic hypergeometric series, 2nd edition, (2004), encyclopedia of mathematics and its applications, 96, cambridge university press, cambridge. [8] gauchman. h., integral inequalities in q-calculus, comput. math. appl. 47 (2-3)(2004), 281300. [9] grueb. w and rheinboldt. w., on generalisation of an inequality of l.v. kantrovich, proc. amer. math. soc., 10 (1959), 407-415. [10] grüss. g., uber das maximum des absoluten betrages von 1/(b − a) ∫ b a f(x)g(x)dx − 1/(b − a)2 ∫ b a f(x)dx ∫ b a g(x)dx, math.z.39(1)(1935), 215-226. [11] jackson. f.h., on a q-definite integrals. quarterly journal of pure and applied mathematics 41, 1910, 193-203. [12] klamkin. d.s and mclenaghan. j.e., an ellipse inequality, math. mag., 50 (1977), 261-263. [13] kac. v.g and cheung. p., quantum calculus, universitext, springer-verlag, new york, (2002). [14] mercer. a.mcd and mercer. p., new proofs of the grüss inequality, aust. j. math. anal. apll. 1 (2) (2004), article id 12. [15] mitrinović. d.s., pečarić. j.e., fink. a.m., classical and new inequalities in analysis, in: mathematics and its applications, vol. 61, kluwer acadmic publishers, dordrecht, the netherlands, 1993. [16] rajković. p.m and marinkonvić. s.d., fractional integrals and derivatives in q-calculus, applicable analysis and discrete mathematics, 1 (2007), 311-323. 58 brahim, taf and nefzi [17] rajković. p., marinkonvić. s.d and stanković. m.s., the inequalities for some types of qintegrals, arxiv. math. ca, 8 may 2006. [18] sarikaya. m.z, aktan. n and yildirim. h., on weighted čebyšev-grüss type inequalities on time scales, j. math. inequal. 2(2008), 185-195. 1departement of mathematics, faculty of science of tunis, tunisia 2department of mathematics, faculty sei, umab university of mostaganem, algeria ∗corresponding author international journal of analysis and applications volume 17, number 4 (2019), 630-651 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-630 received 2019-04-24; accepted 2019-05-22; published 2019-07-01. 2010 mathematics subject classification. 91b02. key words and phrases. production function; holling-type ii function; equilibrium points; economic growth. ©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 630 economic growth by means of interspecific functional response of capital-labour in dynamical system oyoon abdul razzaq1, najeeb alam khan2,*, noor ul ain2 , muhammad ayaz2 1department of humanities & social sciences, bahria university, karachi 75260, pakistan 2department of mathematics, university of karachi, karachi 75270, pakistan *corresponding author: njbalam@yahoo.com abstract. physical capital and labour force are two major factors of any economy, which play a key role in its growth. the association of these two components with each other is also a matter of study, which is carried out in this endeavour by means of an ecological system with holling-type ii function. the governing model is an avantgrade approach for economic theory as its equilibrium states and the stability analysis so obtained, referrer to different economic states with detail information about the capital-labour interaction. this novel assessment also contributes a significant way to scrutinize the capability of labours on consuming time on the capital and the efficiency of capital on processing output. moreover, different patterns and cyclic behaviour of the cobb-douglas and constant-elasticity production functions, for different steady and oscillating states of the system are also provided comparatively. in addition, a numerical example is also discussed graphically with economic significance. these measurements will consequently keep the production cycle moving and so sustain the economic growth. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-630 int. j. anal. appl. 17 (4) (2019) 631 1. introduction there is a large number of literature that has revealed different exogenous and endogenous causes and other active censures, which might destroy or stable the well-ordered economy of a country. the neo-classical theory of economic growth was firstly developed by robert solow as an alternative to the harrod-domar model of growth. along with the labour and capital, he also added that the growth of any country comes from adding more ideas and new technologies. recently, many types of research are found that illustrated these factors in production functions [1-9]. fundamentally, these models accumulate many major assumptions of capital and labour in the production function, and as a consequence in the long-term economic growth. for instance, • sustain rise in capital investment increases the ratio of capital to labour, which temporarily increases the growth rate. • increase in the labour supply in addition to a higher level of productivity of labour and capital raises the trend of growth rate. • the declination of the marginal product of additional units of capital causes a long-term growth path with the same growth rate of gdp of an economy. • when the growth rate of output, capital and labour force is the same and so output per worker and capital per worker are constant, the steady-state growth path is reached. • different pace of technological changes between countries will also show much variation in the growth rates. the present endeavour shows a ground-breaking approach to study the upshots of a production function that stem from the interaction of the capital and the labour force with the help of the ecological assumptions. we structured the dynamical system of predator-prey as the labour force (the output from the workers) and physical capital (machines), respectively. the predator-prey relationship is deep-rooted in nature and now it has become one of the inveterate concept to study the interactions among species and their environment. after the contrivance of theoretical ecology by lotka [10] and volterra [11], the models, named as lotka-volterra equations, played an important role in different fields of applied sciences [12-15]. the novelty in this attempt is the inclusion of holling-type ii function [16-20] in the system, which can effectively measure the functional response between the capital and the labour force. the existence of competitive interaction between these two factors produces a major change in the int. j. anal. appl. 17 (4) (2019) 632 production process that yields a salient change in economic growth. using these moulds, the economic cycle that possibly occurs as a result of the interaction is also sketched in fig. 1. each equilibrium point of the dynamical system is discussed with its stability and economic significance. different plots of some production functions [21-24] are also added to study the theory pictorially. the inventive incorporation of ecological concept in the economic growth theory will definitely make the mainstream economist think diversely while making economic measurements. figure 1: pictorial view of different circumstances with capital-labour interaction 2. mathematical modelling and equilibria nowadays, economic growth is widely studied to estimate the social stability of a country around the globe. most of the models, which are used for this purpose, are based on the production function that comprehends the exponential product of physical capital and labour force. the term physical capital includes things like machines, computers and other equipment that are used in the production process. whereas the labour force is meant to be the performance of skilled and unskilled activities of human workers. the previous models lack int. j. anal. appl. 17 (4) (2019) 633 many repercussions, which occur during the output production. for instance, the effects of death, retirement or termination etc. of labours in a particular time period. similarly, the time consumed operating a machine, its expiry etc. are the key factors that actually affect the performance of producing output in a targeted duration. these upshots affect the production progress on the individual level, which gradually leads to fluctuations in gdp of a country. innovatively, considering all these facts, we take advantage of the well-known ecological systems, which sorts out dynamics of such ups and downs. the novelty in the model is that the labours are assumed as predators, which use the physical capitals as their prey in combination with holling type ii function. 2.1. model manifestation let, ( )tk and ( )tl represent the physical capital and labour force. then the dynamics of these two species can be studied through the following system of equations, in conjunction with the holling type ii function [3-9, 16-20] as, ( ) ( )( ) ( ) ( )tkttkfs td tdk ii  −−= , ( ) ( )( ) ( ) ( )tlttlgr td tdl ii  −+= , (1) with the initial conditions as, ( ) 0 0 kk = and, ( ) 0 0 ll = (2) in system (1), s and r show the capital investment and population of labours, respectively. here, we assume ( )( ) ( ) ( )         −= 1 1  tk tktkf and ( )( ) ( ) ( )         −= 2 1  tl tltlg , as the logistic growth of capital and labours, where 1  and 2  define the maximum populations of capital and labours sustained by the environment. whereas,  andextent to which the are the  availability of the labour force affects the growth of capital and vice versa, accordingly. the holling type ii function, ( )t ii  , is outlined as, ( ) ( ) ( ) ( )tkh tltk t ii   + = 1 , whereor the at which the labour come into contact with equipment the rate examines  learning time and h measures the handling time or the average time spent to process a capital int. j. anal. appl. 17 (4) (2019) 634 for producing output. finally,and tal expiry of the capi or explain the depreciation  nd a  the retirement, dismissal or death of labour, respectively. this advanced concept in the above system can be easily understood by considering different states of economic growth of any country. for instance, let the investment on machinery is increased while keeping the appointments of new labours constant. the existing labours are then allocated on appropriate machines for producing output. each labour has to operate multiple machines at a time. this is because the proportion of machines per labour decreases as capital density increases. additionally, at a very high level of investment on machines, labours require very little time waiting or searching for a machine and spend almost all their time in processing and producing output. effectiveness of labour is then satiated and the number of invested capital reaches a plateau. this will in return brings out a positive effect on the growth rate of the labour force as it increases the labour efficiency to produce output more effectively. 2.2. equilibria and stability illustration equilibria are the constant solutions to the system and are attained by taking the derivatives of the functions equal to zero, coupled with the stability analysis which is investigated through the eigenvalues of the jacobian matrix at a particular equilibrium point. generally, if the matrix has a positive real part or multiple zeroes then the equilibrium point is unstable for both cases. negative real parts of all eigenvalues conclude the point to be locally asymptotically stable. moreover, if the jacobian matrix has negative real eigenvalues along with a pair of purely imaginary eigenvalues, then a hopf bifurcation point might occur. the linearized stability test fails if it has negative real eigenvalues with exactly one zero eigenvalues. now, consider eq. (1) with ( ) 0= td tdk ,and let the jacobian matrix for an equilibrium ( ) 0= td tdl point ( ) lk , be, ( ) ( ) ( )               − + +        − + + −− + −        − =                     kh kl r kh l kh k kh lk s lk 1 2 1 1 11 2 1 , 2 2 2 1 , (3) then the governing system (1) has the following equilibrium points arising in different circumstances. int. j. anal. appl. 17 (4) (2019) 635 2.2.1. trivial equilibrium point on substituting the trivial equilibrium point ( )0,0 0 e in the jacobian matrix and simplifying, we get the eigen values ( ) −− rs , , which shows the 0 e to be locally asymptotically stable if,  s0 and.  r0 2.2.2. predator-free equilibrium point the predator-free equilibrium point is attained as ( )       − 0,1 1 s s e  . on substituting 1 e in the jacobian matrix and simplifying, we get the eigen values ( )         −+ − −−− 11 1,    hshs s rs , which shows the 1 e to be locally asymptotically stable if, s and. 1 11 1   h h r + ++−  2.2.3. prey-free equilibrium point the calculated prey free equilibrium point is ( )       − r r e  2 2 ,0 . for this point, the jacobian matrix produces the eigenvalues ( )       − +−− r r sr   2, , which shows the 2 e to be locally asymptotically stable if ( ) r rr s  −+  20 and. r 2.2.4. coexistence equilibrium point let the coexistence equilibrium point beuch that,, s ( )lke , 3 ( ) ( ) ( )                   +−+ ++         +−+ +−= 3/1 322 2 3/1 322 2 4 4 21 fmgaa fmg fmgaa g f k , ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 2 41 2 41 1 3/1 322 11 1 2 3/1 322 11 1 11 1           +−++++ + +           +−++++ + +++ −−= fmgaa f hshshs fmg fmgaa f hshshs f hshshsg sl         int. j. anal. appl. 17 (4) (2019) 636 where, ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ,129 914 2 2 11121 523 1121111 434 2 111 333 2 1 222 11 333     hhhshhsrh hshshhshssrh shshshsrh hhshsrha ++−+−− −+++−++ +−+− −+−= 22 3 srhf = ,, 1 22 1 22 2  rhsrhsrhg −+−=  21 2 21 2 21 2 11 22 hrhrhsrhsrm +−−−+−= , the 3 e will exists, if ( )fmga + 22 4 or ( ) 02 + fmg . on substituting 3 e in eq. (3), we get the following characteristic equation, 0 21 2 =++ dd  , with ( )         + + + +−−         + ++++−−= 2 2 12 1 11 2 1 2 kh lh kh s k kh r lsrd         and ( ) ( )                 + +−+ + −−         − + +−+ + = 2 2 12 3 2 2 1 2 1 2 11 kh lhs k kh l s lr kh k r kh lk d           which has the roots, 2 4 2 2 11 ddd −− = , sotherwise , o 04 2 2 1 − dd and 2,1=i or f 0id , is locally asymptotically stable if 3e unstable with a complex conjugate pair and then a hopf bifurcation will lead to a limit cycle. 3. numerical discussion with economic significance in this section, we have discussed the aforementioned model for some numerical values of the parameters and attained different graphical demonstration using the mathematica 11.0 software program. the cobb-douglas and constant-elasticity of production functions [21-24] are also portrayed to study the effects on the economic growth that takes place by means of the functional responses of physical capital and labour force. mathematica 11.0 has a built-in program of the well-known fifth-fourth runge kutta method. nowadays, this software has gained significant importance because of its friendly user interface and capability of solving any int. j. anal. appl. 17 (4) (2019) 637 kind of mathematical model. the appendix contains the code of this program made for illustrating the governing model numerically and graphically. a pictorial view of the system (1) is obtained by taking 2=s , 2=r , 3 1 = , 5 2 = ,, 1= 1= , 2= , 4= , 4= ,in shown , as ( ) 10 =l and ( ) 10 =k , with initial conditions 5=h fig. 2. the existence of trivial equilibrium point can be seen clearly in this figure for a long-term. this is assumed as the production phase in the absence of technical machinery and skilled labours, as comprehensively discussed in the cobb-douglas model [21-22]. it also explains that if depreciation of capital and termination of labours become greater than the investment and appointment of new labours, accordingly, both factors will extinct after a particular time. as a result, the economic growth will either stop or reach the underperformance stage. fig. 3 apparently describes the prey free equilibrium stage for the values, 2=s , 4=r , 3 1 = , 5 2 = , 6= , 1= , 8= , 4= , 2= ,. the , with the same initial conditions 15=h condition evidently states that the increase in the number of employees requires more investment in the capital. thus, if further investment is not made, the rate becomes inversely proportional to the increasing number of labours and the deficiency of physical capital occurs. therefore, on continuous usage of the available capitals, after reaching a particular time, it will suffer devastation and no more machinery will exist. it is the plateau when a change in the economy is due to the effective endeavours of labours to produce output without using advanced types of machinery. examples can be taken as the countries that are below the line of underdevelopment, which have a deficiency in the latest technical developments. subsequently, the economic growth will either stop or move slowly as compared to the developed countries that accompany high technologies to boost their production growth. the predator-free equilibrium is shown in fig. 4 for the parameters taken as 5=s , 020r = , 3 1 = , 5 2 = , 6= , 1= , 8= , 4= , 2= ,, with the aforementioned initial 3=h conditions. it elucidates that there is an investment on new latest technologies but due to the lack of skilled and expert labours, this will automatically decrease the efficiency of labour force towards the production, which in return decreases the economic growth. fig. 5 portrays the oscillations that illustrate the unstable state of coexistence equilibrium for the parameters taken as 3=s , 2=r , 1 1 = , 3 2 = , 6= , 1= , 8= , 2= , 3= ,, 5.3=h with the same initial conditions. it states the performance of both the factors with the rate of int. j. anal. appl. 17 (4) (2019) 638 investment on capital greater than the rate of labour appointments. this will create fluctuations in the economic activity of a country. furthermore, figs. 6-9 demonstrate the production for different stages of equilibria of the capital and labour force for the time  100,0t , 2 1 = , 3 2 = , 6= , 1= , 8= , , 5.3=h and different values of s , r , ,known -the wellplotting s are achieved by . these figure cobb-douglas production function, ( ) ( ) ( )  −= 1, tltklkp , (4) with factor productivity 1= and 4.0= . here, the values of ( )tk and ( )tl , generated through the dynamical system (1), are substituted in eq. (4) to attain the curves. to the best of our knowledge, in most of the literatures [21-22] the parametric graph of eq. (4) is usually plotted for constant rate of labour force. evidently, those curves contradict the reality that are the actual cause of rise and fall of economic cycle. here, comparative patterns of the eq. (4) are illustrated for distinct configurations of capital and labour. fig. 5 shows an increasing production process when investment and growth rate of labours are equal and so is the depreciation and retirement rate. in fig. 6 a curvy rise and fall can be seen in the production function, which is attained by taking the investment level less than growth of the labour and depreciation of the capital greater than the retirements of the labours. on the other hand, production process is found increasing in fig. 7 in a nonlinear manner for the case when the investment level is greater than the growth of the labour and also depreciation rate is taken greater than retirements. furthermore, when investment is increased as compared to the growth of labors and depreciation rate is decreased against the retirements of the labors, a cyclic production progress is attained, as shown in fig. 8. in addition, comparison plots of different production functions proposed in the literature [2124] are also demonstrated in figs. 10(a-c)-12(a-c). these graphs are also attained consequently through the solutions of the dynamical system (1) for different values of  and h , with 21 = , 3 2 = , 6= , 1= , 2= , 3= , 2=r ,is production functiondouglas -the cobb . 3=s shown in figs. 10a, 11a and 12a, the constant-elasticity of production function [23], , ( ) ( ) ( ) ( ) ( )( )   /1 10183.1, −−− −+= tltklkp t (5) int. j. anal. appl. 17 (4) (2019) 639 is plotted in figs. 10b, 11b and 12b for 584.0= , 519.0= and 756.0= . while figs. 10c, 11c and 12c represent the production function [23, 24], ( ) ( ) ( ) ( )( )   /11, −−− −+= tltklkp , (6) for 1= , 519.0= and 756.0= , where  is the efficiency parameter,  is functional distribution parameter and  is the elasticity substitution parameter in eqs. (5) and (6). the unstable open regions are clearly found in figs. 10(a-c) of each production function, respectively, for 8= and 1=h . figs. 11a and 11c, illustrate small logarithmic spirals for the parameters 2= and 2=h , whereas fig. 11c starts with a constant line and after some time goes round. furthermore, when 8= and 5.3=h , figs. 12a and 12c portrait a single rotation, while 12b produces multiple rotations, with the contraction in the open unstable region. 4. conclusion in this paper, we studied the economic growth by means of the ecological system in conjunction with holling-type ii function. the discussions were successfully made by relating the biological terms of the system with the economical parameters and defining preys and predators as the capital and labour force, accordingly. all the possible equilibrium points of the ecological structure were analysed according to economic significance. many production functions have been found in literature, which has been used to study economic growth. here, we took some among them and studied the effect of functional response of physical capital and labour force on these production functions. we figured out the following outcomes that are surely beneficial for the mainstream economists for different aspects of economic studies. • each of the constant solutions shows different states of the economy that might occur endogenously or exogenously. • generated production functions show different curves for each case of equilibrium points of physical capital and labour force. • the prey-free steady-state interprets the economy that is less developed technologically. here, exogenous causes might affect the rate of investment on the latest equipment, which can boost the efficiency of labour output. • predator-free state reflects the economy that lacks skilled labours, i.e. in the absence of competent labours, the increasing rate of investment on the latest types of machinery will not be adequately beneficial for economic growth. this state might also be due to int. j. anal. appl. 17 (4) (2019) 640 the migration of educated and proficient labours to other countries for the sack of high wages etc. thus to this end, proponents of endogenous causes might play a key role in this regard. • the coexistence fluctuations are considered as the ups and downs in the growth of capital and efficiency of labours force. the production function of such economies yields cyclic behaviour. these cycles keep the economic ball rolling of any country. • the functional response of physical capital and labour is evidently seen to be a major characteristic of the production function, as these factors together may create a positive effect on economic growth. • the time consumed by the labours on each equipment can modify the output of the labour force factor, which overall enrich the production progress. • the holling-type ii function helps to evaluate the handling time and searching efficiency of the labours for the physical capitals. • stability conditions at each state can balance the investment and depreciation on capital, similarly growth in excessive appointments and retirement or dismiss of professional labours. in future, some more interspecific interactions between different economic components will be discussed in addition to the imprecise theory. appendix solex = ndsolve[{d[k[t], t] == s k[t] (1 k[t]/\[eta]1 ) (\[beta] \[alpha] k[t] l[t])/(1 + \[alpha] h k[t]) \[delta] k[t], d[l[t], t] == r l[t] (1 l[t]/\[eta]2 ) + (\[gamma] \[alpha] k[t] l[t])/(1 + \[alpha] h k[t]) \[sigma] l[t] , k[0] == 1, l[0] == 1}, {k[t], l[t]}, {t, 0, 100}] plot[evaluate[{k[t], l[t]} /. solex], {t, 0, 5}, plotstyle→{blue, dashed}, frame→true, plottheme→"business", axesstyle→directive[black, 12], framelabel→{"time(t)", "population(k[t],l[t])"}, plotlegends→{"k[t]", "l[t]"}] parametricplot3d[evaluate[{(k[t])^0.4 (l[t])^0.6, (k[t]), (l[t])} /. solex], {t, 0, 100}, plottheme→"marketing", plotrange→all, axeslabel→{"p[k,l]", "k[t]", "l[t]"}] int. j. anal. appl. 17 (4) (2019) 641 solh1 = parametricndsolve[{d[k[t], t] == s k[t] (1 k[t]/\[eta]1 ) (\[beta] \[alpha] k[t] l[t])/(1 + \[alpha] h k[t]) \[delta] k[t], d[l[t], t] == r l[t] (1 l[t]/\[eta]2 ) + (\[gamma] \[alpha] k[t] l[t])/(1 + \[alpha] h k[t]) \[sigma] l[t] , k[0] == 1, l[0] == 1}, {k[t], l[t]}, {t, 0, 300}, {h, \[alpha]}, maxsteps →\[infinity]] parametricplot3d[evaluate[{(k[t][1, 8])^0.4 (l[t][1, 8])^0.6, (k[t][1, 8]), (l[t][1, 8])} /. solh1], {t, 0, 100}, plottheme→"marketing", plotrange→all, axeslabel→{"p[k,l]", "k[t]", "l[t]"}] parametricplot3d[evaluate[{0.584 (1.0183)^ t (0.519 (k[t][1, 8])^-0.756 + 0.481 (l[t][1, 8])^0.756)^-1.822, (k[t][1, 8]), (l[t][1, 8])} /. solh1], {t, 0, 100}, plottheme→"marketing", plotrange→all, axeslabel→{"p[k,l]", "k[t]", "l[t]"}] parametricplot3d[evaluate[{(0.519 (k[t][1, 8])^-0.756 + 0.481 (l[t][1, 8])^-0.756)^-1.822, (k[t][1, 8]), (l[t][1, 8])} /. solh1], {t, 0, 100}, plottheme→"marketing", plotrange→all, axeslabel→{"p[k,l]", "k[t]", "l[t]"}] references [1] d. cai, h.ye and l. gu, a generalized solow-swan model, abstr. appl. anal. 2014 (2014), art. id 395089. [2] m. zhou, d. cai and h. chen, a solow-swan model with technological overflow and catch-up, wuhan univ. j. nat. sci. 12(6) (2007), 975-978. [3] r.m. solow, a contribution to the theory of economic growth, quart. j. econ. 70 (1956), 65-94. [4] e. pelinescu, the impact of human capital on economic growth, proc. econ. finance. 22 (2015), 184190. [5] l. gallaway and v. shukla, the neoclassical production function, amer. econ. rev. 64(3) (1974), 348358. [6] a. hochstein, the harrod-domar model in a keynesian framework, int. adv. econ. res. 23 (2017), 349-350. [7] a.n. link and m.v. hasselt, a public sector knowledge production function, econ. lett. 175 (2019), 64-66. [8] m.d.p.p. romero, a.s. braza and a. exposito, industry level production functions and energy use in 12 eu countries, j. clean. prod. 212(1) (2019), 880-892. [9] m.d.p.p. romero and a.s. braza, productive energy use and economic growth: energy, physical and human capital relationships, energy econ. 49 (2015), 420-429. [10] a.j. lotka, elements of physical biology, first ed., williams and wilkins, baltimore, 1925. int. j. anal. appl. 17 (4) (2019) 642 [11] v. volterra, variazioni efluttuazioni del numero di individui in specie animali conviventi, mem. reale accad. naz. lince. 2 (1926), 31-113. [12] l. nie, z. teng, l. hu and j. peng, the dynamics of a lotka–volterra predator–prey model with state dependent impulsive harvest for predator, biosyst. 98 (2009), 67-72. [13] b. chakraborty and n. bairagi, complexity in a prey-predator model with prey refuge and diffusion, ecol. complexity. 37 (2019), 11-23. [14] b. sahoo and s. poria, dynamics of predator-prey system with fading memory, appl. math. comput. 347(15) (2019), 319-333. [15] a. bouskila, games played by predators and prey, encyclopaedia ani. behav. (2019), 382-388. [16] s.l. tilahun, prey predator hyperheuristic, appl. soft comput. 59 (2017), 104-114. [17] v. castellanos and r.e.c. lopez, existence of limit cycles in a three level trophic chain with lotkavolterra and holling type ii functional responses, chaos solitons fractals. 95 (2017), 157-167. [18] m. liu, c. du and m. deng, persistence and extinction of a modified leslie-gower holling-type ii stochastic predator-prey model with impulsive toxicant input in polluted environments, nonlinear anal. hybrid syst. 27 (2018), 177-190. [19] z. xiao, x. xie and y. xue, stability and bifurcation in a holling-type ii predator-prey model with allee effect and time delay, adv. difference equ. 2018 (2018), art. id 288. [20] j. carkovs, j. goldsteine and k. sadurskis, the holling-type ii population model subjected to rapid random attacks of predator, j. appl. math. 2018 (2018), art. id 6146027. [21] g.e. vilcu, a geometric perspective on the generalized cobb-douglas production functions, appl. math. lett. 24 (2011), 777-783. [22] x. wang and y. fu, some characterizations of the cobb-douglas and ces production functions in microeconomics, abstr. appl. anal. (2013), art. id 761832. [23] k.j. arrow, h.b. chenery, b.s. minhas and r.m. solow, capital-labour substitution and economic efficiency, rev. econ. statist. 43(3) (1961), 225-250. [24] k. sato, a two-level constant-elasticity of substitution production function, rev. econ. stud. 34(2) (1967), 201-218. int. j. anal. appl. 17 (4) (2019) 643 figure 2. solutions of eq. (1) showing trivial equilibrium point when 2=s , 2=r , 3 1 = , 5 2 = , 1= , 1= , 2= , 4= , 4= ,. 5=h figure 3. solutions of eq. (1) showing prey free equilibrium point when 2=s , 4=r , 3 1 = , 5 2 = , 6= , 1= , 8= , 4= , 2= ,. 15=h int. j. anal. appl. 17 (4) (2019) 644 figure 4. solutions of eq. (1) showing predator-free equilibrium point when 5=s , 020r = , 3 1 = , 5 2 = , 6= , 1= , 8= , 4= , 2= ,. 3=h figure 5. solutions of eq. (1) showing unstable fluctuations of coexistence equilibrium point when 3=s , 2=r , 1 1 = , 3 2 = , 6= , 1= , 8= , 2= , 3= ,. 5.3=h int. j. anal. appl. 17 (4) (2019) 645 figure 6. cobb-douglas production function [22] for  100,0t , 2 1 = , 3 2 = , 6= , 1= , 8= , 5.3=h when . 4== , 2== rs figure 7. cobb-douglas production function [22] for  100,0t , 2 1 = , 3 2 = , 6= , 1= , 8= , 5.3=h when . 2,4 ==  , 4,2 == rs int. j. anal. appl. 17 (4) (2019) 646 figure 8. cobb-douglas production function [22] for  100,0t , 2 1 = , 3 2 = , 6= , 1= , 8= , 5.3=h when . 2,4 ==  , 316.0,5 == rs figure 9. cobb-douglas production function [22] for  100,0t , 2 1 = , 3 2 = , 6= , 1= , 8= , 5.3=h when . 3,2 ==  , 2,3 == rs int. j. anal. appl. 17 (4) (2019) 647 (a) production function [22] with 1= and 4.0= . (b) production function [23] with 584.0= , 519.0= and 756.0= . int. j. anal. appl. 17 (4) (2019) 648 (c) production function [24] with 1= , 519.0= and 756.0= . figure 10(a-c). different production functions [22-24] for, 2 1 = , 3 2 = , 6= , 1= , 2= , 3= , 2=r ,. 1=h and 8= when 3=s (a) production function [22] with 1= and 4.0= . int. j. anal. appl. 17 (4) (2019) 649 (b) production function [23] with 584.0= , 519.0= and 756.0= . (c) production function [24] with 1= , 519.0= and 756.0= . figure 11(a-c). different production functions [22-24] for , 2 1 = , 3 2 = , 6= , 1= , 2= , 3= , 2=r ,. 2=h and 2= when 3=s int. j. anal. appl. 17 (4) (2019) 650 (a) production function [22] with 1= and 4.0= . (b) production function [23] with 584.0= , 519.0= and 756.0= . int. j. anal. appl. 17 (4) (2019) 651 (c) production function [24] with 1= , 519.0= and 756.0= . figure 12(a-c). different production functions [22-24] for , 2 1 = , 3 2 = , 6= , 1= , 2= , 3= , 2=r ,. 5.3=h and 8= when 3=s int. j. anal. appl. (2022), 20:69 a novel of cubic ideals in γ-semigroups pannawit khamrot1, thiti gaketem2,∗ 1department of mathematics, faculty of science and agricultural technology, rajamangala university of technology lanna of phitsanulok, phitsanulok, thailand 2fuzzy algebras and decision-making problems research unit, department of mathematics school of science, university of phayao, phayao 56000, thailand ∗corresponding author: thiti.ga@up.ac.th abstract. in this paper, we give the concepts of new types of cubic ideals in γ-semigroups. we study properties and relationships between cubic (α,β)-ideals and ideals in semigroups. furthermore, we proved some basic properties of cubic almost ideals in semigroups. 1. introduction the theory for dealing with uncertainty, fuzzy set theory, was discovered by zadeh in 1965 [18], mathematical tool for describing the behavior of the systems that are too complex or illdefined to admit precise mathematical analysis by classical methods and tools. the studies of cubic sets and cubic subgroups were presented by jun et al in 2012 [10,11]. later v. chinnadurai and k. bharathivelan [2] studies cubic ideals in γ-semigroups and proved basic properties of cubic ideals in γ-semigroups. the theory of ideal is structured important in semigroups and many researchers used knowledge of ideals in γ-semigroups discussed in fuzzy semigroup such as chinram et al. [3] discussed almost quasi-γ-ideal and fuzzy almost quasi-γ-ideals in γ-semigroup, m. k. r. marapureddy and prv s. r. doradla [14] discussed weak interior ideals of γ-semigroups, s.k. majumder and m. mandal [9] discussed fuzzy generalized bi-ideal in γ-semigroups. in the study of the concept of cubic ideals, many researchers expanded on this idea [4,6,7,7,8,13,15]. recently, in 2021 [17] a. simuen et al. discussed a novel of ideals and fuzzy ideals of γ-semigroups. received: nov. 14, 2022. 2010 mathematics subject classification. 20m12, 06f05. key words and phrases. (α,β)-cubci ideal; cubic almost ideal; γ-semigroups. https://doi.org/10.28924/2291-8639-20-2022-69 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-69 2 int. j. anal. appl. (2022), 20:69 in this paper we extend new fuzzy ideals to cubic ideals of γ-semigroups and we investigate the properties of new types cubic ideals. of γ-semigroups. 2. preliminaries in this section, we review concepts basic definitions and the theorem used to prove all result in the next section. a sub-γ-semigroup of a γ-semigroup s is a non-empty set k of s such that kγk ⊆ k. a left (right) ideal of a γ-semigroup s is a non-empty set k of s such that sγk ⊆k (kγs ⊆k). by an ideal of a γ-semigroup s it is both a left and a right ideal of s. a quasi-ideal of a γ-semigroup s is a non-empty set k of s such that kγs∩sγk⊆k. a sub-γ-semigroup k of a γ-semigroup s is called a bi-ideal of s if kγsγk⊆k. definition 2.1. [17] let s be a γ-semigroup, k be a non-empty subset of s, for all e ∈ s and α,β ∈ γ. then k is said to be (1) a left (right) almost ideal of γ-semigroup s is a non-empty set k such that (eγk) ∩k 6= ∅ ((kγe) ∩k 6= ∅) (2) an almost bi-ideal of γ-semigroup s is a non-empty set k such that (kγeγk) ∩k 6= ∅. (3) an almost quasi-ideal of γ-semigroup s is a non-empty set k such that (eγk∩kγe)∩k 6= ∅. (4) a left α-ideal of a γ-semigroup s is a non-empty set k such that sαk ⊆k. a right β-ideal of a γ-semigroup s is a non-empty set k such that kβs ⊆k. (5) an (α,β)-ideal of a γ-semigroup s is a non-empty set k such that it is both a left α-ideal and a right β-ideal of s. we see that for any ζ1,ζ2 ∈ [0, 1], we have ζ1 ∨ζ2 = max{ζ1,ζ2} and ζ1 ∧ζ2 = min{ζ1,ζ2}. a fuzzy set υ of a non-empty set t is function from t into unit closed interval [0, 1] of real numbers, i.e., υ : t → [0, 1]. for any two fuzzy sets υ and ν of a non-empty set t , define ≥, =,∧, and ∨ as follows: (1) υ ≥ ν ⇔ υ(e) ≥ ν(e) for all e ∈t , (2) υ = ν ⇔ υ ≥ ν and ν ≥ υ, (3) (υ ∧ν)(e) = min{υ(e),ν(e)} = υ(e) ∧ν(e) for all e ∈t , (4) (υ ∨ν)(e) = max{υ(e),ν(e)} = υ(e) ∨ν(e) for all e ∈t . for the symbol υ ≤ ν, we mean ν ≥ υ. the following definitions are types of fuzzy subsemigroups on γ-semigroups. definition 2.2. [17] a fuzzy set υ of a γ-semigroup s is said to be (1) a fuzzy subsemigroup of s if υ(eγf ) ≥ υ(e) ∧υ(f ) for all e,f ∈s and γ ∈ γ, (2) a fuzzy left (right) ideal of s if υ(eγf ) ≥ υ(f ) (υ(eγf ) ≥ υ(e)) for all e,f ∈s and γ ∈ γ, int. j. anal. appl. (2022), 20:69 3 (3) a fuzzy ideal of s if it is both a fuzzy left ideal and a fuzzy right ideal of s, (4) a fuzzy bi-ideal of s if υ is a fuzzy subsemigroup of s and υ(eγf βh) ≥ υ(e) ∧υ(h) for all e,f ,h ∈s and γ,β ∈ γ. now, we review the concept of interval valued fuzzy sets. let cs[0, 1] be the set of all closed subintervals of [0, 1], i.e., cs[0, 1] = {ω̌ = [ωl,ωu] | 0 ≤ ωl ≤ ωu ≤ 1}, where ωl is a lower interval value of ω̌ and ωu is an upper interval value of ω̌. we note that [ω,ω] = {ω} for all ω ∈ [0, 1]. for ω = 0 or 1, we shall denote [0, 0] by 0̌ and [1, 1] by 1̌. for ω̌ := [ωl,ωu] and ζ̌ := [ζl,ζu] in cs[0, 1], the operations “�", “=", “f", “g" are defined as follows: (1) ω̌ � ζ̌ if and only if ωl ≤ ζl and ωu ≤ ζu (2) ω̌ = ζ̌ if and only if ωl = ζl and ωu = ζu (3) ω̌ f ζ̌ = [(ωl ∧ζl), (ωu ∧ζu)] (4) ω̌ g ζ̌ = [(ωl ∨ζl), (ωu ∨ζu)]. if ω̌ � ζ̌, we mean ζ̌ � ω̌. definition 2.3. [19] an interval valued fuzzy set (shortly, ivf set) of a non-empty set t is a function ω̌ : t → cs[0, 1]. next, we shall give definitions of various types of ivf subsemigroups. definition 2.4. [1] an ivf set ω̌ of a γ-semigroup s is said to be an ivf subsemigroup of s if ω̌(eαf ) % ω̌(e) f ω̌(f ) for all e,f ∈s and α ∈ γ. definition 2.5. [1] an ivf set ω̌ of a semigroup s is said to be an ivf left (right) ideal of s if ω̌(eαf ) % ω̌(e) (ω̌(eαf ) % ω̌(e)) for all e,f ∈s and α ∈ γ. an ivf subset ω̌ of s is called an ivf ideal of s if it is both an ivf left ideal and an ivf right ideal of s. definition 2.6. [1] let k be a subset of a non-empty set t . an interval valued characteristic function (shortly, ivcf) χ̌k of t is defined to be a function χ̌k : t → cs[0, 1] by χ̌k(e) =  1̌ if e ∈k, 0̌ if e /∈k for all e ∈t . for two ivf subsets ω̌ and ζ̌ of a non-empty set t , define (1) ω̌ v ζ̌ ⇔ ω̌(e) � ζ̌(e) for all e ∈t , 4 int. j. anal. appl. (2022), 20:69 (2) ω̌ = ζ̌ ⇔ ω̌ v ζ̌ and ζ̌ v ω̌, (3) (ω̌ u ζ̌)(e) = ω̌(e) f ζ̌(e) for all e ∈t . (4) (ω̌ t ζ̌)(e) = ω̌(e) g ζ̌(e) for all e ∈t . definition 2.7. [10] a cubic set (cb set) c of a non-empty set t is a structure of the form c = {〈e,ω̌(e),υ(r)〉 | e ∈t} and denoted by c = 〈ω̌,υ〉 where ω̌ is an ivf set and υ is a fuzzy set. in this case, we will use c(e) = 〈ω̌(e),υ(e)〉 = 〈[ωl(e),ωu(e)],υ(e)〉 for all e ∈t . definition 2.8. [11] let t be a semigroup and k be a non-empty set of t , the characteristic cb set of k in t is defined to be the structure ≥k = {〈e,ω̌λk(e),υλk(e)〉 : e ∈ t} which is briefly denoted by ≥k = 〈ω̌λk,υλk〉 where ω̌λk(e) =  1̌, if e ∈k, 0̌, if e /∈k and υλk(e) =  0, if e ∈k, 1, if e /∈k. definition 2.9. [2] a cb set c = 〈ω̌,υ〉 of a γ-semigroup s is called (1) a cb subsemigroup of s if ω̌(eαf ) % ω̌(e)f ω̌(f ) and υ(eαf ) ≤ υ(e)∨υ(f ) for all e,f ∈s and α ∈ γ. (2) a cb left(right)ideal of s if ω̌(eαf ) % ω̌(f ) (ω̌eαf ) � ω̌(e)) and υ(eαf ) ≤ υ(f )(υ(eαf ) ≤ υ(e)) for all e,f ∈s and α ∈ γ. a cb ideal of s if it is both a cb left ideal and a cb right ideal of s. for e ∈t , define fe = {(y,z) ∈t ×t | e = yz}. definition 2.10. [11] let c = 〈ω̌,υ〉 and d = 〈ζ̌,ν〉 be two cb set in a semigroup s. then the cb product of c and d is a structure c�d = {〈e, (ω̌�ζ̌)(e, (υ ·ν)(e)〉 : e ∈s} which is briefly denoted by c � d = 〈(ω̌�ζ̌), (υ · ν)〉 where ω̌�ζ̌ and υ · ν are defined as follows, respectively: (ω̌�ζ̌)(e) =  rsup(y,z)∈fe{ω̌(y) f ζ̌(z)} if fe 6= ∅, 0̌, if fe = ∅, int. j. anal. appl. (2022), 20:69 5 and (υ ·ν)(e) =  inf (y,z)∈fe{υ(y) ∨ν(z)} if fe 6= ∅, 1, if fe = ∅. definition 2.11. [11] for two cb st c = 〈ω̌,υ〉 and d = 〈ρ̌,τ〉 in a semigroup s, we define c v d ⇔ ω̌ ρ̌ and υ ≥ τ definition 2.12. [11] let c = 〈ω̌,υ〉 and d = 〈ρ̌,τ〉 be two cb set in a semigroup s. then the intersection of c and d denoted by cud is the cb set cǔd = 〈ω̌ u ζ̌,υ ∨ν〉 where (c u d)(e) = ω̌(e) f ρ̌(e) and (υ ∨τ)(e) = υ(e) ∨τ(e) for all e ∈s. and union of c and d denoted by ctd is the cb set cťd = 〈ω̌ t ρ̌,υ∧τ〉 where (ω̌ t ρ̌)(e) = ω̌(e) g ρ̌(e) and (υ ∧τ)(e) = υ(e) ∧τ(e) for all e ∈s. 3. new types of cubic ideals in this section, we define cubic fuzzy (α,β)-ideal and study basic properties of it. definition 3.1. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ. then c = 〈ω̌,υ〉 is called (1) a cb left α-ideal of s if ω̌(eαf ) % ω̌(f ) and υ(eαf ) ≤ υ(f ) for all e,f ∈s. (2) a cb right β-ideal of s if ω̌(eβf ) % ω̌(e) and υ(eβf ) ≤ υ(f ) for all e,f ∈s. (3) a cb (α,β)-ideal of s if it is both a cb left α-ideal and a cb right β-ideal of s. (4) a cb α-ideal of s if it is a cb (α,α)-ideal of s. theorem 3.1. let k be a nonempty subset of γ-semigroup s. then k is a left α-ideal (right β-ideal, (α,β)-ideal) of s if and only if ≥k = 〈ω̌λk,υλk〉 is a cb left α-ideal (right β-ideal, (α,β)-ideal) of s. proof. suppose that k is a left α-ideal of s and e,f ∈s. if f ∈k, then eαf ∈k. thus ω̌λk(f ) = ω̌λk(eαf ) = 1̌ and υλk(f ) = υλk(eαf ) = 0. hence ω̌λk(eαf ) % ω̌λk(f ) and υλk(eαf ) ≤ υλk(f ). if f /∈k, then eαf ∈k. thus ω̌λk(f ) = 0̌, ω̌λk(eαf ) = 1̌ and υλk(f ) = 1,υλk(eαf ) = 0. hence, ω̌λk(eαf ) % ω̌λk(f ) and υλk(eαf ) ≤ υλk(f ). therefore ≥k = 〈ω̌λk,υλk〉 is a cb left α-ideal of s. conversely, assume that ≥k = 〈ω̌λk,υλk〉 is a cb left α-ideal of s and e,f ∈s with f ∈k. then ω̌λk(f ) = 1̌ and υλk(f ) = 0. by assumption, ω̌λk(eαf ) % ω̌λk(f ) and υλk(eαf ) ≤ υλk(f ). thus, eαf ∈k. hence, k is a left α-ideal of s. � 6 int. j. anal. appl. (2022), 20:69 theorem 3.2. the intersection and union of any two cb left α-ideals (right β-ideals, (α,β)-ideals) of a γ-semigroup s is a cb left α-ideal (right β-ideal, (α,β)-ideal) of s. proof. let c = 〈ω̌,υ〉 and d = 〈ρ̌,τ〉 be cb left α-ideals of s and let e,f ∈s. then (ω̌ u ρ̌)(eαf ) = ω̌(eαf ) f ρ̌(eαf ) % ω̌(f ) f ρ̌(f ) = (ω̌ u ρ̌)(f ) and (υ ∩τ)(eαf ) = υ(eαf ) ∨τ(eαf ) ≤ υ(v) ∨τ(f ) = (υ ∩τ)(f ). similarly, (ω̌ t ρ̌)(eαf ) = ω̌(eαf ) g ρ̌(eαf ) % ω̌(f ) g ρ̌(f ) = (ω̌ u ρ̌)(f ) and (υ ∪τ)(eαf ) = υ(eαf ) ∧τ(eαf ) ≤ υ(v) ∧τ(f ) = (υ ∪τ)(f ). thus, cǔd and cťd are cb left α-ideals of s. � theorem 3.3. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and c(ň,m) = (ω̌ň,υm) be cb point with ω̌ň = {f ∈ s | ω̌ň(f ) % ň} and υm = {f ∈ s | υm(f ) ≤ m}. then c = 〈ω̌,υ〉 is a cb left α-ideal (right β-ideal, (α,β)-ideal) of s if and only if c = 〈ω̌,υ〉 is a nonempty set and c(ň,m) is a left α-ideal (right β-ideal, (α,β)-ideal) of s for all (ň,m) ∈ (0, 1] × [0, 1). proof. suppose that c = 〈ω̌,υ〉 is a cb left α-ideal of s and α ∈ γ. then ω̌(eαf ) % ω̌(f ) and υ(eαf ) ≤ υ(f ) for all e,f ∈ s. let (ň,m) ∈ (0, 1] × [0, 1) be such that c(ň,m) 6= ∅. let f ∈ c(ň,m) and f ∈ s. then ω̌(f ) % 1̌ and υ(f ) ≤ m. thus ω̌(eαf ) % ω̌(f ) % ň and υ(eαf ) ≤ υ(f ) ≤ m so eαf ∈ c(ň,m). hence, c(ň,m) = (ω̌ň,υm) is a left α-ideal of s. conversely, assume that c(ň,m) = (ω̌ň,υm) is a left α-ideal of s if (ň,m) ∈ (0, 1] × [0, 1) and c(ň,m) 6= ∅. let e,f ∈ s and ň = ω̌(f ),m = υ(f ). by assumption ω̌(f ) % ň and υ(f ) ≤ m. then f ∈ c(ň,m). thus c(ň,m) 6= ∅. hence, c(ň,m) is a left α-ideal of s. since f ∈ c(ň,m) and e ∈ s, we have eαf ∈ c(ň,m). thus, ω̌(eαf ) ≥ ň = ω̌(f ) and υ(eαf ) ≤ m = υ(f ). hence, c = 〈ω̌,υ〉 is a cb left α-ideal of s. � next, we will define the (α,β)-product. for cb c = 〈ω̌,υ〉 and d = 〈ρ̌,τ〉, define product ω̌ ◦α ρ̌ and υ ◦α τ as follows: for e ∈s (ω̌�αρ̌)(e) =  rsup(y,z)∈feα{ω̌(y) fα ρ̌(z)} if feα 6= ∅, 0̌, if feα = ∅, and (υ ·α τ)(e) =  inf (y,z)∈fe{υ(y) ∨α τ(z)} if feα 6= ∅, 1, if feα = ∅. where feα = {(y,z) ∈s×s | e = yz}, for e ∈s and α ∈ γ. int. j. anal. appl. (2022), 20:69 7 next, we define cb (α,β)-bi-ideal and study basic properties of it. definition 3.2. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ. then c = 〈ω̌,υ〉 is called a cb (α,β)-bi-ideal of s if ω̌ ◦α ω̌λs ◦β ω̌ % ω̌ and υ ◦α υλs ◦β υ ≤ υ where ≥s = 〈ω̌λs,υλs〉 is cb set mapping every element of s to 〈1̌, 0〉. theorem 3.4. let k be a nonempty subset of γ-semigroup s. then k is an (α,β)-bi-ideal of s if and only if characteristic function ≥k = 〈ω̌λk,υλk〉 is a cb (α,β)-bi-ideal of s. proof. suppose that k is an (α,β)-bi-ideal of s and kαsβk⊆k. if e ∈kαsβk, then ω̌λk(e) = (ω̌λk ◦α ω̌λs ◦β ω̌λk)(e) = 1̌ and υλk(e) = (υλk ◦α υλs ◦β υλk)(e) = 0. hence, (ω̌λk ◦α ω̌λs ◦β ω̌λk)(e) % ω̌λk(e) and (υλk ◦α υλs ◦β υλk) ≤ υλk(e) if e ∈kαsβk, then ω̌λk(e) = (ω̌λk ◦α ω̌λs ◦β ω̌λk)(e) = 0̌ and υλk(e) = (υλk ◦α υλs ◦β υλk)(e) = 1. hence, (ω̌λk ◦α ω̌λs ◦β ω̌λk)(e) % ω̌λk(e) and (υλk ◦α υλs ◦β υλk) ≤ υλk(e). therefore, ≥k = (ω̌λk,υλk) is a cb (α,β)-bi-ideal of s. conversely, assume that ≥k = (ω̌λk,υλk) is a cb (α,β)-bi-ideal of s. and e ∈kαsβk. then (ω̌ ◦α ω̌λk ◦β ω̌)(e) = 1̌ and (υ ◦α ≥s ◦β υ)(e) = 0. by assumption, (ω̌λk ◦α ω̌λs ◦β ω̌λk)(e) % ω̌λk(e) and (υλk ◦α υλs ◦β υλk) ≤ υλk(e). thus, e ∈k. hence, k is an (α,β)-bi-ideal of s. � theorem 3.5. the intersection of any two cb (α,β)-bi-ideals of a γ-semigroup s is a cb (α,β)-biideal of s. proof. let c = 〈ω̌,υ〉 and d = 〈ρ̌,τ〉 be cb (α,β)-bi-ideals of s and e ∈s. then ((ω̌ u ρ̌) ◦α ω̌λs ◦β (ω̌ u ρ̌))(e) % (ω̌ ◦α ω̌λs ◦β ω̌)(e) f (ρ̌◦α ω̌λs ◦β ρ̌)(e) % (ω̌ u ρ̌)(e) and ((υ ∩τ) ◦α υλs ◦β (υ ∩τ))(e) ≤ (υ ◦α υλs ◦β υ)(u) ∨ (τ ◦α υλs ◦β τ)(u) ≤ (υ ∩τ)(e). thus, cǔd is a cb α-bi-ideals of s. � next, we define cb (α,β)-quasi-ideal and study basic properties of it. definition 3.3. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ. then c = 〈ω̌,υ〉 is called a cb (α,β)-quasi-ideal of s if ω̌λs ◦α ω̌ u ω̌ ◦β ω̌λs ω̌ and υλs ◦α υ ∪υ ◦β υλs ≥ υ. theorem 3.6. if c = 〈ω̌,υ〉 and d = 〈ρ̌,η〉 is a cb left α-ideal and a cb right α-ideal of s respectively, then cǔd is a cb α-quasi-ideal of s. 8 int. j. anal. appl. (2022), 20:69 proof. let c = 〈ω̌,υ〉 and d = 〈ρ̌,η〉 is a cb left α-ideal and a cb right α-ideal of s respectively. then ρ̌◦α ω̌ ω̌λs ◦α ω̌ ω̌ and ρ̌◦α ω̌ ρ̌◦α ω̌λs ρ̌. thus, ρ̌◦α ω̌ ω̌ ∩ ρ̌. so, ω̌λs ◦α (ω̌ u ρ̌) u (ω̌ u ρ̌) ◦α ω̌λs v ω̌λs ◦α (ω̌ u ρ̌) ◦α ω̌λs ω̌ ∩ ρ̌. thus, ω̌u ρ̌ is a cb α-quasi-ideal of s. similarly, we can show that υ∩η is a cs α-quasi-ideal of s. hence, cǔd is a cb α-quasi-ideal of s. � theorem 3.7. every cb (α,β)-quasi-ideal of γ-semigroup s is intersection of a cb left α-ideal and a cb right β-ideal of s. proof. let c = 〈ω̌,υ〉 be a cb (α,β)-quasi-ideal of s. consider ρ̌ = ω̌ t (ω̌λs ◦α ω̌) and τ = υ ∪ (υλs ◦α υ) where d = 〈ρ̌,τ〉, $̌ = ω̌ t (ω̌ ◦β ω̌λs ) and ν = υ ∪ (υ ◦β υλs ) where k = 〈$̌,ν〉. then ω̌λs ◦α ρ̌ = ω̌λs ◦α (ω̌ t (ω̌λs ◦α ω̌)) = (ω̌λs ◦α ω̌) t (ω̌λs ◦α (ω̌λs ◦α ω̌)) = (ω̌λs ◦α ω̌) t ((≥s ◦α ω̌λs ) ◦α ω̌) = (ω̌λs ◦α ω̌) t (≥s ◦α ω̌) ω̌ t (ω̌λs ◦α ω̌) = ρ̌. and $̌ ◦β ω̌λs = (ω̌ t (ω̌ ◦β ω̌λs )) ◦α ω̌λs = (ω̌ ◦α ω̌λs ) t (ω̌ ◦β ω̌λs ◦α ω̌λs ) = (ω̌ ◦α ω̌λs ) t ω̌ ◦β (ω̌λs ◦α ω̌λs ) = (ω̌ ◦α ω̌λs ) t (ω̌ ◦β ω̌λs ) ω̌ t (ω̌ ◦β ω̌λs ) = $̌. simlarly, we can show that υλs ◦α τ ≥ τ and ν ◦β υλs ≥ ν. thus d = 〈ρ̌,τ〉 and k = 〈$̌,ν〉 is a cb left α-ideal and a cb right β-ideal of s. now, ω̌ v (ω̌ t (ω̌λs ◦α ω̌)) u (ω̌ t (ω̌ ◦β ω̌λs )) = ρ̌u $̌ and ρ̌∩ $̌ = (ω̌ t (ω̌λs ◦α ω̌)) u (ω̌ t (ω̌ ◦β ω̌λs )) = ω̌ ∩ ((≥s ◦α ω̌) t (ω̌ ◦β ω̌λs )) ω̌ u ω̌ = ω̌. hence, ω̌ = ρ̌u $̌. simlarly, we can show that υ = τ ∩ν. � theorem 3.8. let k be a nonempty subset of γ-semigroup s. then k is a (α,β)-quasi-ideal of s if and only if characteristic function ≥k = (ω̌λk,υλk) is a cb (α,β)-quasi-ideal of s. proof. suppose that k is a (α,β)-quasi-ideal of s and e ∈s. if f ∈ (sαk) ∩ (kβs), then e ∈k. thus ω̌λk(e) = 1̌ and υλk(e) = 0. hence ((ω̌λk ◦α ω̌λs ) f (ω̌λs ◦β ω̌λk))(f ) ω̌λk(u) and ((υλk ◦α υλs ) ∨ (υλs ◦β υλk))(u) ≥ υλk(f ). if f /∈ (sαk) ∩ (kβs), then e ∈k. thus ω̌λk(e) = 0̌ and υλk(e) = 1. hence, ω̌λk))(f ) ω̌λk(u) and ((υλk ◦α υλs ) ∨ (υλs ◦β υλk))(u) ≥ υλk(f ). therefore ≥k = (ω̌λk,υλk) is a cb (α,β)-quasi-ideal of s. int. j. anal. appl. (2022), 20:69 9 conversely, assume that ≥k = (ω̌λk,υλk) is a cb (α,β)-quasi-ideal of s and f ∈ (sαk) ∩ (kβs). then ((ω̌λk ◦α ω̌λs ) f (ω̌λs ◦β ω̌λk))(f ) = 1̌ and ((υλk ◦α υλs ) ∨ (υλs ◦β υλk))(f ) = 0. by assumption, ((ω̌λk ◦α ω̌λs ) f (ω̌λs ◦β ω̌λk))(f ) ω̌λk(f ) and ((υλk ◦α υλs ) ∨ (υλs ◦β υλk))(f ) ≥ υλk(f ) thus e ∈k. hence, k is a (α,β)-quasi-ideal of s. � 4. new types of cubic almost ideals definition 4.1. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ is said to be (1) a cb almost left α-ideal of s if (ω̌n ◦α ω̌) u ω̌ 6= 0̌ and (υm ◦α υ) ∪υ 6= 1. (2) a cb almost right β-ideal of s if (ω̌ ◦β ω̌n) u ω̌ 6= 0̌ and (υ ◦β υm) ∪υ 6= 1. (3) a cb almost (α,β)-ideal of s if it is both a cb almost left α-ideal and a cb almost right β-ideal of s. theorem 4.1. if c = 〈ω̌,υ〉 is a cb almost left α-ideal (right β-ideal, (α,β)-ideal) of a γ-semigroup s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d, then d = 〈ρ̌,τ〉 is a cb left almost α-ideal (right β-ideal, (α,β)-ideal) of s. proof. suppose that c = 〈ω̌,υ〉 is a cb almost left α-ideal of s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d. then (ω̌n ◦α ω̌)u ω̌ 6= 0̌ and (υm ◦α υ)∪υ 6= 1. thus (ω̌n ◦α ω̌)u ω̌ (ρ̌n ◦α ρ̌)u ρ̌ 6= 0̌ and (υm ◦α υ) ∪υ ≥ (τm ◦α τ) ∪τ 6= 0. hence, d = 〈ρ̌,τ〉 is a cb left almost α-ideal of s. � theorem 4.2. let k be a nonempty subset of γ-semigroup s. then k is an almost left α-ideal (right β-ideal, (α,β)-ideal) of s if and only if characteristic function ≥k = (ω̌λk,υλk) is a cb almost left α-ideal (right β-ideal, (α,β)-ideal) of s. proof. suppose that k is an almost left α-ideal of s. then eαk∩k 6= ∅ for all e ∈ s. thus there exists r ∈ eαk and r ∈k. so (ω̌n ◦α ω̌λk)(r) = ω̌λk(r) = 1̌ and (υm ◦α υλk)(r) = υλk(r) = 0. hence, (ω̌n ◦α ω̌λk) u ω̌λk 6= 0̌ and (υm ◦α υλk) ∪υλk 6= 1. therefore, ≥k = (ω̌λk,υλk) is a cb almost left α-ideal of s. conversely, assume that ≥k = (ω̌λk,υλk) is a cb almost left α-ideal of s and e ∈ s. then (ω̌n ◦α ω̌λk)u ω̌λk 6= 0̌ and (υm ◦α υλk)∪υλk 6= 1. thus there exists r ∈s such that ((ω̌n ◦α ω̌λk)f ω̌λk)(r) 6= 0̌ and ((υm ◦α υλk) ∨υλk)(r) 6= 1. hence, r ∈ eαk∩k implies eαk∩k 6= ∅. therefore k is an almost left α-ideal of s. � next, we review definition of supp(c) and we study properties between supp(ξ) and cb almost left α-ideal (right β-ideal, (α,β)-ideal) of γ-semigroups. let c = 〈ω̌,υ〉 be a cb set of a non-empty of s. then the support of c instead of supp(c) = {e ∈s | ω̌(e) 6= 0̌ andυ(e) 6= 0}. 10 int. j. anal. appl. (2022), 20:69 theorem 4.3. let c = 〈ω̌,υ〉 be a cb set of a non-empty of a γ-semigroup s. then c = 〈ω̌,υ〉 is a cb almost left α-ideal (right β-ideal, (α,β)-ideal) of s if and only if supp(c) is an almost left α-ideal (right β-ideal, (α,β)-ideal) of s. proof. let c = 〈ω̌,υ〉 be a cb almost left α-ideal of s and e ∈s. then (ω̌n ◦α ω̌) u ω̌ 6= 0̌and (υm ◦α υλk) ∪ υλk 6= 1. thus there exists r ∈ s such that ((ω̌n ◦α ω̌λk) f ω̌λk)(r) 6= 0̌ and ((υm ◦α υλk) ∨υλk)(r) 6= 1. so there exists k ∈ s such that r = uαk, ω̌(r) 6= 0̌, υ(r) 6= 0 and ω̌(k) 6= 0̌, υ(k) 6= 0. it implies that r,k ∈ supp(c). thus, (ω̌n ◦α ω̌λsupp(c) )(r) 6= 0̌, (υm ◦α υλsupp(c) )(r) 6= 1 and ω̌λsupp(c) 6= 0̌, υλsupp(c) 6= 1. hence, (ω̌m ◦α ω̌λsupp(c) ) u ω̌λsupp(c) 6= 0̌ and (υm ◦α υλsupp(c) ) ∪υλsupp(c) 6= 1. therefore, supp(c) is a cb almost left α-ideal of s. this show that supp(c) is an almost left α-ideal of s. conversely, let supp(c) be an almost left α-ideal of s. then by theorem 4.2, ≥supp(c) is a cb almost left α-ideal of s. thus (ω̌m ◦α ω̌λsupp(c) ) u ω̌λsupp(c) 6= 0̌ and (υm ◦α υλsupp(c) ) ∪υλsupp(c) 6= 1. so there exists r ∈s such that ((ω̌m ◦α ω̌λsupp(c) ) f ω̌λsupp(c) )(r) 6= 0̌ and (υm ◦α υλsupp(c) ) ∨υλsupp(c) (r) 6= 1. it implies that ((ω̌n ◦α ω̌λsupp(c) ))(r) 6= 0̌, ((υm ◦α υλsupp(c) ))(r) 6= 1 and ω̌λsupp(c) (r) 6= 0̌, υλsupp(c) (r) 6= 1. thus there exists k ∈ s such that r = uαk, ω̌n(r) 6= 0̌, υn(r) 6= 0 and ω̌n(k) 6= 0̌, υm(k) 6= 0. hence, (ω̌n ◦α ω̌) u ω̌ 6= 0 and (υm ◦α υ) ∪υ 6= 1. therefore, c = 〈ω̌,υ〉 be a cb almost left α-ideal of s. � definition 4.2. an almost ideal i of a γ-semigroup s is called a minimal if for every almost ideal of j of s such that j ⊆i, we have j = i. definition 4.3. a cb almost left α-ideal (right β-ideal, (α,β)-ideal) c = 〈ω̌,υ〉 of a γ-semigroup s is minimal if for all cb almost left α-ideal (right β-ideal, (α,β)-ideal) d = 〈ρ̌,τ〉 of s such that d v c, then supp(d) = supp(c). theorem 4.4. let k be a nonempty subset of a γ-semigroup s then k is a minimal almost left α-ideal (right β-ideal, (α,β)-ideal) if and only if ≥k = (ω̌λk,υλk) is a minimal cb almost left α-ideal (right β-ideal, (α,β)-ideal) of s. proof. suppose that k is a minimal almost left α-ideal of s. then k is an almost left α-ideal of s. thus by theorem 4.2, ≥k = (ω̌λk,υλk) is a cb left α-ideal of s. let d = 〈ρ̌,τ〉 be a cb left α-ideal of s such that d v c then by theorem 4.3, supp(d) is an almost left α-ideal of s. thus supp(d) v supp(≥k) = k. by assumption, supp(d) = k = supp(≥k). thus, ≥k = (ω̌λk,υλk) is a minimal cb almost left α-ideal of s. conversely, suppose that ≥k = (ω̌λk,υλk) is a minimal cb almost left α-ideal of s. then by theorem 4.2, k is an almost left α-ideal of s. let j be an almost left α-ideal of s such that j ⊆ k. then by theorem 4.2, ≥j = (ω̌λj ,υλj ) is a cs left α-ideal of s such that ≥j v ≥k. thus, j = supp(≥j ) = supp(≥k) = k. hence, k is a minimal almost left α-ideal of s. � int. j. anal. appl. (2022), 20:69 11 corollary 4.1. let s be a γ-semigroup then s has no proper almost left α-ideal (right β-ideal, (α,β)-ideal) if and only if for any cb almost left α-ideal (right β-ideal, (α,β)-ideal) c = 〈ω̌,υ〉 of s, supp(c) = s. next, we define cb almost (α,β)-quasi-ideals and we study properties of it. definition 4.4. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ is said to be cb almost (α,β)-quasi-ideal of s if (ω̌ ◦α ω̌n) u (ω̌n ◦β ω̌) 6= 0̌ and (υ ◦α υm) ∨ (υm ◦β υ) 6= 1. theorem 4.5. if c = 〈ω̌,υ〉 is a cb almost (α,β)-quasi-ideal of a γ-semigroup s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d, then d = 〈ρ̌,τ〉 is a cb (α,β)-quasi-ideal of s. proof. suppose that c = 〈ω̌,υ〉 is a cb almost (α,β)-quasi-ideal of a γ-semigroup s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d. then (ω̌◦αω̌n)u(ω̌n◦βω̌) 6= 0̌ and (υ◦αυm)∪(υm◦βυ) 6= 1. thus, (ω̌◦αω̌n)u(ω̌n◦β ω̌) (ρ̌◦α ρ̌n)u(ρ̌n◦β ρ̌) 6= 0̌ and (υ◦αυm)∪(υm◦β υ) ≥ (τ◦ατm)∪(τm◦β τ) 6= 1. hence, d = 〈ρ̌,τ〉 is a cb (α,β)-quasi-ideal of s. � theorem 4.6. let k be a nonempty subset of γ-semigroup s. then k is an almost (α,β)-quasi-ideal of s if and only if characteristic function ≥k = (ω̌λk,υλk) is a cb almost (α,β)-quasi-ideal of s. proof. suppose that k is an almost (α,β)-quasi-ideal of s. then (kαe) ∩ (eβk) ∩k 6= ∅ for all e ∈s. thus there exists v ∈ (kαe) ∩ (eβk) and v ∈k. so (ω̌n ◦α ω̌λk) f (ω̌λk ◦β ω̌n)(v) 6= 0̌ and (υm ◦α υλk) ∨ (υλk ◦β υm)(v) 6= 1. hence, (ω̌n◦αω̌λk)u(ω̌λk◦β ω̌n) 6= 0̌ and (υm◦αυλk)∪(υλk◦βυm) 6= 1. therefore, ≥k = (ω̌λk,υλk) is a cb almost (α,β)-quasi-ideal of s. conversely, assume that ≥k = (ω̌λk,υλk) is a cb almost (α,β)-quasi-ideal of s and e ∈s. then (ω̌n ◦α ω̌λk) u (ω̌λk ◦β ω̌n) 6= 0̌ and (υm ◦α υλk) ∪ (υλk ◦β υm) 6= 1. thus there exists r ∈s such that (ω̌n◦αω̌λk)u(ω̌λk◦βω̌n)(r) 6= 0̌ and (υm◦αυλk)∪(υλk◦βυm)(r) 6= 1. hence, r ∈ (kαe) ∩ (eβk) ∩k implies (kαe) ∩ (eβk) ∩k 6= ∅. therefore, k is an almost (α,β)quasi-ideal of s. � next, we study properties between supp(c) and cb almost (α,β)-quasi-ideal of γ-semigroups. theorem 4.7. let c = 〈ω̌,υ〉 is a cb sets of a non-empty of a γ-semigroup s. then c = 〈ω̌,υ〉 is a cb almost (α,β)-quasi-ideal of s if and only if supp(c) is an almost (α,β)-quasi-ideal of s. proof. let c = 〈ω̌,υ〉 be a cb almost (α,β)-quasi-ideal of s and e ∈s. then (ω̌n ◦α ω̌) u (ω̌ ◦β ω̌n) 6= 0̌ and (υm ◦α υ) ∪ (υ ◦β υm) 6= 1. thus there exists r ∈ s such that (ω̌n ◦α ω̌) f (ω̌ ◦β ω̌n)(r) 6= 0̌ and (υm ◦α υ) ∨ (υ ◦β υm)(r) 6= 1. so there exists k1,k2 ∈ s such that r = k1αu = uβk2, ω̌(r) 6= 0̌, υ(r) 6= 0 and ω̌(k1) 6= 0̌, υ(k1) 6= 0. it implies that r,k1,k2 ∈ supp(c). thus ((ω̌λsupp(c) ◦α ω̌m) f (ω̌n ◦β ω̌λsupp(c) ))(r) 6= 0 and ω̌λsupp(c) 6= 0̌. similalry ((υλsupp(c) ◦α υm) ∨ (υm ◦β υλsupp(c) ))(r) 6= 1 and 12 int. j. anal. appl. (2022), 20:69 υλsupp(c) 6= 1. hence, (ω̌λsupp(c) ◦α ω̌m) u (ω̌n ◦β ω̌λsupp(c) ) u ω̌λsupp(c) 6= 0̌ and (υλsupp(c) ◦α υm) t (υm ◦β υλsupp(c) ) 6= 1. therefore, ≥supp(c) is a cb almost (α,β)-quasi-ideal of s. this show that supp(c) is an almost (α,β)-quasi-ideal of s. conversely, let supp(c) be an almost (α,β)-quasi-ideal of s. then by theorem 4.6, supp(c) is a cb (α,β)-quasi-ideal of s. thus (ω̌λsupp(c)◦αω̌n)u(ω̌n◦βω̌λsupp(c) )uω̌λsupp(c) 6= 0̌ and (υλsupp(c)◦αυm)∪ (υm◦βυλsupp(c) )∪υλsupp(c) 6= 1. so there exists r ∈s such that ((ω̌n◦αω̌λsupp(c) )f(ω̌λsupp(c)◦βω̌n))(r) 6= 0̌ and ((υm◦αυλsupp(c) )∨(υλsupp(c)◦βυm))(r) 6= 1. it implies that (ω̌n◦αω̌λsupp(c) )f(ω̌λsupp(c) )◦βω̌n)(r) 6= 0̌ and (υm◦αυλsupp(c) )∨(υλsupp(c) ◦βυm)(r) 6= 1. thus there exist k1,k2 ∈s such that r = k1αe = eβk2, ω̌n(r) 6= 0̌, υm(r) 6= 1 and ω̌n(k) 6= 0̌, υm(k) 6= 1. hence, (ω̌n ◦α ω̌) u (ω̌ ◦β ω̌n) 6= 0̌ and (υm ◦α υ) ∪ (υ ◦β υm) 6= 1. therefore, c = 〈ω̌,υ〉 be a cb almost (α,β)-quasi-ideal of s. � definition 4.5. a cb almost (α,β)-quasi-ideal c = 〈ω̌,υ〉 of a γ-semigroup s is minimal if for all cs almost (α,β)-quasi-ideal d = 〈ρ̌,τ〉 of s such that d v c, then supp(d) = supp(c). theorem 4.8. let k be a nonempty subset of a γ-semigroup s then k is a minimal almost (α,β)quasi-ideal if and only if ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-quasi-ideal of s. proof. suppose that k is a minimal almost (α,β)-quasi-ideal of s. then k is an almost (α,β)-quasiideal of s. thus by theorem 4.6, ≥k = (ω̌λk,υλk) is a cs (α,β)-quasi-ideal of s. let d = 〈ρ̌,τ〉 be a cb (α,β)-quasi-ideal of s such that d v ≥k. then by theorem 4.7, supp(d) is an almost (α,β)-quasi-ideal of s. thus supp(d) v supp(≥k) = k. by assumption, supp(d) = k = supp(≥k). thus, ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-quasi-ideal of s. conversely, suppose that ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-quasi-ideal of s. then by theorem 4.7, k is an almost (α,β)-quasi-ideal of s. let j be an almost (α,β)-quasi-ideal of s such that j ⊆k. then by theorem 4.7, ≥j = (ω̌λj ,υλj ) is a cb (α,β)-quasi-ideal of s such that ≥j ⊆≥k. thus, j = supp(≥j ) = supp(≥k) = k. hence, k is a minimal almost (α,β)-quasi-ideal of s. � corollary 4.2. let s be a γ-semigroup then s has no proper almost (α,β)-quasi-ideal if and only if for any cb almost (α,β)-quasi-ideal c = 〈ω̌,υ〉 of s, supp(c) = s. next, we define cb almost (α,β)-bi-ideals and we study properties of it. definition 4.6. let c = 〈ω̌,υ〉 be a cb set of a γ-semigroup s and α,β ∈ γ is said to be cb almost (α,β)-bi-ideal of s if (ω̌ ◦α ω̌n ◦β ω̌) u ω̌ 6= 0̌ and (υ ◦α υm ◦β υ) ∪υ 6= 0. theorem 4.9. if c = 〈ω̌,υ〉 is a cb almost (α,β)-bi-ideal of a γ-semigroup s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d, then d = 〈ρ̌,τ〉 is a cb (α,β)-bi-ideal of s. proof. suppose that c = 〈ω̌,υ〉 is a cb almost (α,β)-bi-ideal of a γ-semigroup s and d = 〈ρ̌,τ〉 is a cb set of s such that c v d. then (ω̌ ◦α ω̌n ◦β ω̌) f ω̌ 6= 0̌ and (υ ◦α υm ◦β υ) ∨υ 6= 0. thus, int. j. anal. appl. (2022), 20:69 13 (ω̌◦α ω̌n ◦β ω̌)f ω̌ (ρ̌◦α ρ̌n ◦β ρ̌)f ω̌ 6= 0̌ and (υ◦α υm ◦β υ)∨υ ≥ (τ ◦α τm ◦β τ)∨τ 6= 0. hence, d = 〈ρ̌,τ〉 is a cb (α,β)-bi-ideal of s. � theorem 4.10. let k be a nonempty subset of γ-semigroup s. then k is an almost (α,β)-bi-ideal of s if and only if characteristic function ≥k = (ω̌λk,υλk) is a cb almost (α,β)-bi-ideal of s. proof. suppose that k is an almost (α,β)-bi-ideal of s. then kαeβk∩k 6= ∅. for all e ∈s. thus there exists f ∈ kαeβk and f ∈ k. so (ω̌λk ◦α ω̌n ◦β ω̌λk)(f ) = ω̌λk(f ) = 1̌ and (υλk ◦α υm ◦β υλk)(f ) = υλk(f ) = 0. hence, (ω̌λk ◦α ω̌n ◦β ω̌λk) u ω̌λk 6= 0̌ and (υλk ◦α υm ◦β υλk) ∨υλk 6= 1. therefore, ≥k = (ω̌λk,υλk) is a cb almost (α,β)-bi-ideal of s. conversely, assume that ≥k = (ω̌λk,υλk) is a cb almost (α,β)-bi-ideal of s and u ∈ s. then (ω̌λk ◦α ω̌n ◦β ω̌λk) u ω̌λk 6= 0 and (υλk ◦α υm ◦β υλk) ∪υλk 6= 1. thus there exists e ∈s such that ((ω̌λk ◦α ω̌n ◦β ω̌λk) f ω̌λk)(e) 6= 0 and ((υλk ◦α υm◦β υλk)∨υλk)(e) 6= 1. hence, r ∈kαeβk∩k implies that kαeβk∩k 6= ∅. therefore, k is an almost (α,β)-bi-ideal of s. � next, we study properties between supp(c) and cb almost (α,β)-bi-ideal of γ-semigroups. theorem 4.11. let c = 〈ω̌,υ〉 be a fuzzy sets of a non-empty of a γ-semigroup s. then c = 〈ω̌,υ〉 is a cb almost (α,β)-bi-ideal of s if and only if supp(c) is an almost (α,β)-bi-ideal of s. proof. let c = 〈ω̌,υ〉 is a cb almost (α,β)-bi-ideal of s and u ∈s. then (ω̌◦α ω̌m ◦β ω̌) u ω̌ 6= 0̌ and (υ◦α υn ◦β υ) ∪υ 6= 1. thus there exists r ∈s such that ((ω̌◦α ω̌m ◦β ω̌) f ω̌)(r) 6= 0 and ((υ ◦α υn ◦β υ) ∨υ)(r) 6= 0. so there exists k1,k2 ∈ s such that r = k1αβk2, ω̌(r) 6= 0̌, υ(r) 6= 1. it implies that r,k1,k2 ∈ supp(c). thus (ω̌λsupp(c) ◦α ω̌m ◦β ω̌λsupp(c) )(r) 6= 0 and ≥̌supp(c) 6= 0̌. similarly (υλsupp(c) ◦α υm ◦β υλsupp(c) )(r) 6= 0 and υλsupp(c) 6= 0. hence, (ω̌λsupp(c)◦αω̌m◦βω̌λsupp(c)uω̌λsupp(c) 6= 0̌ and (υλsupp(c)◦αυm◦βυλsupp(c) )∨υλsupp(c) ) 6= 0. therefore ≥supp(c) is a cb almost (α,β)-bi-ideal of s. this show that supp(c) is an almost (α,β)-bi-ideal of s. conversely, let supp(c) is an almost (α,β)-bi-ideal of s. then by theorem 4.10, ≥supp(c) is a cb almost (α,β)-bi-ideal of s. thus (ω̌λsupp(c) ◦α ω̌m ◦β ω̌λsupp(c) ) u ω̌λsupp(c) 6= 0̌ and (υλsupp(c) ◦α υn ◦β υλsupp(c) )∪υλsupp(c) 6= 0. so there exists r ∈s such that ((ω̌λsupp(c) ◦α ω̌m◦β ω̌λsupp(c) )fω̌λsupp(c) )(r) 6= 0̌ and ((υλsupp(c) ◦α υn ◦β υλsupp(c) ) ∨υλsupp(c) )(r) 6= 0. it implies that (ω̌λsupp(c) ◦α ω̌m ◦β ω̌λsupp(c) )(r) 6= 0̌ and ω̌λsupp(c) (r) 6= 0̌. similarly (υλsupp(c)◦αυn◦βυλsupp(c) )(r) 6= 0 and ≥ n supp(r) 6= 0. thus there exist k1,k2 ∈s such that r = k1αuβk2, ω̌(r) 6= 0, υ(r) 6= 0 and ω̌(k) 6= 0, υ(k) 6= 0. hence, (ω̌◦αω̌m◦βω̌)uω̌ 6= 0̌ and (υ◦αυn◦βυ)∪υ 6= 0. therefore, c = 〈ω̌,υ〉 is a cb almost (α,β)-bi-ideal of s. � definition 4.7. a cb almost (α,β)-bi-ideal c = 〈ω̌,υ〉 of a γ-semigroup s is minimal if for all cb almost (α,β)-bi-ideal d = 〈ρ̌,τ〉 of s such that d v c, then supp(d) = supp(c). 14 int. j. anal. appl. (2022), 20:69 theorem 4.12. let k be a nonempty subset of a γ-semigroup s then k is a minimal almost (α,β)bi-ideal if and only if ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-bi-ideal of s. proof. suppose that k is a minimal almost (α,β)-bi-ideal of s. then k is an almost (α,β)-bi-ideal of s. thus by theorem 4.10, ≥k = (ω̌λk,υλk) is a cb (α,β)-bi-ideal of s. let d = 〈ρ̌,τ〉 be a cb (α,β)-bi-ideal of s such that d v ≥k. then by theorem 4.11, supp(d) is an almost (α,β)bi-ideal of s. thus, supp(d) v supp(≥k) = k. by assumption, supp(d) = k = supp(≥k). thus, ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-bi-ideal of s. conversely, suppose that ≥k = (ω̌λk,υλk) is a minimal cb almost (α,β)-bi-ideal of s. then by theorem 4.10, k is an almost (α,β)-bi-ideal of s. let j be an almost (α,β)-bi-ideal of s such that j ⊆k. then by theorem 4.10, ≥j = (ω̌λj ,υλj ) is a cb (α,β)-bi-ideal of s such that ≥j ⊆≥k. thus, j = supp(≥j ) = supp(≥k) = k. hence, k is a minimal almost (α,β)-bi-ideal of s. � corollary 4.3. let s be a γ-semigroup then s has no proper almost (α,β)-bi-ideal if and only if for any cb almost (α,β)-bi-ideal c = 〈ω̌,υ〉 of s, supp(c) = s. 5. conclusion in this article, we give the concept of a new cubic ideals and cubic almost ideals in a γ-semigroups. we study properites of new cubic ideals and cubic almost ideals. we hope that the study of this topic are useful mathematical tools. in the future we study a new hesitant fuzzy ideal and hesitant fuzzy almost ideals in semigroups or algebric system. acknowledgements: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017) fuzzy algebras and decisionmaking problems research unit, department of mathematics, school of science, university of phayao, phayao 56000, thailand. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. bashir, a. sarwar, characterizations of γ-semigroups by the properties of their interval valued t-fuzzy ideals, ann. fuzzy math. inform. 9 (2015), 441-461. [2] v. chinnadaurai, k. bharathivelan, cubic ideal of γ-semigroups, int. j. current res. modern educ. 1 (2016), 138-150. [3] r. chinram, on quasi-gamma-ideals in gamma-semigroups, scienceasia. 32 (2006), 351-353. https://doi.org/ 10.2306/scienceasia1513-1874.2006.32.351. [4] n. deetae, p. khamrod, q-cubic bi-quasi ideals of semigroups, glob. j. pure appl. math. 16 (2020), 553-566. [5] t. gaketem, cubic (1, 2)-ideals in semigroups, twms j. appl. eng. math. 12 (2022), 1271-1282. [6] t. gaketem, cubic interior ideals in semigroups, azerbaijan j. math. 10 (2020), 85-104. [7] t. gaketem, a. iampan, cubic filters of semigroups, appl. sci. 24 (2022), 131-141. https://doi.org/10.2306/scienceasia1513-1874.2006.32.351 https://doi.org/10.2306/scienceasia1513-1874.2006.32.351 int. j. anal. appl. (2022), 20:69 15 [8] a. hadi, a. khan, cubic generalized bi-ideal in semigroups, discuss. math., gen. algebra appl. 36 (2016), 131146. https://doi.org/10.7151/dmgaa.1254. [9] a. iampan, note on bi-ideal in γ-semigroups, int. j. algebra. 34 (2009), 181-188. [10] y.b. jun, c.s. kim, k.o. yang, cubic sets, ann. fuzzy math. inform. 4 (2012), 83-98. [11] y.b. junand, a. khan, cubic ideals in semigroups, honam math. j. 35 (2013), 607-623. https://doi.org/10. 5831/hmj.2013.35.4.607. [12] n. kuroki, fuzzy bi-ideals in semigroup, comment. math. univ. st. paul. 28 (1980), 17–21. https://doi.org/ 10.14992/00010265. [13] p. khamrot, t. gaketem, cubic bi-quasi ideals of semigroups, j. discrete math. sci. cryptography. 24 (2021), 1113-1126. https://doi.org/10.1080/09720529.2021.1889777. [14] p. kummoon, t. changphas, bi-bases of γ-semigroups, thai j. math. special issue (2017), 75-86. [15] d. mandal, characterizations of γ-semirings by their cubic ideals, int. j. math. comput. sci. 13 (2019), 150-156. [16] al. narayanan, t. manikantan, interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in semigroups, j. appl. math. comput. 20 (2006), 455–464. https://doi.org/10.1007/bf02831952. [17] a. simuen, a. iampan, r. chinram, a novel of ideals and fuzzy ideals of γ-semigroups, j. math. 2021 (2021), 6638299. https://doi.org/10.1155/2021/6638299. [18] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [19] l.a. zadeh, the concept of a linguistic variable and its application to approximate reasoning–i, inform. sci. 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90036-5. https://doi.org/10.7151/dmgaa.1254 https://doi.org/10.5831/hmj.2013.35.4.607 https://doi.org/10.5831/hmj.2013.35.4.607 https://doi.org/10.14992/00010265 https://doi.org/10.14992/00010265 https://doi.org/10.1080/09720529.2021.1889777 https://doi.org/10.1007/bf02831952 https://doi.org/10.1155/2021/6638299 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/0020-0255(75)90036-5 1. introduction 2. preliminaries 3. new types of cubic ideals 4. new types of cubic almost ideals 5. conclusion references international journal of analysis and applications volume 17, number 2 (2019), 275-281 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-275 fixed point theorem of ćirić-pata type ao-lei sima, fei he∗ and ning lu school of mathematical sciences, inner mongolia university, hohhot 010021, china ∗corresponding author: hefei@imu.edu.cn abstract. in this article, we proved a fixed point theorem of ćirić-pata type in metric space. this result extends several results in the existing literature. moreover, an example is given in the support of our result. in particular, the main result provides a complete solution to an open problem raised by kadelburg and radenović (j. egypt. math. soc. 24 (2016) 77-82). 1. introduction throughout this paper, (x,d) will be a complete metric space. fix an arbitrary point x0 ∈ x and denote ‖x‖ = d(x,x0), for each x ∈ x. also, ψ : [0, 1] → [0,∞) is an increasing function, continuous at zero, with ψ(0) = 0. given a function f : x → x. in 2011, pata [1] obtained the following result which is a generalization of the classical banach contraction principle. theorem 1.1. [1] let λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. if the inequality d(fx,fy) ≤ (1 −ε)d(x,y) + λεαψ(ε)[1 + ‖x‖ + ‖y‖]β (1.1) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. received 2018-11-06; accepted 2018-12-14; published 2019-03-01. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. fixed point theorem; pata type contraction; ćirić type quasi-contraction; metric space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 275 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-275 int. j. anal. appl. 17 (2) (2019) 276 afterward many pata type fixed point theorems have been established by various authors; see ( [2], [3], [4], [5], [6], [7], [8], [9]). particularly, kadelburg and radenović [7] proved some fixed point theorems of pata type and raised the following open question on pata-version of ćirić contraction principle (see [10]). problem 1.1. [7] prove or disprove the following. let f : x → x and let λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. if the inequality d(fx,fy) ≤ (1 −ε) max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)} + λεαψ(ε)[1 + ‖x‖ + ‖y‖]β (1.2) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. furthermore, the sequence {fnx0} converges to z. very recently, jacobe et al. give the following result. theorem 1.2. [5] let f : x → x and let λ ≥ 0, α ≥ 1 and β ∈ [0,α] be fixed constants. if the inequality d(fx,fy) ≤ (1 −ε) max { d(x,y), d(x,fx) + d(y,fy) 2 , d(x,fy) + d(y,fx) 2 } + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]β (1.3) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point in x. in this paper, we give a fixed point theorem of ćirić-pata type in metric space. this theorem extends the main results in ( [1], [5], [7]) and provides a complete solution to the above problem 1.1. finally, an example is given to illustrate the superiority of the main results. 2. main results our result of this paper are stated as follows. theorem 2.1. let λ ≥ 0, α ≥ 1 be fixed constants. for x,y ∈ x, we denote m(x,y) = max{d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}. if the inequality d(fx,fy) ≤ (1 −ε)m(x,y) + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α (2.1) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. furthermore, the sequence {fnx0} converges to z. int. j. anal. appl. 17 (2) (2019) 277 proof. starting from x0, construct a sequence {xn} such that xn = fxn−1 = fnx0. if xn0 = xn0+1 for some n0, then xn0 is a fixed point of f. thus, we always assume that xn 6= xn+1 for all n ∈ n. we prove that xn 6= xm for all m,n ∈ n and n 6= m. assume that there exist n0,m0 ∈ n such that n0 < m0 and xn0 = xm0. denote a = max{d(xi,xj) : n0 ≤ i < j ≤ m0} and b = max{‖xi‖ : n0 ≤ i ≤ m0 + 1}. it is obvious that a = max{d(xi,xj) : n0 + 1 ≤ i < j ≤ m0} and a > 0. for each i,j ∈ n such that n0 + 1 ≤ i < j ≤ m0, we have d(xi,xj) ≤ (1 −ε)m(xi−1,yj−1) + λεαψ(ε)[1 + ‖xi−1‖ + ‖xj−1‖ + ‖xi‖ + ‖xj‖]α ≤ (1 −ε)a + λεαψ(ε)(1 + 4b)α. it follows that a ≤ (1 −ε)a + λεαψ(ε)(1 + 4b)α and a ≤ λεα−1ψ(ε)(1 + 4b)α. letting ε → 0, we can see a ≤ 0. this is a contradiction with a > 0. denote dn = max{d(xi,xj) : 0 ≤ i < j ≤ n} and δn = sup{d(xi,xj) : n ≤ i < j}. in order to prove {xn} is a cauchy sequence, we divide into the following three steps. step 1. we show that d(fx,fy) < m(x,y) for all x,y ∈ x and x 6= y. let ε = 0 in (2.1), we have d(fx,fy) ≤ m(x,y) for all x,y ∈ x. assume that there exist x0,y0 ∈ x and x0 6= y0 such that d(fx0,fy0) = m(x0,y0). using (2.1), we get m(x0,y0) = d(fx0,fy0) ≤ (1 −ε)m(x0,y0) + λεαψ(ε)[1 + ‖x0‖ + ‖y0‖ + ‖fx0‖ + ‖fy0‖]α. it follows that m(x0,y0) ≤ λεα−1ψ(ε)[1 + ‖x0‖ + ‖y0‖ + ‖fx0‖ + ‖fy0‖]α. passing to the limit as ε → 0, we see m(x0,y0) ≤ 0, a contradiction. step 2. we prove that {dn} is bounded. by step 1, we see that d(xi,xj) = d(fxi−1,fxj−1) < m(xi−1,xj−1) ≤ dn, for all i,j ∈ n such that 0 < i < j ≤ n. thus there exists `n ∈ n such that 1 ≤ `n ≤ n and dn = d(x0,x`n ). using (2.1), we have dn = d(x0,x`n ) ≤ d(x0,x1) + d(x1,x`n ) ≤ d(x0,x1) + (1 −ε)m(x0,x`n−1) + λε αψ(ε)[1 + ‖x0‖ + ‖x`n−1‖ + ‖x1‖ + ‖x`n‖] α ≤ (1 −ε)dn + λεαψ(ε)(1 + 3dn)α + d(x0,x1). int. j. anal. appl. 17 (2) (2019) 278 this implies that εdn ≤ λεαψ(ε)(1 + 3dn)α + d(x0,x1). suppose that {dn} is unbounded. then there exists a subsequence {dnk} with {dn} such that dnk → ∞ (k →∞) and dnk ≥ 1 + d(x0,x1). let ε = εk = 1+d(x0,x1) dnk . then we get 1 + d(x0,x1) dnk ·dnk ≤ λ[ 1 + d(x0,x1) dnk ]αψ(εk)(1 + 3dnk ) α + d(x0,x1) and 1 ≤ λ( 1 dnk + 3)αψ(εk)[1 + d(x0,x1)] α. letting k →∞, we have εk → 0 and λ( 1 dnk + 3)αψ(εk)[1 + d(x0,x1)] α → 0. this is a contradiction. thus {dn} is bounded and there exists a constant m > 0 such that dn ≤ m. step 3. we show that δn → 0. observe that d(xi,xj) = d(fxi−1,xj−1) < m(xi−1,xj−1) ≤ δn for every i,j ∈ n with n + 1 ≤ i < j. thus we get δn+1 ≤ δn ≤ ··· ≤ δ0 ≤ m. it is easy to see that {δn} is decreasing and bounded sequence. it follows that lim n→∞ δn = δ for some δ ≥ 0. assume that δ > 0. from (2.1), it holds for each i,j ∈ n with n + 1 ≤ i < j, d(xi,xj) ≤ (1 −ε)m(xi−1,xj−1) + λεαψ(ε)(1 + 4m)α. this implies that δn+1 ≤ (1 −ε)δn + λεαψ(ε)(1 + 4m)α. (2.2) letting n →∞ in (2.2), we get δ ≤ (1 −ε)δ + λεαψ(ε)(1 + 4m)α and δ ≤ λεα−1ψ(ε)(1 + 4m)α. from λεα−1ψ(ε)(1 + 4m)α → 0 (ε → 0), we see δ ≤ 0, a contradiction. for each p ∈ n, we get d(xn,xn+p) ≤ δn → 0 (n →∞). hence, {xn} is cauchy sequence. since x is complete, there exists z ∈ x such that xn → z (n →∞). now, we show that fz = z. using (2.1), we get d(fz,xn+1) ≤ (1 −ε) max{d(z,xn),d(z,fz),d(xn,xn+1),d(z,xn+1),d(xn,fz)} + λεαψ(ε)(1 + 4m)α. int. j. anal. appl. 17 (2) (2019) 279 by taking limits on both sides when n →∞, we obtain d(fz,z) ≤ (1 −ε)d(fz,z) + λεαψ(ε)(1 + 4m)α. then d(fz,z) ≤ λεα−1ψ(ε)(1 + 4m)α → 0 (ε → 0). this implies that d(fz,z) = 0 and fz = z. finally, we prove the uniqueness of z. if fu = u, fv = v for any two fixed u,v ∈ x, then we can write (2.1) in the form d(u,v) = d(fu,fv) ≤ (1 −ε) max{d(u,v),d(u,fu),d(v,fv),d(u,fv),d(v,fu)} + λ�αψ(ε)[1 + ‖u‖ + ‖v‖ + ‖fu‖ + ‖fv‖]α ≤ (1 −ε)d(u,v) + λεαψ(ε)[1 + 2‖u‖ + 2‖v‖]α. therefore d(u,v) ≤ λεα−1ψ(ε)[1 + 2‖u‖ + 2‖v‖]α → 0 (ε → 0), which implies that d(u,v) = 0 and u = v. hence, f has a unique fixed point z ∈ x. � remark 2.1. it is easy to see that the condition (2.1) is weaker than the condition (1.2). hence, theorem 2.1 provides a solution to problem 1.1. from theorem 2.1 we get the following corollaries. corollary 2.1. let f : x → x and let λ ≥ 0, α ≥ 1 be fixed constants. if the inequality d(fx,fy) ≤ (1 −ε)d(x,y) + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α (2.3) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. furthermore, the sequence {fnx0} converges to z. remark 2.2. it is easy to see that the condition (2.3) is weaker than the condition (1.1). thus corollary 2.1 is an extension of theorem 1.1. corollary 2.2. let f : x → x and let λ ≥ 0, α ≥ 1 be fixed constants. if the inequality d(fx,fy) ≤ 1 −ε 2 (d(x,fy) + d(y,fx)) + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]α (2.4) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. furthermore, the sequence {fnx0} converges to z. int. j. anal. appl. 17 (2) (2019) 280 remark 2.3. it is easy to see that the condition 2.4 is weaker than the condition 2.1 in [7]. thus corollary 2.2 is an extension of theorem 2.1 in [7]. corollary 2.3. let f : x → x and let λ ≥ 0, α ≥ 1 be fixed constants. if the inequality d(fx,fy) ≤ (1 −ε) max { d(x,y),d(x,fx),d(y,fy), d(x,fy) + d(y,fx) 2 } + λεαψ(ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖]β (2.5) is satisfied for every ε ∈ [0, 1] and all x,y ∈ x, then f has a unique fixed point z ∈ x. furthermore, the sequence {fnx0} converges to z. remark 2.4. it is easy to see that the condition (2.5) is weaker than the condition (1.3). thus corollary 2.3 is an extension of theorem 1.2. the following is an example which can apply to theorem 2.1 but not corollary 2.3 or theorem 1.2. example 2.1. let x = {0, 1, 2, 4, 8, 9} ⋃ { 1 2n : n = 1, 2, · · ·} with the usual metric. it is easily to check that x is a complete metric space. define f : x → x by fx =   8 x = 9,1 2 x others . then mapping f satisfies the condition (2.1) with λ = 8 9 , β = 1 and ψ(ε) = ε 1 8 (for all ε ∈ [0, 1] ). moreover, it is worth mentioning that 8 9 − 1 + ε ≤ 8 9 [1 + 9 8 (ε− 1)] ≤ 8 9 ε 9 8 ≤ 8 9 εε 1 8 . thus we have the following two cases. (1) if x = 9 and y 6= 9, then d(fx,fy) = d(8, 1 2 y) = 8 − 1 2 y ≤ 8 9 (9 − 1 2 y) = 8 9 d(9,fy) ≤ 8 9 m(9,y) = (1 −ε)m(9,y) + ( 8 9 − 1 + ε)m(9,y) ≤ (1 −ε)m(9,y) + ( 8 9 − 1 + ε)[1 + ‖9‖ + ‖y‖ + ‖f9‖ + ‖fy‖] ≤ (1 −ε)m(9,y) + 8 9 εε 1 8 [1 + ‖9‖ + ‖y‖ + ‖f9‖ + ‖fy‖] int. j. anal. appl. 17 (2) (2019) 281 (2) if x 6= 9 and y 6= 9, then d(fx,fy) = 1 2 (x−y) ≤ 8 9 (x−y) = 8 9 d(x,y) ≤ 8 9 m(x,y) = (1 −ε)m(x,y) + ( 8 9 − 1 + ε)m(x,y) ≤ (1 −ε)m(x,y) + ( 8 9 − 1 + ε)[1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖] ≤ (1 −ε)m(x,y) + 8 9 εε 1 8 [1 + ‖x‖ + ‖y‖ + ‖fx‖ + ‖fy‖] hence, f satisfies all conditions of theorem 2.1. this leads to f has a unique fixed point. indeed, 0 is the fixed point for the mapping f. now, let ε = 0, x = 9 and y = 4, we have d(f9,f4) = 6 > 11 2 = max{5, 1, 2, 11 2 } = max{d(9, 4),d(9,f9),d(4,f4), d(9,f4) + d(4,f9) 2 } it is easy to see that f does not satisfy the condition (2.5) of corollary 2.3. also, f does not satisfy the condition (1.3) of theorem 1.2. acknowledgements. this work was supported by the national natural science foundation of china (no. 11561049, 11471236 ) references [1] v. pata, a fixed point theorems in metric spaces, j. fixed point theory appl. 10 (2011), 299–305. [2] m. a. alghamdi, a. petrusel and n. shahzad, correction: a fixed point theorem for cyclic generalized contractions in metric spaces, fixed point theory appl. 2012 (2012), 122. [3] s. balasubramanian, a pata-type fixed point theorem, math. sci. 8 (2014), 65–69. [4] m. eshaghi, s. mohseni, m. r. delavar, m. de la sen, g. h. kim and a. arian, pata contractions and coupled type fixed points, fixed point theory appl. 2014 (2014), 130. [5] g. k. jacob, m. s. khan, c. park and s. jun, on generalized pata type contractions, mathmatics. 6 (2018), 25. [6] z. kadelburg and s. radenović, fixed point and tripled fixed point theorems under pata-type conditions in ordered metric spaces, international journal of analysis and applications. 6 (2014), 113–122. [7] z. kadelburg and s. radenović, fixed points theorems under pata-type conditions in metric spaces, j. egypt. math. soc. 24 (2016), 77–82. [8] s. m. kolagar, m. ramezani and m. eshaghi, pata type fixed point theorems of multivalued operators in ordered metric spaces with applications to hyperbolic differential inclusions, proc. amer. math. soc. 6 (2016), 21–34. [9] m. paknazar, m. eshaghi, y. j. cho and s. m. vaezpour, a pata-type fixed point theorem in modular spaces with application, fixed point theory and appl. 2013 (2013), 239. [10] l. j. ćirić, a generalization of banach’s contraction principle, proc. amer. math. soc. 45 (1974), 267–273. 1. introduction 2. main results references international journal of analysis and applications volume 16, number 6 (2018), 882-893 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-882 ideal convergent sequence spaces with respect to invariant mean and a musielak-orlicz function over n-normed spaces sunil k. sharma∗ department of mathematics, model institute of engineering & technology, kot bhalwal 181122, j & k, india ∗corresponding author: sunilksharma42@gmail.com abstract. in the present paper we defined i-convergent sequence spaces with respect to invariant mean and a musielak-orlicz function m = (mk) over n-normed spaces. we also make an effort to study some topological properties and prove some inclusion relation between these spaces. 1. introduction and preliminaries let σ be an injective mapping from the set of the positive integers to itself such that σp(n) 6= n for all positive integers n and p, where σp(n) = σ(σp−1(n)). an invariant mean or a σ-mean is a continuous linear functional defined on the space `∞ such that for all x = (xn) ∈ `∞: (1) if xn ≥ 0 for all n, then φ(x) ≥ 0, (2) φ(e) = 1, (3) φ(sx) = φ(x), where sx = (xσ(n)). vσ denotes the set of bounded sequences all of whose invariant means are equal which is also called as the space of σ-convergent sequences. in [26], it is defined by vσ = { x ∈ `∞ : lim k tkn(x) = `, uniformly in n,` = σ − lim x } , received 2017-09-21; accepted 2017-12-07; published 2018-11-02. 2010 mathematics subject classification. 40a05,40a35, 46a45. key words and phrases. i-convergent, invariant mean, orlicz function, musielak-orlicz function, n-normed space, atransform. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 882 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-882 int. j. anal. appl. 16 (6) (2018) 883 where tkn(x) = xn+xσ1(n)+···+xσk(n) k+1 . σ-mean is called a banach limit if σ is the translation mapping n → n + 1. in this case, vσ becomes the set of almost convergent sequences which is denoted by ĉ and defined in [11] as ĉ = { x ∈ `∞ : lim k dkn(x) exists uniformly in n } , where dkn(x) = xn+xn+1+···+xn+k k+1 . the space of strongly almost converegnt sequences was introduced by maddox [12] as follow: ĉ = { x ∈ `∞ : lim k dkn(|x− `e|) exists uniformly in n for some ` } . the notion of ideal convergence was first introduced by p. kostyrko [8] as a generalization of statistical convergence which was further studied in topological spaces by das, kostyrko, wilczynski and malik see [1]. more applications of ideals can be seen in ([1], [2]). mursaleen and sharma [19] continue in this direction and introduced i-convergence of generalized sequences with respect to musielak-orlicz function. a family i ⊂ 2x of subsets of a non empty set x is said to be an ideal in x if (1) φ ∈i (2) a,b ∈i imply a∪b ∈i (3) a ∈i, b ⊂ a imply b ∈i, while an admissible ideal i of x further satisfies {x}∈i for each x ∈ x see [8]. a sequence (xn)n∈n in x is said to be i-convergent to x ∈ x, if for each � > 0 the set a(�) = { n ∈ n : ||xn −x|| ≥ � } belongs to i. a sequence (xn)n∈n in x is said to be i-bounded to x ∈ x if there exists an k > 0 such that {n ∈ n : |xn| > k} ∈ i. for more details about ideal convergence sequence spaces (see [7], [9], [15], [16], [17], [18], [21], [25], [26], [27]) and references therein. let a = aij be an infinite matrix of complex numbers aij, where i,j,∈ n. we write ax = (ai(x)) if ai(x) = ∞∑ j=1 aijxj converges for each i ∈ n. throughout the paper, by tkn(ax), we mean tkn(ax) = an(x) + aσ1(n)(x) + · · · ,aσk(n)(x) k + 1 , for all k,n ∈ n. a sequence space x is called as solid (or normal) if (αkxk) ∈ x whenever (xk) ∈ x and (αk) is a sequence of scalars such that |αk| ≤ 1 for all k ∈ n. let x be a sequence space and k = {k1 < k2 < · · ·}⊆ n. the sequence space zxk = {(xkn) ∈ w : (xn) ∈ x} is called k-step space of x. int. j. anal. appl. 16 (6) (2018) 884 a canonical preimage of a sequence (xkn) ∈ zxk is a sequence (yn) ∈ w defined by yn =   xn, if n ∈ n;0, otherwise. a sequence space x is monotone if it contains the canonical preimages of all its step spaces. an orlicz function m is a function, which is continuous, non-decreasing and convex with m(0) = 0, m(x) > 0 for x > 0 and m(x) −→∞ as x −→∞. lindenstrauss and tzafriri [10] used the idea of orlicz function to define the following sequence space. let w be the space of all real or complex sequences x = (xk), then `m = { x ∈ w : ∞∑ k=1 m (|xk| ρ ) < ∞ } which is called as an orlicz sequence space. the space `m is a banach space with the norm ||x|| = inf { ρ > 0 : ∞∑ k=1 m (|xk| ρ ) ≤ 1 } . it is shown in [10] that every orlicz sequence space `m contains a subspace isomorphic to `p(p ≥ 1). the ∆2−condition is equivalent to m(lx) ≤ klm(x) for all values of x ≥ 0, and for l > 1. a sequence m = (mk) of orlicz function is called a musielak-orlicz function see ([13],[20]). a sequence n = (nk) defined by nk(v) = sup{|v|u− (mk) : u ≥ 0}, k = 1, 2, · · · is called the complementary function of a musielak-orlicz function m. for a given musielak-orlicz function m, the musielak-orlicz sequence space tm and its subspace hm are defined as follows tm = { x ∈ w : im(cx) < ∞ for some c > 0 } , hm = { x ∈ w : im(cx) < ∞ for all c > 0 } , where im is a convex modular defined by im(x) = ∞∑ k=1 mk(xk),x = (xk) ∈ tm. we consider tm equipped with the luxemburg norm ||x|| = inf { k > 0 : im (x k ) ≤ 1 } or equipped with the orlicz norm ||x||0 = inf {1 k ( 1 + im(kx) ) : k > 0 } . for more details about sequence spaces defined by orlicz function see ([22], [23], [24]) and reference therein. the concept of 2-normed spaces was initially developed by gähler[3] in the mid of 1960’s, while that of int. j. anal. appl. 16 (6) (2018) 885 n-normed spaces one can see in misiak [14]. since then, many others have studied this concept and obtained various results, see gunawan ([4],[5]) and gunawan and mashadi [6]. let n ∈ n and x be a linear space over the field k, where k is field of real or complex numbers of dimension d, where d ≥ n ≥ 2. a real valued function ||·, · · · , ·|| on xn satisfying the following four conditions: (1) ||x1,x2, · · · ,xn|| = 0 if and only if x1,x2, · · · ,xn are linearly dependent in x; (2) ||x1,x2, · · · ,xn|| is invariant under permutation; (3) ||αx1,x2, · · · ,xn|| = |α| ||x1,x2, · · · ,xn|| for any α ∈ k, and (4) ||x + x′,x2, · · · ,xn|| ≤ ||x,x2, · · · ,xn|| + ||x′,x2, · · · ,xn|| is called a n-norm on x, and the pair (x, ||·, · · · , ·||) is called a n-normed space over the field k. for example, we may take x = rn being equipped with the euclidean n-norm ||x1,x2, · · · ,xn||e = the volume of the n-dimensional parallelopiped spanned by the vectors x1,x2, · · · ,xn which may be given explicitly by the formula ||x1,x2, · · · ,xn||e = |det(xij)|, where xi = (xi1,xi2, · · · ,xin) ∈ rn for each i = 1, 2, · · · ,n. let (x, ||·, · · · , ·||) be a n-normed space of dimension d ≥ n ≥ 2 and {a1,a2, · · · ,an} be linearly independent set in x. then the following function ||·, · · · , ·||∞ on xn−1 defined by ||x1,x2, · · · ,xn−1||∞ = max{||x1,x2, · · · ,xn−1,ai|| : i = 1, 2, · · · ,n} defines an (n− 1)-norm on x with respect to {a1,a2, · · · ,an}. a sequence (xk) in a n-normed space (x, ||·, · · · , ·||) is said to converge to some l ∈ x if lim k→∞ ||xk −l,z1, · · · ,zn−1|| = 0 for every z1, · · · ,zn−1 ∈ x. a sequence (xk) in a n-normed space (x, ||·, · · · , ·||) is said to be cauchy if lim k,p→∞ ||xk −xp,z1, · · · ,zn−1|| = 0 for every z1, · · · ,zn−1 ∈ x. if every cauchy sequence in x converges to some l ∈ x, then x is said to be complete with respect to the n-norm. any complete n-normed space is said to be n-banach space. in the present paper, we define some new sequence spaces by using the concept of ideal convergence, invariant mean, musielak-orlicz function, n-normed and a transform as follows: i− cσ0 (a,m,p, ||·, · · · , ·||) ={ x ∈ w : { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≥ � } ∈i, for all n ∈ n } , int. j. anal. appl. 16 (6) (2018) 886 i− cσ(a,m,p, ||·, · · · , ·||) ={ x ∈ w : { k ∈ n : [ mk ( || tkn(a(x) −l) ρ ,z1, · · · ,zn−1|| )]pk ≥ � } ∈i, for all n ∈ n & for some l ∈ c } , i− `σ∞(a,m,p, ||·, · · · , ·||) ={ x ∈ w : ∃ k > 0 such that { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≥ k } ∈i, for all n ∈ n } . if we take p = (pk) = 1, we get the spaces i− cσ0 (a,m, ||·, · · · , ·||) ={ x ∈ w : { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )] ≥ � } ∈i, for all n ∈ n } , i− cσ(a,m, ||·, · · · , ·||) ={ x ∈ w : { k ∈ n : [ mk ( || tkn(a(x) −l) ρ ,z1, · · · ,zn−1|| )] ≥ � } ∈i, for all n ∈ n & for some l ∈ c } , i− `σ∞(a,m, ||·, · · · , ·||) ={ x ∈ w : ∃ k > 0 such that { k ∈ n : [ mk ( || tkna(x) ρ ,z1, · · · ,zn−1|| )] ≥ k } ∈i, for all n ∈ n } . the following inequality will be used throughout the paper. if 0 ≤ pk ≤ sup pk = h, d = max(1, 2h−1) then |ak + bk|pk ≤ d{|ak|pk + |bk|pk} (1.1) for all k and ak,bk ∈ c. also |a|pk ≤ max(1, |a|h) for all a ∈ c. the main goal of this paper is to introduce the sequence spaces i−cσ0 (a,m,p, ||·, · · · , ·||), i−cσ(a,m,p, ||·, · · · , ·||) and i−`σ∞(a,m,p, ||·, · · · , ·||) defined by a musielak-orlicz function m = (mk) over n-normed spaces. we also make an effort to study some topological properties and prove some inclusion relation between these spaces. 2. main results theorem 2.1 let m = (mk) be a musielak-orlicz function, p = (pk) be a bounded sequence of positive real numbers. then the spaces i − cσ0 (a,m,p, ||·, · · · , ·||), i − cσ(a,m,p, ||·, · · · , ·||) and i − `σ∞(a,m,p, ||·, · · · , ·||) are linear. proof. let x,y ∈ i − cσ0 (a,m,p, ||·, · · · , ·||) and let α,β be scalars. then there exist positive numbers ρ1 and ρ2 such that for every � > 0 d1 = { k ∈ n : [ mk ( || tkn(a(x)) ρ1 ,z1, · · · ,zn−1|| )]pk ≥ � 2d } ∈i, (2.1) d1 = { k ∈ n : [ mk ( || tkn(a(y)) ρ2 ,z1, · · · ,zn−1|| )]pk ≥ � 2d } ∈i, (2.2) int. j. anal. appl. 16 (6) (2018) 887 let ρ3 = max { 2|α|ρ1, 2|β|ρ2 } . since m = (mk) is non-decreasing, convex function and so by using inequality (1.1), we have[ mk ( ||tkn(a(αx+βy)) ρ3 ,z1, · · · ,zn−1|| )]pk ≤ [ mk ( || tkn(αa(x)) ρ3 ,z1, · · · ,zn−1|| )]pk + [ mk ( || tkn(βa(y)) ρ3 ,z1, · · · ,zn−1|| )]pk ≤ [ mk ( || tkn(a(x)) ρ1 ,z1, · · · ,zn−1|| )]pk + [ mk ( || tkn(a(y)) ρ2 ,z1, · · · ,zn−1|| )]pk now by (2.1) and (2.2), we have{ k ∈ n : [ mk ( || tkn(a(αx + βy)) ρ3 ,z1, · · · ,zn−1|| )]pk > � } ⊂ d1 ∪d2. therefore αx+βy ∈i−cσ0 (a,m,p, ||·, · · · , ·||). hence i−cσ0 (a,m,p, ||·, · · · , ·||) is a linear space. similarly we can prove that i− cσ(a,m,p, ||·, · · · , ·||) and i− `σ∞(a,m,p, ||·, · · · , ·||) are linear spaces. � theorem 2.2 let m = (mk) be a musielak-orlicz function. then i− cσ0 (a,m,p, ||·, · · · , ·||) ⊂i− c σ(a,m,p, ||·, · · · , ·||) ⊂i− `σ∞(a,m,p, ||·, · · · , ·||). proof. the first inclusion is obvious. for second inclusion, let x ∈ i − cσ(a,m,p, ||·, · · · , ·||). then there exists ρ1 > 0 such that for every � > 0 a1 = { k ∈ n : [ mk ( || tkn(a(x) −l) ρ1 ,z1, · · · ,zn−1|| )]pk ≥ � } ∈i. let us define ρ = 2ρ1. since m = (mk) is non-decreasing and convex, we have mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) ≤ mk ( || tkn(a(x) −l) ρ1 ,z1, · · · ,zn−1|| ) + mk ( || tkn(l) ρ1 ,z1, · · · ,zn−1|| ) . suppose that k /∈ a1. hence by above inequality and (1.1), we have[ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ d {[ mk ( || tkn(a(x) −l) ρ1 ,z1, · · · ,zn−1|| )]pk + [ mk ( || tkn(l) ρ1 ,z1, · · · ,zn−1|| )]pk} < d { � + [ mk ( || tkn(l) ρ ,z1, · · · ,zn−1|| )]pk} . because of the fact that [ mk ( ||tkn(l) ρ1 ,z1, · · · ,zn−1|| )]pk ≤ max { 1, [ mk ( ||tkn(l) ρ1 ,z1, · · · ,zn−1|| )]h} , we have [ mk ( || tkn(l) ρ ,z1, · · · ,zn−1|| )]pk < ∞. int. j. anal. appl. 16 (6) (2018) 888 put k = d { � + [ mk ( ||tkn(l) ρ ,z1, · · · ,zn−1|| )]pk} . it follows that { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk > k } ∈i which means x ∈i− `σ∞(a,m,p, ||·, · · · , ·||). this completes the proof of the theorem. � theorem 2.3 let m = (mk) be a musielak-orlicz function, p = (pk) be a bounded sequence of positive real numbers. then i− `σ∞(a,m,p, ||·, · · · , ·||) is a paranormed space with paranorm defined by g(x) = inf { ρ > 0 : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ 1 } . proof. it is clear that g(x) = g(−x). since mk(0) = 0, we get g(0) = 0. let us take x,y ∈ i − cσ∞(a,m,p, ||·, · · · , ·||). let b(x) = { ρ > 0 : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ 1 } , b(y) = { ρ > 0 : [ mk ( || tkn(a(y)) ρ ,z1, · · · ,zn−1|| )]pk ≤ 1 } . let ρ1 ∈ b(x) and ρ2 ∈ b(y). if ρ = ρ1 + ρ2, then we have[ mk ( ||tkn(a(x+y)) ρ ,z1, · · · ,zn−1|| )] ≤ ( ρ1 ρ1 + ρ2 )[ mk ( || tkn(a(x)) ρ1 ,z1, · · · ,zn−1|| )] + [ mk ( || tkn(a(y)) ρ2 ,z1, · · · ,zn−1|| )] . thus [ mk ( ||tkn(a(x+y)) ρ1+ρ2 ,z1, · · · ,zn−1|| )]pk ≤ 1 and g(x + y) ≤ inf { (ρ1 + ρ2) > 0 : ρ1 ∈ b(x), ρ2 ∈ b(y) } ≤ inf { ρ1 > 0 : ρ1 ∈ b(x) } + inf { ρ2 > 0 : ρ2 ∈ b(y) } = g(x) + g(y). let ηs → η where η,ηs ∈ c and let g(xs −x) → 0 as s → ∞. we have to show that g(ηsxs −ηx) → 0 as s →∞. let b(xs) = { ρs > 0 : [ m ( || tkn(a(x s)) ρs ,z1, · · · ,zn−1|| )]pk ≤ 1 } , b(xs −x) = { ρ′s > 0 : [ m ( || tkn(a(x s −x)) ρ′s ,z1, · · · ,zn−1|| )]pk ≤ 1 } . int. j. anal. appl. 16 (6) (2018) 889 if ρs ∈ b(xs) and ρ′s ∈ b(xs −x) then we observe that[ mk ( ||tkn(a(η sxs−ηx)) ρs|ηs−η|+ρ ′ s|η| ,z1, · · · ,zn−1|| ) ≤ [ mk ( || tkn(a(η sxs −ηxs)) ρs|ηs −η| + ρ ′ s|η| + |(ηxs −ηx)| ρs|ηs −η| + ρ ′ s|η| ,z1, · · · ,zn−1|| )] ≤ |ηs −η|ρs ρs|ηs −η| + ρ ′ s|η| [ mk ( || tkn(a(x s)) ρs ,z1, · · · ,zn−1|| )] + |η|ρ ′ s ρs|ηs −η| + ρ ′ s|η| [ mk ( || tkn(a(x s −x)) ρ ′ s ,z1, · · · ,zn−1|| ) . from the above inequality, it follows that [ mk ( || tkn(a(η sxs −ηx)) ρs|ηs −η| + ρ ′ s|η| ,z1, · · · ,zn−1|| )]pk ≤ 1 and consequently, g(ηsxs −ηx) ≤ inf {( ρs|ηs −η| + ρ ′ s|η| ) > 0 : ρs ∈ b(xs),ρ ′ s ∈ b(x s −x) } ≤ (|ηs −η|) > 0 inf { ρ > 0 : ρs ∈ b(xs) } + (|η|) > 0 inf { (ρ ′ s) pn h : ρ ′ s ∈ b(x s −x) } −→ 0 as s −→∞. this completes the proof of the theorem. � theorem 2.4 let m′ = (m′k) and m ′′ = (m′′k ) are musielak-orlicz functions that satisfies the ∆2condition. then (i) i− cσ0 (a,m′,p, ||·, · · · , ·||) ⊆i− cσ0 (a,m′ ◦m′′,p, ||·, · · · , ·||) (ii) i− cσ(a,m′,p, ||·, · · · , ·||) ⊆i− cσ(a,m′ ◦m′′,p, ||·, · · · , ·||) (iii) i− lσ∞(a,m′,p, ||·, · · · , ·||) ⊆i− lσ∞θ (a,m ′ ◦m′′,p, ||·, · · · , ·||). proof. (i) we prove the theorem in two parts. firstly, let m′k ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) > δ. since m′ is nondecreasing, convex and satisfies ∆2-condition, we have[ m′′k ( m′k ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| ))]pk ≤ (kδ−1m′′2 (2) pk) [ m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1|| )]pk ≤ max{1, (kδ−1m′′k (2) h))h [ m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1 )]pk , int. j. anal. appl. 16 (6) (2018) 890 where k ≥ 1 and δ < 1. from the last inequality, the inclusion{ k ∈ n : [ m′′k ( m′k ( ||tkn(ax) ρ ,z1, · · · ,zn−1|| ))]pk ≥ � } ⊆ { k ∈ n : [ m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1 )]pk ≥ � max{1, (kδ−1m′′k (2)h)} } is obtained. if x ∈i−cσ0 (m′,a,p, ||·, · · · , ·), then the set in the right side of the above inclusion belongs to the ideal and so { k ∈ n : [ m′′k ( m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1|| ))]pk ≥ � } ∈i. secondly, suppose that m′k ( ||tkn(ax) ρ ,z1, · · · ,zn−1|| ) ≤ δ. since m′′k is continuous, we have m′′k ( m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1|| )) < � for all � > 0 which implies i− lim k [ m′′k ( m′k ( || tkn(ax) ρ ,z1, · · · ,zn−1|| ))]pk = 0 as� → 0. this completes the proof of (i) part. similarly, we can prove other parts. � theorem 2.5 let m′ = (m′k) and m ′′ = (m′′k ) are musielak-orlicz functions that satisfies the ∆2condition. then (i) i− cσ0 (a,m,p, ||·, · · · , ·||) ∩i− cσ0 (a,m′,p, ||·, · · · , ·||) ⊆i− cσ0 (a,m′ + m,p, ||·, · · · , ·||) (ii) i− cσ(a,m,p, ||·, · · · , ·||) ∩i− cσ(a,m′,p, ||·, · · · , ·||) ⊆i− cσ(a,m′ + m,p, ||·, · · · , ·||) (iii) i− lσ∞(a,m,p, ||·, · · · , ·||) ∩i− lσ∞(a,m′,p, ||·, · · · , ·||) ⊆i− lσ∞(a,m′ + m,p, ||·, · · · , ·||). proof. (i) let x ∈ i − cσ0 (a,m,p, ||·, · · · , ·||) ∩i − cσ0 (a,m′,p, ||·, · · · , ·||). then there exists k1 > 0 and k2 > 0 such that a1 = { k ∈ n : [ mk ( || tkn(a(x)) ρ1 ,z1, · · · ,zn−1|| )]pk ≥ k1 } ∈i and a2 = { k ∈ n : [ m′k ( || tkn(a(x)) ρ1 ,z1, · · · ,zn−1|| )]pk ≥ k2 } ∈i for some ρ > 0. let k /∈ a1 ∪a2. then we have[ (mk + m ′ k) ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ d {( mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ))pk + ( m′k ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ))pk < {k1 + k2}. int. j. anal. appl. 16 (6) (2018) 891 k /∈ b = { k ∈ n : [ (m′k + mk) ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ))pk > k}. we have a1 ∪ a2 ∈ i and so b ⊂ a1 ∪a2 which implies b ∈ i. this means that x ∈ i − cσ0 (a,m′ + m,p, ||·, · · · , ·||). this completes the proof of (i) part of the theorem. similarly, we can prove (ii) and (iii) part. � theorem 2.6 if sup k [ mk(t) ]pk < ∞ for all t > 0, then we have i− cσ(a,m,p, ||·, · · · , ·||) ⊆i− `σ∞(a,m,p, ||·, · · · , ·||). proof. let x ∈i− cσ(a,m,p, ||·, · · · , ·||). by using inequality (1.1), we have [ mk ( || tkn(a(x)) ρ )]pk ≤ d {[ mk ( || tkn(a(x) −l) ρ ,z1, · · · ,zn−1|| )]pk + [ mk ( || tkn(l) ρ ,z1, · · · ,zn−1|| )]pk} , where ρ = 2ρ1. hence, we have { k ∈ n : [ mk ( || tkn(a(x)) ρ )]pk ≥ k } ⊆ { k ∈ n : [ mk ( || tkn(a(x) −l) ρ1 ,z1, · · · ,zn−1 )]pk ≥ � } for all n and some k > 0. since the set in the right side of the above inclusion belongs to the ideal, all of its subsets are in the ideal. hence { k ∈ n : [ mk ( || tkn(a(x)) ρ )]pk ≥ k } ∈i which completes the proof. � theorem 2.7 let 0 < pk ≤ qk < ∞ for each k ∈ n and ( qk pk ) be bounded. then following inclusions hold (i) i− cσ0 (a,m,q, ||·, · · · , ·||) ⊆i− cσ0 (a,m,p, ||·, · · · , ·||) (ii) i− cσ(a,m,q, ||·, · · · , ·||) ⊆i− cσ(a,m,p, ||·, · · · , ·||). proof. (i) let x ∈ i − cσ0 (a,m,q, ||·, · · · , ·||). write αk = pk qk . by hypothesis, we have 0 < α ≤ αk ≤ 1. if[ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk ≥ 1, the inequality [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk holds. this implies the inclusion{ k ∈ n : [ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≥ � } ⊆ { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk ≥ � } int. j. anal. appl. 16 (6) (2018) 892 and so the result is obvious. conversely, if [ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk < 1, we obtain the following inclusion{ k ∈ n : [ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≥ � } ⊆ { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk ≥ � 1 α } since then the inequality [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≤ ([ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]qk)α holds. hence we conclude that x ∈i−cσ0 (a,m,p, ||·, · · · , ·||). this completes the proof of (i) part. similarly, we can prove (ii) part. � theorem 2.8 if 0 < inf pk ≤ pk ≤ 1 for each k ∈ n. then the following inclusions hold: (i) i− cσ0 (a,m,p, ||·, · · · , ·||) ⊆i− cσ0 (a,m, ||·, · · · , ·||) (ii) i− cσ(a,m,p, ||·, · · · , ·||) ⊆i− cσ(a,m, ||·, · · · , ·||). proof. let x ∈ i − cσ0 (a,m,p, ||·, · · · , ·||). suppose that k /∈ {[ mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1 )]pk ≥ � } for 0 < � < 1. by hypothesis, the inequality mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) ≤ [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk holds. then we have k /∈ { k ∈ n : mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) ≥ � } which implies{ k ∈ n : mk ( ||tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) ≥ � } ⊆ { k ∈ n : [ mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| )]pk ≥ � } . hence x ∈i− cσ0 (a,m, ||·, · · · , ·||) since the set { k ∈ n : mk ( || tkn(a(x)) ρ ,z1, · · · ,zn−1|| ) ≥ � } ∈i. this completes the proof of (i) part. similarly, we can prove (ii) part. � corollary 2.9 if 0 < inf pk ≤ pk ≤ 1 for each k ∈ n. then the following inclusions hold: (i) i− cσ0 (a,m, ||·, · · · , ·||) ⊆i− cσ0 (a,m,p, ||·, · · · , ·||) (ii) i− cσ(a,m, ||·, · · · , ·||) ⊆i− cσ(a,m,p, ||·, · · · , ·||). proof. the proof is obvious by theorem 2.8. � int. j. anal. appl. 16 (6) (2018) 893 references [1] p. das, p. kostyrko, w. wilczynski and p. malik, i and i* convergence of double sequences, math. slovaca, 58 (2008), 605-620. [2] p. das and p.malik, on the statistical and ivariation of double sequences, real anal. exch. 33 (2)(2007-2008), 351-364. [3] s. gähler, linear 2-normietre rume, math. nachr., 28 (1965), 1-43. [4] h. gunawan, on n-inner product, n-norms, and the cauchy-schwartz inequality, sci. math. jap., 5 (2001), 47-54. [5] h. gunawan, the space of p-summable sequence and its natural n-norm, bull. aust. math. soc., 64 (2001), 137-147. [6] h. gunawan and m. mashadi, on n-normed spaces, int. j. math. math. sci., 27 (2001), 631-639. [7] e. e. kara, m. daştan and m. ïlkhan, on almost ideal convergence with respect to an orlicz function, konuralp j. math. 4 (2016), 87-94. [8] p. kostyrko, t. salat and w. wilczynski, i-convergence, real anal. exch. 26 (2) (2000), 669-686. [9] v. kumar, on i and i* convergence of double sequences, math. commun., 12 (2007), 171-181. [10] j. lindenstrauss and l. tzafriri, on orlicz sequence spaces, israel j. math. 10(1971), 345-355. [11] g. g. lorentz, a contribution to the theory of divergent series, acta math. 80(1948), 167-190. [12] i. j. maddox, spaces of strongly summable sequence, q. j. math; 18(1967), 345-355. [13] l. maligranda,orlicz spaces and interpolation, seminars in mathematics 5, polish academy of science, 1989. [14] a. misiak, n-inner product spaces, math. nachr. 140 (1989), 299-319. [15] m. mursaleen and a. alotaibi, on i-convergence in radom 2-normed spaces, math. slovaca, 61(6)(2011), 933-940. [16] m. mursaleen, s. a. mohiuddine and o. h. h. edely, on ideal convergence of double sequences in intuitioistic fuzzy normed spaces, comput. math. appl., 59 (2010), 603-611. [17] m. mursaleen and s. a. mohiuddine, on ideal convergence of double sequences in probabilistic normed spaces, math. reports, 12(64)(4) (2010), 359-371. [18] m. mursaleen and s. a. mohiuddine, on ideal convergence in probabilistic normed spaces, math. slovaca, 62(2012), 49-62. [19] m. mursaleen and s. k. sharma, spaces of ideal convergent sequences, world sci. j. 2014(2014), art. id 134534. [20] j. musielak, orlicz spaces and modular spaces, lecture notes in mathematics, springer-verlag berlin heidelberg, (1983). [21] k. raj and s. k. sharma, ideal convergent sequence spaces defined by a musielak-orlicz function, thai. j. math., 11 (2013), 577-587. [22] k. raj and s. k. sharma, some sequence spaces in 2-normed spaces defined by musielak-orlicz functions, acta univ. sapientiae math., 3 (2011), 97-109. [23] k. raj and s. k. sharma, some generalized difference double sequence spaces defined by a sequence of orlicz-function, cubo, 14 (2012), 167-189. [24] k. raj and s. k. sharma, some multiplier sequence spaces defined by a musielak-orlicz function in n-normed spaces, n. z. j. math. 42 (2012), 45-56. [25] a. şahiner, m. gürdal, s. saltan and h. gunawan, on ideal convergence in 2-normed spaces, taiwanese j. math., 11 (2007), 1477-1484. [26] p. schaefer, invariant matrices and invariant means, proc. amer. math. soc.; 36(1972), 104-110. [27] b. c. tripathy and b. hazarika, some i-convergent sequence spaces defined by orlicz functions, acta math. appl. sin. engl. ser. 27 (2011), 149-154. 1. introduction and preliminaries 2. main results references international journal of analysis and applications volume 17, number 3 (2019), 329-341 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-329 m-polar fuzzy hyperideals in la-semihypergroups ahmed elmoasry1,2,∗, naveed yaqoob1 1department of mathematics, college of science al-zulfi, majmaah university, al-zulfi, saudi arabia 2department of mathematics, faculty of science, aswan university, aswan, egypt ∗corresponding author: a.elmoasry@mu.edu.sa abstract. in this paper, we initiate a study of m-polar fuzzy sets in hyperstructure theory, particularly in left almost semihypergroups. we define an m-polar fuzzy left (right, two sided) hyperideal in a left almost semihypergroup and provided some results on these m-polar fuzzy left (right, two sided) hyperideals. 1. introduction in 1934, hyperstructure theory was introduced by a french mathematician marty [1] and then, several authors continued their researches in this direction. hyperstructures are now widely studied from theoretical point of view and for their applications in many subjects of pure and applied mathematics. some preliminary results and definitions about hyperstructure theory can be found in the books written by corsini [2] and vougiouklis [3]. the concept of a semihypergroup is a generalization of the concept of a semigroup. many authors studied different aspects of semihypergroups. some principal notions about semihypergroup theory can be found in [4]. in 2011, hila and dine [5] introduced the concept of non-associative semihypergroups (la-semihypergroups) which is a generalization of semigroups, semihypergroups, and la-semigroups. later, yaqoob et al. [6] explored this concept further and studied intra-regular left almost semihypergroups by their hyperideals using pure left identity. other results on la-semihypergroups can be found in [7–9]. received 2019-03-07; accepted 2019-04-01; published 2019-05-01. 1991 mathematics subject classification. 20n20, 03e72. key words and phrases. la-semihypergroups; hyperideals; m-polar fuzzy sets; m-polar fuzzy hyperideals. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 329 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-329 int. j. anal. appl. 17 (3) (2019) 330 in 1965, zadeh [10] introduced the notion of a fuzzy subset of a non-empty set x, as a function from x to [0, 1]. after the introduction of fuzzy sets, several researchers conducted the researches on generalizations of fuzzy sets with huge applications in computer, logics, automata and many branches of pure and applied mathematics. in 1971, rosenfeld [11] defined the concept of a fuzzy subgroup of a group, and provided a tool and a novel approach to establish the fuzzy analogs of many results in group theory. however, in 1979, anthony and sherwood [12] generalized the rosenfeld’s definition of fuzzy group. sherwood [13] defined products of fuzzy subgroups using t-norms and gave some properties of these products. asaad [14] proposed the idea about the relation between structure of a group and the structure of a fuzzy subgroup. mukherjee and bhattacharya, introduced the concept of a fuzzy normal subgroup and a fuzzy coset in [15]. in 1981, das [16] introduced the concept of level subsets and level subgroups. fuzzy set theory has been developed in the context of hyperalgebraic structure theory. in [17], davvaz introduced the concept of fuzzy hyperideals in a semihypergroup. more on fuzzy hyperstructures one can find in [18]. in 1994, zhang [19] generalized the idea of a fuzzy set and gave the concept of a bipolar fuzzy set on a given set x as a map, which associates each element of x, to a real number, in the interval [−1, 1]. in 2014, chen et al. [20] introduced the idea of m-polar fuzzy sets, as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and 2-polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one [20]. recently, akram with others applied the concept of m-polar fuzzy sets in different directions, like, lie algebras [21, 22], graph theory [23, 24], groups [25], lattices [26] and matroids [27]. akrem published a book on m-polar fuzzy graphs [28], in which he provided several types of m-polar fuzzy graphs with applications. al-masarwah and ahmad [29] studied m-polar (α,β)-fuzzy ideals in bck/bci-algebra. 2. preliminaries definition 2.1. a map ∗ : h × h → p∗(h) is called a hyperoperation or a join operation on a set h, where h is a non-empty set and p∗(h) = p(h)\{∅} denotes the set of all non-empty subsets of h. definition 2.2. a hypergroupoid is a pair (h,∗), where ∗ is a hyperoperation on h. definition 2.3. [5, 6] a hypergroupoid (h,∗) is called an la-semihypergroup if for all x,y,z ∈ h, (x∗y) ∗z = (z ∗y) ∗x. the law (x∗y) ∗ z = (z ∗y) ∗x is called a left invertive law. in an la-semihypergroup, the medial law (x∗y) ∗ (z ∗w) = (x∗z) ∗ (y ∗w) holds for all x,y,z,w ∈ h. let a and b be two non-empty subsets of h. then we define int. j. anal. appl. 17 (3) (2019) 331 a∗b = ⋃ a∈a,b∈b a∗ b, a∗a = {a}∗a and a∗b = {a}∗b. definition 2.4. an element e ∈ h is called a left identity (resp., pure left identity) if for all x ∈ h, x ∈ e∗x (resp., x = e∗x). an la-semihypergroup may or may not contains a left identity and a pure left identity. lemma 2.1. [6] let h be an la-semihypergroup with a pure left identity e. then x∗ (y ∗z) = y ∗ (x∗z) holds for all x,y,z ∈ h. lemma 2.2. [6] let h be an la-semihypergroup with a pure left identity e. then the paramedial law (x∗y) ∗ (z ∗w) = (w ∗z) ∗ (y ∗x) holds for all x,y,z,w ∈ h. definition 2.5. [5] let h be an la-semihypergroup. a non-empty subset a of h is called a sub lasemihypergroup of h if x∗y ⊆ a for every x,y ∈ a. definition 2.6. [5] a subset i of an la-semihypergroup h is called a right (resp., left) hyperideal of h if i ∗h ⊆ i (resp., h ∗ i ⊆ i) and is called a hyperideal if it is two-sided hyperideal. definition 2.7. [6] by a bi-hyperideal (resp., generalized bi-hyperideal) of an la-semihypergroup h, we mean a sub la-semihypergroup (resp., non-empty subset) b of h such that (b ∗h) ∗b ⊆ b. definition 2.8. [6] by an interior hyperideal of an la-semihypergroup h, we mean a subset a of h such that (h ∗a) ∗h ⊆ a. definition 2.9. an la-semihypergroup h is called a regular la-semihypergroup if for every x ∈ h, x ∈ (x∗y) ∗x, for some y ∈ h. an element a of h is called a left regular element of h if there exists x ∈ h such that a ∈ x∗ (a∗a) and h is called left regular if every element of h is left regular. definition 2.10. [20] an m-polar fuzzy set (or a [0, 1]m-set) on x is a mapping m̂ : x → [0, 1]m. the membership value of every element x ∈ x is denoted by m̂(x) = ( p1 ◦m̂(x),p2 ◦m̂(x), ...,pm ◦m̂(x) ) , where pi : [0, 1] m → [0, 1] is the i-th projection mapping (i ∈ m). note that [0, 1]m (m-power of [0, 1]) is considered a poset with the point-wise order �, where m is an arbitrary ordinal number (we make an appointment that m = {n|n < m} when m > 0), � is defined by u � v ⇔ pi(u) � pi(v) for each i ∈ m ( u,v ∈ [0, 1]m). also, 0̂ = (0, 0, · · · , 0) is the smallest element in [0, 1]m and 1̂ = (1, 1, · · · , 1) is the largest element in [0, 1]m. int. j. anal. appl. 17 (3) (2019) 332 3. m-polar fuzzy hyperideals in this section, we will provide some results on m-polar fuzzy hyperideals. let m̂1 and m̂2 be two m-polar fuzzy sets in an la-semihypergroup h. then for any x ∈ h, we define the following: (m̂1 ∩m̂2)(x) = min { m̂1(x),m̂2(x) } , (m̂1 ∪m̂2)(x) = max { m̂1(x),m̂2(x) } , and (m̂1 }m̂2)(x) =   sup x∈y∗z {min{m̂1(y),m̂2(z)} if x ∈ y ∗z (0, 0, ..., 0) if x /∈ y ∗z. we denote the set of all m-polar fuzzy subsets in an la-semihypergroup h by fm(h). for any m̂1, m̂2 ∈ fm(h), m̂1 ⊆ m̂2 ⇔ m̂1(x) � m̂2(x), for each x ∈ h. proposition 3.1. let h be an la-semihypergroup. then the set (fm(h),}) is an la-semihypergroup. proof. the proof is straightforward. � proposition 3.2. if h is an la-semihypergroup, then the medial law (m̂1 }m̂2) } (m̂3 }m̂4) = (m̂1 }m̂3) } (m̂2 }m̂4) holds in fm(h), for all m̂1, m̂2, m̂3 and m̂4 in fm(h). proof. the proof is straightforward. � proposition 3.3. let h be an la-semihypergroup with a pure left identity. then the following properties hold in fm(h) : (i): m̂1 } (m̂2 }m̂3) = m̂2 } (m̂1 }m̂3); (ii): (m̂1 }m̂2) } (m̂3 }m̂4) = (m̂4 }m̂3) } (m̂2 }m̂1); for all m̂1,m̂2,m̂3 and m̂4 in fm(h). int. j. anal. appl. 17 (3) (2019) 333 proof. (i) let m̂1, m̂2 and m̂3 be in fm(h). there exist x and y in h such that z ∈ x∗y. then (m̂1 } (m̂2 }m̂3))(z) = sup z∈x∗y { min { m̂1(x), (m̂2 }m̂3)(y) }} = sup z∈x∗y { min { m̂1(x), sup y∈t∗w { min { m̂2(t),m̂3(w) }}}} = sup z∈x∗y sup y∈t∗w { min { m̂1(x),m̂2(t),m̂3(w) }} = sup z∈x∗(t∗w) { min { m̂1(x),m̂2(t),m̂3(w) }} = sup z∈t∗(x∗w) { min { m̂2(t),m̂1(x),m̂3(w) }} , = sup z∈t∗m sup m∈x∗w { min { m̂2(t),m̂1(x),m̂3(w) }} = sup z∈t∗m { min { m̂2(t), sup m∈x∗w { min { m̂1(x),m̂3(w) }}}} = sup z∈t∗m { min { m̂2(t), (m̂1 }m̂3)(m) }} = (m̂2 } (m̂1 }m̂3))(z). let z be an element of h such that z /∈ x∗y, for some x,y ∈ h. then we have (m̂1 } (m̂2 }m̂3))(z) = (0, 0, ..., 0) = (m̂2 } (m̂1 }m̂3))(z). hence this shows that m̂1 } (m̂2 }m̂3) = m̂2 } (m̂1 }m̂3) holds in fm(h). (2) the proof is similar to (1). � lemma 3.1. let h be an la-semihypergroup with a pure left identity. then for all x ∈ h, h}h = h, where h(x) = (1, 1, ..., 1). proof. the proof is straightforward. � definition 3.1. an m-polar fuzzy set m̂ of an la-semihypergroup h is called an m-polar fuzzy sub lasemihypergroup of h if for all x,y ∈ h, the following condition hold: inf z∈x∗y { m̂(z) } � min { m̂(x),m̂(y) } . that is inf z∈x∗y { pi ◦m̂(z) } � min { pi ◦m̂(x),pi ◦m̂(y) } , for each i = 1, 2, 3, · · · ,m. int. j. anal. appl. 17 (3) (2019) 334 example 3.1. consider an la-semihypergroup h = {e,a,b,c,d}, with the hyperoperation ” ∗ ” defined as follow: ∗ e a b c d e e e e e e a e a a a a b e a a {a,d} a c e a {a,b} {a,c} {a,d} d e a a {a,b} a we define a 5-polar fuzzy set m̂ : h → [0, 1]5 as follows: m̂(x) =   (0.6, 0.7, 0.8, 0.8, 0.9) if x = e (0.5, 0.6, 0.8, 0.8, 0.8) if x = a (0.4, 0.4, 0.7, 0.7, 0.7) if x ∈{b,d} (0.1, 0.3, 0.2, 0.4, 0.1) if x = c. by routine calculations, it is easy to verify that m̂ is a 5-polar fuzzy sub la-semihypergroup of h. definition 3.2. an m-polar fuzzy set m̂ of an la-semihypergroup h is called an m-polar fuzzy left (resp., right) hyperideal of h if for all x,y ∈ h, the following condition hold: inf z∈x∗y { m̂(z) } � m̂(y) (resp., inf z∈x∗y { m̂(z) } � m̂(x)). that is inf z∈x∗y { pi ◦m̂(z) } � pi ◦m̂(y) (resp., inf z∈x∗y { pi ◦m̂(z) } � pi ◦m̂(x)), for each i = 1, 2, 3, · · · ,m. definition 3.3. an m-polar fuzzy set m̂ of an la-semihypergroup h is called an m-polar fuzzy hyperideal of h if for all x,y ∈ h, the following condition hold: inf z∈x∗y { m̂(z) } � max { m̂(x),m̂(y) } . that is inf z∈x∗y { pi ◦m̂(z) } � max { pi ◦m̂(x),pi ◦m̂(y) } , for each i = 1, 2, 3, · · · ,m. int. j. anal. appl. 17 (3) (2019) 335 example 3.2. consider an la-semihypergroup h = {e,a,b,c,d}, with the hyperoperation ” ∗ ” defined as follow: ∗ e a b c d e e e e e e a e a {e,b} a a b e {e,b} b {e,b} {e,b} c e a {e,b} {a,c,d} d d e a {e,b} {a,d} {a,d} we define a 3-polar fuzzy set m̂ : h → [0, 1]3 as follows: m̂(x) =   (0.8, 0.9, 0.8) if x = e (0.5, 0.5, 0.5) if x = a (0.7, 0.7, 0.6) if x = b (0.1, 0.2, 0.2) if x = c (0.3, 0.4, 0.3) if x = d. by routine calculations, it is easy to verify that m̂ is a 3-polar fuzzy hyperideal of h. definition 3.4. let h be an la-semihypergroup. let t̂ = (t1, t2, ..., tm) ∈ [0, 1]m and m̂ be an m-polar fuzzy set in h. then, the set (i): m̂t̂ = {x ∈ h |m̂(x) � t̂} is called a t̂-level cut of m̂. (ii): m̂s t̂ = {x ∈ h |m̂(x) � t̂} is called a strong t̂-level cut of m̂. (iii): im(m̂) is called the image of m̂. theorem 3.1. let m̂ be an m-polar fuzzy set in h, such that the least upper bound t̂0 of im(m̂) exist. then the following conditions are equivalent: (i): m̂ is an m-polar fuzzy sub la-semihypergroup of h. (ii): for all t̂ ∈ im(m̂), the non-empty t̂-level subset m̂t̂ of m̂ is a sub la-semihypergroup of h. (iii): for all t̂ ∈ im(m̂)\ t̂0, the non-empty strong t̂-level subset m̂st̂ of m̂ is a sub la-semihypergroup of h. (iv): for all t̂ ∈ [0, 1]m, the non-empty strong t̂-level subset m̂s t̂ of m̂ is a sub la-semihypergroup of h. (v): for all t̂ ∈ [0, 1]m, the non-empty t̂-level subset m̂t̂ of m̂ is a sub la-semihypergroup of h. proof. (i) → (iv). let m̂ be a m-polar fuzzy sub la-semihypergroup of h, t̂ ∈ [0, 1]m, and x,y ∈ m̂s t̂ . then we have m̂(x),m̂(y) � t̂. thus, min{m̂(x),m̂(y)} � t̂. since m̂ is a sub la-semihypergroup of h, so inf z∈x∗y m̂(z) � t̂. thus z ∈ m̂s t̂ for each z ∈ x∗y. hence m̂s t̂ is a sub la-semihypergroup of h. int. j. anal. appl. 17 (3) (2019) 336 (iv) → (iii). it is clear. (iii) → (ii). let t̂ ∈ im(m̂). then m̂t̂ is non-empty. since m̂t̂ = h⋂̂ t≺α̂ m̂α̂, where α̂ ∈ im(m̂) \ t̂0. then by (iii) we get that m̂t̂ is a sub la-semihypergroup of h. (ii) → (v). let t̂ ∈ [0, 1]m and m̂t̂ be non-empty. suppose that x,y ∈ m̂t̂. then, we have m̂(x),m̂(y) � t̂. let α̂ = min{m̂(x),m̂(y)}. it is clear that α̂ � t̂. thus x,y ∈ m̂t̂ and α̂ ∈ im(m̂), by (ii) m̂α̂ is a sub la-semihypergroup of h, hence z ∈ m̂α̂ for each z ∈ x∗ y. then we have inf z∈x∗y m̂(z) � α̂ � t̂. therefore x∗y ⊆ m̂t̂. then m̂t̂ is a sub la-semihypergroup of h. (v) → (i). assume that the non-empty set m̂t̂ is a sub la-semihypergroup of h, for any t̂ ∈ [0, 1] m. let x,y ∈ h. let us take t̂ = min{m̂(x),m̂(y)}. then m̂(x),m̂(y) � t̂. thus x,y ∈ m̂t̂. since m̂t̂ is a sub la-semihypergroup of h, so for each z ∈ x∗y, z ∈ m̂t̂. thus, inf z∈x∗y m̂(z) � t̂ = min{m̂(x),m̂(y)}. this shows that m̂ is an m-polar fuzzy sub la-semihypergroup of h. this completes the proof. � theorem 3.2. let m̂ be an m-polar fuzzy set in h, such that the least upper bound t̂0 of im(m̂) exist. then the following conditions are equivalent: (i): m̂ is an m-polar fuzzy left (resp., right) hyperideal of h. (ii): for all t̂ ∈ im(m̂), the non-empty t̂-level subset m̂t̂ of m̂ is a left (resp., right) hyperideal of h. (iii): for all t̂ ∈ im(m̂) \ t̂0, the non-empty strong t̂-level subset m̂st̂ of m̂ is a left (resp., right) hyperideal of h. (iv): for all t̂ ∈ [0, 1]m, the non-empty strong t̂-level subset m̂s t̂ of m̂ is a left (resp., right) hyperideal of h. (v): for all t̂ ∈ [0, 1]m, the non-empty t̂-level subset m̂t̂ of m̂ is a left (resp., right) hyperideal of h. proof. the proof is similar to the proof of theorem 3.1. � theorem 3.3. an m-polar fuzzy subset m̂ of an la-semihypergroup h is an m-polar fuzzy (1) sub la-semihypergroup of h if and only if m̂ }m̂ ⊆ m̂, (2) left hyperideal of h if and only if h}m̂ ⊆ m̂, (3) right hyperideal of h if and only if m̂ }h⊆ m̂, (4) hyperideal of h if and only if h}m̂ ⊆ m̂ and m̂ }h⊆ m̂. int. j. anal. appl. 17 (3) (2019) 337 proof. (1) let m̂ be an m-polar fuzzy sub la-semihypergroup of h and z ∈ h. let us suppose that z ∈ x∗y for x,y ∈ h. then (m̂ }m̂) (z) = sup z∈x∗y {min{m̂(x),m̂(y)} � sup z∈x∗y { inf z∈x∗y m̂ (z) } � sup z∈x∗y { m̂ (x∗y) } = m̂ (z) . therefore (m̂ }m̂) ⊆ m̂. if there do not exist any x,y ∈ h such that z ∈ x∗y, then (m̂ }m̂) (z) = 0 � m̂ (z) . hence for all cases m̂ }m̂ ⊆ m̂. conversely, let us assume that m̂}m̂ ⊆ m̂ holds for all m-polar fuzzy subsets of h. let x,y ∈ h. then, we have inf z∈x∗y m̂ (z) � inf z∈x∗y { (m̂ }m̂) (z) } = inf z∈x∗y { sup z∈x∗y {min{m̂(x),m̂(y)} } = min{m̂(x),m̂(y)}. this means that m̂ is an m-polar fuzzy sub la-semihypergroup of h. the other cases can be seen in a similar way. � theorem 3.4. if { m̂i } i∈λ is a family of m-polar fuzzy left (resp., right) hyperideals of an la-semihypergroup h, then ⋂ i∈λ m̂i is also a fuzzy left (resp., right) hyperideal of h. proof. let { m̂i } i∈λ be a family of m-polar fuzzy left hyperideals of an la-semihypergroup h and let x,y ∈ h. then we have ⋂ i∈λ m̂i (y) = ∧ i∈λ { m̂i (y) } � ∧ i∈λ { inf z∈x∗y { m̂i (z) }} = inf z∈x∗y { ∧ i∈λ { m̂i (z) }} = inf z∈x∗y { ⋂ i∈λ m̂i (z) } . hence ⋂ i∈λ m̂i is an m-polar fuzzy left hyperideal of la-semihypergroup h. similarly we can prove it for right hyperideals. this completes the proof. � int. j. anal. appl. 17 (3) (2019) 338 let h be an la-semihypergroup and let ∅ 6= a ⊆ h. then the m-polar fuzzy characteristic function χa m̂ of a is defined by χa m̂ (x) =   (1, 1, ..., 1) if x ∈ a(0, 0, ..., 0) if x /∈ a . theorem 3.5. let a be a nonempty subset of an la-semihypergroup h. then a is a sub la-semihypergroup (resp., left hyperideal, right hyperideal) of h if and only if χa m̂ is an m-polar fuzzy sub la-semihypergroup (resp., left hyperideal, right hyperideal) of h. proof. let a be a left hyperideal of h. for any x,y ∈ h, we have the following cases: case (1) : if x,y ∈ a, then x∗y ⊆ a. then inf z∈x∗y χa m̂ (z) = 1̂ and χa m̂ (y) = 1̂. therefore inf z∈x∗y χa m̂ (z) = χa m̂ (y) . case (2) : if x,y /∈ a, then χa m̂ (y) = 0̂. so inf z∈x∗y χa m̂ (z) � χa m̂ (y) . case (3) : if x ∈ a and y /∈ a, then χa m̂ (y) = 0̂. so inf z∈x∗y χa m̂ (z) � χa m̂ (y) . case (4) : if x /∈ a and y ∈ a, then x∗y ⊆ a. then inf z∈x∗y χa m̂ (z) = 1̂ and χa m̂ (y) = 1̂. therefore inf z∈x∗y χa m̂ (z) = χa m̂ (y) . hence χa m̂ is an m-polar fuzzy left hyperideal of h. conversely, suppose χa m̂ is an m-polar fuzzy left hyperideal of h and let x ∈ h and y ∈ a. then we have inf z∈x∗y χa m̂ (z) � χa m̂ (y) = 1̂. but, we know that inf z∈x∗y χa m̂ (z) � 1̂. therefore inf z∈x∗y χa m̂ (z) = 1̂. this implies that z ∈ a for each z ∈ x∗y. hence a is a left hyperideal of h. the other cases can be seen in a similar way. � proposition 3.4. let m̂1 be an m-polar fuzzy right hyperideal of h and m̂2 be an m-polar fuzzy left hyperideal of h. then m̂1 }m̂2 ⊆ m̂1 ∩m̂2. int. j. anal. appl. 17 (3) (2019) 339 proof. let m̂1 be an m-polar fuzzy right hyperideal of h and m̂2 be an m-polar fuzzy left hyperideal of h. let z ∈ h and suppose that there exist x,y ∈ h such that z ∈ x∗y. then (m̂1 }m̂2)(z) = sup z∈x∗y { min{m̂1(x),m̂2(y)} } � sup z∈x∗y { min { inf z∈x∗y { m̂1(z) } , inf z∈x∗y { m̂2(z) }}} = min{m̂1(z),m̂2(z)} = (m̂1 ∩m̂2)(z). let us suppose that there do not exist x,y ∈ h such that z ∈ x∗y. then, (m̂1}m̂2)(z) = 0̂ � (m̂1∩m̂2)(z). this completes the proof. � lemma 3.2. let h be an la-semihypergroup with a pure left identity. then every m-polar fuzzy right hyperideal is an m-polar fuzzy hyperideal. proof. the proof is straightforward. � theorem 3.6. if m̂ is an m-polar fuzzy left hyperideal of h with a pure left identity, then m̂ ∪ (m̂ }h) and m̂ ∪ (m̂ }m̂) are m-polar fuzzy hyperideals of h. proof. let m̂ be an m-polar fuzzy left hyperideal of h. we have( m̂ ∪ (m̂ }h) ) }h = (m̂ }h) ∪ ((m̂ }h) }h) = (m̂ }h) ∪ ((h}h) }m̂) = (m̂ }h) ∪ (h}m̂) (by lemma 3.1) ⊆ (m̂ }h) ∪m̂. hence, m̂ ∪(m̂ }h) is an m-polar fuzzy right hyperideal of h. by lemma 3.2, m̂ ∪(m̂ }h) is an m-polar fuzzy hyperideal of h. in a similar way we can prove that m̂ ∪(m̂ }m̂) is an m-polar fuzzy hyperideal. � proposition 3.5. let h be a regular la-semihypergroup with a pure left identity e. if m̂ is an m-polar fuzzy right hyperideal of h, then m̂(x∗y) = m̂(y ∗x) holds for all x,y ∈ h. proof. let m̂ be an m-polar fuzzy right hyperideal of a regular la-semihypergroup h with a pure left identity e. let x,y ∈ h. since h is regular, x ∈ (x ∗ a) ∗ x and y ∈ (y ∗ b) ∗ y for some a,b ∈ h. now by using the medial and paramedial laws, we get x∗y ⊆ ((x∗a) ∗x) ∗ ((y ∗ b) ∗y) = (y ∗x) ∗ ((y ∗ b) ∗ (x∗a)). int. j. anal. appl. 17 (3) (2019) 340 since m̂ is an m-polar fuzzy right hyperideal, then for every w ∈ x∗y ⊆ (y∗x) ∗ ((y∗b) ∗ (x∗a)), we have inf w∈x∗y {m̂(w)} � inf w1∈(y∗x)∗((y∗b)∗(x∗a)) {m̂(w1)} � inf s∈y∗x {m̂(s)}, again by using the medial and paramedial laws, we get y ∗x ⊆ ((y ∗ b) ∗y) ∗ ((x∗a) ∗x) = (x∗y) ∗ ((x∗a) ∗ (y ∗ b)). since m̂ is an m-polar fuzzy right hyperideal, for every t ∈ y ∗x ⊆ (x∗y) ∗ ((x∗a) ∗ (y ∗ b)), we have inf t∈y∗x {m̂(t)} � inf t1∈(x∗y)∗((x∗a)∗(y∗b)) {m̂(t1)} � inf p∈x∗y {m̂(p)}, this shows that m̂(x∗y) = m̂(y ∗x) holds for all x,y ∈ h. � lemma 3.3. if h is a left regular la-semihypergroup, then every m-polar fuzzy left (resp., right) hyperideal m̂ of h is an m-polar fuzzy idempotent. proof. let m̂ be any m-polar fuzzy left hyperideal of a left regular la-semihypergroup h with a pure left identity. then m̂ }m̂ ⊆h∗m̂ ⊆ m̂. since h is left regular, for every a ∈ h, there exists x ∈ h such that a ∈ x∗ (a∗a). by lemma 2.1, we have a ∈ x∗ (a∗a) = a∗ (x∗a). therefore (m̂ }m̂)(a) = sup a∈a∗(x∗a) { min { m̂(a), inf s∈x∗a m̂(s) }} � min{m̂(a),m̂(a)} = m̂(a). thus we get m̂ ⊆ m̂ } m̂. this implies that m̂ } m̂ = m̂. hence this shows that m̂ is m-polar fuzzy idempotent. the other case can be proved in a similar way. � 4. conclusions in this paper, we studied the hyperideal-structure of m-polar fuzzy sets in left almost semihypergroups. we defined m-polar fuzzy sub la-semihypergroups and m-polar fuzzy left (right, two sided) hyperideal in a left almost semihypergroup and used these m-polar fuzzy left (right, two sided) hyperideals to characterize some classes of left almost semihypergroups. acknowledgements. the authors would like to thank deanship of scientific research at majmaah university for supporting this work under project number 35/37. int. j. anal. appl. 17 (3) (2019) 341 references [1] f. marty, sur une generalization de la notion de group, 8th congres math. scandinaves, stockholm, (1934), 45-49. [2] p. corsini, prolegomena of hypergroup theory, second edition, aviani editor, (1993). [3] t. vougiouklis, hyperstructures and their representations, hadronic press, florida, (1994). [4] b. davvaz, semihypergroup theory. elsevier /academic press, london, 2016. [5] k. hila and j. dine, on hyperideals in left almost semihypergroups, isrn algebra, 2011 (2011), article id 953124. [6] n. yaqoob, p. corsini and f. yousafzai, on intra-regular left almost semihypergroups with pure left identity, j. math. 2013 (2013), art. id 510790. [7] f. yousafzai, p. corsini, some charactrization problems in la-semihypergroups, j. algebra number theory, adv. appl. 10(1-2) (2013), 41-55. [8] v. amjad, k. hila, f. yousafzai, generalized hyperideals in locally associative left almost semihypergroups, new york j. math. 20 (2014), 1063-1076. [9] f. yousafzai, k. hila, p. corsini, a. zeb, existence of non-associative algebraic hyper-structures and related problems, afr. mat. 26(5) (2015), 981-995. [10] l. a. zadeh, fuzzy sets, inform. control, 8 (1965), 338-353. [11] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 512-517. [12] j. m. anthony and h. sherwood, fuzzy subgroups redefined, j. math. anal. appl., 69 (1979), 124-130. [13] h. sherwood, products of fuzzy subgroups, fuzzy sets syst. 11 (1983), 79-89. [14] m. asaad, groups and fuzzy subgroups, fuzzy sets syst. 39 (1991), 323-328. [15] n. p. mukherjee and p. bhattacharya, fuzzy normal subgroups and fuzzy cosets, inform. sci. 34 (1984), 225-239. [16] p. s. das, fuzzy groups and level subgroups, j. math. anal. appl. 84 (1981), 264-269. [17] b. davvaz, fuzzy hyperideals in semihypergroups, italian j. pure appl. math. 8 (2000), 67-74. [18] b. davvaz, i. cristea, fuzzy algebraic hyperstructures. an introduction. studies in fuzziness and soft computing, springer, (2015). [19] w. r. zhang, bipolar fuzzy sets, proc. fuzz-ieee, (1998), 835-840. [20] j. chen, s.-g. li, s. ma and x. w, m-polar fuzzy sets: an extension of m-polar fuzzy sets, sci. world j. 2014 (2014), art. id 416530. [21] m. akram and a. farooq, m-polar fuzzy lie ideals of lie algebras, quasigroups relat. syst. 24(2) (2016), 141-150. [22] m. akram, a. farooq and k. p. shum, on m-polar fuzzy lie subalgebras, italian j. pure appl. math. 36 (2016) 445-454 [23] m. akram and n. waseem, certain metrics in m-polar fuzzy graphs, new math. nat. comput. 12(2) (2016), 135-155. [24] m. akram and h. r. younas, certain types of irregular m-polar fuzzy graphs, j. appl. math. comput. 53(1-2) (2017), 365-382. [25] a. farooq, g. ali and m. akram, on m-polar fuzzy groups, int. j. algebra stat. 5(2) (2016), 115-127. [26] m. sarwar and m. akram, novel applications of m-polar fuzzy concept lattice, new math. nat. comput. 13(3) (2017), 261-287. [27] m. sarwar and m. akram, new applications of m-polar fuzzy matroids, symmetry, 9(12) (2017), art. id 319. [28] m. akram, m-polar fuzzy graphs, studies in fuzziness and soft computing, vol 371, springer, (2019). [29] a. al-masarwah and a. g. ahmad, m-polar (α,β)-fuzzy ideals in bck/bci-algebras, symmetry, 11 (2019), art. id 44. 1. introduction 2. preliminaries 3. m-polar fuzzy hyperideals 4. conclusions references international journal of analysis and applications volume 16, number 1 (2018), 83-96 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-83 a new truncated m-fractional derivative type unifying some fractional derivative types with classical properties j. vanterler da c. sousa∗, and e. capelas de oliveira department of applied mathematics, imecc–unicamp, adress: sérgio buarque de holanda 651, 13083–859, campinas sp, brazil ∗corresponding author: ra160908@ime.unicamp.br abstract. we introduce a truncated m-fractional derivative type for α-differentiable functions that generalizes four other fractional derivatives types recently introduced by khalil et al., katugampola and sousa et al., the so-called conformable fractional derivative, alternative fractional derivative, generalized alternative fractional derivative and m-fractional derivative, respectively. we denote this new differential operator by id α,β m , where the parameter α, associated with the order of the derivative is such that 0 < α < 1, β > 0 and m is the notation to designate that the function to be derived involves the truncated mittag-leffler function with one parameter. the definition of this truncated m-fractional derivative type satisfies the properties of the integer-order calculus. we also present, the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. finally, we obtain the analytical solution of the m-fractional heat equation and present a graphical analysis. 1. introduction the non integer-order calculus or fractional calculus, as it is largely diffused, is as important and ancient as the integer-order calculus, and for many years the scientific community didn’t know it. currently, there are numerous and important definitions of fractional derivatives types, each one of them with its peculiarity received 14th september, 2017; accepted 4th december, 2017; published 3rd january, 2018. 2010 mathematics subject classification. 26a06, 26a24, 26a33, 26a42. key words and phrases. alternative fractional derivative; conformable fractional derivative; m-fractional heat equation; truncated m-fractional derivative type; truncated mittag-leffler function. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 83 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-83 int. j. anal. appl. 16 (1) (2018) 84 and application [1–3]. although, to be highlighted only from 1974, after the first international conference on fractional calculus, it has shown to be important and with great applicability in modeling problems, more precisely natural phenomena. for this growth, it was necessary the contribution of famous mathematicians, such as lagrange, abel, euler, liouville, riemann, as well as recently caputo and mainardi, among others. it is possible to define various integrals and fractional derivatives. each definition has its own peculiarity and thus makes the fractional calculus fruitful in the sense of theory and applications. we report some types of fractional derivatives that have been introduced so far, among which we mention: riemann-liouville, caputo, hadamard, caputo-hadamard, riesz, among others [3, 4]. many of these derivatives are defined from the fractional integral in the riemann-liouville sense. recently, katugampola [5], has presented a new fractional integral unifying six existing fractional integrals, named riemann-liouville, hadamard, erdélyikober, katugampola, weyl and liouville [3, 6]. on the other hand, also recent, khalil et al. [7] proposed the so-called compatible fractional derivative of order α with 0 < α < 1 in order to generalize the classical properties of calculus. more recently, in 2014, katugampola [8] has also proposed an alternative fractional derivative with classical properties, which refers to the leibniz and newton calculus, similar to the conformable fractional derivative. in 2017, sousa and oliveira [9], introduced an m-fractional derivative involving a mittag-leffler function with one parameter [10] that also satisfies the properties of integer-order calculus. in this sense, we are going to introduce a truncated m-fractional derivative type that unifies four existing fractional derivative types mentioned above and which also satisfies the classical properties of integer-order calculus. the study of differential equations has proved very useful over time. one of the main reasons is that the simplest differential equations have the ability to model more complex natural systems [1, 3, 11]. there are a larger number of application involving differential equations of which we mention: the problem of population dynamics, brachistochronous problem, wave equation, heat equation and others [1, 12]. however, natural systems over time, become more complex and more than differential equations, provides a rough and simplified description of the actual process, it is necessary that new and more refined mathematical tools are presented and studied. in this sense, fractional derivatives are used to propose modeling in order to obtain more precise results in the studies and applications involving differential equations [11]. then through the use of properties of a truncated m-fractional derivative type, we will present an analytical study of the heat equation and through graph, we will analyze the behavior of the solution in relation to other types of fractional derivatives the so-called local derivatives. this paper is organized as follows: in section 2, our main result, we introduce the concept of truncated m-fractional derivative type involving a truncated mittag-leffler function, as well as several theorems. also, we introduce the respective m-fractional integral for which we demonstrate the inverse property. in section 3, we present the relationship between a truncated m-fractional derivative type, introduced here, int. j. anal. appl. 16 (1) (2018) 85 and the conformable fractional derivative, generalized and alternative fractional derivative and m-fractional derivative. in section 4, we perform an analytical study of the m-fractional heat equation in order to obtain the analytical solution and present some graphs. concluding remarks close the paper. 2. truncated m-fractional derivative type in this section, we define a truncated m-fractional derivative type and obtain several results that have a great similarity with the results found in the classical calculus. from the definition, we present a theorem showing that this truncated m-fractional derivative type is linear, obeys the product rule and the composition of two α-differentiable functions, the quotient rule and the chain rule. it is also shown that the derivative of a constant is zero, as well as versions for rolle’s theorem, the mean value theorem and an extension of the mean value theorem. further, the continuity of this truncated m-fractional derivative type is shown as in integer-order calculus. also, we introduce the concept of m-fractional integral of a f function. from the definition, we shown the inverse theorem. we define the truncated mittag-leffler function of one parameter by: ieβ (z) = i∑ k=0 zk γ (βk + 1) , (2.1) with β > 0 and z ∈ c. from eq.(2.1), we define a truncated m-fractional derivative type that unifies other four fractional derivatives that refer to classical properties of the integer-order calculus. in this work, if a truncated m-fractional derivative type of order α as defined in eq.(2.2) of a function f exists, we say that the function f is α-differentiable. thus, let us begin with the following definition, which is a generalization of the usual definition of integer order derivative. definition 2.1. let f : [0,∞) → r. for 0 < α < 1 a truncated m-fractional derivative type of f of order α, denoted by id α,β m , is id α,β m f (t) := lim ε→0 f (t ieβ (εt−α)) −f (t) ε , (2.2) ∀t > 0 and ieβ (·), β > 0 is a truncated mittag-leffler function of one parameter, as defined in eq.(2.1). note that, if f is α-differentiable in some open interval (0,a), a > 0, and lim t→0+ ( id α,β m f (t) ) exist, then we have id α,β m f (0) = lim t→0+ ( id α,β m f (t) ) . theorem 2.1. if a function f : [0,∞) → r is α-differentiable for t0 > 0, with 0 < α ≤ 1, β > 0, then f is continuous at t0. int. j. anal. appl. 16 (1) (2018) 86 proof. in fact, let us consider the identity f ( t0 ieβ ( εt−α0 )) −f (t0) = ( f ( t0 ieβ ( εt−α0 )) −f (t0) ε ) ε. (2.3) applying the limit ε → 0 on both sides of eq.(4.1), we get lim ε→0 f ( t0 ieβ ( εt−α0 )) −f (t0) = lim ε→0 ( f ( t0 ieβ ( εt−α0 )) −f (t0) ε ) lim ε→0 ε = id α,β m f (t) lim ε→0 ε = 0. then, f is continuous at t0. � using the definition of truncated mittag-leffler function of one parameter, we have f ( t ieβ ( εt−α )) = f ( t i∑ k=0 (εt−α) k γ (βk + 1) ) . (2.4) applying the limit ε → 0 on both sides of eq.(2.4) and since f is a continuous function, we have lim ε→0 f ( t ieβ ( εt−α )) = lim ε→0 f ( t i∑ k=0 (εt−α) k γ (βk + 1) ) = f ( tlim ε→0 i∑ k=0 (εt−α) k γ (βk + 1) ) . (2.5) further, we have t ieβ ( εt−α ) = t i∑ k=0 (εt−α) k γ (βk + 1) = t + εt1−α γ (β + 1) + t (εt−α) 2 γ (2β + 1) + t (εt−α) 3 γ (3β + 1) + · · · + t (εt−α) i γ (iβ + 1) . (2.6) applying the limit ε → 0 on both sides of eq.(2.6), we have lim ε→0 i∑ k=0 (εt−α) k γ (βk + 1) = 1. in this way, we conclude that lim ε→0 f ( t ieβ ( εt−α )) = f (t) . (2.7) here, we present the theorem that encompasses the main classical properties of integer order calculus. for the chain rule, it is verified through an example, as we will see next. we will do here, only the demonstration of the chain rule, for other items, follow the same steps of theorem 2 found in the paper by sousa and oliveira [9]. int. j. anal. appl. 16 (1) (2018) 87 theorem 2.2. let 0 < α ≤ 1, β > 0, a,b ∈ r and f,g α-differentiable, at a point t > 0. then: (1) id α,β m (af + bg) (t) = a id α,β m f (t) + b id α,β m g (t). (2) id α,β m (f ·g) (t) = f (t) id α,β m g (t) + g (t) id α,β m f (t). (3) id α,β m ( f g ) (t) = g (t) id α,β m f (t) −f (t) id α,β m g (t) [g (t)] 2 . (4) id α,β m (c) = 0, where f(t) = c is a constant. (5) (chain rule)if f is differentiable, then id α,β m (f) (t) = t1−α γ (β + 1) df(t) dt . proof. from eq.(2.6), we have t ieβ ( εt−α ) = t + εt1−α γ (β + 1) + o ( ε2 ) , and introducing the following change, h = εt1−α ( 1 γ (β + 1) + o (ε) ) ⇒ ε = h t1−α ( 1 γ(β+1) + o (ε) ), we conclude that id α,β m f (t) = lim ε→0 f (t + h) −f (t) htα−1 1 γ(β+1) (1 + γ (β + 1)o (ε)) = t1−α γ (β + 1) lim ε→0 f (t + h) −f (t) h 1 + γ (β + 1)o (ε) = t1−α γ (β + 1) df (t) dt , with β > 0 and t > 0. � (6) id α,β m (f ◦g) (t) = f ′(g(t))id α,β m g(t), for f differentiable at g(t). now, it is necessary to know if, in addition to the previous theorem 2 that contains important properties similar to integer-order calculus, this truncated m-fractional derivative type eq.(2.2) also has important theorems related to the classical calculus. we shall now see that: the rolle’s theorem, the mean value theorem and its extension coming from the integer-order calculus can be extended to α-differentiable functions, i.e., that admit truncated m-fractional derivative as introduced in eq.(2.2). theorem 2.3. (rolle’s theorem for fractional α-differentiable functions) let a > 0, and f : [a,b] → r be a function with the properties: (1) f is continuous on [a,b]. (2) f is α-differentiable on (a,b) for some α ∈ (0, 1). (3) f(a) = f(b). then, ∃c ∈ (a,b), such that id α,β m f(c) = 0, with β > 0. int. j. anal. appl. 16 (1) (2018) 88 proof. since f is continuous on [a,b] and f(a) = f(b), there exist c ∈ (a,b), at which the function has a local extreme. then, id α,β m f (c) = lim ε→0− f (c ieβ (εc−α)) −f (c) ε = lim ε→0+ f (c ieβ (εc−α)) −f (c) ε . but, the two limits have opposite signs. hence, id α,β m f (c) = 0. � the proof of theorem 4 and theorem 5, will be omitted, but follow the same reasoning of the respective theorems demonstrated in sousa and oliveira [9]. theorem 2.4. (mean-value theorem for fractional α-differentiable functions) let a > 0 and f : [a,b] → r be a function with the properties: (1) f is continuous on [a,b]. (2) f is α-differentiable on (a,b) for some α ∈ (0, 1). then, ∃c ∈ (a,b), such that id α,β m f (c) = f (b) −f (a) bα α − aα α , with β > 0. theorem 2.5. (extension mean value theorem for fractional α-differentiable functions) let a > 0, and f,g : [a,b] → r functions that satisfy: (1) f,g are continuous on [a,b]. (2) f,g are α-differentiable for some α ∈ (0, 1). then, ∃c ∈ (a,b), such that id α,β m f (c) id α,β m g (c) = f (b) −f (a) g (b) −g (a) , (2.8) with β > 0. definition 2.2. let α ∈ (n,n + 1], for some n ∈ n, β > 0 and f n-differentiable for t > 0. then the α-fractional derivative of f is defined by id α,β;n m f (t) := lim ε→0 f(n) (t ieβ (εtn−α)) −f(n) (t) ε , (2.9) since the limit exist. from definition 2 and the chain rule, that is, from item 5 of theorem 2, by induction on n, we can prove that id α,β;n m f (t) = tn+1−α γ (β + 1) f(n+1)(t), α ∈ (n,n + 1] and f is (n + 1)-differentiable for t > 0. now, we know that: this truncated m-fractional derivative type eq.(2.2) has a corresponding m-fractional integral. then, we will present the definition and a theorem that corresponds to the inverse property. for other results involving integrals, one can consult [9, 13]. int. j. anal. appl. 16 (1) (2018) 89 definition 2.3. let a ≥ 0 and t ≥ a. also, let f be a function defined in (a,t] and 0 < α < 1. then, the m-fractional integral of order α of function f is defined by [9] miα,βa f (t) = γ (β + 1) ∫ t a f (x) x1−α dx, (2.10) with β > 0. theorem 2.6. (inverse) let a ≥ 0 and 0 < α < 1. also, let f be a continuous function such that exist miα,βa f. then id α,β m (mi α,β a f (t)) = f(t), (2.11) with t ≥ a and β > 0. proof. in fact, using the chain rule as seen in theorem 2.2, we have id α,β m ( miα,βa f (t) ) = t1−α γ (β + 1) d dt (miα,βa f (t)) = t1−α γ (β + 1) d dt ( γ (β + 1) ∫ t a f (x) x1−α dx ) = t1−α γ (β + 1) ( γ (β + 1) t1−α f (t) ) = f (t) . (2.12) � with the condition f(a) = 0, by theorem 6, that is, eq.(2.12), we have miα,βa [ id α,β m f(t) ] = f(t). 3. relation with other fractional derivatives types in this section, we will discuss the relationship between the fractional conformable derivative proposed by khalil et al. [7], the alternative fractional derivative and the generalized alternative fractional derivative proposed by katugampola [8] and the m-fractional derivative proposed by sousa and oliveira [9], with our truncated m-fractional derivative type. khalil et al. [7] proposed a definition of a fractional derivative, called conformable fractional derivative that refers to the classical properties of integer order calculus, given by f(α) (t) = lim ε→0 f ( t + εt1−α ) −f (t) ε , (3.1) with α ∈ (0, 1) and t > 0. in 2014, katugampola [8] proposed another definition of a fractional derivative, called an alternative fractional derivative which also refers to the classical properties of integer-order calculus, given by dαf (t) = lim ε→0 f ( teεt −α ) −f (t) ε , (3.2) int. j. anal. appl. 16 (1) (2018) 90 with α ∈ (0, 1) and t > 0. in the same paper, katugampola [8] by means of a truncated exponential function, that is, ke x, proposed another generalized fractional derivative, given by dαkf (t) = lim ε→0 f ( ke εt−αt ) −f (t) ε , (3.3) with α ∈ (0, 1) and t > 0. recently, sousa and oliveira [9] introduced the m-fractional derivative d α,β m where the parameter β > 0 and m is the notation to designate that the function to be derived involves the mittag-leffler function of one parameter, given by d α,β m f (t) := lim ε→0 f (t eβ (εt−α)) −f (t) ε , (3.4) with α ∈ (0, 1) and t > 0. it is clear that our definition of truncated m-fractional derivative type eq.(2.2) is more general than the fractional derivatives eq.(3.1), eq.(3.2), eq.(3.3) and eq.(3.4). we will now study particular cases involving such fractional derivatives. choosing β = 1 and applying the limit i → 0 on both sides of eq.(2.2), we have 1d α,β m f (t) = lim ε→0 f (t 1e1 (εt−α)) −f (t) ε . (3.5) but, it is know that 1e1 ( εt−α ) = 1∑ k=0 (εt−α) k γ (k + 1) = 1 + εt−α. (3.6) thus, we conclude that 1d α,β m f (t) = lim ε→0 f ( t + εt1−α ) −f (t) ε = f(α)(t), (3.7) which is exactly the conformable fractional derivative eq.(3.1). choosing β = 1 and applying the limit i →∞ on both sides of eq.(2.2), we have ∞d α,β m f (t) = lim ε→0 f (t ∞e1 (εt−α)) −f (t) ε . (3.8) but, as we have ∞e1 ( εt−α ) = ∞∑ k=0 (εt−α) k γ (k + 1) = eεt −α , (3.9) we conclude that ∞d α,β m f (t) = lim ε→0 f ( teεt −α ) −f (t) ε = dαf (t) , (3.10) which is exactly the alternative fractional derivative, eq.(3.2). int. j. anal. appl. 16 (1) (2018) 91 choosing β = 1 in eq.(2.2), we have id α,β m f (t) = lim ε→0 f (t ie1 (εt−α)) −f (t) ε . (3.11) remembering that ie1 ( εt−α ) = i∑ k=0 (εt−α) k γ (k + 1) = eεt −α i , (3.12) we have id α,β m f (t) = lim ε→0 f ( eεt −α i t ) −f (t) ε = dαi f (t) , (3.13) exactly the generalized fractional derivative, eq.(3.3). finally, applying the limit i →∞ on both sides of eq.(2.2), we have ∞d α,β m f (t) = lim ε→0 f (t ∞eβ (εt−α)) −f (t) ε , (3.14) since ∞eβ ( εt−α ) = ∞∑ k=0 (εt−α) k γ (k + 1) = eβ ( εt−α ) , (3.15) we conclude that ∞d α,β m f (t) = lim ε→0 f (t eβ (εt−α)) −f (t) ε = d α,β m f (t) , (3.16) exactly the m-fractional derivative, eq.(3.4). 4. application in this section, we obtain the solution of the heat equation using a truncated m-fractional derivative type with 0 < α < 1 and present some graphs about the behavior of the solution. consider the heat equation in one dimension given by ∂u (x,t) ∂t = k ∂2u (x,t) ∂x2 , 0 < x < l, t > 0, where k is a positive constant. using a m-fractional derivative type, we propose an m-fractional heat equation given by ∂αu (x,t) ∂tα = k ∂2u (x,t) ∂x2 , 0 < x < l, t > 0, (4.1) where 0 < α < 1 and with the initial condition and boundary conditions given by u (0, t) = 0 , t ≥ 0, (4.2) u (l,t) = 0 , t ≥ 0, u (x, 0) = f (x) , 0 ≤ x ≤ l. int. j. anal. appl. 16 (1) (2018) 92 we start, considering the so-called m-fractional linear differential equation with constant coefficients ∂αv (x,t) ∂tα ±µ2v (x,t) = 0, (4.3) where µ2 is a positive constant. using the item 5 in theorem 2.2, the eq.(4.1) can be written as follows t1−α γ (β + 1) dv (x,t) dt ±µ2v (x,t) = 0, whose solution is given by v (t) = ce±γ(β+1) µ2tα α , (4.4) with 0 < α < 1 and β > 0. now, we will use separation of variables method to obtain the solution of the m-fractional heat equation. then, considering u (x,t) = p (x) q (t) and replacing in eq.(4.1), we get dα dtα q (t) p (x) = k d2 dx2 p (x) q (t) which implies 1 kq (t) dα dtα q (t) = 1 p (x) d2 dx2 p (x) = ξ. (4.5) from eq.(4.5), we obtain a system of differential equations, given by dα dtα q (t) −kξq (t) = 0 (4.6) and d2 dx2 p (x) − ξp (x) = 0. (4.7) first, let’s find the solution of eq.(4.7). for this, we must study three cases, that is, ξ = 0, ξ = −µ2 e ξ = µ2. case 1: ξ = 0. substituting ξ = 0 into eq.(4.7), we have d2 dx2 p (x) − ξp (x) = 0, (4.8) whose solution is given by p (x) = c1x + c2, with c1 and c2 arbitrary constant. using the initial conditions given by eq.(4.2), we obtain that c1 = c2 = 0. like this, p (x) = 0, which implies u(x,t) = 0 trivial solution. case 2: ξ = −µ2. substituting ξ = −µ2 into eq.(4.7), we get d2 dx2 p (x) + µ2p (x) = 0, int. j. anal. appl. 16 (1) (2018) 93 whose solution is given by p (x) = c2 sin (µx) + c1 cos (µx), with c1 and c2 arbitrary constant. using the initial conditions eq.(4.2), we obtain c1 = 0 and 0 = c2 sin (µx) which implies that µ = nπ l , with n = 1, 2, .... then, we obtain pn (x) = an sin (nπx l ) and µ = nπ l . case 3: ξ = µ2. substituting ξ = µ2 into eq.(4.7), we get d2 dx2 p (x) −µ2p (x) = 0 whose solution is given by p (x) = c1 e µx + c2 e −µx = a cosh (µx) + b sinh (µx), with c1, c2, a, b arbitrary constant. using the boundary conditions eq.(4.2), we have a = 0 and 0 = b sinh (µx). as λ = −µ2 < 0 and λl 6= 0 then sinh (µx) 6= 0. like this, we get b = 0 and then pn (x) = 0, which implies u(x,t) = 0, trivial solution. therefore, the solution of eq.(4.7) is given by pn (x) = an sin (nπx l ) andµ = nπ l . (4.9) using the eq.( 4.3) and eq.(4.4), we have qn (t) = bn exp ( −γ (β + 1) (nπ l )2 k α tα ) , (4.10) where bn are constant coefficients. so, using the eq.(4.9) and eq.(4.10), the partial solutions of eq.(4.1), is given by uβ (x,t) = ∞∑ n=1 cn sin (nπx l ) exp ( −γ (β + 1) (nπ l )2 k α tα ) . using eq.(4.2), we get u (x, 0) = f (x) = ∞∑ n=1 cn sin (nπx l ) which provides cn through cn = 2 l ∫ l 0 f (x) sin (nπx l ) dx. so, we conclude that the solution of m-fractional heat equation eq.(4.1), satisfying the conditions eq.(4.2), is given by uβ (x,t) = ∞∑ n=1 sin (nπx l ) exp ( −γ (β + 1) (nπ l )2 k α tα )( 2 l ∫ l 0 f (x) sin (nπx l ) dx ) . (4.11) taking the limit β → 1 in the last equation and using eq.(2.2), we have u (x,t) = ∞∑ n=1 sin (nπx l ) exp ( − (nπ l )2 k α tα )( 2 l ∫ l 0 f (x) sin (nπx l ) dx ) , (4.12) which is exactly the solution of the fractional heat equation proposed by çenesiz et al. [14]. int. j. anal. appl. 16 (1) (2018) 94 on the other hand, taking the limit β → 1 and α → 1, using eq.(2.2), we recover the solution of heat equation of integer order. u (x,t) = ∞∑ n=1 sin (nπx l ) exp ( − (nπ l )2 kt )( 2 l ∫ l 0 f (x) sin (nπx l ) dx ) . (4.13) next, we will present some plots by choosing values for the parameters β and α, to see the behavior of the solution presented in eq (2.2). the graphics were plotted using matlab 7:10 software (r2010a). for the elaboration of the following plots, we choose the function f(x) = 50x(1 − x) and for each fixed β, we vary the α parameter. figure 1. analytical solution of the m-fractional heat equation eq.(4.11). we consider the values β = 0.5, l = 1, k = 0.003 and at time t = 150. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 x u (x ,t ) α = 0.1 α = 0.6 α = 1.0 figure 2. analytical solution of the m-fractional heat equation eq.(4.11). we take the values β = 1.0, l = 1, k = 0.003 and at time t = 150. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 x u (x ,t ) α = 0.1 α = 0.6 α = 1.0 int. j. anal. appl. 16 (1) (2018) 95 figure 3. analytical solution of the m-fractional heat equation eq.(4.11). we chose the values β = 2.0, l = 1, k = 0.003 and at time t = 150. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 x u (x ,t ) α = 0.1 α = 0.6 α = 1.0 5. concluding remarks we introduced a new truncated m-fractional derivative type for α-differentiable functions and consequently its m-fractional integral, we obtained important results with respect to the properties of the integerorder derivative. for a class of α-differentiable functions, in the context of fractional derivatives, we conclude that this truncated m-fractional derivative type proposed here behaves well with respect to the classical properties of integer-order calculus. using the truncated mittag-leffler function, it was possible to introduce a truncated m-fractional derivative type associated with α-differentiable functions and consequently we obtained a very important relation with the fractional derivatives mentioned in the paper as seen in section 3. we conclude that the presented results contain as particular cases the derivatives proposed by khalil et al. [7], katugampola [8] and sousa and oliveira [9]. in addition, using our truncated m-fractional derivative type we performed and discussed the analytical solution of the heat equation. references [1] r. herrmann, fractional calculus: an introduction for physicists, world scientific publishing company, singapore, 2011. [2] a. a. kilbas, h. m. srivastava, j. j. trujillo, theory and applications of fractional differential equations. north-holland mathematics studies, elsevier, amsterdam, vol. 207, 2006. [3] i. podlubny, fractional differential equations, mathematics in science and engineering, academic press, san diego, vol. 198, 1999. [4] e. capelas de oliveira, j. a. tenreiro machado, a review of definitions for fractional derivatives and integral, math. probl. eng., 2014, (2014) (238459). int. j. anal. appl. 16 (1) (2018) 96 [5] u. n. katugampola, new fractional integral unifying six existing fractional integrals, arxiv.org/abs/1612.08596, (2016). [6] r. figueiredo camargo, e. capelas de oliveira, fractional calculus (in portuguese), editora livraria da f́ısica, são paulo, 2015. [7] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math, 264, (2014) 65–70. [8] u. n. katugampola, a new fractional derivative with classical properties, arxiv:1410.6535v2, (2014). [9] j. vanterler da c. sousa, e. capelas de oliveira, m-fractional derivative with classical properties, submitted, (2017). [10] r. gorenflo, a. a. kilbas, f. mainardi, s. v. rogosin, mittag-leffler functions, related topics and applications, springer, berlin, 2014. [11] a. a. kilbas, h. m.srivastava, j. j. trujillo, theory and applications of the fractional differential equations, vol. 204, elsevier, amsterdam, 2006. [12] m. rafikov, j. m. balthazar, optimal pest control problem in population dynamics, computational & applied mathematics 24 (1) (2005) 65–81. [13] o. s. iyiola, e. r. nwaeze, some new results on the new conformable fractional calculus with application using d’alembert approach, progr. fract. differ. appl. 2, (2016) 115–122. [14] y. cenesiz, a. kurt, the solutions of time and space conformable fractional heat equations with conformable fourier transform, acta univ. sapientiae, math. 7 (2) (2015) 130–140. 1. introduction 2. truncated m-fractional derivative type 3. relation with other fractional derivatives types 4. application 5. concluding remarks references int. j. anal. appl. (2022), 20:33 received: may 16, 2022. 2010 mathematics subject classification. 42c40, 65r10, 44a35. key words and phrases. continuous wavelet transform, q-bessel fourier transform, q-bessel operator. https://doi.org/10.28924/2291-8639-20-2022-33 © 2022 the author(s) issn: 2291-8639 1 the continuous wavelet transform for a q-bessel type operator c.p. pandey1,*, jyoti saikia2 department of mathematics, north eastern regional institute of science and technology, nirjuli, 791109, arunachal pradesh, india *corresponding author: drcppandey@gmail.com abstract. in this paper, we consider a differential operator  on )0, by accomplishing harmonic analysis tools with respect to the operator  we study some definitions and properties of q-bessel continuous wavelet transform. we also explore generalized q-bessel fourier transform and convolution product on  )0, associated with the operator  and finally a new continuous wavelet transform associated with q-bessel operator is constructed and investigated. 1. introduction for a function ( )2f l r , the wavelet transform with respect to the wavelet ( )2l r  is defined by ( ) ( ) ( ) 2 12 1 , 2 1 ( ) , , , 0w f f t t dt r          − =   (1.1) where, 2 1 1/ 2 2 , 1 1 ( ) . t t        −  − =     (1.2) https://doi.org/10.28924/2291-8639-20-2022-33 2 int. j. anal. appl. (2022), 20:33 translation 2  is defined by 2 2 2 ( ) ( ),t t r      = −  and dilation 1 d  is defined by 1 1/ 2 1 1 1 ( ) , 0 t d t       −   =     . we can write ( ) ( ) 2 1 2 1, t d t       = . (1.3) from above equations, we can say that wavelet transform of the function f on r is an integral transform and the dilated translate of  is the kernel. we can also express wavelet transform as the convolution: ( )( ) ( )( ) 12 1 , 2 , * o w f f g     = , (1.4) where, ( ) ( ).g t t= − since there is a special type of convolution for every integral transform, therefore one can define wavelet transform with respect to a integral transform using associated convolution. the concept of wavelet is a collection of function derived from a single function called mother wavelet, after that by applying the two operators known as translation and dilation we get a new type of continuous wavelet transform. here presently, we introduce a q-bessel operator [1] and [2]. ( ) ( ) ( ) ( )( )1 2 2, 2 1 ( ) 1 . v v q v f t f q t q f t q f qt t −  = − + + (1.5) the above q-bessel operator associated with q-bessel function by the eigenvalue equation. ( ) ( )2 2 2, , , .q v v vj x q j x q = − unlike the elementary functions such as trigonometric, exponential etc the bessel wavelets are related to special functions and jachkson introduced the concept of q-analysis at the beginning of the twentieth century. we have arranged this paper as follows: in section 2, we will review briefly the basics of q-bessel fourier transform, here we recall notations, some definitions of q-bessel fourier and inverse fourier transform and the preposition associated with other operators and convolution 3 int. j. anal. appl. (2022), 20:33 product. in section 3, some results of harmonic analysis with respect to q-bessel operator for the generalized q-bessel transform is collected and the definition and properties of convolution product is also discussed. to extend the classical theory of wavelets to the differential operator ,q   is the actual aim of this work. we define a generalized wavelet, which satisfy the below admissibility condition ( )( ) , 2 , , 0 0 . q q q g d c f g        =   (1.6) where ,q f  denotes the generalized q-bessel fourier transform related to operator given by ( )( ) ( ) ( ) , 2 2 1 , 0 , q q q f g c g t j t q t d t        +  =  ( ), , .q p qg l  +   with ( ) ( ) 2 2 2 , 2 2 ;1 , 1 ; q q q c q q q   +   = − and ( );j x q being the normalized bessel function of index  . starting with a single generalized wavelet g, a family of generalized wavelets is constructed by putting ( ) ( )( )   1 2 , , , , 0 , a b q b a q q g x a t g x a b  + + =      where ( ) 2 2 2 1 a n x g x g a a + +   =     and ,q b t  is generalized translation operators related to the differential operator ,q   . the continuous generalized q-bessel wavelet transform of a function  ( ),2, 0q qf l  +   at the scale q a +  and the position  0qb +   is defined by ( )( ) ( ) ( ) 2 1 , , , , 0 , . q g q qa b f a b c f x g x d x       + =  (1.7) in section 4, we develop a relationship between the generalized wavelet transforms and q-bessel continuous wavelet transforms. such a relationship helps us to build certain formulas for the generalized q-bessel continuous wavelet transform (cwt). 4 int. j. anal. appl. (2022), 20:33 in section 5, we study the intertwining operator q  to establish the continuous generalized qbessel wavelet transform in form of classical one. as a result, we got a new inversion formulas for dual operator t q  of q  . 2. preliminaries in the present section we recapitulate some facts about harmonic analysis related to the qbessel operator. we cite here, as briefly as possible, only those properties actually required for the discussion. throughout this section assume 1/ 2  − . let the space , ,q p l  , 1 p   denote the sets of real functions on q + for which ( ) 1/ 2 1 , , 0 , p p qq p f f x x d x    +   =       and ( ) , , . q q x f sup f x  +   =   the q-bessel fourier transform , ,,q n f  in [3] is defined for ,1,q f l   by ( )( ) ( ) ( )2 2 1, , 0 , , , q q q q f f c f t j t q t d t t        + + =   (2.1) where j  is normalized q-bessel function. ( ) ( ) ( ) ( ) ( ) 1 2 2 2 2 2 2 2 0 , 1 . ; , n n n n n n q j x q x q q q q   + + = = − (2.2) theorem 2.1 (i) the q-bessel fourier transform , ,2, ,2, : q q q f l l    → defines an isomorphism and for all functions ,2,q f l   , ( ) ( )2, , ,2,,2,, .q q qqf f f f f f  = = (2.3) (ii) if f , ( ),qf f ,1,ql  then ( ) ( )( ) ( ) ( )2, , 0 , , q n q f x f f j x q d        =  (2.4) for almost all , q x +   where 5 int. j. anal. appl. (2022), 20:33 ( ) ( ) ( )2 2 1 1 1 q q q q d d         − + + = + (2.5) (iii) for all ,1, ,2,q q f l l     we have ( ) ( ) 2 22 1 2 1 , 0 0 . q q q f f d f x x d x        + + =  (iv) the inverse transform is given by ( )( ) ( ) ( ) ( )1 2, , 0 , , q n q f g x g j x q d        − =  the q-bessel translation operators , , 0, q x x    is defined by ( )( ) ( ) ( ) 2 1, , 0 , , , q x q q f y f z d x y z z d z      + =  (2.6) where ( ) ( ) ( ) ( )2 2 2 2 2 1, , 0 , , , , , q q q d x y z c j xs q j ys q j zs q s d s        +   =      (2.7) the convolution product of q-bessel for two functions ,f g is defined as ( ) ( ) ( ) 2 1, , 0 , 0. q q q x q f g x c f y g y y d y x      +  =   (2.8) theorem 2.2 (i) let 1 p   and , ,q p f l   . then 0x  , , , ,q x q pl     and , , ,, , . q x q pq p f f     (ii) for , ,q p f l   , 1 p   , we have ( )( ) ( ) ( )( )2, ,, , , ,,, .q n q x q nf f j x q f f       = (iii) let , [1, )p r   such that 1 1 1 p r + = . if , ,q p f l   and , ,q p g l   , then for every 0x  we have ( ) ( ) ( ) ( )2 1 2 1, , 0 0 q x q q x q f y g y y d y f y g y y d y         + + =  (iv) for , , [1, )p r s   such that 1 1 1 1 p r s + − = . if , ,q p f l   and , ,q p g l   then 6 int. j. anal. appl. (2022), 20:33 , , , ,, , q q p q rq s f g f g     (v) for , ,q p f l   and , ,q p g l   we have ( ) ( ) ( ), ,, , ,, , ,, .q n q q n q nf f g f f f g   = definition 2.1 a function ,2,q g l   is a q-bessel wavelet of order  , if it satisfies the admissibility condition. ( )( ) 2 , , ,, 0 0 . q g q n d c f g        =   (2.9) definition 2.2 let  ( ),2, 0q qg l  +   be a q-bessel wavelet of order  . then continuous q-bessel wavelet transform is defined as follows ( )( ) ( ) ( )   2 1 , , , 0 , , , 0 , q g q q q qa b s f a b c f x g x d x a b      + + + =      (2.10) where ( ) ( ) 1 2 ,, , , q b a qa b g a g a b    + =   (2.11) 2 2 1 a x g g a a +   =     (2.12) the q-bessel continuous wavelet transform has been investigated in detail in [4] from which we see the following basis properties. theorem 2.3 let be  ( ),2, 0q qg l  +   be a q-bessel wavelet. then (i) for all  ( ),2, 0q qf l  +   , the plancherel formula we have ( ) ( )( ) 22 2 1 2 1 , 2 ,0 0 0 1 , q q q g q g d a f x x d x s f a b b d b c a        + + =  . (ii) for all  ( ),2, 0q qf l  +   , we have ( ) ( )( ) ( ) , 2 1 , , 2 , 0 0 , , q q q g q qa b g c d a f x s f a b g b d b x c a        + +   =        . 3. harmonic analysis associated with  and generalized fourier transform let m be the map defined by ( ) ( )2 .nmf x x f x= 7 int. j. anal. appl. (2022), 20:33 let , ,q p l  , 1 p   be the class of measurable functions f on [0, ) for which 1 , , , , , 2 . q p n q p n f m f   − + =   for  and x , put ( ) ( )2 2 2, 2, , . n n n x q x j x q     + = (3.1) where 2n j + is the normalized bessel function with index 2n + is given by equation (2.1). from [4] see the following properties. theorem 3.1 (i) ,n  possess the laplace type integral representation ( ) ( ) ( ) ( ) ( ) 1 2 2 2 2 2 , 0 , 1 : : cos : n n q q q c q x f t q xt q d t     = +  (3.2) when 1q − → and 1 2  −  where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2 2 12 2 2 2 2 2 1 2 0 2 1 ; 1 : , : , cos : 1 . 1 1 ( ; ); 2 2 n nq n n n n n q q x q q q c q f t q x q q x q qx q q       − + =   + − = = = −       +         (ii) ( )2, ,n q  satisfies the differential equation ( ) ( )2 2 2, , , ,, ,q n n nq q       = − . (3.3) (iii) for all  and x ( ) im2 2, , xn n q x e      . (3.4) definition 3.1 the generalized q-bessel fourier transform is defined for a function ,1, ,q n f l   is defined by ( )( ) ( ) ( )2 2 1, , 2 , 0 , q q n n q f f c f x x q x d x        +  + =  (3.5) by (3.1) and (3.5) we observe that 1 , , 2 , q q n f f m  −  + = (3.6) where , 2q n f + is the fourier-bessel transform of order 2n + . (ii) if ,1, ,q n f l   then ( ) ( ), 0 [0, )qf f c   and 8 int. j. anal. appl. (2022), 20:33 , , , 2 ,1, ,, , , ( ) q n q n q nq n f f b f    +  where , 2q n b + is given in [3]. theorem 3.2 let ,1, ,q n f l   such that ( ), ,1, 2q q nf f l  + . then for almost all 0x  , ( ) ( )( ) ( )2 2 1, 2 , , 0 , . q n q n q f x c f f x q d          + +  =  proof. by (3.1), (3.6) and theorem 2.1(ii) we have ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) 2 2 1 2 1 2 1 , 2 , , , 2 , 2 2 0 0 2 1 2 1 , 2 , 2 2 0 2 1 , , n q n q n q q n q n n q n q n q n n q n c f f x q d x c f m f j x d x m c f f j x d x m f x f x                           + − + +  + + +  − + + + + − = = = =    for all 0x  . theorem 3.3 (i) for every ,1, , , , ,q n q p n f l l     space where 2p  we have the plancherel formula ( )( ) ( )( ) ( ) 22 2 1 , 2 0 0 . q q q n f t t d t f f d       +  + =  (ii) the inverse of this transform is given by ( )( ) ( ) ( ) ( )1 2, , 2 0 , q n q n f g x g x q d         −  + =  . proof. (i) let ,1, , , , ,q n q p n f l l     . by (3.6) and theorem 2.1 (iii) we have ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( ) 22 1 , 2 , 2 2 0 0 2 1 2 4 1 0 2 2 1 0 , q q n q n q n n q q f f d f m f d m f x x d x f x x d x             −  + + +  − + +  + = = =     the proof of (ii) is standard. 4. generalized convolution product definition 4.1 the generalized translation operator , ,q x n t  is define by the relation 9 int. j. anal. appl. (2022), 20:33 2 2 1 , , , n n q x n q x t x m m    + − = (4.1) where 2 , n q x   + are the bessel translation operators of order 2n + . definition 4.2 define the generalized convolution product of two functions f and g on [0, ) by ( ) ( ) ( ) 2 1, 2 , , 0 # q q n q x n q f g x c t f y g y y d y     + + =  (4.2) where , 2q n c + is given by (1.6). from by (4.1) we have ( ) ( )1 1, 2# ,q q nf g m m f m g − − +  =    (4.3) where , 2q n+  is the bessel convolution. theorem 4.1 (i) let f be in ,1, , , 1 . q n l p     then 2 , , ,1, ,,1, , n q x n q nq n t x f    . (ii) for ,2, ,q n f l   , we have ( )( ) ( ) ( )( )2, , , , , ,,q q x n n q nf t f x q f f         = . (iii) if ,1, ,q n f l   and ,1, ,q n g l   then ( ) ( ) ( ) ( )2 1 2 1, , , , 0 0 . q x n q q x n q t f y g y y d y f y t g y y d y       + + =  (iv) for ,1, , , q n f g l   then ,1, , # q q n f g l   and ,1, , ,1, ,,1, , # q q n q nq n f g f g    . (v) for ,1, ,q n f l   and ,1, ,q n g l   we have ( )( ) ( )( ) ( )( ), , ,#q q q qf f g f f f g    = . proof. (i) by (4.1) and theorem 2.2(i) we have 2 2 1 , , , ,1, , ,1, , 2 2 1 , ,1, 2 2 1 ,1, 2 2 ,1, , . n n q x n q x q n q n n n q x q n n q n n q n t f x m m f x m f x m f x f           + − + − + − + = =  = 10 int. j. anal. appl. (2022), 20:33 (ii) by (3.1), (3.6), (4.1) and theorem 2.2(ii) we have ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) 1 2 2 2 1 , , , , 2 , 2 2 1 2 1 , 2 , 2 2 1 1 2 , 2 2 1 , , 2 2 , , , , . n n n q q x n q n q x n n n q n q x n n n q n n q n n q f t f f m x m f x m f m f x m j f m f x q f m f x q f f                          − + −  + − + − + − − + + − +  = = = = = (iii) by (4.1) and theorem 2.2(iii) we have ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 4 2 1 1 2 1 , , , 0 0 2 4 1 2 1 2 1 , 0 22 1 2 1 2 1 , 0 2 1 2 1 , , 0 2 1 , , 0 . n n n q x n q q x q n n n q x q nn n q x q n q x n q q x n q t f y g y y d y x y m f y m g y y d y x y m f y m g y y d y y m f y xy m g y y d y y m f y t g y y d y f y t g y y d y                  + + − − +  − + − +  − + − +  − +  + = = = = =       (iv) by (4.3) and theorem 2.2(iv) we have ( )1 ,1, , ,1, 2 1 1 ,1, 2 ,1, 2 ,1, , ,1, , # # . q q q n q n q n q n q n q n f g m f g m f m g f g       − + − − + +   = (v) by (3.6), (4.3) and theorem 2.2(v) we have ( )( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 1 , , , 2 1 1 1 , 1 1 , , , , # * # . q q q q n q q q q q q f f g f m m f m g f m m m f m g f m f f m g f f f g         − −   + − − −  − −      =    =   = = this concludes the proof. 11 int. j. anal. appl. (2022), 20:33 5. transmutation operators definition 5.1 for a bounded function f on [0, ) , define the integral transform q  by ( ) ( ) ( ) ( ) ( ) 1 2 2 2 0 1 : : , n q q f x q c q x f t q f xt d t   = +  (5.1) where ( )2:c q and ( )2:f t q is given theorem 3.1(i). remark 5.1 (i) for n=0, q  reduces to q-riemann liouville integral transform of order  given by ( )( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 0, 1 : : , 0 0 , 0 . n q q q c q x f t q f xt d t if x r f x f x     +  =   =  (ii) it is checked that 2 ,q n q m r   + = (5.2) (iii) from theorem 3.1(i) and (5.1) we have ( ) ( )( )( )2 2, , cos ,n qx q xt q x  = (5.3) definition 5.2 define the integral transform t q  for a differential function f on [0, ) by ( ) ( ) ( ) ( )2 2 21 : : qt q n qy d tx f y q c q f q f t t t      −   = +      remark 5.2 (i) for n=0, t q  reduces to q-weyl integral transform of order  given by ( )( ) ( ) ( ) ( ) 1 2 2 , 0 1 : : , 0 . q q y w f y q c q f q f t t d t y t       = +      (ii) it is seen that 1 2 , t q n q w m   − + = (5.4) theorem 5.1 (i) if ( ), [0, ),qf l dx  then , ,q q nf l   and ,, , , .q qq nf f   (ii) if ,1, ,q n f l   then ( ),1 [0, ), t q q f l dx   and ,1, ,,1 . t q q nq f f    (iii) for any ( ),1 [0, ),qf l dx  and ,1, ,q ng l  we have the duality relation 12 int. j. anal. appl. (2022), 20:33 ( ) ( ) ( ) ( )2 1 0 0 . t q q q q f x g x x d x f y g y d y      + =  (iv) for all ,1, ,q n f l   we have ( ) ( ), , , t q q c q f f f f  = (5.5) where ,q c f is the q-cosine fourier transform given by ( )( ) ( ) ( )2, 0 cos ; , 0. q c q f f f x x q d x    =  (v) let ,1, , , q n f g l   . then ( )# * ,t t tq q q qf g f g  = where * is the convolution product defined by ( ) ( ) ( ) ( ) ( ) 2 1 2 1 1 2 1 2 0 1 1 x q q q f f x f y f y y d y     −  + +  =  +  , with x   is a q-generalized translation given in details in [5]. (vi) let ,1, ,q n f l   and ( ), [0, ),qg l dx  . then ( ) ( )* #tq q q qf g f g  = . (5.6) proof. (i) by (5.1) and [5.2] we have , 2 , ,, , , , , , q q n q qq n q n f m r r f f     +    = =  (ii) by (5.1) and [5.4] we have 1 , ,1, ,,1 ,1, 2 t q q q nq q n f m r f    − +  = (iii) by (4.3), (5.2) we have ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) 2 1 1 2 4 1 2 , 0 0 1 2 , 0 0 . n q q n q q n q q t q q f x g x x d x r f x m g x x d x f y w m g y d y f y g y d y         + − + + +  − +  = = =     13 int. j. anal. appl. (2022), 20:33 (iv) by (3.6), (5.4) we have ( ) ( ) ( ) ( ) 1 , , 2 , 1 , 2 , . t q c q q c n q q n q f f f w m f f m f f f    − + − +  = = = (v) by (4.3), (5.4) we have ( ) ( ) ( ) ( ) ( ) 1 1 2 , , 2 1 1 2 , 2 , # . t q q n q q n n q n q t t q q f g w m f m g w m f w m g f g        − − + + − − + +  =    =  =  (vi) by (3.6), (4.3) ,(5.4) we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 , 2 1 , 2 2 , 1 2 , 2 , # . q q q n q q n n q n q n q t q q f g m m f m g m m f r g mr w m f g f g          − − + − + + − + +  =     =     =    =  this achieves the proof. 6. generalized wavelets definition 6.1 a generalized q-bessel wavelet is a function ,2, ,q n g l   satisfying the admissibility condition ( )( ) 2 , 0 0 . q g q d c f g       =   (6.1) remark 6.1 by (3.6) and (6.1), ,2, ,q n g l   is a generalized q-bessel wavelet if and only if, 1 m g − is a q-bessel wavelet of order 2n + , and we have ( )( ) 1 2 1 2 , 2 0 . q n g q n m g d c f m g c      −  − + + = = (6.2) note 6.1 for ,2, ,q n g l   where qa +  and  0qb +   we have ( ) ( )( )1/ 2, , , , ,a b n q b n ag x a t g x   = (6.3) where a g is given in (2.12) and ,q bt  are the generalized translation operators defined by (4.1). 14 int. j. anal. appl. (2022), 20:33 theorem 6.1 for all q a +  and  0qb +   we have ( ) ( ) ( ) ( ) 22 1 , , , , nn a b n a b g x bx m g x   + − = (6.4) proof. using (2.11), (4.1) and (6.3) we have ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 1/ 2 , , , , , 2 1/ 2 2 1 , 2 1/ 2 2 1 , 22 1 , , a b n q b n a n n a b a n n q b a nn q b g x a t g x bx a m g x bx a m g x bx m g x        + − + − + − = = = = which ends the proof. definition 6.2 let  ( ),2, , 0q n qg l  +   be a generalized a q-bessel wavelet. then for a function  ( ),2, , 0q n qf l  +   , the continuous generalized a q-bessel wavelet transform by ( )( ) ( ) ( )  2 1, , , 2 , , , 0 , , 0 , q g n q n a b n q q q f a b c f x g x x d x a b       + + + + =      (6.5) where ( ) ( )1/ 2, , , , ,a b n q b n ag x a t g   = and ( ) 2 2 1 / a g g x a a + = . it can also be written in the form ( )( ) ( )1/ 2, , , # ,q g n q af a b a f g b   = (6.6) where # q is the generalized convolution product given by (4.2). theorem 6.2 we have ( )( ) ( ) ( )( )1 2 2 1 , , , , , . n n q g n q m g f a b b s m f a b    − + − = (6.7) proof. from (2.10), (6.4) and (6.5) we deduce that ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )1 2 1 , , , 2 , , , 0 22 1 2 2 1 , 2 , 0 22 1 1 2 4 1 , 2 , 0 2 2 1 , , , , q g n q n a b n q nn n q n q a b nn n q n q a b n n q m g f a b c f x g x x d x c f x b m g x x x d x c b m f x m g x x d x b s m f a b             −  + +  + − + +  + − − + + + + − = = = =    15 int. j. anal. appl. (2022), 20:33 which concludes the proof. theorem 6.3 (plancherel formula) let  ( ),2, , 0q n qg l  +   be a generalized wavelet. for every  ( ),2, , 0q n qf l  +   we have the plancherel formula ( ) ( )( ) 22 2 1 2 1 , , 2 0 0 0 1 , q q q g n q g d a f x x d x f a b b d b c a        + + =  . proof. by (6.2) and theorem 2.1(i) we have ( )( ) ( )( ) ( ) ( ) 1 1 22 2 1 2 1 2 4 1 , , 2 2, 0 0 0 0 2 2 1 2 4 1 0 2 2 1 0 , , . q qn n q g n q qq m g n n qm g g q d a d a f a b b d b s m f a b b d b a a c m f x x d x c f x x d x         − −     + + − + +  + − + +  + = = =     theorem 6.4 (calderon’s formula) let ,2, ,q n g l   be a generalized wavelet, such that ( ), , q q f g    . then for ,2, ,q n f l   and 0    , the function ( ) ( )( ) ( ), 2 1, , , , , 2 0 1 , q q g n a b n q g d a f x f a b g x b d b c a         +  =  belongs to ,2, ,q n l  . proof. by (6.2), (6.4), (6.7) and theorem 2.1(ii) we have ( )( ) ( ) ( )( )( ) ( ) 1 1 2 1 1 2 4 12 2 1 2, , , , , , 2 2 0 0 , ,1 , . qn nn q qq m g q g n a b n q n g m g d a d a s m f a b m g b d bx f a b g x b d b a c a c f x           − −   + − − + + + +    = =   theorem 6.5 (inversion formula) let ,2, ,q n g l   be a generalized wavelet. if ,1, ,q n f l   and ( ), ,1, 2q q nf f l  + then we have ( ) ( )( ) ( ) 2 1, , , , , 2 0 0 1 , q q g n a b n q g d a f x f a b g x b d b c a       +   =       for 0x  . proof. by (6.2), (6.4), (6.7 we have 16 int. j. anal. appl. (2022), 20:33 ( )( ) ( ) ( )( )( )1 1 2 2 1 2 1 1 2 4 1 , , , , , 2 2 2, 0 0 0 0 1 , , , n q qn n q g n a b n q qn q m g g m g d a d ax f a b g x b d b s m f a b m g b d b c a c a        − −     + + − − + + +     =            the result shows from theorem 2.1(iii). 7. inversion of the intertwining operator t q  through the generalized wavelet transform to obtain inversion formulas or t q  involving generalized wavelets, we have to establish some preliminary lemmas. lemma 7.1 let ( ),1, , ,2, ,0 [0, [,q n q ng l l dx     such that ( ) ( ),1, , [0, [,c q nf g l dx  and satisfying 2n   + such that ( )( ) ( )ncf g o = (7.1) as 0 → . then , ,2, ,q g q n l    and ( )( ) ( )( ) ( )( ) 22 4 1 , 2 4 1 2 2 1 . n c q g cn n f f g        + + + +  + + = (7.2) proof. we have ( ) ( )( ) ( ) 0 2 cos . c g x f g x d     =  so by (5.3), ( ) ( ) ( ) ( )2 0 , q n g x h x d         + =  where ( ) ( )( ) ( )( ) 22 4 1 2 4 1 2 2 1 n cn n h f g       + + + +  + + = clearly, h ( ),1, , [0, [,q nl dx  . so by (7.2) and theorem 6.3 we have ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) 2 22 4 1 2 0 0 1 22 4 1 0 1 1 2 , , , , n n c n c h d m n f g d m n f g d m n i i                  − − − +  − − − =   = +    = +     17 int. j. anal. appl. (2022), 20:33 where ( ) ( )( ) 24 1 2 , 4 4 1 n m n n     + + − =  + + . by (7.1) there is a positive constant k such that ( ) 2 2 4 1 1 0 . 2 2 n k i k d n        − − −  =   − −  from the plancherel theorem for the cosine transform, it follows that ( )( ) ( )( ) ( ) 2 2 22 4 1 2 1 0 0 , 2 n c c i f g d f g d g x dx           − − − =  =     which achieves the proof. lemma 7.2 let ( ),1, , ,2, ,0 [0, [,q n q ng l l dx     such that ( ) ( ),1, , [0, [,c q nf g l dx  and satisfying 2 4 1n  + + such that ( )( ) ( )cf g o   = (7.3) as 0 → . then ,2, ,q q n g l    is a generalized wavelet and ( ) ( ), , , , [0, [,c q g q nf l dx   . proof. by (7.3) and lemma 7.1 , ,2, ,q q n g l    , ( ),q gf  is bounded and ( )( ) ( )2 4 1, 0. n q g f o as       − − −  = → hence q g satisfies the admissibility condition (6.1). the continuous wavelet transform on  )0, is defined by ( )( ) ( ) ( )( ), 0 1 , , q g b a w f a b f x g x dx a   =  (7.4) where 0, 0a b  and ( ),2, , [0, [,q ng l dx  is a classical wavelet on  )0, , i.e., satisfies the admissibility condition ( ) ( )( ) 2 0 0 . q c d c g f g      =   (7.5) remark 7.1 (ii) by (5.5), (6.1) and (7.5), ( )g d is a generalized wavelet, if and only if , t q g  is a wavelet and ( ), t q g g c c = . lemma 7.3 let g be as in lemma 7.2. then , , ,q p n f l    , p=1 or 2, we have 18 int. j. anal. appl. (2022), 20:33 ( )( ) ( )( ) ( ) , ,2 4 1 1 , , . q g t q q g qn f a b w f a b a      + +  =    proof. by (6.6) we have ( )( ) ( ) ( ) , ,2 4 1 1 , . q g q q gn a f a b f b a     + + =  but ( ) ( ), 2 1 q g q ana g a  = by (2.12) and (5.1). so by (5.6) and (7.4) we get ( )( ) ( ) ( ) ( ) ( )( ) ( ) , 2 4 1 2 4 1 ,2 4 1 1 , 1 1 , , q g q q an t q q an t q q g qn f a b f g b a f g b a w a b a           + + + + + +  =     =     =    which completes the proof. theorem 7.1 let g be as in lemma 7.2. then we have the following inversion formulas for t q  : (i) if ,1, ,q n f l   and ( ), ,1, 2q q nf f l  + then for almost all 0x  we have ( ) ( )( ) ( ) ( ) ( ) , 2 1 , , 2 4 2, 0 0 1 , . q g t q q g q q g na b da f x w f a b x b db c a         + + +    =         (ii) for ,1, , ,2, ,q n q n f l l     and 0    , the function ( ) ( )( ) ( ) ( ) ( ) , , 2 1 , , 2 4 2, 0 1 , q g t q q g q q g na b da f x w f a b x b db c a           + + +   =     satisfies , ,2, ,0, lim 0. q n f f    → → − = conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. 19 int. j. anal. appl. (2022), 20:33 references [1] l. dhaouadi and m. hleili, generalized q-bessel operator, bull. math. anal. appl. 7 (2015), 2037. [2] l. dhaouadi, s. islem, h. elmonser, q-bessel fourier transform and variation diminishing kernel, arxiv:1209.5088v2. (2012). https://doi.org/10.48550/arxiv.1209.5088. [3] l. dhaouadi, m.j. atia, jacobi operator, q-difference equation and orthogonal polynomials, arxiv:1211.0359v1. (2012). https://doi.org/10.48550/arxiv.1211.0359. [4] m.m. dixit, c.p. pandey, d. das, the continuous generalized wavelet transform associated with q-bessel operator, bol. soc. paran. mat. http://www.spm.uem.br/bspm/pdf/next/305.pdf. [5] k. trimeche, generalized harmonic analysis and wavelet packets, gordon and breach science publisher, amsterdam, (2001) [6] r.s. pathak, c.p. pandey, laguerre wavelet transforms, integral transforms and special functions. 20 (2009), 505–518. https://doi.org/10.1080/10652460802047809. [7] j. saikia, c.p. pandey, inversion formula for the wavelet transform associated with legendre transform, in: d. giri, r. buyya, s. ponnusamy, d. de, a. adamatzky, j.h. abawajy (eds.), proceedings of the sixth international conference on mathematics and computing, springer singapore, singapore, 2021: pp. 287–295. https://doi.org/10.1007/978-981-15-8061-1_23. [8] c. p. pandey, pranami phukan, continuous and discrete wavelet transforms associated with hermite transform, int. j. anal. appl. 18 (2020), 531-549. https://doi.org/10.28924/2291-8639-18-2020-531. international journal of analysis and applications volume 18, number 2 (2020), 277-303 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-277 fuzzy ideals on ordered ag-groupoids nasreen kausar∗, meshari alesemi, salahuddin department of mathematics, university of agriculture fsd-pakistan, department of mathematics, jazan university, jazan, kingdom of saudi arabia email addresses: kausar.nasreen57@gmail.com(kausar); malesemi@jazanu.edu.sa(alesemi); drsalah12@hotmail.com(salahuddin) ∗corresponding author: kausar.nasreen57@gmail.com abstract. in this paper, we define the concept of direct product of finite fuzzy normal subrings over nonassociative and non-commutative rings (la-ring) and investigate the some fundamental properties of direct product of fuzzy normal subrings. 1. introduction in 1972, a generalization of commutative semigroups has been established by kazim et al [12]. in ternary commutative law: abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law, that is (ab)c = (cb)a. this law (ab)c = (cb)a is called the left invertive law. a groupoid s is said to be a left almost semigroup (abbreviated as la-semigroup) if it satisfies the left invertive law : (ab)c = (cb)a. this structure is also known as abel-grassmann’s groupoid (abbreviated as ag-groupoid) in [22]. an ag-groupoid is a midway structure between an abelian semigroup and a groupoid. mushtaq et al [21], investigated the concept of ideals of ag-groupoids. in [4] (resp. [1]), a groupoid s is said to be medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)). in [12], an ag-groupoid is medial, but in general an ag-groupoid needs not to be received 2019-07-16; accepted 2019-08-27; published 2020-03-02. 2010 mathematics subject classification. 03f55, 08a72, 20n25. key words and phrases. fuzzy left (right, interior, quasi-, bi-, generalized bi-) ideals; regular (intra-regular) ordered aggroupoids. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 277 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-277 int. j. anal. appl. 18 (2) (2020) 278 paramedial. every ag-groupoid with left identity is paramedial by protic et al [22] and also satisfies a(bc) = b(ac), (ab)(cd) = (dc)(ba). in [13], if (s, ·,≤) is an ordered semigroup and ∅ 6= a ⊆ s, we define a subset of s as follows : (a] = {s ∈ s : s ≤ a for some a ∈ a}. a non-empty subset a of s is called a subsemigroup of s if a2 ⊆ a. a is called a left (resp. right) ideal of s if following hold (1) sa ⊆ a (resp. as ⊆ a). (2) if a ∈ a and b ∈ s such that b ≤ a implies b ∈ a. equivalent definition: a is called a left (resp. right) ideal of s if (a] ⊆ a and sa ⊆ a (resp. as ⊆ a). in [13, 15], an ordered semigroup s is said to be a regular if for every a ∈ s, there exists an element x ∈ s such that a ≤ axa. in [14, 15], an ordered semigroup s is said to be an intra-regular if for every a ∈ s there exist elements x,y ∈ s such that a ≤ xa2y. we will define the concept of fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals of an ordered ag-groupoid s. we will establish a study by discussing the different properties of such ideals. we will also characterize regular (resp. intra-regular, both regular and intra-regular) ordered ag-groupoids by the properties of fuzzy left (right, quasi-, bi-, generalized bi-) ideals. 2. fuzzy ideals on ordered ag-groupoids in [25] an ordered ag-groupoid s, is a partially ordered set, at the same time an ag-groupoid such that a ≤ b, implies ac ≤ bc and ca ≤ cb for all a,b,c ∈ s. two conditions are equivalent to the one condition (ca)d ≤ (cb)d for all a,b,c,d ∈ s. let s be an ordered ag-groupoid and ∅ 6= a ⊆ s, we define a subset (a] = {s ∈ s : s ≤ a for some a ∈ a} of s, obviously a ⊆ (a]. if a = {a}, then we write (a] instead of ({a}]. for ∅ 6= a,b ⊆ s, then ab = {ab | a ∈ a,b ∈ b}, ((a]] = (a], (a](b] ⊆ (ab], ((a](b]] = (ab], if a ⊆ b then (a] ⊆ (b], (a∩b] 6= (a] ∩ (b] in general. for ∅ 6= a ⊆ s. a is called an ag-subgroupoid of s if a2 ⊆ a. a is called a left (resp. right) ideal of s if the following hold (1) sa ⊆ a (resp. as ⊆ a). (2) if a ∈ a and b ∈ s such that b ≤ a implies b ∈ a. equivalent definition : a is called a left (resp. right) ideal of s if (a] ⊆ a and sa ⊆ a (resp. as ⊆ a). a is called an ideal of s if a is both a left ideal and a right ideal of s. in particular, if a and b are any types of ideals of s, then (a∩b] = (a] ∩ (b]. we denote by l(a),s(a),i(a) the left ideal, the right ideal and the ideal of s, respectively generated by a. we have l(a) = {s ∈ s : s ≤ a or s ≤ xa for some x ∈ s} = (a ∪ sa], s(a) = (a ∪ as], i(a) = (a∪sa∪as ∪ (sa)s]. first time, zadeh introduced the concept of fuzzy set in his classical paper [27] of 1965. this concept has provided a useful mathematical tool for describing the behavior of systems that are too complex to admit precise mathematical analysis by classical methods and tools. extensive applications of fuzzy set theory int. j. anal. appl. 18 (2) (2020) 279 have been found in various fields such as artificial intelligence, computer science, management science, expert systems, finite state machines, languages, robotics, coding theory and others. rosenfeld [23], was the first, who introduced the concept of fuzzy set in a group. the study of fuzzy set in semigroups was established by kuroki [18, 19]. he studied fuzzy ideals and fuzzy interior (resp. quasi-, bi-, generalized bi-, semiprime, semiprime quasi-) ideals of semigroups. a systematic exposition of fuzzy semigroups appeared by mordeson et al [20], where one can find the theoretical results on fuzzy semigroups and their use in fuzzy finite state machines and languages. fuzzy sets in ordered semigroups/ordered groupoids were first explored by kehayopulu et al [16, 17]. they also studied fuzzy ideals and fuzzy interior (resp. quasi-, bi-, generalized bi-) ideals in ordered semigroups. by a fuzzy subset µ of an ordered ag-groupoid s, we mean a function µ : s → [0, 1], the complement of µ is denoted by µ′, is a fuzzy subset of s given by µ′(x) = 1 −µ(x) for all x ∈ s. a fuzzy subset µ of s is called a fuzzy ag-subgroupoid of s if µ(xy) ≥ µ(x) ∧ µ(y) for all x,y ∈ s. µ is called a fuzzy left (resp. right) ideal of s if (1) µ(xy) ≥ µ(y) (resp. µ(xy) ≥ µ(x)). (2) x ≤ y, implies µ(x) ≥ µ(y) for all x,y ∈ s. µ is a fuzzy ideal of s if µ is both a fuzzy left ideal and a fuzzy right ideal of s. every fuzzy ideal (whether left, right, two-sided) is a fuzzy ag-subgroupoid of s but the converse is not true in general. a fuzzy subset µ of s is called a fuzzy interior ideal of s if (1) µ ((xy)z) ≥ µ (y) . (2) x ≤ y, implies µ(x) ≥ µ(y) for all x,y,z ∈ s. a fuzzy subset µ of s is called a fuzzy quasi-ideal of s if (1) (µ◦s)∩(s ◦µ) ⊆ µ. (2) x ≤ y, implies µ(x) ≥ µ(y) for all x,y ∈ s. a fuzzy ag-subgroupoid µ of s is called a fuzzy bi-ideal of s if (1) µ((xa)y) ≥ min{µ(x),µ(y)}. (2) x ≤ y, implies µ(x) ≥ µ(y) for all x,a,y ∈ s. a fuzzy subset µ of s is called a fuzzy generalized bi-ideal of s if (1) µ((xa)y) ≥ min{µ(x),µ(y)}. (2) x ≤ y, implies µ(x) ≥ µ(y) for all x,a,y ∈ s. every fuzzy bi-ideal of s is a fuzzy generalized bi-ideal of s. a fuzzy ideal µ of s is called a fuzzy idempotent of s if µ◦µ = µ. we denote by f(s), the set of all fuzzy subsets of s. we define an order relation ”⊆” on f(s) such that µ ⊆ γ if and only if µ(x) ≤ γ(x) for all x ∈ s. then (f(s),◦,⊆) is an ordered ag-groupoid. by the symbols µ∧γ and µ∨γ, we mean the following fuzzy subsets: (µ∧γ)(x) = min{µ(x),γ(x)} and (µ∨γ)(x) = max{µ(x),γ(x)}. for ∅ 6= a ⊆ s, the characteristic function of a is denoted by χa and defined by χa(a) =   1 if a ∈ a0 if a /∈ a an ordered ag-groupoid s can be considered a fuzzy subset of itself and we write s = χs, i.e., s(x) = χs(x) = 1 for all x ∈ s. this implies that s(x) = 1 for all x ∈ s. int. j. anal. appl. 18 (2) (2020) 280 let x ∈ s, we define a set ax = {(y,z) ∈ s ×s | x ≤ yz}. let µ and γ be two fuzzy subsets of s, then product of µ and γ is denoted by µ◦γ and defined by : (µ◦γ)(x) =   ∨(y,z)∈axmin{µ(y),γ(z)} if aa 6= ∅0 if aa = ∅ now we give the imperative properties of such ideals of an odered ag-groupoid s, which will be very helpful in later sections. lemma 2.1. let s be an ordered ag-groupoid. then the following properties hold. (1) (µ◦γ) ◦β = (β ◦γ) ◦µ, (2) (µ◦γ) ◦ (β ◦ δ) = (µ◦β) ◦ (γ ◦ δ) for all fuzzy subsets µ,γ,β and δ of s. proof. let µ,γ and β be fuzzy subsets of an ordered ag-groupoid s. we have to show that (µ◦γ) ◦ β = (β ◦γ) ◦µ. now ((µ◦γ) ◦β)(x) = ∨(y,z)∈axmin{(µ◦γ)(y),β(z)} = ∨(y,z)∈axmin{∨(s,t)∈aymin{µ(s),γ(t)},β(z)} = ∨((s,t),z)∈axmin{min{µ(s),γ(t)},β(z)} = ∨((s,t),z)∈axmin{(µ(s),γ(t)),β(z)} = ∨((z,t),s)∈axmin{(β(z),γ(t)),µ(s)} = ∨((z,t),s)∈axmin{min{β(z),γ(t)},µ(s)} = ∨(w,s)∈axmin{∨(z,t)∈awmin{β(z),γ(t)},µ(s)} = ∨(w,s)∈axmin{(β ◦γ)(w),µ(s)} = ((β ◦γ) ◦µ)(x) similarly, we can prove (2) . � proposition 2.1. let s be an ordered ag-groupoid with left identity e. then the following assertions hold. (1) µ◦ (γ ◦β) = γ ◦ (µ◦β) , (2) (µ◦γ) ◦ (β ◦ δ) = (δ ◦γ) ◦ (β ◦µ), (3) (µ◦γ) ◦ (β ◦ δ) = (δ ◦β) ◦ (γ ◦µ) for all fuzzy subsets µ,γ,β and δ of s. proof. same as lemma 2.1. � theorem 2.1. let a and b be two non-empty subsets of an ordered ag-groupoid s. then the following assertions hold. (1) if a ⊆ b then χa ⊆ χb. (2) χa ◦χb = χ(ab]. int. j. anal. appl. 18 (2) (2020) 281 (3) χa ∪χb = χa∪b. (4) χa ∩χb = χa∩b. proof. straight forward. � theorem 2.2. let a be a non-empty subset of an ordered ag-groupoid s. then the following properties hold. (1) a is an ag-subgroupoid of s if and only if χa is a fuzzy ag-subgroupoid of s. (2) a is a left (resp. right, two-sided) ideal of s if and only if χa is a fuzzy left (resp. right, two-sided) ideal of s. proof. (1) let a be an ag-subgroupoid of s and a,b ∈ s. if a,b ∈ a, then by definition χa(a) = 1 = χa(b). since ab ∈ a, a being an ag-subgroupoid of s, this implies that χa(ab) = 1. thus χa(ab) ≥ χa(a) ∧ χa (b) . similarly, we have χa(ab) ≥ χa(a) ∧ χa(b), when a,b /∈ a. hence χa is a fuzzy agsubgroupoid of s. conversely, suppose that χa is a fuzzy ag-subgroupoid of s and let a,b ∈ a. since χa(ab) ≥ χa(a) ∧ χa (b) = 1, χa being a fuzzy ag-subgroupoid of s. thus χa(ab) = 1, i.e., ab ∈ a. hence a is an agsubgroupoid of s. (2) let a be a left ideal of s and a,b ∈ s. if a,b ∈ a, then by definition χa(b) = 1. since ab ∈ a, a being a left ideal of s, this means that χa(ab) = 1. thus χa(ab) ≥ χa (b) . similarly, we have χa(ab) ≥ χa(b), when a,b /∈ a. therefore χa is a fuzzy left ideal of s. conversely, assume that χa is a fuzzy left ideal of s. let a,b ∈ a and z ∈ s. since χa(zb) ≥ χa (b) = 1, χa being a fuzzy left ideal of s. thus χa(zb) = 1, i.e., zb ∈ a. therefore a is a left ideal of s. � theorem 2.3. let µ be a fuzzy subset of an ordered ag-groupoid s. then the following assertions hold. (1) µ is a fuzzy ag-subgroupoid of s if and only if µ◦µ ⊆ µ. (2) µ is a fuzzy left (resp. right) ideal of s if and only if s ◦µ ⊆ µ (resp. µ◦s ⊆ µ). proof. (1) suppose that µ is a fuzzy ag-groupoid of s and x ∈ s. for µ◦µ ⊆ µ. if (µ◦µ)(x) = 0, then µ◦µ ⊆ µ, otherwise we have (µ◦µ)(x) = ∨(y,z)∈axmin{µ(y),µ(z)} ≤ ∨(y,z)∈axmin{µ(yz)} = µ(x). ⇒ µ◦µ ⊆ µ. int. j. anal. appl. 18 (2) (2020) 282 conversely, assume that µ◦µ ⊆ µ. let y,z ∈ s such that x ≤ yz. now µ(yz) ≥ µ(x) ≥ (µ◦µ)(x) = ∨(s,t)∈axmin{µ(s),µ(t)} ≥ µ(y) ∧µ(z). ⇒ µ(yz) ≥ µ(y) ∧µ(z). hence µ is a fuzzy ag-subgroupoid of s. (2) suppose that µ is a fuzzy left ideal of s and x ∈ s. if (s ◦µ)(x) = 0, then s ◦µ ⊆ µ, otherwise we have (s ◦µ)(x) = ∨(y,z)∈axmin{s(y),µ(z)} = ∨(y,z)∈axmin{1,µ(z)} = ∨(y,z)∈axµ(z) ≤ ∨(y,z)∈axµ(yz) = µ(x). ⇒ s ◦µ ⊆ µ. conversely, assume that s ◦µ ⊆ µ. let y,z ∈ s such that x ≤ yz. now µ(yz) ≥ µ(x) ≥ (s ◦µ)(x) = ∨(s,t)∈axmin{s(s),µ(t)} ≥ s(y) ∧µ(z) = 1 ∧µ(z) = µ(z). ⇒ µ (yz) ≥ µ(z). therefore µ is a fuzzy left ideal of s. similarly, we can prove (3) . � lemma 2.2. if µ and γ are two fuzzy ag-subgroupoids (resp. (left, right, two-sided) ideals) of an ordered ag-groupoid s, then µ∩γ is also a fuzzy ag-subgroupoid (resp. (left, right, two-sided) ideal) of s. int. j. anal. appl. 18 (2) (2020) 283 proof. let µ and γ be two fuzzy ag-subgroupoids of s. we have to show that µ ∩ γ is also a fuzzy agsubgroupoid of s. now (µ∩γ)(xy) = µ(xy) ∧γ(xy) ≥ {µ(x) ∧µ(y)}∧{γ(x) ∧γ(y)} = µ(x) ∧{µ(y) ∧γ(x)}∧γ(y) = µ(x) ∧{γ(x) ∧µ(y)}∧γ(y) = {µ(x) ∧γ(x)}∧{µ(y) ∧γ(y)} = (µ∩γ)(x) ∧ (µ∩γ)(y). hence µ∩γ is a fuzzy ag-subgroupoids of s. similarly, for ideals. � lemma 2.3. if µ and γ are two fuzzy ag-subgroupoids of an ordered ag-groupoid s, then µ◦γ is also a fuzzy ag-subgroupoid of s. proof. let µ and γ be two fuzzy ag-subgroupoids of s. we have to show that µ ◦ γ is also a fuzzy agsubgroupoid of s. now (µ ◦ γ)2 = (µ ◦ γ) ◦ (µ ◦ γ) = (µ ◦ µ) ◦ (γ ◦ γ) ⊆ µ ◦ γ. hence µ ◦ γ is a fuzzy ag-subgroupoid of s. � remark 2.1. if µ is a fuzzy ag-subgroupoid of an ordered ag-groupoid s, then µ ◦ µ is also a fuzzy ag-subgroupoid of s. lemma 2.4. let s be an ordered ag-groupoid with left identity e. then ss = s and es = s = se. proof. since ss ⊆ s and x = ex ∈ ss, i.e., ss = s. since e is a left identity of s, i.e., es = s. now se = (ss) e = (es)s = ss = s. � lemma 2.5. let s be an ordered ag-groupoid with left identity e. then every fuzzy right ideal of s is a fuzzy ideal of s. proof. let µ be a fuzzy right ideal of s and x,y ∈ s. now µ (xy) = µ ((ex) y) = µ ((yx) e) ≥ µ (yx) ≥ µ (y) . thus µ is a fuzzy ideal of s. � lemma 2.6. if µ and γ are two fuzzy left (resp. right) ideals of an ordered ag-groupoid s with left identity e, then µ◦γ is also a fuzzy left (resp. right) ideal of s. proof. let µ and γ be two fuzzy left ideals of s. we have to show that µ◦γ is also a fuzzy left ideal of s. now s◦(µ◦γ) = (s◦s)◦(µ◦γ) = (s◦µ)◦(s◦γ) ⊆ µ◦γ. hence µ◦γ is a fuzzy left ideal of s. similarly, for right ideals. � int. j. anal. appl. 18 (2) (2020) 284 remark 2.2. if µ is a fuzzy left (resp. right) ideal of an ordered ag-groupoid s with left identity e. then µ◦µ is a fuzzy ideal of s. lemma 2.7. if µ and γ are two fuzzy ideals of an ordered ag-groupoid s, then µ◦γ ⊆ µ∩γ. proof. let µ and γ be two fuzzy ideals of s and x ∈ s. if (µ◦γ)(x) = 0, then µ◦γ ⊆ µ∩γ, otherwise we have (µ◦γ) (x) = ∨(y,z)∈axmin{µ(y),γ(z)} = ∨(y,z)∈axmin{µ(yz),γ(yz)} = ∨(y,z)∈ax{µ(yz) ∧γ(yz)} = ∨(y,z)∈ax (µ∩γ)(yz) = (µ∩γ)(x). ⇒ µ◦γ ⊆ µ∩γ. � remark 2.3. if µ is a fuzzy ideal of an ordered ag-groupoid s, then µ◦µ ⊆ µ. lemma 2.8. let s be an ordered ag-groupoid. then µ◦γ ⊆ µ∩γ for every fuzzy right ideal µ and every fuzzy left ideal γ of s. proof. same as lemma 2.7. � theorem 2.4. let a be a non-empty subset of an ordered ag-groupoid s. then the following conditions are true. (1) a is an interior ideal of s if and only if χa is a fuzzy interior ideal of s. (2) a is a quasi-ideal of s if and only if χa is a fuzzy quasi-ideal of s. (3) a is a bi-ideal of s if and only if χa is a fuzzy bi-ideal of s. (4) a is a generalized bi-ideal of s if and only if χa is a fuzzy generalized bi-ideal of s. proof. (1) let a be an interior ideal of s and x,y,a ∈ s. if a ∈ a, then by definition χa(a) = 1. since (xa)y ∈ a, a being an interior ideal of s, this means that χa((xa)y) = 1. thus χa((xa)y) ≥ χa(a). similarly, we have χa((xa)y) ≥ χa(a), when a /∈ a. hence χa is a fuzzy interior ideal of s. conversely, suppose that χa is a fuzzy interior ideal of s. let x,y ∈ s and a ∈ a, so χa(a) = 1. since χa((xa)y) ≥ χa(a) = 1, χa being a fuzzy interior ideal of s. thus χa((xa)y) = 1, i.e., (xa)y ∈ a. hence a is an interior ideal of s. int. j. anal. appl. 18 (2) (2020) 285 (2) let a be a quasi-ideal of s. now (χa ◦s) ∩ (s ◦χa) = (χa ◦χs) ∩ (χs ◦χa) = χas ∩χsa = χas∩sa ⊆ χa, by the theorem 2.1 therefore χa is a fuzzy quasi-ideal of s. conversely, assume that χa is a fuzzy quasi-ideal of s. let x be an element of as ∩sa. now χa(x) ⊇ ((χa ◦s) ∩ (s ◦χa))(x) = min{(χa ◦s)(x), (s ◦χa)(x)} = min{(χa ◦χs)(x), (χs ◦χa)(x)} = min{χas(x),χsa(x)} = (χas ∩χsa)(x) = χas∩sa(x) = 1. this implies that x ∈ a, i.e., as ∩sa ⊆ a. therefore a is a quasi-ideal of s. (3) let a be a bi-ideal of s, this implies that χa is a fuzzy ag-subgroupoid of s by the theorem 2.2. let x,y,a ∈ s. if x,y ∈ a, then by definition χa(x) = 1 = χa(y). since (xa)y ∈ a, a being a bi-ideal of s, this means that χa((xa)y) = 1. thus χa((xa)y) ≥ χa(x)∧χa(y). similarly, we have χa((xa)y) ≥ χa(x)∧χa(y), when x,y /∈ a. hence χa is a fuzzy bi-ideal of s. conversely, suppose that χa is a fuzzy bi-ideal of s, this means that a is an ag-subgroupoid of s by the theorem 2.2. let a ∈ s and x,y ∈ a, so χa(x) = 1 = χa(y). since χa((xa)y) ≥ χa(x) ∧ χa(y) = 1, χa being a fuzzy interior ideal of s. thus χa((xa)y) = 1, i.e., (xa)y ∈ a. hence a is a bi-ideal of s. similarly, we can prove (4) . � theorem 2.5. let µ be a fuzzy subset of an ordered ag-groupoid s. then µ is a fuzzy interior ideal of s if and only if (s ◦µ) ◦s ⊆ µ. proof. suppose that µ is a fuzzy interior ideal of s and x ∈ s. if ((s ◦µ) ◦s)(x) = 0, then (s ◦µ) ◦s ⊆ µ, otherwise there exist a,b,c,d ∈ s such that x ≤ ab and a ≤ cd. since µ is a fuzzy interior ideal of s, this implies that µ((cd)b) ≥ µ(d). now ((s ◦µ) ◦s)(x) = ∨(a,b)∈axmin{(s ◦µ)(a),s(b)} = ∨(a,b)∈axmin{∨(c,d)∈aamin{s(c),µ(d)},s(b)} = ∨(a,b)∈axmin{∨(c,d)∈aamin{1,µ(d)}, 1} = ∨(a,b)∈ax{∨(c,d)∈aaµ(d)} = ∨((c,d),b)∈axµ(d) ≤∨((c,d),b)∈axµ((cd)b) = µ(x). ⇒ (s ◦µ) ◦s ⊆ µ. int. j. anal. appl. 18 (2) (2020) 286 conversely, assume that (s ◦µ) ◦s ⊆ µ and let y,z ∈ s such that a ≤ (xy)z. now µ((xy)z) ≥ µ(a) ≥ ((s ◦µ) ◦s)(a) = ∨(s,t)∈aamin{(s ◦µ)(s),s(t)} ≥ (s ◦µ)(xy) ∧s(z) = ∨(m,n)∈axymin{s(m),µ(n)}∧s(z) ≥ {s(x) ∧µ(y)}∧s(z) = 1 ∧µ(y) ∧ 1 = µ(y). ⇒ µ((xy)z) ≥ µ(y). therefore µ is a fuzzy interior ideal of s. � theorem 2.6. let µ be a fuzzy ag-subgroupoid of an ordered ag-groupoid s. then µ is a fuzzy bi-ideal of s if and only if (µ◦s) ◦µ ⊆ µ. proof. same as theorem 2.5. � theorem 2.7. let µ be a fuzzy subset of an ordered ag-groupoid s. then µ is a fuzzy generalized bi-ideal of s if and only if (µ◦s) ◦µ ⊆ µ. proof. same as theorem 2.5. � lemma 2.9. if µ and γ are two fuzzy bi(resp. generalized bi-, quasi-, interior) ideals of an ordered aggroupoid s, then µ∩γ is also a fuzzy bi(resp. generalized bi-, quasi-, interior) ideal of s. proof. let µ and γ be two fuzzy bi-ideals of s. this implies that µ and γ be two fuzzy ag-subgroupoids of s, then µ∩γ is also a fuzzy ag-subgroupoid of s. we have to show that (µ∩γ)((xa)y) ≥ (µ∩γ)(x)∧(µ∩γ)(y). now (µ∩γ)((xa)y) = µ((xa)y) ∧γ((xa)y) ≥ {µ(x) ∧µ(y)}∧{γ(x) ∧γ(y)} = µ(x) ∧{µ(y) ∧γ(x)}∧γ(y) = µ(x) ∧{γ(x) ∧µ(y)}∧γ(y) = {µ(x) ∧γ(x)}∧{µ(y) ∧γ(y)} = (µ∩γ)(x) ∧ (µ∩γ)(y). ⇒ (µ∩γ)((xa)y) ≥ (µ∩γ)(x) ∧ (µ∩γ)(y). int. j. anal. appl. 18 (2) (2020) 287 hence µ∩γ is a fuzzy bi-ideal of s. � lemma 2.10. if µ and γ are two fuzzy bi(resp. generalized bi-, interior) ideals of an ordered ag-groupoid s with left identity e, then µ◦γ is also a fuzzy bi(resp. generalized bi-, interior) ideal of s. proof. let µ and γ be two fuzzy bi-ideals of s. we have to show that µ◦γ is also a fuzzy bi-ideal of s. since µ and γ are fuzzy ag-subgroupoids of s, then µ◦γ is also a fuzzy ag-subgroupoid of s by the lemma 2.3. now ((µ◦γ) ◦s) ◦ (µ◦γ) = ((µ◦γ) ◦ (s ◦s)) ◦ (µ◦γ) = ((µ◦s) ◦ (γ ◦s)) ◦ (µ◦γ) = ((µ◦s) ◦µ) ◦ ((γ ◦s) ◦γ) ⊆ (µ◦γ). therefore µ◦γ is a fuzzy bi-ideal of s. � lemma 2.11. every fuzzy ideal of an ordered ag-groupoid s is a fuzzy interior ideal of s. the converse is not true in general. proof. straight forward. � proposition 2.2. let µ be a fuzzy subset of an ordered ag-groupoid s with left identity e. then µ is a fuzzy ideal of s if and only if µ is a fuzzy interior ideal of s. proof. let µ be a fuzzy interior ideal of s and x,y ∈ s. now µ(xy) = µ((ex)y) ≥ µ(x), thus µ is a fuzzy right ideal of s. hence µ is a fuzzy ideal of s by the lemma 2.5. converse is true by the lemma 2.11. � lemma 2.12. every fuzzy left (right, two-sided) ideal of an ordered ag-groupoid s is a fuzzy bi-ideal of s. the converse is not true in general. proof. suppose that µ is a fuzzy right ideal of s and x,y,z ∈ s. now µ ((xy)z) = µ (xy) ≥ µ (x) and µ((xy)z) = µ((zy)x) ≥ µ(zy) ≥ µ(z), this implies that µ((xy)z) ≥ µ(x) ∧ µ(z). hence µ is a fuzzy bi-ideal of s. � lemma 2.13. every fuzzy bi-ideal of an ordered ag-groupoid s is a fuzzy generalized bi-ideal of s. the converse is not true in general. proof. obvious. � lemma 2.14. every fuzzy left (right, two-sided) ideal of an ordered ag-groupoid s is a fuzzy quasi-ideal of s. the converse is not true in general. int. j. anal. appl. 18 (2) (2020) 288 proof. straight forward. � proposition 2.3. every fuzzy quasi-ideal of an ordered ag-groupoid s is a fuzzy ag-subgroupoid of s. proof. let µ be a fuzzy quasi-ideal of s. since µ◦µ ⊆ µ◦s and µ◦µ ⊆ s◦µ, i.e., µ◦µ ⊆ µ◦s∩s◦µ ⊆ µ. so µ is a fuzzy ag-subgroupoid of s. � proposition 2.4. let µ be a fuzzy right ideal and γ be a fuzzy left ideal of an ordered ag-groupoid s, respectively. then µ∩γ is a fuzzy quasi-ideal of s. proof. since ((µ∩γ) ◦s) ∩ (s ◦ (µ∩γ)) ⊆ (µ◦s) ∩ (s ◦γ) ⊆ µ∩γ. therefore µ∩γ is a fuzzy quasi-ideal of s. � lemma 2.15. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then every fuzzy quasi-ideal of s is a fuzzy bi-ideal of s. proof. let µ be a fuzzy quasi-ideal of s. since µ◦µ ⊆ µ by the proposition 2.3. now (µ◦s) ◦µ ⊆ (s ◦s) ◦µ ⊆ s ◦µ and (µ◦s) ◦µ ⊆ (µ◦s) ◦s = (µ◦s) ◦ (e◦s) = (µ◦e) ◦ (s ◦s) = (µ◦e) ◦s = µ◦s. ⇒ (µ◦s) ◦µ ⊆ µ◦s ∩s ◦µ ⊆ µ. hence µ is a fuzzy bi-ideal of s. � proposition 2.5. if µ and γ are two fuzzy quasi-ideals of an ordered ag-groupoid s with left identity e, such that (xe)s = xs for all x ∈ s. then µ◦γ is a fuzzy bi-ideal of s. proof. let µ and γ be two fuzzy quasi-ideals of s, this implies that µ and γ be two fuzzy bi-ideals of s, by the lemma 2.15. then µ◦γ is also a fuzzy bi-ideal of s by the lemma 2.10. � 3. regular ordered ag-groupoids an ordered ag-groupoid s will be called a regular if for every x ∈ s, there exists an element a ∈ s such that x ≤ (xa)x. equivalent definitions are as follows: (1) a ⊆ ((as)a] for every a ⊆ s. (2) x ∈ ((xs)x] for every x ∈ s. in this section, we characterize regular ordered ag-groupoids by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals. lemma 3.1. every fuzzy right ideal of a regular ordered ag-groupoid s is a fuzzy ideal of s. int. j. anal. appl. 18 (2) (2020) 289 proof. suppsoe that µ is a fuzzy right ideal of s. let x,y ∈ s, this implies that there exists an element a ∈ s, such that x ≤ (xa)x. thus µ(xy) ≥ µ(((xa)x)y) = µ((yx)(xa)) ≥ µ(yx) ≥ µ(y). hence µ is a fuzzy ideal of s. � lemma 3.2. every fuzzy ideal of a regular ordered ag-groupoid s is a fuzzy idempotent. proof. assume that µ is a fuzzy ideal of s and µ◦µ ⊆ µ. we have to show that µ ⊆ µ◦µ. let x ∈ s, this means that there exists an element a ∈ s such that x ≤ (xa)x. thus (µ◦µ)(x) = ∨(y,z)∈axmin{µ(y),µ(z)} ≥ µ(xa) ∧µ(x) ≥ µ(x) ∧µ(x) = µ(x). ⇒ µ ⊆ µ◦µ. therefore µ = µ◦µ. � remark 3.1. every fuzzy right ideal of a regular ordered ag-groupoid s is a fuzzy idempotent. lemma 3.3. let µ be a fuzzy subset of a regular ordered ag-groupoid s. then µ is a fuzzy ideal of s if and only if µ is a fuzzy interior ideal of s. proof. suppose that µ is a fuzzy interior ideal of s. let x,y ∈ s, then there exists an element a ∈ s, such that x ≤ (xa)x. thus µ(xy) ≥ µ(((xa)x)y) = µ((yx)(xa)) ≥ µ(x), i.e., µ is a fuzzy right ideal of s. so µ is a fuzzy ideal of s by the lemma 3.1. converse is true by the lemma 2.11. � remark 3.2. the concept of fuzzy (two-sided, interior) ideals coincides in regular ordered ag-groupoids. proposition 3.1. let s be a regular ordered ag-groupoid. then (µ◦s) ∩ (s ◦µ) = µ for every fuzzy right ideal µ of s. proof. assume that µ is a fuzzy right ideal of s. then (µ◦s) ∩ (s ◦µ) ⊆ µ, because every fuzzy right ideal of s is a fuzzy quasi-ideal of s by the lemma 2.14. let x ∈ s, this implies that there exists an element a ∈ s, such that x ≤ (xa)x. thus (µ◦s)(x) = ∨(y,z)∈axmin{µ(y),s(z)} ≥ µ(xa) ∧s(x) ≥ µ(x) ∧ 1 = µ(x). ⇒ µ ⊆ µ◦s. similarly, we have µ ⊆ s ◦µ, i.e., µ ⊆ (µ◦s) ∩ (s ◦µ). therefore (µ◦s) ∩ (s ◦µ) = µ. � lemma 3.4. let s be a regular ordered ag-groupoid. then µ◦γ = µ∩γ for every fuzzy right ideal µ and every fuzzy left ideal γ of s. int. j. anal. appl. 18 (2) (2020) 290 proof. since µ◦γ ⊆ µ∩γ for every fuzzy right ideal µ and every fuzzy left ideal γ of s by the lemma 2.8. let x ∈ s, this means that there exists an element a ∈ s such that x ≤ (xa)x. thus (µ◦γ)(x) = ∨(y,z)∈axmin{µ(y),γ(z)} ≥ µ(xa) ∧γ(x) ≥ µ(x) ∧γ(x) = (µ∩γ)(x). ⇒ µ∩γ ⊆ µ◦γ. hence µ◦γ = µ∩γ. � lemma 3.5. let s be an ordered ag-groupoid with left identity e and a ∈ s. then sa is a smallest left ideal of s containing a. proof. let x ∈ sa and s ∈ s, this implies that x = s1a, s1 ∈ s. thus sx = s(s1a) = (es)(s1a) = ((s1a)s)e = ((s1a)(es))e = ((s1e)(as))e = (e(as))(s1e) = (as)(s1e) = ((s1e)s)a ∈ sa. hence sx ∈ sa and (sa] ⊆ sa. now a = ea ∈ sa, so sa is a left ideal of s containing a. let i be another left ideal of s containing a. since sa ∈ i, because i is a left ideal of s. but sa ∈ sa, this means that sa ⊆ i. therefore sa is a smallest left ideal of s containing a. � lemma 3.6. let s be an ordered ag-groupoid with left identity e. then as is a left ideal of s. proof. straight forward. � proposition 3.2. let s be an ordered ag-groupoid with left identity e and a ∈ s. then as∪sa is a smallest right ideal of s containing a. proof. we have to show that as ∪sa is a smallest right ideal of s containing a. now (as ∪sa)s = (as)s ∪ (sa)s = (ss)a∪ (sa)(es) ⊆ sa∪ (se)(as) = sa∪s(as) = sa∪a(ss) ⊆ sa∪as = as ∪sa. thus (as ∪ sa)s ⊆ as ∪ sa and also (as ∪ sa] ⊆ as ∪ sa. therefore as ∪ sa is a right ideal of s. since a ∈ sa, i.e., a ∈ as ∪ sa. let i be another right ideal of s containing a. now as ∈ is ⊆ i and sa = (ss)a = (as)s ∈ (is)s ⊆ is ⊆ i, i.e., as ∪ sa ⊆ i. hence as ∪ sa is a smallest right ideal of s containing a. � int. j. anal. appl. 18 (2) (2020) 291 theorem 3.1. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is a regular. (2) µ∩γ = µ◦γ for every fuzzy right ideal µ and every fuzzy left ideal γ of s. (3) β = (β ◦s) ◦β for every fuzzy quasi-ideal β of s. proof. suppose that (1) holds and β be a fuzzy quasi-ideal of s. then (β ◦s) ◦β ⊆ β, because every fuzzy quasi-ideal of s is a fuzzy bi-ideal of s by the lemma 2.15. let x ∈ s, this implies that there exists an element a of s such that x ≤ (xa)x. thus ((β ◦s) ◦β)(x) = ∨(y,z)∈axmin{(β ◦s)(y),β(z)} ≥ (β ◦s)(xa) ∧β(x) = ∨(s,t)∈axamin{β(s),s(t)}∧β(x) ≥ β(x) ∧s(a) ∧β(x) = β(x). ⇒ β ⊆ (β ◦s) ◦β. therefore β = (β ◦s) ◦β, i.e., (1) implies (3) . assume that (3) holds. let µ be a fuzzy right ideal and γ be a fuzzy left ideal of s. this means that µ and γ be fuzzy quasi-ideals of s by the lemma 2.14, so µ∩γ be also a fuzzy quasi-ideal of s. then by our assumption, µ∩γ = ((µ∩γ)◦s)◦(µ∩γ) ⊆ (µ◦s)◦γ ⊆ µ◦γ, i.e., µ ∩ γ ⊆ µ ◦ γ. since µ ◦ γ ⊆ µ ∩ γ. hence µ ◦ γ = µ ∩ γ, i.e., (3) ⇒ (2). suppose that (2) is true and a ∈ s. then sa is a left ideal of s containing a by the lemma 3.5 and as ∪ sa is a right ideal of s containing a by the proposition 3.2. so χsa is a fuzzy left ideal and χas∪sa is a fuzzy right ideal of s, by the theorem 2.2. then by our supposition χas∪sa ∩χsa = χas∪sa ◦χsa, i.e., χ(as∪sa)∩sa = χ((as∪sa)sa] by the theorem 2.1. thus (as∪sa)∩sa = ((as∪sa)sa]. since a ∈ (as∪sa)∩sa, i.e., a ∈ ((as∪sa)sa], so a ∈ ((as)(sa) ∪ (sa)(sa)]. now (sa)(sa) = ((se)a)(sa) = ((ae)s)(sa) = (as)(sa). this implies that ((as)(sa) ∪ (sa)(sa)] = ((as)(sa) ∪ (as)(sa)] = ((as)(sa)]. thus a ∈ ((as)(sa)]. then a ≤ (ax)(ya) = ((ya)x)a = (((ey)a)x)a = (((ay)e)x)a = ((xe)(ay))a = (a((xe)y))a ∈ (as)a, for any x,y ∈ s. this means that a ∈ ((as)a], i.e., a is regular. hence s is a regular, i.e., (2) ⇒ (1) . � theorem 3.2. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is a regular. int. j. anal. appl. 18 (2) (2020) 292 (2 ) µ = (µ◦s) ◦µ for every fuzzy quasi-ideal µ of s. (3) β = (β ◦s) ◦β for every fuzzy bi-ideal β of s. (4) δ = (δ ◦s) ◦ δ for every fuzzy generalized bi-ideal δ of s. proof. (1) ⇒ (4), is obvious. since (4) ⇒ (3) , every fuzzy bi-ideal of s is a fuzzy generalized bi-ideal of s by the lemma 2.13. since (3) ⇒ (2) , every fuzzy quasi-ideal of s is a fuzzy bi-ideal of s by the lemma 2.15. (2) ⇒ (1) , by the theorem 3.1. � theorem 3.3. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is a regular. (2) µ∩γ = (µ◦γ) ◦µ for every fuzzy quasi-ideal µ and every fuzzy ideal γ of s. (3) β ∩γ = (β ◦γ) ◦β for every fuzzy bi-ideal β and every fuzzy ideal γ of s. (4) δ ∩γ = (δ ◦γ) ◦ δ for every fuzzy generalized bi-ideal δ and every fuzzy ideal γ of s. proof. suppose that (1) holds. let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of s. now (δ ◦γ) ◦ δ ⊆ (s ◦γ) ◦s ⊆ γ ◦s ⊆ γ and (δ ◦γ) ◦ δ ⊆ (δ ◦s) ◦ δ ⊆ δ, i.e., (δ ◦γ) ◦ δ ⊆ δ ∩γ. let x ∈ s, this means that there exists an element a ∈ s such that x ≤ (xa)x. now xa ≤ ((xa)x)a = (ax)(xa) = x((ax)a). thus ((δ ◦γ) ◦ δ)(x) = ∨(y,z)∈axmin{(δ ◦γ)(y),δ(z)} ≥ (δ ◦γ)(xa) ∧ δ(x) = ∨(s,t)∈axamin{δ(s),γ(t)}∧ δ(x) ≥ δ(x) ∧γ((ax)a) ∧ δ(x) ≥ δ(x) ∧γ(x) = (δ ∩γ)(x). ⇒ δ ∩γ ⊆ (δ ◦γ) ◦ δ. hence δ ∩ γ = (δ ◦ γ) ◦ δ, i.e., (1) ⇒ (4) . it is clear that (4) ⇒ (3) and (3) ⇒ (2). assume that (2) is true. then µ∩s = (µ◦s) ◦µ, where s itself is a fuzzy two-sided ideal, so µ = (µ◦s) ◦µ. therefore s is a regular by the theorem 3.1, i.e., (2) ⇒ (1) . � theorem 3.4. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is a regular. (2) µ∩γ ⊆ γ ◦µ for every fuzzy quasi-ideal µ and every fuzzy right ideal γ of s. (3) β ∩γ ⊆ γ ◦β for every fuzzy bi-ideal β and every fuzzy right ideal γ of s. (4) δ ∩γ ⊆ γ ◦ δ for every fuzzy generalized bi-ideal δ and every fuzzy right ideal γ of s. int. j. anal. appl. 18 (2) (2020) 293 proof. (1) ⇒ (4), is obvious. since (4) ⇒ (3) and (3) ⇒ (2) . suppose that (2) is true, this implies that γ ∩µ = µ∩γ ⊆ γ ◦µ, where µ is a fuzzy left ideal of s. since γ ◦µ ⊆ γ ∩µ, so γ ∩µ = γ ◦µ. hence s is a regular by the theorem 3.1, i.e., (2) ⇒ (1) . � theorem 3.5. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is a regular. (2) µ∩γ∩λ ⊆ (µ◦γ)◦λ for every fuzzy quasi-ideal µ, every fuzzy right ideal γ and every fuzzy left ideal λ of s. (3) β ∩ γ ∩ λ ⊆ (β ◦ γ) ◦ λ for every fuzzy bi-ideal β, every fuzzy right ideal γ and every fuzzy left ideal λ of s. (4) δ ∩γ ∩λ ⊆ (δ ◦γ) ◦λ for every fuzzy generalized bi-ideal δ, every fuzzy right ideal γ and every fuzzy left ideal λ of s. proof. suppose that (1) holds. let δ be a fuzzy generalized bi-ideal, γ be a fuzzy right ideal and λ be a fuzzy left ideal of s. let x ∈ s, this implies that there exists an element a ∈ s such that x ≤ (xa)x. now x ≤ (xa)x. xa ≤ ((xa)x)a = (ax)(xa) = x((ax)a). (ax)a ≤ (a((xa)x))a = ((xa)(ax))a = (a(ax))(xa) = x((a(ax))a) = x(((ea)(ax))a) = x(((xa)(ae))a) = x((((ae)a)x)a) = x((nx)a) = x((nx)(ea)) = x((ae)(xn)) = x(x((ae)n)) = x(xm). ⇒ xa ≤ x((ax)a) ≤ x(x(xm)) = (ex)(x(xm)) = ((xm)x)(xe). thus ((δ ◦γ) ◦λ)(x) = ∨(y,z)∈axmin{(δ ◦γ)(y),λ(z)} ≥ (δ ◦γ)(xa) ∧λ(x) = ∨(s,t)∈axamin{δ(s),γ(t)}∧λ(x) ≥ δ((xm)x) ∧γ(xe) ∧λ(x) ≥ δ(x) ∧γ(x) ∧λ(x) = (δ ∩γ ∩λ)(x). ⇒ δ ∩γ ∩λ ⊆ (δ ◦γ) ◦λ. int. j. anal. appl. 18 (2) (2020) 294 hence (1) ⇒ (4) . it is clear that (4) ⇒ (3) and (3) ⇒ (2) . assume that (2) holds. then µ ∩ s ∩ λ ⊆ (µ◦s) ◦λ, where µ is a right ideal of s, i.e., µ∩λ ⊆ µ◦λ. since µ◦λ ⊆ µ∩λ, so µ◦λ = µ∩λ. therefore s is a regular by the theorem 3.1, i.e., (2) ⇒ (1) . � 4. intra-regular ordered ag-groupoids an ordered ag-groupoid s will be called an intra-regular ordered ag-groupoid if for every x ∈ s there exist elements a,b ∈ s such that x ≤ (ax2)b. equivalent definitions are as follows: (1) a ⊆ ((sa2)s] for every a ⊆ s. (2) x ∈ ((sx2)s] for every x ∈ s. in this section, we characterize intra-regular ordered ag-groupoids by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals. lemma 4.1. every fuzzy left (right) ideal of an intra-regular ordered ag-groupoid s is a fuzzy ideal of s. proof. suppose that µ is a fuzzy right ideal of s. let x,y ∈ s, this implies that there exist elements a,b ∈ s, such that x ≤ (ax2)b. thus µ(xy) ≥ µ(((ax2)b)y) = µ((yb)(ax2)) ≥ µ(yb) ≥ µ(y). hence µ is a fuzzy ideal of s. � lemma 4.2. let s be an intra-regular ordered ag-groupoid with left identity e. then every fuzzy ideal of s is a fuzzy idempotent. proof. assume that µ is a fuzzy ideal of s and µ◦µ ⊆ µ. let x ∈ s, this means that there exist elements a,b ∈ s, such that x ≤ (ax2)b. now x ≤ (ax2)b = (a(xx))b = (x(ax))b = (x(ax))(eb) = (xe)((ax)b) = (ax)((xe)b). thus (µ◦µ)(x) = ∨(y,z)∈axmin{µ(y),µ(z)} ≥ µ(ax) ∧µ((xe)b) ≥ µ(x) ∧µ(x) = µ(x). ⇒ µ ⊆ µ◦µ. therefore µ = µ◦µ. � proposition 4.1. let µ be a fuzzy subset of an intra-regular ordered ag-groupoid s with left identity e. then µ is a fuzzy ideal of s if and only if µ is a fuzzy interior ideal of s. int. j. anal. appl. 18 (2) (2020) 295 proof. suppose that µ is a fuzzy interior ideal of s. let x,y ∈ s, then there exist elements a,b ∈ s, such that x ≤ (ax2)b. thus µ(xy) ≥ µ(((ax2)b)y) = µ((yb)(ax2)) = µ((yb)(a(xx))) = µ((yb)(x(ax))) = µ((yx)(b(ax))) ≥ µ(x). so µ is a fuzzy right ideal of s, hence µ is a fuzzy ideal of s by the lemma 4.1. converse is true by the lemma 2.11. � remark 4.1. the concept of fuzzy (two-sided, interior) ideals coincides in intra-regular ordered aggroupoids with left identity. lemma 4.3. let s be an intra-regular ordered ag-groupoid with left identity e. then γ ∩ µ ⊆ µ ◦ γ for every fuzzy left ideal µ and every fuzzy right ideal γ of s. proof. let µ be a fuzzy left ideal and γ be a fuzzy right ideal of s. let x ∈ s, this implies that there exist elements a,b ∈ s such that x ≤ (ax2)b. now x ≤ (ax2)b = (a(xx))b = (x(ax))b = (x(ax))(eb) = (xe)((ax)b) = (ax)((xe)b). thus (µ◦γ)(x) = ∨(y,z)∈axmin{µ(y),γ(z)} ≥ µ(ax) ∧γ((xe)b) ≥ µ(x) ∧γ(x) = γ(x) ∧µ(x) = (γ ∩µ)(x). ⇒ γ ∩µ ⊆ µ◦γ. � theorem 4.1. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is an intra-regular. (2) γ ∩µ ⊆ µ◦γ for every fuzzy left ideal µ and every fuzzy right ideal γ of s. proof. (1) ⇒ (2) is obvious by the lemma 4.3. suppose that (2) holds and a ∈ s. then sa is a left ideal of s containing a by the lemma 3.5 and as ∪ sa is a right ideal of s containing a by the proposition 3.2. so χsa is a fuzzy left ideal and χas∪sa is a fuzzy right ideal of s, by the theorem 2.2. by our int. j. anal. appl. 18 (2) (2020) 296 supposition χas∪sa ∩ χsa ⊆ χsa ◦ χas∪sa, i.e., χ(as∪sa)∩sa ⊆ χ((sa)(as∪sa)] by the theorem 2.1. thus (as ∪sa) ∩sa ⊆ (sa(as ∪sa)]. since a ∈ (as ∪sa) ∩sa, i.e., a ∈ (sa(as ∪sa)] = ((sa)(as) ∪ (sa)(sa)]. now (sa)(as) = (sa)((ea)(ss)) = (sa)((ss)(ae)) = (sa)(((ae)s)s) = (sa)((as)s) = (sa)((ss)a) = (sa)(sa). this implies that ((sa)(as) ∪ (sa)(sa)] = ((sa)(sa) ∪ (sa)(sa)] = ((sa)(sa)] = ((sa)a)s] = (((sa)(ea))s] = (((se)(aa))s] = ((sa2)s]. thus a ∈ (sa2)s, i.e., a is an intra-regular. hence s is an intra-regular, i.e., (2) ⇒ (1) . � theorem 4.2. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is an intra-regular. (2) µ∩γ = (µ◦γ) ◦µ for every fuzzy quasi-ideal µ and every fuzzy ideal γ of s. (3) β ∩γ = (β ◦γ) ◦β for every fuzzy bi-ideal β and every fuzzy ideal γ of s. (4) δ ∩γ = (δ ◦γ) ◦ δ for every fuzzy generalized bi-ideal δ and every fuzzy ideal γ of s. proof. suppose that (1) holds. let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of s. now (δ ◦γ) ◦δ ⊆ (s ◦γ) ◦s ⊆ γ ◦s ⊆ γ and (δ ◦γ) ◦δ ⊆ (δ ◦s) ◦δ ⊆ δ, thus (δ ◦γ) ◦δ ⊆ δ ∩γ. let x ∈ s, this implies that there exist elements a,b ∈ s such that x ≤ (ax2)b. now x ≤ (ax2)b = (a(xx))b = (x(ax))b = (b(ax))x. b(ax) ≤ b(a((ax2)b)) = b((ax2)(ab)) = b((ax2)c) = (ax2)(bc) = (ax2)d = (ax2)(ed) = (de)(x2a) = m(x2a) = x2(ma) = (xx)l = (lx)x = (lx)(ex) = (xe)(xl) = x((xe)l). int. j. anal. appl. 18 (2) (2020) 297 thus ((δ ◦γ) ◦ δ)(x) = ∨(y,z)∈axmin{(δ ◦γ)(y),δ(z)} ≥ (δ ◦γ)(b(ax)) ∧ δ(x) = ∨(s,t)∈ab(ax)min{δ(s),γ(t)}∧ δ(x) ≥ δ(x) ∧γ((xe)l) ∧ δ(x) ≥ δ(x) ∧γ(x) = (δ ∩γ)(x). ⇒ δ ∩γ ⊆ (δ ◦γ) ◦ δ. hence δ ∩ γ = (δ ◦ γ) ◦ δ, i.e., (1) ⇒ (4) . it is clear that (4) ⇒ (3) and (3) ⇒ (2) . assume that (2) is true. let µ be a fuzzy right ideal and γ be a fuzzy two-sided ideal of s. since every fuzzy right ideal of s is a fuzzy quasi-ideal of s by the lemma 2.14, so µ is a fuzzy quasi-ideal of s. by our assumption µ∩γ = (µ◦γ) ◦µ ⊆ (s ◦γ) ◦µ ⊆ γ ◦µ, i.e., µ∩γ ⊆ γ ◦µ. therefore s is an intra-regular by the theorem 4.1, i.e., (2) ⇒ (1) . � theorem 4.3. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is an intra-regular. (2) µ∩γ ⊆ γ ◦µ for every fuzzy quasi-ideal µ and every fuzzy left ideal γ of s. (3) β ∩γ ⊆ γ ◦β for every fuzzy bi-ideal β and every fuzzy left ideal γ of s. (4) δ ∩γ ⊆ γ ◦ δ for every fuzzy generalized bi-ideal δ and every fuzzy left ideal γ of s. proof. assume that (1) holds. let δ be a fuzzy generalized bi-ideal and γ be a fuzzy left ideal of s. let x ∈ s, this means that there exist elements a,b ∈ s such that x ≤ (ax2)b. now x ≤ (a(xx))b = (x(ax))b = (b(ax))x. thus (γ ◦ δ)(x) = ∨(y,z)∈axmin{γ(y),δ(z)} ≥ γ(b(ax)) ∧δ(x) ≥ γ(x) ∧ δ(x) = δ(x) ∧γ(x) = (δ ∩γ)(x). ⇒ δ ∩γ ⊆ γ ◦ δ. hence (1) implies (4) . it is clear that (4) ⇒ (3) and (3) ⇒ (2) . suppose that (2) holds. let µ be a fuzzy right ideal and γ be a fuzzy left ideal of s. since every fuzzy right ideal of s is a fuzzy quasi-ideal of s, this implies that µ is a fuzzy quasi-ideal of s. by our supposition, µ∩γ ⊆ γ ◦µ. thus s is an intra-regular by the theorem 4.1, i.e., (2) ⇒ (1) . � int. j. anal. appl. 18 (2) (2020) 298 theorem 4.4. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is an intra-regular. (2) µ∩γ∩λ ⊆ (γ◦µ)◦λ for every fuzzy quasi-ideal µ, every fuzzy left ideal γ and every fuzzy right ideal λ of s. (3) β ∩ γ ∩ λ ⊆ (γ ◦ β) ◦ λ for every fuzzy bi-ideal β, every fuzzy left ideal γ and every fuzzy right ideal λ of s. (4) δ ∩γ ∩λ ⊆ (γ ◦ δ) ◦λ for every fuzzy generalized bi-ideal δ, every fuzzy left ideal γ and every fuzzy right ideal λ of s. proof. suppose that (1) holds. let δ be a fuzzy generalized bi-ideal, γ be a fuzzy left ideal and λ be a fuzzy right ideal of s. let x ∈ s, then there exist elements a,b ∈ s such that x ≤ (ax2)b. now x ≤ (a(xx))b = (x(ax))b = (b(ax))x. b(ax) ≤ b(a((ax2)b)) = b((ax2)(ab)) = b((ax2)c) = (ax2)(bc) = (ax2)d = (a(xx))d = (x(ax))d = (d(ax))x. thus ((γ ◦ δ) ◦λ)(x) = ∨(y,z)∈axmin{(γ ◦ δ)(y),λ(z)} ≥ (γ ◦ δ)(b(ax)) ∧λ(x) = ∨(s,t)∈ab(ax)min{γ(s),δ(t)}∧λ(x) ≥ γ(d(ax)) ∧δ(x) ∧λ(x) = γ(x) ∧δ(x) ∧λ(x) = (γ ∩ δ ∩λ)(x). ⇒ γ ∩δ ∩λ ⊆ (γ ◦ δ) ◦λ. thus (1) implies (4) . since (4) ⇒ (3) and (3) ⇒ (2) . assume that (2) holds. then µ∩s∩λ ⊆ (s◦µ)◦λ, where µ is a fuzzy left ideal of s, i.e., µ∩λ ⊆ µ◦λ. thus s is an intra-regular, i.e., (2) ⇒ (1) . � 5. regular and intra-regular ordered ag-groupoids in this section, we characterize both regular and intra-regular ordered ag-groupoid by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals. theorem 5.1. let s be an odered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. int. j. anal. appl. 18 (2) (2020) 299 (1) s is both a regular and an intra-regular. (2) µ◦µ = µ for every fuzzy bi-ideal µ of s. (3) µ1 ∩µ2 = (µ1 ◦µ2) ∩ (µ2 ◦µ1) for all fuzzy bi-ideals µ1 and µ2 of s. proof. suppose that (1) holds. let µ be a fuzzy bi-ideal of s and µ ◦ µ ⊆ µ. let x ∈ s, this implies that there exists an element a ∈ s such that x ≤ (xa)x, also there exist elements a,b ∈ s such that x ≤ (ax2)b. now x ≤ (xa)x xa ≤ ((ax2)b)a = (ab)(ax2) = c(a(xx)) = c(x(ax)) = x(c(ax)) = x((ec)(ax)) = x((xa)(ce)) = x((xa)d) = x((da)x) = x(lx) = l(xx) = (el)(xx) = (xx)(le) = (xx)m = (mx)x. mx ≤ m((ax2)b) = (ax2)(mb) = (a(xx))n = (x(ax))n = (x(ax))(en) = (xe)((ax)n) = (xe)((ax)(en)) = (xe)((ae)(xn)) = (xe)(x((ae)n)) = (xe)(xu) = x((xe)u) = xw. ⇒ xa ≤ (mx)x = (xw)x. thus (µ◦µ)(x) = ∨(y,z)∈axmin{µ(y),µ(z)} ≥ µ((xw)x) ∧µ(x) ≥ µ(x) ∧µ(x) ∧µ(x) = µ(x). ⇒ µ ⊆ µ◦µ. hence µ = µ◦µ, i.e., (1) implies (2) . assume that (2) is true. let µ1 and µ2 be two fuzzy bi-ideals of s, then µ1∩µ2 and µ1◦µ2 be also fuzzy bi-ideals of s. by our assumption µ1∩µ2 = (µ1∩µ2)◦(µ1∩µ2) ⊆ µ1◦µ2 and µ1 ∩µ2 = (µ1 ∩µ2) ◦ (µ1 ∩µ2) ⊆ µ2 ◦µ1, this implies that µ1 ∩µ2 ⊆ (µ1 ◦µ2) ∩ (µ2 ◦µ1). again by our int. j. anal. appl. 18 (2) (2020) 300 supposition (µ1 ◦µ2) ∩ (µ2 ◦µ1) = ((µ1 ◦µ2) ∩ (µ2 ◦µ1)) ◦ ((µ1 ◦µ2) ∩ (µ2 ◦µ1)) ⊆ (µ1 ◦µ2) ◦ (µ2 ◦µ1) ⊆ (µ1 ◦s) ◦ (s ◦µ1) = ((s ◦µ1) ◦s) ◦µ1 = (((s ◦e) ◦µ1) ◦s) ◦µ1 = (((µ1 ◦e) ◦s) ◦s) ◦µ1 = ((µ1 ◦s) ◦s) ◦µ1 = ((s ◦s) ◦µ1) ◦µ1 = (s ◦µ1) ◦µ1 = ((s ◦e) ◦µ1) ◦µ1 = ((µ1 ◦e) ◦s) ◦µ1 = (µ1 ◦s) ◦µ1 ⊆ µ1. ⇒ (µ1 ◦µ2) ∩ (µ2 ◦µ1) ⊆ µ1. similarly, we have (µ1 ◦ µ2) ∩ (µ2 ◦ µ1) ⊆ µ2, thus (µ1 ◦ µ2) ∩ (µ2 ◦ µ1) ⊆ µ1 ∩ µ2. hence µ1 ∩ µ2 = (µ1 ◦µ2) ∩ (µ2 ◦µ1), i.e., (2) ⇒ (3) . suppose that (3) holds. let µ be a fuzzy right ideal and γ be a fuzzy ideal of s, then µ and γ be also fuzzy bi-ideals of s, because every fuzzy right ideal and fuzzy ideal of s is a fuzzy bi-ideal of s. by our supposition µ∩γ = (µ◦γ)∩(γ◦µ), this implies that µ∩γ ⊆ (µ◦γ)∩(γ◦µ), i.e., µ∩γ ⊆ µ◦γ and µ∩γ ⊆ γ◦µ, where γ is also a fuzzy left ideal of s. since µ◦γ ⊆ µ∩γ, thus µ∩γ = µ◦γ and µ∩γ ⊆ γ ◦µ. hence s is both a regular and an intra-regular, i.e., (3) ⇒ (1) . � theorem 5.2. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is both a regular and an intra-regular. (2) every fuzzy quasi-ideal of s is a fuzzy idempotent. proof. suppose that s is both a regular and an intra-regular. let µ be a fuzzy quasi-ideal of s. then µ be a fuzzy bi-ideal of s and µ ◦ µ ⊆ µ. let x ∈ s, this means that there exists an element a ∈ s such that x ≤ (xa)x, and also there exist elements a,b ∈ s such that x ≤ (ax2)b. since x ≤ (xa)x = ((xw)x)x by the theorem 5.1. thus (µ◦µ)(x) = ∨(y,z)∈axmin{µ(y),µ(z)} ≥ µ((xw)x) ∧µ(x) ≥ µ(x) ∧µ(x) ∧µ(x) = µ(x). ⇒ µ ⊆ µ◦µ. hence µ = µ◦µ. conversely, assume that every fuzzy quasi-ideal of s is a fuzzy idempotent. let a ∈ s, then sa is a left ideal of s containing a by the lemma 3.5.this implies that sa is a quasi-ideal of s, so int. j. anal. appl. 18 (2) (2020) 301 χsa is a fuzzy quasi-ideal of s by the theorem 2.4. by our assumption χsa = χsa ◦χsa = χ((sa)(sa)], i.e., sa = ((sa)(sa)]. since a ∈ sa, i.e., a ∈ ((sa)(sa)]. thus a is both a regular and an intra-regular by the theorems 3.1 and 4.1, respectively. hence s is both a regular and an intra-regular. � theorem 5.3. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is both a regular and an intra-regular. (2) µ∩γ ⊆ µ◦γ for all fuzzy quasi-ideals µ and γ of s. (3) µ∩γ ⊆ µ◦γ for every fuzzy quasi-ideal µ and every fuzzy bi-ideal γ of s. (4) µ∩γ ⊆ µ◦γ for every fuzzy bi-ideal µ and every fuzzy quasi-ideal γ of s. (5) µ∩γ ⊆ µ◦γ for all fuzzy bi-ideals µ and γ of s. (6) µ∩γ ⊆ µ◦γ for every fuzzy bi-ideal µ and every fuzzy generalized bi-ideal γ of s. (7) µ∩γ ⊆ µ◦γ for every fuzzy generalized bi-ideal µ and every fuzzy quasi-ideal γ of s. (8) µ∩γ ⊆ µ◦γ for every fuzzy generalized bi-ideal µ and every fuzzy bi-ideal γ of s. (9) µ∩γ ⊆ µ◦γ for all fuzzy generalized bi-ideals µ and γ of s. proof. suppose that (1) holds. assume that µ and γ be two fuzzy generalized bi-ideals of s. let x ∈ s, this implies that there exists an element a ∈ s such that x ≤ (xa)x, and also there exist elements a,b ∈ s such that x ≤ (ax2)b. since x ≤ (xa)x = ((xw)x)x by the theorem 5.1. thus (µ◦γ)(x) = ∨(y,z)∈axmin{µ(y),γ(z)} ≥ µ((xw)x) ∧γ(x) ≥ µ(x) ∧µ(x) ∧γ(x) = (µ∩γ)(x). ⇒ µ∩γ ⊆ µ◦γ. hence (1) ⇒ (9) . it is clear that (9) ⇒ (8) ⇒ (7) ⇒ (4) ⇒ (2) and (9) ⇒ (6) ⇒ (5) ⇒ (3) . assume that (2) holds. let µ be a fuzzy right ideal and γ be a fuzzy left ideal of s. since every fuzzy right ideal and every fuzzy left ideal of s is a fuzzy quasi-ideal of s by the lemma 2.14. by our assumption, µ∩γ ⊆ µ◦γ. since µ◦γ ⊆ µ∩γ, so µ∩γ = µ◦γ, i.e., s is a regular. again by our assumption, µ∩γ = γ ∩µ ⊆ γ ◦µ, i.e., s is an intra-regular. hence s is both a regular and an intra-regular, i.e., (2) ⇒ (1) . in similar way, we can prove that (3) ⇒ (1) . � theorem 5.4. let s be an ordered ag-groupoid with left identity e, such that (xe)s = xs for all x ∈ s. then the following conditions are equivalent. (1) s is both a regular and an intra-regular. (2) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy right ideal µ and every fuzzy left ideal γ of s. int. j. anal. appl. 18 (2) (2020) 302 (3) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy right ideal µ and every fuzzy quasi-ideal γ of s. (4) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy right ideal µ and every fuzzy bi-ideal γ of s. (5) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy right ideal µ and every fuzzy generalized bi-ideal γ of s. (6) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy left ideal µ and every fuzzy quasi-ideal γ of s. (7) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy left ideal µ and every fuzzy bi-ideal γ of s. (8) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy left ideal µ and every fuzzy generalized bi-ideal γ of s. (9) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for all fuzzy quasi-ideals µ and γ of s. (10) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy quasi-ideal µ and every fuzzy bi-ideal γ of s. (11) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy quasi-ideal µ and every fuzzy generalized bi-ideal γ of s. (12) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for all fuzzy bi-ideals µ and γ of s. (13) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for every fuzzy bi-ideal µ and every fuzzy generalized bi-ideal γ of s. (14) µ∩γ ⊆ (µ◦γ) ∩ (γ ◦µ) for all fuzzy generalized bi-ideals µ and γ of s. proof. since µ∩γ ⊆ µ◦γ and µ∩γ ⊆ γ ◦µ for all fuzzy generalized bi-ideals µ and γ of s by the theorem 5.3. hence µ∩γ ⊆ (µ◦γ)∩(γ◦µ), i.e., (1) ⇒ (14) . it is clear that (14) ⇒ (13) ⇒ (12) ⇒ (9) ⇒ (6) ⇒ (2) , (14) ⇒ (11) ⇒ (10) ⇒ (9) , (14) ⇒ (8) ⇒ (7) ⇒ (6) and (14) ⇒ (5) ⇒ (4) ⇒ (3) ⇒ (2) . assume that (2) is true. let µ be a fuzzy right ideal and γ be a fuzzy left ideal of s. by our assumption µ∩γ ⊆ (µ◦γ)∩(γ◦µ) ⊆ γ◦µ, i.e., s is an intra-regular. again µ∩γ ⊆ (µ◦γ)∩(γ◦µ) ⊆ µ◦γ. since µ◦γ ⊆ µ∩γ, so µ∩γ = µ◦γ, i.e., s is a regular. hence (2) ⇒ (1) . � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. r. cho, j. jezek and t. kepka, paramedial groupoids, czech. math. j. 49(1999), 277-290. [2] a. lafi, dfig control: a fuzzy approach, int. j. adv. appl. sci. 6(2019), 107-116. [3] s. a. razak, d. mohamad, i. i. abdullah, a group decision making problem using hierarchical based fuzzy soft matrix approach, int. j. adv. appl. sci. 4(2017) 26-32. [4] j. jezek and t. kepka, medial groupoids, rozpravy csav rada mat. a prir. ved., 93/2, 1983, 93 pp. [5] t. kadir, in discrepancy between the traditional fuzzy logic and inductive, int. j. adv. appl. sci. 1(2014), 36-43. [6] n. kausar, m. waqar, characterizations of non-associative rings by their intuitionistic fuzzy bi-ideals, eur. j. pure appl. math. 12(2019), 226-250. [7] n. kausar, characterizations of non-associative ordered semigroups by the properties of their fuzzy ideals with thresholds (α,β], prikl. diskr. mat. 43(2019), 37-59. [8] n. kausar, direct product of finite intuitionistic fuzzy normal subrings over non-associative rings, eur. j. pure appl. math. 12(2)(2019), 622-648. [9] n. kausar, b. islam, m. javaid, s, amjad, u. ijaz, characterizations of non-associative rings by the properties of their fuzzy ideals, j. taibah univ. sci. 13(2019), 820-833. int. j. anal. appl. 18 (2) (2020) 303 [10] n. kausar, b. islam, s. amjad, m. waqar, intuitionistics fuzzy ideals with thresholds(α,β] in la-rings, eur. j. pure appl. math. 12(3)(2019), 906-943. [11] n. kausar, m. waqar, direct product of finite fuzzy normal subrings over non-associative rings, int. j. anal. appl. 17(5)(2019), 752-770. [12] m. a. kazim and m. naseeruddin, on almost semigroups, alig. bull. math. 2(1972), 1-7. [13] n. kehayopulu, on left regular ordered semigroups, math. japon. 35(1990), 1057-1060. [14] n. kehayopulu, on intra-regular ordered semigroups, semigroup forum, 46(1993), 271-278. [15] n. kehayopulu, on completely regular ordered semigroups, sci. math. 1(1998) 27-32. [16] n. kehayopulu and m. tsingelis, fuzzy sets in ordered groupoids, semigroup forum, 65(2002), 128-132. [17] n. kehayopulu and m. tsingelis, fuzzy bi-ideals in ordered semigroups, inform. sci. 171(2005), 13-28. [18] n. kuroki, fuzzy bi-ideals in semigroups, comment. math. univ. st. pauli, 28(1979), 17-21. [19] n. kuroki, on fuzzy semigroups, inform. sci. 53(1991), 203-236. [20] j. n. mordeson, d. s. malik and n. kuroki, fuzzy semigroups, springer, berlin, 2003. [21] q. mushtaq and s. m. yusuf, on la-semigroups, alig. bull. math. 8(1978), 65-70. [22] p. v. protic and n. stevanovic, ag-test and some general properties of abel-grassmann’s groupoids, pure math. appl. 6(1995), 371-383. [23] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35(1971), 512-517. [24] t. shah, n. kausar and i. rehman, intuitionistic fuzzy normal subrings over a non-associative ring, an. st. univ. ovidius constanta, 1(2012), 369-386. [25] t. shah, n. kausar, characterizations of non-associative ordered semigroups by their fuzzy bi-ideals, theor. computer sci. 529(2014), 96-110. [26] o. ozer, s. omran, on the generalized c*valued metric spaces related with banach fixed point theory, int. j. adv. appl. sci. 4(2017), 35-37. [27] l. a. zadeh, fuzzy sets, inform. control, 8(1965), 338-353. 1. introduction 2. fuzzy ideals on ordered ag-groupoids 3. regular ordered ag-groupoids 4. intra-regular ordered ag-groupoids 5. regular and intra-regular ordered ag-groupoids references international journal of analysis and applications volume 17, number 5 (2019), 864-878 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-864 weak non-associative structures of groups with applications shah nawaz1, muhammad gulistan1, naveed yaqoob2 and seifedine kadry3,∗ 1department of mathematics, hazara university, mansehra, kp, pakistan 2department of mathematics, college of science, al-zulfi majmaah university, al-zulfi saudi arabia 3department of mathematics and computer science, faculty of science, beirut arab university, p.o. box 11-5020, beirut 11072809, lebanon ∗corresponding author: s.kadry@bau.edu.lb abstract. inspiring by the weak symmetry occurring in the hv-left invertive structures, in this article we have introduce a new class of hv-la-groups which is a generalization of la-hypergroups. we have investigated different types of homomorphisms of hv-la-groups. moreover, we have constructed the hvla-groups. at the end a useful application of weak symmetry related with hv-left invertive structure has been presented using the chemical redox reaction. 1. introduction kazim and naseerudin [1] laid the idea of left almost semigroup (denoted by la-semigroups). they generalized some handy result of semigroup theory. afterwards, mushtaq [2] and other, went further in the detail of the structure and added various beneficial results to the theory of la-semigroup, see paper [3–9]. an la-semigroup is midway structure between commutative semigroup and groupoid. mushtaq and kamran [10] in 1996 proposed the idea of left almost groups. they proved that if g is left almost group and h is left received 2019-05-15; accepted 2019-06-19; published 2019-09-02. 2010 mathematics subject classification. 20n05. key words and phrases. hyperoperation; hv-la-groups; homomorphisms; chemical redox reaction. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 864 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-864 int. j. anal. appl. 17 (5) (2019) 865 almost subgroup then g/h is a set of left almost group. hyperstucture notion was initiated by marty in 1934, when he [11] elucidated hypergroup and embarked in analyze their properties and applied them to a group. handful papers and books have been written in this direction, see [12–14]. in 1990, in greece, thomas vougiouklis was organized a congress on hyperstructure and name it aha, but also there had been three more congresses organized in italy by corsini, on the same topic with different name. during that congress vougiouklis [17] introduced the concept of weak structure which is now named hv-structures. the impetus of hv-structures is that, the quotient of group with respect to invariant subgroup is a group. marty from 1934 notify that, the quotient of group with respect to any subgroup is a hypergroup. finally, the quotient of a group with respect to any partition (equivalently to any equivalence relation) is an hv-group. let (g,◦) be group and r be equivalence relation, then (g/r,◦) is an hv-group. several authors have studied different aspects of hv-structure. for instance, vougiouklis [18, 19], spratalis [20–24] and davvaz [25]. not long ago, in 2011, hila and dine [15] laid the idea of la-semihypergroups, which is generalization of semigroups, semihypergroups and la-semigroups. yaqoob, corsini and yousfzai [16] extended the work of hila and dine and characterized intera regular left almost hypergroups by their hyperideal by using pure left identity. the idea of hv-la-semigroups was given by gulistan et al., [26] in 2015. another motivation for the study of hyperstructure comes from chemical reaction, such as chain redox and dismutation reaction were provided different example of weak structures. for detail see papers [27–29]. this very article communicates the novel class of hv-la-groups which is a generalization of la-hypergroups. we have defined various types of homomorphisms. additionally, we have constructed the hv-la-groups, and presented the chemical example by making use of redox reactions. 2. preliminaries in this section we recall some helping material from different papers, like [16, 19, 26]. definition 2.1. [16] a hypergroupoid (h,◦) is called la-semihypergroup, if it satisfies the following law (x◦y) ◦z = (z ◦y) ◦x for all x,y,z ∈h example: [16] let h = z if we define x ◦ y = y − x + 3z, where x,y ∈ z. then (h,◦) becomes an la-semihypergroup. definition 2.2. [19] the hyperoperation ∗ : h×h −→ p∗(h) is called weakly associative hyperoperation (abbreviated as wass) if for any a,b,c ∈h (a∗ b) ∗ c∩a∗ (b∗ c) 6= ∅ definition 2.3. [19]the hyperoperation is weakly commutative (abbreviated as cow) if for any a,b ∈h a∗ b∩ b∗a 6= φ int. j. anal. appl. 17 (5) (2019) 866 definition 2.4. [26] let h be non-empty set and ∗ be hyperoperation on h. then (h,∗) is called an hv-la-semigroup, if it satisfies the weak left invertive law, for all x,y,z ∈h (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅ example: [26] let h = (0,∞) we define x ∗ y = { y x+1 , y x } where x,y ∈ h. then for all x,y,z ∈ h satisfies (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅. hence (h,∗) is an hv-la-semigroup. 3. hv-la-groups in this section, we present a new generalize class of non-associative hyperstructures namely hv-la-groups and provided different examples and some basic results. definition 3.1. let h be a non-empty set and ∗ be hyperoperation on h. then (h,∗) is called an hv-lagroup, if it satisfies the following axioms (i) (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅ for all x,y,z ∈h, (ii) h∗x = h = h∗x for all x ∈h. example: let h = {a,b,c} be a finite set. the hyperoperation ∗ is defined as follow ∗ a b c a a {a,b} {a,c} b c {h} {a,b} c b {a,c} {h} here all elements of h satisfy the weak left invertive law. also left invertive law is not hold in h, i.e. h = (a∗ b) ∗ c 6= (c∗ b) ∗a = {a,b} alike, associative law is not hold in h, i.e. h = (c∗ c) ∗a 6= c∗ (c∗a) = {a,c} . even, weak associative law is not true {b} = (b∗a) ∗a∩ b∗ (a∗a) = {c} = φ hence (h,∗) is an hv-la-group. int. j. anal. appl. 17 (5) (2019) 867 example: let h = {a,b,c,d} be a finite set and the hyperoperation is defined in the following table. ∗ a b c d a a b c d b c {a,c} {b,c} d c b {a,c} {a,c} d d d d d {a,b,c} as all elements of h satisfy the weak left invertive law but h do not satisfies the left invertive law, associative law and weak associative law i.e. {a,c} = (b∗ b) ∗ c 6= (c∗ b) ∗ b = {a,b,c} , and {a,c} = (b∗ b) ∗ c 6= b∗ (b∗ c) = {a,b,c} . also {b} = (b∗a) ∗a∩ b∗ (a∗a) = {c} = φ so (h,∗) is an hv-la-group. now we will discuss some very basic results related with hv-la-groups. lemma 3.1. if (h,∗) is an hv-la-group, then (a∗ b) ∗ (c∗d) ∩ (a∗ c) ∗ (b∗d) 6= φ, hold for all a,b,c,d ∈h. proof: let us consider x ∈ (a∗ b) ∗ (c∗d) = ((c∗d) ∗ b) ∗a = ((b∗d) ∗ c) ∗a = (a∗ c) ∗ (b∗d) . this implies that x ∈ (a∗ c) ∗ (b∗d). from this we can say that (a∗ b) ∗ (c∗d) ∩ (a∗ c) ∗ (b∗d) 6= φ, hold for all a,b,c,d ∈h. 2 proposition 3.1. let (h,◦) be an la-hypergroup with left identity e and non-empty set a, such that a ⊆ h. if (a ◦ (a ◦ x)) ◦ y ∩ (a ◦ (a ◦ y)) ◦ x 6= ∅ ∀x,y ∈ h and we define a hyperoperation a⊗r on h as xa⊗ry = (x◦y) ◦a, then (h,a ⊗ r) become an hv-la-group. int. j. anal. appl. 17 (5) (2019) 868 proof: let x,y,z ∈h, we have (xa⊗ry)a ⊗ rz = ((x◦y) ◦a)a ⊗ rz = (((x◦y) ◦a) ◦z) ◦a = ((z ◦a) ◦ (x◦y)) ◦a = (a◦ (a◦z)) ◦ (y ◦x) = y ◦ ((a◦ (a◦z)) ◦x) and on the other hand (za⊗ry)a ⊗ rx = ((z ◦y) ◦a)a ⊗ rx = (((z ◦y) ◦a) ◦z) ◦a = ((x◦a) ◦ (z ◦y)) ◦a = (a◦ (a◦x)) ◦ (y ◦z) = y ◦ ((a◦ (a◦x)) ◦z) but, since y◦((a◦(a◦z))◦x)∩y◦((a◦(a◦x))◦z) 6= ∅ ∀ for all x,y,z ∈h. it follows that (xa⊗ry)a ⊗ rz∩ (za⊗ry)a ⊗ rx 6= ∅. next, we have xa⊗rh = (x◦h) ◦a = h also ha ⊗ rx = (h◦x) ◦a = h. hence (h,a⊗r) becomes an hv-la-group. 2 next we defined some regular relations on hv-la-groups. definition 3.2. the equivalence relation ρ is called regular on right (on the left ), if for all x of h ( h is an hv-la-group), from aρb, it follows that (a◦x) − ρ (b◦x) , (x◦a) − ρ (x◦ b) respectively. lemma 3.2. the relation ρ = φ−1 ∗φ = {(x,y) ∈h1 ×h1 : φ(x) = φ(y)} is regular on h1. proof: the ρ is an equivalence relation on h1 obviously. we have to show that ρ is regular on h1. let x,y,z ∈ h1 such that xρy, this implies that φ(x) = φ(y) =⇒ φ(xz) = φ(yz) and φ(zx) = φ(zy). so (xz)ρ(yz). thus xρy =⇒ (xz)ρ(yz) and (zx)ρ(zy). hence ρ = φ−1 ∗φ = {(x,y) ∈h1 ×h1 : φ(x) = φ(y)} is regular on h1. 2 on hv-la-group h, we are concerned with equivalence relation for which the family of equivalence classes form an hv-la-group under the hyperoperation induced by that on h. for an equivalence relation ρ on h, we may use xρ, and x or ρ(x) to denote the equivalence class of x ∈ h. moreover, generally if a is a int. j. anal. appl. 17 (5) (2019) 869 non-empty subset of h then aρ = u{xρ | x ∈ a}. we let h/ρ denote the family {xρ | x ∈h} of class of ρ. the hyperoperation on h induces a hyperoperation ⊗ on h/ρ defined by xρ ⊗yρ = {zρ / z ∈ xρ ◦yρ} where x, y ∈h. the structure (h/ρ,⊗) is known as quotient structure. theorem 3.1. let (h,◦) be hv-la-group. then (h/ρ,⊗) is an hv-la-group iff (ρ (x) ◦ρ (y)) ◦ρ (z) ∩ (ρ (z) ◦ρ (y)) ◦ρ (x) 6= φ ∀x,y,z ∈h proof: in h/ρ, we have let u ∈ (ρ (x) ⊗ρ (y)) ⊗ρ (z) = {u/u ∈ x◦y}⊗z = {t ∈ u◦z,u ∈ x◦y} = {t ∈ (z ◦y) ◦x} =⇒ u ∈ (ρ (x) ⊗ρ (y)) ⊗ρ (z) this implies that (ρ (x) ⊗ρ (y)) ⊗ρ (z) ⊆ (ρ (z) ⊗ρ (y)) ⊗ρ (x) . so (ρ (x) ⊗ρ (y)) ⊗ρ (z) ∩ (ρ (z) ⊗ρ (y)) ⊗ ρ (x) 6= φ. since x ◦h = h = h ◦ x, =⇒ whence h/ρ ⊗ x = h/ρ = x ⊗h/ρ. hence (h/ρ,⊗) is an hv-la-group. 2 theorem 3.2. let (h,◦) be hv-la-group and ρ be equivalence relation on h. if ρ is a regular relation, then (h/ρ,⊗) is an hv-la-group. proof: first we show that ⊗ is a well defined on h/ρ, consider x = x1 and y = y1. we check that x⊗y = x1⊗y1. we have xρx1 and yρy1. since ρ is regular, it follows that (x◦y) ρ (x1 ◦y) , (x1 ◦y) ρ (x1 ◦y1) whence (x◦y) ρ (x1 ◦y1). this implies that for all n ∈ (x◦y) there exists n1 such that nρn1. which shows that n = n1. it follows that that x⊗y ⊆ x1 ⊗y1 and similarly we obtain converse. hence ⊗ is well defined . next we show weak left invertive property of ⊗. let x,y,z be arbitrary element in h/ρ and l ∈ (x⊗y) ⊗z. this implies that v ∈ x⊗y and l ∈ v⊗z. it means that v1 ∈ x◦y and l1 ∈ v◦z such that vρv1 and lρl1. since ρ is regular relation, it follows that there exists l2 ∈ v1◦z ⊆ (x◦y)◦z ⊆ (z ◦y)◦x ( since h is a hv-la-group) such that l1ρl2. from here we obtain that there exist l3 ∈ z ◦y such that l2 ∈ l3 ◦x. we have l = l1 = l2 ∈ l3 ⊗x ⊆ (z ⊗y) ⊗x =⇒ (x⊗y) ⊗z ⊆ (z ⊗y) ⊗x =⇒ (x⊗y) ⊗z ∩ (z ⊗y) ⊗x 6= φ. int. j. anal. appl. 17 (5) (2019) 870 finally we show the reproductive axiom since x◦h = h = h◦x whence x⊗h/ρ = h/ρ = h/ρ⊗x. hence (h/ρ,⊗) is a hv-la-group. 2 next we defined the homorphisms of hv-la-groups. definition 3.3. a mapping φ : h1 −→ h2 (where h1 and h2 are hv-la-group) is said to be good homomorphism if it satisfies the following property φ(xy) = φ(x)φ(y) ∀ x,y ∈h1 example: let h1 = {a,b,c} and h2 = {l,m,n}, be two hv-la-hypergroups with hyperoperation is defined in the following tables respectively, ∗ l m n l {l} {m} {n} m {n} {l,n} {m} n {m} {l,n} {l,n} and ◦ a b c a {a} {b} {c} b {c} {a,c} {b,c} c {b} {a,c} {a,c} the mapping f : h1 −→h2 is defined by f (a) = l , f (b) = m, f (c) = n. then assuredly f homomorphism is a strong homomorphism. if good homomorphism is 1 − 1 and onto is called isomorphism. if f is an isomorphism, then h1 and h2 are said to be isomorphic, which is denoted by h1 ∼= h2. definition 3.4. let (h1,◦) and (h2,∗) be two hv-la-hypergroups. the map f : h1 −→ h2 is called inclusion homomorphism if for all x,y ∈h1 satisfies the following property f(x◦y) ⊆ f(x) ∗f(y) int. j. anal. appl. 17 (5) (2019) 871 example: let h1 = {l,m,n} and h2 = {a,b,c} be two hv-lahypergroups with hyperoperations defined in the following table: ∗ l m n l {l} {m} {n} m {n} {l,m} {m} n {m} {n} {n,l} and ◦ a b c a {a} {b} {c} b {c} {h} {b} c {b} {c} {h} the mapping f : h1 −→ h2 is defined by f(l) = a , f(m) = b , f(n) = c. then clearly f is an inclusion homomorphism. definition 3.5. let (h1,◦) and (h2,∗) be two hv-la-hypergroup. the map f : h1 −→h2 is called weak homomorphism or hv homomorphism, if for all x,y ∈h1, the condition is hold f(x◦y) ∩f(x) ∗f(y) 6= φ example: let h1 = {l,m,n} and h2 = {a,b,c} are two finite sets, where (h1,∗) and (h2,◦) are hv-la-hypergroups, the hyperoperation is defined in following tables: ∗ l m n l {l} {m} {n} m {n} {l,m} {m} n {m} {n} {n,l} and ◦ a b c a a {a,b} {a,c} b c {h} {a,b} c b {a,c} {h} . the mapping f : h1 −→ h2 is defined by f(l) = a, f(m) = b, f(n) = c. then clearly f is a weak homomorphism or hv-la-homomorphism. theorem 3.3. let φ : h1 −→ h2 be good homomorphism of an hv-la groups. then there exist a monomorphism ψ : h1/ρ −→h2 such that imφ = imψ and diagram int. j. anal. appl. 17 (5) (2019) 872 commutes i.e. ψ ∗ρ• = φ where the mapping ρ• : h1 −→h1/ρ is defined by ρ•(x) = ρ(x) ∀x ∈h1. proof: let ψ : h1/ρ −→ h2 is defined as ψ(ρ(x)) = φ(x) ∀x ∈ h1 since φ : h1 −→ h2. first we show that ψ is well defined. for this let ρ(x1),ρ(x2) ∈h1/ρ such that ρ(x1) = ρ(x2) ρ•(x1) = ρ •(x2) ψ(ρ•(x1)) = ψ(ρ •(x2)) φ(x1) = φ(x2) ψ(ρ(x1)) = ψ(ρ(x2)). next we will show that ψ is one -one. for this let ψ(ρ(x1)), ψ(ρ(x2)) ∈h1/ρ ψ(ρ(x1)) = ψ(ρ(x2)) φ(x1) = φ(x2) ψ(ρ•(x1)) = ψ(ρ •(x2)) ρ•(x1) = ρ •(x2) ρ(x1) = ρ(x2). finally we show that ψ is homomorphism. let x,y ∈h1 we have ψ(ρ(x) ∗ρ(y)) = {ψ(ρ(z)) : z ∈ xy} = {φ (z) : z ∈ xy} = φ(xy) = φ (x) φ (y) = ψ(ρ(x)) ∗ψ(ρ(y)) int. j. anal. appl. 17 (5) (2019) 873 hence ψ is monomorphism and it is easy to prove that imφ = imψ. now for all x ∈h1, we have (ψ∗ρ•)(x) = ψ((ρ•)(x)) = ψ((ρ)(x)) = φ(x). hence diagram commutes. 2 theorem 3.4. let φ : h1 −→h2 be good homomorphism of an hv-la groups. then h1/ρ ∼= im φ. proof: it follows from the theorem 3.3. 2 theorem 3.5. let φ : h1 −→h2 be good homomorphism of an hv-la groups. if k is a regular relation on h1 such that k ⊆ ρ, then there exists a unique monomorphism ψ : h1/k −→h2 such that im φ = im ψ and diagram commute i.e.ψ ∗k? = φ, where mapping k? : h1 −→h2/k is defined as k?(x) = k (x) ∀x ∈h1. proof: straightforward. 2 theorem 3.6. let θ and σ be two regular relation on an hv-la-group h such that θ ⊆ σ. then σ/θ is regular relation on h/θ. proof: we define σ/θ : h/θ ◦ h/θ −→ p?(h/θ) by σ/θ(θ(x)) = θ(x) ∀θ(x) ∈ h/θ. we first show that the mapping is well defined, consider θ (x) = θ (y) =⇒ (x,y) ∈ θ ⊆ σ =⇒ (θ (x) ,θ (y)) ∈ σ/θ and so σ/θ (θ (x)) = σ/θ (θ (y)) . next we show that σ/θ is an equivalence relation. let x ∈ h, then (x,x) ∈ σ =⇒ (θ (x) ,θ (y)) ∈ σ/θ, thus σ/θ is reflexive. also let x,y ∈ h, such that (θ (x) ,θ (y)) ∈ σ/θ. as (x,y) ∈ σ =⇒ (y,x) ∈ σ due to the symmetry of σ. which implies that (θ (y) ,θ (x)) ∈ σ/θ. hence σ/θ is symmetric . again let x,y,z ∈ h, such that (θ (x) ,θ (y)) , (θ (y) ,θ (z)) ∈ σ/θ and (x,y) , (y,z) ∈ σ =⇒ (x,z) ∈ σ due to the transitivity of σ. which implies that (θ (x) ,θ (y)) ∈ σ/θ. hence σ/θ is transitive. thus σ/θ is an equivalence relation. now we have to show that, it is a regular. for it let x,y,z ∈ h, such that (θ (x)) σ/θ (θ (y)) =⇒ (x,y) ∈ σ =⇒ xσy =⇒ (xz) ρ (yz) =⇒ {θ (µ) : µ ∈ xz}σ/θ{θ (ν) : ν ∈ yz}. int. j. anal. appl. 17 (5) (2019) 874 which implies that (θ (x) ⊗θ (z)) σ/θ (θ (y) ⊗θ (z)) and similarly we can show that (θ (x)) σ/θ (θ (y)) =⇒ (θ (z) ⊗θ (x)) σ/θ (θ (z) ⊗θ (z)) . hence σ/θ is regular relation on h/θ. 2 theorem 3.7. let θ and σ be two regular relations on an hv-la-group h such that θ ⊆ σ. then (h/θ) / (σ/θ) ∼= h/σ. proof: let us define ψ : (h/θ) / (σ/θ) −→ h/σ by ψ (σ/θ (θ (x))) = σ (x) ∀x ∈ h. it is easy to show that this map is bijective. we only show that it is homomorphism. suppose x,y ∈h, then ψ(σ/θ (θ (x)) ⊗σ/θ (θ (y))) = ψ({σ/θ (θ (z)) : θ (z) ∈ θ (x) ⊗θ (z)}) = ψ({σ/θ (θ (z)) : z ∈ xy}) = {ψ(σ/θ (θ (z)) : z ∈ xy)} = {σ (z) : z ∈ xy} = σ (x) σ (y) = ψ(σ/θ (θ (x))) ⊗ψ(σ/θ (θ (y))) hence ψ is homomorphism. thus (h/θ) / (σ/θ) ∼= h/σ. 2 3.1. construction of hv-la-groups: in this section we present the construction of hv-la-groups through any non-empty set having more than two elements. consider a finite set h, such that |h| > 2. define the hyperoperation ◦ on h as follows xi ◦xj =   xj for i = 1 xk for j = 1and k ≡ 2 − i mod |h| h for i = j,i 6= 1,j 6= 1 xi otherwise, for i ≺ j or i � j   then h under the hyperoperation ◦ form an hv-la-group. the above construction can be explained with the help of an example. example: let h = {x1,x2,x3} be any set and define the binary hyperoperation ◦ defined above in the following cayley,s table: ◦ x1, x2 x3 x1, x1 x2 x3 x2 x3 h x2 x3 x2 x3 h then clearly h form an hv-la-group. one can see that ◦ satisfy the weak left invertive law with reproductive axiom, also ◦ is non-left invertive and non-associative i.e. h = (x3 ◦x3) ◦x2 6= (x2 ◦x3) ◦x3 = x2 int. j. anal. appl. 17 (5) (2019) 875 and h = (x2 ◦x2) ◦x1 6= x2 ◦ (x2 ◦x1) = x2 also it is not wass (x2 ◦x1) ◦x1 ∩x2 ◦ (x1 ◦x1) = φ. hence (h,◦) is an hv-la-group. the result can easily be generalized to n elements. 3.2. chemical example of hv-lagroup: here in this section we utilize the newly defined structure namely hv-lagroups in applications. for this purpose we study chemical reactions. the best example of hv-la-group in chemical reaction is a redox reaction. 3.3. redox reaction: the chemical reaction in which one specie loss the electron and other specie gain the electron. oxidation mean loss of electron. reduction mean gain of electron. the redox reaction is a vital for biochemical reaction and industrial process. the electron transfer in cell and oxidation of glucose in the human body are the example of redox reaction. the reaction between hydrogen and fluorine is an example of redox reaction i.e. h2 + f −→ 2hf h2 −→ 2h+ + 2e−( oxidation) f2 + 2e − −→ 2f (reduction) each half reaction has standard reduction potential ( e0 ) which is equal to the potential difference at equilibrium under the standard condition of an electrochemical cell in which the cathode reaction is half reaction considered and anode is a standard hydrogen electrode (she). for the redox reaction, the potential of cell is defined as e◦cell = e◦cathod −e◦anode where e◦anode is the standard potential at the anode and e ◦ cathod is the standard potential at the cathode as given in the table of standard electrode potential. now consider the redox reaction of mn mn0 + 2mn+4 + 2mn+3 −→ 3mn+2 + 2mn+4 mn0 −→ mn+2 + mn+4 + 2e− + 2mn+3 + 2mn+4 manganese having a variable oxidation state of 0, +1, +2, +3, +4, +5, +6, +7. if we take mn0,mn+4,mn+3,mn+4 together we will get pure redox reaction. the flow chart is given as int. j. anal. appl. 17 (5) (2019) 876 mn species with different oxidation state react with themselves. all possible reactions are presented in the following table ⊕ mn0 mn+1 mn+2 mn+3 mn+4 mn0 mn0 { mn0,mn+1 } { mn0,mn+2 } { mn0,mn+3 } { mn0,mn+4 } mn+1 { mn0,mn+1 } { mn0,mn+2 } { mn0,mn+3 } { mn+2 } { mn+1,mn+4 } mn+2 mn+1 { mn0,mn+3 } { mn+1,mn+3 } { mn+1,mn+4 } { mn+2,mn+4 } mn+3 { mn0,mn+3 } { mn+1,mn+3 } { mn+2,mn+3 } mn+3 { mn+3,mn+4 } mn+4 { mn0,mn+4 } { mn+1,mn+4 } { mn+2,mn+4 } { mn+3,mn+4 } mn+4 the standard reduction potentials ( e0 ) for conversion of each oxidation state to another are e0 ( mn+4/mn+3 ) = +0.95,e0 ( mn+3/mn+2 ) = +1.542,e0 ( mn+2/mn+1 ) = −0.59, e0 ( mn+1/mn+0 ) = 0.296. if we replace mn0 = a, mn+1 = b, mn+2 = c, mn+3 = d, mn+4 = e, then we obtain the following table ⊕ a b c d e a {a} {a,b} {a,c} {a,d} {a,e} b {a,b} {a,c} {a,d} {c} {b,e} c {a,c} {a,d} {b,d} {b,e} {c,e} d {a,d} {b,d} {c,d} {d} {d,e} e {a,e} {b,e} {c,e} {d,e} {e} int. j. anal. appl. 17 (5) (2019) 877 as all elements of h satisfy the weak left invertive law with the reproductive axiom, but h do not satisfies the left invertive law, associative law and weak associative law {a,c} = (b⊕ b) ⊕a 6= (a⊕ b) ⊕ b = {a,b,c} , {a,b,c,d} = (b⊕ b) ⊕ c 6= b⊕ (b⊕ c) = {a,b,c} , and (b⊕d) ⊕d = {b,e}∩ c = b⊕ (d⊕d) = φ. hence (h,⊕) is an hv-la-group. 4. conclusion and perspectives in this research, we have introduce a new class of hv-la-groups and investigated different types of homomorphisms of hv-la-groups. additionally, we construct the hv-la-groups and applied our result to a chemical redox reaction. in the future work, we will apply our result to different kind of applications. references [1] m. a. kazim and m. naseeruddin, on almost semigroups, aligarh bull. math., 2 (1972), 1-7. [2] q. mushtaq and s. m. yusuf, on la-semigroups, aligarh bull. math., 8 (1978), 65-70. [3] p. holgate, groupoids satisfying a simple invertive law, math. stud., 61(1-4) (1992), 101-106. [4] j. r. cho, j. jezek and t. kepka, paramedial groupoids, czechoslovak math. j., 49(2) (1999), 277-290. [5] m. akram, n. yaqoob and m. khan, on (m, n)-ideals in la-semigroups, appl. math. sci., 7(44) (2013), 2187-2191. [6] m. khan and n. ahmad, characterizations of left almost semigroups by their ideals, j. adv. res. pure math., 2(3) (2010), 61-73. [7] p. v. protic and n. stevanovic, ag-test and some general properties of abelgrassmann’s groupoids, pure math. appl., 6(4) (1995), 371-383. [8] n. stevanovic and p.v. protic, composition of abel-grassmann’s 3-bands, novi sad j. math., 34(2) (2004), 175-182. [9] q. mushtaq and s.m. yusuf, on locally associative la-semigroups, j. nat. sci. math., 19(1) (1979), 57-62. [10] q. mushtaq and m. s. kamran, left almost group, proc. pak. acad sci., 33 (1996), 12 [11] f. marty, sur une generalization de la notion de groupe, 8iem congres des mathematicians scandinaves tenua stockholm, (1934) 45-49. [12] p. corsini, prolegomena of hypergroup theory, aviani editore, (1993). [13] t. vougiouklis, hyperstructures and their representations, hadronic press,palm harbor, flarida, usa, (1994). [14] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic, (2003). [15] k. hila and j. dine, on hyperideals in left almost semihypergroups, isrn algebra, 2011 (2011), article id 953124. [16] n. yaqoob, p. corsini and f. yousafzai, on intra-regular left almost semihypergroups with pure left identity, j. math., 2013 (2013), article id 510790. [17] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, algebraic hyperstructures and applications, proceedings of the fourth international congress, (1991), 203-211. [18] t. vougiouklis, a new class of hyperstructures, journal of combinatorics, inform. syst. sci., 20 (1995), 229-235. [19] t. vougiouklis, ∂-operations and hv-fields, acta math. sin. (engl. ser.), 24(7) (2008), 1067-1078. int. j. anal. appl. 17 (5) (2019) 878 [20] t. vougiouklis, the h/v-structures, algebraic hyperstructures and applications, taru publications, new delhi, (2004), 115-123. [21] s. spartalis, on hv-semigroups, italian j. pure appl. math., 11 (2002), 165-174. [22] s. spartalis, on the number of hv-rings with p-hyperoperations, discr. math., 155 (1996), 225-231. [23] s. spartalis, on reversible hv-group, algebr. hyperstruct. appl., (1994) 163-170. [24] s. spartalis, quoitients of p-hv-rings, new front. hyperstruct., (1996) 167-176. [25] s. spartalis and t. vougiouklis, the fundamental relations on hv-rings, riv. mat. pura appl., 7 (1994), 7-20. [26] m. gulistan, n. yaqoob and m. shahzad, a note on hv-la-semigroup u.p.b. sci. bull., series a, 77 (3) (2015), 93-106 [27] b. davvaz, weak algebraic hyperstructures as a model for interpretation of chemical reactions, iran. j. math. chem. 7 (2) (2016), 267-283. [28] b. davvaz, a. dehghan nezhad, a. benvidi, chemical hyperalgebra: dismutation reactions, match commun. math. comput. chem. 67 (2012), 55-63. [29] b. davvaz, a. d. nezad and a. benvidi, chain reactions as experimental examples of ternary algebraic hyperstructures, match commun. math. comput. chem. 65 (2) (2011), 491-499. 1. introduction 2. preliminaries 3. hv-la-groups 3.1. construction of hv-la-groups: 3.2. chemical example of hv-lagroup: 3.3. redox reaction: 4. conclusion and perspectives references international journal of analysis and applications volume 16, number 6 (2018), 822-841 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-822 new modified method of the chebyshev collocation method for solving fractional diffusion equation h. jaleb, h. adibi∗ department of mathematics, central tehran branch, islamic azad university, tehran, iran ∗corresponding author: adibih@aut.ac.ir abstract. in this article a modification of the chebyshev collocation method is applied to the solution of space fractional differential equations. the fractional derivative is considered in the caputo sense. the finite difference scheme and chebyshev collocation method are used. the numerical results obtained by this approach have been compared with other methods. the results show the reliability and efficiency of the proposed method. 1. introduction the fractional partial differential equations (fpdes) arise in numerous problems of engineering, physics, mathematics, chemistry, biology,and viscoelasticity ( [1], [2], [3], [4]).most fractional differential equations do not have exact analytical solutions, thus many authors are seeking ways to numerically solve these problems( [5], [6]). recently, some different methods to solve fractional differential equations have been given such as variational iteration method [7], homotopy perturbation method [8], adomian decomposition method [9], homotopy analysis method [10], and collocation method [11]. a least square finite element solution of a fractional-order two-point boundary value problems, developed in [12]. sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation as well proposed in [13]. wavelet operational received 2017-10-18; accepted 2017-12-16; published 2018-11-02. 2010 mathematics subject classification. 34a08. key words and phrases. fractional diffusion equation; caputo derivative; fractional riccati differential equation; finite difference; collocation; chebyshev polynomials. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 822 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-822 int. j. anal. appl. 16 (6) (2018) 823 method for solving fractional partial differential equations used in [14]. method of lines to transform the space fractional fokker-planck equation into a system of ordinary differential equations suggested in( [15], [16]) .the space fractional diffusion equations are solved numerically .khader proposed chebyshev collocation method to discretize space fractional diffusion equations to obtain a linear system of ordinary differential equations and he solved the resulting system by finite difference method [17]. saadatmandi and et al. [18]applied tau approach to solve space fractional diffusion equations. 2. basic ideas and definitions definition 2.1. the caputo fractional derivative operator c0 d α x of order α is defined in the following form [4]: c 0 d α xf(x) = 1 γ(m−α) ∫ x 0 f(m)(t) (x−t)α−m+1 dt, α > 0, where m− 1 < α ≤ m, m ∈ n, x > 0. caputo fractional derivative operator is a linear operation and for the caputo derivative we have [19]: c 0 d α xc = 0, (2.1) c 0 d α xx n =   0, n ∈ n0 and n < dαe,γ(n+1) γ(n+1−α)x n−α, n ∈ n0 and n ≥dαe, (2.2) where c is a constant and dαe denotes the smallest integer greater than or equal to α and n0 = {1, 2, ...}. for α ∈ n0, the caputo differential operator coincides with the usual differential of integer order ( [19], [20], [21]). definition 2.2. the weighted−lpnorm is defined in the following form [22]: ‖u‖lpw(−1,1) = ( ∫ 1 −1 |u(x)|pw(x)dx)1/p for 1 ≤ p < ∞, (2.3) and we again set ‖u‖l∞w (−1,1) = sup −1≤x≤1 |u(x)| = ‖u‖l∞(−1,1). (2.4) the space of functions for which a particular norm is finite forms a banach space, indicated bylpw(−1, 1). int. j. anal. appl. 16 (6) (2018) 824 definition 2.3. we define natural sobolev norms as follows [22]: ‖u‖hmw (−1,1) = ( m∑ k=0 ‖u(k)‖2l2w(−1,1)) 1/2. (2.5) the hilbert space associated with this norm is denoted by hmw (−1, 1). we also define the seminorms |u| h m,n w (−1,1) = ( m∑ k=min(m,n+1) ‖u(k)‖2l2w(−1,1)) 1/2. (2.6) 2.2. a review of the chebyshev polynomials the well known chebyshev polynomials are defined on the interval [-1, 1] as [23]: t0(z) = 1, t1(z) = z, tn+1(z) = 2ztn(z) −tn−1(z), n = 1, 2, ... . the analytic form of the chebyshev polynomials tn(z) of degree n is given by the following: tn(z) = n [ n 2 ]∑ i=0 (−1)i2n−2i−1 (n− i− 1)! (i)!(n− 2i)! zn−2i, (2.7) where [n 2 ] denotes the integer part of n/2. the orthogonality condition is ∫ 1 −1 ti(z)tj(z)√ 1 −z2 dz =   π, for i = j = 0, π 2 , for i = j 6= 0, 0, for i 6= j. in order to use these polynomials on the interval x ∈ [0, 1], we define the so called shifted chebyshev polynomials by introducing the change of variable z=2x-1.we denote tn(2x− 1) by t∗n(x), defined as: t∗n(x) = n n∑ k=0 (−1)n−k 22k(n + k − 1)! (2k)!(n−k)! xk, n = 2, 3, ... , (2.8) int. j. anal. appl. 16 (6) (2018) 825 where t∗0 (x) = 1 and t ∗ 1 (x) = 2x− 1. a function u(x), which is squared integrable in [0, 1], may be expressed in terms of shifted chebyshev polynomials as: u(x) = ∞∑ i=0 cit ∗ i (x), where c0 = 1 π ∫ 1 0 u(t)t∗0 (x)√ x−x2 dx, ci = 2 π ∫ 1 0 u(t)t∗i (x)√ x−x2 dx, i = 1, 2, ... . (2.9) theorem 2.1. [19] let u(x) be approximated by shifted chebyshev polynomials as: um(x) = m∑ i=0 cit ∗ i (x), (2.10) and α > 0, then dα(um(x)) = m∑ i=dαe i∑ k=dαe ciw (α) i,k x k−α, (2.11) where w (α) i,k is given by: w (α) i,k = (−1) i−k 2 2ki(i + k − 1)!γ(k + 1) (i−k)!(2k)!γ(k + 1 −α) . (2.12) 3. the process of solving the space fractional diffusion equation and modified method we consider space fractional diffusion equation [17] ∂u(x,t) ∂t = d(x,t) ∂αu(x,t) ∂xα + s(x,t), a < x < b, 0 ≤ t ≤ m, 1 < α ≤ 2, (3.1) with initial condition u(x, 0) = u0(x), a < x < b, (3.2) int. j. anal. appl. 16 (6) (2018) 826 and boundary conditions u(a,t) = u(b,t) = 0, (3.3) where the function s(x,t) is a source term. we use the chebyshev collocation method to discretize 3.1 and to get a linear system of ordinary differential equations and use the finite difference method (fdm) ( [24], [25]) to solve the resulting system, and obtain the coefficients in the approximate solution. so we approximate u(x,t) as: um(x,t) = m∑ i=0 λi(t)t ∗ i (x). (3.4) from eqs. 3.1, 3.4 and using theorem 2.1 we have: m∑ i=0 dλi(t) dt t∗i (x) = m∑ i=dαe i∑ k=dαe λi(t)w (α) i,k x k−α + s(x,t). (3.5) collocating, eq. 3.5 at (m + 1 −dαe) points xp yields: m∑ i=0 dλi(t) dt t∗i (xp) = m∑ i=dαe i∑ k=dαe λi(t)w (α) i,k x k−α p + s(xp, t), p = 0, 1, ...,m−dαe. (3.6) now we use of roots of shifted chebyshev polynomials t∗ m+1−dαe(x) as suitable collocation points. by substituting eqs 3.4 and 2.11 in the boundary conditions 3.3 we get m∑ i=0 (−1)iλi(t) = 0, m∑ i=0 λi(t) = 0. (3.7) if so, dαe equations obtained from 3.7, along with m+1-dαe equations obtained from 3.6 give (m+1) ordinary differential equations which may be solved by using fdm, i=0,1,...,n, τ = m n , 0 ≤ ti ≤ m, ti = iτ, to get the m unknown λi, i=0,1,...,m, in various time levels tn. by determining the unknowns λi(tn) [17],the approximate m degree polynomials different time of tn as obtained as follows: um(x,tn) = m∑ i=0 λi(tn)t ∗ i (x) = λ n ot ∗ 0 (x) + λ n 1 t ∗ 1 (x) + λ n 2 t ∗ 2 (x) + ... + λ n mt ∗ m(x) = λ́no + λ́ n 1 x + λ́ n 2 x 2 + ... + λ́nmx m, (3.8) int. j. anal. appl. 16 (6) (2018) 827 in which t is the final time and λni = λi(tn). to improve the proposed method , firstly, on average, approximate solution um(x,tn) obtained by 3.8 and the exact solution of problem 3.1,so it new approximate first stage is called and the symbol unewapproximate(1)(x,tn) show. namely : unewapprox(1)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)]. (3.9) note that uex(x,tn) and uapprox(x,tn), respectively are exact solution and approximate solution of the problem 3.1. at this stage , if |unewapprox(1)(x,tn) − uex(x,tn)| to obtain,it is observed that the value of the amount |uapprox(x,tn) −uex(x,tn) is smaller. in other words, the error between the first stage approximate and exact solution of the problem, the smaller of ,the error between the approximate solution obtained from 3.8 and the exact solution problem. in the second stage,on average,approximate solution first gain and the exact solution problem and the second stage is called an new approximation solution, and the symbolunewapprox(2)(x,tn) show.namely : unewapprox(2)(x,tn) = 1 2 [unewapprox(1)(x,tn) + uex(x,tn)]. (3.10) at this stage , if the value of |unewapprox(2)(x,tn) − uex(x,tn| to determine, it will be seen that the value of the amount |unewapprox(1)(x,tn) − uex(x,tn| is smaller. in other words,the error between, the exact solution and approximate solution of the second stage, is the first step lower. if so, this trend continue, the average, the approximate solution to the (n-1)th, with the exact solution uex(x,tn) of the problem, it will be obtained new approximate polynomial and (n)th stage new approximate polynomial is called and the symbolunewapprox(n)(x,tn) show , namely: unewapprox(n)(x,tn) = 1 2 [unewapprox(n−1)(x,tn) + uex(x,tn)]. (3.11) it will be seen, that the amount of |unewapprox(n)(x,tn) − uex(x,tn| is much smaller that the amount |uapprox(x,tn)−uex(x,tn)| is.so that uapprox(x,tn) polynomial approximation to the results of the proposed method is [17].this claim with the numerical results obtained by solving the presented examples shown. in fact with this work , the numerical solution of equation3.1 is improved. the results of numerical examples , the absolute errors and the new approximation solutions for the various iterations of the improved method int. j. anal. appl. 16 (6) (2018) 828 , for tables, is presented and compared by the several other numerical methods. in this work,the number of repeat procedures , with the symbol i is shown in the tables. 4. error analysis and convergence this section is concerned with the studying of the convergence analysis and getting an upper bound for the error of the proposed formula. theorem 4.1. [19] the error |et (m)| = |dαu(x) −dαum(x)| in approximating dαu(x) by dαum(x) is bounded as: |et (m)| ≤ | ∞∑ i=m+1 ci( i∑ k=dαe k−dαe∑ j=0 θi,j,k)|, (4.1) where θi,j,k = (−1)i−k2i(i+k−1)!γ(k−α+ 1 2 ) hjγ(k+ 1 2 )(i−k)!γ(k−α−j+1)γ(k+j−α+1), j = 1, 2, ... . theorem 4.2. (chebyshev truncation theorem) .the truncation error u(x)−un (x), where un (x) = ∑n k=0 ckt ∗ k (x), is the truncated chebyshev series of u, satisfies the inequality [22]: ‖u(x) −un (x)‖lpw(−1,1) ≤ cn −m m∑ k=min(m,n+1) ‖u(k)‖lpw(−1,1), for 1 ≤ p < ∞, (4.2) for all functions u whose distributional derivatives of order up to m belong to lpw(−1, 1). c is a constant and depends on m. if so, when n →∞, we have: 0 ≤ lim n−→∞ (‖u(x) −un (x)‖lpw(−1,1)) ≤ lim n−→∞ (cn−m m∑ k=min(m,n+1) ‖u(k)‖lpw(−1,1)), (4.3) in the equation 4.3, if max | ∑m k=min(m,n+1) ‖u (k)‖lpw(−1,1)| ≤ m, weher m dimension is fixed, in the case we have: limn−→∞(cn −m ∑m k=min(m,n+1) ‖u (k)‖lpw(−1,1)) = 0. then, according equation 4.3, and according to the squeeze theorem, we have: limn−→∞(‖u(x) −un (x)‖lpw(−1,1)) = 0. the result is a convergence of approach gives us. int. j. anal. appl. 16 (6) (2018) 829 now, to discuss modified method error analysis is presented, polynomial approximations obtained 3.8 of the proposed approach [17], p0(x,tn) call. namely: p0(x,tn) = um(x,tn) = m∑ i=0 λi(tn)t ∗ i (x). (4.4) so we have: |p0(x,tn) −uex(x,tn)| ≤ ε0, (4.5) if you put unewapprox(1)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)] = p1(tn), (4.6) we have: |p1(x,tn) −uex(x,tn)| ≤ ε1. (4.7) considering the ties 4.5, 4.6 and 4.7, we have: |p1(x,tn) −uex(x,tn)| ≤ ε1 ⇒| 1 2 [p0(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ ε1 ⇒|p0(x,tn) −uex(x,tn)| ≤ 2ε1 ≤ ε0, so the result is: ε1 ≤ ε0 2 . (4.8) for these arrangements, if unewapprox(2)(x,tn) = 1 2 [um(x,tn) + uex(x,tn)] to p2(tn) call, you can write: |p2(x,tn) −uex(x,tn)| ≤ ε2 ⇒| 1 2 [p1(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ ε2 int. j. anal. appl. 16 (6) (2018) 830 ⇒|p1(x,tn) −uex(x,tn)| ≤ 2ε2 ⇒| 1 2 [p0(x,tn) + uex(x,tn)] −uex(x,tn)| ≤ 2ε2 ⇒|p0(x,tn) −uex(x,tn)| ≤ 2 × 2ε2 ≤ ε0, so the result is: ε2 ≤ ε0 22 . (4.9) by following this process, the n-th stage will be: εn ≤ ε0 2n . (4.10) in fact, if pn(x,tn) polynomial approximation is made in step n, we get the following result: |pn(x,tn) −uex(x,tn)| ≤ εn ≤ ε0 2n . (4.11) for 4.11, can be written: 0 ≤ lim n→∞ (|pn(x,tn) −uex(x,tn)|) ≤ lim n→∞ ( ε0 2n ), (4.12) then, according equation 4.12, and according to the squeeze theorem, we have: lim n→∞ (|pn(x,tn) −uex(x,tn|) = 0. the result is a convergence of approach gives us. int. j. anal. appl. 16 (6) (2018) 831 remark 1. the presented method, can be applied for solution of numerical the fractional riccati differential equation.also dαu(t) + u2(t) − 1 = 0, t > 0, 0 < α ≤ 1, with the initial condition u(0) = u0, in next section we illustrated this approach by example 5.1. 5. numerical results example 5.1. consider the fractional riccati differential equation of the form dαu(t) + u2(t) − 1 = 0, t > 0, 0 < α ≤ 1, (5.1) with the initial condition u(0) = u0, (5.2) and the parameter α, refers to the fractional order of the time derivative. for α = 1; the eq.5.1 is the standard riccati differential equation du(t) dt + u2(t) − 1 = 0. the exact solution to this equation is u(t) = e2t − 1 e2t + 1 . now we approximate the function u(t) by using formula ?? and its caputo derivative dαu(t) by using the presented formula 2.11 with m=5.then fractional riccati differential equation 5.1 is transformed to the following approximated form: 5∑ i=1 i∑ k=1 ciw (α) i,k t k−α + ( 5∑ i=0 cit ∗ i (t)) 2 − 1 = 0, (5.3) int. j. anal. appl. 16 (6) (2018) 832 where w (α) i,k is defined in 2.12. also the initial condition 5.2 is given by : 5∑ i=0 ci(t ∗ i (0)) = u0. (5.4) we now collocate eq.5.3 at (m + 1 −dαe) points tp as: 5∑ i=1 i∑ k=1 ciw (α) i,k t k−α p + ( 5∑ i=0 cit ∗ i (tp)) 2 − 1 = 0, p = 0, 1, 2, 3, 4. (5.5) note that t,ps are roots of shifted chebyshev polynomial t ∗ 5 (t), i.e. t0 = 0.5, t1 = 0.206107, t2 = 0.793893, t3 = 0.024471, t4 = 0.975528. by using eqs.5.4 and 5.5, we obtain a system of non-linear algebraic equations which contains 6 equations for the unknowns ci, i = 0, 1, ..., 5. by solving the previous system, utilizing the newton iteration method, we obtain the unknown ci, i = 0, 1, ..., 5, and therefore, the approximate solution is obtained via: u5(t) = 5∑ i=0 cit ∗ i (t). (5.6) for α = 1 , and then determine the coefficients ci about5.6, polynomial approximation as follows: u5(t) = 5∑ i=0 cit ∗ i (t) = 2.66714 × 10 −17 + 0.999372x + 0.0157609x2 − 0.41893x3 + 0.180634x4 − 0.0152477x5. (5.7) in this way, the improved method described for polynomial approximation 5.7 was used. in the table 1, 2 the numerical results and absolute error between the exact solution uex, and the approximate solution uapprox with different values of i, by means of the proposed modified method are given. int. j. anal. appl. 16 (6) (2018) 833 table 1: comparison of absolute errors for u(x)at m=5 with different values of i for example 5.1. by modified method x i=0 i=10 i=20 i=30 i=35 |error(0)| |error(10)| |error(20)| |error(30)| |error(35)| 0.0 2.66714×10−17 0.00000 0.00000 0.00000 0.1 2.58369×10−5 2.52314×10−8 2.46400×10−11 2.39808×10−14 6.93889×10−16 0.2 6.22951×10−5 6.08351×10−8 5.94093×10−11 5.80924×10−14 1.77636×10−16 0.3 3.25723×10−5 3.18089×10−8 3.10634×10−11 3.02536×10−14 9.43690×10−16 0.4 2.16611×10−5 2.11534×10−8 2.06575×10−11 2.02061×10−14 6.10623×10−16 0.5 4.38002×10−5 4.27737×10−8 4.17711×10−11 4.06897×10−14 1.27676×10−15 0.6 1.64887×10−5 1.61022×10−8 1.57249×10−11 1.53211×10−14 4.44089×10−16 0.7 3.04507×10−5 2.97370×10−8 2.90398×10−11 2.84217×10−14 6.66134×10−16 0.8 4.75369×10−5 4.64228×10−8 4.53346×10−11 4.44089×10−14 1.33227×10−15 0.9 1.43902×10−5 1.40529×10−8 1.37235×10−11 1.35447×10−14 4.44089×10−16 1.0 3.98000×10−6 3.88672×10−8 3.79574×10−12 3.77476×10−15 2.22045×10−16 table 2: comparison of absolute errors for u(x)at m=5 with different values of i for example 5.1. by modified method x i=40 |error(40)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 int. j. anal. appl. 16 (6) (2018) 834 example 5.2. in this section, we consider space fractional diffusion equation3.1 with α = 1.8, of the form: ∂u(x,t) ∂t = d(x,t) ∂1.8u(x,t) ∂x1.8 + s(x,t), where, 0 < x < 1, with the diffusion coefficient: d(x,t)= γ(1.2)x1.8, and the source function: s(x,t)=3x2(2x− 1)e−t. the initial and boundary conditions are respectively as: u(x,0)=x2(1 −x), u(0,t)=u(1,t)=0. the exact solution of this problem is u(x,t)= x2(1 −x)e−t. we apply the present method with m=3, and approximate the solution as follows: u3(x,t) = 3∑ i=0 λi(t)t ∗ i (x). (5.8) in 5.8, after determining the coefficients λi(t) for t=2 [17], polynomial approximation is as follows. u3(x, 2) = 3∑ i=0 λi(t800)t ∗ i (x) = λ́ 800 o + λ́ 800 1 x + λ́ 800 2 x 2 + λ́8003 x 3 = − 8.673617−19 + 0.000894x + 0.134649x2 − 0.135543x3 (5.9) in table3, the absolute error, between the exact solution uex and the approximate solution uapprox at m=3 and time step τ = 0.0025, with the final time t=2 is given. also, in the table 4, 5 ,6 the numerical results and absolute error between the exact solution uex, and the approximate solution uapprox with different values of i, by means of the proposed modified method are given. it is notable that by considering τ = 0.0025,and using finite differential method (fdm) about 5.8 [17], we will has 800 (t τ = 2 0.0025 = 800) level time for approximate solutions u(x,tn), 0 < x < 1. in the above example all 800 values of u(x,tn) are calculated by utilizing mathematica. example 5.3. [16] in this example, we consider the following space fractional diffusion equation ∂u(x,t) ∂t = p(x) ∂αu(x,t) ∂xα + s(x,t), 0 < x < 1 (5.10) with initial conditionu(x, 0) = x4, and boundary conditions int. j. anal. appl. 16 (6) (2018) 835 table 3: comparison of absolute errors for u(x,2) at m=3 and t=2 for example 5.2. x modified method method[17] method [26] method [18] 0.0 2.46519×10−32 1.70849 ×10−4 4.483787×10−3 0.00 0.1 2.60209×10−18 2.10940 ×10−5 4.479660×10−3 2.89×10−5 0.2 5.20417×10−18 1.76609 ×10−4 4.201329×10−3 1.09×10−4 0.3 8.67362×10−18 3.01420 ×10−4 3.695172×10−3 2.20×10−4 0.4 1.04083×10−17 4.04138 ×10−4 3.007566×10−3 3.40×10−4 0.5 1.38778×10−17 4.89044 ×10−4 2.184889 ×10−3 4.45×10−4 0.6 2.08167×10−17 4.89044 ×10−4 1.273510 ×10−3 5.15×10−4 0.7 1.38778×10−17 5.63305 ×10−4 0.319831 ×10−3 5.27×10−4 0.8 1.38778×10−17 6.33367 ×10−4 0.629793 ×10−3 4.60×10−4 0.9 2.77556×10−17 7.05677 ×10−4 1.528978 ×10−3 2.91×10−4 1.0 0.00000 8.82821 ×10−4 2.331347 ×10−3 0.00 table 4: comparison of absolute errors for u(x,2)at m=3 and t=2 with different values of i for example 5.2. by modified method x i=0 i=5 i=10 i=15 i=20 |error(0)| |error(5)| |error(10)| |error(15)| |error(20)| 0.0 8.67362×10−19 2.71051×10−20 8.47033 ×10−22 2.49698 ×10−23 8.27181×10−25 0.1 8.23560×10−5 2.57363 ×10−6 8.04258×10−8 2.51331×10−9 7.85408×10−11 0.2 1.49747×10−4 4.67958 ×10−6 1.46237×10−7 4.56991×10−9 1.42810×10−10 0.3 2.00921×10−4 6.27878×10−6 1.96212×10−7 6.13162×10−9 1.91613×10−10 0.4 2.34628×10−4 7.33213 ×10−6 2.29129×10−7 7.16028×10−9 2.23759×10−10 0.5 2.49617×10−4 7.80052×10−6 2.43766×10−7 7.61770×10−9 2.38059×10−10 0.6 2.44636×10−4 7.64488 ×10−6 2.38902×10−7 7.46570×10−9 2.33303×10−10 0.7 2.18435×10−4 6.82609×10−6 2.13315×10−7 6.66611×10−9 2.08316×10−10 0.8 1.69763×10−4 5.30508 ×10−6 1.65784×10−7 5.18075×10−9 1.61898×10−10 0.9 9.73680×10−5 3.04275×10−6 9.50859×10−8 2.97144×10−9 9.28574×10−11 1.0 2.60209×10−18 2.77556 ×10−17 0.00000 0.00000 0.00000 u(0, t) = 0,u(1, t) = e−t, where the function s(x,t) = −2e−tx4 is a source term, andp(x) = 1 24 γ(5 −α). the exact solution to this equation is e−tx4. by applying the proposed method [17] for α = 1.2, polynomial approximation is as follows: int. j. anal. appl. 16 (6) (2018) 836 table 5: comparison of absolute errors for u(x,2)at m=3 and t=2 with different values of i for example 5.2. by modified method x i=25 i=30 i=35 i=40 i=45 |error(25)| |error(30)| |error(35)| |error(40)| |error(45)| 0.0 2.58494×10−26 8.07794×10−28 2.52435×10−29 7.88861×10−31 2.46519×10−32 0.1 2.45440×10−12 7.66997×10−14 2.39674×10−15 7.45931×10−17 2.60209×10−18 0.2 4.46280×10−12 1.39462×10−13 4.35763×10−15 1.35308×10−16 5.20417×10−18 0.3 5.98792×10−12 1.87123×10−13 5.84775×10−15 1.82146×10−16 8.67362×10−18 0.4 6.99246×10−12 2.18513×10−13 6.82440×10−15 2.11636×10−16 1.04083×10−17 0.5 7.43915×10−12 2.32474×10−13 7.25808×10−15 2.22045×10−16 1.38778×10−17 0.6 7.29072×10−12 2.27839×10−13 7.11237×10−15 2.22045×10−16 2.08167×10−17 0.7 6.50988×10−12 2.03434×10−13 6.35603×10−15 1.94289×10−16 1.38778×10−17 0.8 5.05931×10−12 1.58096×10−13 4.92661×10−15 1.52656×10−16 1.38778×10−17 0.9 2.90179×10−12 9.06775×10−14 2.83107×10−15 6.93889×10−17 2.77556×10−17 1.0 0.00000 0.00000 0.00000 0.00000 0.00000 table 6: comparison of absolute errors for u(x,2)at m=3 and t=2 with different values of i for example 5.2. by modified method x i=50 |error(50)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 int. j. anal. appl. 16 (6) (2018) 837 table 7: comparison of absolute errors for u(x,1)at m=4 and t=1 with different values of i for example 5.2. by modified method x i=0 i=10 i=20 i=30 i=40 |error(0)| |error(10)| |error(20)| |error(30)| |error(40)| 0.0 1.38778×10−17 1.35525×10−20 1.32349 ×10−23 1.29247×10−26 1.26218×10−29 0.1 5.01756×10−3 4.89996×10−6 4.78512×10−8 4.67296×10−12 4.56345×10−15 0.2 6.38835×10−3 6.23862×10−6 6.09241×10−9 4.56991×10−12 5.81029×10−15 0.3 5.64747×10−3 5.51510×10−6 5.38584×10−9 5.94962×10−12 5.13605×10−15 0.4 3.99532×10−13 3.90167×10−6 3.81023×10−9 3.72093×10−12 3.63435×10−15 0.5 2.29764×10−3 2.24378×10−6 2.19120×10−9 2.13984×10−12 2.08776×10−15 0.6 1.08549×10−3 1.06005×10−6 1.03521×10−9 1.01094×10−12 9.84182×10−16 0.7 5.55277×10−4 5.42263×10−7 5.29540×10−10 5.17141×10−13 5.09222×10−16 0.8 5.68706×10−4 5.55377×10−7 5.42361×10−10 5.29633×10−13 5.31044×10−16 0.9 6.52824×10−4 6.37523×10−7 6.22582×10−10 6.07962×10−13 5.84742×10−16 1.0 0.00000 3.39934×10−17 0.00000×10−17 4.76800×10−17 4.18183×10−18 u4(x, 1) = 4∑ i=0 λi(t)t ∗ i (x) = 1.38778 × 10 −17 + 0.074363x− 0.274432x2 + 0.339516x3 + 0.228432x4, (5.11) note that, in this example ∆t = 0.001 is considered. now apply improved method for polynomial approximation expression 5.11,absolute error between the exact solution, and new approximate solution obtained based on the number of repetitions of the process, shown in tables 7 and 8. example 5.4. [15] consider the following space fractional diffusion equation ∂u(x,t) ∂t = p(x) ∂1.5u(x,t) ∂x1.5 + s(x,t), 0 < x < 1 (5.12) with the initial condition u(x, 0) = (x2 + 1) sin(1), and boundary conditions u(0, t) sin(t + 1),u(1, t) = 2 sin(t + 1), for t > 0, the source function s(x,t) = (x2 + 1) cos(t + 1) − 2x sin(t + 1), int. j. anal. appl. 16 (6) (2018) 838 table 8: comparison of absolute errors for u(x,1)at m=4 and t=1 with different values of i for example 5.2. by modified method x i=50 i=60 i=70 i=80 |error(50)| |error(60)| |error(70)| |error(80)| 0.0 1.23260×10−32 1.20371×10−35 1.17549 ×10−38 1.14794×10−41 0.1 4.46548×10−18 4.06965×10−21 3.97427×10−24 3.88112×10−27 0.2 5.73658×10−18 3.37867×10−21 3.29948×10−24 3.222151×10−27 0.3 5.24988×10−18 2.07294×10−21 2.02436×10−24 1.97691×10−27 0.4 4.76720×10−18 1.22852×10−20 1.19973×10−23 1.17161×10−26 0.5 3.31277×10−18 2.72581×10−20 2.66192×10−23 2.59953×10−26 0.6 4.52895×10−19 4.69916×10−20 4.58902×10−23 4.48146×10−26 0.7 3.81242×10−18 7.14857×10−20 6.98103×10−23 6.81741×10−26 0.8 3.56196×10−17 1.00740×10−19 9.83794×10−23 9.60736×10−26 0.9 2.85435×10−17 1.34756×10−19 1.31598×10−22 1.28513×10−25 1.0 1.11632×10−17 1.73532×10−19 1.69465×10−22 1.65493×10−25 andp(x) = γ(1.5)x0.5. the exact solution of this problem is u(x,t) = (x2 + 1) sin(t + 1). by applying the proposed method [17] , polynomial approximation is as follows: u2(x, 1) = 2∑ i=0 λi(t)t ∗ i (x) = 0.909297 + 0.00049296x + 0.908804x 2, (5.13) note that, in this example ∆t = 0.001 is considered. now apply improved method for polynomial approximation expression 5.13. absolute error between the exact solution, and new approximate solution obtained based on the number of repetitions of the process, shown in tables 9 and 10. 6. conclusion in this paper, we proposed a new modified of numerical method ,based on the shifted chebyshev collocation method and finite difference scheme, to find the solution of the space fractional diffusion equations and fractional riccati differential equation. in this method, the fractional derivatives are described in the caputo sense. comparison between our proposed method and other methods , shows that this scheme is superior and evidently the error gets smaller. int. j. anal. appl. 16 (6) (2018) 839 table 9: comparison of absolute errors for u(x,1)at m=2 and t=1 with different values of i for example 5.4. by modified method x i=0 i=10 i=20 i=30 i=38 |error(0)| |error(10)| |error(20)| |error(30)| |error(38)| 0.0 2.22×10−18 0.00000 0.00000 0.00000 0.00000 0.1 4.43×10−5 4.33266×10−8 4.23110×10−11 4.13003×10−14 2.22054×10−16 0.2 7.89×10−5 7.70250×10−8 7.52197×10−11 7.33857×10−14 2.22045×10−16 0.3 1.03×10−4 1.01095×10−7 9.87258×10−11 9.63674×10−14 3.33067×10−16 0.4 1.18×10−4 1.15538×10−7 1.12830×10−11 1.10134×10−13 3.33067×10−16 0.5 1.23×10−4 1.20352×10−7 1.17531×10−11 1.14797×10−13 4.44089×10−16 0.6 1.18×10−4 1.15538×10−7 1.12830×10−11 1.10245×10−13 6.66134×10−16 0.7 7.89×10−4 1.01095×10−7 9.87258×10−11 9.62563×10−14 2.22045×10−16 0.8 4.43×10−4 7.70250×10−7 7.52197×10−11 7.32747×10−14 2.22045×10−16 0.9 4.43×10−16 4.33266×10−8 4.23110×10−11 4.13003×10−14 2.22045×10−16 1.0 2.22×10−16 0.00000 0.00000 0.00000 0.00000 table10: comparison of absolute errors for u(x,1)at m=2 and t=1 with different values of i for example 5.4. by modified method x i=40 |error(40)| 0.0 0.00000 0.1 0.00000 0.2 0.00000 0.3 0.00000 0.4 0.00000 0.5 0.00000 0.6 0.00000 0.7 0.00000 0.8 0.00000 0.9 0.00000 1.0 0.00000 int. j. anal. appl. 16 (6) (2018) 840 7. acknowledgements it should be mentioned that the above article has been derived from ph.d thesis, at the islamic azad university central tehran branch. references [1] l. bagley and p. j. torvik, on the appearance of the fractional derivative in the behavior of real materials, j. appl. mech, 51 (19840), 294-298. [2] k. b. oldham and j. spanier, the fractional cvalculus, academic press, new york and london, ( 1974). [3] k. s. miller and b. ross, an introduction to the fractional calculus and fractional differential equations, john wiley, new york, (1993). [4] s. g. samko, a.a. kilbas and o.i. marichev, fractional integrals and derivatives: theory and applications, gordon and breach science publishers, usa, (1993). [5] s. das, fractional calculus for system identification and controls, springer, new york, (2008). [6] h. sweilam and m. m. khader, a chebyshev pseudo-spectral method for solving fractional integro-differential equations, anziam j. 51 (2010), 464-475. [7] m. inc, the approximate and exact solutions of the space-and time-fractional burger,s equations with initial conditions by varational iteration method, j. math. anal. appl. 345 (2008), 476-484. [8] n. h. sweilam, m.m.khader and r.f. al-bar, numerical studies for a multi-order fractional differential equation, phys. lett. a, 371 (2007), 26-33. [9] h. jafari and v. daftardar-gejji, solving linear and nonlinear fractional diffusion and wave equations by adm, appl. math. comput, 180 (2006), 488-497. [10] i. hashim, o. abdulaziz and s. momani, homotopy analysis method for fractional ivps, commun. nonlinear sci. numer. simul. 14 (2009), 674-684. [11] e. a. rawashdeh, numerical solution of fractional integro-differential equations by collocation method, appl. math. comput, 176 (2006), 1-6. [12] g. j. fix, j.p. roop,least squares finite element solution of the fractional order two-point boundary value problem, comput. math. appl. 48 (2004), 1017-1033. [13] r. darzi, b.mohammadzade, s.musavi, r.behshti, sumudu transform method for solving fractional differential equations and fractional diffusion-wave equation, j. math. comput. sci. 6 (2013) 79-84. [14] a. neamaty, b. agheli, r. darzi, solving fractional partial differential equation by using wavelet operational method, j. math. comput. sci. 7 (2013), 230-240 . [15] f. liu, v .anh and i. turner,numerical solution of the space fractional fokker-plank equation, j. comput. appl. math. 166 (2004), 209-219. [16] f. liu, v. anh, i. turner, p. zhuang, time fractional advection dispersion equation, j. appl. math. comput. 13 (2003), 233-245. [17] m. m. khader, on the numerical solutions for the fractional diffusion equation, commun. nonlinear sci. numer. simul. 16 (2011), 2535-2542. int. j. anal. appl. 16 (6) (2018) 841 [18] a. saadatmandi, m. dehghan, a. tau approach for solution of the space fractional diffusion equation, comput. math. appl. 62 (2011), 1135-1142. [19] m. m. khader, n. h. swetlam and a. m. s. mahdy, the chebyshev collection method for solving fractional order kleingordon equation, wseas trans. math. 13 (2014), 2224-2880 . [20] i. podlubny, fractional differential equations, academic press, new york, (1999). [21] m. joseph kimeu, fractional calculus: definitions and applications, western kentucky university, (2009). [22] c. canuto, a. quarteroni, m. y. hussaini, and t. a. zang, spectral methods fundamentals in single domains, springerverlag berlin heidelberg, printed in germany (2006). [23] m. a.snyder, chebyshev methods in numerical approximation, prentice-hall, inc. englewood cliffs, n. j. (1966). [24] m. m. meerschaert and c. tadjeran,finite difference approximations for fractional advection-dispersion flow equations, j. comput. appl. math. 172 (2008), 65-77. [25] m. m. meerschaert and c. tadjeran, finite difference approximations for two-sided space-fractional partial differential equations, appl. numer. math. 56 (2006), 80-90. [26] m. m. khader, n. h. sweilam and a. m. s. mahdy, an efficient numerical method for solving the fractional diffusion equation, j. appl. math. bioinform. 1 (2011), 1-12. 1. introduction 2. basic ideas and definitions 3. the process of solving the space fractional diffusion equation and modified method 4. error analysis and convergence 5. numerical results 6. conclusion 7. acknowledgements references international journal of analysis and applications volume 17, number 4 (2019), 517-529 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-517 bifurcations and invariant sets in a class of two-dimensional endomorphisms djellit ilham1,∗, fakroune yamina1 and selmani wissame2 1laboratory of mathematics, dynamics and modelization, university badji mokhtar, annaba-algeria 2university 20 aout 1955, skikda-algeria ∗corresponding author: ilhem.djelit@univ-annaba.dz abstract. several endomorphisms of the plane have been constructed by simple maps. we study the dynamics occuring in one of them, which is rich in global bifurcations. the invariants sets are stable manifolds of saddle type points or cycles, as well as closed curves issued from hopf bifurcations. the present paper focuses some bifurcations related with attractors or basins which produce other attractors which coexist with invariant sets. 1. introduction multistable systems, i.e. systems with a large number of coexisting stable systems, are very common in nature. they are subject of increasing interest in the last two decades. it has been the discovery of multiple stable states in many systems (agarwal in [1]; arecchi et al. in [2]; djellit and soula in [4]) that triggered the research in many other fields. multistable behaviors were found in discrete systems (hénon in [9]; ikeda in [10]; grebogi et al. in [7]; feudel et al. in [6]; lorenz in [11]). we present and explain numerical results illustrating the mechanism of bifurcations that occurs in a typical dynamical system relative to ”coupled-uncoupled” two-dimensional map. because the non-unique dynamics associated with the degree of the nonlinearity leads to a real interest and a large richness of the received 2019-03-25; accepted 2019-04-22; published 2019-07-01. 2010 mathematics subject classification. 37g10, 37j20, 37j15, 37d10. key words and phrases. bifurcation; endomorphism; attractor. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 517 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-517 int. j. anal. appl. 17 (4) (2019) 518 bifurcations situations, and interesting phenomena are uncovered, due in essence to the presence of invariant sets. this can have a profound effect on our understanding of the dynamical behavior. we provide numerical evidence and show some mechanisms associated with the appearance and disappearance of closed invariant curves. consider this dynamical system generated by a family of two-dimensional continuous noninvertible maps ta,b defined by : ta,b :   x ′ = y y′ = ay − bx + x2 (1) for a = 1 and b real parameter, this map was already studied by razafimandimby [14] and in clerc and hartmann [3]. they considered the influence domains of stable singularities of the quadratic endomorphism and they studied self-intersections of the unstable set in the form of loops. this fact has fundamental and known consequences in bifurcations theory particularly in global bifurcations and chaos. such bifurcations governing the route to chaos have been extensively studied since 1975 in [8, 12]. many of chaotic behaviors that are observed in dynamical systems are intimately associated with the presence of homoclinic and/or heteroclinic points of maps. contact bifurcations may correspond to homoclinic and heteroclinic bifurcations, critical and /or invariant curves are useful for interpreting such problems. several papers have shown the importance of critical curves in the bifurcations of basins, for example gumowski and mira [8] who have developed the role of these curves in bifurcations from simply connected basin to nonconnected basin, ferchichi et al. in [5] and mira et al. in [13] have studied bifurcations of type simply connected basin to multiply connected basin. these basic bifurcations result from the contact of a basin boundary with a critical curve segment or an attracting set leading to modifications of the basin. we study an endomorphism using both analytic bifurcation theory and numerical methods. the dynamics involves various transitions by bifurcations. the analysis of the dynamics of this model as a function of three parameters (a,b,and n ) is not an easy task. the particular case, with a = 1 fixed, as a function of (b,n = 2) has been studied in ref. [3]. there it has been shown that for the map t, the bifurcations of the model and route to chaos could be well determined. the planar system is the following: ta,b,n :   x ′ = y y′ = ay + fn(x) (2) where x,y are real variables, a and b are real parameters. ta,b,n has a nonconstant jacobian determinant detj = −f′n(x). these systems ta,b,n which become uncoupled when a = 0, are of fundamental importance in dynamical systems. two main tools are used by developing the method of critical curves and incorporating invariant int. j. anal. appl. 17 (4) (2019) 519 curves : two well-established methods. we display invariant sets for n = 2,and 4 and we examine their properties. we mainly focused on this research area, the identification and verification of some properties of such maps. first we consider the structure of the one-dimensional endomorphism of the n-degree polynomial fn(x) = x n − bx = x(xn−1 − b) (3) where fn(x) is assumed to depend continuously on the real parameter b. for n even, n = 2k,k > 1, the fixed points of fn are : x = 0 a stable node for b ∈]−1; 1[ and x1 = (b + 1) 1 n−1 stable node for b ∈] n+1 n−1 ;−1[. for n = 2k + 1,the equation fn(x) = x admits a unique trivial solution x = 0 for b ≤−1: and for b > −1;we have three solutions x = 0; x1 = (b + 1) 1 n−1 , and x2 = −(b + 1) 1 n−1 . for n = 2 , the two-dimensional endomorphism (2) has some symmetry property and possesses at most 2 fixed points depending upon the parameter values. we describe a specific family of bifurcations in a region of real parameter plane (a,b) for which the mappings were expected to have simple dynamics. we compute the first few bifurcation curves in this family and we study the bifurcation diagram which consists to these bifurcation curves in the parameter (a,b) plane. the figures 1.(a,b,c,d,e,f) give the parameter value for which at least one fixed point is attractive (blue domain corresponding to the value 1). more generally, these figures give the existence domains of a stable cycle of order k of which is given by the upper colored squares. the black regions (k = 15) corresponds to the existence of bounded iterated sequences. these figures are typical of maps with dominating odd or even degree terms.we can recognize on the diagram period doubling bifurcations, and spring area communication according to mira [13]. the bifurcation structure is a ” boxwithin-a-box ” type, as is well known infinitely many periodic are opened by fold bifurcations and are closed by homoclinic bifurcations by the intriguing ” box-within-a-box ” bifurcation structure. when n is even, there is an interesting passage from symmetric parameter diagram to non symmetric parameter diagram (clearly visible for the existence domain related with fixed points of color blue). and we can remark that for n even or odd, the diagrams are quite similar and then we can argue that some properties are automatically deduced. for n = 2, the two existence domains associated with the two fixed points are symmetric with respect to a black line located in the frontier to each domain, for n even or odd, the existence domains of different k-cycles shrink. for n = 3, the two domains are superimposed and symmetric to the line a = 0. this case (n odd) is treated separately in an other paper. despite the simplicity of this map, the concept ”coupled-uncoupled” is central to our interest and subject. so, in the following we shall focus our attention on the description of the bifurcations which are expected to occur as we vary the parameters along the bifurcation path indicated in fig. 1(a). int. j. anal. appl. 17 (4) (2019) 520 (a) bifurcation structure n=2 (b) bifurcation structure n=3 (c) bifurcation structure n=4 (d) bifurcation structure n=5 (e) bifurcation structure n=6 (f) bifurcation structure n=7 figure 1. bifurcation structure for n even and odd values. this paper intends to give such a study, particularly to consider the map (2) with n = 2. therefore, it is structured in the following way. in section 2 we introduce the language used in [12, 13];to analyze the system, and give some properties. section 3 gives some results concerning the uncoupled map and the main bifurcations. section 4 illustrates properties of invariant curves and connected-nonconnected bifurcation basins. 2. symmetry and fundamental properties first we analyse the dynamics for the case t = ta,b,2 of the form ta,b,2(x,y) = (y,ay + f2(x)) with f2(x) = x 2 − bx, a and b are real parameters. let us consider the case f2(x) a quadratic polynomial (n = 2). we know some results which enable us to detect, predict, determine cycles and fixed points, and locate bifurcation curves in the parameter plane. the fixed points and basic bifurcations of ta,b,2 were analyzed, they are solutions obtained by a trivial manipulation with x′ = x and y′ = y: besides the trivial solution o(0; 0), we can observe that further fixed point p(1 + b−a, 1 + b−a) exists if a 6= 1 + b. in this section we focus our attention on bifurcations playing an important role in the dynamics, those happening for a 6= 0 and b 6= 0. we can easily state the following proposition. int. j. anal. appl. 17 (4) (2019) 521 proposition 2.1. : if b = −1 + a, then o(0; 0) is the unique fixed point of the map ta,b,2 defined by (1). the parameter curve λ(1)0 : b = −1 + a is the transcritical bifurcation curve such that the two fixed points o and p exist, and are symmetric with respect to the parameter curve. consider the variable change that moves the point p to coincide with the origin as follows  x = x ∗ + 1 + b−a y = y∗ + 1 + b−a . after injection of this change in t(= ta,b,2) and some simple simplification, we have   x ∗ = y∗ y = ay∗ −x∗ ∗x∗ − (2a− b− 2)x∗ we put b∗ = 2a− b− 2, and a∗ = a, ta,b,2 becomes ta∗,b∗,2. to simplify the notation, in the following we shall write t∗ instead of ta∗,b∗,2. this new map t ∗ has two fixed points (0, 0) which corresponds to p for t and p∗(1 + b∗ −a∗, 1 + b∗ −a∗) which corresponds to the trivial solution for t. these two maps t and t∗ change their fixed points. let l : r2 → r2 (x,y)(a,b) → (x∗,y∗)(a∗,b∗) = (x− (1 + b−a),y − (1 + b−a))(a,2a−b−2) . proposition 2.2. l is symmetric by respect to ∆, or l◦t = t∗ ◦l. corollary 2.1. if p is a fixed point of t , then l(p) is a fixed point of t∗. corollary 2.2. the parametric curve λ(1)0 : b = −1 + a is invariant by l. the fixed point p(1 + b−a, 1 + b−a) with the parametric vector (a,b) is associated with the fixed point o(0, 0)(a∗,b∗) related to parametric vector (a ∗,b∗) = (a, 2a − b − 2). thus both fixed point o and p can undergo identical sets of bifurcations in parameter space. we can choose the parameters so that one fixed point experiences all the bifurcations, while the other has none. it suffices then to study the nature and bifurcations of o(0, 0) ; those of p are deduced automatically. j = ∣∣∣∣∣∣ 0 1 −b + 2x a ∣∣∣∣∣∣, we consider now the conditions of local stability of the fixed point o(0; 0), in terms of the parameters of the map t . with j(0,0) = ∣∣∣∣∣∣ 0 1 −b a ∣∣∣∣∣∣, is jacobian matrix of t in o(0; 0) which has two eigenvalues λ1 = a2 + √ a2 4 − b; λ2 = a 2 − √ a2 4 − b. we can conclude for a2 − 4b > 0 with considering b < a 2 4 : proposition 2.3. if 1 + a + b = 0 or 1 + b + 3a = 0, the map t undergoes a flip bifurcation at the fixed point o(0, 0) or p(1 + b−a, 1 + b−a), respectively. int. j. anal. appl. 17 (4) (2019) 522 proof. consider the square map t2(the second iterate of t) : t2(x; y) = t(y,ay − bx + x2) = (ay − bx + x2,a(ay − bx + x2) − by + y2). fixed points of t2 are computed by considering t2(x,y) = (x,y),we obtain then : ay − bx + x2 = x and a(ay−bx + x2)−by + y2 = y.we put from the first equation y = ((1 + b)x−x2)/a, and by replacing y with its value in the second equation we obtain a quartic polynomial of x. after simplification of the two fixed points, we have this quadratic polynomial : q(x) = x2 − (1 + a + b)x + a(1 + a + b). roots of q(x) are the abscissa of the two points of the cycle of order 2 : x1,2 = ((1 + a + b)± √ (1 + a + b)(1 + b− 3a))/2 and the ordinates are : y1 = x2 ; y2 = x1. the sufficient condition for the existence of the 2-cycle is given by (1 + a + b)(1 + b− 3a) > 0. if (1 + a + b)(1 + b− 3a) = 0 we have a flip bifurcation. let us denote : λ1 =     a b  / 1 + a + b = 0   and λ′1 =     a b  / 1 + b− 3a = 0   if 1 + a + b = 0 we have x1 = x2 = 0, this flip bifurcation line λ1 is associated with the fixed point o(0, 0). if 1 + b− 3a = 0 we have x1 = x2 = 2a, this flip bifurcation line λ ′ 1is associated with the other fixed point p(1 + b−a, 1 + b−a). also the two curves λ1 and λ ′ 1 are clearly visible in figure 1.(a). � proposition 2.4. the map t undergoes discrete hopf bifurcations at the fixed point o(0, 0) for a = 2cos (αk) and b = 1. proof. we obtain the parameter values related to hopf bifurcations for the fixed point o(0, 0) of focus type if we put the eigenvalues λ1 = exp(iαk) and λ2 = exp(−iαk) with αk = 2πk ; k ∈ n. � proposition 2.5. the parametric line b− 2a + 3 = 0 is the symmetric of the line ∆ : b = 1 with respect to the line λ(1)0 : b−a + 1 = 0. this line ∆′ : b−2a+ 3 = 0 is associated with the fixed point p and plays a key role for global bifurcations. similarly for ∆ which is associated with the trivial fixed point o for global bifurcations. proof. with a simple computation we obtain this line. � from fig1.(a), we can see that the stability domain of the trivial fixed point o = (0, 0) is represented by a blue triangle, bounded by the transcritical bifurcation curve λ(1)0 : b−a + 1 = 0, the flip bifurcation curve 1 + a + b = 0 and the hopf bifurcation curve ∆ : b = 1. and for the fixed point p(1 + b−a, 1 + b−a) another blue triangle is associated, bounded by the transcritical bifurcation curve ∆′ : b−a + 1 = 0, the flip bifurcation curve 1 + b + 3a = 0 and the hopf bifurcation curve ∆′ : b− 2a + 3 = 0 int. j. anal. appl. 17 (4) (2019) 523 3. uncoupled map properties for a = 0, the second iterate of the map, i.e. t2 is a uncoupled map because t2(x; y) = t(y,−bx + x2) = (−bx + x2,−by + y2) = (f2(x),f2(y)) is with separate variables.then, the dynamics of t is associated with those of f2 and their bifurcations also are strictly related. if we consider then f2(x) = −bx+x2, its derivative is f′2(x) = −b + 2x. for −3 < b < −1 : f2 has two points, the trivial point x = 0 which is unstable with an eigenvalue equal to b, and the other point b + 1 stable and its eigenvalue is equal to b + 2. also t has two fixed points o = (0; 0) et p = (b + 1; b + 1). the first point is a saddle with eigenvalues λ1,2= ± √ −b. the second point p has two eigenvalues λ1,2 = ± √ b + 2 is a stable star-node for b ∈]−2;−1[ ; and a stable focus for b ∈] − 3;−2[ . t possesses a 2-cycle which is a saddle c2t = {(0; b + 1); (b + 1; 0)} ; with eigenvalues b; b + 2. for b = −1; f2(x) has a unique fixed point x = 0; such that f′2(0) = 1; that means that b = −1 is a transcritical bifurcation value. for b = −3; the fixed point x = 0 is still unstable, and the other point b + 1 has an eigenvalue equal to −1; b = −3 is a flip bifurcation value. for −1 − √ 6 < b < −3; the second point b + 1 becomes unstable and gives rise to a stable 2−cycle c2f2 : {x1; x2} = { 1 2 [ b− 1 + √ (b− 1)(b + 3) ] ; 1 2 [ b− 1 − √ (b− 1)(b + 3) ]} , with an eigenvalue equal to 1−(b− 1)(b + 3). t has 4−cycle c4t : {(x1; x2); (x2; x2); (x2; x1); (x1; x1)} . for b = −1 − √ 6; the 2− cycle c2f2 : {x1; x2} of f2(x) becomes unstable since f 2′ 2 (x) = −1, and a 4−cycle c4f2 : {x3; x4; x5; x6} appears. two stable nodes 8−cycles of homogene type are generated by the 4−cycle : c8t,1 : { ti(x3; x3); i = 1, ..., 8 } ; c8t,2 : { ti(x4; x3); i = 1, ..., 8 } these two 8−cycles have attraction basins of rectangular form and disjoints. the combinaison of the 2−cycle c2f2 : {x1; x2} with the 4−cycle c 4 f2 : {x3; x4; x5; x6} leads to the creation of four 8−cycles of mixed saddle type. and two others are given by the combinaison with the two fixed points such that c8t,3 : { ti(x1; x3); i = 1, ..., 8 } ; c8t,4 : { ti(x2; x3); i = 1, ..., 8 } ; c8t,5 : { ti(0; x3); i = 1, ..., 8 } ; c8t,6 : { ti(1 + b; x3); i = 1, ..., 8 } . we remark the emergence of a decreasing sequence of the parameter values of b : {b1; b2; b3, ...}which correspond to flip bifurcation values. these values of b tend towards b1 such the attractor becomes a cantor set. table of the parameter values of b : f2(x) = −bx + x2: n 1 2 3 4 5 6 bn -3 -3.44931 -3.54402 -3.56437 -3.56875 -3.56994 int. j. anal. appl. 17 (4) (2019) 524 we give the position of cycles for the map t and precisely cycles which are generated by an infinite sequence of flip bifurcations and provocate chaos. the two fixed points o and p can be stable inside a given region of the space of the parameters of the map, and can lose stability via a hopf bifurcation, as well as a transcritical bifurcation or a flip bifurcation. this last situation is reported in the qualitative fig. 2.(a,b,c) for b < 0, when o is unstable and p a stable point. (a) b < 0; o and p the two fixed points and c2 t = {c12 ; c 2 2} = {(0; b + 1); (b + 1; 0)} (b) different cycles of order 4 (c) different cycles of order 8 figure 2. different cycles for b < 0. other pecular properties that can be deduced from the properties of the one-dimensional map f2(x) for the endomorphism t when a = 0 concern critical curves lc−1, lc, lc1, lc2 which are given here by( lc−1 : x = s−1; where s−1= b 2 is the critical point of rank 0 of f2(x) ) and   lc : y = s0 = f2(s−1); lc1 : x = s0 = −b24 lc2 : y = s1; lc3 : x = s1   with sk = f2(sk−1) of rank (k + 1) for b = −2, the critical points s−1,s0,s1 are equal, b < −2, the interval i = [s0,s1] is an absorbing interval containing an attracting set, but for b > −2, the interval i = [s0,s1] is not an absorbing interval. for selected values of the parameter b and for n = 2, the coexistence of many different local attractors opens an important question on their behavior, and the problem of the delimitation of the boundaries of their basins which are made up of rectangles. in figures 3.(a,b), the immediate basin (containing attractors) is formed by disjoint rectangles bounded by segment of the critical curves which are straight lines parallel to the coordinates axes, and issued from the critical points of f2(x). studying these basins help in understanding the ways of multistability formation. we refer to [13] for its complete description. the bifurcation curves related to the two values of b = 1 and b = −3.7 correspond to homoclinic bifurcation curves for the fixed points o or p and determine also the end of the range in which the quadratic shape of f2(x) plays no role. we can see an explosion of straight lines of different colors bounding rectangular basins. int. j. anal. appl. 17 (4) (2019) 525 (a) a=0; b=1 (b) a=0; b=-3.7 figure 3. basins and critical curves. 4. basins and invariant curves we present some results of numerical simulations and discuss their implications. we have to explain the creation of two invariant curves, one issued from a fixed point of saddle type, constitutes the basin boundary of the second one which is an attracting closed curve. the phase portrait of the quadratic endomorphism t is studied by constructing regular invariant curves associated to saddles and hopf bifurcation. in this section, the regions of parameters chosen for numerical experiments contain parametric values which are not arbitrary but characteristic values with focus on selected properties. thus, in the following we shall focus our attention on the description of the bifurcations which are expected to occur as we vary the parameters a close to 2 in (a) and (b) and −2 in (c) and (d) and b close to the value 1. since b > 0, the fixed point o is stable, inside its attraction basin undergoes a hopf bifurcation and a little closed curve exists inside the basin, together with a closed stable manifoldof the saddle point p . the following figures show the corresponding basin structure of t for n = 2. figures 4.(a,b,c,d,e,f) represent the existing attractors (the two fixed points o; p , an invariant closed curve (icc) around the fixed pointo after a hopf bifurcation) and the stable manifold emanating from the saddle point. in figs 4.(a,b) a ' 2,b ' 1 (λ1 = λ2 = 1), the two fixed points exist, p is a saddle and its invariant stable manifold delimits the basin of o which has undergone a hopf bifurcation when a uncreases from 1.7 to 1.9. the two curves are closed and invariant. in figs 4.(c,d), a '−2,b = 1 (λ1 = λ2 = −1), the fixed point o undergoes the flip bifurcation λ1 (1 + b + a = 0) and the 2−cycle is still merged with o which gives again a icc. in figs 4.(e,f), and for n = 4 we have nearly the same behavior with invariant curves associated with the fixed points. we observe that small changes in the location of the critical curve lc−1 which has some effects on the properties of the attractors, and it may cause remarkable asymmetries in the structure of the basins, which can only be detected from the global properties of the studied model. int. j. anal. appl. 17 (4) (2019) 526 (a) invariant closed curve in a yellow basin n=2 (b) only invariant curves are displayed n = 2, x = {−0.36, .., 1}, y = {−0.28, .., 0.15} (c) invariant closed curve in a yellow basin n=4 (d) only invariant curves are displayed n = 4, x = {−1, .., 1}, y = {−0.28, .., 1} (e) a > 0 (f) a < 0 figure 4. bounded basin delimited by the invariant stable manifold and an icc around the stable fixed point. other invariant sets for n = 2 (and similarly for n even) can be estimated for t2. these invariant sets are obtained by iterating invariant lines in the immediate bounded basin. the case a = −2 is very interesting, because we can put in evidence the existence of such sets. the special character of this kind of sets has been already observed in uncoupled maps. to illustrate the idea, we put l1 : y2 = αx2 + β, x2 and y2 are the second iterates of x and y. we obtain then a(ay − bx + x2) − by + y2 = α(ay − bx + x2) + β. by virtue of invariance of l1 by t ( y = αx + β), we have : for all x, (a + α2 −α)x2 + (a2α−ab + 2αβ −aα2)x + (a2β − bβ + β2 −aαβ −β) = 0. then a + α2 −α = 0 ; β(−b + β −aα2 − 1) = 0 and −ab + 2αβ −aα3 = 0 for a = −2, we have l1 : y = −x + b− 1. if we take b = 1 + ε, then l1 : y = −x + ε the iterates t(l1) are also invariant because t 2(t(l1)) = t(t 2(l1)) = t(l1). t(l1) is a parabola whose equation is y = x2 α2 + (a− b α − 2β α2 )x + β α ( b + β α ) = x2 − bx− b + 1 int. j. anal. appl. 17 (4) (2019) 527 if we take β = 0, we have l2 : y = 0, l3 : y = x. besides the elements seen thus far, there is another particularity in the dynamics of t . the strong dependence on the parameters causes a rich variety of complex patterns on the plane and gives rise to different kinds of basins. taking into account the complexity of the matter and its nature, the study of these phenomena can be carried out only via the association of numerical investigations guided by fundamental considerations that can be found in [13]. the endomorphism (2) with a 6= 0 is of type (z0 − z2), whose three first critical curves lc−1, lc, lc ′ 1 are given here by( lc−1 : { (x,y) /x = b 2 } ) and   lc : t(lc−1) = { (x,y) /y = ax− b 2 4 } lc1 : { (x,y) /y = 1 a2 x2 + (a + b 2 2a2 − b a )x + b 4 16a2 − b 3 4a }   the inverses t−1± are given by: t−1± (x,y) =   x = b 2 ± 1 2 √ b2 + 4(y −ax) y = x (8) attractors constitute an interesting set of study by themselves. their attraction basins are split in two equal parts by lc−1. to understand the behavior of the map when a is close to −2, we follow the evolution in the phase plane when we vary the parameter b (b ≈−5.87, b = 0.95). all these situations have been shown in [13] proposed by mira et al. for a wide variety of reasons and different models, in all they incorporate nonlinear terms and quadratic nonlinearities. in most cases, the basin of attraction undergoes some changes. the geometric structure of the basin is occurring for particular choices of the parameters. recent works dealing with cases of multiple attractors in noninvertible maps have highlighted how noninvertibility can become a source of bifurcations and complex structure in the basins of attraction. however, the global phenomena and the complex structures of the basins shown here are due to the both sets which are critical and invariant sets. for a = −2, the critical curves lc−1, lc, lc1 and lc2 delimit the attractor that occupies the whole basin. the invariant lines of t and t2 respectively (y = x and y = −x + 1) contain fixed points (0, 0) and (5, 5) for y = x and the two points of the 2−cycle {(2,−1); (−1, 2)} for y = −x + 1 (see fig. 6.). 5. conclusion this family of polynomial maps with n = 2 is symmetric with respect to the curve lc−1. and the bifurcations shown in the different sections are related to this curve, but we can observe that basins become asymmetric if n = 2k,k > 1. a variety of interesting properties in this family are pointed out, including degenerate and int. j. anal. appl. 17 (4) (2019) 528 (a) connected basin (b) bifurcation basin to nonconnected basin (c) nonconnected basin, b < 0 (d) nonconnected basin, b > 0 figure 5. the evolution of basin in the phase plane. figure 6. critical and invariant sets for a = −2. global bifurcations. according to the bifurcation diagram shown in fig.1(a) and by exploiting the relation between invariant sets and critical curves, we have considered four parametric points (a,b) which are of two or three codimension bifurcation points, (−2, 1), (−2,−7), (2, 1) and (0,−1). the two first points are of two-degree of complexity and the two last points are of three-degree of complexity, the behavior of the solutions is mainly determined by the invariant sets and critical curves. we are not trying to show the details of the bifurcation mechanisms leading to the dynamic situations described above for n = 2k,k > 1, however, we believe that they have identical behavior to those observed in secs. 2 and 3, and display the same structure associated with the basin bifurcation connected-nonconnected described in sec. 4, even if they occur in a very narrow interval of values of the parameter b. references [1] g. s. agarwal, existence of multistability in systems with complex order parameters, phys. rev. a, 26 (1982), 888–891. [2] f. t. arecchi, r. meucci, g. puccioni and and j. tredicce, experimental evidence of subharmonic bifurcations, multistability, and turbulence in a q-switched gas laser, phys. rev. lett., 49 (1982), 1217–1220. [3] r. l. clerc and c. hartmann, invariant manifolds of separable discrete dynamic systems, dynamics days, la jolla, california, (1982). [4] i. djellit and y. soula, on riddled sets and bifurcations of chaotic attractors, appl. math. sci., 1 (13) (2007), 603–614. int. j. anal. appl. 17 (4) (2019) 529 [5] m. r. ferchichi, i. djellit and j. c. sprott, broken symmetry in modified lorenz model, int. j. dyn. syst. differential equations, 5 (2) (2015), 136-148. [6] u. feudel and c. grebogi, multistability and the control of complexity, chaos, 7 (1997), 597–604. [7] c. grebogi, e. kostelich, e. ott and j. a. yorke, multi-dimensioned intertwined basin boundaries: basin structure of the kicked double rotor, physica d, 25 (1987), 347–360. [8] i. gumowski, c. mira, dynamique chaotique, ed. cepadues, toulouse, (1980). [9] m. henon, a two-dimensional mapping with a strange attractor, commun. math. phys., 50, (1976), 69–77. [10] k. ikeda, multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, opt. commun., 30 (1976), 257–261. [11] h. w. lorenz, multiple attractors, complex basin boundaries, and transient motion in deterministic economic systems, in dynamic economic models and optimal control, ed. feichtinger, g. (elsevier), (1992). [12] c. mira, chaotic dynamics, world scientific, (1987). [13] c. mira, l. gardini, a. barugola and j. c. cathala, chaotic dynamics in two-dimensional non invertible maps, world scientific, (1996). [14] b. razafimandimby, domaine d’influence de certaines singularités stables d’un endomophisme de r2, thèse de 3ème cycle, univ. paul sabatier, toulouse, (1981). 1. introduction 2. symmetry and fundamental properties 3. uncoupled map properties 4. basins and invariant curves 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 179-184 http://www.etamaths.com existence result for nonlinear initial value problems involving the difference of two monotone functions j.a. nanware abstract. in this paper, monotone iterative technique for nonlinear initial value problems involving the difference of two functions is developed. as an application of this technique, existence of solution of nonlinear initial value problems involving the difference of two functions is obtained. 1. introduction in the last few decades many authors pointed out that fractional derivatives and fractional integrals are very suitable for the description of properties of various real materials, e.g. polymers. it has been shown that new fractional order models are more adequate than integer order models. the advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, and in many other fields, like theory of fractals [10]. many dynamical models are described by fractional differential equations. analytical as well as numerical methods are available for studying fractional differential equations such as power series method, compositional method, transform method and adomain methods etc. (see details in [4, 14, 21] and references therein). the method of lower and upper solutions has been effectively used for proving the existence results for a class of variety of nonlinear problems. monotone iterative technique coupled with method of lower and upper solutions is an effective mechanism that offers constructive procedure to obtain existence results in a closed set [5]. the basic theory of fractional differential equation with riemann-liouville fractional derivative is developed in [2, 7, 9]. in 2008, lakshmikantham and vatsala obtained the local and global existence of solution of riemann-liouville fractional differential equation and uniqueness of solution in [6, 8]. recently, mcrae [11] developed monotone method for riemann-liouville fractional differential equation with initial conditions and studied the qualitative properties of solutions of initial value problem. recently, nanware et.al. developed monotone method for system of caputo fractional differential equations with periodic boundary conditions when the function is quasimonotone nondecreasing and mixed quasimonotone[3, 19], riemann-liouville fractional differential equations with integral boundary conditions when the function on the right is sum of nondecreasing and nonincreasing functions [15] and system of riemann-liouville fractional differential equations with 2010 mathematics subject classification. 34a12,34c60. key words and phrases. fractional differential equations; initial value problems; lower and upper solutions, existence result. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 179 180 nanware integral boundary conditions when the function is quasimonotone nondecreasing [12, 16, 20]. monotone method is successfully applied to obtain existence and uniqueness of solutions of these problems [12, 13, 17, 18]. monotone iterative technique for the following initial value problem u′ = f(t,u) −g(t,u), u(0) = u0, where f and g are in c([j × r,r]) and nondecreasing in u, uniformly in t, is developed by bhaskar and mcrae [1]. in this paper, monotone iterative technique is developed for nonlinear initial value problems involving the difference of two monotone functions with riemann-liouville fractional derivative and successfully applied this technique to obtain existence of solution of the problem. the paper is organized in the following manner: basic definitions and results are considered in the second section. monotone iterative technique is developed in the third section and the technique is successively employed to prove existence result. in the last section some remarks are given. 2. definitions and basic results the riemann-liouville fractional derivative of order q, (0 < q < 1) [21] is defined as [0d q t ]u(t) = 1 γ(1 −q) d dt ∫ t 0 u(s) (t−s)q ds.(2.1) consider the following riemann-liouville fractional differential equation (2.2) [0d q t ]u(t) = f(t,u(t)) −g(t,u(t)), t ∈ j = [0,t] with initial condition (2.3) u(0) = u0 where f,g ∈ c(j × r,r) are both nondecreasing in u(t), uniformly in t. this is called a nonlinear initial value problem (ivp). definition 2.1. a pair of functions v(t) and w(t) in cp(j,r) are called ordered lower and upper solutions of the nonlinear ivp (2.2) − (2.3) if [0d q t ]v(t) ≤ f(t,v(t)) −g(t,v(t)), v(0) ≤ u0 [0d q t ]w(t) ≥ f(t,w(t)) −g(t,w(t)), w(0) ≥ u0 definition 2.2. the functions v(t) and w(t) in cp(j,r) are called coupled lower and upper solutions of the nonlinear ivp (2.2) − (2.3) if [0d q t ]v(t) ≤ f(t,v(t)) −g(t,w(t)), v(0) ≤ u0 [0d q t ]w(t) ≥ f(t,w(t)) −g(t,v(t)), w(0) ≥ u0 lemma 2.1. [2] let m ∈ cp(j,r) and for any t1 ∈ (0,t] we have m(t1) = 0 and m(t) < 0 for 0 ≤ t < t1. then it follows that dqm(t1) ≥ 0. lemma 2.2. [6] let {u�(t)} be a family of continuous functions on j, for each � > 0 where dqu�(t) = f(t,u�(t)), u�(t0) = u�(t)(t− t0)1−q}t=t0 and |f(t,u�(t))| ≤ m for t0 ≤ t ≤ t . then the family {u�(t)} is equicontinuous on [t0,t]. theorem 2.1. [11] let v,w ∈ cp(j,r),f ∈ c([t0,t] ×r,r) and nonlinear initial value problems 181 : (i) dqv(t) ≤ f(t,v(t)) and : (ii)dqw(t) ≥ f(t,w(t)), t0 < t ≤ t. assume f(t,u) satisfies the lipschitz condition f(t,x) −f(t,y) ≤ l(x−y), x ≥ y,l > 0. then v0 < w0, where v0 = v(t)(t−t0)1−q|t=t0 and w0 = w(t)(t−t0)1−q|t=t0, implies v(t) ≤ w(t), t ∈ [t0,t]. 3. main results in this section we develop monotone iterative technique for nonlinear ivp (2.2)− (2.3). as an application of the technique we prove the existence of solution of nonlinear initial value problem (2.2) − (2.3). theorem 3.1. assume that: (i): f(t,u(t)) and g(t,u(t)) in c[j ×r,r] are nondecreasing in u(t), (ii): v0(t) and w0(t) in c(j,r) are coupled lower and upper solutions of ivp (2.2) − (2.3) such that v0(t) ≤ w0(t), t ∈ j = [0,t]. then there exist monotone sequences {vn(t)} and {wn(t)} in c(j,r) such that {vn(t)}→ v(t) and {wn(t)}→ w(t) as n →∞, uniformly and monotonically on j and the functions v(t) and w(t) are the coupled minimal and maximal solutions of nonlinear ivp (2.2) − (2.3) respectively. proof : consider the following coupled linear system of fractional differential equations with initial conditions (livp) (3.1) vn+1(t) = f(t,vn) −g(t,wn), vn+1(0) = u0 [0d q t ]wn+1(t) = f(t,wn) −g(t,vn), wn+1(0) = u0 since the functions f(t,u) and g(t,u) are continuous on j × r, the solutions of livp (3.1) exist on j. we claim that v0(t) ≤ v1(t) ≤ w1(t) ≤ w0(t) on j. for this, set p(t) = v1(t) −v0(t) then we have [0d q t ]p(t) = [0d q t ]v1(t) − [0d q t ]v0(t) ≥ f(t,v0) −g(t,w0) −f(t,v0) + g(t,w0) [0d q t ]p(t) ≥ 0 p(0) = 0 by applying theorem 2.1, we get v0(t) ≤ v1(t). similarly, we can prove w1(t) ≤ w0(t) on j. also, we prove that v1(t) ≤ w1(t) on j. set p(t) = w1(t) −v1(t). then we have [0d q t ]p(t) = [0d q t ]w1(t) − [0d q t ]v1(t) ≥ f(t,w0) −g(t,v0) −f(t,v0) + g(t,w0) [0d q t ]p(t) ≥ 0 p(0) ≥ 0 thus,by applying theorem 2.1, we get v1(t) ≤ w1(t). assume that for some k > 1, vk−1(t) ≤ vk(t) ≤ wk(t) ≤ wk−1(t). we claim that 182 nanware vk(t) ≤ vk+1(t) ≤ wk+1(t) ≤ wk(t) on j. to prove this, set p(t) = vk(t) −vk+1(t). since f(t,u) and g(t,u) are nondecreasing in u, we get [0d q t ]p(t) = [0d q t ]vk(t) − [0d q t ]vk+1(t) ≤ f(t,vk−1) −g(t,wk−1) −f(t,vk) + g(t,wk) [0d q t ]p(t) ≤ 0 p(0) = 0 by applying theorem 2.1, we have vk ≤ vk+1 on j. by induction, it follows that vk ≤ vk+1 for all k ≥ 1, t ∈ j. similarly we prove wk+1(t) ≤ wk(t) on j. next we prove vk+1(t) ≤ wk+1(t). consider p(t) = wk+1(t) −vk+1(t). since f(t,u) and g(t,u) are nondecreasing in u, we have [0d q t ]p(t) = [0d q t ]wk+1(t) − [0d q t ]vk+1(t) ≥ f(t,wk) −g(t,vk) −f(t,vk) + g(t,wk) [0d q t ]p(t) ≥ 0 p(0) = 0 hence, by applying theorem 2.1, we get vk+1 ≤ wk+1 on j. by induction, we get vk+1 ≥ wk+1 for all k ≥ 1, t ∈ j. thus we have sequences vn and wn on j such that v0 ≤ v1 ≤ v2 ≤ ... ≤ vn ≤ wn ≤ wn−1 ≤ ... ≤ w2 ≤ w1 ≤ w0. clearly the sequences {vn} and {wn} are nondecreasing and bounded below and nondecreasing and bounded above respectively. by lemma 2.2 it follows that the sequences {vn} and {wn} are equicontinuous and uniformly bounded. applying ascoli-arzela theorem, there exist convergent subsequences {vnk} and {wnk} converging to v and w uniformly and monotonically on j respectively. then we have {vn(t)}→ v(t) and {wn(t)}→ w(t) as n →∞. using corresponding fractional volterra integral equations (3.2) vn+1(t) = u0 + 1 γ(q) ∫ t t0 (t−s)q−1 { f(s,vn(s)) −g(s,wn(s)) } ds wn+1(t) = u0 + 1 γ(q) ∫ t t0 (t−s)q−1 { f(s,wn(s)) −g(s,vn(s)) } ds it follows that v(t) and w(t) are solutions of (3.1). next we claim that v(t) and w(t) are the coupled minimal and maximal solutions of livp (3.1). for this, let u(t)) be any solution of nonlinear ivp (2.2) − (2.3) different from v(t) and w(t), so that there exists k such that vk(t) ≤ u(t) ≤ wk(t) on j and set p(t) = u(t) −vk+1(t) so that [0d q t ]p(t) = [0d q t ]u(t) − [0d q t ]vk+1(t) ≥ f(t,u) −g(t,u) −f(t,vk) + g(t,wk) [0d q t ]p(t) ≥ 0 p(0) = 0. by applying theorem 2.1, we have vk+1(t) ≤ u(t) on j. since v0(t) ≤ u(t) on j, by induction it follows that vk(t) ≤ u(t) for all k. nonlinear initial value problems 183 similarly we prove u(t) ≤ wk(t) for all k on j. thus vk(t) ≤ uk(t) ≤ wk(t) on [0,t]. in limiting case, we have v(t) ≤ u(t) ≤ w(t) on [0,t]. this completes the proof. 4. remarks (1) if f(t,u) and g(t,u) in c(j×r) and if there exists positive constants m,n such that f(t,u) + mu and g(t,u) + nu are both nondecreasing, for t ∈ j and v0 ≤ u ≤ w0 then we may write g(t,u) = f(t,u) −g(t,u) = |f(t,u) + (m + n)u|− |g(t,u) + (m + n)u| = f1(t,u) −g1(t,u). clearly f1 and g1 are both monotone nondecreasing functions and theorem 3.1 can be applied. (2) if g = 0 and as in theorem 2.1, f satisfies for some m > 0, f(t,u1) −f(t,u2) ≥−m(u1 −u2) whenever u1 ≥ u2. define f1 = f(t,u) + mu. then f1 is nondecreasing in u and we may write [0d q t ]u(t) = f(t,u), u(0) = u0 as [0d q t ]u(t) = f1(t,u) −mu, u(0) = u0 and with appropriate modifications we can apply theorem 3.1 to [0d q t ]u(t) = f1(t,u) −mu, u(0) = u0. thus we obtain new result. (3) if f = 0 and g satisfies for some m > 0, g(t,u1) −g(t,u2) ≥−m(u1 −u2), whenever u1 ≥ u2, we define g1(t,u) = g(t,u) + mu. then g1 is nondecreasing in u and we write [0d q t ]u(t) = −g(t,u), u(0) = u0 as [0d q t ]u(t) = mu−g1(t,u), u(0) = u0. theorem 2.1 may be applied to obtain the coupled minimal and maximal solutions of the original problem. (4) theorem 3.1 can easily be modified to include the ivp of the form [0d q t ]u(t) + ku(t) = f(t,u) −g(t,u), u(0) = 0. 184 nanware references [1] t.g.bhaskar, f.a.mcrae, monotone iterative techniques for nonlinear problems involving the difference of two monotone functions, applied mathematics and computation 133 (2002), 187-192. [2] j.vasundhara devi, f.a.mcrae, z. drici, variational lyapunov method for fractional differential equations, computers and mathematics with applications,64 (2012), 2982-2989. [3] d.b.dhaigude, j.a.nanware and v.r.nikam, monotone technique for system of caputo fractional differential equations with periodic boundary conditions, dynamics of continuous, discrete and impulsive systems, 19 (2012), 575-584. [4] a.a. kilbas, h.m.srivastava, and j.j. trujillo, theory and applications of fractional differential equations, north holland mathematical studies vol.204. elsevier(north-holland) sciences publishers, amsterdam, 2006. [5] g.s.ladde, v.lakshmikantham, a.s.vatsala, monotone iterative techniques for nonlinear differential equations, pitman advanced publishing program, london, 1985. [6] v.lakshmikantham, a.s.vatsala, theory of fractional differential equations and applications, communications in applied analysis 11 (2007), 395-402. [7] v.lakshmikantham, a.s.vatsala, basic theory of fractional differential equations and applications, nonlinear analysis, 69 (2008), 2677-2682. [8] v.lakshmikantham, a.s.vatsala, general uniqueness and monotone iterative technique for fractional differential equations, applied mathematics letters, 21 (2008), no.8, 828-834. [9] v. lakshmikantham, s. leela and j.v. devi, theory and applications of fractional dynamical systems, cambridge scientific publishers ltd., 2009. [10] b.mandelbrot, the fractional geometry of nature, freeman, san francisco, 1982. [11] f.a.mcrae, monotone iterative technique and existence results for fractional differential equations, nonlinear analysis, 71 (2009), no.12, 6093-6096. [12] j.a.nanware, existence and uniqueness of solution of fractional differential equations via monotone method, bull. marathwada maths. society, 14 (2013), 39-55. [13] j.a.nanware, monotone method in fractional differential equations and applications, dr.babsaheb ambedkar marathwada university, ph.d thesis, 2013. [14] j.a.nanware, g.a.birajdar, methods of solving fractional differential equations of order α(0 < α < 1), bull. marathwada maths. society, 15 (2014), 40-53. [15] j.a.nanware, d.b.dhaigude, existence and uniqueness of solution of riemann-liouville fractional differential equations with integral boundary conditions, int. jour. nonlinear science, 14 (2012), 410-415. [16] j.a.nanware, d.b.dhaigude, monotone iterative scheme for system of riemann-liouville fractional differential equations with integral boundary conditions, math.modelling scien.computation, springer-verlag, 283 (2012), 395-402. [17] j.a.nanware, d.b.dhaigude, existence and uniqueness of solution of differential equations of fractional order with integral boundary conditions, j. nonlinear. sci. appl., 7 (2014), 246-254. [18] j.a.nanware, d.b.dhaigude, boundary value problems for differential equations of noninteger order involving caputo fractional derivative, adv. stu. contem. math., 24 (2014), 369-376. [19] j.a.nanware, d.b.dhaigude, monotone technique for finite system of caputo fractional differential equations with periodic boundary conditions, dynamics of continuous, discrete and impulsive systems. (accepted) [20] j.a.nanware, n.b.jadhav, d.b.dhaigude, monotone iterative technique for finite system of riemann-liouville fractional differential equations with integral boundary conditions, international conference of mathematical sciences 2014, elsevier, 235-238 (2014). [21] i.podlubny, fractional differential equations, san diego, academic press, 1999. department of mathematics, shrikrishna mahavidyalaya, gunjoti, dist.osmanabad 413 613(m.s), india international journal of analysis and applications volume 17, number 2 (2019), 167-190 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-167 new integral transform: shehu transform a generalization of sumudu and laplace transform for solving differential equations shehu maitama∗, weidong zhao school of mathematics, shandong university, jinan, shandong 250100, p.r. china ∗corresponding author: smusman12@sci.just.edu.jo abstract. in this paper, we introduce a laplace-type integral transform called the shehu transform which is a generalization of the laplace and the sumudu integral transforms for solving differential equations in the time domain. the proposed integral transform is successfully derived from the classical fourier integral transform and is applied to both ordinary and partial differential equations to show its simplicity, efficiency, and the high accuracy. 1. introduction historically, the origin of the integral transforms can be traced back to the work of p. s. laplace in 1780s and joseph fourier in 1822. in recent years, differential and integral equations have been solved using many integral transforms ( [1][11]). the laplace transform, and fourier integral transforms are the most commonly used in the literature. the fourier integral transform [12] was named after the french mathematician joseph fourier. mathematically, fourier integral transform is defined as: z[f(t)] = f(ω) = 1 √ 2π ∫ ∞ −∞ exp (−iωt) f(t)dt. (1.1) received 2018-07-27; accepted 2018-10-06; published 2019-03-01. 2010 mathematics subject classification. 44a10, 44a15, 44a20, 44a30, 44a35. key words and phrases. shehu transform; fourier integral transform; laplace transform; natural transform; sumudu transform; ordinary and partial differential equations. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 167 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-167 int. j. anal. appl. 17 (2) (2019) 168 the fourier transform have many applications in physics and engineering processes [13]. the laplace integral transform is similar with the fourier transform and is defined as: £[f(t)] = f(s) = ∫ ∞ −∞ exp (−st) f(t)dt. (1.2) the laplace transform is highly efficient for solving some class of ordinary and partial differential equations [14]. by replacing the variable iω with the variable s in equ.(1.1), the well-known fourier transform will become a laplace transform and the vice-versa. the only difference between the laplace transform, and the fourier transform is that the laplace transform can be defined for both stable and unstable system while the fourier transform can only be defined on a stable system. in mathematical literature, the discrete-time equivalent of the laplace transform called z-transform [15] converts a discrete-time signal into a complex frequency-domain representation. the basic idea of the z-transform was known to laplace and later it was re-introduced by the jewish-polish mathematician witold hurewicz to treat a sampled-data control systems used with radar in 1947 ( [16][17]). in mathematics and signal processing, the bilateral or two-sided z-transform of a discrete-time signal x[n] is the normal power series x(z) which is defined as: x(z) = z{x[n]} = ∞∑ n=−∞ x[n]z−n, (1.3) where n is an integer and z is in general a complex number [18]. the multiplicative version of the two-sided laplace transform called the mellin integral transform is defined as [19]: m[f(s); s] = f∗(s) = ∫ ∞ 0 xs−1f(x)dx. (1.4) the mellin integral transform is similar with the laplace transform and fourier transform and is widely applied in computer science and number theory due to its invariant property [20, 21]. in railway engineering, the laplace-carson transform [22] which is a laplace-type integral transform named after pierre simon laplace and john renshaw carson is defined as: f̂c(p) = p ∫ ∞ 0 exp (−pt) f(t)dt, t ≥ 0. (1.5) the laplace-carson integral transform have many applications in physics and engineering and can easily be converted into a mellin deconvolution problem, see [23, 24]. in mathematics, the hankel’s integral transform [25] which is similar to the fourier transform was first introduced by the german mathematician hermann hankel and was widely used in physical science and engineering [26]. the hankel’s transform is defined as: fv(s) = hv[f(r)] = ∫ ∞ 0 rf(r)jv(sr)dr, r ≥ 0, (1.6) where jv is the bessel function of the first kind of order v with v ≥−12 . int. j. anal. appl. 17 (2) (2019) 169 in 1993, watugala introduced a laplace-like integral transform called the sumudu integral transform [27]. in recent years, sumudu transform has been applied to many real-life problems because of its scale and unit preserving properties ( [28][31]). the mathematical definition of the sumudu transform is given by: s[f(t)](u) = g(u) = 1 u ∫ ∞ 0 exp ( −t u ) f(t)dt, (1.7) provided the integral exists for some u. based on the basic idea of the laplace and the sumudu integral transform, the elzaki transform was proposed in 2011. the elzaki transform is closely related with the laplace transform, sumudu transform, and the natural transform. elzaki transform is defined as [32]: e[f(t)] = t(u) = u ∫ ∞ 0 exp ( −t u ) f(t)dt, (1.8) provided the integral exists for some u. the natural transform [33] which is similar to laplace and sumudu integral transform was introduced in 2008. in recent years, natural transform was successfully applied to many applications (see [34, 35]). the natural transform is defined by the following integral: n+[f(t)](s,u) = r(s,u) = 1 u ∫ ∞ 0 exp ( −st u ) f(t)dt, s > 0, u > 0, (1.9) provided the integral exists for some variables u and s. recently, a new integral transform called the mtransform which is also similar to natural transform is introduced by srivastava et al. in 2015. mathematically speaking, m-transform is closely connected with the well-known laplace transform and the sumudu integral transform. m-transform was successfully applied to first order initial-boundary value problem (see srivastava et al. [36]). the m-transform is defined as: mρ,m[f(t)](u,v) = ∫ ∞ 0 exp (−ut) f(vt) (tm + vm) ρ dt, (1.10) (ρ ∈ c; <(ρ) ≥ 0, m ∈ z+ = 1, 2, 3, · · ·) , where both u ∈ c and v ∈ r+ are the m-transform variables. in 2013, atangana and kilicman introduced a novel integral transform called the abdon-kilicman integral transform [37] for solving some differential equations with some kind of singularities. the novel integral transform is defined as: mn(s) = mn[f(x)](s) = ∫ ∞ 0 xn exp (−xs) f(x)dx. (1.11) the atangana-kilicman integral becomes laplace transform when n = 0. recently, a laplace-type integral transform called the yang transform ( [38][40]) for solving steady heat transfer problems was introduced in 2016. the integral transform is defined as: y [φ(τ)] = φ(ω) = ∫ ∞ 0 exp ( −τ ω ) φ(τ)dτ, (1.12) provided the integral exists for some ω. int. j. anal. appl. 17 (2) (2019) 170 due to the rapid development in the physical science and engineering models, there are many other integral transforms in the literature. however, most of the existing integral transforms have some limitations and cannot be used directly to solved nonlinear problems or many complex mathematical models. as a result, many authors became highly interested to come up with the alternative approach for solving many real-life problems. in 2016, atangana and alkaltani introduced a new double integral equation and their properties based on the laplace transform and decomposition method. the double integral transform was successfully applied to second order partial differential equation with singularity called the two-dimensional mboctara equation [41]. recently, eltayeb applied double laplace decomposition method to nonlinear partial differential equations [42]. in 2017, belgacem el at. extended the applications of the natural and the sumudu transforms to fractional diffusion and stokes fluid flow realms [43]. motivated by the above-mentioned researches, in this paper we proposed a laplace-type integral transform called shehu transform for solving both ordinary and partial differential equations. the laplace-type integral transform converges to laplace transform when u = 1, and to yang integral transform when s = 1. the proposed integral transform is successfully applied to both ordinary and partial differential equations. all the results obtained in the applications section can easily be verified using the laplace or fourier integral transforms. throughout this paper, the shehu transform is denoted by an operator s[.]. 2. main result definition 1. the shehu transform of the function v(t) of exponential order is defined over the set of functions, a = { v(t) : ∃ n, η1, η2 > 0, |v(t)| < n exp ( |t| ηi ) , if t ∈ (−1)i × [0,∞) } , by the following integral s [v(t)] = v (s,u) = ∫ ∞ 0 exp ( −st u ) v(t)dt = lim α→∞ ∫ α 0 exp ( −st u ) v(t)dt; s > 0, u > 0. (2.1) it converges if the limit of the integral exists, and diverges if not. the inverse shehu transform is given by s−1 [v (s,u)] = v(t), for t ≥ 0. (2.2) equivalently v(t) = s−1 [v (s,u)] = 1 2πi ∫ α+i∞ α−i∞ 1 u exp ( st u ) v (s,u) ds, (2.3) int. j. anal. appl. 17 (2) (2019) 171 where s and u are the shehu transform variables, and α is a real constant and the integral in equ.(2.3) is taken along s = α in the complex plane s = x + iy. theorem 1. the sufficient condition for the existence of shehu transform. if the function v(t) is piecewise continues in every finite interval 0 ≤ t ≤ β and of exponential order α for t > β. then its shehu transform v (s,u) exists. proof. for any positive number β, we algebraically deduce∫ ∞ 0 exp ( − st u ) v(t)dt = ∫ β 0 exp ( − st u ) v(t)dt + ∫ ∞ β exp ( − st u ) v(t)dt. (2.4) since the function v(t) is piecewise continues in every finite interval 0 ≤ t ≤ β, then the first integral on the right hand side exists. besides, the second integral on the right hand side exists, since the function v(t) is of exponential order α for t > α. to verify this claim, we consider the following case∣∣∣∣ ∫ ∞ α exp ( − st u ) v(t)dt ∣∣∣∣ ≤ ∫ ∞ α ∣∣∣∣exp ( − st u ) v(t) ∣∣∣∣dt ≤ ∫ ∞ 0 exp ( − st u ) |v(t)|dt ≤ ∫ ∞ α exp ( − st u ) ne exp (βt) dt = n ∫ ∞ α exp ( − (s−βu)t u ) dt = − un (s−βu) lim γ→∞ [ exp ( − (s−βu)t u )]γ 0 = un s−βu . the proof is complete. � property 1. linearity property of shehu transform. let the functions αv(t) and βw(t) be in set a, then (αv(t) + βw(t)) ∈a, where α and β are nonzero arbitrary constants, and s [αv(t) + βw(t)] = αs [v(t)] + βs [w(t)] . (2.5) proof. using the definition 1 of shehu transform, we get s [αv(t) + βw(t)] = ∫ ∞ 0 exp ( −st u ) (αv(t) + βw(t))dt = ∫ ∞ 0 exp ( −st u ) (αv(t)) dt + ∫ ∞ 0 exp ( −st u ) (βw(t))dt = α ∫ ∞ 0 exp ( −st u ) v(t)dt + β ∫ ∞ 0 exp ( −st u ) w(t)dt = αu ∫ ∞ 0 exp (−st) v(ut)dt + βu ∫ ∞ 0 exp (−st) w(ut)dt = αs [v(t)] + βs [w(t)] . int. j. anal. appl. 17 (2) (2019) 172 the proof is complete. � in particular, using the definition 1 and property 1, we obtain s [3 cos(t) + 5 sin(2t)] = 3s [cos(t)] + 5s [sin(2t)] = 3us s2 + u2 + 5u2 s2 + (2u)2 , see entries of table 1. property 2. change of scale property of shehu transform. let the function v(βt) be in set a, where β is an arbitrary constant. then s [v(βt)] = u β v ( s β ,u ) . (2.6) proof. using the definition 1 of shehu transform, we deduce s [v(βt)] = ∫ ∞ 0 exp ( −st u ) v(βt)dt (2.7) substituting η = βt which implies t = η β and dt = dη β in equ.(2.7) yields s [v(βt)] = 1 β ∫ ∞ 0 exp ( −sη uβ ) v(η)dη = 1 β ∫ ∞ 0 exp ( −st uβ ) v (t) dt = u β ∫ ∞ 0 exp ( −st β ) v(ut)dt = u β v ( s β ,u ) . the proof is complete. � theorem 2. derivative of shehu transform. if the function v(n)(t) is the nth derivative of the function v(t) ∈ a with respect to ′t′, then its shehu transform is defined by s [ v(n)(t) ] = sn un v (s,u) − n−1∑ k=0 (s u )n−(k+1) v(k)(0). (2.8) when n=1, 2, and 3 in equ. (2.8) above, we obtain the following derivatives with respect to t. s [v′(t)] = s u v (s,u) −v(0). (2.9) s [v′′(t)] = s2 u2 v (s,u) − s u v(0) −v′(0). (2.10) s [v′′′(t)] = s3 u3 v (s,u) − s2 u2 v(0) − s u v′(0) −v′′(0). (2.11) int. j. anal. appl. 17 (2) (2019) 173 proof. now suppose equ. (2.8) is true for n = k. then using equ. (2.9) and the induction hypothesis, we deduce s [ (v(k)(t))′ ] = s u s [ v(k)(t) ] −v(k)(0) = s u [ sk uk s [v(t)] − k−1∑ i=0 (s u )k−(i+1) v(i)(0) ] −v(k)(0) = (s u )k+1 s [v(t)] − k∑ i=0 (s u )k−i v(i)(0), which implies that equ. (2.8) holds for n = k+ 1. by induction hypothesis the proof is complete. � the following important properties are obtain using the leibniz’s rule s [ ∂v(x,t) ∂x ] = ∫ ∞ 0 exp ( −st u ) ∂v(x,t) ∂x dt = ∂ ∂x ∫ ∞ 0 exp ( −st u ) v(x,t) dt = ∂ ∂x [v (x,s,u)] ⇒ s [ ∂v(x,t) ∂x ] = d dx [v (x,s,u)] , s [ ∂2v(x,t) ∂x2 ] = ∫ ∞ 0 exp ( −st u ) ∂2v(x,t) ∂x2 dt = ∂2 ∂x2 ∫ ∞ 0 exp ( −st u ) v(x,t) dt = ∂2 ∂x2 [v (x,s,u)] ⇒ s [ ∂2v(x,t) ∂x2 ] = d2 dx2 [v (x,s,u)] , and s [ ∂nv(x,t) ∂xn ] = ∫ ∞ 0 exp ( −st u ) ∂nv(x,t) ∂xn dt = ∂n ∂xn ∫ ∞ 0 exp ( −st u ) v(x,t)dt = ∂n ∂xn [v (x,s,u)] ⇒ s [ ∂nv(x,t) ∂xn ] = dn dxn [v (x,s,u)] . 3. some useful results of shehu transform property 3. let the function v(t) = 1 be in set a. then its shehu transform is given by s [1] = u s . (3.1) proof. using equ.(2.1), we deduce s [1] = ∫ ∞ 0 exp ( −st u ) dt = − u s lim γ→∞ [ exp ( −st u )]γ 0 = u s . this ends the proof. � property 4. let the function v(t) = t be in set a. then its shehu transform is given by s [t] = u2 s2 . (3.2) int. j. anal. appl. 17 (2) (2019) 174 proof. using the definition 1 of the shehu transform and integration by parts, we get s [t] = ∫ ∞ 0 t exp ( −st u ) dt = u s lim γ→∞ [ t exp ( −st u )]γ 0 + u s ∫ ∞ 0 exp ( −st u ) dt = − u2 s2 lim γ→∞ [ exp ( −st u )]γ 0 = u2 s2 . thus the proof ends. � property 5. let the function v(t) = t n n! n = 0, 1, 2.. be in set a. then its shehu transform is given by s [ tn n! ] = (u s )n+1 . (3.3) proof. from the definition 1 of the shehu transform and integration by parts, we deduce s [tn] = ∫ ∞ 0 tn exp ( −st u ) dt = u s n ∫ ∞ 0 tn−1 exp ( −st u ) dt = u2 s2 n(n− 1) ∫ ∞ 0 tn−2 exp ( −st u ) dt = u3 s3 n(n− 1)(n− 2) ∫ ∞ 0 tn−3 exp ( −st u ) dt = u4 s4 n(n− 1)(n− 2)(n− 3) ∫ ∞ 0 tn−4 exp ( −st u ) dt = u5 s5 n(n− 1)(n− 2)(n− 3)(n− 4) ∫ ∞ 0 tn−5 exp ( −st u ) dt = · · · = n! (u s )n+1 . the proof is completed. � property 6. let the function v(t) = t n γ(n+1) n = 0, 1, 2, · · · be in set a. then its shehu transform is given by s [ tn γ(n + 1) ] = (u s )n+1 . (3.4) the proof of property 6 follows immediately from the previous property 5. � property 7. let the function v(t) = exp(αt) be in a. then its shehu transform is given by s [exp(αt)] = u s−αu . (3.5) proof. using equ.(2.1), we get s [exp(αt)] = ∫ ∞ 0 exp ( − (s−αu)t u ) dt = − u s−αu lim γ→∞ [ exp ( − (s−αu)t u )]γ 0 = u s−αu . this ends the proof. � int. j. anal. appl. 17 (2) (2019) 175 property 8. let the function v(t) = t exp(αt) be in set a. then its shehu transform is given by s [t exp(αt)] = u2 (s−αu)2 . (3.6) proof. using the definition 1 of the shehu transform and integration by parts, we get s [t exp(αt)] = ∫ ∞ 0 t exp ( − (s−αu)t u ) dt = − u s−αu lim γ→∞ [ t exp ( − (s−αu)t u )]γ 0 + u s−αu ∫ ∞ 0 exp ( − (s−αu)t u ) dt = − u2 (s−αu)2 lim γ→∞ [ exp ( − (s−αu)t u )]γ 0 = u2 (s−αu)2 . the proof is complete. � property 9. let the function v(t) = tn exp(αt) n! n = 0, 1, 2, ... be in set a. then its shehu transform is given by s [ tn exp(αt) n! ] = un+1 (s−αu)n+1 . (3.7) proof. using the definition 1 of the shehu transform and integration by parts, we deduce s [tn exp(αt)] = ∫ ∞ 0 tn exp ( − (s−αu)t u ) dt = un (s−αu) ∫ ∞ 0 tn−1 exp ( − (s−αu)t u ) dt = u2n(n− 1) (s−αu)2 ∫ ∞ 0 tn−2 exp ( − (s−αu)t u ) dt = · · · = n! (s−αu)n+1 . thus the proof is complete. � property 10. let the function v(t) = t n γ(n+1) exp(αt) n = 0, 1, 2, ... be in set a. then its shehu transform is given by s [ tn exp(αt) γ(n + 1) ] = un+1 (s−αu)n+1 . (3.8) the proof of property 10 follows as a direct consequence of property 9. � property 11. let the function v(t) = sin(αt) be in set a. then its shehu transform is given by s [sin(αt)] = αu2 s2 + α2u2 . (3.9) int. j. anal. appl. 17 (2) (2019) 176 proof. using the definition 1 of the shehu transform and integration by parts, we get s [sin(αt)] = ∫ ∞ 0 exp ( − st u ) sin(αt)dt = − u s lim γ→∞ [ exp ( − st u ) sin(αt) ]γ 0 + uα s ∫ ∞ 0 exp ( − st u ) cos(αt)dt = − αu2 s2 lim γ→∞ [ exp ( − st u ) cos(αt) ]γ 0 − α2u2 s2 ∫ ∞ 0 exp ( − st u ) sin(αt)dt = αu2 s2 − α2u2 s2 ∫ ∞ 0 exp ( − st u ) sin(αt)dt. simplifying the required integrals complete the proof of property 11. � property 12. let the function v(t) = cos(αt) be in set a. then its shehu transform is given by s [cos(αt)] = us s2 + α2u2 . (3.10) proof. using the definition 1 of the shehu transform and integration by parts, we deduce s [cos(αt)] = ∫ ∞ 0 exp ( − st u ) cos(αt)dt = − u s lim γ→∞ [ exp ( − st u ) cos(αt) ]γ 0 − αu s ∫ ∞ 0 exp ( − st u ) sin(αt)dt = u s − αu2 s2 lim γ→∞ [ exp ( − st u ) sin(αt) ]γ 0 − α2u2 s2 ∫ ∞ 0 exp ( − st u ) cos(αt)dt = u s − α2u2 s2 ∫ ∞ 0 exp ( − st u ) cos(αt)dt. simplifying the required integrals complete the proof of property 12. � property 13. let the function v(t) = sinh(αt) α be in set a. then its shehu transform is given by s [ sinh(αt) α ] = u2 s2 −α2u2 . (3.11) proof. from the definition 1 of the shehu transform and integration by parts, we get s [sinh(αt)] = ∫ ∞ 0 exp ( − st u ) sinh(αt)dt = − u s lim γ→∞ [ exp ( − st u ) sinh(αt) ]γ 0 + uα s ∫ ∞ 0 exp ( − st u ) cosh(αt)dt = − αu2 s2 lim γ→∞ [ exp ( − st u ) cos(αt) ]γ 0 + α2u2 s2 ∫ ∞ 0 exp ( − st u ) sinh(αt)dt = αu2 s2 + α2u2 s2 ∫ ∞ 0 exp ( − st u ) sinh(αt)dt. simplifying the required integrals complete the proof of property 13. � int. j. anal. appl. 17 (2) (2019) 177 property 14. let the function v(t) = cosh(αt) be in set a. then its shehu transform is given by s [cosh(αt)] = us s2 −α2u2 . (3.12) proof. applying the definition 1 of the shehu transform and integration by parts, we get s [cosh(αt)] = ∫ ∞ 0 exp ( − st u ) cosh(αt)dt = − u s lim γ→∞ [ exp ( − st u ) cos(αt) ]γ 0 + αu s ∫ ∞ 0 exp ( − st u ) sinh(αt)dt = u s − αu2 s lim γ→∞ [ exp ( − st u ) sinh(αt) ]γ 0 + α2u2 s2 ∫ ∞ 0 exp ( − st u ) cos(αt)dt = u s + α2u2 s2 ∫ ∞ 0 exp ( − st u ) cosh(αt)dt. collecting the required integrals complete the proof of property 14. � property 15. let the function exp(βt) sin(αt) α be in set a. then its shehu transform is given by s [ exp (βt) sin(αt) α ] = u2 (s−βu)2 + α2u2 . (3.13) proof. using the definition 1 of the shehu transform and integration by parts, we deduce s [exp (βt) sin(αt)] = ∫ ∞ 0 exp ( − (s−βu) u t ) sin(αt)dt = −u (s−βu) lim γ→∞ [ exp ( − (s−βu) u t ) sin(αt)dt ]γ 0 + uα s−βu ∫ ∞ 0 exp ( − (s−βu) u t ) cos(αt)dt = − u2α (s−βu)2 lim γ→∞ [ exp ( − (s−βu) u t ) cos(αt)dt ]γ 0 − αu2α2 (s−βu)2 ∫ ∞ 0 exp ( − (s−βu) u t ) sin(αt)dt = u2α (s−βu)2 − u2α2 (s−βu)2 ∫ ∞ 0 exp ( − (s−βu) u t ) sin(αt)dt. simplifying the required integrals complete the proof of property 15. this ends the proof. � property 16. let the function exp (βt) cos(αt) be in set a. then its shehu transform is given by s [exp (βt) cos(αt)] = u(s−αu) (s−βu)2 + α2u2 . (3.14) int. j. anal. appl. 17 (2) (2019) 178 proof. applying the definition 1 of the shehu transform and integration by parts, we get s [exp (βt) cos(αt)] = ∫ ∞ 0 exp ( − (s−βu) u t ) cos(αt)dt = − u s−βu lim γ→∞ [ exp ( − (s−βu) u t ) cos(αt) ]γ 0 + αu s−βu ∫ ∞ 0 exp ( − (s−βu) u t ) sin(αt)dt = u s−βu + αu s−βu ∫ ∞ 0 exp ( − (s−βu) u t ) sin(αt)dt = u s−βu + αu2 (s−αu)2 lim γ→∞ [ exp ( − (s−βu) u t ) sin(αt) ]γ 0 − α2u2 (s−βu)2 ∫ ∞ 0 exp ( − (s−βu) u t ) cos(αt)dt = u s−βu − α2u2 (s−βu)2 ∫ ∞ 0 exp ( − (s−βu) u t ) cos(αt)dt. simplifying the required integrals complete the proof of property 16. � property 17. let the function exp(βt)−exp(αt) β−α be in set a. then its shehu transform is given by s [ exp (αt) β −α ] = u2 (s−αu)(s−βu) . (3.15) proof. using the definition of shehu transform, we get s [ exp (αt) β −α ] = u β −α ∫ ∞ 0 exp ( −st u ) (exp(βt) − exp (αt)) dt = 1 β −α ∫ ∞ 0 e (βu−s) u tdt− 1 β −α ∫ ∞ 0 exp ( (αu−s)t u ) dt = u (β −α)(βu−s) lim γ→∞ [ exp ( − (s−βu)t u )]γ 0 − u (β −α)(αu−s) lim γ→∞ [ exp ( − (s−βu)t u )]γ 0 = − u (β −α)(βu−s) + u (β −α)(αu−s) = −u(αu−s) + u(βu−s) (β −α)(αu−s)(βu−s) = u2 (s−αu)(s−βu) . the proof is complete. � property 18. let the function β exp(βt)−α exp(αt) β−α be in set a. then its shehu transform is given by: s [ β exp (βt) −α exp (αt) β −α ] = us (s−αu)(s−βu) . (3.16) int. j. anal. appl. 17 (2) (2019) 179 proof: using the definition of shehu transform, we get s [ β exp (βt) −α exp (αt) β −α ] = 1 β −α ∫ ∞ 0 exp ( −st u ) (β exp (βt) −α exp (αt)) dt = β β −α ∫ ∞ 0 exp ( (βu−s) u t ) dt− α β −α ∫ ∞ 0 exp ( (αu−s)t u ) dt = uβ (β −α)(βu−s) lim γ→∞ [ exp ( − (s−βu)t u )]γ 0 − uα (β −α)(αu−s) lim γ→∞ [ exp ( − (s−αu)t u )]γ 0 = − uβ (β −α)(βu−s) + uα (β −α)(αu−s) = −uβ(αu−s) + uα(βu−s) (β −α)(αu−s)(βu−s) = us (s−αu)(s−βu) . this ends the proof. � more properties of the shehu transform and their converges to the natural transform, the sumudu transform, and the laplace transform are presented in table 1. the comprehensive summary of shehu transform properties are presented in table 2. 4. applications in this section, the applications of the proposed transform are presented. the simplicity, efficiency and high accuracy of the shehu transform are clearly illustrated. example 1. consider the following first order ordinary differential equation dv(t) dt + v(t) = 0, (4.1) subject to the initial condition v(0) = 1. (4.2) applying the shehu transform on both sides of equ. (4.1), we get s u v (s,u) −v(0) + v (s,u) = 0. (4.3) substituting the given initial condition and simplifying, we deduce v (s,u) = u s + u (4.4) taking the inverse shehu transform of equ. (4.4), yields v(t) = exp(−t). (4.5) int. j. anal. appl. 17 (2) (2019) 180 example 2. consider the following second order ordinary differential equation d2v(t) dt2 + dv(t) dt = 1 (4.6) subject to the initial conditions v(0) = 0, dv(0) dt = 0. (4.7) applying the shehu transform on both sides of equ. (4.6), we obtain s2 u2 v (s,u) − s u v(0) −v′(0) + s u v (s,u) −v(0) = u s . (4.8) substituting the given initial conditions and simplifying, we deduce v (s,u) = − u s + u2 s2 + u s + u . (4.9) taking the inverse shehu transform of equ. (4.9), we get v(t) = −1 + t + exp(−t). (4.10) example 3. consider the following second nonhomogeneous order ordinary differential equation d2v(t) dt2 − 3 dv(t) dt + 2v(t) = exp(3t). (4.11) subject to the initial conditions v(0) = 1, dv(0) dt = 0. (4.12) applying the shehu transform on both sides of equ. (4.11), yields s2 u2 v (s,u) − s u v(0) −v′(0) − 3 (s u v (s,u) −v(0) ) + 2v (s,u) = u s− 3u . (4.13) substituting the given initial conditions and simplifying, we obtain v (s,u) = 5 2 u (s−u) − 2 u s− 2u + 1 2 u (s− 3u) . (4.14) taking the inverse shehu transform of equ. (4.14), we get v(t) = 5 2 exp(t) − 2 exp(2t) + 1 2 exp(3t). (4.15) example 4. consider the following ordinary differential equation d2v(t) dt2 + 2 dv(t) dt + 5v(t) = exp(−t) sin(t). (4.16) subject to the initial conditions v(0) = 0, dv(0) dt = 1. (4.17) applying the shehu transform on both sides of equ. (4.16), we get s2 u2 v (s,u) − s u v(0) −v′(0) + 2 (s u v (s,u) − s u v(0) ) + 5v (s,u) = u2 (s + u)2 + u2 (4.18) int. j. anal. appl. 17 (2) (2019) 181 substituting the given initial conditions and simplifying, we get v (s,u) = 1 3 u2 ((s + u)2 + u2) + 2 3 u2 ((s + u)2 + (2u)2) (4.19) taking the inverse shehu transform of equ. (4.19), we get v(t) = 1 3 exp(−t) sin(t) + 2 3 exp(−t) sin(2t). (4.20) example 5. consider the following homogeneous partial differential equation ∂v(x,t) ∂t = ∂2v(x,t) ∂x2 (4.21) subject to the boundary and initial conditions v(0, t) = 0, v(1, t) = 0, v(x, 0) = 3sin(2πx). (4.22) applying the shehu transform on both sides of equ. (4.21), we get s u v (x,s,u) −v(x, 0) = d2v (x,s,u) dx2 . (4.23) substituting the given initial condition and simplifying, we get d2v (x,s,u) dx2 − s u v (x,s,u) = −3sin(2πx). (4.24) the general solution of equ. (4.24) can be written as v (x,s,u) = vh(x,s,u) + vp(x,s,u), (4.25) where vh(x,s,u) is the solution of the homogeneous part which is given by vh(x,s,u) = α1 exp (√ s u x ) + α2 exp ( − √ s u x ) , (4.26) and vp(x,s,u) is the solution of the nonhomogeneous part which is given by vp(x,s,u) = β1sin(2πx) + β2cos(2πx). (4.27) applying the boundary conditions on equ. (4.26), we get α1 + α2 = 0 and α1 exp (√ s u ) + α2 exp ( − √ s u ) = 0 ⇒ vh(x,s,u) = 0, since α1 = α2 = 0. using the method of undetermined coefficients on the nonhomogeneous part, we get vp(x,s,u) = 3u s + 4π2u sin(2πx), (4.28) since, β1 = 3u s+4π2u , and β2 = 0. then equ. (4.25) will become v (x,s,u) = 3u s + 4π2u sin(2πx), (4.29) int. j. anal. appl. 17 (2) (2019) 182 taking the inverse shehu transform of equ. (4.29), we get v(x,t) = 3 exp(−4π2t) sin(2πx). (4.30) figure 1. 3d and 2d surfaces of the analytical solution of equ. (4.21) in the ranges −1 < x < 1, and −1 < t < 1, when t = 1. example 6. consider the following nonhomogeneous partial differential equation ∂2v(x,t) ∂t2 = β2 ∂2v(x,t) ∂x2 + sin(πx) (4.31) subject to the boundary and initial conditions v(0, t) = 0, v(1, t) = 0, v(x, 0) = 0, ∂v(x, 0) ∂t = 0, β2 = 1. (4.32) applying the shehu transform on both sides of equ. (4.31), we get s2 u2 v (x,s,u) − s u v(x, 0) −v′(x, 0) = d2v (x,s,u) dx2 + u s sin(πx). (4.33) substituting the given initial condition and simplifying, we get d2v (x,s,u) dx2 − s2 u2 v (x,s,u) = − u s sin(πx). (4.34) the general solution of equ. (4.34) can be written as v (x,s,u) = vh(x,s,u) + vp(x,s,u), (4.35) where vh(x,s,u) is the solution of the homogeneous part which is given by vh(x,s,u) = λ1 exp (s u x ) + λ2 exp ( − s u x ) , (4.36) and vp(x,s,u) is the solution of the nonhomogeneous part which is given by vp(x,s,u) = η1 sin(πx) + η2 cos(πx). (4.37) applying the boundary conditions on equ. (4.36), we deduce λ1 + λ2 = 0 and λ1 exp (s u ) + λ2 exp ( − s u ) = 0 ⇒ vh(x,s,u) = 0, int. j. anal. appl. 17 (2) (2019) 183 since λ1 = λ2 = 0. using the method of undetermined coefficients on the nonhomogeneous part, we get vp(x,s,u) = 1 π2 ( u s − us s2 + u2π2 ) sin(πx), (4.38) since, η1 = u3 s(s2 + u2π2) = 1 π2 ( u s − us s2 + u2π2 ) , and η2 = 0. then equ. (4.35) will becomes v (x,s,u) = 1 π2 ( u s − us s2 + u2π2 ) sin(πx). (4.39) taking the inverse shehu transform of equ. (4.39), we get v(x,t) = 1 π2 (1 − cos(πt)) sin(πx). (4.40) figure 2. 3d and 2d surfaces of the analytical solution of equ. (4.31) in the ranges −1 < x < 1, and −1 < t < 1. 5. conclusion we introduced an efficient laplace-type integral transform called the shehu transform for solving both ordinary and partial differential equations. we presented its existence and inverse transform. we presented some useful properties and their applications for solving ordinary and partial differential equations. we provide a comprehensive list of the laplace transform, sumudu transform, and the natural transform in table 1 to show their mutual relationship with the shehu transform. finally, based on the mathematical formulations, simplicity and the findings of the proposed integral transform, we conclude that it is highly efficient because of the following advantages: • it is a generalization of the laplace and the sumudu integral transforms. • its visualization is easier then the sumudu transform, the natural transform and the elzaki transform. int. j. anal. appl. 17 (2) (2019) 184 • the laplace-type integral transform become laplace transform when the variable u = 1 and the yang integral transform when the variable s = 1. • it can easily be applied directly to some class of ordinary and the partial differential equations as demonstrated in the application section. • for advanced research in physical science and engineering, the proposed integral transform can be considered a stepping-stone to the sumudu transform, the natural transform, the elzaki transform, and the laplace transform. 6. acknowledgements the authors are highly grateful to the editor’s and the anonymous referees’ for their useful comments and suggestions in this paper. this research is partially supported by the national natural science foundations of china under grants no. 11571206. the first author also acknowledges the financial support of chinese scholarship council (csc) in shandong university with grand (csc no: 2017gxz025381). references [1] h.a. agwa, f.m. ali, a. kilicman, a new integral transform on time scales and its applications, adv. difference equ., 60(2012), 1–14. [2] c. ahrendt, the laplace transform on time scales, pan. amer. math. j., 19(2009), 1–36. [3] a. atangana, a note on the triple laplace transform and its applications to some kind of third-order differential equation, abstr. appl. anal., 2013(2013), article id 769102, 1–10. [4] h.m. srivastava, a.k. golmankhaneh, d. baleanu, x.y. yang, local fractional sumudu transform with applications to ivps on cantor sets, abstr. appl. anal., 2014(2014), article id 620529, 1–7. [5] g. dattoli, m. r. martinelli, p. e. ricci, on new families of integral transforms for the solution of partial differential equations, integral transforms spec. funct., 8(2005), 661–667. [6] h. bulut, h.m. baskonus, and f.b.m. belgacem, the analytical solution of some fractional ordinary differential equations by the sumudu transform method, abstr. appl. anal., 2013(2013), article id 203875, 1–6. [7] s.weerakoon, the sumudu transform and the laplace transform: reply, int. j. math. educ. sci. technol., 28(1997), 159–160. [8] d. albayrak, s.d. purohit, and f. uçar, certain inversion and representation formulas for q-sumudu transforms, hacet. j. math. stat., 43(2014), 699–713. [9] s. weerakoon, application of sumudu transform to partial differential equations, int. j. math. educ. sci. technol., 25(1994), 277–283. [10] x.y. yang, y. yang, c. cattani, and m. zhu, a new technique for solving 1-d burgers equation, thermal science, 21(2017), s129–s136. [11] a. kilicman, h. eltayeb, on new integral transform and differential equations, j. math. probl. eng, 2010(2010), article id: 463579, 1–13. [12] s. bochner, k. chandrasekharan, fourier transforms, princeton university press, princeton, nj, usa, (1949). [13] r. n. bracewell, the fourier transform and its applications, mcgraw-hill, boston, mass, usa, 3rd edition, (2000). int. j. anal. appl. 17 (2) (2019) 185 [14] r. murray, spiegel. theory and problems of laplace transform. new york, usa: schaum’s outline series, mcgraw–hill, (1965). [15] l. debnath, d. bhatta.. integral transform and their applications. crc press, new york, ny, usa (2010). [16] b. davies, integral transforms and their applications, texts in applied mathematics, springer, new york, ny, usa, (2002). [17] e.i. jury, theory and applications of the z-transform method, john wiley and sons, new york, ny,usa, (1964). [18] k. liu, r.j. hu, c. cattani, g.n. xie, x.j. yang, and y. zhao, local fractional z-transforms with applications to signals on cantor sets, abstr. appl. anal., 2014(2014),article id: 638648, 1–6. [19] p.m. morse, h. feshbach, methods of theoretical physics, mcgraw-hill, new york, (1953), 484–485. [20] p. flajolet, x. gourdon, and p. dumas, mellin transforms and asymptotics: harmonic sums, theor. comput. sci., 144(1995), 3–58. [21] c. donolato, analytical and numerical inversion of the laplace-carson transform by a differential method, comput. phys. commun., 145(2002), 298–309. [22] a.m. makarov, application of the laplace-carson method of integral transformation to the theory of unsteady visco-plastic flows, j. engrg. phys. thermophys 19(1970), 94–99. [23] e. sjntoft, a straightforward deconvolution method for use in small computers, nucl. instrum. methods, 163(1979), 519–522. [24] a.s. vasudeva murthy, a note on the differential inversion method of hohlfield et al., siam j. appl. math., 55(1995), 712–722. [25] i.n. sneddon, the use of integral transform, mcgraw-hill, new york, (1972). [26] k. xie, y. wanga, k. wang, and x. cai, application of hankel transforms to boundary value problems of water flow due to a circular source, appl. math. comput., 216(2010), 1469–1477. [27] watugala gk. sumudu transform–a new integral transform to solve differential equations and control engineering problems. math. engg. indust., 6(1998), 319–329. [28] m.a. asiru, sumudu transform and solution of integral equations of convolution type, int. j. math. edu. sci. tech., 33(2002), 944–949. [29] f.b.m. belgacem, s.l. kalla, a.a. karaballi, analytical investigations of the sumudu transform and applications to integral production equations, math. probl. engg., 3(2003), 103-118. [30] f.b.m. belgacem, a.a. karaballi, sumudu transform fundamental properties, investigations and applications. j. appl. math. stoch. anal., 2006(2006), article id 91083, 1–23. [31] h. eltayeh and a. kilicman, on some applications of a new integral transform, int. j. math. anal., 4(2010), 123–132. [32] t.m. elzaki. the new integral transform ”elzaki transform”. glob. j. pure appl. math., 7(2011), 57–64. [33] z.h. khan, w.a. khan, n-transform-properties and applications. nust j. engg. sci., 1(2008), 127–133. [34] f.b.m. belgacem, r. silambarasan, theory of natural transform. math. engg. sci. aeros., 3(2012), 99–124. [35] f.b.m. belgacem, r. silambarasan, advances in the natural transform. aip conference proceedings; 1493 january 2012; usa: american institute of physics. (2012), 106–110. [36] h.m. srivastava, minjie luo, r.k.raina, a new integral transform and its applications, acta math. sci., 35(2015), 1386–1400. [37] a. atangana, a. kilicman, a novel integral operator transform and its application to some fode and fpde with some kind of singularities, math. probl. eng., 2013(2013), article id 531984, 1–7. int. j. anal. appl. 17 (2) (2019) 186 [38] x.j. yang, a new integral transform method for solving steady heat-transfer problem, thermal science, 20(2016), s639–s642. [39] x.j. yang, a new integral transform operator for solving the heat-diffusion problem, appl. math. lett., 64(2017), 193–197. [40] y.x. jun, f. gao , a new technology for solving diffusion and heat equations, thermal science, 21(2017), 133–140. [41] a. atangana, b.s.t. alkaltani, a novel double integral transform and its applications, j. nonlinear sci. appl., 9(2016), 424–434. [42] h. eltayeb, a note on double laplace decomposition method and nonlinear partial differential equations, new trends math. sci., 5(2017), 156–164. [43] f.b.m. belgacem, r. silambarasan, h. zakia, t. mekkaoui, new and extended applications of the natural and sumudu transforms: fractional diffusion and stokes fluid flow realms. advances real and complex analysis with applications, publisher: birkhuser, singapore, (2017). 107–120. int. j. anal. appl. 17 (2) (2019) 187 appendix table 1: here we present a comprehensive list of the shehu transform of some special functions and their relationship with the natural transform n [v(t)], the sumudu transform s [v(t)], and the laplace transform. s.no. v(t) s [v(t)] n [v(t)] s [v(t)] £ [v(t)] 1 1 u s 1 s 1 1 s 2 t u 2 s2 u s2 u 1 s2 3 exp(α(t)) u s−αu 1 s−αu 1 1−αu 1 s−α 4 sin(αt) α u2 s2+α2u2 u s2+α2u2 u 1+α2u2 1 s2+α2 5 cos(αt) us s2+α2u2 s s2+α2u2 1 1+α2u2 s s2+α2 6 cosh αt us s2−u2 s s2−u2 1 1−u2 s s2−1 7 t n n! n = 0, 1, 2... ( u s )n+1 un sn+1 un 1 sn+1 8 t n γ(n+1) n = 0, 1, 2, ... ( u s )n+1 un sn+1 un 1 sn+1 9 cos(t) us s2+u2 s s2+u2 1 1+u2 1 s2+1 10 sin(t) u 2 s2+u2 u s2+u2 u 1+u2 1 s2+1 11 sinh(αt) α u2 s2−α2u2 αu s2−α2u2 αu2 1−α2u2 α s2−α2 12 cosh(αt) us s2−α2u2 s s2−α2u2 1 1−α2u2 s s2−α2 13 exp(βt)cosh(αt) u(s−βu) (s−βu)2−α2u2 s−βu (s−βu)2−α2u2 1−βu (s−βu)2−α2u2 s−β (s−βu)2−α2 14 exp(βt) sinh(αt) α u2 (s−βu)2−α2u2 u (s−βu)2−α2u2 u (1−βu)2−α2u2 1 (s−β)2−α2 15 t sin(αt) 2α u3s (s2+α2u2)2 u2s (s2+α2u2)2 u3 (1+α2u2)2 s (s2+α2)2 16 t cos(αt) u2(s2−α2u2)2 (s2+α2u2)2 u(s2−α2u2)2 (s2+α2u2)2 u(1−α2u2)2 (1+α2u2)2 (s2−α2)2 (s2+α2)2 17 sin(αt)+αt cos(αt) 2α u2s2 (s2+α2u2)2 us2 (s2+α2u2)2 u (1+α2u2)2 s2 (s2+α2)2 18 cos αt− αt sin(αt) 2 us3 (s2+α2u2)2 s3 (s2+α2u2)2 1 (1+α2u2)2 s3 (s2+α2)2 19 sin(αt)−αt cos(αt) 2α3 u4 (s2+α2u2)2 u3 (s2+α2u2)2 u3 (1+α2u2)2 1 (s2+α2)2 int. j. anal. appl. 17 (2) (2019) 188 20 t sinh(αt) + t cosh(αt) u 2 (s−αu)2 u (s−αu)2 u2 (1−αu)2 1 (s−α)2 21 t sinh(αt) 2α u3s (s2−α2u2)2 u2s (s2−α2u2)2 u2 (1−α2u2)2 s (s2−α2)2 22 si(αt) (sine integral) u s tan−1 ( αu s ) 1 s tan−1 ( αu s ) tan−1 ( u √ α2 ) 1 s tan−1 ( α s ) 23 ci(αt) (cosine integral) − u 2s log ( s2+α2 α2 ) − 1 2s log ( s2+α2u2 α2u2 ) −1 2 log ( α2u2+1 α2u2 ) − 1 2s log ( s2+α2 α2 ) 24 ei(αt) (exp. integral) −u s log ( αu−s αu ) −1 s log ( αu−s αu ) log ( αu−1 αu ) −1 s log ( α−s α ) 25 (3−α2t2) sin(αt)−3αt cos(αt) 8α5 u6 (s2+α2u2)3 u5 (s2+α2u2)3 u5 (1+α2u2)3 1 (s2+α2)3 26 (3−α2t2) sin(αt)+5αt cos(αt) 8α u2s4 (s2+α2u2)3 us4 (s2+α2u2)3 u (1+α2u2)3 s4 (s2+α2)3 27 (8−α2t2) cos(αt)−7αt sin(αt) 8 us5 (s2+α2u2)3 s5 (s2+α2u2)3 1 (1+α2u2)3 s5 (s2+α2)3 28 t2 sin(αt) 2α u4(3s2−α2u2) (s2+α2u2)3 u3(3s2−α2u3) (s2+α2u2)3 u3(−3+α2u2) (1+α2u2)3 (3s2−α2) (s2+α2)3 29 t2 cos(αt) 2 u3(s3−3α2u2s) (s2+α2u2)3 u2(s3−3α2u2s) (s2+α2u2)3 u2(1−3α2u2) (1+α2u2)3 (s3−3α2s) (s2+α2)3 30 t3 sin(αt) 24α su5(s−αu)2 (s2+α2u2)4 su4(s−αu)2 (s2+α2u2)4 u4(1−αu)2 (1+α2u2)4 s(s−α)2 (s2+α2)4 31 exp(αt)−exp(βt) α−β α 6= β u2 (s−αu)(s−βu) u (s−αu)(s−βu) u (1−βu)(1−αu) 1 (s−β)(s−α) 32 α exp(αt)−β exp(βt) α−β α 6= β us (s−βu)(s−αu) s (s−βu)(s−αu) 1 (1−βu)(1−αu) s (s−β)(s−α) 33 i0(αt) u√ s2−α2u2 1√ s2−α2u2 1√ 1−α2u2 1√ s2−α2 34 δ(t−α) u exp (−αs u ) 1 u exp (−αs u ) 1 u exp (−α u ) exp(−αs) 35 j0(αt) u√ s2+α2u2 1√ s2+α2u2 1√ 1+α2u2 1√ s2+α2 int. j. anal. appl. 17 (2) (2019) 189 table 2: general properties of shehu transform s.no. definition/property s [v(t)] = v (s,u) transforms 1 definition s [v(t)] ∫∞ 0 exp (−st u ) v(t) dt; s > 0, u > 0 2 inverse v(t) = s−1 [v (s,u)] 1 2πi ∫α+i∞ α−i∞ 1 u exp ( st u ) v (s,u) ds 3 linearity s [αv(t) + βw(t)] αs [v(t)] + βs [w(t)] 4 change of scale s [v(αt)] u α v ( s α ,u ) 5 derivatives s [v′(t)] s u v (s,u) −v(0) s [v′′(t)] s 2 u2 v (s,u) − s u v(0) −v′(0) s [v′′′(t)] s 3 u3 v (s,u) − s 2 u2 v(0) − s u v′(0) −v′′(0) ... s [ v(n)(t) ] sn un v (s,u) − ∑n−1 k=0 ( s u )n−(k+1) int. j. anal. appl. 17 (2) (2019) 190 table 3: summary of some integral transform and their definitions s.no. integral transform definition 1 laplace transform £[f(t)] = f(s) = ∫∞ 0 exp (−st) f(t)dt 2 fourier transform z[f(t)] = f(ω) = 1√ 2π ∫∞ −∞ exp (−iωt) f(t)dt 3 melling transform m[f(s); s] = f∗(s) = ∫∞ 0 xs−1f(x)dx 4 hankel’s transform fv(s) = hv[f(r)] = ∫∞ 0 rf(r)jv(sr)dr, r ≥ 0 5 sumudu transform s[f(t)] = g(u) = 1 u ∫∞ 0 exp (−t u ) f(t)dt 6 laplace-carson transform f̂c(p) = p ∫∞ 0 exp (−pt) f(t)dt, t ≥ 0 7 atangana-kilicman transform mn(s) = mn[f(x)](s) = ∫∞ 0 xn exp (−xs) f(x)dx 8 el-zaki transform e[f(t)] = t(u) = u ∫∞ 0 exp (−t u ) f(t)dt 9 yang transform y [φ(τ)] = φ(ω) = ∫∞ 0 exp (−τ ω ) φ(τ)dτ 10 natural transform n+[f(t)] = r(s,u) = 1 u ∫∞ 0 exp (−st u ) f(t)dt, s > 0, u > 0 11 z-transform x(z) = z{x[n]} = ∑∞ n=−∞x[n]z −n, n ∈ z, z ∈ c 12 m-transform mρ,m[f(t)](u,v) = ∫∞ 0 exp(−ut)f(vt) (tm+vm)ρ dt, ρ ∈ c, <(ρ) ≥ 0, m ∈ z+ 1. introduction 2. main result 3. some useful results of shehu transform 4. applications 5. conclusion 6. acknowledgements references appendix international journal of analysis and applications volume 18, number 1 (2020), 99-103 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-99 on fixed point theorem in non-archimedean fuzzy normed spaces m.e. egwe∗ department of mathematics, university of ibadan, ibadan, nigeria ∗corresponding author: murphy.egwe@ui.edu.ng abstract. let (x, n) be a non-archimedean fuzzy normed space and (x, ‖.‖), a non-archimedean normed space where x is a linear space over a linearly ordered non-archimedean field k with a non-archimedean valuation. we give a proof of the fixed point theorem in non-archimedean fuzzy normed space. 1. introduction definition 1.1 [9]: a valuation is a map | · | from a field k into a non-negative reals such that (i) |a| = 0 if and only if a = 0 (ii) |ab| = |a||b| (iii) |a + b| ≤ |a| + |b| for all a,b ∈ k (triangle inequality). when a field k carries an absolute value | · |, it is called a valued field (k, | · |). examples of the pair (k, | · |) is called a valued field. examples of valuations are provided by the usual absolute values of r and c. in the definition above, if the triangle inequality is replaced by a strong triangle inequality, i.e, |a+b| ≤ max(|a|, |b|) for all a,b ∈ k, the map | | is then called a non-archimedean or ultrametric valuation. theorem 1.2 [1]: let be a complete space, 0 < λ < 1, and f : x → x be a map such that ‖f(x)−f(y)‖≤ ‖x−y‖ for all x,y ∈ x. then there exists a unique point x◦ such that f(x◦) = x◦. this fixed point in several cases have been obtained in non-archimedean normed and metric spaces (see [2], [4] [5], [6]). in this paper, we shall prove a version given in [4] for non-archimedean fuzzy normed spaces. received 2019-10-08; accepted 2019-11-14; published 2020-01-02. 2010 mathematics subject classification. 46s10, 46s40, 47h10. key words and phrases. fixed point, non-archimedean, fuzzy normed space, spherically complete. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 99 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-99 int. j. anal. appl. 18 (1) (2020) 100 2. main result definition 2.1 [4], [11]: let (x,‖.‖), a non-archimedean normed space. a function n : x × r → [0, 1] is called a nonarhimedean norm on x if for all x,y ∈ x and all s,t ∈ r, (i) n(x,t) = 0 for t ≤ 0, (ii) n(x,t) = 1 if and only if x = 0 for all t > 0, (iii) n(λx,t) = n(x, t |λ| ) for λ 6= 0 (iv) n(x + y, max{s,t}) ≥ min{n(x,s),n(y,t)} (v) n(x,∗) is nondecreasing function of r and lim t→∞ n(x,t) = 1 definition 2.2: let (x,n) be a non-archimedean fuzzy normed space. a closed ball in (x,n) with centre a is the set of points b(x,t) := {n(x−a,t) ≤ r,} where t ∈ r+. definition 2.3: a sequence of n-closed balls b(x1, t) ⊆ b(x2, t) ⊆ ··· ⊂ b(xn, t) is called a sequence of closed balls ordered by inclusion. definition 2.4: let (x,n) be a non-archimedean fuzzy normed space. let {b(xi, t)}n be a sequence of closed balls ordered by inclusion. then, (x,n) is said to be spherically complete if the sequence of closed balls {b(x,t)}n satisfies the finite intersection property in (x,n). i.e., n⋂ i=1 {b(xi, t)} 6= 0. theorem 2.5 [1] let v be a complete normed linear space, 0 < α < 1, and ϕ : v → v such that ‖ϕ(u) −ϕ(ν)‖≤ α‖u−ν‖ for all u,ν ∈ v. then, there exists a fixed point, u◦ ∈ v such that ϕ(u◦) = u◦. a version of this theorem on non-archimedean normed space was proved in [6] as seen in the next result. proposition 2.6 [6]: suppose that x and y are non-archimedean normed over a non-archimedean field k with |p| 6= 1 for some p ∈ n. assume that x or y is spherically complete. if f : x → y is a surjective isometry, then for each x ∈ x, there exists a unique y ∈ x such that f(x) + f(y) = f ( x + y p ) . we now state and prove a version of this result for the non-archimedean fuzzy normed spaces proposition 2.7: let (x,n) and (y,n) be non-archimedean fuzzy normed spaces over a non-archimedean field k with |p| > 1 for some p ∈ n. assume that x or y is spherically complete. if f : (x,n) → (y,n) is a surjective isometry, then for each u ∈ x, there exists a unique ν ∈ x such that f(u) + f(ν) = f ( u + ν p ) . proof: first, we prove that (x,n) or (y,n) is spherically complete. suppose that (x,n) is spherically complete and let {b(yi, t)}n be a sequence od closed balls in (y,n) ordered by inclusion. then by the surjectivity of f, there is a sequence {b(xi, t)}n of closed balls in (x,n) ordered by inclusion with b(x1, t) = f −1(b(y1, t)) ⊆ b(x2, t) = f−1(b(y2, t)) ⊆ ···⊆ b(xn, t) = f−1(b(yn, t)). thus, n⋂ f−1(b(yi, t)) 6= φ as φ 6= n⋂ (b(xi, t)) = f −1(b(yi, t)) because (x,n) is spherically complete. thus, (y,n) is spherically complete if (x,n) is spherically complete. int. j. anal. appl. 18 (1) (2020) 101 conversely, let (y,n) be spherically complete and {b(x,t)}n a sequence of closed balls in (x,n) ordered by inclusion. then there exists a sequence {f(b(xi, t))}n of closed balls in (y,n) such that f(b(x1, t)) ⊆ f(b(x2, t)) ⊆ ···f(b(xn, t)). then, b(x1, t) = f(b(x1, t)) ⊆ f(b(x2, t)) = b(x2, t) ⊆ ···f(b(xn, t)) = b(xn, t) and n⋂ b(xi, t) 6= φ as φ 6= n⋂ f(b(xi, t)) = n⋂ (b(xi, t)) because (y,n) is spherically complete. thus, (x,n) is spherically complete if (y,n) is. this implies that (x,n) or (y,n) is spherically complete. next, we show that there exists a unique ν ∈ x such that f(u) + f(ν) = f ( u + ν p ) for each u ∈ x. to do this,let u ∈ x, and consider the mapping ϕ : x → x : x 7→ px−u. now, n(ϕ(x) −ϕ(y), t) = n(px−u− (py −u), t) = n(px−u−py + u), t) = n((px−py), t) = n(x−y, t |p| ) < n(x−y,t). thus, there exists m > 1 such that m.n(ϕ(x) −ϕ(y), t) < n(x−y,t), i.e., n(ϕ(x) −ϕ(y), t) < 1 m n(x−y,t). obviously, 0 < 1 m < 1, and n(ϕ(x) −ϕ(y), t) ≤ n(x−y,t) which implies that ϕ is a contractive mapping. let ψ : y → y be an isometry defined by ψ(y) = f(u) + y. if h = ϕf−1ψf. then, n(ϕh(x) −h(y), t) = n((ϕf−1ψf)(x) − (ϕf−1ψf)(y), t) = n(p(ϕf−1ψf)(x) −u− (p(ϕf−1ψf)(y)), t) = n(p(ϕf−1ψf)(x) −u−p(ϕf−1ψf)(y) + u,t) = n(p(ϕf−1ψf)(x) −p(ϕf−1ψf)(y), t) = n((ϕf−1ψf)(x) − (ϕf−1ψf)(y), t |p| ) < n((ϕf−1ψf)(x) − (ϕf−1ψf)(y), t) = n((ψf)(x) − (ψf)(y), t) = n(f(x) −f(y), t) = n(x−y,t). similarly, there exists k∗ > 1 such that k∗.n(h(x) −h(y), t) ≤ n(x−y,t). this also implies that n(h(x) −h(y), t) ≤ 1 k∗ n(x−y,t). int. j. anal. appl. 18 (1) (2020) 102 since 1 k∗ < 1 and n(h(x) − h(y), t) ≤ n(x − y,t), then, h is a contraction mapping. by the fixed point theorem, h has a unique fixed point ν such that (ψf−1ψf)(ν) = h(ν) = ν. but ψ ( u + ν p ) = p. u + ν p −u = u + ν −u = ν. therefore, ψ(ν) = ψ(f(ν)) = ψ ( u + ν p ) = f ( u + ν p ) , as f, and ψ are injections. since ψ(f(ν)) = f(u) + f(ν) by definition, it follows that f(u) + f(ν) = f ( u + ν p ) . � remark 2.8: it is necessary for |p| > 1 for (i) if |p| > 1, then n(ϕ(u) −ϕ(ν),p) = n(u−ν,t) as n(u−ν, t |p| ) = n(u−ν,t). also, n(h(u) −h(ν), t) = n(u−ν,t) as n((f−1ψf)(u) − (f−1ψf)(ν), t |p| ) = n((f−1ψf)(u) − (f−1ψf)(ν), t). so, ϕ and h are not contraction mappings. (ii) if |p| = 0, then p = 0 by definition of valuation. this negates the assumption that p ∈ n. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d. burago, y. burago, s. ivanon: a course in metric geometry, amer. math. soc. 2001. [2] y. je cho, t.m. rassias, r. saadati: fuzzy operator theory in mathematical analysis. springer international publishing, 2018 [3] a. granas, j. dugunji: fixed point theory, springer, 2003. [4] d. kangb, h. kohb, i.g. chao: on the mazur-ulam theorem in non-archimedean fuzzy normed spaces, appl. math. lett. 25(2012), 301-304. [5] h. mamghaderi, h.p. masiha: on stationary points of multivalued strongly contractive mappings in partially ordered ultrametric spaces and non-archimedean normed spaces. p-adic numbers, ultr. anal. appl. 9(2)(2017), 144-150. int. j. anal. appl. 18 (1) (2020) 103 [6] m.s. moslehian, g. sadeghi: a mazur-ulam theorem in non-archimedean normed spaces, nonlinear anal., theory methods appl. 69(2008), 3405-3408. [7] a. narayanan, s. vijayabalaji: fuzzy n-normed linear spaces. int. j. math. math. sci. 24(2005), 3963-3977. [8] c. petalas, t. vidalis: a fixed point theorem in non-archimedean vector spaces,proc. amer. math. soc. 118(3)(1993), 819-821. [9] a.c.m. rooij: non-archimedean functional analysis, marcel dekker ny, 1978. [10] f.shi, c. huang: fuzzy bases and the fuzzy dimension of fuzzy vector spaces, math. commun. 15(2010), 303-310. [11] z. wang, p.k. sahoo: stability of an acq-functional equation in various stability of an acq-functional equation in various. j. nonlinear sci. appl. 8(2015), 64-85. 1. introduction 2. main result references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 119-131 http://www.etamaths.com new inequalities of hermite-hadamard type for n-times differentiable s-convex functions with applications muhammad amer latif1,∗, sever s. dragomir2 and ebrahim momoniat1 abstract. in this paper, some new inequalities hermite-hadamard type are obtained for functions whose nth derivatives in absolute value are s-convex functions. from our results, several inequalities of hermite-hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are s-convex functions as special cases. our results may provide refinements of some results already exist in literature. applications to trapezoidal rule and to special means of established results are given. 1. introduction a function f : i → r, ∅ 6= i ⊆ r, is said to be convex on i if the inequality f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) holds for all x,y ∈ i and t ∈ [0, 1]. let f : i ⊆ r → r be a convex mapping and a,b ∈ i with a < b. then f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 . (1.1) the double inequality (1.1) is known as the hermite-hadamard inequality (see [4]). the inequalities (1.1) hold in reversed direction if f is concave. for recent results, refinements, counterparts, generalizations and new hermite-hadamard-type inequalities see [3, 7, 10–12, 14] and the references therein. in [4], hudzik and maligranda considered among others the class of functions which are s-convex in the second sense and is defined as follows. definition 1.1. [4] let s ∈ (0, 1] be a fixed real number. a function f : [0,∞) → [0,∞) is said to be s-convex in the second sense, if f (tx + (1 − t) y) ≤ tsf (x) + (1 − t)s f (y) (1.2) holds for all x, y ∈ [0,∞) and t ∈ [0, 1]. the class of s-convex functions in the second sense is denoted by k2s . if the inequality (1.2) holds in reversed direction, then f is to be an s-concave function in the second sense. it is clear that the definition of s-convexity (s-concavity) coincides with the definition of convexity (concavity) when s = 1. in [2], dragomir and fitzpatrick proved a variant of hadamard’s inequality which holds for s-convex functions in the second sense. theorem 1.1. [2] suppose that f : [0,∞) → [0,∞) is an s-convex function in the second sense, where s ∈ (0, 1), a, b ∈ [0,∞) with a < b. if a, b ∈ l ([a,b]), then the following inequalities hold 2s−1f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) s + 1 . (1.3) received 8th november, 2016; accepted 11th january, 2017; published 1st march, 2017. 2010 mathematics subject classification. primary 26d07; secondary 26d15. key words and phrases. hermite-hadamard’s inequality; convex function; s-convex; hölder integral inequality; trapezoidal rule. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 119 120 latif, dragomir and mamonait for more recent results on hermite-hadamard type inequalities for functions whose derivatives in absolute value are s-convex functions, we refer the interested reader to [1, 8, 9, 13] and the references therein. the main purpose of the present paper is to establish new hermite-hadamard type inequalities for functions whose nth derivatives in absolute value are s-convex. we believe that the results presented in this paper are better than those already exist in the literature concerning the inequalities of hermitehadamard type for s-convex functions. applications of our results to trapezoidal formula and to special means are given in section 3 and section 4. 2. main results we will use the following lemmas to establish our main results in this section. lemma 2.1. let f : i ⊂ r → r be a function such that f(n) exists on i◦ and f(n) ∈ l ([a,b]), where a, b ∈ i◦ with a < b, n ∈ n, we have the identity f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) = (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)n (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 b + 1 + t 2 a ) dt, (2.1) where an empty sum is understood to be nil. proof. suppose in = (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 a + 1 + t 2 b ) dt and jn = (−1)n (b−a)n 2n+1n! ∫ 1 0 (1 − t)n−1 (n− 1 + t) f(n) ( 1 − t 2 b + 1 + t 2 a ) dt. for n = 1, we have i1 = b−a 4 ∫ 1 0 tf ′ ( 1 − t 2 a + 1 + t 2 b ) dt and j1 = (−1) (b−a) 4 ∫ 1 0 tf ′ ( 1 − t 2 a + 1 + t 2 b ) dt. by integration by parts and using the substitution x = 1−t 2 a + 1+t 2 b for i1 and x = 1+t 2 a + 1−t 2 b for j1, we obtain i1 = 1 2 f (b) − 1 b−a ∫ b a+b 2 f (x) and j1 = 1 2 f (a) − 1 b−a ∫ a+b 2 a f (x) . hence i1 + j1 = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx, which coincides with the l.h.s of (2.1) for n = 1. similarly for n = 2, and using similar arguments as above, we have i2 + j2 = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx inequalities for n-times differentiable s-convex functions 121 which coincides with the l.h.s of (2.1) for n = 2. suppose (2.1) holds for n = m− 1 ≥ 3. now for n = m, we have (b−a)m 2m+1m! ∫ 1 0 (1 − t)m−1 (m− 1 + t) f(m) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)m (b−a)m 2m+1m! ∫ 1 0 (1 − t)m−1 (m− 1 + t) f(m) ( 1 − t 2 b + 1 + t 2 a ) dt = − (b−a)m−1 (m− 1) [ 1 + (−1)m−1 ] 2mm! f(m−1) ( a + b 2 ) + (b−a)m−1 2m (m− 1)! ∫ 1 0 (1 − t)m−2 (m− 2 + t) f(m) ( 1 − t 2 a + 1 + t 2 b ) dt + (−1)m−1 (b−a)m−1 2m (m− 1)! ∫ 1 0 (1 − t)m−2 (m− 2 + t) f(m) ( 1 − t 2 b + 1 + t 2 a ) dt = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− m−2∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) − (b−a)m−1 (m− 1) [ 1 + (−1)m−1 ] 2mm! f(m−1) ( a + b 2 ) = f (a) + f (b) 2 − 1 b−a ∫ b a f (x) dx− m−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 ) . this completes the proof of the lemma. � lemma 2.2. [15] let x ≥ 0, y ≥ 0, the inequality (x + y) θ ≤ xθ + yθ holds for 0 < θ ≤ 1 and the inequality (x−y)θ ≤ xθ −yθ holds for θ ≥ 1. now we state and prove some new hermite-hadamard type inequalities for functions whose nth derivatives in absolute value are s-convex and s-concave in the second sense. theorem 2.1. let f : i ⊂ [0,∞) → r be a function such that f(n) exists on i◦ and f(n) ∈ l ([a,b]), where a, b ∈ i◦ with a < b, n ∈ n. if ∣∣f(n)∣∣q is s-convex on [a,b] for some fixed s ∈ (0, 1] and q ∈ [1,∞), we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx − n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a) n 2n+s/q+1n! ( n n + 1 )1−1/q ×   (( n2 + s (n− 1) )∣∣f(n) (a)∣∣q (n + s) (n + s + 1) + ( n n + 1 + n (n + s) n + s + 1 b (s + 1,n) )∣∣∣f(n) (b)∣∣∣q )1/q + (( n2 + s (n− 1) )∣∣f(n) (b)∣∣q (n + s) (n + s + 1) + ( n n + 1 + n (n + s) n + s + 1 b (s + 1,n) )∣∣∣f(n) (a)∣∣∣q )1/q  , (2.2) 122 latif, dragomir and mamonait where b (α,β) = ∫ 1 0 tα−1 (1 − t)β−1 dt,α,β > 0 is the euler beta function. proof. from lemma 2.1, the hölder inequality and s-convexity of ∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)n−1 (n− 1 + t) dt )1−1/q × {(∫ 1 0 (1 − t)n−1 (n− 1 + t) [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + t)s ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (1 − t)n−1 (n− 1 + t) [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + t)s ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.3) since ∫ 1 0 (1 − t)n−1 (n− 1 + t) dt = n n + 1 , ∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 − t)s dt = n2 + s (n− 1) (n + s) (n + s + 1) , by using the property b (x,y + 1) = y x + y b (x,y) of the euler beta function and lemma 2.2, we have∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 + t)s dt ≤ ∫ 1 0 (1 − t)n−1 (n− 1 + t) (1 + ts) dt = n n + 1 + nb (s + 1,n) −b (s + 1,n + 1) = n n + 1 + nb (s + 1,n) − n n + s + 1 b (s + 1,n) = n n + 1 + n (n + s) n + s + 1 b (s + 1,n) . from the above facts and the inequality (2.3), we get the required inequality (2.2). this completes the proof of the theorem. � the following corollaries are direct consequences of theorem 2.1. corollary 2.1. under the assumptions of theorem 2.1, if q = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx − n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a) n 2n+s+1n! × ( n2 + s (n− 1) (n + s) (n + s + 1) + n n + 1 + n (n + s) n + s + 1 b (s + 1,n) )[∣∣∣f(n) (a)∣∣∣ + ∣∣∣f(n) (b)∣∣∣] , (2.4) where b (α,β) is the euler beta function defined as in theorem 2.1. inequalities for n-times differentiable s-convex functions 123 corollary 2.2. under the assumptions of theorem 2.1, if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) ( 1 2 )3+s/q−1/q     ∣∣∣f′ (a)∣∣∣q (1 + s) (2 + s) + ( 1 2 + 1 2 + s )∣∣∣f′ (b)∣∣∣q   1/q +  (1 2 + 1 2 + s )∣∣∣f′ (a)∣∣∣q + ∣∣∣f′ (b)∣∣∣q (1 + s) (2 + s)   1/q   . (2.5) corollary 2.3. if we take q = 1 in corollary 2.2, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) (s + 3) s + 1 ( 1 2 )3+s [∣∣∣f′ (a)∣∣∣ + ∣∣∣f′ (b)∣∣∣] . (2.6) corollary 2.4. suppose the assumptions of theorem 2.1 are fulfilled and if n = 2, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a) 2 24+s/q ( 2 3 )1−1/q × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (a)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (b)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (b)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (a)∣∣∣q]1/q } . (2.7) corollary 2.5. if q = 1 in corollary 2.4, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 24+s ( 2 3 + s2 + 7s + 8 (s + 1) (s + 2) (s + 3) )[∣∣∣f′′ (a)∣∣∣ + ∣∣∣f′′ (b)∣∣∣] . (2.8) theorem 2.2. let f : i ⊂ r → r be an n-times differentiable function on i◦, n ∈ n. if f(n) ∈ l ([a,b]), where a, b ∈ i◦ with a < b. if ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n [ n(2q−1)/(q−1) − (n− 1)(2q−1)/(q−1) ]1−1/q 2n+s/q+1n! ( q − 1 2q − 1 )1−1/q ×   [ ∣∣f(n) (a)∣∣q nq −q + s + 1 + ( 1 nq −q + 1 + b (s + 1,nq −q + 1) )∣∣∣f(n) (b)∣∣∣q ]1/q + [ ∣∣f(n) (b)∣∣q nq −q + s + 1 + ( 1 nq −q + 1 + b (s + 1,nq −q + 1) )∣∣∣f(n) (a)∣∣∣q ]1/q  . (2.9) where b (α,β) is the euler beta function defined as in theorem 2.1. 124 latif, dragomir and mamonait proof. using lemma 2.1, by using the first inequality in lemma 2.2, the hölder inequality and convexity of ∣∣f(n)∣∣q on [a,b], we have ∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (n− 1 + t)q/(q−1) dt )1−1/q × {(∫ 1 0 (1 − t)q(n−1) [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + ts) ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (1 − t)q(n−1) [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + ts) ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.10) by simple computations, we observe that∫ 1 0 (n− 1 + t)q/(q−1) dt = ( q − 1 2q − 1 )[ n(2q−1)/(q−1) − (n− 1)(2q−1)/(q−1) ] , ∫ 1 0 (1 − t)q(n−1)+s dt = 1 nq −q + s + 1 and ∫ 1 0 (1 − t)q(n−1) (1 + ts) dt = 1 nq −q + 1 + b (s + 1,nq −q + 1) . using the above results in (2.10), we obtain the required result. this completes the proof of the theorem. � corollary 2.6. suppose the assumptions of theorem 2.2 are satisfied and if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q ( q − 1 2q − 1 )1−1/q ( 1 s + 1 )1/q × {[∣∣∣f′ (a)∣∣∣q + (s + 2) ∣∣∣f′ (b)∣∣∣q]1/q + [∣∣∣f′ (b)∣∣∣q + (s + 2) ∣∣∣f′ (a)∣∣∣q]1/q} . (2.11) corollary 2.7. under the assumptions of theorem 2.2, if n = 2, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 [ 2(2q−1)/(q−1) − 1 ]1−1/q 24+s/q ( q − 1 2q − 1 )1−1/q × {[ 1 q + s + 1 ∣∣∣f′′ (a)∣∣∣q + ( 1 q + 1 + b (s + 1,q + 1) )∣∣∣f′′ (b)∣∣∣q]1/q + [ 1 q + s + 1 ∣∣∣f′′ (b)∣∣∣q + ( 1 q + 1 + b (s + 1,q + 1) )∣∣∣f′′ (a)∣∣∣q]1/q } , (2.12) where b (α,β) is the euler beta function defined as in theorem 2.1. inequalities for n-times differentiable s-convex functions 125 theorem 2.3. let f : i ⊂ r → r be an n-times differentiable function on i◦, n ∈ n. if f(n) ∈ l ([a,b]), where a, b ∈ i◦ with a < b. if ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+ s q +1n! ( q − 1 nq − 1 )1−1/q {[ p ∣∣∣f(n) (a)∣∣∣q + q∣∣∣f(n) (b)∣∣∣q]1/q + [ p ∣∣∣f(n) (b)∣∣∣q + q ∣∣∣f(n) (a)∣∣∣q]1/q} , (2.13) where p = nq+s+1b ( 1 n ; s + 1,q + 1 ) , q = nq ( s + 2 s + 1 ) − 1 q + 1 −b (s + 1,q + 1) , b (α,β) is the euler beta function defined as in theorem 2.1 and b (x; α,β) = ∫ x 0 tα−1 (1 − t)β−1 dt,α,β > 0, 0 ≤ x ≤ 1 is the incomplete beta function. proof. using lemma 2.1, the first inequality in lemma 2.2, the hölder inequality and convexity of∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)q(n−1)/(q−1) dt )1−1/q × {(∫ 1 0 (n− 1 + t)q [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + ts) ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 (n− 1 + t)q [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 + ts) ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.14) by using the second inequality of lemma 2.2 and simple computation, it is easy to observe that∫ 1 0 (1 − t)q(n−1)/(q−1) dt = q − 1 nq − 1 , ∫ 1 0 (n− 1 + t)q (1 − t)s dt = nq+s+1 ∫ 1 n 0 ts (1 − t)q dt = nq+s+1b ( 1 n ; s + 1,q + 1 ) = p and ∫ 1 0 (n− 1 + t)q (1 + ts) dt ≤ ∫ 1 0 (nq − (1 − t)q) (1 + ts) dt = nq ( s + 2 s + 1 ) − 1 q + 1 −b (s + 1,q + 1) = q. hence (2.13) follows from (2.14) and using the above results. this completes the proof of the theorem. � 126 latif, dragomir and mamonait corollary 2.8. suppose the assumptions of theorem 2.3 are satisfied and if n = 1, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q × {[ b (s + 1,q + 1) ∣∣∣f′ (a)∣∣∣q + (s + 2 s + 1 − 1 q + 1 −b (s + 1,q + 1) )∣∣∣f′ (b)∣∣∣q]1/q + [ b (s + 1,q + 1) ∣∣∣f′ (b)∣∣∣q + (s + 2 s + 1 − 1 q + 1 −b (s + 1,q + 1) )∣∣∣f′ (a)∣∣∣q]1/q } , (2.15) where b (α,β) is the euler beta function defined as in theorem 2.1. corollary 2.9. suppose the assumptions of theorem 2.3 are satisfied and if n = 2, we have the inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 24+ s q ( q − 1 2q − 1 )1−1/q {[ 2q+s+1b ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (a)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 −b (s + 1,q + 1) )∣∣∣f′′ (b)∣∣∣q]1/q + [ 2q+s+1b ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (b)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 − 2q+s+1b (s + 1,q + 1) )∣∣∣f′′ (a)∣∣∣q]1/q } , (2.16) where b (α,β) is the euler beta function defined as in theorem 2.1 and b (x; α,β) is the incomplete beta function defined as in theorem 2.3. a different approach results in the following theorem. theorem 2.4. let f : i ⊂ r → r be a function such that f(n) exists on i◦ and f(n) ∈ l ([a,b]) for n ∈ n, where a, b ∈ i◦ with a < b. if ∣∣f(n)∣∣q is convex on [a,b] for q ∈ (1,∞), we have the inequality ∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ nn+1−1/q (b−a)n 2n+s/q+1n! ( 1 s + 1 )1/q [ b ( 1 n ; nq − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × {[∣∣∣f(n) (a)∣∣∣q + (2s+1 − 1)∣∣∣f(n) (b)∣∣∣q]1/q + [( 2s+1 − 1 )∣∣∣f(n) (a)∣∣∣q + ∣∣∣f(n) (b)∣∣∣q]1/q} , (2.17) where b (x; α,β) = ∫ x 0 tα−1 (1 − t)1−β dt, 0 ≤ x ≤ 1,α > 0,β > 0 is the incomplete beta function. inequalities for n-times differentiable s-convex functions 127 proof. using lemma 2.1, the hölder inequality and the convexity of ∣∣f(n)∣∣q on [a,b], we have∣∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx− n−1∑ k=1 k [ 1 + (−1)k ] (b−a)k 2k+1 (k + 1)! f(k) ( a + b 2 )∣∣∣∣∣∣ ≤ (b−a)n 2n+s/q+1n! (∫ 1 0 (1 − t)q(n−1)/(q−1) (n− 1 + t)q/(q−1) dt )1−1/q × {(∫ 1 0 [ (1 − t)s ∣∣∣f(n) (a)∣∣∣q + (1 + t)s ∣∣∣f(n) (b)∣∣∣q]dt)1/q + (∫ 1 0 [ (1 − t)s ∣∣∣f(n) (b)∣∣∣q + (1 − t)s ∣∣∣f(n) (a)∣∣∣q]dt)1/q } . (2.18) by simple computations of integrals, the inequality (2.17) follows from the inequality (2.18) and using the fact that∫ 1 0 (1 − t)q(n−1)/(q−1) (n− 1 + t)q/(q−1) dt = n nq+q−1 q−1 b ( 1 n ; nq − 1 q − 1 , 2q − 1 q − 1 ) . this completes the proof of the theorem. � corollary 2.10. suppose the assumptions of theorem 2.4 are satisfied and if n = 1, we have∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)22+s/q ( 1 s + 1 )1/q ( q − 1 2q − 1 )1−1/q × {[∣∣∣f′ (a)∣∣∣q + (2s+1 − 1)∣∣∣f′ (b)∣∣∣q]1/q + [(2s+1 − 1)∣∣∣f′ (a)∣∣∣q + ∣∣∣f′ (b)∣∣∣q]1/q} . (2.19) proof. proof follows from the fact that b ( 1; 1, 2q − 1 q − 1 ) = b ( 1, 2q − 1 q − 1 ) = ∫ 1 0 (1 − t) 2q−1 q−1 −1 dt = q − 1 2q − 1 . � corollary 2.11. under the assumptions of theorem 2.4 and n = 2, we have the following inequality∣∣∣∣∣f (a) + f (b)2 − 1b−a ∫ b a f (x) dx ∣∣∣∣∣ ≤ (b−a)2 21+s/q+1/q ( 1 s + 1 )1/q [ b ( 1 2 ; 2q − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × {[∣∣∣f′′ (a)∣∣∣q + (2s+1 − 1)∣∣∣f′′ (b)∣∣∣q]1/q + [(2s+1 − 1)∣∣∣f′′ (a)∣∣∣q + ∣∣∣f′′ (b)∣∣∣q]1/q} , (2.20) where b (x; α,β) is defined as in theorem 2.4. 3. applications to the trapezoidal formula let d be a division of the interval [a,b], i.e. a = x0 < x1 < ... < xn−1 < xn = b, and consider the quadrature formula ∫ b a f(x)dx = t(f,d) + e(f,d), where t(f,d) = n−1∑ i=0 (xi+1 −xi) f (xi) + f (xi+1) 2 128 latif, dragomir and mamonait is the trapezoidal versions and e(f,d) is the associated error. here, we derive some error estimates for the trapezoidal formula in terms of absolute values of the second derivative of f which may be better than those already exist in the literature. theorem 3.1. let f : i ⊆ r → r be a differentiable function on i◦ such that f ′′ ∈ l ([a,b]), where a,b ∈ i◦ with a < b. if ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q ≥ 1, then for every division d of [a,b], we have |e(f,d)| ≤ 1 24+s/q ( 2 3 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi+1)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi)∣∣∣q]1/q } . (3.1) proof. by applying corollary 2.4 on the subinterval [xi,xi+1] (i = 0, 1, . . . ,n− 1) of the division d, we have |e(f,d)| = ∣∣∣∣∣ n−1∑ i=0 { (xi+1 −xi) f (xi) + f (xi+1) 2 − ∫ xi+1 xi f (x) dx }∣∣∣∣∣ ≤ n−1∑ i=0 (xi+1 −xi) ∣∣∣∣f (xi) + f (xi+1)2 − 1xi+1 −xi ∫ xi+1 xi f (x) dx ∣∣∣∣ ≤ 1 24+s/q ( 2 3 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ (s + 4) (s + 2) (s + 3) ∣∣∣f′′ (xi+1)∣∣∣q + (2 3 + 2 (s + 1) (s + 3) )∣∣∣f′′ (xi)∣∣∣q]1/q } . (3.2) � corollary 3.1. under the assumptions of theorem 3.1, for q = 1, we have |e(f,d)| ≤ 1 24+s ( 2 3 + s2 + 7s + 8 (s + 1) (s + 2) (s + 3) ) × n−1∑ i=0 (xi+1 −xi) 3 [∣∣∣f′′ (xi)∣∣∣ + ∣∣∣f′′ (xi+1)∣∣∣] . (3.3) theorem 3.2. let f : i ⊆ r → r be a differentiable function on i◦ such that f ′′ ∈ l ([a,b]), where a,b ∈ i◦ with a < b. if ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |e(f,d)| ≤ [ 2(2q−1)/(q−1) − 1 ]1−1/q 24+s/q ( q − 1 2q − 1 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ 1 q + s + 1 ∣∣∣f′′ (xi)∣∣∣q + ( 1 q + 1 + b (s + 1,q + 1) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ 1 q + s + 1 ∣∣∣f′′ (xi+1)∣∣∣q + ( 1 q + 1 + b (s + 1,q + 1) )∣∣∣f′′ (xi)∣∣∣q]1/q } , (3.4) where b (α,β) is the euler beta function defined as in theorem 2.1. inequalities for n-times differentiable s-convex functions 129 proof. it follows from corollary 2.7. � theorem 3.3. let f : i ⊆ r → r be a differentiable function on i◦ such that f ′′ ∈ l ([a,b]), where a,b ∈ i◦ with a < b. if ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |e(f,d)| ≤ 1 24+ 1 q ( q − 1 2q − 1 )1−1/q n−1∑ i=0 (xi+1 −xi) 3 × {[ 2q+s+1b ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (xi)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 −b (s + 1,q + 1) )∣∣∣f′′ (xi+1)∣∣∣q]1/q + [ 2q+s+1b ( 1 2 ; s + 1,q + 1 )∣∣∣f′′ (xi+1)∣∣∣q + ( 2q ( s + 2 s + 1 ) − 1 q + 1 − 2q+s+1b (s + 1,q + 1) )∣∣∣f′′ (xi)∣∣∣q]1/q } , (3.5) where b (α,β) is the euler beta functions and b (x; α,β) is the incomplete beta function defined as in theorem 2.3. proof. it is a direct consequence of corollary 2.9. � theorem 3.4. let f : i ⊆ r → r be a differentiable function on i◦ such that f ′′ ∈ l ([a,b]), where a,b ∈ i◦ with a < b. if ∣∣∣f′′∣∣∣q is s-convex on [a,b] for q > 1, then for every division d of [a,b], we have |e(f,d)| ≤ 1 21+s/q+1/q ( 1 s + 1 )1/q [ b ( 1 2 ; 2q − 1 q − 1 , 2q − 1 q − 1 )]1−1/q × n−1∑ i=0 (xi+1 −xi) 3 {[∣∣∣f′′ (xi)∣∣∣q + (2s+1 − 1)∣∣∣f′′ (xi+1)∣∣∣q]1/q + [( 2s+1 − 1 )∣∣∣f′′ (xi)∣∣∣q + ∣∣∣f′′ (xi+1)∣∣∣q]1/q} , (3.6) where b (x; α,β) is the incomplete beta function defined as in theorem 2.3. proof. the proof follows by using corollary 2.9. � 4. applications to the special means now, we consider applications of our results to special means. we consider the means for positive real numbers a, b ∈ r+. we consider (1) the arithmetic mean: a (a,b) = a + b 2 ; a,b > 0. (2) generalized log-mean: lp (a,b) = [ bp+1 −ap+1 (p + 1) (b−a) ]1 p ; a, b > 0, p ∈ r\{−1, 0} , a 6= b. now we apply our results from section 2 to give some inequalities for special means. it is shown in [4] that the function f : [0,∞) → r defined by f (t) =   a, t = 0, bts + c, t > 0, where s ∈ (0, 1), a, b, c ∈ r with 0 ≤ c ≤ a, b ≥ 0, is s-convex on [0,∞). hence, for a = c = 0, b ≥ 0, the function f (t) = bts is an s-convex function on [0,∞), where s ∈ (0, 1). 130 latif, dragomir and mamonait theorem 4.1. for a, b ∈ r+, a < b, 0 < s < 1 and q ∈ [1,∞), we have∣∣∣a(as+2,bs+2)−ls+2s+2 (as+2,bs+2)∣∣∣ ≤ (b−a)2 23+s ( 2 3 (s + 2) (s + 1) + s2 + 7s + 8 s + 3 ) a (as,bs) . (4.1) proof. let f (x) = xs+2, x ∈ r+. then ∣∣∣f′′ (x)∣∣∣ = (s + 2) (s + 1) xs is an s-convex function on r+. applying corollary 2.5, we obtain the required result. � theorem 4.2. for a, b ∈ r+, a < b, 0 < s < 1 and q ∈ (1,∞), we have∣∣∣a(as/q+1,bs/q+1)−ls/q+1s/q+1 (as/q+1,bs/q+1)∣∣∣ ≤ (b−a) 22+s/q ( q − 1 2q − 1 )1−1/q ( 1 s + 1 )1/q (s/q + 1) × { [2a (as,bs) + (s + 1) bs] 1/q + [2a (as,bs) + (s + 1) as] 1/q } . (4.2) proof. let f (x) = xs/q+1, x ∈ r+. then ∣∣∣f′ (x)∣∣∣q = [(s/q + 1)]q xs is an s-convex function on r+. applying corollary 2.6, we obtain the required result. � theorem 4.3. for a, b ∈ r+, a < b, 0 < s < 1 and n, q ∈ n, n,q > 1, we have∣∣∣a(as/q+1,bs/q+1)−ls/q+1s/q+1 (as/q+1,bs/q+1)∣∣∣ ≤ (b−a) 22+s/q−1/q ( 1 s + 1 )1/q ( q − 1 2q − 1 )1−1/q (s/q + 1) × { [a (as,bs) + (2s − 1) bs]1/q + [(2s − 1) as + a (as,bs)]1/q } . (4.3) proof. let f (x) = xs/q+1, x ∈ r+. then ∣∣∣f′ (x)∣∣∣q = [(s/q + 1)]q xs is an s-convex function on r+. applying corollary 2.10, we obtain the required result. � remark 4.1. many other interesting inequalities for means can be obtained by applying the other results to some suitable s-convex functions, however the details are left to the interested reader. remark 4.2. for s = 1, we get some of the results from [12]. references [1] m. w. alomari, m. darus and u. s. kirmaci, some inequalities of hadamard type inequalities for s-convex , acts math. sci. ser. b engl. ed. 31 (2011), no. 4, 1643-1652. [2] s. s. dragomir and s. fitzpatric, the hadamard inequalities for s-convex functions in the second sense, demonstratio math. 32 (1999), no. 4 687-696. [3] s. s. dragomir, r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (1998) 91–95. [4] h. hudzik, l. maligranda, some remarks on s-convex functions, aequationes math. 48 (1994) 100–111. [5] j. hadamard, étude sur les propriétés des fonctions entières et en particulier d’une fonction considerée par riemann, j. math pures appl., 58 (1893), 171–215. [6] ch. hermite, sur deux limites d’une intégrale définie, mathesis 3 (1883), 82. [7] d. -y. hwang, some inequalities for n-time differentiable mappings and applications, kyugpook math. j. 43(2003), 335-343. [8] i̇. i̇şcan, generalization of different type integral inequalities for s-convex functions via fractional integrals, appl. anal. (2014). 93 (9) (2014), 1846-1862. [9] w .-d. jiang, d.-w. niu, y. hua, and f. qi, generalizations of hermite-hadamard inequality to n-time differentiable functions which are s-convex in the second sense, analysis (munich) 32 (2012), 1001–1012. [10] u. s. kırmacı and m.e. özdemir, on some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, appl. math. comp., 153 (2004), 361-368. [11] u. s. kırmacı, improvement and further generalization of inequalities for differentiable mappings and applications, comp and math. with appl., 55 (2008), 485-493. inequalities for n-times differentiable s-convex functions 131 [12] m. a. latif and s. s. dragomir, new inequalities of hermite-hadamard type for n-times differentiable convex and concave functions with applications. filomat 30 (10) (2016), 26092621. [13] m. avci, h. kavurmaci, m. e. özdemir, new inequalities of hermite-hadamard type via s-convex functions in the second sense with applications, appl. math. comp., 217 (2011) 5171–5176. [14] c. e. m. pearce and j. pečarić, inequalities for differentiable mappings with application to special means and quadrature formulae, appl. math. lett., 13(2) (2000), 51–55. [15] shan peng, wei wei and jinrong wang, on the hermite-hadamard inequalities for convex functions via hadamard fractional integrals facta universitatis (niš) ser. math. inform. 29 (1) (2014), 55-75. 1school of computer science and applied mathematics, university of the witwatersrand, private bag 3, wits 2050, johannesburg, south africa 2mathematics, college of engineering and science, victoria university, po box 14428, melbourne city, mc 8001, australia; school of computer science and applied mathematics, university of the witwatersrand, private bag 3, wits 2050, johannesburg, south africa ∗corresponding author: m amer latif@hotmail.com 1. introduction 2. main results 3. applications to the trapezoidal formula 4. applications to the special means references international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 63-68 http://www.etamaths.com second hankel determinant for analytic functions defined by ruscheweyh derivative t. yavuz abstract. let s denote the class of analytic and univalent functions in the open unit disk d = {z : |z| < 1} with the normalization conditions. in the present article an upper bound for the second hankel determinant ∣∣a2a4 −a23∣∣ is obtained for the analytic functions defined by ruscheweyh derivative. 1. introduction let d be the unit disk {z : |z| < 1} , a be the class of functions analytic in d, satisfying the conditions (1.1) f(0) = 0 and f′(0) = 1. then each function f in a has the taylor expansion (1.2) f(z) = z + ∞∑ n=2 anz n because of the conditions (1.1) . let s denote class of analytic and univalent functions in d with the normalization conditions (1.1) . the qth determinant for q ≥ 1 and n ≥ 0 is stated by noonan and thomas [13] as (1.3) hq (n) = ∣∣∣∣∣∣∣∣∣ an an+1 · · · an+q+1 an+1 · · · . . . ... ... an+q−1 · · · an+2q−2 ∣∣∣∣∣∣∣∣∣ . this determinant has also been considered by several authors. for example, noor in [14] determined the rate of growth of hq (n) as n →∞ for functions f given by (1.1) with bounded boundary. ehrenborg in [2] stadied the hankel determinant of exponential polynomials. the hankel transform of an integer sequence and some of its properties were discussed by layman’s article [9]. it is well known that [1] that for f ∈ s and given by (1.2) the sharp inequality ∣∣a3 −a22∣∣ ≤ 1 holds. this corresponds to the hankel determinant with q = 2 and k = 1. after that, fekete-szegö further generalized the estimate ∣∣a3 −µa22∣∣ with real µ and f ∈ s. for a given class of functions in a, the sharp bound for the nonlinear functional ∣∣a2a4 −a23∣∣ is known as the second hankel determinant. this corresponds to the hankel determinant 2010 mathematics subject classification. primary 30c45, secondary 33c45. key words and phrases. univalent functions, starlike functions, convex functions, hankel determinant, ruscheweyh derivative. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 63 64 yavuz with q = 2 and k = 2. in particular, sharp bounds on h2 (2) were obtained by several authors of articles [7], [17], [5], [6], [18] and [12] for different subclasses of univalent functions. let f(z) = z + ∞∑ n=2 anz n and g(z) = z + ∞∑ n=2 bnz n be analytic functions in d. the hadamard product (convolution) of f and g, denoted by f ∗g is defined by (1.4) (f ∗g) (z) = z + ∞∑ n=2 anbnz n, z ∈ d. let n ∈ n0 = {0, 1, 2, . . .} . the ruscheweyh derivative [15] of the nth order of f, denoted by dnf (z) , is defined by (1.5) dnf (z) = z (1 −z)n+1 ∗f (z) = z + ∞∑ k=2 γ (n + k) γ (n + 1) (k − 1)! akz k. the ruscheweyh derivative gave an impulse for various generalization of well known classes of functions. by using the ruscheweyh derivative, we can generalize the class of the starlike and convex functions functions, denoted by s∗ and c,which are defined as (1.6) s∗ = { f(z) ∈ s : re ( zf′ (z) f (z) ) > 0, z ∈ d } and (1.7) c = { f(z) ∈ s : re ( 1 + zf′′ (z) f′ (z) ) > 0, z ∈ d } . the class rn was studied by singh and singh [16], which is given by the following definition (1.8) re z (dnf (z)) ′ dnf (z) > 0, z ∈ d. we denote that r0 = s ∗ and r1 = c. in the present paper, we obtain an upper bound for functional ∣∣a2a4 −a23∣∣ in the class rn. 2. preliminary results the following lemmas are required to prove our main results. let p be the family of all functions p analytic in d for which re (p(z)) > 0 and (2.1) p(z) = 1 + c1z + c2z + · · · . lemma 1. (duren, [1]) if p ∈ p, then |ck| ≤ 2 for each k ∈ n. lemma 2. (grenander&szegö, [4]) the power series for p(z) given by (2.1) converges in d to a function in p if and only if the toeplitz determinants (2.2) dn = ∣∣∣∣∣∣∣∣∣ 2 c1 c2 · · · cn c−1 2 c1 · · · cn−1 ... ... ... ... ... c−n c−n+1 c−n+2 · · · 2 ∣∣∣∣∣∣∣∣∣ , n = 1, 2, · · · . univalent functions 65 and c−k = ck, are all nonnegative. they are strictly positive except for p(z) = m∑ k=1 ρkp0 ( eitkz ) , ρk > 0, tk real and tk 6= tj for k 6= j; in this case dn > 0 for n < m− 1 and dn = 0 for n ≥ m. we may assume that without restriction that c1 > 0. on using lemma 2.2, for n = 2 and n = 3 respectively, we get (2.3) d2 = ∣∣∣∣∣∣ 2 c1 c2 c1 2 c1 c2 c1 2 ∣∣∣∣∣∣ = 8 + 2 re { c21c2 } − 2 |c2| 2 − 4c21 ≥ 0, which is equivalent to (2.4) 2c2 = c 2 1 + x ( 4 − c21 ) for some x, |x| ≤ 1. if we consider the determinant (2.5) dn = ∣∣∣∣∣∣∣∣ 2 c1 c2 c3 c1 2 c1 c2 c2 c1 2 c1 c3 c2 c1 2 ∣∣∣∣∣∣∣∣ ≥ 0, we get the following inequality (2.6) ∣∣∣(4c3 − 4c1c2 + c31)(4 − c21) + c1 (2c2 − c21)2∣∣∣ ≤ 2 (4 − c21)2−2 ∣∣(2c2 − c21)∣∣2 . from (2.4) and (2.6), it is obtained that (2.7) 4c3 = c 3 1 + 2c1 ( 4 − c21 ) x− c1 ( 4 − c21 ) x2 + 2c1 ( 4 − c21 )( 1 −|x|2 ) z for some z, |z| ≤ 1. 3. main results we prove the following theorem by using thecniques of libera and zlotkiewicz [10], [11]. theorem 1. let the function f given by (1.2) be in the class in rn. then (3.1) ∣∣a2a4 −a23∣∣ ≤   1, n = 0 1 8 , n = 1 12(n−1) (n+1)2(n+2)2(n+3) , n > 1 proof. since f ∈ rn, there exists an analytic function p ∈ p in the unit disk d with p(0) = 1 and re (p(z)) > 0 such that (3.2) z (dnf (z)) ′ dnf (z) = p(z) let (3.3) f(z) = dnf(z) = z + ∞∑ k=2 akz k, where (3.4) ak = γ (n + k) γ (n + 1) (k − 1)! ak, 66 yavuz then we have (3.5) zf ′(z) f(z) = p(z). by using the series expansion of f(z) and p(z) as in (3.3) and (2.1) , equating coefficients in (3.5) yields a2 = 1 n + 1 c1 a3 = 1 (n + 1) (n + 2) { c2 + c 2 1 } (3.6) a4 = 1 (n + 1) (n + 2) (n + 3) { 2c3 + 3c1c2 + c 3 1 } . hence, we get from (3.6) (3.7) a2a4 −a23 = a(n) { 2c1c3 + 3c 2 1c2 + c 4 1 −b(n) ( c2 + c 2 1 )2} , where (3.8) a(n) = 1 (n + 1) (n + 2) (n + 3) , and (3.9) b(n) = ( n + 3 n + 2 ) , n = 0, 1, 2, · · · . using (2.4) and (2.7) in (3.7) ,we get∣∣a2a4 −a23∣∣ = a(n) ∣∣2c1c3 + 3c21c2 + c41 −b(n) (c22 + 2c1c2 + c41)∣∣ and (3.10) ∣∣a2a4 −a23∣∣ = a(n) ∣∣∣∣3 ( 1 − 3 4 b(n) ) c41 + 3 2 (1 −b(n)) c21x ( 4 − c21 ) − c21 2 ( 4 − c21 ) x2 + c1 ( 4 − c21 )( 1 −|x|2 ) z −b(n) x2 ( 4 − c21 )2 4 ∣∣∣∣∣ since the function p(eiθz), (θ ∈ r) is also in the class p , we assume that without loss of generality that c1 > 0. for convenience of notation, we take c1 = c, c ∈ [0, 2] . applying the triangle inequality with the assumptions c1 = c ∈ [0, 2] , |x| = ρ and |z| ≤ 1, it is obtained that∣∣a2a4 −a23∣∣ ≤ a(n) {3 ∣∣∣∣1 − 34b(n) ∣∣∣∣c4 + 32 (b(n) − 1) c2ρ(4 − c2)(3.11) +ρ2 ( 4 − c2 ) c (c− 2) 2 + c ( 4 − c2 ) + b(n)ρ2 ( 4 − c2 )2 4 } = g(c,ρ). we now maximize the function g(c,ρ) on the closed square [0, 2] × [0, 1] . since (3.12) ∂g(c,ρ) ∂ρ = 3 2 (b(n) − 1) c2 ( 4 − c2 ) −ρ ( 4 − c2 ) (2 − c) { c− b(n) 2 (2 + c) } univalent functions 67 and b(n) ∈ [ 1, 3 2 ] , we get the following inequality (3.13) ∂g(c,ρ) ∂ρ ≥ ρ ( 4 − c2 ) (2 − c) (6 − c) 4 > 0. hence, g(c,ρ) can not have a maximum in the interior of the closed square [0, 2]× [0, 1] . hence, for fixed c ∈ [0, 2] (3.14) max 0≤ρ≤1 g(c,ρ) = g(c, 1) = f(c). one can obtain that (3.15) ∣∣a2a4 −a23∣∣ ≤ a(n)f(c), where (3.16) f(c) = 3 ∣∣∣∣1 − 34b(n) ∣∣∣∣c4 + 32 (b(n) − 1) c2 (4 − c2) + c ( 4 − c2 ) 2 + b(n) ( 4 − c2 )2 4 . since (3.17) f ′(c) =   25 3 c3 + c ( 4 − c2 ) + 3 2 c3, n = 0 8 3 c ( 1 − c2 ) , n = 1 (12 − 9b(n)) c3 + (b(n) − 1) c ( 4 − c2 ) − 3 (b(n) − 2) c3, n > 1 , we have to consider following three cases: case 1. for n = 0, f ′(c) > 0. hence f(c) ≤ f(2). we get the following result (3.18) ∣∣a2a4 −a23∣∣ ≤ a(0) {48 ∣∣∣∣1 − 34b(0) ∣∣∣∣ } = 1. this one coincides with the result in the article [8]. case 2. after necessarly calculations, it is obtained that (3.19) f ′(0) = 0 and f ′(1) = 0. since f ′′(0) > 0 and f ′′(1) < 0, f(c) has a maximum at c = 1. hence, we obtain (3.20) ∣∣a2a4 −a23∣∣ ≤ 18, which is also stated in [8]. case 3. let n > 1. then, f ′(c) can be rewrite as (3.21) f ′(c) = c { (20 − 14b(n)) c2 + 8 (b(n) − 1) } . since 20−14b(n) > 0 and b(n)−1 > 0, we get f ′(0) = 0, f ′′(0) > 0 and f ′(c) > 0 in the interval (0, 2] . therefore, it is obvious that (3.22) ∣∣a2a4 −a23∣∣ ≤ a(n) {48 ∣∣∣∣1 − 34b(n) ∣∣∣∣ } = 12 (n− 1) (n + 1) 2 (n + 2) 2 (n + 3) . this completes the proof of theorem. � 68 yavuz references [1] p. l. duren, univalent functions, springer-verlag, new york, berlin, heidelberg, tokyo, 1983. [2] r. ehrenborg, the hankel determinant of exponantial polynomials. american mathematical monthly, 107 (2000), 557-560. [3] m. fekete and g. szegö, eine bemerkung uber ungerade schlichte funktionen, j. london math. soc, 8 (1933), 85-89. [4] u. grenander and g. szegö, toeplitz forms and their application, univ. of calofornia press, berkely and los angeles, (1958). [5] t. hayami and s. owa, hankel determinant for p-valently starlike and convex functions of order α, general math., 17 (2009), 29-44. [6] t. hayami and s. owa, generalized hankel determinant for certain classes, int. j. math. anal., 4 (2010), 2573-2585. [7] a. janteng, s. a. halim, and m. darus, coefficient inequality for a function whose derivative has positive real part, j. ineq. pure and appl. math, 7 (2) (2006), 1-5. [8] a. janteng, halim, s. a. and darus, m. : hankel determinant for starlike and convex functions, int. journal of math. analysis, i (13) (2007), 619-625. [9] j. w. layman, the hankel transform and some of its properties. j. of integer sequences, 4 (2001), 1-11. [10] r.j. libera, and e.j. zlotkiewicz, early coefficients of the inverse of a regular convex function, proc. amer. math. soc., 85(2) (1982), 225–230. [11] r.j. libera, and e.j. zlotkiewicz, coefficient bounds for the inverse of a function with derivative in p , proc. amer. math. soc., 87(2) (1983), 251–289. [12] g. murugusundaramoorthy and n. magesh, coefficient inequalities for certain classes of analytic functions associated with hankel determinant, bulletin of math. anal. appl., i (3) (2009), 85-89. [13] j. w. noonan and d. k. thomas, on the second hankel determinant of a really mean p valent functions, trans. amer. math. soc, 223 (2) (1976), 337-346. [14] k. i. noor, hankel determinant problem for the class of functions with bounded boundary rotation. rev. roum. math. pures et appl, 28 (8) (1983), 731-739. [15] s. ruscheweyh, new criteria for univalent functions, proc. amer. math. soc. 49 (1975) 109115. [16] r. singh, s. singh, integrals of certain univalent functions, proc. amer. math. soc. 77 (1979) 336-340. [17] s. c. soh and d. mohamad, coefficient bounds for certain classes of close-to-convex functions, int. journal of math. analysis, 2 (27) (2008), 1343-1351. [18] t. yavuz, second hankel determinant problem for a certain subclass of univalent functions, international journal of mathematical analysis vol. 9(10), (2015), 493 498. gebze technical university, department of mathematics, kocaeli, turkey international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 22-37 http://www.etamaths.com existence of quasilinear neutral impulsive integrodifferential equations in banach space b. radhakrishnan abstract. in this paper, we devoted to study the existence of mild solutions for quasilinear impulsive integrodifferential equation in banach spaces. the results are established by using hausdorff’s measure of noncompactness and the fixed point theorems. application is provided to illustrate the theory. 1. introduction in various fields of engineering and physics, many problems that are related to linear viscoelasticity, nonlinear elasticity have mathematical models and are described by the problems of differential or integral equations or integrodifferential equations. our work centers on the problems described by the integrodifferential models. it is important to note that when we describe the systems which are functions of space and time by partial differential equations, in some situations, such a formulation may not accurately model the physical system because, while describing the system as a function at a given time, it may fail to take into account the effect of past history. neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades [1, 2, 3]. a good guide to the literature for neutral functional differential equations is the book by hale and verduyn lunel [4] and the references therein. the existence of solution to evolution equations with nonlocal conditions in banach space was studied first by byszewski [5, 6]. byszewski and lakshmikanthan [7] proved an existence and uniqueness of solutions of a nonlocal cauchy problem in banach spaces. ntouyas and tsamatos [8] studied the existence for semilinear evolution equations with nonlocal conditions. the problem of existence of solutions of evolution equations in banach space has been studied by several authors [9, 10]. however, one may easily visualize that abrupt changes such as shock, harvesting and disasters may occur in nature. these phenomena are short time perturbations whose duration is negligible in comparison with the duration of the whole evolution process. consequently, it is natural to assume, in modeling these problems, that these perturbations act instantaneously, that is in the form of impulses. the theory of impulsive differential equation [11, 12, 13] is much richer than the corresponding theory of differential equations without impulsive effects. the impulsive condition ∆u(ti) = u(t + i ) −u(t − i ) = ii(u(t − i )), i = 1, 2, . . . ,m, 2010 mathematics subject classification. 34a37, 34k05, 34k40, 47h10. key words and phrases. mild solution, neutral differential equation, impulsive condition, hausdorff measure of noncompactness, fixed point theorem. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 22 quasilinear neutral impulsive integrodifferential equations 23 is a combination of traditional initial value problems and short-term perturbations whose duration is negligible in comparison with the duration of the process. liu [14] discussed the iterative methods for the solution of impulsive functional differential systems. measures of noncompactness are a very useful tool in many branches of mathematics. they are used in the fixed point theory, linear operators theory, theory of differential and integral equations and others [15]. there are two measures which are the most important ones. the kuratowski measure of noncompactness σ(x) of a bounded set x in a metric space is defined as infimum of numbers r > 0 such that x can be covered with a finite number of sets of diameter smaller than r. the hausdorff measure of noncompactness χ(x) defined as infimum of numbers r > 0 such that x can be covered with a finite number of balls of radii smaller than r. there exist many formulae on χ(x) in various spaces [15, 18]. let e be a banach space and f be a subspace of e. let χe(x), χf(x), σe(x), σf(x) denote hausdorff and kuratowski measures in spaces e,f, respectively. then, for any bounded x ⊂ f we have χe(x) ≤ χf(x) ≤ σf(x) = σe(x) ≤ 2χe(x). the notion of a measure of weak compactness was introduced by de blasi [16] and was subsequently used in numerous branches of functional analysis and the theory of differential and integral equations. several authors have studied the measures of noncompactness in banach spaces [17, 18, 19]. motivated by [9, 15, 20, 21], in this paper, we study the existence results for quasilinear equation represented by first-order neutral integrodifferential equations using the semigroup theory and the measure of noncompactness. 2. preliminaries we consider the quasilinear integrodifferential equations with impulsive and nonlocal condition of the form d dt [ x(t) + e ( t,x(t), ∫ t 0 k(t,s,x(s))ds )] + a(t,x(t))x(t) = f(t,x(t)) + ∫ t 0 g(t,s,x(s))ds, t ∈ [0,b], t 6= tk,(1) x(0) + h(x) = x0,(2) ∆x(tk) = ik(x(tk)), k = 1, 2, 3, . . . ,n,(3) where a : [0,b] × x → x is a continuous function in banach space x, x0 ∈ x, f : [0,b] × x → x, g : λ × x → x, h : pc([0,b],x) → x, e : [0,b] × x × x → x, k : λ × x → x and ∆x(tk) = x(t+k ) − x(t − k ), for all k = 1, 2, . . . ,m; 0 = t0 < t1 < t2 < ... < tm < tm+1 = b; constitutes an impulsive condition. here λ = {(t,s) : 0 ≤ s ≤ t ≤ b}. let x be a banach space with norm || · ||. let pc([0,b],x) consist of functions u from [0,b] into x, such that x(t) is continuous at t 6= ti and left continuous at t = ti and the right limit x(t + i ) exists, for i = 1, 2, 3, . . . ,n. evidently pc([0,b],x) is a banach space with the norm ‖x‖pc = sup t∈[0,b] ‖x(t)‖, 24 radhakrishnan and denoted l([0,b],x) by the space of x-valued bochner integrable functions on [0,b] with the form ‖x‖l = ∫ b 0 ‖x(t)‖dt. the hausdorff’s measure of noncompactness χy is defined by χ(b) = inf{r > 0, b can be covered by finite number of balls with radii r}, for bounded set b in a banach space y . lemma 2.1 [15]. let y be a real banach space and b,e ⊆ y be bounded, with the following properties: (i) b is precompact if and only if χx (b) = 0. (ii) χy (b) = χy (b̄) = χy (conb), where b̄ and conb mean the closure and convex hull of b respectively. (iii) χy (b) ≤ χy (e), where b ⊆ e. (iv) χy (b + e) ≤ χy (b) + χy (e), where b + e = {x + y : x ∈ b, y ∈ e}. (v) χy (b∪e) ≤ max{χy (b), χy (e)}. (vi) χy (λb) ≤ |λ|χy (b), for any λ ∈ r. (vii) if the map f : d(f) ⊆ y → z is lipschitz continuous with constant r, then χz (fb) ≤ rχy (b), for any bounded subset b ⊆ d(f), where z be a banach space. (viii) χy (b) = inf{dy (b,e); e ⊆ y is precompact} = inf{dy (b,e); e ⊆ y is finite valued}, where dy (b,e) means the nonsymmetric (or symmetric) hausdorff distance between b and e in y . (ix) if {wn}+∞n=1 is decreasing sequence of bounded closed nonempty subsets of y and lim n→∞ χy (wn) = 0, then +∞⋂ n=1 wn is nonempty and compact in y. the map f : w ⊆ y → y is said to be a χy -contraction if there exists a positive constant r < 1 such that χy (f(b)) ≤ rχy (b) for any bounded closed subset b ⊆ w, where y is a banach space. lemma 2.2 (darbo-sadovskii [15]). if w ⊆ y is bounded closed and convex, the continuous map f : w → w is a χy -contraction, the map f has atleast one fixed point in w. we denote by χ the hausdorff’s measure of noncompactness of x and also denote χc by the hausdorff’s measure of noncompactness of pc([0,b],x). before we prove the existence results, we need the following lemmas. lemma 2.3 [22] if w ⊆ pc([0,b],x) is bounded, then χ(w(t)) ≤ χc(w), for all t ∈ [0,b], where w(t) = {u(t); u ∈ w}⊆ x. furthermore if w is equicontinuous on [0,b], then χ(w(t)) is continuous on [0,b] and χc(w) = sup{χ(w(t)), t ∈ [0,b]}. lemma 2.4 [22, 23]. if {un}∞n=1 ⊂ l1([0,b],x) is uniformly integrable, then the function χ({un(t)}∞n=1) is measurable and χ ({∫ t 0 un(s)ds }∞ n=1 ) ≤ 2 ∫ t 0 χ({un(s)}∞n=1)ds.(4) quasilinear neutral impulsive integrodifferential equations 25 lemma 2.5 if w ⊆pc([0,b],x) is bounded and equicontinuous, then χ(w(t)) is continuous and χ (∫ t 0 w(s)ds ) ≤ ∫ t 0 χ(w(s))ds, for all t ∈ [0,b],(5) where ∫ t 0 w(s)ds = {∫ t 0 u(s)ds : u ∈ w } . the c0 semigroup uu(t,s) is said to be equicontinuous if (t,s) → {uu(t,s)u(s) : u ∈ b} is equicontinuous for t > 0, for all bounded set b in x. the following lemma is obvious. lemma 2.6 if the evolution family {uu(t,s)}0≤s≤t≤b is equicontinuous and η ∈ l([0,b],r+), then the set {∫ t 0 uu(t,s)u(s)ds, ||u(s)|| ≤ η(s), for a.e s ∈ [0,b] } , is equicontinuous for t ∈ [0,b]. we know that, for any fixed u ∈ pc([0,b],x) there exist a unique continuous function uu : [0,b] × [0,b] → b(x) defined on [0,b] × [0,b] such that uu(t,s) = i + ∫ t s au(w)uu(w,s)dw,(6) where b(x) denote the banach space of bounded linear operators from x to x with the norm ||f|| = sup {||fu|| : ||u|| = 1}, and i stands for the identity operator on x, au(t) = a(t,u(t)), we have uu(t,t) = i, uu(t,s)uu(s,r) = uu(t,r), (t,s,r) ∈ [0,b] × [0,b] × [0,b], ∂uu(t,s) ∂t = au(t)uu(t,s), for almost all t,s ∈ [0,b]. 3. the existence of mild solution definition 3.1 a function x ∈pc([0,b],x) is said to be a mild solution of (1)−(3) if it satisfies the integral equation x(t) = ux(t, 0)x0 −ux(t, 0)h(x) + ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 a(s,x(s))ux(t,s)e ( s,x(s), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0<tk<t ux(t,tk)ik(x(tk)), 0 ≤ t ≤ b. in this paper, we denote m0 = sup{‖ux(t,s)‖ : (t,s) ∈ [0,b] × [0,b]}, for all x ∈ x. without loss of generality, we let x0 = 0. assume the following conditions: (h1) the evolution family {ux(t,s)}0≤s≤t≤b generated by a(t,x(t)) is equicontinuous and ||ux(t,s)|| ≤ m0, for almost t,s ∈ [0,b]. (h2) (a) the function h : pc([0,b] ×x) → x is continuous and compact. (b) there exists n0 > 0 such that ||h(x)|| ≤ n0, for all u ∈pc([0,b]; x). 26 radhakrishnan (h3) (i) the nonlinear function f : [0,b] ×x → x satisfies the carathèodorytype conditions; that is, f(·,x) is measurable for all x ∈ x and f(t, ·) is continuous, for a.e t ∈ [a,b]. (ii) there exists a function α ∈l([0,b],r+) such that for every x ∈ x, we have ‖f(t,x)‖≤ α(t)(1 + ‖x‖), a.e t ∈ [0,b]. (iii) there exists a function mf ∈l([0,b],r+) such that, for every bounded k ⊂ x, we have χ(f(t,k)) ≤ mf (t)χ(k), a.e t ∈ [0,b]. (h4) (i) the nonlinear function g : [0,b]×[0,b]×x → x satisfies the carathèodorytype conditions; i.e., g(·, ·,x) is measurable, for all x ∈ x and g(t,s, ·) is continuous for a.e t ∈ [a,b]. (ii) there exist two functions β1 ∈l([0,b],r+) and β2 ∈l([0,b],r+) such that for every x ∈ x, we have ‖g(t,s,x(s))‖≤ β1(t)β2(s)(1 + ‖x(s)‖), a.e t ∈ [0,b]. (iii) there exist functions mg,ng ∈l([0,b],r+) such that, for every bounded k ⊂ x, we have χ(g(t,s,k)) ≤ mg(t)ng(s)χ(k), a.e t ∈ [0,b]. assume that the finite bound of ∫ t 0 mg(s)ds is g0. (h5) (i) the function e : [0,b] ×x ×x → x satisfies the carathèodory-type conditions; that is, e(·,x,x1) is measurable, for all x,x1 ∈ x and e(t, ·, ·) is continuous, for a.e t ∈ [0,b]. (ii) there exists a function γ ∈l([0,b],r+) such that for every x,x1 ∈ x, we have ‖e(t,x,x1)‖≤ γ(t)(1 + ‖x‖) + ‖x1‖, a.e t ∈ [0,b]. (iii) the nonlinear function q : [0,b]×[0,b]×x → x satisfies the caratheodorytype conditions; i.e. k(·, ·,x) is measurable, for all x ∈ x and k(t,s, ·) is continuous, for a.e t ∈ [0,b]. (iv) there exist two functions ω1 ∈ l([0,b],r+) and ω2 ∈ l([0,b],r+) such that for every x ∈ x, we have ‖k(t,s,x(s))‖≤ ω1(t)ω2(s)(1 + ‖x(s)‖), a.e t ∈ [0,b]. (v) there exists a function η ∈l([0,b],r+) such that for every x,x1 ∈ x, we have ‖a(t,x(t))e(t,x,x1)‖≤ η(t)‖e(t,x,x1)‖, a.e t ∈ [0,b]. (vi) there exists a function me ∈l([0,b],r+) such that, for every bounded k, k1 ⊂ x, we have χ(e(t,k,k1)) ≤ me(t)(χ(k) + ϕ(k1)), a.e t ∈ [0,b]. assume that the finite bound of ∫ t 0 me(s)ds is g1. (vii) there exist functions mk,nk ∈l([0,b],r+) such that for every bounded k ⊂ x, we have χ(k(t,s,k)) ≤ mk(t)nk(s)χ(k), a.e t ∈ [0,b]. assume that the finite bound of ∫ t 0 mk(s)ds is g2. quasilinear neutral impulsive integrodifferential equations 27 (h6) for every t ∈ [0,b] and there exist positive constants n1 and n2, the scalar equation m(t) = m0n0 + γ1(1 + m(s)) + m0γ0 + m0c1ω(t)(1 + m(s)) + γ(t)c1 ∫ t 0 η(t)ω1(s)ds +m0 ∫ t 0 [ α(s)(1 + m(s))ds + c0 ∫ t 0 β1(s)(1 + m(s))ds + n∑ k=1 dk ] , where c0 = ∫ s 0 β(t)dt. (h7) ik : x → x is continuous. there exist constants dk > 0 k = 1, 2, 3, . . . ,n such that ‖ik(x(tk))‖≤ n∑ k=1 dk, where, k = 1, 2, 3, . . . ,n. for any bounded subset k ⊂ x, and there is a constant lk > 0 such that χ(ik(k)) ≤ n∑ k=1 liχ(k), k = 1, 2 . . . ,n. theorem: 3.1. if assumptions (h1) − (h7) holds, then the quasilinear neutral impulsive problem (1) − (3) has at least one mild solution. proof. let m(t) be a solution of the scalar equation m(t) = m0n0 + γ1(1 + m(s)) + m0γ0 + m0c1ω(t)(1 + m(s)) + γ(t)c1 ∫ t 0 η(t)ω1(s)ds +m0 ∫ t 0 [ α(s)(1 + m(s))ds + c0 ∫ t 0 β1(s)(1 + m(s))ds + n∑ k=1 dk ] .(7) let us assume that the finite bound of ∫ t 0 β2(s)ds is c0, for t ∈ [0,b]. consider the map f : pc([0,b],x) →pc([0,b],x) defined by (fx)(t) = ux(t, 0)h(x) + ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 a(s,x(s))ux(t,s)e ( s,x(s), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0<tk<t ux(t,tk)ik(x(tk)), 0 ≤ t ≤ b, for all x ∈pc([0,b],x).(8) let us take w0 = {x ∈ pc([0,b],x), ||x(t)|| ≤ m(t), for all t ∈ [0,b]}. then w0 ⊆ pc([0,b]; x) is bounded and convex. we define w1 = con k(w0), where con means the closure of the convex hull in pc([0,b],x). as ux(t,s) is equicontinuous, h is compact and w0 ⊆ pc([0,b],x) is bounded, due to lemma 2.6 and using the assumptions, w1 ⊆pc([0,b],x) is bounded closed convex nonempty and equicontinuous on [0,b]. 28 radhakrishnan for any x ∈f(w0), we know ‖x(t)‖ ≤ ‖ux(t, 0)h(x)‖ + ‖ux(t, 0)e(0,x(0), 0)‖ + ‖e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) ‖ + ∫ t 0 ‖a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds + ∫ t 0 ‖ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(s))dτ ] ‖ds + ∑ 0<tk<t ‖ux(t,tk)ii(x(tk))‖ ≤ m0n0 + m0γ0 + γ1(1 + ‖x‖) + ∫ t 0 k(t,s,x(s))ds +m0η(t) ∫ t 0 ‖e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds +m0 [∫ t 0 ||f(s,x(s))||ds + ∫ t 0 ∫ s 0 ||g(s,τ,x(τ))||dτds ] + m0 n∑ k=1 ‖ik(x(tk))‖ ≤ m0n0 + m0γ0 + γ1(1 + ‖x‖) + ω1(t) ∫ t 0 ω2(s)(1 + ‖x‖)ds +m0 [∫ t 0 [ η(s)γ(s)ds + ∫ s 0 ω1(s)ω2(τ)dsdτ ] (1 + ‖x‖) ] +m0 ∫ t 0 α(s)(1 + ‖x(s)‖)ds + m0 ∫ t 0 ∫ s 0 β1(s)β2(τ)(1 + ‖x(τ)‖)dτds+m0 n∑ k=1 dk ≤ m0n0 + γ1(1 + m(s)) + m0γ0 + m0c1ω(t)(1 + m(s)) + η(t)γ(t)c1 ∫ t 0 ω1(s)ds +m0 ∫ t 0 [ α(s)(1 + m(s))ds + c0 ∫ t 0 β1(s)(1 + m(s))ds + n∑ k=1 dk ] = m(t). it follows that w1 ⊂ w0. we define wn+1 = con f(wn), for n = 1, 2, 3, · · · . from above we know that {wn}∞n=1 is a decreasing sequence of bounded, closed, convex, equicontinuous on [0,b] and nonempty subsets in pc([0,b],x). now for n ≥ 1 and t ∈ [0,b], wn(t) and f(wn(t)) are bounded subsets of x, hence, for any � > 0, there is a sequence {xk}∞k=1 ⊆ wn, such that (see, e.g. [24], quasilinear neutral impulsive integrodifferential equations 29 pp.125). χ(wn+1(t)) = χ(fwn(t)) ≤ 2χ ( e(t,{xk(s)}∞k=1, ∫ t 0 k(t,s,{xk(t)}∞k=1)ds) ) +2m0η(t) ∫ t 0 χ ( e(s,{xk(s)}∞k=1, ∫ s 0 k(s,τ,{xk(τ)}∞k=1)dτ) ) ds +2m0 ∫ t 0 χ ( f(s,{xk(s)}∞k=1) ) ds + 4m0 ∫ t 0 ∫ s 0 χ ( g(s,τ,{uk(τ)}∞k=1) ) dτds +2m0 n∑ i=1 χ ( ik({uk(tk)}∞k=1) ) + � ≤ 2me(t)χ{xk(t)}∞k=1 + 2mk(t) ∫ t 0 mk(s)χ{xk(s)}∞k=1ds +2m0η(t) [∫ t 0 me(s)χ{xk(s)}∞k=1ds + 2 ∫ t 0 ∫ s 0 mk(s)mk(τ)χ{xk(τ)}∞k=1dτds ] +2m0 ∫ t 0 mf (s)χ ( {uk(s)}∞k=1 ) ds+4m0 ∫ t 0 ∫ s 0 mg(s)ng(τ)χ ( {uk(τ)}∞k=1 ) dτds +2m0 n∑ i=1 liχ ( {uk(tk)}∞k=1 ) + � ≤ 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] χ(wn(t)) + 2m0 [∫ t 0 {2g2mk(s) +mf (s)}χ(wn(s))ds + 2g0 ∫ t 0 ng(s)χ(wn(s))ds ] + 2m0 n∑ k=1 lkχ(wn(tk))+�. since � > 0 is arbitrary, it follows that from the above inequality that χ(wn+1(t)) ≤ 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] χ(wn(t)) +2m0 [∫ t 0 [2g2mk(s) + mf (s) + 2g0ng(s)]χ(wn(s)) ] ds +2m0 n∑ k=1 lkχ(wn(tk), for all t ∈ [0,b].(9) because wn is decreasing for n, we have λ(t) = lim n→∞ χ(wn(t)), for all t ∈ [0,b]. from (9), we have λ(t) ≤ 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] λ(t) +2m0 [∫ t 0 [2g2mk(s) + mf (s) + 2g0ng(s)]λ(s)ds+ n∑ k=1 lkλ(tk) ] , for t ∈ [0,b], which implies that λ(t) = 0, for all ti ∈ [0,b]. by lemma 2.3, we know that lim n→∞ χ(wn(t)) = 0. using lemma 2.1 we know that w = ∞⋂ n=1 wn is convex 30 radhakrishnan compact and nonempty in pc([0,b],x) and f(w) ⊂ w. by the schauder fixed point theorem, there exist at least one mild solution u of the initial value problem (1) − (3), where x ∈ w is a fixed point of the continuous map f. � remark 3.2. if the functions f, g and ii are compact or lipschitz continuous (see e.g [5, 7]), then (h3) − (h7) is automatically satisfied. in some of the early related results in references and above results, it is supposed that the map h is uniformly bounded. in fact, if h is compact, then it must be bounded on bounded set. here we give an existence result under growth condition of f,g and ii, when h is not uniformly bounded. precisely, we replace the assumptions (h3) − (h6) by (h8) there exists a function p ∈l([0,b],r+) and a increasing function φ : r+ → r+ such that ‖f(t,x)‖≤ lf (t)φ(‖x‖), for a.e t ∈ [0,b], for all x ∈pc([0,b],x). (h9) there exist two functions lg ∈ l([0,b],r+)and l̂g ∈ l([0,b],r+) and a increasing function ψ : r+ → r+ such that ‖g(t,s,x)‖≤ lg(t)l̂g(s)ψ(‖x‖), for a.e t ∈ [0,b] and for all lg ∈pc([0,b],x). assume that the finite bound of ∫ t 0 lg(s)ds is g3. (h10) there exists a function le ∈ l([0,b],r+) and a increasing function γ : r+ → r+ such that ‖e(t,x,x1)‖≤ le(t)γ(‖x‖) + ‖x1‖ for a.e t ∈ [0,b] and for all lg ∈pc([0,b],x). assume that the finite bound of ∫ t 0 le(s)ds is g5. (h11) there exist two functions lk ∈ l([0,b],r+)and l̂k ∈ l([0,b],r+) and a increasing function θ : r+ → r+ such that ‖k(t,s,x)‖≤ lk(t)l̂k(s)θ(‖x‖), for a.e t ∈ [0,b] and for all lk ∈pc([0,b],x). assume that the finite bound of ∫ t 0 lk(s)ds is g4. theorem: 3.2. suppose that the assumptions (h1) − (h2) and (h8) − (h11) are satisfied, then the equation (1) − (3) has at least one mild solution if lim r→∞ sup 1 r { m0 [ ϕ(r) + le(t) ] + le(t)(γ‖x‖) +g3lk(t)θ(r) + η(t)m0 [ g4γ(r) + g3θ(r) ∫ t 0 l̂k(s)ds ] +m0 [ φ(r) ∫ t 0 lf (s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ]} < 1,(10) where ϕ(r) = sup{||h(x)||, ||x|| ≤ r}. quasilinear neutral impulsive integrodifferential equations 31 proof. the inequality (10) implies that there exist a constant r > 0 such that m0 [ ϕ(r) + le(t) ] + le(t)(γ‖x‖) + g3lk(t)θ(r) + η(t)m0 [ g4γ(r) +g3θ(r) ∫ t 0 l̂k(s)ds ] + m0 [ φ(r) ∫ t 0 p(s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ] < r, as in the proof of theorem 3.1, let w0 = {x ∈pc([0,b],x), ||x(t)|| ≤ r} and w1 = con fw0. then for any x ∈ w1, we have ‖x(t)‖ ≤ ‖ux(t, 0)h(x)‖ + ‖ux(t, 0)e(0,x(0), 0)‖ + ‖e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) ‖ + ∫ t 0 ‖a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds + ∫ t 0 ‖ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(s))dτ ] ‖ds + ∑ 0<tk<t ‖ux(t,tk)ii(x(tk))‖ ≤ m0 [ ϕ(r) + le(t) ] + le(t)(γ‖x‖) + ∫ t 0 lk(t)l̂k(s)θ(‖x‖)ds +η(t)m0 ∫ t 0 [ p(s)γ(‖x‖) + ∫ s 0 lk(s)l̂k(τ)θ(‖x‖)dτ ] ds +m0 [∫ t 0 lf (s)φ(‖x(s)‖)ds + ∫ t 0 ∫ s 0 lg(s)l̂g(τ)ψ‖(x(τ))||dτds + n∑ k=1 dk ] ≤ m0 [ ϕ(r) + le(t) ] + le(t)(γ‖x‖) + g3lk(t)θ(‖x‖) +η(t)m0 [ g4γ(‖x‖) + g3 ∫ t 0 l̂k(s)θ(‖x‖)ds ] +m0 [∫ t 0 p(s)φ(‖x(s)‖)ds + g2 ∫ t 0 l̂g(s)ψ(‖x(s)‖)ds + n∑ k=1 dk ] ‖x(t)‖ ≤ m0 [ ϕ(r) + le(t) ] + le(t)(γ‖x‖) + g3lk(t)θ(r) +η(t)m0 [ g4γ(r) + g3θ(r) ∫ t 0 l̂k(s)ds ] +m0 [ φ(r) ∫ t 0 p(s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ] < r, for t ∈ [0,b]. it means that w1 ⊂ w0. so we can complete the proof similarly to theorem 3.1. 4. when h is lipschitz in this section, we discuss the equation (1) − (3) when h is lipschitz and f,g and ik are not lipschitz. assume that 32 radhakrishnan (h12) the function h is a lipschitz continuous in x, there exists a constant l0 > 0 such that ‖h(x) −h(y)‖≤ l0‖x−y‖, x,y ∈pc([0,b],x). theorem: 4.1. suppose that the assumptions (h1)−(h12) are satisfied, then the equation (1) − (3) has at least one mild solution provided that m0[l0 + h4(t)]χc(b) + 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] +2m0 [∫ t 0 {2g2mk(s) + mf (s) + 2g0ng(s)}ds + n∑ k=1 lk ] < 1.(11) proof. consider the map f : pc([0,b],x) → pc([0,b],x) is defined by f = f1 + f2, where (f1x)(t) = ux(t, 0)h(u) + ux(t, 0)e(0,x(0), 0), (f2u)(t) = ∫ t 0 a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ds −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0<tk<t ux(t,tk)ik(x(tk)), for x ∈ pc([0,b],x). as defined in the proof of theorem 3.1. we define w0 = {x ∈ pc([0,b],x) : ||x(t)|| ≤ m(t), for all t ∈ [0,b]} and let w = confw0. then from the proof of theorem 3.1 we know that w is a bounded closed convex and equicontinuous subset of pc([0,b],x) and fw ⊂ w. we shall prove that f is χc-contraction on w. then darbo-sadovskii’s fixed point theorem can be used to get a fixed point of f in w, which is a mild solution of (1) − (2). first, for every bounded subset b ⊂ w, from (h12) and lemma 2.1 we have χc(f1b) = χc(ub(t, 0)h(b)) + ub(t, 0)e(0,b(0), 0) ≤ m0χc [ (h(b)) + e(0,b(0), 0) ] ≤ m0[l0 + h4(t)]χc(b)(12) next, for every bounded subset b ⊂ w, for t ∈ [0,b] and every � > 0, there is a sequence {xk}∞k=1 ⊂ b such that χ(f2(b(t)) ≤ 2χ({f2xi(t)}∞n=1 + �. quasilinear neutral impulsive integrodifferential equations 33 note that b and f2b are equicontinuous, we can get from lemma 2.1, lemma 2.4, lemma 2.5 and using the assumptions we get χ(f2b(t)) ≤ 2χ ( e(t,{xk(s)}∞k=1, ∫ t 0 k(t,s,{xk(t)}∞k=1)ds) ) 2m0η(t) ∫ t 0 χ ( e(s,{xk(s)}∞k=1, ∫ s 0 k(s,τ,{xk(τ)}∞k=1)dτ) ) ds +2m0 ∫ t 0 χ ( f(s,{xk(s)}∞k=1) ) ds + 4m0 ∫ t 0 ∫ s 0 χ ( g(s,τ,{uk(τ)}∞k=1) ) dτds +2m0 n∑ i=1 χ ( ik({uk(tk)}∞k=1) ) + � ≤ 2me(t)χ{xk(t)}∞k=1 + 2mk(t) ∫ t 0 mk(s)χ{xk(s)}∞k=1ds +2m0η(t) [∫ t 0 me(s)χ{xk(s)}∞k=1ds + 2 ∫ t 0 ∫ s 0 mk(s)mk(τ)χ{xk(τ)}∞k=1dτds ] +2m0 ∫ t 0 mf (s)χ ( {uk(s)}∞k=1 ) ds+4m0 ∫ t 0 ∫ s 0 mg(s)ng(τ)χ ( {uk(τ)}∞k=1 ) dτds +2m0 n∑ i=1 liχ ( {uk(tk)}∞k=1 ) + � ≤ 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] χ(b) +2m0 [∫ t 0 {2g2mk(s) + mf (s)}χ(b))ds + 2g0 ∫ t 0 ng(s)χ(b)ds ] +2m0 n∑ k=1 lkχ(b)+�. since � > 0 is arbitrary, it follows that from the above inequality that χc(f2b(t)) ≤ 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] χc(b) +2m0 [∫ t 0 {2g2mk(s) + mf (s) + 2g0ng(s)}ds + n∑ k=1 lk ] χc(b)(13) for any bounded b ⊂ w. now, for any subset b ⊂ w, due to lemma 2.1, (12) and (13) we have χc(fb) ≤ χc(f1b) + χc(f2b) ≤ m0[l0 + h4(t)]χc(b) + 2 [ me(t) + mk(t)g2 + m0η(t)g1 ] χc(b) +2m0 [∫ t 0 {2g2mk(s) + mf (s) + 2g0ng(s)}ds + n∑ k=1 lk ] χc(b)(14) by (14) we know that f is a χc-contraction on w. by lemma 2.2, there is a fixed point x of f in w, which is a solution of (1) − (3). this completes the proof. theorem: 4.2. suppose that the assumptions (h1)−(h12) are satisfied, then the equation (1)−(3) has at least one mild solution if (15) and the following condition 34 radhakrishnan are satisfied. m0l0 + lim r→∞ sup 1 r { m0le(t) + le(t)γ(r) + g3lk(t)θ(r) +η(t)m0 [ g4γ(r) + g3θ(r) ∫ t 0 l̂k(s)ds ] +m0 [ φ(r) ∫ t 0 lf (s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ]} < 1.(15) proof. from the equation (15) and fact that l0 < 1, there exists a constant r > 0 such that m0 ( rl0 + ‖h(0)‖ + le(t) ) + le(t)(γ‖x‖) + g3lk(t)θ(r) +η(t)m0 [ g4γ(r) + g3θ(r) ∫ t 0 l̂k(s)ds ] +m0 [ φ(r) ∫ t 0 lf (s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ] } < r. we define w0 = {x ∈ pc([0,b],x), ‖x(t)‖ ≤ r, for all t ∈ [0,b]}. then for every x ∈ w0, we have ‖fx(t)‖ ≤ ‖ux(t, 0)h(u)‖ + ‖ux(t, 0)e(0,x(0), 0)‖ + ‖e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) ‖ + ∫ t 0 ‖a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ‖ds + ∫ t 0 ‖ux(t,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(s))dτ ] ‖ds + ∑ 0<tk<t ‖ux(t,tk)ii(x(tk))‖ ≤ m0 ( rl0 + ‖h(0)‖ + le(t) ) + le(t)(γ‖x‖) + g3lk(t)θ(r) +η(t)m0 [ g4γ(r) + g3θ(r) ∫ t 0 l̂k(s)ds ] +m0 [ φ(r) ∫ t 0 lf (s)ds + g2ψ(r) ∫ t 0 l̂g(s)ds + n∑ k=1 dk ] } < r. for t ∈ [0,b]. this means that fw0 ⊂ w0. define w = confw0. the above proof also implies that fw ⊂ w. so we can prove the theorem similar with theorem 4.1 and hence we omit it. 5. application the notion of controllability is of great importance in mathematical control theory. many fundamental problems of control theory such as pole-assignment, stabilizability and optimal control may be solved under the assumption that the system is controllable. it means that it is possible to steer any initial state of the system to any final state in some finite time using an admissible control. during the last few decades, several authors [25, 27] have discussed the existence, uniqueness, and asymptotic behavior of the solution of these systems. apart from these, the study of controllability and observability properties of a system in control theory quasilinear neutral impulsive integrodifferential equations 35 is certainly, at present, one of the most active interdisciplinary areas of research. control theory arises in most modern applications. on the other hand, control theory has remained a discipline where many mathematical ideas and methods have fused to produce a new body of important mathematics. as an application of theorem 3.1 we shall consider the system (1) − (3) with a control parameter such as d dt [x(t) + e(t,x(t), ∫ t 0 k(t,s,x(s))ds)] +a(t,x(t))x(t)(16) = f(t,x(t))+cv(t)+ ∫ t 0 g(t,s,x(s))ds, t ∈ [0,b], t 6= ti, x(0) + h(x) = x0,(17) ∆x(tk) = ik(x(tk)), k = 1, 2, 3, . . . ,n,(18) where a,f,g,h and ik are as before and c is a bounded linear operator from a banach space v into x and the control function v ∈ l2(j,v ). the mild solution of (16) − (18) is given by x(t) = ux(t, 0)x0 −ux(t, 0)h(x) + ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 ux(t,s) [ f(s,x(s)) + cv(s) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0<tk<t ux(t,tk)ik(x(tk)), 0 ≤ t ≤ b. definition 5.1 ([26, 27]) system (16) − (18) is said to be controllable on the interval j, if for every x0, x1 ∈ x, there exists a control v ∈ l2(j,v ) such that the mild solution u(·) of (16) − (18) satisfies x(0) = x0 and x(b) = x1. to study the controllability result we need the following additional condition: (h13) the linear operator w : l 2(j,v ) → x, defined by wv = ∫ b 0 ux(b,s)cv(s)ds induces an inverse operator w−1 defined an l2(j,v )/kerw and there exists a positive constant m1 > 0 such that ‖cw−1‖≤m1. theorem: 5.1. if the assumptions (h1) − (h13) are satisfied, then the system (16) − (18) is controllable on j. 36 radhakrishnan proof. using the assumption (h13), for an arbitrary function u(·), define the control v(t) = w−1 [ u1 −ux(b, 0)x0 + ux(b, 0)h(x) −ux(b, 0)e(0,x(0), 0) +e ( b,x(b), ∫ b 0 k(b,s,x(s))ds ) − ∫ b 0 a(b,x(b))ux(b,s)e ( b,x(b), ∫ s 0 k(s,τ,x(τ))dτ ) ds − ∫ b 0 ux(b,s) [ f(s,x(s)) − ∫ s 0 g(s,τ,x(τ))dτ ] ds − ∑ 0<tk<t ux(t,tk)ik(x(tk)) ] (t). we shall show that when using this control, the operator h : z → z defined by (hv)(t) = ux(t, 0)x0 −ux(t, 0)h(x) + ux(t, 0)e(0,x(0), 0) −e ( t,x(t), ∫ t 0 k(t,s,x(s))ds ) + ∫ t 0 a(t,x(t))ux(t,s)e ( t,x(t), ∫ s 0 k(s,τ,x(τ))dτ ) ds + ∫ t 0 ux(t,s) [ f(s,x(s)) + cw−1 [ u1 −ux(b, 0)x0 + ux(b, 0)h(x) −ux(b, 0)e(0,x(0), 0) + e ( b,x(b), ∫ b 0 k(b,s,x(s))ds ) + ∫ b 0 a(b,x(b))ux(b,s)e ( b,x(b), ∫ s 0 k(s,τ,x(τ))dτ ) ds − ∫ b 0 ux(b,s) [ f(s,x(s)) + ∫ s 0 g(s,τ,x(τ))dτ ] ds − ∑ 0<tk<t ux(t,tk)ik(x(tk)) ] (s) + ∫ s 0 g(s,τ,x(τ))dτ ] ds + ∑ 0<tk<t ux(t,tk)ik(x(tk)) has a fixed point. this fixed point is, then a solution of (16) − (18). clearly, (hv)(b) = x(b) = x1, which means that the control v steers the system (16) − (18) from the initial state x0 to the final state x1 at time b, provided we can obtain a fixed point of the nonlinear operator h. the remaining part of the proof is similar to theorem 3.1 and hence, it is omitted. references [1] k. balachandran and e. r. anandhi, neutral functional integrodifferential control systems in banach spaces, kybernetika, 39 (2003), 359-367. [2] k. balachandran and e. r. anandhi, controllability of neutral functional integrodifferential infinite delay systems in banach spaces, taiwanese journal of mathematics, 8 (2004), 689702. [3] e. hernandez and h. r. henriquez, impulsive partial neutral differential equations, applied mathematics letters, 19 (2006), 215-222. quasilinear neutral impulsive integrodifferential equations 37 [4] j. k. hale and s. m. verduyn lunel, introduction to functional-differential equations, springer-verlag, new york, 1993. [5] l. byszewski, theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, j. math. anal. appl. 162 (1992), 494-505. [6] l. byszewski, h. akca, existence of solutions of a semilinear functional-differential evolution nonlocal problem, nonlinear anal. 34 (1998), 65-72. [7] l. byszewski, v. lakshmikanthan, theorems about the existence and uniqueness of solutions of a nonlocal cauchy problem in banach spaces, applicable anal. 40 (1990), 11-19. [8] s. k. ntouyas, p. ch. tsamatos, global existence for semilinear evolution equations with nonlocal conditions, j. math. anal. appl. 210 (1997), 679-687. [9] q. dong, g. li, j. zhang, quasilinear nonlocal integrodifferential equations in banach spaces, electronic j. diff. equa. 19 (2008), 1-8. [10] a. pazy, semigroups of linear operators and applications to partial differential equations, springer, new york, 1983. [11] d. guo, x. liu, external solutions of nonlinear impulsive integrodifferential equations in banach spaces, j. math. anal. appl. 177 (1993), 538-552. [12] j. liang, j. h. liu, t. j. xiao, nonlocal impulsive problems for nonlinear differential equations in banach spaces, math. computer modelling. 49 (2009), 798-804. [13] a. m. samoilenko, n. a. perestyuk, impulsive differential equations, world scientific, singapore, 1995. [14] z. luo, j. j. nieto, new results for the periodic boundary value problem for impulsive integrodifferential equations, nonlinear anal. 70 (2009), 2248-2260. [15] j. banas, k. goebel, measure of noncompactness in banach spaces, lecture notes in pure and applied mathematics, vol. 60, marcle dekker, new york, 1980. [16] f.s. de blasi, on a property of the unit sphere in a banach space, bull. math. soc. sci. math. r.s. roumanie, 21 (1977), 259-262. [17] j. m. ayerbe, t. d. benavides, g. l. acedo, measure of noncmpactness in in metric fixed point theorem, birkhauser, basel, 1997. [18] j. banas, w.g. el-sayed, measures of noncompactness and solvability of an integral equation in the class of functions of locally bounded variation, j. math. anal. appl. 167 (1992), 133151. [19] g. emmanuele, measures of weak noncompactness and fixed point theorems, bull. math. soc. sci. math. r. s. roumanie, 25 (1981), 353-358. [20] q. dong, g. li, the existence of solutions for semilinear differential equations with nonlocal conditions in banach spaces, electronic j. qualitative theory of diff. equa. 47 (2009), 1-13. [21] z. fan, q. dong, g. li, semilinear differential equations with nonlocal conditions in banach spaces, international j. nonlinear sci. 2 (2006), 131-139. [22] h.p. heinz, on the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, nonlinear anal. 7 (1983), 1351-1371. [23] m. kamenskii, v. obukhovskii, p. zecca, condensing multivalued maps and semilinear differential inclusions in banach spaces, de gruyter series. nonlinear anal. appl. vol.7, de gruyter, berlin, 2001. [24] d. both, multivalued perturbation of m-accretive differential inclusions, israel j. math. 108 (1998), 109138. [25] k. balachandran, j. y. park, existence of solutions and controllability of nonlinear integrodifferential systems in banach spaces, mathematical problems in engineering, 2 (2003), 65-79. [26] b. radhakrishnan, k. balachandran, controllability results for semilinear impulsive integrodifferential evolution systems with nonlocal conditions, j. control theory and appl. 10 (2012), 28-34. [27] l. wang and z. wang, controllability of abstract neutral functional differential systems with infinite delay, dynamics of continuous, discrete and impulsive systems ser.b: applications and algorithms, 9 (2002), 59-70. department of mathematics, psg college of technology, coimbatore-641004, india int. j. anal. appl. (2022), 20:27 applications of standard methods for solving the electric train mathematical model with proportional delay s. m. khaled∗ department of mathematics, faculty of science, helwan university, p.o. box 11795, cairo, egypt ∗corresponding author: ksmahmoud@ut.edu.sa abstract. in electric trains, the current is collected via a certain device, called the pantograph. the governing mathematical model of such physical problem is well-known as the pantograph delay differential equation (pdde): y ′(t) = ay(t)+ by (ct), where c is a proportional delay parameter. in the literature, a special case of the pdde was analyzed when c =−1. the objective of this paper is to determine the general solution of the pdde for arbitrary c. in addition, it will be shown that the obtained general solution reduces to exact one at c = −1. such exact solution is expressed in terms of several types of functions, e.g., hyberbolic, mittag-leffler, and trigonometric functions. moreover, it is declared that the exact trigonometric solution is periodic with periodicity 2π√ b2−a2 which agrees with the corresponding results in the literature. furthermore, the solution of pdde is provided at almost all possible cases of the involved parameters a, b, and c. finally, the solution ambartsumian delay equation (ade), which has an application in the theory of surface brightness in the milky way, will be recovered as a special case of our results. 1. introduction the field of delay differential equations (ddes) is a growing area of research due to its wide applications in applied sciences. a specific and famous application of the ddes is well-known as the pantograph by which the current can be collected in electric trains. the mathematical model describing the pantograph technique is called the pdde [1-5]. also, other ddes-models arise in astronomical applications, as an example, the ambartsumian delay equation (ade) [6-14] which is received: apr. 21, 2022. 2010 mathematics subject classification. 93a30. key words and phrases. pantograph; electric trains; mathematical model; series method; mittag-leffler. https://doi.org/10.28924/2291-8639-20-2022-27 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-27 2 int. j. anal. appl. (2022), 20:27 essential in studying the brightness in the milky way. to author’s knowledge, there are no standard methods to solve ddes in contrast to the ordinary differential equations (odes). for examples, the simple ode y ′(t) = ay(t) can be easily solved via applying the separation technique, while the simple dde y ′(t) = ay (ωt) (ω 6= 1) or y ′(t) = ay (t −σ) (σ ∈ r) can’t be solved neither by the separation technique nor by the other standard methods that dealing with odes. on the other hand, the ddes can be viewed as a generalization of the odes because of the existing of delay parameters. for illustration, the ddes y ′(t) = ay (ωt) and y ′(t) = ay (t −σ) reduces to the ordinary version y ′(t) = ay (t) when ω = 1 and σ = 0, respectively. in this paper, an extended version of the above ddes in the form [15-19]: y ′(t) = ay(t) + by (ct) , y(0) = λ, (1) is to be analyzed for reals a,b,λ, and c. the model (1) is well-known in the literature [15-19] and called the pdde. this model reduces to the ade [6-14]]; y ′(t) = −y(t) + 1 q y ( t q ) , y(0) = λ at the particular values a = −1 and b = c = 1 q (q > 1). this last model and its several generalizations have been addressed by numerous researchers [6-14, 20, 21]. in ref. [22], the special case of the pdde, i.e., y ′(t) = ay(t) + by (−t) (c = −1) has been solved. the main purpose of this work is to obtain the solution of the pdde at arbitrary c. our analysis is based on the standard series method (ssm) and the maclaurin series expansion (mse). our results will also be used to derive previous results in the literature as special cases of ours. the convergence of the obtained series will also be addressed. 2. solution by ssm based on the ssm, the solution of eq. (1) is assumed as y(t) = ∞∑ n=0 dnt n. (2) from eq. (3) and eq. (1), it then follows ∞∑ n=1 ndnt n−1 = a ∞∑ n=0 dnt n + b ∞∑ n=0 dnc ntn, (3) int. j. anal. appl. (2022), 20:27 3 i.e., ∞∑ n=0 (n + 1)dn+1t n = ∞∑ n=0 (a + bcn) dnt n, (4) which yields ∞∑ n=0 [(n + 1)dn+1 − (a + bcn) dn] tn = 0. (5) therefore dn+1 = ( a + bcn n + 1 ) dn, n ≥ 0. (6) accordingly, d1 = 1 1! (a + b) d0, d2 = 1 2 (a + bc) d1 = 1 2 (a + b) (a + bc) d0 = 1 2! 1∏ k=0 ( a + bck ) d0, d3 = 1 3 ( a + bc2 ) d2 = 1 6 (a + b) (a + bc) ( a + bc2 ) d0 = 1 3! 2∏ k=0 ( a + bck ) d0, d4 = 1 4 ( a + bc3 ) d3 = 1 24 (a + b) (a + bc) ( a + bc2 )( a + bc3 ) d0 = 1 4! 3∏ k=0 ( a + bck ) d0, ... dn = 1 n! (a + b) (a + bc) . . . ( a + bcn−2 )( a + bcn−1 ) d0 = 1 n! n−1∏ k=0 ( a + bck ) d0. (7) hence y(t) = d0 + ∞∑ n=1 dn t n, = d0 + ∞∑ n=1 1 n! n−1∏ k=0 ( a + bck ) d0 t n, = d0 [ 1 + ∞∑ n=1 n−1∏ k=0 ( a + bck ) tn n! ] . (8) applying the initial condition y(0) = λ, yields y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( a + bck )] . (9) 4 int. j. anal. appl. (2022), 20:27 3. solution by the mse according to the mse, we have y(t) = ∞∑ n=0 y(n)(0) n! tn. (10) at t = 0, we get from eq. (1) that y(1)(0) = ay(0) + by (0) = λ (a + b) . (11) differentiating eq. (11) w.r.t. to t, then y(2)(t) = ay(1)(t) + bcy(1) (ct) , (12) and hence, y(2)(0) = (a + bc) y(1)(0) = λ (a + b) (a + bc) . (13) similarly, we have from eq. (13) that y(3)(t) = ay(2)(t) + bc2y(2) (ct) . (14) accordingly, y(3)(0) = ( a + bc2 ) y(2) (0) = λ (a + b) (a + bc) ( a + bc2 ) . (15) repeating this procedure n-times, we get y(n)(0) = λ (a + b) (a + bc) ( a + bc2 ) . . . . . . ( a + bcn−2 )( a + bcn−1 ) , = λ n−1∏ k=0 ( a + bck ) , n ≥ 1. (16) therefore y(t) = y(0) + ∞∑ n=1 y(n)(0) tn n! , = λ + λ ∞∑ n=1 n−1∏ k=0 ( a + bck ) tn n! , = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( a + bck )] , (17) which is also the same solution obtained in the previous section. int. j. anal. appl. (2022), 20:27 5 4. convergence analysis theorem 1. for a,b ∈r, the closed-form series solution: y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( a + bck )] , (18) has infinite radius of convergence ∀ c ∈ [−1, 1] and hence the series is uniformly convergent on any compact interval on r. proof: eq. (18) can be expressed as y(t) = λ [ 1 + ∞∑ n=1 cn(t) ] , (19) where cn is defined by cn(t) = tn n! n−1∏ k=0 ( a + bck ) , n ≥ 1. (20) let ρ is the radius of convergence, by ratio test we can write 1 ρ = lim n→∞ ∣∣∣∣cn+1cn ∣∣∣∣ = limn→∞ ∣∣∣∣∣∣ tn+1 (n+1)! ∏n k=0 ( a + bck ) tn n! ∏n−1 k=0 (a + bc k) ∣∣∣∣∣∣ , = lim n→∞ ∣∣∣∣∣(a + bc n) ∏n−1 k=0 ( a + bck ) (n + 1) ∏n−1 k=0 (a + bc k) ∣∣∣∣∣ |t| , = lim n→∞ ∣∣∣∣a + bcnn + 1 ∣∣∣∣ |t| . (21) for c ∈ (−1, 1), we have limn→∞cn = 0, then 1 ρ = lim n→∞ ∣∣∣∣a + bcnn + 1 ∣∣∣∣ |t| = 0, ∀ c ∈ (−1, 1), t ∈r. (22) at c = 1 we have cn = 1 (∀n ∈n), then the limit in eq. (22) reduces to 1 ρ = lim n→∞ ∣∣∣∣a + bn + 1 ∣∣∣∣ |t| = 0, where c = 1, t ∈r. (23) at c = −1 we have cn = −1 (if n is odd) and cn = 1 (if n is even), accordingly, the limit (22) becomes 1 ρ = lim n→∞ ∣∣∣∣a±bn + 1 ∣∣∣∣ |t| = 0, where c = −1, t ∈r, (24) which completes the proof. � 6 int. j. anal. appl. (2022), 20:27 5. special cases & comparisons it was shown in the previous section that the solution of the pdde is given by the closed-form: y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( a + bck )] . (25) in this section, the solutions of several special cases are determined and some of them are compared with those in the relevant literature. 5.1. λ = 1. at λ = 1, the pdde (1) becomes y ′(t) = ay(t) + by (ct) , y(0) = 1, (26) which has been solved by fox et. al [19]. the solution of eqs. (26) can be directly obtained from (18) by substituting λ = 1, and this gives y(t) = 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( a + bck ) , (27) which is the corresponding solution in [19]. 5.2. b = 0. at b = 0, the present pdde reduces to y ′(t) = ay(t), y(0) = λ, (28) which has the exact solution: y(t) = λeat. (29) such exact solution can also be derived as a special case of our solution (18). to do that, we substitute b = 0 into (18) to obtain y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 a ] , (30) i.e., y(t) = λ [ 1 + ∞∑ n=1 antn n! ] , (31) where n−1∏ k=0 a = an, ∀ n ≥ 1. (32) eq. (32) can be written as y(t) = λ ∞∑ n=0 (at)n n! = λeat, (33) int. j. anal. appl. (2022), 20:27 7 which agrees with the exact solution of given in eq. (29). 5.3. a = 0. at a = 0, the pdde yields y ′(t) = by (ct) , y(0) = λ. (34) the solution of such case can be deduced from (18) as follows. inserting a = 0 into (18) leads to y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( bck )] = λ [ 1 + ∞∑ n=1 c 1 2 n(n−1) (bt) n n! ] , (35) where n−1∏ k=0 ( bck ) = bnc 1 2 n(n−1), ∀ n ≥ 1. (36) the solution (36) is equivalent to the form: y(t) = λ ∞∑ n=0 c 1 2 n(n−1) (bt) n n! . (37) 5.4. c = 1. at c = 1, we have pdde becomes the ode: y ′(t) = (a + b)y(t), y(0) = λ, (38) and its solution is known as y(t) = λe(a+b)t. (39) this solution can also be obtained by substituting c = 1 into (18) which gives y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 (a + b) ] , (40) i.e., y(t) = λ [ 1 + ∞∑ n=1 (a + b)ntn n! ] , (41) where n−1∏ k=0 (a + b) = (a + b)n, ∀ n ≥ 1. (42) eq. (42) can be written as y(t) = λ ∞∑ n=0 ((a + b)t)n n! = λe(a+b)t, (43) which is the same exact solution of the eqs. (39). 8 int. j. anal. appl. (2022), 20:27 5.5. c = −1, a 6= ±b. at c = −1, the pdde becomes y ′(t) = ay(t) + by(−t), y(0) = λ. (44) indeed, the model (44) is interesting and its exact solution is obtained here in terms of hyperbolic and trigonometric functions. in addition, such exact solution will be expressed in an equivalent form via mittag-leffler functions. substituting c = −1 into (18), we obtain y(t) = λ [ 1 + ∞∑ n=1 tn n! ψn ] , (45) where ψn is defined by ψn = n−1∏ k=0 ( a + b(−1)k ) , n ≥ 1. (46) using the products rules, it can be shown that ψn = n−1∏ k=0 ( a + b(−1)k ) =   (a2 −b2) n 2 , if n even, (a + b) n+1 2 (a−b) n−1 2 , if n odd. . (47) the series solution (45) can be written as y(t) = λ [ 1 + ∞∑ n=1 t2n−1 (2n− 1)! ψ2n−1 + ∞∑ n=1 t2n (2n)! ψ2n ] , (48) where ψ2n−1 and ψ2n are obtained from (47) by ψ2n−1 = (a + b) n(a−b)n−1, ψ2n = (a2 −b2)n. (49) accordingly, eq. (48) gives y(t) = λ [ 1 + ∞∑ n=1 t2n−1 (2n− 1)! (a + b)n(a−b)n−1 + ∞∑ n=1 t2n (2n)! (a2 −b2)n ] , (50) i.e., y(t) = λ [ 1 + ∞∑ n=0 t2n+1 (2n + 1)! (a + b)n+1(a−b)n + ∞∑ n=0 t2n+2 (2n + 2)! (a2 −b2)n+1 ] . (51) int. j. anal. appl. (2022), 20:27 9 5.5.1. hyperbolic functions (a > b). eq. (51) gives y(t) = λ [ 1 + (a + b) ∞∑ n=0 (√ a2 −b2 )2n t2n+1 (2n + 1)! + ∞∑ n=0 (√ a2 −b2 )2n+2 t2n+2 (2n + 2)! ] , = λ [ 1 + a + b √ a2 −b2 ∞∑ n=0 (√ a2 −b2 )2n+1 t2n+1 (2n + 1)! + ∞∑ n=0 (√ a2 −b2 t )2n+2 (2n + 2)! ] , = λ [ 1 + √ a + b a−b ∞∑ n=0 (√ a2 −b2 t )2n+1 (2n + 1)! + ∞∑ n=0 (√ a2 −b2 t )2n+2 (2n + 2)! ] , = λ [ 1 + √ a + b a−b sinh (√ a2 −b2 t ) + ∞∑ n=1 (√ a2 −b2 t )2n (2n)! ] , = λ [√ a + b a−b sinh (√ a2 −b2 t ) + cosh (√ a2 −b2 t )] , a > b. (52) the curves of the hyperbolic solution in eq. (52) are depicted in fig. 1 at three different sets for the values of a and b. 0.0 0.5 1.0 1.5 2.0 t 10 20 30 40 50 60 yhtl a=4, b=3 a=3, b=2 a=2, b=1 figure 1. plots of the hyperbolic solution in eq. (52) at a = 2,b = 1, a = 3,b = 2, and a = 4,b = 3. 5.5.2. mittag-leffler functions. here, it is noted that eq. (51) can be written using the gamma function in the form: y(t) = λ [ 1 + ∞∑ n=0 t2n+1 γ(2n + 2) (a + b)n+1(a−b)n + ∞∑ n=0 t2n+2 γ(2n + 3) (a2 −b2)n+1 ] , (53) 10 int. j. anal. appl. (2022), 20:27 or y(t) = λ [ 1 + (a + b)t ∞∑ n=0 ( (a2 −b2)t2 )n γ(2n + 2) + (a2 −b2)t2 ∞∑ n=0 ( (a2 −b2)t2 )n γ(2n + 3) ] . (54) using the definition of the two-parameter mittag-leffler function: eα,β(z) = ∞∑ n=0 zn γ(αn + β) , (55) then eq. (54) takes the following final form: y(t) = λ [ 1 + (a + b)te2,2 ( (a2 −b2)t2 ) + (a2 −b2)t2e2,3 ( (a2 −b2)t2 )] . (56) this last form can also be used to establish the solution in terms of trigonometric functions via some properties of the mittag-leffler functions as shown below. 5.5.3. trigonometric functions (b > a). suppose that b > a, then we can rewrite (56) as y(t) = λ [ 1 + (a + b)te2,2 ( −( √ b2 −a2 t)2 ) − (b2 −a2)t2e2,3 ( −( √ b2 −a2 t)2 )] . (57) applying the following properties [21]: e2,2(−z2) = sin(z) z , e2,3(−z2) = 1 − cos(z) z2 , (58) for z = √ b2 −a2 t we have e2,2 ( −( √ b2 −a2 t)2 ) = sin (√ b2 −a2 t ) √ b2 −a2 t , (59) e2,3 ( −( √ b2 −a2 t)2 ) = 1 − cos (√ b2 −a2 t ) (b2 −a2)t2 . (60) substituting (59) and (60) into (57) and simplifying, we obtain y(t) = λ [√ b + a b−a sin (√ b2 −a2 t ) + cos (√ b2 −a2 t )] , b > a. (61) this also the same solution reported by ebaid and al-jeaid [22]. it can be easily seen that the solution is periodic with periodicity p = 2π√ b2−a2 . fig. 2 shows three different periodic solutions with p = 2π,π, and π/2. int. j. anal. appl. (2022), 20:27 11 2 4 6 8 10 t -3 -2 -1 1 2 3 yhtl p= π 2 p=π p=2π figure 2. plots of the periodic solution in eq. (61) at a = 1,b = √ 2 (p = 2π), a = √ 3,b = √ 7 (p = π), and a = 1,b = √ 17 (p = π 2 ). 6. the ade (a = −1,b = c = 1 q ,q > 1) at a = −1 and b = c = 1 q (q > 1), the pdde (1) becomes the ade [10-12]: y ′(t) = −y(t) + 1 q y ( t q ) , y(0) = λ, (62) substituting the above values into (18) implies that y(t) = λ [ 1 + ∞∑ n=1 tn n! n−1∏ k=0 ( q−(k+1) − 1 )] , (63) i.e., y(t) = λ [ 1 + ∞∑ n=1 tn n! n∏ k=1 ( q−k − 1 )] . (64) which is in full agreement with the corresponding result in [10]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h.i. andrews, third paper: calculating the behaviour of an overhead catenary system for railway electrification, proc. inst. mech. eng. 179 (1964), 809–846. https://doi.org/10.1243/pime_proc_1964_179_050_02. [2] m.r. abbott, numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, computer j. 13 (1970), 363–368. https://doi.org/10.1093/comjnl/13.4.363. https://doi.org/10.1243/pime_proc_1964_179_050_02 https://doi.org/10.1093/comjnl/13.4.363 12 int. j. anal. appl. (2022), 20:27 [3] g. gilbert, h.e.h. davies, pantograph motion on a nearly uniform railway overhead line, proc. inst. electr. eng. 113 (1966), 485-492. https://doi.org/10.1049/piee.1966.0078. [4] p.m. caine, p.r. scott, single-wire railway overhead system, proc. inst. electr. eng. 116 (1969), 1217-1221. https://doi.org/10.1049/piee.1969.0226. [5] j.r. ockendon, a.b. tayler, the dynamics of a current collection system for an electric locomotive, proc. r. soc. lond. a. 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078. [6] v.a. ambartsumian, on the fluctuation of the brightness of the milky way, dokl. akad. nauk ussr, 44 (1994), 223-226. [7] j. patade, s. bhalekar, on analytical solution of ambartsumian equation, natl. acad. sci. lett. 40 (2017), 291–293. https://doi.org/10.1007/s40009-017-0565-2. [8] f.m. alharbi, a. ebaid, new analytic solution for ambartsumian equation, j. math. syst. sci. 8 (2018), 182-186. https://doi.org/10.17265/2159-5291/2018.07.002. [9] h. bakodah, a. ebaid, exact solution of ambartsumian delay differential equation and comparison with daftardargejji and jafari approximate method, mathematics. 6 (2018), 331. https://doi.org/10.3390/math6120331. [10] n.o. alatawi, a. ebaid, solving a delay differential equation by two direct approaches, j. math. syst. sci. 9 (2019), 54-56. https://doi.org/10.17265/2159-5291/2019.02.003. [11] a. ebaid, a. al-enazi, b.z. albalawi, m.d. aljoufi, accurate approximate solution of ambartsumian delay differential equation via decomposition method, math. comput. appl. 24 (2019), 7. https://doi.org/10.3390/ mca24010007. [12] a.a. alatawi, m. aljoufi, f.m. alharbi, a. ebaid, investigation of the surface brightness model in the milky way via homotopy perturbation method, j. appl. math. phys. 8 (2020), 434–442. https://doi.org/10.4236/jamp. 2020.83033. [13] e. a. algehyne, e. r. el-zahar, f. m. alharbi, a. ebaid, development of analytical solution for a generalized ambartsumian equation, aims math. 5 (2020), 249–258. https://doi.org/10.3934/math.2020016. [14] s.m. khaled, e.r. el-zahar, a. ebaid, solution of ambartsumian delay differential equation with conformable derivative, mathematics. 7 (2019), 425. https://doi.org/10.3390/math7050425. [15] t. kato, j.b. mcleod, the functional-differential equation y ′(x) = ay(λx)+by(x), bull. amer. math. soc. 77 (1971), 891-935. [16] a. iserles, on the generalized pantograph functional-differential equation, eur. j. appl. math. 4 (1993), 1–38. https://doi.org/10.1017/s0956792500000966. [17] g. derfel, a. iserles, the pantograph equation in the complex plane, j. math. anal. appl. 213 (1997), 117–132. https://doi.org/10.1006/jmaa.1997.5483. [18] j. patade, s. bhalekar, erratum to: analytical solution of pantograph equation with incommensurate delay, phys. sci. rev. 3 (2018), 20165103. https://doi.org/10.1515/psr-2016-9103. [19] l. fox, d.f. mayers, j.r. ockendon, a.b. tayler, on a functional differential equation, ima j. appl. math. 8 (1971), 271–307. https://doi.org/10.1093/imamat/8.3.271. [20] d. kumar, j. singh, d. baleanu, s. rathore, analysis of a fractional model of the ambartsumian equation, eur. phys. j. plus. 133 (2018), 259. https://doi.org/10.1140/epjp/i2018-12081-3. [21] a. ebaid, c. cattani, a.s. al juhani, e.r. el-zahar, a novel exact solution for the fractional ambartsumian equation, adv. differ. equ. 2021 (2021), 88. https://doi.org/10.1186/s13662-021-03235-w. [22] a. ebaid, h.k. al-jeaid, on the exact solution of the functional differential equation y ′(t) = ay(t)+ by(−t), adv. differ. equ. control processes, 26 (2022), 39-49. https://doi.org/10.17654/0974324322003. https://doi.org/10.1049/piee.1966.0078 https://doi.org/10.1049/piee.1969.0226 https://doi.org/10.1098/rspa.1971.0078 https://doi.org/10.1007/s40009-017-0565-2 https://doi.org/10.17265/2159-5291/2018.07.002 https://doi.org/10.3390/math6120331 https://doi.org/10.17265/2159-5291/2019.02.003 https://doi.org/10.3390/mca24010007 https://doi.org/10.3390/mca24010007 https://doi.org/10.4236/jamp.2020.83033 https://doi.org/10.4236/jamp.2020.83033 https://doi.org/10.3934/math.2020016 https://doi.org/10.3390/math7050425 https://doi.org/10.1017/s0956792500000966 https://doi.org/10.1006/jmaa.1997.5483 https://doi.org/10.1515/psr-2016-9103 https://doi.org/10.1093/imamat/8.3.271 https://doi.org/10.1140/epjp/i2018-12081-3 https://doi.org/10.1186/s13662-021-03235-w https://doi.org/10.17654/0974324322003 1. introduction 2. solution by ssm 3. solution by the mse 4. convergence analysis 5. special cases & comparisons 5.1. =1 5.2. b=0 5.3. a=0 5.4. c=1 5.5. c=-1, a=b 6. the ade (a=-1,b=c=1q,q>1) references international journal of analysis and applications volume 18, number 1 (2020), 129-148 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-129 on cox-ross-rubinstein pricing formula for pricing compound option javed hussain∗, bareerah khan sukkur iba university, pakistan ∗corresponding author: javed.brohi@iba-suk.edu.pk abstract. the fundamental objective of this paper is twofold. firstly, to derive the cox-ross-rubinstein type new formula for risk neutral pricing of european compound call option, where the underlying asset is also a european call option. thirdly, to prove that our newly derived crr risk neutral pricing formula for compound call option, converges in distribution to the well known, continuous time black-scholes formula for pricing the compound call option on call. 1. introduction a compound option is an option that has further an option as the underlying asset. compound options were first studied by geske (1979, [9]), using a partial differential equation method and fourier integrals. afterward, several other approaches were introduced for pricing methods for compound options. for instance, lajeri-chaherli (2002, [15]) used the martingale method and by computing the expectation of truncated bivariate normal variables, priced the compound options. agliardi (2003, [1]) priced a generalized timedependent compound calls. gukhal (2004, [12]) proposed a model for valuation of the compound option, in which the underlying option follows the log-normal jump-diffusion process. fouque and han (2005, [13]) employed a perturbation techniques to compute the prices of compound options. chiarella and kang (2011, [5]) and (2014, [4]) evaluated american style compound option with stochastic volatility. griebsch (2013, [11]) received 2019-10-25; accepted 2019-11-19; published 2020-01-02. 2010 mathematics subject classification. 91g20. key words and phrases. binomial pricing; compound options; cox-ross-rubinstein framework; convergence in distribution. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 129 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-129 int. j. anal. appl. 18 (1) (2020) 130 used fft (fast fourier transform) to price the european style compound option prices, where volatility assumed to be stochastic. since most studies have been done through continuous time approach so in this paper we emphasize on pricing the compound option through the discrete time cox-ross-rubinstein (crr) approach, we refer the reader to chapter 6 of [7]. they key novelty of the work is the detailed proof that discrete time crr price of the compound option converges to the continuous time pricing formula of the compound option. we now give a brief description of all sections of this paper. section 1 is running introduction. the section 2 gives a short introduction to mechanics of the compound option and its continuous time formula. section 3 has been devoted to deriving, in a greater detail, the crr formula for the risk-neutral pricing the european-style compound option on european call option. in section 4, we have given the proof that the crr price of compound option converges to the well-known continuous time pricing formula, in distribution. options. section 6 comprises of conclusion. 2. crr formula for pricing compound options this section comprises of a brief introduction to compound option and our new result on the derivation of its crr premium of european call option when the underlying asset is a european call option on stock. 2.1. introduction to compound option. here we briefly illustrate a compound option which is a particular type of exotic option. to do that we need a brief time line here. t0 t1 t2 today is time zero, t1 is the point time in future and t2 will be even later than that. if today at t = 0 we purchase a call option we are buying a right to exercise a price of k1 and exchange for that we will receive a call option. so in this case we are purchasing a call option on a call option. mainly, there can be four possible variations of compound options. 1. call with underlying call. 2. call with underlying put. 3. put with underlying call. 4. put with underlying put. so today at t = 0 when we purchase a compound option it gives us a right to going forward to exercise that compound option. so let’s just say that stock increases little bit here at t = 1 and we exercise compound option. we pay the strike price of k1 and get the option not the stock. so we only do that if the value at t = 1 of call option is greater than the strike price c > k1. so after having exercise the compound, we now in fact own the more familiar plain vanilla call option on stock and we do need another strike price k2. the int. j. anal. appl. 18 (1) (2020) 131 compound option has two strike prices, first only to purchase a compound option which is k1 the price we pay in order to exercise initially to purchase a call option and that call option has k2 its own strike price. if we go forward in time, say stock move in our favor such that at t1 , there is an intrinsic value in that call option and if s > k2 then we exercise the underlying option by paying the strike price k2 and receiving the underlying asset (in this case it’s a stock). when assumption of geometric brownian motion is made, european style compound option can analytically be valued in terms of integral of bivariate normal distribution. the value of european call on call option at t = 0 is c = sn ( a1,b1; √ t1 t2 ) −e−rt2k2n ( a2,b2; √ t1 t2 ) −e−rt1k1n (a2) (2.1a) where a1 = ln ( s0 s∗ ) + ( r − σ 2 2 ) t1 σ √ t1 ; a2 = a1 + σ √ t1 b1 = ln ( s0 k2 ) + ( r − σ 2 2 ) t2 σ √ t2 ; b2 = b1 + σ √ t2 n = cumulative bivariate normal distribution function. s∗ = critical stock price at t1 such that compound option is in the money. we refer to [14] for further details. 3. derivation of crr compound option formula now we move towards the key objective of the paper i.e. pricing the compound option through binomial approach. we aim to construct a crr formula for the compound options. let me state the key theorem of the section. theorem 3.1. in a viable cox-ross-rubinstein model with parameters s0,t,r,u and d the fair price at time 0 of a european compound call option with expiry m and strike k1, contingent upon a european call option with expiry n and strike k2, can be given as: c = s0b (a,b; p ′; n,n−m) − (1 + r)−(n−m)k2b (a,b; p′,p; n,n−m) (3.1) −(1 + r)−mk1ψ (a; m,p) where p = r−u u−d, p ′ = pu (1+r) , a = [ ln s ∗ s0 −m ln d ln( ud ) ] + 1 and b = [ ln k2 s0 −n ln d ln( ud ) ] + 1, and b(·, · ; ·) is the complementary distribution function of bi-variate binomial distribution. int. j. anal. appl. 18 (1) (2020) 132 proof. consider the discrete time line, 0 m n where m,n ∈ n. we aim to price a compound call option where underlying asset is also a call option. the compound option will be exercised at m (i.e. after m steps/periods) and the underlying call option (whose value depends on underlying stock prices), will be exercised at n steps/periods. by assuming the absence of arbitrage opportunities in the binomial model, the price of the underlying call option either moves up by a factor u or down by a factor d in each period. the probability of an upward movement is p and the probability of a downward movement is 1 −p. in order to compute the fair premium of the compound option at inception, we will begin by employing theorem 5.47 of [7]. the payoff of the compound option can be given as, payoff = max{cstd(sm,k2,n−m) −k1, 0} where e is the risk neutral expectation, k1 is the strike price for the compound option, and cstd is the blackscholes price of underlying call option at the maturity of compound option i.e. m. thus by the theorem 5.46 of [7], the premium of the compound option can be given as, c = (1 + r)−me (max{cstd(sm,k2,n−m) −k1, 0}) where cstd(sm,k2,n − m) is the price of the underlying option. the expectation can be written more explicitly as, c = (1 + r)−m m∑ i=0 max ( cstd(smu idm−i,k2,n−m) −k1, 0 ) m i  pi (1 −p)m−i . (3.2) the compound option will be exercised i.e. its payoff will be nonzero if sm > s ∗, where s∗ denotes the worth of the stock such that the underlying option is in the money at time m i.e. s∗ solves the equation cstd(sm,k2,n−m) = k1. assume that a is the smallest time period i such that. sm > s ∗. let us try to determine bound on a,. as s0u adm−a > s∗, it follows that, a > ln s ∗ s0 −m ln d ln ( u d ) . int. j. anal. appl. 18 (1) (2020) 133 let us return back to equation (3.2) and write the expectation more explicitly. c = (1 + r)−m m∑ i=a ( cstd(smu idm−i,k2,n−m)uidm−i −k1 ) m i  pi (1 −p)m−i = (1 + r)−m m∑ i=a cstd(smu idm−i,k2,n−m)   m i  pi (1 −p)m−i uidm−i −(1 + r)−m m∑ i=a k1   m i  pi (1 −p)m−i c = (1 + r)−m m∑ i=a cstd(smu idm−i,k2,n−m)   m i  pi (1 −p)m−i uidm−i −(1 + r)−mk1ψ (a,m,p) (3.3) where ψ is the complementary binomial distribution function, ψ (a,m,p) = m∑ i=a   m i  pi (1 −p)m−i . we want to further explore the above first term of equation (3.3). here cstd(sm,k2,n−m) represents the price of an underlying european option and the price is given as: cstd(smu idm−i,k2,n−m) = (1 + r)−(n−m)e ({max (sm −k2, 0)}) cstd(smu idm−i,k2,n−m) = (1 + r)−(n−m) n∑ j=m max ( (smu idm−i)ujdn−j −k2, 0 ) ×   n−m j  pj (1 −p)n−m−j . assume that b < n in first instant such that b = min { a : smu jdn−j } > k2. let us try to find bound on b. by definition smu jdn−j > k2, therefore it follows that b > ln k2 sm −n ln d ln ( u d ) . now cstd(smu idm−i,k2,n−m) = (1 + r)−(n−m) n∑ j=b sm   n−m j  pj (1 −p)n−m−j ujdn−m−j −(1 + r)−(n−m)k2 n∑ j=b   n−m j  pj (1 −p)n−m−j . int. j. anal. appl. 18 (1) (2020) 134 substitute the above equation in equation (3.3), we infer that, c = (1 + r)−m m∑ i=a   (1 + r)−(n−m) n∑ j=b sm   n−m j  pj (1 −p)n−m−j ujdn−m−j −(1 + r)−(n−m)k2 n∑ j=b   n−m j  pj (1 −p)n−m−j   ·   m i  pi (1 −p)m−i uidm−i − (1 + r)−mk1ψ (a,m,p) . c = (1 + r)−n m∑ i=a n∑ j=b sm   m i     n−m j  pi+j (1 −p)n−(i+j) ui+jdn−(i+j) −(1 + r)−nk2 m∑ i=a n∑ j=b   m i     n−m j  pi+j (1 −p)n−(i+j) uidm−i −k1ψ ( a,m,p ′ ) = m∑ i=a n∑ j=b s0   m i     n−m j  ( pu (1 + r) )i+j ( (1 −p) d (1 + r) )n−(i+j) −(1 + r)−(n−m)k2 m∑ i=a n∑ j=b   m i     n−m j  ( pu (1 + r)−(n−m) )i ( (1 −p) d (1 + r)−(n−m) )m−i pj (1 −p)n−j − (1 + r)−(n−m)k1ψ (a,m,p) , where   m i   = m! i!(m−i)! and   n−m j   = (n−m)! j!(n−m−j)!. one can see that the first two terms in the last equation represent the cumulative distributive function of bivariate binomial distribution. thus c = s0b (a,b; p ′; n,n−m) − (1 + r)−(n−m)k2b (a,b; p′,p; n,n−m) − (1 + r)−mk1ψ (a,m,p) (3.4) where p′ = pu (1+r) , a = [ ln s ∗ s0 −m ln d ln( ud ) ] + 1 and b = [ ln k2 s0 −n ln d ln( ud ) ] + 1, and ψ(·, ·; ·) is the complementary distribution function of bi-variate binomial distribution. the formula above gives us the crr premium of the compound european call on european call option. � 4. convergence of empirical means and volatilities of log returns to actual means and volatilities following is the key theorem that we intend to prove in this section. theorem 4.1. assume that we are in the framework of theorem 3.1. then following holds: i) if s∗ stock price at time t1 (i.e. maturity time for compound option) such that the compound option in int. j. anal. appl. 18 (1) (2020) 135 the money then the empirical mean µ̂pm and variance of σ̂ 2 pm of log returns ln ( s∗ s0 ) converges, respectively, to µt1 and σ 2t1. ii) let t be an instant between t1 and t2 and st be the stock price at time t. then the empirical mean µ̂p(n−m) and variance of σ̂2p(n−m) of log returns ln ( st s0 ) converges, respectively, to µt2 and σ 2t2. proof. of the compound option given by formula 2.1a. let s∗ be the stock price at t1 such that. the price of compound option is in the money. so s∗ = s0u idm−i ln ( s∗ s0 ) = ln (u) i + ln (d) m−i ln ( s∗ s0 ) = i ln u + (m− i) ln d = i ln ( u d ) + m ln d. (4.1) here i is the binomial random variable with mean e [i] = mp and variance v ar [i] = mp (1 −p) . as m →∞ e [ ln ( s∗ s0 )] = e [ i ln ( u d )] + e [m ln d] . the expectation of constant remains same , so e [ ln ( s∗ s0 )] = e [i] [ ln ( u d )] + [m ln d] = mp [ ln ( u d )] + [m ln d] = [ p ln ( u d ) + ln d ] m. we suppose that the empirical mean value of ln ( s∗ s0 ) is µ̂pm and represent it as: µ̂pm = [ p ln ( u d ) + ln d ] m (4.2) and var [ ln ( s∗ s0 )] = var [ i ln ( u d )] + var [m ln d] . the variance of (ax + b) is a2v arx, hence var [ ln ( s∗ s0 )] = [ ln ( u d )]2 v ar [i] = mp(1 −p) [ ln ( u d )]2 . we suppose that the empirical variance value of ln ( s∗ s0 ) is σ̂2pm and represent it as σ̂2pm = p(1 −p) [ ln ( u d )]2 m. (4.3) int. j. anal. appl. 18 (1) (2020) 136 if µt1 = e [ ln ( s∗ s0 )] and σt = var [ ln ( s∗ s0 )] represent the values of actual mean and variance, we want µ̂pm converges to µt1 and σ̂ 2 pm approaches to σt1 as m →∞ : lim m→∞ mp ln ( u d ) + ln d = µt1, (4.4) and lim m→∞ mp(1 −p) [ ln ( u d )]2 = σ2t1. (4.5) also keep in view the following relation between u and d, u = 1 d . (4.6) to have these convergence we need to choose the parameters u,d and p, which satisfies last three limiting equations. to do so let solve the system of nonlinear equations (4.4), (4.5) and (4.6 ), for u, d and p. ignore the limits in the mentioned equations. using equation (4.6) into (4.4), we may infer that p ln ( u2 ) + ln 1 u = µt1 m ln u = µt1 m(2p− 1) . (4.7) using equation (4.6) into (4.5) we infer, p(1 −p) [ ln ( u2 )]2 = σt1 m [ln u] 2 = σt1 4p(1 −p)m . substituting the equation (4.7) into last equation we get,( µt1 m(2p− 1) )2 = σt1 4p(1 −p)m . on solving the above quadratic equation for p we get, p = 1 2 + 1 2 µ σ √ t1 m . (4.8) now substituting the value of p from (4.8) into getting the value of u by using equation (4.7) ln u = µt1 m · 1( 2 ( 1 2 + 1 2 µ σ √ t1 m ) − 1 ) ln u = σ √ t1 m u = e σ √ t1 m . (4.9) int. j. anal. appl. 18 (1) (2020) 137 as d = 1 u , then the value of d is d = e −σ √ t1 m . (4.10) after plugging the calculated values of u,d and p in equations of µ̂pm and σ̂ 2 pm, we see how these empirical values approaches to the actual values of mean and variance. µ̂pm = [ p ln ( u d ) + ln d ] m. note that ln ( u d ) = ln u − ln d = σ √ t1 m + σ √ t1 m = 2σ √ t1 m . now µ̂pm = {[ 1 2 + 1 2 µ σ √ t1 m ][ 2σ √ t1 m ] −σ √ t1 m m } = [ σ √ t1 m + µ ( t1 m ) −σ √ t1 m ] m lim m→∞ µ̂pm = µt1. next σ̂2pm = p(1 −p) [ ln ( u d )]2 m = [ 1 2 + 1 2 µ σ √ t1 m ][ 1 − { 1 2 + 1 2 µ σ √ t1 m }][ 2σ √ t1 m ]2 m = ( 1 4 − 1 4 µ2 σ2 t1 m ) 4σ2t1 lim m→∞ σ̂2pm = σ 2t1. (4.11) now, at the second expiry of underlying stock: st = s0u jd(n−m)−j ln ( st s0 ) = ln (u) j + (d) (n−m)−j ln ( st s0 ) = j ln u +(n−m)−j ln d = j ln ( u d ) + (n−m) ln d. here j is the binomial random variable with mean e [j] = (n−m) p and variance v ar [j] = (n−m) (1 −p) p. as (n−m) →∞ e [ ln ( st s0 )] = e [ j ln ( u d )] + e [(n−m) ln d] . int. j. anal. appl. 18 (1) (2020) 138 the expectation of constant remains same , so e [ ln ( st s0 )] = e [j] [ ln ( u d )] + [(n−m) ln d] = (n−m) p [ ln ( u d )] + [(n−m) ln d] = [ p ln ( u d ) + ln d ] (n−m) . we suppose that the empirical mean value of ln ( st s0 ) is µ̂p(n−m) and represent it as: µ̂p(n−m) = [ p ln ( u d ) + ln d ] (n−m) and var [ ln ( st s0 )] = var [ j ln ( u d )] + var [(n−m) ln d] . recall the fact that the v ar (ax + b) = a2v arx. by using this, var [ ln ( st s0 )] = [ ln ( u d )]2 v ar [j] = (n−m) p(1 −p) [ ln ( u d )]2 . we suppose that the empirical variance value of ln ( s∗ s0 ) is σ̂2p (n−m) and represent it as: σ̂2p (n−m) = p(1 −p) [ ln ( u d )]2 (n−m) . if µt2 = e [ ln ( st s0 )] and σt2 = var [ ln ( st s0 )] represent the values of actual mean and variance, we want µ̂p(n−m) approaches to µt2 and σ̂2p (n−m) approaches to σt2 as (n−m) →∞ : p ln ( u d ) + ln d → µt2 (n−m) and p(1 −p) [ ln ( u d )]2 → σt2 (n−m) from the last two equations above, we have to calculate the values of parameters u,d and p subject to constraint u = 1 d due to the balance in the tree structure in the binomial model. calculations are shown below: p = 1 2 + 1 2 µ σ √ t2 (n−m) . using the relation u = 1 d int. j. anal. appl. 18 (1) (2020) 139 p ln ( u d ) + ln d = µt2 (n−m) p ln (u) 2 + ln ( 1 u ) = µt2 (n−m) ln u = µt2 (n−m) . 1 (2p− 1) . now plugging the value of p ln u = µt2 (n−m) · 1( 2 ( 1 2 + 1 2 µ σ √ t2 (n−m) ) − 1 ) ln u = µt2 (n−m) · 1( µ σ √ t2 (n−m) ) ln u = t2 (n−m) · σ(√ t2 (n−m) ). after solving the powers ln u = σ √ t2 (n−m) or u = e σ √ t2 (n−m) (4.12) as d = 1 u , then the value of d is d = e −σ √ t2 (n−m) . (4.13) after plugging the calculated values of u,d and p in the µ̂p(n − m) and σ̂2p (n−m) , we see how these empirical values approaches to the actual values of mean and variance. µ̂p(n−m) = [ p ln ( u d ) + ln d ] (n−m) . note that ln ( u d ) = ln u − ln d = σ √ t2 (n−m) + σ √ t2 (n−m) = 2σ √ t2 (n−m). now, µ̂p(n−m) = [ 1 2 + 1 2 µ σ √ t2 (n−m) ][ 2σ √ t2 (n−m) −σ √ t2 (n−m) ] (n−m) = [ σ √ t2 (n−m) + µ ( t2 (n−m) ) −σ √ t2 (n−m) ] (n−m) lim m→∞ µ̂p(n−m) = µt2. int. j. anal. appl. 18 (1) (2020) 140 next σ̂2p (n−m) = p(1 −p) [ ln ( u d )]2 (n−m) = [ 1 2 + 1 2 µ σ √ t2 (n−m) ][ 1 − { 1 2 + 1 2 µ σ √ t2 (n−m) }] × [ 2σ √ t2 (n−m) ]2 (n−m) = [ 1 2 + 1 2 µ σ √ t2 (n−m) ][ 1 2 − 1 2 µ σ √ t2 (n−m) ][ 4σ2t2 ] = ( 1 4 − 1 4 µ2 σ2 t2 (n−m) ) 4σ2t2 lim m→∞ σ̂2p (n−m) = σ 2t2. (4.14) � 5. main convergence result in this section we prove the second most important result that the established general crr formula for binomial pricing formula of a compound call option, to the well-known continuous-time black-scholes formula for european compound option. we state this assertion in the form of following theorem. theorem 5.1. assume that we are in assumptions of theorem 3.1 and theorem 4.1. the fair price at time 0 of a european compound call option with expiry m and strike k1, contingent upon a european call option with expiry n and strike k2, can be given as: c = s0b (a,b; p ′; n,n−m) − (1 + r)−(n−m)k2b (a,b; p′,p; n,n−m) (5.1) −(1 + r)−mk1ψ (a; m,p) , converges in the distribution to the following continuous time price of european compound call option with expiry t1 and strike k1, contingent upon a european call option with expiry t2 and strike k2, s0n ( d1,d2; √ t2 t1 ) −e−rt2k2n ( d∗1,d ∗ 2; √ t2 t1 ) −e−rt1k1n(d∗1), where p = r−u u−d, p ′ = pu (1+r) , a = [ ln s ∗ s0 −m ln d ln( ud ) ] + 1 and b = [ ln k2 s0 −n ln d ln( ud ) ] + 1, and b(·, · ; ·) is the complementary distribution function of bi-variate binomial distribution and d1 = ln s s∗ + ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 ,d∗1 = d1 + σ √ t1 d2 = ln ( s k2 ) + ( ln(1 + r) − 1 2 σ2 ) t2 σ √ t2 ,d∗2 = d2 + σ √ t2. int. j. anal. appl. 18 (1) (2020) 141 proof. from the last section we know that our choice of parameter (1 + r) t1 m ensures that k1(1 + r) −m = ke−rt1 and k2(1 +r) −(n−m) = ke−rt2. next, we are going to show that the crr price of compound option converges to the continuous time black-scholes price of the compound option. to do let us begin with the following observation about the cdf of bivariate binomial random variables i, j, where i and j denotes the number of times the price of underlying option and compound option, has went up, respectively. 1 −b (a,b; p′; n,n−m) = p (i ≤ a− 1,j ≤ b− 1) = p ( i−mp√ mp(1 −p) ≤ a− 1 −mp√ mp(1 −p) , j − (n−m) p√ (n−m) p(1 −p) ≤ b− 1 − (n−m) p√ (n−m) p(1 −p) ) (5.2) where a = min { i ∈ m; i ≥ ln s ∗ s0dm ln u d } ∈ [ ln s ∗ s0dm ln u d , ln k1 cdm ln u d + 1 ) b = min { j ∈ (n−m) ; j ≥ ln k2 sd(n−m) ln ( u d ) } ∈ [ ln k2 sd(n−m) ln ( u d ) , ln k2sd(n−m) ln ( u d ) + 1 ) . for the convergence to the continuous time, mean and variance of continuously compounded rate of returns of stocks with respect to p = (p, 1 −p) can be given as: µ̂pm = [ p ln ( u d ) + ln d ] m and σ̂2pm = mp(1 −p) [ ln ( u d )]2 (5.3) µ̂p(n−m) = [ p ln ( u d ) + ln d ] (n−m) andσ̂2pm = mp(1 −p) [ ln ( u d )]2 . (5.4) recall that i represents the number of times the stock price goes up before the expiry of the compound option i.e. between 0 and m. also recall the equation (4.1) i.e. ln ( s∗ s0 ) = i ln ( u d ) + m ln d. deducing the value of i from this equation we may infer that i−mp√ mp(1 −p) = ln ( st s0 ) −m ln d ln( ud ) −mp√ mp(1 −p) = ln ( st s0 ) −m(ln d + p ln ( u d ) ) ln ( u d )√ mp(1 −p) . using the equations (5.3) it follows that, i−mp√ mp(1 −p) = ln ( st s0 ) − µ̂pm σ̂p √ m . (5.5) let us fix ε ∈ ([0, 1]) . now since a = min { i ∈ m; i ≥ ln s ∗ s0d m ln u d } ∈ [ ln s ∗ s0d m ln u d , ln k1 cdm ln u d + 1 ) therefore: a− 1 = ln s ∗ s0dm ln ( u d ) −ε = ln s∗s0 −m ln d −ε ln (ud) ln u d int. j. anal. appl. 18 (1) (2020) 142 and therefore, a− 1 −mp√ mp(1 −p) = ln s ∗ s0 −m ln d −ε ln u d −mp ln ( u d ) ln u d √ mp(1 −p) = ln s ∗ s0 −m ( p ln u d − ln d ) −ε ln ( u d ) ln u d √ mp(1 −p) = ln s ∗ s0 − µ̂pm−ε ln ( u d ) σ̂p √ m . (5.6) next, let us turn towards the random variable j over (n−m) time steps, where j is the number of up moves of the stock price in the interval m to n. j − (n−m) p√ (n−m) (1 −p)p = ln ( st s0 ) −(n−m) ln d ln( ud ) − (n−m) p√ (n−m) (1 −p)p = ln ( st s0 ) − (n−m) ln d − ln ( u d ) (n−m) p√ (n−m) (1 −p)p . using the equations (5.3) it follows that j − (n−m) p√ (n−m) (1 −p)p = ln ( st s0 ) − µ̂p(n−m) σ̂p √ n . (5.7) again since b = min { j ∈ (n−m) ; j ≥ ln k2 sd(n−m) ln( ud ) } ∈ [ ln k2 sd(n−m) ln( ud ) , ln k2 sd(n−m) ln( ud ) + 1 ) so it follows that b− 1 = ln k2 sd(n−m) ln u d −ε = ln ( k2 s ) − (n−m) ln d −ε ln ( u d ) ln u d and therefore, b− 1 − (n−m) p√ (n−m) (1 −p)p = ln ( k2 s ) − (n−m) ln d −ε ln ( u d ) − (n−m) p ln ( u d ) ln u d √ (n−m) p(1 −p) = ln ( k2 s ) − (n−m) ( p ln ( u d ) − ln d ) −ε ln ( u d ) ln u d √ (n−m) p(1 −p) = ln ( k2 s ) − µ̂p (n−m) −ε ln ( u d ) σ̂p √ (n−m) . (5.8) using equations (5.5), (5.6), (5.7) and (5.8) into (5.2), it follows that, 1 −b (a,b; p′; n,n−m) = p   ln ( st s0 ) −µ̂pm σ̂p √ m ≤ ln s ∗ s0 −µ̂pm−ε ln( ud ) σ̂p √ m , ln ( st s0 ) −µ̂p(n−m) σ̂p √ n−m ≤ ln( k2 s )−µ̂p(n−m)−ε ln( u d ) σ̂p √ (n−m)   . (5.9) int. j. anal. appl. 18 (1) (2020) 143 to proceed further we want to show that µ̂pm → ( ln(1 + r) − 1 2 σ2 ) t1 and µ̂p (n−m) →( ln(1 + r) − 1 2 σ2 ) t2 as m → ∞ and n → ∞, respectively. we will prove first the convergence, and the second will follow the analogue argument. recall that between the time 0 to m, the value of m can be given as, p = 1 2 + 1 2 µ σ √ t1 m . in order to show that µ̂pm → ( ln(1 + r) − 1 2 σ2 ) t1 as m →∞, it is sufficient to show that, p = 1 2 + 1 2 µ σ √ t1 m → 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ t1 m as m →∞ or 2 √ m ( p− 1 2 ) → ( ln(1 + r) − 1 2 σ2 ) σ √ t1 as m →∞. or it is sufficient to show that, lim m→∞ 2 √ m ( p− 1 2 ) = ( ln(1 + r) − 1 2 σ2 ) σ √ t1. let us consider the term on left hand side of last equation. using value of p from 2 √ m ( p− 1 2 ) = 2 √ m ( (1 + r) t1 m −d u −d − 1 2 ) = 2 √ m ( 2(1 + r) t1 m − 2d −u + d 2 (u −d) ) = √ m  2 (1+r) t1md − ud − 1 u d − 1   . on substituting (1 + r) t1 m = e( t1 m ) ln(1+r), u = e σ √ t1 m and d = e −σ √ t1 m ( or u d = e 2σ √ t1 m ) into last equation 2 √ m ( p− 1 2 ) = √ m  2e( t1m ) ln(1+r)+σ √ t1 m −e2σ √ t1 m − 1 e 2σ √ t1 m − 1   . taking limit m →∞, lim m→∞ 2 √ m ( p− 1 2 ) = lim m→∞ 2e( t1 m ) ln(1+r)+σ √ t1 m −e2σ √ t1 m − 1( e 2σ √ t1 m − 1 ) m− 1 2 , ( 0 0 form ) . using l’hospital rule we may infer that, lim m→∞ 2 √ m ( p− 1 2 ) = = lim m→∞ 2e( t1 m ) ln(1+r)+σ √ t1 m −e2σ √ t1 m − 1( e 2σ √ t1 m − 1 ) m− 1 2 = lim m→∞ 2e( t1 m ) ln(1+r)+σ √ t1 m (m−2t1 ln(1 + r) + 1 2 m− 3 2 σ √ t1) −e2σ √ t1 m m− 3 2 σ √ t1 e 2σ √ t1 m ( m−2t1 ln(1 + r) + 1 2 m− 3 2 ) − 1 2 m− 3 2 . int. j. anal. appl. 18 (1) (2020) 144 on dividing the numerator and the denominator by m− 3 2 e 2σ √ t1 m , we get = lim m→∞ 2e( t1 m ) ln(1+r)+σ √ t1 m (m− 1 2 t1 ln(1 + r) + 1 2 σ √ t1) −σ √ t1 σ √ t1 m + 1 2 − 1 2 e 2σ √ t1 m , ( 0 0 form ) . again using l’hospital rule, we infer, = lim m→∞ 2e( t1 m ) ln(1+r)+σ √ t1 m (−m−2t1 ln(1 + r) + 12m −3 2 σ √ t1) ( m− 1 2 t1 ln(1 + r) + 1 2 σ √ t1 ) −1 2 m− 3 2 t1 ln(1 + r) −1 2 m− 3 2 σ √ t1 − 12m −3 2 e −2σ √ t1 m σ √ t1 . on multiplication of 2m 3 2 into the numerator and denominator and simplifying it follows that, = lim m→∞ 4e( t1 m ) ln(1+r)+σ √ t1 m (m−1 (t1 ln(1 + r)) 2 −m 1 2 t 3 2 1 σ ln(1 + r) + 1 4 t1σ − 12t1 ln(1 + r)) −σ √ t1 ( 1 + e −2σ √ t1 m ) = 4 ( 1 4 σ2 − 1 2 t1 ln(1 + r) ) −2σ √ t1 lim m→∞ 2 √ m ( p− 1 2 ) = ln(1 + r) − σ 2 2 σ √ t1. on returning the value of p, lim m→∞ p = lim m→∞ 1 2 + 1 2 µ σ √ t1 m = 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ t1 m . (5.10) on the same lines we can prove that, between m and n, lim m→∞ 1 2 + 1 2 µ σ √ t2 n−m = 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ t2 n−m . (5.11) now recall from the equation (5.3), µ̂pm = [ p ln ( u d ) + ln d ] m. substitute the values of u = e σ √ t1 m and d = e −σ √ t1 m ( or u d = e 2σ √ t1 m ) , we get, µ̂pm = [ p ln ( u d ) + ln d ] m µ̂pm = [ p ln ( e 2σ √ t1 m ) + ln e −σ √ t1 m ] m = [ 2σp √ t1 m −σ √ t1 m ] m = [2p− 1] mσ √ t1 m . int. j. anal. appl. 18 (1) (2020) 145 taking limit m →∞ and using equation (5.10), it follows that, lim m→∞ µ̂pm = lim m→∞ [ 2 ( 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ t1 m ) − 1 ] mσ √ t1 m = lim m→∞ [( ln(1 + r) − 1 2 σ2 ) σ √ t1 m ] mσ √ t1 m lim m→∞ µ̂pm = ( ln(1 + r) − 1 2 σ2 ) t1. (5.12) now we turn towards the next interval i.e. m to n. recall the equation (5.4), µ̂p(n−m) = [ p ln ( u d ) + ln d ] (n−m) . substitute the values of u = e σ √ t2 n−m and d = e −σ √ t2 n−m ( or u d = e 2σ √ t2 n−m ) , we get, µ̂p(n−m) = [ p ln ( e 2σ √ t2 n−m ) + ln e −σ √ t2 n−m ] (n−m) = [ 2σp √ t2 n−m −σ √ t2 n−m ] (n−m) = [2p− 1] (n−m) σ √ t2 n−m . taking limit (n−m) →∞ and using equation (5.11), it follows that, lim (n−m)→∞ µ̂p(n−m) = [ 2 ( 1 2 + 1 2 ( ln(1 + r) − 1 2 σ2 ) σ √ t2 n−m ) − 1 ] (n−m) σ √ t2 n−m = lim m→∞ [( ln(1 + r) − 1 2 σ2 ) σ √ t2 n−m ] (n−m) σ √ t2 n−m lim (n−m)→∞ µ̂p(n−m) = ( ln(1 + r) − 1 2 σ2 ) t2. (5.13) now let us return to the equation (5.6). take limit m → ∞ and ε → 0, and using the equations (5.12) and (4.11), we consider the following limit. lim m→∞ lim ε→0 ln s ∗ s0 − µ̂pm−ε ln ( u d ) σ̂p √ m = ln s ∗ s0 − ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 = d1. (5.14) similarly consider the equation (5.8). by application of limit (n−m) → ∞ and ε → 0, and using the equation (4.14) and (5.13), we consider the following limit, lim m→∞ lim ε→0 ln ( k2 s ) − µ̂p(n−m) −ε ln ( u d ) σ̂p √ (n−m) = ln ( k2 s ) − ( ln(1 + r) − 1 2 σ2 ) t2 σ √ t2 = d2. (5.15) int. j. anal. appl. 18 (1) (2020) 146 thus we are in a position to claim our key convergence, using bivariate binomial convergence to bivariate binomial from [16] (page 9) and theorem 4.1 we infer that 1 −b (a,b; p′; n,n−m) = p   ln ( st s0 ) −µ̂pm σ̂p √ m ≤ ln s ∗ s0 −µ̂pm−ε ln( ud ) σ̂p √ m , ln ( st s0 ) −µ̂p(n−m) σ̂p √ n ≤ ln( k2 s )−µ̂p(n−m)−ε ln( u d ) σ̂p √ (n−m)   → n ( −d1,−d2; √ t2 t1 ) as (n−m) , m →∞, where ln s ∗ s0 − ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 : = −d1 ln ( k2 s ) − ( ln(1 + r) − 1 2 σ2 ) t2 σ √ t2 : = −d2. since 1 −n ( −d1,−d2; √ t2 t1 ) = n (d1,d2) , hence b (a,b; p′; n,n−m) → n ( d1,d2; √ t2 t1 ) . where, ln s s∗ + ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 : = d1 ln ( s k2 ) + ( ln(1 + r) − 1 2 σ2 ) t2 σ √ t2 : = d2. precisely on the same line of argument we can show that b (a,b; p,p′; n,n−m) → n ( d∗1,d ∗ 2; √ t2 t1 ) where, d∗1 = d1 + σ √ t1 d∗2 = d2 + σ √ t2 and n ( ·, ·; √ t2 t1 ) is the bivariate standard normal cdf with correlation coefficient √ t2 t1 . finally we will deal with the one-dimensional convergence i.e. ψ (i ≥ a,m,p) → n (d∗1) as m →∞. again 1 −ψ (i ≥ a,m,p) = p (i ≤ a− 1) = p ( i−mp√ mp(1 −p) ≤ a− 1 −mp√ mp(1 −p) ) where a = min { i ∈ m; i ≥ ln s ∗ s0dm ln u d } ∈ [ ln s ∗ s0dm ln u d , ln k1 cdm ln u d + 1 ) . int. j. anal. appl. 18 (1) (2020) 147 using equation (5.5) and equation (5.6) into the last equation we infer that 1 −ψ (i ≥ a,m,p) = p  ln ( st s0 ) − µ̂pm σ̂p √ m ≤ ln s ∗ s0 − µ̂pm−ε ln ( u d ) σ̂p √ m   . using the convergence (5.14) and the one dimensional central limit theorem, and arguing in same manner as we argued in the end of the section 4 , we may infer that 1 −ψ (i ≥ a,m,p) = p  ln ( st s0 ) − µ̂pm σ̂p √ m ≤ ln s ∗ s0 − µ̂pm−ε ln ( u d ) σ̂p √ m   → n ( ln s ∗ s0 − ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 −σ √ t1 ) as m →∞, where n (·) is the c.d.f of standard normal distribution. by the use of symmetry property of the standard normal c.d.f i.e. 1 −n (z) = n (−z), it follows that, ψ (i ≥ a,m,p) = n ( ln s s∗ + ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 + σ √ t1 ) = n(d∗1). hence, we are done with the conclusion that c = s0b (i ≥ a,j ≥ b; p′; n,n−m) − (1 + r)−(n−m)k2b (i ≥ a,j ≥ b; p′,p; n,n−m) −(1 + r)−mk1ψ (i ≥ a,m,p) → s0n ( d1,d2; √ t2 t1 ) −e−rt2k2n ( d∗1,d ∗ 2; √ t2 t1 ) −e−rt1k1n(d∗1), (5.16) where d1 = ln s s∗ + ( ln(1 + r) − 1 2 σ2 ) t1 σ √ t1 ,d∗1 = d1 + σ √ t1 d2 = ln ( s k2 ) + ( ln(1 + r) − 1 2 σ2 ) t2 σ √ t2 ,d∗2 = d2 + σ √ t2. thus we have shown that our developed crr formula converges to the standard well-known continuous time black-scholes price of the compound option. � 6. conclusion the paper provides a comprehensive treatment of the binomial pricing of option of financial derivatives, in general, and options in particular. following two new results have been proven. 1. the cox-ross rubinstein(crr) type formula has been derived for risk-neutral pricing of the european style compound call option on european call option. 2. it has been shown explicitly that for a suitable choice of parameters, the crr formula to price the compound call option of on call, converges to the well known, continuous time version of the int. j. anal. appl. 18 (1) (2020) 148 black-scholes price of the same option. simulation of our derived formula will be helpful for financial market. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] agliardi, e., agliardi, r., a closed-form solution for multi-compound options. risk lett. 1 (2) (2004), 12. [2] black, f., & scholes, m. the pricing of options and corporate liabilities. j. politic. econ. 81 (3) (1973), 637-654. [3] cassimon, d., engelen, p.-j., thomassen, l., & van wouwe, m. the valuation of a nda using a 6-fold compound option. res. policy, 33 (1) (2004), 41-51. [4] chiarella, c., griebsch, s., kang, b., a comparative study on time-efficient methods to price compound options in the heston model. comput. math. appl. 67 (6) (2014), 12541270. [5] chiarella, c., kang, b., the evaluation of american compound option prices under stochastic volatility and stochastic interest rates. j. comput. financ. 14 (9) (2011), 121. [6] cortazar, g., & schwartz, e. s. a compound option model of production and intermediate inventories. j. business, 66 (4) (1993), 517-540. [7] cutland, n. j., & roux, a. derivative pricing in discrete time, springer science & business media, 2012. [8] cox, j. c., & ross, s. a. the valuation of options for alternative stochastic processes. j. financ. econ. 3 (1-2) (1976), 145-166. [9] geske, r. the valuation of compound options. j. financ. econ. 7 (1) (1979), 63-81. [10] geske, r. the valuation of corporate liabilities as compound options. j. financ. quant. anal. 12 (4) (1977), 541-552. [11] griebsch, s.a. the evaluation of european compound option prices under stochastic volatility using fourier transform techniques. rev. deriv. res. 16 (2) (2013), 135165. [12] gukhal, c.r. the compound option approach to american options on jump diffusion. j. econ. dyn. control 28 (10) (2004), 20552074. [13] fouque, j.-p., han, c.-h. evaluation of compound options using perturbation approximation. j. comput. financ. 9 (1) (2005), 4161. [14] hull, j. c. options, futures, and other derivatives: pearson education india, 2006. [15] lajeri-chaherli, f., 2002. a note on the valuation of compound options. j. futures markets, 22 (11), 11031115. [16] marshall, a. w., and i. olkin, a family of bivariate distributions generated by the bivariate bernoulli distribution, j. amer. stat. assoc. 80 (1985), 332-338. [17] merton, r. c. option pricing when underlying stock returns are discontinuous. j. financ. econ. 3 (1-2) (1976), 125-144. [18] samuelson p.a. rational theory of warrant pricing. in: grnbaum f., van moerbeke p., moll v. (eds) henry p. mckean jr. selecta. contemporary mathematicians. birkhuser, cham. 2015. [19] z. brzeźniak and t. zastawniak, basic stochastic process. springer, 1999. 1. introduction 2. crr formula for pricing compound options 2.1. introduction to compound option 3. derivation of crr compound option formula 4. convergence of empirical means and volatilities of log returns to actual means and volatilities 5. main convergence result 6. conclusion bibliography references international journal of analysis and applications volume 16, number 5 (2018), 673-688 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-673 µ−values for matrices corresponding to symmetries in control systems mutti-ur rehman∗, m. fazeel anwar department of mathematics, sukkur iba university, 65200 sukkur-pakistan ∗corresponding author: mutti.rehman@iba-suk.edu.pk abstract. in this article we consider numerical approximation of structured singular values (µ−values). the lower bounds for µ−values are approximated by using ordinary differential equations based technique. the structured singular values provide a vital tool to investigate stability of feedback systems. we also compute the lower bounds of µ−values for certain matrices that correspond to symmetries in control systems. 1. introduction the structured singular values known as µ-values is a well-known mathematical tool in control, introduced in 1981 by j. c. doyle [13]. they can be used to discuss stability of linear systems subject to certain perturbations. applications of structured singular values in engineering system are described in [14]. the exact computation of µ−values is known to be an np-hard problem see [2]. a considerable effort has been made to compute the lower and upper bound for structured singular values. the power method [10] provides a lower bound for µ−values when we consider pure complex perturbations. it however fails to converge for pure real uncertainties for more details see [16]. the upper bound of µ-values provides critical information which guarantees the stability of feedback linear systems. the well-known matlab function mussv available in matlab control toolbox approximates an upper bounds for structured singular values by means of diagonal balancing and linear matrix inequality techniques [5]. the methodology proposed in [12] is based on a two level algorithm, inner and outer algorithm. in inner algorithm, we attempt received 2018-05-09; accepted 2018-08-02; published 2018-09-05. 2010 mathematics subject classification. 65f15, 34h05, key words and phrases. µ-values; spectral radius; family of block diagonal perturbations. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 673 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-673 int. j. anal. appl. 16 (5) (2018) 674 to solve an optimization problem while outer algorithm allows us to compute an extremizer by varying a small parameter �. in [4], danielson used symmetric groups to design model predictive controllers with reduced complexity for constrained linear control systems. in model predictive control, the control input is obtained by solving a constrained finite time optimal control problem. for a piecewise affine control law symmetries are statespace and input-space transformations that relate controller pieces. using symmetry he could discard some of the pieces of a given controller. these discarded pieces can also be reconstructed using symmetry. using symmetries of the control system he was able to reduce the complexity of the controller and save memory without sacrificing performance. it was also noted that the amount of reduction in complexity depends on the number of symmetries possessed by the system. for systems with large symmetry groups the techniques presented in [4] can significantly reduce the complexity of the piecewise affine control-law produced using explicit model predictive control.the goal of this article is to compute the µ−values for matrices corresponding to a control system whose symmetry group is s5. we present a comparison between lower bounds of µ-values approximated by mussv and the algorithm presented in [12]. 2. basic framework let cn,n (rn,n) denote the collection of n×n complex (real) matrices and let m ∈ cn,n. we denote a family of block diagonal matrices by θb = {diag(ciii, γj) : ci ∈ c(r), γj ∈ cmj,mj (rmj,mj )}. in the above equation, ii is an identity matrix having dimension i. definition 2.1. [9]. the structured singular values denoted by µ for a given matrix m ∈ cn,n or m ∈ rn,n and a set of block diagonal matrices θb is defined as µθb (m) := 1 min{‖∆‖2 : ∆ ∈ θb, det(i −m∆) = 0} . (2.1) in above definition 2.1, det(·) represent the determinant of a matrix (i −m∆) while minimum is over an admissible perturbation ∆. in this particular case we will denote the set of pure complex uncertainties by θ ′ b. if ∆ ∈ θ ′ b, there is a function exp(iφ)∆ ∈ θ ′ b for any real number φ and as a consequence we have ∆ ∈ θ ′ b such that the spectral radius of m∆ attains the exact value 1 iff there is ∆∗ ∈ θ ′ b such that m∆ ∗ has the eigenvalue 1. the perturbation ∆∗ is constructed in such a way that it possesses a unit 2-norm and as result det(i−m∆∗) = 0. the above construction allows us to write an alternate definition of µ-values when pure complex uncertainties are under consideration. int. j. anal. appl. 16 (5) (2018) 675 θ ′ b = 1 min { ‖∆‖2 : ∆ ∈ θ ′ b, ρ(m∆) = 1 }. (2.2) in equ. (2.2), ρ(·) denotes the spectral radius of a matrix m∆. 2.1. the µ-value based on a structured spectral value set. for a given n-dimensional complex matrix m ∈ cn×n and a perturbation level � the structured spectral value set is the collection of all eigenvalues of matrix (�m∆) defined as λθb�0 (m) = {λ ∈ λ(�m∆) : ∆ ∈ θb}, (2.3) where λ(·) denotes the spectrum of a matrix and ‖∆‖2 = 1. for mixed real and complex uncertainties, we let σθb� (m) = {η = 1 −λ1 : λ1 ∈ λ θb � (m)}. (2.4) the formulation in equ. (2.4) allows us to write down structured singular values defined in equ. (2.2) as follows: µθb (m) = 1 arg min{0 ∈ σθb� (m)} . (2.5) while on the other hand when pure complex uncertainties are under consideration then equ. (2.3) allows us to alternatively express µ-value as µ θ ′ b (m) = 1 arg min{max |λ1| = 1} . (2.6) 2.2. mathematical problem. we consider the following optimization problem, ξ(�0) = arg min |η|, (2.7) where η ∈ σθb�0 (m) for some fixed parameter �1 > 0. it is clear from the above discussion that the µ-value µθb (m) is the reciprocal of the minimum value of �1 for which η(�1) = 0. therefore we suggests a two-level algorithm that is inner and outer algorithm. for inner algorithm, we solve equ. (2.7) by constructing and then solving a gradient system of ordinary differential equations. while for the case of outer algorithm, we make use of an iterative method to first vary the perturbation level �1. this gives the knowledge of the computation of derivative of a local extremizer say ∆(�1) with respect to some fixed parameter �1. we addressed the case of a purely complex uncertainties when θ∗b by taking the inner algorithm in order to compute a local optimum for λ(�1) = arg max |λ1|. (2.8) in equ. (2.8), λ1 ∈ λ θ∗b �1 (m) which yields a lower bound for the µ-value in case of pure complex perturbations that is µ∆∗b (m). int. j. anal. appl. 16 (5) (2018) 676 3. purely complex uncertainties in this section, we give a solution of the maximization problem (2.8) for m ∈ cn,n while considering the set of pure complex uncertainties given below θ∗b = {diag(α1i1, ...,αnin; ∆1, ..., ∆f ) : αi ∈ c, ∆j ∈ c mj,mj}, (3.1) the following lemma describes the behavior of the eigenvalues of a matrix valued function. lemma 3.1. for a family of matrices υ : r → cn,n suppose that λ1(t) is an eigenvalue of a matrix valued function υ(t) which converges to a simple eigenvalue λ ′ of υ0 = υ(0) as t → 0. then λ1(t) is analytic near t = 0 with dλ1 dt = w∗0 υ1v0 w∗0v0 , where υ1 = υ̇(0) and v0,w0 are right and left eigenvectors of υ0 associated to λ ′ , that is, (υ0 −λ ′ i)v0 = 0 and w∗0 (υ0 −λ ′ i) = 0. to deal with the optimization problem (2.8) we need to compute an uncertainty ∆local in such a way that ρ(�1a∆local) has the maximum growth along ∆ ∈ θ∗b with ‖∆‖2 ≤ 1. in the following we call λ1 the greatest eigenvalue if |λ1| equals to the spectral radius. definition 3.2. a matrix valued function ∆ ∈ θ∗b such that ‖∆‖ possesses a unit 2-norm and (�1m∆) has greatest eigenvalue which maximizes the modulus of λ θ∗b �1 (m) is called a local extremizer. in the following theorem 3.3, we give the characterization of local extremizers towards a gradient system of ordinary differential equations. theorem 3.3 [12]. let ∆local = diag(α1i1, ...,αnin; ∆1, ..., ∆f ). in above equation the ∆local possesses a unit 2-norm and is a local extremizer of λ θ∗b �1 (a). further suppose that the matrix (�1m∆local) possesses a simple greatest eigenvalue that is λ1 = |λ1|eiθ, with the right and left eigenvectors v and w which are scaled so that s = eiθw∗v > 0. partitioning of u and v yields v = (vt1 , . . . , v t n , v t n+1, . . . ,v t n+f ) t; u = a∗w = (ut1 , . . . ,u t n, u t n+1, . . . ,u t n+f ) t, (3.2) additionally we assume that u∗kvk 6= 0 ∀ k = 1, . . . ,n, (3.3) ‖un+h‖2 · ‖vn+h‖2 6= 0 ∀ h = 1, . . . ,f. (3.4) then |sk| = 1 ∀ k = 1, . . . ,n and ‖∆h‖2 = 1 ∀h = 1, . . . ,f. int. j. anal. appl. 16 (5) (2018) 677 3.1. system of odes to compute extremal points of λ ∆∗b � (m). we now compute a local maximizer for |λ1| where λ1 ∈ λθb∗�1 (m). for this we first construct a matrix valued function ∆(t) in such a way that the greatest eigenvalue λ(t) of the matrix (�1m∆(t)) has the maximum growth. we then derive and solve a gradient system of ordinary differential equations which satisfies the initial choice ∆(0). 3.2. the local optimization problem. let λ1 = |λ1|eiθ be a simple eigenvalue. further suppose that v,w are normalized so that ‖w‖ = ‖v‖ = 1, w∗v = |w∗v|e−iθ. (3.5) by making use of lemma 3.1, we have d dt |λ1|2 = 2|λ1|re ( u∗∆̇v eiθw∗v ) = 2|λ1| |w∗v| re(u∗∆̇v), (3.6) where u = m∗w. for ∆ ∈ θb we aim to compute the direction ∆̇ = τ that maximizes the local growth of the modulus of λ1. we get τ = diag(ω1ir1, . . . ,ωsirn , ω1, . . . , ωf ) (3.7) as a solution of the optimization problem τ∗ = arg max{re(u∗τx)} subject to re(δiωi) = 0, i = 1 : n, and re〈∆j, ωj〉 = 0, j = 1 : f. (3.8) in the following lemma 3.2.1, we give the solution τ∗ of the optimization problem as discussed in the (3.8). lemma 3.2.1 [12]. τ∗ = diag(ω1ir1, . . . ,ωnirn , ω1, . . . , ωf ), (3.9) with ωi = νi (v ∗ i ui −re (v ∗ i uisi) si) , i = 1, . . . ,n (3.10) ωj = ζj ( un+jv ∗ n+j −re〈∆j,un+jv ∗ n+j〉∆j ) , j = 1, . . . ,f. (3.11) the coefficient νi > 0 is the reciprocal of the absolute value of the expression appearing in the right-hand side in equ. (3.10) when it’s different from zero and νi = 1 else. while the coefficient ζj > 0 is the reciprocal of the frobenius norm of the matrix appearing in the right hand side of equ. (3.11) if it’s different from zero and ζj = 1 else. we write down the result as obtained in the previous lemma 1.2 as: τ∗ = g1pθ∗b (uv ∗) −d2∆. (3.12) int. j. anal. appl. 16 (5) (2018) 678 in above equation pθ∗b (·) is the orthogonal projection while g1,d2 ∈ θ ∗ b are diagonal matrices while the matrix d1 is positive. 3.3. the gradient system of ordinary differential equations. the result in the previous lemma 3.2.1 allows us to consider the following differential equation on θ∗b: ∆̇ = g1pθ∗b (uv ∗) −d2∆. (3.13) in the above equation v(t) is an eigenvector having the unit 2-norm ans is associated to a simple eigenvalue λ(t) of the matrix (�1m∆(t)) for some fixed parameter �1 > 0. the differential equation (3.13) is a gradient system of ordinary differential equations because it’s the right-hand side is the projected gradient of τ 7→ re(u∗τv). 3.4. choice of initial value matrix and �0. in order to compute �0 we choose the initial value matrix ∆0 = dp∆b (wv ∗), (3.14) where d is the positive diagonal matrix such that ∆0 ∈ θb. as a natural choice for the initialization of the perturbation level, we take �0 as �0 = 1 µ̂θb (m) . (3.15) where µ̂θb (m) is the upper bound of µ-value approximated by mussv. 4. numerical experimentation in this section, we present the main contribution which is the numerical approximation of both lower and upper bounds of µ-values. these results are computed by well-known matlab routine mussv and the algorithm [12]. example 1. in table 1, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix a4. the matrix a4 is gievn as below. in first column of table 1, we present the size of the matrix a4. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns, we present both upper and lower bounds of ssv, that is, µ mussv u , µmussvl approximated by matlab routine mussv and the lower bound µ new l approximated by algorithm [12] respectively. a4 =   −0.5 + 1.4434i −0.5774i 0.5 − 0.2887i −0.5 + 0.2887 −0.5 + 0.8660i 0 0 0 −0.5 + 0.8660i 0 −0.5 − 0.8660i −1 −0.5 − 0.2887i −0.5774i 0.5774i −0.8660i   , int. j. anal. appl. 16 (5) (2018) 679 n θb µ mussv u µ mussv l µ new l 04 {diag(∆1) : ∆1 ∈ c4,4} 2.5031 2.5030 2.5030 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ r} 0.6354 0.0000 0.6297 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ c} 2.3780 2.3748 2.3748 04 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ r, ∆1 ∈ c2,2} 1.8114 1.7568 1.7552 04 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ c, ∆1 ∈ c2,2} 2.3832 2.3813 2.3811 04 {diag(∆1, ∆2) : ∆1, ∆2 ∈ c2,2} 2.4047 2.4027 2.4029 04 {diag(δ1i1, ∆2) : δ1 ∈ r, ∆2 ∈ c3,3} 1.8415 1.8415 1.8414 table 1. computation of bounds of µ-values n θb µ mussv u µ mussv l µ new l 04 {diag(∆1) : ∆1 ∈ c4,4} 2.8765 2.8765 2.8763 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ r} 1.5023 0.0000 1.0000 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ c} 2.7375 2.7326 2.7326 04 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ r, ∆1 ∈ c2,2} 1.8732 1.8716 0.5373 04 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ c, ∆1 ∈ c2,2} 2.7375 2.7334 2.7336 04 {diag(∆1, ∆2) : ∆1, ∆2 ∈ c2,2} 2.7375 2.7373 2.7372 table 2. computation of bounds of µ-values in table 2, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix b4. the matrix b4 is gievn as below. in first column of table 2, we present the size of the matrix b4. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by matlab routine mussv and the lower bound µnewl approximated by algorithm [12] respectively. b4 =   0.5 + 0.8660i −1 −0.5 − 0.8660i −1 −0.5 + 0.8660i 0 −0.5 − 0.8660i −1 0.5774i −0.5 + 0.2887i 0.5 − 0.2887i −0.5 + 0.2887i −0.5 + 0.2887i −0.5774i 0.5 − 0.2887i 1 − 0.5774i   . int. j. anal. appl. 16 (5) (2018) 680 example 2. consider the following four dimensional matrix a5. a5 =   −0.5 + 0.8660i 0.2500 − 0.4330i 0.5 0.5 0 −0.75 − 0.4330i −0.5000 0 + 0.8660i 0.5 + 0.8660i 0 0 0 −0.5 − 0.8660i 0.25 + 0.4330i 0.25 + 1.2990i 0.25 − 0.4330i   . consider the perturbations set as θb = {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ r}. by applying matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ = 1.0e + 050   4.9136 0.0000 0.0000 0.0000 0.0000 4.9136 0.0000 0.0000 0.0000 0.0000 4.9136 0.0000 0.0000 0.0000 0.0000 4.9136   , while ‖∆̂‖2 = 4.9136e + 050. for this particular example, we have obtained an upper bound µ upper pd = 1.5797 while the same lower bound as µlowerpd = 0.0000. by making use of our algorithm [12], we have obtained the perturbation �∗∆∗ with ∆∗ =   −0.7049 0.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000 −1.0000 0.0000 0.0000 0.0000 0.0000 0.7490   , and �∗ = 0.7549. while ‖∆∗‖2 = 1. the same lower bound can be obtained µlowernew = 1.3248 as the one obtained by mussv. in the table 3, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix a5. the matrix a5 is gievn as below. in first column of table 3, we present the size of the matrix a5. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by matlab routine mussv and the lower bound µnewl approximated by algorithm [12] respectively. in the following table 4, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix b5. the matrix b5 is gievn as below. in first column of table 4, we present the size of the matrix b5. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is int. j. anal. appl. 16 (5) (2018) 681 n θb µ mussv u µ mussv l µ new l 04 {diag(∆1) : ∆1 ∈ c4,4} 2.2701 2.2701 2.2699 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ c} 1.8679 1.8679 1.8675 04 {diag(δ1i1,δ2i1, ∆1) : δ1,δ2 ∈ c, ∆1 ∈ c2,2} 2.0084 2.0084 2.0076 04 {diag(δ1i1,δ2i1, ∆1) : δ1,δ2 ∈ r, ∆1 ∈ c2,2} 2.1863 2.1863 2.1861 04 {diag(δ1i1, ∆1) : δ1 ∈ r, ∆1 ∈ c3,3} 2.0242 2.0242 0.5000 04 {diag(δ1i1, ∆1) : δ1 ∈ c, ∆1 ∈ c3,3} 2.1956 2.1932 2.1934 table 3. computation of bounds of µ-values n ∆b µ mussv u µ mussv l µ new l 04 {diag(∆1) : ∆1 ∈ c4,4} 2.9458 2.9458 2.9456 04 {diag(δ1i1,δ2i1,δ3i1,δ4i1) : δ1,δ2,δ3,δ4 ∈ r} 1.5155 0.0000 1.0000 04 {diag(δ1i2,δ2i2) : δ1,δ2 ∈ r} 1.2164 0.0000 1.0894 04 {diag(δ1i1, ∆2) : δ1 ∈ r, ∆2 ∈ r3,3} 1.1641 0.0000 1.0000 04 {diag(∆1) : ∆1 ∈ r4,4} 1.2165 0.0000 0.7275 table 4. computation of bounds of µ-values µmussvu , µ mussv l approximated by matlab routine mussv and the lower bound µ new l approximated by algorithm [12] respectively. b5 =   −1 0.75 − 0.4330i 0.25 + 1.2990i −0.75 − 0.4330i 0 −0.5 0.25 + 1.2990i −0.25 − 1.2990i 0 −0.8660i −0.2500 − 1.2990i −0.75 + 0.4330i 0 0.75 + 0.4330i −0.75 − 0.4330i −0.2500 + 0.4330i   . example 3. consider the following four dimensional matrix a6. a6 =   0.3819 0.3819 1.0000 0 0 0 1.2360 −0.3819 0.1680 0 0 0 −0.6180 1.0000 −0.6180 0 0 0 0 0 0 0.3819 −1.2360 0.3819 0 0 0 1.2360 −0.3819 0.6180 0 0 0 −1.0000 0.6180 0.6180   , consider the perturbation set as int. j. anal. appl. 16 (5) (2018) 682 n ∆b µ mussv u µ mussv l µ new l 06 {diag(δ1i6) : δ1 ∈ c} 1.3061 1.3037 1.3037 06 {diag(δ1i6) : δ1 ∈ r}} 0.9560 0.0000 0.9560 06 {diag(δiii) : δi ∈ r, ∀i = 1 : 6} 1.9330 1.9330 1.9330 06 {diag(δiii) : δi ∈ c, ∀i = 1 : 6} 1.1875 0.0000 1.1875 06 {diag(δ1i3,δ2i3) : δ1,δ2 ∈ r} 0.9560 0.0000 0.9560 table 5. computation of bounds of µ-values θb = {diag(∆1) : ∆1 ∈ c6,6}. by applying the matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2136 0.2414 −0.2030 0 0 0 −0.1765 −0.1994 0.1678 0 0 0 0.0386 0.0436 −0.0367   , while ‖∆̂‖2 = 0.4989. for this particular example, we have obtained an upper bound µ upper pd = 2.0043 while the same lower bound as µlowerpd = 2.0043. by making use of our algorithm [12], we have obtained the perturbation �∗∆∗ with ∆∗ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.4282 0.4837 −0.4069 0 0 0 −0.3538 −0.3997 0.3363 0 0 0 0.0773 0.0874 −0.0735   , and �∗ = 0.4989. while ‖∆∗‖2 = 1. the same lower bound can be obtained µlowernew = 2.0043 as the one obtained by mussv. in table 5, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix a6. the matrix a6 is gievn as below. in first column of table 5, we present the size of the matrix a6. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by matlab routine mussv and the lower bound µnewl approximated by algorithm [12] respectively. int. j. anal. appl. 16 (5) (2018) 683 consider the following four dimensional matrix b6. b6 =   0 0 0 −1.0000 0 0 0 0 0 0 −1.0000 0 0 0 0 0.3819 0.3819 1.0000 −1.0000 0 0 0 0 0 0 −1.0000 0 0 0 0 0.3819 0.3819 1.0000 0 0 0   , consider the perturbation set as θb = {diag(∆1) : ∆1 ∈ c6,6}. by applying matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2136 0.2414 −0.2030 0 0 0 −0.1765 −0.1994 0.1678 0 0 0 0.0386 0.0436 −0.0367   , and ‖∆̂‖2 = 0.7658. for this example, one can obtain the upper bound µ upper pd = 1.3059 while the same lower bound as µlowerpd = 1.3059. by making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ with ∆∗ =   0 0 0 −0.1848 −0.1848 0.3413 0 0 0 −0.1848 −0.1848 0.3413 0 0 0 −0.2002 −0.2002 0.3696 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   , and �∗ = 0.7658. while ‖∆∗‖2 = 1. the same lower bound can be obtained µlowernew = 1.3059 as the one obtained by mussv. in the following table 6, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix b6. the matrix b6 is gievn as below. in first column of table 6, we present the size of the matrix b6. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is µ mussv u , µ mussv l int. j. anal. appl. 16 (5) (2018) 684 n θb µ mussv u µ mussv l µ new l 06 {diag(δ1i6) : δ1 ∈ c} 1.0016 1.0000 0.5000 06 {diag(δ1i6) : δ1 ∈ r}} 0.9560 0.0000 1.0000 06 {diag(δiii) : δi ∈ r, ∀i = 1 : 6} 1.0000 0.0000 1.0000 06 {diag(δiii) : δi ∈ c, ∀i = 1 : 6} 1.0041 1.0000 0.7453 06 {diag(δ1i3,δ2i3) : δ1,δ2 ∈ r} 1.0813 0.0000 1.0000 table 6. computation of bounds of µ-values approximated by matlab routine mussv and the lower bound µnewl approximated by algorithm [12] respectively. example 4. in the following example, we consider a five dimensional complex matrix a7 given as, a7 =   0 1 0 1 1 0 0 −1 −1 −1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0   , consider the perturbation set as θb = {diag(∆1) : ∆1 ∈ c5,5}. by applying apply the matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   −0.0000 0.0000 −0.0000 0.0000 −0.0000 0.1028 −0.0978 0.0226 −0.0000 0.0441 0.0802 −0.0763 0.0176 −0.0000 0.0344 0.2006 −0.1909 0.0441 −0.0000 0.0860 0.1644 −0.1565 0.0361 −0.0000 0.0705   , and ‖∆̂‖2 = 0.4244. for this example, one can obtain the upper bound µ upper pd = 2.3563 while the same lower bound as µlowerpd = 2.3563. by making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ int. j. anal. appl. 16 (5) (2018) 685 with ∆∗ =   0.0000 −0.0000 0.0000 0.0000 0.0000 0.2421 −0.2304 0.0532 0.0000 0.1038 0.1889 −0.1798 0.0415 0.0000 0.0810 0.4726 −0.4498 0.1038 0.0000 0.2026 0.3874 −0.3687 0.0851 0.0000 0.1661   , and �∗ = 0.4244 while ‖∆∗‖2 = 1 the same lower bound can be obtained µlowernew = 2.3563 as the one obtained by mussv. in the following example, we consider a five dimensional complex matrix b7 given as, b7 =   1 0 0 0 0 −1 −1 0 −1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0   , consider the perturbation set as ∆b = {diag(∆1) : ∆1 ∈ c5,5}. by applying matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   0.1057 −0.2887 0.0000 0.1057 0.0000 0.0774 −0.2113 0.0000 0.0774 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000 0.1057 −0.2887 0.0000 0.1057 0.0000 0.0000 −0.0000 0.0000 0.0000 0.0000   , and ‖∆̂‖2 = 0.5176. for this example, one can obtain the upper bound µ upper pd = 1.9319 while the same lower bound as µlowerpd = 1.9319. by making use of our algorithm [12], we have obtained the perturbation � ∗∆∗ with ∆∗ =   0.0000 −0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 −0.0000 0.0000 0.0000 0.0000 −0.0000 −0.5000 0.0000 0.5000 0.0000 −0.0000 −0.0000 0.0000 0.0000 −0.0000 0.0000 0.5000 −0.0000 −0.5000   , and �∗ = 1.0000 while ‖∆∗‖2 = 1.0000 the same lower bound can be obtained µlowernew = 1.0000 as the one obtained by mussv. int. j. anal. appl. 16 (5) (2018) 686 example 5. in the following table 7, we give comparison of numerical approximation to both lower and upper bounds of structured singular values approximated by the well-known matlab routine mussv and algorithm [12] for the matrix a8. the matrix a8 is gievn as below. in first column of table 7, we present the size of the matrix a8. while in the next column, we present the family of block diagonal matrices which is denoted by θb. in the third, fourth and fifth columns we present both upper and lower bounds that is µmussvu , µ mussv l approximated by matlab routine mussv and the lower bound µ new l approximated by algorithm [12] respectively. a8 =   0 0 1 1 −1 1 −1 0 1 0 0 1 1 0 −1 0 0 0 0 1 0 0 1 0 0   , in the following example, we consider a five dimensional complex matrix b8 given as, b8 =   −1 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 1 −1 0 1 0 0 −1 −1 0 1   , consider the perturbation set as θb = {diag(∆1) : ∆1 ∈ c5,5}. by applying the matlab function mussv, we have obtained the perturbation ∆̂ with ∆̂ =   −0.0288 0.0576 0.0288 0.1091 0.1091 0.0576 −0.1151 −0.0576 −0.2182 −0.2182 0.0288 −0.0576 −0.0288 −0.1091 −0.1091 −0.0228 0.0455 0.0228 0.0863 0.0863 −0.0228 0.0455 0.0228 0.0863 0.0863   , int. j. anal. appl. 16 (5) (2018) 687 n θb µ mussv u µ mussv l µ new l 05 {diag(∆1) : ∆1 ∈ c5,5} 2.4142 2.4142 2.4142 05 {diag(δiii) : δi ∈ r, ∀i = 1 : 5} 2.1616 2.1516 2.0609 05 {diag(δiii) : δi ∈ c, ∀i = 1 : 5} 2.1628 2.1537 1.9511 05 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ r, ∆1 ∈ c3,3} 2.2182 2.2174 2.2176 05 {diag(δ1i1,δ2i1, ∆2) : δ1,δ2 ∈ c, ∆1 ∈ c3,3} 2.7178 2.7176 0.5000 05 {diag(∆1,δ1i1, ∆2) : δ1 ∈ c, ∆1, ∆2 ∈ c2,2} 2.2592 2.2571 2.2589 05 {diag(∆1,δ1i1, ∆2) : δ1 ∈ r, ∆1, ∆2 ∈ c2,2} 2.2592 2.2592 2.2592 table 7. computation of bounds of µ-values and ‖∆̂‖2 = 0.4569. for this example, one can obtain the upper bound µ upper pd = 2.1889 while the same lower bound as µlowerpd = 2.1889. by making use of our algorithm [12], one can obtain the perturbation � ∗∆∗ with ∆∗ =   −0.0630 0.1260 0.0630 0.2388 0.2388 0.1260 −0.2520 −0.1260 −0.4777 −0.4777 0.0630 −0.1260 −0.0630 −0.2388 −0.2388 −0.0498 0.0997 0.0498 0.1890 0.1890 −0.0498 0.0997 0.0498 0.1890 0.1890   , and �∗ = 0.4569 while ‖∆∗‖2 = 1.0000. the same lower bound can be obtained µlowernew = 2.1889 as the one obtained by mussv. 5. conclusion in this article we have considered the numerical approximation of µ-values for the matrix representations of finite symmetric groups sn over the filed of complex numbers by using well-known matlab function mussv and algorithm [12]. the experimental results indicates the different behaviors of lower bounds of µ-values with once computed by mussv and our algorithm. references [1] bernhardsson, bo and rantzer, anders and qiu, li. real perturbation values and real quadratic forms in a complex vector space. linear algebra appl., 1(1994): 131-154. [2] braatz, richard p and young, peter m and doyle, john c and morari, manfred. computational complexity of µ calculation. ieee trans. autom. control, 39(1994): 1000-1002. [3] chen, jie and fan, michael kh and nett, carl n. structured singular values with nondiagonal structures. i. characterizations. ieee trans. autom. control, 41(1996): 1507-1511. [4] danielson, claus robert. symmetric constrained optimal control: theory, algorithms, and applications. university of california, berkeley, 2014. int. j. anal. appl. 16 (5) (2018) 688 [5] fan, michael kh and tits, andré l and doyle, john c. robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. ieee trans. autom. control, 36(1991): 25-38 [6] hinrichsen, d and pritchard, aj. mathematical systems theory i, vol. 48 of texts in applied mathematics. springerverlag, berlin volume 48(2005). [7] karow, michael and kokiopoulou, effrosyni and kressner, daniel. on the computation of structured singular values and pseudospectra. syst. control lett. 59(2010): 122-129. [8] karow, michael and kressner, daniel and tisseur, françoise. structured eigenvalue condition numbers. siam j. matrix anal. appl. 28(2006): 1052-1068. [9] packard, andrew and doyle, john. the complex structured singular value. automatica 29(1993): 71-109. [10] packard, andy and fan, michael kh and doyle, john. a power method for the structured singular value. decision and control, 1988., proceedings of the 27th ieee conference on, 2132-2137, (1998). [11] qiu, li and bernhardsson, bo and rantzer, anders and davison, ej and young, pm and doyle, jc. a formula for computation of the real stability radius. automatica, 31(1995), 879-890. [12] rehman, mutti-ur and tabassum, shabana numerical computation of structured singular values for companion matrices. j. appl. math. phys., 5(2017), 1057-1072. [13] doyle, john analysis of feedback systems with structured uncertainties. ieee proc. d-control theory appl., 129(1982), 242–250. [14] ferreres, gilles a practical approach to robustness analysis with aeronautical applications. springer science & business media, (1999) [15] fu, minyue the real structured singular value is hardly approximable. ieee trans. autom. control, 42(1997), 1286 1288. [16] newlin, matthew p and glavaski, sonja t advances in the computation of the/spl mu/lower bound. proc. 1995 amer. control conf., (1995). 1. introduction 2. basic framework 2.1. the -value based on a structured spectral value set 2.2. mathematical problem 3. purely complex uncertainties 3.1. system of odes to compute extremal points of b*(m) 3.2. the local optimization problem 3.3. the gradient system of ordinary differential equations 3.4. choice of initial value matrix and 0 4. numerical experimentation 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 93-99 http://www.etamaths.com geometry of a class of generalized cubic polynomials christopher frayer abstract. this paper studies a class of generalized complex cubic polynomials of the form p(z) = (z − 1)(z − r1)k(z − r2)k where r1 and r2 lie on the unit circle and k is a natural number. we completely characterize where the nontrivial critical points of p can lie, and to what extent they determine the polynomial. the main results include (1) a nontrivial critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a ‘desert’ in the unit disk in which critical points cannot occur. several recent papers ([1], [2], [3]) have studied the geometry of cubic polynomials, specifically asking, how the critical points of a cubic polynomial depend upon its roots. frayer, kwon, schafhauser, and swenson [1] studied the critical points of a family of polynomials γ = {q : c → c |q(z) = (z − 1)(z −r1)(z −r2), |r1| = |r2| = 1} . for p ∈ γ the main results of [1] include: • a critical point almost always determines p uniquely. • there is a desert in the unit disk, the open disk {z ∈ c : |z − 2 3 | < 1 3 }, in which critical points of p cannot occur. • if 0 < |g−1 3 | ≤ 2 3 , then there is a unique p ∈ γ with p′′(g) = 0. additionally, if |g − 1 3 | > 2 3 , there is no p ∈ γ with p′′(g) = 0. we will extend the results of [1] to a class of generalized cubic polynomials γk = { q : c → c |q(z) = (z − 1)(z −r1)k(z −r2)k, |r1| = |r2| = 1, k ∈ n } . a polynomial of the form p(z) = (z − 1)(z −r1)k(z −r2)k has 2k critical points; k − 1 critical points at r1 and r2 respectively, and two nontrivial critical points. differentiation gives p′(z) = (z−r1)k−1(z−r2)k−1 [ (2k + 1)z2 − (2k + (k + 1)(r1 + r2))z + k(r1 + r2) + r1r2 ] so that the two nontrivial critical points of p are the roots of q(z) = (2k + 1)z2 − (2k + (k + 1)(r1 + r2))z + k(r1 + r2) + r1r2.(1) 2010 mathematics subject classification. 14g22. key words and phrases. geometry; generalized cubic polynomials. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 93 94 frayer this paper will characterize where the nontrivial critical points of p ∈ γk lie, and to what extent they determine p. preliminary information circles which are internally tangent to the unit circle at 1 will play an important role in what follows. given α > 0, denote by tα the circle of diameter α passing through 1 and 1 −α in the complex plane. that is, tα = { z ∈ c : ∣∣∣z −(1 − α 2 )∣∣∣ = α 2 } . for example, t2 is the unit circle (a circle of diameter 2 centered at the origin). a key result of [1] will be used to establish a geometric relationship between the critical points of a polynomial in γk. theorem 1 ([1]). let f(z) = (z−1)(z−r1) · · ·(z−rn), where |zk| = 1 for each k. let c1, c2, . . ., cn denote the critical points of f(z), and suppose that 1 6= ck ∈ tαk for each k. then n∑ k=1 1 αk = n.(2) a general result related to the geometry of complex polynomials is the gauss-lucas theorem. theorem 2 (gauss-lucas theorem). let p be a complex-valued polynomial. the critical points of p are located in the convex hull of its roots. an additional fact of interest is related to fractional linear transformations. theorem 3 ([4]). a fractional linear transformation t sends the unit circle to the unit circle if and only if t(z) = ᾱz+β̄ βz+α for some α,β ∈ c. critical points we begin by analyzing a few special cases for future reference. example 1. suppose p ∈ γk has nontrivial critical point c = 1. this occurs if and only if z = 1 is a repeated root of p. that is, r1 and/or r2 must be 1. hence, p(z) = (z−1)k+1(z−r)k for some r ∈ t2. conversely, given p(z) = (z−1)k+1(z−r)k for some r ∈ t2, differentiation yields p′(z) = (2k + 1)(z − 1)k−1(z −r)k−1 [ (z − 1) ( z − k 2k + 1 − (k + 1) 2k + 1 r )] . therefore, p ∈ γk has a nontrivial critical point at z = 1 if and only if p(z) = (z − 1)k+1(z − r)k with r ∈ t2. in this case, the other nontrivial critical point is k 2k+1 + (k+1) 2k+1 r ∈ t2k+2 2k+1 . now that we know which polynomials in γk have nontrivial critical point c = 1, we may assume that c 6= 1 throughout the remainder of the paper. geometry of a class of generalized cubic polynomials 95 example 2. suppose p ∈ γk has nontrivial critical point 1 6= c ∈ t2. this occurs if and only if z = c is a repeated root of p with multiplicity greater than k. that is, r1 = r2 = c so that p(z) = (z−1)(z−c)2k. conversely, given p(z) = (z−1)(z−c)2k, differentiation yields p′(z) = (2k + 1)(z − c)2k−2 [ (z − c) ( z − 2k 2k + 1 − 1 2k + 1 c )] . therefore, p ∈ γk has nontrivial critical point c 6= 1 on t2 if and only if p(z) = (z− 1)(z−c)2k. in this case, the other nontrivial critical point is 2k 2k+1 + 1 2k+1 c ∈ t 2 2k+1 . let’s now determine where the nontrivial critical points of p ∈ γk lie. the gauss-lucas theorem guarantees that the nontrivial critical points will lie within the unit disk. but we can say more; there is a desert in the unit disk, the open disk {z | z ∈ tα with 0 < α < 22k+1}, in which nontrivial critical points of p cannot occur. theorem 4. no polynomial p ∈ γk has a nontrivial critical point strictly inside t 2 2k+1 . proof. let c1 6= 1 and c2 6= 1 be nontrivial critical points of p(z) = (z − 1)(z − r1) k(z − r2)k with c1 ∈ tα and c2 ∈ tβ. as the 2k − 2 trivial critical points lie on t2, theorem 1 gives (2k − 2) ( 1 2 ) + 1 α + 1 β = 2k which simplifies to 1 α + 1 β = k + 1.(3) suppose to the contrary that α < 2 2k+1 . then 1 β = k + 1 − 1 α < k + 1 − 2k + 1 2 = 1 2 . but then β > 2 which violates theorem 2. � theorem 5. let c1 6= 1 and c2 6= 1 be nontrivial critical points of p ∈ γk with c1 ∈ tα and c2 ∈ tβ. if c1 lies on t 2 k+1 so does c2. otherwise, c1 and c2 lie on opposite sides of t 2 k+1 . proof. let c1 6= 1 and c2 6= 1 be nontrivial critical points of p ∈ γk with c1 ∈ tα and c2 ∈ tβ. then, from equation (3), 1α + 1 β = k + 1. therefore, α = 2 k+1 if and 96 frayer only if β = 2 k+1 . additionally, if α < 2 k+1 , then 1 β = k + 1 − 1 α < k + 1 − k + 1 2 = k + 1 2 and β > 2 k+1 . � now that we know where the nontrivial critical points lie, let’s investigate to what extent they determine the polynomial. given p ∈ γk with roots at 1, r1 and r2, and a nontrivial critical point c, we have 0 = q′(c) = (2k + 1)c2 − (2k + (k + 1)(r1 + r2))c + k(r1 + r2) + r1r2. direct calculations give r2 = (k − c(k + 1))r1 + (2k + 1)c2 − 2k −r1 + c(k + 1) −k . definition 1. given c ∈ c, define fc(z) = (k − c(k + 1))z + (2k + 1)c2 − 2k −z + c(k + 1) −k and let sc denote the image of the unit circle under fc. that is, fc(t2) = sc and fc(r1) = r2. theorem 6. polynomial p(z) = (z−1)(z−r1)k(z−r2)k ∈ γk has nontrivial critical c 6= 1 if and only if fc(r1) = r2. as fractional linear transformations send circles and lines to circles and lines, sc will be a circle when c /∈ t 2 k+1 . to see this, note that sc is a line when |c(k + 1) −k| = 1 ←→ ∣∣∣∣c− ( 1 − 1 k + 1 )∣∣∣∣ = 1k + 1 which is equivalent to c ∈ t 2 k+1 . we have established the following theorem (see theorem 8). let’s investigate a special case. example 3. suppose 1 6= c ∈ t2. using the fact that fc(c) = c, fc(1) = (2k + 1)c−k k + 1 , and fc(−1) = c2(2k + 1) + c(1 −k) −k c(k + 1) + (1 −k) direct calculations give ∣∣∣∣fc(z) − ( 2k + 1 k + 1 ) c ∣∣∣∣ = kk + 1 for z ∈{c,±1}. therefore, for 1 6= c ∈ t2, sc is a circle with radius kk+1 and center( 2k+1 k+1 ) c, which is externally tangent to t2 at c. geometry of a class of generalized cubic polynomials 97 when 1 6= c ∈ t2, it follows from example 2 that the other critical point of p lies on the boundary of the desert at c2 = 2k 2k+1 + 1 2k+1 c. similar calculations show that sc2 is a circle with radius k k+1 and center ( 1 k+1 ) c, which is internally tangent to t2 at c. when c = 1, fc(z) = −z+1 −z+1 = 1 and (fc) −1 does not exist. if c 6= 1, then (fc) −1 = fc so that fc(r2) = r1. hence, fc restricts to a one-to-one correspondence from sc∩t2 to itself, and if c is a nontrivial critical point of p, then {r1,r2}⊆ sc∩t2. this observation allows us to classify the polynomials in γk which have a critical point at 1 6= c in the unit disk! we simply need to study the intersection of circles t2 and sc. theorem 7. if c /∈ {1,− 1 2k+1 } lies on tα for some α ∈ [ 2 2k+1 , 2 ] , then there is a unique p ∈ γk with nontrivial critical point at c. proof. let c ∈ c. in order to determine if there is a polynomial p ∈ γk with critical point at c we must study the intersection of sc and t2. as sc and t2 are circles, their intersection is disjoint, contains one point, contains two points, or is all of t2. if sc∩t2 = ∅, then there is no polynomial in γk with a nontrivial critical point at c. at a minimum, this occurs when c ∈ tα with α > 2 (theorem 2) and α < 22k+1 (theorem 4). if sc ∩ t2 = {r}, then fc(r) = r and by theorem 6, r is a nontrivial critical point of p(z) = (z − 1)(z − r)2k. conversely, as illustrated in example 3, if p(z) = (z − 1)(z −r)2k, then sc ∩t2 = {r}. if sc∩t2 = {r,s} with r 6= s, there are two possibilities: fc(r) = r and fc(s) = s, or fc(r) = s and fc(s) = r. we will rule out the first possibility. if fc(r) = r and fc(s) = s, then by theorem 6, c is a nontrivial critical point of p(z) = (z−1)(z−r)2k and p(z) = (z − 1)(z −s)2k. by the gauss-lucas theorem, c lies on line segments 1r and 1s. a contradiction. therefore, fc(r) = s and fc(s) = r, and it follows by theorem 6 that p(z) = (z − 1)(z −r)k(z −s)k is the only polynomial in γk with a nontrivial critical point at c. if sc ∩t2 = t2, then fc(t2) = t2. as fc(z) = (k − c(k + 1))z + (2k + 1)c2 − 2k −z + c(k + 1) −k = − (k − c(k + 1))z + (2k + 1)c2 − 2k z + k − c(k + 1) , according to theorem 3, fc(t2) = t2 exactly when k−c(k + 1) = k − c(k + 1) and (2k + 1)c2 − 2k = 1. the first equation implies c ∈ r, and the second equation simplifies to ((2k + 1)c + 1)(c− 1) = 0. since c 6= 1, sc ∩t2 = t2 precisely when c = − 12k+1 . therefore, c = − 1 2k+1 is the nontrivial critical point of p ∈ γk if and only if p(z) = (z−1)(z−r)k ( z −f− 1 2k+1 (r) )k for r ∈ t2. in order to establish uniqueness, we need to show that if c 6= − 1 2k+1 lies on tα with α ∈ ( 2 2k+1 , 2), then |sc ∩ t2| = 2. this claim follows from a simple ‘root 98 frayer dragging’-type argument. without loss of generality, suppose that sc ∩t2 = ∅ and sc lies inside t2. as we ‘drag’ c to t2 along a line segment going away from the origin, sc is continuously transformed into a circle externally tangent to t2. the intermediate value theorem implies that there exists a c0 on the line segment with sc0 internally tangent to t2. as c never crosses t 2 2k+1 , this is a contradiction. � now that we have proven uniqueness, let’s revisit theorem 5. theorem 8. suppose c1 and c2 are nontrivial critical points of p ∈ γk. if 1 6= c1 ∈ t 2 k+1 , then c2 = c̄1. stated differently, if c ∈ t 2 k+1 , then sc is a vertical line passing through fc(1) = (2k+1)c−k k+1 . we use this fact, along with uniqueness, to provide a proof. proof. let c = x + iy ∈ t 2 k+1 . suppose r = eiθ with cos(θ) = ( 2k+1 k+1 x− k k+1 ) and q(z) = (z − 1)(z −r)k(z − r̄)k ∈ γk. then q′(z) = (z −r)k−1(z − r̄)k−1 [ (2k + 1)z2 − ((k + 1)(r + r̄) + 2k) z + k(r + r̄) + rr̄ ] and q has nontrivial critical points when (2k + 1)z2 − ((k + 1)2 cos(θ) + 2k) z + 2k cos(θ) + 1 = 0. using cos(θ) = ( 2k+1 k+1 x− k k+1 ) yields (2k + 1)z2 − (2(2k + 1)x)z + 2k(2k + 1) k + 1 x− (2k + 1)(k − 1) k + 1 = 0 (2k + 1) [ z2 − 2xz + 2k k + 1 x− k − 1 k + 1 ] = 0 z2 − (c + c̄)z + cc̄ = 0 (z − c)(z − c̄) = 0 and q ∈ γk has nontrivial critical points at c and c̄. therefore, by uniqueness, if p ∈ γk has nontrivial critical point at 1 6= c1 ∈ t 2 k+1 , then c2 = c̄1. � centers given p(z) = (z − 1)(z − r1)k(z − r2)k ∈ γk, we saw in equation (1) that the nontrivial critical points are the solutions of q(z) = (2k + 1)z2 − (2k + (k + 1)(r1 + r2))z + k(r1 + r2) + r1r2. we define g ∈ c to be the center of p(z) if q′(g) = 0. since q has degree 2, every p ∈ γk has the unique center g = k 2k + 1 + k + 1 2k + 1 ( r1 + r2 2 ) . as in [1] we will use a geometric construction to show exactly where the center can lie. geometry of a class of generalized cubic polynomials 99 theorem 9. let g ∈ c • p ∈ γk has center k2k+1 if and only if p(z) = (z − 1)(z −r1) k(z −r2)k with r2 = −r1. • if 0 < |g − k 2k+1 | ≤ k+1 2k+1 , then there is a unique polynomial in γk with center g. • if |g − k 2k+1 | > k+1 2k+1 , then there is no polynomial in γk with center g. proof. suppose g is the center of p ∈ γk. by the gasuss-lucas theorem, g is contained in 4r1r21, where r1 and r2 are points to be constructed on t2. even though we do not know r1 and r2, their midpoint, w, lies in the unit disk with g = k 2k+1 + k+1 2k+1 w. therefore |g − k 2k+1 | ≤ k+1 2k+1 . if 0 < |g− k 2k+1 | ≤ k+1 2k+1 , then g 6= k 2k+1 and w 6= 0. as r1r2 is a chord of t2, its perpendicular bisector passes through w and the origin o. since w lies in the unit disk, the line through w perpendicular to ow intersects t2 in two places, r1 and r2. if g = k 2k+1 , then w = 0 is the midpoint of r1r2 and it follows that r2 = −r1. � this proof completes the extension of [1] to the class of generalized cubics γk. this paper completely characterizes where the critical points and centers of a p ∈ γk can lie and to what extent they determine a polynomial in γk. references [1] christopher frayer, myeon kwon, christopher schafahuser, and james a. swenson, the geometry of cubic polynomials, math. magazine 87 (2014), no. 2, 113–124. [2] dan kalman, an elementary proof of marden’s theorem, amer. math. monthly 115 (2008), no. 4, 330–338. [3] sam northshield, geometry of cubic polynomials, math. magazine 86 (april 2013), 136–143. [4] e.b saff and a.d snider, fundamentals of com,plex analysis for mathematics, science, and engineering, prentice-hall, anglewood cliffs, new jersey, 1993. university of wisconsin-platteville, united states international journal of analysis and applications issn 2291-8639 volume 6, number 2 (2014), 205-219 http://www.etamaths.com topological vector-space valued cone banach spaces nayyar mehmood1,∗, akbar azam1 and suzana aleksić2 abstract. in this paper we introduce the notion of tvs-cone normed spaces, discuss related topological concepts and characterize the tvs-cone norm in various directions. we construct generalize locally convex tvs generated by a family of tvs-cone seminorms. the class of weak contractions properly includes large classes of highly applicable contractions like banach, kannan, chatterjea and quasi etc. we prove fixed point results in tvs-cone banach spaces for nonexpansive self mappings and self/non-self weak contractive mappings. we discuss the necessary conditions for t -stability of picard iteration. to ensure the novelty of our work we establish an application in homotopy theory without the assumption of normality on cone and many non-trivial examples. 1. introduction recently beg et al. [1] introduced and studied topological vector space-valued cone metric spaces (tvs-cone metric spaces), which generalized the cone metric spaces [2]. many generalizations and extensions have been made by many researchers, (see [3-6] ). for more details about topological vector spaces we refer to [7, 8]. actually the idea of cone metric space was properly introduced by huang and zhang in [2]. in their setting the set of real numbers was replaced by an ordered banach space and a vector valued metric was defined on a nonempty set. many authors [9-14] studied the properties of cone metric spaces and generalized important fixed point results of complete metric spaces. the concept of cone metric space in the sense of huang-zhang was characterized by al-rawashdeh et al. in [15]. in [16], the author introduced the notion of cone banach spaces with normal cones and proved some results regarding fixed points by using nonexpansive mappings. later on many authors investigated some useful results in fixed and coupled fixed points, (see [17-19] ). weak contractions were considered in [20], to study the fixed point results for self mappings. it has been shown that the banach, kannan, chatterjea, zamfirescue, quasi and many other contractions are weak contractions. the importance of non-self mappings is obvious. in fact fixed point theorems for non-self mappings generalized all the corresponding results presented for self-mappings. a variety of results on nonself mappings and weak contractions can be found in [21-27]. in this article, we introduce tvs-cone banach space and investigate some properties without assumption of normality on cones. we generalize the results of [16] and 2010 mathematics subject classification. 47h10; 54h25. key words and phrases. tvs-cone banach space; non-normal cones; weak contraction; nonself mappings; iteration processes; t-stability; fixed points; homotopy; generalized locally convex tvs. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 205 206 mehmood, azam and aleksić explore some characteristics of norms in cone normed space. we prove fixed point results for picard, mann, ishikawa and krasnoseskij iterations, we also present results for weak contractive non-self mappings. many examples have been given and a homotopy result is established for nonexpansive mappings. we discuss the necessary conditions for t-stability of picard iteration. 2. preliminaries let e be a topological vector space with its zero vector θ. a nonempty subset k of e is called a convex cone if k + k ⊆ k and λk ⊆ k for λ ≥ 0. a convex cone k is said to be pointed (or proper) if k ∩ (−k) = {θ}, and k is normal (or saturated) if e has a base of neighborhoods of zero consisting of order-convex subsets. for a given cone k ⊆ e we define a partial ordering 4 with respect to k by x 4 y if and only if y −x ∈ k, x ≺ y stands for x 4 y and x 6= y, while x � y stands for y−x ∈ int k, where int k denotes the interior of k. the cone k is said to be solid if it has a nonempty interior. definition 1. let v be a vector space over r. a vector-valued function ‖·‖k : v → e; x → v is called a tvs-cone norm on x if the following conditions are satisfied: (n1) ‖x‖k < θ for all x ∈ v , (n2) ‖x‖k = θ if and only if x = θ, (n3) ‖x + y‖k 4 ‖x‖k + ‖y‖k for all x,y ∈ v , (n4) ‖kx‖k = |k|‖x‖k for all k ∈ r. the pair (x,‖·‖k) is called a tvs-cone norm space (in brief tvs-cns). definition 2. let (v,‖·‖k) be a tvs-cone norm space and {xn} a sequence in v . (i) {xn} tvs-cone converges to x ∈ v if for every c ∈ e with θ � c there exists n0 ∈ n such that ‖xn −x‖k � c for all n ≥ n0. (ii) {xn} is a tvs-cone cauchy sequence if for every c ∈ e with θ � c there exists n0 ∈ n such that ‖xn −xm‖k � c for all n,m ≥ n0. (iii) (v,‖·‖k) is a tvs-cone complete or a tvs-cone banach space if every tvs-cone cauchy sequence in v is tvs-cone convergent. using the consequences of lemma 2.4 from [28], we have the following properties. lemma 3. let (e,k) be a locally convex tvs. the following properties hold. (a) for a sequence {vn} in e with θ 4 vn → θ, let θ � c then there exists positive integer n0 such that vn � c for each n > n0. (b) there exists a sequence {vn} in e such that for some positive integer n0 holds θ 4 vn � c for all n > n0, but vn 6→ θ. (c) if there exists v in e such that θ 4 v � c for all c ∈ int k, then v = θ. (d) if a 4 λa, where a ∈ k and 0 ≤ λ < 1, then a = θ. remark 4. for a banach space e with non-normal cone k, with norm ‖·‖ . the following may hold. (a) for sequences {vn}, {un} in e with norm ‖·‖ , it may happen that vn → v, un → u, but ‖vn −un‖ 6→‖v −u‖ (see example 5). in particular, vn → v, n →∞, may imply that ‖vn −v‖ 6→ θ, n →∞ (this is impossible in cns defined in [16] if the cone is normal). (b) if vn → v and vn → u, then v = u. topological vector-space valued cone banach spaces 207 example 5. let v = r and let e be the set of all real-valued functions on v which also have continuous derivatives on v. then e is a vector space over r under the following operations: (x + y) (t) = x (t) + y (t) , (αx) (t) = αx (t) for all x,y ∈ e,α ∈ r. then e with norm ‖x‖ = ‖x‖∞ + ‖x′‖∞, has non-normal solid cone, see [5, 8]: k = {x ∈ e : θ 4 x}, where θ(t) = 0 for all t ∈ x. consider the sequences xn(t) = 1 + sin nt n + 2 , yn(t) = 1 − sin nt n + 2 , n ≥ 0. in e. we have xn → θ, yn → θ, n → +∞, but ‖xn −yn‖ = ∥∥∥∥2 sin ntn + 2 ∥∥∥∥ = sup t∈v { 2 sin nt n + 2 } + sup t∈v { 2n cos nt n + 2 } = 2 sin n n + 2 + 1 6→ θ, n → +∞. also as xn → θ, consider ‖xn −θ‖ = ‖xn‖ = 1 6→ θ. definition 6 ([1]). let x be a nonempty set and (e,k) a tvs. a vector-valued function d : x ×x → e is said to be a tvs-cone metric if the following conditions are satisfied: (c1) θ 4 d(x,y) for all x,y ∈ x and d(x,y) = θ if and only if x = y, (c2) d(x,y) = d(y,x) for all x,y ∈ x, (c3) d(x,z) 4 d(x,y) + d(y,z) for all x,y,z ∈ x. the pair (x,d) is called a tvs-cone metric space. note that each tvs-cns is a tvs-cone metric space with induced tvs-cone metric d : x ×x → e defined by d(x,y) = ‖x−y‖ for all x,y ∈ x. remark 7 ([1]). the concept of cone metric spaces is more general than that of metric spaces, because each metric space is a cone metric space, and a cone metric space in the sense of huang and zhang is a special case of tvs-cone metric spaces when (x,d) is a cone metric space with respect to a normal cone k. if k is a normal cone, then a tvs-cns (v,‖ · ‖k) becomes a cns [16] and with the induced tvs-cone metric [1] this space becomes cone metric space in the sense of [2]. cns in the case of [16] gives us generalized induced norm known as b-norm ‖·‖b : v → r defined by ‖·‖b = ‖‖·‖k‖. the triangular property of cone norm ‖x + y‖k 4 ‖x‖k + ‖y‖k , gives us the following property of b-norm, ‖x + y‖b ≤ k(‖x‖b + ‖y‖b), where k is a constant of normality. 208 mehmood, azam and aleksić obviously every norm is a b-norm, but the contrary is not true, consider the following example example 8. let x = r and ‖·‖b : x → r defined by ‖x‖b = |x|3. for x,y ∈ x we have |x + y|3 ≤ (|x|+ |y|)3 ≤ 23(|x|3 + |y|3), but |x + y|3 6≤ (|x|3 + |y|3). therefore, ‖x‖b is a b-norm, but it is not a norm on x. let us recall the following definitions. definition 9 ([5, 29]). let x be a nonempty set. a vector-valued function d : x ×x → e is said to be cone symmetric if the following conditions are satisfied: (c1) θ 4 d(x,y) for all x,y ∈ x and d(x,y) = θ if and only if x = y; (c2) d(x,y) = d(y,x) for all x,y ∈ x. the pair (x,d) is called a cone symmetric space. it is clear that the cone symmetric space may not be a cone metric space (see example 2.2 in [29]). for a given cone symmetric space (x,d) one can deduce (see [29]) a symmetric metric space with d : x×x → r defined by d(x,y) = ‖d(x,y)‖ for all x,y ∈ x. for a cone metric space (x,d) with normal cone k with normal constant k ≥ 1, we have d(x,y) = ‖d(x,y)‖≤ k‖d(x,z) + d(z,y)‖≤ k(d(x,z) + d(z,y)). in this case, the metric d becomes b-metric and, hence, the concept of b-metric spaces is more general then that of metric spaces and the topology τd generated by d coincides with τb generated by b-metric on x. in the following we explore the concept of tvs-cone seminorm. definition 10. let v be a vector space over scalars f . if a mapping ρk : x → (e,k) satisfies: (sn1) ρk(x) < θ for all x ∈ v , (sn2) ρk(x + y) 4 ρk(x) + ρk(y) for all x,y ∈ v , (sn3) ρk(kx) = |k|ρk(x) for all x ∈ v , k ∈ f . then ρk is called a tvs-cone seminorm on x. note that a tvs-cone seminorm is a norm if ρk(x) = θ implies x = θ. a tvs-cone seminorm on x induces a pseudo tvs-cone metric defined by dp(x,y) = ρk(x−y) which satisfies: (pc1) θ 4 dp(x,y) for all x,y ∈ x, (pc2) dp(x,y) = dp(y,x) for all x,y ∈ x, (pc3) dp(x,z) 4 dp(x,y) + dp(y,z) for all x,y,z ∈ x. note that dp(x,y) = θ does not imply x 6= y. the class of tvs-cone pseudo metric spaces is larger than the class of tvs-cone metric spaces. equivalently, ρk is a tvs-cone seminorm on a vector space v if the following conditions are satisfied: (sn(i)) ρk(v + v) 4 ρk(u) + ρk(v) for all u,v ∈ v, (sn(ii)) ρk(kv) = |k|ρk(v) for all v ∈ v , k ∈ f. this cone seminorm gives us generalized seminorm, so called b-seminorm ‖·‖bs : x → r defined by ‖x‖bs = ‖ρk(x)‖ . topological vector-space valued cone banach spaces 209 using (sn(i)), b-seminorm has the following property ‖x + y‖bs ≤ k(‖x‖bs + ‖y‖bs). note that b-seminorm is a seminorm if k = 1 and every seminorm is a b-seminorm. the next example shows that the contrary is not true, i.e., b-seminorm does not need to be seminorm. example 11. let x = r and ‖ · ‖bs : x → r is defined by ‖x‖bs = |x|3 + 1. for x,y ∈ x, we have |x + y|3 + 1 ≤ (|x| + |y|)3 + 1 ≤ 23(|x|3 + |y|3) + 1 ≤ 23(|x|3 + |y|3) + 16 = 23(|x|3 + 1 + |y|3 + 1), which implies ‖x+y‖bs ≤ 23(‖x‖bs+‖y‖bs). this shows that ‖x‖bs is a b-seminorm, but not a seminorm on x. 3. main results let {ρki : i ∈ i} be a family of tvs-cone seminorms on a vector space v . for θ � ε and i ∈ i = {1, 2, 3, . . . ,n}, define u(u0,ρk1,ρk2,ρk3,...,ρkn,ε) = u(u0,ρkn,ε) = {u ∈ v : ρki(u−u0) � ε, i ∈ i} . note that u(u0,ρkn,ε) = u0 + u(θ,ρkn,ε). lemma 12. the set u(θ,ρkn,ε) is balanced and convex in v. proof. for any w ∈ u(θ,ρkn,ε) and |k| ≤ 1, we have ρki(kw) 4 |k|ρki(w) � ε, i ∈ i. thus u(θ,ρkn,ε) is absorbing. now, for 0 ≤ t ≤ 1 and u,v ∈ u(θ,ρkn,ε), we obtain ρki(tu + (1 − t)v) 4 tρki(u) + (1 − t)ρki(v) � tε + (1 − t)ε = ε, which implies tρki(u) + (1−t)ρki(v) ∈u(θ,ρkn,ε). therefore, u(θ,ρkn,ε) is convex. lemma 13. let {ρki : i ∈ i} be a family of tvs-cone seminorms on a vector space (v,f). for each v ∈ v denote with nv the collection of sets of the form u(v,ρkn,ε) = {u ∈ v : ρki(u−v) � ε,i ∈ i}. let t be the collection of ∅ and all subsets g of x such that for each u ∈ g there exists some u ∈ nv such that u ⊆ g. then t is topology on v and preserves the structure of vector space. the sets nv form an open locally convex neighborhood base at x. the topological space (v,t ) is hausdorff iff the family {ρki : i ∈ i} of tvs-cone seminorms is separating, i.e. for θ 6= u ∈ v there exists some i0 ∈ i such that ρki0 (u) 6= θ. proof. it is clear that v and the union of any number of elements of t belong to t . we will show that a,b ∈t implies a∩b ∈t . the case a∩b = ∅ is obvious. suppose that a∩b 6= ∅ and v ∈ a∩b. by definition of t there exist u1, u2 ∈t such that u1 ⊆ a and u2 ⊆ b. let for comparable ε,δ ∈ int k, we define u1 := u(v,ρkn,ε) = {u ∈ v : ρki(u−v) � ε, 1 ≤ i ≤ n} and u2 = u(v,µkm,δ) = { u ∈ v : µkj(u−v) � δ, 1 ≤ j ≤ m } . if we set u3 = u(v,ρk1,ρk2,ρk3,...,ρkn,µk1,µk2,µk3,...,µkm,γ), 210 mehmood, azam and aleksić where γ = ε if δ − ε ∈ int k and γ = δ if ε − δ ∈ int k, then u3 ∈ nv and u3 ⊆ u1 ∩ u2 ⊆ a ∩ b. hence t is topology on v . let u(v,ρkn,ε) ∈ nv and w ∈ u(v,ρkn,ε). then ρki(w − v) � ε, 1 ≤ i ≤ n. now choose θ � δ such that δ � ε − ρki(w − v) for 1 ≤ i ≤ n. for any 1 ≤ i ≤ n and u ∈ v satisfying ρki(w −u) � δ we have ρki(u−v) 4 ρki(u−w) + ρki(w −v) � δ + ρki(w −v) � ε. we see that u(u,ρkn,δ) ⊆ u(v,ρkn,ε), hence u(v,ρkn,ε) is open. lemma 12 implies that the elements of nv are convex. therefore, nv is an open locally convex neighborhood base at v consisting of the open sets u(v,ρkn,ε). now we will show that the topology t is compatible. let u,v ∈ v and let u(u+v,ρkn,ε) be a basic neighborhood of u + v. let (un,vn) → (u,v) in v × v . then there exists an integer n0 such that (un,vn) ∈ u(u,ρkn,ε2 ) ×u(v,ρkn,ε2 ) for all n ≥ n0. for 1 ≤ i ≤ n and for all n ≥ n0, we have ρki(u + v − (un + vn)) 4 ρki(u−un) + ρki(v −vn) � ε, which gives un + vn ∈ u(u+v,ρkn,ε), and, therefore, un + vn → u + v. now let (kn,vn) → (k,v) in f ×v. let u(kv,ρkn,δ) be a basic neighborhood of kv. choose t > 0 and θ � γ such that for 1 ≤ i ≤ n there exists an integer m0 such that (kn,vn) ∈ {ζ ∈ f : |ζ −k| < t}×u(v,ρkn,γ) for all n ≥ m0, with tρki(v) � δ 2 and (|k| + t)γ � δ 2 . for n ≥ m0 we have ρki(kv −knvn) 4 ρki(kv −knv) + ρki(knv −knvn) 4 |k −kn|ρki(v) + |kn|ρki(v −vn) 4 |t|ρki(v) + |kn|ρki(v −vn) � |t|ρki(v) + (|k| + t)γ � δ 2 + δ 2 = δ, thus knvn ∈u(kv,ρkn,δ). therefore (v,t ) is a tvs. now, suppose that the family p = {ρki : i ∈ i} of tvs-cone seminorms is separating. for any u,v ∈ v with u 6= v there exists some j0 ∈ i such that θ � δ = ρkj0 (v − u). thus, the open sets u(u,ρkj0,δ2 ) and u(v,ρkj0,δ2 ) are disjoint containing u and v and so the space (v,t ) is hausdorff. we conclude that the space (v,t ) is locally convex tvs. definition 14. [20] let x be a tvs-cone normed space and t : x → x an operator. (i) t is an almost weak contraction if for all x,y ∈ e, l ≥ 0 and δ ∈ (0, 1), we have (w1) ‖tu−tv‖k 4 δ · ‖u−v‖k + l · ‖u−tu‖k , ∀u,v ∈ x. (ii) t is a weak contraction if (w2) ‖tu−tv‖k 4 δ · ‖u−v‖k + l · ‖v −tu‖k . definition 15. let x be a tvs-cone normed space and t : x → x an operator. (i) t is a zamfirescue contraction if for all u,v ∈ x and a ∈ [0, 1), b,c ∈ [0, 1 2 ), one of the following conditions is satisfied topological vector-space valued cone banach spaces 211 (z1) ‖tu−tv‖k 4 a · ‖u−v‖k , (z2) ‖tu−tv‖k 4 b (‖u−tu‖k + ‖v −tv‖k) , (z3) ‖tu−tv‖k 4 c (‖u−tv‖k + ‖v −tu‖k) . (ii) t is a quasi contraction if for all u,v ∈ x and α ∈ [0, 1), holds ‖tu−tv‖k 4 c ·m, where m ∈{‖u−v‖k ,‖u−tu‖k ,‖v −tv‖k ,‖u−tv‖k ,‖v −tu‖k} . remark 16. [20](a) every zamfirescue contraction is a weak contraction. (b) every quasi contraction is a weak contraction. definition 17. [19] let x be a tvs-cone normed space, t : x → x an operator and u0 ∈ x. a sequence {un} is called: 1) picard iteration if (p1) un+1 = tun; 2) mann iteration if (m1) un+1 = (1 −αn) un + αntun; 3) ishikawa iteration if un+1 = (1 −αn) un + αntvn, (i1) vn = (1 −βn) xn + βntxn, where {αn}⊆ (0, 1) and {βn}⊆ [0, 1). 4) krasnoselskij iteration if un+1 = (1 −λ) un + λtun, where λ ∈ (0, 1). denote with f(t) the set of all fixed points of t. lemma 18. let x be a tvs-cone banach space and {an} and {bn} be sequences in e satisfying an+1 4 λan + bn, where λ ∈ (0, 1) and bn → θ as n → ∞. then lim n→∞ an = θ. proof. on the contrary, suppose that lim n→∞ an 6= θ and lim n→∞ an = c, for some θ � c. then, by lemma 3 (d), we have an = θ. in the following theorem we obtain a fixed point result for nonself weak contractions in a tvs-cone banach space. theorem 19. let x be a tvs cone banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is a weak contraction (satisfying (w2)), such that δ(1 + l) < 1. if t(∂c) ⊆ c, then t has a fixed point. 212 mehmood, azam and aleksić proof. we construct two sequences {un} and {vn} in the following way. let us choose u0 arbitrary in x and set v1 = tu0. if v1 ∈ c, then set u1 = v1. if not, then there exists u1 ∈ ∂c such that ‖u1 −u0‖k + ‖u1 −v1‖k = ‖u0 −v1‖k . thus u1 ∈ c and let v2 = tu1. we have ‖v2 −v1‖k = ‖tu0 −tu1‖k 4 δ · ‖u1 −u0‖k + l · ‖u1 −tu0‖ . if v2 ∈ c, set u2 = v2. otherwise, there exists u2 ∈ ∂c such that ‖u2 −u1‖k + ‖v2 −u2‖k = ‖v2 −u1‖k . thus u2 ∈ c. let v3 = tu2 and consider ‖v2 −v3‖k = ‖tu1 −tu2‖k 4 δ · ‖u2 −u1‖k + l · ‖u2 −tu1‖k . continuing in the same way, we construct the sequences {un} and {vn} such that (i) vn+1 = tun, (ii) ‖vn −vn+1‖k 4 δ · ‖un−1 −un‖k + l · ‖un −tun−1‖k, where (iii) vn ∈ c implies vn = un. (iv) if vn 6∈ c, then vn 6= un, and then un ∈ ∂c is such that ‖un−1 −un‖k + ‖vn −un‖k = ‖vn −un−1‖k . we will show that {un} is a cauchy sequence. define p = {ui ∈{un} : ui = vi} , q = {ui ∈{un} : ui 6= vi} . it is obvious that if un ∈ q, then un−1 and un+1 are in p. we have the following three possibilities. case 1. if un, un+1 ∈ p, then ‖un −un+1‖k = ‖vn −vn+1‖k 4 δ · ‖un−1 −un‖k + l · ‖un −tun−1‖k 4 δ · ‖un−1 −un‖k . case 2. if un ∈ p, un+1 ∈ q, then ‖un −un+1‖k 4 ‖un −un+1‖k + ‖un+1 −vn+1‖k = ‖un −vn+1‖k = ‖vn −vn+1‖k 4 δ · ‖un−1 −un‖k + l · ‖un −tun−1‖k 4 δ · ‖un−1 −un‖k . topological vector-space valued cone banach spaces 213 case 3. if un ∈ q, un+1 ∈ p, then ‖un −un+1‖k 4 ‖vn −un‖k + ‖vn −vn+1‖k 4 ‖vn −un‖k + δ · ‖un−1 −un‖k + l · ‖un −tun−1‖k 4 ‖vn −un‖k + ‖un−1 −un‖k + l · ‖vn −un‖k = ‖vn −un−1‖k + l · ‖vn −un‖k = ‖vn −un−1‖k + l · ‖vn −un−1‖k −l · ‖un−1 −un‖k 4 (1 + l)‖vn−1 −vn‖k 4 (1 + l)δ · ‖un−2 −un−1‖k + (1 + l)l · ‖un−1 −tun−2‖k 4 (1 + l)δ · ‖un−2 −un−1‖k = h‖un−2 −un−1‖k , where h = (1 + l)δ < 1. taking α = max{δ,h}, and combining all above three cases we have ‖un −un+1‖k 4 { α‖un−1 −un‖k α‖un−2 −un−1‖k . by mathematical induction, for all n > 0, we have ‖un −un+1‖k 4 h (n−1)/2w for w ∈{‖u1 −u0‖k ,‖u2 −u1‖k} . now for n > m, we consider ‖um −un‖k 4 ‖un −un−1‖k + ‖un−1 −un−2‖ + · · · + ‖um−1 −um‖ 4 (h(n−1)/2 + h(n−2)/2 + · · · + h(m−1)/2)w 4 h(m−1)/2 1 −h(n−m)/2 w. as h < 1, we have h(m−1)/2 → 0 when n,m →∞, and this gives us h (m−1)/2 1−h(n−m)/2 w → θ, n → ∞, in the locally convex space e. now, according to lemma 3-(a), we conclude that for every c ∈ e with θ � c there is a natural number k1 such that ‖um −un‖k � c for all m,n ≥ k1, so {un} is a tvs-cone cauchy sequence in c. as c is closed, thus there exists some u ∈ c, such that un → u as n →∞. by construction of {un} there exists a subsequence { unq } such that vnq = unq = tunq−1 and unq → u as q → ∞. so, for a given c ∈ e with θ � c, let us choose a natural number k2 such that ∥∥u−unq∥∥k � c1+l and ∥∥unq−1 −u∥∥k � cδ for all q−1 ≥ k2. now, we have ‖u−tu‖k 4 ∥∥u−unq∥∥k + ∥∥unq −tu∥∥k 4 ∥∥u−unq∥∥k + ∥∥tunq−1 −tu∥∥k 4 ∥∥u−unq∥∥k + δ∥∥unq−1 −u∥∥k + l∥∥u−tunq−1∥∥k 4 (1 + l) ∥∥u−unq∥∥k + δ∥∥unq−1 −u∥∥k , i.e. ‖u−tu‖k � c(k2) for all q − 1 ≥ k2. this completes the proof. 214 mehmood, azam and aleksić theorem 20. let x be a tvs cone banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is a weak contraction (satisfying (w1)), such that δ(1 + l) < 1. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. corollary 21. let x be a cone banach space with normal cone k and c be a nonempty closed and convex subset of x. suppose that t : c → x is a weak contraction/almost weak contraction (satisfying (w1)/(w2)), such that δ(1+l) < 1. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. the following corollaries are due to remark 16. corollary 22. let x be a tvs cone banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is zamfirecue operator. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. corollary 23. let x be a tvs cone banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is quasi operator. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. corollary 24. let x be a banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is zamfirecue operator. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. corollary 25. let x be a banach space and c be a nonempty closed and convex subset of x. suppose that t : c → x is a quasi operator. if t satisfies the condition: u ∈ ∂c ⇒ tu ∈ c, then t has a fixed point. theorem 26. let e be a tvs-normed space, c be a closed and convex subset of e. let t : c → c be an almost weak contractive mapping (satisfying (w1)) with f(t) 6= ϕ. let {un} be ishikawa iteration satisfying (α) ∞∑ j=0 αj = ∞, u0 ∈ c is arbitrary chosen. then {un} converges strongly to a unique fixed point of t. proof. it can be shown that (w1) gives us a unique fixed point. let p ∈ f(t) be a unique fixed point of t and {un} be ishikawa iteration defined in (i1) and u0 ∈ c. we have ‖un+1 −p‖k = ‖(1 −αn)un + αntvn −p‖k = ‖(1 −αn)(un −p) + αn(tvn −p)‖k 4 (1 −αn)‖un −p‖k + αn‖tvn −p‖k 4 (1 −αn)‖un −p‖k + αnδ‖vn −p‖k , by (w1), and ‖vn −p‖k = ‖(1 −βn)un + βntun −p‖k = ‖(1 −βn)(un −p) + βn(tun −p)‖k 4 (1 −βn)‖un −p‖k + βnδ‖un −p‖k , by (w1). topological vector-space valued cone banach spaces 215 so, we obtain ‖un+1 −p‖k 4 ( 1 − (1 − δ)2αn ) ‖un −p‖k 4 e−(1−δ) 2αn ‖un −p‖k 4 ( e −(1−δ)2 n∑ j=0 αj ) · ‖u0 −p‖k . using (α), this implies ( e −(1−δ)2 n∑ j=0 αj ) → 0, n →∞, which gives us ( e −(1−δ)2 n∑ j=0 αj ) · ‖u0 −p‖k → θ, n → ∞, in the locally convex space e. this completes the proof of theorem. the following corollaries are due to remark 16. corollary 27. let e be a tvs-normed space, c be a closed and convex subset of e. let t : c → c be zamfirescue operator, with f(t) 6= ϕ. let {un} be ishikawa iteration satisfying ∞∑ j=0 αj = ∞ where u0 ∈ c is arbitrary chosen. then {un} converges strongly to a unique fixed point of t. corollary 28. let e be a tvs-normed space, c be a closed and convex subset of e. let t : c → c be a quasi operator, with f(t) 6= ϕ. let {un} be ishikawa iteration satisfying ∞∑ j=0 αj = ∞ where u0 ∈ c is arbitrary chosen. then {un} converges strongly to a unique fixed point of t. corollary 29. [24] let e be a normed space, c be a closed and convex subset of e. let t : c → c be a zamfirescue operator, with f(t) 6= ϕ. let {un} be ishikawa iteration satisfying ∞∑ j=0 αj = ∞ and u0 ∈ c is arbitrary chosen. then {un} converges strongly to a unique fixed point of t. corollary 30. [24] let e be a normed space, c be a closed and convex subset of e. let t : c → c be a quasi operator, with f(t) 6= ϕ. let {un} be ishikawa iteration satisfying ∞∑ j=0 αj = ∞ and u0 ∈ c is arbitrary chosen. then {un} converges strongly to a unique fixed point of t. the following theorem is a result for fixed point of non-expansive mappings in tvs-cone banach space for krasnoselskij iteration with λ = 1 2 . 216 mehmood, azam and aleksić theorem 31. let c be a closed and convex subset of a tvs-cone banach space (x,‖·‖k). suppose that the mapping f : c → c satisfies (a) ‖v −fv‖k + ‖u−fu‖k 4 η‖v −u‖k for all u,v ∈ c. then f has at least one fixed point if 2 ≤ η ≤ 4. proof. let us choose v0 ∈ c arbitrary and define sequence {vn} as follows: vn+1 = vn + fvn 2 , n = 0, 1, 2, 3, . . . since vn −fvn = 2 ( vn − vn + fvn 2 ) = 2(vn −vn+1), we have (b) ‖vn −fvn‖k = 2‖vn −vn+1‖k , n = 0, 1, 2, 3, . . . . combining (a) and (b), we have 2‖vn−1 −vn‖k + 2‖vn −vn+1‖k 4 η‖vn−1 −vn‖k , which gives ‖vn −vn+1‖k 4 λ‖vn−1 −vn‖k , n = 0, 1, 2, 3 . . . , for λ = η−2 2 < 1. according to the previous inequality, for m ≥ n, we obtain ‖vn −vm‖k 4 λn 1 −λ ‖v0 −v1‖k . since λn → 0 as n → ∞, then λ n 1−λ ‖v0 −v1‖k → θ, n → ∞, in the locally convex space e. now, according to lemma 3 part (a), we conclude that for every c ∈ e with θ � c there exists a natural number n1 such that ‖vn −vm‖k � c for all m,n ≥ n1. therefore, {vn} is a tvs-cone cauchy sequence in c. since c is closed, there exists some w ∈ c, such that vn → w as n → ∞. now, choose a positive integer m1 such that for every c ∈ e with θ � c we have ‖w −vn‖k � 1 η c for all n ≥ m1. substituting v = w and u = vn in (a), for all n ≥ m1, we obtain ‖w −fw‖k + 2‖vn −vn+1‖k 4 η‖w −vn‖k , ‖w −fw‖k 4 η‖w −vn‖k − 2‖vn −vn+1‖k � c. thus, w = fw is a fixed point of f. corollary 32. [16] let c be a closed and convex subset of a cone banach space (x,‖·‖k). suppose that the mapping f : c → c satisfies ‖v −fv‖k + ‖u−fu‖k 4 η‖v −u‖k for all u,v ∈ c. then f has at least one fixed point if 2 ≤ η ≤ 4. the next theorem is an application of above theorem in topological homotopy theory. theorem 33. let (x,‖·‖k) be a tvs-cone banach space, c a closed and convex subset of x and u an open subset of c. let k : [0, 1] × ū → c be a homotopy mapping with the following conditions: (a) ξ 6= k(t,ξ), for each ξ ∈ ∂u and each t ∈ [0, 1], topological vector-space valued cone banach spaces 217 (b) k(t, ·) : ū → c is a mapping satisfying the conditions of theorem 31, (c) there exists a continuous increasing function g : (0, 1] → p such that∥∥∥k(s,ξ) −k(t, ξ́)∥∥∥ k 4 g(s) −g(t), g(s) ∈ g(t) + p, for all s,t ∈ [0, 1], and each ξ ∈ ū. then k(0, ·) has a fixed point if and only if k(1, ·) has a fixed point. proof. we first suppose that k(0, ·) has a fixed point z, i.e. z = k(0,z). from (a), we obtain z ∈ u. define γ := {(t,ξ) ∈ [0, 1] ×c : ξ = k(ξ,t)}. clearly γ 6= φ. we define the partial ordering in γ as follows: (t,ξ) (s, ξ́) ⇔ t ≤ s and ∥∥∥ξ́ − ξ∥∥∥ k 4 2 η − 2 (g(s) −g(t)). let b be a totally ordered subset of γ and t̊ = sup{t : (t,ξ) ∈ b}. consider a sequence {(tn,ξn)}n≥0 in b such that, (tn,ξn) (tn+1,ξn+1) and tn → t̊ as n →∞. for m > n, we have ‖ξm − ξn‖k 4 2 η − 2 (g(tm) −g(tn)) → θ, as n,m →∞, and conclude that {ξn} is a tvs-cone cauchy sequence. there exists ξ̊ ∈ c such that ξn → ξ̊. choose n0 ∈ n such that for θ � c we have ‖̊ξ − ξn‖k � c η for all n ≥ n0. the mapping k(t, ·) satisfies all the conditions of theorem 31 and substituting v = ξ̊ and u = ξn into (1), for all n ≥ n0, we obtain∥∥∥̊ξ −k(̊t, ξ̊)∥∥∥ k + 2 ∥∥ξn − ξn+1∥∥k 4 η∥∥∥̊ξ − ξn∥∥∥k ,∥∥∥̊ξ −k(̊t, ξ̊)∥∥∥ k 4 η ∥∥∥̊ξ − ξn∥∥∥ k − 2 ∥∥ξn − ξn+1∥∥k � c. we see that ξ̊ = k(̊t, ξ̊) and, hence, ξ̊ ∈ u, which implies (̊t, ξ̊) ∈ γ. thus, (t,ξ) (̊t, ξ̊) for all (t,ξ) ∈b gives us that (̊t, ξ̊) is an upper bound of b. by zorn’s lemma, γ has maximal element (̊t, ξ̊). we claim that t̊ = 1. on the contrary, suppose that t̊ ≤ 1. let us choose θ � r arbitrary and, for any t ≥ t̊, consider br(̊ξ) = { ξ : ∥∥ξ − ξ̊∥∥ k 4 r } ⊂ u, where r = 2 η−2 (g(t) −g(̊t)). using the condition (c), we have∥∥k(t,ξ) −k(̊t, ξ̊)∥∥ k 4 g(t) −g(̊t) = η − 2 2 r � r. hence, for each t ∈ [0, 1], there exists some ξ ∈ br(̊ξ) ⊂ u such that ξ = k(t,ξ). since ∥∥ξ − ξ̊∥∥ k 4 r = 2 η − 2 (g(t) −g(̊t)) implies (̊t, ξ̊) (t,ξ), we obtain a contradiction. therefore, t̊ = 1. from the above it follows that k(1, ·) has a fixed point ξ̊ = k(1, ξ̊). 218 mehmood, azam and aleksić conversely, if k(1, ·) has a fixed point, then, in the same way, we can prove that k(0, ·) has a fixed point. let x be a tvs cone normed space and t be a self operator of x. let u0 be any fixed point and xn+1 = ξ(t,xn) is an iteration process involving t, which computes the sequence {xn} in x. definition 34. (see also [30]) the iteration procedure xn+1 = ξ(t,xn) is said to be t -stable with respect to t if {xn} converges to a unique fixed point q of t and whenever {yn} is a sequence in x with lim n→∞ ‖yn+1 − ξ(t,xn)‖k = θ we have lim n→∞ yn = q. theorem 35. let x be a tvs-cone normed space and t be a weak contraction (satisfying (w1)) with v = q ∈ f(t) 6= ϕ, in addition, whenever {yn} is a sequence with lim n→∞ ‖yn+1 −tyn‖k = θ, then the picard iteration defined in (p1) is t -stable. proof. we will show that the sequence {yn} with lim n→∞ ‖yn+1 − ξ(t,xn)‖k = θ, satisfies lim n→∞ yn = q. we have ‖yn+1 −q‖k 4 ‖yn+1 −tyn‖k + ‖tyn −q‖k 4 ‖yn+1 −tyn‖k + δ‖yn −q‖k + l‖yn −tyn‖k = δ‖yn −q‖k + (‖yn+1 −tyn‖k + l‖yn −tyn‖k) = δan + bn, where an = ‖yn −q‖k and bn = (‖yn+1 −tyn‖k + l‖yn −tyn‖k). using lemma 18, we have an → θ as n →∞. thus, lim n→∞ yn = q. references [1] beg, i, azam, a, arshad, m: common fixed points for maps on topological vector space valued cone metric spaces. int. j. math. math. sci. 2009 (2009), article id 15. [2] huang, l, zhang, x: cone metric spaces and fixed point theorems of contractive mappings. j. math. anal. appl. 332(2007), 1468–1476. [3] azam, a, beg, i, arshad, m: fixed point in topological vector space-valued cone metric spaces. fixed point theory and appl. 2010 (2010), article id 604084. [4] azam, a, mehmood, n: multivalued fixed point theorems in tvs-cone metric spaces. fixed point theory and applications. 2013 (2013), article id 184. [5] radenović, s, kadelburg z, janković, s: on cone metric spaces. a survey, nonlinear anal. 74 (2011), 2591-260. [6] rezapour, sh, khandani, h, vaezpour, sm: efficacy of cones on topological vector spaces and application to common fixed points of multifunctions. rendiconti del circolo matematico di palermo. 59 (2010), 185–197. [7] rudin, w: functional analysis. mcgraw-hill, inc. usa. 1973. [8] schaefer, h, h, wolff, m, p: topological vector spaces, 2nd edition. 1999 springer-verlag new york, inc. [9] arshad, m, azam, a and vetro, p, some common fixed point results in cone metric spaces, fixed point theory appl. 2009 (2009), article id 493965. [10] azam, a, arshad, m, beg, i: common fixed points of two maps in cone metric spaces. rend. circ. mat. palermo 57 (2008), 433–441. [11] haghi, rh, rezapour, s, shahzad, n: some fixed point generalizations are not real generalizations. nonlinear analysis: theory, methods & applications. 74(5)(2011), 1799-1803. topological vector-space valued cone banach spaces 219 [12] khani, m, pourmahdian, m: on the metrizability of cone metric spaces. topology appl. 158(2)(2011), 190–193. [13] rezapour, sh: best approximations in cone metric spaces. mathematica moravica. 11(2007), 85–88. [14] rezapour, sh, hamlbarani, r: some notes on paper ”cone metric spaces and fixed point theorems of contractive mappings”. j. math. anal. appl. 345(2008), 719–724. [15] al-rawashdeh, a., shatanawi, w. and khandaqji, m: normed ordered and e-mertic spaces, international journal of mathematics and mathematical sciences, 2012(2012), article id 272137. [16] karapınar, e: fixed point theorems in cone banach spaces. fixed point theory and applications. 2009(2009), article id 609281. [17] abdeljawad, t, karapinar, e, tas, k: common fixed point theorems in cone banach space. hacettepe journal of mathematics and statistics, volume 40 (2) (2011), 211 – 217. [18] mutlu, a, yolcu, n: fixed point theorems for φp-operator in cone banach spaces. fixed point theory and applications 2013(2013), article id 56. [19] yousefi, b, yadegarnejad, a, kenary, ha, park, c: equivalence of semistability of picard, mann, krasnoselskij and ishikawa iterations. fixed point theory and applications, 2014(2014), article id 5. [20] berinde, v: on the approximation of fixed points of weak contractive mappings. carpathian j. math, 19(1)(2003), 7-22. [21] alghamdi, m a, berinde, v, shahzad, n: fixed points of multivalued nonself almost contractions. journal of applied mathematics, 2013. [22] assad, n. a: a fixed point theorem in banach space. publications de l’institut mathématique (beograd)(ns), 47(61)(1990), 137-140. [23] rhoades, b.e: a fixed point theorem for some non-self-mappings, math. japon. 23 (1978), 457-459. [24] berinde, v: approximating fixed points of weak contractions using the picard iteration. nonlinear analysis forum. vol. 9. 2004. [25] berinde, v: iterative approximation of fixed points. berlin: springer, (2007) [26] berinde, v: a convergence theorem for some mean value fixed point iteration procedures. dem math, 38(1)(2005), 177-184. [27] berinde, v: a convergence theorem for mann iteration in the class of zamfirescu operators. univ. vest timi. ser. mat.-inform, 45(2007), 33-41. [28] kadelburg, z, radenović, s, rakočević, v: topological vector space-valued cone metric spaces and fixed point theorems. fixed point theory and applications, 2010(2010), article id 170253. [29] radenović. s, kadelburg z: quasi-contractions on symmetric and cone symmetric spaces. banach j. math. anal. 5 (2011), no. 1, 38–50. [30] asadi, m., soleimani, h.,be, r: on t-stability of picard iteration in cone metric spaces. fixed point theory and applications, 2009. 1department of mathematics, comsats institute of information technology, chak shahzad, islamabad 44000, pakistan 2faculty of science, university of kragujevac, radoja domanovića 12, 34 000 kragujevac, serbia ∗corresponding author international journal of analysis and applications volume 18, number 2 (2020), 161-171 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-161 analysis of quasistatic frictional contact problem with subdifferential form, unilateral condition and long-term memory a. ourahmoun1,∗, b. bouderah1, t. serrar2 1university of m’sila 28000, m’sila lmpa laboratory, algeria 2university of setif 1, 19000, algeria ∗corresponding author: ourahmounabbes@yahoo.fr abstract. we consider a quasistatic problem which models the contact between a deformable body and an obstacle called foundation. the material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory, thus at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. in contact mechanics, history-dependent operators could arise both in the constitutive law of the material and in the frictional contact conditions. the mathematical analysis of contact models leads to the study of variational and hemivariational inequalities. for this reason a large number of contact problems lead to inequalities which involve history dependent operators, called history dependent inequalities. such inequalities could be variational or hemivariational and variational hemivariational. in this paper we derive a weak formulation of the problem and, under appropriate regularity hypotheses, we stablish an existence and uniqueness result. the proof of the result is based on arguments of variational inequalities monotone operators and banach fixed point theorem. 1. introduction contact mechanics still remain a rich domain of research, and the literature devoted to various aspects of the subject is growing. an early attempt at the study of contact problems for elastic viscoelastic materials received 2019-12-31; accepted 2020-02-03; published 2020-03-02. 2000 mathematics subject classification. primary 74m10, 74m15, 49j40. key words and phrases. elastic material; frictional contact; unilateral condition; weak solution. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 161 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-161 int. j. anal. appl. 18 (2) (2020) 162 within the mathematical analysis framework was introduced in the pioneering reference works [6, 7, 15]. further extensions to non convex contact conditions with non-monotone and possible multi-valued constitutive laws led to the active domain of non-smooth mechanic within the framework of the so-called hemivariational inequalities, for a mathematical as well as mechanical treatment we refer to [10]. there is a growing interest in the study of history-dependent inequalities. for instance, a class of variational inequalities with history dependent operators was considered in [15], where abstract existence, uniqueness and regularity results were proved. these results were extended in[18] to a more general class of variational inequalities and were completed in [6] with error estimate and convergence results. various results on hemivariational and variational-hemivariational inequalities with history dependent operators, formulated in sobolev-type spaces, could befound in [7, 9]. we introduce a new model of frictional contact for viscoelastic materials and to illustrate the use of history dependent variational hemivariational inequality in its variational analysis. thus, in section 2 we introduce the contact problem, in which the material’s behavior is modeled by a nonlinear viscoelastic constitutive law with long memory, the process is quasistatic, the contact is frictional and the contact conditions are in a subdifferential form with unilateral conditions for the displacement. then, in section 3 we list the assumptions on the data and derive the variational formulation of the problem. it is in a form of a historydependent variational-hemivariational inequality in which the unknown is the displacement field.next in section 4 we state our main existence and uniqueness result, theorem (4.2) the proof of the theorem is obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for history dependent operators. 2. the contact model the physical setting we consider is the following. a deformable body occupies a domain ω ⊂ rd (d = 1, 2, 3 in applications) with outer lipschitz surface γ that is divided into three disjoint measurable parts γi (i = 1, 2, 3) such that meas(γ1) > 0. let [0,t] be the time interval of interest, where t > 0. the body is clamped on γ1 × (0,t) and therefore the displacement field vanishes there. a volume force of density f0 acts in ω ×(0,t) and surface tractions of density f1 act on γ2 ×(0,t). the body is in contact on γ3 ×(0,t) with a rigid obstacle, the so-called foundation is in frictional contact. we assume that the process is quasistatic with long term memory and we use (1) as constitutive law. we denote by u, σ and ε(u) the displacement field, the stress field and the linearized strain tensor, respectively, and let v be the unit outward normal vector to γ. here and below, we sometimes do not indicate explicitly the dependence of various functions on the spatial variable x ∈ ω ∪ γ. for a vector field u, we use notation uv = u ·v and uτ = u − uv v for the normal and tangential components of u on γ. similarly, for the stress int. j. anal. appl. 18 (2) (2020) 163 field σ, its normal and tangential components on the boundary are defined by equalities σv = (σv)·v and στ = σv −σvv, respectively. finally, we use sd for the space of second order symmetric tensors on rd and “·” will represent the canonical inner product and the euclidean norm on the spaces rd and sd, respectively. we also use the following notation: h = ( l2(ω) )d ,h = { σ = (σij) | σij = σji ∈ l2(ω), 1 ≤ i ≤ j ≤ d } h1 = {u ∈ h : ε(u) ∈h} ; h1 = {σ ∈h | divσ ∈ h} here ε : h1 → h and div : h1 → h are the deformation and the divergence operators, respectively, defined by: ε(u) = (εij(u)) ; εij(u) = 1 2 (ui,j + uj,i) ; divσ = (σij,j) the spaces h, h, h1 and h1 are real hilbert spaces endowed with the canonical inner products given by: (u,v)h = ∫ ω ui.vidx ,(σ,τ)h = ∫ ω σij.τijdx (u,v)h1 = (u,v)h + (ε(u),ε(v))h (σ,τ)h1 = (σ,τ)h + (divσ,divτ)h we recall that c denotes the class of continuous functions; and cm, m ∈ n∗ the set of m times continuously differentiable functions. finally d(ω) denotes the set of infinitely differentiable real functions with compact support in ω; and w m,p ,m ∈ n, 1 ≤ p ≤ +∞ for the classical sobolev spaces; and hm0 (ω) := {w ∈ w m,2(ω),w = 0 on γ},m ≥ 1. with these assumptions, the classical formulation or mathematical model which describes the equilibrium of the body in the physical setting above is the following. problem p. find a displacement field u : ω×r+ → rd, a stress field σ : ω×r+ →sd and two interface forces ηv : γ3 ×r+ → r and ξv : γ3 ×r+ → r such that σ(t) = aε(u(t)) + t∫ 0 b(t−s)ε(u(s))ds in ω × (0,t) (2.1) int. j. anal. appl. 18 (2) (2020) 164 div(σ(t)) + f0(t) = 0 in ω × (0,t) (2.2) u(t) = 0 on γ1 × (0,t) (2.3) σ(t)v = f2(t) on γ2 × (0,t) (2.4)   uv(t) ≤ g , σv(t) + ξv(t) + ηv(t) ≤ 0 (uv(t) −g)(σv(t) + ξv(t) + ηv(t)) = 0 |uv(t)| ≤ fm   t∫ 0 u+v (s)ds   ηv(t) =   0 if uv(t) ≺ 0 fm   t∫ 0 u+v (s)ds   if uv(t) ≥ 0 ξv(t) ∈ ∂jv(uv(t)) on γ3 × (0,t) on γ3 × (0,t) (2.5)   ‖στ (t)‖≤ fb(uv(t)) −στ (t) = fb(uv(t)) uτ (t) ‖uτ (t)‖ if uτ (t) 6= 0 on γ3 × (0,t) (2.6) first, eq.(2.1) is the constitutive law for viscoelastic materials in which a represent the elasticity operator and b represents the relaxation tensor. various comments and mechanical interpretation related to such kind of equations could be found in [8, 16]. equation (2.2) is the equilibrium equation that we use here since we assume that the process is quasistatic. conditions (2.3) and (2.4) represent the displacement and traction conditions, respectively. condition (2.5) represents the contact condition in which g > 0, jv and fm are given functions and ∂jv represents the clarke subdifferential of jv. finally, relations (2.7) represent the static version of coulomb’s law of dry friction. here fb denotes a positive function, the friction bound assumed to depend on the normal displacement uv. the contact condition (2.5) represents the trait of novelty of our model. note that this condition models the contact with a foundation made of a rigid body covered by a layer made of soft material and a thin crust with memory effects. 3. variational analysis to derive a variational formulation of the problem we use the spaces for the displacement field we use the space v = { v = (vi) ∈ h1(ω) | v = 0 on γ1 } (3.1) int. j. anal. appl. 18 (2) (2020) 165 which is a real hilbert space with inner product (u,v)v = (ε(u),ε(v))h where (u,v)v = ∫ ω ε(u) ·ε(v)dx and associated norm ‖·‖v . we consider the space of fourth-order tensor fields q∞ = { e = (eijkl) | eijkl = eklij = ejikl ∈ l ∞(ω) } which is a real banach space with norm ‖e‖q∞ = max0≤i,j,k,l≤d ‖eijkl‖l∞(ω) finally, we use n for the set of positive integers and r+ for the set of nonnegative real numbers. for a normed space x, we use the notation c(r+; x) for the space of continuous functions defined on r+ with values in x. by the sobolev trace theorem, we have ‖v‖l2(γ3,rd) ≤‖γ‖‖v‖v , ∀v ∈ v (3.2) ‖γ‖ being the norm of the trace operator γ : v → l2(γ3,rd) we now list the assumptions on the data and we assume that 1. the elasticity operator a : ω ×sd →sd satisfies the following properties   (a) there exists la > 0 such that for all ε1,ε2 ∈sd,a.e.x ∈ ω, ‖a(x,ε1) −a(x,ε2)‖≤ la |ε1 − ε2| (b) there exists m > 0 such that for all ε1,ε2 ∈sd,a.e.x ∈ ω, (a(x,ε1) −a(x,ε2)) . (ε1 − ε2) ≥ m‖ε1 − ε2‖ 2 a(·,ε1) is measurable on ω for all ε ∈sd a(x, 0) = 0 for a.e.x ∈ ω (3.3) 2. the relaxation tensor b is such that b ∈c(r+,q∞) (3.4) int. j. anal. appl. 18 (2) (2020) 166 3. the potential function jv : γ3 × r → r,assumed to satisfy the following conditions   (a) jv(·,r) is measurable on γ3for all r ∈ r and there exists − e ∈ l2(γ3) such that jv(·, − e) ∈ l1(γ3) (b) jv(x, ·) is locally lipschitz on r for a.e.x ∈ γ3 (c) | jv(x,r)| ≤ − c0 + − c1 |r| for a.e.x ∈ γ3 and for all r ∈ r with − c0, − c1 ≥ 0 (3.5) next, we assume that the penetration bound g : γ3 → r, the memory function fm :γ3 × r → r+ and the friction bound fb : γ3 × r → r are assumed to satisfy the following conditions. g ∈ l2(γ3) , g(x) ≥ 0 a.e.on γ3. (3.6) and   (a)there exists lfm > 0 such that |fm(x,r1) −fm(x,r1)| ≤ lfm |r1 −r2| for all r1,r2 ∈ r, a.e.x ∈ γ3 (b) fm(·,r) is measurable on γ3 for all r ∈ r (c) x → fm(x, 0) ∈ l2(γ3) (3.7)   (a)there exists lfb > 0 such that |fb(x,r1) −fb(x,r1)| ≤ lfb |r1 −r2| for all r1,r2 ∈ r,a.e.x ∈ γ3 (b) fb(·,r) is measurable on γ3, for all r1,r2 ∈ r,a.e.x ∈ γ3 (c) fb(x,r) = 0 for r ≤ 0 ,fb(x,r) > 0 for all r > 0 a.e.x ∈ γ3 (3.8) we also assume that the densities of body forces and surface tractions have the regularity f0 ∈ c(r+; l2(ω; rd)) , f2 ∈ c(r+; l2(γ2; rd)) (3.9) and, finally, we assume the smallness condition lfb ‖γ‖ + α jv < mf (3.10) we now introduce the set of the admissible displacement fields u ⊂ v and the function f : r+ → v ′ defined by   u = {v ∈ v | vv ≤ g on γ3}〈f(t), v〉v ′×v = (f0(t), v)(l2(ω),rd) + (f2(t), v)(l2(γ3),rd), for all v ∈ v, t ∈ r+ (3.11) int. j. anal. appl. 18 (2) (2020) 167 assume now that (u,σ) represents a couple of regular functions which satisfy (2.1)−(2.6) and let t ∈ r+, v ∈ u. we perform an integration by parts, split the surface integral on three integrals on γ1, γ2 and γ3, and use the equalities (2.2) − (2.4)to deduce that ∫ ω σ(t) · (ε(v) −ε(u(t)))dx = ∫ ω f0(t) · (v −u(t))dx + ∫ γ2 f2(t) · (v −u(t))dγ + ∫ γ3 σv(t)(vv −uv(t))dγ + ∫ γ3 στ (t)(vτ −uτ (t))dγ (3.12) next, we use the contact boundary condition (2.5), the definition (3.12) and the definition of the clarke subdifferential to obtain that ∫ γ3 σv(t)(vv −uv(t))dγ + ∫ γ3 fm   t∫ 0 u+v (s)ds   (v+v −u+v (t))dγ + ∫ γ3 j0v(uv(t), vv −uv(t))dγ ≥ 0 (3.13) note that here and below we use notation j0v(r1; r2) for the generalized directional derivative of jv at r1 in the direction r2, see [1, 2] for details. on the other hand, the friction law (2.6) yields ∫ γ3 στ (t)(vτ −uτ (t))dγ + ∫ γ3 fb(uv(t) (‖vτ‖−‖uτ (t)‖) dγ ≥ 0 (3.14) we now combine equality (3.13) with inequalities (3.14), (3.15) to deduce that ∫ ω σ(t) · (ε(v) −ε(u(t)))dx + ∫ γ3 fb(uv(t) (‖vτ‖−‖uτ (t)‖) dγ+ ∫ γ3 fm   t∫ 0 u+v (s)ds   (v+v −u+v (t))dγ + ∫ γ3 j0v(uv(t), vv −uv(t))dγ ≥ ∫ ω f0(t) · (v −u(t))dx + ∫ γ2 f2(t) · (v −u(t))dγ (3.15) finally, we substitute the consitutive law (2.1) in (3.15) and use notation (3.12) to obtain the following variational formulation of problem p, in terms of displacement. problem pv find a displacement field u : r+ → u such that the unique solvability of problem pv is given by the following existence and uniqueness result, that we state here and prove in the next section. theorem 3.1 assume that (3.7)–(3.11) hold. then, problem pv has a unique solution u ∈ c(r+; u). int. j. anal. appl. 18 (2) (2020) 168 we end this section with some remarks on the weak solvability of the contact problem p. first, a couple of functions (u,σ) defined on the positive real line r+ with values on the product space v ×q is called a weak solution to problem p if u is a solution of the variational problem pv and σ satisfies the constitutive law (2.1). we conclude that, under the assumption of theorem 8.1, problem p has a unique weak solution. moreover, the solution has the regularity u ∈ c(r+; u) and σ ∈ c(r+; q). next, recall that theorem 8.1 provides the weak solvability of the contact problem p under the smallness assumption (24) involving the friction bound fb, and the normal compliance potential jv. finally, note that the unknowns ηv and ξv of problem p cannot be recovered since they cannot be computed when the solution u of problem p is known.actually, these unknowns represent interface forces and, as usual in solving contact problems with unilateral constraints, we do not have information neither on the uniqueness of these functions and on their regularity. 4. an existence and uniqueness result we present in this section an abstract result on history-dependent variational-hemivariational inequalities that we shall use to prove the unique solvability of problem pv. for more details on the material presented in this section, we send the reader to [1, 2]. theorem 4.1.let x be a reflexive banach space and y be a normed space. we denote by x ′ the dual of x and by 〈·, ·〉 x′×x the duality pairing of x and x ′ . let k be a subset of x and a : x → x ′ , ψ : c(r+; x) → c(r+; y ) be given operators ,consider also a function φ : y ×k ×k → r, a locally lipschitz function j : x → r and a function f : r+ → x ′ . with these data we consider the problem of finding a function u : r+ → u such that, for each t ∈ r+, the following inequality holds: 〈au(t),v −u(t))〉 + φ((ψu)(t),u(t),v) −φ((ψu)(t),u(t),u(t)) (4.1) +j0(u(t),v −u(t)) ≥〈f(t),v −u(t)〉 , for all v ∈ k in the study of (4.1), we assume the following hypotheses. k is a nonempty, closed and convex subset of x. a : x → x ′ is an operator such that   (a)a is pseudomonotone and there exist αa > 0,βa,γa ∈ r and u0 ∈ k such that: 〈av,v −u0〉≥ αa‖v‖ 2 x −βa‖v‖ 2 x −γa for all v ∈ x. (b)a is strongly monotone,i.e.,there exists ma > 0 such that 〈av1 −av2,v1 −v2〉≥ ma‖v1 −v2‖ 2 x , ∀v1,v2 ∈ x (4.2) int. j. anal. appl. 18 (2) (2020) 169 φ : y ×k ×k → r is a function such that   (a) φ(y,u, ·) is convex and l.s.c. on k, for all y ∈ y,u ∈ k (b) there exist αφ , βφ > 0 such that φ(y1,u1,v2) −φ(y1,u1,v1) + φ(y2,u2,v1) −φ(y2,u2,v2) ≤ αφ‖u1 −u2‖x ‖v1 −v2‖x + βφ‖y1 −y2‖y ‖v1 −v2‖x for all y1,y2 ∈ y , u1,u2,v1,v2 ∈ k (4.3) j : x → r is a function such that   (a) j is locally lipschitz (b) ‖∂j(v)‖x′ ≤ c0 + c1 ‖v‖x , for all v ∈ v , c0 ,c1 ≥ 0 (c) there exists αj > 0 such that j0(v1,v2 −v1) − j0(v2,v1 −v2) ≤ αj ‖v1 −v2‖ 2 x , for all v1,v2 ∈ x (4.4)   for any n ∈ n , there exists sn > 0 such that ‖(ψu1) (t) − (ψu2) (t)‖y ≤ sn ∫ t 0 ‖u1(s) −u2(s)‖ds for all u1,u2 ∈ c(r+; x), for all t ∈ [0,n]. (4.5) αϕ + αj < ma ; αϕ < αj (4.6) f ∈ c(r+; x∗) (4.7) note that an operator ψ which satisfies condition (4.5) is called a history dependent operator. inequality (4.1) is governed both by the function φ which is assumed to be convex with respect its second argument and by the function j which is locally lipschitz and could be nonconvex. therefore, this inequality is a variational-hemivariational inequality. in addition, the function φ in (4.1) depends on the operator ψ , assumed to be history-dependent.for this problem we have the following existence and uniqueness result. theorem 4.2. let x be a reflexive banach space, y a normed space, and assume that (4.2)–(4.7) hold. then, inequality (4.1) has a unique solution u ∈ c(r+; k). the proof of is obtained by using arguments of elliptic variational-hemivariational inequalities and a fixed point result for history dependent operators. proof (theorem 4.1) we start by defining the operators a : v → v ′ ,f : c(r+; v ) → c(r+; q×l2(γ3)) and the functions φ : l2(γ3) ×v ×v → r and j : v → r by (au, v) = ∫ ω fε(u) ·ε(v)dx for all u, v ∈ v (4.8) int. j. anal. appl. 18 (2) (2020) 170 (fu)(t) =  ∫ ω b(t−s)ε(u(s))ds,fm   t∫ 0 u+v (s)ds     (4.9) for all u ∈ c(r+; v ), t ∈ r+ ϕ(ξ, u, v) = (ξ1,ε(v))q + ( ξ2,v + v ) l2(γ3) + (fb(uv),‖vτ‖)l2(γ3) (4.10) for all ξ = (ξ1,ξ2) ∈ q×l 2(γ3), u, v ∈ v j(v) = ∫ γ3 jv(vv)dγ ; for all v ∈ v. (4.11) then, it is easy to see that problem pv is equivalent to the problem of finding a function u : r+ → u such that for each t ∈ r+, the following inequality holds: 〈au(t), v − u(t)〉 + ϕ((fu)(t), u(t), v)−ϕ((fu)(t), u(t), u)+ (4.12) j0(u(t), v − u(t)) ≥〈f(t), v − u(t))〉 , for all v ∈ v to solve this problem, we use theorem 4.1 with x = v , y = l2(γ3) and k = u and, to this end, we check in what follows that assumptions (4.2)–(4.7) hold. we use arguments similar to those used in our previous works [8, 9] and, for this reason, we skip the details and we resume the proof as follows. first, we note that assumption (3.7) and definition (3.12) imply (4.2). next, a simple calculation based on the definition (4.5) of the operator a and the properties (3.4) of the elasticity operator show that (4.2) holds with m a = αa = mf. moreover, using assumption (3.9) and the trace inequality (3.2), it is easy to see that the function φ defined by (4.5) satisfies condition (4.3) with αφ = l fb ‖γ‖. on the other hand, assumption (3.6) on the function jv and definition (4.8) show that condition (4.4) holds with α j = α jv . and, a simple calculation based on assumptions (3.5), (3.9) imply that the operator (4.9) is a history-dependent operator, i.e., it satisfies condition (4.5). now, keeping in mind that m a = αa = mf,α φ =l fb ‖γ‖ and αj = α jv , we easily deduce that the smallness assumption (3.11) shows that conditions (4.6) hold, too. finally, we note that regularity (3.9) on the densites of the body forces and tractions combined with definition (3.12) show that condition (4.7) is satisfied. we are now in a position to use theorem 4.2 to deduce the existence of a unique function u ∈ c(r+; u) such that (4.12) holds, for each t ∈ r+. and, using notation (4.8)–(4.12), we deduce that u is the unique solution to problem pv which concludes the proof. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (2) (2020) 171 references [1] adly, s. ernst, e., thera, m.: stability of the solution set of non-coercive variational inequalities. commun. contemp. math. 4 (2002), 145–160. [2] barbu, v. nonlinear semigroups and differential equations in banach spaces; springer: heidelberg, germany, 1976. [3] brézis, h. problèmes unilatéraux. j. math. pures appl. 51 (1972), 1–168. [4] carstensen, c., gwinner, j.: a theory of discretization for nonlinear evolution inequalities applied to parabolic signorini problems. ann. mat. pura appl. 177 (1999), 363–394. [5] chau, o. analyse variationnelle et numérique en mécanique du contact. thesis, perpignan, june 2000. [6] ciarlet, p.g.: mathematical elasticity, vol. i: three-dimensional elasticity. north-holland, amsterdam, 1988. [7] duvaut, g., lions, j.l.: les inéquations en mécanique et en physique. dunod, paris, 1972. [8] eck, ch., jarusek, j., krbec, m.: unilateral contact problems, variational methods and existencetheorems. monographs & texbooks in pure & applied mathematics, vol. 270, chapman and hall, london, 2005. [9] glowinski, r numerical methods for nonlinear variational problems. springer, berlin, 1984. [10] goeleven, d, motreanu, d., dumont, y., rochdi, m.: variational and hemivariational inequalities,theory, methods and applications, volume i: unilateral analysis and unilateral mechanics. kluwer, dordrecht, 2003. [11] han, w, sofonea, m.: evolutionary variational inequalities arising in viscoelastic contact problems. siam j. numer. anal. 38 (2000), 556–579. . [12] lions, j.l. quelques méthodes de résolution des problèmes aux limites non linéaires. dunod et gauthier-villars, paris, 1969. [13] lions, j., magenes, e.: problèmes aux limites non homogènes et applications, vol. 1. dunod, paris, 1968. [14] matei, a, sofonea, m.: variational inequalities with applications: a study of antiplane frictional contact problems, advances in mechanics and mathematics, vol. 18. springer, berlin, 2009. [15] kikuchi, n., oden, j.t. contact problems in elasticity. siam, philadelphia, 1988. [16] necas, j., hlavácek, i. mathematical theory of elastic and elastoplastic bodies: an introduction.elsevier, amsterdam, 1981. [17] panagiotopoulos, p.d. inequality problems in meechanics and applications. birkhäuser, basel, 1985. [18] panagiotopoulos, p.d. inequality problems in meechanics and applications. birkhäuser, basel, 1985. [19] sofonea, m., han, w., migórski, s. numerical analysis of history-dependent variational inequalities with applications to contact problems. eur. j. appl. math. 26 (2015), 427–452. [20] sofonea, m., matei, a. history-dependent quasivariational inequalities arising in contact mechanics. eur. j. appl. math. 22 (2011), 471–491. [21] sofonea, m., matei, a.mathematical models in contact mechanics, london mathematical society lecture note series, vol. 398. cambridge university press, cambridge, 2012. [22] sofonea, m., migórski, s. a class of history-dependent variational-hemivariational inequalities. nonlinear differ. equ. appl. 23 (2016), 38. [23] sofonea, m., xiao, y. fully history-dependent quasivariational inequalities in contact mechanics. appl. anal. 95 (2016), 2464–2484. [24] zeidler, e.nonlinear functional analysis and its applications, ii/a, linear monotone operators. springer, berlin, 1997. 1. introduction 2. the contact model 3. variational analysis 4. an existence and uniqueness result references international journal of analysis and applications volume 17, number 1 (2019), 132-166 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-132 received 2018-09-17; accepted 2018-11-09; published 2019-01-04. 2010 mathematics subject classification. 91b02. key words and phrases. ahp; banking system; evaluating, rankings; experts. ©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 132 applying ahp in evaluation of vietnamese commercial banks thanhtuyen tran* scientific research office, lac hong university, no. 10 huynh van nghe, bien hoa city, dong nai, vietnam *corresponding author: copcoi2@gmail.com abstract. bank rankings are one of the ways to rate the bank system and to create competitive advantages, which have emerged as the central issue and considered as one of the most important organizational innovation. this research is objective to explore and to demonstrate utility of analytic hierarchy process (ahp) application in banking for the purpose of proposing suitable model for partners evaluation and selecting banking strategic alliances in vietnam. the ahp is applied to examine what criteria should be encompassed in evaluating and examining the importance weightings of influential criteria when ranking the bank system. in this study, a short review of literature regarding application ahp in banking decision-making is presented, focusing on partner evaluation criteria and methods to propose model for partner evaluation and selecting strategic banking for the current study. after a long process of calculation based on ahp, i have come up with the final rankings according expert’s interview: acb’s percentages have change widely from each subcriterion; finally it gets 12.98% at the top of the list. coming very closely downwards are dab, seabank etc., at the bottom of the rankings is sgb at 7.41%. by this paper, author would contribute to the ranking process of the banking system, in general, and the special case of vietnamese banking a very modern model to apply, then to choose the right alliance for further cooperation, not only for banking system but it can be applied for a lot of industries. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-132 int. j. anal. appl. 17 (1) (2019) 133 i. introduction 1.1 research background bank rankings are one of the ways to rate the bank system and to create competitive advantages, which have emerged as the central issue and considered as one of the most important organizational innovation. the facts of successful bank rankings demonstrate that choosing a right cooperative partner possibly decides the fate of the majority of strategic alliances [1]. in the new era of banking, bank ranking formation has been growing among the financial industry during the last decades [2]. bank rankings often include units like mutual fund managing companies, asset management companies, securities brokerages and corporate finance companies. financial mergers and financial rankings can bring some advantages in terms of improving financial structures, and promoting the operating performance of financial organizations [3]. in the context of related studies focusing on vietnamese industries are increasing; the author explores that studies that examine the bank rankings formation between banks are still rare although synergies between banking are so significant ([4],[5]) the growth of banking system increases in parallel with the monetization degree of markets; thus the development in the banking sector mutually and deeply affects the other sectors of the economy, particularly real economy [6]. over the past two decades, the vietnamese government has undertaken a series of reforms to strengthen and modernize this sector in order to adapt to the rising household incomes and an increasing demand for retail banking services, which resulted from rapid economic growth [7]. soon after the vietnamese government lifted its ban on the establishment of new banks in the years 2000s, numerous new banks began operations. in 2006, the government responded to the excessive number of banks and this leads to the competition for capital among banks. there are various forms of competitive pressure, such as retaining new customers, providing new financial services and holding available businesses [8]. some methods like mergers and alliance formation have now become an emergent issue for vietnamese banks and some banks’ top managers are trying to adopt these philosophies to improve their competitive advantage ([9],[10]). int. j. anal. appl. 17 (1) (2019) 134 bank rankings of financial organizations are closely linked to organizational performance, government policy, shareholder rights and customer satisfaction. it is essential for financial organizations to select their strategies carefully. factors requiring consideration include various internal, external, qualitative and quantitative attributes, indicating that the selected problem is an analytical hierarchy issue [11]. a well-known approach that can effectively deal with this problem is the analytic hierarchy process (ahp) [12]. the ahp methodology involves separating a complex decision issue into elemental problems to establish a hierarchical model. when the decision problem is divided into smaller constituent parts in a hierarchy, pair-wise comparisons of the relative importance of elements are conducted at each level to establish a set of priorities. ahp is widely employed in diverse fields, especially growing its effectiveness among the financial industry ([13], [14], [15]). for example, korhonen and voutilainen (2006) [16] studied alternative alliances between banks and insurance companies. six different possible structure models for such alliances and nine criteria are used to evaluate the models. the use of the ahp focused the discussions on pair-wise comparisons. based on the evaluations of the panel, the alternatives financial conglomerate and cross-selling agreement, and no overlapping service channels are most preferred. seçme et al. (2009) [17] proposed a fuzzy multi-criteria decision model to evaluate the performances of banks. the largest five commercial banks of turkish banking sector are examined and these banks are evaluated in terms of several financial and non-financial indicators. fuzzy analytic hierarchy process (fahp) and technique for order performance by similarity to ideal solution (topsis) methods are integrated in the proposed model. after the weights for a number of criteria are determined based on the opinions of experts using the fahp method, these weights is input to the topsis method to rank the banks. the results show that not only financial performance but also non-financial performance should be taken into account in a competitive environment. financial organizations have been globally studied, but few of these studies have examined the strategies used by banks in vietnam for making decisions regarding bank rankings. basing on the successful experiences of rankings which raises some rules for choosing strategic alliance partners, and gives a description of how to choose the best partner with ahp, in this paper we have studied the bank rankings between 10 top vietnamese banks int. j. anal. appl. 17 (1) (2019) 135 that are already on the financial industry for the purpose of proposinga suitable model for partner evaluation and selecting banking strategic alliance for any financial organizations. the main objective of this study is applying ahp to examine what criteria should be encompassed in evaluating and examining the importance weightings of influential criteria when ranking the bank system. 1.2 research objectives and implications our research objectives are to explore and to demonstrate utility of ahp application in banking for the purpose of proposing suitable model for partners evaluation and selecting banking strategic alliances in vietnam. we want to apply ahp to examine what criteria should be encompassed in evaluating and examining the importance weightings of influential criteria when ranking the bank system. in this study, a short review of literature regarding application ahp in banking decision-making is presented, focusing on partner evaluation criteria and methods to propose model for partner evaluation and selecting strategic banking for the current study. analytic hierarchy process (ahp) application in banking sector is growing most recently and has been seen as a high potential decision support tool in banking sector in the days to come. the use of ahp as a decision support tool is appreciated and interested by the author. this study reviews application of ahp in the finance sector with specific reference to banking. bank ranking is one of the most complex and ill-structured tasks faced by banks. in deriving these strategies bankers usually try to achieve multiple, and sometimes conflicting objectives such as profitability, growth, liquidity, and market share subject to constraints on credit and exchange risks and regulatory requirements [18]. however, it is safe to assume that the ahp methodology can be applied to other complex and ill-defined strategic issues faced by other banking institutions because when compared with existing techniques on the one hand, and with qualitative managerial judgment on the other hand, the ahp provides a useful, simple and powerful tool for dealing with strategic planning in banking [19]. as mentioned above, the alliance with a highly regarded financial services institution may give financial organizations and cooperative industries an opportunity to build a suitable strategic relationship. the proposed strategy may also attract the concerns and preferences int. j. anal. appl. 17 (1) (2019) 136 of bank stakeholders. the results of this study provide a valuable reference for bank administrators. this current study contributes to the two elements of practical application of ahp method and academic application of the field evaluating and then to form suitable business strategy for a financial institution in a developing country. by presenting and applying ahp to researches and analyzing the advantages and disadvantages of this method, it provides top managers in related areas with the ability to integrate the multi-attribute preferences of consumers using a hierarchical model to determine the bank's relative position in the marketplace. the suitability of ahp in examining bank selection by consumers for managerial decision making is demonstrated using an empirical analysis in a major metropolitian area. implications of the findings of this analysis for strategic planning in the areas of marketing mix and organizational characteristics of a bank are explored. suggestions for application of ahp to other areas of financial services management are included. the research method wasapplied in this research includes: (1) research discovery: to explore preliminary research issues that need as well as claims the research problem. (2) method of describing and comparing or the method of decision-making. (3) method of intergrated analysis towards the problem of assessing the quality and selecting suitable model for partner evaluation and bank rankings in vietnam. (4) qualitative amd expert methods: to review evaluation criteria for selecting suitable model for partner evaluation and bank rankings in vietnam. (5) quantitative research method: collecting information and data in quantitative form. this method is used in the process of applying ahp to evaluate and bank rankings in vietnam. (6) data are collected through the process of surveying and interviewing representatives of banks chief executives, managers and staff; practicing outdoor activities; company file documents; journal and newspapers. int. j. anal. appl. 17 (1) (2019) 137 ii. literature review 2.1 basic concepts the application of ahp in banking sector is growing most recently, and it is being combined with conventional bank evaluation parameters in this study. first, the main and sub criteria for the evaluation of banks performance are discussed along with the alternative banks. then the literature for the selection of banks through its performance is given. decision-making problem the availability of more choices makes the process of decision making complicated. thus it becomes very arduous task to select from the array of choices. the problem becomes even more gigantic in case of emerging a fierce competition among banks. decision making process thus becomes a complicated phenomenon when the best alliance can help bank increase competitive advantage and survive through competition. many factors are involved in choosing a partner thus selection of best banks to form alliances will fall into the category of multi-criteria analysis problem. 2.2 decision-making using the analytic hierarchy process in this section, we will describe problem with the analytic hierarchy process which include its concept, functions, basic scales, practical applications, and illustrative examples. finally, we analyze the advantages and limitations of ahp method. 2.2.1 concepts in previous studies, ahp was implemented to help decision maker to choose the best solution among several alternatives across multiple criteria. decision-making is related to the level of intelligence, wisdom and creativity to satisfy basic needs, to have better selective choices and to increase productivity for the enterprises. evaluating a decision requires several considerations such as the benefits derived from making the right decision, the costs, the risks, and losses resulting from the actions taken if the wrong decision is made. decision-making methods range from variety of choices in order to use more suitable decision-making tools. in the 1970s, thomas saaty developed ahp as a way of making decision dealing with weapons tradeoffs, resource and asset allocation when he was a int. j. anal. appl. 17 (1) (2019) 138 professor at the wharton school of business and a consultant with the arms control disarmament agency. 2.2.2 function of ahp ahp is a time-tested method that has been used to decide for many successful businesses worldwide. it uses the judgments of decision makers to form a decomposition of problems into hierarchies. problem complexity is represented by the number of levels in the hierarchy which combine with the decision-makers model of the problem to be solved [12]. the hierarchy, as shown in figure 1, is used to derive ratio-scaled measures for decision alternatives and the relative value that alternatives have against organizational goals (customer satisfaction, product/service, financial, human resource, and organizational effectiveness) and project risks. ahp uses matrix algebra to sort out factors to arrive at a mathematically optimal solution and derives ratio scales from paired comparisons of factors and choice options. ahp consists of four steps [20]. in the first step, the author defines the problem and state the goal or objective. in part two, the criteria or factors that influence the goal are made clear. in this step, the structure of these factors into levels and sublevels are also formed. in part three, the author uses paired comparisons of each factor with respect to each other that forms a comparison matrix with calculated weights, ranked eigen values, and consistency measures. in the final step, synthesize the ranks of alternatives until the final choice is made. figure 2.1: ahp hierarchy objective criterion 1 criterion 2 criterion 3 criterion 4 selection 1 selection 2 selection 3 int. j. anal. appl. 17 (1) (2019) 139 2.2.3 ahp basic scales the paired comparison scales between the comparison pair (aij) of two items (item i and item j) is as follows: (itemi) 9-8-7-6-5-4-3-2-1-2-3-4-5-6-7-8-9 (item j) the preference scale for pair-wise comparisons of two items ranges from the maximum value 9 to 1/9 (0.111 in decimal from). let aij represent the comparison between item-i (left) and item-j (right). if item-i is 5 times (strong importance) more important than item-j for a given criteria or product, then the comparison aji = 1/aij = 1/5 (0.200) or the reciprocal value for the paired comparison between both items. after the comparison matrix is formed, ahp terminates by computing an eigenvector (also called a priority vector) that represents the relative ranking of importance (or preference) attached to the criteria or objects being compared. the largest eigenvalue provides a measure of consistency. consistency is a matrix algebraic property of cardinal transitivity where the equality a(ij) = 1/a(ji) = a(ji)-1, and a(ij) = a(ik) a(kj) for any index i, j, k. inconsistencies arise if the transitive property is not satisfied as determined when the largest eigen value from the comparison matrix far exceeds the number of items being compared. the ahp preference scale shows in table 2.1 to form the comparison matrices [12]. table 2.1: preferences made on 1-9 scale ahp scale of importance for comparison pair (aij) numeric rating reciprocal (decimal) extreme importance 9 1/9 (0.111) very strong to extremely 8 1/8 (0.125) very strong importance 7 1/7 (0.143) strongly to very strong 6 1/6 (0.167) strong importance 5 1/5(0.200) moderately to strong 4 1/4(0.250) moderate importance 3 1/3(0.333) equally to moderately 2 1.2(0.500) equal importance 1 1(1.000) int. j. anal. appl. 17 (1) (2019) 140 the geometric mean is an alternative measure of the priority and was formed by taking the n-th root of the product matrix of row elements divided by the column sum of row geometric means. the geometric mean agrees closely with the priority. lambdamax (4.2385) is an eigen value scalar that solved the characteristic equation of the input comparison matrix. ideally, the lambdamax value should equal the number of factors in the comparison (n=4) for total consistency. the consistency index (ci) measures the degree of logical consistency among pair-wise comparisons. the random index (ri) is the average ci value of randomly-generated comparison matrices using saaty’s preference scale (table 3) sorted by the number of items being considered. if |ci| <0.05, it shows good consistency of pair-wise comparisons. if |ci|>0.05 1 means the pair-wise comparison should be revised. ci = (λ max − n) (n − 1) consistency ratio (cr) indicates the amount of allowed inconsistency (0.10 or 10%). higher numbers mean the comparisons are less consistent. smaller numbers mean comparisons are more consistent. crs above 0.1 means the pair-wise comparison should be revisited or revised. 𝐶𝑅 = |ci| ri random index (ri) is the average value of ci for random matrices using the saaty scale obtained by forman (geoff, 2004). to determine the goodness of ci, ahp compares it by random index (ri), and the result is what we call consistency ratio (cr). random index is the consistency index of a randomly generated reciprocal matrix from the scale 1 to 9 (geoff, 2004). table 2.4 below shows the values r.i. sorted out by order 1 to 15 matrix. the cr can then be calculated. table 2. 2: ri index n= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ri = 0.0 0 0.0 0 0.5 8 0.9 0 1.1 2 1.2 4 1.3 2 1.4 1 1.4 5 1.4 9 1.5 1 1.4 8 1.5 6 1.5 7 1.5 9 int. j. anal. appl. 17 (1) (2019) 141 iii. methodology 3.1 selection of banks for the purpose of the study in this paper expert opinion is collected for the generation of criteria and sub criteria weights through a questionnaire containing fuzzy pair wise comparisons using linguistic terms. further the alternative banks are given weights based on the size of the stock market. ten banks were selected purposively for the purpose of the study. the banks selected for the purpose for the study are traded in hanoi and ho chi minh city stock markets whose selection of the index set is based on the following criteria, which are referenced from thebanker.com – 500 banking brands in 2014: 1. company's market capitalization rank in the universe should be less than 500 2. company's turnover rank in the universe should be less than 500 3. company's trading frequency should be at least 90% in the last six months. 4. company should have a positive net-worth. 5. a company that comes out with an initial public offering (ipo) will be eligible for inclusion in the index, if it fulfills the normal eligibility criteria for the index for a 3 month period instead of a 6 month period. the banks selected for the purpose of the study are bidv, vietinbank, acb, sacombank, dab, hdbank, seabank, sgb, mbb, and shb (listed in table 3.1). moreover, the author has many friends who are current working in the banking system. the banks are selected to administer survey questionnaires are sgb (ho chi minh city branch); dab (ho chi minh city branch); and vietcombank (ho chi minh city branch). customers who have high frequent bank transactions were also invited to participate in this study. the details about participants who are both experts and customers are listed in table 3.4. int. j. anal. appl. 17 (1) (2019) 142 table 3. 1: list of selected banks code full name stock market bidv joint stock commercial bank for investment and development of vietnam http://goo.gl/q4bpq8 vietinbank vietnam joint stock commercial bank for industry and trade http://goo.gl/uoagub acb asia commercial bank http://goo.gl/e06zxg sacombank sai gonthuong tin commercial joint stock bank http://goo.gl/0i5ggg dongabank dong a commercial joint stock bank http://goo.gl/dah42k hdbank ho chi minh development joint stock commercial bank http://goo.gl/b93bej seabank southeast asia commercial joint stock bank http://goo.gl/6cjgbr sgb saigon bank for industry and trade http://goo.gl/cn1cko mbb military commercial joint stock bank http://goo.gl/qywcm5 shb saigon hanoi commercial joint stock bank http://goo.gl/unukss 3.2 evaluation criteria and sub-criteria the first step of the proposed model is to determine all the important criteria and their relationship with the decision variables in the form of a hierarchy. this step is crucial because the selected criteria can influence the final choice. these questions are always raised whenever we have contacts with the people we want to survey on. and these are asked by short interviews. this step is crucial because it can raise that the data used in this study is provided and confirmed by the experts in the field of banking and customers using banking services. 1. are you an expert in this field, working in it daily? 2. do you work in this field occasionally? 3. are you knowledgeable about this field through occasional professional reading? 4. would you classify yourself as an informed layman? 5. are you uninformed about this field? int. j. anal. appl. 17 (1) (2019) 143 the first round questionnaire was implemented in november 4th to november 8th 2014. in the first round, the panellists were asked to suggest new or current criteria in the job of ranking banks. many new suggestions were received. they were asked to review the list below and provide their judgments about their likelihood and impacts in the table 3.2. then these factors are arranged into the 5 main criteria including: income, expenditure, staff, security and atm services. table 3.2: summary of parameters to evaluate bank performance parameters weighted average frequency (expert) percentage safety of funds 1.50 221 2.9404% secured atms 1.60 212 2.8206% atm availability 1.61 201 2.6743% reputation 1.61 120 1.5966% personal attention 1.65 012 0.1597% pleasing manners 1.66 100 1.3305% confidentiality 1.67 021 0.2794% closeness to work 1.69 003 0.0399% timely service 1.70 321 4.2709% friendly staff willing to help 1.71 101 1.3438% clear communication 1.74 102 1.3571% higher rate of intdeposits 1.74 101 1.3438% size of the bank 1.74 121 1.6099% quick/prompt service 1.75 012 0.1597% minimum waiting time 1.75 101 1.3438% convenient working hour 1.75 012 0.1597% more no. of branches 1.78 011 0.1464% int. j. anal. appl. 17 (1) (2019) 144 good complaint handling 1.80 101 1.3438% any branch banking 1.81 013 0.1730% modern looking(building) 1.83 013 0.1730% prompt response 1.83 101 1.3438% ease contact branch manager 1.83 104 1.3837% user friendly atms 1.84 207 2.7541% brand name 1.84 011 0.1464% interest expenditure 1.85 301 4.0048% connectivity to other bank's atms 1.87 344 4.5769% accuracy/absence of errors 1.89 344 4.5769% no breakdown of machine 1.90 011 0.1464% closeness to home 1.90 001 0.0133% delivering what is promised 1.90 015 0.1996% dependability 1.90 025 0.3326% secured internet banking 1.98 358 4.7632% employees dress & appearance 2.01 012 0.1597% user friendly net banking 2.03 106 1.4103% salary account 2.04 014 0.1863% easy connectivity 2.05 296 3.9383% higher rate of intloans 2.06 014 0.1863% int. j. anal. appl. 17 (1) (2019) 145 other income 2.07 345 4.5902% debit card 2.09 104 1.3837% low/reasonable service-charges 2.10 044 0.5854% advertisement 2.11 042 0.5588% investment 2.11 258 3.4327% staff knowledge 2.12 268 3.5657% innovative services 2.14 355 4.7233% error free net banking 2.16 011 0.1464% advances 2.17 233 3.1001% internet banking 2.18 100 1.3305% businesses/employee 2.20 333 4.4305% depository services 2.22 053 0.7052% phone banking 2.25 044 0.5854% profit/employee 2.26 257 3.4194% credit card 2.30 035 0.4657% one stop banking 2.33 111 1.4768% operating expenditure 2.33 333 4.4305% other it based services 2.44 014 0.1863% interest income 2.53 385 5.1224% friend's referral 2.61 014 0.1863% my father's bank 2.62 014 0.1863% after we have 5 main criteria, the experts were asked to list down sub-criteria of each main criterion. this process is called second round selection, which are listed in the following tables. this process will be taken placed right after we summarized the main criteria, which was during the period of 13 november to 16 november, 2014. moreover, they are also asked to fulfil one more point before we go the survey of ahp to do the ranking of the field. from table 3.13, the author takes only the parameters which are evaluated by experts and its frequency of important about more 2%. table 3.3 lists 19 parameters which are more important and used in this study. this process is important, so that the author has seriously int. j. anal. appl. 17 (1) (2019) 146 considered to be used in the ahp method. these 19 parameters are put into 5 main criteria, which are also by survey experts and mentioned earlier. table 3.3: 19 main parameters parameters weighted average frequency (expert) percentage safety of funds 1.50 221 2.9404% secured atms 1.60 212 2.8206% atm availability 1.61 201 2.6743% timely service 1.70 321 4.2709% user friendly atms 1.84 207 2.7541% interest expenditure 1.85 301 4.0048% connectivity to other bank's atms 1.87 344 4.5769% accuracy/absence of errors 1.89 344 4.5769% secured internet banking 1.98 358 4.7632% easy connectivity 2.05 296 3.9383% other income 2.07 345 4.5902% investment 2.11 258 3.4327% staff knowledge 2.12 268 3.5657% innovative services 2.14 355 4.7233% advances 2.17 233 3.1001% businesses per employee 2.20 333 4.4305% profit per employee 2.26 257 3.4194% operating expenditure 2.33 333 4.4305% interest income 2.53 385 5.1224% int. j. anal. appl. 17 (1) (2019) 147 the hierarchy is structured from the top (the overall goal of the problem) through the intermediate levels (criteria and sub-criteria on which subsequent levels depend) to the bottom level (the list of alternatives). the structure of the above-mentioned hierarchy is given in figure 3.1. figure 3.1 just summarizes and visualizes what have mentioned. we have here 5 main criteria, 19 sub-criteria and 10 alternatives i.e. banking brands. figure 3.1: research hierarchy (2) operating ex. ranking vietnamese banks income (1) expenditure (2) staff (3) security (4) atm services (5) (1) investment (1) advances (1) interest income (1) other income (2) interest ex. (3) businesses/employee (3) profit/employee (3) staff knowledge (3) timely services (4) safety of funds (4) secured atms (4) secured i-banking (4) accuracy (5) availability (5) user friendly (5) connectivity (5) innovation (5) atm quality bidv vietinbank acb sacombank dab hdbank seabank sgb mbb shb int. j. anal. appl. 17 (1) (2019) 148 table 3.4: descriptions of participants into selecting research elements positions financial market number gender years of working working and professional experience management board 1 male 8 tracking the evolution of the project and monitor the operation of financial organizations. 2 female 6 being responsible for management of the sales stages, maintaining the operation of financial companies 3 male 7 managing and monitoring the contracts with suppliers, partners and other out sources. group leaders 4 male 3 responsible for reporting status financial products/services to higher levels. responsible for the formulation according to reports in financial companies; criteria given by the managers. 5 male 4 6 male 3 7 female 4 store managers (5people) 8 female male 1-3 time management of shipping – delivering financial products/services. counting and reporting to high levels about the status. sale person (5 people) 9 female male 0.5-3 selling and marketing products/services to customers. monitoring the interaction process. customers 97 female male have used banking services for years int. j. anal. appl. 17 (1) (2019) 149 these experts are working in the financial organizations e.g., prudential; bao viet insurance, aia vietnam etc. they are all anonymous in this study. customers are described to use the banking services for years. they are employees in organizations in hanoi city. they are the researchers’ friends and are willing to participate in this study. these organizations have the connection with banks in this study. they have the salary paying interactions by months. 3.4 the process to select the right target of the ahp method this section presents the process according to the method of calculation process of ahp. start with a hierarchical diagram level 5 main criteria governing the evaluation of the bank industry (see figure 3.1). this matrix shows the relationship between the main criteria according to the scale of the ahp. based on this table can determine the correlation between the level of importance of the variables. the whole process of this study is present in figure 3.2. there are 9 phases to run, select and analysis based on the applied method – ahp. figure 3.2: flowchart of phases to carry research 1. data collection 2. verifying data initially (1) no 3. sorting data based on ahp 4. arranging data into dss of excel 5. verifying data (2): calculating, comparing, ranking… qualified 6. converting into ahp scale 7. creating and putting data into ahp matrix tables 8. calculating matrix by using set-up formulas in excel 9. finalizing and reporting results int. j. anal. appl. 17 (1) (2019) 150 phase 1: data collection delphi method: 1st round (14-18th november, 2014); 2nd round (13-16 december, 2014) ahp questionnaire: 18-25th january, 2015 the process of data collection is carried out according to the method of experts: step 1: based on assessment model has been developed, author use pilot interviews to experts to verify the appropriateness of the 6 main criteria kpi in level one and 16 in level two, together with confirming the identification actual business reality. step 2: based on the results of step 1 to adjust the model and building surveys/questionnaires. step 3: surveys combined with direct interviews to each expert. at the request should have over 30 experts but by the actual situation should be reduced to 26experts. moreover, customers who have used banking services for years are also involved in this study. step 4: collect other data through reports and documents related phase 2: a batch process data after collecting a full range of primary data through surveys and interviews with experts’ opinions, together with the secondary data through reports: financial, production, performance, etc., these data are processed through the steps of a batch filtering criteria of the main criteria, then filter by the kpi side, verify authenticity versus reality and reliability of the data. phase 3: sorting data and kpi main criteria of the model a batch is classified according to six main criteria, and then they are further classified according to 16 kpi sub-criteria of the model phase 4: enter the model created by excel excel is as a dss generator: it is used to construct the computational model,and to handle data for each kpi fitting main criteria in the evaluation model. after setup is complete, the dss model is to conduct verification of scale, the formula to ensure the appropriateness and scientific. then, we enter the processed and classified data prior to the dss model to prepare for phase 5. int. j. anal. appl. 17 (1) (2019) 151 phase 5: data processing secondly after entering the data, we conducted calculations, check, and then evaluate for each scale suitably. phase 6: scale transition to ahp ahp method using pair wise comparison scale separately on a scale from one to nine, so after the calculation results ranking suppliers, we must make the transition scales corresponding to the ahp under the own standards of this method. this stage aims to prepare for entering data into the matrix of pair wise comparison at a later stage. phase 7: creating and putting data into ahp matrix table the first boot after the application of the ahp hierarchy drawing assessment model, enter the criteria in a level floor, then to the sub-criteria floor level to level two to n, and finally enter choice alternatives. then enter the data were processed in each pair wise comparison matrices, respectively. inside is under processing data has been entered into the pair wise comparison matrix between the six main criteria in the evaluation model. phase 8: calculating matrix by using set-up formulas in excel after we have the ahp tables, we use excel as a tool to calculate these matrices. inputting all the data surveyed is a careful step to do to make sure that the calculation is accurate. phase 9: finalizing and reporting results this is the final step of the process. we just see and report results. this will be illustrated carefully in chapter 4. once the hierarchy is established, the fuzzy pair wise comparison takes place. the experts compare all the criteria on the same level of the hierarchy. a pair wise comparison is performed by using fuzzy linguistic terms in the scale of 0 – 10 described by the triangular fuzzy numbers in the table 1.1. in buckley's method, the element of the negative judgment is treated as an inverse and reversed order of the fuzzy number of the corresponding positive judgment. thus it requires not only a rigorous manipulation in the construction of reciprocal matrix but also due to transitivity the result becomes inconsistent. again to reflect pessimistic, most likely and optimistic decision making environment, triangular fuzzy numbers with minimum value, most plausible value & maximum value are considered. int. j. anal. appl. 17 (1) (2019) 152 𝐴 = ( 1 𝑎12 𝑎1𝑛 𝑎21 1 𝑎2𝑛 𝑎31 𝑎32 𝑎3𝑛 ) to simplify the calculation of element weight the fuzzy pair wise comparison matrix is broken into crisp matrices where the crisp matrices are formed by taking the minimum values, most plausible values & maximum values from the triangular fuzzy numbers, which were mentioned in chapter 2. iv. results 4.1 setting stage in this chapter, the whole process of calculation will be analyzed. then the results of each weight of the alternatives will be illustrated. and finally, the final results of selection the right instant coffee supplier will be displayed according to experts’ interview. comparable data are collected by the method of survey experts through interviews and direct the relevant agencies. homogeneity index (incon) 0.05 of ahp is satisfactory. the main criteria are comparable bond correlation pairs separate to produce detailed data calculations. the tables above are typical illustrations for pair wised comparison matrices need to enter the data set gathered from interviews of experts in the relevant industry. there are 23 matrices developed to cater for the processing of the data model. and following authors quote a matrix in which to further illustrate this problem. denoting: income – ic expenditure – exp staff – st security – sec atm service – atm int. j. anal. appl. 17 (1) (2019) 153 table 4.1: matrix of pair wise comparison criteria ic exp st sec atm ic 1.0110 2.00000 2.00000 2.00000 4.00000 exp 0.50 1.00 1.00000 0.20000 2.00000 st 0.50 1.00 1.00 0.33333 1.00000 sec 0.50 5.00 3.00 1.00 4.00000 atm 0.25 0.50 1.00 0.25 1.00 total 2.7500 9.50 8.00 3.7833 12.00 table 4.1 gives a typical example of how to input interviewed data into the matrix of ahp. for example, income is 2 times important than expenditure, then expenditure is equal 0.5 of income. table 4.2: results of first phase calculation criteria ic exp st sec atm weight ic 0.3636 0.2105 0.2500 0.5286 0.3333 33.7% exp 0.1818 0.1053 0.1250 0.0529 0.1667 12.6% st 0.1818 0.1053 0.1250 0.0881 0.0833 11.7% sec 0.1818 0.5263 0.3750 0.2643 0.3333 33.6% atm 0.0909 0.0526 0.1250 0.0661 0.0833 8.4% total 1 1 1 1 1 100% after doing calculation from table 4.1, each cell is done by choice divided to the cell total value of the matrix. for example, we have 0.3243 = 1/3.0833. then, the weight is the average of each row, which is total divided to the number of criteria. this weight will be used to calculated the second phase of the matrix, which is illustrated in the below table. int. j. anal. appl. 17 (1) (2019) 154 table 4.3: results of second phase calculation criteria ic exp st sec atm sum sum/weight ic 0.3372 0.2526 0.2334 0.6723 0.3344 1.83 5.43 exp 0.1686 0.1263 0.1167 0.0672 0.1672 0.65 5.11 st 0.1686 0.1263 0.1167 0.1121 0.0836 0.61 5.20 sec 0.1686 0.6316 0.3501 0.3362 0.3344 1.82 5.42 atm 0.0843 0.0632 0.1167 0.0840 0.0836 0.43 5.17 total 0.93 1.20 0.93 1.27 1.00 5.34 26.33 average 0.19 0.24 0.19 0.25 0.20 1.067 5.27 from table 4.1 and table 4.3, we have each criterion value calculation. then, we come up with sum and sum/weight. sum/weight is an important element to calculation lambda max and ci with cr factors, which then are used to test the consistency of the matrix and calculation. λmax = ∑ sum weight n = 26.33 5 = 5.27 ci = λmax − n n − 1 = 5.27 − 5 5 = 0.066 ci=0.066<0.05, it shows good consistency of pair-wise comparisons. cr = |ci| ri = 0.066 1.12 = 0.059 as mentioned in chapter 2, there are 6 criteria so ri=1.12. cr=0.059 = 5.9% <10%, that means consistent. table 4.4: results from matrix of sub-criteria under income income investment advances interest income other income weight investment 0.7039 0.8077 0.6316 0.5000 66.1% advances 0.1006 0.1154 0.2105 0.3333 19.0% interest income 0.1173 0.0577 0.1053 0.1111 9.8% other income 0.0782 0.0192 0.0526 0.0556 5.1% total 1 1 1 1 100% int. j. anal. appl. 17 (1) (2019) 155 cr=0.083 = 8.3% <10%, that means consistent. from table 4.3, we have the weight of income is 33.7% over 100% of 5 main criteria (1-level). then, we have the weight of each sub-criterion from table 4.4. here, we come to the table showing the percentage of sub-criterion over the whole picture to choose the supplier. table 4. 5: the global percentage of each sub-criterion under income income weight (local) weight of fc weight of subcriterion (global) investment 66.1% 33.7% 22.27% advances 19.0% 33.7% 6.40% interest income 9.8% 33.7% 3.30% other income 5.1% 33.7% 1.73% total 100% -33.7% int. j. anal. appl. 17 (1) (2019) 156 table 4.6: results from matrix of the alternatives under sub-criterion -investment sub-criterion (investment) bidv vietinbank acb sacombank dab hdbank seabank sgb mbb shb weight bidv 0.0727 0.0643 0.2130 0.1094 0.1541 0.0317 0.02 0.10 0.14 0.06 9.6% vietinbank 0.0364 0.0322 0.0106 0.0122 0.0110 0.1270 0.26 0.05 0.18 0.01 7.4% acb 0.0182 0.1609 0.0532 0.0219 0.0193 0.1905 0.18 0.21 0.09 0.02 9.6% sacombank 0.0727 0.2895 0.2662 0.1094 0.1541 0.2540 0.02 0.10 0.14 0.22 16.3% dab 0.0364 0.2252 0.2130 0.0547 0.0771 0.1905 0.03 0.03 0.02 0.22 11.0% hdbank 0.1455 0.0161 0.0177 0.0273 0.0257 0.0635 0.18 0.21 0.09 0.06 8.3% seabank 0.29 0.01 0.03 0.55 0.23 0.03 0.09 0.21 0.14 0.03 16.0% sgb 0.04 0.03 0.01 0.05 0.15 0.02 0.02 0.05 0.14 0.17 6.8% mbb 0.07 0.01 0.03 0.04 0.15 0.03 0.03 0.02 0.05 0.17 5.9% shb 0.22 0.16 0.16 0.03 0.02 0.06 0.18 0.02 0.02 0.06 9.1% total 1 1 1 1 1 1 1 1 1 1 100% int. j. anal. appl. 17 (1) (2019) 157 cr=0.083 = 8.3% <10%, that means consistent. with the same process, from tables 4.5 and 4.6 we can calculate the whole percentage of choice for each supplier under each sub-criterion, which are investment (22.27%); advances: 6.4%; interest income: 3.3%; and other income: 1.73% table 4.7: global percentage of each supplier under investment of income sub-criterion (investment) weight (local) weight of subcriterion (investment) weight of supplier (global) bidv 9.6% 22.27% 2.15% vietinbank 7.4% 22.27% 1.65% acb 9.6% 22.27% 2.13% sacombank 16.3% 22.27% 3.62% dab 11.0% 22.27% 2.44% hdbank 8.3% 22.27% 1.84% seabank 16.0% 22.27% 3.56% sgb 6.8% 22.27% 1.52% mbb 5.9% 22.27% 1.31% shb 9.1% 22.27% 2.03% total 100% -22.27% these are examples what thesis does and gets to have the data from interviews, and surveys of experts. 4.2 results and analyses five suppliers by each criterion as mentioned earlier, the steps to calculate by apply ahp which are from the main criteria to all the alternatives. table 4.8 just summarizes the results of the 1 step which is the weights of main criteria. table 4.8: main criteria weights main criteria sum sum/weight global weight ic 1.83 5.43 33.7% exp 0.65 5.11 12.6% st 0.61 5.20 11.7% sec 1.82 5.42 33.6% atm 0.43 5.17 8.4% int. j. anal. appl. 17 (1) (2019) 158 according experts and customers surveyed in this study, the incomes and security of the bank are the highly important factors at 33.7% and 33.6%. atm services are at lowest percentage (8.4%). this will be discussed in the chapter 5 in which the author would like to make conclusions and research suggestions. under their main criteria, the sub-criteria are calculated their global weights. for example, the main criteria of investment is income (at 33.7%) and the investment local weight is at 66.1% so its global weight is 22.7% (=66.1%*33.7%). table 4.9: sub-criteria weights sub-criteria sum sum/weight local weights main criteria global weights investment 3.04 4.60 66.1% income 33.7% 22.27% advances 0.79 4.15 19.0% 6.40% interest income 0.41 4.15 9.8% 3.30% other income 0.21 4.00 5.1% 1.73% interest ex. 1.33 2.00 66.7% expenditure 12.6% 8.40% operating ex. 0.67 2.00 33.3% 4.20% biz/employee 0.89 4.17 21.4% staff 11.7% 2.51% profit/employee 0.63 4.27 14.9% 1.74% staff knowledge 0.52 4.11 12.6% 1.47% timely service 2.16 4.23 51.2% 5.99% safety of funds 1.53 4.16 36.8% security 33.6% 12.36% secured atms 0.75 4.10 18.4% 6.19% secured ibanking 1.16 4.09 28.5% 9.56% accuracy 0.67 4.13 16.3% 5.48% availability 0.52 5.10 10.1% atm 8.4% 0.85% user friendly 0.80 5.13 15.6% 1.31% connectivity 1.98 5.40 36.6% 3.08% innovation 1.25 5.18 24.1% 2.03% atm quality 0.70 5.14 13.6% 1.14% the sub-criteria are very important to calculate out the evaluations then rankings, which are mentioned in later parts. int. j. anal. appl. 17 (1) (2019) 159 table 4.10: summary of evaluation process bidv vietinbank acb sacombank dab hdbank seabank sgb mbb shb investment (22.7%) 2.11% 1.77% 2.98% 1.97% 3.13% 1.88% 2.87% 1.66% 1.46% 2.44% advances (6.40%) 0.47% 0.67% 0.82% 0.70% 0.88% 0.50% 0.87% 0.44% 0.37% 0.67% interest income (3.30%) 0.31% 0.36% 0.31% 0.45% 0.41% 0.26% 0.49% 0.23% 0.19% 0.30% other income (1.73%) 0.21% 0.15% 0.16% 0.19% 0.27% 0.14% 0.23% 0.14% 0.10% 0.14% interest ex. (8.40%) 1.09% 1.12% 0.93% 0.91% 1.21% 0.55% 0.91% 0.66% 0.48% 0.55% operating ex. (4.20%) 0.44% 0.41% 0.62% 0.47% 0.44% 0.27% 0.35% 0.43% 0.36% 0.41% biz/employee (2.51%) 0.27% 0.17% 0.24% 0.29% 0.37% 0.21% 0.29% 0.24% 0.24% 0.19% profit/employee (1.74%) 0.16% 0.14% 0.19% 0.15% 0.26% 0.20% 0.21% 0.15% 0.12% 0.16% staff knowledge (1.47%) 0.14% 0.13% 0.16% 0.15% 0.14% 0.14% 0.13% 0.12% 0.17% 0.19% timely service (5.99%) 0.60% 0.58% 0.76% 0.82% 0.55% 0.51% 0.64% 0.41% 0.45% 0.66% safety of funds (12.36%) 1.25% 1.13% 1.97% 1.43% 1.26% 1.09% 1.09% 0.80% 1.25% 1.10% secured atms (6.19%) 0.79% 0.37% 0.63% 0.59% 0.80% 0.55% 0.93% 0.44% 0.55% 0.53% secured i-banking (9.56%) 0.98% 0.94% 1.68% 1.09% 0.85% 0.84% 1.01% 0.72% 0.52% 0.92% int. j. anal. appl. 17 (1) (2019) 160 accuracy (5.48%) 0.53% 0.56% 0.64% 0.64% 0.51% 0.50% 0.48% 0.35% 0.63% 0.63% availability (0.85%) 0.11% 0.07% 0.08% 0.09% 0.10% 0.07% 0.11% 0.05% 0.08% 0.07% user friendly (1.31%) 0.17% 0.09% 0.12% 0.13% 0.15% 0.12% 0.19% 0.11% 0.11% 0.12% connectivity (3.08%) 0.38% 0.27% 0.29% 0.33% 0.48% 0.25% 0.41% 0.24% 0.18% 0.25% innovation (2.03%) 0.24% 0.22% 0.25% 0.22% 0.19% 0.18% 0.21% 0.13% 0.18% 0.21% atm quality (1.14%) 0.13% 0.14% 0.15% 0.12% 0.11% 0.09% 0.12% 0.09% 0.08% 0.12% total 10.38% 9.29% 12.98% 10.74% 12.11% 8.35% 11.54% 7.41% 7.52% 9.66% int. j. anal. appl. 17 (1) (2019) 161 4.3 the final rankings after respectively calculating, analysis and evaluating of suppliers through each subcriterion of six main criteria in balanced scorecard of ahp model, we have been solving the second floor of ahp hierarchy. and this is the final calculation results which are obtained after running the data through the two floors of the criteria assessment model according to the method of ahp. the percentages are of banks shown in the table. based on these values, we can rank as well as further analysis of the selected alternatives. plus we can evaluate each bank. besides, to compare the degree of difference between the alternatives, any financial organizations can make a decision in choosing the best suppliers and the most suitable. table 4. 10: the final rankings ranking banks global weight 1 acb 12.98% 2 dab 12.11% 3 seabank 11.54% 4 sacombank 10.74% 5 bidv 10.38% 6 shb 9.66% 7 vietinbank 9.29% 8 hdbank 8.35% 9 mbb 7.52% 10 sgb 7.41% table 4.11 summarizes the final results in evaluating and rankings, which are previously detailed in table 4.10 after applying ahp method. we can see the changes of percentage of banks by criteria. acb’s percentages have change widely from each sub-criterion; finally it gets 12.98% at the top of the list. coming very closely downwards are dab, seabank etc., at the bottom of the table is sgb at 7.41%. this chapter discusses data analysis and the results of the current study. we first conduct setting to categorize the focused characteristics and steps towards this study will take place. then, the selection analysis of each supplier is summarized in detail. int. j. anal. appl. 17 (1) (2019) 162 the purpose is to find the final rankings of vietnamese banking system according to the survey results from experts. from that, the final rankings were set up to get the results, which can be further discussed in the next chapter. v. conclusions 5.1 discussions and managerial implications this final chapter will comment on the results achieved, pointed out the conclusions and recommendations presented by the author, and the limitations encountered. on the other hand, the author gives a number of research directions for the development of the subject in the future and expands the application of ahp in practice. in fact, many scholars and experts have already studied the related subjects of measurement performance, which includes the meaning of performance management, its elements and contents, and the measurement index ([21], [22],[23], [24]). on the contrary, the study of performance management is still not sufficient so far. in this study, the author conceder corporate intangible value and clearly understand performance management ability of each vietnamese banking system by ahp. besides, performance management is the key factor of high-tech companies’ operation outcome, the author hopes those results can offer performance management as reference for the academia and professionals. the results from the model are evaluated using the method of ahp quantification. ahp can compare the tiniest differences between providers through the numbers, charts and graphs. the results of detailed calculations to each level of the ladder system provide multi-faceted perspective. strong ability to synthesize the components of the hierarchy and logic algorithms are not too complicated, but also help managers can examine each aspect and see the overview are all issues are considered. in an organization that has always existed three important lines: the first line of communication throughout the system, the second is financial flows, also known simply as cash flow, and finally the material flow. purchasing is one of the important tasks of the business because it is responsible for the physical input line of the organization. increasing awareness of purchasing should be advanced position and its role in the enterprise is increasing. most organizations now recognize closely related to purchasing strategy should the company access to parts purchasing increasingly more difficult. information security requirements for these departments are increasingly stringent. int. j. anal. appl. 17 (1) (2019) 163 the process of evaluation and selection of suppliers has long held bias in a qualitative sense, dependent on experience and emotions of those who have related responsibilities. therefore, it is necessary to apply the typical methods such as quantitative analysis of this process ahp presented in this study. with the aim of increasing the computational content of the evaluation process suppliers, especially the comparison of suppliers in the same industry as ahp has shown. this enables the analysis of all the providers and more scientific. thus, this thesis would help the facility managers ensure objectivity to the reasonable decision. through the application of analytical methods to process steps or methods to compare providers evaluate other qualitative factors could improve and contribute to the financial organizations which then in the future they can apply and expand their business. moreover, the main evaluation criteria and sub-criteria have been quantified to ensure that most of the stages in the purchasing process. when evaluating partners is well supplied, all stages in the process of purchasing them achieve flawless collaboration. 5.2 limitations and future research this thesis utilizes the interview method access the expert groups and questionnaire surveys with data collected to be slightly biased and subjective experience. the data primarily comes from the documents and reports out there, not yet homogeneous. years missing data so that comparisons between providers and become limp. the process measurement data collected are processed and applied scales also unsettled. the comparison between the criteria in suppliers has not yet met the stringent requirements of the equivalent. the transformation scales to scales ahp has many limitations. it is possible to dig more theoretical model further evaluation. there are many criteria that can be used for model assessment. every type of business and every business will have specific criteria in accordance with the individual's typical enterprise. it is important to note build an assessment model provider in accordance with industry characteristics and distinctions of the business. it should be tried to reach deep to the data source to the enterprise purchasing the thesis topic under direction of this form of anonymous real close to reality than now. int. j. anal. appl. 17 (1) (2019) 164 finally, the different measures provide distinct perspectives which help us have deeper conclusion about the association between working capital management and firm performance. therefore, future researches should fill this research gap by generalizing findings using larger sample size in order to have more general, imperative vision as well as solutions for enterprises in many other fields. more measures of firm performance management as well as measurement performance components should be applied in future researches have better evaluation. 5.3 conclusions by this thesis, author would contribute to the banking system by providing the evaluating by the discussed criteria and sub-criteria. the research results suggest that performance management, which invest technology, improving quality, and structural management, is one of the main sources of competitive advantage for firms. this study argues that performance management is a necessary strategic tool for use against competitors. the emphasis on intellectual capital can help firms implement new initiatives for enhancing their performance. that means the technology on the security should be focused. moreover, many experts and customers rate the incomes of a bank is really important, so that banks should build up the structural and marketing management to boost the imcomes. other factors, including atm and staff, are chosen at the certain level to evaluate a bank. int. j. anal. appl. 17 (1) (2019) 165 references [1] yong-gang, c. u. i. (2004), a method based ahp: to choose the best strategic alliance partner [j]. journal of tianjin university of commerce, 1, 004. [2] korhonen, p., &voutilainen, r. (2006), finding the most preferred alliance structure between banks and insurance companies. european journal of operational research, 175(2), 1285-1299. [3] wang, t. c., & lin, y. l. (2009), applying the consistent fuzzy preference relations to select merger strategy for commercial banks in new financial environments. expert systems with applications, 36(3), 7019-7026. [4] n. t. nguyen and t. t. tran, optimizing mathematical parameters of grey system theory: an empirical forecasting case of vietnamese tourism. neural comput.appl., (2017), https://doi.org/10.1007/s00521-017-3058-9. [5] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci., 3(12) (2016a), 5-20. [6] seçme, n. y., bayrakdaroğlu, a., &kahraman, c. (2009), fuzzy performance evaluation in turkish banking sector using analytic hierarchy process and topsis. expert systems with applications, 36(9), 11699-11709. [7] ho, a., & baxter, r. a. (2011), banking reform in vietnam. asia focus, (june). [8] min, d. m., kim, j. r., kim, w. c., min, d., & ku, s. (1996), ibrs: intelligent bank reengineering system. decision support systems, 18(1), 97-105. [9] n. t. nguyen and t. t. tran, mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dynamics in nature and society, 2015. [10] n. t. nguyen and t. t. tran, raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl., (2018b). https://doi.org/10.1007/s00521-018-3639-2. [11] kurttila, m., pesonen, m., kangas, j., &kajanus, m. (2000), utilizing the analytic hierarchy process (ahp) in swot analysis—a hybrid method and its application to a forest-certification case. forest policy and economics, 1(1), 41-52. [12] saaty, t. l. (2008), decision making with the analytic hierarchy process.international journal of services sciences, 1(1), 83-98. [13] ngai, e. w. t. (2003), selection of web sites for online advertising using the ahp. information & management, 40(4), 233-242. int. j. anal. appl. 17 (1) (2019) 166 [14] salmeron, j. l., &herrero, i. (2005), an ahp-based methodology to rank critical success factors of executive information systems. computer standards & interfaces, 28(1), 1-12. [15] yu, h. c., lee, z. y., & chang, s. c. (2005), using a fuzzy multi-criteria decision making approach to evaluate alternative licensing mechanisms. information& management, 42(4), 517-531. [16] korhonen, p., & voutilainen, r. (2006), finding the most preferred alliance structure between banks and insurance companies. european journal of operational research, 175(2), 1285-1299. [17] seçme, n. y., bayrakdaroğlu, a., &kahraman, c. (2009), fuzzy performance evaluation in turkish banking sector using analytic hierarchy process and topsis. expert systems with applications, 36(9), 11699-11709. [18] arbel, a., &orgler, y. e. (1990), an application of the ahp to bank strategic planning: the mergers and acquisitions process. european journal of operational research, 48(1), 27-37. [19] chen, z., & tan, j. (2011), does bancassurance add value for banks? evidence from mergers and acquisitions between european banks and insurance companies. research in international business and finance, 25(1), 104-112. [20] sevkli, mehmet.,koh, s.c. lenny., zaim, selim., demirbag, mehmet., tatoglu, ekrem. (2008), hybrid analytical hierarchy process model for supplier selection. industrial management & data systems, vol. 108 no. 1. pages 122–142. [21] t. t. tran, forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci., 4(9) (2017), 8695. [22] t. t. tran, a strategic alliance study by performance evaluation and forecasting techniques: a case in the petroleum industry. int. j. adv. appl. sci., 5(2) (2018), 136147. [23] n. t. nguyen and t. t. tran, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci., 5(9) (2018a), 73-81. [24] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci., 3(12) (2016b), 5-20. int. j. anal. appl. (2022), 20:58 new generalizations of sup-hesitant fuzzy ideals of semigroups uraiwan jittburus1, pongpun julatha1,∗, attaphol pumila1, napaporn chunsee2, aiyared iampan3, rukchart prasertpong4 1faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand 2faculty of science and technology, uttaradit rajabhat university, uttaradit 53000, thailand 3fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, phayao 56000, thailand 4division of mathematics and statistics, faculty of science and technology, nakhon sawan rajabhat university, nakhon sawan 60000, thailand ∗corresponding author: pongpun.j@psru.ac.th abstract. as general concepts of sup-hesitant fuzzy right (resp., left, interior, two-sided) ideals of semigroups, the concepts of sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideals and sup − β hesitant fuzzy right (resp., left, interior, two-sided) ideals are introduced and their properties are investigated. then, the concepts are established by fuzzy sets, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets, pythagorean fuzzy sets, hesitant fuzzy sets, hybrid sets, interval-valued fuzzy sets and cubic sets. finally, we characterize which is intra-regular, completely regular, simple semigroups or another type of semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. 1. introduction the fuzzy set theory presented by zadeh [46] has been successfully and widely applied in many areas such as robotics, expert, computer science, finite state machine, control engineering, logic theory, automata theory, group theory, graph theory and semigroup theory. furthermore, in the literature, a number of concepts of fuzzy sets and their generalizations and extensions have been received: aug. 24, 2022. 2010 mathematics subject classification. 03e72, 08a72, 20m12. key words and phrases. semigroup; sup-hesitant fuzzy ideal; generalized sup-hesitant fuzzy ideal; łukasiewicz fuzzy set; fuzzy ideal; hesitant fuzzy ideal; interval-valued fuzzy ideal. https://doi.org/10.28924/2291-8639-20-2022-58 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-58 2 int. j. anal. appl. (2022), 20:58 introduced and studied, for instance, łukasiewicz fuzzy sets [23], łukasiewicz anti-fuzzy sets [22], anti-type of fuzzy sets [25, 37], negative fuzzy sets [18], bipolar fuzzy sets [29, 48], interval-valued fuzzy sets [47], intuitionistic fuzzy sets [4], pythagorean fuzzy sets [44,45], rough sets [2,34], hesitant fuzzy sets [41,42], cubic sets [19] and hybrid sets [1,24]. on semigroups, kuroki [27, 28] applied fuzzy sets to semigroups. mordeson et al. [30] explained semigroup theory to fuzzy semigroup theory and showed their applications in coding theory, languages and fuzzy finite state machines. shabir and nawaz [37], khan and asif [25], julatha and siripitukdet [17] studied anti-type of fuzzy sets based on ideal theory in semigroups. chinnadurai and arulselvam [7] introduced pythagorean fuzzy sets based on ideal theory in semigroups and investigated their properties. narayanan and manikantan [33], and thillaigovindan and chinnadurai [40] studied intervalvalued fuzzy sets in semigroups. jun and khan [20], umar et al. [43], and muhiuddin [31] studied cubic sets in semigroups. anis et al. [1], elavarasan et al. [9] studied hybrid sets in semigroups. jun et al. [21] and talee et al. [39] studied hesitant fuzzy sets in semigroup. studying hesitant fuzzy sets, in the meaning of the supremum of their images, on semigroups, jittburus and julatha [12] introduced sup-hesitant fuzzy ideals of semigroups and investigated properties via sets, fuzzy sets, interval-valued fuzzy sets and hesitant fuzzy sets. phummee et al. [35] introduced sup-hesitant fuzzy interior ideals of semigroups and studied its properties by sets, fuzzy sets, interval-valued fuzzy sets and hesitant fuzzy sets. julatha et al. [13] introduced sup-hesitant fuzzy right (left) ideals of semigroups and studied their characterizations in terms of sets, fuzzy sets, pythagorean fuzzy sets, interval-valued fuzzy sets, hesitant fuzzy sets, cubic sets and hybrid sets. many researchers have taken intense and eager interest in the novel area of hesitant fuzzy sets on algebraic structures in the meaning of the supremum of their images (see [1,10,12,14–16,32,35,36,38]). as previously stated, it motivated us to study hesitant fuzzy sets on semigroups in the meaning of the supremum of their images. we will introduce concepts of sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideals and sup− β -hesitant fuzzy right (resp., left, interior, two-sided) ideals and investigate their properties. also, we will show that every sup-hesitant fuzzy right (resp., left, interior, two-sided) ideal of a semigroup is both a sup+α-hesitant fuzzy right (resp., left, interior, two-sided) ideal and a sup− β -hesitant fuzzy right (resp., left, interior, two-sided) ideal, but the converse is not true. later, the concepts will be established by fuzzy sets, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets, pythagorean fuzzy sets, hesitant fuzzy sets, hybrid sets, interval-valued fuzzy sets and cubic sets. finally, we will characterize which is intra-regular, left (right) regular, completely regular, left (right) simple and simple semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. 2. preliminaries in this section we first give some basic definitions and results which will be used in this paper. int. j. anal. appl. (2022), 20:58 3 in what follows, unless otherwise specified, let a be a semigroup, b be a nonempty set, ℘(b) be the power set of b and h,n ∈ ℘([0, 1]). a nonempty subset b of a is called a right ideal (resp., a left ideal, an interior ideal) of a if ba⊆b (resp., ab ⊆b, aba⊆b) and an ideal of a if b is both a right ideal and left ideal of a. a fuzzy subset (fs) [46] of b is defined to be a function ξ : b → [0, 1] where [0, 1] is the unit interval. for fss ξ and η of b, define ξ ≤ η if ξ(p) ≤ η(p) for all p ∈b. a fs ξ of a is called (1) a fuzzy right ideal (fri) [30] of a if ξ(p) ≤ ξ(pq) for all p,q ∈a, (2) a fuzzy left ideal (fli) [30] of a if ξ(q) ≤ ξ(pq) for all p,q ∈a, (3) a fuzzy ideal (fi) [30] of a if it is both a frl and a fli of a, that is, max{ξ(p),ξ(q)}≤ ξ(pq) for all p,q ∈a, (4) a fuzzy interior ideal (fii) [30] of a if ξ(w) ≤ ξ(pwq) for all p,q,w ∈a, (5) an anti-fuzzy right ideal (afri) [37] of a if ξ(pq) ≤ ξ(p) for all p,q ∈a, (6) an anti-fuzzy left ideal (afli) [37] of a if ξ(pq) ≤ ξ(q) for all p,q ∈a, (7) an anti-fuzzy ideal (afi) [37] of a if it is both an afri and an afli of a, that is, ξ(pq) ≤ min{ξ(p),ξ(q)} for all p,q ∈a, and (8) an anti-fuzzy interior ideal (afii) [25] of a if ξ(pwq) ≤ ξ(w) for all p,q,w ∈a. a pythagorean fuzzy set (pfs) p [44, 45] in b is an object having the form p = {(p,ξ(p),η(p)) |p ∈b} where the functions ξ : b → [0, 1] and η : b → [0, 1] denote the degree of membership and the degree of nonmembership, respectively, and 0 ≤ (ξ(p))2 + (η(p))2 ≤ 1 for all p ∈b. for the sake of simplicity, we shall use the symbol (ξ,η) of the pfs {(p,ξ(p),η(p)) | p ∈b}. a pfs (ξ,η) in a is called (1) a pythagorean fuzzy right ideal (pfri) [7] of a if ξ is a fri and η is an afri of a, (2) a pythagorean fuzzy left ideal (pfli) [7] of a if ξ is a fli and η is an afli of a, (3) a pythagorean fuzzy ideal (pfi) [7] of a if it is both a pfri and a pfli of a, and (4) a pythagorean fuzzy interior ideal (pfii) [7] of a if ξ is a fii and η is an afii of a. by an interval number ă we mean an interval [a−,a+], where a−,a+ ∈ [0, 1] and a− ≤ a+. we denote d([0, 1]) for the set of all interval numbers. then, we obtain d([0, 1]) ⊆ ℘([0, 1]). for ă = [a−,a+], b̆ = [b−,b+] ∈d([0, 1]), the operations -, = and ≺ in case of two elements in d([0, 1]) are defined by: (1) ă b̆ ⇔ a− ≤ b− and a+ ≤ b+, (2) ă = b̆ ⇔ a− = b− and a+ = b+, and (3) ă ≺ b̆ ⇔ ă b̆ and ă 6= b̆. an interval-valued fuzzy set (ivfs) [47] on b is defined to be a function λ̆ : b →d([0, 1]), λ̆(p) 7→ [λ̆l(p), λ̆u(p)] where λ̆l and λ̆u are fss of b such that λ̆l ≤ λ̆u. for fss ξ and η of b with ξ ≤ η, 4 int. j. anal. appl. (2022), 20:58 we define the ivfs [ξ,η] on b by [ξ,η](p) = [ξ(p),η(p)] for all p ∈b. an ivfs λ̆ = [λ̆l, λ̆u] on a is called (1) an interval-valued fuzzy right ideal (ivfri) [33,40] of a if λ̆(p) λ̆(pq) for all p,q ∈a, that is, λ̆l and λ̆u are fris of a, (2) an interval-valued fuzzy left ideal (ivfli) [33,40] of a if λ̆(q) λ̆(pq) for all p,q ∈a, that is, λ̆l and λ̆u are flis of a, (3) an interval-valued fuzzy ideal (ivfi) [33,40] of a if it is both an ivfri and an ivfli of a, that is, λ̆l and λ̆u are fis of a, and (4) an interval-valued fuzzy interior ideal (ivfii) [40] of a if λ̆(w) λ̆(pwq) for all p,q,w ∈a, that is, λ̆l and λ̆u are fiis of a. a cubic set [19] in b is defined to be a function 〈λ̆,η〉 : b →d([0, 1])×[0, 1],p 7→ (λ̆(p),η(p)) where λ̆ : b →d([0, 1]) and η : b → [0, 1]. a cubic set 〈λ̆,η〉 in a is called (1) a cubic right ideal (curi) [20] of a if λ̆ is an ivfri and η is an afri of a, (2) a cubic left ideal (culi) [20] of a if λ̆ is an ivfli and η is an afli of a, (3) a cubic ideal (cui) [20] of a if it is both a curi and a culi of a, and (4) a cubic interior ideal (cuii) [31] of a if λ̆ is an ivfii and η is an afii of a. a hesitant fuzzy set (hfs) [41,42] on b is defined to be a function ε̂ : b → ℘([0, 1]). note that every ivfs on b is a hfs on b. a hfs ε̂ on a is called (1) a hesitant fuzzy right ideal (hfri) [21] of a if ε̂(p) ⊆ ε̂(pq) for all p,q ∈a, (2) a hesitant fuzzy left ideal (hfli) [21] of a if ε̂(q) ⊆ ε̂(pq) for all p,q ∈a, (3) a hesitant fuzzy ideal (hfi) [12, 21] of a if it is both a hfrl and a hfli of a, that is, ε̂(p) ∪ ε̂(q) ⊆ ε̂(pq) for all p,q ∈a, (4) a hesitant fuzzy interior ideal (hfii) [35,39] of a if ε̂(w) ⊆ ε̂(pwq) for all p,q,w ∈a. a hybrid set in a over a set b is defined to be a function (ε̂,η) : a→ ℘(b)×[0, 1],p 7→ (ε̂(p),η(p)) where ε̂ : a → ℘(b) and η : a → [0, 1]. note that every cubic set in a is a hybrid set in a over [0, 1]. a hybrid set (ε̂,η) in a over [0, 1] is called (1) a hybrid right ideal (hyri) [1] of a over [0, 1] if ε̂ is a hfri and η is an afri of a, (2) a hybrid left ideal (hyli) [1] of a over [0, 1] if ε̂ is a hfli and η is an afli of a, (3) a hybrid ideal (hyi) [1] of a over [0, 1] if it is both a hyri and a hyli of a over [0, 1], and (4) a hybrid interior ideal (hyii) [13] of a over [0, 1] if ε̂ is a hfii and η is an afii of a. for a hfs ε̂ on b, a nonempty subset z of b, k ∈ [0, 1] and h∈ ℘([0, 1]), we define (1) the element suph [12,35] of [0, 1] by suph = { suph 0 if h 6= ∅, otherwise, (2) the subset s[ε̂;h] [12,35] of b by s[ε̂;h] = {p ∈b|supε̂(p) ≥ suph}, int. j. anal. appl. (2022), 20:58 5 (3) the hfs h(ε̂,h) sup [14,16] on b by h(ε̂,h) sup (p) = {k ∈h | supε̂(p) ≥ k} for all p ∈b, (4) the characteristic hesitant fuzzy set (chfs) χ̂z of z on b by χ̂z : b → ℘([0, 1]),p 7→ { [0, 1] ∅ if p ∈z, otherwise, (5) the fs fε̂ of b by fε̂(p) = supε̂(p) for all p ∈b, (6) a supremum complement ω̂ [36] of ε̂ onb if ω̂ is a hfs onb such that supω̂(p) = 1−supε̂(p) for all p ∈b, (7) the hfs ε̂∗ by ε̂∗(p) = {1 − supε̂(p)} for all p ∈b. let sc(ε̂) be the set of all supremum complements of ε̂. then, we obtain that (1) ε̂∗ ∈sc(ε̂), (2) fω̂(p) = 1 − supε̂(p) for all ω̂ ∈sc(ε̂) and p ∈b, (3) sup(ε̂∗)∗(p) = supε̂(p) = fε̂(p) for all p ∈b, (2) supε̂(p) = 1 − (1 − supε̂(p)) = 1 − supω̂(p) for all ω̂ ∈sc(ε̂) and p ∈b, (5) supλ̆(p) = sup λ̆(p) = λ̆u(p) for every ivfs λ̆ on b and for all p ∈b, (6) h(ε̂,[0,1]) sup is both a hfs and an ivfs on b. jittburus and julatha [12] introduced a sup-hesitant fuzzy ideal, which is a generalization of the concepts of an ivfi and a hfi, of a semigroup and studied its properties via sets, fss, hfss and ivfss in the following. definition 2.1. [12] a hfs ε̂ on a is called a sup-hesitant fuzzy ideal of a related to h (briefly, h-sup-hesitant fuzzy ideal) of a if the set s[ε̂;h] is an ideal of a. we say that ε̂ is a sup-hesitant fuzzy ideal (sup-hfi) of a if ε̂ is a h-sup-hesitant fuzzy ideal of a for all h∈ ℘([0, 1]) when s[ε̂;h] 6= ∅. theorem 2.1. [12] every hfi of a is a sup-hfi of a. theorem 2.2. [12] every ivfi of a is a sup-hfi of a. theorem 2.3. [12] let ε̂ be a hfs on a. the followings are equivalent: (1) ε̂ is a sup-hfi of a, (2) fε̂ is a fi of a, (3) supε̂(pq) ≥ max{supε̂(p), supε̂(q)} for all p,q ∈a, (4) h(ε̂,h) sup is a hfi of a for all h∈ ℘([0, 1]). theorem 2.4. [12] let b be a nonempty subset of a. then b is an ideal of a if and only if the chfs χ̂b is a sup-hfi of a. phummee et al. [35] introduced a sup-hesitant fuzzy interior ideal, shown a generalization of the concepts of a sup-hfi, an ivfii and a hfii, of a semigroup and studied its properties via sets, fss, hfss and ivfss. 6 int. j. anal. appl. (2022), 20:58 definition 2.2. [35] a hfs ε̂ on a is called a sup-hesitant fuzzy interior ideal of a related to h (briefly, h-sup-hesitant fuzzy interior ideal) of a if the set s[ε̂;h] is an interior ideal of a. we say that ε̂ is a sup-hesitant fuzzy interior ideal (sup-hfii) of a if ε̂ is a h-sup-hesitant fuzzy interior ideal of a for all h∈ ℘([0, 1]) when s[ε̂;h] 6= ∅. theorem 2.5. [35] every sup-hfi of a is a sup-hfii of a. theorem 2.6. [35] every hfii of a is a sup-hfii of a. theorem 2.7. [35] every ivfii of a is a sup-hfii of a. theorem 2.8. [35] let ε̂ be a hfs on a. the followings are equivalent: (1) ε̂ is a sup-hfii of a, (2) fε̂ is a fii of a, (3) supε̂(pwq) ≥ supε̂(w) for all p,q,w ∈a, (4) h(ε̂,h) sup is a hfii of a for all h∈ ℘([0, 1]). theorem 2.9. [35] if b is a nonempty subset of a, then b is an interior ideal of a if and only if χ̂b is a sup-hfii of a. julatha et al. [13] introduced a sup-hesitant fuzzy right (left) ideal, shown a generalization of the concept of a hfri (hfli) and an ivfri (ivfli), of a semigroup and studied its properties via sets, fss, pfss, hfss, ivfss, cubic sets and hybrid sets. definition 2.3. [13] let ε̂ be a hfs on a. (1) ε̂ is called a sup-hesitant fuzzy left ideal (sup-hfli) of a if (∀p,q ∈ a)(supε̂(q) ≤ supε̂(pq)). (2) ε̂ is called a sup-hesitant fuzzy right ideal (sup-hfri) of a if (∀p,q ∈ a)(supε̂(p) ≤ supε̂(pq)). theorem 2.10. [13] let ε̂ be a hfs on a. the followings are equivalent: (1) ε̂ is a sup-hfri (sup-hfli) of a, (2) fε̂ is a fri (fli) of a, (3) h(ε̂,h) sup is a hfri (hfli) of a for all h∈ ℘([0, 1]). theorem 2.11. [13] if b is a nonempty subset of a, then b is a right ideal (resp., left ideal) of a if and only if χ̂b is a sup-hfri (resp., sup-hfli) of a. 3. generalized sup-hesitant fuzzy ideals in what follows, let α and β be elements of [0, 1], unless otherwise specified. we introduce the concepts of sup+α-hesitant fuzzy left (resp., right, interior, two-sided) ideals and sup − β -hesitant fuzzy left int. j. anal. appl. (2022), 20:58 7 (resp., right, interior, two-sided) ideals of semigroups and investigate their properties. the concepts are established by fss, pfss, hfss, ivfss, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets, hybrid sets and cubic sets. 3.1. sup+α-hesitant fuzzy ideals. in this part, we introduce a sup + α-hesitant fuzzy right ideal, a sup + αhesitant fuzzy left ideal, a sup+α-hesitant fuzzy interior ideal and a sup + α-hesitant fuzzy two-sided ideal of a semigroup, and investigate some of their properties. also, it is shown that a sup+α-hesitant fuzzy left (resp., right, interior, two-sided) ideal of a semigroup is a generalization of the concept of a sup-hesitant fuzzy left (resp., right, interior, two-sided) ideal. for a hfs ε̂ on a, and h,n∈ ℘([0, 1]), we define (1) sup+αh = min{suph + α, 1}, (2) hv+α n if and only if sup + αh≤ sup + αn, (3) h@+α n if and only if sup + αh < sup + αn, (4) h∼=+α n if and only if sup + αh = sup + αn. we denote hvn (resp., h@n, h∼= n) for hv+0 n (resp., h@ + 0 n, h ∼=+0 n). then we have (1) sup+0h = suph, (2) hvn if and only if suph≤ supn, (3) h@n if and only if suph < supn, (4) h∼= n if and only if suph = supn, (5) h∼=+α n if and only if hv+α n and nv+α h. for elements ă = [a−,a+] and b̆ = [b−,b+] in d([0, 1]), then the following are true: (1) if ă b̆, then ă v b̆, and (2) if ă = b̆, then ă ∼= b̆. definition 3.1. a hfs ε̂ on a is called (1) a sup+α-hesitant fuzzy right ideal (sup + α-hfri) of a if (∀p,q ∈a)(ε̂(p) v+α ε̂(pq)), (2) a sup+α-hesitant fuzzy left ideal (sup + α-hfli) of a if (∀p,q ∈a)(ε̂(q) v+α ε̂(pq)), (3) a sup+α-hesitant fuzzy two-sided ideal (or a sup + α-hesitant fuzzy ideal (sup + α-hfi)) of a if it is both a sup+α-hfri and a sup + α-hfli of a, (4) a sup+α-hesitant fuzzy interior ideal (sup + α-hfii) of a if (∀p,q,w ∈a)(ε̂(w) v+α ε̂(pwq)). example 3.1. let a = {(1, 1), (0, 1), (0, 0), (1, 0)}. then a is a semigroup with respect to multiplication defined as follows: (p1,p2)(p3,p4) = (p1,p4) for all p1,p2,p3, p4 ∈{0, 1}. (1) a hfs ε̂1 of a is defined by ε̂1((0, 0)) = (0, 0.8), ε̂1((0, 1)) = {0.2, 0.4, 0.9}, ε̂1((1, 0)) = ∅ and ε̂1((1, 1)) = {0}. then ε̂1 is a sup + 0.2-hfri of a but not a sup + 0.2-hfli of a because ε̂1((1, 0)(0, 0)) @ + 0.2 ε̂1((0, 0)). 8 int. j. anal. appl. (2022), 20:58 (2) a hfs ε̂2 of a is defined by ε̂2((0, 0)) = {0.2, 0.4, 0.5}, ε̂2((0, 1)) = (0.3, 0.7), ε̂2((1, 0)) = [0, 0.5] and ε̂2((1, 1)) = {0.7, 0.8, 0.9}. then ε̂2 is a sup + 0.3-hfli of a but not a sup + 0.3-hfri of a because ε̂2((1, 1)(1, 0)) @ + 0.3 ε̂2((1, 1)). example 3.2. let a = {p1,p2,p3,p4} and define the binary operation “ ·” on a as follows: · p1 p2 p3 p4 p1 p1 p1 p1 p1 p2 p1 p1 p1 p1 p3 p1 p1 p2 p1 p4 p1 p1 p2 p2 then a is a be the semigroup under the binary operation “ ·” [30]. now, define hfss ε̂1 and ε̂2 on a by ε̂1(p1) = [0.3, 0.6], ε̂1(p2) = {0.3, 0.5}, ε̂1(p3) = ∅ and ε̂1(p4) = {1}, ε̂2(p1) = [0.3, 0.7], ε̂2(p2) = {0.3, 0.5, 0.8}, ε̂2(p3) = ∅ and ε̂2(p4) = [0.2, 0.5]. thus (1) ε̂1 is a sup + 0.4-hfii but not a sup + 0.4-hfi of a because ε̂1(p4p4) @ + 0.4 ε̂1(p4), (2) ε̂2 is a sup + 0.3-hfi of a. proposition 3.1. every sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a is a sup+α-hfri (resp., sup+α-hfli, sup + α-hfii, sup + α-hfi) of a for all α ∈ [0, 1]. proof. assume that ε̂ is a sup-hfri of a, α ∈ [0, 1] and p,q ∈a. then supε̂(pq) ≥ supε̂(p) and so sup+αε̂(pq) = min{supε̂(pq) + α, 1}≥ min{supε̂(p) + α, 1} = sup + αε̂(p). hence ε̂(p) v+α ε̂(pq). therefore, we obtain that ε̂ is a sup+α-hfri of a. similarly, we can prove the other results. � example 3.3. let a = {p1,p2,p3,p4} be the semigroup defined in example 3.2. we define a hfs ε̂ on a by ε̂(p1) = {0.1, 0.5, 0.8}, ε̂(p2) = [0, 0.9], ε̂(p3) = [0, 0.5] and ε̂(p4) = ∅. then ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a for all α ∈ [0.2, 1] but ε̂ is not a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a. indeed, ε̂ is not a sup-hfri and sup-hfi of a because supε̂(p2p1) < supε̂(p2), ε̂ is not a sup-hfli of a because supε̂(p1p2) < supε̂(p2), and ε̂ is not a sup-hfii of a because supε̂(p1p2p3) < supε̂(p2). int. j. anal. appl. (2022), 20:58 9 from proposition 3.1 and example 3.3, we have that the concept of a sup+α-hfri (resp., sup + αhfli, sup+α-hfii, sup + α-hfi) of a semigroup a is a generalization of the concept of a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a. proposition 3.2. let ε̂ be a hfs on a and k ∈ [0, 1]. if ε̂ is a sup+α-hfri (resp., sup+α-hfli, sup+αhfii, sup+α-hfi) of a for all α ∈ [0,k], then ε̂ is a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a. proof. let ε̂ be a sup+α-hfri of a for all α ∈ [0,k]. suppose ε̂ is not a sup-hfri of a, that is, there exist p,q ∈a such that supε̂(pq) < supε̂(p). choose α = min{ supε̂(p) − supε̂(pq) 2 ,k}. then α ∈ [0,k] and supε̂(pq) + α ≤ supε̂(pq) + ( supε̂(p) − supε̂(pq) 2 ) < supε̂(pq) + (supε̂(p) − supε̂(pq)) = supε̂(p) ≤ 1. thus sup+αε̂(p) = min{supε̂(p) + α, 1} > supε̂(pq) + α = min{supε̂(pq) + α, 1} = sup+αε̂(pq). hence ε̂(pq) @+α ε̂(p). since ε̂ is a sup + α-hfri of a, we hvae ε̂(pq) @+α ε̂(p) v + α ε̂(pq). this is a contradiction. therefore, ε̂ is a sup-hfri of a. similarly, we can prove the other results. � proposition 3.3. if ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, then ε̂ is a sup+ k -hfri (resp., sup+ k -hfli, sup+ k -hfii, sup+ k -hfi) of a for all k ∈ [α, 1]. proof. assume that ε̂ is a sup+α-hfri of a, k ∈ [α, 1] and p,q ∈ a. then ε̂(p) v+α ε̂(pq), that is, min{supε̂(pq) + α, 1}≥ min{supε̂(p) + α, 1}. if supε̂(pq) + α ≥ 1, then supε̂(pq) + k ≥ supε̂(pq) + α ≥ 1 ≥ sup+ k ε̂(p) 10 int. j. anal. appl. (2022), 20:58 and so ε̂(p) v+ k ε̂(pq). on the other hand, suppose that supε̂(pq) + α ≥ supε̂(p) + α. then supε̂(pq) ≥ supε̂(p) and so supε̂(pq) + k ≥ supε̂(p) + k ≥ sup+ k ε̂(p). thus ε̂(p) v+ k ε̂(pq). therefore, ε̂ is a sup+ k -hfri of a. similarly, we can prove the other results. � proposition 3.4. every sup+α-hfi of a is a sup+α-hfii of a. proof. assume that ε̂ is a sup+α-hfi of a. then ε̂(w) v+α ε̂(wq) v+α ε̂(pwq) for all p,q,w ∈ a. therefore, ε̂ is a sup+α-hfii of a. � from proposition 3.4 and example 3.2, we have that the concept of a sup+α-hfii of a semigroup a is a generalization of the concept of a sup+α-hfi of a. 3.2. sup− β -hesitant fuzzy ideals. in this part, we introduce a sup− β -hesitant fuzzy right ideal, a sup− β hesitant fuzzy left ideal, a sup− β -hesitant fuzzy interior ideal and a sup− β -hesitant fuzzy two-sided ideal of a semigroup, and investigate some of their properties. moreover, it is shown that a sup− β -hesitant fuzzy left (resp., right, interior, two-sided) ideal of a semigroup is a generalization of the concept of a sup-hesitant fuzzy left (resp., right, interior, two-sided) ideal. for a hfs ε̂ on a and h,n∈ ℘([0, 1]), we define (1) sup− β h = max{suph−β, 0}, (2) hv− β n if and only if sup− β h≤ sup− β n, (3) h@− β n if and only if sup− β h < sup− β n, (4) h∼=−β n if and only if sup − β h = sup− β n. then we have (1) sup−0h = suph, (2) hvn if and only if hv−0 n, (3) h@n if and only if h@−0 n, (4) h∼= n if and only if h∼=−0 n. definition 3.2. let ε̂ be a hfs on a. (1) ε̂ is called a sup− β -hesitant fuzzy right ideal (sup− β -hfri) of a if (∀p,q ∈a)(ε̂(p) v− β ε̂(pq)), (2) ε̂ is called a sup− β -hesitant fuzzy left ideal (sup− β -hfli) of a if (∀p,q ∈a)(ε̂(q) v− β ε̂(pq)), (3) ε̂ is called a sup− β -hesitant fuzzy two-sided ideal (or a sup− β -hesitant fuzzy ideal (sup− β -hfi)) of a if it is both a sup− β -hfri and a sup− β -hfli of a, (4) ε̂ is called a sup− β -hesitant fuzzy interior ideal (sup− β -hfii) of a if (∀p,q,w ∈ a)(ε̂(w) v− β ε̂(pwq)). example 3.4. let a = {(1, 1), (0, 1), (0, 0), (1, 0)} be a semigroup defined in example 3.1. int. j. anal. appl. (2022), 20:58 11 (1) a hfs ε̂1 of a is defined by ε̂1((0, 0)) = (0, 0.8], ε̂1((0, 1)) = {0.4, 0.8}, ε̂1((1, 0)) = ∅ and ε̂1((1, 1)) = [0, 0.4]. then ε̂1 is a sup − 0.4-hfri of a but not a sup − 0.4-hfli of a because ε̂1((1, 0)(0, 1)) @ − 0.4 ε̂1((0, 1)). (2) a hfs ε̂2 of a is defined by ε̂2((0, 0)) = {0.2, 0.4, 0.5}, ε̂2((0, 1)) = (0.3, 0.6), ε̂2((1, 0)) = ∅ and ε̂2((1, 1)) = {0.4, 0.5, 0.6}. then ε̂2 is a sup − 0.5-hfli of a but not a sup − 0.5-hfri of a because ε̂2((0, 1)(1, 0)) @ − 0.5 ε̂2((0, 1)). example 3.5. let a = {p1,p2,p3,p4} be the semigroup defined in example 3.2. we define a hfs ε̂ on a by ε̂(p1) = [0.3, 0.7], ε̂(p2) = {0.3, 0.5}, ε̂(p3) = ∅, and ε̂(p4) = (0, 0.7). thus (1) ε̂ is a sup− β -hfii of a for all β ∈ [0, 1] but not a sup− k -hfi of a for all k ∈ [0, 0.7) because ε̂(p4p3) @ − k ε̂(p4) for all k ∈ [0, 0.7). (2) ε̂ is a sup− β -hfi of a for all β ∈ [0.7, 1]. proposition 3.5. every sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a for all β ∈ [0, 1]. proof. assume that ε̂ is a sup-hfri of a, β ∈ [0, 1] and p,q ∈a. then supε̂(pq) ≥ supε̂(p) and thus sup− β ε̂(pq) = max{supε̂(pq) −β, 0}≥ max{supε̂(p) −β, 0} = sup− β ε̂(p). hence ε̂(p) v− β ε̂(pq). therefore, ε̂ is a sup− β -hfri of a. similarly, we can prove the other results. � example 3.6. from example 3.3, we get that the hfs ε̂ is a sup− β -hfri, sup− β -hfli, sup− β -hfii and sup− β -hfi of a for all β ∈ [0.9, 1]. however, ε̂ is not a sup-hfri, sup-hfli, sup-hfii and sup-hfi of a. from example 3.6 and proposition 3.5, we obtain that the concept of a sup− β -hfri (resp., sup− β hfli, sup− β -hfii, sup− β -hfi) of a semigroup a is a generalization of the concept of a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a. proposition 3.6. let ε̂ be a hfs on a and k ∈ [0, 1]. if ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β hfii, sup− β -hfi) of a for all β ∈ [0,k], then ε̂ is a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a. 12 int. j. anal. appl. (2022), 20:58 proof. let ε̂ be a sup− β -hfri of a for all β ∈ [0,k]. suppose that supε̂(pq) < supε̂(p) for some p,q ∈a. choose β ∈ [0,k] such that supε̂(p) −β > 0. then sup− β ε̂(p) = max{supε̂(p) −β, 0} = supε̂(p) −β. since ε̂ is a sup− β -hfri of a, we have sup− β ε̂(pq) ≥ sup− β ε̂(p) = supε̂(p) −β > 0. thus sup− β ε̂(pq) = supε̂(pq) −β < supε̂(p) −β = sup− β ε̂(p) ≤ sup− β ε̂(pq), which is a contradiction. hence ε̂ is a sup-hfri of a. similarly, we can prove the other results. � proposition 3.7. if ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, then ε̂ is a sup− k -hfri (resp., sup− k -hfli, sup− k -hfii, sup− k -hfi) of a for all k ∈ [β, 1]. proof. assume that ε̂ is a sup− β -hfri of a, k ∈ [β, 1] and p,q ∈a. then ε̂(p) v− β ε̂(pq), that is, max{supε̂(pq) −β, 0}≥ max{supε̂(p) −β, 0}≥ supε̂(p) −β. if 0 ≥ supε̂(p) −β, then sup− k ε̂(pq) ≥ 0 ≥ supε̂(p) −β ≥ supε̂(p) −k and so sup− k ε̂(pq) ≥ sup− k ε̂(p), which implies that ε̂(p) v− k ε̂(pq). on the other hand, suppose that supε̂(pq) −β ≥ supε̂(p) −β. then supε̂(pq) ≥ supε̂(p). thus supε̂(pq) −k ≥ supε̂(p) −k. hence sup− k ε̂(pq) ≥ sup− k ε̂(p), which implies that ε̂(p) v− k ε̂(pq). therefore, ε̂ is a sup− k -hfri of a. similarly, we can prove the other results. � proposition 3.8. every sup− β -hfi of a is a sup− β -hfii of a. proof. assume that ε̂ is a sup− β -hfi of a. then ε̂(w) v− β ε̂(wq) v− β ε̂(pwq) for all p,q,w ∈ a. therefore, ε̂ is a sup− β -hfii of a. � from proposition 3.8 and example 3.5, we have that the concept of a sup− β -hfii of a semigroup a is a generalization of the concept of a sup− β -hfi of a. proposition 3.9. let ε̂ be a hfs on a. then the followings are true: int. j. anal. appl. (2022), 20:58 13 (1) ε̂ is a sup-hfri of a if and only if ε̂(p) v ε̂(pq) for all p,q ∈a, (2) ε̂ is a sup-hfli of a if and only if ε̂(q) v ε̂(pq) for all p,q ∈a, (3) ε̂ is a sup-hfii of a if and only if ε̂(w) v ε̂(pwq) for all p,q,w ∈a. proof. it follows from proposition 3.6. � proposition 3.10. let ε̂ be a hfs on a. then the followings are equivalent: (1) ε̂ is a sup-hfri (resp., sup-hfli, sup-hfii, sup-hfi) of a, (2) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a for all β ∈ [0, 1], (3) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a for all α ∈ [0, 1]. proof. it follows from propositions 3.1, 3.2, 3.5 and 3.6. � 3.3. fuzzy sets, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets and pythagorean fuzzy sets. in this part, we characterize sup+α-hfris, sup + α-hflis, sup + α-hfiis, sup + α-hfis, sup − β -hfris, sup− β -hflis, sup− β -hfiis, and sup− β -hfis of semigroups in terms of fss, pfss, łukasiewicz fuzzy sets and łukasiewicz anti-fuzzy sets. for a fs ξ of a, consider the fs ξ+α : a→ [0, 1],p 7→ min{ξ(p) + α, 1}, which is called an α-łukasiewicz anti-fuzzy set [22] of ξ in a. in case that 0 ≤ α ≤ 1 − sup{ξ(p) | p ∈a}, the łukasiewicz anti-fuzzy set ξ+α is called a fuzzy α-translation [8] of ξ of type i. for a fs ξ of a, consider the fs ξ− β : a→ [0, 1],p 7→ max{ξ(p) −β, 0}. then ξ− β (p) = max{ξ(p) + (1−β)−1, 0} for all p ∈a and so ξ− β is called an 1−β-łukasiewicz fuzzy set [23] of ξ in a. in case that 0 ≤ β ≤ inf{ξ(p) | p ∈ a}, the łukasiewicz fuzzy set ξ− β is called a fuzzy β-translation [8] of ξ of type ii. then we have the following results: (1) ξ−0 = ξ = ξ + 0 , (2) ξ− β ≤ ξ ≤ ξ+α, (3) ξ− k ≤ ξ− β for all k ∈ [β, 1], (4) ξ+ k ≤ ξ+α for all k ∈ [0,α], (5) the fs (fε̂)+α is an α-łukasiewicz anti-fuzzy set of f ε̂ in a and (fε̂)+α (p) = sup + αε̂(p) for each hfs ε̂ on a and p ∈a, (6) the fs (fε̂)− β is an 1−β-łukasiewicz fuzzy set of fε̂ in a and (fε̂)− β (p) = sup− β ε̂(p) for each hfs ε̂ on a and p ∈a. theorem 3.1. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) (fε̂)+α is a fri (resp., fli, fii, fi) of a, and 14 int. j. anal. appl. (2022), 20:58 (3) (fε̂)+ k is a fri (resp., fli, fii, fi) of a for all k ∈ [α, 1]. proof. (1) ⇒ (3). assume that ε̂ is a sup+α-hfri of a, k ∈ [α, 1] and p,q ∈a. then ε̂(p) v+α ε̂(pq) and so min{supε̂(p) + α, 1} ≤ supε̂(pq) + α. if min{supε̂(p) + α, 1} = supε̂(p) + α, then supε̂(p) ≤ supε̂(pq). thus (fε̂)+ k (pq) = sup+ k ε̂(pq) ≥ sup+ k ε̂(p) = (fε̂)+ k (p) and so (fε̂)+ k (pq) ≥ (fε̂)+ k (p). on the other hand, suppose that min{supε̂(p) + α, 1} = 1. then supε̂(pq) + k ≥ supε̂(pq) + α ≥ 1 and so (fε̂)+ k (pq) = sup+ k ε̂(pq) = 1 ≥ (fε̂)+ k (p). hence (fε̂)+ k (pq) ≥ (fε̂)+ k (p). therefore, (fε̂)+ k is a fri of a for all k ∈ [α, 1]. (3) ⇒ (2). it is directly obtained from taking k = α. (2) ⇒ (1). assume that (fε̂)+α is a fri of a. then (fε̂)+α (pq) ≥ (fε̂)+α (p) for all p,q ∈a. thus sup+αε̂(pq) = (f ε̂)+α (pq) ≥ (f ε̂)+α (p) = sup + αε̂(p) for all p,q ∈a. hence ε̂(p) v+α ε̂(pq) for all p,q ∈a, which implies that ε̂ is a sup+α-hfri of a. � theorem 3.2. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) (fε̂)− β is a fri (resp., fli, fii, fi) of a, and (3) (fε̂)− k is a fri (resp., fli, fii, fi) of a for all k ∈ [β, 1]. proof. (1) ⇒ (3). assume that ε̂ is a sup− β -hfri of a, k ∈ [β, 1] and p,q ∈a. if 0 ≥ supε̂(p)−β, then 0 ≥ supε̂(p) −k and so (fε̂)− k (pq) ≥ 0 = sup− k ε̂(p) = (fε̂)− k (p). thus (fε̂)− k (pq) ≥ (fε̂)− k (p). on the other hand, suppose that supε̂(p) −β > 0. since ε̂ is a sup− β hfri of a, we have ε̂(p) v− β ε̂(pq). then supε̂(pq) − β ≥ supε̂(p) − β and so supε̂(pq) ≥ supε̂(p). thus (fε̂)− k (pq) = sup− k ε̂(pq) ≥ sup− k ε̂(p) = (fε̂)− k (p). hence (fε̂)− k (pq) ≥ (fε̂)− k (p). therefore, (fε̂)− k is a fri of a for all k ∈ [β, 1]. (3) ⇒ (2). it is directly obtained from taking k = β. (2) ⇒ (1). assume that (fε̂)− β is a fri of a and p,q ∈a. then (fε̂)− β (pq) ≥ (fε̂)− β (p) and so sup− β ε̂(pq) = (fε̂)− β (pq) ≥ (fε̂)− β (p) = sup− β ε̂(p). hence ε̂(p) v− β ε̂(pq). therefore, ε̂ is a sup− β -hfri of a. � int. j. anal. appl. (2022), 20:58 15 lemma 3.1. let ε̂ be a hfs on a. then sup+ k ω̂(p) = 1−sup− k ε̂(p) and sup− k ω̂(p) = 1−sup+ k ε̂(p) for all p ∈a, ω̂ ∈sc(ε̂) and k ∈ [0, 1]. proof. let p ∈a, ω̂ ∈sc(ε̂) and k ∈ [0, 1]. then sup+ k ω̂(p) = min{supω̂(p) + k, 1} = min{(1 − supε̂(p)) + k, 1} = min{1 − (supε̂(p) −k), 1} = 1 − max{supε̂(p) −k, 0} = 1 − sup− k ε̂(p) and sup− k ω̂(p) = max{supω̂(p) −k, 0} = max{(1 − supε̂(p)) −k, 1 − 1} = max{1 − (supε̂(p) + k), 1 − 1} = 1 − min{supε̂(p) + k, 1} = 1 − sup+ k ε̂(p). therefore, sup+ k ω̂(p) = 1 − sup− k ε̂(p) and sup− k ω̂(p) = 1 − sup+ k ε̂(p). � theorem 3.3. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) (fε̂ ∗ )−α is an afri (resp., afli, afii, afi) of a, (3) (fω̂)−α is an afri (resp., afli, afii, afi) of a for all ω̂ ∈sc(ε̂), and (4) (fω̂)− k is an afri (resp., afli, afii, afi) of a for all ω̂ ∈sc(ε̂) and k ∈ [α, 1]. proof. (1) ⇒ (4). assume that ε̂ is a sup+α-hfri of a. by proposition 3.3, we have that ε̂ is a sup+ k -hfri of a for all k ∈ [α, 1]. by lemma 3.1, we get (fω̂)− k (p) = sup− k ω̂(p) = 1 − sup+ k ε̂(p) ≥ 1 − sup+ k ε̂(pq) = sup− k ω̂(pq) = (fω̂)− k (pq) for all ω̂ ∈ sc(ε̂), k ∈ [α, 1] and p,q ∈ a. hence (fω̂)− k is an afri of a for all ω̂ ∈ sc(ε̂) and k ∈ [α, 1]. (4) ⇒ (3) and (3) ⇒ (2). they are clear. 16 int. j. anal. appl. (2022), 20:58 (2) ⇒ (1). assume that (fε̂ ∗ )−α is an afri of a and p,q ∈ a. then (fε̂ ∗ )−α (p) ≥ (fε̂ ∗ )−α (pq) and by lemma 3.1, we have sup+αε̂(p) = 1 − sup − αε̂ ∗(p) = 1 − (fε̂ ∗ )−α (p) ≤ 1 − (fε̂ ∗ )−α (pq) = 1 − sup−αε̂ ∗(pq) = sup+αε̂(pq). hence ε̂(p) v+α ε̂(pq). therefore, ε̂ is a sup+α-hfri of a. � theorem 3.4. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) (fε̂ ∗ )+ β is an afri (resp., afli, afii, afi) of a, (3) (fω̂)+ β is an afri (resp., afli, afii, afi) of a for all ω̂ ∈sc(ε̂), and (4) (fω̂)+ k is an afri (resp., afli, afii, afi) of a for all ω̂ ∈sc(ε̂) and for all k ∈ [β, 1]. proof. (1) ⇒ (4). assume that ε̂ is a sup− β -hfri of a. by proposition 3.7 and lemma 3.1, we get (fω̂)+ k (pq) = sup+ k ω̂(pq) = 1 − sup− k ε̂(pq) ≤ 1 − sup− k ε̂(p) = sup+ k ω̂(p) = (fω̂)+ k (p) for all ω̂ ∈ sc(ε̂), k ∈ [β, 1] and p,q ∈ a. hence (fω̂)+ k is an afri of a for all ω̂ ∈ sc(ε̂) and k ∈ [β, 1]. (4) ⇒ (3) and (3) ⇒ (2). they are clear. (2) ⇒ (1). assume that (fε̂ ∗ )− β is an afri of a. by lemma 3.1, we get sup− β ε̂(pq) = 1 − sup+ β ε̂∗(pq) = 1 − (fε̂ ∗ )+ β (pq) ≥ 1 − (fε̂ ∗ )+ β (p) = 1 − sup+ β ε̂∗(p) = sup− β ε̂(p) for all p,q ∈a. thus ε̂(p) v− β ε̂(pq) for all p,q ∈a, which implies that ε̂ is a sup− β -hfri of a. � theorem 3.5. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) ((fε̂)+α, (f ε̂∗)−α ) is a pfri (resp., pfli, pfii, pfi) of a, (3) ((fε̂)+α, (f ω̂)−α ) is a pfri (resp., pfli, pfii, pfi) of a for all ω̂ ∈sc(ε̂), and (4) ((fε̂)+ k , (fω̂)− k ) is a pfri (resp., pfli, pfii, pfi) of a for all ω̂ ∈sc(ε̂) and k ∈ [α, 1]. int. j. anal. appl. (2022), 20:58 17 proof. it follows from theorems 3.1 and 3.3. � theorem 3.6. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) ((fε̂)− β , (fε̂ ∗ )+ β ) is a pfri (resp., pfli, pfii, pfi) of a, (3) ((fε̂)− β , (fω̂)+ β ) is a pfri (resp., pfli, pfii, pfi) of a for all ω̂ ∈sc(ε̂), and (4) ((fε̂)− k , (fω̂)+ k ) is a pfri (resp., pfli, pfii, pfi) of a for all ω̂ ∈sc(ε̂) and k ∈ [β, 1]. proof. it follows from theorems 3.2 and 3.4. � 3.4. hesitant fuzzy sets and hybrid sets. in this part, we characterize sup+α-hfris, sup + α-hflis, sup+α-hfiis, sup + α-hfis, sup − β -hfris, sup− β -hflis, sup− β -hfiis and sup− β -hfis of semigroups in terms of hfss and hybrid sets. theorem 3.7. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, and (2) h(ε̂,h) sup is a hfri (resp., hfli, hfii, hfi) of a for all h∈ ℘([0, 1 −α]). proof. (1) ⇒ (2). assume that ε̂ is a sup+α-hfri of a. let h ∈ ℘([0, 1 − α]), p,q ∈ a and k ∈h(ε̂,h) sup (p). then ε̂(p) v+α ε̂(pq) and supε̂(p) ≥ k ∈h. thus supε̂(pq) = (supε̂(pq) + α) −α ≥ sup+αε̂(pq) −α ≥ sup+αε̂(p) −α = min{supε̂(p), 1 −α} ≥ k which implies that k ∈h(ε̂,h) sup (pq). hence h(ε̂,h) sup (p) ⊆h(ε̂,h) sup (pq). therefore, h(ε̂,h) sup is a hfri of a for all h∈ ℘([0, 1 −α]). (2) ⇒ (1). assume that h(ε̂,h) sup is a hfri of a for all h ∈ ℘([0, 1 − α]). let p,q ∈ a and h = [0, 1 −α]. then sup+αε̂(p) −α = min{supε̂(p), 1 −α}∈h (ε̂,h) sup (p) ⊆h(ε̂,h) sup (pq). thus sup+αε̂(pq) −α = min{supε̂(pq), 1 −α}≥ sup + αε̂(p) −α. hence sup+αε̂(pq) ≥ sup + αε̂(p) which implies that ε̂(p) v+α ε̂(pq). therefore, ε̂ is a sup+α-hfri of a. � theorem 3.8. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, 18 int. j. anal. appl. (2022), 20:58 (2) h(ε̂,h) sup is a hfri (resp., hfli, hfii, hfi) of a for all h∈ ℘((β, 1]). proof. (1) ⇒ (2). assume that ε̂ is a sup− β -hfri ofa. leth∈ ℘((β, 1]), p,q ∈aand k ∈h(ε̂,h) sup (p). then supε̂(p) ≥ k > β, k ∈h and ε̂(p) v− β ε̂(pq). thus max{supε̂(pq),β} = sup− β ε̂(pq) + β ≥ sup− β ε̂(p) + β = max{supε̂(p),β} ≥ k > β, that is, supε̂(pq) ≥ k. hence k ∈h(ε̂,h) sup (pq) and so h(ε̂,h) sup (p) ⊆h(ε̂,h) sup (pq). therefore, h(ε̂,h) sup is a hfri of a for all h∈ ℘((β, 1]). (2) ⇒ (1). assume that h(ε̂,h) sup is a hfri of a for all h∈ ℘((β, 1]) and p,q ∈a. if supε̂(p) ≤ β, then sup− β ε̂(p) = 0 ≤ sup− β ε̂(pq) and so ε̂(p) v− β ε̂(pq). on the other hand, suppose that supε̂(p) > β. let h = (β, 1]. then sup− β ε̂(p) + β = max{supε̂(p),β} = supε̂(p) ∈h(ε̂,h) sup (p) ⊆h(ε̂,h) sup (pq). thus sup− β ε̂(pq) + β ≥ (supε̂(pq) −β) + β = supε̂(pq) ≥ sup− β ε̂(p) + β. hence sup− β ε̂(p) ≤ sup−αε̂(pq) and so ε̂(p) v − β ε̂(pq). therefore, ε̂ is a sup− β -hfri of a. � theorem 3.9. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) (h(ε̂,h) sup , (fε̂ ∗ )−α ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all h∈ ℘([0, 1 −α]), (3) (h(ε̂,h) sup , (fω̂)−α ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all ω̂ ∈ sc(ε̂) and h∈ ℘([0, 1 −α]), and (4) (h(ε̂,h) sup , (fω̂)− k ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all ω̂ ∈sc(ε̂), k ∈ [α, 1] and h∈ ℘([0, 1 −α]). proof. it follows from theorems 3.3 and 3.7. � theorem 3.10. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) (h(ε̂,h) sup , (fε̂ ∗ )+ β ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all h∈ ℘((β, 1]), (3) (h(ε̂,h) sup , (fω̂)+ β ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all ω̂ ∈ sc(ε̂) and h∈ ℘((β, 1]), and int. j. anal. appl. (2022), 20:58 19 (4) (h(ε̂,h) sup , (fω̂)+ k ) is a hyri (resp., hyli, hyii, hyi) of a over [0, 1] for all ω̂ ∈sc(ε̂), k ∈ [β, 1] and h∈ ℘((β, 1]). proof. it follows from theorems 3.4 and 3.8. � 3.5. interval-valued fuzzy sets and cubic sets. in this part, we characterize sup+α-hfris, sup + αhflis, sup+α-hfiis, sup + α-hfis, sup − β -hfris, sup− β -hflis, sup− β -hfiis, and sup− β -hfis of semigroups in terms of ivfss and cubic sets. let ξ and η be fss of a such that ξ ≤ η, the followings are true. (1) [ξ,ξ], [ξ− β ,η], [ξ,η+α ] and [ξ − β ,η+α ] are ivfss on a. (2) if α ≥ β, then [ξ−α,η − β ] is an ivfs on a. (3) if α ≤ β, then [ξ+α,η + β ] is an ivfs on a. (4) if λ̆ is an ivfs on a and λ̆ = [ξ,η], then λ̆l = ξ and λ̆u = η. theorem 3.11. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) [(fε̂)+α, (f ε̂)+ k ] is an ivfri (resp., ivfli, ivfii, ivfi) of a for all k ∈ [α, 1], (3) [(fε̂)+ k , (fε̂)+ β ] is an ivfri (resp., ivfli, ivfii, ivfi) of a for all k,β ∈ [α, 1] with k ≤ β, (4) h(ε̂,[0,1−α]) sup is an ivfri (resp., ivfli, ivfii, ivfi) of a. proof. (1) ⇔ (2) and (1) ⇔ (3). it follows from theorem 3.1. (1) ⇒ (4). assume that ε̂ is a sup+α-hfri of a and p,q ∈ a. then ε̂(p) v+α ε̂(pq) and so sup+αε̂(p) ≤ sup + αε̂(pq). thus h(ε̂,[0,1−α]) sup (p) = [0, sup+αε̂(p) −α] [0, sup + αε̂(pq) −α] = h (ε̂,[0,1−α]) sup (pq). therefore, h(ε̂,[0,1−α]) sup is an ivfri of a. (4) ⇒ (1). assume that h(ε̂,[0,1−α]) sup is an ivfri of a and p,q ∈a. then [0, sup+αε̂(p) −α] = h (ε̂,[0,1−α]) sup (p) -h(ε̂,[0,1−α]) sup (pq) = [0, sup+αε̂(pq) −α]. thus sup+αε̂(p) = (sup + αε̂(p) −α) + α ≤ (sup + αε̂(pq) −α) + α = sup + αε̂(pq). hence ε̂(p) v+α ε̂(pq). therefore, ε̂ is a sup+α-hfri of a. � theorem 3.12. for a hfs ε̂ on a, the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) [(fε̂)− k , (fε̂)− β ] is an ivfri (resp., ivfli, ivfii, ivfi) of a for all k ∈ [β, 1], and (3) [(fε̂)− k , (fε̂)−α ] is an ivfri (resp., ivfli, ivfii, ivfi) of a for all k,α ∈ [β, 1] with α ≤ k. proof. it follows from theorem 3.2. � 20 int. j. anal. appl. (2022), 20:58 theorem 3.13. let ε̂ be a hfs on a and λ̆ = h(ε̂,[0,1−α]) sup . then the followings are equivalent: (1) ε̂ is a sup+α-hfri (resp., sup + α-hfli, sup + α-hfii, sup + α-hfi) of a, (2) 〈λ̆, (fε̂ ∗ )−α〉 is a curi (resp., culi, cuii, cui) of a, (3) 〈λ̆, (fω̂)−α〉 is a curi (resp., culi, cuii, cui) of a for all ω̂ ∈sc(ε̂), (4) 〈λ̆, (fω̂)− k 〉 is a curi (resp., culi, cuii, cui) of a for all ω̂ ∈sc(ε̂) and k ∈ [α, 1]. proof. it follows from theorems 3.3 and 3.11. � theorem 3.14. let ε̂ be a hfs on a, k ∈ [β, 1] and λ̆ = [(fε̂)− k , (fε̂)− β ]. then the followings are equivalent: (1) ε̂ is a sup− β -hfri (resp., sup− β -hfli, sup− β -hfii, sup− β -hfi) of a, (2) 〈λ̆, (fε̂ ∗ )+ β 〉 is a curi (resp., culi, cuii, cui) of a, (3) 〈λ̆, (fω̂)+ β 〉 is a curi (resp., culi, cuii, cui) of a for all ω̂ ∈sc(ε̂). proof. it follows from theorems 3.4 and 3.12. � 4. characterizing semigroups by sup+α-type and sup − β -type of hfss in this section, we characterize intra-regular, completely regular, left (right) regular, left (right) simple and simple semigroups and groups in terms of sup+α-type and sup − β -type of hfss. a semigroup a is called (1) intra-regular if for each w ∈a, there exist p,q ∈a such that w = pw2q, (2) completely regular if for each p ∈a there exists q ∈a such that p = pqp and pq = qp, (3) left regular if for each p ∈a there exists q ∈a such that p = qp2, (4) right regular if for each p ∈a there exists q ∈a such that p = p2q, (5) left simple if a = b for each left ideal b of a, (6) right simple if a = b for each right ideal b of a, (7) simple if a = b for each ideal b of a, (8) group if it is both left simple and right simple. it is well-known that a is completely regular if and only if it is both left and right regular. proposition 4.1. let ε̂ be a hfs on an intra-regular semigroup a. then ε̂ is a sup+α-hfii (resp., sup− β -hfii) of a if and only if ε̂ is a sup+α-hfi (resp., sup − β -hfi) of a. proof. (⇒). assume that ε̂ is a sup+α-hfii of a and p,q ∈ a. then there exist w1,w2,w3,w4 ∈ a such that p = w1p2w2 and q = w3q2w4. thus ε̂(p) v+α ε̂(w1p2w2q) = ε̂(pq) and ε̂(q) v+α ε̂(pw3q 2w4) = ε̂(pq). therefore, ε̂ is a sup+α-hfi of a. (⇐). it follows from proposition 3.4. � theorem 4.1. let a be a semigroup. the followings are equivalent: int. j. anal. appl. (2022), 20:58 21 (1) a is intra-regular, (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfi ε̂ of a and p ∈a, (3) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfii ε̂ of a and p ∈a, (4) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -hfi ε̂ of a and p ∈a, (5) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -hfii ε̂ of a and p ∈a, (6) ε̂(p) ∼= ε̂(p2) for each sup-hfi ε̂ of a and p ∈a, (7) ε̂(p) ∼= ε̂(p2) for each sup-hfii ε̂ of a and p ∈a. proof. (7) ⇒ (6). it follows from proposition 3.4. (2) ⇔ (4) ⇔ (6) and (3) ⇔ (5) ⇔ (7). they follow from proposition 3.10. (1) ⇒ (2). assume that (1) holds, k ∈ [0, 1], ε̂ is a sup+ k -hfi of a and p ∈ a. there exist q,w ∈a such that p = qp2w. thus ε̂(p) v+ k ε̂(p2) v+ k ε̂(p2w) v+ k ε̂(qp2w) = ε̂(p). hence ε̂(p) ∼=+k ε̂(p 2). therefore, ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfi ε̂ of a and p ∈a. (1) ⇒ (3). it is similar to prove (1) ⇒ (2). (7) ⇒ (1). assume that (7) holds and p ∈ a. then j[p2] = {p2} ∪ ap2 ∪ p2a ∪ ap2a is an interior ideal of a and by theorem 2.9, χ̂j[p2] is a sup-hfii of a. since p2 ∈ j[p2], we get χ̂j[p2](p) ∼= χ̂j[p2](p2) = [0, 1] which implies that χ̂j[p2](p) = [0, 1]. thus p ∈ j[p2] and so p ∈ap2a. hence a is intra-regular. (6) ⇒ (1). it is similar to prove (7) ⇒ (1). � proposition 4.2. let ε̂ be a hfs of an intra-regular semigroup a. then the followings are true: (1) ε̂(pq) ∼=+α ε̂(qp) for each sup+α-hfii of a and p,q ∈a, (2) ε̂(pq) ∼=+α ε̂(qp) for each sup+α-hfi of a and p,q ∈a. proof. (1). let ε̂ be a sup+α-hfii of a and p,q ∈a. by theorem 4.1, we have ε̂(qp) v+α ε̂(p(qp)q) = ε̂((pq) 2) ∼=+α ε̂(pq) v + α ε̂(q(pq)p) = ε̂((qp) 2) ∼=+α ε̂(qp). thus ε̂(pq) ∼=+α ε̂(qp). (2). it follows from (1) and proposition 4.1. � similarly we can prove the following theorem. proposition 4.3. let ε̂ be a hfs of an intra-regular semigroup a. then the followings are true: (1) ε̂(pq) ∼=−β ε̂(qp) for each sup − β -hfii of a and p,q ∈a, (2) ε̂(pq) ∼=−β ε̂(qp) for each sup − β -hfi of a and p,q ∈a. theorem 4.2. let a be a semigroup. the followings are equivalent: (1) a is left regular, 22 int. j. anal. appl. (2022), 20:58 (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfli of a and p ∈a, (3) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -hfli of a and p ∈a, (4) ε̂(p) ∼= ε̂(p2) for each sup-hfli of a and p ∈a. proof. (2) ⇔ (3) ⇔ (4). it follows from proposition 3.10. (1) ⇒ (2). assume that (1) holds, k ∈ [0, 1], ε̂ is a sup+ k -hfli of a and p ∈ a. since a is left regular, there exists q ∈a such that p = qp2. thus, by ε̂ is a sup+ k -hfli of a, we obtain ε̂(p) v+ k ε̂(p2) v+ k ε̂(qp2) = ε̂(p). hence ε̂(p) ∼=+k ε̂(p 2). therefore ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfli of a and p ∈a. (4) ⇒ (1). assume that (4) holds and p ∈ a. then l[p2] = {p2}∪ap2 is a left ideal of a and by theorem 2.11, we get that χ̂l[p2] is a sup-hfli of a. since p2 ∈ l[p2], we have χ̂l[p2](p) ∼= χ̂l[p2](p 2) = [0, 1] and so χ̂l[p2](p) = [0, 1]. thus p ∈ l[p2] which implies that p ∈ap2. therefore, a is left regular. � the left-right dual of theorem 4.2 reads as follows: theorem 4.3. let a be a semigroup. the followings are equivalent: (1) a is right regular, (2) ε̂(p) ∼=+k ε̂(p 2) for each k ∈ [0, 1], sup+ k -hfri ε̂ of a and p ∈a, (3) ε̂(p) ∼=−k ε̂(p 2) for each k ∈ [0, 1], sup− k -hfri ε̂ of a and p ∈a, (4) ε̂(p) ∼= ε̂(p2) for each sup-hfri ε̂ of a and p ∈a. theorem 4.4. let a be a semigroup. the followings are equivalent: (1) a is completely regular, (2) ε̂(p) ∼=+k ε̂(p 2) and ω̂(p) ∼=+k ω̂(p 2) for each k ∈ [0, 1], sup+ k -hfri ε̂, sup+ k -hfli ω̂ of a and p ∈a, (3) ε̂(p) ∼=−k ε̂(p 2) and ω̂(p) ∼=−k ω̂(p 2) for each k ∈ [0, 1], sup− k -hfri ε̂, sup− k -hfli ω̂ of a and p ∈a, (4) ε̂(p) ∼= ε̂(p2) and ω̂(p) ∼= ω̂(p2) for each sup-hfri ε̂, sup-hfli ω̂ of a and p ∈a. proof. it follows from theorems 4.2 and 4.3. � a hfs ε̂ on a is called (1) constant if ε̂(p) = ε̂(q) for all p,q ∈a, (2) sup+α-constant if ε̂(p) ∼=+α ε̂(q) for all p,q ∈a, (3) sup− β -constant if ε̂(p) ∼=−β ε̂(q) for all p,q ∈a, and (4) sup-constant if ε̂(p) ∼= ε̂(q) for all p,q ∈a. then it can be easily seen the following conditions: (1) if ε̂ is constant, then ε̂ is sup-constant, int. j. anal. appl. (2022), 20:58 23 (2) if ε̂ is sup-constant, then ε̂ is both sup+α-constant and sup − β -constant. theorem 4.5. let a be a semigroup. the followings are equivalent: (1) a is left simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -hfli ε̂ of a, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -hfli ε̂ of a, (4) ε̂ is sup-constant for every sup-hfli ε̂ of a. proof. (2) ⇔ (3) ⇔ (4). it follows from proposition 3.10. (1) ⇒ (2). assume that (1) holds, k ∈ [0, 1] and ε̂ is a sup+ k -hfli of a. let p,q ∈a. since a is left simple, we have p ∈ a = aq and q ∈ a = ap. thus p = w1q and q = w2p for some p,q ∈ a. since ε̂ is a sup+ k -hfli of a, we get ε̂(p) v+ k ε̂(w2p) = ε̂(q) v+k ε̂(w1q) = ε̂(p). then ε̂(p) ∼=+k ε̂(q). hence ε̂ is sup + k -constant. therefore, we obtain that ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -hfli ε̂ of a. (4) ⇒ (1). assume that (4) holds. let l be a left ideal of a and w ∈ l. then, by theorem 2.11, we have χ̂l is sup-hfli of a. by assumption (4), we get that χ̂l is sup-constant. thus χ̂l(p) ∼= χ̂l(w) = [0, 1] foll p ∈ a which implies that χ̂l(p) = [0, 1] for all p ∈ a. hence a = l. therefore, a is left simple. � the left-right dual of theorem 4.5 reads as follows: theorem 4.6. let a be a semigroup. the followings are equivalent: (1) a is right simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -hfri ε̂ of a, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -hfri ε̂ of a, (4) ε̂ is sup-constant for every sup-hfri ε̂ of a. the following theorem can be seen in a similar way as in the proof of theorem 4.5. theorem 4.7. let a be a semigroup. the followings are equivalent: (1) a is simple, (2) ε̂ is sup+ k -constant for every k ∈ [0, 1] and sup+ k -hfi ε̂ of a, (3) ε̂ is sup− k -constant for every k ∈ [0, 1] and sup− k -hfi ε̂ of a, (4) ε̂ is sup-constant for every sup-hfi ε̂ of a. from theorems 4.5 and 4.6, we have the following theorem. theorem 4.8. let a be a semigroup. the followings are equivalent: (1) a is group, 24 int. j. anal. appl. (2022), 20:58 (2) ε̂ and ω̂ are sup+ k -constant for every k ∈ [0, 1], sup+ k -hfli ε̂ and sup+ k -hfri ω̂ of a, (3) ε̂ and ω̂ are sup− k -constant for every k ∈ [0, 1], sup− k -hfli ε̂ and sup− k -hfri ω̂ of a, (4) ε̂ and ω̂ are sup-constant for every sup-hfli ε̂ and sup-hfri ω̂ of a. 5. conclusions and future works in present paper, we have introduced the concepts of sup+α-hfris (resp., sup + α-hflis, sup + α-hfiis, sup+α-hfis) and sup − β -hfris (resp., sup− β -hflis, sup− β -hfiis, sup− β -hfis) which are generalizations of the concepts of sup-hfris (resp., sup-hflis, sup-hfiis, sup-hfis) of semigroups, and discussed their some properties. furthermore, the concepts have been established by fss, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets, pfss, hfss, hybrid sets, ivfss and cubic sets. finally, we have characterized intra-regular, left (right) regular, completely regular, left (right) simple and simple semigroups in terms of sup+α-type and sup − β -type of hesitant fuzzy sets. the following are objectives for study and research in semigroups and other algebras: • to introduce and study sup+α-type and sup − β -type of hfss based on bi-ideals of semigroups, • to introduce and study sup+α-type and sup − β -type of hfss based on ideal theory in bck/bcialgebras, ternary semigroups, γ-semigroups and la-semigroups, • to introduce and study sup+α-type and sup − β -type of hfss based on substructures of gealgebras, brk-algebras, be-algebras and iup-algebras [5,6,11,26], • to apply this study to the concept of rough sets according to ansari’s study [2,3]. acknowledgment: this work was supported by research development and research management fund of pibulsongkram rajabhat university for 2022 [grant number rdi-2-65-19]. the authors wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. anis, m. khan, y.b. jun, hybrid ideals in semigroups, cogent math. 4 (2017), art. id 1352117. https: //doi.org/10.1080/23311835.2017.1352117. [2] m.a. ansari, rough set theory applied to ju-algebras, int. j. math. comput. sci. 16 (2021), 1371–1384. [3] m.a. ansari, a. iampan, generalized rough (m,n) bi-γ-ideals in ordered la-γ-semigroups, commun. math. appl. 12 (2021), 545–557. [4] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87–96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [5] r.k. bandaru, on brk-algebras, int. j. math. math. sci. 2012 (2012), art. id 952654. https://doi.org/10. 1155/2012/952654. [6] r.k. bandaru, a.b. saeid, y.b. jun, on ge-algebras, bull. sect. logic univ. łódź 50 (2021), 81–96. https: //doi.org/10.18778/0138-0680.2020.20. https://doi.org/10.1080/23311835.2017.1352117 https://doi.org/10.1080/23311835.2017.1352117 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1155/2012/952654 https://doi.org/10.1155/2012/952654 https://doi.org/10.18778/0138-0680.2020.20 https://doi.org/10.18778/0138-0680.2020.20 int. j. anal. appl. (2022), 20:58 25 [7] v. chinnadurai, a. arulselvam, on pythagorean fuzzy ideals in semigroups, j. xi’an univ. archit. technol. 12 (2020), 1005–1012. [8] t. guntasow, s. sajak, a. jomkham, a. iampan, fuzzy translations of a fuzzy set in up-algebras, j. indones. math. soc. 23 (2017), 1–19. https://doi.org/10.22342/jims.23.2.371.1-19. [9] b. elavarasan, k. porselvi, y.b. jun, hybrid generalized bi-ideals in semigroups, int. j. math. comput. sci. 14 (2019), 601–612. [10] h. harizavi, y.b. jun, sup-hesitant fuzzy quasi-associative ideals of bci-algebras, filomat. 34 (2020), 4189– 4197. https://doi.org/10.2298/fil2012189h. [11] a. iampan, p. julatha, p. khamrot, et al. independent up-algebras, j. math. comput. sci. 27 (2022), 65–76. https://doi.org/10.22436/jmcs.027.01.06. [12] u. jittburus, p. julatha, new generalizations of hesitant and interval-valued fuzzy ideals of semigroups, adv. math., sci. j. 10 (2021), 2199–2212. https://doi.org/10.37418/amsj.10.4.34. [13] p. julatha, t. gaketem, p. khamrot, et al. sup-hesitant fuzzy ideals and bi-ideals of semigroups, submitted. [14] p. julatha, a. iampan, a new generalization of hesitant and interval-valued fuzzy ideals of ternary semigroups, int. j. fuzzy log. intell. syst. 21 (2021), 169–175. https://doi.org/10.5391/ijfis.2021.21.2.169. [15] p. julatha, a. iampan, on inf-hesitant fuzzy γ-ideals of γ-semigroups, adv. fuzzy syst. 2022 (2022), 9755894. https://doi.org/10.1155/2022/9755894. [16] p. julatha, a. iampan, sup-hesitant fuzzy ideals of γ-semigroups, j. math. comput. sci. 26 (2022), 148–161. https://doi.org/10.22436/jmcs.026.02.05. [17] p. julatha, m. siripitukdet, some characterizations of anti-fuzzy (generalized) bi-ideals of semigroups, thai j. math. 16 (2018), 335–346. [18] y.b. jun, m.s. kang, c.h. park, n-subalgebras in bck/bci-algebras based on point n-structures, int. j. math. math. sci. 2010 (2010), 303412. https://doi.org/10.1155/2010/303412. [19] y.b. jun, c.s. kim, m.s. kang, cubic subalgebras and ideals of bck/bci-algebras, far east j. math. sci. 44 (2010), 239–250. [20] y.b. jun, a. khan, cubic ideals in semigroups, honam math. j. 35 (2013), 607–623. https://doi.org/10.5831/ hmj.2013.35.4.607. [21] y.b. jun, k.j. lee, s.z. song, hesitant fuzzy bi-ideals in semigroups, commun. korean math. soc. 30 (2015), 143–154. https://doi.org/10.4134/ckms.2015.30.3.143. [22] y.b. jun, łukasiewicz anti fuzzy set and its application in be-algebras, trans. fuzzy sets syst. in press. http: //doi.org/10.30495/tfss.2022.1960391.1037. [23] y.b. jun, łukasiewicz fuzzy subalgebras in bck-algebras and bci-algebras, ann. fuzzy math. inform. 23 (2022), 213–223. http://doi.org/10.30948/afmi.2022.23.2.213. [24] y.b. jun, s.z. song, g. muhiuddin, hybrid structures and applications, ann. commun. math. 1 (2018), 11–25. [25] m. khan, t. asif, characterizations of semigroups by their anti-fuzzy ideals, j. math. res. 2 (2010), 134–143. [26] h.s. kim, y.h. kim, on be-algebras, sci. math. jpn. 66 (2007), 113–116. https://doi.org/10.32219/isms. 66.1_113. [27] n. kuroki, fuzzy bi-ideals in semigroups, comment. math. univ. st. pauli. 28 (1979), 17–21. https://doi.org/ 10.14992/00010265. [28] n. kuroki, on fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy sets syst. 5 (1981), 203–215. https: //doi.org/10.1016/0165-0114(81)90018-x. [29] k.j. lee, bipolar fuzzy subalgebras and bipolar fuzzy ideals of bck/bci-algebras, bull. malays. math. sci. soc. 32 (2009), 361–373. [30] j.n. mordeson, d.s. malik, n. kuroki, fuzzy semigroups, springer, berlin, (2012). https://doi.org/10.22342/jims.23.2.371.1-19 https://doi.org/10.2298/fil2012189h https://doi.org/10.22436/jmcs.027.01.06 https://doi.org/10.37418/amsj.10.4.34 https://doi.org/10.5391/ijfis.2021.21.2.169 https://doi.org/10.1155/2022/9755894 https://doi.org/10.22436/jmcs.026.02.05 https://doi.org/10.1155/2010/303412 https://doi.org/10.5831/hmj.2013.35.4.607 https://doi.org/10.5831/hmj.2013.35.4.607 https://doi.org/10.4134/ckms.2015.30.3.143 http://doi.org/10.30495/tfss.2022.1960391.1037 http://doi.org/10.30495/tfss.2022.1960391.1037 http://doi.org/10.30948/afmi.2022.23.2.213 https://doi.org/10.32219/isms.66.1_113 https://doi.org/10.32219/isms.66.1_113 https://doi.org/10.14992/00010265 https://doi.org/10.14992/00010265 https://doi.org/10.1016/0165-0114(81)90018-x https://doi.org/10.1016/0165-0114(81)90018-x 26 int. j. anal. appl. (2022), 20:58 [31] g. muhiuddin, cubic interior ideals in semigroups, appl. appl. math. 14 (2019), 463–474. https:// digitalcommons.pvamu.edu/aam/vol14/iss1/32. [32] g. muhiuddin, y.b. jun, sup-hesitant fuzzy subalgebras and its translations and extensions, ann. commun. math. 2 (2019), 48–56. [33] a. l. narayanan, t. manikantan, interval-valued fuzzy ideals generated by an interval-valued fuzzy subset in semigroups, j. appl. math. comput. 20 (2006), 455–464. https://doi.org/10.1007/bf02831952. [34] z. pawlak, rough sets, int. j. inform. comput. sci. 11 (1982), 341–356. https://doi.org/10.1007/bf01001956. [35] p. phummee, s. papan, c. noyoampaeng, et al. sup-hesitant fuzzy interior ideals of semigroups and their suphesitant fuzzy translations, int. j. innov. comput. inform. control. 18 (2022), 121–132. https://doi.org/10. 24507/ijicic.18.01.121. [36] n. ratchakhwan, p. julatha, t. gaketem, et al. (inf, sup)-hesitant fuzzy ideals of bck/bci-algebras, int. j. anal. appl. 20 (2022), 34. https://doi.org/10.28924/2291-8639-20-2022-34. [37] m. shabir, y. nawaz, semigroups characterized by the properties of their anti-fuzzy ideals, j. adv. res. pure math. 1 (2009), 42–59. [38] m.m. takallo, r.a. borzooei, y.b. jun, sup-hesitant fuzzy p-ideals of bci-algebras, fuzzy inform. eng. 13 (2021), 460–469. https://doi.org/10.1080/16168658.2021.1993668. [39] a.f. talee, m.y. abassi, s.a. khan, hesitant fuzzy ideals in semigroups with a frontier, arya bhatta j. math. inform. 9 (2017), 163–170. [40] n. thillaigovindan, v. chinnadurai, on interval valued fuzzy quasi-ideals of semigroups, east asian math. j. 25 (2009), 449–467. [41] v. torra, hesitant fuzzy sets, int. j. intell. syst. 25 (2010), 529–539. https://doi.org/10.1002/int.20418. [42] v. torra, y. narukawa, on hesitant fuzzy sets and decision, in: 2009 ieee international conference on fuzzy systems, ieee, jeju island, south korea, 2009: pp. 1378–1382. https://doi.org/10.1109/fuzzy.2009.5276884. [43] s. umar, a. hadi, a. kham, on prime cubic bi-ideals of semigroups, ann. fuzzy math. inform. 9 (2015), 957–974. [44] r. r. yager, a. m. abbasov, pythagorean membership grades, complex numbers, and decision making, int. j. intell. syst. 28 (2013), 436–452. https://doi.org/10.1002/int.21584. [45] r.r. yager, pythagorean fuzzy subsets, in: 2013 joint ifsa world congress and nafips annual meeting (ifsa/nafips), ieee, edmonton, ab, canada, 2013: pp. 57–61. https://doi.org/10.1109/ifsa-nafips. 2013.6608375. [46] l. a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [47] l. a. zadeh, the concept of a linguistic variable and its application to approximate reasoning i, inform. sci. 8 (1975), 199–249. https://doi.org/10.1016/0020-0255(75)90036-5. [48] w.r. zhang, bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in: nafips/ifis/nasa ’94. proceedings of the first international joint conference of the north american fuzzy information processing society biannual conference. the industrial fuzzy control and intelligent systems conference, and the nasa joint technology wo, ieee, san antonio, tx, usa, 1994: pp. 305–309. https://doi.org/10.1109/ijcf.1994.375115. https://digitalcommons.pvamu.edu/aam/vol14/iss1/32 https://digitalcommons.pvamu.edu/aam/vol14/iss1/32 https://doi.org/10.1007/bf02831952 https://doi.org/10.1007/bf01001956 https://doi.org/10.24507/ijicic.18.01.121 https://doi.org/10.24507/ijicic.18.01.121 https://doi.org/10.28924/2291-8639-20-2022-34 https://doi.org/10.1080/16168658.2021.1993668 https://doi.org/10.1002/int.20418 https://doi.org/10.1109/fuzzy.2009.5276884 https://doi.org/10.1002/int.21584 https://doi.org/10.1109/ifsa-nafips.2013.6608375 https://doi.org/10.1109/ifsa-nafips.2013.6608375 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/0020-0255(75)90036-5 https://doi.org/10.1109/ijcf.1994.375115 1. introduction 2. preliminaries 3. generalized sup-hesitant fuzzy ideals 3.1. sup+-hesitant fuzzy ideals 3.2. sup--hesitant fuzzy ideals 3.3. fuzzy sets, łukasiewicz fuzzy sets, łukasiewicz anti-fuzzy sets and pythagorean fuzzy sets 3.4. hesitant fuzzy sets and hybrid sets 3.5. interval-valued fuzzy sets and cubic sets 4. characterizing semigroups by sup+-type and sup--type of hfss 5. conclusions and future works acknowledgment: references international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 1-8 http://www.etamaths.com applications of extremal theorem to a class of p-valent analytic functions liangpeng xiong∗ and xiaoli liu abstract. a subclass j m,l p,λ (ξ,α) of p-valent analytic functions with a generalized multiplier transformation operator is introduced. we discuss the compactness as well as the extreme points of j m,l p,λ (ξ,α) under the topology of uniform convergence. finally, as one of the applications of extremal theorem, we solve the sharp distortion inequalities problem as max f∈j m,l p,λ (ξ,α) |f(χ)(z)|,χ = 0, 1, 2, ... several related basic results and remarks about the old or new classes are also presented. 1. introduction let ap denote the class of functions of the form (1) f(z) = zp + ∞∑ k=p+1 akz k,p ∈ n = {1, 2, 3, ...}, which are analytic in 4 = {z : z ∈ c, |z| < 1}. in fact, ap is a vector space over c with the usual definitions of addition and scalar multiplication for functions. let the topology on ap be given by a metric ρ which is equivalent to the topological of uniform convergence on compact subsets, where the ρ is determined as ρ(f,g) = ∞∑ n=1 1 2n ‖f −g‖n 1 + ‖f −g‖n whenever f and g belong to ap, and ‖f‖n = max{|f(z)| : |z| = rn, 0 < rn < 1, lim n→∞ rn = 1}. it follows from theorems of weierstrass and montel that this topology space is complete(see [11], p38). if f ⊂ ap then f is called locally uniformly bounded if there is a constant m such that |f(z)|6 m whenever f ∈ f. moreover, montel’s theorem implies that f ⊂ ap is compact if and only if f is closed and locally uniformly bounded(see [11], p39). we use the notation hf for the closed convex hull of f, where hf = { ∞∑ k=1 tkfk,fk ∈ f, tk > 0, ∞∑ k=1 tk = 1 } . let v be a subclass of a linear topological space. if v ∈ v and if v = tf1 + (1 − t)f2, 0 < t < 1,f1 ∈ v,f2 ∈ v can make sure that f1 = f2, then we say v ∈ ev, where ev denotes the set of extreme points of v. again, suppose that m is a convex subset of ap and j : m → r, if j(tf + (1 − t)g) 6 tj(f) + (1 − t)j(g) whenever f,g ∈ m and 0 6 t 6 1, then linear functional j is called convex on m. 2010 mathematics subject classification. 30c35, 30c45, 35q30. key words and phrases. p-valent functions; extreme points; linear topological space; distortion inequalities; multiplier transformation operator. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 xiong and liu recently, prajapat [18] and sharma et al. [20] studied a generalized multiplier transformation operator jmp (λ,l) : ap → ap as (2) jmp (λ,l)f(z) =   p + l λ zp− p+l λ ∫ z 0 t p+l λ −p−1jm+1p (λ,l)f(t)dt, m ∈ z−, λ p + l z1+p− p+l λ ( z p+l λ −pjm−1p (λ,l)f(z) )′ , m ∈ z+, f(z), m = 0. where m ∈ z = {..,−2,−1, 0, 1, 2, ..},λ > 0, l > −p. it easily follows from the above definition of the operator that the series expansion of jmp (λ,l)f(z) for f(z) of the form (1.1) is given by (3) jmp (λ,l)f(z) = z p + ∞∑ k=p+1 ( 1 + λ(k −p) p + l )m akz k. we note that operator jmp (λ,l) contains kinds of operators introduced and studied by different mathematicians (for details, see [3, 4, 9, 13]). now, we reproduce here briefly some of these special cases as follows. remark 1.1. (i) for m ∈ z+ ∪{0}, jmp (λ,l) ≡ imp (λ,l), (see cătaş[5]). (ii) for m ∈ z+ ∪{0}, jmp (1, l) ≡ ip(m,l), (see kumar et al. [12]). (iii) for m ∈ z+ ∪{0}, jmp (1, 0) ≡ dmp , (see orhan et al. [16]). (iv) for m ∈ z− ∪{0}, jmp (λ,l) operator, (see el-ashwah and aouf [10]). (v) for m ∈ z− ∪{0}, jmp (1, 1) operator, (see patel and sahoo [17]). for various other special cases studied earlier of the operator jmp (λ,l), one can refer to [8, 14, 15, 18]. let vp be the subclass of ap consisting functions of the form (4) f∇(z) = z p − ∞∑ k=p+1 akz k,ak > 0,p ∈ n = {1, 2, 3, ...}, it is easy to see that (5) jmp (λ,l)f∇(z) = z p − ∞∑ k=p+1 ( 1 + λ(k −p) p + l )m akz k. we introduce the class j m,l p,λ (ξ,α) as a subclass of vp consisting of functions f which satisfy (6) < { z ( jmp (λ,l)f∇(z) )′ + ξz2 ( jmp (λ,l)f∇(z) )′′ ξz ( jmp (λ,l)f∇(z) )′ + (1 − ξ)jmp (λ,l)f∇(z) } > α,z ∈4, where 0 6 α < p, 0 6 ξ 6 1,m ∈ z+ ∪{0},λ > 0, l > −p. here, for reader’s convenience, we depict some of subclasses related the above j m,l p,λ (ξ,α). remark 1.2. (i) j 0,l p,λ(ξ,α) was studied by altintaş et al.[1] and xiong [21]. (ii) j 0,l 1,λ(ξ,α) was studied by altintaş [2]. (iii) j 0,l p,λ(0,α) ≡ s ∗ p(α) was studied by darwish and aouf [6]. (iv) j 0,l p,λ(1,α) ≡ kp(α) was studied by darwish and aouf [6]. (v) j 0,l 1,λ(0,α) ≡ s ∗(α) and j 0,l 1,λ(1,α) ≡ k(α) were studied by srivastava et al. [19] and domokos [7], respectively. in this paper we obtain the extreme points for class j m,l p,λ (ξ,α). furthermore, the sharp distortion inequalities are given by using the extreme points theory. p-valent analytic functions 3 2. preliminary results in this section, we give a sufficient and necessary condition for the functions f∇(z) ∈ vp to be in j m,l p,λ (ξ,α). as a lemma, we find that the class j m,l p,λ (ξ,α) is the closed convex hull of a set m . lemma 2.1. a function f(z) = zp − ∞∑ k=p+1 akz k belongs to j m,l p,λ (ξ,α) if and only if (7) ∞∑ k=p+1 ψ(k)ak 6 (p−α)[ξ(p− 1) + 1] (p ∈ n = {1, 2, ..}). where (8) ψ(k) = (k −α)[ξ(k − 1) + 1] ( 1 + λ(k −p) p + l )m . proof. if we taking n = 1 and bk = ( 1 + λ(k −p) p + l )m ak, then this lemma is an immediate consequence of altintas et al. [1] (also see xiong [21], lemma 1]) in function f∇(z) = z p − ∞∑ k=p+1 bkz k. � lemma 2.2. if m = {fk(z) : k = p,p + 1,p + 2, ...}, then j m,l p,λ (ξ,α) = hm , where fk(z) are determined by   fp(z) = z p, k = p, fk(z) = z p − (p−α)[ξ(p− 1) + 1] ψ(k) zk, k > p + 1. proof. suppose that f(z) ∈ hm and f(z) = λpfp(z) + ∞∑ k=p+1 λkfk(z),λk > 0,λp + ∞∑ k=p+1 λk = 1. making use of the elements in m , we can express f(z) = λpz p + ∞∑ k=p+1 λk [ zp − (p−α)[ξ(p− 1) + 1] ψ(k) zk ] = λpz p + ∞∑ k=p+1 λkz p − ∞∑ k=p+1 λk (p−α)[ξ(p− 1) + 1] ψ(k) zk = ( λp + ∞∑ k=p+1 λk ) zp − ∞∑ k=p+1 λk (p−α)[ξ(p− 1) + 1] ψ(k) zk = zp − ∞∑ k=p+1 λk (p−α)[ξ(p− 1) + 1] ψ(k) zk = zp − ∞∑ k=p+1 bkz k, where (9) bk = λk (p−α)[ξ(p− 1) + 1] ψ(k) . consequently, by using (2.3) and the lemma 2.1, we are lead to ∞∑ k=p+1 ψ(k) (p−α)[ξ(p− 1) + 1] bk = ∞∑ k=p+1 λk = 1 −λp 6 1, thus, it follows that f(z) ∈ j m,lp,λ (ξ,α). this implies hm ⊂ j m,l p,λ (ξ,α). 4 xiong and liu we next consider j m,l p,λ (ξ,α) ⊂ hm . if a function f(z) ∈ j m,l p,λ (ξ,α), recalling lemma 2.1 we see that ak 6 (p−α)[ξ(p− 1) + 1] ψ(k) (k > p + 1). here, taking λk = ψ(k) (p−α)[ξ(p− 1) + 1] ak, where k > p + 1 and λp = 1− ∞∑ k=p+1 λk, then we can know 0 6 λk 6 1,k > p. hence, we conclude that f(z) = zp − ∞∑ k=p+1 akz k = zp − ∞∑ k=p+1 (p−α)[ξ(p− 1) + 1] ψ(k) λkz k = zp − ∞∑ k=p+1 λk [ zp − ( zp − (p−α)[ξ(p− 1) + 1] ψ(k) zk )] = zp − ∞∑ k=p+1 λkz p + ∞∑ k=p+1 λk ( zp − (p−α)[ξ(p− 1) + 1] ψ(k) zk ) = ( 1 − ∞∑ k=p+1 λk ) zp + ∞∑ k=p+1 λkfk(z) = ∞∑ k=p λkfk(z). this proves that j m,l p,λ (ξ,α) ⊂ hm and completes the proof of lemma 2.2. � lemma 2.3. ([11],p44) let x be a locally convex linear topological space and let u be a compact subset of x. if hu be a compact then ehu ⊂ u. 3. compactness and extreme points in this section, we prove that the class j m,l p,λ (ξ,α) is compact subset of ap, which can help us to obtain the extreme points of j m,l p,λ (ξ,α) and to complete the works in section 4. theorem 3.1. the class j m,l p,λ (ξ,α) is a compact subset of ap. proof. we need to prove j m,l p,λ (ξ,α) is closed and locally uniformly bounded. suppose that f(z) = zp − ∞∑ k=p+1 akz k ∈ j m,lp,λ (ξ,α) and |z|6 r, 0 < r < 1, then |f(z)|6 rp + ∞∑ k=p+1 ψ(k)|ak| rk ψ(k) 6 rp + (p−α)[ξ(p− 1) + 1] ∞∑ k=p+1 rk ψ(k) . moreover, it is easy to see that lim k−→∞ sup ( rk ψ(k) )1 k = r lim k−→∞ sup(ψ(k))− 1 k = r < 1, this shows that the series ∞∑ k=p+1 rk ψ(k) is convergent, so the results above assert that j m,l p,λ (ξ,α) is locally uniformly bounded. we next prove that the j m,l p,λ (ξ,α) is sequentially closed and suppose that {fj(z)}⊂ j m,lp,λ (ξ,α) ⊂ vp ⊂ ap p-valent analytic functions 5 and {fj(z)} → f(z) as j → ∞, where fj(z) = zp − ∞∑ k=p+1 ajkz k. in fact, weierstrass’ theorem asserts f(z) ∈ vp([11], p38), so we can set f(z) = zp − ∞∑ k=p+1 akz k, thus, it is easy to know that ajk → ak as j →∞. because of lemma 2.1 and fj(z) ∈ j m,lp,λ (ξ,α), for a positive integer h, then we have h∑ k=p+1 ψ(k)ajk 6 ∞∑ k=p+1 ψ(k)ajk 6 (p−α)[ξ(p− 1) + 1]. choosing j →∞, it follows that h∑ k=p+1 ψ(k)ak 6 (p−α)[ξ(p− 1) + 1]. furthermore, taking h → +∞, this leads to ∞∑ k=p+1 ψ(k)ak 6 (p−α)[ξ(p− 1) + 1], which implies that f(z) ∈ j m,lp,λ (ξ,α). this completes the proof of theorem 3.1. � theorem 3.2. ej m,l p,λ (ξ,α) = m , where m is defined as lemma 2.2. proof. suppose that zp − (p−α)[ξ(p− 1) + 1] ψ(k) zk = tg1(z) + (1 − t)g2(z), where gi(z) = z p − ∞∑ k=p+1 ak,iz k ∈ j m,lp,λ (ξ,α), 0 < t < 1, i = 1, 2, then we have (10) (p−α)[ξ(p− 1) + 1] ψ(k) = tak,1 + (1 − t)ak,2. since gi(z) ∈ j m,l p,λ (ξ,α), lemma 2.1 gives (11) ak,i 6 (p−α)[ξ(p− 1) + 1] ψ(k) , i = 1, 2. the (10) and (11) imply that ak,1 = ak,2 = (p−α)[ξ(p− 1) + 1] ψ(k) . thus, we can know that g1(z) = g2(z), which gives us fk(z) = z p − (p−α)[ξ(p− 1) + 1] ψ(k) zk ∈ ej m,lp,λ (ξ,α), k > p + 1. likewise, we also can obtain fp(z) = z p ∈ ej m,lp,λ (ξ,α). in fact, we get m ⊂ ej m,l p,λ (ξ,α). again, using the lemma 2.2, we know j m,l p,λ (ξ,α) = hm . moreover, theorem 3.1 shows that m ⊂ j m,l p,λ (ξ,α) is a compact set, thus, using the lemma 2.3, it suffices to verify that ej m,l p,λ (ξ,α) = ehm ⊂ m . this completes the proof of theorem 3.2. � 6 xiong and liu 4. applications of extreme points theorem we want to maximize the |f(x)(z)|(x = 0, 1, 2, ...) over j m,lp,λ (ξ,α) by making critical use of the information about extreme points. for this point, let linear functional j : j m,l p,λ (ξ,α) → r be defined as: j(f) = |f(x)(z)|,f(z) ∈ j m,lp,λ (ξ,α),x = {0, 1, 2, ...}. it is easy to see that j is a continuous and convex functional. by using the known krein-milman theorem, ([11], p45, theorem 4.6) gives a important result: let f be a compact subset of a and j is a real-valued, continuous, convex functional on hf , then max{j(f) : f ∈ hf} = max{j(f) : f ∈ f} = max{j(f) : f ∈ ehf}. thus, following the lemma 2.2, theorem 3.1 and theorem 3.2, we can know that max{|f(x)(z)| : f ∈ j m,lp,λ (ξ,α)} = max{|f (x)(z)| : f ∈ hm} = max{|f(x)(z)| : f ∈ m}. hence, in order to solve the extremal problems on |f(x)(z)| over j m,lp,λ (ξ,α), it needs only to solve them over the smaller class m . we next turn to consider the |f(x)(z)| over m . in lemma 2.2, taking the function f(z) = zp − (p−α)[ξ(p− 1) + 1] ψ(k) zk, k > p + 1, and after some simplifications, then we have  f(χ)(z) = p! (p−χ)! zp−χ − (p−α)[ξ(p− 1) + 1]k! (k −χ)!ψ(k) zk−χ, χ = 0, 1, 2, ...,p f(χ)(z) = − (p−α)[ξ(p− 1) + 1]k! (k −χ)!ψ(k) zk−χ, χ = p + 1,p + 2, ...,k f(χ)(z) = 0, χ > k. setting χ ∈ n0 = {0, 1, 2, ...}, 0 < r < 1, we define the sequence {j (χ) k } as follows: case i: if χ = 0, then j (χ) k = { 0, k < p + 1, (p−α)[ξ(p−1)+1] ψ(k) rk, k > p + 1. case ii: if χ ∈ n = {1, 2, 3, ...}, then j (χ) k =   0, k < max{χ,p + 1}, (p−α)[ξ(p− 1) + 1]k! (k −χ)!ψ(k) rk−χ, k > max{χ,p + 1}. we can easily prove that j (χ) k → 0 as k →∞, this implies that there is a kχ ∈{p+1,p+2, ...}(χ ∈ n0), such that (12) j (χ) kχ = max{j (χ)k : k = p + 1,p + 2, ...}. finally, we now present the deserved results according to the analysis above. theorem 4.1. suppose that f(z) = zp − ∞∑ k=p+1 akz k ∈ j m,lp,λ (ξ,α), where |z| = r < 1, then (1) if χ = 0, 1, 2, ...,p, we have n1 6 |f(χ)(z)|6 n2, where n1 = p! (p−χ)! rp−χ − (p−α)[ξ(p− 1) + 1]kχ! (kχ −χ)!ψ(kχ) rkχ−χ and n2 = p! (p−χ)! rp−χ + (p−α)[ξ(p− 1) + 1]kχ! (kχ −χ)!ψ(kχ) rkχ−χ. p-valent analytic functions 7 (2)if χ = p + 1,p + 2, ..., we have − (p−α)[ξ(p− 1) + 1]kχ! (kχ −χ)!ψ(kχ) rkχ−χ 6 |f(χ)(z)|6 (p−α)[ξ(p− 1) + 1]kχ! (kχ −χ)!ψ(kχ) rkχ−χ. all the above kχ is defined as (12), and ψ(kχ) are the values of ψ(k) in (2.2) whenever k = kχ. the results are sharp and the extremal functions are given by the m of lemma 2.2. two special cases of theorem 4.1 when m = 0,ξ = 0 and m = 0,ξ = 1 yield, respectively, corollary 4.1. suppose that f(z) = zp − ∞∑ k=p+1 akz k ∈ s∗p(α), where |z| = r < 1, then (1) if χ = 0, 1, 2, ...,p, we have n1 6 |f(χ)(z)|6 n2, where n1 = p! (p−χ)! rp−χ − (p−α)kχ! (kχ −χ)!(kχ −α) rkχ−χ and n2 = p! (p−χ)! rp−χ + (p−α)kχ! (kχ −χ)!(kχ −α) rkχ−χ (2)if χ = p + 1,p + 2, ..., we have − (p−α)kχ! (kχ −χ)!(kχ −α) rkχ−χ 6 |f(χ)(z)|6 (p−α)kχ! (kχ −χ)!(kχ −α) rkχ−χ, all the above kχ is defined as (12). the results are sharp and the extremal functions are given by the m of lemma 2.2. corollary 4.2. suppose that f(z) = zp − ∞∑ k=p+1 akz k ∈ kp(α), where |z| = r < 1, then (1) if χ = 0, 1, 2, ...,p, we have n1 6 |f(χ)(z)|6 n2, where n1 = p! (p−χ)! rp−χ − (p−α)p ·kχ! (kχ −χ)!(kχ −α)kχ rkχ−χ and n2 = p! (p−χ)! rp−χ − (p−α)p ·kχ! (kχ −χ)!(kχ −α)kχ rkχ−χ (2)if χ = p + 1,p + 2, ..., we have − (p−α)p ·kχ! (kχ −χ)!(kχ −α)kχ rkχ−χ 6 |f(χ)(z)|6 (p−α)p ·kχ! (kχ −χ)!(kχ −α)kχ rkχ−χ. all the above kχ is defined as (12). the results are sharp and the extremal functions are given by the m of lemma 2.2. remark 4.1. in fact, various other interesting consequences of our main results (which are asserted by theorems 3.1, 3.2 and 4.1 and corollaries 4.1 to 4.2 above) can be derived by appropriately choosing special parameters as the remark 1.1 and remark 1.2. the details may be left as an exercise for the interested reader. acknowledgement: the authors are thankful to the referees for reading this paper and this work was supported by scientific research fund of sichuan provincial education department of china, grant no.14zb0364. 8 xiong and liu references [1] o.altintaş, h.irmak and h.m.srivastava, fractional calculus and certain starlike functions with negative coefficients, comput. math. appl., 30(2)(1995), 9-16. [2] o.altintaş, on a subclass of certain starlike functions with negative coefficients, math. japonica, 36(3)(1991), 1-7. [3] y.abu muhanna, l.l.li and s.ponnusamy, extremal problems on the class of convex functions of order −1/2, archiv der mathematik, 103(6)(2014), 461-471. [4] r.m.ali, s.r.mondal and v.ravichandran, on the janowski convexity and starlikeness of the confluent hypergeometric function, bulletin of the belgian mathematical society-simon stevin, 22(2)(2015), 227-250. [5] a.cătaş, on certain classes of p-valent functions defined by multiplier transformations, in proceedings of the international symposium on geometric function theory and applications: gfta 2007 (eds. s. owa, y. polatoglu), tc istanbul university publications, turkey, 2008, pp. 241-250. [6] h.e.darwish and m.k.aouf, generalizations of modified-hadamard products of p-valent functions with negative coefficients, math. comput. modelling 49(1-2)(2009), 38-45. [7] t.domokos, on a subclass of certain starlike functions with negative coefficients, stud. univ. babeş-bolyai math., 36(1991), 29-36. [8] j.dziok, applications of extreme points to distortion estimates, appl.math.comput. 215(2009), 71-77. [9] e.deniz and h.orhan, loewner chains and univalence criteria related with ruscheweyh and salagean derivatives, journal of applied analysis and computation, 5(3)(2015), 465-478. [10] r.m.el-ashwah and m.k.aouf, some properties of new integral operator, acta univ.apulensis math.inform., 24(2010), 51-61. [11] d.hallenbeck and t.h.macgregor, linear problems and convexity techniques in geometric function theory, 39-46. pitman advanced publishing program, boston, pitman, (1984) [12] s.s.kumar, h.c.taneja and v.ravichandran, classes of multivalent functions defined by dzoik-srivastava linear operator and multiplier transformations, kyungpook math. j., 46(2006), 97-109. [13] s.kanas and d.raducanu, some class of analytic functions related to conic domains, mathematica slovaca, 64(5)(2014), 1183-1196. [14] j.l.liu, certain sufficient conditions for strongly starlike functions associated with an integral operator, bull.malays.math.sci.soc., 34(1)(2011), 21-30. [15] g.murugusundaramoorthy and k.vijaya, second hankel determinant for bi-univalent analytic functions associated with hohlov operator, international journal of analysis and applications, 8(1)(2015), 22-29. [16] h.orhan and h.kiziltunc, a generalization on subfamily of p-valent functions with negative coefficients, appl. math.comput., 155(2004), 521-530. [17] j.patel, p.sahoo, certain subclasses of multivalent analytic functions, indian j. pure appl. math., 34(3)(2003), 487-500. [18] j.k.prajapat, subordination and superordination preserving properties for generalized multiplier transformation operator, math. comput. modelling, 55(2012), 1456-1465. [19] h.m.srivastava, s.owa and s.k.chatterjea, a note on certain classes of starlike functions, rend.sem.mat.univ.padova, 77(1987), 115-124. [20] poonam sharma, j.k.prajapat and r.k.raina, certain subordination results involving a generalized multiplier transformation operator, journal of classical analysis, 2(1)(2013), 85-106. [21] l.p.xiong, some general results and extreme points of p-valent functions with negative coefficients, demonstratio mathematica, 44(2)(2011), 261-272. department of mathematics, engineering and technical college of chengdu university of technology, leshan, sichuan, 614007, p.r.china ∗corresponding author: xlpwxf@163.com international journal of analysis and applications volume 19, number 5 (2021), 695-708 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-695 received june 2nd, 2021; accepted july 8th, 2021; published august 2nd, 2021. 2010 mathematics subject classification. 62e10. key words and phrases. discretization; inverted topp-leone distribution; moments; count data; data analysis. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 695 a discrete analog of inverted topp-leone distribution: properties, estimation and applications ahmed sedky eldeeb1,2,*, muhammad ahsan-ul-haq3,4, ayesha babar5 1department of business administration, college of business, king khaled university, saudi arabia 2 department of statistics, mathematics and insurance, alexandria university, egypt 3college of statistical & actuarial sciences, university of the punjab, pakistan 4quality enhancement cell, national college of arts, lahore pakistan 5school of statistics, minhaj university lahore pakistan *corresponding author: asedky@kku.edu.sa abstract. in this study, a discrete inverted topp-leone (ditl) distribution is proposed by utilizing the survival discretization approach. the proposed distribution's mathematical features were derived. the maximum likelihood (ml), method of least squares (ls), weighted least squares (wls), and cramer von-mises (cvm) estimation techniques were used to estimate the parameter. the theoretical results of the ml, ls, wls, and cvm estimators were demonstrated via a comprehensive simulation study. the proposed ditl distribution has been applied to analyze two count data sets number of deaths due to covid-19 in pakistan and india and the findings show the relevance of the proposed distribution. 1. introduction in december 2019, the first incidence of covid-19 was reported in the chinese city of wuhan. covid-19 is an extremely contagious disease. in pakistan, the first case was reported on february 26, 2020 [1]. the first death was reported in pakistan on march 20, 2020. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-695 int. j. anal. appl. 19 (5) (2021) 696 many researchers make efforts to study the patterns of pandemic covid-19 and provide models which better fit the data and can be used to have an idea about the expected number of cases to help the government to take decisions regarding precautionary measures. these efforts include the derivation of different probabilistic models and time series modeling of the data such as a new discrete lindley [2], discrete marshall-olkin generalized exponential distribution to model the daily new cases in egypt [3], a new discrete generalized distribution to analyze the count of daily cases in hong kong and iran [4], a mathematical model known as sir is used to predict the daily new cases in china [5,6], the logistic growth model is used to estimate the final size and its peak time of coronavirus epidemic [7], autoregressive time series model is used to forecast the recovered and confirmed cases [8], and discrete marshall-olkin lomax distribution is used to estimate the daily new cases in australia [9]. practically, lifetime data sets are often recorded as whole numbers of counts. to model the count data, there are few classical distributions as geometric, poisson, negative binomial, etc. these models sometimes do not provide a better fit due to the complex behavior of data. from the last few decades, researchers have paid attention to introduced discrete type distributions which meet the required need to model the complex behavior of data sets. several discretization approaches are available in the literature. a detailed systematic review was conducted on discretization approaches [10]. among all approaches, one of the most important is the survival discretizing approach due to its important feature of keeping the original form of the survival function. “let x a continuous random variable. if x has a survival function 𝑆𝑋(𝑥), then the discrete random variable 𝑌 = [𝑋] , where [𝑋] indicates the smallest integer part or equal to 𝑋, have probability mass function (pmf) written as”: 𝑃(𝑌 = 𝑘) = ∑(−1)𝑖 1 𝑖=0 𝑆𝑋 (𝑘 + 𝑖). 𝑘 = 0,1,2,3, … (1) using survival discretization approach authors have derived discrete distributions. some of these include discrete weibull [11], discrete rayleigh [12], discrete burr and pareto [13], discrete inverse weibull [14], discrete lindley [15], discrete generalized rayleigh [16], discrete bilal [17], discrete nadarajah-haghighi [18], discrete burr-hutke [19] and many others. int. j. anal. appl. 19 (5) (2021) 697 the inverted topp-leone distribution [20] was derived for the analysis of reliability observations. a comprehensive discussion about its mathematical properties, reliability characteristics, stochastic ordering, and parameter estimation via complete and censored samples, among others is also presented in the mentioned paper. the corresponding survival function is given by 𝑆(𝑥) = (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 , 𝑥 > 0, 𝛿 > 0. (2) where 𝛿 is the shape parameter. the goal of this study is to introduce a new one-parameter discrete model called the discrete inverted topp-leone (ditl) distribution, which is based on the survival function approach of discretization. the ditl distribution can be used to model the over-dispersed data sets. we derive some of its properties such as, survival and hazard function, quantile function, moments, and generating function. the maximum likelihood, cramer-von mises, least-square, and weighted least square estimation methods are used to estimate the model's parameter. a simulation study is conducted to elaborate on the performance of these estimation methods. in the end, we will use data sets about the number of deaths due to coronavirus in pakistan and india to illustrate the importance of the proposed distribution. the organization of the article is as follows. section 2 contains the derivation of the proposed distribution and its features. section 3 addressed maximum likelihood estimation, as well as the least-squares, weighted least squares, and cramer von mises approach. in section 4, a complete simulation study is used to assess the behavior of these estimators. the proposed distribution's application is discussed in section 5. the conclusion has been presented in the final section. 2. the ditl distribution and properties using a discretization approach based on the survival function, the discrete inverted topp-leone distribution has been developed. the probability mass function (pmf) of ditl distribution can be represented as 𝑃(𝑥) = (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 , 𝛿 > 0, 𝑥 = 0,1,2, … (3) int. j. anal. appl. 19 (5) (2021) 698 the pmf plots of the ditl distribution with some selected values of parameter δ are presented in figure 1. figure 1: behavior of pmf of ditld for different parameter values the cumulative distribution and survival functions of ditl are given, respectively as 𝐹(𝑥) = 1 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 (4) and, 𝑆(𝑥) = (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 (5) the hazard function of the ditl distribution is obtained using eq. (3) and eq. (5). the behavior of the hazard function is illustrated in figure 2. ℎ(𝑥) = (1+2𝑥)𝛿 (1+𝑥)2𝛿 − (3+2𝑥)𝛿 (2+𝑥)2𝛿 (3+2𝑥)𝛿 (2+𝑥)2𝛿 int. j. anal. appl. 19 (5) (2021) 699 or ℎ(𝑥) = (1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 (1 + 𝑥)2𝛿 (3 + 2𝑥)𝛿 − 1 (6) where 𝛿 > 0 & 𝑥 = 0,1,2, …. note that, for 𝑥 → 0 the hrf turn into ℎ(0) = [ ( 4 3 ) 𝛿 − 1]. figure 2: behavior of hrf of ditld for different parameter values the reverse hazard function of ditl is given as 𝑟∗(𝑥) = 𝑃(𝑥) 𝐹(𝑥) = (1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 (1 + 𝑥)2𝛿 [(2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 ] (7) the second rate of failure of ditl is r∗∗(x) = log [ 𝑆(𝑥) 𝑆(𝑥 + 1) ] = log [ (3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿 (2 + 𝑥)2𝛿 (5 + 2𝑥)𝛿 ] (8) the recurrence relation for generating the probabilities of discrete ditl distribution is given by int. j. anal. appl. 19 (5) (2021) 700 𝑃(𝑥 + 1) 𝑃(𝑥) = (1 + 𝑥)2𝛿 (2 + 𝑥)2𝛿 [(3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿 − (5 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ] (2 + 𝑥)2𝛿 (3 + 𝑥)2𝛿 [(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 ] or 𝑃(𝑥 + 1) = (1 + 𝑥)2𝛿 [(3 + 2𝑥)𝛿 (3 + 𝑥)2𝛿 − (5 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ] (3 + 𝑥)2𝛿 [(1 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 ] 𝑃(𝑥). 2.1. quantile function the pth quantile function of ditl distribution is given by 𝑥𝑝 = √4((1 − 𝑝)(1 𝛿)⁄ − 1)2 − 4(1 − 𝑝)(1 𝛿) ⁄ ((1 − 𝑝)1 𝛿⁄ − 1) − 2((1 − 𝑝)1 𝛿⁄ − 1) 2(1 − 𝑝)(1 𝛿)⁄ . 2.2. the moments of ditl distribution the non-central moments of ditl distribution can be obtained using eq. (3) as follows: 𝜇𝑟 ʹ = ∑ 𝑥𝑟 ∞ 𝑥=0 𝑃(𝑥) 𝜇𝑟 ʹ = ∑ 𝑥𝑟 ∞ 𝑥=0 [ (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ] (9) in particular, the mean of ditl distribution is 𝜇1 ʹ = ∑ 𝑥 ∞ 𝑥=0 [ (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ] the central moments of ditl distribution can be obtained using the following relation 𝜇𝑟 = ∑ ( 𝑟 𝑗 ) 𝑟 𝑗=0 (−1)𝑗 𝜇 𝑗 𝜇𝑟−𝑗 ′ the variance of ditl distribution is given as 𝑉𝑎𝑟(𝑋) = ∑ 𝑥2 ∞ 𝑥=0 [ (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ] − {∑ 𝑥 ∞ 𝑥=0 [ (1 + 2𝑥)𝛿 (1 + 𝑥)2𝛿 − (3 + 2𝑥)𝛿 (2 + 𝑥)2𝛿 ]} 2 the dispersion index can be calculated using the expression 𝐷𝐼 = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝐷𝐼𝑇𝐷 𝑀𝑒𝑎𝑛 𝑜𝑓 𝐷𝐼𝑇𝐿𝐷 since the above equation cannot be solved, so the descriptive measures, i.e., mean and variance are computed numerically. these measures are presented in figure 3. int. j. anal. appl. 19 (5) (2021) 701 figure 3: plots of mean, variance and dispersion index for ditl distribution from figure 3, it is apparent that the mean, variance, and di of the ditl distribution have decreasing behavior with an increase in parameter δ. 3. parameter estimation of ditl distribution this section describes the parameter estimation of ditl distribution using four different estimation methods. these methods are maximum likelihood estimator (mle), cramer vonmises estimator (cvm), least square estimator (ls), and weighted least square estimator (wlse). 3.1. method of maximum likelihood let 𝑥1, 𝑥2, 𝑥3, … , 𝑥𝑛 be a random sample of size ‘n’ from ditl distribution with parameter 𝛿. the likelihood function is given by 𝐿 = ∑ log[(1 + 2𝑥𝑖 ) 𝛿 (2 + 𝑥𝑖 ) 2𝛿 − (3 + 2𝑥𝑖 ) 𝛿 (1 + 𝑥𝑖 ) 2𝛿 ] 𝑛 𝑖=1 − 2𝛿 ∑ log(1 + 𝑥𝑖 ) 𝑛 𝑖=1 − 2𝛿 ∑ log(2 + 𝑥𝑖 ) 𝑛 𝑖=1 (10) now partially differentiate with respect to parameter δ. int. j. anal. appl. 19 (5) (2021) 702 𝜕𝐿 𝜕𝛿 = ∑ { 2(1 + 2𝑥𝑖 ) 𝛿 (2 + 𝑥𝑖 ) 2𝛿 log(2 + 𝑥𝑖 ) + (2 + 𝑥𝑖 ) 2𝛿 (1 + 2𝑥𝑖 ) 𝛿 log(1 + 2𝑥𝑖 ) −2(3 + 2𝑥𝑖 ) 𝛿 (1 + 𝑥𝑖 ) 2𝛿 log(1 + 𝑥𝑖 ) − (3 + 2𝑥𝑖 ) 𝛿 log(3 + 2𝑥𝑖 ) (1 + 𝑥𝑖 ) 2𝛿 } [(1 + 2𝑥𝑖 ) 𝛿 (2 + 𝑥𝑖 ) 2𝛿 − (3 + 2𝑥𝑖 ) 𝛿 (1 + 𝑥𝑖 ) 2𝛿 ] 𝑛 𝑖=1 − 2 ∑ log(1 + 𝑥𝑖 ) 𝑛 𝑖=1 − 2 ∑ log(2 + 𝑥𝑖 ) 𝑛 𝑖=1 (11) the mle of ditl distribution parameter δ can be obtained from the above equation (11). the exact solution is not possible, so we can be obtained numerically. 3.2. method of cramer von-mises the cramèr-von-mises estimators (cvme), can be determined depending on the difference between both the estimated and exact distributions. the cvme estimator (𝛿𝐶𝑉𝑀 ) can be obtained by minimizing 𝐶𝑉𝑀(𝛿) = 1 12𝑛 + ∑ [𝐹(𝑥𝑖 ) − 2𝑖 − 1 2𝑛 ] 2𝑛 𝑖=1 (12) 3.3. method of least squares and weighted least squares the least-square estimator can be obtained by minimizing the sum of the square of residuals. 𝐿𝑆(𝛿) = ∑ [𝐹(𝑥𝑖 ) − 𝑖 𝑛 + 1 ] 2𝑛 𝑖=1 (13) the weighted least squares estimators of the parameter of ditl distribution are obtained by minimizing 𝑊𝐿𝑆(𝛿) = ∑ (𝑛 + 1)2(𝑛 + 2) 𝑛 − 𝑖 + 1 [𝐹(𝑥𝑖 ) − 𝑖 𝑛 + 1 ] 2𝑛 𝑖=1 (14) with respect to parameter δ. 4. simulation in this section, a simulation analysis evaluates the output of four different estimators of the ditl for different values of parameter δ. we consider the different sample sizes n = 10, 20, 50, and 100 for the different values of parameter = (0.8, 1, 1.5, 2, 3, 5, 10). from ditl distribution, we generate 10000 iterations of random samples. for each computation, we get the average estimations (aes) and mean square error (mse). the output of considered estimators is compared in terms of mse, with the lowest mse values indicating the best successful technique of estimation. the r program is used to obtain simulation results. the ae and mse values for the mle, cvm, ls, and wls int. j. anal. appl. 19 (5) (2021) 703 approaches are shown in table 1. methods tend towards the true parameter values, suggesting that all estimators are asymptotically unbiased. table 1: the simulation results for the parameter δ para. n mle cvme lse wse 𝛿 aes mses aes mses aes mses aes mses 0.8 10 0.9138 0.1310 1.0117 0.2847 0.9922 0.3008 0.9590 0.2406 20 0.8460 0.0376 0.9669 0.1519 0.9203 0.1012 0.9341 0.1144 50 0.8137 0.0142 0.9142 0.0476 0.9046 0.0414 0.8965 0.0380 100 0.8042 0.0065 0.8961 0.0224 0.8972 0.0249 0.8923 0.0210 1.0 10 1.1199 0.1786 1.3289 0.5450 1.2527 0.4231 1.2878 0.5169 20 1.0729 0.0675 1.2515 0.2398 1.2063 0.2054 1.2225 0.2150 50 1.0200 0.0212 1.1863 0.0908 1.1841 0.0879 1.1623 0.0745 100 1.0113 0.0101 1.1653 0.0539 1.1742 0.0590 1.1414 0.0433 1.5 10 1.6754 0.4174 2.0778 1.2522 1.9456 0.9305 1.9506 0.9876 20 1.5762 0.1524 1.9852 0.6515 1.9267 0.5357 1.8867 0.4822 50 1.5332 0.0543 1.9298 0.3412 1.9285 0.3425 1.8565 0.2574 100 1.5079 0.0213 1.9242 0.2546 1.8986 0.2330 1.8440 0.1781 2.0 10 2.2598 0.7833 2.8469 2.1167 2.6438 1.5017 2.6569 1.6724 20 2.0934 0.2675 2.7375 1.2770 2.6655 1.0143 2.6394 1.0230 50 2.0488 0.1046 2.6844 0.7219 2.6551 0.6799 2.5891 0.5918 100 2.0202 0.0468 2.6759 0.5741 2.6522 0.5508 2.5782 0.4564 3.0 10 3.6942 29.170 4.2539 3.9863 3.9756 2.8443 3.9646 3.1796 20 3.2014 0.7266 4.2718 2.8786 4.0647 2.2260 4.1605 2.7242 50 3.0637 0.2182 4.1455 1.7956 4.1121 1.7172 4.1168 1.7456 100 3.0366 0.1003 4.1571 1.5673 4.1206 1.4607 4.1275 1.5432 5.0 10 11.356 497.89 6.6005 7.1286 5.7374 2.9089 6.0616 3.5941 20 5.7253 29.774 6.8668 6.2545 6.4529 4.2814 6.6718 4.9023 50 5.1582 0.7127 6.8237 4.3197 6.6485 3.6260 7.0444 5.4704 100 5.0605 0.3035 6.7486 3.5393 6.7070 3.3384 7.1014 5.0482 10.0 10 51.192 3263.8 9.6885 0.9487 9.5835 1.5339 9.0152 2.6714 20 31.884 1671.7 9.8330 0.5029 9.6529 1.0747 9.7720 1.1258 50 15.195 355.64 9.9858 0.0241 9.9235 0.2293 11.7927 4.8997 100 10.609 24.082 9.9994 0.0081 9.9972 0.0099 12.7972 9.2633 int. j. anal. appl. 19 (5) (2021) 704 if 𝛿=0.8, 1, 1.5, and 2, the mle is the best estimation method in all sample sizes. if 𝛿 =3 with sample size n=10, the lse method is the best estimation method while in other sample sizes the mle is the best method of estimation. if 𝛿=5 with sample sizes n=10 and 20 the ls is the best estimation method while the mle is the best method in sample sizes n= 50 and 100. if 𝛿= 10 the cvme is the best estimation method in all sample sizes. 5. application in this section, we illustrate the importance of the proposed distribution using two data sets. both data sets are counts. five one-parameter competitive distributions of the ditl distribution are poison distribution, discrete pareto distribution [13], discrete rayleigh distribution [12], discrete inverse rayleigh distribution [21], and discrete burr-hutke distribution [19]. the first data represents the number of deaths due to coronavirus in pakistan. a sample of 44 deaths is considered from march 18, 2020, to april 30, 2020. the second data set represents the number of deaths due to coronavirus in india. a sample of 51 deaths is considered from march 11, 2020, to april 30, 2020. both data sets are plotted in figure 4. figure 4. plots for covid-19 daily deaths in pakistan and india the results presented in tables 2-3 and figures 5-6 demonstrate the sufficiency and superiority of the proposed distribution in modeling the data sets when compared to the competitive distributions. int. j. anal. appl. 19 (5) (2021) 705 table 2: the mles, standard errors of the competing models for number of deaths in pakistan model mle goodness-of-fit measures 𝐸(𝛿) se ℓ aic bic ks ditld 0.70958 0.10705 -153.04 308.09 309.87 0.1619 dpoi 9.47694 0.46409 -283.94 569.89 571.67 0.3910 dpr 0.50220 0.07576 -162.19 326.38 328.17 0.4010 dr 9.98734 0.75335 -168.85 339.70 341.49 0.3390 dir 7.43010 1.26260 -166.31 334.61 336.40 0.3820 dbhd 0.99483 0.01148 -175.37 352.74 354.52 0.6470 table 3: the mles, standard errors of the competing models for number of deaths in india model mle goodness-of-fit measures 𝐸(𝛿) se ℓ aic bic ks ditld 0.52980 0.07421 -218.93 439.86 441.79 0.2190 dpoi 22.6661 0.66665 -740.90 1483.8 1485.7 0.4930 dpr 0.40864 0.05726 -221.91 445.82 447.75 0.2470 dr 22.8734 1.60190 -259.23 520.46 522.39 0.3780 dir 3.74690 0.59100 -306.05 614.10 616.03 0.5470 dbhd 0.99905 0.00443 -249.99 501.97 503.91 0.5100 figure 5: density plot for the deaths due to covid-19 in pakistan int. j. anal. appl. 19 (5) (2021) 706 figure 6: density plot for the deaths due to covid-19 in india 6. conclusion in this study, a new one-parameter discrete inverted topp-leone distribution has been proposed using a survival discretizing approach. some structural properties of the proposed distribution are discussed. different estimation methods including maximum likelihood estimation, leastsquare and weighted least squares estimation, and cramer von-mises estimation have been presented. we carried out a simulation study to illustrate the performance of the parameter by different methods. among all methods, the maximum likelihood estimator performs better for a large sample size. application of the proposed ditl distribution in the analysis of two discrete data sets about covid-19 has been presented. the ditl distribution, which has flexible features, is expected to make a significant contribution to the field of count data modeling. acknowledgement: authors would like to express their gratitude to king khalid university, saudi arabia for providing administrative and technical support. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (5) (2021) 707 references [1] m. ahsan-ul-haq, m. ahmed, j. zafar, and p. l. ramos, modeling of covid‑19 cases in pakistan using lifetime, ann. data sci. (2021), https://doi.org/10.1007/s40745-021-00338-9. [2] a. a. al-babtain, abdul hadi n. ahmed, and a. z. afify, a new discrete analog of the continuous lindley distribution, with reliability applications, entropy, 22(6) (2020), 603. [3] e. m. almetwally, h. m. almongy and h. a. saleh, managing risk of spreading “covid-19” in egypt: modelling using a discrete marshall-olkin generalized exponential distribution, int. j. probab. stat. 9(2) (2020), 33-41. [4] e. m. almetwally, and g. m. ibrahim, discrete alpha power inverse lomax distribution with application of covid-19 data, int. j. appl. math. 9(6) (2020), 11-22. [5] b. milan, estimation of the final size of the second phase of the coronavirus covid 19 epidemic by the logistic model. medrxiv (preprint), (2020b), https://doi.org/10.1101/2020.03.11.20024901. [6] i. nesteruk, statistics based predictions of coronavirus 2019-ncov spreading in mainland china, medrxiv, (2020), https://doi.org/10.1101/2020.02.12.20021931. [7] b. milan, estimation of the final size of the covid-19 epidemic. medrxiv, (2020), https://doi.org/10.1101/2020.02.16.20023606. [8] m. maleki, m. r. mahmoudi, d. wraith, and k. pho, time series modelling to forecast the confirmed and recovered cases of covid-19, travel med. infect. dis. 37 (2020), 101742. [9] g. m. ibrahim, and e. m. almetwally, discrete marshall-olkin lomax distribution application of covid-19, biomed. j. sci. techn. res. 32(5) (2021), 25381-25390. [10] s. chakraborty, generating discrete analogues of continuous probability distributions-a survey of methods and constructions, j. stat. distrib. appl. 2 (2015), 6. [11] t. nakagawa, and s. osaki, the discrete weibull distribution. ieee trans. reliab. 24(5) (1975), 300-301. [12] roy, dilip. discrete rayleigh distribution, ieee trans. reliab. 53(2) (2004), 255-260. [13] h. krishna, and p. s. pundir, discrete burr and discrete pareto distributions, stat. methodol. 6(2) (2009), 177–88. [14] m. a. jazi, d. l. chin, and h. a. mohammad, a discrete inverse weibull distribution and estimation of its parameters. stat. methodol. 7(2) (2010), 121–32. [15] e. gómez-déniz, and c. enrique, the discrete lindley distribution: properties and applications, j. stat. comput. simul. 81(11) (2010), 1405–16. [16] m. h. alamatsaz, s. dey, t. dey and s. s. harandi, discrete generalized rayleigh distribution, pak. j. stat. 32(1) (2016), 1-20. int. j. anal. appl. 19 (5) (2021) 708 [17] e. altun, m. el-morshedy, and m. s. eliwa, a study on discrete bilal distribution with properties and applications on integer-valued autoregressive process, revstat. stat. j. 18 (2020), 70-99. [18] m. shafqat, s. ali, i. shah and s. dey, univariate discrete nadarajah and haghighi distribution: properties and different methods of estimation, statistica, 80(3) (2020), 301-330. [19] m. el-morshedy, m. s. eliwa, and e. altun, discrete burr-hatke distribution with properties, estimation methods and regression model, ieee access, 8 (2020), 74359–70. [20] a. s. hassan, m. elgarhy and r. ragab, statistical properties and estimation of inverted topp-leone distribution, j. stat. appl. probab. 9(2) (2020), 319-331. [21] t. hussain, and m. ahmad, discrete inverse rayleigh distribution, pak. j. stat. 30(2) (2014), 203-222. international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 1-14 http://www.etamaths.com smoothness to the boundary of biholomorphic mappings steven g. krantz abstract. under a plausible geometric hypothesis, we show that a biholomorphic mapping of smoothly bounded, pseudoconvex domains extends to a diffeomorphism of the closures. 1. introduction the riemann mapping theorem (see [11]) tells us that the function theory of a simply connected, planar domain ω, other than than the entire plane, can be transferred from ω to the unit disc d. but, for many questions, one needs to know the behavior of the riemann mapping at the boundary. the first person to take up this issue was p. painlevé. he proved in his thesis that, if the domain ω has c∞ boundary, then the riemann mapping (and its inverse) extends smoothly to the boundary (see [5] for details of this history). later o. kellogg gave a proof of this result that connected the riemann mapping with potential theory. further on, stefan warschawski refined kellogg’s results and gave substantive local boundary analyses of the riemann mapping. it was quite some time before any progress was made on this question in the context of several complex variables. the first real theorem of a general nature was proved by c. fefferman [7]. he showed that a biholomorphic mapping of smoothly bounded, strongly pseudoconvex domains in cn extends to a diffeomorphism of the closures. fefferman’s work opened up a flood of developments in this subject. we only mention here that bell [2] and bell/ligocka [4] were able to greatly simplify fefferman’s proof by connecting the problem in a rather direct fashion with the bergman projection. the work of bell and bell/ligocka led to a number of simplifications, generalizations, and extensions of fefferman’s result. many different mathematicians have contributed to the development of this work. the big remaining open problem is this: problem: let ω1 and ω2 in cn be smoothly bounded, (weakly levi) pseudoconvex domains. let φ : ω1 → ω2 be a biholomorphic mapping. does φ extend to a diffeomorphism of the closures? there are some counterexamples to this question—see for instance [9]—but these definitely do not have smooth boundary. in fact they do not even have c2 boundary. 2010 mathematics subject classification. 32h40, 32h02. key words and phrases. pseudoconvex domain; biholomorphic mapping; lipschitz condition; smooth extension; diffeomorphism. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 krantz in the present paper we are unable to give a full answer to this main problem. but we present the following somewhat encouraging partial result. theorem 1.1. let ω1, ω2 be smoothly bounded, levi pseudoconvex domains in cn. let φ : ω1 → ω2 be a biholomorphic mapping. assume that φ and φ−1 each satisfy a lipschitz condition of order exceeding (n− 1)/n. then φ continues to a diffeomorphism of the closures of the domains. corollary 1.2. let ω1, ω2 be smoothly bounded, pseudoconvex domains in cn. let φ : ω1 → ω2 be a biholomorphic mapping. assume that φ and φ−1 each satisfy a lipschitz condition of order 1. then φ continues to a diffeomorphism of the closures of the domains. this result is in the nature of a bootstrapping result from partial differential equations. it seems to be the first general result—for all pseudoconvex domains— of its kind. and it has some basis in the history of the subject. for painlevé proved his theorem in dimension one by first establishing a result for c1 boundary smoothness of the mapping, and then bootstrapping. no less an eminence gris than jacques hadamard cast public doubt on painlevé’s bootstrapping argument, and painlevé had to work strenuously to defend his theorem. see [5] for the full history. it may be noted that the hypothesis of lipschitz continuity in the theorem is a nontrivial one. henkin [12] was able to show, prior to fefferman’s celebrated result, that a biholomorphic mapping of smoothly bounded, strongly pseudoconvex domains will extend to be lipschitz 1/2 to the boundary. he did so by analyzing and estimating the carathéodory metric. but there are not many results of this type. we see that the lipschitz condition in theorem 1.1 in case n = 2 meshes nicely with henkin’s result described in the last paragraph. a final, rather significant, comment is this. our arguments here are inspired by those in [2]. bell uses global regularity ideas of kohn which exploit weighed l2 spaces. in the paper [2], a good deal of the work is expended in proving that the complex jacobian determinant u of the mapping φ is bounded. this fact is used in turn to prove that the complex jacobian determinant u of the inverse mapping φ−1 is nonvanishing. as we shall see below, our hypothesis of lipschitz continuity of order exceeding (n − 1)/n obviates these arguments and gets to the necessary result rather quickly. the remaining steps comprise just one paragraph on page 108 of [2]. we have to work a bit harder because we need to set things up in the context of kohn’s weighted l2 spaces. but the spirit of our arguments follows bell. we also warn the reader of the following point. the main thrust of this paper is to prove estimates on the derivatives of the mappings φ and φ−1. however our crucial lemma 4.2, based on an old idea of s. r. bell, entails taking a good many derivatives of φ. so it appears as though there are a number of extraneous terms in our calculations that involve derivatives of φ. but we will go to quite a lot of extra trouble to find a means of absorbing those extra derivatives. in the end they will all be accounted for, and we will obtain valid estimates for the derivatives of φ. 2. condition r and related ideas throughout this paper we shall use the language of sobolev spaces (see [1], [19]). for s a nonnegative integer and 1 ≤ p ≤∞, we let ws,p denote the usual sobolev smoothness to the boundary of biholomorphic mappings 3 space of functions with s weak derivatives in lp. the norm that we use on the sobolev space is standard, and we refer to [1] for details. one of the important innovations that s. r. bell introduced into this subject is his condition r. it says this: condition r: let ω ⊆ cn be a smoothly bounded domain. we say that ω satisfies condition r if the bergman projection p maps c∞(ω) to c∞(ω). equivalently, for each s > 0, there is an m(s) > 0 so that the bergman projection p maps the sobolev space wm(s),2(ω) to the sobolev space ws,2(ω). in what follows we shall suppose that 1 < m(1) < m(2) < · · · → ∞ and that each m(j) is an integer. in the paper [2], bell proves the following elegant result: theorem 2.1. let ω1, ω2 be smoothly bounded, pseudoconvex domains in cn. assume that one (but not necessarily both) of these domains satisfies condition r. then any biholomorphic mapping φ : ω1 → ω2 extends to a diffeomorphism of the closures. 3. ideas of kohn the classical treatment of the ∂-neumann problem is based on the traditional euclidean l2 inner product—see [8]. kohn’s idea in [14] is to use an inner product with a weight. this is inspired by work of hörmander [13], and that in turn comes from old ideas of carleman. kohn’s setup is this (see [14]). fix a smoothly bounded domain ω in cn. let λ be a c∞, nonnegative function on a neighborhood of ω. usually λ will be strictly plurisubharmonic. with λ fixed and t ≥ 0, we shall define the ∂-neumann problem of weight t, with real t > 0. we let a be the space of all forms on ω which have c∞ coefficients up to the boundary. for φ,ψ ∈a, we define 〈φ,ψ〉(t) ≡〈φ,e−tλψ〉 and ‖φ‖2(t) = 〈φ,φ〉(t) . here 〈 , 〉 = 〈 , 〉(0) is the usual l2 inner product. it is an easily verified fact that the norms ‖ ‖(t) are equivalent to the norm ‖ ‖0 = ‖ ‖. hence a function is in the completion under any of these norms if and only if it is square integrable. we let ãt be the hilbert space obtained by completing a under the norm ‖ ‖(t). the ∂-neumann problem may be set up in the 〈 , 〉(t) inner product rather than the usual l2 inner product 〈 , 〉. these are familiar ideas, and the details are provided in [14]]. one of the main points that must be noted is that the formal adjoint of the operator ∂, when calculated in the 〈 , 〉(t) inner product, is it = i− tσ(i,dλ) . here σ is the “symbol” in the usual sense of pseudodifferential operators. also i is the standard formal adjoint of ∂ with respect to the standard euclidean hermitian inner product and it is the formal adjoint of ∂ with respect to the inner product 〈 , 〉(t). we thus see how the parameter t comes into play. if t is chosen positive and large enough, then certain terms in the usual ∂-neumann estimates can be forced to dominate certain others. again see [14] for the details. 4 krantz let ω1, ω2 be smoothly bounded, pseudoconvex domains and φ : ω1 → ω2 a biholomorphic mapping which is bi-lipschitz of order exceeding (n − 1)/n. we shall apply the preceding ideas on ω2 with λ(z) = λ2(z) = |z|2 and on ω1 with λ(z) = λ1(z) = |φ(z)|2. note that we are applying kohn’s construction twice.1 in this context we shall refer to the t-weighted bergman projection as pt,1 (for ω1) and pt,2 (for ω2). we shall also call bell’s regularity condition in the context of kohn’s weighted inner product by the name “condition rt.” we shall denote the bergman kernels by kt,1 and kt,2 . as a result of these ideas, the ∂-neumann problem on ω2, formulated with the indicated weight λ, satisfies favorable estimates (this follows from [14]) as long as t is large enough. hence ω2 satisfies condition rt. bell also makes use of these facts. these are the tools that we shall need in the next section to get to our result. in what follows we shall take it that we are working with the bergman theory for the inner product 〈 , 〉(t) for t sufficiently large, and that ω2 satisfies condition rt. we formulate this last by saying that pt,2 : wm(s),2(ω) → ws,2(ω) for any s ≥ 0 and suitable m(s) ≥ s. sometimes, in what follows, we will talk about (i) a domain ω with weight λ but make no reference to (ii) ω1, ω2, or the mapping φ. we will later apply (i) to (ii). 4. the guts of the proof in this section, ω is a smoothly bounded, pseudoconvex domain equipped with the weight e−tλ. lemma 4.1. let ω ⊂⊂ cn be smoothly bounded and pseudoconvex. suppose that the λ from the weight on ω is smooth on ω. assume that ω satisfies condition rt with respect to the projection pt. let w ∈ ω be fixed. let kt denote the bergman kernel. then there is a constant cw > 0 so that ‖kt(w, ·)‖sup ≤ cw . proof: the function k(z, ·) is harmonic. let ϕ : ω → r be a radial, c∞ function centered at w and supported in ω so that the radius of the support is comparable to half the distance of w to the boundary. assume that ϕ ≥ 0 and ∫ ϕ(ζ) dv (ζ) = 1. then the mean value property implies that kt(z,w) = ∫ ω kt(z,ζ)ϕ(ζ) dv (ζ) = ∫ ω kt(z,ζ) [ ϕ(ζ)etλ(ζ) ] e−tλ(ζ) dv (ζ) . 1it is because the weight e−t|φ(z)| 2 gets differentiated in the proofs below that we must be careful to absorb these error terms. smoothness to the boundary of biholomorphic mappings 5 of course this last displayed expression equals pt ( ϕ( ·)etλ( ·) ) . thus ‖kt(w, ·)‖sup = sup z∈ω |kt(w,z)| = sup z∈ω |kt(z,w)| = sup z∈ω |pt [ ϕ( ·)etλ( ·) ] | . by sobolev’s theorem, this last is ≤ c(ω,w)‖pt [ ϕ( ·)etλ( ·) ] ‖w2n+1,2 . by condition rt, this is ≤ c(ω,w) · ‖ϕ( ·)etλ( ·)‖wm(2n+1),2 ≡ cw . remark: it is worth noting that the estimate obtained in this last proof depends on some derivatives of λ on a compact set. in practice this causes no harm. we only need to know that ‖kt(w, ·)‖sup is bounded so that we can perform an integration by parts in the proof of lemma 4.2 below. lemma 4.2. let u ∈ c∞(ω) be arbitrary. let s ∈ {0, 1, 2, . . .}. then there is a v ∈ c∞(ω) such that ptv = 0 and the functions u and v agree to order s on ∂ω. remark: this lemma in the present formulation is not entirely satisfactory. for, in its statement here, we suppose that the weight λ is smooth across the boundary. yet, in the applications below, the weight is taken to be |φ(z)|2, and that is not known a priori to be smooth up to the boundary (in fact our goal is to prove that it is smooth up to the boundary). the way to address this problem is to use an approximation argument. in the case that the domain ω1 is convex, the approximation is very simple. we simply replace ω1 by (1−�)ω1, � > 0, so that the mapping is smooth across the boundary. the relevant estimates are uniform in �, and the result is correct in the limit. for non-convex ω1, we must take advangage of ideas in [3]. for bell explains there how to localize the smoothness-to-the-boundary arguments that we present here. as a result, if p ∈ ∂ω1, νp is the euclidean unit outward normal vector at p, and u is a small neighborhood of p, then we may apply our arguments on ω�1 ≡ (u∩ω1)−�νp. thus the mapping will be smooth across the boundary and (a localized version of) lemma 4.2 will apply without any problem. all the relevant estimates are uniform in �, and our desired result holds in the limit. proof: this lemma is the key to bell’s approach to these matters. we will need to expend some effort to adapt bell’s ideas to the new weighted context. we of course assume that our domain ω is equipped with an inner product 〈 , 〉(t) based on a weight e−tλ. after a partition of unity, it suffices to prove the assertion in a small neighborhood w of z0 ∈ ∂ω. after a rotation, we may assume that ∂ρ/∂z1 6= 0 on w ∩ ω. [here ρ is a defining function for the domain ω—see [15].] 6 krantz define the differential operator ν = re {∑n j=1 ∂ρ ∂zj ∂ ∂zj } ∑n j=1 ∣∣∣ ∂ρ∂zj ∣∣∣2 . observe that νρ ≡ 1. now we shall define v by induction on s. in what follows, we shall make use of the differential operator t = ∂ ∂ζ1 + t ∂λ ∂ζ1 . for the case s = 0, we set w1 = ρu tρ . if w is small then of course tρ will not vanish. also define v1 = tw1 = u + o(ρ) . then we see immediately that u and v1 agree to order 0 on ∂ω. furthermore, ptv1 = ∫ kt(z,ζ)tw1e −tλ dv = − ∫ tζ [ kt(z,ζ)e −tλ]w1 dv = 0 . the penultimate equality comes from integration by parts. this operation is justified by lemma 4.1. note that tζ annihilates kt(z,ζ)e −tλ(ζ) by a simple calculation (using the fact that kt(z,ζ) is conjugate holomorphic in the ζ variable). now suppose inductively that ws−1 = ws−2 + θs−1 ·ρs−1 , vs−1 = tws−1 (for some smooth function θs−1). we construct ws = ws−1 + θs ·ρs so that vs ≡ tws agrees to order s− 1 with u on ∂ω. by the inductive hypothesis, vs = tws = tws−1 + t(θsρ s) = vs−1 + ρ s−1 [sθstρ + ρtθs] . this expression agrees, by the inductive hypothesis, with u to order s− 1 on ∂ω. we now must examine d(u − vs), where d is any s-order differential operator. there are two cases: case 1:: assume that d involves a tangential derivative d0. then we may write d = d0d1. then d(u−vs) = d0α, where α vanishes on ∂ω. but then it follows that d0α = 0 because d0 is tangential. smoothness to the boundary of biholomorphic mappings 7 case 2:: now assume that d has no tangential derivative in it. so we take d = νs, where ν was defined at the beginning of this discussion. our job is to choose θs so that νs(u−vs) = 0 on ∂ω . so we require that νs(u−vs−1) −νs (t(θsρs)) = 0 on ∂ω . this is the same as νs(u−vs−1) −θs (νstρs) = 0 on ∂ω (because terms that contain ρ must vanish on ∂ω) or νs(u−vs−1) −θs ( νs ∂ ∂ζ1 ρs ) −θs ( νst ∂λ ∂ζ1 ρs ) = 0 on ∂ω . this may be rewritten as νs(u−vs−1) −θs ·s! ∂ρ ∂ζ1 − t ·θs · τ ·ρ, where τ stands for terms that come from the differentiations. the last line may be rewritten as θs = νs(u−vs−1) s! ∂ρ ∂ζ1 + t · τ ·ρ . if w is small enough then the denominator cannot vanish and we see that we have a well-defined choice for θs as desired. we note that, in [2], bell has a particularly elegant way of expressing the content of this last lemma. his formulation will be useful for us later, so we formulate it now. first some notation. if ω ⊆ cn is a domain (a connected, open set), then let w 0s,p(ω) be the closure of c∞c (ω) in ws,p(ω). now we have bell’s formulation of our lemma 4.2: corollary 4.3. let ω be a smoothly bounded, pseudoconvex domain. let s,m ∈ {0, 1, 2, . . .}. then there is a linear operator ψs,m : ws+m,2(ω) → w 0s,2(ω) such that ptψ s,m = id. the norm of this operator depends polynomially on t and on derivatives of λ. 5. a deeper analysis a troublesome feature of lemma 4.2 and corollary 4.3 is that the weight λ gets differentiated s times, and λ (in practice) is defined in terms of the mapping φ. since our job in the end is to estimate the derivatives of φ, this looks problematic. we need to develop a way to absorb these extraneous derivatives of φ. with that thought in mind, we recall the standard sobolev embedding theorem for a smooth domain in rn (see [1], [19] for details). proposition 5.1. [?] let ω ⊆ rn be a smoothly bounded domain. let wm,p be the standard sobolev space of functions on ω having m weak derivatives in the space lp. equip wm,p with the usual norm. then we have the embedding wk,p ⊆ w`,q 8 krantz whenever k > `, 1 ≤ p < q ≤∞, and 1 q = 1 p − k − ` n . we are particularly interested in domains in cn. hence, for us, n = 2n. also we will apply the result in case k = 4 and ` = 2. thus the important inclusion is wk+n/2,2 ⊆ wk,4 . we will generally exploit this embedding in the form of the inequality ‖f‖k,4 ≤ c‖f‖k+n/2,2 . we will also make good use of the following refinement of the sobolev theorem that is due to ehrling, gagliardo, and nirenberg (see [1] for the details): theorem 5.2. let ω ⊆ rn be a smoothly bounded domain. let wm,p be the standard sobolev space of functions on ω having m weak derivatives in the space lp. equip wm,p with the usual norm. let �0 > 0. let m be a positive integer. then there is a constant k, depending on �0, m, p, and ω so that, for any integer j with 0 ≤ j ≤ m− 1, any 0 < � < �0, and any u ∈ wm,p, ‖u‖j,p ≤ k�‖u‖m,p + k�−j/(m−j)‖u‖0,p . it is worth taking some time now to do a little analysis. examine the proof of lemma 4.2. [note that, when we apply lemma 4.2 and corollary 4.3, we do so on ω1 with λ(z) = λ1(z) = |φ(z)|2.] at each stage we integrate by parts, and therefore a derivative falls on λ (and hence on φ). thus we see that the function v that we construct has the form v = q0 + q1t∇φ + q2t2∇2φ + · · · + qsts∇sφ + q′1t∇φ + q ′ 2t 2∇2φ + · · · + q′st s∇sφ . here we use the notation ∇φ or ∇φ to denote some derivative of some component of φ or φ and ∇jφ or ∇jφ to denote some jth derivative of some component of φ or φ. also qj, q ′ j denotes an expression that involves components of φ, derivatives of ρ, and derivatives of θj. thus we see that ‖v‖r,2 ≤ c · [∫ ∣∣q0∣∣2 dv 1/2 + ∫ ∣∣∇rq0∣∣2 dv 1/2 + ∫ |q1t∇φ|2 dv 1/2 + ∫ ∣∣(t∇rq1)φ∣∣2 + ∣∣tq1∇r+1φ∣∣2 dv 1/2 + ∫ |q2t∇2φ|2 dv 1/2 + ∫ ∣∣(t2∇rq2)φ∣∣2 + ∣∣t2q2∇r+2φ∣∣2 dv 1/2 + · · · + ∫ |qst∇sφ|2 dv 1/2 + ∫ ∣∣(ts∇rqs)φ∣∣2 + ∣∣tsqs∇r+sφ∣∣2 dv 1/2] plus similar terms involves q′j and ∇ jφ. smoothness to the boundary of biholomorphic mappings 9 using hölder’s inequality, we see that this last is majorized by c · [∫ |q0|2 dv 1/2 + ∫ ∣∣∇rq0∣∣2dv 1/2 + t ∫ |q1|4 dv 1/4 · ∫ |∇φ|4 dv 1/4 + t ∫ ∣∣∇rq1∣∣4 dv 1/4 ·∫ ∣∣φ∣∣4 dv 1/4 +t ∫ ∣∣q1∣∣4 dv 1/4 ·∫ ∣∣∇r+1φ∣∣4 dv 1/4 + t2 ∫ |q2|4 dv 1/4 · ∫ |∇2φ|4 dv 1/4 + t2 ∫ ∣∣∇rq2∣∣4 dv 1/4 ·∫ ∣∣φ∣∣4 dv 1/4 +t2 ∫ ∣∣q2∣∣4 dv 1/4 ·∫ ∣∣∇r+2φ∣∣4 dv 1/4 + · · · + t ∫ |qs|4 dv 1/4 · ∫ |∇sφ|4 dv 1/4 + ts ∫ ∣∣∇rqs∣∣4 dv 1/4 ·∫ ∣∣φ∣∣4 dv 1/4 +ts ∫ ∣∣qs∣∣4 dv 1/4 ·∫ ∣∣∇r+sφ∣∣4 dv 1/4] plus similar terms involves q′j and ∇ jφ. now we may estimate this more elegantly as c′ · (1 + t)s [ 1 + ‖φ‖r+s,4 ] . and then we apply the sobolev inequality noted above to estimate this at last by c′ · (1 + t)s [ 1 + ‖φ‖r+s+n/2,2 ] . at last we apply the ehrling-gagliardo-nirenberg inequality (with m = r + s + n, j = r + s + n/2) to obtain ≤ c′′(1 + t)s ( 1 + � · ‖φ‖r+s+n,2 + �−(r+s+n/2)/(n/2) · ‖φ‖0,2 ) . (5.2) this inequality is valid for any 0 < � < 1 provided that c′′ is large enough. furthermore, c′′ depends on s. we finally note that, in the proof of lemma 4.2, the definition of w1 involves a division by tρ (and the definition of t entails a derivative of φ). but this can be treated with a neumann series, or just using the quotient rule. now we can reformulate corollary 4.3 as follows: corollary 5.3. let ω be a smoothly bounded, pseudoconvex domain. let s ∈ {0, 1, 2, . . .}. fix any � > 0. then there is a linear operator ψs : wr+s+n,2(ω) → w 0r,2(ω such that qtψ s = id. moreover, the norm of this operator does not exceed c′′(1 + t)s ( 1 + � · ‖φ‖r+s+n,2 + �−(r+s+n/2)/(n/2) · ‖φ‖0,2 ) . 6. the final argument for the rest of this discussion, we let ω1 and ω2 be fixed, smoothly bounded, pseudoconvex domains in cn. we fix a strictly plurisubharmonic function λ(z) = λ2(z) = |z|2, and we equip ω2 with the inner product with weight e−tλ(z). we also, as above, equip ω1 with the inner product having weight λ(z) = λ1(z) = |φ(z)|2. we assume that there is a biholomorphic mapping φ : ω1 → ω2 that extends in a bi-lipschitzian fashion, of order greater than (n − 1)/n, to the boundary, and we equip ω1 with the inner product with weight e −tλ|φ(z)|2 . we let pt,1 and pt,2 10 krantz be the resulting bergman projections for ω1, ω2 respectively. we let u denote the complex jacobian determinant of φ. and we let u denote the complex jacobian determinant of φ−1. so u is a complex-valued holomorphic function on ω1 and u is a complex-valued holomorphic function on ω2. finally, for j = 1, 2, we let δj(z) = δωj (z) = disteuclid(z, cωj) for z ∈ ωj. lemma 6.1. the bergman kernels for the two domains are related by kt,1(z,ζ) = u(z) ·kt,2(φ(z), φ(ζ)) ·u(ζ) . (6.1.1) proof: this is a standard change-of-variables argument, using the canonical relationship between the real jacobian determinant of a biholomorphic mapping and the complex jacobian determinant of that mapping (see lemma 1.4.10 of [15]). now, if f is a bergman space function on ω1, then we have∫ ω1 [ u(z)kt,2(φ(z), φ(ζ))u(ζ) ] f(ζ)e−tλ(φ(ζ) dv (ζ) = ∫ ω2 u(z)kt,2(φ(z),ξ)u(φ−1(ξ)) ×f(φ−1(ξ))u−1(ξ)u−1(ξ)e−tλ(ξ) dv (ξ) = f(z)u(z)u−1(φ(z)) = f(z) . thus we see that the righthand side of (6.1.1) has the reproducing property on ω1. it is also conjugate symmetric and is an element of the bergman space in the first variable. therefore it must equal kt,1(z,ζ). lemma 6.2. for any function g ∈ l2(ω2), we have pt,1 (u · (g ◦ φ)) = u · ((pt,2(g) ◦ φ) . proof: we use the preceding lemma to calculate that u(z) · ((pt,2(g) ◦ φ) (z) = u(z) ∫ ω2 kt,2(φ(z),ζ)g(ζ)e −tλ(ζ) dv (ζ) = u(z) ∫ ω2 u(z)−1kt,1(z, φ −1(ξ))u(φ(ξ))−1 ×g(ζ)e−tλ(ζ) dv (ζ) = u(z) ∫ ω1 u(z)−1kt,1(z,ξ)u(ξ)−1 ×g(φ(ξ))e−tλ(φ(ξ))u(ξ)u(ξ) dv (ξ) = ∫ ω1 kt,1(z,ξ)g(φ(ξ))u(ξ)e −tλ(φ(ξ)) dv (ξ) = pt,1 (u · (g ◦ φ)) (z) . that establishes the result. it will be useful to have the following corollary, in which φ is replaced by φ−1 (and of course the corresponding bergman kernels switch roles): smoothness to the boundary of biholomorphic mappings 11 corollary 6.3. let u denote the complex jacobian determinant of φ−1. then, for any function g ∈ l2(ω1), we have pt,2 ( u · (g ◦ φ−1) ) = u · ( (pt,1(g) ◦ φ−1 ) . lemma 6.4. let h∞(ω1) denote the space of holomorphic functions on ω1 which extend smoothly to ω1. let s ∈ {0, 1, 2, . . .}. if h ∈ h∞(ω1), then let φs = ψsh, where ψs is introduced in corollary 4.3. then u · (h◦ φ−1) = pt,2(u · (φs ◦ φ−1)) . proof: we calculate, using corollary 6.3, that pt,2(u ·(φs◦φ−1)) = u ·(pt,1(φs)◦φ−1) = u ·(pt,1(ψsh)◦φ−1) = u ·(h◦φ−1) . the next lemma has nothing to do with condition r. it is really only calculus. lemma 6.5. suppose that φ−1 : ω2 → ω1 is a biholomorphic mapping between smoothly bounded, pseudoconvex domains in cn. assume that φ is bi-lipschitz of order exceeding (n− 1)/n. let u denote the complex jacobian determinant of φ−1. for each nonnegative integer s, there is an integer n = n(s) such that the operator g 7−→ u · (g ◦ φ−1) is bounded from w 0s+n,2(ω1) to w 0 s,2(ω2). proof: in what follows we let dj(z) denote the euclidean distance of z from the boundary of ωj. since the components of φ−1 are holomorphic and lipschitz of order exceeding (n− 1)/n, the derivatives of φ−1 satisfy finite growth conditions at the boundary (see [10]). that is to say∣∣∣∣∂αφ−1∂wα (w) ∣∣∣∣ ≤ c ·d2(w)−k+(n−1)/n . (6.5.1) here α is a multi-index, k = |α|, and dj is the distance of the argument to the boundary of ωj, j = 1, 2. estimates like this one go back to hardy and littlewood (see [10]). now sobolev’s lemma and taylor’s formula tell us that, for g ∈ w 0 s+|α|+n,2(ω1), |dαg(z)| ≤ c · ‖g‖s+|α|+nd1(z)s . for a given s, in order to show that an n exists so that g 7→ u ·(g◦φ−1) is bounded from w 0s+n,2(ω1) to w 0 s,2(ω2), it will suffice to show that there is an integer m > 0 such that d1(φ −1(w))m ≤ c · d2(w). that such an m exists is proved by range [16]. the proof, naturally, consists of applying hopf’s lemma to ρ◦ φ−1, where ρ is a bounded, plurisubharmonic exhaustion function for ω2 of the form vd 1/m 2 with v ∈ c∞(ω2) and v < 0 on ω2. of course diederich and fornæss [6] have proved the existence of such an exhaustion function. range [17] has given a simpler approach to the matter, with the penalty of assuming greater boundary smoothness. that completes the proof of the lemma. 12 krantz lemma 6.6. let s ∈{0, 1, 2, . . .}. with notation as above, ‖u · (h◦ φ−1)‖s ≤‖h‖s+n . proof: we note that kohn’s theory (see [2] for the details) entails that pt,2 maps ws,2(ω2) to ws,2(ω2) for t sufficiently large and any s. now we apply corollary 6.3 and then lemma 6.4 to see that ‖u · (h◦ φ−1)‖s ≤ ‖pt,2(u · (φs ◦ φ−1))‖s ≤ ‖u · (φs ◦ φ−1)‖m(s) ≤ ‖φs‖m(s)+n = ‖ψn,s+nh‖m(s)+n ≤ (1 + t)2s+nc′′ ( � · ‖φ‖m(s)+2n+n,2 + �−(r+s+n/2)/(n/2) · ‖φ‖0,2 ) ‖h‖s+2n . in the second inequality we use condition rt. in the third inequality here we use lemma 6.4. that completes the argument. proof of theorem 1.1: the last lemma tells us that u · (h◦ φ−1) ∈ h∞(ω2) if h ∈ h∞(ω1). taking h ≡ 1, we conclude immediately that ‖u‖s,2 ≤ c′′(1 + t)s ( 1 + �s · ‖φ‖m(s)+2n+n,2 + �−(m(s)+2n+n/2)/(n/2)s · ‖φ‖0,2 ) . we will later specify �s to mesh nicely with our other estimates. next take h = wj, where wj is the j th coordinate function on ω2. we conclude now that ‖u· ( φ−1 ) j ‖s,2 ≤ c′′(1+t)s ( 1+�s·‖φ‖m(s)+2n+n,2+�−(m(s)+2n+n/2)/(n/2)s ·‖φ‖0,2 ) . (6.7) fix a point z ∈ ω1. the fact that φ : ω1 → ω2 is lipschitz of order greater than (n− 1)/n tells us that |∇φ(z)| ≤ c · δ1(z)−1/n+� for some � > 0. hence the complex jacobian determinant u of φ at z is bounded by d−1+� ′ 1 (z) for some � ′ > 0. we know from results of range [16], proved with a direct application of hopf’s lemma, that d1(φ −1(w))m ≤ c·d2(w) for some positive integer m. but in fact the bi-lipschitz condition of order exceeding (n− 1)/n guarantees that m must be 1. it follows then that u must be of size at least d1−� ′ 2 if it vanishes at some point of ∂ω2 (it cannot vanish in the interior). that contradicts the smoothness of u to the boundary. we conclude then that u cannot vanish. hence it is bounded from 0 in modulus. so we may see from (6.7) that ‖ ( φ−1 ) j ‖s,2 ≤ c′′(1 + t)s ([ �s (1 + 2t)n+n/2 ]−(m(s)+2n+n/2)/(n/2) · ‖φ‖0,2 + �s (1 + 2t)n+n/2 · ‖φ‖2s ) .(6.8) smoothness to the boundary of biholomorphic mappings 13 of course a similar argument may be applied with φ−1 replace by φ and the roles of ω1 and ω2 reversed to see that ‖ ( φ ) j ‖s,2 ≤ c′′(1 + t)s ([ �s (1 + 2t)n+n/2 ]−(m(s)+2n+n/2)/(n/2) · ‖φ−1‖0,2 + �s (1 + 2t)n+n/2 · ‖φ−1‖2s ) .(6.9) now let λ` = 10 −`. in inequality (6.8), replace s by ` and multiply through by λ`. likewise, in inequality (6.9), replace s by ` and multiply through by λ`. now sum over `. the result is ∑ ` [ λ`‖ ( φ−1 ) j ‖`,2 + λ`‖ ( φ ) j ‖`,2 ] ≤ ∑ ` [ c′′ ( 1 + �−0` (·‖φ‖0,2 + � −(m(s)+2n+n/2)/(n/2) ` · ‖φ −1‖0,2 +�`‖φ−1‖2`,2 + �`‖φ‖2`,2 ) λ` ] . what is nice about this inequality is that we can now absorb the two �` terms on the righthand side into the lefthand side. in order to do this, we must note that the term ‖ ‖2`,2 on the righthand side has coefficient �`λ` while the same term on the lefthand side has coefficient λ2`. so we must choose �` so that �`λ` ≤ (1/2)λ2`. clearly �` = (1/2)10 −` will do the job. the result is that∑ ` [ λ`‖ ( φ−1 ) j ‖`,2 + λ`‖ ( φ ) j ‖`,2 ] ≤ ∑ ` c′′′λ` · ( � −(m(s)+2n+n/2)/(n/2) ` · ‖φ‖0,2 + � −(m(s)+2n+n/2)/(n/2) ` · ‖φ −1‖0,2 ) . we may conclude from this last inequality that ‖ ( φ−1 ) )j‖j,2 and ‖ ( φ ) )j‖j,2 are bounded. thus the bihlomorphic mapping extends to a diffeomorphism of the closures. that is our theorem. we remark that, if we strengthen the hypotheses of our theorem to φ and φ−1 both being lipschitz 1, then it is immediate that u and u are bounded and the proof simplifies notably. having a lipschitz condition of order less than 1 makes things a bit trickier. 7. concluding remarks it would be natural to suppose that a theorem like the one that we prove here is actually valid with only the assumption that φ and φ−1 are lipschitz of order � for some � > 0. the methods that we have do not suffice to establish such a result. 14 krantz we repeat that, of course, the hope is that no lipschitz hypothesis should be needed. the conclusion should be true all the time. that question will be a topic for future research. references [1] r. a. adams, sobolev spaces, academic press, new york, 1975. [2] s. r. bell, biholomorphic mappings and the ∂ problem, ann. math., 114(1981), 103-113. [3] s. r. bell, local boundary behavior of proper holomorphic mappings, complex analysis of several variables (madison, wis., 1982), 1–7, proc. sympos. pure math., 41, amer. math. soc., providence, ri, 1984. [4] s. r. bell and e. ligocka, a simplification and extension of fefferman’s theorem on biholomorphic mappings, invent. math. 57(1980), 283–289. [5] r. b. burckel, an introduction to classical complex analysis, academic press, new york, 1979. [6] k. diederich and j. e. fornæss, pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, invent. math. 39(1977), 129–141. [7] c. fefferman, the bergman kernel and biholomorphic mappings of pseudoconvex domains, invent. math. 26(1974), 1–65. [8] g. b. folland and j. j. kohn, the neumann problem for the cauchy-riemann complex, princeton university press, princeton, 1972. [9] b. fridman, biholomorphic transformations that do not extend continuously to the boundary, michigan math. j. 38(1991), 67–73. [10] g. m. goluzin, geometric theory of functions of a complex variable, american mathematical society, providence, 1969. [11] r. e. greene and s. g. krantz, function theory of one complex variable, 3rd ed., american mathematical society, providence, ri, 2006. [12] g. m. henkin, an analytic polyhedron is not holomorphically equivalent to a strictly pseudoconvex domain, (russian) dokl. akad. nauk sssr 210(1973), 1026–1029. [13] l. hörmander, l2 estimates and existence theorems for the ∂ operator, acta math. 113(1965), 89–152. w [14] j. j. kohn, global regularity for ∂ on weakly pseudo-convex manifolds, trans. ams 181(1973), 273–292. [15] s. g. krantz, function theory of several complex variables, 2nd ed., american mathematical society, providenc, ri, 2001. [16] r. m. range, the carathéodory metric and holomorphic maps on a class of weakly pseudoconvex domains, pacific j. math. 78(1978), 173–189. [17] r. m. range, a remark on bounded strictly plurisubharmonic exhaustion functions, proc. ams 81(1981), 220–222. [18] s. roman, the formula of faà di bruno, am. math. monthly 87(1980), 805-809. [19] e. m. stein, harmonic analysis: real variable methods, orthogonality, and oscillatory integrals, princeton university press, princeton, nj, 1993. department of mathematics, washington university in st. louis, st. louis, missouri 63130, united states international journal of analysis and applications volume 19, number 4 (2021), 604-618 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-604 a possible generalized model of the chlorine concentration decay in pipes: exact solution yussri m. mahrous∗ department of studies and basic sciences, faculty of community, university of tabuk, saudi arabia ∗corresponding author: y.mahrous@ut.edu.sa; mahrous1972@yahoo.com abstract. this paper discusses a possible generalization of the transport model describing the chlorine concentration decay in pipes. the proposed generalized model is governed by a second-order fractional partial differential equation. the exact solution of the generalized model is obtained via the laplace transform method and the method of residues. the exact solution reduces to the corresponding published one as the fractional order α tends to one. analytical expression for the dimensionless cup-mixing average concentration is deduced. influences of various parameters on the behavior of the dimensionless cup-mixing average concentration are discussed. it is shown that the physical interpretation of the dimensionless cup-mixing average concentration in view of the fractional calculus is completely different than its interpretation in the classical calculus. 1. introduction chlorine is the most commonly employed disinfectant in most countries and minimum levels of chlorine must be maintained to ensure the disinfection capacity of distributed water [1,2]. studying the chlorine decay reflects its importance in engineering and industrial sciences [3]. in this paper, we propose a generalized received april 18th, 2021; accepted may 25th, 2021; published june 24th, 2021. 2010 mathematics subject classification. 35r11. key words and phrases. fractional partial differential equation; bessel function; boundary value problem; exact solution; laplace transform. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 604 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-604 int. j. anal. appl. 19 (4) (2021) 605 model of the chlorine transport in pipes. the standard model was formulated by biswas et. al [4]. the proposed model, in dimensionless form, is governed by the fractional partial differential equation (fpde): (1) ∂αu ∂xα = a0 r ∂ ∂r ( 1 r ∂u ∂r ) −a1u, α ∈ (0, 1], under the boundary conditions (bcs): u(0,r) = 1, 0 ≤ r ≤ 1,(2) ∂ ∂r u(x, 0) = 0, 0 ≤ x ≤ 1,(3) ∂ ∂r u(x, 1) + a2u(x, 1) = 0, 0 ≤ x ≤ 1,(4) where u(x,r) is the chlorine concentration, α is the order of the fractional derivative in caputo sense. the dimensionless parameters a0, a1 and a2 are related to the chlorine decay. the parameter a0 stands for the radial diffusion. it depends on the pipe length, the effective diffusivity of chlorine, and the flow rate throughout the system. in addition, a1 depends on the reactivity of chlorine with species such as viable cells or chemical compounds in the bulk liquid phase and on the residence time in the system. the parameter a2 reflects the wall consumption and depends on the wall consumption rate v ∗ d , the pipe radius r ∗ 0 and the effective diffusivity of chlorine d, where a2 = v ∗ d r ∗ 0/d. the objective of this paper is to apply the laplace transform (lt) method to obtain the exact solution of the system (1)-(4). the lt is a well-known method for solving ordinary differential equations (odes) and fractional differential equations (pdes). such lt method has been successfully applied on several models such as diffusions [5], heat transfer of nanofluids suspended with carbon-nanotubes [6], singular boundary value problems (sbvps) related to fluid flow of carbon-nanotubes [7,8], and the mhd marangoni convection over a flat plate [9]. furthermore, the lt was successfully applied to solve the ambartsumian delay equation [10]. moreover, one can find in refs. [11-24] other interesting applications of the lt. int. j. anal. appl. 19 (4) (2021) 606 2. the lt method the application of lt, l(·), on both sides of eq. (1) gives (5) l ( ∂αu ∂xα ) = l ( a0 r ∂ ∂r ( 1 r ∂u(x,r) ∂r )) −l (a1u(x,r)) , or (6) sαu(s,r) −sα−1u(0,r) = a0 r d dr ( 1 r du(s,r) dr ) −a1u(s,r), from the bc (2) and eq. (6), we have (7) d2u(s,r) dr2 + 1 r du(s,r) dr − ( sα + a1 a0 ) u(s,r) = − sα−1 a0 . the solution of eq. (8) is (8) u(s,r) = ρ1j0 ( i √ sα + a1 a0 r ) + ρ2y0 ( i √ sα + a1 a0 r ) + sα−1 sα + a1 , where i = √ −1. besides, j0 (·) and y0(·) are bessel functions and ρ1 and ρ2 are unknown constants. physically, u(x,r) is bounded at r = 0, hence ρ2 must be vanishes and therefore, (9) u(s,r) = ρ1j0 ( i √ sα + a1 a0 r ) + sα−1 sα + a1 , and (10) du(s,r) dr = −ρ1i √ sα + a1 a0 j1 ( i √ sα + a1 a0 r ) , where j′0(λr) = −λj1(λr). the bc (4) gives (11) du(s, 1) dr + a2u(s, 1) = 0. from eqs. (9), (10) and (11), we obtain ρ1 as (12) ρ1 = − a2s α−1 (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )]. int. j. anal. appl. 19 (4) (2021) 607 substituting (12) into (9), yields (13) u(s,r) = − a2s α−1j0 ( i √ sα+a1 a0 r ) (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα−1+a1 a0 )] + sα−1 sα + a1 , which can be written as (14) u(s,r) = −a2h(s,r) + sα−1 sα + a1 , where (15) h(s,r) = sα−1j0 ( i √ sα+a1 a0 r ) (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )]. applying the inverse lt on eq. (14) leads to (16) u(x,r) = −a2h(x,r) + eα (−a1xα) , where h(x,r) is the inverse lt of h(s,r) and (17) h(x,r) = l−1   sα−1j0 ( i √ sα+a1 a0 r ) (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )]   . 3. the exact solution in order to find the exact solution, it should be first evaluate the inverse lt of (17). we observe from eq. (15) or eq. (17) that the denominator has simple poles at sα = −a1 and i √ sα+a1 a0 = λ1,λ2, . . .λn, . . . hence, we have simple poles at s = (−a1)1/α and s = (−a1 −a0λ2n)1/α, n = 1, 2, 3, . . . , where λn are the roots of (18) a2j0 (λn) −λnj1 (λn) = 0. int. j. anal. appl. 19 (4) (2021) 608 so, h(x,r) can be evaluated by applying theorem 1, in appendix a, by calculating the residues of esxh(s,r) at s = (−a1)1/α and s = (−a1 −a0λ2n)1/α, and then by taking their sum. at s = (−a1)1/α, we have (res esxh)s =(−a1)1/α = lim s→(−a1)1/α ( s− (−a1)1/α ) esxsα−1j0 ( i √ sα+a1 a0 r ) (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )], = e(−a1) 1/αx lim s→(−a1)1/α sα−1j0 ( i √ sα+a1 a0 r ) a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 ) × lim s→(−a1)1/α s− (−a1)1/α sα + a1 , = e(−a1) 1/αx ( (−a1)(α−1)/αj0 (0) a2j0 (0) − 0 ) . lim s→(−a1)1/α 1 αsα−1 , = e(−a1) 1/αx αa2 , where j0 (0) = 1.(19) at s = (−a1 −a0λ2n)1/α, we have (res esxh)s=(−a1−a0λ2n)1/α = lim s→(−a1−a0λ2n)1/α ( s− (−a1 −a0λ2n)1/α ) esxsα−1j0 ( i √ sα+a1 a0 r ) (sα + a1) [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )], or (res esxh)s=(−a1−a0λ2n)1/α = lim s→(−a1−a0λ2n)1/α s− (−a1 −a0λ2n)1/α a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 ) × lim s→(−a1−a0λ2n)1/α esxsα−1j0 ( i √ sα+a1 a0 r ) sα + a1 , which can be written as (20) (res esxh)s=(−a1−a0λ2n)1/α = e(−a1−a0λ 2 n) 1/αx ( −a1 −a0λ2n )(α−1)/α j0 (−λnr) −a0λ2n . lim s→(−a1−a0λ2n)1/α q(s,r). int. j. anal. appl. 19 (4) (2021) 609 using the l’hospital’s rule, we have lim s→(−a1−a0λ2n)1/α q(s,r) = lims→(−a1−a0λ2n)1/α ( s− (−a1 −a0λ2n)1/α ) lims→(−a1−a0λ2n)1/α [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )] = 0 0 , = lims→(−a1−a0λ2n)1/α d ds ( s + a1 + a0λ 2 n ) lims→(−a1−a0λ2n)1/α d ds [ a2j0 ( i √ sα+a1 a0 ) − i √ sα+a1 a0 j1 ( i √ sα+a1 a0 )], = 1 σ ,(21) where σ is defined by (22) σ = lim s→(−a1−a0λ2n)1/α d ds [ a2j0 ( i √ sα + a1 a0 ) − i √ sα + a1 a0 j1 ( i √ sα + a1 a0 )] . assume that y = i √ sα+a1 a0 , then σ = lim s→(−a1−a0λ2n)1/α d ds [a2j0(y) −yj1(y)] = lim s→(−a1−a0λ2n)1/α [ −a2j1(y) dy ds − y 2 (j0(y) −j2(y)) dy ds −j1(y) dy ds ] , = lim s→(−a1−a0λ2n)1/α dy ds [−a2j1(y) −yj0(y)] ,(23) where the properties of bessel functions are used, see appendix b. the magnitude dy ds is (24) dy ds = − αsα−1 2a0y . we also note at s = (−a1 −a0λ2n)1/α that (25) y = −λn, dy ds = α(−a1 −a0λ2n)(α−1)/α 2a0λn . inserting eqs. (25) into eq. (23), noting that the functions j0 and j2 are even and j1 is odd, we find (26) σ = α(−a1 −a0λ2n)(α−1)/α 2a0λn (a2j1 (λn) + λnj0 (λn)) . from (26) and (21), it then follows (27) lim s→(−a1−a0λ2n)1/α q(s,r) = 2a0λn α(−a1 −a0λ2n)(α−1)/α (a2j1 (λn) + λnj0 (λn)) . int. j. anal. appl. 19 (4) (2021) 610 substituting (27) into (20) and simplifying, gives (28) (res esxh)s=(−a1−a0λ2n)1/α = − 2 α ∞∑ n=1 e(−a1−a0λ 2 n) 1/αxj0 (λnr) λn [a2j1 (λn) + λnj0 (λn)] . hence, h(x,r) in eq. (17) is given by h(x,r) = (res esxh)s=(−a1)1/α + (res e sxh)s=(−a1−a0λ2n)1/α , = e(−a1) 1/αx αa2 − 2 α ∞∑ n=1 e(−a1−a0λ 2 n) 1/αxj0 (λnr) λn [a2j1 (λn) + λnj0 (λn)] .(29) inserting (29) into (14), and after simplifying, we obtain the solution u(x,r) as (30) u(x,r) = − 1 α e(−a1) 1/αx + 2 α ∞∑ n=1 a2e (−a1−a0λ2n) 1/αxj0 (λnr) λn [a2j1 (λn) + λnj0 (λn)] + eα (−a1xα) , eq. (18) implies (31) a2 = λnj1 (λn) j0 (λn) . substituting (31) into (30), yields (32) u(x,r) = − 1 α e(−a1) 1/αx + 2 α ∞∑ n=1 λnj1 (λn) j0 (λnr) e (−a1−a0λ2n) 1/αx (a22 + λ 2 n) j 2 0 (λn) + eα (−a1xα) , as α → 1, eq. (32) reduces to (33) u(x,r) = −e−a1x + 2 ∞∑ n=1 λnj1 (λn) j0 (λnr) e −(a1+a0λ2n)x (a22 + λ 2 n) j 2 0 (λn) + e1 (−a1x) , which can be simplified to (34) u(x,r) = 2 ∞∑ n=1 λnj1 (λn) j0 (λnr) e −(a1+a0λ2n)x (a22 + λ 2 n) j 2 0 (λn) , where e1 (−a1x) = e−a1x. the solution (34) is identical to the same result obtained by biswas et al. [4] for the chlorine decay model with classical partial derivative with respect to x. int. j. anal. appl. 19 (4) (2021) 611 4. results and discussion according to biswas et al. [4], the dimensionless cup-mixing average concentration is defined by (35) uav = 2 ∫ 1 0 u(x,r) rdr. substituting (32) into (35), yields (36) uav = ( eα (−a1xα) − 1 α e(−a1) 1/αx )∫ 1 0 2rdr + 4 ∞∑ n=1 λnj1 (λn) e (−a1−a0λ2n) 1/αx (a22 + λ 2 n) j 2 0 (λn) ∫ 1 0 rj0 (λnr) dr, or (37) uav = eα (−a1xα) − 1 α e(−a1) 1/αx + 4 ∞∑ n=1 j21 (λn) (a22 + λ 2 n) j 2 0 (λn) e(−a1−a0λ 2 n) 1/α x. implementing the relation (31) we have (38) uav = eα (−a1xα) − 1 α e(−a1) 1/αx + 4 ∞∑ n=1 a22 λ2n (a 2 2 + λ 2 n) e(−a1−a0λ 2 n) 1/α x. if the pipe walls act as a perfect sink, i.e., v ∗0 →∞ or a2 →∞, then the cup-mixing average concentration is obtained from eq. (38) by the limit: (39) uav = eα (−a1xα) − 1 α e(−a1) 1/αx + 4 lim a2→∞ ( ∞∑ n=1 a22 λ2n (a 2 2 + λ 2 n) e(−a1−a0λ 2 n) 1/α x ) , which gives (40) uav = eα (−a1xα) − 1 α e(−a1) 1/αx + ∞∑ n=1 4 λ2n e(−a1−a0λ 2 n) 1/α x, where λn’s are the roots of j0 (λn) = 0. moreover, if v ∗ 0 → 0 or a2 → 0 (i.e., the pipe walls are inert and no chlorine consumption takes place at the walls), then u(x,r) in eq. (19) reduces to (41) u(x,r) = eα (−a1xα) , and accordingly, (42) uav = 2 ∫ 1 0 eα (−a1xα) rdr = eα (−a1xα) . int. j. anal. appl. 19 (4) (2021) 612 following biswas et al. [4], we consider the first three terms of the series (38), hence, three roots λ1, λ2, and λ3 of eq. (18) are to be used. table 1 presents the three roots λ1, λ2 and λ3 of eq. (18) at different values of a2 in the range 0.01 ≤ a2 < 1. the roots are calculated using the command “findroot” in mathematica. in tables 2 and 3, the values of λ1, λ2 and λ3 are listed for selected values of a2 in the range 1 ≤ a2 < 10 and in the range 10 ≤ a2 < 1000, respectively. table 1. the first three roots 1 , 2 , and 3 of eq. (27) at different values of a2 in the range .101.0 2  a a2 1 2 3 0.01 0.141245 3.83431 7.01701 0.1 0.441682 3.85771 7.02983 0.2 0.616975 3.88351 7.04403 0.5 0.940771 3.95937 7.08638 table 2. the first three roots 1 , 2 , and 3 of eq. (27) at different values of a2 in the range .101 2  a a2 1 2 3 1 1.25578 4.07948 7.1558 2 1.59945 4.29096 7.28839 5 1.98981 4.71314 7.61771 table 3. the first three roots 1 , 2 , and 3 of eq. (27) at different values of a2 in the range .100010 2  a a2 1 2 3 10 2.1795 5.03321 7.95688 50 2.35724 5.4112 8.48399 100 2.3809 5.46521 8.56783 the curves of the cup-mixing average concentration uav are depicted in figs. 1-4 versus a1, at the outlet x = 1 of a pipe, for several values of a0 and a2 when α = 1/3. figure 1 indicates that the uav is a decreasing int. j. anal. appl. 19 (4) (2021) 613 function in the parameter a1 in the absence of a2 (i.e., a2 = 0). however, the behavior of uav is different in the case a2 6= 0 where uav decreases in two subdomians of a1 and increases in a certain domain. this last conclusion can be also confirmed and seen in figs. 2-4 for the curves of uav when a2 has a specified nonezero value, i.e., a2 doesn’t vanish. the influence of the fractional order α on the cup-mixing average concentration uav is displayed in fig. 5. it can be seen from this figure that uav is a decreasing function in the full domain of the parameter a1 when α = 1 (classical derivative) of while uav is of different behavior when α = {1/3, 1/5, 1/7} (fractional derivative). the discussion above may give some lights about the modeling of chlorine decay in view of the fractional calculus. a2=0.0 a2=0.1 a2=0.3 a2=0.5 2 4 6 8 10 a1 -2.0 -1.5 -1.0 -0.5 0.5 1.0 uav figure 1. the cup-mixing average concentration uav versus a1 at different values of a2 when α = 1/3 and a0= 1.4. int. j. anal. appl. 19 (4) (2021) 614 a2=1 a2=3 a2=5 a2=7 2 4 6 8 10 a1 -1.0 -0.5 uav figure 2. the cup-mixing average concentration uav versus a1 at different values of a2 when α = 1/3 and a0 = 1.4 × 10−3. a2=10 a2=20 a2=30 a2=40 2 4 6 8 10 a1 -1.5 -1.0 -0.5 uav figure 3. the cup-mixing average concentration uav versus a1 at different values of a2 when α = 1/3 and a0 = 1.4 × 10−2. int. j. anal. appl. 19 (4) (2021) 615 a2=100 a2=200 a2=300 a2=400 2 4 6 8 10 a1 -2.0 -1.5 -1.0 -0.5 uav figure 4. the cup-mixing average concentration uav versus a1 at different values of a2 (higher values) when α = 1/3 and a0 = 1.4 × 10−2. α=1 α=1/3 α=1/5 α=1/7 2 4 6 8 10 a1 -6 -5 -4 -3 -2 -1 uav figure 5. influence of the fractional order α on the cup-mixing average concentration uav when a0 = 1.4 and a2 = 0.5. int. j. anal. appl. 19 (4) (2021) 616 5. conclusion a possible generalization of the transport model describing the chlorine concentration decay in pipes was analyzed. the exact solution of the generalized model was obtained using the lt and the method of residues. the obtained exact solutions reduced to the corresponding published solutions as the fractional order α tends to one. analytical expression for the dimensionless cup-mixing average concentration was deduced. the effects of the impeded parameters on the dimensionless cup-mixing average concentration were discussed and analyzed. the results showed that the behavior of the dimensionless cup-mixing average concentration in view of the fractional calculus is completely different than its behavior using the classical calculus. appendices: a. residues method a basic theorem for obtaining the inverse lt using the method of residues is given below. theorem 1: the inverse lt of a function h(s,r) using the method of residues is given by h(x,r) = sum of residues of esxh(s,r) at all poles of h(s,r), see ref. [25] for details. b. properties of bessel functions the bessel functions j0(y), j1(y), and j2(y) are defined by j0(y) = ∞∑ k=0 (−1)k (k!) 2 (y 2 )2k ,(b.1) j1(y) = ∞∑ k=0 (−1)k k!(k + 1)! (y 2 )2k+1 ,(b.2) j2(y) = ∞∑ k=0 (−1)k k!(k + 2)! (y 2 )2k+2 ,(b.3) int. j. anal. appl. 19 (4) (2021) 617 and satisfy the properties: d dy (j0(λy)) = −λj1 (λy)(b.4) d dy (j1(λy)) = λ 2 (j0(λy) −j2(λy)) ,(b.5) yj2(y) + yj0(y) = 2j1(y).(b.6) availability of data and materials: not applicable. funding: na. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.m. clark, e.j. read, j.c. hoff, analysis of inactivation of giardia lamblia by chlorine, j. environ. eng. 115 (1989), 80–90. [2] m.w. lechevallier, c.d. cawthon, r.g. lee, inactivation of biofilm bacteria, appl. environ. microbiol. 54 (1988), 2492–2499. [3] b.f. arnold, j.m. colford, treating water with chlorine at point-of-use to improve water quality and reduce child diarrhea in developing countries: a systematic review and meta-analysis, amer. j. trop. med. hyg. 76 (2007), 354–364. [4] p. biswas, c. lu, r.m. clark, a model for chlorine concentration decay in pipes, water res. 27 (1993), 1715–1724. [5] j. jakubowski, m. wisniewolski, on matching diffusions, laplace transforms and partial differential equations, stoch. proc. appl. 125 (2015), 3663-3690. [6] a. ebaid, m. al sharif, application of laplace transform for the exact effect of a magnetic field on heat transfer of carbon-nanotubes suspended nanofluids, z. naturforsch., a, 70 (2015), 471-475. [7] a. ebaid, a.m. wazwaz, e. alali, b. masaedeh, hypergeometric series solution to a class of second-order boundary value problems via laplace transform with applications to nanouids, commun. theor. phys. 67 (2017), 231. [8] a. ebaid, e. alali, h. saleh, the exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, j. assoc. arab univ. basi appl. sci. 24 (2017), 156-159. [9] s.m. khaled, the exact effects of radiation and joule heating on magnetohydrodynamic marangoni convection over a flat surface, therm. sci. 22 (2018), 63-72. [10] h.o. bakodah, a. ebaid, exact solution of ambartsumian delay differential equation and comparison with daftardar-gejji and jafari approximate method, mathematics, 6 (2018), 331. [11] s. handibag, b.d. karande, laplace substitution method for solving partial differential equations involving mixed partial derivatives, int. j. comput. eng. res. 2 (2012), 1049-1052. int. j. anal. appl. 19 (4) (2021) 618 [12] n. dogan, solution of the system of ordinary differential equations by combined laplace transform-adomian decomposition method, math. comput. appl. 17 (2012), 203-211. [13] p. rai, application of laplace transforms to solve ode using matlab, j. inform. math. sci. 7 (2015), 93-97. [14] s.s. handibag, b.d. karande, laplace substitution method for nth order linear and non-linear pde’s involving mixed partial derivatives, int. res. j. eng. technol. 2 (2015), 378-388. [15] a.a. alshikh, m.m.a. mahgob, a comparative study between laplace transform and two new integrals “elzaki” transform and “aboodh” transform, pure appl. math. j. 5 (2016), 145-150. [16] a. atangana, b.s.t. alkaltani, a novel double integral transform and its applications, j. nonlinear sci. appl. 9 (2016), 424-434. [17] x. lianga, f. gao, y.-n. gao, x.-j. yang, applications of a novel integral transform to partial differential equations, j. nonlinear sci. appl. 10 (2017), 528-534. [18] p.v. pavani, u.l. priya, b.a. reddy, solving differential equations by using laplace transforms, int. j. res. anal. rev. 5 (2018), 1796-1799. [19] b.m. faraj, f.w. ahmed, on the matlab technique by using laplace transform for solving second order ode with initial conditions exactly, matrix sci. math. 3 (2019), 8-10. [20] a. mousa, t.m. elzaki, solution of volterra integro-differential equations by triple laplace transform, irish interdiscip. j. sci. res. 3 (2019), 67-72. [21] r.r. dhunde, g.l. waghmare, double laplace iterative method for solving nonlinear partial differential equations, new trends math. sci. 7 (2019), 138-149. [22] d. ziane, m.h. cherif, c. cattani, k. belghaba, yang-laplace decomposition method for nonlinear system of local fractional partial differential equations, appl. math. nonlinear sci. 4 (2019), 489-502. [23] s. mastoi, w.a.m. othman, n. kumaresan, randomly generated grids and laplace transform for partial differential equations, int. j. disaster recovery bus. contin. 11 (2020), 1694-1702. [24] h. zhang, m. nadeem, a. rauf, z. guo hui, a novel approach for the analytical solution of nonlinear time-fractional differential equations, int. j. numer. meth. heat fluid flow, 31 (2021) 1069–1084. [25] m.r. spiegel, laplace transforms, mcgraw-hill. inc., new york, 1965. 1. introduction 2. the lt method 3. the exact solution 4. results and discussion 5. conclusion appendices a. residues method b. properties of bessel functions references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 162-170 http://www.etamaths.com on the iterated exponent of convergence of solutions of linear differential equations abdallah el farissi abstract. in this paper, we investigate the relationship between solutions and their derivatives of the differential equation f(k)+ak−1f (k−1)+...+a0f = 0 for k ≥ 2 and small functions, where aj (j = 0, 1, ..., k − 1) are meromorphic functions of finite iterated p-order. 1. introduction and main results in this paper, a meromorphic function will mean meromorphic in the whole complexe plan. we shall use the standard notations in nevanlinna value distribution of meromorphic functions [9, 11] such as t (r,f) ,n (r,f) ,m (r,f). for the definition of the iterated order of a meromorphic function, we use the same definition as in [10] , [2, p. 317] , [9, p. 129] . for all r ∈ r, we define exp1 r := er and expp+1 r := exp ( expp r ) , p ∈ n. we also define for all r sufficiently large log1 r := log r and logp+1 r := log ( logp r ) , p ∈ n. moreover, we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r. definition 1.1: ([10] , [11]) let f be a meromorphic function. the iterated p−order ρp (f) of f is defined by (1.1) ρp (f) = lim r→+∞ logp t (r,f) log r (p > 1 is an integer) , for p = 1, this notation is called order, and for p = 2 hyper-order. definition 1.2: (see [10]) the finiteness degree of the order of a meromorphic function f is defined by (1.2) i (f) =   0 for f rational, min{n ∈ n : ρj (f) < +∞} for f transcendental for wich same n ∈ n with ρn (f) < +∞ exists ∞ for f with ρn (f) = +∞ for all n ∈ n. 2010 mathematics subject classification. 34m10, 30d35. key words and phrases. linear differential equations; meromorphic functions; iterated p−order; iterated exponent of convergence of the sequence of distinct zeros. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 162 linear differential equations 163 definition 1.3: (see [10]) let f be a meromorphic function. the iterated convergence exponent of the sequence of zeros of f (z) is defined by (1.3) λp (f) = lim r→+∞ logp n ( r, 1 f ) log r ; p > 1 is an integer, where n ( r, 1 f ) is the counting function of zeros of f (z) in {|z| < r}. similary the iterated convergence exponent of the sequence of distinct zeros of f (z) is defined by (1.3) λp (f) = lim r→+∞ logp n ( r, 1 f ) log r ; p > 1 is an integer, where n ( r, 1 f ) is the counting function of distinct zeros of f (z) in {|z| < r}. definition 1.5 [10] the growth index of the iterated convergence exponent of the sequence of zero points of a meromorphic function f with iterated order is defined by iλ (f) =   0 if n ( r, 1 f ) = o (log r) , min{n ∈ n : λn (f) < ∞} if λn (f) < ∞ for some n ∈ n, ∞ if λn (f) = ∞ for all n ∈ n. similarly, we can define the growth index iλ (f) of λp (f) . for k > 2, we consider the linear differential equation (1.5) f(k) + a (z) f = 0, where a (z) is a transcendental meromorphic function of finite iterated order ρp (a) = ρ > 0. in [14] , wang and lü have investigated the fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients and their derivatives. they have obtained the following result: theorem a [14] suppose that a (z) is a transcendental meromorphic function satisfying δ (∞,a) = lim r→+∞ m(r,a) t(r,a) = δ > 0, ρ (a) = ρ < +∞. then every meromorphic solution f (z) 6≡ 0 of the equation (1.6) f ′′ + a (z) f = 0 satisfies that f,f ′ and f ′′ have infinitely many fixed points and (1.7) τ (f) = τ ( f ′ ) = τ ( f ′′ ) = ρ (f) = +∞, (1.8) τ2 (f) = τ2 ( f ′ ) = τ2 ( f ′′ ) = ρ2 (f) = ρ. theorem a has been generalized to higher order differential equations by liu and zhang as follows (see [12]): 164 abdallah el farissi theorem b [12] suppose that k > 2 and a (z) is a transcendental meromorphic function satisfying δ (∞,a) = δ > 0, ρ (a) = ρ < +∞. then every meromorphic solution f (z) 6≡ 0 of (1.4) , satisfies that f and f ′ ,f ′′ , ...,f(k) all have infinitely many fixed points and (1.9) τ (f) = τ ( f ′ ) = τ ( f ′′ ) = ... = τ ( f(k) ) = ρ (f) = +∞, (1.10) τ2 (f) = τ2 ( f ′ ) = τ2 ( f ′′ ) = ... = τ2 ( f(k) ) = ρ2 (f) = ρ. theorem a and b have been generalized by b. beläıdi for iterated p-order (see [1]): theorem c [1] let k > 2 and a (z) be transcendental meromorphic function of finite iterated order ρp (a) = ρ > 0 such that δ (∞,a) = δ > 0. suppose, moreover, that either: (i) all poles of f are of uniformly multiplicity or that (ii) δ (∞,f) > 0. if ϕ 6≡ 0 is a meromorphic function with finite iterated p−order ρp (ϕ) < +∞, then every meromorphic solution f (z) 6≡ 0 of (1.5), satisfies (1.11) λp (f −ϕ) = λp ( f ′ −ϕ ) = ... = λp ( f(k) −ϕ ) = ρp (f) = +∞, and (1.12) λp+1 (f −ϕ) = λp+1 ( f ′ −ϕ ) = ... = λp+1 ( f(k) −ϕ ) = ρp+1 (f) = ρ. for k > 2, we consider the linear differential equation (1.13) f(k) + ak−1f (k−1) + ... + a0f = 0,k > 2, recently bouabdelli and beläıdi [3] investigate the relationship between small functions and derivative of solutions of equation (1.13) and obtain some theorems which extended the previous results given by xu, tu and zheng (see [15]). theorem d [3] let k > 2 and (aj)j=0,1,2,...k−1 be entire functions of finite iterated order with i (a0) = p (0 < p < ∞) and satisfy one of the following conditions: (i) max{ρp (aj) , (j = 1, ...,k − 1)} < ρp (a0) = ρ < +∞, (ii) max{ρp (aj) , (j = 1, ...,k − 1)} 6 ρp (a0) = ρ (0 < ρ < ∞) and max{σp (aj) ,ρp (aj) = ρp (a0)} < σp (a0) = σ, (0 < σ < ∞) , then for every solution f 6≡ 0 of (1.13) and for any entire function ϕ 6≡ 0 satisfying ρp+1 (ϕ) < ρ, we have λp+1 ( f(i) −ϕ ) = λp+1 ( f(i) −ϕ ) = ρp+1 (f) = ρ,i ∈ n. theorem e [3] let k > 2 and (aj)j=0,1,2,...k−1 be meromorphic functions of finite iterated order with i (a0) = p (0 < p < ∞) satisfying max{ρp (aj) , (j = 1, ...,k − 1)} < ρp (a0) = ρ < +∞ and δ (∞,a0) > 0. then for every meromorphic solution f 6≡ 0 of (1.13) whose poles are of uniformly bounded multiplicity and for any meromorphic function ϕ 6≡ 0 satisfying ρp+1 (ϕ) < ρ, we have λp+1 ( f(i) −ϕ ) = λp+1 ( f(i) −ϕ ) = ρp+1 (f) = ρ,i ∈ n. linear differential equations 165 in all previous theorems we note that, the conditions on the coefficients gives us that any solution of the equation (1.13) is of infinite p−order and the same (p + 1)−order. and there are several papers where the authors show that on certain conditions all solutions of the equation of infinite p−order and the same (p + 1)−order (see [2] , [6] , [7] , [12] ...). the question that arises is: if any solution of the equation is of infinite p−order and the same (p + 1)−order, is that we have the same results?. in this paper we give an answer of above question and we prove the following theorems: theorem 1.1 let k > 2 and (aj)j=0,1,2,...k−1 be meromorphic functions of finite p−order. suppose that all solution of the equation (1.13) of infinite p−order and ρp+1 (f) = ρ. if ϕ 6≡ 0 is a meromorphic function with i (ϕ) < p+1 or ρp+1 (ϕ) < ρ, then every meromorphic solution f 6≡ 0 of (1.13) satisfies (1.14) iλ ( f(i) −ϕ ) = iλ ( f(i) −ϕ ) = i (f) = p + 1, i ∈ n and (1.15) λp+1 ( f(i) −ϕ ) = λp+1 ( f(i) −ϕ ) = ρp+1 (f) = ρ, i ∈ n. theorem 1.2 let k > 2 and (aj)j=0,1,2,...k−1 be meromorphic functions of finite p−order. suppose that all solution of the equation (1.13) of infinite p−order. if ϕ 6≡ 0 is an meromorphic function with ρp (ϕ) < +∞, then every meromorphic solution f 6≡ 0 of (1.13) satisfies (1.16) λp ( f(i) −ϕ ) = λp ( f(i) −ϕ ) = ρp (f) = ∞, i ∈ n. remark 1.2 the proof of theorems 1.1, 1.2 are quite different from that in the proof of theorem d and e (see [3]) we give a simple proof of theorems in the paper. the main ingredient in the proof is lemma 2.5. corollary 1.1 under the assumptions of theorem 1.1, if ϕ (z) = z, then for every meromorphic solution f of (1.13), we have (1.17) iτ ( f(i) ) = iτ ( f(i) ) = i (f) = p + 1, i ∈ n and (1.18) τp+1 ( f(i) ) = τp+1 ( f(i) ) = ρp+1 (f) = ρp (a0) = ρ, i ∈ n. corollary 1.2 suppose that k > 2 and a (z) is a transcendental meromorphic function such that 0 < ρp (a) = ρ < +∞. if ϕ 6≡ 0 is meromorphic function with i (ϕ) < p + 1 or ρp+1 (ϕ) < ρ, then every solution f 6≡ 0 of (1.5) satisfies (1.14) and (1.15). 166 abdallah el farissi corollary 1.3 let k > 2 and (aj)j=0,1,2,...k−1 be entire functions of finite iterated p-order such that i (a0) = p; 0 < p < ∞. suppose that max{i (aj) , (j = 1, ...,k − 1)} < i (a0) or max{ρp (aj) , (j = 1, ...,k − 1)} < ρp (a0) < +∞. if ϕ 6≡ 0 is an entire function with i (ϕ) < p + 1 or ρp+1 (ϕ) < ρp (a0), then every solution f 6≡ 0 of (1.13) satisfies (1.14) and (1.15) . for p = 1 in theorem 1.1 we gat the following corollary (see [7]) corollary 1.4 [7] let k > 2 and aj (j = 0, 1, ...,k − 1) be meromorphic functions of finite order such that all solution of equation (1.13) satisfy ρ (f) = +∞ and ρ2 (f) = ρ. then if ϕ 6≡ 0 is an meromorphic function with ρ2 (ϕ) < ρ, then every solution f 6≡ 0 of (1.13) satisfies (1.19) λ ( f(i) −ϕ ) = λ ( f(i) −ϕ ) = ρ (f) = +∞, i ∈ n and (1.20) λ2 ( f(i) −ϕ ) = λ2 ( f(i) −ϕ ) = ρ2 (f) = ρ, i ∈ n. remark 1.3: theorem 1.1 is the improvement of theorems a, b, c and d and theorem 1.2 is the improvement of theorem e. 2. auxiliary lemmas to prove our main results, we need the following lemmas. lemma 2.1 [5] suppose that a0, a1, ..., ak−1, f ( 6≡ 0) are meromorphic functions and let f be a meromorphic solution of the equation (2.1) f(k) + ak−1f (k−1) + ... + a1f ′ + a0f = f, such that i (f) = ρ + 1 (0 < p < ∞) . if either max{i (f) , i (aj) j = 0, 1, ...,k − 1} < p + 1 or max{ρp+1 (f) ,ρp+1 (aj) j = 0, 1, ...,k − 1} < ρp+1 (f) , then we have iλ (f) = iλ (f) = i (f) = p + 1 and λp+1 (f) = λp+1 (f) = ρp+1 (f) . lemma 2.2 (see remark 1.3 of [10]). if f is a meromorphic function with i (f) = p, then ρp ( f ′ ) = ρp (f). lemma 2.3 [10] let k > 2 and aj (j = 0, 1, ...,k − 1) be entire functions of finite iterated p-order such that i (a0) = p, (0 < p < ∞). suppose that max{i (aj) , (j = 1, ...,k − 1)} < i (a0) or max{ρp (aj) , (j = 1, ...,k − 1)} < ρp (a0) < +∞, then every solution f 6≡ 0 of (1.13) satisfies i (f) = p + 1 and ρp+1 (f) = ρp (a0) . let aj (j = 0, 1, ...,k − 1) be a functions. we define the following sequence of functions: linear differential equations 167 (2.2)   a0j = aj, j = 0, 1, ...,k − 1 aik−1 = a i−1 k−1 − ( ai−10 )′ ai−10 , i ∈ n aij = a i−1 j + a i−1 j+1 ( ψi−1j+1 )′ ψi−1j+1 , j = 0, 1, ...,k − 2, i ∈ n, where ψi−1j+1 = ai−1j+1 ai−10 . remark 2.1: in the case where one of functions aij (j = 0, 1, ...,k − 1) is equal to zero then ai+1j = a i j−1 (j = 0, 1, ...,k − 1) . lemma 2.4 suppose that f is a solution of (1.13) . then gi = f (i) is a solution of the equation (2.3) g (k) i + a i k−1g (k−1) i + ... + a i 0gi = 0, where aij (j = 0, 1, ...,k − 1) are given by (2.2). proof: assume that f is a solution of equation (1.13) and let gi = f (i). we prove that gi is an entire solution of the equation (2.3) . our proof is by induction: for i = 1, differentiating both sides of (1.13) , we obtain (2.4) f(k+1) + ak−1f (k) + ( a ′ k−1 + ak−2 ) f(k−1) + ... + ( a ′ 1 + a0 ) f ′ + a ′ 0f = 0, and replacing f by f = − (f(k) + ak−1f (k−1) + ... + a1f ′ ) a0 , we get f(k+1) + ( ak−1 − a ′ 0 a0 ) f(k) + ( a ′ k−1 + ak−2 −ak−1 a ′ 0 a0 ) f(k−1)... + ( a ′ 1 + a0 −a1 a ′ 0 a0 ) f ′ = 0. that is g (k) 1 + a 1 k−1g (k−1) 1 + a 1 k−2g (k−2) 1 ... + a 1 0g1 = 0. suppose that the assertion is true for the values which are strictly smaller than a certain i. we suppose gi−1 is a solution of the equation (2.5) g (k) i−1 + a i−1 k−1g (k−1) i−1 + a i−1 k−2g (k−2) i−1 ... + a i−1 0 gi−1 = 0. differentiating both sides of (2.5) , we can write g (k+1) i−1 + a i−1 k−1g (k) i−1 + (( ai−1k−1 )′ + ak−2 ) g (k−1) i−1 + ... (2.6) + (( ai−11 )′ + ai−10 ) g ′ i−1 + a ′ 0gi−1 = 0. 168 abdallah el farissi in (2.6) , replacing gi−1 by gi−1 = − (g (k) i−1 + a i−1 k−1g (k−1) i−1 + a i−1 k−2g (k−2) i−1 ... + a (gi−1) ′ ) ai−10 , yields g (k+1) i−1 +  ai−1k−1 − ( ai−10 )′ ai−10  g(k)i−1 +  (ai−1k−1)′ + ak−2 −ai−1k−1 ( ai−10 )′ ai−10  g(k−1)i−1 ...+ (2.7) +  (ai−11 )′ + ai−10 −ai−11 ( ai−10 )′ ai−10  g′i−1 = 0. that is g (k) i + a i k−1g (k−1) i + a i k−2g (k−2) i ... + a i 0gi = 0. lemma 2.4 is thus proved. lemma 2.5 let aj (j = 0, 1, ...,k − 1) be meromorphic functions of finite order such that all solution of equation (1.13) has infinit p−order and ρp+1 (f) = ρ. and let aij, (j = 0, 1, ...,k − 1) be defined as in (2.2). then all nontrivial meromorphic solution g of the equation (2.8) g(k) + aik−1g (k−1) + ... + ai0g = 0, k > 2 satisfies: ρp (g) = +∞ and ρp+1 (g) = ρ. proof: let {f1,f2, ...,fk} be a fundamental system of solutions of (1.13). we show that { f (i) 1 ,f (i) 2 , ...,f (i) k } is a fundamental system of solutions of (2.8). by lemma 2.4, it follows that f (i) 1 ,f (i) 2 , ...,f (i) k is solutions of (2.8) . let α1,α2, ...,αk be constants such that α1f (i) 1 + α2f (i) 2 + ... + αkf (i) k = 0. then, we have α1f1 + α2f2 + ... + αkfk = p (z) , where p (z) is a polynomial of degree less than i. since α1f1 + α2f2 + ... + αkfk is a solution of (1.13), then p is a solution of (1.13), and by the conditions of the lemma 2.5, we conclude that p is an infinite solution of (1.13); this leads to a contradiction. therefore, p is a trivial solution. we deduce that α1f1 + α2f2 + ... + αkfk = 0. using the fact that {f1,f2, ...,fk} is a fundamental solution of (1.13), we get α1 = α2 = ... = αk = 0. now, let g be a non trivial solution of (2.8). then, using the fact that { f (i) 1 ,f (i) 2 , ...,f (i) k } is a fundamental solution of (2.8) , we claim that there exist constants α1,α2, ...,αk not all equal to zero, such that g = α1f (i) 1 + α2f (i) 2 + ... + αkf (i) k . let h = α1f1 + α2f2 + ... + αkfk, h be a solution of (1.13) and h(i) = g. hence, by lemma 2.2, we have ρp+1 (h) = ρp+1 (g) , and by the conditions of the lemma 2.5, we have ρp (h) = ρp (g) = +∞ and ρp+1 (h) = ρp+1 (g) = ρ. linear differential equations 169 3. proof of theorems and corollary 1.3 firstily we proof the theorem 1.1 proof of theorem 1.1 assume that f is a solution of equation (1.13) . by the conditions of theorem 1.1, we can write i (f) = p + 1, ρp+1 (f) = ρ. taking gi = f (i), then, using lemma 2.2, we have i (gi) = p + 1, ρp+1 (gi) = ρ. now, let w (z) = gi (z) −ϕ (z) , where ϕ is a meromorphic function with ρp+1 (ϕ) < ρp (a0) . then i (w) = i (gi) = p + 1, and ρp+1 (w) = ρp+1 (gi) = ρp+1 (f) = ρ (a0) . in order to prove iλ (gi −ϕ) = iλ (gi −ϕ) = p+1 and λp+1 (gi −ϕ) = λp+1 (gi −ϕ) = ρ (a0), we need to prove only iλ (w) = iλ (w) = p + 1 and λp+1 (w) = ρ (a0) . using the fact that gi = w + ϕ, and by lemma 2.4 we get (3.1) w(k) + aik−1w (k−1) + ... + ai0w = − ( ϕ(k) + aik−1ϕ (k−1) + ... + ai0ϕ ) = f. by ρp ( aij ) < ∞, ρp+1 (ϕ) < ρ and lemma 2.5, we get f 6≡ 0 and ρp+1 (f) < ∞. by lemma 2.1 iλ (w) = iλ (w) = p + 1 and λp+1 (w) = λp+1 (w) = ρp+1 (w) = ρ (a0) . the proof of theorem 1.1 is complete. proof of theorem 1.2 by the same reasoning as before we can prove theorem 1.2. proof of corollary 1.3 by lemma 2.3 we get i (f) = p + 1 and ρp+1 (f) = ρp (a0) appliying theorem 1.2 we can easily get the conclusions of corollary 1.3 references [1] b. beläıdi, oscillation of fixed points of solutions of some linear differential equations, acta. math. univ. comenianae, 77 (2008), 263-269. [2] l. g. bernal, on growth k-order of solutions of a complex homogeneous linear differential equation, proceedings of the american mathematical society, 101 (1987), 317–322. [3] r. bouabdelli and b. beläıdi, on the iterated exponent of convergence of solutions of linear differential equations with entire and meromorphic coefficients, journal of mathematics, 2013 (2013), article id 429083. [4] z. x. chen, the fixed points and hyper-order of solutions of second order complex differential equations, acta mathematica scientia, 20 (2000), 425-432. (in chinese). [5] t. b. cao, z. x. chen, x. m. zheng, and j. tu, on the iterated order of meromorphic solutions of higher order linear differential equations, annals of differential equations, 21 (2005), 111–122. [6] a. el farissi, and m. benbachir, oscillation of fixed points of solutions to complex linear differential equations, electron. j. diff. equ., 2013 (2013), article id 41. [7] a. el farissi, value distribution of meromorphic solutions and their derivatives of complex differential equations, hindawi publishing corporation, isrn mathematical analysis, 2013 (2013), article id 497921. [8] g. g. gundersen, estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, j. london math. soc., 2 (1988), 88-104. [9] w. k. hayman, meromorphic functions, clarendon press, oxford, 1964 [10] l. kinnunen, linear differential equations with solutions of finite iterated order, southeast asian bull. math., 22 (1998), 385-405. [11] i. laine, nevanlinna theory and complex differential equations, walter de gruyter, berlin, new york, 1993. [12] m. s. liu and x. m. zhang, fixed points of meromorphic solutions of higher order linear differential equations, ann. acad. sci. fenn. ser. a. i. math., 31 (2006), 191-211. [13] r. nevanlinna, eindeutige analytische funktionen, zweite auflage. reprint. die grundlehren der mathematischen wissenschaften, band 46. springer-verlag, berlin-new york, 1974. 170 abdallah el farissi [14] j. wang and w. r. lü, the fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients, acta math. appl. sin. 27 (2004), 72-80. (in chinese). [15] h. y. xu, j. tu, x. m. zheng, on the hyper exponent of convergence of zeros of f(j) − ϕ of higher order linear differential equations, advances in difference equations, 2012 (2012), article 114. [16] h. x. yi and c. c. yang, the uniqueness theory of meromorphic functions, science press, beijing, 1995 (in chinese). [17] q. t. zhang and c. c. yang, the fixed points and resolution theory of meromorphic functions, beijing university press, beijing, 1988 (in chinese). department of mathematics and informatics, faculty of exact sciences, university of bechar, algeria international journal of analysis and applications volume 17, number 3 (2019), 369-387 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-369 on approximation solutions of the cauchy-jensen and the additive-quadratic functional equation in paranormed spaces prondanai kaskasem1 and chakkrid klin-eam1,2,∗ 1department of mathematics, faculty of science, naresuan university, phitsanulok, 65000, thailand 2research center for academic excellence in mathematics, naresuan university, thailand ∗corresponding author: chakkridk@nu.ac.th abstract. in this paper, we prove the generalized hyers-ulam-rassias stability of the bi-cauchy-jensen functional equation and the bi-additive-quadratic functional equation in paranormed spaces. moreover, we investigate the hyers-ulam-rassias stability of the generalized cauchy-jensen equation in such spaces. 1. introduction and preliminaries the stability problem of functional equations was initiated by ulam in 1940 [17] arising from concerning the stability of group homomorphisms. these question form is the object of the stability theory. in 1941, hyers [7] provided a first affirmative partial answer to ulam’s problem for the case of approximately additive mapping in banach spaces. in 1978, rassias [16] gave a generalization of hyers’s theorem for linear mapping by considering an unbounded cauchy difference. a generalization of rassias’s result was developed by găvruţa [6] in 1994 by replacing the unbounded cauchy difference by a general control function. for more information on that subject and further references we refer to a survey paper [3] and to a recent monograph on ulam stability [4]. received 2018-12-21; accepted 2019-01-30; published 2019-05-01. 2010 mathematics subject classification. 39b82; 39b52. key words and phrases. hyers-ulam-rassias stability; bi-cauchy-jensen functional equation; bi-additive-quadratic functional equation; generalized cauchy-jensen functional equation; paranormed space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 369 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-369 int. j. anal. appl. 17 (3) (2019) 370 let r and n be the set of real numbers and the set of natural numbers, respectively. next, let x and y be vector spaces and k be a positive integer, a function f : xk → y is called k-additive functional equation if and only if f satisfies the equation f(x1,x2, . . . ,xi−1,xi + y,xi+1, . . . ,xk) =f(x1,x1, . . . ,xi) + f(x1,x2, . . . ,xi−1,y,xi+1, . . . ,xk) for all i ∈ n, 1 ≤ i ≤ k and for every x1,x2, . . . ,xk,y ∈ x, that is, f is additive in each of its variables xi ∈ x for all i = 1, 2, . . . ,k. some fundamental properties on such mappings be mentioned in [10]. in particular, a 2-additive functional equation is called bi-additive functional equation. a mapping f : x × x → y is called a bi-additive-quadratic functional equation (bi-aqe, shortly) if f satisfies the system equations f(x + y,z) = f(x,z) + f(y,z), f(x,y + z) + f(x,y −z) = 2f(x,y) + 2f(x,z) (1.1) for all x,y,z ∈ x. when x = y = r, the solution of (1.1) is given by the function f(x,y) = cxy2 where x,y,c ∈ r. for mapping f : x ×x → y satisfies f(x + y,z + w) + f(x + y,z −w) = 2f(x,z) + 2f(x,w) + 2f(y,z) + 2f(y,w) (1.2) for all x,y,z,w ∈ x. in 2005, park, bae and chung [13] proved that the mapping f : x × x → y satisfies (1.1) if and only if it satisfies (1.2) and provided the general solution of (1.1) which is given by f(x,y) = m(x,y,y) and m(x,y,z) = m(x,z,y) for all x,y,z ∈ x where m : x × x × x → y is a multi-additive mapping. a mapping f : x×x → y is called a bi-cauchy-jensen functional equation (bi-cje, shortly) if f satisfies the system equations f(x + y,z) = f(x,z) + f(y,z) 2f ( x, y + z 2 ) = f(x,y) + f(x,z) (1.3) for all x,y,z ∈ x. in particular, for x = y = r, the solution of (1.3) is given by the function f(x,y) = axy + bx where x,y,a,b ∈ r. for mapping f : x ×x → y satisfies 2f ( x + y, z + w 2 ) = f(x,z) + f(x,w) + f(y,z) + f(y,w) (1.4) for all x,y,z,w ∈ x. in 2006, park and bae [12] showed that the mapping f : x ×x → y satisfies (1.3) if and only if it satisfies (1.4) and gave the general solution of (1.4) which is given by f(x,y) = b(x,y) + a(x) for all x,y ∈ x where b : x ×x → y is a bi-additive mapping and a : x → y is an additive mapping. int. j. anal. appl. 17 (3) (2019) 371 next, we recall the concepts of paranormed space and some basic facts on the fréchet spaces. definition 1.1 ([18]). let x be a vector space. a paranorm on x is a function p : x → r such that for all x,y ∈ x, the following conditions hold : (i) p(0) = 0; (ii) p(−x) = p(x); (iii) p(x + y) ≤ p(x) + p(y) (triangle inequality); (iv) if {tn} is a sequence of scalars with tn → t and {xn}⊆ x with p(xn−x) → 0, then p(tnxn−tx) → 0. (continuity of scalar multication) the pair (x,p) is called a paranormed space if p is a paranorm on x. note that p(nx) ≤ np(x) for all n ∈ n and all x ∈ x. the paranorm p on x is called total if, in addition, p satisfies (v) p(x) = 0 implies x = 0. a fréchet space is a total and complete paranormed space. in 2015, bae and park [2] proved the hyers-ulam stability of the functional equation (1.2) and (1.4) in paranormed spaces in the sense of rassias [16]. we refer to some works of stability of the functional equation (1.2) and (1.4) and various functional equations in paranormed spaces with [1, 8, 9, 11, 13–15]. in the first section of main results, we investigate stability of the functional equation (1.2) and (1.4) in paranormed spaces in the sense of găvruţa [6]. in 2009, gao et al. [5] introduced the generalized cauchy-jensen functional equation and gave some useful properties. let g be an n-divisible abelian group where n ∈ n and x be a normed space with norm ‖·‖x. for a mapping f : g → x is called a generalized cauchy-jensen functional equation (gcje, shortly) if it satisfies the equation αf ( x + y α + z ) = f(x) + f(y) + αf(z) (1.5) for all x,y,z ∈ x and for any fixed positive integer α ≥ 2. in particular, when α = 2, it is called a cauchy-jensen functional equation (cje, shortly). proposition 1.1 ([5]). let g be an n-divisible abelian group for some positive integer n and x be a normed space with norm ‖·‖x. then a mapping f : g → x is additive if and only if it satisfies ‖f(x) + f(y) + nf(z)‖x ≤ ∥∥∥∥nf(x + yn + z) ∥∥∥∥ x for all x,y,z ∈ g. the following corollary is an immediate consequence of proposition 1.1. int. j. anal. appl. 17 (3) (2019) 372 corollary 1.1 ([5]). for a mapping f : g → x, the following statements are equivalent. (a) f is additive. (b) f(x) + f(y) + nf(z) = nf(x+y n + z), for all x,y,z ∈ g. (c) ‖f(x) + f(y) + nf(z)‖x ≤‖nf(x+yn + z)‖x, for all x,y,z ∈ g. clearly, a vector space is n-divisible abelian group, so corollary 1.1 is right when g is a vector space. in the second section of main results, we proved the stability of the functional equation (1.5) in paranormed spaces in the sense of rassias [16]. throughtout this paper, assume that (x,p) is a fréchet space and that (e,‖·‖) is a banach space. 2. the stability of the bi-cauchy-jensen functional equation and bi-additive-quadratic functional equation in paranormed spaces the following result is the generalized hyers-ulam-rassias stability of the functional equation (1.4). theorem 2.1. let ϕ : e ×e ×e ×e → [0,∞) be a function and f : e ×e → x be a mapping satisfying f(x, 0) = 0 for all x ∈ e such that p ( 2f ( x + y, z + w 2 ) −f(x,z) −f(x,w) −f(y,z) −f(y,w) ) ≤ϕ(x,y,z,w) (2.1) for all x,y,z,w ∈ e. then there exists a unique mapping f : e ×e → x satisfying (1.4) such that p (2f(x,y) −f(x,y)) ≤ ϕ̃(x,x,y,y) (2.2) for all x,y ∈ e where ϕ̃(x,y,z,w) (2.3) := ∞∑ j=0 6j [ 6ϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) +ϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j )] < ∞ for all x,y,z,w ∈ e and the mapping f : e ×e → x is given by f(x,y) = lim j→∞ 2·6jf ( x 2j , y 3j ) for all x,y ∈ e. int. j. anal. appl. 17 (3) (2019) 373 proof. letting y = x in (2.1), we obtain that p ( 2f ( 2x, z + w 2 ) − 2f(x,z) − 2f(x,w) ) ≤ ϕ(x,x,z,w) (2.4) for all x,z,w ∈ e. letting w = −z in (2.4), we get that p (2f(x,z) + 2f(x,−z)) ≤ ϕ(x,x,z,−z) (2.5) for all x,z ∈ e. subtituting z by −z and w by −z in (2.4), we get p(2f(2x,−z) − 4f(x,−z)) ≤ ϕ(x,x,−z,−z) (2.6) for all x,z ∈ e. it follows from (2.5) and (2.6) that p(4f(x,z) + 2f(2x,−z)) (2.7) = p(4f(x,z) + 4f(x,−z) − 4f(x,−z) + 2f(2x,−z)) ≤ p(4f(x,z) + 4f(x,−z)) + p(2f(2x,−z) − 4f(x,−z)) ≤ 2p(2f(x,z) + 2f(x,−z)) + p(2f(2x,−z) − 4f(x,−z)) ≤ 2ϕ(x,x,z,−z) + ϕ(x,x,−z,−z) for all x,z ∈ e. letting w = −3z in (2.4), we have p(2f(2x,−z) − 2f(x,−3z) − 2f(x,z)) ≤ ϕ(x,x,z,−3z) for all x,z ∈ e. by (ii) of definition 1.1, we have p(2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤ ϕ(x,x,z,−3z) (2.8) for all x,z ∈ e. by (2.7) and (2.8), we have p(6f(x,z) + 2f(x,−3z)) (2.9) =p(4f(x,z) + 2f(2x,−z) + 2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤p(4f(x,z) + 2f(2x,−z)) + p(2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤2ϕ(x,x,z,−z) + ϕ(x,x,−z,−z) + ϕ(x,x,z,−3z) for all x,z ∈ e. putting z = 3z in (2.6), we obtain that p(2f(2x,−3z) − 4f(x,−3z)) ≤ ϕ(x,x,−3z,−3z) (2.10) int. j. anal. appl. 17 (3) (2019) 374 for all x,z ∈ e. it follows from (2.9) and (2.10) p(12f(x,z) + 2f(2x,−3z)) (2.11) =p(12f(x,z) + 4f(x,−3z) − 4f(x,−3z) + 2f(2x,−3z)) ≤p(12f(x,z) + 4f(x,−3z)) + p(2f(2x,−3z) − 4f(x,−3z)) ≤2p(6f(x,z) + 2f(x,−3z)) + p(2f(2x,−3z) − 4f(x,−3z)) ≤4ϕ(x,x,z,−z) + 2ϕ(x,x,−z,−z) + 2ϕ(x,x,z,−3z) + ϕ(x,x,−3z,−3z) for all x,z ∈ e. replacing z by −z in the above inequality, we get that p(12f(x,−z) + 2f(2x, 3z)) (2.12) ≤4ϕ(x,x,−z,z) + 2ϕ(x,x,z,z) + 2ϕ(x,x,−z, 3z) + ϕ(x,x, 3z, 3z) for all x,z ∈ e. by (2.5) and the above inequality, we have p(12f(x,z) − 2f(2x, 3z)) (2.13) =p(12f(x,z) + 12f(x,−z) − 12f(x,−z) − 2f(2x, 3z)) ≤p(12f(x,z) + 12f(x,−z)) + p(−12f(x,−z) − 2f(2x, 3z)) =p(12f(x,z) + 12f(x,−z)) + p(12f(x,−z) + 2f(2x, 3z)) ≤6p(2f(x,z) + 2f(x,−z)) + p(12f(x,−z) + 2f(2x, 3z)) ≤6ϕ(x,x,z,−z) + 4ϕ(x,x,−z,z) + 2ϕ(x,x,z,z) + 2ϕ(x,x,−z, 3z) + ϕ(x,x, 3z, 3z) for all x,z ∈ e. replacing x by x 2j+1 and z by z 3j+1 in (2.13), we obtain that p ( 12f ( x 2j+1 , z 3j+1 ) − 2f ( x 2j , z 3j )) (2.14) ≤6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) + ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j ) int. j. anal. appl. 17 (3) (2019) 375 for all x,z ∈ e. by (2.14), for any integers l,m such that 0 ≤ l < m, we get that p ( 2·6mf ( x 2m , z 3m ) − 2·6lf ( x 2l , z 3l )) (2.15) =p ( 2·6mf ( x 2m , z 3m ) − 2·6m−1f ( x 2m−1 , z 3m−1 ) + 2·6m−1f ( x 2m−1 , z 3m−1 ) − 2·6m−2f ( x 2m−2 , z 3m−2 ) + 2·6m−2f ( x 2m−2 , z 3m−2 ) + · · · + 2·6l+1f ( x 2l+1 , z 3l+1 ) − 2·6lf ( x 2l , z 3l )) ≤p ( 2·6mf ( x 2m , z 3m ) − 2·6m−1f ( x 2m−1 , z 3m−1 )) + p ( 2·6m−1f ( x 2m−1 , z 3m−1 ) − 2·6m−2f ( x 2m−2 , z 3m−2 )) + · · · + p ( 2·6l+1f ( x 2l+1 , z 3l+1 ) − 2·6lf ( x 2l , z 3l )) ≤6m−1p ( 12f ( x 2m , z 3m ) − 2f ( x 2m−1 , z 3m−1 )) + 6m−2p ( 12f ( x 2m−1 , z 3m−1 ) − 2f ( x 2m−2 , z 3m−2 )) + · · · + 6lp ( 12f ( x 2l+1 , z 3l+1 ) − 2f ( x 2l , z 3l )) = m−1∑ j=l 6jp ( 12f ( x 2j+1 , z 3j+1 ) − 2f ( x 2j , z 3j )) ≤ ∞∑ j=l 6j [ 6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) +ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j )] for all x,z ∈ e. it follows from (2.3) that lim l→∞ ∞∑ j=l 6j [ 6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) +ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j )] = 0 for all x,z ∈ e. this implies that the sequence {2·6jf ( x 2j , z 3j ) }∞j=0 is a cauchy sequence in x for all x,z ∈ e. since x is complete paranormed space, the sequence {2·6jf ( x 2j , z 3j ) }∞j=0 converges for all x,z ∈ e. define f : e ×e → x by f(x,z) = lim j→∞ 2·6jf ( x 2j , z 3j ) (2.16) int. j. anal. appl. 17 (3) (2019) 376 for all x,z ∈ e. by (2.3), we get that ∞∑ j=1 1 6 ·2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = ∞∑ j=0 1 6 ·2·6j+1ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) = ∞∑ j=0 2·6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) ≤ ∞∑ j=0 6·6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) + ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j ) ≤ ∞∑ j=0 6j [ 6ϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) +ϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j )] =ϕ̃(x,y,z,w) < ∞ for all x,y,z,w ∈ e. this implies that lim j→∞ 1 6 ·2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 for all x,y,z,w ∈ e, which implies lim j→∞ 2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 (2.17) for all x,y,z,w ∈ e. it follows from (2.1), (2.16) and (2.17) that we have p ( 2f ( x + y, z + w 2 ) −f(x,z) −f(x,w) −f(y,z) −f(y,w) ) ≤p ( 2 lim j→∞ 2·6jf ( x + y 2j , z+w 2 3j ) − lim j→∞ 2·6jf ( x 2j , z 3j ) − lim j→∞ 2·6jf ( x 2j , w 3j ) − lim j→∞ 2·6jf ( y 2j , z 3j ) − lim j→∞ 2·6jf ( y 2j , w 3j )) ≤ lim j→∞ 2·6jp ( 2f ( x 2j + y 2j , z 3j + w 3j 2 ) −f ( x 2j , z 3j ) −f ( x 2j , w 3j ) −f ( y 2j , z 3j ) −f ( y 2j , w 3j )) ≤ lim j→∞ 2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 int. j. anal. appl. 17 (3) (2019) 377 for all x,y,z,w ∈ e. since x is total, we have 2f ( x + y, z + w 2 ) = f(x,z) + f(x,w) + f(y,z) + f(y,w) for all x,y,z,w ∈ e. setting l = 0 and taking m → ∞ in (2.15), this implies that the inequality (2.2). next, we will show that f is unique. let g : e × e → x be another mapping satisfying (1.4) and (2.2). by [12], there exists bi-additive mapping b,b′ : e×e → x and additive mapping a,a′ : e → x such that f(x,y) = b(x,y) + a(x) and g(x,y) = b′(x,y) + a′(x) for all x,y ∈ e. since b is bi-additive mapping, a is additive mapping and f(x, 0) = 0 for all x ∈ e, we have f(x,y) − 6f (x 2 , x 3 ) = [b(x,y) + a(x)] − 6 [ b (x 2 , y 3 ) + a (x 2 )] = b(x,y) + a(x) − 6b (x 2 , y 3 ) − 6a (x 2 ) = b(x,y) + a(x) −b(x,y) − 3a (x) = −2a(x) = −2b(x, 0) − 2a(x) = −2f(x, 0) = −2 lim j→∞ 2 · 6jf ( x 2j , 0 ) = 0 for all x,y ∈ e, that is, f(x,y) = 6f (x 2 , y 3 ) (2.18) for all x,y ∈ e. replacing x by x 2 and y by y 3 in (2.18), we have f (x 2 , y 3 ) = 6f ( x 22 , y 32 ) for all x,y ∈ e. continuing this process, we have f(x,y) = 6nf ( x 2n , y 3n ) for all x,y ∈ e and for all n ∈ n. similarly, we get that g(x,y) = 6ng ( x 2n , y 3n ) for all x,y ∈ e and for all n ∈ n. for any n ∈ n, we obtain int. j. anal. appl. 17 (3) (2019) 378 that p(f(x,y) −g(x,y)) (2.19) =p ( 6nf ( x 2n , y 3n ) − 6ng ( x 2n , y 3n )) =p ( 6nf ( x 2n , y 3n ) − 2·6nf ( x 2n , y 3n ) + 2·6nf ( x 2n , y 3n ) − 6ng ( x 2n , y 3n )) ≤p ( 6nf ( x 2n , y 3n ) − 2·6nf ( x 2n , y 3n )) + p ( 2·6nf ( x 2n , y 3n ) − 6ng ( x 2n , y 3n )) ≤6np ( f ( x 2n , y 3n ) − 2f ( x 2n , y 3n )) + 6np ( 2f ( x 2n , y 3n ) −g ( x 2n , y 3n )) ≤2·6nϕ̃ ( x 2n , x 2n , y 3n , y 3n ) =2·6n ∞∑ j=0 6j [ 6ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j+1 ,− y 3n 3j+1 ) + 4ϕ ( x 2n 2j+1 , x 2n 2j+1 ,− y 3n 3j+1 , y 3n 3j+1 ) + 2ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j+1 , y 3n 3j+1 ) + 2ϕ ( x 2n 2j+1 , x 2n 2j+1 ,− y 3n 3j+1 , y 3n 3j ) +ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j , y 3n 3j )] =2 ∞∑ j=0 6n+j [ 6ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j+1 ,− y 3n+j+1 ) + 4ϕ ( x 2n+j+1 , x 2n+j+1 ,− y 3n+j+1 , y 3n+j+1 ) + 2ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j+1 , y 3n+j+1 ) +2ϕ ( x 2n+j+1 , x 2n+j+1 ,− y 3n+j+1 , y 3n+j ) + ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j , y 3n+j )] =2 ∞∑ i=n 6i [ 6ϕ ( x 2i+1 , x 2i+1 , y 3i+1 ,− y 3i+1 ) + 4ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 , y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i ) +ϕ ( x 2i+1 , x 2i+1 , y 3i , y 3i )] for all x,y ∈ e. by (2.3), we obtain that lim n→∞ ∞∑ i=n 6i [ 6ϕ ( x 2i+1 , x 2i+1 , y 3i+1 ,− y 3i+1 ) + 4ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i+1 ) (2.20) + 2ϕ ( x 2i+1 , x 2i+1 , y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i ) +ϕ ( x 2i+1 , x 2i+1 , y 3i , y 3i )] = 0 for all x,y ∈ e. from (2.20), taking limit n →∞ in (2.19), we obtain that lim n→∞ p(f(x,y) −g(x,y)) = 0 for all x,y ∈ e. since paranorm p on x is total, we have f(x,y) −g(x,y) = 0 for all x,y ∈ e. hence f is a unique mapping satisfying (1.4) and (2.2). � int. j. anal. appl. 17 (3) (2019) 379 remark 2.1. let r,θ be positive real numbers with r > log2 6. if we set ϕ(x,y,z,w) = θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r) for all x,y,z,w ∈ e, then theorem 2.1 recovers theorem 2.1 in [2]. the following result is the generalized hyers-ulam-rassias stability of the functional equation (1.2). theorem 2.2. let ϕ : e ×e ×e ×e → [0,∞) be a function and f : e ×e → x be a mapping satisfying f(x, 0) = 0 for all x ∈ e such that p (f(x + y,z + w) + f(x + y,z −w) − 2f(x,z) − 2f(x,w) − 2f(y,z) − 2f(y,w)) (2.21) ≤ϕ(x,y,z,w) for all x,y,z,w ∈ e. then there exists a unique mapping f : e ×e → x satisfying (1.2) such that p (f(x,y) −f(x,y)) ≤ ϕ̃(x,x,y,y) (2.22) for all x,y ∈ e where ϕ̃(x,y,z,w) := ∞∑ j=0 8jϕ ( x 2j+1 , y 2j+1 , z 2j+1 , w 2j+1 ) < ∞ (2.23) for all x,y,z,w ∈ e where the mapping f : e ×e → x is given by f(x,y) = lim j→∞ 8jf ( x 2j , y 2j ) for all x,y ∈ e. proof. letting y = x and w = z in (2.21), we obtain that p(f(2x, 2z) − 8f(x,z)) ≤ ϕ(x,x,z,z) for all x,z ∈ e. replacing x by x 2j+1 and z by z 2j+1 in the above inequality, we get that p ( f ( x 2j , z 2j ) − 8f ( x 2j+1 , z 2j+1 )) ≤ ϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) (2.24) for all nonnegative integer j and for all x,z ∈ e. it follows from (2.24) that we have p ( 8jf ( x 2j , z 2j ) − 8j+1f ( x 2j+1 , z 2j+1 )) (2.25) ≤8jp ( f ( x 2j , z 2j ) − 8f ( x 2j+1 , z 2j+1 )) ≤8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) int. j. anal. appl. 17 (3) (2019) 380 for all nonnegative integer j and for all x,z ∈ e. by (2.25), for any integers l and m such that 0 ≤ l < m, we have p ( 8lf ( x 2l , y 2l ) − 8mf ( x 2m , y 2m )) (2.26) =p ( 8lf ( x 2l , y 2l ) − 8l+1f ( x 2l+1 , y 2l+1 ) + 8l+1f ( x 2l+1 , y 2l+1 ) + · · · + 8m−1f ( x 2m−1 , y 2m−1 ) − 8mf ( x 2m , y 2m )) ≤p ( 8lf ( x 2l , y 2l ) − 8l+1f ( x 2l+1 , y 2l+1 )) + p ( 8l+1f ( x 2l+1 , y 2l+1 ) − 8l+2f ( x 2l+2 , y 2l+2 )) + · · · + p ( 8m−1f ( x 2m−1 , y 2m−1 ) − 8mf ( x 2m , y 2m )) = m−1∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) ≤ ∞∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) for all x,z ∈ e. it follows from (2.23) that we obtain that lim l→∞ ∞∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) = 0 for all x,z ∈ e. this implies that {8jf ( x 2j , z 2j ) } is cauchy seqeunce in x for all x,z ∈ e. by completeness of x, the sequence {8jf ( x 2j , z 2j ) } is convergent sequence for all x,y ∈ e. define f : e ×e → x by f(x,z) = lim j→∞ 8jf ( x 2j , z 2j ) for all x,z ∈ e. by (2.21), we obtain that p(f(x + y,z + w) + f(x + y,z −w) − 2f(x,z) − 2f(x,w) − 2f(y,z) − 2f(y,w)) =p ( lim j→∞ 8jf ( x + y 2j , z + w 2j ) + lim j→∞ 8jf ( x + y 2j , z −w 2j ) − 2· lim j→∞ 8jf ( x 2j , z 2j ) −2· lim j→∞ 8jf ( x 2j , w 2j ) − 2· lim j→∞ 8jf ( y 2j , z 2j ) − 2· lim j→∞ 8jf ( y 2j , w 2j )) = lim j→∞ p ( 8jf ( x + y 2j , z + w 2j ) + 8jf ( x + y 2j , z −w 2j ) − 2·8jf ( x 2j , z 2j ) −2·8jf ( x 2j , w 2j ) − 2·8jf ( y 2j , z 2j ) − 2·8jf ( y 2j , w 2j )) ≤ lim j→∞ 8jp ( f ( x + y 2j , z + w 2j ) + f ( x + y 2j , z −w 2j ) − 2f ( x 2j , z 2j ) − 2f ( x 2j , w 2j ) −2f ( y 2j , z 2j ) − 2f ( y 2j , w 2j )) ≤ lim j→∞ 8jϕ ( x 2j , y 2j , z 2j , w 2j ) = 0 int. j. anal. appl. 17 (3) (2019) 381 for all x,y,z,w ∈ e. since x is total, we have f(x + y,z + w) + f(x + y,z −w) = 2f(x,z) + 2f(x,w) + 2f(y,z) + 2f(y,w) for all x,y,z,w ∈ e. setting l = 0 and letting m → ∞ in (2.26), the inequality (2.26) holds. next, we will show that f is unique. let g : e × e → x be a another mapping satisfying (1.2) and (2.22). it follows from theorem 3 in [13] that there exists multi-additive mapping m,m′ : e × e × e → x such that f(x,y) = m(x,y,y), g(x,y) = m′(x,y,y), m(x,y,z) = m(x,z,y) and m′(x,y,z) = m′(x,z,y) for all x,y,z ∈ e. for any n ∈ n, we get that p(f(x,y) −g(x,y)) =p (m(x,y,y) −m′(x,y,y)) =p ( m ( 2nx 2n , 2ny 2n , 2ny 2n ) −m′ ( 2nx 2n , 2ny 2n , 2ny 2n )) =p ( 8n [ m ( x 2n , y 2n , y 2n )] − 8n [ m′ ( x 2n , y 2n , y 2n )]) =p ( 8n [ m ( x 2n , y 2n , y 2n ) −m′ ( x 2n , y 2n , y 2n )]) ≤8np ( m ( x 2n , y 2n , y 2n ) −m′ ( x 2n , y 2n , y 2n )) ≤8np ( f ( x 2n , y 2n ) −f ( x 2n , y 2n ) + f ( x 2n , y 2n ) −g ( x 2n , y 2n )) ≤8n [ p ( f ( x 2n , y 2n ) −f ( x 2n , y 2n )) + p ( f ( x 2n , y 2n ) −g ( x 2n , y 2n ))] ≤2·8nϕ̃ ( x 2n , x 2n , y 2n , y 2n ) =2·8n ∞∑ j=0 8jϕ ( x 2n 2j+1 , x 2n 2j+1 , y 2n 2j+1 , y 2n 2j+1 ) =2· ∞∑ j=0 8n+jϕ ( x 2n+j+1 , x 2n+j+1 , y 2n+j+1 , y 2n+j+1 ) =2· ∞∑ i=n 8iϕ ( x 2i+1 , x 2i+1 , y 2i+1 , y 2i+1 ) for all x,y ∈ e. by (2.23), we get that lim n→∞ ∞∑ i=n 8iϕ ( x 2i+1 , x 2i+1 , y 2i+1 , y 2i+1 ) = 0 for all x,y ∈ e. hence lim n→∞ p(f(x,y) −g(x,y)) = 0 for all x,y ∈ e. since paranorm p on x is total, we have f(x,y) −g(x,y) = 0 for all x,y ∈ e. hence f is a unique mapping satisfying (1.2) and (2.22). � int. j. anal. appl. 17 (3) (2019) 382 remark 2.2. let r,θ be positive real numbers with r > 3. if we set ϕ(x,y,z,w) = θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r) for all x,y,z,w ∈ e, then theorem 2.2 recovers theorem 3.1 in [2]. 3. stability of the generalized cauchy-jensen functional equation in paranormed space the following result is the hyers-ulam-rassias stability of the functional equation (1.5). theorem 3.1. let r be a positive real number with r > 1, and let f : e → x be a mapping satisfying p ( αf ( x + y α + z ) −f(x) −f(y) −αf(z) ) ≤ θ (‖x‖r + ‖y‖r + ‖z‖r) (3.1) for all x,y,z ∈ e. then there exists a unique mapping f : e → x satisfying (1.5) such that p (f(x) −f(x)) ≤ ( 3αr + 1 αr −α ) θ‖x‖r (3.2) for all x ∈ e where the mapping f : e → x is given by f(x) = lim n→∞ αnf ( x αn ) for all x ∈ e. proof. putting x = y = z = 0 in (3.1), we have p(f(0)) ≤ 0. since x is total, we obtain that f(0) = 0. letting x = −x α , y = x α and z = 0 in (3.1), we obtain that p ( f ( − x α ) + f (x α )) (3.3) =p ( −f ( − x α ) −f (x α )) =p ( αf ( −x α + x α α + 0 ) −f ( − x α ) −f (x α ) −αf(0) ) ≤θ (∥∥∥−x α ∥∥∥r + ∥∥∥x α ∥∥∥r + ‖0‖r) = 2θ αr ‖x‖r for all x ∈ e. replacing x by αx in the inequality (3.3), we get that p (f(−x) + f(x)) ≤ 2θ‖x‖r (3.4) int. j. anal. appl. 17 (3) (2019) 383 for all x ∈ e. replacing x = −x, y = 0, and z = x α in (3.1), we have p ( f(−x) + αf (x α )) (3.5) =p ( −f(−x) −αf (x α )) =p ( αf ( −x + 0 α + ( − x α )) −f(−x) −f(0) −αf (x α )) ≤θ ( ‖−x‖r + ‖0‖r + ∥∥∥x α ∥∥∥r) = ( 1 + 1 αr ) θ‖x‖r for all x ∈ e. it follows from (3.4) and (3.5) that we have p ( αf (x α ) −f(x) ) = p ( αf (x α ) + f(−x) −f(−x) −f(x) ) (3.6) ≤ p ( αf (x α ) + f(−x) ) + p (f(−x) + f(x)) ≤ ( 1 + 1 αr ) θ‖x‖r + 2θ‖x‖r ≤ ( 3 + 1 αr ) θ‖x‖r for all x ∈ e. for i ∈ n, replacing x = x αi in (3.6), we get that p ( αi+1f ( x αi+1 ) −αif ( x αi )) ≤ αip ( αf ( x αi+1 ) −f ( x αi )) (3.7) ≤ αi ( 3 + 1 αr ) θ ∥∥∥ x αi ∥∥∥r = ( 1 αr−1 )i ( 3 + 1 αr ) θ‖x‖r for all x ∈ e. for given nonnegative integer l,m such that l < m, we have p ( αmf ( x αm ) −αlf ( x αl )) (3.8) =p ( αmf ( x αm ) −αm−1f ( x αm−1 ) + αm−1f ( x αm−1 ) + · · · + αl+1f ( x αl+1 ) −αlf ( x αl )) ≤ m−1∑ j=l p ( αj+1f ( x αj+1 ) −αjf ( x αj )) ≤ m−1∑ j=l ( 1 αr−1 )i ( 3 + 1 αr ) θ‖x‖r ≤ ( 3 + 1 αr ) θ‖x‖r ∞∑ j=0 ( 1 αr−1 )i int. j. anal. appl. 17 (3) (2019) 384 for all x ∈ e. since r > 1, we have 1 αr−1 < 1. since 1 αr−1 < 1, the sequence {αnf ( x αn ) } is cauchy sequence for all x ∈ e. by completeness of x, the sequence {αnf ( x αn ) } converges. define f : e → x by f(x) = lim n→∞ αnf ( x αn ) (3.9) for all x ∈ e. moreover, letting l = 0 and taking limit m →∞ in (3.8), we can obtain that inequality (3.2). it follows from (3.1) and (3.9) that p ( αf ( x + y α + z ) −f(x) −f(y) −αf(z) ) =p ( α· lim n→∞ αnf ( x+y α + z αn ) − lim n→∞ αnf ( x αn ) − lim n→∞ αnf ( y αn ) −α lim n→∞ αnf ( z αn )) = lim n→∞ αnp ( αf ( x αn + y αn α + z αn ) −f ( x αn ) −f ( y αn ) −αf ( z αn )) ≤ lim n→∞ αnθ (∥∥∥ x αn ∥∥∥r + ∥∥∥ x αn ∥∥∥r + ∥∥∥ x αn ∥∥∥r) =θ‖x‖r lim n→∞ ( 1 αr−1 )n = 0 for all x,y,z ∈ e. since x is total, we have αf ( x + y α + z ) = f(x) + f(y) + αf(z) for all x,y,z ∈ e. by corollary 1.1, f is additive. next, we will show that f is unique. let g be another mapping satisfying (1.5) and (3.2). then, we consider p (f(x) −g(x)) = p ( nf (x n ) −nf (x n ) + nf (x n ) −ng (x n )) (3.10) ≤ n ( p ( f (x n ) −f (x n )) + p ( f (x n ) −g (x n ))) ≤ 2nθ ( 3αr + 1 αr −α )∥∥∥x n ∥∥∥r = ( 1 nr−1 ) 2θ ( 3αr + 1 αr −α ) ‖x‖r for all x ∈ e. since r − 1 > 0, taking limit n → ∞ in (3.10), we have p(f(x) − g(x)) = 0 for all x ∈ e. since x is total, we have f(x) = g(x) for all x ∈ e, that is f is unique. � theorem 3.2. let r be a positive real number with r < 1 and let f : x → e be a mapping satisfying∥∥∥∥αf ( x + y α + z ) −f(x) −f(y) −αf(z) ∥∥∥∥ ≤ p (x)r + p (y)r + p (z)r (3.11) for all x,y,z ∈ x. then there exists a unique mapping f : x → e satisfying (1.5) such that ‖f(x) −f(x)‖≤ 2 + 3αr α−αr p (x) r (3.12) int. j. anal. appl. 17 (3) (2019) 385 for all x ∈ x where the mapping f : x → e is given by f(x) = lim n→∞ 1 αn f (αnx) for all x ∈ x. proof. letting x = y = z = 0 in (3.11), we get that ‖2f(0)‖ = ∥∥∥∥αf ( 0 + 0 α + 0 ) −f(0) −f(0) −αf(0) ∥∥∥∥ ≤p (0)r + p (0)r + p (0)r =0 so f(0) = 0. subtituting x = −αx, y = 0 and z = x in (3.11), we obtain that ‖f(−αx) + αf(x)‖ = ∥∥∥∥αf ( −αx + 0 α + x ) −f(−αx) −f(0) −αf(x) ∥∥∥∥ ≤ p (−αx)r + p (0)r + p (x)r ≤ (1 + αr)p(x)r for all x ∈ x. letting x = −αx, y = αx and z = x, we get that ‖f(−αx) + f(αx)‖ = ∥∥∥∥αf ( −αx + αx α + x ) −f(−αx) −f(αx) −αf(x) ∥∥∥∥ ≤ p (−αx)r + p (αx)r + p (x)r ≤ (1 + 2αr)p (x)r for all x ∈ x. then we have ‖f(αx) −αf(x)‖ = ‖f(αx) + f(−αx) −f(−αx) −αf(x)‖ = ‖f(αx) + f(−αx)‖ + ‖f(−αx) + αf(x)‖ ≤ (1 + αr)p(x)r + (1 + 2αr)p (x)r = (2 + 3αr)p(x)r and so ∥∥∥∥ 1αf(αx) −f(x) ∥∥∥∥ ≤ 2 + 3αrα p(x)r (3.13) for all x ∈ x. replacing x = αix and multiplying by 1 αi in (3.13), we have∥∥∥∥ 1αi+1 f(αi+1x) − 1αif(αix) ∥∥∥∥ ≤ 1αi · 2 + 3α r α p(αix)r (3.14) ≤ 2 + 3αr α p(x)r· ( 1 α1−r )i int. j. anal. appl. 17 (3) (2019) 386 for all x ∈ x. by (3.14), for any integers l,m such that 0 ≤ l < m, we obtain that∥∥∥∥ 1αmf(αmx) − 1αlf(αlx) ∥∥∥∥ (3.15) = ∥∥∥∥ 1αmf(αmx) + 1αm−1 f(αm−1x) − 1αm−1 f(αm−1x) + · · · + 1αl+1 f(αl+1x) − 1 αl f(αlx) ∥∥∥∥ ≤ m−1∑ i=l 2 + 3αr α p(x)r· ( 1 α1−r )i ≤ 2 + 3αr α p(x)r· ∞∑ i=0 ( 1 α1−r )i for all x ∈ x. since 1 α1−r < 1, we have ∑∞ i=0 ( 1 α1−r )i < ∞. it follows from (3.15) that the sequence { 1 αn f(αnx)} is cauchy sequence for all x ∈ x. since e is complete, the sequence { 1 αn f(αnx)} is convergent sequence. we define a mapping f : x → e by f(x) = lim n→∞ 1 αn f (αnx) (3.16) for all x ∈ x. moreover, letting l = 0 and taking limit m → ∞ in (3.15), we can obtain that inequality (3.12). it follows from (3.11) and (3.16) that we have∥∥∥∥αf ( x + y α + z ) −f(x) −f(y) −αf(z) ∥∥∥∥ = ∥∥∥∥α limn→∞ 1αnf ( αn ( x + y α + z )) − lim n→∞ 1 αn f (αny) − lim n→∞ 1 αn f (αnz) −α lim n→∞ 1 αn f (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn ∥∥∥∥αf ( αnx + αny α + αnz ) −f (αny) −f (αnz) −αf (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn ∥∥∥∥αf ( αnx + αny α + αnz ) −f (αny) −f (αnz) −αf (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn (p(αnx)r + p(αny)r + p(αnz)r) ≤ lim n→∞ αnr αn (p(x)r + p(y)r + p(z)r) ≤(p(x)r + p(y)r + p(z)r) lim n→∞ ( 1 α1−r )n = 0 for all x,y,z ∈ x. hence αf ( x + y α + z ) = f(x) + f(y) + αf(z) for all x,y,z ∈ x, that is f is the generalized cauchy-jensen functional equation. by tha same reasoning as in the proof of theorem 3.1, f is unique. � competing interests. the author declare that they have no competing interests. int. j. anal. appl. 17 (3) (2019) 387 author’s contributions. all authors contributed equally to writing of this paper. all authors read and approved the final manuscript. acknowledgement. the authors would like to thank naresuan university and science achievement scholarship of thailand, which provides facilities and funding for this research. references [1] j.h. bae and w.g. park, approximate quadratic forms in paranormed spaces, j. comput. anal. appl., 19 (4) (2015), 740–750 [2] j.h. bae and w.g. park, on the ulam stability of the cauchy-jensen equation and the additive-quadratic equation, j. nonlinear sci. appl., 8 (2015), 710–718 [3] j. brzdȩk, w. fechner, m.s. moslehian and j. silorska, recent developments of the conditional stability of the homomorphism equation, banach j. math. anal., 9 (2015), 278–326. [4] j. brzdȩk, d. popa, i. rasa and b. xu, ulam stability of operators, mathematical analysis and its applications vol.1, academic press, elsevier, oxford, 2018. [5] z.x. gao, h.x. cao, w.t. zheng and l. xu, generalized hyers-ulam-rassias stability of functional inequalities and functional equations, j. math. inequal., 3 (1) (2009) 63–77 [6] p. găvruţa, a generalization of the hyers-ulam-rassias stability of approximately additive mapping, j. math. anal. appl., 184 (1994), 431–436 [7] d.h. hyers, on the stability of the linear functional equation, proceedings of the national academy of sciences of the united states of america, 27 (4) (1941), 222–224 [8] k.w. jun, y.h. lee and y.s. cho , on the stability of cauchy-jensen functional equation, commun. korean math. soc., 23 (3) (2008) 377–386 [9] k.w. jun, y.h. lee and j.a. son, on the stability of cauchy-jensen functional equation iii, korean j. math., 16 (2) (2008) 205–214 [10] m. kuczma, an introduction to the theory of functional equational equations and inequalties: cauchy’s equation and jensen’s inequality, birkhäuser verlag basel, 2009. [11] s.j. lee, c. park and j.r. lee, functional inequalities in paranormed spaces, j. chungcheong math. soc., 26 2 (2013), 287–296. [12] w.g. park and j.h. bae, on a cauchy-jensen functional equation and its stability, j. math.anal. appl., 323 (1) (2006), 634–643 [13] w.g. park, j.h. bae and b.h. chung, on an additive-quadratic functional equation and its stability, j. appl. math. & computing, 18 (1-2) (2005), 563–572 [14] c. park and j.r. lee, functional equations and inequalities in paranormed spaces, j. inequal. appl., 2013 (2013), art. id 198. [15] c. park and d.y. shin, functional equations in paranormed spaces, adv. differ. equ., 2012 (2012), art. id 123. [16] th.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72 (2) (1978), 297–300 [17] s.m. ulam, a collection of mathematical problems, interscience tracts in pure and appliedmathematics, no. 8, interscience publishers, new york, ny, usa, 1960. [18] a. wilansky, modern methods in topological vector space, mcgraw-hill international book co., new york, 1, 1978. 1. introduction and preliminaries 2. the stability of the bi-cauchy-jensen functional equation and bi-additive-quadratic functional equation in paranormed spaces 3. stability of the generalized cauchy-jensen functional equation in paranormed space competing interests author's contributions acknowledgement references international journal of analysis and applications volume 19, number 6 (2021), 970-983 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-970 on truly nonlinear oscillator equations of ermakov-pinney type marcellin nonti, kolawole kegnide damien adjaï, jean akande, marc delphin monsia∗ department of physics, university of abomey-calavi, abomey-calavi, 01.bp.526, cotonou, benin ∗corresponding author: monsiadelphin@yahoo.fr abstract. in this paper we present a general class of differential equations of ermakov-pinney type which may serve as truly nonlinear oscillators. we show the existence of periodic solutions by exact integration after the phase plane analysis. the related quadratic lienard type equations are examined to show for the first time that the jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations. 1. introduction in the research field of periodic solution to lienard nonlinear differential equations of the form (1.1) ẍ + f(x) = 0 where the overdot stands for the derivative with respect to time, and f(x) is a nonlinear function of x, it is less usual to notice differential equations with exact periodic solutions. it is again very less usual to find differential equations with exact periodic solutions in terms of trigonometric functions. this makes the ermakov-pinney equation (1.2) ẍ + ax + b x3 = 0 received october 5th, 2021; accepted november 2nd, 2021; published november 19th, 2021. 2010 mathematics subject classification. 34c15, 34c25. key words and phrases. lienard equations; ermakov-pinney equation; truly nonlinear oscillators; periodic solution. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 970 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-970 int. j. anal. appl. 19 (6) (2021) 971 unusual and underlines its high usefulness in science and engineering. in this way a lot of applications in classical mechanics as well as in quantum mechanics for example, has been carried out during these decades ( [1], [2]). the ermakov-pinney equation (1.2) has been studied in [2] to show the existence of new periodic solutions and non-periodic solutions. in [3] an exceptional lienard equation with strong and high order nonlinearity is presented. a lienard equation with a periodic solution in terms of a single trigonometric function, which may lead to a quadratic lienard type equation with a periodic solution exhibiting harmonic oscillations, and contains several well-known equations like the ermakov-pinney equation [4], the mickens truly nonlinear oscillators and the cubic duffing equation as special cases, has never been highlighted in the literature despite the well established theory of differential equations, as it is carried out in [3]. according to ( [5], [6]) all ermakov-pinney equations may be reduced to (1.3) ẍ + b x3 = 0 using a variable change. one may say that the cubic singularity defines the nonlinear property of the ermakov-pinney equation. so a differential equation with cubic nonlinearity may be said of ermakov-pinney type. in this perspective consider the lienard differential equation ( [2], [3]) (1.4) ẍ + 1 2 (α−q)axα−q−1 + qb 2 x−q−1 = 0 making q = 2, yields as equation (1.5) ẍ + 1 2 (α− 2)axα−3 + b x3 = 0 in view of the above, the equation (1.5) may be characterized of ermakov-pinney type. for α = 2, the equation (1.5) reduces to (1.3). the equation (1.5) may be reduced to a quadrature. using the corresponding first integral ( [2], [3]) (1.6) ẋ2x2 + axα = b the general solution of (1.5) is written as the quadrature defined by (1.7) ±(t + k) = ∫ xdx √ b−axα let (1.8) j = ∫ xdx √ b−axα int. j. anal. appl. 19 (6) (2021) 972 the change of variable x2 = b − axα, and dx = − 2 aα ( b−x2 a )1−α α xdx, where a 6= 0, and α 6= 0, turns j into (1.9) j = − 2 αa 2 α ∫ ( b−x2 )2−α α dx a new change of variable φ = x√ b , where b 6= 0, reduces j to the form (1.10) j = − 2 √ b bα ( b a ) 2 α ∫ ( 1 −φ2 )2−α α dφ as can be seen, the value of the integral in j could not be known exactly. a new change of variable in terms of trigonometric or hyperbolic functions may be also performed but this does not solve the problem. however, it shows that the general solutions of some specific equations of the equation (1.5) are not periodic. the equation (1.5) may be reduced to the form (1.11) ẍ + (n− 1)ax2n−3 + b x3 = 0 where α = 2n, n is an integer. the equation (1.11) may be of physical importance since it has the structure of truly nonlinear oscillators formulated by mickens in his book [7], and contains the famous ermakovpinney equation as special case. it is known also that differential equations with power nonlinearities are often encountered in mathematical modeling of physical problems. a vast literature exists on the topic of truly nonlinear oscillators. during the last decades many authors investigated these nonlinear differential equations. as nonlinear differential equations, they have no exact explicit solutions in general. moreover they could not be solved by the well known standard approximate analytical techniques [7]. so the existence of periodic solutions of these equations is yet under some debate. this particularly, becomes an attractive research problem when the second order autonomous truly nonlinear equation has a singularity at the origin and can have no critical point, necessary condition, according to [8], for a planar autonomous systems to have a periodic solution. it was the case of the so-called pseudo-oscillator investigated in [8]. the authors [8] concluded that such a differential equation has no periodic solution. in contrast to this the author in [9] showed that periodic solution exists, at least a non-smooth solution. the author [9] carried out a theory to build such periodic solutions. in [10] the two general solutions predicted in [8] have been exactly calculated and the authors [10] concluded also to the non existence of smoth periodic solution. the equation (1.11) has a cubic singularity at the origin for a positive integer n, but may have fixed points. for n negative, singularities appear also. choosing n = 1, that is α = 2, reduces the equation (1.11) to the ermakov-pinney int. j. anal. appl. 19 (6) (2021) 973 type equation (1.3). for n = 2, that is α = 4, the equation (1.11) becomes the ermakov-pinney equation (1.2). using n = 3, in other words α = 6, the equation (1.11) transforms into (1.12) ẍ + 2ax3 + b x3 = 0 for b = 0, the equation (1.12) reduces to the well-known restricted cubic duffing equation ( [7], [11], [12]) for which it is said that all the solutions are periodic. contrary to these authors and to several others, it has been shown that such an equation may exhibit non-periodic solution, precisely complex-valued solutions [13]. putting n = 4, into the equation (1.11), yields (1.13) ẍ + 3ax5 + b x3 = 0 when b = 0, the equation (1.13) reduces to the restricted quintic duffing equation. the equations (1.12), (1.13) and others, which may be obtained for various values of n, have not been previously studied in the literature. the above underlines the mathematical importance to ask whether the equation (1.11) has periodic solutions for n � 2. the case n = 2, corresponding to the ermakov-pinney equation has been investigated in several papers. periodic solutions have been carried out for this equation in ( [2], [4]). the case of negative n will be considered separately in a paper. in the present work we show that the equation (1.11) may exhibit periodic and explicit general solutions for a b = 1, and 2 � n � 4. the choice of negative b has no physical sense as we can see below. in this regard the qualitative properties of solutions of the equation (1.11) are investigated in section (2), and the section (3) is devoted to exhibit exact periodic solutions of (1.11) under the preceding conditions. the general solutions of the equation (1.3) are also exhibited in this section. finally a conclusion is formulated for the work. 2. qualitative properties of solutions the qualitative properties of solutions to (1.11) are investigated in this section using the phase plane method. therefore the equation (1.11) is equivalent to the planar autonomous dynamical system (2.1)   ẋ = y ẏ = −(n− 1)ax2n−3 − b x3 the fixed point is defined by y = 0 and x = [ − b (n−1)a ] 1 2n . as one may see, for b a = −1, the critical point is real, but for b a = 1, the coordinate x may become complex. from (2.1) one may write int. j. anal. appl. 19 (6) (2021) 974 (2.2) dy dx = − (n− 1)ax2n + b yx3 the separation of variable leads to ydy = − [ (n− 1)ax2n−3 + b x3 ] dx by integration, one may obtain the integral curves given by (2.3) y2 = −ax2n−2 + b x2 + c this means, according to the equation (1.6) that the integration constant can be choosen as c = 0, so that the hamiltonian of the system can be written (2.4) h = ẋ2x2 + ax2n the hamiltonian (2.4) is time independent such that the equation (1.11) defines a conservative system from physical point of view. in this context, one can expect the existence of periodic solutions to the equation (1.11). now the objective is to calculate the exact and explicit general solutions of the equations (1.3), (1.12) and (1.13). 3. general solutions 3.1. general solution of (1.3). the reduced ermakov-pinney equation (1.3) has been the object of a high consideration in the literature since the ermakov-pinney equations can be reduced to (1.3) as mentioned before. in order to study its properties, the ermakov-pinney equation (1.3) for b = −1, in ( [5], [6]) is transformed into (3.1) 2zz̈ − 3ż2 + 4z4 = 0 using the point transformation (3.2) z = 1 x2 however, no explicit general solution is given. in [6] the equation (3.1) is examined from symmetry group point of view and the authors [6] arrived to examine (3.3) 2zz̈ − 3ż2 + bz4 = 0 int. j. anal. appl. 19 (6) (2021) 975 where b is the arbitrary constant. the authors [6] observe that the analysis of lie point symmetries is not adequate for (3.3) which requires, rather than nonlocal symmetry calculation. as the evaluation of nonlocal symmetries may be complicated, the authors [6] apply the jacobi last multiplier approach to find the solution of (3.1) in terms of time dependent integral. on the other hand, the equation (1.3) is also investigated in [14]. the authors [14] succeed to calculate a general solution of (1.3) where b = 1, in terms of the so-called c invariant related to the ermakov-pinney invariant. however, the point transformation of the equation (1.3) into (3.1) shows clearly that such an equation has two general solutions which differ only by a sign. the objective in this paragraph is to determine in a direct fashion the two general solutions and by exactly solving the auxiliary equation (3.1), in terms of arbitrary constants. 3.1.1. direct method. according to (1.10) the equation (1.3) may be reduced to the quadrature (3.4) ± ( t + k ) = − √ b b ( b a )∫ dφ which leads to (3.5) − √ b b ( b a ) φ = ± ( t + k ) using the change φ = x√ b = x √ b b , the equation (3.5) gives (3.6) x = ±a(t + k) from which, using the previous relation x2 = b−ax2, one may secure the general solutions of (1.3) as (3.7) x(t) = ± [ b a −a(t + k)2 ]1 2 where k is an integration constant. 3.1.2. solution using the auxiliary equation (3.1). let us consider the generalized sundman transformation theory introduced recently in the literature by akande and coworkers [15]. in fact the generalized sundman transformation is a powerfull change of variables which allows solving differential equations with a few mathematical manipulations. in the theory introduced by akande et al. [15] the oscillator harmonic equation (3.8) ü + ω2u = 0 where ω is a constant, ü means the second derivative with respect to τ, and (3.9) u(τ) = a0sin(ωτ + β) is transformed, under the change of variables int. j. anal. appl. 19 (6) (2021) 976 (3.10) u(τ) = f(t,z), dτ = g(t,z)dt where (3.11) f(t,z) = ∫ g(z)ldz, g(t,z) = eγϕ(z), and ∂f(t,z) ∂z 6= 0 to the second order differential equation (3.12) z̈ + ( l g′(z) g(z) −γϕ′(z) ) ż2 + ω2 exp(2γϕ(z)) ∫ g(z)ldz g(z)l = 0 where a0, β, l, γ are arbitrary parameters, g(z) 6= 0, and ϕ(z) are arbitrary functions of z, and prime denotes differentiation with respect to z. the application of ϕ(z) = ln(f(z)), leads to (3.13) z̈ + ( l g′(z) g(z) −γ f′(z) f(z) ) ż2 + ω2 f(z)2γ ∫ g(z)ldz g(z)l = 0 putting g(z) = z, and f(z) = z2, into (3.13), allows one to obtain (3.14) z̈ + (l− 2γ) ż2 z + ω2 l + 1 z4γ+1 = 0 which may reduce, choosing γ = −l = 1 2 , to (3.15) z̈ − 3 2 ż2 z + 2ω2z3 = 0 the equation (3.15) may be identified to (3.3) when 4ω2 = b and to (3.1) when ω2 = 1. so from the solutions of (3.15) one may deduce those of (1.3) with b = −1. in this context the transformation defined by (3.10) and (3.11) becomes u(τ) = ∫ z− 1 2 dz = 2z 1 2 that is, z = u 2 4 , and dτ = zdt , so that the equation (3.15) may be reduced to the quadrature determined by (3.16) a20 4 (t + k1) = ∫ dτ sin2(ωτ + β) where k1 is an integration constant. the change of variable s = ωτ + β, reduces the integral in (3.16) to int. j. anal. appl. 19 (6) (2021) 977 (3.17) j = 1 ω ∫ ds sin2(s) which yields (3.18) −cot(s) = ωa20 4 (t + k1) such that (3.19) ωτ + β = cot−1 [ − ωa20 4 (t + k1) ] substituting (3.19) into (3.9) yields the general solution to (3.15) as (3.20) z(t) = 1 4 a20sin 2 [ cot−1 ( − ωa20 4 (t + k1) )] using (3.20) one may deduce the solution of (3.1) in the form (3.21) z(t) = 1 4 a20sin 2 [ cot−1 ( − a20 4 (t + k1) )] therefore the solution of (1.3) where b = −1, becomes (3.22) x(t)2 = 4 a20sin 2 [ cot−1 ( − a 2 0 4 (t + k1) )] that is (3.23) x(t) = ± 2 a0sin [ cot−1 ( − a 2 0 4 (t + k1) )] knowing −cot−1(ν) = sin−1 ( 1√ 1+ν2 ) , for ν � 0, and cot−1(ν) = π−sin−1 ( 1√ 1+ν2 ) , for ν ≺ 0, then the general solutions (3.23) reduce to (3.24) x(t) = ± 2 √ 1 + a40 16 (t + k1)2 a0 now the problem to be solved is to calculate the general solution of the equations (1.12) and (1.13) to show analytically and explicitly the existence of periodic solutions to the equation (1.11). int. j. anal. appl. 19 (6) (2021) 978 3.2. exact periodic and complex-valued solutions. 3.2.1. periodic and complex-valued solutions of (1.12). the equation (1.12) is obtained when n = 3, that is when α = 6, from the equation (1.11). two cases may be investigated. periodic solution for reason of simplicity we choose a = b = 1. in this case the integral j becomes (3.25) j = − 1 3 ∫ dφ 3 √ (1 −φ2)2 which may be rewritten as [17] (3.26) j = − 1 3 ∫ φ 0 dξ 3 √ (1 − ξ2)2 = ±(t + c1) where 0 ≺ φ ≺∞, and c1 is an arbitrary parameter. by integration, (3.26) reduces to [17] (3.27) j = 1 3 3 2 4 √ 3 f(ψ,k) where ψ = cos−1 [√ 3−1− 3 √ φ2−1 √ 3+1+ 3 √ φ2−1 ] , and k2 = 2+ √ 3 4 . using (3.27), one may write (3.28) j = 1 2 4 √ 3 f(ψ,k) = ±(t + c1) from which one may get cosψ = cn[±2 4 √ 3(t + c1),k] that is (3.29) √ 3 − 1 − 3 √ φ2 − 1 √ 3 + 1 + 3 √ φ2 − 1 = cn [ ± 2 4 √ 3(t + c1),k ] in this situation φ may be written as (3.30) φ(t) = ±  1 + (√ 3 − 1 − ( √ 3 + 1)cn [ ± 2 4 √ 3(t + c1),k ] 1 + cn [ ± 2 4 √ 3(t + c1),k ] )3 1 2 knowing φ = x√ b = x, that is xα = b−x 2 a , which becomes xα = 1 −x2, as a = b = 1, the solution x takes the definitive form (3.31) x(t) = ( (1 − √ 3) + ( √ 3 + 1)cn [ ± 2 4 √ 3(t + c1),k ] 1 + cn [ ± 2 4 √ 3(t + c1),k ] )12 int. j. anal. appl. 19 (6) (2021) 979 complex-valued solution: a = −b = 1 this case corresponds to (3.32) j = i 3 ∫ (1 −φ2) −2 3 dφ = i 3 ∫ dφ 3 √ (1 −φ2)2 where i is the purely imaginary number. the equation (3.32) gives [17] (3.33) ∫ φ 0 dξ 3 √ (1 − ξ2)2 = ±3i(t + c2) where 0 ≺ φ ≺∞, and c2 is an arbitrary parameter. the evaluation of the integral in (3.33) leads to (3.34) 3 2 4 √ 3 f(ψ,k) = ±3i(t + c2) the equation (3.34) may be rewritten in the form (3.35) cosψ = cn [ ± 2i 4 √ 3(t + c2),k ] such that (3.36) √ 3 − 1 − 3 √ φ2 − 1 √ 3 + 1 + 3 √ φ2 − 1 = cn [ ± 2i 4 √ 3(t + c2),k ] from (3.36) one may get φ as (3.37) φ(t) = ± [ 1 + ( ( √ 3 − 1)cn [ ± 2 4 √ 3(t + c2),k ′ ] − ( √ 3 + 1) 1 + cn [ ± 2 4 √ 3(t + c2),k′ ] )3]1 2 with k′ = √ 1 −k2. in the present case, φ = x√−1 , which is rewritten as φ = −ix, that is x = iφ, so that (3.38) xα = −1 −x2 = φ2 − 1 from which one may secure the complex-valued solution x(t) in the form (3.39) x(t) = [ ( √ 3 − 1)cn [ ± 2 4 √ 3(t + c2),k ′ ] − ( √ 3 + 1) 1 + cn [ ± 2 4 √ 3(t + c2),k′ ] ]12 int. j. anal. appl. 19 (6) (2021) 980 3.2.2. periodic and complex-valued solutions of (1.13). the equation (1.13) corresponds to n = 4, which gives α = 8. two cases may be studied. periodic solution: a = b = 1 in this case, the integral j is written in this form (3.40) j = − 1 4 ∫ dφ 4 √ (1 −φ2)3 using (1.7) one may get the equation ( [16], [17]) (3.41) j = − 1 4 ∫ φ 0 dξ 4 √ (1 − ξ2)3 = ±(t + c3) where c3 is an arbitrary parameter. therefore, by integration, the equation (3.41) reduces to (3.42) √ 2f(ψ, √ 2 2 ) = ∓4(t + c3) where 0 ≺ φ � 1. from (3.42) one may ensure the following equation (3.43) cosψ = cn [ ∓ 2 √ 2(t + c3), √ 2 2 ] which is written in this form (3.44) 4 √ 1 −φ2 = cn [ ∓ 2 √ 2(t + c3), √ 2 2 ] from the equation (3.44), one may secure (3.45) φ2 = 1 − cn4 [ 2 √ 2(t + c3), √ 2 2 ] using the relation φ = x√ b , that is x = φ, the equation (3.45) is rewritten as (3.46) x2 = 1 − cn4 [ 2 √ 2(t + c3), √ 2 2 ] from which the relation xα = b−x 2 a , that is x8 = 1 −x2, gives (3.47) x(t) = [ cn ( 2 √ 2(t + c3), √ 2 2 )]12 complex-valued solution: a = −b = 1 this condition leads to the equation [17] int. j. anal. appl. 19 (6) (2021) 981 (3.48) ∫ x 0 ξdξ√ 1 + ξ8 = ±i(t + c4) where 0 ≺ x � 1. the integral in (3.48) is known as hyperelliptic integral and its evaluation gives [18] (3.49) 1 4 cn−1 [ 1 −x4 1 + x4 , √ 2 2 ] = ±i(t + c4) which may be rearranged in the form (3.50) 1 −x4 1 + x4 = cn [ ±4i(t + c4), √ 2 2 ] from (3.50) one may get (3.51) x4 = 1 − cn [ 4i(t + c4), √ 2 2 ] 1 + cn [ 4i(t + c4), √ 2 2 ] so that the general solution x(t) may take the expression (3.52) x(t) =  1 − cn [ 4i(t + c4), √ 2 2 ] 1 + cn [ 4i(t + c4), √ 2 2 ]   1 4 or in the definitive form (3.53) x(t) =  cn [ 4(t + c4), √ 2 2 ] − 1 cn [ 4(t + c4), √ 2 2 ] + 1   1 4 thus, the above shows that under the conditions that a b = 1, and 2 � n � 4, the explicit general solutions of (1.11) are periodic. in the sequel of this work, the related quadratic lienard type equations to the equation (1.13) is examined. 4. quadratic lienard type equations to determine the quadratic lienard type equations related to (1.13), consider the change of variable (4.1) ϑ = xp thus one may obtain (4.2) dx dt = 1 p ϑ̇ϑ 1−p p int. j. anal. appl. 19 (6) (2021) 982 and (4.3) d2x dt2 = 1 p ϑ̈ϑ 1−p p + 1 −p p2 ϑ̇2ϑ 1−2p p substituting (4.3) into (1.13) and taking into account (4.1), yields (4.4) ϑ̈ + 1 −p p ϑ̇2 ϑ + 3apϑ p+4 p + bpϑ p−4 p = 0 the solution (3.47) ensures the general periodic solution of (4.4) in the form (4.5) ϑ(t) = ( cn [ 2 √ 2(t + c3), √ 2 2 ])p 2 where a = b = 1. by application of p = 2, the equation (4.4) reduces to (4.6) ϑ̈− 1 2 ϑ̇2 ϑ + 6ϑ3 + 2 ϑ = 0 and its general solution becomes (4.7) ϑ(t) = cn [ 2 √ 2(t + c3), √ 2 2 ] which shows for the first time that the jacobi elliptic function cn [18] may be solution of second order autonomous non-polynomial differential equations. now a conclusion of this work may be addressed. 5. conclusion in this paper a general class of truly nonlinear oscillator equations is presented. the conditions of existence of periodic solutions are shown and periodic and explicit general solutions are examined. the general solutions of a well known ermakov-pinney type equation are also calculated. finally it has been shown that the jacobi elliptic function cn may be solution of second-order autonomous non-polynomial differential equations. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (6) (2021) 983 references [1] f. güngör, p.j. torres, lie point symmetry analysis of a second order differential equation with singularity, (2017). http://arxiv.org/abs/1612.07080. [2] m. d monsia, analysis of a purely nonlinear generalized isotonic oscillator equation, (2020). https://vixra.org/pdf/ 2010.0195v1.pdf. [3] m. d. monsia, on a nonlinear differential equation of lienard type, (2020). https://vixra.org/pdf/2011.0050v3.pdf. [4] e. pinney, the nonlinear differential equation y′′ + p(x)y + cy−3 = 0, proc. amer. math. soc. 1 (1950), 681. [5] m. euler, n. euler and p. euler, the riccati and ermakov-pinney hierarchies, j. nonlinear math. phys. 14(2) (2007), 290 – 310. [6] m. c. nucci and p.g. l. leach, jacobi’s last multiplier and the complete symmetry group of the ermakov-pinney equation, j. nonlinear math. phys. 12 (2) (2005), 305 – 320. [7] r. e. michens, truly nonlinear oscillators, world scientific, singapore, (2010). [8] m. gadella and l. p. lara, on the solutions of a nonlinear pseudo-oscillator equation, phys. scripta, 89 (2014), 105205. [9] v. r. gorder, continuous periodic of a nonlinear pseudo-oscillator equation in which the restoring force is inversely proportional to the dependent variable,phys. scripta, 90 (2015), 085208. [10] e. a. doutetien, a. r. yehossou, p. mallick, b. rath and m. d. monsia, on the general solutions of a nonlinear pseudooscillator equation and related quadratic lienard systems, pinsa. 86 (2020). https://doi.org/10.16943/ptinsa/2020/ 154987. [11] l. cveticanin, oscillator with fraction order restoring force, j. sound vibration, 320 (2009), 1064 – 1077. [12] y. wang, unboundedness in a duffing equation with polynomial potentials, j. differ. equations, 160 (2) (2000), 467 – 479. [13] m. d. monsia, the non-periodic solution of a truly nonlinear oscillator with power nonlinearity, (2020). https://vixra. org/pdf/2009.0174v1.pdf. [14] s. c. mancas and h. rosu, existence of periodic orbits in nonlinear oscillators of emden-fowler form, phys. lett. a, 380 (3) (2016), 422–428. [15] j. akande, d. k. k. adjäı and m. d. monsia, theory of exact trigonometric periodic solutions to quadratic lienard type equations, j. math. stat. 14 (1) (2018), 232 – 240. [16] i. s. gradshteyn and i. m. ryzhik, table of integrals, series, and products, academic press, ed. elsevier, california, 2007. [17] p.f. byrd and m. d. friedman, handbook of elliptic integrals for engineers and physicists, springer, berlin, 1954. [18] w. a. schwalm, lectures on selected topics in mathematical physics: elliptic functions and elliptic integrals, iop publishing, 2015. http://arxiv.org/abs/1612.07080 https://vixra.org/pdf/2010.0195v1.pdf https://vixra.org/pdf/2010.0195v1.pdf https://vixra.org/pdf/2011.0050v3.pdf https://doi.org/10.16943/ptinsa/2020/154987 https://doi.org/10.16943/ptinsa/2020/154987 https://vixra.org/pdf/2009.0174v1.pdf https://vixra.org/pdf/2009.0174v1.pdf 1. introduction 2. qualitative properties of solutions 3. general solutions 3.1. general solution of (1.3) 3.2. exact periodic and complex-valued solutions 4. quadratic lienard type equations 5. conclusion references international journal of analysis and applications volume 17, number 5 (2019), 752-770 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-752 direct product of finite fuzzy normal subrings over non-associative rings nasreen kausar1,∗, muhammad azam waqar2 1department of mathematics, university of agriculture fsd, pakistan 2department of school of business mangment, nfc iefr fsd, pakistan ∗corresponding author: kausar.nasreen57@gmail.com abstract. in this paper, we define the concept of direct product of finite fuzzy normal subrings over nonassociative and non-commutative rings (la-ring) and investigate the some fundamental properties of direct product of fuzzy normal subrings. 1. introduction a generalization of commutative semigroups has been established by kazim et al [10]. in ternary commutative law: abc = cba, they introduced the braces on the left side of this law and explored a new pseudo associative law (ab)c = (cb)a. this law (ab)c = (cb)a is called the left invertive law. a groupoid s is left almost semigroup (abbreviated as la-semigroup) if it satisfies the left invertive law: (ab)c = (cb)a. a groupoid s is medial (resp. paramedial) if (ab)(cd) = (ac)(bd) (resp. (ab)(cd) = (db)(ca)), in [5] (resp. [1]). in [10], an la-semigroup is medial, but in general an la-semigroup needs not to be paramedial. every la-semigroup with left identity is paramedial in [19] and also satisfies a(bc) = b(ac) and (ab)(cd) = (dc)(ba). received 2019-05-15; accepted 2019-06-24; published 2019-09-02. 2010 mathematics subject classification. 03f55, 08a72, 20n25. key words and phrases. direct product of fuzzy sets, direct product of fuzzy la-subrings, direct product of fuzzy normal la-subrings. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 752 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-752 int. j. anal. appl. 17 (5) (2019) 753 s. kamran [6], extended the concept of la-semigroup to the left almost group (la-group). an lasemigroup s is left almost group, if there exists left identity e ∈ s such that ea = a for all a ∈ s and for every a ∈ s, there exists b ∈ s such that ba = e. rehman et al [23], discussed the left almost ring (la-ring) of finitely nonzero functions which is a generalization of commutative semigroup ring. by a left almost ring, we mean a non-empty set r with at least two elements such that (r, +) is an la-group, (r, ·) is an la-semigroup, both left and right distributive laws hold. for example, from a commutative ring (r, +, ·) , we can always obtain an la-ring (r,⊕, ·) by defining for all a,b ∈ r, a⊕b = b−a and a ·b is same as in the ring. in fact an la-ring is a non-associative and non-commutative ring. a non-empty subset a of an la-ring r is an la-subring of r if a− b and ab ∈ a for all a,b ∈ a. a is called a left (resp. right) ideal of r, if (a, +) is an la-group and ra ⊆ a (resp. ar ⊆ a). a is called an ideal of r if it is both a left ideal and a right ideal of r. first time, the concept of fuzzy set introduced by zadeh in his classical paper [26]. this concept has provided a useful mathematical tool for describing the behavior of systems that are too complex to admit precise mathematical analysis by classical methods and tools. extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science, management science, expert systems, finite state machines, languages, robotics, coding theory and others. liu [14], introduced the concept of fuzzy subrings and fuzzy ideals of a ring. many authors have explored the theory of fuzzy rings (for example [2–4, 13, 15, 16, 25]). gupta et al [4], gave the idea of intrinsic product of fuzzy subsets of a ring. kuroki [13], characterized regular (intra-regular, both regular and intra-regular) rings in terms of fuzzy left (right, quasi, bi-) ideals. sherwood [24], introduced the concept of product of fuzzy subgroups. after this, further study on this concept continued by osman [17, 18] and ray [20]. zaid [27], gave the idea of normal fuzzy subgroups. kausar et al [21] initiated the idea of intuitionistic fuzzy normal subrings over a non-associative ring and also characterized the non-associative rings by their intuitionistic fuzzy bi-ideals in [7]. recently kausar [9] explored the direct product of finite intuitionistic anti fuzzy normal subrings over non-associative rings. in this paper we explore the concept of [9, 21] in finite fuzzy normal subrings over non-associative and noncommutative rings. recently kausar et al [11] studied the fuzzy ideals in la-rings and also kausar et al [12], investigated a study on intuitionistic fuzzy ideals with thresholds (α, β] in la-rings. in this paper, we give the concept of direct product of fuzzy normal la-subrings. in the first section, we investigate the some basic properties of fuzzy normal la-subrings of an la-ring r. in the second section, we provide the some elementary properties of direct product of fuzzy normal la-subrings of an la-ring r1 × r2. in the third section, we define the direct product of fuzzy subsets µ1,µ2, ...,µn of la-rings r1,r2, ...,rn, int. j. anal. appl. 17 (5) (2019) 754 respectively and examine the some fundamental properties of direct product of fuzzy normal la-subrings of an la-ring r1 ×r2 × ...×rn. specifically we will show that: (1) let a and b be two la-subrings of an la-ring r. then a∩b is an la-subring of r if and only if the characteristic function χz of z = a∩b is a fuzzy normal la-subring of r. (2) let x = a×b and y = c ×d be two la-subrings of an la-ring r1 ×r2. then x ∩y is an lasubring of r1 ×r2 if and only if the characteristic function χz of z = x ∩y is a fuzzy normal la-subring of r1 ×r2. (3) let a = a1×a2×...×an and b = b1×b2×...×bn be two la-subrings of an la-ring r1×r2×...×rn. then a∩b is an la-subring of r1×r2×...×rn if and only if the characteristic function χz of z = a∩b is a fuzzy normal la-subring of r1 ×r2 × ...×rn. 2. fuzzy normal la-subrings in this section, we investigate the some basic properties of fuzzy normal la-subrings of an la-ring r. by a fuzzy subset µ of an la-ring r, we mean a function µ : r → [0, 1] and the complement of µ is denoted by µ′, is a fuzzy subset of r defined by µ′(x) = 1 −µ(x) for all x ∈ r. a fuzzy subset µ of an la-ring r is said to be a fuzzy la-subring of r if µ(x−y) ≥ min{µ(x),µ(y)} and µ(xy) ≥ min{µ(x),µ(y)} for all x,y ∈ r. a fuzzy la-subring of an la-ring r is said to be a fuzzy normal la-subring of r if µ(xy) = µ(yx) for all x,y ∈ r. let a be a non-empty subset of an la-ring r. the characteristic function of a is denoted by χa and defined by χa : r → [0, 1] | x → χa (x) =   1 if x ∈ a0 if x /∈ a lemma 2.1. let a be a non-empty subset of an la-ring r. then a is an la-subring of r if and only if the characteristic function χa of a is a fuzzy normal la-subring of r. proof. let a be an la-subring of r and a,b ∈ r. if a,b ∈ a, then by definition of characteristic function χa(a) = 1 = χa(b). since a− b,ab ∈ a, a being an la-subring of r. this implies that χa(a− b) = 1 = 1 ∧ 1 = χa(a) ∧χa(b) and χa(ab) = 1 = 1 ∧ 1 = χa(a) ∧χa(b). int. j. anal. appl. 17 (5) (2019) 755 thus χa(a−b) ≥ min{χa(a),χa(b)} and χa(ab) ≥ min{χa(a),χa(b)}. since ab and ba ∈ a, so χa(ab) = 1 = χa(ba), i.e., χa(ab) = χa(ba). similarly we have χa(a− b) ≥ min{χa(a),χa(b)}, χa(ab) ≥ min{χa(a),χa(b)}, χa(ab) = χa(ba), when a,b /∈ a. hence the characteristic function χa of a is a fuzzy normal la-subring of r. conversely, suppose that the characteristic function χa of a is a fuzzy normal la-subring of r. let a,b ∈ a, then by definition χa(a) = 1 = χa(b). by the supposition χa(a− b) ≥ χa(a) ∧χa(b) = 1 ∧ 1 = 1 and χa(ab) ≥ χa(a) ∧χa(b) = 1 ∧ 1 = 1. thus χa(a− b) = 1 = χa(ab), i.e., a− b,ab ∈ a. hence a is an la-subring of r. � lemma 2.2. if a and b are two la-subrings of an la-ring r, then their intersection a ∩ b is also an la-subring of r. proof. straight forward. � theorem 2.1. let a and b be two la-subrings of an la-ring r. then a ∩ b is an la-subring of r if and only if the characteristic function χz of z = a∩b is a fuzzy normal la-subring of r. proof. let z = a ∩ b be an la-subring of r and a,b ∈ r. if a,b ∈ z = a ∩ b, then by definition of characteristic function χz(a) = 1 = χz(b). since a − b,ab ∈ a,b, a and b being la-subrings of r. this implies that χz(a− b) = 1 = 1 ∧ 1 = χz(a) ∧χz(b) and χz(ab) = 1 = 1 ∧ 1 = χz(a) ∧χz(b). thus χz(a− b) ≥ min{χz(a),χz(b)} and χz(ab) ≥ min{χz(a),χz(b)}. as ab and ba ∈ z, so χz(ab) = 1 = χz(ba), i.e., χz(ab) = χz(ba). similarly we have χz(a− b) ≥ min{χz(a),χz(b)}, χz(ab) ≥ min{χz(a),χz(b)}, χz(ab) = χz(ba), when a,b /∈ z. hence the characteristic function χz of z is a fuzzy normal la-subring of r. int. j. anal. appl. 17 (5) (2019) 756 conversely, assume that the characteristic function χz of z = a∩b is a fuzzy normal la-subring of r. let a,b ∈ z = a∩b, then by definition of characteristic function χz(a) = 1 = χz(b). by our assumption χz(a− b) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1 and χz(ab) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1. thus χz(a− b) = 1 = χz(ab), i.e., a− b and ab ∈ z. hence z is an la-subring of r. � corollary 2.1. let {ai}i∈i be a family of la-subrings of an la-ring r. then a = ∩ai is an la-subring of r if and only if the characteristic function χa of a = ∩ai is a fuzzy normal la-subring of r. lemma 2.3. if µ and γ are two fuzzy normal la-subrings of an la-ring r, then their intersection µ∩γ is also a fuzzy normal la-subring of r. proof. let µ and γ be two fuzzy normal la-subrings of an la-ring r. we have to show that β = µ∩γ is also a fuzzy normal la-subring of r. now β(z1 −z2) = (µ∩γ)(z1 −z2) = min{µ(z1 −z2),γ(z1 −z2)} ≥ {{µ(z1) ∧µ(z2)}∧{γ(z1) ∧γ(z2)}} = {µ(z1) ∧{µ(z2) ∧γ(z1)}∧γ(z2)} = {µ(z1) ∧{γ(z1) ∧µ(z2)}∧γ(z2)} = {{µ(z1) ∧γ(z1)}∧{µ(z2) ∧γ(z2)}} = min{(µ∩γ)(z1), (µ∩γ)(z2)} = min{β(z1),β(z2)}. ⇒ β(z1 −z2) ≥ min{β(z1),β(z2)}. similarly, we have β(z1 ◦z2) ≥ min{β(z1),β(z2)}. thus β is a fuzzy la-subring of an la-ring r. now β(z1 ◦z2) = (µ∩γ)(z1 ◦z2) = min{µ(z1 ◦z2),γ(z1 ◦z2)} = min{µ(z2 ◦z1),γ(z2 ◦z1)} = (µ∩γ)(z2 ◦z1) = β(z2 ◦z1). hence β = µ∩γ is a fuzzy normal la-subring of r. � int. j. anal. appl. 17 (5) (2019) 757 corollary 2.2. if {µi}i∈i is a family of fuzzy normal la-subrings of an la-ring r, then µ = ∩µi is also a fuzzy normal la-subring of r. 3. direct product of fuzzy normal la-subrings in this section, we define the direct product of fuzzy subsets µ1,µ2 of la-rings r1,r2, respectively and investigate the some elementary properties of direct product of fuzzy normal la-subrings of an la-ring r1 ×r2. let µ1,µ2 be fuzzy subsets of la-rings r1,r2, respectively. the direct product of fuzzy subsets µ1,µ2 is denoted by µ1 ×µ2 and defined as (µ1 ×µ2)(x1,x2) = min{µ1(x1),µ2(x2)}. a fuzzy subset µ1 ×µ2 of an la-ring r1 ×r2 is said to be a fuzzy la-subring of r1 ×r2 if (1) (µ1 ×µ2)(x−y) ≥ min{(µ1 ×µ2)(x), (µ1 ×µ2)(y)}, (2) (µ1 ×µ2)(xy) ≥ min{(µ1 ×µ2)(x), (µ1 ×µ2)(y)} for all x = (x1,x2) ,y = (y1,y2) ∈ r1 ×r2. a fuzzy la-subring of an la-ring r1 × r2 is said to be a fuzzy normal la-subring of r1 × r2 if (µ1 ×µ2)(xy) = (µ1 ×µ2)(yx) for all x = (x1,x2) ,y = (y1,y2) ∈ r1 ×r2. let a×b be a non-empty subset of an la-ring r1 ×r2. the characteristic function of a×b is denoted by χa×b and defined as χa×b : r1 ×r2 → [0, 1] | x = (x1,x2) → χa×b (x) =   1 if x ∈ a×b0 if x /∈ a×b lemma 3.1. [21, lemma 4.2] if a and b are la-subrings of la-rings r1 and r2, respectively, then a×b is an la-subring of an la-ring r1 ×r2 under the same operations defined as in r1 ×r2. proposition 3.1. let a and b be la-subrings of la-rings r1 and r2, respectively. then a × b is an la-subring of an la-ring r1 × r2 if and only if the characteristic function χz of z = a × b is a fuzzy normal la-subring of an la-ring r1 ×r2. proof. let z = a×b be an la-subring of r1×r2 and a = (a1,a2),b = (b1,b2) ∈ r1×r2. if a,b ∈ z = a×b, then by definition of characteristic function χz(a) = 1 = χz(b). since a − b and ab ∈ z, z being an lasubring of an la-ring r1 ×r2. this implies that χz(a− b) = 1 = 1 ∧ 1 = χz(a) ∧χz(b) and χz(ab) = 1 = 1 ∧ 1 = χz(a) ∧χz(b). int. j. anal. appl. 17 (5) (2019) 758 thus χz(a−b) ≥ min{χz(a),χz(b)} and χz(ab) ≥ min{χz(a),χz(b)}. since ab and ba ∈ z, so µχz (ab) = 1 = µχz (ba), i.e., χz(ab) = χz(ba). similarly we have χz(a− b) ≥ min{χz(a),χz(b)}, χz(ab) ≥ min{χz(a),χz(b)}, χz(ab) = χz(ba), when a,b /∈ z. hence the characteristic function χz of z = a×b is a fuzzy normal la-subring of r1×r2. conversely, suppose that the characteristic function χz of z = a × b is a fuzzy normal la-subring of r1 ×r2. we have to show that z = a×b is an la-subring of r1 ×r2. let a,b ∈ z, where a = (a1,a2) and b = (b1,b2) , a1,b1 ∈ a, a2,b2 ∈ b. by definition, we have χz(a) = 1 = χz(b). by our supposition χz(a− b) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1 and χz(ab) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1. thus χz(a−b) = 1 = χz(ab), i.e., a−b and ab ∈ z. hence z = a×b is an la-subring of r1 ×r2. � lemma 3.2. if x = a × b and y = c × d are two la-subrings of an la-ring r1 × r2, then their intersection x ∩y is also an la-subring of r1 ×r2. proof. straight forward. � theorem 3.1. let x = a×b and y = c ×d be two la-subrings of an la-ring r1 ×r2. then x ∩y is an la-subring of r1 × r2 if and only if the characteristic function χz of z = x ∩ y is a fuzzy normal la-subring of r1 ×r2. proof. let z = x ∩y be an la-subring of an la-ring r1 ×r2 and a = (a1,a2),b = (b1,b2) ∈ r1 ×r2. if a,b ∈ z = x∩y, then by definition of characteristic function χz(a) = 1 = χz(b). since a−b and ab ∈ z, z being an la-subring of r1 ×r2. this implies that χz(a− b) = 1 = 1 ∧ 1 = χz(a) ∧χz(b) and χz(ab) = 1 = 1 ∧ 1 = χz(a) ∧χz(b). thus χz(a − b) ≥ min{χz(a),χz(b)} and χz(ab) ≥ min{χz(a),χz(b)}. since ab and ba ∈ z, then by definition χz(ab) = 1 = χz(ba), i.e., χz(ab) = χz(ba). similarly we have χz(a− b) ≥ min{χz(a),χz(b)}, χz(ab) ≥ min{χz(a),χz(b)}, χz(ab) = χz(ba), int. j. anal. appl. 17 (5) (2019) 759 when a,b /∈ z. hence the characteristic function χz of z is a fuzzy normal la-subring of r1 ×r2. conversely, assume that the characteristic function χz of z = x ∩y is a fuzzy normal la-subring of an la-ring r1 ×r2. let a,b ∈ z = x ∩y, then by definition χz(a) = 1 = χz(b). by our assumption χz(a− b) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1 and χz(ab) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1. thus χz(a−b) = 1 = χz(ab), i.e., a−b and ab ∈ z. hence z is an la-subring of an la-ring r1×r2. � corollary 3.1. let {ci}i∈i = {ai ×bi}i∈i be a family of la-subrings of an la-ring r1 × r2. then c = ∩ci is an la-subring of r1 ×r2 if and only if the characteristic function χc of c = ∩ci is a fuzzy normal la-subring of r1 ×r2. lemma 3.3. if µ and γ are fuzzy normal la-subrings of la-rings r1 and r2, respectively, then µ×γ is a fuzzy normal la-subring of an la-ring r1 ×r2. proof. let µ and γ be fuzzy normal la-subrings of la-ring r1 and r2, respectively. we have to show that β = µ×γ is a fuzzy normal la-subring of an la-ring r1 ×r2. now β((a,b) − (c,d)) = (µ×γ)(a− c,b−d) = min{µ(a− c),γ(b−d)} = µ(a− c) ∧γ(b−d) ≥ {µ(a) ∧µ(c)}∧{γ(b) ∧γ(d)} = µ(a) ∧{µ(c) ∧γ(b)}∧γ(d) = µ(a) ∧{γ(b) ∧µ(c)}∧γ(d) = {µ(a) ∧γ(b)}∧{µ(c) ∧γ(d)} = min{(µ×γ)(a,b), (µ×γ)(c,d)} = min{β(a,b),β(c,d)}. ⇒ β((a,b) − (c,d)) ≥ min{β(a,b),β(c,d)}. int. j. anal. appl. 17 (5) (2019) 760 similarly, we have β((a,b) ◦ (c,d)) ≥ min{β(a,b),β(c,d)}. thus µ×γ is a fuzzy la-subring of r1 ×r2. now β((a,b) ◦ (c,d)) = (µ×γ)(ac,bd) = min{µ(ac),γ(bd)} = min{µ(ca),γ(db)} = (µ×γ)(ca,db) = β((c,d) ◦ (a,b)). hence µ×γ is a fuzzy normal la-subring of r1 ×r2. � proposition 3.2. if µ = µ1×µ2 and γ = γ1×γ2 are two fuzzy normal la-subrings of an la-ring r1×r2, then their intersection β = µ∩γ is also a fuzzy normal la-subring of r1 ×r2. proof. let µ = µ1 ×µ2 and γ = γ1 ×γ2 be two fuzzy normal la-subrings of an la-ring r1 ×r2. we have to show that β = µ∩γ is also a fuzzy normal la-subring of r1 ×r2. now β((z1,z2) − (z3,z4)) = (µ∩γ)((z1,z2) − (z3,z4)) = min{µ((z1,z2) − (z3,z4)),γ((z1,z2) − (z3,z4))} ≥ {{µ(z1,z2) ∧µ(z3,z4)}∧{γ(z1,z2) ∧γ(z3,z4)}} = {µ(z1,z2) ∧{µ(z3,z4) ∧γ(z1,z2)}∧γ(z3,z4)} = {µ(z1,z2) ∧{γ(z1,z2) ∧µ(z3,z4)}∧γ(z3,z4)} = {{µ(z1,z2) ∧γ(z1,z2)}∧{µ(z3,z4) ∧γ(z3,z4)}} = min{(µ∩γ)(z1,z2), (µ∩γ)(z3,z4)} = min{β(z1,z2),β(z3,z4)}. ⇒ β((z1,z2) − (z3,z4)) ≥ min{β(z1,z2),β(z3,z4)}. similarly, we have β((z1,z2) ◦ (z3,z4)) ≥ min{β(z1,z2),β(z3,z4)}. thus β = µ∩γ is a fuzzy la-subring of an la-ring r1 ×r2. now β((z1,z2) ◦ (z3,z4)) = (µ∩γ)((z1,z2) ◦ (z3,z4)) = min{µ((z1,z2) ◦ (z3,z4)),γ((z1,z2) ◦ (z3,z4))} = min{µ((z3,z4) ◦ (z1,z2)),γ((z3,z4) ◦ (z1,z2))} = (µ∩γ)((z3,z4) ◦ (z1,z2)) = β((z3,z4) ◦ (z1,z2)). int. j. anal. appl. 17 (5) (2019) 761 hence β = µ∩γ is a fuzzy normal la-subring of an la-ring r1 ×r2. � corollary 3.2. if {βi}i∈i = {µi ×γi}i∈i is a family of fuzzy normal la-subrings of an la-ring r1 ×r2, then β = ∩βi is also a fuzzy normal la-subring of r1 ×r2. theorem 3.2. if µ = µ1 ×µ2 and γ = γ1 ×γ2 are fuzzy normal la-subrings of la-rings r′ = r1 ×r2 and r′′ = r3 × r4, respectively, then β = µ × γ is a fuzzy normal la-subring of an la-ring r′ × r′′ = (r1 ×r2) × (r3 ×r4). proof. let µ = µ1 × µ2 and γ = γ1 × γ2 be fuzzy normal la-subrings of la-rings r′ = r1 × r2 and r′′ = r3 ×r4, respectively. we have to show that β = µ×γ is a fuzzy normal la-subring of an la-ring r′ ×r′′. now β(((z1,z2), (z3,z4)) − ((z5,z6), (z7,z8))) = µ×γ(((z1,z2), (z3,z4)) − ((z5,z6), (z7,z8))) = µ×γ(((z1,z2) − (z5,z6)), ((z3,z4) − (z7,z8))) = min{µ((z1,z2) − (z5,z6)),γ((z3,z4) − (z7,z8))} ≥ min{(µ(z1,z2) ∧µ(z5,z6)), (γ(z3,z4) ∧γ(z7,z8))} = ((µ(z1,z2) ∧µ(z5,z6)) ∧ (γ(z3,z4) ∧γ(z7,z8))) = ((µ(z1,z2) ∧γ(z3,z4)) ∧ (µ(z5,z6) ∧γ(z7,z8))) = min{(µ(z1,z2) ∧γ(z3,z4)), (µ(z5,z6) ∧γ(z7,z8))} = min{µ×γ((z1,z2), (z3,z4)),µ×γ((z5,z6), (z7,z8))} = min{β((z1,z2), (z3,z4)),β((z5,z6), (z7,z8))}, . similarly, we have β(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) ≥ min{β((z1,z2), (z3,z4)),β((z5,z6), (z7,z8))}. int. j. anal. appl. 17 (5) (2019) 762 thus β = µ∩γ is a fuzzy la-subring of an la-ring r′×r′′. now β(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) = µ×γ(((z1,z2), (z3,z4)) ◦ ((z5,z6), (z7,z8))) = µ×γ(((z1,z2) ◦ (z5,z6)), ((z3,z4) ◦ (z7,z8))) = min{µ((z1,z2) ◦ (z5,z6)),γ((z3,z4) ◦ (z7,z8))} = min{µ((z5,z6) ◦ (z1,z2)),γ((z7,z8) ◦ (z3,z4))} = µ×γ(((z5,z6) ◦ (z1,z2)), ((z7,z8) ◦ (z3,z4))) = µ×γ(((z5,z6), (z7,z8)) ◦ ((z1,z2), (z3,z4))) = β(((z5,z6), (z7,z8)) ◦ ((z1,z2), (z3,z4))). hence β = µ∩γ is a fuzzy normal la-subring of an la-ring r′×r′′. � lemma 3.4. let µ and γ be fuzzy subsets of la-rings r1 and r2 with left identities e1 and e2, respectively. if µ×γ is a fuzzy la-subring of an la-ring r1 ×r2, then at least one of the following two statements must hold. (1) µ (x) ≤ γ (e2) , for all x ∈ r1. (2) µ (x) ≤ γ (e1) , for all x ∈ r2. proof. let µ×γ be a fuzzy la-subring of r1 ×r2. by contraposition, suppose that none of the statements (1) and (2) holds. then we can find a and b in r1 and r2, respectively such that µ (a) ≥ γ (e2) and µ (b) ≥ γ (e1) . thus, we have (µ×γ)(a,b) = min{µ(a),γ(b)} ≥ min{µ(e1),γ(e2)} = (µ×γ)(e1,e2). so µ×γ is not a fuzzy la-subring of r1×r2. hence either µ (x) ≤ γ (e2) , for all x ∈ r1 or µ (x) ≤ γ(e1) for all x ∈ r2. � lemma 3.5. let µ and γ be fuzzy subsets of la-rings r1 and r2 with left identities e1 and e2, respectively and µ×γ is a fuzzy normal la-subring of an la-ring r1 ×r2, then the following conditions are true. (1) if µ (x) ≤ γ(e2), for all x ∈ r1, then µ is a fuzzy normal la-subring of r1. (2) if µ (x) ≤ γ(e1), for all x ∈ r2, then γ is a fuzzy normal la-subring of r2. int. j. anal. appl. 17 (5) (2019) 763 proof. (1) let µ (x) ≤ γ (e2) for all x ∈ r1, and y ∈ r1. we have to show that µ is a fuzzy normal la-subring of r1. now µ(x−y) = µ(x + (−y)) = min{µ(x + (−y)),γ(e2 + (−e2))} = (µ×γ)(x + (−y),e2 + (−e2)) = (µ×γ)((x,e2) + (−y,−e2)) = (µ×γ)((x,e2) − (y,e2)) ≥ (µ×γ)(x,e2) ∧ (µ×γ)(y,e2) = min{min{µ(x),γ(e2)},min{µ(y),γ(e2)}} = µ(x) ∧µ(y). and µ(xy) = min{µ(xy),γ(e2e2)} = (µ×γ)(xy,e2e2) = (µ×γ)((x,e2) ◦ (y,e2)) ≥ (µ×γ)(x,e2) ∧ (µ×γ)(y,e2) = min{min{µ(x),γ(e2)},min{µ(y),γ(e2)}} = µ(x) ∧µ(y). thus µ is a fuzzy la-subring of r1. now µ(xy) = min{µ(xy),γ(e2e2)} = (µ×γ) (xy,e2e2) = (µ×γ) ((x,e2) ◦ (y,e2)) = (µ×γ) ((y,e2) ◦ (x,e2)) = (µ×γ)(yx,e2e2) = min{µ(yx),γ(e2e2)} = µ(yx). hence µ is a fuzzy normal la-subring of r1. (2) is same as (1) . � int. j. anal. appl. 17 (5) (2019) 764 4. direct product of finite fuzzy normal la-subrings in this section, we define the direct product of fuzzy subsets µ1,µ2, ...,µn, of la-rings r1,r2, ...,rn, respectively and examine the some fundamental properties of direct product of fuzzy normal la-subrings of an la-ring r1 ×r2 × ...×rn. let µ1,µ2, ...,µn be fuzzy subsets of la-rings r1,r2, ...,rn, respectively. the direct product of fuzzy subsets µ1,µ2, ...,µn is denoted by µ1 × µ2 × ... × µn and defined by (µ1 × µ2 × ... × µn)(x1,x2, ...,xn) = min{µ1(x1),µ2 (x2) , ...,µn(xn)}. a fuzzy subset µ1 × µ2 × ... × µn of an la-ring r1 × r2 × ... × rn is said to be a fuzzy la-subring of r1 ×r2 × ...×rn if (1) (µ1 ×µ2 × ...×µn)(x−y) ≥ min{(µ1 ×µ2 × ...×µn)(x), (µ1 ×µ2 × ...×µn)(y)}, (2) (µ1×µ2×...×µn)(xy) ≥ min{(µ1×µ2×...×µn)(x), (µ1×µ2×...×µn)(y)} for all x = (x1,x2, ...,xn) ,y = (y1,y2, ...,yn) ∈ r1 ×r2 × ...×rn. a fuzzy la-subring of an la-ring r1×r2×...×rn is said to be a fuzzy normal la-subring of r1×r2× ...×rn if (µ1 ×µ2 × ...×µn)(xy) = (µ1 ×µ2 × ...×µn)(yx) for all x = (x1,x2, ...,xn) ,y = (y1,y2, ...,yn) ∈ r1 ×r2 × ...×rn. let a1 ×a2 × ...×an be a non-empty subset of an la-ring r = r1 ×r2 × ...×rn. the characteristic function of a = a1 ×a2 × ...×an is denoted by χa and defined as χa : r → [0, 1] | x = (x1,x2, ...,xn) → χa (x) =   1 if x ∈ a0 if x /∈ a lemma 4.1. if a1,a2, ...,an are la-subrings of la-rings r1,r2, ...,rn, respectively, then a1×a2×...×an is an la-subring of an la-ring r1 ×r2 × ...×rn under the same operations defined as in [21]. proof. straight forward. � proposition 4.1. let a1,a2, ...,an be la-subrings of la-rings r1,r2, ...,rn, respectively. then a1 × a2 × ...×an is an la-subring of an la-ring r1 ×r2 × ...×rn if and only if the characteristic function χa of a = a1 ×a2 × ...×an is a fuzzy normal la-subring of r1 ×r2 × ...×rn. proof. let a = a1 × a2 × ... × an be an la-subring of r1 × r2 × ... × rn and a = (a1,a2, ...,an),b = (b1,b2, ...,bn) ∈ r1 × r2 × ... × rn. if a,b ∈ a = a1 × a2 × ... × an, then by definition of characteristic function χa(a) = 1 = χa(b). since a − b and ab ∈ a, a being an la-subring of r1 × r2 × ... × rn. this implies that χa(a− b) = 1 = 1 ∧ 1 = χa(a) ∧χa(b) and χa(ab) = 1 = 1 ∧ 1 = χa(a) ∧χa(b). int. j. anal. appl. 17 (5) (2019) 765 thus χa(a − b) ≥ min{χa(a),χa(b)} and χa(ab) ≥ min{χa(a),χa(b)}. since ab and ba ∈ a, then by definition χa(ab) = 1 = χa(ba), i.e., µχa (ab) = µχa (ba). similarly we have χa(a− b) ≥ min{χa(a),χa(b)}, χa(ab) ≥ min{χa(a),χa(b)}, χa(ab) = χa(ba), when a,b /∈ a. hence the characteristic function χa of a = a1×a2×...×an is a fuzzy normal la-subring of r1 ×r2 × ...×rn. conversely, suppose that the characteristic function χa of a = a1 ×a2 × ...×an is a fuzzy normal lasubring of r1×r2×...×rn. we have to show that a = a1×a2×...×an is an la-subring of r1×r2×...×rn. let a,b ∈ a, where a = (a1,a, ...,an) and b = (b1,b2, ...,bn) , then by definition χa(a) = 1 = χa(b). by our supposition χa(a− b) ≥ χa(a) ∧χa(b) = 1 ∧ 1 = 1 and χa(ab) ≥ χa(a) ∧χa(b) = 1 ∧ 1 = 1. thus χa(a− b) = 1 = χa(ab), i.e., a− b and ab ∈ a. hence a = a1 ×a2 × ...×an is an la-subring of an la-ring r1 ×r2 × ...×rn. � lemma 4.2. if a = a1 × a2 × ... × an and b = b1 × b2 × ... × bn are two la-subrings of an la-ring r1 ×r2 × ...×rn, then their intersection a∩b is also an la-subring of r1 ×r2 × ...×rn. proof. straight forward. � theorem 4.1. let a = a1 ×a2 × ...×an and b = b1 ×b2 × ...×bn be two la-subrings of an la-ring r1 × r2 × ... × rn. then a ∩ b is an la-subring of r1 × r2 × ... × rn if and only if the characteristic function χz of z = a∩b is a fuzzy normal la-subring of r1 ×r2 × ...×rn. proof. let z = a∩b be an la-subring of r1 ×r2 × ...×rn and a = (a1,a2, ...,an),b = (b1,b1, ...,bn) ∈ r1 × r2 × ... × rn. if a,b ∈ z = a ∩ b, then by definition of characteristic function χz(a) = 1 = χz(b). since a− b and ab ∈ z, z being an la-subring. this implies that χz(a− b) = 1 = 1 ∧ 1 = χz(a) ∧χz(b) and χz(ab) = 1 = 1 ∧ 1 = χz(a) ∧χz(b). int. j. anal. appl. 17 (5) (2019) 766 thus χz(a − b) ≥ min{χz(a),χz(b)} and χz(ab) ≥ min{χz(a),χz(b)}. as ab and ba ∈ z, then by definition χz(ab) = 1 = χz(ba), i.e., χz(ab) = χz(ba). similarly, we have χz(a− b) ≥ min{χz(a),χz(b)}, χz(ab) ≥ min{χz(a),χz(b)}, χz(ab) = χz(ba), when a,b /∈ z. hence the characteristic function χz of z is a fuzzy normal la-subring of r1×r2×...×rn. conversely, assume that the characteristic function χz of z = a ∩ b is a fuzzy normal la-subring of r1 ×r2 × ...×rn. let a,b ∈ z = a∩b, then by definition χz(a) = 1 = χz(b). by our assumption χz(a− b) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1 and χz(ab) ≥ χz(a) ∧χz(b) = 1 ∧ 1 = 1. thus χz(a−b) = 1 = χz(ab), i.e., a−b and ab ∈ z. hence z is an la-subring of r1 ×r2 × ...×rn. � corollary 4.1. let {ai}i∈i = {ai1 ×ai2 × ...×ain}i∈i be a family of la-subrings of an la-ring r1 × r2 × ...×rn, then a = ∩ai is an la-subring of r1 ×r2 × ...×rn if and only if the characteristic function χa of a = ∩ai is a fuzzy normal la-subring of r1 ×r2 × ...×rn. theorem 4.2. if µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn are two fuzzy normal la-subrings of an la-ring r1 × r2 × ... × rn, then their intersection β = µ ∩ γ is also a fuzzy normal la-subring of r1 ×r2 × ...×rn. proof. let µ = µ1 ×µ2 × ...×µn and γ = γ1 ×γ2 × ...×γn be two fuzzy normal la-subrings of an la-ring r1 ×r2 × ...×rn. we have to show that β = µ∩γ is also a fuzzy normal la-subring of r1 ×r2 × ...×rn. let z = (z1,z2, ...,zn) and w = (w1,w2, ...,wn) ∈ r1 ×r2 × ...×rn. now β(z −w) = (µ∩γ)(z −w) = min{µ(z −w),γ(z −w)} ≥ {{µ(z) ∧µ(w)}∧{γ(z) ∧γ(w)}} = {µ(z) ∧{µ(w) ∧γ(z)}∧γ(w)} = {µ(z) ∧{γ(z) ∧µ(w)}∧γ(w)} = {{µ(z) ∧γ(z)}∧{µ(w) ∧γ(w)}} = min{(µ∩γ)(z), (µ∩γ)(w)} = min{β(z),β(w)}. thus β((z1,z2, ...,zn) − (w1,w2, ...wn)) ≥ min{β(z1,z2, ...,zn),β(w1,w2, ...wn)}. int. j. anal. appl. 17 (5) (2019) 767 similarly, we have β((z1,z2, ...,zn) ◦ (w1,w2, ...wn)) ≥ min{β(z1,z2, ...,zn),β(w1,w2, ...wn)}. therefore β = µ∩γ is a fuzzy la-subring of an la-ring r1 ×r2 × ...×rn. now β((z1,z2, ...,zn) ◦ (w1,w2, ...,wn)) = (µ∩γ)(z1w1,z2w2, ...,znwn) = min{µ(z1w1,z2w2, ...,znwn),γ(z1w1,z2w2, ...,znwn)} = min{µ(w1z1,w2z2, ...,wnzn),γ(w1z1,w2z2, ...,wnzn)} = (µ∩γ)(w1z1,w2z2, ...,wnzn) = β((w1,w2, ...,wn) ◦ (z1,z2, ...,zn)). hence β = µ∩γ is a fuzzy normal la-subring of an la-ring r1 ×r2 × ...×rn. � corollary 4.2. if {µi}i∈i = {µi1 ×µi2 × ...×µin}i∈i is a family of fuzzy normal la-subrings of an la-ring r1 ×r2 × ...×rn, then µ = ∩µi is also a fuzzy normal la-subring of r1 ×r2 × ...×rn. proposition 4.2. let µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn be fuzzy subsets of la-rings r = r1 × r2 × ... × rn and r′ = r′1 × r′2 × ... × r′n with left identities e = (e1,e2, ...,en) and e′ = (e1′,e2′, ...,en′), respectively. if µ×γ is a fuzzy la-subring of an la-ring r×r′. then at least one of the following two statements must hold. (1) µ (x) ≤ γ (e′) , for all x ∈ r. (2) µ (x) ≤ γ (e) , for all x ∈ r′. proof. let µ×γ be a fuzzy la-subring of r×r′. by contraposition, suppose that none of the statements (1) and (2) holds. then we can find a and b in r and r′, respectively such that µ (a) ≥ γ (e′) and µ (b) ≥ γ (e) . thus, we have µ×γ(a,b) = min{µ(a),γ(b)} ≥ min{µ(e),γ(e′)} = (µ×γ)(e,e′). therefore µ × γ is not a fuzzy la-subring of r × r′. hence either µ (x) ≤ γ (e′) , for all x ∈ r or µ (x) ≤ γ(e) for all x ∈ r′. � int. j. anal. appl. 17 (5) (2019) 768 proposition 4.3. let µ = µ1 × µ2 × ... × µn and γ = γ1 × γ2 × ... × γn be fuzzy subsets of la-rings r = r1 × r2 × ... × rn and r′ = r′1 × r′2 × ... × r′n with left identities e = (e1,e2, ...,en) and e′ = (e1′,e2′, ...,en′), respectively and µ×γ is a fuzzy normal la-subring of an la-ring r×r′. then the following conditions are true. (1) if µ (x) ≤ γ (e′) , for all x ∈ r, then µ is a fuzzy normal la-subring of r. (2) if µ (x) ≤ γ(e), for all x ∈ r′, then γ is a fuzzy normal la-subring of r′. proof. (1) let µ (x) ≤ γ (e′) , for all x ∈ r, and y ∈ r. we have to show that µ is a fuzzy normal la-subring of r. now µ(x−y) = µ(x + (−y)) = min{µ(x + (−y)),γ(e′ + (−e′))} = (µ×γ)(x + (−y),e′ + (−e′)) = (µ×γ)((x,e′) + (−y,−e′)) = (µ×γ)((x,e′) − (y,e′)) ≥ (µ×γ)(x,e′) ∧ (µ×γ)(y,e′) = min{min{µ(x),γ(e′)},min{µ(y),γ(e′)}} = µ(x) ∧µ(y). and µ(xy) = min{µ(xy),γ(e′e′)} = (µ×γ)(xy,e′e′) = (µ×γ)((x,e′) ◦ (y,e′)) ≥ (µ×γ)(x,e′) ∧ (µ×γ)(y,e′) = min{min{µ(x),γ(e′)},min{µ(y),γ(e′)}} = µ(x) ∧µ(y). int. j. anal. appl. 17 (5) (2019) 769 thus µ is a fuzzy la-subring of r. now µ(xy) = min{µ(xy),γ(e′e′)} = (µ×γ) (xy,e′e′) = (µ×γ) ((x,e′) ◦ (y,e′)) = (µ×γ) ((y,e′) ◦ (x,e′)) = (µ×γ)(yx,e′e′) = min{µ(yx),γ(e′e′)} = µ(yx). hence µ is a fuzzy normal la-subring of r. (2) is same as (1) . � references [1] r. j. cho, j. jezek and t. kepka, paramedial groupoids, czechoslovak math. j., 49 (1999) 277-290. [2] k. a. dib, n. galhum and a. a. m. hassan, fuzzy rings and fuzzy ideals, fuzzy math., 4 (1996) 245-261. [3] v. n. dixit, r. kumar and n. ajmal, fuzzy ideals and fuzzy prime ideals of a ring, fuzzy set syst., 44 (1991) 127-138. [4] k. c. gupta and m. k. kantroo, the intrinsic product of fuzzy subsets of a ring, fuzzy set syst., 57 (1993) 103-110. [5] j. jezek and t. kepka, medial groupoids, rozpravy csav rada mat. a prir. ved., 93/2, 1983, 93 pp. [6] m. s. kamran, conditions for la-semigroups to resemble associative structures, ph.d. thesis, quaid-i-azam university, islamabad, 1993. [7] n. kausar, m. waqar, characterizations of non-associative rings by their intuitionistic fuzzy bi-ideals, eur. j. pure appl. math. 12 (2019), 226-250. [8] n. kausar, characterizations of non-associative ordered semigroups by the properties of their fuzzy ideals with thresholds (α,β], prikl. diskr. mat. 43 (2019), 37-59. [9] n. kausar, direct product of finite intuitionistic fuzzy normal subrings over non-associative rings, eur. j. pure appl. math., 12 (2019), 622-648. [10] m. a. kazim and m. naseeruddin, on almost semigroups, alig. bull. math., 2 (1972), 1-7. [11] n. kausar, b. islam, m. javaid, s, amjad, u. ijaz, characterizations of non-associative rings by the properties of their fuzzy ideals, j. taibah univ. sci. 13 (2019), 820-833. [12] n. kausar, b. islam, s. amjad, m. waqar, intuitionistics fuzzy ideals with thresholds(,] in la-rings, eur. j. pure appli. math. 12 (2019) 906-943. [13] n. kuroki, regular fuzzy duo rings, inform. sci., 94 (1996), 119-139. [14] w. j. liu, fuzzy invariant subgroups and ideals, fuzzy sets syst., 8 (1982), 133-139. [15] t. k. mukherjee and m. k. sen, on fuzzy ideals of a ring 1, fuzzy sets syst., 21 (1987), 99-104. [16] t. k. mukherjee and m. k. sen, prime fuzzy ideals in rings, fuzzy sets syst., 32 (1989), 337-341. [17] m. t. a. osman, on the direct product of fuzzy subgroups, fuzzy sets syst., 12 (1984), 87-91. [18] m. t. a. osman, on some product of fuzzy subgroups, fuzzy sets syst., 24 (1987), 79-86. int. j. anal. appl. 17 (5) (2019) 770 [19] p. v. protic and n. stevanovic, ag-test and some general properties of abel-grassmann’s groupoids, pure math. appl., 6 (1995) 371-383. [20] a. k. ray, product of fuzzy subgroups, fuzzy sets syst., 105 (1999), 181-183. [21] t. shah, n. kausar and i. rehman, intuitionistic fuzzy normal subrings over a non-associative ring, an. st. univ. ovidius constanta, 1 (2012) 369-386. [22] t. shah, n. kausar, characterizations of non-associative ordered semigroups by their fuzzy bi-ideals, theor. comput. sci. 529 (2014), 96-110. [23] t. shah and i. rehman, on la-rings of finitely non-zero functions, int. j. contemp. math. sci., 5 (2010) 209-222. [24] h. sherwood, product of fuzzy subgroups, fuzzy sets syst., 11 (1983) 65-77. [25] u. m. swamy and k. l. n. swamy, fuzzy prime ideals of rings, j. math. anal. appl., 134 (1988) 94-103. [26] l. a. zadeh, fuzzy sets, inform. control, 8 (1965) 338-353. [27] s. a. zaid, on normal fuzzy subgroups, j. fac. educ. ain shams univ. cairo, 13 (1988), 115-125. 1. introduction 2. fuzzy normal la-subrings 3. direct product of fuzzy normal la-subrings 4. direct product of finite fuzzy normal la-subrings references international journal of analysis and applications volume 19, number 6 (2021), 826-835 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-826 on new subclass of harmonic univalent functions associated with modified q-operator shujaat ali shah1,∗, asghar ali maitlo1, muhammad afzal soomro1, khalida inayat noor2 1department of mathematics and statistics, quaid-i-awam university of engineering, science and technology, nawabshah, 67480 sindh, pakistan 2 department of mathematics, comsats university islamabad, islamabad, 45550, pakistan ∗corresponding author: shahglike@yahoo.com abstract. in this article, we introduce new subclasses of harmonic univalent functions associated with the q-difference operator. the modified q-srivastava-attiya operator is defined and certain applications of this operator are discussed. we investigate the sufficient condition, distortion result, extreme points and invariance of convex combination of the elements of the subclasses. 1. introduction a real-valued function u (x,y) is said to be harmonic in a domain d if it has continuous second order partial derivatives in d and satisfies uxx + uyy = 0. we say that a continuous f : ω (⊂ c) → c defined by f (z) = u (x,y) +iv (x,y) is harmonic if both u (x,y) and v (x,y) are real harmonic in ω. it is observed that every harmonic function f in any simply connected domain ω can be written as f (z) = h(z) +g(z), where h and g are analytic in ω, and are called, respectively, the analytic and co-analytic parts of f. received august 25th, 2021; accepted october 11th, 2021; published october 28th, 2021. 2010 mathematics subject classification. 30c45, 30c55. key words and phrases. univalent functions; harmonic function; q-difference operator; the q-srivastava-attiya operator. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 826 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-826 int. j. anal. appl. 19 (6) (2021) 827 we denote by h the class of complex-valued harmonic functions f = h + g defined in the open unit disc u = {z : |z| < 1} and normalized by h(0) = g(0) = h′(0) − 1 = 0. such mappings have the following power series representation (1.1) f(z) = z + ∞∑ n=2 anz n + ∞∑ n=2 bnzn, |b1| < 1. it is clear that, when g(z) is identically zero, the class h reduces to the class a of normalized analytic functions in u. due to lewy [14], a function f ∈h is locally univalent and sense-preserving in u if and only if |h′ (z)| > |g′ (z)| , for z ∈u. we denote by sh the subclass of h consisting of all sense-preserving univalent harmonic functions f. firstly, clunie et al. [5] was discussed some geometric properties of the class sh and its subclasses. later on, several authors contributed in the study of subclasses of the class sh, for example, see [1,3,6,7,9,11,19,20]. the theory of q-calculus operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, q-difference and q-integral equations, as well as geometric function theory of complex analysis. the fractional q-calculus is the q-extension of the ordinary fractional calculus and dates back to early 20th century (e.g. see [10] or [2]). for 0 < q < 1, the q-difference operator was introduced by jackson [10] and is defined by (1.2) ∂qh(z) = h(z) −h(qz) (1 −q)z ; q 6= 1, z 6= 0, for q ∈ (0, 1) and h ∈a with h(z) = z + ∑∞ n=2 anz n. it is clear that lim q→1− ∂qh(z) = h ′(z), where h′(z) is the ordinary derivative of the function. it can easily be seen that for n ∈ n = {1, 2, 3, ..} and z ∈u ∂q { ∞∑ n=1 anz n } = ∞∑ n=1 [n]q z n−1, where [n]q = 1 −qn 1 −q = 1 + q + q2 + · · · . recently, in [18], shah and noor introduced the q-analogue of srivastava-attiya operator jsq,b : a→a by (1.3) jsq,bh(z) = z + ∞∑ n=2 ( [1 + b]q [n + b]q )s anz n, where h ∈ a, s ∈ c and b ∈ c\z−0 . it is noted that for q → 1 − in (1.3), then the integral operator studied by the authors in [21] is deduced. moreover, for particular choices of s and b, the operator jsq,b reduces to the q-alexander, q-libera and q-bernardi operators defined in [17]. int. j. anal. appl. 19 (6) (2021) 828 jahangiri [12] was the first who introduced the q-analogue of complex harmonic functions and studied various geometric properties. nowadays, certain subclasses of sh associated with operators and q-operators were discussed by the prominent researchers, like [8, 12, 13, 15, 16, 22]. in motivation of above said literature, first we modify the q-srivastava-attiya operator and then we define some new subclasses of sh. for f = h + g given by (1.1), we define (1.4) jsq,bf(z) = j s q,bh(z) + j s q,bg(z), where jsq,bh(z) is given by (1.3) and jsq,bg(z) = ∞∑ n=1 ( [1 + b]q [n + b]q )s bnz n. it is observed that, if co-analytic part of f = h + g is identically zero, then the modified q-srivastava-attiya operator defined by (1.4) turn out to be the q-srivastava-attiya operator introduced in [18]. for f = h + g ∈sh, we define a new class hsq (γ,λ,β) as follows: definition 1.1. let f = h + g ∈sh and is given by (1.1). then f ∈hsq (γ,λ,β) if < [ 1 + 1 γ { (1 −λ) f(z) z + λ (∂qf(z)) − 1 }] ≥ β, where β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). particularly, if q → 1−, then the class hsq (γ,λ,β) reduces to the class, denoted by hs (γ,λ,β), of functions f ∈sh satisfies < [ 1 + 1 γ { (1 −λ) f(z) z + λf′(z) − 1 }] ≥ β, where β ∈ [0, 1), γ ∈ c\{0} and λ ∈ [0, 1]. moreover, we denote by hs (1,λ,β) = hs (λ,β) and hs (1, 1,β) = hs (β) the classes of functions f = h + g ∈sh satisfies < [ (1 −λ) f(z) z + λf′(z) ] ≥ β and <(f′(z)) ≥ β, respectively. we further define hsq (γ,λ,β) = hsq (γ,λ,β) ∩sh, where sh denote the subclass of sh consisting of functions of the type f(z) = h(z) + g (z), where (1.5) h(z) = z − ∞∑ n=2 |an|zn and g(z) = − ∞∑ n=1 |bn|zn. now, by using modified q-srivastava-attiya operator given by (1.4), we define the following. definition 1.2. let f = h + g ∈sh, and is given by (1.1). then, for s ∈ r, b > 1, β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). f ∈hss,bq (γ,λ,β) if (1.6) < [ 1 + 1 γ { (1 −λ) jsq,bf(z) z + λ ( jsq,b∂qf(z) ) − 1 }] ≥ β, int. j. anal. appl. 19 (6) (2021) 829 where jsq,bf(z) is given by (1.4). also, we define hs s,b q (γ,λ,β) = hs s,b q (γ,λ,β) ∩sh, where sh denote the subclass of sh consisting of functions given by (1.5). it is noted that, for s = 0 we have hss,bq (γ,λ,β) = hsq (γ,λ,β) and hs s,b q (γ,λ,β) = hsq (γ,λ,β). 2. main results theorem 2.1. let a function f = h + g ∈sh given by (1.1) and satisfies (2.1) ∞∑ n=1 {{ 1 + ( [n]q − 1 ) λ } [|an| + |bn|] }( [1 + b]q [n + b]q )s ≤ 1 + (1 −β) γ, where s ∈ r, b > 1, β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). then f ∈hss,bq (γ,λ,β). this result is sharp. proof. we wish to show that f = h + g ∈sh satisfies (1.6), whenever the coefficients of f satisfies (2.1). we use the fact that <(w) ≥ ξ if and only if |1 − ξ + w| ≥ |1 + ξ −w|. so it suffices to show that ∣∣∣∣1 −β + 1 + 1γ { (1 −λ) jsq,bf(z) z + λ ( jsq,b∂qf(z) ) − 1 }∣∣∣∣ ≥ ∣∣∣∣1 + β − 1 − 1γ { (1 −λ) jsq,bf(z) z + λ ( jsq,b∂qf(z) ) − 1 }∣∣∣∣ , or equivalently, ∣∣∣∣(2 −β) γ + (1 −λ) jsq,bf(z)z + λ(jsq,b∂qf(z))− 1 ∣∣∣∣ − ∣∣∣∣βγ − (1 −λ) jsq,bf(z)z −λ(jsq,b∂qf(z))−z ∣∣∣∣ ≥ 0. using (1.3) and (1.4), from the left hand side, we get ∣∣∣∣∣∣∣ (2 −β) γ + (1 −λ) { 1 + ∑∞ n=2 ( [1+b] q [n+b] q )s anz n−1 + ∑∞ n=1 ( [1+b] q [n+b] q )s bnzn−1 } +λ { 1 + ∑∞ n=2 ( [1+b]q [n+b]q )s [n]q anz n−1 + ∑∞ n=1 ( [1+b]q [n+b]q )s [n]q bnz n−1 } − 1 ∣∣∣∣∣∣∣ − ∣∣∣∣∣∣∣ βγ + (1 −λ) { 1 + ∑∞ n=2 ( [1+b] q [n+b] q )s anz n−1 + ∑∞ n=1 ( [1+b] q [n+b] q )s bnzn−1 } +λ { 1 + ∑∞ n=2 ( [1+b]q [n+b]q )s [n]q anz n−1 + ∑∞ n=1 ( [1+b] q [n+b]q )s [n]q bnz n−1 } − 1 ∣∣∣∣∣∣∣ , int. j. anal. appl. 19 (6) (2021) 830 this implies ∣∣∣∣∣∣∣ (2 −β) γ + ∑∞ n=2 { (1 −λ) + λ [n]q }( [1+b]q [n+b] q )s anz n−1 + ∑∞ n=1 { (1 −λ) + λ [n]q }( [1+b] q [n+b] q )s bnzn−1 ∣∣∣∣∣∣∣ − ∣∣∣∣∣∣∣ βγ + ∑∞ n=2 { (1 −λ) + λ [n]q }( [1+b]q [n+b]q )s anz n−1 + ∑∞ n=1 { (1 −λ) + λ [n]q }( [1+b] q [n+b] q )s bnzn−1 ∣∣∣∣∣∣∣ ≥ (2 −β) γ − ∞∑ n=2 { (1 −λ) + λ [n]q }( [1 + b]q [n + b]q )s |an| |z| n−1 − ∞∑ n=1 { (1 −λ) + λ [n]q }( [1 + b]q [n + b]q )s |bn| |z| n−1 −βγ − ∞∑ n=2 { (1 −λ) + λ [n]q }( [1 + b]q [n + b]q )s |an| |z| n−1 − ∞∑ n=1 { (1 −λ) + λ [n]q }( [1 + b]q [n + b]q )s |bn| |z| n−1 = 2 (1 −β) γ   1 − 2 ∑∞ n=2 { 1 + ( [n]q − 1 ) λ }( [1+b] q [n+b]q )s |an| |z| n−1 −2 ∑∞ n=1 { 1 + ( [n]q − 1 ) λ }( [1+b] q [n+b] q )s |bn| |z| n−1   the above expression is nonnegative by (2.1). hence f ∈hss,bq (α,β). the coefficient bound, given by (2.1), is sharp for the harmonic function f(z) = z + ∞∑ n=2 (1 −β) γ 1 + ( [n]q − 1 ) λ ( [n + b]q [1 + b]q )s xnz n + ∞∑ n=1 (1 −β) γ 1 + ( [n]q − 1 ) λ ( [n + b]q [1 + b]q )s ynz n, with ∑∞ n=2 |xn| + ∑∞ n=1 |yn| = 1. � for different choices of parameters, we deduce certain new results as following. if s = 0 in theorem 2.1, then we have a following new result. corollary 2.1. let a function f(z) = h(z) + g(z) ∈sh given by (1.1) and satisfies ∞∑ n=1 {{ 1 + ( [n]q − 1 ) λ } [|an| + |bn|] } ≤ 1 + (1 −β) γ, where β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). then f ∈hsq (γ,λ,β). this result is sharp. if q → 1−, then corollary 2.1 reduces to a new result as follows: int. j. anal. appl. 19 (6) (2021) 831 corollary 2.2. let a function f(z) = h(z) + g(z) ∈sh given by (1.1) and satisfies ∞∑ n=1 {{1 + (n− 1) λ} [|an| + |bn|]}≤ 1 + (1 −β) γ, where β ∈ [0, 1), γ ∈ c\{0} and λ ∈ [0, 1]. then f ∈hs (γ,λ,β). this result is sharp. if we take γ = 1 in corollary 2.2 then we obtain. corollary 2.3. let a function f(z) = h(z) + g(z) ∈sh given by (1.1) and satisfies ∞∑ n=1 {{1 + (n− 1) λ} [|an| + |bn|]}≤ 2 −β, where β ∈ [0, 1) and λ ∈ [0, 1]. then f ∈hs (λ,β). this result is sharp. moreover, when λ = 1 in corollary 2.3, we get the sufficient condition for f in hs (β). now, we state and prove the necessary and sufficient conditions for the harmonic functions f = h + g to be in hs s,b q (γ,λ,β) as following. theorem 2.2. let f = h + g ∈sh given by (1.5). then f ∈hs s,b q (γ,λ,β) if and only if (2.2) ∞∑ n=1 {{ 1 + ( [n]q − 1 ) λ } [|an| + |bn|] }( [1 + b]q [n + b]q )s ≤ 1 + (1 −β) γ, where s ∈ r, b > 1, β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). proof. the sufficient condition is obvious from the theorem 2.1, because hs s,b q (γ,λ,β) ⊂ hs s,b q (γ,λ,β). we need to prove the necessary condition only, that is, if f ∈ hs s,b q (γ,λ,β), then the coefficients of the function f = h + g satisfy the inequality 2.2. let f ∈hs s,b q (γ,λ,β). then, by the definition of hst s,b q (γ,λ,β), we have (2.3) < [ 1 + 1 γ { (1 −λ) jsq,bf(z) z + λ ( jsq,b∂qf(z) ) − 1 } −β ] ≥ 0, where s ∈ r, b > 1, β ∈ [0, 1), γ ∈ c\{0}, λ ∈ [0, 1] and q ∈ (0, 1). equivalently, we can write (2.3) as (2.4) < { (1 −β) γ + (1 −λ) jsq,bf(z) z + λ ( jsq,b∂qf(z) ) − 1 } ≥ 0. substituting f = h + g in (2.4) and employing (1.4) along with (1.5), and also some computation yields <   (1 −β) γ + (1 −λ) { 1 − ∑∞ n=2 ( [1+b]q [n+b]q )s |an|zn−1 − ∑∞ n=1 ( [1+b] q [n+b]q )s |bn|zn−1 } +λ { 1 − ∑∞ n=2 ( [1+b] q [n+b] q )s [n]q |an|z n−1 − ∑∞ n=1 ( [1+b] q [n+b] q )s [n]q |bn|z n−1 } − 1   ≥ 0. this implies < [ (1 −β) γ − ∞∑ n=2 { 1 + ( [n]q − 1 ) λ } |an|zn−1 − ∞∑ n=1 { 1 + ( [n]q − 1 ) λ } |bn|zn−1 ] ≥ 0. int. j. anal. appl. 19 (6) (2021) 832 the above required condition must hold for all values of z in u. upon choosing the values of z on the positive real axis where 0 ≤ z = r < 1, we must have (2.5) (1 −β) γ − ∞∑ n=2 { 1 + ( [n]q − 1 ) λ } |an|rn−1 − ∞∑ n=1 { 1 + ( [n]q − 1 ) λ } |bn|rn−1 ≥ 0. if the inequality (2.2) does not hold, then the numerator in (2.5) is negative for r sufficiently close to 1. hence there exists z0 = r0 in (0, 1) for which the quotient in (2.5) is negative. this contradicts the required condition for f ∈hst s,b q (γ,λ,β) and so the proof is complete. � next, we want to discuss the distortion bounds for the function f ∈hs s,b q (γ,λ,β), which yields a covering result for this class. theorem 2.3. if f ∈hs s,b q (γ,λ,β) and |z| = r < 1, then (1 −|b1|) r −tr2 ≤ |f(z)| ≤ (1 + |b1|) r + tr2, with (2.6) t = ( [2 + b]q [1 + b]q )s  (1 −β) γ( [2]q − 1 ) λ + 1 − 1( [2]q − 1 ) λ + 1 |b1|   . proof. let f ∈hs s,b q (γ,λ,β). taking absolute value of f, we get |f(z)| ≤ (1 + |b1|) r + ∞∑ n=2 (|an| + |bn|) r2 ≤ (1 + |b1|) r + (1 −β) γ [2 + b]sq{( [2]q − 1 ) λ + 1 } [1 + b] s q × ∞∑ n=2     {( [2]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [2 + b]sq   (|an| + |bn|)  r2 ≤ (1 + |b1|) r + (1 −β) γ [2 + b]sq{( [2]q − 1 ) λ + 1 } [1 + b] s q ∞∑ n=2     {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq   (|an| + |bn|)  r2 ≤ (1 + |b1|) r + (1 −β) γ [2 + b]sq{( [2]q − 1 ) λ + 1 } [1 + b] s q × { 1 − |b1| (1 −β) γ } r2, (by (2.2)) ≤ (1 + |b1|) r + tr2, where t is given by (2.6). hence this is the required right hand inequality. similarly, one can easily prove the required left hand inequality. � int. j. anal. appl. 19 (6) (2021) 833 by making use of the left hand inequality of the above theorem and letting r → 1, we obtain corollary 2.4. (covering result) if f ∈hs s,b q (γ,λ,β), then {w : |w| < (1 −l) − (1 −m) |b1|}⊂ f (e) , where l = (1−β)γ[2+b]s q {([2]q−1)λ+1}[1+b]sq and m = [2+b]s q {([2]q−1)λ+1}[1+b]sq . in particular, we obtain the covering results for the subclasses of harmonic functions defined in definition 1.1 and its special cases. now, our task is to determine the extreme points of closed convex hulls of hs s,b q (γ,λ,β) denoted by clcohs s,b q (γ,λ,β). theorem 2.4. a function f ∈hs s,b q (γ,λ,β) if and only if (2.7) f(z) = ∞∑ n=1 (xnhn (z) + yngn (z)) , where h1(z) = z, hn (z) = z − (1 −β) γ 1 + ( [n]q − 1 ) λ ( [n + b]q [1 + b]q )s zn; (n = 2, 3, ...) and gn(z) = z − (1 −β) γ 1 + ( [n]q − 1 ) λ ( [n + b]q [1 + b]q )s zn; (n = 1, 2, 3, ...) , with ∑∞ n=1 (xn + yn) = 1 and xn,yn ≥ 0. particularly, {hn} and {gsn} are the extreme points of hs s,b q (γ,λ,β). proof. we assume function f as given by (2.7) f(z) = ∞∑ n=1 (xnhn (z) + yngn (z)) = ∞∑ n=1 (xn + yn) z − ∞∑ n=2 xnrnz n − ∞∑ n=1 ynrnz n,(2.8) where rn = (1−β)γ[n+b]s q {([n]q−1)λ+1}[1+b]sq . equating (2.8) with (1.5), we get |an| = xnrn and |bn| = ynrn. now, ∞∑ n=2 {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq |an| + ∞∑ n=1 {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq |bn| = ∞∑ n=1 (xn + yn) −x1 = 1 −x1 ≤ 1, int. j. anal. appl. 19 (6) (2021) 834 this implies ∞∑ n=1 [{( [n]q − 1 ) λ + 1 } [|an| + |bn|] ]( [1 + b]q [n + b]q )s ≤ 1 + (1 −β) γ thus, by theorem 2.2, f ∈hs s,b q (γ,λ,β). conversely, let f ∈hs s,b q (γ,λ,β). we take xn = {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq |an| ; (n = 2, 3, ...) and yn = {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq |bn| ; (n = 1, 2, ...) , with ∑∞ n=1 (xn + yn) = 1. we follows our required result by substituting the values of |an| and |bn| from the above relations in (1.5). � finally, we wish to show that the class hs s,b q (γ,λ,β) is closed under the convex combination of its elements. theorem 2.5. the class hs s,b q (γ,λ,β) is closed under the convex combination. proof. let fis ∈hs s,b q (γ,λ,β), (i = 1, 2, ...), with fi = z − ∞∑ n=2 |ai,n|zn − ∞∑ n=1 |bi,n|zn. on using theorem 2.2, we can write[ ∞∑ n=1 {{ 1 + ( [n]q − 1 ) λ } [|an| + |bn|] }]( [1 + b]q [n + b]q )s ≤ 1 + (1 −β) γ. now, (2.9) ∞∑ i=1 κifi = z − ∞∑ n=2 ( ∞∑ i=1 κi |ai,n| ) zn − ∞∑ n=2 ( ∞∑ i=1 κi |ai,n| ) zn. to prove our result, we use (2.8) and (2.9) ∞∑ n=2 {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq ( ∞∑ i=1 κi |ai,n| ) + ∞∑ n=1 {( [n]q − 1 ) λ + 1 } [1 + b] s q (1 −β) γ [n + b]sq ( ∞∑ i=1 κi |bi,n| ) ≤ ∞∑ i=1 κi = 1. therefore ∑∞ i=1 κifis ∈hs s,b q (γ,λ,β). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (6) (2021) 835 references [1] o.p. ahuja, j.m. jahangiri, noshiro-type harmonic univalent functions, sci. math. jpn. 56 (2002), 1–7. [2] g.e. andrews, r. askey, r. roy, special functions, encyclopedia of mathematics and its applications, cambridge university press, cambridge, uk, 1999. [3] y. avci, e. zlotkiewicz, on harmonic univalent mappings, ann. univ. marie curie-sklodowska sect. a 44 (1990), 1–7. [4] n.e. cho, t.h. kim, multiplier transformation and strongly close-to-convex functions, bull. korean math. soc. 40 (2003), 399–410. [5] j. clunie, t. sheil-small, harmonic univalent functions. ann. acad. sci. fenn., ser. a 1 math. 9 (1984), 3–25. [6] m. darus, k. al-shaqsi, on certain subclass of harmonic univalent functions, j. anal. appl. 6 (2008), 17–28. [7] p.l. duren, a survey of harmonic mappings in the plane, texas tech. univ. math. ser. 18 (1992), 1–15. [8] j. dziok, classes of harmonic functions associated with ruscheweyh derivatives, rev. r. acad. cienc. exactas f́ıs. nat. ser. a mat. (racsam) 113 (2019), 1315–1329 [9] j. dziok, m. darus, j. sokó l, t. bulboaca, generalizations of starlike harmonic functions, c. r. math. acad. sci. paris 354 (2016), 13–18. [10] f.h. jackson, on q-functions and a certain difference operator, earth environ. sci. trans. r. soc. edinburgh. 46 (1908), 253–281. [11] j.m. jahangiri, harmonic functions starlike in the unit disc, j. math. anal. appl. 235 (1999), 470–477. [12] j.m. jahangiri, harmonic univalent functions defined by q-calculus operators, int. j. math. anal. appl. 5 (2018), 39–43. [13] j.m. jahangiri, g. murugusundaramoorthy, k. vijaya, starlikeness of rucheweyh type harmonic univalent functions, j. indian acad. math. 26 (2004), 191–200. [14] h. lewy, on the non-vanishing of the jacobian in certain one-to-one mappings, bull. amer. math. soc. 42 (1936), 689–692. [15] g. murugusundramoorthy, j.m. jahangiri, ruscheweyh-type harmonic functions defined by q-differential operators, khayyam j. math. 5 (2019), 79–88. [16] k.i. noor, b. malik, s.z.h. bukhari, harmonic functions defined by a generalized fractional differential operator, j. adv. math. stud. 2 (2009), 41–52. [17] k.i. noor, s. riaz, m.a. noor, on q-bernardi integral operator, twms j. pure appl. math. 8 (2017), 3–11. [18] s.a. shah, k.i. noor, study on q-analogue of certain family of linear operators, turk. j. math. 43 (2019), 2707–2714. [19] t. sheil-small, constants for planar harmonic mappings, j. london math. soc. 2 (1990), 237–248. [20] h. silverman, harmonic univalent functions with negative coefficients, j. math. anal. appl. 220 (1998), 283–289. [21] h.m. srivastava, a.a. attiya, an integral operator associated with the hurwitz-lerch zeta function and differential subordination, integral trans. spec. funct. 18 (2007), 207–216. [22] s. yalçın, a new class of sălăgean-type harmonic univalent functions, appl. math. lett. 18 (2005), 191–198. 1. introduction 2. main results references int. j. anal. appl. (2023), 21:1 essential bipolar fuzzy ideals in semigroups pannawit khamrot1, thiti gaketem2,∗ 1department of mathematics, faculty of science and agricultural technology, rajamangala university of technology lanna of phitsanulok, phitsanulok, thailand 2fuzzy algebras and decision-making problems research unit, department of mathematics school of science, university of phayao, phayao 56000, thailand ∗corresponding author: thiti.ga@up.ac.th abstract. in this paper, we give the concepts of essential bipolar fuzzy ideals in semigroups. we discuss the basic properties and relationships between essential bipolar fuzzy ideals and essential ideals in semigroups finally, we extend to 0-essential bipolar fuzzy ideals in semigroups. 1. introduction the theory for dealing with uncertainty, fuzzy set theory, was discovered by zadeh in 1965 [12], which it has applied in many areas such as medical science, robotics, computer science, information science, control engineering, measure theory, logic, set theory, topology etc. in 2000, lee [6] developed theory of fuzzy set to theory of bipolar fuzzy set which function from intrval [−1, 0]∪[0, 1]. the theory of bipolar set applied in information affects the effectiveness and efficiency of decision making. it is used in decision-making problems, organization problems, economic problems, and evaluation, risk management, environmental and social impact assessments. later in 2012, s.k. majumder [2] studies bipolar fuzzy set in γ-semigroups and integration properties of bipolar fuzzy ideals in γ-semigroups in 1971 u. medhi et al. [7] was introduced essential fuzzy ideals of ring. in 2013, u. medhi and h.k. saikia [8] discussed the concept of t-fuzzy essential ideals and studied the properties of t-fuzzy essential ideals. in 2017 s. wani and k. pawar [11] extend the concept of essential ideals in semigroups go to ternary semiring and studied essential ideals in ternary semiring. in 2020, s. baupradist et al. [1] received: nov. 14, 2022. 2020 mathematics subject classification. 20m12, 06f05. key words and phrases. bipolar fuzzy ideals; bipolar fuzzy ideals; 0-essential bipolar fuzzy ideals. https://doi.org/10.28924/2291-8639-21-2023-1 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-1 2 int. j. anal. appl. (2023), 21:1 studied essential ideals and essential fuzzy ideals in semigroups. together with 0-essential ideals and 0-essential fuzzy ideals in semigroups. in 2022 t. gaketem et al. [10] studied essential bi-ideals and fuzzy essential bi-ideals in semigroups. moreover, t. gaketem and a. iampan [3,4] used knowledge of essential ideals in semigroups go to studied essential ideals in up-algebra. in this papar, we used knowledge of essential fuzzy ideals in semigroups go study in bipolar valued fuzzy ideal in semigroup and we investigate it properties. moreover, we characterize essential bipolar valued fuzzy ideals and 0-essential bipolar valued fuzzy ideals of semigroups. 2. preliminaries in this section, we review concepts basic definitions and the theorem used to prove all result in the next section. a non-empty subset i of a semigroup s is called a subsemigroup of s if i2 ⊆ i. a non-empty subset i of a semigroup s is called a left (right) ideal of s if si ⊆ i (is ⊆ i). an ideal i of a semigroup s is a non-empty subset which is both a left ideal and a right ideal of s. an essential ideal i of a semigroup s if i is an ideal of s and i∩j 6= ∅ for every ideal j of s. we see that for any ζ1,ζ2 ∈ [0, 1], we have ζ1 ∨ζ2 = max{ζ1,ζ2} and ζ1 ∧ζ2 = min{ζ1,ζ2}. a fuzzy set ζ of a non-empty set t is function from t into unit closed interval [0, 1] of real numbers, i.e., ζ : t → [0, 1]. for any two fuzzy sets ζ and % of a non-empty set t, define ≥, =,∧, and ∨ as follows: (1) ζ ≥ % ⇔ ζ(k) ≥ %(k) for all k ∈ t, (2) ζ = % ⇔ ζ ≥ % and % ≥ ζ, (3) (ζ ∧%)(k) = min{ζ(k),%(k)} = ζ(k) ∧%(k) for all k ∈ t, (4) (ζ ∨%)(k) = max{ζ(k),%(k)} = ζ(k) ∨%(k) for all k ∈ t. for the symbol ζ ≤ %, we mean % ≥ ζ. for any element k in a semigroup s, define the set fk by fk := {(y,z) ∈ s×s | k = yz}. for two fuzzy sets ζ and % on a semigroup s, define the product ζ ◦% as follows: for all k ∈ s, (ζ ◦%)(k) =   ∨ (y,z)∈fk {ζ(y) ∧%(z) if fk 6= ∅, 0 if fk = ∅. the following definitions are types of fuzzy subsemigroups on semigroups. definition 2.1. [9] a fuzzy set ζ of a semigroup s is said to be a fuzzy ideal of s if ζ(uv) ≥ ζ(u)∨ζ(v) for all u,v ∈ s. int. j. anal. appl. (2023), 21:1 3 definition 2.2. [1] an essential fuzzy ideal ζ of a semigroup s if ζ is a nonzero fuzzy ideal of s and ζ ∧% 6= 0 for every nonzero fuzzy ideal % of s. now, we reivew definition of bipolar valued fuzzy set and basic properties used in next section. definition 2.3. [6] let s be a non-empty set. a bipolar fuzzy set (bf set) ζ on s is an object having the form ζ := {(k,ζp(k),ζn(k)) | k ∈ s}, where ζp : s → [0, 1] and ζn : s → [−1, 0]. remark 2.1. for the sake of simplicity we shall use the symbol ζ = (s; ζp,ζn) for the bf set ζ = {(k,ζp(k),ζn(k)) | k ∈ s}. the following example of a bf set. example 2.1. let s = {21, 22, 23...}. define ζp : s → [0, 1] is a function ζp(u) =  0 if u is old number 1 if u is even number and ζn : s → [−1, 0] is a function ζn(u) =  −1 if u is old number 0 if u is even number. then ζ = (s; ζp,ζn) is a bf set. for bf sets ζ = (s; ζp,ζn) and % = (s; %p,%n), define products ζp◦%p and ζn◦%n as follows: for u ∈ s (ζp ◦%p)(k) =   ∨ (y,z)∈fk {ζp(y) ∧%p(z)} if k = yz 0 if otherwise. and (ζn ◦%n)(k) =   ∧ (y,z)∈fk {ζn(y) ∨%n(z)} if k = yz 0 if otherwise. definition 2.4. [2] let i be a non-empty set of a semigroup s. a positive characteristic function and a negative characteristic function are respectively defined by λ p i : s → [0, 1],k 7→ p i(u) := { 1 k ∈ i, 0 k /∈ i, and 4 int. j. anal. appl. (2023), 21:1 λni : s → [−1, 0],k 7→ n i (k) := { −1 k ∈ i, 0 k /∈ i. remark 2.2. for the sake of simplicity we shall use the symbol λi = (s; λ p i,λ n i) for the bf set i := {(k,λ p i(k),λ n i(k)) | k ∈ i}. definition 2.5. [2] a bf set ζ = (s; ζp,ζn) on a semigroup s is called a bf ideal on s if it satisfies the following conditions: ζp(uv) ≥ ζp(u) ∨ζp(v) and ζn(uv) ≤ ζn(v) ∧ζn(u) for all u,v ∈ s. the following theorems are true. theorem 2.1. [2] let k be a nonempty subset of semigroup s. then k is an ideal of s if and only if characteristic function λk = (s; λ p k,λ n k) is a bf ideal of s. theorem 2.2. [2] let l and j be subsets of a non-empty set s. then the following holds. (1) λpl∩j = λ p l ∧λ p j. (2) λnl∪j = λ n l ∨λ n j. (3) λpl ◦λ p j = λ p lj. (4) λnl ◦λ n j = λ n lj. let ζ = (s; ζp,ζn) be a bf set of a non-empty of s. then the support of ζ instead of supp(ζ) = {u ∈ s | ζ(u) 6= 0} where ζp(u) 6= 0 and ζn(u) 6= 0 for all u ∈ s. theorem 2.3. let ζ = (s; ζp,ζn) be a nonzero bf set of a semigroup s. then ζ = (s; ζp,ζn) is a bf ideal of s if and only if supp(ζ) is an ideal of s. proof. supposet that ζ = (s; ζp,ζn) is a bf ideal of s and let u,v ∈ s, with u,v ∈ supp(ζ) then ζp(u) 6= 0, ζp(v) 6= 0 and ζn(u) 6= 0, ζn(v) 6= 0 . since ζ = (s; ζp,ζn) is a bf ideal of s we have ζp(uv) ≥ ζp(u) ∨ζp(v) and ζn(uv) ≤ ζn(u) ∧ζn(v) thus, ζp(uv) 6= 0 and ζn(uv) 6= 0. it implies that uv ∈ supp(ζ). hence, supp(ζ) is an ideal of s. conversely, suppose that supp(ζ) is an ideal of s and let u,v,∈ s. if u,v ∈ supp(ζ), then uv ∈ supp(ζ). thus ζp(v) 6= 0 and ζp(uv) 6= 0. hence ζp(uv) ≥ ζp(u) ∨ζp(v). if u /∈ supp(ζ) or v /∈ supp(ζ) then ζp(uv) ≥ ζp(u) ∨ζp(v). similarly, we can show that, ζn(uv) ≤ ζp(u) ∧ζn(v). thus, ζ = (s; ζp,ζn) is a bf ideal of s. � 3. essential bipolar valued fuzzy ideals in a semigroup. definition 3.1. an essential bf ideal ζ = (s; ζp,ζn) of a semigroup s if ζ = (s; ζp,ζn) is a nonzero bf ideal of s and ζp ∧%p 6= 0 and ζn ∨%n 6= 0 for every nonzero bf ideal % = (s; %p,%n) of s. int. j. anal. appl. (2023), 21:1 5 theorem 3.1. let i be an ideal of a semigroup s. then i is an essential ideal of s if and only if λi = (s; λ p i,λ n i) is an essential bf ideal of s. proof. suppose that i is an essential ideal of s and let % = (s; %p,%n) be a nonzero bf ideal of s. then by theorem 2.3 supp(%) is an ideal of s. since i is an essential ideal of s we have i is an ideal of s. thus i ∩ supp(%) 6= ∅. so there exists u ∈ i ∩ supp(%). since i is an ideal of s we have λi = (s; λ p i,λ n i) is a bf ideal of s. since % = (s; % p,%n) is a nonzero bf ideal of s we have (λpi ∧ % p)(u) 6= 0 and (λni ∨ % n)(u) 6= 0 thus, λpi ∧ % p 6= 0 and λni ∨ % n 6= 0. therefore, λi = (s; λ p i,λ n i) is an essential bf ideal of s. conversely, assume that λi = (s; λ p i,λ n i) is an essential bf ideal of s and let j be an ideal of s. then λj = (s; λ p j,λ n j) is a nonzero bf ideal of s. since λi = (s; λ p i,λ n i) is an essential bf ideal of s we have λi = (s; λ p i,λ n i) is a bf ideal of s. thus, λpi ∧λ p j 6= 0 and λ n i ∨λ n j 6= 0 so by theorem 2.2, λ p i∩j 6= 0 and λ n i∪j 6= 0. hence, i ∩ j 6= ∅. therefore, i is an essential ideal of s. � theorem 3.2. let ζ = (s; ζp,ζn) be a nonzero bf ideal of a semigroup s. then ζ = (s; ζp,ζn) is an essential bf ideal of s if and only if supp(ζ) is an essential ideal of s. proof. assume that ζ = (s; ζp,ζn) is an essential bf ideal of s and let j be an ideal of s. then by theorem 2.1, λj = (s; λ p j,λ n j) is a bf ideal of s. since ζ = (s; ζ p,ζn) is an essential bf ideal of s we have ζ = (s; ζp,ζn) is a bf ideal of s. thus, ζp ∧λpj 6= 0 and ζ n ∨λnj 6= 0. so there exists u ∈ s such that (ζp ∧λpj)(u) 6= 0 and (ζ n ∨λnj)(u) 6= 0. it implies that ζ p(u) 6= 0, λpj(u) 6= 0 and ζn(u) 6= 0, λnj(u) 6= 0. hence, u ∈supp(ζ) ∩j so supp(ζ) ∩j 6= ∅. therefore, supp(ζ) is an essential ideal of s. conversely, assume that supp(ζ) is an essential ideal of s and let % = (s; %p,%n) be a nonzero bf ideal of s. then by thoerem 2.3 supp(%) is an ideal of s. since supp(ζ) is an essential ideal of s we have supp(ζ) is an ideal of s. thus supp(ζ)∩supp(%) 6= ∅. so, there exists u ∈ supp(ζ)∩supp(%). it implies that ζp(u) 6= 0, ζn(u) 6= 0 and %p(u) 6= 0, %n(u) 6= 0. for all u ∈ s. hence, (ζp ∧%p)(u) 6= 0 and (ζn ∨ %n)(u) 6= 0 for all u ∈ s. therefore, ζp ∧ %p 6= 0 and ζn ∨ %n 6= 0. we conclude that ζ = (s; ζp,ζn) is an essential bf ideal of s. � theorem 3.3. let ζ = (s; ζp,ζn) be an essential bf ideal of a semigroup s. if % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n, then % = (s; %p,%n) is also an essential bf ideal of s. proof. let % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n and let ξ = (s; ξp,ξn) be any bf ideal of s. since ζ = (s; ζp,ζn) is an essential bf ideal of s we have ζ = (s; ζp,ζn) is a bf ideal of s. thus ζp ∧ ξp 6= 0 and ζn ∨ ξn 6= 0. so %p ∧ ξp 6= 0 and %n ∨ ξn 6= 0. hence % = (s; %p,%n) is an essential bf ideal of s. � next, we study the intersection and union of bf sets as define. let ζ = (s; ζp,ζn) and % = (s; %p,%n) are bf sets of a semigroup s. 6 int. j. anal. appl. (2023), 21:1 define ζ ∩% = (ζp ∩%p,ζn ∩%n) where (ζp ∩%p)(k) = ζp(k) ∧%p(k) and (ζn ∩%n)(k) = ζn(k) ∨%n(k) for all k ∈ s. define ζ ∪% = (ζp ∪%p,ζn ∪%n) where (ζp ∪%p)(k) = ζp(k) ∨%p(k) and (ζn ∪%n)(k) = ζn(k) ∧%n(k) for all k ∈ s. theorem 3.4. let ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) be essential bf ideals of a semigroup s. then ζ1 ∪ζ2 and ζ1 ∩ζ2 are essential bf ideals of s. proof. by theorem 3.3, we have ζ1 ∪ζ2 is an essential bf ideal of s. since ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) are essential bf ideals of s we have ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) are bf ideals of s. thus ζ1 ∩ζ2 is a bf ideal of s. let ξ = (s; ξp,ξn) be a bf ideal of s. then ζp1 ∧ξ p 6= 0 and ζn1 ∨ξ n 6= 0. thus there exists u ∈ s such that (ζp1 ∧ ξ p)(u) 6= 0 and (ζn1 ∨ ξ n)(u) 6= 0. so ζp1(u) 6= 0, ζ n 1(u) 6= 0 and ξ p(u) 6= 0 and ξn(u) 6= 0. since ζp2 6= 0 and ζ n 2 6= 0 and let v ∈ s such that ζ p 2(v) 6= 0 and ζ n 2(v) 6= 0. since ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) are bf ideals of s we have ζ p 1(uv) ≥ ζ p 1(u) ∧ ζ p 1(v) ≥ 0, ζn1(uv) ≤ ζ n 1(u) ∨ ζ n 1(v) ≤ 0 and ζ p 2(uv) ≥ ζ p 2(u) ∧ ζ p 2(v) ≥ 0, ζ n 2(uv) ≤ ζ n 2(u) ∨ ζ n 2(v) ≤ 0. thus (ζ p 1∧ζ p 2)(uv) = ζ p 1(uv)∧ζ p 2(uv) 6= 0 and (ζ n 1∨ζ n 2)(uv) = ζ n 1(uv)∨ζ n 2(uv) 6= 0. since ξ = (s; ξ p,ξn) is a bf ideal of s and ξp(u) 6= 0 and ξn(u) 6= 0 we have ξp(uv) 6= 0 and ξn(uv) 6= 0 for all u,v ∈ s. thus ((ζp1 ∧ ζ p 2) ∧ ξ p)(uv) 6= 0 and ((ζn1 ∨ ζ n 2) ∨ ξ n)(uv) 6= 0. hence ((ζp1 ∧ ζ p 2) ∧ ξ p) 6= 0 and ((ζn1 ∨ζ n 2) ∨ξ n) 6= 0. therefore, ζ1 ∩ζ2 is an essential bf ideal of s. � definition 3.2. [1] an essential ideal i of a semigroup s is called (1) a minimal if for every essential ideal of j of s such that j ⊆ i, we have j = i, (2) a prime if uv ∈ i implies u ∈ i or v ∈ i, (3) a semiprime if u2 ∈ i implies u ∈ i, for all u,v ∈ s. example 3.1. [1] let s be a semigroup with zero. then {0} is a unique minimal essential ideal of s, since {0} is an eseential ideal of s. definition 3.3. an essential bf ideal ζ = (s; ζp,ζn) of a semigroup s is called (1) a minimal if for every essential bf ideal of % = (s; %p,%n) of s such that ζp ≤ %p and ζn ≥ %n, we have supp(ζ) = supp(%), (2) a prime if ζp(uv) ≤ ζp(u) ∨ζp(v) and ζn(uv) ≥ ζn(u) ∧ζn(v) , (3) a semiprime if ζp(u2) ≤ ζp(u) and ζn(u2) ≥ ζn(u), for all u,v ∈ s. theorem 3.5. let i be a non-empty subset of a semigroup s. then the following statement holds. (1) i is a minimal essential ideal of s if and only if λi = (s; λ p i,λ n i) is a minimal essential bf ideal of s, (2) i is a prime essential ideal of s if and only if λi = (s; λ p i,λ n i) is a prime essential bf ideal of s, int. j. anal. appl. (2023), 21:1 7 (3) i is a semiprime essential ideal of s if and only if λi = (s; λ p i,λ n i) is a semiprime essential bf ideal of s. proof. (1) suppose that i is a minimal essential ideal of s. then i is an essential ideal of s. by theorem 3.1, λi = (s; λ p i,λ n i) is an essential bf ideal of s. let ζ = (s; ζ p,ζn) be an essential bf ideal of s such that ζp ≤ λpi and ζ n ≥ λni. then supp(ζ) ⊆ supp(λi). thus, supp(ζ) ⊆ supp(λi) = i. hence, supp(ζ) ⊆ i. since ζ = (s; ζp,ζn) is an essential bf ideal of s we have supp(ζ) is an essential ideal of s. by assumption, supp(ζ) = i = supp(λi). hence, λi = (s; λ p i,λ n i) is a minimal essential bf ideal of s. conversely, λi = (s; λ p i,λ n i) is a minimal essential bf ideal of s and let b be an essential ideal of s such that b ⊆ i. then b is an ideal of s. thus by theorem 3.1, λb = (s; λ p b,λ n b) is an essential bf ideal of s such that λpb ≥ λ p i and λ n b ≤ λ n i. so λb = λi. hence b = supp(λb) = supp(λi) = i. therefore i is a minimal essential ideal of s. (2) suppose that i is a prime essential ideal of s. then i is an essential ideal of s. thus by theorem 3.1 λi = (s; λ p i,λ n i) is an essential bf ideal of s. let u,v ∈ s. if uv ∈ i, then u ∈ i or v ∈ i. thus λpi(u) ∨λ p i(v) = 1 ≥ λ p i(uv) and λ n i(u) ∧λ n i(v) = −1 ≤ λni(uv). if uv /∈ i, then λpi(u) ∨λ p i(v) ≥ λ p i(uv) and λ n i(u) ∧λ n i(v) ≤ λ n i(uv). thus λi = (s; λ p i,λ n i) is a prime essential bf ideal of s. conversely, suppose that λi = (s; λ p i,λ n i) is a prime essential bf ideal of s. then λi = (s; λ p i,λ n i) is an essential bf ideal. thus by theorem 3.1, i is an essential ideal of s. let u,v ∈ s. if uv ∈ i, then λpi(uv) = 1 and λ n i(uv) = −1. by assumption, λ p i(uv) ≤ λ p i(u) ∨ λ p i(v) and λ n i(uv) ≥ λ n i(u) ∧ λ n i(v). thus λ p i(u) ∨ λ p i(v) = 1 and λni(u) ∧λ n i(v) = −1 so u ∈ i or v ∈ i. hence i is a prime essential ideal of s. (3) suppose that i is a semiprime essential ideal of s. then i is an essential ideal of s. thus by theorem 4.1, λi = (s; λ p i,λ n i) is an essential bf ideal of s. let u ∈ s. if u2 ∈ i, then u ∈ i. thus, λpi(u) = λ p i(u 2) = 1 and λni(u) = λ n i(u 2) = −1. hence, λ p i(u 2) ≤ λpi(u) and λ n i(u 2) ≥ λni(u). if u2 /∈ i, then λpi(u 2) = 0 ≤ λpi(u) and λ n i(u 2) = 0 ≥ λni(u). thus λi = (s; λ p i,λ n i) is a semiprime essential bf ideal of s. conversely, suppose that λi = (s; λ p i,λ n i) is a semiprime essential bf ideal of s. then λi = (s; λ p i,λ n i) is an essential bf ideal of s. thus by theorem 4.1, i is an essential ideal of s. let u ∈ s with u2 ∈ i. then λpi(u 2) = 1 and λni(u 2) = −1. by assumption, λ p i(u 2) ≤ λpi(u) and λ n i(u 2) ≥ λni(u). thus λ p i(u) = 1 and λ n i(u) = −1 so u ∈ i. hence, i is a semiprime essential ideal of s. � 8 int. j. anal. appl. (2023), 21:1 theorem 3.6. let ζ = (s; ζp,ζn) be a minimal essential bf ideal of a semigroup s. if % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n, then % = (s; %p,%n) is also a minimal essential bf ideal of s. proof. let % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n and let ξ = (s; ξp,ξn) be any bf ideal of s. since ζ = (s; ζp,ζn) is a minimal essential bf ideal of s we have ζ = (s; ζp,ζn) is a bf ideal of s. thus, ζp ∧ξp 6= 0 and ζn ∨ξn 6= 0. so %p ∧ξp 6= 0 and %n ∨ξn 6= 0. hence, % = (s; %p,%n) is a minimal essential bf ideal of s. � corollary 3.1. let ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) be minimal essential bf ideals of a semigroup s. then ζ1 ∪ζ2 is a minimal essential bf ideals of s. 4. 0-essential bf ideal. in this section, we let s be a semigroup with zero. begin we review the definition 0-essential ideal of s as follows: definition 4.1. [1] a nonzero ideal i of a semigroup with zero s is called a 0-essential ideal of s if i∩j 6= {0} for every nonzero ideal of j of s. example 4.1. [1] let (z12, +) be semigroup. then {0, 2, 4, 6, 8, 10} and z12 are 0-essential ideal of z12. definition 4.2. a bf ideal ζ = (s; ζp,ζn) of a semigroup with zero s is called a nontrivial bf ideal of s if there exists a nonzero element u ∈ s such that ζp(u) 6= 0 and ζn(u) 6= 0. we define the definition of 0-essential bf ideals of a semigroup with zero as follows: definition 4.3. a 0-essential bf ideal ζ = (s; ζp,ζn) of a semigroup with zero s if ζ = (s; ζp,ζn) is a nonzero bf ideal of s and supp(ζ ∧%) 6= {0} for every nonzero bf ideal % = (s; %p,%n) of s. theorem 4.1. let i be a nonzero ideal of a semigroup with zero s. then i is a 0-essential ideal of s if and only if λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s. proof. suppose that i is a 0-essential ideal of s and let % = (s; %p,%n) be a nontrivial bf ideal of s. then by theorem 2.3, supp(%) is a nonzero ideal of s. since i is a 0-essential ideal of s we have i is a nonzero ideal of s. thus i∩supp(%) 6= {0}. so there exists u ∈ i∩ supp(%). since i is a nonzero ideal of s we have λi = (s; λ p i,λ n i) is a nonzero bf ideal of s. since % = (s; % p,%n) is a nonzero bf ideal of s we have supp(λi ∧%)(u) 6= 0. thus, λ p i ∧% p 6= 0 and λni ∨% n 6= 0. therefore, λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s. conversely, assume that λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s and let j be a nonzero ideal of s. then λj = (s; λ p j,λ n j) is a nonzero bf ideal of s. sicne λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s we have λi = (s; λ p i,λ n i) is a nontrivial bf ideal of s. thus, supp(λi ∧λj) 6= {0}. int. j. anal. appl. (2023), 21:1 9 so by theorem 2.2, λpl∩j 6= 0 and λ n l∪j 6= 0. hence, i ∩ j 6= {0}. therefore i is a 0-essential ideal of s. � theorem 4.2. let ζ = (s; ζp,ζn) be a nonzero bf ideal of a semigroup with zero s. then ζ is a 0-essential bf ideal of s if and only if supp(ζ) is a 0-essential ideal of s. proof. assume that ζ = (s; ζp,ζn) is a 0-essential bf ideal of s and let j be a nontrivial ideal of s. then by theorem 2.1, λj = (s; λ p j,λ n j) is a nonzero bf ideal of s. since ζ = (s; ζ p,ζn) is a 0-essential bf ideal of s we have ζ = (s; ζp,ζn) is a nonzero bf ideal of s. thus ζp ∧ λpj 6= 0 and ζn ∨ λnj 6= 0. so there exists a nonzero element u ∈ s such that (ζ p ∧ λpj)(u) 6= 0 and (ζn ∨ λnj)(u) 6= 0. it implies that ζ p(u) 6= 0, ζn(u) 6= 0 and λpj(u) 6= 0 , λ n j(u) 6= 0. hence, u ∈ supp(ζ) ∩j so supp(ζ) ∩j 6= {0}. therefore supp(ζ) is a 0-essential ideal of s. conversely, assume that supp(ζ) is a 0-essential ideal of s and let % = (s; %p,%n) be a nonzero bf ideal of s. then by theorem 2.3 supp(%) is a nontrivial zero ideal of s. since supp(ζ) is a 0-essential ideal of s we have supp(ζ) is a nonzero ideal of s. thus supp(ζ) ∩ supp(%) 6= {0}. so there exists u ∈ supp(ζ)∩supp(%), this implies that ζp(u) 6= 0, ζn(u) 6= 0 and %p(u) 6= 0, %n(u) 6= 0 for all u ∈ s. hence, (ζp ∧%p)(u) 6= 0 and (ζn ∨%n)(u) 6= 0 for all u ∈ s. therefore, ζp ∧%p 6= 0 and ζn ∨%n 6= 0. we conclude that ζ = (s; ζp,ζn) is a 0-essential bf ideal of s. � theorem 4.3. let ζ = (s; ζp,ζn) be a 0-essential bf ideal of a semigroup s. if % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n, then % = (s; %p,%n) is also a 0-essential bf ideal of s. proof. let % = (s; %p,%n) is a bf ideal of s such that ζp ≤ %p and ζn ≥ %n and let ξ = (s; ξp,ξn) be any nonzero bf ideal of s. since ζ = (s; ζp,ζn) is a 0-essential bf ideal of s we have ζ = (s; ζp,ζn) is a bf ideal of s. thus supp(ζ ∧ξ) 6= 0. so %p ∧ξp 6= 0 and %n ∨ξn 6= 0. hence % = (s; %p,%n) is a 0-essential bf ideal of s. � theorem 4.4. let ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) be 0-essential bf ideals of a semigroup s. then ζ1 ∪ζ2 and ζ1 ∩ζ2 are 0-essential bf ideals of s. proof. by theorem 4.3, we have ζ1 ∪ζ2 is a 0-essential bf ideal of s. since ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) are 0-essential bf ideals of s we have ζ1 = (s; ζ p 1,ζ n 1) and ζ2 = (s; ζ p 2,ζ n 2) are bf ideals of s. thus ζ1 ∩ ζ2 is a bf ideal of s. let ξ = (s; ξp,ξn) be a nontrival bf ideal of s. since ζ1 = (s; ζ p 1,ζ n 1) is a bf ideal of s we havesupp(ζ1) is an ideal of s. thus supp(ζ1 ∧ξ) 6= {0}. thus there exists u ∈ s such that (ζ p 1 ∧ξ p)(u) 6= 0 and (ζn1∨ξ n)(u) 6= 0. since ζ2 = (s; ζ p 2,ζ n 2) is a 0-essential bf ideal of s we have supp(ζ2) is a 0-essential bf ideal of s. thus, supp(ζ2 ∧ ξ) 6= {0}. so, there exists a nonzero element v ∈ supp(ζ2 ∧ ξ)(u) implies ζp2(v) 6= 0 and ζ n 2(v) 6= 0. since ζ1 = (s; ζ p 1,ζ n 1) and ξ = (s; ξ p,ξn) are bf ideals of s we have ζp1(v) ≥ ζ p 1(u), ξ p(v) ≥ ξp(u) and ζn1(v) ≤ ζ n 1(u), ξ n(v) ≤ ξn(u). so ((ζp1 ∧ ζ p 2) ∧ ξ p)(v) 6= 0 and ((ζn1 ∨ζ n 2) ∨ξ n)(v) 6= 0. thus, supp((ζ1 ∧ζ2) ∧ξ) 6= {0}. therefore, ζ1 ∩ζ2 is a 0-essential bf ideals of s. � 10 int. j. anal. appl. (2023), 21:1 definition 4.4. [1] a 0-essential ideal i of a semigroup with zero s is called (1) a minimal if for every 0-essential ideal of j of s such that j ⊆ i, we have j = i, (2) a prime if uv ∈ i implies u ∈ i or v ∈ i, (3) a semiprime if u2 ∈ i implies u ∈ i, for all u,v ∈ s. example 4.2. let (z12, +) be a semigroup with zero. then {0, 2, 4, 6, 8, 10} is a minimal 0-essential ideal of s. definition 4.5. a 0-essential bf ideal ζ = (s; ζp,ζn) of a semigroup s is called (1) a minimal if for every 0-essential bf ideal of % = (s; %p,%n) of s such that ζp ≤ %p and ζn ≥ %n, we have supp(ζ) = supp(%), (2) a prime if ζp(uv) ≤ ζp(u) ∨ζp(v) and ζn(uv) ≥ ζn(u) ∧ζn(v) , (3) a semiprime if ζp(u2) ≤ ζp(u) and ζn(u2) ≥ ζn(u), for all u,v ∈ s. theorem 4.5. let i be a non-empty subset of a semigroup s. then the following statement holds. (1) i is a minimal 0-essential ideal of s if and only if λi = (s; λ p i,λ n i) is a minimal 0-essential bf ideal of s, (2) i is a prime 0-essential ideal of s if and only if λi = (s; λ p i,λ n i) is a prime 0-essential bf ideal of s, (3) i is a semiprime 0-essential ideal of s if and only if λi = (s; λ p i,λ n i) is a semiprime 0-essential bf ideal of s. proof. (1) suppose that i is a minimal 0-essential ideal of s. then i is a 0-essential ideal of s. by theorem 4.1, λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s. let ζ = (s; ζ p,ζn) be a 0-essential bf ideal of s such that ζp ≤ λpi and ζ n ≥ λni. then supp(ζ) ⊆ supp(λi). thus supp(ζ) ⊆ supp(λi) = i. thus supp(ζ) ⊆ i. since ζ = (s; ζp,ζn) is a 0-essential bf ideal of s we have supp(ζ) is a 0-essential ideal of s. by assumption, supp(ζ) = i = supp(λi). hence, λi = (s; λ p i,λ n i) is a minimal 0-essential bf ideal of s. conversely, λi = (s; λ p i,λ n i) is a minimal 0-essential bf ideal of s and let b be a 0essential ideal of s such that b ⊆ i. then b is an ideal of s. thus by theorem 4.1, λb = (s; λ p b,λ n b) is an essential bf ideal of s such that λ p b ≥ λ p i and λ n b ≤ λ n i. so λb = λi. hence b = supp(λb) = supp(λi) = i. therefore i is a minimal 0-essential ideal of s. (2) suppose that i is a prime 0-essential ideal of s. then i is a 0-essential ideal of s. thus by theorem 4.1 λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s. let u,v ∈ s. if uv ∈ i, then u ∈ i or v ∈ i. thus λpi(u) ∨λ p i(v) = 1 ≥ λ p i(uv) and λ n i(u) ∧λ n i(v) = −1 ≤ λni(uv). if uv /∈ i, then λpi(u) ∨λ p i(v) ≥ λ p i(uv) and λ n i(u) ∧λ n i(v) ≤ λ n i(uv). int. j. anal. appl. (2023), 21:1 11 thus λi = (s; λ p i,λ n i) is a prime 0-essential bf ideal of s. conversely, suppose that λi = (s; λ p i,λ n i) is a prime 0-essential bf ideal of s. then λi = (s; λ p i,λ n i) is a 0-essential bf ideal. thus by theorem 4.1, i is a 0-essential ideal of s. let u,v ∈ s. if uv ∈ i, then λpi(uv) = 1 and λ n i(uv) = −1. by assumption λ p i(uv) ≤ λ p i(u) ∨ λ p i(v) and λ n i(uv) ≥ λ n i(u) ∧ λ n i(v). thus λ p i(u) ∨ λ p i(v) = 1 and λni(u) ∧λ n i(v) = −1 so u ∈ i or v ∈ i. hence i is a prime 0-essential ideal of i. (3) suppose that i is a semiprime 0-essential ideal of s. then i is a 0-essential ideal of s. thus by theorem 4.1, λi = (s; λ p i,λ n i) is a 0-essential bf ideal of s. let u ∈ s. if u2 ∈ i, then u ∈ i thus λpi(u) = λ p i(u 2) = 1 and λni(u) = λ n i(u 2) = −1. hence, λ p i(u 2) ≤ λpi(u) and λ n i(u 2) ≥ λni(u). if u2 /∈ i, then λpi(u 2) = 0 ≤ λpi(u) and λ n i(u 2) = 0 ≥ λni(u). thus, λi = (s; λ p i,λ n i) is a semiprime 0-essential bf ideal of s. conversely, suppose that λi = (s; λ p i,λ n i) is a semiprime 0-essential bf ideal of s. then λi = (s; λ p i,λ n i) is a 0-essential bf ideal. thus by theorem 4.1, i is a 0-essential ideal of s. let u ∈ s with u2 ∈ i then λpi(u 2) = 1 and λni(u 2) = −1. by assumption, λpi(u 2) ≤ λpi(u) and λni(u 2) ≥ λni(u). thus λ p i(u) = 1 and λ n i(u) = −1 so u ∈ i. hence i is a semiprime 0-essential ideal of s. � acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-rim024) the authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. baupradist, b. chemat, k. palanivel, r. chinram, essential ideals and essential fuzzy ideals in semigroups, j. discrete math. sci. cryptography. 24 (2020), 223–233. https://doi.org/10.1080/09720529.2020.1816643. [2] c.s. kim, j.g. kang, j.m. kang, ideal theory of semigroups based on the bipolar valued fuzzy set theory, ann. fuzzy math. inform. 2 (2012), 193-206. [3] t. gaketem, a. iampan, essential up-ideals and t-essential fuzzy up-ideals of up-algebras, icic express lett. 15 (2021), 1283-1289. https://doi.org/10.24507/icicel.15.12.1283. [4] t. gaketem, p. khamrot, a. iampan, essential up-filters and t-essential fuzzy up-filters of up-algebras, icic express lett. 16 (2021), 1057-1062. https://doi.org/10.24507/icicel.16.10.1057. [5] k. nobuaki, fuzzy bi-ideals in semigroups, comment. math. univ. st. paul 5 (1979), 128-132. [6] k. lee, bipolar-valued fuzzy sets and their operations, in: proceeding international conference on intelligent technologies bangkok, thailand (2000), 307-312. [7] u. medhi, k. rajkhowa, l.k. barthakur, h.k. saikia, on fuzzy essential ideals of rings, adv. fuzzy sets syst. 5 (2008), 287-299. https://doi.org/10.1080/09720529.2020.1816643 https://doi.org/10.24507/icicel.15.12.1283 https://doi.org/10.24507/icicel.16.10.1057 12 int. j. anal. appl. (2023), 21:1 [8] u. medhi, h.k. saikia, on t-fuzzy essential ideals of rings, j. pure appl. math. 89 (2013), 343-353. https: //doi.org/10.12732/ijpam.v89i3.5. [9] j.n. mordeson, d.s. malik, n. kuroki, fuzzy semigroup, springer, berlin, (2003). [10] n. panpetch, t. muangngao, t. gaketem, some essential fuzzy bi-ideals and essential fuzzy bi-ideals in a semigroup, j. math. computer sci. 28 (2023), 326-334. http://dx.doi.org/10.22436/jmcs.028.04.02. [11] s. wani, k. pawar, on essential ideals of a ternary semiring, sohag j. math. 4 (2017), 65-69. https://doi.org/ 10.18576/sjm/040301. [12] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.12732/ijpam.v89i3.5 https://doi.org/10.12732/ijpam.v89i3.5 http://dx.doi.org/10.22436/jmcs.028.04.02 https://doi.org/10.18576/sjm/040301 https://doi.org/10.18576/sjm/040301 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. essential bipolar valued fuzzy ideals in a semigroup. 4. 0-essential bf ideal. references international journal of analysis and applications volume 18, number 6 (2020), 1029-1036 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1029 received march 31st, 2020; accepted june 1st, 2020; published october 13th, 2020. 2010 mathematics subject classification. 54a05. key words and phrases. topological spaces; semi generalized open sets; semi generalized closed sets. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1029 semi generalized open sets and generalized semi closed sets in topological spaces abdelgabar adam hassan1,2,* 1department of mathematics, college of science and arts in tabrjal, jouf university, kingdom of saudi arabia 2department of mathematics, university of nyala, nyala, sudan *corresponding author: aahassan@ju.edu.sa abstract. in this paper we introduce a new class of semi generalized open sets, generalized semi closed sets in topological spaces, and studied some of its basic properties. moreover we define approximately semi generalized open sets and approximately generalized semi closed sets in topological spaces. further we obtained some properties of closure, semi generalized open sets and generalized semi closed sets in topological spaces. 1. introduction the study of generalized closed sets in topological space was initiated by levine in [23]. biswas [18], njasted[15], mashhour[12], robert[19], bhattacharya [13], arya and nour [12 ]. introduced and investigated semi closed, -open and -closed, pre-open, semi*-open, sg-closed, gsclosed, gp-closed, g-closed, g* closed, s*g-closed, w-closed, g*-closed respectively. topology is an important and interesting area of mathematics, the study of which will not only introduce to new concepts and theorems but also put into context old ones like continuous functions [1]. however, to say just this is to understate the significance of topology. it is so fundamental that its influence is evident in almost every other branch of mathematics [3]. topological notions like compactness, connectedness and denseness areas basic to mathematicians of today as sets and functions were https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1029 int. j. anal. appl. 18 (6) (2020) 1030 to those of last century. topology has several different branches, genera l topology, algebraic topology, differential topology and topological algebra, the first, general topology, being the door to the study of the others [3,5]. the aim of this paper is to introduce the concept of semi generalized open sets and generalized semi closed sets in topological spaces, and provide semi generalized open sets and generalized semi closed sets in topological spaces. 2. preliminaries definition 2.1. let 𝑋 be a non – empty sets. a collection of subsets of 𝑋 is said to be a topology on 𝑋 if (i) 𝑋 and the empty set, ∅, belong to τ, (ii) the union of any (finite or infinite) number of sets in τ belong to τ, (iii) the intersection of any two sets in τ belongs to τ. the pair (𝑋, 𝜏) is called a topological space. definition 2.2 let 𝑋 be a non – empty sets and let τ be the collection of all subsets of 𝑋. then τ is called discrete topology on the set 𝑋. the topological space (𝑋, 𝜏) is called a discrete space. observe that the set 𝑋 in definition 1.2. can be any non – empty set. so, there is an infinite number of discrete spaces – one for each𝑋. definition 2.3. let 𝑋 be any non – empty set and 𝜏 = {𝑋, ∅}. then τ is called indiscrete topology and (𝑋, 𝜏) is said to be indiscrete space. definition 2.4. let (𝑋, 𝜏) be a topological space. a subset 𝐴 of 𝑋 is said to be generalized closed if 𝑐𝑙(𝐴) ⊆ ⋃ when ever 𝐴 ⊆ ⋃ and ⋃ is an open in (𝑋, 𝜏). definition 2.5. let (𝑋, 𝜏) be a topological space and 𝐴 ⊆ 𝑋. the generalized closure of 𝐴, denote by 𝑐𝑙∗(𝐴) and is defined by the intersection of all 𝑔 closed sets containing 𝐴 an generalized interior of 𝐴, denoted by 𝑖𝑛𝑡∗(𝐴) and is defined by union of all 𝑔open sets contained in 𝐴. definition 2.6. a subset 𝐴 of a topological space (𝑋, 𝜏) is said to be (i) a semi – open set [9] if 𝐴 ⊆ 𝑐𝑙(𝑖𝑛𝑡(𝐴)) and a semi closed if 𝑖𝑛𝑡(𝑐𝑙(𝐴)) ⊆ 𝐴, (ii) a preopen set [11] if 𝐴 ⊆ 𝑖𝑛𝑡(𝑐𝑙(𝐴)) and a pre closed if 𝑐𝑙(𝑖𝑛𝑡(𝐴)) ⊆ 𝐴, (iii) an ∝ open sets [12] if 𝐴 ⊆ 𝑖𝑛𝑡 (𝑐𝑙(𝑖𝑛𝑡(𝐴))) and ∝ closed sets if 𝑐𝑙 (𝑖𝑛𝑡(𝑐𝑙(𝐴))) ⊆ 𝐴, (iv) a regular set [13] if 𝑖𝑛𝑡(𝑐𝑙(𝐴)) = 𝐴 and a regular closed set if 𝑐𝑙(𝑖𝑛𝑡(𝐴)) = 𝐴, int. j. anal. appl. 18 (6) (2020) 1031 (v) a 𝑄 – set [10] if 𝑐𝑙(𝑖𝑛𝑡(𝐴)) = 𝑖𝑛𝑡(𝑐𝑙(𝐴)). the intersection of all semi – closed (resp. to pre closed, ∝ closed) sets containing a subset 𝐴 of (𝑋, 𝜏) is called semi – closure [14] (resp. pre closure, ∝ closed, 𝑐𝑙𝛼 (𝐴). the semi – interior of 𝐴 is the largest semi – open set contained in 𝐴 and denoted by 𝑆 − 𝑖𝑛𝑡(𝐴). 3. open sets and closed sets in topological definition 3.1. let (𝑋, 𝜏) be any topological space. then the members of 𝜏 are said to be open sets. proposition 3.2. if (𝑋, 𝜏) is any topological space, then (i) 𝑋 and ∅ are open sets, (ii) the union of any (finite or infinite) number of open sets is an open set, (iii) the intersection of any finite number of open sets is an open set. proof. clearly (i) and (ii) are trivial consequences of definition 2.1. and definition 2.1.(i) and (ii). the condition (iii) follows from definition 2.1. definition 3.3. let (𝑋, 𝜏) be a topological space. a subset 𝑆 of 𝑋 is said to be a closed set in (𝑋, 𝜏) if its complement in 𝑋, namely 𝑋\𝑆, is open in (𝑋, 𝜏). proposition 3.4. if (𝑋, 𝜏) is any topological space, then (i) ∅ and 𝑋 are closed sets, (ii) the intersection of any (finite or infinite) number of closed sets is a closed set, (iii) the union of any finite number of closed sets is a closed set. proof. (i) follows immediately proposition 2.2. and definition 2.3., as the complement of 𝑋 is ∅ and the complement of ∅ is 𝑋. to prove that (iii) is true, let 𝑆1, 𝑆2, … , 𝑆𝑛 be closed sets. we are required to prove that 𝑆1 ∪ 𝑆2 ∪ … ∪ 𝑆𝑛is a closed set. it suffices to show, by definition 2.3., that 𝑋\(𝑆1 ∪ 𝑆2 ∪ … ∪ 𝑆𝑛 ) is an open set. as 𝑆1, 𝑆2, … , 𝑆𝑛 are closed sets, their complement 𝑋\𝑆1, 𝑋\𝑆2, … , 𝑋\𝑆𝑛 are open sets. but 𝑋\(𝑆1 ∪ 𝑆2 ∪ … ∪ 𝑆𝑛) = (𝑋\𝑆1) ∩ (𝑋\𝑆2) ∩ … ∩ (𝑋\𝑆𝑛 ) (1) as the right hand side of (1) is a finite intersection of open sets, it is an open set. so that the left hand side of (1) is an open set. hence 𝑆1 ∪ 𝑆2 ∪ … ∪ 𝑆𝑛 is a closed set, as required. so (iii) is true. int. j. anal. appl. 18 (6) (2020) 1032 the proof of (ii) is similar to that of (iii). example 3.5. on any set x there is the trivial topology {∅, x}. there is also the discrete topology whereas any subset of x is open. thus, on a set there can be many topologies. example 3.6. (euclidean topology). in ℝ𝑛 = {(𝑥1, 𝑥2, … , 𝑥𝑛 )|𝑥𝑖 ∈ ℝ}, the euclidean norm of a point 𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛 ) is ‖𝑥‖ = [∑ 𝑥𝑖 2𝑛 𝑖=1 ] 1 2⁄ . the topology generated by this norm is called the euclidean topology of ℝ𝑛. 4. semi generalized * b open sets in this part, we introduce semi generalized* b open sets in topological spaces by using the notion of semi generalized bopen sets, and study some of their properties. definition 4.1. a subset 𝐴 of a topological space (𝑋, 𝜏), is called semi generalized * b open set if 𝐴𝑐 is semi generalized – b closed in 𝑋. we denote the family of all semi generalized – b open sets in 𝑋 by 𝑠𝑔 𝑏 − 𝑂(𝑋). theorem 4.2. if 𝐴 and 𝐵 are 𝑠𝑔 𝑏 – open sets in a space 𝑋. then 𝐴 ∩ 𝐵 is also 𝑠𝑔 𝑏 – open set in 𝑋. proof. if 𝐴 and 𝐵 are 𝑠𝑔 𝑏 – open sets in a space 𝑋. then 𝐴𝑐 and 𝐵𝑐 are 𝑠𝑔 𝑏 – closed sets in a space 𝑋. therefore 𝐴𝑐 ∪ 𝐵𝑐 is also 𝑠𝑔 𝑏 – closed set in 𝑋. (i.e.) 𝐴𝑐 ∪ 𝐵𝑐 = (𝐴 ∩ 𝐵)𝑐 is a 𝑠𝑔 𝑏 – closed set in 𝑋. therefore 𝐴 ∩ 𝐵 𝑠𝑔 𝑏 – open set in 𝑋. 5. semi generalized * b closed sets in this part, we introduce semi generalized* b – closed set and investigate some of its properties. definition 5.1. (closed set). let (𝑋, 𝜏) topological space. the set 𝐴 ⊆ 𝑋 is called closed set if and only if 𝐴𝐶 be open set. i.e. 𝐴 closed set 𝐴 ⇔ 𝐴𝐶 open set 𝐴 open set 𝐴 ⇔ 𝐴𝐶 closed set and denote that by 𝔍𝑋 or 𝔍. example 5.2. let the set 𝑋 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒}. define topology 𝜏 into 𝑋 by the flowing: 𝜏 = {𝑋, ∅, {𝑎}, {𝑐, 𝑑}, {𝑎, 𝑐, 𝑑}, {𝑏, 𝑐, 𝑑, 𝑒}} then 𝔍𝑋 = {𝑋, ∅, {𝑏, 𝑐, 𝑑, 𝑒}, {𝑎, 𝑏, 𝑒}, {𝑏, 𝑒}, {𝑎}}. int. j. anal. appl. 18 (6) (2020) 1033 example 5.3. let(𝑋, 𝐷) discrete topological space. we find that each subset from (𝑋, 𝐷) is open and closed at the same time because: 𝑃(𝑋) = 𝐷, ∀ 𝐴 ⊆ 𝑋 ⇔ 𝐴 ∈ 𝐷 ⇔ 𝐴𝐶 closed definition 5.4. a subset 𝐴 of a topological space (𝑋, 𝜏), is called semi generalized * b closed set if 𝑐𝑙(𝐴) ⊂ ∪ whenever 𝐴 ⊂ ∪ and ∪ is semi generalized – b open in 𝑋. theorem 5.5. every closed set is 𝑠𝑔 𝑏 – closed. proof. let 𝐴 be any closed set in 𝑋 such that 𝐴 ⊂ ∪, where ∪ is 𝑠𝑔 open. since 𝑏 𝑐𝑙(𝐴) ⊂ 𝑐𝑙(𝐴) = 𝐴. therefore 𝑏 𝑐𝑙(𝐴) ⊂ ∪. hence 𝐴 is 𝑠𝑔 𝑏closed set in 𝑋. is example 5.6. let 𝑋 = {𝑎, 𝑏, 𝑐} with 𝜏 = {𝑋, ∅, {𝑏}, {𝑎, 𝑏}}. the set {𝑎, 𝑏} is 𝑠𝑔 𝑏closed set but not a closed set. theorem 5.7. every semi closed set is 𝑠𝑔 𝑏closed set. proof. let 𝐴 be any closed set in 𝑋 and ∪ be any 𝑠𝑔 open set containing 𝐴. since 𝐴 is semi closed set, 𝑏 𝑐𝑙(𝐴) ⊂ 𝑆 𝑐𝑙(𝐴) ⊂ ∪. therefore 𝑏𝑐𝑙(𝐴) ⊂ ∪. hence 𝐴 is 𝑠𝑔 𝑏closed set. corollary 5.8. let 𝑋 = {𝑎1, 𝑎2, 𝑎3} with 𝜏 = {𝑋, ∅, {𝑎1, 𝑎2}}. the set {𝑎1, 𝑎2} is 𝑠𝑔 𝑏closed set but not a semi closed set. theorem 5.9. every ∝closed set is 𝑠𝑔 𝑏closed set. proof. let 𝐴 be any ∝-closed set in, and 𝑋 and ∪ be any 𝑠𝑔 set containing 𝐴. since 𝐴 is ∝closed 𝑏 𝑐𝑙(𝐴) ⊂∝ 𝑐𝑙(𝐴) ⊂ ∪. therefore 𝑏𝑐𝑙(𝐴) ⊂ ∪. hence 𝐴 is 𝑠𝑔 𝑏closed set. theorem 5.10. every 𝑠𝑔 𝑏closed set is 𝑔 𝑏closed set. proof. let 𝐴 be any 𝑠𝑔 𝑏-closed set in, such that ∪ be any open set containing 𝐴. since every open set is 𝑠𝑔 open, we have 𝑏 𝑐𝑙(𝐴).. hence 𝐴 is 𝑔 𝑏closed set. 6. the closure and interior definition 6.1. let 𝑋 be a topological space and 𝐴 ⊆ 𝑋 a subset. the closure of 𝐴 denote�̅� is the intersection of all the closed subsets of 𝑋 that contain 𝐴. the interior of 𝐴 is the union of all the open subsets of 𝑋 that are contained in a. example 6.2. let 𝑋 = ℝ and a = [a, b) with a < b. the closure of 𝐴 is the closed interval [a, b], and the interior of 𝐴 is the open interval (a, b). the closure cannot be smaller, since [a, b) is not closed, and the interior cannot be larger, since[a, b) is not open. theorem 6.3. let 𝑋 be a topological space, 𝑌 ⊂ 𝑋 a subspace, and 𝐴 ⊂ 𝑌 a subset. let �̅� denote the closure of 𝐴 in 𝑋. then the closure of 𝐴 in 𝑌 equals y ∩ a̅ int. j. anal. appl. 18 (6) (2020) 1034 proof. let 𝐵 denote the closure of 𝐴 in 𝑌. to see that b ⊂ y ∩ a̅, note that �̅� is closed in 𝑋, so y ∩ a̅ is closed in 𝑌 and contains 𝐴. hence it contains the closure 𝐵 of a in 𝑌. to prove the opposite inclusion, note that 𝐵 is closed in 𝑌, hence has the form b = y ∩ c for some c that is closed in 𝑋. then a ⊂ b ⊂ c, so c is closed in 𝑋 and contains 𝐴. hence a̅ ⊂ c and y ∩ a̅ ⊂ y ∩ c = b. example 6.4. topologist ’s sine curve. the closure in the euclidean plane of the graph of the function 𝑦 = 𝑠𝑖𝑛 1 𝑥 , 𝑥 > 0 is often called the topologist’s sine curve. figure 1: topologist’s sine curve. denote 𝐴 = {𝑠𝑖𝑛 1 𝑥 |𝑥 > 0} and 𝐵 = {0} × [−1,1]. then the topologist’s sine curve is 𝑋 = 𝐴 ∪ 𝐵. 7. conclusion the aim of this paper is to introduce the concepts of semi generalized open sets and generalized semi closed sets in topological spaces sets and we study some of their properties. furthermore, we discuss the conditions which are added to these concepts in order to coincide with the concept of semi-closed. int. j. anal. appl. 18 (6) (2020) 1035 data availability: no data were used to support this study. acknowledgment: i’m forever indebted my family for their endless patience and encouragement, also i want to recognize and express my thank to anyone helped me. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] j. dugundji, topology, allyn and bacon, inc., boston. 1966. [2] s.a. morris, topology without tears, unisia, 2001. [3] s. sekar, b. jothilakshmi, on semi generalized star b – closed set in topological spaces, int. j. pure appl. math. 113 (2017), 93-102. [4] s. lipschutz, schaum’s outline of theory and problems of general topology, mcgraw-hill, new york. 1965. [5] d. lyappan, n. nagaveni, on semi generalized b-continuous maps, semi generalized b-closed maps in topological space, int. j. math. anal. 6 (2012), 1251-1264. [6] s. s. benchalli, p. g. patil, p. m. nalwad, generalized ωα-closed sets in topological spaces, j. new results sci. 7 (2014), 7–19. [7] n. biswas, on characterization of semi continuous functions, atti accad. naz. lincei. rend. cl. sci. fsi. mat. natur. 48 (1970), 399–402. [8] p. bhattacharyya, b. k. lahiri, semi-generalized closed sets in topology, ind. j. math. 29 (1987), 375382. [9] n. biswas, on characterizations of semi-continuous functions, atti. accad. naz. lincei rend. cl. sci. fis. mat. natur. 8 (1970), 399–402. [10] s.g. crossly, s.k. hildebrand, semiclosure, texas j. sci. 22(1971), 99-112. [11] s.g. crossly, s.k. hildebrand, semitopological properties, fund. math. 74 (1974), 233-254. [12] s.p. arya, t. nour, characterizations of snormal spaces, indian j. pure. appl. math. 21 (8) (1990), 717719. [13] p. bhattacharya, b. k. lahiri, semigeneralized closed sets in topology, indian j. math. 29 (1987), 376382. [14] r. devi, studies on generalizations of closed maps and homeomorphisms in topological spaces, ph.d. thesis, bharathiar university, coimbatore (1994). int. j. anal. appl. 18 (6) (2020) 1036 [15] n. levine, on the commutivity of the closure and interior operator in topological spaces, amer. math. mon. 68 (1961), 474-477. [16] n. levine, semi-open sets and semi-continuity in topological spaces, amer math. mon. 70 (1963), 3641. [17] n. levine, generalized closed sets in topology, rend. circ. mat. palermo, 19 (1970), 89–96. [18] n. biswas, on characterizations of semicontinuous functions, atti. accad. naz. lincei. rend. cl. sci. fis. mat. natur. 48 (1970), 399-402. [19] s. s. benchalli, p. g. patil, t. d. rayanagaudar, ωα-closed sets is topological spaces, glob. j. appl. math. math. sci. 2 (2009), 53-63. [20] d. andrijivic semi, pre-open sets, mat, 1986, vesnic, pp. 24–32. [21] j. weidmann, spectral theory of ordinary differential operators, springer-verlag berlin heidelberg new york, 1987. [22] p. bhattacharya, b.k. lahiri, semi-generalized closed sets in topology, indian j. math. 29 (1987), 376382. [23] n. levine, generalized closed sets in topology, rand. circ. mat. palermo, 19 (1970), 8996. [24] y. gnanambal, on generalized pre-regular closed sets in topological spaces, indian j. pure appl. math. 28(1997), 351–360. international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 96-103 http://www.etamaths.com on the stabilization of the linear kawahara equation with periodic boundary conditions patricia n. da silva and carlos f. vasconcellos∗ abstract. we study the stabilization of global solutions of the linear kawahara equation (k) with periodic boundary conditions under the effect of a localized damping mechanism. the kawahara equation is a model for small amplitude long waves. using separation of variables, the ingham inequality, multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (k) model. 1. introduction in this paper we study the stabilization of global solutions of the linear kawahara equation (k) with periodic boundary conditions under the effect of a localized damping mechanism, that is, we consider the following problem:  ut + βux + κuxxx + ηuxxxxx + a(x)u = 0 x ∈ (0, 2π), t > 0 u(0, t) = u(2π,t), t > 0 ux(0, t) = ux(2π,t), t > 0 uxx(0, t) = uxx(2π,t), t > 0 uxxx(0, t) = uxxx(2π,t), t > 0 uxxxx(0, t) = uxxxx(2π,t), t > 0 u(x, 0) = u0(x), x ∈ (0, 2π) (1.1) the parameter η is a negative real number, κ 6= 0, β is a real number and a ∈ l∞(0, 2π), a ≥ 0 a.e. in (0, 2π) and we assume that a(x) ≥ a0 > 0 a.e. in an open subinterval ω of (0, 2π), where the damping is effectively acting . in the kawahara equation (1.2) ut + ux + κuxxx + ηuxxxxx + uux = 0, the conservative dispersive effect is represented by the term (κuxxx + ηuxxxxx). this equation is a model for plasma wave, capilarity-gravity water waves and other dispersive phenomena when the cubic kdv-type equation is weak. kawahara [10] pointed out that it happens when the coefficient of the third order derivative in the kdv equation becomes very small or even zero. it is then necessary to take into account the higher order effect of dispersion in order to balance the nonlinear effect. kakutani and ono [9] showed that for a critical value of angle between the magneto-acoustic wave in a cold collision-free plasma and the external magnetic 2010 mathematics subject classification. 35q35, 35b40, 35q53. key words and phrases. exponential decay; periodic boundary; kawahara equation. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 96 stabilization of the linear kawahara equation 97 field, the third order derivative term in the kdv equation vanishes and may be replaced by the fifth order derivative term. following this idea, kawahara [10] studied a generalized nonlinear dispersive equation which has a form of the kdv equation with an additional fifth order derivative term. this equation has also been obtainded by hasimoto [8] for the shallow wave near critical values of surface tension. more precisely, in this work hasimoto found these critical values when the bond number is near to one third. while analyzing the evolution of solutions of the water wave-problem, schneider and wayne [19] also showed that the coefficient of the third order dispersive term in nondimensionalized statements of the kdv equation vanishes when the bond number is equal to one third. the bond number is proportional to the strength of the surface tension and in the kdv equation it is related to the leading order dispersive effects in the water-waves problem. with its disappearance, the resulting equation is just burger’s equation whose solutions typically form shocks in finite time. thus, if we wish to model interesting behavior in the water-wave problem it is necessary to include higher order terms. that is, it is necessary to consider the kawahara equation. in any case, the inclusion of the fifth order derivative term takes into account the comparative magnitude of the coefficients of the third and fifth power terms in the linearized dispersion relation. berloff and howard [3] presented the kawahara equation as the purely dispersive form of the following nonlinear partial differential equation ut + u rux + auxx + buxxx + cuxxxx + duxxxxx = 0. the above equation describes the evolution of long waves in various problems in fluid dynamics. the kawahara equation corresponds to the choice a = c = 0 and r = 1 and describes water waves with surface tension. bridges and derks [6] presented the kawahara equation – or fifth-order kdv-type equation – as a particular case of the general form (1.3) ut + κuxxx + ηuxxxxx = ∂ ∂x f(u,ux,uxx) where u(x,t) is a scalar real valued function, κ and η 6= 0 are real parameters and f(u,ux,uxx) is some smooth function. the form (1.2) occurs most often in applications and corresponds to the choice of f in (1.3) with the form f(u,ux,uxx) = −u 2 2 . as noted by kawahara [10], we may assume without loss of generality that η < 0 in (1.2). in fact, if we introduce the following simple transformations u →−u, x →−x and t → t we can obtain an equation of the form of equation (1.2) in which κ and η are replaced, respectively, by −κ and −η. hagarus et al. pointed out that the kawahara equation (1.4) ut = uxxxxx −εuxxx + uux in which ε is a real parameter models water waves in the long-wave regime for moderate values of surface tension, weber numbers close to 1/3; and that for such weber numbers the usual description of long water waves via the korteweg-de vries (kdv) equation fails since the cubic term in the linear dispersion relation vanishes and fifth order dispersion becomes relevant at leading order, ω(k) = k5 + εk3. 98 silva and vasconcellos positive (resp. negative) values of the parameter ε in (1.4) correspond to weber numbers larger (resp. smaller) than 1/3. dispersive problems have been object of intensive research (see, for instance, the classical paper of benjamin-bona-mahoni [2], biagioni-linares [4], bona-chen [5], menzala et al. [15], rosier [16], and references therein). recently global stabilization of the generalized kdv system have been obtained by rosier-zhang [17] and linares-pazoto[12] with critical exponents. for the stabilization of global solutions of the kawahara under the effect of a localized damping mechanism, see vasconcellos and silva [20, 21]. for controllability problems involving dispersive systems, we can consider the works of russel-zhang [18] and laurent et al. [12] about the kdv system; the paper by linares-ortega [14], where the benjamin-ono equation has been analyzed and the paper of zhang and zhao [22] for the kawahara equation. the total energy associated with the (1.1) system is defined by e(t) = 1 2 ∫ 2π 0 |u(x,t)|2dx = 1 2 ‖u(t)‖2. using the above boundary conditions we prove that de dt = η 2 |uxx(0, t)|2 − ∫ 2π 0 a(x)|u(x,t)|2dx ≤ 0, ∀t > 0. so, e(t) is a nonincreasing function of time. this paper is devoted to analyze the following questions: does the energy e(t) → 0 as t → ∞? is it possible to find a rate of decay of the energy? then, we can state our main result: theorem 1.1. there exist c > 0 and γ > 0 such that the energy e(t) associated to the problem (1.1) satisfies e(t) ≤ ce−γt‖u0‖2l2(0,2π) for all u0 ∈ l2(0, 2π). to prove the above theorem we need some generalizations of ingham inequality (see for instance [1],[7] and [11]), multiplier techniques and compactness arguments. we organize this work as follows. in section 2, we present some auxiliary lemmas, useful to demonstrate our main result. in section 3, we prove theorem 1.1 and in section 4, we present our final remarks. 2. auxiliary lemmas lemma 2.1. consider the problem:  vt + βvx + κvxxx + ηvxxxxx = 0 x ∈ (0, 2π), t > 0 v(0, t) = v(2π,t), t > 0 vx(0, t) = vx(2π,t), t > 0 vxx(0, t) = vxx(2π,t), t > 0 vxxx(0, t) = vxxx(2π,t), t > 0 vxxxx(0, t) = vxxxx(2π,t), t > 0 v(x, 0) = u0(x), x ∈ (0, 2π) (2.5) stabilization of the linear kawahara equation 99 the parameter η is a negative real number, κ 6= 0 and β is a real number. then, for t > 0, there exists a constant c1 = c1(t) > 0 such that ‖u0‖2l2(0,2π) ≤ c1 ∫ t 0 ∫ ω |v(x,t)|2dxdt, where ω is an open subinterval of (0, 2π). proof. we assume a solution v of the system (2.5) can be written as v(x,t) = x(x)t(t). then xt ′ + βtx′ + κtx′′′ + ηtx′′′′′ = 0 that is t ′ t = − βx′ + κx′′′ + ηx′′′′′ x = λ for some constant λ. thus, we obtain  βx′ + κx′′′ + ηx′′′′′ + λx = 0 x ∈ (0, 2π), x(0) = x(2π), x′(0) = x′(2π), x′′(0) = x′′(2π), x′′′(0) = x′′′(2π), x′′′′(0) = x′′′′(2π), (2.6) and (2.7) t ′ −λt = 0 to solve (2.6), we use the characteristic equation ηr5 + κr3 + βr + λ = 0. we can show that the eigenvalues λ are pure imaginary numbers. notice that for each k ∈ z, the function φk(x) = 1 √ 2π eikx is an eigenfunction of (2.6) associated with the eigenvalue λk = (−ηk5 + κk3 −βk)i. furthermore, for any l ∈ z, let ml = #{k ∈ z, λk = λl}. then, ml ≤ 5 for any l and in particular m(l) = 1, if |l| is large enough. moreover, (2.8) lim |k|→∞ |λk −λk+1| = ∞. we have (2.9) x(x) = cke ikx, k ∈ z. then, by (2.7) and (2.9), it follows that (2.10) v(x,t) = ∑ k∈z cke i(kx+σkt), σk = −ηk5 + κk3 −βk where u0(x) = ∑ k∈z cke ikx. 100 silva and vasconcellos as pointed out by, jaffard and micu [7], lim sup n |λn+1 −λn| > 2π t gives a sufficient condition for the validity of an ingham type inequality for each t , since we have (2.8), from an ingham inequality (see for instance theorem 3.5 in baiocchi, komornik and loreti [1] for ingham inequalities for sequences with repeated eigenvalues and with weak gap conditions.), it follows that there exists a constant c = c(t) > 0 such that (2.11) ‖u0‖2l2(0,2π) = ∑ k∈z |ck|2 ≤ c(t) ∫ t 0 ∣∣∣∣∣∑ k∈z cke iσkt ∣∣∣∣∣ 2 dt therefore, using (2.11) and the fubini theorem, we have∫ t 0 ∫ ω |v(x,t)|2dxdt = ∫ ω ∫ t 0 ∣∣∣∣∣∑ k∈z cke ikxeiσkt ∣∣∣∣∣ 2 dtdx ≥ 1 c(t) ∫ ω ∑ k∈z ∣∣ckeikx∣∣2 dx = 1 c(t) ∫ ω ∑ k∈z |ck| 2 dx = l(ω) c(t) ∑ k∈z |ck| 2 = l(ω) c(t) ‖u0‖2l2(0,2π)· � here, we denote by l(ω) the length of subset ω lemma 2.2. let w be a solution of the following problem:  wt + βwx + κwxxx + ηwxxxxx = −a(x)u(x,t) x ∈ (0, 2π), t > 0 w(0, t) = w(2π,t), t > 0 wx(0, t) = vw(2π,t), t > 0 wxx(0, t) = wxx(2π,t), t > 0 wxxx(0, t) = wxxx(2π,t), t > 0 wxxxx(0, t) = wxxxx(2π,t), t > 0 w(x, 0) = 0, x ∈ (0, 2π) (2.12) where a = χω, ω ⊂ (0, 2π) and u is the solution of (1.1). the parameter η is a negative real number, κ 6= 0 and β is a real number. then, for t > 0, there exists a constant c2 = c2(t) > 0 such that ‖w(t)‖2l2(0,2π) ≤ c2 ∫ t 0 ∫ ω |u(x,t)|2dxdt. proof. if we multiply the equation (2.12) by w, integrate in (0, 2π) and use the periodic boundary conditions, we have 1 2 d dt ‖w(t)‖2l2(0,2π) = − ∫ ω u(x,t)w(x,t)dx, t > 0. thus d dt ‖w(t)‖2l2(0,2π) ≤ ∫ ω |u(x,t)|2dx + ∫ 2π 0 |w(x,t)|2dx. stabilization of the linear kawahara equation 101 now, if γ(t) = ‖w(t)‖2 l2(0,2π) , we obtain{ γ′(t) ≤ g(t) + γ(t) γ(0) = 0 where g(t) = ∫ ω |u(x,t)|2dx. hence, by gronwall inequality, there exists a constant c2 = c2(t) > 0, such that γ(t) ≤ c2(t) ∫ t 0 g(t)dt, t ∈ (0,t) and the lemma follows. � lemma 2.3. for each t > 0, there exists a constant c3 = c3(t) > 0 such that 1 2 ‖u0‖2l2(0,2π) ≤ c3 ∫ t 0 ∫ ω |u(x,t)|2dxdt, where u is the solution of (1.1). proof. let v and w be respectively the solutions of the problems (2.5) and (2.12). so we have u = v + w (or v = u−w). now using lemmas 2.1 and 2.2, we obtain ‖u0‖2l2(0,2π) ≤ c1 ∫ t 0 ∫ ω |v(x,t)|2dxdt ≤ 2c1 [∫ t 0 ∫ ω |u(x,t)|2dxdt + ∫ t 0 ∫ ω |w(x,t)|2dxdt ] ≤ 2c1 [∫ t 0 ∫ ω |u(x,t)|2dxdt + c2t ∫ t 0 ∫ ω |u(x,t)|2dxdt ] = 2c1(1 + c2t) ∫ t 0 ∫ ω |u(x,t)|2dxdt. the inequality stated in the lemma holds with c3 = c1(1 + c2t). � 3. proof of theorem 1.1 now, we are able to prove theorem 1.1. in fact, if we multiply the equation in system (1.1) by u and integrate in (0, 2π), we have 1 2 d dt ‖u(t)‖2l2(0,2π) = − ∫ ω |u(x,t)|2dxdt ≤ 0 so, e(t) = 1 2 ‖u(t)‖2 l2(0,2π) is a decreasing function of time and moreover e(t) −e(0) = − ∫ t 0 ∫ ω |u(x,t)|2dxdt. thus (1 + c3)e(t) = −c3 ∫ t 0 ∫ ω |u(x,t)|2dxdt + c3e(0) + e(t). since e(t) ≤ e(0) = 1 2 ‖u0‖2l2(0,2π), it follows, by lemma 2.3, that: (1 + c3)e(t) ≤ c3e(0). 102 silva and vasconcellos therefore e(t) ≤ c3 1 + c3 e(0), t > 0. finally, we use the semigroup property to obtain theorem 1.1. remark 3.1. in the lemma 2.2 and in the theorem 1.1, we can consider a ∈ l∞(0, 2π), a ≥ 0 a.e. in (0, 2π) and assume that a(x) ≥ a0 > 0 a.e. in an open subinterval ω of (0, 2π) and the proofs follow in the same way. 4. final remarks we can observe that, if we consider the parameter β = 0 in the system (1.1) the theorem 1.1 follows similarly. now, we will make some comments concerning the exact controllability for kawahara system: in the linear case, boundary exact controllability is proved, using hum method and multipliers techniques, by vasconcellos-silva [20]. in the nonlinear case, internal exact controllability can be found in zhang-zhao [22], where was considered periodic domain with an internal control acting on an arbitrary small nonempty subdomain of [0, 2π]. aided by the bourgain smoothing property of the kawahara equation on a periodic domain, it was showed that the system is locally exactly controllable. we believe that it is possible to show the boundary exact controllability for linear kawahara system in periodic domain that is, we consider the following problem: given u0 and ut in l 2(0,l), find hj ∈ l2(0,l), j = 0, 1, 2, 3, 4 such that the solution of the bellow system:  ut + βux + κuxxx + ηuxxxxx = 0 x ∈ (0, 2π), t > 0 u(0, t) −u(2π,t) = h0, t > 0 ux(0, t) −ux(2π,t) = h1, t > 0 uxx(0, t) −uxx(2π,t) = h2, t > 0 uxxx(0, t) −uxxx(2π,t) = h3, t > 0 uxxxx(0, t) −uxxxx(2π,t) = h4, t > 0 u(., 0) = u0 (4.13) satisfies u(·,t) = ut . as proved by rosier in [16] for linear kdv system, we could use hum method and generalizations of ingham inequalities to obtain the above problem. references [1] c. baiocchi, v. komornik, p. loreti, ingham-beurling type theorems with weakened gap conditions acta. math. hungar., 97 (2002), 55-95 [2] t.b.benjamin, j.l.bona and j.j.mahony, model equations for long waves in nonlinear dispersive systems, philos. trans. roy. soc. london, ser. a 272 (1972), 47-78. [3] n.g.berloff, and l.n. howard solitary and periodic solutions for nonlinear nonintegrable equations, studies in applied mathematics, 99 (1997), 1-24. [4] h.a.biagioni and f.linares, on the benney-lin and kawahara equations, j. math. anal. appl., 211 (1997), 131-152. stabilization of the linear kawahara equation 103 [5] j.l.bona and h.chen, comparision of model equations for small-amplitude long waves, nonlinear anal. 38 (1999), 625-647. [6] t.j.bridges and g. derks, linear instability of solitary wave solutions of the kawahara equation and its generalizations. siam j.math. anal. 33 (2002), 1356-1378. [7] s. jaffard and s. micu, estimates of the constants in generalized ingham’s inequality and applications to the control of the wave equation, asymptot. anal., 28 (2001), 181-214. [8] h.hasimoto, water waves, kagaku, 40 (1970), 401-408 [japanese]. [9] t. kakutani and h. ono, weak non-linear hydromagnetic waves in a cold collision-free plasma, j. phys. soc. japan, 26 (1969), 1305-1318. [10] t.kawahara, oscillatory solitary waves in dispersive media, phys. soc. japan 33 (1972), 260-264. [11] v.komornik and p.loreti, fourier series in control theory, springer-verlag, new york, 2005. [12] c. laurent, l. rosier and b.-y. zhang, control and stabilization of the korteweg-de vries equation on a periodic domain, comm. in p. d. e., 35 (2010), 707-744. [13] f.linares and a.f.pazoto, on the exponential decay of the critical generalized kortewegde vries with localized damping, proc. amer. math. soc. 135 (2007), 1515-1522. [14] f.linares and j.h.ortega, on the controllability and stabilization of the benjamin-ono equation, esaim control optim. calc.var. 11 (2005), 204-218. [15] g.p.menzala, c.f.vasconcellos and e.zuazua, stabilization of the korteweg-de vries equation with localized damping, quarterly applied math., 60 (2002), 111-129. [16] l.rosier, exact boundary controllability for the korteweg-de vries equation on a bounded domain, esaim, control. optimization and calculus of variations, 2 (1997), 33-55. [17] l.rosier and b.y.zhang, global stabilization of the generalized korteweg-de vries equation posed on a finite domain, siam journal on control and optimization 45 (2006), 927-956. [18] d.l.russell and b.y.zhang, exact controllability and stabilization of the korteweg-de vries equation, trans.amer.math.soc., 348 (1996), 1515-1522. [19] g.schneider and c.e.wayne, the rigorous approximation of long-wavelenght capillarygravity waves, arch.rational mech. anal. 162 (2002), 247-285. [20] c.f. vasconcellos and p.n. silva, stabilization of the linear kawahara equation with localized damping, asymptotic analysis 58 (2008), 229-252. [21] c.f. vasconcellos and p.n. silva, stabilization of the kawahara equation with localized damping, esaim control optim. calc. var. 17 (2011), 102-116. [22] bing-yu zhang and xiangqing zhao, control and stabilization of the kawahara equation on a periodic domain. (english) commun. inf. syst. 12 (2012), 77-95. instituto de matemática e estat́ıstica (ime)uerj, r. são francisco xavier, 524, sala 6016, bloco d cep 20550-013, rio de janeiro, rj, brasil ∗corresponding author international journal of analysis and applications volume 18, number 1 (2020), 71-84 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-71 strong convergence theorem for finite family of generalised asymptotically nonexpansive maps agatha chizoba nnubia1, sheila amina bishop2,∗ 1department of mathematics, nnamdi azikiwe university, awka, nigeria 2department of mathematics, convenant university, ota, nigeria ∗corresponding author: sheilabishop95@yahoo.com; sheila.bishop@covenantuniersity.edu.ng abstract. let k be a nonexpansive retract of a uniformly convex banach space x with retraction p and ti:1··· ,m : k −→ x a finite family of uniformly continuous generalised asymptotically nonexpansive maps with a nonempty common fixed point set f. we provided and proved sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of f . 1. introduction let k be a nonempty subset of a normed linear space e. k is said to be (sequentially) compact if every closed bounded sequence in k has a subsequence that converges in k. k is said to be boundedly compact if every bounded subset of k is compact. finite dimensional spaces are boundedly compact. given a subset s of k, we shall denote by co(s) and ccl(s) the convex hull and the closed convex hull of s respectively. if k is boundedly compact convex and s is bounded, then co(s) and hence ccl(s) are compact convex subsets of k. a map t : k → e is said to be semi-compact if for any bounded sequence {xn} ⊂ k such that received 2019-06-14; accepted 2019-07-18; published 2020-01-02. 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. common fixed point set; generalised asymptotically nonexpansive maps; strong convergence; uniformly convex banach space. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 71 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-71 int. j. anal. appl. 18 (1) (2020) 72 ‖xn − txn‖ → 0 as n → ∞ there exists a subsequence {xnj} of {xn} such that xnj converges strongly to some x∗ ∈ k as j → ∞. the map t is said to be demi-compact at z ∈ e if for any bounded sequence {xn} ⊂ k such that ‖xn − txn‖ → z as n → ∞ there exist a subsequence {xnj} of{xn} and a point p ∈ k such that xnj converges strongly to p as j →∞. (observe that if t is additionally continuous, then p−tp = z). a nonlinear map t : k → e is said to be completely continuous if it maps bounded sets into relatively compact sets. a mapping t : k −→ e is called nonexpansive if and only if for all x,y ∈ k, we have that ‖tx−ty‖≤‖x−y‖. (1.1) k is called a nonexpansive retract of e if there exists a nonexpansive map p : e −→ k which is onto and such that p2 = i. the map p is called the nonexpansive retraction of e onto k. let k be a nonempty subset of a real normed space e. let p : e −→ k be a nonexpansive retraction of e onto k. a nonself map t : k −→ e is called asymptotically nonexpansive mapping if and only if there exists a sequence {µn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 such that for all x,y ∈ k, ‖t(pkt)n−1x−t(pkt)n−1y‖≤ (1 + µn)‖x−y‖ ∀ n ∈ n (1.2) where pk : x −→ k is nonexpansive retraction of e onto k. t is called generalised asymptotically nonexpansive mapping if and only if there exist a sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 1 and lim n→∞ ηn = 0 such that for all x,y ∈ d(t), ‖t(pkt)n−1x−t(pkt)n−1y‖≤ µn‖x−y‖ + ηn n ≥ 1. (1.3) goebel and kirk [3] introduced the class of asymptotically nonexpansive mappings as a generalisation of nonexpansive mappings, zegeye and shahzad [13] introduced the class of generalised asymptotically nonexpansive mappings as a generalization of asymptotically nonexpansive maps. as further generalisation, alber, chidume and zegeye [1] introduced the class of total asymptotically nonexpansive mappings, where t : k −→ h is called total asymptotically nonexpansive if and only if there exist two sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continuous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ k, ‖t(pkt)n−1x−t(pkt)n−1y‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1. (1.4) ofoedu and nnubia [8] gave an example to show that the class of asymptotically nonexpansive mappings is a proper subset of the class of total asymptotically nonexpansive mappings. the class of total asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense. these classes of mappings had been studied by several authors (see e.g., [3], [5], [11], [14]). int. j. anal. appl. 18 (1) (2020) 73 2. preliminaries we shall make use of the following result: a banach space e is said to satisfy opial’s condition if for each sequence {xn}n≥1 ∈ e which converges weakly to a point z ∈ e, we have that lim inf n→∞ ‖xn − z‖ < lim inf n→∞ ‖xn − y‖, ∀y ∈ e, such that y 6= z. it is well known that every hilbert space satisfies opial’s condition(see e.g., [9]). a map t is said to satisfy condition b if there exists f : [0,∞) → [0,∞) strictly increasing, continuous, f(0) = 0 such that for all x ∈ d(t),‖x−tx‖ ≥ f(d(x,f)) where f = f(t) = {x ∈ d(t) : x = tx} and d(x,f) = inf{‖x−y‖ : y ∈ f}. lemma 2.1. [2] let h be a real hilbert space. then for all x,y ∈ h the following inequality holds. ‖x + y‖2 ≤‖x‖2 + 2〈y,x + y〉. lemma 2.2. [2] for any x,y,z in a real hilbert space h and a real number λ ∈ [0, 1], ‖λx + (1 −λ)y −z‖2 = λ‖x−z‖2 + (1 −λ)‖y −z‖2 −λ(1 −λ)‖x−y‖2. lemma 2.3. [12] let {xn} be sequence of nonegative real numbers satisfying the following relation: xn+1 ≤ xn −αnxn + δn, n ≥ n0, where {αn}n≥1 ⊂ (0, 1) and {δn}n≥1 ⊂ r satisfying the following conditions: ∞∑ n=0 αn = ∞, lim n→∞ αn = 0 and lim sup n→∞ δn ≤ 0, then lim n→∞ xn = 0. lemma 2.4. let {µn},{νn} and{ηn} be nonnegative sequences such that∑ n≥0 νn < ∞ , ∑ n≥0 ηn < ∞ and µn+1 ≤ (1 + νn)µn + ηn. then lim n→∞ µn exists. the nearest point projection pk : h −→ k defined from h onto k is the function which assign to each x ∈ h its nearest point denoted by pkx ∈ k. thus pkx is the unique point in k such that ‖x−pkx‖≤‖x−y‖ for all y ∈ k and we have the following lemmas. lemma 2.5. [12]. let k be a closed convex nonempty subset of a real hilbert space h. let x ∈ h, then z = pkx if and only if 〈x−z,y −z〉≤ 0 ∀ y ∈ k. lemma 2.6. let k be a closed convex nonempty subset of a banach space e and let ti∈i : k −→ k where i ∈ i = {1, 2, ...,m}. be finite family of continous nonlinear maps in k such that f = ⋂m i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence of successive approximation satisfying (1) lim n→∞ ‖xn −tixn‖ = 0 ∀i ∈ i int. j. anal. appl. 18 (1) (2020) 74 (2) ‖xn+1 −x∗‖≤ (1 + τn)‖xn −x∗‖ + νn where ∑ n≥0 νn < ∞ and ∑ n≥0 τn < ∞. then, {xn} converges strongly to a common fixed point of ti’s if and only if lim infn→∞d(xn,f) = 0. proof now, d(xn+1,f) ≤ (1 + τn)d(xn,f) + νn (2.1) hence lim infn→∞d(xn,f) = 0 ⇒ limn→∞d(xn,f). also, ∀ k > 0 ‖xn+k+1 −x∗‖ ≤ πkj=0(1 + τn+j)‖xn −x ∗‖ + k∑ j=0 νn+k−jπ j−1 r=0(1 + τn+k−r) ≤ πkj=0(1 + τn+j)(‖xn −x ∗‖ + k∑ j=0 νn+k−j) ≤ q(‖xn −x∗‖ + k∑ j=0 νn+j) so that given any ε > 0 there exists an integer n0 > 0, such that for all n ≥ n0, d(xn,f) < ε4(q+1) and νn+j < ε 4m(q+1) ∀ j = 1, 2, ...,m. so ∃ x∗ ∈ f such that d(xn0,x∗) < ε 4(q+1) that is, ‖xn0 − x∗‖ < ε 4(q+1) now, ‖xn+k −xn‖≤‖xn+k −x∗‖ + ‖xn −x∗‖ ≤ 2q(‖xn0 −x ∗‖ + k∑ j=0 νn0+j) ≤ 2q( ε 4(q + 1) + m ε 4m(q + 1) ) = 2q( ε 2(q + 1) < ε. so, {xn} is a cauchy sequence in e and so it converges to some u∗ ∈ k. but, xn −tixn → 0 as n →∞∀ i and ti is continuous ∀ i. hence, 0 = lim n→∞ (xn −tixn) = lim n→∞ xn −ti( lim n→∞ xn) = u ∗−tiu∗. so that u∗ ∈ f. i.e u∗ = x∗ ∈ f. hence, xn → x∗ as n →∞. this completes the proof since the other part is obvious. lemma 2.7. let k be a closed convex nonempty subset of a banach space e and let ti : k −→ k where i ∈ i = {1, 2, ...,m}. be finite family of continuous nonlinear maps in k such that f = ⋂m i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence of successive approximation satisfying (1) lim n→∞ ‖xn −tixn‖ = 0 ∀i ∈ i (2) ‖xn+1 −x∗‖≤ (1 + τn)‖xn −x∗‖ + νn where ∑ n≥0 νn < ∞ and ∑ n≥0 τn < ∞ then, {xn} converges strongly to a common fixed point of ti’s if one of the ti’s satisfy condition b. int. j. anal. appl. 18 (1) (2020) 75 proof let ti0 satisfy condition b. then ∃f : [0,∞) → [0,∞) with f(0) = 0 and f(d(xn,f)) ≤‖xn −ti0xn‖, hence lim n→∞ f(d(xn,f)) ≤ lim n→∞ ‖xn −ti0xn‖ = 0. therefore, lim n→∞ d(xn,f) = 0. thus by lemma 2.6, {xn} converges strongly to a common fixed point of ti’s, this concludes the proof. lemma 2.8. let k be a closed convex nonempty subset of a banach space e and let ti : k −→ kwhere i ∈ i = {1, 2, ...,m}. be finite family of continuous nonlinear maps in k such that f = ⋂n i=1 f(ti) 6= ∅. suppose the sequence {xn}n≥1 of successive approximation satisfies the following conditions (1) lim n→∞ ‖xn −x∗‖ exists withx∗ ∈ f, (2) lim n→∞ ‖xn −tixn‖ = 0 ∀i ∈ i, (3) {xn}n≥1 has a convergent subsequence {xnj}n≥1. then, {xn} converges strongly to a point of f. proof suppose that {xn} has a convergent subsequence {xnj} and let xnj → p as j →∞, since xn−tixn → 0 as n →∞ for all i ∈{1, 2, ...,n}. it implies that xnj −tixnj → 0 as j →∞ for all i ∈ i. also, by continuity of ti .tixnj → tip as j → ∞ for all i ∈ i . so, ‖p−tip‖ = lim n→∞ ‖xnj −tixnj‖ = 0,∀i which implies that p ∈ f. now, lim n→∞ ‖xn −p‖ exists from our hypothesis and lim n→∞ ‖xnj −p‖ = 0, so lim n→∞ ‖xn −p‖ = 0. thus, {xn} converges strongly to a point of f. remark 2.1. the conditions for which {xn} has a convergent subsequence includes (1) ti is completely continuous ∀i ∈{1, ...,n}. (2) ti is demicompact ∀i ∈{1, ...,n}. (3) ti is semicompact for some i ∈{1, ...,n}. (4) k is compact. (5) k is boundedly compact. proposition 2.1. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let t : k −→ x uniformly continuous generalised asymptotically nonexpansive map with associated sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µn − 1) < ∞ ∞∑ n=0 ηn < ∞ , suppose that f(t) 6= ∅. then f(t) is closed and convex (where f(t) is the fixed point set of t ). int. j. anal. appl. 18 (1) (2020) 76 proof. let {xn} be a sequence in f(t) converging to x∗ ∈ k, then xn = txn∀ n ≥ 0. by continuity of t,x∗ = lim n→∞ xn = lim n→∞ txn = t( lim n→∞ xn) = tx ∗. thus, x∗ ∈ f(t) and f(t) is closed. next, we show that f(t) is convex. for t ∈ (0, 1) and x,y ∈ f(t), put z = (1 − t)x + ty, we show that z = tz. ‖z −t(pkt)n−1z‖2 = ‖z‖2 − 2〈z,t(pkt)n−1z〉 + ‖t(pkt)n−1z‖2 = ‖z‖2 − 2(1 − t)〈x,t(pkt)n−1z〉− 2t〈y,t(pkt)n−1z〉 +‖t(pkt)n−1z‖2 = ‖z‖2 + (1 − t)‖x−t(pkt)n−1z‖2 + t‖y −t(pkt)n−1z‖2 −(1 − t)‖x‖2 − t‖y‖2 ≤ ‖z‖2 + (1 − t)(µn‖x−z‖ + ηn)2 + t(µn‖y −z‖ + ηn)2 −(1 − t)‖x‖2 − t‖y‖2 = ‖z‖2 + (1 − t) ( µ2n‖x−z‖ 2 + (2µn‖x−z‖ + ηn)ηn ) +t ( µ2n‖y −z‖ 2 + (2µn‖y −z‖ + ηn)ηn ) − (1 − t)‖x‖2 − t‖y‖2 = ‖z‖2 + (1 − t)µ2n(‖x‖ 2 −‖z‖2 − 2〈x,z〉) +tµ2n(‖y‖ 2 −‖z‖2 − 2〈y,z〉) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n − (1 − t)‖x‖ 2 − t‖y‖2 = (1 + µ2n)‖z‖ 2 + (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 ) − 2µn ( (1 − t)〈x,z〉 +t〈y,z〉 ) + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n ≤ (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) + 2µ2n‖z‖ 2 −2µ2n ( (1 − t)〈x,z〉 + t〈y,z〉 ) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n ( (1 − t)〈x,z〉 +t〈y,z〉− (1 − t)‖z‖2 − t‖z‖2 ) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n ( (1 − t)〈x−z,z〉 +t〈y −z,z〉 ) + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n〈(1 − t)(x−z) + t(y −z),z〉 +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n int. j. anal. appl. 18 (1) (2020) 77 = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2nt(1 − t)(〈x−y + y −x,z〉 +η2n + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) + 4t(1 − t)‖x−y‖µnηn + η2n. thus, lim n→∞ ‖z − t(pkt)n−1z‖ = 0, which implies that lim n→∞ t(pkt) n−1z = z and hence z = lim n→∞ t(pkt) n−1z = tpk ( lim n→∞ t(pkt) n−2z) = tpkz = tz. thus, z ∈ f(t). this completes the proof. proposition 2.2. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let ti : k −→ x (i = 1,...,m) be a finite family of uniformly continuous generalised asymptotically nonexpansive map with associated sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µin − 1) < ∞, ∞∑ n=0 ηin < ∞. suppose that f = ⋂m i=1 f(ti) 6= ∅. then f is closed and convex. proof. by lemma 2.1, we have that f(ti) is closed for each i. now, f = ⋂m i=1 f(ti) is a finite intersection of closed sets, hence closed. also, by lemma 2.1, we have that f(ti) is convex for each i. since f = ⋂m i=1 f(ti) is a finite intersection of convex set, we have that f is convex. proposition 2.3. suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct for all ≥ k, then t is total asymptotically nonexpansive if and only if t is generalised asymptotically nonexpansive. proof it is known that every generalised asymptotically nonexpansive map is total asymptotically nonexpansive, so it suffices to show that that every total asymptotically nonexpansive with the condition of our hypothesis is generalised asymptotically nonexpansive. now, let t be such that ‖tnx−tny‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1 (2.2) since φ is continuous, it follows that φ reaches its maximum (say c0) on the interval [0,k]; moreover, φ(t) ≤ ct whenever t > k. thus, φ(t) ≤ c0 + ct ∀ t ∈ [0, +∞). (2.3) so, we have, ‖tnx−tny‖ ≤ ‖x−y‖ + µn(c0 + c‖x−y‖) + ηn n ≥ 1 = (1 + µnc)‖x−y‖ + µnc0 + ηn = (1 + νn)‖x−y‖ + γn where νn = µnc and γn = µnc0 + ηn. this completes the proof. int. j. anal. appl. 18 (1) (2020) 78 corollary 2.1. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let t : k −→ x be a uniformly continuous total asymptotically nonexpansive map with associated sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µn − 1) < ∞ ∞∑ n=0 ηn < ∞ . suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct∀ t ≥ k, and that f(t) 6= ∅ then f(t) is closed and convex. corollary 2.2. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let ti : k −→ x (i = 1,...,m) be a finite family of uniformly continuous total asymptotically nonexpansive maps with associated sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µin−1) < ∞ ∞∑ n=0 ηin < ∞ . suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct for all t ≥ k, and that f =⋂m i=1 f(ti) 6= ∅ then f is closed and convex. 3. main results proposition 3.1. let h be a normed linear space, let k be a closed convex nonempty subset of h and let tik −→ h (i ∈ i = {1, ...,m}) be a finite family of continuous generalised asymptotically nonexpansive map with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 1 and lim n→∞ ηn = 0 with ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ . suppose that f(t) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by starting with an arbitrary x0 ∈ k, define by yn,i = pk [(1 −αn)xn + αnti(pti)n−1yn,i−1], yn,0 = xn; yn,m = xn+1 = yn+1 n ≥ 0 , (3.1) where {αn}n≥1, is a sequence in (0, 1) such that 0 < ζ < βn < � < 1 ∀ n ≥ 1. let x∗ ∈ f, then lim n→∞ ‖xn−x∗‖ exist proof. let x∗ ∈ f, j ∈ i then from (3.1) we have that ‖yn,j −x∗‖ = ‖pk [(1 −αn)xn + αntj(ptj)n−1yn,j−1] −px∗‖ (3.2) ≤ (1 −αn)‖xn −x∗‖ + αn‖tj(ptj)n−1yn,j−1 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αn(µn,j)‖yn,j−1 −x∗‖ + ηin). = (1 −αn)‖xn −x∗‖ + αnµn,j(‖yn,j−1 −x∗‖ + αnηn,j) (3.3) int. j. anal. appl. 18 (1) (2020) 79 ‖yn,1 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αnµn,1(‖yn,0 −x∗‖ + αnηn,1) ≤ ( 1 + αn(µn,1−1) ) ‖xn −x∗‖ + αnηn,1 (3.4) ‖yn,2 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αnµn,2 ( (1 + α(µn, 1 − 1))‖xn −x∗‖ + αnηn,1 ) + αnηn,2. ≤ ( 1 + αn(µn,2 − 1) + α2nµ(n, 2)(µn,1 − 1) ) ‖xn −x∗‖ + α2nµn,2ηn,1 + αnηn,2. (3.5) ‖yn,3 −x∗‖ ≤ ( 1 + αn(µn,3 − 1) + α2nµ(n, 3)(µn,2 − 1) + α 3 nµ(n, 3)µn,2(µn,1 − 1) ) ‖xn −x∗‖ +αnηn,3 + α 2 nµn,3ηn,2 + α 3 nµn,3µn,2ηn,1. hence, ‖yn,j −x∗‖ ≤ ( 1 + j∑ t=1 αtnπ t−1 s=1µn,j−s+1(µn,j−t+1 − 1) ) ‖xn −x∗‖ + j∑ t=1 αtnπ t−1 s=1µn,j−s+1ηn,j−t+1. ‖xn+1 −x∗‖ ≤ ( 1 + m∑ t=1 αtnπ t−1 s=1µn,m−s+1(µn,m−t+1 − 1) ) ‖xn −x∗‖ + m∑ t=1 αtnπ t−1 s=1µn,m−s+1ηn,m−t+1 ≤ ( 1 + qm−1b m∑ j=1 (µn,j − 1) ) ‖xn −x∗‖ + qm−1b m∑ j=1 ηn,j. (since there exists n0 such that µn,i ≤ q for all n ≥ n0,∀ j ∈ i) so, lim n→∞ ‖xn − x∗‖ exist; and hence {xn},{yn,j} are bounded. theorem 3.1. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let ti : k −→ e be a finite family of uniformly continuous generalised asymptotically nonexpansive maps with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 1, lim n→∞ ηin = 0, ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ int. j. anal. appl. 18 (1) (2020) 80 suppose that f = ⋂n i=1 f(ti) is not empty and let {xn}n≥1 be a sequence generated iteratively by (3.1 ) where {αn}n≥1 is a sequence in (0, 1) satisfying the following conditions: ∞∑ n=1 αn < ∞, 0 < ζ < αn < � < 1 ∀ n ≥ 1 , then ∀j ∈{1, 2, ...,m}, lim n→∞ ‖xn −tjxn‖ = 0 and {xn}n≥1 converges weakly to a point of f . proof. let x∗ ∈ f ‖yn,j −x∗‖2 ≤ (1 −αn)‖xn −x∗‖2 + αn‖tj(ptj)n−1yn,j−1 −x∗‖ (3.6) −αn(1 −αn)g(‖xn −tj(ptj)n−1yn,j−1‖) ≤ (1 −αn)‖xn −x∗‖2 + αn(µn,j‖yn,j−1 −x∗‖ + ηn,j)2 −αn(1 −αn)g(‖xn −tj(ptj)n−1yn,j−1‖) ≤ (1 −αn)‖xn −x∗‖2 + αnµ2n,j‖yn,j−1 −x ∗‖2 +αn(2µn,j‖yn,j−1 −x∗‖ + ηn,j)ηn,j −αn(1 −αn)g(‖xn −tj(ptj)n−1yn,j−1‖) so ‖yn,1 −x∗‖2 ≤ ( 1 + αn(µ 2 n,1 − 1) ) ‖xn −x∗‖2 + αn(2µn,1‖xn −x∗‖ + ηn,1)ηn,1 −αn(1 −αn)g(‖xn −t1(pt1)n−1xn‖) ‖yn,2 −x∗‖2 ≤ ( 1 + αn(µ 2 n,2 − 1) + α 2 nµ 2 n,2(µ 2 n,1 − 1) ) ‖xn −x∗‖2 +αn(2µn,2‖yn,1 −x∗‖ + ηn,2)ηn,2 + α2nµ 2 n,2(2µn,1‖xn −x ∗‖ + ηn,1)ηn,1 −αn(1 −αn)g(‖xn −t2(pt2)n−1yn,1‖) −α2nµ 2 n,2(1 −αn)g(‖xn −t1(pt1) n−1xn‖) so, ‖yn,j −x∗‖2 ≤ ( 1 + j∑ t=1 αtnπ t−1 s=1µ 2 n,j−s+1(µ 2 n,j−t+1 − 1) ) ‖xn −x∗‖2 + j∑ t=1 αtnπ t−1 s=1µ 2 n,j−s+1(2µn,j−t+1‖yn,j−t−1 −x ∗‖ + ηn,j−t+1)ηn,j−t+1πt−1s=0µ 2 n,j−s −(1 −αn) j∑ t=1 αtng(‖xn −tj−t+1(ptj−t+1) n−1yn,j−t‖)πt−1s=1µ 2 n,j−s+1 int. j. anal. appl. 18 (1) (2020) 81 hence, ‖xn+1 −x∗‖2 ≤ ( 1 + m∑ t=1 α t nπ t−1 s=1µ 2 n,m−s+1(µ 2 n,m−t+1 − 1) ) ‖xn −x∗‖2 + m∑ t=1 α t nπ t−1 s=1µ 2 n,m−s+1(2µn,m−t+1‖yn,m−t −x ∗‖ + ηn,m−t+1)ηn,m−t+1 −(1 −αn) m∑ t=1 α t nπ t−1 s=1µ 2 n,m−s+1g ( ‖xn −tm−t+1(ptm−t+1)n−1yn,m−t‖ ) ≤ ( 1 + q 2(m−1) b m∑ j=1 (µ 2 n,j − 1) ) ‖xn −x∗‖ + q2(m−1)b m∑ j=1 (2µn,j‖yn,j−1 −x∗‖ +ηn,j )ηn,j‖−am(1 −αn) m∑ j=1 g(‖xn −tm−j+1(ptm−j+1)n−1yn,m−j‖) ≤ ( 1 + q 2(m−1) b m∑ j=1 (µ 2 n,j − 1) ) ‖xn −x∗‖ +q 2(m−1) b m∑ j=1 ηn,j −am(1 −αn) m∑ j=1 g(‖xn −tm−j+1(ptm−j+1)n−1yn,m−j‖) so, ‖xn+1 −x∗‖2 ≤ ( 1 + d0 m∑ j=1 (µ2n,j − 1) ) ‖xn −x∗‖ +d1 m∑ j=1 ηn,j −d2 m∑ j=1 g(‖xn −tj(ptj)n−1yn,j−1‖) so, lim n→∞ g(‖xn −tj(ptj)n−1yn,j−1‖) = 0, thus lim n→∞ ‖xn −tj(ptj)n−1yn,j−1‖ = 0 ∀ j = 1, ...,m. now, ‖xn −tj(ptj)n−1xn‖ ≤ ‖xn −tj(ptj)n−1yn,j−1‖ + ‖tj(ptj)n−1yn,j−1 −tj(ptj)n−1xn‖ ≤ ‖xn −tj(ptj)n−1yn,j−1‖ + µn,j‖yn,j−1 −xn‖ + ηn,j ≤ ‖xn −tj(ptj)n−1yn,j−1‖ + µn,jαn‖xn −tj−1(ptj−1)n−1yn,j−2‖ + ηn,j hence, lim n→∞ ‖xn −tj(ptj)n−1xn‖ = 0 ∀ j = 1, ...,m. further, ‖xn −tjxn‖≤‖xn −tj(ptj)n−1yn,j−1‖ + ‖tj(ptj)n−1yn,j−1 −tjxn‖ ‖(ptj)n−1yn,j−1 −xn‖ ≤ ‖tj(ptj)n−2yn,j−1 −xn‖ ≤ ‖tj(ptj)n−2yn,j−1 −tj(ptj)n−2yn−1,j−1‖ +‖tj(ptj)n−2yn−1,j−1 −xn−1‖ + ‖xn−1 −xn‖ ≤ µn−1,j‖yn,j−1 −yn−1,j−1‖ + ηn−1,i +‖xn−1 −tj(ptj)n−2yn−1,j−1‖ + ‖xn −xn−1‖ int. j. anal. appl. 18 (1) (2020) 82 ‖yn,j −yn−1,j‖ ≤ ‖yn,j −xn‖ + ‖xn −xn−1‖ + ‖xn−1 −yn−1,j ≤ αn‖xn −tj(ptj)n−1yn,j−1‖ + αn−1‖xn−1 −tm(ptm)n−2yn−1,m−1‖ +αn−1‖xn−1 −tj(ptj)n−2yn−1,j−1‖ so, lim n→∞ ‖yn,j −yn−1,j‖ = 0. also, lim n→∞ ‖xn −xn−1‖ = 0 so that lim n→∞ ‖(ptj)n−1yn,j−1 −xn‖ = 0. hence lim n→∞ ‖xn −tjxn‖ = 0 ∀ j = 1, ...,m. by reflexivity ∃ z ∈ k and {xnj}⊂{xn} such that, {xnj}→w z as j →∞. since, xnj−tixnj → 0 as j →∞ ∀i then z ∈ f(ti) ∀i and so z ∈ f = m⋂ i=1 f(ti). let ωw(xn) be subsequential limit set of the sequence {xn}. let q ∈ ωw(xn) arbitrary. then ∃ {xnr} ⊂ {xn} 3 {xnr} converges weakly q and xnr − tixnr → 0 as r → ∞ ∀i. thus, ωw(xn) ⊆ f. thus {xn}n≥1 converges weakly to a point of f . theorem 3.2. let k,x,p, t ′is,f,{xn} be as in theorem 3.1 then, {xn} converges strongly to a fixed point of t if and only if lim infn→∞d(xn,f) = 0 (where f= f(t)). the proof follows from lemma 2.6, since from theorem 3.1 and it’s proof, the conditions of the lemma are satisfied. theorem 3.3. let k,x,p, t ′is,f,{xn} be as in theorem 3.1 then, {xn} converges strongly to a common fixed point of ti’s if one of the ti’s satify condition b. the proof follows from lemma 2.7, since from the proof of theorem 3.1, the conditions of the lemma are satisfied. theorem 3.4. let k,x,p,t ′is,f,{xn} be as in theorem 3.1 then, {xn} converges strongly to a common fixed point of ti’s if {xn}n≥1 has a convergent subsequence {xnj}n≥1 the proof follows from lemma 2.8, since from the proof of theorem 3.1, the conditions of the lemma are satisfied. as a result of the proposition 2.3 we have the following results. theorem 3.5. let k be a nonexpansive retract of a uniformly convex banach space x with nonexpansive retraction p . let ti : k −→ e be a finite family of uniformly continuous total asymptotically nonexpansive maps with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin, ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ and with function φ : [0, +∞) −→ [0, +∞) satisfying φ(t) ≤ m0t for all t > m1, for some constants m0,m1 > 0. suppose that f = ⋂m i=1 f(ti) is not empty and let {xn}n≥1 be a sequence generated iteratively int. j. anal. appl. 18 (1) (2020) 83 by (3.1 ) where {αn}n≥1 is a sequence in (0, 1) satisfying the following conditions: ∞∑ n=1 αn < ∞, 0 < ζ < αn < � < 1 ∀ n ≥ 1, then for all j ∈{1, 2, ...,m}, lim n→∞ ‖xn −tjxn‖ = 0 and {xn}n≥1 converges weakly to a point of f . theorem 3.6. let k,x,p,t ′is,f,{xn} be as in theorem 3.5 then, {xn} converges strongly to a fixed point of t if and only if lim infn→∞d(xn,f) = 0. theorem 3.7. let k,x,p, t ′is,f,{xn} be as in theorem 3.5. then, {xn} converges strongly to a common fixed point of ti’s if one of the ti’s satify condition b. theorem 3.8. let k,x,p,t ′is,f,{xn} be as in theorem 3.5 then, {xn} converges strongly to a common fixed point of ti’s if {xn}n≥1 has a convergent subsequence {xnj}n≥1. our iterative process generalise some of the existing ones, our theorems improves, generalise and extend several known results and our method of proof is of independent interest. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] alber, y., chidume, c. e. and zegeye, h., approximating fixed points of total asymptotically nonexpansive mappings.fixed point theory appl., 2006(2006), article id 10673. [2] chidume, c. e.; geometric properties of banach spaces and nonlinear iterations. springer verlag series: lecture notes in mathematics vol. 1965(2009) xvii, 326p. [3] goebel, k. and kirk, w.a., a fixed point theorem for asymptotically nonexpansive mappings. proc. amer. math. soc. 35(1972), 171-174. [4] isiogugu, f.o. and osilike, m.o., fixed point and convergence theorems for certain classes of mappings.j. nigerian math. soc., 31(2013), 147-165. [5] lim, t.c. and xu, h.k., fixed point theorems for asymtotically nonexpansive mappings. nonlinear anal., theory methods appl., 22(1994), 1345-1355. [6] moore, c. and nnoli, b.v.c., strong convergence of averaged approximants for lipschitz pseudocontrative maps. j. math. anal. appl., 260(2006), 169-178. [7] moore, c. and ofoedu, e. u., convergence of iteration process to random fixed points of uniformly llipschitzian asymptotically hemi-contractive random maps in banach spaces. j. nigerian math. soc., 25(2006), 87-94. [8] ofoedu, e.u. and nnubia, a.c., approximation of minimum-norm fixed point of total asymptotically nonexpansive mapping. afrika matematika, 26(2015), 699-715. [9] osilike, m.o. and shehu,y., explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps. appl. math. comput., 213(2009), 584-553. [10] osilike, m.o. and shehu,y., explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps in banach spaces. computers math. appl., 57(2009), 1502-1510. int. j. anal. appl. 18 (1) (2020) 84 [11] okeke, g.a., bishop, s. a. and khan, s.h., iterative approximation of fixed point of multivalued ρ-quasi-nonexpansive mappings in modular function spaces with applications. j. funct. spaces, 2018(2018), article id 1785702. [12] takahashi, w., nonlinear functional analysisfixed point theory and applications. yokohanna publisher inc. yokohanna, (2000). [13] zegeye, h. and shahzad, n., strong convergence of an implicit iteration process for a finite family of generalised asymptotically quasigonexpansive maps, appl. math. comput., 189(2007), 1058–1065. [14] zegeye, h. and shahzad n., approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings, fixed point theory appl., 2013(2013), article id 1. 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 3, number 1 (2013), 60-67 http://www.etamaths.com fixed points of expansive type mappings in 2-banach spaces prabha chouhan1,∗, neeraj malviya2 abstract. in present paper, we define expansive mappings in 2-banach space and prove some common unique fixed point theorems which are the extension of results of wang et al. [12] and rhoades [9] in 2-banach space. 1. introduction the research about fixed points of expansive mapping was initiated by machuca (see [6]). later jungck discussed fixed points for other forms of expansive mapping (see [5]). in 1982, wang et al. (see [12]) presented some interesting work on expansive mappings in metric spaces which correspond to some contractive mapping in [10]. also, zhang has done considerable work in this field. in order to generalize the results about fixed point theory, zhang (see [14]) published his work fixed point theory and its applications, in which the fixed point problem for expansive mapping is systematically presented in a chapter. as applications, he also investigated the existence of solutions of equations for locally condensing mapping and locally accretive mapping. on the other hand gahler ([2],[3]) investigated the idea of 2-metric and 2-banach spaces and proved same results. subsequently several authors including iseki [4], rhoades [8], white [13], panja and baisnab [7] and saha et al [11] studied various aspects of the fixed point theory and proved fixed point theorems in 2-metric spaces and 2-banach spaces. recently, the study about fixed point theorem for expansive mapping is deeply explored and has extended too many others directions. motivated and inspired by the above work, in this paper we investigate fixed point for expansive mapping in 2-banach spaces. the presented theorems extend, generalize and improve many existing results in the literature [9] [12]. 2. preliminaries definition 2.1. let x be a real linear space and ‖ ., . ‖ be a non-negative real valued function defined on x ×x satisfying the following conditions: i) ‖x,y‖ = 0 if and only if x and y are linearly dependent in x, ii) ‖x,y‖ = ‖y,x‖ for all x,y ∈ x iii) ‖x,ay‖ =| a | ‖x,y‖ a being real, x,y ∈ x iv) ‖x,y + z‖ = ‖x,y‖ + ‖x,z‖ for all x,y,z ∈ x 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. 2-normed space, 2-banach space, expansive mapping, fixed point. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 60 fixed points of expansive type mappings 61 then ‖ ., . ‖ is called a 2-norm and the pair (x,‖ ., . ‖) is called a linear 2normed space. some of the basic properties of 2-norms are that they are non negative satisfying ‖x,y + ax‖ = ‖x,y‖, for all x,y ∈ x and all real numbers a. definition 2.2. a sequence {xn} in a linear 2-normed space (x,‖ ., . ‖) is called cauchy sequence if lim m,n→∞ ‖ xm −xn,y ‖= 0 for all y in x. definition 2.3. a sequence {xn} in a linear 2-normed space (x,‖ ., . ‖) is said to be convergent if there is a point x in x such that lim n→∞ ‖ xn −x,y ‖= 0 for all y in x. if {xn} converges to x, we write {xn}→ x as n →∞. definition 2.4. a linear 2-normed space x is said to be complete if every cauchy sequence is convergent to an element of x. we then call x to be a 2banach space. definition 2.5. let x be a 2-banach space and t be a self-mapping of x. t is said to be continuous at x if for every sequence {xn} in x, {xn} → x as n → ∞ implies {t(xn)}→ t(x) as n →∞ . example 2.6. let x is r3 and consider the following 2-norm on x as ‖ x, y ‖ = ∣∣∣∣∣∣det   i j kx1 x2 x3 y1 y2 y3   ∣∣∣∣∣∣ , where x = (x1,x2,x3) and y = (y1,y2,y3). then (x,‖ ., . ‖) is a 2-banach space. example 2.7. let pn denotes the set of all real polynomials of degree ≤ n, on the interval [0,1]. by considering usual addition and scalar multiplication, pn is a linear vector space over the reals. let {xo,x1, ..........,x2n} be distinct fixed points in [0,1] and define the following 2-norm on pn:‖ f,g ‖= ∑2n k=0 | f(xk) g(xk) |, whenever f and g are linearly independent and ‖ f,g ‖= 0, if f, g are linearly dependent. then (pn,‖ ., . ‖) is a 2-banach space. example 2.8. let x is q3, the field of rational number and consider the following 2-norm on x as: ‖ x, y ‖ = ∣∣∣∣∣∣det   i j kx1 x2 x3 y1 y2 y3   ∣∣∣∣∣∣ , where x = (x1,x2,x3) and y = (y1,y2,y3). then (x,‖ ., . ‖) is not a 2-banach space but a 2-normed space. 62 chouhan and malviya 3. main results definition 3.1. let (x,‖ ., . ‖) be a 2-banach space with 2-norm ‖ ., . ‖ . a mapping t of x into itself is said to be expansive if there exists a constant h > 1 such that ‖ tx−ty,a ‖≥ h ‖ x−y,a ‖ for all x,y ∈ x example 3.2. let x = r2 and consider the following 2-norm on x as ‖ x,y ‖ = | x1y2 −x2y1 | where x = (x1,x2) and y = (y1,y2). then (x,‖ ., . ‖) is a 2-banach space. define a self map t on x as follows tx = βx where β > 1 for all x ∈ x, clearly t is an expansive mapping. theorem 3.3. let (x,‖ ., . ‖) be a 2-banach space and s, t be surjective mapping of x into itself satisfying. ‖ sx−ty,a ‖ + k [ ‖ x−ty,a ‖ + ‖ y −sx,a ‖ ] ≥ a1 ‖ x−sx,a ‖ + b1 ‖ y −ty,a ‖ +c1 ‖ x−y,a ‖(3.3.1) for all x,y,a ∈ x, x 6= y where a1,b1,c1,k ≥ 0, satisfy a1 < k + 1, b1 < k + 1,c1 > 2k + 1. then s and t have a common unique fixed point in x. proof: we define a sequence {xn} as follows for n = 0, 1, 2, 3, ... (3.3.2) x2n = sx2n+1, x2n+1 = tx2n+2 if x2n = x2n+1 = x2n+2 for some n then we see that x2n is a fixed point of s and t. therefore, we suppose that no two consecutive terms of sequence {xn} are equal. now we put x = x2n+1 and y = x2n+2 in (3.3.1) we get ‖ sx2n+1 −tx2n+2,a ‖ +k [ ‖ x2n+1 −tx2n+2,a ‖ + ‖ x2n+2 −sx2n+1,a ‖ ] ≥ a1 ‖ x2n+1 −sx2n+1,a ‖ +b1 ‖ x2n+2 −tx2n+2,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖ ⇒‖ x2n −x2n+1,a ‖ + k [ ‖ x2n+1 −x2n+1,a ‖ + ‖ x2n+2 −x2n,a ‖ ] ≥ a1 ‖ x2n+1 −x2n,a ‖ + b1 ‖ x2n+2 −x2n+1,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖ ⇒‖ x2n −x2n+1,a ‖ + k [ ‖ x2n+2 −x2n,a ‖ ] ≥ a1 ‖ x2n+1 −x2n,a ‖ +b1 ‖ x2n+2 −x2n+1,a ‖ + c1 ‖ x2n+1 −x2n+2,a ‖ ⇒‖ x2n −x2n+1,a ‖ + k [ ‖ x2n+2 −x2n+1,a ‖ + ‖ x2n+1 −x2n,a ‖ ] ≥ a1 ‖ x2n+1 −x2n,a ‖ + b1 ‖ x2n+2 −x2n+1,a ‖ +c1 ‖ x2n+1 −x2n+2,a ‖ ⇒ (1 + k −a1) ‖ x2n −x2n+1,a ‖≥ (b1 + c1 −k) ‖ x2n+1 −x2n+2,a ‖ ⇒‖ x2n+1 −x2n+2,a ‖≤ (1+k−a1) (b1+c1−k) ‖ x2n −x2n+1,a ‖ ⇒‖ x2n+1 −x2n+2,a ‖≤ k1 ‖ x2n −x2n+1,a ‖ where k1 = (1+k−a1) (b1+c1−k) < 1 (as a1 + b1 + c1 > 1 + 2k) fixed points of expansive type mappings 63 similarly, we can calculate ‖ x2n+2 −x2n+3,a ‖≤ k2 ‖ x2n+1 −x2n+2,a ‖ for n = 0, 1, 2, ... where k2 = (1+k−a1) (b1+c1−k) < 1 (as a1 + b1 + c1 > 1 + 2k) and so on so, in general ⇒‖ xn −xn+1,a ‖≤ k ‖ xn−1 −xn,a ‖ for n = 1, 2, 3... where k = max{k1,k2} then k < 1 (3.3.3) ⇒‖ xn −xn+1,a ‖≤ kn ‖ x0 −x1,a ‖ we can prove that {xn} is a cauchy sequence (using (3.3.3)). so there exists a point x in x such that (3.3.4) {xn}→ x as n →∞ existence of fixed point: since s and t are surjective maps, so there exist two points y and y′ in x such that (3.3.5) x = sy and x = ty′ consider ‖ x2n −x,a ‖=‖ sx2n+1 −ty′,a ‖ ≥−k [ ‖ x2n+1 −ty′,a ‖ + ‖ y′ −sx2n+1,a ‖ ] + a1 ‖ x2n+1 −sx2n+1,a ‖ +b1 ‖ y′ −ty′,a ‖ + c1 ‖ x2n+1 −y′,a ‖ ⇒‖ x2n −x,a ‖≥−k [ ‖ x2n+1 −ty′,a ‖ + ‖ y′ −x2n,a ‖ ] + a1 ‖ x2n+1 −x2n,a ‖ + b1 ‖ y′ −ty′,a ‖ +c1 ‖ x2n+1 −y′,a ‖ as {x2n}, {x2n+1} are subsequences of {xn} as n → ∞, {x2n} → x, {x2n+1} → x (using 3.3.4) therefore ‖ x−x,a ‖≥−k [ ‖ x−x,a ‖ + ‖ y′ −x,a ‖ ] + a1 ‖ x−x,a ‖ +b1 ‖ y′ −x,a ‖ + c1 ‖ x−y′,a ‖ ⇒ 0 ≥ (b1 + c1 −k) ‖ x−y′,a ‖ ⇒‖ x−y′,a ‖= 0 (as 2k + 1 < a1 +b1 +c1 < k + 1 +b1 +c1, so that k < b1 +c1) (3.3.6) ⇒ x = y′ in an exactly similar way (using b1 < k + 1) we can prove that, (3.3.7) x = y the fact (3.3.5) along with (3.3.6) and (3.3.7) shows that x is a common fixed point of s and t . 64 chouhan and malviya uniqueness: let z be another common fixed point of s and t , that is sz = z and tz = z ‖ x−z,a ‖ = ‖ sx−tz,a ‖ ≥−k [ ‖ x−tz,a ‖ + ‖ z −sx,a ‖ ] + a1 ‖ x−sx,a ‖ +b1 ‖ z −tz,a ‖ + c1 ‖ x−z,a ‖ ⇒‖ x−z,a ‖≥−k [ ‖ x−z,a ‖ + ‖ z −x,a ‖ ] + a1 ‖ x−x,a ‖ +b1 ‖ z −z,a ‖ +c1 ‖ x−z,a ‖ ⇒‖ x−z,a ‖≥ (−2k + c1) ‖ x−z,a ‖ ⇒ (1 + 2k − c1) ‖ x−z,a ‖≥ 0 ⇒‖ x−z,a ‖= 0 (as c1 > 2k + 1) ⇒ x = z this completes the proof of the theorem 3.3 corollary 3.4. let (x,‖ ., . ‖) be a 2-banach space and s and t be two surjective mappings of x into itself such that for every x,y,a ∈ x ‖ sx−ty,a ‖≥ a1 ‖ x−y,a ‖(3.4.1) where a1 > 1. then s and t have a common unique fixed point in x. proof: if we put k,a1,b1 = 0 and c1 = a1 in theorem 3.3 then we get above corollary 3.4. corollary 3.5. let (x,‖ ., . ‖) be a 2-banach space and t be a surjective mapping of x into itself such that for every x,y,a ∈ x ‖ tx−ty,a ‖≥ a1 ‖ x−y,a ‖(3.5.1) where a1 > 1. then t has a unique fixed point in x. proof: if we put s = t in corollary 3.4 then we get above corollary 3.5 which is an extension of theorem 1 of wang et al. [12] in 2-banach space. corollary 3.6. let (x,‖ ., . ‖) be a 2-banach space and s, t be two surjective mappings of x into itself satisfying. ‖ sx−ty,a ‖≥ a1 ‖ x−sx,a ‖ +a2 ‖ y −ty,a ‖ +a3 ‖ x−y,a ‖(3.6.1) for each x,y ∈ x, with x 6= y where a1,a2,a3 ≥ 0 and a3 > 1. then s and t have a common unique fixed point in x. proof: if we put k = 0 and b1 = a2, c1 = a3 in theorem 3.3 then we get above corollary 3.6. fixed points of expansive type mappings 65 corollary 3.7. let (x,‖ ., . ‖) be a 2-banach space and t be a surjective mapping of x into itself satisfying. ‖ tx−ty,a ‖≥ a1 ‖ x−tx,a ‖ +a2 ‖ y −ty,a ‖ +a3 ‖ x−y,a ‖(3.7.1) for each x,y ∈ x with x 6= y where a1,a2,a3 ≥ 0 and a3 > 1. then t has a unique fixed point in x. proof: if we put s = t in corollary 3.6 then we get above corollary 3.7 which is an extension of theorem 2 of wang et al. [12] in 2-banach space. the following theorem 3.8 is an example in which the fixed point need not be unique and continuity of self maps are required and it extend the theorem 3 of rhoades [9] in 2-banach spaces. theorem 3.8. let (x,‖ ., . ‖) be a 2-banach space and s, t be two continuous mappings of x into itself satisfying. ‖ sx−ty,a ‖≥ a1 min{‖ x−y,a ‖, ‖ x−sx,a ‖, ‖ y −ty,a ‖}(3.8.1) for every x,y ∈ x, x 6= y where a1 > 1. then s and t have a common fixed point in x. proof: we define a sequence {xn} as follows for n = 0, 1, 2, 3, ... x2n = sx2n+1, x2n+1 = tx2n+2 now we put x = x2n+1 and y = x2n+2 in (3.8.1) we get ‖sx2n+1 −tx2n+2,a‖≥ a1 min { ‖x2n+1 −x2n+2,a‖ ,‖x2n+1 −sx2n+1,a‖ ,‖x2n+2 −tx2n+2,a‖ } = a1 min { ‖x2n+1 −x2n+2,a‖ ,‖x2n+1 −x2n,a‖ ,‖x2n+2 −x2n+1,a‖ } ⇒‖x2n −x2n+1,a‖≥ a1 min { ‖x2n+1 −x2n+2,a‖ ,‖x2n −x2n+1,a‖ } case i ‖ x2n −x2n+1,a ‖≥ a1 ‖ x2n+1 −x2n,a ‖ ⇒ 1 ≥ a1 which is contradiction. case ii ‖ x2n+1 −x2n+2,a ‖≤ 1a1 ‖ x2n −x2n+1,a ‖ ‖ x2n+1 −x2n+2,a ‖≤ k ‖ x2n −x2n+1,a ‖ 66 chouhan and malviya where k = 1 a1 < 1 (as a1 > 1) so, in general ⇒‖ xn −xn+1,a ‖≤ k ‖ xn−1 −xn,a ‖ for n = 1, 2, 3... (3.8.2) ⇒‖ xn −xn+1,a ‖≤ kn ‖ x0 −x1,a ‖ we can prove that {xn} is a cauchy sequence (using (3.8.2). so there exists a point x in x such that (3.8.3) {xn}→ x as n →∞ existence of fixed point: if s and t are continuous then existence part follows very easily. as shown below x = lim n→∞ x2n = lim n→∞ sx2n+1 = s lim n→∞ x2n+1 = sx (as n →∞{x2n+1}→ x) similarly x = lim n→∞ x2n+1 = lim n→∞ tx2n+2 = t lim n→∞ x2n+2 = tx (as n →∞{x2n+2}→ x) this completes the proof of the theorem 3.8 corollary 3.9. let (x,‖ ., . ‖) be a 2-banach space and t be a continuous mapping of x into itself satisfying. ‖ tx−ty,a ‖≥ a1 min{‖ x−y,a ‖, ‖ x−tx,a ‖, ‖ y −ty,a ‖}(3.9.1) for every x,y ∈ x, x 6= y where a1 > 1. then t has a fixed point in x. proof: if we put s = t in theorem 3.8 then we get above corollary 3.9 which is an extension of theorem 3 of wang et al. [12] in 2-banach space. remark 1: if mappings are continuous in theorem 3.3 then existence of fixed point follows very easily as proved in theorem 3.8. remark 2 : in corollary 3.9, we proved the fixed point is unique by using only a3 > 1 and there is no need of a1 < 1 and a2 < 1, so it extend and unify the theorem 2 of wang et al. [12]. references [1] cho y.j. khan m.s. and sing s.l. common fixed points of weakly commuting mappings, univ.u. novom sadu, zb.rad. period.-mat.fak.ser.mat, 18 1(1988)129-142. [2] gahler s. 2-metric raume and ihre topologische strucktur, math.nachr., 26(1963), 115-148. [3] gahler s. uber die unifromisieberkeit 2-metrischer raume, math.nachr. 28(1965), 235 244. [4] iseki k. fixed point theorems in 2-metric space, math.seminar.notes, kobe univ.,3(1975), 133 136. [5] jungck g., commuting mappings and fixed points the american mathematical monthly, vol.83, no.4,pp.261-263, 1976. [6] machuca r., a coincidence theorem the american mathematical monthly, vol.74, no.5,p. 569, 1967. [7] panja c. and baisnab a.p., asymptotic regularity and fixed point theorems, the mathematics student, 46 1(1978), 54-59. [8] rhoades b.e. contractive type mappings on a 2-metric space, math.nachr., 91(1979), 151155. fixed points of expansive type mappings 67 [9] rhoades b.e., some fixed point theorems for pairs of mappings, jnanabha 15 (1985), 151-156. [10] rhoades b.e., a comparison of various definitions of contractive mappings. tran. am. math.soc.226, 257-290 (1977) [11] saha m. dey d. ganguly a. and debnath l. asymptotic regularity and fixed point theorems on a 2-banach space surveys in mathematics and its applications vol.7(2012), 31-38. [12] wang s. z., li b. y., gao z. m., and iseki k., some fixed point theorems on expansion mappings, math. japonica 29 (1984), 631-636. [13] white a. 2-banach spaces, math.nachr., 42(1969), 43 60. [14] zhang s., fixed point theory and its applications, chongqing press, chongqing china, 1984. 1scope college of engineering bhopal (m.p.), india 2nri institute of information science and technology, bhopal (m.p.) india ∗corresponding author international journal of analysis and applications volume 17, number 2 (2019), 282-302 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-282 received 2018-11-12; accepted 2019-01-09; published 2019-03-01. 2010 mathematics subject classification. 91b02. key words and phrases. grey system theory, taiwan, trade policy, top commodities, forecasting. ©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 282 applying grey system theory to forecast the total value of importsand exports of top traded commodities in taiwan thanh-tuyen tran* research center of applied sciences, lac hong university, no. 10 huynh van nghe, bien hoa city, dong nai province, vietnam *corresponding author: copcoi2@gmail.com abstract. export contributes to a large extent to economic growth of an island-type economyliketaiwan. the scientific forecasting on the total value of imports and exports of top traded commodities in taiwan are needed as the essential inputs to determine whether new top traded commodities should be imported or exported, and to make right decision toward activities in various functional areas such as building new container terminals, operation plans, marketing strategies, as well as finance and accounting [1]. taking the original data of the amount of import and export commodity during the years from 2007 to 2013, the author tries to establish a mathematical model of grey forecasting to make a prediction of the total value of imports and exports of top commodities in taiwan for the next 05 coming years from 2014 to 2018.the analysis results show that the usage of grey forecasting models resulted in a very low mean absolute percentage error, which demonstrate its applicability in practice to provide accurate forecasts. this research also indicates that for the future period of time (2014-2017), there will be a steady increase in both exports and imports value of all top commodities. the current study may offer a good idea for the control and scheduling for the terminal operators in decision making and planning. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-282 int. j. anal. appl. 17 (2) (2019) 283 i. introduction international trade refers to the exchange of capital, commodities, and services across international borders or territories [2]. import and export trade is a major component of international trade [3] and the positive contribution of import and export trade is one of the primary factors keeping the national economy growth.imports can be a channel for long-run economic growth because it provides domestic firms with access to needed intermediate factors and foreign technology ([4]; [5]; [6]).growth in imports can serve as a medium for the transfer of growth-enhancing foreign research and development knowledge from developed to developing countries [7]. export expansion and openness to foreign markets is viewed as a key determinant of economic growth because of the positive externalities it provides. for example, firms in a thriving export sector can enjoy the following benefits: efficient resource allocation, greater capacity utilization, exploitation of economies of scale, and increased technological innovation stimulated by foreign market competition [8]. furthermore, exports can provide foreign exchange that allows for more imports of intermediate goods which in turn raises capital formation and thus stimulate output growth and lead to an expansion of production and employment ([9]; [10]). it has been proven that, there is relationship between import and export trade for the growth of cross-country economies and as stated in the works of liu [11], azgun and sevinc [12], alam et al. [13], zhang and zhao [14], and wong [15]. the causal relationship between trade and economic growth for bulgaria, czech republic, and poland was examined in the paper of awokuse [8] and this analystconfirmed that trade stimulates economic growth. these developing countries depend on exports for economic growth, which gives rise to the ‘exportled growth’ hypothesis. the export-led growth hypothesis entails pursuing policies to promote exports and acquire foreign currency reserves, and countries achieve economic growth by importing high-technology goods.causality between exports, imports, and economic growth is not limited to developing countries like bulgaria [16], and previous studies have been examined this reality at developed countries, for example taiwan by chuang [17], portugal by ramos [18], china by mah [19], and korea by awokuse [4]. some studies have been conducted of newly industrialized asian countries by thangavelu and rajaguru [20]; of india by joseph and harilal [21], of malaysia by milad et al. [22]. int. j. anal. appl. 17 (2) (2019) 284 the import and export trade normally requires organizational cooperation among different international sectors ranging from purchasing, manufacturing, transporting, inventory, distribution, etc., especially the engagement of the customs authorities in both the country of export and the country of import ([23]; [24]). for this reason, it is necessary to build suitable plans and strategy to guarantee the flow of international trade across nations become smoothly. in order to have proper plans, accurately forecast the volume of imported-exported goods is the core issue. grey forecasting models have recently popularly used in time-series forecasting due to their simplicity and ability to characterize an unknown system with few data points ([25]; [26]). grey system has been widely employed in different areas due to its ability to deal with the problems of uncertainty with few data points and/or “partial known, partial unknown” information. for having accurately forecasting results of the value of imported and exported goods, the author considers that it is very suitable to apply grey system theory with its gm (1,1) in this study. grey system theory is a useful research tool for scientific world, which was formulated by professor deng julong in 1982 to study the problems of less data, poor information and uncertainty ([27]; [28]; [29]). in order to make positive analysis about the changes inthe total value of imports and exports of top traded commodities in taiwan educational system from 2007 to 2013, and to make a prediction of these problemsfor the next 05 coming years from 2014 and 2018, grey system theory with its gm (1,1) is applied. there will be some suggestions provided in the last chapters to the government and related organizations to deal with the changing in forecasted numbers. furthermore, future works would continue to apply this method, and then if it’s still successful, there would be very good for any nation. one of the main reasons for this objective is that there is a few prediction methods applied to forecast the top traded commodities value of import/export, so this would contribute a lot of things to the whole country education development in long-term. ii. methodology 2.1calculation and applying gm(1,1) based on matlab the researchers use gm (1,1) model to predict the realistic factors for the next 4 years (2014 to 2017). the study takes the exports of food and live animals in the data set of taiwan as example to understand how to compute in gm (1,1) model in period 2007-2013,and other variables are calculated in the same way. the procedure is carried out step by step as following. int. j. anal. appl. 17 (2) (2019) 285 first, the researchers use the gm (1,1) model for trying to forecast the variance of primitive series creating the primitive series: x(o) = (1836; 2259; 1914; 2401; 2942; 3286; 3215) step 1. grey generation (by the method of ago) first ago series: 𝑥(1) = (𝑥(1)(1),𝑥(1)(2),𝑥(1)(3),𝑥(1)(4),𝑥(1)(5),𝑥(1)(6),𝑥(1)(7)) x(1)= (1836; 4095; 6009; 8410; 11352; 14638;17853) x(1)= x(0)(1)= 1836 )2( )1()2( (0))0()1( =+= xxx 4095 )3()2( )1()3( )0((0))0()1( =++= xxxx 6009 )4()3()2( )1()4( )0()0((0))0()1( =+++= xxxxx 8410 )5()4()3()2( )1()5( )0()0()0((0))0()1( =++++= xxxxxx 11352 𝒙(𝟏)(𝟔) = 𝑥(0)(1)+ 𝑥(0)(2)+ 𝑥(0)(3)+ 𝑥(0)(4)+ 𝑥(0)(5)+ 𝑥(0)(6)=14638 𝒙(𝟏)(𝟕) = 𝑥(0)(1)+ 𝑥(0)(2)+ 𝑥(0)(3)+ 𝑥(0)(4)+ 𝑥(0)(5)+ 𝑥(0)(6)+ 𝑥(0)(7)=17853 step 2. create the different equations of gm (1, 1) to find ( )1x series, and the following mean obtained by the mean equation is: z(1)(2) = 1 2 (1836 + 4095) =2965.5 z(1)(3) = 1 2 (4095 + 6009) =5052 z(1)(4) = 1 2 (6009 + 8410) =7209.5 z(1)(5) = 1 2 (8410 + 11352) =9881 z(1)(6) = 1 2 (11352 + 14638) =12995 z(1)(7) = 1 2 (14638 + 17853) =16245.5 step 3. solve equations: to find aandb, the primitive series values are substituted into the grey differential equation to obtain: int. j. anal. appl. 17 (2) (2019) 286 { 2259 + 𝑎 × 2965.5 = 𝑏 1914 + 𝑎 × 5052 = 𝑏 2401 + 𝑎 × 7209.5 = 𝑏 2942 + 𝑎 × 9881 = 𝑏 3286 + 𝑎 × 12995 = 𝑏 3215 + 𝑎 × 16245.5 = 𝑏 convert the linear equations into the form of a matrix: let b= [ −2965.5 −5052 −7209.5 −9881 −12995 −16245.5] ,θ̂ = [ a b ] ,yn = [ 2259 1914 2401 2942 3286 3215] and then use the least square method to find a and b       ===      − 41752.48484 80.10123721)( 1 ^ n tt ybbb b a  use the two coefficients a andb to generate the whitening equation of the differential equation: 484844.1752218)(-0.101237 )1( )1( = x dt dx find the prediction model from equation: ( ) ( ) ( ) ( ) 1 0 1 1 akb b x k x e a a −  + = − +    80.1012372141752.48484 80.1012372141752.48484 8361)1( 8k0.10123721)1( +      −=+ ekx substitute different values of k into the equation: k=0 ( ) ( ) 1 1x = 1836 k=1 ( ) ( ) 1 2x = 3875.869833 k=2 ( ) ( ) 1 3x =6133.065564 k=3 ( ) ( ) 1 4x = 8630.74090 k=4 ( ) ( )1 5x = 11394.5163 k=5 ( ) ( ) 1 6x = 14452.741 k=6 𝑋(1)(7) = 17836.7880 k=7 𝑋(1)(8) = 21581.3674 k=8 𝑋(1)(9) =25724.89103 int. j. anal. appl. 17 (2) (2019) 287 k=9 𝑋(1)(10) =30309.8618 k=10 𝑋(1)(11) =35383.3113 derive the predicted value of the original series according to the accumulated generating operation and obtain: year k value actual value forecasted value residual error error 2007 0 1,836 1836 0 0.0000% 2008 1 2,259 2040 (219) -9.7003% 2009 2 1,914 2257 343 17.9308% 2010 3 2,401 2498 97 4.0265% 2011 4 2,942 2764 (178) -6.0579% 2012 6 3,286 3058 (228) -6.9317% 2013 7 3,215 3384 169 5.2580% 2014 8 -3745 -? 2015 9 -4144 -? 2016 10 -4585 -? 2017 11 -5073 -? after next, this paper uses the matlab with the forecasting method on grey system model. 2.2 matlab introduction matlab is built on the matrix for the data unit even basic, and it has the ability to do straightly calculations or operations as product of matrix, involution of matrix, division of matrix and sparse matrix, etc. in the language system of matlab, nearly all operations are run on matrix operation, and it also can apply the method analogous to mathematical formula to compile a program and realize algorithm, which can ease the problems with less time on doing operations. nevertheless, the process of grey forecasting requires a large amount of matrix operations and progression to be done well, which matlab can handle them usefully and conveniently. it is believed to make the right decision to combine matlab with gm(1,n) to realize algorithm of grey forecasting. 2.2.1program of grey system forecasting based on matlab the basic mathematical model of grey system gm(1,1) is the foundation to make progress for the program of grey forecasting based matlab. this program can handle such kinds of data like data of a single point, set up a forecasting model automatically, dynamically output and display curvilinear figure of the model, forecast all parameters and model equation error terms, and int. j. anal. appl. 17 (2) (2019) 288 predict future value of one time series or some time series based on the predicted time series set.following is the flow chart of gm(1,1) based on matlab (figure 2.1). figure 2.1: the flow chart of gm (1,1) based on matlab start calculation of 𝑋(1) calculation of matrix b yn solution of â = [a,b] = (btb)−1bt𝑌 establishment of time function of model gm solution {𝜀(0)(𝑖)} to test whether 𝜀(0) meets the test calculation of predicting value over solution of 𝑋(0)(𝑖) 𝜀 yes t no input of data 𝑋(0) int. j. anal. appl. 17 (2) (2019) 289 2.2.2 accuracy test and optimization of the model sample of forecasting of model gm (1,1) this paper conducts a practical forecasting on the exports of food and live animals in the data set of taiwan by adopting the above model gm(1,1), and checks the predicted results by means of relative error test. accuracy inspection analysis of forecasting ability numerous methods exist for judging forecasting model accuracy, and no single recognized inspection method exists for forecasting ability. mean absolute percentage error (mape) is often used to measure forecasting accuracy. (mape) is measure of accuracy in a fitted time series value in statistics, specifically trending. it usually expresses accuracy as a percentage (1)wikipedia.orghttp://en.wikipedia.org/wiki/mean_absolute_percentage_error smaller mape value indicates better forecasting ability.   − = 100 1 actual forecastactual n mape n forecasting number of step evaluation of mape forecasting ability is divided forecasting ability is evaluated as follows: • <10 excellent forecasting ability • 10~20 good forecasting ability • 20~50 reasonable forecasting ability • >50 poor forecasting in order to ensure that the gm (1,1) based on matlab has high accuracy for application in predicting the number in reality, this paper takes out the exports of food and live animals in the data set of taiwan estimated by gm (1,1) based on matlab as 3384 which is very close to the original number 3,215 and the error is so small in table 2.1. moreover, repeating above processes in the table 2.1 showing the sample calculation by gm(1,1) based on matlab, it points out that the forecasting error ranging from only 4.0265% to17.9308%in the sample predicted academic years. the result reveals that grey prediction is a good method for prediction. int. j. anal. appl. 17 (2) (2019) 290 table 2.1: the original and prediction values and errors and ago (2007~2014) year actual value forecasted value residual error error % ago 2007-08 1,836 1836 0 0.0000% 1,836 2008-09 2,259 2040 (219) 9.7003% 4,095 2009-10 1,914 2257 343 17.9308% 6,009 2010-11 2,401 2498 97 4.0265% 8,410 2011-12 2,942 2764 (178) 6.0579% 11,352 2012-13 3,286 3058 (228) 6.9317% 14,638 2013-14 3,215 3384 169 5.2580% 17,853 iii. results and analyses 3.1 calculations on the exports and imports in this section, the calculations on the total the value exports of top commodities in taiwan during the recent and future years are mentioned; moreover, prediction values for the 4 years 2014 to 2017 are typically focused. food and live animals the following grey prediction gm (1,1) model estimates the export value of food and live animal. the total value is rising from $us3745 million in 2014 to $us 5073 million in 2017 respectively. the forecasting errors are ranging from only 0.00% to 17.93%, which indicates good predicting ability of gm(1,1).figure 3.1 clearly indicates the steadily growth of the value of exporting thiscommodity in taiwan. through the 10-year time (2007 to 2017), the value of export has been slightly increasingbyseveral hundreds of $us million per year as forecasted. int. j. anal. appl. 17 (2) (2019) 291 table 3.1: food and live animals as predicted values years original (1) prediction (2) error [(2-1)/1] % residual error ago 2007 1,836 1836 0.0000% 0 1,836 2008 2,259 2040 9.7003% (219) 4,095 2009 1,914 2257 17.9308% 343 6,009 2010 2,401 2498 4.0265% 97 8,410 2011 2,942 2764 6.0579% (178) 11,352 2012 3,286 3058 6.9317% (228) 14,638 2013 3,215 3384 5.2580% 169 17,853 2014 ? 3745 ? ? ? 2015 ? 4144 ? ? ? 2016 ? 4585 ? ? ? 2017 ? 5073 ? ? ? figure 3.1: the sequence step of food and live animals as predicted beverages and tobacco the next below is the prediction for the export value of beverages and tobacco. table 3.2 and figure 3.2 indicate a steady increase in the export value of these commodities in the next following four year. the prediction results for next 4 years from 2014 to 2017 show taiwan export for beverages and tobacco amounted to around: $us 564 million; $us 11622 million; $us 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 0 1,000 2,000 3,000 4,000 5,000 6,000 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 value (v) prediction (p) error (%) int. j. anal. appl. 17 (2) (2019) 292 11940 million and $us 12267 million. we also notice the prediction errors which are ranged from only 0.0000% to 18.4191%, interpreted good results. table 3.2: the numbers of beverages and tobacco as predicted values years original (1) prediction (2) error [(2-1)/1] % residual error ago 2007 101 101 0.0000% 0 101 2008 143 169 18.4191% 26 244 2009 197 207 5.0347% 10 441 2010 270 253 6.3567% (17) 711 2011 338 309 8.5960% (29) 1,049 2012 370 378 2.0285% 8 1,419 2013 450 461 2.5066% 11 1,869 2014 ? 564 ? ? ? 2015 ? 689 ? ? ? 2016 ? 842 ? ? ? 2017 ? 1028 ? ? ? figure 3.2: the sequence step of beverages and tobacco as predicted 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 0 200 400 600 800 1,000 1,200 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 value (v) prediction (p) error (%) int. j. anal. appl. 17 (2) (2019) 293 crude materials, inedible, except fuels the next below is the prediction for the export value of crude materials, inedible, and except fuel. table 3.3 and figure 3.3 indicate a steady increase in the export value of this kind of goods in the next year. the total value of exporting this merchandise is rising from$us 4226 million in 2014 to $us 4780 million in 2017 respectively. we also notice the prediction errors which are ranged from only 0.0000% to 27.496%, interpreted reasonable results.apparently, we can recognize the upward trend in figure 3.3 table 3.3: the numbers of crude materials, inedible, except fuels as predicted values years original (1) prediction (2) error [(2-1)/1] % residual error ago 2007 3,450 3450 0.0000% 0 3,450 2008 3,624 3303 8.8678% (321) 7,074 2009 2,699 3441 27.4965% 742 9,773 2010 3,527 3585 1.6569% 58 13,300 2011 4,443 3736 15.9173% (707) 17,743 2012 3,960 3892 1.7056% (68) 21,703 2013 3,761 4056 7.8355% 295 25,464 2014 ? 4226 ? ? ? 2015 ? 4403 ? ? ? 2016 ? 4588 ? ? ? 2017 ? 4780 ? ? ? int. j. anal. appl. 17 (2) (2019) 294 figure 3.3: the sequence step of crude materials, inedible, except fuels predicted mineral fuels, lubricants and related materials table 3.4 shows a significant increase in the export value of mineral fuels, lubricants and related materials in the past-present-future time. the export amounted to around $us 24757 million in 2014, and the amount continues to rise over $us 2000 million by each year. in 2017 the export amounted to around $us 33464 million. figure 3.4 also illustrates the upward trend. the forecasting errors are pretty good (below32.55%), which indicate the method used in the paper is doing well. table 3.4: the numbers of mineral fuels, lubricants and related materials as predicted values years original (1) prediction (2) error [(2-1)/1] % residual error ago 2007 13,749 13749 0.0000% 0 13,749 2008 18,628 13549 27.2630% (5079) 32,377 2009 11,302 14981 32.5544% 3679 43,679 2010 14,397 16564 15.0551% 2167 58,076 2011 17,436 18315 5.0411% 879 75,512 2012 21,608 20250 6.2827% (1358) 97,120 2013 23,173 22390 3.3770% (783) 120,293 2014 ? 24757 ? ? ? 2015 ? 27373 ? ? ? 2016 ? 30265 ? ? ? 2017 ? 33464 ? ? ? 0% 5% 10% 15% 20% 25% 30% 0 1,000 2,000 3,000 4,000 5,000 6,000 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 value (v) prediction (p) error (%) int. j. anal. appl. 17 (2) (2019) 295 figure 3.4: the sequence step of mineral fuels, lubricants and related materials predicted animal and vegetable oils, fats and waxes in table 3.5, gm (1,1) model gives us the predicted export value of thesemerchandiseslike animal and vegetable oils, fats and waxes in 2014, 2015, 2016, and 2017 and the future numbers for these future years will be $us 117 million; $us 127 million; $us 138 million; and $us 150 million respectively. figure 3.5 shows this upward trend in the mount of export for the next four years. the residual errors of forecasting are ranging from only 0.00% to 15.89% show excellent forecasting ability. table 3.5: the numbers of animal and vegetable oils, fats and waxes as predicted values years original (1) prediction (2) error [(2-1)/1] % residual error ago 2007 62 62 0.0000% 0 62 2008 79 72 9.4872% (7) 141 2009 67 78 15.8969% 11 208 2010 82 84 2.8353% 2 290 2011 100 92 8.4274% (8) 390 2012 97 99 2.5187% 2 487 2013 108 108 0.0091% (0) 595 2014 ? 117 ? ? ? 2015 ? 127 ? ? ? 2016 ? 138 ? ? ? 2017 ? 150 ? ? ? 0% 5% 10% 15% 20% 25% 30% 35% 0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 value (v) prediction (p) error (%) int. j. anal. appl. 17 (2) (2019) 296 figure 3.5: the sequence step of animal and vegetable oils, fats and waxes predicted 4.2 balance of trade the commercial balance is the difference between the monetary value of exports and imports of output in an economy over a certain period, measured in the currency of that economy. it is the relationship between a nation's imports and exports. a positive balance is known as a trade surplus if it consists of exporting more than is imported; a negative balance is referred to as a trade deficit or, informally, a trade gap. trademeans the purchase and sales of commodities. in international trade, purchase and sale are replaced by imports and exports. balance of trade is simply the difference between the value of exports and value of imports. thus, the balance of trade denotes the differences of imports and exports of a merchandise of a country during the course of year. it indicates the value of exports and imports of the country in question. if the value of its exports over a period exceeds its value of imports, it is called favourable balance of trade and, conversely, if the value of total imports exceeds the total value of exports over a period, it is unfavourable balance of trade. the favourable balance of trade indicates good economic condition of the country.measuring the balance of trade can be problematic because of problems with recording and collecting data. factors that can affect the balance of trade include: ⚫ the cost of production (land, labor, capital, taxes, incentives, etc.) ⚫ the cost and availability of raw materials, intermediate goods and other inputs; 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 0 20 40 60 80 100 120 140 160 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 value (v) prediction (p) error (%) int. j. anal. appl. 17 (2) (2019) 297 ⚫ exchange rate movements; ⚫ multilateral, bilateral and unilateral taxes or restrictions on trade; ⚫ non-tariff barriers such as environmental, health or safety standards; ⚫ the availability of adequate foreign exchange with which to pay for imports; and ⚫ prices of goods manufactured at home (influenced by the responsiveness of supply) foreign trade has been the engine of taiwan's rapid growth during the past 40 years. main trading partners are mainland china & hong kong, usa, japan, europe and asean countries. generally, export-oriented industrialization was particularly characteristic of the development of the national economies of taiwan, as well as hong kong, south korea, and the asian tigers [30]. taiwan’s main export products consisted of electronics (28% of total), basic metals (9%), plastics & rubber (8%), optical and photographic instruments (8%) and chemicals (7%). main exports partners are mainland china & hong kong (42% of total), usa (12%), japan (7%), europe (11%) and asean countries (15%). a lack of natural resources had made taiwan dependent on imports. taiwan imports mostly mineral products and basic metals, electronic products, chemicals, machinery. main import partners are japan (21% of total), mainland china & hong kong (14%), usa (10%), europe (10%) and asean countries (11%). collected data over a period of year from 2007 to 2013 in our study shown that towards the top 06 commodities likefood and live animals; beverages and tobacco; crude materials, inedible, except fuels; mineral fuels, lubricants and related materials; animal and vegetable oils, fats and waxes; as well as other commodities and transactions not classified elsewhere in the sitc, the value of total imports exceeded the total value of exports. therefore, it reveals the situation of unfavourable balance of trade of these top commodities as listed above. conversely, collected data also from that period of time (2013-2017) indicated that towards the rest of 04 top commodities like chemicals and related products; manufactured goods classified chiefly by material; machinery and transport equipment; and miscellaneous manufactured articles, the value of its exports over a period exceeds its value of imports, thus it is called favourable balance of trade for the situation of these merchandises. for the future period of time, there will be a steady increase in both exports and imports value of all 10 items. therefore, for the top 06 commodities likefood and live animals; beverages and tobacco; crude materials, inedible, except fuels; mineral fuels, lubricants and related materials; animal and vegetable oils, fats and waxes; and other commodities and transactions not int. j. anal. appl. 17 (2) (2019) 298 classified elsewhere in the sitc, the value of total imports of these merchandise will continue to exceed the total value of exports. in many ways, some extend import would be beneficial for the economy, especially for countries in the developing stage. however, the trade deficit is too high will critically affect the economy.on the contrary, mentioned the rest of 04 top commodities like chemicals and related products; manufactured goods classified chiefly by material; machinery and transport equipment; and miscellaneous manufactured articles, the value of total exports over the period of year from 2014 to 2017 will continue toexceed its value of imports, thusfor these merchandises will experienced favourable balance of trade. taiwan, as standing as a developed country, usually imports a lot of raw materials from developing countries, e.g. asean countries, etc. typically, these imported materials are transformed into finished products, and might be exported after adding value. after the further analysis of these indexes, we can find out some important information. 1. in this study, the author used the original data of the amount of import and export commodity during the years from 2007 to 2013, the author proposed a mathematical model of grey forecasting to make a prediction of the total value of imports and exports of top commodities in taiwan for the next 05 coming years from 2014to 2018. 2. with the usage of grey forecastingmodels resulted in a very low mean absolute percentage error (mape),ranging from only 0.00% to 30% (see chapter 4), demonstrate its applicability in practice to provide accurate forecasts. precise forecasting results obtained from this model are essential for the control and scheduling for the terminal operators in decision making and planning. 3. it is shown that there would be the stable trend to the predicted numbers on the total the value of import/exports of top 10 commodities in taiwan during the recent and future years. 4. for the future period of time (2014-2018), there will be a steady increase in both exports and imports value of all 10 items. the value of total imports of the below 06 types of merchandises will continue to exceed the total value of exports as they did in the past period (2007-2013). these merchandises will experienceunfavourable balance of trade. they are: ◼ food and live animals ◼ beverages and tobacco ◼ crude materials, inedible, except fuels int. j. anal. appl. 17 (2) (2019) 299 ◼ mineral fuels, lubricants and related materials ◼ animal and vegetable oils, fats and waxes ◼ and other commodities and transactions not classified elsewhere in the sitc on the contrary,the rest of 04 top commodities show the value of total exportswill continue toexceed its value of imports. these merchandises will experience favourable balance of trade. they are: ◼ chemicals and related products ◼ manufactured goods classified chiefly by material ◼ machinery and transport equipment ◼ miscellaneous manufactured articles v. conclusions this article applied grey forecasting to construct a forecasting model. precise forecasting results obtained from grey system forecasting model is essential for the controlling, scheduling and making decision aboutwhichappropriate items for future import and export activities. our approach helps to foreseen and determines which kinds of merchandise (items that on top and prior import and export selection) will experience favourable balance of trade or unfavourable balance of trade. the findings by grey system theory, evaluated as a good and efficient model of forecasting with satisfactory results, may offer a valuable reference for government in drafting relevant policies for import and export activities and drawing up relevant policies for specified products. the proposed forecasting model was supported by a rich and valid resource of data (collected from the official website of national statistics 2014 (taiwan r.o.c)), compensating for the limitations of earlier studies. the results show that the residual error of the forecasting model is quite low, thus it can conclude that the model has high prediction validity and is clearly a viable means of forecasting the output value of an industry. however, this study had one primary limitation that we only focused on the value of import/exports trade in taiwan as a test case. in order to address this limitation, more research on data of other nations with different topic as well as with larger sample sizes, would be required. to sum up, the effectiveness of grey forecasting model verifies that the fitting degree of predicted value and actual value is very high. the combination grey forecasting model with int. j. anal. appl. 17 (2) (2019) 300 other forecasting model, like arima model, sarima model, neural network, etc. may be more efficient and interesting. however, applying grey forecasting model with the forecasting accuracy of the gm(1,1) to predict future value of import/exports trade in taiwan can be used as an effective prediction method to provide a scientific reference for operators in decision making and planning. references [1] n. t. nguyen and t. t. tran, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9) (2018), 73-81. [2] lim, s. y., & ho, c. m. nonlinearity in asean-5 export-led growth model: empirical evidence from nonparametric approach. econ. model. 3(2) (2013), 136-145. [3] awokuse, t. o. is the export‐led growth hypothesis valid for canada?.can. j. econ./revue canadienned’économique, 36(1) (2003), 126-136. [4] awokuse, t. o. exports, economic growth and causality in korea. appl. econ. lett. 12(11) (2005), 693-696. [5] t. t. tran, a strategic alliance study by performance evaluation and forecasting techniques: a case in the petroleum industry. int. j. adv. appl. sci. 5(2) (2018), 136-147. [6] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci. 3(12) (2016), 520. [7] mazumdar, j. imported machinery and growth in ldcs. j. develop. econ. 65(1) (2001), 209224. [8] awokuse, t. o. causality between exports, imports, and economic growth: evidence from transition economies. econ. lett. 94(3) (2007), 389-395. [9] chou, c. c., chu, c. w., & liang, g. s. a modified regression model for forecasting the volumes of taiwan’s import containers. math. computer model. 47(9) (2008), 797-807. [10] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci. 3(12) (2016), 520. int. j. anal. appl. 17 (2) (2019) 301 [11] x. liu, a positive analysis on our import-export increase and economic growth. modern econ. sci. 3 (2001), 007. [12] azgun, s., &sevinc, h. are imports a reason of growth? evidence from turkey. soc. sci. 5(2) (2010), 66-69. [13] alam, m. m., uddin, m. g. s., &taufique, k. m. r. import inflows of bangladesh: the gravity model approach. int. j. econ. finance, 1(1) (2014), p131. [14] zhang, w., &amp; zhao, s. forecasting research on the total volume of import and export trade of ningbo port by gray forecasting model. j. software, 8(2) (2013), 466-471. [15] wong, h. t. terms of trade and economic growth in malaysia. labuan bull. int. bus. finance, 2(2) (2004), 105-122. [16] djankov, s., &hoekman, b. trade reorientation and post-reform productivity growth in bulgarian enterprises. j. policy reform, 2(2) (1998), 151-168. [17] chuang, y. c. human capital, exports, and economic growth: a causality analysis for taiwan, 1952–1995. rev. int. econ. 8(4) (2000), 712-720. [18] ramos, f. f. r. exports, imports, and economic growth in portugal: evidence from causality and cointegration analysis. econ. model. 18(4) (2001), 613-623. [19] mah, j. s. export expansion, economic growth and causality in china. appl. econ. lett. 12(2) (2005), 105-107. [20] thangavelu, s. m., &rajaguru, g. is there an export or import-led productivity growth in rapidly developing asian countries? a multivariate var analysis. appl. econ. 36(10) (2004), 1083-1093. [21] joseph, k. j., &harilal, k. n. structure and growth of india's it exports: implications of an export-oriented growth strategy. economic and political weekly, 36(34)(2001), 3263-3270. [22] milad, m. a., ross, i. b. i., &marappan, s. modeling and forecasting the volumes of malaysia’s import. int. conf. glob. trends acad. res. bali, indonesia global illuminators, kuala lumpur, malaysia. (2014). [23] t. t. tran, forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci. 4(9) (2017), 86-95. [24] fradinata e, suthummanon s, and sunthiamorntut w. comparison of hybrid ann models: a case study of instant noodle industry in indonesia. int. j. adv. appl. sci. 4(8) (2017), 19-28, int. j. anal. appl. 17 (2) (2019) 302 [25] n. t. nguyen and t. t. tran, optimizing mathematical parameters of grey system theory: an empirical forecasting case of vietnamese tourism. neural comput. appl. (2017), https://doi.org/10.1007/s00521-017-3058-9. [26] n. t. nguyen and t. t. tran, raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl. (2018b). https://doi.org/10.1007/s00521-018-3639-2. [27] deng, j. l. introduction to grey system theory. j. grey syst. 1(1) (1989), 1–24. [28] lin, y., & liu, s. (2004a, october). a historical introduction to grey systems theory. in systems, man and cybernetics, 2004 ieee international conference on, 3, 2403-2408. [29] nguyen, n. t., & tran, t. t. facilitating an advanced product layout to prioritize hot lots in 450 mm wafer foundry in the semiconductor industry. int. j. adv. appl. sci. 3(6) (2016), 14-23. [30] ariff, m., & hill, h. export-oriented industrialisation: the asean experience (vol. 49). routledge, london/new york. (2010). https://doi.org/10.1007/s00521-017-3058-9 https://doi.org/10.1007/s00521-018-3639-2 international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 53-62 http://www.etamaths.com the dhage iteration principle for coupled pbvps of nonlinear second order differential equations bapurao c. dhage abstract. the present paper proposes a new monotone iteration principle for the existence as well as approximations of the coupled solutions for a coupled periodic boundary value problem of second order ordinary nonlinear differential equations. an algorithm for the coupled solutions is developed and it is shown that the sequences of successive approximations defined in a certain way converge monotonically to the coupled solutions of the related differential equations under some suitable hybrid conditions. a numerical example is also indicated to illustrate the abstract theory developed in the paper. we claim that the method as well as the results of this paper are new to literature on nonlinear analysis and applications. 1. introduction given a closed and bounded interval j = [0,t] of the real line r , consider the coupled periodic boundary value problems (in short cpbvps) of nonlinear second order ordinary nonlinear differential equations (in short des) of the form (1.1) −x′′(t) + λ2x(t) = f(t,x(t),y(t)), x(0) = x(t) , x′(0) = x′(t),   and (1.2) −y′′(t) + λ2x(t) = f(t,y(t),x(t)), y(0) = y(t) , y′(0) = y′(t),   for all t ∈ j, where λ ∈ r, λ > 0 and f : j ×r×r → r is a continuous function. by a coupled solution of the cpbvps (1.1) and (1.2) we mean an ordered pair of differentiable functions (u,v) ∈ c(j,r)×c(j,r) that satisfy the des (1.1) and (1.2), where c(j,r) is the space of continuous real-valued functions defined on j. the coupled pbvps (1.1) and (1.2) are well-known and the existence of the coupled solutions for them have been proved using the coupled fixed point theorems based on the properties of cones in the solution space c(j,r). see guo and lakshmikantham [12], heikkilä and lakshmikantham [13] and the references therein. 2010 mathematics subject classification. 34a12, 34a38. key words and phrases. coupled periodic boundary value problems; coupled fixed point theorem; dhage iteration principle; approximate coupled solutions. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 53 54 dhage recnetly, bhaskar and lakshmikantham [11] proved the existence and uniqueness results for the coupled solutions of the cbvps (1.1) and (1.2) without using the properties of the cones, however in this case the nonlinearity f involved in (1.1) and (1.2) is required to satisfy a weak lipschitz condition which is considered to be strong in the theory of nonlinear differential and integral equations. very recently, dhage and dhage [7] proved the existence as well as approximations of the coupled solutions for the coupled initial value problems (in short civps) of the nonlinear first order ordinary differential equations using the dhage iteration principle which does not require any type of lipschitz condition as well as any property of the cones in a appropriate banach space. the aim of the present paper is to extend the method involving the dhage iteration principle to the cpbvps (1.1) and (1.2) for approximating the coupled solutions. therefore, our approach to the considered cpbvps (1.1) and (1.2) is different from the earlier ones discussed in the literature. moreover, when λ = 0, f(t,x,x) = −f1(t,x) and x = y in (1.1) or (1.2) for all t ∈ j and x ∈ r, the results of this paper include the existence and approximations results of dhage et al. [10] as special cases. 2. auxiliary results let (e,�) be a partial ordered set and let d be a metric on e such that (e,�,d) becomes a partially ordered metric space. by e×e we denote a metric space with the metric d∗ defined by (2.1) d∗ ( (x,y), (w,z) ) = d(x,w) + d(y,z) for (x,y), (w,z) ∈ e × e. we define a partial order � in e × e as follows. let (x1,x2), (y1,y2) ∈ e ×e. then, (2.2) (x1,x2) � (y1,y2) ⇐⇒ x1 � y1 and x2 � y2. then, the triplet (e×e,�,d∗) again becomes a partially ordered metric space. let f : e ×e → e and consider the coupled mapping equations, (2.3) f(x,y) = x and f(y,x) = y. a point (x∗,y∗) ∈ e ×e is said to be a coupled solution or coupled fixed point for the coupled mapping equation (2.3) if (2.4) f(x∗,y∗) = x∗ and f(y∗,x∗) = y∗. we need the following definitions in what follows. definition 2.1. a partially ordered normed metric space (e,�,d) is called regular if every nondecreasing (resp. nonincreasing) sequence {xn} converges to x∗, then xn � x∗ (resp. xn � x∗) for all n ∈ n. the details of the regularity property of the ordered sets may be found in heikkilä and lakshmikantham [13] and the references therein. definition 2.2 (dhage [2]). a mapping f : e ×e → e is called partially continuous at a point (a,b) ∈ e ×e if for � > 0 there exists a δ > 0 such that d∗ ( f(x,y),f(a,b) ) < � whenever (x,y) is comparable to (a,b) and d∗ ( (x,y), (a,b) ) < δ. nonlinear second order differential equations 55 if f is partially continuous at every point of e × e, we say that f is partially continuous on e ×e. remark 2.1. if f is partially continuous on e×e, then it is continuous on every totally ordered set or chain in e ×e. definition 2.3. a mapping f : e×e → e is called partially compact if f(c1×c2) is a relatively compact subset of e for all chains c1 and c2 in e. the details of compact and continuous operators may be found in the monograph by heikkilä and lakshmikantham [13] and the references therein. definition 2.4. a mapping f is called mixed monotone if f(x,y) is nondecreasing in x for each y ∈ e and nonincreasing in y for each x ∈ e with respect to the order relation � in e. remark 2.2. if f is mixed monotone, then it is a nondecreasing mapping on e ×e with respect to the order relation � defined in e ×e. definition 2.5 (dhage [2, 3]). the order relation � and the metric d on a nonempty set e are said to be compatible if {xn}n∈n is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk}n∈n of {xn}n∈n converges to x ∗ implies that the whole sequence {xn}n∈n converges to x∗. similarly, given a partially ordered normed linear space (e,�,‖ · ‖), the order relation � and the norm ‖·‖ are said to be compatible if � and the metric d defined through the norm ‖ ·‖ are compatible. clearly, the set r of real numbers with usual order relation ≤ and the metric defined by the absolute value function has this property. similarly, every finite dimensional euclidean space rn is compatible with respect to usual componentwise order relation and the standard norm in it. the dhage iteration principle which states that the sequence of successive approximations of a nonlinear equation beginning with a lower or an upper solution converges monotonically to its solution forms a powerful tool in th existence theory of such equations. the details of dhage iteration principle are given in dhage [2, 3, 4] and dhage and dhage [9]. the following applicable coupled hybrid fixed point theorem is a slight improvement of the coupled hybrid fixed point theorem proved in dhage and dhage [8] containing the dhage iteration principle. theorem 2.1. let (e,�,d) be a regular partially ordered complete metric space such that the metric d and the order relation � are compatible in every compact chain c of e. let f : e × e → e is a mixed monotone, partially continuous and partially compact mapping. if there exist elements x0 ∈ e and y0 ∈ e such that x0 � f(x0,y0) and y0 � f(y0,x0), then f has a coupled fixed point (x∗,y∗) and the sequences {xn} and {yn} defined by xn = f(xn−1,yn−1) = fn(x0,y0) and yn = f(yn−1,xn−1) = fn(y0,x0) converge monotonically to x∗ and y∗respectively. proof. define the sequences {xn} and {yn} of points in e as follows. choose x1 = f(x0,y0) and y1 = f(y0,x0). then, x0 � x1 and y1 � y0. again, choose x2 = f2(x0,y0) = f(x1,y1) = f(f(x0,y0),f(y0,x0)) �f(x0,y0) = x1. 56 dhage similarly, choose y2 = f2(y0,x0) = f(y1,x1) so that y2 �f(f(y0,x0),f(x0,y0)) �f(y0,x0) = y1. proceeding in this way, by induction, define (2.5) xn+1 = f(xn,yn) = fn(x0,y0) and yn+1 = f(yn,xn) = fn(y0,x0), for n = 0, 1, 2, ..., so that (2.6) x0 � x1 � ···� xn � ··· , and (2.7) y0 � y1 � ···� yn � ··· . thus, {xn} and {yn} are respectively monotone nondecreasing and monotone nonincreasing sequences and so are chains in e. from the construction of {xn} and {yn}, it follows that {xn}⊆f ({xn},{yn}) ⊆f ({xn}×{yn}) . since f is partially compact on e ×e, one has f({xn}×{yn}) is a relatively compact subset of e. as a result, f({xn}×{yn}) is compact and that {xn} has a convergent subsequence converging to a point, say x∗ ∈ e. since d and � are compatible in every compact chain c of e, the whole sequence {xn} converges to x∗. similarly, the sequence {yn} converges to a point say y∗ ∈ e. equivalently, (xn,yn) → (x∗,y∗) in the topology of the norm in e×e. as e is a regular, we have that xn � x∗ and yn � y∗ for all n ∈ n. therefore, we obtain (xn,yn) � (x∗,y∗) for all n ∈ n. finally, by the partial continuity of f, we obtain x∗ = lim n→∞ xn+1 = lim n→∞ f(xn,yn) = f(x∗,y∗) and y∗ = lim n→∞ yn+1 = lim n→∞ f(yn,xn) = f(y∗,x∗). thus (x∗,y∗) is a coupled fixed point of the mapping f on e × e into itself. this completes the proof. � remark 2.3. the regularity of the partially ordered metric space e may be replaced with a stronger condition of continuity than the partial continuity of the mappings f on e ×e. again, the condition of compatibility of the order relation � and the norm ‖·‖ in every compact chain of e holds if every partially compact subset of e possesses the compatibility property with respect to � and ‖ ·‖. the simple fact concerning the compactibility of the order relation and the norm mentioned in remark 2.3 has been used in formulating the main results of this paper. in the following sectin we prove the main existence and approximation results for the cbvp (1.1) and (1.2) defined on j. 3. existence and approximations results we place our considerations of the cbvps (1.1) and (1.2) in the function space c(j,r). we define a norm ‖ ·‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) nonlinear second order differential equations 57 for all t ∈ j. clearly, (c(j,r),‖ · ‖,≤) is a partially ordered complete normed linear space and has compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in certain subsets of of it. the following lemma in this connection is useful in what follows. lemma 3.1 (dhage [4]). let ( c(j,r),≤,‖·‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2). then ‖ · ‖ and ≤ are compatible in every partially compact subset s of c(j,r). proof. let s be a partially compact subset of c(j,r) and let {xn}n∈n be a monotone nondecreasing sequence of points in s. then we have x1(t) ≤ x2(t) ≤ ···≤ xn(t) ≤ ··· , (nd) for each t ∈ j. suppose that a subsequence {xnk}k∈n of {xn}n∈n is convergent and converges to a point x in s. then the subsequence {xnk (t)}k∈n of the monotone real sequence {xn(t)}n∈n is convergent. by monotone characterization, the whole sequence {xn(t)}n∈n is convergent and converges to a point x(t) in s for each t ∈ j. this shows that the sequence {xn(t)}n∈n converges to x(t) point-wise on j. to show the convergence is uniform, it is enough to show that the sequence {xn(t)}n∈n is equicontinuous. since s is partially compact, every chain or totally ordered set and consequently {xn}n∈n is an equicontinuous sequence by arzelá-ascoli theorem. hence {xn}n∈n is convergent and converges uniformly to x. as a result, ‖ · ‖ and ≤ are compatible in s. this completes the proof. � we need the following definition in the sequel. definition 3.1. an ordered pair of differentiable functions (u,v) ∈ c(j,r) × c(j,r) is said to be a coupled lower solution of the cpbvps of coupled differential equations (1.1) and (1.2) if −u′′(t) + λ2u(t) ≤ f(t,u(t),v(t)), u(0) ≤ u(t) , u′(0) ≤ u′(t),   , and −v′′(t) + λ2v(t) ≥ f(t,v(t),u(t)), v(0) ≥ v(t) , v′(0) ≥ v′(t),   , for all t ∈ j. similarly, an ordered pair of differentiable functions (p,q) ∈ c(j,r)× c(j,r) is said to be a coupled upper solution of the cpbvps (1.1) and (1.2) if the above inequalities are satisfied with reverse sign. we consider the following set of hypotheses in what follows. (h1) f is bounded on j ×r×r with bound m. (h2) the function f(t,x,y) is nondecreasing in x and nonincreasing in y for each t ∈ j. (h3) the cpbvps (1.1) and (1.2) have a lower coupled solution (u,v) ∈ c(j,r)× c(j,r. (h4) the cpbvps (1.1) and (1.2) have a lower coupled solution (p,q) ∈ c(j,r)× c(j,r). 58 dhage the following useful lemma is obvious and may be found in dhage [1] and the references therein. lemma 3.2. for any σ ∈ l1(j,r), x is a solution to the differential equation (3.3) −x′′(t) + λ2x(t) = σ(t), t ∈ j, x(0) = x(t) , x′(0) = x′(t),   if and only if it is a solution of the integral equation (3.4) x(t) = ∫ t 0 g(t,s) σ(s) ds where, g(t,s) is the green’s function associated to the pbvp (3.5) −x′′(t) + λ2x(t) = 0, t ∈ j, x(0) = x(t) , x′(0) = x′(t).   notice that the green’s function g is continuous and nonnegative on j ×j and therefore, the number k := max{|g(t,s)| : t,s ∈ [0,t]} exists. an application of above lemma 3.2 we obtain lemma 3.3. a pair of function (u,v) ∈ c(j,r)×c(j,r) is a coupled solution of the cpbvps (1.1) and (1.2) if and only if u and v are the solutions of the nonlinear integral equations, (3.6) x(t) = ∫ t 0 g(t,s)f(s,x(s),y(s)) ds and (3.7) y(t) = ∫ t 0 g(t,s)f(s,y(s),x(s)) ds for all t ∈ j, where the green’s function g(t,s) is given by (3.5). theorem 3.1. assume that the hypotheses (h1) through (h3). then the cpbvps (1.1) and (1.2) have a coupled solution (x∗,y∗) defined on j and the sequences {xn} and {yn} defined by (3.8) xn+1(t) = ∫ t 0 g(t,s)f(s,x(s),yn(s)) ds and (3.9) yn+1(t) = ∫ t 0 g(t,s)f(s,yn(s),xn(s)) ds for each t ∈ j converge monotonically to x∗ and y∗ respectively. proof. set e = c(j,r). then, by lemma 3.1, every compact chain in e possesses the compatibility property with respect to the norm ‖ · ‖ and the order relation ≤ in e. consider the mapping f on e ×e defined as (3.10) f(x,y)(t) = ∫ t 0 g(t,s)f(s,x(s),y(s)) ds, t ∈ j nonlinear second order differential equations 59 and (3.11) f(y,x)(t) = ∫ t 0 g(t,s)f(s,y(s),x(s)) ds, t ∈ j. since green’s function g is continuous on j×j, we have that f(x,y),f(y,x) ∈ e. as a result, f defines a mapping f : e × e → e. we shall show that f satisfies the conditions of theorem 2.1. this will be achieved in a series of following steps. step i : f is a mixed monotone operator on e ×e. let x1,x2 ∈ s be such that x1 ≤ x2. then, by hypothesis (h2), f(x1,y)(t) = ∫ t 0 g(t,s)f(s,x1(s),y(s)) ds ≤ ∫ t 0 g(t,s)f(s,x2(s),y(s)) ds = f(x2,y)(t) for all t ∈ j. this shows that f(x,y) is monotone nondecreasing in x for all t ∈ j and y ∈ s. next, let y1,y2 ∈ e be such that y1 ≤ y2. then, f(x,y1)(t) = ∫ t 0 g(t,s)f(s,x(s),y1(s)) ds ≥ ∫ t 0 g(t,s)f(s,x(s),y2(s)) ds = f(x,y2)(t) for all t ∈ j and x ∈ s. hence f(x,y) is monotone nonincreasing in y for all x ∈ e. thus f is a mixed monotone mapping on e ×e. step ii: f is partially continuous mixed monotone operator on e ×e. let {xn}n∈n = {(xn,yn)} be a monotone nondecreasing sequence in a chain c = c1 ×c2 of e ×e such that xn = (xn,yn) → (x,y) = x and xn ≤ x for all n ∈ n. then, by dominated convergence theorem, lim n→∞ f(xn)(t) = ∫ t 0 g(t,s) [ lim n→∞ f(s,xn(s),yn(s)) ] ds = ∫ t 0 g(t,s)f(s,x(s),y(s)) ds = f(x)(t), for all t ∈ j. this shows that f(xn) converges monotonically to f(x) pointwise on j. next, we will show that {f(xn)}n∈n is an equicontinuous sequence of functions in e. let t1, t2 ∈ j be arbitrary. then, by hypothesis (b2), |f(xn)(t2) −f(xn)(t1)| 60 dhage ≤ ∣∣∣∣∣ ∫ t 0 g(t2,s)f(s,xn(s),yn(s)) ds− ∫ t 0 g(t1,s)g(s,xn(s),yn(s)) ds ∣∣∣∣∣ ≤ ∫ t 0 ∣∣g(t2,s) −g(t1,s)∣∣ ∣∣f(s,xn(s),yn(s))∣∣ds ≤ mf ∫ t 0 ∣∣g(t2,s) −g(t1,s)∣∣ds → 0 as t2 − t1 → 0 uniformly for all n ∈ n. this shows that the convergence f(xn) → f(x) is uniform and hence f is a partially continuous on e ×e. step iii: f is a partially compact mixed monotone operator on e ×e. let c1 and c2 be two arbitrary chains in e. we show that f(c1 × c2) is a relatively compact subset of e. to finish it is enough to prove that f(c1 × c2) is uniformly bounded and equicontinuous set in e. let x ∈ c1 and y ∈ c2 be arbitrary. then, by (h1), |f(x,y)(t)| ≤ ∫ t 0 g(t,s)|f(s,x(s),y(s))|ds ≤ mfk t = r for all t ∈ j. taking the supremum over t, we obtain ‖f(x,y)‖≤ r for all x ∈ c1 and y ∈ c2. hence, f(c1×c2) is a uniformly bounded subset of e. next, we show that f(c1 ×c2) is an equicontinuous set in e. let t1, t2 ∈ j be arbitrary. then, for any z ∈ f(c1 × c2), there exist x ∈ c1 and y ∈ c2 such that z = f(x,y). without loss of generality, we may assume that x(t1) ≥ x(t2) and y(t1) ≤ y(t2). therefore, by the definition of f, |z(t1) −z(t2)| = |f(x,y)(t1) −f(x,y)(t2)| = ∣∣∣∣ ∫ t1 0 f(s,x(s),y(s)) ds− ∫ t2 0 f(s,x(s),y(s)) ds ∣∣∣∣ ≤ ∣∣∣∣ ∫ t1 t2 |f(s,x(s),y(s))|ds ∣∣∣∣ ≤ mf |t1 − t2| −→ 0 as t1 → t2, uniformly for all x ∈ c1 and y ∈ c2. as a result, we have |f(x,y)(t1) −f(x,y)(t2)| −→ 0 as t1 → t2, uniformly for all (x,y) ∈ c1 ×c2. consequently f(c1 ×c2) is an equi-continuous set of e. we apply arzeli-ascoli theorem and deduce that f(c1×c2) is a relatively compact subset of e. hence f is partially relatively compact on e ×e. now f is a partially continuous and partially compact mixed monotone operator on e ×e into e. again, by hypothesis (h3), there exist elements x0 and y0 in s such that x0 ≤ f(x0,y0) and y0 ≥ f(y0,x0). thus all the conditions of theorem 2.1 are satisfied and hence the coupled equations f(x,y) = x and y = f(y,x) have a coupled solution (x∗,y∗) and the sequences {xn} and {yn} defined by (3.11) and nonlinear second order differential equations 61 (3.12) converge monotonically to x∗ and y∗ respectively. this completes the proof. � remark 3.1. the conclusion of theorem 3.1 also remains true if we replace the hypothesis (h3) with (h4). the proof of theorem 3.1 under this new hypothesis is obtained using similar arguments with appropriate modifications. example 3.1. given a closed and bounded interval j = [0, 1] in r, consider the coupled pbvps, (3.12) −x′′(t) + x(t) = tanh x(t) − tanh y(t), x(0) = x(1) , x′(0) = x′(1),   and (3.13) −y′′(t) + y(t) = tanh y(t) − tanh x(t), y(0) = y(1) , y′(0) = y′(1),   for all t ∈ [0, 1]. here, the function f is given by f(t,x,y) = tanh x− tanh y. for all t ∈ [0, 1] and x,y ∈ r. clearly, f is uniformly continuous and bounded on j × r × r with bound mf = 2. furthermore, f(t,x,y) is nondecreasing in x for each t ∈ j and y ∈ r and nonincreasing in y for each t ∈ j and x ∈ r. finally, there exist functions x0(t) = − [ e2(e−t −et) (e− 1) + e(1 −e−t) (e− 1) ] and y0(t) = [ e2(e−t −et) (e− 1) + e(1 −e−t) (e− 1) ] such that (3.14) −x′′0 (t) + x0(t) ≤ tanh x0(t) − tanh y0(t), x0(0) ≤ x0(1) , x′0(0) ≤ x ′ 0(1),   and (3.15) −y′′0 (t) + y0(t) ≥ tanh y0(t) − tanh x0(t), y0(0) ≥ y0(1) , y′0(0) ≥ y ′ 0(1),   for all t ∈ j. thus, the nonlinearity f satisfies all the hypotheses (h1) through (h3) of theorem 3.1. hence, the cpbvps (3.12) and (3.13) have a coupled solution (x∗,y∗) defined on [0, 1] and the sequences {xn}∞n=0 and {yn}∞n=0 of successive approximations defined by xn+1(t) = ∫ 1 0 g(t,s) [ tanh xn(s) − tanh yn(s) ] ds, t ∈ [0, 1], and yn+1(t) = ∫ 1 0 g(t,s) [ tanh yn(s) − tanh xn(s) ] ds, t ∈ [0, 1], 62 dhage where g(t,s) is a green’s function associated with the pbvp (3.16) −x′′(t) + x(t) = 0, t ∈ j, x(0) = x(1) , x′(0) = x′(1),   given by g(t,s) = 1 2(e− 1) { e1+s−t + et−s, 0 ≤ s ≤ t ≤ 1 e1+t−s + es−t, 0 ≤ t ≤ s ≤ 1, converge monotonically to x∗ and y∗ respectively. remark 3.2. finally, we mention that theorem 2.1 may be applied to various nonlinear initial and boundary value problems of ordinary coupled differential equations for proving the existence as well as algorithms for the coupled solutions under suitable mixed monotonic and partial compactness type conditions. references [1] b.c. dhage, periodic boundary value problems of first order carathéodory and discontinuous differential equations, nonlinear funct. anal. & appl. 13(2) (2008), 323-352. [2] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ. appl. 5 (2013), 155-184. [3] b.c. dhage, partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45 (4) (2014), 397-426. [4] b.c. dhage, nonlinear d-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, malaya j. mat. 3(1)(2015), 62-85. [5] b.c. dhage, approximating coupled solutions of coupled pbvps of nonlinear first order ordinary differential equations, taiwnese j. math. (submitted) [6] b.c. dhage, s.b. dhage, approximating solutions of nonlinear first order ordinary differential equations, global jour. math. sci. 3 (2014), (in press). [7] b.c. dhage, s.b. dhage, coupled hybrid fixed point theorems in partially ordered metric spaces with application, nonlinear studies 21(4)(2014), 675-686. [8] b.c. dhage, s.b. dhage, approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, indian j. math. 57(1) (2015), 103-119. [9] b.c. dhage, s.b. dhage, approximating positive solutions of pbvps of nonlinear first order ordinary quadratic differential equations, appl. math. lett. 46 (2015), 133-142. [10] b.c. dhage, s.b. dhage, s.k. ntouyas, approximating solutions of nonlinear second order ordinary differential equations, malaya j. mat. 3(3) (2015), 00-00. [11] t. gnana bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear analysis: tma 65 (2006), 1379-1393. [12] d. guo, v, lakshmikantham, coupled fixed point of nonlinear operators with applicatons, nonlinear anal. 11 (1987), 623-632. [13] s. heikkilä, v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. kasubai, gurukul colony, ahmedpur-413 515, dist: latur maharashtra, india international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 198-205 http://www.etamaths.com best proximity points for a new class of generalized proximal mappings tayyab kamran1, muhammad usman ali2 mihai postolache3,4,∗, adrian ghiura4 and misbah farheen1 abstract. the best proximity points are usually used to find the optimal approximate solution of the operator equation tx = x, when t has no fixed point. in this paper, we prove some best proximity point theorems for nonself multivalued operators, following the foot steps of basha and shahzad [best proximity point theorems for generalized proximal contractions, fixed point theory appl., 2012, 2012:42]. 1. introduction fixed point theory have an important role in many branches of mathematics such as differential and integral equations, optimization and variational analysis. this theory mainly concerns with the fixed point equation tx = x, where t : a → b is some nonlinear operator. the solution of this equation is called a fixed point of the operator t . it is not necessary that the equation has a solution for every nonlinear operator t . for example this one has no solution when a ∩ b = ∅. in this case we may find a point x ∈ a which is closest to tx, that is, the distance between tx and x is least as compare to other elements of a. such a point is called the best proximity point of t . the notion of best proximity point was initiated by fan [1] for normed spaces. eldred and veeramani [2] generalized this notion in the context of metric spaces. in literature there are many important best proximity point theorems in different settings: jleli et al. and ali et al. [3, 4], for α-ψ-proximal mappings; akbar and gabeleh [5, 6], derafshpour et al. [7], di bari et al. [8], rezapour et al. [9], vetro [10], for cyclic mappings; alghamdi et al. [11] for mappings in geodesic metric spaces; al-thagafi and shahzad [12], for kakutani multimaps; markin and shahzad [13], for relatively u-continuous mappings; nashine et al. [14], for rational proximal contractions; akbar and gabeleh [15], for multivalued non-self mappings; choudhury et al. [16] for best proximity point and coupled best proximity point in partially ordered metric spaces; shatanawi and pitea [17], for best proximity points and best proximity coupled points in complete metric spaces with (p)-property; jamali and vaespour [18], for best proximity point for nonlinear contractions in menger probabilistic metric spaces; bejenaru and pitea [19], for fixed point and best proximity point theorems in partial metric spaces. motivated and inspired by the research introduced above, in this paper we introduce our best proximity point theorems for nonself multivalued operators, following the foot steps method of basha and shahzad [20]. 2. previous results now, we recollect some basic notions, definitions and results which we require subsequently. let (x,d) be a metric space. for a,b ⊆ x, dist(a,b) = inf{d(a,b) : a ∈ a, b ∈ b}, d(x,b) = inf{d(x,b) : b ∈ b}, a0 = {a ∈ a : d(a,b) = dist(a,b) for some b ∈ b}, b0 = {b ∈ b : d(a,b) = dist(a,b) for some a ∈ a}, while cb(b) is the set of all nonempty closed and bounded subsets of b. a point x∗ ∈ x is said to be a best proximity point of t : a → cb(b) if d(x∗,tx∗) = dist(a,b). the set b is said to be approximatively compact with respect to the set a, if each {vn} in b with d(x,vn) → d(x,b) for some x in a, has a convergent subsequence [20]. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. f-contraction; f-contraction of hardy rogers and ciric type; best proximity point. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 198 best proximity points for a new class of generalized proximal mappings 199 a class of all functions f : (0,∞) → r satisfying the conditions: (f1) f is strictly increasing, that is, for each a1,a2 ∈ (0,∞) with a1 < a2, we have f(a1) < f(a2), (f2) for each sequence {dn} of positive real numbers we have limn→∞ dn = 0 if and only if limn→∞f(dn) = −∞, (f3) for each sequence {dn} of positive real numbers with limn→∞ dn = 0, there exists k ∈ (0, 1) such that limn→∞ dn kf(dn) = 0, is called class f. a contraction involving a function f ∈ f is called an f-contraction. this class was introduced by wardowski in [21]. in time, the functions from this class were used by various authors to generalize their contractive conditions: cosentino and vetro [22]; minak et al. [23]; sgroi and vetro [24]; paesano and vetro [25]; piri and kumam [26]; acar et al. [27]; batra and vashistha [28]. recently, basha and shahzad [20] proved the following best proximity point theorem: theorem 2.1. let a and b be nonempty closed subsets of a complete metric space (x,d). assume that a0 is nonempty and t : a → b is a mapping such that for each x1,x2,u1,u2 ∈ a with d(u1,tx1) = dist(a,b) = d(u2,tx2), we have d(u1,u2) ≤ a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] (2.1) where a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 < 1. further assume that the following conditions hold: (i) t(a0) is contained in b0; (ii) b is approximatively compact with respect to a. then t has a best proximity point. in this paper we introduce some new f type proximal contractions and prove some best proximity point theorems for such contractions. our results generalize some existing best proximity point results. in particular theorem 2.1 becomes a special case of one of our results (theorem 3.1). 3. main results we begin this section with the following definition. definition 3.1. let a and b be two nonempty subsets of a metric space (x,d). a mapping t : a → cb(b) is called αf -proximal contraction of hardy rogers type if there exist two functions α: a×a → [0,∞), f ∈ f and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ a and v1 ∈ tx1, v2 ∈ tx2 with α(x1,x2) ≥ 1 and d(u1,v1) = dist(a,b) = d(u2,v2), we have α(u1,u2) ≥ 1 and τ + f(d(u1,u2)) ≤ f(n(x1,x2)) (3.1) whenever min{d(u1,u2),n(x1,x2)} > 0, where n(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 = 1 and a3 6= 1. remark 3.1. by taking f(x) = ln x for each x ∈ (0,∞), one can see that (3.1) reduces to (2.1). therefore, (3.1) is a proper generalization/extension of (2.1). theorem 3.1. let a and b be nonempty closed subsets of a complete metric space (x,d). assume that a0 is nonempty and t : a → cb(b) is an αf -proximal contraction of hardy rogers type and satisfying the following conditions: (i) for each x ∈ a0, we have tx ⊆ b0; (ii) there exist x1,x2 ∈ a0 and v1 ∈ tx1 such that α(x1,x2) ≥ 1 and d(x2,v1) = dist(a,b); (iii) t is continuous, or, for any sequence {xn} ⊆ a such that xn → x as n → ∞ and α(xn,xn+1) ≥ 1 for each n ∈ n, we have α(xn,x) ≥ 1 for each n ∈ n; (iv) b is approximatively compact with respect to a. then t has a best proximity point. 200 kamran, ali, postolache, ghiura and farheen proof. by hypothesis (ii), we have x1,x2 ∈ a0 and v1 ∈ tx1 for which α(x1,x2) ≥ 1 and d(x2,v1) = dist(a,b). as v2 ∈ tx2 ⊆ b0, there is x3 ∈ a0 satisfying d(x3,v2) = dist(a,b). from (3.1), we get α(x2,x3) ≥ 1 and τ + f(d(x2,x3)) ≤ f(a1d(x1,x2) + a2d(x1,x2) + a3d(x2,x3) + a4[d(x1,x3) + d(x2,x2)]) ≤ f(a1d(x1,x2) + a2d(x1,x2) + a3d(x2,x3) + a4[d(x1,x2) + d(x2,x3)]) = f((a1 + a2 + a4)d(x1,x2) + (a3 + a4)d(x2,x3)). (3.2) as f is strictly increasing, from (3.2), we get d(x2,x3) < (a1 + a2 + a4)d(x1,x2) + (a3 + a4)d(x2,x3). that is, (1 −a3 −a4)d(x2,x3) < (a1 + a2 + a4)d(x1,x2). as a1 + a2 + a3 + 2a4 = 1, the above inequality implies that d(x2,x3) < d(x1,x2). thus by (3.2), we have τ + f(d(x2,x3)) ≤ f(d(x1,x2)). (3.3) from above we have x2,x3 ∈ a0 and v2 ∈ tx2 satisfying α(x2,x3) ≥ 1 and d(x3,v2) = dist(a,b). as v3 ∈ tx3 ⊆ b0, there is x4 ∈ a0 such that d(x4,v3) = dist(a,b). from (3.1), we get α(x3,x4) ≥ 1 and τ + f(d(x3,x4)) ≤ f(a1d(x2,x3) + a2d(x2,x3) + a3d(x3,x4) + a4[d(x2,x4) + d(x3,x3)]) ≤ f(a1d(x2,x3) + a2d(x2,x3) + a3d(x3,x4) + a4[d(x2,x3) + d(x3,x4)]) = f((a1 + a2 + a4)d(x2,x3) + (a3 + a4)d(x3,x4)). after simplification we get τ + f(d(x3,x4)) ≤ f(d(x2,x3)). (3.4) from (3.4) and (3.3), we obtain f(d(x3,x4)) ≤ f(d(x1,x2)) − 2τ. continuing the same process we get sequences {xn} in a0 and {vn} in b0 such that vn ∈ txn, α(xn,xn+1) ≥ 1, d(xn+1,vn) = dist(a,b) and f(d(xn,xn+1)) ≤ f(d(x1,x2)) −nτ for each n ∈ n\{1}. (3.5) letting n → ∞ in (3.5), we get limn→∞f(d(xn,xn+1)) = −∞. thus, by property (f2), we have limn→∞d(xn,xn+1) = 0. let dn = d(xn,xn+1) for each n ∈ n. from (f3) there exists k ∈ (0, 1) such that lim n→∞ dknf(dn) = 0. from (3.5) we have dknf(dn) −d k nf(d1) ≤−d k nnτ ≤ 0 for each n ∈ n. (3.6) letting n →∞ in (3.6), we get lim n→∞ ndkn = 0. this implies that there exists n1 ∈ n such that ndkn ≤ 1 for each n ≥ n1. thus, we have dn ≤ 1 n1/k , for each n ≥ n1. (3.7) best proximity points for a new class of generalized proximal mappings 201 to prove that {xn} is a cauchy sequence in a, consider m,n ∈ n with m > n > n1. by using the triangular inequality and (3.7), we have d(xn,xm) ≤ d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xm−1,xm) = m−1∑ i=n di ≤ ∞∑ i=n di ≤ ∞∑ i=n 1 i1/k . since ∑∞ i=1 1 i1/k is convergent series, we get limn→∞d(xn,xm) = 0, which implies that {xn} is a cauchy sequence in a. since a is closed subset of a complete metric space, there exists x∗ in a such that xn → x∗ as n → ∞. as d(xn+1,vn) = dist(a,b), we have limn→∞d(x∗,vn) = dist(a,b). as b is approximatively compact with respect to a, we get a subsequence {vnk} of {vn} with vnk ∈ txnk that converges to v∗. thus, d(x∗,v∗) = lim k→∞ d(xnk,vnk ) = dist(a,b). by hypothesis (iii), when t is continuous, we get v∗ ∈ tx∗. hence dist(a,b) ≤ d(x∗,tx∗) ≤ d(x∗,v∗) = dist(a,b). this implies that dist(a,b) = d(x∗,tx∗). now we prove the theorem for second assumption of hypothesis (iii), that is, α(xn,x ∗) ≥ 1 for each n ∈ n. since x∗ ∈ a0, then tx∗ ⊆ b0. this implies that for z∗ ∈ tx∗, we have w∗ ∈ a0 such that d(w∗,z∗) = dist(a,b). further note that d(xn+1,vn) = dist(a,b). we claim that d(x∗,w∗) = 0. suppose on contrary that d(x∗,w∗) 6= 0 now from (3.1), we get d(xn+1,w ∗) < a1d(xn,x ∗) + a2d(xn,xn+1) + a3d(x ∗,w∗) + a4[d(xn,w ∗) + d(x∗,xn+1)]. letting n →∞, we get d(x∗,w∗) ≤ (a3 + a4)d(x∗,w∗), which is only possible when d(x∗,w∗) = 0. thus we get dist(a,b) ≤ d(x∗,tx∗) ≤ d(x∗,z∗) = dist(a,b), and this completes the proof. � example 3.1. let x = r × r be endowed with a metric d((x1,x2), (y1,y2)) = |x1 −y1| + |x2 −y2| for each x,y ∈ x. take a = {(0,x) : −1 ≤ x ≤ 1} and b = {(1,x) : −1 ≤ x ≤ 1}. define t : a → cb(b), t(0,x) =   {( 1, x + 1 2 )} if x ≥ 0 {(1,x), (1,x2)} otherwise, and α: a×a → [0,∞), α((0,x), (0,y)) = { 1 if x,y ∈ [0, 1] 0 otherwise. take f(x) = ln x for each x ∈ (0,∞) and τ = 1 2 . it is easy to see that t is αf -proximal contraction of hardy rogers type with a0 = 1 and a2 = a3 = a4 = 0. for each x ∈ a0, we have tx ⊆ b0. also for x1 = (0, 1 2 ) ∈ a0 and v1 = (1, 34 ) ∈ tx1, we have x2 = (0, 3 4 ) such that α(x1,x2) = 1 and d(x2,v1) = dist(a,b). moreover, for any sequence {xn} ⊆ a such that xn → x as n → ∞ and α(xn,xn+1) = 1 for each n ∈ n, we have α(xn,x) = 1 for each n ∈ n. further note that b is approximatively compact with respect to a, therefore, by theorem 3.1, t has a best proximity point. remark 3.2. note that theorem 2.1 is not applicable in the above example. therefore, our theorem properly generalizes/extends theorem 2.1. definition 3.2. let a and b be two nonempty subsets of a metric space (x,d). a mapping t : a → cb(b) is called αf -proximal contraction of ciric type if there exist two functions α: a×a → [0,∞), continuous f in f and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ a and v1 ∈ tx1, v2 ∈ tx2 with α(x1,x2) ≥ 1 and d(u1,v1) = dist(a,b) = d(u2,v2), we have α(u1,u2) ≥ 1 and τ + f(d(u1,u2)) ≤ f(m(x1,x2)) (3.8) 202 kamran, ali, postolache, ghiura and farheen whenever min{d(u1,u2),m(x1,x2)} > 0, where m(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . theorem 3.2. let a and b be nonempty closed subsets of a complete metric space (x,d). assume that a0 is nonempty and t : a → cb(b) is an αf -proximal contraction of ciric type satisfying the following conditions: (i) for each x ∈ a0, we have tx ⊆ b0; (ii) there exist x1,x2 ∈ a0 and v1 ∈ tx1 such that α(x1,x2) ≥ 1 and d(x2,v1) = dist(a,b); (iii) t is continuous, or, for any sequence {xn} ⊆ a such that xn → x as n → ∞ and α(xn,xn+1) ≥ 1 for each n ∈ n, we have α(xn,x) ≥ 1 for each n ∈ n; (iv) b is approximatively compact with respect to a. then t has a best proximity point. proof. by hypothesis (ii), we have x1,x2 ∈ a0 and v1 ∈ tx1 for which α(x1,x2) ≥ 1 and d(x2,v1) = dist(a,b). as v2 ∈ tx2 ⊆ b0, there is x3 ∈ a0 satisfying d(x3,v2) = dist(a,b). from (3.8), we get α(x2,x3) ≥ 1 and τ + f(d(x2,x3)) ≤ f ( max { d(x1,x2),d(x1,x2),d(x2,x3), d(x1,x3) + d(x2,x2) 2 }) = f ( max{d(x1,x2),d(x2,x3)} ) = f(d(x1,x2)), (3.9) otherwise we have a contradiction. from above we have x2,x3 ∈ a0 and v2 ∈ tx2 satisfying α(x2,x3) ≥ 1 and d(x3,v2) = dist(a,b). as v3 ∈ tx3 ⊆ b0, there is x4 ∈ a0 such that d(x4,v3) = dist(a,b). from (3.8), we get α(x3,x4) ≥ 1 and τ + f(d(x3,x4)) ≤ f ( max { d(x2,x3),d(x2,x3),d(x3,x4), d(x2,x4) + d(x3,x3) 2 }) = f ( max{d(x2,x3),d(x3,x4)} ) = f(d(x2,x3)), (3.10) otherwise we have a contradiction. from (3.9) and (3.10), we have f(d(x3,x4)) ≤ f(d(x1,x2)) − 2τ. continuing the same process we get sequences {xn} in a0 and {vn} in b0 such that vn ∈ txn, α(xn,xn+1) ≥ 1, d(xn+1,vn) = dist(a,b) and f(d(xn,xn+1)) ≤ f(d(x1,x2)) −nτ for each n ∈ n−{1}. working on the same lines as the proof of theorem 3.1 is done. we prove that {xn} is a cauchy sequence in a. since a is closed subset of a complete metric space, there exists x∗ in a such that xn → x∗ as n → ∞. as d(xn+1,vn) = dist(a,b). thus, we have limn→∞d(x∗,vn) = dist(a,b). since b is approximatively compact with respect to a, we get a subsequence {vnk} of {vn} with vnk ∈ txnk that converges to v∗. thus, d(x∗,v∗) = lim k→∞ d(xnk+1,vnk ) = dist(a,b). by hypothesis (iii), when t is continuous, we get v∗ ∈ tx∗. hence dist(a,b) ≤ d(x∗,tx∗) ≤ d(x∗,v∗) = dist(a,b). now assume that we have α(xn,x ∗) ≥ 1 for each n ∈ n. since x∗ ∈ a0, then best proximity points for a new class of generalized proximal mappings 203 tx∗ ⊆ b0. this implies that for z∗ ∈ tx∗, we have w∗ ∈ a0 such that d(w∗,z∗) = dist(a,b). further note that d(xn+1,vn) = dist(a,b). we claim that d(x∗,w∗) = 0. on contrary assume that d(x∗,w∗) 6= 0. now, from (3.8), we get τ + f(d(xn+1,w ∗)) < f ( max { d(xn,x ∗),d(xn,xn+1),d(x ∗,w∗), d(xn,w ∗) + d(xn+1,x ∗) 2 ) . letting n →∞, we obtain τ + f(d(x∗,w∗)) ≤ f(d(x∗,w∗)), which is not possible. hence, we have d(x∗,w∗) = 0. thus we get dist(a,b) ≤ d(x∗,tx∗) ≤ d(x∗,z∗) = dist(a,b), and this completes the proof. � example 3.2. let x = r × r be endowed with a metric d((x1,x2), (y1,y2)) = |x1 −y1| + |x2 −y2| for each x,y ∈ x. take a = {(0,x) : −1 ≤ x ≤ 1} and b = {(1,x) : −1 ≤ x ≤ 1}. define t : a → cb(b), t(0,x) = { {(1, x 3 ), (1, x 2 )} if x ≥ 0 {(1,x), (1,x2)} otherwise, and α: a×a → [0,∞) α((0,x), (0,y)) = { 1 if x,y ∈ [0, 1] 0 otherwise. take f(x) = ln x for each x ∈ (0,∞) and τ = 1 2 . it is easy to see that t is αf -proximal contraction of ciric type. for each x ∈ a0, we have tx ⊆ b0. also for x1 = (0, 13 ) ∈ a0 and v1 = (1, 1 6 ) ∈ tx1, we have x2 = (0, 1 6 ) such that α(x1,x2) = 1 and d(x2,v1) = dist(a,b). moreover, for any sequence {xn} ⊆ a such that xn → x as n → ∞ and α(xn,xn+1) = 1 for each n ∈ n, we have α(xn,x) = 1 for each n ∈ n. further, note that b is approximatively compact with respect to a. therefore, by theorem 3.2, t has a best proximity point. 4. consequences by taking α(x,y) = 1 for each x,y ∈ a, the following two theorems immediately follow from our results. theorem 4.1. let a and b be nonempty closed subsets of a complete metric space (x,d). assume that a0 is nonempty and t : a → cb(b) is a mapping for which there exist a function f ∈ f and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ a and v1 ∈ tx1, v2 ∈ tx2 with d(u1,v1) = dist(a,b) = d(u2,v2), we have τ + f(d(u1,u2)) ≤ f(n(x1,x2)) whenever min{d(u1,u2),n(x1,x2)} > 0, where n(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 + a2 + a3 + 2a4 = 1 and a3 6= 1. further assume that the following conditions hold: (i) for each x ∈ a0, we have tx ⊆ b0; (ii) b is approximatively compact with respect to a. then t has a best proximity point. theorem 4.2. let a and b be nonempty closed subsets of a complete metric space (x,d). assume that a0 is nonempty and t : a → cb(b) is a mapping for which there exist a continuous function f ∈ f and a constant τ > 0 such that for each x1,x2,u1,u2 ∈ a and v1 ∈ tx1, v2 ∈ tx2 with d(u1,v1) = dist(a,b) = d(u2,v2), we have τ + f(d(u1,u2)) ≤ f(m(x1,x2)) 204 kamran, ali, postolache, ghiura and farheen whenever min{d(u1,u2),m(x1,x2)} > 0, where m(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . further assume that the following conditions hold: (i) for each x ∈ a0, we have tx ⊆ b0; (ii) b is approximatively compact with respect to a. then t has a best proximity point. when we take x = a = b, we get the following fixed point theorems from our results: theorem 4.3. let (x,d) be a complete metric space. assume t : x → cb(x) is a mapping for which there are two functions α: a × a → [0,∞), f ∈ f and a constant τ > 0 such that for each x1,x2 ∈ x and u1 ∈ tx1, u2 ∈ tx2 with α(x1,x2) ≥ 1, we have α(u1,u2) ≥ 1 and τ + f(d(u1,u2)) ≤ f(n(x1,x2)) whenever min{d(u1,u2),n(x1,x2)} > 0, where n(x1,x2) = a1d(x1,x2) + a2d(x1,u1) + a3d(x2,u2) + a4[d(x1,u2) + d(x2,u1)] with a1,a2,a3,a4 ≥ 0 satisfying a1 +a2 +a3 +2a4 = 1 and a3 6= 1. further assume that t is continuous, or, for any sequence {xn}⊆ x such that xn → x as n →∞ and α(xn,xn+1) ≥ 1 for each n ∈ n, we have α(xn,x) ≥ 1 for each n ∈ n. then t has a fixed point. theorem 4.4. let (x,d) be a complete metric space. assume t : x → cb(x) is a mapping for which there is α: a×a → [0,∞), continuous function, f in f and τ > 0 such that for each x1,x2 ∈ x and u1 ∈ tx1, u2 ∈ tx2 with α(x1,x2) ≥ 1, we have α(u1,u2) ≥ 1 and τ + f(d(u1,u2)) ≤ f(m(x1,x2)) whenever min{d(u1,u2),m(x1,x2)} > 0, where m(x1,x2) = max { d(x1,x2),d(x1,u1),d(x2,u2), d(x1,u2) + d(x2,u1) 2 } . further assume that t is continuous, or, for any sequence {xn}⊆ x such that xn → x as n →∞ and α(xn,xn+1) ≥ 1 for each n ∈ n, we have α(xn,x) ≥ 1 for each n ∈ n. then t has a fixed point. references [1] k. fan, extensions of two fixed point theorems of f. e. browder. math. z., 112 (1969), 234-240. [2] a. eldred, p. veeramani, existence and convergence of best proximity points, j. math. anal. appl., 323 (2006), 1001-1006. [3] m. jleli, b. samet, best proximity point for α-ψ-proximal contraction type mappings and applications, bull. sci. math., 137 (2013), 977-995. [4] m. u. ali, t. kamran, n. shahzad, best proximity point for α-ψ-proximal contractive multimaps, abstr. appl. anal., 2014 (2014), art. id 181598. [5] a. abkar, m. gabeleh, best proximity points for asymptotic cyclic contraction mappings, nonlinear anal., 74 (2011), 7261-7268. [6] a. abkar, m. gabeleh, best proximity points for cyclic mappings in ordered metric spaces, j. optim. theory appl., 151 (2011), 418-424. [7] m. derafshpour, s. rezapour, n. shahzad, best proximity points of cyclic ϕ-contractions in ordered metric spaces, topol. meth. nonlin. anal., 37 (2011), 193-202. [8] c. di bari, t. suzuki, c. vetro, best proximity point for cyclic meir-keeler contraction, nonlinear anal., 69 (2008), 3790-3794. [9] s. rezapour, m. derafshpour, n. shahzad, best proximity points of cyclic φ-contractions on reflexive banach spaces, fixed point theory appl., 2010 (2010), art. id 946178. [10] c. vetro, best proximity points: convergence and existence theorems for p-cyclic mappings, nonlinear anal. 73 (7) (2010), 2283-2291. [11] m. a. alghamdi, m. a. alghamdi, n. shahzad, best proximity point results in geodesic metric spaces, fixed point theory appl., 2012 (2012), art. id 234. [12] m. a. al-thagafi, n. shahzad, best proximity pairs and equilibrium pairs for kakutani multimaps, nonlinear anal., 70 (3) (2009), 1209-1216. [13] j. markin, n. shahzad, best proximity points for relatively u-continuous mappings in banach and hyperconvex spaces, abstr. appl. anal. 2013 (2013), art. id 680186. best proximity points for a new class of generalized proximal mappings 205 [14] h. k. nashine, p. kumam, c. vetro, best proximity point theorems for rational proximal contractions, fixed point theory appl., 2013 (2013), art. id 95. [15] a. abkar, m. gabeleh, the existence of best proximity points for multivalued non-self mappings, racsam, 107 (2) (2013), 319-325. [16] b. s. choudhury, n. metiya, m. postolache, p. konar, a discussion on best proximity point and coupled best proximity point in partially ordered metric spaces. fixed point theory appl., 2015 (2015), art. id 170. [17] w. shatanawi, a. pitea, best proximity point and best proximity coupled point in a complete metric space with (p)-property, filomat, 29 (1) (2015), 63-74. [18] m. jamali, s.m. vaezpour, best proximity point for certain nonlinear contractions in menger probabilistic metric spaces, j. adv. math. stud., 9 (2) (2016), 338-347. [19] a. bejenaru, a. pitea, fixed point and best proximity point theorems in partial metric spaces, j. math. anal., 7 (4) (2016), 25-44. [20] s. s. basha, n. shahzad, best proximity point theorems for generalized proximal contractions, fixed point theory appl., 2012 (2012), art. id 42. [21] d. wardowski, fixed points of a new type of contractive mappings in complete metric spaces, fixed point theory appl., 2012 (2012), art. id 94. [22] m. cosentino, p. vetro, fixed point results for f-contractive mappings of hardy-rogers-type, filomat 28 (4) (2014), 715-722. [23] g. minak, a. helvac, i. altun, ciric type generalized f-contractions on complete metric spaces and fixed point results, filomat, 28 (6) (2014), 1143-1151. [24] m. sgroi, c. vetro, multi-valued f-contractions and the solution of certain functional and integral equations, filomat 27 (7) (2013), 1259-1268. [25] d. paesano, c. vetro, multi-valued f-contractions in 0-complete partial metric spaces with application to volterra type integral equation,rev. r. acad. cienc. exactas fs. nat., ser. a mat., 108 (2) (2014), 1005-1020. [26] h. piri, p. kumam, some fixed point theorems concerning f-contraction in complete metric spaces, fixed point theory appl., 2014 (2014), art. id 210. [27] o. acar, i. altun, a fixed point theorem for multivalued mappings with δ-distance, abstr. appl. anal., 2014 (2014), art. id 497092. [28] r. batra, s. vashistha, fixed points of an f-contraction on metric spaces with a graph, inter. j. comput. math., 91 (12) (2014), 2483-2490. 1department of mathematics, quaid-i-azam university, islamabad-pakistan 2department of mathematics, school of natural sciences, national university of sciences and technology islamabad-pakistan 3department of medical research, china medical university hospital, china medical university, taichung, taiwan 4department of mathematics & computer science, university politehnica of bucharest, 313 splaiul independenţei, 060042 bucharest, romania ∗corresponding author: mi.postolache@mail.cmuh.org.tw, mihai@mathem.pub.ro 1. introduction 2. previous results 3. main results 4. consequences references international journal of analysis and applications volume 16, number 4 (2018), 528-541 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-528 on left almost semihyperrings shah nawaz1, inayatur rehman2, muhammad gulistan1,∗ 1department of mathematics, hazara university, mansehra, pakistan 2department of mathematics and science, college of arts and applied sciences, dhofar university, salalah, oman ∗corresponding author: gulistanmath@hu.edu.pk abstract. the purpose of this article is to introduce the notion of left almost semihyperrings which is a generalization of left almost semirings. we investigate the basic properties of left almost semihyperrings. by using the concept of hyperideal and regular relations we prove some useful results on it. 1. introduction kazim and naseeudin [10] studied left almost semigroup (abbreviated as la-semigroup). they generalized some handy sequal of semigroup theroy. mushtaq and others [14–16] added many useful result of theory of la-semigroups, also see [2, 8, 9]. la-semigroup is the midway structure between a commutative semigroup and a groupoid. on the other hand it posses many interesting properties which we usully find in commutativ and associative algebric structure. hyperstructures were introduced in 1934, when marty [13] defined hypergroups, began to study their properties, and applied them to groups. a number of papers and several book have been written on hyperstructure theory; see [3,19]. currently a book published on hyperstructure [4] points out on its applications in rough set theory, cryptography, automata, codes automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. hila and dine [7] introduced the notion of left almost semihypergroups. yaqoob received 2018-02-18; accepted 2018-04-27; published 2018-07-02. 2010 mathematics subject classification. 20n25. key words and phrases. la-semihypergroups; la-semihyperrings; regular relations. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 528 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-528 int. j. anal. appl. 16 (4) (2018) 529 et al. [20] extended the work of hila and dine and characterized intra-regular left almost semihypergroups by their hyperideals using pure left identity. gulistan et al. [20] introduced the notion of hv-la-semigroups. rehman et al. [20] studied hyperideals and hypersystems in la-hyperrings. yaqoob and gulistan [21] introduced the concept of partial ordering on left almost semihypergroups. yaqoob et al. [22] introduced the idea of left almost polygroups. fuzzy set theory and rough set theory are also applied to la-semihypergroups, see [1, 6, 12, 23, 24, 26]. in [25], yusuf extended the concept of la-groups to a non-associative structure with respect to the binary operations ‘+’ and ‘×’ namely left almost ring (la-ring). further m. shah and t. shah studied some basic properties of la-ring in [18]. kellil [11] studied left almost semirings. the purpose of this article is to introduce a new and more gerenal class of left almost semihyperrings which is of cource a generalization of left almost semiring. we have investigated the basic properties of left almost semihyperrings. by using the concept of hyperideal and regular relations we have proved some useful results on it. 2. la-semihypergroups in this section we recalled some basic ideas from litrature about an la-semihypergroup which helped in further development of this article. definition 2.1. a map ◦ : x ×x −→ p∗(x) is called an hyperoperation or join operation on the set x, where x is a non-empty set and p∗(x) = p(x)\{∅} shows the all non empty subset of x. a hypergroupoid is a set x with the binary operation is a hyperoperation. if a and b be two non-empty subsets of x then we write product as fallow a◦b = ⋃ x1∈a,x2∈b x1 ◦x2, x1 ◦a = {x1}◦a and x1 ◦b = {x1}◦b. definition 2.2. [7, 20] a hypergroupoid (x,◦) is called an la-semihypergroup if for all x1,x2,x3 ∈ x, (x1 ◦x2) ◦x3 = (x3 ◦x2) ◦x1, the law (x1 ◦x2) ◦x3 = (x3 ◦x2) ◦x1 is known as left invertive law. every la-semihypergroup satisfies the law (x1 ◦x2) ◦ (x3 ◦x4) = (x1 ◦x3) ◦ (x2 ◦x4) for all x1,x2,x3,x4 ∈ x. this law is known as medial law (cf. [7]). lemma 2.1. [20] let x be an la-semihypergroup with pure left idetity e, then x1◦(x2◦x3) = x2◦(x1◦x3) holds for all x1,x2,x3 ∈ x. int. j. anal. appl. 16 (4) (2018) 530 lemma 2.2. [20] let x be an la-semihypergroup with pure left identity e, then (x1 ◦ x2) ◦ (x3 ◦ x4) = (x4 ◦x2) ◦ (x3 ◦x1) holds for all x1,x2,x3,x4 ∈ x. the law (x1 ◦x2) ◦ (x3 ◦x4) = (x4 ◦x2) ◦ (x3 ◦x1) is called a paramedial law. 3. left almost semihyperrings in this section we define the notion of left almost semihyperrings and provided some examples with some basic properties. definition 3.1. an algebraic hyperstructure (r,⊕,⊗) is said to be an la-semihyperring if it satisfies the following axioms: (1) (r,⊕) is an la-semihypergroup; (2) (r,⊗) is an la-semihypergroup, with an absorbing element 0 ∈ r, that is, 0 ⊗x = x⊗ 0 = 0 for all x ∈ r. (3) the operation ⊗ is distributive with respect to the hyperoperation ⊕, that is x1 ⊗ (x2 ⊕x3) = (x1 ⊗x2) ⊕ (x1 ⊗x3), (x1 ⊕x2) ⊗x3 = (x1 ⊗x3) ⊕ (x2 ⊗x3), for all x1,x2,x3 ∈ r. example 3.1. let r = {0,a,b} be a set with the hyperoperations ⊕ and ⊗ defined as follow: ⊕ 0 a b 0 0 r r a {a,b} {a,b} {a,b} b r r r ⊗ 0 a b 0 0 0 0 a 0 r b b 0 r r then (r,⊕,⊗) is an la-semihyperring. definition 3.2. let r be an la-semihyperring. an element e ∈ r is called (i) left identity (resp., pure left identity) if for all x ∈ r, x ∈ e⊗x (resp., x = e⊗x), (ii) right identity (resp., pure right identity) if for all x ∈ r, x ∈ x⊗e (resp., x = x⊗e), (iii) identity (resp., pure identity) if for all x ∈ r, x ∈ e⊗x∩x⊗e (resp., x = e⊗x∩x⊗e). lemma 3.1. if an la-semihypering r has a pure left identity e, then it is unique. int. j. anal. appl. 16 (4) (2018) 531 proof. let us consider that there exist another pure left identity fl such that e⊗fl = fl and fl ⊗e = e. consider e = e⊗e = (fl ⊗e) ⊗e = (e⊗e) ⊗fl = e⊗fl = fl. � definition 3.3. an element x ∈ (r,⊕,⊗) is called additively idempotent if x ∈ x⊕x. set of all additively idempotent element is denoted by i⊕ (r) . if every element of r is additively idempotent then r is said to be additively idempotent. definition 3.4. an element x ∈ (r,⊕,⊗) is called multiplicatively idempotent if x ∈ x⊗x. set of all multiplicatively idempotent element is denoted by i⊗ (r) . lemma 3.2. in an la-semihyperring the following laws hold . (i) (x⊗y) ⊗ (z ⊗w) = (x⊗z) ⊗ (y ⊗w) ,∀x,y,z,w ∈ r, known as medial law. (ii) with left identity of an la-semihyperring r, (x⊗y) ⊗ (z ⊗w) = (w ⊗y) ⊗ (z ⊗x) ,∀x,y,z,w ∈ r, known as paramedial law. proof. (i) consider (x⊗y) ⊗ (z ⊗w) = ((z ⊗w) ⊗y) ⊗x = ((y ⊗w) ⊗z) ⊗x = (x⊗z) ⊗ (y ⊗w) . int. j. anal. appl. 16 (4) (2018) 532 (ii) let e ∈ r be the left identity. consider (x⊗y) ⊗ (z ⊗w) = (e⊗ (x⊗y)) ⊗ (z ⊗w) = ((z ⊗w) ⊗ (x⊗y)) ⊗e = ((z ⊗x) ⊗ (w ⊗y)) ⊗e = (e⊗ (w ⊗y)) ⊗ (z ⊗x) = (w ⊗y) ⊗ (z ⊗x) . � remark 3.1. by medial law we have (x⊗y) ⊗ (z ⊗w) = (w ⊗y) ⊗ (z ⊗x) = (w ⊗z) ⊗ (y ⊗x) . theorem 3.1. let r is an la-semihyperring with pure left identity e then x⊗(y ⊗z) = y⊗(x⊗z) for all x,y,z ∈ r. proof. consider x⊗ (y ⊗z) = (e⊗x) ⊗ (y ⊗z) = (e⊗y) ⊗ (x⊗z) by medial law = y ⊗ (x⊗z) , for all x,y,z,w ∈ r. � theorem 3.2. an la-semihyperring r is said to be semihyperring if and only if x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ r. proof. let r is semihyperring then (x⊗y) ⊗z = x⊗ (y ⊗z) , but (x⊗y) ⊗z = (z ⊗y) ⊗x, so x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ r. conversely let x⊗ (y ⊗z) = (z ⊗y) ⊗x for all x,y,z ∈ r. int. j. anal. appl. 16 (4) (2018) 533 since r is la-semihyperring so (x⊗y) ⊗z = (z ⊗y) ⊗x = x⊗ (y ⊗z) . � lemma 3.3. in an la-semihyperring r, with pure left identity e, x⊗y = z ⊗w ⇒ y ⊗x = w ⊗ z for all x,y,z,w ∈ r. proof. consider y ⊗x = (e⊗y) ⊗x = (x⊗y) ⊗e = (z ⊗w) ⊗e since x⊗y = z ⊗w = (e⊗w) ⊗z by left invertive law = w ⊗z for all x,y,z,w ∈ r. � definition 3.5. let ∅ 6= k ⊆ r, then k is called sub la-semihyperring if k itself form the la-semihyperring. proposition 3.1. let ∅ 6= k ⊆ r, then k is called sub la-semihyperring if ∀ x,y ∈ k we get x⊕y ∈ k and x⊗y ∈ k. proof. straightforward. � definition 3.6. let ∅ 6= a ⊆ r, then a is called a left hyperideal (resp., right hyperideal) of an lasemihyperring r if a⊕a ⊆ a and r⊗a ⊆ a (resp., a⊗r ⊆ a) . if a is both a left and a right hyperideal of r, then it is called a hyperideal of r. definition 3.7. let ∅ 6= b ⊆ r, then b is called a bi-hyperideal of an la-semihyperring r if b ⊕b ⊆ b, b ⊗b ⊆ b and (b ⊗r) ⊗b ⊆ b. definition 3.8. let ∅ 6= i ⊆ r, then i is called an interior hyperideal of an la-semihyperring r if i⊕i ⊆ i and (r⊗ i) ⊗r ⊆ i. definition 3.9. let ∅ 6= q ⊆ r, then q is called a quasi-hyperideal of an la-semihyperring r if q⊕q ⊆ q and q⊗r∩r⊗q ⊆ q. int. j. anal. appl. 16 (4) (2018) 534 definition 3.10. a hyperideal a of an la-semihyperring r is called prime hyperideal of r, if for hyperideals i and j of r satisfying, i ⊗j ⊆ a, implies, either i ⊆ a, or j ⊆ a. definition 3.11. for any non-empty subsets x,y of an la-semihyperring (r,⊕,⊗) we define x ⊕y = ⋃ l1∈x l2∈y (l1 ⊕ l2) and x ⊗y = ⋃ l1i∈x l2i∈y ( n∑ i=1 l1i ⊗ l2i ) . proposition 3.2. let x and y be any two hyperideals of an la-semihyperring (r,⊕,⊗) then x⊗y ⊆ x∩y. proof. let x ∈ x ⊗ y = ⋃ l1∈x l2∈y ( n∑ =1 l1 ⊗ l2 ) ⇒ x ∈ ⋃ l1∈x l2∈y l3, where l3 = n∑ =1 l1 ⊗ l2 for each l1 ∈ x and l2 ∈ y. now since x is an hyperideal so l1 ⊗ l2 ∈ x for l2 ∈ y ⊆ r. this implies that n∑ =1 l1 ⊗ l2 ⊆ x. we x ∈ x. similarly by using the fact that y is also hyperideal we get x ∈ y, and thus x ∈ x ∩y. hence x ⊗y ⊆ x ∩y. � proposition 3.3. any left hyperideal of an la-semihyperring r is a sub la-semihyperring. proof. let i is a left hyperideal of an la-semihyperring r. then obviously ∀ x,y ∈ i we get x⊕y ∈ i. also for any l1,x ∈ i and since l1 ∈ i ⊆ r so we have l1 ⊗x ∈ i. thus i is a sub la-simihyperring. � in particular every right hyperideal becomes sub la-semihyperring and so does the hyperideal. theorem 3.3. intersection of any family of hyperideal of an la-semihyperring r is an hyperideal of r. proof. let {ii}i∈∧ be a family of hyperideals of an la-semihyperring r and we have to show that ⋂ i∈∧ ii is also an hyperideal of r. let x,y ∈ ⋂ i∈∧ ii, then x,y ∈ ii. now since each ii is an hyperideal so x⊕y ∈ ii for all i ∈∧. thus x⊕y ∈ ⋂ i∈∧ ii. again let x ∈ ⋂ i∈∧ ii and l1 ∈ r. from x ∈ ⋂ i∈∧ ii we have x ∈ ii for all i ∈∧. since each ii is an hyperideal so l1 ⊕x ∈ ii for all i ∈∧. which implies that l1 ⊕x ∈ ⋂ i∈∧ ii. thus ⋂ i∈∧ ii is an left hyperideal of r. similarly it can easily be proved for right hyperideals and hence ⋂ i∈∧ ii is an hyperideal of r. � corollary 3.1. intersection of any family of sub la-semihyperring of an la-semihyperring r is again sub la-semihyperring of r. proof. straightforward. � theorem 3.4. if i and j are hyperideals of an la-semihyperring r, then i ⊕ j are hyperideals of r. moreover it is the smallest hyperideal of r containing both i and j. int. j. anal. appl. 16 (4) (2018) 535 proof. let us define i ⊕ j = ⋃ l1∈i l2∈j (l1 ⊕ l2) . let x,y ∈ i ⊕ j then ∃ l11, l12 ∈ x and l21, l22 ∈ j such that x ∈ l11⊕l21 and y ∈ l12⊕l22. consider x⊕y ⊆ (l11 ⊕ l21)⊕(l12 ⊕ l22) = (l11 ⊕ l12)⊕(l21 ⊕ l22) by medial law. since i and j are hyperideals so (l11 ⊕ l12) ⊆ i and (l21 ⊕ l22) ⊆ j. thus x⊕y ⊆ (l11 ⊕ l12) ⊕ (l21 ⊕ l22) ⊆ i ⊕j. hence x⊕y ⊆ i ⊕j for all x ∈ i, and y ∈ j. again consider x ∈ i ⊕ j and r ∈ r. for x ∈ i ⊕ j there exist some l1 ∈ i and l2 ∈ j such that x ∈ l1 ⊕ l2. now r⊗x ∈ r⊗ (l1 ⊕ l2) = (r ⊗ l1) ⊕ (r ⊗ l2) by distributive law. since i and j are hyperideals so (r ⊗ l1) ⊆ i and (r ⊗ l2) ⊆ j for any r ∈ r, l1 ∈ i and l2 ∈ j. thus (r ⊗ l1) ⊕ (r ⊗ l2) ⊆ i ⊕j and hence r⊗x ∈ i⊕j for x ∈ i⊕j and r ∈ r. which shows that i⊕j is left hyperideal of r. similarly it can easily be proved for right ideals and thus i ⊕j is an hyperideal of r. now we will show that i ⊕j contains both i and j i.e. i ∪j ⊆ i ⊕j. let x ∈ i ∪j ⇒ x ∈ i or x ∈ j. since i and j are hyperideals so 0 ∈ i and 0 ∈ j. now if l1 ∈ i, since x = x⊕0 ⊆ i ⊕j. and if x ∈ j, since x = 0 ⊕x ⊆ i ⊕j. hence x ∈ i ⊕j and thus i ∪j ⊆ i ⊕j. next we will show that i ⊕j is the smallest hyperideal. let m be any other hyperideal containing both i and j and we have to show that i ⊕j ⊆ m. for this let x ∈ i ⊕j then there exist l1 ∈ i and l2 ∈ j such that x ∈ l1 ⊕ l2. since l1 ∈ i ⊆ i ∪j ⊆ m and l2 ∈ j ⊆ i ∪j ⊆ m. which implies that l1, l2 ∈ m, but m is the hyperideal so l1 ⊕ l2 ⊆ m. thus x ∈ m, and so i ⊕j ⊆ m. hence i ⊕j is the smallest hyperideal of r containing both i and j. � proposition 3.4. let ∅ 6= k ⊆ r and ∅ 6= i ⊆ r such that k is sub la-semihyperring and i is an hyperideal then (i) k ⊕ i is a sub la-semihyperring of r. (ii) k ∩ i is a hyperideal of r. proof. (i) let us define k ⊕ i = ⋃ l1∈k l2∈i (l1 ⊕ l2) . since 0 ∈ k and 0 ∈ i. therefore we have {0} = 0 ⊕ 0 ⊆ k ⊕ i. thus k ⊕ i is non-empty. let x,y ∈ k ⊕ i then there exist l11, l12 ∈ k and l21, l22 ∈ i such that x ∈ l11 ⊕ l21 and y ∈ l12 ⊕ l22. consider x⊕y ⊆ (l11 ⊕ l21) ⊕ (l12 ⊕ l22) = (l11 ⊕ l12) ⊕ (l21 ⊕ l22) by medial law. since k is sub la-semihyperring and i is an hyperideal so (l11 ⊕ l12) ⊆ k and (l21 ⊕ l22) ⊆ i. thus x⊕y ⊆ (l11 ⊕ l12)⊕(l21 ⊕ l22) ⊆ k⊕i. hence x⊕y ⊆ k⊕i for all x,y ∈ k⊕i. and again consider x⊗y ⊆ (l11 ⊕ l21)⊗(l12 ⊕ l22) = (l11 ⊗ l12)⊕(l11 ⊗ l22)⊕(l21 ⊗ l12)⊕(l21 ⊗ l22) by distributive law. now since k is sub la-semihyperring so (l11 ⊗ l12) ∈ k, and i is an hyperideal so ((l11 ⊗ l22) ⊕ (l21 ⊗ l12) ⊕ (l21 ⊗ l22)) ∈ i. eventually x⊗y ⊆ k ⊕ i. hence k ⊕ i is a sub la-semihyperring of r. (ii) let x,y ∈ k ∩ i ⇒ x,y ∈ k and x,y ∈ i. since k is sub la-semihyperring and i is an hyperideal so x⊕y ∈ k and x⊕y ∈ i. which implies that x⊕y ∈ k ∩ i. now again let x ∈ k ∩ i and l1 ∈ r. from x ∈ k∩i we have x ∈ k and x ∈ i. since k is sub la-semihypergroup so l1⊗x ∈ k and i is an hyperideal int. j. anal. appl. 16 (4) (2018) 536 so l1 ⊕ x ∈ i. which implies that l1 ⊕ x ∈ k ∩ i. thus k ∩ i is an left hyperideal of r. similarly it can easily be proved for right hyperideals and hence k ∩ i is an hyperideal of r. � theorem 3.5. let r be an la-semihyperring with pure left identity then right distributive implies left distributive. proof. let r is right distributive, then (l1 ⊕ l2) ⊗ l3 = (l1 ⊗ l3) ⊕ (l2 ⊗ l3) . consider ((l1 ⊗ l3) ⊕ (l2 ⊗ l3)) ⊗e = ((l1 ⊗ l3)⊗) e⊕ ((l2 ⊗ l3) ⊗e) by right distributive law = ((e⊗ l3) ⊗ l1) ⊕ ((e⊗ l3) ⊗ l2) by left invertive law = (l3 ⊗ l1) ⊕ (l3 ⊗ l2) as e is the left identity = l3 ⊗ (l1 ⊕ l2) which shows that it is left distributive. � theorem 3.6. let e ∈ r be an la-semihyperring with pure left identity then every right hyperideal is also a left hyperideal. proof. let l1 be any right hyperideal of r, then it is a sub la-semihyperring of r. now let l1 ∈ x and h ∈ r then h⊗ l1 = (e⊗h) ⊗ l1 = (l1 ⊗h) ⊗e ∈ x. which shows that x is left hyperideal of r and hence hyperideal of r. � lemma 3.4. let x is an right hyperideal of r with pure left identity e then x ⊗x is an hyperideal of r. proof. let x ∈ x ⊗x then l1 = l2 ⊗ l3 where l2, l3 ∈ x. consider l1 ⊗h = (l2 ⊗ l3) ⊗h = (h⊗ l3) ⊗ l2 ∈ x ⊗x, for all h ∈ r. hence x ⊗ x is an right hyperideal of r. as e is the left identity of r so by theorem 3.6 ,x ⊗ x is left hyperideal of r and hence hyperideal of r. � lemma 3.5. let r is an la-semihyperring with pure left identity e. if x is a proper ideal of r then e /∈ x. int. j. anal. appl. 16 (4) (2018) 537 proof. suppose that e ∈ x. let h ∈ r and consider h = e⊗h ∈ r⊗x ⊆ x for all h ∈ r. which implies that r ⊆ x. but x ⊆ r is obvious and thus x = r. which is contradiction to the fact that x is a proper ideal of r. hence e /∈ x. � 4. homomorphisms on la-semihyperrings definition 4.1. a map γ : r1 → r2 where both r1 and r2 are la-semihyperring is called inclusion homorphism if (i) γ(x⊕y) ⊆ γ(x) ⊕γ(y) (ii) γ(x⊗y) ⊆ γ(x) ⊗γ(y) for all x,y ∈ r1. definition 4.2. a map γ : r1 → r2 where both r1 and r2 are la-semihyperring is called strong homorphism if (i) γ(x⊕y) = γ(x) ⊕γ(y) (ii) γ(x⊗y) = γ(x) ⊗γ(y) for all x,y ∈ r1. definition 4.3. let σ be an equivalence relation on r, then σ is said to be left regular if for x,y,z ∈ r such that (x,y) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ z ⊕ x,l6 ∈ z ⊕ y and (z ⊗ x,z ⊗ y) ∈ σ. σ is said to be right regular if for x,y,z ∈ r such that (x,y) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ x ⊕ z,l6 ∈ y ⊕ z and (x ⊗ z ∈ y ⊗ z) ∈ σ. σ is said to be regular if for x,y,z,w ∈ r such that (x,y) ∈ σ and (z,w) ∈ σ, then (l5, l6) ∈ σ for all l5 ∈ x⊕z,l6 ∈ y ⊕z and (x⊗z,y ⊗z) ∈ σ. proposition 4.1. let (r,⊕,⊗) an la-semihyperring and σ is an equivalence relation on r. then σ is a regular relation on r if and only if σ is a left and right regular respectively. proof. suppose that σ is a regular relation on r. let x,y,z ∈ r such that (x,y) ∈ σ and (z,z) ∈ σ then (l5, l6) ∈ σ for l5 ∈ z⊕x,l6 ∈ z⊕y and (z⊗x,z⊗y) ∈ σ. hence σ is a left regular relation on r. similarly σ is a right regular relation on r. conversely let σ is a left and right regular respectively, and let for x,y,z,w ∈ r such that (x,y) ∈ σ and (z,w) ∈ σ. then by left regularity we have (l5, l6) ∈ σ for l5 ∈ x⊕z,l6 ∈ x⊕w and (x⊗z,x⊗w) ∈ σ. now by right regularity we have (l6, l4) ∈ σ for l6 ∈ x⊕w,l4 ∈ y⊕w and (x⊗w,y⊗w) ∈ σ. now by using transitivity of σ we have (l5, l4) ∈ σ for l5 ∈ x⊕z,l4 ∈ y ⊕w and (x⊗z,y ⊗w) ∈ σ. thus σ is a regular relation on r. � proposition 4.2. let γ : r1 → r2 where both r1 and r2 are la-semihyperring is called inclusion hommorphism, then this inclusion hommorphism defines a regular relation σ on r1 given by (h1,h2) ∈ σ if and only if γ(h1) = γ(h2) for all h1,h2 ∈ r1. int. j. anal. appl. 16 (4) (2018) 538 proof. straightforward. � definition 4.4. let x be an hyperideal of la-semihyperring. define a relation % on r as (x,y) ∈ % if and only if x = y or x ∈ a and y ∈ a, then % is a regular relation on r and it is known as rees regular relation. lemma 4.1. % is a regular relation on r. proof. it is obviously an equivalence relation. now let (x,y) ∈ % and z ∈ r. case(i) if x = y then z⊕x = z⊕y and z⊗x = z⊗y so (l5, l5) ∈ % for l5 ∈ z⊕x and (z⊗x,z⊗y) ∈ %. so % is left regular relation on r. similarly % is right regular relation on r. hence % is a regular relation on r. case(ii) if both x and y ∈ a then for z ∈ r,l5 ∈ z⊕x ⊆ i and l6 ∈ z⊕y ⊆ i and z⊗x ∈ i and z⊗y ∈ i so we have (l5, l6) ∈ % for l5 ∈ z ⊕ x, l6 ∈ z ⊕ y and (z ⊗ x,z ⊗ y) ∈ %. so % is left regular relation on r. similarly % is right regular relation on r. hence % is a regular relation on r. � lemma 4.2. let σ be a regular relation on la-semihyperring r, then {σ(l3) : l3 ∈ σ(l1)⊕σ(l2) ∈ σ(l1⊕l2)} and σ(l1) ⊗σ(l2) = σ(l1 ⊗ l2) for all l1, l2 ∈ r. proof. straightforward. � theorem 4.1. let σ be a regular relation on la-semihyperring r, then (r/σ,⊕,⊗) is an la-semihyperring with the mapping ⊕ : r/σ ×r/σ → p∗(r/σ) and ⊗ : r/σ ×r/σ → r/σ by σ(l1) ⊕σ(l2) = {σ(l3) : l3 ∈ σ(l1) ⊕σ(l2) ∈ σ(l1 ⊕ l2)} and σ(l1) ⊗σ(l2) = σ(l1 ⊗ l2) for all σ(l1),σ(l2) ∈ r/σ. proof. indeed by proposition 4.2, the hyperoperation ⊕ and binary operation ⊗ are well defined. now let σ(x),σ(y),σ(z) ∈ r/σ then (1) (σ(x) ⊕σ(y)) ⊕σ(z) = ({σ(w) : w ∈ σ(x) ⊕σ(y) ∈ σ(x⊕y)}) ⊕σ(z) = {σ(e) : e ∈ (σ(x) ⊕σ(y)) ⊕σ(z) ∈ σ((x⊕y) ⊕z)} = {σ(z51) : e ∈ (σ(z) ⊕σ(y)) ⊕σ(x) ∈ σ((z ⊕y) ⊕x)} = (σ(z) ⊕σ(y)) ⊕σ(x). int. j. anal. appl. 16 (4) (2018) 539 (2) (σ(x) ⊗σ(y)) ⊗σ(z) = σ(x⊗y) ⊗σ(z) = σ((x⊗y) ⊗z) = σ((z ⊗y) ⊗x) = σ(z ⊗y) ⊗σ(x) = (σ(z) ⊗σ(y)) ⊗σ(x). (3) σ(x) ⊗ (σ(y) ⊕σ(z)) = σ(x) ⊗ ({σ(w) : w ∈ σ(y) ⊕σ(z) ∈ σ(y ⊕z)}) = σ(x⊗w) = σ(x⊗ (y ⊕z)) = σ((x⊗y) ⊕ (x⊗z)) = {σ(e) : e ∈ σ(x⊗y) ⊕σ(x⊗z) ∈ σ((x⊗y) ⊕ (x⊗z))} = {σ(e) : e ∈ ((σ(x) ⊗σ(y)) ⊕ (σ(x) ⊗σ(z)))} = (σ(x) ⊗σ(y)) ⊕ (σ(x) ⊗σ(z))), which shows that r/σ is left distributive and similarly it is right distributive. hence (r/σ,⊕,⊗) is an la-semihyperring. � theorem 4.2. (r/%,⊕,⊗) is an la-semihyperring. proof. it follows from the proof of the theorem, 4.1. � proposition 4.3. let (r,⊕,⊗) be an la-semihyperring and ∅ 6= n ⊆ r. if we define a well defined hyperoperation � and binary operation � on r/n = {n(x)|x ∈ r} as (n(x))�(n(y)) = {n(n)|n ∈ x⊕y} , and (n(x)) � (n(y)) = n(x⊗y) ∀ x,y ∈ r. then (r/n,�,�) is an la-semihyperring. proof. let (n(x)) , (n(y)) , (n(z)) ∈ r/n, ∀ x,y,z ∈ r. (1)consider int. j. anal. appl. 16 (4) (2018) 540 (n(x) � n(y)) � n(z) = ({n(n)|n ∈ x⊕y}) � (n(z)) = {n(z)|z ∈ n⊕z} = {n(z)|z ∈ (x⊕y) ⊕z} = {n(z)|z ∈ (z ⊕y) ⊕x} = {n(z)|z ∈ n⊕x} = ({n(n)|n ∈ z ⊕y}) � (n(x)) = ((n(z)) � (n(y))) � (n(x)) . hence (r/n,�) is an la-semihypergroup. (2) consider for (n(x)) , (n(y)) , (n(z)) ∈ r/n, ∀ x,y,z ∈ r, we have (n(x) � n(y)) � n(z) = (n(x⊗y)) � (n(z)) = n((x⊗y) ⊗z) = n((z ⊗y) ⊗x) = (n(z ⊗y)) � (n(x)) = (n(z) � n(y)) � n(x). hence (r/n,�) is an la-semigroup. (3) now let n(x),n(y),n(z) ∈ r/n, ∀ x,y,z ∈ r, then consider n(x) � (n(y) � n(z)) = n(x) � ({n(n)|n ∈ y ⊕z}) = n(x⊗n) = n(x⊗ (y ⊕z)) = n((x⊗y) ⊕ (x⊗z)) = n(x⊗y) � n(x⊗z) = (n(x) � n(y)) � (n(x) � n(z)), and similarly (n(x) � n(y)) � n(z) = (n(x) � n(z)) � (n(y) � n(z)). thus the operation � is distributive with respect to the hyperoperation � for all n(x),n(y),n(z) ∈ r/n. hence (r/n,�,�) is an la-semihyperring. � int. j. anal. appl. 16 (4) (2018) 541 references [1] m. azhar, m. gulistan, n. yaqoob and s. kadry, on fuzzy ordered la-semihypergroups, int. j. anal. appl., 16(2) (2018) 276-289. [2] j. r. cho, j. jezek and t. kepka, paramedial groupoids, czechoslovak math. j., 49(2) (1999) 277-290. [3] p. corsini, prolegomena of hypergroup theory, aviani editore, (1993). [4] p. corsini and v. leoreanu, applications of hyperstructure theory, kluwer academic, (2003). [5] m. gulistan, n. yaqoob and m. shahzad, a note on hv-la-semigroups, upb sci. bull., ser. a, 77(3) (2015) 93-106. [6] m. gulistan, m. khan, n. yaqoob and m. shahzad, structural properties of cubic sets in regular la-semihypergroups, fuzzy inf. eng., 9 (2017) 93-116. [7] k. hila and j. dine, on hyperideals in left almost semihypergroups, isrn algebra, article id 953124 (2011) 8 pages. [8] p. holgate, groupoids satisfying a simple invertive law, math. stud., 61(1-4) (1992) 101-106. [9] m. iqbal, and i. ahmad, on further study of ca-ag-groupoids, proc. pakistan acad. sci., 53(3) (2016) 325–337. [10] m.a. kazim and m. naseeruddin, on almost semigroups, aligarh bull. math., 2 (1972) 1-7. [11] r. kellil, on inverses of left almost semirings and strong left almost semirings, j. math. sci., adv. appl., 26 (2014) 29-39. [12] m. khan, m. gulistan, n. yaqoob and f. hussain, general cubic hyperideals of la-semihypergroups, afrika mat., 27(3-4) (2016) 731-751. [13] f. marty, sur une generalization de la notion de groupe, 8im5z congres mathematicians scandinaves tenua stockholm, (1934) 45-49. [14] q. mushtaq and s.m. yusuf, on la-semigroups, aligarh bull. math., 8 (1978) 65-70. [15] q. mushtaq and s. m. yousaf, on locally associative la-semigroups, j. nat. sci. math., 19 (1979), 57-62. [16] q. mushtaq and m. s. kamran: on left almost groups, proc. pakistan acad. sci., 331 (1996) 53-55. [17] i. rehman, n. yaqoob and s. nawaz, hyperideals and hypersystems in la-hyperrings, songklanakarin j. sci. techn., 39(5) (2017) 651-657. [18] m. shah and t. shah, some basic properties of la-ring, int. math. forum, 6(44) (2011) 2195-2199. [19] t. vougiouklis, hyperstructures and their representations, hadronic press, palm harbor, flarida, usa, (1994). [20] n. yaqoob, p. corsini and f. yousafzai, on intra-regular left almost semihypergroups with pure left identity, j. math., 2013 (2013) article id 510790, 10 pages. [21] n. yaqoob and m. gulistan, partially ordered left almost semihypergroups, j. egyptian math. soc., 23 (2015) 231-235. [22] n. yaqoob, i. cristea, m. gulistan and s. nawaz, left almost polygroups, italian j. pure appl. math., 39 (2018) 465-474. [23] n. yaqoob, m. gulistan, v. leoreanu-fotea and k. hila, cubic hyperideals in la-semihypergroups, j. intell. fuzzy syst., 34(4) (2018) 2707-2721. [24] n. yaqoob, applications of rough sets to γ-hyperideals in left almost γ-semihypergroups, neural comput. appl., 21(1) (2012) 267-273. [25] s. m. yusuf, on left almost ring, proc. of 7th international pure math. conference, islamabad, (2006). [26] j. zhan, n. yaqoob and m. khan, roughness in non-associative po-semihyprgroups based on pseudohyperorder relations, j. multiple-valued logic soft comput., 28(2-3) (2017) 153-177. 1. introduction 2. la-semihypergroups 3. left almost semihyperrings 4. homomorphisms on la-semihyperrings references international journal of analysis and applications volume 18, number 2 (2020), 262-276 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-262 semiclassical resonances via meromorphy of the resolvent and the s-matrix soumia belmouhoub1,2,∗, bekkai messirdi2 and abderrahmane senoussaoui1,2 1department of mathematics, faculty of exact and applied sciences, university of oran1 ahmed ben bella, algeria 2laboratory of fundamental and applicable mathematics of oran (lmfao), university of oran1 ahmed ben bella, algeria corresponding author: belmsou@yahoo.fr abstract. the purpose of this paper is to describe the basic problems of resonances via meromorphic continuation of the resolvent and the scattering matrix. an example from mathematical physics is given by investigating the poles of the resolvent of semiclassical schrödinger operators and born-oppenheimer hamiltonians. mathematical techniques, dilation-analyticity and feshbach reduction are used here for the characterization of resonances of these hamiltonians. 1. introduction the spectrum in the complex plane of schrödinger operators p(h) = −h2∆ + v, is often the union of the line imz = 0 and at most finitely-many points of the form iλj(h) on the positive imaginary axis λj(h) > 0. these points correspond to the negative eigenvalues of p(h) so that z(h) = −λ2j(h) belongs to the discrete spectrum of p(h). the resolvent ( p(h) −λ2 )−1 is an operator-valued function defined for imλ > 0 and λ 6= λj(h). we would like to find the largest region in the complex λ-plane on which the received 2020-01-11; accepted 2020-01-28; published 2020-03-02. 2010 mathematics subject classification. 35j10, 35p25, 47a56, 47a75. key words and phrases. schrödinger operators; born-oppenheimer hamiltonians; meromorphic continuation; resonances; scattering matrix; feshbach reduction; dilation operator. c©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 262 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-262 int. j. anal. appl. 18 (2) (2020) 263 resolvent can be defined. for several types of potential v , the spectrum of −h2∆ + v is continuous and equals [0, +∞[, and hence contains no (further) information about v. in this setting resonances replace the discrete data of eigenvalues. precisely, the poles of the meromorphic continuation of the resolvent are called resonances or scattering poles. they constitute a natural remplacement of discrete spectral data for problems on non-compact domains. the multiplicity of a pole λ0 is given in terms of multiplicity of the corresponding resonance z0 = λ 2 0, multiplicity of λ0 = dim imπλ20 ( l2comp(r n) ) where: πλ20 = 1 2πi ∮ γ (z −p(h))−1dz : l2comp(r n) −→ l2loc(r n) γ : [0, 2π[ 3 t 7→ z0 + εeit. the resonances are shown to be the same as the poles of the meromorphically continued scattering matrix. meromorphic extensions of resolvents have been studied in many frameworks and their finite rank poles or resonances, serve in a sense as discrete data similar in character to eigenvalues of a compact manifold. however, if the manifold has constant negative sectional curvature away from a compact, guillopé and zworski [1] showed the meromorphic continuation of the resolvent to c with finite rank poles. for n odd they are defined as the poles of the meromorphic continuation of ( p(h) −λ2 )−1 : l2comp(r n) −→ l2loc(r n) from {imλ > 0} to c or to the riemann surface (logarithmic covering of c) if n is even. the main advantage of odd dimensions greater than one is the strong huyghens principle for the wave equation. effectively, one consequence of the strong huyghens principle is the analytic continuation of ( −h2∆ −λ2 )−1 from {imλ > 0} to c. under suitable assumptions on v, the operator p(h) extends as a selfadjoint family of operators on l2(rn) with continuous spectrum [0, +∞[, for example if v is real-valued and lim |x|→∞ v (x) = 0. for λ ∈ c�[0, +∞[, the resolvent rv (λ) = (p(h) −λ) −1 is an holomorphic function from c�[0, +∞[ to b(l2(rn)) the algebra of bounded operators on l2(rn). as operator on l2(rn), rv (λ) has no analytic extension across its spectrum. but, if we replace l2(rn) by a smaller dense subspace, like c∞0 (r n), then rv (λ) might have some continuation across [0, +∞[ to some riemann surface above c�[0, +∞[. if the continuation turns out to be meromorphic, we then obtain the resonance of p(h) which are exactly the poles of this continuation. when v = 0, i.e. p(h) = −h2∆, is the free hamiltonian, the resonances can be accessible using fourier analysis. if v 6= 0, many effective approaches combine the known extension of the free resolvent to properties of v. the mathematical study of resonances initiated for schrödinger operators on rn. later, it was extended to more geometric situations, such as the laplacian on hyperbolic and asymptotically hyperbolic manifolds, symmetric or locally symmetric spaces, and damek-ricci spaces, see e.g. [1] and [2]. in a typical situation, int. j. anal. appl. 18 (2) (2020) 264 one works on a complete riemannian manifold x, for which the positive laplacian −∆ is an essentially self-adjoint operator on the hilbert space l2(x) of square integrable functions on x. the basic problems of resonances are described here for schrödinger operators and born-oppenheimer hamiltonians with regular and singular potentials. we first show an holomorphic extension result via fredholm operator theory in hilbert spaces. in section 2, we review basic situations for meromorphic continuation of the resolvent. we study in section 3, the meromorphy of the scattering matrix, it follows that the poles of the meromorphic continuation of the s-matrix are exactly the poles of the continuation of the resolvent and conversely. some interesting characterizations of the resonances of semiclassical schrödinger operators and hamiltonians in the born-oppenheimer approximation are obtained in section 4, by dilation-analyticity and feshbach reduction. we start with the following definition. definition 1.1. let ω ⊆ c be open and connected and h a complex hilbert space. suppose that a(λ) is a b(h)-valued analytic function on ω except for isolated singularities. then a(λ) is said meromorphic in ω, if for each λ0 ∈ ω, there exist a neighbourhood uλ0 of λ0, an integer p > 0 and some (ai)1≤i≤p ⊂b(h) such that for all λ ∈ uλ0�{λ0} , we have the finite laurent expansion: a(λ) = p∑ i=1 ai (λ−λ0)i + b(λ) where b(λ) is an holomorphic function on uλ0 with values in the algebra b(h) of bounded linear operators on h. it is easy to see that a(λ) is holomorphic in u�s where s is a discrete set of u whose elements are the poles of a(λ). p is the order of the pole, a1 is the residue of a(λ) at λ0. we essentially have the following result: theorem 1.1. let ω ⊆ c be a connected open set and (a(z))z∈ω a holomorphic family of fredholm operators on h. if a(z0) −1 exists at some z0 ∈ ω, then z 7→ a(z)−1 is a meromorphic family in ω of operators with poles of finite rank. proof. for any z ∈ ω, let n+ = dim ker a(z) and n− = dim hima(z) and set h+ = c n+ and h− = cn−. let{ e1, ...,en+ } be a basis of h+. so define r+ : h −→ h+ x 7→ r+x = ( 〈x,e1〉 , ..., 〈 x,en+ 〉) . int. j. anal. appl. 18 (2) (2020) 265 next choose y1, ...,yn− whose images in h ima(z) form a basis of h ima(z) and define r− : h− −→ h x 7→ r− ( a1, ...,an− ) = n−∑ k=1 akyk. we produce a grushin problem for a(z) as decribed by belmouhoub and messirdi in [3]: a(z) =   a(z) r− r+ 0   . a(z) is well-posed grushin problem for z in some sufficiently small neighborhood v (z) of z with inverse e(z) =   e(z) e+(z) e−(z) e−+(z)   : h ⊕h+ −→ h ⊕h− such that a(z) is invertible if and only if e−+(z) is invertible and  a −1(z) = e(z) −e+(z)e−1−+(z)e−(z) e−1−+(z) = −r+a(z)−1r− (1.1) in particular, this is true for z = z0 and for z ∈ v (z). we know that the index of a(z) is constant in v (z). since the index vanishes at z0, then at any z we have n+ = n− = n and e−+(z) is an n × n matrix with holomorphic coefficients. so for any z ∈ ω the function fz(λ) = det e−+(λ) is holomorphic in a neighborhood of z such that a(λ) is invertible if and only if fz(λ) 6= 0. as ω is connected and since a(z0) is invertible for at least one z0 ∈ ω, none of the functions fz can be identically zero. since det e−+(z) is not identically zero, e−+(z) is a meromorphic family of matrices. from the schur complement (1.1), then z 7→ a(z)−1 is a meromorphic family of operators with poles of finite rank. � 2. meromorphic continuation of the resolvent 2.1. helmholtz operator. consider the resolvent of laplacian ∆ = n∑ k=1 ∂2 ∂x2 k defined by: r0(λ) = (−h2∆ −λ2)−1 : l2(rn) −→ l2(rn) for imλ > 0, h > 0. since the classical theorems on the usual fourier transform extend to the semiclassical case, it suffices to consider h = 1. then the existence of r0(λ) follows from using the fourier transform which provides an explicit diagonalization of −∆ : r0(λ)u(x) = 1 (2π)n ∫ rn eixξ |ξ|2 −λ2 û(ξ)dξ, imλ > 0. (2.1) (2.1) is of course valid in all dimension but the resolvent operator r0(λ) has much nice properties when n is odd. we establish the following important result concerning the meromorphic continuation of the resolvent in odd dimensions. int. j. anal. appl. 18 (2) (2020) 266 theorem 2.1. ( [4]) suppose that n ≥ 3 is odd, then the resolvent operator r0(λ) defined on l2(rn) for imλ > 0, extends analytically to an entire family of operators from l2comp(r n) to l2loc(r n). for any χ ∈ c∞0 (rn) : ‖χr0(λ)χ‖l2(rn)→hk(rn) = o((1 + |λ|) k−1 emax(−imλ,0)r), k = 0, 1, 2 and supp(χ) ⊂ b(0,r) (the open ball of radius r centered at 0). proof. by the functional calculus, we have (see e.g. [4]): r0(λ) = +∞∫ 0 eiλtu(t)dt, imλ > 0 where u(t) = sin(t √ −∆) √ −∆ . now ∂2u ∂t2 = ∆u(t) so u(t) is a operator solution of the wave equation. more precisely, u(x,t) = u(t)φ1(x) + u ′(t)φ0(x) = sin ( t √ −∆ ) √ −∆ φ1(x) + cos ( t √ −∆ ) φ0(x) is the solution of the wave equation with the initial conditions u(x, 0) = φ0(x) and ∂u ∂t (x, 0) = φ1(x). in odd dimensions, the strong huyghens principle (see [5]) implies that: suppf ⊂ b(0,r) =⇒ (u(t)f) (x) = 0, |x| < t−r. so, if χ ∈ c∞0 (rn) with supp(χ) ⊂ b(0,r), then: χr0(λ)χ = 2r∫ 0 eiλtχu(t)χdt, imλ > 0. the right hand side is now defined and, as an operator l2(rn) −→ l2(rn), holomorphic for λ ∈ c. in fact, λχr0(λ)χ = 2r∫ 0 dt ( eiλt ) χu(t)χdt, where dt = 1 i ∂ ∂t . since dtu(t) = 1 i cos ( t √ −∆ ) , integration by parts shows that the right hand side is bounded with a bound depending on r and α = max (−imλ, 0) . so, ‖χr0(λ)χ‖l2→l2 ≤ eαr |λ| . we can consider u(t) as a map from l2(rn) to the sobolev space h1(rn). indeed, since sup λ∈r ∣∣sin tλ λ ∣∣ = |t| , we have: ‖u(t)‖l2→h1 = ‖u(t)‖l2→l2 + ∥∥∥√−∆u(t)∥∥∥ l2→l2 = o (|t|) + o(1) and integrating shows that: ‖χr0(λ)χ‖l2→h1 = o(e αr). int. j. anal. appl. 18 (2) (2020) 267 we also get a bound for χr0(λ)χ as a map from l 2(rn) to h2(rn). recall that the norm on h2(rn) can be taken as ‖u‖l2 + ‖∆u‖l2 . so, we have: ‖χr0(λ)χ‖l2→h2 ≤ ‖∆ (χr0(λ)χ)‖l2→l2 + ‖χr0(λ)χ‖l2→l2 ≤ ‖χ∆ (r0(λ)χ)‖l2→l2 + ‖[∆,χ] (χ1r0(λ)χ1) χ‖l2→l2 +‖χr0(λ)χ‖l2→l2 where χ1 ∈ c∞0 (rn) such that χ1 = 1 near supp(χ) and with suppχ1 in a ball of radius r1 > r. since (−∆ −λ2)r0(λ) = il2(rn) (as operators on functions of compact support), we have: ‖χ∆ (r0(λ)χ)‖l2→l2 = o(λ 2). since [∆,χ] is a first order operator, we obtain: ‖[∆,χ] (χ1r0(λ)χ1) χ‖l2→l2 ≤ c‖χ1r0(λ)χ1‖l2→h1 , c > 0. so ‖χr0(λ)χ‖l2→h2 ≤ |λ| 2 ‖χr0(λ)χ‖l2→l2 + c‖χ1r0(λ)χ1‖l2→h1 +‖χr0(λ)χ‖l2→l2 . χr0(λ)χ is a bounded operator from l 2(rn) to l2(rn) and its image consists of functions supported in b(0,r). by rellich’s lemma, the embedding of this space in l2(rn) is compact. hence: ‖χr0(λ)χ‖l2→h2 = o(|λ|e αr1 ). � remark 2.1. suppose n is odd and r0(λ) : l 2(rn) −→ l2(rn) for imλ > 0. then the analytic continuation of the schwartz kernel r0(λ,x,y) is given by stone’s formula [6]: r0(λ,x,y) −r0(−λ,x,y) = i 2 λn−2 (2π) n−1 ∫ sn eiλω(x−y)dω, λ ∈ c where dω denotes the standard measure on the unit sphere sn of rn. 2.2. schrödinger operators. let’s study now the resolvent of the schrödinger operator p = −∆ + v on l2(rn) with domain h2(rn) where v ∈ l∞(rn,c), n ≥ 3, odd. the resolvent operator rv (λ) = ( p −λ2 )−1 exists at points λ ∈ c such that imλ � 0. theorem 2.2. (meromorphic continuation of the resolvent) suppose that v ∈ l∞comp(rn,c) (ie v is a.e. bounded potential of compact support), n ≥ 3 is odd. then int. j. anal. appl. 18 (2) (2020) 268 1) rv (λ) : l 2(rn) −→ l2(rn) for imλ > 0, is a meromorphic family of operators with finitely many poles. 2) rv (λ) extends to a meromorphic family of operators rv (λ) : l 2 comp(r n) −→ l2loc(r n) for λ ∈ c. 3) if χ ∈ c∞0 (rn) then ‖χrv (λ)χ‖l2→l2 ≤ c λ , λ > 0. proof. we write: rv (λ) −r0(λ) = ( −∆ + v −λ2 ) − ( −∆ −λ2 ) = −rv (λ)v r0(λ) where χ ∈ c∞0 (rn) such that χv = v. multiply the above equation on the right by χ to get rv (λ)χ−r0(λ)χ = −rv (λ)χv r0(λ)χ, so rv (λ)χ (i + v r0(λ)χ) = r0(λ)χ. but, ‖r0(λ)‖l2→l2 ≤ 1 |imλ|2 , so for large values of imλ, (i + v r0(λ)χ) is invertible and rv (λ) −r0(λ) = −r0(λ) (i + v r0(λ)χ) −1 v r0(λ). now v r0(λ)χ = v χr0(λ)χ is compact being the product of a bounded operator with a compact operator. then, v r0(λ)χ is analytic family of compact operators and we can apply the analytic fredholm theory (theorem 1.2) to conclude that rv (λ) extends as a meromorphic operator valued family of operators l2comp(r n) −→ l2loc(r n) for λ ∈ c. on the other hand, we have shown that rv (λ)χ = r0(λ) (i + v r0(λ)χ) −1 is a meromorphic family of operators. for λ � 1, we have ∥∥∥(i + v r0(λ)χ)−1∥∥∥ l2→l2 ≤ 2 and hence the estimate on r0(λ) implies part (3) of the theorem. � remark 2.2. in the even-dimensional case similar results are valid, except that the resolvent operator only extends to be entire on the logarithmic covering of the complex plane. precisely, for n even, rv (λ) extends to be entire as a function of log λ, i.e. entire on the logarithmic covering λ of c. 3. meromorphy of the scattering matrix we have just studied above the meromorphic continuation of the resolvent to c for odd dimensions and to λ for even dimensions. from this it can be deduced that the scattering matrix has a similar continuation. indeed, the meromorphic continuation of the cut-off resolvent χrv (λ)χ permits us to mermorphically continue the scattering matrix s(λ) as a bounded operator on l2loc(s n) on c or on λ depending on the parity of n. in this section, we will define and describe the scattering matrix of p(h) = −h2∆ + v for v ∈ l∞comp(rn,r), n ≥ 3, where l∞comp(rn,r) is the space of real essentially bounded functions of compact int. j. anal. appl. 18 (2) (2020) 269 support. p(h) is self-adjoint with domain h2(rn) and generates a one-parameter strongly continuous unitary group r 3 t 7→ uv (t) = e−itp(h). the unitary group uv (t) provides solutions ψ(t) = uv (t)ψ0, to the initial value problem:   i ∂ψ ∂t = p(h)ψ ψ(0) = ψ0 ∈ h2(rn) u0(t) is the one-parameter strongly continuous unitary group associated to −h2∆. proposition 3.1. 1) for any f ∈ l2(rn), n ≥ 3, the limits lim t→±∞ uv (t) ∗u0(t)f exist and define bounded transformations ω±(p(h),−∆) or ω± called wave operators: ω±f = lim t→±∞ uv (t) ∗u0(t)f ‖ω±‖l2→l2 = 1 2) for any f,g ∈ l2(rn), 〈ω±f, ω±g〉l2 = 〈f,g〉l2( ω∗± )∗ ω∗± = ω±ω ∗ ±. 3) the operator f± = ω±ω ∗ ± satisfies: f 2± = f±, f ∗ ± = f±, f±ω ∗ ± = ω ∗ ±, ∥∥ω∗±f∥∥ = ‖f±f‖ , imω± = imf±. ω±u0(t) = uv (t)ω± and u0(t)ω ∗ ± = ω ∗ ±uv (t). 4) the pair (−∆,p(h)) is asymptotically complete in the sense that the wave operators ω±(−∆,p(h)) exist. proof. the existence of wave operators comes from an explicit estimate for the free propagation given by u0(t) (see e.g. [7]). the relations (2), (3) and (4) follow from the existence of ω± and the simple properties of the unitary evolution groups. � the existence of the wave operators ω± gives the limits lim t→±∞ uv (t) ∗u0(t)f = f±. the scattering operator s maps f− to f+. it is a bounded operator on l 2(rn) since sf− = f+ = ω ∗ +f = ω ∗ +ω−f−, s = ω∗+ω− : l 2(rn) −→ l2(rn). furthermore, the s-operator commutes with the free time evolution u0(t) : su0(t) = ω ∗ +ω−u0(t) = ω ∗ +uv (t)ω− = (uv (−t)ω+) ∗ ω− = (ω+u0(−t)) ∗ ω− = u0(t)s. int. j. anal. appl. 18 (2) (2020) 270 remark 3.1. it’s simple to show that if imω− (p(h),−∆) = imω+ (p(h),−∆) , then, the s-operator is a unitary operator on l2(rn). hence, the s-operator is invertible and s−1 = s∗. this allows for a reduction of the s-operator to a family of operators s(λ) defined on l2(sn) called the s-matrix. effectively, for λ ∈ r we can define the scattering operator s (λ) : l2(sn) −→ l2(sn) by (see [2]): s(λ) = il2(sn) + a(λ) with a trace class operator a(λ), and it continues meromorphically to the entire complex plane. its poles coincide with the poles of the resolvent with multiplicities, ms(λ), related to the multiplicities of the poles of the resolvent, mr(λ), by the formula ms(λ) = mr(λ) −mr(−λ). which gives the following fundamental result: theorem 3.1. the scattering matrix s(λ) admits a meromorphic continuation to c if n is odd, or to the riemann surface λ, if n is even, with poles precisely at the resonances of p(h). the multiplicity of the poles are the same as the multiplicity of the poles for the resolvent of p(h) and the residues at these poles have the same finite rank. remark 3.2. for the semiclassical schrödinger operator p(h), the resonances may be defined as the poles of the meromorphic continuation of the resolvent rv (λ) or in terms of the meromorphic continuation of the s-matrix s(λ). from [8], it follows that the poles of the meromorphic continuation of the s-matrix are exactly the poles of the continuation of the resolvent and conversely. however, the scattering poles differ from the resolvent poles for example for hyperbolic spaces. 4. resonances of semiclassical schrödinger operators and born-oppenheimer hamiltonians there are various models for which one can prove the existence of resonances for example that of stark hydrogen schrödinger operator, and also those of semiclassical approximation and especially the resonances in the born-oppenheimer approximation. 4.1. resonances of semiclassical schrödinger operators. the theory developed by hunziker [9], identify the resonances with the eigenvalues of the deformed hamiltonian pθ(h), in the lower complex half-plane, of the schrödinger operator p(h) = −h2∆+v (x) defined on l2(rn) with domain d(p(h)) = h2(rn)∩d(v ). the resonances do not depend on θ and they are associated with the poles of the meromorphic extension from the upper complex half-plane of the resolvent of pθ(h). in order to prove the existence of such continuation we operate an explicit construction assuming appropriate conditions on the potential. let the potential v (x) be smooth real function, extends analytically in |imθ| < δ0, δ0 > 0, and such that v (−∆ + 1)−1 is compact. we introduce the resonances of p(h) = −h2∆ + v (x) on l2(rn) by using the int. j. anal. appl. 18 (2) (2020) 271 analytic dilation operator: (uθϕ) (x) = e nθ/2ϕ(eθx), ϕ ∈ c∞0 (r n). uθ has a unitary extension on l 2(rn). it follows that v (eθx)(−∆ + 1)−1 is a compact operator-valued analytic function of θ in the strip |imθ| < δ0. then, pθ(h) = u ∗ θ p(h)uθ = −h 2e−2θ∆ + v (eθx) is an analytic family of non selfadjoint operators where θ runs in the strip |imθ| < δ0, since for z ∈ c+ and ϕ,ψ ∈ l2(rn), 〈r(z)ϕ,ψ〉 = 〈uθr(z)ϕ,uθψ〉 = 〈[uθr(z)u∗θ ] uθϕ,uθψ〉 , (4.1) where uθr(z)u ∗ θ = rθ(z) = (pθ(h) −z) −1. let σess = σ�σdisc be the essential spectrum where the discrete spectrum σdisc is the set of isolated points of the spectrum such that the corresponding riesz projectors are finite dimensional. then, by weyl theorem: σess (pθ(h)) = e −2θσess ( −h2∆ + e2θv (eθx) ) = e−2θσess(−h2∆) = e−2θr+. the definition of resonances is adapted here as follows: definition 4.1. a complex number ρ is a resonance of p(h) if reρ > inf σess(p(h)) and if there exists θ small enough, imθ > 0, such that ρ ∈ σdisc(pθ(h)). σ(h) denotes the set of resonances of p(h). it is well known, see for example the works of messirdi [10], [11], that the resolvent operator r(z) = (p(h)−z)−1, z ∈ c�r, admits an analytic extension in c+ = {z ∈ c : imz > 0} and under the assumption that v (eθx) is analytic, we can extend r(z) to a meromorphic function in a larger domain, the set of poles of this extension is precisely σ(h). effectively, pθ(h) −z = −h2e−2θ∆ + v (eθx) −z = [ i + v (eθx) ( −h2e−2θ∆ −z )−1] (−h2e−2θ∆ −z).[ i + v (eθx) ( −h2e−2θ∆ −z )−1] is invertible for |imz| → ∞ and ( −h2e−2θ∆ −z )−1 ∈ b(l2(rn)) if z ∈ c�e−2θr+. furthermore, it is easily shown that ∂ ∂θ [〈 rθ(z)uθϕ,uθψ 〉] = 0 (4.2) (4.2) implies that 〈 rθ(z)uθϕ,uθψ 〉 is an holomorphic function, for all ϕ,ψ ∈ h2(rn) and z ∈ ωδ an non-empty open subset of c+ such that ωδ ∩ σ(pθ(h)) = ∅, |θ| < δ0. however, (4.1) is true for θ ∈ r and z ∈ c+, the two sides of this equality are holomorphic with respect to θ ∈{|θ| < δ0} . the two holomorphic functions 〈r(z)ϕ,ψ〉 and 〈uθr(z)ϕ,uθψ〉 coincide on a subset of r, so equality is still true in {|θ| < δ0} . now consider (4.1) with respect to the variable z, the function z 7−→ rθ(z) is meromorphic on c�σess (pθ(h)) = int. j. anal. appl. 18 (2) (2020) 272 c�e−2θr+, and the poles of rθ(z) are exactly the elements of σdisc (pθ(h)) . more precisely, we have for all θ complex such that |θ| < δ0, σdisc (pθ(h)) =⋃ ϕ,ψ {poles of z 7−→ 〈r(z)ϕ,ψ〉}∩ { z ∈ c : −2imθ < arg z < π 2 } (4.3) and σ(h) = ⋃ imθ>0, |θ|<δ0 σdisc (pθ(h)) . 4.2. resonances of born-oppenheimer hamiltonians. the quantum hamiltonian in the bornoppenheimer approximation is written as: p(h) = −h2∆x + q(x) ; q(x) = −∆y + v (x,y) on l2(rnx ×rpy) when h tends to 0+. ∆x (resp. ∆y) is the laplace operator with respect to x (resp. y), x ∈ rn and y ∈ rp, n = 3m,p = 3q, m,q ≥ 1. q(x) is the electronic hamiltonian defined on l2(rpy). the purpose of this section is to show that using a general dilation operator: uθϕ(x,y) = e nθ/2ϕ(eθx,y),ϕ ∈ c∞0 (r n ×rp) and the feshbach reduction scheme, the study of resonances of p(h) is reduced to the discrete spectrum of a matrix of operators fθ(z) defined on l2(rnx) ⊕l2(rnx) (the so-called effective hamiltonian) such that: z is a resonance of p(h) ⇐⇒∃θ ∈ c, imθ > 0, z ∈ σdisc ( fθ(z) ) . the dilated hamiltonian is pθ(h) = u ∗ θ p(h)uθ = −h 2e−2θ∆x + q(e θx). assume that: (h1) v ∈ l∞(rnx ×rpy,r) and can be analytically extended on the complex strip: dδ0 = { x ∈ cn : |imx| < δ0 ( 1 + |rex|2 )1/2} . thus, p(h) and q(x) are selfadjoint on their respective natural domains h2(rnx ×rpy) and h2(rpy). in particular, the domain of q(x) is independent of x. we suppose furthermore: (h2) for every x ∈ rn, the spectrum of q(x) contains at least two eigenvalues λ1(x) and λ2(x) where λ2(x) is simple and satisfies: inf λ∈σ(q(x))�{λ2(x)} |λ−λ2(x)| ≥ δ. in particular, this last assumption implies that the spectral projector π(x) of q(x) associated to to the wave packet {λ1(x),λ2(x)} is c2-regular with respect to x (see [12]). furthermore, by the mini-max principle, int. j. anal. appl. 18 (2) (2020) 273 we deduce that λ1(x) and λ2(x) are uniformly bounded with respect to x and can be analytically extended on dδ0. we also assume that λ2(x) has a potential well above the maximum level of λ1(x) : (h3) λ2(x) ≥ 0, sup x∈rn λ1(x) < 0, lim |x|→rn λ2(x) > 0, λ −1 2 (0) = {0} , λ ′′ 2 (0) > 0. by virtue of (h1), q(eθx) and pθ(h) can be extended to small enough complex values of θ as analytic families. we then put a viriel hypothesis to avoid the resonances coming from the effective potential λ1(x) near level 0 : (h4) sup x∈rn (2λ1(x) + x.∇λ1(x)) < 0. let uj(x,y), j = 1, 2, the two eigenfunctions of q(x) associated to λ1(x) and λ2(x) respectively, real and normalized in l2(rpy). we then consider a grushin problem that will lead to the feshbach reduction ( [3]). for z,θ ∈ r, let aθz(h) the matrix operator defined from h2(rnx × rpy) ⊕ l2 (rnx) ⊕ l2 (rnx) to l2(rnx ×rpy) ⊕l2 (rnx) ⊕l2 (rnx) by: aθz(h) =   pθ(h) −z uθ1 uθ2〈 .,uθ1 〉 y 0 0〈 .,uθ2 〉 y 0 0   where uθj (x,y) = uj(xe θ,y), j = 1, 2, and 〈., .〉y denotes the inner product in l 2(rpy). we shall study the extension of aθz(h) to z and θ complexes, for this we have the following results: q(eθx) −q(x) = ( v (xeθ,y) −v (x,y) ) ∈b(l2(rn)) and by the assumption (h1) : ∂ ∂θ ( v (xeθ,y) −v (x,y) ) = eθx.∇xv (xeθ,y) = o(1), uniformly with respect to x,y and θ complex such that |θ| small enough. furthermore, for j = 1, 2, λj(xe θ) −λj(x) = o(|θ|), and uj extends into a holomorphic function on dδ0 with values in h 2(rpy), such that: ∥∥∂αxuθj∥∥y = o((1 + |x|2)−|α|/2),∥∥∂αxuθj −∂αxuj∥∥y = o(|θ|(1 + |x|2)−|α|/2) uniformly with respect to x and θ complex, |θ| small enough. int. j. anal. appl. 18 (2) (2020) 274 we now define on l2(rnx ×rpy), for θ complex, |θ| small enough, the projector πθ by: πθu = 〈 u,uθ1 〉 y uθ1 + 〈 u,uθ2 〉 y uθ2 and π̂θ = 1−πθ. it is therefore simple to show, using previous results, that there is a constant c > 0 such that for θ complex, |θ| small enough, re〈π̂θ (pθ(h) −z) π̂θu,π̂θu〉y ≥ c‖π̂θu‖ 2 (4.4) for all u ∈ l2(rnx ×rpy) and |z| small enough. in particular, the estimate (4.4) shows the existence of a bounded inverse for (p ′θ(h) −z) the restriction of π̂θ (pθ(h) −z) to { u ∈ l2(rnx ×rpy) : π̂θu = u } . it is then elementary to verify that aθz(h) is invertible, and its inverse a θ z(h) −1 is given by ( [3]):  xθ(z) u θ 1 −xθ(z)pθ(h) ( .uθ1 ) uθ2 −xθ(z)pθ(h) ( .uθ2 )〈 (1 −pθ(h)xθ(z)) (.),uθ1 〉 y z − 〈 yθ(z) ( .uθ1 ) ,uθ1 〉 y 〈 yθ(z) ( .uθ2 ) ,uθ1 〉 y〈 (1 −pθ(h)xθ(z)) (.),uθ2 〉 y 〈 yθ(z) ( .uθ1 ) ,uθ2 〉 y z − 〈 yθ(z) ( .uθ2 ) ,uθ2 〉 y   (4.5) where xθ(z) = (p ′ θ(h) −z) −1 π̂θ and yθ(z) = pθ(h) −pθ(h)xθ(z)pθ(h). fθ(z) =   〈 yθ(z) ( .uθ1 ) ,uθ1 〉 y 〈 yθ(z) ( .uθ2 ) ,uθ1 〉 y〈 yθ(z) ( .uθ1 ) ,uθ2 〉 y 〈 yθ(z) ( .uθ2 ) ,uθ2 〉 y   (4.6) is called the feshbach operator, it reduces the initial spectral problem to a problem in l2 (rnx)⊕l2 (rnx) . it will also serve to show that we have a theory of resonances for p(h). indeed, we have: fθ(z) = uθfu −1 θ + r θ(z,h) f =   −h2∆x + λ1(x) 0 0 −h2∆x + λ2(x)   and rθ(z,h) =  −h2 〈 ∆xu θ 1,u θ 1 〉 y − 〈 zθ(z) ( .uθ1 ) ,uθ1 〉 y 〈 yθ(z) ( .uθ2 ) ,uθ1 〉 y〈 yθ(z) ( .uθ1 ) ,uθ2 〉 y −h2 〈 ∆xu θ 2,u θ 2 〉 y − 〈 zθ(z) ( .uθ2 ) ,uθ2 〉 y   where zθ(z) = pθ(h) −yθ(z) = pθ(h)xθ(z)pθ(h), and for all m ∈ z,∥∥rθ(z,h)∥∥b(hm(rn)⊕hm(rn),hm−1(rn)⊕hm−1(rn)) = o(h2). on the other hand, using (h3) and (h4), ( f̃θ −z ) is boundedly invertible from l2 (rnx) ⊕ l2 (rnx) to h2 (rnx) ⊕ h2 (rnx) , for z complex, |z| small enough, where f̃θ = uθfu −1 θ + w, w ∈ c ∞ 0 (r n) such that re ( w + λθ2(x) ) > 0. int. j. anal. appl. 18 (2) (2020) 275 thus, kθ(z,h) = ( rθ(z,h) −w(x) )( f̃θ −z )−1 is a compact operator on l2 (rnx)⊕l2 (rnx) , for θ and z complex small enough. kθ(z,h) depends analytically on z and lim z∈r,z→−∞ ∥∥kθ(e−2θz,h)∥∥ = 0. by theorem 1.2, we deduce that (i + kθ(z,h))−1 is a z-meromorphic family for z in a complex neighborhood of 0. so it is the same for ( fθ(z) −z )−1 = ( f̃θ −z ) (i + kθ(z,h))−1 and (see [3]): (pθ(h) −z)−1 = xθ(z) + aθ+(z) ( fθ(z) −z )−1 aθ−(z) (4.7) with aθ+(z) = (( 1 −xθ(z)pθ(h))(.uθ1 ) , ( 1 −xθ(z)pθ(h))(.uθ2 )) aθ−(z) = a θ +(z) ∗. we can also deduce by construction that the spectra of pθ(h) is discrete near 0 and z ∈ σ(h) if and only if there exists θ ∈ c, imθ > 0, |θ| small enough such that z ∈ σdisc ( fθ(z) ) . (4.7) shows that (pθ(h)−z)−1 extends into a meromorphic function in z near 0, and we have exactly like in (4.3) : ⋃ θ∈c,|θ|<δ ⋃ ϕ,ψ { poles of 〈 (pθ(h) −z)−1ϕ,ψ 〉 l2 ⋂ (]−ε,ε[ + i ]−ε,ε[) } = ⋃ θ∈c,|θ|<δ {σ (pθ(h)) ∩ (]−ε,ε[ + i ]−ε,ε[)} is the set of resonances σ(h) of the hamiltonian p(h) in ]−ε,ε[ + i ]−ε,ε[ , ε > 0. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] guillopé, l. & zworski, m., polynomial bounds on the number of resonances for some complete spaces of constant negative curvature near infinity, asymptotic anal. 11 (1995), 1-22. [2] zworski, m., poisson formulae for resonaces, séminaire e.d.p., ecole polytechnique, exposé xiii, 1996-1997. [3] belmouhoub, s. & messirdi, b., singular schrödinger operators via grushin problem method, anal. univ. oradea fasc. matematica, tom xxiv (1) (2017), 83-91. [4] dyatlov, s. & zworski, m., mathematical theory of scattering resonances, amer. math. soc. providence, rhode island, 2019. [5] hörmander, l., the analysis of linear partial differential operators i. distribution theory and fourier analysis, springer verlag, 1983. [6] reed, m. & simon, b., methods of modern mathematical physics, academic press, new york, 1978. [7] hislop, p.d., fundamentals of scattering theory and resonances in quantum mechanics, cubo (temuco), 14 (03) (2012), 01-39. int. j. anal. appl. 18 (2) (2020) 276 [8] hellfer, b. & martinez, a., comparaison entre les diverses notions de résonances, helv. phys. acta, 60 (1987), 992-1003. [9] hunziker, w., distorsion analyticity and molecular resonance curves, ann. inst. henri poincaré, 45 (1986), 339-358. [10] messirdi, b., asymptotique de born-oppenheimer pour la prédissociation moléculaire (cas de potentiels réguliers), ann. inst. henri poincaré, 61 (1992), 255-292. [11] messirdi, b., asymptotique de born-oppenheimer pour la prédissociation moléculaire, ph. d. thesis, university of paris 13, 1993. [12] combes, j.m. & seiler, r., regularity and asymptotic properties of the discrete spectrum of electronic hamiltonians, int. j. quantum chem. xiv (1978), 213-229. 1. introduction 2. meromorphic continuation of the resolvent 2.1. helmholtz operator 2.2. schrödinger operators 3. meromorphy of the scattering matrix 4. resonances of semiclassical schrödinger operators and born-oppenheimer hamiltonians 4.1. resonances of semiclassical schrödinger operators 4.2. resonances of born-oppenheimer hamiltonians references international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 38-49 http://www.etamaths.com opial type integral inequalities for widder derivatives and linear differential operators ghulam farid1,∗ and josip pečarić2 abstract. in this paper we establish opial type integral inequalities for widder derivatives and linear differential operator. also, for applications we construct some related inequalities as special cases. 1. introduction the following inequality established in 1960. by opial [15] : let x(t) ∈ c(1)[0,h] be such that x(0) = x(h) = 0, and x(t) > 0 in (0,h). then (1.1) ∫ h 0 |x(t)x′(t)|dt ≤ h 4 ∫ h 0 (x′(t)) 2 dt, where constant h 4 is the best possible. over the last 50 years, opial inequality (1.1) is studied by many mathematicians and extended, generalized in different ways. it is recognized as fundamental result in the theory of differential equations (see the monograp[1]). opial inequality and its generalizations, extensions and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations, (see, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19]). following theorems by mitrinović and pečarić, include such generalizations of opial’s inequality given in [16, page, 237–238] and for them we need next characterization: we say that a function u : [a,b] −→ r belongs to the class u1(v,k) if it admits the representation u(x) = ∫ x a k(x,t)v(t) dt where v is a continuous function and k is an arbitrary non-negative kernel such that v(x) > 0 implies u(x) > 0 for every x ∈ [a,b]. we also assume that all integrals under consideration exist and are finite. theorem 1.1. let φ : [0,∞) −→ r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let u ∈ u1(v,k) where 2010 mathematics subject classification. 26a24, 26d15. key words and phrases. convex functions; opial–type inequality; widder derivatives; fractional derivative. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 38 opial type integral inequalities 39 (∫ x a (k(x,t))p dt )1 p ≤ m and 1 p + 1 q = 1. then (1.2)∫ b a |u(x)|1−q φ′(|u(x)|)|v(x)|q dx ≤ q mq (b−a) ∫ b a φ ( (b−a) 1 q m|v(x)| ) dx. if the function φ(x 1 q ) is concave, then the reverse inequality holds. a similar result follows by using another class u2(v,k) of functions u : [a,b] −→ r which admits representation u(x) = ∫ b x k(x,t)v(t)dt. theorem 1.2. let φ : [0,∞) −→ r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let u ∈ u2(v,k) where(∫ b x (k(x,t))p dt )1 p ≤ n and 1 p + 1 q = 1. then (1.3)∫ b a |u(x)|1−q φ′(|u(x)|)|v(x)|qdx ≤ q nq(b−a) ∫ b a φ ( (b−a) 1 q n|v(x)| ) dx. if the function φ(x 1 q ) is concave, then reverse inequality holds. in [5] we gave extensions of above mitrinović–pečarić inequalities which are stated in the following theorems. theorem 1.3. let φ : [0,∞) −→ r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let u ∈ u1(v,k) where(∫ x a (k(x,t))p dt )1 p ≤ m and 1 p + 1 q = 1. then∫ b a |u(x)|1−q φ′(|u(x)|)|v(x)|q dx ≤ q mq φ ( m (∫ b a |v(x)|q dx )1 q ) ≤ q mq (b−a) ∫ b a φ ( (b−a) 1 q m|v(x)| ) dx.(1.4) if the function φ(x 1 q ) is concave, then reverse inequalities hold. theorem 1.4. let φ : [0,∞) −→ r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let u ∈ u2(v,k) where(∫ b x (k(x,t))p dt )1 p ≤ m and 1 p + 1 q = 1. then∫ b a |u(x)|1−qφ′(|u(x)|)|v(x)|qdx ≤ q mq φ ( m (∫ b a |v(x)|q dx )1 q ) ≤ q mq(b−a) ∫ b a φ ( (b−a) 1 q m|v(x)| ) dx.(1.5) if the function φ(x 1 q ) is concave, then reverse inequalities hold. in view of the great importance of taylor’s formula in analysis, it may be regarded as extremely surprising that so few attempts at generalization have been made. 40 farid and pečarić the problem of the representation of an arbitrary function by means of linear combinations of prescribed functions has received no small amount of attention (see the introduction of [17]). therefore, in 1927 widder gave generalization of taylor’s formula [17]. in this paper we are interested to give opial type integral inequalities by mitrinović and pečarić for widder derivatives and linear differential operators. also, we give their extensions and as applications discuss their special cases and examples. 2. mitrinović-pečarić inequalities for widder derivatives in this section we give opial type integral inequalities for widder derivatives. we extend these inequalities, also give their special cases and provide some examples. the following are taken from [17]. let f,u0,u1, ...,un ∈ cn+1([a,b]),n ≥ 0, and the wronskians (2.1) wi(x) := w [u0(x),u1(x), ...,un(x)] := ∣∣∣∣∣∣∣∣∣∣∣∣ u0(x) u1(x) . . . ui(x) u ′ 0(x) u ′ 1(x) . . . u ′ i(x) . . . ui0(x) u i 1(x) . . . u i i(x) ∣∣∣∣∣∣∣∣∣∣∣∣ , i = 1, ...,n. here w0(x) = u0(x). assume w0(x) > 0 over [a,b], i = 1, ...,n. for i ≥ 0, the differential operator of order i (widder derivative): (2.2) lif(x) := w [u0(x),u1(x), ...,ui−1(x),f(x)] wi−1(x) i = 1, ...,n + 1; l0f(x) := f(x),∀x ∈ [a,b]. consider also (2.3) gi(x,t) := 1 wi(t) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ u0(t) u1(t) . . . ui(t) u ′ 0(t) u ′ 1(t) . . . u ′ i(t) . . . ui−10 (t) u i−1 1 (t) . . . u i i−1(x) u0(x) u1(x) . . . ui(x) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , i = 1, ...,n; g0(x,t) := u0(x) u0(t) ,∀x,t ∈ [a,b]. example 2.1. sets of the form {u0,u1, ...,un} are {u0,u1, ...,un}, {1,sinx,−cosx,−sin2x,cos2x,..., (−1)nsinnx, (−1)ncosnx}, etc. we also mention the generalized widder-talylor’s formula, see, [17] also [3]. theorem 2.2. let the functions f,u0,u1, ...,un ∈ cn+1([a,b]), and the wronskians w0(x),w1(x), ...,wn(x) > 0 on [a,b], x ∈ [a,b]. then for t ∈ [a,b] we have (2.4) f(x) = y(t) u0(x) u0(t) + l1f(t)g1(x,t) + ... + lnf(t)gn(x,t) + rn(x), where rn(x) := ∫ x t gn(x,s)ln+1f(s)ds opial type integral inequalities 41 for example [17] one could take u0(x) = c > 0. if ui(x) = x i, i = 0, 1, ...,n, defined on [a,b], then liy(t) = f i(t) and gi(x,t) = (x−t)i i! , t ∈ [a,b]. we need the following result. corollary 2.3. by additionally assuming for fixed x0 ∈ [a,b] that lif(x0) = 0, i = 0, 1, ...,n, we get that (2.5) f(x) = ∫ x x0 gn(x,s)ln+1f(s)ds. note that all results of this section are under the assumptions of theorem 2.2 and corollary 2.3. theorem 2.4. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≥ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x x0 |gn(x,t)|pdt )1 p ≤ m, 1 p + 1 q = 1. then (2.6)∫ b x0 |f(x)|1−qφ′(|f(x)||ln+1f(x)|q dx ≤ q mq φ  m (∫ b x0 |ln+1f(t)|qdt )1 q   . if the function φ(x 1 q ) is concave, then the reverse inequality holds. proof. as f has representation f(x) = ∫ x x0 gn(x,t)ln+1f(t)dt. by applying holder’s inequality we have |f(x)| = ∣∣∣∣ ∫ x x0 gn(x,t)ln+1f(t)dt ∣∣∣∣ ≤ (∫ x x0 |gn(x,t)|pdt )1 p (∫ x x0 |ln+1f(t)|qdt )1 q ≤ m (∫ x x0 |ln+1f(t)|qdt )1 q (2.7) let p(x) = ∫ x x0 |ln+1f(t)|qdt. then p′(x) = |ln+1f(x)|q. now from (2.7) we can have |f(x)| ≤ m (p(x)) 1 q . from the convexity of φ(x 1 q ) it follows that the function x1−qφ′(x) is increasing. thus 42 farid and pečarić we have∫ b x0 |f(x)|1−qφ′(|f(x)|)|ln+1f(x)|qdx ≤ ∫ b x0 m1−q (p(x)) 1 q −1 φ′ ( m (p(x)) 1 q ) p′(x)dx = q mq ∫ b x0 φ′ ( m(p(x)) 1 q ) d(m(p(x) 1 q ) = q mq φ ( m (p(b)) 1 q ) = q mq φ  m (∫ b x0 |ln+1f(t)|qdt )1 q   . � next we give extension of above theorem. theorem 2.5. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≥ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x x0 |gn(x,t)|pdt )1 p ≤ m, 1 p + 1 q = 1. then ∫ b x0 |f(x)|1−qφ′(|f(x)||ln+1f(x)|q dx ≤ q mq φ  m (∫ b x0 |ln+1f(t)|qdt )1 q   ≤ q mq (b−x0) ∫ b x0 φ ( (b−x0) 1 q m|ln+1f(t)| ) dt.(2.8) if the function φ(x 1 q ) is concave, then reverse inequalities hold. proof. inequality (2.6) holds by theorem 2.4. since φ(x 1 q ) is convex, the following jensen’s inequality holds (2.9) φ (( 1 b−a ∫ b x0 g(t) dt )1 q ) ≤ 1 b−a ∫ b x0 φ ( g 1 q (t) ) dt. applying (2.9) on (2.6) we get (2.8). � the counter part of above theorem is given in the following. theorem 2.6. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≤ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x0 x |gn(x,t)|pdt )1 p ≤ m, 1 p + 1 q = 1. then (2.10)∫ x0 a |f(x)|1−qφ′(|f(x)||ln+1f(x)|q dx ≤ q mq φ ( m (∫ x0 a |ln+1f(t)|qdt )1 q ) . if the function φ(x 1 q ) is concave, then the reverse inequality holds. proof. as f has representation f(x) = ∫ x x0 gn(x,t)ln+1f(t)dt. opial type integral inequalities 43 by applying holder’s inequality we have |f(x)| = ∣∣∣∣ ∫ x0 x gn(x,t)ln+1f(t)dt ∣∣∣∣ ≤ (∫ x0 x |gn(x,t)|pdt )1 p (∫ x0 x |ln+1f(t)|qdt )1 q ≤ m (∫ x0 x |ln+1f(t)|qdt )1 q (2.11) let p(x) = ∫ x0 x |ln+1f(t)|qdt. then −p′(x) = |ln+1f(x)|q ≥ 0. now from (2.11) we can have |f(x)| ≤ m (p(x)) 1 q . from the convexity of φ(x 1 q ) it follows that the function x1−qφ′(x) is increasing. thus we have∫ x0 a |f(x)|1−qφ′(|f(x)|)|ln+1f(x)|qdx ≤− ∫ x0 a m1−q (p(x)) 1 q −1 φ′ ( m (p(x)) 1 q ) p′(x)dx = − q mq ∫ x0 a φ′ ( m(p(x)) 1 q ) d(m(p(x) 1 q ) = q mq φ ( m (p(a)) 1 q ) = q mq φ ( m (∫ x0 a |ln+1f(t)|qdt )1 q ) . � next we give extension of above theorem. theorem 2.7. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≤ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x0 x |gn(x,t)|pdt )1 p ≤ m, 1 p + 1 q = 1. then∫ x0 a |f(x)|1−qφ′(|f(x)||ln+1f(x)|q dx ≤ q mq φ ( m (∫ x0 a |ln+1f(t)|qdt )1 q ) ≤ q mq (b−x0) ∫ x0 a φ ( (b−x0) 1 q m|ln+1f(t)| ) dt.(2.12) if the function φ(x 1 q ) is concave, then reverse inequalities hold. proof. as in proof of the theorem 2.7, inequalities follow from theorem 2.6 and jensen’s inequality (2.9). � remark 2.8. if directly we replace u by f, v by ln+1f and general kernal k(x,t) by gn(x,t) in (1.2), (1.3), (1.4) and (1.5) we can see above results. next we discuss extreme cases. theorem 2.9. let p = 1,q = ∞ and x ≥ x0 ∈ [a,b]. then we have 1 b−x0 ∫ b x0 |f(x)||ln+1f(x)|dx ≤ m||ln+1f||2∞.(2.13) 44 farid and pečarić proof. as f has representation f(x) = ∫ x x0 gn(x,t)ln+1f(t)dt. from this we get |f(x)| ≤ (∫ x x0 |gn(x,t)|dt ) ||ln+1f||∞ ≤ m||ln+1f||∞ and using |ln+1f(t)| ≤ ||ln+1f||∞ we have |f(x)||ln+1f(t)| ≤ m||ln+1f||2∞. now integrating the last inequality on [x0,b], we get (2.13). � theorem 2.10. let p = 1,q = ∞ and x ≤ x0 ∈ [a,b]. then we have 1 x0 −a ∫ x0 a |f(x)||ln+1f(x)|dx ≤ m||ln+1f||2∞.(2.14) proof. as f has representation f(x) = ∫ x0 x gn(x,t)ln+1f(t)dt. from this we get |f(x)| ≤ (∫ x x0 |gn(x,t)|dt ) ||ln+1f||∞ ≤ m||ln+1f||∞ and using |ln+1f(t)| ≤ ||ln+1f||∞ we have |f(x)||ln+1f(t)| ≤ m||ln+1f(t)||2∞. now integrating the last inequality on [a,x0], we get (2.14). � corollary 2.11. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∫ x x0 |f(x)|q|ln+1f(x)|q dx ≤ mq 2 (∫ x x0 |ln+1f(t)|qdt )2 .(2.15) proof. by setting φ(t) = t2q and b = x ≥ x0 in theorem 2.4 one can get (2.15). � corollary 2.12. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∫ x0 x |f(x)|q|ln+1f(x)|q dx ≤ mq 2 (∫ x0 x |ln+1f(t)|qdt )2 .(2.16) proof. by setting φ(t) = t2q and a = x ≤ x0 in theorem 2.5 one can get (2.16). � corollary 2.13. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∣∣∣∣ ∫ x x0 |f(x)|q|ln+1f(x)|q dx ∣∣∣∣ ≤ mq2 (∫ x x0 |ln+1f(t)|qdt )2 .(2.17) proof. from (2.15) and (2.16) one can easily get (2.17). � opial type integral inequalities 45 example 2.14. if we take u0(x) = c > 0 and un(x) = x n,n = 0, 1, 2, ...,n defined on [a,b], then lnf(x) = f n(x) and gn(x,t) = (x−t)n n! , t ∈ [a,b],. here we can have m = (b−x0) np+1 p n!(np+1) 1 p and the inequality (2.6) becomes ∫ b x0 |f(x)|1−qφ′ ( |f(x)||fn+1(x)|q ) dx ≤ n!q(np + 1) q p (b−x0) q(np+1) p φ  (b−x0) np+1p n!(np + 1) 1 p (∫ b x0 |fn+1(t)|qdt )1 q   . also extension of (2.6) becomes∫ b x0 |f(x)|1−qφ′ ( |f(x)||fn+1(x)|q ) dx ≤ n!q(np + 1) q p (b−x0) q(np+1) p φ  (b−x0) np+1p n!(np + 1) 1 p (∫ b x0 |fn+1(t)|qdt )1 q   ≤ n!q(np + 1) q p (b−x0)q(n+1) ∫ b x0 φ ( (b−x0)n+1 n!(np + 1) 1 p |ln+1f(t)| ) dt. remark 2.15. examples similar to example 2.14 can be obtained by using other inequalites given in above results. we omit here such examples. 3. mitrinović-pečarić inequalities for linear differential operators in this section we give opial type integral inequalities for linear differential operators. we extend these inequalities, also give some special cases. here we follow [12, page, 145–154]. let i be a closed interval of r. let ai(x), i = 0, 1, ...,n−1(n ∈ n),h(x) be continuous functions on i and let l = dn + an−1(x)d n−1 + ... + a0(x) be a fixed linear differential operator on cn(i). let y1(x), ...,yn(x) be a set of linear independent solutions to ly = 0. here the associated green’s function for l is (3.1) h(x,t) := ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ y1(t) . . . yn(t) y ′ 1(t) . . . y ′ n(t) . . . yn−21 (t) . . . y n−2 n (t) y1(x) . . . yn(x) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ / ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ y1(t) . . . yn(t) y ′ 1(t) . . . y ′ n(t) . . . yn−21 (t) . . . y n−2 n (t) y1(t) . . . yn(t) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , which is a continuous function on i2. consider fixed x0 ∈ i, then (3.2) y(x) = ∫ x x0 h(x,t)h(t)dt,x ∈ i is the unique solution to the initial value problem (3.3) ly = h; y(i)(x0) = 0, i = 0, 1, ...,n− 1. 46 farid and pečarić theorem 3.1. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≥ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x x0 (h(x,t))pdt )1 p ≤ m, 1 p + 1 q = 1. then ∫ b x0 |y(x)|1−qφ′(|y(x)||(ly)(x)|q dx ≤ q mq φ  m (∫ b x0 |(ly)(t)|qdt )1 q   .(3.4) if the function φ(x 1 q ) is concave, then the reverse inequality holds. proof. here y has representation y(x) = ∫ x x0 h(x,t)h(t)dt. rest of proof follows from the proof of theorem 2.4. � next we give extension of above theorem. theorem 3.2. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≥ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x x0 (h(x,t))pdt )1 p ≤ m, 1 p + 1 q = 1. then ∫ b x0 |y(x)|1−qφ′(|y(x)||(ly)(x)|q dx ≤ q mq φ  m (∫ b x0 |(ly)(t)|qdt )1 q   ≤ q mq (b−x0) ∫ b x0 φ ( (b−x0) 1 q m|(ly)(t)| ) dt. if the function φ(x 1 q ) is concave, then reverse inequalities hold. proof. proof is similar to the proof of theorem 2.5. � the counter part of above theorem is given in the following. theorem 3.3. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≤ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x0 x (h(x,t))pdt )1 p ≤ m, 1 p + 1 q = 1. then (3.5)∫ x0 a |y(x)|1−qφ′(|y(x)||(ly)(x)|q dx ≤ q mq φ ( m (∫ x0 a |(ly)(t)|qdt )1 q ) . if the function φ(x 1 q ) is concave, then the reverse inequality holds. proof. proof is similar to the proof of theorem 2.6. � next we give extension of above theorem. opial type integral inequalities 47 theorem 3.4. let φ : [0,∞) → r be a differentiable function such that for q > 1 the function φ(x 1 q ) is convex and φ(0) = 0. let x ≤ x0 ∈ [a,b] and p ∈ (1,∞) such that (∫ x0 x (h(x,t))pdt )1 p ≤ m, 1 p + 1 q = 1. then∫ x0 a |y(x)|1−qφ′(|y(x)||(ly)(x)|q dx ≤ q mq φ ( m (∫ x0 a |(ly)(t)|qdt )1 q ) ≤ q mq (b−x0) ∫ x0 a φ ( (b−x0) 1 q m|(ly)(t)| ) dt.(3.6) if the function φ(x 1 q ) is concave, then reverse inequalities hold. proof. proof is similar to the proof of theorem 2.7. � remark 3.5. if directly we replace u by y, v by ly, and general kernal k(x,t) by h(x,t) in (1.2), (1.3), (1.4) and (1.5) we can see above results. next we discuss extreme cases. theorem 3.6. let p = 1,q = ∞ and x ≥ x0 ∈ [a,b]. then we have 1 b−x0 ∫ b x0 |y(x)||(ly)(x)|dx ≤ m||ly||2∞.(3.7) proof. as f has representation y(x) = ∫ x x0 h(x,t)(ly)(t)dt. from this we get |y(x)| ≤ (∫ x x0 |h(x,t)|dt ) ||ly||∞ ≤ m||ly||∞ and using |(ly)(t)| ≤ ||ly||∞ we have |y(x)||(ly)(t)| ≤ m||ly||2∞. now integrating the last inequality on [x0,b], we get (3.7). � theorem 3.7. let p = 1,q = ∞ and x ≤ x0 ∈ [a,b]. then we have 1 x0 −a ∫ x0 a |y(x)||(ly)(x)|dx ≤ m||ly||2∞.(3.8) proof. as y has representation y(x) = ∫ x0 x h(x,t)(ly)(t)dt. from this we get |y(x)| ≤ (∫ x x0 |h(x,t)|dt ) ||ly||∞ ≤ m||ly||∞ and using |(ly)(t)| ≤ ||ly||∞ we have |y(x)||(ly)(t)| ≤ m||ly||2∞. now integrating the last inequality on [a,x0], we get (3.8). � 48 farid and pečarić corollary 3.8. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∫ x x0 |y(x)|q|(ly)(x)|q dx ≤ mq 2 (∫ x x0 |(ly)(t)|qdt )2 .(3.9) proof. by setting φ(t) = t2q and b = x ≥ x0 in theorem 3.1 one can get (3.9). � corollary 3.9. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∫ x0 x |y(x)|q|(ly)(x)|q dx ≤ mq 2 (∫ x0 x |(ly)(t)|qdt )2 .(3.10) proof. by setting φ(t) = t2q and a = x ≤ x0 in theorem 3.2 one can get (3.10). � corollary 3.10. let p,q ∈ (1,∞) such that 1 p + 1 q = 1. then we have∣∣∣∣ ∫ x x0 |y(x)|q|(ly)(x)|q dx ∣∣∣∣ ≤ mq2 (∫ x x0 |(ly)(t)|qdt )2 .(3.11) proof. from (3.9) and (3.10) one can easily get (3.11). � references [1] r. p. agarwal and p. y. h. pang, opial inequalities with applications in differential and difference equations, kluwer academic publishers, dordrecht, boston, london 1995. [2] r. p. agarwal and v. lakshmikantham, uniqueness and non uniqueness criteria for ordinary differential equations, world scientific, singapore, 1993. [3] g. a. anastassiou, advanced inequalities, vol. 11. world scientific, 2011. [4] m. andrić, a. barbir, g. farid and j. pečarić, more on certain opial-type inequality for fractional derivatives and exponentially convex functions, nonlinear funct. anal. appl., to appear. [5] m. andrić, a. barbir, g. farid and j. pečarić, opial–type inequality due to agarwal–pang and fractional differential inequalities, integral transforms spec. funct., 25 (4) (2014), 324–335. [6] m. andrić, j. pečarić and i. perić, improvements of composition rule for the canavati fractional derivatives and applications to opial–type inequalities, dynam. systems. appl., 20 (2011), 383–394. [7] m. andrić, j. pečarić and i. perić, a multiple opial type inequality for the riemann–liouville fractional derivatives, j. math. inequal., 7 (1) (2013), 139–150. [8] m. andrić, j. pečarić and i. perić, composition identities for the caputo fractional derivatives and applications to opial–type inequalities, math. inequal. appl., 16 (3) (2013), 657–670. [9] d. bainov and p. simeonov, integral inequalities and applications, kluwer academic publishers, dordrecht, 1992. [10] g. farid and j. pečarić, opial type integral inequalities for fractional derivatives, fractional differ. calc., 2 (1) (2012), 31–54. [11] g. farid and j. pečarić, opial type integral inequalities for fractional derivatives ii, fractional differ. calc., 2 (2) (2012), 139–155. [12] d. kreider, r. kuller, and f. perkins, an introduction to linear analysis, addison-wesley publishing company, inc., reading, mass., usa, 1966. [13] j. d. li, opial-type integral inequalities involving several higher order derivatives, j. math. anal. appl., 167 (1) (1992), 98–110. [14] d. s. mitrinović, j. e. pečarić and a. m. fink, inequalities involving functions and their integrals ang derivatives, kluwer academic publishers, dordrecht, 1991. [15] z. opial, sur une inégalité, ann. polon. math., 8 (1960), 29–32. [16] j. e. pečarić, f. proschan and y. l. tong, convex functions, partial orderings and statistical applications, academic press, inc., 1992. [17] d. v. widder, a generalization of taylor’s series, transactions of ams, 30 (1) (1928), 126–154. [18] d. v. widder, the laplace transform, princeton uni. press, new jersey, 1941. opial type integral inequalities 49 [19] d. willett, the existence–uniqueness theorems for an nth order linear ordinary differential equation, amer. math. monthly, 75 (1968), 174–178. 1department of mathematics, comsats institute of information technology, attock campus, pakistan 2faculty of textile technology, university of zagreb, prilaz baruna filipovića 28a, 10000 zagreb, croatia ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 159-173 http://www.etamaths.com properties of solutions of complex differential equations in the unit disc zinelâabidine latreuch and benharrat belaïdi∗ abstract. in this paper, we investigate the growth and oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc ∆ = {z : |z| < 1} generated by solutions of the linear differential equation f(k) + a (z) f = 0 (k ≥ 2) , where a (z) is a meromorphic function of finite iterated p−order in ∆. 1. introduction and main results throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the nevanlinna’s value distribution theory on the complex plane and in the unit disc ∆ = {z : |z| < 1} (see [14] , [15] , [18] , [20] , [22]). we need to give some definitions and discussions. firstly, let us give two definitions about the degree of small growth order of functions in ∆ as polynomials on the complex plane c. there are many types of definitions of small growth order of functions in ∆ (see [11] , [12]) . definition 1.1 let f be a meromorphic function in ∆, and d (f) := lim sup r→1− t (r,f) log 1 1−r = b. if b < ∞, we say that f is of finite b degree (or is non-admissible). if b = ∞, we say that f is of infinite degree (or is admissible), both defined by characteristic function t(r,f). definition 1.2 let f be an analytic function in ∆, and dm (f) := lim sup r→1− log+ m (r,f) log 1 1−r = a < ∞ (or a = ∞) , then we say that f is a function of finite a degree (or of infinite degree) defined by maximum modulus function m(r,f) = max |z|=r |f (z)| . now we give the definitions of iterated order and growth index to classify generally the functions of fast growth in ∆ as those in c (see [5] , [17] , [18]). let us define 2010 mathematics subject classification. 34m10, 30d35. key words and phrases. iterated p−order, linear differential equations, iterated exponent of convergence of the sequence of distinct zeros, unit disc, differential polynomials. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 159 160 latreuch and belaïdi inductively, for r ∈ [0, 1) , exp1 r = er and expp+1 r = exp ( expp r ) , p ∈ n. we also define for all r sufficiently large in (0, 1) , log1 r = log r and logp+1 r = log ( logp r ) , p ∈ n. moreover, we denote by exp0 r = r, log0 r = r, exp−1 r = log1 r, log−1 r = exp1 r. definition 1.3 [6] the iterated p−order of a meromorphic function f in ∆ is defined as ρp (f) = lim sup r→1− log+p t (r,f) log 1 1−r (p ≥ 1) . for an analytic function f in ∆, we also define ρm,p (f) = lim sup r→1− log+p+1 m (r,f) log 1 1−r (p ≥ 1) . remark 1.1 it follows by m. tsuji in [22] that if f is an analytic function in ∆, then ρ1 (f) ≤ ρm,1 (f) ≤ ρ1 (f) + 1. however, it follows by proposition 2.2.2 in [18] ρm,p (f) = ρp (f) (p ≥ 2) . definition 1.4 [6] the growth index of the iterated order of a meromorphic function f(z) in ∆ is defined as i (f) =   0, if f is non-admissible, min{j ∈ n : ρj (f) < ∞} , if f is admissible and ρj (f) < ∞ for some j ∈ n, +∞, if ρj (f) = ∞ for all j ∈ n. for an analytic function f in ∆, we also define im (f) =   0, if f is of finite degree, min{j ∈ n : ρm,j (f) < ∞} , if f is of infinite degree and ρm,j (f) < ∞ for some j ∈ n, +∞, if ρm,j (f) = ∞ for all j ∈ n. definition 1.5 ([13] , [16]) the iterated p−type of a meromorphic function f of iterated p−order ρp (f) (0 < ρp (f) < ∞) in ∆ is defined as τp (f) = lim sup r→1− (1 −r)ρp(f) log+p−1 t (r,f) . definition 1.6 [7] let f be a meromorphic function in ∆. then the iterated exponent of convergence of the sequence of zeros of f (z) is defined as λp (f) = lim sup r→1− log+p n ( r, 1 f ) log 1 1−r , where n ( r, 1 f ) is the counting function of zeros of f (z) in {z ∈ c : |z| < r} . similarly, the iterated exponent of convergence of the sequence of distinct zeros of f (z) is defined as λp (f) = lim sup r→1− log+p n ( r, 1 f ) log 1 1−r , properties of solutions of complex differential equations 161 where n ( r, 1 f ) is the counting function of distinct zeros of f (z) in {z ∈ c : |z| < r}. definition 1.7 [7] the growth index of the convergence exponent of the sequence of zeros of a meromorphic f(z) in ∆ is defined as iλ (f) =   0, if n ( r, 1 f ) = o ( log 1 1−r ) , min{j ∈ n : λj (f) < ∞} , if some j ∈ n with λj (f) < ∞ exists, +∞, if λj (f) = ∞ for all j ∈ n. remark 1.2 similarly, we can define the finiteness degree iλ (f) of λp(f). consider for k ≥ 2 the complex differential equation (1.1) f(k) + a (z) f = 0 and the differential polynomial (1.2) gf = dkf (k) + dk−1f (k−1) + · · · + d1f′ + d0f, where a and dj (j = 0, 1, · · · ,k) are meromorphic functions in ∆. let l(g) denote a differential subfield of the field m(g) of meromorphic functions in a domain g ⊂ c. throughout this paper, we simply denote l instead of l(∆) . special case of such differential subfield lp+1,ρ={g meromorphic in ∆: ρp+1 (g) < ρ} , where ρ is a positive constant. recently, t. b. cao, h. y. xu and c. x. zhu [8], t. b. cao, l. m. li, j. tu and h. y. xu [10] have studied the complex oscillation of differential polynomial generated by meromorphic and analytic solutions of second order linear differential equations with meromorphic coefficients and obtained the following results. theorem a [10] let a (z) be an analytic function of infinite degree and of finite iterated order ρm,p (a) = ρ > 0 in the unit disc ∆, and let f 6≡ 0 be a solution of the equation (1.3) f′′ + a (z) f = 0. moreover, let (1.4) p [f] = p ( f,f′, · · · ,f(m) ) = m∑ j=0 pjf (j) be a linear differential polynomial with analytic coefficients pj ∈ lp+1,ρ, assuming that at least one of the coefficients pj does vanish identically. if ϕ (z) ∈ lp+1,ρ is a non-zero analytic function in ∆, and neither p [f] nor p [f] − ϕ vanishes identically, then we have iλ (p [f] −ϕ) = i (f) = p + 1 and λp+1 (p [f] −ϕ) = ρm,p+1 (f) = ρm,p (a) = ρ. theorem b [8] let a be an admissible meromorphic function of finite iterated order ρp (a) = ρ > 0 (1 ≤ p < ∞) in the unit disc ∆ such that δ (∞,a) = 162 latreuch and belaïdi lim inf r→1− m(r,a) t(r,a) = δ > 0, and let f be a non-zero meromorphic solution of equation (1.3) such that δ (∞,f) > 0. moreover, let be a linear differential polynomial (1.4) with meromorphic coefficients pj ∈ lp+1,ρ, assuming that at least one of the coefficients pj does not vanish identically. if ϕ ∈ lp+1,ρ is a non-zero meromorphic function in ∆, and neither p [f] nor p [f] −ϕ vanishes identically, then we have i (f) = iλ (p [f] −ϕ) = p + 1 and λp (p [f] −ϕ) = ρp+1 (f) = ρp (a) = ρ if p > 1, while ρp (a) ≤ λp (p [f] −ϕ) ≤ ρp+1 (f) ≤ ρp (a) + 1 if p = 1. remark 1.3 the idea of the proofs of theorems a-b is borrowed from the paper of laine, rieppo [19] with the modifications reflecting the change from the complex plane c to the unit disc ∆. before we state our results, we define the sequence of meromorphic functions αi,j (j = 0, · · · ,k − 1) in ∆ by (1.5) αi,j = { α′i,j−1 + αi−1,j−1, for all i = 1, · · · ,k − 1, α′0,j−1 −aαk−1,j−1, for i = 0 and (1.6) αi,0 = { di, for all i = 1, · · · ,k − 1, d0 −dka, for i = 0. we define also h and ψ (z) by (1.7) h = ∣∣∣∣∣∣∣∣∣∣ α0,0 α1,0 . . αk−1,0 α0,1 α1,1 . . αk−1,1 . . . . . . . . . . α0,k−1 α1,k−1 . . αk−1,k−1 ∣∣∣∣∣∣∣∣∣∣ , (1.8) ψ (z) = c0ϕ + c1ϕ ′ + · · · + ck−1ϕ(k−1), where cj (j = 0, · · · ,k − 1) are finite iterated p−order meromorphic functions in ∆ depending on αi,j and ϕ 6≡ 0 is a meromorphic function in ∆ with ρp (ϕ) < ∞. the main purpose of this paper is to study the growth and oscillation of differential polynomial (1.2) generated by meromorphic solutions of equation (1.1) in the unit disc ∆. theorem 1.1 suppose that a (z) is a meromorphic function of finite iterated p−order in ∆ and that dj (z) (j = 0, 1, · · · ,k) are finite iterated p−order meromorphic functions in ∆ that are not all vanishing identically such that h 6≡ 0. if f (z) is an infinite iterated p−order meromorphic solution of (1.1) with ρp+1 (f) = ρ, then the differential polynomial (1.2) satisfies ρp (gf ) = ρp (f) = ∞ properties of solutions of complex differential equations 163 and ρp+1 (gf ) = ρp+1 (f) = ρ. furthermore, if f is a finite iterated p−order meromorphic solution of (1.1) such that (1.9) ρp (f) > max{ρp (a) ,ρp (dj) (j = 0, 1, · · · ,k)} , then ρp (gf ) = ρp (f) . remark 1.4 in theorem 1.1, if we do not have the condition h 6≡ 0, then the conclusions of theorem 1.1 cannot hold. for example, if we take dk = 1,d0 = a and dj ≡ 0 (j = 1, · · · ,k − 1) , then h ≡ 0. it follows that gf ≡ 0 and ρp (gf ) = 0. so, if f (z) is an infinite iterated p−order meromorphic solution of (1.1) , then ρp (gf ) = 0 < ρp (f) = ∞, and if f is a finite iterated p−order meromorphic solution of (1.1) such that (1.9) holds, then ρp (gf ) = 0 < ρp (f). corollary 1.1 suppose that a (z) is admissible meromorphic function in ∆ such that i (a) = p (1 ≤ p < ∞) and δ (∞,a) = δ > 0. let dj (z) (j = 0, 1, · · · ,k) be finite iterated p−order meromorphic functions in ∆ that are not all vanishing identically such that h 6≡ 0, and let f be a nonzero meromorphic solution of (1.1) . if δ (∞,f) > 0, then the differential polynomial gf satisfies i (gf ) = p + 1 and ρp+1 (gf ) = ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρp (a) + 1 if p = 1. theorem 1.2 under the assumptions of theorem 1.1, let ϕ (z) 6≡ 0 be a meromorphic function with finite iterated p−order in ∆ such that ψ (z) is not a solution of (1.1) . if f (z) is an infinite iterated p−order meromorphic solution of (1.1) with ρp+1 (f) = ρ, then the differential polynomial (1.2) satisfies λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞ and λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρ. furthermore, if f is a finite iterated p−order meromorphic solution of (1.1) such that (1.10) ρp (f) > max{ρp (a) ,ρp (ϕ) ,ρp (dj) (j = 0, 1, · · · ,k)} , then λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) . corollary 1.2 under the assumptions of corollary 1.1, let ϕ (z) 6≡ 0 be a meromorphic function with finite iterated p−order in ∆ such that ψ (z) 6≡ 0. then the differential polynomial (1.2) satisfies λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρp (a) + 1 if p = 1. 164 latreuch and belaïdi remark 1.5 the ideas of the proofs of theorems 1.1 and 1.2 are from [21] with modification from the complex plane c to the unit disc ∆. for some papers related in the complex plane see [19, 21, 4] . 2. auxiliary lemmas lemma 2.1 [9] let a0, a1, · · · , ak−1, f 6≡ 0 be meromorphic functions in ∆, and let f be a meromorphic solution of the equation (2.1) f(k) + ak−1 (z) f (k−1) + · · · + a1 (z) f′ + a0 (z) f = f (z) such that max{ρp (aj) (j = 0, 1, · · · ,k − 1) ,ρp (f)} < ρp (f) ≤ +∞. then λp (f) = λp (f) = ρp (f) and λp+1 (f) = λp+1 (f) = ρp+1 (f) . lemma 2.2 [6] let p ≥ 1 be an integer, and let a0(z), · · · ,ak−1(z) be analytic functions in ∆ such that i (a0) = p. if max{i (aj) : j = 1, · · · ,k − 1} < p or max{ρp (aj) : j = 1, · · · ,k − 1} < ρp (a0) , then every solution f 6≡ 0 of the equation (2.2) f(k) + ak−1 (z) f (k−1) + · · · + a1 (z) f′ + a0 (z) f = 0, satisfies i (f) = p + 1 and ρp (f) = ∞, ρp (a0) ≤ ρp+1 (f) = ρm,p+1 (f) ≤ max{ρm,p (aj) : j = 0, 1, · · · ,k − 1}. lemma 2.3 [3] let f and g be meromorphic functions in the unit disc ∆ such that 0 < ρp (f) ,ρp (g) < ∞ and 0 < τp (f) ,τp (g) < ∞. then we have (i) if ρp (f) > ρp (g) , then we obtain (2.3) τp (f + g) = τp (fg) = τp (f) . (ii) if ρp (f) = ρp (g) and τp (f) 6= τp (g) , then we get (2.4) ρp (f + g) = ρp (fg) = ρp (f) = ρp (g) . lemma 2.4 [17, 2] let f be a meromorphic function in the unit disc for which i (f) = p ≥ 1 and ρp (f) = β < ∞, and let k ∈ n. then for any ε > 0, (2.5) m ( r, f(k) f ) = o ( expp−2 ( log 1 1 −r )β+ε) for all r outside a set e ⊂ [0, 1) with ∫ e dr 1−r < ∞. lemma 2.5 [8] let a (z) be an admissible meromorphic function in ∆ such that i (a) = p (1 ≤ p < ∞) and δ (∞,a) = δ > 0, and let f be a nonzero meromorphic properties of solutions of complex differential equations 165 solution of (1.1) . if δ (∞,f) > 0, then i (f) = p + 1 and ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ ρp+1 (f) ≤ ρp (a) + 1 if p = 1. lemma 2.6 [1] let g : (0, 1) → r and h : (0, 1) → r be monotone increasing functions such that g (r) ≤ h (r) holds outside of an exceptional set e1 ⊂ [0, 1) for which ∫ e1 dr 1−r < ∞. then there exists a constant d ∈ (0, 1) such that if s (r) = 1 −d (1 −r) , then g (r) ≤ h (s (r)) for all r ∈ [0, 1). 3. proofs of the theorems and the corollaries proof of theorem 1.1 suppose that f is an infinite iterated p−order meromorphic solution of (1.1) with ρp+1 (f) = ρ. by (1.1) we have (3.1) f(k) = −af which implies gf = dkf (k) + dk−1f (k−1) + · · · + d0f (3.2) = dk−1f (k−1) + · · · + (d0 −dka) f. we can rewrite (3.2) as (3.3) gf = k−1∑ i=0 αi,0f (i), where αi,0 are defined in (1.6) . differentiating both sides of equation (3.3) and replacing f(k) with f(k) = −af, we obtain g′f = k−1∑ i=0 α′i,0f (i) + k−1∑ i=0 αi,0f (i+1) = k−1∑ i=0 α′i,0f (i) + k∑ i=1 αi−1,0f (i) = α′0,0f + k−1∑ i=1 α′i,0f (i) + k−1∑ i=1 αi−1,0f (i) + αk−1,0f (k) = α′0,0f + k−1∑ i=1 ( α′i,0 + αi−1,0 ) f(i) −αk−1,0af (3.4) = k−1∑ i=1 ( α′i,0 + αi−1,0 ) f(i) + ( α′0,0 −αk−1,0a ) f. we can rewrite (3.4) as (3.5) g′f = k−1∑ i=0 αi,1f (i), where (3.6) αi,1 = { α′i,0 + αi−1,0, for all i = 1, · · · ,k − 1, α′0,0 −aαk−1,0, for i = 0. 166 latreuch and belaïdi differentiating both sides of equation (3.5) and replacing f(k) with f(k) = −af, we obtain g′′f = k−1∑ i=0 α′i,1f (i) + k−1∑ i=0 αi,1f (i+1) = k−1∑ i=0 α′i,1f (i) + k∑ i=1 αi−1,1f (i) = α′0,1f + k−1∑ i=1 α′i,1f (i) + k−1∑ i=1 αi−1,1f (i) + αk−1,1f (k) = α′0,1f + k−1∑ i=1 ( α′i,1 + αi−1,1 ) f(i) −αk−1,1af (3.7) = k−1∑ i=1 ( α′i,1 + αi−1,1 ) f(i) + ( α′0,1 −αk−1,1a ) f which implies that (3.8) g′′f = k−1∑ i=0 αi,2f (i), where (3.9) αi,2 = { α′i,1 + αi−1,1, for all i = 1, · · · ,k − 1, α′0,1 −aαk−1,1, for i = 0. by using the same method as above we can easily deduce that (3.10) g (j) f = k−1∑ i=0 αi,jf (i), j = 0, 1, · · · ,k − 1, where (3.11) αi,j = { α′i,j−1 + αi−1,j−1, for all i = 1, · · · ,k − 1, α′0,j−1 −aαk−1,j−1, for i = 0 and (3.12) αi,0 = { di, for all i = 1, · · · ,k − 1, d0 −dka, for i = 0. by (3.3) − (3.12) we obtain the system of equations (3.13)   gf = α0,0f + α1,0f ′ + · · · + αk−1,0f(k−1), g′f = α0,1f + α1,1f ′ + · · · + αk−1,1f(k−1), g′′f = α0,2f + α1,2f ′ + · · · + αk−1,2f(k−1), · · · g (k−1) f = α0,k−1f + α1,k−1f ′ + · · · + αk−1,k−1f(k−1). by cramer’s rule, and since h 6≡ 0 we have (3.14) f = ∣∣∣∣∣∣∣∣∣∣ gf α1,0 . . αk−1,0 g′f α1,1 . . αk−1,1 . . . . . . . . . . g (k−1) f α1,k−1 . . αk−1,k−1 ∣∣∣∣∣∣∣∣∣∣ h . properties of solutions of complex differential equations 167 so, we obtain (3.15) f = c0gf + c1g ′ f + · · · + ck−1g (k−1) f , where cj are finite iterated p−order meromorphic functions in ∆ depending on αi,j, where αi,j are defined in (3.11) and (3.12) . if ρp (gf ) < +∞, then by (3.15) we obtain ρp (f) < +∞, and this is a contradiction. hence ρp (gf ) = ρp (f) = +∞. now, we prove that ρp+1 (gf ) = ρp+1 (f) = ρ. by (3.2), we get ρp+1 (gf ) ≤ ρp+1 (f) and by (3.15) we have ρp+1 (f) ≤ ρp+1 (gf ). this yield ρp+1 (gf ) = ρp+1 (f) = ρ. furthermore, if f is a finite iterated p−order meromorphic solution of equation (1.1) such that (3.16) ρp (f) > max{ρp (a) ,ρp (dj) (j = 0, 1, · · · ,k)} , then (3.17) ρp (f) > max{ρp (αi,j) : i = 0, · · · ,k − 1,j = 0, · · · ,k − 1} . by (3.2) and (3.16) we have ρp (gf ) ≤ ρp (f) . now, we prove ρp (gf ) = ρp (f) . if ρp (gf ) < ρp (f) , then by (3.15) and (3.17) we get ρp (f) ≤ max{ρp (cj) (j = 0, · · · ,k − 1) ,ρp (gf )} < ρp (f) and this is a contradiction. hence ρp (gf ) = ρp (f) . remark 3.1 from (3.15) , it follows that the condition h 6≡ 0 is equivalent to that gf,g ′ f,g ′′ f , ...,g (k−1) f are linearly independent over the field of meromorphic functions of finite iterated p−order in ∆. proof of corollary 1.1 suppose f 6≡ 0 is a meromorphic solution of (1.1) . then, by lemma 2.5, we have i (f) = p + 1 and ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ ρp+1 (f) ≤ ρp (a) + 1 if p = 1. thus, by theorem 1.1 we obtain that the differential polynomial gf satisfies i (gf ) = p + 1 and ρp+1 (gf ) = ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρp (a) + 1 if p = 1. proof of theorem 1.2 suppose that f is an infinite iterated p−order meromorphic solution of equation (1.1) with ρp+1 (f) = ρ. set w (z) = gf −ϕ. since ρp (ϕ) < ∞, then by theorem 1.1 we have ρp (w) = ρp (gf ) = ∞ and ρp+1 (w) = ρp+1 (gf ) = ρ. to prove λp (gf −ϕ) = λp (gf −ϕ) = ∞ and λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρ we need to prove λp (w) = λp (w) = ∞ and λp+1 (w) = λp+1 (w) = ρ. by gf = w + ϕ and (3.15) , we get (3.18) f = c0w + c1w ′ + · · · + ck−1w(k−1) + ψ (z) , where (3.19) ψ (z) = c0ϕ + c1ϕ ′ + · · · + ck−1ϕ(k−1). 168 latreuch and belaïdi substituting (3.18) into (1.1) , we obtain (3.20) ck−1w (2k−1) + 2k−2∑ i=0 φiw (i) = − ( ψ(k) + a (z) ψ ) = h, where φi (i = 0, · · · , 2k − 2) are meromorphic functions in ∆ with finite iterated p−order. since ψ (z) is not a solution of (1.1) , it follows that h 6≡ 0. then by lemma 2.1, we obtain λp (w) = λp (w) = ∞ and λp+1 (w) = λp+1 (w) = ρ, i. e., λp (gf −ϕ) = λp (gf −ϕ) = ∞ and λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρ. suppose that f is a finite iterated p−order meromorphic solution of equation (1.1) such that (1.10) holds. set w (z) = gf −ϕ. since ρp (ϕ) < ρp (f) , then by theorem 1.1 we have ρp (w) = ρp (gf ) = ρp (f) . to prove λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) we need to prove λp (w) = λp (w) = ρp (f) . using the same reasoning as above, we get ck−1w (2k−1) + 2k−2∑ i=0 φiw (i) = − ( ψ(k) + a (z) ψ ) = f, where ck−1, φi (i = 0, · · · , 2k − 2) are meromorphic functions in ∆ with finite iterated p−order ρp (ck−1) < ρp (w) , ρp (φi) < ρp (w) (i = 0, · · · , 2k − 2) and ψ (z) = c0ϕ + c1ϕ ′ + · · · + ck−1ϕ(k−1), ρp (f) < ρp (w) . since ψ (z) is not a solution of (1.1) , it follows that f 6≡ 0. then by lemma 2.1, we obtain λp (w) = λp (w) = ρp (f) , i. e., λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) . proof of corollary 1.2 suppose that f 6≡ 0 is a meromorphic solution of (1.1) . then, by lemma 2.5, we have i (f) = p + 1 and ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ ρp+1 (f) ≤ ρp (a) + 1 if p = 1. since ψ 6≡ 0 and ρp (ψ) < ∞, then ψ cannot be a solution of equation (1.1) . thus, by theorem 1.2 we obtain that the differential polynomial gf satisfies λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρp (a) if p > 1, while ρp (a) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρp (a) + 1 if p = 1. 4. discussions and applications in this section, we consider the differential equation (4.1) f′′′ + a (z) f = 0, where a (z) is a meromorphic function of finite iterated p−order in ∆. it is clear that the difficulty of the study of the differential polynomial generated by solutions lies in the calculation of the coefficients αi,j. we explain here that by using our method, the calculation of the coefficients αi,j can be deduced easily. we study for example the growth of the differential polynomial (4.2) gf = f ′′′ + f′′ + f′ + f. properties of solutions of complex differential equations 169 we have (4.3)   gf = α0,0f + α1,0f ′ + α2,0f ′′, g′f = α0,1f + α1,1f ′ + α2,1f ′′, g′′f = α0,2f + α1,2f ′ + α2,2f ′′. by (1.6) we obtain (4.4) αi,0 = { 1, for all i = 1, 2, 1 −a, for i = 0. now, by (3.6) we get αi,1 = { α′i,0 + αi−1,0, for all i = 1, 2 α′0,0 −aα2,0, for i = 0. hence (4.5)   α0,1 = α ′ 0,0 −aα2,0 = −a′ −a, α1,1 = α ′ 1,0 + α0,0 = 1 −a, α2,1 = α ′ 2,0 + α1,0 = 1. finally, by (3.11) we have αi,2 = { α′i,1 + αi−1,1, for all i = 1, 2, α′0,1 −aα2,1, for i = 0. so, we obtain (4.6)   α0,2 = α ′ 0,1 −aα2,1 = −a′′ −a′ −a, α1,2 = α ′ 1,1 + α0,1 = −2a′ −a, α2,2 = α ′ 2,1 + α1,1 = 1 −a. hence (4.7)   gf = (1 −a) f + f′ + f′′, g′f = (−a ′ −a) f + (1 −a) f′ + f′′, g′′f = (−a ′′ −a′ −a) f + (−2a′ −a) f′ + (1 −a) f′′ and h = ∣∣∣∣∣∣ 1 −a 1 1 −a′ −a 1 −a 1 −a′′ −a′ −a −2a′ −a 1 −a ∣∣∣∣∣∣ (4.8) = 3a′ −a−aa′ −aa′′ + a2 −a3 + 2(a′)2 + 1. suppose that h 6≡ 0, by simple calculations we have (4.9) f = ag′′f + (−1 − 2a ′) g′f + ( 1 −a + 2a′ + a2 ) gf h and by different conditions on the solution f we can ensure that ρp (f ′′′ + f′′ + f′ + f) = ρp (f) . turning now to the problem of oscillation, for that we consider a meromorphic function ϕ (z) 6≡ 0 of finite iterated p−order in ∆. from (4.9) we get (4.10) f = aw′′ + (−1 − 2a′) w′ + ( 1 −a + 2a′ + a2 ) w h + ψ (z) , 170 latreuch and belaïdi where w = gf −ϕ and (4.11) ψ (z) = aϕ′′ + (−1 − 2a′) ϕ′ + ( 1 −a + 2a′ + a2 ) ϕ h . hence (4.12) f = a h w′′ + c1w ′ + c0w + ψ, where c1 = − 1 + 2a′ h , c0 = 1 −a + 2a′ + a2 h . substituting (4.12) into (4.1) , we obtain a h w(5) + 4∑ i=0 φiw (i) = − ( ψ(3) + a (z) ψ ) , where φi (i = 0, · · · , 4) are meromorphic functions in ∆ with finite iterated p−order. suppose that all meromorphic solutions f 6≡ 0 of (4.1) are of infinite iterated p−order and ρp+1 (f) = ρ. if ψ 6≡ 0, then by lemma 2.1 we obtain (4.13) λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = +∞ and (4.14) λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) = ρ. suppose that f is a meromorphic solution of (4.1) of finite iterated p−order such that ρp (f) > max{ρp (a) ,ρp (ϕ)} . if ψ(3) + a (z) ψ 6≡ 0, then by lemma 2.1 we obtain λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) . finally, we can state the following two results without the additional conditions h 6≡ 0 and ψ is not a solution of (4.1). theorem 4.1 suppose that a (z) is analytic function in ∆ of finite iterated p−order 0 < ρp (a) < ∞ and 0 < τp (a) < ∞, and that dj (z) (j = 0, 1, 2, 3) are finite iterated p−order analytic functions in ∆ that are not all vanishing identically such that max{ρp (dj) (j = 0, 1, 2, 3)} < ρp (a) . if f is a nontrivial solution of (4.1), then the differential polynomial (4.15) gf = d3f (3) + d2f ′′ + d1f ′ + d0f satisfies ρp (gf ) = ρp (f) = ∞ and ρp (a) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρm,p (a) . theorem 4.2 under the assumptions of theorem 4.1, let ϕ (z) 6≡ 0 be an analytic function in ∆ with finite iterated p−order. if f is a nontrivial solution of (4.1) , then the differential polynomial gf = d3f (3) + d2f ′′ + d1f ′ + d0f (d3 6≡ 0) satisfies (4.16) λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞ properties of solutions of complex differential equations 171 and (4.17) ρp (a) ≤ λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρm,p (a) . remark 4.1 the results obtained in theorems 4.1 and 4.2 and refinement of corollaries 1.1 and 1.2 respectively. proof of theorem 4.1 suppose that f is a nontrivial solution of (4.1). then by lemma 2.2, we have ρp (f) = ∞, ρp (a) ≤ ρp+1 (f) ≤ ρm,p (a) . first, we suppose that d3 6≡ 0. by the same reasoning as before we obtain that h = ∣∣∣∣∣∣ h0 h1 h2 h3 h4 h5 h6 h7 h8 ∣∣∣∣∣∣ , where h0 = d0 − d3a, h1 = d1, h2 = d2,h3 = d′0 − (d2 + d′3) a − d3a′, h4 = d0 + d ′ 1 −d3a, h5 = d1 + d′2,h6 = d′′0 − (d1 + 2d′2 + d′′3 ) a− (d2 + 2d′3) a′ −d3a′′, h7 = 2d ′ 0 + d ′′ 1 − (d2 + 2d′3) a− 2d3a′, h8 = d0 + 2d′1 + d′′2 −d3a. then h = (3d0d1d2 + 3d0d1d ′ 3 + 3d0d2d ′ 2 − 6d0d3d ′ 1 + 3d1d2d ′ 1 + 3d1d3d ′ 0 +d0d2d ′′ 3 − 2d0d3d ′′ 2 + d1d2d ′′ 2 + d1d3d ′′ 1 + d2d3d ′′ 0 + 2d0d ′ 2d ′ 3 + 2d1d ′ 1d ′ 3 − 4d2d ′ 0d ′ 3 +2d2d ′ 1d ′ 2 + 2d3d ′ 0d ′ 2 −d1d ′ 2d ′′ 3 + d1d ′ 3d ′′ 2 + d2d ′ 1d ′′ 3 −d2d ′′ 1d ′ 3 −d3d ′ 1d ′′ 2 +d3d ′ 2d ′′ 1 −d 3 1 − 3d 2 0d3 − 2d1(d ′ 2) 2 − 3d21d ′ 2 −2d3(d ′ 1) 2 −d22d ′′ 1 −d 2 1d ′′ 3 − 3d 2 2d ′ 0 ) a + (2d0d2d ′ 3 + 2d0d3d ′ 2 −d1d2d ′ 2 + 2d1d3d ′ 1 − 4d2d3d ′ 0 + d1d3d ′′ 2 −d2d3d′′1 − 2d1d ′ 2d ′ 3 + 2d2d ′ 1d ′ 3 + 3d0d1d3 + d0d 2 2 −d 2 1d2 + d 2 2d ′ 1 − 2d 2 1d ′ 3 ) a′ +(d2d3d ′ 1 + d0d2d3 −d1d3d ′ 2 −d 2 1d3)a ′′ + (2d2d3d ′ 3 − 3d1d 2 3 + 2d 2 2d3 − 2d 2 3d ′ 2)aa ′ + ( d32 − 3d1d2d3 − 3d1d3d ′ 3 − 3d2d3d ′ 2 −d2d3d ′′ 3 − 2d3d ′ 2d ′ 3 +3d0d 2 3 + 3d 2 3d ′ 1 + 2d2(d ′ 3) 2 + 3d22d ′ 3 + d 2 3d ′′ 2 ) a2 −d33a 3 + 2d2d 2 3(a ′)2 −d2d23aa ′′ − 3d0d1d′0 −d0d1d ′′ 1 −d0d2d ′′ 0 − 2d0d ′ 0d ′ 2 +d1d ′′ 0d ′ 2 + d2d ′ 0d ′′ 1 −d2d ′ 1d ′′ 0 + d 3 0 + 2d0(d ′ 1) 2 + 3d20d ′ 1 + 2d2(d ′ 0) 2 +d21d ′′ 0 + d 2 0d ′′ 2 − 2d1d ′ 0d ′ 1 + d0d ′ 1d ′′ 2 −d0d ′ 2d ′′ 1 −d1d ′ 0d ′′ 2. by d3 6≡ 0, a 6≡ 0 and lemma 2.3, we have ρp (h) = ρp (a), hence h 6≡ 0. for the cases (i) d3 ≡ 0, d2 6≡ 0; (ii) d3 ≡ 0,d2 ≡ 0 and d1 6≡ 0 by using a similar reasoning as above we get h 6≡ 0. finally, if d3 ≡ 0, d2 ≡ 0, d1 ≡ 0 and d0 6≡ 0, then we have h = d30 6≡ 0. hence h 6≡ 0. by h 6≡ 0, we obtain f = 1 h ∣∣∣∣∣∣ gf d1 d2 g′f d0 + d ′ 1 −d3a d1 + d′2 g′′f 2d ′ 0 + d ′′ 1 − (d2 + 2d′3) a− 2d3a′ d0 + 2d′1 + d′′2 −d3a ∣∣∣∣∣∣ , which we can write (4.18) f = 1 h ( d0gf + d1g ′ f + d2g ′′ f ) , where d0 = (d1d2 − 2d0d3 + 2d1d′3 + d2d ′ 2 − 3d3d ′ 1 −d3d ′′ 2 + 2d ′ 2d ′ 3) a + (2d1d3 + 2d3d ′ 2) a ′ + a2d23 + 3d0d ′ 1 − 2d1d ′ 0 + d0d ′′ 2 −d1d ′′ 1 172 latreuch and belaïdi −2d′0d ′ 2 + d ′ 1d ′′ 2 −d ′ 2d ′′ 1 + d 2 0 + 2(d ′ 1) 2, d1 = ( d1d3 − 2d2d′3 −d 2 2 ) a + d2d ′′ 1 −d0d1 − 2d1d ′ 1 + 2d2d ′ 0 −d1d ′′ 2, d2 = d2d3a + d 2 1 −d2d ′ 1 + d1d ′ 2 −d0d2. if ρp (gf ) < +∞, then by (4.18) we obtain ρp (f) < +∞, and this is a contradiction. hence ρp (gf ) = ρp (f) = +∞. now, we prove that ρp+1 (gf ) = ρp+1 (f) . by (4.15), we get ρp+1 (gf ) ≤ ρp+1 (f) and by (4.18) we have ρp+1 (f) ≤ ρp+1 (gf ). this yield ρp (a) ≤ ρp+1 (gf ) = ρp+1 (f) ≤ ρm,p (a) . proof of theorem 4.2 by setting w = gf −ϕ in (4.18) , we have (4.19) f = 1 h (d0w + d1w ′ + d2w ′′) + ψ, where (4.20) ψ = d2ϕ ′′ + d1ϕ ′ + d0ϕ h . since d3 6≡ 0, then h 6≡ 0. it follows by theorem 4.1 that gf is of infinite iterated p−order analytic function and ρp (a) ≤ ρp+1 (gf ) ≤ ρm,p (a) . since ρp (ϕ) < ∞, then we have ρp (w) = ρp (gf ) = ρp (f) = ∞ and ρp (a) ≤ ρp+1 (w) = ρp+1 (gf ) = ρp+1 (f) ≤ ρm,p (a) . substituting (4.19) into (4.1) , we obtain d2 h w(5) + 4∑ i=0 φiw (i) = − ( ψ(3) + a (z) ψ ) , where φi (i = 0, · · · , 4) are meromorphic functions in ∆ with finite iterated p−order. we prove first that ψ 6≡ 0. suppose that ψ ≡ 0, then (4.20) can be rewritten as (4.21) d2ϕ ′′ + d1ϕ ′ + d0ϕ = 0 and by lemma 2.3, we have (4.22) ρ (d0) > max{ρ (d1) ,ρ (d2)} . by (4.21) we obtain d0 = − ( d2 ϕ′′ ϕ + d1 ϕ′ ϕ ) . since ρp (ϕ) = β < ∞, then by lemma 2.4 we have t (r,d0) ≤ t (r,d1) + t (r,d2) + o ( expp−2 ( log 1 1 −r )β+ε) , r /∈ e, where e ⊂ [0, 1) is a set with ∫ e dr 1−r < ∞. then, by using lemma 2.6, we get ρp (d0) ≤ max{ρp (d1) ,ρp (d2)} , which is a contradiction. it is clear now that ψ 6≡ 0 cannot be a solution of (4.1) because ρp (ψ) < ∞. then, by lemma 2.1 we obtain λp (w) = λp (w) = λp (gf −ϕ) = λp (gf −ϕ) = ρp (f) = ∞ and ρp (a) ≤ λp+1 (w) = λp+1 (w) = λp+1 (gf −ϕ) = λp+1 (gf −ϕ) = ρp+1 (f) ≤ ρm,p (a) . properties of solutions of complex differential equations 173 references [1] s. bank, general theorem concerning the growth of solutions of first-order algebraic differential equations, compositio math. 25 (1972), 61-70. [2] b. beläıdi, oscillation of fast growing solutions of linear differential equations in the unit disc, acta univ. sapientiae math. 2 (2010), no. 1, 25–38. [3] b. beläıdi, a. el farissi, fixed points and iterated order of differential polynomial generated by solutions of linear differential equations in the unit disc, j. adv. res. pure math. 3 (2011), no. 1, 161–172. [4] b. beläıdi and z. latreuch, relation between small functions with differential polynomials generated by meromorphic solutions of higher order linear differential equations, submitted. [5] l. g. bernal, on growth k-order of solutions of a complex homogeneous linear differential equation, proc. amer. math. soc. 101 (1987), no. 2, 317–322. [6] t. b. cao and h. x. yi, the growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, j. math. anal. appl. 319 (2006), no. 1, 278–294. [7] t. b. cao, the growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, j. math. anal. appl. 352 (2009), no. 2, 739-748. [8] t. b. cao, h. y. xu and c. x. zhu, on the complex oscillation of differential polynomials generated by meromorphic solutions of differential equations in the unit disc, proc. indian acad. sci. math. sci. 120 (2010), no. 4, 481–493. [9] t. b. cao and z. s. deng, solutions of non-homogeneous linear differential equations in the unit disc, ann. polo. math. 97(2010), no. 1, 51-61. [10] t. b. cao, l. m. li, j. tu and h. y. xu, complex oscillation of differential polynomials generated by analytic solutions of differential equations in the unit disc, math. commun. 16 (2011), no. 1, 205–214. [11] z. x. chen and k. h. shon, the growth of solutions of differential equations with coefficients of small growth in the disc, j. math. anal. appl. 297 (2004), no. 1, 285–304. [12] i. e. chyzhykov, g. g. gundersen and j. heittokangas, linear differential equations and logarithmic derivative estimates, proc. london math. soc. (3) 86 (2003), no. 3, 735–754. [13] a. el farissi, b. beläıdi and z. latreuch, growth and oscillation of differential polynomials in the unit disc, electron. j. diff. equ., vol. 2010(2010), no. 87, 1-7. [14] w. k. hayman, meromorphic functions, oxford mathematical monographs clarendon press, oxford, 1964. [15] j. heittokangas, on complex differential equations in the unit disc, ann. acad. sci. fenn. math. diss. 122 (2000), 1-54. [16] j. heittokangas, r. korhonen and j. rättyä, fast growing solutions of linear differential equations in the unit disc, results math. 49 (2006), no. 3-4, 265–278. [17] l. kinnunen, linear differential equations with solutions of finite iterated order, southeast asian bull. math. 22 (1998), no. 4, 385-405. [18] i. laine, nevanlinna theory and complex differential equations, de gruyter studies in mathematics, 15. walter de gruyter & co., berlin-new york, 1993. [19] i. laine and j. rieppo, differential polynomials generated by linear differential equations, complex var. theory appl. 49 (2004), no. 12, 897–911. [20] i. laine, complex differential equations, handbook of differential equations: ordinary differential equations. vol. iv, 269–363, handb. differ. equ., elsevier/north-holland, amsterdam, 2008. [21] z. latreuch and b. beläıdi, growth and oscillation of differential polynomials generated by complex differential equations, electron. j. diff. equ., vol. 2013 (2013), no. 16, 1-14. [22] m. tsuji, potential theory in modern function theory, chelsea, new york, (1975), reprint of the 1959 edition. department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem-(algeria) ∗corresponding author international journal of analysis and applications volume 19, number 5 (2021), 794-811 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-794 common fixed point theorems for six self-mappings on s− metric spaces thangjam bimol singh1,∗, g. a. hirankumar sharma2, y. mahendra singh2 and m. ranjit singh3 1department of mathematics, jadonang memorial college, longmai(noney)-795159, manipur, india 2department of basic sciences and humanities, manipur institute of technology (a constituent college of manipur university), takyelpat -795004, manipur, india 3department of mathematics, manipur university, canchipur-795003, manipur, india ∗corresponding author: btsalun29@gmail.com abstract. in this paper, we introduce the concepts of common property −(e.a) and common limit range property for six self-mappings and prove common fixed point theorems of such mappings satisfying (ψ,ϕ)− weak contraction on an s−metric space. examples are given to illustrate our results. 1. introduction and preliminaries in 2006, mustafa and sims [21] introduced g− metric space to overcome fundamental flaws in b. c. dhage’s theory of generalized metric spaces ( [10–12]) and discussed the topological properties of g− metric spaces. in 2012, sedghi et al. [26] introduced the concept of s− metric space as a modification of d∗− metric space [27] and g− metric space [21]. but, in 2014, dung et al. [14] showed by giving examples that the class of s− metric spaces and the class of g− metric spaces are distinct. received july 5th, 2021; accepted august 23rd, 2021; published september 20th, 2021. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. fixed point; coincidence point; (a,b,c)(ψ,ϕ)− weak contraction; property −(e.a); common limit range property. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 794 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-794 int. j. anal. appl. 19 (5) (2021) 795 before going to our main work, let us recall some basic definitions, lemmas, and preliminaries that will be used in this paper. definition 1.1. [26] let x be a non-empty set. a function s : x ×x ×x → [0,∞) is said to be an s− metric on x if it satisfies the following properties: (s1) s(x,y,z) = 0 if and only if x = y = z; (s2) s(x,y,z) ≤ s(x,x,a) + s(y,y,a) + s(z,z,a), for all x,y,z,a ∈ x. the pair (x,s) is called an s− metric space. example 1.1. [26] let x = rn and ‖ · ‖ be a norm on x. define s(x,y,z) = ‖2x−y −z‖ + ‖y −z‖, for all x,y,z ∈ x. then (x,s) is an s− metric space. example 1.2. [26] let x = r. define s(x,y,z) = |x−z| + |y −z|, for all x,y,z ∈ x. then (x,s) is an s− metric space. definition 1.2. [26] let (x,s) be an s− metric space. (i) a sequence {xn} in x is called a cauchy sequence if and only if s(xn,xn,xm) → 0 as n,m →∞. (ii) a sequence {xn} in x converges to x ∈ x if and only if s(xn,xn,x) → 0 as n → ∞. in this case, we write lim n→∞ xn = x. (iii) the s− metric space (x,s) is said to be complete if every cauchy sequence in it is convergent. lemma 1.1. [26] in an s− metric space, we have s(x,x,y) = s(y,y,x). lemma 1.2. [26] let (x,s) be an s− metric space. if sequence {xn} in x converges to x, then x is unique. lemma 1.3. [26] let (x,s) be an s− metric space. if sequence {xn} in x converges to x, then {xn} is a cauchy sequence. lemma 1.4. [26] let (x,s) be an s− metric space. if there exist sequences {xn} and {yn} such that lim n→∞ xn = x and lim n→∞ yn = y, then lim n→∞ s(xn,xn,yn) = s(x,x,y). definition 1.3. [3] let x 6= ∅ and p,q : x → x be two self-mappings. if u = px = qx, for some x ∈ x, then x is called a coincidence point of p and q, and u is called a point of coincidence (briefly, poc) of p and q. lemma 1.5. [3] suppose that p and q be weakly compatible self-mappings on a non-empty set x. if p and q have a unique point of coincidence u = px = qx, then u is the unique common fixed point p and q. int. j. anal. appl. 19 (5) (2021) 796 in 1997, alber and guere-delabriere [5] introduced the concept of weak contraction, wherein the authors introduced the following notion for mappings defined on a hilbert space x. consider the following set of real functions φ = { ϕ : [0,∞) → [0,∞) : ϕ is a lower semi-continuous and ϕ(t) = 0 if and only if t = 0 } . a mapping t : x → x is called a ϕ− weak contraction if there exists a function ϕ ∈ φ such that d ( t x,t y ) ≤ d(x,y) −ϕ ( d(x,y) ) , for all x,y ∈ x. dutta and choudhury [15] proved a fixed point theorem for a self-mapping satisfying (ψ,ϕ)−weak contractive condition as follows. theorem 1.1. let (x,d) be a complete metric space and t : x → x be a self-mapping satisfying ψ ( d(t x,t y) ) ≤ ψ ( d(x,y) ) −ϕ ( d(x,y) ) , for some ϕ ∈ φ and ψ ∈ ψ = { ψ : [0,∞) → [0,∞) : ψ is continuous non-decreasing and ψ(0) = 0 } . then, t has a common fixed point in x. many researchers utilized (ψ,ϕ)− weak contractive conditions to prove a number of metrical fixed point theorems (e.g., [2, 4–9, 13], [20], [30]). recently, singh and bimol singh [29] proved some coincidence and common fixed point theorems involving ψ ∈ ψ and ϕ ∈ φ in s− metric spaces. definition 1.4. [28] a pair (a,b) of self-mappings of an s− metric space (x,s) is said to be compatible if lim n→∞ s(abxn,abxn,baxn) = 0, whenever {xn} is a sequence in x such that lim n→∞ axn = lim n→∞ bxn = t, for some t ∈ x. in 1998, jungck and rhoades [18] introduced the following concept of weakly compatibility. definition 1.5. a pair (a,b) of self-mappings of an s− metric space (x,s) is said to be weakly compatible if they commute at each coincidence point (i.e., abx = bax, x ∈ x whenever ax = bx). in 2002, aamri and moutawakil [1] introduced the concept of property −(e.a) in metric spaces. in the same line, we use this concept in s− metric space as follows. definition 1.6. a pair (a,p) of self-mappings of an s− metric space (x,s) is said to satisfy the property −(e.a) if there exists a sequence {xn} in x such that lim n→∞ axn = lim n→∞ pxn = t, for some t ∈ x. int. j. anal. appl. 19 (5) (2021) 797 any pair of compatible as well as non-compatible self-mappings of an s− metric space (x,s) satisfy the property −(e.a), but a pair of mappings satisfying the property −(e.a) need not be non-compatible (see example 1 of [16]). in 2005, liu et al. [19] introduced the notion of common property −(e.a) for hybrid pairs of mappings, which contain the property −(e.a). for more details on various type of compatible mappings and their relation, one may refer to ( [8], [22–25], [31], [32]) and references therein. definition 1.7. two pairs (a,p) and (b,q) of self-mappings of an s− metric space (x,s) are said to satisfy the common property −(e.a) if there exist two sequences {xn} and {yn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = t, for some t ∈ x. in a similar way, we define the notion of common property −(e.a) for six self-mappings on s−metric space. definition 1.8. three pairs (a,p), (b,q) and (c,r) of self-mappings of an s− metric space (x,s) are said to satisfy the common property −(e.a) if there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, for some t ∈ x. it can be observed that the fixed point results usually require closeness of the underlying subspaces for the existence of common fixed points under the property −(e.a) and common property −(e.a). in 2011, sintunavarat and kumam [33] coined the idea of ‘common limit range property’. in 2012, imdad et al. [17] extended the notion of common limit range property to two pairs of self-mappings of a metric space which relax the closeness requirements of the underlying subspaces. definition 1.9. a pair (a,p) of self-mappings of an s− metric space (x,s) is said to satisfy the common limit range property with respect to p, (briefly, (clrp)− property), if there exists a sequence {xn} in x such that lim n→∞ axn = lim n→∞ pxn = t, where t ∈px. thus, one can infer that a pair (a,p) satisfying the property −(e.a) along with the closeness of the subspace px always enjoys the (clrp)− property with respect to the mapping p (see examples 2.16–2.17 of [17]). definition 1.10. two pairs (a,p) and (b,q) of self-mappings of an s− metric space (x,s) are said to satisfy the common limit range property (briefly, (clrpq)− property) with respect to mappings p and q, if there exist two sequences {xn} and {yn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = t, where t ∈px ∩qx. int. j. anal. appl. 19 (5) (2021) 798 example 1.3. [20] let x = [0, 12) endow with s− metric s(x,y,z) = |x−z|+|y−z|. define self-mappings a,b,p,q : x → x by ax =   6, 0 ≤ x ≤ 6 9, 6 < x < 12 ; bx =   0, 0 ≤ x < 6 6, 6 ≤ x < 12 ; px =   6, 0 ≤ x ≤ 6 3, 6 < x < 12 ; qx =   4, 0 ≤ x < 6 12 −x, 6 ≤ x < 12. consider two sequences {xn} and {yn} of x such that xn = 1n and yn = 6+ 1 n , n ∈ n. note that px = {3, 6} and qx = (0, 6]. also, we have lim n→∞ axn = lim n→∞ pxn = 6 ∈ x and lim n→∞ byn = lim n→∞ qyn = 6 ∈qx. it follows that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = t, where t = 6 ∈px ∩qx. therefore the pairs (a,p) and (b,q) satisfy (clrpq)− property. in a similar mode, we give the concept of the common limit range property for six self-mappings as follows. definition 1.11. three pairs (a,p), (b,q) and (c,r) of self-mappings of an s−metric space (x,s) are said to satisfy the common limit range property with respect to mappings p, q and r (briefly, (clrpqr)− property), if there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, where t ∈px ∩qx ∩rx, for some t ∈ x. example 1.4. let x = [0, 5]. define a mapping s : x3 → [0,∞) by s(x,y,z) = |x−y|+|y −z| , ∀x,y,z ∈ x. clearly, (x,s) is an s−metric space. let a,b,c,p,q,r : x → x be six self-mappings defined by ax =   1, if x = [0, 1] 2, if x ∈ (1, 5] ; bx =   0, if x = [0, 1) 1, if x ∈ [1, 5] ; cx =   1, if x = [0, 1] 5, if x ∈ (1, 5] ; px =   1, if x = [0, 1] 3, if x ∈ (1, 5] ; qx =   1 2 , if x = [0, 1) 1, if x ∈ [1, 5] ; rx =   1, if x = [0, 1] 4, if x ∈ (1, 5]. consider the three sequences {xn} = { 1 n } , {yn} = { 1 + 1 2n } , {zn} = { 1 − 1 n } ,∀n ∈ n. now, we have lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = 1 ∈ px ∩qx ∩rx. the pairs (a,p), (b,q) and (c,r) satisfy the (clrpqr)−property. int. j. anal. appl. 19 (5) (2021) 799 definition 1.12. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be six self-mappings. then the mappings a,b,c,p,q and r are called an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) if there exist two functions ψ ∈ ψ and ϕ ∈ φ such that ψ ( m(x,y,z) ) ≤ ψ ( ∆(x,y,z) ) −ϕ ( ∆(x,y,z) ) , (1.1) for all x,y,z ∈ x, where m(x,y,z) = max { s(ax,ax,by),s(by,by,cz) } and ∆(x,y,z) = max { s(px,px,qy),s(ax,ax,rz),s(px,px,by),s(qy,qy,cz) } . in the present paper, we discuss some common fixed point theorems for three pairs of self-mappings employing the common property −(e.a) and common limit range property in s−metric spaces. 2. main results before we start to prove our main theorems, we discuss the following lemmas. lemma 2.1. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) satisfying the following conditions: (i) bx ⊂rx (resp. ax ⊂rx); (ii) the pairs (a,p) and (b,q) satisfy the common property −(e.a). then the pairs (a,p), (b,q) and (c,r) share the common property −(e.a). proof. suppose the pair (a,p) and (b,q) satisfy the common property −(e.a), then there exist two sequences {xn} and {yn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = t, for some t ∈ x. since bx ⊂rx and lim n→∞ byn = t, then there exist n0 ∈ n∪{0} and a sequence {zn} in rx such that byn = rzn, for all n ≥ n0. therefore lim n→∞ byn = lim n→∞ rzn = t. now we claim that lim n→∞ czn = t. on contrary, we suppose that lim n→∞ czn 6= t, then there exists ε > 0 and k ≥ n0 for all k ∈ n∪{0} such that lim k→∞ s(t,t,cznk ) = ε. for this, from (1.1), we obtain ψ ( m(xnk,ynk,znk ) ) ≤ ψ ( ∆(xnk,ynk,znk ) ) −ϕ ( ∆(xnk,ynk,znk ) ) , where m(xnk,ynk,znk ) = max { s(axnk,axnk,bynk ),s(bynk,bynk,cznk ) } int. j. anal. appl. 19 (5) (2021) 800 and ∆(xnk,ynk,znk ) = max { s(pxnk,pxnk,qynk ),s(axnk,axnk,rznk ),s(pxnk,pxnk,bynk ), s(qynk,qynk,cznk ) } taking limit as n →∞, we obtain lim k→∞ ψ ( m(xnk,ynk,znk ) ) ≤ lim k→∞ ψ ( ∆(xnk,ynk,znk ) ) − lim k→∞ ϕ ( ∆(xnk,ynk,znk ) ) , where lim k→∞ m(xnk,ynk,znk ) = lim k→∞ max{s(t,t,t),s(t,t,cznk )} = lim k→∞ s(t,t,cznk ) = ε and lim k→∞ ∆(xnk,ynk,znk ) = max{0, 0, 0,ε} = ε. since ϕ is lower semi-continuous function, so we obtain ϕ(ε) ≤ lim k→∞ inf ϕ ( ∆(xnk,ynk,znk ) ) . consequently, we obtain ψ(ε) ≤ ψ(ε) −ϕ(ε)), gives ϕ(ε)) = 0 implies ε = 0. this is a contradiction. � lemma 2.2. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) satisfying the following conditions: (i) bx ⊂rx and rx is closed; (ii) the pairs (a,p) and (b,q) satisfy the (clrpq)− property. then the pairs (a,p), (b,q) and (c,r) share the common property −(e.a). proof. by lemma 2.1, the pairs (a,p), (b,q) and (c,r) satisfy the common property −(e.a). then there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, for some t ∈px ∩qx. also by (ii), we obtain t ∈rx. this completes the proof. � theorem 2.1. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be six self-mappings. suppose the mappings a,b,c,p,q, and r be (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) satisfying the following conditions: (i) the pairs (a,p), (b,q) and (c,r) share the common property −(e.a); (ii) px, qx and rx are closed subsets of x. int. j. anal. appl. 19 (5) (2021) 801 then the pairs (a,p), (b,q) and (c,r) have their coincidence points in x. further, a,b,c,p,q and r have a unique common fixed point, provided the pairs (a,p) (b,q) and (c,r) are weakly compatible. proof. from (i), the pairs (a,p), (b,q) and (c,r) share the common property −(e.a), then there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, for some t ∈ x. since px is a closed subset of x and lim n→∞ pxn = t, then there exists a point u ∈ x such that pu = t. now, we assert that au = pu. using inequality (1.1) with x = u, y = yn and z = zn, we get ψ ( m(u,yn,zn) ) ≤ ψ ( ∆(u,yn,zn) ) −ϕ ( ∆(u,yn,zn) ) , (2.1) where m(u,yn,zn) = max{s(au,au,byn),s(byn,byn,czn)} and ∆(u,yn,zn) = max { s(pu,pu,qyn),s(au,au,rzn),s(pu,pu,byn), s(qyn,qyn,czn) } . taking the limit as n →∞ in (2.1), we obtain ψ ( s(au,au,t) ) ≤ lim n→∞ ψ ( ∆(u,yn,zn) ) − lim n→∞ ϕ ( ∆(u,yn,zn) ) , (2.2) where lim n→∞ m(u,yn,zn) = max { s(au,au,t),s(t,t,t) } = s(au,au,t) and lim n→∞ ∆(u,yn,zn) = max { s(pu,pu,t),s(au,au,t),s(pu,pu,t),s(t,t,t) } (2.3) = max { 0,s(au,au,t), 0, 0 } =s(au,au,t). since ϕ is lower semi-continuous, we obtain ϕ ( s(au,au,t) ) ≤ lim n→∞ inf ϕ ( ∆(u,yn,zn) ) . (2.4) from (2.2), (2.3) and (2.4), we obtain ψ ( s(au,au,t) ) ≤ ψ ( s(au,au,t) ) − lim n→∞ inf ϕ ( ∆(u,yn,zn) ) (2.5) ≤ ψ ( s(au,au,t) ) −ϕ ( s(au,au,t) ) . int. j. anal. appl. 19 (5) (2021) 802 consequently, ϕ ( s(au,au,t) ) = 0 implies s(au,au,t) = 0. hence au = t = pu. this shows that the pair (a,p) has a coincidence point in x. since qx is a closed subset of x, then lim n→∞ qyn = t ∈ qx. then there exists a point v ∈ x such that qv = t. now, we assert that bv = qv. otherwise from (1.1) with x = u, y = v and z = zn, we obtain ψ ( m(u,v,zn) ) ≤ ψ ( ∆(u,v,zn) ) −ϕ ( ∆(u,v,zn) ) (2.6) where m(u,v,zn) = max { s(au,au,bv),s(bv,bv,czn) } and ∆(u,v,zn) = max { s(pu,pu,qv),s(au,au,rzn),s(pu,pu,bv), s(qv,qv,czn) } taking the limit as n →∞ in (2.6), we get lim n→∞ ψ ( m(u,v,zn) ) ≤ lim n→∞ ψ ( ∆(u,v,zn) ) − lim n→∞ ϕ ( ∆(u,v,zn) ) (2.7) where lim n→∞ m(u,v,zn) = max { s(t,t,bv),s(bv,bv,t) } = s(t,t,bv) and lim n→∞ ∆(u,v,zn) = max { s(t,t,t),s(t,t,t),s(t,t,bv),s(t,t,t) } (2.8) = s(t,t,bv) moreover, lower semi-continuity of ϕ, we have ϕ ( s(t,t,bv) ) ≤ lim n→∞ ϕ ( ∆(u,v,zn) ) (2.9) from (2.7), (2.8) and (2.9), we obtain ψ ( s(t,t,bv) ) ≤ ψ ( s(t,t,bv) ) −ϕ ( s(t,t,bv) ) , so ϕ(s(t,t,bv)) = 0 and it implies s(t,t,bv) = 0. hence bv = qv = t. this shows that v is a coincidence point of the pair (b,q) in x. also since rx is a closed subset of x and lim n→∞ rzn = t. then there exists a point w ∈ x such that rw = t. we show that rw = cw. using inequality (1.1) with x = u, y = v and z = w, we get ψ ( m(u,v,w) ) ≤ ψ ( ∆(u,v,w) ) −ϕ ( ∆(u,v,w) ) , int. j. anal. appl. 19 (5) (2021) 803 where m(u,v,w) = max { s(au,au,bv),s(bv,bv,cw) } = max { s(t,t,t),s(t,t,cw) } = s(t,t,cw) and ∆(u,v,w) = max { s(pu,pu,qv),s(au,au,rw),s(pu,pu,bv),s(qv,qv,cw) } = max { s(t,t,t),s(t,t,t),s(t,t,t),s(t,t,cw) } = s(t,t,cw). from the above inequality, we obtain ψ ( s(t,t,cw) ) ≤ ψ ( s(t,t,cw) ) −ϕ ( s(t,t,cw) ) . so ϕ ( s(t,t,cw) ) = 0, then s(t,t,cw) = 0. hence cw = t = rw. this shows that w is a coincidence point of the pair (c,r). thus the pairs (a,p), (b,q) and (c,r) have their coincidence points in x. it remains to prove that the pairs (a,p), (b,q) and (c,r) have a unique common fixed point in x. since the pairs (a,p), (b,q) and (c,r) are weakly compatible. then au = pu = t implies at = apu = pau = pt. similarly, bt = bqv = qbv = qt and ct = crw = rcw = rt. therefore, t is a coincidence point of the pairs (a,p), (b,q) and (c,r). one can show that at = pt = t by taking x = t,y = v and z = w in (1.1). also at = bt, this can be proved by putting x = y = t and z = w in (1.1). similarly, by putting x = u,y = v and z = t in (1.1), we obtain bt = ct. thus, at = bt = ct = pt = qt = rt. now, we show that the point of coincidence of the pairs (a,p), (b,q) and (c,r) is unique. if the point of coincidence of the pairs (a,p), (b,q) and (c,r) is not unique, then there exist ξ,ξ∗ ∈ x,ξ 6= ξ∗ such that at = pt = bt = qt = ξ and ct = rt = ξ∗. using inequality (1.1), we obtain ψ ( m(t,t,t) ) ≤ ψ ( ∆(t,t,t) ) −ϕ ( ∆(t,t,t) ) . where m(t,t,t) = max { s(at,at,bt),s(bt,bt,ct) } = max { s(ξ,ξ,ξ),s(ξ,ξ,ξ∗) } = s(ξ,ξ,ξ∗) int. j. anal. appl. 19 (5) (2021) 804 and ∆(t,t,t) = max { s(pt,pt,qt),s(at,at,rt),s(pt,pt,bt),s(qt,qt,ct) } = max { s(ξ,ξ,ξ),s(ξ,ξ,ξ∗),s(ξ,ξ,ξ),s(ξ,ξ,ξ∗) } =s(ξ,ξ,ξ∗) therefore, the above inequality becomes ψ ( s(ξ,ξ,ξ∗) ) ≤ ψ ( s(ξ,ξ,ξ∗) ) −ϕ ( s(ξ,ξ,ξ∗) ) , so ϕ ( s(ξ,ξ,ξ∗) ) = 0 i.e., s(ξ,ξ,ξ∗) = 0 which implies ξ = ξ∗. therefore, the point of coincidence of the pairs (a,p), (b,q) and (c,r) is unique and hence by lemma 1.5, the pairs (a,p), (b,q) and (c,r) have a unique common fixed point in x. � example 2.1. let x = [0, 1]. define a mapping s : x3 → [0,∞) by s(x,y,z) =   0, if x = y = z max{x,y,z}, otherwise for all x,y,z ∈ x. clearly, (x,s) is an s− metric space. consider the self-mappings ax = x 4 , bx = x 4 , cx = x 4 , px = x, qx = rx = x 2 , for all x ∈ x. setting ψ(t) = t and ϕ(t) = t 4 for t ∈ [0,∞). (a) in order to check the inequality (1.1), consider the following four cases: (i) x = y = z, (ii) x ≤ y < z, (iii) x ≤ z < y, (iv) y ≤ z < x. case (i): if x = y = z, we get m(x,y,z) = 0, so the condition is trivially satisfied. case (ii): if x ≤ y < z. then, we have m(x,y,z) = max { s (x 4 , x 4 , y 4 ) ,s (y 4 , y 4 , z 4 )} = z 4 and ∆(x,y,z) = max { s ( x,x, y 2 ) ,s (x 4 , x 4 , z 2 ) ,s ( x,x, y 4 ) ,s (y 2 , y 2 , z 4 )} = x or z 2 if x < z 2 , then ψ (z 4 ) = z 4 ≤ 3z 8 = ψ (z 2 ) −ϕ (z 2 ) if z 2 < x =⇒ z 4 < x 2 , so ψ (z 4 ) < ψ (x 2 ) ≤ 3x 4 = ψ(x) −ϕ(x). similarly, the inequality (1.1) is also satisfied for case (iii). case (iv): if y ≤ z < x, we have m(x,y,z) = x 4 and ∆(x,y,z) = x, so the inequality (1.1) reduces to ψ (x 4 ) = x 4 ≤ 3x 4 = ψ(x) −ϕ(x). int. j. anal. appl. 19 (5) (2021) 805 thus, for all x,y,z ∈ x, we obtain ψ ( m(x,y,z) ) ≤ ψ ( ∆(x,y,z) ) −ϕ ( ∆(x,y,z) ) . (b) now, let us show that the pairs (a,p), (b,q) and (c,r) are weakly compatible. for this, let ax = px =⇒ x 4 = x =⇒ x = 0. now, ap0 = a0 = 0 = p0 = pa0. therefore, (a,p) is weakly compatible. similarly, (b,q) and (c,r) are also weakly compatible mappings. (c) now, we show that the pairs (a,p), (b,q) and (c,r) share the common property −(e.a). for this, let xn = 1 n , yn = 1 n + 2 and zn = 1 2n + 3 for n ∈ n. clearly, {xn}, {yn} and {zn} are in x. then, we have s(axn,axn, 0) = s ( 1 4n , 1 4n , 0 ) = max { 1 4n , 1 4n , 0 } = 1 4n → 0 as n →∞. also, s(pxn,pxn, 0) = s ( 1 n , 1 n , 0 ) = max { 1 n , 1 n , 0 } = 1 n → 0 as n →∞. similarly, we get that byn, qyn, czn and rzn → 0 as n →∞. therefore, there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, therefore, (a,p), (b,q) and (c,r) share the common property −(e.a). (d) as px = [0, 1], qx = rx = [0, 1 2 ], then px, qx and rx are closed subsets of x. therefore, all the conditions of theorem 2.1 are satisfied and 0 is the unique common fixed point of the self-mappings. theorem 2.2. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r). if the pairs (a,p), (b,q) and (c,r) satisfy the (clrpqr)− property, then (a,p), (b,q) and (c,r) have their coincidence points. moreover, a,b,c,p,q and r have a unique common fixed point provided the pairs (a,p), (b,q) and (c,r) are weakly compatible. proof. suppose the pairs (a,p), (b,q) and (c,r) satisfy the (clrpqr)− property, then there exist three sequences {xn}, {yn} and {zn} in x such that lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = t, for some t ∈ px ∩qx ∩rx. it follows that t ∈ px and there exists u ∈ x such that pu = t. now we assert that au = pu. using inequality (1.1) with x = u, y = yn, z = zn, we get ψ ( m(u,yn,zn) ) ≤ ψ ( ∆(u,yn,zn) ) −ϕ ( ∆(u,yn,zn) ) , (2.10) int. j. anal. appl. 19 (5) (2021) 806 where m(u,yn,zn) = max { s(au,au,byn),s(byn,byn,czn) } ∆(u,yn,zn) = max { s(pu,pu,qyn),s(au,au,rzn),s(pu,pu,byn), s(qyn,qyn,czn) } . taking the limit as n →∞ in (2.10), we get lim n→∞ ψ ( m(u,yn,zn) ) ≤ lim n→∞ ψ ( ∆(u,yn,zn) ) − lim n→∞ ϕ ( ∆(u,yn,zn) ) where lim n→∞ m(u,yn,zn) = max { s(au,au,t),s(t,t,t) } = s(au,au,t) lim n→∞ ∆(u,yn,zn) = max { s(t,t,t),s(au,au,t),s(t,t,t),s(t,t,t), } = s(au,au,t). from the above inequality, we obtain ψ ( s(au,au,t) ) ≤ ψ ( s(au,au,t) ) −ϕ ( s(au,au,t) ) , so ϕ ( s(au,au,t) ) = 0, i.e., s(au,au,t) = 0. hence au = t = pu, which shows that u is a coincidence point of the pair (a,p). as t ∈qx, there exists a point v ∈ x such that qv = t. we show that bv = qv. using inequality (1.1) with x = u, y = v and z = zn, we have ψ ( m(u,v,zn) ) ≤ ψ ( ∆(u,v,zn) ) −ϕ ( ∆(u,v,zn) ) (2.11) where m(u,v,zn) = max { s(au,au,bv),s(bv,bv,czn) } = max { s(t,t,bv),s(bv,bv,czn) } and ∆(u,v,zn) = max { s(pu,pu,qv),s(au,au,rzn),s(pu,pu,bv), s(qv,qv,czn) } = max { s(t,t,t),s(t,t,rzn),s(t,t,bv),s(t,t,czn) } taking the limit as n →∞ in (2.11), we get lim n→∞ ψ ( m(u,v,zn) ) ≤ lim n→∞ ψ ( ∆(u,v,zn) ) − lim n→∞ ϕ ( ∆(u,v,zn) ) int. j. anal. appl. 19 (5) (2021) 807 where lim n→∞ m(u,v,zn) = max { s(t,t,bv),s(bv,bv,t) } = s(bv,bv,t) and lim n→∞ ∆(u,v,zn) = max { s(t,t,t),s(t,t,t),s(t,bv,bv),s(t,t,t) } = s(bv,bv,t), the above equation gives ψ ( s(bv,bv,t) ) ≤ ψ ( s(bv,bv,t) ) −ϕ ( s(bv,bv,t) ) , so ϕ(s(bv,bv,t)) = 0, i.e., s(bv,bv,t) = 0. hence, bv = qv = t, which shows that v is a coincidence point of the pair (b,q). as t ∈ rx, there exists a point w ∈ x such that rw = t. we show that rw = cw. using inequality (1.1) with x = u, y = v and z = w, we get ψ ( m(u,v,w) ) ≤ ψ ( ∆(u,v,w) ) −ϕ ( ∆(u,v,w) ) where m(u,v,w) = max { s(au,au,bv),s(bv,bv,cw) } = s(t,t,cw) and ∆(u,v,w) = max { s(pu,pu,qv),s(au,au,rw),s(pu,pu,bv),s(qv,qv,cw) } = max { s(t,t,t),s(t,t,t),s(t,t,t),s(t,t,cw) } = s(t,t,cw). follows from the above inequality, we obtain ψ ( s(t,t,cw) ) ≤ ψ ( s(t,t,cw) ) −ϕ ( s(t,t,cw) ) , so ϕ ( s(t,t,cw) ) = 0, i.e., s(t,t,cw) = 0. hence, cw = t = rw, which shows that w is a point of coincidence of the pair (c,r). thus the pairs (a,p), (b,q) and (c,r) have their coincidence points in x. it remains to prove that the pairs (a,p), (b,q) and (c,r) have a unique common fixed point in x. since the pairs (a,p), (b,q) and (c,r) are weakly compatible. then au = pu = t implies at = apu = pau = pt. similarly, bt = bqv = qbv = qt and ct = crw = rcw = rt. therefore, t is a coincidence point of the pairs (a,p), (b,q) and (c,r). following the same steps as in theorem 2.1, one can show that at = bt = ct = pt = qt = rt. now, we show that the point of coincidence of the pairs (a,p), (b,q) and (c,r) is unique. int. j. anal. appl. 19 (5) (2021) 808 if the point of coincidence of the pairs (a,p), (b,q) and (c,r) is not unique, then there exist ξ,ξ∗ ∈ x,ξ 6= ξ∗ such that at = pt = bt = qt = ξ and ct = rt = ξ∗. using inequality (1.1), we obtain ψ ( m(t,t,t) ) ≤ ψ ( ∆(t,t,t) ) −ϕ ( ∆(t,t,t) ) , where m(t,t,t) = max { s(at,at,bt),s(bt,bt,ct) } = max { s(ξ,ξ,ξ),s(ξ,ξ,ξ∗) } = s(ξ,ξ,ξ∗) and ∆(t,t,t) = max { s(pt,pt,qt),s(at,at,rt),s(pt,pt,bt),s(qt,qt,ct) } = max { s(ξ,ξ,ξ),s(ξ,ξ,ξ∗),s(ξ,ξ,ξ),s(ξ,ξ,ξ∗) } =s(ξ,ξ,ξ∗) therefore, the above inequality becomes ψ ( s(ξ,ξ,ξ∗) ) ≤ ψ ( s(ξ,ξ,ξ∗) ) −ϕ ( s(ξ,ξ,ξ∗) ) , so ϕ ( s(ξ,ξ,ξ∗) ) = 0 i.e., s(ξ,ξ,ξ∗) = 0 which implies ξ = ξ∗. therefore, the point of coincidence of the pairs (a,p), (b,q) and (c,r) is unique and hence by lemma 1.5, the pairs (a,p), (b,q) and (c,r) have a unique common fixed point in x. � example 2.2. let x = [0, 20]. define a mapping s : x3 → [0,∞) by s(x,y,z) = |x−y|+|y −z| , ∀x,y,z ∈ x. clearly, (x,s) is an s−metric space. let a,b,c,p,q,r : x → x be six self-mappings defined by ax =   2, if x ∈ [0, 2]3, if x ∈ (2, 20] ; bx =   1, if x ∈ [0, 2)2, if x ∈ [2, 20] ; cx =   2, if x ∈ [0, 2]1, if x ∈ (2, 20] px =   2, if x ∈ [0, 2]6, if x ∈ (2, 20] , qx =   4, if x ∈ [0, 2)2, if x ∈ [2, 20] ; rx =   2, if x ∈ [0, 2]8, if x ∈ (2, 20]. consider three sequences {xn} = {2 − 1 n }, {yn} = {2 + 1 n + 1 }, {zn} = { 1 n },∀n ∈ n. lim n→∞ axn = lim n→∞ pxn = lim n→∞ byn = lim n→∞ qyn = lim n→∞ czn = lim n→∞ rzn = 2, where 2 ∈px ∩qx ∩rx. therefore, the pairs (a,p), (b,q) and (c,r) satisfy (clrpqr)− property. consider ψ(t) = t and ϕ(t) = t 4 . in order to check the inequality (1.1), we have the following eight cases: int. j. anal. appl. 19 (5) (2021) 809 (i) x,z ∈ [0, 2],y ∈ [0, 2), (ii) x ∈ [0, 2],y ∈ [0, 2), z ∈ (2, 20], (iii) x ∈ [0, 2], y ∈ [2, 20], z ∈ [0, 2], (iv) x ∈ [0, 2], y ∈ [2, 20], z ∈ (2, 20], (v) x ∈ (2, 20],y ∈ [0, 2),z ∈ [0, 2], (vi) x ∈ (2, 20]y ∈ [0, 2),z ∈ (2, 20], (vii) x ∈ (2, 20],y ∈ [2, 20],z ∈ [0, 2], (viii) x ∈ (2, 20],y ∈ [2, 20],z ∈ (2, 20], in case (i), we have m(x,y,z) = 1 and ∆(x,y,z) = 2, so the inequality (1.1) reduces to ψ(1) = 1 ≤ 3 2 = ψ(2) −ϕ(2) in case (ii) and (vi), we have m(x,y,z) = 1 and ∆(x,y,z) = 6, so (1.1) reduces to ψ(1) = 1 ≤ 9 2 = ψ(6) −ϕ(6). in case (iii), we have m(x,y,z) = 0, so the inequality (1.1) is trivially satisfied. in case (v) and (vi), we have m(x,y,z) = 2 and ∆(x,y,z) = 5, so the inequality (1.1) reduces to ψ(2) = 2 ≤ 15 4 = ψ(5) −ϕ(5) in case (vii), we have m(x,y,z) = 1 and ∆(x,y,z) = 4, so the inequality (1.1) reduces to ψ(1) = 1 ≤ 3 = ψ(4) −ϕ(4) in case (viii), we have m(x,y,z) = 1 and ∆(x,y,z) = 5, so the inequality (1.1) reduces to ψ(1) = 1 ≤ 15 4 = ψ(5) −ϕ(5) thus, the inequality (1.1) holds true for all x,y,z ∈ x. hence, all the conditions of theorem 2.2 are satisfied, and 2 is a unique common fixed point of the pairs (a,p), (b,q) and (c,r) which also remains a point of coincidence. here, one may notice that all the involved mappings are discontinuous at their unique common fixed point 2. theorem 2.3. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) satisfying the following conditions: (i) bx ⊂rx (resp. ax ⊂rx); (ii) the pairs (a,p) and (b,q) satisfy the common property −(e.a); (iii) px, qx and rx are closed subsets of x. then the pairs (a,p), (b,q) and (c,r) have their coincidence points in x. further, a,b,c,p,q and r have a unique common fixed point, provided the pairs (a,p), (b,q) and (c,r) are weakly compatible. proof. it follows from lemma 2.1 and theorem 2.1. � theorem 2.4. let (x,s) be an s− metric space and a,b,c,p,q,r : x → x be an (a,b,c)(ψ,ϕ)− weak contraction with respect to (p,q,r) satisfying the following conditions: (i) bx ⊂rx and rx is closed; int. j. anal. appl. 19 (5) (2021) 810 (ii) the pairs (a,p) and (b,q) satisfy the (clrpq)− property. then the pairs (a,p), (b,q) and (c,r) have their coincidence points in x. further, a,b,c,p,q and r have a unique common fixed point, provided the pairs (a,p), (b,q) and (c,r) are weakly compatible. proof. it follows from lemma 2.2 and theorem 2.2. � 2.1. conclusion. the concepts of the property −(e.a) and the common limit range property for six selfmappings are discussed to obtain common fixed point theorems of (ψ,ϕ)− weak contraction with illustrative examples on s−metric space. the main advantages of this work are, the mappings and the space used in our results do not require continuity and completeness to obtain the fixed point. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. aamri, d. el moutawakil, some new common fixed point theorems under strict contractive conditions, j. math. anal. appl., 270 (2002), 181–188. [2] m. abbas, d. doric, common fixed point theorem for four mappings satisfying generalized weak contractive conditions, filomat. 24(2) (2010), 1–10. [3] m. abbas, g. jungck, common fixed point results for non-commuting mappings without continuity in cone metric spaces, j. math. anal. appl. 341 (2008), 416–420. [4] m. abbas, m. s. khan, common fixed point theorem of two mappings satisfying a generalized weak contractive condition, int. j. math. math. sci. 2009 (2009), 131068. [5] i.y. alber, s. guerre-delabriere, principle of weakly contractive maps in hilbert space, in: i. gohberg and y. lyubich, (eds.): new results in operator theory and its appl., birkhnuser, basel, switzerland, 98 (1997), 7–22. [6] i. beg, m. abbas, coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, fixed point theory appl. 2006 (2006), 74503. [7] v. berinde, approximating fixed points of weak φ−contractions, fixed point theory. 4 (2003), 131–142. [8] y. j. cho, p. p. murthy, g. jungck, a common fixed point theorems of meir-keeler type, int. j. math. math. sci. 16 (4) (1993), 669–674. [9] b.s. choudhury, p. konor, b.e. rhoades, n. metiya, fixed point theorems for generalized weakly contractive mapping, nonlinear anal.: theory meth. appl. 74 (2011), 2116–2126. [10] b.c. dhage, generalized metric space and mapping with fixed point, bull. cal. math. soc. 84 (1992), 329–336. [11] b.c. dhage, generalized metric space and topological structure i, an. stiint. univ. al. i. cuza iasi. mat. (n.s) 46 (2000), 3–24. [12] b.c. dhage, on generalized metric spaces and topological structure ii, pure. appl. math. sci. 40 (1994), 37–41. [13] h. ding, z. kadelburg, e. karapinar, s. radenovic, common fixed points of weak contractions in cone metric spaces, abstr. appl. anal. 2012 (2012), 793862. [14] n.v. dung, n.t. hieu, s. radojevic, fixed point theorems for g−monotone maps on partially ordered s−metric spaces, filomat, 28 (9) (2014), 1885–1898. int. j. anal. appl. 19 (5) (2021) 811 [15] p.n. dutta, b.s. choudhury, a generalisation of contraction principle in metric spaces, fixed point theory appl. 2008 (2008), 406368. [16] j.x. fang, y. gao, common fixed point theorems under strict contractive conditions in menger spaces, nonlinear anal.: theory meth. appl. 70 (1) (2009), 184–193. [17] m. imdad, b.d. pant, s. chauhan, fixed point theorems in menger spaces using the (clrst ) property and applications, j. nonlinear anal. optim. 3 (2) (2012), 225–237. [18] g. jungck, b.e. rhoades, fixed points for set valued functions without continuity, indian j. pure appl. math. 29 (3) (1998), 227–238. [19] y. liu, j. wu, z. li, common fixed points of single-valued and multi-valued maps, int. j. math. math. sci. 19 (2005), 3045–3055. [20] y. mahendra singh, g.a. hirankumar sharma, m. r. singh, common fixed point theorems for (ψ,ϕ)− weak contractive conditions in metric spaces, hacet. j. math. stat. 48 (5) (2019), 1398–1408. [21] z. mustafa, b. sims, a new approach to generalized metric spaces, j. nonlinear convex anal. 7 (2006), 289–297. [22] r.p. pant, r− weakly commutativity and common fixed points, soochow j. math. 25 (1999), 37–42. [23] h.k. pathak, s.s. chang, y.j. cho, fixed point theorems for compatible mappings of type (p), indian j. math. 36 (2) (1994), 151–166. [24] h.k. pathak, y.j. cho, s.m. kang, b. madharia, compatible mappings of type (c) and common fixed point theorem of greguš type, demonstr. math. 31 (3) (1998), 499–517. [25] h.k. pathak, m.s. khan, compatible mappings of type (b) and common fixed point theorems of greguš type, czechoslovak math. j. 45 (120) (1995), 685–698. [26] s. sedghi, n. shobe, a. aliouche, a generalization of fixed point theorems in s−metric spaces, mat. vesnik, 64 (3) (2012), 258–266. [27] s. sedghi, n. shobe, h. zhou, a common fixed point theorem in d∗−metric spaces, fixed point theory appl. 2007 (2007), 27906. [28] s. sedghi, n. shobkolaei, m. shahraki, t. došenovic, common fixed point of four maps in s−metric spaces, math. sci. 12 (2018), 137–143. [29] m.r. singh, th. bimol singh, some results for α−(ψ, ϕ)− contractive mappings in s−metric spaces, j. adv. math. stud. 14 (2) (2021), 279–293. [30] m.r. singh, g.a. hirankumar sharma, y. mahendra singh, common fixed points for weak contraction occasionally weakly biased mappings, adv. fixed point theory, 7 (4) (2017), 458–467. [31] m.r. singh, y. mahendra singh, compatible mappings of type (e) and common fixed point theorems of meir-keeler type, int. j. math. sci. engg. appl. 1 (2) (2007), 299–315. [32] m.r. singh, y. mahendra singh, on various types of compatible maps and common fixed point theorems for non-continuous maps, hacet. j. math. stat., 40 (4) (2011), 503–513. [33] w. sintunavarat, p. kumam, common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces, j. appl. math. 2011 (2011), 637958. 1. introduction and preliminaries 2. main results 2.1. conclusion references international journal of analysis and applications volume 17, number 2 (2019), 234-243 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-234 multi-objective optimization using local fractional differential operator rabha w. ibrahim1,∗, maslina darus2 1cloud computing center, university of malaya, malaysia 2centre of modelling and data sciences, faculty of science and technology, universiti kebangsaan malaysia, 43600 ukm bangi, selangor, malaysia ∗corresponding author: rabhaibrahim@yahoo.com abstract. in this effort, we aim to generalize the concept of univex functions by utilizing a local fractional differential-difference operator, based on different types of local fractional calculus (fractal calculus). this study leads to a new class of these functions in some optimal problems by illustrating conditions on the generalized functions. we call it the class of local fractional univex functions. strong, weak, converse, and strict converse duality theorems are given. multi-objective optimal problem involves the new process is solved (local optimal problem). the main tool employed in the analysis is based on the local fractional derivative operators. 1. introduction the notion of local fractional calculus (also labeled fractal calculus), which was first suggested by kolwankar and gangal [1] using the riemann-liouville fractional derivative [2]. it was employed to deal with non-differentiable issues from science and engineering [3][5]. local fractional derivative of φ(χ) of order 0 < α ≤ 1 is specified by received 2018-09-24; accepted 2018-11-20; published 2019-03-01. 2010 mathematics subject classification. 44a45. key words and phrases. fractional calculus; fractional operator; fractional differential equation; univex function. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 234 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-234 int. j. anal. appl. 17 (2) (2019) 235 dα φ(χ) = dα φ(χ) χα ∣∣∣ χ=χ0 = lim χ→χ0 dα[φ(χ) −φ(χ0)] [d(χ−χ0)]α , (1.1) where the expression dα[φ(χ) −φ(χ0)]/[d(χ−χ0)]α is the riemann-liouville fractional derivative given by dαφ(χ) dχα = 1 γ(1 −α) d dχ ∫ χ 0 φ(t) (χ− t)α dt; corresponding to the integral operator (iαφ)(χ) = 1 γ(α) ∫ χ 0 (χ− t)α−1φ(t) dt. this operator is well-defined and it is represented to the classical fractional calculus. the local fractional derivative utilizing the fractal geometry is defined by the formula [5] dα φ(χ) = dα φ(χ) χα ∣∣∣ χ=χ0 = lim χ→χ0 ∆α[φ(χ) −φ(χ0)] [(χ−χ0)]α , (1.2) where ∆α[φ(χ) −φ(χ0)] ∼= γ(α + 1)[φ(χ) −φ(χ0)]. dunkl operator (see [6,7]) is a structure for a diff-difference operator λκφ(χ) = d dχ φ(χ) + κ (φ(χ) −φ(−χ) 2χ ) , κ ≥ 0. (1.3) it generalized some special functions and integral transforms in several variables connected with reflection groups. this class of operators has developed many other operators. it applied in the analysis of quantum many body systems. recently, this operator is given in term of fractional calculus [8]. by employing the local fractional differential operator in (1.1) or (1.2), we introduce a generalization of (1.3) as follows: λακφ(χ) = d αφ(χ) + κ (φ(χ) −φ(−χ) 2χ ) , (1.4) ( κ ≥ 0,α ∈ (0, 1],χ 6= 0 ∈ r ) in this study, we aim to generalize the concept of univex functions by utilizing a local fractional differentialdifference operator (1.4). this study leads to a new class of these functions in some optimal problems by illustrating conditions on the generalized functions. we call it the class of local fractional univex functions. strong, weak, converse, and strict converse duality theorems are given, with examples in the sequel. int. j. anal. appl. 17 (2) (2019) 236 2. univex function in this section, we generalize the concept of the univex function, by using the local fractional dunkl operator. define the following functions η : [a,b] × [a,b] → r\{0}, φ : [a,b] → r and φ : r → r. definition 1. a differential function φ is said to be a local fractional univex function of order α ∈ (0, 1] in the direction of ξ ∈ j := [a,b] if for all χ ∈ j, we have λακ φ(χ) ≤ φ ( φ(χ) −φ(ξ) ) η(χ,ξ) . note that, this concept is one of significant tool for optimization. also, we confirm that there are many other techniques for optimization which are generalized by fractional formal operators (see [9][12]). the advantage of using the fractional dunkl operator, is that can be acted on multi-dimensional euclidean spaces. therefor, it can be employed in non-linear multi-objective problem minimize ψ(χ) = ( ψ1(χ), ...,ψn(χ) ) subjectto θ(χ) ≤ 0, (2.1) where ψ : j → rn and θ : j → rn and 0 is the zero vector in rn. the function ψ(χ) has many applications in various studies. it may represent a multi-agent function in cloud computing systems. definition 2. a point ξ ∈ j := {χ ∈ j : θ(χ) ≤ 0} is said to be an efficient outcome of (2.1), if there is no point χ ∈ j, with ψ(χ) ≤ ψ(ξ). moreover, it is known as a weak efficient outcome when ψ(χ) < ψ(ξ). definition 3. the couple (ψ, θ) is called a local fractional univex of order α, if for all χ ∈ j we have η1(χ,ξ).d α κ ψ(χ) ≤ φ1 ( ψ(χ) − ψ(ξ) ) and η2(χ,ξ).d α θ(χ) ≤ −φ2 ( θ(χ) − θ(ξ) ) , where η1 : j ×j → rn, η2 : j ×j → rn, φ1 : rn → r, φ2 : rn → r and dακψ(χ) = ( λακψ1(χ), ..., λ α κψn(χ) ) . this class of local fractional univex functions is denoted by α−type univex. int. j. anal. appl. 17 (2) (2019) 237 3. results in this section, we investigate some sufficient optimality conditions for a point to be an efficient solution of (1.3) under the generalized (α,ρ,η,ϑ)-type univex. theorem 3.1. let ξ be an initial solution of the multi-objective problem (1.3) and c1 and c2 be two nonnegative constants such that (a) θ(ξ) = 0; (b) c1 ( η1(x,ξ).d α ψ(x) ) + c2 ( η2(x,ξ).d α θ(x) ) ≥ 0; (c) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ ω; (d) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (e) c1ρ1 + c2ρ2 ≥ 0. then ξ is an efficient solution of (1.3). proof. suppose that ξ is not an efficient solution of (1.3), then there exists x ∈ λ such that ψ(x) ≤ ψ(ξ). by the assumptions (a) and (d), we have φ1(ψ(x) − ψ(ξ)) ≤ 0, and φ2(θ(ξ)) ≥ 0. (3.1) in view of the assumption (c), we get c1 ( η1(x,ξ).d α ψ(x) ) < −c1ρ1‖ϑ(x,ξ)‖2 (3.2) and c2 ( η2(x,ξ).d α θ(x) ) ≤−c2ρ2‖ϑ(x,ξ)‖2. (3.3) summing the above inequalities and utilizing (e), we conclude that c1 ( η1(x,ξ).d α ψ(x) ) + c2 ( η2(x,ξ).d α θ(x) ) < − ( c1ρ1 + c2ρ2 ) ‖ϑ(x,ξ)‖2 ≤ 0, which contradicts the assumption (b). hence, ξ is an efficient solution of (1.3). this completes the proof. � theorem 3.2. if the following conditions are satisfied: (a) ξ is a weakly efficient solution of (1.3); (b) θ is continuous in ξ; (c) the functions ψ and θ are fractional univex functions of order α ∈ (0, 1) in the direction of ξ ∈ λ. moreover, for some x̄ ∈ λ, we have θ(x̄) < 0. int. j. anal. appl. 17 (2) (2019) 238 then there are two constants c1 ≥ 0 and c2 ≥ 0 such that c1 ( η1(x,ξ).d α ψ(x) ) + c2 ( η2(x,ξ).d α θ(x) ) ≥ 0, ( x ∈ ω, c2θ(ξ) = 0, η1 : ω × ω → rm, η2 : ω × ω → rp ) . proof. our aim is to show that the system η1(x,ξ).d α ψ(x) < 0, η2(x,ξ).d α θ(x) < 0, has no solution for x ∈ ω. let the system has a solution y ∈ ω. by the assumption (a), we have ψ(ξ + �1y) < ψ(ξ) and θ(ξ + �2y) < θ(ξ), for sufficient small arbitrary constants �1, �2 > 0. now, we let x̄ := ξ +�2y; which implies that x̄ ∈ λ∩n�2 (ξ) thus by (b) and (c), we have θ(ξ + �2y) = θ(x̄) < 0; which contradicts (a), where ξ is a weak solution. therefore, the above inequalities are non-negative. hence, in view of (c) these are two constants c1 and c2 satisfy the inequality c1 ( η1(x,ξ).d α ψ(x) ) + c2 ( η2(x,ξ).d α θ(x) ) ≥ 0, with the property c2θ(ξ) = 0. this completes the proof. � next, we consider the dual problem of (1.3) as follows: max ψ(χ) = ( ψ1(χ), ...,ψm(χ) ) subjectto c1 ( η1(x,χ).d α ψ(x) ) + c2 ( η2(x,χ).d α θ(x) ) ≥ 0, c2θ(χ) ≥ 0, (3.4) where χ ∈ ω, c1 and c2 be two non negative constants. theorem 3.3. let x,χ be initial solutions of the multi-objective problems (1.3) and (3.2) respectively. if (a) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ ω; (b) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (c) c1ρ1 + ρ2 ≥ 0; then ψ(x) � ψ(χ). proof. suppose that ψ(x) ≤ ψ(χ). since c1ρ1 + ρ2 ≥ 0 then by (b), we obtain φ1(ψ(x) − ψ(χ)) ≤ 0 φ2(θ(χ)) ≥ 0. int. j. anal. appl. 17 (2) (2019) 239 in virtue of the assumption (a) the above inequalities yield( η1(x,χ).d α ψ(χ) ) < −ρ1‖ϑ(x,χ)‖2( η2(x,χ).d α θ(χ) ) ≤−ρ2‖ϑ(x,χ)‖2, consequently, we obtain c1 ( η1(x,ξ).d α ψ(x) ) < −c1ρ1‖ϑ(x,χ)‖2 and c2 ( η2(x,χ).d α θ(x) ) ≤−ρ2‖ϑ(x,χ)‖2. summing the above inequalities and utilizing (c), we conclude that c1 ( η1(x,χ).d α ψ(χ) ) + c2 ( η2(x,χ).d α θ(χ) ) < − ( c1ρ1 + ρ2 ) ‖ϑ(x,χ)‖2 ≤ 0, which contradicts the assumption (c). this completes the proof. � theorem 3.4. let x0 and χ0 be initial solution for the problems (1.3) and (3.2) respectively. if ψ(x0) = ψ(χ0) then the (weak or strong) duality problems (1.3) and (3.2) has efficient solutions x0 and χ0 respectively. proof. suppose that x0 is not efficient for (1.3), then for some x ∈ λ ψ(x) ≤ ψ(x0) = ψ(χ0), which contradicts weak (strong) duality theorems as χ0 is initial solution for (3.2). therefore, x0 is efficient for (1.3). similarly χ0 is efficient solution for (3.2). hence the proof. � theorem 3.5. let χ0 be an initial solution of the multi-objective problem (3.2) and c1 and c2 be two non negative constants such that (a) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ ω; (b) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (c) c1ρ1 + ρ2 ≥ 0. then χ0 is an efficient solution of (3.2). proof. suppose that χ0 is not an efficient solution of (3.2), then there exists x0 ∈ λ such that ψ(x0) ≤ ψ(χ0). now going on as in theorem 3.3, we have a contradiction. hence, χ0 is an efficient solution of (3.2). � theorem 3.6. let x0,χ0 be initial solutions of the multi-objective problems (1.3) and(3.2) respectively. if (a) ψ(x0) ≤ ψ(χ0); (b) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,ρ,η,ϑ)-type univex at ξ ∈ ω; (c) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; int. j. anal. appl. 17 (2) (2019) 240 (d) c1ρ1 + ρ2 ≥ 0; then x0 = χ0. proof. suppose that x0 6= χ0. since χ0 is an initial solution for (3.2) then by (a) and (c), we have φ1(ψ(x0) − ψ(χ0)) ≤ 0 φ2(θ(χ0)) ≥ 0. in virtue of the assumption (b) the above inequalities imply that( η1(x0,χ0).d α ψ(χ0) ) < −ρ1‖ϑ(x0,χ0)‖2( η2(x0,χ0).d α θ(χ0) ) ≤−ρ2‖ϑ(x0,χ0)‖2, which on summing yields c1 ( η1(x0,χ0).d α ψ(χ0) ) + c2 ( η2(x0,χ0).d α θ(χ0) ) < − ( c1ρ1 + ρ2 ) ‖ϑ(x0,χ0)‖2 ≤ 0, which contradicts to initially of χ0. then we obtain x0 = χ0. this completes the proof. � 4. simulation in this section, we illustrate a simulation to show how the fractional calculus is effected on the multiobjective functions. let ψ, θ : r → r2 such that ψ(x) = ( x2,x3 ) ; θ(x) = ( x,x2 ) . our aim is to show that the couple (ψ, θ) is (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. to determine the fractional dunkl operator on these functions, we shall introduce three cases depending on the value of kv for v = 1. 4.1. case (i) kv = 0. the fractional dunkl operator acts on the functions ψ and θ as follows: dαψ(x) = ( γ(3) γ(3 −α) x2−α, γ(4) γ(4 −α) x3−α ) ; dαθ(x) = ( γ(2) γ(2 −α) x1−α, γ(3) γ(3 −α) x2−α ) . now, by letting η1,2(x,ξ) = (x− ξ 2 , x− ξ 2 ) , ξ = 0, we have η1(x,ξ).d αψ(x) = x3−α γ(3 −α) + 3x4−α γ(4 −α) ; η2(x,ξ).d αθ(x) = x2−α 2γ(2 −α) + x3−α γ(3 −α) . consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain ‖ϑ(x,ξ)‖2 = x4, ξ = 0. int. j. anal. appl. 17 (2) (2019) 241 table 1. fractional multi-objective function, kv = 0 (α) eq. (3.3) eq.(3.4) 0.25 1.6 1.9 0.5 2.6 2.4 0.75 3.1 3.2 it is clear that ψ(ξ) = ψ(0) = (0, 0); θ(ξ) = θ(0) = (0, 0), then by assuming φ1 ( ψ(x) − ψ(ξ) ) = 5x, φ2 ( θ(x) − θ(ξ) ) = −5x, x ∈ [0, 1], we conclude that η1(x,ξ).d α ψ(x) + ρ1‖ϑ(x,ξ)‖2 = x3−α γ(3 −α) + 3x4−α γ(4 −α) + x4 < 5x, x ∈ [0, 1] = φ1 ( ψ(x) − ψ(ξ) ) (4.1) and η2(x,ξ).d α θ(x) + ρ2‖ϑ(x,ξ)‖2 = x2−α 2γ(2 −α) + x3−α γ(3 −α) + x4 < 5x, x ∈ [0, 1] = −φ2 ( θ(x) − θ(ξ) ) (4.2) hence, the couple (ψ, θ) is (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. table 1 shows that for various values of α ∈ (0, 1), the outcomes yield the fractional univexty of the couple (ψ, θ). to apply the conditions of theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 + c2ρ2 = 2 > 0 with the inequalities (3.3) and (3.4). this leads to all the conditions of theorem 3.1 are achieved and hence, ξ = 0 is an efficient solution. note that if we let φ1(y ) = 3y and φ2(y ) = −3y, the couple (ψ, θ) is not (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. 4.2. case (ii) kv = 1. to evaluate the fractional dunkl operator, a calculation implies that σx2 = x 2 − 2 v.x2 v.v = −x2, σx3 = −x3. therefore, one can attain η1(x,ξ).d αψ(x) = x3−α γ(3 −α) + x(2x2)α 2 + 3x4−α γ(4 −α) + x(2x3)α 2 int. j. anal. appl. 17 (2) (2019) 242 and η2(x,ξ).d αθ(x) = x2−α 2γ(2 −α) + x(2x)α 2 + x3−α γ(3 −α) + x(2x2)α 2 . table 2 shows the evaluation of the fractional multi-objective functions for different values of α. table 2. fractional multi-objective function, kv = 1 (α) eq.(3.3) eq. (3.4) 0.25 2.7 2.9 0.5 5 3.8 0.75 4.7 4.8 thus, we conclude that the conditions of theorem 3.1 are satisfied when c1 = c2 = 1; such that c1ρ1 +c2ρ2 = 2 > 0 with the inequalities (3.3) and (3.4). consequently, we obtain ξ = 0 is an efficient solution. 4.3. case (iii) kv = 2. by applying (1.2), we have η1(x,ξ).d αψ(x) = x3−α γ(3 −α) + x(2x2)α + 3x4−α γ(4 −α) + x(2x3)α and η2(x,ξ).d αθ(x) = x2−α 2γ(2 −α) + x(2x)α + x3−α γ(3 −α) + x(2x2)α. table 3 shows the evaluation of the fractional multi-objective functions for different values of α. it is clear that the couple (ψ, θ) is not (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. it is of (α,ρ,η,ϑ)-type univex at ξ ∈ [0, 1], when α ∈ (0, 0.25]. hence, theorem 3.1 can be applied only for this value of α. table 3. fractional multi-objective function, kv = 2 (α) eq. (3.3) eq. (3.4) 0.25 3.5 4.1 0.5 5.4 5.2 0.75 6.4 5.5 5. conclusion this effort is generalized, for the first time, two important concepts in science. the dunkl operator and the univex function, by utilizing the riemann-liouville fractional differential operator. these two generalizations are combined to deliver the fractional multi-objective problems. we studied the duality cases by minimize and maximize the desired function in the rn. simulation is provided to apply the existing solutions. it has been found that the fractional case converges to the ordinary case. these problems can be employed in many studies not only in mathematics, but also in the economy; such as the utility function int. j. anal. appl. 17 (2) (2019) 243 the cost function and the entropy function. one can replace the riemann-liouville fractional differential operator of any type of fractional calculus. acknowledgement the work here is partially supported by ukm grant: gup-2017-064. references [1] k.m. kolwankar, a.d. gangal, fractional differentiability of nowhere differentiable functions and dimensions, chaos 6 (4) (1996), 505-513. [2] a.a. kilbas, h.m. srivastava, j.j. trujiilo, theory and applications of fractional differential equations. amsterdam, netherlands: elsevier, 2006. [3] g. jumarie, maximum entropy, information without probability and complex fractals: classical and quantum approach (vol. 112). springer science & business media, 2013. [4] x-jun yang, d. baleanu and h. m. srivastava, local fractional integral transforms and their applications, elsevier ltd, 2016. [5] x.-jun. yang, local fractional functional analysis and its applications, asian academic publisher limited, hong kong, 2011. [6] c.f. dunkl, differential-difference operators associated to reflection groups, trans. amer. math. soc. 311 (1989), 167–183. [7] m.rosler, m. voit, markov processes related with dunkl operators. adv. appl. math. 21(4)(1998), 575–643. [8] r. w. ibrahim, optimality and duality defined by the concept of tempered fractional univex functions in multi-objective optimization, int. j. anal. appl. 15 (2017), 75-85. [9] r. w. ibrahim, abdullah gani, a new algorithm in cloud computing of multi-agent fractional differential economical system, computing 98 (11) (2016), 1061-1074. [10] r. w. ibrahim, hamid a. jalab, and abdullah gani, perturbation of fractional multi-agent systems in cloud entropy computing, entropy 18 (2016), 31. [11] r. w. ibrahim, abdullah ghani, hybrid cloud entropy systems based on wiener process, kybernetes 45 (7) (2016), 1072-1083. [12] r. w. ibrahim, and abdullah gani, a mathematical model of cloud computing in the economic fractional dynamic system, iran. j. sci. technol. trans. a, sci. 42 (2018), 65-72. 1. introduction 2. univex function 3. results 4. simulation 4.1. case (i) kv=0. 4.2. case (ii) kv=1. 4.3. case (iii) kv=2. 5. conclusion acknowledgement references international journal of analysis and applications volume 19, number 1 (2021), 153-164 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-153 prediction intervals for the first and last point in future sample having from a new bathtub shape failure rate life time model in the presence of outliers ayed r.a. alanzi∗ department of mathematics, college of science and human studies at hotat sudair, majmaah university, majmaah 11952, saudi arabia ∗corresponding author: a.alanzi@mu.edu.sa abstract. acquiring bayesian prediction intervals for the first and final points of observation with the bathtub-shape distribution of the failure rate life-time type under the conditions of available outliers is the focus of the research. these bounds of predication acquired on the basis of the right type-ii censored sample. the procedure is presented with the help of a wide range of illustrative example. 1. introduction the researchers have to apply the same type of distribution in the context of numerous statistical problems to use the previously obtained data in order to predict the future data. this need has been the subject of a number of academic researches and studies with the analysis of the corresponding practical application ( [1]; [2]; [3]). simultaneously, the research which offered the most significant applications was conducted and further analyzed [4]. more particularly, the researcher devoted his work to the increasing function of failure rate or two parameters of the bathtub-shape life-time distribution. chen states that the distribution has λ and β as parameters; in that case, the following equations denote the functions of cumulative distribution and probability density: received november 5th, 2020; accepted november 30th, 2020; published january 7th, 2021. 2010 mathematics subject classification. 62e17. key words and phrases. bathub-shaped model; bayesian prediction; censored samples; outliers; bivariate prior. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 153 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-153 int. j. anal. appl. 19 (1) (2021) 154 f(x; λ, β) = λβ xβ−1ex β −λ (ex β −1), x > 0, λ > 0, β > 0,(1.1) and f(x; λ, β) = 1 − e−λ (e xβ−1).(1.2) according to the calculation of selim [5], the bayesian estimations are based on the bathbut-shape lifetime distribution with two parameters founded on the record values. in the studies of niazi and abdelrahman [6], bayesian prediction boundaries are acquired for bathbut-shape life-time distribution with two parameters and a failure rate function. derivation of bayesian prediction bounds is done for new model of life-time bathbut-shape failure rate with the doubly right type-ii censored samples [7]. the theoretical and practical value of the research was huge, considering the life-time distribution function based on the bayesian prediction intervals [8]. there was also an investigation devoted to the chen distribution in the e-bayesian assessment on the basis of the censoring scheme type-i [9]. th present research deals with model of chen(λ, β) in the framework of two different sampling plans to obtain bayesian intervals for prediction needed in the future studies: provided that x1, x2, · · · , xn is an order random sample from model (1.2) with size n and xr, xr+1, · · · , xn, the sample has (n − r) as the largest for the sample observations. the statistical analysis uses merely the ordered observations that remained, i.e. x = (x1, · · · , xr). it is clear that this sampling type includes a complete ordered sample r = n as a special case. in the first case, a censored sample type-ii is x1 < · · · < xr with r < n defined as an observed sample, while the unobserved sample is identified with xr+1 < xr+2 < · · ·xn as the remaining values. the study of all the rest values of n−r is done with the use of an observed sample. in the second case, the sample of type-ii, identical to that in the previous case, is presented with x1 < · · · < xr while z1 < z2 < · · · < zm is viewed as a future sample of unobserved type with the use of the same population. in terms of using the selected configurations of sampling, it is essential to specify the prediction intervals, which apply previous observations with the aim to identify the future observations. 2. function of likelihood provided that a random sample with the derivation from the population with specified probabilities in (1.1) and (1.2) is x1 < x2 < · · · < xn, it is time to assign x1,x2, · · · ,xn to the life test. recording of the failure times is done merely beginning from the failure timeframe rth with r < n. the analyzed study outputs are identified in the research as the censored data type-ii with the estimated function of likelihood presented below: int. j. anal. appl. 19 (1) (2021) 155 l(λ, β; x) ∝ [1 −fx(x(r); λ, β)]n−r r∏ i=1 [fx(x(i); λ, β)] = (λβ)r exp { r∑ i=1 (β ln x(i) + x β (i) ) −λt1(β; x) } , x(k) > 0 (2.1) where x = (x(1), · · · ,x(r)), t1(β; x) = r∑ i=1 (e x β (i) − 1) + (n−r)(ex β (r) − 1).(2.2) 3. density functions of prior and posterior types in the present work, it is taken that the researcher applies a simple prior density function for making proper measurement. according to the previous assumptions for the λ and β parameters, they are presented as: π(λ, β) = π1(λ) π2(β)(3.1) where π1(λ) is a conjugate prior given by π1(λ) = ba11 γ(a1) λa1−1 e−b1 λ, λ > 0 (a1, b1 > 0).(3.2) and π2(β) = ba22 γ(a2) βa2−1 e−b2 β, β > 0 (a2, b2 > 0).(3.3) according to sarhan et all [10], the function analyzed in the points (3.2) and (3.3) should be applied with the corresponding two parameters to identify the bathtub-shaped distribution. further summarization of joint posterior density function of λ and β parameters is given below with the use of the joint prior density function estimated in (4.1) and likelihood function estimated in (5.4): π∗2 (λ, β, |x) ∝ λ r+a1−1βr +a2−1 exp { r∑ i=1 (β ln x(i) + x β (i) ) − b2 β − λ [t1(β; x) + b1] } ,(3.4) therefor, the λ and β posterior density function can presented as π∗(λ, β|x) ∝ g1(λ|data) g2(β|data) g3(λ,β|data),(3.5) int. j. anal. appl. 19 (1) (2021) 156 with g1(λ|β,data) as gamma density under the constraints of r shape and t1(β; x) scale, while a proper density function of g2(β|data) is presented below as: g2(β|data) ∝ 1 [t1(β; x)]r βr−1 exp { r∑ i=1 β(ln x(i) − b2) } (3.6) while g3(λ,β|data)) can be presented as follows: g3(λ, β|data) = λa1 βa2 e−λb1+x β .(3.7) thereby, taking into consideration the squared error loss function, all functions of λ and β can be interpreted through the bayesian estimation as follows: ĝ(λ, β) = ∫∞ 0 ∫∞ 0 g(λ, β)g1(λ|data) g2(β|λ, data)g3(λ, β|data)dλdβ∫∞ 0 ∫∞ 0 g1(λ|data) g2(β|λ, data) g3(λ, β|data)dλdβ .(3.8) analysis of the equation (5.6) makes it possible to claim that its conversion into a simple closed form is impossible. thus, estimation of the bayesian predications in terms of λ and β as the inputs denoted above is not possible in this form either. consequently, one of the assumptions covers the potential efficiency of applying the technique of importance sampling. it should be done in accordance with the idea suggested by chen and shao [11], which implies approximation of (5.6) to find a solution to the restrictions related to simple closed forms. 3.1. technique of importance sampling. the methodology of importance sampling technique is applied to compute and validate the λ, β bayes estimates as well as a number of constructed relevant functions, for instance g(λ, β). algorithm presents the process of approximating the function of posterior density. algorithm: (1) using g1(β|data) for the estimation of β. (2) using g2(λ|β, data) for the estimation of λ. (3) repeating the stage 1 and 2 consecutively for generation of (λ1, β1), (λ2, β2), · · · , (λm, βm ). the following equation presents the process of approximating in the context of the procedure of importance sampling under the restrictions of bayesian estimates for g(λ, β) along with the relevant control for squared error loss: ĝbs(λ, β) = m∑ i=1 g(λi, βi) g3(λi, βi|data)∑m i=m0 g3(λi, βi|data) ,(3.9) int. j. anal. appl. 19 (1) (2021) 157 4. prediction with outliers presence in the analysis of the process of predicting the future observations with the outliers presence, it is appropriate to use the formal definitions given for (1.1). besides, i apply the random sample of x1, x2, · · · , xn created on the basis of chen (λ, β) on the basis of the given function of population density. the following stage is using the independent unobserved sample as y1, y2, · · · , ym as a result of using the same data to form a future sample. the next stage covers further testing of the boundaries of bayesian prediction for sth with a single outlier in the range of future estimates for ys, s = 1, 2, · · · , m. the following equation presents the ys density function for a provided θ under the conditions described above: h(ys|θ) = d(s) [(s− 1)fs−2(1 −f)m−sf?f + fs−1(1 −f)m−sf? +(m−s)fs−1(1 −f)m−s−1(1 −f?)f],(4.1) with d(s) = ( m− 1 s− 1 ) (4.2) the function of density is presented as f = f(y|θ) and the function of cumulative distribution is given as f = f(y|θ) for all ys that can’t be defined as outliers. balakrishnan and ambagabpitiya [12] refer to f∗ = f∗(y|θ) and f∗ = f∗(y|θ) as to outliers. acquiring the f∗ and f∗ functions is done for the chen (λ, β) model via using a different parameter λ by λ λ0, or λ + λ0 according to the classification of the outliers. the study of alanzi and niazi [8] is devoted the analysis of prediction interval on the basis of using doubly type-ii censored sample for the future to have λ replaced by λ λ0, or λ + λ0. furthermore, the research conducted by alanzi [13] was devoted to study of a right type-ii censored sample to have λ replaced with λ λ0 with further calculation of the interval for prediction aimed at first and final future observation. the present study implies replacement of λ with λ + λ0 as well as calculating the interval for predication needed for the first and final observation that involve using the right type-ii censored samples. 5. prediction of the first observation the case of prediction of the first implies having distribution in the first y1 in the m-size future sample via adding s = 1 in (4.1) with only one outlier presence (type λ + λ0); it is done as follow: h(y1|θ) = (1 −f)m−1f? + (m− 1)(1 −f)m−2(1 −f?)f,(5.1) it is possible to acquire y1 density function with a single type λ + λ0 outlier presence in the case of chen(λ, β) via changing of (1.1) for f and (1.2) for f in (5.1). on replacement of λ for λ + λ0. f ∗ and f∗ have the same int. j. anal. appl. 19 (1) (2021) 158 values as they do in (1.1) and (1.2). it is possible to present a density function in the following simplified form: h1(y1|λ, β) = f(y1; (λm + λ0), β),(5.2) with cdf of y1 presented as follows: h1(y1|λ, β) = f(y1; (λm + λ0), β).(5.3) estimation of the predictive density of y = y1, with x, (λm + λ0) and β is as follows: h∗1(y|x) = ∫ ∞ 0 ∫ ∞ 0 h1(y1|λ, β) π∗(λ, β|x) dλdβ,(5.4) estimation of the y = y1, predictive distribution function with x, λ and β is as follows: h∗1 (y |x) = ∫ ∞ 0 ∫ ∞ 0 h1(y1|x, λ, β) π∗(λ, β|x) dλdβ,(5.5) {(λi, βi); i = 1, 2, · · · ,m} are assumed to be mcmc samples obtained after generation from π∗(λ, β|x) and corresponding parameters of estimation to ensure consistency of h∗1(y1|x, λ, β) and h∗(y1|x, λ, β). thus, ĥ∗1(y |x) = m∑ i=1 h1(y1|λi, βi) hi(5.6) and ĥ∗1 (y |x) = m∑ i=1 h1(y1|λi, βi) hi(5.7) with gi = g3(λi, βi|data) m∑ i=1 g3(λi, βi|data) ; i = 1, 2, · · · , m.(5.8) on the basis of the above-mentioned analysis, the bayesian estimation for y1,(1−τ) 100 % implies having p [l(x) ≤ y1 ≤ u(x)] = 1 − τ, with l(x) as the highest limit for y1 and u(x) as the lowest one. the following estimation is made on the basis of prior estimates for (5.7), 1 − τ 2 and τ 2 , thus: p[y ≥ l(x)|x] = 1 − τ 2 ⇒ ĥ∗1 (l(x)|x) = τ 2 (5.9) and p [y ≤ u(x)|x] = τ 2 ⇒ ĥ∗1 (u(x)|x) = 1 − τ 2 .(5.10) calculation of the prediction limits of y1 is done using the equations (5.9) and (5.10). int. j. anal. appl. 19 (1) (2021) 159 6. prediction of the last observation distribution of the last in a m-size sample with only a single outlier presence is ensured when s = m is added in (4.1). it is possible to present the ym density function for a provided θ with a single outlier presence as follows: h2(ym|θ) = (m− 1)fm−2f?f + fm−1f?,(6.1) provided that a single outlier of type λ + λ0 is presence, it is possible to obtain the ym density function in the case chen (λ, β) via replacing (1.1) for f and (1.2) for f in (6.1). the research uses the f∗ value from (1.1) and f∗ value from (1.2) after λ is replaced with λ + λ0. it is possible to present the mentioned density function in the following simplified form: h2(ym|λ, β) = [ (λ + λ0) m−1∑ j=0 b1j(ym) + λ(m− 1) m−2∑ j=0 b2j(ym) ] , ym > 0, (6.2) with b1j(ym) = a1j(m)f(ym; λ (j + 1) + λ0, β), b2j(ym) = a2j(m) [ f(ym; λ (j + 1), β) −f(ym; λ (j + 2) + λ0, β),(6.3) with ` = 1, 2, a`j(m) = (−1)j ( m− ` j ) ,(6.4) the cdf that is related to pdf h2(ym|λ, β) is as follows: h2(ym|λ, β) = d(s) [ (λ + λ0) m−1∑ j=0 b∗1j(ym) + β(m− 1) m−2∑ j=0 b∗2j(ym) ] ,(6.5) with b∗1j(ym) = a1j(m) λ (j + 1) + λ0 f(ym; λ (j + 1) + λ0, β), b∗2j(ym) = a2j(m) λ (j + 1) f(ym; λ (j + 1), β) − a2j(m) λ (j + 2) + λ0 f(ym; λ (j + 2) + λ0, β),(6.6) with f(ym; λ (j+1)+λ0, β) presented via (1.2). the ym predictive density provided that there are constraints for x from the outlier λ + λ0 can be estimated via using the equations (6.2) in (5.4) and the algorithm. int. j. anal. appl. 19 (1) (2021) 160 h∗2(ym|x) = ∫ ∞ 0 ∫ ∞ 0 h2(ym|λ, β) π∗(λ, β|x) dλdβ,(6.7) with the predictive cdf of ym, g ∗ 2(ym|x) can be defined as follows: h∗2 (ym|x) = ∫ ∞ 0 ∫ ∞ 0 h2(ym|λ, β) π∗(λ, β|x) dλdβ,(6.8) with π∗(λ, β|x) presented in (6.5) and h2(ym|λ, β) presented in (3.5). clearly, it is not possible to present either (6.7) or (6.8) in a closed form. consequently, evaluation can’t be done with the use of analytical approch. on the basis of the mcmc samples {(λi, βi), i = 1, 2, · · · , m}, along with a consistent estimator used for g∗2(ym|x) and g∗2 (ym|x) simulation of data, it is reasonable to state: ĥ∗2(ym|x) = m∑ i=1 h2(ym|λi, βi) hi,(6.9) and ĥ∗2 (ym|x) = m∑ i=1 h2(ym|λi, βi) hi,(6.10) with estimation of hi from (5.8). moreover, it is possible to estimate ĝ ∗ 2(ym|x) and ĝ∗2 (ym|x) for all ym with the use of mcmc samples {(λi, βi), i = 1, 2, · · · , m} that bear certain similarity to them. furthermore, the bayesian prediction boundaries of a (1−τ)100% type for ym implies having p [l(x) ≤ ym ≤ u(x)] = 1−τ with l(x) as the lowest bayesian prediction limit for ym and u(x) is the highest. it is possible to acquire them and identify as l(x) and u(x), which can be show via non-linear equations solutions as follows for ym by non-linear equations solutions. p[y ≥ l(x)|x] = 1 − τ 2 ⇒ ĥ∗2 (l(x)|x) = τ 2 (6.11) and p [y ≤ u(x)|x] = τ 2 ⇒ ĥ∗2 (u(x)|x) = 1 − τ 2 .(6.12) iterative statistical methodologies could apply the equations (6.11) and (6.12) mentioned above to ensure advanced regression that implies also implemented control for ym multicellularity or backward analysis and other factors in other cases. for instance, generating λ = 0.983884 for the prior parameters a1 = 1.3 and b1 = 2.1 using the equation (3.2) for the prior density. also, generating β = 3.90261 for the prior parameters a2 = 3.2 and b2 = 1.4 using the equation (3.3) for the prior density. further, chen distribution with λ = 0.983884 and β = 1, 3.90261 on the basis of using a different value of r allows generating a random n = 30 size sample. int. j. anal. appl. 19 (1) (2021) 161 for an illustration of the example, i can make an assumption that there is different m = 10 size sample with a single outlier λ + λ0 presence. there is a set goal to obtain prediction limits y1 and y15 estimated as 95% for the provided λ0 value in terms of the first and last of future sample. table 1,2,3 demonstrate the limits with the provided λ + λ0 values. table 1. bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0 outlier presence, n = 30,r = 20. λ0 observations y1 y15 0 lower and upper limits (0.199913, 0.690383) 0.991503, 1.20199) length 0.490471 0.210491 percentage of coverage 95.09 % 94.84 % 1 lower and upper limits (0.196435, 0.679618) (0.986197, 1.20059) length 0.483183 0.214388 percentage of coverage 94.57 % 95.24 % 2 lower and upper limits (0.193236, 0.669636) (0.985184, 1.20058) length 0.4764 0.2154 percentage of coverage 94.06 % 95.33 % 3 lower and upper limits (0.190279 , 0.660341) (0.984974 , 1.20058) length 0.470062 0.21561 percentage of coverage 93.41% 95.33% 4 lower and upper limits (0.187532 , 0.651651) (0.98493, 1.20058) length 0.46412 0.215654 percentage of coverage 92.93% 95.33% int. j. anal. appl. 19 (1) (2021) 162 table 2. bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0, outlier presence n = 30, r = 25. λ0 observations y1 y15 0 lower and upper limits (0.193992, 0.679774) (0.983196, 1.19774) length 0.485782 0.214544 percentage of coverage 94.72 % 95.29% 1 lower and upper limits (0.190741, 0.669518) (0.977914, 1.1963) length 0.478776 0.218387 percentage of coverage 94.18 % 95.62% 2 lower and upper limits (0.187741, 0.659979) (0.976825, 1.1963) length 0.472238 0.219473 percentage of coverage 93.57% 95.74% 3 lower and upper limits (0.184957, 0.651072) (0.976582, 1.1963) length 0.466115 0.219716 percentage of coverage 93.01 % 95.7 % 4 lower and upper limits (0.182364, 0.642726) (0.976527, 1.1963) length 0.460362 0.219771 percentage of coverage 92.02 % 95.7% int. j. anal. appl. 19 (1) (2021) 163 table 3. bayesian prediction intervals 95 % for y1 and y15 with a single λ + λ0 outlier presence, n = r = 30. λ0 observations y1 y15 0 lower and upper limits (0.197556, 0.685817) (0.987722, 1.1998) length 0.488261 0.212081 percentage of coverage 94.99 % 95.06% 1 lower and upper limits (0.194182, 0.675296) (0.982433, 1.19838) length 0.481114 0.521755 percentage of coverage 94.43% 95.44% 2 lower and upper limits (0.191073, 0.665526) 0.981386, 1.19838) length 0.474453 0.21595 percentage of coverage 93.9% 95.59% 3 lower and upper limits (0.188195, 0.656417) (0.981161, 1.19838) length 0.468223 0.216995 percentage of coverage 93.27 % 95.59% 4 lower and upper limits (0.185517, 0.647892) (0.981112, 1.19838) length 0.462375 0.21722 percentage of coverage 92.57% 95.6% int. j. anal. appl. 19 (1) (2021) 164 7. conclusion on the findings the research analyzes a single λ + λ0 outlier as multiple outliers can be studied only on the basis of a more profound analysis. it is possible to acquire the bayesian prediction limits of the first y1 and last y15 in the homogeneous case for the future observation with no outliers by λ + λ0 setting in (5.7) and (6.2) equation. table 1, 2 and 3 show the potential impact of λ value on the future observation boundaries and restrictions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] i. r. dunsmor, the bayesian predictive distribution in life testing models, technometrics. 16 (1975), 455-460. [2] a. aitchison, i.r. dunsmore, statistical prediction analysis, cambridge university press, (1975). [3] s. geisser, predictive inference: an introduction, chapman and hall, london, (1993). [4] z. chen, a new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. stat. probab. lett. 49 (2000), 155-162. [5] m. a. selim, bayesian estimations from the two-parameter bathtub-shaped lifetime distribution based on record values. pak j. stat. oper. res. 8(2) (2012), 155-165. [6] s. f. niazi, a. m. abd-elrahman, bayesian prediction bounds for a two-parameter lifetime distribution with bathtubshaped failure rate function. sylwan 159(6) (2015), 34-50. [7] s. f. niazi, a. m. abd-elrahman, bayesian prediction bounds of doubly type-ii censored samples for a new bathtub shape failure rate life time model. j. appl. math. stat. sci. 1 (2016), 21-30. [8] a. r. alanzi, s. f. niazi, bayesian prediction intervals for a new bathtub shape failure rate lifetime model in the presence of outliers. adv. appl. stat. 63 (2020), 1-22. [9] a. algarni, a. m. almarashi, h. okasha, h. k. t. ng, e-bayesian estimation of chen distribution based on type-i censoring scheme. entropy, 22(6) (2020), 636. [10] a. m. sarhan, d. c. hamliton, b. smith, parameter estimation for a twoparameter bathtub-shaped lifetime distribution, appl. math. model. 36 (2012), 5380-5392. [11] m.-h. chen, q.-m. shao, monte carlo estimation of bayesian credible and hpd intervals, j. comput. graph. stat. 8(1) (1999), 2327-2341. [12] n. balakrishnan, r. s. ambagaspitiya, relationships among moments of order statistics in samples from two related outlier models and some applications, commun. stat., theory meth. 17(7) (1988), 2327-2341. [13] a. r. alanzi, bathtub-shape failure rate life time model in a new version with the use of bayesian prediction bounds for the presence of outliers. adv. sci. technol. eng. syst. j. 5(4) (2020), 710-714. 1. introduction 2. function of likelihood 3. density functions of prior and posterior types 3.1. technique of importance sampling 4. prediction with outliers presence 5. prediction of the first observation 6. prediction of the last observation 7. conclusion on the findings references international journal of analysis and applications volume 19, number 5 (2021), 674-694 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-674 applications on the boundary behaviour of the derivative of conformal mapping shatha s. alhily1, vishnu narayan mishra2,∗ 1department of mathematics, college of science, mustansiriyah university, baghdad, iraq shathamaths@uomustansiriyah.edu.iq, shathamaths@yahoo.co.uk 2department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur 484 887, india ∗corresponding author: vishnunarayanmishra@gmail.com, vnm@igntu.ac.in abstract. the objective of this research paper is to describe the behaviour of the boundary derivative of conformal maps from polygon domain onto unit disk through construct some an interesting cases, and its inverse maps. moreover, we study the existence and finiteness the integrability of the derivative of conformal maps over an infinite sector w. 1. introduction let φ be a conformal map from d ⊂ c onto a simply connected domain ω, with its inverse ψ = φ−1 : ω −→ d. brennan’s conjecture states that, for all such φ, (1.1) ∫ ∫ d |φ′|2−pdxdy = ∫ ∫ ω |ψ′|pdxdy < ∞, for 4 3 < p < 4. in connection with this conjecture which is associated with estimating the integral means of derivatives of univalent functions, which is so related with the behaviour of the boundary derivative of conformal maps from polygon domain onto unit disk and the behaviour of the boundary of inverse maps received march 7th, 2020; accepted august 3rd, 2020; published august 2nd, 2021. 2010 mathematics subject classification. 30b30. key words and phrases. conformal mapping; derivative of conformal maps; infinite sector; polygon domain. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 674 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-674 int. j. anal. appl. 19 (5) (2021) 675 an existence and finiteness the integrability of the derivative of conformal maps through constructing some interesting examples when engages the infinite sector w. we started with basic definition of the polygonal domain and we present some typical examples to examine the boundary behaviour of the derivatives of conformal mapping of polygon to unit disk, as well as the boundary behaviour of the derivatives of the inverse maps. in the following some preliminary on the boundary derivative of conformal maps, with basic definition of the polygonal domain that serves as motivation to the main results of this work to give a full support, the reader is referred to references [2, 3, 7–9, 11, 12]. definition 1.1 (polygon). a polygon p is usually defined as a collection of n vertices v1,v2, ...,vn and n edges v1v2,v2v3, ...,vn−1vn,vnv1 such that no pair of nonconsecutive edges share a point. we deviate from the usual practice by defining a polygon as the closed finite connected region of the plane bounded by these vertices and edges. the collection of vertices and edges will be referred to as the boundary of p , denoted by ∂p, a polygon of n vertices will sometimes be called an n-gon. riemann mapping theorem [6]. guarantees the existence of a conformal map φ from polygonal domain p ⊂ c conformally onto the unit disk d (|w| < 1), which can be extended continuously to the boundary ( cf. carathéodory’s theorem [9]). worthy to mention that it is not yet possible to write down a simple formula for the conformal map from one region to another. hence, in case of a map from the upper half-plane or unit disk d to a polygon, then schwarzchristoffel formula [5] allows to compute the conformal map φ defined as follows: consider φ : p −→ d, be a conformal mapping, where ∂p be a circular arc or straight line segment γ, normalized by the conditions φ(z0) = 0 and φ ′(z0) > 0 ( where z0 is some point in p), and observe that φ maps γ onto an arc γ̂ of the unit circle |w| = 1 with ψ(w) = φ−1(z) : d −→ p. φ(z) = a + c ∫ z n−1∏ k=1 (ζ −zk)αk−1dζ where a and c are suitably chosen constants. so,we deliberately construct φ as the composition of one schwarz-christoffel map from p into upper half-plane (by applying schwarzchristoffel transformation) and another map of the upper half-plane to unit disk, in the examples (2.1), (2.2) where φ maps the p to unit disk. the term “polygon” is often modified by “simple” to distinguish it from polygons that cross themselves, simple polygons are also called jordan polygons, because such a polygon divides the plane into two regions. the boundary of a polygon is a “jordan curve” : it separates the plane into two disjoint regions, the interior and the exterior of the polygon. int. j. anal. appl. 19 (5) (2021) 676 this technique could help to study the behavior of conformal mapping by estimating some quantities which belong to interior upper half-plane, which in turn will help to analyze the boundary behaviour of the derivative of this map. 2. main results in the light of reported above, we discussed the theoretical underpinnings of the behaviour of the boundary derivative of conformal maps through of some main results obtained that have been yielded useful information in this regards. (cf. [1]) theorem 2.1. the derivative of the conformal mapping defined on rectangular domain to the unit disk is bounded but the derivative of the inverse maps is unbounded. h+ -1 -a a 1 rectangle unit disk d ζ = snz z = arcsn(ζ,k) φ = sn z− i sn z+ i ψ = sn z+ i 2i cn zd n z ζ = i w + 1 w− 1 w = ζ− iζ+ i figure 1. conformal mappings from rectangular domain onto unit disk proof. map the interior of the rectangular domain with vertices at points z1 = 1, z2 = −1, z3 = −a, and z4 = a with a > 1 into upper half plane needs to define schwarzchristoffel transformtion which maps h+ int. j. anal. appl. 19 (5) (2021) 677 into rectangle as follows: z = a + b ∫ ζ ζo ds (s− 1) 1 2 (s + 1) 1 2 (s−a) 1 2 (s + a) 1 2 z = a + b ∫ ζ ζo ds (s2 − 1) 1 2 (s2 −a2) 1 2 suppose a = 0 and b = 1 for convenience. z = ∫ ζ ζo ds√ (s2 − 1)(s2 −a2) (2.1) suppose ζo = 0 and let a = 1 k , then the integral (2.10) is transformed to z = ∫ ζ ζo ds√ (s2 − 1)(s2 − ( 1 k )2) = k ∫ ζ 0 ds√ (s2 − 1)(k2s2 − 1) z = k ∫ ζ 0 ds√ (1 −s2)(1 −k2s2) (2.2) the integral (2.2) is called an elliptic integral of first kind and k is a modulus of the elliptic integral with 0 < k < 1, denote by z = sn−1(ζ,k).(2.3) the inverse mapping of integral (2.2) is known as the jacobi elliptic function denoted by ζ = sn(z,k). ⇒ φ = ζ − i ζ + i = snz − i snz + i : rectangular domain −→ d φ′ = (snz + i)(cnz dnz) − (snz − i)( dnz) (snz + i)2 = 2i cnz dnz (snz + i)2 , such that, dnz = √ 1 −k2sn2z , cnz = √ 1 −sn2z. |φ′| = ∣∣∣∣2i cnz dnz(snz + i)2 ∣∣∣∣ = 2|cnz dnz||snz + i|2 snz ∈ h+ ⇒ snz + i ∈ h+ that is, |snz + i| ≥ 1 ⇒ 1 |snz + i| ≤ 1 |cnz dnz| = | √ 1 −k2sn2z| | √ 1 −sn2z| |1 −k2sn2z| ≤ 1 + |snz|2 |1 −sn2z| ≤ 1 + |k snz|2 ⇒ |φ′| = 2|cnz dnz| |snz + i|2 ≤ 2(1 + |snz|2)(1 + |k snz|2) ⇒ |φ′| is bounded. it remains to show that the inverse of the derivative of such function φ(z) is unbounded as follows: int. j. anal. appl. 19 (5) (2021) 678 ψ = φ−1 : d −→ rectangular domain ψ′ = 1 φ′ = (snz + i)2 2i cnz dnz ⇒ ψ = (snz + i)2 2i cnz dnz |ψ′| = |snz + i|2 2|cnz dnz| snz ∈ h+ ⇒ (snz + i) ∈ h+ ⇒ |snz + i|2 ≥ 1. |ψ′| = |snz + i|2 2|cnz dnz| ≥ 1 2|cnz dnz| ⇒ |ψ′| is unbounded as |z| −→ 0. � theorem 2.2. the derivative of the conformal mapping defined on triangular domain m to the unit disk is bounded but the derivative of the inverse maps is unbounded. h+ z-plane w1 = 0 b w2 w3 = b w-plane ζ-plane d φ1 φ−11 h( w )φ 2 figure 2. conformal mappings from triangular domain onto unit disk int. j. anal. appl. 19 (5) (2021) 679 proof. to construct conformal mapping defined on triangle domain to unit disk d. we have to define conformal mapping on triangular domain to upper half plane h+ and then define another mapping from h+ to unit disk d to achieve our aim. to do so, first : we have to establish conformal mapping that maps upper halfplane h+ onto triangular domain m by schwarz-christoffel transformation as follows: let φ1 = a ∫ z zo (s−x1)−k1 (s−x2)−k2ds + b. such that −ki = αiπ − 1 ; ∀i = 1, 2 to be schwars-christoffel transformation that maps h + into the interior of the equilateral triangle m such that αi = π 3 ; ∀i = 1, 2, 3. now, by assisstance that zo = 1, a = 1 and b = 0, we obtain schwars-christoffel transformation defined as follows: φ1 = ∫ z 1 (s + 1) −2 3 (s− 1) −2 3 ds.(2.4) which maps x1 = −1,x2 = 1 and x3 = ∞ into m w1w2w3 as follows: (i) in case z = 1 ⇒ φ1(1) = 0 ; that is , w1 = 0 in h+. (ii) in case z = −1, we have φ1(−1) = ∫ −1 1 (s + 1) −2 3 (s− 1) −2 3 ds = w2.(2.5) (iii) in case z = ±∞, we have lim z→±∞ φ1 = ∫ ±∞ 1 (s + 1) −2 3 (s− 1) −2 3 ds = w3.(2.6) to solve these integrals, let us consider first the equation (2.5) by choosing a path of the integration z = x along the real axis in the positive sense, that is s− 1 = |s− 1| eiθ1 s + 1 = |s + 1| eiθ2. the argument (θ1 + θ2) remains constant throughout integration from -1 to 1 since (s + 1) stays positive with zero argument, and (s− 1) has constant argument π. therefore equation (2.5) yields w2 = φ1(−1) = − ∫ 1 −1 (x + 1) −2 3 (1 −x) −2 3 (−e− 2πi 3 )ds. w2 = φ1(−1) = e πi 3 ∫ 1 0 2dx (1 −x2) −2 3 .(2.7) int. j. anal. appl. 19 (5) (2021) 680 by letting x = √ t in equation (2.7), we obtain beta function b( 1 2 , 1 3 ) and w2 = φ1(−1) = e πi 3 b( 1 2 , 1 3 ). ⇒ w2 = b e πi 3 . where b is the value of b( 1 2 , 1 3 ) now, the vertex w3 lies on the positive uaxis. so, w3 must be represented by the boundary integral 1 to ∞ as follows, w3 = φ1(∞) = ∫ ∞ 1 (x + 1) −2 3 (1 −x) −2 3 dx w3 = φ1(∞) = ∫ ∞ 1 dx (x2 − 1) −2 3 .(2.8) but w3 is also represented by integral (2.4) when z = −∞ along the negative real axis. so, w3 = φ1(−∞) = ∫ −∞ 1 (x + 1) −2 3 (1 −x) −2 3 dx = ∫ −1 1 (x + 1) −2 3 (1 −x) −2 3 e −2πi 3 dx + ∫ −∞ −1 (x + 1) −2 3 (1 −x) −2 3 e −4πi 3 dx. w3 = w1 + e −πi 3 ∫ ∞ 1 dx (x2 − 1) 2 3 ⇒ w3 = w1 + e −πi 3 w3.(2.9) solving (2.9) for w3 we obtain: ⇒ w3 −e −πi 3 w3 = w1. w3 (1 −e −πi 3 ) = b e −πi 3 w3 = b ; since (1 −e −πi 3 ) = e −πi 3 . in the end, we found the conformal mapping that maps upper halfplane h+ onto triangular domain m. it is known that φ2 = z−i z+i maps upper halfplane h+ onto unit disk d. hence, we have φ1 = ∫ z 1 (s2 − 1) −2 3 ds : upper half plane h+ −→ triangular domain m and φ2 = z − i z + i : upper halfplane h+ → unit diskd. let h(w) : triangular domain m−→ unit disk d. int. j. anal. appl. 19 (5) (2021) 681 defined as follows: h(w) = (φ2 ◦φ−11 )(w) = φ2(φ −1 1 )(w) h′(w) = φ′2(φ −1 1 )(φ ′−1 1 ) = ( φ−11 − i φ−11 + i ) ′ ( 1 φ′1 ) = (φ−11 + i)(φ ′−1 1 ) − (φ −1 1 − i)(φ ′−1 1 ) (φ−11 + i) 2 1 φ′1 . ⇒ h′(w) = 2i(z2 − 1) 2 3 (φ−11 + i) 2 (z2 − 1) 2 3 = 2i(z2 − 1) 4 3 (φ−11 + i) 2 . |h′(w)| = | 2i(z2 − 1) 4 3 (φ−11 + i) 2 | = 2i|z2 − 1| 4 3 |φ−11 + i|2 , where (φ−11 ) ′ = 1 φ′1 = (z2 − 1) 2 3 and φ−11 ∈ h +, so that (φ−11 + i) ∈ h +. this implies to |φ−11 + i| 2 ≥ 1 ⇒ 1 |φ−11 + i|2 ≤ 1 ⇒ |h′(w)| ≤ 2(|z|2 + 1) 4 3 . |h′(w)| is bounded. what remains is to prove that the inverse of the derivative of h(w) is unbounded. note that; φ−12 = −i ζ+1 ζ−1 is the möbius transformation, it maps the unit disk d to upper half-plane h +, thus we have (φ−12 ) ′ = (ζ − 1)(−i) − (−iζ − i) (ζ − 1)2 = 2i (ζ − 1)2 . we use schwarz-christoffel transformation for mapping h+ into triangular domain m. φ1 = ∫ z 1 (s2 − 1) −2 3 ds. int. j. anal. appl. 19 (5) (2021) 682 therefore, h−1(ζ) = (φ1 ◦φ−12 )(ζ) : d −→m . (h−1(ζ)) ′ = φ′1(φ −1 2 (ζ)) (φ −1 2 (ζ)) ′ = [(φ−12 (ζ)) 2 − 1] −2 3 2i (ζ − 1)2 . = 2i [ −( ζ+1 ζ−1 ) 2 − 1 ]−2 3 (ζ − 1)2 = 2i [ −2ζ2−2 (ζ−1)2 ]−2 3 (ζ − 1)2 . = 2i (ζ − 1)2 [ −2ζ2−2 (ζ−1)2 ]−2 3 = 2i (ζ − 1)2 (−2ζ2 − 2) 2 3 (ζ − 1) −4 3 = 2i (ζ − 1) 2 3 (−2ζ2 − 2) 2 3 . |(h−1(ζ)) ′ | = ∣∣∣∣ 2i (ζ − 1) 2 3 (−2ζ2 − 2) 2 3 ∣∣∣∣ . |(h−1(ζ)) ′ | = ∣∣∣∣ 2 (ζ − 1) 2 3 (2ζ2 + 2) 2 3 ∣∣∣∣ −→∞ as ζ −→ 1. � theorem 2.3. the derivative of the conformal mapping defined on crescent domain to unit disk is bounded but the derivative of the inverse maps is unbounded. proof. compute the conformal mapping of a crescent domain onto unit disk by setting a sequence of functions as follows: let φ1 = 1 z maps c1 → l1 and c2 → l2. φ2 = i z rotates the stripe in the left plane onto stripe in the lower half plane. φ3 = 4πi z extends the stripe in the lower half plane h− between −π,−2π φ4 = e 4iπ z maps the stripe in the lower half plane h−into h+. φ5 = e 4iπ z − i e 4iπ z + i maps the h+ onto unit disk d. int. j. anal. appl. 19 (5) (2021) 683 −2 c2 −4 crescent domain c1 l1 −1 4 l2 −1 2 −1 4 −1 2 −π −4π h+ unit disk d φ1 φ2 φ3 φ4 φ5 figure 3. conformal mappings from crescent domain onto unit disk so; φ(z) = e 4iπ z − i e 4iπ z + i : crescent domain −→ unit disk d. using short calculation we obtain φ′(z) = (e 4iπ z + i)(−4iπ z2 e 4iπ z ) − (e 4iπ z − i)(−4iπ z2 e 4iπ z ) (e 4iπ z + i)2 . ⇒ φ′(z) = 8π z2 e 4iπ z (e 4iπ z + i)2 = 8πe 4iπ z z2(e 4iπ z + i)2 . |φ′(z)| = 8π|e 4iπ z | |z|2|e 4iπ z + i|2 . int. j. anal. appl. 19 (5) (2021) 684 now; |e 4iπ z | = 1 ⇒ e 4iπ z ∈ h+ ⇒ (e 4iπ z + i) ∈ h+. ⇒ |e 4iπ z + i| ≥ 1 ⇒ |e 4iπ z + i|2 ≥ 1 ⇒ 1 |e 4iπ z + i| 2 ≤ 1. ⇒ |φ′(z)| = 8π|e 4iπ z | |z|2|e 4iπ z + i|2 . ≤ 8π |z|2 hence, |φ′(z)| is bounded. we show that the inverse of the derivative of such function φ(z) is unbounded as follows: φ−1(w) = ψ(w) : d −→ crescent domain. ψ = φ−1 = 1 φ = e 4πi z + i e 4πi z − i . ⇒ ψ′ = 1 φ′ ⇒ ψ′ = z2(e 4iπ z + i) 8πe 4iπ z ⇒ |ψ′| = |z|2|e 4iπ z + i| 8π|e 4iπ z | . it is known, e 4iπ z ∈ h+ ⇒ (e 4iπ z + i) ∈ h+. hence |e 4iπ z + i| ≥ 1 & |e 4iπ z | = 1. ⇒ |ψ′| = |z|2|e 4iπ z + i| 8π|e 4iπ z | ≥ |z|2 8π . in the end, we obtain |ψ′| is unbounded. � theorem 2.4. the derivative of the conformal mapping defined on lensshaped domain to the unit disk is bounded but the derivative of the inverse maps is unbounded. proof. compute the conformal mapping a lens domain onto unit disk by setting a sequence of functions as follows: int. j. anal. appl. 19 (5) (2021) 685 α β 0 h+ 0-1 1 unit disk d -1 1 i −i φ1 φ2 φ3 figure 4. conformal mappings from lens domain onto unit disk let φ1 = z −α z −β maps lens-shaped domain to the first quarter plane φ2 = z 2 maps the first quarter plane to the upper halfplane. φ3 = z2 − i z2 + i maps upper half plane h+ to the unit disk d. so, φ(z) = (z−α z−β ) 2 − i (z−α z−β ) 2 + i : lens-shaped domain −→ unit disk d. ⇒ φ(z) = (z −α)2 − i(z −β)2 (z −α)2 + i(z −β)2 . therefore when z = α ⇒ φ(z) = −1 in d z = β ⇒ φ(z) = 1 in d. int. j. anal. appl. 19 (5) (2021) 686 when (z −α)2 = −(z −β)2 ⇒ φ(z) = −(z −β)2 − i(z −β)2 −(z −β)2 + i(z −β)2 = i in d. in the end, if (z −α)2 = (z −β)2 ⇒ φ(z) = (z −β)2 − i(z −β)2 (z −β)2 + i(z −β)2 = −i in d. ⇒ φ′(z) = 4i(z −β)2(z −α) − 4i(z −β)(z −α)2 [(z −α)2 + i(z −β)2]2 ⇒ |φ′(z)| = 4|α−β||z −β||z −α| (z −α)4 + (z −β)4 . where, 4|α−β| = c; c is a constant; (α < β). also, |z −β| ≤ |z| + |β| = m1 & |z −α| ≤ |z| + |α| = m2 ⇒ |φ′(z)| ≤ m (z −α)4 + (z −β)4 ⇒ |φ′(z)| is a bounded for every z in lens-shaped domain. again, we can show that the inverse of the derivative of such function φ(z) is unbounded as follows: ψ′ = 1 φ′ = [(z −α)2 + i(z −β)2]2 4i(z −β)2(z −α) − 4i(z −β)(z −α)2 ⇒ |ψ′| = |(z −α)2 + i(z −β)2|2 4|(z −β)(z −α)(α−β)| ⇒ |ψ′| = (z −α)4 + (z −β)4 4|z −β||z −α||α−β| . if z −→ α or z −→ β, then |ψ′| −→∞, so |ψ′| is unbounded. � the following examples show the integrability of the derivative of conformal maps on infinite sector w exists and is finite for some pth-power integrable function φ when α = π n is a number for some integer n. further details can be found in the books of di francesco [4] and of m. stein [10]. theorem 2.5. let φ(z) be a conformal mapping defined on infinite sector w for the angle α = π 2 onto unit disk d. int. j. anal. appl. 19 (5) (2021) 687 let φ(z) = (φ2 ◦φ1)(z) = z2 − i z2 + i : w → d so that φ1(z) = z2. if maps the infinite sector onto upper half plane h+ and φ2(w) = w−iw+i maps the upper half-plane h + onto unit disk d (see figure 5). then the integrability of the derivative of conformal mapping φ, is as follows:∫ ∫ w |φ′(z)|p dxdy < ∞; for each p > 2 3 . !"#$%&#'(%)*+,-."/% 0%1234% 56% 54% 7+,-."/% figure 5. infinite sector w for the angle α = π 2 proof. given φ(z) = z2 − i z2 + i : w → d ⇒ φ′(z) = (z2 + i)(2z) − (z2 − i)(2z) (z2 + i)2 = i4z (z2 + i)2 ⇒ |φ′(z)| = 4|z| |z2 + i|2 . now, w-plane is an infinite sector. that is; r = |z|→ 0 −∞. (i) so, r = |z|→∞ (i.e; |z| is large). we know that |z2 + i| ≥ |z|2 − 1 ≥ 1 2 |z|2. ⇒ 1 |z2 + i| ≤ 2 |z|2 . ⇒ 1 |z2 + i|2 ≤ 4 |z|4 , int. j. anal. appl. 19 (5) (2021) 688 refer to the behaviour of |z2 + i| at ∞ with respect to the region. ⇒ |φ′(z)| = 4|z| |z2 + i|2 ≤ 16|z| |z|4 = 16|z|−3. (ii) and r = |z| ∼ 0 ⇒ |φ′(z)| = 4|z| |z2 + i|2 ⇒ |φ′(z)| = 4|z| |i|2 = 4|z|. ⇒ |φ′(z)| ≤   16|z| −3 ; |z| is large 4|z| ; |z| ∼ 0 ∫ π 2 0 ∫ ∞ 0 |φ′|p dxdy ≤ 4 ∫ π 2 0 ∫ 1 0 |z|p r drdθ + 16 ∫ π 2 0 ∫ ∞ 1 |z|−3p r drdθ. ∫ π 2 0 ∫ ∞ 0 |φ′|p dxdy = 2π [∫ 1 0 rp+1 dr ] + 8π [∫ ∞ 1 r−3p+1 dr ] . ⇒ ∫ π 2 0 ∫ ∞ 0 |φ′|p dxdy ≤ 2π [ rp+2 p + 2 ∣∣∣∣1 0 + 2π [ r−3p+2 −3p + 2 ∣∣∣∣∞ 1 (2.10) there are two definite integrals on the right-hand side of inequality (2.10). the first one is clearly finite, and the second one is: −3p + 2 < 0 ⇒ −3p < −2 ⇒ p > 2 3 . � theorem 2.6. let φ(z) be a conformal mapping defined on infinite sector w onto unit disk d. let φ(z) = (φ2 ◦φ1)(z) = z4 − i z4 + i : w → d such that φ1(z) = z 4 maps the infinite sector onto upper half plane h+ and φ2(w) = w−iw+i maps the upper half plane h+ onto unit disk d (see figure 6). then the integrability of the derivative of conformal mapping is: ∫ ∫ w |φ′(z)|p dxdy < ∞ ; for each p > 2 5 . int. j. anal. appl. 19 (5) (2021) 689 !"##!$# %&'()*+# ,-&'()*+# .*/0#1/23# 45678# figure 6. infinite sector w for the angle α = π 4 proof. given φ(z) = z4 − i z4 + i : w → d ⇒ φ′(z) = (z4 + i)(4z3) − (z4 − i)(4z3) (z4 + i)2 = i8z3 (z4 + i)2 ⇒ |φ′(z)| = 8|z|3 |z4 + i|2 . now, w-plane is an infinite sector. that is; r = |z|→ 0 −∞. • so,when r = |z|→∞ (that is; |z| be large). we know that |z4 + i| ≥ |z|4 − 1 ≥ 1 2 |z|4. ⇒ 1 |z4 + i| ≤ 2 |z|4 . ⇒ 1 |z4 + i|2 ≤ 4 |z|8 . this is referring to the behaviour of |z4 + i| at ∞ with respect to the region. ⇒ |φ′(z)| = 8|z|3 |z4 + i|2 ≤ 8|z|3 |z|8 = 32|z|−5. • and when r = |z| ∼ 0 ⇒ |φ′(z)| = 8|z|3 |z4 + i|2 ⇒ |φ′(z)| = 8|z|3 |i|2 = 8|z|3. ⇒ |φ′(z)| ≤   8|z| −5 ; |z| is large 8|z|3 ; |z| ∼ 0 int. j. anal. appl. 19 (5) (2021) 690∫ π 4 0 ∫ ∞ 0 |φ′|p dxdy ≤ 8 ∫ π 4 0 ∫ 1 0 r3pr drdθ + 8 ∫ π 4 0 ∫ ∞ 1 r−5pr drdθ = 2π [∫ 1 0 r3p+1 dr ] + 2π [∫ ∞ 1 r−5p+1 dr ] ⇒ ∫ π 4 0 ∫ ∞ 0 |φ′|p dxdy ≤ 2π [ r3p+2 3p + 2 |10 + 2π [ r−5p+2 −5p + 2 |∞1(2.11) the first term on the right -handside of (2.11) is finite, and the second one is: −5p + 2 < 0 ⇒ −5p < −2 ⇒ p > 2 5 . � theorem 2.7. let φ(z) be a conformal mapping defined on infinite sector w onto unit disk d. let φ(z) = (φ2 ◦φ1)(z) = zn − i zn + i : w → d such that φ1(z) = z n maps the infinite sector onto upper halfplane h+ where α is of the form α = π n for some integer n and φ2(w) = w−i w+i maps the upper halfplane h+ onto unit disk d (see figure 7). then the integrability of the derivative of conformal mapping is as follows:∫ ∫ w |φ′(z)|p dxdy < ∞ ; for each p > 2 3π α + 1 . !"##!$# %&'()*+# ,-&'()*+# .*/0#1/23# 4567*# figure 7. infinite sector w for the angle α = π n proof. let α = π n be the angle of the infinite sector w which is mapped by φ1 = z n onto upper half plane h+. one can write φ1 = z π α . we define the power function φ1 = z π α to be the multivalued function z π α = e π α log z ; z 6= 0. int. j. anal. appl. 19 (5) (2021) 691 ⇒ z π α = e π α log |z|+iargz = r π α ei π α θ e±i 2π2 α k various values of z π α are obtained from the principal value e π α log z by multiplying by the integral power (ei 2π2 α ) k of ei 2π2 α . let α = π n is a number for some integer n, then the integral powers ei 2π2 α k of ei 2π2 α are exactly the nth roots of unity, and the values of z π α are the n nth roots of z. ⇒ φ(z) = z π α − i z π α + i = r π α ei π α − i r π α ei π α + i φ(z) = r π α cos π α θ + ir π α sin π α θ − i r π α cos π α θ + ir π α sin π α θ + i . (2.12) simplify last equation (2.12) we get: φ(z) = r 2π α − 2ir π α cos π α θ − 1 r 2π α + 2r π α sin π α θ + 1 = r 2π α − 1 r 2π α + 2r π α sin π α θ + 1 + i −2r π α cos π α θ r 2π α + 2r π α sin π α θ + 1 . when θ = 0 or θ = 2π, it implies that: φ(z) = r 2π α − 1 r 2π α + 1 + i −2r π α r 2π α + 1 = r 2π α − 1 − 2ir π α r 2π α + 1 the derivative of φ(z) can be calculated: φ′(z) = (r 2π α + 1)[ 2π α r 2π α −1 − 2iπ α r π α −1] − (r 2π α − 1 − 2ir π α )[ 2π α r 2π α −1] (r 2π α + 1)2 = 2iπ α r 3π α −1 + 4π α r 4π α −1 − 2iπ α r π α −1 (r 2π α + 1)2 = 4π α r 4π α −1 + i2π α (r 3π α −1 −r π α −1) (r 2π α + 1)2 |φ′(z)| = ∣∣∣4πα r 4πα −1 + i2πα (r 3πα −1 −rπα−1)∣∣∣ (r 2π α + 1)2 |φ′(z)| = √ 16π2 α2 r 8π α −2 + 4π 2 α2 (r 3π α −1 −r π α −1)2 (r 2π α + 1)2 int. j. anal. appl. 19 (5) (2021) 692 again, w-plane is an infinite sector. that is; |z|→ 0 −∞. (i) in case r = |z|→∞ (that is; |z| be large). |φ′(z)| = √ 16π2 α2 |z| 8π α −2 + 4π 2 α2 (|z| 3π α −1 −|z| π α −1 )2 (|z| 2π α + 1)2 . we know that, |z| 2π α + 1 ≥ |z 2π α + 1| ≥ |z| 2π α − 1 ≥ 1 2 |z| 2π α ⇒ |z| 2π α + 1 ≥ 1 2 |z| 2π α ⇒ 1 (|z| 2π α + 1)2 ≤ 4 |z| 4π α . hence, |φ′(z)| = √ 16π2 α2 |z| 8π α −2 + 4π 2 α2 (|z| 3π α −1 −|z| π α −1 )2 (|z| 2π α + 1)2 ≤ 4 √ 16π2 α2 |z| 8π α −2 + 4π 2 α2 (|z| 3π α −1 −|z| π α −1 )2 |z| 4π α . |φ′(z)| ≤ 4 √ 16π2 α2 r 8π α −2 + 4π 2 α2 (r 3π α −1 −r π α −1)2 r 4π α = 4 √ 16π2 α2 r 8π α −2 + 4π 2 α2 r 2π α −2(r 2π α − 1)2 r 4π α = 4 √ 4π2 α2 r 2π α −2 [ 4r 6π α + (r 2π α − 1)2 ] r 4π α |φ′(z)| ≤ 4 √ 4π2 α2 r 2π α −2 [ 4r 6π α + (r 2π α − 1)2 ] r 4π α = 8π α r π α −1 √ 4 r 6π α + (r 2π α − 1)2 r 4π α = 8π α r −3π α −1 √ 4 r 6π α + (r 2π α − 1)2 (ii) in case r = |z| ∼ 0 |φ′(z)| = √ 16π2 α2 |z| 8π α −2 + 4π2 α2 (|z| 3π α −1 −|z| π α −1 )2. ⇒ |φ′(z)| = √ 16π2 α2 r 8π α −2 + 4π2 α2 (r 3π α −1 −r π α −1)2 = √ 16π2 α2 r 8π α −2 + 4π2 α2 r 2π α −2(r 2π α − 1)2 = √ 4π2 α2 r 2π α −2 [ 4r 6π α + (r 2π α − 1)2 ] = 2π α r π α −1 √ 4 r 6π α + (r 2π α − 1)2. int. j. anal. appl. 19 (5) (2021) 693 in the end, ⇒ |φ′(z)| ≤   8π α r −3π α −1 √ 4 r 6π α + (r 2π α − 1)2 ; |z| is large 2π α r π α −1 √ 4 r 6π α + (r 2π α − 1)2. ; |z| ∼ 0 ∫ π n 0 ∫ ∞ 0 |φ′(z)|pdxdy ≤ ∫ π n 0 ∫ 1 0 ( 2π α )p r( π α −1)p [ 4 r 6π α + (r 2π α − 1)2 ]p 2 .rdrdθ + ∫ π n 0 ∫ ∞ 1 ( 8π α )p r( −3π α −1)p [ 4 r 6π α + (r 2π α − 1)2 ]p 2 rdrdθ. ∫ π n 0 ∫ ∞ 0 |φ′(z)|pdxdy ≤ ( 2π α )p( π n ) ∫ 1 0 r( π α −1)p+1dr ∫ 1 0 [ 4 r 6π α + (r 2π α − 1)2 ]p 2 .rdr +( 8π α )p( π n ) ∫ ∞ 1 r( −3π α −1)p+1dr ∫ ∞ 1 [ 4 r 6π α + (r 2π α − 1)2 ]p 2 dr. such that, we have four terms . the first, second and fourth terms on the right -hand-side of (2.4) becomes finite only when ( −3π α − 1)p + 2 ≤ 0 ⇒ ( −3π α − 1)p ≤−2 ⇒ −( 3π α + 1)p ≤−2 ⇒ ( 3π α + 1)p ≥ 2 ⇒ p > 2 3π α + 1 . � acknowledgment: the author would like to express a deep thanks and gratitude to department of mathematics , college of science, mustansiriyah university for deep supporting to appear this research paper as it is now. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. s. alhily, higher integrability of the gradient of conforaml maps, phd thesis, university of sussex, 2013. [2] s. s. alhily and deepmala, boundary behaviour of holomorphic functions on the cardioid domain with some applications, bol. soc. paran. mat. 38(7) (2020), 203-218. [3] p. l. duren, univalent functions, third ed., springer-verlag, new york, 1983. [4] p. h. difrancesco, p. mathieu, d. sénéchal, conformal field theory, first ed., springer-verlag berlin heidelberg, new york, 1997. int. j. anal. appl. 19 (5) (2021) 694 [5] t. a. driscoll, l. n. trefethen, schwarz-christoffel mapping. cambridge monographs on applied and computational mathematics, vol. 8. cambridge university press, cambridge, 2002. [6] s. g. krantz, complex analysis: the geometric viewpoint, vol.23, mathematical association of america, washington, d.c., 1990. [7] v.n. mishra and g. tomar, exiistance of wandering and periodic domain in given angular region, math. slovaca, 70(4) (2020), 839-848. [8] ch. pommerenke, univalent functions, first ed., vandenhoeck and ruprecht, göttingen, germany, 1975. [9] ch. pommerenke, boundary behaviour of conformal mappings, second ed., springer-verlag berlin heidelberg, 1992. [10] e. m. stein, r. shakarchi, complex analysis. princeton university press, princeton, 2003. [11] h. a. saleh, s. s. alhily, growth and bounded solution of second-order of complex differential equations through of coefficient function, iop conf. ser.: mater. sci. eng. 871 (2020) 012057. [12] g. tomar and v.n. mishra, maximum term of transcendental entire function and spider’s web, math. slovaca, 70(4) (2020), 81-86. 1. introduction 2. main results references int. j. anal. appl. (2022), 20:43 generalized stability additive λ-functional inequalities with 3k-variable in α-homogeneous f-spaces ly van an∗ faculty of mathematics teacher education, tay ninh university, ninh trung, ninh son, tay ninh province, vietnam ∗corresponding author: lyvanan145@gmail.com abstract. in this paper, we study to solve two additive λ-functional inequalities with 3k-variables in α-homogeneous f spaces. then we will show that the solutions of the first and second inequalities are additive mappings. 1. introduction let x and y be a normed spaces on the same field k, and f : x → y. we use the notation ∥∥ ·∥∥ for all the norm on both x and y. in this paper, we investisgate some additive λ-functional inequality in α-homogeneous f-spaces. in fact, when x is a α1-homogeneous f-spaces and that y is a α2-homogeneous f-spaces we solve and prove the hyers-ulam-rassias type stability of two forllowing additive α-functional inequality. ∥∥∥∥∥f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj ))∥∥∥∥∥ y (1.1) received: jun. 10, 2022. 2010 mathematics subject classification. 39b62, 39b72, 39b52. key words and phrases. additive β-functional inequality; α-homogeneous f-space; hyers-ulam stability. https://doi.org/10.28924/2291-8639-20-2022-43 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-43 2 int. j. anal. appl. (2022), 20:43 and∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y (1.2) where λ is a fixed complex number with ∣∣λ∣∣ < 1, α1,α2 ∈ r+,α1,α2 ≤ 1 and m is a fixed integer with m > 1. the hyers-ulam stability was first investigated for functional equation of ulam in [19] concerning the stability of group homomorphisms. the functional equation f (x + y) = f (x) + f (y) is called the cauchy equation. in particular, every solution of the cauchy equation is said to be an additive mapping. the hyers [9] gave firts affirmative partial answer to the equation of ulam in banach spaces. after that, hyers’theorem was generalized by aoki [1] additive mappings and by rassias [18] for linear mappings considering an unbouned cauchy diffrence. ageneralization of the rassias theorem was obtained by gǎvruta [6] by replacing the unbounded cauchy difference by a general control function in the spirit of rassias’ approach. the hyers-ulam stability for functional inequalities have been investigated such as in [5], [10], [13], [16], [17], [18]. gilány showed that is if satisfies the functional inequality∥∥2f(x) + 2f(y)− f(x −y)∥∥ ≤ ∥∥f(x + y)∥∥ (1.3) then f satisfies the jordan-von newman functional equation 2f ( x ) + 2f ( y ) = f ( x + y ) + f ( x −y ) (1.4) . gilányi [8] and fechner [5] proved the hyers-ulam stability of the functional inequality (1.3). next chookil [16] proved the of additive β-functional inequalities in non-archimedean banach spaces and in complex banach spaces, and harin leea [11] proved the hyers-ulam stability of additive βfunctional inequalities in ρ-homogeneous f space. recently, the author has studied the additive inequalities of mathematicians around the world, on spaces complex banach spaces , non-archimedan banach spaces or additive β-functional inequalities in p-homogeneous f-space.. so in this paper, we solve and proved the hyers-ulam stability for two ff-functional inequalities (1.1)-(1.2), ie the α-functional inequalities with 3k-variables. under suitable assumptions on spaces x and y, we will prove that the mappings satisfying the α-functional inequatilies (1.1) or (1.2). thus, the results in this paper are generalization of those in [2], [11] for α-functional inequatilies with int. j. anal. appl. (2022), 20:43 3 3kvariables. the paper is organized as followns: in section preliminarier we remind a basic property such as we only redefine the solution definition of the equation of the additive function and f∗-space . section 3: is devoted to prove the hyers-ulam stability of the addive λfunctional inequalities (1.1) when when x is a α1-homogeneous f-spaces and that y is a α2-homogeneous f-spaces. section 4: is devoted to prove the hyers-ulam stability of the addive λfunctional inequalities (1.2) when when x is a α1-homogeneous f-spaces and that y is a α2-homogeneous f-spaces. 2. preliminaries 2.1. f∗spaces. definition 2.1. let x be a ( complex ) linear space. a nonnegative valued function ∥∥ ·∥∥ is an f-norm if it satisfies the following conditions: (1) ∥∥∥x∥∥∥ = 0 if and only if x = 0; (2) ∥∥∥λx∥∥∥ = ∥∥∥x∥∥∥ for all x ∈ x and all λ with ∣∣∣λ∣∣∣ = 1; (3) ∥∥∥x + y∥∥∥ ≤ ∥∥∥x∥∥∥ + ∥∥∥y∥∥∥ for all x,y ∈ x; (4) ∥∥∥λnx∥∥∥ → 0, λn → 0; (5) ∥∥∥λnx∥∥∥ → 0, xn → 0. then ( x, ∥∥∥ · ∥∥∥ ) is called an f∗-space. an f-space is a complete f∗-space. an f-norm is called β-homgeneous ( β > 0 ) if ∥∥∥tx∥∥∥ = ∣∣∣t∣∣∣β∥∥∥x∥∥∥ for all x ∈ x and for all t ∈ c and (x,∥∥∥ ·∥∥∥) is called α-homogeneous f-space. 2.2. solutions of the inequalities. the functional equation f (x + y) = f (x) + f (y) is called the cauchuy equation. in particular, every solution of the cauchuy equation is said to be an additive mapping. 4 int. j. anal. appl. (2022), 20:43 3. hyers-ulam-rassias stability additive λ-functional inequalities (1.1) in α-homogeneous f-spaces now, we first study the solutions of (1.1). note that for these inequalities, when x is a α1homogeneous f-spaces and that y is a α2-homogeneous f-spaces. under this setting, we can show that the mapping satisfying (1.1) is additive. these results are give in the following. lemma 3.1. let m ∈n and a mapping f : x → y satilies ∥∥∥∥∥f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj ))∥∥∥∥∥ y (3.1) for all xj,yj,zj ∈ x for j = 1 → n, then f : x → y is additive proof. assume that f : g → y satisfies (3.1). we replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0, 0, ..., 0 ) in (3.1), we have ∥∥∥2kf(0)∥∥∥ ≤ ∥∥∥λ(2k − 1)f (0)∥∥∥ y ≤ 0 therefore (∣∣∣2k∣∣∣α2 − ∣∣∣λ(2k − 1)∣∣∣α2)∥∥∥f(0)∥∥∥ y ≤ 0 so f ( 0 ) = 0. replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0,z, 0, ..., 0) ) , in (3.1), we get ∥∥∥f(−z)− f(−z)∥∥∥ y ≤ 0 and so f is an odd mapping. replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( x1, ...,xk,y1, ...,yk,m · x1+y12k −v1, ...,m · xk+yk 2k −vk ) in (3.1), we have ∥∥∥f( k∑ j=1 xj + yj 2k + k∑ j=1 vj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( vj )∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 vj ) − k∑ j=1 f ( m xj + yj 2k −vj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y (3.2) int. j. anal. appl. (2022), 20:43 5 for all x1, ...,xk,y1, ...,yk,m x1+y1 2k −v1, ...,mxk+yk2k −vk ∈ g. from (3.1) and (3.2) we infer that∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y ≤ ∥∥∥∥∥λ2 ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj ))∥∥∥∥∥ y (3.3) and so f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) = k∑ j=1 f (xj + yj 2k ) + k∑ j=1 f ( zj ) for all xj,yj,zj ∈ g for j = 1 → n, as we expected. � theorem 3.2. let r > α2 α1 ,m ∈ z,m > 1, θ be nonngative real number, and let f : x → y be a mapping such that ∥∥∥∥∥f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj ))∥∥∥∥∥ y + θ ( k∑ j=1 ∥∥xj∥∥rx + k∑ j=1 ∥∥yj∥∥rx + k∑ j=1 ∥∥zj∥∥rx) (3.4) for all xj,yj,zj ∈ x for all j = 1 → n. then there exists a unique mapping φ : x → y such that∥∥∥f(x)−h(x)∥∥∥ y ≤ ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)(mα1r −mα2)θ∥∥x∥∥rx. (3.5) for all x ∈ x proof. assume that f : x → y satisfies (3.4). replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0, 0, ..., 0 ) in (3.4), we have ∥∥∥2kf(0)∥∥∥ y ≤ ∥∥∥λ(2k − 1)f (0)∥∥∥ y ≤ 0 therefore (∣∣∣2k∣∣∣α2 − ∣∣∣λ(2k − 1)∣∣∣α2)∥∥∥f(0)∥∥∥ y ≤ 0 sof ( 0 ) = 0. next we replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( kx, 0, ..., 0,kx, 0, ..., 0, 0, ..., 0 ) in (3.4), we get 6 int. j. anal. appl. (2022), 20:43 ∥∥∥f((m + 1)x)− f(mx)− f(x)∥∥∥ y ≤ 2kα1rθ ∥∥∥x∥∥∥r x (3.6) for all x ∈ x. thus for q ∈n, we replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( kx, 0, ..., 0,kx, 0, ..., 0,qx, 0, ..., 0 ) in (3.4), we have∥∥∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x) ∥∥∥∥∥ y ≤ ∥∥∥∥∥λ(f((q + 1)x)− f(qx)− f(x) ∥∥∥∥∥ y + θ ( 2kα1r + qα1r )∥∥∥x∥∥∥r x (3.7) for all x ∈ x. for (3.6) and (3.7) m−1∑ q=1 ∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x)∥∥∥ y ≤ m−1∑ q=1 ∥∥∥λ(f((q + 1)x)− f(qx)− f(x)∥∥∥ y + θ (m−1∑ q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) (3.8) for all x ∈ x. from (3.7) and (3.8) and triangle inequality, we have ( 1 − ∣∣λ∣∣α2)∥∥∥f(mx)−mf(x)∥∥∥ y = ( 1 − ∣∣λ∣∣α2)m−1∑ q=1 ∥∥∥f((q + 1)x)− f(qx)− f(x)∥∥∥ y ≤ m−1∑ q=1 ( 1 − ∣∣λ∣∣α2)∥∥∥(f((q + 1)x)− f(qx)− f(x)∥∥∥ y ≤ m−1∑ q=1 ∥∥∥f((q + 1)x)− f(qx)− f(x)∥∥∥ y − m−1∑ q=1 ∥∥∥λ(f((q + 1)x)− f(qx)− f(x))∥∥∥ y ≤ θ (m−1∑ q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) (3.9) for all x ∈ x. from m−1∑ q=1 ∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x)∥∥∥ y = m−1∑ q=1 ∥∥∥(f((q + 1)x)− f(qx)− f(x)∥∥∥ y int. j. anal. appl. (2022), 20:43 7 since ∣∣λ∣∣ < 1, the mapping f satisfies the inequalities ∥∥∥f(mx)−mf(x)∥∥∥ y ≤ θ (∑m−1 q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) ( 1 − ∣∣λ∣∣α2) for all x ∈ x. therefore ∥∥∥f(x)−mf( x m )∥∥∥ y ≤ θ (∑m−1 q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) ( 1 − ∣∣λ∣∣α2)mα1r (3.10) for all x ∈ x. so ∥∥∥mlf( x ml ) −mpf ( x mp )∥∥∥ y ≤ p−1∑ j=l ∥∥∥mjf( x mj ) −mj+1f ( x mj+1 )∥∥∥ y ≤ θ (∑m−1 q=1 ( 2kα1r + qα1r )) ( 1 − ∣∣λ∣∣)mα1r p−1∑ j=l mα2j mα1rj ∥∥∥x∥∥∥r x (3.11) for all nonnegative integers p, l with p > l and all x ∈ x. it follows from (3.11) that the sequence{ mnf ( x mn )} is a cauchy sequence for all x ∈ x. since y is complete, the sequence {mnf ( x mn )} coverges. so one can define the mapping φ : x → y by φ ( x ) := limn→∞m nf ( x mn ) for all x ∈ x. moreover, letting l = 0 and passing the limit m →∞ in (3.11), we get (3.5). it follows from (3.4) that∥∥∥∥∥φ ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 φ ( m xj + yj 2k −zj ) − k∑ j=1 φ (xj + yj 2k )∥∥∥∥∥ y = lim n→∞ ∥∥∥∥∥mn ( f ((m + 1) mn k∑ j=1 xj + yj 2k − 1 mn k∑ j=1 zj ) − k∑ j=1 f ( m mn xj + yj 2k − 1 mn zj ) − k∑ j=1 f ( 1 mn xj + yj 2k ))∥∥∥∥∥ y ≤ lim n→∞ ∥∥∥∥∥mnλ (( f ( 1 mn k∑ j=1 xj + yj 2k + 1 mn k∑ j=1 zj ) − k∑ j=1 f ( 1 mn xj + yj 2k ) − k∑ j=1 f ( 1 mn zj ))∥∥∥∥∥ y + lim n→∞ mα2n mα1nr θ( k∑ j=1 ‖xj‖rx + k∑ j=1 ‖yj‖rx + k∑ j=1 ‖zj‖rx) = ∣∣λ∣∣α2 ∥∥∥∥∥φ ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 φ (xj + yj 2k ) − k∑ j=1 φ ( zj )∥∥∥∥∥ y (3.12) 8 int. j. anal. appl. (2022), 20:43 for all xj,yj,zj ∈ x for all j = 1 → n.∥∥∥∥∥φ ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 φ ( m xj + yj 2k −zj ) − k∑ j=1 φ (xj + yj 2k )∥∥∥∥∥ y ≤ ∣∣λ∣∣α2 ∥∥∥∥∥ y φ ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 φ (xj + yj 2k ) − k∑ j=1 φ ( zj )∥∥∥∥∥ y for all xj,yj,zj ∈ x for all j = 1 → n. so by lemma 3.1 it follows that the mapping φ : x → y is additive. now we need to prove uniqueness, suppose φ′ : x → y is also an additive mapping that satisfies (3.5). then we have ∥∥∥φ(x)−φ′(x)∥∥∥ y = mα2n‖φ ( x mn ) −φ′ ( x mn ) ‖y ≤ mα2n (∥∥∥φ( x mn ) − f ( x mn )∥∥∥ y + ‖φ′ ( x mn ) − f ( x mn )∥∥∥ y ) ≤ 2.mα2n · ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)mα1nr (mα1r −mα2)θ∥∥x∥∥rx (3.13) which tends to zero as n →∞ for all x ∈ x. so we can conclude that φ ( x ) = φ′ ( x ) for all x ∈ x.this proves thus the mapping φ : x → y is a unique mapping satisfying(3.5) as we expected. � theorem 3.3. let r < α2 α1 ,m ∈ z,m > 1, θ be nonngative real number, and let f : x → y be a mapping such that ∥∥∥∥∥f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj ))∥∥∥∥∥ y + θ ( k∑ j=1 ∥∥xj∥∥rx + k∑ j=1 ∥∥yj∥∥rx + k∑ j=1 ∥∥zj∥∥rx) (3.14) for all xj,yj,zj ∈ x for all j = 1 → n. then there exists a unique mapping φ : x → y such that ∥∥∥f(x)−φ(x)∥∥∥ y ≤ ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)(mα2 −mα1r)θ∥∥x∥∥rx. (3.15) for all x ∈ x. the rest of the proof is similar to the proof of theorem 3.2. int. j. anal. appl. (2022), 20:43 9 4. hyers-ulam-rassias stability additive λ-functional inequalities (1.2) in α-homogeneous f-spaces additive β-functional inequality in complex banach space now, we study the solutions of (1.2). note that for these inequalities, when x is a α1-homogeneous f-spaces and that y is a α2homogeneous f-spaces . under this setting, we can show that the mapping satisfying (1.2) is additive. these results are give in the following. lemma 4.1. let m ∈n and a mapping f : y → y satisfies ∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y (4.1) for all xj,yj,zj ∈ x for j = 1 → n, then f : x → y is additive proof. assume that f : x → y satisfies (4.1). replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0, 0, ..., 0 ) in (4.1), we have ∥∥∥(2k − 1)f(0)∥∥∥ y ≤ ∥∥∥kλf (0)∥∥∥ y ≤ 0 (∣∣∣2k − 1∣∣∣α2 − ∣∣∣λk∣∣∣α2)∥∥∥f(0)∥∥∥ y ≤ 0 so f ( 0 ) = 0. replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0,−z, 0, ..., 0) ) , in (4.1), we get ∥∥∥f(−z)− f(−z)∥∥∥ y ≤ 0 and so f is an odd mapping. replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( x1, ...,xk,y1, ...,yk,m · x1+y12k −v1, ...,m · xk+yk 2k −vk ) in (4.1), we have ∥∥∥f((m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 vj ) − k∑ j=1 f ( m xj + yj 2k −vj ) − k∑ j=1 f (xj + yj 2k )∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 vj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( vj ))∥∥∥∥∥ y (4.2) 10 int. j. anal. appl. (2022), 20:43 for all x1, ...,xk,y1, ...,yk,m x1+y1 2k −v1, ...,mxk+yk2k −vk ∈ g. from (4.1) and (4.2) we infer that∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 vj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( vj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 vj ) − k∑ j=1 f ( m xj + yj 2k −vj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y ≤ ∥∥∥∥∥λ2 ( f ( k∑ j=1 xj + yj 2k + k∑ j=1 vj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( vj ))∥∥∥∥∥ y (4.3) and so f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) = k∑ j=1 f (xj + yj 2k ) + k∑ j=1 f ( zj ) for all xj,yj,zj ∈ g for j = 1 → n, as we expected. � theorem 4.2. let r > α2 α1 ,m ∈ z,m > 1, θ be nonngative real number, and let f : x → y be a mapping such that ∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y + θ ( k∑ j=1 ∥∥xj∥∥rx + k∑ j=1 ∥∥yj∥∥rx + k∑ j=1 ∥∥zj∥∥rx) (4.4) for all xj,yj,zj ∈ x for all j = 1 → n. then there exists a unique mapping φ : x → y such that∥∥∥f(x)−h(x)∥∥∥ y ≤ ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)(mα1r −mα2)θ∥∥x∥∥rx. (4.5) for all x ∈ x proof. assume that f : x → y satisfies (4.4). replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( 0, ..., 0, 0, ..., 0, 0, ..., 0 ) in (4.4), we have e ∥∥∥2kf(0)∥∥∥ y ≤ ∥∥∥λ(2k − 1)f (0)∥∥∥ y ≤ 0 therefore (∣∣∣2k∣∣∣α2 − ∣∣∣λ(2k − 1)∣∣∣α2)∥∥∥f(0)∥∥∥ y ≤ 0 sof ( 0 ) = 0. next we replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( kx, 0, ..., 0,kx, 0, ..., 0, 0, ..., 0 ) in (4.4), we get int. j. anal. appl. (2022), 20:43 11 ∥∥∥f((m + 1)x)− f(mx)− f(x)∥∥∥ y ≤ 2kα1rθ ∥∥∥x∥∥∥r x (4.6) for all x ∈ x. thus for q ∈n, we replacing ( x1, ...,xk,y1, ...,yk,z1, ...,zk ) by ( kx, 0, ..., 0,kx, 0, ..., 0,qx, 0, ..., 0 ) in (4.4), we have∥∥∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x) ∥∥∥∥∥ y ≤ ∥∥∥∥∥λ(f((q + 1)x)− f(qx)− f(x) ∥∥∥∥∥ y + θ ( 2kα1r + qα1r )∥∥∥x∥∥∥r x (4.7) for all x ∈ x. for (4.6) and (4.7) m−1∑ q=1 ∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x)∥∥∥ y ≤ m−1∑ q=1 ∥∥∥λ(f((q + 1)x)− f(qx)− f(x)∥∥∥ y + θ (m−1∑ q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) (4.8) for all x ∈ x. from (4.7) and (4.8) and triangle inequality, we have ( 1 − ∣∣λ∣∣α2)∥∥∥f(mx)−mf(x)∥∥∥ y = ( 1 − ∣∣λ∣∣α2)m−1∑ q=1 ∥∥∥f((q + 1)x)− f(qx)− f(x)∥∥∥ y ≤ m−1∑ q=1 ( 1 − ∣∣λ∣∣α2)∥∥∥(f((q + 1)x)− f(qx)− f(x)∥∥∥ y ≤ m−1∑ q=1 ∥∥∥f((q + 1)x)− f(qx)− f(x)∥∥∥ y − m−1∑ q=1 ∥∥∥λ(f((q + 1)x)− f(qx)− f(x))∥∥∥ y ≤ θ (m−1∑ q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) (4.9) for all x ∈ x. from m−1∑ q=1 ∥∥∥f((m−q + 1)x)− f((m−q)x)− f(x)∥∥∥ y = m−1∑ q=1 ∥∥∥(f((q + 1)x)− f(qx)− f(x)∥∥∥ y 12 int. j. anal. appl. (2022), 20:43 since ∣∣λ∣∣ < 1, the mapping f satisfies the inequalities ∥∥∥f(mx)−mf(x)∥∥∥ ≤ θ (∑m−1 q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) ( 1 − ∣∣λ∣∣α2) for all x ∈ x. therefore ∥∥∥f(x)−mf( x m )∥∥∥ y ≤ θ (∑m−1 q=1 ( 2kα1r + qα1r )∥∥∥x∥∥∥r x ) ( 1 − ∣∣λ∣∣α2)mα1r (4.10) for all x ∈ x. so ∥∥∥mlf( x ml ) −mpf ( x mp )∥∥∥ y ≤ p−1∑ j=l ∥∥∥mjf( x mj ) −mj+1f ( x mj+1 )∥∥∥ y ≤ θ (∑m−1 q=1 ( 2kr + qr )) ( 1 − ∣∣λ∣∣α2)mα1r p−1∑ j=l mα2j mα1rj ∥∥∥x∥∥∥r x (4.11) for all nonnegative integers p, l with p > l and all x ∈ x. it follows from (4.11) that the sequence{ mnf ( x mn )} is a cauchy sequence for all x ∈ x. since y is complete, the sequence {mnf ( x mn )} coverges. so one can define the mapping φ : x → y by φ ( x ) := limn→∞m nf ( x mn ) for all x ∈ x. moreover, letting l = 0 and passing the limit m →∞ in (4.11), we get (4.5). it follows from (4.4) that∥∥∥∥∥φ ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) −φ ( k∑ j=1 xj + yj 2k ) − k∑ j=1 φ ( zj )∥∥∥∥∥ y = lim n→∞ ∥∥∥∥∥mn ( f ( 1 mn k∑ j=1 xj + yj 2k + 1 mn k∑ j=1 zj ) − f ( 1 mn k∑ j=1 xj + yj 2k ) − k∑ j=1 f ( 1 mn zj ))∥∥∥∥∥ y + lim n→∞ mα2n mα1nr θ( k∑ j=1 ‖xj‖rx + k∑ j=1 ‖yj‖rx + k∑ j=1 ‖zj‖rx) ≤ lim n→∞ ∣∣λ∣∣α2 ∥∥∥∥∥mn ( f ((m + 1) mn k∑ j=1 xj + yj 2k − 1 mn k∑ j=1 zj ) − k∑ j=1 f ( m mn xj + yj 2k − 1 mn zj ) − k∑ j=1 f ( 1 mn zj ))∥∥∥∥∥ y = ∣∣λ∣∣α2 ∥∥∥∥∥φ (( m + 1 ) k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 φ ( m xj + yj 2k −zj ) − k∑ j=1 φ ( zj )∥∥∥∥∥ y (4.12) int. j. anal. appl. (2022), 20:43 13 for all xj,yj,zj ∈ x for all j = 1 → n. so∥∥∥∥∥φ ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 φ (xj + yj 2k ) − k∑ j=1 φ ( zj )∥∥∥∥∥ y ≤ ∣∣λ∣∣α2 ∥∥∥∥∥φ ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 φ ( m xj + yj 2k −zj ) − k∑ j=1 φ (xj + yj 2k )∥∥∥∥∥ y for all xj,yj,zj ∈ x for all j = 1 → n. so by lemma 4.1 it follows that the mapping φ : x → y is additive. now we need to prove uniqueness ,suppose φ′ : x → y is also an additive mapping that satisfies (4.5) . then we have ∥∥∥φ(x)−φ′(x)∥∥∥ y = mα2n‖φ ( x mn ) −φ′ ( x mn ) ‖y ≤ mα2n (∥∥∥φ( x mn ) − f ( x mn )∥∥∥ y + ‖φ′ ( x mn ) − f ( x mn )∥∥∥ y ) ≤ 2.mα2n · ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)mnα1r (mα1r −mα2)θ∥∥x∥∥rx (4.13) which tends to zero as n →∞ for all x ∈ x. so we can conclude that φ ( x ) = φ′ ( x ) for all x ∈ x.this proves thus the mapping φ : x → y is a unique mapping satisfying(4.5) as we expected. � theorem 4.3. let r < α2 α1 ,m ∈ z,m > 1, θ be nonngative real number, and let f : x → y be a mapping such that ∥∥∥∥∥f ( k∑ j=1 xj + yj 2k + k∑ j=1 zj ) − k∑ j=1 f (xj + yj 2k ) − k∑ j=1 f ( zj )∥∥∥∥∥ y ≤ ∥∥∥∥∥λ ( f ( (m + 1) k∑ j=1 xj + yj 2k − k∑ j=1 zj ) − k∑ j=1 f ( m xj + yj 2k −zj ) − k∑ j=1 f (xj + yj 2k ))∥∥∥∥∥ y + θ ( k∑ j=1 ∥∥xj∥∥rx + k∑ j=1 ∥∥yj∥∥rx + k∑ j=1 ∥∥zj∥∥rx) (4.14) for all xj,yj,zj ∈ x for all j = 1 → n. then there exists a unique mapping φ : x → y such that ∥∥∥f(x)−h(x)∥∥∥ y ≤ ∑m−1 q=1 ( qα1r + 2kα1r )( 1 − ∣∣λ∣∣α2)(mα2 −mα1r )θ∥∥x∥∥rx. (4.15) for all x ∈ x. the proof is similar to theorem 4.2. 14 int. j. anal. appl. (2022), 20:43 5. conclusion in this paper, i have shown that the solutions of the first and second k − variable β-functional inequalities are additive mappings. the hyers-ulam stability for these given from theorems. these are the main results of the paper , which are the generalization of the results [2], [11]. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] t. aoki, on the stability of the linear transformation in banach spaces, j. math. soc. japan. 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064. [2] l.v. an, generalized hyers-ulam type stability of the 2k-variable additive β-functional inequalities and equations in complex banach spaces, int. j. math. trends technol. 66 (2020), 134–147. https://doi.org/10.14445/ 22315373/ijmtt-v66i7p518. [3] a. bahyrycz, m. piszczek, hyers stability of the jensen function equation, acta math. hungar. 142 (2014), 353-365. [4] m. balcerowski, on the functional equations related to a problem of z. boros and z. daróczy, acta math hung. 138 (2013), 329–340. https://doi.org/10.1007/s10474-012-0278-4. [5] w. fechner, stability of a functional inequality associated with the jordan-von neumann functional equation, aequ. math. 71 (2006), 149–161. https://doi.org/10.1007/s00010-005-2775-9. [6] p. gavruta, a generalization of the hyers-ulam-rassias stability of approximately additive mappings, j. math. anal. appl. 184 (1994), 431–436. https://doi.org/10.1006/jmaa.1994.1211. [7] a. gilányi, on a problem by k. nikodem, math. inequal. appl. 5 (2002), 707–710. https://doi.org/10.7153/ mia-05-71. [8] a. gilányi, eine zur parallelogrammgleichung äquivalente ungleichung, aequat. math. 62 (2001), 303–309. https: //doi.org/10.1007/pl00000156. [9] d.h. hyers, on the stability of the linear functional equation, proc. natl. acad. sci. u.s.a. 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222. [10] j.r. lee, c. park, d.y. shin, additive and quadratic functional in equalities in non-archimedean normed spaces, int. j. math. anal. 8 (2014), 1233-1247. https://doi.org/10.12988/ijma.2014.44113. [11] h. lee, j.y. cha, m.w. cho, et al. additive ρ-functional inequalities in β-homogeneous f-spaces, j. korean soc. math. educ. ser. b: pure appl. math. 23 (2016), 319–328. https://doi.org/10.7468/jksmeb.2016.23.3.319. [12] l. maligranda, tosio aoki (1910–1989). in: proceedings of the international symposium on banach and function spaces ii kitakyushu, japan, pp. 1–23 (2006). [13] c. park, y.s. cho, m.-h. han, functional inequalities associated with jordan-von neumann-type additive functional equations, j. inequal. appl. 2007 (2007), 41820. https://doi.org/10.1155/2007/41820. [14] w. prager, j. schwaiger, a system of two inhomogeneous linear functional equations, acta math. hung. 140 (2013), 377–406. https://doi.org/10.1007/s10474-013-0315-y. [15] c. park, functional inequalities in non-archimedean normed spaces, acta. math. sin.-english ser. 31 (2015), 353–366. https://doi.org/10.1007/s10114-015-4278-5. [16] j. rätz, on inequalities associated with the jordan-von neumann functional equation, aequat. math. 66 (2003), 191–200. https://doi.org/10.1007/s00010-003-2684-8. https://doi.org/10.2969/jmsj/00210064 https://doi.org/10.14445/22315373/ijmtt-v66i7p518 https://doi.org/10.14445/22315373/ijmtt-v66i7p518 https://doi.org/10.1007/s10474-012-0278-4 https://doi.org/10.1007/s00010-005-2775-9 https://doi.org/10.1006/jmaa.1994.1211 https://doi.org/10.7153/mia-05-71 https://doi.org/10.7153/mia-05-71 https://doi.org/10.1007/pl00000156 https://doi.org/10.1007/pl00000156 https://doi.org/10.1073/pnas.27.4.222 https://doi.org/10.12988/ijma.2014.44113 https://doi.org/10.7468/jksmeb.2016.23.3.319 https://doi.org/10.1155/2007/41820 https://doi.org/10.1007/s10114-015-4278-5 https://doi.org/10.1007/s00010-003-2684-8 int. j. anal. appl. (2022), 20:43 15 [17] s. rolewicz, metric linear spaces, pwn-polish scientific publishers, warsaw, 1972. [18] t.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72 (1978), 297–300. https://doi.org/10.1090/s0002-9939-1978-0507327-1. [19] s.m. ulam, a collection of mathematical problems, volume 8, interscience publishers, new york, 1960. https://doi.org/10.1090/s0002-9939-1978-0507327-1 1. introduction 2. preliminaries 2.1. f*spaces. 2.2. solutions of the inequalities. 3. hyers-ulam-rassias stability additive -functional inequalities (1.1) in -homogeneous f-spaces 4. hyers-ulam-rassias stability additive -functional inequalities (1.2) in -homogeneous f-spaces 5. conclusion references international journal of analysis and applications volume 18, number 3 (2020), 381-395 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-381 on double shehu transform and its properties with applications suliman alfaqeih∗, emine misirli ege university, faculty of science, department of mathematics, 35100, turkey ∗corresponding author: alfaqeihsuliman@gmail.com abstract. in the current paper, we have generalized the concept of one dimensional shehu transform into two dimensional shehu transform namely, double shehu transform (dht). further, we have established some main properties and theorems related to the (dht). to show the efficiency, high accuracy and applicability of the proposed transform, we have implemented the new transform to solve integral equations and partial differential equations. 1. introduction many problems in the fields of most applied science and engineering encounter double integral equations or partial differential equations describing the physical phenomena [20–22]. solving such equations using single transforms is more difficult than using the double transforms. in the past few decades, integral transforms have been the focus of many authors due to their huge appearance in various applications in modern sciences and engineering, in the current years, a very extensive literature on integral transforms of a function of one but there is a little work available on double integral transform. consequently, great attention has been given to deal with the double integral transform. among the solution methods, integral transform methods are rather and popular, hence in the literature, there are many different types of integral transforms such as fourier transform [1] laplace [2, 3] transform, sumudu transform [4–6,19], natural transform [7], ezaki transform [8], and so on. these kinds of transforms received february 16th, 2020; accepted march 13th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 44a10, 44a20, 44a30, 44a35. key words and phrases. partial differential equation (pde); telegraph equation; integral equations; sumudu transform (st); laplace transforms (lt); shehu transform (ht). ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 381 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-381 int. j. anal. appl. 18 (3) (2020) 382 have a wide variety of applications in various areas in applied physics, applied mathematics, statistics, engineering and in most of other sciences [9, 10]. shehu transform (ht) of single variable [11], is a new transform which was recently introduced by shehu maitama and weidong zhao in 2019. (ht) is a generalization of laplace and sumudu transforms. this transform is used to solve both ordinary and partial differential equations. after the appearance of the (ht), many authors applied this transform to solve partial differential equations including ordinary and fractional, for instance, see [12–15, 20]. the main objective of this paper is to extend the one dimensional shehu integral transform to two dimensional shehu integral transform namely double shehu transform and to get the solution of initial and boundary value problems in different areas of real life science and engineering. this work is organized as follows. in section 2, we present some notations about laplace, sumudu, and single shehu transforms. we then introduce the definition of double shehu transform and its inverse with examples, and we prove the existence and uniqueness theorems of the new transform in section 3. in section 4, we discuss some properties and theorems related to the double shehu transform. in section 5, we implement the double shehu transform method to some examples of double integral equations with convolution and partial differential equations. section 6 is for the conclusions of this paper. 2. preliminaries in this section, we present some basic notations about the laplace, sumudu and the shehu transforms. definition 2.1.let g : (0,∞) → < be a real valued function. the single laplace transform of g is defined by: g(p) = l (g (x) : p) = ∫ ∞ 0 e−pxg (x) dx, p ∈ c. (2.1) definition 2.2. [16] let g(x,t), x, t > 0 be a real valued function. the double laplace transform of g is defined by: lxlt (g (x,t) : (p,q)) = g(p,q) = ∫ ∞ 0 ∫ ∞ 0 e−(px+qt)g (x,t) dxdt, p, q ∈ c. (2.2) definition 2.3. let g : (0,∞) →< be a real valued function. the single sumudu transform of g is defined by: g(p) = s (g (x) : p) = ∫ ∞ 0 e−xg (px) dx, p ∈ c. (2.3) definition 2.4. [17] let g(x,t) be a real valued function. the double sumudu transform of g is defined by: int. j. anal. appl. 18 (3) (2020) 383 sxst (g(x,τ) : (p,q)) = g(p,q) = ∫ ∞ 0 ∫ ∞ 0 e−x−tg(px,qt)dxdt. (2.4) definition 2.5. [11] the single shehu transforms (st) of a real valued function g(x,t) with respect to the variables x and t respectively, are defined by: hx (g (x,t) : p,u) = ∫ ∞ 0 e− px u g (x,t) dx, (2.5) ht (g (x,t) : q,v) = ∫ ∞ 0 e− qt v g (x,t) dt. (2.6) definition 2.6. [11] the inverse shehu transform g(x,t) = h−2xt [ h2xt (g (x,t)) ] is defined by the complex integral formula g(x,t) = h−2xt [ h2xt (g (x,t)) ] = 1 2πi ∫ α+i∞ α−i∞ 1 u e( px u )dp 1 2πi ∫ γ−i∞ γ−i∞ 1 v e( qt v )hxt (g (x,t)) dq. (2.7) 3. double shehu transforms (dht) in this section, we introduce the definitions of (dht) and the inverse of (dht). definition 3.1. the (dht) of the function g(x,t) is defined by the double integral as : h2xt (g (x,t)) = g [(p,q) , (u,v)] = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x,t) dxdt, (3.1) on the set of functions ω = { g (x,t) : ∃k,λ1, λ2 > 0, |g (x,t)| < k exp ( |x + t| λ2j ) , j = 1, 2 and (x,t) ∈ r2+ } , provided that the integral exist. definition 3.2. the inverse double shehu transform g(x,t) = h−2xt [ h2xt (g (x,t)) ] is defined by the complex double integral formula g(x,t) = h−2xt [ h2xt (g (x,t)) ] = 1 2πi ∫ α+i∞ α−i∞ 1 u e( px u )dp 1 2πi ∫ γ−i∞ γ−i∞ 1 v e( qt v )h2xt (g (x,t)) dq. (3.2) 3.1. existence and uniqueness of (dht). theorem 3.1. let g(x,t) be a continuous function on every finite intervals (0,x) and (0,t), and of exponential order, that is for some a,b ∈< sup x,t>0 |g(x,t)| e(ax+bt) < ∞ then the (dht) of g(x,t) exists. int. j. anal. appl. 18 (3) (2020) 384 proof. using definition of dht, we have ∣∣h2xt (g (x,t))∣∣ = ∣∣∣∣ ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x,t) dxdt ∣∣∣∣ ≤ ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v ) |g (x,t)|dxdt ≤ k ∫ ∞ 0 ∫ ∞ 0 e−( px u −ax)−( qtv −bt)dxdt = k ∫ ∞ 0 e−( p u −a)xdx ∫ ∞ 0 e−( q v −b)tdt = kuv (p−au) (q − bv) . theorem 3.2. let h(x,t)and l(x,t) be continuous functions and having the double shehu transformsh2xt (h (x,t)) and h 2 xt (l (x,t)) respectively. if h 2 xt (h (x,t)) = h 2 xt (l (x,t)) then h(x,t) = l(x,t). proof. assume α, and γ to be sufficiently large, then since g(x,t) = h−2xt [ h2xt (g (x,t)) ] = 1 2πi ∫ α+i∞ α−i∞ 1 u e( px u )dp 1 2πi ∫ γ−i∞ γ−i∞ 1 v e( qt v )h2xt (g (x,t)) dq, we deduce that h (x,t) = 1 2πi ∫ α+i∞ α−i∞ 1 u e( px u )dp 1 2πi ∫ γ−i∞ γ−i∞ 1 v e( qt v )h2xt (h (x,t)) dq = 1 2πi ∫ α+i∞ α−i∞ 1 u e( px u )dp 1 2πi ∫ γ−i∞ γ−i∞ 1 v e( qt v )h2xt (l (x,t)) dq = l (x,t) , and the theorem is established. 3.2. double shehu transform of some functions: (1) if g(x,t) = 1, then h2xt (g (x,t)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )dxdt = uv pq . (2) if g(x,t) = e(ax+bt), then h2xt (g (x,t)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )e(ax+bt)dxdt = ∫ ∞ 0 ∫ ∞ 0 e−( (p−au)x u + (q−bv)t v )dxdt = uv (p−au) (q − bv) . int. j. anal. appl. 18 (3) (2020) 385 (3) if g(x,t) = ei(ax+bt), then h2xt (g (x,t)) − uv (p−aui) (q − bvi) = uv (pq + abuv) + (bpv + aqu) uvi (p2 + a2u2) (q2 + b2v2) . as a consequence of property (3) h2xt (cos (ax + bt)) = uv(pq+abuv) (p2+a2u2) , h2xt (sin (ax + bt)) = uv(bpv+aqu) (q2+b2v2) . (4) if g(x,t) = cosh (ax + bt) , then h2xt (g (x,t)) = 1 2 [ h2xt ( e(ax+bt) ) + h2xt ( e−(ax+bt) )] = 1 2 [ uv (p−au) (q − bv) + uv (p + au) (q + bv) ] . similarly, h2xt (sinh (ax + bt)) = 1 2 [ uv (p−au) (q − bv) − uv (p + au) (q + bv) ] . (5) if g(x,t) = xntm, n,m = 0, 1, 2, · · · , then h2xt (g (x,t)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )xntmdxdt = ∫ ∞ 0 e−( px u )xndx ∫ ∞ 0 e−( qt v )tmdxdt, = n!m! ( u p )n+1 ( v q )m+1 . (6) if g(x,t) = xαtγ, α ≥−1,γ ≥−1 then h2xt (g (x,t)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )xαtγdxdt = ∫ ∞ 0 e−( px u )xαdx ∫ ∞ 0 e−( qt v )tγdt by letting y = (px u ) , and z = ( qt v ) = ( u p )α+1 ∫ ∞ 0 e−yyαdy ( v q )γ+1 ∫ ∞ 0 e−zzγdz = γ (α + 1) ( u p )α+1 γ (γ + 1) ( v q )γ+1 . where, γ (.) is the euler gamma function. int. j. anal. appl. 18 (3) (2020) 386 4. properties of the double shehu transform (1) the double shehu transform h2xt (.) is a linear operator, that is h2xt [(ag + bh) (x,t)] = ah 2 xt [g(x,t)] + bh 2 xt [h(x,t)] . proof. h2xt [(ag + bh) (x,t)] = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v ) (ag + bh) dxdt = a ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x,t) dxdt + b ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )h (x,t) dxdt = ah2xt [g(x,t)] + bh 2 xt [h(x,t)] . (2) changing of scale property if h2xt (g (x,t)) = g ((p,q), (u,v)) , then h 2 xt (g (ax,bt)) = 1 ab g (( p a , q b ) , (u,v) ) . proof. using the definition of (dht), we deduce h2xt (g (ax,bt)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (ax,bt) dxdt (4.1) substituting y = ax, and z = bt in equation (4.1), we get = 1 ab ∫ ∞ 0 ∫ ∞ 0 e−( py ua + qz vb )g (y,z) dydz = 1 ab ∫ ∞ 0 ∫ ∞ 0 e − ( ( pa )y u + ( qb )z v ) g (y,z) dydz = 1 ab g ((p a , q b ) , (u,v) ) . (3) shifting property if h2xt (g (x,t)) = g ((p,q), (u,v)) , then h 2 xt ( e−(ax+bt)g (x,t) ) = g ((p + au,q + bv) , (u,v)) . proof. using the definition of (dht), we deduce h2xt ( e−(ax+bt)g (x,t) ) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )e−(ax+bt)g (x,t) dxdt = ∫ ∞ 0 ∫ ∞ 0 e−( (p+au)x u + (q+bv)t v )g (x,t) dxdt = g ((p + au,q + bv) , (u,v)) theorem 4.1. if h2xt (g (x,t)) = g ((p,q), (u,v)) , then h2xt [g (x−a,t− b) h (x−a,t− b)] = e −a p u −b q v g ((p,q), (u,v)) , int. j. anal. appl. 18 (3) (2020) 387 where h (x,t) is the heaviside unit step function defined as follows : h (x−a,t− b) =   1 x > a,t > b0 otherwise proof : by using the definition of (dht), we have h2xt [g (x−a,t− b) h (x−a,t− b)] = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x−a,t− b) h (x−a,t− b) dxdt = ∫ ∞ a ∫ ∞ b e−( px u + qt v )g (x−a,t− b) dxdt by substituting x−a = y , t− b = z , we get = ∫ ∞ a ∫ ∞ b e−( p(y+a) u + q(z+b) v )g (y,z) dydz = e−( ap u + bq v ) ∫ ∞ a ∫ ∞ b e−( py u + qz v )g (y,z) dydz = e−( ap u + bq v )h2xt (g (x,t)) . (4) the double shehu transform of the first and the second order partial derivatives with respect to x h2xt ( ∂g ∂x ) = ( p u ) g2 ((p,q), (u,v)) −ht (g (0, t)) , h2xt ( ∂2g ∂x2 ) = ( p u )2 g2 ((p,q), (u,v)) − ( p u ) ht (g (0, t)) −ht ( ∂ ∂x g (0, t) ) . similarly, with respect to t h2xt ( ∂g ∂t ) = ( q v ) g ((p,q), (u,v)) −hx (g (x, 0)) , h2xt ( ∂2g ∂t2 ) = ( q v )2 g ((p,q), (u,v)) − ( q v ) hx (g (x, 0)) −hx ( ∂ ∂t g (x, 0) ) . moreover, the double shehu transform for the mixed double order partial derivative of function of two variables is given by h2xt ( ∂2g ∂x∂t ) = (pq uv ) g ((p,q), (u,v)) − (q v ) ht (g (0, t)) − (p u ) hx (g (x, 0)) + g (0, 0) . theorem 4.2. the double shehu transforms of the, n,m ∈ n times partial derivatives ∂ ng ∂xn , ∂ mg ∂tm , of the function g(x,t) are given by : h2xt ( ∂ng ∂xn ) = (p u )n g ((p,q), (u,v)) − (p u )n−1 ht (g (0, t)) − n−1∑ j=1 (p u )n−1−j ht ( ∂j ∂xj g (0, t) ) , (4.2) h2xt ( ∂mg ∂tm ) = (q v )n g ((p,q), (u,v)) − (q v )m−1 hx (g (x, 0)) − m−1∑ j=1 (q v )m−1−j hx ( ∂j ∂tj g (x, 0) ) . proof. the proof can be done by mathematical induction. int. j. anal. appl. 18 (3) (2020) 388 theorem 4.3. if the double shehu transform of ∂ ng ∂xn , ∂ mg ∂tm is given by equations (4.2) then the double shehu transforms of xm ∂ng (x,t) ∂xn , and tm ∂mg (x,t) ∂tm are given by h2xt ( xn ∂ng (x,t) ∂xn ) = (−1)nun ∂n ∂pn ( h2xt (g (x,t)) ) , (4.3) h2xt ( tm ∂mg (x,t) ∂tm ) = (−1)mvm ∂m ∂qm ( h2xt (g (x,t)) ) , where m,n = 1, 2, 3, ... (4.4) proof. by using the definition of (dst), we get h2xt (g (x,t)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x,t) dxdt, (4.5) taking the nth partial derivative w.r.t p for both sides of equation (4.5), we have ∂n ∂pn ( h2xt (g (x,t)) ) = ∫ ∞ 0 ∫ ∞ 0 ∂n ∂pn [ e−( px u + qt v ) ] g (x,t) dxd = (−1)n ∫ ∞ 0 ∫ ∞ 0 (x u )n e−( px u + qt v )g (x,t) dxd = (−1)n 1 un ∫ ∞ 0 ∫ ∞ 0 xne−( px u + qt v )g (x,t) dxd = (−1)n 1 un ( h2xt (x ng (x,t)) ) , simplifying, we obtain (−1)nun ∂n ∂pn ( h2xt (g (x,t)) ) = h2xt (x ng (x,t)) . similarly, equation (4.4) can be proven. theorem 4.4. (double shehu-double laplace duality). if the (dht) of a function g (x, t) exists, then h2xt (g (x,t) (p,q) , (u,v)) = l 2 xt ( g (x,t) , (p u , q v )) , where l2xt denotes the (dlt) of the function g(x,t). proof. the proof can be done directly from the definition of (dht) that is h2xt (g (x,t) (p,q) , (u,v)) = ∫ ∞ 0 ∫ ∞ 0 e−( px u + qt v )g (x,t) dxdt = l2xt ( g (x,t) , (p u , q v )) . (5) int. j. anal. appl. 18 (3) (2020) 389 theorem 4.5. (double shehu-double sumudu duality). if the (dht) of a function g (x, t) exists, then h2xt (g (x,t) (p,q) , (u,v)) = uv pq s2xt ( g (x,t) , ( u p , v q )) , where s2xt denotes the (dst) of the function g(x,t). proof. by letting φ = px u ,ϕ = qt v in the (dht) formula, we get h2xt (g (x,t) (p,q) , (u,v)) = uv pq ∫ ∞ 0 ∫ ∞ 0 e−(y+y)g ( uy p , vz q ) dydz = uv pq s2xt ( g (x,t) , ( u p , v q )) . theorem 4.6. let g (x, t) , and h (x, t) be of exponential order, having double shehu transforms h2xt (g (x,t)) , and h 2 xt (h (x,t)), respectively. the the double shehu transform of the convolution of g (x, t) and h (x, t) [g ∗∗h] (x, t) = ∫ x 0 ∫ t 0 g (x−κ,t−λ) h (κ,λ) dκdλ, is given by h2xt ([g ∗∗h] (x, t)) = h 2 xt (g (x, t)) h 2 xt (h (x, t)) proof. using theorem (4.5), we have h2xt ([g ∗∗h] (x, t)) = uv pq s2xt ([g ∗∗h] (x, t)) = ( uv pq )2 s2xt (g (x, t)) s 2 xt (h (x, t)) = ( uv pq ) s2xt (g (x, t)) ( uv pq ) s2xt (h (x, t)) = h2xt (g (x, t)) h 2 xt (h (x, t)) . 5. application of (dht) method to integral and partial differential equations in this section, we illustrate the applicability of the (dht) method, by constructing some examples. example 5.1. consider the following equation of volterra integro pde. ∂ ∂x g(x,t) + ∂ ∂t g(x,t) + 1 −ex −et −ex+t = ∫ x 0 ∫ t 0 g (κ,λ) dκdλ, (5.1) with respect to the intial conditions g(x, 0) = ex, g(0, t) = et. (5.2) applying (dht) for equation (5.1) , we get,(p u ) g2 ((p,q), (u,v)) −ht (g (0, t)) + (q v ) g2 ((p,q), (u,v)) −hx (g (x, 0)) (5.3) + uv pq − uv (p−u) q − uv (q −v) p − uv (p−u) (q −v) = h2xt ([g ∗∗1] (x, t)) . int. j. anal. appl. 18 (3) (2020) 390 the single shehu transform of the initial conditions hx (g (x, 0)) = u (p−u) , ht (g (0, t)) = v (q −v) . (5.4) substituting equation (5.4) into equation(5.3), we get(p u ) g2 ((p,q), (u,v)) − u (p−u) + (q v ) g2 ((p,q), (u,v)) − v (q −v) (5.5) + uv pq − uv (p−u) q − uv (q −v) p − uv (p−u) (q −v) = ( uv pq ) g2 ((p,q), (u,v)) . simplifying, we get g2 ((p,q), (u,v)) = uv (p−u) (q −v) , (5.6) taking the inverse double shehu transform of equation (5.6) g (x,t) = h−2xt [ uv (p−u) (q −v) ] = ex+t. (5.7) example 5.2. consider the following partial double integro-differential equation. ∂2 ∂t2 g(x,t) − ∂2 ∂x2 g(x,t) + g(x,t) + ∫ x 0 ∫ t 0 ex−κ+t−λg (κ,λ) dκdλ = xtex+t + ex+t, (5.8) with respect to the initial conditions g(x, 0) = ex, ∂ ∂x g(x, 0) = ex, g(0, t) = et, ∂ ∂t g(0, t) = et. (5.9) taking the (dht) to both side of equation (5.8), we get (q v )2 g2 ((p,q), (u,v)) − (q v ) hx (g (x, 0)) −hx ( ∂ ∂t g (x, 0) ) − (p u )2 g2 ((p,q), (u,v)) (5.10) − (p u ) ht (g (0, t)) −ht ( ∂ ∂x g (0, t) ) + g2 ((p,q), (u,v)) + h2xt ([ ex+t ∗∗g ] (x, t) ) = u2v2 (p−u)2 (q −v)2 + uv (p−u) (q −v) , substituting the single shehu transform of initial and boundary conditions, hx (g (x, 0)) = u (p−u) ,hx ( ∂ ∂x g (x, 0) ) = u (p−u) ,ht (g (0, t)) = v (q −v) , ht ( ∂ ∂t g (0, t) ) = v (q −v) , in equation (5.10) we get[(q v )2 − (p u )2 + 1 + uv (p−u) (q −v) ] g2 ((p,q), (u,v)) = qu (p−u) v + u (p−u) + pv (q −v) u + v (q −v) (5.11) + u2v2 (p−u)2 (q −v)2 + uv (p−u) (q −v) , int. j. anal. appl. 18 (3) (2020) 391 simplifying, we get g2 ((p,q), (u,v)) = uv (p−u) (q −v) , (5.12) taking the inverse double shehu transform of equation (5.12) g (x,t) = h−2xt [ uv (p−u) (q −v) ] = ex+t. (5.13) example 5.3. consider the following partial integro-differential equation. ∂2 ∂x∂t g(x,t) + g(x,t) = −1 4 x2t2 + xt + 1 + ∫ x 0 ∫ t 0 g (κ,λ) dκdλ, (5.14) with respect to the initial conditions g(x, 0) = 0, g(0, t) = 0. (5.15) taking the (dht) to both side of equation (5.15), we get (pq uv ) g ((p,q), (u,v)) − (q v ) ht (g (0, t)) − (p u ) hx (g (x, 0)) + g (0, 0) + (5.16) g2 ((p,q), (u,v)) = − u3v3 p3q3 + u2v2 p2q2 + uv pq + h2xt ([1 ∗∗g] (x, t)), substituting the single shehu transform of initial and boundary conditions, ht (g (0, t)) = 0, hx (g (x, 0)) = 0, (5.17) in equation (5.16) we get [ pq uv + 1 − uv pq ] g2 ((p,q), (u,v)) = − u3v3 p3q3 + u2v2 p2q2 + uv pq , simplifying, we get g2 ((p,q), (u,v)) = u2v2 p2q2 , (5.18) taking the inverse double shehu transform of equation (5.18) g (x,t) = h−2xt [ u2v2 p2q2 ] = xt. (5.19) example 5.4. consider the following partial differential telegraph equation ∂2g (x,t) ∂x2 − ∂2g (x,t) ∂t2 − ∂g (x,t) ∂t −g + x2 + t = 1, (5.20) with respect to the initial and boundary conditions g (x, 0) = x2 , ∂ ∂t g (x, 0) = 1, g (0, t ) = t, ∂ ∂x g (0, t ) = 0. (5.21) int. j. anal. appl. 18 (3) (2020) 392 applying the (dht) to both side of equation (5.20) and single (ht) to the initial and boundary conditions (5.21), we get(p u )2 g2 ((p,q), (u,v)) − (p u ) ht (g (0, t)) −ht ( ∂ ∂x g (0, t) ) − (q v )2 g2 ((p,q), (u,v)) + (q v ) hx (g (x, 0)) (5.22) +hx ( ∂ ∂t g (x, 0) ) − (q v ) g2 ((p,q), (u,v)) + hx (g (x, 0)) −g2 ((p,q), (u,v)) + u3v p3q + uv2 pq2 = u p . substituting the single (ht) of initial and boundary conditions hx (g (x, 0)) = u3 p3 , hx ( ∂ ∂t g (x, 0) ) = u p ,ht (g (0, t)) = v2 q2 , ht ( ∂ ∂x g (0, t) ) = 0, in equation (5.22), we get[(p u )2 − (q v )2 − (q v ) − 1 ] g2 ((p,q), (u,v)) = pv2 uq2 − qu3 vp3 − u p − u3 p3 − u3v p3q − uv2 pq2 + u p , simplifying, we obtain g2 ((p,q), (u,v)) = uv2 pq2 + 2 u3 p3 v3 q3 , taking the inverse of (dht), we get g (x,t) = h−2xt [ uv2 pq2 + 2 u3 p3 v3 q3 ] = t + x2. (5.23) example 5.5. consider the kortewegde vries pde. ∂3g ∂x3 + ∂g ∂x + ∂g ∂t = 0, (5.24) with respect to the initial and boundary conditions g (x, 0) = e−x, g (0, t ) = e2t, ∂ ∂x g (0, t ) = −e2t, ∂2 ∂x2 g (0, t ) = e2t. (5.25) applying the (dht) to both side of equation (5.24) and single (ht) to initial and boundary conditions (5.25), we get(p u )3 g2 ((p,q), (u,v)) − (p u )2 ht (g (0, t)) − (p u ) ht ( ∂ ∂x g (0, t) ) −ht ( ∂2 ∂x2 g (0, t) ) (5.26) + (p u ) g2 ((p,q), (u,v)) −ht (g (0, t)) + (q v ) g2 ((p,q), (u,v)) −hx (g (x, 0)) = 0, substituting the single (ht) of the initial and boundary conditions hx (g (x, 0)) = u p + u , ht (g (0, t)) = v q − 2v ,ht ( ∂ ∂x g (0, t) ) = −v q − 2v ,ht ( ∂2 ∂x2 g (0, t) ) = v q − 2v , in equation (5.26), we obtain[(p u )3 + (p u ) + (q v )] g2 ((p,q), (u,v)) = (p u )2 v q − 2v − (p u ) v q − 2v + v q − 2v + v q − 2v + u p + u , simplifying, g2 ((p,q), (u,v)) = ( u p + u v q − 2v ) , int. j. anal. appl. 18 (3) (2020) 393 taking the inverse of (dht), we get g (x,t) = h−2xt [ u p + u v q − 2v ] = e−x+2t. (5.27) example 5.6. consider the partial differential euler –bernoulli equation ∂4g ∂x4 + ∂2g ∂t2 = t2 + xt, (5.28) with respect to the initial and boundary conditions g (x, 0) = 0, , ∂ ∂t g (x, 0) = 1 120 x5, g (0, t ) = 1 12 t4, ∂ i ∂xi g (0, t ) = 0, for, i = 1, 2, 3. (5.29) applying the (dht) to both side of equation (5.28) and single (ht) to initial and boundary conditions (5.29), we get(p u )4 g2 ((p,q), (u,v)) − (p u )3 ht (g (0, t)) − (p u )2 ht ( ∂ ∂x g (0, t) ) ht (g (0, t)) − (p u ) ht ( ∂2 ∂x2 g (0, t) ) (5.30) −ht ( ∂3 ∂x3 g (0, t) ) + (q v )2 g2 ((p,q), (u,v)) − (q v ) hx (g (x, 0)) −hx ( ∂ ∂t g (x, 0) ) = 2uv2 pq2 + uv pq , substituting, hx (g (x, 0)) = 0, ht ( ∂ ∂t g (x, 0) ) = ( u p )6 ,ht (g (0, t)) = ( v q )5 ,ht ( ∂i ∂xi g (0, t) ) = 0, i = 1, 2, 3, in equation (5.30), we obtain[(p u )4 + (q v )2] g2 ((p,q), (u,v)) = (p u )3 (v q )5 + ( u p )6 + 2uv2 pq2 + uv pq , simplifying, g2 ((p,q), (u,v)) = ( 2uv4 pq4 + v2u6 q2p6 ) . taking the inverse of (dht), we get g(x,t) = h−2xt [ 2uv4 pq4 + v2u6 q2p6 ] = 2t4 4! + tx5 5! . (5.31) 6. conclusions in this study, we have extended the work of [11], to the double shehu transform (dht). first, we have discussed and proved the existence and uniqueness of the double shehu transform. next, some fundamental properties and theorems using the new double shehu transform have also been presented. it is analyzed that the devolved method is well suited for use in integral and partial differential equations involving two variables. therefore, the proposed method is an efficient, accurate and reliable technique for the integrals and partial differential equations. finally, it is worthwhile to mention that the (dht) can be coupled with some int. j. anal. appl. 18 (3) (2020) 394 other methods to solve non-linear (pdes) arising in applied mathematics, applied physics and engineering, which will be discussed in subsequent articles. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. n. bracewell, the fourier transform and its applications (3rd ed.), mcgraw-hill, new york, (1986). [2] d. v. widder, the laplace transform, princeton university press, princeton, (1946). [3] h. eltayeb, a. kılıc¸man, a note on double laplace transform and telegraphic equations, abstr. appl. anal. 2013 (2013), article id 932578. [4] g. k. watugala, sumudu transform: a new integral transform to solve differential equations and control engineering problems, int. j. math. educ. sci. technol. 24 (1993), 35–43. [5] f. b. m. belgacem, a. a. karaballi, sumudu transform fundamental properties investigations and applications, j. appl. math. stoch. anal. 2006 (2006), article id 91083. [6] m. a. asiru, sumudu transform and the solution of integral equations of convolution type, int. j. math. educ. sci. technol. 32 (2001), 906–910. [7] s. k. q. al-omari, on the application of natural transforms, int. j. pure appl. math. 85 (2013), 729–744. [8] t. m. elzaki, the new integral transform ”elzaki transform”, glob. j. pure appl. math. 7(1) (2011), 57–64. [9] p. k. g. bhadane, v.h. pradhan and s. v. desale, elzaki transform solution of one dimensional groundwater recharge through spreading, int. j. eng. res. appl.3 (6) (2013), 1607–1610. [10] l. debnath and d. bhatta, integral transforms and their applications, crc press, taylor francis group, boca raton, fla, usa, 3rd edition, (2015). [11] s. maitama and w. zhao, new integral transform: shehu transform a generalization of sumudu and laplace transform for solving differential equations, int. j. anal. appl. 17 (2) (2019), 167-190. [12] a. khalouta and a. kade, a new method to solve fractional differential equations: inverse fractional shehu transform method, appl. appl. math. 14 (2) (2019), 926–941. [13] aggarwal, gupta, a.r and s., sharma, n. a new application of shehu transform for handling volterra integral equations of first kind, int. j. res. advent technol. 7 (4) (2019), 438–445. [14] a. bokhari, d. baleanu, and r. belgacem. application of shehu transform to atangana-baleanu derivatives. j. math. computer sci. 20 (2019), 101-107. [15] r. belgacem, d. baleanu, and a. bokhari , shehu transform and applications to caputo-fractional differential equations. int. j. anal. appl. 17 (6) (2019), 917-927. [16] dhunde, ranjit r. and waghmare, g.l. . solving partial integro-differential equations using double laplace transform method, amer. j. comput. appl. math. 5 (1) (2015), 7–10. [17] eltayeb, hassan and kilicman, adem, on double sumudu transform and double laplace transform, malaysian j. math. sci. 4 (1) (2010), 17–30. [18] h. eltayeb, a. kılıçman, a note on the sumudu transforms and differential equations, appl. math. sci. 4 (22) (2010), 1089–1098. [19] g. k. watugala, the sumudu transform for functions of two variables, math. eng. ind. 8 (2002), 293–302. int. j. anal. appl. 18 (3) (2020) 395 [20] a. kılıçman, h. eltayeb, a note on integral transforms and partial differential equations, appl. math. sci. 4 (2010), 109–118. [21] wazwaz, abdul majid . partial differential equations and solitary waves theory, higher education press beijing and springer-verlag, berlin heidelberg (2009). [22] g. l. lamb jr, introductory applications of partial differential equations with emphasis on wave propagation and diffusion, john wiley & sons, new york, ny, usa, (1995). 1. introduction 2. preliminaries 3. double shehu transforms (dht) 3.1. existence and uniqueness of (dht) 3.2. double shehu transform of some functions: 4. properties of the double shehu transform 5. application of (dht) method to integral and partial differential equations 6. conclusions references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 153-161 http://www.etamaths.com on a type of projective semi-symmetric connection s. k. pal1,∗, m. k. pandey2 and r. n. singh1 abstract. in the present paper, we have studied some properties of curvature tensors of special projective semi-symmetric connection. we have shown that curvature tensor of such a connection satisfies bianchi’s identities. 1. introduction the idea of semi-symmetric connection was introduced by a. friedmann and j. a. schouten [2] in 1924. in 1932, h. a. hayden [4] studied semi-symmetric metricconnection. it was k. yano [10] who started systematic study of semi-symmetric metric connection and this was further studied by t. imai [6], r. s. mishra and s. n. pandey [9], u. c. de and b. k. de [1] and several other mathematicians ([7], [11]). in 2001, p. zhao and h. song [12] studied a semi-symmetric connection which is projectively equivalent to levi-civita connection and such a connection is called as projective semi-symmetric connection. they found an invariant under the transformation of projective semi-symmetric connection and showed that this invariant could degenerate into the weyl projective curvature tensor under certain conditions. after this various papers ([3], [5], [13]) on projective semi-symmetric metric connection have appeared. the organization of the paper is as follows. after introduction we give some preliminary results in section 2. in sections 3, we present a brief account of special projective semi-symmetric connection. section 4 is devoted to the study of special projective semi symmetric connection with recurrent curvature tensor. 2. preliminaries let mn be an n-dimensional (n > 2) riemannian manifold equipped with a riemannian metric g and ∇ be the levi-civita connection associated with metric g. a linear connection ∇̄ on mn is called the semi symmetric metric connection [10 ], if the torsion tensor t̄ of the connection ∇̄, given by (2.1) t̄(x,y ) = ∇̄xy −∇̄y x − [x,y ] satisfies the condition (2.2) t̄(x,y ) = π(y )x −π(x)y 2010 mathematics subject classification. 53c12. key words and phrases. projective semi-symmetric connection; curvature tensor. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 153 154 pal, pandey and singh and (2.3) (∇̄xg)(y,z) = 0, where π is a 1 form on mn associated with vector field ρ, i.e., (2.4) π(x) = g(x,ρ). if the geodesic with respect to ∇̄ are always consistent with those of ∇, then ∇̄ is called a connection projectively equivalent to ∇. if ∇̄ is projective equivalent connection to ∇ as well as the semi-symmetric, then ∇̄ is called projective semisymmetric connection. we also call ∇̄ as projective semisymmetric transformation. in this paper, we study a type of projective semi-symmetric connection ∇̄ introduced by p. zhao and h. song [12]. the connection is given by (2.5) ∇̄xy = ∇xy + ψ(y )x + ψ(x)y + φ(y )x −φ(x)y, where 1-forms φ and ψ are given as (2.6) φ(x) = 1 2 π(x) and ψ(x) = n− 1 2(n + 1) π(x). it is easy to observe that torsion tensor of projective semisymmetric transformation is same as given by the equation (2.2) and also that (2.7) (∇̄xg)(y,z) = 1 n + 1 [2π(x)g(y,z) −nπ(y )g(z,x) −nπ(z)g(x,y ), i.e., the connection ∇̄ is a non metric one. let r̄ and r be the curvature tensor of the manifold relative to the projective semi-symmetric connection ∇̄ and levi-civita connection ∇ respectively. it is known that [12] (2.8) r̄(x,y,z) = r(x,y,z) + β(x,y )z + α(x,z)y −α(y,z)x, where β(x,y ) and α(x,y ) are given by the following relations (2.9) β(x,y ) = ψ′(x,y ) − ψ′(y,x) + φ′(y,x) − φ′(x,y ), (2.10) α(x,y ) = ψ′(x,y ) + φ′(y,x) −ψ(x)φ(y ) −φ(x)ψ(y ), (2.11) ψ′(x,y ) = (∇xψ)(y ) −ψ(x)ψ(y ) and (2.12) φ′(x,y ) = (∇xφ)(y ) −φ(x)φ(y ). contracting x in the equation (2.8), we get a relation between ricci tensors r̄ic(y,z) and ric(y,z) of manifold with respect to connections ∇̄ and ∇ respectively (2.13) r̄ic(y,z) = ric(y,z) + β(y,z) − (n− 1)α(y,z). if r̄ and r are scalar curvatures of manifold with respect to connection ∇̄ and ∇ respectively, then from the equation (2.13), we get (2.14) r̄ = r + b− (n− 1)a, on a type of projective semi-symmetric connection 155 where b = n∑ i = 1 β(ei,ei) and a = n∑ i = 1 α(ei,ei). the weyl-projective curvature tensor w , conharmonic curvature tensor p and concircular curvature tensor i are given by [9] (2.15) w(x,y,z) = r(x,y,z) + 1 n− 1 {ric(x,z)y −ric(y,z)x}, p(x,y,z) = r(x,y,z)− 1 n− 2 [ric(y,z)x −ric(x,z)y + g(y,z)qx −g(x,z)qy ], (2.16) where (2.17) g(qx,y ) = ric(x,y ) and (2.18) i(x,y,z) = r(x,y,z) − r n− 1 [g(y,z)x −g(x,z)y ]. 3. special projective semi-symmetric connection in this section, we consider a projective semi-symmetric connection ∇̄ given by the equation (2.5) whose associated 1-form π is closed, i.e., (3.1) (∇̄xπ)y = (∇̄y π)x. such a connection ∇̄ is called special projective semi-symmetric connection [12]. it is easy to verify that both the 1-forms φ and ψ are closed as the 1-form π is closed and also that the tensors φ′ and ψ′ are symmetric. consequently, we get (3.2) β(x,y ) = 0 and (3.3) α(x,y ) = α(y,x). in view of the equations (3.1) and (3.2), the expressions (2.8), (2.13) and (2.14) reduces to (3.4) r̄(x,y,z) = r(x,y,z) + α(x,z)y −α(y,z)x, (3.5) r̄ic(y,z) = ric(y,z) − (n− 1)α(y,z) and (3.6) r̄ = r − (n− 1)a. it is easy to observe that the ricci tensor r̄ic(y,z) is symmetric. now, we prove the following theorems: theorem 3.1. curvature tensor of special projective semi-symmetric connection satisfies bianchi’s first identity. 156 pal, pandey and singh proof : writing two more equations by cyclic permutations of x, y and z from equation (3.4), we get r̄(y,z,x) = r(y,z,x) + α(y,x)z −α(z,x)y, and r̄(z,x,y ) = r(z,x,y ) + α(z,y )x −α(x,y )z. adding these equations to the equation (3.4), we get result. theorem 3.2. curvature tensor of special projective semi-symmetric connection satisfies bianchi’s second identity if α is parallel tensor with respect to levi-civita connection ∇. proof : suppose α is a parallel tensor with respect to levi-civita connection ∇, i.e., ∇α = 0. now differentiating the equation (3.4) covariantly with respect to the connection ∇, we have (3.7) (∇xr̄)(y,z,u) = (∇xr)(y,z,u). writing two more equations by cyclic permutations of x, y and z in above equation, we get (3.8) (∇y r̄)(z,x,u) = (∇y r)(z,x,u), and (3.9) (∇zr̄)(x,y,u) = (∇zr)(x,y,u). adding the equations (3.7), (3.8) and (3.9), we get (∇xr̄)(y,z,u) + (∇y r̄)(z,x,u) + (∇zr̄)(x,y,u) = 0. this shows that the curvature tensor of special projective semi-symmetric connection satisfies bianchi’s second identity. theorem 3.3. the weyl-projective curvature tensor of riemannian manifold with respect to the special projective semi-symmetric connection ∇̄ satisfies w(x,y,z) + w(y,z,x) + w(z,x,y ) = 0. proof : the weyl-projective curvature tensor of riemannian manifold with respect to special projective semi-symmetric connection ∇̄ is given by (3.10) w(x,y,z) = r̄(x,y,z) − 1 n− 1 [r̄ic(y,z)x − r̄ic(x,z)y ]. writing two more equations by cyclic permutations of x, y and z in above equation, we get (3.11) w(y,z,x) = r̄(y,z,x) − 1 n− 1 [r̄ic(z,x)y − r̄ic(y,x)z], (3.12) w(z,x,y ) = r̄(z,x,y ) − 1 n− 1 [r̄ic(x,y )z − r̄ic(z,y )x]. adding the equations (3.10), (3.11) and (3.12), we get w(x,y,z) + w(y,z,x) + w(z,x,y ) = 0. on a type of projective semi-symmetric connection 157 4. special projective semi-symmetric connection with recurrent curvature tensor in this section, we consider a special projective semi-symmetric connection ∇̄ whose curvature tensor r̄ is recurrent with respect to the levi-civita connection ∇, i.e., (4.1) (∇ur̄)(x,y,z) = b(u)r̄(x,y,z), where b is a non-zero 1-form. differentiating the equation (3.4) covariantly with respect to the levi-civita connection ∇, we get (4.2) (∇ur̄)(x,y,z) = (∇ur)(x,y,z) + (∇uα)(x,z)y − (∇uα)(y,z)x. contracting x in the above equation, we have (4.3) (∇ur̄ic)(y,z) = (∇uric)(y,z) − (n− 1)(∇uα)(y,z). putting y = z = ei in the above equation and taking summation over i, 1 ≤ i ≤ n, we get (4.4) (∇u r̄) = (∇ur) − (n− 1)(∇ua). now the equations (3.4) and (4.2) together give (∇ur̄)(x,y,z) −b(u)r̄(x,y,z) =(∇ur)(x,y,z) −b(u)r(x,y,z) +[(∇uα)(x,z) −b(u)α(x,z)]y −[(∇uα)(y,z) −b(u)α(y,z)]x, (4.5) which, in view of the equation (4.1), reduces to (∇ur)(x,y,z) −b(u)r(x,y,z) =[(∇uα)(y,z) −b(u)α(y,z)]x −[(∇uα)(x,z) −b(u)α(x,z)]y. (4.6) contracting x in above, we get (4.7) (∇uric)(y,z) −b(u)ric(y,z) = (n− 1){(∇uα)(y,z) −b(u)α(y,z)}. further, we obtain (4.8) (∇ur) −b(u)r = (n− 1){(∇ua) −b(u)a}. also, from the equation (2.17), we have (4.9) g((∇uq)x,y ) = (∇uric)(x,y ), which can be written as (4.10) g((∇uq)x −b(u)qx,y ) = (∇uric)(x,y ) −b(u)ric(x,y ). now we prove following theorems: theorem 4.1. if the curvature tensor of special projective semi-symmetric connection on a riemannian manifold mn is recurrent with respect to the levi-civita connection then manifold mn is projectively recurrent. proof : differentiating the projective curvature tensor w given by (2.15) covariantly with respect to levi-civita connection ∇, we have (4.11) (∇uw)(x,y,z) = (∇ur)(x,y,z)+ 1 n− 1 {(∇uric)(x,z)y −(∇uric)(y,z)x}. 158 pal, pandey and singh the above equation gives (∇uw)(x,y,z) −b(u)w(x,y,z) =(∇ur)(x,y,z) −b(u)r(x,y,z) + 1 n− 1 [{(∇uric)(x,z) −b(u)ric(x,z)}y −{(∇uric)(y,z) −b(u)ric(y,z)}x]. (4.12) using equation (4.6) and (4.7) in above, we get (∇uw)(x,y,z) = b(u)w(x,y,z), which proves the statement. theorem 4.2. : a riemannian manifold mn admitting a special projective semisymmetric connection whose curvature tensor and tensor α are recurrent with respect to the levi-civita connection, is conharmonically recurrent. proof: differentiating covariantly the equation (2.16) with respect to the levicivita connection, we get (∇up)(x,y,z) =(∇ur)(x,y,z) − 1 n− 2 [(∇uric)(y,z)x − (∇uric)(x,z)y. +g(y,z)(∇uq)x −g(x,z)(∇uq)y ], (4.13) from above, we have (∇up)(x,y,z) −b(u)p(x,y,z) =(∇ur)(x,y,z) −b(u)r(x,y,z) − 1 n− 2 [{(∇uric)(y,z) −b(u)ric(y,z)}x −{(∇uric)(x,z) −b(u)ric(x,z)}y +g(y,z){(∇uq)x −b(u)qx} −g(x,z){(∇uq)y −b(u)qy}]. (4.14) if the tensor α and the curvature tensor of the special projective semi-symmetric connection ∇̄ are recurrent with respect to the levi-civita connection ∇, then from the equations (4.6), (4.7) and (4.10), we get (∇up)(x,y,z) = b(u)p(x,y,z), which shows that manifold is conharmonically recurrent. theorem 4.3. a riemannian manifold mn admitting a special projective semisymmetric connection whose curvature tensor and tensor α are recurrent with respect to levi-civita connection, is concircular recurrent. proof: differentiating the concircular curvature tensor i of mn given by the equation (2.18) covariantly with respect to the levicivita connection ∇, we have (4.15) (∇ui)(x,y,z) = (∇ur)(x,y,z) − ∇ur (n− 1) {g(y,z)x −g(x,z)y}. on a type of projective semi-symmetric connection 159 from this, we have (∇ui)(x,y,z) −b(u)i(x,y,z) =(∇ur)(x,y,z) −b(u)r(x,y,z) − ∇ur −b(u)r (n− 1) {g(y,z)x −g(x,z)y}. (4.16) if the tensor α and the curvature tensor of the special projective semi-symmetric connection ∇̄ are recurrent with respect to the levi-civita connection ∇, then from the equations (4.6), (4.7) and (4.8), we get (∇ui)(x,y,z) = b(u)i(x,y,z). theorem 4.4. let mn be a riemannian manifold admitting a special projective semi-symmetric connection whose ricci-tensor is recurrent with respect to the levi-civita connection. if the manifold is projectively recurrent with respect to levi-civita connection, then the curvature tensor of the special projective semisymmetric connection is recurrent. proof: let the manifold mn be projectively recurrent with respect to levi civita connection ∇. then from the equation (4.12), we have (∇ur)(x,y,z) −b(u)r(x,y,z) = 1 n− 1 [{(∇uric)(y,z) −b(u)ric(y,z)x} −{(∇uric)(x,z) −b(u)ric(x,z)y}]. (4.17) now, from equations (3.5) and (4.3), we get (∇ur̄ic)(y,z) −b(u)r̄ic(y,z) =(∇uric)(y,z) −b(u)ric(y,z) −(n− 1){(∇uα)(y,z) −b(u)α(y,z)}. (4.18) since the ricci tensor of the special projective semi-symmetric connection ∇̄ is recurrent with respect to the levi-civita connection ∇, hence the above equation gives (4.19) (∇uric)(y,z) −b(u)ric(y,z) = (n− 1){(∇uα)(y,z) −b(u)α(y,z)}. thus, from the equations (4.17) and (4.19), we get (∇ur)(x,y,z) −b(u)r(x,y,z) ={(∇uα)(y,z) −b(u)α(y,z)}x −{(∇uα)(x,z) −b(u)α(x,z)}y, (4.20) which, on using in the equation (4.5), gives (4.21) (∇ur̄)(x,y,z) = b(u)r̄(x,y,z). this proves the statement. theorem 4.5. let mn be a riemannian manifold admitting a special projective semi-symmetric connection whose ricci-tensor is recurrent with respect to the levicivita connection. if the manifold is of constant curvature, then the curvature tensor of the special projective semi-symmetric connection is recurrent with respect to the levi-civita connection. 160 pal, pandey and singh proof: if the riemannian manifold mn is of constant curvature, then we have [9] (4.22) r(x,y,z) = 1 n− 1 {ric(y,z)x −ric(x,z)y}. using the above equation in the equation (3.4), we have (4.23) r̄(x,y,z) = 1 n− 1 [{ric(y,z)−(n−1)α(y,z)}x−{ric(x,z)−(n−1)α(x,z)}y ], which, on using the equation (3.5), gives (4.24) r̄(x,y,z) = 1 n− 1 {r̄ic(y,z)x − r̄ic(x,z)y}. differentiating the above equation covariantly with respect to the levi-civita connection, we have (∇ur̄)(x,y,z) = 1 n− 1 {(∇ur̄ic)(y,z)x − (∇ur̄ic)(x,z)y}, which can be written as (∇ur̄)(x,y,z) −b(u)r̄(x,y,z) = 1 n− 1 [{(∇ur̄ic)(y,z) −b(u)r̄ic(y,z)}x −{(∇ur̄ic)(x,z) −b(u)r̄ic(x,z)}y ]. (4.25) since the ricci tensor of special projective semi-symmetric connection is recurrent with respect to the levi-civita connection ∇, hence from the above equation, we have (∇ur̄)(x,y,z) = b(u)r̄(x,y,z), which proves the statement. references [1] de, u.c. and de, b. k., on a type of semi-symmetric connection on a riemannian manifold, ganita, 47(2), (1996), 11-24. [2] friedmann, a. and schouten, j. a., uber die geometrie der halbsymmetrischen ubertragungen, math. zeitschr., 21(1), (1924), 211-223. [3] fengyun, fu. and zhao. p., a property on geodesic mappings of pseudo-symmetric riemannian manifolds, bull. malays. math. sci. soc.(2), 33(2), (2010), 265-272. [4] hayden, h. a., subspaces of space with torsion, proc. london math. soc., 34, (1932), 27-50. [5] han, y., yun, h. and zhao, p., some invariants of quarter-symmetric metric connections under the projective transformation, filomat, 27(4), (2013), 679-691. [6] imai, t., notes on semi-symetric metric connections, tensor, n. s., 24, (1972), 293-296. [7] liang y., some properties of the semi-symmetric metric connection, j. of xiamen university (natural science), 30(1), (1991), 22-24. [8] mishra, r. s., structures on a differentiable manifolds and their applications, chandrama prakashan, allahabad, 1984. [9] mishra, r. s. and pandey, s. n., semi-symmetric metric connection in an almost contact manifold, indian j. pure appl. math., 9(6), (1978), 570-580. [10] yano, k., on semi-symmetric metric connection, revue roumaine de math. pure et appliquees, 15, (1970), 1579-1581. [11] zhao, p. and shangguan l., on semi-symmetric connection, j. of henan normal university (natural science), 19(4), (1994), 13-16. [12] zhao, p. and song h., an invariant of the projective semisymmetric connection, chinese quarterly j. of math., 17(4), (2001), 48-52. [13] zhao, p., some properties of projective semi-symmetric connections, int. math.forum, 3(7), (2008), 341-347. on a type of projective semi-symmetric connection 161 1department of mathematical sciences, a. p. s. university, rewa, (m.p.), india, 486003 2department of mathematics, university institute of technology, rajiv gandhi proudyogiki vishwavidyalaya bhopal, (m.p.), india, 462036 ∗corresponding author international journal of analysis and applications volume 19, number 1 (2021), 123-137 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-123 on subordination results for certain classes of analytic functions of reciprocal order muhammet kamali∗, alina riskulova kyrgyz-turkish manas university, faculty of sciences, department of mathematics, chyngz aitmatov avenue, bishkek, kyrgyz republic ∗corresponding author: muhammet.kamali@manas.edu.kg abstract. in this paper, we introduce the subclass sβ (α,λ) of analytic functions and obtain coefficient inequality for functions belong to this class. furthermore, we give sufficient conditions for starlikeness of reciprocal order of analytic functions. in the last part, we obtain the subordination results of a new subclass of analytic functions of reciprocal order, which are defined here by means of a hadamard product of analytic functions. the results presented in this work improve or generalize the recent works of other authors and also give rise to several new results. 1. introduction and preliminaries let u = {z : |z| < 1} . we denote bya the class of analytic functions on the unit disc u having the following taylor series representation: (1.1) f (z) = z + ∞∑ n=2 anz n. let s denote the subclass of a consisting of all analytic functions f (z) which are also univalent in u. a function f ∈ a is said to be starlike of order α if it satisfies (1.2) re ( zf ′ (z) f (z) ) > α (0 ≤ α < 1, z ∈ u) . received october 12th, 2020; accepted november 18th, 2020; published december 17th, 2020. 2010 mathematics subject classification. 30c45, 30c80. key words and phrases. analytic function; starlike function; convex function; analytic function of reciprocal order; subordinating factor sequence; hadamard product. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 123 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-123 int. j. anal. appl. 19 (1) (2021) 124 we denote by s∗ (α) the subclass of a consisting of functions which are starlike of order α in u. a function f ∈ a is said to be starlike of reciprocal order α in u if (1.3) re ( f (z) zf ′ (z) ) > α (z ∈ u) for some α (0 ≤ α < 1) . we denote the class such functions by s−1∗ (α) ( [1], [4], [10]). a function f ∈ a is said to be convex of order α in u if it satisfies the condition (1.4) re ( 1 + zf′′ (z) f ′ (z) ) > α (z ∈ u) for some 0 ≤ α < 1.we denote by k (α) the subclass of a consisting of functions which are convex of orderα in u. furthermore, a function f ∈ a is said to be convex of reciprocal order α in u if (1.5) re   1 1 + zf ′′ (z) f ′ (z)   > α (0 ≤ α < 1, z ∈ u) we denote the class such functions by k−1 (α) [10]. clearly, we have s∗ (α) ⊆ s∗ (0) = s∗,k (α) ⊆ k (0) = k and f (z) ∈ k (α) if and only if zf ′ (z) ∈ s∗ (α) for 0 ≤ α < 1. for |β| < π 2 and 0 ≤ α < 1, a function f ∈ a is said to be β−spirallike of order α in u if it satisfies (1.6) re ( eiβ zf ′ (z) f (z) ) > α cos β. the class of all such functions is denote by sβ (α) ( [8], [10]). definition 1.1. let h (z) = zf ′ (z) f(z) for f (z) ∈ s. a function f (z) ∈ s is said to be in the class denote by sβ (α) if it satisfies the inequality (1.7) ∣∣∣∣ 1eiβh (z) − 12α ∣∣∣∣ < 12α for some real β and 0 < α < 1. owa et al. [9] gave the following coefficient inequality for the function class sβ (α) . theorem 1.1. [9] if f (z) ∈ a satisfies (1.8) ∞∑ n=2 { n + ∣∣n− 2αe−iβ∣∣} |an| ≤ 1 − ∣∣1 − 2αe−iβ∣∣ for some real |β| < π 2 and 0 < α < cos β, then f (z) ∈ sβ (α) . for f ∈ a, salagean [2] has introduced the following operator called the salagean operator: d0f (z) = f (z) d1f (z) = df (z) = zf ′ (z) = z + ∑∞ n=2 nanz n ... dωf (z) = d ( dω−1f (z) ) = z + ∑∞ n=2 n ωanz n int. j. anal. appl. 19 (1) (2021) 125 where ω ∈ n0 = {0, 1, 2, ...} . definition 1.2. [5] a function f (z) ∈ a is said to be in the class m (α,λ, ω) if it satisfies the inequality (1.9) ∣∣∣∣∣ (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) (1 −λ) z (dωf (z)) ′ + λz (dω+1f (z)) ′ − 1 + λ 2α ∣∣∣∣∣ < 1 + λ2α for 0 < α < 1, 0 ≤ λ < 1, ω ∈ n0 and z ∈ u. m.kamali [5] gave the following coefficients inequality for the function class m (α,λ, ω) . theorem 1.2. [5] let 0 < α < 1 and 0 ≤ λ < 1. if f (z) ∈ a satisfies the following coefficient inequality: ∞∑ n=2 nω (λn + 1 −λ){|2α− (1 + λ) n| + (1 + λ) n}|an| ≤ (1 + λ) −|2α− (1 + λ)| =   2α; 0 < α ≤ 1+λ 2 2 (1 + λ−α) ; 1+λ 2 ≤ α < 1 + λ (1.10) then f (z) ∈ m (α,λ, ω) . 2. some results and coefficient inequality for functions in the class sβ (α,λ) definition 2.1. let g (z) = (1−λ)z(dωf(z)) ′ +λz(dω+1f(z)) ′ (1−λ)(dωf(z))+λ(dω+1f(z)) for f (z) ∈ s. a function f (z) ∈ s is said to be in the class denote by sβ (α,λ) if it satisfies the inequality (2.1) ∣∣∣∣ 1eiβg (z) − 1 + λ2α ∣∣∣∣ < 1 + λ2α for some real β and 0 < α < 1, 0 ≤ λ < 1, ω ∈ n0, z ∈ u. theorem 2.1. if f (z) ∈ sβ (α,λ) iff (2.2) re  eiβ (1 −λ) z ( dωf (z) )′ + λz ( dω+1f (z) )′ (1 −λ) (dωf (z)) + λ (dω+1f (z))   > α1 + λ. proof. let g (z) = (1−λ)z(dωf(z)) ′ +λz(dω+1f(z)) ′ (1−λ)(dωf(z))+λ(dω+1f(z)) forf (z) ∈ s. if f (z) ∈ sβ (α,λ) ,we can write∣∣∣∣ 1eiβg (z) − 1 + λ2α ∣∣∣∣ < 1 + λ2α . then, we can obtain∣∣∣ 1eiβg(z) − 1+λ2α ∣∣∣ < 1+λ2α ⇔ ∣∣∣2α−(1+λ)eiβg(z)2αeiβg(z) ∣∣∣2 < (1+λ2α )2 ⇔ [ 2α− (1 + λ) eiβg (z) ][ 2α− (1 + λ) eiβg (z) ] < (1 + λ) 2 [ eiβg (z) ][ eiβg (z) ] ⇔ 2α− 2 (1 + λ) re [ eiβg (z) ] < 0 ⇔ re [ eiβg (z) ] > α 1+λ ⇔ re { eiβ (1−λ)(dωf(z)) ′ +λ(dω+1f(z)) ′ (1−λ)(dωf(z))+λ(dω+1f(z)) } > α 1+λ . int. j. anal. appl. 19 (1) (2021) 126 � theorem 2.2. if f (z) ∈ a satisfies ∞∑ n=2 nω (1 −λ + λn) { (1 + λ) n + ∣∣(1 + λ) n− 2αe−iβ∣∣} |an| ≤ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣(2.3) for some |β| < π 2 and 0 < α 1+λ < cos β, then f (z) ∈ sβ (α,λ) . proof. it suffices to show that ∣∣∣∣2αe−iβ − (1 + λ) g (z)(1 + λ) g (z) ∣∣∣∣ < 1 for some |β| < π 2 and 0 < α 1+λ < cos β, where g (z) = (1−λ)z(dωf(z)) ′ +λz(dω+1f(z)) ′ (1−λ)(dωf(z))+λ(dω+1f(z)) . note that (2.4) ∣∣∣2αe−iβ−(1+λ)g(z)(1+λ)g(z) ∣∣∣ = ∣∣∣∣2αe−iβ{z+∑∞n=2 nω(1−λ+λn)anzn}−(1+λ){z+∑∞n=2 nω+1(1−λ+λn)anzn}(1+λ){z+∑∞n=2 nω+1(1−λ+λn)anzn} ∣∣∣∣ ≤ | (1+λ)−2αe−iβ|+∑∞n=2 nω(1−λ+λn)|(1+λ)n−2αe−iβ||an||z|n−1 (1+λ){1−∑∞n=2 nω+1(1−λ+λn)δn|an||z|n−1} < |(1+λ)−2αe−iβ|+∑∞n=2 nω(1−λ+λn)|(1+λ)n−2αe−iβ||an| (1+λ){1−∑∞n=2 nω+1(1−λ+λn)|an|} . therefore, if ∞∑ n=2 nω (1 −λ + λn) { (1 + λ) n + ∣∣(1 + λ) n− 2αe−iβ∣∣} |an| ≤ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣ for some |β| < π 2 and 0 < α 1+λ < cos β, then ∞∑ n=2 nω (1 −λ + λn) ∣∣(1 + λ) n− 2αe−iβ∣∣ |an| ≤ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣− ∞∑ n=2 (1 −λ + λn) (1 + λ) nω+1 |an| . using the inequality in (2.4), we obtain∣∣∣∣2αe−iβ − (1 + λ) g (z)(1 + λ) g (z) ∣∣∣∣ < 1. therefore, f (z) ∈ sβ (α,λ) for some |β| < π2 and 0 < α 1+λ < cos β. � taking λ = 0 and ω = 0 in theorem 2.2, we get corollary 2.1 given by owa et al. [9]. corollary 2.1. if f (z) ∈ a satisfies ∞∑ n=2 { n + ∣∣n− 2αe−iβ∣∣} |an| ≤ 1 − ∣∣1 − 2αe−iβ∣∣ for some |β| < π 2 and 0 < α < cos β , then f (z) ∈ sβ (α) . int. j. anal. appl. 19 (1) (2021) 127 taking β = π 4 in theorem 2.2, we have corollary 2.2. corollary 2.2. [9] if f (z) ∈ a satisfies ∞∑ n=2 (1 −λ + λn) { (1 + λ) n + √ (1 + λ) 2 n2 − 2 √ 2 (1 + λ) nα + 4α2 } nω |an| ≤ (1 + λ) − √ (1 + λ) 2 − 2 √ 2 (1 + λ) α + 4α2 for some 0 < α 1+λ < √ 2 2 , then f (z) ∈ sπ 4 (α,λ) . 3. sufficient conditions for starlikeness of reciprocal order first we give following example. example 3.1. let us define the function f (z) by (3.1) f (z) = ze(1− α 1+λ )z (z ∈ u) with 0 < α < 1, 0 ≤ λ < 1. from (3.1) after taking the logarithmical differentiation we have that ln f (z) = ln z + ( 1 − α 1 + λ ) z ⇒ f ′ (z) f (z) = 1 z + ( 1 − α 1 + λ ) ⇒ zf ′ (z) f (z) = 1 + ( 1 − α 1 + λ ) z. this gives us that re { zf ′ (z) f (z) } = re { 1 + ( 1 − α 1 + λ ) z } > α 1 + λ (z ∈ u) . therefore , we see that f (z) ∈ s∗ ( α 1+λ ) . furthermore, we have that zf ′ (z) f (z) = 1 + ( 1 − α 1 + λ ) z ⇒ f (z) zf ′ (z) = 1 1 + ( 1 − α 1+λ ) z . it follows that f (z) zf ′ (z) = 1 (z = 0) and (3.2) re { f (z) zf ′ (z) } = re   11 + (1 − α 1+λ ) eiθ   > 1 + λ2 (1 + λ) −α (z = eiθ) . therefore, we conclude that f (z) ∈ s∗ ( α 1+λ ) and starlike of reciprocal order 1+λ 2(1+λ)−α in u. in order to establish our main results, we require the following lemma due to nunokama et al. [6]. int. j. anal. appl. 19 (1) (2021) 128 lemma 3.1. let p (z) = 1 + ∑∞ n=1 cnz n be analytic in u and suppose that there exists a point z0 ∈ u such that re{p (z)} > 0 for |z| < |z0| and re{p (z0)} = 0. then we have (3.3) z0p ′ (z0) ≤− 1 2 ( 1 + |p (z0)| 2 ) , where z0p ′ (z0) is a negative real number. theorem 3.1. let f (z) = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) ∈ a satisfies f (z) f ′ (z) 6= 0 in 0 < |z| < 1 and (3.4) re [ (1−λ)(dωf(z))+λ(dω+1f(z)) (1−λ)(dω+1f(z))+λ(dω+2f(z)) × { 1 − α 1+λ λ(dω+3f(z))+(1−2λ)(dω+2f(z))−(1−λ)(dω+1f(z)) (1−λ)(dω+1f(z))+λ(dω+2f(z)) }] > − α 2(1+λ) { 3 + ∣∣∣∣ (1−λ)(dωf(z))+λ(dω+1f(z))(1−λ)(dω+1f(z))+λ(dω+2f(z)) ∣∣∣∣2 } (α ≥ 0) . then f (z) is starlike of reciprocal order 0 in u and thus, f (z) is starlike in u. proof. let us define the function p (z) by (3.5) p (z) = f (z) zf ′ (z) . then p (z) is analytic in u and p (0) = 1. differentiating (3.5) logarithmically we obtain p′ (z) p (z) = f ′ (z) f (z) − 1 z − f ′′ (z) f ′ (z) ⇒ α 1 + λ z p ′ (z) p (z) = { f ′ (z) f (z) − 1 z − f ′′ (z) f ′ (z) } α 1 + λ z ⇒ α 1 + λ zp ′ (z) + α 1 + λ p (z) = { zf ′ (z) f (z) − zf ′′ (z) f ′ (z) } α 1 + λ p (z) ⇒ α 1 + λ zp ′ (z) + ( α 1 + λ + 1 ) p (z) − α 1 + λ = { 1 − α 1 + λ zf ′′ (z) f ′ (z) } p (z) = { 1 − α 1 + λ z2f ′′ (z) zf ′ (z) } p (z) . furthermore, we can write f (z) = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) ⇒ zf ′ (z) = (1 −λ) z ( dωf (z) )′ + λz ( dω+1f (z) )′ ⇒ zf ′ (z) = (1 −λ) ( dω+1f (z) ) + λ ( dω+2f (z) ) int. j. anal. appl. 19 (1) (2021) 129 and ( zf ′ (z) )′ = (1 −λ) ( dω+1f (z) )′ +λ ( dω+2f (z) )′ ⇒ f ′ (z) + zf ′′ (z) = (1 −λ) ( dω+1f (z) )′ +λ ( dω+2f (z) )′ ⇒ zf ′ (z) + z2f ′′ (z) = (1 −λ) z ( dω+1f (z) )′ +λz ( dω+2f (z) )′ ⇒ z2f ′′(z) = (1 −λ) ( dω+2f (z) ) + λ ( dω+3f (z) ) − (1 −λ) ( dω+1f (z) ) −λ ( dω+2f (z) ) ⇒ z2f ′′(z) = λ ( dω+3f (z) ) + (1 − 2λ) ( dω+2f (z) ) − (1 −λ) ( dω+1f (z) ) thus,we obtain α 1 + λ zp ′ (z) + ( α 1 + λ + 1 ) p (z) − α 1 + λ = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) (1 −λ) (dω+1f (z)) + λ (dω+2f (z)) × { 1 − α 1 + λ λ ( dω+3f (z) ) + (1 − 2λ) ( dω+2f (z) ) − (1 −λ) ( dω+1f (z) ) (1 −λ) (dω+1f (z)) + λ (dω+2f (z)) } .(3.6) suppose that there exists a point z0 ∈ u such that re{p (z)} > 0 for |z| < |z0| and re{p (z0)} = 0, then from lemma 3.1,we have, z0p ′ (z0) ≤− 1 2 ( 1 + |p (z0)| 2 ) . therefore from (3.6), we have re [ (1 −λ) ( dωf (z0) ) + λ ( dω+1f (z0) ) (1 −λ) (dω+1f (z0)) + λ (dω+2f (z0)) × { 1 − α 1 + λ λ ( dω+3f (z0) ) + (1 − 2λ) ( dω+2f (z0) ) − (1 −λ) ( dω+1f (z0) ) (1 −λ) (dω+1f (z0)) + λ (dω+2f (z0)) }] = re { α 1 + λ z0p ′ (z0) + ( α 1 + λ + 1 ) p (z0) − α 1 + λ } ≤ α 1 + λ { − 1 2 ( 1 + |p (z0)| 2 )} − α 1 + λ = − α 2 (1 + λ) { 3 + |p (z0)| 2 } = − α 2 (1 + λ)  3 + ∣∣∣∣∣ (1 −λ) ( dωf (z0) ) + λ ( dω+1f (z0) ) (1 −λ) (dω+1f (z0)) + λ (dω+2f (z0)) ∣∣∣∣∣ 2   . which contradicts the hypothesis (3.4) of theorem 3.1.thus we complete the proof of theorem 3.1. � taking λ = 0 and ω = 0 in theorem 3.1, we get corollary 3.1 given by b.a.frasin and m.ab.sabri [1]. corollary 3.1. [1] let f (z) ∈ a satisfies f (z) f ′ (z) 6= 0 in 0 < |z| < 1 and re [ zf ′ (z) f (z) { 1 −α zf ′′ (z) f ′ (z) }] > − α 2 { 3 + ∣∣∣∣ f (z)zf′ (z) ∣∣∣∣2 } (z ∈ u; α ≥ 0) . then f (z) is starlike of reciprocal order 0 in u and thus, f (z) is starlike in u. int. j. anal. appl. 19 (1) (2021) 130 theorem 3.2. let f (z) = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) ∈ a satisfies re [ f (z) zf ′ (z) { 1 − ( 1 + λ 2 (1 + λ) −α ) zf ′′ (z) f ′ (z) }] > − 1 2 ( 1 + λ 2 (1 + λ) −α )∣∣∣∣ f (z)zf ′ (z) − ( 1 + λ 2 (1 + λ) −α )∣∣∣∣2 + 1 2 { 3 ( 1 + λ 2 (1 + λ) −α )2 − ( 1 + λ 2 (1 + λ) −α )} (3.7) then f (z) is starlike of reciprocal order 1+λ 2(1+λ)−α in u. proof. let us define the function f(z) zf ′ (z) by (3.8) f (z) zf ′ (z) = ( (1 + λ) −α 2 (1 + λ) −α ) p (z) + 1 + λ 2 (1 + λ) −α ; p (0) = 1. differentiating (3.8) we obtain 1 z − f (z) z2f ′ (z) − f (z) .f ′′ (z) z [f ′ (z)] 2 = ( (1 + λ) −α 2 (1 + λ) −α ) p ′ (z) ⇒ ( (1 + λ) 2 (1 + λ) −α ) − ( (1 + λ) 2 (1 + λ) −α ) f (z) zf ′ (z) − ( (1 + λ) 2 (1 + λ) −α ) f (z) .f ′′ (z) [f ′ (z)] 2 = ( (1 + λ) 2 (1 + λ) −α )( (1 + λ) −α 2 (1 + λ) −α ) zp ′ (z) ⇒ f (z) zf ′ (z) { 1 − ( (1 + λ) 2 (1 + λ) −α ) zf ′′ (z) f ′ (z) } = ( (1 + λ) 2 (1 + λ) −α )( (1 + λ) −α 2 (1 + λ) −α ) zp ′ (z) + { 1 + ( (1 + λ) 2 (1 + λ) −α )}( (1 + λ) −α 2 (1 + λ) −α ) p (z) + ( (1 + λ) 2 (1 + λ) −α )2 .(3.9) suppose that there exists a point z0 ∈ u such that re{p (z)} > 0 for |z| < |z0| and re{p (z0)} = 0, then from lemma 3.1,we have, z0p ′ (z0) ≤− 1 2 ( 1 + |p (z0)| 2 ) . therefore from (3.9), we have re [ f (z0) z0f ′ (z0) { 1 − ( 1 + λ 2 (1 + λ) −α ) z0f ′′ (z0) f ′ (z0) }] ≤ − 1 2 ( 1 + λ 2 (1 + λ) −α )( 1 + λ−α 2 (1 + λ) −α ){ 1 + |p (z0)| 2 } + ( 1 + λ 2 (1 + λ) −α )2 int. j. anal. appl. 19 (1) (2021) 131 ⇒ re [ f (z0) z0f ′ (z0) { 1 − ( 1 + λ 2 (1 + λ) −α ) z0f ′′ (z0) f ′ (z0) }] ≤ − 1 2 ( 1 + λ 2 (1 + λ) −α )( 1 + λ−α 2 (1 + λ) −α ) |p (z0)| 2 + 1 2 { 3 ( 1 + λ 2 (1 + λ) −α )2 − ( 1 + λ 2 (1 + λ) −α )} and thus we write re [ f (z0) z0f ′ (z0) { 1 − ( 1 + λ 2 (1 + λ) −α ) z0f ′′ (z0) f ′ (z0) }] ≤ − 1 2 ( 1 + λ 2 (1 + λ) −α )∣∣∣∣ f (z0)z0f ′ (z0) − ( 1 + λ 2 (1 + λ) −α )∣∣∣∣2 + 1 2 { 3 ( 1 + λ 2 (1 + λ) −α )2 − ( 1 + λ 2 (1 + λ) −α )} which contradicts the hypothesis (3.7) of theorem 3.2. it follow that re [ f (z) zf ′ (z) { 1 − ( 1 + λ 2 (1 + λ) −α ) zf ′′ (z) f ′ (z) }] > − 1 2 ( 1 + λ 2 (1 + λ) −α )∣∣∣∣ f (z)zf ′ (z) − ( 1 + λ 2 (1 + λ) −α )∣∣∣∣2 + 1 2 { 3 ( 1 + λ 2 (1 + λ) −α )2 − ( 1 + λ 2 (1 + λ) −α )} . thus we complete the proof of theorem 3.2. � taking λ = 0 and ω = 0 in theorem 3.2, we get the following corollary 3.2. corollary 3.2. let f (z) ∈ a satisfies re [ f (z) zf ′ (z) { 1 − ( 1 2 −α ) zf ′′ (z) f ′ (z) }] > − 1 2 ( 1 2 −α )∣∣∣∣ f (z)zf′ (z) − ( 1 2 −α )∣∣∣∣2 + 12 { 3 ( 1 2 −α )2 − ( 1 2 −α )} then f (z) is starlike of reciprocal order 1 2−α in u. 4. subordination results and coefficient inequality for functions in the class s−1 (ω,λ) (φ,ψ; α,β) definition 4.1. (hadamard product or convolution).the hadamard product of two power series f (z) =∑∞ n=0 anz n and g (z) = ∑∞ n=0 bnz n analytic in u, is defined as then their hadamard product (or convolution), f ∗g is defined by the power series (4.1) (f ∗g) (z) = ∞∑ n=0 anbnz n = (g ∗f) (z) . int. j. anal. appl. 19 (1) (2021) 132 the functionf ∗g is also analytic in u. tariq al-hawary and b.a.frasin [10] introduce the following subclass of a by making use of the hadamard product. definition 4.2. [10] let φ (z) = z + ∑∞ n=2 δnz nand ψ (z) = z + ∑∞ n=2 µnz nbe analytic in u, such that δn ≥ 0,µn ≥ 0 and δn ≥ µnfor n ≥ 2, we say that f (z) ∈ a is in the class s−1 (φ,ψ; α,β) if f (z)∗φ (z) 6= 0, f (z) ∗ψ (z) 6= 0 and (4.2) ∣∣∣∣∣∣ 1eiβ (f(z)∗φ(z) f(z)∗ψ(z) ) − 1 2α ∣∣∣∣∣∣ < 12α (β ∈ r, 0 < α < 1, z ∈ u) . by making use of the hadamard product (4.1), we now introduce the following subclasses of a. definition 4.3. let f (z) = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) . furthermore, let φ (z) = z + ∑∞ n=2 δnz n and ψ (z) = z + ∑∞ n=2 µnz n be analytic in u, such that δn ≥ 0,µn ≥ 0 and δn ≥ µn for n ≥ 2, we say that f (z) ∈ a is in the class s−1 (ω,λ) (φ,ψ; α,β) if f (z) ∗φ (z) 6= 0, f (z) ∗ψ (z) 6= 0 and (4.3) ∣∣∣∣∣∣ 1eiβ (f(z)∗φ(z) f(z)∗ψ(z) ) − 1 + λ 2α ∣∣∣∣∣∣ < 1 + λ2α (β ∈ r, 0 < α < 1, 0 ≤ λ < 1, ω ∈ n0, z ∈ u) . theorem 4.1. let f (z) = (1 −λ) ( dωf (z) ) + λ ( dω+1f (z) ) . iff (z) ∈ a satisfies ∞∑ n=2 nω (1 −λ + λn) { (1 + λ) δn + ∣∣(1 + λ) δn − 2αe−iβµn∣∣} |an| ≤ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣(4.4) for some that |β| < π 2 and that 0 < α 1+λ < cos β, then f (z) ∈ s−1 (ω,λ) (φ,ψ; α,β) . proof. it suffices to show that∣∣∣∣∣∣ 2α(1 + λ) eiβ (f(z)∗φ(z) f(z)∗ψ(z) ) − 1 ∣∣∣∣∣∣ < 1 ⇒ ∣∣∣∣2αe−iβ (f (z) ∗ψ (z)) − (1 + λ) (f (z) ∗φ (z))(1 + λ) (f (z) ∗φ (z)) ∣∣∣∣ < 1. we observe that ∣∣∣∣2αe−iβ (f (z) ∗ψ (z)) − (1 + λ) (f (z) ∗φ (z))(1 + λ) (f (z) ∗φ (z)) ∣∣∣∣ = ∣∣∣∣(1 + λ) (f (z) ∗φ (z)) − 2αe−iβ (f (z) ∗ψ (z))(1 + λ) (f (z) ∗φ (z)) ∣∣∣∣ = ∣∣∣∣∣ { (1 + λ) − 2αe−iβ } z + ∑∞ n=2 n ω (1 −λ + λn) [ (1 + λ) δn − 2αe−iβµn ] anz n (1 + λ){z + ∑∞ n=2 n ω (1 −λ + λn) δnanzn} ∣∣∣∣∣ ≤ ∣∣(1 + λ) − 2αe−iβ∣∣ + ∑∞n=2 nω (1 −λ + λn) ∣∣(1 + λ) δn − 2αe−iβµn∣∣ |an| |z|n−1 (1 + λ) { 1 − ∑∞ n=2 n ω (1 −λ + λn) δn |an| |z| n−1 } . int. j. anal. appl. 19 (1) (2021) 133 it follows that the last term is bounded by 1 if ∞∑ n=2 nω (1 −λ + λn) { (1 + λ) δn + ∣∣(1 + λ) δn − 2αe−iβµn∣∣} |an| ≤ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣ for some that |β| < π 2 and that 0 < α 1+λ < cos β. � taking λ = 0 and ω = 0 in theorem 4.1, we get corollary 4.1 given by tariq al-hawary et al. [10]. corollary 4.1. [10] if f (z) ∈ a satisfies ∞∑ n=2 { δn + ∣∣δn − 2αe−iβµn∣∣} |an| ≤ 1 − ∣∣1 − 2αe−iβ∣∣ for some that |β| < π 2 and that 0 < α < cos β, then f (z) ∈ s−1 (φ,ψ; α,β) . now, to proceed our subordination results in this section, let us first recall the following definition and lemma. definition 4.4. (subordination principle) given two functions f (z) , g (z) ∈ a in u, g be univalent in u ,f (0) = g (0) and f (u) ⊂ g (u) , then we say that the function f (z) is subordinate to g (z) in u, and write f (z) ≺ g (z) ,z ∈ u. moreover, we say that g (z) is superordinate to f (z) in u. definition 4.5. a sequence {bn} ∞ n=1 of complex numbers is said to be a subordinating factor sequence if, whenever f (z) of the form 1.1, a1 = 1 is analytic, univalent and convex in u ,we have the subordination given by z + ∞∑ k=1 akbkz k ≺ f (z) , (z ∈ u). the following lemma is due to wilf [3]. lemma 4.1. ( [3], [7]) the sequence {bn} ∞ n=1 is a subordinating factor sequence if and only if re { 1 + 2 ∞∑ n=1 bnz n } > 0 (z ∈ u) . let s∗ −1 (ω,λ) (φ,ψ; α,β) ⊆ s−1 (ω,λ) (φ,ψ; α,β) is denote the subclass of functions f ∈ s whose coefficients an satisfy the inequalities (4.4). employing the techniques used by t.al-hawary and b.a.frasin [10], we state and prove the following theorem. theorem 4.2. let f (z) ∈ s∗ −1 (ω,λ) (φ,ψ; α,β) and nω (1 −λ + λn) { (1 + λ) δn + ∣∣(1 + λ) δn − 2αe−iβµn∣∣} is increasing function for n ≥ 2, |β| < π 2 ,0 < α 1+λ < cos β. then (4.5) (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣ 2 [ 2−ω + (1 + λ) δ2 − ∣∣∣2−ω − 21−ω1+λ αe−iβ∣∣∣ + |(1 + λ) δ2 − 2αe−iβµ2|] (f ∗g) (z) ≺ g (z) int. j. anal. appl. 19 (1) (2021) 134 for every function g (z) in the class k and (4.6) re f (z) > − [ 2−ω + (1 + λ) δ2 − ∣∣∣2−ω − 21−ω1+λ αe−iβ∣∣∣ + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣] (1 + λ) δ2 + |(1 + λ) δ2 − 2αe−iβµ2| for z ∈ u. the constant (4.7) (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣ 2 [ 2−ω + (1 + λ) δ2 − ∣∣∣2−ω − 21−ω1+λ αe−iβ∣∣∣ + |(1 + λ) δ2 − 2αe−iβµ2|] cannot be replace by any larger one. proof. let f (z) = z + ∑∞ n=2 anz n ∈ s∗ −1 (ω,λ) (φ,ψ; α,β) and suppose that g (z) = z + ∑∞ n=2 dnz n ∈ k.then 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] (f ∗g) (z) = 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] (z + ∑∞ n=2 andnz n) ; = ∑∞ n=1 { 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] } andnz n; (a1 = 1,d1 = 1). thus, by definition 4.5, the assertion of our theorem will hold if the sequence (4.8) { 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] an }∞ n=1 is a subordinating factor sequence with a1 = 1. in view of lemma 4.1, this will be the case if and only if re { 1 + 2 ∑∞ n=1 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] anz n } > 0 for z ∈ u. since, nω (1 −λ + λn) { (1 + λ) δn + ∣∣(1 + λ) δn − 2αe−iβµn∣∣} is increasing for all n ≥ 2, |β| < π2 , 0 < α 1+λ < cos β, we obtain re { 1 + ∑∞ n=1 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] anz n } = re   1 + 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] z+ 1 [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}]∑∞ n=2 2 ω (1 + λ) { (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}anzn   ≥   1 − 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] r− 1 [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}]∑∞ n=2 n ω (1 −λ + λn) { (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣} |an|rn   > 1 − 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] r − (1+λ)−|(1+λ)−2αe−iβ| [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] r = 1 −r since |z| = r < 1, therefore we obtain re { 1 + 2 ∑∞ n=1 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] anz n } > 0, int. j. anal. appl. 19 (1) (2021) 135 which by lemma 4.1 shows that{ 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] an }∞ n=1 (a1 = 1) is a subordinating factor sequence and hence also the subordination result (4.5). the inequality (4.6) follows from (4.5) by taking g (z) = z 1−z . to prove the sharpness of the constant 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] , we consider the function f0 (z) = z − (1+λ)−|(1+λ)−2αe−iβ| 2ω(1+λ)[(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|] z2(|β| < π 2 , 0 < α 1+λ < cos β), which is a member of the class s∗ −1 (ω,λ) (φ,ψ; α,β) . thus from the relation (4.5), we obtain 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] f0 (z) ≺ z1−z . since re ( z 1−z ) > −1 2 , |z| = r, this implies (4.9) re { 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] ×f0 (z) ∗ z1−z } > −1 2 . therefore, we have re{f (z)} > −[ (1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} which is equation (4.6). now to show that sharpness of the constant factor 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} [(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] . we consider the function f0 (z) = z − (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣ 2ω (1 + λ) [(1 + λ) δ2 + |(1 + λ) δ2 − 2αe−iβµ2|] z2. let ζ = [ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣ + 2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}] . applying equation (4.5) with g (z) = z 1−z , we have 1 2ζ [{ 2ω (1 + λ) { (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z −{(1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2] ≺ z1−z . using the fact that |re (z)| ≤ |z| , we now show that the (4.10) min z∈u  re 12ζ .   {2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z−{ (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2     = −12 int. j. anal. appl. 19 (1) (2021) 136 we have∣∣∣re 12ζ .[{2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z −{(1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2]∣∣∣ ≤ ∣∣∣ 12ζ .[{2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z −{(1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2]∣∣∣ ≤ ∣∣∣ 12ζ .[{2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}]∣∣∣ + 12ζ . ∣∣{(1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}∣∣ = 1 2 { 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}+{(1+λ)−|(1+λ)−2αe−iβ|} ζ } = 1 2 for |z| = 1. this implies that∣∣∣re 12ζ .[{2ω (1 + λ) {(1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z − { (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2]∣∣ ≤ 1 2 and therefore −1 2 ≤ re 1 2ζ . [{ 2ω (1 + λ) { (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z − { (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2] ≤ 1 2 . hence, we have that min z∈u { re 1 2ζ . [{ 2ω (1 + λ) { (1 + λ) δ2 + ∣∣(1 + λ) δ2 − 2αe−iβµ2∣∣}}z] − { (1 + λ) − ∣∣(1 + λ) − 2αe−iβ∣∣}z2]} = −1 2 . that is re { 2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|} 2[(1+λ)−|(1+λ)−2αe−iβ|+2ω(1+λ){(1+λ)δ2+|(1+λ)δ2−2αe−iβµ2|}] (f0 ∗g) (z) } = −1 2 which shows the equation (4.10). � taking λ = 0 and ω = 0 in theorem 4.2, we get corollary 4.2 given by tariq al-hawary et al. [10]. corollary 4.2. let f (z) ∈ s∗ −1 (φ,ψ; α,β) and δn + ∣∣δn − 2αe−iβµn∣∣ is increasing function for n ≥ 2, |β| < π 2 ,0 < α < cos β. then δ2 + ∣∣δ2 − 2αe−iβµ2∣∣ 2{1 + δ2 −|1 − 2αe−iβ| + |δ2 − 2αe−iβµ2|} (f ∗g) (z) ≺ g (z) for every function g (z) in the class k and re f (z) > − 1 + δ2 − ∣∣1 − 2αe−iβ∣∣ + ∣∣δ2 − 2αe−iβµ2∣∣ δ2 + |δ2 − 2αe−iβµ2| for z ∈ u. the constant δ2 + ∣∣δ2 − 2αe−iβµ2∣∣ 2{1 + δ2 −|1 − 2αe−iβ| + |δ2 − 2αe−iβµ2|} cannot be replace by any larger one. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (1) (2021) 137 references [1] b. frasin, m. abd sabri, sufficient conditions for starlikeness of reciprocal order. eur. j. pure appl. math. 10(4) (2017), 871-876. [2] g.s. salagean, subclasses of univalent functions, in: c.a. cazacu, n. boboc, m. jurchescu, i. suciu (eds.), complex analysis — fifth romanian-finnish seminar, springer berlin heidelberg, berlin, heidelberg, 1983: pp. 362–372. [3] h.s. wilf, subordinating factor sequences for convex maps of the unit circle, proc. amer. math. soc. 12 (1961), 689-693. [4] j. nishiwaki, s. owa, coefficient inequalities for starlike and convex functions of reciprocal order, electron. j. math. anal. appl. 1(2) (2013), 212-216. [5] m. kamali, some results associated with distortion bounds and coefficient inequalities for certain new subclasses of analytic functions, tbilisi math. j. 6 (2013), 21-27. [6] m. nunokawa, s.p. goyal, r. kumar, sufficient conditions for starlikeness, j. class. anal. 1(1) (2012), 85-90. [7] r. a. bello, on a subordination result of a subclass of analytic functions, adv. pure math. 7 (2017), 641-646. [8] r.j. libera, univalent spiral functions, canad. j. math. 19 (1967), 725-733. [9] s. owa, f. sağsöz, m. kamali, on some results for subclass of β− spirallike functions of order, tamsui oxford j. inform. math. sci. 28(1) (2012), 79-93. [10] t. al-hawar, b.a. frasin, coefficient estimates and subordination properties for certain classes of analytic functions of reciprocal order, stud. univ. babeş-bolyai math. 63(2) (2018), 203-212. 1. introduction and preliminaries 2. some results and coefficient inequality for functions in the class s0=x"010c( 0=x"010b,0=x"0115) 3. sufficient conditions for starlikeness of reciprocal order 4. subordination results and coefficient inequality for functions in the class s( ,0=x"0115) -1( 0=x"011e,0=x"0120;0=x"010b,0=x"010c) references int. j. anal. appl. (2022), 20:30 some hermite-hadamard inequalities via generalized fractional integral on the interval-valued coordinates jen chieh lo∗ general education center, national taipei university of technology, taipei, taiwan ∗corresponding author: jclo@mail.ntut.edu.tw abstract. in this paper, we established the hermite-hadamard inequalities via generalized fractional. meanwhile, interval analysis is a particular case of set-interval analysis. we established the fractional inequalities and these results are an extension of a previous research. 1. introduction the hermite-hadamard inequality, which is the first basic result of convex mappings with a nature geometric interpretation and extensive use, has attracted attention with great interest in elementrary mathematics. the hermite-hadamard type inequality, which is defined by: f ( a + b 2 ) ≤ 1 b−a ∫ b a f (x) dx ≤ f (a) + f (b) 2 , where f : i ⊂r→r is a convex function on the closed bounded interval i of r,and a,b ∈ i with a < b. in the past decade, fractional calculus has been regarded as one of best tools to describe longmemory processes. many researchers are interested in such a model. the subject of fractional calculus has gained considerable popularity and importance due mainly to its demonstrated applications in numerous seemly diverse and widespread fields of science and engineering. the most important of these models are described by differential equations with fractional derivatives. received: may 2, 2022. 2010 mathematics subject classification. 34a40. key words and phrases. hermite-hadamard inequalities; generalized fractional; interval-valued. https://doi.org/10.28924/2291-8639-20-2022-30 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-30 2 int. j. anal. appl. (2022), 20:30 2. fractional integrals in [28], sarikaya et al. obtained the following hermite-hadamard’s inequalities in fractional integral form: f ( a + b 2 ) ≤ γ (α + 1) 2 (b−a)α [ iαa+f (b) + i α b−f (a) ] ≤ f (a) + f (b) 2 , where g : [a,b] ⊂ r → r is assumed to be a positive convex function on [a,b] , g ∈ l1 [a,b] with a < b, and iα a+ + iα b− are the left-sided and right-sided riemann-liouville fractional integrals of order α > 0, these are respectively defined as []: iα a+ f (x) = 1 γ(α) ∫ x a (x − t)α−1 f (t) dt, x > a, and iα b− f (x) = 1 γ(α) ∫b x (t −x)α−1 f (t) dt, x < b. in [16],katugampola introduced a new fractional which generalizes the riemann-liouville and the hadamard fractional integrals into a single form as follow. definition 2.1 let [a,b] ⊂r be a finite interval. then, the leftand right-side katugampola fractional integralsod order α > 0 of f ∈ xpc (a,b) are defined by ρiαa+f (x) = ρ1−α γ (α) ∫ x a tρ−1 (xρ − tρ)1−α f (t) dt and ρiαb−f (x) = ρ1−α γ (α) ∫ b x tρ−1 (tρ −xρ)1−α f (t) dt where a < x < b and ρ > 0, if the integral exists. theorem 2.2 let α > 0 and ρ > 0. then for x > a, 1. lim ρ→1 ρiα a+ f (x) = jα a+ f (x) , 2. lim ρ→0+ ρiα a+ f (x) = hα a+ f (x) . similar results also hold for right-sided operators. theorem 2.3 let α > 0 and ρ > 0. let f : [aρ,bρ] →r be a positive function with 0 ≤ a ≤ b and f ∈ xpc (a,b) . if f is also a convex function on [a,b] , then the following inequalities hold: f ( aρ + bρ 2 , cρ + dρ 2 ) ≤ ραγ (α + 1) 2 (bρ −aρ)α [ ρiαa+f (b ρ) +ρ iαb−f (a ρ) ] ≤ f (aρ) + f (bρ) 2 int. j. anal. appl. (2022), 20:30 3 where the fractional integral are considered for the function f (xρ) and evaluated at a and b, respectively. in [31], sarikaya and ertuğral gave the definition of generalized fractional integrals (gfis) as following: definition 2.4 the left-sided and right-sided gfis are denoted by a+iϕ and b−iϕ as followings: a+iϕf (x) = ∫ x a ϕ(x−t) x−t dt, x > a, and b−iϕf (x) = ∫b x ϕ(t−x) t−x dt, x < b, where a function ϕ : [0,∞) → [0,∞) satisfies the condition ∫ 1 0 ϕ(t) t dt < ∞. in [31], sarikaya et al. obtained the following hermite-hadamard’s inequalities for gfis under the condition of convexity as follows: theorem 2.5 for a convex function f : [a,b] →r on [a,b] with a < b, then the following inequalities hold: f ( a + b 2 ) ≤ 1 λ (1) [a+iϕf (b) +b− iϕf (a)] ≤ f (a) + f (b) 2 , where λ (x) = ∫ x 0 ϕ((b−a)t) t dt < ∞. the most important feature of generalized fractional integrals is that they generalize some type of fractional integrals such as the riemann-liouville fractional integral, k-riemann-liouville fractional integral, katugampola fractional integrals, conformable fractional, and hadamard feractional integrals. these important special cases of integral operator are mentioned below. (1) if we choose ϕ (x) = x, the operators a+iϕf (x) and b−iϕf (x) are reduce to the riemann integral. (2) considering ϕ (x) = x α γ(α) and α > 0, the operators a+iϕf (x) and b−iϕf (x) are reduce to the riemann-liouville fractioal integrals iα a+ f (x) and iα b− f (x), respectively. here, γ is a gamma function. (3) for ϕ (x) = 1 kγ(α) x α k and α,k > 0, the operators a+iϕf (x) and b−iϕf (x) are reduce to the k-riemann-liouville fractional integrals iα a+,k f (x) and iα b−,k f (x), respectively. here, γk is a k-gamma function. on the other hand, interval analysis is a particular case of set-valued analysis which is the study of sets in the spirit of mathematical analysis and general topology. it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. an old example of interval enclosure is archimede’s method which is related to the computation of the circumference of circle. in 1966, the first book related to interval analysis was given by moore who is known as the first user of intervals in computational mathematics. 4 int. j. anal. appl. (2022), 20:30 after this book, several scientists started to investigate theory and application of interval arithmetic. nowadays, because of its application, interval analysis is a useful tool in various areas related to uncertain data. we can see applications in computer graphics, experimental and computational physics, error analysis, robotics and many others. 3. interval calculus a real valued interval x is bounded, closed subset of r and is defined by x = [ x,x ] = { t ∈r : x ≤ t ≤ x } where x,x ∈ r and x ≤ x. the number x and x are called the left and right endpoints of interval x, respectively. when x = x = a, the interval x is said to be degenerate and we use the form x = a = [a,a] . also we call x positive if x> 0 or negative if x < 0. the set of all closed intervals of r, the sets of all closed positive intervals of r and closed negative intervals of r is denoted by ri,r+i and r − i , respectively. the pompeiu-hausdorff distance between the intervals x and y is defined by d (x,y ) = d ([ x,x ] , [ y ,y ]) = max { |x −y | , ∣∣x −y ∣∣} . it is known that (ri,d) is a complete metric space. now, we give the definitions of basic interval arithmetic operations for the intervals x and y as follows: x + y = [ x + y ,x + y ] , x −y = [ x −y ,x −y ] , x ·y = [min s, max s] where s = { xy ,xy ,xy ,xy } , x/y = [min t, max t ] where t = { x/y ,x/y ,x/y ,x/y } and 0 /∈ y. scalar multiplication of the interval x is defined by λx = λ [ x,x ] =   [ λx,λx ] , λ > 0, 0, λ = 0,[ λx,λx ] , λ < 0, where λ ∈r. the opposits of the interval x is int. j. anal. appl. (2022), 20:30 5 −x := (−1) x = [ −x,−x ] , where λ = −1. the subtraction is given by x −y = x + (−y ) = [ x −y ,x −y ] . in general, −x is not additive inverse for x, i.e. x −x 6= 0. use of monotonic functions f (x) = [ f (x) ,f ( x )] . the definitons of operations lead to a number of algebraic properties which allows ri to be quasilinear space. they can be listed as follows (1)(associativity of addition) (x + y ) + z = x + (y + z) for all x,y,z ∈ri, (2)(additivity elemant) x + 0 = 0 + x = x for all x ∈ri, (3)(commutativity of addition) x + y = y + x for all x,y ∈ri, (4)(cancellation law) x + z = y + z =⇒ x = y for all x,y,z ∈ri, (5)(associativity of multiplication) (x ·y ) ·z = x · (y ·z) for all x,y,z ∈ri, (6)(commutativity of multiplication) x ·y = y ·x for all x,y ∈ri, (7)(unity element) x · 1 = 1 ·x for all x ∈ri, (8)(associativity law) λ (µx) = (λµ) x for all x ∈ri, and for all λ,µ ∈r, (9)(first distributiviyu law) λ (x + y ) = λx + λy for all x,y ∈ri, and for all λ ∈r, (10)(second distributiviyu law) (λ + µ) x = λx + µx for all x ∈ri, and for all λ,µ ∈r. but, this law holds in certain cases. if y ·z > 0, then x ·y + z = x ·y + x ·z. what’s more, one of the set property is the inclusion ⊆ that is given by x ⊆ y ⇐⇒ y ≤ x and x ≤ y . considering together with arthmetic operations and inclusion, one has the following property which is called inclusion isotone of interval operations: let � be the addition, multiplication, subtraction or division. if x,y,z and t areintervals such that x ⊆ y and z ⊆ t, then the following relation is valid 6 int. j. anal. appl. (2022), 20:30 x �z ⊆ y �t. 4. intgral of interval-valued functions in this section, the notion of integral is mentioned for interval-valued functions. before the definition of integral, the necessary concepts will be given as the following: a function f is said to be an interval-valued function of t on [a,b] , if it assigns a nonempty interval to each t ∈ [a,b] , f (t) = [ f (t) ,f (t) ] . a partition of [a,b] is any finite ordered subset p having the form: p : a = t0 < t1 < ... < tn = b. the mesh of a partition p defined by mesh (p ) = max{ti − ti−1 : i = 1, 2, ...,n} . we denoted by p ([a,b]) the set of all partition of [a,b] . let p (δ, [a,b]) be the set of all p ∈ p ([a,b]) such that mesh (p ) < δ. choose an arbitrary point ξi in interval [ti−1,ti ] , (i = 1, 2, ...,n) and let us define the sum s (f,p,δ) = n∑ i=1 f (ξi ) [ti − ti−1] , where f : [a,b] →ri. we call s (f,p,δ) a riemann sum of f corresponding to p ∈ p (δ, [a,b]) . definition 4.1 a function f : [a,b] →ri is called interval riemann intrgrable ( (ir)-integrable) on [a,b] , if there exists a ∈ri such that, for each � > 0, there exists δ > 0 such that d (s (f,p,δ) ,a) < � for every riemann sum s of f corresponding to each p ∈ p (δ, [a,b]) and independent from choice of ξi ∈ [ti−1,ti ] for all 1 ≤ i ≤ n. in this case, a is called the (ir)-integral of f on [a,b] and is denoted by a = (ir) b∫ a f (t) dt. the collection of all functions that are (ir)-integrable on [a,b] will be denoted by ir([a,b]). int. j. anal. appl. (2022), 20:30 7 the following theorem gives relation between (ir)−integrable and riemann integrable (r)integrable. theorem 4.2 let f : [a,b] → ri be an interval-valued function such that f (t) = [ f (t) ,f (t) ] . f ∈ ir([a,b]) if and only if f (t) ,f (t) ∈ r([a,b]) and (ir) b∫ a f (t) dt =  (r) b∫ a f (t) dt, (r) b∫ a f (t) dt   , where r([a,b]) denoted the all r-integrable functions. it is seen easily that, if f (t) ⊆ g (t) for all t ∈ [a,b] , then (ir) b∫ a f (t) dt ⊆ (ir) b∫ a g (t) dt. furthermore, if {ti−1,ti} m i=1 is a δ-fine p1 of [a,b] and if { sj−1,sj }n j=1 is a δ-fine p2 of [c,d] , then retangles 4i,j = [ti−1,ti ] × [ sj−1,sj ] are the partition of retangle 4 = [a,b] × [c,d] and the point ( ξi,ηj ) are inside the retangles [ti−1,ti ] × [ sj−1,sj ] . and we denote the set of all δ-fine partition p of 4 with p1 ×p2, where p1 ∈ p (δ, [a,b]) and p2 ∈ p (δ, [c,d]) . let 4ai,j be the area retangle 4i,j, where 1 ≤ i ≤ m, 1 ≤ j ≤ n, choose arbitrary ( ξi,ηj ) and get s (f,p,δ,4) = m∑ i=1 n∑ j=1 f ( ξi,ηj ) 4ai,j. definition 4.3 a function f : 4 → ri is called interval double riemann integrable ( (id)-integrable) on 4 = [a,b] × [c,d] with the id-integral i = (id) ∫∫ 4 f (t,s) da, if there exists i ∈ri such that, for each � > 0, there exists δ > 0 such that d (s (f,p,δ,4) , i) < � for each p ∈ p (δ,4) . we denote by ir(4) the set of all id-integrable function on 4, and by r([a,b]), ir([a,b]), the set of all r-integrable and ir-integrable functions on [a,b] ,respectively. theorem 4.4 let 4 = [a,b] × [c,d] . if f : 4→ri is id-integrable on 4, then we have (id) ∫∫ 4 f (t,s) da = (ir) ∫ b a (ir) ∫ d c f (s,t) dsdt. 8 int. j. anal. appl. (2022), 20:30 in [25], sadowska obtained the following hermite-hadamard inequality for interval-valued functions: theorem 4.5 let f : [a,b] → r+ i be an interval-valued function such that f (t) = [ f (t) ,f (t) ] and f ∈ ir([a,b]). then f (a) + f (b) 2 ⊆ 1 b−a (ir) b∫ a f (t) dt ⊆ f ( a + b 2 ) . 5. hermite-hadamard inequalities for generalized fractional on the interval-value coordinates throughout this study, we hope to generalize the hermite-hadamard inequalities for generalized fractional on the interval-value coordinates. for bievity, we define a+,c+iϕf (x,y) = ∫ y c ∫ x a ϕ (x − t) ϕ (y − s) (x − t) (y − s) dtds, a+,d−iϕf (x,y) = ∫ d y ∫ x a ϕ (x − t) ϕ (s −y) (x − t) (s −y) dtds, b−,c+iϕf (x,y) = ∫ y c ∫ b x ϕ (t −x) ϕ (y − s) (t −x) (y − s) dtds, and b−,d−iϕf (x,y) = ∫ d y ∫ b x ϕ (t −x) ϕ (s −y) (t −x) (s −y) dtds. theorem 5.1 let f : i × i → r be an interval-valued convex function such that f (t) = [ f (t) , f (t) ] and a,b,c,d ∈ i with a < b and c < d. if f ∈ id([a,b]×[c,d]), then the following inequalities for generalized fractional integral hold: f ( a + b 2 , c + d 2 ) ⊇ 1 4ω (1, 1) [ a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) ] ⊇ f (a,c) + f (a,d) + f (b,c) + f (b,d) 4 , where ω (x,y) = ∫ x 0 ∫ y 0 ϕ((b−a)t)ϕ((d−c)s) ts dtds. proof : for t ∈ [0, 1] , let x = ta + (1 − t) b,y = (1 − t) a + tb,z = sc + (1 − s) d,w = (1 − s) c + sd. due to the convexity of f , int. j. anal. appl. (2022), 20:30 9 f ( x + y 2 , z + w 2 ) ⊇ f (x,z) + f (x,w) + f (y,z) + f (y,w) 4 we get 4f ( a + b 2 , c + d 2 ) ⊇ f (ta + (1 − t) b,sc + (1 − s) d) + f ((1 − t) a + tb,sc + (1 − s) d) +f (ta + (1 − t) b, (1 − s) c + sd) + f ((1 − t) a + tb, (1 − s) c + sd) . multiplying both side by ϕ((b−a)t)ϕ((d−c)s) ts , then integrating the resulting inequality with reapect to t over (0, 1] , we obtain 4f ( a + b 2 , c + d 2 ) (id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts dtds ⊇ (id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts f (ta + (1 − t) b,sc + (1 − s) d) dtds +(id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts f ((1 − t) a + tb,sc + (1 − s) d) dtds +(id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts f (ta + (1 − t) b, (1 − s) c + sd) dtds +(id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts f ((1 − t) a + tb, (1 − s) c + sd) dtds = a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) . and then, since f is convex, for every t ∈ [0, 1] , we have f (ta + (1 − t) b,sc + (1 − s) d) + f ((1 − t) a + tb,sc + (1 − s) d) +f (ta + (1 − t) b, (1 − s) c + sd) + f ((1 − t) a + tb, (1 − s) c + sd) ⊇ tsf (a,c) + t (1 − s) f (a,d) + (1 − t) (1 − s) f (b,c) + (1 − t) sf (b,d) +t (1 − s) f (a,c) + tsf (a,d) + (1 − t) sf (b,c) + (1 − t) (1 − s) f (b,d) + (1 − t) sf (a,c) + (1 − t) (1 − s) f (a,d) + tsf (b,c) + t (1 − s) f (b,d) + (1 − t) (1 − s) f (a,c) + (1 − t) sf (a,d) + t (1 − s) f (b,c) + tsf (b,d) = f (a,c) + f (a,d) + f (b,c) + f (b,d) . multiplying both side by ϕ((b−a)t)ϕ((d−c)s) ts , then integrating the resulting inequality with reapect to t over (0, 1] , we obtain 10 int. j. anal. appl. (2022), 20:30 a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) ⊇ [f (a,c) + f (a,d) + f (b,c) + f (b,d)] (id) ∫ 1 0 ∫ 1 0 ϕ ((b−a) t) ϕ ((d −c) s) ts dtds. lemma 5.2 let f : i × i → r be an interval-valued convex function such that f (t) = [ f (t) , f (t) ] and a,b,c,d ∈ i with a < b and c < d.if f and ∂ 2 ∂t∂s f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂s f ∣∣∣ is convex, then the following equalities for generalized fractional integrals hold: f (a,c) + f (a,d) + f (b,c) + f (b,d) 4 + 1 4ω (1, 1) [ a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) ] = (b−a) (d −c) 4ω (1, 1) ×(id) ∫ 1 0 ∫ 1 0 ω (t,s) [ ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) − ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) − ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) + ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) ] dsdt + ∆ (1) 4ω (1, 1) [a+iϕf (b,d) +a+ iϕf (b,c) +b− iϕf (a,d) +b− iϕf (a,c)] + λ (1) 4ω (1, 1) [c+iϕf (b,d) +d− iϕf (b,c) +c+ iϕf (a,d) +d− iϕf (a,c)] where λ (t) = (ir) ∫ t 0 ϕ((b−a)u) u du and ∆ (s) = (ir) ∫ s 0 ϕ((d−c)λ) λ dλ. proof : here, we apply integration by parts, then we have s1 = (b−a) (d −c) 4ω (1, 1) (id) ∫ 1 0 ∫ 1 0 λ (t,s) ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) dsdt s2 = (b−a) (d −c) 4ω (1, 1) (id) ∫ 1 0 ∫ 1 0 −λ (t,s) ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) dsdt s3 = (b−a) (d −c) 4ω (1, 1) (id) ∫ 1 0 ∫ 1 0 λ (t,s) ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) dsdt and int. j. anal. appl. (2022), 20:30 11 s4 = (b−a) (d −c) 4ω (1, 1) (id) ∫ 1 0 ∫ 1 0 −λ (t,s) ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) dsdt. if we add from s1 to s4 and multiply by (b−a) (d −c) , we obtain the proof. theorem 5.3 let f : i × i → r be an interval-valued convex function such that f (t) = [ f (t) , f (t) ] and a,b,c,d ∈ i with a < b and c < d. if f and ∂ 2 ∂t∂s f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂s f ∣∣∣ is convex, then the following inequalities for generalized fractional integral hold: ∣∣∣∣f (a,c) + f (a,d) + f (b,c) + f (b,d)4 − 1 4ω (1, 1) [ a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) ]∣∣∣∣ ⊇ (b−a) (d −c) ω (1, 1) (id) ∫ 1 0 ∫ 1 0 ts |ω (t,s) − ω (t, 1 − s) − ω (1 − t,s) + ω (1 − t, 1 − s)|dsdt ×   ∣∣∣ ∂2∂t∂s f (a,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (a,d)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,d)∣∣∣ 4   + ∆ (1) 4ω (1, 1) [a+iϕf (b,d) +a+ iϕf (b,c) +b− iϕf (a,d) +b− iϕf (a,c)] + λ (1) 4ω (1, 1) [c+iϕf (b,d) +d− iϕf (b,c) +c+ iϕf (a,d) +d− iϕf (a,c)] . proof : using lemma 5.2 and the convexity of ∣∣∣ ∂2∂t∂s f ∣∣∣ , then we have ∣∣∣∣f (a,c) + f (a,d) + f (b,c) + f (b,d)4 + 1 4ω (1, 1) [ a+,c+iϕf (b,d) +a+,d− iϕf (b,c) +b−,c+ iϕf (a,d) +b−,d− iϕf (a,c) ]∣∣∣∣ ⊇ (b−a) (d −c) 4ω (1, 1) ×(id) ∫ 1 0 ∫ 1 0 {[ω (t,s) − ω (t.1 − s) − ω (1 − t,s) + ω (1 − t, 1 − s)] × ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) } dsdt + ∆ (1) 4ω (1, 1) [a+iϕf (b,d) +a+ iϕf (b,c) +b− iϕf (a,d) +b− iϕf (a,c)] + λ (1) 4ω (1, 1) [c+iϕf (b,d) +d− iϕf (b,c) +c+ iϕf (a,d) +d− iϕf (a,c)] 12 int. j. anal. appl. (2022), 20:30 ⊇ (b−a) (d −c) ω (1, 1) (id) ∫ 1 0 ∫ 1 0 ts |λ (1 − t,s) − λ (t,s) + λ (t, 1 − s) − λ (1 − t, 1 − s)|dsdt ×   ∣∣∣ ∂2∂t∂s f (a,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (a,d)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,d)∣∣∣ 4   + ∆ (1) 4ω (1, 1) [a+iϕf (b,d) +a+ iϕf (b,c) +b− iϕf (a,d) +b− iϕf (a,c)] + λ (1) 4ω (1, 1) [c+iϕf (b,d) +d− iϕf (b,c) +c+ iϕf (a,d) +d− iϕf (a,c)] . this completes the proof. lemma 5.4 let f : i × i → r be an interval-valued convex function such that f (t) = [ f (t) , f (t) ] and a,b,c,d ∈ i with a < b and c < d. if f and ∂ 2 ∂t∂s f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂s f ∣∣∣ is convex, then the following equalities for generalized fractional integrals hold: 4f ( a + b 2 , c + d 2 ) − f ( a, c + d 2 ) − f ( b, c + d 2 ) − f ( a + b 2 ,c ) − f ( a + b 2 ,d ) +2f (a,c) + 2f (a,d) + 2f (b,c) + 2f (b,d) + 1 ω (1, 1) [ b−,d−iϕf (a,c) +b−,c+ iϕf (a,d) +a+,d− iϕf (b,c) +a+,c+ iϕf (b,d) −b− iϕf (a,c) −b− iϕf (a,d) −a+ iϕf (b,c) −a+ iϕf (b,d) −d−iϕf (a,c) −c+ iϕf (a,d) −d− iϕf (b,c) −c+ iϕf (b,d)] = 1 ω (1, 1) 16∑ k=1 jk where j1 = (id) ∫ 1 2 0 ∫ 1 2 0 λ1 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) dsdt, j2 = (id) ∫ 1 2 0 ∫ 1 2 0 −λ1 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) dsdt, j3 = (id) ∫ 1 2 0 ∫ 1 2 0 −λ1 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) dsdt, j4 = (id) ∫ 1 2 0 ∫ 1 2 0 λ1 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) dsdt, j5 = (id) ∫ 1 2 0 ∫ 1 1 2 −λ2 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) dsdt, j6 = (id) ∫ 1 2 0 ∫ 1 1 2 λ2 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) dsdt, int. j. anal. appl. (2022), 20:30 13 j7 = (id) ∫ 1 2 0 ∫ 1 1 2 λ2 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) dsdt, j8 = (id) ∫ 1 2 0 ∫ 1 1 2 −λ2 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) dsdt, j9 = (id) ∫ 1 1 2 ∫ 1 2 0 −λ3 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) dsdt, j10 = (id) ∫ 1 1 2 ∫ 1 2 0 λ3 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) dsdt, j11 = (id) ∫ 1 1 2 ∫ 1 2 0 λ3 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) dsdt, j12 = (id) ∫ 1 1 2 ∫ 1 2 0 −λ3 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) dsdt, j13 = (id) ∫ 1 1 2 ∫ 1 1 2 λ4 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b, (1 − s) c + sd) dsdt, j14 = (id) ∫ 1 1 2 ∫ 1 1 2 −λ4 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb, (1 − s) c + sd) dsdt, j15 = (id) ∫ 1 1 2 ∫ 1 1 2 −λ4 (t,s) ∂2 ∂t∂s f (ta + (1 − t) b,sc + (1 − s) d) dsdt, j16 = (id) ∫ 1 1 2 ∫ 1 1 2 λ4 (t,s) ∂2 ∂t∂s f ((1 − t) a + tb,sc + (1 − s) d) dsdt, and λ1 (t,s) = (id) ∫ t 0 ∫ s 0 ϕ((b−a)u)ϕ((d−c)λ) uλ dudλ, λ2 (t,s) = (id) ∫ t 0 ∫ 1 s ϕ((b−a)u)ϕ((d−c)λ) uλ dudλ, λ3 (t,s) = (id) ∫ 1 t ∫ s 0 ϕ((b−a)u)ϕ((d−c)λ) uλ dudλ, λ4 (t,s) = (id) ∫ 1 t ∫ 1 s ϕ((b−a)u)ϕ((d−c)λ) uλ dudλ. proof : here, we apply integration by parts, then we completes the proof. theorem 5.5 let f : i × i → r be an interval-valued convex function such that f (t) = [ f (t) , f (t) ] and a,b,c,d ∈ i with a < b and c < d. if f and ∂ 2 ∂t∂s f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂s f ∣∣∣ is convex, then the following inequalities for generalized fractional integral hold: ∣∣∣∣4f ( a + b 2 , c + d 2 ) − f ( a, c + d 2 ) − f ( b, c + d 2 ) − f ( a + b 2 ,c ) − f ( a + b 2 ,d ) +2f (a,c) + 2f (a,d) + 2f (b,c) + 2f (b,d) 14 int. j. anal. appl. (2022), 20:30 + 1 ω (1, 1) [ b−,d−iϕf (a,c) +b−,c+ iϕf (a,d) +a+,d− iϕf (b,c) +a+,c+ iϕf (b,d) −b− iϕf (a,c) −b− iϕf (a,d) −a+ iϕf (b,c) −a+ iϕf (b,d) −d−iϕf (a,c) −c+ iϕf (a,d) −d− iϕf (b,c) −c+ iϕf (b,d)]| ⊇ 1 ω (1, 1)   ∣∣∣ ∂2∂t∂s f (a,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (a,d)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,d)∣∣∣ 4   × [ (id) ∫ 1 2 0 ∫ 1 2 0 |λ1 (t,s)|dtds + (id) ∫ 1 2 0 ∫ 1 1 2 |λ2 (t,s)|dtds +(id) ∫ 1 1 2 ∫ 1 2 0 |λ3 (t,s)|dtds + (id) ∫ 1 1 2 ∫ 1 1 2 |λ4 (t,s)|dtds ] . proof : using lemma 5.4 and the convexity of ∣∣∣ ∂2∂t∂s f ∣∣∣ , then we have ∣∣∣∣4f ( a + b 2 , c + d 2 ) − f ( a, c + d 2 ) − f ( b, c + d 2 ) − f ( a + b 2 ,c ) − f ( a + b 2 ,d ) +2f (a,c) + 2f (a,d) + 2f (b,c) + 2f (b,d) + 1 ω (1, 1) [ b−,d−iϕf (a,c) +b−,c+ iϕf (a,d) +a+,d− iϕf (b,c) +a+,c+ iϕf (b,d) −b− iϕf (a,c) −b− iϕf (a,d) −a+ iϕf (b,c) −a+ iϕf (b,d) −d−iϕf (a,c) −c+ iϕf (a,d) −d− iϕf (b,c) −c+ iϕf (b,d)]| ⊇ 1 ω (1, 1) { (id) ∫ 1 2 0 ∫ 1 2 0 λ1 (t,s) [ ts ∂2 ∂t∂s f (a,c) + t (1 − s) ∂2 ∂t∂s f (a,d) + (1 − t) s ∂2 ∂t∂s f (b,c) + (1 − t) (1 − s) ∂2 ∂t∂s f (b,d) ] dtds +(id) ∫ 1 2 0 ∫ 1 1 2 λ2 (t,s) [ −(1 − t) s ∂2 ∂t∂s f (a,c) − (1 − t) (1 − s) ∂2 ∂t∂s f (a,d) −ts ∂2 ∂t∂s f (b,c) − t (1 − s) ∂2 ∂t∂s f (b,d) ] dtds +(id) ∫ 1 1 2 ∫ 1 2 0 λ3 (t,s) [ −t (1 − s) ∂2 ∂t∂s f (a,c) − ts ∂2 ∂t∂s f (a,d) −(1 − t) (1 − s) ∂2 ∂t∂s f (b,c) − (1 − t) s ∂2 ∂t∂s f (b,d) ] dtds +(id) ∫ 1 1 2 ∫ 1 1 2 λ4 (t,s) [ (1 − t) (1 − s) ∂2 ∂t∂s f (a,c) + (1 − t) s ∂2 ∂t∂s f (a,d) +t (1 − s) ∂2 ∂t∂s f (b,c) + ts ∂2 ∂t∂s f (b,d) dtds ]} int. j. anal. appl. (2022), 20:30 15 ⊇ 1 ω (1, 1)   ∣∣∣ ∂2∂t∂s f (a,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (a,d)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,c)∣∣∣ + ∣∣∣ ∂2∂t∂s f (b,d)∣∣∣ 4   × [ (id) ∫ 1 2 0 ∫ 1 2 0 |λ1 (t,s)|dtds + (id) ∫ 1 2 0 ∫ 1 1 2 |λ2 (t,s)|dtds +(id) ∫ 1 1 2 ∫ 1 2 0 |λ3 (t,s)|dtds + (id) ∫ 1 1 2 ∫ 1 1 2 |λ4 (t,s)| ] dtds. conclusion: in this work, the author established hermite-hadamard type inequalities via generalized fractional integral. furthermore, the author extend the inequalities on interval-valued coordinated. acknowledgment: the author would like to express their sincere to the editor and the anonmous reviewers for their helpful comments and suggestions. funding: the work was supportes by the ministry of science and technology of taiwan (most1102115-m-027-003-my2). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.a. ali, h. budak, a. akkurt, y.-m. chu, quantum ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus, open math. 19 (2021), 440–449. https://doi.org/10.1515/math-2021-0020. [2] m.a. ali, h. budak, g. murtaza, y.-m. chu, post-quantum hermite–hadamard type inequalities for interval-valued convex functions, j. inequal. appl. 2021 (2021), 84. https://doi.org/10.1186/s13660-021-02619-6. [3] m.a. ali, h. budak and m.z. sarikaya, new inequalities of hermite-hadamard type for h-convex functions via generalized fractional integrals, progr. fract. differ. appl. 7 (2021), 307–316. https://doi.org/10.18576/pfda/ 070409. [4] m.z. sarikaya, f. ertuğral, on the generalized hermite-hadamard inequalities, ann. univ. craiova, math. computer sci. ser. 47 (2020), 193–213. [5] h. budak, f. hezenci, h. kara, on parameterized inequalities of ostrowski and simpson type for convex functions via generalized fractional integrals, math. meth. appl. sci. 44 (2021), 12522–12536. https://doi.org/10.1002/ mma.7558. [6] h. budak, e. pehlivan, p. kösem, on new extensions of hermite-hadamard inequalities for generalized fractional integrals, sahand commun. math. anal. 18 (2021), 73-88. https://doi.org/10.22130/scma.2020.121963.759. [7] h. chen, u.n. katugampola, hermite-hadamard and hermite-hadamard-fejér type inequalities for generalized fractional integrals, j. math. anal. appl. 446 (2017), 1274–1291. https://doi.org/10.1016/j.jmaa.2016.09. 018. [8] f. ertuğral, m.z. sarikaya, simpson type integral inequalitites for generalized fractional integral, racsam, rev. r. acad. cienc. exactas fís. nat., ser. a mat. 113 (2019), 3115-3124. https://doi.org/10.1007/ s13398-019-00680-x. [9] f. ertuğral, m.z. sarikaya, h. budak, on hermite-hadamard type inequalities associated with the generalized fractional integrals. preprint, 2019. https://www.researchgate.net/publication/334634529. https://doi.org/10.1515/math-2021-0020 https://doi.org/10.1186/s13660-021-02619-6 https://doi.org/10.18576/pfda/070409 https://doi.org/10.18576/pfda/070409 https://doi.org/10.1002/mma.7558 https://doi.org/10.1002/mma.7558 https://doi.org/10.22130/scma.2020.121963.759 https://doi.org/10.1016/j.jmaa.2016.09.018 https://doi.org/10.1016/j.jmaa.2016.09.018 https://doi.org/10.1007/s13398-019-00680-x https://doi.org/10.1007/s13398-019-00680-x https://www.researchgate.net/publication/334634529 16 int. j. anal. appl. (2022), 20:30 [10] r. gorenflo, f. mainardi, fractional calculus, in: a. carpinteri, f. mainardi (eds.), fractals and fractional calculus in continuum mechanics, springer vienna, vienna, 1997: pp. 223–276. https://doi.org/10.1007/ 978-3-7091-2664-6_5. [11] a. gozpinar, e. set, s.s. dragmir, some generalized hermite-hadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are s-convex, acta math. univ. comen. 88 (2019), 87-100. [12] j. hadamard, étude sur les propriét és des fonctions entières et en particulier d’une fonction considérée par riemann, j. math. pures. appl. 58 (1893), 171-216. http://eudml.org/doc/234668. [13] i̇. i̇şan, s. wu, hermite–hadamard type inequalities for harmonically convex functions via fractional integrals, appl. math. comput. 238 (2014), 237–244. https://doi.org/10.1016/j.amc.2014.04.020. [14] m. iqbal, m.i. bhatti, k. nazeer, generalization of inequalities analogous to hermite-hadamard inequality via fractional integrals, bull. korean math. soc. 52 (2015), 707-716. https://doi.org/10.4134/bkms.2015.52.3. 707. [15] m. jleli, d. o’regan, b. samet, on hermite-hadamard type inequalities via generalized fractional integrals, turkish j. math. 40 (2016), 1221–1230. https://doi.org/10.3906/mat-1507-79. [16] u.n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218 (2011), 860–865. https://doi.org/10.1016/j.amc.2011.03.062. [17] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002. [18] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, 1st ed, elsevier, amsterdam, boston, 2006. [19] k. liu, j. wang, d. o’regan, on the hermite–hadamard type inequality for ψ-riemann–liouville fractional integrals via convex functions, j. inequal. appl. 2019 (2019), 27. https://doi.org/10.1186/s13660-019-1982-1. [20] m. matloka, hermite-hadamard type inequalities for fractional integrals, rgmia res. rep. collect. 20 (2017), article 69. https://rgmia.org/papers/v20/v20a69.pdf. [21] n. mehreen, m. anwar, some inequalities via ψ-riemann-liouville fractional integrals, aims math. 4 (2019), 1403–1415. https://doi.org/10.3934/math.2019.5.1403. [22] s. miller, b. ross, an introduction to the fractional calculus and fractional differential equations, john wiley & sons, hoboken, 2003. [23] p.o. mohammed, new generalized riemann-liouville fractional integral inequalities for convex functions, j. math. inequal. 15 (2021), 511–519. https://doi.org/10.7153/jmi-2021-15-38. [24] r.e. moore, interval analysis, prentice-hall, inc., englewood cliffs, 1966. [25] r.e. moore, r.b. kearfott, m.j. cloud, introduction to interval analysis, siam, philadelphia, pa, 2009. [26] s. mubeen, g.m. habibullah, k-fractional integrals and application, int. j. cont. math. sci. 7 (2012), 89-94. [27] e. sadowska, hadamard inequality and a refinement of jensen inequality for set—valued functions, results. math. 32 (1997), 332–337. https://doi.org/10.1007/bf03322144. [28] m.z. sarikaya, e. set, h. yaldiz, n. başak, hermite-hadamard’s inequalities for fractional integrals and related fractional integral, math. computer model. 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12. 048. [29] m.z. sarikaya, h. yaldiz, on generalization integral inequalities for fractional integrals, nihonkai math. j. 25 (2014), 93-104. [30] m.z. sarikaya, h. yaldiz, on hermite-hadamard type inequalities for ϕ-convex functions via fractional integrals, malaysian j. math. sci. 9 (2015), 243-258. https://doi.org/10.1007/978-3-7091-2664-6_5 https://doi.org/10.1007/978-3-7091-2664-6_5 http://eudml.org/doc/234668 https://doi.org/10.1016/j.amc.2014.04.020 https://doi.org/10.4134/bkms.2015.52.3.707 https://doi.org/10.4134/bkms.2015.52.3.707 https://doi.org/10.3906/mat-1507-79 https://doi.org/10.1016/j.amc.2011.03.062 https://doi.org/10.1016/j.cam.2014.01.002 https://doi.org/10.1186/s13660-019-1982-1 https://rgmia.org/papers/v20/v20a69.pdf https://doi.org/10.3934/math.2019.5.1403 https://doi.org/10.7153/jmi-2021-15-38 https://doi.org/10.1007/bf03322144 https://doi.org/10.1016/j.mcm.2011.12.048 https://doi.org/10.1016/j.mcm.2011.12.048 int. j. anal. appl. (2022), 20:30 17 [31] m.z. sarikaya, f. ertuğral, on the generalized hermite-hadamard inequalities, ann. univ. craiova math. computer sci. ser. 47 (2020), 193-213. [32] x.x. you, m.a. ali, h. budak, et al. extensions of hermite–hadamard inequalities for harmonically convex functions via generalized fractional integrals, j. inequal. appl. 2021 (2021), 102. https://doi.org/10.1186/ s13660-021-02638-3. [33] d. zhao, m.a. ali, a. kashuri, h. budak, m.z. sarikaya, hermite–hadamard-type inequalities for the intervalvalued approximately h-convex functions via generalized fractional integrals, j. inequal. appl. 2020 (2020), 222. https://doi.org/10.1186/s13660-020-02488-5. [34] y. zhao, h. sang, w. xiong, z. cui, hermite–hadamard-type inequalities involving ψ-riemann–liouville fractional integrals via s-convex functions, j. inequal. appl. 2020 (2020), 128. https://doi.org/10.1186/ s13660-020-02389-7. https://doi.org/10.1186/s13660-021-02638-3 https://doi.org/10.1186/s13660-021-02638-3 https://doi.org/10.1186/s13660-020-02488-5 https://doi.org/10.1186/s13660-020-02389-7 https://doi.org/10.1186/s13660-020-02389-7 1. introduction 2. fractional integrals 3. interval calculus 4. intgral of interval-valued functions 5. hermite-hadamard inequalities for generalized fractional on the interval-value coordinates references international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 39-52 http://www.etamaths.com strong and ∆-convergence of modified two-step iterations for nearly asymptotically nonexpansive mappings in hyperbolic spaces g. s. saluja abstract. the aim of this article is to establish a ∆-convergence and some strong convergence theorems of modified two-step iterations for two nearly asymptotically nonexpansive mappings in the setting of hyperbolic spaces. our results extend and generalize the previous work from the current existing literature. 1. introduction the class of asymptotically nonexpansive mapping, introduced by goebel and kirk [7] in 1972, is an important generalization of the class of nonexpansive mapping. they proved that if c is a nonempty closed and bounded subset of a uniformly convex banach space, then every asymptotically nonexpansive self mapping of c has a fixed point. there are number of papers dealing with the approximation of fixed points / common fixed points of asymptotically nonexpansive and asymptotically quasinonexpansive mappings in uniformly convex banach spaces using modified mannn and ishikawa iteration processes and have been studied by many authors (see, e.g., [17, 18, 24, 28, 29, 31, 34, 35]). the concept of ∆-convergence in a general metric space was introduced by lim [16]. in 2008, kirk and panyanak [14] used the notion of ∆-convergence introduced by lim [16] to prove in the cat(0) space and analogous of some banach space results which involve weak convergence. further, dhompongsa and panyanak [6] obtained ∆-convergence theorems for the picard, mann and ishikawa iterations in a cat(0) space. since then, the existence problem and the ∆-convergence problem of iterative sequences to a fixed point for nonexpansive mapping, asymptotically nonexpansive mapping, nearly asymptotically nonexpansive mapping, asymptotically quasi-nonexpansive mapping in the intermediate sense, total asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping through picard, mann [19], ishikawa[10], modified agarwal et al. [2] have been rapidly developed in the framework of cat(0) space and many papers have appeared in this 2010 mathematics subject classification. 47h10. key words and phrases. nearly asymptotically nonexpansive mapping; modified two-step iteration scheme; common fixed point; strong convergence; ∆-convergence; hyperbolic space. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 39 40 saluja direction (see, e.g., [1, 5, 6, 11, 20, 25]). the purpose of this paper is to establish some strong convergence theorems of modified two-step iteration process for two nearly asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces which include both uniformly convex banach spaces and cat(0) spaces. our results extend and improve the previous work from the current existing literature. 2. preliminaries let f(t) = {x ∈ k : tx = x} denotes the set of fixed points of the mapping t. we begin with the following definitions. definition 2.1. let (x,d) be a metric space and k be its nonempty subset. then t : k → k said to be (1) nonexpansive if d(tx,ty) ≤ d(x,y) for all x,y ∈ k; (2) asymptotically nonexpansive if there exists a sequence {un} ⊂ [0,∞) with limn→∞un = 0 such that d(t nx,tny) ≤ (1 + un)d(x,y) for all x,y ∈ k and n ≥ 1; (3) asymptotically quasi-nonexpansive if f(t) 6= ∅ and there exists a sequence {un} ⊂ [0,∞) with limn→∞un = 0 such that d(tnx,p) ≤ (1 + un)d(x,p) for all x ∈ k, p ∈ f(t) and n ≥ 1; (4) uniformly l-lipschitzian if there exists a constant l > 0 such that d(tnx,tny) ≤ ld(x,y) for all x,y ∈ k and n ≥ 1; (5) semi-compact if for a sequence {xn} in k with limn→∞d(xn,txn) = 0, there exists a subsequence {xnk} of {xn} such that xnk → p ∈ k as k →∞. (6) a sequence {xn} in k is called approximate fixed point sequence for t (afps, in short) if limn→∞d(xn,txn) = 0. the class of nearly lipschitzian mappings is an important generalization of the class of lipschitzian mappings and was introduced by sahu [26] (see, also [27]). definition 2.2. let k be a nonempty subset of a metric space (x,d) and fix a sequence {an} ⊂ [0,∞) with limn→∞an = 0. a mapping t : k → k said to be nearly lipschitzian with respect to {an} if for all n ≥ 1, there exists a constant kn ≥ 0 such that d(tnx,tny) ≤ kn[d(x,y) + an] for all x, y ∈ k. the infimum of the constants kn for which the above inequality holds is denoted by η(tn) and is called nearly lipschitz constant of tn. a nearly lipschitzian mapping t with sequence {an,η(tn)} is said to be: (i) nearly nonexpansive if η(tn) = 1 for all n ≥ 1; strong and ∆-convergence of modified two-step iterations 41 (i) nearly asymptotically nonexpansive if η(tn) ≥ 1 for all n ≥ 1 and limn→∞η(t n) = 1. (ii) nearly uniformly k-lipschitzian if η(tn) ≤ k for all n ≥ 1. throughout this paper, we work in the setting of hyperbolic space introduced by kohlenbach [15]. it is worth noting that they are different from gromov hyperbolic space [4] or from other notions of hyperbolic space that can be found in the literature (see for example [8, 13, 23]. a hyperbolic space [15] is a triple (x,d,w) where (x,d) is a metric space and w : x2 × [0, 1] → x is such that (i) d(u,w(x,y,α)) ≤ αd(u,x) + (1 −α)d(u,y) (ii) d ( w(x,y,α),w(x,y,β) ) = |α−β|d(x,y) (iii) w(x,y,α) = w(x,y, (1 −α)) (iv) d ( w(x,z,α),w(y,w,β) ) ≤ αd(x,y) + (1 −α) d(z,w) for all x,y,z,w ∈ x and α,β ∈ [0, 1]. the class of hyperbolic spaces in the sense of kohlenbach [15] contains all normed linear spaces and convex subsets thereof as well as hadamard manifolds and cat(0) spaces in the sense of gromov [9]. an important example of a hyperbolic space is the open unit ball bh in a real hilbert space h is as follows. let bh be the open unit ball in h. then kbh (x,y) = arg tanh(1 −σ(x,y)) 1/2, where σ(x,y) = ( 1 −‖x‖2 )( 1 −‖y‖2 ) |1 −〈x,y 〉|2 for all x, y ∈ bh, defines a metric on bh (also known as kobayashi distance). a metric space (x,d) is called a convex metric space introduced by takahashi in [33] if it satisfies only (i). a subset k of a hyperbolic space x is convex if w(x,y,α) ∈ k for all x,y ∈ k and α ∈ [0, 1]. a hyperbolic space (x,d,w) is said to be uniformly convex [32] if for all u,x,y ∈ x, r > 0 and ε ∈ (0, 2], there exists a δ ∈ (0, 1] such that d(w(x,y, 1 2 ),u) ≤ (1−δ)r whenever d(x,u) ≤ r, d(y,u) ≤ r and d(x,y) ≥ εr. a mapping η : (0,∞)×(0, 2] → (0, 1] which provides such a δ = η(r,ε) for given r > 0 and ε ∈ (0, 2], is known as modulus of uniform convexity. we call η monotone if it decreases with r (for a fixed ε). 42 saluja let k be a nonempty subset of hyperbolic space x. let {xn} be a bounded sequence in a hyperbolic space x. for x ∈ x, define a continuous functional r(., {xn}) : x → [0,∞) by r(x, {xn}) = lim supn→∞d(x,xn). the asymptotic radius ρ = r({xn}) of {xn} is given by ρ = inf{r(x,{xn}) : x ∈ x}. the asymptotic center ak({xn}) of a bounded sequence {xn} with respect to a subset k of x is defined as follows: ak({xn}) = { x ∈ x : r(x, {xn}) ≤ r(y, {xn}) } for any y ∈ k. the set of all asymptotic center of {xn} is denoted by a({xn}). it has been shown in [32] that bounded sequences have unique asymptotic center with respect to closed convex subsets in a complete and uniformly hyperbolic space with monotone modulus of uniform convexity. a sequence {xn} in x is said to ∆-converge to x ∈ x if x is the unique asymptotic center of {un} for every subsequence {un} of {xn} [14]. in this case, we write ∆-limn xn = x and call x is the ∆-limit of {xn}. recall that ∆-convergence coincides with weak convergence in banach space with opial’s property [21]. in the sequel we need the following lemmas. lemma 2.3. [12] let (x,d,w) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. let x ∈ x and {αn} be a sequence in [b,c] for some b, c ∈ (0, 1). if {xn} and {yn} are sequences in x such that lim supn→∞d(xn,x) ≤ r, lim supn→∞d(yn,x) ≤ r and limn→∞d(w(xn,yn,αn),x) = r for some r ≥ 0, then limn→∞d(xn,yn) = 0. lemma 2.4. [12] let k be a nonempty closed convex subset of a uniformly convex hyperbolic space x and {xn} a bounded sequence in k such that a({xn}) = {y} and r({xn}) = ρ. if {ym} is another sequence in k such that limm→∞r(ym,{xn}) = ρ, then limm→∞ym = y. lemma 2.5. (see [34]) let {pn}∞n=1, {qn}∞n=1 and {rn}∞n=1 be sequences of nonnegative numbers satisfying the inequality pn+1 ≤ (1 + qn)pn + rn, ∀n ≥ 1. if ∑∞ n=1 qn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞pn exists. first, we define the modified two-step iteration scheme in hyperbolic space as follows. let k be a nonempty closed convex subset of a hyperbolic space x and s, t : k → k be two nearly asymptotically nonexpansive mappings. then, for an arbitrary chosen x1 ∈ k, we construct the sequence {xn} in k such that (2.1) { xn+1 = w(t nxn,s nyn,αn), yn = w(s nxn,t nxn,βn), n ≥ 1, strong and ∆-convergence of modified two-step iterations 43 where {αn} and {βn} are appropriate sequences in (0,1) is called modified two-step iteration scheme. iteration scheme (2.1) is independent of modified ishikawa iteration and modified mann iteration schemes. if βn = 0 for all n ≥ 1 and s = i, where i is the identity mapping, then iteration scheme (2.1) reduces to the following. (2.2) { xn+1 = w(t nxn,xn,αn), n ≥ 1, where {αn} is an appropriate sequence in (0,1) is called modified mann iteration scheme in hyperbolic space. 3. main results lemma 3.1. let k be a nonempty convex subset of a hyperbolic space x and let s, t : k → k be two nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2− 1 ) < ∞. let {xn} be a sequence in k defined by (2.1). then limn→∞d(xn,p) exists for each p ∈ f = f(s) ∩f(t). proof. let p ∈ f = f(s) ∩ f(t), ρ = supn∈n η(sn) ∨ supn∈n η(tn) and an = max{a′n,a′′n} for all n. from (2.1), we have d(yn,p) = d(w(s nxn,t nxn,βn),p) ≤ (1 −βn)d(snxn,p) + βnd(tnxn,p) ≤ (1 −βn)[η(sn)(d(xn,p) + a′n)] + βn[η(t n)(d(xn,p) + a ′′ n)] ≤ (1 −βn)[η(sn)(d(xn,p) + an)] + βn[η(tn)(d(xn,p) + an)] = (1 −βn)η(sn) d(xn,p) + βnη(tn) d(xn,p) + ( η(sn) + η(tn) ) an ≤ η(sn)η(tn)[(1 −βn)d(xn,p) + βnd(xn,p)] + 2ρan = η(sn)η(tn)d(xn,p) + 2ρan.(3.1) again, using (2.1) and (3.1), we get d(xn+1,p) = d(w(t nxn,s nyn,αn),p) ≤ (1 −αn)d(tnxn,p) + αnd(snyn,p) ≤ (1 −αn)[η(tn)(d(xn,p) + a′′n)] + αn[η(s n)(d(yn,p) + a ′ n)] ≤ (1 −αn)[η(tn)(d(xn,p) + an)] + αn[η(sn)(d(yn,p) + an)] = (1 −αn)η(tn)d(xn,p) + αnη(sn)d(yn,p) + ( η(sn) + η(tn) ) an ≤ (1 −αn)η(tn)d(xn,p) + 2ρan +αnη(s n)[η(sn)η(tn)d(xn,p) + 2ρan] ≤ η(sn)2η(tn)2d(xn,p) + (1 + η(sn))2ρan ≤ η(sn)2η(tn)2d(xn,p) + 2ρ(1 + ρ)an = (1 + µn)d(xn,p) + νn(3.2) where µn = ( η(sn)2η(tn)2 − 1 ) and νn = 2ρ(1 + ρ)an. since ∑∞ n=1 ( η(sn)2 η(tn)2−1 ) < ∞ and ∑∞ n=1 an < ∞, it follows that ∑∞ n=1 µn < ∞ and ∑∞ n=1 νn < 44 saluja ∞. hence by lemma 2.5, we get that limn→∞d(xn,p) exists. this completes the proof. � lemma 3.2. let k be a nonempty closed convex subset of a uniformly convex hyperbolic space x with monotone modulus of uniform convexity η and let s, t : k → k be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2 − 1 ) < ∞. let {xn} be a sequence in k defined by (2.1). assume that f = f(s) ∩ f(t) 6= ∅. suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). if d(x,tnx) ≤ d(snx,tnx) and d(x,snx) ≤ d(tnx,snx) for all x ∈ k, then limn→∞d(xn,sxn) = 0 and limn→∞d(xn,txn) = 0. proof. from lemma 3.1, we obtain limn→∞d(xn,p) exists for each p ∈ f. suppose that limn→∞d(xn,p) = r ≥ 0. since d(snxn,p) ≤ η(sn)(d(xn,p) + an) for all n ≥ 1, we have lim sup n→∞ d(snxn,p) ≤ r. also, since d(tnxn,p) ≤ η(tn)(d(xn,p) + an) for all n ≥ 1, we have lim sup n→∞ d(tnxn,p) ≤ r. also (3.1) yields lim sup n→∞ d(yn,p) ≤ r.(3.3) hence lim sup n→∞ d(snyn,p) ≤ lim sup n→∞ η(sn)(d(yn,p) + an) ≤ r.(3.4) since r = lim n→∞ d(xn+1,p) = lim n→∞ d(w(tnxn,s nyn,αn),p), it follows from lemma 2.3 that lim n→∞ d(tnxn,s nyn) = 0.(3.5) from (2.1) and (3.5), we have d(xn+1,t nxn) = d(w(t nxn,s nyn,αn),t nxn) ≤ αn d(tnxn,snyn) ≤ d(tnxn,snyn) → 0 as n →∞.(3.6) hence from (3.5) and (3.6), we have d(xn+1,s nyn) ≤ d(xn+1,tnxn) + d(tnxn,snyn) → 0 as n →∞.(3.7) now using (3.7), we have d(xn+1,p) ≤ d(xn+1,snyn) + d(snyn,p) ≤ d(xn+1,snyn) + η(sn)(d(yn,p) + an).(3.8) strong and ∆-convergence of modified two-step iterations 45 the inequality (3.8) gives r ≤ lim inf n→∞ d(yn,p).(3.9) from (3.3) and (3.9), we get r = lim n→∞ d(yn,p) = lim n→∞ d(w(snxn,t nxn,βn),p).(3.10) applying lemma 2.3 in (3.10), we obtain lim n→∞ d(snxn,t nxn) = 0.(3.11) now using (3.11) and hypothesis of the theorem d(x,tnx) ≤ d(snx,tnx) for all x ∈ k, we get d(xn,s nxn) ≤ d(xn,tnxn) + d(tnxn,snxn) ≤ d(snxn,tnxn) + d(tnxn,snxn) = 2 d(snxn,t nxn) → 0 as n →∞.(3.12) again using (3.11) and hypothesis of the theorem d(x,snx) ≤ d(tnx,snx) for all x ∈ k, we get d(xn,t nxn) ≤ d(xn,snxn) + d(snxn,tnxn) ≤ d(tnxn,snxn) + d(snxn,tnxn) = 2 d(snxn,t nxn) → 0 as n →∞.(3.13) by uniform continuity of s and t , limn→∞d(xn,s nxn) = 0 implies that limn→∞d(sxn,s n+1xn) = 0 and limn→∞d(xn,t nxn) = 0 implies that limn→∞d(txn,t n+1xn) = 0. note that d(xn+1,xn) = d(w(t nxn,s nyn,αn),xn) ≤ (1 −αn)d(xn,tnxn) + αn d(snyn,xn) ≤ (1 −αn)d(xn,tnxn) + αn d(snyn,xn+1) + αn d(xn+1,xn) ≤ (1 −αn)d(xn,tnxn) + αn d(snyn,xn+1) + md(xn+1,xn) this implies that (1 −m) d(xn+1,xn) ≤ (1 −αn)d(xn,tnxn) + αn d(snyn,xn+1). (3.14) since (1 −m) > 0, using (3.7) and (3.13) in (3.14), we get lim n→∞ d(xn+1,xn) = 0.(3.15) also d(xn,sxn) ≤ d(xn,xn+1) + d(xn+1,sn+1xn+1) +d(sn+1xn+1,s n+1xn) + d(s n+1xn,sxn) ≤ ( 1 + η(sn+1) ) d(xn,xn+1) + d(xn+1,s n+1xn+1) +d(sn+1xn,sxn) + an+1.(3.16) using (3.12), (3.15) and uniform continuity of s, equation (3.16) gives lim n→∞ d(xn,sxn) = 0.(3.17) 46 saluja similarly d(xn,txn) ≤ d(xn,xn+1) + d(xn+1,tn+1xn+1) +d(tn+1xn+1,t n+1xn) + d(t n+1xn,txn) ≤ ( 1 + η(tn+1) ) d(xn,xn+1) + d(xn+1,t n+1xn+1) +d(tn+1xn,txn) + an+1.(3.18) the above inequality gives lim n→∞ d(xn,txn) = 0.(3.19) this completes the proof. � we now establish a ∆-convergence and some strong convergence theorems of modified two-step iteration scheme for non-lipschitzian mappings in the framework of uniformly convex hyperbolic spaces. theorem 3.3. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic space x with monotone modulus of uniform convexity η and let s, t : k → k be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2−1 ) < ∞. let {xn} be a sequence in k defined by (2.1). assume that f = f(s)∩f(t) 6= ∅. suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). then {xn} is ∆-convergent to an element of f . proof. it follows from lemma 3.1 that {xn} is bounded, therefore {xn} has a unique asymptotic center (see, [32]), that is, a({xn}) = {x} (say). let a({yn}) = {v}. then by lemma 3.2, limn→∞d(yn,syn) = 0 and limn→∞d(yn,tyn) = 0. s and t are nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))}. by uniform continuity of s and t , we have lim n→∞ d(siyn,s i+1yn) = 0 for i = 1, 2, . . . .(3.20) and lim n→∞ d(tjyn,t j+1yn) = 0 for j = 1, 2, . . . .(3.21) now we claim that v is a common fixed point of s and t. for this, we define a sequence {zn} in k by zm = smv and zm = tmv, m ∈ n. for integers m,n ∈ n, we have d(zm,yn) ≤ d(smv,smyn) + d(smyn,sm−1yn) + · · · + d(syn,yn) ≤ η(sn)(d(v,yn) + a′m) + m−1∑ i=0 d(siyn,s i+1yn).(3.22) then from (3.20) and (3.22), we have r(zm,{yn}) = lim sup m→∞ d(zm,yn) ≤ η(sm)[r(v,{yn}) + a′m]. hence lim sup m→∞ r(zm,{yn}) ≤ r(v,{yn}).(3.23) strong and ∆-convergence of modified two-step iterations 47 since ak({yn}) = {v}, by definition of asymptotic center ak({yn}) of a bounded sequence {yn} with respect to k ⊂ x, we have r(v,{yn}) ≤ r(y,{yn}), ∀ y ∈ k. this implies that lim inf m→∞ r(zm,{yn}) ≥ r(v,{yn}),(3.24) therefore, from (3.23) and (3.24), we have lim m→∞ r(zm,{yn}) = r(v,{yn}). it follows from lemma 2.4 that smv → v. by uniform continuity of s, we have sv = s( lim m→∞ smv) = sm+1v = v, which implies that v is a fixed point of s, that is, v ∈ f(s). similarly, we can show that v ∈ f(t). thus v ∈ f = f(s) ∩f(t). next, we claim that v is the unique asymptotic center for each subsequence {yn} of {xn}. assume contrarily, that is, x 6= v. since limn→∞d(xn,v) exists by lemma 3.1, therefore, by the uniqueness of asymptotic centers, we have lim sup n→∞ d(yn,v) < lim sup n→∞ d(yn,x) ≤ lim sup n→∞ d(xn,x) < lim sup n→∞ d(xn,v) = lim sup n→∞ d(yn,v), a contradiction and hence x = v. since {yn} is an arbitrary subsequence of {xn}, therefore, ak({yn}) = {v} for all subsequence {yn} of {xn}. this proves that {xn} ∆-converges to an element of f. this completes the proof. � theorem 3.4. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic space x with monotone modulus of uniform convexity η and let s, t : k → k be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2−1 ) < ∞. let {xn} be a sequence in k defined by (2.1). assume that f = f(s) ∩f(t) 6= ∅ is a closed set. then {xn} converges strongly to a point in f if and only if lim infn→∞d(xn,f) = 0. proof. necessity is obvious. conversely, suppose that lim infn→∞d(xn,f) = 0. as proved in lemma 3.1, for all p ∈ f, limn→∞d(xn,f) exists. thus by hypothesis limn→∞d(xn,f) = 0. 48 saluja next, we show that {xn} is a cauchy sequence in k. with the help of inequality 1 + x ≤ ex, x ≥ 0. for any integer m ≥ 1, we have from (3.2) d(xn+m,p) ≤ (1 + µn+m−1)d(xn+m−1,p) + νn+m−1 ≤ eµn+m−1d(xn+m−1,p) + νn+m−1 ≤ eµn+m−1 [eµn+m−2d(xn+m−2,p) + νn+m−2] +νn+m−1 ≤ e(µn+m−1+µn+m−2)d(xn+m−2,p) + e(µn+m−1+µn+m−2) × [νn+m−1 + νn+m−2] ≤ . . . ≤ ( e ∑n+m−1 k=n µk ) d(xn,p) + ( e ∑n+m−1 k=n µk )n+m−1∑ k=n νk = w d(xn,p) + w n+m−1∑ k=n νk,(3.25) where w = e ∑∞ n=1 µn < ∞. since limn→∞d(xn,f) = 0, without loss of generality, we may assume that a subsequence {xnk} of {xn} and a sequence {pnk} ⊂ f such that d(xnk,pnk ) → 0 as k →∞. then for any ε > 0, there exists kε > 0 such that d(xnk,pnk ) < ε 4w and ∞∑ k=nkε νk < ε 4w ,(3.26) for all k ≥ kε. for any m ≥ 1 and for all n ≥ nkε , by (3.25) and (3.26), we have d(xn+m,xn) ≤ d(xn+m,pnk ) + d(xn,pnk ) ≤ w d(xn,pnk ) + w ∞∑ k=nkε νk +w d(xn,pnk ) + w ∞∑ k=nkε νk = 2w d(xn,pnk ) + 2w ∞∑ k=nkε νk < 2w. ε 4w + 2w. ε 4w = ε.(3.27) this proves that {xn} is a cauchy sequence in closed subset k of a complete hyperbolic space x and so it must converge to a point z in k, that is, limn→∞xn = z. now, limn→∞d(xn,f) = 0 gives d(z,f) = 0. since f is closed, we have z ∈ f. thus {xn} converges strongly to a point in f . this completes the proof. � theorem 3.5. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic space x with monotone modulus of uniform convexity η and let s, t : k → k be two uniformly continuous nearly asymptotically nonexpansive strong and ∆-convergence of modified two-step iterations 49 mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2−1 ) < ∞. let {xn} be a sequence in k defined by (2.1). assume that f = f(s)∩f(t) 6= ∅. suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). if either sm or tm for some m ≥ 1 is semi-compact, then {xn} converges strongly to a point in f . proof. suppose tm for some m ≥ 1 is semi-compact. by lemma 3.2, we have limn→∞d(xn,txn) = 0. by the uniform continuity of t , we get d(xn,txn) → 0 ⇒ d(txn,t 2xn) → 0 ⇒ ···⇒ d(tixn,ti+1xn) → 0 for all i = 1, 2, 3, . . . , it follows that d(xn,t mxn) ≤ m−1∑ i=0 d(tixn,t i+1xn) → 0 as n →∞. since d(xn,t mxn) → 0 and tm is semi-compact, there exists a subsequence {xnj} of {xn} such that limj→∞tmxnj = x ∈ k. note that d(xnj,x) ≤ d(xnj,t mxnj ) + d(t mxnj,x) → 0 as j →∞. since limn→∞d(xn,txn) = 0, we get x ∈ f(t). similarly, we can show that x ∈ f(s). thus x ∈ f = f(s) ∩ f(t). since limn→∞d(xn,x) exists by lemma 3.1 and limj→∞d(xnj,x) = 0, we conclude that xn → x ∈ f. this shows that the sequence {xn} converges strongly to a point in f. this completes the proof. � senter and dotson [30] introduced the concept of condition (a) as follows. definition 3.6. (see [30]) a mapping t : k → k is said to satisfy condition (a) if there exists a non-decreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(r) > 0 for all r > 0 such that d(x,tx) ≥ f(d(x,f(t))), for all x ∈ k. we modify this definition for two mappings. definition 3.7. two mappings s,t : k → k, where k is a subset of a metric space (x,d), is said to satisfy condition (a′) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0,∞) such that ad(x,sx) + bd(x,tx) ≥ f(d(x,f)) for all x ∈ k where d(x,f) = inf{d(x,p) : p ∈ f = f(s) ∩ f(t) 6= ∅} and a and b are two nonnegative real numbers such that a + b = 1. it is to be noted that condition (a′) is weaker than compactness of the domain k. remark 3.8. condition (a′) reduces to condition (a) when s = t. as an application of theorem 3.3, we establish another strong convergence result employing condition (a′). theorem 3.9. let k be a nonempty closed convex subset of a complete uniformly convex hyperbolic space x with monotone modulus of uniform convexity η and let s, t : k → k be two uniformly continuous nearly asymptotically nonexpansive mappings with sequences {(a′n,η(sn))} and {(a′′n,η(tn))} such that ∑∞ n=1 an < ∞ and ∑∞ n=1 ( η(sn)2η(tn)2−1 ) < ∞. let {xn} be a sequence in k defined by (2.1). 50 saluja assume that f = f(s)∩f(t) 6= ∅. suppose that {αn} and {βn} are real sequence in [l,m] for some l,m ∈ (0, 1). suppose that s and t satisfy the condition (a′). then {xn} converges strongly to a point in f . proof. by lemma 3.2, we know that lim n→∞ d(xn,sxn) = 0 and lim n→∞ d(xn,txn) = 0.(3.28) from condition (a′) and (3.28), we get lim n→∞ f(d(xn,f)) ≤ a. lim n→∞ d(xn,sxn) + b. lim n→∞ d(xn,txn) = 0, that is, lim n→∞ f(d(xn,f)) = 0. since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f(0) = 0, f(t) > 0 for all t ∈ (0,∞), therefore we obtain lim n→∞ d(xn,f) = 0. the conclusion now follows from theorem 3.4. this completes the proof. � example 3.10. (see [26]) let e = r, k = [0, 1] and t : k → k be a mapping defined by t(x) = { 1 2 , if x ∈ [0, 1 2 ], 0, if x ∈ ( 1 2 , 1]. here f(t) = {1 2 }. clearly, t is discontinuous and a non-lipschitzian mapping. however, it is a nearly nonexpansive mapping and hence nearly asymptotically nonexpansive mapping with sequence {an,η(tn)} = { 12n , 1}. indeed, for a sequence {an} with a1 = 12 and an → 0, we have d(tx,ty) ≤ d(x,y) + a1 for all x, y ∈ k and d(tnx,tny) ≤ d(x,y) + an for all x, y ∈ k and n ≥ 2, since tnx = 1 2 for all x ∈ [0, 1] and n ≥ 2. 4. conclusion 1. we prove a ∆-convergence and some strong convergence theorems of modified two-step iteration process which contains modified mann iteration process in the framework of uniformly convex hyperbolic spaces. 2. lemma 3.2 extends theorem 3.8 of agarwal et al. [2] to the case of modified two-step iteration scheme for two mappings and from uniformly convex banach space to a uniformly convex hyperbolic space considered in this paper. 3. our results also extend and generalize the corresponding results of [3, 22, 24, 34] to the case of more general class of nonexpansive and asymptotically nonexpansive mappings, modified two-step iteration scheme for two mappings and from uniformly convex metric space and banach space to a uniformly convex hyperbolic space considered in this paper. strong and ∆-convergence of modified two-step iterations 51 references [1] m. abbas, z. kadelburg and d.r. sahu, fixed point theorems for lipschitzian type mappings in cat(0) spaces, math. comput. model. 55 (2012), 1418-1427. [2] r.p. agarwal, donal o’regan and d.r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, nonlinear convex anal. 8(1) (2007), 61-79. [3] i. beg, an iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces, nonlinear anal. forum, 6 (2001), 27-34. [4] m.r. bridson and a. haefliger, metric spaces of non-positive curvature, vol. 319 of grundlehren der mathematischen wissenschaften, springer, berlin, germany, 1999. [5] s.s. chang, l. wang, h.w. joesph lee, c.k. chan, l. yang, demiclosed principle and ∆-convergence theorems for total asymptotically nonexpansive mappings in cat(0) spaces, appl. math. comput. 219(5) (2012), 2611-2617. [6] s. dhompongsa and b. panyanak, on 4-convergence theorem in cat(0) spaces, comput. math. appl. 56 (2008), 2572-2579. [7] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35 (1972), 171-174. [8] k. goebel and w.a. kirk, iterations processes for nonexpansive mappings, contemp. math. 21 (1983), 115-123. [9] m. gromov, hyperbolic groups. essays in group theory (s. m. gersten, ed). springer verlag, msri publ. 8 (1987), 75-263. [10] s. ishikawa, fixed points by a new iteration method, proc. amer. math. soc. 44 (1974), 147-150. [11] s.h. khan and m. abbas, strong and 4-convergence of some iterative schemes in cat(0) spaces, comput. math. appl. 61 (2011), 109-116. [12] a.r. khan, h. fukhar-ud-din and m.a.a. khan, an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, fixed point theory appl. 2012 (2012), article id 54. [13] w.a. kirk, krasnoselskii’s iteration process in hyperbolic space, numer. funct. anal. optim 4 (1982), 371-381. [14] w.a. kirk and b. panyanak, a concept of convergence in geodesic spaces, nonlinear anal. 68 (2008), 3689-3696. [15] u. kohlenbach, some logical metatheorems with applications in functional analysis, trans. amer. math. soc. 357 (2005), 89-128. [16] t.c. lim, remarks on some fixed point theorems, proc. amer. math. soc. 60 (1976), 179-182. [17] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 259 (2001), 1-7. [18] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings with error member, j. math. anal. appl. 259 (2001), 18-24. [19] w.r. mann, mean value methods in iteration, proc. amer. math. soc. 4 (1953), 506-510. [20] b. nanjaras and b. panyanak, demiclosed principle for asymptotically nonexpansive mappings in cat(0) spaces, fixed point theory appl. 2010 (2010), art. id 268780. [21] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc. 73(1967), 591-597. 52 saluja [22] m.o. osilike, s.c. aniagbosor, weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, math. and computer modelling 32(2000), 1181-1191. [23] s. reich and i. shafrir, nonexpansive iterations in hyperbolic spaces, nonlinear anal.: tma, series a, theory methods, 15(6)(1990), 537-558. [24] b.e. rhoades, fixed point iteration for certain nonlinear mappings, j. math. anal. appl. 183(1994), 118-120. [25] a. şahin and m. başarir, on the strong convergence of a modified s-iteration process for asymptotically quasi-nonexpansive mappings in a cat(0) space, fixed point theory appl. 2013 (2013), article id 12. [26] d.r. sahu, fixed points of demicontinuous nearly lipschitzian mappings in banach spaces, comment. math. univ. carolinae 46(4) (2005), 653-666. [27] d.r. sahu and i. beg, weak and strong convergence for fixed points of nearly asymptotically nonexpansive mappings, int. j. mod. math. 3 (2008), 135-151. [28] g.s. saluja, strong convergence theorem for two asymptotically quasinonexpansive mappings with errors in banach space, tamkang j. math. 38(1) (2007), 85-92. [29] g.s. saluja, convergence result of (l,α)-uniformly lipschitz asymptotically quasi-nonexpansive mappings in uniformly convex banach spaces, jñānābha 38 (2008), 41-48. [30] h.f. senter, w.g. dotson, approximating fixed points of nonexpansive mappings, proc. amer. math. soc. 44 (1974), 375-380. [31] n. shahzad, a. udomene, approximating common fixed points of two asymptotically quasi-nonexpansive mappings in banach spaces, fixed point theory and applications, 2006 (2006), article id 18909. [32] t. shimizu and w. takahashi, fixed points of multivalued mappings in certain convex metric spaces, topol. methods nonlinear anal. 8(1) (1996), 197-203. [33] w. takahashi, a convexity in metric spaces and nonexpansive mappings, kodai math. semin. rep. 22 (1970), 142-149. [34] k.k. tan and h.k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178, 301-308, 1993. [35] k.k. tan and h.k. xu, fixed point iteration processes for asymptotically nonexpansive mappings, proc. amer. math. soc. 122(1994), 733-739. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492001 (c.g.), india international journal of analysis and applications volume 19, number 5 (2021), 709-724 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-709 image restoration by a fractional reaction-diffusion process hana matallah∗, messouad maouni, hakim lakhal laboratory of applied mathematics and history and didactics of mathematics ”lamahis”, department of mathematics, university 20 august 1955 skikda, algeria ∗corresponding author: hanamaatallah5@gmail.com abstract. we propose new approaches to the investigation of a reaction-diffusion model of fractional order in which we apply the fractional derivative in the sense of the caputo by contribution to time on the model proposed by nourddine alaa in 2014, this study is based on the restoration of digital image such that a digital result is given on a noisy image in which this model is found to be effective in eliminating noise. 1. introduction fractional order calculus is a topic which provides a good tool to describe physical memory and heredity, this topic has been applied to many fields such as flabby, oscillation, stochastic diffusion theory and wave propagation, biological materials, control and robotics, quantum mechanics, where it has also become used in the field of image processing. that field has become hot in recent years, and one of the topics recently included in this field is image processing or digital image processing, the latter is has become an important problem thanks to its wide importance in several fields and because this methode used to perform some operations on the digital image to get an enhanced image and to recovery lost information from it, and in this field we find a topic of the restoration of the image, which has become of interest to many researchers and scientists, where we find in 2019 [ [11], [12]] s. lecheheb, m. maouni and h. lakhal, they proved the existence of the solution of a quasilinear equation and they give its application to image denoising and in the received april 22nd, 2021; accepted may 26th, 2021; published august 2nd, 2021. 2010 mathematics subject classification. 35r11. key words and phrases. image restoration; reaction diffusion; fractional order derivative. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 709 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-709 int. j. anal. appl. 19 (5) (2021) 710 same year, they used a nonlinear eliptic model in image restoration. in 2013 [14] m. maouni and nouri, used a new model based on p-gradient using to restore a digital image. leheheb, maouni and lakhal, in 2021 [13], they used a novel model combining the perona-malik equation and the heat equation in image restoration, but the most problem in this topic that always arises in the form of question is how to treat the image while preserving edges but the answer of this quesion is given in 1987 [16] by perona-malik in his model, which is one of the first attempts to derive a model that incorporates local information from an image with in a partial differential equations (pdes), which the main reason for using (pdes) in this domain is that it is possible to analyse the image in continuous spaces. a nonlinear diffusion model, which they called’s anisotropic diffusion was can ducted by perona and malik [16] in to reduce noise and enhance contrast while preserving the edge, but the basic perona-malik [16] pde model is ill-posed in the sens of hadamard. from where the idea of catté, lions, morel and coll in 1992 [5] to integrate directly the regularisation into the equation by convolving the image with the gaussian filter on the gradient of the noisy image to smooth the image first in order to avoid the dependence of the numerical scheme between the solution and the regularization procedure to make the problem well posed and they prove existence, uniqueness and regularity for the related model and demonstrate experimentally that the related model gives similar results to the perona-malik equation [16]. in 2006 [15] morfu proposed a model on the contrast enhancement and noise filtering where he combined the nonlinear diffusion process ruled by fisher equation, his model is given as follows (1.1) ∂u ∂t −div(g(|ou|)ou) = f(u) in qt , u(0,x) = u0(x) in ω, ∂u ∂υ = 0 on σt , with ω is the domaine of the image, qt =]0,t[×ω, σt =]0,t[×∂ω, where t > 0, u0 is the original image and f(s) = s(s−a)(1 −s) with 0 < a < 1. the model of morfu [15] has two disadvantage, the first one is the sensitivity to noise and the second is that no results of existence and consistency in proved. to overcome this problem, in 2014 [2] alaa and all combining the regulaisation procedure in catté [5] and morfu [15] model, they proposed to modify the model of morfu [15] by applying a gaussian filter on the gradient of the noisy image during the calculation of coefficient of anisotropic diffusion, the authors we reable to demonstrate the existence and consistency of the their proposed model, which is given by: (1.2) ∂u ∂t −div(g(|ouσ|)ou) = f(t,x,u) in qt , u(0,x) = u0(x) in ω, ∂u ∂υ = 0 on σt , int. j. anal. appl. 19 (5) (2021) 711 with ω =]0, 1[×]0, 1[ the picture domain with boundary ∂ω, with neumann boundary conditions,u(t,x) is the restored image and u0 is the original image to be processed. qt =]0,t[×ω, σt =]0,t[×∂ω, (t > 0), σ > 0 and gσ is the gaussian filter where: gσ(x) = 1 √ 2πσ exp(− |x|2 4σ ), x ∈ r2. they consider the gradient norm of ω as: |oω| = ( i=2∑ i=1 ( ∂ω ∂xi )2 )1/2 , oωσ is the smoothed version of gradient norme where : oωσ = o(ω ∗gσ) = ω ∗ogσ. the diffusivity g is smooth decreasing function defined by (1.3) g(0) = 1, lim s−→∞ g(s) = 0, one of the diffusivities perona and malik [16] proposed is: g(s) = d√ 1 + η ( s λ )2 , with η ≥ 0, d > 0 and λ is a parameter that separates forward and backward diffusion [20]. in 2016, bassam al-hamzah and naji yabari [4] proposed a new reaction-diffusion model in image processing, which they proved the existence of global solution for the nonlinear reaction-diffusion model. this study deal with the equation: (1.4) ∂u ∂t −div(g(|ouσ|)ou) = f(t,x,u,∇u) in qt , u(0,x) = u0(x) ≥ 0 in ω, ∂u ∂υ = 0 on σt , with ω =]0, 1[×]0.1[, qt =]0,t[×ω and σt =]0,t[×∂ω. the results f(t,x,u,∇u) and f = f(t,x,u) is a generalization of the work f = 0 presented by catté [5], and alaa [2]. in 2018 aaraba, alaa, and khalfi [1] provided the existence of global solution to a reaction-diffusion generic system with application in image restoration and anhancement. this study is a generalization of the work presented by [ [2], [5], [16]] in the case of reaction-diffusion equations. they give an example of application demonstrated on a novel bio-inspired image restoration model [1]. in the same year, alaa and zirhem [3], proposed a new model of nonlinear and anisotropic reaction diffudion system applied to image restoration and to contrast enhancement. this model is based on a system of partial differential equations of type fitzhugh-nagumo. they compared the performance of their alghorithm with the classical fizhughnagumo model. int. j. anal. appl. 19 (5) (2021) 712 the aim of this work is to show how fractional order differential equations are used to restore a digital image. it is afact to apply the fractional derivative in the sens of caputo on the model the reaction-diffusion proposed with alaa and all in 2014 [2], the proposed model is as follows: (1.5) c0 d α t u(t,x) −div(g(|ouσ|)ou) = f(t,x,u) in qt , with the conditions given by: u(0,x) = u0 in ω, ∂u ∂ν = 0 in σt , with 0 < α < 1, qt =]0,t[×ω and ω =]0, 1[×]0, 1[ is the image domain with limit ∂ω, u(t,x) is the solution to the problem, u0 is the original image, ν is an outward normal to domain ω. let σ > 0, ∇uσ be a regularization by convolution of ou, σt =]0,t[×∂ω, where (t > 0) and 0 < t < t, the diffusivity g check the same properties provided by alaa [2], which is given in the equation (1.3) and the function f(t,x,u) is used to represent sources. c0 dαt u(t,x) is the fractional derivative in the caputo sense of order α of u(t,x) defined as [10]: (1.6) c0 d α t u(t,x) = 1 γ(1 −α) ∫ t 0 ∂ ∂t u(s,x)(t−s)−αds 0 < α < 1. in this study, we need the following assumptions and properties: (h1)f : qt ×r −→ r mesurable for (t,x) and continous for u. (h2)∀(t,x) ∈ qt , f(t,x, 0) is a positive function. (h3)∀u ∈ r and for all (t,x) ∈ qt , uf(t,x,u) is negative. (h4)assumed that u(t,x) is differentiable in the sence of the gâteau ( see [8], page 67 ) , so c 0 d α t u(t,x) = t 1−α∂u(t,x) ∂t . (h5)let b(t) = t α−1 and sup 0<t<t | b(t) |≤ cb, where cb > 0 and 0 < α < 1. the equation(1.5) is given as follows: (1.7) ∂u ∂t −b(t)div(g(|ouσ|)ou) = b(t)f(t,x,u) in qt , u(0,x) = u0(x) in ω, ∂u ∂υ = 0 in σt , in this case, we will recall some functional spaces that willbe used throughout this paper. ∀k ∈ n, hk(ω) is the set of functions u defined in ω such as u and its order dmu derivatives where |m| = ∑j=1 n mj ≤ k are int. j. anal. appl. 19 (5) (2021) 713 in l2(ω). hk(ω) is hilbert space with the norm (1.8) ‖u‖hk(ω) = ( ∑ |m|≤k ∫ ω |dmu|2dx )1 2 . by setting (h1(ω))′ the dual of h1(ω). lp(0,t,hk(ω)) is the space of functions u such that, ∀ t ∈ (0,t), u(t) belongs to hk(ω) with the norm (1.9) ‖u‖lp(0,t,hk(ω)) = (∫ t 0 ‖u(t)‖p hk(ω) dt )1 p , 1 < p < ∞, k ∈ n. l∞(0,t,l2(ω)) is the space of functions u such that, ∀ t ∈ (0,t), u(t) belongs to l2(ω) with the norm (1.10) ‖u‖l∞(0,t,l2(ω)) = ( sup 0<t<t ‖u(t)‖2l2(ω) )1 2 . l∞(0,t,c∞(ω)) is the space of funcions u such that, for all every t ∈ (0,t), u(t) belongs to c∞(ω) with the norm (1.11) ‖u‖l∞(0,t,c∞(ω)) = inf{c,‖u(t)‖c∞(ω) ≤ c in (0,t)}. this study is based on the existence of the solution for our model, we truncate the equation and show that it can be solved in the sens of the schauder fixed point theorem. finally by making some estimations, we prove that the solution of the approximate problem converge to the solution of the aour our problem. we state this paper first by an introduction, then we give a definition of solution with the presentation of the most important results of this work, followed by a description of the existence of the reaction diffusion equation problem, finally we give a straightforward application of our result in the fractional reaction diffusion model for image restoration. 2. the main result first, we clearly state our definition of weak solution to the reaction-diffusion equation, we define the folowing spaces: x = {u in l∞(0,t; l2(ω)) ∩l2(0,t; h1(ω)), u(0, .) = u0}. z = {ϕ in c1(qt ), where ϕ(t,.) = 0}. d = {u in c([0,t]; l2(ω)) ∩l2(0,t; h1(ω)} definition 2.1. let u the weak solution of the problem (1.7) if for all u ∈ x and ϕ ∈ z with f(t,x,u) ∈ l1(qt ) (2.1) ∫ qt −u ∂ϕ ∂t dxdt + ∫ qt b(t)g(|ouσ|)ouoϕdxdt = ∫ qt b(t)f(t,x,u)ϕdxdt + ∫ ω u0ϕ(0,x)dx. int. j. anal. appl. 19 (5) (2021) 714 theorem 2.1. under the assumption (h1) − (h5) and for all r ≥ 0, (2.2) sup |u|≤r ( |f(t,x,u)| ) ∈ l1(qt ). then with fixed t > 0, σ > 0 and 0 < α < 1 and for any 0 ≤ u0 ∈ l2(ω), the equation (2.1) admits a weak positive solution. if moreover for all r ≥ 1, f(t,x,r) ≤ 0 and u0(x) ≤ 1, we have 0 ≤ u(t,x) ≤ 1 dans qt . proof of theorem. the proof of theorem(2.1) is done in four step: step1: the positivity of the solution. let the function sign− defined as: (2.3) sign−(r) =   −1 if r < 0, 0 if r ≥ 0. we build a sequence of convex function jε(r) where j ′ ε(r) is bounded and for all r ∈ r, j′ε(r) converge to sign−(r) when ε −→ 0. we consider u the solution of (2.1), we multiply the equation by j′ε(u) and by integration on qt =]0, t[×ω for t ∈ [0,t[ we get:∫ qt ∂u ∂t j′ε(u)dxds + ∫ qt b(t)g(|∇uσ|)∇u∇j′ε(u)dxds = ∫ qt b(t)f(s,x,u)j′ε(u)dxds, we set a(t,x) = g(|∇uσ|) and with ‖∇uσ‖l∞(qt) ≤ c0, for the properties of g, we have a = g(c0) where c0 depend to σ and ‖u0‖l∞ (ω), such that a(t,x) ≥ a ∀(t,x) ∈ qt . ∫ ω [jε(u(t)) − jε(u(0))]dx + a ∫ qt b(t)|∇u|2j′′ε (u)dxdt ≤ ∫ qt b(t)f(t,x,u)j′ε(u)dxdt, then ∫ ω jε(u(t))dx ≤ cb ∫ qt f(t,x,u)j′ε(u)dxdt, ≤ cb ∫ [u<0] f(t,x,u)j′ε(u)dxdt + cb ∫ [u≥0] f(t,x,u)j′ε(u)dxdt, then (2.4) ∫ ω jε(u(t)) ≤ cb ∫ [u<0] f(t,x,u)j′ε(u)dxdt. by crossing in the limit, when ε → 0 (2.5) ∫ ω (u)−(t)dx ≤−cb ∫ [u≤0] f(t,x,u)dxdt, int. j. anal. appl. 19 (5) (2021) 715 then (u)− ≥ 0, so (u)− = 0, hence u ≥ 0. then, we have to prove the following lemma: lemma 2.1. we consider u the weak solution of (2.1), and assume that 0 ≤ u0 ≤ 1 in ω then 0 ≤ u ≤ 1 in qt . proof. in the previous results, we have obtained the positivity of the weak solution if the initial data is positive, so, we assume that u0 ≤ 1 and prove that u ≤ 1. we take ū = 1 −u, where ∇ū = ∇u. for all ū ∈ x and ϕ ∈ z with f(t,x, 1 − ū) ∈ l1(qt ) − ∫ qt ū ∂ϕ ∂t dxdt + ∫ qt b(t)g(|oūσ|)oūoϕdxdt = ∫ qt b(t)f(t,x, 1 − ū)ϕdxdt. let jε(r) a sequence of convex function, where j ′ ε(r) is bounded and ∀r ∈ r, j′ε(r) → sing−(r) when ε → 0, let j′ε(ū) = ϕ∫ qt ∂jε(ū) ∂t dxds + ∫ qt b(t)g(|∇ūσ|)∇ū∇j′ε(ū)dxds = ∫ qt b(t)f(t,x, 1 − ū)j′ε(ū)dxds, ∫ ω jε(ū(t))dx ≤ cb ∫ qt f(t,x, 1 − ū)j′ε(ū)dxdt, ≤ cb ∫ [ū<0] f(t,x, 1 − ū)j′ε(ū)dxdt + cb ∫ [ū≥1] f(t,x, 1 − ū)j′ε(ū)dxdt, pass to the limit when ε → 0 (2.6) − ∫ ω (ū)−(t,x) ≤ cb ∫ [ū≥1] f(t,x, 1 − ū)j′ε(ū)dxdt. hence (2.7) ∫ ω (ū)(t,x)dx ≥ 0, hence (ū) ≥ 0, so u = 1 − ū ≤ 1. � step2: existence result for bounded nonlinearity. first, we will show the existence result for bounded source term f. lemma 2.2. under the above assumption of the nonlinearity f and (h5), if there exists mf ≥ 0, for almost (t,x) ∈ qt and every r ∈ r, we have (2.8) |f(t,x,r)| ≤ mf, then for all u0 ∈ l2(ω), the problem (2.1) admits a weak solution. moreover, there exists c = c ( mf,a,t,‖u‖l2(ω) ) where: (2.9) ‖u(t)‖l∞(0,t;l2(ω)) + ‖u‖l2(0,t;h1(ω)) ≤ c. int. j. anal. appl. 19 (5) (2021) 716 proof. firt, we introduce the space w(0,t) to show the existence of a weak solution with the classical schauder fixed point theorem: w(0,t) = {v ∈ l2(0,t; h1(ω)) ∩l∞(0,t; l2(ω)) : ∂v ∂t ∈ l2(0,t; (h1(ω))′)}, let v ∈ w(0,t) and u be the solution of a linearization of problem (1.7) given by • ∀u ∈ d and ∀ϕ ∈ z (2.10) ∫ qt ( −u ∂ϕ ∂t + b(t)g(|ovσ|)ouoϕ ) dxdt = ∫ qt b(t)f(t,x,v)ϕdxdt + ∫ ω u0ϕ(0,x)dx. we take u as a test function ϕ in (2.10), with 0 < t < t 1 2 ∫ ω u2(t)dx + ∫ qt b(t)g(|ovσ|)|ou|2dxdt = ∫ qt b(t)f(t,x,v)udxdt + 1 2 ∫ ω u20dx. with (3.1) and a(t,x) = g(|ouσ|) ≥ a, ∀(t,x) ∈ qt (2.11) ∫ ω u2(t)dx + 2a ∫ qt |b(t)||ou|2dxdt ≤ mf ∫ qt u2dxdt + ∫ ω u20dx, using gronwall lemma ∫ ω u2(t)dx ≤‖u0‖2l2(ω) ( exp(mft) − 1 ) , sup 0<t<t ∫ ω u2(t)dx ≤ c1, then (2.12) ‖u‖l∞(0,t,l2(ω)) ≤ c1. from (2.11), we have 2a ∫ qt |b(t)||ou|2dxdt + mf ∫ qt |u|2dxdt ≤ ∫ ω |u(t)|2dx + ∫ ω u20dx using (h5) we obtain that ∫ qt |ou|2dxdt + ∫ qt |u|2dxdt ≤ ‖u0‖2l2(ω) min(2a,mf ) , (2.13) ‖u‖l2(0,t,h1(ω)) ≤ c2, c2 > 0. from the previous estimates we introduce the space w0(0,t) = {v ∈ l2(0,t,h1(ω))∩l∞(0,t,l2(ω)) , v(0) = u0 and ‖u‖l2(0,t,h1(ω)) +‖u‖l∞(0,t,l2(ω)) ≤ c}, where c(mf,t,a,‖u0‖l2(ω)). int. j. anal. appl. 19 (5) (2021) 717 the space w0(0,t) is nonemply closed convex in w(0,t), moreover it injects with a compact way in l2(0,t,l2(ω)), we define the application (2.14) f : w0(0,t) −→ w0(0,t) w 7−→ f(w). f is well defined, to apply the schauder fixed point theorem, we have to show that the application f is weakly continuous from w0(0,t) in w0(0,t) we consider a sequence vn ∈ w0(0,t) where vn ⇀ v in w0(0,t) and let un = f(vn). according to the classical result of compactness, we can extract from the sequence (un) a subsequence yet denoted (un) such taht • un ⇀ u weakly in l2(0,t; l2(ω)). • un −→ u strongly in l2(0,t; l2(ω)) and almost every where in qt . • oun ⇀ ou weakly in l2(0,t; l2(ω)). • vn −→ v strongly in l2(0,t; l2(ω)) and almost every where in qt . • ogσ ∗vn −→ogσ ∗v strongly in l2(0,t; l2(ω)) and almost every where in qt . • b(t)g(|ogσ ∗vn|) −→ b(t)g(|ogσ ∗v|) strongly in l2(0,t; l2(ω)). • f(t,x,vn) → f(t,x,v) strongly in l1(qt ) . • b(t)f(t,x,v) → b(t)f(t,x,v) strongly in l1(qt ). the latter is obtained by applying the dominated convergence theorem. we can then pass to the limit, then the sequene un = f(vn) converges weakly to u = f(v) in w0(0,t), then we deduce the existence of u ∈ w0(0,t) such that u = f(u) and thus the existence of u ∈ w(0,t). � step3: the truncated problem and a priori estimates. we consider the truncated function ψn in c∞c (r), such that 0 ≤ ψn ≤ 1 and defined by: (2.15) ψn(r) =   1 if |r| ≤ n, 0 if |r| ≥ n + 1. we truncate the nonlinearity f by ψn fn(t,x,un) = ψn(|u|)f(t,x,u), thus, we can earily check that fn satisfies (h1) − (h3) with mf = mfn and for every (t,x) ∈ qt , ∀r ∈ r fn(t,x,un) → f(t,x,u), since u0 ∈ l2(ω) and |fn(t,x,un)| ≤ mfn, so lemma (2.2) is applied, then we get the existence of a weak int. j. anal. appl. 19 (5) (2021) 718 solution of the problem (2.16) ∂un ∂t −b(t)div(g(|o(unσ|)oun) = b(t)fn(t,x,un) in qt , un(0,x) = un0 in ω, ∂un ∂ν = 0 in σt . now, we are in the case to prove that a subsequence un converge to the weak solution u of the problem (1.7), for this we have to prove that lemma: lemma 2.3. we consider un as sequence of weak solutions given in (2.1): (i) ∫ qt |fn(t,x,un)|dxdt ≤ ∫ ω |un0|dx. (ii) un is bounded in l 2(0,t,h1(ω)) and ∫ qt |unfn(t,x,un)|dxdt ≤ ∫ ω u2n0dx . (iii) un is relatively compact in l 2(qt ). proof. (i) we have: ∂un ∂t −b(t)div(g(|o(unσ|)oun) = b(t)fn(t,x,un),∫ qt ∂un ∂t dxdt− ∫ qt b(t)div(g(|o(unσ|)oun)dxdt = ∫ qt b(t)fn(t,x,un)dxdt, (2.17) ∫ ω |un(t)|dx + cb ∫ qt |fn(t,x,un)|dxdt ≤ ∫ ω |un0|dx, (2.18) ∫ qt |fn(t,x,un)|dxdt ≤ ∫ ω |un0|dx. (ii) first, we prove that un is bounded in l 2(qt ), for this, we consider un as a test function ϕ in the approximate problem 1 2 ∫ qt |un|2(t)dx + cba ∫ qt |∇un|2dxdt ≤ cb ∫ qt |un||f(t,x,un)|dxdt + 1 2 ∫ ω |un0|2dx, we have that cba ∫ qt |∇un|2dxdt ≥ 0 then (2.19) ∫ qt |unfn(t,x,un)|dxdt ≤ 1 2 ∫ ω |un0|dx, where (2.20) sup 0<t<t ‖un(ω)‖l2(ω) ≤‖un0‖2l2(ω). int. j. anal. appl. 19 (5) (2021) 719 let from the previous result 1 2 ∫ ω u2n(t)dx + acb ∫ qt |∇un|2dxdt ≤ cb 2 ∫ qt u2ndxdt + (cb 2 + 1 2 )∫ ω u2n0, min(acb, cb 2 ) ∫ qt |∇un|2dxdt + ∫ qt |un|2dxdt ≤ ( cb + 1 )∫ ω u2n0dx. setting c1 = min(acb, cb 2 ) (2.21) ∫ qt |∇un|2dxdt + ∫ qt |un|2dxdt ≤ c2 ∫ ω u2n0dx, (2.22) ‖un‖l2(0,t,h1(ω)) ≤ c2‖un0‖2l2(ω). (iii) let fn(t,x,un) bounded in l 1(qt ) and ∂un ∂t bounded in l1(0,t, (h1(ω))′), with simon [19], un is relatively compact in l2(qt ). � step4: convergence. according to the previous result of lemma (2.3), un is relatively compact in l 2(qt ), we can extract a subsequence still denoted un where: • un ⇀ u weakly in l2(0,t; l2(ω)) and almost every where in qt . • ogσ ∗un −→ogσ ∗u strongly in l2(qt ) and almost every where in qt . • b(t)g(|ogσ ∗un|) −→ b(t)g(|ogσ ∗u|) strongly in l2(qt ). • f(t,x,vn) ⇀ f(t,x,v) for almost every where qt . • b(t)f(t,x,v) ⇀ b(t)f(t,x,v) for almost every where in qt . to prove that u is a weak solution of (1.7), it suffices to prove that fn(t,x,un) → f(t,x,u) in l1(qt ), since fn(t,x,un) ⇀ f(t,x,u) almost every where in qt . we will prove that fn(t,x,un) is uniformly integrable in l1(qt ). for this we use the vitali theorem where: ∀ε > 0, ∃δ > 0, such that ∀e ⊂ qt measurable with |e| < δ we have:∫ e |fn(t,x,un)| ≤ ε. ∀k ≥ 0: ∫ e |fn(t,x,un)|dxdt ≤ ∫ [e∩|un|≤k] |fn(t,x,un)|dxdt + ∫ [e∩|un|>k] |fn(t,x,un)|dxdt. where: (2.23) ∫ [e∩|un|≤k] |fn(t,x,un)|dxdt ≤ ∫ e sup |un|≤k |fn(t,x,un)|dxdt. ∫ e |fn(t,x,un)|dxdt ≤ ∫ e sup |un|≤k |fn(t,x,un)|dxdt + ∫ [e∩|un|>k] |fn(t,x,un)|dxdt, int. j. anal. appl. 19 (5) (2021) 720 we have that sup |un|≤k |fn(t,x,un)| ∈ l1(qt ), ∀ε > 0, ∃δ > 0 such as |e| < δ then: (2.24) ∫ e sup |un|≤k |fn(t,x,u)|dxdt ≤ ε 2 . we have |un| > k∫ [e∩|un|>k] |fn(t,x,un)|dxdt ≤ 1 k ∫ qt unfn(t,x,un)dxdt, ≤ ∫ e sup |un|≤k |fn(t,x,un)|dxdt + 1 k ∫ e∩|un|>k |unfn(t,x,un)|dxdt, ≤ ε 2 + 1 k ∫ e |unfn(t,x,un)|dxdt, if k ≥ ‖u‖22(ω) ε then (2.25) ∫ [e∩|un|>k] |fn(t,x,un)| ≤ ε 2 , hence (2.26) ∫ e |fn(t,x,un)| ≤ ε. 3. the application if u(t,x) is differentiable in the sense of the gâteau, so c0 dαt u(t,x) = t1−α ∂u ∂t , with 0 < α < 1. (see[ [8], page 67]). so the problem (1.5) became: (3.1) t1−α ∂u ∂t −div(g(|ouσ|)ou) = 2 γ(2.3) exp(x)(2 −x)t1,3 −u(t,x) − 2 exp(x)t2, with 0 < α < 1. the explicit finite difference approximation for (3.1) is t1−α un+1(i,j) −un(i,j) dt −div ( g ( |o(gσ ∗un(i,j))|,λn ) oun(i,j) ) = ( 2 γ(2.3) t−0.7 − 1)un(i,j) − 2 exp(i,j)t2, with dt is the time step, 0 < t < t with t is the processing time and uni,j is the approximation of u(t,x) in the pixel (i,j) in time ndt. first of all, we consider an original image without noise then we apply the new model on a noisy image with an additive gaussian noise. figure 1. cameraman image without noise. int. j. anal. appl. 19 (5) (2021) 721 we apply an additive gaussian noise on the binary image (1) with variance σ = 0.09. figure 2. noisy image with σ = 0.09. then, we apply the fractional model on the noisy image (2), where we set the parameters of the model as follows: we set the processing time t at 0.009, t = 0.001 and α = 0.7. figure 3. restored image with proposed model. int. j. anal. appl. 19 (5) (2021) 722 this fractional model can also be applied to the processing of color images. figure 4. peppers image without noise. figure 5. noisy image with σ = 0.09. figure 6. restored image with proposed model. int. j. anal. appl. 19 (5) (2021) 723 4. the conclusions in the case of summary, we demonstrated the existence of a global weak solution of the proposed model. also, we proved that the truncated problem admits a weak solution according to schauder fixed point theorem. for the nonlinear function satisfying suitable conditions, we established the equi-integrability and we derived a compactness result to be able to pass to the limit to get the desired result. to show the importance of the obtained result, a new application in the field of image restoration. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. aarab, n. alaa, h. khalfi. generic reaction-diffusion model with application to image restoration and enhancement, electron. j. differ. equ. 2018 (2018), 125. [2] n. alaa, m. aitoussous, w. bouarifi, d. bensikaddour, image restoration using a reaction-diffussion process, electron. j. differ. equ. 2014 (2014), 197. [3] n.e. alaa, m. zirhem, bio-inspired reaction diffusion system applied to image restoration, int. j. bio-inspired comput. 12 (2018), 128-137. [4] b. al-hamzah, n. yebri. global existence in reaction diffusion nonlinear parabolic partial differential equation in image procrssing, glob. j. adv. eng. technol. sci. 3 (2016), 5. [5] f. catté, p.l. lions, j-m. morel, t. coll. image selective smoothing and edge detection by nonlinear diffusion. siam j. numer. anal. 29 (1992), 182-193. [6] j. chen. an implicit approximation for the caputo fractional reaction-dispersion equation. j. xiamen univ. (nat. sci.) 46 (2007), 616-619. (in chinese). [7] c. gong, w. bao, g. tang, y. jiang, j. liu, a domain decomposition method for time fractional reaction-diffusion equation, sci. world j. 2014 (2014), 681707. [8] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65–70. [9] a. kilbas, h.m. srivastava and j.j. trujillo. theory and applications of fractional differential equations. volume 204 of north-holland mathematics studies. elsevier, amesterdam, 2006. int. j. anal. appl. 19 (5) (2021) 724 [10] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65–70. [11] s. lecheheb, m. maouni, h. lakhal, existence of the solution of a quasilinear equation and its application to image denoising, int. j. comput. sci. commun. inform. technol. 7 (2019), 1-6. [12] s. lecheheb, m. maouni, h. lakhal, image restoration using nonlinear elliptic equation, int. j. comput. sci. commun. inform. technol. 6 (2019), 32-37. [13] s. lecheheb, m. maouni, h. lakhal, image restoration using a novel model combining the perona-malik equation and the heat equation, int. j. anal. appl. 19 (2021), 228-238. [14] m. maouni, f.z. nouri, image restoration based on p-gradient model, int. j. appl. math. stat. 41 (2013), 48-57. [15] s. morfu. on some applications of diffusion processes for image processing. phys. lett. a, 373 (2009), 24-44. [16] p. perona, j. malik, scale-space and edge detection using anisotropic diffusion, ieee trans. pattern anal. mach. intell. 12 (1990), 629–639. [17] b. tellab, résolution des équations différentielles fractionnaires. université des frères mentouri constantine-1, 2018. [18] feng xing. méthode de décomposition de domaines pour l’équation de schrödinger. analyse numérique [math.na]. université lille 1 sciences et technologies, 2014. [19] j. simon, compact sets in the spacel p (o,t; b), ann. mat. pura appl. 146 (1986), 65–96. [20] j. weickert. anisotropic diffusion in image processing. phd thesis, kaiserslautern university, kaiserslautern, germany, 1996. [21] f.z. zeghbib, m. maouni, f.z. nouri, overlapping and nonoverlapping domain decomposition methods for image restoration, int. j. appl. math. stat. 40 (2013), 123-128. 1. introduction 2. the main result proof of theorem step1: the positivity of the solution step2: existence result for bounded nonlinearity step3: the truncated problem and a priori estimates step4: convergence 3. the application 4. the conclusions references international journal of analysis and applications volume 19, number 4 (2021), 503-511 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-503 trace result for sobolev extension domains djamel ait-akli∗, abdelkader merakeb l2csp, mouloud mammeri university of tizi-ouzou, 15000, tizi-ouzou, algeria ∗corresponding author: djamel.aitakli@ummto.dz abstract. in this paper, we establish the existence and continuity of a trace operator for functions of the sobolev space w 1,p(ω) with 1 < p < ∞ on the boundary of a domain ω that has the sobolev w 1,p−extension property. first, we prove the existence and the continuity of such an operator when it is applied to the elements of the subspace of the up to boundary smooth functions by using a uniform estimate. the essential ingredients used in the proof of this estimate are green’s representation of a function on a disk as well as banach’s isomorphism theorem. finally, we conclude the trace result using the density of smooth functions in w 1,p(ω). the presented proof fully exploits the extensibility hypothesis of the domain ω. the relevance of the result lies in the existence of extension domains which are not lipschitz and under this point of view it constitutes a generalization of the usual trace theorem. 1. introduction the trace operator, when applied to the functions of the sobolev space w 1,p(ω) defined on a lipschitz domain ω, is a standard notion in the theory of sobolev spaces. more precisely, it is well established that the trace operator is well defined on the boundary of a lipschitz domain and, moreover, it is continuous, cf. [1]. the result established in this paper generalizes this fact to the case of a non-lipschitz domain, namely to w 1,p(ω)-extension domains. the regularity of this class of domains is intermediate between lipschitz regularity and jordan regularity. received march 7th, 2021; accepted april 12th, 2021; published 0, 2021. 2010 mathematics subject classification. 46e35, 35j57, 26a16, 46b25. key words and phrases. sobolev spaces; w 1,p−extension domains; green representation; trace inequality; density of smooth functions. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 503 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-503 int. j. anal. appl. 19 (4) (2021) 504 the functions of the sobolev space w 1,p(ω) defined on such a domain constitute a relevant class of functions in the following sense: there exists domains whose boundary is parameterized by a continuous function that is not lipschitz but admit the property of (1,p)− extension. indeed, maz’ya has constructed, cf. [2], an example of a jordan domain, ω, such that the boundary ∂ω is not lipschitz in the neighborhood of at least one of its points and such that the property of (1,p)− extension is only valid for p < 2. we emphasize that a lipschitz domain is always an extension domain. we give, herein, a proof of the existence and the boundedness of a trace operator defined on the sobolev space w 1,p(ω) where ω is a w 1,p−extension domain, 1 < p < ∞. the proof that we present in the framework of extension domains fully exploits the domain’s extensibility hypothesis. we start by introducing some preliminary notions as well as the tools and results that are essential to carry out the proof of the main result. 2. classical facts and tools let ω ⊂ r2 be a domain. we denote by int(ω) the topological interior of ω. throughout this section, and only otherwise explicitly stated, p is a real number with 1 < p < ∞. the usual sobolev space w 1,p(ω) is defined by w 1,p(ω) := {u ∈ lp(ω), ∇u ∈ lp(ω)}. we further assume that ω has the property of w 1,p−extension. recall that the property of w 1,p−extension of a domain ω means that a sobolev extension operator can be defined on w 1,p(ω), we recall its definition: definition 2.1. we say that a domain ω ⊂ r2 is a w 1,p−extension domain, if there is a linear operator p defined by p : w 1,p(ω) → w 1,p(r2) u → pu. such that p is continuous i.e. there exists cext > 0 such that ∀u ∈ w 1,p(ω) we have (2.1) ||pu||w1,p(r2) ≤ cext||u||w1,p(ω) where cext depends only on ω and (pu)(x) = u(x) almost everywhere in ω. the reader may refer to [2] for a detailed discussion of this class of domains. we recall a density result established in ( [3], p.261). this result is valid in the case where ω is an extension domain or more generally a jordan domain: theorem 2.1. let ω ⊂ r2 be a domain whose boundary is a jordan curve, then c∞(ω) is dense in w 1,p(ω) with 1 < p < ∞. int. j. anal. appl. 19 (4) (2021) 505 we recall the property of continuity of the trace operator, on the lipschitz (or weakly lipschitz) boundary of a domain d ⊂ r2, defined for the sobolev space w 1,p(d), cf. [1]: proposition 2.1. let d ⊂ r2 be a lipschitz domain. there exists cd > 0, depending merely on d and p, such that (2.2) ||u|| w 1− 1 p ,p (∂ω) ≤ cd||u||w1,p(d), for all u ∈ w 1,p(d). it should be noted that proposition 2.1 does not apply in principle neither to an extension domain nor to the more general jordan domain given the primordial assumption of the lipschitz character of the ambient domain d. let us also recall a trace result for the functions of the sobolev space w 1,1(dr) where dr ⊂ r2 is a disk of radius r > 0, cf. ( [4], estimate 7.1): proposition 2.2. for all u ∈ w 1,1(dr) we have (2.3) ∫ ∂dr |u|dσ ≤ 2 r ∫ dr |u|d x + ∫ dr |∇u|d x. in addition, we recall the integral representation of a function u ∈ c2(dr) ∩ c0(∂dr) on the disk dr, cf. [5]. for such a function, this representation writes: for all x ∈ int(dr) we have u(x) = ∫ dr ∆u(y)g(x,y)dy + ∫ ∂dr u(y)k(x,y)dσ(y),(2.4) where g is the green function of the disk, dr, associated with the laplace operator and defined by: g(x,y) := γ(x−y) + hx(y), with γ(x) = 1 2π ln |x| being the fundamental solution of the laplace operator in dimension two and hx is the harmonic function in y that equals −γ(x − y) for x ∈ int(dr) and y ∈ ∂dr. on the other hand k(x,y) = ∂g(x,y) ∂νy is the poisson kernel which in the case of the disk is written: k(x,y) = r2 −|x|2 2ω2r|x−y|2 , where ω2 is the measure of the two-dimensional unit ball. this integral representation is interesting insofar as it makes it possible to point-wisely express a function using its laplacian and the values it takes on the boundary of the disk dr. int. j. anal. appl. 19 (4) (2021) 506 3. principal results the main result of this paper is stated in the following theorem, its proof will be given after establishing an auxiliary lemma. theorem 3.1. let ω be a domain having the w 1,p−extension property such that 1 < p < ∞, then we can define a trace operator t : w 1,p(ω) → lp(∂ω) u → tu that coincides with the restriction operator on the boundary ∂ω for continuous functions. in addition, t is continuous i.e. there exists ct > 0 independent on u such that for all u ∈ w 1,p(ω) we have (3.1) ||tu||p,∂ω ≤ ct||u||w1,p(ω) , where we denote || ||p,∂ω the usual norm of the lebesgue space of p−integrable functions. 3.1. auxiliary lemma. we now state with a proof an auxiliary lemma essential for establishing the main theorem: lemma 3.1. let 1 < p < 2 and fix a point x0 ∈ r2. we denote dr0 := d(x0,r0) ⊂ r2 the disk with center x0 and radius r0 > 0. there exists a constant c(r0,p) > 0 which is independent of both u and x such that for all u ∈ c∞(dr0 ) (3.2) ∀x ∈ dr0 2 , |u(x)| ≤ c(r0,p)||u||w1,p(dr0 ). we denote by |x−y|2 the euclidean distance between the points x and y. proof. let u ∈ c∞(dr0 ). there exists f ∈ c∞(dr0 ) and ud ∈ c∞(∂dr0 ) such that the function u solves the problem (3.3)   −∆u(y) = f(y) in dr0, u(y) = ud(y) on ∂dr0. let x ∈ dr0 2 . by using the integral representation (2.4) applied to the function u at x, we have: u(x) = ∫ dr0 f(y)g(x,y) d y + ∫ ∂dr0 ud(y)k(x,y) d σ(y).(3.4) given that ∂dr0 is c ∞−regular, then the green’s function satisfies g(x,.) ∈ w 1,p(dr0 ) for 1 < p < 2, cf. ( [6], estimate 1.5). by using the continuity of the linear form associated with f ∈ w−1,p(ω), we obtain int. j. anal. appl. 19 (4) (2021) 507 from estimate (3.4) the following |u(x)| ≤ ||f||w−1,p(dr0 )||g(x,.)||w1,p(dr0 ) + maxy∈∂dr0 |k(x,y)| ∫ ∂dr0 |ud(y)|d σ(y). the last term of this inequality does make sense since |x − y|2 ≥ r02 for x ∈ dr02 and y ∈ ∂dr0 . by applying the trace inequality on the boundary of the disc dr0 , cf. estimate (2.3), we get |u(x)| ≤||f||w−1,p(dr0 )||g(x,.)||w1,p(dr0 ) + max y∈∂dr0 |k(x,y)| ( 2 r0 ||u||l1(dr0 ) + ||∇u||l1(dr0 ) ) , the holder inequality then yields |u(x)| ≤||f||w−1,p(dr0 )||g(x,.)||w1,p(dr0 ) + max y∈∂dr0 |k(x,y)|cr0 ( ||u||p,dr0 + ||∇u||p,dr0 ) . we apply banach’s isomorphism theorem to the continuous bijective operator l1 defined by l1 : ( w−1,p(dr0 ), || ||w−1,p(dr0 ) ) → ( w 1,p 0 (dr0 ), || ||w1,p(dr0 ) ) f → l(f) = u1, with ∆u1 = f. we then deduce the existence of a constant c ′ 1 independent of f such that |u(x)| ≤c′1||u1||w1,p0 (dr0 )||g(x,.)||w1,p(dr0 ) + max y∈∂dr0 |k(x,y)|cr0||u||w1,p(dr0 )(3.5) with u1 = u−u2 and u2 solves the problem (3.6)   −∆u2 = 0 in dr0, u2 = u d on ∂dr0. by using the poincaré inequality, the estimate (3.5) then becomes |u(x)| ≤c1 ( ||∇u||p,dr0 + ||∇u2||p,dr0 ) ||g(x,.)||w1,p(dr0 )(3.7) + max y∈∂dr0 |k(x,y)|cr0||u||w1,p(dr0 ). using proposition 2.1, we can apply the isomorphism theorem to the operator associated with the problem (3.6) to establish the existence of a constant c2 such that ||∇u2||p,dr0 ≤ c2||u2||w1− 1 p ,p (∂dr0 ) . int. j. anal. appl. 19 (4) (2021) 508 where c2 is independent of u2 and ∆u2 = 0. the estimate (3.7) becomes |u(x)| ≤c1 ( ||∇u||p,dr0 + c2||u||w1− 1 p ,p (∂dr0 ) ) ||g(x,.)||w1,p(dr0 ) + max y∈∂dr0 |k(x,y)|cr0 ( ||u||p,dr0 + ||∇u||p,dr0 ) . applying the trace theorem in w 1,p(dr0 ), cf. proposition 2.1, allows to infer the existence of a constant c3 > 0 such that |u(x)| ≤c1 ( ||∇u||p,dr0 + c3(||u||p,dr0 + ||∇u||p,dr0 ) ) ||g(x,.)||w1,p(dr0 ) + max y∈∂dr0 |k(x,y)|cr0||u||w1,p(dr0 ), then we infer that there exists a constant c(r0,p) which is independent of both u and x such that ∀x ∈ dr0 2 , |u(x)| ≤c(r0,p)||u||w1,p(dr0 ). this last estimate is obtained by using the fact that ||g(x,.)||w1,p(dr0 ) is uniformly bounded for x ∈ dr0 2 . � we now present a proof of theorem 3.1: 3.2. proof of the main result. proof. let u ∈ c∞(ω). the function u is obviously lipschitz on ω. if we denote l the lipschitz constant of the function x → u(x) relatively to the domain ω, then for all x ∈ ∂ω and all y ∈ int(ω) we have |u(x) −u(y)| ≤ l|x−y|2, which immediately implies (3.8) |u(x)| ≤ l|x−y|2 + |u(y)|, for all x ∈ ∂ω and all y ∈ int(ω). let x ∈ ∂ω be fixed and let (yδ)δ>0 be a sequence of point such that yδ ∈ int(ω) and |yδ −x|2 → 0 when δ → 0. let’s fix δ > 0. the estimate (3.8) yields |u(x)| ≤ l|x−yδ|2 + |u(yδ)|,(3.9) for all δ > 0. pose ũ = p u with p being an extension operator defined on w 1,p(ω) whose range is w 1,p(r2) ⊂ w 1,p(ω), cf. definition 2.1. such an extension operator is well defined. it should be noted that the function ũ is not necessarily continuous over r2 − ω i.e. it does not have a representative continuous function. indeed, this is due to the fact that 1 − 2 p < 0 for 1 < p < 2; therefore the classical sobolev embedding into holder spaces doesn’t apply in this case, namely when 1 < p < ∞. since ω is bounded, there exists r0 > 0 and x0 ∈ ω such that if we denote dr0 the disk with center x0 and radius r0 then we have ω ⊂ dr0 . the set, c∞(d2r0 ), of smooth functions up to the boundary defined int. j. anal. appl. 19 (4) (2021) 509 on the lipschitz domain d2r0 ⊂ r2 being dense in w 1,p(d2r0 ), cf. theorem 2.1, therefore there exists a sequence (vn)n, vn ∈ c∞(d2r0 ), such that (3.10) ||vn − ũ||w1,p(d2r0 ) → 0 n →∞. the estimate (3.10) implies that there exists a subsequence (φ(n))n such that vφ(n)(x) → ũ(x) a.e. in d2r0 . as vn and ũ are continuous in ω, this almost everywhere convergence is valid everywhere in ω i.e. (3.11) ∀y ∈ ω, vφ(n)(y) → ũ(y) , when n →∞. using (3.9) and (3.11) we get: ∀δ > 0 and ∀� > 0, ∃n(�,yδ) > 0 such that: |u(x)| ≤ l|x−yδ|2 + |vφ(n(�,yδ))(yδ)| + �,(3.12) where � > 0 is intended to tend towards zero and (n(�,yδ))�>0 is a sequence of integers which tends to +∞ when � → 0 for all δ > 0. on the other hand, the formula (3.12) and the regularity of vφ(n(�,yδ)) allow to write |u(x)| ≤ l|x−yδ|2 + |vφ(n(�,yδ))|∞,dr0 + �, for all δ > 0. by applying the lemma 3.1 in the disk d2r0 with center x0, we have |u(x)| ≤ l|x−yδ|2 + c(r0,p)||vφ(n(�,yδ)||w1,p(d2r0 ) + �(3.13) where c(r0,p) is the constant appearing in the estimate (3.2). the estimate (3.13) is valid for all � > 0 independently of δ. so, passing to the limit � → 0 and using estimate (3.10), we obtain |u(x)| ≤ l|x−yδ|2 + c(r0,p)||ũ||w1,p(d2r0 ),(3.14) for all δ > 0. given that all the elements involved in the estimate (3.14) are independent of δ then by letting δ → 0 we find |u(x)| ≤ c(r0,p)||ũ||w1,p(d2r0 ).(3.15) so the estimate (3.15) yields us |u(x)| ≤ c(r0,p)||pu||w1,p(r2).(3.16) using (2.1), the estimate (3.16) in turn gives |u(x)| ≤ c(r0,p)cext||u||w1,p(ω).(3.17) note that the estimate (3.17) is valid for all x ∈ γ := ∂ω independently of the quantities present at the right side of this inequality. int. j. anal. appl. 19 (4) (2021) 510 now, let us denote t ∈ [a,b] → x(t) = (t,γ(t)) the parametric function representation of the curve γ. so, according to (3.17), we have: |u(t,γ(t))|p ≤ (c(r0,p)cext)p||u|| p w1,p(ω) ,(3.18) for all t ∈ [a,b]. by definition of the line integral, we have∫ γ |u|p ds := ∫ b a |u◦γ|pdsγ.(3.19) thus, by integrating the two sides of (3.18) with respect to the curvilinear abscissa dsγ and using (3.19), we find ∫ γ |u|p ds ≤ (c(r0,p)cext)p||u|| p w1,p(ω) ∫ b a 1 dsγ, it yields ∫ ∂ω |u|p ds ≤ (c(r0,p)cext)p||u|| p w1,p(ω) |∂ω|.(3.20) set ct := [(c(r0,p)cext) p|∂ω|] 1 p . this constant does not depend on u but merely on ω and on the exponent p. from (3.20), we have for all u ∈ c∞(ω) ||u||p,∂ω ≤ ct||u||w1,p(ω).(3.21) currently, we conclude the estimate (3.1). according to theorem 2.1, for any u ∈ w 1,p(ω) there exists (ul)n∈n, ul ∈ c∞(ω), such that ||ul −u||w1,p(ω) → 0. thus, by applying (3.21) to the elements of the sequence (ul)l, we have for all l ∈ n the following estimate: ||ul||p,∂ω ≤ ct||ul||w1,p(ω).(3.22) since (ul)l is a cauchy sequence in the normed space w 1,p(ω) then estimate (3.22) implies that it is also a cauchy sequence in the normed space lp(∂ω). but lp(∂ω) is complete, then there exists u∗ ∈ lp(∂ω) such that ||u∗||p,∂ω ≤ ct||u||w1,p(ω). finally, by setting tu := u∗, we have ||tu||p,∂ω ≤ ct||u||w1,p(ω) int. j. anal. appl. 19 (4) (2021) 511 for all u ∈ w 1,p(ω). this defines a continuous trace operator on the space lp(∂ω) on the boundary ∂ω t : w 1,p(ω) → lp(∂ω) u → tu � remark 3.1. the relevance of the established result lies in the fact that it is valid for functions of the space w 1,p(ω) which are not necessarily continuous for 1 < p < 2. the existence of the trace for this category of functions is not obvious. the example of the domain constructed by mazya in [2] illustrates perfectly not only the relevance but also the generality which results from the fact that we have considered the domain ω having the w 1,p−extension property. acknowledgment. the authors would like to thank very much the anonymous referees for reviewing the article as well as the editorial staff for their support during the submission process. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] e. gagliardo, caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili, ren. sem. mat. univ. padova. 27 (1957), 284–305. [2] v.g. maz’ya, extension of functions from sobolev spaces, j. math. sci. 22 (1983), 1851–1855. [3] j.l. lewis, approximation of sobolev functions in jordan domains, ark. mat. 25 (1987), 255-264. [4] g. auchmuty, sharp boundary trace inequalities, proc. r. soc. edinburgh sect. a: math. 144 (2014), 1-12. [5] l.c. evans, partial differential equations, graduate studies in mathematics, americam mathematical society, providence, 1998. [6] d. mitrea, i. mitrea, on the regularity of green functions in lipschitz domains, commun. part. differ. equ. 36 (2010), 304–327. 1. introduction 2. classical facts and tools 3. principal results 3.1. auxiliary lemma 3.2. proof of the main result acknowledgment references international journal of analysis and applications volume 19, number 6 (2021), 812-825 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-812 stability result for a weakly nonlinearly damped porous system with distributed delay khoudir kibeche1, lamine bouzettouta2,∗, abdelhak djebabla3, fahima hebhoub2 1university of badji mokhtar, annaba, algeria 2laboratory applied mathematics and history and didactics of mathematic, university 20 august 1955, skikda, algeria 3laboratory of applied mathematics, badji mokhtar university-annaba, p.o. box 12, 23000 annaba, algeria ∗corresponding author: lami 750000@yahoo.fr, l.bouzettouta@univ-skikda.dz abstract. in this paper, we consider a one-dimensional porous system damped with a single weakly nonlinear feedback and distributed delay term. without imposing any restrictive growth assumption near the origin on the damping term, we establish an explicit and general decay rate, using a multiplier method and some properties of convex functions in case of the same speed of propagation in the two equations of the system. the result is new and opens more research areas into porous-elastic system. received june 5th, 2021; accepted june 30th, 2021; published october 28th, 2021. 2010 mathematics subject classification. 35b35, 35b40, 93d20. key words and phrases. porous system; general decay; nonlinear damping; distributed delay. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 812 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-812 int. j. anal. appl. 19 (6) (2021) 813 1. introduction in this paper, we consider the following porous system:  ρutt −µuxx − bφx + µ1ut + ∫ τ2 τ1 µ2(s)ut (x,t−s) ds = 0, x ∈ (0, 1), t > 0, jφtt − δφxx + bux + ξφ + α (t) g (φt) = 0, x ∈ (0, 1), t > 0, u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ (0, 1), φ (x, 0) = φ0 (x) , φt (x, 0) = φ1 (x) , x ∈ (0, 1), ux (0, t) = ux (1, t) , φ (0, t) = φ (1, t) = 0 ut (x,−t) = f0 (x,t) in (0, 1) × (0,τ2) (1.1) firstly, to deal with the delay term, we introduce the new variable [17] z (x,ρ,s,t) = ut (x,t−ρs) , x ∈ (0, 1) , ρ ∈ (0, 1) , ρ ∈ (τ1,τ2) , t > 0 then we obtain szt (x,ρ,s,t) + zρ (x,ρ,s,t) = 0, x ∈ (0, 1) , ρ ∈ (0, 1) , ρ ∈ (τ1,τ2) , t > 0 then problem (1.1) is equivalent to  ρutt −µuxx − bφx + µ1ut + ∫ τ2 τ1 µ2(s)z (x, 1, t,s) ds = 0, x ∈ (0, 1), t > 0, jφtt − δφxx + bux + ξφ + α (t) g (φt) = 0, x ∈ (0, 1), t > 0, szt (x,ρ,s,t) + zρ (x,ρ,s,t) = 0, x ∈ (0, 1) , ρ ∈ (0, 1) , ρ ∈ (τ1,τ2) , t > 0 u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ (0, 1), φ (x, 0) = φ0 (x) , φt (x, 0) = φ1 (x) , x ∈ (0, 1), ux (0, t) = ux (1, t) , φ (0, t) = φ (1, t) = 0 z (x,ρ,s, 0) = f0 (x,ρs) , (x,ρ,s) ∈ (0, 1) × (0, 1) × (τ1,τ2) (1.2) in recent paper, apalara in [2] considered the following on-dimensional porous system damped with a single weakly nonlinear feedback  ρutt −µuxx − bφx = 0, x ∈ (0, 1), t > 0, jφtt − δφxx + bux + ξφ + α (t) g (φt) = 0, x ∈ (0, 1), t > 0, u (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , x ∈ (0, 1), φ (x, 0) = φ0 (x) , φt (x, 0) = φ1 (x) , x ∈ (0, 1), ux (0, t) = ux (1, t) , φ (0, t) = φ (1, t) = 0 without in pasing an explicit and general decay rate, he used a multiplier method and some proprieties of convex functions in case of the same speed of propagation in the both equation of the system. the same author, in [3] considered a porous-elastic system with memory term acting only on the porous equation, with int. j. anal. appl. 19 (6) (2021) 814 the mixed boundary neumann-direchlet conditions, he proved a general decay result, for which exponential and polynomial decay results are special cases. back to system (1.1), it is to be noted that when µ1 = µ2 = 0 and replacing the term α (t) g (φt) by the term ∫ t 0 g(t−s)uxx (x,s) ds then (1.1) is equivalent to the well-known timoshenko system of memory type which is exponentially stable depending of the relaxation function g and provided that the wave speeds of the system are equal (see [1, 15]). messaoudi and fareh [16] investigated the following system:  ρutt = µuxx + bφx −βθx, in (0, 1) × (0,∞), jφtt = αφxx − bux + ξφ + mθ + τφt, in (0, 1) × (0,∞), cφt = −qx −βutx −mφt, in (0, 1) × (0,∞), τ0qt −q + kθx = 0, in (0, 1) × (0,∞), and established, using the energy method, an exponential decay result. for more results on the subject, we refer the reader to [5, 10, 11, 19]. concerning the weight of the delay, we assume that∫ τ2 τ1 |µ2(s)|ds < µ1 and establish the well-posedness as well as the exponential stability results of the energy e (t), defined by e (t) = 1 2 ∫ 1 0 [ ρu2t + µu 2 x + ξφ 2 + δφ2x + jφ 2 t + 2bφux ] dx + 1 2 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z 2 (x,ρ,s,t) dsdρdx (1.3) 2. preliminaries in this section, we present some materials needed in the proof of our result. we assume α and g satisfy the following hypotheses: (h1) α : r+ → r+∗ is a non-increasing differentiable function; (h2) g : r → r is a non-decreasing c0-function such that there exist positive constants c1, c2, η and g ∈ c1 ([0,∞)) , with g (0) = 0, and g is linear or strictly convex c2−function on (0,η] such that  s 2 + g2 (s) ≤ g−1 (sg (s)) for all |s| ≤ η c1 |s| ≤ |g (s)| ≤ c2 |s| for all |s| ≥ η remark 2.1. hypothesis (h2) implies that sg (s) > 0 for all s 6= 0. * according to our knowledge, hypothesis (h2) with η = 1 was first introduced by lasiecka and tataru [13]. they established a decay result, which depends on the solution of an explicit nonlinear ordinary differential equation. furthermore, they proved that the monotonicity and continuity of g guarantee the existence of the function g defined in (h2). int. j. anal. appl. 19 (6) (2021) 815 for completeness purpose we state, without proof, the existence and regularity result of system (1.1). first, we introduce the following spaces: h = h1∗ (0, 1) ×l 2 ∗ (0, 1) ×h 1 (0, 1) ×l2 (0, 1) ×l2 ((0, 1) × (0, 1) × (τ1,τ2)) , (2.1) and h̃ = φ0 ∈ [ h2∗ (0, 1) ∩h 1 ∗ (0, 1) ] ×h1∗ (0, 1) × [ h2 (0, 1) ∩h1 (0, 1) ] ×h1 (0, 1) ×l2 ((0, 1) × (0, 1) × (τ1,τ2)) , where l2∗ (0, 1) = { ψ ∈ l2 (0, 1) : ∫ 1 0 ψ (x) dx = 0 } , h1∗ (0, 1) = h 1 (0, 1) ×l2∗ (0, 1) , h2∗ (0, 1) = { ψ ∈ h2 (0, 1) : ψx (0) = ψx (1) = 0 } . for u = (u,ut,φ,φt,z) , we have the following existence and regularity result: proposition 2.1. assume that (h1) and (h2) are satisfied. then for all u0 ∈ h, the system (1.1) has a unique global (weak) solution u ∈ c ( r+; h1∗ (0, 1) ) ∩c1 ( r+; l2∗ (0, 1) ) , φ ∈ c ( r+; h1 (0, 1) ) ∩c1 ( r+; l2 (0, 1) ) . moreover, if u0 ∈ h̃, then the solution satisfies u ∈ l∞ ( r+; h2∗ (0, 1) ∩h 1 ∗ (0, 1) ) ∩w1,∞ ( r+; h1∗ (0, 1) ) ∩w2,∞ ( r+; l2∗ (0, 1) ) , φ ∈ l∞ ( r+; h2 (0, 1) ∩h10 (0, 1) ) ∩w1,∞ ( r+; h10 (0, 1) ) ∩w2,∞ ( r+; l2 (0, 1) ) remark 2.2. this result can be proved using the theory of maximal nonlinear monotone operators (see [8]). 3. technical lemmas in this section, we state and prove our stability results for the energy of system (1.1) by using the multiplier technique. to achieve our goal, we need the following lemmas. lemma 3.1. let (u,φ,z) be the solution of(1.2), then we have e′ (t) ≤−me ∫ 1 0 u2tdx− ∫ 1 0 α (t) φtg (φt) dx ≤ 0 (3.1) int. j. anal. appl. 19 (6) (2021) 816 proof. multiplying (1.2)1, and (1.2)2 by ut, φt respectively, and integrating over (0, 1), using integration by parts and the boundary conditions, we obtain 1 2 d dt ∫ 1 0 ( ρu2t + µu 2 x + ξφ 2 + δφ2x + jφ 2 t + 2bφux ) dx = (3.2) − ∫ 1 0 α (t) φtg (φt) dx−µ1 ∫ 1 0 u2tdx− ∫ 1 0 ut ∫ τ2 τ1 µ2(s)z (x, 1, t,s) dsdx multiplying (1.2)3 by |µ2(s)|z, integrating the product over (0, 1)×(0, 1)×(τ1,τ2) , and recall that z (x, 0,s,t) = ut, we get 1 2 d dt ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z 2 (x,ρ,t,s) dsdρdx = − 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx + 1 2 ∫ 1 0 ut ∫ τ2 τ1 |µ2(s)|dsdx. (3.3) a combination of (3.2) and (3.3) gives e′ (t) = − ∫ 1 0 α (t) φtg (φt) dx−µ1 ∫ 1 0 u2tdx− ∫ 1 0 ut ∫ τ2 τ1 µ2(s)z (x, 1, t,s) dsdx with − ∫ 1 0 ut ∫ τ2 τ1 µ2(s)z (x, 1, t,s) dsdx ≤ 1 2 ∫ τ2 τ1 |µ2(s)| ∫ 1 0 u2tdx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx then e′ (t) ≤− ∫ 1 0 α (t) φtg (φt) dx− ( µ1 − ∫ τ2 τ1 |µ2(s)| )∫ 1 0 u2tdx taking ( µ1 − ∫ τ2 τ1 |µ2(s)| ) = me we obtain (3.1). � lemma 3.2. assume that (h1) and (h2) hold. then, for all u0 ∈h, the functional f1 (t) = j ∫ 1 0 φtφdx + bρ µ ∫ 1 0 φ ∫ x 0 ut (y) dydx (3.4) satisfies, for any ε1 > 0 f ′1 (t) ≤ ( j + ε1bρ µ )∫ 1 0 φ2tdx− jδ ∫ 1 0 φ2xdx + bjε1 ∫ 1 0 u2xdx + bρ 4ε1µ ∫ 1 0 u2tdx + ( jα (0) ε1 + bj 4ε1 − ξj )∫ 1 0 φ2dx + jα (0) 4ε1 ∫ 1 0 g2 (φt) dx (3.5) proof. differentiating f1 (t) , taking into account (1.2) using integrating by parts, and young’s inequality, we obtain f ′1 (t) ≤ j ∫ 1 0 φ2tdx− jδ ∫ 1 0 φ2xdx + bjε1 ∫ 1 0 u2xdx + cpbj 4ε1 ∫ 1 0 φ2xdx− ξjcp ∫ 1 0 φ2xdx −j ∫ 1 0 α (t) φg (φt) dx + bρ µ ∫ 1 0 φt ∫ x 0 ut (y) dydx + bρ µ ∫ 1 0 φ d dt (∫ x 0 ut (y) dy ) dx int. j. anal. appl. 19 (6) (2021) 817 by caucy-schwartz inequality, it is clear that ∫ 1 0 (∫ x 0 ut (y) dy )2 dx ≤ ∫ 1 0 (∫ 1 0 utdx )2 dx ≤ ∫ 1 0 u2tdx then f ′1 (t) ≤ j ∫ 1 0 φ2tdx− jδ ∫ 1 0 φ2xdx + bjε1 ∫ 1 0 u2xdx + cpbj 4ε1 ∫ 1 0 φ2xdx− ξjcp ∫ 1 0 φ2xdx −j ∫ 1 0 α (t) φg (φt) dx + ε1bρ µ ∫ 1 0 φ2tdx + bρ 4ε1µ ∫ 1 0 (∫ x 0 ut (y) dy )2 dx + bρ µ ∫ 1 0 φ d dt (∫ x 0 ut (y) dy ) dx thus we obtain f ′1 (t) ≤ ( j + ε1bρ µ )∫ 1 0 φ2tdx− jδ ∫ 1 0 φ2xdx + bjε1 ∫ 1 0 u2xdx + bρ 4ε1µ ∫ 1 0 u2tdx + ( jα (t) ε1 + bj 4ε1 − ξj )∫ 1 0 φ2dx + jα (t) 4ε1 ∫ 1 0 g2 (φt) dx � lemma 3.3. assume that (h1), (h2) and (3.8) hold. then, for all u0 ∈h, the functional f2 (t) = b ∫ 1 0 φxutdx + b ∫ 1 0 φtuxdx (3.6) satisfies, for any ε2 > 0 f ′2 (t) ≤ ( b2 ρ + ε2 bµ1 ρ + bn0 2ρ )∫ 1 0 φ2xdx− ( b2 j − bξ 4ε2j − b j α (t) )∫ 1 0 u2xdx + bµ1 4ε2ρ ∫ 1 0 u2tdx + ε2 bξ j ∫ 1 0 φ2dx + b j α (t) ∫ 1 0 g2 (φt) dx + 1 2 b ρ ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1,s,t) dsdx (3.7) proof. simple computaions give f ′2 (t) = b2 ρ ∫ 1 0 φ2xdx− b2 j ∫ 1 0 u2xdx + bµ ρ ∫ 1 0 uxxφxdx− bµ1 ρ ∫ 1 0 φxutdx + bδ j ∫ 1 0 φxxuxdx− bξ j ∫ 1 0 φuxdx − b ρ ∫ 1 0 φx ∫ τ2 τ1 µ2(s)ut (x, 1, t,s) dsdx− b j ∫ 1 0 α (t) uxg (φt) dx int. j. anal. appl. 19 (6) (2021) 818 taking into account the fact that µ ρ = δ j (3.8) and using young’s inequality f ′2 (t) ≤ ( b2 ρ + ε2 bµ1 ρ )∫ 1 0 φ2xdx + ( bξ 4ε2j − b2 j + b j α (t) )∫ 1 0 u2xdx + bµ1 4ε2ρ ∫ 1 0 u2tdx + ε2 bξ j ∫ 1 0 φ2dx + b j α (t) ∫ 1 0 g2 (φt) dx − b ρ ∫ 1 0 φx ∫ τ2 τ1 µ2(s)z (x, 1, t,s) dsdx − b ρ ∫ 1 0 φx ∫ τ2 τ1 µ2(s)z (x, 1, t,s) dsdx ≤ 1 2 b ρ ∫ τ2 τ1 |µ2(s)|ds ∫ 1 0 φ2xdx + 1 2 b ρ ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1,s,t) dsdx f ′2 (t) ≤ ( b2 ρ + ε2 bµ1 ρ + bn0 2ρ )∫ 1 0 φ2xdx− ( b2 j − bξ 4ε2j − b j α (t) )∫ 1 0 u2xdx + bµ1 4ε2ρ ∫ 1 0 u2tdx + ε2 bξ j ∫ 1 0 φ2dx + b j α (t) ∫ 1 0 g2 (φt) dx + 1 2 b ρ ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1,s,t) dsdx with ∫ τ2 τ1 |µ2(s)|ds = n0 � lemma 3.4. the functional f3 (t) = −ρ ∫ 1 0 utudx (3.9) satisfies, for any ε3 > 0 f ′3 (t) = + ( µ + n0cp 2 + cpbε3 )∫ 1 0 u2xdx + b 4ε3 ∫ 1 0 φ2xdx− ( ρ− µ1 4ε3 )∫ 1 0 u2tdx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx (3.10) proof. a simple differentiation of f3 (t), using the first equation in (1.2), give f ′3 (t) = −ρ ∫ 1 0 u2tdx + µ ∫ 1 0 u2xdx +cpbε3 ∫ 1 0 u2xdx + b 4ε3 ∫ 1 0 φ2xdx +µ1ε3cp ∫ 1 0 u2xdx + µ1 4ε3 ∫ 1 0 u2tdx + ∫ 1 0 ∫ τ2 τ1 µ2(s)uut (x, 1, t,s) dsdx ∫ 1 0 u ∫ τ2 τ1 µ2(s)ut (x, 1, t,s) dsdx ≤ cp 2 ∫ τ2 τ1 |µ2(s)| ∫ 1 0 u2xdx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx int. j. anal. appl. 19 (6) (2021) 819 then f ′3 (t) = + ( µ + µ1ε3cp + n0cp 2 + cpbε3 )∫ 1 0 u2xdx + b 4ε3 ∫ 1 0 φ2xdx− ( ρ− µ1 4ε3 )∫ 1 0 u2tdx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx � lemma 3.5. the functional f4 (t) = ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2(s)|z 2 (x,ρ,t,s) dsdρdx (3.11) satisfies, for some positive constant m1, the following estimate f ′4 (t) ≤ −m1 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx + ∫ τ2 τ1 |µ2(s)|ds ∫ 1 0 u2tdx −m1 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z 2 (x,ρ,t,s) dsdρdx (3.12) proof. with szt (x,ρ,t,s) + zρ (x,ρ,t,s) = 0 in (0, 1) × (0, 1) × (τ1,τ2) × (0,∞) (3.13) zt (x,ρ,t,s) = − 1 s zρ (x,ρ,t,s) differentiating f4 (t), and using the equation (3.13), we obtain f ′4 (t) = 2 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2(s)|zzt (x,ρ,t,s) dsdρdx = − ∂ ∂ρ ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 e−sρ |µ2(s)|z 2 (x,ρ,t,s) dsdρdx − ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2(s)|z 2 (x,ρ,t,s) dsdρdx f ′4 (t) = − ∫ 1 0 ∫ τ2 τ1 |µ2(s)| [ e−sρz2 (x, 1, t,s) −z2 (x, 0, t,s) ] dsdx − ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2(s)|z 2 (x,ρ,t,s) dsdρdx using the fact that z (x, 0, t,s) = ut and e −s ≤ e−sρ ≤ 1, for all ρ ∈ [0, 1] , we obtain f ′4 (t) ≤ − ∫ 1 0 ∫ τ2 τ1 |µ2(s)|e −sρz2 (x, 1, t,s) dsdx + ∫ τ2 τ1 |µ2(s)|ds ∫ 1 0 u2tdx − ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2(s)|z 2 (x,ρ,t,s) dsdρdx because −se−s is an increasing function, we have −se−s ≤−se−τ2 , for all s ∈ [τ1,τ2] int. j. anal. appl. 19 (6) (2021) 820 finally, setting m1 = e −τ2 , with ∫ τ2 τ1 |µ2(s)| < µ1, we obtain f ′4 (t) ≤ −m1 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1, t,s) dsdx + ∫ τ2 τ1 |µ2(s)|ds ∫ 1 0 u2tdx −m1 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z 2 (x,ρ,t,s) dsdρdx � lemma 3.6. suppose (h1), (h2), and eq. (3.8) hold. let u0 ∈h. then, for n,n1,n2,n3 > 0 sufficiently large, the lyapunov functional defined by l(t) := ne (t) + n1f1 (t) + n2f2 (t) + f3 (t) + n3f4 (t) satisfies, for some positive constants d1,d2 and k1 d1l(t) ≤ e (t) ≤ d2l(t) , ∀t ≥ 0 (3.14) and l′ (t) ≤−k1e (t) + c ∫ 1 0 ( φ2t + g 2 (φt) ) dx, ∀t ≥ 0 (3.15) with l′ (t) ≤ [ bρ 4ε1µ n1 −nme + n3µ1 + bµ1 4ε2ρ n2 − ( ρ− µ1 4ε3 )]∫ 1 0 u2tdx + ( n1 ( j + ε1bρ µ ))∫ 1 0 φ2tdx + ( bjε1n1 + ( µ + n0 2 + bε3 ) −n2 ( b2 j − bξ 4ε2j − b j α (t) ))∫ 1 0 u2xdx + ( n2 1 2ρ ( 2b2 + 2ε2bµ1 + bn0 ) − jδn1 + b 4ε3 )∫ 1 0 φ2xdx + ( ε2 bξ j n2 + n1 ( jα (t) ε1 + bj 4ε1 − ξj ))∫ 1 0 φ2dx + ( n1 jα (t) 4ε1 + b j α (t) n2 )∫ 1 0 g2 (φt) dx + ( 1 2 ( bn2 ρ + 1 ) −m1n3 )∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1,s,t) dsdx −m1n3 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z 2 (x,ρ,t,s) dsdρdx−n ∫ 1 0 α (t) φtg (φt) dx at this point, we have to choose our constants very carefully. first, choosing ε3 << 1, and ε1,ε2 small enough such that ε1 ≤ bρn1 4µ (nme −n3µ1) , ε2 ≤ bµ1n2 4ρ int. j. anal. appl. 19 (6) (2021) 821 moreover, we pick ni i = 1, 2, 3 large enough so that n2 ≥ bjε1n1 + ( µ + n0 2 + bε3 ) b2 j − bξ 4ε2j − b j α (t) and n3 ≥ ( bn2 ρ + 1 ) 2m1 . after that, we can choose n large enough such that n > 1 me [ bρn1 4ε1µ + n3µ1 + n2bµ1 4ε2ρ − ( ρ− µ1 4ε3 )] . consequently, there exists a positive constant η1 such that (3.15) becomes d dt l(t) ≤ −c1 ∫ 1 0 ( u2t + ux + ϕ 2 x + φ 2 ) dx + c2 ∫ 1 0 ( φ2t + g 2 (φt) ) dx −c3 ∫ 1 0 ∫ τ2 τ1 |µ2(s)|z 2 (x, 1,s,t) dsdx. (3.16) in this section, we state and prove our stability result. 4. stability result theorem 4.1. suppose (h1), (h2), and (3.8) hold. let u0 ∈h. there exist positive constants a1,a2,a3 and η0 such that the solution of (1.2) satisfies e (t) ≤ a1g−11 ( a2 ∫ t 0 α (s) ds + a3 ) , t ≥ 0, (4.1) where g−11 = ∫ 1 t 1 g0 (s) ds and g0 (s) = tg ′ (η0t) . remark 4.1. g1 strictly decreases and is convex on (0, 1] and lim t→0 g1 (t) = +∞. proof. we multiply (3.15) by α (t) to get α (t)l′ (t) ≤−k1α (t) e (t) + cα (t) ∫ 1 0 ( φ2t + g 2 (φt) ) dx. (4.2) now, we discuss two cases: case i: g is linear on [0,η]. in this case, using (h2) and eq.(3.1), we deduce that α (t)l′ (t) ≤−k1α (t) e (t) + cα (t) ∫ 1 0 ( φ2t + g 2 (φt) ) dx = −k1α (t) e (t) − ce′ (t) , which can be rewritten as (α (t)l(t) + ce (t))′ −α′ (t)l(t) ≤−k1α (t) e (t) . int. j. anal. appl. 19 (6) (2021) 822 using (h1), we obtain (α (t)l(t) + ce (t))′ ≤−k1α (t) e (t) . by exploiting (3.14), it can easily be shown that s0 (t) := α (t)l(t) + ce (t) ∼ e (t) . (4.3) so, for some positive constant λ1, we obtain s′0 (t) + λ1α (t)s0 (t) ≤ 0, ∀t ≥ 0 (4.4) the combination of eq. (4.3) and (4.4), gives e (t) ≤ e (0) e−λ1 ∫ t 0 α(s)ds = e (0) g−11 ( λ1 ∫ t 0 α (s) ds ) . (4.5) case ii: g is nonlinear on [0,η]. in this case, we first choose 0 < η1 < η such that sg (s) ≤ min{η,g (η)} , ∀|s| ≤ η1. (4.6) using (h2) along with fact that g is continuous and |g (s)| > 0, for s 6= 0, it follows that  s 2 + g2 (s) ≤ g−1 (sg (s)) , ∀|s| ≤ η1 c1 |s| ≤ |sg (s)| ≤ c2 |s| , ∀|s| ≥ η1 (4.7) to estimate the last integral in eq. (4.2), we consider the following partition of (0, 1): i1 = {x ∈ (0, 1) : |φt| ≤ η1} , i2 = {x ∈ (0, 1) : |φt| > η1} . now, with i (t) defined by i (t) = ∫ i1 φtg (φt) dx, we have, using jensen inequality (note that g−1 is concave and recall (4.6)) g−1 (i (t)) ≥ c ∫ i1 g−1 (φtg (φt)) dx. (4.8) the combination of eq. (4.7) and (4.8) yields α (t) ∫ 1 0 ( φ2t + g 2 (φt) ) dx = α (t) ∫ i1 ( φ2t + g 2 (φt) ) dx + α (t) ∫ i2 ( φ2t + g 2 (φt) ) dx ≤ α (t) ∫ i1 g−1 (φtg (φt)) dx + cα (t) ∫ i2 φtg (φt) dx ≤ cα (t) g−1 (i (t)) − ce′ (t) . (4.9) so, by substituting (4.9) into (4.2) and using (4.3) and (h1), we have s′0 (t) ≤−k1α (t) e (t) + cα (t) g −1 (i (t)) (4.10) int. j. anal. appl. 19 (6) (2021) 823 now, for η1 < η and δ0 > 0, using (4.10) and the fact that e ′ ≤ 0, g′ > 0, g′′ > 0 on (0,η) , we find that the functional s1, defined by s1 (t) := g′ ( η0 e (t) e (0) ) s0 (t) + δ0e (t) , satisfies, for some b1, b2 > 0, b1s1 (t) ≤ e (t) ≤ b2s1 (t) (4.11) and s′0 (t) : = η0 e′ (t) e (0) g′′ ( η0 e (t) e (0) ) s0 (t) + g′ ( η0 e (t) e (0) ) s′0 (t) + δ0e ′ (t) ≤ −k1α (t) e (t) g′ ( η0 e (t) e (0) ) + cα (t) g′ ( η0 e (t) e (0) ) g−1 (i (t)) + δ0e ′ (t) (4.12) let g∗ be the convex conjugate of g defined by g∗ (s) = s ( g ′ )−1 (s) −g [( g ′ )−1 (s) ] , if s ∈ (0,g′ (η)] , satisfying the following general young’s inequality ab ≤ g∗ (a) + g (b) , if a ∈ (0,g′ (η)] , b ∈ (0,η] . with a = g′ ( η0 e (t) e (0) ) and b = g−1 (i (t)) , using (4.6), we obtain cα (t) g′ ( η0 e (t) e (0) ) g−1 (i (t)) ≤ cα (t) g∗ ( g′ ( η0 e (t) e (0) )) + cα (t) i (t) . by exploiting (3.1) and the fact that g∗ (s) ≤ s (g′)−1 (s) , we get cα (t) g′ ( η0 e (t) e (0) ) g−1 (i (t)) ≤ cα (t) η0 e (t) e (0) g′ ( η0 e (t) e (0) ) − ce′ (t) (4.13) by substituting (4.12) into eq. (4.13), we obtain s′1 (t) ≤−kα (t) e (t) e (0) g′ ( η0 e (t) e (0) ) = −k1α (t) g0 ( e (t) e (0) ) (4.14) where k > 0 and g0 (t) = tg ′ (η0t) . note that g′0 (t) = g ′ (η0t) + η0tg ′′ (η0t) . so, using the strict convexity of g on (0,η] , we find that g0 (t) ,g ′ 0 (t) > 0 on (0, 1] .with s (t) := b1s1(t) e(0) it is obvious that s (t) ≤ e(t) e(0) ≤ 1. now, using (4.11) and (4.14), we have s (t) ∼ e (t) (4.15) int. j. anal. appl. 19 (6) (2021) 824 and, for some a2 > 0 s′ (t) ≤−a2α (t) g0 (s (t)) . (4.16) inequality (4.16) implies that d dt g1 (s (t)) ≥ a2α (t) , where g1 (t) = ∫ t 1 1 g0 (s) ds. thus, by integrating over [0, t] , we obtain, for some a3 > 0, s (t) ≤ g−11 ( a2 ∫ t 0 α (s) ds + a3 ) . (4.17) here, we used, based on the properties of g0, the fact that g1 is strictly decreasing on (0, 1] .finally, using (4.17) and (4.15), we obtain (4.1). � acknowledgement 4.1. the author wish to thank deeply the anonymous referee for useful remarks and careful reading of the proofs presented in this paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] f. ammar-khodja, a. benabdallah, j.e. muñoz rivera, r. racke, energy decay for timoshenko systems of memory type, j. differ. equ. 194 (2003), 82–115. [2] t.a. apalara, a general decay for a weakly nonlinearly damped porous system, j. dyn. control syst. 25 (2019), 311–322. [3] t.a. apalara, general decay of solutions in one-dimensional porous-elastic system with memory, j. math. anal. appl. 469 (2019), 457–471. [4] l. bouzettouta, s. zitouni, kh. zennir, and h. sissaoui, well-posedness and decay of solutions to bresse system with internal distributed delay, int. j. appl. math. stat. 56 (2017), 153–168. [5] l. bouzettouta. a. djebabla, exponential stabilization of the full von kármán beam by a thermal effect and a frictional damping and distributed delay, j. math. phys. 60 (2019), 041506. [6] l. bouzettouta, f. hebhoub, k. ghennam and s. benferdi, exponential stability for a nonlinear timoshenko system with distributed delay, int. j. anal. appl. 19 (2021), 77-90. [7] p.s. casas, r. quintanilla, exponential stability in thermoelasticity with microtemperatures, int. j. eng. sci. 43 (2005), 33–47. [8] a. haraux, nonlinear evolution equations–q-global behavior of solutions, lecture notes in mathematics 841. springer, berlin, 1981. [9] d. ieşan and r. quintanilla, on thermoelastic bodies with inner structure and microtemperatures, j. math. anal. appl. 354 (2009), 12–23. [10] h. e. khochemane, l. bouzettouta, a. guerouah, exponential decay and well-posedness for a one-dimensional porouselastic system with distributed delay, appl. anal. (2019), 1–15. https://doi.org/10.1080/00036811.2019.1703958. int. j. anal. appl. 19 (6) (2021) 825 [11] h. e. khochemane, a. djebabla, s. zitouni, l. bouzettouta, well-posedness and general decay of a nonlinear damping porous-elastic system with infinite memory, j. math. phys. 61 (2020), 021505. [12] h. e. khochemane, l. bouzettouta and s. zitouni, general decay of a nonlinear damping porous-elastic system with past history. ann. univ. ferrara. 65 (2019), 249–275. [13] i. lasiecka, d. tataru, uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. differ. integral equ. 6 (1993), 507–533. [14] a. magaña and r. quintanilla, on the exponential decay of solutions in one-dimensional generalized porousthermoelasticity, asymptotic anal. 49 (2006), 183–187. [15] s. a. messaoudi and m. i. mustafa, a stability result in a memory-type timoshenko system, dyn. syst. appl. 18 (2009), 457–468. [16] s. a. messaoudi and a. fareh, exponential decay for linear damped porous thermoelastic systems with second sound, discrete contin. dyn. syst. ser. b 20 (2015), 599–612. [17] s. nicaise and c. pignotti, stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, siam j. control optim. 45 (2006), 1561–1585. [18] p. x. pamplona, j. e. muñoz rivera and r. quintanilla, on the decay of solutions for porous-elastic systems with history, j. math. anal. appl. 379 (2011), 682–705. [19] a. soufyane, energy decay for porous-thermo-elasticity systems of memory type, appl. anal. 87 (2008), 451–464. [20] a. soufyane, m. afilal, m. aouam and m. chacha, general decay of solutions of a linear one-dimensional porous thermoelasticity system with a boundary control of memory type, nonlinear anal., theory meth. appl. 72 (2010), 3903–3910. [21] a. soufyane, m. afilal and m. chacha, boundary stabilization of memory type for the porous-thermo-elasticity system, abstr. appl. anal. 2009 (2009), article id 280790. [22] s. zitouni, l. bouzettouta, kh. zennir and d. ouchenane, exponential decay of thermo-elastic bresse system with distributed delay term, hacettepe j. math. stat. 47 (2018), 1216-1230. 1. introduction 2. preliminaries 3. technical lemmas 4. stability result references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 87-92 http://www.etamaths.com a real paley-wiener theorem for the generalized dunkl transform a. abouelaz, a. achak , r. daher, el. loualid∗ abstract. in this article, we prove a real paley-wiener theorem for the generalized dunkl transform on r. 1. introduction in [3] n.b andersen proved a real paley-wiener theorem for the dunkl transform. in this paper, we first prove a real paley-wiener theorem for the generalized dunkl transform. let λα denote the dunkl operator and fα,n the dunkl transform, chettaoui, c., trimèche proved in [4] the following theorem: theorem 1.1. let 1 ≤ p ≤∞. let f ∈s(r) (the schwartz space on r). then lim m→∞ ‖λmα f‖ 1 m p = sup{|λ|,λ ∈ suppfα(f)}. n.b andersen in [3] gave a simple proof of the above theorem by using the real paleywiener theorem for the dunkl transform. our second result is to prove the above theorem for the generalized dunkl transform. the structure of the paper is as follows: in section 2 we set some notations and collect some basic results about the dunkl operator and the dunkl transform, and we give also some facts about harmonic analysis related to the first-order singular differential-difference operator λα,n, and the generalized dunkl transform. in section 3 we state and prove a real paley-wiener theorem for the generalized dunkl transform. in section 4 we give a characterization of the support of the generalized dunkl transform on r 2. preliminaries throughout this paper we assume that α > −1 2 , and we denote by • e(r) the space of functions c∞ on r, provided with the topology of compact convergence for all derivatives. that is the topology defined by semi-norms pa,m(f) = supx∈[−a,a] m∑ k=0 | dk dxk f(x) |, a > 0, m = 0, 1, ... • da(r), the space of c∞ function on r, which are supported in [−a,a], equipped with the topology induced by e(r). • d(r) = ⋃ a>0 da(r), endowed with inductive limit topology. • en(r) (resp dn(r)) stand for the subspace of e(r) (resp d(r)) consisting of functions f such that f(0) = ... = f(2n−1)(0). 2010 mathematics subject classification. 65r10. key words and phrases. real paley-wiener theorem; generalized dunkl transform. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 87 88 ozturk • lpα the class of measurable functions f on r for which ‖f‖p,α < ∞, where ‖f‖p,α = (∫ r |f(x)|p|x|2α+1dx )1 p , ifp < ∞, and ‖f‖∞,α = ‖f‖∞ = esssupx≥0|f(x)|. • lpα,n the class of measurable functions f on r for which ‖f‖p,α,n = ‖m−1f‖p,α+2n < ∞. • dpα,n(r) = dn(r) ⋂ lpα,n(r). • ha, a > 0, the space of entire rapidly decreasing functions of exponential type a ; that is, f ∈ ha, a > 0 if and only if, f is entire on c and for all j=0,1...... qj(f) = sup λ∈c |(1 + λ)mf(λ)e−a|imλ| < ∞ ha, a > 0 is equipped with the topology defined by the semi-norms qj, j = 0, 1..... • h = ⋃ a>0 ha, equipped with the inductive limit topology. 2.1. dunkl transform. in this subsection we recall some facts about harmonic analysis related to dunkl operator λα associated with reflection group z2 on r. we cite here, as briefly as possible, only some properties. for more details we refer to [2, 4, 5]. the dunkl operator λα is defined as follow: (1) λαf(x) = f′(x) + (α + 1 2 ) f(x) −f(−x) x . the dunkl kernel eα is defined by (2) eα(z) = jα(iz) + z 2(α + 1) jα+1(z) (z ∈ c) where jα(z) = γ(α + 1)σ ∞ n=0 (−1)n(z 2 )2n n! γ(n + α + 1) (z ∈ c). is the normalized spherical bessel function of index α. the functions eα(λ) λ ∈ c, are solutions of the differential-difference equation λαu = λu, u(0) = 1. furthermore, dunkl kernel eα possesses the laplace type integral representation eα(z) = aα ∫ 1 −1 (1 − t2)α− 1 2 (1 + t)eztdt, where (3) aα = γ(α + 1) √ πγ(α + 1 2 ). the dunkl transform of a function f ∈ d(r) is defined by (4) fα(f)(λ) = ∫ r f(x)eα(−iλx)|x|2α+1dx, λ ∈ c. theorem 2.1. (i): the dunkl transform fα is a topological automorphism from d(r) onto h. more precisely f ∈ da(r) if, and only if, fα(f) ∈ ha a real paley-wiener theorem 89 (ii): for every f ∈ d(r) , f(x) = ∫ r fα(f)(λ)eα(iλx)|λ|2α+1dλ,∫ r |f(x)|2|x|2α+1dx = mα ∫ r |fα(f)(λ)|2|λ|2α+1dλ, where (5) mα = 1 22(α+1)(γ(α + 1))2 . 2.2. generalized dunkl transform. in this section, we recall some properties about generalized dunkl transform. we refer to [1] for more details and references. the first-order singular differential-difference operator on r is defined as follow (6) λα,nf(x) = f′(x) + (α + 1 2 ) f(x) −f(−x) x − 2n f(−x) x , lemma 2.2. (i): the map mn(f)(x) = x 2nf(x) is a topological isomorphism • from e(r) onto en(r); • from d(r) onto dn(r). (ii): for all f ∈ e(r), λα,n ◦mn(f) = mn ◦ λα+2n(f), where λα+2n is the dunkl operator of order α + 2n given by (1) (iii): let f ∈ en(r) and g ∈ dn(r). then (7) ∫ r λα,nf(x)g(x)|x|2α+1dx = − ∫ r f(x)λα,ng(x)|x|2α+1dx. 2.3. generalized dunkl transform. for λ ∈ c and x ∈ r put (8) ψλ,α,n(x) = x2neα+2n(iλx), where eα+2n is the dunkl kernel of index α + 2n given by (2). proposition 2.3. (i): ψλ,α,n satisfies the differential-difference equation (9) λα,nψλ,α,n = iλψλ,α,n. definition 2.4. the generalized dunkl transform of a function f ∈ dn(r) is defined by (10) fα,n(f)(λ) = ∫ r f(x)ψ−λ,α,n(x)|x|2α+1dx, λ ∈ c. proposition 2.5. for every f ∈ dn(r), (11) fα,n(λα,nf)(λ) = iλfα,n(f)(λ), theorem 2.6. (i): for all f ∈ dn(r), we have the inversion formula f(x) = mα+2n ∫ r fα,n(f)(λ)ψλ,α,n(x)|λ|2α+4n+1dλ, where mα+2n is given by (5). (ii): for every f ∈ dn(r), we have the plancherel formula (12) ∫ r |f(x)|2|x|2α+1dx = mα+2n ∫ r |fα,n(f)(λ)|2|λ|2α+4n+1dλ. 90 ozturk 3. a real paley-wiener theorem in this section, we give a short and simple proof of a real paley-wiener theorem for the dunkl transform. we define the real paley-wiener space pwr(r) as the space of all f ∈ s(r) such that, for n ∈ n0 = n∪{0} (13) sup x∈r,m∈n0 r−mm−n (1 + |x|)n|λmα,nf(x)| < ∞. our real paley-wiener theorem is the following: theorem 3.1. let r > 0. the generalized dunkl transform fα,n is a bijection from pwr(r) onto c∞r (r), and by symmetry a bijection from c ∞ r (r) onto pwr(r). proof. let f ∈ pwr(r), and λ outside [−r,r]. then (7) and (9) yield fα,nf(λ) = ∫ r f(x)ψ−λ,α,n(x)|x|2α+1dx, = (−iλ)−m ∫ r f(x)λmα,nψ−λ,α,n(x)|x| 2α+1dx, = (−iλ)−m(−1)m ∫ r λmα,nf(x)ψ−λ,α,n(x)|x| 2α+1dx, hence, for a positive c, |fα,nf(λ)| = |(−iλ)−m(−1)m ∫ r λmα,nf(x)ψ−λ,α,n(x)|x| 2α+1dx|, ≤ |λ|−m ∫ r |λmα,nf(x)ψ−λ,α,n(x)||x| 2α+1dx|, ≤ c|λ|−m ∫ r rmmn (1 + |x|)−n|x|2α+2n+1dx|, = c( r |λ| )mmn ∫ r (1 + |x|)−n|x|2α+2n+1dx|→ 0 for m →∞. and thus suppfα,nf ⊂ [−r,r]. conversely, let f ∈c∞r (r). fix n ∈ n0. f−1α,nf(λ) := mα+2n ∫ r f(λ)ψλ,α,n(x)|λ|2α+4n+1dλ, xn λmα,nf −1 α,nf(λ) = mα+2n ∫ r f(λ)xn λmα,nψλ,α,n(x)|λ| 2α+4n+1dλ, = mα+2n(−i)m ∫ r λmf(λ)xn x2n λ2n ψx,α,n(λ)|λ|2α+4n+1dλ, = (−i)mmα+2n ∫ r λmf(λ)xn+2nψx,α,n(λ)|λ|2α+2n+1dλ, = (−i)m−n−2nmα+2n ∫ r λmf(λ)λn+2nα,n ψx,α,n(λ)|λ| 2α+2n+1dλ, = (−i)m+n+2nmα+2n ∫ r λn+2nα,n (λ mf(λ))ψx,α,n(λ)|λ|2α+2n+1dλ. a small calculation give λα,n(λ mf(λ) = mλm−1[f(λ) + 1 m λ d dλ f(λ) + 1 m (α + 1 2 (f(λ) − (−1)mf(−λ)) − 2n m f(−λ) λm ] a real paley-wiener theorem 91 let f̃ denote the function in square bracket. an induction argument with f1 = f̃ and f̃i+1 = f̃i, show that we can write, for m > n + 2n λα,n(λ n+2n(λmf(λ)) = λm−n−2nmn+2nf̃n+2n(λ), where f̃n+2n ∈c∞r (r) with suppf̃n+2n ⊂ suppf, and ‖ f̃n+2n ‖∞≤ c n+2n∑ k=0 ‖ dk dxk f ‖∞ where c is a positive constant only depending on f,α,n and n not on m. we get thus |xn λmα,nf −1 α,nf(x)| ≤ cmα+2nr 2α+2n+1mn n+2n∑ k=0 ‖ dk dxk f ‖∞ for all x ∈ r, and m > n + 2n, and thus f−1α,nf ∈ pwr(r) 4. a characterization of the support of the generalized dunkl transform on r theorem 4.1. let 1 ≤ p ≤∞. let f ∈s(r). then lim m→∞ ‖λmα,nf‖ 1 m p,α,n = sup{|λ|,λ ∈ suppfα,n(f)}. proof. define rf = sup{|λ|,λ ∈ suppfα,n(f)}. assume that fα,n has a compact support. then f ∈ pwr(r) by theorem 3.1 and lim m→∞ ‖λmα,nf‖ 1 m p,α,n ≤ rf lim m→∞ m n m = rf for all 1 ≤ p ≤∞, using (13) with n ≥ 2α + 2m + 3. now consider an arbitrary f ∈ .., using (7) ‖ λmα,nf ‖ 2 2,α,n = ∫ r |λmα,nf(x)| 2|x|2α+1dx, = ∫ r λmα,nf(x)λ m α,nf(x)|x| 2α+1dx, = (−1)m ∫ r λ2mα,nf(x)f(x)|x| 2α+1dx. hölder’s inequality with 1 p + 1 q = 1 (14) ‖λmα,nf‖ 2 2,α,n ≤‖λ 2m α,nf‖p,α,n‖f‖q,α,n. similarly, we get ‖λm+1α,n f‖ 2 2,α,n ≤‖λ 2m+1 α,n f‖p,α,n‖λα,nf‖q,α,n. let r < rf. from (11) and (12) ‖ λmα,nf ‖ 2 2,α,n = ∫ r |λmα,nf(x)| 2|x|2α+1dλ, = mα+2n ∫ r |fα,n(λmα,nf(λ))| 2|λ|2α+4n+1dλ, = mα+2n ∫ r |λ|2m|fα,nf(λ)|2|λ|2α+4n+1dλ, ≥ mα+2nr2m ∫ r |fα,nf(λ)|2|λ|2α+4n+1dλ, 92 ozturk where the last integral is positive. combining (14) with the above inequality yields lim inf m→∞ ‖λ2mα,nf‖ 1 2m p,α,n ≥ lim inf m→∞ ‖λmα,nf‖ 1 m 2,α,n ≥ r for any 1 ≤ p ≤∞, and similarly lim inf m→∞ ‖λ2m+1α,n f‖ 1 2m+1 p,α,n ≥ rf. we thus conclude, for any 0 < r < rf r ≤ lim inf m→∞ ‖λmα,nf‖ 1 m p,α,n ≤ lim sup m→∞ ‖λmα,nf‖ 1 m p,α,n ≤ rf this complect the proof of the theorem. references [1] al sadhan, s.a., al subaie, r.f. and mourou, m.a. harmonic analysis associated with a first-order singular differential-difference operator on the real line. current advances in mathematics research, 1(2014), 23-34. [2] m.a. mourou and k. trimèche, "transmutation operators and paley-wiener theorem associated with a singular differential-difference operator on the real line", analysis and applications, 1(2003), 43-70. [3] nils byrial andersen real paley-wiener theorems for the dunkl transform on r, integral transforms and special functions, 17(2006), 543-547 [4] chettaoui, c., trimèche, k., new type paley-wiener theorems for the dunkl transform on r. integral transforms and special functions, 14(2003), 97-115. [5] trimèche, k., paley-wiener theorems for the dunkl transform and dunkl translation operators. integral transforms and special functions, 13(2002), 17-38. department of mathematics, faculty of sciences aïn chock, university of hassan ii, casablanca, morocco ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 24-39 http://www.etamaths.com on weak and strong convergence theorems of modified sp-iteration scheme for total asymptotically nonexpansive mappings g. s. saluja∗ abstract. in this paper, we study modified sp -iteration scheme for three total asymptotically nonexpansive mappings and also establish some weak and strong convergence theorems for mentioned mappings and scheme to converge to common fixed points in the framework of banach spaces. our results extend and generalize the previous works from the current existing literature. 1. introduction let c be a nonempty subset of a banach space e and t : c → c a nonlinear mapping. we denote the set of all fixed points of t by f(t). the set of common fixed points of three mappings t1, t2 and t3 will be denoted by f = ∩3i=1f(ti). definition 1.1. let t : c → c be a mapping. then (1) t is said to be nonexpansive if ‖tx−ty‖ ≤ ‖x−y‖(1.1) for all x, y ∈ c. (2) t is said to be asymptotically nonexpansive if there exists a positive sequence hn ∈ [1,∞) with limn→∞hn = 1 such that ‖tnx−tny‖ ≤ hn ‖x−y‖(1.2) for all x, y ∈ c and n ≥ 1. the class of asymptotically nonexpansive mappings was introduced by goebel and kirk [6] as a generalization of the class of nonexpansive mappings. they proved that if c is a nonempty closed convex subset of a real uniformly convex banach space and t is an asymptotically nonexpansive mapping on c, then has a fixed point. t is said to be asymptotically noneexpansive in the intermediate sense if it is continuous and the following inequality holds: lim sup n→∞ sup x,y∈c ( ‖tnx−tny‖−‖x−y‖ ) ≤ 0.(1.3) observe that if we define cn = lim sup n→∞ sup x,y∈c ( ‖tnx−tny‖−‖x−y‖ ) and νn = max{0,cn}, then νn → 0 as n →∞. it follows that (1.3) is reduced to ‖tnx−tny‖ ≤ ‖x−y‖ + νn,(1.4) 2010 mathematics subject classification. 47h09, 47h10, 47j25. key words and phrases. total asymptotically nonexpansive mapping; modified sp -iteration scheme; common fixed point; strong convergence; weak convergence; banach space. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 24 modified sp -iteration scheme 25 for all x, y ∈ c and n ≥ 1. the class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by bruck, kuczumow and reich [3]. it is known [10] that if c is a nonempty closed convex bounded subset of a uniformly convex banach space e and t is asymptotically nonexpansive in the intermediate sense mapping, then t has a fixed point. it is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate contains properly the class of asymptotically nonexpansive mappings. in 2006, albert et al. [2] introduced the notion of total asymptotically nonexpansive mappings. definition 1.2. ([2]) the mapping t is said to be total asymptotically nonexpansive if ‖tnx−tny‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(1.5) for all x, y ∈ c and n ≥ 1, where {µn} and {νn} are nonnegative real sequences such that µn → 0 and νn → 0 as n →∞ and a strictly increasing continuous function ψ : [0,∞) → [0,∞) with ψ(0) = 0. from the definition, we see that the class of total asymptotically nonexpansive mappings include the class of asymptotically nonexpansive mappings as a special case; see also [4] for more details. remark 1.3. from the above definition, it is clear that each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with νn = 0, µn = kn − 1 for all n ≥ 1, ψ(t) = t, t ≥ 0. (1) mann iteration [12]: chose x1 ∈ c and define xn+1 = (1 −αn)xn + αntxn, n ≥ 1,(1.6) where {αn} is a sequence in (0,1). (2) ishikawa iteration [9]: chose x1 ∈ c and define yn = (1 −βn)xn + βntxn xn+1 = (1 −αn)xn + αntyn, n ≥ 1,(1.7) where {αn} and {βn} are sequences in (0,1). (3) s-iteration [1]: chose x1 ∈ c and define yn = (1 −βn)xn + βntxn xn+1 = (1 −αn)txn + αntyn, n ≥ 1,(1.8) where {αn} and {βn} are sequences in (0,1). note that (1.8) is independent of (1.7) (and hence (1.6)). agarwal, o’regan and sahu [1] showed that their process independent of those of mann and ishikawa and converges faster than both of these (see [[1], proposition 3.1]). (4) modified s-iteration [1]: chose x1 ∈ c and define yn = (1 −βn)xn + βntnxn xn+1 = (1 −αn)tnxn + αntnyn, n ≥ 1,(1.9) where {αn} and {βn} are sequences in (0,1). (5) noor iteration [13]: chose x1 ∈ c and define zn = (1 −γn)xn + γntxn yn = (1 −βn)xn + βntzn xn+1 = (1 −αn)xn + αntyn, n ≥ 1,(1.10) where {αn}, {βn} and {γn} are sequences in [0,1]. 26 saluja (6) modified noor iteration [21]: chose x1 ∈ c and define zn = (1 −γn)xn + γntnxn yn = (1 −βn)xn + βntnzn xn+1 = (1 −αn)xn + αntnyn, n ≥ 1,(1.11) where {αn}, {βn} and {γn} are sequences in [0,1]. recently, phuengrattana and suantai [16] introduced the following iteration scheme. (7) sp-iteration [16]: chose x1 ∈ c and define zn = (1 −γn)xn + γntxn yn = (1 −βn)zn + βntzn xn+1 = (1 −αn)yn + αntyn, n ≥ 1,(1.12) where {αn}, {βn} and {γn} are sequences in [0,1]. inspired and motivated by [16], we modify iteration scheme (1.12) for three total asymptotically nonexpansive self mappings of c as follows: (8) modified sp-iteration: chose x1 ∈ c and define zn = (1 −γn)xn + γntn3 xn yn = (1 −βn)zn + βntn2 zn xn+1 = (1 −αn)yn + αntn1 yn, n ≥ 1,(1.13) where {αn}, {βn} and {γn} are sequences in [0,1]. remark 1.4. if we take tn1 = t n 2 = t n 3 = t for all n ≥ 1, then (1.13) reduces to the sp-iteration scheme (1.12). the three-step iterative approximation problems were studied extensively by noor [13, 14], glowinsky and le tallec [7], and haubruge et al [8]. it has been shown [7] that three-step iterative scheme gives better numerical results than the two step and one step approximate iterations. thus we conclude that three step scheme plays an important and significant role in solving various problems, which arise in pure and applied sciences. the purpose of this paper is to study modified sp-iteration scheme (1.13) and establish some strong and weak convergence theorems for total asymptotically nonexpansive mappings in the setting of banach spaces. our results extend and generalize the previous works from the current existing literature. 2. preliminaries for the sake of convenience, we restate the following definitions and lemmas. let e be a banach space with its dimension greater than or equal to 2. the modulus of convexity of e is the function δe(ε) : (0, 2] → [0, 1] defined by δe(ε) = inf { 1 −‖ 1 2 (x + y)‖ : ‖x‖ = 1, ‖y‖ = 1, ε = ‖x−y‖ } . a banach space e is uniformly convex if and only if δe(ε) > 0 for all ε ∈ (0, 2]. we recall the following: let s = {x ∈ e : ‖x‖ = 1} and let e∗ be the dual of e, that is, the space of all continuous linear functionals f on e. definition 2.1. (i) opial condition: the space e has opial condition [15] if for any sequence {xn} in e, xn converges to x weakly it follows that lim supn→∞‖xn −x‖ < lim supn→∞‖xn −y‖ for all y ∈ e with y 6= x. examples of banach spaces satisfying opial condition are hilbert spaces and all spaces modified sp -iteration scheme 27 lp(1 < p < ∞). on the other hand, lp[0, 2π] with 1 < p 6= 2 fail to satisfy opial condition. (ii) a mapping t : c → c is said to be demiclosed at zero, if for any sequence {xn} in k, the condition xn converges weakly to x ∈ c and txn converges strongly to 0 imply tx = 0. (iii) a banach space e has the kadec-klee property [19] if for every sequence {xn} in e, xn → x weakly and ‖xn‖→‖x‖ it follows that ‖xn −x‖→ 0. definition 2.2. condition (a): the mapping t : c → e with f(t) 6= ∅ is said to satisfy condition (a) [18] if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(t) > 0 for all t ∈ (0,∞) such that ‖x−tx‖≥ f(d(x,f(t))) for all x ∈ c, where d(x,f(t)) = inf{‖x−p‖ : p ∈ f(t)}. now, we modify condition (a) for three mappings. definition 2.3. condition (b): three mappings t1, t2, t3 : c → c are said to satisfy condition (b) if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(t) > 0 for all t ∈ (0,∞) such that a1 ‖x−t1x‖ + a2 ‖x−t2x‖ + a3 t3x ≥ f(d(x,f)) for all x ∈ c, where d(x,f) = inf{‖x−p‖ : p ∈ f = ∩3i=1f(ti)}, where a1, a2 and a3 are nonnegative real numbers such that a1 + a2 + a3 = 1. note that condition (b) reduces to condition (a) when t1 = t2 = t3 = t and hence is more general than the demicompactness of t1, t2 and t3 [18]. a mapping t : c → c is called: (1) demicompact if any bounded sequence {xn} in c such that {xn − txn} converges has a convergent subsequence; (2) semicompact (or hemicompact) if any bounded sequence {xn} in c such that {xn −txn}→ 0 as n →∞ has a convergent subsequence. every demicompact mapping is semicompact but the converse is not true in general. senter and dotson [18] have approximated fixed points of a nonexpansive mapping t by mann iterates whereas maiti and ghosh [11] and tan and xu [20] have approximated the fixed points using ishikawa iterates under the condition (a) of [18]. tan and xu [20] pointed out that condition (a) is weaker than the compactness of c. we shall use condition (b) instead of compactness of c to study the strong convergence of {xn} defined by iteration scheme (1.13). lemma 2.4. (see [20]) let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of nonnegative numbers satisfying the inequality αn+1 ≤ (1 + βn)αn + rn, ∀n ≥ 1. if ∑∞ n=1 βn < ∞ and ∑∞ n=1 rn < ∞, then (i) limn→∞αn exists; (ii) in particular, if {αn}∞n=1 has a subsequence which converges strongly to zero, then limn→∞αn = 0. lemma 2.5. (see [17]) let e be a uniformly convex banach space and 0 < α ≤ tn ≤ β < 1 for all n ∈ n. suppose further that {xn} and {yn} are sequences of e such that lim supn→∞‖xn‖ ≤ a, lim supn→∞‖yn‖≤ a and limn→∞‖tnxn+(1−tn)yn‖ = a hold for some a ≥ 0. then limn→∞‖xn−yn‖ = 0. lemma 2.6. (see [19]) let e be a real reflexive banach space with its dual e∗ has the kadec-klee property. let {xn} be a bounded sequence in e and p, q ∈ ww(xn) (where ww(xn) denotes the set of all weak subsequential limits of {xn}). suppose limn→∞‖txn + (1 − t)p − q‖ exists for all t ∈ [0, 1]. then p = q. lemma 2.7. (see [19]) let k be a nonempty convex subset of a uniformly convex banach space e. then there exists a strictly increasing continuous convex function φ: [0,∞) → [0,∞) with φ(0) = 0 such that for each lipschitzian mapping t : c → c with the lipschitz constant l, ‖ttx + (1 − t)ty −t(tx + (1 − t)y)‖≤ lφ−1 ( ‖x−y‖− 1 l ‖tx−ty‖ ) for all x, y ∈ k and all t ∈ [0, 1]. 28 saluja proposition 2.8. let c be a nonempty subset of a banach space e and t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings. then there exist nonnegative real sequences {µn} and {νn} in [0,∞) with µn → 0 and νn → 0 as n → ∞ and a strictly increasing continuous function ψ : r+ → r+ with ψ(0) = 0 such that ‖tn1 x−t n 1 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.1) ‖tn2 x−t n 2 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.2) and ‖tn3 x−t n 3 y‖ ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn,(2.3) for all x, y ∈ c and n ≥ 1. proof. since t1, t2, t3 : c → c are three total asymptotically nonexpansive mappings, there exist nonnegative real sequences {µn1}, {µn2}, {µn3}, {νn1}, {νn2} and {νn3} in [0,∞) with µn1,µn2,µn3 → 0 and νn1,νn2,νn3 → 0 as n → ∞ and strictly increasing continuous functions ψ1,ψ2,ψ3 : r+ → r+ with ψi(0) = 0 for i = 1, 2, 3 such that ‖tn1 x−t n 1 y‖ ≤ ‖x−y‖ + µn1ψ1(‖x−y‖) + νn1,(2.4) ‖tn2 x−t n 2 y‖ ≤ ‖x−y‖ + µn2ψ2(‖x−y‖) + νn2,(2.5) and ‖tn3 x−t n 3 y‖ ≤ ‖x−y‖ + µn3ψ3(‖x−y‖) + νn3,(2.6) for all x, y ∈ c and n ≥ 1. setting µn = max{µn1, µn2, µn3}, νn = max{νn1, νn2, νn3} and ψ(r) = max{ψi(r), for i = 1, 2, 3 and for r ≥ 0}, then we get that there exist nonnegative real sequences {µn} and {νn} with µn → 0 and νn → 0 as n →∞ and strictly increasing continuous function ψ : r+ → r+ with ψ(0) = 0 such that ‖tn1 x−t n 1 y‖ ≤ ‖x−y‖ + µn1ψ1(‖x−y‖) + νn1 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, ‖tn2 x−t n 2 y‖ ≤ ‖x−y‖ + µn2ψ2(‖x−y‖) + νn2 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, and ‖tn3 x−t n 3 y‖ ≤ ‖x−y‖ + µn3ψ3(‖x−y‖) + νn3 ≤ ‖x−y‖ + µnψ(‖x−y‖) + νn, for all x, y ∈ c and n ≥ 1. � 3. strong convergence theorems in this section, we prove some strong convergence theorems for three total asymptotically nonexpansive mappings in the framework of real banach spaces. first, we shall need the following lemmas. lemma 3.1. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. modified sp -iteration scheme 29 then limn→∞‖xn −p‖ and limn→∞d(xn,f) both exist for all p ∈ f . proof. let p ∈ f. then from (1.13), we have ‖zn −p‖ = ‖(1 −γn)xn + γntn3 xn −p‖ ≤ (1 −γn)‖xn −p‖ + γn‖tn3 xn −p‖ ≤ (1 −γn)‖xn −p‖ + γn[‖xn −p‖ +µnψ(‖xn −p‖) + νn] ≤ (1 −γn)‖xn −p‖ + γn[‖xn −p‖ +µnm‖xn −p‖ + νn] ≤ ‖xn −p‖ + µnm‖xn −p‖ + νn = (1 + µnm)‖xn −p‖ + νn.(3.1) again from (1.13) and (3.1), we have ‖yn −p‖ = ‖(1 −βn)zn + βntn2 zn −p‖ ≤ (1 −βn)‖zn −p‖ + βn‖tn2 zn −p‖ ≤ (1 −βn)‖zn −p‖ + βn[‖zn −p‖ +µnψ(‖zn −p‖) + νn] ≤ (1 −βn)‖zn −p‖ + βn[‖zn −p‖ +µnm‖zn −p‖ + νn] ≤ ‖zn −p‖ + µnm‖zn −p‖ + νn = (1 + µnm)‖zn −p‖ + νn ≤ (1 + µnm)[(1 + µnm)‖xn −p‖ + νn] + νn ≤ (1 + µnm)2‖xn −p‖ + (2 + µnm)νn.(3.2) finally, using (1.13) and (3.2), we have ‖xn+1 −p‖ = ‖(1 −αn)yn + αntn1 yn −p‖ ≤ (1 −αn)‖yn −p‖ + αn‖tn1 yn −p‖ ≤ (1 −αn)‖yn −p‖ + αn[‖yn −p‖ +µnψ(‖yn −p‖) + νn] ≤ (1 −αn)‖yn −p‖ + αn[‖yn −p‖ +µnm‖yn −p‖ + νn] ≤ ‖yn −p‖ + µnm‖yn −p‖ + νn = (1 + µnm)‖yn −p‖ + νn ≤ (1 + µnm)[(1 + µnm)2‖xn −p‖ +(2 + µnm)νn] + νn ≤ (1 + µnm)3‖xn −p‖ + (1 + µnm) × (2 + µnm)νn + νn ≤ (1 + µnq1)‖xn −p‖ + νnq2(3.3) for some q1,q2 > 0. for any p ∈ f, from (3.3), we obtain the following inequality d(xn+1,f) ≤ (1 + µnq1)d(xn,f) + νnq2.(3.4) since ∑∞ n=1 µn < ∞ and ∑∞ n=1 νn < ∞, therefore applying lemma 2.4(i) in (3.3) and (3.4), we have limn→∞‖xn −p‖ and limn→∞d(xn,f) both exist. this completes the proof. � lemma 3.2. let e be a uniformly convex banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let 30 saluja {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. then limn→∞‖xn −tixn‖ = 0 for i = 1, 2, 3. proof. by lemma 3.1, limn→∞‖xn−p‖ exists for all p ∈ f , so we can assume that limn→∞‖xn−p‖ = c. then c > 0 otherwise there is nothing to prove. now (3.1) and (3.2) implies that lim sup n→∞ ‖zn −p‖ ≤ c,(3.5) and lim sup n→∞ ‖yn −p‖ ≤ c.(3.6) also ‖tn1 yn −p‖ ≤ ‖yn −p‖ + µnψ(‖yn −p‖) + νn ≤ ‖yn −p‖ + µnm‖yn −p‖ + νn = (1 + µnm)‖yn −p‖ + νn, and so lim sup n→∞ ‖tn1 yn −p‖ ≤ c.(3.7) since c = ‖xn+1 −p‖ = ‖(1 −αn)(yn −p) + αn(tn1 yn −p)‖. it follows from lemma 2.5 that lim n→∞ ‖tn1 yn −yn‖ = 0.(3.8) again note that ‖tn3 xn −p‖ ≤ ‖xn −p‖ + µnψ(‖xn −p‖) + νn ≤ ‖xn −p‖ + µnm‖xn −p‖ + νn = (1 + µnm)‖xn −p‖ + νn, ‖tn2 zn −p‖ ≤ ‖zn −p‖ + µnψ(‖zn −p‖) + νn ≤ ‖zn −p‖ + µnm‖zn −p‖ + νn = (1 + µnm)‖zn −p‖ + νn. hence, from above inequalities, we obtain lim sup n→∞ ‖tn3 xn −p‖ ≤ c,(3.9) and lim sup n→∞ ‖tn2 zn −p‖ ≤ c.(3.10) further, note that ‖yn −p‖ ≤ ‖yn −tn1 yn‖ + ‖t n 1 yn −p‖ ≤ ‖yn −tn1 yn‖ + ‖yn −p‖ + µnψ(‖yn −p‖) + νn ≤ ‖yn −tn1 yn‖ + ‖yn −p‖ + µnm‖yn −p‖ + νn ≤ ‖yn −tn1 yn‖ + (1 + µnm)‖yn −p‖ + νn. modified sp -iteration scheme 31 it follows from (3.6) and (3.8) that c ≤ lim inf n→∞ ‖yn −p‖.(3.11) from (3.6) and (3.11), we get lim n→∞ ‖yn −p‖ = c.(3.12) now, we have c = lim n→∞ ‖yn −p‖ = ‖(1 −βn)(zn −p) + βn(tn2 zn −p)‖.(3.13) it follows from (3.5), (3.10) and lemma 2.5 that lim n→∞ ‖tn2 zn −zn‖ = 0.(3.14) again note that ‖zn −p‖ ≤ ‖zn −tn2 zn‖ + ‖t n 2 zn −p‖ ≤ ‖zn −tn2 zn‖ + ‖zn −p‖ + µnψ(‖zn −p‖) + νn ≤ ‖zn −tn2 zn‖ + ‖zn −p‖ + µnm‖zn −p‖ + νn ≤ ‖zn −tn2 zn‖ + (1 + µnm)‖zn −p‖ + νn. it follows from (3.5) and (3.14) that c ≤ lim inf n→∞ ‖zn −p‖.(3.15) from (3.5) and (3.15), we get lim n→∞ ‖zn −p‖ = c.(3.16) now, we see that c = lim n→∞ ‖zn −p‖ = ‖(1 −γn)(xn −p) + γn(tn3 xn −p)‖.(3.17) it follows from lemma 2.5 that lim n→∞ ‖tn3 xn −xn‖ = 0.(3.18) again note that ‖xn −zn‖ = γn‖xn −tn3 xn‖ ≤ (1 −δ)‖xn −tn3 xn‖.(3.19) using (3.18) in (3.19), we get lim n→∞ ‖xn −zn‖ = 0.(3.20) further, note that ‖xn −yn‖ = βn‖zn −tn2 zn‖ ≤ (1 − δ)‖zn −tn2 zn‖.(3.21) using (3.14) in (3.21), we get lim n→∞ ‖xn −yn‖ = 0.(3.22) note that ‖xn −tn2 zn‖ ≤ ‖xn −zn‖ + ‖zn −t n 2 zn‖.(3.23) using (3.14) and (3.20) in (3.23), we get lim n→∞ ‖xn −tn2 zn‖ = 0.(3.24) 32 saluja hence ‖xn −tn2 xn‖ ≤ ‖xn −t n 2 zn‖ + ‖t n 2 zn −t n 2 xn‖ ≤ ‖xn −tn2 zn‖ + ‖zn −xn‖ + µnψ(‖zn −xn‖) + νn ≤ ‖xn −tn2 zn‖ + ‖zn −xn‖ + µnm‖zn −xn‖ + νn = ‖xn −tn2 zn‖ + (1 + µnm)‖zn −xn‖ + νn.(3.25) using (3.20) and (3.24) in (3.25), we get lim n→∞ ‖xn −tn2 xn‖ = 0.(3.26) again notice that ‖xn −tn1 yn‖ ≤ ‖xn −yn‖ + ‖yn −t n 1 yn‖.(3.27) using (3.8) and (3.22) in (3.27), we get lim n→∞ ‖xn −tn1 yn‖ = 0.(3.28) hence ‖xn −tn1 xn‖ ≤ ‖xn −t n 1 yn‖ + ‖t n 1 xn −t n 1 yn‖ ≤ ‖xn −tn1 yn‖ + ‖xn −yn‖ + µnψ(‖xn −yn‖) + νn ≤ ‖xn −tn1 yn‖ + ‖xn −yn‖ + µnm‖xn −yn‖ + νn = ‖xn −tn1 yn‖ + (1 + µnm)‖xn −yn‖ + νn.(3.29) using (3.22) and (3.28) in (3.29), we get lim n→∞ ‖xn −tn1 xn‖ = 0.(3.30) by the definitions of xn+1, we have ‖xn −xn+1‖ ≤ ‖xn −yn‖ + ‖tn1 yn −yn‖.(3.31) using (3.8) and (3.22) in (3.31), we get lim n→∞ ‖xn −xn+1‖ = 0.(3.32) by (3.30), (3.31) and uniform continuity of t1, we have ‖xn −t1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −tn+11 xn+1‖ +‖tn+11 xn+1 −t n+1 1 xn‖ + ‖t n+1 1 xn −t1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −tn+11 xn+1‖ + ‖xn+1 −xn‖ +µn+1ψ(‖xn+1 −xn‖) + νn+1 + ‖tn+11 xn −t1xn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −tn+11 xn+1‖ + ‖xn+1 −xn‖ +µn+1m‖xn+1 −xn‖ + νn+1 + ‖tn+11 xn −t1xn‖ = (2 + µn+1m)‖xn −xn+1‖ + ‖xn+1 −tn+11 xn+1‖ +‖tn+11 xn −t1xn‖ + νn+1 → 0 as n →∞.(3.33) similarly, we can prove that ‖xn −t2xn‖ = 0 and ‖xn −t3xn‖ = 0.(3.34) this completes the proof. � theorem 3.3. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) is closed. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. modified sp -iteration scheme 33 then {xn} converges strongly to a common fixed point of the mappings t1, t2 and t3 if and only if lim infn→∞d(xn,f) = 0, where d(x,f) = inf{‖x−p‖ : p ∈ f}. proof. the necessity is obvious. indeed, if xn → q ∈ f as n →∞, then d(xn,f) = inf q∈f d(xn,q) ≤‖xn −q‖→ 0 (n →∞). this shows that lim infn→∞d(xn,f) = 0. conversely, suppose that lim infn→∞d(xn,f) = 0. by lemma 3.1, we have that limn→∞d(xn,f) exists. further, by assumption lim infn→∞d(xn,f) = 0, from (3.4) and lemma 2.4(ii), we conclude that limn→∞d(xn,f) = 0. next, we show that {xn} is a cauchy sequence. from (3.3), we know that ‖xn+1 −p‖ ≤ (1 + µnq1)‖xn −p‖ + νnq2 = (1 + dn)‖xn −p‖ + q2νn,(3.35) where dn = q1µn and for some q1,q2 > 0. since ∑∞ n=1 µn < ∞, it follows that ∑∞ n=1 dn < ∞. since 1 + x ≤ ex for all x ≥ 0, therefore from (3.35), we have ‖xn+m −p‖ ≤ (1 + dn+m−1)‖xn+m−1 −p‖ + q2νn+m−1 ≤ edn+m−1‖xn+m−1 −p‖ + q2νn+m−1 ≤ e[dn+m−1+dn+m−2]‖xn+m−2 −p‖ + edn+m−1q2νn+m−2 +q2νn+m−1 ≤ e[dn+m−1+dn+m−2]‖xn+m−2 −p‖ + edn+m−1q2[νn+m−2 +νn+m−1] ... ≤ ( e ∑n+m−1 j=n dj ) ‖xn −p‖ + ( e ∑n+m−1 j=n dj ) q2 n+m−1∑ j=n νj ≤ ( e ∑∞ j=1 dj ) ‖xn −p‖ + ( e ∑∞ j=1 dj ) q2 n+m−1∑ j=n νj ≤ q3 ‖xn −p‖ + q2q3 n+m−1∑ j=n νj(3.36) for all natural numbers m,n, where q3 = e ∑∞ j=1 dj < ∞. now, given ε > 0, since limn→∞d(xn,f) = 0 and ∑∞ n=1 νn < ∞, there exists a natural number n1 > 0 such that for all n ≥ n1, d(xn,f) < ε8q3 and ∑∞ j=1 νj < ε 4q2q3 . so, we get d(xn1,f) < ε 4q3 and ∑∞ j=n1 νj < ε 4q2q3 . this means that there exists a p1 ∈ f such that ‖xn1 −p1‖ ≤ ε 4q3 . hence, 34 saluja for all integers n ≥ n1 and m ≥ 1, we obtain from (3.36) that ‖xn+m −xn‖ ≤ ‖xn+m −p1‖ + ‖xn −p1‖ ≤ q3 ‖xn1 −p1‖ + q2q3 n+m−1∑ j=n1 νj +q3 ‖xn1 −p1‖ + q2q3 n+m−1∑ j=n1 νj = 2 ( q3 ‖xn1 −p1‖ + q2q3 n+m−1∑ j=n1 νj ) ≤ 2 ( q3 ‖xn1 −p1‖ + q2q3 ∞∑ j=n1 νj ) < 2 ( q3. ε 4q3 + q2q3. ε 4q2q3 ) = ε. this proves that {xn} is a cauchy sequence in c. thus, the completeness of e implies that {xn} must be convergent. assume that limn→∞xn = z. we will prove that z is a common fixed point of t1, t2 and t3, that is, we will show that z ∈ f = ∩3i=1f(ti). since c is closed, therefore z ∈ c. next, we show that z ∈ f. now limn→∞d(xn,f) = 0 gives that d(z,f) = 0. since f is closed, z ∈ f. thus, z is a common fixed point of the mappings t1, t2 and t3. this completes the proof. � we deduce the following result as corollary from theorem 3.3 as follows. corollary 3.4. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) is closed. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 −δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. then {xn} converges strongly to a point p ∈ f if and only if there exists some subsequence {xnj} of {xn} which converges to p ∈ f . theorem 3.5. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. then lim infn→∞d(xn,f) = lim supn→∞d(xn,f) = 0 if {xn} converges to a unique point in f . proof. let p ∈ f. since {xn} converges to p, limn→∞d(xn,p) = 0. so, for a given ε > 0, there exists n1 ∈ n such that d(xn,p) < ε for all n ≥ n1. taking the infimum over p ∈ f(s,t), we obtain that d(xn,f) < ε for all n ≥ n1. this means that limn→∞d(xn,f) = 0. thus we obtain that lim inf n→∞ d(xn,f) = lim sup n→∞ d(xn,f) = 0. this completes the proof. � as an application of theorem 3.3, we establish some strong convergence results as follows. modified sp -iteration scheme 35 theorem 3.6. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. if one of the mappings in {ti : i = 1, 2, 3} is demicompact, then {xn} converges strongly to a common fixed point of the mappings t1, t2 and t3. proof. without loss of generality, we can assume that t1 is demicompact. it follows from (3.33) in lemma 3.2 that limn→∞‖xn − t1xn‖ = 0 and {xn} is bounded, by demicompactness of t1, there exists a subsequence {xnk} of {xn} that converges strongly to some q ∈ c as k →∞. from (3.33) in lemma 3.2 we have lim k→∞ ‖xnk −t1xnk‖ = ‖q −t1q‖ = 0. this implies that q ∈ f(t1). similarly, we can prove that q ∈ f(t2) and q ∈ f(t3). thus, we obtain that q ∈ f = ∩3i=1f(ti). it follows from lemma 3.1 and theorem 3.3 that {xn} must converges strongly to a common fixed point of the mappings t1, t2 and t3. this completes the proof. � theorem 3.7. let e be a real banach space and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1 − δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. if t1, t2 and t3 satisfy condition (b), then {xn} converges strongly to a common fixed point of the mappings t1, t2 and t3. proof. by lemma 3.2, we know that lim n→∞ ‖xn −tixn‖ = 0, for i = 1, 2, 3.(3.37) from condition (b) and (3.37), we get f(d(xn,f) ≤ a1.‖xn −t1xn‖ + a2.‖xn −t2xn‖ + a3.‖xn −t3xn‖ = 0, that is, f(d(xn,f) = 0. since f : [0,∞) → [0,∞) is a nondecreasing function satisfying f(0) = 0, f(t) > 0 for all t ∈ (0,∞), therefore we obtain lim n→∞ d(xn,f) = 0. now all the conditions of theorem 3.3 are satisfied, therefore by its conclusion {xn} converges strongly to a common fixed point of the mappings t1, t2 and t3. this completes the proof. � 4. weak convergence theorems in this section, we prove some weak convergence theorems of iteration scheme (1.13) for three total asymptotically nonexpansive mappings in a uniformly convex banach space such that either it satisfies the opial property or its dual space has the kadec-klee property (kk-property). theorem 4.1. let e be a uniformly convex banach space satisfying opial’s condition and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1−δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. if the mappings i −ti for all i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then {xn} converges weakly to a common fixed point of the mappings t1, t2 and t3. 36 saluja proof. let q ∈ f, from lemma 3.1 the sequence {‖xn −q‖} is convergent and hence bounded. since e is uniformly convex, every bounded subset of e is weakly compact. thus there exists a subsequence {xnk}⊂{xn} such that {xnk} converges weakly to q ∗ ∈ c. from lemma 3.2, we have lim k→∞ ‖xnk −t1xnk‖ = 0, lim k→∞ ‖xnk −t2xnk‖ = 0, lim k→∞ ‖xnk −t3xnk‖ = 0. since the mappings i − ti for all i = 1, 2, 3 are demiclosed at zero, therefore tiq∗ = q∗ for all i = 1, 2, 3, which means q∗ ∈ f . finally, let us prove that {xn} converges weakly to q∗. suppose on contrary that there is a subsequence {xnj} ⊂ {xn} such that {xnj} converges weakly to p∗ ∈ c and q∗ 6= p∗. then by the same method as given above, we can also prove that p∗ ∈ f . from lemma 3.1 the limits limn→∞‖xn − q∗‖ and limn→∞‖xn −p∗‖ exist. by virtue of the opial condition of e, we obtain lim n→∞ ‖xn −q∗‖ = lim nk→∞ ‖xnk −q ∗‖ < lim nk→∞ ‖xnk −p ∗‖ = lim n→∞ ‖xn −p∗‖ = lim nj→∞ ‖xnj −p ∗‖ < lim nj→∞ ‖xnj −q ∗‖ = lim n→∞ ‖xn −q∗‖ which is a contradiction, so q∗ = p∗. thus {xn} converges weakly to a common fixed point of the mappings t1, t2 and t3. this completes the proof. � lemma 4.2. under the conditions of lemma 3.2 and for any p, q ∈ f , limn→∞‖txn + (1 − t)p−q‖ exists for all t ∈ [0, 1]. proof. by lemma 3.1, limn→∞‖xn −z‖ exists for all z ∈ f and therefore {xn} is bounded. letting an(t) = ‖txn + (1 − t)p−q‖ for all t ∈ [0, 1]. then limn→∞an(0) = ‖p− q‖ and limn→∞an(1) = ‖xn − q‖ exists by lemma 3.1. it, therefore, remains to prove the lemma 4.2 for t ∈ (0, 1). for all x ∈ c, we define the mapping wn : c → c by: un(x) = (1 −γn)x + γntn3 x vn(x) = (1 −βn)un(x) + βntn2 un(x) and wn(x) = (1 −αn)vn(x) + αntn1 vn(x)). then it follows that zn = unxn, yn = vnxn, xn+1 = wnxn and wnp = p for all p ∈ f. now from (3.1), (3.2) and (3.3) of lemma 3.1, we see that ‖un(x) −un(y)‖ ≤ (1 + µnm)‖x−y‖ + νn ‖vn(x) −vn(y)‖ ≤ (1 + µnm)2‖x−y‖ + (2 + µnm)νn and ‖wn(x) −wn(y)‖ ≤ (1 + µnq1)‖x−y‖ + q2νn = kn ‖x−y‖ + q2νn,(4.1) for some q1,q2 > 0 and for all x,y ∈ c, where kn = 1 + µnq1 with ∑∞ n=1 νn < ∞ and kn → 1 as n →∞. setting hn, m = wn+m−1wn+m−2 . . .wn, m ≥ 1(4.2) modified sp -iteration scheme 37 and bn, m = ‖hn, m(txn + (1 − t)p) − (thn, mxn + (1 − t)hn,mq)‖. from (4.1) and (4.2), we have ‖hn, m(x) −hn, m(y)‖ = ‖wn+m−1wn+m−2 . . .wn(x) −wn+m−1wn+m−2 . . .wn(y)‖ ≤ kn+m−1‖wn+m−2 . . .wn(x) −wn+m−2 . . .wn(y)‖ +q2νn+m−1 ≤ kn+m−1kn+m−2‖wn+m−3 . . .wn(x) −wn+m−3 . . .wn(y)‖ +q2νn+m−1 + q2νn+m−2 ... ≤ (n+m−1∏ j=n kj ) ‖x−y‖ + q2 n+m−1∑ j=n νj = mn‖x−y‖ + q2 n+m−1∑ j=n νj(4.3) for all x,y ∈ c, where mn = ∏n+m−1 j=n kj and hn, mxn = xn+m, hn, mp = p for all p ∈ f. thus an+m(t) = ‖txn+m + (1 − t)p−q‖ ≤ bn, m + ‖hn, m(txn + (1 − t)p) −q‖ ≤ bn, m + mnan(t) + q2 n+m−1∑ j=n νj ≤ bn, m + mnan(t) + q2 ∞∑ j=1 νj.(4.4) by using [ [5], theorem 2.3], we have bn,m ≤ ϕ−1(‖xn −u‖−‖hn,mxn −hn,mu‖) ≤ ϕ−1(‖xn −u‖−‖xn+m −u + u−hn,mu‖) ≤ ϕ−1(‖xn −u‖− (‖xn+m −u‖−‖hn,mu−u‖)) and so the sequence {bn,m} converges uniformly to 0, i.e., bn,m → 0 as n →∞. since limn→∞mn = 1, q2 > 0 and νj → 0 as j →∞, therefore from (4.4), we have lim sup n→∞ an(t) ≤ lim n,m→∞ bn,m + lim inf n→∞ an(t) + 0 = lim inf n→∞ an(t). this shows that limn→∞an(t) exists, that is, limn→∞‖txn + (1− t)p−q‖ exists for all t ∈ [0, 1]. this completes the proof. � theorem 4.3. let e be a real uniformly convex banach space such that its dual e∗ has the kadec-klee property and c be a nonempty closed convex subset of e. let t1, t2, t3 : c → c be three uniformly continuous and total asymptotically nonexpansive mappings with sequences {µn} and {νn} as defined in proposition 2.8 and f = ∩3i=1f(ti) 6= ∅. let {xn} be the iteration scheme defined by (1.13), where {αn}, {βn} and {γn} are sequences in [δ, 1−δ] for all n ∈ n and for some δ ∈ (0, 1) and the following conditions are satisfied: (i) ∑∞ n=1 µn < ∞, ∑∞ n=1 νn < ∞; (ii) there exists a constant m > 0 such that ψ(t) ≤ m t, t ≥ 0. if the mappings i −ti for all i = 1, 2, 3, where i denotes the identity mapping, are demiclosed at zero, then {xn} converges weakly to a common fixed point of the mappings t1, t2 and t3. proof. by lemma 3.1, {xn} is bounded and since e is reflexive, there exists a subsequence {xnj} of {xn} which converges weakly to some p ∈ c. by lemma 3.2, we have lim j→∞ ‖xnj −tixnj‖ = 0 for all i = 1, 2, 3. 38 saluja since by hypothesis the mappings i −ti for all i = 1, 2, 3 are demiclosed at zero, therefore tip = p for all i = 1, 2, 3, which means p ∈ f. now, we show that {xn} converges weakly to p. suppose {xni} is another subsequence of {xn} converges weakly to some q ∈ c. by the same method as above, we have q ∈ f and p, q ∈ ww(xn). by lemma 4.2, the limit lim n→∞ ‖txn + (1 − t)p−q‖ exists for all t ∈ [0, 1] and so p = q by lemma 2.6. thus, the sequence {xn} converges weakly to p ∈ f. this completes the proof. � example 4.4. let e be the real line with the usual norm |.|, c = [0,∞). assume that t1(x) = x, t2(x) = x 3 and t3(x) = sin x for all x ∈ c. let φ be the strictly increasing continuous function such that φ: r+ → r+ with φ(0) = 0. let {µn}n≥1 and {νn}n≥1 be two nonnegative real sequences defined by µn = 1 n2 and νn = 1 n3 for all n ≥ 1 with µn → 0 and νn → 0 as n → ∞. then t1, t2 and t3 are total asymptotically nonexpansive mappings with common fixed point 0, that is, f = f(t1) ∩f(t2) ∩t3 = {0}. 5. conclusion in this paper, we establish some weak and strong convergence theorems for modified sp iteration scheme for three total asymptotically nonexpansive mappings in the framework of real banach spaces. the results presented in this paper extend and generalize several results from the current existing literature to the case of more general class of mappings, spaces and iteration schemes considered in this paper. references [1] r. p. agarwal, donal o’regan, d. r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, nonlinear convex anal. 8(1)(2007), 61–79. [2] ya. i. albert, c. e. chidume, h. zegeye, approximating fixed point of total asymptotically nonexpansive mappings, fixed point theory appl. (2006) art. id 10673. [3] r. e. bruck, t. kuczumow, s. reich, convergence of iterates of asymptotically nonexpansive mappings in banach spaces with the uniform opial property, colloq. math. 65(1993), 169–179. [4] c. e. chidume, e. u. ofoedu, approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, j. math. anal. appl. 333(2007), 128–141. [5] j. garcia falset, w. kaczor, t. kuczumow, s. reich, weak convergence theorems for asymptotically nonexpansive mappings and semigroups, nonlinear anal., tma, 43(3)(2001), 377–401. [6] k. goebel, w. a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1)(1972), 171–174. [7] r. glowinski, p. le tallec, augemented lagrangian and operator-splitting methods in nonlinear mechanics siam, philadelphia, (1989). [8] s. haubruge, v. h. nguyen, j. j. strodiot, convergence analysis and applications of the glowinski le tallec splitting method for finding a zero of the sum of two maximal monotone operators, j. optim. theory appl. 97(1998), 645–673. [9] s. ishikawa, fixed point by a new iteration method, proc. amer. math. soc. 44(1974), 147–150. [10] w. a. kirk, fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, israel j. math. 17 (1974), 339–346. [11] n. maiti, m. k. ghosh, approximating fixed points by ishikawa iterates, bull. aust. math. soc. 40(1989), 113–117. [12] w. r. mann, mean value methods in iteration, proc. amer. math. soc. 4(1953), 506–510. [13] m. a. noor, new approximation schemes for general variational inequalities, j. math. anal. appl. 251(1),(2000), 217–229. [14] m. a. noor, three-step iterative algorithms for multivalued quasi variational inclusions, j. math. anal. appl. 255(2001), 589–604. [15] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc. 73(1967), 591–597. [16] w. phuengrattana, s. suantai, on the rate of convergence of mann, ishikawa, noor and sp iterations for continuous functions on an arbitrary interval, j. comput. appl. math. 235(2011), 3006–3014. [17] j. schu, weak and strong convergence to fixed points of asymptotically nonexpansive mappings, bull. austral. math. soc. 43(1)(1991), 153–159. [18] h. f. senter, w. g. dotson, approximating fixed points of nonexpansive mappings, proc. amer. math. soc. 44(1974), 375–380. [19] k. sitthikul, s. saejung, convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings, acta univ. palack. olomuc. math. 48(2009), 139–152. modified sp -iteration scheme 39 [20] k. k. tan, h. k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178(1993), 301–308. [21] b. l. xu, m. a. noor, fixed point iterations for asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 267(2002), 444–453. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india ∗corresponding author: saluja1963@gmail.com international journal of analysis and applications volume 18, number 5 (2020), 748-773 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-748 the existence result of renormalized solution for nonlinear parabolic system with variable exponent and l1 data fairouz souilah1, messaoud maouni2,∗ and kamel slimani2 1university 20th august 1955, skikda, algeria 2laboratory of applied mathematics and history and didactics of maths ”lamahis”, algeria ∗corresponding author: m.maouni@univ-skikda.dz; maouni21@gmail.com abstract. in this paper, we prove the existence result of a renormalized solution to a class of nonlinear parabolic systems, which has a variable exponent laplacian term and a leary lions operator with data belong to l1. 1. introduction let ω is bounded open domain of rn, (n ≥ 2) with lipschiz boundary ∂ω, t is a positive number oure aime is to study the existence of renormalized solution for a class of nonlinear parabolic systeme with variable exponent and l1 data. more precisely, we study the asymptotic behavrior of the problem  (b1(u))t − diva(x,t,∇u) + γ(u) = f1(x,t,u,v) in q = ω×]0,t[, (b2(v))t − ∆v+ = f2(x,t,u,v) in q = ω×]0,t[, u = v = 0 on σ = ∂ω×]0,t[, b1(u)(t = 0) = b1(u0) in ω, b2(v)(t = 0) = b2(v0) in ω, (1.1) where diva(x,t,∇u) = div(|∇u|p(x)−2 ∇u) is a leary lions operator (see assumptions (3.1)-(3.3)) with p : ω −→ [1, +∞) be a continuous real-valued function and let p− = minx∈ω p(x) and p + = maxx∈ω p(x) received january 27th, 2020; accepted february 25th, 2020; published june 25th, 2020. 2010 mathematics subject classification. 35j70, 35d05. key words and phrases. nonlinear parabolic systems; variable exponent; renormalized solutions; l1 data. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 748 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-748 int. j. anal. appl. 18 (5) (2020) 749 with 1 < p− ≤ p+ < n. let γ : r → r with γ(s) = λ |s|p(x)−2 s is a continuous increasing function for λ > 0 and γ(0) = 0 such that γ is assumed to belong to l1(q). the function fi : q× r × r → r for i = 1, 2 be a carathéodory function (see assumptions (3.5)-(3.7)). finally the function b : r → r is a strictly increasing c1-function lipchizienne with bi(0) = 0 (see (3.4) ), the data fi and (b1(u0),b2(v0)) is in (l 1)2, for i = 1, 2. the study of differential equations and variational problems with nonstandard growth conditions arouses much interest with the development of elastic mechanics, electro-rheological fluid dynamics and image processing, etc ( see [9], [19] ) . problems of this type have been studied by serval a authors. in 2007 h. redwane, studied the existence of solutions for a class of nonlinear parabolic systems see [18], in 2013 youssef. b and all studied the existence of a renormalized solution for the nonlinear parabolic systems with unbounded nonlinearities see [2] agin in 2016 b . el hamdaoui and all in [11] studied the renormalized solutions for nonlinear parabolic systems in the lebesgue sobolev space with variable exponent and l1 data. in 2016 [17] authors proved the existence and uniqueness of renormalized solution of a reaction diffusion systems which has a variable exponent laplacian term and could be applied to image denoising for the case of parabolic equations. in 2010 t. m. bendahmane, p. wittbold and a.zimmermann [7] have proved the existence and uniqueness of renormalized solution to nonlinear parabolic equations with variable exponent and l1 data. c. zhang and s. zhou studied the renormalized and entropy solution for nonlinear parabolic equation with variable exponent and l1 data. moreover, they obtain the equivalence of renormalized solution and entropy solution(see [23]). in the present paper we prove the existence of renormalized solution for nonlinear parabolic systems with variable exponent and l1 data of (1.1). the notion of renormalized solution was introduced by diperna and lions [10] in their study of the boltzmann equation, and this result can be seen as a generalization of the results obtained by f. souilah and all in [12]. the paper is organized as follows: section 2, to recall some basic notations and properties of variable exponent lebesgue sobolev space. section 3, is devoted to specify the assumptions on, a(x,t,ξ), γ, b1, b2, f1, f2, b1(u0) and b2(v0) needed in the present study. section 4, to give the definition of a renormalized solution of (1.1), and we establish (theorem (4.1) ) the existence of such a solution. 2. the mathematical preliminaries on variable exponent sobolev spaces in this section, we first recall some results on generalized lebesgue-sobolev spaces lp(.)(ω), w 1,p(.)(ω) and w 1,p(.) 0 (ω) where ω is an open subset of r n . we refer to [13] for further properties of lebesgue sobolev spaces with variable exponents. let p : ω −→ [1, +∞) be a continuous real-valued function and let p− = minx∈ω p(x), p + = maxx∈ω p(x) with 1 < p(.) < n. int. j. anal. appl. 18 (5) (2020) 750 2.1. generalized lebesgue-sobolev spaces. first, denote the variable exponent lebesgue space lp(.)(ω) by lp(.)(ω) = {u measurable function in ω : ρp(.)(u) = ∫ ω |u|p(x) dx} , which is equipped with the luxemburg norm ‖u‖lp(.)(ω) = inf  µ > 0, ∫ ω ∣∣∣∣u(x)µ ∣∣∣∣p(x) dx ≤ 1   . (2.1) the space lp(x)(ω) is also called a generalized lebesgue space. the space (lp(.)(ω);‖.‖p(.)) is a separable banach space. moreover, if 1 < p − ≤ p+ < +∞, then lp(.)(ω) is uniformly convex, hence reflexive and its dual space is isomorphic to lp′(.)(ω), where 1 p(x) + 1 p′(x) = 1, for x ∈ ω . the following inequality will be used later: min { ‖u‖p − lp(.)(ω) ,‖u‖p + lp(.)(ω) } ≤ ∫ ω |u(x)|p(x) dx ≤ max { ‖u‖p − lp(.)(ω) ,‖u‖p + lp(.)(ω) } . (2.2) finally, the hölder type inequality∣∣∣∣∣∣ ∫ ω uvdx ∣∣∣∣∣∣ ≤ ( 1 p− + 1 p+ ) ‖u‖ p(.) ‖v‖ p′(.) , (2.3) for all u∈ lp(.)(ω) and v∈ lp ′(.)(ω). next, define the variable exponent sobolev space w 1,p(.)(ω) by w 1,p(.)(ω) = { u ∈ lp(.)(ω), |∇u| ∈ lp(.)(ω) } , (2.4) which is banach space equiped with the following norm ‖u‖ 1,p(.) = ‖u‖ p(.) + ‖∇u‖ p(.) . (2.5) the space (w 1,p(.)(ω);‖.‖1,p(.)) is a separable and reflexive banach space. an important role in manipulating the generalized lebesgue and sobolev spaces is played by the modular ρp(.) of the space l p(.)(ω). to have the following result: proposition 2.1. if un,u ∈ lp(.)(ω) and p+ < +∞, the following properties hold true. (i) ‖u‖ p(.) > 1 =⇒‖u‖p + p(.) < ρp(.)(u) < ‖u‖ p− p(.) , (ii) ‖u‖ p(.) < 1 =⇒‖u‖p − p(.) < ρp(.)(u) < ‖u‖ p+ p(.) , (iii) ‖u‖ p(.) < 1 (respectively = 1; > 1)⇐⇒ ρp(.)(u) < 1 (respectively = 1; > 1), (iv) ‖un‖ p(.) −→ 0 (respectively −→ +∞)⇐⇒ ρp(.)(un) < 1(respectively −→ +∞), int. j. anal. appl. 18 (5) (2020) 751 (v) ρp(.) ( u ‖u‖ p(.) ) = 1. for a measurable function u : ω −→ r, we introduce the following notation ρ1,p(.) = ∫ ω |u|p(x) dx + ∫ ω |∇u|p(x) dx. proposition 2.2. if u ∈ w 1,p(.)(ω) and p+ < +∞, the following properties hold true. (i)‖u‖ 1,p(.) > 1 =⇒‖u‖p + 1,p(.) < ρ1,p(.)(u) < ‖u‖ p− 1,p(.) , (ii)‖u‖ 1,p(.) < 1 =⇒‖u‖p − 1,p(.) < ρ1,p(.)(u) < ‖u‖ p+ 1,p(.) , (iii)‖u‖ 1,p(.) < 1 (respectively = 1; > 1)⇐⇒ ρ1,p(.)(u) < 1(respectively = 1; > 1). extending a variable exponent p : ω −→ [1, +∞) to q = [0,t]×ω by setting p(x,t) = p(x) for all (x,t) ∈ q. we may also consider the generalized lebesgue space lp(.)(q) =  u : q −→ r mesurable such that ∫ q |u(x,t)|p(x) d(x,t) < ∞   , endowed with the norm ‖u‖lp(.)(q) = inf  µ > 0, ∫ q ∣∣∣∣u(x,t)µ ∣∣∣∣p(x) d(x,t) ≤ 1   , which share the same properties as lp(.)(ω). 3. the assumptions on the data this paper, we assume that the following assumptions hold true: let ω be a bounded open set of rn (n ≥ 2), t > 0 is given and we set q = ω×]0,t[, and a : q×rn → rn be a carathéodory function such that for all ξ,η ∈ rn,ξ 6= η a(x,t,ξ).ξ > α |ξ|p(x) , (3.1) |a(x,t,ξ)|6 β [ l(x,t) + |ξ|p(x)−1 ] , (3.2) (a(x,t,ξ) −a(x,t,η)).(ξ −η) > 0, (3.3) where 1 < p− ≤ p+ < +∞, α,β are positives constants and l is a nonnegative function in lp ′(.)(q), γ : r → r is a continuous increasing function with γ(0) = 0. let bi : r → r is a strictly increasing c1−function lipchizienne with bi(0) = 0 and for any ρ,τ are positives constants and for i = 1, 2 such that ρ ≤ b′i(s) ≤ τ, ∀s ∈ r, (3.4) int. j. anal. appl. 18 (5) (2020) 752 fi : q× r × r → r be a carathéodory function such that for any k > 0, there exists σk > 0, ck ∈ l1(q) such that |f1(x,t,s1,s2)| ≤ ck(x,t) + σk|s2|2, (3.5) for almost every (x,t) ∈ (q), for every s1 such that |s1| ≤ k, and for every s2 ∈ r. for any k > 0, there exists ζk > 0 and gk ∈ lp ′(.)(q) such that |f2(x,t,s1,s2)| ≤ gk(x,t) + ζk|s1|p(x)−1, (3.6) for almost every (x,t) ∈ (q), for every s2 such that |s2| ≤ k, and for every s1 ∈ r. f1(x,t,s1,s2)s1 ≥ 0 and f2(x,t,s1,s2)s2 ≥ 0, (3.7) (b1(u0),b2(v0)) ∈ (l1(ω))2. (3.8) 4. the main results in this section, we study the existence of renormalized solutions to problem (1.1). definition 4.1. let 2 − 1 n + 1 < p− ≤ p+ < n and (b1(u0),b2(v0)) ∈ (l1 (ω))2. a measurable functions (u,v) ∈ (c(]0,t[; l1(ω)))2 is a renormalized solution of the problem (1.1) if , tk(u) ∈ lp − (]0,t[; w 1,p(.) 0 (ω)),tk(v) ∈ l 2(]0,t[; h10 (ω)) for any k > 0 , (4.1) γ(u) ∈ l1 (q) and fi(x,t,u,v) ∈ (l1 (q))2, ∀i = 1, 2, b1(u) ∈ l∞ ( ]0,t[; l1 (ω) ) ∩lq − (]0,t[; w 1,q(.) 0 (ω)) (4.2) and b2(v) ∈ l∞ ( ]0,t[; l1 (ω) ) ∩l2(]0,t[; h10 (ω)), for all continuous functions q(x) on ω satisfying q(x) ∈ [ 1,p(x) − n n+1 ) for all x ∈ ω, lim n→∞ ∫ {n≤|u|≤n+1} a(x,t,∇u)∇udxdt + lim n→∞ ∫ {n≤|v|≤n+1} |∇v|2dxdt = 0, (4.3) and if, for every function s ∈ w 2,∞(r) which is piecewise c1 and such that s′ has compact support on r, to have, (b1s(u))t −div(a(x,t,∇u)s ′(u)) + s′′(u)a(x,t,∇u)∇u + γ(u)s′(u) (4.4) = f1(x,t,u,v)s ′(u) in d′(q), (b2s(v))t −div(∇vs ′(v)) + s′′(v)∇v = f2(x,t,u,v)s′(v) in d′(q), (4.5) int. j. anal. appl. 18 (5) (2020) 753 b1s(u)(t = 0) = s(b1(u0)) in ω, (4.6) b2s(v)(t = 0) = s(b2(v0)) in ω, (4.7) where bis(z) = ∫ z 0 b′i(r)s ′(r)dr, for i = 1, 2. the following remarks are concerned with a few comments on definition (4.1). remark 4.1. note that, all terms in (4.4) are well defined. indeed, let k > 0 such that supp(s′) ⊂ [k,k], we have bis(u) belongs to l ∞(q) for all i = 1, 2 because |b1s(u)| ≤ ∫ u 0 |b′1(r)s ′(r)|dr ≤ τ‖s′‖l∞(r), and |b2s(v)| ≤ ∫ v 0 |b′2(r)s ′(r)|dr ≤ τ‖s′‖l∞(r), and s(u) = s(tk(u)) ∈ lp−(]0,t[; w 1,p(.) 0 (ω)),s(v) = s(tk(v)) ∈ l 2(]0,t[; h10 (ω)) and ∂bis(u) ∂t ∈ (d′(q))2 for i = 1, 2. the term s′(u)a(x,t,∇tk(u)) identifes with s′(tk(u))a(x,t,∇(tk(u))) a.e. in q, where u = tk(u) in {|u| ≤ k}, assumptions (3.2) imply that |s′(tk(u))a(x,t,∇tk(u))| ≤ β‖s′‖l∞(r) [ l(x,t) + |∇(tk(u))| p(x)−1 ] a.e in q. (4.8) using (3.2) and (4.1), it follows that s′(u)a(x,t,∇u) ∈ (lp ′(.)(q))n . the term s′′(u)a(x,t,∇u)∇(u) identifes with s′′(u)a(t,x,∇(tk(u)))∇tk(u) and in view of (3.2), (4.1) and (4.8), to obtain s′′(u)a(x,t,∇u)∇(u) ∈ l1(q) and s′(u)γ(u) ∈ l1(q). finally f1(x,t,u,v) s′(u) = f1(x,t,tk(u),v)s′(u) a.e in q . since |tk(u)| ≤ k and s′(u) ∈ l∞(q), ck(x,t) ∈ l1(q), to obtain from (3.5) that f1(x,t,tk(u),v)s ′(u) ∈ l1(q), and f2(x,t,u,v) s′(v) = f2(x,t,u,tk(v))s′(v) a.e in q. since |tk(v)| ≤ k and s′(v) ∈ l∞(q), gk(x,t) ∈ lp ′(.)(q) , to obtain from (3.6) that f2(x,t,u,tk(v))s ′(v) ∈ l1(q). also ∂b1s(u) ∂t ∈ l(p −)′(]0,t[; w−1,p ′(.)(ω)) + l1(q) and b1s(u) ∈ l p−(]0,t[; w 1,p(.) 0 (ω)) ∩ l ∞(q), and ∂b2s(v) ∂t ∈ l2(]0,t[; h−1(ω)) + l1(q) and b2s(v) ∈ l 2(]0,t[; h10 (ω)) ∩ l∞(q), which implies that (b1s(u),b 2 s(v)) ∈ (c(]0,t[; l1(ω)))2. int. j. anal. appl. 18 (5) (2020) 754 4.1. the existence theorem. theorem 4.1. let (b1(u0),b2(v0)) ∈ (l1(ω))2, assume that (3.1)-(3.8) hold true, then there exists at least one renormalized solution (u,v) ∈ (c(]0,t[,l1(ω)))2 of problem (1.1) ( in the sens of definition (4.1) ). proof. of theorem (4.1) the above theorem is to be proved in five steps. • step 1: approximate problem and a priori estimates. let us define the following approximation of b and f for ε > 0 fixed and for i = 1, 2 biε(r) = t1 ε (bi(r)) a.e in ω for ε > 0, ∀r ∈ r, (4.9) biε(u ε 0) are a sequence of (c ∞ c (ω)) 2 functions such that (4.10) (b1ε(u ε 0),b 2 ε(v ε 0)) → (b1(u0),b2(v0)) in (l 1(ω))2 as ε tends to 0. fε1 (x,t,r1,r2) = f1(x,t,t1 ε (r1),r2), (4.11) fε2 (x,t,r1,r2) = f2(x,t,r1,t1 ε (r2)), in view of (3.5), (3.6) and (3.7), there exist gεk ∈ l p′(.)(q), cεk ∈ l 1(q) and σεk,ζ ε k > 0 such that |fε1 (x,t,s1,s2)| ≤ c ε k(x,t) + σ ε k|s2| 2, (4.12) |fε2 (x,t,s1,s2)| ≤ g ε k(x,t) + ζ ε k|s1| p(x)−1, (4.13) for almost every (x,t) ∈ (q), s1,s2 ∈ r, fε1 (x,t,s1,s2)s1 ≥ 0 and f ε 2 (x,t,s1,s2)s2 ≥ 0. (4.14) let us now consider the approximate problem: ( b1ε(u ε) ) t −diva(x,t,∇uε) + γ (uε) = fε1 (x,t,u ε,vε) in q, (4.15) ( b2ε(v ε) ) t − ∆vε = fε2 (x,t,u ε,vε) in q, (4.16) uε = vε = 0 on ]0,t[ ×∂ω, (4.17) b1ε(u ε) (t = 0) = b1ε(u ε 0) in ω, (4.18) b2ε(v ε) (t = 0) = b2ε(v ε 0) in ω. (4.19) int. j. anal. appl. 18 (5) (2020) 755 as a consequence, proving existence of a weak solution uε ∈ lp − (]0,t[; w 1,p(.) 0 (ω)) and v ε ∈ l2(]0,t[; h10 (ω)) of (4.15)-(4.18) is an easy task (see [15]). we choose tk(u ε)χ(0,t) as a test function in (4.15), to get ∫ ω b 1,ε k (u ε)(t)dx + t∫ 0 ∫ ω a(x,t,∇uε)∇tk(uε) + t∫ 0 ∫ ω γ (uε) tk(u ε)dxds = t∫ 0 ∫ ω fε1 (x,t,u ε,vε)tk(u ε)dxds + ∫ ω b 1,ε k (u ε 0)dx, (4.20) for almost every t in (0,t), and where b i,ε k (r) = ∫ r 0 tk(s) ∂biε(s) ∂s ds.∀i = 1, 2. under the definition of b i,ε k (r) the inequality 0 ≤ ∫ ω b 1,ε k (u ε 0)(t)dx ≤ k ∫ ω |b1ε(u ε 0)|dx, k > 0. using (3.1), fε1 (x,t,u ε,vε)tk(u ε) ≥ 0, and we have γ(uε) = λ|uε|p(x)−1uε ≥ 0 because 1 < p− ≤ p(x) ≤ +∞ and the definition of bεk(r) in (4.20), to obtain∫ ω bεk(u 1,ε)(t)dx + α ∫ ek |∇uε|p(x) dxds ≤ k ∥∥b1ε(uε0)∥∥l1(q) , (4.21) where ek = {(x,t) ∈ q : |uε| ≤ k}, using b ε k(u ε)(t) ≥ 0 and inequality (2.2) in (4.21), to get t α ∫ 0 min { ‖∇tk(uε)‖ p− lp(x)(ω) ,‖∇tk(uε)‖ p+ lp(x)(ω) } ≤ α ∫ {(x,t)∈q: |uε|≤k} |∇uε|p(x) dxdt ≤ c, (4.22) then is tk(u ε) is bounded in lp−(]0,t[ ; w 1,p(x) 0 (ω)). similarly, we choose tk(v ε)χ(0,t) as a test function in (4.16), to get∫ ω b 2,ε k (v ε)(t)dx + α ∫ fk |∇vε|2 dxds ≤ k ∥∥b2ε(vε0)∥∥l1(q) , (4.23) where fk = {(x,t) ∈ q : |vε| ≤ k}, then is tk(vε) is bounded in l2(]0,t[ ; h10 (ω)). adding (4.21) and (4.23), one gets∫ ω b 1,ε k (u ε)(t)dx + ∫ ω b 2,ε k (v ε)(t)dx ≤ k ∥∥(b1ε(uε0),b2ε(vε0))∥∥l1(q)×l1(q) . (4.24) also, to obtain k ∫ {(t,x)∈q:|uε|>k} |γ(uε)|dxdt ≤ k‖bε(uε0)‖l1(q) . (4.25) int. j. anal. appl. 18 (5) (2020) 756 hence k ∫ {(x,t)∈q:|uε|>k} |fε1 (x,t,u ε,vε)|dxdt + k ∫ {(x,t)∈q:|vε|>k} |fε2 (x,t,u ε,vε)|dxdt ≤ k ∥∥(b1ε(uε0),b2ε(vε0))∥∥l1(q)×l1(q) . (4.26) now, let t1(s − tk(s)) = tk,1(s) and take tk,1(b1ε(uε)) as test function in (4.15). reasoning as above, by ∇tk,1(s) = ∇sχ{k≤|s|≤k+1} and the young’s inequality, to obtain α ∫ {k≤|b1ε(uε)|≤k+1} b′1,ε(u ε) |∇(uε)|p(x) dxdt ≤ k ∫ {|b1ε(uε0)|>k} ∣∣b1ε(uε0)∣∣dx + ck ∫ {|b1ε(uε)|>k} |γ(uε)|dxdt + ck ∫ {|b1ε(uε)|>k} |fε1 (x,t,u ε,vε)|dxdt ≤ c1, inequality (2.2) implies that t∫ 0 αχ{k≤|b1ε(uε)|≤k+1} min {∥∥∇(b1ε(uε))∥∥p−lp(x)(ω) ,∥∥∇(b1ε(uε))∥∥p+lp(x)(ω)} ≤ α ∫ {k≤|b1ε(uε)|≤k+1} b′1,ε(u ε) |∇(uε)|p(x) dxdt ≤ c1. (4.27) similarly, we choose tk(b 2 ε(v ε)) as test function in (4.16), to have∫ {|b2ε(vε)|≤k} b′2,ε(v ε) |∇(vε)|2 dxdt ≤ k ∫ {|b2ε(vε0 )|>k} ∣∣b2ε(vε0)∣∣dx +ck ∫ {|b2ε(vε)|>k} |fε2 (x,t,u ε,vε)|dxdt ≤ c2, we know that properties of b i,ε k (u ε), (b i,ε k (r ε) ≥ 0, bi,εk (r ε)) ≥ ρ(|r|−1), for all i = 1, 2, to obtain∫ ω ∣∣∣b1,εk (uε)(t)∣∣∣dx + ∫ ω ∣∣∣b2,εk (vε)(t)∣∣∣dx ≤ k∫ ω ∣∣b1ε(uε)(t)∣∣dx + k∫ ω ∣∣b2ε(vε)(t)∣∣dx ≤ ρ ( 2meas(ω) + k ∥∥(b1ε(uε0),b2ε(vε0))∥∥l1(q)×l1(q)) . (4.28) from the estimation (4.22), (4.23), (4.27) , (4.28) and the properites of b i,ε k and b 1 ε(u ε 0), b 2 ε(v ε 0), we deduce that b1ε(u ε) and b2ε(v ε) is bounded in l∞ ( ]0,t[; l1 (ω) ) , (4.29) uε and vε is bounded in l∞ ( ]0,t[; l1 (ω) ) , (4.30) int. j. anal. appl. 18 (5) (2020) 757 and b1ε(u ε) is bounded in lp−(]0,t[ ; w 1,p(x) 0 (ω)), (4.31) and b2ε(v ε) is bounded in l2(]0,t[ ; h10 (ω)), (4.32) by (4.27), (4.28) and lemma 2.1 in [7] by and if 2 − 1 n + 1 < p(.) < n, to obtain b1ε(u ε) is bounded in lq−(]0,t[ ; w 1,q(x) 0 (ω)), (4.33) for all continuous variable exponents q ∈ c(ω) satisfying 1 ≤ q(x) < n(p(x) − 1) + p(x) n + 1 , for all x ∈ ω. and tk (u ε) is bounded in lp − ( ]0,t[; w 1,p(.) 0 (ω) ) , (4.34) and tk (v ε) is bounded in l2 ( ]0,t[; h10 (ω) ) . (4.35) by (4.25) and (4.26), we may conclude that γ(uε) is bounded in l1 ( ]0,t[; l1 (ω) ) , (4.36) and fε1 (x,t,u ε,vε) and fε2 (x,t,u ε,vε) is bounded in l1 ( ]0,t[; l1 (ω) ) , (4.37) independently of ε. proceeding as in [3], [4] that for any s ∈ w 2,∞(r) such that s′ is compact (supp s′ ⊂ [−k,k]), s (uε) is bounded in lp− ( ]0,t[; w 1,p(.) 0 (ω) ) , (4.38) and s (vε) is bounded in l2 ( ]0,t[; h10 (ω) ) , (4.39) and (s (uε))t is bounded in l 1 (q) + l(p−) ′ ( ]0,t[; w−1,p ′(.) (ω) ) , (4.40) and (s (vε))t is bounded in l 1 (q) + l2 ( ]0,t[; h−1 (ω) ) . (4.41) int. j. anal. appl. 18 (5) (2020) 758 in fact, as a consequence of (4.34), by stampacchia’s theorem, we obtain (4.38). to show that (4.40) holds true, we multiply the equation (4.15) by s′(uε) and the equation (4.16) by s′(vε), to obtain ( b1s (u ε) ) t = div(s′ (uε)a(x,t,∇uε)) −a(x,t,∇uε)∇(s′ (uε)) (4.42) −γ (uε) s′ (uε) + fε1 (x,t,u ε,vε)s′ (uε) in d′ (q) . and ( b2s (v ε) ) t = div(s′ (vε)∇vε) −∇(s′ (vε)) (4.43) +fε2 (x,t,u ε,vε)s′ (vε) in d′ (q) . since supp(s′) and supp(s′′) are both included in [−k; k]; uε may be replaced by tk(uε) in {|uε| ≤ k}. to have |s′ (uε)a(x,t,∇uε)| (4.44) ≤ β‖s′‖l∞ [ l(x,t) + |∇tk(uε)| p(x)−1 ] , as a consequence, each term in the right hand side of (4.42) is bounded either in l(p−) ′ ( ]0,t[; w−1,p ′(.) (ω) ) or in l1(q), and obtain (4.40). now we look for an estimate on a sort of energy at infinity of the approximating solutions. for any integer n ≥ 1, consider the lipschitz continuous function θn defined through θn (s) = tn+1 (s) −tn (s) =   0 if |s| ≤ n, (|s|−n) sign(s) if n ≤ |s| ≤ n + 1, sign(s) if |s| ≥ n. remark that ||θn||l∞ ≤ 1 for any n ≥ 1 and that θn (s) → 0, for any s when n tends to infinity. using the admissible test function θn(u ε) in (4.15) leads to ∫ ω θ̃n (u ε) (t) dx + ∫ q a(x,t,∇uε)∇(θn(uε)) dxdt + ∫ q γ (uε) θn(u ε)dxdt = ∫ q fε(x,t,uε)θn(u ε)dxdt + ∫ ω θ̃n (u ε 0) dx, (4.45) where θ̃n (r) (t) = ∫ r 0 θn(s) ∂biε(s) ∂s ds, for all i = 1, 2, for almost any t in ]0,t[ and where θ̃n(r) = r∫ 0 θn(s)ds ≥ 0. hence, dropping a nonnegative term ∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt (4.46) int. j. anal. appl. 18 (5) (2020) 759 ≤ ∫ q γ (uε) θn(u ε)dxdt + ∫ q fε1 (x,t,u ε,vε)θn(u ε)dxdt + ∫ ω θ̃n (u ε 0) dx ≤ ∫ {|uε|≥n} |γ (uε)|dxdt + ∫ {|uε|≥n} |fε1 (x,t,u ε,vε)|dxdt + ∫ {|b1ε(uε0)|≥n} ∣∣b1ε(uε0)∣∣dx. similarly, we take test function θn(v ε) in (4.16) leads to ∫ {n≤|vε|≤n+1} |∇vε|2dxdt (4.47) ≤ ∫ q fε2 (x,t,u ε,vε)θn(v ε)dxdt + ∫ ω θ̃n (v ε 0) dx ≤ ∫ {|vε|≥n} |fε2 (x,t,u ε,vε)|dxdt + ∫ {|b2ε(vε0 )|≥n} ∣∣b2ε(vε0)∣∣dx. next, we study the convergence of (un)n∈n and (vn)n∈n in c(]0,t[; l 1(ω)). lemma 4.1. both (uεn)n∈n and (v εn)n∈n are cauchy sequences in c(]0,t[; l 1(ω)). proof. let εn and εm two positive integers. it follows frome (4.15) and (4.16) that ∫ ω ∂b1εn(u εn −uεm) ∂t ϕdx + t∫ 0 ∫ ω (a(x,t,∇uεn) −a(x,t,∇uεm))∇ϕdxdt + t∫ 0 ∫ ω λ [ |uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm ] φdxds = t∫ 0 ∫ ω [fεn1 (x,t,u εn,vεn) −fεn1 (x,t,u εm,vεm)] ϕdxds, (4.48) and ∫ ω ∂b2εn(v εn −vεm) ∂t φdx + t∫ 0 ∫ ω (∇vεn −∇vεm)∇φdxdt (4.49) = t∫ 0 ∫ ω [fεn2 (x,t,u εn,vεn) −fεn2 (x,t,u εm,vεm)] φdxds, int. j. anal. appl. 18 (5) (2020) 760 where ϕ ∈ l∞(]0,t[; w 1,p(.)(ω)) and φ ∈ l2(]0,t[; h10 (ω)). to do this fix τ ∈ [0,t]. taking ϕ = 1 k tk(u εn −uεm)1{[0,τ[} in (4.48) and φ = 1ktk(v εn −vεm)1{[0,τ[} in (4.49), one gets 1 k ∫ ω b 1,εn k (u εn(τ) −uεn(τ))dx− 1 k ∫ ω b 1,εn k (u εn(0) −uεm(0))dx + τ∫ 0 ∫ ω 1 k (a(x,t,∇uεn) −a(x,t,∇uεm))∇tk(uεn −uεm)dxdt (4.50) + τ∫ 0 ∫ ω λ k [ |uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm ] tk(u εn −uεm)dxds = t∫ 0 ∫ 1 k ω [fεn1 (x,t,u εn,vεn) −fεn1 (x,t,u εm,vεm)] tk(u εn −uεm)dxds, and 1 k ∫ ω b 2,εn k (v εn(τ) −vεm(τ))dx− 1 k ∫ ω b 2,εn k (v εn(0) −vεm(0))dx + 1 k t∫ 0 ∫ ω ∇(vεn −vεm)∇tk(vεn −vεm)dxdt (4.51) = t∫ 0 ∫ ω 1 k [fεn2 (x,t,u εn,vεn) −fεn2 (x,t,u εm,vεm)] tk(v εn −vεm)dxds, where b i,εn k (r) = ∫ r 0 tk(s) ∂biεn(s) ∂s ds. ∀i = 1, 2, adding (4.50) and (4.51), we get 1 k ∫ ω b 1,εn k (u εn(τ) −uεm(τ))dx + 1 k ∫ ω b 2,εn k (v εn(τ) −vεm(τ))dx ≤ τ∫ 0 ∫ ω λ [ |uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm ] dxdt + τ∫ 0 ∫ ω [fεn1 (x,t,u εn,vεn) −fεn1 (x,t,u εm,vεm)] dxdt + τ∫ 0 ∫ ω [fεn2 (x,t,u εn,vεn) −fεn2 (x,t,u εm,vεm)] dxdt + ∫ ω ∣∣b1εn(uεn0 −uεm0 )∣∣dx + ∫ ω ∣∣b2εn(vεn0 −vεm0 )∣∣dx, int. j. anal. appl. 18 (5) (2020) 761 since b i,εn k (r) ≥ ρ ∫ r 0 tk(s)ds ≥ ρ (|s|− 1) .∀i = 1, 2∫ ω |uεn(τ) −uεm(τ)|dx + ∫ ω |vεn(τ) −vεm(τ)|dx ≤ 2k meas(ω) + τ∫ 0 ∫ ω kλ [ |uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm ] dxdt +k τ∫ 0 ∫ ω [fεn1 (x,t,u εn,vεn) −fεn1 (x,t,u εm,vεm)] dxdt +k τ∫ 0 ∫ ω [fεn2 (x,t,u εn,vεn) −fεn2 (x,t,u εm,vεm)] dxdt +k ∫ ω ∣∣b1εn(uεn0 −uεm0 )∣∣dx + k ∫ ω ∣∣b2εn(vεn0 −vεm0 )∣∣dx, letting εn, εm →∞ and them k → 0, to obtain sup τ∈[0,t] ∫ ω |uεn(τ) −uεm(τ)|dx + sup τ∈[0,t] ∫ ω |vεn(τ) −vεm(τ)|dx ≤ τ∫ 0 ∫ ω kλ [ |uεn|p(x)−2 uεn −|uεm|p(x)−2 uεm ] dxdt +k τ∫ 0 ∫ ω [fεn1 (x,t,u εn,vεn) −fεn1 (x,t,u εm,vεm)] dxdt +k τ∫ 0 ∫ ω [fεn2 (x,t,u εn,vεn) −fεn2 (x,t,u εm,vεm)] dxdt +k ∫ ω ∣∣b1εn(uεn0 −uεm0 )∣∣dx + k ∫ ω ∣∣b2εn(vεn0 −vεm0 )∣∣dx. � • step 2: the limit of the solution of the approximated problem. arguing again as in [ [3], [4], [5]] estimates (4.38), (4.40), (4.39) and (4.41) imply that, for a subsequence still indexed by ε, (uε,vε) converge almost every where to (u,v), (4.52) using (4.15), (4.34), (4.35) and (4.44), to get tk(u ε) converge weakly to tk(u) in l p− ( ]0,t[ ; w 1,p(.) 0 (ω) ) , (4.53) and tk(v ε) converge weakly to tk(v) in l 2 ( ]0,t[ ; h10 (ω) ) , (4.54) int. j. anal. appl. 18 (5) (2020) 762 χ{|uε|≤k}a(x,t,∇uε) ⇀ ηk weakly in ( lp ′(.) (q) )n , (4.55) as ε tends to 0 for any k > 0 and any n ≥ 1 and where for any k > 0, ηk belongs to ( lp ′(.) (q) )n . since γ(uε) is a continuous incrassing function, from the monotone convergence theorem and (4.25) and by (4.52), to obtain that γ(uε) converge weakly to γ(u) in l1(q). (4.56) we now establish that (b1(u),b2(v)) belongs to (l ∞ ( ]0,t[ ; l1 (ω) ) )2. indeed using (4.20) and∣∣∣bi,εk (s)∣∣∣ ≥ ρ(|s|− 1), ∀i = 1, 2, leads to∫ ω ∣∣b1ε(uε)∣∣ (t)dx + ∫ ω ∣∣b2ε(vε)∣∣ (t)dx ≤ ρ(2meas(ω) + ‖(fε1 (x,t,u ε,vε),fε2 (x,t,u ε,vε))‖(l1(q))2 + k‖γ (uε)‖l1(q) + k ∥∥(b1ε(uε0),b2ε(vε0))∥∥(l1(ω))2 ). by lemma (4.1) and (4.46), (4.47), we conclude that there exist two subsequences of uεn and vεn, still denoted by themselves for convenience, such that uεn converges to a function u in c(]0,t[; l1(ω)), vεnconverges to a function v in c(]0,t[; l1(ω)). using (4.25) and (4.10),(4.26), we have (b1(u),b2(v)) belongs to (l∞ ( ]0,t[ ; l1 (ω) ) )2. we are now in a position to exploit (4.46) and (4.47). since (uε,vε) is bounded in (l∞ ( ]0,t[ ; l1 (ω) ) )2, to get lim n→+∞ ( sup ε meas{|uε| ≥ n} ) = 0. (4.57) and lim n→+∞ ( sup ε meas{|vε| ≥ n} ) = 0. (4.58) the equi-integrability of the sequence fεi (x,t,u ε,vε) in (l1(q))2. we shall now prove that fεi (x,t,u ε,vε) converges to fi(x,t,u,v) strongly in (l 1(q))2, for all i = 1, 2 by using vitali’s theorem. since fεi (x,t,u ε,vε) → fi(x,t,u,v) a.e in q it suffices to prove that fεi (x,t,u ε,vε) are equi-integrable in q. let δ1 > 0 and a be a measurable subset belonging to ω×]0,t[, we define the following sets gδ1 = {(x,t) ∈ q : |un| ≤ δ1}, (4.59) fδ1 = {(x,t) ∈ q : |un| > δ1}. (4.60) using the generalized hölder’s inequality and poincaré inequality, to have∫ a |fε1 (x,t,u ε,vε)|dxdt = ∫ a∩gδ1 |fε1 (x,t,u ε,vε)|dxdt + ∫ a∩fδ1 |fε1 (x,t,u ε,vε)|dxdt, int. j. anal. appl. 18 (5) (2020) 763 therfore ∫ a |fε1 (x,t,u ε,vε)|dxdt ≤ ∫ a∩gδ1 ( ck,ε(x,t) + σk,ε |vε| 2 ) dxdt + ∫ a∩fδ1 |fε1 (x,t,u ε,vε)|dxdt ≤ ∫ a ck,ε(x,t)dxdt + σk,ε ∫ q |∇tδ1 (v ε)|2 dxdt + ∫ a∩fδ1 |fε1 (x,t,u ε,vε)|dxdt ≤ ∫ a ck,ε(x,t)dxdt + σk,ε (meas(q) + 1) 1 2  ∫ qt |∇tδ1 (v ε)|2 dxdt   1 2 + ∫ a∩fδ1 |fε1 (x,t,u ε,vε)|dxdt ≤ k1 + c2 ( k α ∥∥b2ε(vε0)∥∥l1(ω) )1 2 + ∫ a∩fδ1 1 |uε| |uεfε1 (x,t,u ε,vε)|dxdt ≤ k2 + ∫ a∩fδ1 1 δ1 |uεfε1 (x,t,u ε,vε)|dxdt ≤ k2 + 1 δ1 ( 1 p− + 1 p′− ) ∫ a∩fδ1 |uε|p(x) dxdt   1 p−   ∫ a∩fδ1 |fε1 (x,t,u ε,vε)|p ′(x)(p(x)−1) dxdt   1 p′− → 0 when meas(a) → 0. which shows that fε1 (x,t,u ε,vε) is equi-integrable. by using vitali’s theorem, to get fε1 (x,t,u ε,vε) → f1(x,t,u,v) strongly in l1(q). (4.61) now we prove that fε2 (x,t,u ε,vε) → f2(x,t,u,v) strongly in l1(q). (4.62) int. j. anal. appl. 18 (5) (2020) 764 let δ2 > 0 and a be a measurable subset belonging to ω×]0,t[, we define the following sets gδ2 = {(x,t) ∈ q : |vn| ≤ δ2}, (4.63) fδ2 = {(x,t) ∈ q : |vn| > δ2}. (4.64) using the generalized hölder’s inequality and poincaré inequality, to get∫ a |fε2 (x,t,u ε,vε)|dxdt = ∫ a∩gδ2 |fε2 (x,t,u ε,vε)|dxdt + ∫ a∩fδ2 |fε2 (x,t,u ε,vε)|dxdt, therfore∫ a |fε2 (x,t,u ε,vε)|dxdt ≤ ∫ a∩gδ2 ( gεk(x,t) + ξ ε k |u ε|p(x)−1 ) dxdt + ∫ a∩fδ2 |fε2 (x,t,u ε,vε)|dxdt ≤ ∫ a gεk(x,t)dxdt + ξ ε k ∫ q |∇tδ2 (u ε)|p(x)−1 dxdt + ∫ a∩fδ2 |fε2 (x,t,u ε,vε)|dxdt ≤ ∫ a gεk(x,t)dxdt + ξ ε k ( 1 p− + 1 p′− ) (meas(q) + 1) 1 p−  ∫ qt |∇tδ2 (u ε)|(p(x)−1)p ′(x) dxdt   1 p′− + ∫ a∩fδ2 |fε2 (x,t,u ε,vε)|dxdt ≤ k3 + c4 ( k α ∥∥b1ε(uε0)∥∥l1(ω) )1 2 + ∫ a∩fδ2 1 |vε| |vεfε2 (x,t,u ε,vε)|dxdt ≤ k4 + ∫ a∩fδ2 1 δ2 |vεfε2 (x,t,u ε,vε)|dxdt ≤ k4 + 1 δ2   ∫ a∩fδ2 |vε|2 dxdt   1 2   ∫ a∩fδ2 |fε2 (x,t,u ε,vε)|2 dxdt   1 2 → 0 when meas(a) → 0. which shows that fε2 (x,t,u ε,vε) is equi-integrable. by using vitali’s theorem, to get fε2 (x,t,u ε,vε) → f2(x,t,u,v) strongly in l1(q). (4.65) int. j. anal. appl. 18 (5) (2020) 765 using (4.56), (4.61) and the equi-integrability of the sequence |b1ε(uε0)| in l1(ω) and |b2ε(vε0)| in l1(ω), we deduce that lim n→+∞  sup ε   ∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt + ∫ {n≤|vε|≤n+1} |∇vε|2dxdt     = 0. (4.66) • step 4: strong convergence. the specifie time regularization of tk(u) (for fixed k ≥ 0) is defined as follows. let (v µ 0 )µ be a sequaence in l ∞ (ω)∩w 1,p(.)0 (ω) such that ‖v µ 0‖l∞(ω) ≤ k, ∀µ > 0, and v µ 0 → tk(u0) a.e in ω with 1 µ ‖vµ0‖lp(.)(ω) → 0 as µ → +∞. for fixed k ≥ 0 and µ > 0, let us consider the unique solution tk(u)µ ∈ l∞ (ω) ∩ lp− ( ]0,t[; w 1,p(.) 0 (ω) ) of the monotone problem ∂tk(u)µ ∂t + µ (tk(u)µ −tk(u)) = 0 in d′ (q) , (4.67) tk(u)µ(t = 0) = v µ 0 . (4.68) the behavior of tk(u)µ as µ → +∞ is investigated in [9] and we just recall here that (4.67)-(4.68) imply that tk(u)µ → tk(u) strongly in lp− ( ]0,t[; w 1,p(.) 0 (ω) ) a.e in q, as µ → +∞, (4.69) with ‖tk(u)µ‖l∞(ω) ≤ k, for any µ, and ∂tk(u)µ ∂t ∈ l(p−) ′ ( ]0,t[; w−1,p ′(.) (ω) ) . the main estimate is the following lemma 4.2. let s be an increasing c∞ (r)− function such that s(r) = r for r ≤ k, and supps′ is compact. then lim inf µ→+∞ lim ε→0 t∫ 0 〈 ∂b1s(u ε) ∂t , (tk(u ε)µ −tk(u)) 〉 dt ≥ 0, where here 〈., .〉 denotes the duality pairing between l1(ω) + w−1,p ′(.) (ω) and l∞ (ω) ∩w 1,p(.)0 (ω), and where b1s(z) = ∫ z 0 b′1(r)s ′(r)dr. proof. see [5], lemma 1. � now we are to prove that the weak limit ηk and we prove the weak l 1 convergence of the ”truncted” energy a(x,t,∇tk(uε)) as ε tends to 0. in order to show this result we recall the lemma below. lemma 4.3. the subsequence of uε defined in step 3 satisfies lim sup ε→0 ∫ q a(x,t,∇uε)∇tk(uε)dxdt ≤ ∫ q ηk∇tk(u)dxdt, (4.70) int. j. anal. appl. 18 (5) (2020) 766 lim ε→0 ∫ q [ a ( x,t,∇uεχ{|uε|≤k} ) −a ( x,t,∇uχ{|u|≤k} )] × [ ∇uεχ{|uε|≤k} −∇uχ{||≤k} ] dxdt = 0 (4.71) ηk = a ( x,t,∇uχ{|u|≤k} ) a.e in q, for any k ≥ 0, as ε tends to 0. a(x,t,∇uε)∇tk(uε) →a(x,t,∇u)∇tk(u) weakly in l1 (q) . (4.72) proof. let us introduce a sequence of increasing c∞(r)-functions sn such that, for any n ≥ 1  sn(r) = r if |r| ≤ n, supp (s′n) ⊂ [−(n + 1), (n + 1)] , ‖s′′n‖l∞(r) ≤ 1. (4.73) for fixed k ≥ 0, we consider the test function s′n(uε) ( tk(uε) − (tk(u))µ ) in (4.15), we use the definition (4.73) of s′n and we definie w ε µ = tk(uε) − (tk(u))µ, to get t∫ 0 〈( b1s(u ε) ) t ,wεµ 〉 dt + ∫ q s′n(u ε)a(x,t,∇uε)∇wεµdxdt (4.74) + ∫ q s′′n(u ε)a(x,t,∇uε)∇uεwεµdxdt + ∫ q γ(uε)s′n(u ε)wεµdxdt = ∫ q fε1 (x,t,u ε,vε)s′n(u ε)wεµdxdt. now we pass to the limit in (4.74) as ε → 0, µ → +∞, n → +∞ for k real number fixed. in order to perform this task, we prove below the following results for any k ≥ 0 : lim inf µ→+∞ lim ε→0 t∫ 0 〈( b1s(u ε) ) t ,wεµ 〉 dt ≥ 0 for any n ≥ k, (4.75) lim n→+∞ lim µ→+∞ lim ε→0 ∫ q s′′n(u ε)a(x,t,∇uε)∇uεwεµdxdt = 0, (4.76) lim µ→+∞ lim ε→0 ∫ q γ(uε)s′n(u ε)wεµdxdt = 0, for any n ≥ 1, (4.77) lim µ→+∞ lim ε→0 ∫ q fε1 (x,t,u ε,vε)s′n(u ε)wεµdxdt = 0, for any n ≥ 1. (4.78) proof of (4.75). in view of the definition wεµ, we apply lemma (4.2) with s = sn for fixed n ≥ k. as a consequence, (4.75) hold true. � int. j. anal. appl. 18 (5) (2020) 767 proof of (4.76). for any n ≥ 1 fixed, we have supp(s′′n) ⊂ [−(n + 1),−n]∪ [n,n + 1] , ∥∥wεµ∥∥l∞(q) ≤ 2k and ‖s′′n‖l∞(r) ≤ 1, to get ∣∣∣∣∣∣ ∫ q s′′n(u ε)a(x,t,∇uε)∇uεwεµdxdt ∣∣∣∣∣∣ (4.79) ≤ 2k ∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt, for any n ≥ 1, by (4.66) it possible to etablish (4.76) � proof of (4.77). for fixed n ≥ 1 and in view (4.56) . lebesgue’s convergence theorem implies that for any µ > 0 and any n ≥ 1 lim ε→0 ∫ q γ(uε)s′n(u ε)wεµ dxdt = ∫ q γ(u)s′n(u)(tk(u) −tk (u)µ)dxdt. (4.80) appealing now to (4.69) and passing to the limit as µ → +∞ in (4.80) allows to conclude that (4.77) holds true. � proof of (4.78). by (4.11), (4.61) and lebesgue’s convergence theorem implies that for any µ > 0 and any n ≥ 1, it is possible to pass to the limit for ε → 0 lim ε→0 ∫ q fε1 (x,t,u ε,vε)s′n(u ε)wεµ dxdt = ∫ q f1(x,t,u,v)s ′ n(u)(tk(u) −tk (u)µ)dxdt, using (4.69) permits to the limit as µ tends to +∞ in the above equality to obtain (4.78). � now turn back to the proof of lemma (4.3), due to (4.75)-(4.78), we are in a position to pass to the limit-sup when ε → 0, then to the limit-sup when µ → +∞ and then to the limit as n → +∞ in (4.74). using the definition of wεµ, we deduce that for any k ≥ 0, lim n→+∞ lim sup µ→+∞ lim sup ε→0 ∫ q a(x,t,∇uε)s′n(u ε)∇(tk(uε) −tk(u)µ) dxdt ≤ 0. since a(x,t,∇uε)s′n(uε)∇tk(uε) = a(x,t,∇uε)∇tk(uε) fo k ≤ n, the above inequality implies that for k ≤ n, lim sup ε→0 ∫ q a(x,t,∇uε)∇tk(uε)dxdt (4.81) ≤ lim n→+∞ lim sup µ→+∞ lim sup ε→0 ∫ q a(t,x,∇uε)s′n(u ε)∇tk(u)µdxdt. due to (4.55), to have a(x,t,∇uε)s′n(u ε) → ηn+1s′n(u) weakly in ( lp ′(.) (q) )n as ε → 0, int. j. anal. appl. 18 (5) (2020) 768 and the strong convergence of tk(u)µ to tk(u) in l p−(]0,t[; w 1,p 0 (ω)) as µ → +∞, to get lim µ→+∞ lim ε→0 ∫ q a(x,t,∇uε)s′n(u ε)∇tk(u)µdxdt (4.82) = ∫ q s′n(u)ηn+1∇tk(u)dxdt = ∫ q ηn+1∇tk(u)dxdt, as soon as k ≤ n, since s′n(s) = 1 for |s| ≤ n. now, for k ≤ n, to have s′n(u ε)a(x,t,∇uε)χ{|uε|≤k} = a(x,t,∇u ε)χ{|uε|≤k} a.e in q. letting ε → 0, to obtain ηn+1χ{|u|≤k} = ηkχ{|u|≤k} a.e in q−{|u| = k} for k ≤ n. recalling (4.81) and (4.82) allows to conclude that (4.70) holds true. � proof of (4.71). let k ≥ 0 be fixed. we use the monotone character (3.3) of a(x,t,ξ) with respest to ξ, to obtain iε = ∫ q ( a(x,t,∇uεχ{|uε|≤k}) −a(x,t,∇uχ{|u|≤k}) )( ∇uεχ{|uε|≤k} −∇uχ{|u|≤k} ) dxdt ≥ 0. (4.83) inequality (4.83) is split into iε = iε1 + i ε 2 + i ε 3 where iε1 = ∫ q a(x,t,∇uεχ{|uε|≤k})∇uεχ{|uε|≤k}dxdt, iε2 = − ∫ q a(x,t,∇uεχ{|uε|≤k})∇uχ{|u|≤k}dxdt, iε3 = − ∫ q a(x,t,∇uχ{|u|≤k}) ( ∇uεχ{|uε|≤k} −∇uχ{|u|≤k} ) dxdt. we pass to the limit-sup as ε → 0 in iε1, iε2 and iε3. let us remark that we have uε = tk(uε) and ∇uεχ{|uε|≤k} = ∇tk(uε) a.e in q, and we can assume that k is such that χ{|uε|≤k} almost everywhere converges to χ{|u|≤k}(in fact this is true for almost every k, see lemma 3.2 in [6]). using (4.70), to obtain lim ε→0 iε1 = lim ε→0 ∫ q a(x,t,∇uε)∇tk(uε)dxdt (4.84) ≤ ∫ q ηk∇tk(u)dxdt. int. j. anal. appl. 18 (5) (2020) 769 in view of (4.53) and (4.55), to have lim ε→0 iε2 = −lim ε→0 ∫ q a(x,t,∇uεχ{|uε|≤k}) (∇tk(u)) dxdt (4.85) = − ∫ q ηk (∇tk(u)) dxdt. as a consequence of (4.53), we have for all k > 0 lim ε→0 iε3 = − ∫ q a(x,t,∇uχ{|u|≤k}) (∇tk(uε) −∇tk(u)) dxdt = 0. (4.86) taking the limit-sup as ε → 0 in (4.83) and using (4.84), (4.85) and (4.86) show that (4.71) holds true. � proof of (4.72). using (4.71) and the usual minty argument applies it follows that (4.72) holds true. lemma 4.4. ∇tk(vε) converges to ∇tk(v) in (l2(q))n . proof. denote v εµ = tk(vε) − (tk(v))µ and choose s ′ n(v ε) ( tk(vε) − (tk(v))µ ) the test function in (4.16). one can get that t∫ 0 〈( b2s(v ε) ) t ,v εµ 〉 dt + ∫ q s′n(v ε)∇vε∇v εµdxdt (4.87) + ∫ q s′′n(v ε)|∇vε|2v εµdxdt = ∫ q fε2 (x,t,u ε,vε)s′n(v ε)v εµdxdt. by a similar discussion, one has lim inf µ→+∞ lim ε→0 t∫ 0 〈( b2s(v ε) ) t ,v εµ 〉 dt ≥ 0 for any n ≥ k, (4.88) lim n→+∞ lim µ→+∞ lim ε→0 ∫ q s′′n(v ε)|∇vε|2v εµdxdt = 0, (4.89) and lim µ→+∞ lim ε→0 ∫ q fε2 (x,t,u ε,vε)s′n(v ε)v εµdxdt = 0, for any n ≥ 1. (4.90) hence lim n→+∞ lim µ→+∞ lim ε→0 ∫ q s′n(v ε)∇vε∇v εµdxdt ≤ 0. (4.91) � similarly, one gets that ∇tk(vε) converges to ∇tk(v) in (l2(q))n . � int. j. anal. appl. 18 (5) (2020) 770 • step 5: in this step we prove that (u,v) satisfies (4.3), (4.4)-(4.7) . for any fixed n ≥ 0 one has∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt = ∫ q a(x,t,∇uε)∇tn+1(uε)dxdt− ∫ q a(x,t,∇uε)∇tn(uε)dxdt. according to (4.55) and (4.72) one is at liberty to pass to the limit as ε tends to 0 for fixed n ≥ 1 and to obtain lim ε→0 ∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt (4.92) = ∫ q a(x,t,∇u)∇tn+1(u)dxdt− ∫ q a(x,t,∇u)∇tn(u)dxdt = ∫ {n≤|uε|≤n+1} a(x,t,∇u)∇udxdt. letting n tends to +∞ in (4.92), it follows from estimate (4.66), that lim ε→0 lim ∫ {n≤|uε|≤n+1} a(x,t,∇uε)∇uεdxdt = 0. similarly, one can prove lim ε→0 lim ∫ {n≤|vε|≤n+1} |∇vε|2dxdt = 0. let s be a function in w 2,∞(r) such that s′ has a compact. let k be a positive real number such that supp(s′) ⊂ [−k,k]. pontwise multiplication of that approximate equation (4.15) by (s′(uε),s′(vε)) leads to ( b1s(u ε) ) t −div(s′(uε)a(x,t,∇uε)) (4.93) +s′′(uε)a(x,t,∇uε)∇(uε) + γ(uε)s′(uε) = fε1 (x,t,u ε,vε)s′(uε) in d′(q), and ( b2s(v ε) ) t −div(s′(vε)∇vε) (4.94) +s′′(vε)|∇(vε)|2 = fε2 (x,t,u ε,vε)s′(vε) in d′(q). in what follows to pass to the limit as ε tends to 0 in each term of (4.93). since s is bounded, and (s(uε),s(vε)) converges to (s(u),s(v)) a.e in q and in (l∞(q))2 *-weak, then int. j. anal. appl. 18 (5) (2020) 771 ( ( b1s(u ε) ) t , ( b2s(v ε) ) t ) converges to ( ( b1s(u) ) t , ( b1s(v) ) t ) in d′(q) as ε tends to 0. since supp(s′) ⊂ [−k,k], s′(uε)a(t,x,∇uε) = s′(uε)a(x,t,∇uε)χ{|uε|≤k} a.e in q. the pointwise convergence of uε to u as ε tends to 0, the bounded character of s and (4.72) of lemma(4.3) imply that s′(uε)a(x,t,∇uε) converges to s′(u)a(x,t,∇u) weakly in ( lp ′(.)(q) )n as ε tends to 0, because s′(u) = 0 for |u| ≥ k a.e in q and s′(vε)∇vε converges to s′(v)∇v weakly in l2(q) as ε tends to 0. the pointwise convergence of uε to u, the bounded character of s′, s′′ and (4.72) of lemma (4.3) allow to conclude that s′′(uε)a(x,t,∇uε)∇tk(uε) → s′′(u)a(x,t,∇u)∇tk(u) weakly in l1(q) as ε → 0, and lemma (4.1) shows that s′′(vε)∇εv∇tk(vε) → s′′(v)∇v∇tk(v) weakly in l1(q). the use of (4.56) to obtain that γ(uε)s′(uε) converges to γ(u)s′(u) in l1(q), and we use (4.11), (4.53) and we obtain that fε1 (x,t,u ε,vε)s′(uε) converges to f1(x,t,u,v)s ′(u) in l1(q) and fε2 (x,t,u ε,vε)s′(vε) converges to f2(x,t,u,v)s ′(v) in l1(q). as a consequence of the above convergence result, the position to pass to the limit as ε tends to 0 in equation (4.93) and (4.94), we conclude that (u,v) satisfies (4.4) and (4.5). it remains to show that s(u) satisfies the initial condition (4.6) and s(v) satisfies the initial condition (4.7). to this end, firstly remark that, s being bounded, (s(uε),s(vε)) is bounded in (l∞(q))2, (b1s (u ε) ,b2s (v ε)) is bounded in l∞(q) × l∞(q). secondly, (4.93) and (4.94), the above considerations on the behavior of the terms of this equation show that ∂b1s(u ε) ∂t is bounded in l1(q) + l(p−) ′ (]0,t[; w−1,p ′(.)(ω)) and ∂b2s(v ε) ∂t is bounded in l1(q) + l2(]0,t[; h10 (ω)). as a consequence, an aubin’s type lemma ( [20], corollary 4) implies that (b1s(u ε),b2s(v ε)) lies in a compact set of (c(]0,t[; l1(ω)))2. it follows that, on the one hand, b1s(u ε)(t = 0) converges to b1s(u)(t = 0) strongly in l1(ω) and b2s(v ε)(t = 0) converges to b2s(v)(t = 0) strongly in l 1(ω). due to (4.10), to conclude that (4.6) and (4.7) holds true. as a conclusion of step 3 and step 5, the proof of theorem (4.1) is complete. � int. j. anal. appl. 18 (5) (2020) 772 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. akdim, j.bennouna, m.mekkour, h.redwane, existence of a renormalised solutions for a class of nonlinear degenerated parabolic problems with l1 data, j. partial differ. equ. 26 (2013), 76-98. [2] e. azroula, h. redwane, m. rhoudaf, existence of solutions for nonlinear parabolic systems via weak convergence of truncations, electron. j. differ. equ. 2010 (2010), 68. [3] d. blanchard, and f. murat, renormalised solutions of nonlinear parabolic problems with l1 data, existence and uniqueness, proc. r. soc. edinb., sect. a, math. 127 (6) (1997), 1137-1152. [4] d. blanchard , f. murat, and h. redwane, existence et unicité de la solution reormalisée d’un probléme parabolique assez général, c. r. acad. sci. paris sér., 329 (1999), 575-580. [5] d. blanchard, f. murat, and h. redwane, existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, j. differ. equ. 177 (2001), 331-374 . [6] l. boccardo, a. dall’aglio, t. gallouët , and l. orsina, nonlinear parabolic equations with measure data, j. funct. anal., 147 (1997), 237-258 . [7] t. m. bendahmane, p. wittbold, a.zimmermann, renormalized solutions for a nonlinear parabolic equation with variable exponents and l1-data . j. differ. equ. 249 (2010), 1483-1515. [8] m. b. benboubker, h. chrayteh, m. el moumni and h. hjiaj, entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data, acta math. sin. engl. ser. 31 (1) (2014), 151-169. [9] y.m. chen, s. levine, m. rao, variable exponent, linear growth functionals in image restoration, siam j. appl. math. 66 (2006) 1383-1406. [10] di perna, r.-j. and p.-l. lions, p.-l., on the cauchy problem for boltzmann equations : global existence and weak stability, ann. math. 130 (1989), 321-366 . [11] b. el hamdaoui, j. bennouna, and a. aberqi, renormalized solutions for nonlinear parabolic systems in the lebesgue sobolev spaces with variable exponents, j. math. phys. anal. geom. 14 (1) (2018), 27-53. [12] s. fairouz, m. messaoud, s. kamel, quasilinear parabolic problems in the lebesgue-sobolev space with variable exponent and l1-data. communicated. [13] x.l. fan and d. zhao, on the spaces lp(x)(u) and w m;p(x)(u), j. math. anal. appl. 263 (2001), 424-446. [14] r. landes, on the existence of weak solutions for quasilinear parabolic initial-boundary problems, proc. r. soc. edinb., sect. a, math. 89 (1981), 321-366. [15] j.-l. lions, quelques méthodes de résolution des problémes aux limites non linéaires. dunod et gauthier-villars, 1969. [16] s. ouaro and a. ouédraogo, nonlinear parabolic equation with variable exponent and l1 data. electron. j. differ. equ. 2017 (2017), 32. [17] q. liu and z. guo, c. wang, renormalized solutions to a reaction-diffusion system applied to image denoising, discrete contin. dyn. syst., ser. b, 21 (6) (2016), 1839-1858. [18] h. redwane, existence of solution for a class of nonlinear parabolic systems, electron. j. qual. theory differ. equ. 2007 (2007), 24. [19] m. sanchón, j.m. urbano, entropy solutions for the p(x)-laplace equation, trans. amer. math. soc. 361 (2009), 6387-6405. [20] j. simon, compact sets in lp(0, t;b), ann. mat. pura appl. 146 (1987), 65-96. int. j. anal. appl. 18 (5) (2020) 773 [21] a. youssef. b. jaouad. b. abdelkader. m. mounir, existence of a renormalized solution for the nonlinear parabolic systems with unbounded nonlinearities, int. j. res. rev. appl. sci. 14 (2013), 75-89. [22] c. zhang, entropy solutions for nonlinear elliptic equations with variable exponents. electron. j. differ. equ. 2014 (2014), 92. [23] c. zhang, s. zhou , renormalized and entropy solution for nonlinear parabolic equations with variable exponents and l1 data, j. differ. equ. 248 (2010), 1376-1400. 1. introduction 2. the mathematical preliminaries on variable exponent sobolev spaces 2.1. generalized lebesgue-sobolev spaces. 3. the assumptions on the data 4. the main results 4.1. the existence theorem references international journal of analysis and applications volume 19, number 2 (2021), 205-227 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-205 downhill zagreb topological indices of graphs bashair al-ahmadi, anwar saleh∗, wafa al-shammakh department of mathematics, faculty of science, university of jeddah, jeddah, saudi arabia ∗corresponding author: asaleh1@uj.edu.sa abstract. topological indices are graph invariants determined by the distance or degree of vertices of the molecular graph. topological indices have been used effectively in chemical graph theory in explaining the structures and predicting certain physicochemical properties of chemical compounds. in this research, we introduce the first, second, and forgotten downhill zagreb indices and calculate those topological indices for some standard families of graphs and the join of graphs. also, the downhill topological indices for the firefly graph, book graph, and stacked book graph are established. finally, the downhill indices of graphene and honeycomb network are obtained. 1. introduction a graph is non empty set of vertices together with a number of edges connecting a subset of them. v (g) and e(g) denoted for vertex set and edge set respectively. if we consider the molecules as special chemical structures, and if we replace atoms and bonds with vertices and edges, respectively, the obtained graph is called a molecular graph. that means a molecular graph is a simple graph such that its vertices correspond to the atoms and its edges to the bonds. we note that hydrogen atoms are often omitted and the remaining part of the graph is sometimes called as the carbon graph of the corresponding molecule. chemical graph theory which deals with the above mentioned relations between molecules and corresponding graphs is a branch of mathematical chemistry which has an important effect on the development of received december 24th, 2020; accepted january 25th, 2021; published february 12th, 2021. 2010 mathematics subject classification. 05c35, 05c07, 05c40. key words and phrases. first downhill zagreb index (of a graph); second downhill zagreb index; forgetten downhill zagreb index. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 205 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-205 int. j. anal. appl. 19 (2) (2021) 206 the molecular chemistry along with quantitative structure-property relationship (qspr) and quantitative structure-activity relationship (qsar) studies. in this research work, we are concerned with simple graphs which they are finite, undirected with no loops and multiple edges. the open and closed neighborhoods of a vertex v in a graph g are denoted by n(v) = {u ∈ v : uv ∈ e} and n[v] = n(v) ∪{v}, respectively. the degree of a vertex v in g is denoted by degg(v) or briefly by dg(v), where degg(v) = |n(v)|. when there is no confusion, one can also omit g and use d(v) instead of dg(v). the minimum degree and maximum degree of g are denoted by δ, and ∆ respectively. in a graph g if δ = ∆ = k, then the graph g is called regular graph of degree k. also a graph with the property that ∆ ≤ 4 is called a chemical graph. the following notations and different types of graphs well known in the literature [9] and [2]. a double star is the graph obtained from k2 by joining s pendent edges to one end and r pendent edges to the other end of k2. a wheel wn+1, n > 3 is the join of cn and k1. a helm graph, denoted by hn, is a graph obtained from wn+1 by attaching an end edge to each rim vertex of wn+1, where the vertices corresponding to cn are known as rim vertices. the gear graph is a wheel graph with a vertex added between each pair adjacent graph vertices of the outer cycle. the gear graph gn has 2n + 1 vertices and 3n edges. the sierpinski sieve graph sn is the graph obtained from the connectivity of the sierpinski sieve. the graph has 3(3n−1+1) 2 vertices and 3n edges. the tadpole graph tr,s is obtained by joining a cycle cr and a path ps by a bridge, where r > 3 and s > 1. the cartesian product g of two graphs g1 and g2, denoted g1×g2, has vertex set v (g) = v (g1) × v (g2), and two distinct vertices (a,b) and (c,d) of g1 × g2 are adjacent if either a = c and bd ∈ e(g2), or b = d and ac ∈ (g1). the join g = g1 ∨g2 of two graphs g1 and g2 has vertex set v (g) = v (g1) ⋃ v (g2) and edge set e(g) = e(g1) ⋃ e(g2){uv : u ∈ v (g1),v ∈ v (g2)}. the corona product g◦h is defined as the graph obtained from g and h by taking one copy of g and n1 copies of h and joining by an edge each vertex from the ith-copy of h with the ith-vertex of g. book graph is a cartesian product of a star and single edge, denoted by bm. the m-book graph is defined as the graph cartesian product sm+1 ×p2 , where sm+1 is a star graph and p2 is the path graph . the stacked book graph of order (m,n) is defined as the graph cartesian product sm+1 ×pn, where sm is a star graph and pn is the path graph on n nodes, and it is denoted by bm,n. as it is known, the path, cycle, and complete graphs with n vertices are denoted by pn, cn, kn; and the complete bipartite graph is denoted by kr,s. int. j. anal. appl. 19 (2) (2021) 207 a topological index of a graph is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. in recent decades, a large number of topological indices have been defined and utilized for chemical documentation, isomer discrimination, study of molecular complexity, chirality, similarity/dissimilarity, qsar/qspr, drug design and database selection, lead optimization, etc. as an example, the boiling point of a molecule is directly related to the forces between the atoms. when a solution is heated, the temperature is increased and as it is increased, the kinetic energy between molecules increases. this means that the molecular motion becomes so intense that the bonds between molecules break and become a gas. the moment the liquid turns to gas is labeled as the boiling point. the boiling point can give important clues about the physical properties of chemical structures. molecules which strongly interact or bond with each other through a variety of inter-molecular forces cannot move easily or rapidly and therefore, do not achieve the kinetic energy necessary to escape the liquid state. that is why the boiling points of the alkanes increase with molecular size. the most useful and famous topological indices of a graph are the first and second zagreb indices which have been introduced by gutman and trinajstic in [7]. they are denoted by m1(g) and m2(g) and were defined as m1(g) = ∑ u∈v (g) [ d(u) ]2 and m2(g) = ∑ uv∈e(g) d(u)d(v), respectively. the forgotten topological index was introduced by furtula and gutman [4] and it is stated that the forgotten topological index is a special case of first zagreb index. shehnaz akhter, muhammad imran extended the study of forgotten topological index and determined the close formula of f-index for some graph operations. the zagreb indices have been studied extensively. many new reformulated and extended versions of the zagreb indices have been introduced. for more discussion on these indices, we encourage the readers to consult the articles ( [1], [3], [5], [6], [11], [12], [13], [14], [16], [17], [18], [19]). the downhill domination is introduced for the first time in [10]. for more details about the downhill and uphill domination, we refer to [8, 15]. in this research work, motivated by the downhill domination and the huge applications of topological indices, we define the downhill degree and introduce the first, second and forgotten downhill zagreb indices int. j. anal. appl. 19 (2) (2021) 208 and calculate these topological indices for, some standard families of graphs, join of two graph. also, the downhill topological indices for firefly graph, book graph stacked book graph are established. finally, the downhill indices of graphene and honeycomb network are obtained. 2. downhill zagreb indices in this section, we define the first, second and forgotten downhill zagreb indices and calculate these topological indices for some standard families of graphs. definition 2.1. [10] let g = (v,e) be a graph. a u−v path p in g is a sequence of vertices in g, starting with u and ending at v, such that consecutive vertices in p are adjacent, and no vertex is repeated. a path π = v1,v2, ...vk+1 in g is a downhill path if for every i,1 6 i 6 k, deg(vi) > deg(vi+1). definition 2.2. in a graph g = (v,e). a vertex v is downhill dominates a vertex u if there exists a downhill path originated from v to u. the downhill neighborhood of a vertex v is denoted by ndn(v) and defined as: ndn(v) = {u : v downhill dominates u} . the downhill degree of the vertex v, denotes by ddn(v), is the number of downhill neighbors of v, that means ddn(v) = |ndn(v)|. the downhill closed neighborhood, ndn[v], of a vertex v is the downhill open neighborhood of v taken together with v. it follows that ndn[v] = ndn(v) ∪{v}. definition 2.3. let g = (v,e) be a graph. then the first, second and forgotten downhill zagreb indices are defined as: dwm1(g) = ∑ v∈v (g) (ddn(v)) 2, dwm2(g) = ∑ vu∈e(g) ddn(v)ddn(u) and dwf(g) = ∑ v∈v (g) (ddn(v)) 3. example 2.1. let g be a house graph as in figure 1. then dwm1(g) = ∑ v∈v (g) (ddn(v)) 2 = 0 + 25 + 25 + 25 + 1 + 1 = 77, dwm2(g) = ∑ vu∈e(g) ddn(v)ddn(u) = 0 × 5 + 3 × 5 × 5 + 2 × 5 × 1 + 1 × 1 = 86, dwf(g) = ∑ v∈v (g) (ddn(v)) 3 = 0 + 125 + 125 + 125 + 1 + 1 = 377. int. j. anal. appl. 19 (2) (2021) 209 figure 1. the house graph. proposition 2.1. let g be the connected k-regular graph of n vertices. then, dwm1(g) = n(n− 1)2, dwm2(g) = nk(n− 1)2 2 , dwf(g) = n(n− 1)3. proof. let g be the connected k-regular graph of n vertices. it is easy to see that for any vertex in g has downhill degree n− 1. hence, dwm1(g) = n(n− 1)2 and dwf(g) = n(n− 1)3. similarly, there are nk2 edges in g, where each edge has endpoints of downhill degree n− 1. hence, dwm2(g) = nk(n−1)2 2 . � corollary 2.1. (1) for any cycle cn, dwm1(cn) = dwm2(cn) = n(n− 1)2 and dwf(cn) = n(n− 1)3. (2) for any complete graph kn, dwm1(kn) = n(n − 1)2, dwm2(kn) = n(n−1)3 2 and dwf(kn) = n(n− 1)3. proposition 2.2. let g ∼= pn be a path of n ≥ 3 vertices. then dwm1(g) = (n− 2)(n− 1)2, dwm2(g) = (n− 3)(n− 1)2, dwf(g) = (n− 2)(n− 1)3. proof. let g ∼= pn be a path of n vertices, where n ≥ 3, the path has n−2 vertices of downhill degree n−1 and 2 vertices of downhill degree 0. hence, dwm1(g) = (n−2)(n−1)2 and dwf(g) = (n−2)(n−1)3. there are n−1 edges in g of which n−3 edges have endpoints of downhill degree n−1 and 2 edges have one endpoint of downhill degree n−1 and the another endpoint of downhill degree 0. hence, dwm2(g) = (n−3)(n−1)2. � int. j. anal. appl. 19 (2) (2021) 210 proposition 2.3. let g ∼= sn be a star with n + 1 vertices, where n ≥ 2. then, dwm1(g) = n 2, dwm2(g) = 0, dwf(g) = n3. proof. let g ∼= sn be a star with n + 1 vertices, where n ≥ 2, clearly there are n + 1 vertices of downhill degree n and n vertices of downhill degree 0. hence,dwm1(g) = n 2 and dwf(g) = n3. the star graph g has n edges, where each edge has one endpoint of downhill degree n and the another endpoint of downhill degree 0. hence, dwm2(g) = 0. � proposition 2.4. let g ∼= ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. then, dwm1(g) =   2(s + r + 1)2 if s = r; (s + r + 1)2 + r2 if s > r. proof. let g ∼= ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. we have two cases: case 1. if s = r. then the graph has two vertices of downhill degree s + r + 1 and s + r vertices of downhill degree 0. hence, dwm1(g) = 2(s + r + 1) 2. case 2. if s > r, then there are one vertex of downhill degree s + r + 1, one vertex of downhill degree r and s + r vertices of downhill degree 0. hence, dwm1(g) = (s + r + 1) 2 + r2. � proposition 2.5. let g ∼= ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. then, dwm2(g) =   (s + r + 1)2 if s = r; r(s + r + 1) if s > r. proof. let g ∼= ss,r be a double star with s + r + 2 vertices and s + r + 1 edges, where s,r ≥ 2. we have two cases: case 1. if s = r. it has one edge has endpoints of downhill degree s + r + 1. also, there are s + r edges in g, where each edge has one endpoint of downhill degree s + r + 1 and the another endpoint of downhill degree 0. hence, dwm2(g) = (s + r + 1) 2. case 2. if s > r. it has one edge has one endpoint of downhill degree s + r + 1 and the another endpoint of downhill degree r. also, there are s + r edges where each edge has endpoint of downhill degree s + r + 1 or r and the another endpoint of downhill degree 0. hence, dwm2(g) = r(s + r + 1). � in the same way we can get the following result. int. j. anal. appl. 19 (2) (2021) 211 proposition 2.6. let g ∼= ss,r be a double star with s + r + 2 vertices, where s,r ≥ 2. then, dwf(g) =   2(s + r + 1)3 if s = r; (s + r + 1)3 + r3 if s > r. proposition 2.7. let g ∼= ks,r be the complete bipartite graph, where s < r. then, dwm1(g) = sr 2, dwm2(g) = 0, dwf(g) = sr3. proof. let g ∼= ks,r be the complete bipartite graph, where s < r. there are s+r vertices of which s vertices of downhill degree r and r vertices of downhill degree 0. hence, dwm1(g) = sr 2 and dwf(g) = sr3. there are sr edges in g, where each edge has one endpoint of downhill degree r and the another endpoint of downhill degree 0. hence, dwm2(g) = 0. � proposition 2.8. let g ∼= wn be a wheel graph of n + 1 vertices, where n ≥ 4. then, dwm1(g) = n(n 2 −n + 1), dwm2(g) = n(2n 2 − 3n + 1), dwf(g) = n(n3 − 2n2 + 3n− 1). proof. let g ∼= wn be a wheel graph of n + 1 vertices, where n ≥ 4. there is one vertex of downhill degree n and n vertices of downhill degree n − 1. hence, dwm1(g) = n(n2 − n + 1) and dwf(g) = n(n3 −2n2 + 3n−1). there are 2n edges in g of which n edges have one endpoint of downhill degree n and the another endpoint of downhill degree n− 1 and n edges have endpoints of downhill degree n− 1. hence, dwm2(g) = n(2n 2 − 3n + 1). � proposition 2.9. let g ∼= gn be a gear graph of 2n + 1 vertices, where n ≥ 4. then, dwm1(g) = 4n(n + 1), dwm2(g) = 4n 2, dwf(g) = 8n(n2 + 1). proof. let g ∼= gn be a gear of 2n + 1 vertices, where n ≥ 4. then the graph has one vertex of downhill degree 2n, n vertices of downhill degree 2 and n vertices of downhill degree 0. hence, dwm1(g) = 4n(n+ 1) and dwf(g) = 8n(n2 + 1). there are 3n edges in g of which n edges have one endpoint of downhill degree 2n and the another endpoint of downhill degree 2 and 2n edges have one endpoint of downhill degree 2 and the another endpoint of downhill degree 0. hence, dwm2(g) = 4n 2. � int. j. anal. appl. 19 (2) (2021) 212 proposition 2.10. let g ∼= hn be a helm graph of 2n + 1 vertices, where n ≥ 5. then, dwm1(g) = n(4n 2 + 1), dwm2(g) = n(8n 2 − 6n + 1), dwf(g) = n(8n3 − 4n2 + 6n− 1). proof. let g ∼= hn be a helm graph of 2n + 1 vertices, where n ≥ 5. then there are one vertex of downhill degree 2n, n vertices of downhill degree 2n − 1 and n vertices of downhill degree 0. hence, dwm1(g) = n(4n 2 + 1) and dwf(g) = n(8n3 − 4n2 + 6n − 1). there are 3n edges of which n edges have one endpoint of downhill degree 2n and the another endpoint of downhill degree 2n− 1, n edges have endpoints of downhill degree 2n−1 and n edges have one endpoint of downhill degree 2n−1 and the another one of downhill degree 0. hence, dwm2(g) = n(8n 2 − 6n + 1). � proposition 2.11. let g ∼= sn be the sierpinski of m vertices, where m = 3(3n−1+1) 2 . then, dwm1(g) = (m− 3)(m− 1)2, dwm2(g) = (2m− 9)(m− 1)2, dwf(g) = (m− 3)(m− 1)3. proof. let g ∼= sn be the sierpinski graph of m vertices, where m = 3(3n−1+1) 2 . there are m − 3 vertices of downhill degree m − 1 and 3 vertices of downhill degree 0. hence, dwm1 = (m − 3)(m − 1)2 and dwf(g) = (m − 3)(m − 1)3. there are 2m − 3 edges of which 2m − 9 has endpoints of downhill degree m− 1 and 6 edges have one endpoint of downhill degree m− 1 and the another endpoint of downhill degree 0. hence, dwm2(g) = (2m− 9)(m− 1)2. � theorem 2.1. let g ∼= tn,m be the tadpole graph with n + m vertices, where n,m ≥ 3. then, dwm1(g) = (n + m− 1)2 + (n− 1)(n− 2)2 + (m− 1)3, dwm2(g) = (n + m− 1)(2n + m− 5) + (n− 2)3 + (m− 2)(m− 1)2, dwf(g) = (n + m− 1)3 + (n− 1)(n− 2)3 + (m− 1)4. proof. let g ∼= tn,m be the tadpole graph with n + m vertices. then the graph has one vertex of downhill degree n + m− 1, n− 1 vertices of downhill degree n− 2, m− 1 vertices of downhill degree m− 1 and one vertex of downhill degree 0. hence, dwm1(g) = (n + m− 1)2 + (n− 1)(n− 2)2 + (m− 1)3, dwf(g) = (n + m− 1)3 + (n− 1)(n− 2)3 + (m− 1)4. int. j. anal. appl. 19 (2) (2021) 213 there are m + n edges of which one edge has one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree m−1, 2 edges have one endpoint of downhill degree n + m−1 and the another endpoint n−2, n−2 edges have endpoints of downhill degree n−2, m−2 edges have endpoints of downhill degree m − 1 and one edge has one endpoint of downhill degree m − 1 and another endpoint of downhill degree 0. hence, dwm2(g) = (n + m− 1)(2n + m− 5) + (n− 2)3 + (m− 2)(m− 1)2. � 3. the downhill zagreb indices of graphs under some binary operations theorem 3.1. let g ∼= cn ∨pm be the join graph with n + m vertices, where n,m ≥ 3. then, dwm1(g) =   2(n− 1)(2n− 1)2 if n = m; (m− 2)(n + m− 1)2 + (n + 2)(n + 1)2 if n = m + 1; (m− 2)(n + m− 1)2 + 2n2 + n(n− 1)2 if n > m + 1; n(n + m− 1)2 + (m− 2)(m− 1)2 if n < m. proof. let g ∼= cn ∨pm be the join graph with n + m vertices, where m ≥ 3. we have four cases: case 1. if n = m. in this case, there are 2n−2 vertices of downhill degree 2n−1 and 2 vertices of downhill degree 0. hence, dwm1(g) = 2(n− 1)(2n− 1)2. case 2. if n = m + 1. in this case, there are m−2 vertices of downhill degree n + m−1 and n + 2 vertices of downhill degree n + 1. hence, dwm1(g) = (m− 2)(n + m− 1)2 + (n + 2)(n + 1)2. case 3. if n > m + 1. in this case, there are m − 2 vertices of downhill degree n + m − 1, 2 vertices of downhill degree n and n vertices of downhill degree n− 1. hence, dwm1(g) = (m− 2)(n + m− 1)2 + 2n2 + n(n− 1)2. case 4. if n < m. in this case, there are n vertices of downhill degree n + m−1, m−2 vertices of downhill degree m− 1 and 2 vertices of downhill degree 0. hence, dwm1(g) = n(n + m− 1)2 + (m− 2)(m− 1)2. � int. j. anal. appl. 19 (2) (2021) 214 theorem 3.2. let g ∼= cn ∨pm be the join graph with n + m vertices, where n,m ≥ 3. then, dwm2(g) =   (n2 − 3)(2n− 1)2 if n = m; a if n = m + 1; b if n > m + 1; c if n < m, where, a = (m− 3)(n + m− 1)2 + 3n(n + 1)2 + (nm− 2n + 2)(n + m− 1)(n + 1), b = (m− 3)(n + m− 1)2 + n(n + m− 1)(nm−m− 2n + 4) + n(n− 1)(3n− 1) and c = n(n + m− 1)2 + (m− 3)(m− 1)2 + n(m− 2)(m− 1)(n + m− 1). proof. let g ∼= cn ∨pm be the join graph with n + m vertices, where n,m ≥ 3. clearly the graph g has nm + n + m− 1 edges and we have four cases: case 1. if n = m. in this case, there are n2−3 edges have endpoints of downhill degree 2n−1 and 2(n + 1) edges have one endpoint of downhill degree 2n− 1 and the another endpoint of downhil degree 0. hence, dwm2(g) = (n 2 − 3)(2n− 1)2. case 2. if n = m + 1. in this case, there are m− 3 edges have endpoints of downhill degree n + m− 1, 3n edges have endpoints of downhill degree n + 1 and nm− 2n + 2 edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree n + 1. hence, dwm2(g) = (m− 3)(n + m− 1)2 + 3n(n + 1)2 + (nm− 2n + 2)(n + m− 1)(n + 1). case 3. if n > m + 1. in this case, there are m− 3 edges have endpoints of downhill degree n + m− 1, n edges have endpoints of downhill degree n−1, nm−2n edges have one endpoint of downhill degree n+m−1 and the another endpoint of downhill degree n−1, 2n edges have one endpoint of downhill degree n and the another endpoint of downhill degree n− 1 and 2 edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree n. hence, dwm2(g) = (m− 3)(n + m− 1)2 + n(n + m− 1)(nm−m− 2n + 4) + n(n− 1)(3n− 1). case 4. if n < m. in this case, there are n edges have endpoints of downhill degree n + m− 1, m− 3 have endpoints downhill degree m− 1, n(m− 2) edges have one endpoint of downhill degree n + m− 1 and the another endpoint of downhill degree m−1 and 2(n+ 1) edges have one endpoint of downhill degree n+m−1 or m− 1 and the another endpoint of downhill degree 0. hence, dwm2(g) = n(n + m− 1)2 + (m− 3)(m− 1)2 + n(m− 2)(m− 1)(n + m− 1). � int. j. anal. appl. 19 (2) (2021) 215 theorem 3.3. let g ∼= cn ∨pm be the join graph with n + m vertices, where n,m ≥ 3. then, dwf(g) =   2(n− 1)(2n− 1)3 if n = m; (m− 2)(n + m− 1)3 + (n + 2)(n + 1)3 if n = m + 1; (m− 2)(n + m− 1)3 + 2n3 + n(n− 1)3 if n > m + 1; n(n + m− 1)3 + (m− 2)(m− 1)3 if n < m. proof. the proof similar to the proof of theorem 3.1. � proposition 3.1. let g ∼= bm be a book graph of 2(m + 1) vertices, where m ≥ 2. then, dwm1(g) = 2(4m 2 + 5m + 1), dwm2(g) = m(8m + 7) + 1, dwf(g) = 16m3 + 24m2 + 14m + 2. proof. let g ∼= bm be a book of 2(m + 1) vertices, where m ≥ 2. then there are 2 vertices of downhill degree 2m + 1 and 2m vertices of downhill degree 1. hence, dwm1(g) = 2(4m 2 + 5m + 1), dwf(g) = 16m3 + 24m2 + 14m + 2. there are 3m + 1 edges in g of which one edges has endpoints of downhill degree 2m + 1, m edges have endpoints of downhill degree 1 and 2m edges have one endpoint of downhill degree 2m + 1 and the another endpoint of downhill degree 1. hence, dwm2(g) = m(8m + 7) + 1. � theorem 3.4. let g ∼= bm,t be a stacked book graph with t(m + 1) vertices, where m ≥ 2 and t ≥ 3. then, dwm1(g) = (t− 2)(t(m + 1) − 1)2 + m(t− 2)(t− 1)2 + 2m2, dwm2(g) = (t(m + 1) − 1)((t− 3)(t(m + 1) − 1) + m(t2 − 3t + 4)) + m(t− 3)(t− 1)2, dwf(g) = (t− 2)(t(m + 1) − 1)3 + m(t− 2)(t− 1)3 + 2m3. proof. let g ∼= bm,t be a stacked book graph with t(m + 1) vertices, where m ≥ 2 and t ≥ 3. there are t−2 vertices of downhill degree t(m + 1)−1, m(t−2) vertices of downhill degree t−1, 2 vertices of downhill degree m and 2m vertices of downhill degree 0. hence, dwm1(g) = (t− 2)(t(m + 1) − 1)2 + m(t− 2)(t− 1)2 + 2m2, dwf(g) = (t− 2)(t(m + 1) − 1)3 + m(t− 2)(t− 1)3 + 2m3. int. j. anal. appl. 19 (2) (2021) 216 suppose that ea,b = {uv ∈ e(g) : ddn(u) = a and ddn(v) = b}. the stacked book graph contains 6 types of edges e0,t−1,e0,m,et(m+1)−1,m,et(m+1)−1,t(m+1)−1,et−1,t−1 and et−1,t(m+1)−1 edges. in the figure 2, the types of edges, e0,t−1,e0,m,et(m+1)−1,m, et(m+1)−1,t(m+1)−1,et−1,t−1 and et−1,t(m+1)−1 are colored in red, blue, green, yellow, pink and black, respectively. figure 2. stacked book graph bm,t. table 1 gives the number of edges in each type. type number of edges e0,t−1 2m e0,m 2m em,t(m+1)−1 2 et(m+1)−1,t(m+1)−1 t− 3 et−1,t−1 m(t− 3) et−1,t(m+1)−1 m(t− 2) table 1. the number of edges in the different types of edges of stacked book graph. thus, we get dwm2(g) = m(t− 3)(t− 1)2 + (t− 3)(t(m + 1) − 1)2 + m(t− 2)(t− 1)(t(m + 1) − 1) + 2m(t(m + 1) − 1) = (t(m + 1) − 1)((t− 3) + m(t− 2)(t− 1) + 2m) + m(t− 3)(t− 1)2. int. j. anal. appl. 19 (2) (2021) 217 hence, dwm2(g) = (t(m + 1) − 1)((t− 3)(t(m + 1) − 1) + m(t2 − 3t + 4)) + m(t− 3)(t− 1)2. � proposition 3.2. let g ∼= fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. then, dwm1(g) = (2a + 2b + c) 2 + 2a + b, dwm2(g) = (2a + 2b + c)(2a + b) + a, dwf(g) = (2a + 2b + c)3 + 2a + b. proof. let g ∼= fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. it has one vertex of downhill degree 2a + 2b + c, 2a + b vertices of downhill degree 1 and b + c vertices of downhill degree 0. hence, dwm1(g) = (2a + 2b + c) 2 + 2a + b, dwf(g) = (2a + 2b + c)3 + 2a + b. the firefly graph contains 4 types of edges e0,1,e1,1,e2a+2b+c,0 and e2a+2b+c,1 edges. in the figure 3, the different types of edges e0,1,e1,1,e2a+2b+c,0 and e2a+2b+c,1 are colored in blue, red, green and black, respectively. figure 3. firefly graph fa,b,c. table 2, gives the number of edges in each type. int. j. anal. appl. 19 (2) (2021) 218 type number of edges e0,1 b e1,1 a e0,2a+2b+c c e1,2a+2b+c 2a + b table 2. the number of edges in the different types of edges of firefly graph. hence, dwm2(g) = (2a + 2b + c)(2a + b) + a. � 4. the downhill zagreb indices of honeycomb network and graphene in this section, we obtain exact values for first, second and forgotten downhill indices for honeycomb network and graphene. honeycomb network the honeycomb network very much important in computer graphics, cellular base stations, image processing and representation of benzene hydrocarbons in chemistry. the recursive use hexagonal tiling in a particular pattern, honeycomb networks are formed. definition 4.1. the honeycomb network hc(1) is a hexagon and hc(2) is obtain by dding 6 hexagons to the boundary edges of hc(1). the honeycomb network hc(n) is obtained from hc(n− 1) by adding a layer of hexagons around the boundary of hc(n− 1). the number of vertices and edges of hc(n) are 6n2 and 9n2 − 3n respectively. int. j. anal. appl. 19 (2) (2021) 219 figure 4. honeycomb network hc(4) with downhill degree of the vertices. theorem 4.1. let g ∼= hc(n) be a honeycomb network of dimension n, where n ≥ 3. then, dwm1(g) = (6n 2 − 6n)(6n2 − 1)2 + 12, dwm2(g) = (9n 2 − 15n + 6)(6n2 − 1)2 + 72n2 − 6, dwf(g) = (6n2 − 6n)(6n2 − 1)3 + 12. proof. let g ∼= hc(n) be a honeycomb network of dimension n, where n ≥ 3. there are 2n lines in g as in figure 4. by labeling the lines from up to down l1,l2, ...,l2n, it is clear to see that, l1 symmetric with l2n. l2 is symmetric with l2n−1 ... ln is symmetric with ln+1. the first line l1 has 2n + 1 vertices in which 4 vertices are of downhill degree 1, n− 2 vertices of downhill degree 0 and n− 1 vertices of downhill degree 6n2 − 1. the line ln has 4n− 1 vertices in which 2 vertices of downhill degree 1 and 4n− 3 vertices of downhill degree 6nn − 1. any line between l1 and ln has two vertices of downhill degree 0 and the others of downhill degree 6n2 − 1. the number of vertices of downhill degree 1 is 12, the number of vertices of downhill degree 0 is 6(n− 2). therefore, the number of vertices of downhill degree 6n2 − 1 is 6n2 − 6n. hence, dwm1(g) = (6n 2 − 6n)(6n2 − 1)2 + 12, dwf(g) = (6n2 − 6n)(6n2 − 1)3 + 12. int. j. anal. appl. 19 (2) (2021) 220 in a honeycomb network there are four types of edges based on the downhill degree of the vertices of each edge. the following table gives the four types and gives the number of edges in each type. type number of edges e1,1 6 e1,6n2−1 12 e0,6n2−1 12(n− 2) e6n2−1,6n2−1 9n 2 − 15n + 6 thus, we get dwm2(g) = ∑ uv∈e(g) ddn(u)ddn(v) = |e1,1|(1)(1) + |e1,6n2−1|(1)(6n2 − 1) + |e0,6n2−1|(0)(6n2 − 1) + |e6n2−1,6n2−1|(6n2 − 1)(6n2 − 1) = 6 + 12(6n2 − 1) + (9n2 − 15n + 6)(6n2 − 1)2 = (9n2 − 15n + 6)(6n2 − 1)2 + 72n2 − 12 + 6. hence, dwm2(g) = (9n 2 − 15n + 6)(6n2 − 1)2 + 72n2 − 6. � graphene graphene is a single layer of carbon atoms which are tightly bound in a hexagonal honeycomb lattice. graphene is 200 times stronger than steel, one million times thinner than a human hair and word’s most conductive material and such properties attracted researchers and scientists to study more about graphene. graphene has unique properties which unlocks various applications from electronics to optics, sensors, and bio-devices. theorem 4.2. let g ∼= gt,s be the graph of graphene with t rows of benzene rings and s benzene rings in each row. the first downhill zagreb index is given by dwm1(g) =   150 if t = 1,s = 1; 2(t− 1)(4t + 1)2 + 2(t + 34) if t > 1,s = 1; 2(s− 1)(4s + 1)2 + 72 if t = 1,s > 1; 2(st− 1)(2st + 2s + 2t− 1)2 + 2(t + 12) if t > 1,s > 1. int. j. anal. appl. 19 (2) (2021) 221 proof. let g ∼= gt,s be the graph of graphene with t rows of benzene rings and s benzene rings in each row. let v1,v2, ...,v2s+1 be the vertices of the first line l1. there are t + 1 lines in which two lines with 2s + 1 vertices and the others lines have 2s + 2 vertices as in figure 5. we have four cases: figure 5. graphene gt,s. case 1. if t = 1 and s = 1. the graph g become cycle with 6 vertices and by corollary 2.1, dwm1(g) = 6(5)2 = 150. case 2. if t > 1 and s = 1.the number of vertices in this case is 4t + 2. the first line l1 has 3 vertices of downhill degree 3. by symmetry, it can be seen that the line lt+1 has the same vertices as l1. the second line l2 has one vertex of downhill degree 3, one vertex of downhill degree 1 and 2 vertices of downhill degree 4t + 1. by symmetry, the line lt has the same vertices as l2. also, there are lk lines, where k = t − 3, having 2 vertices of downhill degree 1 and 2 vertices of downhill degree 4t + 1. thus, we get dwm1(g) = 2(t− 1)(4t + 1)2 + 2(t + 34). case 3. if t = 1 and s > 1. in this case, there are only two lines and each line has 2s + 1 vertices. the first line l1 has 4 vertices of downhill degree 3, s− 2 vertices of downhill degree 0 and s− 1 vertices of downhill degree 4s + 1. by symmetry, it can be seen that the line l2 has the same vertices as l1. thus, we get dwm1(g) = 2(s− 1)(4s + 1)2 + 72. case 4. if t > 1 and s > 1. the number of vertices in this case is 2st + 2s + 2t. let n = 2st + 2s + 2t. the first line l1 has two vertices of downhill degree 2, two vertices of downhill degree 1, s−2 vertices of downhill degree 0 and s−1 vertices of downhill degree n−1. by symmetry, it can be seen that the line lt+1 has the int. j. anal. appl. 19 (2) (2021) 222 same vertices as l1. the second line l2 has one vertex of downhill degree 2, one vertex of downhill degree 1 and 2s vertices of downhill degree n− 1. by symmetry, the line lt has the same vertices as l2. also, there are k = t− 3 lines, each line have 2 vertices of downhill degree 1 and 2s vertices of downhill degree n− 1. thus, the first downhill zagreb index of a graphene is given by, dwm1(g) = 2(st− 1)(2st + 2s + 2t− 1)2 + 2(t + 12). � theorem 4.3. let g ∼= gt,s be a graph of graphene with t rows of benzene rings and s benzene rings in each row. the second downhill zagreb index is given by dwm2(g) =   150 if t = 1,s = 1; (4t + 1)(8t2 − 8t + 5) + t + 52 if t > 1,s = 1; (4s + 1)(4s2 − 3s + 11) + 54 if t = 1,s > 1; α if t > 1,s > 1, where α = (2st + 2s + 2t− 1) ((s(3t− 2) − (1 + t))(2st + 2s + 2t− 1) + 2(t + 4)) + t + 16. proof. let g ∼= gt,s be the graph of graphene with t rows of benzene rings and s benzene rings in each row. so the graph has 2st + 2s + 2t vertices and 3st + 2s + 2t− 1 edges. we have four cases: case 1. if t = 1 and s = 1. the graph g become cycle with 6 vertices and by corollary 2.1, dwm2(g) = 6(5)2 = 150. case 2. if t > 1 and s = 1. the number of vertices in this case is 4t + 2 and the number of edges is 5t + 1. two dimensional structure of graphene contains the following types of edges e1,1,e3,3,e1,4t+1,e3,4t+1 and e4t+1,4t+1. in figure 6 the edge types, e1,1,e3,3,e1,4t+1,e3,4t+1 and e4t+1,4t+1 are colored in red, blue, green, yellow and black, respectively. int. j. anal. appl. 19 (2) (2021) 223 figure 6. graphene gt,s with t > 1,s = 1. the number of edges in e1,1,e3,3,e1,4t+1,e3,4t+1 and e4t+1,4t+1 in each row is mention in the following table. row |e1,1| |e3,3| |e1,4t+1| |e3,4t+1| |e4t+1,4t+1| 1 0 3 1 2 1 2 1 0 2 0 2 3 1 0 2 0 2 4 1 0 2 0 2 . . . . . . . . . . . . . . . . . . t− 1 1 0 1 1 2 t 0 3 0 1 0 total t− 2 6 2 + 2(t− 3) 4 1 + 2(t− 2) therefore, we have |e1,1| = t− 2 edges, |e3,3| = 6 edges, |e1,4t+1| = 2t− 4 edges, |e3,4t+1| = 4 edges and |e4t+1,4t+1| = 2t− 3 edges. int. j. anal. appl. 19 (2) (2021) 224 dwm2(g) = ∑ uv∈e(g) ddn(u)ddn(v) = |e1,1|(1)(1) + |e3,3|(3)(3) + |e1,4t+1|(1)(4t + 1) + |e3,4t+1|(3)(4t + 1) + |e4t+1,4t+1|(4t + 1)(4t + 1) = t− 2 + 6 × 9 + (2t− 4)(4t + 1) + 4 × 3(4t + 1) + (2t− 3)(4t + 1)(4t + 1) = (4t + 1)(2t− 4 + 12 + 8t2 + 2t− 12t− 3) + t + 52. hence, dwm2(g) = (4t + 1)(8t 2 − 8t + 5) + t + 52. case 3. if t = 1 and s > 1. the number of vertices in this case is 4s + 2 and the number of edges is 5s + 1. two dimensional structure of graphene, we have e3,3,e3,4s+1,e0,4s+1 and e4s+1,4s+1 edges. in figure 7, we have colored e3,3,e3,4s+1,e0,4s+1 and e4s+1,4s+1 edges in red, blue, green and black, respectively. figure 7. graphene gt,s with t = 1,s > 1. therefore, we have |e3,3| = 6 edges, |e3,4s+1| = 4 edges, |e0,4s+1| = 4(s−2) edges and |e4s+1,4s+1| = s−1 edges. dwm2(g) = ∑ uv∈e(g) ddn(u)ddn(v) = |e3,3|(3)(3) + |e3,4s+1|(3)(4s + 1) + |e0,4s+1|(0)(4s + 1) + |e4s+1,4s+1|(4s + 1)(4s + 1) = 6 × 9 + 4 × 3(4s + 1) + 4(s− 2)(0)(4s + 1) + (s− 1)(4s + 1)(4s + 1) = (4s + 1)(12 + 4s2 + s− 4s− 1) + 54. hence, dwm2(g) = (4s + 1)(4s 2 − 3s + 11) + 54. case 4. if t > 1 and s > 1. let n = 2st + 2s + 2t. two dimensional structure of graphene contains the following types of edges e1,1,e2,2,e0,n−1,e1,n−1,e2,n−1 and en−1,n−1. in figure 8, the edges int. j. anal. appl. 19 (2) (2021) 225 e1,1,e2,2,e0,n−1,e1,n−1,e2,n−1 and en−1,n−1 are colored in red, blue, green, yellow, pink and black, respectively. figure 8. graphene gt,s with t > 1,s > 1. the number of edges in e1,1,e2,2,e0,n−1,e1,n−1,e2,n−1 and en−1,n−1 in each row is mention in the following table. row |e1,1| |e2,2| |e0,n−1| |e1,n−1| |e2,n−1| |en−1,n−1| 1 1 2 2(s− 2) 3 2 3s− 2 2 1 0 0 2 0 3s− 1 3 1 0 0 2 0 3s− 1 . . . . . . . . . . . . . . . . . . . . . t− 1 1 0 0 1 1 3s− 1 t 1 2 2(s− 2) 2 1 s− 1 total t 4 4(s− 2) 2t 4 s(3t− 2) − (1 + t) therefore, we have |e1,1| = t edges, |e2,2| = 4 edges, |e0,n−1| = 4(s − 2) edges, |e1,n−1| = 2t edges, |e2,n−1| = 4 edges and |en−1,n−1| = s(3t− 2) − (1 + t)edges. int. j. anal. appl. 19 (2) (2021) 226 dwm2(g) = ∑ uv∈e(g) ddn(u)ddn(v) = |e1,1|(1)(1) + |e2,2|(2)(2) + |e0,n−1|(0)(n− 1) + |e1,n−1|(1)(n− 1) + |e2,n−1|(2)(n− 1) + |en−1,n−1|(n− 1)(n− 1) = t + 4 × 9 + 2t(n− 1) + 4 × 2(n− 1) + s(3t− 2) − (1 + t))(n− 1)(n− 1) = (n− 1)(2t + 8 + (s(3t− 2) − (1 + t))(n− 1)) + t + 16. hence, dwm2(g) = (2st + 2s + 2t− 1)((s(3t− 2) − (1 + t))(2st + 2s + 2t− 1) + 2(t + 4)) + t + 16. � proposition 4.1. let g ∼= gt,s be the graph of graphene with t rows of benzene rings and s benzene rings in each row. the forgotten downhill zagreb index is given by dwf(g) =   750 if t = 1,s = 1; 2(t− 1)(4t + 1)3 + 2(t + 106) if t > 1,s = 1; 2(s− 1)(4s + 1)3 + 216 if t = 1,s > 1; 2(st− 1)(2st + 2s + 2t− 1)3 + 2(t + 24) if t > 1,s > 1. proof. the proof similar to the proof theorem 4.2. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. alwardi, a. alqesmah, r. rangarajan, i.n. cangul, entire zagreb indices of graphs, discrete math. algorithm. appl. 10 (2018), 1850037. [2] j.a. bondy, u.s.r. murty, graph theory, springer, berlin, germany, 2008. [3] j. braun, a. kerber, m. meringer, c. rucker, similarity of molecular descriptors: the equivalence of zagreb indices and walk counts, match commun. math. comput. chem. 54 (2005), 163-176. [4] b. furtula, i. gutman, a forgotten topological index, j. math. chem. 53 (2015), 1184–1190. [5] m. ghorbani, m.a. hosseinzadeh, the third version of zagreb index, discrete math. algorithm. appl. 05 (2013), 1350039. [6] i. gutman, k.c. das, the first zagreb index 30 years after, match commun. math. comput. chem. 50 (2004), 83-92. [7] i. gutman, n. trinajstic, graph theory and molecular orbitals, total π−electron energy of alternant hydrocarbons, chem. phys. lett. 17 (1972), 535-538. [8] j. deering, uphill & downhill domination in graphs and related graph parameters. thesis, east tennessee state university, 2013. int. j. anal. appl. 19 (2) (2021) 227 [9] f. harary, graph theory, addison-wesley, reading mass, 1969. [10] s.t. hedetniemi, t.w. haynes, j.d. jamieson, w.b. jamieson, downhill domination in graphs, discuss. math., graph theory. 34 (2014), 603–612. [11] m.h. khalifeh, h. yousefi-azari, a.r. ashrafi, the first and second zagreb indices of some graph operations, discrete appl. math. 157 (2009), 804-811. [12] j. kok, n.k. sudev, u. mary, on chromatic zagreb indices of certain graphs, discrete math. algorithm. appl. 09 (2017), 1750014. [13] s. nikolić, g. kovačević, a. miličević, n. trinajstić, the zagreb indices 30 years after, croatica chemica acta, 76 (2003), 113-124. [14] a. saleh, a. aqeel, i. n. cangul, on the entire abc index of graphs, proc. jangjeon math. soc. 23 (2020), 39-51. [15] anwar saleh, najat muthana, wafa al-shammakh, hanaa alashwali, monotone chromatic number of graphs, int. j. anal. appl. 18 (2020), 1108-1122. [16] b. zhou, zagreb indices, match commun. math. comput. chem. 52 (2004), 113-118. [17] b. zhou, i. gutman, relations between wiener, hyper-wiener and zagreb indices, chem. phys. lett. 394 (2004), 93-95. [18] b. zhou, i. gutman, further properties of zagreb indices, match commun. math. comput. chem. 54 (2005), 233-239. [19] s. wang, b. wei, multiplicative zagreb indices of cacti, discrete math. algorithm. appl. 8 (2016), 1650040. 1. introduction 2. downhill zagreb indices 3. the downhill zagreb indices of graphs under some binary operations 4. the downhill zagreb indices of honeycomb network and graphene references international journal of analysis and applications volume 16, number 6 (2018), 783-792 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-783 analytic functions related with mocanu class akhter rasheed1,∗, saqib hussain2, muhammad asad zaighum1 and zahid shareef 3 1department of mathematics and statistics, riphah international university, islamabad, pakistan 2department of mathematics, comsats university islamabad, abbottabad campus, pakistan 3division of engineering, higher colleges of technology, p.o. box 4114, fujairah, uae ∗corresponding author: akhter@ciit.net.pk abstract. in this article, we define a new class of analytic functions. this class generalizes the mocanu class. we obtain relationships of this class with other subclasses of analytic functions and derived many interesting results. 1. introduction let a denote the class of functions f analytic in ∆ = {z ∈ c : |z| < 1}, normalized by f (0) = 0 and f ′ (0) = 1. so each f ∈a has series representation of form f (z) = z + ∞∑ n=2 anz n. (1.1) a function f ∈a is in class s if and only if f (z1) = f (z2) implies z1 = z2, for all z1, z2 ∈ ∆. an analytic function f is subordinate to an analytic function g (written as f ≺ g) if and only if there exists an analytic function w with w (0) = 0 and |w (z)| < 1 for z ∈ ∆ such that f (z) = g (w (z)) . received 2018-01-24; accepted 2018-03-14; published 2018-11-02. 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. convex functions; strongly starlike functions; subordination. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 783 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-783 int. j. anal. appl. 16 (6) (2018) 784 for 0 ≤ α ≤ 1, mocanu [15] introduced the class mα of functions f ∈ a such that f(z)f ′ (z) z 6= 0 for all z ∈ ∆ and re { (1 −α) zf′ (z) f (z) + α (zf′ (z)) ′ f′ (z) } > 0, z ∈ ∆. (1.2) mocanu defined the class mα geometrically as a class of functions that maps the circle centered at the origin on α−convex arcs and derived the condition (1.2). the class mα was extensively studied in literature by several authors, for instance, see [4–6, 18–21]. for particular values of α, we obtain a number of interesting classes of analytic functions having nice geometry, for instance m0 = s∗ and m1 = c are well known classes of starlike and convex univalent functions introduced by alexander [1]. by s∗ (δ) and c (δ) , 0 ≤ δ < 1, we mean the subclasses of starlike and convex function of order δ given by (1.3) and (1.4) respectively. re { zf′ (z) f (z) } > δ, z ∈ ∆, (1.3) and re { 1 + zf′′ (z) f′ (z) } > δ, z ∈ ∆. (1.4) we denote the classes of uniformly starlike and uniformly convex functions by ust and ucv, see [7,13,14]. a function f ∈s is uniformly starlike if f maps every circular arc γ contained in ∆ with center ζ ∈ ∆ onto a starlike arc with respect to f (ζ). a function f ∈ s is uniformly convex if f maps every circular arc γ contained in ∆ with center ζ ∈ ∆ onto a convex arc. in 1999, kanas and wisnoiska [9] introduced the class k −ucv, (k ≥ 0) of k-uniformly convex functions as: f ∈ k −ucv ⇐⇒ f ∈a and re { 1 + zf′′ (z) f′ (z) } > k ∣∣∣∣zf′′ (z)f′ (z) ∣∣∣∣ , z ∈ ∆, (1.5) where its geometric definition and connections with the conic domains were considered. it is worth mentioning that 1 −ucv = ucv. in recent years many authors investigated interesting properties of these classes. for some details see [2, 8–12, 26, 30] and references cited there in. let ss∗ (λ) denote the class of strongly starlike functions of order λ as: ss∗ (λ) = { f ∈a : ∣∣∣∣arg zf′ (z)f (z) ∣∣∣∣ < λπ2 } , (1.6) where λ ∈ (0, 1). this class of functions was introduced and discussed by [3, 27]. in our current investigation, we extend the work of j. sokol [25] and introduced a new class of analytic function as: definition 1.1. let f ∈a and k ≥ 0, 0 ≤ α ≤ 1. then f ∈ k −umα if and only if re { (1 −α) zf′ (z) f (z) + α (zf′ (z)) ′ f′ (z) } > k ∣∣∣∣∣(zf ′ (z)) ′ f′ (z) − 1 ∣∣∣∣∣ , z ∈ ∆. (1.7) int. j. anal. appl. 16 (6) (2018) 785 for special values of parameters k and α, we obtain a number of known classes of analytic functions. here we present few of them. (i) 0 −umα = mα, [15]. (ii) 0 −um0 = s∗, [1]. (iii) 1 −um1 = ucv, [7]. (iv) k −um1 = k −ucv, [9]. lemma 1.1. let u = u1 + iu2 and v = v1 + iv2 and let ψ (u,v) : c2 ×∆ → c be a complex-valued function satisfying the conditions: (i) ψ (u,v) is continuous in a domain d ⊂ c2, (ii) (1, 0) ∈ d and ψ (1, 0) > 0. (iii) re ψ (iu2,v1) ≤ 0 whenever (iu2,v1) ∈ d and v1 ≤−12 ( 1 + u22 ) . if h (z) = 1 + c1z + c2z 2 + · · · is a function that is analytic in ∆ such that (h(z),zh′(z)) ∈ d and re ψ (h(z),zh′(z)) > 0 for z ∈ ∆, then re h(z) > 0. this result is due to miller [16]. lemma 1.2. let β > 0 and 0 ≤ γ < 1. let p (z) + βzp′ (z) p (z) ≺ 1 + (1 − 2γ) 1 −z . then p (z) ≺ 1 + (1 − 2δ) 1 −z , where δ = (2γ −β) + √ (2γ −β)2 + 8β 4 . (1.8) lemma 1.3. [17] if −1 ≤ b < a ≤ 1, λ > 0 and the complex number γ satisfies re{γ}≥−λ (1 −a) / (1 −b) , then the differential equation q (z) + zq′ (z) λq (z) + γ = 1 + az 1 + bz , z ∈ ∆, has a univalent solution in e given by q (z) =   zλ+γ(1+bz)λ(a−b)/b λ z∫ 0 tλ+γ−1(1+bt)λ(a−b)/bdt − γ λ , b 6= 0, zλ+γeλaz λ z∫ 0 tλ+γ−1eλatdt − γ λ , b = 0. int. j. anal. appl. 16 (6) (2018) 786 if h (z) = 1 + c1z + c2z 2 + . . . is analytic in ∆ and satisfies h (z) + zh′ (z) λh (z) + γ ≺ 1 + az 1 + bz , z ∈ ∆, then h (z) ≺ q (z) ≺ 1 + az 1 + bz , and q (z) is the best dominant. lemma 1.4. [29] let ε be a positive measure on [0, 1] . let g be a complex-valued function defined on ∆×[0, 1] such that g (., t) is analytic in ∆ for each t ∈ [0, 1] and g (z, .) is ε-integrable on [0, 1] for all z ∈ ∆. in addition, suppose that re g (z,t) > 0, g (−r,t) is real and re{1/g (z,t)} ≥ 1/g (−r,t) for |z| ≤ r < 1 and t ∈ [0, 1] . if g (z) = 1∫ 0 g (z,t) dε (t) , then re{1/g (z)}≥ 1/g (−r) . lemma 1.5. [28, chapter 14] let a, b and c 6= 0,−1,−2 . . . be complex numbers. then, for re c > re b > 0, (i) 2f1 (a,b,c; z) = γ (c) γ (c− b) γ (b) 1∫ 0 tb−1 (1 − t)c−b−1 (1 − tz)−a dt. (ii) 2f1 (a,b,c; z) = 2f1 (b,a,c; z) . (iii) 2f1 (a,b,c; z) = (1 −z) −a 2f1 ( a,c− b,c; z z − 1 ) . lemma 1.6. let p be analytic in ∆ of the form p (z) = 1 + ∞∑ n=m cnz n, cm 6= 0, with p (z) 6= 0 in ∆. if there exists a point z0, |z0| < 1 such that |arg p (z)| < πϕ2 (|z| < |z0|) and |arg p (z0)| = πϕ 2 for some ϕ > 0, then we have z0p ′(z0) p(z0) = ilϕ, where  l ≥ m 2 ( a + 1 a ) , when arg p (z0) = πϕ 2 , l ≤−m 2 ( a + 1 a ) , when arg p (z0) = −πϕ2 , where (p (z0)) 1/ϕ = ±ia (a > 0) . this result is generalization of the nunokawa’s lemma [23]. 2. results and discussion theorem 2.1. let f ∈ k −umα. then f ∈s∗ (δ) , where δ = (2γ −β) + √ (2γ −β)2 + 8β 4 , (2.1) with β = α+k 1+k and γ = k 1+k . int. j. anal. appl. 16 (6) (2018) 787 proof. let zf′ (z) f (z) = p (z) , where p is analytic in · with p (0) = 1. we obtain re { (1 −α) p (z) + α ( p (z) + zp′ (z) p (z) )} > k ∣∣∣∣p (z) + zp′ (z)p (z) − 1 ∣∣∣∣ = k ∣∣∣∣1 −p (z) − zp′ (z)p (z) ∣∣∣∣ re { (1 −α) p (z) + α ( p (z) + zp′ (z) p (z) )} > re k { (1 −p (z)) − zp′ (z) p (z) } , hence re { p (z) + zp′ (z) 1 β p (z) } > γ, (2.2) where β = α+k 1+k and γ = k 1+k . the above relation can be written in the following briot-bouquet differential subordination p (z) + zp′ (z) 1 β p (z) ≺ 1 + (1 − 2γ) z 1 −z . now using lemma 2, we have p ∈s∗ (δ) , where δ is given by (2.1) . � special cases (i) for α = 0, k = 1,we have β = γ = 1 2 . then for f ∈ 1 −um0, we have f ∈s∗ (δ) , where δ ' 0.64. (ii) for α = 0, we have β = k 1+k and γ = k 1+k . then δ1 = k + √ 9k2 + 8k 4(k + 1) . in other words for f ∈ k −um0, we have f ∈s∗ (δ1) . (iii) if α = 1, β = 1, γ = k 1+k , then δ2 = (k − 1) + √ (k − 1)2 + 8 (k + 1)2 4(k + 1) . in other words f ∈ k −ucv implies f ∈s∗ (δ2) . theorem 2.2. let f ∈ k −umα and of the form f (z) = z + ∞∑ n=m+1 anz n, am+1 6= 0. then f is strongly starlike of order β0, where β0 = min β∈(0,1)   1 − 2aβ cos βπ 2 + ( k2 − 1 )( aβ cos βπ 2 )2 +( βm(a2+1) 2a + aβ sin βπ 2 )2   . (2.3) int. j. anal. appl. 16 (6) (2018) 788 proof. let zf′ (z) f (z) = p (z) . then p is the form p (z) = 1 + ∞∑ n=m cnz n, cm 6= 0. now using the definition of the class k −umα, we have re { (1 −α) p (z) + α ( p (z) + zp′ (z) p (z) )} > k ∣∣∣∣p (z) + zp′ (z)p (z) − 1 ∣∣∣∣ . (2.4) if there exists a point z0 ∈ ∆ such that |arg p (z)| < βπ 2 , |z| < |z0| and |arg p (z0)| = βπ 2 , then from lemma 1.6, we have z0p ′(z0) p(z0) = ilβ, where (p (z0)) 1/β = ±ia (a > 0) and l :   l ≥ m 2 ( a + 1 a ) , when arg p (z0) = βπ 2 , l ≤−m 2 ( a + 1 a ) , when arg p (z0) = −βπ2 . (2.5) for the case arg p (z0) = βπ 2 , we obtain re { p (z0) + αz0p ′ (z0) p (z0) } = re { (ia) β + iαlβ } = aβ cos βπ 2 . (2.6) also, we have k ∣∣∣∣p (z0) − 1 + z0p′ (z0)p (z0) ∣∣∣∣ = k ∣∣∣(ia)β + ilβ − 1∣∣∣ = k ∣∣∣∣aβ cos βπ2 − 1 + i ( aβ sin βπ 2 + lβ )∣∣∣∣ = k √( aβ cos βπ 2 − 1 )2 + ( aβ sin βπ 2 + lβ )2 . (2.7) then from l ≥ m ( a2 + 1 ) /2a for β ≥ β0, we have 0 ≤ 1 − 2aβ cos β π 2 + ( k2 − 1 )( aβ cos β π 2 )2 + ( βm ( a2 + 1 ) 2a + aβ sin β π 2 )2 . (2.8) therefore 0 ≤ 1 − 2aβ0 cos β0 π 2 + ( k2 − 1 )( aβ0 cos β0 π 2 )2 + ( β0m ( a2 + 1 ) 2a + aβ0 sin β0 π 2 )2 . (2.9) by using (2.6) and (2.7) , we have re { (1 −α) p (zo) + α { p (zo) + zp′ (zo) p (zo) }} < k ∣∣∣∣p (z0) + zp′ (z0)p (z0) − 1 ∣∣∣∣ . which is contradiction, therefore |arg p (z)| < βπ 2 for |z| < 1. int. j. anal. appl. 16 (6) (2018) 789 similarly we can prove the case arg p (z0) = −βπ2 by using the same method as the above we will get a contradiction. this proves that f is strongly starlike of order β0. � corollary 2.1. let f ∈ucv and of the form f (z) = z + ∞∑ n=m+1 anz n, am+1 6= 0. then f is strongly starlike of order β0, where β0 = min β∈(0,1)  1 − 2aβ cos βπ2 + ( βm ( a2 + 1 ) 2a + aβ cos β π 2 )2  . this result is due to nunokawa and sokol [24]. theorem 2.3. if f ∈ k −umα, then zf′ (z) f (z) ≺ q (z) = 1 g (z) , (2.10) where g (z) = [ 2f1 ( 1 β (1 −γ) , 1; 1 β + 1; z z−1 )] with β = k+α 1+k and γ = k 1+k . proof. let zf′ (z) f (z) = p (z) , where p is analytic in ∆ with p (0) = 1. now using (1.7), we obtain re { (1 −α) p (z) + α ( p (z) + zp′ (z) p (z) )} > k ∣∣∣∣p (z) + zp′ (z)p (z) − 1 ∣∣∣∣ = k ∣∣∣∣1 −p (z) − zp′ (z)p (z) ∣∣∣∣ re { (1 −α) p (z) + α ( p (z) + zp′ (z) p (z) )} > re k { (1 −p (z)) − zp′ (z) p (z) } . this implies that re { p (z) + zp′ (z) 1 β p (z) } > γ, where β = k+α 1+k and γ = k 1+k . the above relation can be written in the following briot-bouquet differential subordination p (z) + zp′ (z) 1 β p (z) ≺ 1 + (1 − 2γ) z 1 −z . (2.11) using lemma 1.3 for λ = 1 β and γ = 0, we have p (z) ≺ q (z) = 1 g (z) = 1 1/β 1∫ 0 t1/β−1 ( 1−tz 1−z )−2(γ−1)/β dt = { 2f1 ( 2 β (1 −γ) , 1; 1 β + 1; z z − 1 )}−1 . which is the required result. � int. j. anal. appl. 16 (6) (2018) 790 theorem 2.4. if f ∈ k −umα. then re zf′ (z) f (z) > 1 g (−1) = γ0 = { 2f1 ( 2 β (1 −γ) , 1; 1 β + 1; 1 2 )}−1 . in other words k −umα ⊂s∗ (γ0) , where γ0 = { 2f1 ( 2 β (1 −γ) , 1; 1 β + 1; 1 2 )}−1 . (2.12) proof. to prove k −umα ⊂ s∗ (γ0) , we show that inf |z|<1 {re q (z)} = q (−1) . now for a = 2 β (1 −γ) , b = 1 β , c = 1 β + 1, we have g (z) = (1 + bz) a 1∫ 0 tb−1 (1 + btz) −a dt. = γ (b) γ (c) 2f1 ( 1,a,c; z z − 1 ) . (2.13) to prove that inf |z|<1 {re q (z)} = q (−1) , we need to show that re{1/g (z)}≥ 1/g (−1) . now by using lemma 1.4, (2.13) it can easily follows that g (z) = 1∫ 0 g (z,t) dε (t) , where g (z,t) = 1 −z 1 − (1 − t) z , (0 ≤ t ≤ 1) , dε (t) = γ (b) γ (a) γ (c−a) ta−1 (1 − t)c−a−1 dt, which is a positive measure on [0, 1] . it is clear that re g (z,t) > 0 and g (−r,t) is real for 0 ≤ |z| ≤ r < 1 and t ∈ [0, 1] . also re { 1 g (z,t) } = re { 1 − (1 − t) z 1 −z } ≥ 1 + (1 − t) r 1 + r = 1 g (−r,t) for |z| ≤ r < 1. therefore using lemma 1.4, we have re{1/g (z)}≥ 1/g (−r) . now letting r → 1−, it follows re{1/g (z)}≥ 1/g (−1) . therefore k −umα ⊂s∗ (γ0) . � int. j. anal. appl. 16 (6) (2018) 791 references [1] j. w. alexander, functions which map the interior of the unit circle upon simple regions, ann. math., 17(1915), 12–22. [2] m. arif, a. ali and j. muhammad, some sufficient conditions for alpha convex functions of order beta, vfast trans. math., 1(2)(2013), 8-12. [3] d. a. brannan and w. e. kirwan, on some classes of bounded univalent functions, j. london math. soc., 1(2)(1969), 431–443. [4] j. dziok, characterizations of analytic functions associated with functions of bounded variation, ann. polon. math., 109(2013), 199–207. [5] j. dziok, classes of functions associated with bounded mocanu variation, j. inequal. appl., 2013, article id 349. [6] j. dziok, generalizations of multivalent mocanu functions, appl. math. computation., 269(2015), 965–971. [7] a. w. goodman, on uniformly convex functions, ann. polon. math., 56(1991), 87–92. [8] s. kanas and a. wísniowska, conic regions and k-uniform convexity ii, folia sci. univ. tech. resov., 22(1998), 65–78. [9] s. kanas and a. wísniowska, conic regions and k-uniform convexity, j. comput. appl. math., 105(1999), 327–336. [10] s. kanas and h. m. srivastava, linear operators associated with k-uniformly convex functions, integral transform. spec.funct., 9(2)(2000), 121–132. [11] a. lecko and a. wísniowska, geometric properties of subclasses of starlike functions, j. comp. appl. math., 155(2003), 383–387. [12] x. li, d. ding, l. xu, c. qin and s. hu, j. funt. spac, certain subclasses of multivalent functions defined by higherorder derivative., 2017, article id 5739196. [13] w. ma and d. minda, uniformly convex functions, ann. polon. math., 2(57)(1992), 165–175. [14] w. ma and d. minda, uniformly convex functions ii, ann. polon. math., 3(58)(1993), 275–285. [15] p. t. mocanu, une propriété de convexité g énéralisée dans la théorie de la représentation conforme. mathematica (cluj, 1929) 11(34)(1969), 127-133. [16] s. s. miller, differential inequalities and carathéodory functions, bull. amer. math. soc., 81(1975), 79 81. [17] s. s. miller and p. t. mocanu, univalent solutions of briot-bouquet differential subordination, j. differential eqns., 56(1985), 297-309. [18] k. i. noor and w. ul-haq, on some implication type results involving generalized bounded mocanu variations, comput. math. appl., 63(10)(2012), 1456-1461. [19] k. i. noor and s. hussain, on certain analytic functions associated with ruscheweyh derivatives and bounded mocanu variation, j. math. anal. appl., 340(2008), 1145–1152. [20] k. i. noor and a. muhammad, on analytic functions with generalized bounded mocanu variation, appl. math. comput., 196(2) (2008), 802-811. [21] k. i. noor and s. n. malik, on generalized bounded mocanu variation associated with conic domain, math. comput. model., 55(2012), 844-852. [22] m. nunokawa and j. sokol, remarks on some starlike functions, j. inequal. appl., 593(2013), 1-8. [23] m. nunokawa, on the order of strongly starlikeness of strongly convex functions, proc. japan acad. ser. a., 69(7)(1993), 234-237. [24] m. nunokawa, j. sokó l, on order of strongly starlikeness in the class of uniformly convex functions, math. nachr., (2015), 1-6.. [25] j. sokó l and m. nunokawa, on some class of convex functions, c. r. acad. sci. paris, ser. i., 353(2015), 427-431. int. j. anal. appl. 16 (6) (2018) 792 [26] j. soko l and a.wísniowska, on some classes of starlike functions related with parabola, folia sci. univ. tech. resov., 28 (1993), 35-42. [27] j. stankiewicz, quelques problémes extrémaux dans les classes des fonctions α-angulairement étoilées, ann. univ. mariae curie–sk lodowska, sect. a., 20(1966), 59-75. [28] e. t. whittaker and g. n. watson, a course of modern analysis, 4th ed. cambridge univ. press., 1958. [29] d. r. wilken and j. feng, a remark on convex and starlike functions, j. london math. soc., 21(1980)., 287-290. [30] a. wísniowska wajnryb, some extremal bounds for subclasses of univalent functions, appl. math. comput,. 215(2009), 2634-2641. 1. introduction 2. results and discussion references int. j. anal. appl. (2022), 20:41 picture n-sets and applications in semigroups anusorn simuen1, ronnason chinram1,∗, winita yonthanthum1, aiyared iampan2 1division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 2department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand ∗corresponding author: ronnason.c@psu.ac.th abstract. in this paper, we study picture n-structures and apply it to semigroups. moreover, we define picture n-ideals in semigroups and investigate several properties of these ideals in semigroups. 1. introduction fuzzy sets were introduced by zadeh [8] in 1965 as an extension of the classical notion of sets. fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the unit interval [0,1]. next, fuzzy sets were generalized to other concepts. atanassov [1] generalized fuzzy sets into intuitionistic fuzzy sets in 1986 by considering for each element of the sets is a degree of membership and a degree of nonmembership. the notion of the classical sets, fuzzy sets, and intuitionistic fuzzy sets were extended into neutrosophic sets which is the tool for dealing with incomplete, inconsistent, and indeterminate information by smaradache [6]. in 2009, jun et al. [4] gave the concept of a negative-valued function and constructed n-structures. later, smaradache et al. [7] introduced the notion of neutrosophic n-structures and applied it to semigroups in 2017. next, elavarasan et al. [3] introduced neutrosophic n-ideals in semigroups and investigated its several properties. in 2014, the concept of picture fuzzy set was first introduced by cuong [2] in 2014, which is a generalization of the concept of fuzzy sets and intuitionistic fuzzy sets. this concept is based on adequate in situations when we face human received: jul. 4, 2022. 2010 mathematics subject classification. 03e72. key words and phrases. n-sets; picture n-structures; picture n-ideals. https://doi.org/10.28924/2291-8639-20-2022-41 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-41 2 int. j. anal. appl. (2022), 20:41 opinions involving more answers of types: yes, abstain, no, refusal. picture fuzzy sets were focused on the degree of positive memberships, the degree of neutral memberships, and the degree of negative memberships. picture fuzzy sets are generalizations of fuzzy sets and intuitionistic fuzzy sets. the concept of picture fuzzy sets will differ from the concept of neutrosophic sets. the applications of picture fuzzy sets in semigroups were studied by yiarayong [5] in 2020. in this paper, we study picture n-structures and apply it to semigroup. moreover, we define picture n-ideals in semigroups and investigate several properties of these ideals in semigroups. 2. notations in this section, we introduce the concept of picture n-structures of sets. definition 2.1. a picture n-structure over a set s defined to be the structure: pn := { x tn(x), in(x),fn(x) | x ∈ s } where tn : s → [−1,0] is called the negative positive membership function, in : s → [−1,0] is called the negative neutral membership function, and fn : s → [−1,0] is called the negative false membership function with the condition −1 ≤ tn(x)+ in(x)+ fn(x) ≤ 0 for all x ∈ s. we denote this structure by pn = s (tn,in,fn) . definition 2.2. let pn = s(tn,in,fn) and pm = s (tm,im,fm) be picture n-structures over a set s. then (1) pn is called a picture n-substructure of pm over s, denote by pn ⊆ pm, if (a) tn(x)≥ tm(x), (b) in(x)≤ im(x), (c) fn(x)≤ fm(x) for all x ∈ s. (2) the union of pn and pm is defined to be a picture n-structure over s pn∪m =(s;tn∪m, in∪m,fn∪m) where tn∪m(x)=min{tn(x),tm(x)}, in∪m(x)=max{in(x), im(x)}, fn∪m(x)=max{fn(x),fm(x)} for all x ∈ s. (3) the intersection of pn and pm is defined to be a picture n-structure over s pn∩m =(s;tn∩m, in∩m,fn∩m) where tn∩m(x)=max{tn(x),tm(x)}, in∩m(x)=min{in(x), im(x)}, int. j. anal. appl. (2022), 20:41 3 fn∩m(x)=min{fn(x),fm(x)} for all x ∈ s. definition 2.3. for a subset a of a set s, consider the picture n-structure χa(pn)= s (χa(t)n,χa(i)n,χa(f)n) where χa(t)n(x)=  −1 if x ∈ a, 0 otherwise, χa(i)n(x)=  0 if x ∈ a,−1 otherwise, χa(f)n(x)=  0 if x ∈ a,−1 otherwise for all x ∈ s. the picture n-structure χa(pn) is called the characteristic picture n-structure of a over s. definition 2.4. let pn be a picture n-structure over s and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. consider the following sets: tαn = {x ∈ s | tn(x)≤ α}, i β n = {x ∈ s | in(x)≥ β}, f γ n = {x ∈ s | fn(x)≥ γ}. the set pn(α,β,γ) := {x ∈ s | tn(x)≤ α,in(x)≥ β,fn(x)≥ γ} is called an (α,β,γ)-level set of pn. note that pn(α,β,γ)= tαn ∩ i β n ∩fγ n . 3. applications of picture n-sets in semigroups the picture n-product of pn and pm is defined to be a picture n-structure over a semigroup s pn �pm := { x tn◦m(x), in◦m(x),fn◦m(x) | x ∈ s } where tn◦m(x)=  infx=abmax{tn(a),tm(b)} for x = ab for some a,b ∈ s, 0 otherwise, in◦m(x)=  supx=abmin{in(a), im(b)} for x = ab for some a,b ∈ s, 0 otherwise, 4 int. j. anal. appl. (2022), 20:41 fn◦m(x)=  supx=abmin{fn(a),fm(b)} for x = ab for some a,b ∈ s, 0 otherwise. we denote the picture n-product of pn and pm by pn �pm = s(tn◦m,in◦m,fn◦m). for x ∈ s, the element x (tn◦m,in◦m,fn◦m) is simply denoted by (pn �pm)(x)= (tn◦m(x), in◦m(x),fn◦m(x)) for the sake of convenience. definition 3.1. a picture n-structure pn over a semigroup s is called a picture n-subsemigroup of s if it satisfies: (1) tn(xy)≤max{tn(x),tn(y)}, (2) in(xy)≥min{in(x), in(y)}, (3) fn(xy)≥min{fn(x),fn(y)} for all x,y ∈ s. definition 3.2. a picture n-structure pn over a semigroup s is called a picture n-left ideal of s if it satisfies: (1) tn(xy)≤ tn(y), (2) in(xy)≥ in(y), (3) fn(xy)≥ fn(y) for all x,y ∈ s. definition 3.3. a picture n-structure pn over a semigroup s is called a picture n-right ideal of s if it satisfies: (1) tn(xy)≤ tn(x), (2) in(xy)≥ in(x), (3) fn(xy)≥ fn(x) for all x,y ∈ s. we called pn a picture n-ideal if it is both a picture n-left ideal and a picture n-right ideal of s. theorem 3.1. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1 ≤ α + β + γ ≤ 0. if pn is a picture n-left ideal of s, then (α,β,γ)-level set of pn is a picture n-left ideal of s whenever it is nonempty. proof. assume that pn(α,β,γ) 6= ∅ for α,β,γ ∈ [−1,0] with −1 ≤ α + β + γ ≤ 0. let pn be a picture n-left ideal of s, and let x,y ∈ pn(α,β,γ). then tn(xy)≤ tn(y)≤ α, in(xy)≥ in(y)≥ β, and fn(xy) ≥ fn(y) ≥ γ which imply xy ∈ pn(α,β,γ). hence pn(α,β,γ) is a picture n-left ideal of s. � int. j. anal. appl. (2022), 20:41 5 theorem 3.2. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1 ≤ α + β + γ ≤ 0. if pn is a picture n-right ideal of s, then (α,β,γ)-level set of pn is a picture n-right ideal of s whenever it is nonempty. proof. it is similar to theorem 3.1. � theorem 3.3. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1 ≤ α+β +γ ≤ 0. if pn is a picture n-ideal of s, then (α,β,γ)-level set of pn is a picture n-ideal of s whenever it is nonempty. proof. it follows from theorem 3.1 and 3.2. � theorem 3.4. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. if tαn , i β n , and fγ n are left ideals of s, then pn is a picture n-left ideal of s whenever it is nonempty. proof. let a,b ∈ s such that tn(ab) > tn(b). then tn(ab) > tα ≥ tn(b) for some tα ∈ [−1,0). thus b ∈ ttα n (b) but ab /∈ ttα n (b), this is a contradiction. so tn(ab)≤ tn(b). similarly way we can get in(ab)≥ in(b) and fn(ab)≥ fn(b). therefore pn is a picture n-left ideal of s. � theorem 3.5. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1≤ α+β +γ ≤ 0. if tαn , i β n , and fγ n are right ideals of s, then pn is a picture n-right ideal of s whenever it is nonempty. proof. it is similar to theorem 3.4. � theorem 3.6. let pn be a picture n-structure over a semigroup s and let α,β,γ ∈ [−1,0] be such that −1≤ α+β+γ ≤ 0. if tαn , i β n , and fγ n are ideals of s, then pn is a picture n-ideal of s whenever it is nonempty. proof. it follows from theorem 3.4 and 3.5. � theorem 3.7. let s be a semigroup. then intersection of two picture n-left ideals of s is also a picture n-left ideal of s. proof. let pn := s (tn,in,fn) and pm := s (tm,im,fm) be picture n-left ideals of s. then tn∩m(xy)=max{tn(xy),tm(xy)}≤max{tn(y),tm(y)}= tn∩m(y), in∩m(xy)=min{in(xy), im(xy)}≥min{in(y), im(y)}= in∩m(y), fn∩m(xy)=min{fn(xy),fm(xy)}≥min{fn(y),fm(y)}= fn∩m(y) for all x,y ∈ s. then pn∩m is a picture n-left ideal of s. � 6 int. j. anal. appl. (2022), 20:41 theorem 3.8. let s be a semigroup. then intersection of two picture n-right ideals of s is also a picture n-right ideal of s. proof. it is similar to theorem 3.7. � theorem 3.9. let s be a semigroup. then intersection of two picture n-ideals of s is also a picture n-ideal of s. proof. it follows from theorem 3.7 and 3.8. � theorem 3.10. for any nonempty subset a of a semigroup s, the following conditions are equivalent: (1) a is a left ideal of s. (2) the characteristic picture n-structure χa(pn) over s is a picture n-left ideal of s. proof. assume that a is a left ideal of s. let x,y ∈ s. if y /∈ a, then χa(t)n(xy)≤ 0= χa(t)n(y), χa(i)n(xy)≥−1= χa(i)n(y), χa(f)n(xy)≥−1= χa(f)n(y). otherwise y ∈ a. then xy ∈ a, we have χa(t)n(xy)=−1= χa(t)n(y), χa(i)n(xy)=0= χa(i)n(y), χa(f)n(xy)=0= χa(f)n(y). therefore χa(pn) is a picture n-left ideal of s. conversely, suppose that χa(pn) is a picture n-left ideal of s. let y ∈ a and x ∈ s. then χa(t)n(xy)≤χa(t)n(y)=−1, χa(i)n(xy)≥χa(i)n(y)=0, χa(f)n(xy)≥χa(f)n(y)=0. hence xy ∈ a. therefore a is a left ideal of s. � theorem 3.11. for any nonempty subset a of a semigroup s, the following conditions are equivalent: (1) a is a right ideal of s. (2) the characteristic picture n-structure χa(pn) over s is a picture n-right ideal of s. proof. it is similar to theorem 3.10. � theorem 3.12. for any nonempty subset a of a semigroup s, the following conditions are equivalent: int. j. anal. appl. (2022), 20:41 7 (1) a is an ideal of s. (2) the characteristic picture n-structure χa(pn) over s is a picture n-ideal of s. proof. it follows from theorem 3.11 and 3.12. � theorem 3.13. let χa(pn) and χb(pn) be characteristic picture n-structures over a semigroup s for subsets a and b of s. then (1) χa(pn)∩χb(pn)= χa∩b(pn). (2) χa(pn)�χb(pn)= χab(pn). proof. (1) let s ∈ s. if s ∈ a∩b, then s ∈ a and s ∈ b. thus (χa(t)n ∩χb(t)n)(s)=max{χa(t)n(s),χb(t)n(s)}=−1= χa∩b(t)n(s), (χa(i)n ∩χb(i)n)(s)=min{χa(i)n(s),χb(i)n(s)}=0= χa∩b(i)n(s), (χa(f)n ∩χb(f)n)(s)=min{χa(f)n(s),χb(f)n(s)}=0= χa∩b(f)n(s). hence χa(pn)∩χb(pn)= χa∩b(pn). if s /∈ a∩b, then s /∈ a or s /∈ b. thus (χa(t)n ∩χb(t)n)(s)=max{χa(t)n(s),χb(t)n(s)}=0= χa∩b(t)n(s), (χa(i)n ∩χb(i)n)(s)=min{χa(i)n(s),χb(i)n(s)}=−1= χa∩b(i)n(s), (χa(f)n ∩χb(f)n)(s)=min{χa(f)n(s),χb(f)n(s)}=−1= χa∩b(f)n(s). hence χa(pn)∩χb(pn)= χa∩b(pn). (2) let x ∈ s. if x /∈ ab, then (χa(t)n ◦χb(t)n)(x)=0= χab(t)n(x), (χa(i)n ◦χb(i)n)(x)=0= χab(i)n(x), (χa(f)n ◦χb(f)n)(x)=0= χab(f)n(x). if x ∈ ab, then x = ab for some a ∈ a and b ∈ b. we have (χa(t)n ◦χb(t)n)(x)= inf x=ab max{χa(t)n(a),χb(t)n(b)} ≤max{χa(t)n(a),χb(t)n(b)} =−1 = χab(t)n(x), 8 int. j. anal. appl. (2022), 20:41 (χa(i)n ◦χb(i)n)(x)= sup x=ab min{χa(i)n(a),χb(i)n(b)} ≥min{χa(i)n(a),χb(i)n(b)} =0 = χab(i)n(x), (χa(f)n ◦χb(f)n)(x)= sup x=ab min{χa(f)n(a),χb(f)n(b)} ≥min{χa(f)n(a),χb(f)n(b)} =0 = χab(f)n(x). therefore χa(pn)�χb(pn)= χab(pn). � theorem 3.14. let pm be a picture n-structure over a semigroup s. then pm is a picture n-left ideal of s if and only if pn �pm ⊆ pm for any picture n-structure pn over s. proof. assume that pm is a picture n-left ideal of s and let s,t,u ∈ s. if s = tu, then we have (i) tm(s) = tm(tu) ≤ tm(u) ≤ max{tm(t),tm(u)} which implies tm(s) ≤ tn◦m(s). otherwise s 6= tu. then tm(s)≤ 0= tn◦m(s). (ii) im(s) = im(tu) ≥ im(u) ≥ min{im(t), im(u)} which implies im(s) ≥ in◦m(s). otherwise s 6= tu. then im(s)≥−1= in◦m(s). (iii) fm(s)= fm(tu)≥ fm(u)≥min{fm(t),fm(u)} which implies fm(s)≥ fn◦m(s). otherwise s 6= tu. then fm(s)≥−1= fn◦m(s). conversely, assume that pm is a picture n-structure over s such that pn � pm ⊆ pm for any picture n-structure pn over s. let x,y ∈ s. if a = xy, then tm(xy)= tm(a) ≤ (χx(t)n ◦tm)(a) = inf a=st max{χx(t)n(s),tm(t)} ≤max{χx(t)n(x),tm(y)} = tm(y), im(xy)= im(a) ≥ (χx(i)n ◦ im)(a) = sup a=st min{χx(i)n(s), im(t)} int. j. anal. appl. (2022), 20:41 9 ≥min{χx(i)n(x), im(y)} = im(y), fm(xy)= fm(a) ≥ (χx(f)n ◦fm)(a) = sup a=st min{χx(f)n(s),fm(t)} ≥max{χx(f)n(x),fm(y)} = fm(y). therefore pm is a picture n-left ideal of s. � theorem 3.15. let pm be a picture n-structure over a semigroup s. then pm is a picture n-right ideal of s if and only if pm �pn ⊆ pm for any picture n-structure pn over s. proof. it is similar to theorem 3.14. � theorem 3.16. let pm be a picture n-structure over s. then pm is a picture n-ideal of s if and only if pm �pn ⊆ pm for any picture n-structure pn over s. proof. it follows from theorem 3.14 and 3.15. � theorem 3.17. let pm and pn be picture n-structure over s. if pm is a picture n-left ideal of s, then so is pm �pn. proof. assume that pm is a picture n-left ideal of s, and let x,y ∈ s. if there exist a,b ∈ s such that y = ab, then xy = x(ab)= (xa)b. we have (tn ◦tm)(y)= inf y=ab max{tn(a),tm(b)} ≤ inf xy=(xa)b max{tn(xa),tm(b)} = inf xy=cb max{tn(c),tm(b)} =(tn ◦tm)(xy), (in ◦ im)(y)= sup y=ab min{in(a), im(b)} ≥ sup xy=(xa)b min{in(xa), im(b)} = sup xy=cb min{in(c), im(b)} =(in ◦ im)(xy), 10 int. j. anal. appl. (2022), 20:41 (fn ◦fm)(y)= sup y=ab min{fn(a),fm(b)} ≥ sup xy=(xa)b min{fn(xa),fm(b)} = sup xy=cb min{fn(c),fm(b)} =(fn ◦fm)(xy). therefore pm �pn is a picture n-left ideal of s. � theorem 3.18. let pm and pn be picture n-structure over a semigroup s. if pm is a picture n-right ideal of s, then so is pm �pn. proof. it is similar to theorem 3.17. � theorem 3.19. let pm and pn be picture n-structure over a semigroup s. if pm is a picture n-ideal of s, then so is pm �pn. proof. it follows from theorem 3.17 and 3.18. � acknowledgment: this work was supported by the psu-tuyf charitable trust fund, prince of songkla university, contract no. 2-2564-01. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] k.t. atanassov, intuitionistic fuzzy sets, vii itkr session, sofia, 20-23 june 1983 (deposed in centr. sci.-techn. library of the bulg. acad. of sci., 1697/84) (in bulgarian). reprinted: int. j. bioautom. 20(s1) (2016), s1–s6. https://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_02.pdf. [2] b.c. cuong, picture fuzzy sets, j. computer sci. cybernetics, 30 (2014), 409–420. https://doi.org/10.15625/ 1813-9663/30/4/5032. [3] b. elavarasan, f. smarandache, y.b. jun, neutrosophic ℵ–ideals in semigroups, neutrosophic sets syst. 28 (2019), 273–280. https://digitalrepository.unm.edu/nss_journal/vol28/iss1/21/. [4] y.b. jun, k.j. lee, s.z. song, n-ideals of bck/bci-algebras, j. chungcheong math. soc. 22 (2009), 417–437. http://www.ccms.or.kr/data/pdfpaper/jcms22_3/22_3_417.pdf. [5] p. yiarayong, semigroup characterized by picture fuzzy sets, int. j. innov. comput. inform. control, 16 (2020), 2121–2130. https://doi.org/10.24507/ijicic.16.06.2121. [6] f. smarandache, a unifying field in logics: neutrosophic logic. neutrosophy, neutrosophic set, probability, american research press, rehoboth, 1999. p. 1-141. [7] f. smarandache, m. khan, s. anis, y. b. jun, neutrosophic n-structures and their applications in semigroups, ann. fuzzy math. inform. 14 (2017), 583–598, https://doi.org/10.30948/afmi.2017.14.6.583. [8] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_02.pdf https://doi.org/10.15625/1813-9663/30/4/5032 https://doi.org/10.15625/1813-9663/30/4/5032 https://digitalrepository.unm.edu/nss_journal/vol28/iss1/21/ http://www.ccms.or.kr/data/pdfpaper/jcms22_3/22_3_417.pdf https://doi.org/10.24507/ijicic.16.06.2121 https://doi.org/10.30948/afmi.2017.14.6.583 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. notations 3. applications of picture n-sets in semigroups references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 145-152 http://www.etamaths.com best approximation of the dunkl multiplier operators tk,`,m fethi soltani abstract. we study some class of dunkl multiplier operators tk,`,m; and we give for them an application of the theory of reproducing kernels to the tikhonov regularization, which gives the best approximation of the operators tk,`,m on a hilbert spaces h s k` . 1. introduction in this paper, we consider rd with the euclidean inner product 〈., .〉 and norm |y| := √ 〈y,y〉. for α ∈ rd\{0}, let σα be the reflection in the hyperplane hα ⊂ rd orthogonal to α: σαx := x− 2〈α,x〉 |α|2 α. a finite set < ⊂ rd\{0} is called a root system, if < ∩ r.α = {−α,α} and σα< = < for all α ∈<. we assume that it is normalized by |α|2 = 2 for all α ∈<. for a root system <, the reflections σα, α ∈ <, generate a finite group g. the coxeter group g is a subgroup of the orthogonal group o(d). all reflections in g, correspond to suitable pairs of roots. for a given β ∈ rd\ ⋃ α∈<hα, we fix the positive subsystem <+ := {α ∈ < : 〈α,β〉 > 0}. then for each α ∈ < either α ∈<+ or −α ∈<+. let k,` : < → c be two multiplicity functions on < (a functions which are constants on the orbits under the action of g). as an abbreviation, we introduce the index γk := ∑ α∈<+ k(α) and γ` := ∑ α∈<+ `(α). throughout this paper, we will assume that k(α),`(α) ≥ 0 for all α ∈ <, and γ` ≥ γk. moreover, let wk denote the weight function wk(x) := ∏ α∈<+ |〈α,x〉| 2k(α), for all x ∈ rd, which is g-invariant and homogeneous of degree 2γk. let ck be the mehta-type constant given by ck := (∫ rd e−|x| 2/2wk(x)dx )−1 . 2010 mathematics subject classification. 42b10; 42b15; 46e35. key words and phrases. hilbert spaces; dunkl multiplier operators; tikhonov regularization; extremal functions. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 145 146 fethi soltani we denote by µk the measure on rd given by dµk(x) := ckwk(x)dx; and by lp(µk), 1 ≤ p ≤∞, the space of measurable functions f on rd, such that ‖f‖lp(µk) := (∫ rd |f(x)|pdµk(x) )1/p < ∞, 1 ≤ p < ∞, ‖f‖l∞(µk) := ess sup x∈rd |f(x)| < ∞. for f ∈ l1(µk) the dunkl transform is defined (see [2]) by fk(f)(y) := ∫ rd ek(−ix,y)f(x)dµk(x), y ∈ rd, where ek(−ix,y) denotes the dunkl kernel (for more details, see the next section). let s > 0. we consider the hilbert hsk` consisting of functions f ∈ l 2(µ`) such that es|z| 2/2f`(f) ∈ l2(µk). the space hsk` is endowed with the inner product 〈f,g〉hs k` := ∫ rd es|z| 2 f`(f)(z)f`(g)(z)dµk(z). let m be a function in l2(µk). the dunkl multiplier operators tk,`,m, are defined for f ∈ hsk` by tk,`,mf(x,a) := f−1k (m(a.)f`(f))(x), (x,a) ∈ k := r d × (0,∞). these operators are studied in [14] where the author established some applications (calderón’s reproducing formulas, best approximation formulas, extremal functions....). in particular, when k = ` these operators are studied in [13]. for m ∈ l2(µk) satisfying the admissibility condition: ∫∞ 0 |m(ax)|2 da a = 1, a.e. x ∈ rd, then the operators tk,`,m satisfy, for f ∈ hsk`: ‖tk,`,mf‖2l2(ωk) = ‖f`(f)‖ 2 l2(µk) , where ωk is the measure on k given by dωk(x,a) := daa dµk(x). building on the ideas of matsuura et al. [5], saitoh [9, 11] and yamada et al. [18], and using the theory of reproducing kernels [8], we give best approximation of the operator tk,`,m on the hilbert spaces h s k`. more precisely, for all λ > 0, g ∈ l2(ωk), the infimum inf f∈hs k` { λ‖f‖2hs k` + ‖g −tk,`,mf‖2l2(ωk) } , is attained at one function f∗λ,g, called the extremal function, and given by f∗λ,g(y) = ∫ rd e`(iy,z) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] dµ`(z). next we show for f∗λ,g the following properties. (i) ‖f∗λ,g‖hsk` ≤ 1 2 √ λ ‖g‖l2(ωk). (ii) tk,`,mf ∗ λ,g(y,a) = ∫ rd m(az)ek(iy,z) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] dµk(z). in the dunkl setting, the extremal functions are studied in several directions [12, 13, 14, 15, 16]. this paper is organized as follows. in section 2 we define and study the dunkl multiplier operators tk,`,m on the hilbert space h s k`. the last section of this paper is best approximation of the dunkl multiplier operators tk,`,m 147 devoted to give an application of the theory of reproducing kernels to the tikhonov regularization, which gives the best approximation of the operators tk,`,m on the hilbert space hsk`. 2. dunkl type multiplier operators the dunkl operators dj; j = 1, ...,d, on rd associated with the finite reflection group g and multiplicity function k are given, for a function f of class c1 on rd, by djf(x) := ∂ ∂xj f(x) + ∑ α∈<+ k(α)αj f(x) −f(σαx) 〈α,x〉 . for y ∈ rd, the initial problem dju(.,y)(x) = yju(x,y), j = 1, ...,d, with u(0,y) = 1 admits a unique analytic solution on rd, which will be denoted by ek(x,y) and called dunkl kernel [1, 3]. this kernel has a unique analytic extension to cd ×cd (see [7]). in our case (see [1, 2]), |ek(ix,y)| ≤ 1, x,y ∈ rd. (2.1) the dunkl kernel gives rise to an integral transform, which is called dunkl transform on rd, and was introduced by dunkl in [2], where already many basic properties were established. dunkl’s results were completed and extended later by de jeu [3]. the dunkl transform of a function f in l1(µk), is defined by fk(f)(y) := ∫ rd ek(−ix,y)f(x)dµk(x), y ∈ rd. we notice that f0 agrees with the fourier transform f that is given by f(f)(y) := (2π)−d/2 ∫ rd e−i〈x,y〉f(x)dx, x ∈ rd. some of the properties of dunkl transform fk are collected bellow (see [2, 3]). theorem 2.1 (i) l1 −l∞-boundedness. for all f ∈ l1(µk), fk(f) ∈ l∞(µk) and ‖fk(f)‖l∞(µk) ≤‖f‖l1(µk). (ii) inversion theorem. let f ∈ l1(µk), such that fk(f) ∈ l1(µk). then f(x) = fk(fk(f))(−x), a.e. x ∈ rd. (iii) plancherel theorem. the dunkl transform fk extends uniquely to an isometric isomorphism of l2(µk) onto itself. in particular, ‖fk(f)‖l2(µk) = ‖f‖l2(µk). let s > 0. we define the hilbert space hsk`, as the set of all f ∈ l 2(µ`) such that es|z| 2/2f`(f) ∈ l2(µk). the space hsk` provided with the inner product 〈f,g〉hs k` := ∫ rd es|z| 2 f`(f)(z)f`(g)(z)dµk(z), and the norm ‖f‖hs k` = √ 〈f,f〉hs k` . the space hsk` satisfies the following properties. (i) the hsk` has the reproducing kernel hsk`(x,y) = c` ck ∫ rd e−s|z| 2 e`(ix,z)e`(−iy,z)w`−k(z)dµ`(z). 148 fethi soltani if k = `, then hskk is the dunkl-type heat kernel [6, 12] and this kernel is given by hskk(x,y) = 1 (2s)γk+d/2 e−(|x| 2+|y|2)/4sek ( x √ 2s , y √ 2s ) . (ii) the space hsk` is continuously contained in l 2(µ`) and ‖f‖2l2(µ`) ≤ c` ck (2 e )γ`−γk(γ` −γk s )γ`−γk ‖f‖2hs k` . (iii) if f ∈ hsk` then f`(f) ∈ l 1(µ`) and ‖f`(f)‖l1(µ`) ≤ ck,`‖f‖hsk` , where ck,` = ( c` ck ∫ rd e−s|z| 2 w`−k(z)dµ`(z) )1/2 . (2.2) (iv) if f ∈ hsk`, then f`(f) ∈ l 1 ∩l2(µ`) and f(x) = ∫ rd e`(ix,z)f`(f)(z)dµ`(z), a.e. x ∈ rd. let λ > 0. we denote by 〈., .〉λ,hs k` the inner product defined on the space hsk` by 〈f,g〉λ,hs k` := λ〈f,g〉hs k` + 〈f`(f),f`(g)〉l2(µk), (2.3) and the norm ‖f‖λ,hs k` := √ 〈f,f〉λ,hs k` . on hsk` the two norms ‖.‖hsk` and ‖.‖λ,hsk` are equivalent. this (hsk`,〈., .〉λ,hsk` ) is a hilbert space with reproducing kernel given by ksk`(x,y) = c` ck ∫ rd e`(ix,z)e`(−iy,z) 1 + λes|z| 2 w`−k(z)dµ`(z). (2.4) let m be a function in l2(µk). the dunkl multiplier operators tk,`,m, are defined for f ∈ hsk` by tk,`,mf(x,a) := f−1k (m(a.)f`(f))(x), (x,a) ∈ k. (2.5) we denote by ωk the measure on k given by dωk(x,a) := daa dµk(x); and by l2(ωk), the space of measurable functions f on k, such that ‖f‖l2(ωk) := (∫ rd ∫ ∞ 0 |f(x,a)|2dωk(x,a) )1/2 < ∞. let m be a function in l2(µk) satisfying the admissibility condition∫ ∞ 0 |m(ax)|2 da a = 1, a.e. x ∈ rd. (2.6) then from theorem 2.1 (iii) , for f ∈ hsk`, we have ‖tk,`,mf‖l2(ωk) = ‖f`(f)‖l2(µk) ≤‖f‖hsk`. (2.7) best approximation of the dunkl multiplier operators tk,`,m 149 3. extremal functions for the operators tk,`,m in this section, by using the theory of extremal function and reproducing kernel of hilbert space [8, 9, 10, 11] we study the extremal function associated to the dunkl multiplier operators tk,`,m. in the particular case when k = ` this function is studied in [16, 17]. the main result of this section can be stated as follows. theorem 3.1. let m ∈ l2(µk) satisfying (2.6). for any g ∈ l2(ωk) and for any λ > 0, there exists a unique function f∗λ,g, where the infimum inf f∈hs k` { λ‖f‖2hs k` + ‖g −tk,`,mf‖2l2(ωk) } (3.1) is attained. moreover, the extremal function f∗λ,g is given by f∗λ,g(y) = ∫ rd ∫ ∞ 0 g(x,a)qs(x,y,a)dωk(x,a), where qs(x,y,a) = ∫ rd m(az)ek(−ix,z)e`(iy,z) 1 + λes|z| 2 dµ`(z). proof. let s,λ > 0. since m ∈ l2(µk) and satisfying (2.6), then by (2.7), the inner product 〈., .〉λ,hs k` defined by (2.3) is written by 〈f,g〉λ,hs k` = λ〈f,g〉hs k` + 〈tk,`,mf,tk,`,mg〉l2(ωk). then, the existence and unicity of the extremal function f∗λ,g satisfying (3.1) is obtained in [4, 5, 10]. especially, f∗η,g is given by the reproducing kernel of h s k` with ‖.‖λ,hs k` norm as f∗λ,g(y) = 〈g,tk,`,m(k s k`(.,y))〉l2(ωk), (3.2) where ksk` is the kernel given by (2.4). then, we obtain the result by theorem 2.1 (ii) and the fact that f`(ksk`(.,y))(z) = c` ck e`(−iy,z) 1 + λes|z| 2 w`−k(z), z ∈ r d. (3.3) � theorem 3.2. let λ > 0 and g ∈ l2(ωk). the extremal function f∗λ,g satisfies (i) |f∗λ,g(y)| ≤ ck,` 2 √ λ ‖g‖l2(ωk), where ck,` is the constant given by (2.2). (ii) ‖f∗λ,g‖ 2 l2(µ`) ≤ dk,` λ ‖m‖2l2(µk) ∫ rd ∫ ∞ 0 |g(x,a)|2 e(|x| 2+a2)/2 a2γk+d+1 dωk(x,a), where dk,` = ck √ π 4c` √ 2a2γk+d (2 e )γ`−γk(γ` −γk s )γ`−γk . proof. (i) from (2.7) and (3.2), we have |f∗λ,g(y)| ≤ ‖g‖l2(ωk)‖tk,`,m(k s k`(.,y))‖l2(ωk) ≤ ‖g‖l2(ωk)‖f`(k s k`(.,y))‖l2(µk). then, by (3.3) we deduce |f∗λ,g(y)| ≤ ‖g‖l2(ωk) ( c` ck ∫ rd w`−k(z)dµ`(z) [1 + λes|z| 2 ]2 )1/2 . 150 fethi soltani using the fact that [1 + λes|z| 2 ]2 ≥ 4λes|z| 2 , we obtain the result. (ii) we write f∗λ,g(y) = ∫ rd ∫ ∞ 0 √ ae−(|x| 2+a2)/4 e (|x|2+a2)/4 √ a g(x,a)qs(x,y,a)dωk(x,a). applying hölder’s inequality, we obtain |f∗λ,g(y)| 2 ≤ √ π 2 ∫ rd ∫ ∞ 0 |g(x,a)|2 e(|x| 2+a2)/2 a |qs(x,y,a)|2dωk(x,a). thus and from fubini-tonnelli’s theorem, we get ‖f∗λ,g‖ 2 l2(µ`) ≤ √ π 2 ∫ rd ∫ ∞ 0 |g(x,a)|2 e(|x| 2+a2)/2 a ‖qs(x,.,a)‖2l2(µ`)dωk(x,a). (3.4) the function z → m(az)ek(−ix,z) 1+λes|z| 2 belongs to l 1 ∩l2(µ`), then by theorem 2.1 (ii), qs(x,y,a) = f−1` (m(az)ek(−ix,z) 1 + λes|z| 2 ) (y). thus, by theorem 2.1 (iii) we deduce that ‖qs(x,.,a)‖2l2(µ`) = ∫ rd |f`(qs(x,.,a))(z)|2dµ`(z) ≤ ∫ rd |m(az)|2dµ`(z) [1 + λes|z| 2 ]2 . then ‖q(x,.,a)‖2l2(µ`) ≤ ck 4λc` ∫ rd e−s|z| 2 |m(az)|2w`−k(z)dµk(z) ≤ ck 4λc`a2γk+d (2 e )γ`−γk(γ` −γk s )γ`−γk ‖m‖2l2(µk). from this inequality we deduce the result. � theorem 3.3. let s,λ > 0. for every g ∈ l2(ωk), we have (i) f∗λ,g(y) = ∫ rd e`(iy,z) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] dµ`(z). (ii) f`(f∗λ,g)(z) = 1 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] . (iii) ‖f∗λ,g‖hsk` ≤ 1 2 √ λ ‖g‖l2(ωk). proof. (i) from (3.2) we have f∗λ,g(y) = ∫ rd ∫ ∞ 0 g(x,b)tk,`,m(k s k`(.,y))(x,b)dωk(x,b). since∫ rd ∫ ∞ 0 |g(x,b)tk,`,m(ksk`(.,y))(x,b)|dωk(x,b) ≤‖g‖l2(ωk)‖f`(k s k`(.,y))‖l2(µk) < ∞, best approximation of the dunkl multiplier operators tk,`,m 151 then, by fubini’s theorem, theorem 2.1 (iii) and (3.3) we obtain f∗λ,g(y) = ∫ ∞ 0 ∫ rd g(x,b)tk,`,m(k s k`(.,y))(x,b)dµk(x) db b = ∫ ∞ 0 ∫ rd m(bz)fk(g(.,b))(z)f`(ksk`(.,y))(z)dµk(z) db b = ∫ ∞ 0 ∫ rd m(bz)fk(g(.,b))(z)e`(iy,z) 1 + λes|z| 2 dµ`(z) db b . since ∫ ∞ 0 ∫ rd ∣∣∣m(bz)fk(g(.,b))(z)e`(iy,z) 1 + λes|z| 2 ∣∣∣dµ`(z) db b ≤ ck,` 2 √ λ ‖g‖l2(ωk) < ∞, then, by fubini’s theorem we deduce that f∗λ,g(y) = ∫ rd e`(iy,z) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] dµ`(z). (ii) the function z → 1 1+λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] belongs to l1 ∩ l2(µ`). then by theorem 2.1 (ii) and (iii), it follows that f ∗ λ,g belongs to l 2(µ`), and f`(f∗λ,g)(z) = 1 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] . (iii) from (ii), hölder’s inequality and (2.6) we have |f`(f∗η,g)(z)| 2 ≤ 1 [1 + ηes|z| 2 ]2 [∫ ∞ 0 |fk(g(.,b))(z)|2 db b ] . thus, ‖f∗λ,g‖ 2 hs k` ≤ ∫ rd es|z| 2 [1 + λes|z| 2 ]2 [∫ ∞ 0 |fk(g(.,b))(z)|2 db b ] dµk(z) ≤ 1 4λ ∫ rd [∫ ∞ 0 |fk(g(.,b))(z)|2 db b ] dµk(z) = 1 4λ ‖g‖2l2(ωk), which ends the proof. � theorem 3.4. let s,λ > 0. for every g ∈ l2(ωk), we have tk,`,mf ∗ λ,g(y,a) = ∫ rd m(az)ek(iy,z) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] dµk(z). proof. from (2.5) and theorem 3.3 (ii), we have tk,`,mf ∗ λ,g(y,a) = f −1 k ( m(az) 1 + λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ]) (y). the function z → m(az) 1+λes|z| 2 [∫ ∞ 0 m(bz)fk(g(.,b))(z) db b ] belongs to l1(µk). then by theorem 2.1 (ii), we obtain the result. � acknowledgments the author is partially supported by the dgrst research project lr11es11 and cmcu program 10g/1503 152 fethi soltani references [1] c.f. dunkl, integral kernels with reflection group invariance, canad. j. math. 43 (1991) 1213– 1227. [2] c.f. dunkl, hankel transforms associated to finite reflection groups, contemp. math. 138 (1992) 123–138. [3] m.f.e.de jeu, the dunkl transform, invent. math. 113 (1993) 147–162. [4] g.s. kimeldorf and g. wahba, some results on tchebycheffian spline functions, j. math. anal. appl. 33 (1971) 82–95. [5] t. matsuura, s. saitoh and d.d. trong, inversion formulas in heat conduction multidimensional spaces, j. inv. ill-posed problems 13 (2005) 479–493. [6] m. rösler and m. voit, markov processes related with dunkl operators, adv. appl. math. 21 (1998) 575–643. [7] e.m. opdam, dunkl operators, bessel functions and the discriminant of a finite coxeter group, compositio math. 85(3) (1993) 333–373. [8] s. saitoh, hilbert spaces induced by hilbert space valued functions, proc. amer. math. soc. 89 (1983) 74–78. [9] s. saitoh, the weierstrass transform and an isometry in the heat equation, appl. anal. 16 (1983) 1–6. [10] s. saitoh, approximate real inversion formulas of the gaussian convolution, appl. anal. 83 (2004) 727–733. [11] s. saitoh, best approximation, tikhonov regularization and reproducing kernels, kodai math. j. 28 (2005) 359–367. [12] f. soltani, inversion formulas in the dunkl-type heat conduction on rd, appl. anal. 84 (2005) 541–553. [13] f. soltani, best approximation formulas for the dunkl l2-multiplier operators on rd, rocky mountain j. math. 42 (2012) 305–328. [14] f. soltani, multiplier operators and extremal functions related to the dual dunkl-sonine operator, acta math. sci. 33b(2) (2013) 430–442. [15] f. soltani, operators and tikhonov regularization on the fock space, int. trans. spec. funct. 25(4) (2014) 283–294. [16] f. soltani, uncertainty principles and extremal functions for the dunkl l2-multiplier operators, j. oper. 2014 (2014), article id 659069. [17] f. soltani and a. nemri, analytical and numerical applications for the fourier multiplier operators on rn × (0,∞), appl. anal. 2014, doi:10.1080/00036811.2014.937432. [18] m. yamada, t. matsuura and s. saitoh, representations of inverse functions by the integral transform with the sign kernel, frac. calc. appl. anal. 2 (2007) 161–168. department of mathematics, faculty of science, jazan university, p.o.box 277, jazan 45142, saudi arabia international journal of analysis and applications volume 17, number 4 (2019), 479-502 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-479 computing special smarandache curves according to darboux frame in euclidean 4-space m. khalifa saad1,2,∗ and m. a. abd-rabo3 1department of mathematics, faculty of science, islamic university of madinah, 170 madinah, ksa 2department of mathematics, faculty of science, sohag university, 82524 sohag, egypt 3school of mathematics and statistics, zhengzhou university, 450001 zhengzhou, china ∗corresponding author: m khalifag@yahoo.com, mohammed.khalifa@iu.edu.sa abstract. in this paper, we study some special smarandache curves and their differential geometric properties according to darboux frame in euclidean 4-space e4. also, we compute some of these curves which lie fully on a hypersurface in e4. moreover, we defray some computational examples in support our main results. 1. introduction the geometric modeling of free-form curves and surfaces is of central importance for sophisticated cad/cam systems. among all space curves, smarandache curves have special emplacement regarding their properties, because of this, they deserve especial attention in euclidean geometry as well as in other geometries. it is known that smarandache geometry is a geometry which has at least one smarandache denied axiom [1]. an axiom is said to be smarandache denied, if it behaves in at least two different ways within the same space. smarandache geometries are connected with the theory of relativity and the parallel universes. smarandache curves are the objects of smarandache geometry. by definition, if the position received 2019-03-27; accepted 2019-04-24; published 2019-07-01. 2010 mathematics subject classification. 53a04, 53a07, 53a35. key words and phrases. smarandache curves; hypersurfaces; darboux frame; euclidean 4-space. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 479 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-479 int. j. anal. appl. 17 (4) (2019) 480 vector of a curve δ is composed by frenet frame’s vectors of another curve β, then the curve δ is called a smarandache curve [2]. in differential geometry, frame fields constitute an important tool while studying curves and surfaces. the most familiar frame fields are frenet frame along a space curve and darboux frame along a surface curve [3]. it is analogous to the frenet frame as applied to surface geometry. the darboux frame exists at any nonumbilic point of a surface embedded in euclidean space. in [4] m. düldü et al. extend the darboux frame field into euclidean 4-space e4. in the light of the existing studies in the field of geometry, many interesting results on smarandache curves have been obtained by many mathematicians, see for example, [2, 3, 5–12]. turgut and yilmaz [11] have introduced a particular circumstance of such curves, they entitled it smarandache tb2 curves in the space e14. they studied special smarandache curves which are defined by the tangent and second binormal vector fields. in [8], the author has illustrated certain special smarandache curves in the euclidean 3-space. recently, h.s. abdel-aziz and m. khalifa saad [2, 5] have studied special smarandache curves of an arbitrary curve such as tn, tb and tnb with respect to frenet frame in the three-dimensional galilean and pseudo-galilean spaces. also, in [6, 7] they have investigated smarandache curves according to darboux frame in the three-dimensional minkowski space e31. the main goal of this article is to investigate smarandache curves in e4 for a given curve with reference to its darboux frame of first kind. the results presented in this paper generalize and refine some of the existing results in the literature and significant in mathematical modeling and other applications. 2. basic notions and properties we now review some basic concepts on classical differential geometry of space curves and surfaces in euclidean 4-space e4 [4, 13–15]. let {ei | i = 1, 2, 3, 4} be the standard basis of e4, therefore e4 = {x =∑4 i=1 xiei | xi ∈ r}. the scalar product and vector product of vectors x,y,z ∈ e 4 are respectively defined by 〈x,y 〉 = 4∑ i=1 xiyi , x ×y ×z = ∣∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 e4 x1 x2 x3 x4 y1 y2 y3 y4 z1 z2 z3 z4 ∣∣∣∣∣∣∣∣∣∣∣∣ . definition 2.1. let f : d ⊂ e4 → r be a differentiable mapping of an open set d. given c ∈ r, we recall that the level set c of the f is the set defined as f−1(c) which is the set of solutions in d of the equations f(x,y,z,w) = c. int. j. anal. appl. 17 (4) (2019) 481 proposition 2.1. consider that f : u ⊂ e4 → r is a differentiable function and c ∈ f(u) is a regular value of f, then f−1(c) is a regular surface in e4. the implicit surface f is regular if ∇f = (fx,fy,fz,fw) 6= 0. the unit surface normal vector of the implicit surface f is given by n = ∇f‖∇f‖. definition 2.2. suppose that f : rn → r, if all second partial derivatives of f exist and are continuous on the domain of f, then the hessian matrix hf is a square n×n matrix, usually defined as hij = ∂ 2f ∂xi∂xj . 2.1. curves on a hypersurface in e4. let r : i ⊂ r → m be a regular curve in e4, where ‖r′‖ = ‖dr/du‖ = 1, ∀ u ∈ i , we will omit u for simplification. then the frenet frame is defined by   t′ n′ b′1 b′2   =   0 κ1 0 0 −κ1 0 κ2 0 0 −κ2 0 κ3 0 0 κ3 0     t n b1 b2   , ( ′ = d du ) (2.1) where t, n, b1 and b2 are the tangent, principal normal, first binormal and the second binormal vector fields of r, respectively and the functions κi | i = 1, 2, 3, are the curvature functions of r. theorem 2.1. [2] let r : i → e4 be a regular curve. then t = r ′ ‖r′‖ , b2 = r ′ ×r ′′ ×r(3) ‖r′ ×r′′ ×r(3)‖ , b1 = b2 ×r ′ ×r ′′ ‖b2 ×r ′ ×r′′‖ , n = b1 × b2 ×r ′ ‖b1 × b2 ×r ′‖ , (2.2) and κ1 = 〈n,r ′′ 〉 ‖r′‖2 , κ2 = 〈b1,r(3)〉 ‖r′‖3κ1 , κ3 = 〈b2,r(4)〉 ‖r′‖4κ1κ2 . (2.3) definition 2.3. let r be a regular curve in e4, which is parameterized by arc length and n-times continuously differentiable. then r is called a frenet curve, if at every point the vectors r ′ ,r ′′ , ...,r(n−1) are linearly independent. 2.2. darboux frame field of first kind in e4. let m be an oriented hypersurface in e4 and r be a frenet curve of class cn(n ≥ 4) with arc-length parameter u lying on m. we denote the unit tangent and unit normal vector fields of the curve by t and n, and p, u ∈ tum. definition 2.4. let r : i → e4 be a regular parameterized curve lying on m in e4, then the frame field {t, p, u, n} along r is called extended darboux frame field of first kind if {t, n,r′′} is linearly independent, therefore t = r ′ ‖r′‖, n = ∇f ‖∇f‖, p = r′′−(r′′.n)n ‖r′′−(r′′.n)n‖, u = [t, n, e] int. j. anal. appl. 17 (4) (2019) 482 the derivatives of the darboux frame field of first kind in e4 are given by [16]  t′ p′ u′ n′   =   0 κ1g 0 κn −κ1g 0 κ2g τ1g 0 −κ2g 0 τ2g −κn −τ1g −τ2g 0     t p u n   , (2.4) where κn,κ 1 g,κ 2 g,τ 1 g and τ 2 g are real valued functions denote the normal curvature, geodesic curvatures and the geodesic torsions, respectively. these functions are given by κn = 〈t′, n〉 = 1 ‖∇f‖ 〈r ′′ ,∇f 〉 = −1 ‖∇f‖ (r′hf (r ′)t), κ1g = 〈t ′, p〉 = ( r ′′ (r ′′ )t − 1 ‖∇f‖2 (r ′ hf (r ′ )t)2 )1 2 , κ2g = 〈p ′, u〉 = 1 (κ1g) 2∇f ( m1 + 1 ‖∇f‖2 (r ′ hf (r ′ )t) m2 ) , τ1g = 〈p ′, n〉 = −1 (κ1g) ( 1 ‖∇f‖ (r ′ hf (r ′′ )t) + 1 ‖∇f‖3 (r ′ hf (∇f)t)(r ′ hf (r ′ )t) ) , τ2g = 〈u ′, n〉 = ( −1 (κ1g)‖∇f‖2 ) m3, (2.5) where m1 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ x ′′ y ′′ z ′′ w ′′ x(3) y(3) z(3) w(3) fx fy fz fw ∣∣∣∣∣∣∣∣∣∣∣∣ , m2 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ x ′′ y ′′ z ′′ w ′′ a b c d fx fy fz fw ∣∣∣∣∣∣∣∣∣∣∣∣ , m3 = ∣∣∣∣∣∣∣∣∣∣∣∣ x ′ y ′ z ′ w ′ fx fy fz fw a b c d p q t v ∣∣∣∣∣∣∣∣∣∣∣∣ , and [a,b,c,d] = (∇f) ′ = r ′ hf , [p,q,t,v] = (∇f) ′′ = α ′′ hf + α ′ dhf du ′ , dhf du = [ ∂hf ∂x (r ′ )t ... ∂hf ∂w (r ′ )t]. remark 2.1. [2] let r : i → e4 be a regular parameterized curve in e4, then • r is called asymptotic curve if and only if kn = 0. • r is called a line of curvature if and only if τ1g = τ2g = 0. 3. first kind smarandache curves in e4 consider r = r(u) is a curve lying fully on an oriented hypersurface m in e4. let {t, p, u, n} be a darboux frame field of first kind along r(u) and κn,κ 1 g,κ 2 g,τ 1 g ,τ 2 g are real valued functions in arc length parameter u of r. so, we have the following definition. int. j. anal. appl. 17 (4) (2019) 483 definition 3.1. [11] a regular curve α(s(u)) in e4, whose position vector is obtained by extended darboux frame vectors of another regular curve r(u) is called smarandache curve . in the following we continue our studies of special smarandache curves that we started in [2, 5, 6]. here we investigate some special smarandache curves of first kind called tp, tu, pu and pn ( the other special smarandache curves can be computed in the same manner) and then obtain some of their differential geometric properties which represent the main results. let r(u) = (x(u),y(u),z(u),w(u)) be a curve of class cn(n ≥ 4) lying on m. then, by using proposition 2.1, the unit normal vector field along r is given by n̄ = ∇f ‖∇f‖ . (3.1) 3.1. tp-smarandache curves. definition 3.2. let m be an oriented hypersurface in e4 and the frent curve r = r(u) lying fully on m with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the tp-smarandache curve of r is defined as α(s) = 1 √ 2 (t + p). (3.2) theorem 3.1. let r = r(u) be a frenet curve lying on a hypersurface m in e4 with darboux frame {t, p, u, n} and non-zero constant curvatures κn,κ1g,κ2g,τ1g and τ2g . then the curvature functions of the tp− smarandache curve of r satisfy the following equations: κ̄n = 1 λ1‖∇f‖   (κ1g(−κn + τ1g ) + κ2gτ2g )fw − ((κ1g)2+ κn(κn + τ 1 g ))fx − ((κ1g)2 + (κ2g)2+ τ1g (κn + τ 1 g ))fy + (κ 1 gκ 2 g − (κn + τ1g )τ2g )fz   , κ̄1g = 1 λ1λ6   (κ1g(−κn + τ1g ) + κ2gτ2g )λ5 + ((κ1g)2+ κn(κn + τ 1 g ))λ2 + ((κ 1 g) 2 + (κ2g) 2+ τ1g (κn + τ 1 g ))λ3 + (κ 1 gκ 2 g − (κn + τ1g )τ2g )λ4   , κ̄2g = 1 λ26λ11   −(λ2λ7 + λ3λ8 + λ4λ9 + λ5λ10)λ′6+ λ6((κ 1 gλ2 −κ2gλ4)λ8 −λ5(κnλ7 + τ1g λ8+ τ2g λ9) + λ3(−κ1gλ7 + κ2gλ9 + τ1g λ10) + λ7λ′2+ λ8λ ′ 3 + λ9λ ′ 4 + λ10(κnλ2 + τ 2 g λ4 + λ ′ 5))   , int. j. anal. appl. 17 (4) (2019) 484 τ̄1g = 1 λ26‖∇f‖   −(fxλ2 + fyλ3 + fzλ4 + fwλ5)λ′6+ λ6(−κ1gfxλ3 + κ2gfzλ3 −κnfxλ5− τ2g fzλ5 + fxλ ′ 2 + fy(κ 1 gλ2 −κ2gλ4 − τ1g λ5+ λ′3) + fzλ ′ 4 + fw(κnλ2 + τ 1 g λ3 + τ 2 g λ4 + λ ′ 5))   , τ̄2g = 1 λ211‖∇f‖   −(fxλ7 + fyλ8 + fzλ9 + fwλ10)λ′11+ λ11(−κ1gfxλ8 + κ2gfzλ8 −κnfxλ10− τ2g fzλ10 + fxλ ′ 7 + fy(κ 1 gλ7 −κ2gλ9 − τ1g λ10+ λ′8) + fzλ ′ 9 + fw(κnλ7 + τ 1 g λ8 + τ 2 g λ9 + λ ′ 10))   . (3.3) proof. because α = α(s) is a tp− smarandache curve reference to frenet curve r, then differentiating eq. (3.2), we get α′ = 1 √ 2 ( −κ1gt + κ 1 gp + κ 2 gu + (τ 1 g + κn)n ) . (3.4) again, differentiating eq. (3.4), we obtain α′′ = 1 √ 2   (−(κ1g)2 −κn(κn + τ1g ))t − ((κ1g)2 + (κ2g)2 + τ1g (κn + τ1g ))p +(κ1gκ 2 g − (κn + τ1g )τ2g )u + (−κ1gκn + κ1gτ1g + κ2gτ2g )n   . also, eq. (3.4) leads to t̄ = −κ1gt + κ1gp + κ2gu + (τ1g + κn)n λ1 , where λ1 = √ 2(κ1g) 2 + (κ2g) 2 + (τ1g + κn) 2, and we get n̄ = fxt + fyp + fzu + fwn ‖∇f‖ . on the other hand, we have p̄ = λ2t + λ3p + λ4u + λ5n λ6 , where λ2 = 1 √ 2 ( (κ1g) 2 + (κn) 2 + κnτ 1 g ) + σ1 ‖∇f‖ fx , λ3 = 1 √ 2 ( (κ1g) 2 + (κ2g) 2 + κnτ 1 g + (τ 1 g ) 2 ) + σ1 ‖∇f‖ fy , λ4 = 1 √ 2 ( −κ1gκ 2 g + κnτ 2 g + τ 1 g τ 2 g ) + σ1 ‖∇f‖ fz , λ5 = 1 √ 2 ( κ1gκn −κ 1 gτ 1 g −κ 2 gτ 2 g ) + σ1 ‖∇f‖ fw , λ6 = √ (λ2)2 + (λ3)2 + (λ4)2 + (λ5)2, int. j. anal. appl. 17 (4) (2019) 485 σ1 = 〈α′′, n〉 = 1 √ 2‖∇f‖   (−(κ1g)2 −κn(κn + τ1g ))fx − ((κ1g)2 + (κ2g)2+ τ1g (κn + τ 1 g ))fy + (κ 1 gκ 2 g − (κn + τ1g )τ2g )fz+ (−κ1gκn + κ1gτ1g + κ2gτ2g )fw   . also, we get ū = λ7t + λ8p + λ9u + λ10n λ11 , where, λ7 =   κ2gfwλ3 −κnfzλ3 − τ1g fzλ3 −κ1gfwλ4+ κnfyλ4 + τ 1 g fyλ4 −κ2gfyλ5 + κ1gfzλ5   , λ8 =   −κ2gfwλ2 + κnfzλ2 + τ1g fzλ2 −κ1gfwλ4− κnfxλ4 − τ1g fxλ4 + κ2gfxλ5 + κ1gfzλ5   , λ9 =   κ1gfwλ2 −κnfyλ2 − τ1g fyλ2 + κ1gfwλ3+ κnfxλ3 + τ 1 g fxλ3 −κ1gfxλ5 −κ1gfyλ5   , λ10 = ( κ2gfyλ2 −κ1gfzλ2 −κ2gfxλ3 −κ1gfzλ3 + κ1gfxλ4 + κ1gfyλ4 ) , λ11 = λ1 λ6 ‖∇f‖. in the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of α are computed as in eqs. (3.3). � corollary 3.1. if r is an asymptotic curve. then, the following equations hold: t̄ = −κ1gt + κ1gp + κ2gu + τ1g n λ1 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = λ13t + λ14p + λ15u + λ16n λ17 , ū = λ18t + λ19p + λ20u + λ21n λ22 , and then the curvature functions are computed as follows κ̄n = (κ1gτ 1 g + κ 2 gτ 2 g )fw − (κ1g)2fx − ((κ1g)2 + (κ2g)2 + (τ1g )2)fy + (κ1gκ2g − τ1g τ2g )fz λ12‖∇f‖ , κ̄1g = (κ1gτ 1 g + κ 2 gτ 2 g )λ16 − (κ1g)2λ13 − ((κ1g)2 + (κ2g)2 + (τ1g )2)λ14 + (κ1gκ2g − τ1g τ2g )λ15 λ12λ17 , κ̄2g = 1 λ22λ 2 17   λ17(κ 1 gλ13λ19 −κ2gλ15λ19 − τ1g λ16λ19− τ2g λ16λ20 + λ14(−κ1gλ18 + κ2gλ20) + λ18λ′13+ λ19λ ′ 14 + λ20λ ′ 15 + λ21(τ 1 g λ14 + τ 2 g λ15+ λ′16)) − (λ21λ16 + λ13λ18 + λ14λ19 + λ15λ20)λ′17   , int. j. anal. appl. 17 (4) (2019) 486 τ̄1g = 1 λ217‖∇f‖   −fx(κ1gλ14λ17 −λ17λ′13 + λ13λ′17) + fy(λ17(κ1gλ13− κ2gλ15 − τ1g λ16 + λ′14) −λ14λ′17)+ fz(λ17(κ 2 gλ14 − τ2g λ16 + λ′15) −λ15λ′17)+ fw(λ17(τ 1 g λ14 + τ 2 g λ15 + λ ′ 16) −λ16λ′17)   , τ̄2g = 1 λ222‖∇f‖   −(λ21fw + λ18fx + λ19fy + λ20fz)λ′22 + λ22((−τ1g λ21+ κ1gλ18)fy + λ20(τ 2 g fw −κ2gfy) + λ19(τ1g fw −κ1gfx+ κ2gfz) + fwλ ′ 21 + fxλ ′ 18 + fyλ ′ 19 + fz(−τ2g λ21 + λ′20))   , where λ12 = √ 2(κ1g) 2 + (κ2g) 2 + (τ1g ) 2, λ13 = √ 2(κ1g) 2‖∇f‖ + 2σ1fx, λ14 = √ 2((κ1g) 2 + (κ2g) 2 + (τ1g ) 2)‖∇f‖ + 2σ1fy, λ15 = √ 2(−κ1gκ 2 g + τ 1 g τ 2 g )‖∇f‖ + 2σ1fz, λ16 = √ 2(−κ1gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ1fw, λ17 = √ (λ13)2 + (λ14)2 + (λ15)2 + (λ16)2, λ18 = κ 2 gfwλ14 − τ 1 g fzλ14 −κ 1 gfwλ15 + τ 1 g fyλ15 −κ 2 gfyλ16 + κ 1 gfzλ16, λ19 = −κ2gfwλ13 + τ 1 g fzλ13 −κ 1 gfwλ15 − τ 1 g fxλ15 + κ 2 gfxλ16 + κ 1 gfzλ16, λ20 = k 1 gλ13fw + k 1 gλ14fw + τ 1 g λ14fx −k 1 gλ16fx − τ 1 g λ13fy −k 1 gλ16fy, λ21 = κ 2 gfyλ13 −κ 1 gfzλ13 −κ 2 gfxλ14 −κ 1 gfzλ14 + κ 1 gfxλ15 + κ 1 gfyλ15, λ22 = λ12 λ17 ‖∇f‖, σ1 = 1 √ 2‖∇f‖   −(κ1g)2fx − ((κ1g)2 + (κ2g)2 + (τ1g )2)fy+ (κ1gκ 2 g − τ1g τ2g )fz + (κ1gτ1g + κ2gτ2g )fw   . corollary 3.2. if r is a line of curvature. then, the following equations hold: t̄ = −κ1gt + κ1gp + κ2gu + κnn λ1 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = λ24t + λ25p + λ26u + λ27n λ28 , ū = λ29t + λ30p + λ31u + λ32n λ33 , and then the curvature functions are computed as follows κ̄n = − κ1gκnfw + ((κ 1 g) 2 + κ2n)fx + ((κ 1 g) 2 + (κ2g) 2)fy −κ1gκ2gfz λ23‖∇f‖ , κ̄1g = − κ1gκnλ27 + ((κ 1 g) 2 + κ2n)λ24 + ((κ 1 g) 2 + (κ2g) 2)λ25 −κ1gκ2gλ26 λ23λ28 , int. j. anal. appl. 17 (4) (2019) 487 κ̄2g = 1 λ33λ 2 28   λ28(λ25(−κ1gλ29 + κ2gλ31) + λ29(−κnλ27 + λ′24)+ λ30(κ 1 gλ24 −κ2gλ26 + λ′25) + λ31λ′26 + λ32(κnλ24+ λ′27)) − (λ32λ27 + λ24λ29 + λ25λ30 + λ26λ31)λ′28   , τ̄1g = 1 λ228‖∇f‖   −fx(λ28(κ1gλ25 + κnλ27 −λ′24) + λ24λ′28)+ fy(λ28(κ 1 gλ24 −κ2gλ26 + λ′25) −λ25λ′28)+ fz(λ28(κ 2 gλ25 + λ ′ 26) −λ26λ′28)+ fw(λ28(κnλ24 + λ ′ 27) −λ27λ′28)   , τ̄2g = 1 λ233‖∇f‖   −(λ32fw + λ29fx + λ30fy + λ31fz)λ′33+ λ33(−κnλ32fx −κ1gλ30fx+ λ29(κnfw + κ 1 gfy) + κ 2 gλ30fz + fwλ ′ 32+ fxλ ′ 29 + fy(−κ2gλ31 + λ′30) + fzλ′31)   , where λ23 = √ 2(κ1g) 2 + (κ2g) 2 + (κn)2, λ24 = √ 2((κ1g) 2 + κ2n) 2‖∇f‖ + 2σ1fx, λ25 = √ 2((κ1g) 2 + (κ2g) 2)‖∇f‖ + 2σ1fy, λ26 = − √ 2κ1gκ 2 g‖∇f‖ + 2σ1fz, λ27 = √ 2κ1gκn‖∇f‖ + 2σ1fw, λ28 = √ (λ24)2 + (λ25)2 + (λ26)2 + (λ27)2, λ29 = κ 2 gfwλ25 −κnfzλ25 −κ 1 gfwλ26 + κnfyλ26 −κ 2 gfyλ27 + κ 1 gfzλ27, λ30 = −κ2gfwλ24 + κnfzλ24 −κ 1 gfwλ26 −κnfxλ26 + κ 2 gfxλ27 + κ 1 gfzλ27, λ31 = κ 1 gfwλ24 −κnfyλ24 + κ 1 gfwλ25 + κnfxλ25 −κ 1 gfxλ27 −κ 1 gfyλ27, λ32 = κ 2 gfyλ24 −κ 1 gfzλ24 −κ 2 gfxλ25 −κ 1 gfzλ25 + κ 1 gfxλ26 + κ 1 gfyλ26, λ33 = λ23 λ28 ‖∇f‖, σ1 = 1 √ 2‖∇f‖ (−κ1gκnfw − ((κ 1 g) 2 + κ2n)fx − ((κ 1 g) 2 + (κ2g) 2)fy + κ 1 gκ 2 gfz). 3.2. tu-smarandache curves. definition 3.3. let m be an oriented hypersurface in e4 and the frenet curve r = r(u) lying fully on m with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the tu-smarandache curve of r is defined as β(u) = 1 √ 2 (t + u). (3.5) theorem 3.2. let r = r(u) be a frenet curve lying on a hypersurface m in e4 with darboux frame {t, p, u, n} and non-zero constant curvatures; κn,κ1g,κ2g,τ1g and τ2g . then the curvature functions of the int. j. anal. appl. 17 (4) (2019) 488 tu− smarandache curve of r satisfy the following equations: κ̄n = 1 µ1‖∇f‖   (κ1g −κ2g)τ1g fw − (κ1g(κ1g −κ2g) + κn(κn + τ2g ))fx+ (κ1g −κ2g)κ2gfy − (κn + τ2g )(τ1g fy + τ2g fz)   , κ̄1g = 1 µ1µ6   (κ1g −κ2g)τ1g µ5 − (κ1g(κ1g −κ2g) + κn(κn + τ2g ))µ2+ (κ1g −κ2g)κ2gµ3 − (κn + τ2g )(τ1g fy + τ2g µ4)   , κ̄2g = 1 µ26µ11   −(µ2µ7 + µ3µ8 + µ4µ9 + µ5µ10)µ′6 + µ6((κ1gµ2− κ2gµ4)µ8 −µ5(κnµ7 + τ1g µ8 + τ2g µ9) + µ3(−κ1gµ7+ κ2gµ9 + τ 1 g µ10) + µ7µ ′ 2 + µ8µ ′ 3+ µ9µ ′ 4 + µ10(κnµ2 + τ 2 g µ4 + µ ′ 5))   , τ̄1g = 1 µ26‖∇f‖   −(fxµ2 + fyµ3 + fzµ4 + fwµ5)µ′6 + µ6(fxµ′2 −κ1gfxµ3 + κ2gfzµ3 −κnfxµ5 − τ2g fzµ5+ fy(κ 1 gµ2 −κ2gµ4 − τ1g µ5 + µ′3) + fzµ′4+ fw(κnµ2 + τ 1 g µ3 + τ 2 g µ4 + µ ′ 5))   , τ̄2g = 1 µ211‖∇f‖   −(fxµ7 + fyµ8 + fzµ9 + fwµ10)µ′11 + µ11(fxµ′7 −κ1gfxµ8 + κ2gfzµ8 −κnfxµ10 − τ2g fzµ10+ fy(κ 1 gµ7 −κ2gµ9 − τ1g µ10 + µ′8)+ fzµ ′ 9 + fw(κnµ7 + τ 1 g µ8 + τ 2 g µ9 + µ ′ 10))   . (3.6) proof. since β = β(s) is a tu− smarandache curve reference to frenet curve r. then, by differentiating eq. (3.5), we get β′ = 1 √ 2 ( (κ1g −κ 2 g)p + (κn + τ 2 g )u ) , (3.7) again, by differentiating eq. (3.7), we have β′′ = 1 √ 2   (−κ1g(κ1g −κ2g) −κn(κn + τ2g ))t − τ1g (κn + τ2g )p +((κ1g −κ2g)κ2g − (κn + τ2g )τ2g )u + (κ1g −κ2g)τ1g n   . also, from eq. (3.7), we obtain t̄ = (κ1g −κ2g)p + (κn + τ2g )u µ1 , where µ1 = √ 2(κ1g −κ2g)2 + (κn + τ2g )2, and we have n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = µ2t + µ3p + µ4u + µ5n µ6 , int. j. anal. appl. 17 (4) (2019) 489 where µ2 = 1 √ 2 ( ((κ1g) 2 −κ1gκ 2 g + κn(κn + τ 2 g )) ) + σ2 ‖∇f‖ fx , µ3 = 1 √ 2 ( τ1g (κn + τ 2 g ) ) + σ2 ‖∇f‖ fy , µ4 = 1 √ 2 ( (κ2g(−κ 1 g + κ 2 g) + τ 2 g (κn + τ 2 g )) ) + σ2 ‖∇f‖ fz , µ5 = 1 √ 2 ( (−κ1g + κ 2 g)τ 1 g ) + σ2 ‖∇f‖ fw , µ6 = √ (µ2)2 + (µ3)2 + (µ4)2 + (µ5)2 σ2 = 〈β′′, n〉 = 1 √ 2‖∇f‖   (−(κ1g)2 + κ1gκ2g −κn(κn + τ2g ))fx + τ1g (κn + τ2g )fy +((κ1g −κ2g)κ2g − τ2g (κn + τ2g ))fz + (κ1g −κ2g)τ1g fw   . also, we get ū = µ7t + µ8p + µ3u + µ10n δ , where µ7 = −κnfzµ3 − τ2g fzµ3 −κ 1 gfwµ4 + κ 2 gfwµ4 + κnfyµ4 + τ 2 g fyµ4 + κ 1 gfzµ5 −κ 2 gfzµ5, µ8 = κnfzµ2 + τ 2 g fzµ2 −κnfxµ4 − τ 2 g fxµ4, µ9 = κ 1 gfwµ2 −κ 2 gfwµ2 −κnfyµ2 − τ 2 g fyµ2 + κnfxµ3 + τ 2 g fxµ3 −κ 1 gfxµ5 + κ 2 gfxµ5, µ10 = −κ1gfzµ2 + κ 2 gfzµ2 + κ 1 gfxµ4 −κ 2 gfxµ4, µ11 = µ1 µ6 ‖∇f‖. in the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of β are computed as in eqs. (3.6). � corollary 3.3. if r is an asymptotic curve. then, the following equations hold: t̄ = (κ1g −κ2g)p + τ2g u µ1 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = µ13t + µ14p + µ15u + µ16n µ17 , ū = µ18t + µ19p + µ20u + µ21n µ22 , int. j. anal. appl. 17 (4) (2019) 490 and the curvature functions are obtained as follows: κ̄n = (κ1g −κ2g)τ1g fw + κ1g(−κ1g + κ2g)fx − τ1g τ2g fy − (−κ1gκ2g + (κ2g)2 + (τ2g )2)fz µ12‖∇f‖ , κ̄1g = κ1g(−κ1g + κ2g)µ13 + (κ1g −κ2g)κ2gµ15 − τ2g (τ1g µ14 + τ2g µ15) + (κ1g −κ2g)τ1g µ16 µ12µ17 , κ̄2g = 1 µ217µ22   µ17(κ 1 gµ13µ19 −κ2gµ15µ19 − τ1g µ16µ19 − τ2g µ16µ20+ τ2g µ15µ21 + µ14(−κ1gµ18 + κ2gµ20 + τ1g µ21) + µ18µ′13 + µ19µ′14+ µ20µ ′ 15 + µ21µ ′ 16) − (µ13µ18 + µ14µ19 + µ15µ20 + µ16µ21)µ′17   , τ̄1g = 1 µ217‖∇f‖   −fx(κ1gµ14µ17 −µ17µ′13 + µ13µ′17) + fy(µ17(κ1gµ13− κ2gµ15 − τ1g µ16 + µ′14) −µ14µ′17) + fz(µ17(κ2gµ14 − τ2g µ16+ µ′15) −µ15µ′17) + fw(µ17(τ1g µ14 + τ2g µ15 + µ′16) −µ16µ′17)   , τ̄2g = 1 µ222‖∇f‖   −fx(κ1gµ19µ22 −µ22µ′18 + µ18µ′22) + fy(µ22(κ1gµ18 −κ2gµ20− τ1g µ21 + µ ′ 19) −µ19µ′22) + fz(µ22(κ2gµ19 − τ2g µ21 + µ′20)− µ20µ ′ 22) + fw(µ22(τ 1 g µ19 + τ 2 g µ20 + µ ′ 21) −µ21µ′22)   , where µ12 = √ (κ1g −κ2g)2 + (τ2g )2, µ13 = √ 2((κ1g) 2 −κ1gκ 2 g)‖∇f‖ + 2σ2fx , µ14 = √ 2τ1g τ 2 g‖∇f‖ + 2σ2fy , µ15 = √ 2(κ2g(−κ 1 g + κ 2 g) + (τ 2 g ) 2)‖∇f‖ + 2σ2fz, µ16 = √ 2(−κ1g + κ 2 g)τ 1 g‖∇f‖ + 2σ2fw, µ217 = (µ13) 2 + (µ14) 2 + (µ15) 2 + (µ16) 2, µ18 = −(κ1gµ15fw −κ 2 gµ15fw − τ 2 g µ15fy + τ 2 g µ14fz −κ 1 gµ16fz + κ 2 gµ16fz), µ19 = −(τ2g µ15fx − τ 2 g µ13fz), µ20 = −(−κ1gµ13fw + κ 2 gµ13fw − τ 2 g µ14fx + κ 1 gµ16fx −κ 2 gµ16fx + τ 2 g µ13fy), µ21 = −(−κ1gµ15fx + κ 2 gµ15fx + κ 1 gµ13fz −κ 2 gµ13fz), µ22 = µ12 µ17 ‖∇f‖. int. j. anal. appl. 17 (4) (2019) 491 corollary 3.4. if r is a line of curvature curve. then, the following equations hold: t̄ = (κ1g −κ2g)p + κnn µ23 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = µ24t + µ25p + µ26u + µ27n µ28 , ū = µ29t + µ30p + µ31u + µ32n µ33 , and the curvature functions are obtained as follows: κ̄n = −((κ1g)2 −κ1gκ2g + κ2n)fx + (κ1g −κ2g)κ2gfz µ23‖∇f‖ , κ̄1g = −((κ1g)2 −κ1gκ2g + κ2n)µ24 + (κ1g −κ2g)κ2gµ26 µ23µ28 , κ̄2g = 1 µ217µ22   µ28(−κnµ27µ29 + µ25(−κ1gµ29 + κ2gµ31) + µ29µ′24+ µ30(κ 1 gµ24 −κ2gµ26 + µ′25) + µ31µ′26 + µ32(κnµ24+ µ′27)) − (µ24µ29 + µ25µ30 + µ26µ31 + µ27µ32)µ′28   , τ̄1g = 1 µ228‖∇f‖   −fx(µ28(κ1gµ25 + κnµ27 −µ′24) + µ24µ′28)+ fy(µ28(κ 1 gµ24 −κ2gµ26 + µ′25) −µ25µ′28)+ fz(µ28(κ 2 gµ25 + µ ′ 26) −µ26µ′28)+ fw(µ28(κnµ24 + µ ′ 27) −µ27µ′28)   , τ̄2g = 1 µ233‖∇f‖   −fx(µ33(κ1gµ30 + κnµ32 −µ′29) + µ29µ′33)+ fy(µ33(κ 1 gµ29 −κ2gµ31 + µ′30) −µ30µ′33)+ fz(µ33(κ 2 gµ30 + µ ′ 31) −µ31µ′33)+ fw(µ33(κnµ29 + µ ′ 32) −µ32µ′33)   , where µ23 = √ (κ1g −κ2g)2 + κ2n, µ24 = √ 2((κ 1 g) 2 −κ1gκ 2 g + κ 2 n)‖∇f‖ + 2σ2fx , µ25 = 2σ2fy , µ26 = √ 2κ 2 g(−κ 1 g + κ 2 g)‖∇f‖ + 2σ2fz, µ27 = 2σ2fw, µ 2 28 = (µ24) 2 + (µ25) 2 + (µ26) 2 + (µ27) 2 , µ29 = −(κ1gµ26fw −κ 2 gµ26fw −κnµ26fy + κnµ25fz −κ 1 gµ27fz + κ 2 gµ27fz), µ30 = −(κnµ26fx −κnµ25fz), µ31 = −(−κ1gµ24fw + κ 2 gµ24fw −κnµ25fx + κ 1 gµ27fx −κ 2 gµ27fx + κnµ24fy), µ32 = −(−κ1gµ26fx + κ 2 gµ26fx + κ 1 gµ24fz −κ 2 gµ24fz), µ33 = µ24 µ28 ‖∇f‖. int. j. anal. appl. 17 (4) (2019) 492 3.3. pu-smarandache curves. definition 3.4. let m be an oriented hypersurface in e4and the frenet curve γ = γ(s) lying fully on m with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the pu-smarandache curve of γ is defined as γ(u) = 1 √ 2 (p + u). (3.8) theorem 3.3. let r = r(u) be a frenet curve lying on a hypersurface m in e4 with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the curvature functions of the pu− smarandache curve of r satisfy the following equations: κ̄n = −1 ν1‖∇f‖   (κ1gκn + κ 2 g(τ 1 g − τ2g ))fw + (−κ1gκ2g+ κn(τ 1 g + τ 2 g ))fx + ((κ 1 g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ2g ))fy + ((κ 2 g) 2 + τ2g (τ 1 g + τ 2 g ))fz   , κ̄1g = 1 ν1ν6   (κ1gκn + κ 2 g(τ 1 g − τ2g ))ν5 + (−κ1gκ2g+ κn(τ 1 g + τ 2 g ))ν2 + ((κ 1 g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ2g ))ν3 + ((κ 2 g) 2 + τ2g (τ 1 g + τ 2 g ))ν4   , κ̄2g = 1 ν26ν11   −(ν2ν7 + ν3ν8 + ν4ν9 + ν5ν10)ν′6 + ν6((κ1gν2 −κ2gν4)ν8 −ν5(κnν7 + τ1g ν8 + τ2g ν9) + ν3(−κ1gν7 + κ2gν9 + τ1g ν10)+ ν7ν ′ 2 + ν8ν ′ 3 + ν9ν ′ 4 + ν10(κnν2 + τ 2 g ν4 + ν ′ 5))   , τ̄1g = 1 ν26‖∇f‖   −(fxν2 + fyν3 + fzν4 + fwν5)ν′6 + ν6(−κ1gfxν3 + κ2gfzν3 −κnfxν5 − τ2g fzν5 + fxν′2 + fy(κ1gν2 −κ2gν4 − τ1g ν5 +ν′3) + fzν ′ 4 + fw(κnν2 + τ 1 g ν3 + τ 2 g ν4 + ν ′ 5))   , τ̄2g = 1 ν211‖∇f‖   −(fxν7 + fyν8 + fzν9 + fwν10)ν′11 + ν11(−κ1gfxν8 + κ2gfzν8 −κnfxν10 − τ2g fzν10 + fxν′7 + fy(κ1gν7 −κ2gν9 − τ1g ν10 +ν′8) + fzν ′ 9 + fw(κnν7 + τ 1 g ν8 + τ 2 g ν9 + ν ′ 10))   . (3.9) proof. since γ = γ(s) is a pu− smarandache curve reference to frenet curve r. then, by differentiating eq. (3.8), we get γ′ = −κ1gt + κ2gp + κ2gu + (τ1g + τ2g )n√ 2 , γ′′ = 1 √ 2 (κ1gκ 2 g −κn(τ 1 g + τ 2 g ))t + (−(κ 1 g) 2 − (κ2g) 2 − τ1g (τ 1 g + τ 2 g ))p +(−(κ2g) 2 − τ2g (τ 1 g + τ 2 g ))u + (−κ 1 gκn −κ 2 gτ 1 g + κ 2 gτ 2 g )n. (3.10) int. j. anal. appl. 17 (4) (2019) 493 using eqs. (3.10), we have t̄ = −κ1gt + κ2gp + κ2gu + (τ1g + τ2g )n ν1 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , where ν1 = √ 2(κ2g) 2 + (τ1g + τ 2 g ) 2 + (κ1g) 2. on the other hand, we get p̄ = ν2t + ν3p + ν4u + ν5n ν6 , where ν2 = √ 2(−κ1gκ 2 g + κn(τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fx , ν3 = √ 2((κ1g) 2 + (κ2g) 2 + τ1g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fy , ν4 = √ 2((κ2g) 2 + τ2g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fz , ν5 = √ 2(κ1gκn + κ 2 gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ3fw , ν26 = ν 2 2 + ν 2 3 + ν 2 4 + ν 2 5 , σ3 = 〈γ′′, n〉 = 1 2‖∇f‖   −(κ1gκn + κ2g(τ1g − τ2g ))fw + (κ1gκ2g− κn(τ 1 g + τ 2 g ))fx − ((κ1g)2 + (κ2g)2 + τ1g (τ1g + τ2g ))fy − ((κ2g)2 + τ2g (τ1g + τ2g ))fz   . also, we get ū = ν7t + ν8p + ν9u + ν10n ν11 , where ν7 = κ 2 gfwν3 − τ 1 g fzν3 − τ 2 g fzν3 + κ 2 gfwν4 + τ 1 g fyν4 + τ 2 g fyν4 −κ 2 g(fy + fz)ν5, ν8 = −fw(κ2gν2 + κ 1 gν4) + fz((τ 1 g + τ 2 g )ν2 + κ 1 gν5) + fx(−(τ 1 g + τ 2 g )ν4 + κ 2 gν5), ν9 = fw(−κ2gν2 + κ 1 gν3) −fy((τ 1 g + τ 2 g )ν2 + κ 1 gν5) + fx((τ 1 g + τ 2 g )ν3 + κ 2 gν5), ν10 = κ 2 gfyν2 + κ 2 gfzν2 −κ 2 gfxν3 −κ 1 gfzν3 −κ 2 gfxν4 + κ 1 gfyν4, ν11 = ν1‖∇f‖ν6. in the light of the above calculations, the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of γ are computed as in eqs. (3.9). � int. j. anal. appl. 17 (4) (2019) 494 corollary 3.5. if r is an asymptotic curve. then, the following equations hold: t̄ = −κ1gt −κ2gp + κ2gu + (τ1g + τ2g )n ν12 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = ν13t + ν14p + ν15u + ν16n ν17 , ū = ν18t + ν19p + ν20u + ν21n ν22 , and the curvature functions are obtained as follows: κ̄n = 1 ν17‖∇f‖   κ2g(−τ1g + τ2g )fw + κ1gκ2gfx − ((κ1g)2 + (κ2g)2+ τ1g (τ 1 g + τ 2 g ))fy − ((κ2g)2 + τ2g (τ1g + τ2g ))fz   , κ̄1g = 1 ν12ν17   κ1gκ2gν13 − ((κ1g)2 + (κ2g)2 + τ1g (τ1g + τ2g ))ν14− ((κ2g) 2 + τ2g (τ 1 g + τ 2 g ))ν15 + κ 2 g(−τ1g + τ2g )ν16   , κ̄2g = 1 ν217ν22   ν17(κ 1 gν13ν19 −κ2gν15ν19 − τ1g ν16ν19 − τ2g ν16ν20+ τ2g ν15ν21 + ν14(−κ1gν18 + κ2gν20 + τ1g ν21) + ν18ν′13 + ν19ν′14+ ν20ν ′ 15 + ν21ν ′ 16) − (ν13ν18 + ν14ν19 + ν15ν20 + ν16ν21)ν′17   , τ̄1g = 1 ν217‖∇f‖   −fx(κ1gν14ν17 −ν17ν′13 + ν13ν′17) + fy(ν17(κ1gν13 −κ2gν15− τ1g ν16 + ν ′ 14) −ν14ν′17) + fz(ν17(κ2gν14 − τ2g ν16 + ν′15)− ν15ν ′ 17) + fw(ν17(τ 1 g ν14 + τ 2 g ν15 + ν ′ 16) −ν16ν′17)   , τ̄2g = 1 ν222‖∇f‖   −fx(κ1gν19ν22 −ν22ν′18 + ν18ν′22) + fy(ν22(κ1gν18− κ2gν20 − τ1g ν21 + ν′19) −ν19ν′22) + fz(ν22(κ2gν19 − τ2g ν21+ ν′20) −ν20ν′22) + fw(ν22(τ1g ν19 + τ2g ν20 + ν′21) −ν21ν′22)   , where ν12 = √ 2(κ2g) 2 + (τ1g + τ 2 g ) 2 + (κ1g) 2, ν13 = √ 2(−κ1gκ 2 g)‖∇f‖ + 2σ3fx, ν14 = √ 2((κ 1 g) 2 + (κ 2 g) 2 + τ 1 g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fy, ν15 = √ 2((κ 2 g) 2 + τ 2 g (τ 1 g + τ 2 g ))‖∇f‖ + 2σ3fz, ν16 = √ 2(κ 2 gτ 1 g −κ 2 gτ 2 g )‖∇f‖ + 2σ3fw, ν 2 17 = ν 2 12 + ν 2 13 + ν 2 14 + ν 2 15, ν18 = −κ2gν16fy + ν15(κ 2 gfw + (τ 1 g + τ 2 g )fy) −κ 2 gν16fz + ν14(κ 2 gfw − (τ 1 g + τ 2 g )fz), ν19 = −κ2gν13fw + κ 2 gν16fx −ν15(κ 1 gfw + (τ 1 g + τ 2 g )fx) + (τ 1 g + τ 2 g )ν13fz + κ 1 gν16fz, ν20 = −κ2gν13fw + κ 2 gν16fx + ν14(κ 1 gfw + (τ 1 g + τ 2 g )fx) − τ 1 g ν13fy − τ 2 g ν13fy −κ 1 gν16fy, ν21 = κ 2 gν13fy + ν15(−κ 2 gfx + κ 1 gfy) + κ 2 gν13fz −ν14(κ 2 gfx + κ 1 gfz), ν22 = ν12 ν17 ‖∇f‖. int. j. anal. appl. 17 (4) (2019) 495 corollary 3.6. if r is a line of curvature. then, the following equations hold: t̄ = − κ1gt + κ 2 gp −κ2gu ν12 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = ν13t + ν14p + ν15u + ν16n ν17 , ū = ν18t + ν19p + ν20u + ν21n ν22 , and the curvature functions are obtained as follows: κ̄n = −1 ν23‖∇f‖ ( κ1gκnfw −κ 1 gκ 2 gfx + ((κ 1 g) 2 + (κ2g) 2)fy + (κ 2 g) 2fz ) , κ̄1g = −1 ν23ν28 ( −κ1gκ 2 gν24 + ((κ 1 g) 2 + (κ2g) 2)ν25 + (κ 2 g) 2ν26 + κ 1 gκnν27 ) , κ̄2g = 1 ν228ν33   ν28(−κnν27ν29 + ν25(−κ1gν29 + κ2gν31) + ν29ν′24+ ν30(κ 1 gν24 −κ2gν26 + ν′25) + ν31ν′26 + ν32(κnν24 + ν′27))− (ν24ν29 + ν25ν30 + ν26ν31 + ν27ν32)ν ′ 28   , τ̄1g = 1 ν228‖∇f‖   −fx(ν28(κ1gν25 + κnν27 −ν′24) + ν24ν′28) + fy(ν28(κ1gν24− κ2gν26 + ν ′ 25) −ν25ν′28) + fz(ν28(κ2gν25 + ν′26) −ν26ν′28)+ fw(ν28(κnν24 + ν ′ 27) −ν27ν′28)   , τ̄2g = 1 ν233‖∇f‖   −fx(ν33(κ1gν30 + κnν32 −ν′29) + ν29ν33) + fy(ν33(κ1gν29− κ2gν31 + ν ′ 30) −ν30ν′33) + fz(ν33(κ2gν30 + ν′31) −ν31ν′33)+ fw(ν33(κnν29 + ν ′ 32) −ν32ν′33)   , where ν23 = √ 2(κ2g) 2 + (κ1g) 2, ν24 = − √ 2κ1gκ 2 g‖∇f‖ + 2σ3fx, ν25 = √ 2((κ1g) 2 + (κ2g) 2)‖∇f‖ + 2σ3fy, ν26 = √ 2(κ2g) 2‖∇f‖ + 2σ3fz, ν27 = √ 2κ1gκn‖∇f‖ + 2σ3fw, ν 2 28 = ν 2 12 + ν 2 13 + ν 2 14 + ν 2 15, ν29 = κ 2 gν25fw + κ 2 gν26fw −κ 2 gν27fy −κ 2 gν27fz, ν19 = −κ2gν24fw −κ 1 gν26fw + κ 2 gν27fx + κ 1 gν27fz, ν20 = −κ2gν24fw + κ 1 gν25fw + κ 2 gν27fx −κ 1 gν27fy, ν21 = κ 2 gν24fy + ν26(−κ 2 gfx + κ 1 gfy) + κ 2 gν24fz −ν25(κ 2 gfx + κ 1 gfz), ν22 = ν23 ν28 ‖∇f‖. int. j. anal. appl. 17 (4) (2019) 496 3.4. pn-smarandache curves. definition 3.5. let m be an oriented hypersurface in e4and the frenet curve δ = δ(s) lying fully on m with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the pn-smarandache curve of δ is defined as δ(u) = 1 √ 2 (p + n). (3.11) theorem 3.4. let r = r(u) be a frenet curve lying on a hypersurface m in e4 with darboux frame {t, p, u, n} and non-zero curvatures κn,κ1g,κ2g,τ1g and τ2g . then the curvature functions of the pn− smarandache curve of r satisfy the following equations: κ̄n = −1 ξ1‖∇f‖   (κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))fw + (−κ1g + κn)τ1g fx +((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ2g )fy + τ1g (κ2g + τ2g )fz   , κ̄1g = −1 ξ1ξ6   (κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))ξ5 + (−κ1g + κn)τ1g ξ2+ ((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ2g )ξ3 + τ1g (κ2g + τ2g )ξ4   , κ̄2g = 1 ξ26ξ11   −(ξ2ξ7 + ξ3ξ8 + ξ4ξ9 + ξ5ξ10)ξ′6 + ξ6((κ1gξ2 −κ2gξ4)ξ8 −ξ5(κnξ7 + τ1g ξ8 + τ2g ξ9) + ξ3(−κ1gξ7 + κ2gξ9 + τ1g ξ10)+ ξ7ξ ′ 2 + ξ8ξ ′ 3 + ξ9ξ ′ 4 + ξ10(κnξ2 + τ 2 g ξ4 + ξ ′ 5))   , τ̄1g = 1 ξ26‖∇f‖   −(fxξ2 + fyξ3 + fzξ4 + fwξ5)ξ′6 + ξ6(−κ1gfxξ3 + κ2gfzξ3 −κnfxξ5 − τ2g fzξ5 + fxξ′2 + fy(κ1gξ2 −κ2gξ4 − τ1g ξ5 +ξ′3) + fzξ ′ 4 + fw(κnξ2 + τ 1 g ξ3 + τ 2 g ξ4 + ξ ′ 5))   , τ̄2g = 1 ξ211‖∇f‖   −(fxξ7 + fyξ8 + fzξ9 + fwξ10)ξ′11 + ξ11(−κ1gfxξ8+ κ2gfzξ8 −κnfxξ10 − τ2g fzξ10 + fxξ′7 + fy(κ1gξ7− κ2gξ9 − τ1g ξ10 + ξ′8) + fzξ′9 + fw(κnξ7 + τ1g ξ8 + τ2g ξ9 + ξ′10))   . (3.12) proof. let δ = δ(s) be a pn− smarandache curve reference to frenet curve r. then, by differentiating eq. (3.11), we obtain δ′ = −(κ1g + κn)t − τ1g p + (κ2g − τ2g )u + τ1g n√ 2 , δ′′ = 1 √ 2 (κ1g −κn)τ 1 g t + (−(κ 2 g) 2 −κ1g(κ 1 g + κn) − (τ 1 g ) 2 + κ2gτ 2 g )p −τ1g (κ 2 g + τ 2 g )u + (−κn(κ 1 g + κn) − (τ 1 g ) 2 + κ2gτ 2 g − (τ 2 g ) 2)n. (3.13) therefore, from eqs. (3.13), we get t̄ = −(κ1g + κn)t − τ1g p + (κ2g − τ2g )u + τ1g n ξ1 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , int. j. anal. appl. 17 (4) (2019) 497 where ξ1 = √ 2(τ1g ) 2 + (κ1g + κn) 2 + (κ2g − τ2g )2. on the other hand, we obtain p̄ = ξ2t + ξ3p + ξ4u + ξ5n ξ6 , where ξ2 = √ 2(−κ1g + κn)τ 1 g‖∇f‖ + 2σ4fx , ξ3 = √ 2((κ2g) 2 + κ1g(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ 2 g )‖∇f‖ + 2σ4fy , ξ4 = √ 2τ1g (κ 2 g + τ 2 g )‖∇f‖ + 2σ4fz , ξ5 = √ 2(κn(κ 1 g + κn) + (τ 1 g ) 2 −κ2gτ 2 g + (τ 2 g ) 2)‖∇f‖ + 2σ4fw , ξ26 = ξ 2 2 + ξ 2 3 + ξ 2 4 + ξ 2 5 , σ4 = 〈δ′′, n〉 = 1 2‖∇f‖   −(κn(κ1g + κn) + (τ1g )2 + τ2g (−κ2g + τ2g ))fw+ (κ1g −κn)τ1g fx − ((κ2g)2 + κ1g(κ1g + κn)+ (τ1g ) 2 −κ2gτ2g )fy − τ1g (κ2g + τ2g )fz   . also, we have ū = ξ7t + ξ8p + ξ9u + ξ10n ξ11 , where, ξ7 = κ 2 gfwξ3 − τ 2 g fwξ3 − τ 1 g fzξ3 + τ 1 g fwξ4 + τ 1 g fyξ4 + (−(κ 2 g − τ 2 g )fy − τ 1 g fz)ξ5, ξ8 = fw((−κ2g + τ 2 g )ξ2 − (κ 1 g + κn)ξ4) + fz(τ 1 g ξ2 + (κ 1 g + κn)ξ5) −fx(τ 1 g ξ4 + (−κ 2 g + τ 2 g )ξ5), ξ9 = fw(−τ1g ξ2 + (κ 1 g + κn)ξ3) + τ 1 g fx(ξ3 + ξ5) −fy(τ 1 g ξ2 + (κ 1 g + κn)ξ5), ξ10 = (κ 2 gfyξ2 − τ 2 g fyξ2 + τ 1 g fzξ2 −κ 2 gfxξ3 + τ 2 g fxξ3 −κ 1 gfzξ3 −κnfzξ3 − τ1g fxξ4 + κ 1 gfyξ4 + κnfyξ4), ξ11 = ξ1‖∇f‖ξ6. in the light of the above calculations , the curvature functions κ̄n, κ̄1g, κ̄ 2 g, τ̄ 1 g and τ̄ 2 g of δ are computed as in eqs. (3.12). � int. j. anal. appl. 17 (4) (2019) 498 corollary 3.7. if r be asymptotic curve. then, the following equations hold: t̄ = −κ1gt − τ1g p + (κ2g − τ2g )u + τ1g n ξ12 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = ξ13t + ξ14p + ξ15u + ξ16n ξ17 , ū = ξ18t + ξ19p + ξ20u + ξ21n ξ22 . thus, the curvature functions can be computed as follows: κ̄n = −1 ‖∇f‖ξ12   ((τ1g )2 + τ2g (−κ2g + τ2g ))fw −κ1gτ1g fx + ((κ1g)2 + (κ2g)2+ (τ1g ) 2 −κ2gτ2g )fy + τ1g (κ2g + τ2g )fz   , κ̄1g = −1 ξ12ξ17   ((τ1g )2 + τ2g (−κ2g + τ2g ))ξ16 −κ1gτ1g ξ13 + ((κ1g)2+ (κ2g) 2 + (τ1g ) 2 −κ2gτ2g )ξ14 + τ1g (κ2g + τ2g )ξ15   , κ̄2g = 1 ξ217 ξ22   ξ17(κ 1 gξ13ξ19 −κ2gξ15ξ19 − τ1g ξ16ξ19 − τ2g ξ16ξ20+ τ2g ξ15ξ21 + ξ14(−κ1gξ18 + κ2gξ20 + τ1g ξ21) + ξ18ξ′13 + ξ19ξ′14+ ξ20ξ ′ 15 + ξ21ξ ′ 16) − (ξ13ξ18 + ξ14ξ19 + ξ15ξ20 + ξ16ξ21)ξ′17   , τ̄1g = 1 ξ217‖∇f‖   −fx(κ1gξ14ξ17 − ξ17ξ′13 + ξ13ξ′17) + fy(ξ17(κ1gξ13 −κ2gξ15− τ1g ξ16 + ξ ′ 14) − ξ14ξ′17) + fz(ξ17(κ2gξ14 − τ2g ξ16 + ξ′15)− ξ15ξ ′ 17) + fw(ξ17(τ 1 g ξ14 + τ 2 g ξ15 + ξ ′ 16) − ξ16ξ′17)   , τ̄2g = 1 ξ222‖∇f‖   −fx(κ1gξ19ξ22 − ξ22ξ′18 + ξ18ξ′22) + fy(ξ22(κ1gξ18 −κ2gξ20− τ1g ξ21 + ξ ′ 19) − ξ19ξ′22) + fz(ξ22(κ2gξ19 − τ2g ξ21 + ξ′20)− ξ20ξ ′ 22) + fw(ξ22(τ 1 g ξ19 + τ 2 g ξ20 + ξ ′ 21) − ξ21ξ′22)   , where ξ12 = √ 2(τ1g ) 2 + (κ1g) 2 + (κ2g − τ2g )2, ξ13 = √ 2(−κ1g)τ 1 g‖∇f‖ + 2σ4fx , ξ14 = √ 2((κ 2 g) 2 + κ 1 g(κ 1 g) + (τ 1 g ) 2 −κ2gτ 2 g )‖∇f‖ + 2σ4fy , ξ15 = √ 2τ 1 g (κ 2 g + τ 2 g )‖∇f‖ + 2σ4fz , ξ16 = √ 2((τ 1 g ) 2 −κ2gτ 2 g + (τ 2 g ) 2 )‖∇f‖ + 2σ4fw , ξ217 = ξ 2 2 + ξ 2 3 + ξ 2 4 + ξ 2 5 , ξ18 = κ 2 gfwξ14 − τ 2 g fwξ14 − τ 1 g fzξ14 + τ 1 g fwξ15 + τ 1 g fyξ15 + (−(κ 2 g − τ 2 g )fy − τ 1 g fz)ξ16, ξ19 = fw((−κ2g + τ 2 g )ξ14 − (κ 1 g)ξ15) + fz(τ 1 g ξ14 + (κ 1 g)ξ16) −fx(τ 1 g ξ15 + (−κ 2 g + τ 2 g )ξ16), ξ20 = fw(−τ1g ξ14 + (κ 1 g)ξ14) + τ 1 g fx(ξ14 + ξ16) −fy(τ 1 g ξ14 + (κ 1 g)ξ16), ξ21 = κ 2 gfyξ13 − τ 2 g fyξ13 + τ 1 g fzξ13 −κ 2 gfxξ14 + τ 2 g fxξ14 −κ 1 gfzξ14 − τ 1 g fxξ15 + κ 1 gfyξ15, ξ22 = ξ12‖∇f‖ξ17. int. j. anal. appl. 17 (4) (2019) 499 corollary 3.8. if r be line of curvature. then, the following equations hold: t̄ = −(κ1g + κn)t + k2gu ξ23 , n̄ = fxt + fyp + fzu + fwn ‖∇f‖ , p̄ = ξ24t + ξ25p + ξ26u + ξ27n ξ28 , ū = ξ29t + ξ30p + ξ31u + ξ32n ξ33 . thus, the curvature functions can be computed as follows: κ̄n = − κn(κ 1 g + κn)fw + ((κ 2 g) 2 + κ1g(κ 1 g + κn))fy ‖∇f‖ξ23 , κ̄1g = − κn(κ 1 g + κn)ξ25 + ((κ 2 g) 2 + κ1g(κ 1 g + κn))ξ27 ξ23ξ28 , κ̄2g = 1 ξ228 ξ33   ξ28(−κnξ27ξ29 + ξ25(−κ1gξ29 + κ2gξ31) + ξ29ξ′24+ ξ30(κ 1 gξ24 −κ2gξ26 + ξ′25) + ξ31ξ′26 + ξ32(κnξ24+ ξ′27)) − (ξ24ξ29 + ξ25ξ30 + ξ26ξ31 + ξ27ξ32)ξ′28   , τ̄1g = 1 ξ228‖∇f‖   −fx(ξ28(κ1gξ25 + κnξ27 − ξ′24) + ξ24ξ′28) + fy(ξ28(κ1gξ24− κ2gξ26 + ξ ′ 25) − ξ25ξ′28) + fz(ξ28(κ2gξ25 + ξ′26) − ξ26ξ′28)+ fw(ξ − 28(κnξ24 + ξ′27) − ξ27ξ′28)   , τ̄2g = 1 ξ233‖∇f‖   −fx(ξ33(κ1gξ30 + κnξ32 − ξ′29) + ξ29ξ′33) + fy(ξ33(κ1gξ29− κ2gξ31 + ξ ′ 30) − ξ30ξ′33) + fz(ξ33(κ2gξ30 + ξ′31) − ξ31ξ′33)+ fw(ξ − 33(κnξ29 + ξ′32) − ξ32ξ′33)   , where ξ23 = √ (κ1g + κn) 2 − (κ2g)2, ξ24 = 2σ4fx , ξ25 = √ 2((κ2g) 2 + κ1g(κ 1 g + κn))‖∇f‖ + 2σ4fy , ξ26 = 2σ4fz , ξ27 = √ 2κn(κ 1 g + κn)‖∇f‖ + 2σ4fw , ξ 2 28 = ξ 2 24 + ξ 2 25 + ξ 2 26 + ξ 2 27 , ξ29 = (kg2ξ25fw −κ2gξ27fy), ξ30 = −κ2gξ24fw −κ 1 gξ26fw −κnξ26[u]fw + κ 2 gξ27fx + (κ 1 g + κn)ξ27fz, ξ31 = (κ 1 g + κn)ξ25fw −κ 1 gξ27fy −κnξ27fy, ξ32 = κ 2 gξ24fy + κ 1 gξ26fy + κnξ26fy − ξ25(κ 2 gfx + (κ 1 g + κn)fz), ξ33 = ξ23‖∇f‖ξ28. int. j. anal. appl. 17 (4) (2019) 500 4. examples example 4.1. consider the curve r(u) given by r(u) = (cos(u), sin(u), cos(2u), sin(2u)) . by using the definition 2.6, we can calculate the darboux frame {t, p, u, n} as follows: t = ( − sin(u) 5 , cos(u) 5 ,− 2 5 sin(2u), 2 5 cos(2u) ) , n = ( cos(u) √ 2 , sin(u) √ 2 , cos(2u) √ 2 , sin(2u) √ 2 ) , p = ( cos(u) √ 2 , sin(u) √ 2 ,− cos(2u) √ 2 ,− sin(2u) √ 2 ) , u = ( 2 sin(u) √ 5 ,− 2 cos(u) √ 5 ,− 1 √ 5 sin(2u), 1 √ 5 cos(2u) ) . the curvature functions of the curve r can be computed as follows: κn = − √ 5 2 ,κ1g = 3 √ 10 , κ2g = −2 √ 2 5 ,τ1g = τ 2 g = 0. therefore, we can obtain tu−smarndache curve as φ = 1 √ 2 (t + u) = ( sin(u) √ 10 , −cos(u) √ 10 , −3 sin(2u) √ 10 , 3 cos(2u) √ 10 ) . by using the definition 2.6, we can obtain t̄ = 1 √ 37 (cos(u), sin(u),−6 cos(2u),−6 sin(2u)) , n̄ = 1 2 √ 5 (sin(u), cos(u), 3 sin(2u), 3 cos(2u)) , p̄ = 1 √ 16930 (17 sin(u),−17 cos(u), 129 sin(2u),−129 cos(2u)) , ū = 1 √ 8465 (−54 cos(u),−54 sin(u),−9 cos(2u),−9 sin(2u)) . the curvature functions of the smarandache curve φ can be computed as follows: κ̄n = −1 2 √ 37 5 , κ̄1g = 1531 √ 626410 , κ̄2g = −324 √ 2 1693 , τ̄1g = τ̄ 2 g = 0. example 4.2. consider the unit speed curve given by r(u) = ( cos(u) 2 , sin(u) 2 , u 2 , u √ 2 ) . by using the definition 2.6 we can calculate the darboux frame {t, p, u, n} as follow: t = ( −sin(u) 2 , cos(u) 2 , 1 2 , 1 √ 2 ) , n = ( cos(u) 2 , sin(u) 2 , 1 2 , 1 √ 2 ) , int. j. anal. appl. 17 (4) (2019) 501 p = ( − √ 3 2 cos(u), − √ 3 2 sin(u), 1 2 √ 3 , −1 √ 6 ) , u = ( − √ 2 3 sin(u), √ 2 3 cos(u), −1 √ 6 ,− 1 2 √ 3 ) , the curvature functions of the curve r can be computed as follows κn = − 1 4 , κ1g = √ 3 4 , κ2g = − 1 √ 2 ,τ1g = 0, τ 2 g = − 1 √ 6 . therefore we can obtain tn−smarndache curve as β = ( cos(u) − sin(u) 2 √ 2 , cos(u) + sin(u) 2 √ 2 , 1 √ 2 , 0 ) . by using the definition 2.6, we can obtain t̄ = ( − cos(u) + sin(u) 2 √ 2 , cos(u) − sin(u) 2 √ 2 , 0, 0 ) , n̄ = ( cos(u) − sin(u) 2 √ 2 , cos(u) + sin(u) 2 √ 2 , 1 2 , 1 2 ) , p̄ = ( −3(cos(u) − sin(u)) √ 22 , 3(cos(u) + sin(u)) √ 22 , 1 √ 11 , 1 √ 11 ) , ū = ( 0, 0, 1 √ 11 , 1 √ 11 ) . the curvature functions of the smarandache curve β can be computed as κ̄n = −1 2 , κ̄1g = 3 √ 11 , κ̄2g = 0, τ̄ 1 g = τ̄ 2 g = 0. 5. conclusion in the four dimensional euclidean space e4, some special smarandache curves lying on a hypersurface are investigated. also, the differential geometric properties of these curves are obtained. furthermore, some computational examples in support of our main results are given. acknowledgment this research was supported by islamic university of madinah. we would like to thank our colleagues from deanship of scientific research who provided insight and expertise that greatly assisted the research. references [1] c. ashbacher, smarandache geometries, smarandache notions j. 8(1-3) (1997), 212–215. [2] m. khalifa saad, spacelike and timelike admissible smarandache curves in pseudo-galilean space, j. egypt. math. soc. 24 (2016), 416–423. [3] m. çetin, h. kocayiğit, on the quaternionic smarandache curves in euclidean 3space, int. j. contemp. math. sci. 8(3) (2013), 139–150. [4] m. do carmo, differential geometry of curves and surface, englewood cliffs, nj, usa, prentice hall, 1976. int. j. anal. appl. 17 (4) (2019) 502 [5] h.s. abdel-aziz, m. khalifa saad, smarandache curves of some special curves in the galilean 3-space, honam math. j. 37(2) (2015), 253–264. [6] h.s. abdelaziz, m. khalifa saad, computation of smarandache curves according to darboux frame in minkowski 3-space, j. egypt. math. soc. 25 (2017), 382–390. [7] h.s. abdelaziz, m. khalifa saad, some geometric invariants of pseudo-spherical evolutes in the hyperbolic 3-space, comput. mater. continua 57(3) (2018), 389–415. [8] a.t. ali, special smarandache curves in the euclidean space, int. j. math. comb. 2 (2010), 30–36. [9] o. bektas, s. yuce, smarandache curves according to darboux frame in euclidean space, rom. j. math. comput. sci. 3(1) (2013), 48–59. [10] m. çetin, y. tunçer, m. k. karacan, smarandache curves according to bishop frame in euclidean space, gen. math. notes 20(2) (2014), 50–66. [11] m. turgut, s. yilmaz, smarandache curves in minkowski space-time, int. j. math. comb. 3 (2008), 51–55. [12] k. taşk prü, m. tosun, smarandache curves according to sabban frame on s2, bol. soc. paran. mat. 32(1) (2014), 51–59. [13] o. aléssio, differential geometry of intersection curves in r4 of three implicit surfaces, comput. aided geom. des. 26 (2009), 455–471. [14] h.s. abdel-aziz, m. khalifa saad, a. a. abdel-salam, on implicit surfaces and their intersection curve in euclidean 4-space, houston j. math. 40(2) (2014), 339–352. [15] b. o’neill, elementary differential geometry, burlington, ma, usa, academic press, 1966. [16] m. düldü, b. düldül, n. kuruoǧlu and e. özdamar, extension of the darboux frame into euclidean 4-space and its invariants, turk. j. math. 41 (2017), 1628–1639. 1. introduction 2. basic notions and properties 2.1. curves on a hypersurface in e4 2.2. darboux frame field of first kind in e4 3. first kind smarandache curves in e4 3.1. tp-smarandache curves 3.2. tu-smarandache curves 3.3. pu-smarandache curves 3.4. pn-smarandache curves 4. examples 5. conclusion acknowledgment references international journal of analysis and applications volume 16, number 6 (2018), 894-903 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-894 on integrated and differentiated c2-sequence spaces lakshmi narayan mishra1,2,∗, sukhdev singh3, vishnu narayan mishra4 1department of mathematics, school of advanced sciences, vellore institute of technology, university, vellore 632014, tn, india 2l. 1627 awadh puri colony beniganj, phase -iii, opposite industrial training institute (i.t.i.), ayodhya main road faizabad-224 001, up, india 3department of mathematics, lovely professional university, jalandhar-delhi road, phagwara-144411, punjab, india 4department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur, madhya pradesh 484887, india ∗corresponding author: lakshminarayanmishra04@gmail.com abstract. the integrated and differentiated c2-sequence spaces are defined and studied by using the norm on the bicomplex space c2, infinite matrices of the bicomplex number and the orlicz functions. we also studied some topological properties of the c2-sequence spaces we define the α-duals of the integrated and differentiated c2-sequence spaces. received 2018-02-28; accepted 2018-05-24; published 2018-11-02. 2010 mathematics subject classification. 06f30, 54a99. key words and phrases. bicomplex numbers; bicomplex net; c2-sequence spaces. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 894 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-894 int. j. anal. appl. 16 (6) (2018) 895 1. introduction the set of bicomplex numbers [8] is denoted by c2 and sets of real and complex numbers are denoted as c0 and c1, respectively.the set of bicomplex number is defined as (cf. [8], [9]) c2 := {a1 + i1a2 + i2a3 + i1i2a4 : ak ∈ c0, 1 ≤ k ≤ 4} := {w1 + i2w2 : w1,w2 ∈ c1} where i21 = i 2 2 = −1, i1i2 = i2i1. the set of bicomplex numbers c2 have exactly two non-trivial idempotent elements denoted by e1 and e2 give as e1 = (1 + i1i2)/2 and e2 = (1 − i1i2)/2. note that e1 + e2 = 1 and e1.e2 = 0. the number ξ = w1 + i2w2 can be uniquely expressed as a complex combination of e1 and e2 [8]. ξ = w1 + i2w2 = 1ξe1 + 2ξe2, (1.1) where 1ξ = w1 − i1w2 and 2ξ = w1 + i1w2. the complex coefficients 1ξ and 2ξ are called the idempotent components of ξ, and 1ξe1 + 2ξe2 is known as idempotent representation of bicomplex number ξ. the auxiliary complex spaces a1 and a2 are defined as follows: a1 = { 1ξ : ξ ∈ c2 } and a2 = { 2ξ : ξ ∈ c2 } . the norm in c2 is defined as follows: ||ξ|| = √ a21 + a 2 2 + a 2 3 + a 2 4 = √ |w1|2 + |w2|2 = √ |1ξ|2 + |2ξ|2 2 (1.2) further, the norm of the product of two bicomplex numbers and the product of their norms are connected by means of the following inequality: ||ξ .η|| ≤ √ 2 ||ξ|| . ||η|| (1.3) the inequality given in (1.3) is the best possible relation . for this reason, we call c2 as modified complex banach algebra [8]. throughout the paper, the ω4, c, c0 and ` ∞ c2 denote the space of all bicomplex sequences, convergent sequences, null sequences and all bounded sequences. we denote the zero sequence (0, 0, 0, . . . , 0, . . .) by π. refer the book by mursaleen [?] for details about summability methods. the orlicz function m is defined as m : [0,∞) → [0,∞). it is continuous, non-decreasing and m(0) = 0,m(x) > 0 for x > 0. also, for λ ∈ (0, 1) it satisfies the condition m(λx + (1 −λ)y) ≤ λm(x) + (1 −λ)m(y) (1.4) and if the condition of convexity of the orlicz function m is replaced by m(x + y) ≤m(x) + m(y), then the function m is called the modulus function. int. j. anal. appl. 16 (6) (2018) 896 the notations (x : y ) denote the class of all matrices m, such that m : x → y . therefore, m ∈ (x : y ) if and only if m(x) = {(mx)n}n∈n ∈ y . a sequence {ξn} in c2 is said to be m-summable to the bicomplex number ξ if m(ξn) converges to ξ which is called m-limit of {ξn}. in [1], the sequence space bvp is defined, which have all sequences such that their ∆-transform is in `p, where ∆ denotes the matrix ∆ = {δnm} as δnm :=   (−1)n−k , n− 1 ≤ m ≤ n 0 , 0 ≤ m ≤ n− 1 or k > n (1.5) we consider following matrices for our c2-sequence spaces. ωnm :=   ξ , 1 ≤ m ≤ nξ 0 , ξ ≺id η (1.6) γnm :=   ξ , m = n −ξ , n− 1 = m 0 , otherwise (1.7) πnm :=   ξ−1 , 1 ≤ m ≤ n 0 , m > n (1.8) and πnm :=   ξ−1 , n = m −ξ−1 , n− 1 = m 0 otherwise (1.9) here we must note that ξ−1 exists if and only if ξ ∈ c2/o2. the integrated and differentiated sequence space were first studied by goes and goes [3]. in this paper, we define and study some c2-sequence space. in the last section we studied the α-dual of these sequence spaces. 2. bicomplex integrated (int) and differentiated (diff ) c2-sequence spaces goes and goes [3] has given the concept of the integrated sequence space. in this section we will obtain the matrix domains of the sequence space `1 by using the bicomplex matrices. we shall show that the integrated and differentiated c2-sequence spaces are banach spaces, bk-spaces, norm isomorphic to `1, separable these int. j. anal. appl. 16 (6) (2018) 897 spaces have ak-property. the spaces ∫ bv and ∫ `1 have monotone norms and therefore the spaces ∫ bv and∫ `1 have ak-property. let ω4 denote the family of bicomplex sequences. now we are giving the definitions of some c2-sequence spaces as follows: definition 2.1 (integrated c2-sequence spaces). `1(c2,m,‖.‖) = { {ξn}∈ ω4 : ∞∑ n=1 m ( ‖nξn‖ k ) < ∞, for some k > 0 } and bv(c2,m,‖.‖) := { ξ = {ξn}∈ ω4 : ∞∑ n=2 m ( ‖∆(nξn)‖ k ) < ∞, for some k > 0 } definition 2.2 (differentiated c2-sequence spaces). `1(c2,m,‖.‖) := { ξ = {ξn}∈ ω4 : ∞∑ n=1 m ( ‖ξn/n‖ k ) < ∞, for some k > 0 } bv(c2,m,‖.‖) := { ξ = {ξn}∈ ω4 : ∞∑ n=2 m ( ‖∆(ξn/n)‖ k ) < ∞, for some k > 0 } we can redefine the spaces `(c2,m,‖.‖), bv(c2,m,‖.‖), `(c2,m,‖.‖) and bv(c2,m,‖.‖) by (`1)ω = `1(c2,m,‖.‖), (`1)γ = bv(c2,m,‖.‖), (`1)π = `1(c2,m,‖.‖), (`1)λ = bv(c2,m,‖.‖). let ξ = {ξn}∈ `1(c2,m,‖.‖). then the ω-transform of ξ is defined as ζn := (ω(ξ))n = n∑ m=1 m ( ‖(mξm)‖ k ) for some k > 0 or equivalently, 1ζn := (ω( 1ξ))n = n∑ m=1 m ( |m 1ξm| k ) and 2ζn := (ω( 2ξ))n = n∑ m=1 m ( |m 2ξm| k ) let ξ = {ξn}∈ bv(c2,m,‖.‖). the γ-transform of {ξn} is defined as ζn := (ω(ξ))n =   ξ1 , p = 1 ∆(pξp) , p ≥ 2 let ξ = {ξn}∈ `1(c2,m,‖.‖). the π-transform of {ξn} is defined as ζn = (π ξ)n = n∑ p=1 m ( ‖ξp/p‖ k ) for some k > 0 let ξ = {ξn}∈ bv(c2,m,‖.‖). the σ-transform of {ξn} is defined as int. j. anal. appl. 16 (6) (2018) 898 ζn := (σ(ξ))n =   ξ1 , p = 1 ∆(p−1 ξp) , p ≥ 2 for the convenience, we use the following notations. k1 = `1(c2,m,‖.‖), k2 = bv(c2,m,‖.‖), k3 = `1(c2,m,‖.‖), k4 = bv(c2,m,‖.‖). proposition 2.1. a sequence {ξn} is in x(c2,m,‖.‖) if and only if {1ξn} ∈ s(a1,m,‖.‖) and {2ξn} ∈ s(a2,m,‖.‖), where x = k1,k2,k3 and k4. theorem 2.1. the space `1(c2,m,‖.‖) is a linear space over c0. proof. let {ξn},{ηn}∈ `1(c2,m,‖.‖). then there exist p1 > 0 and p2 > 0 such that ∞∑ n=1 m ( ‖nξn‖ p1 ) < ∞ and ∞∑ n=1 m ( ‖nξn‖ p2 ) < ∞ now let α,β ∈ c2 \ o2 and p = max{2‖α‖p1, 2‖β‖p2}. then ∞∑ k=1 m ( ‖α∆(k ξk) + β∆(k ηk)‖ p ) ≤ ∞∑ k=1 m ( ‖α ∆(k ξk)‖ p1 ) + ∞∑ k=1 m ( ‖β ∆(k ηk)‖ p2 ) . therefore, {αξn +β ηn}∈ `1(c2,m,‖.‖). hence, the space `1(c2,m,‖.‖) is a linear space over c2\o2. � lemma 2.1. the functions ‖ξ‖`1(c2,m,‖.‖) = ∑∞ m=1 ‖ωnmξm‖ and ‖ξ‖`1(c2,m,‖.‖) = ∑∞ m=1 ‖πnmξm‖ are norms on the spaces `1(c2,m,‖.‖) and `1(c2,m,‖.‖), respectively. theorem 2.2. the spaces `1(c2,m,‖.‖) and `1(c2,m,‖.‖) are banach spaces with norms ‖ξ‖`1(c2,m,‖.‖) =∑∞ m=1 ‖ωnmξm‖ and ‖ξ‖`1(c2,m,‖.‖) = ∑∞ m=1 ‖πnmξm‖, respectively. proof. let {ξnk} be a cauchy sequence in `1(c2,m,‖.‖). then for given � > 0, ∃ m0 ∈ n such that ‖ξnk − ξ m k ‖ < �, ∀n,m > m0 (2.1) therefore, ∑ k ‖ω(ξm)k − ω(ξn)k‖ < �, ∀n,m > m0 ⇒ {ω(ξ1)k, ω(ξ2)k, ω(ξ3)k, . . . , ω(ξn)k, . . .} is a cauchy sequence of bicomplex numbers. since, c2 is a modified banach space. therefore, {ω(ξn)k} is convergence in c2. suppose that ω(ξn)k → ω(ξ), n →∞,∀k using all these limits, we define a sequence {ω(ξ)1, ω(ξ)2, ω(ξ)3, . . . ,}. int. j. anal. appl. 16 (6) (2018) 899 and from equation (2.1), we have p∑ k=1 ‖ω(ξm)k − ω(ξn)k‖ < � (2.2) for any n > m0, by letting m →∞ and p →∞, we have ‖ξn − ξ‖`1(c2,m,‖.‖) ≤ � in particular, ‖ξ‖`1(c2,m,‖.‖) ≤ k + ‖ξ n‖`1(c2,m,‖.‖), for some k ≥ �. hence, ξ ∈ `1(c2,m,‖.‖). further, ξn → ξ. therefore, `1(c2,m,‖.‖) is complete. � corollary 2.1. the space `1(c2,m,‖.‖) is a banach space. theorem 2.3. the spaces `1(c2,m,‖.‖) and `1(c2,m,‖.‖) are bk-spaces with norms ‖ξ‖`1(c2,m,‖.‖) =∑∞ m=1 ‖ωnmξm‖ and ‖ξ‖`1(c2,m,‖.‖) = ∑∞ m=1 ‖πnmξm‖, respectively. proof. let {ξn}∈ `1(c2,m,‖.‖). define fp(ξn) = ξp,∀n ∈ n. then ‖ξn‖`1(c2,m,‖.‖) = ∑ ‖nξn‖ so that ‖nξn‖≤‖ξn‖`1(c2,m,‖.‖) ⇒‖ξn‖≤ k‖ξn‖`1(c2,m,‖.‖) ⇒‖fn(ξp)‖≤ k‖ξn‖`1(c2,m,‖.‖). therefore, fn is a continuous linear functional for each n. so, `1(c2,m,‖.‖) is a bk-space. � in the similar manner, we can prove that `1(c2,m,‖.‖) is a bk-space. theorem 2.4. the space bv(c2,m,‖.‖) is a bk-space with the norm ‖ξ‖bv(c2,m,‖.‖) = ∑∞ m=1 ‖∆(mξm)‖. proof. as we know, bv(c2,m,‖.‖) = (`1)σ is true and `1 is a bk-space with respect to the norm ‖ξ‖`1 and also the matrix σ is a triangular matrix.then by wilansky [?], the space bv is a bk-space. � theorem 2.5. the function ‖ξ‖bv(c2,m,‖.‖) = ∑∞ m=1 ‖∆(mξm)‖ is a norm on bv(c2,m,‖.‖). theorem 2.6. the spaces bv(c2,m,‖.‖) and bv have ak-property. proof. let {ξnk}∈ bv(c2,m,‖.‖) and [ξ n k ] = {ξ n 1 ,ξ n 2 ,ξ n 3 , . . . , . . . ,ξ n k , 0, 0, 0, . . .}. ξnk − [ξ n k ] = {0, 0, 0, . . . ,ξ n k+1,ξ n k+2, . . . ,}. ⇒ ‖ξnk − [ξ n k ]‖bv(c2,m,‖.‖) = ‖0, 0, 0, . . . ,ξ n k+1,ξ n k+2, . . . ,‖bv(c2,m,‖.‖). = ∑ p≥k+1 m ( ‖ξnp /p‖ k ) → 0, as p → 0. ⇒ [ξnk ] → ξ n k as k →∞ int. j. anal. appl. 16 (6) (2018) 900 then, the space bv(c2,m,‖.‖) has ak-property. � theorem 2.7. the spaces `1(c2,m,‖.‖), bv(c2,m,‖.‖), `1(c2,m,‖.‖) and bv(c2,m,‖.‖) are norm isomorphic to `1. proof. we must show that there is a one-one and onto linear mapping between bv(c2,m,‖.‖) and `1. suppose that t : bv(c2,m,‖.‖) → `l be a mapping defined as ξ 7→ tξ. clearly, for ξ = θ ⇒ tξ = θ. now, let η ∈ `1. define a sequence {ξk}∈ bv(c2,m,‖.‖) by ξk = 1 k k∑ p=1 yp then ‖ξk‖bv(c2,m,‖.‖) = ∑ k ∆(k ξk) = ∑ k ∥∥∥∥ k∑ p=1 pηp − (p− 1) k−1∑ p=1 ηp ∥∥∥∥ = ∑ k ‖ηk‖ = ‖η‖`1 therefore, ξn ∈ bv(c2,m,‖.‖). hence, the spaces bv(c2,m,‖.‖) and `1 are isomorphic. � in the similar way, we can prove the isomorphism of remaining spaces. theorem 2.8. the spaces `1(c2,m,‖.‖) and bv(c2,m,‖.‖) have monotone norm. proof. let {ξn}∈ bv(c2,m,‖.‖). define ‖ξn‖bv(c2,m,‖.‖) = ∑ k=1 ∆(kξk) and ‖[ξp]‖bv(c2,m,‖.‖) = ∑n k=1 ‖∆(pξp)‖, ∀{ξk}∈ bv(c2,m,‖.‖). now, suppose q > p, then ‖[ξp]‖bv(c2,m,‖.‖) = p∑ k=1 ‖∆(k ξk)‖ ≤ q∑ k=1 ‖∆(k ξk)‖ ≤ ‖[ξq]‖bv(c2,m,‖.‖) also, sup‖[ξn]‖bv(c2,m,‖.‖) = sup ( n∑ k=1 ‖∆(k ξk)‖ ) = ‖ξn‖bv(c2,m,‖.‖). therefore, the space bv(c2,m,‖.‖) has the monotone norm. � int. j. anal. appl. 16 (6) (2018) 901 remark 2.1. the spaces `1 and bv(c2,m,‖.‖) have ab-property. theorem 2.9. the following statements hold for bv(c2,m,‖.‖) and bv(c2,m,‖.‖) given as : (1) if ζ(m) = {ζ(m)n } is sequence where {ζ (m) n }∈ bv(c2,m,‖.‖) of elements of bv(c2,m,‖.‖), defined as ζ(m)n :=   1/m , n ≥ m 0 , n < m this sequence is the basis for the space bv(c2,m,‖.‖) and select bm = (mξ)m, for all m ∈ n and matrix m defined in equation (??), then ξ ∈ bv(c2,m,‖.‖) has the unique representation of the type: ξ = ∑ m (mξ)m ζ (m) n (2) define a sequence {ηmn } with ηmn ∈ bv(c2,m,‖.‖) as η(m)n :=   m , n ≥ m 0 , n < m then this sequence ζ(m) is a basis for the space bv and for em = (ax)m, for all m ∈ n, where the matrix a is defined by γ = [γnm], every sequence ξ ∈ bv have unique representation as ξ = ∑ m emζ (m) corollary 2.2. the spaces bv(c2,m,‖.‖) and bv(c2,m,‖.‖) are separable. 3. α−duals of the c2-sequence spaces in this section, we determine the α−duals of the spaces k2 and k4. let ξ = {ξn} and η = {ηn} be sequences, and a and b be two subsets of ω4. now let m = (amk) be an infinite matrix of bicomplex numbers. define ξη = (ξnηn), ξ−1n ? b = {ζ ∈ ω4 : ζ ξ ∈ b}. n(a,b) = ∩ξ∈aξ−1 ? b = {ζ ∈ ω4 : ζ ξ ∈ b, for ξ ∈ a}. in particular, for b = `1,cs or bs, we have ξ α = ξ−1 ? `1, ξ β = ξ1 ? cs and ξγ = ξ−1 ? bs. the α− dual of a are given by aα = m(a,`1). suppose that mm = (amk) ∞ k=0 denotes the m-th row of the matrix m. let mm(ξ) = ∑∞ k=0 amkξk, ∀n = 0, 1, 2, . . ., and m(ξ) = [mm(ξ)]∞m=0, where mm ∈ ξβ lemma 3.1. [?] let a1,a2 be to bk-spaces, and m = [ηnm] be a triangular matrix where ξnm ∈ c2/o2, then for matrix sma1 = [ξnm] defined with ν = {νm}∈ a1 as int. j. anal. appl. 16 (6) (2018) 902 ξnm = n∑ i=1 νi ηnm µim then a2a1(m) ⊂ a1(m) holds if and only if the matrix sma1 = mdνm −1 ∈ (a1 : a1), where dν is a diagonal matrix such that [dν]nn = νn, ∀n ∈ n. lemma 3.2. [?] let {γk} be a sequence in ω4 and m = [ηnm] be an invertible triangular matrix. define a matrix sma1 = [ξnm] as ξnm = n∑ i=m ηi µim then a β 1 (m) = {ηm ∈ ω4 : s(m) ∈ (a1 : c)} and a γ 1 (m) = {ηm ∈ ω4 : s(m) ∈ (a1 : `∞)} lemma 3.3. let m = [ξnm] be an infinite matrix of bicomplex numbers. then (1) m ∈ (`1 : `1) ⇐⇒ sup ∑ k∈n ‖ξnm‖ < ∞. (2) m ∈ (`1 : `∞) ⇐⇒ supk,n∈n ‖ξnm‖ < ∞ (3) m ∈ (`1,c) ⇐⇒ supk,n∈n ‖ξnm‖ < ∞ and for some sequence {κm} such that lim n→∞ ξnm = κm theorem 3.1. for the space bv(c2,m,‖.‖), we have bv(c2,m,‖.‖)α = α1 where α1 = { ξ = {ξn}∈ ω4 : ∑ k ∥∥∥∥ n∑ m=1 m ( ‖∆(ξm/m)‖ k ) ηk ∥∥∥∥ < ∞, (ηk) ∈ bv(c2,m,‖.‖) for some k > 0 } proof. {ξn} be any sequence in ω4. assume the following relation ξnηn = n∑ k=1 m ( ‖∆(ξn/n)‖ k ) ηk = (eη)k where e = {enk} is defined by enm =   m ( ‖∆(ξn/n)‖ k ) , 1 ≤ m ≤ n 0 , n < m (3.1) therefore, from the equation (3.1) and the lemma (3.3) we have{ m ( ‖∆(ξn/n)‖ k ) ζn } ∈ `1 if and only if eη ∈ `1, whenever η ∈ `1. so, ξ = {ξns}∈ bv(c2,m,‖.‖)α if and only if e ∈ (bv(c2,m,‖.‖) : `1). hence proved. � int. j. anal. appl. 16 (6) (2018) 903 analogously, we can prove the following theorems. theorem 3.2. for the space bv(c2,m,‖.‖) bv(c2,m,‖.‖)α = α2 where α1 = { ξ = {ξn}∈ ω4 : ∑ k ∥∥∥∥ n∑ m=1 m ( ‖(ξm/m)‖ k ) ηk ∥∥∥∥ < ∞, (ηk) ∈ bv(c2,m,‖.‖) for some k > 0 } theorem 3.3. for the space `1(c2,m,‖.‖) `1(c2,m,‖.‖)α = α1 where α1 = { ξ = {ξn}∈ ω4 : ∑ k ∥∥∥∥ n∑ m=1 m ( ‖(mξm)‖ k ) ηk ∥∥∥∥ < ∞, (ηk) ∈ bv(c2,m,‖.‖) for some k > 0 } theorem 3.4. for the space `1(c2,m,‖.‖) `1(c2,m,‖.‖)α = α2 where α1 = { ξ = {ξn}∈ ω4 : ∑ k ∥∥∥∥ n∑ m=1 m ( ‖(∆(mξm))‖ k ) ηk ∥∥∥∥ < ∞, (ηk) ∈ bv(c2,m,‖.‖) for some k > 0 } references [1] b. atlay, f. basar, summability theory and its applications, bentham science publishers, e-books, monographs, istambul, (2012). [2] b. atlay, f. basar, certain topological properties and duals of the domain of a triangle matrix in a sequence spaces, j. math. anal. appl., 336 (2007), 632-645. [3] goes and goes, sequences of bounded variation and sequences of fourier coefficients i, math. z, 118 (1970) 93-102. [4] h. kizmaz, on certain sequence spaces, canad. math. bull., 24(2),169-176, (1981). [5] m. et, r. colak, on some generalized difference sequence spaces, soochow j. math., 21(4), 377-386, (1995). [6] h. nakano, modular sequence spaces, proc. japan acad., 27, 508-512, (1951). [7] s.d. prashar and b chaudhary, sequence spaces defined by orlicz functions, indian j. pure appl. mathematics, 25, 419-478, (1994). [8] g.b. price, an introduction to multicomplex space and functions, marcel dekker, inc., new york, (1991). [9] rajiv k srivastava, bicomplex numbers: analysis and applications, math. student, 72 (1-4), 63-87, (2007). 1. introduction 2. bicomplex integrated (int) and differentiated (diff) c2-sequence spaces 3. -duals of the c2-sequence spaces references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 79-86 http://www.etamaths.com fixed point theorems for α−ψ−quasi contractive mappings in metric-like spaces vildan ozturk abstract. in this paper, we give fixed point theorems for α − ψ−quasi contractions and α−ψ−p−quasi contractions in complete metric-like spaces. 1. introduction and preliminaries fixed point theory became one of the most interesting area of research in the last fifty years. many authors studied contractive type mappings on a complete metric space which are generalizations of banach contraction principles. recently, samet et al. [17] introduced the notion of α − ψ contractive mappings and established some fixed point theorems in complete metric spaces. later some other authors generalized α−ψ contractions ([5-7][9-14],[18]). in last years, many generalizations of the concept of metric spaces are defined and some fixed point theorems was proved in these spaces.in particular, in 1994, matthews introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the banach contraction principle can be generalized to the partial metric context for applications in program verification ([15]). later on, many researchers studied fixed point theorems in partial metric spaces ([1],[2],[8],[16],[20]). recently, amini-harandi generalized the partial metric spaces by introducing the metric-like spaces and proved some fixed point theorems in such spaces ([3]). after authors gived some fixed point theorems in metric-like spaces ([19]). in this paper, we introduce the notion of α−ψ−quasi contractive mappings in complete metric-like spaces and in last parts we give α−ψ−p−quasi contraction in metric like spaces. our results are generalisations of the many existing results in the literature. first we give some definitions and facts about metric-like spaces. definition 1. ([3]) a mapping σ : x × x → r+ , where x is a nonempty set, is said to be metric-like on x if for any x,y,z ∈ x, the following three conditions hold true: 2010 mathematics subject classification. 47h10. key words and phrases. α − ψ−quasi contraction, α − ψ − p−quasi contraction fixed point, metric-like spaces. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 79 80 ozturk (σ1) σ (x,y) = 0 ⇒ x = y; (σ2) σ (x,y) = σ (y,x) , (σ3) σ (x,z) ≤ σ (x,y) + σ (y,z) . the pair (x,σ) is called a metric-like space.then a metric-like on x satisfies all of the conditions of a metric except that σ (x,x) may be positive for x ∈ x. each metric-like σ on x generates a topology τσ on x whose base is the family of open σ−balls bσ (x,ε) = {y ∈ x : |σ (x,y) −σ (x,x)| < ε} for all x ∈ x and ε > 0. then the sequence {xn} in the metric-like space (x,σ) converges to a point x ∈ x if and only if lim n→∞ σ (xn,x) = σ (x,x) . let (x,σ) and (y,τ) be metric-like spaces and let f : x → y be a continuous mapping. then lim n→∞ xn = x =⇒ lim n→∞ f(xn) = f(x). a sequence {xn} ∞ n=0 of elements of x is called σ−cauchy if limn,m→∞ σ (xn,xm) exists and is finite.the metric-like space (x,σ) is called complete if for each σ−cauchy sequence {xn} ∞ n=0 , there is some x ∈ x such that lim n→∞ σ (xn,x) = σ (x,x) = lim n,m→∞ σ (xn,xm) . every partial metric space is a metric-like space. below we give another example of a metric-like space. example 1. ([3]) let x = {0, 1} , and let σ(x,y) = { 2, if x = y = 0 1, otherwise then (x,σ) is a metric-like space, but since σ (0, 0) � σ (0, 1) , then (x,σ) is not a partial metric space. 2. fixed point results for α−ψ contractive mappings denote by ψ the family of nondecreasing functions ψ : [0,∞) → [0,∞) such that limn→∞ψ n (t) = 0 for all t > 0. lemma 1. if ψ ∈ ψ, then the following are satisfied. (a) ψ (t) < t for all t > 0 (b) ψ (0) = 0 (c) ψ is right continuous at t = 0. remark 1. (a) if ψ : [0,∞) → [0,∞) is nondecreasing such that ∞∑ n=1 ψn (t) < ∞ for each t > 0, then ψ ∈ ψ. (b) if ψ : [0,∞) → [0,∞) is upper semicontinuous such that ψ (t) < t for all t > 0, then limn→∞ψ n (t) = 0 for all t > 0. fixed point theorems 81 definition 2. ([16])let t : x → x and α : x ×x → [0,∞) . we say that t is α-admissible if x,y ∈ x, α (x,y) ≥ 1 ⇒ α (tx,ty) ≥ 1. definition 3. let (x,σ) be a complete metric-like space and t : x → x be a given mapping. we say that t is an α − ψ−quasi contractive mapping if there exist α : x ×x → [0,∞) and ψ ∈ ψ such that (1) α (x,y) σ (tx,ty) ≤ ψ(m(x,y)) for all x,y ∈ x where m(x,y) = max{σ (x,y) ,σ (x,tx) ,σ (y,ty) ,σ (x,ty) ,σ (y,tx) ,σ (x,x) ,σ (y,y)} . theorem 2. let (x,σ) be a complete metric-like space and t : x → x be an α − ψ−quasi contractive mapping. assume that there exists x0 ∈ x such that o (x0,∞) = {tnx0 : n = 0, 1, 2...} is bounded and (i) α ( tix0,t jx0 ) ≥ 1 for all i,j ≥ 0 with i < j, (ii) t is σ−continuous or limn→∞ inf α (t nx0,x) ≥ 1 for any cluster point x of {tnx0} . then t has a fixed point. proof. let x0 ∈ x be such that o (x0,∞) = {tnx0 : n = 0, 1, 2...} is bounded and α ( tix0,t jx0 ) ≥ 1 for all i,j ≥ 0 with i < j. define the sequence {xn} in x by xn+1 = txn for all n ∈ n∪{0} . if xn = xn+1 for some n ∈ n, then x∗ = xn is a fixed point of t. assume that xn 6= xn+1 for all n ∈ n∪{0} . now we shall show {xn} is a σ−cauchy sequence. let δ (xn) = diam ({txn,txn+1, ...}) for n = 0, 1, 2, .... since δ (xn) ≤ δ (x0) and δ (x0) < ∞ we assert that for n = 0, 1, 2, ... (2) δ (xn) ≤ ψn (δ (x0)) . for n = 0, (2) holds. suppose that (2) holds for n = k. we will show that ( 2) holds when n = k + 1. let txr−1,txs−1 ∈ {txk,txk+1, ...} for any r,s ≥ k + 1. then σ (xr,xs) = σ (txr−1,txs−1) ≤ α (xr−1,xs−1) σ (txr−1,txs−1) ≤ ψ(m (xr,xs))(3) 82 ozturk where m (xr,xs) = max   σ (xr−1,xs−1) ,σ (xr−1,txr−1) ,σ (xs−1,txs−1) , σ (xr−1,txs−1) ,σ (xs−1,txr−1) , σ (xr−1,xr−1) ,σ (xs−1,xs−1)   = max   σ (txr,txs) ,σ (txr,txr+1) ,σ (txs,txs+1) ,σ (txr,txs+1) ,σ (txs,txr+1) , σ (txr,txr) ,σ (txs,txs)   ≤ δ (xk) . then by (3), σ (xr,xs) = σ (txr−1,txs−1) ≤ ψ(δ (xk)) ≤ ψ ( ψk (δ (x0)) ) = ψk+1 (δ (x0)) . thus (2) is proved for n = 0, 1, 2, .... hence from (2) we have limn→∞δ (xn) = 0. thus {xn} is a σ−cauchy sequence in (x,σ) . by the completeness of x , there exists z ∈ x such that limn→∞xn = z, that is, (4) lim n→∞ σ (xn,z) = σ (z,z) = lim n,m→∞ σ (xn,xm) = 0. if t is σ−continuous, lim n→∞ σ (txn,tz) = lim n→∞ σ (xn+1,tz) = σ (z,tz) = 0. this proves z is a fixed point. if limn→∞ inf α (t nx0,x) ≥ 1 for any cluster point x of {tnx0} , there exists n0 ∈ n such that α (xn,z) ≥ 1, for all n > n0. thus, σ (xn+1,tz) ≤ σ (txn,tz) ≤ α (xn,z) σ (txn,tz) ≤ ψ (m (xn,z))(5) where m (xn,z) = max { σ (xn,z) ,σ (xn,txn) ,σ (z,tz) ,σ (xn,tz) , σ (z,txn) ,σ (xn,xn) ,σ (z,z) } if σ (z,tz) > 0, using upper semicontinuity of ψ, σ (z,tz) = limn→∞ sup σ (xn+1,tz) ≤ limn→∞ sup ψ(m (xn,z)) ≤ ψ (σ(z,tz)) < σ(z,tz) which is a contradiction. thus, we obtain σ (tz,z) = 0.so tz = z. � fixed point theorems 83 example 2. let x = {0, 1, 2} . define σ : x ×x → r+ as follows: σ (0, 0) = 0 σ(1, 1) = 3 σ(2, 2) = 1 σ (0, 1) = σ (1, 0) = 7 σ (0, 2) = σ (2, 0) = 3 σ (1, 2) = σ (2, 1) = 4. then (x,σ) is a complete metric-like space. define the mapping t : x → x by t0 = 0, t1 = 2 and t2 = 0 and α : x ×x → [0,∞) by α (x,y) = { 1 4 , if (x,y) 6= (0, 0) 1, (x,y) = (0, 0) then t is an α− ψ-quasi contractive mapping with ψ (t) = t 1+t . moreover, there exists x0 ∈ x such that α ( tix0,t jx0 ) ≥ 1, for all i,j ≥ 0 with i < j. so for x0 = 0, we have α ( ti0,tj0 ) = α (0, 0) = 1. obviously (1) is satisfied for all x,y ∈ x. all hypotheses of theorem 2 are satisfied. consequently t has a fixed point. and x0 = 0 is fixed point of t. taking in theorem 2, α (x,y) = 1 for all x,y ∈ x,we obtain immediately the following corollaries. corollary 3. let (x,σ) be a complete metric-like space and t : x → x be a given mapping. suppose that there exists a function ψ ∈ ψ such that σ (tx,ty) ≤ ψ(m(x,y)) where m(x,y) = max{σ (x,y) ,σ (x,tx) ,σ (y,ty) ,σ (x,ty) ,σ (y,tx) ,σ (x,x) ,σ (y,y)} for all x,y ∈ x. then t has a unique fixed point. corollary 4. let (x,σ) be a complete metric-like space and t : x → x be a given mapping. suppose that there exists a constant c ∈ (0, 1) such that σ (tx,ty) ≤ cm(x,y) where m(x,y) = max{σ (x,y) ,σ (x,tx) ,σ (y,ty) ,σ (x,ty) ,σ (y,tx) ,σ (x,x) ,σ (y,y)} for all x,y ∈ x. then t has a unique fixed point. 84 ozturk 3. fixed point results for α−ψ −p−quasi contractive mappings in this section we give α−ψ −p−quasi contraction in consideration of aminiharandi [4]. definition 4. let (x,σ) be a complete metric-like space and t : x → x be a given mapping. we say that t is an α−ψ−p−quasi contractive mapping if there exist α : x ×x → [0,∞) and ψ ∈ ψ such that (6) α (x,y) σ (tpx,tpy) ≤ ψ(m(x,y)) for all x,y ∈ x where m(x,y) = max { σ ( tiu,tjv ) : u,v ∈{x,y} , 0 ≤ i,j ≤ p and i + j < 2p } . theorem 5. let (x,σ) be a complete metric-like space and let p ∈ n. suppose that t : x → x be an α−ψ−p−quasi contractive map. assume that there exists x0 ∈ x such that o (x0,∞) = {tnx0 : n = 0, 1, 2...} is bounded and (i) there exists x0 ∈ x such that α ( tix0,t jx0 ) ≥ 1 for all i,j ∈ n∪{0} , (ii) tm : x → x is σ−continuous for some m ∈ n. then t has a fixed point. proof. let x0 ∈ x be such that o (x,∞) = {x,tx,...} is bounded and α ( tix0,t jx0 ) ≥ 1 for all i,j ∈ n∪{0} . define the sequence {xn} in x by xn+1 = txn for all n ≥ 0. let n be a positive integer with n ≥ p, and let i,j ∈ {p,p + 1, ...,n} . since t is an α−ψ −p−quasi contractive map, then σ ( tix,tjx ) = σ ( tpti−px,tptj−px ) ≤ α ( ti−px,tj−px ) σ ( tpti−px,tptj−px ) ≤ ψ{max { σ ( tkti−px,t ltj−px ) : 0 ≤ k,l ≤ p and k + l < 2p } } ≤ ψ(δ [o (x,n)])(7) hence by lemma 1 (a) , σ ( tix,tjx ) < δ [o (x,n)] . thus for sufficiently large n ∈ n there exist positive integers k,l with k < p and p ≤ l ≤ n such that σ ( tkx,t lx ) = δ [o (x,n)] . we show that {tnx} is a σ−cauchy sequence. without loss of generality assume that p ≤ n < m.then, from (7) σ (tnx,tmx) = σ ( tptn−px,tm−n+ptn−px ) ≤ α ( tn−px,tm−px ) σ ( tptn−px,tm−n+ptn−px ) ≤ ψ ( δ [ o ( tn−px,m−n + p )]) . by (7), there exists positive integers k1 and l1 with k1 < p and p ≤ l1 ≤ m−n + p such that δ [ o ( tn−px,m−n + p )] = σ ( tk1tn−px,t l1tn−px ) . fixed point theorems 85 similarly we have σ ( tk1tn−px,t l1tn−px ) = σ ( tk1+ptn−2px,t l1tn−px ) ≤ ψ(δ [ o ( tn−2px,m−n + 2p )] ). thus, σ (tnx,tmx) ≤ ψ(δ [ o ( tn−px,m−n + p )] ) ≤ ψ2(δ [ o ( tn−2px,m−n + 2p )] ). proceeding in this manner, we obtain σ (tnx,tmx) ≤ ψ[ n p ] ( δ [ o ( tn−[ n p ]px,m−n + [ n p ] p )]) ≤ ψ[ n p ] (δ [o (x,m + p)]) . hence σ (tnx,tmx) ≤ ψ[ n p ] (δ [o (x,∞)]) . by definition of ψ, limn→∞σ (t nx,tmx) = 0. hence we conclude that {tnx} is a σ−cauchy sequence. by the completeness of x, there is some u ∈ x such that lim n→∞ σ (tnx,u) = lim n→∞ σ (tnx,tmx) = σ (u,u) = 0 for each x ∈ x. now we show that tu = u. by the continuity of tm, lim n→∞ σ ( tm+nx,tu ) = σ (u,tu) = 0. hence, u = tu. � example 3. let x = [0,∞) and σ : x ×x → [0,∞) be defined σ (x,y) = { 0, x = y max{x,y} , otherwise . then, (x,σ) is a complete metric-like space. let q and q′denote respectively the set of rational numbers and irrational numbers. let t : [0,∞) → [0,∞) and α : x ×x → [0,∞) be defined by t (x) = {√ 2, x ∈ q√ 3, otherwise α (x,y) = { 1, x ∈ q′ 0, otherwise then t is an α− ψ-2−contractive mapping with ψ (t) = t 1+t . then t2 (x) = √ 3 for each x ∈ x. moreover t is discontinuous and t2 is continuous. then all conditions of theorem 5 are satisfied. and x = √ 3 is fixed point of t. 86 ozturk references [1] t. abdeljawad, fixed points for generalized weakly contractive mappings in partial metric spaces, mathematical and computer modelling 54 (2011), no.11-12, 2923–2927. [2] i. altun, f. sola, h. simsek, generalized contractions on partial metric spaces. topol. appl. 157 (2010), no 18, 2778-2785. [3] a. amini-harandi, metric-like spaces, partial metric spaces and fixed points. fixed point theory appl. 2012 (2012), article id 204. [4] a. amini-harandi, p−quasi contraction maps and fixed points in metric spaces. journal of nonlinear and convex analysis 15 (2014), no. 4, 727-731. [5] g.v.r. babu, k.t. kidane, fixed points of almost generalized α−ψ contractive maps. int. j. math. sci. comput. 3 (2013), no. 1, 30-38. [6] s.h. cho, j.s. bae, fixed points of weak α−contraction type maps. fixed point theory appl. 2014 (2014) article id 175. [7] s.h. cho, j.s. bae, fixed points theorems for α − ψ-quasi contractive mappings in metric spaces. fixed point theory appl. 2013 (2013), article id 268. [8] l. cirić, b. samet, h. aydi, c. vetro, common fixed points of generalized contractions on partial metric spaces and an application. appl. math. comput. 218 (2011), no. 6, 2398-2406. [9] s. fathollahi, n. hussain, l.a. khan, fixed point results for modified weak and rational α − ψ−contractions in ordered 2-metric spaces. fixed point theory appl. 2014 (2014), article id 6. [10] j. hassanzadeasl, common fixed point theorems for α − ψ contractive type mappings. international journal of analysis 2013 (2013), article id 654659. [11] n. hussain, e. karapınar, p. salimi, f. akbar, α−admissible mappings and related fixed point theorems. j. i̇neq. appl. 2013 (2013),article id 114. [12] m. jleli, e. karapınar, b. samet, fixed point results for α − ψλ-contractions on gauge spaces and applications. abstract appl. anal. 2013 (2013) article id 730825. [13] e. karapınar, b. samet, generalized α − ψ contractive mappings and related fixed point theorems with applications. abstract appl. anal. 2012 (2012), article id 793486. [14] p. kumam, c. vetro, p. vetro, fixed points for weak α − ψ−contractions in partial metric spaces. abstract appl. anal. 2013 (2013) article id 986028. [15] s.g. matthews, partial metric topology. in: proc. 8th summer conference on general topology and applications. ann. new york acad. sci. 728 (1994),183-197. [16] s. oltra, o. valero, banach’s fixed theorem for partial metric spaces. rend. ist. mat. univ. trieste 36 (2004),17-26. [17] b. samet, c. vetro, p. vetro, fixed point theorems for α − ψ−contractive type mappings. nonlinear analysis 75 (2012), no. 4, 2154–2165. [18] w. sintunavarat, s. plubtieng, p. katchang, fixed point result and applications on a b−metric space endowed with an arbitrary binary relation. fixed point theory appl. 2013 (2013) article id 296. [19] s. shukla, s. radenović, v.ć. rajić, some common fixed point theorems in 0-σ−complete metric-like spaces. vietnam j math. 41 (2013) 341-352. [20] o. valero, on banach fixed point theorems for partial metric spaces. applied general topology 6 (2005), no. 2, 229–240. department of mathematics, faculty of science and arts, artvin coruh university, artvin, turkey international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 1-9 http://www.etamaths.com generalized norms inequalities for absolute value operators ilyas ali∗, hu yang, abdul shakoor abstract. in this article, we generalize some norms inequalities for sums, differences, and products of absolute value operators. our results based on minkowski type inequalities and generalized forms of the cauchy-schwarz inequality. some other related inequalities are also discussed. 1. introduction in this article, notations are same as in [3], for reader convenience we recall that let h be a complex separable hilbert space and b(h) denote the c∗-algebra of all bounded linear operators on h. let |a| denote the absolute value of a ∈ b(h), and is defined as |a| = (a∗a) 1 2 , where a∗ is the adjoint operator of a. if a is compact operator on complex separable hilbert space h, then the singular values of a enumerated as s1(a) ≥ s2(a) ≥ ... which are the eigenvalues of positive operator |a|. a norm |||.||| stand for untarily invariant norm i.e., a norm with the property that |||uav ||| = |||a||| for all a and for all unitary operators u, v in b(h). operator norm and schatten p-norms are denoted as ||.|| and ||.||p respectively. except the operator norm, which is defined on all of b(h), each unitarily invariant norm is defined on an ideal in b(h). when we use the symbol |||a||| it is implicit understood that operator a is in this ideal. for 0 < p < 1, a norm ‖.‖p defines a quasi-norm. for this norm it is well-known that ‖a + b‖p ≤ 2 1 p −1 (‖a‖p + ‖b‖p) .(1.1) by the definition of the schatten p-norm, we have ‖ | a |r ‖p ≤‖a‖prp,(1.2) where r,p are real numbers. also, since the singular values of | a |r and | a∗ |r are same, so ||| | a |r ||| = ||| | a∗ |r |||.(1.3) the unitarily invariant norms for differences of the absolute values of hilbert space operators have attracted the attention of several mathematicians. it has been 2000 mathematics subject classification. 47a30; 47a63; 47b10. key words and phrases. unitarily invariant norm; schatten p-norm; cauchy-schwarz inequality; minkowski inequality; absolute value operators. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 ali, yang and shakoor proved by k. shebrawi and h. albadawi in [3] that if ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and 0 < r ≤ 1 . then ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2n1− r 2 n∑ i=1 |||(a∗i | xi | ai) r ||| 1 2 |||(b∗i | xi | bi) r ||| 1 2 ,(1.4) which leads to the following inequality ||| | a |2r − | b |2r ||| ≤ 21−r||| | a + b |2r ||| 1 2 ||| | a−b |2r ||| 1 2 .(1.5) inequality (1.5) generalize the result presented by bhatia in [5] as follows: ||| | a | − | b | ||| ≤ √ 2||| | a + b | ||| 1 2 ||| | a−b | ||| 1 2 .(1.6) k. shebrawi and h. albadawi also proved in [3] that if a,b,x be operators in b(h) such that x is self adjoint operator and 0 < r ≤ 1 2 , 1 ≤ p ≤ 2, then ‖ | a∗xb + b∗xa |r ‖p ≤ 2 1 p −r+ 1 2‖(a∗ | x | a)r ‖ 1 2 p‖(b∗ | x | b) r ‖ 1 2 p ,(1.7) this leads to the following inequality ‖ | a |2r − | b |2r ‖p ≤ 2 1 p −2r+ 1 2‖ | a + b |2r ‖ 1 2 p‖ | a−b |2r ‖ 1 2 p .(1.8) inequality (1.8) generalize the following result in [5] ‖ | a | − | b | ‖p ≤ 2 1 p −1 2‖ | a + b | ‖ 1 2 p‖ | a−b | ‖ 1 2 p ,(1.9) where 1 ≤ p ≤ 2. this article we have organized as: in section 2, we generalize the inequality (1.5) and also we discuss some other related results. in section 3, we present some schatten p-norms inequalities, one of which generalize the inequality (1.8). 2. generalized unitarily invariant norms inequalities for absolute value operators in this section, we generalize some unitarily invariant norms inequalities for absolute value operators. our results based on several lemmas. first two lemmas contain norm inequalities of minkowski type and generalized forms of the cauchyschwarz inequality, see [4] and [2] respectively. lemma 2.1. let ai,bi ∈ b(h), i = 1, 2, ...,n. then n 1 2 −1 r ||| n∑ i=1 | ai + bi |r ||| 1 r ≤ 2 1 r −1 ( ||| n∑ i=1 | ai |r ||| 1 r + ||| n∑ i=1 | bi |r ||| 1 r ) (2.1) for 0 < r ≤ 1, n−| 1 r −1 2 |||| n∑ i=1 | ai + bi |r ||| 1 r ≤ ||| n∑ i=1 | ari | ||| 1 r + ||| n∑ i=1 | bi |r ||| 1 r(2.2) generalized norms inequalities for absolute value operators 3 for r ≥ 1, and n−(1− 1 p )/r||| n∑ i=1 | ai + bi |r ||| 1 r p ≤ ||| n∑ i=1 | ari | ||| 1 r p + ||| n∑ i=1 | bi |r ||| 1 r p(2.3) for 1 ≤ p,r < ∞. lemma 2.2. for a,b,x ∈ b(h), for all unitarily invariant norms and for all positive real numbers µ1, µ2 and r such that µ −1 1 + µ −1 2 = 1, we have ||| | a∗xb |r ||| ≤ ||| | aa∗x | µ1r 2 ||| 1 µ1 ||| | xbb∗ | µ2r 2 ||| 1 µ2 ,(2.4) and also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ||| | a∗xb |r ||| ≤ ||| ( a∗f2(| x∗ |)a )µ1r 2 ||| 1 µ1 ||| ( b∗g2(| x |)b )µ2r 2 ||| 1 µ2 .(2.5) for following two lemmas see [1] and [6, pp. 293, 294]. lemma 2.3. let a be a positive operator in b(h). then for every normalized unitarily invariant norm (i.e.,|||diag(1, 0, 0, ..., 0)||| = 1), we have |||a|||r ≤ |||ar|||(2.6) for 0 ≤ r ≤ 1 and |||ar||| ≤ |||a|||r(2.7) for r ≥ 1. lemma 2.4. let a and b be a positive operator in b(h). then |||ar −br||| ≤ ||| | a−b |r |||(2.8) for 0 ≤ r ≤ 1 and ||| | a−b |r ||| ≤ |||ar −br|||(2.9) for r ≥ 1. last lemma is a consequence of the concavity (convexity) of the function f(t) = tr, 0 ≤ r ≤ 1 (r ≥ 1). lemma 2.5. let a and b be two positive real numbers (a + b)r ≤ ar + br(2.10) for 0 ≤ r ≤ 1 and (a + b)r ≤ 2r−1(ar + br)(2.11) for r ≥ 1. theorem 2.1. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers, such that µ −1 1 +µ −1 2 = 1 4 ali, yang and shakoor and 0 < r ≤ 1 . then ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2n1− r 2 n∑ i=1 ||| | aia∗ixi | µ1r 2 ||| 1 µ1 ||| | xibib∗i | µ2r 2 ||| 1 µ2 ,(2.12) also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2n1− r 2 n∑ i=1 ||| ( a∗if 2(| xi |)ai )µ1r 2 ||| 1 µ1 ||| ( b∗i g 2(| xi |)bi )µ2r 2 ||| 1 µ2 . (2.13) proof. by applying (2.1), the triangle inequality, (1.3) and (2.4), respectively, we obtain ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| 1 r ≤ 2 1 r −1n 1 r −1 2 ( ||| n∑ i=1 | a∗ixibi | r ||| 1 r + ||| n∑ i=1 | b∗i xiai | r ||| 1 r ) ≤ 2 1 r −1n 1 r −1 2  ( n∑ i=1 ||| | a∗ixibi | r ||| )1 r + ( n∑ i=1 ||| | b∗i xiai | r ||| )1 r   ≤ 2 1 r n 1 r −1 2 ( n∑ i=1 ||| | a∗ixibi | r ||| )1 r ≤ 2 1 r n 1 r −1 2 ( n∑ i=1 ||| | aia∗ixi | µ1r 2 ||| 1 µ1 ||| | xibib∗i | µ2r 2 ||| 1 µ2 )1 r . the proof is completed. by applying (2.5) and the proof of the first part of theorem 2.1, we can obtain (2.13). � replace ai, bi by ai + bi, ai −bi respectively and also take f(t) = g(t) = t 1 2 in theorem 2.1, then, we obtain the following result. corollary 2.1. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers, such that µ −1 1 +µ −1 2 = 1 and 0 < r ≤ 1. then 2r−1n r 2 −1||| n∑ i=1 | a∗ixiai −b ∗ i xibi | r ||| ≤ n∑ i=1 ||| | (ai + bi)(ai + bi)∗xi | µ1r 2 ||| 1 µ1 ||| | xi(ai −bi)(ai −bi)∗ | µ2r 2 ||| 1 µ2 , generalized norms inequalities for absolute value operators 5 and 2r−1n r 2 −1||| n∑ i=1 | a∗ixiai −b ∗ i xibi | r ||| ≤ n∑ i=1 |||((ai + bi)∗ | xi | (ai + bi)) µ1r 2 ||| 1 µ1 |||((ai −bi)∗ | xi | (ai −bi)) µ2r 2 ||| 1 µ2 . by similar way applying to the proof of theorem 2.1, based on the inequality (2.2), we can obtain the following result. theorem 2.2. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers, such that µ −1 1 +µ −1 2 = 1 and r ≥ 1. then ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2rn|1− r 2 | n∑ i=1 ||| | aia∗ixi | µ1r 2 ||| 1 µ1 ||| | xibib∗i | µ2r 2 ||| 1 µ2 ,(2.14) also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2rn|1− r 2 | n∑ i=1 ||| ( a∗if 2(| xi |)ai )µ1r 2 ||| 1 µ1 ||| ( b∗i g 2(| xi |)bi )µ2r 2 ||| 1 µ2 . (2.15) corollary 2.2. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers, such that µ −1 1 +µ −1 2 = 1 and r ≥ 1. then n−|1− r 2 |||| n∑ i=1 | a∗ixiai −b ∗ i xibi | r ||| ≤ n∑ i=1 ||| | (ai + bi)(ai + bi)∗xi | µ1r 2 ||| 1 µ1 ||| | xi(ai −bi)(ai −bi)∗ | µ2r 2 ||| 1 µ2 , and n−|1− r 2 |||| n∑ i=1 | a∗ixiai −b ∗ i xibi | r ||| ≤ n∑ i=1 |||((ai + bi)∗ | xi | (ai + bi)) µ1r 2 ||| 1 µ1 |||((ai −bi)∗ | xi | (ai −bi)) µ2r 2 ||| 1 µ2 . remark 2.1. if we take f(t) = tα and g(t) = t(1−α) for α ∈ [0, 1], then from the inequality (2.13) we can obtain important special case. also, if we take f(t) = 6 ali, yang and shakoor g(t) = t 1 2 then from (2.13) we have ||| n∑ i=1 | a∗ixibi + b ∗ i xiai | r ||| ≤ 2n1− r 2 n∑ i=1 |||(a∗i | xi | ai) µ1r 2 ||| 1 µ1 |||(b∗i | xi | bi) µ2r 2 ||| 1 µ2 , which is the more general form of the inequality (1.4). similar remark we can give for the inequality (2.15), which is more general form of the inequality (2.19) in [3]. our following result contains a promised generalization of (1.5). corollary 2.3. let a and b be operators in b(h) and µ1, µ2 are positive real numbers such that µ−11 + µ −1 2 = 1 then ||| | a |2r − | b |2r ||| ≤ 21−r||| | a + b |µ1r ||| 1 µ1 ||| | a−b |µ2r ||| 1 µ2(2.16) for 0 < r ≤ 1, and ||| | a∗a−b∗b |r ||| ≤ ||| | a + b |µ1r ||| 1 µ1 ||| | a−b |µ2r ||| 1 µ2(2.17) for r ≥ 1. proof. by (2.8) and from second result of corollary (2.1), we have ||| | a |2r − | b |2r ||| ≤ ||| || a |2 − | b |2|r ||| ≤ 21−r||| | a + b |µ1r ||| 1 µ1 ||| | a−b |µ2r ||| 1 µ2 , for 0 < r ≤ 1. and inequality (2.17) is a special case of the first result of corollary (2.2). remark 2.2. special cases of second results in corollary 2.1 and 2.2 respectively are: let a,b,x be operators in b(h) such that x is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 + µ −1 2 = 1, then ||| | a∗xa−b∗xb |r ||| ≤ 21−r|||((a + b)∗ | x | (a + b)) µ1r 2 ||| 1 µ1 |||((a−b)∗ | x | (a−b)) µ2r 2 ||| 1 µ2 . for 0 < r ≤ 1, and ||| | a∗xa−b∗xb |r ||| ≤ |||((a + b)∗ | x | (a + b)) µ1r 2 ||| 1 µ1 |||((a−b)∗ | x | (a−b)) µ2r 2 ||| 1 µ2 . for r ≥ 1. corollary 2.4. let a be an operator in b(h) and if µ1, µ2 are positive real numbers such that µ−11 + µ −1 2 = 1, then ||| | a∗a−aa∗ |r ||| ≤ 21+r||| | rea |µ1r ||| 1 µ1 ||| | ima |µ2r ||| 1 µ2 , for 0 < r ≤ 1, and ||| | a∗a−aa∗ |r ||| ≤ 22r||| | rea |µ1r ||| 1 µ1 ||| | ima |µ2r ||| 1 µ2 , for r ≥ 1. generalized norms inequalities for absolute value operators 7 3. generalized norm inequalities for the schatten p-norm schatten p-norm for absolute value operators are discussed in this section. our these results refine some of the results in section 2 and also, our first result leads to a generalization of (1.8). theorem 3.1. let a,b,x be operators in b(h) such that x is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 + µ −1 2 = 1, then ‖ | a∗xb + b∗xa |r ‖p ≤ 2 1 p −r+ 1 2‖ | aa∗x | µ1r 2 ‖ 1 µ1 p ‖ | xbb∗ | µ2r 2 ‖ 1 µ2 p ,(3.1) also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ‖ | a∗xb + b∗xa |r ‖p ≤ 2 1 p −r+ 1 2‖ ( a∗f2(| x |)a )µ1r 2 ‖ 1 µ1 p ‖ ( b∗g2(| x |)b )µ2r 2 ‖ 1 µ2 p .(3.2) for 0 < r ≤ 1 2 and 1 ≤ p ≤ 2. proof. by applying (1.2), (1.1), (2.10), (1.3) and (2.4) respectively, we obtain ‖ | a∗xb + b∗xa |r ‖p = ‖a∗xb + b∗xa‖rrp ≤ ( 2 1 rp −1 (‖a∗xb‖rp + ‖b∗xa‖rp) )r ≤ 2 1 p −r (‖a∗xb‖2rrp + ‖b∗xa‖2rrp)12 ≤ 2 1 p −r (‖ | a∗xb |r ‖2p + ‖ | b∗xa |r ‖2p)12 = 2 1 p −r+ 1 2‖ | a∗xb |r ‖p ≤ 2 1 p −r+ 1 2‖ | aa∗x | µ1r 2 ‖ 1 µ1 p ‖ | xbb∗ | µ2r 2 ‖ 1 µ2 p . the proof is completed. by applying (2.5) and the proof of the first part of theorem 3.1, we can obtain (3.2). � corollary 3.1. let a,b,x be operators in b(h) such that x is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 + µ −1 2 = 1, then ‖ | a∗xa−b∗xb |r ‖p ≤ 2 1 p −2r+ 1 2‖ | (a + b)(a + b)∗x | µ1r 2 ‖ 1 µ1 p ‖ | x(a−b)(a−b)∗ | µ2r 2 ‖ 1 µ2 p , and ‖ | a∗xa−b∗xb |r ‖p ≤ 2 1 p −2r+ 1 2‖((a + b)∗ | x | (a + b)) µ1r 2 ‖ 1 µ1 p ‖((a−b)∗ | x | (a−b)) µ2r 2 ‖ 1 µ2 p , for 0 < r ≤ 1 2 and 1 ≤ p ≤ 2. remark 3.1. by using (2.8) and from second inequality in corollary (3.1), we can obtain ‖ | a |2r + | b |2r ‖p ≤ ‖ || a |2 − | b |2|r ‖p ≤ 2 1 p −2r+ 1 2‖ | a + b |µ1r| ‖ 1 µ1 p ‖ | a−b |µ2r ‖ 1 µ2 p , 8 ali, yang and shakoor which is the generalized form of the inequality (1.8). similarly to the proof of theorem 3.1, based on (2.11), we can obtain the following result. theorem 3.2. let a,b,x be operators in b(h) such that x is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 + µ −1 2 = 1, then ‖ | a∗xb + b∗xa |r ‖p ≤ 2r‖ | aa∗x | µ1r 2 ‖ 1 µ1 p ‖ | xbb∗ | µ2r 2 ‖ 1 µ2 p ,(3.3) also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ‖ | a∗xb + b∗xa |r ‖p ≤ 2r‖ ( a∗f2(| x |)a )µ1r 2 ‖ 1 µ1 p ‖ ( b∗g2(| x |)b )µ2r 2 ‖ 1 µ2 p ,(3.4) for r ≥ 1 2 and p ≥ 2. corollary 3.2. let a,b,x be operators in b(h) such that x is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 + µ −1 2 = 1, then ‖ | a∗xa−b∗xb |r ‖p ≤ ‖ | (a + b)(a + b)∗x | µ1r 2 ‖ 1 µ1 p ‖ | x(a−b)(a−b)∗ | µ2r 2 ‖ 1 µ2 p , and ‖ | a∗xa−b∗xb |r ‖p ≤ ‖((a + b)∗ | x | (a + b)) µ1r 2 ‖ 1 µ1 p ‖((a−b)∗ | x | (a−b)) µ2r 2 ‖ 1 µ2 p , for r ≥ 1 2 and p ≥ 2. similarly to the proof of theorem 3.1, based on (2.3), we can also obtain the following result. theorem 3.3. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 +µ −1 2 = 1, then ‖ n∑ i=1 | a∗ixibi + b ∗ i xiai | r ‖p ≤ 2rn1− 1 p n∑ i=1 ‖ | aia∗ixi | µ1r 2 ‖ 1 µ1 p ‖ | xibib∗i | µ2r 2 ‖ 1 µ2 p ,(3.5) also, if f and g are nonnegative continuous functions on [0,∞) satisfying f(t)g(t) = t, for all t ∈ [0,∞), then we have ‖ n∑ i=1 | a∗ixibi + b ∗ i xiai | r ‖p ≤ 2rn1− 1 p n∑ i=1 ‖ ( a∗if 2(| xi |)ai )µ1r 2 ‖ 1 µ1 p ‖ ( b∗i g 2(| xi |)bi )µ2r 2 ‖ 1 µ2 p ,(3.6) generalized norms inequalities for absolute value operators 9 for r,p ≥ 1. corollary 3.3. let ai,bi,xi (i = 1, 2, ...,n) be operators in b(h) such that xi is self adjoint operator and if µ1, µ2 are positive real numbers such that µ −1 1 +µ −1 2 = 1, then n 1 p −1‖ n∑ i=1 | a∗ixiai −b ∗ i xibi | r ‖p ≤ n∑ i=1 ||| | (ai + bi)(ai + bi)∗xi | µ1r 2 ‖ 1 µ1 p ‖ | xi(ai −bi)(ai −bi)∗ | µ2r 2 ‖ 1 µ2 p , and n 1 p −1‖ n∑ i=1 | a∗ixiai −b ∗ i xibi | r ‖p ≤ n∑ i=1 ‖((ai + bi)∗ | xi | (ai + bi)) µ1r 2 ‖ 1 µ1 p ‖((ai −bi)∗ | xi | (ai −bi)) µ2r 2 ‖ 1 µ2 p , for r,p ≥ 1. remark 3.2. for r ≤ 2 and p ≤ 2 r or r ≥ 2 and p(4 − r) ≤ 2, the results in corollary 3.3 refine the results in corollary 2.2 respectively. acknowledgements the authors thank the referees for their careful reading of the manuscript. this work was supported by the national natural science foundation of china (no. 11171361). references [1] hiai, f, zhan, x: inequalities involving unitarily invariant norms and operator monotone functions. linear algebra appl. 341, 151-169 (2002). [2] albadawi, h: holder-type inequalities involving unitarily invariant norms. positivity. 16, 255270 (2012). [3] shebrawi, k, albadawi, h: norm inequalities for the absolute value of hilbert space operators. linear and multilinear algebra. 58, 453-463 (2010). [4] shebrawi, k, albadawi, h: operator norm inequalities of minkowski type. j. ineq. pure appl. math. 9, issue 1, article 26, 11 pp (2008). [5] bhatia, r: perturbation inequalities for the absolute value map in norm ideals of operators. j. oper. theory. 19, 129-136 (1988). [6] bhatia, r: matrix analysis, springer-verlag, new york, 1997. college of mathematics and statistics, chongqing university, chongqing 401331, p.r.china ∗corresponding author international journal of analysis and applications volume 18, number 5 (2020), 689-698 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-689 some notes on error analysis for kernel based regularized interpolation qing zou∗ applied mathematics and computational sciences, the university of iowa, ia 52242, usa ∗corresponding author: zou-qing@uiowa.edu abstract. kernel based regularized interpolation is one of the most important methods for approximating functions. the theory behind the kernel based regularized interpolation is the well-known representer theorem, which shows the form of approximation function in the reproducing kernel hilbert spaces. because of the advantages of the kernel based regularized interpolation, it is widely used in many mathematical and engineering applications, for example, dimension reduction and dimension estimation. however, the performance of the approximation is not fully understood from the theoretical perspective. in other word, the error analysis for the kernel based regularized interpolation is lacking. in this paper, some error bounds in terms of the reproducing kernel hilbert space norm and sobolev space norm are given to understand the behavior of the approximation function. 1. introduction approximating functions in high dimensional spaces is one of the central problems in both mathematics and engineering. many real-world problems can be viewed as a function approximation problem. for example, the classification problem in engineering can be viewed as approximating a function whose function values give the classes that the inputs belong to [7] and in image processing, the patch-based image denoising problem can be seen as approximating a function from noisy patches to clean pixels. received may 12th, 2020; accepted june 4th, 2020; published june 17th, 2020. 2010 mathematics subject classification. 65d05, 41a05, 41a10, 65j05, 65g99. key words and phrases. error bound; kernel based regularized interpolation; reproducing kernel hilbert space; sobolev space. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 689 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-689 int. j. anal. appl. 18 (5) (2020) 690 mathematically speaking, the function approximation problem is to approximate an unknown continuous function f : x → r from the knowledge of some observations {(xi,yi)}ni=1 ⊂ x × r, where x ⊂ r d,d ≥ 1 is the input space and n ∈ n is the number of observations. one of the classical ways for approximating functions to explain the real-world phenomenon is called kriging (a.k.a gaussian process regression) [4]. however, to use kriging, we need a large amount of observations, which is expensive to achieve. when only a few samples are given, kriging will yield non-stationary behavior [10]. another alternative for approximating multivariate functions is the inverse distance weighting (idw) method, which was originally proposed in [9]. the key assumption of idw is that the points that are close to each other are more alike than those that are far away from each other. therefore, idw produces prediction at a point relying on observed points close to that point. in other word, the measured values close to the prediction location have more effects on the predicted value than those which are far away. based on which, the approximation function can be represented as f̂(x) = ∑n i=1 ||x−xi|| −αyi∑n i=1 ||x−xi||−α , where α is a parameter and usually it takes the value 2. idw can produce good accuracy for the points that are near the observations. however, for the points that are away from the observations, idw cannot work well. this is one drawback of idw. another drawback of idw is that the gradient vanishes at the observations. considering the drawbacks of these methods, the kernel based regularized interpolation (see (2.2) for details) was proposed. the kernel based regularized interpolation in 1d is similar to the spline interpolation. it has several advantages. first of all, the kernel based regularized interpolation works for any dimensions with many different situations [1, 2]. it also enjoys solid theoretical understandings — the representer theorem — which we will show it below. furthermore, comparing to other methods, one can have the same approximation accuracy using kernel based regularized interpolation by solving a better conditioned linear system. the kernel based regularized interpolation also works for the case that the observations have some noise (see section 2 for analysis). this is one of the biggest advantages that other interpolation methods do not have. last but not least, we have much flexibility in choosing kernels when using kernel based regularized interpolation. kernel based regularized interpolation is also widely used in other engineering problems, for example, clustering, dimension reduction [1] and dimension estimation [11]. in view of the advantages of kernel based regularized interpolation, there is pretty much work on using kernel based regularized interpolation to deal with different situations, as we mentioned above. however, the work on the error analysis for the kernel based regularized interpolation is very limited. it is actually worthwhile to reflect on the nature of error bounds that are needed to characterize the behavior of the int. j. anal. appl. 18 (5) (2020) 691 learned functions from observations. in this paper, we would like to provide some error analysis for the kernel based regularized interpolation from the function space point of view. in other word, we focus specifically on deriving hilbert space type and sobolev space type error estimates for the kernel based regularized interpolation. 2. preliminary let us recall here some basic facts from kernel methods for our analysis. we consider a positive definite kernel k : x ×x → r, i.e., for n ∈ n, x1, · · · ,xn ∈ x and a1, · · · ,an ∈ r, n∑ i=1 n∑ j=1 aiajk(xi,xj) ≥ 0. associated with the kernel k, there exists a unique native hilbert space hk of functions from x to r. the elements in hk are of the form f(·) = ∑ i∈i αik(·,xi), where i is a countable set. the norm defined for hk is given by 〈f,g〉hk =: ∑ i∈i ∑ j∈j αiβjk(xi,xj), where g = ∑ j∈j βjk(·,xj) ∈hk. the hilbert space hk is called reproducing kernel hilbert space (rkhs) because the kernel k acts as a reproducing kernel on hk, i.e., f(x) = 〈f(·),k(·,x)〉hk ∀f ∈hk. we require the kernel to be positive definite because when we approximate the functions, linear systems will be involved. the positive-definite property guarantees the linear systems are well conditioned and the approximation problem can be solved in hk using the given observations {(xi,yi)}ni=1. indeed, one can solve the functions approximation problem by considering the regularization problem with the regularization parameter λ > 0, min f∈hk n∑ i=1 ||f(xi) −yi||2 + λ||f||2hk. (2.1) the solution of this regularization problem is characterized by the representer theorem [8], which states that the solution of (2.1) is given as f = n∑ i=1 αik(x,xi). (2.2) this expression is called kernel based regularized interpolation. in the rest of this paper, we use the notation ixf to stand for the kernel based regularized interpolation, i.e., ixf = n∑ i=1 αik(x,xi). int. j. anal. appl. 18 (5) (2020) 692 to obtain error analysis, we need to have the true function. we use the notation f to represent the true function that is only known at the sampling locations, i.e., we only have the information yi = f(xi) about the true function f. the goal of this work is to obtain the bounds for the error between the true function f and the kernel based regularized interpolation ixf. it is worth mentioning that the kernel based regularized interpolation is well defined for positive definite kernels, but it is not a pure interpolation. this means that we may not have ixf(xi) = yi = f(xi). but in the case of strictly positive definite kernels, the regularization problem (2.1) will become an interpolation problem which exactly interpolates f at the observations and hence the resulted kernel interpolation is a pure interpolation. however, there is no need to consider strictly positive definite kernels. using positive definite kernel to obtain kernel based regularized interpolation is more general. this is because we have a tunable parameter λ in the regularization problem and it provides a trade-off between pointwise accuracy and stability. another important reason is that in real-world applications, the observations may be corrupted by noise and it makes no sense to have pure interpolations anymore. 3. hilbert space type error analysis in this section, we would like to provide a special case about the error analysis for the kernel based regularized interpolation. the term “special case” is used here because we will assume in this section that the true function f is living in the rkhs. while in some cases, we cannot guarantee that the true function is within the rkhs. we will show the error analysis for the general case in the next section. before proceeding to the error analysis, we first recall an important concept in numerical analysis: best approximation [5]. given f ∈hk, define ρ := inf p∈s ||f −p||hk, where s ⊂hk. then the number ρ is called the minimax error for the approximation of the function f by functions in s. in fact [5], there is a unique p̂ ∈ s such that ||f − p̂||hk = ρ = inf p∈s ||f −p||hk. the function p̂ ∈ s is called the best approximation of f w.r.t. the hk-norm. now, let us look at the inner product of the rkhs. by cauchy-schwarz inequality [3], we have that |〈h,k(·,x)〉hk| ≤ ||h||hk · ||k(·,x)||hk, ∀h ∈hk. (3.1) based on which, we can claim from (3.1) that there exists a constant 0 < β ≤ 1 such that sup x∈x |〈h,k(·,x)〉hk| ||k(·,x)||hk ≥ β · ||h||hk. (3.2) we then have the first error estimation. int. j. anal. appl. 18 (5) (2020) 693 theorem 3.1. let s = span{k(x,xi)}ni=1 ⊂ hk. assume that the true function f ∈ hk. ixf is the kernel based regularized interpolation. then we have the following error bound ||f − ixf||hk ≤ (1 + c β ) ·ρ, where c > 1, 0 < β ≤ 1 are two constants and ρ is the minimax error for the approximation of the function f by functions in s in terms of the hk-norm. proof. first of all, let us consider the trivial case that the true function f ∈ s ⊂hk. in this case, the kernel based regularized interpolation is exact, meaning that we have ixf = f. then both ||f − ixf||hk and ρ are zero. so the conclusion becomes trivial in this case. next, we assume that f ∈hk but f 6∈ s. then for any g ∈ s, we have ||f − ixf||hk ≤ ||f −g||hk + ||ixf −g||hk. (3.3) replacing h by ixf −g in (3.2), we have β||ixf −g||hk ≤ sup x∈x |〈ixf −g,k(·,x)〉hk| ||k(·,x)||hk . because we have f 6= g, then for the right-hand side (rhs) of the above inequality, there exists a constant c > 1 such that rhs = sup x∈x |〈f −g + ixf −f,k(·,x)〉hk| ||k(·,x)||hk ≤ sup x∈x [ |〈f −g,k(·,x)〉hk| ||k(·,x)||hk + |〈ixf −f,k(·,x)〉hk| ||k(·,x)||hk ] = sup x∈x |〈f −g,k(·,x)〉hk| ||k(·,x)||hk + sup x∈x |〈ixf −f,k(·,x)〉hk| ||k(·,x)||hk ≤ c · sup x∈x |〈f −g,k(·,x)〉hk| ||k(·,x)||hk ≤ c · ||f −g||hk. the last inequality is due to the cauchy-schwarz inequality. therefore, we have ||ixf −g||hk ≤ c β ||f −g||hk. then by (3.3), we can obtain that ||f − ixf||hk ≤ (1 + c β )||f −g||hk, ∀g ∈ s, which implies that ||f − ixf||hk ≤ (1 + c β ) inf g∈s ||f −g||hk = (1 + c β ) ·ρ. this completes the proof. � int. j. anal. appl. 18 (5) (2020) 694 in fact, this error bound can be further tightened. the improvement is due to the observation that if the true function f is within hk, the kernel based regularized interpolation is actually a projection. let s = span{k(x,xi)}ni=1 ⊂hk as defined in theorem 3.1. then if f ∈hk, we can see that the kernel based regularized interpolation is a projection ϕ : hk → s. the most important property of a projection is the idempotence. if the function f is already in s, the projection does nothing but to keep the function, i.e., ϕ(g) = g, ∀g ∈ s. this implies the idempotence of the projector: ϕ2(f) = ϕ(ϕ(f)) = ϕ(f), ∀f ∈hk. if we define ||ϕ|| to be the operator norm, which is given by ||ϕ|| =: max 06=g∈hk ||ϕ(g)||hk ||g||hk . then we have the identity ||ϕ|| = ||i −ϕ||, where i is the identity operator. this conclusion is due to the following result given in [6]. lemma 3.1. let h be a hilbert space and u ⊂h. let p : h→ u be an (oblique) projection (p2 = p ) and 0 6= p 6= i. then ||p || = ||i −p ||. using the observation that the kernel based regularized interpolation is a projection if f ∈ hk, we then have the following tightened error bound. theorem 3.2. suppose s = span{k(x,xi)}ni=1 ⊂ hk. the kernel based regularized interpolation is given by ixf = ϕ(f), where ϕ is a projector from hk to s. then we have the following error bounds ||f − ixf||hk ≤ ρ β , where 0 < β ≤ 1 is a constant and ρ is the minimax error for the approximation of the function f by functions in s in terms of the hk-norm. proof. since ϕ is a projection, we have ||ϕ|| = ||i −ϕ||. based on which, we can derive that for all g ∈ s, ||f − ixf||hk =||f −g − ixf + g||hk =||f −g −ϕ(f −g)||hk =||(i −ϕ)(f −g)||hk ≤||i −ϕ|| · ||f −g||hk =||ϕ|| · ||f −g||hk. int. j. anal. appl. 18 (5) (2020) 695 to show the error bound, we need to show that ||ϕ|| ≤ 1 β . first, it is straightforward that if f ∈hk, we have ||ixf||hk ≤ ||f||hk. since 0 < β ≤ 1, we can get 1 β ≥ 1. so for f ∈hk, ||ixf||hk ≤ 1 β ||f||hk. therefore, we can obtain that ||ϕ|| = max 06=f∈hk ||ϕ(f)||hk ||f||hk = max 06=f∈hk ||ixf||hk ||f||hk ≤ 1 β ||f||hk ||f||hk = 1 β . this gives us the desired bound. � 4. sobolev space type error analysis in the previous section, we have introduced a special case for the error analysis about kernel based regularized interpolation, where we assumed that the true function f lives in the reproducing kernel hilbert space hk. however, in general we do not have this guarantee. the true functions are usually within a larger function space. in this section, we consider the case that the true function f is in the sobolev spaces, which are in fact the spaces that consist of all f ∈ lp(x) with certain properties. the formal definition of the sobolev space is given as follows. definition 4.1. let k be a nonnegative integer, p ∈ [1,∞]. the sobolev space wk,p(x) is the set of all the functions f ∈ lp(x) such that for each multi-index α with |α| ≤ k, the αth weak derivative ∂αf exists and ∂αf ∈lp(x). the norm of the sobolev space is defined as ||f||wk,p(x) =   [∑ |α|≤k ||∂ αf||plp(x) ]1/p , 1 ≤ p < ∞ max|α|≤k||∂ αf||l∞(x), p = ∞ . when p = 2, we usually write wk,2(x) := hk(x). for simplicity, we replace ||f||wk,p(x) by ||f||k,p,x when no confusion and we ignore the p when p = 2. except for the standard norm defined for the sobolev space, we also have the definition of the seminorm. the standard seminorm over the space wk,p(x) is given as |f|wk,p(x) =   [∑ |α|=k ||∂ αf||plp(x) ]1/p , 1 ≤ p < ∞ max|α|=k||∂ αf||l∞(x), p = ∞ . int. j. anal. appl. 18 (5) (2020) 696 similarly, we write |f|wk,p(x) as |f|k,p,x when no confusion and ignore the p when p = 2. we now define the continuous embedding between two banach spaces. let w,v be two banach spaces with v ⊂ w . we say the space v is continuously embedded in w and write v ↪→ w , if there exists a constant c > 0 such that for all v ∈ v , we have ||v||w ≤ c||v||v . with these concepts in the theory of sobolev spaces, we can then have the error bound for the kernel based regularized interpolation. we assume that the true function f is within hk(x). theorem 4.1. let k and m be nonnegative integers with k ≥ 0,k + 1 ≥ m. suppose hk ⊃ s = span{k(x,xi)}. then there exists a constant c > 0 such that |f − ixf|m,x ≤ c inf g∈s ||f + g||k+1,x. when the positive definite kernel is chosen to be the polynomial kernel with degree d and k + 1 > d, we have |f − ixf|m,x ≤ c|f|k+1,x. proof. since k > 0, we then have hk+1(x) ↪→ c(x), where c(x) is the set of all the continuous functions defined on x. so f ∈ hk+1(x) is continuous and ixf is well defined. besides, there exists a constant c1 > 0 such that ||ixf||m,x ≤ n∑ i=1 |αi| · ||k(x,xi)||m,x ≤ c1||f||c(x). since hk+1(x) ↪→ c(x), we then have that there exists a constant c2 > 0 such that ||f||c(x) ≤ c2||f||k+1,x. therefore, there exists some c3 > 0 such that ||ixf||m,x ≤ c3||f||k+1,x. (4.1) now, for all f ∈ hk+1 and g ∈ s, we can obtain from (4.1) that there exist some constant c > 0, |f − ixf|m,x ≤ ||f − ixf||m,x = ||f − ixf + g − ixg||m,x = ||(f + g) − ix(f + g)||m,x ≤ ||f + g||m,x + ||ix(f + g)||m,x ≤ c||f + g||k+1,x. by which we have |f − ixf|m,x ≤ c inf g∈s ||f + g||k+1,x. int. j. anal. appl. 18 (5) (2020) 697 we now consider the case that the positive definite kernel k is the polynomial kernel. first, we have the following inequality using the norm equivalence theorem [12] ||f||k+1,x ≤ c4  |f|k+1,x + ∑ |α|≤k ∣∣∣∣ ∫ x ∂αf(x)dx ∣∣∣∣   , (4.2) ∀f ∈ hk+1(x). since the kernel is a polynomial kernel and k + 1 > d, we have that for all g ∈ s, ∂αg = 0 for |α| = k + 1. replacing f by f + g in (4.2) gives us that ∀f ∈ hk+1(x),g ∈ s, ||f + g||k+1,x ≤c4  |f + g|k+1,x + ∑ |α|≤k ∣∣∣∣ ∫ x ∂α(f + g)dx ∣∣∣∣   =c4  |f|k+1,x + ∑ |α|≤k ∣∣∣∣ ∫ x ∂α(f + g)dx ∣∣∣∣   . for the term ∫ x ∂α(f + g)dx, we should note that for any f ∈ hk+1(x), one can always have a g ∈ s such that for |α| ≤ k, ∫ x ∂α(f + g)dx = 0. (4.3) this is because when we set |α| = k,k−1,k−2, · · · , we will have a set of equations from (4.3). we can then solve the set of equations to obtain the coefficients of the polynomial g ∈ s. besides, we note that such set of equations is always solvable because we assumed that k + 1 > d. then the g which was constructed by such way will annihilate the integral ∫ x ∂α(f + g)dx for |α| ≤ k. therefore, we have that inf g∈s ||f + g||k+1,x ≤ c4|f|k+1,x, for some constant c4 > 0. then by the conclusion we obtained from the first part, we know that there exists some c > 0 such that |f − ixf|m,x ≤ c|f|k+1,x. this completes the proof of the result. � from this error bound, we can see that if we consider the polynomial kernel and the true function is in the rkhs hk, we will have |f − ixf|m,x ≤ c||f||hk, for some constant c > 0. this is because the rkhs hk is continuously embedded in hk+1(x) and we then have |f|k+1,x ≤ c||f||hk . int. j. anal. appl. 18 (5) (2020) 698 5. conclusion in this paper, some error bounds for the kernel based regularized interpolation are provided. we derived these error bounds in terms of the reproducing kernel hilbert space norm and sobolev space norm. we also introduced a error bound for a special type of a commonly used kernel: polynomial kernel. the error bounds then provide theoretical understandings of the approximation performance. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. belkin and p. niyogi, semi-supervised learning on riemannian manifolds, mach. learn. 56 (2004), 209–239. [2] m. belkin, p. niyogi, and v. sindhwani, manifold regularization: a geometric framework for learning from labeled and unlabeled examples, j. mach. learn. res. 7 (2006), 2399–2434. [3] r. bhatia and c. davis, a cauchy-schwarz inequality for operators with applications, linear algebra appl. 223 (1995), 119–129. [4] n. cressie, the origins of kriging, math. geol. 22 (1990), 239–252. [5] f. r. deutsch, best approximation in inner product spaces, springer science & business media, 2012. [6] t. kato, estimation of iterated matrices, with application to the von neumann condition, numer. math. 2 (1960), 22–29. [7] h.-j. rong, g.-b. huang, n. sundararajan, and p. saratchandran, online sequential fuzzy extreme learning machine for function approximation and classification problems, ieee trans. syst. man cybern. part b (cybernetics), 39 (2009), 1067–1072. [8] b. schölkopf, r. herbrich, and a. j. smola, a generalized representer theorem, in international conference on computational learning theory, springer, 2001, 416–426. [9] d. shepard, a two-dimensional interpolation function for irregularly-spaced data, in proceedings of the 1968 23rd acm national conference, 1968, 517–524. [10] m. thakur, b. samanta, and d. chakravarty, a non-stationary geostatistical approach to multigaussian kriging for local reserve estimation, stoch. environ. res. risk assess. 32 (2018), 2381–2404. [11] j. j. thiagarajan, p.-t. bremer, and k. n. ramamurthy, multiple kernel interpolation for inverting non-linear dimensionality reduction and dimension estimation, in 2014 ieee international conference on acoustics, speech and signal processing (icassp), ieee, 2014, 6751–6755. [12] a. wannebo, equivalent norms for the sobolev space w m,p 0 (ω), ark. mat. 32 (1994), 245–254. 1. introduction 2. preliminary 3. hilbert space type error analysis 4. sobolev space type error analysis 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 70-78 http://www.etamaths.com fixed point theorems for ciric’s and generalized contractions in b-metric spaces muhammad sarwar∗ and mujeeb ur rahman abstract. in this article we obtained b-metric variant of common fixed point results for ciric’s and generalized contractions. we have also proved some fixed point results for rational contractive type conditions in the context of b-metric space. a particular example is also given in the support of our established result regarding ciric’s type contraction. 1. introduction fixed point theory is one of the most important topic in the development of non linear analysis. in this area, the first important and significant result was proved by banach in 1922 for contraction mapping in complete metric space. a comprehensive literature and generalization of the banach’s contraction theorem can be found in [7] and [11]. some problems particularly the problem of the convergence of measurable functions with respect to measure leads czerwik (see [5],[6]) to the generalization of metric space and introduce the concept of b-metric space after czerwik([5], [6]) many papers have been published containing fixed point results on b-metric spaces for single value and multivalued functions(see[1], [2], [10]). in this paper we proved b-metric variant of fixed point results for ciric’s and generalized contraction. we also studied some results involving rational contractive type condition. through out the paper r+ will represent the set of non negative real numbers. 2. preliminaries definition 1. [8]. let x be a non-empty set and let d : x×x → r+ be a function satisfying the conditions, d1) d(x,y) = 0 ⇔ x = y; d2) d(x,y) = d(y,x); d3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ x. then d it is called metric on x, and the pair (x,d) is called metric space. definition 2. [6]. let x be a non empty set, let k ≥ 1 be a real number then a mapping b : x×x → r+ is called b-metric if ∀ x,y,z ∈ x, the following conditions are satisfied: b1) b(x,y) = 0 ⇔ x = y; 2010 mathematics subject classification. 47h10, 54h25, 55m20. key words and phrases. complete b-metric space, contraction mapping, self mappings, cauchy sequence, fixed point. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 70 fixed point theorems 71 b2) b(x,y) = b(y,x); b3) b(x,y) ≤ k[b(x,z) + b(z,y)]. and the pair (x,b) is called b-metric space. it is clear from the definition of b-metric that every metric space is b-metric for k = 1, but the converse is not true as clear from the following example. example 2.1. [2]. let x = {0, 1, 2}. defined b : x ×x → r+ as follows b(0, 0) = b(1, 1) = b(2, 2) = 0, b(1, 2) = b(2, 1) = b(0, 1) = b(1, 0) = 1,b(2, 0) = b(0, 2) = m ≥ 2 for k = m 2 where m ≥ 2 the function defined above is a b-metric space but not a metric for m > 2. for more examples of b-metric space (see [2], [10]). in our main work we will use the following definitions which can be found in [2] and [10]. definition 3. a sequence {xn} in b-metric space (x,b) is called cauchy sequence if for � > 0 there exist a positive integer n such that for m,n ≥ n we have b(xm,xn) < �. definition 4. a sequence {xn} is called convergent in b-metric space (x,b) if for � > 0 and n ≥ n we have b(xn,x) < � where x is called the limit point of the sequence {xn}. definition 5. a b-metric space (x,b) is said to be complete if every cauchy sequence in x converge to a point of x. definition 6. [4]. let (x,d) be a metric space, a self mapping t : x → x is called generalized contraction if and only for all x,y ∈ x, there exist c1,c2,c3,c4 such that sup{c1 + c2 + c3 + 2c4} < 1 and d(tx,ty) ≤ c1.d(x,y) + c2.d(x,tx) + c3.d(y,ty) + c4.[d(x,ty) + d(y,tx)] definition 7. [11]. let (x,d) be a metric space, a self mapping t : x → x is called ciric’s type contraction if and only if for all x,y ∈ x, there exist h < 1 and, d(tx,ty) ≤ h. max { d(x,y),d(x,tx),d(y,ty), d(x,ty) + d(y,tx) 2 } definition 8. [4]. let (x,d) be a metric space, a self mapping t : x → x is called quasi contraction if and only if for all x,y ∈ x, there exist h < 1 and, d(tx,ty) ≤ h. max { d(x,y),d(x,tx),d(y,ty),d(x,ty) + d(y,tx) } remark. • in [4] the author shown by an example that every quasi contraction need not to be generalized contraction in metric spaces. theorem 2.1. [10]. let (x,b) be a complete bmetric space with k ≥ 1 and let t : x → x be a contraction with α ∈ [0, 1) and kα < 1 then t has a unique fixed point in x. definition 9. [3]. let (x,d) be a metric space a mapping t : x → x is called weak contraction if there exist constants α ∈ (0, 1) and some β ≥ 0 such that (1) d(tx,ty) ≤ α.d(x,y) + β.d(y,tx) for all x,y ∈ x. 72 sarwar and rahman due to the symmetry of distance the weak contraction clearly include the following (2) d(tx,ty) ≤ α.d(x,y) + β.d(x,ty) for all x,y ∈ x. therefore to check the weak contraction of the mapping, we have to check both (1) and (2). remarks. • it is clear that every contraction in metric space is necessarily weak contraction but the converse is not always true(see[3]). • unlike metric space every mapping satisfying the contractive condition in b-metric space need not to be weak contraction necessarily(see[10]). on the other hand khan[9] proved the following fixed point result for complete metric spaces. theorem 2.2. let (x,d) be a complete metric space and t be a self mapping on x satisfying the following condition d(tx,ty) ≤ µ. d(x,tx).d(x,ty) + d(y,ty).d(y,tx) d(x,ty) + d(y,tx) ∀ x,y ∈ x and µ ∈ [0, 1), then t has a unique fixed point. 3. main results lemma 1. let (x,b) be a b-metric space and {xn} be a sequence in b-metric space such that (3) b(xn,xn+1) ≤ α.b(xn−1,xn) for n = 1, 2, 3, ... and 0 ≤ αk < 1, α ∈ [0, 1), and k is defined in b-metric space then {xn} is a cauchy sequence in x. proof. let n,m ∈ n and m > n we have b(xn,xm) ≤ k[b(xn,xn+1) + b(xn+1,xn+2)..................] ≤ kb(xn,xn+1) + k2[b(xn+1,xn+2) + b(xn+2,xn+3)..........] ≤ kb(xn,xn+1) + k2b(xn+1,xn+2) + k3b(xn+2,xn+3).......................... now using (3) we have b(xn,xm) ≤ kαnb(x0,x1) + k2αn+1b(x0,x1) + k3αn+2b(x0,x1) + ............ ≤ (1 + kα + (kα)2 + ........)kαnb(x0,x1) ≤ kαn ( 1 1 −kα ) b(x0,x1). since kα < 1 therefore taking limit m,n →∞ we have lim m,n→∞ b(xn,xm) = 0. hence {xn} is a cauchy sequence in b-metric space x. � the next theorem is about to hold a unique fixed point for generalized contraction in complete b-metric space. fixed point theorems 73 theorem 3.1. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition (4) b(tx,ty) ≤ α.b(x,y) + β.b(x,tx) + γ.b(y,ty) + µ.[b(x,ty) + b(y,tx)] ∀ x,y ∈ x, where α,β,γ,µ ≥ 0, with (5) kα + kβ + γ + (k2 + k)µ < 1 then t has a unique fixed point. proof. let x0 be arbitrary in x we define a sequence {xn} in x by the rule x0,x1 = tx0,x2 = tx1, ........,xn+1 = txn, consider b(xn,xn+1) = b(txn−1,txn) by using condition (4) we have b(xn,xn+1) ≤ α.b(xn−1,xn)+β.b(xn−1,xn)+γ.b(xn,xn+1)+µ.[b(xn−1,xn+1)+b(xn,xn)] ≤ α.b(xn−1,xn) + β.b(xn−1,xn) + γ.b(xn,xn+1) + µ.k[b(xn−1,xn) + b(xn,xn+1] so b(xn,xn+1) ≤ (α + β + kµ) (1 − (γ + kµ)) b(xn−1,xn) ≤ λb(xn−1,xn) where λ = α + β + kµ 1 − (γ + kµ) from (5) it is clear that λ < 1/k. now from lemma 1 we can say that {xn} is a cauchy sequence in complete b-metric space, so there exist u ∈ x such that limn→∞xn = u. now we have to show that u is the fixed point of t for this consider, b(tu,txn) ≤ α.b(u,xn) + β.b(u,tu) + γ.b(xn,txn) + µ.[b(u,txn) + b(xn,tu)] ≤ α.b(u,xn) + β.b(u,tu) + γ.b(xn,xn+1) + µ.[b(u,xn+1) + b(xn,tu)]. taking limit n →∞ we have b(tu,u) ≤ α.b(u,u) + β.b(u,tu) + γ.b(u,u) + µ.[b(u,u) + b(u,tu)] ≤ (β + µ).b(tu,u) the above inequality is possible only if b(tu,u) = 0 ⇒ tu = u. hence u is the fixed point of t . uniqueness. let u 6= v be two fixed points of t then b(u,v) = b(tu,tv) ≤ α.b(u,v) + β.b(u,tu) + γ.b(v,tv) + µ.[b(u,tv) + b(v,tu)] = (α + 2µ).b(u,v) since u,v are fixed points of t so finally we get using (5) the above inequality is possible only if b(u,v) = 0 ⇒ u = v. hence fixed point of t is unique in x. � theorem 3.1 yields the following corollaries. 74 sarwar and rahman corollary 3.1. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition b(tx,ty) ≤ α.b(x,y) + β.b(x,tx) + γ.b(y,ty) ∀ x,y ∈ x, where α,β,γ ≥ 0 with kα+kβ +γ < 1, then t has a unique fixed point. proof. putting µ = 0 in theorem(3.1) we get the required result easily. � corollary 3.2. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition b(tx,ty) ≤ α.b(x,y) + β.b(x,tx) ∀ x,y ∈ x, where α,β ≥ 0 with kα + kβ < 1, then t has a unique fixed point. proof. by putting µ = γ = 0 in theorem(3.1) we get the required result. � corollary 3.3. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition b(tx,ty) ≤ α.b(x,y) ∀ x,y ∈ x, where α ≥ 0 with kα < 1 then t has a unique fixed point. proof. by putting β = γ = µ = 0 in theorem(3.1) we get the require result. � remark. • corollary (3.3) is the result of [10]. now we present the modified form of khan’s theorem[9] in the context of b-metric spaces. theorem 3.2. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition (6) b(tx,ty) ≤ β.b(x,y) + µ. b(x,tx).b(x,ty) + b(y,ty).b(y,tx) b(x,ty) + b(y,tx) ∀ x,y ∈ x and β,µ ≥ 0, b(x,ty) + b(y,tx) 6= 0 with k(β + µ) < 1, then t has a unique fixed point. proof. let x0 be arbitrary in x, we define a sequence {xn} by the rule, x0, x1 = tx0, x2 = tx1, ........., xn+1 = txn for all n ∈ n now to show that {xn} is a cauchy sequence in x then consider, b(xn,xn+1) = b(txn−1,txn) from equation (6) we have b(xn,xn+1) ≤ β.b(xn−1,xn) + µ. b(xn−1,txn−1).b(xn−1,txn) + b(xn,txn).b(xn,txn−1) b(xn−1,txn) + b(xn,txn−1) ≤ β.b(xn−1,xn) + µ. b(xn−1,xn).b(xn−1,xn+1) + b(xn,xn+1).b(xn,xn) b(xn−1,xn+1) + b(xn,xn) ≤ (β + µ). b(xn−1,xn). fixed point theorems 75 since β + µ < 1 k , therefore by lemma 1 {xn} is a cauchy sequence in complete b-metric space x, so there must exist u ∈ x such that limn→∞xn = u. now to show that u is the fixed point of t for this consider, b(txn,tu) ≤ β.b(xn,u) + µ. b(xn,txn).b(xn,tu) + b(u,tu).b(u,txn) b(xn,tu) + b(u,txn) ≤ β.b(xn,u) + µ. b(xn,xn+1).b(xn,tu) + b(u,tu).b(u,xn+1) b(xn,tu) + b(u,xn+1) taking limit n →∞ we have, β.b(xn,u) + µ. b(xn,xn+1).b(xn,tu) + b(u,tu).b(u,xn+1) b(xn,tu) + b(u,xn+1) → 0. hence b(u,tu) = 0 ⇒ tu = u thus u is the fixed point of t. uniqueness. let u,v are two distinct fixed points of t , for u 6= v consider, b(u,v) = b(tu,tv) ≤ β.b(u,v) + µ. b(u,tu).b(u,tv) + b(v,tv).b(v,tu) b(u,tv) + b(v,tu) since u,v are fixed points of t we get b(u,v) = b(tu,tv) ≤ β.b(u,v) + µ. b(u,u).b(u,v) + b(v,v).b(v,u) b(u,v) + b(v,u) = β.b(u,v) using the restriction in the theorem the above inequality is possible only if, b(u,v) = 0 ⇒ u = v. therefore fixed point of t is unique. this complete the proof of the theorem. remark. • in order to hold khan’s theorem in b-metric space we made no restriction. � theorem 3.3. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition (7) b(tx,ty) ≤ α. b(x,y) + β. b(y,ty)[1 + b(x,tx)] 1 + b(x,y) + γ. b(y,ty) + b(y,tx) 1 + b(y,ty).b(y,tx) ∀ x,y ∈ x, where α,β,γ ≥ 0 and kα + β + γ < 1 then t has a unique fixed point. proof. let x0 be arbitrary in x, we define a sequence {xn} by the rule, x0,x1 = tx0,x2 = tx1, .........,xn+1 = txn for all n ∈ n now to show that {xn} is a cauchy sequence in x then consider, b(xn,xn+1) = b(txn−1,txn) 76 sarwar and rahman from equation (7) we have b(xn,xn+1) ≤ α. b(xn−1,xn) + β. b(xn,xn+1)[1 + b(xn−1,xn)] 1 + b(xn−1,xn) + γ. b(xn,xn+1) + b(xn,xn) 1 + b(xn,xn+1).b(xn,xn) ≤ α. b(xn−1,xn) + β. b(xn,xn+1)[1 + b(xn−1,xn)] 1 + b(xn−1,xn) + γ. b(xn,xn+1) therefore b(xn,xn+1) ≤ α 1 − (β + γ) . b(xn−1,xn) = h. b(xn−1,xn) where h = α 1−(β+γ) with h < 1 k , because kα + β + γ < 1 similarly we have b(xn,xn+1) ≤ h2. b(xn−2,xn−1) continuing the same process we get b(xn,xn+1) ≤ hn. b(x0,x1) since 0 ≤ h < 1 so hn → 0 as n →∞ , which shows that {xn} is a cauchy sequence in complete b-metric space so there exist z ∈ x such that xn → z as n →∞ now to show that z is the fixed point of t for this consider, (8) b(txn,tz) ≤ α. b(xn,z) + β. b(z,tz)[1 + b(xn,txn)] 1 + b(xn,z)) + γ. b(z,tz) + b(z,txn) 1 + b(z,tz).b(z,txn) from the construction it is clear that txn = xn+1 and also {xn} is a cauchy sequence converges to z. therefore taking limit n →∞ equation (8) become b(z,tz) ≤ (β + γ)b(z,tz) which is possible only if b(z,tz) = 0, thus tz = z. hence z is the fixed point of t . uniqueness. suppose that t has two fixed points z and w for z 6= w consider, b(z,w) = b(tz,tw) ≤ α. b(z,w)+β. b(w,tw)[1 + b(z,tz)] 1 + b(z,w) +γ. b(w,tw) + b(w,tz) 1 + b(w.tw).b(w,tz) so the above inequality become b(z,w) ≤ (α + γ). b(z,w) the above inequality is possible only if b(z,w) = 0 ⇒ z = w. hence fixed point of t is unique. � our next theorem is about b-metric variant of ciric’s type contraction. theorem 3.4. let (x,b) be a complete b-metric space with coefficient k ≥ 1 and t be a self mapping t : x → x satisfying the condition (9) b(tx,ty) ≤ h. max { b(x,y),b(x,tx),b(y,ty), 1 2k [b(x,ty) + b(y,tx)] } ∀ x,y ∈ x, where h ∈ [0, 1) and kh < 1 then t has a unique fixed point. fixed point theorems 77 proof. let x0 be arbitrary in x we define a sequence {xn} in x by the rule x0, x1 = tx0, x2 = tx1, ........, xn+1 = txn. consider b(xn,xn+1) = b(txn−1,txn) using (11) we have b(xn,xn+1) ≤ h. max { b(xn−1,xn),b(xn−1,txn−1),b(xn,txn), 1 2k [b(xn−1,txn)+b(xn,txn−1)] } = h. max { b(xn−1,xn),b(xn−1,xn),b(xn,xn+1), 1 2k [b(xn−1,xn+1) + b(xn,xn)] } ≤ h. max { b(xn−1,xn),b(xn−1,xn),b(xn,xn+1), 1 2 [b(xn−1,xn) + b(xn,xn+1)] } (10) = h. max { b(xn−1,xn),b(xn,xn+1), 1 2 [b(xn−1,xn) + b(xn,xn+1)] } if b(xn−1,xn) < b(xn,xn+1). then b(xn−1,xn) < 1 2 [b(xn−1,xn) + b(xn,xn+1)] < b(xn,xn+1). so (10) implies that b(xn,xn+1 ≤ h.b(xn,xn+1) which is impossible since h < 1 for the same reason we also ignored the term b(xn,xn+1) thus (10) become b(xn,xn+1) ≤ h.b(xn−1,xn). by use of lemma(1) we get that {xn} is a cauchy sequence in complete b-metric space x so there exist u ∈ x such that limn→∞xn = u, now we have to show that u is the fixed point of t for this consider b(tu,txn) ≤ h. max { b(u,xn),b(u,tu),b(xn,txn), 1 2k [b(u,txn) + b(xn,tu)] } ≤ h. max { b(u,xn),b(u,tu),b(xn,xn+1), 1 2k [b(u,xn+1) + b(xn,tu)] } . taking limit n →∞ we have b(tu,u) ≤ h. max { b(u,tu), 1 2k b(u,tu) } ≤ h.b(tu,u) the above inequality is possible only if b(tu,u) = 0 ⇒ tu = u. hence u is the fixed point of t . uniqueness. let u 6= v be two fixed points of t then b(u,v) = b(tu,tv) ≤ h. max { b(u,v),b(u,tu),b(v,tv), 1 2k [b(u,tv) + b(v,tu)] } since u,v are the fixed points of t so finally we have b(u,v) ≤ h.b(u,v) 78 sarwar and rahman the above inequality is possible only if b(u,v) = 0 ⇒ u = v. therefore fixed point of t in x is unique. � the above theorem produce the following corollary for k = 1. corollary 3.4. let (x,b) be a complete metric space with coefficient and t be a self mapping t : x → x satisfying the condition (11) b(tx,ty) ≤ h. max { b(x,y),b(x,tx),b(y,ty), 1 2 [b(x,ty) + b(y,tx)] } for all x,y ∈ x, where h ∈ [0, 1), then t has a unique fixed point. remark. corollary 3.4 is the result of b. e. rohades [11]. example 3.1. let x = [0, 1] and b(x,y) = |x−y|2 be a b-metric with coefficient k = 2 ∀ x,y ∈ x, we define the mapping t by tx = 2 3 if x ∈ [0, 1) and t1 = 0 then t satisfy all the conditions of the theorem(3.4) for h ∈ [ 4 9 , 1 2 ) having x = 2 3 is its unique fixed point in x. references [1] mohammad akkouchi, a common fixed point theorem for expansive mappings under strict implicit conditions on b-metric spaces, acta univ. palacki. olomuc, fac. rer. nat. mathematica , 50(2011), 5-15,. [2] h. ayadi et al., a fixed point theorem for set valued quasi contraction in b-metric spaces, fixed point theorey and applications ,2012(2012), article id 88,. [3] v. berinde, iterative approximation of fixed poins of weak contraction using picard iteration, non linear analysis forum, 9(2004), 43-53. [4] lj. b. ciric, a generalization of banach’s contraction principle, proc.amer.math. soc., 45 (1974), 267-273. [5] s. czerwik, contraction mappings in b-metric space, acta math inf univ ostravinsis, 1(1993), 5-11. [6] s. czerwik, non linear set value contraction mappings in b-metric spaces, atti sem math fis univ modina, 46(2)(1998), 263-273. [7] b. k. dass and s. gupta, an extension of banach’s contraction principle through rational expression, indian journal of pure and applied mathematics, 6 (1975), 1455-1458. [8] p. hitzler, generalized metrics and topology in logic programming semantics. ph.d. thesis, national univeristy of ireland, university college cork, (2001). [9] m. s. khan, a fixed point theorem for metric spaces, rendiconti dell’ istituto di mathematica dell’ universtia di tresti, vol. 8 (1976), 69-72,. [10] mehmet kir, hukmi kizitune, on some well known fixed point theorems in b-metric space, turkish journal of analysis and number theory, vol, 1(2013), 13-16. [11] b. e. rohades, a comparison of various definition of contractive mappings, transfer, amer. soc., 226(1977), 257-290. [12] madhu shrivastava , q. k. qureshi and a. d. singh, a fixed point theorem for continuous mapping in dislocated quasi metric space. international journal of theoretical and applied sciences, 4(1)(2012), 39 40. department of mathematics, university of malakand, chakdara dir(l), pakistan ∗corresponding author international journal of analysis and applications volume 19, number 5 (2021), 784-793 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-784 on mocanu-type functions with generalized bounded variations shujaat ali shah∗, muhammad afzal soomro and asghar ali maitlo department of mathematics and statistics, quaid-i-awam university of engineering, science and technology, nawabshah, 67480 sindh, pakistan ∗corresponding author: shahglike@yahoo.com abstract. the main focus of this article is the study of classes mδµ (ϕ, h) and qδµ (ϕ, g1, h). we present various inclusion relationships and some applications of our investigations are considered. also, we include radius problem. 1. introduction let a be the class of analytic functions of the form f(z) = z + ∞∑ n=2 anz n, (1.1) in the open unit disk u = {z : |z| < 1}. if f and g are analytic in u, we say that f is subordinate to g, written f ≺ g or f(z) ≺ g(z), if there exists a schwartz function w in u such that f(z) = g(w(z)). the convolution or hadamard product of two functions f,g ∈a is denoted by f ∗g and is defined as (f ∗g)(z) = z + ∞∑ n=2 anbnz n, z ∈u. (1.2) analytic functions p in the class p[a,b] can be defined by using subordination as follows [3]. let p be analytic in u with p(0) = 1. then p ∈p[a,b], if and only if, p(z) ≺ 1 + az 1 + bz , − 1 ≤ b < a ≤ 1, z ∈u. received july 12th, 2021; accepted august 17th, 2021; published september 14th, 2021. 2010 mathematics subject classification. 30c45, 30c55. key words and phrases. analytic functions; janowski functions; conic regions; bounded boundary rotations. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 784 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-784 int. j. anal. appl. 19 (5) (2021) 785 for k > 0, the conic domains ωk, defined as; ωk = { u + iv : u > k √ (u− 1)2 + v2 } . the domains ωk (k = 0) represents right half plane, ωk (0 < k < 1) represents hyperbola, ωk (k = 1) represents a parabola and ωk (k > 1) represents an ellipse. the extremal functions for these conic regions are given as; pk(z) =   1+z 1−z , k = 0 1 + 2 π2 ( log 1+ √ z 1− √ z )2 , k = 1 1 + 2 1−k2 [( 2 π arccos k ) arctan h √ z ] , 0 < k < 1 1 + 1 k2−1 sin ( π 2r(t) ∫ u(z)√t 0 1√ 1−x2 √ 1−(tx)2 dx ) + 1 k2−1,k > 1, (1.3) where u(z) = z− √ t z− √ tz , t ∈ (0, 1), z ∈ u and z is chosen such that k = cosh ( πr′(t) 4r(t) ) , r(t) is legendre’s complete elliptic integral of the first kind and r′(t) is complementary integral of r(t). see [4, 5] for more information. these conic regions are being studied by several authors, see [6, 9, 12]. in 2017, dziok and noor [2] introduced and studied the concepts of some general classes given as below. definition 1.1. let µ ≥ 0, φ = (φ1(z),φ2(z)) and h = (h1(z),h2(z)) where hi ∈ a with hi(0) = 1, (i = 1, 2). then pµ(h) = {µq1 + (1 −µ) q2 : q1 ∈p (h1) , q2 ∈p (h2)} , where p (h) = {q ∈a : q ≺ h with q(0) = 1} . some special cases: (i) pµ(h) = pµ((h,h)). if µ = m4 + 1 2 , (m ≥ 2), then pµ(h) = pm(h). (ii) if µ = m 4 + 1 2 , (m ≥ 2) , and h(z) = 1+(1−2ρ)z 1−z , then pµ(h) = pm(ρ), this class was introduced by padmanabhan et al. [13]. (iii) if µ = m 4 + 1 2 , (m ≥ 2) and h(z) = 1+az 1+bz (−1 ≤ b < a ≤ 1), then pµ(h) = pm [a,b], this class was introduced by noor [10]. moreover, for a = 1 and b = −1 we have pµ(h) = pm; see [14]. (iv) if µ = m 4 + 1 2 , (m ≥ 2) and h(z) = pκ(z) (κ ≥ 0), then pµ(h) = pm (pκ), this class was defined by noor et al. [11]. definition 1.2. let f ∈a and δ ≥ 0. then f ∈ mδµ (φ,ξ,h) if and only if jδ (f ((z))) ∈pµ(h), where jδ (f ((z))) = (1 −δ) (ξ ∗φ2) ∗f (ξ ∗φ1) ∗f + δ φ2 ∗f φ1 ∗f . if ξ1(z) = z + ∞∑ n=2 1 n zn, φ1(z) = zϕ ′(z) and φ2(z) = zφ ′ 1(z), then we have the following special cases. mδ (φ,ξ,h) = mδ1 (φ,ξ, (h,h)) , m δ µ (φ,h) = m δ µ (φ,ξ1,h) , int. j. anal. appl. 19 (5) (2021) 786 mδµ (ϕ,h) = m δ µ ((φ2,φ1) ,h) , (1.4) s∗µ (ϕ,h) = m 0 µ (ϕ,h) , s ∗ (ϕ,h) = m01 (ϕ,h) . (1.5) definition 1.3. let f ∈ a, g = (g1,g2), where gi ∈ a with gi(0) = 1 (i = 1, 2), and δ,ϑ ≥ 0. then f ∈qδµ,ϑ (φ,ξ,g,h) if there exists g ∈ s ∗ ϑ (ϕ,g) such that (1 −δ) (ξ ∗φ2) ∗f (ξ ∗φ1) ∗g + δ φ2 ∗f φ1 ∗g ∈pµ (h) . if ξ1(z) = z + ∞∑ n=2 1 n zn, φ1(z) = zϕ ′(z) and φ2(z) = zφ ′ 1(z), then we have the following special cases. qδ (φ,ξ,g1,h1) = mδ1,1 (φ,ξ, (g1,g2) , (h1,h2)) , qδµ,ϑ (φ,g,h) = m δ µ,ϑ (φ,ξ1,g,h) , qδµ (ϕ,g1,h) = q δ µ,1 ((φ2,φ1) , (g1,g1) ,h) . (1.6) from (1.4), we denote the class mδµ (ϕ,h) of functions f ∈a satisfies jδ (f(z)) ∈pµ(h), where jδ (f(z)) = (1 − δ) z (ϕ∗f)′ (ϕ∗f) + δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ , and pµ(h) is given by definition 1.1. similarly, from (1.6), we denote the class qδµ (ϕ,h,h) of functions f ∈a satisfies jδ (f(z),g(z)) ∈pµ(h), where jδ (f(z),g(z)) = (1 −δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ , for g ∈ s∗ (ϕ,h), the class s∗ (ϕ,h) is given by (1.5). 2. preliminary results lemma 2.1. [2] let h = (h1,h2), where hi (i = 1, 2) are analytic, univalent convex functions with hi(0) = 1 (i = 1, 2) and let κ : u → c (set of complex numbers) with <(κ) > 0. if p(z) is analytic, with p(0) = 1 in u, satisfies p(z) + κzp′(z) ∈pµ(h), then p(z) ∈pµ(h). int. j. anal. appl. 19 (5) (2021) 787 lemma 2.2. [8] let h be analytic, univalent convex function in u with h(0) = 1 and re (γh(z) + σ) > 0, σ,γ ∈ c and γ 6= 0. if p(z) is analytic in u and p(0) = h(0), then{ p(z) + zp′(z) γp(z) + σ } ≺ h(z), implies p(z) ≺ q(z) ≺ h(z), where q(z) is best dominant and is given as, q(z) = [{∫ 1 0 ( exp ∫ tz t h(u) − 1 u du ) dt }−1 − σ γ ] . lemma 2.3. [15] if f ∈ c,g ∈ s∗, then for each h analytic in u with h(0) = 1, (f ∗hg) (u) (f ∗g) (u) ⊂ coh(u), where coh(u) denotes the convex hull of h(u). 3. main results 3.1. inclusion results. theorem 3.1. let δ ≥ 0, ϕ ∈a and h be any convex univalent function in u. then mδ1 (ϕ,h) ⊂ m 0 1 (ϕ,h) . proof. let f ∈ mδ1 (ϕ,h). then, by definition, (1 −δ) z (ϕ∗f)′ (ϕ∗f) + δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ ∈p(h), or (1 − δ) z (ϕ∗f)′ (ϕ∗f) + δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ ≺ h(z). (3.1) consider z (ϕ∗f)′ (ϕ∗f) = p(z). (3.2) on logarithmic differentiation of (3.2), we have( z (ϕ∗f)′ )′ (ϕ∗f)′ = z (ϕ∗f)′ (ϕ∗f) + zp′(z) p(z) . (3.3) from (3.2) and (3.3), we get ( z (ϕ∗f)′ )′ (ϕ∗f)′ = p(z) + zp′(z) p(z) . (3.4) on making use of (3.2) and (3.4) in (3.1), we obtain (1 −δ) p(z) + δ [ p(z) + zp′(z) p(z) ] ≺ h(z), this implies p(z) + δ zp′(z) p(z) ≺ h(z). by using lemma 2.2, we conclude p(z) ≺ h(z). hence f ∈ m01 (ϕ,h). � int. j. anal. appl. 19 (5) (2021) 788 remark 3.1. following different choices of ϕ and h give certain inclusion results for the above theorem. (i) ϕ ∈ a, h(z) = 1+az 1+bz , where −1 ≤ b < a ≤ 1. (ii) ϕ ∈ a, h(z) = pk(z), where pk(z) is given by (1.3). corollary 3.1. let δ ≥ 1. then mδ1 (ϕ,h) ⊂ m 1 1 (ϕ,h) . proof. let f ∈ mδ1 (ϕ,h). then , by definition, (1 − δ) z (ϕ∗f)′ (ϕ∗f) + δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ = s1(z) ≺ h(z), from previous theorem, we can write z (ϕ∗f)′ (ϕ∗f) = s2(z) ≺ h(z). now, δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ = [ (1 −δ) z (ϕ∗f)′ (ϕ∗f) + δ ( z (ϕ∗f)′ )′ (ϕ∗f)′ ] + (δ − 1) z (ϕ∗f)′ (ϕ∗f) = s1(z) + (δ − 1) s2(z). implies that ( z (ϕ∗f)′ )′ (ϕ∗f)′ = ( 1 − 1 δ ) s2(z) + 1 δ s1(z). (3.5) since s1,s2 ≺ h(z), (3.5) gives us ( z (ϕ∗f)′ )′ (ϕ∗f)′ ≺ h(z). hence f ∈ mδ1 (ϕ,h). � remark 3.2. the different choices of ϕ and h given in remark 3.1 hold the inclusion result proved in above theorem. theorem 3.2. let δ, µ ≥ 0, ϕ ∈a, h = (h1,h2) where hi,h ∈a with hi(0) = h(0) = 1 (i = 1, 2). then qδµ (ϕ,h,h) ⊂q 0 µ (ϕ,h,h) . proof. let f ∈qδµ (ϕ,h,h). then, by definition, (1 − δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ ∈pµ(h), (3.6) for g ∈ s∗ (ϕ,h). consider z (ϕ∗f)′ (ϕ∗g) = p(z), (3.7) int. j. anal. appl. 19 (5) (2021) 789 where p(z) is analytic with p(0) = 1 in u. on logarithmic differentiation of (3.7), we get( z (ϕ∗f)′ )′ (ϕ∗f)′ = z (ϕ∗g)′ (ϕ∗g) + zp′(z) p(z) , ( z (ϕ∗f)′ )′ (ϕ∗g)′ = z (ϕ∗f)′ (ϕ∗g)′  z (ϕ∗g)′ (ϕ∗g) + zp′(z) z(ϕ∗f)′ (ϕ∗g)   , this implies ( z (ϕ∗f)′ )′ (ϕ∗g)′ = z (ϕ∗f)′ (ϕ∗g) + zp′(z) z(ϕ∗g)′ (ϕ∗g) . (3.8) from (3.7) and (3.8), we have( z (ϕ∗f)′ )′ (ϕ∗g)′ = p(z) + zp′(z) p0(z) ; with p0(z) = z (ϕ∗g)′ (ϕ∗g) . (3.9) now, from (3.6), (3.7) and (3.9), we obtain (1 − δ) p(z) + δ ( p(z) + zp′(z) p0(z) ) ∈pµ(h), or equivalently, p(z) + δ p0(z) zp′(z) ∈pµ(h). if g ∈ s∗ (ϕ,h), then z(ϕ∗g) ′ (ϕ∗g) ≺ h(z); h ∈ p. this implies <(p0(z)) > 0 in u. thus, by lemma 2.1, we conclude p(z) ∈pµ(h). consequently, z(ϕ∗f)′ (ϕ∗g) ∈pµ(h). hence, f ∈q 0 µ (ϕ,h,h). � remark 3.3. it is easy to see that the inclusion in theorem 3.2 is true for different choices of ϕ, h and h = (h1,h2) given as following. (i) ϕ ∈ a, h1(z) = 1+az1+bz = h2(z), where −1 ≤ b < a ≤ 1. (ii) ϕ ∈ a, h1(z) = pk(z) = h2(z), where pk(z) is given by (1.3). (iii) ϕ ∈ a, h1(z) = 1+az1+bz , h2(z) = pk(z). (iv) ϕ ∈ a, h1(z) = pk(z), h2(z) = 1+az1+bz . corollary 3.2. let δ ≥ 1. then qδµ (ϕ,h,h) ⊂q 1 µ (ϕ,h,h) . proof. let f ∈qδµ (ϕ,h,h). then, by definition, (1 −δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ = p1(z) ∈pµ(h), where g ∈ s∗ (ϕ,h). from previous theorem, we can write z (ϕ∗f)′ (ϕ∗g) = p2(z) ∈pµ(h). int. j. anal. appl. 19 (5) (2021) 790 now, δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ = [ (1 − δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ ] + (δ − 1) z (ϕ∗f)′ (ϕ∗g) = p1(z) + (δ − 1) p2(z). this implies ( z (ϕ∗f)′ )′ (ϕ∗g)′ = ( 1 − 1 δ ) p2(z) + 1 δ p1(z). since p1, p2 ∈pµ(h) and pµ(h) is convex set, then( z (ϕ∗f)′ )′ (ϕ∗g)′ ∈pµ(h). hence f ∈q1µ (ϕ,h,h). � theorem 3.3. let 0 ≤ δ1 < δ. then qδµ (ϕ,h,h) ⊂q δ1 µ (ϕ,h,h) . proof. if δ1 = 0, then it is obvious from theorem 3.2. for δ1 > 0. let f ∈qδµ (ϕ,h,h). then, from theorem 3.2 z (ϕ∗f)′ (ϕ∗g) = p2(z) ∈pµ(h). (3.10) as we can write (1 − δ1) z (ϕ∗f)′ (ϕ∗g) + δ1 ( z (ϕ∗f)′ )′ (ϕ∗g)′ = δ1 δ [( δ δ1 − 1 ) z (ϕ∗f)′ (ϕ∗g) + (1 − δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ ] . (3.11) since f ∈qδµ (ϕ,h,h), from definition of qδµ (ϕ,h,h), we have (1 −δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ = p1(z) ∈pµ(h). (3.12) from (3.10-3.12) and the convexity of pµ(h) implies (1 − δ1) z (ϕ∗f)′ (ϕ∗g) + δ1 ( z (ϕ∗f)′ )′ (ϕ∗g)′ ∈pµ(h). hence f ∈qδ1µ (ϕ,h,h). � remark 3.4. it is easy to see that the inclusion in theorem 3.3 is true for all choices given in remark 3.3. theorem 3.4. the class qδµ (ϕ,h,h) is closed under the convex convolution. int. j. anal. appl. 19 (5) (2021) 791 proof. let f ∈qδµ (ϕ,h,h). then, by definition, (1 − δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ ∈pµ(h). (3.13) first, we need to prove ς ∗f ∈q0µ (ϕ,h,h) for ς ∈ c. we take δ = 0, then (3.13) implies z (ϕ∗f)′ (ϕ∗g) ∈pµ(h). (3.14) let z (ϕ∗ (ς ∗f))′ (z) (ϕ∗ (ς ∗g)) (z) = ς ∗ z(ϕ∗f) ′ (ϕ∗g) ((ϕ∗g)) (z) ς ∗ (ϕ∗g) (z) = ς ∗h0(z) ((ϕ∗g)) (z) ς ∗ (ϕ∗g) (z) , where h0(z) = z(ϕ∗f)′ (ϕ∗g) ∈ pµ(h). since g ∈ s ∗(ϕ,h) implies ϕ ∗ g ∈ s∗(h) ⊂ s∗; h ∈ p. thus, by lemma 2.3, we conclude z (ϕ∗ (ς ∗f))′ (z) (ϕ∗ (ς ∗g)) (z) ∈pµ(h). (3.15) similarly, for δ = 1, we can easily prove z ( ϕ∗ (ς ∗f)′ )′ (z) (ϕ∗ (ς ∗g))′ (z) ∈pµ(h). (3.16) our required result follows from (3.15) and (3.16). � corollary 3.3. the class qδµ (ϕ,h,h) is closed under the following operators. (i) f1(z) = ∫ z 0 f(t) t dt. (ii) f2(z) = 2 z ∫ z 0 f(t)dt, (libera’s operator [7]). (iii) f3(z) = ∫ z 0 f(t)−f(xt) t−xt dt, |x| ≤ 1, x 6= 1. (iv) f4(z) = c+1 zc ∫ z 0 tc−1f(t), re(c) ≥ 0, (generalized bernardi operator [1]). proof. we may write, fi(z) = f(z) ∗φi(z), where φi(z), i = 1, 2, 3, 4, are convex and given by φ1(z) = − log (1 −z) = ∞∑ n=1 1 n zn, φ2(z) = −2[z−log(1−z)] z = ∞∑ n=1 2 n+1 zn, φ3(z) = 1 1−x log ( 1−xz 1−z ) = ∞∑ n=1 1−xn (1−x)n z n, |x| ≤ 1, x 6= 1, φ4(z) = ∞∑ n=1 1+c n+c zn, re(c) ≥ 0. the proof follows easily by using theorem 3.4. � int. j. anal. appl. 19 (5) (2021) 792 3.2. radius problem. theorem 3.5. let f ∈ m01 ( ϕ, 1+az 1+bz ) . then, f ∈ mδ1 ( ϕ, 1+z 1−z ) for |z| < rδ, where rδ = 2a2 {δ (a−b) + 2a} + √ δ2 (a−b)2 + 4aδ (a−b) . proof. let f ∈ m01 ( ϕ, 1+az 1+bz ) . then, by definition, z (ϕ∗f)′ (ϕ∗f) = p(z) ≺ 1 + az 1 + bz . (3.17) on logrithmic differentiation of (3.17), we get ( z (ϕ∗f)′ )′ (ϕ∗f)′ = z (ϕ∗f)′ (ϕ∗f) + zp′(z) p(z) . (3.18) by (3.17) and (3.18), we obtain ( z (ϕ∗f)′ )′ (ϕ∗f)′ = p(z) + zp′(z) p(z) . (3.19) now, (1 − δ) z (ϕ∗f)′ (ϕ∗g) + δ ( z (ϕ∗f)′ )′ (ϕ∗g)′ = p(z) + δ zp′(z) p(z) . <(jδ (f(z))) ≥ a2r2 −{δ (a−b) + 2a}r + 1 (1 −ar) (1 −br) . for <(jδ (f(z))) > 0 in u, we get rδ = 2a2 {δ (a−b) + 2a} + √ δ2 (a−b)2 + 4aδ (a−b) . � corollary 3.4. let f ∈ m01 ( z 1−z , 1+z 1−z ) = s∗. then f ∈ mδ1 ( z 1 −z , 1 + z 1 −z ) = m(δ), for |z| < rδ = 1(1+δ)+√δ2+2δ . moreover, for δ = 1, we have well known result s∗ ⊂ c, for |z| < r1 = 1 2 + √ 3 . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (5) (2021) 793 references [1] s.d. bernardi, convex and starlike univalent functions, trans. amer. math. soc. 135 (1969), 429–446. [2] j. dziok, k.i. noor, classes of analytic functions related to a combination of two convex functions, j. math. inequal. 11 (2017), 413–427. [3] w. janowski, some extremal problems for certain families of analytic functions i, ann. polon. math. 28 (1973), 297–326. [4] s. kanas, a. wisniowska, conic regions and k-uniform convexity, j. comput. appl. math. 105 (1999), 327–336. [5] s. kanas and a. wisniowska, conic domain and starlike functions, rev. roumaine math. pures appl. 45 (2000), 647–657. [6] h.a. al-kharsani and a. sofo, subordination results on harmonic k-uniformly convex mappings and related classes, comput. math. appl. 59 (2010), 3718–3726. [7] r.j. libera, some classes of regular univalent functions, proc. amer. math. soc. 16 (1965), 755–758. [8] s.s. miller, p.t. mocanu, second order differential inequalities in the complex plane, j. math. anal. appl. 65(2) (1978), 289–305. [9] k.i. noor, m. arif, w. ul-haq, on k-uniformly close-to-convex functions of complex order, appl. math. comput. 215 (2009), 629–635. [10] k.i. noor, some properties of analytic functions with bounded radius rotations, compl. var. ellipt. eqn. 54 (2009), 865–877. [11] k.i. noor, m.a. noor, higher order close-to-convex functions related with conic domains, appl. math. inf. sci. 8 (2014), 2455–2463. [12] h. orhan, e. deniz d. raducanu, the fekete-szego problem for subclasses of analytic functions de ned by a di erential operator related to conic domains, comput. math. appl. 59 (2010), 283–295. [13] k.s. padmanabhan, r. parvatham, properties of a class of functions with bounded boundary rotation, ann. polon. math. 31 (1975), 311–323. [14] b. pinchuk, functions with bounded boundary rotation, israel j. math. 10 (1971), 7–16. [15] s. ruscheweyh, t. sheil-small, hadamard product of schlicht functions and the polya-schoenberg conjecture, comment. math. helv. 48 (1973), 119–135. 1. introduction 2. preliminary results 3. main results 3.1. inclusion results 3.2. radius problem references international journal of analysis and applications volume 19, number 6 (2021), 949-969 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-949 topological sensitivity analysis for the anisotropic laplace problem imen kallel1,2,∗ 1northern border university, college of science, arar, p.o. box 1631, saudi arabia 2ur analysis and control of pde’s, ur 13e64, department of mathematics, faculty of sciences of monastir, university of monastir, 5019 monastir, tunisie ∗corresponding author: imenkallel16@gmail.com abstract. this paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. the aim of this paper is to propose an alternative approach based on the kohnvogelius formulation and the topological sensitivity analysis method. the idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. we derive a topological asymptotic expansion for the anisotropic laplace operator. the unknown object is reconstructed using level-set curve of the topological gradient. we make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm. 1. introduction in this work we will establish a topological sensitivity analysis for the anisotropic laplace operator. the topological sensitivity analysis consists of studying the variation of a given cost functional with respect to the presence of a small domain perturbation, such as the insertion of inclusions, cavities, cracks or source-terms. in our paper we concentrate in a small dirichlet geometric perturbation. let us briefly discuss the history of this method. its main idea was originally introduced by schumacher [22] in the context of compliance minimization in linear elasticity. in the same context sokolowski and received august 28th, 2021; accepted october 12th, 2021; published november 16th, 2021. 2010 mathematics subject classification. 49q12, 65n21, 35n10. key words and phrases. topological gradient; kohn-vogelius formulation; laplace problem; reconstruction problem. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 949 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-949 int. j. anal. appl. 19 (6) (2021) 950 zochowski [13], who studied the effect of an extract infinitesimal part of the material in structural mechanics. then in [17] masmoudi worked out a topological sensitivity analysis framework based on a generalization of the adjoint method and on the use of a truncation technique. by using this framework the topological sensitivity is obtained for several equations [8, 18, 20, 21]. for other works on the topological sensitivity concept, we refer to the book by novotny and sokolowski [19]. the general idea of the proposed method is to rephrase the inverse problem as a topology optimization problem, where the object immersed in the anisotropic media is the unknown variable. our aim is to detect this unknown object immersed from over-determined boundary data. let h be an unknown object immersed inside the background domain ω and having a smooth boundary σ = ∂h. the geometric inverse problem that we consider here can be formulated as follows: • giving two boundary data on γ ; an imposed flux φ ∈ h−1/2(γ) and a measured datum ϕd ∈ h1/2(γ). • find the unknown location of the object h inside the domain ω such that the solution ϕ of the anisotropic laplace equation satisfies the following over-determined boundary value problem   −div(a∇ϕ) = f in ω\h, a∇ϕ · n = φ on γ, ϕ = ϕd on γ, ϕ = 0 on ∂h. where a is a symmetric positive definite matrix, n is the exterior unit normal vector and f ∈ l2(ω) is a given source term. in this formulation the domain ω\h is unknown since the free boundary ∑ is unknown. this problem is ill-posed in the sense of hadamard [10]. the majority of works dealing with this kind of problems fall into the category of shape optimization and based on the shape differentiation technics. it is proved in [3, 7] that the studied inverse problems, treated as a shape optimization problems, are severely ill-posed (i.e. unstable), for both dirichlet and neumann conditions on the boundary ∑ . thus they have to use some regularization methods to solve them numerically. to solve this inverse problem, we extend the topological sensitivity analysis notion to the anisotropic case and we suggest an alternative approach based on the kohn-vogelius formulation [6] and the topological gradient method [1, 2, 5, 8, 9, 16, 17]. we combine here the advantages of the kohn-vogelius formulation as a self regularization technique and the topological gradient approach as an accurate and fast method. the first step of our approach is based on the kohn-vogelius formulation which rephrase the considered geometrical inverse problem into a topology optimization one. it leads to define for any given permutation int. j. anal. appl. 19 (6) (2021) 951 h two forward problems: the first one, called neumann problem, is associated to the neumann datum φ (pn)   −div(a∇ϕn ) = f in ω\h, a∇ϕn · n = φ on γ, ϕn = 0 on ∂h. the second one is associated to the measured data ϕd, it is called the dirichlet problem: (pd)   −div(a∇ϕd) = f in ω\h, ϕd = ϕd on γ, ϕd = 0 on ∂h. one can observe that if h coincides with the exact obstacle h∗ , the misfit between the solutions of(pn) and (pd) vanishes: ϕn = ϕd . starting from this observation, the inverse problem can be formulated as a topological optimization one. the unknown object will be characterized as the minimum of the following kohn-vogelius type functional [6] j(ω\h) = ∫ ω\h a|∇ϕn (x) −∇ϕd(x)|2dx where the kohn-vogelius function is exactly j(ω\h) = ∫ ω\h [∇ϕn (x) −∇ϕd(x)]t ·a[∇ϕn (x) −∇ϕd(x)]dx more precisely, the identification problem can be formulated as follow: (p)   find h∗ ⊂ ω such that j(ω\h∗) = min h∈dad j(ω\h) where dad is a given set of admissible domains. to solve the topological optimization problem (p) and detect the location of the unknown object we will derive a topological sensitivity analysis for the kohn-vogelius function j which gives the variation of a criterion with respect to the presence of a small dirichlet geometric perturbation in the domain. a one-shot reconstruction algorithm is proposed. the main advantage of this algorithm is that, it provides fast and accurate results for detection. the paper is organized as follows. in the next section, we present the perturbed neumann and dirichlet problems. in section 3 we study the topological sensitivity analysis for the function j . the obtained results are based on a preliminary estimate describing the perturbation caused by the presence of a small geometry modification of the background domain ω. a simplified formulation of the shape function variation with respect to the creation of the hole hz,ε in ω is derived in section 4. the section 5 is devoted to the kohnvogelius type function variation. the proposed numerical algorithm and the detection results are described in section 6. int. j. anal. appl. 19 (6) (2021) 952 2. the perturbed problems in this section, we present the neumann and dirichlet problems in the perturbed domain. in the presence of a small geometry perturbation hz,ε inside the domain ω, the neumann problem consists in finding ϕεn ∈ h 1(ω\hz,ε) solution to   −div(a∇ϕεn ) = f in ω\hz,ε, a∇ϕεn · n = φ on γn, ϕεn = 0 on ∂hz,ε. the neumann problem in the non perturbed domain is: find ϕ0n ∈ h 1(ω) solution to  −div(a∇ϕ 0 n ) = f in ω, a∇ϕ0n · n = φ on γn. similarly, the perturbed dirichlet problem consists in finding ϕεd ∈ h 1(ω\hz,ε) solution to  −div(a∇ϕεd) = f in ω\hz,ε, ϕεd = ϕd on γd, ϕεd = 0 on ∂hz,ε. in the absence of any perturbation (i.e ε = 0), the dirichlet problem is: find ϕ0d ∈ h 1(ω) solution to  −div(a∇ϕ 0 d) = f in ω, ϕ0d = ϕd on γd. we introduce the considered shape functional j . given a small geometrical perturbation hz,ε inside the initial domain ω, the function j measures the difference between the neumann and dirichlet perturbed solutions. we define j as j(ω\hz,ε) = ∫ ω\hz,ε a|∇ϕεn (x) −∇ϕ ε d(x)| 2dx, ∀hz,ε ⊂ ω in the non perturbed domain (ε = 0), the function j is expressed as j(ω) = ∫ ω a|∇ϕ0n (x) −∇ϕ 0 d(x)| 2dx. our aim is to derive an asymptotic expansion for the function j and calculate the topological sensitivity function δj . the variation of the function j with respect to the presence of a small perturbation is given by j(ω\hz,ε) −j(ω) = ∫ ω\hz,ε a|∇ϕεn (x) −∇ϕ ε d(x)| 2dx − ∫ ω a|∇ϕ0n (x) −∇ϕ 0 d(x)| 2dx. in the following, we will derive a topological sensitivity analysis valid for all function jε verifying the following hypothesis: int. j. anal. appl. 19 (6) (2021) 953 hypothesis 2.1. the function j0 is differentiable in h1(ω). there exist a real number δj ∈ r, independent of ε and a scalar function ρ : r+ → r+ such that ∀ε ≥ 0 jε(ϕε) −j0(ϕ0) = dj0(ϕ0)(ϕε −ϕ0) + ρ(ε)δj + o(ρ(ε)), lim ε→0 ρ(ε) = 0 with ϕε is the solution of the perturbed anisotropic laplace problem inside the perturbed domain ω\hz,ε (2.1) (pε)   −div(a∇ϕε) = f in ω\hz,ε, a∇ϕε · n = φ on γ, ϕε = ϕd on γ, ϕε = 0 on ∂hz,ε. here δj is called the topological sensitivity function. 3. topological asymptotic expansion in this section, we derive a topological asymptotic expansion for the anisotropic laplace operator. we start our analysis by establishing a variational formulation associated to the anisotropic laplace system. from the weak formulation of 2.1, we deduce that ϕε ∈vε is the unique solution to aε(ϕε,v) = lε(v) ∀v ∈vε, where the function space vε, the bilinear form aε and the linear form lε are defined by vε { w ∈ h1(ω\hz,ε); w = 0 in γd ∪∂hz,ε } and for all ϕ,v ∈vε   aε(v,w) = ∫ ω\hz,ε [∇w]t ·a∇vdx, lε(w) = ∫ ω\hz,ε fwdx + ∫ γn φwds. under the hypothesis 2.1, the variation of the shape function j reads j(ω\hz,ε) −j(ω) = jε(ϕε) −j0(ϕ0) = dj0(ϕ0)(ϕε −ϕ0) + ρ(ε)δj + o(ρ(ε)) let v0 ∈v0 be the solution to the associated adjoint problem a0(w,v0) = dj0(ϕ0)(w),∀w ∈v0. int. j. anal. appl. 19 (6) (2021) 954 then, the shape function variation rewritten as j(ω\hz,ε) −j(ω) = a0(ϕ0 −ϕε,v0) + ρ(ε)δj + o(ρ(ε)). aiming to derive an asymptotic expansion for j , we examine in the next section the asymptotic behavior with respect to ε of the term a0(ϕ0 −ϕε,v0). 3.1. asymptotic formula for the anisotropic laplace problem. this section is devoted to the main theoretical result. a topological asymptotic expansion is derived for the anisotropic laplace operator with respect to the presence of a small topological perturbation hz,ε in the initial domain. the obtained results are general and valid for large class of cost functions. more precisely, the derived asymptotic expansion is valid for all cost function j satisfying the assumption 2.1. the main result of this section is summarized in the following theorem. theorem 3.1. let j a design function of the form j(ω\hz,ε) = j(ϕε). if j satisfies the assumption 2.1 , then j has the following asymptotic expansion j(ω\hz,ε) − j(ω) = ρ(ε)(δa + δj) + o(ρ(ε)) where   ρ(ε) = −1 log(ε) , δa = 2π √ |a|ϕ0(z)v0(z) if d = 2, ρ(ε) = ε, δa = −4π √ |a|ϕ0(z)v0(z) if d = 3. the term δj is the variation of the considered cost function j . in order to check the hypothesis 2.1, we derive an asymptotic expansion the variation of the bilinear form. we have a0(ϕ0 −ϕε,v0) = ∫ ω [∇(ϕ0 −ϕε)]t ·a∇v0dx.(3.1) using the green formula, we obtain (3.2) a0(ϕ0 −ϕε,v0) = ∫ hz,ε [∇ϕ0]t ·a∇v0dx + ∫ ∂hz,ε [a∇(ϕ0 −ϕε)]t · nv0dx next, we shall examine each term on the right hand side of 3.2 separately. the following lemma gives an estimate for the first term. lemma 3.1. we have ∫ hz,ε [∇ϕ0]t ·a∇v0dx = εd|h|[∇ϕ0(z)]t ·a∇v0(z) + o(εd). int. j. anal. appl. 19 (6) (2021) 955 proof using the change of variable x = z + εy one obtains ∫ hz,ε [∇ϕ0]t ·a∇v0dx = εd|h|[∇ϕ0(z)]t ·a∇v0(z) + εd ∫ h { [∇ϕ0(x + εy)]t ·a∇v0(z + εy) − [∇ϕ0(z)]t ·a∇v0(z) } dy due the the smoothness of ϕ0 and v0 in hz,ε, we derive lim ε→0 ∫ h { [∇ϕ0(x + εy)]t ·a∇v0(z + εy) − [∇ϕ0(z)]t ·a∇v0(z) } dy = 0 then it follows ∫ hz,ε [∇ϕ0]t ·a∇v0dx = εd|h|[∇ϕ0(z)]t ·a∇v0(z) + o(εd). next to examine the second term of 3.2 we introduce the variable χε = ϕ0 −ϕε. it is easily to show that χε satisfies the following system (3.3)   −div(a∇χε) = 0 in ω\hz,ε, a∇χε · n = 0 on γn, χε = 0 sur γd, χε = ϕ0 on ∂hz,ε. we can write χε as χε = hε + rε where hε(x) = e(x−z) e(z) ϕ0(z), x ∈ ω, with e is the fundamental solution of the anisotropic laplace operator [14]: e(x) =   1 2π √ |a| log(||a∗x||), d = 2, −1 4π √ |a|||a∗x|| , d = 3. int. j. anal. appl. 19 (6) (2021) 956 where |a| is the determinant of a, a∗ be the positive-definite symmetric matrix such that a2∗ = a−1. then rε is solution of (3.4)   −div(a∇rε) = div(a∇hε) in ω\hz,ε, a∇rε · n = −a∇hε · n on γn, rε = −hε sur γd, rε = ϕ0 −hε on ∂hz,ε. we set r1(ε) = ∫ ∂hz,ε [a∇hε]t · n(v0 −v0(z))ds, r2(ε) = ∫ ∂hz,ε [a∇rε]t · nv0ds. lemma 3.2. we have∫ ∂hz,ε [a∇(ϕ0 −ϕε)]t · nv0ds = − ϕ0(z)v0(z) e(ε) + r1(ε) + r2(ε) proof using the green formulation, we obtain ∫ ∂hz,ε [a∇(ϕ0 −ϕε)]t · nv0ds = ∫ ∂hz,ε [a∇χε]t · nv0ds = ∫ ∂hz,ε [a∇hε]t · nv0ds + ∫ ∂hz,ε [a∇rε]t · nv0ds = ∫ ∂hz,ε [a∇hε]t · n(v0 −v0(z))ds + ∫ ∂ωε [a∇hε]t · nv0(z)ds + r2(ε) = ∫ ∂hz,ε [a∇hε]t · nv0(z)ds + r1(ε) + r2(ε) = ϕ0(z)v0(z) e(ε) ∫ ∂ωε [a∇e(x−z)]t · nds + r1(ε) + r2(ε). such that ∫ ∂hz,ε [a∇e(x−z)]t · nds = −1 hence ∫ hz,ε [∇χε]t ·a∇v0dx = − ϕ0(z)v0(z) e(ε) + r1(ε) + r2(ε) assuming that ri(ε) = o (ρ(ε)) , i = 1, 2. we will give the proof for the two dimensional case in section 4.3. besides thanks to the fundamental solution, we obtain the main result presented in the following sections concerns the topological asymptotic expansion of an arbitrary design function j. some cost function examples are presented in section 4. int. j. anal. appl. 19 (6) (2021) 957 4. a particular class of cost function in this section, we present some useful examples of shape functions and we gives their variations δj . 4.1. first example. this example is concerned with the l2−norm. we consider the shape function defined by j(ω\hz,ε) = ∫ ω\hz,ε |ϕε −ϕd|2dx, ∀ϕε ∈ h1(ω\hz,ε). where ϕd ∈ h1(ω) is a given desired (objective) function. proposition 4.1. the cost function jε defined by jε(ϕ) = ∫ ω\hz,ε |ϕ−ϕd|2dx, ∀ϕ ∈ h1(ω\hz,ε). satisfies the hypothesis 2.1 with dj0(ϕ0) = 2 ∫ ω (ϕ0 −ϕd) w dx, ∀w ∈ h1(ω) and δj(x) = 0,∀x ∈ ω. 4.2. second example. here, we are dealing with the h1−semi-norm. we consider the shape function j(ω\hz,ε) = ∫ ω\hz,ε a|∇ϕε −∇ϕd|2dx with ϕd ∈ h2(ω) is a given desired function. proposition 4.2. the cost function jε defined by jε(ϕ) = ∫ ω\hz,ε a|∇ϕ−∇ϕd|2dx, ∀ϕ ∈ h1(ω\hz,ε), satisfies the hypothesis 2.1 with dj0(ϕ0) = 2 ∫ ω (∇ϕ0 −∇ϕd)t ·a∇w dx, ∀w ∈ h1(ω) and ∀x ∈ ω, δj(x) =   2π √ |a||ϕ0(x)|2, d = 2 −4π √ |a||ϕ0(x)|2, d = 3. 4.3. proofs. int. j. anal. appl. 19 (6) (2021) 958 4.3.1. preliminary results. the aim of this section is to give some technical results which will be used in section 4.3. lemma 4.1. [4] consider ψ ∈ h1/2(∂hz,ε),gd ∈ h1/2(γd) and gn ∈ h−1/2(γn) if x is solution of the following system (4.1)   −∆x = 0 in ω\hz,ε, ∇x ·n = gn on γn, x = gd on γd, x = ψ on ∂hz,ε. there exists a non negative constant c such that ||x||1,ω\hz,ε ≤ c [ 1√ − log(ε) ||ψ(z + εy)||1/2,∂h + ||gd||1/2,γd + ||gn||−1/2,γn ] lemma 4.2. the function defined by hε(x) = e(x−z) e(ε) ϕ0(z), ∀x ∈ ω\hz,ε admits the following estimates ||hε||1,ωr ≤ −c log ε ||hε||1,ω\hz,ε ≤ c √ − log ε where r is a strictly positive real number such that ωε ⊂ b(z,r) ⊂ ω, we set ωr = ω\b(z,r). proof of lemma 4.2 using the definition of hε, we have ||hε||1,ωr = −1 log ε ||ϕ0(z) log(|x−z|)||1,ωr since that z /∈ ωr, we deduce that ||hε||1,ωr ≤ −c log ε . then we have hε is solution of (4.2)   −∆hε = 0 in ω\hz,ε, ∇hε ·n = ϕ0(z) e(ε) ∇e(x−z) ·n on γn, hε = e(x−z) e(ε) ϕ0(z) on γd, hε = e(x−z) e(ε) ϕ0(z) on ∂hz,ε. int. j. anal. appl. 19 (6) (2021) 959 note that, by lemma 4.1, ‖hε‖1,ω\hz,ε ≤ c { 1√ − log(ε) ‖ e(εy) e(ε) ϕ0(z)‖1 2 ,∂h + ‖ ϕ0(z) e(ε) ∇e(x−z).n‖−1 2 ,γn +|| e(x−z) e(ε) ϕ0(z)||1 2 ,γd } ≤ c1√ − log(ε) + c2 − log(ε) . then, ‖hε‖1,ω\hz,ε ≤ c√ − log(ε) this completes the proof. � lemma 4.3. there exists a constant c strictly positive such that ‖ϕ0 −hε‖1 2 ,∂hz,ε ≤ −c log(ε) . proof of lemma 4.3 using the change of variables x = z + εy, we obtain ϕ0(x) −hε(x) = ϕ0(z + εy) −hε(z + εy) = ϕ0(z + εy) − e(εy) e(ε) ϕ0(z) = ϕ0(z + εy) − ( 1 + e(y) e(ε) ) ϕ0(z). the smoothness of ϕ0 and e in hz,ε and h gives that ϕ0(z + εy) −ϕ0(z) = o(ε) and e(y)ϕ0(z) e(ε) = o ( −1 log(ε) ) . hence ϕ0(x) −hε(x) = o ( −1 log(ε) ) . thus the proof is complete. � lemma 4.4. we have the following estimation ‖rε‖1,ω\hz,ε ≤ c√ − log(ε) . int. j. anal. appl. 19 (6) (2021) 960 proof of lemma 4.4 we can write rε solution of the system 3.4 as follows rε = r 1 ε + r 2 ε where r 1 ε satisfies (4.3)   ∆r1ε = 0 in ω\hz,ε, ∇r1ε · n = 0 on γ, r1ε = 0 sur γ, r1ε = ϕ0 −hε on ∂hz,ε, and r2ε is solution of (4.4)   ∆r2ε = −∆hε in ω\hz,ε, ∇r2ε · n = −∇hε · n on γ, r2ε = −hε sur γ, r2ε = 0 on ∂hz,ε, it then follows from 4.3 and the green formulation that∫ ω\hz,ε |∇r1ε| 2dx =< ∇r1ε · n,ϕ0 −hε >−1/2,1/2,∂h, note that, by the theorem of the normal trace, we obtain∫ ω\hz,ε |∇r1ε|2dx ≤ ||∇r1ε||−1/2,∂hz,ε||(ϕ0 −hε)||1/2,∂hz,ε ≤ ||r1ε||1,ω\hz,ε||ϕ0 −hε||1/2,∂hz,ε ≤ −c log ε ||r1ε||1,ω\hz,ε the poincare inequality gives us ||r1ε||1,ω\hz,ε ≤ c ∫ ω\hz,ε |∇r1ε| 2dx ||r2ε||1,ω\hz,ε ≤ c { ||∇hε.n||−1/2,γn + ||hε||1/2,γd + ||∆hε||0,ω\hz,ε } ≤ ||hε||1,ω\hz,ε then, ||r2ε||1,ω\hz,ε ≤ c √ − log ε since ||rε||1,ω\hz,ε ≤ ||r 1 ε||1,ω\hz,ε + ||r 2 ε||1,ω\hz,ε we conclude that ||rε||1,ω\hz,ε ≤ c √ − log ε . � int. j. anal. appl. 19 (6) (2021) 961 proof of theorem 3.1 we only need to prove that r1(ε) = o ( −1 log(ε) ) and r2(ε) = o ( −1 log(ε) ) . remember that r1(ε) = ∫ ∂hz,ε [a∇hε]t · n(v0 −v0(z))ds we have |r1(ε)| ≤ ||a||∞||∇hε · n||−1 2 ,∂hz,ε||v0 −v0(z)||12 ,∂hz,ε ≤ ||a||∞||hε||1,ω\hz,ε||v0 −v0(z)||12 ,∂hz,ε changing variables x = z + εy and using the lemma 4.2 |r1(ε)| ≤ cε√ − log(ε) = o ( −1 log(ε) ) . on the other hand r2(ε) = ∫ ∂hz,ε [a∇rε]t · nv0ds. we have |r2(ε)| ≤ c||∇rε||−1 2 ,∂hz,ε||v0||12 ,∂hz,ε ≤ c||rε||1,ω\hz,ε||v0||1,hz,ε. likewise, using the same change of variables and due to lemma 4.4, it follows that |r2(ε)| ≤ cε√ − log(ε) = o ( −1 log(ε) ) . which completes the proof. � proof of proposition 4.1 the function j is differentiable with respect to ϕ and we have dj0(ϕ0)(w) = 2 ∫ ω (ϕ0 −ϕd)w dx, ∀w ∈v0. int. j. anal. appl. 19 (6) (2021) 962 computing the variation j(ω\hz,ε) −j(ω) j(ω\hz,ε) −j(ω) = ∫ ω\hz,ε |ϕε −ϕd|2dx− ∫ ω |ϕ0 −ϕd|2dx = ∫ ω [|ϕε −ϕd|2 −|ϕ0 −ϕd|2]dx− ∫ hz,ε |ϕd|2dx = ∫ ω [|ϕ0 −ϕε|2 − 2(ϕ0 −ϕε)(ϕ0 −ϕd)]dx− ∫ ω\hz,ε |ϕd|2dx = dj0(ϕ0)(ϕε −ϕ0) + ∫ ω\hz,ε |ϕ0 −ϕε|2dx + ∫ hz,ε |ϕ0|2dx− ∫ hz,ε |ϕd|2dx. by the divergence formula and the system 3.3, we have∫ ω\hz,ε |ϕ0 −ϕε|2dx = o ( −1 log(ε) ) . a change of variable and the fact that ϕ0 and ϕd are of class c 2 in a neighborhood of the origin yield∫ hz,ε |ϕd|2dx ≤ cε2 and ∫ hz,ε |ϕ0|2dx ≤ cε2. hence j(ω\hz,ε) −j(ω) = dj0(ϕ0)(ϕε −ϕ0) + o ( −1 log(ε) ) . finally, by theorem 3.1, we deduce j(ω\hz,ε) − j(ω) = −1 log(ε) 2π √ |a|ϕ0(x0)v0(x0) + o ( −1 log(ε) ) . � proof of proposition 4.2 the function j is differentiable and we have dj0(ϕ0)(w) = 2 ∫ ω a(∇ϕ0 −∇ϕd)t .∇w dx, ∀w ∈v0. moreover, we have j(ω\hz,ε) −j(ω) = ∫ ω\hz,ε a|∇ϕε −∇ϕd|2dx− ∫ ω a|∇ϕ0 −∇ϕd|2dx = dj0(ϕ0)(ϕε −ϕ0) + ∫ hz,ε a|∇ϕ0|2dx− ∫ hz,ε a|∇ϕd|2dx + ∫ ω\hz,ε a|∇ϕ0 −∇ϕε|2dx as ϕ0 and ϕd are sufficiently regular in hz,ε, we obtain∫ hz,ε a|∇ϕ0|2dx = o(ε2) = o ( −1 log(ε) ) int. j. anal. appl. 19 (6) (2021) 963 and ∫ hz,ε a|∇ϕd|2dx = o(ε2) = o ( −1 log(ε) ) . by the divergence formula, we have∫ ω\hz,ε a|∇ϕ0 −∇ϕε|2dx = ∫ ∂hz,ε [a(∇ϕ0 −∇ϕε)]t .nϕ0dx = −2π log(ε) √ |a||ϕ0(x0)|2 + o ( −1 log(ε) ) . hence j(ω\hz,ε) −j(ω) = dj0(ϕ0)(ϕε −ϕ0) + −2π √ |a| log(ε) |ϕ0(z)|2 + o ( −1 log(ε) ) . finally, by theorem 3.1, we deduce j(ω\hz,ε) −j(ω) = −1 log(ε) 2π √ |a| [ ϕ0(z)v0(z) + |ϕ0(z)|2 ] + o ( −1 log(ε) ) . � 5. the kohn-vogelius norms the kohn-vogelius criterion [15] is used like a cost functional. since the boundary conditions (ϕd, φ) are overspecified, one can define for any hole h two forward problems: • the ”dirichlet” problem:   −div(a∇ϕd) = 0 in ω\h, ϕd = ϕd on γd, ϕd = 0 on ∂h. • the ”neumann” problem:   −div(a∇ϕn ) = 0 in ω\h, a∇ϕn · n = φ on γn, ϕn = 0 on ∂h. the optimal hole h∗ coincides with the actual boundary h when the misfit between the solutions vanishes: ϕd = ϕn . therefore, we propose an identification process based on the minimization of the following energy functional j(ω\h) = ∫ ω\h a|∇ϕn −∇ϕd|2dx. this is the so-called kohn-vogelius criterion [15]. our approach concerns the derived topological optimization problem: min h⊂ω j(ω\h). int. j. anal. appl. 19 (6) (2021) 964 we will use the topological gradient method to solve this problem. it provides an asymptotic expansion of the function j with respect to a small topological perturbation of the domain ω. 5.1. asymptotic expansion of the cost functional. the following theorem describes the variation of the function j when creating a small hole hz,ε inside the domain ω with a dirichlet boundary condition on ∂hz,ε. for all ε ≥ 0,j(ω\hz,ε) = ∫ ω\hz,ε a|∇ϕεn −ϕ ε d| 2dx where ϕεn and ϕ ε d are the solutions to the systems  −div(a∇ϕεn ) = 0 in ω\hz,ε, a∇ϕεn · n = φ on γn, ϕεn = 0 on ∂hz,ε. ;   −div(a∇ϕεd) = 0 in ω\hz,ε, ϕεd = ϕd on γd, ϕεd = 0 on ∂hz,ε. 5.1.1. the three dimensional case. in this paragraph, we present the topological asymptotic expansion for the anisotropic laplace equations in the three dimensional case. in this case the fundamental solution of the anisotropic laplace operator e is given by e(x) = −1 4π √ |a|||a∗x|| , ∀x ∈ ω. theorem 5.1. under the same hypotheses of theorem 3.1, the function j has the following asymptotic expansion j(ω\hz,ε) −j(ω) = ε { 4π √ |a| [ ϕ0n (z)v 0 n (z) + ϕ 0 d(z)v 0 d(z) ] + δj(z) } + o(ε), with δj(x) = 4π √ |a| { |ϕ0n (x)| 2 + |ϕ0d(x)| 2 } , ∀x ∈ ω. 5.1.2. the two dimensional case. in this paragraph, the result is obtained using the same technique described in the previous paragraph. the unique difference comes from the expression of the fundamental solution of the anisotropic laplace equations. in this case e is given by e(x) = 1 2π √ |a| log(||a∗x||), ∀x ∈ ω. theorem 5.2. under the same hypotheses of theorem 3.1, the function j has the following asymptotic expansion j(ω\hz,ε) −j(ω) = −1 log(ε) { 2π √ |a| [ ϕ 0 n (z)v 0 n (z) + ϕ 0 d(z)v 0 d(z) ] + δj(z) } + o ( −1 log(ε) ) , with δj(x) = 2π √ |a| { |ϕ0n (x)| 2 + |ϕ0d(x)| 2 } , ∀x ∈ ω. int. j. anal. appl. 19 (6) (2021) 965 6. numerical result this section is concerned with some numerical investigations. we consider the bidimentional case and we present a fast and simple one-iteration identification algorithm. the unknown object h is identified using a level set curve of the topological gradient δj . more precisely, the unknown object h is likely to be located at zone where the topological gradient δj is more negative. one-iteration algorithm: [11, 12] • solve the two problems (p0n ) and (p 0 d) in initial domain ω, • compute the topological gradient function δj(x), x ∈ ω, • determine the unknown object h = {x ∈ ω, such that δj(x) < cmin}, where cmin is a negative constant chosen in such a way that the cost function j has the most negative value. next, we will present some numerical simulations using the proposed algorithm. in figure 1, we test our algorithm on circular shape. in figure 2, we consider the case of an elliptical shape. in figure 3, we can notice that, when the shape is non-regular, the reconstruction is quite efficient. in the case of non trivial shape, yet we applied a one-iteration algorithm, we obtain an interesting reconstruction result (see figure 4). the obtained result can serve as a good initial estimate for an iterative optimization process based on the shape derivative. the considered model can be viewed as a prototype of a geometric inverse problem valid in many applications. 6.0.1. reconstruction of circular-shaped objects. in this case, we test our procedure to detect an object having circular-shaped. we reconstruct in this case the object h described by a disk centered at z = (2, 0) with different radius: r ∈ {0.2, 0.4, 0.6}. the obtained results are illustrated in figure 1. one can easily observe in figure 1, the unknown object is located in the region where the topological sensitivity function δj is the most negative (red zone). the boundary of h∗ is approximated by an iso-value curve. our one-iteration algorithm gives an efficient reconstruction results for the different chosen sizes. 6.0.2. reconstruction of ellipse-shaped objects. in this example, we reconstruct an object described by an ellipse inserted in the disc d = b((2, 0), 1) and centered at (2, 0). we represent the results in figure 2. in this case, we examine the numerical reconstruction of various ellipses having different directions and sizes. as one can observe in figure 2, the boundary of the object is again detected and located in the zone where int. j. anal. appl. 19 (6) (2021) 966 the topological gradient is most negative (red lines). also, our one-iteration algorithm gives quite effecient reconstruction results for different chosen ellipse-shaped objects. 6.0.3. reconstruction of geometry with corners. we tried to apply our proposed algorithm to detect more complicated geometry. our objective is to reconstruct an object with corners. more precisely, we are trying to detect a square and rectangle shape . we can see in the figure 3 (a), that the unknown square h∗ is located in the zone where the topological gradient function δj is the most negative (red zone) and also its boundary is approximated by an iso-value curve. so here, our one-iteration algorithm detects the location and the shape of the square. but in the case of a rectangle shape (see figure 3 (b)) the boundary h∗ cannot be approached by any iso-value curve of the topological gradient function. we can remark in this case, that the one-iteration algorithm detects the zone containing the unknown geometry but the reconstruction result is not good. 6.0.4. reconstruction of a non trivial-shaped objects. we apply now our proposed algorithm to detect a non trivial shapes. we can see in figure 4 that the unknown shape h∗ is located in the zone where the topological gradient function δj is the most negative (red iso-values) but we cannot approximate the boundary of h∗ by any iso-value curve of the topological sensitivity function δj . we can improve these reconstruction results by suggesting an iterative algorithm. r=0.2:negative zone(red zone) iso-value of δj zoom showing the iso-value of δj approximating δh∗(black line) r=0.4:negative zone(red zone) iso-value of δj zoom showing the iso-value of δj approximating δh∗(black line) int. j. anal. appl. 19 (6) (2021) 967 r=0.6:negative zone(red zone) iso-value of δj zoom showing the iso-value of δj approximating δh∗(black line) figure 1. reconstruction of circle shaped objects figure 2. reconstruction of an ellipse shaped objects (a) square shape (b) rectangle shape figure 3. reconstruction of objects with corners. acknowledgement the author gratefully acknowledge the approval and the support of this research study by the grant no. 7772-sci-2018-3-9-f from the deanship of scientific research at northern border university, arar, saudi arabia. int. j. anal. appl. 19 (6) (2021) 968 figure 4. reconstruction of a non trivial shape. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. abdelwahed, m. hassine and m. masmoudi, optimal shape design for fluid flow using topological perturbation technique, j. math. anal. appl. 356(2) (2009), 548-563. [2] m. abdelwahed and m. hassine, topological optimization method for a geometric control problem in stokes flow, appl. numer. math. 59(8) (2009), 1823-1838. [3] l. afraites, m. dambrine, k. eppler and d. kateb, detecting perfectly insulated obstacles by shape optimization techniques of order two, discrete contin. dyn. syst. ser. b, 8(2) (2007), 389-416. [4] s. amstutz, aspects théoriques et numériques en optimisation de forme topologique, phd thesis, toulouse, insa, 2003. [5] s. amstutz, the topological asymptotic for the navier-stokes equations, esaim: control, optim. calc. var. 11(3) (2005), 401-425. [6] s. andrieux, t. baranger and a. ben abda, solving cauchy problems by minimizing an energy-like functional. inverse probl. 22(1) (2006), 115-133. [7] m. badra, f. caubet and m. dambrine, detecting an obstacle immersed in a fluid by shape optimization methods, math. models meth. appl. sci. 21(10) (2011), 2069-2101. [8] s. garreau, p. guillaume and m. masmoudi, the topological asymptotic for pde systems: the elasticity case, siam j. control optim. 39(6) (2001), 1756-1778. [9] p. guillaume and k. sid idris, the topological asymptotic expansion for the dirichlet problem, siam j. control optim. 41(4) (2002), 1042-1072. [10] a. hamdi, a non-iterative method for identifying multiple unknown timedependent sources compactly supported occurring in a 2d parabolic equation,inverse probl. sci. eng. 26(5) (2018), 744-772. [11] m. hassine and i.kallel, kohn-vogelius formulation and topological sensitivity analysis based method for solving geometric inverse problems, arab j. math. sci. 24(1) (2018), 43-62. [12] m. hassine and i. kallel, one-iteration reconstruction algorithm for geometric inverse problems, appl. math. e-notes, 18 (2018), 43-50. int. j. anal. appl. 19 (6) (2021) 969 [13] l. jackowska-strumillo, j. sokolowski and a. zochowski, the topological derivative method in shape optimization, in: proceedings of the 38th ieee conference on decision and control (cat. no. 99ch36304), volume 1, pages 674-679. ieee, 1999. [14] e. kim, h. kang and k. kim, anisotropic polarization tensors and detection of an anisotropic inclusion, siam j. appl. math. 63(4) (2003), 1276-1291. [15] r. v. kohn and m. vogelius, relaxation of a variational method for impedance computed tomography, commun. pure appl. math. 40(6) (1987), 745-777. [16] m. hassine, shape optimization for the stokes equations using topological sensitivity analysis, rev. afr. rech. inform. math. appl. 5 (2016), 216-229. [17] m. masmoudi, the topological asymptotic, computational methods for control applications, ed. h. kawarada and j. périaux. international series, gakuto, 2002. [18] m. masmoudi, j. pommier and b. samet, the topological asymptotic expansion for the maxwell equations and some applications, inverse probl. 21(2) (2005), 547. [19] a. novotny and j. soko lowski, topological derivatives in shape optimization, springer science & business media, 2012. [20] j. pommier and b. samet, the topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole, siam j. control optim. 43(3) (2004), 899-921. [21] b. samet, s. amstutz and m. masmoudi, the topological asymptotic for the helmholtz equation, siam j. control optim. 42(5) (2003), 1523-1544. [22] w. yan, m. liu and f. jing, shape inverse problem for stokesbrinkmann equations, appl. math. lett. 88 (2019), 222-229. 1. introduction 2. the perturbed problems 3. topological asymptotic expansion 3.1. asymptotic formula for the anisotropic laplace problem 4. a particular class of cost function 4.1. first example 4.2. second example 4.3. proofs 5. the kohn-vogelius norms 5.1. asymptotic expansion of the cost functional. 6. numerical result acknowledgement references international journal of analysis and applications volume 16, number 5 (2018), 763-774 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-763 generalized r-convex functions and integral inequalities muhammad aslam noor∗, khalida inayat noor, farhat safdar department of mathematics, comsats university islamabad, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we introduce and investigate a new class of generalized convex functions, known as generalized r-convex function. some new hermite-hadamard integral inequalities via generalized r-convex functions have been established. results proved in this paper can be viewed as significant new contributions in this area of research. 1. introduction in order to obtain some desirable results in general framework related to pure and applied sciences, the concept of convexity has been extended and generalized in several directions, see [1, 2, 4, 5, 7, 11–17]. several branches of mathematical and engineering sciences has been developed by using the crucial and significant concepts of convex analysis and hence it becomes one of the most interesting and useful concept of mathematics for last few decades. hermite-hadamard inequality is one of the most important inequality related to convex function, see [9, 10]. in recent years, much attention has been given to derive the hermite-hadamard type inequalities for various types of convex functions, see [8–10, 17, 18, 22]. gill et al [8] introduced and investigated the concept of r-convex functions. they established some new hermite-hadamard integral inequalities for rconvex functions. for the applications of r-convex functions in statistics and probability theory, see pecaric et al. [25–27, 30, 31] and the references therein. received 2018-02-07; accepted 2018-04-27; published 2018-09-05. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. generalized convex functions; generalized r-convex functions; hermite-hadamard type inequalities. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 763 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-763 int. j. anal. appl. 16 (5) (2018) 764 gordji et al. [7] has introduced an other class of convex functions, which is called generalized convex( ϕ-convex ) functions. the generalized convex functions are nonconvex functions. for recent developments, see [3, 6, 18–21, 25–31] and the references therein. inspired and motivated by the ongoing research, we introduce a new class of generalized convex functions, which is called generalized r-convex functions. we derive some new hermite-hadamard integral inequalities for these nonconvex functions. some special cases are also discussed, which can be obtained from our main results. the ideas and techniques of this may be starting point for further research in this field. 2. preliminaries let i = [a,b] be an interval in real line r. let f : i → r be a continuous function and η(·, ·) : r×r → r be a continuous bifunction. we need the following well known results and concepts. definition 2.1. [7] a function f : i = [a,b] → r is said to be generalized convex function with respect to a bifunction η : r×r → r, if f((1 − t)a + tb) ≤ (1 − t)f(a) + t[f(a) + η(f(b),f(a))],∀a,b ∈ i,t ∈ [0, 1]. definition 2.2. a function f : i = [a,b] → r is said to be generalized r-convex function with respect to a bifunction η : r×r → r, if ∀a,b ∈ i, t ∈ [0, 1] f((1 − t)a + tb) ≤   { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r , r 6= 0, [f(a)]1−t[f(a) + η(f(b),f(a))]t, r = 0. 1-convex is called generalized convex functions and 0-convex is called generalized log convex functions. if t = 1 2 , then f( a + b 2 ) ≤   { [f(a)]r+[f(a)+η(f(b),f(a))]r 2 }1 r , r 6= 0, √ [f(a)][f(a) + η(f(b),f(a))], r = 0. the function f is called generalized jensen r-convex function. if η(f(b) −f(a)) = f(b) −f(a), then definition 2.3. [8] a function f : i = [a,b] → r is said to be r-convex function, if f((1 − t)a + tb) ≤   { (1 − t)[fr(a)] + t[fr(b)] }1 r , r 6= 0, [f(a)]1−t[f(b)]t, r = 0. int. j. anal. appl. 16 (5) (2018) 765 note that for r = 1, we have classical convex functions and for r = 0, we have log convex functions. for the recent applications of convex functions and their generalizations, see [1–9, 12–15, 17, 18, 22–27, 31]. the generalized logarithmic means of order r of positive numbers a,b is defined by lr(a,b) =   r r+1 ar+1−br+1 ar−br , r 6= 0,−1,a 6= b, a−b log a−log b, r = 0,a 6= b, ab log a−log b a−b , r = −1,a 6= b, a, a = b. definition 2.4. the beta function, also called the euler integral of the first kind, is defined as β(x,y) = ∫ 1 0 tx−1(1 − t)y−1dt = γ(x)γ(y) γ(x + y) , x,y > 0 where γ(.) is a gamma function. 3. main results in this section, we establish several new integral inequalities of hermite-hadamard type for generalized r-convex functions. theorem 3.1. let f : i → r be generalized r-convex function on i. then for 0 < r ≤ 1, we have 1 b−a ∫ b a f(x)dx ≤ ( r r + 1 ){( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r . proof. let f be generalized r-convex function on i. then, ∀a,b ∈ i,t ∈ [0, 1], we have f((1 − t)a + tb) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r . using minkowski’s inequality and the fact that f is generalized r-convex function, we have 1 b−a ∫ b a f(x)dx = ∫ 1 0 f((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r dt ≤ {(∫ 1 0 (1 − t) 1 r [f(a)]dt )r + (∫ 1 0 t 1 r [f(a) + η(f(b),f(a))]dt )r}1 r = {( r r + 1 )r( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r = ( r r + 1 ){( [f(a)]r + [f(a) + η(f(b),f(a))]r )}1r , which is the required result. � int. j. anal. appl. 16 (5) (2018) 766 corollary 3.1. [29] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.7, we have 1 b−a ∫ b a f(x)dx ≤ ( r r + 1 ){( [fr(a) + fr(b)] )}1r . theorem 3.2. let f : i → r be generalized r-convex function on i. then for 0 < r ≤ 1, we have fr( a + b 2 ) − 1 2(b−a) ∫ b a η(f(a + b−x),f(x)]rdx ≤ 1 2(b−a) ∫ b a fr(x)dx ≤ {{ [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r + [f(b) + η(f(a),f(b))]r )} . proof. let f be generalized r-convex function on i. then for t = 1 2 , we have fr( x + y 2 ) ≤ { [f(x)]r + [f(x) + η(f(y),f(x))]r 2 } . this implies that fr( a + b 2 ) ≤ { [f((1 − t)a + tb)]r + [f((1 − t)a + tb) + η(f(ta + (1 − t)b),f((1 − t)a + tb))]r 2 } . integrating the above inequality with respect to t on [0,1], we have fr( a + b 2 ) ≤ 1 2 ∫ 1 0 { [f((1 − t)a + tb)]r + [f((1 − t)a + tb) +η(f(ta + (1 − t)b),f((1 − t)a + tb))]r } dt = 1 2(b−a) ∫ b a { [f(x)]r + [f(x) + η(f(a + b−x),f(x))]r } dx. this implies fr( a + b 2 ) − 1 2(b−a) ∫ b a [f(x) + η(f(a + b−x),f(x))]rdx ≤ 1 2(b−a) ∫ b a fr(x)dx. (3.1) consider, fr((1 − t)a + tb) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r } , (3.2) fr((1 − t)b + ta) ≤ { (1 − t)[f(b)]r + t[f(b) + η(f(a),f(b))]r } . (3.3) adding (3.2) and (3.3), we have fr((1 − t)a + tb) + fr((1 − t)b + ta) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r } + { (1 − t)[f(b)]r + t[f(b) + η(f(a),f(b))]r } . int. j. anal. appl. 16 (5) (2018) 767 integrating the above inequality with respect to t on [0,1], we have 2 b−a ∫ b a fr(x)dx ≤ ∫ 1 0 { (1 − t) ( [f(a)]r + [f(b)]r ) + t ( [f(a) + η(f(b),f(a))]r +[f(b) + η(f(a),f(b))]r )} dt, which implies that 1 b−a ∫ b a fr(x)dx ≤ {{ [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r +[f(b) + η(f(a),f(b))]r )} . (3.4) combining (3.1) and (3.4), we have fr( a + b 2 ) − 1 2(b−a) ∫ b a [f(x) + η(f(a + b−x),f(x))]rdx ≤ 1 2(b−a) ∫ b a fr(x)dx ≤ {{ [f(a)]r + [f(b)]r } 4 + 1 4 ( [f(a) + η(f(b),f(a))]r + [f(b) + η(f(a),f(b))]r )} , which is the required result. � corollary 3.2. [29] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.2, we have fr( a + b 2 ) ≤ 1 b−a ∫ b a fr(x)dx ≤ { [fr(a) + fr(b)] 2 } . theorem 3.3. let f : i → r be generalized r-convex function on i and r ≥ 0, then 1 b−a ∫ b a f(x)dx ≤   ( r r+1 ){ η(fr+1(b),fr+1(a)) η(fr(b),fr(a)) } , r 6= 0{ [f(a)+η(f(b),f(a))]−[f(a)] log[f(a)+η(f(b),f(a))]−log[f(a)] } , r = 0. proof. first, let r > 0 and f be generalized r-convex function on i. then, ∀a,b ∈ i,t ∈ [0, 1], we have 1 b−a ∫ b a f(x)dx = ∫ 1 0 f((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r dt. (3.5) subsituting u = [(1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r] in (3.5), we have 1 b−a ∫ b a f(x)dx ≤ 1 [f(a) + η(f(b),f(a))]r − [f(a)]r ∫ [f(a)+η(f(b),f(a))]r [f(a)]r u 1 r du = ( r r + 1 ){( [f(a) + η(f(b),f(a))]r+1 − [f(a)]r+1 ) [f(a) + η(f(b),f(a))]r − [f(a)]r } int. j. anal. appl. 16 (5) (2018) 768 for r = 0, we have f((1 − t)a + tb) ≤ { [f(a)]1−t[f(a) + η(f(b),f(a))]t } . hence we have, 1 b−a ∫ b a f(x)dx = ∫ 1 0 f((1 − t)a + tb)dt ≤ ∫ 1 0 { [f(a)]1−t[f(a) + η(f(b),f(a))]t } dt = [f(a)] ∫ 1 0 { [f(a) + η(f(b),f(a))] [f(a)] }t dt = { [f(a) + η(f(b),f(a))] − [f(a)] log[f(a) + η(f(b),f(a))] − log[f(a)] } , which is the required result. � corollary 3.3. [8] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.3, we have 1 b−a ∫ b a f(x)dx ≤   ( r r+1 ){ (fr+1(b)−fr+1(a)) (fr(b)−fr(a)) } = lr ( f(a),f(b) ) , r 6= 0{ [f(b)−f(a)] log[f(b)]−log[f(a)] } = l ( f(a),f(b) ) , r = 0. theorem 3.4. let f : i → r be generalized r-convex function on i and r ≥ 0, then 1 b−a ∫ b a f(x)dx ≤   { [fr(a)] + ( r r+1 )r η(fr(a),fr(a))] )}1r ,r 6= 0,f(a) = f(b), 1 η(f−1(b),f−1(a)) [ log[f−1(a) + η(f−1(b),f−1(a))] − log[f−1(a)] ] , r = −1. proof. first, let r > 0, f be generalized r-convex function on i and f(a) = f(b). then, ∀a,b ∈ i,t ∈ [0, 1], we have 1 b−a ∫ b a f(x)dx = ∫ 1 0 f((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r + t[f(a) + η(f(a),f(a))]r }1 r dt. = f(a) for r = −1 and f(a) 6= f(b), we have int. j. anal. appl. 16 (5) (2018) 769 1 b−a ∫ b a f(x)dx = ∫ 1 0 f((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]−1 + t[f(a) + η(f(b),f(a))]−1 }−1 dt = 1 [f(a) + η(f(b),f(a))]−1 − [f(a)]−1 ∫ [f(a)+η(f(b),f(a))]−1 [f(a)]−1 1 u du = [ log[f(a) + η(f(b),f(a))]−1 − log[f(a)]−1 ] [f(a) + η(f(b),f(a))]−1 − [f(a)]−1 , which is the required result. � corollary 3.4. [29] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.4, we have 1 b−a ∫ b a f(x)dx ≤   f(a),r 6= 0,f(a) = f(b), f(a)f(b) log[f(b)]−log[f(a)] f(b)−f(a) = l−1(f(a),f(b), r = −1. theorem 3.5. let f,g : i → r be generalized r1-convex function and generalized r2-convex function respectively on i. then for r1 > 0,r2 > 0 >, we have 1 b−a ∫ b a f(x)g(x)dx ≤ ( r r + 1 ){( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + ( r r + 1 ){( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 . proof. let f,g : i → r be generalized r1-convex function and generalized r2-convex function respectively on i with (r1 > 0,r2 > 0). then, ∀a,b ∈ i,t ∈ [0, 1], we have f((1 − t)a + tb) ≤ { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 , g((1 − t)a + tb) ≤ { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 . int. j. anal. appl. 16 (5) (2018) 770 using cauchy’s and minkowski’s inequalities and the fact that f and g are generalized r1 and r2-convex functions, we have 1 b−a ∫ b a f(x)g(x)dx = ∫ 1 0 f((1 − t)a + tb)g((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 dt ≤ 1 2 ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 2 r1 dt + 1 2 ∫ 1 0 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 2 r2 dt ≤ 1 2 {(∫ 1 0 (1 − t) 2 r1 [f(a)]2dt )r1 2 + (∫ 1 0 t 2 r1 [f(a) + η(f(b),f(a))]2dt )r1 2 } 2 r1 + 1 2 {(∫ 1 0 (1 − t) 2 r2 [g(a)]2dt )r2 2 + (∫ 1 0 t 2 r2 [g(a) + η(g(b),g(a))]2dt )r2 2 } 2 r2 = {( r r + 1 )r1 2 ( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + {( r r + 1 )r2 2 ( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 = ( r r + 1 ){( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 )} 2r1 + ( r r + 1 ){( [g(a)]r2 [g(a) + η(g(b),g(a))]r2 )} 2r2 , which is the required result. � corollary 3.5. [29] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.5, we have 1 b−a ∫ b a f(x)g(x)dx ≤ ( r r + 1 ){( [fr1 (a) + fr1 (b)] )} 2r1 + ( r r + 1 ){( [gr2 (a) + gr2 (b)] )} 2r2 . int. j. anal. appl. 16 (5) (2018) 771 theorem 3.6. let f,g : i → r be generalized r1-convex function and generalized r2-convex function respectively on i. then for r1 > 0,r2 > 0 and 1 r1 + 1 r2 = 1 , we have 1 b−a ∫ b a f(x)dx ≤ 1 2 {( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 ) 1 r1 ( [g(a)]r2 + [gr2 (a) + η(g(b),g(a))]r2 ) 1 r1 } . proof. let f,g : i → r be generalized r1-convex function and generalized r2-convex function respectively on i with (r1 > 0,r2 > 0). then, ∀a,b ∈ i,t ∈ [0, 1], we have f((1 − t)a + tb) ≤ { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 g((1 − t)a + tb) ≤ { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 . using holder’s inequality and the fact that f and g are generalized r1 and r2-convex functions, we have 1 b−a ∫ b a f(x)g(x)dx = ∫ 1 0 f((1 − t)a + tb)g((1 − t)a + tb)dt ≤ ∫ 1 0 { (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 } 1 r1 { (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 } 1 r2 dt ≤ {∫ 1 0 (1 − t)[f(a)]r1 + t[f(a) + η(f(b),f(a))]r1 dt } 1 r1 {∫ 1 0 (1 − t)[g(a)]r2 + t[g(a) + η(g(b),g(a))]r2 dt } 1 r2 = {( [f(a)]r1 ∫ 1 0 (1 − t)dt + [f(a) + η(f(b),f(a))]r1 ∫ 1 0 tdt )} 1 r1 {( [g(a)]r2 ∫ 1 0 (1 − t)dt + [g(a) + η(g(b),g(a))]r2 ∫ 1 0 tdt )} 1 r2 = 1 2 {( [f(a)]r1 + [f(a) + η(f(b),f(a))]r1 ) 1 r1 ( [g(a)]r2 + [gr2 (a) + η(g(b),g(a))]r2 ) 1 r1 } , which is the required result. � corollary 3.6. [29] if η(f(b),f(a)) = f(b) −f(a), then, under the assumptions of theorem 3.6, we have 1 b−a ∫ b a f(x)dx ≤ {( [fr1 (a) + fr1 (b)] ) 1 r1 ( [gr2 (a) + gr2 (b) ) 1 r1 } 2 . int. j. anal. appl. 16 (5) (2018) 772 theorem 3.7. let f,g : i → r be generalized r-convex function on i. then for r > 0, we have ( 1 b−a ∫ b a f(x)g(x)dx )r ≤ { m(a,b) ( r r + 2 )r + n(a,b) ( β( 1 r + 1, 1 r + 1) )r} . where m(a,b) = ( [fr(a)gr(a)] + [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r ) n(a,b) = ( [fr(a)][g(a) + η(g(b),g(a))]r + [gr(a)][f(a) + η(f(b),f(a))]r ) , and β(·, ·) is the beta function. proof. let f,g be two generalized r-convex functions on i. then ∀a,b ∈ i,t ∈ [0, 1], we have f((1 − t)a + tb) ≤ { (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r }1 r g((1 − t)a + tb) ≤ { (1 − t)[g(a)]r + t[g(a) + η(g(b),g(a))]r }1 r . using minkowski’s inequality and the fact that f and g are generalized r-convex functions, we have ( 1 b−a ∫ b a f(x)g(x)dx )r = (∫ 1 0 f((1 − t)a + tb)g((1 − t)a + tb)dt )r ≤ {∫ 1 0 ( (1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r (1 − t)[g(a)]r + t[g(a) + η(g(b),g(a))]r )1 r dt }r = {∫ 1 0 ( (1 − t)2[fr(a)gr(a)] + t(1 − t)( [fr(a)][g(a) + η(g(b),g(a))]r + [gr(a)][f(a) + η(f(b),f(a))]r ) +t2 ( [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r )}r int. j. anal. appl. 16 (5) (2018) 773 ≤ { [fr(a)gr(a)] (∫ 1 0 (1 − t) 2 r dt )r + ( [fr(a)][g(a) + η(g(b),g(a))]r + [gr(a)][f(a) + η(f(b),f(a))]r ) (∫ 1 0 [t(1 − t)] 1 r dt )r + ( [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r )(∫ 1 0 t 2 r dt )r = ( [fr(a)gr(a)] + [f(a) + η(f(b),f(a))]r[g(a) + η(g(b),g(a))]r ) (∫ 1 0 t 2 r dt )r + ( [fr(a)][gr(a) + η(gr(b),gr(a))] + [gr(a)][fr(a) + η(fr(b),fr(a))] ) (∫ 1 0 [t(1 − t)] 1 r dt )r = { m(a,b) ( r r + 2 )r + n(a,b) ( β( 1 r + 1, 1 r + 1) )r} , which is the required result. � acknowledgements the authors would like to thank the rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl, 335(2007),1294-1308. [2] m. alomari, m. darus and s. s. dragomir, new inequalities of simpson’s type for s-convex functions with applications, rgmia res. rep. coll, 12 (4) (2009). [3] g. cristescu, l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrechet, holland, (2002). [4] m. r. delavar and s. s. dragomir, on η-convexity, math. inequal. appl., 20(1)(2017), 203-216. [5] s. s. dragomir and c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university, australia, (2000). [6] m. e. gordji, m. r. delavar and m. d. sen, on ϕ convex functions, j. math. inequal, 10(1)(2016), 173-183. [7] m. e. gordji, m. r. delavar and s. s. dragomir, an inequality related to η-convex functions (ii), int. j. nonlinear. anal. appl, 6(2)(2015), 27-33. [8] p. m. gill, c. e. m. pearce , j. pecaric, hadamards inequality for r-convex functions, j. math. anal. appl, 215(1997), 461-470. [9] j. hadamard, etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann, j. math. pure. appl., 58(1893), 171-215. int. j. anal. appl. 16 (5) (2018) 774 [10] c. hermite, sur deux limites d’une integrale definie, mathesis, 3(1883), 82. [11] d. h. hyers and s. m. ulam, approximately convex functions, proc. amer. math. soc, 3(1952), 821-828. [12] c. p. niculescu and l. e. persson, convex functions and their applications. springer-verlag, new york, (2006). [13] m. a. noor, general variational inequalities, appl. math. letters,1(1988), 119-121. [14] m. a. noor, some develpments in general variational inequalities, appl. math. comput. 152(2004), 199-277. [15] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inform. sci. 10(5)(2016), 1811-1814. [16] m. a. noor and k. i. noor, some implicit methods for solving harmonic variational inequalities , inter. j. anal. appl. 12(1)(2016), 10-14. [17] m. a. noor, k. i. noor and m. u. awan, some new estimates of hermite-hadamard inequalities via harmonically r-convex functions, le mathematiche, lxxi(ii)(2016), 117-127. [18] m. a. noor, k. i. noor, m. u. awan and f. safdar, on strongly generalized convex functions, filomat, 31(18)(2017), 5783-5790. [19] m. a. noor, k. i. noor and f. safdar, generalized geometrically convex functions and inequalities, j. inequal. appl, 2017(2017), article id 22. [20] m. a. noor, k. i. noor and f. safdar, integral inequaities via generalized convex functions, j. math. computer, sci, 17(4)(2017), 465-476. [21] m. a. noor, k. i. noor, s. iftikhar, f. safdar, integral inequaities for relative harmonic (s,η)-convex functions, appl. math. comp. sci, 1(1)(2015), 27-34. [22] m. a. noor, k. i. noor and f. safdar, integral inequaities via generalized (α,m)-convex functions, j. nonlinear. funct. anal, 2017(2017), article id 32. [23] m. a. noor, k. i. noor and f. safdar, new inequalities for generalized log h-convexd functions, j. appl. math. inform. 36(3-4)(2018), 245-256. [24] m. a. noor, k. i. noor and f. safdar,inequalities via generalized beta m-convex functions, j. math. anal. 9((2018). [25] m. a. noor, k. i. noor and s. iftikhar, inequaities via (p,r)-convex functions, rad, (2018). [26] m. a. noor, k. i. noor and s. iftikhar, on harmonic (h,r)-convex functions, proced. jangj. math. soc. 21(2)(2018), 239-251. [27] m. a. noor, k. i. noor, f. safdar, m. u. awan and s. ullah, inequaities via generalized log m-convex functions, j. nonlinear. sci. appl, 10(2017), 5789-5802. [28] m. a. noor, k. i. noor, s. iftikhar and f. safdar, generalized (h,r)-harmonic convex functions and inequalities, int. j. math. anal. 16(4)(2018). [29] n. p. n. ngoc, n.v. vinh, p. t. t. hien, integral inequalities of hadamard type for r-convex functions, int. math. forum, 4(35)(2009), 1723-1728. [30] j. pecaric, f. proschan and y. t. tong, convex functions, partial ordering and statistical applications, academic pres, new york, (1992). [31] g. s. yang, refinement of hadamard’s inequality for r-convex functions, indian j. pure appl. math. 32(10)(2001), 15711579. 1. introduction 2. preliminaries 3. main results acknowledgements references international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 16-21 http://www.etamaths.com a note on fixed point theory for cyclic weaker meir–keeler function in complete metric spaces stojan radenović abstract. in this paper we consider, discuss, improve and complement recent fixed points results for so-called cyclical weaker meir-keeler functions, established by chi-ming chen [chi-ming chen, fixed point theory for the cyclic weaker meir-keeler function in complete metric spaces, fixed point theory appl., 2012, 2012:17]. in fact, we prove that weaker meir-keeler notion is superfluous in results. 1. introduction and preliminaries the banach contraction principle [1] has various applications in many branches of applied science. it ensures the existence and uniqueness of fixed point of a contraction on a complete metric space. after this interesting principle, several authors generalized it by introducing the various contractions on metric spaces (see, e.g., [2]-[14]). rhoades [19], in his work compare several contractions defined on metric spaces. cyclic representations and cyclic contractions were introduced by kirk et al. [9] and further used by several authors to obtain various interesting and significant fixed point results (see, e.g., [2], [3], [8], [11],[12], [13], [14], [16]-[18]). however, we have proved ([16]-[18]) the following result: • if some ordinary fixed point theorem in the setting of complete metric spaces has a true cyclic-type extension, then these both theorems are equivalent. in this paper we prove the similar things. namely, we consider, discuss, improve and complement recent fixed points results for so-called cyclical weaker meir-keeler functions, established by chi-ming chen in [4]. in fact, we prove that weaker meirkeeler notion introduced in [4], is superfluous in results. it is well known that a function ψ : [0, +∞) → [0, +∞) is said to be a meirkeeler function if for each η > 0, there exists δ > 0 such that for t ∈ [0, +∞) with η ≤ t < η + δ, we have ψ (t) < η. chi-ming chen introduced weaker meir-keeler function: definition 1.1. [4] the function ψ : [0, +∞) → [0, +∞) is said to be a weaker meir-keeler function for each η > 0, there exists δ > 0 such that for t ∈ [0, +∞) with η ≤ t < η + δ, there exists n0 ∈ n such that ψn0 (t) < η. also in [4], the author assume the following conditions for a weaker meir-keeler function ψ : [0, +∞) → [0, +∞) : 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. fixed point theory; weaker meir-keeler function; cyclic typecontraction; cauchy sequence. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 16 fixed point theory for cyclic weaker meir–keeler function 17 (ψ1) ψ (t) > 0 for t > 0 and ψ (0) = 0; (ψ2) for all t ∈ [0,∞),{ψ n (t)}n∈n is decreasing; (ψ3) for tn ∈ [0,∞), we have that: (a) if limn→∞ tn = γ > 0, then limn→∞ψ (tn) < γ, and (b) if limn→∞ tn = 0, then limn→∞ψ (tn) = 0. chi-ming chen in [4] suppose that ϕ : [0, +∞) → [0, +∞) is a non-decreasing and continuous function satisfying: (ϕ1) ϕ (t) > 0 for t > 0 and ϕ (0) = 0; (ϕ2) ϕ is subadditive, that is, for every µ1,µ2 ∈ [0, +∞),ϕ (µ1 + µ2) ≤ ϕ (µ1) + ϕ (µ2) ; (ϕ3)for all t ∈ (0,∞) , limn→∞ tn = 0 if and only if limn→∞ϕ (tn) = 0. author state the notion of cyclic weaker (ψ �ϕ)−contraction as follows: definition 1.2. [4] let (x,d) be a metric space, m ∈ n, a1, ...,am be nonempty subsets of x and x = ∪mi=1ai. an operator f : x → x is called a cyclic weaker (ψ �ϕ)−contraction if: (i) x = ∪mi=1ai is a cyclic representation of x with respect to f; (ii) for any x ∈ ai,y ∈ ai+1, i ∈{1, 2, ...,m} , (1.1) ϕ (d (fx,fy)) ≤ ψ (ϕ (d (x,y))) , where am+1 = a1. in [4] author proved the following: theorem 1.3. let (x,d) be a complete metric space, m ∈ n,a1, ...,am be nonempty closed subsets of x and x = ∪mi=1ai. let f : x → x be a cyclic weaker (ψ �ϕ)−contraction. then, f has a unique fixed point z ∈∩mi=1ai.: the cyclic weaker (ψ,ϕ)−contraction is defined in [4]: definition 1.4. let ψ : [0,∞) → [0,∞) be a weaker meir-keeler function satisfying conditions (ψ1) , (ψ2) and (ψ3) . also, let ϕ : [0,∞) → [0,∞) be a nondecreasing and continuous function satisfying (ϕ1) . definition 1.5. let (x,d) be a metric space, m ∈ n, a1, ...,am be nonempty subsets of x and x = ∪mi=1ai. an operator f : x → x is called a cyclic weaker (ψ,ϕ)−contraction if: (i) x = ∪mi=1ai is a cyclic representation of x with respect to f; (ii) for any x ∈ ai,y ∈ ai+1, i ∈{1, 2, ...,m} , (1.2) d (fx,fy) ≤ ψ (d (x,y)) −ϕ (d (x,y)) , where am+1 = a1. in [4] author proved the following result for this type of operator: theorem 1.6. let (x,d) be a complete metric space, m ∈ n, a1, ...,am be nonempty closed subsets of x and x = ∪mi=1ai. let f : x → x be a cyclic weaker (ψ,ϕ)−contraction. then, f has a unique fixed point z ∈∩mi=1ai. here we will use the following (new, useful and very significant) result for the proofs of cyclic-type results (see also [15]-[18]): lemma 1.7. let (x,d) be a metric space, f : x → x be a mapping and let x = ∪pi=1ai be a cyclic representation of x w.r.t. f. assume that (1.3) lim n→∞ d (xn,xn+1) = 0, where xn+1 = fxn,x1 ∈ a1. if {xn} is not a cauchy sequence then there exist ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that the 18 radenović following sequences tend to ε+ when k →∞ : (1.4) d ( xm(k)−j(k),xn(k) ) , d ( xm(k)−j(k)+1,xn(k) ) , d ( xm(k)−j(k),xn(k)+1 ) , d ( xm(k)−j(k)+1,xn(k)+1 ) , where j (k) ∈{1, 2, ...,p} is chosen so that n (k)−m (k)+ j (k) ≡ 1 (mod p) , for each k ∈ n. 2. main results in this section, first of all, we announce the following remarks: (a) author in [4] has not the assumption that the function ψ is a non-decreasing. however, from the proof of both theorems follows that he use this fact (page 3, lines 22-25; page 6, lines 15-18). (b) further, from (ψ2) and (ψ3) , (a) we follows that ψ n (t) → 0 (as n →∞) for all t ∈ [0,∞). proof. indeed, there exists limn→∞ψ n (t) = γ ≥ 0. if γ > 0, then (2.1) γ = lim n→ ψn+1 (t) = lim n→∞ ψ (ψn (t)) < γ (by (ψ3) , (a)). a contradiction. � (c) since, must non-decreasing and ψn (t) ↓ 0 as n → ∞ for all t ∈ [0,∞) we easy obtain that ψ (t) < t for t > 0. (d) further, we have that d (xn+1,xn) → 0 (as n →∞)without using the notion of a weaker meir-keeler function. that is, lines 26-33 on page 3 are superfluous. (e) now, according to lemma 1.7. one can obtain much shorter proof of theorem 1.3. namely, we do not use the property (ϕ2) of the function ϕ. proof. indeed, putting x = xm(k)−j(k),y = xn(k) in (1.1) we obtain a contradiction: (2.2) ϕ ( d ( fxm(k)−j(k),,fxn(k) )) ≤ ψ ( ϕ ( d ( xm(k)−j(k),,xn(k) ))) that is., (2.3) ϕ ( d ( xm(k)−j(k)+1,xn(k)+1 )) ≤ ψ ( ϕ ( d ( xm(k)−j(k),xn(k) ))) . now, passing to limit as k →∞ and using the properties of ϕ and ψ, follows (2.4) ϕ (ε) ≤ lim k→∞ ψ ( ϕ ( d ( xm(k)−j(k),xn(k) ))) < ϕ (ε) . hence, {xn} is a cauchy sequence. � (e’) similarly, putting x = xm(k)−j(k),y = xn(k) in (1.2) we obtain again a contradiction: (2.5) d ( xm(k)−j(k)+1,xn(k)+1 ) ≤ ψ ( d ( xm(k)−j(k),xn(k) )) −ϕ ( d ( xm(k)−j(k),xn(k) )) . letting to limit as k →∞ and using again the properties of ϕ and ψ, we have (2.6) ε ≤ lim k→∞ ψ ( d ( xm(k)−j(k),xn(k) )) −ϕ (ε) < ε−ϕ (ε) . this means that {xn} is a cauchy sequence. � by the same method as in [16]-[18] one can prove the following two results: theorem 2.1. theorem 1.3. is a equivalent with the following: • let (x,d) be a complete metric space and let f : x → x be a weaker (ψ �ϕ)−contraction, that is., (2.7) ϕ (d (fx,fy)) ≤ ψ (ϕ (d (x,y))) , for all x,y ∈ x. then, f has a unique fixed point z ∈ x. fixed point theory for cyclic weaker meir–keeler function 19 theorem 2.2. theorem 1.6. is a equivalent with the following: • let (x,d) be a complete metric space and let f : x → x be a weaker (ψ,ϕ)−contraction, that is., (2.8) d (fx,fy) ≤ ψ ((d (x,y))) −ϕ (d (x,y)) , for all x,y ∈ x. then, f has a unique fixed point z ∈ x. conclusion: in all previous results, that is in theorems 3 and 4 of [4] it is sufficient that the functions ψ and ϕ satisfy the following conditions: 1. ψ : [0,∞) → [0,∞) is a non-decreasing function satisfying (ψ1) , (ψ2)and (ψ3) ; 2. ϕ : [0, +∞) → [0, +∞) is a non-decreasing and continuous function satisfying (ϕ1) and (ϕ3) . hence, without weaker meir-keeler property for ψ as well as without the subadditivity for ϕ. in the sequel we announce the following two results generalizing theorems 1.3. and 1.6. above, that is., theorems 3 and 4 from [4]. firstly, we define: definition 2.3. let (x,d) be a metric space, m ∈ n, a1, ...,am be nonempty subsets of x and x = ∪mi=1ai. an operator f : x → x is called a cyclic generalized (ψ �ϕ)−contraction (resp. cyclic generalized (ψ,ϕ)−contraction) if: (i) x = ∪mi=1ai is a cyclic representation of x with respect to f; (ii) for any x ∈ ai,y ∈ ai+1, i ∈{1, 2, ...,m} , (2.9) ϕ (d (fx,fy)) ≤ ψ (ϕ (m (x,y))) , where am+1 = a1 (2.10) (resp. d (fx,fy) ≤ ψ (m (x,y)) −ϕ (m (x,y))), where m (x,y) = max { d (x,y) ,d (x,fx) ,d (y,fy) , d(x,fy)+d(y,fx) 2 } (iii) ψ,ϕ : [0,∞) → [0,∞) are functions satisfying 1. and 2. from above conclusion. theorem 2.4. let (x,d) be a complete metric space, m ∈ n,a1, ...,am be nonempty closed subsets of x and x = ∪mi=1ai. let f : x → x be a cyclic generalized (ψ �ϕ)−contraction (resp. cyclic generalized (ψ,ϕ)−contraction). then, f has a unique fixed point z ∈∩mi=1ai. proof. given x0 ∈ x and let xn+1 = fxn, for n ∈ {0, 1, ..} . picard sequence. if there exists n0 ∈ {0, 1, ...} such that xn0+1 = xn0, then we finished the proof. therefore, let xn+1 6= xn for all n ∈{0, 1, ...} . it is clear, that for any n ∈{1, 2, ...} there exists in ∈{1, 2, ...,m} such that xn−1 ∈ ain and xn ∈ ain+1. since f : x → x is a cyclic generalized (ψ �ϕ)−contraction, we have that for all n ∈{0, 1, ...} (2.11) ϕ (d (xn,xn+1)) = ϕ (d (fxn−1,fxn)) ≤ ψ (ϕ (m (xn−1,xn))) , where m (xn−1,xn) = max { d (xn−1,xn) ,d (xn−1,xn) ,d (xn,xn+1) , d (xn−1,xn+1) + d (xn,xn) 2 } (2.12) = max { d (xn−1,xn) ,d (xn,xn+1) , d (xn−1,xn+1) 2 } ≤ max{d (xn−1,xn) ,d (xn,xn+1)} . if d (xn,xn+1) > d (xn−1,xn) then from (2.11) follows (because ψ (t) < t,t > 0): (2.13) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn,xn+1))) < ϕ (d (xn,xn+1)) . 20 radenović a contradiction. therefore, for all n ∈{0, 1, ...} we obtain (because ψ is nondecreasing): (2.14) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn−1,xn))) . that is., we have that (2.15) ϕ (d (xn,xn+1)) ≤ ψ (ϕ (d (xn−1,xn))) ≤ ... ≤ ψn (ϕ (d (x0,x1))) . hence, ϕ (d (xn,xn+1)) → 0, i.e., d (xn,xn+1) → 0 as n →∞. next, we claim that {xn} is a cauchy sequence. if this is not case, then according to lemma 1.7. by putting in x = xm(k)−j(k),y = xn(k) in (2.9) we have: (2.16) ϕ ( d ( xm(k)−j(k)+1,xn(k)+1 )) ≤ ψ ( ϕ ( m ( xm(k)−j(k),xn(k) ))) , where (2.17) m ( xm(k)−j(k),xn(k) ) = max { d ( xm(k)−j(k),xn(k) ) ,d ( xm(k)−j(k),xm(k)−j(k)+1 ) ,d ( xn(k),xn(k)+1 ) , d ( xm(k)−j(k),xn(k)+1 ) + d ( xm(k)−j(k)+1,xn(k) ) 2 } . first of all, we have (2.18) lim k→∞ m ( xm(k)−j(k),xn(k) ) = max { ε, 0, 0, ε + ε 2 } = ε, that is., (2.19) lim k→∞ ϕ ( m ( xm(k)−j(k),xn(k) )) = ϕ ( lim k→∞ m ( xm(k)−j(k),xn(k) )) = ϕ (ε) . further from (2.17) as well as by the properties of the functions ψ and ϕ follows: (2.20) 0 < ϕ (ε) ≤ lim k→∞ ψ ( ϕ ( m ( xm(k)−j(k),xn(k) ))) < lim k→∞ m ( xm(k)−j(k),xn(k) ) = ϕ (ε) . a contradiction. hence {xn} is a cauchy sequence. the rest of the proof is further as in any of papers [16]-[18]. the proof for the case of cyclic generalized (ψ,ϕ)−contraction is very similar. � finaly, we announce the following important and significant remark regarding several proofs that picard sequence {xn} is a cauchy: remark 2.5. using our lemma 1.7. we can obtain much shorter proofs that picard sequence xn+1 = fxn, in each of the papers [2], [3], [6], [7], [9], [12], [11] and [13] is a cauchy. for this, it is sufficient putting x = xm(k)−j(k),y = xn(k) in the contractive condition of cyclic type theorem in each of the papers. references [1] s. banach, sur les operations dans les ensembles abstraits et leur applications aux equations integrales, fund. math. 3 (1922) 133-181. [2] m. a. alghamdi, a. petrusel and n. shahzad, a fixed point theorem for cyclic generalized contractions in metric spaces, fixed point theory appl., 2012 (2012), article id 122. [3] r. p. agarwal, m. a. alghamdi, d. o’regan and n. shahzad, fixed point theory for cyclic weak kannan type mappings, journal of indian math. soc. 81 (2014), 01-11. fixed point theory for cyclic weaker meir–keeler function 21 [4] chi-ming chen, fixed point theory for the cyclic weaker meir-keeler function in complete metric spaces, fixed point theory appl., 2012 (2012), article id 17. [5] m. s. jovanović, generalized contractive mappings on compact metric spaces, third mathematical conference of the republic of srpska, trebinje 7 and 8 june 2013. [6] e. karapinar, fixed point theory for cyclic weak φ−contraction, appl. math. lett., 24 (2011) 822-825. [7] e. karapinar, k. sadarangani, corrigendum to ”fixed point theory for cyclic weak φ−contraction” [appl. math. lett. 24 (6)(2011) 822-825], appl. math. lett., 25 (2012) 15821584. [8] s. karpagam, s. agarwal, best proximity point theorems for cyclic orbital meir-keeler contractions maps, nonlinear anal., 74 (2011) 1040-1046. [9] w. a. kirk, p. s. srinavasan, p. veeramani, fixed points for mapping satisfying cyclical contractive conditions, fixed point theory 4 (2003), 79-89. [10] l. milićević, contractive families on compact spaces, arxiv:1312.0587v1 [math.mg], 2, december 2013. [11] h. k. nashine, cyclic generalized ψ−weakly contractive mappings and fixed point results with applications to integral equations, nonlinear anal., 75 (2012) 6160-6169. [12] h. k. nashine, z. kadelburg, and p. kumam, implicit-relation-type cyclic contractive mappings and applications to integral equations, abstr. appl. anal., 2012 (2012), article id 386253, 15 pages. [13] m. pacurar, ioan a. rus, fixed point theory for cyclic ϕ−contractions, nonlinear anal., 72 (2010) 1181-1187. [14] m. a. petric, some results concerning cyclical contractive mappings, general math., 18 (2010), 213-226. [15] s. radenović, z. kadelburg, d. jandrlić and a. jandrlić, some results on weak contraction maps, bull. iranian math. soc. 38 (2012), 625-645. [16] s. radenović, some remarks on mappings satisfying cyclical contractive conditions, fixed point theory appl. submitted. [17] s. radenović, a note on fixed point theory for cyclic ϕ−contractions, demonstratio matematica, submitted. [18] s. radenović, some results on cyclic generalized weakly c-contractions on partial metric spaces, in bull. allahabad math. soc. submitted. [19] b. e. rhoades, a comparison of various definitions of contractive mappings, trans. amer. math. soc. 226 (1977), 257-290. faculty of mathematics and information technology, teacher education, dong thap university, cao lanch city, dong thap province, viet nam international journal of analysis and applications volume 18, number 5 (2020), 738-747 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-738 new approach of mhd boundary layer flow towards a porous stretching sheet via symmetry analysis and the generalized exp-function method a.a. gaber1,2∗, m.h. shehata1 1department of mathematics, college of science, and human studies at hotat sudair, majmaah university, saudi arabia 2department of mathematics, faculty of education, ain shams university, roxy, hiliopolis, cairo, egypt ∗corresponding author: aagaber6@gmail.com, a.gaber@mu.edu.sa abstract. due to importance of the slip effect on modeling the boundary layer flows, symmetries and exact solution investigations have been introduced in this paper for studying the effect of a slip boundary layer on the stretching sheet through a porous medium. the exact solution of the investigating model is obtained in term of exponential via the generalized exp-function method. this solution satisfies the boundary conditions. finally, the effect of parameters on the velocity field is studied. 1. introduction symmetry group analysis based on the transformation groups, now known as lie groups, is the most important solution method for the nonlinear problems in the literature. this approach is used to analysis the symmetries of the differential equations. then, the corresponding symmetry groups can be used to simplify the analysis of the problems governing by the differential equations in the engineering science, mathematical physics, and mechanics. lie groups characterize the symmetry of the differential equations and may be a point, a contact, and a potential or a nonlocal symmetry. it has also been verified that these kinds of groups can be represented by their infinitesimals that contain dependent variables, independent received march 8th, 2020; accepted may 12th, 2020; published june 24th, 2020. 1991 mathematics subject classification. 35q35, 34b15. key words and phrases. boundary layer; exact solutions; generalized exp-function method. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 738 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-738 int. j. anal. appl. 18 (5) (2020) 739 variables and the derivatives of dependent variables as arguments. in the last century, the application of the lie groups has been developed by a number of mathematicians. ovsiannikov [1], olver [2], ibragimov [3], and bluman and kumei [4] are some of the mathematicians who have huge number of studies in that field [5-9]. the boundary layer [10-13] equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions i.e. basically closed-form solutions. in the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation (ode). prandtl’s boundary layer equations admit more and different symmetry groups. symmetry groups or simply symmetries are invariant transformations which do not alter the structural form of the equation under investigation (bluman and kumei [1]). this work is organized as follows. the problem is formulated in section 2 and in section 3 we calculate the symmetries of the thermal boundary layer equations. all invariant solutions of the thermal boundary layer equations in section 4. finally, we show the effect of parameters on the velocity field. 2. formulation of the problem we consider the steady state 2d magnetohydrodynamic (mhn) boundary layer, incompressible and viscous flow on stretching sheet through a porous medium, where m is the magnetic parameter, kp is the permeability parameter and fw is the mass transfer parameter, which is positive for suction and negative for injection. ∂u ∂x + ∂υ ∂y = 0, u ∂u ∂x + υ ∂u ∂y = v ∂2u ∂y2 − v k0 u− α0b 2 0 ρ u. (1) in (1) u and υ are the components of velocity respectively in the x and y directions, k0 is the permeability of the porous medium, b0 is magnetic field of uniform strength and σ0 is electrical conductivity, v = µ ρ is the kinematic viscosity, µ is the coefficient of fluid viscosity and ρ is the fluid density. by using the boundary layer approximations and neglecting viscous dissipation. the appropriate boundary conditions for the problem are given by u = bx, υ = υw at y = 0, u −→ 0 at y = ∞. (2) where b is the stretching rate, υw is the wall velocity and the velocity components along x, y coordinates, respectively, are u = ∂ψ ∂y and υ = −∂ψ ∂x (3) int. j. anal. appl. 18 (5) (2020) 740 where ψ is the stream function. using the relations (3) in the boundary layer (2) and in the energy (1) we get the following equations ∂ψ ∂y ∂2ψ ∂y∂x − ∂ψ ∂x ∂2ψ ∂y2 = v ∂3ψ ∂y3 − v k ∂ψ ∂y − α0b 2 0 ρ ∂ψ ∂y , (4) the boundary conditions (2) then become ∂ψ ∂y = bx, ∂ψ ∂x = υw at y = 0, ∂ψ ∂y −→ 0 at y = ∞. (5) 3. symmetry analysis for the boundary layer equations firstly, we shall derive the similarity solutions using the lie-group method [11] under which (1) is invariant. consider the one-parameter (ε) lie group of infinitesimal transformations in (x, y, ψ) given by lie point symmetries x∗ = x∗(x,t,ψ; ε), y∗ = y∗(x,y,ψ; ε) ψ∗ = ψ∗(x,y,ψ; ε). (6) with associated infinitesimal form x∗ = x + εη(x,y,ψ; ε) + o(ε2), y∗ = y + εζ(x,y,ψ; ε) + o(ε2), ψ∗ = ψ + εψ(x,y,ψ; ε) + o(ε2), (7) where “ε” is a small parameter. if we set: ∆1 = ∂ψ ∂y ∂2ψ ∂y∂x − ∂ψ ∂x ∂2ψ ∂y2 −v ∂3ψ ∂y3 + ( v k + α0b 2 0 ρ ) ∂ψ ∂y , (8) the invariance conditions[1-4] γ(3)(∆α) = 0 whenever ∆α = 0, α = 1, 2, (9) where γ(3) is given by γ(3) = χ + gx ∂ ∂ψx + gxx ∂ ∂ψxx + gxt ∂ ∂ψxt + gxxx ∂ ∂ψxxx . (10) where χ = ζ ∂ ∂x + τ ∂ ∂y + g ∂ ∂ψ (11) int. j. anal. appl. 18 (5) (2020) 741 the components ζx, ζy, τx, τy, gx, gxx, gxxy....can be determined from the following expressions: gs = dsg −ψtdsζ −ψxdsτ, g sj = djg −ψtsdjζ −ψxsdjτ (12) equation (9) gives the following system of linear partial differential equations: ζy = 0, ζψ = 0, gy = 0, gx = 0, τy = 0, τψ = 0, gψψ = 0, ζx −gψ = 0, (13) solving the system (13), after substitution from (12) into(13), and using the invariance of the boundary conditions (6), yields ζ = λ1x + λ2 τ = λ3(x) g = λ1ψ + λ4 (14) in order to study the group theoretic structure, the vector field operator v is written as v = v1(λ1) + v2(λ2) + v3(λ3) + v4(λ4), (15) where v1 = x ∂ ∂x + ψ ∂ ∂ψ , v2 = ∂ ∂x , v3 = λ3(x) ∂ ∂y , v4 = ∂ ∂ψ . (16) it is easy to verify, that the vector fields are closed under the lie bracket as follows [v1,v1] = [v2,v2] = [v3,v3] = [v4,v4] = [v1,v3] = 0 [v2,v3] = [v2,v4] = 0, [v3,v1] = [v3,v2] = [v3,v4] = 0 [v4,v2] = [v4,v3] = 0, [v1,v2] = −[v2,v1] = −v2 [v1,v4] = −[v4,v1] = −v4 further, from the symmetries given in (16) the following possibilities exist for the solution of (9). (i)v1 (ii)v2 + v3 (iii)v2 + v3 + v4 int. j. anal. appl. 18 (5) (2020) 742 having determined the infinitessimals, the symmetry variables are found by solving the auxiliary equation dx ζ = dy τ = dψ g . (17) 4. reductions and exact solutions now we look the similarity solutions with respect to the generators v1 η∗ = y, ψ = xf(η∗), (18) the reduced system of odes is f 82 −ff ′′ −vf ′′′ + ( v k + α0b 2 0 ρ )f ′ = 0, (19) the boundary condition take the following forms f ′ = b, f = υw at η ∗ = 0 f ′ = 0 at η∗ →∞. (20) we look for a similarity solution of (19) ,and boundary condition (20) as the following form: f = √ bv f(η) and η = √ b v η∗ (21) using (21) we obtain the following self-similar equations f′′′ −f82 + ff′′ − (kp + m)f′ = 0, (22) subject to the boundary conditions f(0) = fw, f ′(0) = 1 f′(∞) = 0 (23) where m = α0b 2 0 ρb is the magnetic field, kp = v k0b is the permeability of the porous medium and fw = υw√ bv where fw > 0 corresponds to suction and fw < 0 for injection. equation (22) is nonlinear differential equation which can be solved by the generalized he’s exp-function method. in view of the generalized exp-function method [14-16], we assume that the solution of (22) can be expressed in the form f(η) = a−c[φ(η) −c] + ... + ap[φ(η) p] r−d[φ(η)−d] + ... + rq[φ(τ)q] , (24) where c, d, p and q are positive integers which are unknown to be further determined, an and rm are unknown constants. in addition, φ(η) satisfies riccati equation, φ′(η) = a + bφ(η) + cφ(η)2. (25) int. j. anal. appl. 18 (5) (2020) 743 in order to determine values of c and p, we balance the linear term of the highest order in eq. (24) with the highest order nonlinear term f′′′ and f82, we have f′′′(η) = a1φ −c−8d−3 + ... + a2φ p+8q+3 r1φ −9d + ... + r2φ 9q , (26) f82(η) = a3φ −2c−6d−2 + ... + a4φ 2p+6q+2 r3φ −9d + ... + r4φ 9q , (27) where ai and ri are determined coefficients only for simplicity. from balancing the lowest order and highest order of φ (26) and (27), we obtain −7d − c − 3 = −6d − 2c − 2, which leads to the limit c = d + 1,and 7q + p + 3 = 6q + 2p + 2,which leads to the limit p = q + 1, for simplicity d = q = 0, the function in eq. (24), becomes f(η) = γ−1φ −1 + γ0 + γ1φ (28) substituting (28) into (22), equating to zero the coefficients of all powers of φ(η) yields a set of algebraic equations for γ0, γ1 and γ−1, we obtain the following system γ21bc + 12γ1bc 2 + 2γ0γ1c 2 = 0, − 6γ−1a 3 + γ2−1a 2, γ21c 2 + 6γ1c 3 = 0, − 12γ−1a 2b + γ2−1ab + 2γ0γ−1a 2 = 0, − 6γ−1a 3 + γ2−1a 2 = 0, γ21bc + 12γ1bc 2 + 2γ0γ1c 2 = 0, − 12γ−1a 2b + γ2−1ab + 2γ0γ−1a 2 −mγ1b + 5γ−1γ1bc − (2(−γ−1c + γ1a))γ1b + γ1(8abc + b 3) + γ1(γ−1bc + γ1ab) + γ0γ1(2ac + b 2) = 0, − (2(−γ−1c + γ1a))γ1c −γ 2 1b 2 + 2γ−1γ1c 2 + 3γ0γ1bc + γ 2 1(2ac + b2) + γ1(8ac 2 + 7b2c) −mγ1c, 5γ1γ−1ab + 2γ−1b(−γ−1c + γ1a) + mγ−1b −γ−1(8abc + b 3) + γ0γ−1(2ac + b 2) + γ−1(γ−1bc + γ1ab) + 24γ−1abc + 6γ−1b(2ac + b 2) − 6γ−1(4abc + b(2ac + b 2)) = 0 (29) solving the system of algebraic equations with the aid of maple, we obtain the following results: γ−1 = ((kp + m) −b2 + γ0b) c , γ1 = 0 at a = 0. (30) γ−1 = −6c, γ1 = 6a at b = 0. (31) int. j. anal. appl. 18 (5) (2020) 744 substituting (30) into (28), the solutions of (1) can be written as: f(η) = γ0 − ((kp + m) −b2 + γ0b) c c exp(bη) − 1 b exp(bη) , (32) where γ = a−1 r−1 . now we have to apply the boundary conditions to the solution (19), noting that the third one is already satisfied. on using the first two boundary conditions we then need to solve the system: γ0 − ((kp + m) −b2 + γ0b) c c − 1 b = fw, −bγ + b2 −m + ((kp + m) −b2 + γ0b)(c − 1) c = 1. (33) by solving eq. (33) then substituting in eq. (32), we obtain the closed form solution f(η) = γ0 − ((kp + m) −b2 + γ0b) b + ((kp + m) −b2 + γ0b) cb exp(bη) , (34) where c 6= 0, γ0 = −c+1+fw( 12 fw+ 1 2 √ f2w+4+4(kp+m)) 1 2 fw+ 1 2 √ f2w+4+4(kp+m) and b = 1 2 fw + 1 2 √ f2w + 4 + 4(kp + m). 5. results and discussion figs. 1–3 have been made in order to see the effects of the permeability of the porous medium kp, suction/injection parameter fw and the mhd parameter m on the velocity field. fig. (1) from this figure, rise in m indicates the raise of magnetic field which acts like a resistive force and consequently fluid flow slowdowns relatively and hence boundary layer thickness increases. fig. (2) the effect of the influence of the porous medium on horizontal velocity. it is found that the horizontal velocity decreases with the increase of k i.e. increased permeability parameter (kp) caues an increase in resistance to fluid along the surface, and this leads to increase the thickness of the boundary layer. fig. (3) show the effects of suction (fw > 0) and injection (fw < 0) on the horizontal velocity f /(η) the effect of suction is to decrease the horizontal velocity whereas the effect of injection is to increase this. 6. conclusion in this paper, the couple system of mhd boundary layer flow towards a porous stretching sheet have been reduced by symmetry method to ordinary differential equations. the exact solutions of ordinary differential equations is obtained by the generalized exp-function method. finally, some plots have been given for study the effects of various parameters on velocity of fluid . 7. acknowledgements the authors would like to thank the deanship of scientific research of majmaah niversity for the financial grant received for conducting this research. int. j. anal. appl. 18 (5) (2020) 745 figure 1. velocity profile for different values of magnatic field m figure 2. velocity profile for different values of porous medium kp figure 3. velocity profile for several values of fw int. j. anal. appl. 18 (5) (2020) 746 appendix a: the exact solutions for riccati equation. cases a b c solutions of ricati equation 1 6=0 6= 0 6= 0 −b+ √ 4ac−b2 tan( 1 2 ( √ 4ac−b2(ζ+d0))) 2c 2 free 6= 0 0 −a b + exp(bζ) 3 0 6= 0 6= 0 −b exp(bζ+bd0) c exp(bζ+bd0)−1 4 0 -1 -1 −d0 exp(ζ)+d0 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] l.v. ovsiannikov, group analysis of differential equations. nauka, moscow, 1978. [2] p.j. olver, application of lie groups to differential equations. springer-verlag, 1986. [3] n.k. ibragimov, crc handbook of lie group analysis of differential equations, crc press, boca raton, 1994. [4] g.w. bluman, s. kumei, symmetries and differential equations, springer-verlag, new york, 1989. [5] m.f. el-sayed, g.m. moatimid, m.h.m. moussa, r.m. el-shiekh, f.a.h. el-shiekh, a.a. el-satar, a study of integrability and symmetry for the (p + 1)th boltzmann equation via painlevé analysis and lie-group method, math. meth. appl. sci. 38 (2015), 3670–3677. [6] m.h.m. moussa, a.a. gaber, symmetry analysis and solitary wave solutions of nonlinear ion-acoustic waves equation, int. j. anal. appl. 18 (3) (2020), 448-460. [7] m. rosa, j.c. camacho, m.s. bruzón, m.l. gandarias, classical and potential symmetries for a generalized fisher equation, j. comput. appl. math. 318 (2017), 181–188. [8] j.c. camacho, m. rosa, m.l. gandarias, m.s. bruzón, classical symmetries, travelling wave solutions and conservation laws of a generalized fornberg–whitham equation, j. comput. appl. math. 318 (2017), 149–155. [9] r. sinuvasan, k.m. tamizhmani, p.g.l. leach, symmetries, travelling-wave and self-similar solutions of the burgers hierarchy, appl. math. comput. 303 (2017), 165–170. [10] m. f. el-sayed, g. m. moatimid, m. h. m. moussa, r. m. el-shiekh and a. a. el-satar, similarity reductions and new exact solutions for b-family equations, amer. j. math. stat. 2 (3) (2012), 40-43 [11] b.c. sakiadis, boundary-layer behavior on continuous solid surfaces: i. boundary-layer equations for two-dimensional and axisymmetric flow, aiche j. 7 (1961), 26–28. [12] m. hatami, d.d. ganji, heat transfer and nanofluid flow in suction and blowing process between parallel disks in presence of variable magnetic field, j. mol. liquids. 190 (2014), 159–168. [13] k. bhattacharyya, s. mukhopadhyay, g.c. layek, i. pop, effects of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet, int. j. heat mass transfer. 55 (2012), 2945–2952. [14] a. ebaid and abdul-majid wazwaz, on the generalized exp-function method and its application to boundary layer flow at nano-scale j. comput. theor. nanoscience, (2014), 11, 1–7 [15] a.e.-h. ebaid, generalization of he’s exp-function method and new exact solutions for burgers equation, z. naturforsch., a. 64 (2009), 604–608. int. j. anal. appl. 18 (5) (2020) 747 [16] e.m.e. zayed, a.-g. al-nowehy, exact solutions and optical soliton solutions for the (2 + 1)-dimensional hyperbolic nonlinear schrödinger equation, optik. 127 (2016), 4970–4983. 1. introduction 2. formulation of the problem 3. symmetry analysis for the boundary layer equations 4. reductions and exact solutions 5. results and discussion 6. conclusion 7. acknowledgements references international journal of analysis and applications volume 18, number 3 (2020), 366-380 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-366 reproducing formulas for the fourier-like multipliers operators in q-rubin setting ahmed saoudi1,2,∗ 1northern border university, college of science, arar, p.o. box 1631, saudi arabia 2 université de tunis el manar, faculté des sciences de tunis, tunisie ∗corresponding author: ahmed.saoudi@ipeim.rnu.tn abstract. the aim of this work is to study of the q2-fourier multiplier operators on rq and we give for them calderón’s reproducing formulas and best approximation on the q2-analogue sobolev type space hq using the theory of q2-fourier transform and reproducing kernels. 1. introduction the q2-analogue differential-difference operator ∂q, also called q-rubin’s operator defined on rq in [11, 12] by ∂qf(z) =   f(q−1z) + f(−q−1z) −f(qz) + f(−qz) − 2f(−z) 2(1 −q)z if z 6= 0 lim z→0 ∂qf(z) in rq if z = 0. this operator has correct eigenvalue relationships for analogue exponential fourier analysis using the functions and orthogonalities of [9]. the q2-analogue fourier transform we employ to make our constructions and results in this paper is based on analogue trigonometric functions and orthogonality results from [9] which have important applications to received january 13th, 2020; accepted january 30th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 46e35; 43a32. key words and phrases. q-fourier analysis; q-rubin’s operator; l2-multiplier operators; calderón’s reproducing formulas; extremal functions. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 366 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-366 int. j. anal. appl. 18 (3) (2020) 367 q-deformed quantum mechanics. this transform generalizing the usual fourier transform, is given by fq(f)(x) := k ∫ +∞ −∞ f(t)e(−itx; q2)dqt, x ∈ r̃q. in this paper we study the fourier multiplier operators tm defined for f ∈ l2q by tmf(x) := f−1q (mafq(f)) (x), x ∈ rq, where the function ma is given by ma(x) = m(ax). these operators are a generalization of the multiplier operators tm associated with a bounded function m and given by tm(ϕ) = f−1(mf(ϕ)), where f(ϕ) denotes the ordinary fourier transform on rn. these operators made the interest of several mathematicians and they were generalized in many settings, (see for instance [1, 2, 14, 18]). this paper is organized as follows. in section 2, we recall some basic harmonic analysis results related with the q-rubin’s operator ∂q and we introduce preliminary facts that will be used later. in section 3, we study the q2-fourier l2-multiplier operators tq and we give for them a plancherel formula and pointwise reproducing formulas. afterward, we give calderón’s reproducing formulas by using the theory of q2-analogue fourier transform. the last section of this paper is devoted to giving best approximation for the operators tq and good estimates of the associated extremal function on the q2-analogue sobolev type space hq studied in [15–17]. 2. notations and preliminaries throughout this paper, we assume 0 < q < 1 and we refer the reader to [5, 7] for the definitions and properties of hypergeometric functions. in this section we will fix some notations and recall some preliminary results. we put rq = {±qn : n ∈ z} and r̃q = rq ∪{0}. for a ∈ c, the q-shifted factorials are defined by (a; q)0 = 1; (a; q)n = n−1∏ k=0 (1 −aqk),n = 1, 2, ...; (a; q)∞ = ∞∏ k=0 (1 −aqk). we denote also [a]q = 1 −qa 1 −q , a ∈ c and [n]q! = (q; q)n (1 −q)n , n ∈ n. a q-analogue of the classical exponential function is given by (see [11, 12]) e(z; q2) = cos(−iz; q2) + i sin(−iz; q2), (2.1) where cos(z; q2) = +∞∑ n=0 qn(n+1) (−1)nz2n [2n]q! , sin(z; q2) = +∞∑ n=0 qn(n+1) (−1)nz2n+1 [2n + 1]q! , (2.2) int. j. anal. appl. 18 (3) (2020) 368 satisfying the following inequality for all x ∈ rq |cos(x; q2)| ≤ 1 (q; q)∞ , sin(x; q2)| ≤ 1 (q; q)∞ and |e(ix; q2)| ≤ 2 (q; q)∞ . (2.3) the q-differential-difference operators is defined as (see [11, 12]) ∂qf(z) =   f(q−1z) + f(−q−1z) −f(qz) + f(−qz) − 2f(−z) 2(1 −q)z if z 6= 0 lim z→0 ∂qf(z) in rq if z = 0 and we denote a repeated application by ∂0qf = f, ∂ n+1 q f = ∂q(∂ n q f). the q-jackson integrals are defined by (see [6]) ∫ a 0 f(x)dqx = (1 −q)a +∞∑ n=0 qnf(aqn), ∫ b a f(x)dqx = (1 −q) +∞∑ n=0 qn(bf(bqn) −af(aqn)) and ∫ +∞ −∞ f(x)dqx = (1 −q) +∞∑ n=−∞ qn{f(qn) + f(−qn)} , provided the sums converge absolutely. in the following we denote by • cq,0 the space of bounded functions on rq, continued at 0 and vanishing a ∞. • cpq the space of functions p-times q-differentiable on rq such that for all 0 ≤ n ≤ p. ∂pqf is continuous on rq, • dq the space of functions infinitely q-differentiable on rq with compact supports. • sq stands for the q-analogue schwartz space of smooth functions over rq whose q-derivatives of all order decay at infinity. sq is endowed with the topology generated by the following family of semi-norms: ‖u‖m,sq (f) := sup x∈r;k≤m (1 + |x|)m|∂kq u(x)| for all u ∈sq and m ∈ n. • s′q the space of tempered distributions on rq, it is the topological dual of sq. • lpq = { f : ‖f‖q,p = (∫ +∞ −∞ |f(x)| pdqx )1 p < ∞ } . • l∞q = { f : ‖f‖q,∞ = supx∈rq |f(x)| < ∞ } . int. j. anal. appl. 18 (3) (2020) 369 the q2-fourier transform was defined by r. l. rubin defined in [11], as follow fq(f)(x) = k ∫ +∞ −∞ f(t)e(−itx; q2)dqt, x ∈ r̃q where k = (q; q2)∞ 2(q2; q2)∞(1 −q)2 . to get convergence of our analogue functions to their classical counterparts as q ↑ 1 as in [9,12], we impose the condition that 1 − q = q2m for some integer m. therefore, in the remainder of this paper, letting q ↑ 1 subject to the condition log(1 −q) log(q) ∈ 2z. it was shown in ( [4, 11]) that the q2-fourier transform fq verifies the following properties: (a) if f, uf(u) ∈ l1q, then ∂q(fq)(f)(x) = fq(−iuf(u))(x). (b) if f, ∂qf ∈ l1q, then fq(∂q(f))(x) = ixfq(f)(x). (2.4) (c) if f ∈ l1q, then fq(f) ∈cq,0 and we have ‖fq(f)‖q,∞ ≤ 2k (q; q)∞ ‖f‖q,1. (2.5) (d) if f ∈ l1q, then, we have the reciprocity formula ∀t ∈ rq, f(t) = k ∫ +∞ −∞ fq(f)(x)e(itx; q2)dqx. (2.6) (e) the q2-fourier transform fq is an isomorphism from sq onto itself and we have, for all f ∈sq f−1q (f)(x) = fq(f)(−x) = fq(f)(x). (2.7) (f) fq is an isomorphism from l2q onto itself, and we have ‖fq(f)‖2,q = ‖f‖q,2, ∀f ∈ l2q (2.8) and ∀t ∈ rq, f(t) = k ∫ +∞ −∞ fq(f)(x)e(itx; q2)dqx. the q-translation operator τq;x,x ∈ rq is defined on l1q by (see [11]) τq,y(f)(x) = k ∫ +∞ −∞ fq(f)(t)e(itx; q2)e(ity; q2)dqt, y ∈ rq, τq,0(f)(x) = (f)(x). it was shown in [11] that the q-translation operator can be also defined on l2q. furthermore, it verifies the following properties int. j. anal. appl. 18 (3) (2020) 370 (a) for f,g ∈ l1q, we have τq,yf(x) = τq,xf(y), ∀x,y ∈ rq and ∫ +∞ −∞ τq,y(f)(−x)g(x)dqx = ∫ +∞ −∞ f(x)τq,y(g)(−x)dqx, ∀y ∈ r̃q. (b) for all f ∈ l1q and all y ∈ rq, we have(see [3])∫ +∞ −∞ τq,y(f)(x)dqx = ∫ +∞ −∞ f(x)dqx. (2.9) (c) for all y ∈ rq and for all f ∈ lpq ,1 ≤ p ≤∞, we have τq,y(f) ∈ lpq (see [3]) and ‖τq,yf‖q,p ≤ m‖f‖q,p, (2.10) where m = 4(−q,q)∞ (1 −q)2q(q,q)∞ + 2c, with c = k2‖e(·,q2)‖∞,q‖e(·,q2)‖1,q. (2.11) (d) τq;yf is an isomorphism for f ∈ l2q onto itself and we have ‖τq,yf‖q,2 ≤ 2 (q,q)∞ ‖f‖q,2, ∀y ∈ r̃q. (2.12) (e) let f ∈ l2q, then fq(τq,yf)(λ) = e(iλy; q2)fq(f)(λ), ∀y ∈ r̃q. (2.13) the q-convolution product is defined by using the q-translation operator, as follow for f ∈ l2q and g ∈ l1q, the q-convolution product is given by f ∗g(y) = k ∫ +∞ −∞ τq,yf(x)g(x)dqx. the q-convolution product satisfying the following properties: (a) f ∗g = g ∗f. (b) ∀f,g ∈ l1q ∩l2q, fq(f ∗q g) = fq(f)fq(g). (c) ∀f,g ∈sq, f ∗q g ∈sq. (d) f ∗g ∈ l2q if and only if fq(f)fq(g) ∈ l2q and we have fq(f ∗g) = fq(f)fq(g). (e) let f,g ∈ l2q. then we have ‖f ∗g‖2q,2 = k‖fq(f)fq(g)‖ 2 q,2, (2.14) and f ∗g = f−1q (fq(f)fq(g)) . (2.15) int. j. anal. appl. 18 (3) (2020) 371 (f) if f,g ∈ l1q then f ∗g ∈ l1q and ‖f ∗g‖q,1 = km‖f‖q,1‖g‖q,1. (2.16) 3. l2-multiplier operators for the q-rubin-fourier transform in this section we study the q2-fourier-multiplier operators and we establish theirs calderón’s reproducing formulas in l2-case. definition 3.1. let a ∈ r+q , m ∈ l2q and f a smooth function on rq. we define the q2-fourier l2-multiplier operators tm for a regular function f on rq as follow tmf(x) = f−1q (mafq(f)) (x), x ∈ rq, (3.1) where the function ma is given by ma(x) = m(ax). remark 3.1. let a ∈ r+q , m ∈ l2q and f, we can write the operator tm as tmf(x) = f−1q (ma) ∗f(x), x ∈ rq, (3.2) where f−1q (ma)(x) = 1 a f−1q (m)( x a ). proposition 3.1. (i) if m ∈ l2q and f ∈ l1q, then tmf ∈ l2q, and we have ‖tmf‖q,2 ≤ 2k √ a(q,q)∞ ‖m‖q,2‖f‖q,1. (ii) if m ∈ l∞q and f ∈ l2q, then tmf ∈ l2q, and we have ‖tmf‖q,2 ≤‖m‖∞,q‖f‖q,2. (iii) if m ∈ l2q and f ∈ l2q, then tmf ∈ l∞q , and we have tmf(x) = k ∫ ∞ −∞ m(aξ)fq(f)(ξ)e(iξx; q2)dqξ, x ∈ rq and ‖tmf‖q,∞ ≤ 2k √ a(q,q)∞ ‖m‖q,2‖f‖q,2. int. j. anal. appl. 18 (3) (2020) 372 proof. i) let m ∈ l2q, and f ∈ l1. from the definition of the q2-fourier l2-multiplier operators (3.1) and relations (2.5) and (2.8) we get that the function tmf belongs to l2q, and we have ‖tmf‖q,2 = ‖mafq(f)‖q,2 ≤ 1 √ a ‖m‖q,2‖fq(f)‖q,∞ ≤ 2k √ a(q,q)∞ ‖m‖q,2‖f‖q,1. ii) the result follows from the plancherel theorem for the rubin operator. iii) let m ∈ l2q, and f ∈ l2q, then from inversion formula we get tmf ∈ l∞q , and by relation (2.5) we obtain ‖tmf‖q,∞ ≤ 2k (q,q)∞ ‖mafq(f)‖q,1 then, using hölder’s inequality, we get ‖tmf‖q,∞ ≤ 2k √ a(q,q)∞ ‖m‖q,2‖f‖q,2. � in the following, we give plancherel and pointwise reproducing inversion formulas for the q2-fouriermultiplier operators tm. theorem 3.1. let m be a function in l2q satisfying the admissibility condition:∫ ∞ 0 |ma(x)|2 dqa a = 1, x ∈ rq. (3.3) i)plancherel formula: for all f in l2q, we have∫ ∞ 0 ‖tmf‖2q,2 dqa a = k ∫ ∞ −∞ |f(x)|2dq(x). ii) first calderón’s formula: let f be a function in l1q such that fqf in l1q then we have f(x) = ∫ ∞ 0 ( tmf ∗f−1q (ma) ) (x) dqa a , x ∈ rq. proof. i) according to identity (2.14) and relation (3.2) we have∫ ∞ 0 ‖tmf‖2q,2 dqa a = ∫ ∞ 0 ‖f−1q (ma) ∗f‖ 2 q,2 dqa a = k ∫ ∞ 0 ‖mafq(f)‖2q,2 dqa a = k ∫ ∞ −∞ |fq(x)| 2 (∫ ∞ 0 |ma| 2 dqa a ) dqx. int. j. anal. appl. 18 (3) (2020) 373 the result follows from plancherel theorem (2.8) and the assumption (3.3). ii) let f be a function in l1q, then∫ ∞ 0 ( tmf ∗f−1q (ma) ) (x) dqa a = ∫ ∞ 0 ( k ∫ ∞ −∞ tmf(y)τq,x ( f−1q (ma) ) (y)dqy ) dqa a . from proposition 3.1 i), relation (2.12) and plancherel theorem, it is obvious that tmf,τq,x ( f−1q (ma) ) ∈ l2q. after that, according to relation (2.13), identity (3.1) and plancherel theorem of the q2-fourier transform, we obtain∫ ∞ 0 ( tmf ∗f−1q (ma) ) (x) dqa a = k ∫ ∞ 0 (∫ ∞ −∞ e(ixy; q2)fq(f)(y)|ma(y)|2dqy ) dqa a . since ∫ ∞ 0 (∫ ∞ −∞ |e(ixy; q2)fq(f)(y)||ma(y)|2dqy ) dqa a ≤‖fq(f)‖q,1 ≤∞, then, by fubini’s theorem, we have∫ ∞ 0 ( tmf ∗f−1q (ma) ) (x) dqa a = k ∫ ∞ −∞ e(ixy; q2)fq(y) (∫ ∞ 0 |ma(y)|2 dqa a ) dqy = k ∫ ∞ −∞ e(ixy; q2)fq(y)dqy = f(x). � we need the following technical lemma to establish the calderón’s reproducing formulas for the q2-fourier l2-multiplier operators. lemma 3.1. let m be a function in l2q ∩l∞q satisfy the admissibility condition (3.3). then the function φγ,δ(x) = ∫ δ γ |m(ax)|2 dqa a belongs to l2q for all 0 < γ < δ < ∞ and we have φγ,δ(x) ∈ l2q ∩l ∞ q . proof. using hölder’s inequality for the measure dqa a , we get |φγ,δ(x)|2 ≤ ln (δ/γ) ∫ δ γ |m(ax)|4 dqa a , x ∈ rq. therefore, ‖φγ,δ‖2q,2 ≤ ln (δ/γ) ∫ δ γ (∫ ∞ −∞ |m(ax)|4dqx ) dqa a ≤ ln (δ/γ) ∫ δ γ (∫ ∞ −∞ |m(x)|4dqx ) da a2 ≤ ( 1 γ − 1 δ ) ln (δ/γ)‖m‖2q,2‖m‖ 2 q,∞ < ∞. int. j. anal. appl. 18 (3) (2020) 374 on the other hand, from the admissibility condition (3.3), we get ‖φγ,δ‖q,∞ ≤ 1, which completes the proof. � theorem 3.2. (second calderón’s formula) let f ∈ l2q, m ∈ l2q ∩ l∞q satisfy the admissibility condition (3.3) and 0 < γ < δ < ∞. then the function fγ,δ(x) = ∫ δ γ ( tmf ∗f−1q (ma) ) (x) dqa a , x ∈ rq belongs to l2q and satisfies lim (γ,δ)→(0,∞) ‖fγ,δ −f‖q,2 = 0. (3.4) proof. let f be a function in l2q, and m ∈ l2q ∩l∞q , then∫ ∞ 0 ( tmf ∗f−1q (ma) ) (x) dqa a = ∫ ∞ 0 ( k ∫ ∞ −∞ tmf(y)τq,x ( f−1q (ma) ) (y)dqy ) dqa a . according to proposition 3.1, relation (2.12) and plancherel theorem, it is obvious that tmf,τq,x ( f−1q (ma) ) ∈ l2q. then, from relation (2.13) and the identity (3.1), we obtain fγ,δ(x) = k ∫ δ γ (∫ ∞ −∞ e(ixy,q2)fq(f)(y)|ma(y)|2dqy ) dqa a . by fubini-tonnelli’s theorem, hölder’s inequality and lemma 3.1, we get∫ δ γ (∫ ∞ −∞ |e(ixy,q2)fq(f)(y)||ma(y)|2dqy ) dqa a ≤ 2 (q,q)∞ ∫ ∞ −∞ |fq(f)(y)|φγ,δ(y)dqy ≤ 2 (q,q)∞ ‖f‖q,2‖φγ,δ‖q,2 < ∞. then, according to fubini’s theorem and the inversion formula, we have fγ,δ(x) = k ∫ ∞ −∞ e(ixy,q2)fq(f)(y) (∫ δ γ |ma(y)|2 dqa a ) dqy = k ∫ ∞ −∞ e(ixy,q2)fq(f)(y)φγ,δ(y)dqy = f−1q [fq(f)φγ,δ] (x). on the other hand, the function φγ,δ belongs to l ∞ q which allows to see that fγ,δ belongs to l 2 q and using the identity (2.15), we obtain fq(fγ,δ) = fq(f)φγ,δ. by the plancherel formula we get ‖fγ,δ −f‖2q,2 = ∫ ∞ −∞ |fq(f)(y)|2(1 − φγ,δ(y))2dqy. int. j. anal. appl. 18 (3) (2020) 375 the the admissibility condition (3.3) leads to lim (γ,δ)→(0,∞) φγ,δ(y) = 1, y ∈ rq and |fq(f)(y)|2(1 − φγ,δ(y))2 ≤ |fq(f)(y)|2. finally, the relation (3.4) follows from the dominated convergence theorem. � 4. the extremal function associated with q2-fourier l2-multiplier operators in this section, we study the extremal function associated to the q2-fourier l2-multiplier operators. let s ∈ r and 1 ≤ p < ∞, the q2-analogue sobolev type spaces is defined in [15] by ws,pq = { u ∈s′q : (1 + |ξ| 2) s 2fq(u) ∈ lpq } . in the particular case p = 2, we denote ws,pq by hsq which provided with the inner product 〈u,v〉hsq = ∫ +∞ −∞ (1 + ξ2)sfq(u)(ξ)fq(v)(ξ)dqξ and the norm ‖u‖hsq := √ 〈u,u〉hsq. hsq is a hilbert space satisfying the following properties (a) h0q = l2q. (b) for all s > 0 the space hsq is continuously contained in l2q and we have ‖f‖q,2 ≤‖f‖hsq. (4.1) proposition 4.1. let m be a function in l∞q . then the q 2-fourier l2-multiplier operators tm are bounded and linear from hsq into l2q and we have for all f ∈hsq ‖tmf‖q,2 ≤‖m‖q,∞‖f‖hsq. proof. let f ∈hsq. according to proposition 3.1 (ii), the operator tm belongs to l2q and we have ‖tmf‖q,2 ≤‖m‖q,∞‖f‖q,2. on the other hand, by the inequality (4.1) we have ‖f‖q,2 ≤‖f‖hsq , which gives the result. � int. j. anal. appl. 18 (3) (2020) 376 definition 4.1. let η > 0 and let m be a function in l∞q . we denote by 〈u,v〉hsq,η the inner product defined on the space hsq by 〈f,g〉hsq,η = η〈f,g〉hsq + 〈tmf,tmg〉q,2 (4.2) and the norm ‖f‖hsq,η = √ 〈f,f〉hsq,η. it is easy to show the following results. proposition 4.2. let m be a function in l∞q and f in hsq (i) the norm ‖ ·‖hsq,η satisfies: ‖f‖2hsq,η = η‖f‖ 2 hsq + ‖tmf‖2q,2. (ii) the norms ‖ ·‖hsq,η and ‖ ·‖hsq are equivalent and we have √ η ‖f‖hsq ≤‖f‖hsq,η ≤ √ η + ‖m‖2q,∞ ‖f‖hsq. theorem 4.1. let s > 1 2 and m be a function in l∞q . then the hilbert space (hsq,〈·, ·〉hsq,η) has the following reproducing kernel ψs,η(x,y) = ∫ ∞ −∞ e(ixξ,q2)e(−iyξ,q2) η(1 + |ξ|2)s + |ma(ξ)|2 dq(ξ), (4.3) such that (i) for all y ∈ rq, the function x 7→ ψs,η(x,y) belongs to hsq. (ii) for all f ∈hsq and y ∈ rq, we have the reproducing property 〈f, ψs,η(·,y)〉hsq,η = f(y). (iii) the hilbert space (hsq,〈·, ·〉hsq ) has the following reproducing kernel ψs(x,y) = ∫ ∞ −∞ e(ixξ,q2)e(−iyξ,q2) (1 + |ξ|2)s dq(ξ). (4.4) proof. (i) let y ∈ rq and s > 12 . from the relation (2.3), we show that the function ϕy : ξ −→ e(−iyξ,q2) η(1 + |ξ|2)s + |ma(ξ)|2 belongs to l1q ∩l2q. hence the function ψs,η is well defined and by the inversion formula, we obtain ψs,η(x,y) = f−1q (ϕy)(x), x ∈ rq. on the other hand, using plancherel theorem, we get that ψs,η(·,y) belongs to l2q and we have fq (ψs,η(·,y)) (ξ) = e(−iyξ,q2) η(1 + |ξ|2)s + |ma(ξ)|2 , ξ ∈ rq. (4.5) therefore, by the identity (2) we obtain int. j. anal. appl. 18 (3) (2020) 377 |fq (ψs,η(·,y)) (ξ)| ≤ (q,q)−1∞ 2η(1 + |ξ|2)s , and ‖ψs,η(·,y)‖2hsq ≤ (2η(q,q)∞) −2‖(1 + | · |2)−s‖q,1 < ∞. this proves that for every y ∈ rq, the function ψs,η(·,y) belongs to hsq. (ii) let f ∈ hsq and y ∈ rq. according to the definition of inner product (4.2) and identity (4.5), we obtain 〈f, ψs,η(·,y)〉hsq,η = ∫ ∞ −∞ e(ixξ,q2)fq(ξ)dq(ξ). on the other hand, the function ξ 7−→ (1 + |ξ|2)−s/2 belongs to l2q for all s > 1/2. therefore, the function fq(f) belongs to l1q and we have 〈f, ψs,η(·,y)〉hsq,η = f(y). (iii) the result is obtained by taking m a null function and η = 1. � the main result of this section can be stated as follows. theorem 4.2. let s > 1 2 and m be a function in l∞q and a > 0. for any h ∈ l2q and for any η > 0, there exists a unique function f∗η,h,a, where the infimum inf f∈hsq { η‖f‖2hsq + ‖h−tmf‖ 2 q,2 } (4.6) is attained. moreover the extremal function f∗η,h,a is given by f∗η,h,a(y) = ∫ ∞ ∞ h(x)θs,η(x,y)dqx, (4.7) where θs,η(x,y) = ∫ ∞ −∞ ma(ξ)e(ixξ,q 2) η(1 + |ξ|2)s + |ma(ξ)|2 e(−iyξ,q2)dqξ. proof. the existence and unicity of the extremal function f∗η,h,a satisfying (4.6) is given by [8, 10, 13]. on the other hand from theorem 4.1 we have f∗η,h,a(y) = 〈h,tm(ψs,η(·,y))〉q,2. from proposition 3.1 and identity (4.5) we obtain θs,η(x,y) = tm(ψs,η(·,y))(x) = ∫ ∞ −∞ ma(ξ)e(ixξ,q 2) η(1 + |ξ|2)s + |ma(ξ)|2 e(−iyξ,q2)dqξ. � int. j. anal. appl. 18 (3) (2020) 378 theorem 4.3. let s > 1 2 and m be a function in l∞q and h ∈ l2q. then the extremal function f∗η,h,a satisfies the following properties: fq(f∗η,h,a)(ξ) = ma(ξ) η(1 + |ξ|2)s + |ma(ξ)|2 fq(h)(ξ), ξ ∈ rq and ‖f∗η,h,a‖ 2 hsq ≤ 1 4η ‖h‖2q,2. proof. let y ∈ rq, then the function gy : ξ 7−→ ma(ξ)e(−iyξ,q2) η(1 + |ξ|2)s + |ma(ξ)|2 belongs to l1q ∩l2q and by the inversion formula we obtain θs,η(x,y) = f−1q (gy)(x), x ∈ rq. hence, by plancherel formula, we have θs,η(·,y) belongs to l2q and f∗η,h,a(y) = ∫ ∞ −∞ fq(h)(ξ)gy(ξ)dqξ = ∫ ∞ −∞ ma(ξ)fq(h)(ξ) η(1 + |ξ|2)s + |ma(ξ)|2 e(iyξ,q2)dqξ. on the other hand, the function f : ξ 7−→ ma(ξ)fq(h)(ξ) η(1 + |ξ|2)s + |ma(ξ)|2 belongs to l1q ∩l2q and by the inversion formula we obtain f∗η,h,a(y) = f −1 q (f)(y). afterwards, by plancherel formula, it follows that f∗η,h,a belongs to l 2 q and we have fq(f∗η,h,a)(ξ) = ma(ξ)fq(h)(ξ) η(1 + |ξ|2)s + |ma(ξ)|2 , ξ ∈ rq. hence (1 + |ξ|2)s ∣∣fq(f∗η,h,a)(ξ)∣∣2 = (1 + |ξ|2)s ∣∣∣∣∣ ma(ξ)fq(h)(ξ)η(1 + |ξ|2)s + |ma(ξ)|2 ∣∣∣∣∣ 2 ≤ (1 + |ξ|2)s ∣∣∣ma(ξ)fq(h)(ξ)∣∣∣2 4η(1 + |ξ|2)s|ma(ξ)|2 ≤ 1 4η |fq(h)(ξ)| 2 . finally, using plancherel theorem, we obtain ‖f∗η,h,a‖ 2 hsq ≤ 1 4η ‖h‖2q,2. int. j. anal. appl. 18 (3) (2020) 379 � theorem 4.4. (third calderón’s formula). let s > 1 2 , m be a function in l∞q and f ∈ hsq. the extremal function given by f∗η,a(y) = ∫ ∞ −∞ tmf(x)θs,η(x,y)dqx (4.8) satisfies lim η→0+ ‖f∗η,a −f‖hsq = 0. moreover, {f∗η,a}η>0 converges uniformly to f when η converge to 0+. proof. let f ∈ hsq, h = tmf and f∗η,a = f∗η,h,a. according to proposition 4.1 the function h belongs to l 2 q. from the definition of the q2-fourier-multiplier operators tm and theorem 4.3, we obtain fq(f∗η,a)(ξ) = |ma(ξ)|2 η(1 + |ξ|2)s + |ma(ξ)|2 fq(f)(ξ), ξ ∈ rq. hence, it follows that fq(f∗η,a −f)(ξ) = −η(1 + |ξ|2)s η(1 + |ξ|2)s + |ma(ξ)|2 fq(f)(ξ), ξ ∈ rq. (4.9) therefore, ‖f∗η,a −f‖ 2 hsq = ∫ ∞ −∞ η2(1 + |ξ|2)3s(ξ)|fq(f)(ξ)|2 (η(1 + |ξ|2)s + |ma(ξ)|2) 2 dqx. then, from the dominated convergence theorem and the following inequality η2(1 + |ξ|2)3s|fq(f)(ξ)|2 (η(1 + |ξ|2)s + |ma(ξ)|2) 2 ≤ (1 + |ξ|2)s|fq(f)(ξ)|2, we deduce that lim η→0+ ‖f∗η,a −f‖hsq = 0. on the other hand, the function ξ 7−→ (1 + |ξ|2)−s/2 belongs to l2q for all s > 1/2. therefore, the function fq(f) belongs to l1q∩l2q for all f ∈hsq. then, according to (4.9) and the inversion formula for the q2-fourier transform, we get f∗η,a(y) −f(y) = k ∫ ∞ −∞ −η(1 + |ξ|2)sfq(f)(ξ) η(1 + |ξ|2)s + |ma(ξ)|2 e(iyξ,q2)dqx. by using the dominated convergence theorem and the fact η(1 + |ξ|2)s|fq(f)(ξ)|2 η(1 + |ξ|2)s + |ma(ξ)|2 ≤ |fq(f)(ξ)|, we deduce that lim η→0+ sup y∈rq ‖f∗η,a(y) −f(y)‖ = 0. which completes the proof of the theorem. � int. j. anal. appl. 18 (3) (2020) 380 acknowledgement 4.1. the author gratefully acknowledge the approval and the support of this research study by the grant no. 7909-sci-2018-3-9-f from the deanship of scientific research at northern border university, arar, k.s.a. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.-p. anker, lp fourier multipliers on riemannian symmetric spaces of the noncompact type, ann. math. 132 (1990), 597-628. [2] j. j. betancor, ó. ciaurri, and j. l. varona, the multiplier of the interval [−1, 1] for the dunkl transform on the real line, j. funct. ana. 242 (1) (2007), 327-336. [3] n. bettaibi, k. mezlini, and m. el guénichi, on rubin’s harmonic analysis and its related positive definite functions, acta math. sci. 32 (5) (2012), 1851-1874. [4] a. fitouhi and r. h. bettaieb, wavelet transforms in the q2-analogue fourier analysis, math. sci. res. j. 12(9) (2008), 202-214. [5] g. gasper and m. rahman, basic hypergeometric series, cambridge university press, 2004. [6] f. jackson, on a q-definite integrals, quart. j. pure appl. math. 41 (1910), 193-203. [7] v. kac and p. cheung, quantum calculus, springer science & business media, 2001. [8] g. kimeldorf and g. wahba, some results on tchebycheffian spline functions, j. math. anal. appl. 33(1) (1971), 82-95. [9] t. h. koornwinder and r. f. swarttouw, on q-analogues of the fourier and hankel transforms, trans. amer. math. soc. 333 (1) (1992), 445-461. [10] t. matsuura, s. saitoh, and d. trong, approximate and analytical inversion formulas in heat conduction on multidimensional spaces, j. inverse ill-posed probl. 13 (5) (2005), 479-493. [11] r. rubin, duhamel solutions of non-homogeneous q2-analogue wave equations, proc. amer. math. soc. 135 (3) (2007), 777-785. [12] r. l. rubin, a q2-analogue operator for q2-analogue fourier analysis, j. math. anal. appl. 212(2) (1997), 571-582. [13] s. saitoh, approximate real inversion formulas of the gaussian convolution, appl. anal. 83 (7) (2004), 727-733. [14] a. saoudi, calderón’s reproducing formulas for the weinstein l2-multiplier operators, asian-eur. j. math. https://doi. org/10.1142/s1793557121500030 (2019). [15] a. saoudi and a. fitouhi, on q2-analogue sobolev type spaces, le mat. 70 (2) (2015), 63-77. [16] a. saoudi and a. fitouhi, three applications in q2-analogue sobolev spaces, appl. math. e-notes. 17 (2017), 1-9. [17] a. saoudi and a. fitouhi, littlewood-paley decomposition in quantum calculus, appl. anal. https://doi.org/10.1080/ 00036811.2018.1555321 (2018). [18] a. saoudi and i. a. kallel, l2-uncertainty principle for the weinstein-multiplier operators, int. j. anal. appl. 17 (1) (2019), 64-75. https://doi.org/10.1142/s1793557121500030 https://doi.org/10.1142/s1793557121500030 https://doi.org/10.1080/00036811.2018.1555321 https://doi.org/10.1080/00036811.2018.1555321 1. introduction 2. notations and preliminaries 3. l2-multiplier operators for the q-rubin-fourier transform 4. the extremal function associated with q2-fourier l2-multiplier operators references international journal of analysis and applications volume 19, number 3 (2021), 296-318 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-296 generalized petrović’s inequalities for coordinated exponentially m-convex functions wasim iqbal1, muhammad aslam noor1,∗, khalida inayat noor1, farhat safdar2 1comsats university islamabad, islamabad, pakistan 2department of mathematics, sbk women university, quetta, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we introduce a new class of convex function, which is called coordinated exponentially m-convex functions. some new petrović’s type inequalities for exponentially m-convex functions and coordinated exponentially m-convex functions are derived. lagrange-type and cauchy-type mean value theorems for exponentially m−convex and coordinated exponentially m-convex functions are also derived. several special cases are discussed. we also prove the lagrange type and cauchy type mean value theorems for exponentially m-convex and coordinated exponentially m−convex functions. results proved in this paper may stimulate further research in different areas of pure and applied sciences. 1. introduction convex functions and their variant forms are being used to study a wide class of problems which arises in various branches of pure and applied sciences. for recent applications, generalizations and other aspects of convex functions, see [1–5, 11–23, 28, 33, 34] and the references therein. one of the most significant inequality is the petrović’s inequality [25]. petrović’s type inequality have been obtained by several authors, see [7, 10, 24–31] and reference therein. in recent years, the convexity theory have been generalized in several directions using novel and innovative received february 2nd, 2021; accepted march 8th, 2021; published april 1st, 2021. 2010 mathematics subject classification. primary 26a51; secondary 26d15, 49j40. key words and phrases. petrović’s inequality; exponentially m−convex functions; exponentially m−convex functions on coordinates. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 296 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-296 int. j. anal. appl. 19 (3) (2021) 297 techniques. toader [34] introduced the concepts of m-convex sets and m-convex functions, which appeared to be an interesting generalization of the convex sets and convex functions. exponentially convex functions were introduced by bernstein [6], which have applications in covariance analysis. avriel [4] investigated this concept by imposing the condition of r-convex functions. it is well known that logconvex functions is closely related to exponentially convex functions, which have important and interesting applications in information theory and machine learning. motivated and inspired by these applications, noor et. al. [14] considered exponentially convex functions and explored their basic characterizations. they have shown that the optimality conditions of the differentiable exponentially convex functions can be characterized by variational inequalities, which have appeared an interesting field with applications in various fields of pure and applied sciences. for the applications and other aspects of the variational inequalities, see noor et al.[12, 21] and references therein. pal and wong [23] provided the application of exponentially convex functions in information, optimization, statistical theory and related areas. for other aspects of exponentially convex functions and their generalizations, see [2, 5, 12, 14, 16, 18, 19, 21–23, 33]. it is worth mentioning that the exponentially convex functions and m-convex functions are clearly two different generalizations of the convex functions. it is natural to unify these classes. motivated by these facts, rashid et al [33] introduced the exponentially m-convex functions and derived some hermite-hadamrd type inequalities. these integral inequalities can be used to obtain the upper and lower bounds for the integrand, which have applications in material sciences and numerical analysis. petrović’s [25] derives some integral inequalities for convex functions, which are known as petrović’s type inequalities. for the applications and other aspects of petrović’s inequalities, see [10, 24–27, 29, 31]. in this paper, we introduce some new concepts of coordinated exponentially m-convex functions. we derive petrović’s type inequality for exponentially mconvex and coordinated exponentially m-convex functions. the lagrange and cauchy mean value results for the exponentially m-convex functions are derived. several important cases are discussed as applications of the obtained results. we expect that the ideas and techniques of this paper may be staring point for further research in this areas. 2. preliminaries in this section, we recall the basic definitions and concepts of the exponentially convex functions. definition 2.1. a nonempty set ω ⊆ r is convex, if σu + (1 −σ)v ∈ ω, ∀u,v ∈ ω, σ ∈ [0, 1]. definition 2.2. a function f : ω → r is convex, if f(σu + (1 −σ)v) ≤ σf(u) + (1 −σ)f(v), ∀u,v ∈ ω, σ ∈ [0, 1]. int. j. anal. appl. 19 (3) (2021) 298 toader [34] introduced m-convex functions as follows: definition 2.3. the function f : [0,b] → r,b > 0, is said to be m-convex, where m ∈ [0, 1], if we have f (σu + m(1 −σ)v) 6 σf(u) + m(1 −σ)f(v), ∀u,v ∈ [0,b],σ ∈ [0, 1]. remark 2.1. one can note that the notion of m-convexity reduces to convexity for m = 1. for m = 0, we obtain starshaped functions. noor et al. [12, 14] introduced exponentially convex function as follows: definition 2.4. a function f is called exponentially convex on ω, if ef(u+σ(v−u)) ≤ (1 −σ)ef(u) + σef(v), ∀u,v ∈ ω, σ ∈ [0, 1],(2.1) which can be written in the following equivalent form, which is due to avriel [4]. definition 2.5. a function f is called exponentially convex function on ω, if f(u + σ(v −u)) ≤ log[(1 −σ)ef(u) + σef(v)], ∀u,v ∈ ω, σ ∈ [0, 1].(2.2) for the applications of the exponentially convex functions in information theory and mathematical programming, see antczak [3] and alirezaei and mathar [2]. rashid el al.[33] introduced exponentially m−convex function as follows: definition 2.6. a function f : ω → r on an interval of real line is exponentially m-convex, where m ∈ (0, 1], if (2.3) ef(σu+m(1−σ)v) ≤ σef(u) + m(1 −σ)ef(v),∀u,v ∈ ω,σ ∈ [0, 1]. from now onwards, we take i1 = [a1,b1] and i2 = [c1,d1] as intervals in r. dragomir [8] introduced coordinated convex functions as follows: definition 2.7. “ let us consider the bidimensional interval ∆ = i1 × i2 in r2 with a1 < b1 and c1 < d1. also, let f : i1 × i2 → r be a mapping. define partial mappings as” (2.4) fv : i1 → r defined by fv(x) = f(x,v) and “ (2.5) fu : i2 → r defined by fu(y) = f(u,y). the function f is called coordinated convex, if the partial mappings defined in (2.7) and (2.8) are convex on i1 and i2 respectively, for all v ∈ i2 and u ∈ i1. int. j. anal. appl. 19 (3) (2021) 299 definition 2.8. the function f : ∆ → r is convex in ∆, if f(σu + (1 −σ)z1,σv + (1 −σ)w1) ≤ σf(u,v) + (1 −σ)f(z1,w1),(2.6) ∀(u,v), (z1,w1) ∈ ∆,σ ∈ [0, 1]. farid et al.[9] introduced coordinated m-convex functions as follows: definition 2.9. let ∆1 = [0,b] × [0,d] ⊂ [0,∞)2, then a function f : ∆ → r is m−convex on coordinates if the partial mappings (2.7) fv : [0,b] → r defined by fv(x) = f(x,v) and (2.8) fu : [0,d] → r defined by fu(y) = f(u,y) are m− convex on [0,b] and [0,d] respectively for all v ∈ [0,d] and u ∈ [0,b]. we now introduce the concept of exponentially m−convex functions on coordinates, which is the main motivation of this paper. definition 2.10. let f : ∆1 → r be a positive mapping. the function f is coordinated exponentially m−convex, if the partial mappings defined in (2.7) and (2.8) are exponentially m−convex on [0,b] and [0,d] respectively, for all v ∈ [0,d] and u ∈ [0,b]. definition 2.11. a positive mapping f : ∆1 → r is exponentially m−convex in ∆1, if ef(σu+m(1−σ)z1,σv+m(1−σ)w1) ≤ σef(u,v) + m(1 −σ)ef(z1,w1),(2.9) ∀(u,v), (z1,w1) ∈ ∆1,σ ∈ [0, 1],m ∈ (0, 1]. lemma 2.1. “ every exponentially m−convex mapping f : ∆1 → r is coordinated exponentially m−convex, but converse is not true in general.” proof. let a positive mapping f : ∆1 → r be an exponentially m−convex in ∆1. also, let fu : [0,d] → r defined as fu(v1) := f(u,v1). then \efu(σv1+m(1−σ)w1) = ef(u,σv1+m(1−σ)w1) = ef(σu+m(1−σ)z1,σv1+m(1−σ)w1) ≤ σef(u,v1) + m(1 −σ)ef(z1,w1) = σefu(v1) + m(1 −σ)efz1 (w1), ∀σ ∈ [0, 1],v1,w1 ∈ [0,d], ” int. j. anal. appl. 19 (3) (2021) 300 which shows the exponentially m−convexity of fu. similarly, one can show the exponentially m−convexity of fv. now, consider the positive mapping f0 : [0, 1]2 → [0,∞) given by ef0(u,v1) = uv1. clearly f is coordinated exponentially m−convex. but it is not exponentially m−convex on [0, 1]2. indeed, if (u, 0), (0,w1) ∈ [0, 1]2 and σ ∈ [0, 1]. then ef(σ(u,0)+m(1−σ)(0,w1)) = ef(σu,m(1−σ)w1) = mσ(1 −σ)uw1 and σef(u,0) + m(1 −σ)ef(0,w1) = 0. thus, ∀ σ ∈ (0, 1), u,w1 ∈ (0, 1), one has ef(σ(u,0)+m(1−σ)(0,w1)) > σef(u,0) + m(1 −σ)ef(0,w1), which shows that f is not exponentially m−convex. � petrović [25] derived some inequality for convex functions. theorem 2.1. let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul for each l = 1, ...,n. consider the function f is convex on [0,a1], then n∑ κ=1 pκf(uκ) ≤f ( n∑ κ=1 pκuκ ) + ( n∑ κ=1 pκ − 1 ) f(0).(2.10) “ rehman et al. [31] gave the petrović’s inequality on coordinated convex functions.” theorem 2.2. “ let (ui, i = 1, 2, ...,n) and (vj,j = 1, 2, ...,n) be non-negative n-tuples and (pk,k = 1, ...,n) and (ql, l = 1, ...,n) be positive n-tuples such that ” “ n∑ κ=1 pκ ≥ 1, 0 6= n∑ κ=1 pκuκ ≥ uj for every j = 1, 2, ...,n, ” and “ n∑ l=1 ql ≥ 1, 0 6= n∑ l=1 qlvl ≥ vj for every i = 1, 2, ...,n. int. j. anal. appl. 19 (3) (2021) 301 ” “let f : [0,a1) × [0,b1) → r be a convex on coordinates, then ” \ n∑ κ=1 n∑ l=1 pκqlf(uκ,vl) 6f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) f ( n∑ κ=1 pκuκ, 0 ) (2.11) + ( n∑ κ=1 pκ − 1 )( f ( 0, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) f(0, 0) ) .” 3. main results “ in this section, we prove an important lemma, which plays a key role for proving our next results. lemma 3.1. “ let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, θ ∈ [0,a1],” \ n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul > mθ for each l = 1, ...,n.” “ suppose a positive function f : [0,a1] → r is exponentially m−convex. if e f(u) u−mθ is increasing on [0,a1], then” \e f ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκe f(uκ).”(3.1) proof. since ∑n κ=1 pκuκ ≥ ul > mθ for all l = 1, ...,n and ef(u) u−mθ is increasing on [0,a1], we have \ e f ( n∑ κ=1 pκuκ ) ( n∑ κ=1 pκuκ −mθ ) ≥ ef(uκ) (uκ −mθ) . ” this implies (uκ −mθ)e f ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) ef(uκ). multiplying above inequality by pκ and taking sum for κ = 1, ...,n, one has n∑ κ=1 pκ(uκ −mθ)e f ( n∑ κ=1 pκuκ ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ l=1 pκe f(uκ), from which, one has the required result. � “ we now derive the petrović’s type inequality for exponentially m−convex functions. ” int. j. anal. appl. 19 (3) (2021) 302 theorem 3.1. “let (ui, i = 1, 2, ...,n) be non-negative n-tuples and (pj,j = 1, 2, ...,n) be positive n-tuples such that ∑n j=1 pj ≥ 1, θ ∈ [0,a1], ” \ n∑ κ=1 pκuκ ∈ [0,a1] and n∑ κ=1 pκuκ ≥ ul > θ for each l = 1, ...,n.” let a positive function f : [0,∞) → r be an exponentially m−convex. then (3.2) n∑ l=1 ple f(uκ) ≤ ae f ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl −a ) ef(θ) , where a =   n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ   . proof. let a function f be an exponentially m−convex and ℵ(x) = ef(x) −mef(a) x−ma . we take y > x > ma and x = ma + σ(y −ma), where σ ∈ (0, 1). then ℵ(x) = ef(σy+m(1−σ)a) −mef(a) σy + m(1 −σ)a−ma) ≤ σef(y) + m(1 −σ)ef(a) −mef(a) σ(y −ma) = ef(y) −mef(a) y −ma . this implies ℵ(x) ≤ℵ(y). hence ℵ(x) is increasing on [0,a]. as we have proved that, if f is exponentially m−convex, then e f(x)−mef(a) x−ma is increasing for x > mθ. substituting ef(x) by ef(x) −mef(θ) in lemma 3.1, one has e f ( n∑ κ=1 pκuκ ) −mef(θ) ≥ ( n∑ κ=1 pκuκ −mθ ) n∑ l=1 pl(ul −mθ) n∑ l=1 pl ( ef(uκ) −mef(θ) ) . this gives us n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ ( e f ( n∑ κ=1 pκuκ ) −mef(θ) ) ≥ n∑ l=1 ple f(uκ) −m n∑ l=1 ple f(θ).” int. j. anal. appl. 19 (3) (2021) 303 this leads to n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ e f ( n∑ κ=1 pκuκ ) ≥ n∑ l=1 ple f(uκ) −m n∑ l=1 ple f(θ) +m   n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ  ef(θ). finally, we have n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ e f ( n∑ κ=1 pκuκ ) ≥ n∑ l=1 ple f(uκ)− m   n∑ l=1 pl − n∑ l=1 pl(ul −mθ) n∑ κ=1 pκuκ −mθ  ef(θ), which is the required result. � if θ = 0, then theorem 3.1 reduces to the following new result. it can be considered as petrović’s type inequality for exponentially m−convex function. theorem 3.2. let the conditions given in theorem 3.1 be satisfied and let a positive function f : [0,∞) → r be an exponentially m−convex. then (3.3) n∑ l=1 ple f(uκ) ≤ e f ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl − 1 ) ef(0). if m = 1, then theorem 3.1 reduces to the following new result. it can be viewed as a new generalized petrović’s type inequality for exponentially convex function. theorem 3.3. let the conditions given in theorem 3.1 be satisfied. also, let a positive function f : [0,∞) → r be an exponentially convex. then (3.4) n∑ l=1 ple f(uκ) ≤   n∑ l=1 pl(ul −θ) n∑ κ=1 pκuκ −θ  ef ( n∑ κ=1 pκuκ ) +   n∑ l=1 pl −   n∑ l=1 pl(ul −θ) n∑ κ=1 pκuκ −θ    ef(θ). if m = 1 and θ = 0, then theorem 3.1 reduces to the following new result. it can be considered as petrović’s type inequality for exponentially convex function. int. j. anal. appl. 19 (3) (2021) 304 theorem 3.4. let the conditions given in theorem 3.1 be satisfied and let a positive function f : [0,∞) → r be an exponentially convex. then (3.5) n∑ l=1 ple f(uκ) ≤ e f ( n∑ κ=1 pκuκ ) + ( n∑ l=1 pl − 1 ) ef(0). now, we derive the generalized petrović’s type inequality for coordinated exponentially m−convex functions. theorem 3.5. “ let (ui, i = 1, 2, ...,n) and (vj,j = 1, 2, ...,n) be non-negative n-tuples and (pk,k = 1, 2, ...,n) and (ql, l = 1, ...,n) be positive n-tuples such that θ ∈ [0,a1], ∑n κ=1 pκ ≥ 1, ∑n l=1 ql ≥ 1, ” \ n∑ κ=1 pκuκ ∈ [0,a1), 0 6= n∑ κ=1 pκuκ ≥ uj > θ for every j = 1, 2, ...,n” and \ n∑ l=1 qlvl ∈ [0,b1), 0 6= n∑ l=1 qlvl ≥ vi > θ for every i = 1, 2, ...,n.” let a positive function f : [0,∞)2 → r be coordinated exponentially m−convex function. then (3.6) n∑ κ=1 n∑ l=1 pκqle f(uj,vl) ≤ a { be f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql −b ) e f ( n∑ κ=1 pκuκ,θ )} + m ( n∑ κ=1 pκ −a ){ be f ( θ, n∑ l=1 qlvl ) + m ( n∑ l=1 ql −b ) ef(θ,θ) } , where a =   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  (3.7) and b =   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ   .(3.8) proof. “ consider the partial mappings fu : [0,a1] → r and fv : [0,b1] → r defined by fu(v1) = f(u,v) and fv(u) = f(u,v). as f is coordinated exponentially m−convex on [0,∞)2. therefore, the partial mapping fv is exponentially m−convex on [0,b1]. by theorem 3.1, we have int. j. anal. appl. 19 (3) (2021) 305 n∑ κ=1 pκe fv(uj) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  efv ( n∑ κ=1 pκuκ ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  efv(θ). “this is equivalent to ” n∑ κ=1 pκe f(uj,v) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  ef ( n∑ κ=1 pκuκ,v ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  ef(θ,v). by setting v = vl, we get n∑ κ=1 pκe f(uj,vl) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  ef ( n∑ κ=1 pκuκ,vl ) +m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ  ef(θ,vl). multiplying above inequality by ql and taking sum for l = 1, ...,n, one has (3.9) n∑ κ=1 n∑ l=1 pκqle f(uj,vl) ≤   n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ   n∑ l=1 qle f ( n∑ κ=1 pκuκ,vl ) + m   n∑ κ=1 pκ − n∑ κ=1 pκ(uκ −mθ) n∑ κ=1 pκuκ −mθ   n∑ l=1 qle f(θ,vl). now again by theorem 3.1, we have n∑ l=1 qle f ( n∑ κ=1 pκuκ,vl ) ≤   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + m   n∑ l=1 ql − n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef ( n∑ κ=1 pκuκ,θ ) int. j. anal. appl. 19 (3) (2021) 306 and n∑ l=1 qle f(θ,vj) ≤   n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef ( θ, n∑ l=1 qlvl ) +m   n∑ l=1 ql − n∑ l=1 ql(vl −mθ) n∑ l=1 qlvl −mθ  ef(θ,θ). putting these values in inequality (3.9) and using the notations given in (3.7) and (3.8), we get the required result. � if m = 1, then theorem 3.5 reduces to the following new result. theorem 3.6. “ let the conditions given in 3.5 be satisfied. also, let a positive function f : [0,∞)2 → r be coordinated exponentially m−convex function, then (3.10) n∑ κ=1 n∑ l=1 pκqle f(uj,vl) ≤ c { de f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql −d ) e f ( n∑ κ=1 pκuκ,θ )} + ( n∑ κ=1 pκ −c ){ de f ( θ, n∑ l=1 qlvl ) + ( n∑ l=1 ql −d ) ef(θ,θ) } , where c =   n∑ κ=1 pκ(uκ −θ) n∑ κ=1 pκuκ −θ   and d =   n∑ l=1 ql(vl −θ) n∑ l=1 qlvl −θ   . if θ = 0, then theorem 3.5 reduces to the following new result. theorem 3.7. let the conditions given in theorem 3.5 be satisfied. if f : [0,∞)2 → r be coordinated exponentially m−convex, then (3.11) n∑ κ=1 n∑ l=1 pκqle f(uj,vl) ≤ { e f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql − 1 ) e f ( n∑ κ=1 pκuκ,0 )} + m ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + m ( n∑ l=1 ql − 1 ) ef(0,0) } . int. j. anal. appl. 19 (3) (2021) 307 if θ = 0 and m = 1, then theorem 3.5 reduces to the following new result. theorem 3.8. let the conditions given in theorem 3.5 be satisfied. if f : [0,∞)2 → r be coordinated exponentially convex, then (3.12) n∑ κ=1 n∑ l=1 pκqle f(uj,vl) ≤ { e f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) e f ( n∑ κ=1 pκuκ,0 )} + ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + ( n∑ l=1 ql − 1 ) ef(0,0) } . by considering non-negative difference of (3.3), we define the following functional. (3.13) p(ef) = e f ( n∑ κ=1 pκuκ ) + m ( n∑ l=1 pl − 1 ) ef(0) − n∑ l=1 ple f(uκ). also by considering non-negative difference of (3.11), we define the following functional. (3.14) υ(ef) = { e f ( n∑ κ=1 pκuκ, n∑ l=1 qlvl ) +m ( n∑ l=1 ql − 1 ) e f ( n∑ κ=1 pκuκ,0 )} + m ( n∑ κ=1 pκ − 1 ){ e f ( 0, n∑ l=1 qlvl ) + m ( n∑ l=1 ql − 1 ) ef(0,0) } − n∑ κ=1 n∑ l=1 pκqle f(uj,vl). we need the following lemma. lemma 3.2. let a positive function f : [0,b1] → r be an exponentially m−convex such that n1 6 (u−ma1)ef(u)f′(u) −ef(u) + mef(a1) u2 − 2ma1u + ma21 6 n1, ∀u ∈ [0,b1]\{a1} and a1 ∈ (0,b1). let γ1,γ2 : [0,b1] → r be positive functions defined as γ1(u) = log[n1u 2 −ef(u)] and γ2(u) = log[e f(u) −n1u2], then γ1 and γ2 are exponentially m−convex on [0,b1]. int. j. anal. appl. 19 (3) (2021) 308 proof. suppose pγ1 (u) = eγ1(u) −meγ1(a1) u−ma1 = n1u 2 −ef(u) −mn1a12 + mef(a1) u−ma1 = n1(u 2 −ma12) u−ma1 − ef(u) −mef(a1) u−ma1 . by differentiating with respect to u, one has p ′γ1 (u) = n1 (u−ma1)2u− (u2 −ma21) (u−ma1)2 − (u−ma1)ef(u)f′(u) −ef(u) + mef(a1) (u−ma1)2 . since u2 − 2ma1u + m2a21 −m 2a21 + ma 2 1 = (u−ma1) 2 −m(m− 1)a21 > 0, by the given condition, one has n1(u 2 −ma12u + ma21) ≥ (u−ma1)e f(u)f′(u) −ef(u) + mef(a1). this implies n1 u2 − 2ma1u + ma21 (u−ma1)2 ≥ (u−ma1)ef(u)f(u) −ef(u) + mef(a1) (u−ma1)2 , n1 u2 − 2ma1u + ma21 (u−ma1)2 − (u−ma1)ef(u)f(u) −ef(u) + mef(a1) (u−ma1)2 ≥ 0. this implies p ′γ1 (u) ≥ 0, ∀u ∈ [0,a1) ∪ (a1,b1]. similarly, one can show that p ′γ2 (u) ≥ 0, ∀u ∈ [0,a1) ∪ (a1,b1]. this implies that pγ1 and pγ2 are increasing on u ∈ [0,a1) ∪ (a1,b1] for all a ∈ (0,b1). hence by (??), γ1(u) and γ2(u) are exponentially m−convex in [0,b1]. � here we prove the mean value theorems related to functional defined for petrović’s inequality for exponentially m−convex functions. theorem 3.9. “ let (u1, ...,un) ∈ [0,b1], and (p1, ...,pn) be positive n-tuples such that ∑n k=1 pkuk ≥ uj for each j = 1, 2, ...,n. also let φ(u) = log u2.” if a positive exponentially m−convex function f ∈ c1([0,b1]), then there exist γ ∈ (0,b1) such that p(ef) = (γ −ma1)ef(γ)f′(γ) −ef(γ) + mef(a1) (γ2 − 2ma1γ + ma21) p(eφ),(3.15) int. j. anal. appl. 19 (3) (2021) 309 “ provided that p(eφ) is non zero and a ∈ (0,b1).” proof. “as f ∈ c1([0,b1]), so there exist real numbers n1 and n1 such that “ n1 6 (u−ma1)ef(u)f′(u) −ef(u) + mef(a1) (u2 − 2ma1u + ma21) 6 n1, ∀u ∈ [0,b1] and a1 ∈ (0,b1). consider the functions γ1 and γ2 defined in lemma 3.2. as γ1 is exponentially m−convex in [0,b1], so p(eγ1 ) ≥ 0, that is p ( n1u 2 −ef(u) ) ≥ 0, this gives (3.16) n1p(eφ) ≥p(ef). similarly γ2 is exponentially m−convex [0,b1], therefore one has (3.17) n1p(φ) 6p(ef). by assumption p(eφ) is non zero, combining inequalities (3.16) and (3.17), one has n1 6 p(ef) p(eφ) 6 n1. hence there exist v ∈ (0,b1) such that p(ef) p(eφ) = (γ −ma1)ef(γ)f′(γ) −ef(γ) + mef(a1) (γ2 − 2ma1γ + ma21) , which is the required result. � if we take m = 1, then theorem 3.9 reduces to the following result. theorem 3.10. let the conditions given in theorem 3.9 be satisfied. if f ∈ c1([0,b1]) is a positive exponentially convex function, then there exist γ ∈ (0,b1) such that p(ef) = (γ −a1)ef(γ)f′(γ) −ef(γ) + ef(a1) (γ −a1)2 p(eφ),(3.18) provided that p(eφ) is non zero and a ∈ (0,b1). theorem 3.11. let the conditions given in theorem 3.9 be satisfied. suppose the positive exponentially m−convex functions f1,f2 ∈ c1([0,b1]), then there exist γ ∈ (0,b1) such that p(ef1 ) p(ef2 ) = (γ −ma1)ef1(γ)f′1(γ) −ef1(γ) + mef1(a) (γ −ma1)ef2(γ)f2′(γ) −ef2(γ) + mef2(a) , “provided that the denominators are non-zero and a1 ∈ (0,b1).” int. j. anal. appl. 19 (3) (2021) 310 proof. suppose k ∈ c1([0,b1]) be a function defined as k = log (c1e f1 − c2ef2 ), where c1 and c2 are defined as c1 = p(ef2 ), c2 = p(ef1 ). then using theorem 3.9 with f = k, one has (γ −ma1)elog(c1e f1(γ)−c2ef2(γ))(log(c1e f1(γ) − c2ef2(γ)))′ − (c1ef1(γ) − c2ef2(γ)) + m(c1e f1(a) − c2ef2(a)) = 0, this gives (γ −ma1)(c1ef1(γ) − c2ef2(γ))′ − c1ef1(γ) + c2ef2(γ) + mc1ef1(a) −mc2ef2(a) = 0, that is (γ −ma1)(c1ef1(γ)f′1(γ) − c2e f2(γ)f′2(γ)) − c1e f1(γ) + c2e f2(γ) + mc1e f1(a) −mc2ef2(a) = 0, this gives (γ −ma1)c1ef1(γ)f′1(γ) − (γ −ma1)c2e f2(γ)f′2(γ) − c1e f1(γ) + c2e f2(γ) + mc1e f1(a) −mc2ef2(a) = 0, which implies c1 {(γ −ma1)f′1(γ) −f1(γ) + mf1(a)}− c2 {(γ −ma1)f ′ 2(γ) + f2(γ) −mf2(a)} = 0 c1 { (γ −ma1)ef1(γ)f′1(γ) −e f1(γ) + mef1(a) } = c2 { (γ −ma1)ef2(γ)f′2(γ) −e f2(γ) + mef2(a) } . this gives c2 c1 = (γ −ma1)ef1(γ)f′1(γ) −ef1(γ) + mef1(a) (γ −ma1)ef2(γ)f′2(γ) −ef2(γ) + mef2(a) . putting the values of c1 and c2, one has the required result. � if we take m = 1, then theorem 3.11 reduces to the following result. int. j. anal. appl. 19 (3) (2021) 311 theorem 3.12. let the conditions given in theorem 3.11 be satisfied. suppose the positive exponentially convex functions f1,f2 ∈ c1([0,b1]), then there exist γ ∈ (0,b1) such that p(ef1 ) p(ef2 ) = (γ −a1)ef1(γ)f′1(γ) −ef1(γ) + ef1(a) (γ −a1)ef2(γ)f2′(γ) −ef2(γ) + ef2(a) , “provided that the denominators are non-zero and a1 ∈ (0,b1).” here we state an important lemma that is helpful in proving mean value theorems related to the nonnegative functional of petrovic̀’s inequality for coordinated exponentially m−convex functions. lemma 3.3. let ∆ = [0,b1]× [0,d1]. also, let f : ∆ → r be a positive coordinated exponentially m−convex function such that n1 6 (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u2 − 2ma1u + ma21)v2 6 n1 and n2 6 (v −mc1)ef(u,v) ∂∂vf(u,v) −e f(u,v) + mef(u,c1) (v2 − 2ma1v + mc21)u2 6 n2 ∀u ∈ [0,b1]\{a1}, a1 ∈ (0,b1) and v ∈ [0,d1]\{c1}, c ∈ (0,d1). consider the functions αv : [0,b1] → r, and αu : [0,d1] → r, defined as α(u,v) = log[max{n1,n2}u2v2 −ef(u,v)] and β(u,v) = log[ef(u,v) − min{n1,n2}u2v2]. then α and β are coordinated exponentially m−convex. proof. suppose the partial mappings αv : [0,b1] → r and αu : [0,d1] → r defined as αv(u) := α(u,v) for all u ∈ (0,b1] and αu(v) := α(u,v) for all v ∈ (0,d]. pαv (u) = eαv(u) −meαv(a1) u−ma1 = eα(u,v) −meα(a1,v) u−ma1 = elog[max{n1,n2}u 2v2−mef(u,v)] −melog[max{n1,n2}a 2 1v 2−ef(a1,v)] u−ma1 = n1u 2v2 −ef(u,v) −mn1a21v2 + mef(a1,v) u−ma1 = n1 (u2 −ma21)v2 u−ma1 − ef(u,v) −mef(a1,v) u−ma1 . int. j. anal. appl. 19 (3) (2021) 312 differentiating partially with respect to u, one has p ′αv (u) = n1v 2 (u−ma1)2u− (u 2 −ma21) (u−ma1)2 − (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u−ma1)2 = n1v 2 (u 2 − 2ma1u + ma21) (u−ma1)2 − (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u−ma1)2 by the given condition, one has n1 ≥ (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u2 − 2ma1u + ma21)v2 . since (u2 − 2ma1u + ma21)v 2 > 0. this implies n1 (u2 − 2ma1u + ma21)v2 (u−ma1)2 ≥ (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u−ma1)2 , n1 (u2 − 2ma1u + ma21)v2 (u−ma1)2 − (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + mef(a1,v) (u−ma1)2 ≥ 0. this implies p ′αv (u) ≥ 0, ∀u ∈ [0,ma1) ∪ (ma1,b1]. similarly, one can show that p ′αu(v) ≥ 0, ∀u ∈ [0,mc1) ∪ (mc1,d1]. this ensure that pαv is increasing on [0,ma1) ∪ (ma1,b1] for all a1 ∈ [0,b1] and pαu is increasing on [0,mc1) ∪ (mc1,d1] for all c1 ∈ [0,d1]. by (??), α is exponentially m−convex. hence by lemma 2.1, α is coordinated exponentially m−convex. similarly, one can show that β is coordinated exponentially m−convex. � “here we give mean value theorems related to the functional defined for petrovic̀’s type inequality for coordinated exponentially m−convex functions.” int. j. anal. appl. 19 (3) (2021) 313 theorem 3.13. let (u1, ...,un) ∈ [0,b1], (v1, ...,vn) ∈ [0,d1] be non-negative n-tuples and (q1, ...,qn), (p1, ...,pn) be positive n-tuples such that∑n k=1 pkuk ≥ uj for each j = 1, 2, ...,n. also let ϕ(u,v) = log (u 2v2). let a positive coordinated exponentially m−convex function f ∈ c1(∆), then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that υ(ef) = (γ1 −ma)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ21 − 2maγ1 + ma2)ζ21 υ(eϕ)(3.19) and υ(ef) = (γ2 −ma)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) (γ22 − 2maγ2 + ma2)ζ22 υ(eϕ),(3.20) provided that υ(eϕ) is non-zero and a ∈ (0,b1). proof. as f has continuous first order partial derivative in ∆, so there exist real numbers n1,n2,n1 and n2 such that n1 6 (u−ma1)ef(u,v) ∂∂uf(u,v) −e f(u,v) + ef(a,v) (u2 − 2ma1u + ma21)v2 6 n1 and n2 6 (v −ma1)ef(u,v) ∂∂vf(u,v) −e f(u,v) + ef(u,a) (v2 − 2ma1v + ma21)u2 6 n2, ∀u ∈ (0,b1], v ∈ (0,d] and a ∈ (0,b1). consider the functions α and β defined in lemma 3.3. as α is coordinated exponentially m−convex, then υ(eα) ≥ 0, that is υ ( n1u 2v2 −ef(u,v) ) ≥ 0, this gives (3.21) n1υ(e ϕ) ≥ υ(ef). similarly β is coordinated exponentially m−convex, therefore one has (3.22) n1υ(e ϕ) 6 υ(ef). by assumption υ(eϕ) is non-zero, so combining inequalities (3.21) and (3.22), one has n1 6 υ(ef) υ(eϕ) 6 n1. int. j. anal. appl. 19 (3) (2021) 314 hence there exists (γ1,ζ1) in the interior of ∆, such that υ(ef) = (γ1 −ma)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ21 − 2maγ1 + ma2)ζ21 υ(eϕ). similarly, one can show that υ(ef) = (γ2 −ma)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) (γ22 − 2maγ2 + ma2)ζ22 υ(eϕ), which is the required result. � if we take m = 1, then theorem 3.13 reduces to the following result. theorem 3.14. let the conditions given in theorem 3.13 be satisfied. also, let a positive coordinated exponentially convex function f ∈ c1(∆), then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that υ(ef) = (γ1 −a)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + ef(a,ζ1) (γ21 − 2aγ1 + a2)ζ21 υ(eϕ)(3.23) and υ(ef) = (γ2 −a)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + ef(a,ζ2) (γ22 − 2aγ2 + a2)ζ22 υ(eϕ),(3.24) provided that υ(eϕ) is non-zero and a ∈ (0,b1). theorem 3.15. let the conditions given in theorem 3.13 be satisfied. also let the positive coordinated exponentially m−convex functions f1,f2 ∈ c1(∆), “then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that” υ(ef1 ) υ(ef2 ) = (γ1 −ma)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ2 −ma)ef(γ2,ζ2) ∂∂uf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) and υ(ef1 ) υ(ef2 ) = (γ1 −ma)ef(γ1,ζ1) ∂∂vf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ2 −ma)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) , “provided that the denominators are non-zero and a ∈ (0,b1).” proof. suppose k = log (c1e f1 − c2ef2 ), “where c1 and c2 are defined as ” c1 = υ(e f2 ), c2 = υ(e f1 ). int. j. anal. appl. 19 (3) (2021) 315 using theorem 3.13 with f = k, one has (γ −ma)elog(c1e f1−c2ef2 )(γ,ζ) ∂ ∂u log(c1e f1 − c2ef2 )(γ,ζ) −elog(c1e f1−c2ef2 )(γ,ζ) + melog (c1e f1−c2ef2 )(a,ζ) = 0, (γ −ma) ∂ ∂u (c1e f1 − c2ef2 )(γ,ζ) − (c1ef1 − c2ef2 )(γ,ζ) + m(c1e f1 − c2ef2 )(a,ζ) = 0, (γ1 −ma)c1ef1(γ1,ζ1) ∂ ∂u f1(γ1,ζ1) − (γ2 −ma)c2ef2(γ2,ζ2) ∂ ∂u f2(γ2,ζ2) − c1ef1(γ1,ζ1) + c2ef2(γ2,ζ2) + mc1ef1(a,ζ1) −mc2ef2(a,ζ2) = 0, c1 { (γ1 −ma)ef1(γ1,ζ1) ∂ ∂u f1(γ1,ζ1) −ef1(γ1,ζ1) + mef1(a,ζ1) } − c2 { (γ2 −ma)ef2(γ2,ζ2) ∂ ∂u f2(γ2,ζ2) −ef2(γ2,ζ2) + mef2(a,ζ2) } = 0, c1 { (γ1 −ma)ef1(γ1,ζ1) ∂ ∂u f1(γ1,ζ1) −ef1(γ1,ζ1) + mef1(a,ζ1) } = c2 { (γ2 −ma)ef2(γ2,ζ2) ∂ ∂u f2(γ2,ζ2) −ef2(γ2,ζ2) + mef2(a,ζ2) } , c1 { (γ1 −ma) ∂ ∂u ef1(v,u) −ef1(v,u) + ef1(a,u) } = c2 { (γ1 −ma) ∂ ∂u ef2(v,u) −ef2(v,u) + mef2(a,u) } , c2 c1 = (γ1 −ma)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ2 −ma)ef(γ2,ζ2) ∂∂uf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) . similarly, one can show that c2 c1 = (γ1 −ma)ef(γ1,ζ1) ∂∂vf(γ1,ζ1) −e f(γ1,ζ1) + mef(a,ζ1) (γ2 −ma)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + mef(a,ζ2) . putting the values of c1 and c2, one has the required result. � if we take m = 1, then theorem 3.15 reduces to the following result. theorem 3.16. let the conditions given in theorem 3.13 be satisfied. also let the positive coordinated exponentially convex functions f1,f2 ∈ c1(∆), “then there exist (γ1,ζ1) and (γ2,ζ2) in the interior of ∆, such that” υ(ef1 ) υ(ef2 ) = (γ1 −a)ef(γ1,ζ1) ∂∂uf(γ1,ζ1) −e f(γ1,ζ1) + ef(a,ζ1) (γ2 −a)ef(γ2,ζ2) ∂∂uf(γ2,ζ2) −e f(γ2,ζ2) + ef(a,ζ2) and υ(ef1 ) υ(ef2 ) = (γ1 −a)ef(γ1,ζ1) ∂∂vf(γ1,ζ1) −e f(γ1,ζ1) + ef(a,ζ1) (γ2 −a)ef(γ2,ζ2) ∂∂vf(γ2,ζ2) −e f(γ2,ζ2) + ef(a,ζ2) , “provided that the denominators are non-zero and a ∈ (0,b1).” int. j. anal. appl. 19 (3) (2021) 316 4. conclusion we have defined the coordinated exponentially m-convex functions. petrović’s type inequality for exponentially m-convex and coordinated exponentially m-convex functions have been derived. we obtained lagrange-type and cauchy-type mean value theorems for exponentially m−convex and coordinated exponentially m-convex functions. some new special cases are discovered. it is expected the ideas and techniques of this paper may motivate the researchers working in functional analysis, information theory and statistical theory to find some applications. this is a new path for future research. 5. acknowledgments we wish to express our deepest gratitude to our teachers, colleagues, collaborators and friends, who have direct or indirect contributions in the process of this paper. the authors would like to thank the rector, comsats university islamabad, islamabad, pakistan, for providing excellent research facilities. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. a. latif, m. alomar, on hadmard-type inequalities for h-convex functions on the co-ordinates, int. j. math. anal. 3(33)(2009), 1645–1656. [2] g. alirezaei, r. mathar, on exponentially concave functions and their impact in information theory, in: 2018 information theory and applications workshop (ita), ieee, san diego, ca, 2018: pp. 1–10. [3] t. antczak, (p,r)-invex sets and functions, j. math. anal. appl. 263 (2001), 355–379. [4] m. avriel, r-convex functions, math. program. 2 (1972), 309–323. [5] m.u. awan, m.a. noor, k.i. noor, hermite-hadamard inequalities for exponentially convex functions, appl. math. inform. sci. 12(2) (2018), 405–409. [6] s.n. bernstein, sur les fonctions absolument monotones, acta math. 52 (1929), 1–66. [7] s. butt, j. pečarić, a.u. rehman, exponential convexity of petrović and related functional, j. inequal. appl. 2011 (2011), 89. [8] s.s. dragomir, on hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, taiwan. j. math. 5(4) (2001), 775–788. [9] g. farid, m. marwan, a.u. rehman, new mean value theorems and generalization of hadamard inequality via coordinated m-convex functions, j. inequal. appl. 2015 (2015), 283. [10] w. iqbal, k.m. awan, a.u. rehman, g. farid, an extension of petrović’s inequality for h−convex (h−concave) functions in plane, open j. math. sci. 3(2019), 398–403. int. j. anal. appl. 19 (3) (2021) 317 [11] v.g. mihesan, a generalization of the convexity, seminar on functional equations, approximation and convexity, cluj-napoca, romania, 1993. [12] m.a. noor, advanced convex analysis and variational inequalities, lecture notes, comsats university islamabad, islmabad, pakistan, (2006-2020). [13] m.a. noor, some new classes of nonconvex functions, nonlinear funct. anal. appl. 11(1)(2006), 165– 171. [14] m.a. noor, k.i. noor, exponentially general convex functions, transylvanian j. math. mech. 11(1-2) (2019), 141-153. [15] m.a. noor, k.i. noor, on exponentially convex functions, j. orissa math. soc. 38(1-2)(2019), 33-51. [16] m.a. noor, k.i. noor, new classes of strongly exponentially preinvex functions, aims math. 4(6) (2019), 1554–1568. [17] m.a. noor, k.i. noor, strongly exponentially convex functions and their properties, j. adv. math. stud. 12(2) (2019), 177–185. [18] m.a. noor, k.i. noor, some properties of exponentially preinvex functions, facta univ. (nis), 34(5) (2019), 941–955. [19] m.a. noor, k.i. noor, strongly exponentially convex functions, u.p.b. sci. bull. ser. a. 81(4) (2019), 75–84. [20] m.a. noor, k.i. noor, m.u. awan, fractional ostrowski inequalities for s-godunova-levin functions, int. j. anal. appl. 5(2)(2014), 167–173. [21] m.a. noor, k.i. noor, m.th. rassias, new trends in general variational inequalities, acta appl. math. 170(1) (2020), 986-1046. [22] c. niclulescu, l.e. persson, convex functions and their applications, springer, new york, 2018. [23] s. pal, t.k. wong, on exponentially concave functions and a new information geometry, ann. probab. 46(2) (2018), 1070-1113. [24] j.e. pečarić, on the petrović’s inequality for convex functions, glasnik mat. 18(38) (1983), 77–85. [25] m. petrović’s, sur une fontionnelle, publ. math. univ. belgrade, 1 (1932), 146–149. [26] j. pečarić, v. čuljak, inequality of petrović and giaccardi for convex function of higher order, southeast asian bull. math. 26(1) (2003), 57–61. [27] j. pečarić, j. peric, improvements of the giaccardi and the petrovic inequality and related stolarsky type means, ann. univ. craiova, math. computer sci. ser. 39(1)(2012), 65–75. [28] j.e. pečarić, f. proschan, y.l. tong, convex functions, partial orderings, and statistical applications, academic press, boston, 1992. [29] a.u. rehman, g. farid, v.n. mishra, generalized convex function and associated petrović’s inequality, int. j. anal. appl. 17(1)(2019), 122–131. int. j. anal. appl. 19 (3) (2021) 318 [30] a.u. rehman, g. farid, w. iqbal, more about petrovic’s inequality on coordinates via m-convex functions and related results, kragujevac j. math. 44(3)(2020), 335–351. [31] a.u. rehman, m. mudessir, h.t. fazal, g. farid, petrović’s inequality on coordinates and related results, cogent math. 3 (2016), 1227298. [32] s. rashid, m. a. noor, k. i. noor, f. safdar, integral inequalities for generalized preinvex function, punjab univ. j. math. 51(10) (2019), 77–91. [33] s. rashid, m. a. noor, k. i. noor, f. safdar, fractional exponentially m-convex functions and inequalities, int. j. anal. appl. 17(3)(2019), 464–478. [34] g. toader, some generalizations of the convexity, in: proceedings of the colloquium on approximation and optimization, univ. cluj-napoca, cluj-napoca, 1985. 1. introduction 2. preliminaries 3. main results 4. conclusion 5. acknowledgments references international journal of analysis and applications volume 19, number 2 (2021), 193-204 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-193 global existence and uniqueness of the weak solution in thixotropic model amira rahai, amar guesmia∗ laboratory of applied mathematics and history and didactics of mathematics (lamahis), department of mathematics, university 20 august 1955 skikda, algeria ∗corresponding author: guesmiaamar19@gmail.com abstract. in this paper, we study global existence, uniqueness and boundedness of the weak solution for the system (p ) which is formulated by two subsystems (p1) and (p2), the first describes the thixotropic problem and the second describes the diffusion degradation of c, using galerkin’s method, lax-milgran’s and maximum principle. moreover we show that the unique solution is positive. 1. introduction the phenomenon of thixotropy has recently attracted a great deal of attention. the term was first applied [3] to an ”isothermal reversible sol-gel transformation”. as the gel state is often merely one of high viscosity, the definition has been made more general, and the term is then applied [5] to any ” isothermal reversible decrease of viscosity with increase of rate of shear”. colloidal solutions provide the more common examples of thixotropy and may be divided into three important classes : • solutions in newtonian liquids of lyophilic substances whose molecules are of great length, e.g., gelatine, starch and many synthetic polymers. • suspensions of solid particles such as pigments in oils, or clays in water. received november 10th, 2020; accepted december 7th, 2020; published february 1st, 2021. 2010 mathematics subject classification. primary 90c57, 90c59 secondary 90c49. key words and phrases. thixotropic; global solution; boundedness; positive solution. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 193 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-193 int. j. anal. appl. 19 (2) (2021) 194 • concentrated emulsions ( [11], [12]) of oil droplets in water; foams of gas bubbles in water (with, of course, stabilising agents). thixotropic fluids are used widely in civil engineering, food, cosmetic as well as pharmaceutical industries, and impact every aspect of our lives. as emulsions, suspensions, or polymeric gels, they are very different from each other compositionally, but most of them have one thing in common, i.e., the existence of microstructures. the microstructures are changeable and may comprise a network of flocculated colloidal particles, tangles of polymers, or a spatial arrangement of suspended particles or drops [1]. thixotropic fluids have a lot of special characters, such as aging, rejuvenation, and viscosity bifurcation [14] and by rate dependent properties associated to their structural level. the behavior of these substances under rheological tests have been analyzed in many scientific works ( [2], [9], [10], [13], [15]), which was firstly proposed by moore [8] in 1959. all of these scientific works were presented a qualitative explanation of the break down and build-up processes of the structure. in this paper, we are interested in the study of the global existence and uniqueness of weak positive solution for the elliptic-parabolic model’s. our model is defined as follows: (p)   (p1)   ut + ∆u−λdiv [ u ∇(c−u0)√ β+|∇(c−u0)|2 ] + u = u0 (t,x) ∈ r+ × ω u = 0 ∂ω u (0,x) = u0 x ∈ ω (p2)   −∆c + τc = 0 x ∈ ωc = g ∂ω where u (t,x) is a function denotes the speed of fluid in the position x ∈ ω ⊂ r2 or r3, ω is a bounded convex domain with smooth boundary ∂ω ∈ h 3 2 (∂ω), λ > 0 is the viscosity of the fluid, β > 0 is a parameter constant, c denotes the concentration of chemical signal that stimulates the fluid. the parameter τ is a time constant and it is expressed on the one hand the movement of fluid and secondly the diffusion degradation of c. to simplify the solution of the system (p), a decomposition of (p) into two subsystem (p1) and (p2) are adopted. galerkin’s method is very important to help us to demonstrate the existence and uniqueness of a weak solution for system (p1) . to prove the existence and uniqueness of a weak solution for system (p2), we use lax-milgram’s theorem and maximum principle. however this theorem can not be applied directly because it is nonhomogenous system. for this reason an adoptation of trace theorem it used to simplify the system(p2) . therefore we have the existence and uniqueness of a weak solution for system (p). moreover we show that the solution is positive. int. j. anal. appl. 19 (2) (2021) 195 the following initial-boundary conditions on u0 and g assumptions are used to prove the proposed solution of (p) • h1 : g ∈ l 1 2 (∂ω) . • h2 : g ∈ l 3 2 (∂ω) . • h3 : u0 ∈ l2 (ω) . • h4 : u0 ≥ 0 and g ≥ 0. if the hypothesis h1 is satisfies and using the theorem of trace, one can find a lifting of this trace which we denote r (g) ∈ h10 (ω) . thus by definition it verefies γ0 (r (g)) = g. now we looking for c having the form c = c̃ + r (g) reduves the problem (p2) to c̃ . ( p̃2 )  −∆c̃ + τc̃− ∆r (g) + τr (g) = 0 x ∈ ωc̃ = 0 on ∂ω definition 1.1. we say (u, c̃) ∈ l2 ( 0,t,h10 (ω) ) ×h10 (ω) with ut ∈ l2 ( 0,t,h−1(ω) ) is a weak solution of the problem (p) if and only if (1.1) 〈ut,v〉 + b (u,v,t) = (u0,v) (1.2) a (c̃,q,t) = l (q) where   b (u,v,t) = − ∫ ω ∇u∇vdx + ∫ ω (δτc + 1) uvdx− δ ∫ ω u∇u0∇vdx a (c̃,q,t) = ∫ ω (∇c̃∇q + τc̃q) dx l (q) = − ∫ ω (∇r (g)∇q + τr (g) q) dx for all (v,q) ∈ ( h10 (ω) )2 , 0 ≤ t ≤ t, (1.3) u (0,x) = u0 ∈ l2 (ω) and (1.4) δ = λ√ β + |∇(c−u0)| 2 . remark 1.1. note that u ∈ c ( [0,t] ,l2 (ω) ) as u ∈ l2 ( 0,t,h10 (ω) ) and ut ∈ l2 ( 0,t,h−1(ω) ) . then equation 1.3 makes sense. int. j. anal. appl. 19 (2) (2021) 196 2. existence of weak solution of the problem (p) in this section, we are interested in the study of the existence and uniqueness of weak solution of the problem (p1), which its variational formulat is given by equation 1.1 using galerkin’s method and use the theorem of lax-milgram to study the existence and uniqueness of weak solution of the problem (p2), which its variational formulat is given by equation 1.2 . so we have the existence and uniqueness of weak solution of the problem (p) . 2.1. existence of weak solution of the problem (p2). theorem 2.1. if the hypothesis h1 holds. then the problem (p2) has only one solution c ∈ h1 (ω) for any q ∈ h1 (ω) . by applying the theorem of lax-milgram, the solution c̃ of the problem 1.2 exists and it is unique. so (p2) has unique solution. remark 2.1. elliptic regularity theorem remains valid provided that the boundary condition g is in the space l 3 2 (∂ω) which is the image by the operator trace space h2 (ω) . remark 2.2. [17] if c ∈ h2 (ω) and ( c is a solution of problem (p2)) this implies that c ∈ w 1,q (ω) ( h2 (ω) ↪→ w 1,q (ω) for 1 ≤ q ≤ 2 ). using the maximum principle one can show that the solution of the problem (p2) is positive as follows. multiplying the first equation of (p2) by q ∈ h10 (ω) , we obtain other variational formulat for problem (p2) ( p̃3 )∫ ω (∇c∇q + τcq) dx = 0. proposition 2.1. [16] if g ∈ l 3 2 (∂ω) and c ∈ h1 (ω) ∩ c ( ω ) then the problem ( p̃3 ) have a positive solution c. proof. as ∂ω is smooth enough and g ∈ l 3 2 (∂ω) then c ∈ h2 (ω) . and as ω ⊂ r2 or r3, by embedding of sobolev spaces ( h2 ( ω ) ↪→ c ( ω ) ) this implies that c ∈ c ( ω ) . if c = g ≥ 0 on ∂ω, then c− = min (c, 0) ∈ h10 (ω) . so, we have ∫ ω cc−dx = ∫ ω ( c− )2 dx ∫ ω ∇c∇c−dx = ∫ ω ( ∇c− )2 dx, since the support of functions c− and c+ = max (c, 0) is set a (x) = {x/u (x) = 0} . this implies that ∇u = 0 on a (x) . as c = c+ + c−, thus we have int. j. anal. appl. 19 (2) (2021) 197 0 = ∫ (( ∇c− )2 + τ ( c− )2) dx ≥ min (1,τ) ∥∥c−∥∥2 h10 (ω) finally, we find c− = 0. � 2.2. existence of weak solution of the problem (p1). lemma 2.1. i) for all v ∈ h10 (ω) then b (., ., t) is continuous in h10 (ω) ×h10 (ω) , there exists a constant positive m such that (2.1) |b (u,v,t)| ≤ m ‖u‖h1(ω) ‖v‖h1(ω) ii) for any u ∈ h10 (ω) and h2 is hold. then exists a constant positive α such that (2.2) b (u,u,t) ≥ α‖u‖2h10 (ω) . proof. i) we use the cauchy-shwartz inequality and c ∈ h1 (ω) ↪→ lq (ω) for any q ∈ [ 1, 2n n−2 [ with n = 2 or n = 3, we obtain i) as follows |b (u,v,t)| ≤ ‖∇u‖l2(ω) ‖∇v‖l2(ω) + [ |δτ|‖c‖l2(ω) + 1 ] ‖u‖l2(ω) ‖v‖l2(ω) + |δ|‖u‖l2(ω) ‖u0‖l2(ω) ‖∇v‖l2(ω) ≤ m ‖u‖h1(ω) ‖v‖h1(ω) . ii) making use of −∆c + τc = 0 the expression of b (u,u,t) becomes b (u,u,t) = − ∫ (∇u)2 dx + ∫ (δτc + 1) u2dx− δ ∫ u∇u∇u0dx = − ∫ (∇u)2 dx + ∫ (δτc + 1) u2dx− δ 2 ∫ (∇u)2∇u0dx = ∫ ( −1 − δ 2 ∇u0 ) (∇u)2 dx + ∫ (δτc + 1) u2dx ≥‖∇u‖2l2(ω) . finally, by poincarre inequality yields, b (u,u,t) ≥ α‖u‖2h10 (ω) � 2.2.1. galerkin approximations. to demonstrate the existence of weak solution of the problem (p1) via the method of galerkin, we assume wk = wk (x) are smooth function verifying (2.3) {wk} ∞ k=1 is an arthogonal basis of h 1 0 (ω) and (2.4) {wk} ∞ k=1 is an arthonormal basis of l 2 (ω) . int. j. anal. appl. 19 (2) (2021) 198 consider a positive integer m. we will look for a function um : [0,t] → h10 (ω) of the form (2.5) um : = m∑ k=1 dkm (t) wk which satisfies (2.6) dkm (0) = (u0,wk) and (2.7) 〈 u ′ m,wk 〉 + b (um,wk, t) = (u0,wk) , 0 ≤ t ≤ t and k = 1, ...,m where u ′ = ut and here (., .) denotes the scalar product in l 2 (ω) . theorem 2.2. (construction of the approximate solution ) for each integer m, there exists a unique function um of the form equation 2.5 satisfying equation 2.6 and equation 2.7 . proof. assuming um has the structure equation 2.5. substituting equation 2.5 into equation 2.7 and using equation 2.4 we obtained (2.8) d′km (t) + ∑m l=1 d l mb (wl,wk, t) = d k m (0) , 0 ≤ t ≤ t and k = 1, ...,m according to standard existence theory for ordinary differential equations, there exists a unique absolutely continuous functions dm (t) = ( d1m,d 2 m, ...,d m m, ) satisfying equation 2.6 and equation 2.8. so um of the form equation 2.5 satisfies equation 2.6 and equation 2.7 for all t ∈ [0,t] . � 2.2.2. energy estimates. we propose now to send m to infinity and show a subsequence of our solutions um of the approximation problems equation 2.6 and equation 2.7 converges to a weak solution of (p1). for this we will need some uniform estimates. theorem 2.3. ( energy estimates ) [17]. there exists a constant c, depending only on ω, t and c, such that (2.9) max0≤t≤t ‖um‖l2(ω) + ‖um‖l2(0,t,h10 (ω)) + ‖u ′ m‖l2(0,t,h−1(ω)) ≤ c‖u0‖l2(ω) for m = 1, 2, ... proof. (1) multiplying equation 2.7 by dkm (t), summing for k = 1, ...,m, and then recalling equation 2.5 we find (2.10) (u′m,um) + b (um,um, t) = (u0,um) int. j. anal. appl. 19 (2) (2021) 199 for all 0 ≤ t ≤ t . from lemme 2.1, there exists constant α > 0 such that (2.11) α‖um‖ 2 h10 (ω) ≤ b (um,um, t) for all 0 ≤ t ≤ t, m = 1, ... furthermore |(u0,um)| ≤ 12 ‖u0‖ 2 l2(ω) + 1 2 ‖um‖ 2 l2(ω) , and (u ′ m,um) = d dt ( ‖um‖ 2 l2(ω) ) for a.e. 0 ≤ t ≤ t . consequently equation 2.10 yields the inequality (2.12) d dt ( ‖um‖ 2 l2(ω) ) + 2α‖um‖ 2 h10 (ω) ≤ c1 ‖um‖ 2 l2(ω) + c2 ‖u0‖ 2 l2(ω) for all 0 ≤ t ≤ t and appropriate constants c1 and c2. (2) now write (2.13) ϕ (t) : = ‖um‖ 2 l2(ω) and (2.14) ζ (t) : = ‖u0‖ 2 l2(ω) . then equation 2.12 implies (2.15) ϕ′ (t) ≤ c1ϕ (t) + c2ζ (t) for a.e. 0 ≤ t ≤ t. thus the differential form of gronwall’s inequality yields the estimate (2.16) ϕ (t) ≤ ec1t ( ϕ (0) + c2 ∫ t 0 ζ (s) ds ) (0 ≤ t ≤ t) . since ϕ (0) = ‖um (0)‖ 2 l2(ω) ≤ ‖u0‖ 2 l2(ω) by equation 2.6, we obtain from equations 2.13 2.16 the estimate (2.17) max 0≤t≤t ‖um‖l2(ω) ≤ c‖u0‖l2(ω) . (3) integrate inequality equation 2.12 from 0 to t and we employ the inequality equation 2.17 to find ‖um‖ 2 l2(0,t,h10 (ω)) = ∫ t 0 ‖um‖ 2 h10 (ω) dt ≤ c‖u0‖ 2 l2(ω) . int. j. anal. appl. 19 (2) (2021) 200 (4) fix any v ∈ h10 (ω), with ‖v‖ 2 h10 (ω) ≤ 1, and write v = v1 + v2, where v1 ∈ span (wk) k=m k=1 , and( v2,wk ) = 0 (k = 1, ...,m) . we use equation 2.7, we deduce for all 0 ≤ t ≤ t that ( u′m,v 1 ) + b ( um,v 1, t ) = ( u0,v 1 ) then equation 2.5 implies 〈u′m,v〉 = (u ′ m,v) = ( u′m,v 1 ) = ( u0,v 1 ) −b ( um,v 1, t ) , consequently |〈u′m,v〉| ≤ c ( ‖u0‖ 2 l2(ω) + ‖um‖h10 (ω) ) . simce ∥∥v1∥∥2 h10 (ω) ≤‖v‖2h10 (ω) ≤ 1. thus ‖u′m‖h−1(ω) ≤ c ( ‖u0‖ 2 l2(ω) + ‖um‖h10 (ω) ) , and therefore ‖u′m‖ 2 l2(0,t,h−1(ω)) = ∫ t 0 ‖u′m‖ 2 h−1(ω) dt ≤ c ∫ t 0 ( ‖u0‖ 2 l2(ω) + ‖um‖h10 (ω) ) dt ≤ c‖u0‖l2(ω) . � 2.2.3. existence and uniqueness. next we pass to limit as m → ∞, to build a weak solution of our initial boundary-value problem (p1) . theorem 2.4. (existence of weak solution). under hypothesis h2 and h3, there exists a weak solution of (p1) . proof. (1) according to the energy estimates equation 2.9, we see that the sequence {um} ∞ m=1 is bounded in l2 ( 0,t,h10 (ω) ) and {u′m} ∞ m=1is bounded in l 2 ( 0,t,h−1(ω) ) . consequently there exists a subsequence which is also noted by {um} ∞ m=1 and a function u ∈ l 2 ( 0,t,h10 (ω) ) , with u′ ∈ l2 ( 0,t,h−1(ω) ) , such that (2.18) um ⇀ u weakly in l 2 ( 0,t,h10 (ω) ) u′m ⇀ u ′ weakly in l2 ( 0,t,h−1(ω) ) . int. j. anal. appl. 19 (2) (2021) 201 (2) next fix an integer n and choose a function v ∈ c1 ( 0,t,h10 (ω) ) having the form (2.19) v (t) = n∑ k=1 dk (t) wk where { dk }n k=1 are given smooth functions. we choose m ≥ n, multiply equation 2.7 by dk (t) , sum for k = 1, ...,n, and then integrate with respect to t to find (2.20) ∫ t 0 〈u′m,v〉 + b (um,v,t) dt = ∫ t 0 (u0,v) dt. we recall equation 2.18 to find upon passing to weak limits that (2.21) ∫t 0 〈u′,v〉 + b (u,v,t) dt = ∫t 0 (u0,v) dt ∀v ∈ l2 ( 0,t,h10 (ω) ) . as functions of the from equation 2.19 are dense in l2 ( 0,t,h10 (ω) ) . hence in particular (2.22) 〈u′,v〉 + b (u,v,t) = (u0,v) ∀v ∈ h10 (ω) and∀t ∈ [0,t] , and from remark 1.1 we have u ∈ c ( 0,t,l2(ω) ) . (3) in order to prove u (0) = u0, we first note from equation 2.21 that (2.23) ∫ t 0 −〈u,v′〉 + b (u,v,t) dt = ∫ t 0 (u0,v) dt + (u (0) ,v (0)) for each v ∈ c1 ( 0,t,h10 (ω) ) with v (t) = 0. similary, from equation 2.20 we deduce (2.24) ∫ t 0 −〈um,v′〉 + b (um,v,t) dt = ∫ t 0 (u0,v) dt + (um (0) ,v (0)) . we use again equation 2.18, we obtain (2.25) ∫ t 0 −〈u,v′〉 + b (u,v,t) dt = ∫ t 0 (u0,v) dt + (u0,v (0)) , since um (0) → u0 in l2(ω). comparing equation 2.23 and equation 2.25, we conclude u (0) = u0. � theorem 2.5. (uniqueness of a weak solutions) a weak solution of (p1) is unique. int. j. anal. appl. 19 (2) (2021) 202 proof. we suppose there exists two weak solutions u1 and u2. we put u = u2 −u1 then u is also a solution of (p1) with u0 = (u2 −u1) (0) ≡ 0. setting v = u in identity equation 2.19 we have d dt ( 1 2 ‖u‖2l2(u) + b (u,u,t) ) = 0. from lemma 2.1 we have b (u,u,t) ≥ α‖u‖2h10 (u) ≥ 0, so d dt ( 1 2 ‖u‖2l2(u)) ≤ 0, then integrate with respect to t to find ‖u‖2l2(u) ≤‖u0‖ 2 l2(u) = 0, thus u ≡ 0. � 2.3. global solution of problem (p). our main results in this paper are stated as follows. theorem 2.6. i) if c > c0 > 0 and b (u,u,t) ≥ ∫ ω u0udx. then the solution (u,c) of problem (p) is global. ii) if c > c0 > 0 and b (u,u,t) ≥ ∫ ω u0udx. then the solution (u,c) of problem (p) is global. furthermore there exists σ > 0 such that ‖u‖l2(ω) ≤ e σt‖u0‖l2(ω) . proof. we put (2.26) z (t) = 1 2 ∫ ω u2dx we derivate the equation 2.26 and we use first equations of (p1) and (p2) to find i) we have dz dt = ∫ ω u0udx−b (u,u,t) ≤ 0 therefore z (t) ≤ z (0) . ii) we have dz dt = ∫ ω u0udx−b (u,u,t) = ∫ ω u0udx + ∫ ω ∇u2dx− ∫ ω (δτc + 1) u2dx + δ ∫ ω u∇u∇u0dx = ∫ ω u0udx + ∫ ω ( 1 + δ∇u0 2 ) ∇u2dx− ∫ ω (δτc + 1) u2dx ≤ |δτc0 + 1|‖u‖ 2 l2(ω) = σz (t) . this implies that int. j. anal. appl. 19 (2) (2021) 203 z (t) ≤ z (0) eσt. � proposition 2.2. [16] let u0 ∈ l2 (ω) and u ∈ c ( [0,t] ; l2 (ω) ∩l2 ( [0,t] ; h10 (ω) )) is the unique weak positive solution of (p1). if u0 ≥ 0 in ω, then u ≥ 0 in ]0,t[ × ω. proof. if u0 ≥ 0 on ∂ω. therefore u− = min (u, 0) ∈ l2 ( [0,t] ; h10 (ω) ) . we obtain for all 0 ≤ t ≤ t 1 2 d dt ∫ ω ( u− )2 dx + ∫ ω b ( u−,u−, t ) dx = ∫ ω u0u −dx using the lemma 2.1 and integrating with respect to t from 0 to t, we get 1 2 d dt ∫ ω ( u− )2 dx + α ∫ t 0 ‖u (s)‖2h10 (ω) ds ≤ 1 2 d dt ∫ ω ( u− (0) )2 dx = 0. since u− (0) = (u0) − = 0. so u− = 0. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h.a. barnes, thixotropy a review, j. non-newtonian fluid mech. 70 (1997), 1-33. [2] h. el-gendy, m. alcoutlabi, m. jemmett, m. deo, j. magda, r. venkatesan, a. montesi, the propagation of pressure in a gelled waxy oil pipeline as studied by particle imaging velocimetry, aiche j. 58 (2012), 302–311. [3] h. freundlich, thixotropy, paris, 1935. [4] c.f. goodeve, a general theory of thixotropy and viscosity, trans. faraday soc. 35 (1939), 342-358. [5] c.f. goodeve, g.w. whitfield, the measurement of thixotropy in absolute units. trans. faraday soc. 34 (1938), 511-520. [6] t. hillen, k. painter, global existence for a parabolic chemotaxis model with prevention of overcrowding, adv. appl. math. 26 (2001), 280–301. [7] c.mesikh, a. guesmia, and s.saadi, global existence and uniqueness of the weak solution in keller segel model. glob. j. sci. front. res. f, 14 (2014), 46-55. [8] f. moore, the rheology of cereamic slips and bodies. trans. br. ceramic soc. 58 (1959), 470–484 [9] a. mujumdar, a.n. beris, a.b. metzner, transient phenomena in thixotropic systems, j. non-newtonian fluid mech. 102 (2002), 157–178. [10] q.d. nguyen, d.v. boger, thixotropic behaviour of concentrated bauxite residue suspensions. rheol. acta, 24(4) (1985), 427-437. [11] j.o. sibree, the viscosity of emulsions. part i. trans. faraday soc. 26 (1930), 26-36. [12] j.o. sibree, the viscosity of emulsions. part ii. trans. faraday soc. 27 (1931), 161-176. [13] p.r. de souza mendes, modeling the thixotropic behavior of structured fluids, j. non-newtonian fluid mech. 164 (2009), 66–75. int. j. anal. appl. 19 (2) (2021) 204 [14] j. de vicente, c. berli, aging, rejuvenation, and thixotropy in yielding magneto-rheological fluids. rheol. acta, 52 (2013), 467–483. [15] x. zhang, w. li, x. gong, thixotropy of mr shear-thickening fluids. smart materials and structures, 19(12) (2010), 125012. [16] g. allaire, analyse numérique et optimisation. éd. de l’ecole polytechnique, paris, 2005. [17] l.c. evans, partial differential equations. graduate studies in mathematics, 2nd edn. american mathematical society, providence, ri, 2010. 1. introduction 2. existence of weak solution of the problem ( p) 2.1. existence of weak solution of the problem ( p2) 2.2. existence of weak solution of the problem ( p1) 2.3. global solution of problem ( p) references int. j. anal. appl. (2022), 20:6 on the uphill zagreb indices of graphs anwar saleh∗, sara bazhear and najat muthana department of mathematics, faculty of science, university of jeddah, jeddah, saudi arabia ∗corresponding author: asaleh1@uj.edu.sa abstract. one of the tools, to research and investigation the structural dependence of various properties and some activities of chemical structures and networks is the topological indices of graphs. in this research work, we introduce novel indices of graphs which they based on the uphill degree of the vertices termed as uphill zagreb topological indices. exact formulae of these new indices for some important and famous families of graphs are established. 1. introduction in this research, by graphs, we mean undirected finite simple graph. we denote g = (v,e) for a graph, where v is the set of vertices and e is the set of edges. for a vertex v ∈ v (g) the degree of v, d(v) is the number of edges incident with v. any terminology or notation which, we did not mention its definition, we refer the reader to [3]. topological indices have a widespread position specifically in pharmacology, chemistry, networks and many others. (see [8, 9, 13–15, 18, 24, 25]). almost of the indices of contemporary interesting in mathematical chemistry are introduced based on vertex degrees of the chemical graph. the two well-known topological indices of graphs are the zagreb indices that have been introduced by gutman and trinajstic by their work in [16], and described as m1 (g) = ∑ u∈v (g) (d (u)) 2 and m2 (g) =∑ uv∈e(g)d (u) d (v), respectively. the forgotten topological index was introduced by furtula and gutman [10] as f (g) = ∑ u∈v (g) (d (u)) 3 . zagreb indices were studied considerably due to their numerous applications inside the area of present chemical methods which want extra time and more charges. many new reformulated and prolonged versions of the zagreb indices have been delivered for several similar reasons (cf. [1,4,12,19,22,27–29]). received: nov. 14, 2021. 2010 mathematics subject classification. 05c35, 05c07, 05c40. key words and phrases. first uphill zegrib index; second uphill zegrib index; forgotten uphill zegrib index. https://doi.org/10.28924/2291-8639-20-2022-6 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-6 2 int. j. anal. appl. (2022), 20:6 the uphill domination and some related concepts are introduced and studied in ( [7,23]). in this paper motivated by the large applications of topological indices and the concept of uphill domination, we introduce novel indices of graphs based on a new degree (uphill degree) of the vertices termed as uphill zagreb topological indices. some properties and exact formulae of these new topological indices for some standard and famous families of graphs are established. 2. some results on the uphill zagreb indices of graphs definition 2.1. [7] for any graph g = (v,e). a path u −v is a sequence of vertices in g, initialing with u and terminal at v, such that sequential vertices are adjacent, and no vertex is repeated. a path p = v1,v2, ...vk+1 in g is an uphill path if for every i,1 ≤ i ≤ k, deg(vi ) ≤ deg(vi+1). for any vertices u and v in g, if there is an uphill path from u to v we say that u is uphill adjacent to v. definition 2.2. a vertex v is uphill dominates a vertex u in a graph g if v uphill adjacent to u. an uphill neighborhood of the vertex v is denoted by nup(v) and described as: nup(v) = {u : v uphill adjacent to u }. the uphill degree of the vertex v, denoted by dup(v), is the number of vertices which v uphill adjacent them, that means dup(v) = |nup(v)|. the uphill closed neighborhood, nup[v], of the vertex v is the uphill open neighborhood of v together with the vertex v. the maximum and minimum uphill degrees in the graph g denoted by ∆up(g) and δup(g), respectively. the vertex with uphill degree equal to zero is called uphill isolated vertex. in this paper by ex,y, we mean that ex,y = {uv ∈ e(g) : dup(u) = x and dup(v) = y}. definition 2.3. for any graph g = (v,e) the first uphill zagreb, second uphill zagreb, forgotten uphill zagreb index and modified first uphill zagreb are defined as: upm1(g) = ∑ v∈v (g) (dup(v)) 2, upm2(g) = ∑ vu∈e(g) dup(v)dup(u), upm∗1(g) = ∑ vu∈e(g) (dup(v) + dup(u)), and upf (g) = ∑ v∈v (g) (dup(v)) 3. lemma 2.1. suppose g be a graph, for any two vertices u and v, where u is uphill adjacent to v, if d(u) < d(v). then dup(u) > dup(v). int. j. anal. appl. (2022), 20:6 3 proof. if g is a graph, where u and v be any two vertices in g, where u is uphill adjacent to v. then there are two cases: case 1. when u is adjacent with v. without loss of generality, suppose that nup(v) = {v1,v2, ...,vr}, where r = dup(v). in this case, nup[v] ⊆ nup(u). hence, dup(u) > dup(v). case 2. when u not adjacent with v. then, there is an uphill path γ from u to v. clearly, dup(u) ≥ dup(v) + k + 1, where k is the number of internal vertices in γ. hence, dup(u) > dup(v). � definition 2.4. a graph g is called k-uphill regular graph if ∆up(g) = δup(g) = k. theorem 2.1. let g be any graph. then, the graph g is regular if and only if g is uphill regular. proof. if g is regular graph of n vertices. then it is straightforward, that g is n − 1 uphill regular graph. to prove the other direction of the theorem, we will prove that if g is not regular then g is not uphill regular. suppose that g is not regular. then, there exist at least two vertices say u and v such that d(u) < d(v). we have two cases: case 1. if u is uphill adjacent to v. then, by lemma 2.1, the graph g is not uphill regular. case 2. if u is not uphill adjacent to v. then, there exist at least two adjacent vertices in some paths from u to v say w and w1, where d(w) > d(w1). by lemma 2.1, we get dup(w1) > dup(w). hence, the graph g is not uphill regular. � proposition 2.1. for any graph g, upm2(g) = 0 if and only if for each edge e = uv, either u or v is uphill isolated vertex. proposition 2.2. for any graph g, ∑ v∈v (g)dup(v) ≤ τ(g), where τ(g) is the number of uphill paths in g. furthermore, the equality holds for the graph without cycle (acylic). proof. let g be any graph and let τ(g) be the number of uphill paths in g, for any vertex v in g it is obviously from the definition of uphill degree of the vertex that, dup(v) is less than or equal the 4 int. j. anal. appl. (2022), 20:6 number of paths in g which originated from v. also if we denote by γv as the number of uphill paths in g originated from v, then τ(g) = ∑ v∈v (g) γv. therefore,∑ v∈v (g) dup(v) ≤ ∑ v∈v (g) γv = τ(g). furthermore, it is easy to check that if g is acyclic, then dup(v) = γv. hence the equality holds. � proposition 2.3. let g ∼= pn, be a path of n ≥ 3 vertices. then, i. upm1(g) = n3 − 6n2 + 13n− 10, ii. upm2(g) = n3 − 7n2 + 17n− 15, iii. upm∗1(g) = 2n 2 − 8n + 8, iv. upf (g) = n4 − 9n3 + 33n2 − 57n + 38. proof. suppose g ∼= pn be a path of n ≥ 3 vertices. then there are two vertices with uphill degree n− 2 and n− 2 vertices of uphill degree n− 3. so by using the definition of first uphill index we get, upm1(g) = 2(n− 2)2 + (n− 2)(n− 3)2 = n3 − 6n2 + 13n− 10. similarly, upf (g) = 2(n− 2)3 + (n− 2)(n− 3)3 = n4 − 9n3 + 33n2 − 57n + 38. there are two edges of the type en−2,n−3 and n− 3 edges of the type en−3,n−3. then, upm2(g) = 2[(n− 2)(n− 3)] + (n− 3)3 = n3 − 7n2 + 17n− 15. in the same way we get upm∗1(g) = 2(2n− 5) + (n− 3)(2n− 6) = 2n2 − 8n + 8. � proposition 2.4. let g = (v,e) be a regular graph of degree k and has n vertices. then, i. upm1(g) = n(n− 1)2, ii. upm2(g) = nk(n−1)2 2 , iii. upm∗1(g) = nk(n− 1), iv. upf (g) = n(n− 1)3. proof. let g = (v,e) be k− regular graph with n vertices. then it isn’t difficult to see that between any two vertices of g there exists an uphill path, so for any v ∈ v (g), we have dup(v) = n−1. hence, upm1(g) = n(n−1)2, upm2(g) = nk(n−1)2 2 , upm∗1(g) = nk(n−1) and upf (g) = n(n−1) 3. � int. j. anal. appl. (2022), 20:6 5 corollary 2.1. for any complete graph kn, we have i. upm1(kn) = upm∗1(g) = n(n− 1) 2, ii. upm2(kn) = n(n−1)3 2 , iii. upf (kn) = n(n− 1)3. corollary 2.2. for any cycle cn, where n ≥ 3, i. upm1(cn) = upm2(cn) = n(n− 1)2, ii. upm∗1(cn) = 2n(n− 1), iii. upf (cn) = n(n− 1)3. by a graph wn, we mean a wheel graph of n + 1 vertices. proposition 2.5. let g ∼= wn, where n ≥ 3 be a wheel graph. then, i. upm1(g) = upm2(g) = n3, ii. upm∗1(g) = 2n 2, iii. upf (g) = n4. proof. let g ∼= wn, where n ≥ 3 be a wheel graph. the graph g has one vertex of uphill degree zero and n vertices of uphill degree n. then, upm1(g) = n(n) 2 + 0 = n3. in the same way, we get the forgotten uphill index upf (g) = n(n)3 + 0 = n4. there are n edges of the type en,n and n edges of the type en,0. then, upm∗1(g) = 2n 2 and upm2(g) = n(n) 2 = n3. � a tadpole graph tm,n is constructed by joining between cm and pn by a bridge [17]. proposition 2.6. let g ∼= tm,n, where m,n ≥ 3 be a tadpole graph of m + n vertices. then, i. upm1(g) = m3 − 3m2 + 3m− 2 + n3 − 2n2 + 3n, ii. upm2(g) = m3 − 4m2 + 5m− 4 + n3 − 3n2 + 4n, 6 int. j. anal. appl. (2022), 20:6 iii. upm∗1(g) = 2m 2 − 4m + 2n2 − 5n + 5, iv. upf (g) = m4 − 4m3 + 6m2 − 4m + 2 + n4 − 3n3 + 6n2 − 4n. proof. let g ∼= tm,n, where m,n ≥ 3 be a tadpole graph of m + n vertices. there are m− 1 vertices of uphill degree m− 1, one vertex of uphill degree zero, n− 1 vertices of uphill degree n− 1 and one vertex of uphill degree n. then, we get upm1(g) = (m− 1)3 + (n− 1)3 + (n)2 = m3 − 3m2 + 3m− 2 + n3 − 2n2 + 3n. similarly, upf (g) = (m− 1)4 + (n− 1)4 + (n)3 = m4 − 4m3 + 6m2 − 4m + 2 + n4 − 3n3 + 6n2 − 4n. type number of edges em−1,0 2 em−1,m−1 m− 2 e0,n−1 1 en−1,n−1 n− 2 en−1,n 1 table 1. edge partition of tadpole graph based on uphill degree of end vertices. now, by using the partition given in table1, we get upm2(g) = (m− 1)2(m− 2) + (n− 1)2(n− 2) + n(n− 1) = m3 − 4m2 + 5m− 4 + n3 − 3n2 + 4n. also, upm∗1(g) = 2 (m− 1) + (m− 2) (2m− 2) + (n− 1) + (2n− 2) (n− 2) = 2m2 − 4m + 2n2 − 5n + 5. � the graph which obtained from a wheel graph with extra vertex between each pair of adjacent vertices of the outer cycle is called gear graph gn [17]. proposition 2.7. let g ∼= gn, where n ≥ 4 be a gear graph. then, i. upm1(g) = 10n, int. j. anal. appl. (2022), 20:6 7 ii. upm2(g) = 6n, iii. upm∗1(g) = 9n, iv. upf (g) = 28n. proof. let g ∼= gn, where n ≥ 4 be a gear graph. then, there are n vertices of uphill degree one, one vertex of uphill degree zero and n vertices of uphill degree three. clearly, we get upm1(g) = n + 9n = 10n. in the same way, we get upf (g) = n + 27n = 28n. there are n edges of the type e1,0 and 2n edges of the type e1,3. then, upm∗1(g) = 9n, upm2(g) = 6n. � the windmill graph wd(s,k), where s,k ≥ 2, is a graph of k copies of complete graph ks at a shared common vertex [17]. proposition 2.8. let g ∼= wd(s,k), where s ≥ 3 and k ≥ 2 be a windmill graph of k(s − 1) + 1 vertices. then, i. upm1(g) = k(s − 1)3, ii. upm2(g) = k(s − 1)3(s−22 ), iii. upm∗1(g) = k (s − 1) 3 , iv. upf (g) = k(s − 1)4. proof. let g ∼= wd(s,k), where s ≥ 3 and k ≥ 2 be a windmill graph of k(s − 1) + 1 vertices. the graph g has one vertex of uphill degree zero and k(s − 1) vertices of uphill degree s − 1. so, upm1(g) = k(s − 1)(s − 1)2 = k(s − 1)3. similarly, upf (g) = k(s − 1)(s − 1)3 = k(s − 1)4. there are sk(s−1) 2 −k(s−1) edges of the type es−1,s−1 and k(s−1) edges of the type es−1,0 . then, upm2(g) = (s − 1)2[ sk(s − 1) 2 −k(s − 1)] 8 int. j. anal. appl. (2022), 20:6 = k(s − 1)3[ s − 2 2 ]. also, upm∗1(g) = k (s − 1) (s − 2) 2 (2s − 2) + k (s − 1) (s − 1) = (s − 1) (k (s − 1) (s − 2) + k (s − 1)) = k (s − 1)3 . � the graph which is obtained from wn by adding an end edge to each outer vertex of wn, is called helm graph and denoted by hn [17]. proposition 2.9. let g ∼= hn, where n ≥ 3 be a helm graph of 2n + 1 vertices. then, i. upm1(g) = 2n3 + 2n2 + n, ii. upm2(g) = 2n3 + n2, iii. upm∗2(g) = 5n 2 + n, iv. upf (g) = 2n4 + 3n3 + 3n2 + n. proof. let g ∼= hn, where n ≥ 3 be a helm graph. the graph g has only one vertex of uphill degree zero, n vertices of uphill degree n + 1 and n vertices of uphill degree n. then, upm1(g) = n(n + 1) 2 + n(n)2 = 2n3 + 2n2 + n. similarly, upf (g) = n(n + 1)3 + n(n)3 = 2n4 + 3n3 + 3n2 + n. there are n edges of the type en+1,n, n edges of the type en,n and n edges of the type en,0. then, upm2(g) = n 2(n + 1) + n3 = 2n3 + n2. similarly, upm∗1(g) = n(2n + 1) + 2n(n) + n 2 = 5n2 + n. � the double star graph sr,t which obtained from complete graph of two vertices by joining r pendent edges to one vertex and t pendent edges to the other vertex of the complete graph k2 [17]. theorem 2.2. let g ∼= sr,t, where r,t ≥ 2 be a double star graph. then, int. j. anal. appl. (2022), 20:6 9 i. upm1(g) =  8r + 2 if r = t; 4r + t + 1 if r < t. ii. upm2(g) =  4r + 1 if r = t; 2r if r < t. iii. upm∗1(g) =  6r + 2 if r = t; 3r + t + 1 if r < t. iv. upf (g) =  16r + 2 if r = t; 8r + t + 1 if r < t. proof. let g ∼= sr,t, where r,t ≥ 2 be a double star with r + t + 2 vertices. then, i. there are two cases: case 1. if r = t, it has two vertices of uphill degree one and there are 2r vertices of uphill degree 2, then upm1(g) = 8r + 2. case 2. if r < t, there are r vertices of uphill degree 2 and s + 1 vertices of uphill degree one, also one uphill isolated vertex, then upm1(g) = 4r + t + 1. ii. we have two cases: case 1. if r = t, it has one edge of the type e1,1 and there are 2r edges in g, where each edge of the type e1,2,then upm2(g) = 4r + 1. case 2. when r < t, there are t + 1 edges where each edge of the type e1,0. also, there are r edges where each edge of the type e2,1,then upm2(g) = 2r. iii. similarly as in part ii, if r = t, then upm∗1(g) = 6r + 2. if r < t, then upm ∗ 1(g) = 3r + t + 1. iv. the proof is similarly to part i. � the subdivision graph s(g) of the graph g is a graph obtained from g by replacing each of its edge by a path of length 2. by simple calculation, we get the uphill zagreb indices for the subdivision graphs of path, cycle and complete graph. proposition 2.10. let g ∼= s(pn), where n ≥ 3. then, i. upm1(g) = 8n3 − 36n2 + 56n− 30, ii. upm2(g) = 8n3 − 40n2 + 68n− 40, iii. upm∗1(g) = 8n 2 − 24n + 18, iv. upf (g) = 16n4 − 104n3 + 264n2 − 308n + 138. proposition 2.11. let g ∼= s(cn), where n ≥ 3. then, i. upm1(g) = upm2(g) = 2n(2n− 1)2, ii. upm∗1(g) = 2n(4n− 2), 10 int. j. anal. appl. (2022), 20:6 iii. upf (g) = 2n(2n− 1)3. proposition 2.12. let g ∼= s(kn), where n ≥ 4. then, i. upm1(g) = 2n(n− 1), ii. upm2(g) = 0, iii. upm∗1(g) = 2n(n− 1), iv. upf (g) = 4n(n− 1). for each p ≥ 0, the p-sun tree, denoted by sup, is the tree of order n = 2p + 1 formed by taking the star on p + 1 vertices and subdividing each edge. for p,q ≥ 0, the (p,q)-double sun, denoted by dsup,q, is the tree of order n = 2(p + q + 1) obtained by connecting the centers of dsup and dsuq with an edge [11]. proposition 2.13. let g ∼= sup, where p ≥ 3 be a sun graph of 2p + 1 vertices. then, i. upm1(g) = 5p, ii. upm2(g) = 2p, iii. upm∗1(g) = 4p, iv. upf (g) = 9p. proposition 2.14. let g ∼= sup,q, where p,q ≥ 2 be a double sun graph of 2(p + q + 1) vertices. then, i. upm1(g) =  26p + 2 if p = q; 13p + 5q + 1 if p < q. ii. upm2(g) =  16p + 1 if p = q; 8p + 3q + 1 if p < q. iii. upm∗1(g) =  16p + 2 if p = q; 8p + 3q + 1 if p < q. iv. upf (g) =  70p + 2 if p = q; 35p + 9q + 1 if p < q. the central graph of a graph g is obtained by subdividing each edge of g exactly once and joining all the non-adjacent vertices of g and denoted by c(g) [2]. proposition 2.15. let g ∼= c(pn), where n ≥ 4 be a central graph of a path with 2n − 1 vertices. then, i. upm1(g) = n(n− 1)(2n− 1), int. j. anal. appl. (2022), 20:6 11 ii. upm2(g) = n4−n3+n2−3n+2 2 , iii. upm∗1(g) = n 3 −n, iv. upf (g) = n(n− 1)((n− 1)2 + n2). proof. from the definition of the central graph of a graph obviously in c(pn) there are n vertices of uphill degree n− 1 and n− 1 vertices of uphill degree n. so, upm1(g) = n(n− 1)(2n− 1). then clearly, upf (g) = n(n− 1)((n− 1)2 + n2). type number of edges en−1,n 2(n− 1) en−1,n−1 2(n−2)+(n−2)(n−3) 2 table 2. edge partition of c(pn) graph based on uphill degree of end vertices. now, by using the partition in table 2, we get upm2(g) = 2n (n− 1)2 + (n− 1)3 (n− 2) 2 = n4 −n3 + n2 − 3n + 2 2 . by using the same partition in table 2, we get upm∗1(g) = 2 (2n− 1) (n− 1) + (2n− 2) 2 (n− 2) + (n− 2) (n− 3) 2 = n3 −n. � proposition 2.16. let g ∼= c(cn), where n ≥ 5 be a central graph of a cycle with 2n vertices. then, i. upm1(g) = n(n− 1)2 + n3, ii. upm2(g) = 2n2(n− 1) + n(n−3)(n−1)2 2 , iii. upm∗1(g) = n 3 + n, iv. upf (g) = n(n− 1)3 + n4. proposition 2.17. for any graph g ∼= c(kn), i. upm1(g) = upm∗1(g) = upf(g) 2 , 12 int. j. anal. appl. (2022), 20:6 ii. upm2(g) = 0. proof. clearly, there are n(n−1) 2 vertices of uphill degree 2 and all other vertices of uphill degree zero. also there are n(n− 1) edges of type e2,0. then we have upm1 = 2n(n− 1), upf (g) = 4n(n− 1), upm∗1(g) = 2n(n− 1), upm2(g) = 0. � an (r,s) banana tree denoted by br,s, is a graph obtained by connecting one leaf of each of r copies of an star graph of s vertices with a single root vertex that is distinct from all the stars [6]. theorem 2.3. let g ∼= br,s, where s ≥ 4 be a banana tree graph with rs + 1 vertices. then, i. upm1(g) =  2s + 44 if r = 2; r(s + 2) if r ≥ 3. ii. upm2(g) =  32 if r = 2; 0 if r ≥ 3. iii. upm∗1(g) =  rs − 2r + 24 if r = 2; rs + 2r if r ≥ 3. iv. upf (g) =  2s + 188 if r = 2; 8r + s − 2 if r ≥ 3. proof. let g ∼= br,s, where s ≥ 4 be a banana tree graph with rs + 1 vertices. then, i. we have two cases: case 1. if r = 2, the graph g has two vertices of uphill degree zero, three vertices of uphill degree four and 2(s − 2) vertices of uphill degree one, then upm1(g) = 2s + 44. case 2. if r ≥ 3, there are r + 1 vertices of uphill degree zero, r vertices of uphill degree two and r(s − 2) vertices of uphill degree one, then upm1(g) = r(s + 2). ii. we have two cases: case 1. if r = 2, it has two edges of the type e4,4, two edges of the type e0,4 and there are r(s − 2) edges in g, where each edge of the type e1,0, then upm2(g) = 32. case 2. if r ≥ 3, there are 2r edges where each edge of the type e0,2. also, there are r(s − 2) edges where each edge of the type e1,0, then upm2(g) = 0. iii. as part ii, we get upm∗1(g) = rs + 2r if r ≥ 3 and if r = 2, upm ∗ 1(g) = rs − 2r + 24 . iv. in the same way as part i. int. j. anal. appl. (2022), 20:6 13 � a firecracker graph fr,s is a graph obtained by the concatenation of n stars, each consists of s vertices, by linking one leaf from each star [6]. theorem 2.4. let g ∼= fr,s , where s ≥ 5 be a firecracker graph with rs vertices. then, i. upm1(g) =  2s + 14 if r = 2; 2(2r − 3)2 + (r − 2)(2r − 5)2 + r(s − 2) if r ≥ 3. ii. upm2(g) =  9 if r = 2; 2(2r − 3)(2r − 5) + (r − 3)(2r − 5)2 if r ≥ 3. iii. upm∗1(g) =  2s + 8 if r = 2; 6r2 + rs − 21r + 18 if r ≥ 3. iv. upf (g) =  50 + 2s if r = 2; 2(2r − 3)3 + (r − 2)(2r − 5)3 + r(s − 2) if r ≥ 3. proof. let g ∼= fr,s , where s ≥ 5 be a firecracker graph with rs vertices. then, i. there are two cases: case 1. if r = 2, the graph g has two vertices of uphill degree zero, two vertices of uphill degree three and 2(s − 2) vertices of uphill degree one, then upm1(g) = 2s + 14. case 2. if r ≥ 3, there are two vertices of uphill degree 2r − 3, r − 2 vertices of uphill degree 2r−5,r vertices of uphill degree zero and r(s−2) vertices of uphill degree one, then upm1(g) = 2(2r − 3)2 + (r − 2)(2r − 5)2 + r(s − 2). ii. there are two cases: case 1. if r = 2, the graph g has 2(s − 2) edges of type e1,0, two edges of the type e0,3 and one edge of type e3,3. so, upm2(g) = 9. case 2. if r ≥ 3, in this case, there are r(s−2) of type e1,0, r −2 edges of the type e0,2r−5, two edges of the type e0,2r−3, two edges of the type e2r−3,2r−5 and r − 3 edges of type e2r−5,2r−5, then upm2(g) = 2(2r − 3)(2r − 5) + (r − 3)(2r − 5)2. iii. as the method in ii. iv. as the method in i. � book graph is a cartesian product of a star and single edge, denoted by br. the r-book graph is defined as the graph cartesian product sr+1 × p2 , where sr+1 is a star graph and p2 is the path graph. the stacked book graph of order (r,t) is defined as the graph cartesian product sr+1 ×pt, where sr is a star graph and pt is the path graph on t nodes, and it is denoted by br,t [17]. proposition 2.18. let g ∼= br , where r ≥ 2 be a book graph of 2(r + 1) vertices. then, 14 int. j. anal. appl. (2022), 20:6 i. upm1(g) = 2(9r + 1), ii. upm2(g) = 15r + 1, iii. upm∗1(g) = 14r + 2, iv. upf (g) = 2(27r + 1). proof. let g ∼= br , where r ≥ 2 be a book graph of 2(r + 1) vertices. there are two vertices of uphill degree one and 2r vertices of uphill degree three. then, upm1(g) = 2(9r + 1), in the same way, upf (g) = 2(27r + 1). the graph g has three kinds of edges, one edge of the type e1,1 , 2r edges of the type e3,1 and r edges of the type e3,3. then, upm2(g) = 15r + 1. also, upm∗1(g) = 14r + 2. � theorem 2.5. let g ∼= br,t , where r ≥ 2 and t ≥ 3 be a stacked book graph with t(r + 1) vertices. then, i. upm1(g) = 2r(2t − 3)2 + (t − 2)[(t − 3)2 + 2(t − 2) + r(2t − 5)2], ii. upm2(g) = 2r(2t − 3)(3t − 7) + (t − 3)[2(t − 2) + (t − 3)2 + r(2t − 5)2 + r(t − 2)(2t − 5)], iii. upm∗1(g) = −22rt + 20r + 2t 2 − 8t + 7rt2 + 8, iv. upf (g) = 2r(2t − 3)3 + (t − 2)[(t − 3)3 + 2(t − 2) + r(2t − 5)3]. proof. let g ∼= br,t , where r ≥ 2 and t ≥ 3, be a stacked book graph with t(r + 1) vertices. in figure 1, we can see the graph g has 2r vertices are labeled by (v1,1,v2,1, ...,vr,1) and (v1,t,v2,t, ...,vr,t) of uphill degree 2t − 3, r(t − 2) vertices are labeled by (v1,2,v1,3, ...,v1,t−1), (v2,2,v2,3, ...,v2,t−1), ..., (vr,2,vr,3, ...,vr,t−1) of uphill degree 2t−5, two vertices are labeled by (v0,1) and (v0,t) of uphill degree t−2, t−2 vertices are labeled by (v0,2,v0,3, ...,v0,t−1) of uphill degree t − 3. then, upm1(g) = 2r(2t − 3)2 + (t − 2)[(t − 3)2 + 2(t − 2) + r(2t − 5)2]. similarly, upf (g) = 2r(2t − 3)3 + (t − 2)[(t − 3)3 + 2(t − 2) + r(2t − 5)3]. there are 6 types of edges. int. j. anal. appl. (2022), 20:6 15 type number of edges e2t−3,2t−5 2r e2t−3,t−2 2r et−2,t−3 2 et−3,t−3 t − 3 e2t−5,2t−5 r(t − 3) et−3,2t−5 r(t − 2) table 3. edge partition of br,t graph based on uphill degree of end vertices. in figure 1, the types of edges, e2t−3,2t−5,e2t−3,t−2,et−2,t−3, et−3,t−3,e2t−5,2t−5 and et−3,2t−5 are colored by green, purple, red, yellow, blue and black, respectively. figure 1. stacked book graph br,t now, by using the partition in table 3, we get upm2(g) = 2r(2t − 3)(3t − 7) + (t − 3)[2(t − 2) + (t − 3)2 + r(2t − 5)2 + r(t − 2)(2t − 5)]. also, upm∗1(g) = −22rt + 20r + 2t 2 − 8t + 7rt2 + 8. 16 int. j. anal. appl. (2022), 20:6 � a firefly graph fa,b,c is a graph of n = 2a + 2b + c + 1 vertices that consists of c pendant edges, a triangles, and b pendant paths of length 2, all of them sharing a common vertex [5]. proposition 2.19. let g ∼= fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. then, i. upm1(g) = 8a + 5b + c, ii. upm2(g) = 2(2a + b), iii. upm∗1(g) = 8a + 4b + c, iv. upf (g) = 16a + 9b + c. proof. let g ∼= fa,b,c be the firefly graph with 2a + 2b + c + 1 vertices. in figure 2, we can see the graph g has one uphill isolated vertex, 2a vertices are labeled by (v1,v2, ...,v2a) of uphill degree two, b vertices are labeled by (u1,u2, ...ub) of uphill degree one, b vertices are labeled by (u′1,u ′ 2, ...,u ′ b) of uphill degree two and c vertices are labeled by (w1,w2, ...wc) of uphill degree one. then, upm1(g) = 8a + 5b + c. similarly, upf (g) = 16a + 9b + c. there are 4 types of edges. type number of edges e2,2 a e2,0 2a e1,0 b + c e2,1 b table 4. edge partition of fa,b,c graph based on uphill degree of end vertices. in figure 2, the types of edges, e2,2,e2,0,e1,0 and e2,1 are colored by blue, red, black blue and green, respectively. int. j. anal. appl. (2022), 20:6 17 figure 2. firefly graph fa,b,c now, by using the edge partition in table 4, we get upm2(g) = 2(2a + b), and upm∗1(g) = 8a + 4b + c. � corollary 2.3. let g ∼= bf, be the butterfly graph with 2a + c + 1 vertices. then, i. upm1(g) = upm∗1(g) = 8a + c, ii. upm2(g) = 4a, iii. upf (g) = 16a + c. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. alwardi, a. alqesmah, r. rangarajan, i.n. cangul, entire zagreb indices of graphs, discrete math. algorithm. appl. 10 (2018), 1850037. https://doi.org/10.1142/s1793830918500374. [2] j.a. aruldoss and g. gurulakshmi, the dominator coloring of central and middle graph of some special graphs, int. j. math. appl. 4 (2016), 67–73. [3] j. a. bondy and u. s. r. murty, graph theory, springer, berlin, 2008. [4] j. braun, a. kerber, m. meringer, c. rucker, similarity of molecular descriptors: the equivalence of zagreb indices and walk counts, match commun. math. comput. chem. 54 (2005), 163-176. [5] a.e. brondani, c.s. oliveira, f.a.m. frança, l. de lima, aα-spectrum of a firefly graph, electron. notes theor. computer sci. 346 (2019), 209–219. https://doi.org/10.1016/j.entcs.2019.08.019. https://doi.org/10.1142/s1793830918500374 https://doi.org/10.1016/j.entcs.2019.08.019 18 int. j. anal. appl. (2022), 20:6 [6] w.c. chen, h.i. lu, y.n. yeh, operations of interlaced trees and graceful trees, southeast asian bull. math. 21 (1997), 337-348. [7] j. deering, uphill and downhill domination in graphs and related graph parameters, thesis, east tennessee state university, johnson, (2013). [8] m.v. diudea, nanomolecules and nanostructures-poynomial and indices, univ. kragujevac, 2010. [9] m.v. diudea, m.s. florescu, p.v. khadikar, molecular topology and its applications, eficon, bucarest, 2006. [10] b. furtula, i. gutman, a forgotten topological index, j math chem. 53 (2015), 1184–1190. https://doi.org/ 10.1007/s10910-015-0480-z. [11] w. gao, the randić energy of generalized double sun, czech. math. j. (2021), 1–28. https://doi.org/10. 21136/cmj.2021.0463-20. [12] i. gutman, k. c. das, the first zagreb index 30 years after, match commun. math. comput. chem. 50 (2004), 83-92. [13] i. gutman, b. furula (eds.), novel molecular structure descriptors-theory and applications i, univ. kragujevac, kragujevac, 2010 . [14] i. gutman, b. furula (eds.), novel molecular structure descriptors-theory and applications ii, univ. kragujevac, kragujevac, 2010. [15] i. gutman, b. furula (eds.), distance in molecular graphs, univ. kragujevac, kragujevac, 2012. [16] i. gutman, n. trinajstić, graph theory and molecular orbitals. total φ-electron energy of alternant hydrocarbons, chem. phys. lett. 17 (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1. [17] f. harary, graph theory, addison-wesley, reading mass, 1969. [18] m. karelson, molecular descriptors in qsar-qspr, wiley, new york, 2000. [19] m.h. khalifeh, h. yousefi-azari, a.r. ashrafi, the first and second zagreb indices of some graph operations, discr. appl. math. 157 (2009), 804–811. https://doi.org/10.1016/j.dam.2008.06.015. [20] j. kok, n.k. sudev, u. mary, on chromatic zagreb indices of certain graphs, discrete math. algorithm. appl. 9 (2017), 1750014. https://doi.org/10.1142/s1793830917500148. [21] s. nikolić, g. kovačević, a. miličević, n. trinajstić, the zagreb indices 30 years after, croat. chem. acta, 76 (2003), 113-124. [22] a. saleh, a. aqeel and i. n. cangul, on the entire abc index of graphs, proc. jangjeon math. soc. 23 (2020), 39-51. [23] a. saleh, n. muthana, w. al-shammakh, h. alashwali, monotone chromatic number of graphs, int. j. anal. appl. 18 (2020), 1108-1122. https://doi.org/10.28924/2291-8639-18-2020-1108. [24] r. todeschini, v. consonni, handbook of molecular descriptors, wiley-vch, weinheim, 2000. [25] r. todeschini, v. consonni, molecular descriptors for chemoinformatics, wiley-vch, weinheim, 2009. [26] s. wang, b. wei, multiplicative zagreb indices of cacti, discrete math. algorithm. appl. 8 (2016), 1650040. https://doi.org/10.1142/s1793830916500403. [27] b. zhou, zagreb indices, match commun. math. comput. chem. 52 (2004), 113-118. [28] b. zhou, i. gutman, relations between wiener, hyper-wiener and zagreb indices, chem. phys. lett. 394 (2004), 93–95. https://doi.org/10.1016/j.cplett.2004.06.117. [29] b. zhou, i. gutman, further properties of zagreb indices, match commun. math. comput. chem. 54 (2005), 233-239. https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.1007/s10910-015-0480-z https://doi.org/10.21136/cmj.2021.0463-20 https://doi.org/10.21136/cmj.2021.0463-20 https://doi.org/10.1016/0009-2614(72)85099-1 https://doi.org/10.1016/j.dam.2008.06.015 https://doi.org/10.1142/s1793830917500148 https://doi.org/10.28924/2291-8639-18-2020-1108 https://doi.org/10.1142/s1793830916500403. https://doi.org/10.1016/j.cplett.2004.06.117 1. introduction 2. some results on the uphill zagreb indices of graphs references international journal of analysis and applications volume 19, number 4 (2021), 512-517 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-512 on firmly non-expansive mappings joseph frank gordon1,∗, esther opoku gyasi2 1department of mathematics education, akenten appiah-menka university of skills training and entrepreneurial development, kumasi, ghana 2department of mathematical sciences, university of mines and technology, tarkwa, ghana ∗corresponding author: josephfrankgordon@gmail.com abstract. in this paper, it is shown that for a closed convex subset c and to every non-expansive mapping t : c → c, one can associate a firmly non-expansive mapping with the same fixed point set as t in a given banach space. 1. introduction the study of non-expansive mappings in the sixties have experimented a boost, basically motivated by browder’s work on the relationship between monotone operators, non-expansive mappings [1–3, 3–5] and the seminal paper by kirk [6], where the significance of the geometric properties of the norm for the existence of fixed points for non-expansive mappings was highlighted. now the history of firmly non-expansive mappings goes back to the paper by minty [7], where he implicitly used this class of mappings to study the resolvent of a monotone operator. browder [3] first introduced firmly non-expansive mappings in the concept of hilbert spaces h. that is, given a c closed convex subset of a hilbert space h, a mapping f : c →h is firmly non-expansive if for all x,y ∈c ‖fx−fy‖2 ≤〈x−y,fx−fy〉.(1.1) received march 13th, 2021; accepted april 20th, 2021; published may 11th, 2021. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. non-expansive mappings; firmly non-expansive mappings; metric projection; fixed points. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 512 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-512 int. j. anal. appl. 19 (4) (2021) 513 in his study of non-expansive projections on subsets of banach spaces, bruck [8] defined a firmly nonexpansive mapping f : c →x , where c is a closed convex subset of a real banach space x , to be a mapping such that for all x,y ∈c and α ≥ 0, ‖fx−fy‖≤‖α(x−y) + (1 −α)(fx−fy)‖.(1.2) it is clear that equation (1.2) reduces to equation (1.1) when in hilbert spaces and also when α = 1, f becomes a non-expansive mappings, that is, for each x and y in c, we have ‖fx−fy‖≤‖x−y‖. a trivial example of equation (1.2) is the identity mapping. a non-trivial example of equation (1.2) in a hilbert space is given by the metric projection pcx = argminy∈c{‖x−y‖}.(1.3) to see this, recall that in a real hilbert space h, ∀x,y ∈h, then 〈x,y〉≥ 0 if and only if ‖x‖≤‖x + ay‖,(1.4) for all a ≥ 0. now equation (1.2) can be written as ‖fx−fy‖≤‖fx−fy + α(x−y −fx + fy)‖.(1.5) now applying equation (1.4) on equation (1.5), we obtain the following 〈fx−fy,x−y −fx + fy〉≥ 0, 〈x−y −fx + fy,fx−fy〉≥ 0, 〈x−y − (fx−fy),fx−fy〉≥ 0, 〈x−y,fx−fy〉−〈fx−fy,fx−fy〉≥ 0, 〈x−y,fx−fy〉≥ 〈fx−fy,fx−fy〉, 〈x−y,fx−fy〉≥ ‖fx−fy‖2. hence we have that in a real hilbert space, a firmly non-expansive mapping f can be written as ‖fx−fy‖2 ≤〈x−y,fx−fy〉.(1.6) but in a real hilbert space, equation (1.3) satisfies the following inequality ‖pcx−pcy‖2 ≤〈x−y,pcx−pcy〉.(1.7) this means that from equations (1.6) and (1.7), we can simply conclude that f = pc and so the metric projection pc is a firmly non-expansive mapping in a real hilbert space. int. j. anal. appl. 19 (4) (2021) 514 in this paper, we give a simple proof showing that to any non-expansive self-mapping t : c → c that has fixed points, one can associate a large family of firmly non-expansive mappings having the same fixed point set as t. that is, from the point of view of the existence of fixed points on closed convex sets, non-expansive and firmly non-expansive mappings exhibit a similar behavior. however, this is no longer true in non-convex domains [9]. 2. main results let t be a non-expansive mapping defined on a closed convex subset c of a normed space x , thus, t : c →c. for a fixed r ∈ r>1, we can define the following mapping tr : c →c by x 7→ ( 1 − 1 r ) x + 1 r t(trx).(2.1) now we observe that equation (2.1) (the new mapping tr) always exist. to see this, one can create an internal contraction f : c →c such that f(y) = ( 1 − 1 r ) x + 1 r ty, where x is fixed. now ‖f(y) − f(z)‖ = 1 r ‖ty − tz‖ ≤ 1 r ‖y − z‖. hence f is a contraction mapping and by the banach contraction mapping theorem [10], there exists u ∈c such that f(u) = u, thus, u = (1 − 1 r )x + 1 r tu. since for every x ∈c, we can find a unique u such that u = trx, then equation (2.1) always exists. now we have the following claims. claim 1: tr is a non-expansive mapping. to see this, we have the following: ‖trx−try‖ = ‖(1 − 1 r )(x−y) + 1 r (t(trx) −t(try))‖, ≤ (1 − 1 r )‖x−y‖ + 1 r ‖t(trx−t(try))‖, ≤ (1 − 1 r )‖x−y‖ + 1 r ‖trx−try)‖, ‖trx−try‖− 1 r ‖trx−try‖≤ (1 − 1 r )‖x−y‖, (1 − 1 r )‖trx−try‖≤‖(1 − 1 r )‖‖x−y‖. so we have that tr is a non-expansive mapping since r > 1. claim 2: now we prove that trx is a firmly non-expansive mapping. now for r > 1,α ∈ (0, 1) and β > 0, we have the following evaluation: ‖trx−try‖ = ‖β[α(x−y) + (1 −α)(trx−try)] −βα(x−y) + (trx−try) −β(1 −α)(trx−try)‖. int. j. anal. appl. 19 (4) (2021) 515 but (trx−try) −β(1 −α)(trx−try) = ( 1 − 1 r ) (x−y) + 1 r ( t(trx) −t(try) ) −β(1 −α) [( 1 − 1 r ) (x−y) + 1 r ( t(trx) −t(try) )] , = ( 1 − 1 r ) (x−y) −β(1 −α) ( 1 − 1 r ) (x−y) + 1 r ( t(trx) −t(try) ) −β(1 −α) 1 r ( t(trx) −t(try) ) , = ( 1 − 1 r ) (x−y)[1 −β(1 −α)] + 1 r ( 1 −β(1 −α) ) (t(trx) −t(try)). hence ‖trx−try‖ = ‖β[α(x−y) + (1 −α)(trx−try)] −βα(x−y) + ( 1 − 1 r ) (x−y)[1 −β(1 −α)] + 1 r ( 1 −β(1 −α) ) (t(trx) −t(try))‖, = ‖β[α(x−y) + (1 −α)(trx−try)] + [−βα + ( 1 − 1 r ) (1 −β(1 −α))](x−y) + 1 r ( 1 −β(1 −α ) (t(trx) −t(try))‖. now let −βα + ( 1 − 1 r ) (1 −β(1 −α)) = 0. this implies that β = r − 1 αr + (α− 1)(r − 1) , 1 r (1 −β(1 −α)) = α αr + (1 −α)(r − 1) . hence we have ‖trx−try‖≤ β‖α(x−y) + (1 −α)(trx−try)‖ + α αr + (1 −α)(r − 1) ‖t(trx) −t(try)‖. so by the non-expansiveness of t ,the above inequality becomes ‖trx−try‖≤ β‖α(x−y) + (1 −α)(trx−try)‖ + α αr + (1 −α)(r − 1) ‖trx−try‖, = r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(trx−try)‖ + α αr + (1 −α)(r − 1) ‖trx−try‖. int. j. anal. appl. 19 (4) (2021) 516 after simplifying we obtain the following results ‖trx−try‖− α αr + (1 −α)(r − 1) ‖trx−try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(trx−try)‖, [1 − α αr + (1 −α)(r − 1) ]‖trx−try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(trx−try)‖, r − 1 αr + (1 −α)(r − 1) ‖trx−try‖≤ r − 1 αr + (1 −α)(r − 1) ‖α(x−y) + (1 −α)(trx−try)‖, ‖trx−try‖≤‖α(x−y) + (1 −α)(trx−try)‖, since β = r−1 αr+(1−α)(r−1) > 0. hence tr is firmly non-expansive mapping. claim 3: we have that z is a fixed point of t if and only if it is also a fixed point of tr. proof. now suppose that tz = z. then we have the following evaluation: ‖trz −z‖ = ∥∥∥(1 − 1 r ) z + 1 r t(trz) −z ∥∥∥, = ∥∥∥1 r t(trz) − 1 r z ∥∥∥, = 1 r ∥∥∥t(trz) −z∥∥∥, = 1 r ∥∥∥t(trz) −tz∥∥∥, ≤ 1 r ∥∥∥trz −z∥∥∥. hence ‖trz−z‖≤ 1r‖trz−z‖ which is not possible since r > 1. it is possible when ‖trz−z‖ = 0 ⇒ trz = z. so z is a fixed point of trz. on the other hand, let us suppose that trz = z, that is z is a fixed point of tr. then z = trz, = ( 1 − 1 r ) z + 1 r t(trz), = ( 1 − 1 r ) z + 1 r t(z). this gives us [ 1 − ( 1 − 1 r )] z = 1 r t(z). hence z is a fixed point of t and that concludes our main result. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (4) (2021) 517 references [1] f.e. browder, nonlinear monotone operators and convex sets in banach spaces. bull. amer. math. soc. 71(5) (1965), 780–785. [2] f.e. browder, nonexpansive nonlinear operators in a banach space. proc. nat. acad. sci. usa, 54(4) (1965), 1041-1044. [3] f.e. browder, convergence theorems for sequences of nonlinear operators in banach spaces. math. zeitschr. 100(3) (1976), 201–225. [4] f.e. browder, w.v. petryshyn. the solution by iteration of nonlinear functional equations in banach spaces. bull. amer. math. soc. 72(3) (1966), 571–575. [5] f.e. browder, w.v. petryshyn. construction of fixed points of nonlinear mappings in hilbert space. j. math. anal. appl. 20(2) (1967), 197–228. [6] w. kirk. a fixed point theorem for mappings which do not increase distances. amer. math. mon. 72(9) (1965), 1004–1006. [7] j. george, minty et al. monotone (nonlinear) operators in hilbert space. duke math. j. 29(3) (1962), 341–346. [8] r. bruck. nonexpansive projections on subsets of banach spaces. pac. j. math. 47(2) (1973), 341–355. [9] r. smarzewski. on firmly nonexpansive mappings. proc. amer. math. soc. 113(3) (1991), 723–725. [10] s. banach, h. steinhaus. sur le principe de la condensation de singularités. fundam. math. 1(9) (1927), 50–61. 1. introduction 2. main results references int. j. anal. appl. (2022), 20:37 on intuitionistic fuzzy β generalized α normal spaces abdulgawad a. q. al-qubati∗, hadba f. al-qahtani department of mathematics, college of science and arts, najran university, saudi arabia ∗corresponding author: gawad196999@yahoo.com abstract. in this paper a new concept of generalized intuitionistic fuzzy topological space called intuitionistic fuzzy β generalized α normal space is introduced. several characterizations of intuitionistic fuzzy β generalized α normal space, intuitionistic fuzzy strongly β generalized α normal and intuitionistic fuzzy strongly β generalized α regular spaces are studied. moreover, the related intuitionistic fuzzy functions with intuitionistic fuzzy β generalized α normal spaces are investigated. 1. introduction the notion of intuitionistic fuzzy set was first defined by atanassov [7, 8] as a generalization of zadeh [21] fuzzy set. this notion of intuitionistic fuzzy set has been developed by the same author and appeared in the literature [7,8]. using the notion of intuitionistic fuzzy sets, coker [13] introduced the notion of intuitionistic fuzzy topological spaces as a generalization of chang [11] fuzzy topological spaces. recently many concepts of fuzzy topological space have been extended in intuitionistic fuzzy topological spaces. separation axioms in intuitionistic fuzzy topological space have been studied by some authors [1–4, 6, 9, 10]. jayanthi [17] introduced the generalized β closed set in intuitionistic fuzzy topological spaces and intuitionistic fuzzy generalized closed sets are introduced by saranya and jayanthi [18]. then gomathi and jayanthi [15, 16] have studied intuitionistic fuzzy β generalized α closed sets and intuitionistic fuzzy β generalized α continuous functions respectively. thanh and quang [19] have studied πgp-normality in topological spaces by using πgp-closed and πgp-open sets. in this paper, we introduce a new class of spaces called an intuitionistic fuzzy β generalized α normal received: jun. 13, 2022. 2010 mathematics subject classification. 54a05, 54a08, 54d10. key words and phrases. intuitionistic fuzzy topology; intuitionistic fuzzy β generalized α closed sets; intuitionistic fuzzy β generalized α normal spaces. https://doi.org/10.28924/2291-8639-20-2022-37 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-37 2 int. j. anal. appl. (2022), 20:37 spaces, and investigate some of their properties. some interesting characterizations such as an if β generalized α-normality is hereditary property with respect to an open and if β generalized α-closed subspace, and equivalently of intuitionistic fuzzyβ generalized α-normal space and other meanings are introduced. in addition, we introduce the concept of intuitionistic fuzzy β generalized α-regular spaces, some of their properties are investigated such as showing that an if strongly-regular β∗t1/2 space is an if β generalized α-regular, and every if β generalized α-normal r0 space is an if β generalized α-regular space. in the last section of this paper, we study some properties of related intuitionistic fuzzy functions with intuitionistic fuzzy β generalized α normal spaces. 2. preliminaries definition 2.1 [7] let t be a non empty fixed set. an intuitionistic fuzzy set h (ifs for short )in t is an object having the form h = {〈t,µh(t),γh(t)〉 : t ∈ t} where the function µh : t → i and γh : t → i denote the degree of membership (namely µh(t)) and the degree of non-membership (namely γh(t)) of each element t ∈ t to the set h respectively, and 0 ≤ µh(t) + γh(t) ≤ 1, for each t ∈ t. definition 2.2 [7] let h and j be if sets of the form h = {〈t,µh(t),γh(t)〉 : t ∈ t} and j = {〈x,µj(t),γj(t)〉 : t ∈ t}. then (a) h ⊆ j if and only if µh(t) ≤ µj(t) and γh(t) ≥ γj(t) for all t ∈ t . (b) h = j if and only if h ⊆ j and j ⊆ h. (c) hc = {〈t,γh(t),µh(t)〉 : t ∈ t}. (d) h ⋂ j = {〈t,µh(t) ∧µj(t),γh(t) ∨γj(t)〉 : t ∈ t}. (e) h ⋃ j = {〈t,µh(t) ∨µj(t),γh(t) ∧γj(t)〉 : t ∈ t}. (f) 0∼t = {〈t, , 0, 1〉 : t ∈ t} and 1 ∼ t = {〈t, 1, 0〉 : t ∈ t}. definition 2.3 [5] let {hi : i ∈ i} be an arbitrary family of ifs in t. then (a) ⋂ hi = {〈t,∧µhi (t),∨γhi (t) : t ∈ t}. (b) ⋃ hi = {t,∨µhi (t),∧γhi (t) : t ∈ t}. definition 2.4 [12] let α,β ∈ [0, 1],α + β ≤ 1 an intuitionistic fuzzy point (ifp for short) of nonempty set t is an ifs of t denoted by p = t(α,β) and defined by p = t(α,β)(y) =  (α,β)if t = y (0, 1)if t 6= y (2.1) int. j. anal. appl. (2022), 20:37 3 in this case, t is called the support of t(α,β) and α,β are called the value and no value of t(α,β) respectively. clearly an intuitionistic fuzzy point can be represented by an ordered pair of fuzzy point as follows: t(α,β) = (tα, 1 − t(1−β)). an ifp,t(α,β) is said to belong to an ifs h = {〈t,µh(t),γh(t)〉 : t ∈ t} denoted by p = t(α,β) ∈ h(orp ⊆ h), if α ≤ µh(t) and β ≥ γh(t). we identify a fuzzy point tr in t by the if point t(r,(1−r)) in t . for more details about operations on if-sets, if points and if functions, we can see [7,8,12–14]. definition 2.5 [13] an intuitionistic fuzzy topology (briefly ift) on a nonempty set t is a family ψ of intuitionistic fuzzy sets in t satisfy the following axioms: (t1) 0∼t , 1 ∼ t ∈ ψ. (t2) if h1,h2 ∈ ψ, then h1 ⋂ h2 ∈ ψ. (t3) if hλ ∈ ψ for each λinλ, then ⋃ λ∈λ hλ ∈ ψ. in this case the pair (t,ψ) is called an intuitionistic fuzzy topological space (briefly ifts) denoted by t, and each intuitionistic fuzzy set in ψ is known as an intuitionistic fuzzy open set (briefly ifos ) of t. the complement hc of an ifos h in ifts (t,ψ) is an intuitionistic fuzzy closed set (briefly ifcs) in t. definition 2.6. an ifs h of an ifts (t,ψ) is an (i) if α-open set (resp.α closed ) if h ⊆ (int(cl(int(h))) ( resp.cl(int(cl(h))) ⊆ h. [14] (ii) if β −open( resp.β closed ) if h ⊆ (cl(int(cl(h))) ( resp.int(cl(int(h))) ⊆ h. [14] (iii) if semi-open if h ⊆ cl(int(h)). [14] (iv) if pre-open if h ⊆ (int(cl(h))). [14] (v) if generalized closed (briefly ifgc) if cl(h) ⊆ j whenever h ⊆ j and j is an ifo. [18] 4 int. j. anal. appl. (2022), 20:37 definition 2.7 [5] let h be any ifs in ifts (t,ψ). then the if β closure and if interior of h are defined as follows, ifβcl(h) = ⋂ {f : h ⊆ f,f is ifβcs in t} . ifβint(h) = ⋃ {j : j ⊆ h,j is ifβos in t} . definition 2.8 [15] an ifs h of an ifts (t,ψ) is said to be an if β generalized α -closed set (ifβgαcs for short) if βcl(h) ⊆ j whenever h ⊆ j and j is an ifαos in (t,ψ). the complement hc of an ifβgαcs h in an ifts (t,ψ) is called an if β generalized α open set(ifβgαos for short) in t. the family of all ifβgα cs of an ifts (t,ψ) is denoted by ifβgαcs(t). definition 2.9 let h be any ifs in ifts (t,ψ). then the if β generalized α closure and if β generalized α interior of h are defined as follows, ifβgαcl(h) = ⋂ {f : h ⊆ f,f is ifβgαcs in t}. ifβgαint(h) = ⋃ {j : j ⊆ h,j is ifβgαos in t}. 3. intuitionistic fuzzy β generalized α normal spaces in this section, we have introduced if β generalized α -normal space and studied some of its characterizations. definition 3.1 an if topological space t is said to be if β generalized α-normal(in short if β g α-normal) if for every pair of if disjoint β generalized αclosed sets h1 and h2 of t, there exist if disjoint β-open sets r1, r2 of t such that h1 ⊆ r1 and h2 ⊆ r2. example 3.2 let t = {a,b} and r1, r2 are if sets on t defined as follows: r1 = 〈t, ( a1.0, b 0.0 ), ( a 0.0 , b 1.0 )〉. r2 = 〈t, ( a0.0, b 1.0 ), ( a 1.0 , b 0.0 )〉. then the family ψ = {0∼t , 1 ∼ t ,r1,r2} is an ift on t. the if sets r1,r2 in t are if disjoint β open sets and the if sets h1 = 〈t, ( a0.7, b 0.0 ), ( a 0.3 , b 1.0 )〉 ,h2 = 〈t, ( a0.0, b 0.6 ), ( a 1.0 , b 0.4 )〉 are if β g α -css such tha h1 ⋂ h2 = 0 ∼ t and h1 ⊆ r1 and h2 ⊆ r2. int. j. anal. appl. (2022), 20:37 5 then t is an if β g α-normal space. theorem 3.3 for an ifts (t,ψ), the following are equivalent: (1) t is β g α-normal. (2) for any pair of if disjoint β g α oss r1 and r2 of t whose union is 1∼t , there exist an if disjoint β-css h1 and h2 of t such that h1 ⊆ r1 and h2 ⊆ r2 and h1 ⋃ h2 = 1 ∼ t . (3) for each ifβ g α-cs h and an ifβ g α-os k containing h, there exists an ifβ-os r2 such that h ⊆ r2 ⊆ ifβ −cl(r2) ⊆ k. (4) for any pair of if disjoint β g α-css h and k of t there exists an ifβ-os r2 of t such that h ⊆ r2 and ifβ −cl(r2) ⋂ k = 0∼t . (5) for any pair of if disjoint β gα-css h and k of t there exists an ifβ-oss r1 and r2 of t such that h ⊆ r1, k ⊆ r2 and ifβ −cl(r1) ⋂ ifβ −cl(r2) = 0∼t . proof.(1) ⇒ (2) let r1 and r2 be two if β g αoss in an if β g α-normal space t such that r1 ⋃ r2 = 1 ∼ t . then r c 1, r c 2 are if disjoint β g αcss. since t is an if β g α-normal space there exist if disjoint β-oss r and q such that rc1 ⊆ r and r c 2 ⊆ q. let h1 = r c ,h2 = qc. then h1 and h2 are if β-css h1 ⊆ r1, h2 ⊆ r2 and h1 ⋃ h2 = 1 ∼ t . (2) ⇒ (3) let h be an ifβ g αcs and k be an ifβ g αos containing h. then hc and k are ifβ g α -oss such that hc ⋃ k = 1∼t . then by (2) there exist an if β-css h1 and h2 such that h1 ⊆ hc and h2 ⊆ k and h1 ⋃ h2 = 1 ∼ t . thus, we obtain h ⊆ h c 1, k c ⊆ hc2 and h c 1 ⋂ hc2 = 0 ∼ t . let r2 = hc1 and r1 = h c 2. then r1 and r2 are if disjoint β -oss such that h ⊆ h c 1 ⊆ r2 ⊆ k. as rc2 an if β-cs, we have h ⊆ r2 ⊆ ifβ −cl(r2) ⊆ k. (3) ⇒ (4) let h and k be if disjoint β g α -css of t. then h ⊆ kc where kc is ifβ g α -open. by the part (3), there exists an if β-os r2 of t such that h ⊆ r2 ⊆ β − cl(r2) ⊆ kc. thus, if β −c1(r2) ⋂ k = 0∼t . (4) ⇒ (5) let h and k be any if disjoint β g α -css of t. then by the part (4), there exists an if β-os r1 containing h such that ifβ−c1(r1) ⋂ k = 0∼t . since ifβ−c1(r1) is an ifβ g α closed, then it is ifβ g α -closed. thus ifβ − c1(r1) and k are if disjoint β g α-css of t. again by the part (4), there exists an if β-os r2 in t such that k ⊆ r2 and ifβ−c1(r1) ⋂ ifβ−c1(r2) = 0∼t . (5) ⇒ (1) let h and k be any if disjoint β g α-css of t. then by the part (5), there exist if β-oss r1 and r2 such that h ⊆ r1, k ⊆ r2, and ifβ −c1(r1) ⋂ ifβ −c1(r2) = 0∼t . therefore, 6 int. j. anal. appl. (2022), 20:37 we obtain that r1 ⋂ r2 = 0 ∼ t . hence t is ifβ g α-normal. � definition 3.4 [16] a space (t,ψ) is called β∗ t1/2 if every if β g α-cs in t is β closed. definition 3.5 [15] if every if β g α-cs is an ifcs in (t,ψ), then the space can be called as an if β g α t1/2 space. for the if regularity we give the following definition. definition 3.6 an ifts t is said to be if β g α-regular if for every β g α-cs f of t and an if point p = t(α,β) not in f there exist if disjoint β-oss r1, r2 of t such that p ∈ r1 and f ⊆ r2. example 3.7 let t = {a,b,c} and m,n are if sets on t defined as follows: r1 = 〈t, ( a1.0, b 0.0 , c 1.0 ), ( a 0.0 , b 1.0 , c 0.0 )〉. r2 = 〈t, ( a0.0, b 1.0 , c 0.0 ), ( a 1.0 , b 0.0 , c 1.0 )〉. rc1 = 〈t, ( a 0.0 , b 1.0 , c 0.0 ), ( a 1.0 , b 0.0 , c 1.0 )〉. rc2 = 〈t, ( a 1.0 , b 0.0 , c 1.0 ), ( a 0.0 , b 1.0 , c 0.0 )〉. let p = t(α,β) = a(0.5, 0.3) with p * rc1. then there exist if disjoint β-oss r1, r2 of t such that p ⊆ r1, rc1 ⊆ r2, and r1 ⋂ r2 = 0 ∼ t . then the family ψ = {0∼t , 1 ∼ t ,r1,r2} is an ift on t. which is an β g α-regular space. definition 3.8 [20] an ifts t is called if strongly-regular if for each if β g α-cs h and an if point p = t(α,β) not in h, there exist an if β g α-oss u, v of t such that p ∈ u and h ⊆ v . definition 3.9 [20] an if topological space t is called if strongly-normal if for each if β g α-css h1 and h2, there exist an if β g α-oss u, v such that h1 ⊆ u and h2 ⊆ v . since every if β -os is an if β g α-os then we have. if β g α normal (resp.regular) space ⇒ if strongly-normal (resp. regular) space. lemma 3.10 an if strongly-regularβ∗-t1/2 space is an if β g α-regular. int. j. anal. appl. (2022), 20:37 7 proof let (t,ψ) be an if strongly-regular space as well as β∗ —t1/2 space. since, (t,ψ) is a β∗-t1/2 space, then every if β g α-cs in t is β closed i.e. the class of if β g α-css and β-closed sets coincide. now, (t,ψ) is strongly regular space which provides that for each if β g α-cs h of t and an if point p = t(α,β) not in h there exist if disjoint β-oss u, v such that p ∈ u and h ⊆ v . combining these facts, it is concluded that for each if β g α-cs h and each if point p = t(α,β) there exist an if disjoint β-oss u and v such that h ⊆ u and p ∈ v , which turns (t,ψ) to be an if β g α-regular.� definition 3.11 an if β g α space is said to be if r0 if for if β os r and each if point p = t(α,β) ∈ r, then if βcl{p}⊆ r. theorem 3.12 every if β g α-normal-r0 space is an if β g α-regular space. proof let h be an if β gα-cs in t and an if point p = t(α,β) in t such that p is not in h. then, p ∈ hc, where hc is an if β g α os in t. since t is an if β g α-normal r0 space, we have ifβcl{p} ⊆ hc, then h ⋂ ifβcl{p} = 0∼t . thus h and ifβcl{p} are if disjoint β g α-css in t. by β g α-normality of t, there exist if disjoint β-oss r1, r2 of t such that h ⊆ r1 and if βcl{p}⊆ r2. therefore, there exist an ifβ-oss r1, r2 of t such that h ⊆ r1 and p ∈ r2. hence, t is an if β g α-regular space.� lemma (3.13) [17] suppose h ⊆ y ⊆ t and (t,ψ) is an if β g α space. if y is open and an if β g α-closed in (t,ψ) and h is an if β g α-closed in (y,ty ), then h is also β g α-closed in (t,ψ). if β generalized α-normality is hereditary property with respect to an open and if β generalized α-closed subspace. theorem 3.14 if (t,ψ) is an if β generalized α-normal space and y is an if open and β g α-cs of (t,ψ), then (y,ty ) is an if β generalized α-normal subspace. proof let h1 and h2 be any two if disjoint β g α-css of (y,ty ). since y is an if open and β g α-cs of (t,ψ), hence, in view of lemma (3.13), h1 and h2 are if β g α -closed in (t,ψ), and since (t,ψ) is an if β g α-normal, then there exist an if disjoint β-oss r1 and r2 of (t,ψ) such that h1 ⊆ r1 and h2 ⊆ r2. as y is also an if open so y is an if αopen and then we get r1 ⋂ y and r2 ⋂ y as an if disjoint β-oss of the if subspace (y,ty ) such that h1 ⊆ r1 ⋂ y and h2 ⊆ r2 ⋂ y . hence, (y,ty ) is an if β g α-normal space.� 8 int. j. anal. appl. (2022), 20:37 4. the related intuitionistic fuzzy functions with intuitionistic fuzzy β generalized α normal spaces we start by the following definition. definition 4.1 a function f : (t,ψ) → (y,δ) is called: (1) if β g α-closed if f (h) is if β -gα-closed in y for each if β g α-cs h of t. (2) if m-β-open if f (h) is an ifβ -open in y for each if β-os h of t. (3) if β g α-irresolute if f−1(h) is if β g α-closed in t for each ifβ g α cs h in y. definition 4.2 [16] an if function f : (t,ψ) → (y,δ) is said to be an if β g α-continuous function if f−1(f ) is an if β g α-cs in t for every if-cs h in y. theorem 4.3 let f : (t,ψ) → (y,δ) be an if continuous β gα -closed injection and if (y,δ) is an if β g α-normal, then (t,ψ) is an if β g α -normal. proof let h1 and h2 are if disjoint β g α-css in (t,ψ), since f is injective, f (h1) and f (h2) are if β g α -css in (y,δ), there exist an if disjoint β -oss r1 and r2 such that f (hi ) ⊆ ri for i = 1, 2. since f is an if β g α-continuous,f−1(r1) andf−1(r2) are if β g α -css in (t,ψ) and hi ⊆ f−1(ri ) for i = 1, 2. put qi = ifβ − int(f−1(ri ) for i = 1, 2. then q1 and q2 are ifβ oss with h1 ⊆ q1 and h2 ⊆ q2, and q1 ⋂ q2 = 0 ∼ t . then (t,ψ) is an if β g α -normal.� the following important lemma can be proved easily. lemma 4.4 (a) the image of if β g α-os under an if-open continuous function is β g α-open. (b) the inverse image of if β g α-o(resp.β gα-c) set under an open continuous function is if β g α-o (resp. β g α-c) set. proposition 4.5 the image of if β g α-os under if-open and if-closed continuous function is if β g α-open. proof clearly.� int. j. anal. appl. (2022), 20:37 9 theorem 4.6 if f : (t,ψ) → (y,δ) be an if-open and if-closed continuous bijection function and h be a if β g α -cs in (y,δ), then f−1(h) is if β g α-cs in (t,ψ). proof let h be an if β g α-cs in (y,δ) and r be any if β g α -os of (t,ψ) such that f−1(h) ⊆ r. then by the proposition (4.5), we have f (r) is if β g α-os of (y,δ) such that h ⊆ f (r). since h is an if β g α -cs of (y,δ) and f (r) is ifβ g α-os in (y,δ), thus if β − cl(h) ⊆ r. by lemma (4.4) we obtain that f−1(h) ⊆ f−1(ifβ − cl(h)) ⊆ r, where f−1(ifβ − cl(h)) is βclosed in (t,ψ). this implies that ifβ − cl(f−1(h)) ⊂ r. therefore f−1(h) is if β g α-cs in (t,ψ).� we show that an ifβ generalized α-normality is a topological property with respect to an if open-and-closed bijection continuous function. theorem 4.7 an if β g α-normality is a topological property. proof let (t,ψ) be an if β g α-normal space and be an open-and-closed bijection continuous function. we need to show that (y,δ) is if β generalized α-normal. let h1 and h2 be any if disjoint β generalized α-css in (y,δ). then by the theorem (4.6) f−1(h1) and f−1(h1) are if disjoint β generalized α-css of (t,ψ). by if β g α-normality of (t,ψ), there exist β-oss r1 and r2 of (t,ψ) such that f−1(h1) ⊆ r1, f−1(h2) ⊆ r2 and r1 ⋂ r2 = 0 ∼ t . then, we have h1 ⊆ f (r1), h2 ⊆ f (r2) and f (r1) ⋂ f (r2) = 0 ∼ t . thus, f (r1) and f (r2) are if disjoint β-oss of (y,δ) such that h1 ⊆ f (r1) and h2 ⊆ f (r2). hence, (y,δ) is if β g α-normal. �. theorem 4.8 if f : (t,ψ) → (y,δ) be an ifβ g α -irresolute, m-β-open bijection function from an if β g α -normal (t,ψ) to an if space (y,δ), then (y,δ) is an if β g α -normal space. proof let h1 and h2 be any two if disjoint β g α-css in (y,δ). since f is an if β g α -irresolute, we have f−1(h1) and f−1(h2) are if disjoint β g α-css in (t,ψ). by if β gα-normality of (t,ψ), there exist β -oss r1 and r2 in (t,ψ) such that f−1(h1) ⊆ r1, f−1(h2) ⊆ r2 and r1 ⋂ r2 = 0 ∼ t . since f is an if m-β-open and bijection function, we have f (r1) and f (r2) are if β -oss in (y,δ) such that h1 ⊆ f (r1), h2 ⊆ f (r2) and f (r1) ⋂ f (r2) = 0 ∼ t . therefore, (y,δ) is an if β g α -normal.� theorem 4.9 if f : (t,ψ) → (y,δ) is an if β g α closed continuous surjection and (t,ψ) is an ifβ normal, then (y,δ) is an if β g α-normal. 10 int. j. anal. appl. (2022), 20:37 proof since every if β normal is an if β g α -normal, the proof is clear.� 5. conclusion in this paper, we introduced the concept of if β g α normal spaces with study some of its properties. we also investigated the related intuitionistic fuzzy functions with intuitionistic fuzzy β g α normal spaces. in the future, based on some recent intuitionistic fuzzy β g α spaces studies, we will expand the research content of this paper further. also,the entire content will be a successful tool for the researchers for finding the path to obtain the results in the context of intuitionistic fuzzy strongly-regular and strongly-normal spaces. authors’ contributions: all authors read and approved the final manuscript. acknowledgment: the authors would like to express their gratitudes to the ministry of education and the deanship of scientific research-najran university -kingdom of saudi arabia for their financial and technical support under code number [nu/serc/10/564]. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. al-qubati, h.f. al-qahtani, on b-separation axioms in intuistionistic fuzzy topological spaces, int. j. math. trends technol. 21 (2015), 83-93. [2] a. al-qubati, on b-regularity and normality in intuistionistic fuzzy topological spaces, j. inform. math. sci. 9 (2017), 89-100. [3] a. al-qubati, on intuitionistic fuzzy β and β∗-normal spaces, int. j. math. anal. 12 (2018), 517 531. https: //doi.org/10.12988/ijma.2018.8859. [4] a. al-qubati, m.e. sayed, h.f. al-qahtani, small and large inductive dimensions of intuitionistic fuzzy topological spaces, nanosci. nanotechnol. lett. 12 (2020), 413–417. https://doi.org/10.1166/nnl.2020.3113. [5] m.e. abd el-monsef, r.a. mahmoud, e.r. lashin, β-closure and β-interior, rep. j. fac. edu. ain shams univ. 10 (1986), 235-245. [6] h.a. al-qahtani, a. al-qubati, on fuzzy pre-separation axioms, j. adv. stud. topol. 4 (2013), 1. https: //doi.org/10.20454/jast.2013.663. [7] k.t. atanassov, intuitionistic fuzzy sets, in: v. sgurev (ed.), vii itkr’s session, sofia, (1983). [8] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87–96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [9] g. balasubramanian, fuzzy β open sets and fuzzy β separation axioms, kybernetika, 35 (1999), 215-223. http://dml.cz/dmlcz/135282. [10] s. bayhan, d. coker, pairwise separation axioms in intuitionistic topological spaces, hacettepe j. math. stat. 34 (2005), 101-114. https://doi.org/10.12988/ijma.2018.8859 https://doi.org/10.12988/ijma.2018.8859 https://doi.org/10.1166/nnl.2020.3113 https://doi.org/10.20454/jast.2013.663 https://doi.org/10.20454/jast.2013.663 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 http://dml.cz/dmlcz/135282 int. j. anal. appl. (2022), 20:37 11 [11] c.l. chang, fuzzy topological spaces, j. math. anal. appl. 24 (1968), 182-190. [12] d. coker, m. demirci, on intuitionistic fuzzy points, notes ifs. 1 (1995), 79-84. [13] d. coker, an introduction to intuitionistic fuzzy topological spaces, fuzzy sets syst. 88 (1997), 81-89. https: //doi.org/10.1016/s0165-0114(96)00076-0. [14] h. gurcay, d. coker, a.h. es, on fuzzy continuity in intuitionistic fuzzy topological spaces, j. fuzzy math. 5 (1997), 365–378. [15] m. gomathi, d. jayanthi, on intuitionistic fuzzy β generalized α closed sets, glob. j. pure appl. math. 13 (2017), 2439-2455. [16] m. gomathi, d. jayanthi, on intuitionistic fuzzy β generalized α continuous mappings, adv. fuzzy math. 12 (2017), 499-513. [17] d. jayanthi, generalize β-closed sets in intuitionistic fuzzy topological spaces, int. j. adv. found. res. sci. eng. 1 (2014), 39-44. [18] m. saranya, d. jayanthi, on intuitionistic fuzzy β generalized closed sets, int. j. comput. eng. res. 6 (2016), 37-42. [19] l.n.t. nhon, b.q. thinh, pigp-normal topological spaces, j. adv. stud. topol. 4 (2012), 48-54. https://doi. org/10.20454/jast.2013.458. [20] k. vidyottama, r.k.c. thakur, g-pre regular and g-pre normal topological spaces, int. j. adv. res. computer sci. software eng. 5 (2015), 397-400. [21] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.1016/s0165-0114(96)00076-0 https://doi.org/10.1016/s0165-0114(96)00076-0 https://doi.org/10.20454/jast.2013.458 https://doi.org/10.20454/jast.2013.458 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. intuitionistic fuzzy generalized normal spaces 4. the related intuitionistic fuzzy functions with intuitionistic fuzzy generalized normal spaces 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 15-21 http://www.etamaths.com an analog of titchmarsh’s theorem for the jacobi-dunkl transform in the space l2α,β(r) a. abouelaz, a. belkhadir∗ and r. daher abstract. in this paper, using a generalized jacobi-dunkl translation operator, we prove an analog of titchmarsh’s theorem for functions satisfying the jacobi-dunkl lipschitz condition in l2(r,aα,β(t)dt) , α ≥ β ≥−12 ,α 6= − 1 2 . 1. introduction titchmarsh’s theorem characterizes the set of functions satisfying the cauchylipschitz condition by means of an asymptotic estimate growth of the norm of their fourier transform, namely we have: theorem 1.1. [10] let α ∈ (0, 1) and assume that f ∈ l2(r) . then the following are equivalents: (1) ‖f(t + h) −f(t)‖ = o(hα) , as α → 0 ; (2) ∫ |λ|≥r |f̂(λ)|2dλ = o(r−2α) , as r →∞ . where f̂ stands for the fourier transform of f . in this paper, we prove an analog of theorem 1.1 for the jacobi-dunkl transform for functions satisfying the jacobi-dunkl lipschitz condition in the space l2(r,aα,β(t)dt) . for this purpose, we use the generalized translation operator. similar results have been established in the context of noncompact rank one riemannian symetric spaces [9]. in section 2 below, we recapitulate from [1, 2, 3, 5] some results related to the harmonic analysis associated with jacobi-dunkl operator λα,β . section 3 is devoted to the main result after defining the class lip(δ, 2,α,β) of functions in l2α,β(r) satisfying the lipschitz condition correspondent to the generalized jacobi-dunkl translation. 2. notations and preliminaries the jacobi-dunkl function with parameters (α,β) , α ≥ β ≥ −1 2 ,α 6= −1 2 , is defined by the formula : (1) ∀x ∈ r, ψ(α,β)λ (x) = { ϕ (α,β) µ (x) − i λ d dx ϕ (α,β) µ (x) , if λ ∈ c\{0}; 1 , if λ = 0. 2010 mathematics subject classification. 33c45. key words and phrases. titchmarsh’s theorem; jacobi-dunkl transform; generalized jacobidunkl translation; jacobi-dunkl lipschitz condition. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 15 16 abouelaz, belkhadir and daher with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕ (α,β) µ is the jacobi function given by: (2) ϕ(α,β)µ (x) = f ( ρ + iµ 2 , ρ− iµ 2 ; α + 1,−(sinh(x))2 ) , f is the gauss hypergeometric function (see [1, 6, 7]). ψ (α,β) λ is the unique c ∞-solution on r of the differentiel-difference equation (3) { λα,βu = iλu , λ ∈ c; u(0) = 1. where λα,β is the jacobi-dunkl operator given by: λα,βu(x) = du dx (x) + [(2α + 1) coth x + (2β + 1) tanh x] × u(x) −u(−x) 2 the operator λα,β is a particular case of the operator d given by: du(x) = du dx (x) + a′(x) a(x) ( u(x) −u(−x) 2 ) where a(x) = |x|2α+1b(x) , and b a function of class c∞ on r , even and positive. the operator λα,β corresponds to the function a(x) = aα,β(x) = 2 ρ(sinh |x|)2α+1(cosh |x|)2β+1. using the relation d dx ϕ(α,β)µ (x) = − µ2 + ρ2 4(α + 1) sinh(2x)ϕ(α+1,β+1)µ (x) , the function ψ (α,β) λ can be written in the form above (see [2]), (4) ψ (α,β) λ (x) = ϕ (α,β) µ (x) + i λ 4(α + 1) sinh(2x)ϕ(α+1,β+1)µ (x) , ∀x ∈ r , where λ2 = µ2 + ρ2 , ρ = α + β + 1. denote by l2α,β(r) = l 2(r,aα,β(t)dt) the space of measurable functions g on r such that ||g||l2 α,β (r) = (∫ r |g(t)|2aα,β(t)dt )1/2 < +∞ . using the eigenfunctions ψ (α,β) λ of the operator λα,β called the jacobi-dunkl kernels, we define the jacobi-dunkl transform of a function f ∈ l2α,β(r) by: (5) fα,β(f)(λ) = ∫ r f(x)ψ (α,β) λ (x)aα,β(x)dx, ∀λ ∈ r . and the inversion formula (6) f(t) = ∫ r fα,β(f)(λ)ψ (α,β) −λ (t)dσ(λ) , where: dσ(λ) = |λ| 8π √ λ2 −ρ2|cα,β( √ λ2 −ρ2)| ir\]−ρ,ρ[(λ)dλ an analog of titchmarsh’s theorem for jacobi-dunkl transform 17 here, cα,β(µ) = 2ρ−iµγ(α + 1)γ(iµ) γ( 1 2 (ρ + iµ))γ( 1 2 (α−β + 1 + iµ)) , µ ∈ c\ (in) . and ir\]−ρ,ρ[ is the characteristic function of r\] −ρ,ρ[ . denote l2σ(r) = l 2(r,dσ(λ)). the jacobi-dunkl transform is a unitary isomorphism from l2α,β(r) onto l 2 σ(r), i.e. (7) ||f|| = ||f||l2 α,β (r) = ||fα,β(f)||l2σ(r) . the operator of jacobi-dunkl translation is defined by: (8) txf(y) = ∫ r f(z)dνα,βx,y (z) , ∀ x,y ∈ r . where να,βx,y , x,y ∈ r are the signed measures given by (9) dνα,βx,y (z) =   kα,β(x,y,z)aα,β(z)dz , if x,y ∈ r?; δx , if y = 0; δy , if x = 0. here, δx is the dirac measure at x. and, kα,β(x,y,z) = mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αiix,y × ∫π 0 ρθ(x,y,z) ×(gθ(x,y,z)) α−β−1 + sin 2β θdθ. ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x| + |y||, |x| + |y|] , ρθ(x,y,z) = 1 −σθx,y,z + σ θ z,x,y + σ θ z,y,x σθx,y,z =   cosh(x) + cosh(y) − cosh(z) cos(θ) sinh(x) sinh(y) , if xy 6= 0; 0 , if xy = 0. , ∀x,y,z ∈ r , ∀θ ∈ [0,π] . gθ(x,y,z) = 1 − cosh2 x− cosh2 y − cosh2 z + 2 cosh x cosh y cosh z cos θ . t+ = { t , if t > 0; 0 , if t ≤ 0. and, mα,β =   2−2ργ(α + 1) √ πγ(α−β)γ(β + 1 2 ) , if α > β; 0 , if α = β. in [2], we have (10) fα,β(thf)(λ) = ψ α,β λ (h).fα,β(f)(λ) ; h,λ ∈ r . for α ≥ −1 2 , we introduce the bessel normalized function of the first kind defined by jα(z) = γ(α + 1) ∞∑ n=0 (−1)n(z 2 )2n n!γ(n + α + 1) , z ∈ c. 18 abouelaz, belkhadir and daher moreover, we see that lim z→0 jα(z) − 1 z2 6= 0 , by consequence, there exists c1 > 0 and η > 0 satisfying (11) |z| ≤ η ⇒ |jα(z) − 1| ≥ c1|z|2 . lemma 2.1. the following inequalities are valids for jacobi functions ϕα,βµ (t) : (1) |ϕ(α,β)µ (t)| ≤ 1 ; (2) |1 −ϕ(α,β)µ (t)| ≤ t2(µ2 + ρ2) . proof. (see [8], lemma 3.1-3.2) � lemma 2.2. let α ≥ β ≥ −1 2 , α 6= −1 2 . then for |ν| ≤ ρ , there exists a positive constant c2 such that |1 −ϕ(α,β)µ+iν (t)| ≥ c2|1 − jα(µt)| . proof. (see [4], lemma 9) � 3. main result in this section we introduce and prove an analog of theorem 1.1. firstly we have to define, for functions in l2α,β(r) , the condition of cauchy-lipschitz related to the jacobi-dunkl translation operator given in (8). definition 3.1. let δ ∈ (0, 1) . a function f ∈ l2α,β(r) is said to be in the jacobi-dunkl-lipschitz class, denoted by lip(δ, 2,α,β) , if ||thf + t−hf − 2f|| = o(hδ) , as h → 0 . theorem 3.2. let f ∈ l2α,β(r) . then the following are equivalents: (1) f ∈ lip(δ, 2,α,β) ; (2) ∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = o(r−2δ) , as r →∞ . proof. 1) ⇒ 2) . assume that f ∈ lip(δ, 2,α,β); then we have: ||thf + t−hf − 2f|| = o(hδ) , as h → 0 . fα,β(thf + t−hf − 2f)(λ) = (ψ (α,β) λ (h) + ψ (α,β) λ (−h) − 2).fα,β(f)(λ). since ψ (α,β) λ (h) = ϕ (α,β) µ (h) + i λ 4(α + 1) sinh(2h)ϕ (α+1,β+1) µ (h), ψ (α,β) λ (−h) = ϕ (α,β) µ (−h) − i λ 4(α + 1) sinh(2h)ϕ (α+1,β+1) µ (−h), and ϕ (α,β) µ is even [see (2)]; then: fα,β(thf + t−hf − 2f)(λ) = 2(ϕ(α,β)µ (h) − 1).fα,β(f)(λ). from parseval’s identity (7) we write: (12) ||thf + t−hf − 2f||2 = 4 ∫ r |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ). by (11) and lemma 2.2, we get: an analog of titchmarsh’s theorem for jacobi-dunkl transform 19 ∫ η 2h ≤|λ|≤η h |1−ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≥ ∫ η 2h ≤|λ|≤η h c21c 2 2|µh| 4|fα,β(f)(λ)|2dσ(λ) , from η 2h ≤ |λ| ≤ η h we have,( η 2h )2 −ρ2 ≤ µ2 ≤ (η h )2 −ρ2 ⇒ µ2h2 ≥ η2 4 −ρ2h2 take h ≤ η 3ρ , then we have µ2h2 ≥ c3 = c3(η). so, ∫ η 2h ≤|λ|≤η h |1−ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≥ c21c 2 2c 2 3 ∫ η 2h ≤|λ|≤η h |fα,β(f)(λ)|2dσ(λ) . there exists then a positive constant c such that:∫ η 2h ≤|λ|≤η h |fα,β(f)(λ)|2dσ(λ) ≤ c ∫ r |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≤ ch2δ , for all 0 < h ≤ η 3ρ , (see (12)). then we have,∫ r≤|λ|≤2r |fα,β(f)(λ)|2dσ(λ) ≤ cr−2δ , as r →∞. furthermore, we obtain: ∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = ∞∑ i=0 ∫ 2ir≤|λ|≤2i+1r |fα,β(f)(λ)|2dσ(λ) ≤ c ∞∑ i=0 ( 2ir )−2δ ≤ cr−2δ. this proves that:∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = o(r−2δ) , as r →∞. 2) ⇒ 1) . suppose now that∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = o(r−2δ) , as r →∞, and write ∫ r |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) = ∫ |λ|< 1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) + ∫ |λ|≥1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) 20 abouelaz, belkhadir and daher — using the inequality (1) of lemma 2.1, we get: ∫ |λ|≥1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≤ 4 ∫ |λ|≥1 h |fα,β(f)(λ)|2dσ(λ) then, (13) ∫ |λ|≥1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) = o(h2δ) , as h → 0. — set φ(x) = ∫ ∞ x |fα,β(f)(λ)|2dσ(λ) . an integration by parts gives:∫ x 0 λ2|fα,β(f)(λ)|2dσ(λ) = ∫ x 0 −λ2φ′(λ)dλ = −x2φ(x) + 2 ∫ x 0 λφ(λ)dλ ≤ 2 ∫ x 0 o(λ1−2δ)dλ = o(x2−2δ). from the second inequality of lemma 2.1, we get ∫ |λ|< 1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≤ ∫ |λ|< 1 h (µ2 + ρ2)h2|fα,β(f)(λ)|2dσ(λ) ≤ h2 ∫ |λ|< 1 h λ2|fα,β(f)(λ)|2dσ(λ) = o(h2.h−2+2δ). hence, (14) ∫ |λ|< 1 h |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) = o(h2δ). finally, we conclude from (13) and (14) that∫ r |1 −ϕ(α,β)µ (h)| 2|fα,β(f)(λ)|2dσ(λ) = ∫ |λ|< 1 h + ∫ |λ|≥1 h = o(h2δ) + o(h2δ) = o(h2δ). and this ends the proof. � references [1] ben mohamed. h and mejjaoli. h, distributional jacobi-dunkl transform and application, afr. diaspora j. math 1 (2004), 24–46. [2] ben mohamed. h, the jacobi-dunkl transform on r and the convolution product on new spaces of distributions , ramanujan j. 21 (2010), 145–175. [3] ben salem. n and ould ahmed salem. a , convolution structure associated with the jacobidunkl operator on r., ramanujan j. 12(3) (2006), 359–378. an analog of titchmarsh’s theorem for jacobi-dunkl transform 21 [4] bray. w o and pinsky. m a, growth properties of fourier transforms via moduli of continuity , journal of functional analysis. 255 (2008), 2256–2285. [5] chouchane. f, mili. m and trimèche. k, positivity of the intertwining opertor and harmonic analysis associated with the jacobi-dunkl operator on r, j. anal. appl. 1(4) (2003), 387–412. [6] koornwinder. t h , jacobi functions and analysis on noncompact semi-simple lie groups, in: askey. r a , koornwinder. t h and schempp. w (eds) special functions: group theoritical aspects and applications. d. reidel, dordrecht (1984). [7] koornwinder. t h , a new proof of a paley-wiener type theorems for the jacobi transform, ark. math. 13 (1975), 145–159. [8] platonov. s s, approximation of functions in l2-metric on noncompact rank 1 symetric spaces, algebra analiz. 11 (1) (1999), 244–270. [9] platonov. s s, the fourier transform of functions satisfying the lipschitz condition on rank 1 symetric spaces, siberian math. j. 46(6) (2005), 1108–1118. [10] titchmarsh. e c, introduction to the theory of fourier integrals , claredon, oxford. 1948, komkniga, moscow. 2005. department of mathematics and informatics, faculty of science ain chock, university of hassan ii, casablanca, morocco, ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 100-103 http://www.etamaths.com gelfand triple isomorphisms for weighted banach spaces on locally compact groups s.s. pandey and ashish kumar∗ abstract. as in [1], we use the concept of wavelet transform on a locally compact group g to construct weighted banach spaces h1w(g), we being a submultiple weight function on g. the main result of this paper provides an extension of a unitary mapping u from h(g1) to h(g2) under suitable conditions to an isomorphism between the gelfand triple (h1w, h, h1∼w )(g1) and (h1w, h, h1∼w )(g2); where g1, g2 are any two locally compact groups, h a hilbert space and h1∼w is the space of all continuous-conjugate linear functional on h1w. this paper paves the way for the study of some other properties of gelfand triples. 1. introduction i.m. gelfand introduced a triple of abstract space consisting of a frechet space f of test functions continuously and densely embedded in hilbert space h while h itself is continuously and densely embedded in the dual space f′ of f (for details see [3]). feichtinger and kozek [2] have studied a number of properties of gelfand triple replacing the frechet space f by a suitable banach space. in particular, they have discussed in detail the extensions of isomorphisms between l2-spaces on elementary locally compact abelian groups g to gelfand triples of the form (s0,l 2,s′0)(g) where s0(g) is the well known feichtinger algebra, which has a number of highly useful functional properties ([2], p. 237). also, they have studied some important properties of the operator gelfand triple (b, h, b′), where b is the banach space of all bounded linear operators from s′0(g) to s0(g) with respect to the operator norm ‖ ·‖op. in the present paper, following feichtinger and gröchenig ([1], p. 309), we define wavelet transform vgf of a function f with respect to g, both as elements of a hilbert space h, on a locally compact group g. using these wavelet transforms, we construct a weighted banach spaces h1w(g) as in ([1], p. 317), where w is a submultiplicative weight function on g. we prove that any unitary map u from h(g1) to h(g2) extends as an isomorphism from the gelfand triple (h1w,h,h1∼w )(g1) to (h1w,h,h1∼w )(g2) if and only if the restrictions of u and u∗ are bounded linear operators from h1w(g1) to h1w(g2) respectively, where h1∼w (g1) is the banach space of all continuous-conjugate linear functionals on h1w(g). this paper paves the way for the study of some properties associated with these gelfand triples. 2. notations and basic concepts. let g be a locally compact group and dx the normalized haar measure on it. we assume that w : g → r+ is a submultiplicative weight function on g such that w(x◦y) ≤ w(x) w(y) for all x, y ∈ g 2010 mathematics subject classification. 43a15, 43a32, 43b65 and 47a67. key words and phrases. wavelet transform on locally compact groups; unitary mappings and gelfand triples. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 100 gelfand triple isomorphisms 101 we suppose that all weight on g are symmetrical, i.e., w(x) = w(−x), ∀ x ∈ g. we denote by lpw(g), 1 ≤ p < ∞, the banach space of functions on g with respect to the norm ‖f‖p,w = (∫ g |f(x)|p wp(x) dx )1/p < ∞. (2.1) in case p = ∞, we define the space l∞w (g) as the banach space of all measurable functions f on g such that ‖f‖∞,w = ess sup{|f(x)| w(x) : x ∈ g} < ∞. (2.2) the conjugate space of lpw(g) is the space l p′ w−1 (g), where 1/p + 1/p′ = 1.. it is well known that l1w(g) is a commutative banach algebra under convolution , which is usually known as beurling algebra, and we have the properties: lpw(g) ∗l 1 w(g) ⊆ l p w(g) and ‖f ∗g‖≤‖f‖p,w ‖g‖1,w for all f ∈ lpw(g) and g ∈ l1w(g). 3. wavelet transform on g. let (π,h) be an irreducible continuous unitary representation of a locally compact group g on a hilbert space h(g). if f,g ∈h, then the wavelet transform of f with respect to g is given by ([1], p. 317): vgf : x →〈π(x)g, f〉, where vgf(x) = ∫ g π(x) g(x)f̄(y) dy and f̄(y) is the complex conjugate of f(y). the representation π is called square integrable provided vgf ∈ l1(g), ∀ g ∈ h. it is known that if π is square-integrable, i.e., vgg ∈ l2(g), then ∃ a unique, positive, self-adjoint and densely defined operator a on h satisfying the following orthogonality relation ([1], pp. 309-310): ∫ g vg1f1(x) vg2f2(x) dx = 〈ag2, ag1〉〈f1,f2〉 for all f1,f2 ∈h and g1,g2 ∈ dom a. in case f1 = f2 = g1 = g2 = g ∈ dom a, f1 = f and ‖g1‖2 = 1, then we have vg f ∗vg g = vg f〈g,g〉 = vg. now, on the line of feichtinger and gröchenig ([1], p.317),we define the set of analyzing vector h1w(g) by h1w(g) = {g : g ∈h,vg g ∈ l 1 w(g)}. since π is irreducible ,h1w is a dense linear subspace of h. we suppose that h1w(g) is non-trivial and g is a non-zero fixed element of h1w(g). we define h1w(g) by h1w(g) = {f; f ∈h,vg f ∈ l 1 w}, which is a banach space under the norm ‖f|h1w‖ = ‖vg f|l 1 w‖. 102 pandey and kumar as mention by feichtinger and gröchenig (loc. cit.),h1w(g) is a π-invariant banach space dense in h and the set {π(x)g,x ∈ g} is a total subset of h1w(g) we denote by h1 ∼ w (g) the banach space of all continuous-conjugate linear functionals on h1w(g). hence h1 ∼ w (g) is a π-invariant banach space with the continuous dense embeddings h1w ↪→h ↪→h 1∼ w which insure that (h1w, h, h 1∼ w ) forms a gabor triple (for detail see [3]) 4. extension of unitary gelfand triple isomorphisms in a recent paper feichtinger and kozek ([2], pp.239-240) have shown that a unitary mapping u acting from l2(g1) to l 2(g2) extends to an isomorphism between the gelfand triples (s0, l 2, s′0)(g1) and (s0, l 2, s′0)(g2) if an only if the restrictions of u and u ∗ are bounded linear operators between s0(g1)) and s0(g2), where s0(g1) and s0(g2) denote the feichtinger algebras on elementary locally compact abelian groups g1 and g2 respectively and s ′ 0(g1)) and s′0(g)1) their topological duals. also, they have pointed out some applications of the above isomorphism ([2], p.239). in this section, on the lines of feichtinger and kozek, we study an extension of a unitary mapping u acting from h to h to an isomorphism between the gelfand triples (h1w,h,h1∼w )(g1) and (h1w,h,h1∼w )(g2), where g1, g2 are any two locally compact groups. precisely, we prove the following : theorem 4.1. if u is an unitary operator from h(g1) to h(g2), then it extends isomorphism from the gelfand triples where g1, g2 are any two locally compact groups (h1w,h,h1∼w )(g1) and (h1w,h,h1∼w )(g2) if and only if there exists a positive constant c such that ‖uf|h1w(g2)‖≤ c ‖f|h 1 w(g1)‖, ∀ f ∈h 1 w(g1) (4.1) and ‖u∗f|h1w(g1)‖≤ c ‖f|h 1 w(g2)‖, ∀ f ∈h 1 w(g2) (4.2) where u∗ is the adjoint operator of u. proof: the proof follows on the lines of feichtinger and kozek ([2], p. 240). but, since our settings are different, it is necessary to give the proof. let us assume that (4.1) holds true. then by virtue of the relation 〈ūg,f〉 = 〈g,u∗f〉, ∀ g ∈h1∼w (g1) and f ∈h 1∼ w (g2), we see that g →ū g is a bounded linear mapping, which extends the unitary map u on h(g1). next, since h(g1) is boundedly dense in h1∼w (g1), ū is a continuous and bounded linear mapping from h1∼w (g1) to h1∼w (g2). also, ū is unique and it coincides with u on h(g1). in the same way it is clear that ū∗ is a unique, continuous and bounded linear mapping from h1∼w (g2) to h1∼w (g1), which coincides with u∗ on h(g2). thus we infer that ū defines an isomorphism between h1∼w (g1) and h1∼w (g2) with respect to their norm topologies. conversely, we suppose that u is a unitary operator, which extends as an isomorphism from (h1w,h,h1∼w )(g1) and (h1w,h,h1∼w )(g2). hence the restrictions of u and u∗ are bounded gelfand triple isomorphisms 103 linear mapping on the spaces h1w(g1) and h1w(g2) respectively. this completes the proof of the theorem. as a corollary of the above theorem, we show that the gelfand triples isomorphisms holds true provided there exists a bijection mapping v between h(g1) and h(g2) such that 〈f1 f2〉h(g1) = 〈v f1, v f2〉h(g2) for all f1, f2 ∈h(g1). as in [2], p.240), we prove the following : corollary: if v : h1w(g1) →h1w(g2) is an isomorphism, then it extends to a unitary gelfand triples isomorphism between (h1w,h,h1∼w )(g1) and (h1w,h,h1∼w )(g2) if and only if 〈f1, f2〉h(g1) = 〈v f1, v f2〉h(g2) (4.3) for all f1, f2 ∈h1w(g1). proof: the isomorphism v as defined above is a unitary operator isomorphism from (h1w,h,h1∼w )(g1) to (h1w,h,h1∼w )(g2) provided the condition (4.3) holds true. conversely, we suppose that the condition (4.3) holds. then we have ‖f‖2h(g1) = 〈f, f〉h(g1) = 〈v f, v f〉h(g2) = ‖v f‖2h(g2), ∀ f ∈h 1 w(g1). ⇒ ‖f‖2h(g1) = ‖v f‖h(g2) say u, ⇒ v is extends to an isomorphism mapping, from h(g1) to h(g2). next, since v (h1w(g1)) = h1w(g1) is dense in h(g2) u has a dense range in h(g2). ⇒ u is an isomorphism from h(g1) to h(g2). hence, by the duality condition, u is an isomorphism between the gelfand triples (h1w,h,h1∼w )(g1) and (h1w,h,h1∼w )(g2) this complete the proof the theorem references [1] h. g. feichtinger and k. gröchenig. banach spaces related to integrable group representations and their atomic decompositions i, journal of functional analysis, 86 (1989), 307-340. [2] h. g. feichtinger and w. kozek. quantization of tf lattice-invariant operators on elementary lca groups. in h.g. feichtinger and t. ströhmer, editors, gabor analysis and alorithms: theory and applications, birkhaüser, boston, 1998, 233-266. [3] i.m. gelfand and n.j. wilenkin. genralized functions, vol. iv: some applications of harmonic analysis. rigged hilbert spaces. academic press, new york, 1964. department of mathematics, r.d. university ,jabalpur, india ∗corresponding author international journal of analysis and applications volume 18, number 4 (2020), 644-662 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-644 received april 1st, 2020; accepted april 20th, 2020; published may 28th, 2020. 2010 mathematics subject classification. 47l15. key words and phrases. commutative; compact; operation; w*-algebras. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 644 a commutative and compact derivations for w* algebras abdelgabar adam hassan1,2,*, mohammad jawed1 1jouf university, college of science and arts in tabrjal, department of mathematics, kingdom of saudi arabia 2university of nyala, department of mathematics, sudan *corresponding author: aahassan@ju.edu.sa abstract. in this paper, we study the compact derivations on w* algebras. let m be w*-algebra, let ( )ls m be algebra of all measurable operators with m , it is show that the results in the maximum set of orthogonal predictions. we have found that w* algebra a contains the center of a w* algebra ß and is either a commutative operation or properly infinite. we have considered derivations from w* algebra two-sided ideals. 1. introduction let m be a w*-algebra and let ( )z m be the center of m . fix a m and consider the inner derivation a  on m generated by the component a , which is  :( ) ,a a =  . the norm closing two sided ideal ( )f b generated by the finite projections of a w* algebra b behaves somewhat similar to the idealized compact operators of ( )b h (see [11],[8],[9]). therefore, it is natural to ask about any sub-algebras d of b that is any derivation from  into ( )f b implemented from an element of ( )y b . https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-644 int. j. anal. appl. 18 (4) (2020) 645 we perform two main difficulties: the presence of the center of b and the fact that the main characteristic in [8] proof (that is, if n q , is a sequence of mutually orthogonal projections and ( )t b h hence 0n nq tq   for all n implies that t is not compact) failure to generalize to the case in which g is of type ii . finally, we have considered derivations from d at the two-sided ( ) ( )( )11 , , 1 1c b b l b      + + =   +   to obtain faithful finite normal trace  on b . 2. notations preliminary lemma (1). let b be a semi-finite algebra, let ( )0q p b and 0 0x q be such that 0x , is a faithful trace on 0q b . assume there are ( )nq p b , ( )nf p and nu b for 1, n ,...i in n += , such that the projections n q are mutually orthogonal and * *, n n n o n n n q u u q f u u= = for all n (i.e., 0n n q q f ). let 0n n n x u f x= . then n jrw x o→ . proof. assume that i n in n q n  = = . let  be a faithful semi-finite normal ( )fsn trace on b + to be agreed on 0q b with 0x  . then for all nq b b +  we have ( ) ( ) ( ) ( ) ( ) ( ) 0 * * * * * * . n n n n n n n n n o n n n n o x n n n n x b u u bu u u u u bu q f u bu f q f u bu f b       = = = = = let ( )p p b be any semi-finite projection. then by [11] there is a central decomposition of the identity ( )1, , 0e e p e e     =  = for   such that ( )pe   for all   . then ( ) ( ) ( ) 1 1 n n n n x n n n pe q pe q q pe q        =  = = =   2 1 n n pe x   =   int. j. anal. appl. 18 (4) (2020) 646 whence 0 n pe x   for all   . let 0  and let   be a finite index set such that 2 0 e x     . then for all n , 2 2 0 2 0 2 0 n n n n n pe x pe u f x pu f e x e x              = =       hence from 22 n n px pe x     + where 0npx → , to completes the proof. lemma (2). let ( )t f p , then there is an 0  and ( )0 e p  such that for every ( )0 f p  with f e we have ( )tf  . proof. let ( )12 0t =  and let g be the sum of a maximal family of mutually orthogonal central projections g  such that ( )tg  . then ( ) ( )suptg tg   =  , hence 1g  . let e z g= − and let ( )0 f p  with f e . since 0fg = , by the maximally of the family we have ( )tf  . 3. relatively compact derivation let m be a w  -algebra and let ( )z m be the center of m . fix a m and consider the inner derivation a  on m generated by the element a , that is  :( ) ,a a  =  . obviously, a there is a linear bounded operator on ( ), m m  , where m  is a c  -norm on m . it is known that there exists ( )c z m such that the following estimate holds: a ma c  − . in view of this result, it is natural to ask whether there exists is an element y m with 1y  and ( )c z m such that  , a y a c − . definition (3). a linear subspace i in the w* algebra m equipped with a norm i  is said to be a symmetric operator ideal if int. j. anal. appl. 18 (4) (2020) 647 (i) , i s s for all s i  (ii) , ii s s for all s i  =  (iii) , , i i asb a s b for all s i a b m   . observe, that every symmetric operator ideal i is a two-sided ideal in m , and therefore by [13], it follows from 0 s t  and t i that s i and i i s t . corollary (4). let m be a w  -algebra and let i be an ideal in m . let : m i → be a derivation. then there exists an element a i , such that  ,a a = =  . proof. since  is a derivation on a w  -algebra, it is necessarily inner [8]. thus, there exists an element d m , such that ( ) ( )  · ,d d  = =  . it follows from the hypothesis that  ,d m i . using [22] (or [20]), we obtain  , ,d m d m i i    = −  =  and  , , 1, 2kd m i k = , where 1 2 , k d d id dk d m  = + =  , for 1, 2k = . it follows now, that there exist ( )1 2, c c z m and ( )1 2,u u u m , such that  , 1 / 2k k k kd u d c − for 1, 2k = . again applying [20], we obtain k k d c i−  , for 1, 2k = . setting ( ) ( )1 1 2 2:a d c i d c= − + − , we deduce that a i and  ,a =  . corollary (5). let be a semi-finite w* -algebra and let be a symmetric operator space. fix and consider inner derivation on the algebra given by . if , then there exists satisfying the inequality and such that . proof. the existence of such that . now, if , then . hence, if , then , where and for , and so , that is . m e ( )a a s m=  a = ( )ls m ( )   ( ), , x a x x ls m =  ( )m e  d e e m e d  →  ( )  ,x d x = d e ( )  ,x d x = ( )u u m ( ) 2 e e e ee u du ud du ud d = −  + =  1 : 1x m x m x =   4 1 i ii x u = = ( )iu u m 1i  1, 2, 3, 4i = ( ) ( ) 4 1 8 i i eie e x u d   =   8m e ed →    int. j. anal. appl. 18 (4) (2020) 648 4. a commutative operation on w* sub-algebras when a a commutative operation is is crucial because it provides the following explicit way to find an operator t b implementing the derivation. for the rest of this section let  be any a commutative operation sub-algebras of b and : b  → be any derivation. let u be the unitary group of  and m be a given invariant mean on u , i.e., a linear functional on the algebra of bounded complex-valued functions on u such that (i) for all real ( )  ( ) , f inf f u u u mf sup f u u u    (ii) for all ( ) ( ), , u uu u mf ms where f v f uv for v u = =  . thus m is bounded and ( )  f u umf sup u  for all f (see [8] for the existence and properties of m ). for each b   the map ( )( )m u u  → is linear and bounded and hence defines an element ( )t b    . explicitly, ( ) ( )( ) t m u u for all b    →  the same easy computation as in [8] shows that a t =  . notice that for all a b the map ( ) ( )( )m u bu e b  → = defines an element ( )e b which clearly belongs to b . moreover it is easy to see that e is a conditional expectation (i.e., a projection of norm one) from b onto b (see [6]). theorem (6). let  be a commutative operation w* sub-algebras of b containing the center of b . for every derivation ( ): f b  → there is a ( )t f b such that a t =  . we have seen that given an invariant mean m on u there is a unique t b such that a t =  and ( ) 0e t = . we are going to show that ( )t b . reasoning by contradiction assume that ( )t b . we proof requires several reductions to the restricted derivation int. j. anal. appl. 18 (4) (2020) 649 ( ):e e f b  → for some ( )0 e p  . to simplify notations we shall assume each time that 1e = . let us start by noticing that if ( )iq p  for 1, 1, , 0n ni n n q q += + = and 1n np q q += + , then ( ) ( ) 1 1 1 n i i n n n n i n ptp q tq q q q q  + + + = = + + hence ( ) ( ) ( ) 1 max n i i i i i i n ptp q tq q tq   + = = + definition (7). for every ( )q p  define    , q q = to be the central projection. set ( )   1p p p p=   = . thus p p iff ( ) ( )ptpg tg = for all ( )g p . we collect several properties of  q . corollary (8). let b be a semi-finite w* algebra with a trace  , let  be a properly infinite w* sub-algebras of b and let 1 1  +   . then for every derivation ( )1: ,c b +→ there is ( )1 ,a t c b + such that a t =  . in the notations introduced there, it is easy to see that ( )( ) ( )1 1, ,c b c b   + += ), where 0   =  and 0  is the usual trace on ( )0b h . we can actually simplify the proof by choosing n i  = since the condition  is no longer required. corollary (9). let 1n n p q q + = + . then there is a largest central projection  1,n nq q + such that for every ( )g p with  1,n ng q q + , we have ( ) ( )1 1q tq g ptpg = . proof. let ( ) ( ) ( ) i i ig g p q tq g ptpg =  = and ( )g p = +  if ( )g p and 0  then ng g . since ( ) ( )maxi i iptpg q tq g = for all ( )g p , we see that ( )1n ng g p+ = . notice that  is hereditary (i.e., g −  and ( ), ff p g   + imply f  ). int. j. anal. appl. 18 (4) (2020) 650 let  1, supn nq q + =  . we have only to show that  1,n nq q +  . let ( )g g  + = + be the sum of a maximal collection of mutually orthogonal projections ( )g  +  . then for every f  we have   ( )( )1, 0n nq q g f+ − + = because of the maximal of the collection of  . then  1,n nq q g + = + . consider now any ( ), 0g p   , then ( )g g g  = + and since ( ) ( )g g g    +  +  , we have ( )( ) ( )( )n nq tq g g ptpg g    + = + for all  . since ( ) ( )( ). n nq tq g resp ptpg  is the direct sum of then ( )( ) ( )( )( ). n nq tq g g resp ptpg g    + + , then we have ( ) ( )( ) ( )( ) ( ) sup sup n n n n q tq g q tq g g ptpg g ptpg           += = = + whence n g g . since 0  is arbitrary, we have  1,n ng q q ++ =  which completes the proof. corollary (10). (i) if 1 0 n n q q + = with ( )iq p  then  11 ,n nq q +−  1,n nq q + . (ii) if 1n n q q +  with ( )iq p  then    1n nq q + . (iii) if 0  with ( ), qq p p  +  then    , q q = and    1 q −  if ( ) 0tg  for all ( )0 e p  then the following hold: (iv) if ( )e p then  e e= . (v) if ( )q p  then   ( )q qc , where ( )c q is the central support of q . proof. we have to show that for every ( )  1, 1 , n ng p g q q +  − we have 1ng g + . let e + be the sum e  of a maximal collection of mutually orthogonal projections of 1ng + that are majored by g . then int. j. anal. appl. 18 (4) (2020) 651 ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 sup sup n n n n n n n n n n n n q tq f q tq f tq q q q q q q q f t f         + + + + + + + + + + = + + = = whence 1n e g + +  . by the maximalist of the collection, ( )0 g e  − + does not majority any nonzero projection of 1n g + and since ( ) 1n np g g +=  , any central projection ( )g g e   − + must be in n g . by definition of  , this implies that ( )g e − +  whence ( )  1,n ng e q q +− +  . so, ( )  11 ,n ng e g q q +− +   − and hence 1ng e g += +  which completes the proof. (ii) let ( )g p and  ng q . then ( ) ( ) ( )1 1n n n ntg q tq g q tq g   + +=  ( )tg whence equality holds and    1n nq q + by the maximalist of  1nq + . (iii)  , q  is maximal under the condition: if ( )g p and  , g q  then ( ) ( ) ( )( ) ( )qtqg q t q g tg     + + = which is the same condition defining    ,q i q q− = . thus    , q q = . applying this to  we have    ,q = and thus by (i) we have      1 , 1q q  − = − . (ii) let ( ),e e p+  then ( )( ) ( )( )ete e te e   + = + . this implies that if 0  , then  e e+  so  e e and if    e e e e+ = −  then ( )( ) ( )( )0 ete e t e   = + = + whence  e e= . (v) follows at once from (ii) and (iv). the condition that ( ) 0te  for all ( )0 e p  is of course meaningless unless b is properly infinite. hence, we may assume without loss of generality that: b is properly infinite and semi-finite. there is an 0  such that ( )te  for all ( )0 e p  . int. j. anal. appl. 18 (4) (2020) 652 lemma (11). let p p and  ) ( 1, , ,n ptp n ptpr x r x +=  = − − , where ( )ptpx denotes the spectral measure of the self-adjoint operator ptp . then there is an ( )ne p , with n e i e= − such that i i r e are properly infinite and ( ) i i jc r e e= for , n 1i n= + . proof. let  )1 ,n n ptpr r r x += + =  and let 0f  be any central projection. if rf were finite, we would have ( ) ( ) ( )( ) ( )( ) 1 1 tf ptpf ptp r f ptp r f      = = − = −  thus rf is infinite and nonzero. hence r is properly infinite and ( )c r n= . now let 1e be the maximal central projection majored by ( )nc r , such that n nr f is properly infinite. then ( ), n n nc r e e= and ( )n nr n e− is finite, hence ( ) 1 1n n n nr n e r e+ + +− = is properly infinite and ( )1 1 1, n n nc r e e+ + += . end of the proof of theorem (6). take any ( )00 q p b  such that 0q b has a faithful trace 0x  with 0 0x q h and assume 0 1x = . let ,p p   be the not decreasing to zero. we are going to construct inductively a sequence ( ) ( ), , ,n n n nf p q p b u    partial isometrics in b, n x h such that (a) * * 0 0 , , . ., n n n n n n n n u u q u u q f i e q q f= = (b) 0n n n n x u f x q h=  (c) 0 n m q q for n m=  (d) ( ) n mn m hence p p for n m       (e) nn q p   (f) 1 1 n nn p x  +  (g) 2,n ntx x  . int. j. anal. appl. 18 (4) (2020) 653 the induction can be started with an arbitrary p  ; assume we have the construction for 1n − . let us apply lemma(11) to n p p  = and obtain ( ) ( ), = , 1i ie p r p b for i n n  + as defined there. then 2 2 2 0 1 0 1 o n n x e x e x + = = + let n f be (any of) the projection n e or 1n e + for which 2 1 0 2i e x  and let i be the corresponding index. then i n r f is properly infinite and has central support n f . now 0 q is finite having a finite faithful trace 0x  , hence so is 0 0 1 1 j j q f q q for j n   − and ( ) 1 1 n j nj q f − = . let ( )  1 1 , 1 n n i n j nj s inf r f q f − = = − . by the parallelogram law (see [2]) applied to n f we have that ( ) 1 1 1 1 inf , 1 n n i n n j n j n i n j j r f s q f q f r f − − = =       − − −             whence i n n r f s− is finite and hence n s is properly infinite and ( )n nc s f= . since 0 nq f is finite and ( )0 n nc q f f we have 0 n nq f s , i.e., there is a partial isometry nu b and a ( ),n n nq p b q s  such that (a) holds. let nx be defined by (b) and choose 1n n + so that (d) and (f) hold. -since nn i q r p    we have (e), since ( ) 1 1 1 n n j nj q q f − =  − we have (c). finally nn i n n x r x p x  = = hence (g) follows from ( ) ( ) ( ) ( ) 2 2 0 1 2 , , , , . n n n n n n n n i n i n i n i n n n tx x p tp x x p tp r x r x r x r x x f x         = =  = =  let now 1nn n n y x p x  + = − . b is semi-finite, hence we can apply lemma (1) to obtain that 0 n bew x → . since 1 0 n n p x  + → we thus have 0 n bew y → and n n y p h , where int. j. anal. appl. 18 (4) (2020) 654 ( ) 1n nn p p p p d   + = −  and are mutually orthogonal by (d). clearly for n large enough, ( ) ( ) 14, nn n yty y t =  . since ( ) ( )( )n ny yt u um    = , by the properties of the invariant mean mentioned, we have that ( )( )  14sup ny u u u u      . thus we can find for every n , a unitary n v u such that ( )( ) 14,n n n nv v y y    . let 1 n nn a v p  = = , then a d and ( ) ( )( ) ( )( ) ( )( ) ( )( ) 1 4 , , , , , n n n n n n n n n n n n n n n n n n n n a a y y p a a p y y p a at a ta p y y p v v p y y v v y y            = = − = = = for all n . therefore ( ) 0na y → . but because of (π), we have ( ) ( )a f b  , which completes the proof. 5. the property of infinite w* sub-algebra lemma (12). let be a properly infinite projection and . let projection be finite or properly infinite, and . let . for every we denote by such a projection that is the largest central projection, for which holds. we have and for the following relations and . moreover, if all projections are finite then is a finite projection as well. proof. since, we have ( ) ( ) ( )0 , 1; 0,azb z m s b e  =  ( )( )0, 1azc e  = ( )q p m ( ) 1c q = ( )0,azq e  0n  n nz 1 nz− ( ) ( )1 1 an n zz q z e−  − ( ),nb + 1n nz  ( )1 1 1 1 1 : n n n n d z z z b   + + =    −  = +    ( ) 1: , , 0 a z hold q e d d b+   ( ) 1s d = ( ), , 1az ne b n +  ( ), a z e d + ( ) ( )1 , , a a z n z n e b e b  + +  + int. j. anal. appl. 18 (4) (2020) 655 . hence, for every . in addition, and is properly infinite projection. hence, in the case when is finite projection, it follows that . let us consider the case when is a properly infinite projection with and such that . in this case, with and deduce . all other statements follow from the form of element . since, and for every . observe also that . finally, let all projections be finite. since , we have for every . there projections standing on the right-hand sides are finite. hence, is finite projection as a sum of the left-hand sides [22]. we shall use a following well-known implication . we supply here a straightforward argument. let be such that . then and therefore . this means . as in [6] we can use theorem (6) to extend the result to the properly infinite case. theorem (13). let  be a properly infinite w* sub-algebra of b containing the center of b . for every derivation ( ): f b  → there is a ( )t f b such that a t =  . before we start the proof let us recall that if  is properly infinite there is an infinite countable decomposition of the identity into mutually orthogonal projections of  , all ( ) ( ) ( ) ( ) ( )1 1 1 11 1 , 1 , a a a z n n z n n z n e z q z e b z e b  + + + + −  − +  − + 1n n z z +  n ( ) ( ), 0,a az n n ze b e +  + ( )0, a z e + q 1 n n z  q ( ) 1c q = ( )0,azq e  ( ) ( ), 0, , ,a az n z np q q e q e b= = + = + ( )1 1nn z c q  =  = d 1 1 1 ,z d z b= ( ) ( )1 1 1n n n n nz z z z b+ + +− = − ( ), a n n z n z q z e b + n ( ) ( ) ( )( )1 11 1n nns d s b z z z  += = + − = ( ), , 1az ne b n +  ( )1 1 1, n ndz b d z z += − = ( )1 1n n nb z z + + − ( ) ( )1 1 1, , , a a z z e d z e b z+ = + ( )( ) ( )( )1 1 1, , a a z n n z n n n e d z z e b z z + + + + − = + − n ( ),aze d + ( )( ) ( ) ( ), , 0p q zp zq z p z m z c p c q      ( )z z z m  ( ) ( ) ( ) ( )( )0 z c pz c qz z c p c q    ( ) ( )z c p c q   ( ) ( )z zp z p z q z zq   = = zp zq int. j. anal. appl. 18 (4) (2020) 656 equivalent in  to i, and thus a fortify equivalent in b to 1 [8]. therefore there is a spatial isomorphism ( )0: b b b b h → =  with ( )10 n h l + = and ( ) ( )0b h  =  =  [5]. recall also that the elements b of b (or  ) are represented by bounded matrices , , ij b i j    with entries in b (or  ) by the formula ( ) ( )ij kl jk ili e t i e t e  =  where ij e is the canonical matrix unit of ( )0b h . in particular if ,  are the maximal a commutative operation subalgebras of ( )0b h of laurent (resp. diagonal) matrices, then ( ).b b resp b b    iff ijb   is a laurent matrix with entries in b , i.e., ij i jb b −= ,where k b , denotes the entry along the kth diagonal( ). ij ij iiresp b b= for all ,i j  . proof. let 1    − = then ( )( ) ( ): d f b f b → = is a relative compact derivation. let us define the following w* algebras: ( )1 1, , ,n n n nb b  − + =   =   =   =  , and 2n n+  =  . first, let us notice that ( ) ( ) ( )( ) ( ) ( ) ( )   0 0 n f b b b h f b b f b   =     =   = by [22]. therefore ( ) ( ) ( ) ( )( )  1 1 0n n nf b a f b f b − −    =  =   = because  is spatial now ( )( ) ( )0 . n b b h b i i =    =      int. j. anal. appl. 18 (4) (2020) 657 thus we can apply theorem(6) to the derivation  restricted to the a commutative operation sub-algebra n  of b and we obtain a ( )nt f b such that n na t = −  vanishes on n . now 1n n b +       =  . therefore, for all n n a  and 1 1n n a + +  we have ( ) ( ) ( ) ( )1 1 1 1n n n n n n n n n n n na a a a a a a a   + + + += = = i.e., ( )1n na + and na commute and hence ( ) ( )  1 0n n n f b +    = thus n  also vanishes on 1n+  . now n  is a commutative operation and hence so are n and 2n+  . moreover, n b     implies ( )1 n b − =      and hence 2n n i i b + =          thus we can apply again theorem(6) to the relative compact derivation n  restricted to 2n+  . let ( )1nt f b+  be such that n agrees with ad 1nt + on 2n+ . since 1n n i i +       =  and n  vanishes on 1n+  , we see that ad 1nt + vanishes on n i  , i.e., ( ) ( ) ( )( ) ( )1 0n n nt i f b b h fg b+      =    then for all ( )1, , n niji j t +   and ( ) ( ) ( ) ( )2 1nn ni n jnijt e i e t i e f b+ =    int. j. anal. appl. 18 (4) (2020) 658 whence by lemma(12)(a) ( ) ( )1n ijt f b+  . but we saw that ( )  0nd f b  = , hence ( )1 0n ijt + = for all ,i j  , so 1 0 n t + = . therefore n  vanishes also on 2n+  and hence on i . now and  generate ( )0b h , whence 1n+ =  and i  generate  . thus by the  -weak continuity of n  (see [6]) we see that 0, . ., n n n a t i e a t  − = = =  . clearly ( )1 nad t  − = and ( ) ( )1 nt b −  . let us assume in this part that b is semi-finite and let  be a fsn trace on it. beside the closed ideal ( )f b we can also consider the (non closed) two-sided norm-ideals ( )1 , 1 1c b for  +  +   defined by ( ) ( )  ( ) ( ) 1 1 1 1 1 11 , , . c b b b b b b for b c b          + + + + ++ =    =  obviously, ( ) ( )11 , , ,c b b l b     + + =  where the latter is the non commutative 1 l + -space of b relative to  (see [14]). recall the following facts about ( )1l m+ spaces in the case of a general w* algebra m and 1 1  +   ( ( )l m is identified with m ): ( )1l m+ is a banach space, its dual is isomorphic to ( )1l m  + (with 111 1    + + + = ), and the duality is established by the functional tr on ( ) 1 l m , where if ( ) ( )11 ,a l m b l m   ++  we have ( ), nab ba l m and ( ) ( ) ( ) 1 1 ,tr ab tr ba tr ab a b    + + =  , ( ) ( )  1 1 1 1 1 1 max , 1a tr a trab b l m b       + + + + + = =   (see [14]). of course, if m b= we can identify ( )1l m+ with ( )1 ,l b + and tr with  . the following inequality will be used here only in the semi-finite case and in the context of 1 c + ideals, but since the same proof holds for 1 l + -spaces, we shall consider the general case. int. j. anal. appl. 18 (4) (2020) 659 corollary (14). let m be a w* algebra, ( )10, a l m +  and ( )1 1 1, , 0, 1n n n n n nq q p m q q q q+ + + = + = . then 1 1 1 1 1 1 1 1n n n n a q aq q aq      + + + + + + + +  + proof. let us first note that 1 1 1 1n n i i i i i n i n q aq q aq   + + + + = = =  and 1 1 1 1 1 1 i i n n i i i n i n q aq q aq     + + + + + = =+ =  consider first 1 n+ = and take the polar decomposition's , , 1 i i i i i q aq u q aq i n n= = + . then i i u u  and i i u u  are majored by i q and hence i u commutes with j q . therefore ( ) * 1n n b u u + = + commutes with i q and 1b = . then 1 1 1 1 . n i i i n n i i i n n i i i n n a trab tr q baq tr q aq q aq + = + = + =    =       =     =    consider now 0  . let ( )1b l m  + be such that 1 1b  +  and 1 1 1 . n n i i i i i n i n q a q tr q a q b  + + = =+    =         take the polar decomposition's a u a= and b v b= , then vu are in m and ,a b are in ( ) ( )1 1,l m l m   + + , respectively. let int. j. anal. appl. 18 (4) (2020) 660 ( ) ( ) ( ) 1 1 1 1 . n z z i i i n f z tr q u a q v b    +   + −  +  =   =      then by standard arguments, it is easy to see that f is analytic on 0 re z n  and continuous and bounded on 0 re z n  . then by the three-line theorem (see [4]) we have ( ) ( ) ( ) 1 1 11 1 1 t t if ma t itx f max f     + + +   + now 1 1 1 1 n i ii n q a qf  + = +   =  +   and by holder’s inequality ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 n i t i t j j j n n i t i t j j j n f it tr q u a q v b b q u a q v b b           +   + −   + +  = +   + −   + +  =   =        ( )( )1 11 1 max . i t i t j j j q u a q v b b n        + −   ++  +   again by holder’s inequality applied twice and by the result already obtained in the 0 = case, ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n i t i t j j j n n i t i t j j j n i t i t f it tr q u a a q v b q u a a q v b u a a u a a a               +   + + −   +  = +   + + −   +  = + + + + + +   + =           thus ( )11 1f a + + whence by the second equality in this proof, 1 1 1 1 1 1 1 1 n n i i i i i n i n a q aq q aq       + + + + + + + = =+  =  data availability no data were used to support this study. int. j. anal. appl. 18 (4) (2020) 661 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] i. chifan, s. popa, j.o. sizemore, some oeand w∗-rigidity results for actions by wreath product groups, j. funct. anal. 263 (2012), 3422–3448 [2] m. breijer, fredholm theories in von algebras i, math. ann. 178 (1968), 243-254. [3] e. christensenex, extension of derivations, j. funct. anal. 27 (1978), 234-247. [4] j. conway, functions of one complex variable, 2nd ed., springerverlag, new york,1978. [5] j. dixmier, les algebres d’operateurs dans 1’espace hilbertien, 2nd ed., gauthier-villars, paris, 1969. [6] f. gilfeather and d. larsinn, nest-subalgebras of von neumann algebras: commutants modulo compacts and distance estimates, j. oper. theory, 7 (1982), 279-302. [7] a. connes, e. blanchard, institut henri poincaré, institut des hautes études scientifiques (paris, france), institut de mathématiques de jussieu, eds., quanta of maths: conference in honor of alain connes, non commutative geometry, institut henri poincaré, institut des hautes études scientifiques, institut de mathématiques de jussieu, paris, france, march 29-april 6, 2007, american mathematical society ; clay mathematics institute, providence, r.i. : cambridge, ma, 2010. [8] s. albeverio, sh. ayupov, k. kudaybergenov, structure of derivations on various algebras of measurable operators for type i von neumann algebras, j. funct. anal. 256 (9) (2009), 2917–2943. [9] v. kaftal, relative weak convergence in semifinite von neumann algebras, proc. amer. math. soc. 84 (1982), 89-94. [10] s. sakal, c*-algebras and w*-algebras (ergebnisse der mathematik und ihrer grenzgebiete, vol. 60), springer-verlag, berlin, new york, 1971. [11] m. takesaki, theory of operator algebras i, springer-verlag, new york, 1979. [12] n. higson, e. guentner, group c*-algebras and k-theory, in noncommutative geometry (martina franca, 2000), pp. 137-251. lecture notes in math., 1831. [13] d. voiculescu, free non-commutative random variables, random matrices and the ii1–factors of free groups, quantum probability and related topics vi, l. accardi, ed., world scientific, singapore, 1991, pp. 473–487. [14] a.f. ber, f.a. sukochev, commutator estimates in w∗-factors, trans. amer.math. soc. 364(2012), 5571-5587. [15] f. murray, j. von neumann: rings of operators, iv, ann. math. 44(1943), 716-808. int. j. anal. appl. 18 (4) (2020) 662 [16] j. peterson, l2-rigidity in von neumann algebras, invent. math. 175 (2009), 417–433. [17] b.e. johnson, s.k. parrott, operators commuting with a von neumann algebra modulo the set of compact operators, j. funct. anal. 11 (1972), 39–61. [18] r. kadison, a note on derivations of operator algebras, bull. lond. math. soc. 7 (1975), 41–44. [19] k. dykema, free products of hyperfinite von neumann algebras and free dimension, duke math. j. 69 (1993), 97-119. [20] c. consani, m. marcolli, noncommutative geometry, dynamics, and ∞-adic arakelov geometry, selecta math. 10 (2004), 167. [21] a.f. ber, f.a. sukochev, commutator estimates in w∗-algebras, j. funct. anal. 262 (2012), 537–568. [22] d. pask, a. rennie, the noncommutative geometry of graph c∗-algebras i: the index theorem, j. funct. anal. 233 (2006), 92–134. [23] s. popa, f. radulescu, derivations of von neumann algebras into the compact ideal space of a semifinite algebra, duke math. j. 57(2)(1988), 485–518. [24] i.e. segal, a non-commutative extension of abstract integration, ann. math. 57 (1953), 401–457. international journal of analysis and applications volume 19, number 5 (2021), 773-783 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-773 new numerical solution for two parametric surfaces intersection dragging problem ramadhan a. m. alsaidi∗ department of mathematics, college of science and arts in gurayat, jouf university, gurayat 77454, saudi arabia ∗corresponding author: rsaidi@ju.edu.sa abstract. the problem of intersecting two parametric surfaces has been one of the main technical challenges in computer-aided design, computer graphics, solid modeling, and geometrics. this paper aims at reducing and minimizing time and space required for the computations process of parametric surface intersection. to do this, a new numerically accelerating method based on continuation technique was utilized first by calculating a starting point, and second by tracing sequential points along the intersection curve following broyden’s method. two factors have been identified as influential in controlling component jumping: initial points and step size. test examples of intersecting two parametric surfaces demonstrated that this method was highly efficient with high-speed parametric solution. the intersection results are often given as curve’s points. 1. introduction the intersection between two parametric surfaces has been an incessantly fascinating yet challenging topic in algebraic geometry because it has significant applications in computer graphics, modeling and computeraided geometric design. the applications of this geometrical algebraic notion cover a wide-range of varying computerized processes including simulation of manufacturing processes, boundary evaluations, finite element of mesh generations, and contouring of scattered data. for its graphically-oriented signification, the topic received april 24th, 2021; accepted june 15th, 2021; published august 26th, 2021. 2010 mathematics subject classification. 68u05. key words and phrases. broyden’s method; continuation method; curve tracing; initial point; parametric surface; surface intersection. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 773 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-773 int. j. anal. appl. 19 (5) (2021) 774 has been of a great appeal to young and old researchers in computing arena. with the mounting interest in the field of computing designs, intersecting two surfaces in particular became the focus of several studies in the last decade of the millennium. a cataloguing survey of these studies may include, among many others, chapter 12 of [1] hoschek and lasser which presented a comprehensive overview to the problem of intersecting free-form curves and surfaces; patrikalakis’s [2] review of the surface-surface intersection (ssi) methods which developed between (1988-1992); farin’s [3] ssi bibliography of 50 references, including techniques developed in the era of 1968–1990. patrikalakis and maekawa [4, 5] proposed a new results developed in the late 90s, containing an extensive body of ssi literature. on the other hand, special styles of surfaces such as sweep surfaces or torus were studied in [6]. a new algorithm for torus/torus intersection was introduced in [7]. ku-jinkim [8] suggested a system to calculate all the circles in the intersection curve of two tori established on the geometric properties of the circles included in a torus. recent studies include [9] in which an efficient and robust technique using a bounding volume hierarchy (bvh) for computing the intersection curve of two freeform surfaces has been discussed. numerous methods have been suggested to ray trace parametric surfaces, such as nurbs or bézier surfaces. nishita et al. [10] examined the clipping algorithm for computing the points at which a ray intersects a rational bezier surface patch. campagna et al. in [11] extended and optimized the nishita et al. algorithm. to speed up nishita’s method, wanget al. [12] combined bezier-clipping ray coherency and newton iterations. for computing the self–intersection and intersection curve(s) of two biquadratic b´ezier surface patches, chau, stéphane, et al. [13] introduced three symbolic–numeric procedures. in [14], sylvester matrix was used to construct a 9th degree polynomial coefficients of two cubic b´ezier curves for getting the intersection points of the aforementioned curves through computing the real roots of this polynomial. alsaidi, r. a. m [15] presented two methods for computing the intersection points of two parametric surfaces and one technique for computing the start point based on extended newton method. these methods gave an acceptable accurate result for computing the intersection curve between two surfaces. continuation technique among these methods uses extended newton method frequently. however, newton method has many disadvantages such as both the jacobian matrix and its inverse actions should be calculated for each iteration which consumes quite time depending on the size of problem system. to overcome this problem, this work uses new accelerating methods known as quasi-newton methods or broyden’s method [16] instead of newton method. the paper aims to solve the dragging problem of intersection parametric surfaces including biquadratic b´ezier form surface intersection by optimizing alsaidi’s work [15] through minimizing the required memory and rendering the computations as simple as possible. in the next section, the problem will be stated. in section 3, newton method and broyden’s method are recalled and compared. the developed technique is explained in section 4. section 5 deals with the numerical problems so as to compare the develop method with alsaidi’s work [15]. section 6 contains summary and the concluding remarks of this work. int. j. anal. appl. 19 (5) (2021) 775 2. problem statement let s1(u, v) and s2( s, t) be two parametric surfaces defined as: s1 =   x1(u,v) y 1(u,v) z1(u,v)   and s2 =   x2(s,t) y 2(s,t) z2(s,t)   then the problem of intersection between them involves the setting s1(u,v) = s2(s,t) ⇒ s1(u,v) −s2(s,t) = 0 which can be interpreted as three nonlinear polynomial equations in four unknowns (u,v,s,t) as: (2.1) c(u,v,s,t) =   x1(u,v) −x2(s,t) y 1(u,v) −y 2(s,t) z1(u,v) −z2(s,t)   =   f1(u,v,s,t) f2(u,v,s,t) f3(u,v,s,t)   =   0 0 0   this system can be solved by computing the set p = { (u,v,s,t)�[0, 1]4||c(u,v,s,t) = s1(u,v) −s2(s,t) = 0 } which is defined as a curve in four-dimension space c(u,v,s,t) = (u(α),v(α),s(α), t(α)) on the other hand, two biquadratic b’ezier surfaces x(u, v) and y(s, t) are defined by x(u,v) = ∑2 i=0 ∑2 j=0 pijbi(u)bj(v), y (s,t) = ∑2 i=0 ∑2 j=0 cijbi(s)bj(t), where pij and cij are restricted to be in q3, and the quadratic bernstein polynomials is defined as bi(u) =   2 i  ui(1 − u)2−i consequently, the intersection curve is computed by solving the following system x(u, v) = y(s, t) which is represented by three non-linear equations and four unknown variables. note that the intersection curve is as in [0, 1]4. however, there are many numerical techniques for solving these equations. this paper focuses on continuation method [15, 17, 18] which is efficient in both cases of parametric surface and biquadratic bezier surface. 3. developed method in this work, the computation system of the intersection points between two parametric surfaces consists of two different and successive phases: first, calculating the starting point, then tracing sequential points along the intersection curve [15, 19]. 3.1. computing a starting point. as mentioned above, the intersection problem can be viewed as solving the system of equation 2.1. there are many numerical methods to solve this problem such as newton method which is used in many works such as alsaidi’s [15]. however, the newton’s method has the disadvantage that the jacobian matrix and its inverse should be calculated at each iteration. this implies that the form of int. j. anal. appl. 19 (5) (2021) 776 the iterative procedure for newton’s method takes longer time. this work, consequently, utilizes broyden’s technique to avoid this problem. in this approach, an approximation matrix is employed instead of the jacobian matrix. the approximation matrix can advantageously be changed at each iteration. in this way, the broyden’s iterative expression is derived as: xn+1 = xn −a−1c (xn) here is an defined as: an = an−1 + yn −an−1ansn ‖sn‖ 2 2 stn where yn = cn − cn−1 and sn = xn − xn−1. for a more detailed treatment the reader is encouraged to consult [16]. for the intent of simplicity, the following algorithm summarizes the steps of the broyden’s technique to compute the starting point: algorithm 1 (broyden’s method). input: the function matrix c of 2.1. an initial point given x0 = (u0,v0,s0, t0). a given tolerance ε. number of iterations m. (1) evaluating c (x0). (2) computing the jacobian matrix j(u,v,s,t) =   ∂f1 ∂u ∂f1 ∂v ∂f1 ∂s ∂f1 ∂t ∂f2 ∂u ∂f2 ∂v ∂f2 ∂s ∂f2 ∂t ∂f3 ∂u ∂f3 ∂v ∂f3 ∂s ∂f3 ∂t   (3) since in this stage there is not enough information to calculate a0, set l = 04 and aa = j (x0) ⇒ a−10 = i −1 (x0), viz the jacobian matrix j is computed at initial guess x0. (4) extracting the new point by x1 = x0 −a−10 c (x0) (5) evaluating c ( x1 ) . (6) calculating y1 = c (x1) −c (x0) and s1 = x1 −x0. (7) estimating st1a −1 0 y1. (8) computing a−11 = a −1 0 + ( 1 st1a −1 0 y1 )[( s1 −a−10 y1 ) st1a −1 0 ] (9) extracting next point via x2 = x1 −a−11 c (x1) ,l = l + 1 (10) the previous process is repeated by updating initial guess (x0 = x2) in the next stage until the threshold is created (i.e. c (x2) − c (x1) ≤ ε or l > m). this indicates that the starting point (i.e.x1 = (u1,v1,s1, t1)) is computed. int. j. anal. appl. 19 (5) (2021) 777 the sequential intersection points can be traced along the tangent direction by the following procedures: (i) determination of the tangent unit vector to the path which is uniquely defined by a · λ = 0 and represented by the solution by q. (ii) selection the desired distance 4 and the tangent vector direction. (iii) extraction of the initial guess via x0 = x1 + (q ×4), where 4 is the step size. (iv) to find next point, solve eq. 2.1 using broyden’s technique. (v) evaluation of the relation error (e). if ≤ ε, then the next point is computed and then go to step iii. (vi) else, go to step (iv). 3.2. type convergence of broyden’s method. it should be noted that broyden’s technique converges superlinearlly. this indicates that limn→∞ ‖xn+1−p‖ ‖xn−p‖ = 0, where p is the solution to the system c(x) = 0, and xn and xn+1 are successive approximations to p. 4. numerical implementations the develop system has been applied and its performance was measured by some representative numerical applications. all the computations were performed on a computer with 8 gb ram and 2.50 ghz intel core i5 processor and the code was applied in matlab. the average of accuracies for every experiments was 100%. figure 1. two biquadratic bezier surfaces b1 and b2 int. j. anal. appl. 19 (5) (2021) 778 4.1. first example: consider two biquadratic surfaces (b1 and b2 ) having the following control points: b1(u,v) =   ( 1 7 , 0, 3 5 ) ( 3 5 , 1 5 , 3 4 ) ( 1, 0, 7 10 ) ( 3 8 , 4 9 , 2 3 ) ( 2 3 , 3 4 , 1 3 ) ( 6 7 , 3 8 , 5 7 ) ( 1 5 , 6 7 , 4 7 ) ( 3 4 , 7 8 , 3 4 ) ( 7 8 , 7 9 , 5 8 )   b2(s,t) =   2 7 , 1 7 , 2 5 ) ( 3 5 , 1 10 , 2 3 ) ( 1, 0, 4 5 ) ( 3 8 , 4 9 , 2 3 ) ( 1 3 , 1 2 , 1 ) ( 5 7 , 3 8 , 2 7 ) ( 1 5 , 6 7 , 3 7 ) ( 3 4 , 7 8 , 5 8 ) ( 7 8 , 4 7 , 1 2 )   the two bezier’s surfaces are displayed in figure 1. it is worth pointing out here that the performance of the proposed system needs to select the appropriate initial guess which was chosen as x0 = (0.9994, 0.2501, 0.9994, 0.2502) and the step size was determined as 0.02. the resulting intersection curve between b1 and b2 as points is shown in figure 2. figure 2. the intersection curve points between b1 and b2 on the u,v space a comparison between the time performance of proposed system and alsaidi’s work is done. it is provided for some points in table 1. int. j. anal. appl. 19 (5) (2021) 779 kth point develope’s point coordinates alsaidi’s point coordinates number of iterations time performance in seconds developed alsaidi developed alsaidi 1st (0.7307, 0.2501, 0.7936, 0.3184) (0.7274, 0.2327, 0.7849, 0.2990) 12 6 0.0479425 5.1307965 30th (0.5193, 0.5305, 0.6851, 0.6779) (0.5367, 0.5272, 0.7021, 0.6699) 3 3 0.010677 0.1008304 61at (0.1332, 0.6325, 0.2210, 0.7251) (0.1474, 0.6275, 0.2400, 0.7278) 3 3 0.013063 0.066004 100in (0.2801, 0.2146, 0.2897, 0.2206) (0.2643, 0.2274, 0.2758, 0.2336) 4 3 0.017218 0.071702 127th (0.6313, 0.1290, 0.6533, 0.1722) (0.6160, 0.1248, 0.6355, 0.1653) 3 3 0.01235 0.062806 166th (0.5656, 0.5205, 0.7289, 0.6544) (0.5791, 0.5164, 0.7409, 0.6460) 4 3 0.017066 0.067136 200th (0.1389, 0.6306, 0.2286, 0.7936) (0.1493, 0.6268, 0.2425, 0.7281) 3 3 0.018479 0.078431 table 1. a comparison between the time performance table 1 shows that the present system runs faster and achieves the best performance in the first example. 4.2. second example: given the two surfaces s1(u,v) =   u v u2 + v2   and s2(s,t) =   s t 45−s2−t2 5  . figure 3 shows the two surfaces. figure 3. two parametric surfaces s1(u,v) and s2(s,t) ideally specify an initial point[u0,v0,s0,u0] = [1, 1, 1, 1]. after the introducing system has been applied, the outcome intersection curve represented as points (i.e. 151 points) is illustrated in figure 4. int. j. anal. appl. 19 (5) (2021) 780 figure 4. the intersection curve points between s1 and s2 on the u,v space it is to be noted that the experiment was performed under the same environment which alsaidi [15] used. the comparison results indicate that the developed system consistently outperformed the alsaidi’s methods as shown in table 2. 2* kth point develope’s point coordinates 2* alsaidi’s point coordinates number of iterations time performance in seconds developed alsaidi developed alsaidi 1st (1, 2.5495, 1, 2.5495) (1.936, 1.936, 1.936, 1.936) 9 6 0.0097 0.299215 25th (−2.092, 1.767,−2.092, 1.767) (−1.198, 2.463,−1.198, 2.463) 4 3 0.003846 0.110219 71st (0.257,−2.726, 0.257,−2.726) (−0.85,−2.603,−0.85,−2.603) 4 3 0.003947 0.126905 105th (2.633, 0.754, 2.633, 0.754) (2.7148,−0.3605, 2.7148,−0.3605) 4 3 0.004042 0.118836 127th (0.433, 2.704, 0.433, 2.704) (1.473, 2.31, 1.473, 2.31) 4 3 0.003772 0.129153 151st (−2.419, 1.284,−2.419, 1.284) (−1.702, 2.146,−1.702, 2.146) 4 3 0.004643 0.132114 table 2. showing the results of the comparison between in example2. significantly observed, alsaidi’s work-run took an average of 0.121591624 sec which is considerably more than the average of 0.004146306 sec of the proposed system. hence, it can be said that the developed system offers the best quality–runtime ratio. 5. analysis the execution of the proposed system is significantly affected by the following factors: initial guess and step lengths through tracing the control component jumping. the effect of these parameters is investigated in the previous example. int. j. anal. appl. 19 (5) (2021) 781 5.1. effect of initial guess: it is well-known that the select of an initial guess is very efficient in the computation of any proposed method. however, there is no common solution that fits all concrete problems or cases. there are many works that set some criteria to determine the initial guess but they did not reach the efficiency in problem solving. consequently, the best choice of the initial guess in this work was determined experimentally by taking into consideration that the initial guess is close to the solution (see table 3 . initial guess run-time(average) [1, 1, 1, 1] 0.004146305960265 [0, 1, 1, 1] 0.003603891390728 [1, 0, 1, 1] 0.003603891390728 [1, 1, 0, 1] 0.004146305960265 [1, 1, 1, 0] 0.004146305960265 [1, 1, 0, 0] 0.003347392052980 [1, 0, 0, 0] 0.002750917880795 [0, 0, 0, 0] convergence not achieved table 3. influence of initial guess on run-time and convergence 5.2. effect of step size: the efficiency of the tracing in a proposed system is dependent on getting good step size to avert bisection. nevertheless, too small step size increases the number of segments that are required for approximating the intersection curve within a given tolerance [20]. in this work, the effect was investigated through the performance of the develop system with different size of step lengths while another parameter was kept fixed. the result shows in table 4. size length run-time(average) 0.02 0.003288 0.2 0.004146306 0.6 0.003288048344371 table 4. effect of step size on run-time and convergence int. j. anal. appl. 19 (5) (2021) 782 (a) step size=0.2 (b) step size=0.06 (c) step size=0.02 figure 5. plots of effect of step size 6. conclusion this work introduces a develop continuation algorithm method based on broyden’s technique to compute points of curve yielded from an intersection between two parametric surfaces. it has experimentally been found that the proposed algorithm can be applied to a variety of parametric surfaces efficiently. the comparisons between the developed technique and alsaisi’s work has been applied to measure out and validate the efficacy of the developed method. moreover, the effect of factors as well as possible restrictions has been numerically examined. the main contribution of this research work has been the reducing of the dragging intersection problem by speeding up the computation process. the developed system offers the best quality–runtime ratio and achieves the best performance in biquadratic bezier surfaces. it is recommended to extend the application of the current intersection technique on implicit surfaces and rational b-spline surfaces to more general and complex surfaces. acknowledgement: the author feels grateful to the reviewers and editors for their suggestions and constructively insightful comments that refine and improve the paper. this work has been supported by jouf university, ksa. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. hoschek, d. lasser, fundamentals of computer aided geometric design, ak peters, ltd., wellesley, ma, 1993. [2] n. m. patrikalakis, surface-to-surface intersections, ieee computer graph. appl. 13 (1) (1993), 89–95. [3] r. e. barnhill, geometry processing for design and manufacturing, siam, philadelphia, 1992. [4] n. m. patrikalakis, t. maekawa, shape interrogation for computer aided design and manufacturing, vol. 15, springer, 2002. [5] n. m. patrikalakis, t. maekawa, intersection problems, in: shape interrogation for computer aided design and manufacturing, springer, 2010, pp. 109–160. int. j. anal. appl. 19 (5) (2021) 783 [6] j.-k. seong, k.-j. kim, m.-s. kim, g. elber, r. r. martin, intersecting a freeform surface with a general swept surface, computer-aided design, 37 (5) (2005), 473–483. [7] x.-m. liu, c.-y. liu, j.-h. yong, j.-c. paul, torus/torus intersection, computer-aided design appl. 8 (3) (2011), 465–477. [8] k.-j. kim, circles in torus–torus intersections, j. comput. appl. math. 236 (9) (2012), 2387–2397. [9] y. park, s.-h. son, m.-s. kim, g. elber, surface–surface-intersection computation using a bounding volume hierarchy with osculating toroidal patches in the leaf nodes, computer-aided design, 127 (2020), 102866. [10] t. nishita, t. w. sederberg, m. kakimoto, ray tracing trimmed rational surface patches, in: proceedings of the 17th annual conference on computer graphics and interactive techniques, 1990, pp. 337–345. [11] s. campagna, p. slusallek, h. p. seidel, ray tracing of parametric surfaces: bezier clipping, chebyshev boxing and bounding volume hierarchy. a critical comparison with new results, computer graphics group, university of erlangen, germany (1997). [12] s.-w. wang, z.-c. shih, r.-c. chang, et al., an efficient and stable ray tracing algorithm for parametric surfaces, j. inform. sci. eng. 18 (4) (2002), 541–561. [13] s. chau, m. oberneder, a. galligo, b. jüttler, intersecting biquadratic bézier surface patches, in: geometric modeling and algebraic geometry, springer, 2008, pp. 161–180. [14] b. bi ly, the method of finding points of intersection of two cubic bezier curves using the sylvester matrix, silesian j. pure appl. math. 6 (2016), 155–176. [15] r. a. m. alsaidi, a. musleh, two methods for surface/surface intersection problem comparative study, int. j. computer appl. 92 (2014), 1-8. [16] r. burden, j. faires, a. burden, numerical solutions of nonlinear systems of equations, in: numerical analysis. cengage learning, boston, (1997) pp 544–576. [17] w. c. rheinboldt, numerical analysis of parametrized nonlinear equations, wiley-interscience, 1986. [18] h. b. keller, lectures on numerical methods in bifurcation problems, springer verlag, heidelberg, germany, 1987. [19] k. abdel-malek, h.-j. yeh, determining intersection curves between surfaces of two solids, computer-aided design 28 (6-7) (1996), 539–549. [20] t. dokken, aspects of intersection algorithms and approximation, phd thesis, university of oslo, (1997). 1. introduction 2. problem statement 3. developed method 3.1. computing a starting point 3.2. type convergence of broyden’s method 4. numerical implementations 4.1. first example: 4.2. second example: 5. analysis 5.1. effect of initial guess: 5.2. effect of step size: 6. conclusion references international journal of analysis and applications volume 19, number 2 (2021), 228-238 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-228 image restoration using a novel model combining the perona-malik equation and the heat equation samira lecheheb∗, messaoud maouni, hakim lakhal laboratory of applied mathematics and history and didactics of maths ”lamahis”, department of mathematics, university of 20 august 1955, skikda, algeria ∗corresponding author: lecheheb.samira24@gmail.com; s.lecheheb@univ-skikda.dz abstract. this article is devoted to the mathematical study of a new proposed model based on a peronamalik equation combined with a heat equation. this study shows how system of partial differential equations can be used to restore a digital image. by using compactness method and the monotonicity arguments, with suitable assumptions on the nonlinearities, we prove the existence of the weak solution for the proposed model which its consistency is given in our work. 1. introduction in recent years, the application of partial differential equations in image processing have attracted attention of many authors in computer vision. various techniques have been developed in image processing during the last decades. now these techniques are used for all kinds of tasks in all kinds of domains: industrial inspection, medical visualization, human computer interfaces, artistic effects, etc. texture extraction and image restoration are the two fundamental problems that have made a significant contribution to this discipline, as can be ascertained from recent survey papers [1,4,5,8,10,11,16,17,19,21]. the question here is: how to preserve the contours of an image while the elimination of the noise. in 1990, perona and malik [20] answered this question in their model which is one of the first attempts to derive a model that incorporates local information from an image within a pde framework. the numerical results of this model represent received january 6th, 2021; accepted february 1st, 2021; published february 22nd, 2021. 2010 mathematics subject classification. 35q30, 65n12, 76m25. key words and phrases. topological degree; nonlinear system; homotopy. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 228 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-228 int. j. anal. appl. 19 (2) (2021) 229 an efficient and effective tool for image denoising. nevertheless, two important phenomena have been observed. the first one proved by kichenassamy [14] that the perona-malik model is an ill-posed problem in the sense of hadamard, this paradoxical result is sometimes referred to as the perona-malik paradox [14]. the second one is the phenomenon known as the staircase effect. to make this problem well posed, catté et all [9] suggested introducing the regularization in space and time directly into the continuous equation. there is an extensive literature for the purpose of denoising in image processing. let us mention the work of aboulaich et al [2], concerning new diffusion models for the image processing. the proposed model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. in their valuable monograph [7], atlas et al. have presented a new model for image restoration. the proposed model is an interpolation of two classical models, perona-malik and p-laplacian. by using the monotonicity arguments, they proved the existence and the uniqueness of solutions for p large enough. they also studied the asymptotic behavior of the solution as p →∞ and they proved that the limit problem coincides with the perona-malik model in the some subregion. in 2016, afraites et al [3] proposed a model removing noise while preserving the edges and reducing staircase effect where they combined a nonlinear regularization of total variation (tv) operator’s with a decomposition approach of h−1 norm suggested by guo et al. ( [12, 13]). the generalization of this work was made by atlas et al [6]. until the work of lecheheb et al. [15] combining the perona-malik equation with the heat equation, the authors were able to demonstrate the existence and consistency of the their proposed model. we build up on their works by providing a generalization to the case of systems. in this article, we present a novel model for image denoising, which combines the peronamalik equation and the heat equation. our model is well-posed. by using the compactness method and the monotonicity arguments, we prove the existence of solutions for the following system (1.1)   −div ( g1(|∇v|)∇u ) − 1 λ21 ∆u = f(x,u,v) in ω, −div ( g2(|∇u|)∇v ) − 1 λ22 ∆v = h(x,u,v) in ω, ( g1(|∇v|) + 1 λ21 ) ∇u ·~n = ( g2(|∇u|) + 1 λ22 ) ∇v ·~n = 0 on ∂ω, where ω ⊆ rn is a bounded domain with smooth boundary ∂ω, with neumann boundary conditions. ~n is the unit outward normal at the image boundaries ∂ω and 0 < λ ≤ 1 suth that λ = (λ1,λ2). the function g(·) = (g1,g2) is defined by one of the following expressions: g(k) = 1 1 + ( k λ )2 or g(k) = exp ( − k2 2λ2 ) , int. j. anal. appl. 19 (2) (2021) 230 it is clear that the function g(k) is a decreasing non-negative function satisfying the following conditions (1.2)   lim g(k) k→0 = 1, lim g(k) k→+∞ = 0. we remark that, if gi = 1 for i = 1, 2 we recover the linear diffusion. for the rest of this article, we assume that f,h : ω × r × r → r are carathéodory functions satisfying the growth condition (1.3)   |f(x,s,z)| ≤ d1(x) + 1 2λ21 |s| + 1 2λ22 |z|, |h(x,s,z)| ≤ d2(x) + 1 2λ21 |s| + 1 2λ22 |z|, where d = (d1,d2) is in (l 2(ω))2 and 0 < λ ≤ 1, (λ = (λ1,λ2)). the content of this paper is arranged as follows. in the next section we will present the main results. in the section 3, we will study the existence of the solutions of the problem (1.1) under some different conditions on the nonlinear terms. this existence is obtained by using the compactness method [18] and the monotonicity arguments. 2. main results in this section, we are ready to present the definition of a weak solution for problem (1.1) and the main result. at beginning, let v = h10 (ω) ×h 1 0 (ω), which is a banach space endowed with the norm ‖(u,v)‖2v = ‖u‖ 2 h10 (ω) + ‖v‖2h10 (ω), and let w = l2(ω) ×l2(ω). in the sequel, ‖ · ‖h10 (ω) and ‖ · ‖l2(ω) will denote the usual norms of h 1 0 (ω) and l2(ω), respectively. we give now the definition of a weak solution for problem (1.1) definition 2.1. we say that (u,v) ∈ v is a weak solution for the system (1.1) if for any (φ,ψ) ∈ v we have (2.1) ∫ ω (g1(|∇v|) + 1 λ21 )∇u∇φ dx + ∫ ω (g2(|∇u|) + 1 λ22 )∇v∇ψ dx = ∫ ω f(x,u,v)φ dx + ∫ ω h(x,u,v)ψ dx. the main result of this article read as follows. int. j. anal. appl. 19 (2) (2021) 231 theorem 2.1. assume conditions (1.2) and (1.3) are fulfilled. then problem (1.1) has at least one solution. 3. proof of main result in the present section, we will use the compactness method to obtain existence results from finitedimensional approximations. proof. let x be a finite-dimensional subspace of v endowed with the v-norm, and x∗ its dual. define de mappings l : x × [0, 1] −→ x∗ by (3.1) 〈l(u,v,t), (φ,ψ)〉v = ∫ ω ( g1(t|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ ω ( g2(t|∇u|) + 1 λ22 ) ∇v∇ψ dx − ∫ ω f(x,tu,tv)φ dx− ∫ ω h(x,tu,tv)ψ dx, for all (φ,ψ) ∈ x, l is well defined. next, we divide the proof into four steps: step 1:: in this step, we show{ (u,v) ∈ x : l(u,v,t) = 0 for some t ∈ [0, 1] } ⊂ b̄ ( 2 min(k1,k2) ‖(d1,d2)‖w ) is established. indeed, if l(u,v,t) = 0 for some (u,v,t) ∈ x × [0, 1], then 0 = 〈l(u,v,t),u,v〉≥ min(k1,k2)‖(u,v)‖v − 2‖(d1,d2)‖w, which implies that ‖(u,v)‖v ≤ 2 min(k1,k2) ‖(d1,d2)‖w. consequently, for any r > 2 min(k1,k2) ‖(d1,d2)‖w, we have (3.2) l(u,v,t) 6= 0 if (u,v,t) ∈ ∂bx(r) × [0, 1]. step 2: in this step, we show l ( b̄x(r) × [0, 1] ) ⊂ b̄x ∗ ( mr + 2‖(d1,d2)‖w ) is established. if (u,v,t) ∈ b̄x(r) × [0, 1], we have |〈l(u,v,t), (φ,ψ)〉| ≤ ( max (2λ21 + 3 2λ21 , 2λ22 + 3 2λ22 ) ‖(u,v)‖v + 2‖(d1,d2)‖w ) ‖(ϕ,ψ)‖v ≤ ( max (2λ21 + 3 2λ21 , 2λ22 + 3 2λ22︸ ︷︷ ︸ m ) r + 2‖(d1,d2)‖w ) ‖(ϕ,ψ)‖v ≤ ( mr + 2‖(d1,d2)‖w ) ‖(ϕ,ψ)‖v , for all (ϕ,ψ) ∈ v, and hence (3.3) l ( b̄x(r) × [0, 1] ) ⊂ b̄x ∗ ( mr + 2‖(d1,d2)‖w ) . int. j. anal. appl. 19 (2) (2021) 232 step 3: in this step, we prove that l is a continuous mapping on b̄x(r) × [0, 1]. let (un,vn, tn) ∈ b̄x(r)×[0, 1] converge to (u,v,t) in x×[0, 1], i.e in v×[0, 1]. since (l(un,vn, tn)) is bounded because of (3.3), to prove that l(un,vn, tn) → l(u,v,t), it is sufficient to show that l(u,v,t) is the unique cluster point of (l(un,vn, tn)). let l ∈ x∗ be such a cluster point, still we denote by (tn), (un) and (vn) a subsequence of (tn), (un) and (vn) respectively such that l(un,vn, tn) → l in x∗. since (un,vn) → (u,v) in v, it follows that (un,vn) → (u,v) in w, and hence, going if necessary to a subsequence, we may assume that (un,vn) → (u,v) a.e. in ω . on the other hand, (∂iun,∂ivn) → (∂iu,∂iv) in w, therefore (∇un,∇vn) → (∇u,∇v) a.e in ω. this implies that g1(tn|∇vn|) → g1(t|∇v|) a.e. in ω, g2(tn|∇un|) → g2(t|∇u|) a.e. in ω, and hence, for any (φ,ψ) ∈ x, g1(tn|∇vn|)∇φ → g1(t|∇v|)∇φ in l2(ω), g2(tn|∇un|)∇ψ → g2(t|∇u|)∇ψ in l2(ω). for the last term, f(x,tnun, tnvn) → f(x,tu,tv) a.e.. using lebesgue dominated convergence theorem and (1.3), we arrive at f(x,tnun, tnvn) → f(x,tu,tv) in l2(ω). consequently ∫ ω f(x,tnun, tnvn)φ dx → ∫ ω f(x,tu,tv)φ dx. similarly we have ∫ ω h(x,tnun, tnvn)φ dx → ∫ ω h(x,tu,tv)φ dx. we conclude that 〈l(un,vn, tn), (φ,ψ)〉v = ∫ ω ( g1(tn|∇vn|) + 1 λ21 ) ∇un∇φ dx + ∫ ω ( g2(tn|∇un|) + 1 λ22 ) ∇vn∇ψ dx − ∫ ω f(x,tnun, tnvn)φ dx− ∫ ω h(x,tnun, tnvn)ψ dx → ∫ ω ( g1(t|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ ω ( g2(t|∇u|) + 1 λ22 ) ∇v∇ψ dx − ∫ ω f(x,tu,tv)φ dx− ∫ ω h(x,tu,tv)ψ dx = 〈l(u,v,t), (ϕ,ψ)〉v . int. j. anal. appl. 19 (2) (2021) 233 thus l = l(u,v,t). all those properties allow us to apply the homotopy invariance property to (3.4) degb ( l(·, ·, 1),b(r), 0 ) = degb ( l(·, ·, 0),b(r), 0 ) . but l(u,v, 0) = 0 is equivalant to the problem (1 + 1 λ21 ) ∫ ω ∇u∇φ dx + (1 + 1 λ22 ) ∫ ω ∇v∇ψ dx = ∫ ω f(x)φ dx + ∫ ω h(x)ψ dx, for all (φ,ψ) ∈ x, whose solution is unique because of the boundedness of the set of its possible solutions. consequently, degb ( l(·, ·, 0),b(r), 0 ) = ±1, and from (3.4) and the existence property of degree, there exists (u,v) ∈ bx(r) which satisfies (3.5) ∫ ω ( g1(|∇v|) + 1 λ21 ) ∇u∇φ dx + ∫ ω ( g2(|∇u|) + 1 λ22 ) ∇v∇ψ dx = ∫ ω f(x,u,v)φ dx + ∫ ω h(x,u,v)ψ dx, |(u,v)‖v ≤ 2 min(k1,k2) ‖(d1,d2)‖w, for all (φ,ψ) ∈ x. step 4: we now show the passage to the limit. consider the function bi : rn → rn defined by bi(ζi) = ( gi(ζi) + 1 λ2i ) ζi for any ζi ∈ rn and i = 1, 2. to prove the passage to the limit, we need the following lemma: lemma 3.1. [16] let 0 < λi ≤ 1, for any ζi,ζ′i ∈ r n such that ζi 6= ζ′i we have (bi(ζi) − bi(ζ′i))(ζi − ζ ′ i) > 0 for i = 1, 2. the proof of the above lemma can be found in [16]. lemma 3.2. if b ∈ c(rn,rn ), b(ζ) ≤ (1 + 1 λ2 )ζ for all ζ ∈ rn and if un → u in h10 (ω) then b(∇un) → b(∇u) in l2(ω). lemma (3.2) is proved by the dominated convergence theorem of lebesgue. now, it is well known that one can write v = ⋃ n≥1 xn where xn ⊂ xn+1(n ≥ 1) and xn has dimension n. consequently, given any (φ,ψ) ∈ v , there exists a sequence (φn,ψn) with (φn,ψn) ∈ xn int. j. anal. appl. 19 (2) (2021) 234 which converges to (φ,ψ). on the other hand, by (3.5) applied to x = xn, there exists, for each n ≥ 1, some (un,vn) ∈ xn such that∫ ω b1(∇un)∇ϕ1 dx + ∫ ω b2(∇vn)∇ϕ2 dx = ∫ ω f(x,un,vn)ϕ1 dx + ∫ ω h(x,un,vn)ϕ2 dx, ‖(un,vn)‖v ≤ 2 min(k1,k2) ‖(d1,d2)‖w, for all (ϕ1,ϕ2) ∈ xn. in particular, taking (ϕ1,ϕ2) = (φn,ψn) introduced above, (3.6) ∫ ω b1(∇un)∇φn dx + ∫ ω b2(∇vn)∇ψn dx = ∫ ω f(x,un,vn)φn dx + ∫ ω h(x,un,vn)ψn dx, ‖(un,vn)‖v ≤ 2 min(k1,k2) ‖(d1,d2)‖w, for all n ≥ 1. the estimate in (3.6) implies that, going if necessary to subsequences, we can assume that there exists (u,v) ∈ v such that (un,vn) → (u,v) weakly in v, (un,vn) → (u,v) strongly in w and (un,vn) → (u,v) a.e. in ω. as ( b1(∇un) ) n∈n is bounded in l 2(ω), then there exists ξ1 ∈ l2(ω) such that b1(∇un) → ξ1 weakly in l2(ω). similarly, we obtain b2(∇vn) → ξ2 weakly in l2(ω), and (∇φn,∇ψn) → (∇φ,∇ψ) strongly in w. on the other hand, as f(x,un,vn) → f(x,u,v) in l2(ω) and h(x,un,vn) → h(x,u,v) in l2(ω) , one can let n →∞ in (3.6) to obtain (3.7) ∫ ω ξ1∇φ dx + ∫ ω ξ2∇ψ dx = ∫ ω f(x,u,v)φ dx + ∫ ω h(x,u,v)ψ dx. it remains to show that (3.8) ∫ ω ξ1∇φ dx = ∫ ω b1(∇u)∇φ dx, and (3.9) ∫ ω ξ2∇ψ dx = ∫ ω b2(∇v)∇ψ dx. to prove the two equalities, we use the trick of minty [16]; we begin by studying the limit of∫ ω b1(∇un)∇un dx, and ∫ ω b2(∇vn)∇vn dx. int. j. anal. appl. 19 (2) (2021) 235 indeed ∫ ω b1(∇un)∇un dx = ∫ ω f(x,un,vn)un dx → ∫ ω f(x,u,v)u dx,∫ ω b2(∇vn)∇vn dx = ∫ ω h(x,un,vn)vn dx → ∫ ω h(x,u,v)v dx, because (un,vn) → (u,v) weakly in v. but we know that (u,v) satisfies (3.7), and hence∫ ω f(x,u,v)u dx = ∫ ω ξ1∇u dx, and ∫ ω h(x,u,v)v dx = ∫ ω ξ2∇v dx. therefore (3.10) lim n→+∞ ∫ ω b1(∇un)∇un dx = ∫ ω f(x,u,v)u dx = ∫ ω ξ1∇u dx, and (3.11) lim n→+∞ ∫ ω b2(∇vn)∇vn dx = ∫ ω h(x,u,v)v dx = ∫ ω ξ2∇v dx. let (φ,ψ) ∈ v, it exists (φn,ψn)n∈n such that (φn,ψn) ∈ xn for all n ∈ n and (φn,ψn) → (ϕ,ψ) in v when n → +∞. thanks to lemma 3.1, we will pass to the limit in the two terms∫ ω b1(∇un)∇φn dx, and ∫ ω b2(∇vn)∇ψn dx. indeed, for the first equation 0 ≤ ∫ ω (b1(∇un) − b1(∇φn))(∇un −∇φn) dx =∫ ω b1(∇un)∇un dx− ∫ ω b1(∇un)∇φn dx− ∫ ω b1(∇φn)∇un dx + ∫ ω b1(∇φn)∇φn dx = t1,n − t2,n − t3,n + t4,n, we saw in (3.10) that t1,n → ∫ ω ξ1∇u dx when n →∞. we have lim n→+∞ t2,n = ∫ ω ξ1∇φ dx. similarly lim n→+∞ t3,n = ∫ ω b1(∇φ)∇u dx. int. j. anal. appl. 19 (2) (2021) 236 finally, we also have lim n→+∞ t4,n = ∫ ω b1(∇φ)∇φ dx, when n → +∞. the passage to the limit therefore gives:∫ ω (ξ1 − b1(∇φ))(∇u−∇φ) dx ≥ 0 for all φ ∈ h10 (ω). similarly, we obtain∫ ω (ξ2 − b2(∇ψ))(∇u−∇ψ) dx ≥ 0 for all ψ ∈ h10 (ω). we now choose judicious test functions φ and ψ. we take φ = u + 1 n w, with w ∈ h10 (ω) and n ∈ n ∗, and ψ = v + 1 n w̃, with w̃ ∈ h10 (ω) and n ∈ n ∗. we thus obtain: − 1 n ∫ ω ( ξ1 − b1(∇u + 1 n ∇w) ) ∇w dx ≥ 0, and − 1 n ∫ ω ( ξ2 − b2(∇v + 1 n ∇w̃) ) ∇w̃ dx ≥ 0, then ∫ ω ( ξ1 − b1(∇u + 1 n ∇w) ) ∇w dx ≤ 0, and ∫ ω ( ξ2 − b2(∇v + 1 n ∇w̃) ) ∇w̃ dx ≤ 0. but u + 1 n w → u in h10 (ω), v + 1 n w̃ → v in h10 (ω), from lemma 3.2, we get b1 ( ∇u + 1 n ∇w ) → b1(∇u) in l2(ω), and b2 ( ∇v + 1 n ∇w̃ ) → b2(∇v) in l2(ω). passing to the limit when n → +∞, we then obtain∫ ω (ξ1 − b1(∇u))∇w dx ≤ 0, ∀w ∈ h10 (ω), int. j. anal. appl. 19 (2) (2021) 237 and ∫ ω (ξ2 − b2(∇v))∇w̃ dx ≤ 0, ∀w̃ ∈ h10 (ω). by linearity (can change w into −w and w̃ into −w̃), we have∫ ω (ξ1 − b1(∇u))∇w dx = 0, ∀w ∈ h10 (ω), and ∫ ω (ξ2 − b2(∇v))∇w̃ dx = 0, ∀w̃ ∈ h10 (ω). we deduce that ∫ ω ξ1∇w dx = ∫ ω b1(∇u)∇w dx, ∀w ∈ h10 (ω),∫ ω ξ2∇w̃ dx = ∫ ω b2(∇v)∇w̃ dx, ∀w̃ ∈ h10 (ω). hence we have showed that (u,v) is a solution of (1.1). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. aarab, n.e. alaa, h. khalfi, generic reaction-diffusion model with application to image restoration and enhancement. electron. j. differ. equ. 2018 (2018), 125. [2] r. aboulaich, d. meskine, a. souissi, new diffusion models in image processing, computers math. appl. 56(4) (2008), 874-882. [3] l. afraites, a. atlas, f. karami, d. meskine, some class of parabolic systems applied to image processing. discr. contin. dyn. syst. ser. b. 21(6) (2016), 1671-1687. [4] n.e. alaa, m. aitoussous, w. bouarifi, d. bensikaddour, image restoration unisg a reaction-diffusion process. electron. j. differ. equ. 2014 (2014), 197. [5] n.e. alaa, m. zirhem, bio-inspired reaction diffusion system applied to image restoration. int. j. bio-inspired comput. 12(2) (2018), 128-137. [6] a. atlas, m. bendahmane, f. karami, d. meskine, o. oubbih, a nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. discr. contin. dyn. syst. ser. b. http://dx.doi.org/10.3934/dcdsb.2020321. [7] a. atlas, f. karami, d. meskine, the perona-malik inequality and application to image denoising. nonlinear anal., real world appl. 18 (2014), 57-68. [8] g. aubert, p. kornprobst, mathematical problems in image processing. springer-verlag, new-york. (2002). [9] h.f. catte, p.l. lions, j.m. morel, t. coll, image selective smoothing and edge detection by nonlinear diffusion. siam j. numer. anal. 29 (1) (1992), 182-193. [10] a. chambolle, p.l. lions, image recovery via total variation minimization and related problems. numer. math. 74 (2) (1997), 147-188. [11] y. chen, m. levine rao, variable exponent, linear growth functionals in image restoration. siam j. appl. math. 44 (4) (2004), 1383-1404. int. j. anal. appl. 19 (2) (2021) 238 [12] z. guo, q. liu, j. sun, b. wu, reaction-diffusion systems with p(x)-growth for image denoising. nonlinear anal., real world appl. 12 (5) (2011), 2904-2918. [13] z. guo, q. liu, on a reaction-diffusion system applied to image decomposition and restoration. math. computer model. 53 (2011), 1336-1350. [14] s. kichenassamy, the perona–malik paradox. siam j. appl. math. 57 (5) (1997), 1328-1342. [15] s. lecheheb, m. maouni, h. lakhal, existence of solutions of a quasilinear problem with neumann boundary conditions. bol. soc. parana. mat. in press. [16] s. lecheheb, m. maouni, h. lakhal, existence of the solution of a quasilinear equation and its application to image denoising. international journal of computer science, int. j. computer sci. commun. inform. technol. 7 (2) (2019), 1-6. [17] s. lecheheb, m. maouni, h. lakhal, image restoration using nonlinear elliptic equation. int. j. computer sci. commun. inform. technol. 6 (2) (2019), 32-37. [18] j.l. lions, quelques méthodes de résolution des problèmes aux limites non linéaires. dunod, paris, (1969). [19] s. osher, l. rudin, e. fatemi, nonlinear total variation based noise removal algorithms. physica d, 40 (1992), 259-268. [20] p. perona, j. malik, scale-espace and edge detection using anisotropic diffusion. ieee trans. pattern anal. mach. intell. 12 (1990), 429-439. [21] m. zirhem, n.e. alaa, existence and uniqueness of an entropy solution for a nonlinear reaction-diffusion system applied to texture analysis. j. math. anal. appl. 484 (1) (2020), 123-719. 1. introduction 2. main results 3. proof of main result references international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 64-70 http://www.etamaths.com new fixed point results for rational type contractions in partially ordered b−metric spaces reza arab∗, kolsoum zare abstract. the purpose of this paper is to establish some fixed point theorems for a mapping having a monotone property satisfying a contractive condition of rational type in the partially ordered b−metric spaces. the results presented in the paper generalize and extend several well-known results in the literature. an example is given to support the usability of our results. 1. introduction in [11, 12], s. czerwik introduced the notion of a b-metric space which is a generalization of usual metric space and generalized the banach contraction principle in the context of complete b-metric spaces. after that, many authors have carried out further studies on b-metric space and their topological properties (see e.g., [2, 3, 4, 5, 8, 17]) and the references therein. the following definitions and results will be needed in what follows. definition 1.1. [11] let x be a (nonempty) set and s ≥ 1 be given a real number. a function d : x × x −→ r+ is said to be a b-metric space if and only if for all x,y,z ∈ x, the following conditions are satisfied: (i) d(x,y) = 0 if and only if x = y; (ii) d(x,y) = d(y,x); (iii) d(x,y) ≤ s[d(x,z) + d(z,y)]. then the triplet (x,d,s) is called a b−metric space with the parameter s. clearly, a (standard)metric space is also a b−metric space, but the converse is not always true. example 1.1. let x = [0, 1] and d : x ×x −→ r+ be defined by b(x,y) = |x−y|2 for all x,y ∈ x. clearly, (x,d,s = 2) is a b−metric space that is not a metric space. also, the following example of a b−metric space is given in [6]. example 1.2. let p ∈ (0, 1). then the space lp([0, 1]) of all real functions f : [0, 1] −→ r such that∫ 1 0 |f(x)|pdx < ∞ endowed with the functional d : lp([0, 1]) ×lp([0, 1]) −→ r given by d(f,g) = ( ∫ 1 0 |f(x) −g(x)|pdx) 1 p , for all f,g ∈ lp([0, 1]) is a b−metric space with s = 2 1 p . since in general a b-metric is not continuous, we need the following simple lemma about the bconvergent sequences in the proof of our main result. lemma 1.1. [1] let (x,d) be a b−metric space with s ≥ 1 and suppose that {xn} and {yn} are b−convergent to x,y, respectively. then we have, 1 s2 d(x,y) ≤ lim inf d(xn,yn) ≤ lim sup d(xn,yn) ≤ s2d(x,y). 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. fixed point; rational type generalized contraction mappings; b−metric space; partially ordered set. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 64 new fixed point results for rational type contractions 65 in particular, if x = y, then we have limd(xn,yn) = 0. moreover, for each z ∈ x, we have, 1 s d(x,z) ≤ lim inf d(xn,z) ≤ lim sup d(xn,z) ≤ sd(x,z). in [13], dass and gupta presented the following fixed point theorem. theorem 1.2. let (x,d) be a complete metric space and t : x −→ x a mapping such that there exist α,β ≥ 0 with α + β < 1 satisfying d(tx,ty) ≤ α d(y,ty)(1 + d(x,tx) 1 + d(x,y) + βd(x,y), for all x,y ∈ x. then t has a unique fixed point. in [9], cabrera, harjani and sadarangani proved the above theorem in the context of partially ordered metric spaces. theorem 1.3. let (x,≤) be a partially ordered set and suppose that there exists a metric d in x such that (x,d) is a complete metric space. let t : x −→ x be a continuous and non-decreasing mapping such that there exist α,β ≥ 0 with α + β < 1 satisfying d(tx,ty) ≤ α d(y,ty)(1 + d(x,tx) 1 + d(x,y) + βd(x,y), for all x,y ∈ x with x ≤ y. if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. notice that theorem 1.3 is theorem 1.2 in the context of ordered metric spaces. harjani et al. [14] extended the result of jaggi [13] and established a fixed point result in partially ordered metric spaces. recently, chandok et al. [10] proved the following theorem. theorem 1.4. let (x,≤) be a partially ordered set and suppose that there exists a metric d on x such that (x,d) is a complete metric space. suppose that t is a continuous self-mapping on x, t is monotone nondecreasing mapping and d(tx,ty) ≤ α d(y,ty)d(x,tx) d(x,y) + βd(x,y) + γ(d(x,tx) + d(y,ty)) + δ(d(y,tx) + d(x,ty)), for all x,y ∈ x with x ≥ y,x 6= y and for some α,β,γ,δ ∈ [0, 1) with α + β + 2γ + 2δ < 1. if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. the purpose of this paper is to establish some fixed point results satisfying a generalized contraction mapping of rational type in b−metric spaces endowed with partial order. also, we establish a result for existence and uniqueness of fixed point for such class of mappings. 2. main results in this section, we will present some fixed point theorems for contractive mappings in the setting of b-metric spaces. furthermore, we will give examples to support our main results. the first result in this paper is the following fixed point theorem. theorem 2.1. suppose that (x,d,≤) is a partially ordered complete b-metric space. let t : x −→ x be a continuous and nondecreasing mapping. suppose there exist mappings ai : x ×x −→ [0, 1) such that for all x,y ∈ x and i = 1, 2, · · · , 7 ai(tx,ty) ≤ ai(x,y). also, for all x,y ∈ x with x ≤ y, d(tx,ty) ≤a1(x,y)d(x,y) + a2(x,y)[d(x,tx) + d(y,ty)] + a3(x,y) d(y,tx) + d(x,ty) s + a4(x,y)d(y,ty)ϕ(d(x,y),d(x,tx)) + a5(x,y)d(y,tx)ϕ(d(x,y),d(x,ty)) + a6(x,y)d(x,y)ϕ(d(x,y),d(x,tx) + d(y,tx)) + a7(x,y)d(y,tx), (2.1) where ϕ : r+ ×r+ −→ r+ is a function such that ϕ(t,t) = 1 for all t ∈ r+ and sup x,y∈x {a1(x,y) + a2(x,y) + a3(x,y) + a4(x,y) + a6(x,y)}≤ 1 s + 1 . 66 arab and zare if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. proof 1. if x0 = tx0, then we have the result. suppose that x0 < tx0. then we construct a sequence {xn} in x such that xn+1 = txn for every n = 0, 1, 2, · · · .(2.2) since t is a nondecreasing mapping, we obtain by induction that x0 ≤ tx0 = x1 ≤ tx1 = x2 ≤ ···≤ txn−1 = xn ≤ txn = xn+1 ≤ ··· .(2.3) if there exists some k ∈ n such that xk+1 = xk, then from (2.2), xk+1 = txk = xk, that is, xk is a fixed point of t and the proof is finished. so, we suppose that xn+1 = xn, for all n inn. since xn < xn+1, for all n ∈ n, we set x = xn and y = xn+1 in (2.1), we have d(txn,txn+1) ≤a1(xn,xn+1)d(xn,xn+1) + a2(xn,xn+1)[d(xn,xn+1) + d(xn+1,xn+2)] + a3(xn,xn+1) d(xn+1,xn+1) + d(xn,xn+2) s + a4(xn,xn+1)d(xn+1,xn+2)ϕ(d(xn,xn+1),d(xn,xn+1)) + a5(xn,xn+1)d(xn+1,xn+1)ϕ(d(xn,xn+1),d(xn,xn+2))) + a6(xn,xn+1)d(xn,xn+1)ϕ(d(xn,xn+1),d(xn,xn+1) + d(xn+1,xn+1))) + a7(xn,xn+1)d(xn+1,xn+1). (2.4) that is, d(xn+1,xn+2) ≤a1(xn,xn+1)d(xn,xn+1) + a2(xn,xn+1)[d(xn,xn+1) + d(xn+1,xn+2)] + a3(xn,xn+1)[d(xn,xn+1) + d(xn+1,xn+2)] + a4(xn,xn+1)d(xn+1,xn+2) + a6(xn,xn+1)d(xn,xn+1) =[a1(xn,xn+1) + a2(xn,xn+1) + a3(xn,xn+1) + a6(xn,xn+1)]d(xn,xn+1) + [a2(xn,xn+1) + a3(xn,xn+1) + a4(xn,xn+1)]d(xn+1,xn+2) =[a1 + a2 + a3 + a6](txn−1,txn)d(xn,xn+1) + [a2 + a3 + a4](txn−1,txn)d(xn+1,xn+2) ≤[a1 + a2 + a3 + a6](xn−1,xn)d(xn,xn+1) + [a2 + a3 + a4](xn−1,xn)d(xn+1,xn+2) ... ≤[a1 + a2 + a3 + a6](x0,x1)d(xn,xn+1) + [a2 + a3 + a4](x0,x1)d(xn+1,xn+2), which implies that d(xn+1,xn+2) ≤ (a1 + a2 + a3 + a6)(x0,x1) 1 − (a2 + a3 + a4)(x0,x1) d(xn,xn+1). now, (a1 + 2a2 + 2a3 + a4 + a6)(x0,x1) ≤{(s + 1) a1 + (s + 1) a2 + (s + 1) a3 + (s + 1) a4 + (s + 1) a6}(x0,x1) ≤ sup x,y∈x {(s + 1) a1(x,y) + (s + 1) a2(x,y) + (s + 1) a3(x,y) + (s + 1) a4(x,y) + (s + 1) a6(x,y)} < 1. thus we get d(xn+1,xn+2) ≤ λ d(xn,xn+1), where λ = (a1 + a2 + a3 + a6)(x0,x1) 1 − (a2 + a3 + a4)(x0,x1) < 1. obviously, 0 ≤ λ < 1 s . then by repeated application (2.4), we have d(xn+1,xn+2) ≤ λ d(xn,xn+1) ≤ λ2 d(xn−1,xn) ≤ ···≤ λn+1 d(x0,x1).(2.5) new fixed point results for rational type contractions 67 thus, setting any positive integers m and n (m > n), we have d(xn,xm) ≤sd(xn,xn+1) + s2d(xn+1,xn+2) + · · · + sm−nd(xm−1,xm) ≤[sλn + s2λn+1 + · · · + sm−nλm−1]d(x0,x1) =sλn[1 + sλ + (sλ)2 + · · · + (sλ)m−n−1]d(x0,x1) ≤sλn[1 + sλ + (sλ)2 + · · · ]d(x0,x1) ≤ sλn 1 −sλ d(x0,x1). since 0 ≤ λ < 1 s , we notice that sλn 1 −sλ −→ 0 as n −→ ∞ for any m ∈ n. so {xn} is cauchy in a complete b-metric space x, there exist x ∈ x such that lim n−→∞ xn+1 = x.(2.6) letting n −→∞ in (2.2) and from the continuity of t , we get x = lim n−→∞ xn+1 = lim n−→∞ t(xn) = t( lim n−→∞ xn) = t(x). this implies that x is a fixed point of t . example 2.1. let x = [0, 1] with the usual order ≤. define d(x,y) = |x−y|2. then d is a b−metric with s = 2. also define a1(x,y) = x + y + 1 32 and tx = 1 16 x2. we observe that a1(tx,ty) = 1 32 ( 1 16 x2 + 1 16 y2 + 1) = 1 32 ( 1 16 x.x + 1 16 y.y + 1) ≤ x + y + 1 32 = a1(x,y), and for all comparable x,y ∈ x, we get d(tx,ty)) =|tx−ty|2 = | 1 16 x2 − 1 16 y2|2 = 1 162 |x + y||x + y||x−y|2 ≤ 1 16 × 8 |x + y||x−y|2 ≤ 1 8 x + y + 1 16 |x−y|2 = 1 8 a1(x,y)d(x,y) ≤ a1(x,y)d(x,y) moreover, t is a nondecreasing continuous mapping with respect to the usual order ≤ . hence, all conditions of theorem 2.1 are satisfied. therefore, t has a fixed point x = 0. corollary 2.2. suppose that (x,d,≤) is a partially ordered complete b-metric space. let t : x −→ x be a continuous and nondecreasing mapping such that the following conditions hold: d(tx,ty) ≤a1d(x,y) + a2[d(x,tx) + d(y,ty)] + a3 d(y,tx) + d(x,ty) s + a4d(y,ty)ϕ(d(x,y),d(x,tx)) + a5d(y,tx)ϕ(d(x,y),d(x,ty)) + a6d(x,y)ϕ(d(x,y),d(x,tx) + d(y,tx)) + a7d(y,tx), for all x,y ∈ x with x ≤ y, where ai are nonnegative coefficients for i = 1, 2, · · · , 7 with a1 + a2 + a3 + a4 + a6 ≤ 1 s + 1 , and ϕ : r+ ×r+ −→ r+ is a function such that ϕ(t,t) = 1 for all t ∈ r+. if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. 68 arab and zare example 2.2. let x = [0, 1] with the usual order ≤. define d(x,y) = |x−y|2. then d is a b−metric with s = 2. also define tx = 1 2 x− 1 4 x2. for all comparable x,y ∈ x, we get d(tx,ty)) =|tx−ty|2 = | 1 2 x− 1 4 x2 − 1 2 y + 1 4 y2|2 = | 1 2 (x−y) − 1 4 (x−y)(x + y)|2 =|x−y||2.| 1 2 − 1 4 (x + y)|2 ≤ 1 4 |x−y||2 =a1d(x,y). moreover, t is a nondecreasing continuous mapping with respect to the usual order ≤ and a1 = 1 4 < 1 s + 1 . hence, all conditions of corollary 2.2 are satisfied. therefore, t has a fixed point x = 0. if we take ϕ(t,s) = 1 + s 1 + t for all t,s ∈ r+ in theorem 2.1 and corollary 2.2, we have the following theorem and corollary. theorem 2.3. suppose that (x,d,≤) is a partially ordered complete b-metric space. let t : x −→ x be a continuous and nondecreasing mapping. suppose there exist mappings ai : x ×x −→ [0, 1) such that for all x,y ∈ x and i = 1, 2, · · · , 7 ai(tx,ty) ≤ ai(x,y). also, for all x,y ∈ x with x ≤ y, d(tx,ty) ≤a1(x,y)d(x,y) + a2(x,y)[d(x,tx) + d(y,ty)] + a3(x,y) d(y,tx) + d(x,ty) s + a4(x,y) d(y,ty)[1 + d(x,tx)] 1 + d(x,y) + a5(x,y) d(y,tx)[1 + d(x,ty)] 1 + d(x,y) + a6(x,y) d(x,y)[1 + d(x,tx) + d(y,tx)] 1 + d(x,y) + a7(x,y)d(y,tx), and sup x,y∈x {a1(x,y) + a2(x,y) + a3(x,y) + a4(x,y) + a6(x,y)}≤ 1 s + 1 . if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. corollary 2.4. suppose that (x,d,≤) is a partially ordered complete b-metric space. let t : x −→ x be a continuous and nondecreasing mapping such that the following conditions hold: d(tx,ty) ≤a1d(x,y) + a2[d(x,tx) + d(y,ty)] + a3 d(y,tx) + d(x,ty) s + a4 d(y,ty)[1 + d(x,tx)] 1 + d(x,y) + a5 d(y,tx)[1 + d(x,ty)] 1 + d(x,y) + a6 d(x,y)[1 + d(x,tx) + d(y,tx)] 1 + d(x,y) + a7d(y,tx), for all x,y ∈ x with x ≤ y, where ai are nonnegative coefficients for i = 1, 2, · · · , 7 with a1 + a2 + a3 + a4 + a6 ≤ 1 s + 1 . if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. in our next theorem we relax the continuity assumption of the mapping t in theorem 2.1 by imposing the following order condition of the metric space x: if {xn} is a non-decreasing sequence in x such that xn −→ x, then xn ≤ x for all n ∈ n. new fixed point results for rational type contractions 69 theorem 2.5. suppose that (x,d,≤) is a partially ordered complete b-metric space. assume that if {xn} is a non-decreasing sequence in x such that xn −→ x, then xn ≤ x for all n ∈ n. let t : x −→ x be a nondecreasing mapping. suppose there exist continuous mappings ai : x ×x −→ [0, 1) such that for all x,y ∈ x and i = 1, 2, · · · , 7 ai(tx,ty) ≤ ai(x,y). also, for all x,y ∈ x with x ≤ y, d(tx,ty) ≤a1(x,y)d(x,y) + a2(x,y)[d(x,tx) + d(y,ty)] + a3(x,y) d(y,tx) + d(x,ty) s + a4(x,y)d(y,ty)ϕ(d(x,y),d(x,tx)) + a5(x,y)d(y,tx)ϕ(d(x,y),d(x,ty)) + a6(x,y)d(x,y)ϕ(d(x,y),d(x,tx) + d(y,tx)) + a7(x,y)d(y,tx), (2.7) where ϕ : r+ ×r+ −→ r+ is a continuous function such that ϕ(t,t) = 1 for all t ∈ r+ and sup x,y∈x {a1(x,y) + a2(x,y) + a3(x,y) + a4(x,y) + a6(x,y)}≤ 1 s + 1 . if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. proof 2. we take the same sequence {xn} as in the proof of theorem 2.1. then we have x0 ≤ x1 ≤ x2 ≤ ··· ≤ xn ≤ xn+1 ≤ ··· . that is, {xn} is a nondecreasing sequence. also, this sequence converges to x. then xn ≤ x, for alln ∈ n. suppose that tx 6= x, that is, d(x,tx) > 0. since xn ≤ x for all n, applying (2.7) and using lemma 1.1, we have 1 s d(x,tx) ≤ lim sup n→∞ d(xn+1,tx) = lim sup n→∞ d(txn,tx) ≤ lim sup n→∞ { a1(xn,x)d(xn,x) + a2(xn,x)[d(xn,xn+1) + d(x,tx)] + a3(xn,x) d(x,xn+1) + d(xn,tx) s + a4(xn,x)d(x,tx)ϕ(d(xn,x),d(xn,xn+1)) + a5(xn,x)d(x,xn+1)ϕ(d(xn,x),d(xn,tx))) + a6(xn,x)d(xn,x)ϕ(d(xn,x),d(xn,xn+1) + d(x,xn+1))) + a7(xn,x)d(x,xn+1) } ≤[a2(x,x) + a3(x,x) + a4(x,x)]d(x,tx) ≤ 1 s + 1 d(x,tx) < 1 s d(x,tx), which is a contradiction. hence, tx = x, that is, x is a fixed point of t . corollary 2.6. suppose that (x,d,≤) is a partially ordered complete b-metric space. assume that if {xn} is a non-decreasing sequence in x such that xn −→ x, then xn ≤ x for all n ∈ n. let t : x −→ x be a nondecreasing mapping such that the following conditions hold: d(tx,ty) ≤a1d(x,y) + a2[d(x,tx) + d(y,ty)] + a3 d(y,tx) + d(x,ty) s + a4d(y,ty)ϕ(d(x,y),d(x,tx)) + a5d(y,tx)ϕ(d(x,y),d(x,ty)) + a6d(x,y)ϕ(d(x,y),d(x,tx) + d(y,tx)) + a7d(y,tx), for all x,y ∈ x with x ≤ y, where ai are nonnegative coefficients for i = 1, 2, · · · , 7 with a1 + a2 + a3 + a4 + a6 ≤ 1 s + 1 , and ϕ : r+ ×r+ −→ r+ is a function such that ϕ(t,t) = 1 for all t ∈ r+. if there exists x0 ∈ x such that x0 ≤ tx0, then t has a fixed point. remark 2.1. since a b−metric space is a metric space when s = 1, so our results can be viewed as the generalization and the extension of several comparable results. 70 arab and zare references [1] a. aghajani, m. abbas and j. r. roshan, common fixed point of generalized weak contractive mappings in partially ordered b−metric spaces, mathematica slovaca, 64 (2014), 941-960. [2] a. aghajani, r. arab, fixed points of (ψ,ϕ,θ)-contractive mappings in partially ordered b−metric spaces and application to quadratic integral equations, fixed point theory and applications, 2013 (2013), article id 245. [3] r. allahyari, r. arab, a. shole haghighi, a generalization on weak contractions in partially ordered b−metric spaces and its application to quadratic integral equations, journal of inequalities and applications, 2014 (2014), article id 355. [4] r. allahyari, r. arab, a. shole haghighi, fixed points of admissible almost contractive type mappings on b− metric spaces with an application to quadratic integral equations, journal of inequalities and applications, 2015 (2015), article id 32. [5] h. aydi, m. f. bota, e. karapinar and s. moradi, a common fixed point for weak ϕ−contractions on b−metric spaces, fixed point theory, 13 (2012), 337-346. [6] v. berinde, generalized contractions in quasimetric spaces, seminar on fixed point theory, 1993, 3-9. [7] tg. bhaskar, v. lakshmikantham, fixed point theory in partially ordered metric spaces and applications, nonlinear anal. 65 (2006), 1379-1393. [8] m. boriceanu, m. bota and a. petrusel, multivalued fractals in b−metric spaces, cent. eur. j. math., 8 (2010), 367-377. [9] i. cabrera, j. harjani, k. sadarangani, a fixed point theorem for contractions of rational type in partially ordered metric spaces. ann. univ. ferrara, 59 (2013), 251-258. [10] s. chandok, t. d. narang, m. taoudi,fixed point theorem for generalized contractions satisfying rational type expressions in partially ordered metric spaces, gulf journal of mathematics, 2 (2014), 87-93. [11] s. czerwik, nonlinear set-valued contraction mappings in b−metric spaces, atti sem. mat. fis. univ. modena, 46 (1998), 263-276. [12] s. czerwik, contraction mappings in b−metric spaces, acta mathematica et informatica universitatis ostraviensis, 1(1993), 5-11. [13] b.k. dass, s. gupta, an extension of banach contraction principle through rational expressions. indian j. pure appl. math., 6 (1975), 1455-1458. [14] j.harjani, b. lopez, and k. sadarangani, a fixed point theorem for mappings satisfying a contractive condition of rational type on a partially orderedmetric space, abstract and applied analysis, 2010 (2010), article id 190701. [15] d. s. jaggi, some unique fixed point theorems, indian journal of pure and applied mathematics, 8 (1977), 223-230. [16] m. pacurar, sequences of almost contractions and fixed points in b−metric spaces, anal. univ. de vest, timisoara seria matematica informatica, 48 (2010), 125-137. [17] m-a. kutbi, e. karapnar, j. ahmad, a. azam, some fixed point results for multi-valued mappings in b−metric spaces. j. inequal. appl. 2014 (2014), article id 126. department of mathematics, sari branch, islamic azad university, sari, iran ∗corresponding author: mathreza.arab@iausari.ac.ir international journal of analysis and applications volume 19, number 1 (2021), 77-90 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-77 exponential stability for a nonlinear timoshenko system with distributed delay lamine bouzettouta∗, fahima hebhoub, karima ghennam, sabrina benferdi university of 20 august 1955, skikda, algeria ∗corresponding author: lami-750000@yahoo.fr abstract. this paper is concerned with a nonlinear timoshenko system modeling clamped thin elastic beams with distributed delay time. the distributed delay is defined on feedback term associated to the equation for rotation angle. under suitable assumptions on the data, we establish the exponential stability of the system under the usual equal wave speeds assumption. 1. introduction in this work, we consider the following non linear timoshenko system with distributed delay, (1.1)   ρ1ϕtt −k(ϕx + ψ)x = 0, ρ2ψtt − bψxx + k(ϕx + ψ) + µ1ψt + ∫ τ2 τ1 µ2(s)ψt(x,t−s)ds + f(ψ) = 0, where t denotes the time variable and x the space variable along a beam of length 1 in its equilibrium configuration. here, ϕ = ϕ(x,t) and ψ = ψ(x,t) denotes the transverse displacement of the beam and the rotation angle of its filament, respectively. the term µ1ψt represents a frictional damping and f(ψ) is a forcing term. the coefficients, ρ1,ρ2,k are positive constants represent the density, the polar momentum received october 13th, 2020; accepted november 5th, 2020; published december 4th, 2020. 2010 mathematics subject classification. 93d15, 93d20, 35b37, 35b40, 74d05. key words and phrases. timoshenko system, distributed delay time, exponential stability, lyapunov functional. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 77 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-77 int. j. anal. appl. 19 (1) (2021) 78 of inertia of a cross section, shear modulus respectively, and b = ei where e is the young’s modulus of elasticity, i is the moment of inertia cross-section. system (1.1) is supplemented with the following initial conditions (1.2) ϕ(x, 0) = ϕ0, ϕt(x, 0) = ϕt, ψ(x, 0) = ψ0, ψt(x, 0) = ψ1 ψt(x,−t) = f0(x,t), and dirichlet boundary conditions (1.3) ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t), where x ∈ (0, 1), t ∈ (τ1,τ2). the initial data (ϕ0,ψ0,ϕ1,ψ1,f0) belongs to a suitable functional spacial. this type of problems (without delay), has been considered, first in [15] where µ1 = µ2 = f = 0. the stability of this problems has received much attention in last years, we can find in the literature many results about different stability of timoshenko systems depending, in particular, on the weights µ1 and µ2 (see [14]) recently also a great consideration ha been addressed to time delay effects. on such problems, it was showed that a small delay acted on a boundary control, or internal can destabilize a system which is uniformly asymptotically stable in the absence of delays. see for instance ( [5]) . in [13] s. nicaise and c. pignotti examined a system of wave equation with initial feedback (1.4)   utt −uxx + µ0ut + ∫ τ2 τ1 a(x)µ(s)ut(t−s)ds u = 0 on γ0(0,α) ∂u ∂ν = 0 on γ1(0,α) u(x, 0) = u0(x) and ut(x, 0) = u1(x) in ω ut(x,−t) = f0(x,−t) in ω(0,τ2) where a ∈ l2(ω) is a function chosen with some assumptions. they proved that the above system is exponentially stable under the condition µ0 > ||a||α ∫ τ2 τ1 µ(s)ds similarly result was obtained by the authors when the distributed delay acted on the part of boundary. in [11] mustapha considered a timoshenko system of thermoelasticity of type iii with distributed delay and establish the stability for the case of equal and non equal speeds of wave propagation .appalara [1] int. j. anal. appl. 19 (1) (2021) 79 investigated a thermo-elastic system of timoshenko type with second sound and distributed delay (1.5)   ρ1ϕtt −k(ϕx + ψ)x + γ1ϕt + ∫ τ2 τ1 γ2(s)ϕt(x,t−s) = 0 ρ2ψtt − bψxx + k(ϕx + ψ) + δθx = 0 ρ3θt + qx + δψtx = 0 τqt + βq + θx = 0. in (0, 1)(0,α), this system is exponentially stable regardless the speeds of wave propagation. the same author studied in [2] a one dimensional timoshenko system with linear frictional damping and a distributed delay acting on the displacement equation,he showed that dissipation through the frictional damping is strong enough to uniformly stabilize the system. for other results about different types of time delay (discrete and continuos delay) we refer the reader to see ( [1–4, 8, 10]). b. feng and h. l. pelier [6] considered a following non linear timoshenko system with constant delay and forcing term: (1.6)   ρ1ϕtt −k(ϕx + ψ)x = 0, ρ2ψtt − bψxx + k(ϕx + ψ) + µ1ψt + µ2(s)ψt(x,t− τ) + f(ψ) = 0, and obtained an exponential stability under equal wave speeds. recently s. a. messaoudi, b. said-houari [10] established the stability of a thermoelastic timoshenko system of type iii with past history and distributed delay for the cases of equal and non equal speeds of wave propagation respectively. in the present work, we extend the result of feng and pelier, [6] where constant delay is replaced by distributed delay. 2. preliminaries in this section we present the some assumptions needed later to prove our results. as in [12], we introduce the following new dependent variable z (x,ρ,s,t) = ψt (x,t−ρs) , x ∈ (0, 1) , ρ ∈ (0, 1) , t,s ∈ (τ1,τ2). then, the above variable z satisfies szt (x,ρ,s,t) + zρ (x,ρ,s,t) = 0, (x,ρ,s,t) ∈ (0, 1) × (0, 1) × (τ1,τ2) × (0, +∞) . int. j. anal. appl. 19 (1) (2021) 80 therefore, the problem (1.1) is equivalent to (2.1)   ρ1ϕtt −k(ϕx + ψ)x = 0, x ∈ (0, 1) , t > 0, ρ2ψtt − bψxx + k(ϕx + ψ) + µ1ψt + ∫ τ2 τ1 µ2(s)z(x, 1,s,t)ds + f(ψ) = 0, x ∈ (0, 1) , t > 0, szt (x,ρ,s,t) + zρ (x,ρ,s,t) = 0, ρ ∈ (0, 1) , s ∈ (τ1,τ2) , t > 0, with the following initial and boundary conditions (2.2)   ϕ(x, 0) = ϕ0, ϕt(x, 0) = ϕ1, x ∈ (0, 1) , ψ(x, 0) = ψ0, ψt(x, 0) = ψ1, x ∈ (0, 1) , z(x,ρ,s, 0) = f0(x,ρs), x ∈ (0, 1), ρ ∈ (0, 1) , s ∈ (0,τ2), ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t) = 0, t > 0. concerning the weight of the delay, we only assume that (2.3) ∫ τ2 τ1 |µ2 (s)|ds < µ1. in addition, we give some hypothesis on the forcing term f(ψ(x,t)). we assume that f : ir → ir satisfies the following condition (2.4) ∣∣f(ψ1) −f(ψ2)∣∣ ≤ k0 (∣∣ψ1∣∣θ − ∣∣ψ2∣∣θ)∣∣ψ1 −ψ2∣∣ for all ψ1,ψ2 ∈ ir, where k0 > 0, θ > 0. also (2.5) 0 ≤ f̃(ψ) ≤ f(ψ)ψ, for all ψ ∈ ir, with f̃(y) = ∫ y 0 f(s)ds. we introduce the hilbert space, h = h10 (0, 1) ×l 2 (0, 1) ×h10 (0, 1) ×l 2 (0, 1) ×l2 ((0, 1) × (0, 1) × (τ1,τ2)) for u = (ϕ,u,ψ,v,z) t , ( ϕ̃, ũ, ψ̃, ṽ, z̃ )t equipped with the scalar product 〈u,ũ〉h = ∫ 1 0 [ ρ1uũ + ρ2vṽ + k (ϕx + ψ) ( ϕ̃x + ψ̃ ) + bψxψ̃x ] dx + ∫ 1 0 ∫ τ2 τ1 s |µ2 (s)| ∫ 1 0 z (x,ρ,s,t) z̃ (x,ρ,s,t) dρdsdx. we introduce two new dependent variables ϕt = u and ψt = v, then the system (2.1)-(2.2) can be written as (2.6)   ∂u ∂t = au + f, t > 0 u (x, 0) = u0 (x) = ( ϕ0,ϕ1,ψ0,ψ1,f0 )t , int. j. anal. appl. 19 (1) (2021) 81 and (2.7) au =   u k ρ1 (ϕxx + ψx) v b ρ2 ψxx − kρ2 (ϕx + ψ) − µ1 ρ2 v − µ1 ρ2 ∫ τ2 τ1 µ2 (s) z (x,ρ,s,t) ds −1 τ zρ (x,ρ,s,t)   , f =   0 0 0 −1 ρ2 f(ψ) 0   with the domain d (a) = { (ϕ,u,ψ,v,z) t ∈ h : v = z (x, 0,s,t) in (0, 1) } , where h = ( h2 (0, 1) ∩h10 (0, 1) ) ×h10 (0, 1) × ( h2 (0, 1) ∩h10 (0, 1) ) ×h10 (0, 1) ×l 2 ((0, 1) × (0, 1) × (τ1,τ2)) . clearly, d(a) is dense in h, we have the following existence and uniqueness result (see [6]). theorem 2.1. let u0 ∈ h and assume that (2.4)-(2.5) and µ2 < µ1 hold. then, there exists a unique solution u ∈ c (r+,h) of problem (2.1). moreover, if u0 ∈ d(a), then u ∈ c (r+,d(a)) ∩c (r+,h) . 3. stability result in this section, we use the energy method to show that the solution of problem (2.1)–(2.2) decays exponentially, below we shall give the stability result. theorem 3.1. assume that (2.4)-(2.5) and µ2 < µ1 hold. assume that ρ1 ρ2 = k b also holds. then, with respect to mild solutions, there exist $1 > 0 and $2 > 0 such that (3.1) e (t) ≤ $1e−$2t, t ≥ 0. to achieve our goal we state and prove the following lemmas. lemma 3.1. the energy functional e (t) of problem (2.1)–(2.2), defined by e (t) = 1 2 ∫ 1 0 ( ρ1ϕ 2 t + ρ2ψ 2 t ) dx + 1 2 ∫ 1 0 { k (ϕx + ψ) 2 + bψ2x } dx + ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2 (s)|z (x,ρ,s,t) dsdρdx + ∫ 1 0 f̃(ψ)dx(3.2) int. j. anal. appl. 19 (1) (2021) 82 satisfies (3.3) de (t) dt ≤−m1 ∫ 1 0 ψ2t dx ≤ 0, where m1 = µ1 − ∫ τ2 τ1 |µ2 (s)|ds. proof. multiplying the first equation in (2.1) by ϕt, the second equation by ψt, integrating over (0, 1) and summing them up we get 1 2 d dt ∫ 1 0 ( ρ1ϕ 2 t + ρ2ψ 2 t ) dx + 1 2 d dt ∫ 1 0 { k (ϕx + ψ) 2 + bψ2x } dx = −µ1 ∫ 1 0 ψ2t dx−µ1 ∫ 1 0 f (ψ) ψtdx− ∫ 1 0 ∫ τ2 τ1 ψtµ2 (s) z (x, 1,s,t) dsdx.(3.4) multiplying the third equation of (2.1) by |µ2 (s)|z (x,ρ,s,t) and integrating over (0, 1) × (0, 1) × (τ1,τ2), we obtain 1 2 d dt ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2 (s)|z2 (x,ρ,s,t) dsdρdx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx − 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 0,s,t) dsdx = 0,(3.5) by summing (3.5), (3.4) and using the fact that z (x, 0,s,t) = ϕt (x,t) , we have de (t) dt = − ( µ1 − 1 2 ∫ τ2 τ1 |µ2 (s)|ds )∫ 1 0 ψ2t dx − 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx − ∫ 1 0 ψt ∫ τ2 τ1 µ2 (s) z(x, 1,s,t)dsdx.(3.6) now, using young’s inequality, we arrive at − ∫ 1 0 ψt ∫ τ2 τ1 µ2 (s) z (x, 1,s,t) dsdx ≤ 1 2 ∫ τ2 τ1 |µ2 (s)|ds ∫ 1 0 ψ2t dx + 1 2 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx.(3.7) inserting (3.7) in (3.6) and using (2.3), we have (3.2) and (3.3). the proof is complete. � lemma 3.2. let (ϕ,ψ,z) be the solution of (2.1)–(2.2). then, the functional (3.8) i1 (t) := − ∫ 1 0 (ρ1ϕϕt + ρ2ψψt)dx− µ1 2 ∫ 1 0 ψ2dx. int. j. anal. appl. 19 (1) (2021) 83 satisfies di1 (t) dt ≤− ∫ 1 0 (ρ1ϕ 2 t + ρ2ψ 2 t )dx + c0 ∫ 1 0 ψ2xdx + k ∫ 1 0 (ϕx + ψ) 2dx + µ1 4 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)dsdx,(3.9) proof. differentiating i1 (t), we obtain di1 (t) dt = −ρ1 ∫ 1 0 ϕ2tdx−ρ1 ∫ 1 0 ϕϕttdx−ρ2 ∫ 1 0 ψ2t dx −ρ2 ∫ 1 0 ψψttdx−µ1 ∫ 1 0 ψψtdx, and using (2.1)1, (2.1)2, we get di1 (t) dt = −ρ1 ∫ 1 0 ϕ2tdx−ρ2 ∫ 1 0 ψ2t dx + b ∫ 1 0 ψ2xdx + k ∫ 1 0 (ϕx + ψ) 2dx + ∫ 1 0 f(ψ)ψdx + ∫ 1 0 ψ ∫ τ2 τ1 µ2 (s) z(x, 1,s,t)dsdx.(3.10) applying young’s and poincaré inequalities, we have∫ 1 0 ψ ∫ τ2 τ1 |µ2 (s)|z (x, 1,s,t) dsdx ≤ µ1 ∫ 1 0 ψ2xdx + µ1 4 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx,(3.11) ∫ 1 0 |f(ψ)ψ|dx ≤ ∫ 1 0 |ψ|θ |ψ| |ψ|dx ≤‖ψ‖θ2(θ+1) ‖ψ‖2(θ+1) ‖ψ‖ ≤ c1 ∫ 1 0 ψ2xdx.(3.12) by substituting (3.11), (3.12) in (3.10), we obtain (3.9). � now, let w be the solution of (3.13) −wxx = ψx, w (0) = w (1) = 0, then we get w (x,t) = − ∫ x 0 ψ (y,t) dy + x (∫ 1 0 ψ (y,t) dy ) . we have the following inequalities. lemma 3.3. the solution of (3.13) satisfies∫ 1 0 w2xdx ≤ ∫ 1 0 ψ2dx and ∫ 1 0 w2tdx ≤ ∫ 1 0 ψ2t dx. int. j. anal. appl. 19 (1) (2021) 84 proof. we multiply equation (3.13) by w, integrate by parts and use the cauchy–schwarz inequality to obtain (3.14) ∫ 1 0 w2xdx ≤ ∫ 1 0 ψ2dx next, we differentiate (3.13) with respect to t and by the same procedure as above, we obtain (3.15) ∫ 1 0 w2tdx ≤ ∫ 1 0 ψ2t dx. this completes the proof of lemma (3.3). � lemma 3.4. let (ϕ,ψ,z) be the solution of (2.1)–(2.2). then, for any ε2 > 0, the functional (3.16) i2 (t) := ∫ 1 0 ( ρ2ψtψ + ρ1ϕtw + µ1 2 ψ2 ) dx, satisfies di2 (t) dt ≤− b 2 ∫ 1 0 ψ2xdx + ( ρ1 4ε2 + ρ2 )∫ 1 0 ψ2t dx + ρ1ε2 ∫ 1 0 ϕ2tdx(3.17) + µ1 4ε2 ∫ 1 0 (∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)ds ) dx− ∫ 1 0 f̃(ψ)dx proof. by differentiation i2 (t) , we obtain and by using (2.1)1, (2.1)2, we have di2 (t) dt = ρ2 ∫ 1 0 ψ2t dx− b ∫ 1 0 ψ2xdx + ρ1 ∫ 1 0 ϕtwtdx−k ∫ 1 0 ψ2dx + k ∫ 1 0 w2xdx − ∫ 1 0 f(ψ)ψdx− ∫ 1 0 ψ (∫ τ2 τ1 µ2 (s) z(x, 1,s,t)ds ) dx(3.18) using young’s inequality and (3.15), we have ρ1 ∫ 1 0 ϕtwtdx ≤ ρ1ε2 ∫ 1 0 ϕ2tdx + ρ1 4ε2 ∫ 1 0 w2tdx ≤ ρ1ε2 ∫ 1 0 ϕ2tdx + ρ1 4ε2 ∫ 1 0 ψ2t dx(3.19) using young’s, cauchy-schwarz, poincaré inequalities, we get − ∫ 1 0 ψ (∫ τ2 τ1 |µ2 (s) z(x, 1,s,t)|ds ) dx ≤ δ1 ∫ 1 0 ψ2xdx + µ1 4δ1 ∫ 1 0 (∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)ds ) dx(3.20) cauchy-schwarz and poincaré’s inequalities, give∫ 1 0 |f(ψ)ψ|dx ≤ ∫ 1 0 |ψ|θ |ψ| |ψ|dx ≤‖ψ‖θ2(θ+1) ‖ψ‖2(θ+1) ‖ψ‖ ≤ c1 ∫ 1 0 ψ2xdx.(3.21) int. j. anal. appl. 19 (1) (2021) 85 by substituting (3.19), (3.20), (3.21) in (3.18), recalling (3.14), (3.15), (2.5) and letting δ1 = b 2 , we obtain (3.17). the proof is now complete. � lemma 3.5. let (ϕ,ψ,z) be the solution of (2.1)–(2.2). then, the functional (3.22) i3(t) := ρ2 ∫ 1 0 ψt(ϕx + ψ) + ρ2 ∫ 1 0 ψxϕtdx, satisfies di3 (t) dt ≤ b [ψxϕx] 1 0 dx + ( ρ2 + µ21 k )∫ 1 0 ψ2t dx− k 4 ∫ 1 0 (ϕx + ψ) 2dx + c1 ∫ 1 0 ψ2xdx + µ1 k ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)dsdx− ∫ 1 0 f̃(ψ)dx + ( ρ2k −ρ1b ρ1 )∫ 1 0 ψx (ϕx + ψ)x dx,(3.23) where c1 is a positive constant. proof. by differentiation i3 (t) and using (2.1)1, (2.1)2, we obtain di3 (t) dt = b [ψxϕx] 1 0 + ρ2 ∫ 1 0 ψ2t dx−k ∫ 1 0 (ϕx + ψ) 2dx−µ1 ∫ 1 0 ψt(ϕx + ψ)dx − ∫ 1 0 ∫ τ2 τ1 µ2 (s) (ϕx + ψ)z(x, 1,s,t)dsdx− ∫ 1 0 f(ψ)(ϕx + ψ)dx,(3.24) by using young’s inequality, we have (3.25) µ1 ∫ 1 0 |ψt(ϕx + ψ)|dx ≤ k 4 ∫ 1 0 (ϕx + ψ) 2dx + µ21 k ∫ 1 0 ψ2t dx. using young’s and cauchy schwarz inequalities, we get∫ 1 0 (ϕx + ψ) ∫ τ2 τ1 |µ2 (s) z(x, 1,s,t)|dsdx ≤ k 4 ∫ 1 0 (ϕx + ψ) 2dx + µ21 k ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)dsdx.(3.26) young’s, cauchy schwarz and poincaré inequalities lead to∫ 1 0 f(ψ)ϕxdx ≤‖ϕx‖‖ψ‖ θ 2(θ+1) ‖ψ‖2(θ+1) ≤ δ0 2b2 ∫ 1 0 ϕ2xdx + b2 2δ0λ1 ∫ 1 0 ψ2xdx ≤ δ0 2b2 ∫ 1 0 (ϕx + ψ) 2dx + δ0 2b2 ∫ 1 0 ψ2dx + b2 2δ0λ1 ∫ 1 0 ψ2xdx ≤ δ0 2b2 ∫ 1 0 (ϕx + ψ) 2dx + ( δ0 2λ1b2 + b2 2δ0λ1 )∫ 1 0 ψ2xdx.(3.27) inserting (3.25)-(3.27) in (3.24) and letting δ0 = 1 2 kb2, we obtain (3.23). � int. j. anal. appl. 19 (1) (2021) 86 next, in order to handle the boundary terms, appearing in (3.23) , we define the function q(x) = −4x + 2, x ∈ (0, 1) so, we have the following result. lemma 3.6. let (ϕ,ψ,z) be the solution of (2.1)–(2.2), then for any ε1 > 0, the following estimate holds b [ψxϕx] 1 0 ≤− bρ2 4ε1 d dt ∫ 1 0 qψtψxdx− ρ1ε1 k d dt ∫ 1 0 qϕtϕxdx + 3ε1 ∫ 1 0 ϕ2xdx + ( 2ρ1ε1 k + bρ2 2ε1 ) ∫ 1 0 ψ2t dx + ( k2ε21 4 + ε1 4b2 ) ∫ 1 0 (ϕx + ψ) 2dx + b 4ε1 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dx + ( b2 2ε21 + 1 4λ1b2 + b2 8ε21λ1 + µ1b 4ε1 + b2 4ε31 + ε1) ∫ 1 0 ψ2xdx(3.28) proof. by using young’s inequality, we easily see that, for ε1 > 0, (3.29) b [ψxϕx] 1 0 ≤ ε1 [ ϕ2x (1) + ϕ 2 x (0) ] + b2 4ε1 [ ψ2x (1) + ψ 2 x (0) ] , we need the following fact d dt ∫ 1 0 bρ2qψtψxdx = bρ2 ∫ 1 0 qψttψxdx + bρ2 ∫ 1 0 qψtψxtdx. on the other hand bρ2 ∫ 1 0 qψttψxdx = b 2 ∫ 1 0 qψxxψxdx−kb ∫ 1 0 q(ϕt + ψ)ψxdx − b ∫ 1 0 ∫ τ2 τ1 qψxµ2 (s) z (x, 1,s,t) dsdx− b ∫ 1 0 qf(ψ)ϕxdx ≤−b2 [ ψ2x (1) + ψ 2 x (0) ] + 2b2 ∫ 1 0 ψ2xdx + (k 2ε2 + ε b2 ) ∫ 1 0 (ϕx + ψ) 2dx + ( b2 ε2 + ε 2λ1b2 + b2 2ελ1 + µ1b) ∫ 1 0 ψ2xdx + b ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx.(3.30) therefore bρ2 ∫ 1 0 qψtψxtdx = 2ρ2b ∫ 1 0 ψ2t dx similarly d dt ∫ 1 0 ρ1qϕtϕxdx = ∫ 1 0 q(ϕt + ψ)ϕxdx + ∫ 1 0 ρ1qϕtϕxtdx ≤−k [ ϕ2x (1) + ϕ 2 x (0) ] + 3k ∫ 1 0 ϕ2xdx + k ∫ 1 0 ψ2xdx + 2ρ1 ∫ 1 0 ψ2t dx int. j. anal. appl. 19 (1) (2021) 87 which, along with (3.29)-(3.30), gives us (3.28). the proof is now complete. � lemma 3.7. let (ϕ,ψ,z) be the solution of (2.1)–(2.2). then, for η1 > 0, the functional (3.31) f4 (t) = ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2 (s)|z2 (x,ρ,s,t) dsdρdx, satisfies f ′4 (t) ≤−η1 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2(s)|z2(x,ρ,s,t)dsdρdx −η1 ∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2 (x, 1,s,t) dsdx + µ1 ∫ 1 0 ψ2t dx.(3.32) proof. differentiating f4 (t) and using (2.1)3, we obtain f ′4(t) = −2 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 e−sρ |µ2(s)|z(x,ρ,s,t)zρ(x,ρ,s,t)dsdρdx = − ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 e−sρ |µ2(s)| ∂ ∂ρ [z2(x,ρ,s,t)]dsdρdx integration by parts gives and using the fact that z(x, 0,s,t) = ψt and e −s ≤ e−sρ ≤ 1, we get for all ρ ∈ [0, 1] f ′4(t) ≤− ∫ 1 0 ∫ τ2 τ1 e−s |µ2(s)|z2(x, 1,s,t)dsdx + ∫ τ2 τ1 |µ2(s)|ds ∫ 1 0 ψ2t dx − ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−s |µ2(s)|z2(x,ρ,s,t)dsdρdx. since −e−s is an increasing function, we have −e−s ≤ −e−τ2 for all s ∈ [τ1,τ2] . finally, setting η1 = e−τ2 and recalling (2.3), we obtain (3.32). � now, we define the lyapunov functional l(t) by (3.33) l (t) := ne (t) + 1 8 i1 (t) + n1i2 (t) + i3 (t) + n2i4 (t) , where n1,n2 and n are positive constants. lemma 3.8. let (ϕ,ψ,z) be the solution of (2.1)–(2.2). then, there exists two positive constants β1 and β2 such that the lyapunov functional l(t) satisfies (3.34) β1e (t) ≤ l (t) ≤ β2e (t) , ∀t ≥ 0, and (3.35) l′ (t) ≤−λ1e (t) + ( ρ2k −ρ1b ρ1 )∫ 1 0 ψx (ϕx + ψ)x dx. int. j. anal. appl. 19 (1) (2021) 88 proof. let l (t) := ne (t) + 1 8 i1 (t) + n1i2 (t) + i3 (t) + n2i4 (t) , then |l (t) −ne (t)| ≤ ρ1 8 ∫ 1 0 |ϕϕt|dx + ρ2 8 ∫ 1 0 |ψψt|dx + µ1 16 ∫ 1 0 ψ2dx + n1ρ2 ∫ 1 0 |ψtψ|dx + n1ρ1 ∫ 1 0 |ϕtw|dx + n1 µ1 2 ∫ 1 0 ψ2dx + ρ2 ∫ 1 0 |ψt(ϕx + ψ)|dx + ρ2 ∫ 1 0 |ψxϕt|dx + n2 ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 se−sρ |µ2 (s)|z2 (x,ρ,s,t) dsdρdx. exploiting young’s, poincaré and cauchy–schwarz inequalities, we obtain |l (t) −ne (t)| ≤ c ∫ 1 0 ( ψ2x + ψ 2 t + ϕ 2 t + (ϕx + ψ) 2 + ∫ 1 0 ∫ τ2 τ1 s |µ2 (s)|z2 (x, 1,s,t) dsdρ ) dx + ∫ 1 0 f̃(ψ)dx ≤ ce (t) , now, combining (3.3), (3.9), (3.17), (3.23) and (3.32), we get by differentiating l (t), exploiting (3.3), (3.9), (3.17), (3.23), (3.28), (3.32) and setting ε2 = ρ1 16n1 , we get dl (t) dt = − ( nm1 −n1 (4n1 + ρ2) −n2µ1 − ( ρ2 + µ21 k ) − ( 2ρ1ε1 k + bρ2 2ε1 ) + ρ2 )∫ 1 0 ψ2t dx − ( b 2 n1 − c0 8 − ( b2 2ε21 + 1 4b2 + b2 8ε21 + µ1b 4ε1 + b2 4ε31 + ε1) )∫ 1 0 ψ2xdx − ( k 8 −ε1 ( k2ε1 + 1 b2 ))∫ 1 0 (ϕx + ψ) 2dx −n2β ∫ 1 0 ∫ 1 0 ∫ τ2 τ1 s |µ2 (s)|z2 (x,ρ,s,t) dsdρdx − ( n2β − 4n21 µ1 − µ1 32 − µ1 k − b 4ε1 )∫ 1 0 ∫ τ2 τ1 |µ2 (s)|z2(x, 1,s,t)dsdx − ρ1 16 ∫ 1 0 ϕ2tdx− (n1 + 1) ∫ 1 0 f̃(ψ)dx, + ( ρ2k −ρ1b ρ1 )∫ 1 0 ψx (ϕx + ψ)x dx. first, we choose ε1 small enough such that k 8 −ε1 ( k2ε1 + 1 b2 ) > 0. after, we take n1 large so that b 2 n1 − c0 8 − ( b2 2ε21 + 1 4b2 + b2 8ε21 + µ1b 4ε1 + b2 4ε31 + ε1 > 0. int. j. anal. appl. 19 (1) (2021) 89 then, we select n2 large to satisfies n2β − 4n21 µ1 − µ1 32 − µ1 k − b 4ε1 > 0. by finally choose n large enough (even larger so that 1.1 remains valid) such that nm1 −n1 ( ρ1 4ε2 + ρ2 ) −n2µ1 − ( ρ2 + µ21 k ) − ( 2ρ1ε1 k + bρ2 2ε1 ) + ρ2 > 0, we obtain (3.35). the proof is complete. � acknowledgments: the authors wish to thank deeply the anonymous referee for his/her useful remarks and his/her careful reading of the proofs presented in this paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] t.a. apalara, well-posedness and exponential stability for a linear damped timoshenko system with second sound and internal distributed delay. electron. j. differ. equ. 2014 (2014), 254. [2] t.a. apalara, uniform decay in weakly dissipative timoshenko system with internal distributed delay feedbacks, acta math. sci. 36 (2016), 815–830. [3] l. bouzettouta, d. abdelhak, exponential stabilization of the full von kármán beam by a thermal effect and a frictional damping and distributed delay, j. math. phys. 60 (2019), 041506. [4] l. bouzettouta, s. zitouni, kh. zennir. and h. sissaoui, stability of bresse system with internal distributed delay. j. math. comput. sci. 7(1) (2017), 92–118. [5] r. datko, j. lagnese and m. p. polis, an example on the effect of time delays in boundary feedback stabilization of wave equations. siam j. control optim. 24(1) (1986), 152–156. [6] b.w. feng and m. l. pelicer, global existence and exponential stability for a nonlinear timoshenko system with delay. bound. value probl. 2015 (2015), article id 206. [7] h.e. khochemane, a. djebabla, s. zitouni, l. bouzettouta, well-posedness and general decay of a nonlinear damping porous-elastic system with infinite memory, j. math. phys. 61 (2020), 021505. [8] h.e. khochemane, l. bouzettouta, a. guerouah, exponential decay and well-posedness for a one-dimensional porouselastic system with distributed delay, appl. anal. (2019), 1–15. https://doi.org/10.1080/00036811.2019.1703958. [9] h. e. khochemane , s. zitouni and l. bouzettouta, stability result for a nonlinear damping porous-elastic system with delay term, nonlinear studies, 27(2) (2020), 1–17. [10] s. a. messaoudi and b. said-houari, energy decay in a timoshenko-type system for thermoelasticity of type iii with distributed delay and past history, electron. j. differ. equ. 2018 (2018), 75. [11] m.i. mustafa, a uniform stability result for thermoelasticity of type iii with boundary distributed delay, j. math. anal. appl. 415 (2014), 148–158. [12] s. nicaise and c. pignotti, stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. siam j. control optim. 45(5) (2006), 1561–1585. [13] s. nicaise and c. pignotti, interior feedback stabilization of wave equations with time dependent delay. electron. j. differ. equ. 2011 (2011), 41. int. j. anal. appl. 19 (1) (2021) 90 [14] c. pignotti, a note on stabilization of locally damped wave equations with time delay, syst. control lett. 61 (2012), 92–97. [15] s. p. timoshenko, on the correction for shear of the differential equation for transverse vibrations of prismatic bars. phil. mag. ser. 6(41) (1921), 744–746. [16] s. zitouni, l. bouzettouta, kh. zennir and d. ouchenane, exponential decay of thermo-elastic bresse system with distributed delay term, hacettepe j. math. stat. 47(5) (2018), 1216–1230. 1. introduction 2. preliminaries 3. stability result references international journal of analysis and applications volume 16, number 4 (2018), 454-461 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-454 stability conditions of a class of linear retarded differential systems serbun ufuk değer1,∗ and yaşar bolat2 1institute of sciences, kastamonu university, kastamonu, turkey 2department of mathematics, faculty of art & science, kastamonu university, kastamonu, turkey ∗corresponding author: sudeger@kastamonu.edu.tr abstract. in this paper, we give some new necessary and sufficient conditions for the asymptotic stability of a linear retarded differential system with two delays x′ (t) + (1 − a) x (t) + a (x (t − k) + x (t − l)) = 0, t ≥ 0, where a < 1 is a real number, a is a 2 × 2 real constant matrix, and k, l are positive numbers such that k > l. 1. introduction and preliminaries retarded differential equations are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. stability of these equation has a wide range of applications in science and engineering. recently, these equations have been investigated by many authors; for example, matsunaga [1], cooke and van den driessche [2], kuang [3] , cooke and grossman [4], ruan and wei [5], hale and lunel [6], khokhlovaa, kipnis and malygina [8], cermák and jánsky [9], hrabalova [10], nakajima [11], hara and sakata [12], smith [13], freedman and kuang [14] and bellman and cooke [15] which have studied the asymptotic stability of linear retarded received 2018-03-09; accepted 2018-05-09; published 2018-07-02. 2010 mathematics subject classification. 39a13, 39a30. key words and phrases. differential equations; characteristic equation; asymptotic stability. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 454 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-454 int. j. anal. appl. 16 (4) (2018) 455 differential equations. in this paper, we give some new necessary and sufficient conditions for the asymptotic stability of the following system x′ (t) + (1 −a) x (t) + a (x (t−k) + x (t− l)) = 0, t ≥ 0, (1.1) where a < 1 is a real number, a is a 2 × 2 real constant matrix, and k, l are positive numbers such that k > l. system (1.1) is called a retarded or delay differential system if the highest derivative term does not have a delay.the characteristic equations of retarded differential equations are polynomials. these polynomials are exponential polynomials or quasi-polynomials as named in bellman and cooke [15]. we know that for the linear retarded differential equation, the zero solution being asymptotically stable is equivalent to all solutions having limit zero as t → ∞ which in turn is true if and only if all roots of the associated characteristic equation have negative real parts. the purpose of this paper is to obtain new results for the asymptotic stability of zero solution of system (1.1) when a is a constant matrix. now we will give some basic information that we use for the lemmas. if we get x (t) = py (t) for a regular matrix p in (1.1), then we obtain the following system; y′ (t) + (1 −a) y (t) + p−1ap (y (t−k) + y (t− l)) = 0, t ≥ 0. thus, matrix a can be given in one of the following two matrices in jordan form[7]: (i) a =   q1 p 0 q2   , b1,b2 and p are real constants, (ii) a = q   cos θ −sin θ sin θ cos θ   , q,θ are real constants and |θ| < π 2 . here we discuss the case (ii), the other case should be discussed similarly. the characteristic equation of system (1.1) is given as f(λ) := det ( λi2 + (1 −a) i2 + a ( e−λk + e−λl )) = 0, (1.2) where i2 is the 2 × 2 identity matrix. by the case (ii), we have f(λ) as follows f(λ) ≡ fθ (λ) fθ ( λ ) = 0, where fθ (λ) = ( λ + (1 −a) + q ( e−λk+i|θ| + e−λl+i|θ| )) , and λ is the complex conjugate of any complex λ. note that fθ ( λ ) = 0 implies fθ ( λ ) = 0. int. j. anal. appl. 16 (4) (2018) 456 2. some auxiliary lemmas lemma 2.1. the zero solution of (1.1) is asymptotically stable if and only if all the roots of equation fθ (λ,q) = λ + (1 −a) + q ( e−λk+i|θ| + e−λl+i|θ| ) (2.1) lie in the left half of the complex plane. since fθ is an analytic function of λ and q for the fixed numbers k,l,a and θ, one can regard the root λ = λ (q) of (2.1) as a continuous function of q. the next lemma plays very important role for the main theorem. lemma 2.2. as q varies, the sum of the multiplicities of the roots of (2.1) in the open right half-plane can change only if a root appears on or crosses the imaginary axis. consequently, we claim that (2.1) has only imaginary roots ±iω. we will determine the value of q as equation (2.1) has roots on the imaginary axis. now, we can write the characteristic equation (2.1) as follows; λ + (1 −a) + q ( e−λk+iθ + e−λl+iθ ) = 0. (2.2) let λ = iω is a root (2.2) such that ω ∈ r. firsty, since fθ (0) 6= 0, we see that ω 6= 0. if ω 6= 0, then we write iω + (1 −a) + q ( e−iωk+iθ + e−iωl+iθ ) = 0, and from this equation, we have  ω = q (sin (ωk −θ) + sin (ωl−θ))a− 1 = q (cos (ωk −θ) + cos (ωl−θ)) , (2.3) which is equivalent to   ω = 2q sin ( ω(k+l) 2 −θ ) cos ( ω(k−l) 2 ) a− 1 = 2q cos ( ω(k+l) 2 −θ ) cos ( ω(k−l) 2 ) . (2.4) from (2.4) , we get ω a− 1 = tan ( ω (k + l) 2 −θ ) . (2.5) we know that the function tanjant is defined as on the region h = { (t, tan t) : t ∈ r, t = π 2 + ρπ, ρ ∈ z } . thus, (2.5) has only a sequence of the roots {ωj : j ≥ 1} , where ωj ∈ ( (2j − 1) π + 2θ k + l , (2j + 1) π + 2θ k + l ) for ωj > 0 and ωj ∈ ( −(2j + 1) π + 2θ k + l , −(2j − 1) π + 2θ k + l ) for ωj < 0. int. j. anal. appl. 16 (4) (2018) 457 lemma 2.3. suppose that {qj : j ≥ 1} > 0 and 0 < θ < π 2 . let λ = iωj be a root of (2.1) where ωj ∈( (3−4j)π k−l , (4j−3)π k−l ) − { −nπ+2θ k+l , nπ+2θ k+l } is a real number for n ∈ n. then the following conditions hold: (i) if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 ≤ 0, then there exists no real number ωj. (ii) if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 > 0, then there exist the real numbers ωj and qj, qj is as follows: qj = a− 1 2 cos ( ωj(k+l) 2 −θ ) cos ( ωj(k−l) 2 ). remark 2.1. in case ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 > 0, for the equality ωj = ±ϕj = ± √( 2qj cos ( ωj (k − l) 2 ))2 − (a− 1)2, the sum of delays k and l is as follows; (kn + ln) + = 2 ϕj  −(2n + 2) π + arccos   a− 1 2qj cos ( ωj(k−l) 2 )   + θ   (kn + ln) − = 2 ϕj  −2nπ + arccos   a− 1 2qj cos ( ωj(k−l) 2 )  −θ   , for n ∈ n. also, iϕj or −iϕj is a root of (2.1) for the sum of delays (kn + ln) + or (kn + ln) − for n ∈ n. proof of lemma 2.3. by (2.4), we can write ω2 + (a− 1)2 = 2q cos ( ω (k − l) 2 ) . substituting ω = {ωj}j≥1 and q = {qj}j≥1 into the above equation, we obtain ω2j + (a− 1) 2 = 2qj cos ( ωj (k − l) 2 ) . (2.6) if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 ≤ 0, then statement (2.6) implies ω2j < 0, contradicts ω 2 j > 0; thus, condition (i) is verified; that is, (2.1) has no root on the imaginary axis for all k > l > 0. on the other hand, if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 > 0, statement (2.6) implies ωj = ±ϕj for ϕj =√( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2. from (2.4), we get qj = a− 1 2 cos ( ωj(k+l) 2 −θ ) cos ( ωj(k−l) 2 ). now we will show that iϕj is a root of (2.1) . in case ωj = ( 2qj cos ( ωj (k − l) 2 ))2 − (a− 1)2 , int. j. anal. appl. 16 (4) (2018) 458 (2.4) and cos ( ωj(k−l) 2 ) > 0 implies sin ( ωj(k+l) 2 −θ ) > 0. thus, we can write ωj (k + l) 2 −θ = −(2n + 2) π + arccos   a− 1 2qj cos ( ωj(k−l) 2 )   for n ∈ n (2.7) which yields (kn + ln) + . after that, we have sin  arccos   a− 1 2qj cos ( ωj(k−l) 2 )     = ϕj 2qj cos ( ωj(k−l) 2 ) (2.8) because of arccos   a− 1 2qj cos ( ωj(k−l) 2 )   =   arcsin   ϕj 2qj cos ( ωj(k−l) 2 )   if a− 1 > 0 π − arcsin   ϕj 2qj cos ( ωj(k−l) 2 )   if a− 1 < 0. for the case k + l = (kn + ln) + , from (2.1) we have f (iω) = iω + (1 −a) + q ( ei(ωk−θ) + ei(ωl−θ) ) = i √( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 + (1 −a) + + 2qj cos ( ωj(k−l) 2 ) e −i ( ωj (k+l) 2 −θ ) , = i √( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 + (1 −a) + + 2qj cos ( ωj(k−l) 2 ) e −i  −(2n+2)π+arccos   a−1 2qj cos ( ωj (k−l) 2 )     , = i √( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 + (1 −a) + + 2qj cos ( ωj(k−l) 2 ){ cos ( arccos ( a−1 2qj cos ( ωj (k−l) 2 ) ))} − − 2qj cos ( ωj(k−l) 2 ){ i sin ( arccos ( a−1 2qj cos ( ωj (k−l) 2 ) ))} , = i √( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 + (1 −a) + + (1 −a) − i √( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 = 0; thus, we can see that iϕj is a root of (2.1) . similarly, when ωj ∈ ( (3 − 4j) π k − l , 0 ] , −iϕj is a root of (2.1) for the sum of delays (kn + ln) − . the proof is completed. � int. j. anal. appl. 16 (4) (2018) 459 when ωj < 0, we have the following analogous result. lemma 2.4. suppose that {qj : j ≥ 1} < 0 and 0 < θ < π2 . let λ = iωj be a root of (2.1) where ωj ∈( (4j−3)π k−l , (4j−1)π k−l ) ∪ ( (1−4j)π k−l , (3−4j)π k−l ) − { −nπ+2θ k+l , nπ+2θ k+l } for n ∈ n. then the following conditions hold: (i) if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 ≤ 0, then there exists no real number ωj. (ii) if ( 2qj cos ( ωj(k−l) 2 ))2 − (a− 1)2 > 0, then there exist the real numbers ωj and qj, qj is as follows: qj = a− 1 2 cos ( ωj(k+l) 2 −θ ) cos ( ωj(k−l) 2 ). proof. the proof is similar of the lemma 2.3. � lemma 2.5. suppose that a < 1. let λ (q) = re (λ (q))+i im (λ (q)) the root of (2.1) satisfying re (λ (qj)) = 0, im (λ (qj)) = ωj. then the following equation is provided: si gn re ( λ′ (qj) ) . si gn qj > 0 proof. taking the derivative of λ with respect to q on (2.1) , we have dλ dq + ( e−λk+iθ + e−λl+iθ ) + q ( −ke−λk+iθ − le−λl+iθ ) dλ dq = 0; dλ dq = − e−λk+iθ + e−λl+iθ 1 −q (ke−λk+iθ + le−λl+iθ) = λ + 1 −a q (1 −q (ke−λk+iθ + le−λl+iθ)) . substituting ω = ωj and q = qj into the above equation, we get dλ dq | λ=iω = iωj + 1 −a qj ( 1 −qj ( ke−i(ωjk−θ) + le−i(ωjl−θ) )) thus, it follows that; re dλ dq | λ=iω = (1 −a) qj (1 −qj (k cos (ωjk −θ) + l cos (ωjl−θ))) m + (2.9) + ωjq 2 j (k sin (ωjk −θ) + l sin (ωjl−θ)) m , (2.10) where m = q2j (1 −qj (k cos (ωjk −θ) + l cos (ωjl−θ))) 2 + q4j (k sin (ωjk −θ) + l sin (ωjl−θ)) 2 . let a1 = sin (ωk −θ) + sin (ωl−θ) and a2 = cos (ωk −θ) + cos (ωl−θ) . by (2.4) , we have a1(ω) a2(ω) = tan ( ω(k+l) 2 −θ ) , thus d dω ( a1 (ω) a2 (ω) ) = a′1 (ω) a2 (ω) −a′2 (ω) a1 (ω) (a2 (ω)) 2 > 0, is obtained, which implies a′1 (ω) a2 (ω) −a′2 (ω) a1 (ω) > 0. int. j. anal. appl. 16 (4) (2018) 460 since a′1 = k cos (ωk −θ) + l cos (ωl−θ) and a′2 = −k sin (ωk −θ) − l sin (ωl−θ), (2.9) can written re dλ dq | λ=iω = qj [((1 −a) − (1 −a) qja′1 (ω)) −ωjqja′2 (ω)] m , we use (2.4) for above equation, then we get re dλ dq | λ=iω = qj [ (1 −a) + q2j (a ′ 1 (ω) a2 (ω) −a′2 (ω) a1 (ω)) ] m . hence, the proof is completed. � 3. main results theorem 3.1. suppose that a < 1, 0 < θ < π 2 and the matrix a of the system (1.1) is written as the form (ii) . we define q−θ = max j≥1 {qj : qj < 0} , q+θ = min j≥1 {qj : qj > 0} , and let a neighborhood of q = 0 is ( q−θ ,q + θ ) . then system (1.1) is asymptotically stable if and only if q−θ < q < q + θ . proof. in case of q = 0, the root of (2.1) is only λ (0) = a−1 < 0. thus, the root of the equation (2.1) has a negative real part. by the continuity of the roots with respect to q and by the asymptotic stability of (1.1), we can claim that the roots of equation (2.1) are inside a neighborhood ( q−θ ,q + θ ) of q = 0. since λ (0) < 0, in case of q−θ < ∞, by lemma 2.4, equation (2.1) has roots on the imaginary axis because of q−θ is the first value q < 0. by lemma 2.2, all roots of the equation (2.1) have negative real parts for ( q−θ , 0 ] . similarly, in case of q+θ < ∞, by lemma 2.3, equation (2.1) has roots on the imaginary axis because of q+θ is the first value q > 0. by lemma 2.2, all roots of the equation (2.1) have negative real parts for [ 0,q+θ ) . also, by lemma 2.5, equation (2.1) has at least one root positive real part for ( −∞,q−θ ) and( q+θ ,∞ ) . thus, the proof is completed. � theorem 3.2. suppose that a < 1, θ = 0 and the matrix a of the system (1.1) is written as the form (ii) . we define q−0 = max j≥1 { a− 1 2 : a < 1 } , q+0 = min j≥1 {qj : qj > 0} , and let a neighborhood of q = 0 is ( a−1 2 ,q+0 ) . then system (1.1) is asymptotically stable if and only if a− 1 2 < q < q+0 . proof. when θ = 0, equation (2.1) has only root λ = 0 as q = a−1 2 , that is λ ( a−1 2 ) = 0. since dλ ( a− 1 2 ) dq < 0, int. j. anal. appl. 16 (4) (2018) 461 we can write q−0 = max j≥1 { qj = a− 1 2 : a < 1, j ≥ 1 } . since the rest of the proof is similar to the theorem 3.11, it is obvious. � theorem 3.3. suppose that a < 1, and the matrix a of the system (1.1) is written as the form (i) . we define q− = max j≥1 {qj : qj < 0} , q+ = min j≥1 {qj : qj > 0} , and let a neighborhood of q = 0 is (q−,q+). then system (1.1) is asymptotically stable if and only if q− < q1,q2 < q +. proof. the proof is similar to the theorem 3.1. � references [1] h. matsunaga, delay dependent and delay independent stability criteria for a delay differential system, american mathematical society, 136 fields inst. commun. 42 (2008), 4305-4312. [2] k. l. cooke and p. van den driessche, on zeroes of some transcendental equations, funkcial.ekvacioj, 29 (1986), 77–90. [3] y. kuang, delay differential equations with applications in population dynamics, academic press, boston, 1993. [4] k. l. cooke and z. grossman, discrete delay, distributed delay and stability switches, j. math. anal. appl. 86 (1982), 592-627. [5] s.ruan and j.wei, on the zeros of transcendental functions with applications to stability of delay differential equations with two delays, dynamic of continuous, discr. impuls. syst. (2003), 863-874. [6] j.k.hale and s. m. verduyn lunel, introduction to functional differential equations, springer-verlag, new york, 1993. [7] s. elaydi, an introduction to difference equations, 3rd ed., springer-verlag, new york, 2005. [8] t. khokhlovaa, m. kipnis, v. malygina, discrete delay, the stability cone for a delay differential matrix equation, appl. math. lett. 24 (2011), 742-745. [9] j. cermák, j.jánsky, stability switches in linear delay difference equations, appl. math. comput. 243 (2014) 755–766. [10] jana hrabalova, stability properties of a discrrtized neutral delay differential equation, tatra mt. math. publ. 54 (2013), 83–92 [11] h. nakajima, on the stability of a linear retarded differential-difference equation, funkcialaj ekvacioj. 57 (2014), 43-56 [12] t. hara, s. sakata, an application of the hurwitz theorem to the root analysis of the characteristic equation, appl. math. lett. 24 (2011) 12–15 [13] h. smith, an introduction to delay differential equations with applications to the life science, springer, new york 2010. [14] h. i .freedman, y.kuang, stability switches in linear scalar neutral delay equation, funkcial. ekvac. 34 (1991), 187–209. [15] r. bellman and k. l. cooke, differential-difference equations, academic press, new york, 1963. 1. introduction and preliminaries 2. some auxiliary lemmas 3. main results references international journal of analysis and applications volume 19, number 6 (2021), 858-889 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-858 stochastic chemotaxis model with fractional derivative driven by multiplicative noise ali slimani∗, amira rahai, amar guesmia, lamine bouzettouta laboratory of applied mathematics and history and didactics of mathematics (lamahis) university of 20 august 1955, skikda, algeria ∗corresponding author: alislimani21math@gmail.com abstract. we introduce stochastic model of chemotaxis by fractional derivative generalizing the deterministic keller segel model. these models include fluctuations which are important in systems with small particle numbers or close to a critical point. in this work, we study of nonlinear stochastic chemotaxis model with dirichlet boundary conditions, fractional derivative and disturbed by multiplicative noise. the required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model. 1. introduction in this study, we consider on the following generalized sksm with time-space fractional derivative on a bounded domain d ⊂ rd(1 ≤ d ≤ 3) : (1.1)   cd β t u + (−∆) α 2 u−∇(u∇c) = g(u)ẇ(t), (t,x) ∈ [0,t] ×d, cd β t c + (−∆) α 2 c− c∇c = f(c)ẇ(t), (t,x) ∈ [0,t] ×d, with subject to the initial conditions: (1.2)   u(0,x) = u0(x), x ∈ d,c(0,x) = c0(x), x ∈ d, received april 15th, 2021; accepted may 7th, 2021; published october 28th, 2021. 2010 mathematics subject classification. 92c17, 35k58, 82c22. key words and phrases. stochastic keller-segel model; chemotaxis; fractional derivative; mild solution; regularity properties. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 858 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-858 int. j. anal. appl. 19 (6) (2021) 859 and the dirichlet boundary conditions: (1.3)   u(t,x) |∂d = 0, t ∈ [0,t],c(t,x) |∂d = 0, t ∈ [0,t], where u = u(t,x) denots the population density of biological individuals, c = c(t,x) denots the concentration of chemical substance, and ∇(u∇c) is called a chemotactic term that is used to model the fact that cells are attracted by chemical stimulus. in which the terms g(u)ẇ(t) = g(u) dw(t) dt , and f(c)ẇ(t) = f(c) dw(t) dt they describe the case-dependent random noise, where w(t)t∈[0t] is ft− adapted wiener process defined on a completed probability space (ω,f,p) with the expectation e and associate with the normal filtration ft = σ{w(s) : 0 ≤ s ≤ t}, the operator (−∆) α 2 , α ∈ (1, 2) stands for the fractional power of the laplacian (see [1]). we denote by cd β t the caputo derivative of order β, which is defined by (see [17]) (1.4) cd β t u(t,x) =   1 γ(1−β) t∫ 0 ∂u(s,x) ∂s ds (t−s)β , 0 < β < 1, ∂u(s,x) ∂s , β = 1, cd β t c(t,x) =   1 γ(1−β) t∫ 0 ∂c(s,x) ∂s ds (t−s)β , 0 < β < 1, ∂c(s,x) ∂s , β = 1, where γ(.) stands for the gemma function γ(β) = ∫ ∞ 0 tβ−1e−tdt. the rest of the paper is organized as follows. in section 2, we will introduce some notations and preliminaries, which play a crucial role in our theorem analysis. in section 3, the existence and uniqueness of mild solution to the problem of time-space fractional (2.1) and in section 4, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.1) are proved. in section 5, the existence and uniqueness of mild solution to the problem of time-space fractional (2.6). finally, the spatial and temporal regularity properties of mild solution to this time-space fractional (2.6) are proved. we use stochastic analysis techniques, fractional calculus and semigroup theory. next, we mention some notations and preliminaries the task at work. 2. notations and preliminaries denote the basic functional space lp(d), 1 ≤ p < ∞ and hs(d) by the usual lebesgue and sobolev spaces, respectively. we assume that a is the negative laplacian −∆ in a bounded domain d with zero dirichlet boundary conditions in a hilbert space h = l2(d), which are given by a = −∆, d(a) = h10 (d) ∩h 2(d). int. j. anal. appl. 19 (6) (2021) 860 since the operator a is self-adjoint on h with discrete spectral, i.e., there exists the eigenvectors en with corresponding eigenvalues λn such that aen = λnen,en = √ 2 sin(nπ),λn = π 2n2,n ∈ n+. for any s > 0, let ḣs be the domain of the fractional power a s 2 = (−∆) s 2 , which can be defined by a s 2 en = λ s 2 n, n = 1, 2, ..., and ḣs = d(a s 2 ) = {v ∈ l2(d),s.t. ‖ v ‖2 ḣs = ∞∑ n=1 λ s 2 nv 2 n < ∞}, where vn := 〈v,en〉 with the inner product 〈., .〉 in l2(d). we denote that ‖ v ‖ḣs=‖ a s 2 v ‖, and the corresponding dual space ḣ−s with the inverse operator a− s 2 .we also denote as for a s 2 and the bilinear operators b(u,c) = o(uoc), and d(b) = h10 and l(c,v) = cov, and d(l) = h 1 0 with a slight abuse of notation l(c,c) = l(c). then the eqs (1.1) and (1.3) can be rewritten as the following abstract formulation: (2.1)   cdβu(t) = −aαu(t) + b(u(t),c(t)) + g(u(t)) dw(t) dt , t > 0, u(0) = u0, and (2.2)   cdβc(t) = −aαc(t) + l(c(t)) + f(c(t)) dw(t) dt , t > 0, c(0) = c0, where {w(t)}t≥0 is a q− wiener process with linear bounded covariance operator q such that tr(q) < ∞. further, there exists the eigenvalues λn and corresponding eigenfunctions en satisfy qn = λnen, n = 1, 2, ..., then the wiener process is given by w(t) = ∞∑ n=1 λ 1 2 nβn(t)en, in which {βn}n≥1 is a sequence of real-valued standard brownian motions. let l20 = l2(q 1 2 (h),h) denote the space of hillbert-schmidt operators from q 1 2 (h) to h with the norm ‖ φ ‖l20 =‖ φq 1 2 ‖hs= ( ∞∑ n=1 φq 1 2 en) 1 2 , i.e., l20 = {φ ∈ l(h) : ∞∑ n=1 ‖ φq 1 2 ‖2< ∞}, where l(h) is the space of bounded linear operators from h to h. for an arbitrary banach space b, we denote ‖ . ‖lp(ω;b) by the norm in lp(ω,f,p; b) , which defined as ‖ v ‖lp(ω;b)= (e[‖ v ‖ p b]) 1 p , ∀v ∈ lp(ω,f,p; b), for any p ≥ 2. we shall also need the following result with respect to the fractional operator aα (see ref. [18]). int. j. anal. appl. 19 (6) (2021) 861 lemma 2.1. for any α > 0, an analytic semigroup sα(t) = e −taα, t ≥ 0 is generated by the operator −aα on lp, and for any ν ≥ 0, there exists a constant cαν dependent on α and ν such that (2.3) ‖ aνsα(t) ‖£(lp)≤ cανt− ν α , t > 0, in which £(b) denotes the banach space of all linear bounded operators from b to itself. next, we will introduce the following lemma to estimate the stochastic integrals, which contains the burkholder-davis-gundy’s inequality. lemma 2.2. ( [8]) for any 0 ≤ t1 < t2 ≤ t and p ≥ 2, and for any predictable stochastic process v : [0,t] × ω → l20, which satisfies e[( t∫ 0 ‖ v(s) ‖2l20 ds) p 2 ] < ∞, then we have (2.4) e[‖ t2∫ t1 ‖ v(s)dw(s) ‖p ds] ≤ c(p)e[( t2∫ t1 ‖ v(s) ‖2l20 ds) p 2 ], where c(p) = [ p(p−1) 2 ] p 2 ( p p−1 ) p( p 2 −1) is a constant. now, we give the following definition of mild solution for our time-space fractional stochastic keller-segel model. definition 2.1. a ft adapted process (u(t),c(t))t∈[0,t] is called a mild solution (1.1), if (u(t),c(t))t∈[0,t] ∈ c ( [0,t]; ḣν ) p a. e, and it holds, (2.5) u(t) = eβ(t)u0 + ∫ t 0 (t−s)β−1eββ(t−s)b (u(s),c(s)) ds + ∫ t 0 (t−s)β−1eββ(t−s)g(u(s))dw(s), and (2.6) c(t) = eβ(t)c0 + ∫ t 0 (t−s)β−1eββ(t−s)l(c(s))ds + ∫ t 0 (t−s)β−1eββ(t−s)f(c(s))dw(s), respectefily for a. s. ω ∈ ω, where the generalized mittag-leffler operators eβ(t) and eββ(t) are defined as eβ(t) = ∫ ∞ 0 mβ(θ)sα(t βθ)dθ, and eββ(t) = ∫ ∞ 0 βθmβ(θ)sα(t βθ)dθ, int. j. anal. appl. 19 (6) (2021) 862 which contain the mainardi’s wright-type function with β ∈ (0, 1) given by mβ(θ) = ∞∑ n=0 (−1)nθn n!γ(1 −β(1 + n)) , in which the mainardi function mβ(θ) act as a bridge between the classical integral-order and fractional derivatives of differential equations, for more details see [19, 20]. here, the derivation of mild solution (2.5) and (2.6) can be found in appendix (7) and appendix (8) (respectively). lemma 2.3. [2] for any β ∈ (0, 1) and −1 < ε < ∞, it is not difficult to verity that (2.7) mβ(θ) ≥ 0, and ∫ ∞ 0 θεmβ(θ)dθ = γ(1 + ε) γ(1 + βε) , for all θ ≥ 0. theorem 2.1. for any t > 0, eβ(t) and eββ(t) are linear and bounded operators. moreover, for 0 ≤ ν < α < 2, there exist constants cα = c(α,β,ν) > 0 and cβ = c(α,β,ν) > 0 such that (2.8) ‖ eβ(t)v ‖ḣν≤ cαt −βν α ‖ v ‖, ‖ eββ(t)v ‖ḣν≤ cβt −βν α ‖ v ‖ . proof. for t > 0 and 0 ≤ ν < α < 2, by means of the lemma (2.1) and lemma (2.3), we have ‖ eβ(t)v ‖ḣν ≤ ∫∞ 0 mβ(θ) ‖ aνsα(tβθ)v ‖ dθ ≤ ∫∞ 0 cανt −βν α θ −ν α mβ(θ) ‖ v ‖ dθ = cανγ(1−να ) γ(1−βν α ) t −βν α ‖ v ‖ v ∈ l2(d), and ‖ eββ(t)v ‖ḣν ≤ ∞∫ 0 βθmβ(θ) ‖ aνsα(tβθ)v ‖ dθ ≤ ∞∫ 0 cανβt −βν α θ1− ν α mβ(θ) ‖ v ‖ do = cανβγ(2−να ) γ(1+β(1−ν α )) t −βν α ‖ v ‖, v ∈ l2(d), which imply that the estimates (2.8) hold, so it is easy to know that eβ(t) and eββ(t) are linear and bounded operators. � theorem 2.2. for any t > 0, the operators eβ(t) and eββ(t) are strongly continuous. moreover, for any 0 ≤ t1 < t2 ≤ t and for 0 < ν < α < 2, there exist constants cαν = c(α,β,ν) > 0 and cβν = c(α,β,ν) > 0 such that (2.9) ‖ (eβ(t2) −eβ(t1))v ‖ḣν≤ cαν(t2 − t1) βν α ‖ v ‖, int. j. anal. appl. 19 (6) (2021) 863 and (2.10) ‖ (eββ(t2) −eββ(t1))v ‖ḣν≤ cβν(t2 − t1) βν α ‖ v ‖ . proof. for any 0 ≤ t1 < t2 ≤ t, it is easy to deduce that (2.11) t2∫ t1 dsα(t βθ) dt dt = sα(t β 2 θ) −sα(t β 1 θ) = − t2∫ t1 βtβ−1θaαsα(t βθ)dt, for 0 < ν < α < 2, making use of the above expression, the lemma (2.1) and lemma (2.3), we can arrive at ‖ (eβ(t2) −eβ(t1))v ‖ḣν = ‖ aν(eβ(t2) −eβ(t1))v ‖ = ‖ ∞∫ 0 mβ(θ)aν((sα(t β 2 θ) −sα(t β 1 θ))vdθ)vdθ ‖ ≤ ∞∫ 0 βθmβ(θ) t2∫ t1 tβ−1 ‖ aα+νsα(tβθ)v ‖l2 dtdθ ≤ ∞∫ 0 cανβθ −ν α mβ(θ)( t2∫ t1 t −βν α −1dt) ‖ v ‖ dθ = αcανγ(1−να ) νγ(1−βν α ) (t −βν α 2 − t −βν α 1 ) ‖ v ‖ ≤ αcανγ(1− ν α ) νt 2βν α γ(1 βν α ) (t2 − t1) βν α ‖ v ‖, v ∈ l2(d), and ‖ (eββ(t2) −eββ(t1))v ‖ḣν = ‖ aν(eββ(t2) −eββ(t1))v ‖ = ‖ ∞∫ 0 βθmβ(θ)aν(sα(t β 2 θ) −sα(t β 1 θ))vdθ ‖ ≤ ∞∫ 0 β2θ2mβ(θ) t2∫ t1 tβ−1 ‖ aα+νsα(tβθ)v ‖l2 dtdθ ≤ ∞∫ 0 cανβ 2θ1− ν α mβ(θ)( t2∫ t1 t −βν α −1dt) ‖ v ‖ dθ = αcανγ(2−να ) νγ(1+β(1−ν α )) (t −βν α 2 − t −βν α 1 ) ‖ v ‖ ≤ αcανγ(1− ν α ) νt 2βν α 0 γ(1+β(1− ν α )) (t2 − t1) βν α ‖ v ‖, v ∈ l2(d), int. j. anal. appl. 19 (6) (2021) 864 it is obviously to see that the term ‖ (eβ(t2) −eβ(t1))v ‖ḣν→ 0 and ‖ (eββ(t2) −eββ(t1))v ‖ḣν→ 0 as t1 → t2. which mean that the operators eβ(t) and eββ(t) are strongly continuous. � remark 2.1. assume ν = 0 in theorem (2.2), then there exist constants cα = c(α,β) > 0 and cβ = c(α,β) > 0 such that (2.12) ‖ (eβ(t2) −eβ(t1))v ‖ḣν≤ cα(t2 − t1) ‖ v ‖, and (2.13) ‖ (eββ(t2) −eββ(t1))v ‖ḣν≤ cβ(t2 − t1) ‖ v ‖ . proof. for any 0 < t0 ≤ t1 < t2 ≤ t, the same as the proof of theorem (2.2), we get ‖ (eβ(t2) −eβ(t1))v ‖ḣν = ‖ ∞∫ 0 mβ(θ)((sα(t β 2 θ) −sα(t β 1 θ))vdθ)vdθ ‖l2 ≤ ∞∫ 0 βθmβ(θ) t2∫ t1 tβ−1 ‖ aαsα(tβθ)v ‖l2 dtdθ ≤ ∞∫ 0 cααβθmβ(θ)( t2∫ t1 t−1dt) ‖ v ‖l2 dθ ≤ cααβ(ln t2 − ln t1) ‖ v ‖ = cααβ t0 (t2 − t1) ‖ v ‖, v ∈ l2(d), and ‖ (eββ(t2) −eββ(t1))v ‖ḣν = ‖ ∞∫ 0 βθmββ(θ)((sα(t β 2 θ) −sα(t β 1 θ))vdθ)vdθ ‖l2 ≤ ∞∫ 0 β2θ2mβ(θ) t2∫ t1 tβ−1 ‖ aαsα(tβθ)v ‖l2 dtdθ ≤ ∞∫ 0 cααβ 2θmβ(θ)( t2∫ t1 t−1dt) ‖ v ‖ dθ ≤ cααβ 2γ(2) γ(1+β) (ln t2 − ln t1) ‖ v ‖ ≤ cααβ 2γ(2) t0γ(1+β) (t2 − t1) ‖ v ‖, v ∈ l2(d). this completes the proof. � int. j. anal. appl. 19 (6) (2021) 865 3. existence and uniqueness of mild solution our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.1). to do this, the following assumptions are imposed. 3.1. assumption . the measurable function g : ω × h → l20 satisfies the following globallipschitz and growth conditions: (3.1) ‖ g(v) ‖l20≤ c ‖ v ‖, ‖ g(u) −g(v) ‖l20≤ c ‖ u−v ‖, for all u,v ∈ h. 3.2. assumption . let c, c1 are a positive real number, then the bounded bilinear operator b : l20(d) → h−1(d) satisfies the following properties: ‖ b(u,c) ‖ḣ−1 ≤ c ‖ u ‖‖ c ‖ ≤ c1 ‖ u ‖2, and (3.2) ‖ b(u,c) −b(v,c) ‖ḣ−1≤ cc1(‖ u ‖ + ‖ v ‖) ‖ u−v ‖, where c1 depend a norm the c in l 2 0(d), and for all u,v,c ∈ l20(d). 3.3. assumption . assume that the initial value u0 : ω → ḣν is a f0-measurable random variable, it holds that (3.3) ‖ u0 ‖lp(ω,ḣν)< ∞, for any 0 ≤ ν < α < 2. theorem 3.1. let assumptiom (3.1) to (3.3) be satisfied for some p ≥ 2, then there exists a unique mild solution (u(t))t∈[0,t] in the space l p(ω,ḣν) with 0 ≤ ν < α < 2. proof. we fix an ω ∈ ω and use the standard picard’s iteration argument to prove the existence of mild solution. to begin with, the sequence of stochastic process {u(t)}n≥0 is constructed as (3.4)   un+1(t) = eβ(t)u0 + n1(un(t)) + n2(un(t)),u0(t) = u0, int. j. anal. appl. 19 (6) (2021) 866 where (3.5)   n1(un(t)) = t∫ 0 (t−s)β−1eβ,β(t−s)b(un(s),c(s))ds, n2(un(t)) = t∫ 0 (t−s)β−1eβ,β(t−s)g(un(s))dw(s). the proof will be split into three steps. step1 for each n ≥ 0, we show that sup e[‖ un(t) ‖ p ḣν ] < ∞, note that (3.6) e[‖ un+1(t) ‖ p ḣν ] ≤ 3p−1e[‖ eβ(t)u0 ‖ p ḣν ] + 3p−1e[‖ n1(un(t)) ‖ p ḣν ] + 3p−1e[‖ n2(um(t)) ‖ p ḣν ]. the application of the lemma (2.1) gives (3.7) e[‖ eβ(t)u0 ‖ p ḣν ] ≤ e[ ∞∫ 0 mβ(θ)(‖ aνsα(tβθ)u0 ‖2) 1 2 dθ] = e[ ∞∫ 0 mβ(θ)( ∑∞ n=1〈aαe −tβθaαu0,en〉2) 1 2 dθ] = e[ ∞∫ 0 mβ(θ)( ∑∞ n=1〈aαu0,e −tβθλ α 2 en〉2) 1 2 dθ] ≤ e[ ∞∫ 0 mβ(θ) ‖ u0 ‖ḣν dθ] = e[‖ u0 ‖ḣν ] applying the following hölder inequality to the second term of the right-hand side of (3.6) (3.8) e[‖ n1(un(t)) ‖ p ḣν ] ≤ e[( t∫ 0 ‖ (t−s)β−1a1eβ,β(t−s)aν−1b(un(s),c(s)) ‖ ds)p] ≤ cpβ( t∫ 0 (t−s) p(β−1− β α ) p−1 ds)p−1 t∫ 0 e[‖ aν−1b(un(s),c(s)) ‖p]ds ≤ k1 t∫ 0 e[‖ un(s) ‖ p ḣν ]ds, where k1 = c p βc pc p 1 [ p−1 pβ(1− 1 alpha )−1 ] p−1tpβ(1 1 α )−1( max t∈[0,t] e[‖ un(t) ‖ p ḣν . making use of the hölder inequality and lemma (2.2) to the third term of the right-hand side of (3.6), we get (3.9) e[‖ n2(un(t)) ‖ p ḣν ] ≤ c(p)e[( t∫ 0 ‖ (t−s)β−1eβ,β(t−s) ‖2 aνg(un(s)) ‖2l20 ds) p 2 ] ≤ k2 t∫ 0 e[‖ un(s) ‖2ḣν ]ds, where k2 = c(p)c p βc pc p 1 [ p−2 p(2β−1)−2 ] p−2 2 t p(2β−1)−2 2 . int. j. anal. appl. 19 (6) (2021) 867 using the above estimates (3.6) and (3.9), we have e[‖ un+1(t) ‖ p ḣν ] ≤ 3p−1e[‖ u0 ‖ p ḣν ] + 3p−1(k1 + k2) t∫ 0 e[‖ un(s) ‖ p ḣν ]ds by means of the extension of gronwall’s lemma, it holds that sup t∈[0,t] e[‖ un+1(t) ‖ p ḣν ] < ∞, for each n ≥ 0. step2 show that the sequence {un(t)}n≥0 is a cauchy sequence in the space lp(ω; ḣν). for any n ≥ m ≥ 1, applying the similar arguments employed to obtain (3.8) and (3.9), we get (3.10) e[‖ un(t) −um(t) ‖ p ḣν ] ≤ 2p−1e[‖ n1(un−1(t)) −n1(um−1(t)) ‖ p ḣν ] + 2p−1e[‖ n2(un−1(t)) −n2(um−1(t)) ‖ p ḣν ] ≤ k t∫ 0 e[‖ un−1(s) −um−1(s) ‖ p ḣν ]ds, in wich k = 2p−1{cpβc pc p 1}[ p−1 p(β−β ν )−1 ]p−1tp(β− β α )( max t∈[0,t] e[‖ un−1(t) ‖ p ḣν ]) + max t∈[0,t] e[‖ um−1(t) ‖ p ḣν ]) + c(p)c p βc pc p 1 [ p−2 p(2β−1)−2 ] p−2 2 t p(2β−1)−2 2 }. a direct application of gronwall’s lemma yields sup t∈[0,t] e[‖ un(t) −um(t) ‖ p ḣν ] = 0, for all t > 0. taking limits to the stochastic sequence {un(t)}n≥0 in (3.4) as n →∞, we finish the proof of the existence of mild solution to (2.1). step3 we show the uniqueness of mild solution. assume u and v are two mild solutions of the problem (2.1), using the similar calculations as in step 2, we can obtain (3.11) sup t∈[0,t] e[‖ u(t) −v(t) ‖p ḣν ] = 0, for all t > 0, which implies that u = v, it follows that the uniqueness of mild solution. obviously, when ν = 0, the above three steps still work. thus the proof of theorem 3.1 is completed. � 4. regularity of mild solution in this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional sksm based on the analytic semigroup. int. j. anal. appl. 19 (6) (2021) 868 theorem 4.1. let assumptions (3.1) to (3.3) hold with 1 ≤ ν < α < 2 and p ≥ 2 let u(t) be a unique mild solution of the problem (2.1) with p(u(t) ∈ ḣν) = 1 for any t ∈ [0,t], then there exists a constant c such that (4.1) sup t∈[0,t] ‖ u(t) ‖lp(ω;ḣν)≤ c(‖ u0 ‖lp(ω;h) + sup t∈[0,t] ‖ u(t) ‖lp(ω;ḣ1)). proof. for any 0 ≤ t ≤ t and 1 ≤ ν < α < 2, we have (4.2) ‖ u(t) ‖lp(ω;ḣν) = (e[‖ u(t) ‖ p ḣν ]) 1 p =‖ aνu(t) ‖lp(ω;h) ≤ ‖ aνeβ(t)u0 ‖lp(ω;h) + ‖ aν t∫ 0 (t−s)β−1eβ,β(t−s)b(u(s),c(s))ds ‖lp(ω;h) + ‖ aν t∫ 0 (t−s)β−1eβ,β(t−s)g(u(s))dw(s) ‖ lp(ω;h) = i + ii + iii. using theorem (2.1), the first term can be estimated by (4.3) i =‖ aνeβ(t)u0 ‖lp(ω;h)≤ cαt− βν α ‖ u0 ‖lp(ω;h)< ∞. it is easy to know that (4.4) t∫ 0 cαt −βν α ‖ u0 ‖lp(ω;h) dt = αcα α−βν t α−βν α ‖ u0 ‖lp(ω;h) the application of theorem (2.1) and assumptions (3.2), we get (4.5) (ii)p ≤ e[(‖ aν t∫ 0 (t−s)β−1aνeβ,β(t−s)b(u(s),c(s)) ‖ ds)p] ≤ cpβ( t∫ 0 (t−s) p[β−1− β(ν+1) α ] p−1 ds)p−1 t∫ 0 e[‖ a−1b(u(s),c(s)) ‖ p ḣ1 ]ds ≤ c2 sup t∈[0,t] e[‖ u(s) ‖p ḣ1 ], int. j. anal. appl. 19 (6) (2021) 869 where c2 = c p βc pc p 1{ p−1 p[β−β(ν+1) α ]−1 }p−1tp[β− β(ν+1) α ]−1 + ( max t∈[0,t] e[‖ u(t) ‖ḣ1 ]). by means of theorem (2.1), assumptions (3.1) and lemma (2.2), we can deduce (4.6) (iii)p ≤ c(p)e[(‖ aν t∫ 0 ‖ (t−s)β−1aν−1eβ,β(t−s) ‖2‖ a1g(u(s)) ‖2l02 ds) p 2 ] ≤ c(p)cpβ( t∫ 0 (t−s) 2p[β−1− β(ν−1) α ] p−2 ds) p−2 2 t∫ 0 e ‖ a1g(u(s)) ‖ p l02 ds ≤ c3 sup t∈[0,t] e[‖ u(s) ‖p ḣ1 ], where c3 = c(p)c p βc pc p 1 [ p−2 p[2β−1−β(ν−1) α ]−2 ] p−2 2 t p[2β−1− β(ν−1) α ]−2 2 . thus, we conclude the proof of theorem (4.2) by combining with the estimates (4.2)(4.6). � next, we will devote to the temporal regularity of the mild solution. theorem 4.2. let assumptions (3.1) to (3.3) hold with 0 < ν < α < 2 and p ≥ 2 for any 0 ≤ t1 < t2 ≤ t , the unique mild solution u(t)to the problem (2.1) is hölder continuous with respect to the norm ‖ . ‖lp(ω;ḣν) and satisfies (4.7) ‖ u(t2) −u(t1) ‖lp(ω;ḣν)≤ c(t2 − t1) γ. proof. foor any 0 ≤ t1 < t2 ≤ t, for the mild solution (2.5), we have (4.8) u(t2) −u(t1) = eβ(t2)u0 −eβ(t1)u0 + t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)b(u(s),c(s))ds − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)b(u(s),c(s))ds + t2∫ 0 (t2 −s)β−1β−1eβ,β(t2 −s)g(u(s))dw(s) − t1∫ 0 (t2 −s)β−1β−1eβ,β(t1 −s)g(u(s))dw(s) = i1 + i2 + i3, where i1 = eβ(t2)u0 −eβ(t1)u0, int. j. anal. appl. 19 (6) (2021) 870 and (4.9) i2 = t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)b(u(s),c(s))ds − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)b(u(s),c(s))ds = t1∫ 0 (t1 −s)β−1[eβ,β(t2 −s) −eβ,β(t1 −s)]b(u(s),c(s))ds + t1∫ 0 [(t1 −s)β−1 − (t2 −s)β−1]eβ,β(t2 −s)b(u(s),c(s))ds + t2∫ t1 (t2 −s)β−1eβ,β(t2 −s)b(u(s),c(s))ds = i21 + i22 + i23, and (4.10) i3 = t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)g(u(s))dw(s) − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)g(u(s))dw(s) = t1∫ 0 (t1 −s)β−1[eβ,β(t2 −s) −eβ,β(t1 −s)]g(u(s))dw(s) + t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]eβ,β(t2 −s)g(u(s))dw(s) + t2∫ t1 (t2 −s)β−1eβ,β(t2 −s)g(u(s))dw(s) = i31 + i32 + i33. for any 0 < ν < α < 2 and p ≥ 2, by by virtue of theorem (2.2), it follows that (4.11) e[‖ i1 ‖ p ḣν ] = e[‖ aν[eβ(t2) −eβ(t1)]u0 ‖p] ≤ cpα,ν(t2 − t1) pβν α e[‖ u0 ‖p]. int. j. anal. appl. 19 (6) (2021) 871 for the first term i21 in (4.9), applying the assumption (3.2) and theorem (2.2) and hölder’s inequality, we have (4.12) e[‖ i21 ‖ p ḣν ] = e[‖ t1∫ 0 (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)]b(u(s),c(s))ds ‖p] ≤ cpβν(t2 − t1) pβ(ν+1) α ( t1∫ 0 (t1s) p(β−1) p−1 ds)p−1 t1∫ 0 e[‖ a−1b(u(s),c(s)) ‖ p ḣ1 ]ds ≤ cpcp1 c p βνt p( p−1 pβ−1 ) p−1(supt∈[0,t] e[‖ u(s) ‖ 2p ḣ1 ])(t2 − t1) pβ(ν+1) α . using the assumptions (3.2), theorem 2.1 and hölder’s inequality, we get (4.13) e[‖ i22 ‖ p ḣν = e[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]aνeβ,β(t2 −s)b(u(s),c(s))ds ‖p] ≤ cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− β(ν+1) α } p p−1 ds)p−1 × t1∫ 0 e[‖ a1b(u(s),c(s)) ‖ p ḣ1 ]ds ≤ cpcp1 c p βt{ p−1 p[β−β(ν+1) α ]−1 }p−1(supt∈[0,t] e[‖ u(s) ‖ 2p ḣ1 ])(t2 − t1) pβ(α−v−1)−α α , and (4.14) e[‖ i23 ‖ p ḣν ] = e[‖ t2∫ t1 (t2s) β−1aνeββ(t2 −s)b(u(s),c(s))ds ‖p] ≤ cpβ( t2∫ t1 [(t2 −s)β−1− β(ν+1) α ] p p−1 ds)p−1 × t2∫ t1 e[‖ a1b(u(s),c(s)) ‖ p ḣ1 ]ds ≤ cpcp1 c p β{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,t] e[‖ u(s) ‖2p ḣ1 ])(t2 − t1) pβ(α−v−1) α . int. j. anal. appl. 19 (6) (2021) 872 next, by following the similar arguments as in the proof of (4.12)(4.14) and using the lemma (2.2), there holds e[‖ i31 ‖ p ḣν ] = e[‖ t1∫ 0 (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)]g(u(s))dw(s) ‖p] ≤ c(p)e[( t1∫ 0 ‖ (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)] ‖2‖ g(u(s)) ‖2l20 ds) p 2 ] ≤ c(p)cpβν(t2 − t1) pβν α ( t1∫ 0 (t1 −s) 2p(β−1) p−2 ds) p−2 2 t1∫ 0 e ‖ g(u(s)) ‖p l20 ds ≤ c(p)cpβνt 2pβ−p−1 2 ( p−1 2pβ−p−2 ) p−1( sup t∈[0,t] e[‖ u(t) ‖p])(t1 − t2) pβν α , and (4.15) e[‖ i32 ‖] = e[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]aνeββ(t2 −s)g(u(s))dw(s) ‖p] ≤ c(p)e[( t1∫ 0 ‖ ((t2 −s)β−1 − (t1 −s)β−1)aνeββ(t2 −s) ‖2‖ g(u(s)) ‖2l20 ) p 2 ]ds ≤ c(p)cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− βν 2 } 2p p−2 ds) p−2 2 × t1∫ 0 e ‖ g(u(s)) ‖p l20 ds ≤ c(p)cpβc pt[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,t] e[‖ u(t) ‖p])(t2 − t1) 2pβ(α−ν)−(p+2)α 2α , and (4.16) e[‖ i33 ‖] = e[‖ t2∫ t1 (t2 −s)β−1aνeββ(t2 −s)g(u(s))dw(s) ‖p] ≤ c(p)e[ t2∫ t1 (t2 −s)β−1aνeββ(t2 −s) ‖2‖ g(u(s)) ‖2l20 ds) p 2 ] ≤ c(p)cpβ( t2∫ t1 [(t2 −s)β−1− βν α ] 2p p−2 ds) p−2 2 t2∫ t1 e ‖ g(u) ‖p l20 ds ≤ c(p)cpβc ρ[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,t] e[‖ u(t) ‖p])(t2 − t1) 2pβ(α−ν)−pα 2α . � int. j. anal. appl. 19 (6) (2021) 873 taking expectation on the both side of (4.8), and in view of the estimates (4.11)(4.16), we conclude that (4.17) ‖ u(t2) −u(t1) ‖lp(ω;ḣv)≤ c(t2 − t1) γ, in which we tak γ = min{βν α , pβ(α−ν−1)−α pα , 2pβ(α−ν)−(p+2)α 2pα }, whene 0 < t2 − t1 < 1. otherwise, if t2 − t1 ≥ 1 then we set γ = max{ β(ν+1) α , β(α−ν−1) α , 2pβ(α−ν)−pα 2pα }. this completes the proof of theorem (4.2) 5. existence and uniqueness of mild solution our main purpose of this section is to prove the existence and uniqueness of mild solution to the problem (2.6). to do this, the following assumptions are imposed. 5.1. assumption. the measurable function f : ω × h → l20 satisfies the following global lipschitz and growth conditions: (5.1) ‖ f(v) ‖l20≤ c ‖ v ‖, ‖ f(u) −f(v) ‖l20≤ c ‖ u−v ‖, for all u,v ∈ h. 5.2. assumption. let c, is a positive real number, then the bounded bilinear operator l : l20(d) → h−1(d) satisfies the following properties: (5.2) ‖ l(c) ‖ḣ−1≤ c ‖ c ‖ 2, and (5.3) ‖ l(c) −l(v) ‖ḣ−1≤ c(‖ c ‖ + ‖ v ‖) ‖ c−v ‖, and for all v,c ∈ l20(d). 5.3. assumption. assume that the initial value c0 : ω → ḣν is a f0-measurable random variable, it holds that (5.4) ‖ c0 ‖lp(ω,ḣν)< ∞, for any 0 ≤ ν < α < 2. theorem 5.1. let assumptiom (5.1) to (5.3) be satisfied for some p ≥ 2, then there exists a unique mild solution (c(t))t∈[0,t] in the space l p(ω,ḣν) with 0 ≤ ν < α < 2. int. j. anal. appl. 19 (6) (2021) 874 proof. we fix an ω ∈ ω and use the standard picard’s iteration argument to prove the existence of mild solution. to begin with, the sequence of stochastic process {cn(t)}n≥0 is constructed as (5.5)   cn+1(t) = eβ(t)c0 + n1(cn(t)) + n2(cn(t)),c0(t) = c0, where (5.6)   n1(cn(t)) = t∫ 0 (t−s)β−1eβ,β(t−s)l(cn(s))ds, n2(cn(t)) = t∫ 0 (t−s)β−1eβ,β(t−s)f(cn(s))dw(s). the proof will be split into three steps. step1 for each n ≥ 0, we show that sup e[‖ cn(t) ‖ p ḣν ] < ∞, note that (5.7) e[‖ cn+1(t) ‖ p ḣν ] ≤ 3p−1e[‖ eβ(t)c0 ‖ p ḣν ] + 3p−1e[‖ n1(cn(t)) ‖ p ḣν ] + 3p−1e[‖ n2(cn(t)) ‖ p ḣν ]. the application of the lemma (2.1) gives (5.8) e[‖ eβ(t)c0 ‖ p ḣν ] ≤ e[ ∞∫ 0 mβ(θ)(‖ aνsα(tβθ)c0 ‖2) 1 2 dθ] = e[ ∞∫ 0 mβ(θ)( ∑∞ n=1〈aαe −tβθaαc0,en〉2) 1 2 dθ] = e[ ∞∫ 0 mβ(θ)( ∑∞ n=1〈aαu0,e −tβθλ α 2 en〉2) 1 2 dθ] ≤ e[ ∞∫ 0 mβ(θ) ‖ c0 ‖ḣν dθ] = e[‖ c0 ‖ḣν ]. applying the following hölder inequality to the second term of the right-hand side of (5.7) (5.9) e[‖ n1(cn(t)) ‖ p ḣν ] ≤ e[( t∫ 0 ‖ (t−s)β−1a1eβ,β(t−s)aν−1l(cn(s)) ‖ ds)p] ≤ cpβ( t∫ 0 (t−s) p(β−1− β α ) p−1 ds)p−1 t∫ 0 e[‖ aν−1l(cn(s)) ‖p]ds ≤ k1 t∫ 0 e[‖ cn(s) ‖ p ḣν ]]ds, int. j. anal. appl. 19 (6) (2021) 875 where k1 = c p βc p[ p−1 pβ(1− 1 α )−1 ] p−1tpβ(1− 1 α )−1( max t∈[0,t] e[‖ cn(t) ‖ p ḣν ). making use of the hölder inequality and lemma (2.2) to the third term of the right-hand side of (5.7), we get (5.10) e[‖ n2(cn(t)) ‖ p ḣν ] ≤ c(p)e[( t∫ 0 ‖ (t−s)β−1eβ,β(t−s) ‖2 aνf(cn(s)) ‖2l20 ds) p 2 ] ≤ k2 t∫ 0 e[‖ cn(s) ‖2ḣν ]ds, where k2 = c(p)c p βc p[ p−2 p(2β−1)−2 ] p−2 2 t p(2β−1)−2 2 . using the above estimates (5.7)(5.10), we have e[‖ cn+1(t) ‖ p ḣν ] ≤ 3p−1e[‖ c0 ‖ p ḣν ] + 3p−1(k1 + k2) t∫ 0 e[‖ cn(s) ‖ p ḣν ]ds . by means of the extension of gronwall’s lemma, it holds that sup t∈[0,t] e[‖ cn+1(t) ‖ p ḣν ] < ∞, for each n ≥ 0. step1: show that the sequence {cn(t)}n≥0 is a cauchy sequence in the space lp(ω; ḣν). for any n ≥ m ≥ 1, applying the similar arguments employed to obtain (5.9) and (5.10), we get (5.11) e[‖ cn(t) − cm(t) ‖ p ḣν ] ≤ 2p−1e[‖ n1(cn−1(t)) −n1(cm−1(t)) ‖ p ḣν ] + 2p−1e[‖ n2(cn−1(t)) −n2(cm−1(t)) ‖ p ḣν ] ≤ k t∫ 0 e[‖ cn−1(s) − cm−1(s) ‖ p ḣν ]ds, in wich (5.12) k = 2p−1{cpβc p}[ p−1 p(β−β ν )−1 ]p−1tp(β− β α )( max t∈[0,t] e[‖ cn−1(t) ‖ p ḣν ]) + max t∈[0,t] e[‖ cm−1(t) ‖ p ḣν ]) + c(p)c p βc p[ p−2 p(2β−1)−2 ] p−2 2 t p(2β−1)−2 2 }. a direct application of gronwall’s lemma yields sup t∈[0,t] e[‖ cn(t) − cm(t) ‖ p ḣν ] = 0, for all t > 0. taking limits to the stochastic sequence {cn(t)}n≥0 in (5.5) as n → ∞, we finish the proof of the existence of mild solution to (2.6). step 3: we show the uniqueness of mild solution. assume c and v are two mild solutions of the problem int. j. anal. appl. 19 (6) (2021) 876 (2.6), using the similar calculations as in step 2, we can obtain (5.13) sup t∈[0,t] e[‖ c(t) −v(t) ‖p ḣν ] = 0, for all t > 0, which implies that c = v, it follows that the uniqueness of mild solution. obviously, when ν = 0, the above three steps still work. thus the proof of theorem (6.1) is completed. � 6. regularity of mild solution in this section, we will prove the spatial and temporal regularity properties of mild solution to time-space fractional sksm based on the analytic semigroup. theorem 6.1. let assumptions (5.1) to (5.3) hold with 1 ≤ ν < α < 2 and p ≥ 2, let c(t) be a unique mild solution of the problem (2.6) with p(c(t) ∈ ḣν) = 1 for any t ∈ [0,t], then there exists a constant c such that (6.1) sup t∈[0,t] ‖ c(t) ‖lp(ω;ḣν)≤ c(‖ c0 ‖lp(ω;h) + sup t∈[0,t] ‖ c(t) ‖lp(ω;ḣ1)). proof. for any 0 ≤ t ≤ t and 1 ≤ ν < α < 2, we have (6.2) ‖ c(t) ‖lp(ω;ḣν) = (e[‖ c(t) ‖ p ḣν ]) 1 p =‖ aνc(t) ‖lp(ω;h) ≤ ‖ aνeβ(t)c0 ‖lp(ω;h) + ‖ aν t∫ 0 (t−s)β−1eβ,β(t−s)l(c(s))ds ‖lp(ω;h) + ‖ aν t∫ 0 (t−s)β−1eβ,β(t−s)f(c(s))dw(s) ‖ lp(ω;h) = i + ii + iii. using theorem (2.1), the first term can be estimated by (6.3) i =‖ aνeβ(t)c0 ‖lp(ω;h)≤ cαt− βν α ‖ c0 ‖lp(ω;h)< ∞. it is easy to know that (6.4) t∫ 0 cαt −βν α ‖ c0 ‖lp(ω;h) dt = αcα α−βν t α−βν α ‖ c0 ‖lp(ω;h) . int. j. anal. appl. 19 (6) (2021) 877 the application of theorem (2.1) and assumptions (5.2), we get (6.5) (ii)p ≤ e[(‖ aν t∫ 0 (t−s)β−1aνeβ,β(t−s)l(c(s)) ‖ ds)p] ≤ cpβ( t∫ 0 (t−s) p[β−1− β(ν+1) α ] p−1 ds)p−1 t∫ 0 e[‖ a−1l(c(s)) ‖ p ḣ1 ]ds ≤ c2 sup t∈[0,t] e[‖ c(s) ‖p ḣ1 ], where c2 = c p βc p{ p−1 p[β−β(ν+1) α ]−1 }p−1tp[β− β(ν+1) α ]−1 + ( max t∈[0,t] e[‖ c(t) ‖ḣ1 ]). by means of theorem (2.1), assumptions (5.1) and lemma (2.2), we can deduce (6.6) (iii)p ≤ c(p)e[(‖ aν t∫ 0 ‖ (t−s)β−1aν−1eβ,β(t−s) ‖2‖ a1f(c(s)) ‖2l02 ds) p 2 ] ≤ c(p)cpβ( t∫ 0 (t−s) 2p[β−1− β(ν−1) α ] p−2 ds) p−2 2 t∫ 0 e ‖ a1f(c(s)) ‖ p l02 ds ≤ c3 sup t∈[0,t] e[‖ c(s) ‖p ḣ1 ], where c3 = c(p)c p βc p[ p−2 p[2β−1−β(ν−1) α ]−2 ] p−2 2 t p[2β−1− β(ν−1) α ]−2 2 . thus, we conclude the proof of theorem (6.1) by combining with the estimates (6.2)(6.6). � next, we will devote to the temporal regularity of the mild solution. theorem 6.2. let assumptions (5.1) to (5.3) hold with 0 < ν < α < 2 and p ≥ 2, for any 0 ≤ t1 < t2 ≤ t , the unique mild solution c(t)to the problem (2.6) is hölder continuous with respect to the norm ‖ . ‖lp(ω;ḣν) and satisfies (6.7) ‖ c(t2) − c(t1) ‖lp(ω;ḣν)≤ c(t2 − t1) γ. int. j. anal. appl. 19 (6) (2021) 878 proof. foor any 0 ≤ t1 < t2 ≤ t, for the mild solution (2.6), we have (6.8) c(t2) − c(t1) = eβ(t2)c0 −eβ(t1)c0 + t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)l(c(s))ds − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)l(c(s))ds + t2∫ 0 (t2 −s)β−1β−1eβ,β(t2 −s)f(c(s))dw(s) − t1∫ 0 (t2 −s)β−1β−1eβ,β(t1 −s)g(u(s))dw(s) = i1 + i2 + i3, where i1 = eβ(t2)c0 −eβ(t1)c0, and (6.9) i2 = t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)l(c(s))ds − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)l(c(s))ds = t1∫ 0 (t1 −s)β−1[eβ,β(t2 −s) −eβ,β(t1 −s)]l(c(s))ds + t1∫ 0 [(t1 −s)β−1 − (t2 −s)β−1]eβ,β(t2 −s)l(c(s))ds + t2∫ t1 (t2 −s)β−1eβ,β(t2 −s)l(c(s))ds = i21 + i22 + i23, int. j. anal. appl. 19 (6) (2021) 879 and (6.10) i3 = t2∫ 0 (t2 −s)β−1eβ,β(t2 −s)f(c(s))dw(s) − t1∫ 0 (t1 −s)β−1eβ,β(t1 −s)f(c(s))dw(s) = t1∫ 0 (t1 −s)β−1[eβ,β(t2 −s) −eβ,β(t1 −s)]f(c(s))dw(s) + t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]eβ,β(t2 −s)g(u(s))dw(s) + t2∫ t1 (t2 −s)β−1eβ,β(t2 −s)f(c(s))dw(s) = i31 + i32 + i33. for any 0 < ν < α < 2 and p ≥ 2, by virtue of theorem (2.2), it follows that (6.11) e[‖ i1 ‖ p ḣν ] = e[‖ aν[eβ(t2) −eβ(t1)]c0 ‖p] ≤ cpαν(t2 − t1) pβν α e[‖ c0 ‖p]. for the first term i21 in (6.9), applying the assumption (5.2) and theorem (6.2) and hölder’s inequality, we have (6.12) e[‖ i21 ‖ p ḣν ] = e[‖ t1∫ 0 (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)]l(c(s))ds ‖p] ≤ cpβν(t2 − t1) pβ(ν+1) α ( t1∫ 0 (t1 −s) p(β−1) p−1 ds)p−1 t1∫ 0 e[‖ a−1l(c(s)) ‖ p ḣ1 ]ds ≤ cpcpβνt p( p−1 pβ−1 ) p−1( sup t∈[0,t] e[‖ c(s) ‖2p ḣ1 ])(t2 − t1) pβ(ν+1) α . int. j. anal. appl. 19 (6) (2021) 880 using the assumptions (5.2), theorem (6.2) and hölder’s inequality, we get (6.13) e[‖ i22 ‖ p ḣν = e[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]aνeβ,β(t2 −s)l(c(s))ds ‖p] ≤ cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− β(ν+1) α } p p−1 ds)p−1 × t1∫ 0 e[‖ a1l(c(s)) ‖ p ḣ1 ]ds ≤ cpcpβt{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,t] e[‖ c(s) ‖2p ḣ1 ])(t2 − t1) pβ(α−v−1)−α α , and (6.14) e[‖ i23 ‖ p ḣν ] = e[‖ t2∫ t1 (t2 −s)β−1aνeββ(t2 −s)l(c(s))ds ‖p] ≤ cpβ( t2∫ t1 [(t2 −s)β−1− β(ν+1) α ] p p−1 ds)p−1 × t2∫ t1 e[‖ a1l(c(s)) ‖ p ḣ1 ]ds ≤ cpcpβ{ p−1 p[β−β(ν+1) α ]−1 }p−1( sup t∈[0,t] e[‖ c(s) ‖2p ḣ1 ])(t2 − t1) pβ(α−v−1) α . next, by following the similar arguments as in the proof of (6.12)(6.14) and using the lemma (2.2), there holds e[‖ i31 ‖ p ḣν ] = e[‖ t1∫ 0 (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)]f(u(s))dw(s) ‖p] ≤ c(p)e[( t1∫ 0 ‖ (t1 −s)β−1aν[eβ,β(t2 −s) −eβ,β(t1 −s)] ‖2‖ f(c(s)) ‖2l20 ds) p 2 ] ≤ c(p)cpβν(t2 − t1) pβν α ( t1∫ 0 (t1 −s) 2p(β−1) p−2 ds) p−2 2 t1∫ 0 e ‖ f(c(s)) ‖p l20 ds ≤ c(p)cpβνt 2pβ−p−1 2 ( p−1 2pβ−p−2 ) p−1( sup t∈[0,t] e[‖ c(t) ‖p])(t1 − t2) pβν α , int. j. anal. appl. 19 (6) (2021) 881 and (6.15) e[‖ i32 ‖] = e[‖ t1∫ 0 [(t2 −s)β−1 − (t1 −s)β−1]aνeββ(t2 −s)f(c(s))dw(s) ‖p] ≤ c(p)e[( t1∫ 0 ‖ ((t2 −s)β−1 − (t1 −s)β−1)aνeββ(t2 −s) ‖2‖ f(c(s)) ‖2l20 ) p 2 ]ds ≤ c(p)cpβ( t1∫ 0 {[(t2 −s)β−1 − (t1 −s)β−1] × (t2 −s)− βν 2 } 2p p−2 ds) p−2 2 × t1∫ 0 e ‖ f(c(s)) ‖p l20 ds ≤ c(p)cpβc pt [ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,t] e[‖ c(t) ‖p])(t2 − t1) 2pβ(α−ν)−(p+2)α 2α , and (6.16) e[‖ i33 ‖] = e[‖ t2∫ t1 (t2 −s)β−1aνeββ(t2 −s)f(c(s))dw(s) ‖p] ≤ c(p)e[ t2∫ t1 (t2 −s)β−1aνeββ(t2 −s) ‖2‖ f(c(s)) ‖2l20 ds) p 2 ] ≤ c(p)cpβ( t2∫ t1 [(t2 −s)β−1− βν α ] 2p p−2 ds) p−2 2 t2∫ t1 e ‖ f(u) ‖p l20 ds ≤ c(p)cpβc ρ[ α(p−2) 2pβ(α−ν)−(p+2)α] p−2 2 ( sup t∈[0,t] e[‖ c(t) ‖p])(t2 − t1) 2pβ(α−ν)−pα 2α taking expectation on the both side of (6.8), and in view of the estimates (6.11)(6.16), we conclude that (6.17) ‖ c(t2) − c(t1) ‖lp(ω;ḣv)≤ c(t2 − t1) γ, in which we tak γ = min{βν α , pβ(α−ν−1)−α pα , 2pβ(α−ν)−(p+2)α 2pα }, whene 0 < t2 − t1 < 1. otherwise, if t2 − t1 ≥ 1 then we set γ = max{ β(ν+1) α , β(α−ν−1) α , 2pβ(α−ν)−pα 2pα }. � this completes the proof of theorem (6.2) int. j. anal. appl. 19 (6) (2021) 882 7. appendix a considering the following abstract formulation of time-space fractional stochastic of equation (2.1) (7.1)   cd β t u(t) = −aαu(t) + b(u(t),c(t)) + g(u(t)) dw(t) dt , t > 0, u(0) = u0. we derive the mild solution to (7.1) by means of laplace transform, which denoted by a. let λ > 0, and we define that û(λ) = ∞∫ 0 e−λsu(s)ds, b̂(λ) = ∞∫ 0 e−λsb(u(s),c(s))ds, and ĝ(λ) = ∞∫ 0 e−λs[g(u(s)) dw(s) ds ]ds = ∞∫ 0 e−λsg(u(s))dw(s). upon laplace transform, using the formula cd̂ β t u(λ) = λ βû−λβ−1u0. then applying the laplace transform to (7.1), we obtain (7.2) û(λ) = 1 λ u0 + 1 λβ (−aα)û(λ) + 1λβ [b̂(λ) + ĝ(λ)] = λβ−1(λβi + aα) −1u0 + (λ βi + aα) −1[b̂(λ) + ĝ(λ)] = λβ−1 ∞∫ 0 e−λ βssα(s)u0ds + ∞∫ 0 e−λ βssα(s)[b̂(λ) + ĝ(λ)]ds, in which i is the identity operator, and sα(t) = e −taα is an analytic semigroup generated by the operator −aα. we introduce the following one-sided stable probability density function: (7.3) wβ = 1 π ∞∑ n=1 (−1)n−1θβn−1 γ(βn + 1) n! sin(nπβ), θ ∈ (0,∞), whose laplace transform is given by (7.4) ∞∫ 0 e−λθwβ(θ)dθ = e −λβ, 0 < β < 1. int. j. anal. appl. 19 (6) (2021) 883 making use of above expression (7.4), then the terms on the right-hand side of (7.2) can be written as (7.5) λβ−1 ∞∫ 0 e−λ βssα(s)u0ds = ∞∫ 0 λβ−1e−λ βtβsα(t β)u0dt = ∞∫ 0 β(λt)β−1e−(λt) β sα(t β)u0dt = ∞∫ 0 1 λ d dt [e−(λt) β ]sα(t β)u0dt = ∞∫ 0 ∞∫ 0 θwβ(θ)e −λtθsα(t β)u0dθdt = ∞∫ 0 e−λt[ ∞∫ 0 wβ(θ)sα( tβ θβ )u0dθ]dt, and (7.6) ∞∫ 0 e−λ βssα(s)b̂(λ)ds = ∞∫ 0 βtβ−1e−(λt) β sα(t β)b̂(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β sα(t β)e−λstβ−1b(u(s),c(s))dsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λtθsα(t β)e−λstβ−1b(u(s),c(s))dθdsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λ(t+s)sα( tβ θβ )t β−1 θβ b(u(s),c(s))dθdsdt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ b(u(s),c(s))dθds]dt, and ∞∫ 0 e−λ βssα(s)ĝ(λ)ds = ∞∫ 0 βtβ−1e−(λt) β sα(t β)ĝ(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β sα(t β)e−λsg(u(s))dw(s)dt int. j. anal. appl. 19 (6) (2021) 884 (7.7) = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λtθsα(t β)e−λstβ−1g(u(s))dθdw(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λ(t+s)sα( tβ θβ )t β−1 θβ g(u(s))dθdw(s)dt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdw(s)]dt. together with (7.2) and (7.5)(7.7) helps us to get (7.8) û(λ) = ∞∫ 0 e−λt[ ∞∫ 0 wβ(θ)sα( tβ θβ )u0dθ]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ b(u(s),c(s))dθds]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdw(s)]dt, now, by means of inverse laplace transform to (7.8), we have achieved that (7.9) u(t) = ∞∫ 0 wβ(θ)sα( tβ θβ )u0dθ + β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ b(u(s),c(s))dθds + β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ g(u(s))dθdw(s) = ∞∫ 0 1 β θ− 1 β −1wβ(θ − 1 β )sα(t βθ)u0dθ + t∫ 0 ∞∫ 0 θ− 1 β wβ(θ − 1 β )sα((t−s)βθ)(t−s)β−1b(u(s),c(s))dθds t∫ 0 ∞∫ 0 θ− 1 β wβ(θ − 1 β )sα((t−s)βθ)(t−s)β−1g(u(s))dθdw(s). here, we also introduce the mainardi’s wright-type function mβ(θ) = ∞∑ n=0 (−1)nθn nθ!γ(1 −β(1 + n)) = 1 π ∞∑ n=1 (−1)n−1θn−1 (n− 1)! γ(nβ) sin(nπβ), int. j. anal. appl. 19 (6) (2021) 885 where 0 < β < 1 and θ ∈ (0,∞). further, the relationships between the probability density function wβ(θ) and mainardi’s wright-type function mβ(θ) are shown that mβ(θ) = 1 β θ− 1 β −1wβ(θ − 1 β ). we denote the generalized mittag-leffler operators eα(t) and eββ(t) as eα(t) = ∞∫ 0 mβ(θ)sα(t βθ)dθ, and eββ(t) = ∞∫ 0 βθmβ(θ)sα(t βθ)dθ. therefore, the equation (7.9) can be written as (7.10) u(t) = eβ(t)u0 + t∫ 0 (t−s)β−1eββ(t−s)b (u(s),c(s)) ds + t∫ 0 (t−s)β−1eββ(t−s)g(u(s))dw(s), up to now, we have deduced the mild solution (7.10) to the time-space fractional stochastic equation (2.1). 8. appendix b considering the following abstract formulation of time-space fractional stochastic of equation (2.6) (8.1)   cd β t c(t) = −aαc(t) + l(c(t)) + f(c(t)) dw(t) dt , t > 0, c(0) = c0, we derive the mild solution to (8.1) by means of laplace transform, which denoted by ˆ. λ > 0, and we define that ĉ(λ) = ∞∫ 0 e−λsc(s)ds, l̂(λ) = ∞∫ 0 e−λsl(c(s))ds, and ĥ(λ) = ∞∫ 0 e−λs[f(c(s)) dw(s) ds ]ds = ∞∫ 0 e−λsf(c(s))dw(s). upon laplace transform, using the formula cd̂ β t c(λ) = λ βĉ−λβ−1c0. then applying the laplace transform to (8.1), we obtain (8.2) ĉ(λ) = 1 λ c0 + 1 λβ (−aα)ĉ(λ) + 1λβ [l̂(λ) + ĥ(λ)] = λβ−1(λβi + aα) −1c0 + (λ βi + aα) −1[l̂(λ) + ĥ(λ)] = λβ−1 ∞∫ 0 e−λ βssα(s)c0ds + ∞∫ 0 e−λ βssα(s)[l̂(λ) + ĥ(λ)]ds int. j. anal. appl. 19 (6) (2021) 886 in which i is the identity operator, and sα(t) = e −taα is an analytic semigroup generated by the operator −aα. we introduce the following one-sided stable probability density function: (8.3) wβ = 1 π ∞∑ n=1 (−1)n−1θβn−1 γ(βn + 1) n! sin(nπβ), θ ∈ (0,∞), whose laplace transform is given by (8.4) ∞∫ 0 e−λθwβ(θ)dθ = e −λβ, 0 < β < 1. making use of above expression (8.4), then the terms on the right-hand side of (8.2) can be written as (8.5) λβ−1 ∞∫ 0 e−λ βssα(s)c0ds = ∞∫ 0 λβ−1e−λ βtβsα(t β)c0dt = ∞∫ 0 β(λt)β−1e−(λt) β sα(t β)c0dt = ∞∫ 0 1 λ d dt [e−(λt) β ]sα(t β)c0dt = ∞∫ 0 ∞∫ 0 θwβ(θ)e −λtθsα(t β)c0dθdt = ∞∫ 0 e−λt[ ∞∫ 0 wβ(θ)sα( tβ θβ )c0dθ]dt, and (8.6) ∞∫ 0 e−λ βssα(s)l̂(λ)ds = ∞∫ 0 βtβ−1e−(λt) β sα(t β)l̂(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β sα(t β)e−λstβ−1l(c(s))dsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λtθsα(t β)e−λstβ−1l(c(s))dθdsdt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λ(t+s)sα( tβ θβ )t β−1 θβ l(c(s))dθdsdt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ l(c(s))dθds]dt, int. j. anal. appl. 19 (6) (2021) 887 and (8.7) ∞∫ 0 e−λ βssα(s)ĥ(λ)ds = ∞∫ 0 βtβ−1e−(λt) β sα(t β)ĥ(λ)dt = ∞∫ 0 ∞∫ 0 βtβ−1e−(λt) β sα(t β)e−λsf(c(s))dw(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λtθsα(t β)e−λstβ−1f(c(s))dθdw(s)dt = ∞∫ 0 ∞∫ 0 ∞∫ 0 βwβ(θ)e −λ(t+s)sα( tβ θβ )t β−1 θβ f(c(s))dθdw(s)dt = ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdw(s)]dt. together with (8.2) and (8.5)(8.7) helps us to get (8.8) ĉ(λ) = ∞∫ 0 e−λt[ ∞∫ 0 wβ(θ)sα( tβ θβ )c0dθ]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ l(c(s))dθds]dt + ∞∫ 0 e−λt[β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdw(s)]dt. now, by means of inverse laplace transform to (8.8), we have achieved that c(t) = ∞∫ 0 wβ(θ)sα( tβ θβ )c0dθ + β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ l(c(s))dθds + β t∫ 0 ∞∫ 0 wβ(θ)sα( (t−s)β θβ ) (t−s)β−1 θβ f(c(s))dθdw(s) int. j. anal. appl. 19 (6) (2021) 888 (8.9) = ∞∫ 0 1 β θ− 1 β −1wβ(θ − 1 β )sα(t βθ)c0dθ + t∫ 0 ∞∫ 0 θ− 1 β wβ(θ − 1 β )sα((t−s)βθ)(t−s)β−1l(c(s))dθds t∫ 0 ∞∫ 0 θ− 1 β wβ(θ − 1 β )sα((t−s)βθ)(t−s)β−1f(c(s))dθdw(s). here, we also introduce the mainardi’s wright-type function mβ(θ) = ∞∑ n=0 (−1)nθn nθ!γ(1 −β(1 + n)) = 1 π ∞∑ n=1 (−1)n−1θn−1 (n− 1)! γ(nβ) sin(nπβ), where 0 < β < 1 and θ ∈ (0,∞). further, the relationships between the probability density function wβ(θ) and mainardi’s wright-type function mβ(θ) are shown that mβ(θ) = 1 β θ− 1 β −1wβ(θ − 1 β ). we denote the generalized mittag-leffler operators eα(t) and eββ(t) as eα(t) = ∞∫ 0 mβ(θ)sα(t βθ)dθ, and eββ(t) = ∞∫ 0 βθmβ(θ)sα(t βθ)dθ. therefore, the equation (7.9) can be written as (8.10) c(t) = eβ(t)c0 + t∫ 0 (t−s)β−1eββ(t−s)l(c(s))ds + t∫ 0 (t−s)β−1eββ(t−s)f(c(s))dw(s). up to now, we have deduced the mild solution (8.10) to the time-space fractional stochastic equation (2.6). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (6) (2021) 889 references [1] l. debbi, well-posedness of the multidimensional fractional stochastic navier–stokes equations on the torus and on bounded domains, j. math. fluid mech. 18 (2016) 25–69. [2] p.m. de carvalho-neto, g. planas, mild solutions to the time fractional navier–stokes equations in rn , j. differ. equ. 259 (2015), 2948–2980. [3] s. zitouni, k. zennir, l. bouzettouta, uniform decay for a viscoelastic wave equation with density and time-varying delay in rn, filomat. 33 (2019), 961–970. [4] l. bouzettouta, f. hebhoub, k. ghennam, s. benferdi, exponential stability for a nonlinear timoshenko system with distributed delay, int. j. anal. appl. 19 (2021), 77-90. [5] a. guesmia, n. daili, existence and uniqueness of an entropy solution for burgers equations, appl. math. sci. 2 (2008), 1635-1664, . [6] a. guesmia, n. daili, about the existence and uniqueness of solution to fractional burgers equation, acta univ. apul. 21(2010), 161-170. [7] e.f. keller, l.a. segel, model for chemotaxis, j. theor. biol. 30 (1971), 225–234. [8] r. kruse, strong and weak approximation of semilinear stochastic evolution equations, springer, new york, 2014. [9] f. mainardi, the fundamental solutions for the fractional diffusion-wave equation, appl. math. lett. 9 (1996), 23–28. [10] k.a. khelil, f. bouchelaghem, l. bouzettouta, exponential stability of linear levin-nohel integro-dynamic equations on time scales. int. j. appl. math. stat. 56 (2017), 138-149. [11] g. zou, b. wang, stochastic burgers’ equation with fractional derivative driven by multiplicative noise, computers math. appl. 74 (2017), 3195–3208. [12] n. dib, a. guesmia, n. daili, on the solution of stochastic generalized burgers equation, commun. math. appl. 9 (2018), 521-528. [13] s. momani, non-perturbative analytical solutions of the spaceand time-fractional burgers equations, chaos solitons fractals. 28 (2006), 930–937. [14] c. mesikh, a. guesmia, s.saadi, global existence and uniqueness of the weak solution in keller segel model, glob. j. sci. front. res. f, 14 (2014), 46-55. [15] t. nagai, t. senba, t. suzuki, chemotactic collapse in a parabolic system of mathematical biology. hiroshima math. j. 30 (2000), 463-497. [16] a. rahai, a. guesmia, global existence and uniqueness of the weak solution in thixotropic model, int. j. anal. appl. 19 (2021), 193-204. [17] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [18] d. yang, m-dissipativity for kolmogorov operator of a fractional burgers equation with space-time white noise, potential anal. 44 (2016), 215–227. [19] x.-j. yang, h.m. srivastava, t. machado, a new fractional derivative without singular kernel: application to the modelling of the steady heat flow, therm. sci. 20 (2016), 753–756. [20] y. zhou, f. jiao, existence of mild solutions for fractional neutral evolution equations, computers math. appl. 59 (2010), 1063–1077. 1. introduction 2. notations and preliminaries 3. existence and uniqueness of mild solution 3.1. assumption 3.2. assumption 3.3. assumption 4. regularity of mild solution 5. existence and uniqueness of mild solution 5.1. assumption 5.2. assumption 5.3. assumption 6. regularity of mild solution 7. appendix a 8. appendix b references international journal of analysis and applications volume 18, number 6 (2020), 1037-1047 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1037 meromorphic starlike functions with respect to symmetric points muhammad ghaffar khan1,∗, maslina darus2, bakhtiar ahmad3, gangadharan murugusundaramoorthy4, raees khan5, nasir khan5 1department of mathematics, abdul wali khan university mardan, pakistan 2department of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, 43600, bangi, selangor, malaysia 3govt. degree college mardan, 23200 mardan, pakistan 4department of mathematics, sas, vellore institute of technology, deemed to be university, vellore 632014, india 5department of mathematics, fata university, dara adam khel, n.m.d. kohat, kp, pakistan ∗corresponding author: ghaffarkhan020@gmail.com abstract. the main purpose of this article is to introduce a class of meromorphic functions associated with the symmetric points in circular domain. we investigate the necessary and sufficient conditions, distortions theorem for this class. furthermore, we obtain closure and convolutions properties, radii of starlikeness and partial sum results for these functions. 1. introduction denoted by m, the class of functions f which are analytic in the d∗ = d�{0} , where d = {z ∈ c : |z| < 1} and having the following series expansion form (1.1) f(z) = 1 z + ∞∑ n=1 an z n, z ∈ d∗. received august 5th, 2020; accepted september 4th, 2020; published october 14th, 2020. 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. meromorphic functions; subordinations; convolutions; janowski functions. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1037 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1037 int. j. anal. appl. 18 (6) (2020) 1038 we say that an analytic function f1 (z) is subordinate to f2 (z) in d, symbolically represented as f1 (z) ≺ f2 (z), if there exists an analytic function w(z) with conditions |w(z)| < 1 and w(0) = 1 such that f1 (z) = f2 (w(z)). moreover, if f2 (z) is univalent, then we have the following equivalency from [1] and [2], f1 (0) = f2 (0) and f1 (d) ⊆ f2 (d) . for two functions f1 (z) = 1 z + ∑∞ n=1 an,1 z n and f2 (z) = 1 z + ∑∞ n=1 an,2 z n in d∗ the convolution or hadamard product is defined by (f1 ∗f2) (z) = 1 z + ∞∑ n=1 an,1an,2z n. a function f ∈ m is said to be in the class ms∗ (α) of meromorphic starlike functions of order α if it satisfies the inequality (1.2) < ( zf′(z) f(z) ) < −α, z ∈ d∗ 0 ≤ α < 1. for some recent investigation of meromorphic functions see [3–13]. motivated by aforementioned and recent work of [14], we define the functions’ class as below: let −1 ≤ b < a ≤ 1. then the function f is in the class ms∗∗ [a,b] if it satisfies (1.3) − 2zf′(z) f(z) −f(z) ≺ 1 + az 1 + bz , (z ∈ d∗), or equivalently (1.4) ∣∣∣∣∣∣ 2zf′(z) f(z)−f(−z) + 1 b 2zf′(z) f(z)−f(−z) + a ∣∣∣∣∣∣ < 1 (z ∈ d∗) . 2. coefficient inequalities theorem 2.1. let f ∈m and assumed as in (1.1) then f ∈ms∗∗ [a,b], if and only if (2.1) ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an| ≤ a−b. this inequality is sharp. proof. let us assume that condition (2.1) holds. to show f ∈ ms∗∗ [a,b] , we only need to show the inequality (1.4) holds. for this consider∣∣∣∣∣∣ 2zf′(z) f(z)−f(−z) + 1 b 2zf′(z) f(z)−f(−z) + a ∣∣∣∣∣∣ = ∣∣∣∣∣ zf ′(z) + f(z)−f(−z) 2 bzf′(z) + a f(z)−f(−z) 2 ∣∣∣∣∣ = ∣∣∣∣ ∑∞n=1 ( n+ 1−(−1)n 2 ) an (b−a)+ ∑∞ n=1(bn+a 1−(−1)n 2 )an ∣∣∣∣ ≤ ∑∞ n=1 ( n+ 1−(−1)n 2 ) an (b−a)− ∑∞ n=1(bn+a 1−(−1)n 2 )an < 1. int. j. anal. appl. 18 (6) (2020) 1039 now for other part let suppose f ∈ ms∗∗ [a,b] . we are to show that the inequality (2.1) , holds true. consider ∣∣∣∣∣∣ 2zf′(z) f(z)−f(−z) + 1 b 2zf′(z) f(z)−f(−z) + a ∣∣∣∣∣∣ = ∣∣∣∣∣ zf ′ (z) + f(z)−f(−z) 2 bzf′ (z) + a f(z)−f(−z) 2 ∣∣∣∣∣ = ∣∣∣∣ ∑∞n=1 ( n+ 1−(−1)n 2 ) an (a−b)+ ∑∞ n=1(bn+a 1−(−1)n 2 )an ∣∣∣∣ , since the <(z) ≤ |z| , we have (2.2) < { ∑∞ n=1 ( n+ 1−(−1)n 2 ) anz n−1 (a−b)+ ∑∞ n=1(bn+a 1−(−1)n 2 )anzn−1 } < 1 , now if we choose the value of z on real axis then 2zf ′ (z) f(z)−f(−z) is real. letting z → 1 − on real axis and some simple calculation in (2.2) , lead us to (2.1) . � theorem 2.2. let f ∈m and assumed as in (1.1) then f ∈ms∗∗ [a,b], if and only if (2.3) z [ f(z) ∗ ( (1 − 2z) ( 1 + beiθ ) z (1 −z)2 − z ( 1 + aeiθ ) 1 −z2 )] 6= 0, (z ∈ d) . proof. it is easy to verify the relations (2.4) f(z) ∗ z 1 −z2 = f(z) −f (−z) 2 and f(z) ∗ [ 1 z (1 −z)2 − 2 (1 −z)2 ] = −zf ′ (z). to prove (2.3) , if f ∈ms∗∗ [a,b] then we write (1.3), by using definition of subordination as (2.5) − 2zf ′ (z) f (z) −f (−z) = 1 + aw (z) 1 + bw (z) , which is equivalent to − 2zf ′ (z) f(z) −f (−z) 6= 1 + aeiθ 1 + beiθ , z ∈ d, θ ∈ [0, 2π] , which implies that (2.6) −zf ′ (z) ( 1 + beiθ ) − f(z) −f (−z) 2 ( 1 + aeiθ ) 6= 0. using the relation (2.4), (2.6) become z [ f(z) ∗ ( (1 − 2z) ( 1 + beiθ ) z (1 −z)2 − z ( 1 + aeiθ ) 1 −z2 )] 6= 0, for z ∈ d. conversly, suppose that the condition (2.3) hold, it follows that zf (z) 6= 0 for all z ∈ d. hence φ (z) = − 2zf ′ (z) f(z)−f(−z) is analytic in d with φ (0) = 1. since (2.7) − 2zf ′ (z) f(z) −f (−z) 6= 1 + aeiθ 1 + beiθ . if we denote ψ (z) = 1 + az 1 + bz , int. j. anal. appl. 18 (6) (2020) 1040 the relation (2.7) , show that φ (d)∩ψ (d) = ∅. therefore the simply connected domain φ (d) is contained in connected component of c�ψ (∂d) . the univalence of ”φ” togather with the fact ψ (0) = φ (0) = 1, this show that φ ≺ ψ which shows that f ∈ms∗∗ [a,b] . � theorem 2.3. the class ms∗∗ [a,b] is closed under convex combination. proof. let fi (z) ∈ms∗∗ [a,b] , such that fi (z) = 1 z + ∞∑ n=1 an,iz n, i ∈ n. then by equation (2.1) , we have ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an,i| ≤ a−b. for ∑∞ i=1 δi = 1, 0 ≤ δ ≤ 1, we have ∞∑ i=1 δifi (z) = 1 z + ∞∑ n=1 ( ∞∑ i=1 δian,i ) zn. using 2.1, we have ∞∑ n=1 ( ∞∑ i=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) δi |an,i| ) ≤ ∞∑ i=1 δi { ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an,i| } ≤ (a−b) ∞∑ i=1 δi = a−b. hence ms∗∗ [a,b] is convex. � theorem 2.4. let f ∈ms∗∗ [a,b] , |z| = r. then (2.8) 1 r − a−b a + b + 2 r ≤ |f (z)| ≤ 1 r + a−b a + b + 2 r. proof. as |f (z)| = ∣∣∣∣∣1z + ∞∑ n=1 anz n ∣∣∣∣∣ ≤ 1 r + ∞∑ n=1 |an| |r| n ≤ 1 r + a−b a + b + 2 r. where we have used theorem 2.1, on similar argument we have int. j. anal. appl. 18 (6) (2020) 1041 |f (z)| = ∣∣∣∣∣1z + ∞∑ n=1 anz n ∣∣∣∣∣ ≥ 1 r − ∞∑ n=1 |an| |r| n ≥ 1 r − a−b a + b + 2 r. thus prove the result. � theorem 2.5. let f (z) ∈ms∗∗ [a,b] , |z| = r. then (2.9) 1 r2 − 2 (a−b) a + b + 2 r ≤ |f′(z)| ≤ 1 r2 + 2 (a−b) a + b + 2 r. proof. as |f′(z)| = ∣∣∣∣∣− 1z2 + ∞∑ n=1 nanz n−1 ∣∣∣∣∣ ≤ 1 r2 + ∞∑ n=1 |an| |r| n−1 ≤ 1 r2 + 2 (a−b) a + b + 2 r. where we have used theorem 2.1, and |f′(z)| = ∣∣∣∣∣− 1z2 + ∞∑ n=1 nanz n−1 ∣∣∣∣∣ ≥ 1 r2 − ∞∑ n=1 |an| |r| n ≥ 1 r2 − 2 (a−b) a + b + 2 r. thus prove the result. � theorem 2.6. let f (z) ∈ms∗∗ [a,b] of the form (1.1) , and h (z) = 1 z + ∑∞ n=1 bnz n ∈ms∗∗ (a,b) with |bn| ≤ 1, then f (z) ∗h (z) ∈ms∗∗ [a,b] . proof. since by theorem 2.1, we have ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an| ≤ a−b. int. j. anal. appl. 18 (6) (2020) 1042 since ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |anbn| = ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an| |bn| ≤ ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an| ≤ a−b. thus f (z) ∗h (z) ∈ms∗∗[a,b]. � theorem 2.7. if f ∈ms∗∗[a,b]. then f ∈ms∗ (α) for |z| < r1, where (2.10) r1 = ( (1−α) ( (1+b)n+(1+a) 1−(−1)n 2 ) (n+α)(a−b) ) 1 n+1 . proof. let f ∈ms∗∗[a,b]. to prove f ∈ms∗ (α) , we only need to show∣∣∣∣ zf′(z) + f(z)zf′(z) − (1 − 2α) f(z) ∣∣∣∣ < 1. using (1.1) along with some simple computation yields (2.11) ∞∑ n=1 n + α 1 −α |an| |z| n+1 ≤ 1. as f is in the class ms∗∗[a,b] so we have from (2.1) , ∞∑ n=1 (1 + b) n + (1 + a) 1−(−1)n 2 a−b |an| ≤ 1. now inequality (2.11) will be true, if the following holds ∑∞ n=1 n+α 1−α |an| |z| n+1 < ∑∞ n=1 (1+b)n+(1+a) 1−(−1)n 2 a−b |an| , which implies that |z|n+1 < (1−α) ( (1+b)n+(1+a) 1−(−1)n 2 ) (n+α)(a−b) , and so |z| < ( (1−α) ( (1+b)n+(1+a) 1−(−1)n 2 ) (n+α)(a−b) ) 1 n+1 = r1, we get the required condition. � int. j. anal. appl. 18 (6) (2020) 1043 theorem 2.8. if f0 (z) = 1 z and for n ≥ 1 fn (z) = 1 z + a−b (1 + b) n + (1 + a) 1−(−1)n 2 zn. then f ∈ms∗∗[a,b] if and only if (2.12) f (z) = ∞∑ n=0 δnfn (z) , where δn ≥ 0 and ∞∑ n=1 δn = 1. proof. let f (z) be expressed in the form (2.12) , then f (z) = 1 z + ∞∑ n=1 δn a−b (1 + b) n + (1 + a) 1−(−1)n 2 zn, and for above function, we have ∞∑ n=1 [ (1 + b) n + (1 + a) 1 − (−1)n 2 ] ×δn a−b (1 + b) n + (1 + a) 1−(−1)n 2 = (a−b) (1 − δ0) ≤ a−b. thus by theorem 2.1, f (z) ∈ms∗∗[a,b]. conversly, let f (z) ∈ms∗∗[a,b], since by theorem 2.1, we have |an| ≤ a−b (1 + b) n + (1 + a) 1−(−1)n 2 , n ≥ 1, we set δn = (1 + b) n + (1 + a) 1−(−1)n 2 a−b |an| , n ≥ 1, and δ0 = 1 − ∞∑ n=1 δn, so, it follows that f (z) = ∞∑ n=0 δnfn (z) . hence proof is complete. � int. j. anal. appl. 18 (6) (2020) 1044 3. partial sums silverman [17] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. as a natural extension, one is interested to search results analogous to those of silverman for meromorphic univalent functions. in this section, motivated essentially by the work of silverman [17] and cho and owa [15]( also see [16, 18]) we will investigate the ratio of a function of the form (3.1) f(z) = 1 z + ∞∑ n=1 anz n, to its sequence of partial sums (3.2) f1(z) = 1 z and fk(z) = 1 z + k∑ n=1 anz n when the coefficients are sufficiently small to satisfy the condition analogous to ∞∑ n=1 ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) |an| ≤ a−b. for the sake of brevity we rewrite it as (3.3) ∞∑ n=1 dn|an| ≤ a−b, where (3.4) dn(a,b) := ( (1 + b) n + (1 + a) 1 − (−1)n 2 ) more precisely we will determine sharp lower bounds for <{f(z)/fk(z)} and <{fk(z)/f(z)}. in this connection we make use of the well known results that< { 1+w(z) 1−w(z) } > 0 (z ∈ d∗) if and only if ω(z) = ∞∑ n=1 cnz n satisfies the inequality |ω(z)| ≤ |z|. unless otherwise stated, we will assume that f is of the form (1.1) and its sequence of partial sums is denoted by fk(z) = 1 z + ∑k n=1 anz n. theorem 3.1. let f ∈ms∗∗ [a,b] be given by (1.1)satisfies condition (2.1),then (3.5) re { f(z) fk(z) } ≥ dk+1(a,b) + b −a dk+1(a,b) (z ∈ d∗) where (3.6) dn(a,b) ≥   a−b, if n = 1, 2, 3, . . . ,kdk+1(a,b), if n = k + 1,k + 2, . . . . the result (3.5) is sharp with the function given by (3.7) f(z) = 1 z + a−b dk+1(a,b) zk+1. int. j. anal. appl. 18 (6) (2020) 1045 proof. define the function w(z) by 1 + w(z) 1 −w(z) = dk+1(a,b) a−b [ f(z) fk(z) − dk+1(a,b) + b −a dk+1(a,b) ] (3.8) = 1 + k∑ n=1 anz n+1 + ( dk+1(a,b) a−b ) ∞∑ n=k+1 anz n+1 1 + k∑ n=1 anzn+1 . it suffices to show that |w(z)| ≤ 1. now, from (3.8) we can write w(z) = ( dk+1(a,b) a−b ) ∞∑ n=k+1 anz n+1 2 + 2 k∑ n=1 anzn+1 + ( dk+1(a,b) a−b ) ∞∑ k=n+1 anzn+1 . hence we obtain |w(z)| ≤ ( dk+1(a,b) a−b ) ∞∑ k=n+1 |an| 2 − 2 k∑ n=1 |an|− ( dk+1(a,b) a−b ) ∞∑ n=k+1 |an| . now |w(z)| ≤ 1 if 2 ( dk+1(a,b) a−b ) ∞∑ n=k+1 |an| ≤ 2 − 2 k∑ n=1 |an| or, equivalently, k∑ n=1 |an| + dk+1(a,b) a−b ∞∑ n=k+1 |an| ≤ 1. from the condition (2.1), it is sufficient to show that k∑ n=1 |an| + dk+1(a,b) a−b ∞∑ n=k+1 |an| ≤ ∞∑ n=1 dn(a,b) a−b |an| which is equivalent to k∑ n=1 ( dn(a,b) + b −a a−b ) |an| + ∞∑ n=k+1 ( dn(a,b) −dk+1(a,b) a−b ) |an| ≥ 0.(3.9) int. j. anal. appl. 18 (6) (2020) 1046 to see that the function given by (3.7) gives the sharp result, we observe that for z = reiπ/k f(z) fk(z) = 1 + a−b dk+1(a,b) zn → 1 − a−b dk+1(a,b) = dk+1(a,b) + b −a dk+1(a,b) when r → 1−. which shows the bound (3.5) is the best possible for each k ∈ n. � we next determine bounds for fk(z)/f(z). theorem 3.2. if f of the form (3.2) satisfies the condition (2.1), then (3.10) re { fk(z) f(z) } ≥ dk+1(a,b) dk+1(a,b) + b −a (z ∈ d∗), where (3.11) dk(a,b) ≥   a−b, if k = 1, 2, 3, . . . ,ndk+1(a,b), if k = n + 1,n + 2, . . . . . the result (3.10) is sharp with the function given by (3.7). proof. we write 1 + w(z) 1 −w(z) = dk+1(a,b) + b −a a−b [ fk(z) f(z) − dk+1(a,b) dk+1(a,b) + b −a ] = 1 + k∑ n=1 anz n+1 − ( dk+1(a,b) a−b ) ∞∑ n=k+1 anz n+1 1 + ∞∑ n=1 anzn+1 , where |w(z)| ≤ ( dk+1(a,b)+b−a a−b ) ∞∑ n=k+1 |an| 2 − 2 k∑ n=1 |an|− ( dk+1(a,b)+b−a a−b ) ∞∑ n=k+1 |an| ≤ 1. this last inequality is equivalent to k∑ n=1 |an| + dk+1(a,b) a−b ∞∑ n=k+1 |an| ≤ 1. make use of (2.1) to get (3.9). finally, equality holds in (3.10) for the extremal function f(z) given by (3.7). � acknowledgment: the second author is supported by ukm grant: gup-2019-032. authors contributions: all authors jointly worked on the results and they read and approved the final manuscript. int. j. anal. appl. 18 (6) (2020) 1047 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. s. miller and p. t.mocanu, second order differential inequalities in the complex plane, j. math. anal. appl. 65 (1978), 289–305. [2] s. s. miller and p. t.mocanu, differential subordinations, theory and applications, series on monographs and textbooks in pure and appl. math. no. 225, marcel dekker inc., new york, 2000. [3] r. m. ali and v. ravichandran, class of meromorphic α-convex functions, taiwan. j. math. 14 (2010), 1479–1490. [4] a. a. catas, note on meromorphic m-valent starlike functions, j. math. inequal. 4 (2010), 601–607. [5] o. s. kwon and n. e. cho, a class of nonlinear operators preservimg double subordinations, abstr. appl. anal. 2018 (2008), 792160. [6] k. i. noor and f. amber, on certain classes of meromorphic functions associated with conic domains, abstr. appl. anal. 2012 (2012), id 7921601. [7] m. nunokawa and o. p. ahuja, on meromorphic starlike and convex functions, indian j. pure appl. math. 32 (2001), 1027–1032. [8] m. nunokawa, n. owa, n. uyanik and h. shiraishi, sufficient conditions for starlikeness of order α for meromorphic functions, mathematical and computer modelling, indian j. pure appl. math. 55 (2012), 1245–1250. [9] h. tang, g. t. deng and s. h. li, on a certain new subclass of meromorphic close-to-convex functions, j. inequal. appl. 2013 (2013), 164. [10] z. g. wang, z. h. liu and r. g. xaing, some criteria for meromorphic multivalent starlike functions, appl. math. comput. 218 (2011), 1107–1111. [11] z. g. wang, y. sun and n. xu, certain properties of meromorphic close-to-convex functions, appl. math. lett. 25 (2012), 454–460. [12] z. g. wang, h. m. serivastava and s. m yuan, some basic properties of certain subclasses of meromorphically starlike functions, j. inequal. appl. 2014 (2014), 29. [13] d. j. hallenbeck and s. ruscheweyh, subordination by convex functions, proc. amer. math. soc. 52 (1975), 191–195. [14] k. sakaguchi, on a certain univalent mapping, j. math. soc. japan. 2 (1959), 72–75. [15] n. e. cho and s. owa, partial sums of certain meromorphic functions, j. inequal. pure appl. math. 5 (2004), 30. [16] s. sivaprasad, v. ravichandran and g. murugusundaramoorthy, classes of meromorphic p-valent parabolic starlike functions with positive coefficients, aust. j. math. anal. appl. 2 (2005), 3. [17] h. silverman, partial sums of starlike and convex functions, j. math. anal. appl., 209 (1997), 221–227. [18] b. a. frasin and g. murugusundaramoorthy, partial sums of meromorphic multivalent functions with positive coefficients, studia univ. babes-balyai math. annul. (2005), 33–40. 1. introduction 2. coefficient inequalities 3. partial sums references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 20-26 http://www.etamaths.com some new inequalities of qi type for definite integrals bo-yan xi1 and feng qi2,∗ abstract. in the paper, the authors establish some new integral inequalities, from which some integral inequalities of qi type may be derived. 1. introduction in [11] and its preprint [12], an interesting integral inequality below was obtained. theorem 1.1 ([11, 12]). let n ∈ n and the n-th order derivative of f be continuous on [a,b] ⊆ r = (−∞,∞), satisfying f(i)(a) ≥ 0 and f(n)(x) ≥ n! for 0 ≤ i ≤ n−1. then (1.1) ∫ b a fn+2(x) d x ≥ [∫ b a f(x) d x ]n+1 . at the end of [11, 12], the following open problem was posed. open problem 1.1 ([11, 12]). under what conditions does the inequality (1.2) ∫ b a ft(x) d x ≥ [∫ b a f(x) d x ]t−1 hold for some t > 1? thereafter, the following answer to open problem 1.1 was confirmed. theorem 1.2 ([14, 15]). let t ≥ 1 and f be a continuous function on [a,b] ⊆ r such that (1.3) ∫ b a f(x) d x ≥ (b−a)t−1. then the inequality (1.2) is valid. to the best of our knowledge, till now there have been many mathematicians and articles devoted to generalizing and applying the integral inequality (1.1) and to answering open problem 1.1. in these investigations, different and various tools, ideas, methods, and techniques, such as jensen’s inequality [6], convexity method [4], functional inequalities in abstract spaces [1, 4, 6], probability measures viewpoint [1, 7, 8], hölder inequality and its reversed variants [10, 19], analytical 2010 mathematics subject classification. 26d15. key words and phrases. generalization; integral inequality; qi type integral inequality; convex function; jensen’s inequality; r-mean convex function; geometrically convex function; logarithmically convex function; open problem. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 20 some new inequalities of qi type for definite integrals 21 methods [15], cauchy’s mean value theorem [3, 13], and q-integral [2, 9, 20], have been created. recently, this type of inequalities were generalized in [18] to double integrals. importantly, the mathematical meanings in probability and statistics was found in [4]. for a much complete list of references appeared in recent years on this topic, please refer to [20]. the aim of this paper is to establish some new integral inequalities, from which some integral inequalities of qi type may be derived. in other words, the integral inequality (1.1) will be generalized and some more answers to open problem 1.1 will be supplied in this paper. 2. definitions and lemmas before establishing some new inequalities of qi type, we state several definitions and lemmas. let i ⊆ r be an interval and n ∈ n. for f : i → r+ = (0,∞), xk ∈ i for 1 ≤ k ≤ n, and λk ≥ 0 satisfying ∑n i=1 λk = 1, let (2.1) mn(f(x),λ,r) =   [ n∑ k=1 λkf r(xk) ]1/r , r 6= 0, n∏ k=1 fλk (xk), r = 0. especially, for xk ∈ i ⊆ r+, let (2.2) mn(x,λ,r) =   ( n∑ k=1 λkx r k )1/r , r 6= 0, n∏ k=1 x λk k , r = 0. definition 2.1 ([5, p. 348]). let i ⊆ r be an interval. a function f : i → r is said to be convex if (2.3) f(λx + (1 −λ)y) ≤ λf(x) + (1 −λ)f(y) for all x,y ∈ i and λ ∈ [0, 1]. if the above inequality is reversed, then f is said to be concave on i. definition 2.2 ([16, 17]). let i ⊆ r+ be an interval and r ∈ r. a function f : i → r+ is said to be r-mean convex on i if (2.4) f(m2(x,λ,r)) ≤ m2(f(x),λ,r) for all x1,x2 ∈ i and λ ∈ [0, 1]. if the above inequality is reversed, then we say that the function f is r-mean concave on i. when r = 0, the r-mean convex (r-mean concave, respectively) functions are called geometrically convex (geometrically concave, respectively) functions. definition 2.3 ([5, p. 349]). let i ⊆ r be an interval. a function f : i → r+ is said to be logarithmically convex on i if (2.5) f(λx + (1 −λ)y) ≤ [f(x)]λ[f(y)]1−λ for all x,y ∈ i and λ ∈ [0, 1]. if the above inequality is reversed, then the function f is said to be logarithmically concave on i. 22 xi and qi lemma 2.1 (jensen’s inequality). let i ⊆ r+ be an interval, r ∈ r, and f : i → r+. then f is r-mean convex (r-mean concave, respectively) on i if and only if (2.6) f(mn(x,λ,r)) ≤ mn(f(x),λ,r) holds for all x = (x1,x2, . . . ,xn) ∈ in and λk ≥ 0 satisfying ∑n k=1 λk = 1. proof. this may be found in [16, 17]. � the following lemmas are useful for us. lemma 2.2. for x,y ∈ r+, if either xy ≤ 4, or x ≤ 1, or y ≤ 1, then xy ≤ x + y. proof. the proof is elementary. � 3. some new integral inequalities of qi type now we are in a position to establish some new integral inequalities of qi type. theorem 3.1. suppose i ⊆ r0 = [0,∞) is an interval, f : [a,b] → i is continuous and not identically zero, g : i → r0 is convex (or concave, respectively), and (3.1) g((b−a)u) q g(b−a)g(u) for u ∈ i, and (3.2) ∫ b a f(x) d x r g(b−a) b−a . then we have (3.3) ∫ b a g(f(x)) d x r g (∫ b a f(x) d x ) ∫ b a f(x) d x . proof. let (3.4) xk = a + k n (b−a), 1 ≤ k ≤ n. if g(u) is a convex function on i, then it is easy to see that mn ( f(x), 1 n , 1 ) ∈ i, and, by jensen’s inequality (2.6) and corresponding conditions, g [∫ b a f(x) d x ] = g ( (b−a) lim n→∞ mn ( f(x), 1 n , 1 )) ≤ g(b−a) lim n→∞ g ( mn ( f(x), 1 n , 1 )) ≤ g(b−a) lim n→∞ mn ( g ( f(x), 1 n , 1 )) = g(b−a) b−a ∫ b a g(f(x)) d x. therefore, it follows that∫ b a f(x) d x ∫ b a g(f(x)) d x−g [∫ b a f(x) d x ] = ∫ b a g(f(x)) d x [∫ b a f(x) d x− g(b−a) b−a ] ≥ 0. some new inequalities of qi type for definite integrals 23 thus, the inequality (3.3) in the direction ≥ is true. if g(u) is a concave function on i, the proof is similar. this completes the proof of theorem 3.1. � applying theorem 3.1 to special cases of g(u) result in the following corollaries, which show that theorem 3.1 and theorem 3.2 and 3.3 below are generalizations of the inequality (1.1) and answers of open problem 1.1. corollary 3.1. let f(x) is a positive continuous function on an interval [a,b] ⊆ r. (1) if t 6∈ [0, 1) and ∫ b a f(x) d x ≥ (b−a)t−1, then the inequality (1.2) is valid; (2) if 0 < t ≤ 1 and ∫ b a f(x) d x ≤ (b − a)t−1, then the inequality (1.2) is reversed. corollary 3.2. let f(x) be a positive continuous function on [a,b] ⊆ r. (1) if t 6∈ [0, 1) and f(x) ≥ (b−a)t−2, then the inequality (1.2) is valid; (2) if 0 < t ≤ 1 and f(x) ≤ (b−a)t−2, then the inequality (1.2) is reversed. corollary 3.3. suppose f(x) is a positive continuous function on [a,b] ⊆ r. (1) if t ≥ 2 and f(x) ≥ (t− 1)(x−a)t−2, then the inequality (1.2) is valid; (2) if 2 < t ≤ 3 and f′(x) ≥ (b − a)(t − 1)(x − a)t−2 on [a,b], then the inequality (1.2) is also valid; (3) if t > 3 and f′(x) ≥ (t−1)(t−2)(x−a)t−3 on [a,b], then the inequality (1.2) is still valid. corollary 3.4. suppose f(x) is a positive continuous function on [a,b] ⊆ r, and suppose that either 0 < f(x) ≤ 4 b−a, or 0 < f(x) ≤ 1, or 0 < b−a ≤ 1. if c > 1 and ∫ b a f(x) d x ≥ cb−a b−a , then ∫ b a cf(x) d x ≥ c ∫ b a f(x) d x∫ b a f(x) d x .(3.5) in particular, ∫ b a ef(x) d x ≥ exp [∫ b a f(x) d x ] ∫ b a f(x) d x .(3.6) proof. from lemma 2.2, when x,y > 0 and either xy ≤ 4 or x ≤ 1, it follows that cx+y ≥ cxy. by choosing g(u) = cu in theorem 3.1, corollary 3.4 follows. � theorem 3.2. suppose i ⊆ r+ is an interval, f : [a,b] → i is a continuous function and not identically zero, and g : i → r+. (1) for r 6= 0, if g(u) is r-mean convex (or r-mean concave, respectively) on i, and (3.7) g ( (b−a)1/ru ) q g ( (b−a)1/r ) g(u) for u ∈ i, and (3.8) ∫ b a f(x) d x r g ( (b−a)1/r ) (b−a)1/r , 24 xi and qi then [∫ b a gr(f(x)) d x ]1/r r g ((∫ b a fr(x) d x )1/r) ∫ b a f(x) d x .(3.9) (2) if g(u) is a geometrically convex (or geometrically concave, respectively) on i, satisfying (3.10) g ( e(b−a)u ) q g ( eb−a ) g(eu), u ∈ i and (3.11) ∫ b a f(x) d x r g ( eb−a ) , then exp ( 1 b−a ∫ b a ln g(f(x)) d x ) r g ( exp (∫ b a ln f(x) d x )) ∫ b a f(x) d x .(3.12) proof. let g(u) be a r-mean convex function on i and adopt the notations in (3.4). utilizing mn ( f(x), 1 n , 1 ) ∈ i and jensen’s inequality (2.6) leads to g ([∫ b a fr(x) d x ]1/r) = g ( (b−a)1/r lim n→∞ mn ( f(x), 1 n ,r )) ≤ g ( (b−a)1/r ) lim n→∞ g ( mn ( f(x), 1 n ,r )) ≤ g ( (b−a)1/r ) lim n→∞ mn ( g(f(x)), 1 n ,r ) = g ( (b−a)1/r ) (b−a)1/r [∫ b a gr(f(x)) d x ]1/r , hence, the inequality (3.9) is true. let g(u) be a geometrically convex function on i. making use of jensen’s inequality (2.6) results in g ( exp (∫ b a ln f(x) d x )) = g ( exp ( (b−a) lim n→∞ mn ( ln f(x), 1 n , 1 ))) ≤ g ( eb−a ) lim n→∞ g ( mn ( f(x), 1 n , 0 )) ≤ g ( eb−a ) lim n→∞ mn ( g(f(x)), 1 n , 0 ) = g ( eb−a ) exp ( 1 b−a ∫ b a ln g(f(x)) d x ) , therefore, the inequality (3.12) is true. the rest can be proved similarly. the proof of theorem 3.2 is complete. � theorem 3.3. suppose i ⊆ r+ is an interval, f : [a,b] → i is a continuous function and not identically zero, and g : i → r+ is a logarithmically convex (or logarithmically concave, respectively) function, satisfying (3.13) g((b−a)u) q g(b−a)g(u), u ∈ i some new inequalities of qi type for definite integrals 25 and (3.14) ∫ b a f(x) d x r g(b−a). then we have (3.15) exp ( 1 b−a ∫ b a ln g(f(x)) d x ) r g (∫ b a f(x) d x ) ∫ b a f(x) d x . proof. when g(u) is a logarithmically convex function on i, jensen’s inequality (2.6) gives g (∫ b a f(x) d x ) = g ( (b−a) lim n→∞ mn ( f(x), 1 n , 1 )) ≤ g(b−a) lim n→∞ g ( mn ( f(x), 1 n , 1 )) ≤ g(b−a) lim n→∞ mn ( g(f(x)), 1 n , 0) ) = g ( eb−a ) exp ( lim n→∞ mn ( ln g(f(x)), 1 n , 1 )) = g ( eb−a ) exp ( 1 b−a ∫ b a ln g(f(x)) d x ) , as a result, the inequality (3.15) is true. the rest can be proved similarly. the proof of theorem 3.3 is complete. � acknowledgements the author was partially supported by the nnsf under grant no. 11361038 of china and by the foundation of the research program of science and technology at universities of inner mongolia autonomous region under grant no. njzy13159, china. references [1] m. akkouchi, on an integral inequality of feng qi, divulg. mat. 13 (2005), no. 1, 11–19. [2] k. brahim, n. bettaibi, and m. sellemi, on some feng qi type q-integral inequalities, j. inequal. pure appl. math. 9 (2008), no. 2, art. 43; available online at http://www.emis.de/journals/jipam/article975.html. [3] y. chen and j. kimball, note on an open problem of feng qi, j. inequal. pure appl. math. 7 (2006), no. 1, art. 4; available online at http://www.emis.de/journals/jipam/article621.html. [4] v. csiszár and t. f. móri, the convexity method of proving moment-type inequalities, statist. probab. lett. 66 (2004), no. 3, 303–313; available online at http://dx.doi.org/10.1016/j.spl.2003.11.007. [5] j.-c. kuang, applied inequalities, 3rd edition, shandong science and technology press, ji’nan city, china, 2004. (chinese) [6] s. mazouzi and f. qi, on an open problem regarding an integral inequality, j. inequal. pure appl. math. 4 (2003), no. 2, art. 31; available online at http://www.emis.de/journals/jipam/article269.html. [7] y. miao, further development of qi-type integral inequality, j. inequal. pure appl. math. 7 (2006), no. 4, art. 144; available online at http://www.emis.de/journals/jipam/article763.html. 26 xi and qi [8] y. miao and j.-f. liu, discrete results of qi-type inequality, bull. korean math. soc. 46 (2009), no. 1, 125–134; available online at http://dx.doi.org/10.4134/bkms.2009.46.1.125. [9] y. miao and f. qi, several q-integral inequalities, j. math. inequal. 3 (2009), no. 1, 115–121; available online at http://dx.doi.org/10.7153/jmi-03-11. [10] t. k. pogány, on an open problem of f. qi, j. inequal. pure appl. math. 3 (2002), no. 4, art. 54; available online at http://www.emis.de/journals/jipam/article206.html. [11] f. qi, several integral inequalities, j. inequal. pure appl. math. 1 (2000), no. 2, art. 19; available online at http://www.emis.de/journals/jipam/article113.html. [12] f. qi, several integral inequalities, rgmia res. rep. coll. 2 (1999), no. 7, art. 9, 1039–1042; available online at http://rgmia.org/v2n7.php. [13] f. qi, a.-j. li, w.-z. zhao, d.-w. niu, and j.-cao, extensions of several integral inequalities, j. inequal. pure appl. math. 7 (2006), no. 3, art. 107; available online at http://www.emis.de/journals/jipam/article706.html. [14] f. qi and k.-w. yu, note on an integral inequality, j. math. anal. approx. theory 2 (2007), no. 1, 96–98. [15] n. towghi, notes on integral inequalities, rgmia res. rep. coll. 4 (2001), no. 2, art. 12, 277–278; available online at http://rgmia.org/v4n2.php. [16] s.-h. wu, rp -convex function and jensen’s type inequality, math. pract. theory 35 (2005), no. 3, 220–228. (chinese) [17] b.-y. xi and t.-y. bao, on some properties of r-mean convex function, math. pract. theory 38 (2008), no. 12, 113–119. (chinese) [18] b.-y. xi and f. qi, some inequalities of qi type for double integrals, j. egyptian math. soc. (2014), in press; available online at http://dx.doi.org/10.1016/j.joems.2013.11.002. [19] l. yin, q.-m. luo, and f. qi, several integral inequalities on time scales, j. math. inequal. 6 (2012), no. 3, 419–429; available online at http://dx.doi.org/10.7153/jmi-06-39. [20] l. yin and f. qi, some integral inequalities on time scales, results math. 64 (2013), no. 3, 371–381; available online at http://dx.doi.org/10.1007/s00025-013-0320-z. 1college of mathematics, inner mongolia university for nationalities, tongliao city, inner mongolia autonomous region, 028043, china 2department of mathematics, school of science, tianjin polytechnic university, tianjin city, 300387, china ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 3, number 2 (2013), 81-92 http://www.etamaths.com the complementary hankel type transformations of arbitrary order b.b. waphare∗ and s.b. gunjal abstract. in this paper four self reciprocal integral transformations of hankel type are defined through (hi,α,βf)(y) = fi(y) = ∫ ∞ 0 pi(x)gi,α,β(xy)f(x)dx,h −1 i,α,β = hi,α,β, where i = 1, 2, 3, 4; (α−β) ≥ 0, p1(x) = x4α, g1,α,β(x) = x−(α−β)jα−β(x), jα−β(x) being the bessel function of the first kind of order (α−β), p2(x) = x4β, g2,α,β(x) = (−1)α−βx2(α−β)g1,α,β(x), p3(x) = x−4α, g3,α,β(x) = x4αg1,α,β(x) and p4(x) = x −4β, g4,α,β(x) = (−1)α−βxg1,α,β(x). the simultaneous use of transformations h1,α,β and h2,α,β (which are denoted by hα,β) allows us to solve many problems of mathematical physics involving the differential operator ∆α,β = d 2 + 4αx−1d, whereas the pair of transformations h3,α,β and h4,α,β (which we express by hα,β) permits us to tackle those problems containing its adjoint operator ∆∗ α,β = d2 − (4α)x−1d + 4αx−2, no matter what the real value of α − β be. these transformations are also investigated in a space of generalized functions according to the mixed parseval equation∫ ∞ 0 f(x)g(x)dx = ∫ ∞ 0 (hα,βf)(y)(hα,βg)(y)dy, which is now valid for all real α−β. 1. introduction: following zemanian [15, 17], it can be proved that the hankel type transformation of order (α−β) ≥−1 2 (1.1) (hα,βf)(y) = ∫ ∞ 0 (xy)α+βjα−β(xy)f(x)dx, where jα−β(x) denotes the bessel type function of the first kind is an automorphism on the space hα,β of infinitely differentiable complex-valued functions φ(x), x ∈ (0,∞) such that ρ α,β m,k(φ) = sup 0<x<∞ ∣∣xm(x−1d)kx2β−1φ(x)∣∣ exists for each pair of non-negative integers m and k. then the generalized hankel type transformation h′α,β is defined on the dual space h ′ α,β by means of the adjoint 2010 mathematics subject classification. 46f12, 44a05. key words and phrases. complementary hankel type transformation, parseval equation, generalized functions. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 81 82 waphare and gunjal operator of hα,β that is,〈 h′α,βf, φ 〉 = 〈 f,hα,βφ 〉 , f ∈ h′α,β, φ ∈ hα,β. we emphasize that the last expression appears as a generalization of a well-known parseval equation for the hankel type transformation ([17, p. 127],[6]). consequently, the generalized transformation h′α,β is also an automorphism on the space h′α,β provided that (α − β) ≥ − 1 2 . zemanian [16], koh [4] later on remove the restriction (α−β) ≥−1 2 and assume that (α−β) be any fixed real number. for it, a positive integer r is chosen such that α−β + r ≥−1 2 and then the hankel type transformation of arbitrary order α−β is given by (hα,β,rφ)(y) = (−1)ry−r[hα,β,r(nα,β,r−1 . . . . . .nα,β,1nα,βφ)](y), where the operator nα,β = x 2αdx2β−1 generates an isomorphism from the space hα,β onto hα,β,1 and the mapping φ(x) → xrφ(x) is an isomorphism between the spaces hα,β and hα,β,r as well. the main idea of this method consists in leading any member φ(x) ∈ hα,β by means of the applications of the operator nα,β,nα,β,1, . . . . . . and nα,β,r−1 one after another to the space hα,β,r where the inversion formula of the hankel type transform has already a sense since α−β + r ≥ 1 2 . this procedure is employed to extend to an arbitrary order α −β. the following variant of the hankel type transformation (see [8]): (1.2) (h1,α,βf)(y) = f1(y) = ∫ ∞ 0 x4αg1,α,β(xy)f(x)dx. where g1,α,β(x) = x −(α−β)jα−β(x). the transform (1.2) is called in the available literature the schwartz’s hankel type transformation. the main objective of this paper is to extend the transformation (1.2) to any real value of α −β through a technique quite different from that used in the previous works [4],[8],[16]. to attain a more symmetrical expression for our results, from now on we assume that (α − β) ≥ 0. l(i) denotes the space of all functions f(x) that are lebesgue integrable on the real positive axis i = (0,∞). d(i) stands for the space of infinitely differentiable functions whose supports are contained in i and its dual d′(i) is the space of schwartz distributions [11]. finally, e(i) represents the space of all infinitely differentiable functions on i and its dual e′(i) is the space of distributions with compact supports [17, p. 36]. 2. classical results on schwartz’s hankel type transformations: we establish the following in relation with the transformation (1.2). theorem 2.1. let (α−β) ≥ 0. if y2αf(y) ∈ l(i), then∫ ∞ 0 y4αg1,α,β(xy)f1(y)dy = 1 2 [f(x + 0) + f(x− 0)], in a neighborhood of every point y = x where f(y) is of bounded variation. the complementary hankel type transformations 83 proof. the outcome of the theorem follows from the relation (h1,α,βf)(y) = y2β−1 { hα,β ( x2αf(x) )} (y) between (1.1) and (1.2) and from the inversion theorem for the hankel type transformation [13, p. 240]. � other conditions under which theorem 2.1 holds were proposed by a.l. schwartz [12]. note that the function g1,α,β(xy) satisfies the equation. (2.1) (∆1,α,β,x + y 2)g1,α,β(xy) = 0. where ∆1,α,β,x = ∆1,α,β = d 2 + (4α)x−1d, d ≡ d dx . if f(x) and g1(y) are functions defined on i such that x 2αg1(y) ∈ l(i) and if f1(y) = (h1,α,βf)(y) and g(x) = (h−11,α,βg1)(x) = (h1,α,βg1)(x), where (α−β) ≥ 0, then we easily obtain the parseval equation (2.2) ∫ ∞ 0 x4αf(x)g(x)dx = ∫ ∞ 0 y4αf1(y)g1(y)dy. in past years, schwartz’s hankel transformation has been investigated in certain spaces of distributions, amongst other authors, by l.s. dubey and j.n. pandey [2], w.y.lee [5], a. schuitman [10] and g. altenburg [1]. further, we introduce the transformation (2.3) (h2,α,βg)(y) = g2(y) = ∫ ∞ 0 x4βg2,α,β(xy)g(x)dx. where g2,α,β(x) = (−1)α−βx2(α−β)g1,α,β(x) fullfils the equation (2.4) (∆2,α,β,x + y 2)g2,α,β(xy) = 0, ∆2,α,β denotes the differential operator ∆2,α,β,x = ∆2,α,β = d 2 + 4βx−1d. observe that the multiplication x2(α−β) only implies the change of sign for the parameter (α − β) in (2.1) to convert it into equation (2.4). on the other hand transformations (1.2) and (2.3) are closely related as it is made evident by (2.5) (h2,α,βg)(y) = (−1)α−βy2(α−β) [ h1,α,β ( x−2(α−β)g(x) )] (y). then from theorem 2.1 and (2.5) it is immediately inferred the classical inversion formula for the transformation h2,α,β: theorem 2.2. let (α−β) ≥ 0. if y2βg(y) ∈ l(i) and g2(y) is defined as in (2.3), then we have ∫ ∞ 0 y4βg2,α,β(xy)g2(y)dy = 1 2 [g(x + 0) + g(x− 0)] 84 waphare and gunjal in a neighborhood of every point y = x where g(y) is of bounded variation. for a pair functions f(x) and g2(y) such that x 2βf(x) and y2βg2(y) ∈ l(i), the following parseval equation (2.6) ∫ ∞ 0 x4βf(x)g(x)dx = ∫ ∞ 0 y4βf2(y)g2(y)dy is valid, where f2(y) = (h2,α,βf)(y), g(x) = (h−12,α,βg2)(x) = (h2,α,βg2)(x) and (α−β) ≥ 0. according to the relation (2.5) the transforms h1,α,β and h2,α,β coincide seemingly when and only when α − β = 0. however, it is worth to remark that there are other values of α−β which make equal the transformations h1,α,β and h2,α,β. infact, from the relation [13, p. 16] g1,−n(x) = g2,n, n = 0, 1, 2, 3, . . . . . . , we quickly deduce that (h1,−nf)(y) = (h2,nf)(y), in other words, the transformation h2,n of positive integer order might be used to replace the transformation h1,α,β of negative integer index. this fact and above considerations suggest to adopt the notation (2.7) ∆α,β = ∆α,β,x = d 2 + (4α)x−1d = x−4αdx4αd, −∞ < (α−β) < ∞ and bα−β(x) = { g1,α,β(x), if (α−β) ≥ 0 g2,−α,β(x), if (α−β) < 0 . now we assemble (1.2) and (2.3) in the unique expression (2.8) (hα,βf)(y) = f(y) = ∫ ∞ 0 x4αbα−β(xy)f(x)dx. thus theorem 2.1 and 2.2 can be enunciated together: theorem 2.3. let (α−β) be an arbitrary real number. if the function f(y) is such that x2αf(x) ∈ l(i), if f(y) is of bounded variation in a neighborhood of y = x ∈ i and if f(y) is given by (2.8), then∫ ∞ 0 y4αbα−β(xy)f(y)dy = 1 2 [f(x + 0) + f(x− 0)]. following [7], this other variant of the hankel type transformation (2.9) (h3,α,βf)(y) = f3(y) = ∫ ∞ 0 x−4αg3,α,β(xy)f(x)dx, where g3,α,β(x) = x 4αg1,α,β turns out to be a solution of the equation (2.10) (∆3,α,β,x + y 2)g3,α,β(xy) = 0, ∆3,α,β being the differential operator ∆3,α,β,x = ∆3,α,β = d 2 − (4α)x−1d + (4α)x−2. the complementary hankel type transformations 85 the following inversion formula follows from [7]: theorem 2.4. let (α − β) ≥ 0. if f(y) is a function defined on i such that y2β−1f(y) ∈ l(i), then∫ ∞ 0 y−4αg3,α,β(xy)f3(y)dy = 1 2 [f(x + 0) + f(x− 0)] in a neighborhood of every point y = x where f(y) is of bounded variation. moreover, if f(x) and g3(y) are two functions defined over the positive real axis such that x2β−1f(x) and y2β−1g3(y) ∈ l(i), we obtain the parseval equation (2.11) ∫ ∞ 0 x−4αf(x)g(x)dx = ∫ ∞ 0 y−4αf3(y)g3(y)dy. where f3(y) = (h3,α,βf)(y), g(x) = (h−13,α,βg3)(x) = (h3,α,βg3)(x) and (α−β) ≥ 0. finally, we introduce a fourth integral transformation by means of, (2.12) (h4,α,βg)(y) = g4(y) = ∫ ∞ 0 x−4βg4,α,β(xy)g(x)dx, in those kernel appears the function g4,α,β(x) = (−1)α−βxg1,α,β(x) = (−1)α−βx−2(α−β)g3,α,β(x), (2.13) (∆4,α,β,x + y 2)g4,α,β(xy) = 0. here ∆4,α,β symbolizes the differential operator ∆4,α,β,x = ∆4,α,β = d 2 − (4β)x−1d + (4β)x−2. from the relation (2.14) (h4,α,βg)(y) = (−1)α−βy [ h1,α,β ( x−1g(x) )] (y) between transformations (1.2) and (2.12), by virtue of theorem 2.1, we can easily state the inversion formula for the new transform: theorem 2.5. let (α −β) ≥ 0. if g(y) is a function on i such that y−2βg(y) ∈ l(i), if g(y) is of bounded variation in a neighborhood of the point y = x ∈ i and g4(y) is given by (2.12), then∫ ∞ 0 y−4βg4,α,β(xy)g4(y)dy = 1 2 [g(x + 0) + g(x− 0)]. another result, which we shall need, is the corresponding parseval equation that now takes the form (2.15) ∫ ∞ 0 x−4βf(x)g(x)dx = ∫ ∞ 0 y−4βf4(y)g4(y)dy. where f4(y) = (h4,α,βf)(y), g(x) = (h−14,α,βg4)(x) = (h4,α,βg4)(x) and (α−β) ≥ 0,f(x) and g4(y) being a pair of functions such that x −2βf(x) and y−2βg4(y) ∈ 86 waphare and gunjal l(i). by pursuing theorem 2.4 and 2.5, parseval relations (2.11) and (2.15) and differential operators ∆3,α,β and ∆4,α,β the unique apparent difference lies in the change of the sign on the parameter (α−β). in the same way it is convenient to point that the multiplication of g3,α,β(x) by (−1)α−βx−2(α−β) to get the function g4,α,β(x) exactly implies the said change of sign. furthermore, it is easily seen that (h3,−ng)(y) = (h4,ng)(y), since g3,−n(x) = g4,n(x), n = 0, 1, 2, 3, . . . . . . ., now, it is wholly justified to adopt the following notation (2.16) ∆∗α,β = ∆4,α,β = d 2 − (4α)x−1d + (4α)x−2 = dx4αdx−4α, −∞ < (α−β) < ∞ and b∗α−β(x) = { g3,α,β(x), if (α−β) ≥ 0 g4,−α,−β(x), if (α−β) ≤ 0 . thus, the transformation (2.9) and (2.12) can be rewritten in the unique expression (2.17) (h∗α,βg)(y) = g ∗(y) = ∫ ∞ 0 x−4αb∗α−β(xy)g(x)dx. now, we summarize theorems 2.4 and 2.5. theorem 2.6. if g(y) is a function defined on i such that y2β−1g(y) ∈ l(i), if g(y) is of bounded variation in a neighborhood of a point y = x ∈ i and if g∗(y) is given by (2.17), then∫ ∞ 0 y−4αb∗α−β(xy)g ∗(y)dy = 1 2 [g(x + 0) + g(x− 0)], for any real value of (α−β). by an application of fubini’s theorem, we can establish another parseval equations involving a pair of these four transformations hi,α,β(i = 1, 2, 3, 4). theorem 2.7. let (α−β) ≥ 0. (a) if x2αf(x) ∈ l(i) and y2β−1g3(y) ∈ l(i), then (2.18) ∫ ∞ 0 f(x)g(x)dx = ∫ ∞ 0 f1(y)g1(y)dy, where f1(y) = (h1,α,βf)(y) and g3(y) = (h3,α,βg)(y). (b) the pair of transformations h2,α,β and h4,α,β verifies also the mixed parseval equation (2.19) ∫ ∞ 0 f(x)g(x)dx = ∫ ∞ 0 f2(y)g4(y)dy where now f2(y) = (h2,α,βf)(y) and g4(y) = (h4,α,βg)(y), provided that x2βf(x) ∈ l(i) and y−2βg4(y) ∈ l(i). the complementary hankel type transformations 87 remark 1. note that ∆3,α,β and ∆4,α,β are formally the adjoints operators ∆1,α,β and ∆2,α,β respectively. by this reason, h3,α,β will be called the adjoint or complementary transform of h1,α,β and we shall refer to h4,α,β as the adjoint or complementary transform of h2,α,β. remark 2. the parseval equality (2.18) involves the transformation h1,α,β and its complementary h3,α,β where as the parseval formula (2.19) relates both transforms h2,α,β and h4,α,β. observe that, in comparison with (2.2), (2.6), (2.11) and (2.15), expressions (2.18) and (2.19) do not contain any weight function. remark 3. according to the notation we have just adopted, theorem 2.7 admits this statement more concisely. theorem 2.8. let (α−β) be any fixed real number. if x2αf(x) and y2β−1g∗(y) ∈ l(i) and if we put f(y) = (hα,βf)(y) and g(x) = (h∗−1α,β g ∗)(x) = (h∗α,βg ∗)(x), then ∫ ∞ 0 f(x)g(x)dx = ∫ ∞ 0 f(y)g∗(y)dy, that is, (2.20) ∫ ∞ 0 f(x)g(x)dx = ∫ ∞ 0 (hα,βf)(y)(h∗α,βg)(y)dy. 3. the generalized schwartz’s hankel transformation of arbitrary order: let (α − β) be any fixed real number and i an integer, 1 ≤ i ≤ 4. the space hi,α,β consists of all infinitely differentiable complex-valued functions φ(x) defined on i for which η α,β,i m,k (φ) = sup x∈i ∣∣xm(x−1d)kwi(x)φ(x)∣∣ exist for each pair of non-negative integers m and k, where w1(x) = 1, w2(x) = x−2(α−β), w3(x) = x −4α and w4(x) = x −1 with the topology generated by the collections of seminorms { η α,β,i m,k } ,hi,α,β are frechet spaces. we denote h1,α,β = h1 and h4,α,β = h4, since the definition of these spaces is independent of the particular choice of the parameter α−β. theorem 3.1. a function φ(x) defined on i is a member of hi,α,β if and only if, (a) φ(x) is infinitely differentiable on i, (b) φ(x) has the form φ(x) = w−1i (x)[a0 + a1x 2 + a2x 4 + · · · · · · + akx2k + o(x2k)] in some vicinity of the origin, and (c) dkφ(x) is of rapid descent when x → ∞ for each k = 0, 1, 2, 3, . . . . . .. h′i,α,β symbolizes the dual space of hi,α,β and its members are generalized functions of slow growth. we assign to h′i,α,β the weak topology generated by the multinorm {ξi,φ} defined through ξi,φ(f) = ∣∣〈f,φ〉∣∣ , f ∈ h′i,α,β, φ ∈ hi,α,β. thus, h′i,α,β is also sequentially complete. 88 waphare and gunjal now we introduce the new spaces (3.1) hα,β = { h1, if (α−β) ≥ 0 h2,−α,−β if (α−β) ≤ 0 , and (3.2) h∗α,β = { h3,α,β, if (α−β) ≥ 0 h4, if (α−β) ≤ 0 . h′α,β and h ∗′ α,β, denote the dual spaces of hα,β and h ∗ α,β respectively some properties of these spaces are listed below: (a) d(i) ⊂ hi,α,β and the restriction of every f ∈ h′i,α,β to d(i) is a member of d′(i). moreover, hi,α,β is a dense subspace of e(i). therefore, e′(i) is a subspace of h′i,α,β. consequently, d(i) ⊂ hα,β ⊂ e(i) and d(i) ⊂ h∗α,β ⊂ e(i). (b) the operations φ → ∆α,βφ and ψ → ∆∗α,βψ, where ∆α,β and ∆ ∗ α,β represent the operators given by (2.7) and (2.16), are continuous linear mappings from the testing function spaces hα,β and h ∗ α,β into themselves, respectively. infact, the inequality η α,β,i m,k (∆i,α,βφ) ≤ |2k + 6α + 2β|η α,β,i m,k+1(φ) + η α,β,i m+2,k+2(φ), i = 1, 2, 3, 4, is satisfied for every φ ∈ hi,α,β by taking into account that ∆i,α,β = w −1 i (x)[x 2(x−1d)2 + (6α + 2β)(x−1d)]wi(x). hence, the generalized differential operator ∆α,β defined on the distributional space h∗α,β as the adjoint operator of ∆ ∗ α,β, that is to say,〈 ∆α,βf,ψ 〉 = 〈 f, ∆∗α,βψ 〉 , f ∈ h∗α,β, ψ ∈ h ∗ α,β, is also a continuous linear mapping of h∗α,β into itself. conversely, the generalized differential operator ∆∗α,β will be defined on h ′ α,β by means of〈 ∆∗α,βf,ψ 〉 = 〈 f, ∆α,βψ 〉 , f ∈ h′α,β, ψ ∈ hα,β and produces a continuous linear mapping of the space h′α,β into itself. (c) assume that (α−β) is any real number. then, hα,β may be identified with a subspace of h∗ ′ α,β, that is, hα,β ⊂ h ∗′ α,β, the topology of hα,β being stronger than that induced on it by h∗ ′ α,β. indeed, any f ∈ h1 generates a regular distribution f in the space h′3,α,β by (3.3) 〈 f,φ 〉 = ∫ i f(x)φ(x)dx, φ ∈ h3,α,β. provided that (α −β) ≥ 0. the linearity of (3.3) being obvious, the continuity is inferred from ∣∣〈f,φ〉∣∣ ≤ ηα,β,30,0 (φ) ∫ i x4α|f(x)|dx. notice that last integral exists since f(x) = o(1) when x → 0, and f(x) is of rapid descent at infinity, by virtue of theorem 3.1. to see the second part of the assertion, the complementary hankel type transformations 89 observe that ξ3,φ(f) ≤ ρ α,β,1 0,0 (f) ∫ ∞ 0 |φ(x)|dx. on the other hand, two members f and g of h1 giving rise to the same regular distribution in h′3,α,β must be identical. these considerations justify the inclusion h1 ⊂ h′3,α,β. analogously, h2,−α,−β may be identified one-to-one with a subspace of h′4. in other words, h2,−α,−β ⊂ h′4 whenever (α−β) ≤ 0. we can now conclude, in view of notation (3.1) and (3.2), that hα,β ⊂ h∗α,β. finally, we can proceed in similar way to get the inclusions h3,α,β ⊂ h′1 and h4 ⊂ h′2,−α,−β for (α − β) ≥ 0 and (α − β) ≤ 0, respectively, from which one deduces that h∗α,β ⊂ h ′ α,β. theorem 3.2. (a) the hankel type transformation hα,β defined by (2.8) is an automorphism on the space hα,β, no matter what the real value of (α−β) be. (b) the complementary hankel type transformation h∗α,β, as given by (2.17) is as well an automorphism on h∗α,β whatever be the real number α−β. proof. (a) in [1, th.5] it was proven that h1,α,β is an automorphism on h1. to study the transformation h2,α,β, firstly we note that the operation φ → x2(α−β)φ is an isomorphism from h1 into h2,α,β and then we take into account the relation (2.5) between transformations h1,α,β and h2,α,β. (b) since the operation ψ → x4αψ is an isomorphism from h1 into h3,α,β the expression (h3,α,βφ)(y) = y2α [ h1,α,β(x−4αφ) ] (y), φ ∈h3,α,β, which connects the transforms (2.9) and (1.2), implies that h3,α,β is an automorphism on h3,α,β. finally in view that the mapping φ → xφ defines an isomorphism from h1 into h4 and (2.14), we deduce that h4,α,β is an automorphism on the space h4 as well. according to convention (2.17), we can now conclude the validity of statement (b). � let (α − β) be any fixed real number. we define the generalized schwartz’s hankel transformation hα,β of arbitrary order in the distributional space h∗ ′ α,β, as the adjoint operator of h∗α,β acting on h ∗ α,β, namely, (3.4) 〈 h ′ α,βf,φ 〉 = 〈 f,h∗α,βφ 〉 , for every f ∈ h∗ ′ α,β and φ ∈ h ∗ α,β. if we put φ = h ∗ α,βφ, the application of part (b) in theorem 3.2 allows us to rewrite (3.4) as follows (3.5) 〈 hα,βf,h∗α,βφ 〉 = 〈 f,φ 〉 , f ∈ h∗ ′ α,β, φ ∈ h ∗ α,β. this expression can be understood as an extension of the parseval equation (2.20) to distributions. theorem 3.3. the generalized schwartz’s hankel type transformation h′α,β given by (3.4) or (3.5) is an automorphism on h∗ ′ α,β, whatever be the real value of the parameter α−β. proof. it is a consequence of part (b) in theorem 3.2 and [17, th.1.10-2]. � 90 waphare and gunjal remark 4. note that the generalized transformation h ′ α,β is defined on the space h∗ ′ α,β as the adjoint operator of the complementary transform, and not over the space h ′ α,β by means of the adjoint operator of itself, as usual in the available literature. this is suggested and justified, at once, by the inclusion hα,β ⊂ h∗ ′ α,β. indeed, because of this inclusion if hα,β acts on hα,β, the most natural would be to define hα,β on h∗ ′ α,β instead of doing it on the space h ′ α,β. on the other hand, when h ′ α,β is defined as usual, that is, (3.6) 〈 h ′ α,βf, φ 〉 = 〈 f,hα,βφ 〉 , f ∈ h ′ α,β, φ ∈ hα,β. a schuitman [10] found the relation h ′ α,βf = y 4αhα,β(x−4αf), f ∈ hα,β, between classical and generalized transformations, which means that the conventional transformation hα,β is not a particular case of generalized one. on the contrary, if definitions (3.4) or (3.5) are adopted, f ∈ hα,β implies that hα,βf ∈ hα,β in view of part(i) in theorem 3.2. therefore, hα,βf generates a regular member in h∗ ′ α,β through (3.3), (3.7) 〈 hα,βf, φ 〉 = ∫ i (hα,βf)(y)φ(y)dy, φ ∈ h∗α,β. by virtue of parseval relation (2.20), the right-hand side of (3.7) takes the form∫ i f(x) ( h∗α,β(φ) ) (x)dx = 〈 f,h∗α,βφ 〉 = 〈 h ′ α,βf, φ 〉 , by definition (3.4). we have〈 hα,βf, φ 〉 = 〈 h ′ α,βf, φ 〉 , for all φ ∈ h∗α,β. in other words, the classical transformation hα,β coincides now with the generalized one hα,β, when this acts on the testing-function space h∗α,β. this proves that the definition (3.4) is more appropriate than (3.6). remark 5. note that the definitions of the generalized operators introduces in (ii) are consistent with the rules of distributional calculus, in particular, with the definitions of multiplication by an infinitely differentiable function, differentiation of distribution which does not occur in [1], [5] and [10]. for every real α − β, it is easily seen that ∆α,β acts as a classical differential operator on the space hα,β and as a generalized one on h∗ ′ α,β, while that the conventional operator ∆∗α,β acting on h ∗ α,β is understood as a distributional operator on h′α,β. these facts are justified in view of the inclusions hα,β ⊂ h ∗′ α,β and h∗α,β ⊂ h ′ α,β, since ∆ ∗ α,β, is the adjoint operator of ∆α,β. the complementary hankel type transformations 91 remark 6. assume that (α − β) is any real number. the generalized transformation h ′ α,β can be similarly defined on the space h ′ α,β as the adjoint operator of hα,β on hα,β, namely, (3.8) 〈 h ′ α,βg, ψ 〉 = 〈 g,hα,βψ 〉 , g ∈ h ′ α,β, ψ ∈ hα,β. finally, set ψ = hα,βψ. then, because of theorem 3.2(a), (3.8) becomes〈 h ′ α,βg,hα,βψ 〉 = 〈 g,ψ 〉 , g ∈ h ′ α,β, ψ ∈ hα,β. again this expression can be considered as an exact transcription to generalized functions of the mixed parseval relation (2.20). it is fulfilled that h ′ α,βg = hα,βg whatever be g ∈ h∗α,β as well. an analogous result to theorem 3.3 can be inferred from theorem 3.2(a): theorem 3.4. the generalized complementary hankel transformation h ′ α,β, as given by (3.8) is an automorphism on h ′ α,β independently of the value of (α−β) ∈ r. next, we collect some operation-transform formulas. theorem 3.5. let (α−β) be an arbitrary real number. for all φ ∈ hα,β, hα,β{∆α,βφ} = −y2hα,βφ ∆α,βhα,βφ = hα,β(−x2φ). if ψ ∈ h∗α,β, then hα,β{∆∗α,βψ} = −y 2hα,βψ ∆∗α,βhα,βφ = hα,β(−x 2φ). for every f ∈ h∗ ′ α,β one has hα,β{∆α,βf} = −y2hα,βf whereas h ′ α,β{∆ ∗ α,βg} = −y 2h ′ α,βg holds for any g ∈ h′α,β. remark 7. the cauchy problem posed by w.y. lee [5] ∂u ∂t = p(∆α,β,x)u(x,t) u(x, 0) = f(x) where f(x) is a known member of h∗ ′ α,β, p denotes a polynomial with constant coefficients and the unknown function u(x,t) ∈ h∗ ′ α,β, can now be solved by applying the transformation hα,β not only for (α−β) ≥−12 , but whatever be the real value of the parameter α−β. this and above results justify that the transformation hα,β be called the generalized schwartz hankel type transformation of arbitrary order. 92 waphare and gunjal references [1] altenburg g; bessel transformation in raumen von grundfunktionen uber dem interval ω = (0,∞) und deven dualraumen, math. nachr, 108 (1982), 197-218. [2] dubey l.s. and pandey j.n., on the hankel transform of distribution, tohoku math. j. 27, (1975), 337-354. [3] gray a., mathews, g.b. and macrobert t.m., a treatise on bessel functions and their applications to physics, macmillan, london, 1952. [4] koh e.l., the hankel transformation of negative order for distributions of rapid growth, siam j. math. anal. 1(1970), 322-327. [5] lee w.y., on schwartz’s hankel transformation of certain spaces of distributions, siam j. math. anal. 6 (1975), 427-432. [6] macauley-owen p, parseval theorem for hankel transform, proc. london math. soc. 45 (1939), 458-474. [7] mendez perez, j.m.r., on the bessel transforms, jnanabha 17(1987), 79-88. [8] mendez perez j.m.r., on the bessel transformation of arbitrary order, math. nachr, 136 (1988), 233-239. [9] mendez perez j.m.r, a mixed parseval equation and the generalized hankel transformation, proc. amer. math. soc., 102(1988), 619-624. [10] shuitman a, on a certain test function space for schwartz’s hankel transform, delft prog, rep.2 (1977), 192-206. [11] schwartz l., theorie des distributions, herman paris 1966. [12] schwartz a.l, an inversion theorem for hankel transform, proc. amer. math. soc., 22 (1969), 713-717. [13] titchmarsh e.c., introduction to the theory of fourier integrals, oxford univ. press, london, 1959. [14] watson g.n., a treatise on the theory of bessel functions, cambridge univ. press, cambridge , 1958. [15] zemanian a.h., a distributional hankel transformation, siam j. appl. math. 14(1966), 561-576. [16] zemanian a.h., hankel transforms of arbitrary order, duke math. j. 34 (1967), 761-769. [17] zemanian a.h., generalized integral transformations interscience. n.y. 1968. mit acsc, alandi, tal. khed, dist: pune, maharashtra, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 121-128 http://www.etamaths.com ps-ro fuzzy open(closed) functions and ps-ro fuzzy semi-homeomorphism pankaj chettri∗, subodh gurung and khusboo katwal abstract. the aim of this paper is to introduce and characterize some new class of functions in a fuzzy topological space termed as ps-ro fuzzy open(closed) functions, ps-ro fuzzy pre semiopen functions and ps-ro fuzzy semi-homeomorphism. the interrelation among these concepts and also their relations with the parallel existing concepts are established. it is also shown with the help of examples that these newly introduced concepts are independent of the well known existing allied concepts. 1. introduction and preliminaries the concept of fuzzy open (closed) functions were introduced by c.l. chang [1] and their characterizations were studied by s.r. malghan and s.s. benchalli [9]. in [4], a new idea of fuzzy topology termed as pseudo regular open fuzzy topology (in short, ps-ro fuzzy topology) was introduced. the members of this topology are named as ps-ro open fuzzy sets and their complement as ps-ro closed fuzzy sets. interms of above fuzzy sets, a new class of functions called ps-ro fuzzy continuous functions were introduced and explored in [5], [6]. in [2], a notion of ps-ro semiopen(closed)fuzzy sets, ps-ro fuzzy semiopen functions and ps-ro fuzzy semicontinuous functions were introduced. also, in [3], a new idea of ps-ro fuzzy irresolute function was initiated and studied. the concept of fuzzy pre semiopen and fuzzy semi-homeomorphism were introduced by yalvac [10]. in this paper, a new class of functions called ps-ro fuzzy open(closed) functions are defined and their different characterizations are studied. interestingly, it is shown that the concept of ps-ro fuzzy open (closed) functions are independent of the well known concept of fuzzy open (closed) functions. also, introducing a new class of functions called ps-ro fuzzy pre semiopen function and ps-ro fuzzy semi-homeomorphism, their different properties and interrelations with the existing allied concepts has been established. to make this paper self content, we state a few known definitions and results here that we require subsequently. let x be a non-empty set and i be the closed interval [0, 1]. a fuzzy set µ on x is a function on x into i. if f is a fuction from x into a set y and a, b are fuzzy sets on x and y respectively, then 1 −a (called complement of a), f(a) and f−1(b) are fuzzy sets on x, y and x respectively, defined by (1−a)(x) = 1−a(x)∀x ∈ x, 2010 mathematics subject classification. 03e72, 54a40, 54c08. key words and phrases. ps-ro open(semiopen) fuzzy set, ps-ro fuzzy open(closed) function, ps-ro fuzzy semi-homeomorphism. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 121 122 chettri, gurung and katwal f(a)(y) = { supz∈f−1(y) a(z),whenf −1(y) 6= ∅ 0, otherwise and f−1(b)(x) = b(f(x)) [11]. a collection τ ⊆ ix is called a fuzzy topology on x if (i) 0, 1 ∈ τ (ii) ∀µ1,µ2, ...,µn ∈ τ ⇒∧n i=1 µi ∈ τ (iii) µα ∈ τ, ∀ α ∈ λ (where λ is an index set) ⇒∨µα ∈ τ. then, (x,τ) is called a fts [1]. let f be a function from a set x into a set y . then the following holds: (i) f−1(1 −b) = 1 −f−1(b), for any fuzzy set b on y . (ii) a1 ≤ a2 ⇒ f(a1) ≤ f(a2), for any fuzzy sets a1 and a2 on x. also, b1 ≤ b2 ⇒ f−1(b1) ≤ f−1(b2), for any fuzzy sets b1 and b2 on y . (iii) ff−1(b) ≤ b, for any fuzzy set b on y and the equality holds if f is onto. also, f−1f(a) ≥ a, for any fuzzy set a on x, equality holds if f is one-to-one [1]. for a fuzzy set µ in x, the set µα = {x ∈ x : µ(x) > α} is called the strong α-level set of x. in a fts (x,τ), the family iα(τ) = {µα : µ ∈ τ} for all α ∈ i1 = [0, 1) forms a topology on x called strong α-level topology on x [8], [7]. a fuzzy open(closed) set µ on a fts (x,τ) is said to be pseudo regular open(closed) fuzzy set if the strong αlevel set µα is regular open(closed) in (x,iα(τ)),∀α ∈ i1. the family of all pseudo regular open fuzzy sets form a fuzzy topology on x called ps-ro fuzzy topology on x,members of which are called ps-ro open fuzzy sets and their complements as ps-ro closed fuzzy sets on (x,τ) [4]. a function f from fts (x,τ1) to fts (y,τ2) is pseudo fuzzy ro continuous (in short, ps-ro fuzzy continuous) if f−1(u) is ps-ro open fuzzy set on x for each pseudo regular open fuzzy set u on y [5]. equivalently, f is ps-ro fuzzy continuous if f−1(a) is ps-ro open fuzzy set on x for each ps-ro open fuzzy set a on y [6]. a fuzzy set a on a fts (x,τ) is said to be ps-ro semiopen fuzzy set if there exist a ps-ro open fuzzy set u such that u ≤ a ≤ ps-cl(u), where ps-cl(u) is ps-closure of u and the complement of a is called ps-ro semiclosed fuzzy set [2]. the fuzzy operators termed as fuzzy ps-closure(interior), ps-semiclosure(interior) are denoted by ps-cl(ps-int) and ps-scl(ps-sint) respectively. ps-int(ps-sint) of a fuzzy subset a the union of all ps-ro open (ps-ro semiopen) fuzzy set on x contained in a and ps-cl(ps-scl) of a fuzzy subset a the intersection of all ps-ro closed (ps-ro semiclosed) fuzzy set on x containing a [5], [6], [2]. a function f from a fts (x,τ1) to another fts (y,τ2) is called ps-ro fuzzy semiopen function [2] if f(a) is ps-ro semiopen fuzzy set on y for each ps-ro open fuzzy set a on x. the function f is called ps-ro fuzzy irresolute [3] if f−1(u) is ps-ro semiopen fuzzy set on x for each ps-ro semiopen fuzzy set u on y . if a function f be bijective, then f is ps-ro fuzzy irresolute function iff for every fuzzy set a of x, ps-sint(f(a)) ≤ (ps-sint(a))[3]. for a function f : x → y , the following are equivalent: (a)f is ps-ro fuzzy continuous. (b) inverse image of each ps-ro open fuzzy sets on y under f is ps-ro open on x. (c)for all fuzzy set a on x, f(ps-cl(a)) ≤ ps-cl(f(a)). (d) for all fuzzy set b on y , ps-cl(f−1(b)) ≤ f−1(ps-cl(b)) [6]. 2. ps-ro fuzzy open and closed functions definition 2.1. let (x,τ1) and (y,τ2) be two fts. a function f : (x,τ1) → (y,τ2) is said to be ps-ro fuzzy open(closed) if f(a) is ps-ro open(closed) fuzzy set on y for each ps-ro open(closed) fuzzy set a on x. theorem 2.1. if f is ps-ro continuous and ps-ro fuzzy open and a be any ps-ro semiopen fuzzy set on x then f(a) is ps-ro semiopen fuzzy set on y . ps-ro fuzzy open(closed) functions 123 proof: let a be any ps-ro semiopen fuzzy set on x, there exist ps-ro open fuzzy set u on x such that u ≤ a ≤ ps-cl(u). so, f(u) ≤ f(a) ≤ f(ps-cl(u)) ≤ pscl(f(u)) and f(u) is ps-ro open fuzzy set. hence, f(a) is ps-ro semiopen fuzzy set on y . example 2.1. let x = {a,b,c} and y = {x,y,z}. let a and b be two fuzzy sets on x defined by a(a) = 0.1,a(b) = 0.2,a(c) = 0.2 and b(x) = 0.3 ∀x ∈ x. let c,d and e be fuzzy set on y defined by c(t) = 0.3 ∀t ∈ y , d(x) = 0.3,d(y) = 0.3,d(z) = 0.4 and e(t) = 0.4 ∀t ∈ y . clearly, τ1 = {0, 1,a,b} and τ2 = {0, 1,c,d,e} are fuzzy topologies on x and y respectively. clearly, for 0.1 ≤ α < 0.2, a is not pseudo regular open fuzzy set on (x,τ1). therefore, ps-ro fuzzy topology on x is {0, 1,b}. again, d is not pseudo regular open fuzzy set for 0.3 ≤ α < 0.4 on (y,τ2). so, ps-ro fuzzy topology on y is {0, 1,c,e}. define a function f from (x,τ1) to (y,τ2) by f(a) = x,f(b) = y and f(c) = y. b is ps-ro open fuzzy set on y and f(b)(t) = 0.3 = c(t) ∀ t ∈ y . therefore, f(b) is ps-ro open fuzzy set on y . also, f(0) = 0,f(1) = 1. hence, f is ps-ro fuzzy open. now, f(a)(x) = 0.1,f(a)(y) = 0.2,f(a)(z) = 0. clearly, f(a) is not open fuzzy set on y . hence, f is not fuzzy open function. again, here f is ps-ro fuzzy closed as f(1 − b)(t) = 0.7 = (1 − c)(t), ∀t ∈ y is ps-ro closed fuzzy set on y but f is not fuzzy closed function since f(1 − a)(t) = 0.9, 0.8 and 0 for t = x,y and z respectively, is not fuzzy closed set on y . remark 2.1. let f be fuzzy open (closed) from a fts (x,τ1) to a fts (y,τ2) and a be a open (closed) fuzzy set on x. then, f(a) is fuzzy open (closed) on y which is not necessarily ps-ro open (closed) fuzzy set on y , for an example in example( 2.1), a is fuzzy open but not ps-ro fuzzy opnen on x. hence, a fuzzy open (closed) function may not be ps-ro fuzzy open (closed). in the view of this and example ( 2.1) we conclude that ps-ro fuzzy open (closed) functions and fuzzy open (closed) functions do not imply each other. theorem 2.2. let f be a function from a fts (x,τ1) to a fts (y,τ2). then the following statements are equivalent: (a)f is ps-ro fuzzy open. (b)f(ps-int(a)) ≤ ps-int(f(a)), for each fuzzy set a on x. (c)f−1(ps-cl(b)) ≤ ps-cl(f−1(b)), for each fuzzy set b on y . (d)ps-int(f−1(b)) ≤ f−1(ps-int(b)), for each fuzzy set b on y . proof: (a) ⇒ (b) let f be ps-ro fuzzy open function. let a be any fuzzy set on x. f(ps-int(a)) is ps-ro open fuzzy set on y . now, f(ps-int(a)) = ps-int(f(psint(a))) ≤ ps-int(f(a)). (b) ⇒ (a) let a be a ps-ro open fuzzy set on x. then a = ps-int(a). so, f(a) = f(ps-int(a)) ≤ ps-int(f(a)) ≤ f(a). so, f(a) = ps-int(f(a)), proving f(a) is ps-ro open fuzzy set on y . thus, f is ps-ro fuzzy open. (b) ⇒ (c) let b be any fuzzy sets on y . let a = f−1(1−b) be a fuzzy set on x. we have f(ps-int(a)) ≤ ps-int(f(a)) ≤ ps-int(1 − b). hence, ps-int(f−1(1 − b)) ≤ f−1(ps-int(1 − b)). then, f−1(ps-cl(b)) = 1 − f−1(ps-int(1 − b)) ≤ 1 − psint(f−1(1−b)) = ps-cl(1−f−1(1−b)) = ps-cl(f−1(b)). so, f−1(ps-cl(b)) ≤ pscl(f−1(b)) (c) ⇒ (d) let b be any fuzzy set on y and c = 1 − b. then, c is also fuzzy set on y . we have f−1(ps-cl(c)) ≤ ps-cl(f−1(c)). so, ps-int(f−1(b)) = 1 −pscl(f−1(c)) ≤ 1 − f−1(ps-cl(c)) = f−1(1 − ps-cl(c)) = f−1(ps-int(1 − c)) = 124 chettri, gurung and katwal f−1(ps-int(b)). hence, ps-int(f−1(b)) ≤ f−1(ps-int(b)). (d) ⇒ (b) let a be any fuzzy set on x and let b = f(a). then we have psint(a) ≤ ps-int(f−1f(a)) = ps-int(f−1(b)) ≤ f−1(ps-int(b)). so, f(ps-int(a)) ≤ f(f−1(ps-int(b))) ≤ ps-int(b) = ps-int(f(a)). hence, f(ps-int(a)) ≤ ps-int(f(a)). corollary 2.1. if f : (x,τ1) → (y,τ2) is a ps-ro fuzzy open and ps-ro fuzzy continuous then f−1(ps-cl(b)) = ps-cl(f−1(b)), for each fuzzy set b on y . proof: straightforward and hence omitted. theorem 2.3. let f be a function from a fts (x,τ1) to a fts (y,τ2). then f is ps-ro fuzzy closed (open) iff for each fuzzy set a on y and for any ps-ro open (closed) fuzzy set b on x such that f−1(a) ≤ b, there is a ps-ro open(closed) fuzzy set c on y such that a ≤ c and f−1(c) ≤ b. proof: let f be ps-ro fuzzy closed(open). let a be any fuzzy set on y and let b be a ps-ro open(closed) fuzzy set on x such that f−1(a) ≤ b. let c = 1 − f(1 − b). then c is a ps-ro open(closed) fuzzy set on x, since f is ps-ro fuzzy closed(open) and 1 − b is ps-ro closed(open) fuzzy set on x, f(1 − b) is ps-ro closed(open) fuzzy set on y . hence, 1 −b ≤ 1 −f−1(a) = f−1(1 −a). so, f(1 − b) ≤ f(f−1(1 − a)) ≤ 1 − a. hence, a ≤ 1 − f(1 − b) = c. further, f−1(c) = f−1(1 −f(1 −b)) = 1 −f−1(f(1 −b)) ≤ 1 − (1 −b) = b. conversely, let f satisfies the given condition. let b be a ps-ro closed(open) fuzzy set on x. then, a = 1 − b is ps-ro open(closed) fuzzy set on x. so, f−1(1 − f(b)) = 1−f−1(f(b)) ≤ 1−b = a. by hypothesis, there is a ps-ro open(closed) fuzzy set c on y such that 1 −f(b) ≤ c and f−1(c) ≤ a = 1 −b. hence, 1 −c ≤ f(b). also, b ≤ 1 −f−1(c) = f−1(1 −c). so, f(b) ≤ f(f−1(1 −c)) ≤ 1 −c. thus, we have f(b) = 1 −c, which is a ps-ro closed(open) fuzzy set on y . hence, f is ps-ro fuzzy closed(open). theorem 2.4. let f be a function from a fts (x,τ1) to a fts (y,τ2). then f is ps-ro fuzzy closed iff for each fuzzy set a on x, ps-cl(f(a)) ≤ f(ps-cl(a)). proof: let f be ps-ro fuzzy closed and a be any fuzzy set on x. since ps-cl(a) is ps-ro closed fuzzy set on x and f is ps-ro fuzzy closed, f(ps-cl(a)) is ps-ro closed fuzzy set on y . as, a ≤ ps-cl(a), f(a) ≤ f(ps-cl(a)). so, ps-cl(f(a)) ≤ pscl(f(ps-cl(a))) = f(ps-cl(a)). conversely, let a be any ps-ro closed fuzzy set on x. then f(a) = f(ps-cl(a)) ≥ ps-cl(f(a)). as, f(a) ≤ ps-cl(f(a)), f(a) = pscl(f(a)), i.e. f(a) is ps-ro closed fuzzy set on y . hence, f is ps-ro fuzzy closed. theorem 2.5. for a bijective function f from a fts (x,τ1) to a fts (y,τ2), the following are equivalent. (a)f−1 : y → x is ps-ro fuzzy continuous. (b) f is ps-ro fuzzy open. (c) f is ps-ro fuzzy closed. proof: (a) ⇒ (b) let f−1 be ps-ro fuzzy continuous. let u be a ps-ro open fuzzy set on x. since, f−1 is ps-ro fuzzy continuous, (f−1)−1(u) = f(u) is ps-ro open fuzzy set on y . hence, f is ps-ro fuzzy open. (b) ⇒ (c) let f be bijective and ps-ro fuzzy open. let v be a ps-ro closed fuzzy set on x. then, 1 −v = a is ps-ro open fuzzy set on x. since f is ps-ro fuzzy open and bijective, f(a) = f(1−v ) = 1−f(v ) is ps-ro open fuzzy set on y . therefore, f(v ) is ps-ro closed fuzzy set on y . hence, f is ps-ro fuzzy closed. (c) ⇒ (a) let f be ps-ro fuzzy closed and bijective. let v be a ps-ro closed fuzzy ps-ro fuzzy open(closed) functions 125 set on x. then f(v ) is ps-ro closed fuzzy set on y . but f(v ) = (f−1)−1(v ) and hence f−1 is ps-ro fuzzy continuous. 3. ps-ro fuzzy semi-homeomorphism definition 3.1. let (x,τ1) and (y,τ2) be two fts and f : (x,τ1) → (y,τ2). then f is said to be (i) ps-ro fuzzy pre semiopen function if f(a) is ps-ro semiopen fuzzy set on y , for each ps-ro semiopen fuzzy set a on x. (ii) ps-ro fuzzy homeomorphism if f is bijective, ps-ro fuzzy continuous and ps-ro fuzzy open function. (iii) ps-ro fuzzy semi-homeomorphism if f is bijective, ps-ro fuzzy pre semiopen and ps-ro fuzzy irresolute. theorem 3.1. let (x,τ1) and (y,τ2) be two fts. if f : (x,τ1) → (y,τ2) is ps-ro fuzzy continuous and ps-ro fuzzy open, then f is ps-ro fuzzy irresolute. proof: let f be ps-ro fuzzy continuous and ps-ro fuzzy open function. let u be a ps-ro semiopen fuzzy set on y . then ∃ ps-ro open fuzzy set v on y such that v ≤ u ≤ ps-cl(v ). now, f−1(v ) is ps-ro open fuzzy on x. hence, f−1(v ) ≤ f−1(u) ≤ f−1(ps-cl(v )). f is ps-ro fuzzy open and v is fuzzy set on y , f−1(pscl(v )) ≤ ps-cl(f−1(v )). so, f−1(v ) ≤ f−1(u) ≤ ps-cl(f−1(v )). thus, f−1(u) is ps-ro semiopen fuzzy set on x and hence f is ps-ro fuzzy irresolute. theorem 3.2. let (x,τ1) and (y,τ2) be two fts. if f : (x,τ1) → (y,τ2) is ps-ro fuzzy continuous and ps-ro fuzzy open, then f is ps-ro fuzzy pre semiopen. proof: let f be ps-ro fuzzy continuous and ps-ro fuzzy open function. let a be a ps-ro semiopen fuzzy set on x. then ∃ ps-ro open fuzzy set v on x such that v ≤ a ≤ ps-cl(v ). now, since f is ps-ro fuzzy continuous and v is a fuzzy set on x, f(ps-cl(v )) ≤ ps-cl(f(v )). hence f(v ) ≤ f(a) ≤ f(ps-cl(v )) ≤ ps-cl(f(v )). also, f(v ) is ps-ro open fuzzy set on y . so, f(a) is ps-ro semiopen fuzzy set on y . thus, f is ps-ro fuzzy pre semiopen function. remark 3.1. from theorem( 3.1) and ( 3.2) it follows that ps-ro fuzzy homeomorphism implies ps-ro fuzzy semi-homeomorphism. however, the converse is not true follows from the example below: example 3.1. let x = {a,b,c} and y = {x,y,z}. let a and b be two fuzzy sets on x defined by a(a) = 0.2,a(b) = 0.2,a(c) = 0.3 and b(x) = 0.2 ∀x ∈ x. let c,d and e be fuzzy set on y defined by c(x) = 0.2,c(y) = 0.3,c(z) = 0.3, d(x) = 0.4,d(y) = 0.4,d(z) = 0.5 and e(t) = 0.3 ∀t ∈ y . clearly, τ1 = {0, 1,a,b} and τ2 = {0, 1,c,d,e} are fuzzy topologies on x and y respectively. in the corresponding topological space (x,iα(τ1)), ∀α ∈ i1 = [0, 1), the open sets are φ,x,aα and bα, where aα =   x, for α < 0.2 {c}, for 0.2 ≤ α < 0.3 φ, for α ≥ 0.3 and bα = { x, for α < 0.2 φ, for α ≥ 0.2 for 0.2 ≤ α < 0.3, the closed sets on (x,iα(τ1)) are φ,x and x −{c}. therefore, int(cl(aα)) = x. so, aα is not regular open on (x,iα(τ1)) for 0.2 ≤ α < 0.3. thus, a is not pseudo regular open fuzzy set on (x,τ1). clearly, b α is regular 126 chettri, gurung and katwal open on (x,iα(τ1)), ∀α ∈ i1. hence b is pseudo regular open fuzzy set on (x,τ1). therefore, ps-ro fuzzy topology on x is {0, 1,b}. similarly, it can be seen that c and d are not pseudo regular open fuzzy set on (y,τ2) and thus, ps-ro fuzzy topology on y is {0, 1,e}. define a function f from the fts (x,τ1) to the fts (y,τ2) by f(a) = x,f(b) = y and f(c) = y. then, f −1(0) = 0,f−1(1) = 1 and f−1(e)(t) = 0.3 ∀t ∈ x. since f−1(e) is not ps-ro open fuzzy set on x, f is not ps-ro fuzzy continuous. now, ps-cl(e) = 1 − e where, (1 − e)(t) = 0.7 ∀t ∈ y . so, e ≤ d ≤ ps-cl(e). thus, d is ps-ro semiopen fuzzy set on y . we have, f−1(d)(t) = 0.4 ∀t ∈ x and ps-cl(b) = 1 −b where, (1 −b)(t) = 0.8 ∀t ∈ x. so, b ≤ f−1(d) ≤ ps-cl(b). so, f−1(d) is ps-ro semiopen fuzzy set on x. again, e is ps-ro open and hence ps-ro semiopen fuzzy set on y . f−1(e) is ps-ro semiopen fuzzy set on x, as b ≤ f−1(e) ≤ ps-cl(b). hence, f−1(u) is ps-ro semiopen fuzzy set on x, for every ps-ro semiopen fuzzy set u on y . thus, f is ps-ro fuzzy irresolute function. theorem 3.3. let (x,τ1), (y,τ2) and (z,τ3) be three fts and f : (x,τ1) → (y,τ2), g : (y,τ2) → (z,τ3). then the following statements are valid: (a) if f and g are ps-ro fuzzy pre semiopen functions then g ◦f is so. (b) if f is ps-ro fuzzy semiopen function and g is ps-ro fuzzy pre semiopen function then g ◦f is a ps-ro fuzzy semiopen function. proof:(a) let u be ps-ro semiopen fuzzy set on x. since, f and g are ps-ro fuzzy pre semiopen functions, f(u) and hence g(f(u)) are ps-ro semiopen fuzzy sets on y and z respectively. hence, (g ◦f)(u) = g(f(u)) is ps-ro semiopen fuzzy set on z for each ps-ro semiopen fuzzy set u on x. thus, g ◦f is ps-ro fuzzy semiopen function. (b) let u be a ps-ro open fuzzy set on x. since, f and g are both ps-ro fuzzy semiopen functions, g(f(u)) is ps-ro semiopen fuzzy set on z. thus, g◦f is ps-ro fuzzy semiopen function. theorem 3.4. let a function f from a fts (x,τ1) to a fts (y,τ2) be bijective. f is ps-ro fuzzy semi-homeomorphism iff f and f−1 are both ps-ro fuzzy irresolute functions and ps-ro fuzzy pre semiopen functions. proof: let f be ps-ro fuzzy semi-homeomorphism. now, since f is bijective, f−1 exist. let f−1 = g. as, f is ps-ro fuzzy irresolute, for each ps-ro semiopen fuzzy set a on y , f−1(a) is ps-ro semiopen fuzzy set on x. but, f−1 = g, so, g(a) is ps-ro semiopen fuzzy set on x, for each ps-ro semiopen fuzzy set a on y . thus, g is ps-ro fuzzy pre semiopen. again, f is ps-ro fuzzy pre semiopen. therefore, for each ps-ro semiopen fuzzy set b on x, f(b) is ps-ro semiopen fuzzy set on y . but, f−1 = g, so, f = g−1 and g−1(b) is ps-ro semiopen fuzzy set on y , for each ps-ro semiopen fuzzy set b on x. hence, g is ps-ro fuzzy irresolute. conversely, straightforward. theorem 3.5. a bijective function f from a fts (x,τ1) to a fts (y,τ2) is psro fuzzy semi-homeomorphism iff for each fuzzy set a on x, f(ps-scl(a)) = psscl(f(a)). proof: let f be ps-ro fuzzy semi-homeomorphism. then, f is ps-ro fuzzy irresolute. so, for each fuzzy set a on x, f(ps-scl(a)) ≤ ps-scl(f(a)). again, since f is ps-ro fuzzy semi-homeomorphism, f−1 is ps-ro fuzzy irresolute. as, ps-scl(a) is ps-ro semiclosed fuzzy set on x, (f−1)−1(ps-scl(a)) = f(ps-scl(a)) is ps-ro ps-ro fuzzy open(closed) functions 127 semiclosed fuzzy set on y . now, a ≤ ps-scl(a). so, f(a) ≤ f(ps-scl(a)), psscl(f(a)) ≤ f(ps-scl(a)). hence, f(ps-scl(a)) = ps-scl(f(a)). conversely, let f be bijective and f(ps-scl(a)) = ps-scl(f(a)), for each fuzzy set a on x. then, clearly f(ps-scl(a)) ≤ ps-scl(f(a)). hence, f is ps-ro fuzzy irresolute function. let a be any ps-ro semiclosed fuzzy set on x. then b = 1 −a is ps-ro semiopen fuzzy set on x. now, a = ps-scl(a). so, f(a) = f(ps-scl(a)) = ps-scl(f(a)). 1 − f(a) = 1 − ps-scl(f(a)) so, f(1 − a) = ps-sint(1 − f(a))(as f is bijective, f(1 − a) = 1 − f(a)). f(b) = ps-sint(f(1 − a)) = ps-sint(f(b)). this implies that f(b) is ps-ro semiopen fuzzy set on y . hence, f is ps-ro fuzzy pre semiopen function. therefore, f is ps-ro fuzzy semi-homeomorphism. corollary 3.1. let f : (x,τ1) → (y,τ2) be bijective. f is a ps-ro fuzzy semihomeomorphism iff for each fuzzy set b on y , f−1(ps-scl(b)) = ps-scl(f−1(b)). proof: since, f is a ps-ro fuzzy semi-homeomorphism, f−1 is also so. theorem 3.6. let a function f from a fts (x,τ1) to a fts (y,τ2) be bijective. f is ps-ro fuzzy semi-homeomorphism iff for each fuzzy set a on x, f(ps-sint(a)) = pssint(f(a)). proof: let f be ps-ro fuzzy semi-homeomorphism. then, f is bijective and both f and f−1 are ps-ro fuzzy irresolute. so, for each fuzzy set a on x, pssint(f(a)) ≤ f(ps-sint(a)). ps-sint(a) being ps-ro semiopen fuzzy set on x, (f−1)−1(ps-sint(a)) = f(ps-sint(a)) is ps-ro semiopen fuzzy set on y . now, ps-sint(a) ≤ a, f(ps-sint(a)) ≤ f(a). so, f(ps-sint(a)) ≤ ps-sint(f(a)). hence, f(ps-sint(a)) = ps-sint(f(a)). conversely, let f be bijective and f(ps-sint(a)) = ps-sint(f(a)), for each fuzzy set a on x. then, clearly ps-sint(f(a)) ≤ f(pssint(a)). also, f is bijective. hence, f is ps-ro fuzzy irresolute function. now, let b be a ps-ro semiopen fuzzy set on x. then, by given condition we have f(pssint(b)) = ps-sint(f(b)). so, f(b) = ps-sint(f(b)). this implies that f(b) is ps-ro semiopen fuzzy set on y . hence, f is ps-ro fuzzy pre semiopen function. therefore, f is ps-ro fuzzy semi-homeomorphism. corollary 3.2. let f : (x,τ1) → (y,τ2) be bijective. f is a ps-ro fuzzy semihomeomorphism iff for each fuzzy set b on y , f−1(ps-sint(b)) = ps-sint(f−1(b)). proof: since, f is a ps-ro fuzzy semi-homeomorphism, f−1 is also so. references [1] chang, c. l., fuzzy topological spaces, j. math. anal. appl.,24(1968), 182-190. [2] chettri, p., gurung, s. and halder, s., on ps-ro semiopen fuzzy set and ps-ro fuzzy semicontinuous, semiopen functions, tbilisi mathematical journal., 7(1)(2014) 87-97. [3] chettri, p., katwal, k. and gurung, s., on ps-ro fuzzy irresolute functions, pre print. [4] deb ray, a. and chettri, p., on pseudo δ-open fuzzy sets and pseudo fuzzy δ-continuous functions, international journal of contemporary mathematical sciences., 5(29)(2010), 14031411. [5] deb ray, a and chettri, p., fuzzy pseudo nearly compact spaces and ps-ro continuous functions, the journal of fuzzy mathematics., 19(3) (2011), 737-746. [6] deb ray, a. and chettri, p., further on fuzzy pseudo near compactness and ps-ro fuzzy continuous functions, pre print. [7] kohli, j. k. and prasannan, a. r., starplus-compactness and starplus-compact open fuzzy topologies on function spaces, j. math. anal. appl., 254(2001), 87-100. [8] lowen, r., fuzzy topological spaces and fuzzy compactness, j. math. anal. appl., 56(1976), 621-633. [9] malghan, s. r. and benchalli, s.s., open maps, closed maps and local compactness in fuzzy topological spaces, j. math. anal. appl., 99(1984), 338-349. 128 chettri, gurung and katwal [10] yalvac,t.h., semi-interior and semi-closure of a fuzzy set, j. math. anal. appl., 132(1988), 356-364. [11] zadeh, l. a., fuzzy sets, information and control., 8(1965), 338-353. department of mathematics, sikkim manipal institute of technology, majitar, rangpoo, sikkim, pin 737136, india ∗corresponding author international journal of analysis and applications volume 18, number 5 (2020), 859-875 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-859 new types of bipolar fuzzy ideals of bck-algebras g. muhiuddin1,∗, d. al-kadi2, a. mahboob3, k.p. shum4 1department of mathematics, university of tabuk, tabuk 71491, saudi arabia 2department of mathematics and statistic, taif university, taif 21974, saudi arabia 3department of mathematics, madanapalle institute of technology & science, madanapalle-517325, india 4institute of mathematics, yunnan university, kunming 650091, people’s republic of china ∗corresponding author: chishtygm@gmail.com abstract. the notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of bck-algebras are introduced, and related properties are investigated. characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of bck-algebras are given, and several properties are discussed. finally, we prove that if t is an implicative bck-algebra, then a fuzzy subset µ of t is a bipolar fuzzy ideal of t if and only if it is a bipolar fuzzy implicative ideal of t. 1. introduction bck-algebras and bci-algebras are two classes of non-classical logic algebras which were introduced by y. imai and k. iseki in 1966 (see [9, 10]). they are algebraic formulation of bck-system and bci-system in combinatory logic. fuzzy sets, which were introduced by zadeh [29], deal with possibilistic uncertainty, connected with imprecision of states, perceptions and preferences. after the introduction of fuzzy sets by zadeh, fuzzy set theory has become an active area of research in various fields. these are widely scattered over many received june 1st, 2020; accepted july 1st, 2020; published july 29th, 2020. 2010 mathematics subject classification. 06d72, 06f35, 16d25, 94d05. key words and phrases. bck-algebras; fuzzy ideal; bipolar fuzzy ideal; bipolar fuzzy closed ideal; bipolar fuzzy positive implicative ideal; bipolar fuzzy implicative ideal. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 859 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-859 int. j. anal. appl. 18 (5) (2020) 860 disciplines such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others. the elements in fuzzy sets have degrees of belonging which range between 0 and 1. if the membership degree of an element is 0 then the element does not belong to the fuzzy set and it completely belongs to the correspondent fuzzy set if the membership is 1. if the membership lies between (0, 1) then it belongs partially to the fuzzy set. among such elements some have irrelevant characteristics to the property corresponding to a fuzzy set and the others have contrary characteristics to the property and the traditional fuzzy set representation cannot be described it. moreover, muhiuddin et al. studied the fuzzy set theoretical approach to the bck/bci-algebras on various aspects (see for e.g., [22–25]). also, some related concepts of fuzzy sets in different algebras have been studied in [1, 3, 4, 6, 26–28]. as a generalization of traditional fuzzy sets, zhang [30] first introduced the bipolar fuzzy sets concept . also, lee [16, 17] studied the bipolar fuzzy sets in which a negative (resp. positive) membership degree is given for each element in the fuzzy set that ranges over the interval [−1, 0] (resp. [0, 1]). later on, a number of research papers have been devoted to the study of bipolar fuzzy set theory in several algebraic structures (see for e.g., [7, 12, 14, 15, 20, 21]. in this paper, we introduce the notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of bck-algebras and investigate related properties. we give characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of bck-algebras and several properties are discussed. finally, we prove that if t is an implicative bck-algebra, then a fuzzy subset µ of t is a bipolar fuzzy ideal of t if and only if it is a bipolar fuzzy implicative ideal of t. 2. preliminaries we review some definitions and properties that will be useful in our results. definition 2.1. an algebra (t ;∗, 0) of kind (2, 0) is called a bck-algebra if it satisfies the following conditions: (k1) ((t∗u) ∗ (t∗v)) ∗ (v ∗u) = 0, (k2) (t∗ (t∗u)) ∗u = 0, (k3) t∗ t = 0, (k4) 0 ∗ t = 0, (k5) t∗u = 0 and u∗ t = 0 ⇒ t = u, for all t,u,v ∈ t in a bck-algebra, the following are true. (k6) t∗ 0 = t, int. j. anal. appl. 18 (5) (2020) 861 (k7) (t∗u) ∗v = (t∗v) ∗u. a nonempty subset x of a bck-algebra t is called an ideal of t if it satisfies (i1) 0 ∈ x, (i2) ∀t,u ∈ t,t∗u ∈ x,u ∈ x ⇒ t ∈ x. a nonempty subset x of a bck-algebra t is called an implicative ideal of t if it satisfies (i1) and (i3) ∀t,u,v ∈ t, ((t∗ (u∗ t)) ∗v ∈ x,v ∈ x ⇒ t ∈ x. a fuzzy set µ in t is called a fuzzy ideal of t if it satisfies (f1) µ(0) ≥ µ(t), (f2) µ(t) ≥ µ(t∗u) ∧µ(u). a fuzzy positive implicative ideal of t is a fuzzy set µ in t which satisfies (f1) and (f3) µ(t∗v) ≥ µ((t∗u) ∗v) ∧µ(u∗v). a fuzzy implicative ideal of t is a fuzzy set µ in t which satisfies (f1) and (f4) µ(t) ≥ µ((t∗ (u∗ t)) ∗v) ∧µ(v). for more information regarding bck-algebras, we refer the reader to [19] lemma 2.1. [15] in a bck-algebra t , every bipolar fuzzy ideal of t is a bipolar fuzzy subalgebra of t . 3. bipolar fuzzy ideal in the following sections, t denotes a bck-algebra unless otherwise specified. for any family {γi | i ∈ γ} of real numbers, we define ∨{γi | i ∈ γ} :=   max{γi | i ∈ γ} if γ is finite, sup{γi | i ∈ γ} otherwise, ∧{γi | i ∈ γ} :=   min{γi | i ∈ γ} if γ is finite, inf{γi | i ∈ γ} otherwise. moreover, if γ = {1, 2, ...,n}, then ∨{γi | i ∈ γ} and ∧{γi | i ∈ γ} are denoted by γ1 ∨ γ2 ∨ ... ∨ γn and γ1 ∧γ2 ∧ ...∧γn, respectively. for a bipolar fuzzy set µ = (t ; µn,µp) and (α,β) ∈ [−1, 0) × (0, 1], we define (n) n(µ; α) := {t ∈ t : µn(t) ≤ α} is called the negative α− cut of µ. (p) p(µ; β) := {t ∈ t : µp(t) ≥ β} is called positive β − cut of µ. int. j. anal. appl. 18 (5) (2020) 862 the set c(µ; (α,β)) := n(µ; α) ∩p(µ; β) is called the (α,β) − cut of µ = (t ; µn,µp). for every k ∈ (0, 1), if (α,β) = (−k,k), then the set c(µ; k) := n(µ;−k) ∩p(µ; k) is called the k − cut of µ = (t ; µn,µp). definition 3.1. [15] a bipolar fuzzy set µ = (t ; µn,µp) in a bck-algebra t is called a bipolar fuzzy ideal of t if it satisfies the following assertions: (bf1) (∀t ∈ t) (µn(0) ≤ µn(t), µp(0) ≥ µp(t)), (bf2) (∀t,u ∈ t) (µn(t) ≤ µn(t∗u) ∨µn(u), µp(t) ≥ µp(t∗u) ∧µp(u)). example 3.1. consider the bck-algebra (t ;∗, 0) given in table 1. ∗ 0 c d e 0 0 0 0 0 c c 0 0 c d d c 0 d e e e e 0 table 1: cayley table define a bipolar fuzzy set µ = (t ; µn,µp) in t by ∗ 0 c d e µn -0.2 -0.3 -0.3 -0.8 µp 0 0.5 0.5 1 then by routine calculations, µ = (t ; µn,µp) is a bipolar fuzzy ideal of t . int. j. anal. appl. 18 (5) (2020) 863 example 3.2. consider the bck-algebra (t ;∗, 0) given in table 2. ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 0 0 d d d d 0 0 e e d e a 0 table 2: cayley table define µ by ∗ 0 a c d e µn -0.7 -0.6 -0.7 -0.6 -.06 µp 0.6 0.1 0.6 0.1 0.1 then by routine calculations, µ = (t ; µn,µp) is a bipolar fuzzy ideal of t . theorem 3.1. if µ = (t ; µn,µp) is a bipolar fuzzy ideal in t and i(0) = {t ∈ t : µn(t) = µn(0),µp(t) = µp(0)}. then i(0) is an ideal of t . proof. let t,u ∈ t be such that t ∗ u ∈ i(0) and u ∈ i(0). then we have µn(t ∗ u) = µn(0),µp(t ∗ u) = µp(0),µn(u) = µn(0) and µp(u) = µp(0). thus using definition 3.1 (bf2), we get µn(t) ≤ µn(0) ∨µn(0) = µn(0) and µp(t) ≥ µn(0) ∧ µn(0) = µn(0). on the other hand, we know from definition 3.1 (bf1) that µn(0) ≤ µn(t), µp(0) ≥ µp(t) and so µn(t) = µn(0), µp(t) = µp(0). hence, t ∈ i(0). it is obvious that 0 ∈ i(0). therefore, i(0) is an ideal of t . � if µ = (t ; µn,µp) is a bipolar fuzzy set in t which satisfies definition 3.1 (bf1) then the following example shows that definition 3.1 (bf2) is sufficient for i(0) to be an ideal of t. example 3.3. consider the bck-algebra (t ;∗, 0) given in table 3. ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 c 0 d d d d 0 d e e e e e 0 table 3: cayley table int. j. anal. appl. 18 (5) (2020) 864 define µ by ∗ 0 a c d e µn -0.7 -0.7 -0.5 -0.3 -.07 µp 0.8 0.8 0.4 0.2 0.8 note that µ = (t ; µn,µp) is a bipolar fuzzy set in t which satisfies definition 3.1 (bf1) and not (bf2) as there exist c,e ∈ t where µn(c) = −0.5 6≤−0.7 = µn(c∗e) ∨µn(e). then i(0) = {0,a,e} is not an ideal of t as c∗e = 0 ∈ i(0) and e ∈ i(0), but c 6∈ i(0). theorem 3.2. let µ = (t ; µn,µp) be a bipolar fuzzy set in t and let ∅ 6= i ⊆ t such that 0 ∈ i. define µn(t) = α1 or α2 whenever t ∈ i or t 6∈ i respectively, and µp(t) = β1 or β2 whenever t ∈ i or t 6∈ i respectively where α1,α2 ∈ [−1, 0] such that α1 ≤ α2 and β1,β2 ∈ [0, 1] such that β1 ≥ β2. then definition 3.1 (bf2) and the implication (∀t,u ∈ t) (t∗u ∈ i and u ∈ i ⇒ t ∈ i) are equivalent. proof. let t ∗ u ∈ i and u ∈ i. then µn(t ∗ u) = µn(u) = α1, and µp(t ∗ u) = µp(u) = β1. therefore, µn(t) ≤ µn(t ∗ u) ∨ µn(u) = α1 ∨ α1 = α1 and so µn(t) = α1 as α1 ≤ α2. similarly, µp(t) ≥ β1 and so µp(t) = β1 as β1 ≥ β2. so whenever t∗u ∈ i and u ∈ i we have t ∈ i. conversely, if the implication is valid then for t,u ∈ t we have µn(t∗u) = µn(u) = µn(t) = α1, µp(t∗u) = µp(u) = µp(t) = β1 . then it is obvious to see that definition 3.1 (bf2) is satisfied. � having 0 ∈ i in theorem 6.1 and as (∀t ∈ t) (µn(0) = µn(t ∗ t) ≤ µn(t), µp(0) = µp(t ∗ t) ≥ µp(t)). thus (∀t ∈ t) (µn(0) ≤ µn(t), µp(0) ≥ µp(t)). that is definition 3.1 (bf1) is satisfied. hence, we have the following theorem. theorem 3.3. let µ = (t ; µn,µp) be a bipolar fuzzy set in t and let ∅ 6= i ⊆ t such that 0 ∈ i. define µn(t) = α1 or α2 whenever t ∈ i or t 6∈ i respectively. define µp(t) = β1 or β2 whenever t ∈ i or t 6∈ i respectively where α1,α2 ∈ [−1, 0] such that α1 ≤ α2 and β1,β2 ∈ [0, 1] such that β1 ≥ β2. then µ = (t; µn,µp) is a bipolar fuzzy ideal of t if and only if i is an ideal of t . theorem 3.4. [15] a bipolar fuzzy set µ = (t ; µn,µp) in t is a bipolar fuzzy ideal of t if both the nonempty negative α-cut and the nonempty positive β-cut of µ = (t ; µn,µp) are ideals of t for all (α,β) ∈ [−1, 0) × (0, 1]. corollary 3.1. if µ = (t ; µn,µp) in t is a bipolar fuzzy ideal of t , then the k-cut of µ = (t ; µn,µp) is an ideal of t for all k ∈ (0, 1). int. j. anal. appl. 18 (5) (2020) 865 the following example shows the converse of corollary 3.1 may not be true. example 3.4. let t = {0, 1, 2, 3, 4} be a set in which the operation ∗ is defined by the following cayley table which is given in table 4. ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 1 0 0 2 2 2 0 2 0 3 3 3 3 0 0 4 4 4 3 2 0 table 4: cayley table then (t ;∗, 0) is a bck-algebra. let µ = (t; µn,µp) be a bipolar fuzzy set in t given by ∗ 0 1 2 3 4 µn -0.7 -0.4 -0.6 -0.4 -0.2 µp 0.8 0.2 0.6 0.3 0.4 by routine calculations, we know that c(µ; 0.4) := n(µ;−0.4) ∩p(µ; 0.4) = {0, 2}. is an ideal of t , but µ = (t; µn,µp) is not an bipolar fuzzy ideal of t because µn(4) = −0.2 6≤−0.4 = µn(4 ∗ 2) ∨µn(2). lemma 3.1. [15] every bipolar fuzzy ideal µ = (t ; µn,µp) of t satisfies the following implication. (∀t,u ∈ t)(t ≤ u ⇒ µn(t) ≤ µn(u),µp(t) ≥ µp(u)). proposition 3.1. [15] a bipolar fuzzy set µ = (t ; µn,µp) in t is a bipolar fuzzy ideal of t if and only if for all t,u,v ∈ t , (t∗u) ∗v = 0 implies µn(t) ≤ µn(u) ∨µn(v) and µp(t) ≥ µp(u) ∨µp(v). 4. bipolar fuzzy closed ideal definition 4.1. a bipolar fuzzy ideal µ = (t ; µn,µp) in t is said to a bipolar fuzzy closed ideal if µn(0∗t) ≤ µn(t) and µp(0 ∗ t) ≥ µp(t) for all t ∈ t . example 4.1. consider the bci-algebra (t ;∗, 0) given in table 5. int. j. anal. appl. 18 (5) (2020) 866 ∗ 0 a c d e 0 0 0 c d e a a 0 c d e c c c 0 e d d d d e 0 c e e e d c 0 table 5: cayley table define µ by ∗ 0 a c d e µn -0.8 -0.6 -0.8 -0.4 -.03 µp 0.6 0.6 0.4 0.3 0.1 then µ = (t ; µn,µp) is a bipolar fuzzy closed ideal of t . as an extension to theorem 3.3 we give the following theorem. theorem 4.1. let µ = (t ; µn,µp) be a bipolar fuzzy set in t and let ∅ 6= i ⊆ t such that 0 ∈ i. define µn(t),µp(t) as defined in theorem 3.3. then µ = (t ; µn,µp) is a bipolar fuzzy closed ideal of t if and only if i is a closed ideal of t . proof. assume µ = (t; µn,µp) is a bipolar fuzzy closed ideal of t and we show that 0 ∗ t ∈ i, ∀t ∈ i. for t ∈ i, we have µn(t) = α1,µp(t) = β1. as 0 ∗ t ≤ 0 then µn(0 ∗ t) ≤ µn(0), µp(0 ∗ t) ≥ µp(0). therefore, µn(0 ∗ t) ≤ µn(0) = α1 and so µn(0 ∗ t) = α1 as α1 ≤ α2. also µn(0 ∗ t) = β1 as β1 ≥ β2. that is, i is a closed ideal. for the converse, let i be a closed ideal of t and show that µn(0∗t) ≤ µn(t), µp(0∗t) ≥ µp(t). for t ∈ t we know that 0 ∗ t ≤ 0 and so µn(0 ∗ t) ≤ µn(0) ≤ µn(t). also µp(0 ∗ t) ≥ µp(0) ≥ µp(t). this proves that µ = (t; µn,µp) is a bipolar fuzzy closed ideal of t. � theorem 4.2. let µ = (t ; µn,µp) be a bipolar fuzzy set in t . if one of the following assertion is satisfied (1) (∀t,u ∈ t) (µn(t) ≤ µn(t∗u) ∨µn(u),µp(t) ≥ µp(t∗u) ∧µp(u)) (2) (∀t,u ∈ t) (µn(t) ≤ µn(u∗ t) ∨µn(u),µp(t) ≥ µp(u∗ t) ∧µp(u)) then µ = (t ; µn,µp) is a bipolar fuzzy closed ideal of t . proof. we will show that using any of the assertions above we get µn(0 ∗ t) ≤ µn(t), µp(0 ∗ t) ≥ µp(t). using (1), let u = 0 then we have µn(0 ∗ t) ≤ µn((0 ∗ t) ∗ 0) ∨ µn(0) = µn(0). as 0 ∗ t = 0 we know that 0 ≤ t and so µn(0) ≤ µn(t). therefore, µn(0 ∗ t) ≤ µn(t). also knowing that µp(0) ≥ µp(t) and µp(0 ∗ t) ≥ µp((0 ∗ t) ∗ 0) ∧µp(0) = µp(0) we get µp(0 ∗ t) ≥ µp(t). use the same approach with assertion (2) and let u = 0 to have µn(0∗ t) ≤ µn(0∗(0∗ t))∨µn(0) = µn(0) ≤ µn(t). therefore, µn(0 ∗ t) ≤ µn(t), µp(0 ∗ t) ≥ µp(t). � int. j. anal. appl. 18 (5) (2020) 867 5. bipolar fuzzy positive implicative ideal definition 5.1. a bipolar fuzzy set µ = (t ; µn,µp) in t is called bipolar fuzzy positive implicative ideal of t if it satisfies (bf1) and the following assertions: (bf3) (∀t,u,v ∈ t) µn(t∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) (bf4) (∀t,u,v ∈ t) µp(t∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v). example 5.1. let t = {0, 1, 2, 3} be a set in which the operation ∗ is given by the following cayley table in table 6. ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 1 0 2 3 3 3 3 0 table 6: cayley table let µ = (t ; µn,µp) be a bipolar fuzzy set in t defined as follows ∗ 0 1 2 3 µn -0.6 -0.6 -0.6 -0.4 µp 0.8 0.8 0.8 0.6 then by routine calculations µ = (t ; µn,µp) is a bipolar fuzzy positive implicative ideal of t . note that every bipolar fuzzy positive implicative ideal is a bipolar fuzzy ideal, but converse is not true. example 5.2. consider the bck-algebra (t,∗, 0) given in example 5.2. let µ = (t ; µn,µp) be a bipolar fuzzy set in t given by: ∗ 0 1 2 3 µn -0.6 -0.5 -0.5 -0.4 µp 0.8 0.7 0.7 0.6 by direct calculations we know that µ = (t ; µn,µp) is a bipolar fuzzy ideal of t but is not a bipolar fuzzy positive implicative ideal as µn(2 ∗ 1) = µ(1) = −0.5 6≤−0.6 = µn((2 ∗ 1) ∗ 1) ∨µn(1 ∗ 1). we provide a condition for a bipolar fuzzy ideal to be a bipolar fuzzy positive implicative ideal. int. j. anal. appl. 18 (5) (2020) 868 theorem 5.1. if µ = (t ; µn,µp) is a bipolar fuzzy ideal of t satisfying for all t,u,v ∈ t (bf5) µn(t∗v) ≤ µn(((t∗u) ∗u) ∗v) ∨µn(v), (bf6) µp(t∗v) ≥ µp(((t∗u) ∗u) ∗v) ∧µp(v), then µ is a bipolar fuzzy positive implicative ideal of t . proof. using (k1) and (k7), we have ((t∗v) ∗v) ∗ (u∗v) ≤ (t∗v) ∗u = (t∗u) ∗v, ∀t,u,v ∈ t. since µ is order reversing, it follows from (bf3) and (bf4) that µn(t∗v) ≤ µn(((t∗v) ∗v) ∗ (u∗v)) ∨µn(u∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) and µp(t∗v) ≥ µp(((t∗v) ∗v) ∗ (u∗v)) ∧µp(u∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v) for all t,u,v ∈ t . hence, µ is a bipolar fuzzy positive implicative ideal of t . � theorem 5.2. let w ∈ t . if µ = (t; µn,µp) is a bipolar fuzzy positive implicative ideal of t , then i(w) is a positive implicative ideal of t . proof. recall that 0 ∈ i(w). let t,u ∈ t be such that (t ∗ u) ∗ v ∈ i(w) and u ∗ v ∈ i(w). then µn(w) ≥ µn((t∗u)∗v), µp(w) ≤ µp((t∗u)∗v), µn(w) ≥ µn(u∗v) and µp(w) ≤ µp(u∗v). since µ = (t; µn,µp) is a bipolar fuzzy positive implicative ideal of t , it follows from (bf3) and (bf4) that µn(t∗v) ≤ µn((t∗u) ∗v) ∨µn(u∗v) ≤ µn(w) and µp(t∗v) ≥ µp((t∗u) ∗v) ∧µp(u∗v) ≥ µp(w) so that t∗v ∈ i(w). therefore i(w) is a positive implicative ideal of t . � lemma 5.1. let µ = (t ; µn,µp) be a bipolar fuzzy ideal of t . then • µ is a bipolar fuzzy positive implicative ideal of t if and only if it satisfies (bf7) ∀t,u ∈ t,µn(t∗u) ≤ µn((t∗u) ∗u) and µp(t∗u) ≥ µp((t∗u) ∗u). • µ is a bipolar fuzzy positive implicative ideal of t if and only if it satisfies (bf8) ∀t,u,v ∈ t,µn((t∗v) ∗ (u∗v)) ≤ µn((t∗u) ∗v) and µp((t∗v) ∗ (u∗v)) ≥ µp((t∗u) ∗v). theorem 5.3. if µ is a bipolar fuzzy positive implicative ideal of t , then (bf9) ∀t,u,a,b ∈ t, ((t∗u) ∗u) ∗a ≤ b ⇒ µn(t∗u) ≤ µn(a) ∨µn(b) and µp(t∗u) ≥ µp(a) ∧µp(b). (bf10) ∀t,u,v,a,b ∈ t, ((t∗u) ∗v) ∗a ≤ b ⇒ µn((t∗v) ∗ (u∗v)) ≤ µn(a) ∨µn(b) int. j. anal. appl. 18 (5) (2020) 869 and µp((t∗v) ∗ (u∗v)) ≥ µp(a) ∧µp(b). proof. let t,u,a,b ∈ t be such that ((t∗u) ∗u) ∗a ≤ b. using proposition 3.1, we have µn((t∗u) ∗u) ≤ µn(a) ∨µn(b) and µp((t∗u) ∗u) ≥ µp(a) ∧µp(b). it follows that from (bf3), (bf4), (k3) and (bf1), µn(t∗u) ≤ µn((t∗u) ∗u) ∨µn(u∗u) = µn((t∗u) ∗u) ∨µn(0) = µn((t∗u) ∗u) µn(t∗u) ≤ µn(a) ∨µn(b) and µp(t∗u) ≥ µp((t∗u) ∗u) ∧µp(u∗u) = µp((t∗u) ∗u) ∧µp(0) = µp((t∗u) ∗u) µp(t∗u) ≥ µp(a) ∧µp(b). now let t,u,v,a,b ∈ t be such that ((t∗u) ∗v) ∗a ≤ b, that is, (((t∗u) ∗v) ∗a) ∗ b = 0. since µ is a bipolar fuzzy positive implicative ideal of t , it follows from proposition 3.1 and lemma 5.1 that µn((t∗v) ∗ (u∗v)) ≤ µn((t∗u) ∗v) ≤ µn(a) ∨µn(b) and µp((t∗v) ∗ (u∗v)) ≥ µp((t∗u) ∗v) ≥ µp(a) ∧µp(b). this completes the proof. � we now give conditions for a bipolar fuzzy set to be a bipolar fuzzy positive implicative ideal. theorem 5.4. let µ be a bipolar fuzzy set in t satisfying the condition (bf9). then µ is a bipolar fuzzy positive implicative ideal of t . proof. we first prove that µ is a bipolar fuzzy ideal of t. let t,u,v ∈ t be such that t∗u ≤ v. then (((t∗ 0) ∗ 0) ∗u) ∗v = (t∗u) ∗v = 0, that is, ((t∗ 0) ∗ 0) ∗u ≤ v, which implies from (k6) and (bf9) that µn(t) = µn(t∗ 0) ≤ µn(u) ∨µn(v) and µp(t) = µp(t∗ 0) ≥ µp(u) ∧µp(v). therefore, by proposition 3.1, we know that µ is a bipolar fuzzy ideal of t . note that (((t∗u) ∗u) ∗ ((t∗ u) ∗u)) ∗ 0 = 0 for all t,u ∈ t . using (bf9) and (bf1), we have µn(t∗u) ≤ µn((t∗u) ∗u) ∨µn(0) = µn((t∗u) ∗u) int. j. anal. appl. 18 (5) (2020) 870 and µp(t∗u) ≥ µp((t∗u) ∗u) ∧µp(0) = µp((t∗u) ∗u). so µ is a bipolar fuzzy positive implicative ideal of t by lemma 5.1. � theorem 5.5. let µ be a fuzzy set in t satisfying the condition (bf10). then µ is a bipolar fuzzy positive implicative ideal of t . proof. let t,u,a,b ∈ t be such that ((t∗u) ∗u) ∗a = b, that is (((t∗u) ∗u) ∗a) ∗ b = 0, which implies from (k6), (k3) and (bf10) that µn(t∗u) = µn((t∗u) ∗ 0) = µn((t∗u) ∗ (u∗u)) ≤ µn(a) ∨µn(b) and µp(t∗u) = µp((t∗u) ∗ 0) = µp((t∗u) ∗ (u∗u)) ≥ µn(a) ∧µp(b). so µ is a bipolar fuzzy positive implicative ideal of t by theorem 5.4. � theorem 5.6. let µ and λ be bipolar fuzzy ideals of t such that µn(0) = λn(0),µp(0) = λp(0) and µ ⊆ λ, that is µn(t) ≤ λn(t) and µp(t) ≥ λp(t) for all t ∈ t . if µ is bipolar fuzzy positive implicative ideal of t , then so is λ. proof. assume that µ is a bipolar fuzzy positive implicative ideal of t. for any t,u,v ∈ t, we have λn(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) = λn(((t∗v) ∗ ((t∗u) ∗v)) ∗ (u∗v)) [by (k7)] = λn(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [by (k7)] ≥ µn(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [since µ ⊆ λ] ≥ µn(((t∗ ((t∗u) ∗v)) ∗u) ∗v) = µn(((t∗u) ∗ ((t∗u) ∗v)) ∗v) [by (k7)] = µn(((t∗u) ∗v) ∗ ((t∗u) ∗v)) [by (k7)] = µn(0) = λn(0). [by (k3) and assumption] it follows from (bf1) and (bf2) that λn((t∗v) ∗ (u∗v)) ≤ λn(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) ∨λn((t∗u) ∗v) ≤ λn(0) ∨λn((t∗u) ∗v) = λn((t∗u) ∗v) and λp(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) = λp(((t∗v) ∗ ((t∗u) ∗v)) ∗ (u∗v)) [by (k7)] = λp(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [by (k7)] int. j. anal. appl. 18 (5) (2020) 871 ≤ µp(((t∗ ((t∗u) ∗v)) ∗v) ∗ (u∗v)) [since µ ⊆ λ] ≤ µp(((t∗ ((t∗u) ∗v)) ∗u) ∗v) = µp(((t∗u) ∗ ((t∗u) ∗v)) ∗v) [by (k7)] = µp(((t∗u) ∗v) ∗ ((t∗u) ∗v)) [by (k7)] = µp(0) = λp(0). [by (k3) and assumption] it follows from (bf1) and (bf2) that λp((t∗v) ∗ (u∗v)) ≥ λp(((t∗v) ∗ (u∗v)) ∗ ((t∗u) ∗v)) ∧λp((t∗u) ∗v) ≥ λp(0) ∧λp((t∗u) ∗v) = λp((t∗u) ∗v) for all t,u,v ∈ t . hence, by lemma 5.1, λ is a bipolar fuzzy positive implicative ideal of t . � 6. bipolar fuzzy implicative ideal definition 6.1. a bipolar fuzzy set µ = (t ; µn,µp) in t is called bipolar fuzzy implicative ideal of t if it satisfies (bf1) and the following assertions: (bf11) (∀t,u,v ∈ t) µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) (bf12) (∀t,u,v ∈ t) µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v). example 6.1. let t = {0, 1, 2, 3} be a set in which the operation ∗ is defined by table 7: ∗ 0 1 2 3 0 0 0 0 0 1 1 0 0 1 2 2 1 0 2 3 3 3 3 0 table 7: cayley table hence, (t;∗, 0) is a bck-algebra. let µ = (t ; µn,µp) be a bipolar fuzzy set in t defined by 0 1 2 3 µn -0.5 -0.5 -0.5 -0.4 µp 0.7 0.7 0.7 0.6 by routine calculations, µ = (t; µn,µp) is a bipolar fuzzy implicative ideal of t . now, we give a relation between a bipolar fuzzy ideal and a bipolar fuzzy implicative ideal. theorem 6.1. every bipolar fuzzy implicative ideal of t is a bipolar fuzzy ideal of t . int. j. anal. appl. 18 (5) (2020) 872 proof. let µ be a bipolar fuzzy implicative ideal of t .then for all t,u,v ∈ t , µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) and µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v). putting u = t and v = u, µn(t) ≤ µn((t∗ (t∗ t)) ∗u) ∨µn(u) µn(t) ≤ µn((t∗ 0) ∗u) ∨µn(u) µn(t) ≤ µn(t∗u) ∨µn(u) and µp(t) ≥ µp((t∗ (t∗ t)) ∗u) ∧µp(u) µp(t) ≥ µp((t∗ 0) ∗u) ∧µp(u) µp(t) ≥ µp(t∗u) ∧µp(u). thus µ satisfies (bf2). consequently, µ is a bipolar fuzzy ideal of t by (bf1). � in view of lemma 2.1 and theorem 6.1, we conclude the following corollary. corollary 6.1. every bipolar fuzzy implicative ideal of t is a bipolar fuzzy subalgebra of t . the following example shows that the converse of theorem 6.1 is not true in general. example 6.2. let t = {0,a,c,d,e} be a set in which the operation ∗ is defined by table 8: ∗ 0 a c d e 0 0 0 0 0 0 a a 0 a 0 0 c c c 0 0 0 d d d d 0 0 e e d e a 0 table 8: cayley table hence, (t;∗, 0) is a bck-algebra. let µ = (t ; µn,µp) be a bipolar fuzzy set in t defined by ∗ 0 a c d e µn -0.8 -0.7 -0.8 -0.7 -.07 µp 0.7 0.2 0.7 0.2 0.2 by routine calculations, we know that µ = (t ; µn,µp) is a bipolar fuzzy ideal of t , but not a bipolar fuzzy implicative ideal of t since p(µ; 0.5) = {0, c} is not an implicative ideal of t . we provide a condition for bipolar fuzzy ideal to be a bipolar fuzzy implicative ideal. int. j. anal. appl. 18 (5) (2020) 873 theorem 6.2. if t is an implicative bck-algebra, then every bipolar fuzzy ideal of t is a bipolar fuzzy implicative ideal of t . proof. since t is an implicative bck-algebra, it follows that t = t ∗ (u ∗ t) for all t,u ∈ t . let µ be a bipolar fuzzy ideal of t . then by (bf2) µn(t) ≤ µn(t∗v) ∨µn(v) µn(t) ≤ µn((t∗ (u∗ t) ∗v)) ∨µn(v) and µp(t) ≥ µp(t∗v) ∧µp(v) µp(t) ≥ µp((t∗ (u∗ t) ∗v)) ∧µp(v). hence, µ is a bipolar fuzzy implicative ideal of t . the proof is complete. � in view of theorem 6.1 and theorem 6.2, we have the following theorem. theorem 6.3. if t is an implicative bck-algebra, then a fuzzy subset µ of t is a bipolar fuzzy ideal of t if and only if it is a bipolar fuzzy implicative ideal of t . theorem 6.4. let µ be a bipolar fuzzy implicative ideal of t . then the set tµ = {t ∈ t : µn(t) = µn(0) and µp(t) = µp(0)} is an implicative ideal of t . proof. clearly 0 ∈ t . let t,u,v ∈ tµ be such that (t∗ (u∗ t)) ∗v ∈ tµ and v ∈ tµ. then, we have µn((t∗ (u∗ t)) ∗v) = µn(v) = µn(0) and µp((t∗ (u∗ t)) ∗v) = µp(v) = µp(0). it follows that µn(t) ≤ µn((t∗ (u∗ t)) ∗v) ∨µn(v) ≤ µn(0) ∨µn(0) µn(t) ≤ µn(0) and µp(t) ≥ µp((t∗ (u∗ t)) ∗v) ∧µp(v) ≥ µp(0) ∨µp(0) µp(t) ≥ µp(0). by using (bf1), we get µn(t) = µn(0) and µp(t) = µp(0) and hence t ∈ tµ. consequently, tµ is an implicative ideal of t . � acknowledgement: the authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments that improved the overall presentation of this paper. int. j. anal. appl. 18 (5) (2020) 874 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d. al-kadi, g. muhiuddin, bipolar fuzzy bci-implicative ideals of bci-algebras, ann. commun. math. 3 (1) (2020), 88–96. [2] a. al-masarwah, a.g. ahmad, on some properties of doubt bipolar fuzzy h-ideals in bck/bci-algebras, eur. j. pure appl. math. 11 (2018), 652-670. [3] anas al-masarwah, abd ghafur ahmad, g. muhiuddin, doubt n-ideals theory in bck-algebras based on n-structures, ann. commun. math. 3 (1) (2020), 54–62. [4] a.m. al-roqi, g. muhiuddin, s. aldhafeeri, normal unisoft filters in r0-algebras. cogent math. 4 (2017), 1383006. [5] j. chen, s. li, s. ma, x. wang, m-polar fuzzy sets: an extension of bipolar fuzzy sets, sci. world j. 2014 (2014), article id 416530. [6] p.a. ejegwa, j. a. otuwe, frattini fuzzy subgroups of fuzzy groups ann. commun. math. 2 (1) (2019), 24-31 [7] k. hayat, t. mahmood, b.y. cao, on bipolar anti fuzzy h-ideals in hemirings, fuzzy inf. eng. 9 (2017), 1–19. [8] m. ibrara, a. khana, b. davvazb, characterizations of regular ordered semigroups in terms of (α,β)-bipolar fuzzy generalized bi-ideals, j. intell. fuzzy syst. 33 (2017), 365-376. [9] y. imai, k. iseki, on axiom systems of propositional calculi, proc. japan acad. 42 (1966), 19-22. [10] k. iséki, an algebra related with a propositional calculus, proc. japan acad. 42 (1966) 26-29. [11] c. jana, m. pal, a.b. saeid, (∈,∈ ∨q)-bipolar fuzzy bck/bci-algebras, missouri j. math. sci. 29 (2017), 139-160. [12] y.b. jun, m.s. kang, h.s. kim, bipolar fuzzy hyper bck-ideals in hyper bck-algebras, iran. j. fuzzy syst. 8 (2) (2011), 105–120. [13] y.b. jun, closed fuzzy ideals in bci-algebras, math. japon. 38 (1) (1993), 199-202. [14] k. kawila, c. udomsetchai, a. iampan, bipolar fuzzy up-algebras, math. comput. appl. 23 (4) (2018), 69. [15] k.j. lee, bipolar fuzzy subalgebras and bipolar fuzzy ideals of bck/bci-algebras, bull. malays. math. sci. soc. 32 (2009), 361–373. [16] k.m. lee, bipolar-valued fuzzy sets and their operations, proc. int. conf. intell. technol. bangkok, thailand (2000), 307–312. [17] k.m. lee, comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets, j. fuzzy log. intell. syst. 14 (2004), 125–129. [18] j. meng, y.b. jun, h. s. kim, fuzzy implicative ideals of bck-algebras, fuzzy sets syst. 89 (1997), 243-248. [19] j. meng, y.b. jun, bck-algebras, kyung moon sa co., seoul korea 1994. [20] g. muhiuddin, habib harizavi, y.b. jun, bipolar-valued fuzzy soft hyper bck ideals in hyper bck algebras, discrete math. algorithms appl. 12 (2) (2020), 2050018. [21] g. muhiuddin, bipolar fuzzy ku-subalgebras/ideals of ku-algebras, ann. fuzzy math. inform. 8, (3) (2014), 409-418. [22] g. muhiuddin, k. p. shum, new types of (α,β) fuzzy subalgebras of bck/bci-algebras, int. j. math. computer sci. 14 (2) (2019), 449–464. [23] g. muhiuddin, s. aldhafeeri, characteristic fuzzy sets and conditional fuzzy subalgebras, j. comput. anal. appl. 25 (8) (2018), 1398–1409. int. j. anal. appl. 18 (5) (2020) 875 [24] g. muhiuddin, a.m. al-roqi, classifications of (alpha, beta)-fuzzy ideals in bck/bci–algebras, j. math. anal. 7 (6) (2016), 75–82. [25] g. muhiuddin, a.m. al-roqi, subalgebras of bck/bci-algebras based on (α,β)-type fuzzy sets, j. comput. anal. appl. 18 (6) (2015), 1057–1064. [26] t. senapati, k.p. shum, cubic subalgebras of bch-algebras, ann. commun. math. 1 (1) (2018), 65-73. [27] a.f. talee, m.y. abbasi, a. basar, on properties of hesitant fuzzy ideals in semigroups, ann. commun. math. 3 (1) (2020), 97-106 [28] s. thongarsa, p. burandate, a. iampan, some operations of fuzzy sets in up-algebras with respect to a triangular norm, ann. commun. math. 2 (1) (2019), 1-10 [29] l.a. zadeh, fuzzy sets, inform. control 8 (1965), 338-353. [30] w.r. zhang, bipolar fuzzy sets, proc. fuzzy-ieee (1998), 835-840. 1. introduction 2. preliminaries 3. bipolar fuzzy ideal 4. bipolar fuzzy closed ideal 5. bipolar fuzzy positive implicative ideal 6. bipolar fuzzy implicative ideal references int. j. anal. appl. (2022), 20:5 riemann-liouville fractional versions of hadamard inequality for strongly m-convex functions ghulam farid1, saira bano akbar2, laxmi rathour3, lakshmi narayan mishra4, vishnu narayan mishra5,∗ 1department of mathematics, comsats university islamabad, attock campus, pakistan 2department of mathematics, comsats university islamabad, lahore campus, pakistan 3ward number – 16, bhagatbandh, anuppur 484 224, madhya pradesh, india 4department of mathematics, school of advanced sciences, vellore institute of technology (vit) university, vellore 632 014, tamil nadu, india 5department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur, madhya pradesh 484 887, india ∗corresponding author: vishnunarayanmishra@gmail.com, vnm@igntu.ac.in abstract. this paper deals with hadamard inequalities for strongly m-convex functions via riemannliouville fractional integrals. these inequalities provide refinements of well known fractional integral inequalities for convex functions. further, by applying an identity error estimations are obtained and compared with already known error estimations. 1. introduction first, we will give some definitions and well known results which are needful and connected with the findings of this paper. definition 1.1. [8] a function f : i →r will be called convex if f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y), (1.1) received: jun. 7, 2021. 2010 mathematics subject classification. 26a51, 26a33, 33e12. key words and phrases. m-convex function; strongly m-convex function; hadamard inequality; riemann–liouville fractional integrals. https://doi.org/10.28924/2291-8639-20-2022-5 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-5 2 int. j. anal. appl. (2022), 20:5 ∀ x,y ∈ i = [x,y] and t ∈ [0, 1]. the function f will be strictly convex if f (xt + (1 − t)y) < tf (x) + (1 − t)f (y), (1.2) holds ∀ x 6= y ∈ i and t ∈ (0, 1). definition 1.2. [7] let (x, ||.||) be a normed space. a function f : e ⊂x→r will be called strongly convex function with modulus c if f (xt + (1 − t)y) ≤ tf (x) + (1 − t)f (y) −ct(1 − t)||y −x||2 (1.3) holds ∀ x,y ∈ e ⊆x,t ∈ [0, 1]. definition 1.3. [12] a function f : [0,b] →r ,b > 0 will be called m-convex function if f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y), (1.4) holds ∀ x,y ∈ [0,b] and t,m ∈ [0, 1]. definition 1.4. [6] a function f : [0,b] →r will be called strongly m-convex function if f (tx + m(1 − t)y) ≤ tf (x) + m(1 − t)f (y) −cmt(1 − t)|y −x|2, (1.5) holds ∀ x,y ∈ [0,b],b > 0 and t,m ∈ [0, 1]. the hadamard inequality is another way of representing convex function stated in the upcoming theorem. theorem 1.1. [2] let f : i → r be a convex function on interval i ⊂ r and x,y ∈ i where x < y. then the following inequality holds: f ( x + y 2 ) ≤ 1 y −x ∫ y x f (u)du ≤ f (x) + f (y) 2 . (1.6) if order in inequality (1.6) is reversed, then it holds for concave function. the riemann-liouville fractional integrals are defined as follows: definition 1.5. [10] let f ∈ l1[x,y]. the riemann-liouville fractional integral operators of order β > 0 are defined as follows: j β x+ f (u) = 1 γ(β) ∫ u x (u − t)β−1f (t)dt,u > x, (1.7) j β y− f (u) = 1 γ(β) ∫ y u (t −u)β−1f (t)dt,u < y. (1.8) the following version of the hadamard inequality for convex functions via riemann-liouville fractional integrals was proved by sarikaya et al. in [9]: int. j. anal. appl. (2022), 20:5 3 theorem 1.2. [9] let f : [x,y] →r be a positive function with 0 ≤ x < y and f ∈ l1[x,y]. if f is convex function on [x,y], then the following inequality for fractional integrals holds: f ( x + y 2 ) ≤ γ(β + 1) 2(y −x)β [ (j β x+ f )(y) + (j β y− f )(x) ] ≤ f (x) + f (y) 2 (1.9) with β > 0. in [9], they also studied the error estimations of this fractional hadamard inequality by establishing an identity. another version of the hadamard inequality was proved by sarikaya and yildirim in [11]. theorem 1.3. [11] let f : [x,y] → r be a positive function with 0 ≤ x < y. if f ∈ l1[x,y], then the following fractional integrals equality holds: f ( x + y 2 ) ≤ 2β−1γ(β + 1) (y −x)β [ j β ( x+y 2 )+ f (y) + j β ( x+y 2 )− f (x) ] ≤ f (x) + f (y) 2 . (1.10) our aim in this paper is to study all of the above inequalities and their error estimations by applying definition of strongly m-convex functions. in the upcoming section we will prove the version of hadamard inequality for strongly m-convex functions, which simultaneously will represent refinement as well as generalization of theorem 1.2. another version of the hadamard inequality will be proved which will provide refinement and generalization of theorem 1.3 at the same time. also error estimations of the fractional hadamard inequality are given in refined form. 2. main results theorem 2.1. let f ∈ l1[x,y] be a positive function with 0 ≤ x < my. if f is strongly m-convex function on [x,my], m 6= 0 with modulus c, then the following fractional integral inequality holds: f ( x + my 2 ) + cmβ 4 ( (x −y)2 β + 2 + 2(my − x m )2 β(β + 1)(β + 2) + 2(x −y)(my − x m ) (β + 1)(β + 2) ) (2.1) ≤ γ(β + 1) 2(my −x)β [ j β x+ f (my) + mβ+1j β y− f ( x m ) ] ≤ ( f (x) −m2f ( x m2 )) β 2(β + 1) + m 2 ( f (y) + mf ( x m2 )) − cmβ ( (y −x)2 + m(y − x m2 )2 ) 2(β + 1)(β + 2) , with β,c > 0. proof. since the function f is strongly m-convex function, for u,v ∈ [x,y] we have f ( u + mv 2 ) ≤ f (u) + mf (v) 2 − cm 4 |u −v|2. (2.2) 4 int. j. anal. appl. (2022), 20:5 by setting u = xt + m(1 − t)y and v = yt + (1 − t) x m , we have f ( x + my 2 ) ≤ 1 2 f (xt + m(1 − t)y) + m 2 f ( yt + (1 − t) x m ) (2.3) − cm 4 ∣∣∣t(x −y) + (1 − t) (my − x m )∣∣∣2 . by multiplying inequality (2.3) with tβ−1 on both sides and then integrating over the interval [0, 1], we get f ( x + my 2 )∫ 1 0 tβ−1dt ≤ 1 2 ∫ 1 0 f (xt + m(1 − t)y)tβ−1dt (2.4) + m 2 ∫ 1 0 f ( yt + (1 − t) x m ) tβ−1dt − cm 4 ∫ 1 0 ∣∣∣t(x −y) + (1 − t) (my − x m )∣∣∣2 tβ−1dt. by change of variables we will get 1 β f ( x + my 2 ) (2.5) ≤ γ(β) 2(my −x)β [ 1 γ(β) ∫ my x (my −u)β−1f (u)du + mβ+1 γ(β) ∫ y x m ( v − x m )α−1 f (v)dv ] − cm 4 ( (x −y)2 β + 2 + 2(my − x m )2 β(β + 1)(β + 2) + 2(x −y)(my − x m ) (β + 1)(β + 2) ) . therefore, the above inequality takes the following form: f ( x + my 2 ) ≤ γ(β + 1) 2(my −x)β [ j β x+ f (my) + mβ+1j β y− f ( x m )] (2.6) − cmβ 4 ( (x −y)2 β + 2 + 2(my − x m )2 β(β + 1)(β + 2) + 2(x −y)(my − x m ) (β + 1)(β + 2) ) . from the definition of strongly m-convex function with modulus c, for t ∈ [0, 1] we have the following inequality: f (tx + m(1 − t)y) + mf ( yt + (1 − t) x m ) (2.7) ≤ t ( f (x) −m2f ( x m2 )) + m ( f (y) + mf ( x m2 )) −cmt(1 − t) ( (y −x)2 + m ( y − x m2 )2) . by multiplying inequality (2.7) with tβ−1 on both sides and then integrating over the interval [0, 1], we get ∫ 1 0 f (tx + m(1 − t)y)tβ−1dt + m ∫ 1 0 f ( yt + (1 − t) x m ) tβ−1dt (2.8) ≤ ( f (x) −m2f ( x m2 ))∫ 1 0 tβdt + m ( f (y) + mf ( x m2 ))∫ 1 0 tβ−1dt −cm ( (y −x)2 + m ( y − x m2 )2)∫ 1 0 tβ(1 − t)dt. int. j. anal. appl. (2022), 20:5 5 by change of variables we will get γ(β) (my −x)β [ 1 γ(β) ∫ my x (my −u)β−1f (u)du + mβ+1 γ(β) ∫ y x m ( v − x m )β−1 f (v)dv ] (2.9) ≤ ( f (x) −m2f ( x m2 )) 1 (β + 1) + m β ( f (y) + mf ( x m2 )) − cm ( (y −x)2 + m(y − x m2 )2 ) (β + 1)(β + 2) . therefore, the above inequality takes the following form: γ(β + 1) 2(my −x)β [ j β x+ f (my) + mβ+1j β y− f ( x m )] (2.10) ≤ ( f (x) −m2f ( x m2 )) β 2(β + 1) + m 2 ( f (y) + mf ( x m2 )) − cmβ ( (y −x)2 + m(y − x m2 )2 ) 2(β + 1)(β + 2) . from inequalities (2.6) and (2.10), one can get inequality (2.1). � remark 2.1. (i) for c = 0 in inequality (2.1), we get [4, theorem 2.1], for c 6= 0 (2.1) is the refinement. (ii) for m = 1 and c = 0 in inequality (2.1), we get theorem 1.2. (iii) for m = 1 in inequality (2.1) we get the hadamard inequality for strongly convex function. (iv) for β = 1 and m = 1 in (2.1) we will get the refinement of the hadamard inequality. the next result is the refinement of another version of the hadamard inequality for riemann-liouville fractional integrals stated in theorem 1.3. theorem 2.2. let f ∈ l1[x,y] be a positive function with 0 ≤ x < my. if f is a strongly m-convex function on [x,my], m 6= 0 with modulus c, then the following fractional integral inequality holds: f ( x + my 2 ) + cmβ 4 [ (x −y)2 4(β + 2) + (my − x m )2(β2 + 5β + 8) 4β(β + 1)(β + 2) + (x −y)(my − x m )(β + 3) 2(β + 1)(β + 2) ] (2.11) ≤ 2β−1γ(β + 1) (my −x)β [( j β ( x+my 2 )+ f ) (ym) + mβ+1 ( j β ( x+ym 2m )− f )( x m )] ≤ β ( f (x) −m2f ( x m2 )) 4(β + 1) + m 2 ( f (y) + mf ( x m2 ) ) − cmβ((y −x)2 + m(y − x m2 )2)(β + 3) 8(β + 1)(β + 2) , with β,c > 0. proof. for t ∈ [0, 1] and strongly m-convexity of function, let u = x t 2 +m(2−t 2 )y and v = (2−t 2 ) x m +y t 2 in inequality (2.2), we have 6 int. j. anal. appl. (2022), 20:5 f ( x + my 2 ) ≤ 1 2 f ( x t 2 + m ( 2 − t 2 ) y ) + m 2 f (( 2 − t 2 ) x m + y t 2 ) (2.12) − cm 4 ∣∣∣∣t2 (x −y) + 2 − t2 ( my − x m )∣∣∣∣2 . by multiplying (2.12) with tβ−1 on both sides and making integration over [0, 1] we get f ( x + my 2 )∫ 1 0 tβ−1dt ≤ 1 2 ∫ 1 0 f ( x t 2 + m ( 2 − t 2 ) y ) tβ−1dt (2.13) + m 2 ∫ 1 0 f (( 2 − t 2 ) x m + y t 2 ) tβ−1dt − cm 4 ∫ 1 0 ∣∣∣∣t2 (x −y) + 2 − t2 ( my − x m )∣∣∣∣2 tβ−1dt. by using change of variables and computing the last integral, from (2.13) we get 1 β f ( x + my 2 ) (2.14) ≤ 2β−1γ(β) (my −x)β [ 1 γ(β) ∫ my x+my 2 (my −u)β−1f (u)du + mβ+1 γ(β) ∫ ym+x 2m x m ( v − x m )β−1 f (v)dv ] − cm 4 [ (x −y)2 4(β + 2) + (my − x m )2(β2 + 5β + 8) 4β(β + 1)(β + 2) + (x −y)(my − x m )(β + 3) 2(β + 1)(β + 2) ] . further it takes the following form f ( x + my 2 ) ≤ 2β−1γ(β + 1) (my −x)β [( j β ( x+my 2 )+ f ) (ym) + mβ+1 ( j β ( ym+x 2m )− f )( x m )] (2.15) − cmβ 4 [ (x −y)2 4(β + 2) + (my − x m )2(β2 + 5β + 8) 4β(β + 1)(β + 2) + (x −y)(my − x m )(β + 3) 2(β + 1)(β + 2) ] . the first inequality of (2.11) can be seen in (2.15). now we prove the second inequality of (2.11). since f is strongly m-convex function and t ∈ [0, 1], we have the following inequality: f ( x t 2 + m ( 2 − t 2 ) y ) + mf (( 2 − t 2 ) x m + y t 2 ) ≤ t 2 ( f (x) −m2f ( x m2 )) (2.16) + m ( f (y) + mf ( x m2 )) − cmt(2 − t) 4 [ (y −x)2 + m ( y − x m2 )2] . int. j. anal. appl. (2022), 20:5 7 by multiplying inequality (2.16) with tβ−1 on both sides and making integration over [0, 1] we get∫ 1 0 f ( x t 2 + m ( 2 − t 2 ) y ) tβ−1dt + m ∫ 1 0 f (( 2 − t 2 ) x m + y t 2 ) tβ−1dt (2.17) ≤ 1 2 ( f (x) −m2f ( x m2 ))∫ 1 0 tβdt + m ( f (y) + mf ( x m2 ))∫ 1 0 tβ−1dt − cm 4 [ (y −x)2 + m ( y − x m2 )]∫ 1 0 tβ(2 − t)dt. by using change of variables and computing the last integral, from (2.17) we get 2βγ(β) (my −x)β [ 1 γ(β) ∫ my x+my 2 (my −u)β−1f (u)du + mβ+1 γ(β) ∫ my+x 2m x m ( v − x m )β−1 f (v)dv ] (2.18) ≤ ( f (x) −m2f ( x m2 ) ) 2(β + 1) + m ( f (y) + mf ( x m2 ) ) β − cm((y −x)2 + m(y − x m2 )2)(β + 3) 4(β + 1)(β + 2) . further it takes the following form 2β−1γ(β + 1) (my −x)β [( j β ( x+my 2 )+ f ) (ym) + mβ+1 ( j β ( x+ym 2m )− f )( x m )] (2.19) ≤ β ( f (x) −m2f ( x m2 ) ) 4(β + 1) + m 2 ( f (y) + mf ( x m2 )) − cmβ((y −x)2 + m(y − x m2 )2)(β + 3) 8(β + 1)(β + 2) . from inequalities (2.15) and (2.19), we have inequality (2.11). � remark 2.2. (i) for c = 0 in (2.1), we will get [3, theorem 2.1], and for c 6= 0 its refinement is obtained. (ii) for β = 1 and m = 1 in (2.1) we will get refinement of the hadamard inequality. (iii) for m = 1 and c = 0 in (2.1), we will get theorem 1.3. (iv) for m = 1 in inequality (2.11) we will get the hadamard inequality for strongly convex function. (v) for m = 1, c = 0 and β = 1 in inequality (2.11), we get the inequality (1.6). 3. error estimations lemma 3.1. [9] let f : [x,y] → r be a differentiable mapping on (x,y) with x < y. if f ′ ∈ [x,y], then the following fractional integrals equality holds: f (x) + f (y) 2 − γ(β + 1) 2(y −x)β [ (j β x+ f )(y) + (j β y− f )(x) ] (3.1) = y −x 2 ∫ 1 0 [ (1 − t)β − tβ ] f ′ (tx + (1 − t)y) dt. theorem 3.1. let f : [x,y] → r be a differentiable mapping on (x,y) with x < my. if |f ′| is a strongly m-convex function on [x,my], m 6= 0, with modulus c, then the following fractional integrals 8 int. j. anal. appl. (2022), 20:5 inequality holds:∣∣∣∣f (x) + f (y)2 − γ(β + 1)2(y −x)β [ (j β x+ f )(y) + (j β y− f )(x) ]∣∣∣∣ (3.2) ≤ y −x 2 [ 1 (β + 1) ( 1 − 1 2β )[ |f ′(x)| + m|f ′ ( y m ) | ] − 2cm( y m −x)2 (β + 2)(β + 3) ( 1 − β + 4 2β+2 )] , with β,c > 0. proof. since |f ′| is strongly m-convex function on [x,my] and t ∈ [0, 1], we have∣∣f ′ (tx + (1 − t)y)∣∣dt = ∣∣∣f ′(tx + m(1 − t) y m )∣∣∣dt (3.3) ≤ t|f ′(x)| + m(1 − t) ∣∣∣f ′( y m )∣∣∣−cmt(1 − t) ( y m −x )2 . by using lemma 3.1 and (3.3) we have∣∣∣∣f (x) + f (y)2 − γ(β + 1)2(y −x)β [ (j β x+ f )(y) + (j β y− f )(x) ]∣∣∣∣ (3.4) ≤ y −x 2 ∫ 1 0 ∣∣(1 − t)β − tβ∣∣ ∣∣∣f ′(tx + m(1 − t) y m )∣∣∣dt ≤ y −x 2 ∫ 1 0 ∣∣(1 − t)β − tβ∣∣( t|f ′(x)| + m(1 − t) ∣∣∣f ′( y m )∣∣∣−cmt(1 − t) ( y m −x )2) dt ≤ y −x 2[∫ 1/2 0 ( (1 − t)β − tβ )( t|f ′(x)| + m(1 − t) ∣∣∣f ′( y m )∣∣∣−cmt(1 − t) ( y m −x )2) dt + ∫ 1 1/2 ( tβ − (1 − t)β )( t|f ′(x)| + m(1 − t) ∣∣∣f ′( y m )∣∣∣−cmt(1 − t) ( y m −x )2) dt ] ≤ y −x 2 [ |f ′(x)| ( 1 (β + 1)(β + 2) − (1/2)β+1 β + 1 ) + m ∣∣∣f ′( y m )∣∣∣( 1 (β + 2) − (1/2)β+1 β + 1 ) − cm( y m −x)2 (β + 2)(β + 3) ( 1 − β + 4 2β+2 ) + |f ′(x)| ( 1 β + 2 − (1/2)β+1 β + 1 ) + m|f ′( y m )| ( 1 (β + 1)(β + 2) − (1/2)β+1 β + 1 ) − cm( y m −x)2 (β + 2)(β + 3) ( 1 − β + 4 2β+2 )] . after simplify the last inequality of (3.4), we get the inequality (3.2). � remark 3.1. (i) if m = 1, c = 0 in inequality (3.2), we get [9, theorem 3], for c > 0 we get its refinement. (ii) if m = 1, c = 0 and β = 1 in inequality (3.2), we get [1, theorem 2.2], for c > 0 we get its refinement. corollary 3.1. for β = 1∣∣∣∣f (x) + f (y)2 − 1(y −x) ∫ y x f (u)du ∣∣∣∣ ≤ y −x8 [ |f ′(x)| + m|f ′( y m )| ] − cm( y m −x)3 32 , with c > 0. int. j. anal. appl. (2022), 20:5 9 theorem 3.2. let f : [x,y] → r be a differentiable mapping on (x,y) with x < my. if |f ′|q is strongly m-convex on [x,y] for q ≥ 1, then the following inequality for fractional integrals holds: ∣∣∣∣2β−1γ(β + 1)(my −x)β [ (j β ( x+my 2 )+ f )(my) + mβ+1(j β ( x+my 2m )− f ) ( x m )] (3.5) − 1 2 [ f ( x + my 2 ) + mf ( x + my 2m )]∣∣∣∣ ≤ my −x4(β + 1) ( 1 2(β + 2) )1 q [( |f ′(x)|q(β + 1) + m|f ′(y)|q(β + 3) − cm(y −x)2(β + 1)(β + 4) 2(β + 3) )1 q + ( |f ′(y)|q(β + 1) + m(β + 3) ∣∣∣f ′( x m2 )∣∣∣q − cm( xm2 −y)2(β + 1)(β + 4) 2(β + 3) )1 q ] , with β > 0. proof. by applying [3, lemma 2.3] and strong m-convexity of |f ′|, let q = 1 we have ∣∣∣∣2β−1γ(β + 1)(my −x)β [ (j β ( x+my 2 )+ f )(my) + mβ+1(j β ( x+my 2m )− f ) ( x m )] (3.6) − 1 2 [ f ( x + my 2 ) + mf ( x + my 2m )]∣∣∣∣ ≤ my −x 4 [∫ 1 0 ∣∣∣∣tβf ′ ( t 2 x + m ( 2 − t 2 ) y )∣∣∣∣dt + ∫ 1 0 ∣∣∣∣tβf ′ (( 2 − t 2 ) x m + t 2 y )∣∣∣∣dt ] ≤ my −x 4 [( |f ′(x)|−m|f ′(y)| + |f ′(y)|−m|f ′( x m2 )| 2 )∫ 1 0 tβ+1dt + m ( |f ′(y)| + ∣∣∣f ′( x m2 )∣∣∣)∫ 1 0 tβdt − cm 4 ( (y −x)2 + m ( x m2 −y )2)∫ 1 0 tβ+1(2 − t)dt ] = my −x 4 [( |f ′(x)|−m|f ′(y)| + |f ′(y)|−m|f ′( x m2 )| 2(β + 2) ) + m ( |f ′(y)| + |f ′( x m2 )| ) β + 1 − cm(β + 4) 4(β + 2)(β + 3) ( (y −x)2 + m ( x m2 −y )2)] . now, for strongly m-convexity of |f ′|q, q > 1, using power mean inequality we get ∣∣∣∣2β−1γ(β + 1)(my −x)β [ (j β ( x+my 2 )+ f )(my) + mβ+1(j β ( x+my 2m )− f ) ( x m )] − 1 2 [ f ( x + my 2 ) (3.7) + mf ( x + my 2m )]∣∣∣∣ ≤ my −x4 (∫ 1 0 tβdt )1−1 q [(∫ 1 0 tβ ∣∣∣∣f ′ ( t 2 x + m ( 2 − t 2 ) y )∣∣∣∣qdt )1 q + (∫ 1 0 tβ ∣∣∣∣f ′ (( 2 − t 2 ) x m + t 2 y )∣∣∣∣qdt )1 q ] ≤ my −x 4(β + 1) 1 p ( |f ′(x)|q −m|f ′(y)|q 2 ∫ 1 0 tβ+1dt 10 int. j. anal. appl. (2022), 20:5 +m|f ′(y)|q ∫ 1 0 tβdt − cm(y −x)2 4 ∫ 1 0 tβ+1(2 − t)dt )1 q + ( |f ′(y)|q −m|f ′( x m2 )|q 2∫ 1 0 tβ+1dt + m ∣∣∣f ′( x m2 )∣∣∣q ∫ 1 0 tβdt − cm( x m2 −y)2 4 ∫ 1 0 tβ+1(2 − t)dt )1 q ≤ my −x 4(β + 1) 1 p[( |f ′(x)|q 2(β + 2) + m |f ′(y)|q(β + 3) 2(β + 1)(β + 2) − cm(y −x)2(β + 4) 4(β + 2)(β + 3) )1 q + ( |f ′(y)|q 2(β + 2) +m |f ′( x m2 )|q(β + 3) 2(β + 1)(β + 2) − cm( x m2 −y)2(β + 4) 4(β + 2)(β + 3) )1 q ] = my −x 4(β + 1) ( 1 2(β + 2) )1 q [( |f ′(x)|q(β + 1) + m|f ′(y)|q(β + 3) − cm(y −x)2(β + 4)(β + 1) 2(β + 3) )1 q + ( |f ′(y)|q(β + 1) + m|f ′( x m2 )|q(β + 3) − cm( x m2 −y)2(β + 4)(β + 1) 2(β + 3) )1 q ] . so, we have inequality 3.5. � remark 3.2. (i) if c = 0 in in inequality (3.5), we have [3, theorem 2.4]. (ii) if m = 1 and c = 0 in inequality (3.5), we have [11, theorem 5]. (iii) if m = 1, c = 0 and β = 1 in inequality 3.5 we get the inequality which is proved by kirmaci [5]. corollary 3.2. if m = 1, q = 1 and β = 1 in 3.5, we get∣∣∣∣ 1y −x ∫ y x f (u)du − f ( x + y 2 )∣∣∣∣ ≤ y −x8 [ |f ′(x)| + |f ′(y)|− 5c(y −x)2 12 ] . theorem 3.3. let f : [x,y] →r be a differentiable mapping on (x,y) with x < y. if |f ′|q is strongly m-convex function on [x,y] for q > 1, then the following fractional integrals inequality holds: ∣∣∣∣2β−1γ(β + 1)(my −x)β [ (j β ( x+my 2 )+ f )(my) + mβ+1(j β ( x+my 2m )− f ) ( x m )] (3.8) − 1 2 [ f ( x + my 2 ) + mf ( x + my 2m )]∣∣∣∣ ≤ my −x16 ( 4 βp + 1 )1 p [( |f ′(x)|q + 3m|f ′(y)|q − 2cm(y −x)2 3 )1 q + ( 3m|f ′( x m2 )|q + |f ′(y)|q − 2cm( x m2 −y)2 3 )1 q   ≤ my −x 16 ( 4 βp + 1 )1 p [ |f ′(x)| + |f ′(y)| + 3m ( |f ′ ( x m2 ) | + |f ′(y)| ) − 2cm 3 ( (y −x)2 + ( x m2 −y)2 )] , where 1 p + 1 q = 1, β > 0. int. j. anal. appl. (2022), 20:5 11 proof. by applying [3, lemma 2.3], using holder inequality and strong m-convexity of |f ′|q, we get∣∣∣∣2β−1γ(β + 1)(my −x)β [ (j β ( x+my 2 )+ f )(my) + mβ+1(j β ( x+my 2m )− f ) ( x m )] (3.9) − 1 2 [ f ( x + my 2 ) + mf ( x + my 2m )]∣∣∣∣ ≤ my −x4 (∫ 1 0 tpβdt )1 p [(∫ 1 0 ∣∣∣∣f ′ ( x t 2 + m ( (2 − t) 2 ) y )∣∣∣∣q dt )1 q + (∫ 1 0 ∣∣∣∣f ′ ( y t 2 + ( 2 − t 2 ) x m )∣∣∣∣q dt )1 q ] ≤ my −x 4 ( 1 βp + 1 )1 p [( |f ′(x)|q ∫ 1 0 t 2 dt + m|f ′(y)|q ∫ 1 0 2 − t 2 dt − cm(y −x)2 4 ∫ 1 0 t(2 − t)dt )1 q + ( |f ′(y)|q ∫ 1 0 t 2 dt + m ∣∣∣f ′( x m2 )∣∣∣q ∫ 1 0 ( 2 − t 2 ) dt − cm( x m2 −y)2 4 ∫ 1 0 t(2 − t)dt )1 q ] . = my −x 16 ( 4 βp + 1 )1 p [( |f ′(x)|q + 3m|f ′(y)|q − 2cm(y −x)2 3 )1 q + ( 3m|f ′ ( x m2 ) |q + |f ′(y)|q − 2cm( x m2 −y)2 3 )1 q ] ≤ my −x 16 ( 4 βp + 1 )1 p [ |f ′(x)| + |f ′(y)| + 3m ( |f ′(y)| + |f ′ ( x m2 ) | ) − 2cm 3 ( (y −x)2 + ( x m2 −y)2 )] . we have used aq + bq ≤ (a + b)q, for a ≥ 0,b ≥ 0. � remark 3.3. (i) if c = 0 in in 3.8, we have [3, theorem 2.7] . (ii) if m = 1 and c = 0 in 3.8, we have [11, theorem 6]. corollary 3.3. for q = 1 and β = 1 and m = 1 , we have∣∣∣∣ 1(y −x) ∫ y x f (u)du − f ( x + y 2 )∣∣∣∣ ≤ y −x4 ( 4 p + 1 )1 p [ |f ′(x)| + |f ′(y)|− c(y −x)2 3 ] . corollary 3.4. for β = 1 and m = 1, we have∣∣∣∣ 1(y −x) ∫ y x f (u)du − f ( x + y 2 )∣∣∣∣ ≤ y −x16 ( 4 p + 1 )1 p [( |f ′(a)| + 3|f ′(y)|− 2c(y −x)2 3 )1 q + ( 3|f ′(x)| + |f ′(y)|− 2c(y −x)2 3 )1 q ] . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s.s. dragomir, r.p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (1998), 91–95. https://doi.org/10.1016/ s0893-9659(98)00086-x. https://doi.org/10.1016/s0893-9659(98)00086-x https://doi.org/10.1016/s0893-9659(98)00086-x 12 int. j. anal. appl. (2022), 20:5 [2] s.s. dragomir, c.e.m. pearce, selected topics on hermite-hadamard inequalities and applications, rgmia monographs, victoria university, 2000. http://rgmia.org/monographs/hermite_hadamard.html. [3] g. farid, a. ur. rehman, b. tariq, on hadamard-type inequalities for m-convex functions via riemann-liouville fractional integrals, stud. univ. babes, -bolyai math. 62 (2017), 141-150. https://doi.org/10.24193/subbmath. 2017.2.01. [4] g. farid, a. ur. rehman, b. tariq, a. waheed, on hadamard type inequalities for m-convex functions via fractional integrals, j. inequal. spec. funct. 7 (2016), 150-167. [5] u. kirmaci, inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, appl. math. comput. 147 (2004), 137-146. https://doi.org/10.1016/s0096-3003(02)00657-4. [6] t. lara, n. merentes, r. quintero, e. rosales, on strongly m-convex functions, math. aeterna, 5 (2015), 521-535. [7] b.t. polyak, existence theorems and convergence of minimizing sequences in extremum problems with restrictions, soviet math. doklady, 7 (1966), 72-75. [8] a.w. roberts, d.e. varberg, convex functions, academic press, new york, 1973. [9] m. z. sarikaya, e. s. h. yaldiz, n. başak, hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities, math. comput. model. 57 (2013), 2403-2407. https://doi.org/10.1016/j.mcm.2011. 12.048. [10] h.m. srivastava, z. tomovski, fractional calculus with an integral operator containing generalized mittag-leffler function in the kernel, appl. math. comput. 211 (2009), 198-210. https://doi.org/10.1016/j.amc.2009.01. 055. [11] m.z. sarikaya, h. yildirim, on hermit-hadamard type inequalities for rieman-liouville fractional integrals, miskolc math. notes, 17 (2017), 1049-1059. [12] g. toader, some generalizations of the convexity, in: proceedings of the colloquium on approximation and optimization, univ. cluj-napoca, cluj-napoca, (1985), 329-338. http://rgmia.org/monographs/hermite_hadamard.html https://doi.org/10.24193/subbmath.2017.2.01 https://doi.org/10.24193/subbmath.2017.2.01 https://doi.org/10.1016/s0096-3003(02)00657-4 https://doi.org/10.1016/j.mcm.2011.12.048 https://doi.org/10.1016/j.mcm.2011.12.048 https://doi.org/10.1016/j.amc.2009.01.055 https://doi.org/10.1016/j.amc.2009.01.055 1. introduction 2. main results 3. error estimations references international journal of analysis and applications volume 18, number 3 (2020), 337-355 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-337 absolute value variational inequalities and dynamical systems safeera batool, muhammad aslam noor∗ and khalida inayat noor department of mathematics, comsats university islamabad, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we consider the absolute value variational inequalities. we propose and analyze the projected dynamical system associated with absolute value variational inequalities by using the projection method. we suggest different iterative algorithms for solving absolute value variational inequalities by discretizing the corresponding projected dynamical system. the convergence of the suggested methods is proved under suitable constraints. numerical examples are given to illustrate the efficiency and implementation of the methods. results proved in this paper continue to hold for previously known classes of absolute value variational inequalities. 1. introduction variational inequalities theory was introduced earlier by stampacchia [45] and now it is developed as a well-established branch of nonlinear analysis and optimization. variational inequalities theory is widely applied in industry, economics, social, pure and applied sciences, see [17, 20, 27–29, 31, 32, 35–38]. in fact, variational inequalities theory provides us the direct, natural, unified and dynamic framework for the natural analysis of a wide range of unrelated linear and nonlinear problems, see [5, 7, 11–13, 23]. since the discovery of variational inequalities theory, a number of numerical methods including projection method, wiener-hopf equations, auxiliary principle and dynamical systems has been developed for solving the variational inequalities and the related optimization problems, see the references therein [1-49]. received january 29th, 2020; accepted february 24th, 2020; published may 1st, 2020. 2000 mathematics subject classification. 49j40, 65n30, 26d10, 90c23. key words and phrases. absolute value variational inequalities; dynamical systems; iterative methods; convergence. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 337 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-337 int. j. anal. appl. 18 (3) (2020) 338 the relationship of variational inequalities and complementarity problems was proved by karamardian [15], who proved that, if the involved set is a convex cone, then variational inequalities are equivalent to complementarity problem. complementarity problem was introduced by lemke [19]. the equivalence between variational inequalities and complementarity problem has been used in suggesting many iterative algorithms for solving complementarity problems, see [17,26,27,37]. absolute value complementarity problem is an important and useful generalization of the complementarity problem. absolute value complementarity problem was introduced by noor et al. [41] and he also shown that absolute value complementarity problem is equivalent to absolute value variational inequalities. absolute value variational inequalities include the variational inequalities as a special case. it is also proved that if the underlying set is the whole space, then absolute value variational inequalities transform into absolute system of equations, which were introduced and studied by mangasarian [23], rohn [43] and hu and huang [13]. for the applications of absolute system of equations, see [13, 14, 24, 25, 41, 42]. fixed point theory played an important role in developing various types of algorithms for finding the solution of variational inequalities. the equivalence between variational inequalities and fixed point problem can be established by using the projection technique, see [17]. this alternative formulation has been used to suggest the projected dynamical systems related to variational inequalities. projected dynamical systems were introduced by dupuis and nagurney, see [7]. the projected dynamical systems are identified by the discontinuous right hand side. an innovative aspect of the projected dynamical system is that the set of its stationary points corresponds to the solution set of the associated variational inequality problems. hence the equilibrium problems in different branches of pure and applied sciences which are studied in the framework of variational inequalities, can now be considered in the more general framework of projected dynamical systems. projected dynamical systems are effective in the development of many efficient numerical techniques for approximating the solutions of variational inequalities and related nonlinear problems, see [3, 6, 10, 11]. the global asymptotic stability of the projected dynamical systems has also been studied by noor [33] and xia and wang [46]. in this paper, we consider the absolute value variational inequalities. we propose and analyze the projected dynamical system associated with absolute value variational inequalities by using projection method. it is important to mention that such type of the dynamical systems for absolute value variational inequalities have not been investigated earlier. we suggest different iterative algorithms for the solution of absolute value variational inequalities by dicretizing the corresponding projected dynamical system. the convergence of the suggested methods is proved under suitable conditions. two examples are given to illustrate the implementation and efficiency of the proposed iterative methods. since the absolute value int. j. anal. appl. 18 (3) (2020) 339 variational inequalities include the classical variational inequalities, complementarity problems and absolute value equations as special cases, the results obtained in this paper continue to hold for these problems. 2. formulations and basic results let h be a real hilbert space, whose norm and inner product are denoted by ‖.‖ and 〈., .〉 respectively. let k be a closed and convex set in h. for given operators t,b : h → h, consider the problem of finding u ∈ k such that 〈tu + b|u|−f,v −u〉≥ 0, ∀v ∈ k, (1) where f is a continuous functional defined on h and |u| contains the absolute values of components of u ∈ h. the inequality (1) is called absolute value variational inequality. we will now discuss some special cases of the problem (1). (i). if b|u| = 0,∀u ∈ h, then (1) is equivalent to find u ∈ k such that 〈tu−f,v −u〉≥ 0, ∀v ∈ k. (2) inequalities of type equations (2) are known as variational inequalities, which were introduced by lions and stampacchia [20] and have been studied extensively in recent years, see [20, 28, 29, 32, 35–38]. (ii). if k∗ = {u ∈ k : 〈u,v〉 ≥ 0,v ∈ k} is the polar cone of the closed and convex cone k in h, then problem (1) is equivalent to find u ∈ k such that u ∈ k, tu + b|u|−f ∈ k∗, 〈tu + b|u|−f,u〉 = 0, (3) which is called an absolute value complementarity problem. if b|u| = 0, then problem (3) is known as the complementarity problem, the origin of which can be traced back to lemke [19] and have been studied by cottle and dantzig [4] who restated the linear and quadratic programming problem as a complementarity problem. (iii). if k = h, then problem (1) is equivalent to find u ∈ h such that tu + b|u|−f = 0, (4) which is called the system of absolute value equations. absolute system of equations was introduced by mangasarian and meyer [24] and was investigated by hu and huang [13] in a more general context. a generalized newton method was also suggested to find the solution of absolute system of equations, see [14]. an algorithm to compute all the solutions of (4) was proposed by rohn [44] and equivalent formulations of int. j. anal. appl. 18 (3) (2020) 340 (4) are presented by prokopyev [43]. for the applications of absolute system of equations in various areas of engineering and mathematics, see [11, 12, 23–25]. hence, it is clear that the absolute value variational inequality (1) is more general and includes previously known classes of variational inequalities and the system of absolute value equations as special cases. for the recent applications of absolute value variational inequalities, see [40, 41]. in order to derive the main results of this paper, we recall some standard definitions and results. definition 2.1. an operator t : h → h is said to be strongly monotone, if there exists a constant α > 0 such that 〈tu−tv,u−v〉≥ α‖u−v‖2, ∀u,v ∈ h. definition 2.2. an operator t : h → h is said to be lipchitz continuous, if there exists a constant β > 0 such that ‖tu−tv‖≤ β‖u−v‖, ∀u,v ∈ h. if t is strongly monotone and lipchitz continuous operator, then from definitions (2.1) and (2.2), we have α ≤ β. definition 2.3. an operator t : h → h is said to be monotone, if 〈tu−tv,u−v〉≥ 0, ∀u,v ∈ h. definition 2.4. an operator t : h → h is said to be pseudomonotone, if 〈tu,v −u〉≥ 0, implies 〈tv,v −u〉≥ 0 ∀u,v ∈ h. we now consider the well-known projection lemma [17]. this result is useful to reformulate the variational inequalities as a fixed point problem. lemma 2.1. [17] let k be a closed and convex set in h. then for a given z ∈ h,u ∈ k satisfies 〈u−z,v −u〉≥ 0, ∀v ∈ k, (5) int. j. anal. appl. 18 (3) (2020) 341 if and only if u = pkz, where pk is the projection of h onto a closed and convex set k in h. the above lemma plays an important role in obtaining the main results of this paper. the projection operator pk has following properties: (1) the projection operator is non-expansive, that is, ‖pku−pkv‖≤‖u−v‖, ∀u,v ∈ h. (2) the projection operator pk is co-coercive map, that is, 〈pku−pkv,u−v〉≥ ‖pku−pkv‖2, ∀u,v ∈ h. 3. existence theory in this section, we consider a new type of projected dynamical system associated with absolute value variational inequalities and discuss the existence of the solution, stability and convergence of that dynamical system to the solution of absolute value variational inequality (1). using lemma 2.1, we can now state the following result which shows that problem (1) is equivalent to a fixed point problem. lemma 3.1. let k be a closed convex set in h. then u ∈ k is a solution of absolute value variational inequality (1), if and only if, u ∈ k satisfies the relation u = pk[u−ρtu−ρb|u| + ρf], (6) where ρ > 0 is a constant. this alternative equivalent formulation is very important from the theoretical as well as from the numerical point of view. this equivalent provides important tools to approximate the solution of variational inequalities. this equivalent formulation played an important part in establishing the various explicit and implicit iterative methods. koperlevich [18] developed the the extragradient method for solving the variational inequalities, the convergence of which requires only the lipschitz continuity. we now define the residue vector r(u) by the following relation r(u) = u−pk[u−ρtu−ρb|u| + ρf]. (7) int. j. anal. appl. 18 (3) (2020) 342 using lemma 3.1, one can conclude that u ∈ k is a solution of the absolute value variational inequality (1), if and only if, u ∈ k is a zero of the equation r(u) = 0. (8) using (7), we suggest a new projected dynamical system related to absolute value variational inequality (1) as follows: find u ∈ k such that du dt = −γr(u) = γ{pk[u−ρtu−ρb|u| + ρf] −u}, u(t0) = u0 ∈ k, (9) where γ > 0 is a parameter. the expression on the right hand side of (9) is discontinuous on the boundary of k. it can be observed from the definition that the solution of (9) always belongs to k. hence, it is meaningful to discuss the existence, uniqueness and continuous dependence of the solution of (9). these type of projected dynamical systems, corresponding to variational inequalities, have been investigated widely, see [3, 6, 7, 9–11, 33, 46–48]. the dynamical system is said to be stable in the lyapunov sense, if the small initial perturbation does not grow in time for a phase trajectory and asymptotically stable, if the small perturbation vanishes as the time goes by. if the dynamical system is asymptotically stable, then it is lyapunov stable but not conversely, in general. definition 3.1. the dynamical system is said to converge globally to the solution set k̂ of (1), if and only if, irrespective of the initial point, the trajectory of the dynamical system satisfies limn→∞ dist(u(t),k̂) = 0, (10) where dist(u,k̂) = inf v∈k̂‖u−v‖. it can be observed that if the set k̂ contains a unique point û, then (10) implies that limn→∞ u(t) = û. here, the lyapunov stability of the dynamical system at û implies the global exponential stability of the dynamical system at û. int. j. anal. appl. 18 (3) (2020) 343 definition 3.2. the dynamical system is said to be globally exponentially stable with degree λ > 0, at û ∈ k̂ if and only if, irrespective of the initial point, the trajectory of the system u(t) satisfies ‖u(t) − û‖≤ η1‖u(t0) − û‖e−λ(t−t0), ∀t ≥ t0, where η1 > 0 and λ > 0, are independent of the initial point. it can also be noted that global exponential stability always implies the global asymptotic stability and hence the dynamical system converges arbitrarily fast. lemma 3.2. [34] (gronwall lemma) let u and v be real valued non-negative continuous function with domain {t : t ≥ t0} and let α(t) = α0|t− t0|, where α0 is a monotonically increasing function. if for t ≥ t0 u(t) ≤ α(t) + ∫ t t0 u(x)v(x)dx, then u(t) ≤ α(t).exp{ ∫ t t0 v(x)dx}. now onwards, we assume that the nonempty set k̂ is bounded, unless otherwise specified. we now discuss the existence of the solution of absolute value variational inequality (1) via dynamical system (9), mainly using the technique of noor [33] and xia and wang [46]. theorem 3.1. let the operators t and b be lipchitz continuous with constants β1 > 0 and β2 > 0, respectively. if γ > 0 and lemma 6 holds, then, for each u0 ∈ k, there exists a unique and continuous solution u(t) of the dynamical system (9) with u(t0) = u0 over [t0,∞]. proof. let h(u) = γ{pk[u−ρtu−ρb|u| + ρf] −u} where γ > 0 is a constant. for all u,w ∈ k, consider ‖h(u) −h(w)‖ ≤ γ{‖u−w‖ +‖pk[u−ρtu−ρb|u| + ρf] −pk[w −ρtw −ρb|w| + ρf]‖} ≤ γ{‖u−w‖ + ‖u−w‖ρ‖tu−tw‖ + ρ‖b|u|−b|w|‖ ≤ γ{‖u−w‖ + ‖u−w‖ρβ1‖u−w‖ + ρβ2‖|u|− |w|‖ ≤ γ{‖u−w‖ + ‖u−w‖ρβ1‖u−w‖ + ρβ2‖u−w‖ ≤ γ(2 + ρ(β1 + β2))‖u−w‖. (11) where β1 > 0 and β2 > 0 are lipchitz constants of the operators t and b respectively. this shows that the operator h(u) is lipchitz continuous in k. so, for each u0 ∈ k, there exists a continuous and unique int. j. anal. appl. 18 (3) (2020) 344 solution u(t) of the dynamical system (9), which is defined in an interval t0 ≤ t ≤ t with initial condition u(t0) = u0. let its maximal interval of existence be [t0,t). for any u ∈ h, consider ‖ du dt ‖ = ‖h(u)‖ = γ{‖pk[u−ρtu−ρb|u| + ρf] −u} ≤ γ{‖pk[u−ρtu−ρb|u| + ρf] −pk[u]‖ + ‖pk[u] −pk[û]‖ +‖pk[û] −u‖} ≤ γ{ρ‖tu‖ + ρ‖b|u|‖ + ‖ρf‖ + ‖u− û‖ + ‖pk[û]‖ + ‖u‖} ≤ γ{ρβ1‖u‖ + ρβ2‖|u|‖ + ‖ρf‖ + ‖u‖ + ‖û‖ + ‖pk[û]‖ + ‖u‖} ≤ γ(2 + ρ(β1 + β2))‖u‖ + γ{‖pk[û]‖ + ‖û‖ + ‖ρf‖}, (12) where the lipchitz continuity of the operators t and b with constants β1 > 0,β2 > 0, is used. integrating (12) from t0 to t, we obtain ‖u(t)‖ ≤ ‖u0‖ + ∫ t t0 ‖su(z)‖dz ≤ (‖u0‖ + k1(t− t0)) + k2 ∫ t t0 ‖u(z)‖dz, where, k1 = γ{‖pk[û]‖ + ‖û‖ + ‖ρf‖}, and k2 = γ(2 + ρ(β1 + β2)). hence, by using gronwall’s lemma 3.2, we have ‖u(t)‖≤{‖u0‖ + k1(t− t0)}ek2(t−t0), t ∈ [t0,s]. this proves that the solution u(t) is bounded on the interval [t0,s). hence, s = ∞. � we now study the stability of dynamical system (9) by using the technique of noor [33] and xia and wang [46]. theorem 3.2. let t and b be pseudomonotone and lipchitz continuous operators with constants β1 > 0,β2 > 0, respectively. then the dynamical system (9) is stable in the lyapunov sense and converges globally to the solution of the absolute value variational inequality (1). proof. since the operators t and b are lipchitz continuous with constants β1 > 0,β2 > 0. it follows from theorem 3.1, that the dynamical system (9) has a continuous and unique solution u(t) over the interval int. j. anal. appl. 18 (3) (2020) 345 [t0,s) for any fixed u0 ∈ k. let u(t) = u(t,t0; u0) be the solution of the initial value problem (9). for a given û ∈ k, consider the following lyapunov function l(u) = ‖u− û‖2, u ∈ k. (13) it can be observed that limn−→∞ l(un) = +∞ whenever the sequence un ⊂ k and limn→∞ un = +∞. therefore, it can be concluded that the level sets of l are bounded. let u ∈ k be a solution of (1). then 〈tû + b|û|−f,w − û〉≥ 0, ∀w ∈ k. (14) using the pseudomonotonicity of the operators t and b in (14), we obtain 〈tw + b|w|−f,w − û〉≥ 0. (15) taking w = pk[u−ρtu−ρb|u| + ρf] in (15), we have 〈tpk[u−ρtu−ρb|u| + ρf],pk[u−ρtu−ρb|u| + ρf] − û〉≥ 0. (16) setting v = û, u = pk[u−ρtu−ρb|u| + ρf], and z = u−ρtpk[u−ρtu−ρb|u| + ρf] in (6), we get 〈pk[u−ρtu−ρb|u| + ρf] −u + ρtpk[u−ρtu−ρb|u| + ρf], û−pk[u] −ρtu−ρb|u| + ρf]〉≥ 0. (17) adding (16) and (17) 〈−ρtpk[u−ρtu−ρb|u| + ρf] + pk[u−ρtu−ρb|u| + ρf] −u+ ρtpk[u−ρtu−ρb|u| + ρf], û−pk[u]〉≥ 0, (18) and using (8), we get 〈−r(u), û−u + r(u)〉≥ 0, (19) which shows that 〈u− û,r(u)〉≥ ‖r(u)‖2. (20) int. j. anal. appl. 18 (3) (2020) 346 hence from (9), (13) and (19), we get d dt l(u) = dl du du dt = 〈2(u− û),γ{pk[u−ρtu−ρb|u| + ρf] −u}〉 = 2γ〈u− û,r(u)〉 ≤−2γ‖r(u)‖2 ≤ 0. this proves that l(u) is global lyapunov function for the dynamical system (9) and hence the dynamical system (9) is lyapunov stable. since, we have {u(t) : t ≥ t0} ⊂ k0, where k0 = {u ∈ k : l(u) ≤ l(u0)} and l(u) is continuously differentiable function on the closed and bounded set k, lasalle’s invariance principle shows the convergence of the trajectory to the largest subset ϕ of the following subset: x = {u ∈ k; dl dt = 0}. if dl dt = 0, then, from (20), we have ‖r(u)‖2 = 0 that is ‖u−pk[u−ρtu−ρb|u| + ρf]‖ = 0. (21) using (21) in (9), we get du dt = 0, which implies that u is an equilibrium point of the system (9). conversely, if du dt = 0. then from(20), we have dl dt = 0. thus, we conclude that x = {u ∈ k; dl dt = 0} = k0 ∩ k̂, where the nonempty set x is convex and invariant set which is contained in the solution set k̂. so, limn→∞ dis(u(t),x) = 0. (22) using definition 7 and (22), we obtain the global convergence of the dynamical system (9) to the solution set of (1). in particular, if x = {û}, then limn→∞u(t) = û, int. j. anal. appl. 18 (3) (2020) 347 which proves that the dynamical system (9) is globally asymptotically stable. � theorem 3.3. let the operators t and b be lipchitz continuous with constants β1 > 0 and β2 > 0, respectively. if γ < 0, then the projected dynamical system (9) globally exponentially converges to the unique solution of the absolute value variational inequality (1). proof. it can be seen from theorem 3.1 that a unique and continuously differentiable solution of the dynamical system (9) exists over the interval [t0,∞). so, dl dt = 2γ〈u(t) − û,pk[u(t) −ρtu(t) −ρb|u(t)| + ρf] −u(t)〉 = −2γ‖u(t) − û‖2 + 2γ〈u(t) − û,pk[u(t) −ρtu(t) −ρb|u(t)| + ρf] − û〉, (23) where û ∈ k is the solution of the absolute value variational inequality (1). so, we have û = pk[û−ρtû−ρb|û| + ρf]. from the nonexpansivity of pk and the lipchitz continuity of the operators t and b, we have ‖pk[u−ρtu−ρb|u| + ρf] −pk[û−ρtû−ρb|û| + ρf]‖ ≤ ‖u− û‖ + ρ‖tu−tû‖ + ρ‖b|u|−b|û|‖ ≤ (1 + ρ(β1 + β2))‖u− û‖. (24) from (23) and (24), we obtain d dt ‖u− û‖2 ≤−2γ‖u(t) − û‖2 + 2γ(1 + ρ(β1 + β2))‖u− û‖2 = −2γθ‖u− û‖2. (25) where θ = 1 + ρ(β1 + β2). thus, for γ = −γ1, where γ1 is a positive constant, we get ‖u− û‖2 ≤‖u(t0) −u‖e−θγ1(t−t0). hence, using definition 8, it is proved that the trajectory of the solution of the system (9) will converge globally exponentially to the unique solution of the absolute value variational inequality (1). � int. j. anal. appl. 18 (3) (2020) 348 4. iterative methods in this section, we use the projected dynamical system (9) associated with absolute variational inequality (1) to suggest and analyze some iterative schemes, which will be used to obtain the solution of absolute value variational inequalities. for the numerical computations of the suggested algorithms, the projection operator, pk, is defined by pk(ui) =   u, ‖u−s‖≤ r s + r(u−s) ‖u−s‖ , ‖u−s‖ > r, if k = {u ∈ rn : ‖u−s‖≤ r,s ∈ rn,r > 0}. consider the dynamical system (9) with γ = 1, du dt + u = pk[u−ρtu−ρb|u| + ρf], u(t0) = u0. (26) for a given η ∈ [0, 1], using forward difference schemes, we discretize (24) and suggest the following iterative algorithms to find the solution of absolute value variational inequalities. algorithm 4.1. for a given u0 ∈ k, compute un+1 by the iterative scheme un+1 = pk[η(un+1 −un) + 1 + h h un − un+1 h −ρtun+1 −ρb|un+1| + ρf], n = 0, 1, 2, ... (27) which is an implicit method. for the implementation of algorithm 4.1, we suggest the following two-step, iterative method. algorithm 4.2. for a given u0 ∈ k, compute un+1 by the iterative schemes wn = pk[un −ρtun −ρb|un| + ρf] un+1 = pk[η(wn −un) + 1 + h h un −ρtwn −ρb|wn| + ρf − wn h ]. algorithm 4.2 is a new two-step iterative method for finding the solution of absolute value variational inequality (1). we also consider some special cases of algorithm 4.2. (1) for η = 0, algorithm 4.2 collapses to the following iterative method. int. j. anal. appl. 18 (3) (2020) 349 algorithm 4.3. for a given u0 ∈ k, compute un+1 by the iterative scheme wn = pk[un −ρtun −ρb|un| + ρf] un+1 = pk[ 1 + h h un − wn h −ρtwn −ρb|wn| + ρf], algorithm 4.3 is extragradient type methods in the sense of korpelevich [18]. (2) for η = 1, algorithm 4.2 reduces to the following iterative method. algorithm 4.4. for a given u0 ∈ k, compute un+1 by the iterative scheme wn = pk[un −ρtun −ρb|un| + ρf] un+1 = pk[wn −ρtwn −ρb|wn| + ρf]. algorithm 4.4 is modified extragradient type iterative method in the sense of noor [35, 36]. (3) for η = 1 2 , algorithm 4.2 transforms into the following iterative method. algorithm 4.5. for a given u0 ∈ k, compute un+1 by the iterative scheme wn = pk[un −ρtun −ρb|un| + ρf] un+1 = pk[ h + 2 2h un − h− 2 2h wn −ρtwn −ρb|wn| + ρf]. clearly, algorithm 4.2 contains the known iterative methods such as extragradient method by korpelevich [18], modified extragradient method by noor [35, 36] as special cases. this shows that algorithm 4.2 is quite general and unifying one. we now discuss the convergence of algorithm 4.2 which is the main motivation of the next result. theorem 4.1. let u ∈ k be a solution of absolute value variational inequality (1). let un+1 be the approximate solution obtained from (26). if t and b are monotone operators, then ‖u−un+1‖2 ≤‖u−un‖2 −‖un −un+1‖2. (28) proof. let u ∈ k be a solution of absolute value variational inequality (1) i.e; < 〈u + b|u|−f,v −u〉≥ 0,∀v ∈ k. int. j. anal. appl. 18 (3) (2020) 350 using monotonicity of the operators t and b, we get 〈tv + b|v|−f,v −u〉≥ 0,∀v ∈ k. (29) taking v = un+1 in (28), we obtain 〈tun+1 −b|un+1|−f,un+1 −u〉≥ 0. (30) using lemma 2.1, algorithm 4.1 can be rewritten in the following equivalent form, that is 〈(ρtun+1 + ρb|un+1|−ρf + 1 + h h −η)un+1 −( 1 + h h −η)un,v −un+1〉≥ 0. (31) subtituting v=u in (30) to have, 〈(ρtun+1 + ρb|un+1| − ρf + ( 1 + h h −η)un+1 −( 1 + h h −η)un,u−un+1〉≥ 0. (32) from (29) and (31), we obtain ( 1 + h h −η)〈un+1 −un,u−un+1〉 ≥ ρ〈tun+1 + |un+1|−f,un+1 −u〉 ≥ 0. (33) from (32) and using inequality 2〈u,v〉 = ‖u + v‖2 −‖u‖2 −‖v‖2, ∀u,v ∈ h, we get ‖u−un+1‖2 ≤‖u−un‖2 −‖un −un+1‖2, which is (27), the required result. � theorem 4.2. let u ∈ k be the solution of absolute value variational inequality (1). let un+1 be the approximate solution obtained from algorithm 4.1. if t and b are monotone operators, then un+1 converges to u ∈ k satisfying (1). proof. let t and b be a monotone operators. then, from (27), it follows that sequence {ui}∞i=1 is a bounded sequence and ∞∑ n=1 ||un+1 −u||2 ≤ ||u−u0||2. which shows that limn→∞||un+1 −un||2 = 0. (34) int. j. anal. appl. 18 (3) (2020) 351 since {uj}∞j=1 is bounded, so there exists a limit point u ∈ k such that subsequence {ujl} ∞ l=j converges to. replacing un by unj in (32) and taking the limit ui −→∞ , we obtain 〈tu + b|u|−f,v −u〉≥ 0, ∀v ∈ k. which implies that u is the solution of (1)and ‖un+1 −u‖2 ≤‖u−un‖2, it follows from the above inequality that the limit point un is unique and limn→∞un+1 = u, which completes the proof. � 5. computational results in this section, we consider two numerical examples to examine the efficiency and implementation of the suggested algorithm 4.2 from the aspects of the number of iterations and the elapsed cpu time in seconds (denoted by toc). the experiments in both the examples are performed with intel(i7) 2.2ghz, 8gb ram, and the codes are written in matlab r2010. example 5.1. [8] consider random matrix t and f, for absolute value variational inequality (1) in matlab code as n = input(dimensionofmatrixt =); rand(state, 0); r = rand(n,n); f = rand(n, 1); t = r′ ∗r + n∗eye(n); b = eye(n); with random initial guess. the comparison between algorithm 16, the yong method [49] and algorithm 2.1 [8] is presented in table 1. the average times taken by cpu for every order of n are presented by toc, in table 1. note that for any size of dimension n, algorithm 16 converges faster than both the yong method [49] and algorithm 2.1 [8]. int. j. anal. appl. 18 (3) (2020) 352 table 1. numerical results for example 5.1. order yong method algorithm 2.1 [8] algorithm 4.2 heading no. of iter. toc no. of iter. toc no. of iter. toc 4 2 2.230 3 .00050 3 .001 8 2 3.340 3 .00065 3 .00099 16 3 3.790 3 .00078 3 .00099 32 2 4.120 3 .00101 3 .00099 64 3 6.690 3 .01224 3 .001 128 3 12.450 3 .04209 3 .001 256 3 34.670 3 .06714 3 .00099 512 5 79.570 3 .32896 3 .00099 1024 5 157.12 3 .83559 3 .004 example 5.2. [22] we randomly select a matrix a according to the following structure: a = round(100 ∗ (eye(n; n) − .02 ∗ (2 ∗rand(n; n) − 1)); b = eye(n); b(i) = (−i)i, i = 1, 2, ...,n. table 2. numerical results for example 5.2. order p-ssor p-cg p-jacobi algorithm 4.2 500 linter number 6 6 18 18 cpu time .0205 .0397 .0040 .0069 1000 linter number 6 6 18 38 cpu time .0954 .1953 .0573 .0560 1500 linter number 6 6 31 38 cpu time .2427 .4852 .1613 .1230 2000 linter number 6 6 38 38 cpu time .4888 .9111 .3541 .2100 from table 2, all the iteration steps (linter number) and the elapsed cpu times show that algorithm 16 is more efficient than the p-ssor, p-jacobi and p-cg iteration methods. int. j. anal. appl. 18 (3) (2020) 353 conclusion in this paper, we have considered and analyzed dynamical system associate with absolute value variational inequalities. we have discussed the existence of a solution of the absolute value variational inequalities using only the lipschitz continuity of the underlying operators. asymptotic stability of the solution is investigated using the lyapunov functions. dynamical system is used to consider some extrgradient type methods for solving the absolute value equations. some numerical examples are given, which shows that the proposed methods perform better then the previous ones. ideas and techniques of this paper may inspire further research in this dynamical field. acknowledgement: the author would like to thank the rector, comsats university islamabad, islamabad, pakistan, for providing excellent research and academic environments conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. h. ahn, iterative methods for linear complementarity problems with upper bounds on primary variables, math. program. 26(3)(1983), 295-315. [2] c. bakxchi and a. capello, disequazioni variationalies quasi varionali, applicantioni a problemi di frontiera libera, vols. i and ii, bologna, italia. (1978). [3] b. b. bin-mohsin, m. a. noor, k. i. noor and r. latif, projected dynamical system for variational inequalities, j. adv. math. stud. 11(1)(2018), 1-9. [4] r. w. cottle and g. b. dantzig, complementarity pivot theory of mathematical programming. linear algebra appl. 1(1968), 103-125. [5] i. c. dolcetta and u. mosco, implicit complementarity problems and quasi variational inequalities. var. ineq. and compl. prob. th. app. (eds. r.w. cottle, f. giannessi and j.l. lions) john wiley and sons, new york, new jersey, 1980. [6] j. dong, d. h. zhang and a. nagurney, a projected dynamical systems model of general financial equilibrium with stability analysis. math. comput. model. 24(2)(1996), 35-44. [7] p. dupuis and a. nagurney, dynamical systems and variational inequalities. ann. oper. res. 44(1)(1993), 7-42. [8] h. esmaeili, m. mirzapour and e. mahmoodabadi, a fast convergent two-step iterative method to solve the absolute value equation. u.p.b. sci. bull., ser. a. 78(1)(2016), 25-32. [9] g. fichera, g. problemi elastostatistic con vincoli unilaterali il prolema di signorini con ambigue condizone as contorno atti. acad. naz. lincei. mem. cl. sci. fis. mat. nature. sez. la., 8(7)(1964), 91-140. [10] t. l. friesz, d. h. berstein, n. j. mehta, r. l. tobin and s. ganjliazadeh, day-to day dynamic network disequilibrium and idealized traveler information systems. oper. res. 42(6)(1994), 1120-1136. [11] t. l. friesz, d. h. berstein and r. stough, dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows. transport. sci. 30(1)(1996), 14-31. [12] g. glowinski, j. l.lions and r. tremolieres, numerical analysis of variational inequalities. northholland, amsterdam, 1981. int. j. anal. appl. 18 (3) (2020) 354 [13] s. l. hu and z. h. huang, a note on absolute value equations. optim. lett. 4(3)(2010), 417-424. [14] s. l. hu and z. h. huang, a generalized newton method for absolute value equations associated with second order cones. j. comput. appl. math. 235(5)(2011), 1490-1501. [15] s. karamardian, the complementarity problem. math. program. 2(1)(1972), 107-109. [16] y. f. ke and c. f. ma, sor-like iteration method for solving absolute value equations. appl. math. comput. 311(2017), 195-202. [17] d. kinderlehrer and g. stampacchia, an introduction to variational inequalities and their applications. siam, philadelphia, 1980. [18] g. m. korpelevich, an extragradient method for finding saddle points and for other problems. ekonomika mat. metody, 12(4)(1976), 747-756. [19] c. e. lemke, bimatrix equilibrium points and mathematical programming. manage. sci. 11(7)(1965), 681-689. [20] j. l. lions and g. stampacchia, variational inequalities. commun. pure appl. math. 20(3)(1967), 493-519. [21] j. l. lions, optimal control of systems governed by partial differential equations. springer-verlag, berlin. 1971. [22] c. q. lv and c. f. ma, picard splitting method and picard cg method for solving the absolute value equation. j. nonlinear sci. appl. 10(2017), 3643-3654. [23] o. l. mangasarian, the linear complementarity problem as a separable bilinear program. j. glob. optim. 6(2)(1995), 153-161. [24] o. l. mangasarian and r. r. meyer, absolute value equations, linear algebra appl. 419(2-3)(2006), 359-367. [25] o. l. mangasarian, absolute value programming. addison-wesley publishing, boston. 2007. [26] k. g. murty, linear complementarity, linear and nonlinear programming. heldermann verlag, berlin. 1988. [27] m. a. noor, on variational inequalities. ph.d. thesis. burnel university, u.k. (1975). [28] m. a. noor, mildly nonlinear variational inequalities. mathematica. 24(47)(1982), 99-110. [29] m. a. noor, strongly nonlinear variational inequalities. c. r. math. rep. acad. sci. canada. 4(4)(1982), 213-218. [30] m. a. noor, iterative methods for a class of complementarity problems. j. math. anal. appl. 133(2)(1988), 366-382. [31] m. a. noor, some recent advances in variational inequalities. part i. basic concepts. new zealand j. math. 26(1)(1997), 53-80. [32] m. a. noor, some recent advances in variational inequalties. part ii. other concepts. new zealand j. math. 26(2)(1977), 229-255. [33] m. a. noor, stability of the modified projected dynamical systems. computer math. appl. 44(2002), 1-5. [34] m. a. noor, resolvent dynamical systems for mixed variational inequalities. korean j. comput. appl. math. 9(1)(2002), 15-26. [35] m. a. noor, some developments in general variational inequalities. appl. math. comput. 152(2004), 199-277. [36] m. a. noor, on an implicit method for nonconvex variational inequalities,.j. optim. theory appl. 147(2)(2010), 411-417. [37] m. a. noor and k. i. noor, iterative methods for variational inequalities and nonlinear programming. oper. res. verf. 31(1979), 455-463. [38] m. a. noor, y. j. wang and n. xiu, some new projection methods for variational inequalities. appl. math. comput. 137(2)(2003), 423-435. [39] m. a. noor, k. i. noor and a. bnouhachem, on a unified implicit method for variational inequalities. j. comput. appl. math. 249(2013), 69-73. [40] m. a. noor, m, k. i. noor and s. batool, on generalized absolute value equations. u.p.b. sci. bull., series a, 80(4)(2018), 63-70. int. j. anal. appl. 18 (3) (2020) 355 [41] m. a. noor, j. iqbal, k. i. noor and e. al-said, generalized aor method for solving absolute value complementarity problems j. appl. math. 2012(2012), 743861. [42] o. prokopyev, o. on equivalent reformulation for absolute value equations. comput. optim. app., 44(3)(2009), 363-372. [43] j. rohn, a theorem of the alternatives for the equation ax + b|x| = b. linear and multilinear algebra, 52(6)(2004), 421-426. [44] j. rohn, an algorithm for computing all solutions of an absolute value equation. optim. lett. 6(5)(2011), 851-856. [45] g. stampacchia, formes bilineaires coercivites sur les ensembles convexes. c. r. acad. sci. paris. 258(1964), 4413-4416. [46] y. s. xia and j. wang, on the stability of globally projected dynamical systems. j. optim. theory appl. 106(1)(2000), 129-150. [47] y. s. xia and j. wang, a recurrent neural network for solving linear projection equation. neural networks 13(3)(2000), 337-350. [48] y. s. xia, on convergence conditions of an extended projection neural network. neural comput. 17(3)(2005), 515-525. [49] l. yong, particle swarm optimization for absolute value equations. j. comput. inform. syst. 6(7)(2010), 2359-2366. 1. introduction 2. formulations and basic results 3. existence theory 4. iterative methods 5. computational results conclusion references international journal of analysis and applications volume 19, number 2 (2021), 280-287 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-280 fixed point results for ω-interpolative chatterjea type contraction in quasi-partial b-metric space pragati gautam1,∗, swapnil verma1, manuel de la sen2, sejal sundriyal1 1department of mathematics, kamala nehru college, university of delhi, august kranti marg, new delhi 110049, india 2institute of research and development of processes, university of basque country, campus of leioa(bizkaia)-aptdo, 644-bilbao, bilbao, 48080, spain ∗corresponding author: pgautam@knc.du.ac.in abstract. the purpose of this paper is to revisit chatterjea type contraction and determine some fixed point results for interpolative chatterjea type contraction mapping in the setting of quasi-partial b-metric space using the concept of ω-admissibility introduced by popescu. also we present some useful examples to elucidate relevance of the concept. 1. introduction in the diversified field of non-linear analysis, banach [1] contraction principle holds a significant position. the fixed point theorems are used to demonstrate the uniqueness of a solution of differential equations, fredholm integral equations and picard theorem etc. forging ahead banach’s approach, many celebrated authors [2–7] introduced distinctive concepts. in the year 1972, chatterjea [8] inaugurated his contraction defined as received february 7th, 2021; accepted march 1st, 2021; published march 17th, 2021. 2010 mathematics subject classification. 46t99, 47h10, 54h25. key words and phrases. quasi-partial b-metric space; fixed point; ω admissible; interpolation; chatterjea contraction. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 280 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-280 int. j. anal. appl. 19 (2) (2021) 281 let (x, d) be a complete metric space. a self mapping h : x → x is called chatterjea type contraction if d(hσ, hθ) ≤ δ[d(σ, hθ) + d(θ, hσ)] for all σ, θ ∈ x, where δ ∈ (0, 1/2). then he interestingly proved that it has a unique fixed point in complete metric space. additionally this result was proved by mishra et al. [9] in complete quasi-partial b-metric space for interpolative chatterjea type contraction, which can be designated as, let (x, qpb) be a complete quasi-partial b-metric space. a self mapping h : x → x is called interpolative chatterjea type contraction if there exists δ ∈ [0, 1 s ),ρ ∈ (0, 1) such that qpb(hσ, hθ) ≤ δ[qpb(σ, hθ)]ρ [ 1 s2 qpb(θ, hσ) ]1−ρ for all σ, θ ∈ x. afterwards as a modification in the concept of α-admissible maps, popescu [10] introduced ω-orbital admissible maps. in this research field, several authors [13–15] have made valuable contributions. in this paper, we commence the concept of ω-interpolative chatterjea contraction in quasi-partial b-metric space and deliver relevant examples. 2. preliminaries definition 2.1 [10]: let ω : x × x → [0, ∞) be a mapping and x 6= φ. a self mapping h : x → x is called ω-admissible if, ω(σ, hσ) ≥ 1 =⇒ ω(hσ, h2σ) ≥ 1 for all σ ∈ x. in order to ignore the continuity of contractive mappings, we often consider the following condition. (m) if we take a sequence θn in x such that ω(θn, θn+1) ≥ 1, for all n. also as n →∞, θn → θ ∈ x, then from θn there exists θn(k) such that for all k, ω(θn(k), θ) ≥ 1. definition 2.2 [11] : a quasi-partial metric space on a set non-empty set x is a function qp: x×x → [0, ∞) that satisfies the following properties : [qp1] if qp(σ, σ) = qp(σ, θ) = qp(θ, θ) then σ = θ [qp2] qp(σ, σ) ≤ qp(σ, θ) [qp3] qp(σ, σ) ≤ qp(θ, σ) int. j. anal. appl. 19 (2) (2021) 282 [qp4] qp(σ, θ) + qp(δ, δ) ≤ qp(σ, δ) + qp(δ, θ) for all σ, θ, δ ∈ x. definition 2.3 [12] : a quasi-partial b-metric space on a set x 6= φ is a function qpb : x × x → [0, ∞) such that for a real number s ≥ 1 satisfies the following properties : [qpb1] if qpb(σ, σ) = qpb(θ, θ) = qpb(σ, θ) then σ = θ [qpb2] qpb(σ, σ) ≤ qpb(σ, θ) [qpb3] qpb(σ, σ) ≤ qpb(θ, σ) [qpb4] qpb(σ, θ) ≤ s[qpb(σ, δ) + qpb(δ, θ)] −qpb(δ, δ) for all σ, θ, δ ∈ x. here, we present an example to show the usability of the concept. example 2.1 : let x= [ 0, π 2k ] . define qpb(σ, θ) = sink|σ2 −θ2|, where k ≥ 1 here, qpb(σ, σ) = sink|σ2 −σ2| = 0 = qpb(θ, θ), qpb(σ, θ) = sink|σ2 −θ2| then 0 = sink|σ2 −θ2| =⇒ σ = θ i.e [qpb1] satisfied. qpb(σ, σ) = 0 ≤ qpb(σ, θ) as 0 ≤ sink|σ2 −θ2| i.e [qpb2] satisfied. qpb(σ, σ) = 0 ≤ qpb(θ, σ) as 0 ≤ sink|θ2 −σ2| i.e [qpb3] satisfied. as σ, θ, δ ∈ x |σ2 −θ2| ≤ π 2k , |σ2 −θ2| + |δ2 −θ2| ≤ π 2k k[|σ2 − δ2| + |δ2 −θ2|] ≤ π 2 , since sin k is increasing. qpb(σ, θ) + qpb(δ, δ) = sink|σ2 −θ2| ≤ sink(|σ2 −δ2| + |δ2 −θ2|) ≤ k(|σ2 − δ2| + |δ2 −θ2|) ≤ k(sink(|σ2 − δ2| + |δ2 −θ2|)) ≤ k[qpb(σ, δ) + qpb(δ, θ)] ≤ s[qpb(σ, δ) + qpb(δ, θ)] where, s ≥ k ≥ 1 i.e. [qpb4] satisfied. therefore, (x,qpb) is a quasi-partial-b metric space. int. j. anal. appl. 19 (2) (2021) 283 3. main results here, we introduce the concept of ω-interpolative chatterjea type contractions in quasi-partial b-metric space. definition 3.1 : let (x, qpb) be a complete quasi-partial b-metric space. a self mapping h : x → x is called ω-interpolative chatterjea type contraction if there exists δ ∈ [0, 1 s ), ρ ∈ (0, 1) such that ω(σ, θ)qpb(hσ, hθ) ≤ δ[qpb(σ, hθ)]ρ [ 1 s2 qpb(θ, hσ) ]1−ρ for all σ, θ ∈ x. our main result is as follows : theorem 3.1. let h : x → x be an ω-admissible self mapping which forms ω-interpolative chatterjea type contraction on a complete quasi-partial b-metric space (x, qpb). if there exists σ0 ∈ x such that ω(σ0, hσ0) ≥ 1, then h has a fixed point in x. proof : let σ0 ∈ (x, qpb) such that ω(σ0, hσ0) ≥ 1. let us consider a sequence σn defined as σn = h n(σ0), n ≥ 0. considering for some n0, if σn0 = σn0+1, this implies that σn0 is a fixed point of h. if σn0 6= σn0+1, for all n ≥ 0 then, qpb(σn, hσn) = qpb(σn, hσn+1) > 0 also h is ω-admissible, ω(σ1, σ2) = ω(hσ0, hσ1) ≥ 1. =⇒ ω(σn, σn+1) ≥ 1, for all n ≥ 0. taking σ = σn and θ = σn−1, we get qpb(σn+1, σn) ≤ ω(σn, σn+1)qpb(hσn, hσn−1) ≤ δ[qpb(σn, hσn−1)]ρ [ 1 s2 qpb(σn−1, hσn) ]1−ρ ≤ δ[qpb(σn, σn)]ρ [ 1 s2 qpb(σn−1, σn+1) ]1−ρ ≤ δ[qpb(σn+1,σn)]ρ [ 1 s2 [s [qpb(σn−1, σn) + qpb(σn, σn+1)] −qpb(σn, σn)] ]1−ρ ≤ δ[qpb(σn+1, σn)]ρ [ s s2 [qpb(σn−1, σn) + qpb(σn, σn+1)] ]1−ρ (3.1) ≤ δ[qpb(σn+1, σn)]ρ [ 1 s [qpb(σn−1, σn) + qpb(σn, σn+1)] ]1−ρ int. j. anal. appl. 19 (2) (2021) 284 if qpb(σn−1, σn) ≤ qpb(σn, σn+1) for all n ≥ 1, then 1 s [qpb(σn−1, σn) + qpb(σn, σn+1)] 1−ρ ≤ qpb(σn, σn+1)1−ρ that is, 1 s [qpb(σn−1, σn) + qpb(σn, σn+1)] ≤ qpb(σn, σn+1) but qpb(σn+1, σn) ≤ qpb(σn−1, σn), which is a contradiction as per equation 3.1. thus qpb(σn−1, σn) is a decreasing sequence. now, let limn→∞qpb(σn−1,σn) = l. as we have, from equation 3.1 qpb(σn+1, σn) ≤ ω(σn, σn+1)qpb(hσn, hσn−1) ≤ δ[qpb(σn+1, σn)]ρ [ 1 s2 qpb(σn−1, σn) ]1−ρ ω(σ, θ)qpb(σn+1, σn) 1−ρ ≤ δ [ [ 1 s2 qpb(σn−1, σn) ]1−ρ ω(σn, σn−1)qpb(σn+1, σn) ≤ δ 1 1−β [ [ 1 s2 qpb(σn−1, σn) ] (3.2) ω(σn, σn−1)qpb(σn+1,σn) ≤ δqpb(σn−1, σn) ≤ ϕnqpb(σ0, σ1) so putting n →∞ in equation 3.2, we get l = 0. now, to show σn is cauchy sequence. let n, t ∈ n. qpb(σn, σn+t) ≤ sqpb(σn, σn+1) + s2qpb(σn+1, σn+2) + . . . + stqpb(σn+t−1, σn+t) ≤ [sδn + s2δn+1 + . . . + stδn+t−1]qpb(σ0, σ1) (3.3) ≤ st n+t−1∑ i=n δiqpb(σ0, σ1) ≤ st ∞∑ i=n δiqpb(σ0, σ1) · · · from equation 3.3, lim n→∞ qpb(σn, σn+t) = lim m→∞, n→∞ qpb(σn+m), σn+m+t) ≤ st lim m→∞ ∞∑ i=m lim n→∞ δiqpb(σn, σn+1) = 0 now, if σn 6= hσn. qpb(σn+1, hη) = qpb(hσn, hη) ≤ δ[qpb(σn, hη)]ρ [ 1 s2 qpb(η, hµn) ]1−ρ ≤ δ[qpb(σn, hη)]β[qpb(η, σn+1)]1−ρ int. j. anal. appl. 19 (2) (2021) 285 for η ∈ x. here, for n →∞, qpb(η, hη) = 0. this is a contradiction and hence hη = η. corollary 3.1 let (x,qpb) be a complete quasi-partial b-metric space whose subsets ξ1 and ξ2 are closed. suppose that h : ξ1 ∪ ξ2 → ξ1 ∪ ξ2 satisfies : ω(σ, θ)qpb(hσ, hθ) ≤ δ[qpb(σ, hθ)]ρ [ 1 s2 qpb(θ, hσ) ]1−ρ for all σ ∈ ξ1 and θ ∈ ξ2 such that σ, θ ∈ x�fix(h), where ρ ≥ 0, s ≥ 1. if h(ξ1) ⊆ ξ2 and h(ξ2) ⊆ ξ1, then there exists a fixed point of h in ξ1 ∩ ξ2. proof : in theorem 3.1, it is enough to take, ω(σ, θ) =   1 if(ξ1 × ξ2) ∪ (ξ2 × ξ1) 0 otherwise corollary 3.2 suppose (x, qpb, �) be a complete partially-ordered quasi-partial b-metric space. let h : x → x be the mapping such that: ω(σ, θ)qpb(hσ, hθ) ≤ δ[qpb(σ, hθ)]ρ [ 1 s2 qpb(θ, hσ) ]1−ρ such that σ, θ ∈ x�fix(h) where ρ ≥ 0, s ≥ 1. let us assume the following : a) h is non decreasing with respect to partial order �; b) there exists σ0 ∈ x such that σ0 � hσ0; c) h is continuous on (x, qpb). then h has a fixed point in x . proof : in theorem 3.1, it is enough to take, ω(σ,θ) =   1 if(σ � θ)or(θ � σ) 0 otherwise example 3.1 : let us consider the set x = [0, 3] with quasi-partial b-metric defined as qpb(σ, θ) = sink|σ2 −θ2| and h be a self-mapping on x which is defined as hσ =   5 2 ifσ ∈ [2, 3] 2 7 ifσ ∈ [0, 2] and taking, ω(σ, θ) =   1 if(σ, θ) ∈ [2, 3] 0 otherwise let σ, θ ∈ x be such that σ 6= hσ, θ 6= hθ and ω(σ, θ) ≥ 1 σ, θ ∈ [2, 3], and we have hσ = hθ = 5 2 . therefore, definition 3.1 holds, for σ0 = 3. ω(3, h3) = ω(3, 5 2 ) = 1, ω(σ, θ) ≥ 1 for σ, θ ∈ x. so σ, θ ∈ [2, 3] and hσ = hθ ∈ [2, 3]. thus h is ω-orbital admissible as ω(hσ, hθ) ≥ 1. now we show that condition (m) holds, let us take a sequence θn in x such that ω(θn, θn+1) ≥ 1 for all n, int. j. anal. appl. 19 (2) (2021) 286 figure 1. the fixed points of h are 5 2 and 2 7 . then θn ⊂ [2, 3]. also as n →∞, θn → v ∈ x, we get |θn −v|→ 0. therefore v ∈ [2, 3], ω(θn, v) = 1. hereby theorem 3.1 holds true and the fixed points of h are 5 2 and 2 7 as shown in figure 1. 4. conclusion the accession of this study is to commence the proposition of interpolative chatterjea type contraction on ω-admissible mapping in quasi-partial b-metric space. ω-admissibility finds it’s real world applications in varying fields be it in classical game theory for finding behaviour in multi-player games or even infinite games that are played on graphs. apart from this it is also used in deciding the extensions of dls by concrete domains. this concept has been conceded in many researches earlier. the current research can also be exercised effectively in all these areas of study. acknowledgements: all authors are grateful to the spanish government for grant rti2018-094366b-i00 (mciu/aei/feder, ue) and to the basque government for grant it1207-19. funding: no external funding has been received in this research. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. banach, sur les operations dans les ensembles abstraits et leur application aux equations integrales, fund. math. 3 (1922), 133-181. [2] r. kannan, some results on fixed points–ii, amer. math. mon. 76 (1969), 405–408. [3] e. karapinar, revisiting the kannan type contractions via interpolation, adv. theory nonlinear anal. appl. 2 (2018), 85–87. [4] a. gupta, p. gautam, some coupled fixed point theorems on quasi-partial b-metric spaces, int. j. math. anal. 9 (2015), 293–306. int. j. anal. appl. 19 (2) (2021) 287 [5] i.a. bakhtin, the contraction principle in quasi metric spaces, funct. anal. 30 (1989), 26–37. [6] s. czerwik, contraction mappings in b-metric spaces, acta math. inform. univ. ostrav. 1 (1993), 5–11. [7] s.g. matthews, partial metric topology, ann. n. y. acad. sci. 728 (1994), 183–197. [8] s.k. chatterjea, fixed-point theorems, c. r. acad. bulg. sci. 25 (1972), 727-730. [9] p. gautam, v.n. mishra, r. ali, s. verma, interpolative chatterjea and cyclic chatterjea contraction on quasi-partial b-metric space, aims math. 6 (2020), 1727-1742. [10] o. popescu, some new fixed point theorems for α-geraghty contraction type maps in metric spaces, fixed point theory appl. 2014 (2014), 190. [11] e. karapınar, i.m. erhan, a. öztürk, fixed point theorems on quasi-partial metric spaces, math. comput. model. 57 (2013), 2442-2448. [12] a. gupta, p. gautam, quasi-partial b-metric spaces and some related fixed point theorems, fixed point theory appl. 2015 (2015), 18. [13] v.n. mishra, l.m. sánchez ruiz, p. gautam, s. verma, interpolative reich–rus–ćirić and hardy–rogers contraction on quasi-partial b-metric space and related fixed point results, mathematics. 8 (2020), 1598. [14] p. gautam, l.m. sánchez ruiz, s. verma, fixed point of interpolative rus–reich–ćirić contraction mapping on rectangular quasi-partial b-metric space, symmetry. 13 (2021), 32. [15] p. gautam, s. verma, fixed point via implicit contraction mapping on quasi-partial b-metric space, j. anal. (2021). https://doi.org/10.1007/s41478-021-00309-6. https://doi.org/10.1007/s41478-021-00309-6 1. introduction 2. preliminaries 3. main results 4. conclusion references international journal of analysis and applications volume 18, number 6 (2020), 981-988 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-981 received may 14th, 2020; accepted june 26th, 2020; published september 29th, 2020. 2010 mathematics subject classification. 62p20, 91g70. key words and phrases. energy productivity; globalization; cointegration; error correction model; iran. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 981 how globalization is related to energy productivity? (a mathematical analysis on iran’s agricultural data) maryam ghasemi, reza moghaddasi*, yaaghoob zeraatkish department of agricultural economics, extension and education, science and research branch, islamic azad university, tehran, iran *corresponding author: r.moghaddasi@srbiau.ac.ir abstract. agriculture is recognized as a key economic sector for almost all developing nations including iran. nowadays, world economy has been highly integrated as economic performance of a given country is more susceptible to policies taken by other countries than four decades ago. so, investigating the possible impacts of this new worldwide phenomenon (globalization) would be a vital issue. this paper is an empirical examination of the association between globalization and energy productivity in the iran's agriculture context by using time series econometrics techniques including causality and cointegration tests for the period 1967-2018. main results confirm existence of a unidirectional causality from globalization to agricultural energy productivity. other findings revealed that every one percent increase in globalization index is expected to lead in 0.75 percent rise in agricultural energy productivity in the long-run. the error correction coefficient is estimated at -0.25 implying that the effect of every shock imposed on the long-run equation would be vanished in just four years. 1. introduction energy is categorized among the essential drivers of global development in general and developing nations’ economic growth in particular [19,13,30,31,32]. some ecological economists believe that in the biophysical growth model, energy acts as the most influential factor [27,23,24,26,27,29]. also many studies have reported the impact of economic growth on export https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-981 int. j. anal. appl. 18 (6) (2020) 982 (growth-driven export) [20,1,14,33,34]. so, one may think of an association between energy use and export. in other words, trade is an important driver of economic growth and export enhancement promotes the economic activities and, thus, the energy demand [25,2,17,18,20,22]. on the other hand, globalization is currently a popular and controversial issue, though often remaining a loose and poorly-defined concept. sometimes too comprehensively, the term is used to encompass increases in trade and liberalization policies as well as reductions in transportation costs and technology transfer. as far as its impact is concerned, discussion of globalization tends to consider simultaneously its effects on economic growth, employment and income distribution. as many other developing nations, agriculture plays an important role in the iranian economy. based on recent data, it accounts for about 9.8% of gross domestic product, 22% of employment and more than 20% of non-oil exports (cbi1, 2018). also as a member of opec2, iran is an example of oil-dominant economies that has provided cheap energy to different sectors, including agriculture, for decades. it has been seeking non-oil export promotion and low dependency on oil export for three decades. so from policy making point of view it would be very informative to know how energy use has affected major economic variables (growth and export) of the sector. little empirical studies can be found dealing with the nexus between energy productivity (ep3), and globalization (g) in the iranian agriculture context. considering the vital role of agriculture in the iranian economy and rapid development of globalization in the world, the main motivation of current study can be defined as how globalization affects energy productivity in the sector. the remainder of the paper is structured as follows. next section deals with methodology employed. section 3 provides main obtained results and discusses, and finally section 4 presents conclusion. 1 central bank of the islamic republic of iran 2 organization of petroleum exporting countries 3 energy used per unit of output (agricultural value added in this study) int. j. anal. appl. 18 (6) (2020) 983 2. materials and methods 2.1. econometric techniques in this study the commonly used method in examination of the causal association between two series, namely; granger causality test is applied. considering two variables of interest including ep and g following equations are specified: 𝐸𝑃𝑡 = ∑ 𝑎𝑖 𝑝 1 𝐺𝑡−𝑖 + ∑ 𝑏𝑗 𝑞 1 𝐸𝑃𝑡−𝑗 + 𝑒1𝑡 (1) 𝐺𝑡 = ∑ 𝑚𝑖 𝑟 1 𝐸𝑃𝑡−𝑖 + ∑ 𝑛𝑗 𝑠 1 𝐺𝑡−𝑗 + 𝑒2𝑡 (2) where p, q, r, and s are lags, and 𝑒𝑖's are residuals assumed to possess all favorite statistical features. according to estimation results following alternatives are expected: ∑ 𝑎𝑖 𝑝 1 ≠ 0 → unidirectional causality from g to ep ∑ 𝑚𝑖 𝑟 1 ≠ 0 → unidirectional causality from ep to g ∑ 𝑎𝑖 𝑝 1 ≠ 0 and ∑ 𝑚𝑖 𝑟 1 ≠ 0 → bidirectional causality between g and ep ∑ 𝑎𝑖 𝑝 1 = 0 and ∑ 𝑚𝑖 𝑟 1 = 0 → independent variables after detecting causal possible causal relationship one should check existence of long-run association between variables. here the johansen-juselius (jj) test is applied. it lets to estimate not only magnitude of association but also presence of long-run equilibrium relationship between variables. based on jj test, if zt is a vector containing two non-stationary time series (𝐸𝑃𝑡 , 𝐺𝑡 ) then behavior of z can be stated via a vector autoregressive process: 𝑍𝑡 = 𝐶1𝑍𝑡−1 + … + 𝐶𝑘 𝑍𝑡−𝑘 + 𝑉𝑡 (3) equation 3 is commonly estimated by the ml1 methodology. in the case of existence of longrun association, the short-run dynamics can be related to long-run stable relationship via vectorerror-correction model (vecm). this model has a very important policy making implication as it reveals how the impact of a shock would be corrected as time passes [13,6,7,8,11,12,15]. 2.2. data all required information for estimation of equations 1-3 is taken from relevant sources including ministry of energy, ministry of agriculture and cbi. furthermore annual data for the period spanning from 1967 to 2018 is used in estimation process. 1 maximum likelihood int. j. anal. appl. 18 (6) (2020) 984 3. results and discussion descriptive statistics of variables of interest are presented in table 1. table 1. descriptive statistics variable description unit of measurement average min max standard deviation growth rate (%) ep agricultural energy productivity billion rials per one barrel oe energy 12321 8925 16539 1023 0.08 g kof globalization index 32.02 21.32 49.28 4.07 17.5 note: oe denotes oil equivalent as table 1 shows, both variables have followed upward trend. also, according to figures presented in table 2 both two variables are non-stationary, so we are allowed to examine possible long-run equilibrium relationship between variables. table 2. results of three unit root tests adf pp kpss variable first difference level first difference level first difference level -8.49** -1.02 -5.33** -3.12 0.18 0.32* ep -8.69** -1.13 -6.75** -1.27 0.28 0.85* g note: variables are in natural logarithm table 3 portrays the output of jj test of cointegration that clearly confirmed existence of one long-run equilibrium relationship. table 3. the results of jj test trace statistic maximum eigenvalue statistic null hypothesis value critical value 5% value critical value 5% r=0* 29.36 15.49 19.31 14.26 r=1 1.11 6.02 2.93 3.84 int. j. anal. appl. 18 (6) (2020) 985 table 4 reports results of granger causality test. existence of one way causal relationship from globalization to energy productivity in agriculture is confirmed. it means that by getting iranian economy more integrated into the world economy, energy productivity in agriculture will change. this is consistent with some previous studies in the field [9,28,14,3,4,5]. table 4. results of causality test null hypothesis no of observations f-statistic probability g doesn't granger cause ep 48 7.51 0.01 ep doesn't granger cause g 0.36 0.78 note: variables are in natural logarithm moreover, the long-run cointegrating coefficient is estimated at 0.75 implying that by one percent rise in globalization index one could expect 0.75 percent increase in energy productivity in the iran’s agriculture. some other studies including adebola, 2011; apergis and payne, 2010 have reported same results. in addition, the error correction coefficient is estimated at -0.25 meaning that the impact of any external shock (stemming from policies such as energy price liberalization) is expected to be completely vanished in just four years. 4. conclusion this study empirically dealt with the causal association between energy productivity and globalization in the context of iranian agriculture. annual information on two variables of interest is gathered from official national sources. results obtained from application of granger causality test reveals existence of a unilateral causal association from globalization to energy productivity in the agriculture sector. further examinations showed presence of a long-run equilibrium relationship between globalization and energy productivity. based on estimation results it is expected that one percent increase in globalization index lead to 0.75 raise in energy productivity. considering direct and relatively strong effect of globalization on energy productivity at one hand and undeniable role of agriculture in iranian economy at the other hand, taking measures to reduce susceptibility of domestic agriculture from foreign shocks (such as border measures and bilateral trade agreements) is highly recommended. int. j. anal. appl. 18 (6) (2020) 986 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. abbas. causality between exports and economic growth: investigating suitable trade policy for pakistan. eurasian j. bus. econ. 5 (2012), 91-98. [2] s.s. adebola. electricity consumption and economic growth: trivariate investigation in botswana with capital formation. int. j. energy econ. policy, 1 (2011), 32-46. [3] a. adewuyi, o. adeniyi. trade and consumption of energy varieties: empirical analysis of selected west africa economies. renew. sustain. energy rev. 47 (2015), 354-366. [4] m. ahmed, m. azam. causal nexus between energy consumption and economic growth for high, middle and low income countries using frequency domain analysis. renew. sustain. energy rev. 60 (2016), 653-678. [5] g. altinay, e. karagol. electricity consumption and economic growth: evidence from turkey. energy econ. 27 (2005), 49-56. [6] n. apergis, j.e. payne. energy consumption and growth in south america: evidence from a panel error correction model. energy econ. 32 (2010), 1421-1426. [7] v.c.r. chandran, s. sharma, k. madhavan. electricity consumption-growth nexus: the case of malaysia. energy policy, 38 (2010), 600-612. [8] j. chontanawat, l.c. hunt, r. pierse. does energy consumption cause economic growth? evidence from a systematic study of over 100 countries, j. policy model. 30 (2008), 209–220. [9] m.a. destek. renewable energy consumption and economic growth in newly industrialized countries: evidence from asymmetric causality test. renew. energy, 95 (2016), 478-484. [10] n. doytch, s. narayan. an investigation of renewable and non-renewable energy consumption and economic growth nexus using industrial and residential energy consumption. energy econ. 68 (2017), 160-176. [11] c. dritsaki. causal nexus between economic growth, exports and government debt: the case of greece. proc. econ. finance, 5 (2013), 251-259. [12] c.y. ee. export-led growth hypothesis: empirical evidence from selected sub-saharan african countries, proc. econ. finance, 35 (2016), 232-240. [13] a.a.g. raeeni, s. hosseini, r. moghaddasi. how energy consumption is related to agricultural growth and export: an econometric analysis on iranian data, energy reports, 5 (2019), 50-53. int. j. anal. appl. 18 (6) (2020) 987 [14] k.k. gokmenoglu, z. sehnaz, n. taspinar. the export-led growth: a case study of costa rica. proc. econ. finance, 25 (2015), 471-477. [15] w.s. jung, p.j. marshall, exports, growth and causality in developing countries, j. develop. econ. 18 (1985), 1-12. [16] k. krisna, a. ozyildirim, n.r. swanson. trade, investment and growth: nexus, analysis and prognosis, j. develop. econ. 70 (2003), 479-499. [17] j. lee. export specialization and economic growth around the world. econ. syst. 35 (2011), 45-63. [18] c.c. lee, c.p. chang. energy consumption and economic growth in asian economies: a more comprehensive analysis using panel data. resource energy econ. 30 (2008), 50–65. [19] k. makun. cointegration relationship between economic growth, export and electricity consumption: evidence from fiji. adv. energy. 2 (2015), 1-7. [20] p.k. mishra. the dynamics of relationship between exports and economic growth in india. int. j. econ. sci. appl. res. 4 (2011), 53-70. [21] r. moghaddasi, a.a. pour. energy consumption and total factor productivity growth in iranian agriculture. energy rep. 2 (2016), 218-220. [22] m. mutascu. a bootstrap panel granger causality analysis of energy consumption and economic growth in the g7 countries. renew. sustain. energy rev. 63 (2016), 166-171. [23] s. narayan. predictability within the energy consumption-economic growth nexus: some evidence from income and regional groups. econ. model. 54 (2016), 515-521. [24] s. paul, r.n. bhattacharya. causality between energy consumption and economic growth in india: a note on conflicting results. energy econ. 26 (2004), 977-983. [25] p. sadorsky. energy consumption, output and trade in south america. energy econ. 34 (2012), 476488. [26] m. shahbaz, m. zakaria, s.j.h. shahzad, m.k. mahalik. the energy consumption and economic growth nexus in top ten energy-consuming countries: fresh evidence from using the quantileon-quantile approach. energy econ. 71 (2018), 282-301. [27] d.i. stern. energy use and economic growth in the usa: a multivariate approach. energy econ. 15 (1993), 137-150. [28] d. streimikiene, r. kasperowicz. review of economic growth and energy consumption: a panel cointegration analysis for eu countries. renew. sustain. energy rev. 59 (2016), 1545-1549. [29] r. sultan. an econometric study of economic growth, energy and exports in mauritius: implications for trade and climate policy. int. j. energy econ. policy, 2 (2012), 225-237. int. j. anal. appl. 18 (6) (2020) 988 [30] r.b. tekin. economic growth, exports and foreign direct investment in least developed countries: a panel granger causality analysis. econ. model. 29 (2012), 868-878. [31] m. trost, s. bojnec. causality between public wage bill, exports and economic growth in slovenia. econ. res. j. 28 (2015), 119-131. [32] g.a. vamvoukas. trade liberalization and economic expansion: a sensitivity analysis. south-east. eur. j. econ. 1 (2007), 71-88. [33] h.y. yang. a note of the causal relationship between energy and gdp in taiwan. energy econ. 22 (2000), 309–317. [34] m. zibaei, m.h. tarazkar. a study of the short-term and long-term relationship of the value added and energy consumption in the agriculture sector. bank agric. q. j. 6 (2004), 157-171. international journal of analysis and applications volume 18, number 3 (2020), 493-512 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-493 exact solutions of kupershmidt equation, approximate solutions for time-fractional kupershmidt equation: a comparison study medjahed djilali1,∗, ali hakem2 and abdelkader benali3 1oran higher school of economics, laboratory acedp, djillali liabes university, 22000 sidi-bel-abbes, algeria 2laboratory acedp, djillali liabes university, 22000 sidi-bel-abbes, algeria 3faculty of the exact sciences and computer, mathematics department, university of hassiba benbouali, chlef 02000, algeria ∗corresponding author: djilalimedjahed@yahoo.fr abstract. in this article, a technique namely tanh method is applied to obtain some traveling wave solutions for kupershmidt equation, and by using ladm we obtain an approximate solution to timefractional kupershmidt equation. a comparison between the traveling wave solution (exact solution) and the approximate one of equation under study, indicate that laplace adomian decomposition method (ladm) is highly accurate and can be considered a very useful and valuable method. 1. introduction the study of nonlinear evolution equations have attracted attention of many mathematicians and physicists. many authors are interested to the research of the exact solutions [9, 21, 30], because the exact solutions to nonlinear evolution equations are the key tool to understand the various physical phenomena that govern the real world today. hence, searching for exact traveling wave solutions to nonlinear evolution received february 19th, 2020; accepted march 16th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 83c15, 35c07, 81q05, 35l05, 47j35. key words and phrases. adomian polynomials; caputo’s fractional derivative; kupershmidt equation; laplace transform adomian decomposition method; tanh method. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 493 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-493 int. j. anal. appl. 18 (3) (2020) 494 equations plays an important role in the study of nonlinear physical phenomena in many fields such as fluid dynamics, water wave mechanics, meteorology, electromagnetic theory, plasma physics and nonlinear optics [9,21]. in this paper, we will study an important nonlinear evolution equation called kueprshmidt equation (see [12,13] ) in the form ut −u5x − 5 2 uu3x − 25 4 uxu2x − 5 4 u2ux = 0. many researchers have studied the general fifth order kdv equation in different contexts: ut + ωu5x + αuu3x + βuxu2x+ + γu 2ux = 0, where ω,α,β and γ real constants. this class includes the generalized kaup-kupershmidt equation [34] ut + 20a 2bu5x + 10abuu3x + 25abuxu2x + bu 2ux = 0. as the constants a 6= 0,b 6= 0 take different values, we retrieve different types of kaup-kupershmidt equation. for examples, in the case a = 1 20 ,b = 30 see [11,16,39], for a = 1 60 ,b = 180, see [38], reyes [32] studied the case a = 1 10 ,b = −5, if we take a = − 1 30 ,b = 45 we will find the equation studied by parker [19, 28], and when a = 1 30 ,b = 5 we get the equation treated in [7,17]. while, we obtain the kupershmidt equation (the equation under study) by taking a = 1 5 ,b = −5 4 . to investigate the traveling wave solutions (soliton solutions) [10, 18], we propose in this work the tanh method (or hyperbolic tangent method), because it is a powerful technique to search for traveling waves coming out from one-dimensional nonlinear wave and evolution equations. in particular, in those problems where dispersive effects, reaction, diffusion and/or convection play an important role. to show the strength of the method, an overview is given to find out which kind of problems are solved with this technique and how in some nontrivial cases this method, adapted to the problem at hand, still can be applied. single as well as coupled equations, arising from wave phenomena which appears in different scientific domains such as physics, chemical kinetics, geochemistry and mathematical biology [15,24,25,45]. but some evolution problems do not admit the traveling wave solutions, due to that, we propose a semianalytical method called laplace adomian decomposition method (ladm), it is a combination of the adomian decomposition method (adm) and laplace transforms. this method was successfully used for solving different problems in [5,8,14,18,20,23,37,40]. the adm was introduced by adomian [1–4] and has been applied to a wide class of problems in physics, biology and chemical reactions. the method provides the solution in a rapid convergent series with computable terms. the underlying idea of the technique is to assume an infinite solution of the form u = ∑∞ n=0 un, then apply laplace transformation to the differential equation. the nonlinear terms are then decomposed in terms of adomian polynomials [6, 41, 42] and an int. j. anal. appl. 18 (3) (2020) 495 iterative algorithm is constructed for the determination of the un in a recursive manner. our goal is to obtained the approximate solutions of the time-fractional kupershmidt equation, and compare this solution (in particular case) with the traveling wave solution of the equation to show that the proposed algorithm (ladm) is suitable for such problems and is very efficient. 2. preliminaries before the beginning of this research, we are trying in a hurry to get to know the supporting materials to accomplish this work. 2.1. the tanh method. the non-linear wave and evolution equations (in principle, in one dimension) are commonly written as: ut = [u,ux,uxx, . . . ] or utt = [u,ux,uxx, . . . ] (2.1) we like to know whether traveling waves (or stationary waves) are solutions of (2.1). the first step is to unite the independent variables x and t into one particular variable through the definition ζ = c(x − µt). here c(> 0) represents the wave number and µ is the (unknown) velocity of the traveling wave. accordingly, the quantity u(x; t) is replaced by u(ζ), so that we deal with odes, rather than with pdes. in this way, equations like (2.1) are transformed into −cµ du dζ = [u,c du dζ , d2u dζ2 , . . . ] or c2µ2 d2u dζ2 = [u,c du dζ , d2u dζ2 , . . . ] (2.2) our main goal is to derive exact or at least approximate solutions, if possible, for these odes. so we introduce a new variable φ = tanh ζ in the ode. the latter equation then solely depends on φ, because all derivatives d dζ in (2.2) are now replaced by (1 − φ2) d dζ the solution(s) we are looking for, will be written as a finite power series in φ f(φ) = n∑ n=0 anφ n (2.3) to determine n (highest order of φ), the following balancing procedure is used.at least two terms proportional to φn must appear after substitution of ansatz (2.3) into the equation under study. as a result of this analysis, we definitely require an+1 = 0 and an 6= 0 for a particular n. it turns out that n = 1 or 2 in most cases. this balance (and thus n) is obtained by comparing the behavior of φn in the highest derivative against its counterpart within the nonlinear term(s). as soon as n is determined in this way, we get after substitution of (2.3) into(2.2) (transformed to the φ variable) algebraic equations for an (n = 0; 1; . . . ; n). depending on the problem under study, the wave number c will remain fixed or undetermined. as already mentioned, the velocity µ of the traveling wave is always a function of c. if one is able to find nontrivial values for an (n = 0; 1; . . . ; n), in terms of known quantities, a solution is ultimately obtained (see [24]). int. j. anal. appl. 18 (3) (2020) 496 2.2. laplace transform. given a suitable function f(t) the laplace transform [35, 36], written f(s) is defined by l[f(t)] = f(s) = ∫ ∞ 0 f(t)e−stdt, (2.4) the inverse laplace transform is defined by l−1[f(s)] = f(t). (2.5) the important properties of laplace transform and it’s inverse that will be used in this paper are : • if f1(t) and f2(t) are two functions whose laplace transform exists, then • l[af1(t) + bf2(t)] = al[f1(t)] + bl[f2(t)], • l(tα) = γ(α + 1)s−α−1, α > 0, • l(tn) = n! sn+1 , n a positive integer. • the inverse laplace transform is linear, i.e l−1[af1(s) + bf2(s)] = al−1[f1(s)] + bl−1[f2(s)], • l−1( 1 sα ) = t α−1 γ(α) , α > 0 2.3. caputo derivative. there exists a vast literature on different definitions of fractional derivatives. the most popular ones are the riemann-liouville and the caputo derivatives. the caputo derivative of order α is defined by the formula [22,27,29]: dα∗f(t) =   1 γ(m−α) ∫ t 0 (t− τ)m−α−1f(m)(τ)dτ, if m− 1 < α < m dm dtm f(t), if α = m, (2.6) where m ∈ n∗ and γ(.) denotes the gamma function defined by γ(x) = ∫∞ 0 tx−1e−tdt, x > 0. the important properties of the caputo derivative that will be used in this paper are [23,26,31,33,43,44]: dαtβ = γ(1 + β) γ(1 + β −α) tβ−α (2.7) dαc = 0 (2.8) the laplace transform of the caputo derivative is: l[dαt u(x,t)] = s αu(x,s) − n−1∑ i=0 u(i)(x, 0+)sα−1−i, n− 1 < α ≤ n (2.9) 2.4. the adomian decomposition method combined with laplace transform. the adm is a method to solve ordinary and nonlinear differential equations. using this method is possible to express analytic solutions in terms of a series. in a nutshell, the method identifies and separates the linear and nonlinear parts of a differential equation. inverting and applying the highest order differential operator that is contained in the linear part of the equation, it is possible to express the solution in terms of the rest of the equation affected by the inverse operator. at this point, the solution is proposed by means of a series int. j. anal. appl. 18 (3) (2020) 497 with terms that will be determined and that give rise to the adomian polynomials. the nonlinear part can also be expressed in terms of these polynomials. the initial (or the border conditions) and the terms that contain the independent variables will be considered as the initial approximation. in this way and by means of a recurrence relations, it is possible to find the terms of the series that give the approximate solution of the differential equation (see [14]). given a partial (or ordinary) differential equation fu(x,t) = h(x,t) with initial condition u(x, 0) = f(x), (2.10) where f is a differential operator that could, in general, be nonlinear and therefore includes some linear and nonlinear terms. in general, eq. (2.10) could be written as ltu(x,t) = ru(x,t) + nu(x,t) + h(x,t) (2.11) where lt = ∂ α ∂tα , 0 < α ≤ 1 (in this paper) , r is a linear operator that includes partial derivatives with respect to x , n is a nonlinear operator and h is a non-homogeneous term that is u -independent. the ladm consists of applying laplace transform first on both sides of eq. (2.11), obtaining l{ltu(x,t)} = l{ru(x,t) + nu(x,t) + h(x,t)} . (2.12) an equivalent expression to (5 )is sαu(x,s) −u(x, 0)sα−1 = l{ru(x,t) + nu(x,t) + h(x,t)} . (2.13) in the homogeneous case, h(x,t) = 0, we have u(x,s) = f(x) s + 1 sα l{ru(x,t) + nu(x,t)} . (2.14) now, applying the inverse laplace transform to e q. (2.14) u(x,t) = f(x) + l−1[ 1 sα l{ru(x,t) + nu(x,t)}]. (2.15) the adm method proposes a series solution u (x , t ) given by, u(x,t) = ∞∑ n=0 un(x,t). (2.16) the nonlinear term nu(x,t) is given by nu(x,t) = ∞∑ n=0 pn(u0,u1,u2, . . . ,un). (2.17) where {an} ∞ n=0 is the so-called adomian polynomials sequence established in [42] , in general, give us term to term: int. j. anal. appl. 18 (3) (2020) 498 p0 = n(u0) p1 = u1n ′(u0) p2 = u2n ′(u0) + 1 2 u21n ′′(u0) p3 = u3n ′(u0) + u1u2n ′′(u0) + 1 3! u31n (3)(u0) p4 = u4n ′(u0) + ( 1 2 u22 + u1u3)n ′′(u0) + 1 2! u21u2n (3)(u0) + 1 4! u41n (4)(u0) ... other polynomials can be generated in a similar way. some other approaches to obtain adomian’s polynomials can be found in [42]. using (2.16) and (2.17) into e q. (2.15), we obtain, ∞∑ n=0 un(x,t) = f(x) + l−1 [ 1 sα l { r ∞∑ n=0 un(x,t) + ∞∑ n=0 pn(u0,u1,u2, . . . ,un) }] . (2.18) we deduce the following recurrence formulas  u0(x,t) = f(x) un+1(x,t) = l−1 [ 1 sα l{run(x,t) + pn(u0,u1,u2, . . . ,un)} ] , n = 0, 1, 2 . . . (2.19) using (2.19) we can obtain an approximate solution of (2.10), using u(x,t) ≈ k∑ n=0 un(x,t), where lim t→∞ k∑ n=0 un(x,t) = u(x,t) (2.20) remark 2.1. all results and plots bellow are obtained by using mathematica software. 3. main results 3.1. kupershmidt equation solutions by using tanh method. in this section, we will apply the tanh method to find the axact solutions of kupershmidt equation in the form, ∂u(x,t) ∂t − ∂5u(x,t) ∂x5 − 5 2 u(x,t) ∂3u(x,t) ∂x3 − 25 4 ∂u(x,t) ∂x ∂2u(x,t) ∂x2 − 5 4 u(x,t)2 ∂u(x,t) ∂x = 0. (3.1) we consider the traveling wave transformation defined by, u(ζ) = u(x,t), ζ = c(x−µt). (3.2) using traveling wave eqs. (3.2), then (3.1) transform into the following ordinary differential equations µu(1) + c4u(5) + 5 2 c2u(3)u + 25 4 c2u(2)u(1) + 5 4 u2u(1) = 0, (3.3) int. j. anal. appl. 18 (3) (2020) 499 now balancing the highest order derivative u(5) and nonlinear term u(2)u(1), we get 2n + 3 = n + 5 or equivalent to n = 2. therefore, eq. (2.3) reduces to u(ζ) = a0 + a1 tanh(ζ) + a2 tanh 2(ζ), (3.4) substituting eq. (3.4) into eq. (3.3) and using mathematica software we get a polynomial of tanh(ζ)k, (k = 0, 1, 2, ...). equating the coefficients of this polynomial of the same powers of tanh(ζ) to zero, we obtain a system of algebraic equations for a0,a1,a2,µ and c. − 16a1c5 + 5a0a1c3 − 25 2 a1a2c 3 −a1cµ− 5 4 a20a1c = 0 − 272a2c5 + 35 2 a21c 3 − 25a22c 3 + 40a0a2c 3 − 2a2cµ− 5 2 a0a 2 1c− 5 2 a20a2c = 0 136a1c 5 − 20a0a1c3 + 265 2 a1a2c 3 + a1cµ− 5a31c 4 + 5 4 a20a1c− 15 2 a0a1a2c = 0, 1232a2c 5 − 45a21c 3 + 165a22c 3 − 100a0a2c3 + 2a2cµ + 5 2 a0a 2 1c− 5a0a 2 2c + 5 2 a20a2c− 5a 2 1a2c = 0, − 240a1c5 + 15a0a1c3 − 515 2 a1a2c 3 + 5a31c 4 − 25 4 a1a 2 2c + 15 2 a0a1a2c = 0, − 1680a2c5 + + 55 2 a21c 3 − 275a22c 3 + 60a0a2c 3 − 5a32c 2 + 5a0a 2 2c + 5a 2 1a2c = 0, 120a1c 5 + 275 2 a1a2c 3 + 25 4 a1a 2 2c = 0, 720a2c 5 + 135a22c 3 + 5a32c 2 = 0, where a2 6= 0. solving them by means of mathematica gives:{ a0 → 4c2,a1 → 0,a2 →−6c2,µ →−c4 } ,{ a0 → 32c2,a1 → 0,a2 →−48c2,µ →−176c4 } , substituting into eq. (3.4), it follows u1(x,t) = 4c 2 − 6c2 tanh2(cx + c5t), (3.5) u2(x,t) = 32c 2 − 48c2 tanh2(cx + 176c5t), (3.6) 3.2. the approximate solution of time-fractional kupershmidt equation by ladm. consider the time-fractional kupershmidt equation ∂α ∂tα u(x,t) = ∂5u(x,t) ∂x5 + 5 2 u(x,t) ∂3u(x,t) ∂x3 + 25 4 ∂u(x,t) ∂x ∂2u(x,t) ∂x2 + 5 4 u(x,t)2 ∂u(x,t) ∂x (3.7) subject to the initial conditions u(x, 0) = f(x) = 4c2 − 6c2 tanh2(cx), (3.8) int. j. anal. appl. 18 (3) (2020) 500 where 0 < α ≤ 1 and ∂ α ∂tα = dαt the derivatives in the sens of caputo. comparing (3.7) with eq. (2.11) we have that h(x,t) = 0, lt and r becomes: ltu = d α t u = ∂α ∂tα u, r(u) = ∂5u(x,t) ∂x5 = u5x(x,t), (3.9) while the nonlinear term are given by nu = 5 2 u(x,t) ∂3u(x,t) ∂x3 + 25 4 ∂u(x,t) ∂x ∂2u(x,t) ∂x2 + 5 4 u(x,t)2 ∂u(x,t) ∂x := 5 2 u(x,t)u3x(x,t) + 25 4 ux(x,t)u2x(x,t) + 5 4 u(x,t)2ux(x,t), (3.10) by using now eq. (2.19) through the ladm method we obtain recursively  u0(x,t) = f(x) un+1(x,t) = l−1 [ 1 sα l{r(un) + pn(u0,u1,u2, . . . ,un)} ] , n = 0, 1, 2 . . . (3.11) from this, we will consider the decomposition of the nonlinear terms into adomian polynomials as nu = n1u + n2u + n3u = ∞∑ n=0 pn(u0,u1,u2, . . . ,un). (3.12) let n1u = 5 2 u(x,t)u3x(x,t) = 5 2 ∞∑ n=0 un ∞∑ n=0 un3x = ∞∑ n=0 an(u0,u1,u2, . . . ,un), (3.13) n2u = 25 4 ux(x,t)u2x(x,t) = 25 4 ∞∑ n=0 unx ∞∑ n=0 un2x = ∞∑ n=0 bn(u0,u1,u2, . . . ,un), (3.14) n3u = 5 4 u(x,t)2ux(x,t) = 5 4 ( ∞∑ n=0 un )2 ∗ ∞∑ n=0 unx = ∞∑ n=0 cn(u0,u1,u2, . . . ,un), (3.15) where pn = an + bn + cn. using adm, eq.(2.16) gives u(x,t) = ∞∑ n=0 un(x,t), (3.16) thus, the adomian polynomials an are in the forms a0 = 5 2 u0u03x a1 = 5 2 u1u03x + 5 2 u0u13x a2 = 5 2 u2u03x + 5 2 u1u13x + 5 2 u0u23x a3 = 5 2 u3u03x + 5 2 u2u13x + 5 2 u1u23x + 5 2 u0u33x a4 = 5 2 u4u03x + 5 2 u3u13x + 5 2 u2u23x + 5 2 u1u33x + 5 2 u0u43x, ... (3.17) int. j. anal. appl. 18 (3) (2020) 501 b0 = 25 4 u0xu02x b1 = 25 4 u1xu02x + 25 4 u0xu12x b2 = 25 4 u2xu02x + 25 4 u1xu12x + 25 4 u0xu22x b3 = 25 4 u3xu02x + 25 4 u2xu12x + 25 4 u1xu22x + 25 4 u0xu32x b4 = 25 4 u4xu02x + 25 4 u3xu12x + 25 4 u2xu22x + 25 4 u1xu32x + 25 4 u0xu42x, ... (3.18) and c0 = 5 4 u20u0x, c1 = 5 4 u20u1x + 5 2 u1u0u0x, c2 = 5 4 u20u2x + 5 2 u2u0u0x + 5 4 u21u0x c3 = 5 4 u20u3x + 5 2 u3u0u0x + 5 2 u2u0u1x + 5 2 u1u0u2x + 5 2 u1u2u0x + 5 4 u21u1x c4 = 5 4 u20u4x + 5 2 u4u0u0x + 5 2 u2u0u2x + 5 4 u22u0x + 5 2 u1u3u0x + 5 2 u1u2u1x + 5 4 u21u2x. ... (3.19) through the ladm we obtain recursively u0(x,t) = f(x), u1(x,t) = l−1 [ 1 sα l{u05x + a0 + b0 + c0} ] , u2(x,t) = l−1 [ 1 sα l{u15x + a1 + b1 + c1} ] , u3(x,t) = l−1 [ 1 sα l{u25x + a2 + b2 + c2} ] , ... ... un+1(x,t) = l−1 [ 1 sα l{un5x + an + bn + cn} ] . (3.20) int. j. anal. appl. 18 (3) (2020) 502 besides a0 = − 1440c7 tanh3(cx)sech4(cx) + 960c7 tanh(cx)sech4(cx) + 720c7 tanh5(cx)sech2(cx) − 480c7 tanh3(cx)sech2(cx) b0 =900c 7 tanh(cx)sech6(cx) − 1800c7 tanh3(cx)sech4(cx) c0 = − 540c7 tanh5(cx)sech2(cx) + 720c7 tanh3(cx)sech2(cx) − 240c7 tanh(cx)sech2(cx). with the above, we have u0(x,t) =4c 2 − 6c2 tanh2(cx) u1(x,t) = − 732c7tα tanh(cx)sech6(cx) γ(α + 1) − 744c7tα tanh3(cx)sech4(cx) γ(α + 1) + 960c7tα tanh(cx)sech4(cx) γ(α + 1) − 12c7tα tanh5(cx)sech2(cx) γ(α + 1) + 240c7tα tanh3(cx)sech2(cx) γ(α + 1) − 240c7tα tanh(cx)sech2(cx) γ(α + 1) , (3.21) and proceeding in a similar way we get a1 = 101760c12tαsech10(cx) γ(α + 1) − 328320c12tα tanh2(cx)sech10(cx) γ(α + 1) + 1188720c12tα tanh4(cx)sech8(cx) γ(α + 1) − 120000c12tαsech8(cx) γ(α + 1) − 442560c12tα tanh2(cx)sech8(cx) γ(α + 1) + 833760c12tα tanh6(cx)sech6(cx) γ(α + 1) 19200c12tαsech6(cx) γ(α + 1) − 2111040c12tα tanh4(cx)sech6(cx) γ(α + 1) + 950400c12tα tanh2(cx)sech6(cx) γ(α + 1) − 684720c12tα tanh8(cx)sech4(cx) γ(α + 1) + 990240c12tα tanh6(cx)sech4(cx) γ(α + 1) − 187200c12tα tanh4(cx)sech4(cx) γ(α + 1) − 105600c12tα tanh2(cx)sech4(cx) γ(α + 1) − 1440c12tα tanh10(cx)sech2(cx) γ(α + 1) + 29760c12tα tanh8(cx)sech2(cx) γ(α + 1) − 48000c12tα tanh6(cx)sech2(cx) γ(α + 1) + 19200c12tα tanh4(cx)sech2(cx) γ(α + 1) , b1 = 54900c12tαsech12(cx) γ(α + 1) − 72000c12tαsech10(cx) γ(α + 1) + 18000c12tαsech8(cx) γ(α + 1) − 1035000c12tα tanh2(cx)sech10(cx) γ(α + 1) + 202500c12tα tanh4(cx)sech8(cx) γ(α + 1) + 1278000c12tα tanh2(cx)sech8(cx) γ(α + 1) + 1299600c12tα tanh6(cx)sech6(cx) γ(α + 1) − 1224000c12tα tanh4(cx)sech6(cx) γ(α + 1) − 216000c12tα tanh2(cx)sech6(cx) γ(α + 1) + 7200c12tα tanh8(cx)sech4(cx) γ(α + 1) − 144000c12tα tanh6(cx)sech4(cx) γ(α + 1) + 144000c12tα tanh4(cx)sech4(cx) γ(α + 1) , int. j. anal. appl. 18 (3) (2020) 503 c1 = − 14640c12tαsech8(cx) γ(α + 1) + 19200c12tαsech6(cx) γ(α + 1) − 4800c12tαsech4(cx) γ(α + 1) − 164700c12tα tanh4(cx)sech8(cx) γ(α + 1) + 131760c12tα tanh2(cx)sech8(cx) γ(α + 1) − 36720c12tα tanh6(cx)sech6(cx) γ(α + 1) + 175680c12tα tanh4(cx)sech6(cx) γ(α + 1) − 129600c12tα tanh2(cx)sech6(cx) γ(α + 1) + 129060c12tα tanh8(cx)sech4(cx) γ(α + 1) − 270720c12tα tanh6(cx)sech4(cx) γ(α + 1) + 162720c12tα tanh4(cx)sech4(cx) γ(α + 1) − 19200c12tα tanh2(cx)sech4(cx) γ(α + 1) + 1080c12tα tanh10(cx)sech2(cx) γ(α + 1) − 23040c12tα tanh8(cx)sech2(cx) γ(α + 1) + 50880c12tα tanh6(cx)sech2(cx) γ(α + 1) − 38400c12tα tanh4(cx)sech2(cx) γ(α + 1) + 9600c12tα tanh2(cx)sech2(cx) γ(α + 1) , thus, u2 = 24c12t2αsech4(cx) γ(2α + 1) + 12c12t2α cosh(2cx)sech4(cx) γ(2α + 1) , (3.22) a2 = 115200c17sech4(cx) tanh11(cx)t2α γ(α + 1)2 + 32371200c17sech6(cx) tanh9(cx)t2α γ(α + 1)2 + 11520000c17sech10(cx) tanh7(cx)t2α γ(α + 1)2 + 149932800c17sech8(cx) tanh7(cx)t2α γ(α + 1)2 + 2304000c17sech4(cx) tanh7(cx)t2α γ(α + 1)2 + 249523200c17sech12(cx) tanh5(cx)t2α γ(α + 1)2 + 18316800c17sech8(cx) tanh5(cx)t2α γ(α + 1)2 + 20736000c17sech6(cx) tanh5(cx)t2α γ(α + 1)2 + 11520c17 cosh(2cx)sech4(cx) tanh5(cx)t2α γ(2α + 1) + 137164320c17sech14(cx) tanh3(cx)t2α γ(α + 1)2 + 308563200c17sech10(cx) tanh3(cx)t2α γ(α + 1)2 + 6336000c17sech6(cx) tanh3(cx)t2α γ(α + 1)2 + 23040c17sech6(cx) tanh3(cx)t2α γ(2α + 1) + 15360c17sech4(cx) tanh3(cx)t2α γ(2α + 1) + 960c17 cosh(2cx)sech4(cx) tanh3(cx)t2α γ(2α + 1) + 4320c17sech6(cx) sinh(2cx) tanh2(cx)t2α γ(2α + 1) + 10080c17sech4(cx) sinh(2cx) tanh2(cx)t2α γ(2α + 1) + 960c17sech4(cx) sinh(2cx)t2α γ(2α + 1) + 46382400c17sech14(cx) tanh(cx)t2α γ(α + 1)2 + 11808000c17sech10(cx) tanh(cx)t2α γ(α + 1)2 + 2880c17 cosh(2cx)sech8(cx) tanh(cx)t2α γ(2α + 1) + 6720c17 cosh(2cx)sech6(cx) tanh(cx)t2α γ(2α + 1) int. j. anal. appl. 18 (3) (2020) 504 − 2880c17sech4(cx) tanh13(cx)t2α γ(α + 1)2 − 1550880c17sech6(cx) tanh11(cx)t2α γ(α + 1)2 − 83764800c17sech8(cx) tanh9(cx)t2α γ(α + 1)2 − 1267200c17sech4(cx) tanh9(cx)t2α γ(α + 1)2 − 57772800c17sech6(cx) tanh7(cx)t2α γ(α + 1)2 − 294724800c17sech10(cx) tanh5(cx)t2α γ(α + 1)2 − 1152000c17sech4(cx) tanh5(cx)t2α γ(α + 1)2 − 366019200c17sech12(cx) tanh3(cx)t2α γ(α + 1)2 − 86400000c17sech8(cx) tanh3(cx)t2α γ(α + 1)2 − 18622080c17sech16(cx) tanh(cx)t2α γ(α + 1)2 − 38419200c17sech12(cx) tanh(cx)t2α γ(α + 1)2 − 1152000c17sech8(cx) tanh(cx)t2α γ(α + 1)2 − 23040c17sech4(cx) tanh5(cx)t2α γ(2α + 1) − 17280c17sech4(cx) sinh(2cx) tanh4(cx)t2α γ(2α + 1) − 11520c17 cosh(2cx)sech6(cx) tanh3(cx)t2α γ(2α + 1) − 2880c17sech6(cx) sinh(2cx)t2α γ(2α + 1) − 5760c17sech8(cx) tanh(cx)t2α γ(2α + 1) − 13440c17sech6(cx) tanh(cx)t2α γ(2α + 1) − 5760c17 cosh(2cx)sech4(cx) tanh(cx)t2α γ(2α + 1) , b2 = 288000c17sech4(cx) tanh11(cx)t2α γ(α + 1)2 + 54576000c17sech6(cx) tanh9(cx)t2α γ(α + 1)2 + 431856000c17sech8(cx) tanh7(cx)t2α γ(α + 1)2 + 5760000c17sech4(cx) tanh7(cx)t2α γ(α + 1)2 + 220665600c17sech12(cx) tanh5(cx)t2α γ(α + 1)2 + 43200000c17sech6(cx) tanh5(cx)t2α γ(α + 1)2 + 143305200c17sech14(cx) tanh3(cx)t2α γ(α + 1)2 + 360144000c17sech10(cx) tanh3(cx)t2α γ(α + 1)2 + 7200000c17sech6(cx) tanh3(cx)t2α γ(α + 1)2 + 43200c17sech6(cx) tanh3(cx)t2α γ(2α + 1) + 18000c17sech6(cx) sinh(2cx) tanh2(cx)t2α γ(2α + 1) + 115956000c17sech14(cx) tanh(cx)t2α γ(α + 1)2 + 29520000c17sech10(cx) tanh(cx)t2α γ(α + 1)2 + 7200c17 cosh(2cx)sech8(cx) tanh(cx)t2α γ(2α + 1) − 7200c17sech4(cx) tanh13(cx)t2α γ(α + 1)2 − 2602800c17sech6(cx) tanh11(cx)t2α γ(α + 1)2 − 210794400c17sech8(cx) tanh9(cx)t2α γ(α + 1)2 − 3168000c17sech4(cx) tanh9(cx)t2α γ(α + 1)2 − 177393600c17sech10(cx) tanh7(cx)t2α γ(α + 1)2 − 102096000c17sech6(cx) tanh7(cx)t2α γ(α + 1)2 int. j. anal. appl. 18 (3) (2020) 505 − 154908000c17sech10(cx) tanh5(cx)t2α γ(α + 1)2 − 128736000c17sech8(cx) tanh5(cx)t2α γ(α + 1)2 − 2880000c17sech4(cx) tanh5(cx)t2α γ(α + 1)2 − 416520000c17sech12(cx) tanh3(cx)t2α γ(α + 1)2 − 95040000c17sech8(cx) tanh3(cx)t2α γ(α + 1)2 − 46555200c17sech16(cx) tanh(cx)t2α γ(α + 1)2 − 96048000c17sech12(cx) tanh(cx)t2α γ(α + 1)2 − 2880000c17sech8(cx) tanh(cx)t2α γ(α + 1)2 − 21600c17 cosh(2cx)sech6(cx) tanh3(cx)t2α γ(2α + 1) − 1800c17sech8(cx) sinh(2cx)t2α γ(2α + 1) − 14400c17sech8(cx) tanh(cx)t2α γ(2α + 1) − 3600c17 cosh(2cx)sech6(cx) tanh(cx)t2α γ(2α + 1) , c2 = 86400c17sech6(cx) tanh9(cx)t2α γ(α + 1)2 + 5702400c17sech8(cx) tanh7(cx)t2α γ(α + 1)2 + 26697600c17sech10(cx) tanh5(cx)t2α γ(α + 1)2 + 1728000c17sech6(cx) tanh5(cx)t2α γ(α + 1)2 + 4320c17sech4(cx) tanh5(cx)t2α γ(2α + 1) + 1080c17sech4(cx) sinh(2cx) tanh4(cx)t2α γ(2α + 1) + 21081600c17sech12(cx) tanh3(cx)t2α γ(α + 1)2 + 6912000c17sech8(cx) tanh3(cx)t2α γ(α + 1)2 + 4320c17sech6(cx) tanh3(cx)t2α γ(2α + 1) + 2880c17 cosh(2cx)sech4(cx) tanh3(cx)t2α γ(2α + 1) + 480c17sech4(cx) sinh(2cx)t2α γ(2α + 1) + 1440c17 cosh(2cx)sech6(cx) tanh(cx)t2α γ(2α + 1) + 1920c17sech4(cx) tanh(cx)t2α γ(2α + 1) − 2160c17sech6(cx) tanh11(cx)t2α γ(α + 1)2 − 267840c17sech8(cx) tanh9(cx)t2α γ(α + 1)2 − 8566560c17sech10(cx) tanh7(cx)t2α γ(α + 1)2 − 950400c17sech6(cx) tanh7(cx)t2α γ(α + 1)2 − 16338240c17sech12(cx) tanh5(cx)t2α γ(α + 1)2 − 12268800c17sech8(cx) tanh5(cx)t2α γ(α + 1)2 − 8037360c17sech14(cx) tanh3(cx)t2α γ(α + 1)2 − 19094400c17sech10(cx) tanh3(cx)t2α γ(α + 1)2 − 864000c17sech6(cx) tanh3(cx)t2α γ(α + 1)2 − 2160c17 cosh(2cx)sech4(cx) tanh5(cx)t2α γ(2α + 1) − 2160c17 cosh(2cx)sech6(cx) tanh3(cx)t2α γ(2α + 1) − 5760c17sech4(cx) tanh3(cx)t2α γ(2α + 1) − 1440c17sech4(cx) sinh(2cx) tanh2(cx)t2α γ(2α + 1) − 2880c17sech6(cx) tanh(cx)t2α γ(2α + 1) − 960c17 cosh(2cx)sech4(cx) tanh(cx)t2α γ(2α + 1) , int. j. anal. appl. 18 (3) (2020) 506 u3 = − 1260c17γ(2α + 1)t3α cosh(4cx) tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) − 29700c17γ(2α + 1)t3α tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) + 2169c17t3α cosh(4cx) tanh(cx)sech8(cx) γ(3α + 1) + 20880c17γ(2α + 1)t3α cosh(2cx) tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) − 73341c17t3α cosh(2cx) tanh(cx)sech8(cx) 2γ(3α + 1) − 3c17t3α cosh(6cx) tanh(cx)sech8(cx) 2γ(3α + 1) + 51879c17t3α tanh(cx)sech8(cx) γ(3α + 1) . (3.23) thus, the approximate solution of time-fractional kupershmidt equation (3.7) with the first four terms is: u(x,t) = 20880c17γ(2α + 1)t3α cosh(2cx) tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) − 73341c17t3α cosh(2cx) tanh(cx)sech8(cx) 2γ(3α + 1) − 29700c17γ(2α + 1)t3α tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) − 1260c17γ(2α + 1)t3α cosh(4cx) tanh(cx)sech8(cx) γ(α + 1)2γ(3α + 1) + 2169c17t3α cosh(4cx) tanh(cx)sech8(cx) γ(3α + 1) − 3c17t3α cosh(6cx) tanh(cx)sech8(cx) 2γ(3α + 1) + 51879c17t3α tanh(cx)sech8(cx) γ(3α + 1) + 12c12t2α cosh(2cx)sech4(cx) γ(2α + 1) − 732c7tα tanh(cx)sech6(cx) γ(α + 1) − 744c7tα tanh3(cx)sech4(cx) γ(α + 1) + 960c7tα tanh(cx)sech4(cx) γ(α + 1) − 24c12t2αsech4(cx) γ(2α + 1) − 12c7tα tanh5(cx)sech2(cx) γ(α + 1) + 240c7tα tanh3(cx)sech2(cx) γ(α + 1) − 240c7tα tanh(cx)sech2(cx) γ(α + 1) − 6c2 tanh2(cx) + 4c2. (3.24) set u(x,t) = uα(x,t) and take in particular c = 12 , we have: u1(x,t) = − 2507t3 tanh ( x 2 ) sech8 ( x 2 ) 262144 + 3393t3 cosh(x) tanh ( x 2 ) sech8 ( x 2 ) 524288 − t3 cosh(3x) tanh ( x 2 ) sech8 ( x 2 ) 524288 − 117t3 cosh(2x) tanh ( x 2 ) sech8 ( x 2 ) 262144 + 3t2 cosh(x)sech4 ( x 2 ) 2048 − 183 32 t tanh (x 2 ) sech6 (x 2 ) − 93 16 t tanh3 (x 2 ) sech4 (x 2 ) + 15 2 t tanh (x 2 ) sech4 (x 2 ) − 3 32 t tanh5 (x 2 ) sech2 (x 2 ) − 3t2sech4 ( x 2 ) 1024 + 15 8 t tanh3 (x 2 ) sech2 (x 2 ) − 15 8 t tanh (x 2 ) sech2 (x 2 ) − 3 2 tanh2 (x 2 ) + 1, (3.25) int. j. anal. appl. 18 (3) (2020) 507 u1 2 (x,t) = − 17293t3/2 tanh ( x 2 ) sech8 ( x 2 ) 32768 √ π − 2475t3/2 tanh ( x 2 ) sech8 ( x 2 ) 2048π3/2 + 435t3/2 cosh(x) tanh ( x 2 ) sech8 ( x 2 ) 512π3/2 + 723t3/2 cosh(2x) tanh ( x 2 ) sech8 ( x 2 ) 32768 √ π − 105t3/2 cosh(2x) tanh ( x 2 ) sech8 ( x 2 ) 2048π3/2 − 3 512 tsech4 (x 2 ) − 24447t3/2 cosh(x) tanh ( x 2 ) sech8 ( x 2 ) 65536 √ π − t3/2 cosh(3x) tanh ( x 2 ) sech8 ( x 2 ) 65536 √ π + 3t cosh(x)sech4 ( x 2 ) 1024 − 183 √ t tanh ( x 2 ) sech6 ( x 2 ) 16 √ π − 93 √ t tanh3 ( x 2 ) sech4 ( x 2 ) 8 √ π + 15 √ t tanh ( x 2 ) sech4 ( x 2 ) √ π − 3 √ t tanh5 ( x 2 ) sech2 ( x 2 ) 16 √ π + 15 √ t tanh3 ( x 2 ) sech2 ( x 2 ) 4 √ π − 15 √ t tanh ( x 2 ) sech2 ( x 2 ) 4 √ π − 3 2 tanh2 (x 2 ) + 1, (3.26) u3 4 (x,t) = t3/2 cosh(x)sech4 ( x 2 ) 256 √ π + 51879t9/4 tanh ( x 2 ) sech8 ( x 2 ) 131072γ ( 13 4 ) − 22275√πt9/4 tanh (x2)sech8 (x2) 131072γ ( 7 4 )2 γ ( 13 4 ) − 183t3/4 tanh ( x 2 ) sech6 ( x 2 ) 32γ ( 7 4 ) − 93t3/4 tanh3 (x2)sech4 (x2) 16γ ( 7 4 ) + 15t3/4 tanh (x2)sech4 (x2) 2γ ( 7 4 ) − 3t3/4 tanh5 ( x 2 ) sech2 ( x 2 ) 32γ ( 7 4 ) + 15t3/4 tanh3 (x2)sech2 (x2) 8γ ( 7 4 ) − 15t3/4 tanh (x2)sech2 (x2) 8γ ( 7 4 ) + 2169t9/4 cosh(2x) tanh ( x 2 ) sech8 ( x 2 ) 131072γ ( 13 4 ) + 3915√πt9/4 cosh(x) tanh (x2)sech8 (x2) 32768γ ( 7 4 )2 γ ( 13 4 ) − 73341t9/4 cosh(x) tanh ( x 2 ) sech8 ( x 2 ) 262144γ ( 13 4 ) − 3t9/4 cosh(3x) tanh (x2)sech8 (x2) 262144γ ( 13 4 ) − 945 √ πt9/4 cosh(2x) tanh ( x 2 ) sech8 ( x 2 ) 131072γ ( 7 4 )2 γ ( 13 4 ) − t3/2sech4 ( x 2 ) 128 √ π − 3 2 tanh2 (x 2 ) + 1. (3.27) remark 3.1. under the initial conditions (3.8) with c = 1 2 , if α = 1 then (3.7) becomes (3.1), and an exact solution is (3.5). int. j. anal. appl. 18 (3) (2020) 508 t= 1 t = 3 t= 5 x u l a d m u e x a c t e r r o r u l a d m u e x a c t e r r o r u l a d m u e x a c t e r r o r -5 -0 .4 57 57 3 -0 .4 57 57 6 2 .2 3 6 7 4 ∗ 1 0 − 6 -0 .4 51 95 9 -0 .4 52 01 8 0. 00 00 59 24 62 -0 .4 45 47 7 -0 .4 45 74 6 0. 00 02 68 78 4 -4 -0 .3 87 43 8 -0 .3 87 45 0. 00 00 12 38 11 -0 .3 72 77 9 -0 .3 73 11 1 0. 00 03 32 43 7 -0 .3 55 51 -0 .3 57 04 0. 00 15 30 52 -3 -0 .2 13 17 9 -0 .2 13 21 5 0. 00 00 36 14 89 -0 .1 78 36 5 -0 .1 79 34 2 0. 00 09 77 48 6 -0 .1 37 53 3 -0 .1 42 06 8 0. 00 45 34 46 -2 0. 16 03 5 0. 16 03 97 0. 00 00 46 97 8 0. 22 26 02 0. 22 38 52 0. 00 12 49 45 0. 28 47 25 0. 29 04 17 0. 00 56 91 99 -1 0. 71 35 55 0. 71 32 99 0. 00 02 55 82 3 0. 78 42 49 0. 77 73 34 0. 00 69 14 62 0. 86 78 67 0. 83 58 36 0. 03 20 30 6 0 0. 99 85 35 0. 99 85 36 9 .5 3 1 4 7 ∗ 1 0 − 7 0. 98 68 16 0. 98 68 93 0. 00 00 76 86 46 0. 96 33 79 0. 96 39 67 0. 00 05 87 90 4 1 0. 64 49 6 0. 64 52 16 0. 00 02 55 43 4 0. 56 76 43 0. 57 45 26 0. 00 68 83 48 0. 47 07 78 0. 50 25 74 0. 03 17 95 8 2 0. 10 04 83 0. 10 04 36 0. 00 00 47 64 04 0. 04 55 15 8 0. 04 42 12 8 0. 00 13 03 07 -0 .0 02 03 72 9 -0 .0 08 14 25 2 0. 00 61 05 23 3 -0 .2 43 92 9 -0 .2 43 89 3 0. 00 00 36 11 07 -0 .2 72 56 9 -0 .2 71 59 4 0. 00 09 74 34 7 -0 .3 01 05 1 -0 .2 96 54 2 0. 00 45 09 47 4 -0 .4 00 24 -0 .4 00 22 7 0. 00 00 12 44 94 -0 .4 11 93 8 -0 .4 11 6 0. 00 03 37 96 8 -0 .4 23 28 5 -0 .4 21 71 2 0. 00 15 73 06 5 -0 .4 62 5 -0 .4 62 49 8 2 .2 7 7 6 2 ∗ 1 0 − 6 -0 .4 66 91 8 -0 .4 66 85 6 0. 00 00 62 55 81 -0 .4 71 00 6 -0 .4 70 71 2 0. 00 02 94 35 5 table 1. a comparison between approximate solution and exact solution of (3.1) for t = 1, 3, 5. int. j. anal. appl. 18 (3) (2020) 509 exact approx ...for α=1 approx ...for α=0.5 approx ...for α=0.75 -10 -5 5 10 x -0.5 0.5 1.0 u (a) uexact and uladm figure 1. plot of the exact solution (3.5) of eq. (3.1) and approximate solutions of eq. (3.1), when c = 1 2 for t = 5 and x ∈ [−10, 10]. (a) u(x, t) (b) u1(x, t) figure 2. plot of the exact solution (3.5) given by eq. (3.1) and approximate solution uladm given by eq. (3.7) when α = 1 and c = 12 , for (x,t) ∈ [−10, 10] × [0, 5] . 4. conclusion in this paper, we discussed three stages related to the study of kupershmidt equation. first we used the tanh method to get the exact solution of the equation under study. in the second stage, thanks to the ladm method (adm in combination with the laplace transform), we obtain the approximate solutions to the int. j. anal. appl. 18 (3) (2020) 510 (a) u 1 2 (x, t) (b) u 3 4 (x, t) figure 3. plot of the approximate solution uladm given by eq. (3.7) when α = 0.5,α = 0.75 and c = 1 2 , for (x,t) ∈ [−10, 10] × [0, 5]. time-fractional kupershmidt equation. finally, in order to show the accuracy and efficiency of our method, compare our results with the exact solution of the equation obtained by the tanh method. furthermore, we conclude that the ladm is a powerful tool that produces high quality approximate solutions for nonlinear partial differential equations using simple calculations and that attains converge with only few terms. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. abdelrazec, d. pelinovsky , convergence of the adomian decomposition method for initial-value problems, numer. methods partial differ. equations, 27 (2011), 749–766. [2] g. adomian , a review of the decomposition method in applied mathematics , j. math. anal. appl. 135 (1988), 501–544. [3] g. adomian, system of nonlinear partial differential equations , j. math. anal. appl. 115 (1) (1986), 235-238. [4] g. adomian, solving frontier problems of physics: the decomposition method, kluwer academic publication, boston, 1994. [5] e. babolian a, j. biazar b, a.r. vahidi, a new computational method for laplace transforms by decomposition method, appl. math. comput. 150 (2004), 841–846. [6] e. babolian, s. javadi , new method for calculating adomian polynomials , appl. math. comput. 153 (2004), 253–259. [7] k. charalambous , a. k. halder and peter g. l. leach, a note on analysis of the kaup-kupershmidt equation, aip conf. proc. 2153 (2019), 020006. [8] j. fadaei, application of laplace-adomian decomposition method on linear and nonlinear system of pdes, appl. math. sci. 5 (27) (2011), 1307 1315. int. j. anal. appl. 18 (3) (2020) 511 [9] e. fan, uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, chaos solitons fractals, 16 (2003), 819–839. [10] e. g. fan, traveling wave solutions for nonlinear equations using symbolic computation, computers math. appl. 42 (6-7) (2002), 671-680. [11] a. fordy and j. gibbons, some remarkable nonlinear transfomations, phys. lett. a, 75 (5) (1980), 325. [12] b. fuchssteiner, w. oevel, the bi-hamiltonlan structure of some nonlinear fifthand seventh-order differential equatlons and recursion formulas for their symmetries and conserved covarlants, j. math. phys. 23 (1982), 358-363. [13] b. fuchssteiner, w. oevel and w. wiwianka, computer-algebra methods for investigation of hereditary operators of higher order soliton equations, computer phys. commun. 44 (1987), 47-55. [14] o. g-gaxiola · j. r. chávez, r. b-jaquez, solution of the nonlinear kompaneets equation through the laplace-adomian decomposition method, int. j. appl. comput. math. 3 (2017), 489–504. [15] m. a. helal, m. s. mehanna , the tanh method and adomian decomposition method for solving the foam drainage equation, appl. math. comput. 190 (2007), 599-609. [16] w. hereman , a. nuseir , symbolic methods to construct exact solutions of nonlinear partial differential equations , math. computers simul. 43 (1997), 13-27. [17] x. b. hu , d. l. wang and x. m. qian , soliton solutions and symmetries of the 2 + 1 dimensional kaup–kupershmidt equation, phys. lett. a, 262 (1999), 409–415. [18] m. hussain and m. khan, modified laplace decomposition method, appl. math. sci. 36 (4) (2010), 1769 1783. [19] m. inc, on numerical soliton solution of the kaup–kupershmidt equation and convergence analysis of the decomposition method, appl. math. comput. 172 (2006), 72–85. [20] k. jaradat , d. aloqali, w. alhabashene, using laplace decomposition method to solve nonlinear klien-gordan equation, u.p.b. sci. bull., ser. d, 80 (2) (2018), 213–222. [21] k. khan, m. ali akbar, a. h. arnous, exact traveling wave solutions for system of nonlinear evolution equations, springer plus, 5 (2016), 663. [22] a. a. kilbas, h. m. srivastava, and j. j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, the netherlands, 2006. [23] s. kumar & a. yildirim & y. khanc & l. weid, a fractional model of the diffusion equation and its analytical solution using laplace transform sci. iran. 19 (4) (2012), 1117-1123. [24] w. malfliet, the tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, j. comput. appl. math. 164-165 (2004), 529-541. [25] w. malfliet, the tanh method: a tool for solving certain classes of non-linear pdes, math. meth. appl. sci. 28 (2005), 2031–2035. [26] z. m. odibat, s. momani, approximate solutions for boundary value problems of time-fractional wave equation, appl. math. comput. 181 (2006), 767-774. [27] k. b. oldham and j. spanier, the fractional calculus, academic press, new york, ny, usa, 1974. [28] a. parker, on soliton solutions of the kaup–kupershmidt equation. ii. ’anomalous’ n-soliton solutions, physica d 137 (2000), 34–48. [29] i. podlubny, fractional differential equations, academic press, new york, ny, usa, 1999. [30] g. qingling, exact solutions of the mbbm equation, appl. math. sci. 5 (25) (2011), 1209 1215. [31] s. s. ray, r. k. bera, an approximate solution of nonlinear fractional differential equation by adomians decomposition method, appl. math. comput. 167 (2005), 561-571. int. j. anal. appl. 18 (3) (2020) 512 [32] e. g. reyes, nonlocal symmetries and the kaup–kupershmidt equation, j. math. phys. 46 (2005), 073507. [33] s. s. ray, r. k. bera , analytical solution of a fractional diffusion equation by adomian decomposition method, appl. math. comput. 174 (2006), 329-336. [34] a. h. salas, solving the generalized kaup–kupershmidt equation, adv. studi. theor. phys. 6 (18) (2012), 879 885. [35] j. l. schiff, the laplace tranform, theory and applications, springer-verlag, new york, 1999. [36] m. r. spiegel, laplace tranforms, mcgraw-hill, new york, 1965. [37] t. sumbal shaikh, n. ahmed, n. shahid, z. iqbal, solution of the zabolotskaya-khokholov equation by laplace decomposition method, int. j. sci. eng. res. 9 (2) (2018), 1811–1816. [38] p. wang, bilinear form and soliton solutions for the fifth-order kaup-kupershmidt, mod. phys. lett. b, 31 (6) (2017), 1750057. [39] p. wang and s. h. xiao, soliton solutions for the fifth-order kaup-kupershmidt equation, phys. scr. 93 (10), 105201. [40] a. m. wazwaz, the combined laplace transform-adomian decomposition method for handling nonlinear volterra integrodifferential equations, appl. math. comput. 216 (4) (2010), 1304-1309. [41] a. m. wazwaz, a new algorithm for calculating adomian polynomials for nonlinear operators , appl. math. comput. 111 (2000), 53-69. [42] a. m. wazwaz, partial differential equations and solitary waves theory, higher education press, berlin (2009). [43] l. yan, numerical solutions of fractional fokker-planck equations using iterative laplace transform method, abstr. appl. anal. 2013(2013), article id 465160. [44] q. yu , f. liu, v. anh and i. turner, solving linear and non-linear space–time fractional reaction–diffusion equations by the adomian decomposition method, int. j. numer. methods eng. 74 (2008), 138-158. [45] s. a. zarea, the tanh method: a tool for solving some mathematical models, chaos solitons fractals, 41 (2009), 979–988. 1. introduction 2. preliminaries 2.1. the tanh method 2.2. laplace transform 2.3. caputo derivative 2.4. the adomian decomposition method combined with laplace transform 3. main results 3.1. kupershmidt equation solutions by using tanh method 3.2. the approximate solution of time-fractional kupershmidt equation by ladm 4. conclusion references int. j. anal. appl. (2022), 20:2 an affirmative result on banach space v. srinivas1,2, t. thirupathi2,∗ 1department of mathematics, university college of science, saifabad,hyderabad, telangana, india 2department of mathematics, sreenidhi institute of science and technology, hyderabad, telangana, india ∗corresponding author: thotathirupathi1986gmail.com abstract. the aim of this paper is to establish a common fixed point theorem on banach space using occasionally weakly compatible (owc) mappings. 1. introduction fixed point theory is one of the most powerful topics of modern mathematics and might be taken as main subject of analysis.for the past many years, fixed point theory has been evolved as the area of research for many researchers. banach contraction principle is one such result proposed by banach to name a few.for the study of discontinuous and noncompatible mappings in fixed point theory we refer the literature like [4] and [5]. pathak and others [1] proved a fixed point theorem on complete metric space using continuity and weakly compatible mappings.thereafter sushil sharma, bhavana deshpande, and alok pandey [2] proved some more results on banach space.further several theorems [3], [6], [7], [8] , [9] and [10]are being generated on banach space using various conditions.the focus of this work is now on proving a result in banach space without a continuity constraint using owc mappings to prove a common fixed point theorem. before we prove our theorem, we’ll present some definitions and examples. received: aug. 28, 2021. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. banach space; weakly compatible mapping; occasionally weakly compatible (owc) mappings. https://doi.org/10.28924/2291-8639-20-2022-2 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-2 2 int. j. anal. appl. (2022), 20:2 2. preliminaries a pair (i,j) of a banach space is said to be definition 2.1 weakly commuting iff ‖ijα−jiα‖≤‖iα−jα‖ for all α ∈ x. definition 2.2 cmpatible iff ‖ijαj −jiαj‖=0 as j→∞ whenever {αj} is a sequence in x such that ‖iαj −jαj‖=0 as j →∞ for some η ∈ x. definition 2.3 weakly compatible if iη = jη for some η ∈ x such that ijη = jiη definition 2.4 owc if and only if there exists a point η ∈ x such that iη = jη implies ijη = jiη. we now discuss some examples to find the relation among the above definitions. example 2.5 let x = [0,1] is a banach space with ‖u −v‖= |u −v|, u,v ∈ x. define t and s as t(α)= { 1+2α 2 if 0≤ α ≤ 2 3 ; 5α+2 8 if 2 3 < α ≤ 1. s(α)= { 1−2α if 0≤ α ≤ 2 3 ; α if 2 3 ≤ α ≤ 1. take a sequence αk as αk = 1 6 − 1 k t(αk)= t( 1 6 − 1 k )= 1+2(1 6 −1 k ) 2 =2 3 s(αk)= s( 1 6 − 1 k )=1−2(1 6 − 1 k )=2 3 tαk=sαk= 2 3 ask →∞. now ts(αk)= gj( 1 6 − 1 k )=g(1−2(1 6 − 1 k )) =g(2 3 + 2 k )= 3(2 3 )+2 k +1 5 =3 5 as k →∞. stαk = s[t( 1 6 − 1 k )]=s[1−2(1 6 − 1 k )] =s(2 3 + 2 k )=2 3 + 2 k =2 3 as k →∞. limk →∞‖(tsαk −stαk)‖ 6=0 therefore the pair (t,s) is not compatible. but t(1 6 )=s(1 6 )=(2 3 ). ts(1 6 )=t[1−2(1 6 )]=g(2 3 )=2 3 and st(1 6 )= j(g(1 6 ))= 2(2 3 )+1 2 = 2 3 . hence the pair (t,s) weakly compatiable. example 2.6 let x = [0,1] is a banach space with ‖u −v‖= |u −v|, ∀u,v ∈ x. define four maps g,j,h and i as follows g(α)= i(α)= { α+2 8 if 0≤ α < 1 2 ; 1−α if α ≥ 1 2 . int. j. anal. appl. (2022), 20:2 3 j(α)= h(α)= { α+1 4 if 0≤ α < 1 2 ; 4α−1 2 if α ≥ 1 2 . here g(0)= j(0)= 1 4 and g(1 2 )= j(1 2 )= 1 2 . clearly α =0 and α = 1 2 are two coincidence points. if α =0 then gj(0)= g(1 4 )= (1 4 +2) 8 = 9 32 and jg(0)= j(1 4 )= (1 4 )+1 4 = 5 16 . therefore gj(0) 6= jg(0). now jg(1 2 )= j(1− 1 2 )= j(1 2 )= 4(1 2 )−1 2 = 1 2 gj(1 2 )= g(1 2 )=1− 1 2 = 1 2 therefore gj(1 2 )= jg(1 2 ). thus g and j are owc, but not weakly compatible. the following theorem was proved in metric space [1]. 3. theorem let x be a comlete metic space and self mappings g,h,i and j are satisfying (b1) g(x)⊆ h(x) and i(x)⊆ j(x) (b2) d(gu,iv)2p ≤ [aφ0(d(ju,hv)2p)+(1−a)max{φ1(d(ju,hv)2p), φ2(d(ju,gv) qd(hv,iv)q ′ ),φ3(d(ju,iv) rd(hv,gv)r ′ ), φ4(d(ju,gv) sd(hv,gu)s ′ ),φ5(d(ju,iv) ld(hv,iv)l ′ )}] for all u,v∈x where φj ∈ φ,j =0,1,2,3,4,5and a takes values from0 to1 inclusive and p,q,q ′ , r, r ′ .s,s ′ , l, l ′ takes from 0 exclusive to 1 inclusive such that 2p = q +q′ = r + r ′ = s + s′ = l + l′. (b3) the mapping g or i is a continuous (b4) the pairs (g,j) and (i,h) are weakly compatible mappings. then g,h,i and j have a unique fixed point which is common. now we’ll present a list of key lemmas that are useful to our main result. lemma 3.1 [10] if φj ∈ φ and j ∈ {0,1,2,3,4,5} ,φ is upper semicontinuous and also contractive modulus such that max{φj(t)}≤ φ(t) for all t > 0 and also φ(t) < t for t > 0. lemma 3.2 [1] let φj ∈ φ and βj be a non-negative real sequence.if βj+1 ≤ φ(βj) for j ∈ n, then the sequence converges to 0. now we genealize the existence of the above theorem by extending it on to banach space under the following modified conditions. 4 int. j. anal. appl. (2022), 20:2 theorem 3.3 let (x,‖.‖) be a banach space and self mappings g,h,i and j are satisfying (b1) g(x)⊆ h(x) and i(x)⊆ j(x) (b2) ‖(gu − iv)‖2p ≤ [aφ0(‖(ju −hv)‖2p)+(1−a)max{φ1(‖ju −hv‖2p), φ2(‖ju −gu‖q‖hv − iv‖q ′ ),φ3(‖ju − iv‖r‖hv −gu‖r ′ ), φ4( 1 2 ‖ju −gu‖s‖hv − iv‖s ′ ),φ5( 1 2 ‖ju − iv‖l‖hv − iv‖l ′ )}] for all u,v∈x where φj ∈ φ,j =0,1,2,3,4,5and a takes values from0 to1 inclusive and p,q,q′, r, r ′,s,s′, l, l′ takes from 0 exclusive to 1 inclusive such that 2p = q +q′ = r + r ′ = s + s′ = l + l′. (b3) two pairs (g,j) and (i,h) have a coincidence point (b4) the pairs maps (g,j) and (i,h) are owc. then g,h,i and j have a unique common fixed point. proof using the condition (b1), there is a point u0 ∈ x such that gu0 = hu1.for this point u1 ∈ x there exists a point u2 in x such that iu1 = ju2 and so on. continuing this process it is possible to construct a sequence {vj} for j = 1,2,3......in x such that v2j = gu2j = hu2j+1,v2j+1 = iu2j+1 = ju2j+2 for j ≥ 0. we now prove {vj} is a cauchy sequence. putting u = u2j and v = v2j+1 in (b2), we get ‖v2j −v2j+1‖2p ≤ [aφ0(‖v2j−1 −v2j‖2p)+(1−a)max{φ1(‖v2j−1 −v2j‖2p), φ2(‖v2j−1 −v2j‖q‖v2j −v2j+1‖q ′ ),φ3(‖v2j−1 −v2j+1‖r‖v2j −v2j‖r ′ ), φ4( 1 2 ‖v2j−1 −v2j‖s‖v2j −v2j‖s ′ ),φ5( 1 2 ‖v2j−1 − iv2j+1‖l‖v2j −v2j+1‖l ′ )}]. denote ρj = ‖vj −vj+1‖ (ρ2j) 2p ≤ [aφ0(ρ2j−1)2p)+(1−a)max{φ1(ρ2j−1)2p,φ2((ρ2j−1)q(ρ2j)q ′ ),φ3(0), φ4(0),φ5( 1 2 [(ρ2j−1) l +(ρ2j) l′)](ρ2j) l)}]. (ρ2j) 2p ≤ [aφ0(ρ2j−1)2p)+(1−a)max{φ1(ρ2j−1)2p,φ2((ρ2j−1)q(ρ2j)q ′ ),φ3(0), φ4(0),φ5( 1 2 [(ρ2j−1) l(ρ2j) l +(ρl ′ 2j)(ρ2j) l)])}]. if ρ2j > ρ2j−1 then we have (ρ2j) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2(ρ2j)q+q ′ ,φ3(0),φ4(0), φ5( 1 2 [(ρ2j) l+l′(ρ2j) l+l′ +(ρl+l ′ 2j )])}(ρ2j)2p)]. ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2(ρ2j)2p,φ3(0),φ4(0),φ5(ρ2j)2p)]. using lemma (3.2) (ρ2j) 2p ≤ φ(ρ2j)2p < (ρ2j)2p. a contradiction. int. j. anal. appl. (2022), 20:2 5 this implies ρ2j ≤ ρ2j−1 then using this inequality the condition (b2) yields ρ2j ≤ φ(ρ2j−1). (3.1) similarly taking u = u2j+2 and v = u2j+1 in (b2), we get ‖v2j+1 −v2j+2‖2p ≤ [aφ0(‖v2j −v2j+1‖2p)+(1−a)max{φ1(‖v2j −v2j+1‖2p), φ2(‖v2j+1 −v2j+2‖q‖v2j −v2j+1‖q ′ ),φ3(‖v2j+1 −v2j+1‖r‖v2j −v2j+1‖r ′ ), φ4( 1 2 ‖v2j+11 −v2j+2‖s‖v2j −v2j+2‖s ′ ),φ5( 1 2 ‖v2j+2 − iv2j+1‖l‖v2j −v2j+1‖l ′ )}]. (ρ2j+1) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2((ρ2j+1)q(ρ2j)q ′ ),φ3(0), φ4( 1 2 [(ρ2j+1) s(ρ2j) s′ +(ρ2j+1) s′],φ5(0)}]. (ρ2j+1) 2p ≤ [aφ0(ρ2j)2p)+(1−a)max{φ1(ρ2j)2p,φ2((ρ2j+1)q(ρ2j)q ′ ),φ3(0), φ4( 1 2 [(ρ2j+1) s(ρ2j) s′ +(ρ2j+1) s′(ρ2j+1) s′]),φ5(0)}]. if ρ2j+1 > ρ2j,then we have (ρ2j+1) 2p ≤ [aφ0(ρ2j+1)2p)+(1−a)max{φ1(ρ2j+1)2p,φ2((ρ2j+1)q+q ′ ),φ3(0), φ4(ρ2j+1),φ5(0)}]. using lemma(3.2) (ρ2j+1) 2p ≤ φ(ρ2j+1)2p < (ρ2j+1)2p which is a contradiction. thus we must have ρ2j+1 ≤ ρ2j. again applying (b2) to the above inequality,we obtain ρ2j+1 ≤ φ(ρ2j). (3.2) from (2.1) and (2.2), in general ρj+1 ≤ φ(ρj), for j=0,1,2,3.... by lemma 3.3 we get ρj → 0 as j →∞. this shows that ρj = ‖vj −vj+1‖→ 0 as j →∞. hence {vj} is a cauchy sequence. since x banach space, ∃ a point η ∈ x such that vj → η as j →∞. consequently, the subsequences gα2j,jα2j, iα2j+1 and hα2j of {vj} also converge to the same point η ∈ x. since the pair (g,j) is owc ,there exists u ∈ c(g,j) such that gu = ju = η(say) and gju = jgu = η ′ (say). hence we have gη = jη = η ′ (say) (3.3) since the pair (i,h) is owc ,there exists v ∈ c(i,h) such that iv = hv = δ(say) and ihv = hiv = δ′(say). 6 int. j. anal. appl. (2022), 20:2 hence we have iδ = hδ = δ ′ (say) (3.4) now we claim that η ′ = δ′ substitute u = η,v = δ in (b2) ‖(gη − iδ)‖2p ≤ [φ0(‖(jη −hδ)‖2p)+(1−a)max{φ1(‖jη −hδ‖2p), φ2(‖jη −gη‖q‖hδ − iδ‖q ′ ),φ3(‖jη − iδ‖r‖hδ −gη‖r ′ ), φ4( 1 2 ‖jη −gη‖s‖hδ − iδ‖s ′ ),φ5( 1 2 ‖jη − iδ‖l‖hδ − iδ‖l ′ )}]. using (3) and (4), we get ‖η′ −δ′‖2p ≤ [φ0(‖η′ −δ′‖2p)+(1−a)max{φ1(‖η′ −δ′‖2p), φ2(‖η′ −η′‖q‖δ′ −δ‖q ′ ),φ3(‖η′ −δ′‖r‖δ′ −η′‖r ′ ), φ4( 1 2 ‖η′ −η′‖s‖δ′ −η′‖s ′ ),φ5( 1 2 ‖η′ −δ′‖l‖δ′ −δ′‖l ′ )}]. ‖η′ −δ′‖2p ≤ [φ0(‖η′ −δ′‖2p)+(1−a)max{φ1(‖η′ −δ′‖2p),φ2(0),φ3(‖η′ −δ′‖2p),φ4(0),φ5(0)}]. since by lemma (3.2) ‖η′ −δ′‖2p ≤ φ(‖η′ −δ′‖)2p < ‖η′ −δ′‖2p a contradiction. therefore η′ = δ′. hence from (2.3) we get gη = jη = δ ′ . (3.5) next claim that η = δ′ substitute u = u and v = δ in (b2) ‖(gu − iδ)‖2p ≤ [φ0(‖(ju −hδ)‖2p)+(1−a)max{φ1(‖ju −hδ‖2p), φ2(‖ju −gu‖q‖hδ − iδ‖q ′ ),φ3(‖ju − iδ‖r‖hδ −gu‖r ′ ), φ4( 1 2 ‖ju −gu‖s‖hδ − iδ‖s ′ ),φ5( 1 2 ‖ju − iδ‖l‖hδ − iδ‖l ′ )}]. using gu = ju = η and iδ = hδ = δ ′ , we get ‖η−δ′‖2p ≤ [φ0(‖η−δ′‖2p)+(1−a)max{φ1(‖η−δ′‖2p),φ2(‖η−η‖q‖δ−δ‖q ′ ),φ3(‖η−δ′‖r‖δ′−η‖r ′ ), φ4( 1 2 ‖η −η‖s‖δ′ −η‖s ′ ),φ5( 1 2 ‖η −δ′‖l‖δ′ −δ′‖l ′ )}]. ‖η −δ′‖2p ≤ [φ0(‖η −δ′‖2p)+(1−a)max{φ1(‖η −δ′‖2p),φ2(0),φ3(‖η −δ′‖2p),φ4(0),φ5(0)}]. by lemma(3.2) ‖η −δ′‖2p ≤ φ(‖η −δ′‖)2p < ‖η −δ′‖2p which is a contradiction, and hence η = δ′. from (2.5),we get gη = jη = η (3.6) and iδ = hδ = η. (3.7) now we claim that η = δ. again in (b2) putting u = η and v = v ‖gη − iv‖2p ≤ [φ0(‖(jη −hv)‖2p)+(1−a)max{φ1(‖jη −hv‖2p), int. j. anal. appl. (2022), 20:2 7 φ2(‖jη −gη‖q‖hv − iv‖q ′ ),φ3(‖jη − iv‖r‖hv −gη‖r ′ ), φ4( 1 2 ‖jη −gη‖s‖hv − iv‖s ′ ),φ5( 1 2 ‖ju − iv‖l‖hv − iv‖l ′ )}]. using gη = jη = η and iv = hv = δ ‖η−δ‖2p ≤ [φ0(‖η−δ‖2p)+(1−a)max{φ1(‖η−δ‖2p),φ2(‖η−η‖q‖δ−δ‖q ′ ),φ3(‖η−δ‖r‖η−δ‖r ′ ), φ4( 1 2 ‖η −η‖s‖δ −η‖s ′ ),φ5( 1 2 ‖η −δ‖l‖δ −δ‖l ′ )}]. ‖η −δ‖2p ≤ [φ0(‖η −δ‖2p)+(1−a)max{φ1(‖η −δ‖2p),φ2(0),φ3(‖η −δ‖2p),φ4(0),φ5(0)}]. by lemma(3.2) ‖η −δ‖2p ≤ φ(‖η −δ‖)2p < ‖η −δ‖2p which is a contradiction. therefore η = δ. from (2.7), we get iη = hη = η. (3.8) from (2.6 and (2.8), we get gη = jη = iη = hη = η. hence this gives that η is a common fixed point for g, h, i and j. for uniqueness: suppose η and η∗ (η 6= η∗) are two common fixed points. then put u = η and v = η∗ in the inequality (b2) ‖gη − iη∗‖2p ≤ [aφ0(‖jη −hη∗‖2p)+(1−a)max{φ1(‖jη −hη∗‖2p), φ2(‖jη −gη‖q‖hη∗ − iη∗‖q ′ ),φ3(‖jη − iη∗‖r‖hη∗ −gη‖r ′ ), φ4( 1 2 ‖jη −gη‖s‖hη∗ −gη‖s ′ ),φ5( 1 2 ‖jη − iη∗‖l‖hη∗ − iη∗‖l ′ )}] ‖η−η∗‖2p ≤ [aφ0(‖η−η∗‖2p)+(1−a)max{φ1(‖η−η∗‖2p),φ2(0),φ3(‖η−η∗‖2p),φ4(0),φ5(0)}] ‖η −η∗‖2p ≤ [φ(‖η −η∗‖2p) < ‖η −η∗‖2p a contradiction. which gives η = η∗, this proves the uniqueness. example now we continue to discuss the example (2.6) to justyfy our theorem(3.3). now g(x)=i(x)=[1 4 , 5 16 )∪ (1 2 ) while j(x)=h(x)= [1 4 , 3 8 )∪ (1 2 ) clearly g(x)⊆ h(x), i(x)⊆ j(x) therefore (b1) is satisfied. now we verify the condition (b2) case(i). if u,v ∈ [0, 1 2 ),then we have ‖(gu − iv)‖= |gα− iv| put u = 1 3 ,v = 1 5 . then the inequality (b2) implies ‖g(1 3 )− i(1 5 )‖2p ≤ [aφ0(‖j(13)−h( 1 5 ))‖2p)+(1−a)max{φ1(‖j(13)−h( 1 5 )‖2p), 8 int. j. anal. appl. (2022), 20:2 φ2(‖j(13)−g( 1 3 )‖q‖h(1 5 )− i(1 5 )‖q ′ ),φ3(‖j(13)− i( 1 3 )‖r‖h(1 5 )−g(1 3 )‖r ′ ), φ4( 1 2 ‖j(1 3 )−g(1 3 )‖s‖h(1 5 )− i(1 5 )‖s ′ ),φ5( 1 2 ‖j(1 3 )− i(1 5 )‖l‖h(1 5 )− i(1 5 )‖l ′ )}] for a = 1 2 and p = q = q′ = r = r ′ = s = s′ = l = l′ = 1 2 ‖0.0166‖≤ [1 2 φ0‖(0.033)‖+(1− 12)max{φ1‖(0.01665)‖, φ2‖(0.0322)‖,φ3‖(0.01870)‖,φ4‖(0.0161)‖,φ5‖(0.019)}] |0.0166| < |0.0327|. hence the inequality (b2) holds. also the verification in the remaining intervals is also simple. here it is evident that 1 2 is the unique common fixed point for the four self mappings. 4. conclusion in this paper we proved a common fixed point theorem on banach space using owc mappings.it is also clear from the example proved that the mappings are neither compatible nor weakly compatible but owc mappings.moreover the continuity condition is dropped.hence we conclude that our theorem stands as an improvement of theorem (3). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h.k. pathak, m.s. khan, r. tiwari, a common fixed point theorem and its application to nonlinear integral equations, computers math. appl. 53 (2007), 961–971. https://doi.org/10.1016/j.camwa.2006.08.046. [2] s. sharma, b. deshpande, a. pandey, common fixed point point theorem for a pair of weakly compatible mappings on banach spaces, east asian math. j. 27 (2011), 573–583. https://doi.org/10.7858/eamj.2011.27.5.573. [3] a. djoudi, l. nisse, greguš type fixed points for weakly compatible maps, bull. belg. math. soc. simon stevin. 10 (2003), 369–378. https://doi.org/10.36045/bbms/1063372343. [4] s. sharma, b. deshpande, common fixed point theorems for finite number of mappings without continuity and compatibility on intuitionistic fuzzy metric spaces, chaos solitons fractals. 40 (2009), 2242–2256. https://doi. org/10.1016/j.chaos.2007.10.011. [5] r.a. rashwan, a common fixed point theorem in uniformly convex banach spaces. ital. j. pure appl. math. 3 (1998), 117–126. [6] n. shahzad, s. sahar, fixed points of biased mappings in complete metric spaces, radovi math. 11 (2002), 249-261. [7] s. sharma, p. tilwankar, some fixed point theorems in intuitionistic fuzzy metric spaces, tamkang j. math. 42 (2011), 405–414. https://doi.org/10.5556/j.tkjm.42.2011.683. [8] v. srinivas, a result on banach space using property e.a, indian j. sci. technol. 13 (2020), 4490–4499. https: //doi.org/10.17485/ijst/v13i44.1909. [9] v. srinivas, t. thirupathi, a result on banach space using e.a like property, malaya j. mat. 8 (2020), 903–908. https://doi.org/10.26637/mjm0803/0029. [10] h.k. pathak, s.n. mishra, a.k. kalinde, common fixed point theorems with applications to nonlinear integral equations, demonstr. math. 32 (1999), 547-564. https://doi.org/10.1515/dema-1999-0310. https://doi.org/10.1016/j.camwa.2006.08.046 https://doi.org/10.7858/eamj.2011.27.5.573 https://doi.org/10.36045/bbms/1063372343 https://doi.org/10.1016/j.chaos.2007.10.011 https://doi.org/10.1016/j.chaos.2007.10.011 https://doi.org/10.5556/j.tkjm.42.2011.683 https://doi.org/10.17485/ijst/v13i44.1909 https://doi.org/10.17485/ijst/v13i44.1909 https://doi.org/10.26637/mjm0803/0029 https://doi.org/10.1515/dema-1999-0310 1. introduction 2. preliminaries 3. theorem 4. conclusion references international journal of analysis and applications volume 18, number 4 (2020), 586-593 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-586 on some new contractive conditions for asymptotically regular set-valued mappings pradip debnath1,∗, manuel de la sen2 1department of applied science and humanities, assam university, silchar, cachar, assam 788011, india 2institute of research and development of processes, university of the basque country, campus of leioa, 48940-leioa (bizakaia), spain ∗corresponding author: debnath.pradip@yahoo.com abstract. in this paper, we introduce two new contractive conditions of proinov-type for asymptotically regular set-valued mappings and prove the existence of their fixed points. our results extend some results due to nadler and boyd and wong. 1. preliminaries the notion of asymptotic regularity was introduced by browder and petryshyn [6]. this notion has been exploited by several authors to obtain fixed points of the concerned map. in this context, fixed points of dissipative multivalued maps were studied by aubin and siegel [1]. weak convergence of asymptotically regular (in short, ar) sequences for nonexpansive mappings was studied by engl [11] and its relation with chebychef-centers was established. fixed points of ar mappings and their applications were studied by guay and singh [12], rhoades et al. [17], beg and azam [2], singh et. al. [19], debnath and de la sen [9]. the connection between fixed points and asymptotic regularity may easily be understood from the fact that contraction maps are ar. nadler [14] introduced the set-valued version of banach’s contraction principle with received march 18th, 2020; accepted april 27th, 2020; published may 11th, 2020. 2010 mathematics subject classification. 47h10, 54h25, 54e50. key words and phrases. fixed point; asymptotically regular map; set-valued map; metric space. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 586 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-586 int. j. anal. appl. 18 (4) (2020) 587 the help of hausdorff metric. aubin and siegel [1] has discussed about the applications of ar multivalued maps in control theory, optimisation and system theory. sequences of ar multivalued maps have been used by itoh and takahashi [13] and rhoades et al. [18]. in [16], rhoades compared several contractive conditions and found that most of the conditions imply asymptotic regularity. as such, study of ar maps is a well motivated area of research in fixed point theory. we recall the definition of pompeiu-hausdorff metric which plays a crucial role in set-valued analysis. let γ(x) denote the class of all nonempty closed and bounded subsets of a non-empty set x and (γ(x),ph) denote the pompeiu-hausdorff metric in a ms (x,δ). the metric function ph : γ(x)×γ(x) → [0,∞) is defined by ph(u,v ) = max{sup ξ∈v ∆(ξ,u), sup η∈u ∆(η,v )}, for all u,v ∈ γ(x), where ∆(η,v ) = infξ∈v δ(η,ξ). definition 1.1. [14] let r : x → γ(x) be a set-valued map. µ ∈ x is called a fixed point of r if µ ∈ rµ. following results are important in the present context. lemma 1.1. [4, 7] let (x,δ) be a ms and u,v,w ∈ γ(x). then (1) ∆(µ,v ) ≤ δ(µ,γ) for any γ ∈ v and µ ∈ x; (2) ∆(µ,v ) ≤ph(u,v ) for any µ ∈ u; (3) ∆(µ,u) ≤ δ(µ,ν) + ∆(ν,u) for all µ,ν ∈ x. lemma 1.2. [14] let {un} be a sequence in γ(x) and limn→∞ph(un,u) = 0 for some u ∈ γ(x). if µn ∈ un and limn→∞δ(µn,µ) = 0 for some µ ∈ x, then µ ∈ u. definition 1.2. [14] let r : x → γ(x) be a set-valued map. r is said to be a set-valued contraction if ph(rµ,rν) ≤ λδ(µ,ν) for all µ,ν ∈ x, where λ ∈ [0, 1). orbital sequence is one of the important components in the investigation of fixed points for set-valued maps (see [3, 8]). definition 1.3. [10] let (x,δ) be a ms and r : x → γ(x) a set-valued map. an r-orbital (or, simply orbital) sequence of r at a point µ ∈ x is a set o(µ,r) of points in x defined by o(µ,r) = {µ0 = µ,µn+1 ∈ rµn,n = 0, 1, 2, . . .}. inspired by the recent work of proinov [15], in the present paper we introduce some new contractive definitions for set-valued maps and show that our results extend some results due to nadler [14] and boyd and wong [5]. the rest of the paper is organised as follows. section 2 contains three lemmas on the asymptotic regularity of the set-valued map under consideration, while in section 3 we present our main results with the help of those lemmas. section 4 contains conclusions and future work. int. j. anal. appl. 18 (4) (2020) 588 2. some lemmas on asymptotic regularity in this section we present three lemmas. the first lemma provides us with conditions on the two auxiliary control functions g and h those ensure that the set-valued map under consideration is ar. the recent proofs due to proinov [15] will be taken as a framework and his proofs will be extended to their set-valued analogues using the function ∆ and the pompeiu-hausdorff metric ph. definition 2.1. [9] let (x,δ) be a ms. a set-valued map r : x → γ(x) is said to be asymptotically regular (in short, ar) at a point µ0 ∈ x, if for any orbital sequence {µn} = o(µ0,r), we have lim n→∞ δ(µn,µn+1) = 0. if r is ar at all points of its domain, then it is called ar. lemma 2.1. let (x,δ) be a ms and r : x → γ(x) be a set-valued map satisfying h(ph(rµ,rν)) ≤ g(δ(µ,ν)) for all µ,ν ∈ x, where g,h : (0,∞) → r are functions such that (1) g(t) < h(t) for all t > 0; (2) inft>� h(t) > −∞ for any � > 0. further, suppose that at least one of the following is true: (1) h is nondecreasing and lim supt→�+ g(t) < h(�+) for any � > 0; (2) if {h(tn)} and {g(tn)} are convergent sequences with the same limit and {h(tn)} is strictly decreasing, then limn→∞ tn = 0. then r is ar. proof. fix µ0 ∈ x and consider the orbital sequence {µn} = o(µ0,r). let en = δ(µn,µn+1). to prove that en → 0. rest of the proof follows verbatim from the proof of lemma 3.2 in [15]. � in our second lemma, we find conditions on g and h which show that if the map r is ar, then the orbital sequence {µn} = o(µ0,r) is cauchy. lemma 2.2. let (x,δ) be a ms and r : x → γ(x) be a set-valued map satisfying h(ph(rµ,rν)) ≤ g(δ(µ,ν)) for all µ,ν ∈ x, where the functions g,h satisfy at least one of the following conditions: (1) h is nondecreasing, g(t) < h(t) for all t > 0 and lim supt→�+ g(t) < h(�+) for any � > 0; (2) lim supt→� g(t) < lim inft→�+ h(t) for any � > 0; (3) lim supt→�+ g(t) < lim inft→� h(t) for any � > 0. if r is ar at a point µ0 ∈ x, then the orbital sequence {µn} = o(µ0,r) is a cauchy sequence. int. j. anal. appl. 18 (4) (2020) 589 proof. let r be ar at the point µ0 ∈ x. construct the orbital sequence {µn} = o(µ0,r). suppose that the sequence {µn} is not cauchy. next, using exactly similar arguments as in the proof of lemma 3.3 in [15], we arrive at a contradiction. therefore, {µn} is a cauchy sequence. � in the third lemma, we establish conditions on the functions g and h which guarantee the existence of a fixed point if the orbital sequence {µn} = o(µ0,r) is convergent. in fact, the limit of the orbital sequence happens to be a fixed point of r. we recall the definition of closed graph for a set-valued map. definition 2.2. let (x,δ) be a ms and r : x → γ(x) be a set-valued map. r is said to have a closed graph g(r) if g(r) = {(µ,ν) : ν ∈ rµ,µ ∈ x} is a closed subset of x ×x with the product topology. lemma 2.3. let (x,δ) be a ms and r : x → γ(x) be a set-valued map satisfying h(ph(rµ,rν)) ≤ g(δ(µ,ν)), (2.1) for all µ,ν ∈ x, where g,h : (0,∞) → r are functions satisfying at least one of the following conditions: (1) r has a closed graph; (2) h is nondecreasing, g(t) < h(t) for all t > 0; (3) lim supt→0+ g(t) < lim inft→� h(t) for any � > 0. if {µn} = o(µ0,r) is an orbital sequence for some µ0 ∈ x and limn→∞µn = θ, then θ is a fixed point of r. proof. we split the proof into three parts. part i: suppose condition 1 above is true, i.e., for all sequences {µn} and {νn}, where νn ∈ rµn for each n = 0, 1, 2, . . ., such that µn → µ and νn → ν as n →∞, we have ν ∈ rµ. since {µn} = o(µ0,r) is an orbital sequence and limn→∞µn = θ, we have µn+1 ∈ rµn for each n = 0, 1, 2, . . . and limn→∞µn+1 = θ. thus by condition 1, we have θ ∈ rθ. part ii: consider the orbital sequence {µn} = o(µ0,r). if ph(rµn,rθ) = 0 for all but finitely many values of n, then ∆(θ,rθ) ≤ δ(θ,µn+1) + ∆(µn+1,rθ) ≤ δ(θ,µn+1) + ph(rµn,rθ), (using property 3 of lemma 1.1) = δ(θ,µn+1), for those values of n. taking limit in the above inequality, as n →∞, we have ∆(θ,rθ) = 0, i.e., θ ∈ rθ. int. j. anal. appl. 18 (4) (2020) 590 further suppose that ph(rµn,rθ) > 0 for infinitely many n and let condition 2 above be true. then using (2.1) from hypothesis with µ = µn and ν = θ, we have h(ph(rµn,rθ)) ≤ g(δ(µn,θ)) (2.2) =⇒ h(∆(µn+1,rθ)) ≤ g(δ(µn,θ)), ( since ∆(µn+1,rθ) ≤ph(rµn,rθ) and h is nondecreasing). (2.3) since condition 2 is true, we have h(∆(µn+1,rθ)) ≤ g(δ(µn,θ)) < h(δ(µn,θ)). thus we have ∆(µn+1,rθ) < δ(µn,θ) and further, taking limit as n →∞, we obtain, ∆(θ,rθ) ≤ 0, i.e, θ ∈ rθ. part iii: suppose condition 3 in the hypothesis is true. let αn = ∆(µn+1,rθ) and βn = δ(µn,θ). then from (2.2), we have h(αn) ≤ g(βn) (2.4) for infinitely many values of n. clearly, αn → � and βn → 0 as n →∞, where � = ∆(θ,rθ) > 0. it follows from (2.4) that lim inf t→� h(t) ≤ lim inf n→∞ h(αn) ≤ lim sup n→∞ g(βn), using (2.4) ≤ lim sup t→0 g(t). (2.5) but (2.5) is a contradiction to condition 3 in the hypothesis if � > 0. hence we must have � = ∆(θ,rθ) = 0, i.e., θ ∈ rθ. � 3. main results in this section we present our two main results with the help of the lemmas established in the previous section. theorem 3.1. let (x,δ) be a ms and r : x → γ(x) be a set-valued map satisfying h(ph(rµ,rν)) ≤ g(δ(µ,ν)), (3.1) for all µ,ν ∈ x, where the functions g,h : (0,∞) → r are functions satisfying the following conditions: (1) h is nondecreasing; (2) g(t) < h(t) for all t > 0; (3) lim supt→�+ g(t) < h(�+) for any � > 0; (4) inft>� h(t) > −∞ for any � > 0. then r has a fixed point. int. j. anal. appl. 18 (4) (2020) 591 proof. using condition (1)-(4) and lemma 2.1, we have that r is ar. fix µ0 ∈ x and construct the orbital sequence {µn} = o(µ0,r). from conditions (1)-(3) and lemma 2.2, it follows that the orbital sequence {µn} is cauchy. since (x,δ) is complete, we have µn → θ (for some θ ∈ x) as n → ∞. again, using conditions (1) and (2) and lemma 2.3, we can conclude that θ is a fixed point of r. � theorem 3.2. let (x,δ) be a ms and r : x → γ(x) be a set-valued map satisfying h(ph(rµ,rν)) ≤ g(δ(µ,ν)), (3.2) for all µ,ν ∈ x, where the functions g,h : (0,∞) → r satisfy the following conditions: (1) g(t) < h(t) for all t > 0; (2) inft>� h(t) > −∞ for any � > 0; (3) if {h(tn)} and {g(tn)} are convergent sequences with the same limit and {h(tn)} is strictly decreasing, then tn → 0 as n →∞; (4) lim supt→�+ g(t) < lim inft→� h(t) or lim supt→� g(t) < lim inft→�+ h(t) for any � > 0; (5) r has a closed graph or lim supt→0+ g(t) < lim inft→� h(t) for any � > 0. then r has a fixed point. proof. fix µ0 ∈ x and construct the orbital sequence {µn} = o(µ0,r). from conditions (1)-(3) and lemma 2.1, we have that r is ar at µ0. from condition (4) and lemma 2.2, we can prove that the orbital sequence {µn} is cauchy. since (x,δ) is complete, we have µn → θ (for some θ ∈ x) as n →∞. next by condition (5) and lemma 2.3, we conclude that θ ∈ rθ. � the remarks below show that our results extend some results due to nadler and boyd-wong. remark 3.1. (1) if h(t) = t and g(t) = λt, where 0 ≤ λ < 1, then theorem 3.1 and theorem 3.2 both reduce to set-valued version of banach contraction principle due to nadler [14]. (2) if h(t) = t, then both the theorems reduce to set-valued versions of boyd-wong’s fixed point theorem [5]. conclusion. in the present paper, we extended some recent results due to proinov to their set-valued or multi-valued counterparts. inspired by the work of proinov, we introduced some new contractive definitions on the map under consideration with the help of two control functions. asymptotic regularity of the map has been established and a scheme to obtain a fixed point has been discussed. common fixed points and coincidence points may be investigated in future in this context. int. j. anal. appl. 18 (4) (2020) 592 authors’ contributions: all authors contributed equally and significantly in writing this article. all authors read and approved the final manuscript. acknowledgment: research of the first author p. debnath is supported by ugc (ministry of hrd, govt. of india) through ugc-bsr start-up grant vide letter no. f.30-452/2018(bsr) dated 12 feb 2019. the author m. de la sen acknowledges the grant it 1207-19 from basque government. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. p. aubin and j. siegel. fixed points and stationary points of dissipative multivalued maps. proc. amer. math. soc., 78 (1980), 391–398. [2] i. beg and a. azam. fixed points of asymptotically regular multivalued mappings. j. austral. math. soc. (series a), 53 (1992), 284–289. [3] m. berinde and v. berinde. on a general class of multi-valued weakly picard mappings. j. math. anal. appl., 326 (2007), 772–782. [4] m. boriceanu, a. petrusel, and i.a. rus. fixed point theorems for some multivalued generalized contraction in b-metric spaces. int. j. math. stat., 6 (2010), 65–76. [5] d. w. boyd and j. s. wong. on nonlinear contractions. proc. amer. math. soc., 20 (1969), 458–464. [6] f. e. browder and w. v. petryshyn. the solution by iteration of nonlinear functional equation in banach spaces. bull am. math. soc., 72 (1966), 571–576. [7] s. czerwik. nonlinear set-valued contraction mappings in b-metric spaces. atti sem. mat. univ. modena, 46 (1998), 263–276. [8] p. z. daffer and h. kaneko. fixed points of generalized contractive multi-valued mappings. j. math. anal. appl., 192 (1995), 655–666. [9] p. debnath and m. de la sen. contractive inequalities for some asymptotically regular set-valued mappings and their fixed points. symmetry, 12 (2020), 411. [10] p. debnath and m. de la sen. fixed points of eventually δ-restrictive and δ(�)-restrictive set-valued maps in metric spaces. symmetry, 12 (2020), 127. [11] h. w. engl. weak convergence of asymptotically regular sequences for nonexpansive mappings and connections with certain chebyshef–centers. nonlinear anal., theory methods appl., 1 (5) (1977), 495–501. [12] m. d. guay and k. l. singh. fixed points of asymptotically regular mappings. math. vesnik, 35 (1983), 101–106. [13] s. itoh and w. takahashi. single valued mappings, multivaluued mappings and fixed point theorems. j. math. anal. appl., 59 (1977), 514–521. [14] s. b. nadler. multi-valued contraction mappings. pac. j. math., 30 (2) (1969), 475–488. [15] p. d. proinov. fixed point theorems for generalized contractive mappings in metric spaces. j. fixed point theory appl., 22 (2020), 21. [16] b. e. rhoades. a comparison of various definitions of contractive mappings. trans. amer. math. soc., 226 (1977), 257–290. [17] b. e. rhoades, s. sessa, m. s. khan, and m. swaleh. on fixed points of asymptotically regular mappings. j. austral. math. soc. (ser. a), 43 (1987), 328–346. int. j. anal. appl. 18 (4) (2020) 593 [18] b. e. rhoades, s. l. singh, and c. kulshrestha. coincidence theorem for some multivalued mappings. int. j. math. and math. sci., 7 (3) (1984), 429–434. [19] s. l. singh, s. n. mishra, and r. pant. new fixed point theorems for asymptotically regular multi-valued maps. nonlinear anal., theory methods appl., 71 (2009), 3299–3304. 1. preliminaries 2. some lemmas on asymptotic regularity 3. main results references int. j. anal. appl. (2022), 20:16 weighted ostrowski’s type integral inequalities for mapping whose first derivative is bounded s. fahad1, m. a. mustafa2, z. ullah3, t. hussain2, a. qayyum2,∗ 1bahauddin zakriya university multan-pakistan 2institute of southern punjab multan-pakistan 3department of mathematics, division of science and technology, university of education lahore-pakistan ∗corresponding author: atherqayyum@isp.edu.pk abstract. the aim of paper is to develop the inequalities for l∞, lp and l1 norms. applications for some special weight functions and perturbed expressions are also determined via chebychev functional. we recaptured the previous results for different weights. 1. introduction in 1938, ostrowski established the interesting integral inequality for differentiable mappings with bounded derivative [10]. cerone [3] also worked on this inequality. different authors worked on the generalization of ostrowski’s type inequalities that is [1][2] and [9]. further work done by iftikhar et al. [6], mustafa et al. [7] and qayyum et al. [12][14]. let the functional s ( f ; $; ĵ, ǩ ) be defined as: s ( f ; $; ĵ, ǩ ) = f (z̈) −m̈ ( f ; $; ĵ, ǩ ) , (1.1) where f (z̈) : [ ĵ, ǩ ] →r be a continuous mapping, m̈ ( f ; $; ĵ, ǩ ) is weighted integral mean and is defined as: m̈ ( f ; $; ĵ, ǩ ) = 1 ǩ − ĵ ∫ ǩ ĵ f (ř) $ (ř) dř. (1.2) the functional s ( f ; $; ĵ, ǩ ) represents the deviation of f (z̈) from its integral mean over [ ĵ, ǩ ] . received: jan. 12, 2022. 2010 mathematics subject classification. 35a23. key words and phrases. ostrowski inequality; weight function; numerical integration. https://doi.org/10.28924/2291-8639-20-2022-16 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-16 2 int. j. anal. appl. (2022), 20:16 we assume non-negative weight function $ : (ĵ, ǩ) → [0,∞) is integrable∫ ǩ ĵ $(ř)dř < ∞. (1.3) we define m, m1 and µ as m ( ĵ, ǩ ) = ∫ ǩ ĵ $(ř)dř, m1 ( ĵ, ǩ ) = ∫ ǩ ĵ ř$(ř)dř and µ ( ĵ, ǩ ) = m1 ( ĵ, ǩ ) m ( ĵ, ǩ ) . (1.4) 2. main result theorem 2.1. let f : [ ĵ, ǩ ] → r be continuous on [ ĵ, ǩ ] and differentiable mapping on ( ĵ, ǩ ) , then the following weighted peano kernel, define ġ (., .) : [ ĵ, ǩ ] →r as: ġ (z̈, ř) =   � (�+δ)(z̈−ĵ) ∫ ř ĵ $ (u) du, if ř ∈ [ ĵ, z̈ ] δ (�+δ)(ǩ−z̈) ∫ ř ǩ $ (u) du, if ř ∈ ( z̈, ǩ ] (2.1) ∀ ř ∈ [ ĵ, ǩ ] , z̈ ∈ [ ĵ, ǩ ] , $ is weight function as stated in (1.3) and �,δ ∈ r non-negative and both are not zero at a time. then the following weighted integral identity τ ($; z̈; �,δ) = ∫ ǩ ĵ ġ (z̈, ř) f ′ (ř) dř = bf (z̈) − 1 � + δ [ �m̈ ( f ; $; ĵ, z̈ ) + δm̈ ( f ; $; z̈, ǩ )] , (2.2) holds, where b = 1 � + δ [ � z̈ − ĵ m ( ĵ, z̈ ) + δ ǩ − z̈ m ( z̈, ǩ )] , m̈ ( f ; $; ĵ, ǩ ) is weighted integral mean as defined in (1.2). proof. from (2.1), we have∫ ǩ ĵ ġ (z̈, ř) f ′ (ř) dř = 1 � + δ { � z̈ − ĵ ∫ z̈ ĵ $ (ř) dř + δ ǩ − z̈ ∫ ǩ z̈ $ (ř) dř } f (z̈) − 1 � + δ { � z̈ − ĵ ∫ z̈ ĵ f (ř) $ (ř) dř + δ ǩ − z̈ ∫ ǩ z̈ f (ř) $ (ř) dř } , where the integration by parts formula has been utilized on the separate interval [ ĵ, z̈ ] and ( z̈, ǩ ] . simplification of the expressions readily produces the identity as stated in (2.2). � int. j. anal. appl. (2022), 20:16 3 theorem 2.2. let f : [ ĵ, ǩ ] →r be continuous on [ ĵ, ǩ ] and differentiable mapping on ( ĵ, ǩ ) , whose first derivative f ′ : [ ĵ, ǩ ] → r is bounded on ( ĵ, ǩ ) , then following weighted integral inequalities |τ ($; z̈; �,δ)| ≤   ( �m(ĵ,z̈) z̈−ĵ { z̈ −µ ( ĵ, z̈ )} + δm(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )}) ∥∥∥f ′∥∥∥ ∞ �+δ for f ′ ∈ l∞ [ ĵ, ǩ ] ∥∥∥f ′∥∥∥ p $(z̈) (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] ϑ �+δ [ 1 + |ρ| ϑ ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ l1 [ ĵ, ǩ ] (2.3) are hold for all ř ∈ [ ĵ, ǩ ] , z̈ ∈ [ ĵ, ǩ ] , $ is weight function as stated in (1.3) and �,δ ∈r non-negative and both are not zero at a time, where ϑ = 1( z̈ − ĵ )( ǩ − z̈ ) (�m(ĵ, z̈)(ǩ − z̈) + δm(z̈, ǩ)(z̈ − ĵ)) and ρ = 1( z̈ − ĵ )( ǩ − z̈ ) (�m(ĵ, z̈)(ǩ − z̈)−δm(z̈, ǩ)(z̈ − ĵ)) . proof. taking the modulus of (2.2) and using (1.2) |τ ($; z̈; �,δ)| = ∣∣∣∣∣ ∫ ǩ ĵ ġ (z̈, ř) f ′ (ř) dř ∣∣∣∣∣ ≤ ∫ ǩ ĵ ∣∣ġ (z̈, ř)∣∣ ∣∣∣f ′ (ř)∣∣∣dř, (2.4) where we use properties of the integral and modulus. thus for f ′ ∈ l∞ [ ĵ, ǩ ] from (2.4) |τ ($; z̈; �,δ)| ≤ ∥∥∥f ′∥∥∥ ∞ ∫ ǩ ĵ ∣∣ġ (z̈, ř)∣∣dř from which a simple calculation using (2.1), gives∫ ǩ ĵ ġ (z̈, ř) dř = 1 � + δ [ � z̈ − ĵ { z̈m ( ĵ, z̈ ) −m1 ( ĵ, z̈ )} + δ ǩ − z̈ { z̈m ( z̈, ǩ ) −m1 ( z̈, ǩ )}] . from above, first inequality given in (2.3) is obtained. 4 int. j. anal. appl. (2022), 20:16 further, using hölder’s inequality, we have for f ′ ∈ lp [ ĵ, ǩ ] from (2.4) |τ ($; z̈,�,δ)| ≤ ∥∥∥f ′∥∥∥ p (∫ ǩ ĵ ∣∣ġ (z̈, ř)∣∣q̈ dř )1 q̈ , where 1 p + 1 q̈ = 1, p > 1. with the help of mean value theorem and by using the technique qayyum et al. [11], we get (∫ ǩ ĵ ġ |(z̈, ř)|q̈ dř )1 q̈ = $ (z̈) (� + δ) (q̈ + 1) 1 q̈ [ �q̈ ( z̈ − ĵ ) − (−1)q̈+1 δq̈ ( ǩ − z̈ )]1q̈ . so the second inequality given in (2.3) is obtained. finally, for f ′ ∈ l1 [ ĵ, ǩ ] we have from (2.4) and using (2.1) |τ ($; z̈; �,δ)| ≤ sup ř∈[ĵ,ǩ] ∣∣ġ (z̈, ř)∣∣∥∥∥f ′∥∥∥ 1 , where sup ř∈[ĵ,ǩ] ∣∣ġ (z̈, ř)∣∣ = 1 � + δ max ( � z̈ − ĵ m ( ĵ, z̈ ) , δ ǩ − z̈ m ( z̈, ǩ )) = 1 2 (� + δ) ( z̈ − ĵ )( ǩ − z̈ ) [�m(ĵ, z̈)(ǩ − z̈) +δm ( z̈, ǩ )( z̈ − ĵ )] ×  1 + ∣∣∣∣ 1(z̈−ĵ)(ǩ−z̈) [�m(ĵ, z̈)(ǩ − z̈)−δm(z̈, ǩ)(z̈ − ĵ)] ∣∣∣∣ 1 (z̈−ĵ)(ǩ−z̈) [ �m ( ĵ, z̈ )( ǩ − z̈ ) + δm ( z̈, ǩ )( z̈ − ĵ )]   . hence proved. � remark 2.1. triangular inequality from (2.2) and (1.1) is (� + δ) τ ($; z̈; �,δ) = �s ( f ; $; ĵ, z̈ ) + δs ( f ; $; z̈, ǩ ) int. j. anal. appl. (2022), 20:16 5 then using triangular inequality in (2.3), we get |(� + δ) τ ($; z̈; �,δ)| ≤   � 2 ( m(ĵ,z̈) z̈−ĵ { z̈ −µ ( ĵ, z̈ )})∥∥f ′∥∥∞,[ĵ,z̈] + δ 2 ( m(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )})∥∥f ′∥∥∞,[z̈,ǩ] for f ′ ∈ l∞ [ ĵ, ǩ ] �$(z̈) ( z̈−ĵ q̈+1 )1 q̈ ∥∥f ′∥∥ p,[ĵ,z̈] +δ$(z̈) ( ǩ−z̈ q̈+1 )1 q̈ ∥∥f ′∥∥ p,[z̈,ǩ] for f ′ ∈ lp [ ĵ, ǩ ] ϑ 2 ∥∥f ′∥∥ 1,[ĵ,z̈] + |ρ| 2 ∥∥f ′∥∥ 1,[z̈,ǩ] for f ′ ∈ l1 [ ĵ, ǩ ] . (2.5) remark 2.2. since we may write (2.2) as �m̈ ( f ; $; ĵ, z̈ ) + δm̈ ( f ; $; z̈, ǩ ) = �m̈ ( f ; $; ĵ, z̈ ) + δ ǩ − z̈ (∫ ǩ ĵ $ (ř) f (ř) dř − ∫ z̈ ĵ $ (ř) f (ř) dř ) = [ �−δ ( z̈ − ĵ ǩ − z̈ )] m̈ ( f ; $; ĵ, z̈ ) + δ ( ǩ − ĵ ǩ − z̈ ) m̈ ( f ; $; z̈, ǩ ) , thus τ ($; z̈; �,δ) is 1 b τ ($; z̈; �,δ) = f (z̈) − 1 b [( 1 − δ � + δ λ ) m̈ ( f ; $; ĵ, z̈ ) + δ � + δ λm̈ ( f ; $; z̈, ǩ )] , where λ = ǩ − ĵ ǩ − z̈ , same as [ ĵ, ǩ ] , m̈ ( f ; $; ĵ, ǩ ) is also fixed. 6 int. j. anal. appl. (2022), 20:16 corollary 2.1. let the conditions of theorem 2.2 holds. then the results for δ = � |τ ($; z̈; �,�)| ≤   ( m(ĵ,z̈) z̈−ĵ { z̈ −µ ( ĵ, z̈ )} + m(z̈,ǩ) ǩ−z̈ { z̈ −µ ( z̈, ǩ )}) ∥∥∥f ′∥∥∥ ∞ 2 for f ′ ∈ l∞ [ ĵ, ǩ ] ( ǩ−ĵ q̈+1 )1q̈ ∥∥∥f ′∥∥∥ p $(z̈) 2 for f ′ ∈ lp [ ĵ, ǩ ] ζ [ 1 + |η| ζ ] ∥∥∥f ′∥∥∥ 1 4 for f ′ ∈ l1 [ ĵ, ǩ ] (2.6) where τ ($; z̈; �,�) := 1 2 [( 1 z̈ − ĵ m ( ĵ, z̈ ) + 1 ǩ − z̈ m ( z̈, ǩ )) f (z̈) − { m̈ ( f ; $; ĵ, z̈ ) + m̈ ( f ; $; z̈, ǩ )}] , ζ = 1( z̈ − ĵ )( ǩ − z̈ ) [m(ĵ, z̈)(ǩ − z̈) + m(z̈, ǩ)(z̈ − ĵ)] and η = 1( z̈ − ĵ )( ǩ − z̈ ) [m(ĵ, z̈)(ǩ − z̈)−m(z̈, ǩ)(z̈ − ĵ)] . proof. the result is readily obtained on allowing � = δ in (2.3) so that the left hand side is τ ($; z̈; �,�) from (2.4). � corollary 2.2. according to theorem 2.2, then mid point ( z̈ = ď ⇒ ĵ+ǩ 2 ) inequality from (2.2)∣∣τ ($; ď,�,δ)∣∣ ≤   2 ǩ−ĵ [ �m ( ĵ, ď ){ ď−µ ( ĵ, ď )} +δm ( ď, ǩ ){ ď−µ ( ď, ǩ )}] ∥∥∥f ′∥∥∥ ∞ �+δ for f ′ ∈ l∞ [ ĵ, ǩ ] [ �q̈ + δq̈ ]1 q̈ ( ǩ−ĵ 2(q̈+1) )1 q̈ ∥∥∥f ′∥∥∥ p $(ď) (�+δ) for f ′ ∈ lp [ ĵ, ǩ ] % (�+δ) [ 1 + |ψ| % ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ l1 [ ĵ, ǩ ] (2.7) where % = 1 ǩ − ĵ [ �m ( ĵ, ď ) + δm ( ď, ǩ )] int. j. anal. appl. (2022), 20:16 7 and ψ = 1 ǩ − ĵ [ �m ( ĵ, ď ) −δm ( ď, ǩ )] . proof. placing z̈ = ď ⇒ ĵ+ǩ 2 in (2.4) and (2.3) produces the results as stated in (2.7). � corollary 2.3. when the conditions of theorem 2.2 hold and � = δ using in (2.7) is evaluated at mid point ( z̈ = ď ⇒ ĵ+ǩ 2 ) ∣∣τ ($; ď,�,�)∣∣ ≤   [ m ( ĵ, ď ){ ď−µ ( ĵ, ď )} +m ( ď, ǩ ){ ď−µ ( ď, ǩ )}] ∥∥∥f ′∥∥∥ ∞ ǩ−ĵ for f ′ ∈ l∞ [ ĵ, ǩ ] ( ǩ−ĵ q̈+1 )1q̈ ∥∥∥f ′∥∥∥ p $(ď) 2 for f ′ ∈ lp [ ĵ, ǩ ] ζ 2 [ 1 + |η| ζ ] ∥∥∥f ′∥∥∥ 1 2 for f ′ ∈ l1 [ ĵ, ǩ ] (2.8) where ζ = 2( ǩ − ĵ ) [m(ĵ, ď) + m(ď, ǩ)] and η = 2( ǩ − ĵ ) [m(ĵ, ď)−m(ď, ǩ)] . proof. putting � = δ in (2.7), we get (2.8). � remark 2.3. for $ (z̈) = 1 in (2.3) and (2.5) − (2.8), we get cerone’s results [3]. 3. application for some special means now we discuss application for some special means by taking different weights. remark 3.1. for uniform (legendre) mean: let $ (ř) = 1 put in (2.3) and in (2.4), we get cerone’s results [3]. remark 3.2. for logarithm mean: let $ (ř) = ln (1/ř) , ĵ = 0, ǩ = 1, then µ ( ĵ, ǩ ) is µ (0, 1) = 1 4 , 8 int. j. anal. appl. (2022), 20:16 then ∣∣∣∣ [ 1 � + δ { � z̈ − ĵ + δ ǩ − z̈ }]( f (z̈) − ∫ 1 0 ln (1/ř) f (ř) dř )∣∣∣∣ ≤   [ 1 �+δ { � z̈−ĵ + δ ǩ−z̈ }]( z̈ − 1 4 )∥∥f ′∥∥∞ for f ′ ∈ l∞ [ĵ, ǩ] ∥∥∥f ′∥∥∥ p ln(1/ř) (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] || f ′′||1 2(�+δ) ( � z̈−ĵ + δ ǩ−z̈ )∣∣∣ � z̈−ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ l1 [ĵ, ǩ] . holds. remark 3.3. for jacobi mean: let $ (ř) = 1/ √ ř, ĵ = 0, ǩ = 1, in (1.4), we get µ (0, 1) = 1 3 , then ∣∣∣∣ [ 1 � + δ { � z̈ − ĵ + δ ǩ − z̈ }]( 2f (z̈) − ∫ 1 0 f (ř) 1/ √ řdř )∣∣∣∣ ≤   [ 2 �+δ { � z̈−ĵ + δ ǩ−z̈ }]( z̈ − 1 3 )∥∥f ′∥∥∞ for f ′ ∈ l∞ [ĵ, ǩ] ∥∥∥f ′∥∥∥ p 1/ √ ř (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] || f ′′||1 (�+δ) ( � z̈−ĵ + δ ǩ−z̈ )∣∣∣ � z̈−ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ l1 [ĵ, ǩ] . remark 3.4. for chebyshev mean: let $ (ř) = 1/ √ 1 − ř2, ĵ = −1, ǩ = 1, then µ (−1, 1) = 0, int. j. anal. appl. (2022), 20:16 9 thus ∣∣∣∣ 1� + δ { � z̈ − ĵ + δ ǩ − z̈ }( πf (z̈) − ∫ 1 −1 1/ √ 1 − ř2f (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−ĵ + δ ǩ−z̈ } (πz̈) ∥∥f ′∥∥∞ for f ′ ∈ l∞ [ĵ, ǩ] ∥∥∥f ′∥∥∥ p 1/ √ 1−ř2 (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] π|| f ′′||1 2(�+δ) ( � z̈−ĵ + δ ǩ−z̈ )∣∣∣ � z̈−ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ l1 [ĵ, ǩ] . remark 3.5. for laguerre mean: let $ (ř) = e−ř ĵ = 0, ǩ = ∞, then µ (0,∞) = 1, and ∣∣∣∣ 1� + δ { � z̈ − ĵ + δ ǩ − z̈ }( f (z̈) − ∫ ∞ 0 e−řf (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−ĵ + δ ǩ−z̈ } {z̈ − 1} ∥∥f ′∥∥∞ for f ′ ∈ l∞ [ĵ, ǩ] ∥∥∥f ′∥∥∥ p e−ř (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] || f ′′||1 2(�+δ) ( � z̈−ĵ + δ ǩ−z̈ )∣∣∣ � z̈−ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ l1 [ĵ, ǩ] . holds. remark 3.6. for hermite mean: let $ (ř) = e−ř 2 ĵ = −∞, ǩ = ∞, then µ (−∞,∞) = 0, 10 int. j. anal. appl. (2022), 20:16 and ∣∣∣∣ 1� + δ { � z̈ − ĵ + δ ǩ − z̈ }( √ πf (z̈) − ∫ ∞ −∞ e−ř 2 f (ř) dř )∣∣∣∣ ≤   1 �+δ { � z̈−ĵ + δ ǩ−z̈ }(√ πz̈ )∥∥f ′∥∥∞ for f ′ ∈ l∞ [ĵ, ǩ] ∥∥∥f ′∥∥∥ p e−ř 2 (�+δ)(q̈+1) 1 q̈ [ �q̈ ( z̈ − ĵ ) + δq̈ ( ǩ − z̈ )]1 q̈ for f ′ ∈ lp [ ĵ, ǩ ] √ π|| f ′′||1 2(�+δ) ( � z̈−ĵ + δ ǩ−z̈ )∣∣∣ � z̈−ĵ − δ ǩ−z̈ ∣∣∣ for f ′ ∈ l1 [ĵ, ǩ] . 4. perturbed results for weighted ostrowski type inequalities perturbed versions of the results of the previous section may be obtained by using grüss type results involving chebychev functional ť (f ,g; $) = m̈ (f g; $) −m̈ (f ; $) m̈ (g; $) , (4.1) where m̈ (f ; $) is the weighted integral mean as defined in (1.2). for f ,g : [ ĵ, ǩ ] −→r and integrable on [ ĵ, ǩ ] , as is their product, then ∣∣ť (f ,g)∣∣ ≤ ť 1 2 (f , f ) ť 1 2 (g,g) , dragomir [4] for f , g ∈ l2 [ ĵ, ǩ ] ≤ γ −γ 2 ř 1 2 (f , f ) , matic et al. [8] for γ ≤ g (ř) ≤ γ, ř ∈ [ ĵ, ǩ ] ≤ (γ −γ) (φ −φ) 4 , grüss [5] for φ ≤ g (ř) ≤ ψ, ř ∈ [ ĵ, ǩ ] . (4.2) we obtain following theorem: theorem 4.1. let f : [ ĵ, ǩ ] −→r be an absolutely continuous mapping and � ≥ 0, δ ≥ 0, � + δ 6= 0, then ∣∣∣∣τ ($; z̈; �,δ) − (z̈ −γ)2 s ∣∣∣∣ ≤ ( ǩ − ĵ ) κ (z̈) [ 1 ǩ − ĵ ∥∥∥f ′∥∥∥2 2 −s2 ]1 2 , f ′ ∈ l2 [ ĵ, ǩ ] ≤ ( ǩ − ĵ ) κ (z̈) γ −γ 2 , γ ≤ f ′ (ř) ≤ γ, ř ∈ [ ĵ, ǩ ] ≤ ( ǩ − ĵ ) γ −γ 4 . (4.3) int. j. anal. appl. (2022), 20:16 11 the constant 1 4 is the best possible, where τ ($; z̈; �,δ) is as given in (2.2), γ = �ĵ + δǩ � + δ , s = f ( ǩ ) − f ( ĵ ) ǩ − ĵ , (4.4) κ2 = $ (z̈) 2 [ 1 3 (( � � + δ )2 ( z̈ − ĵ ) + ( δ � + δ )2 ( ǩ − z̈ )) − ( (z̈ −γ) 2 ( ǩ − ĵ ) )2 . (4.5) proof. associating f (ř) with ġ (z̈, ř) and g (ř) with f ′ (ř), then from (2.1) and (4.1) , we get ť ( ġ (z̈, .) , f ′ (.) ) = m̈ ( ġ (z̈, .) , f ′ (.) ) −m̈ ( ġ (z̈, .) ) m̈ ( , f ′ (.) ) . by using (2.1) ( ǩ − ĵ ) ť ( ġ (z̈, .) , f ′ (.) ) (4.6) = τ ($; z̈; �,δ) − ( ǩ − ĵ ) m̈ ( ġ (z̈, .) ) s. now from (2.1) ( ǩ − ĵ ) m̈ ( ġ (z̈, .) ) = ∫ ǩ ĵ ġ (z̈, ř) dř = $ (z̈) � + δ [ � z̈ − ĵ ( z̈ − ĵ )2 2 − δ ǩ − z̈ ( ǩ − z̈ )2 2 ] = $ (z̈) 2 (z̈ −γ) (4.7) (4.7) and (4.5) gives the left hand side of (4.3). now, for the bounds on (4.6) from (4.2), we have to find ť 1 2 ( ġ (z̈, .) , ġ (z̈, .) ) and φ ≤ ġ (z̈, .) ≤ φ. firstly, we note however that 0 ≤ ť 1 2 ( f ′ (.) , f ′ (.) ) = [ m̈ ( f ′ (.) )2 −m̈2 ( f ′ (.) )]12 =   1 ǩ − ĵ ∫ ǩ ĵ [ f ′ (ř) ]2 dř − ( 1 ǩ − ĵ ∫ ǩ ĵ f ′ (ř) dř )2 1 2 = [ 1 ǩ − ĵ ∥∥∥f ′ (ř)∥∥∥2 2 −s2 ]1 2 ≤ γ −γ 2 , where γ ≤ f ′ (ř) ≤ γ, ř ∈ [ ĵ, ǩ ] . (4.8) 12 int. j. anal. appl. (2022), 20:16 now from (2.1), the definition of ġ (z̈, ř), we have κ (z̈) 2 = ť ( ġ (z̈, .) , ġ (z̈, .) ) = m̈ ( ġ2 (z̈, .) ) −m̈2 ( ġ (z̈, .) ) , (4.8-1) from (4.7) m̈ ( ġ (z̈, .) ) = $ (z̈) (z̈ −γ) 2 ( ǩ − ĵ ) and m̈ ( ġ2 (z̈, .) ) = ( � (� + δ) ( z̈ − ĵ ) )2 ∫ z̈ ĵ (∫ ř ĵ $ (u) du )2 dř + ( δ (� + δ) ( ǩ − z̈ ))2 ∫ ǩ z̈ (∫ ř ǩ $ (u) du )2 dř = $ (z̈) 2 3 [( � � + δ )2 ( z̈ − ĵ ) + ( δ � + δ )2 ( ǩ − z̈ )] . by substituting the derived results into (4.8-1), gives 0 ≤ κ (z̈) = ť 1 2 ( ġ (z̈, .) , ġ (z̈, .) ) , (4.9) which is given explicitly by (4.5). we observe from (2.1), that for �,δ ≥ 0 and both are not zero at a time give φ = sup ř∈[ĵ,ǩ] ġ (z̈, ř) and φ = inf ř∈[ĵ,ǩ] ġ (z̈, ř) , giving φ = � �+δ and φ = δ �+δ . hence, proved the result. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. s. burnett, p. cerone, s. s. dragomir, j. roumeliotis, a. sofo, a survey on ostrowski type inequalities for twice differentiable mappings and applications, inequal. theory appl. 1 (2001), 24-30. [2] h. budak, m. z. sarikaya, a. qayyum, new refinements and applications of ostrowski type inequalities for mappings whose nth derivatives are of bounded variation, twms j. appl. eng. math. 11 (2021), 424-435. [3] p. cerone, a new ostrowski type inequality involving integral means over end intervals, tamkang j. math. 33 (2002), 109–118. https://doi.org/10.5556/j.tkjm.33.2002.290. [4] s. s. dragomir, better bounds in some ostrowski-gruss type inequalities, tamkang j. math. 32 (2001), 211–216. https://doi.org/10.5556/j.tkjm.32.2001.376. [5] d. s. mitrinović, j. e. pečarić, a. m. fink, classical and new inequalities in analysis, kluwer academic publishers, dordrecht, 1993. https://doi.org/10.5556/j.tkjm.33.2002.290 https://doi.org/10.5556/j.tkjm.32.2001.376 int. j. anal. appl. (2022), 20:16 13 [6] m. iftikhar, a. qayyum, s. fahad, m. arslan, a new version of ostrowski type integral inequalities for different differentiable mapping, open j. math. sci. 5 (2021), 353–359. https://doi.org/10.30538/oms2021.0170. [7] m. a. mustafa, a. qayyum, t. hussain, m. saleem, some integral inequalities for the quadratic functions of bounded variations and application, turk. j. anal. numb. theory 10 (2022), 1–3. https://doi.org/10.12691/ tjant-10-1-1. [8] m. matic, j. e. pečarić, and n. ujević, on new estimation of the remainder in generalized taylor’s formula, math. inequal. appl. 2 (1999), 343-361. [9] j. nasir, s. qaisar, s. i. butt, a. qayyum, some ostrowski type inequalities for mappings whose second derivatives are preinvex function via fractional integral operator, aims math. 7 (2022), 3303–3320. https://doi.org/10. 3934/math.2022184. [10] a. ostrowski, uber die absolutabweichung einer differentienbaren funktionen von ihren integralmittelwert, comment. math. helv. 10 (1938), 226–227. [11] a. qayyum, a weighted ostrowski gruss type inequality for twice differentiable mappings and applications, int. j. math. comput. 1 (2008), 63-71. [12] a. qayyum, m. shoaib, m. amer latif, a generalized inequality of ostrowski type for twice differentiable bounded mappings and applications, appl. math. sci. 8 (2014), 1889-1901. [13] a. qayyum, i. faye, m. shoaib, m. a. latif, a generalization of ostrowski type inequality for mappings whose second derivatives belong to l 1 (ĵ, ǩ) and applications, int. j. pure appl. math. sci. 98 (2015), 169-180. [14] a. qayyum, a. r. kashif, m. shoaib, i. faye, derivation and applications of inequalities of ostrowski type for n-times differentiable mappings for cumulative distribution function and some quadrature rules, j. nonlinear sci. appl. 9 (2016), 1844-1857. https://doi.org/10.30538/oms2021.0170 https://doi.org/10.12691/tjant-10-1-1 https://doi.org/10.12691/tjant-10-1-1 https://doi.org/10.3934/math.2022184 https://doi.org/10.3934/math.2022184 1. introduction 2. main result 3. application for some special means 4. perturbed results for weighted ostrowski type inequalities references international journal of analysis and applications volume 18, number 3 (2020), 396-408 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-396 on the mixture of weighted exponential and weighted gamma distribution taswar iqbal∗, muhammad zafar iqbal department of mathematics and statistics, university of agriculture faisalabad, faisalabad, pakistan ∗corresponding author: therisers1@gmail.com abstract. in practice, finite mixture models were often used to fit various type of observed phenomena, specifically those which are random in nature. in this paper, a finite mixture model based on weighted versions of exponential and gamma distribution is considered and studied. some mathematical properties of the resulting model are discussed including moment generating function, skewness, kurtosis, survival function, hazard rate function, stochastic ordering, order statistics, bonferroni and lorenz curves, renyi entropy measure and estimation of the model parameters. two real-life data applications from different fields exhibit the fact that in certain situations, the proposed mixture model might be a better alternative than the existing popular models. 1. introduction the mixture distributions over time have provided a mathematical based way to model a wide range of random phenomena. the mixture models are effective and flexible to analyze and interpret random situations in a possibly heterogeneous populations. in many situations, observed data may be assumed to have come from a mixture population of two or more distributions. the mixture models are used in medicine, psychology, finance, engineering, fisheries research, economics, life testing and reliability among others. in this paper, we perpend a finite mixture of two continuous distributions, a one-parameter exponential distribution and a two-parameter gamma distribution. in mixture model, the distribution of interest is received january 15th, 2020; accepted february 10th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 60e05, 62e15. key words and phrases. finite mixture models; weighted exponential distribution; weighted gamma distribution. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 396 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-396 int. j. anal. appl. 18 (3) (2020) 397 modeled as a mixture of two or more distributions in varying proportions. thus a mixture model can be used to model complex situations through proper choice of its mixing components. it can handle situations where a single parametric family is unable to provide a satisfactory model for local variation in the observed data. the flexibility and high degree of accuracy of finite mixture models have been the main reason for their successful applications in a wide range of fields. the concept of finite mixture distribution was pioneered by newcomb [1] as a model for outliers. however, the credit for the introduction of statistical modeling using finite mixture of distributions goes to pearson [2] while applying the technique in an analysis of crab morphometry data provided by weldon [3] and [4] for a comprehensive survey, readers are referred to titterington et al. [5] lindsay [6] bohning [18] and mclachlan and peel [19] and the references therein. the main objective here is to induce greater flexibility in modeling various types of data, specially in situations where these individual distributions fail to adequately fit the data separately. in the recent years a number of new life time distributions have been developed and proposed by many researchers which are in general modification or an extension or generalization of lindley distribution. the range of such distributions include modified one parameter lindley distribution [7], improved second-degree lindley distribution [8], amarendra distribution [9], sujatha distribution [10], shankar distribution [11] and akash distribution [12]. each model has its own limitations and advantages for modeling the lifetime data set. as an extension of these models, in the context to find a new distribution which is more flexible we have introduced a new continuous one-parameter distribution. the new proposed model is a two-component mixture of length biased exponential distribution with parameter θ and length biased gamma distribution with shape parameter 2 and scale parameter θ with their mixing proportions θ θ+1 and 1 θ+1 respectively. suppose we have the following model f (x) = pf1 (x) + (1 −p) f2 (x) , where fj (x) , j = 1, 2 are densities. clearly, p = θ θ+1 and 1−p = 1 θ+1 , are mixture weights (0 < p < 1) , and f (x) indeed a valid density. next, we have f1 (x) = θ 2xe−θx f2 (x) = θ3 2 x2e−θx our weighted distribution will have density f(x) = θ3 θ + 1 [1 + x 2 ]xe−θx x > 0,θ > 0 (1.1) the density in (1.1) is called the mixture of weighted exponential and weighted gamma (hereafter mweg in short) distribution. the cumulative distribution function of mweg distribution can be obtained as f(x) = 1 − [θ2x2 + (θ + 1)2θx + 2(θ + 1)]e−θx 2(θ + 1) x > 0,θ > 0 (1.2) int. j. anal. appl. 18 (3) (2020) 398 (a) (b) figure 1. probability density function of mweg distribution at different values of θ (a) (b) figure 2. cumulative distribution function of mweg distribution at different values of θ 2. properties of the mweg distribution the mathematical properties of the proposed mweg distribution are discussed as follows 2.1. hazard function. the hazard function is defined as the ratio of probability density function to the survival function, it is also known as the hazard rate or force of mortality. the hazard function associated with the mweg distribution is hf (x) = f(x) 1 −f(x) hf (x) = 2θ3x ( 1 + x 2 ) (θ2x2 + (2θ2 + 2θ) x + 2θ + 2) hf (x) = θ3x(2 + x) θ2x2 + (θ + 1)2θx + 2(θ + 1) (2.1) int. j. anal. appl. 18 (3) (2020) 399 (a) (b) figure 3. hazard function of mweg distribution at different values of θ 2.2. survival function. let x be the non-negative random variable with pdf, f(x) given by (1.1). the reliability function s(x) corresponding to the finite mixture of 2 components of (1.1) is given by s(x) = ps1(x) + (1 −p)s2(x) s(x) = [θ2x2 + (θ + 1)2θx + 2(θ + 1)]e−θx 2(θ + 1) where sj(x) , is the reliability function corresponding to the j-th component in the mixture, j = 1, 2. one can write the hazard rate function of a mixture in terms of the hazard rate functions of the two components as follows. hf (x) = b(x)h1(x) + [1 −b(x)]h2(x), where b(x) = ps1(x)/ [ps1(x) + (1 −p)s2(x)] and hj(x) are hazard rate function for the j-th component, j=1,2. on differentiating the hazard function, we get h′f (x) = b(x)h ′ 1(x) + (1 −b(x))h ′ 2(x) −b(x)(1 −b(x)) (h1(x) −h2(x)) 2 where prime denote the derivatives with respect to x. now, from the above, it follows that if h′j(x) < 0, for all x, (j = 1, 2) then h′(x) < 0, for all x. therefore, a mixture with decreasing hazard rate components has decreasing hazard rate. however, if the components have increasing hazard rates, their mixture need not have increasing hazard rate. 2.3. moments and related measures. moments plays a very important role to understand some important features of a distribution. in statistics, moments can be used to study central tendency, dispersion, skewness and kurtosis of any distribution. the moment generating function of mweg distribution is mx(t) = θ3 θ + 1 ∫ ∞ 0 [ 1 + x 2 ] xe−(θ−t)xdx mx(t) = θ3(θ + t + 1) (θ + 1) (θ3 + 3tθ2 + 3t2θ + t3) int. j. anal. appl. 18 (3) (2020) 400 mx(t) = θ3(θ + t + 1) (θ + 1)(θ + t)3 (2.2) by using the equation (2.2) the first four moments about the origin of mweg distribution are given by µ′1 = 2θ + 3 θ(θ + 1) µ′2 = 6(θ + 2) θ2(θ + 1) µ′3 = 12(2θ + 5) θ3(θ + 1) µ′4 = 120(θ + 3) θ4(θ + 1) by using the relationship between moments about origin and moments about mean, the moments about mean of mweg distribution are given by µ2 = 2θ2 + 6θ + 3 θ2(θ + 1)2 µ3 = 2 ( 2θ3 + 9θ2 + 9θ + 3 ) θ3 (θ3 + 3θ2 + 3θ + 1) µ4 = 24θ4 + 144θ3 + 252θ2 + 180θ + 45 ) θ4 (θ4 + 4θ3 + 6θ2 + 4θ + 1) skewness and kurtosis plays a vital role in explaining the shape and tail property of a distribution. the expressions of skewness and kurtosis are given by √ β1 = µ3 µ 3/2 2 = 2 ( 2θ3 + 9θ2 + 9θ + 3 ) (2θ2 + 6θ + 3) 3/2 β2 = µ4 µ22 = 3 ( 8θ4 + 48θ3 + 84θ2 + 60θ + 15 ) θ2(θ + 1)2 (2θ2 + 6θ + 3) the index of dispersion and coefficient of variation are thus obtained as c.v = σ µ′1 = √ 2θ2 + 6θ + 3 (2θ + 3) γ = σ2 µ′1 = 2θ2 + 6θ + 3 θ(θ + 1)(2θ + 3) the condition under which mweg distribution is over-dispersed, under-dispersed, and equi-dispersed has been given along amarendra, ishita, akash, lindley and exponential distributions in table 1 int. j. anal. appl. 18 (3) (2020) 401 table 1. over-dispersion, under-dispersion and equi-dispersion mweg, ishita, amarendra, akash, lindley and exponential distributions for parameterθ. distribution over-dispersion under-dispersion equi-dispersion mweg θ < 1.10073 θ > 1.10073 θ = 1.10073 ishita θ < 1.535653152 θ > 1.535653152 θ = 1.535653152 amarendra θ < 1.525763580 θ > 1.525763580 θ = 1.525763580 akash θ < 1.515400063 θ > 1.515400063 θ = 1.515400063 lindley θ < 1.170086487 θ > 1.170086487 θ = 1.170086487 exponential θ < 1 θ > 1 θ = 1 2.4. mean residual life function. the mean residual function of the mweg distribution is given as m(x) = 1 1 −f(x) ∫ ∞ x [1 −f(t)]dt m(x) = 2(θ + 1) [θ2x2 + (θ + 1)2θx + 2(θ + 1)] e−θx ∫ ∞ x [ θ2t2 + (θ + 1)2θt + 2(θ + 1) 2(θ + 1) ] e−θtdt m(x) = θ2x2 + (θ + 2)2θx + 2(2θ + 3) θ [θ2x2 + (θ + 1)2θx + 2(θ + 1)] (2.3) (a) (b) figure 4. mean residual life function of mweg distribution at different values of θ 2.5. stochastic orderings. in probability theory, stochastic ordering is considered an important tool for assessing the comparative behavior of a positive continuous random variable. a random variable x is said to be smaller than a random variable z, in the i. stochastic order (x ≤st z) , if fx(x) ≥ fz(x) for all x. ii. hazard rate order (x ≤hr z) , if hx(x) ≥ hz(x) for all x. iii. mean residual life order (x ≤mrl z) , if mx(x) ≤ mz(x) for all x. int. j. anal. appl. 18 (3) (2020) 402 iv. likelihood ratio order (x ≤lr z) , if fx(x) fz(x) decreases in x. shaked and shanthikumar [13] proposed the following well-known results for demonstrating the stochastic ordering of distributions x ≤lr z ⇒ x ≤hr z ⇒ x ≤mrl z ⇒ x ≤st z (2.4) in the following theorem, it has shown that mweg distribution is being ordered with respect to the strongest “likelihood ratio” ordering theorem let x ∼ mweg distribution (θ1) and z ∼ mweg distribution (θ2) , if θ1 ≥ θ2, then x ≤lr z and hence x ≤hr z ⇒ x ≤mrl z and x ≤st z proof we have fx(x) = θ31 θ1 + 1 [ x + x2 2 ] e−θ1x fz(x) = θ32 θ2 + 1 [ x + x2 2 ] e−θ2x fx(x) fz(x) = θ31 θ32 [ x + x 2 2 ] e−θ1x θ32 θ2+1 [ x + x 2 2 ] e−θ2x fx(x) fz(x) = θ31 (θ2 + 1) θ32 (θ1 + 1) e−(θ1−θ2)x log [ fx(x) fz(x) ] = log [ θ31 (θ2 + 1) θ32 (θ1 + 1) ] − (θ1 −θ2) x this implies that d dx log fx(x) fz(x) = −(θ1 −θ2) hence for θ1 ≥ θ2, ddx log fx(x) fz(x) < 0, it means that x ≤lr z and hence x ≤hr z ⇒ x ≤mrl z and x ≤st z 2.6. order statistics. let x1,x2,x3 . . .xn be a random sample of size n from mweg distribution. let x(1),x(2),x(3) . . .x(n) denotes the corresponding order statistics. the probability density function and the int. j. anal. appl. 18 (3) (2020) 403 cumulative distribution function of the kth order statistic, say z = x(k) are given below fz(x) =n   n− 1 k − 1  f(x)k−1(1 −f(x))n−kf(x) and fz(x) = n∑ j=k n−j∑ l=0   n j     n− j l   (−1)lf(x)j+l respectively for different values of k = 1, ...,n hence the probability density function and cumulative distribution function of kth order statistics are given as fz(x) = θ3 θ+1 [ 1 + x 2 ] xe−θxn   n− 1 k − 1   ( 1 − ( θ2x2 + ( 2θ2 + 2θ ) x + 2θ + 2 ) e−θx 2(θ + 1) )k−1 (( θ2x2 + ( 2θ2 + 2θ ) x + 2θ + 2 ) e−θx 2(θ + 1) )n−k it can be written as fz(x) = n!θ3(2+θ)xe−θx 2(θ+1)(n−k)!(k−1)! ∑n−k l=0   n−k l   (−1)l (1 − (θ2x2+(2θ2+2θ)x+2θ+2)e−θx)e−θx 2(θ+1) )k+l−1 similarly fz(x) = ∑n j=k ∑n−j l=0   n− j j     n− j l   (−1)l (1 − (θ2x2+(2θ2+2θ)x+2θ+2)e−θx 2(θ+1) )j+l 2.7. bonferroni and lorenz curves. the bonferroni and lorenz curves [14] and bonferroni and gini indices have applications in many fields like insurance, medicines, demography and also in economics for studying the patterns of income and poverty. the bonferroni and lorenz curves may be defined as bp = 1 pµ ∫ q 0 xf(x)dx bp = 1 pµ ∫ ∞ 0 xf(x)dx− ∫ ∞ q xf(x)dx b(p) = 1 pµ [ µ− ∫ ∞ q xf(x)dx ] b(p) = 1 p [ 1 − 1 µ ∫ ∞ q xf(x)dx ] (2.5) and lp = 1 µ ∫ q 0 xf(x)dx = 1 µ [ xf(x)dx− ∫ ∞ q xf(x)dx ] int. j. anal. appl. 18 (3) (2020) 404 l(p) = 1 µ [ µ− ∫ ∞ q xf(x)dx ] l(p) = 1 − 1 µ ∫ ∞ q xf(x)dx (2.6) where e(x) = µ and q = f−1(p) by using (1.1) and (1.2) , we can define the bonfernoni and gini indices as b = 1 − ∫ 1 0 b(p)dp (2.7) b = 1 − ∫ 1 0 b(p)dp and g = 1 − 2 ∫ 1 0 l(p)dp (2.8) now by using pdf of mweg distribution, we get ∫ ∞ q xf(x)dx = ∫ ∞ q θ3 θ + 1 [ 1 + x 2 ] xe−θxdx ∫ ∞ q xf(x)dx = θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2θ(θ + 1) e−θq (2.9) using above (2.9) in (2.5) and (2.6), we get 1 µ ∫ ∞ q xf(x)dx = θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2(2θ + 3) e−θq b(p) = 1 p [ 1 − θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2(2θ + 3) e−θq ] (2.10) and l(p) = 1 − θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2(2θ + 3) e−θq (2.11) by using (2.10) and (2.11) in (2.7) and (2.8), we get b = 1 − θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2(2θ + 3) e−θq (2.12) and g = −1 + θ3 ( q3 + 2q2 ) + θ2 ( 3q2 + 4q ) + θ(6q + 4) + 6 2θ + 3 e−θq (2.13) int. j. anal. appl. 18 (3) (2020) 405 2.8. renyi entropy. entropy of a random variable x can be defined as a measure of variation of uncertainty. renyi entropy [15] is considered as a very popular entropy measure. if x is a continuous random variable having probability density function f(x), then renyi entropy is defined as tr(y) = 1 1 −γ log {∫ fγ(x)dx } where γ > 0 also γ 6= 1 thus, the renyi entropy for the mweg distribution ( 1) can be obtained as tr(γ) = 1 1 −γ log ∫ ∞ 0 θ3γ (θ + 1)γ [ 1 + x 2 ]γ xγe−sγxdx = 1 1 −γ log   θ3γ (θ + 1)γ ∫ ∞ 0 ∞∑ j=0   γ j  (x 2 )j xγe−θγxdx   = 1 1 −γ log   θ3γ (θ + 1)γ ∞∑ j=0   γ j  (x 2 )j ∫ ∞ 0 xγe−θγxdx   = 1 1 −γ log   θ3γ (θ + 1)γ ∞∑ j=0   γ j  (1 2 )j γ(γ + j + 1) (θγ)γ+j+1   = 1 1 −γ log  θ2γ−j−1 (θ + 1)γ ∞∑ j=0 γ(γ + j + 1) 2jγγ+j+1   γ j     3. estimation of parameters this section consists of estimation of the unknown parameters of proposed model by using the methods of moments and maximum likelihood. 3.1. maximum likelihood estimation. let x1,x2,x3 . . .xn be an iid random sample from mweg, then the likelihood function of the mweg distribution is given by, l = ( θ3 θ + 1 )n n∏ i=1 [ xi + x2i 2 ] e−θ ∑n i=1 xi and the log-likelihood function can be written as ln l = n ln ( θ3 θ + 1 ) + n∑ i=1 [ xi + x2i 2 ] −nθx̄ d dθ (ln l) = 3n θ − n θ + 1 −nx̄ where x̄ is the sample mean. we can find the maximum likelihood estimate of θ by simply equating d dθ (ln l) = 0, and it can be find by solving the following nonlinear equation nx̄θ(θ + 1) − 2nθ − 3n = 0 (3.1) int. j. anal. appl. 18 (3) (2020) 406 3.2. method of moment. let x1,x2,x3 . . .xn be a random sample of size n from mweg distribution. by equating the first population moment about origin to the sample mean, the method of moment estimate θ̂ of θ is the same as the ml estimate as given in (3.1) 4. real data application the goodness of fit of mweg distribution has been checked by using several lifetime data sets from engineering and medical science. in this section, we have used two real-life data sets to compare the goodness of fit by using ml estimate of the parameter of mweg distribution with the exponential, akash, lindley, ishita and modified one parameter lindley distributions and have proved that mweg distribution provides better estimate for modeling lifetime data sets as compared to its competing models. data set 1: the following dataset acts the breaking stress of carbon fibers of 50 mm in length nichols and padgett [16] 3.70, 2.12, 2.95, 4.70, 1.25, 3.22, 1.69, 3.27, 2.87, 1.47, 3.11, 3.65, 2.74, 3.15, 2.97, 2.03, 4.38, 3.39, 3.28, 3.09, 1.87, 3.15, 4.90, 4.42, 2.73, 1.08, 3.39, 1.89, 1.84, 2.81, 4.20, 3.33, 2.55, 3.31, 1.57, 2.41, 2.50, 2.56, 2.96, 2.88, 0.39, 3.68, 2.48, 0.85, 1.61, 3.31, 2.67, 3.19, 3.60, 1.80, 2.35, 2.82, 2.05, 3.65, 3.75, 2.43, 2.79, 2.85, 2.93, 3.22, 3.11, 2.53, 2.55, 2.59, 2.03, 1.61 data set 2: the following data represent the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by bjerkedal [17]. the data are as follows: 0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08,1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63,1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54,2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55 in order to compare mweg, exponential, lindley, akash, ishita and modified one parameter lindley distributions, values of −2ln l , aic (akaike information criterion), aicc (akaike information criterion corrected), bic (bayesian information criterion) and k-s statistic ( kolmogorov-smirnov statistic) for two real data sets have been computed and presented in the following table. int. j. anal. appl. 18 (3) (2020) 407 table 2. mle’s,−2lnl, aic, aicc, bic, and k-s statistics of the fitted distributions of data sets 1 and 2 model mle of θ̂ se(θ̂) -2lnl aic caic bic k-s d a t a s e t 1 mweg 1.2832 0.1751 50.8754 52.8754 53.0976 53.8711 0.3058 mopld 1.5213 0.1523 59.7066 61.7066 61.9288 62.7023 0.3569 ishita 1.0948 0.1217 60.1647 62.1647 62.3869 63.1604 0.3507 exponential 0.5263 0.1177 65.4742 67.6742 67.8964 68.6699 0.4395 lindely 0.8162 0.1361 60.4991 62.4991 62.7213 63.4948 0.3911 akash 1.1569 0.1456 59.5226 61.5226 61.7448 62.5183 0.3705 d a t a s e t 2 mweg 1.3697 0.0989 193.103 195.102 195.16 197.379 0.153 mopld 1.5852 0.0839 220.004 222.004 222.061 224.281 0.231 ishita 1.1598 0.0677 216.632 218.632 218.689 220.909 0.229 exponential 0.5655 0.0666 226.074 228.074 228.131 230.351 0.295 lindely 0.8683 0.0766 213.857 215.857 215.914 218.134 0.247 akash 1.2159 0.081 214.678 216.678 216.735 218.954 0.234 the best distribution corresponds to lower values of −2lnl,aic,aicc,bic and k-s statistic. it can be easily seen from above table that mweg distribution provide better fit as compare to exponential, lindley, akash, ishita and modified one parameter lindley distributions. 5. conclusion in this paper, we consider a simple mixture of two absolutely continuous distributions weighted exponential and weighted gamma distribution. some structural properties of the resulting distribution are discussed. the resulting model appears to be a reasonable choice in the sense of modeling lifetime data sets, in particular, where the popular choices (e.g., exponential, gamma, lindley, ishita and/or akash distribution) fail to adequately model the observed phenomena. we sincerely hope that this mixture model will find many more applications in different fields affecting human life. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. newcomb, a generalized theory of the combination of observations so as to obtain the best result. amer. j. math. 8(4)(1886), 343-366. [2] k. pearson, contributions to the mathematical theory of evolution. philos. trans. r. soc. lond., a. 185 (1894), 71-110. int. j. anal. appl. 18 (3) (2020) 408 [3] w. f. r. weldon, i. certain correlated variations in crangon vulgaris. proceedings of the royal society of london, 51(308314)(1892), 1-21. [4] w. f. r. weldon, ii. on certain correlated variations in carcinus mænas. proceedings of the royal society of london, 54(326-330)(1894), 318-329. [5] d. m. titterington, a. f. smith, and u. e. makov, statistical analysis of finite mixture distributions. wiley, (1985). [6] b. g. lindsay, mixture models: theory, geometry and applications. in nsf-cbms regional conference series in probability and statistics (pp. i-163). institute of mathematical statistics and the american statistical association, (1995). [7] s. karuppusamy, v. balakrishnan, and k. sadasivan, modified one-parameter lindley distribution and its applications. int. j. eng. res. appl. 8(1)(2018), 50-56. [8] s. karuppusamy, v. balakrishnan, k. sadasivan, improved second-degree lindley distribution and its applications. iosr j. math. 13(6)(2018), 1-10. [9] r. shanker, amarendra distribution and its applications. amer. j. math. stat. 6(1)(2016), 44-56. [10] r. shanker, sujatha distribution and its applications. stat. trans. new ser. 17(3)(2016), 391-410. [11] r. shanker, shanker distribution and its applications. int. j. stat. appl. 5(6)(2015), 338-348. [12] r. shanker, akash distribution and its applications. int. j. probab. stat. 4(3)(2015), 65-75. [13] j. g. shanthikumar, stochastic orders and their applications. academic press, (1994). [14] c. e. bonferroni, elementi di statistica generale,(ristampa con aggiunte): anno accademico 1932/33; bari, r. istit. super. di scienze economiche. gili. (1933). [15] a. rényi, on measures of entropy and information. in proceedings of the fourth berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics. the regents of the university of california. (1961). [16] m. d. nichols and w. j. padgett, a bootstrap control chart for weibull percentiles. qual. reliab. eng. int. 22(2)(2006), 141-151. [17] t. bjerkedal, acquisition of resistance in guinea pies infected with different doses of virulent tubercle bacilli. amer. j. hygiene, 72(1)(1960), 130-48. [18] d. böhning, computer-assisted analysis of mixtures and applications: meta-analysis, disease mapping and others (vol. 81). crc press, (1999). [19] g. j. mclachlan, d. peel, finite mixture models. john wiley & sons. (2004). 1. introduction 2. properties of the mweg distribution 2.1. hazard function 2.2. survival function 2.3. moments and related measures 2.4. mean residual life function 2.5. stochastic orderings 2.6. order statistics 2.7. bonferroni and lorenz curves 2.8. renyi entropy 3. estimation of parameters 3.1. maximum likelihood estimation 3.2. method of moment 4. real data application 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 1, number 1 (2013), 33-39 http://www.etamaths.com on the degree of approximation of a function by (c, 1)(e,q) means of its fourier-laguerre series xhevat z. krasniqi abstract. in this note a theorem on the degree of approximation of a function by (c, 1)(e, q) means of its fourier-laguerre series at the frontier point x = 0 is proved. 1. introduction let us consider the infinite series ∑∞ n=0 un with the sequence of its n-th partial sums s := {sn}. if for q > 0 (1.1) eqn(s) = 1 (1 + q)n n∑ k=0 ( n k ) qksk → s1 as n →∞, then it is said that s := {sn} is summable by (e,q) means (see hardy [3]), and we write sn → s1(e,q). the fourier-laguerre expansion of a function f(x) ∈ l(0,∞) is given by (1.2) f(x) ∼ ∞∑ n=0 anl (α) n (x), where (1.3) an = 1 γ(α + 1) ( n+α n ) ∫ ∞ 0 e−yyαl(α)n (y)dy, l (α) n (x) denotes the n-th laguerre polynomial of order α > −1, defined by generating function (1.4) ∞∑ n=0 l(α)n (x)ω n = e xω ω−1 (1 −ω)α+1 , and it is assumed that the integral (1.3) exists. in 1971, d. p. gupta [2] estimated the order of the function by cesáro means of series (1.2) at the point x = 0, after replacing the continuity condition in szegö’s theorem [6] by a much lighter condition. he proved the following theorem. 2010 mathematics subject classification. 42c10, 40g05, 41a25. key words and phrases. (c, 1)(e, q) summability, fourier-laguerre series, degree of approximation. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 33 34 krasniqi theorem 1.1 ([2]). if f(t) = ∫ t 0 |f(y)| y dy = o ( log ( 1 t ))1+p , t → 0,−1 < p < ∞, and ∫ ∞ 1 e−y/2y(3α−3k−1)/3|f(y)|dy < ∞, are fulfilled, then σkn(0) = o (log n) p+1 provided that k > α + 1/2, α > −1, with σkn(0) being the n-th cesàro mean of order k. further, we use the notation (1.5) φ(y) = e−yyα[f(y) −f(0)] γ(α + 1) , and denote by tn harmonic means of the series (1.2). t. singh [5] estimated the deviation tn(x) −f(x) at the point x = 0 by some weaker conditions than those of theorem 1.1. namely, he verified the following theorem. theorem 1.2 ([5]). for α ∈ (−5/6,−1/2) tn(0) −f(0) = o (log n) p+1 provided that ∫ δ t |φ(y)| yα+1 dy = o ( log ( 1 t ))1+p , t → 0,−1 < p < ∞, (1.6) ∫ n δ ey/2y−(2α+3)/4|φ(y)|dy = o ( n−(2α+1)/4 (log n) p+1 ) , and ∫ ∞ n ey/2y−1/3|φ(y)|dy = o ( (log n) p+1 ) , n →∞, where δ is a fixed positive constant. very recently, nigam and sharma [4] proved a theorem of such type using (e, 1) means which is entirely different from (c,k) and harmonic means of the series (1.2), they employed a condition which is weaker than condition (1.6), and increased the range of α to (−1,−1/2) which is more appropriate for applications. in their paper they established the following statement. theorem 1.3 ([4]). if (1.7) e1n = 1 2n n∑ k=0 ( n k ) sk →∞ as n →∞, then the degree of approximation of fourier-laguerre expansion (1.2) at the point x = 0 by (e, 1) means e1n is given by (1.8) e1n(0) −f(0) = o (ξ(n)) provided that (1.9) φ(t) = ∫ t 0 |φ(y)|dy = o ( tα+1ξ ( 1 t )) , t → 0, on the degree of approximation 35 (1.10) ∫ n δ ey/2y−(2α+3)/4|φ(y)|dy = o ( n−(2α+1)/4ξ (n) ) , and (1.11) ∫ ∞ n ey/2y−1/3|φ(y)|dy = o (ξ (n)) , n →∞, where δ is a fixed positive constant, α ∈ (−1,−1/2), and ξ(t) is a positive monotonic increasing function of t such that ξ(n) →∞ as n →∞. as is pointed out in [1] the infinite series 1 − 4 ∞∑ n=1 (−3)n−1 is not (e, 1) summable nor (c, 1) summable. however, it is proved that the above series is (c, 1)(e, 1) summable. therefore the product summability (c, 1)(e, 1) is more powerful than the individual methods (c, 1) and (e, 1). thus, (c, 1)(e, 1) mean gives an approximation for a wider class of fourier-laguerre series than the individual methods (c, 1) and (e, 1). the main aim of this paper is to prove the counterpart of the theorem 1.3 using the product mean (c, 1)(e,q), which obviously, based on what we discussed above, will give more general results. to achieve this aim we need an auxiliary result (see [6], page 175). lemma 1.1. let α be arbitrary and real, c and d be fixed positive constants, and let n →∞. then (1.12) l(α)n (x) = o (n α) , if 0 ≤ x ≤ c n and (1.13) l(α)n (x) = o ( x−(2α+1)/4n(2α−1)/4 ) if c n ≤ x ≤ d. 2. main result we prove the following theorem. theorem 2.1. te degree of approximation of fourier-laguerre expansion (1.2) at the point x = 0 by (c, 1)(e,q), q ≥ 1 means [(c, 1)(e,q)]n is given by [(c, 1)(e,q)]n(0) −f(0) = o (ξ(n)) provided that (2.1) φ(t) = ∫ t 0 |φ(y)|dy = o ( tα+1ξ ( 1 t )) , t → 0, (2.2) ∫ n δ ey/2y−(2α+3)/4|φ(y)|dy = o ( n−(2α+1)/4ξ (n) ) , and (2.3) ∫ ∞ n ey/2y−1/3|φ(y)|dy = o (ξ (n)) , n →∞, where δ is a fixed positive constant, α ∈ (−1,−1/2), and ξ(t) is a positive monotonic increasing function of t such that ξ(n) →∞ as n →∞. 36 krasniqi proof. based on the equality (2.4) l(α)n (0) = ( n + α α ) , we obtain sn(0) = n∑ k=0 akl (α) n (0) = 1 γ(α + 1) ∫ ∞ 0 e−yyαf(y) n∑ k=0 l (α) k (y)dy = 1 γ(α + 1) ∫ ∞ 0 e−yyαf(y)l(α+1)n (y)dy.(2.5) thus, [(e,q)]n(0) = 1 (1 + q)n n∑ k=0 ( n k ) qksk(0) = 1 (1 + q)n n∑ k=0 ( n k ) qk γ(α + 1) ∫ ∞ 0 e−yyαf(y)l (α+1) k (y)dy, and [(c, 1)(e,q)]n(0) = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qksk(0) = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qk γ(α + 1) ∫ ∞ 0 e−yyαf(y)l (α+1) k (y)dy.(2.6) therefore, using (1.5) we have (c, 1)(eqn)(0) −f(0) = = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qk ∫ ∞ 0 φ(y)l (α+1) k (y)dy = (∫ 1/n 0 + ∫ δ 1/n + ∫ n δ + ∫ ∞ n ) 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qkφ(y)l (α+1) k (y)dy := 4∑ m=0 rm.(2.7) on the degree of approximation 37 using the property of the orthogonality, condition (2.1) and lemma 1.1, we obtain r1 = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qko ( kα+1 )∫ 1/n 0 |φ(y)|dy = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qko ( nα+1 ) o ( ξ (n) nα+1 ) = o ( 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qkξ (n) ) = o (ξ (n)) ,(2.8) since n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qk = n + 1. again, using the property of the orthogonality and lemma 1.1, we have r2 = 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qko ( k(2α+1)/4 )∫ δ 1/n y(2α+3)/4|φ(y)|dy. since v∑ k=0 ( v k ) qkk(2α+1)/4 = [ v2 ]∑ k=0 ( v k ) qkk(2α+1)/4 + v∑ k=[ v2 ]+1 ( v k ) qkk(2α+1)/4 ≤ v∑ k=0 ( v k ) qkk(2α+1)/4 + ( v[ v 2 ]) v∑ k=[ v2 ]+1 qkk(2α+1)/4 ≤ (1 + q)v v(2α+1)/4 + ( v[ v 2 ])v(2α+5)/4qv = (1 + q) v v(2α+1)/4 + ( v[ v 2 ])v(2α+1)/4vqv q ≥ 1. and (1 + q)v = v∑ k=0 ( v k ) qk = ( v 0 ) q0 + ( v 1 ) q1 + · · · + ( v[ v 2 ])q[ v2 ] + ( v[v 2 ] + 1 ) q[ v 2 ]+1 + · · · + ( v v ) qv ≥ ( v[ v 2 ])q[ v2 ] + ( v[v 2 ] + 1 ) q[ v 2 ]+1 + · · · + ( v v ) qv ≥ [( v[ v 2 ]) + ( v[v 2 ]) + · · · + ( v[v 2 ])]q[ v2 ] ≥ k ([v 2 ] + 1 )( v[ v 2 ])qv ≥ k 2 v ( v[ v 2 ])qv, (for k ≤ 1/q), 38 krasniqi then 1 (1 + q)v v∑ k=0 ( v k ) qkk(2α+1)/4 ≤ ( 1 + 2 k ) v(2α+1)/4. and moreover, 1 n + 1 n∑ v=0 1 (1 + q)v v∑ k=0 ( v k ) qkk(2α+1)/4 = o ( n(2α+1)/4 ) . using latter estimation, and doing the same reasoning as in [4] page 6, we obtain (2.9) r2 = o ( n(2α+1)/4 )∫ δ 1/n y(2α+3)/4|φ(y)|dy = o (ξ(n)) . further we estimate r3: r3 ≤ 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v ∫ n δ ey/2y−(2α+3)/4|φ(y)|e−y/2y(2α+3)/4|l(α+1)k (y)|dy = 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v o ( k(2α+1)/4 ∫ n δ ey/2y−(2α+3)/4|φ(y)|dy ) = 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v o ( k(2α+1)/4o ( n−(2α+1)/4ξ(n) )) = o (ξ(n)) .(2.10) finally, we have r4 ≤ 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v ∫ ∞ n ey/2y−(3α+5)/6|φ(y)|e−y/2y(3α+5)/6|l(α+1)k (y)|dy = 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v o ( k(α+1)/4 ∫ ∞ n ey/2|φ(y)| y(α+1)/2+1/3 dy ) = 1 n + 1 n∑ v=0 ∑v k=0 ( v k ) qk (1 + q)v o ( k(α+1)/2k−(α+1)/2o (ξ(n)) ) = o (ξ(n)) .(2.11) now, putting estimations (2.8)-(2.11) into (2.7) we obtain [(c, 1)(e,q)]n(0) −f(0) = o (ξ(n)) . the proof of the theorem is completed. � references [1] v. n. mishra et al.: approximation of signals by product summability transform, asian journal of mathematics and statistics, 6(1): 12–22, 2013. [2] d. p. gupta: degree of approximation by cesàro means of fourier-laguerre expansions, acta sci. math. (szeged), vol. 32, pp. 255–259, 1971. [3] g. h. hardy: divergent series, oxford university press, oxford, uk, 1st edition, 1949. [4] h. k. nigam, a. sharma: a study on degree of approximation by (n, p, q)(e, 1) summability means of the fourier-laguerre expansion summability of fourier series, int. j. math. math. sci. volume 2010, article id 351016, 7 pages doi:10.1155/2010/351016. on the degree of approximation 39 [5] t. singh: degree of approximation by harmonic means of fourier-laguerre expansions, publ. math. debrecen, vol. 24, no. 1–2, pp. 53–57, 1977. [6] g. szegö: orthogonal polynomials. colloquium publications american mathematical society, new york, ny, usa, 1959. state university of prishtina ”hasan prishtina”, faculty of education, department of mathematics and informatics, avenue ”mother theresa” 5, 10000 prishtina, kosova international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 81-90 http://www.etamaths.com growth properties of wronskians in the light of relative order sanjib kumar datta1,∗, tanmay biswas2, golok kumar mondal3 abstract. in this paper we study the comparative growth properties of composition of entire and meromorphic functions on the basis of relative order (relative lower order) of wronskians generated by entire and meromorphic functions. 1. introduction, definitions and notations. let f be an entire function defined in the open complex plane c. the function m (r,f) on |z| = r known as maximum modulus function corresponding to f is defined as follows: m(r,f) = max|z|=r |f (z)| . when f is meromorphic, m (r,f) can not be defined. in this situation one may define another function t (r,f) known as nevanlinna’s characteristic function of f, playing the same role as m (r,f) in the following manner: t (r,f) = n (r,f) + m (r,f) . when f is an entire function, t (r,f) reduces to m (r,f) . we call the function n (r,a; f) ( − n (r,a; f) ) as counting function of a-points (distinct a-points) of f. in many occasions n (r,∞; f) and − n (r,∞; f) are denoted by n (r,f) and − n (r,f) respectively.we put n (r,a; f) = r∫ 0 n (t,a; f) −n (0,a; f) t dt + − n (0,a; f) log r , where we denote by n (t,a; f) ( − n (t,a; f) ) the number of a-points (distinct a-points) of f in |z| ≤ t and an ∞ -point is a pole of f . 2010 mathematics subject classification. 30d20, 30d30, 30d35. key words and phrases. entire function; meromorphic function; relative order (relative lower order); wronskian. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 81 82 datta, biswas abd mondal on the other hand m ( r, 1 f−a ) is denoted by m (r,a; f) and we mean m (r,∞; f) by m (r,f) , which is called the proximity function of f. we also put m (r,f) = 1 2π 2π∫ 0 log+ ∣∣f (reiθ)∣∣dθ, where log+ x = max (log x, 0) for all x > 0 . if the entire function g is non-constant then tg (r) is strictly increasing and continuous and its inverse t−1g : (tg (0) ,∞) → (0,∞) exists and is such that lim s→∞ t−1g (s) = ∞. lahiri and banerjee [4] introduced the definition of relative order of a meromorphic function with respect to an entire function which is as follows: definition 1.1. [4] let f be meromorphic and g be entire. the relative order of f with respect to g denoted by ρg (f) is defined as ρg (f) = inf {µ > 0 : tf (r) < tg (rµ) for all sufficiently large r} = lim sup r→∞ log t−1g tf (r) log r . analogously, one can define the relative lower order of a meromorphic function f with respect to an entire function g denoted by λg (f) in the following manner : λg (f) = lim inf r→∞ log t−1g tf (r) log r . if we consider g (z) = exp z, definition 1.1 coincides {cf.[4]} with the classical definition of order and lower order of meromorphic function which are as follows: definition 1.2. the order ρf and lower order λf of a meromorphic function f are defined as ρf = lim sup r→∞ log tf (r) log r and λf = lim inf r→∞ log tf (r) log r where log[k] x = log ( log[k−1] x ) , k = 1, 2, 3, ... and log[0] x = x. the following definitions are also well known: definition 1.3. a meromorphic function a ≡ a (z) is called small with respect to f if t (r,a) = s (r,f) where s (r,f) = o{t (r,f)} i.e., s(r,f) t(r,f) → 0 as r →∞ . definition 1.4. let a1,a2, ....ak be linearly independent meromorphic functions and small with respect to f .we denote by l (f) = w (a1,a2, ....ak; f) , the wronskian determinant of a1,a2, ....,ak, f i.e., l (f) = ∣∣∣∣∣∣∣∣∣∣∣∣ a1 a2 . . . ak f a ′ 1 a ′ 2 . . . a ′ k f ′ . . . . . . . . . . . . . . . . . . . . . a (k) 1 a (k) 2 . . . a (k) k f (k) ∣∣∣∣∣∣∣∣∣∣∣∣ . growth properties of wronskians in the light of relative order 83 definition 1.5. if a ∈ c∪{∞},the quantity δ (a; f) = 1 − lim sup r→∞ n (r,a; f) tf (r) = lim inf r→∞ m (r,a; f) tf (r) is called the nevanlinna’s deficiency of the value “a”. from the second fundamental theorem it follows that the set of values of a ∈ c ∪ {∞} for which δ (a; f) > 0 is countable and ∑ a6=∞ δ (a; f) + δ (∞; f) ≤ 2 (cf [3],.p.43 ). if in particular, ∑ a 6=∞ δ (a; f) + δ (∞; f) = 2, we say that f has the maximum deficiency sum. in this connection the following two definitions are also relevant : definition 1.6. [1] a non-constant entire function f is said have the property (a) if for any δ > 1 and for all large r, [mf (r)] 2 ≤ mf ( rδ ) holds. for exapmles of functions with or without the property (a), one may see [1]. definition 1.7. two entire functions g and h are said to be asymptotically equivalent if there exists l (0 < l < ∞) such that mg (r) mh (r) → l as r →∞ and in this case we write g ∼ h . clearly if g ∼ h then h ∼ g. in this paper we establish some newly developed results based on the growth properties of relative order and relative lower order of wronskians generated by entire and meromorphic functions. we do not explain the standard notations and definitions in the theory of entire and meromorphic functions because those are available in [3] and [5]. 2. lemmas. in this section we present some lemmas which will be needed in the sequel. lemma 2.1. [1] let g be an entire function and α > 1, 0 < β < α. then mg (αr) > βmg (r) for all sufficiently large r. lemma 2.2. [1] let f be an entire function which satisfies the property (a). then for any positive integer n and for all large r, [mf (r)] n ≤ mf ( rδ ) holds where δ > 1. lemma 2.3. let g be entire. then for all sufficiently large values of r, tg (r) ≤ log mg (r) ≤ 3tg (2r) . lemma 3 follows from theorem 1.6 {cf. p.18, [3]} on putting r = 2r. lemma 2.4. [2] if f be a transcendental meromorphic function with the maximum deficiency sum and g be a transcendental entire function of regular growth having non zero finite order and ∑ a6=∞ δ (a; g) + δ (∞; g) = 2, then the relative order and relative lower order of l(f) with respect to l(g) are same as those of f with respect to g i.e., ρl[g] (l [f]) = ρg (f) and λl[g] (l [f]) = λg (f) . 84 datta, biswas abd mondal lemma 2.5. let g and h be any two transcendental entire functions of regular growth having non zero finite order with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2 and ∑ a6=∞ δ (a; h)+ δ (∞; h) = 2 respectively. then for any transcendental meromorphic function f with the maximum deficiency sum, ρl[g] (l[f]) = ρl[h] (l[f]) and λl[g] (l[f]) = λl[h] (l[f]) , if g and h have the property (a) and g ∼ h. proof. let ε > 0 be arbitrary. now we get from definition 1.7 and lemma 2.1 for all sufficiently large values of r that (1) mg (r) < (l + ε) mh (r) ≤ mh (αr) , where α > 1 is such that l + ε < α. now from lemma 2.3 and in view of definition 1.1, we obtain for all sufficiently large values of r that tf (r) ≤ tg [ (r) (ρg(f)+ε) ] i.e., tf (r) ≤ log mg [ (r) (ρg(f)+ε) ] . therefore in view of (1), for any δ > 1 it follows from above by using lemma 2.2 and lemma 2.3 that tf (r) ≤ 1 3 log [ mh [ (αr) (ρg(f)+ε) ]]3 i.e., tf (r) ≤ 1 3 log mh [ (αr) δ(ρg(f)+ε) ] i.e., tf (r) ≤ th [ (2αr) δ(ρg(f)+ε) ] i.e., log t−1h tf (r) log r ≤ δ (ρg (f) + ε) log (2αr) log r . letting δ → 1+ we get from above that (2) ρh (f) ≤ ρg (f) . since h ∼ g, we also obtain that (3) ρg (f) ≤ ρh (f) . now in view of lemma 2.4 we obtain from (2) and (3) that ρl[g] (l[f]) = ρl[h] (l[f]) . similarly we have λl[g] (l[f]) = λl[h] (l[f]) . thus the lemma follows. � growth properties of wronskians in the light of relative order 85 3. theorems. in this section we present the main results of the paper. theorem 3.1. suppose f be a transcendental meromorphic function having the maximum deficiency sum. also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (f) ≤ ρh (f) < ∞ .then λh (f ◦g) ρh (f) ≤ lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ λh (f ◦g) λh (f) ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ ρh (f ◦g) λh (f) . proof. from the definition of ρh (f) and λh (f ◦g) and lemma 2.4 we have for arbitrary positive ε and for all sufficiently arge values of r that (4) log t−1h tf◦g (r) > (λh (f ◦g) −ε) log r and log t−1 l[h] tl[f] (r) ≤ ( ρl[h] (l[f]) + ε ) log r i.e., log t−1 l[h] tl[f] (r) ≤ (ρh (f) + ε) log r .(5) now from (4) and (5) it follows for all sufficiently large values of r that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) > (λh (f ◦g) −ε) log r (ρh (f) + ε) log r . as ε (> 0) is arbitrary , we obtain that (6) lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) > λh (f ◦g) ρh (f) . again for a sequence of values of r tending to infinity , (7) log t−1h tf◦g (r) ≤ (λh (f ◦g) + ε) log r and for all sufficiently large values of r , log t−1 l[h] tl[f] (r) > ( λl[h] (l[f]) −ε ) log r i.e., log t−1 l[h] tl[f] (r) > (λh (f) −ε) log r .(8) combining (7) and (8) we get for a sequence of values of r tending to infinity that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ (λh (f ◦g) + ε) log r (λh (f) −ε) log r . since ε (> 0) is arbitrary, it follows that (9) lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ λh (f ◦g) λh (f) . 86 datta, biswas abd mondal also for a sequence of values of r tending to infinity, log t−1 l[h] tl[f] (r) ≤ ( λl[h] (l[f]) + ε ) log r i.e., log t−1 l[h] tl[f] (r) ≤ (λh (f) + ε) log r .(10) now from (4) and (10) we obtain for a sequence of values of r tending to infinity that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≥ (λh (f ◦g) −ε) log r (λh (f) + ε) log r . as ε (> 0) is arbitrary, we get from above that (11) lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≥ λh (f ◦g) λh (f) . also for all sufficiently large values of r , (12) log t−1h tf◦g (r) ≤ (ρh (f ◦g) + ε) log r . now it follows from (8) and (12) for all sufficiently large values of r that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ (ρh (f ◦g) + ε) log r (λh (f) −ε) log r . since ε (> 0) is arbitrary, we obtain that (13) lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ ρh (f ◦g) λh (f) . thus the theorem follows from (6) , (9) , (11) and (13) . � the following theorem can be proved in the line of theorem 3.1 and so its proof is omitted. theorem 3.2. let g be a transcendental entire function with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2. also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and f be any meromorphic function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (g) ≤ ρh (g) < ∞ .then λh (f ◦g) ρh (g) ≤ lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) ≤ λh (f ◦g) λh (g) ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) ≤ ρh (f ◦g) λh (g) . theorem 3.3. suppose f be a transcendental meromorphic function with ∑ a6=∞ δ (a; f) +δ (∞; f) = 2. also let g be entire and h be a transcendental entire function of regular growth having non zero finite order with ∑ a 6=∞ δ (a; h) + δ (∞; h) = 2, 0 < ρh (f ◦g) < ∞ and 0 < ρh (f) < ∞ .then lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ ρh (f ◦g) ρh (f) ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) . growth properties of wronskians in the light of relative order 87 proof. from the definition of ρl[h] (l[f]) and in view of lemma 2.4 we get for a sequence of values of r tending to infinity that log t−1 l[h] tl[f] (r) > ( ρl[h] (l[f]) −ε ) log r i.e., log t−1 l[h] tl[f] (r) > (ρh (f) −ε) log r .(14) now from (12) and (14) it follows for a sequence of values of r tending to infinity that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ (ρh (f ◦g) + ε) log r (ρh (f) −ε) log r . as ε (> 0) is arbitrary, we obtain that (15) lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ ρh (f ◦g) ρh (f) . again for a sequence of values of r tending to infinity , (16) log t−1h tf◦g (r) > (ρh (f ◦g) −ε) log r . so combining (5) and (16) we get for a sequence of values of r tending to infinity that log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) > (ρh (f ◦g) −ε) log r (ρh (f) + ε) log r . since ε (> 0) is arbitrary, it follows that (17) lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) > ρh (f ◦g) ρh (f) . thus the theorem follows from (15) and (17) . � the following theorem can be carried out in the line of theorem 3.3 and therefore we omit its proof. theorem 3.4. let f be meromorphic and g,h be both transcendental entire functions with the maximum deficiency sums and 0 < ρh (f ◦g) < ∞ , 0 < ρh (g) < ∞. in addition, let h of regular growth having non zero finite order. then lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) ≤ ρh (f ◦g) ρh (g) ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) . the following theorem is a natural consequence of theorem 3.1 and theorem 3.3 : theorem 3.5. suppose f be a transcendental meromorphic function having the maximum deficiency sum. also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ and 0 < λh (f) ≤ ρh (f) < ∞ .then lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) ≤ min { λh (f ◦g) λh (f) , ρh (f ◦g) ρh (f) } ≤ max { λh (f ◦g) λh (f) , ρh (f ◦g) ρh (f) } ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) . 88 datta, biswas abd mondal the proof is omitted. analogously one may state the following theorem without its proof. theorem 3.6. let f be meromorphic and g,h be both transcendental entire functions with the maximum deficiency sums and 0 < λh (f ◦g) ≤ ρh (f ◦g) < ∞ , 0 < λh (g) ≤ ρh (g) < ∞. in addition, let h of regular growth having non zero finite order. then lim inf r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) ≤ min { λh (f ◦g) λh (g) , ρh (f ◦g) ρh (g) } ≤ max { λh (f ◦g) λh (g) , ρh (f ◦g) ρh (g) } ≤ lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) . theorem 3.7. suppose f be a transcendental meromorphic function having the maximum deficiency sum. also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and g be any entire function such that ρh (f) < ∞ and λh (f ◦g) = ∞ .then lim r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[f] (r) = ∞ . proof. let us suppose that the conclusion of the theorem do not hold. then we can find a constant β > 0 such that for a sequence of values of r tending to infinity, (18) log t−1h tf◦g (r) ≤ β log t −1 l[h] tl[f] (r) . again from the definition of ρl[g] (l [f]) it follows for all sufficiently large values of r and in view of lemma 2.4 that log t−1 l[h] tl[f] (r) ≤ ( ρl[h] (l[f]) + ε ) log r i.e., log t−1 l[h] tl[f] (r) ≤ (ρh (f) + ε) log r .(19) thus from (18) and (19) , we have for a sequence of values of r tending to infinity that log t−1h tf◦g (r) ≤ β (ρh (f) + ε) log r i.e., log t−1h tf◦g (r) log r ≤ β (ρh (f) + ε) log r log r i.e., lim inf r→∞ log t−1h tf◦g (r) log r = λh (f ◦g) < ∞. this is a contradiction. hence the theorem follows. � remark 3.8. theorem 3.7 is also valid with “limit superior” instead of “limit” if λh (f ◦g) = ∞ is replaced by ρh (f ◦g) = ∞ and the other conditions remain the same. corollary 3.9. under the assumptions of theorem 3.7 and remark 3.8, lim r→∞ t−1h tf◦g (r) t−1 l[h] tl[f] (r) = ∞ and lim sup r→∞ t−1h tf◦g (r) t−1 l[h] tl[f] (r) = ∞ . respectively hold. growth properties of wronskians in the light of relative order 89 the proof is omitted. analogously one may also state the following theorem and corollaries without their proofs as those may be carried out in the line of remark 3.8, theorem 3.7 and corollary 3.9 respectively. theorem 3.10. let g be a transcendental entire function with ∑ a6=∞ δ (a; g)+δ (∞; g) = 2. also let h be a transcendental entire function of regular growth having non zero finite order with ∑ a6=∞ δ (a; h) + δ (∞; h) = 2 and f be any meromorphic function such that ρh (g) < ∞ and ρh (f ◦g) = ∞ .then lim sup r→∞ log t−1h tf◦g (r) log t−1 l[h] tl[g] (r) = ∞ . corollary 3.11. theorem 3.10 is also valid with “limit” instead of “limit superior” if ρh (f ◦g) = ∞ is replaced by λh (f ◦g) = ∞ and the other conditions remain the same. corollary 3.12. under the assumptions of theorem 3.7 and corollary 3.11, lim sup r→∞ t−1h tf◦g (r) t−1 l[h] tl[g] (r) = ∞ and lim r→∞ t−1h tf◦g (r) t−1 l[h] tl[g] (r) = ∞ respectively hold. theorem 3.13. suppose f be a transcendental meromorphic function with ∑ a 6=∞ δ (a; f) +δ (∞; f) = 2. also let h be a transcendental entire function of regular growth having non zero finite order with the maximum deficiency sum and g be any entire function such that 0 < ρh (f ◦g) < ∞ and 0 < ρh (f) < ∞ and g ∼ h. then lim inf r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) ≤ 1 ≤ lim sup r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) . proof. from the definition of ρg (f) we get for all sufficiently large values of r that (20) log t−1g tf (r) ≤ (ρg (f) + ε) log r and for a sequence of values of r tending to infinity that (21) log t−1g tf (r) ≥ (ρg (f) −ε) log r . now from (14) and (20) it follows for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1 l[h] tl[f] (r) ≤ (ρg (f) + ε) log r (ρh (f) −ε) log r . as ε (> 0) is arbitrary we obtain that (22) lim inf r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) ≤ ρg (f) ρh (f) . now as g ∼ h , in view of lemma 2.4 and lemma 2.5 we obtain from (22) that (23) lim inf r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) ≤ 1 . 90 datta, biswas abd mondal again combining (5) and (21) we get for a sequence of values of r tending to infinity that log t−1g tf (r) log t−1 l[h] tl[f] (r) > (ρg (f) −ε) log r (ρh (f) + ε) log r . since ε (> 0) is arbitrary, it follows that (24) lim sup r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) > ρg (f) ρh (f) . now as g ∼ h , in view of lemma 2.4 and lemma 2.5 we obtain from (24) that (25) lim sup r→∞ log t−1g tf (r) log t−1 l[h] tl[f] (r) > 1 . thus the theorem follows from (23) and (25) . � references [1] bernal, l. : orden relative de crecimiento de funciones enteras , collect. math., vol. 39 (1988), pp.209-229. [2] datta, s. k. , biswas, t. and ali, s.: some growth properties of wronskians using their relative order, journal of classical analysis, vol. 3, no. 1 (2013), pp. 91-99. [3] hayman, w. k. : meromorphic functions, the clarendon press, oxford, 1964. [4] lahiri, b. k. and banerjee, d. :relative order of entire and meromorphic functions, proc. nat. acad. sci. india, vol. 69(a) iii(1999), pp.339-354. [5] valiron, g. : lectures on the general theory of integral functions, chelsea publishing company, 1949. 1department of mathematics,university of kalyani, kalyani, dist-nadia,pin741235, west bengal, india 2rajbari, rabindrapalli, r. n. tagore road, p.o.krishnagar, dist-nadia, pin741101, west bengal, india 3dhulauri rabindra vidyaniketan (h.s.), vill +p.o.dhulauri , p.s.domkal, distmurshidabad , pin742308, west bengal, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 2, number 2 (2013), 124-136 http://www.etamaths.com value distribution and uniqueness theorems for difference of entire and meromorphic functions subhas s. bhoosnurmath∗ and smita r kabbur abstract. we investigate the value distribution and uniqueness problems of difference polynomials of entire and meromorphic functions. 1. introduction and main results let f(z) be a meromorphic functions of finite order. we define difference operator as , ∆cf = f(z + c) −f(z), and ∆nc f = ∆ n−1 c (∆cf), n ≥ 2, where c is a non-zero constant. in particular, if c = 1 we use the usual difference notation ∆cf = ∆f. certain estimates involving the derivative f 7→ f′ of a meromorphic function play key roles in the construction and applications of classical nevanlinna theory. recently, there has been an increasing interest in studying difference equations in the complex plane. halburd and korhonen [2] established a version of nevanlinna theory based on difference operator. bergweiler and langley [ 12] considered the value distribution of zeros of difference operators that can be viewed as discrete analogues of the zeros of f′(z). ishizaki and yanagihara [13] developed a version of wiman valiron theory for difference equations of entire functions of small growth. growth estimates for the difference analogue of logarithmic derivative f(z+c) f(z) were given by halburd and korhonen [1] . this result has potentially large number of applications in the study of difference equation. many ideas and methods from the theory of differential equations are utilized to obtain information about meromorphic solutions of difference equations. the analogue of clunie lemma used to ensure that finite order meromorphic solutions of certain non-linear difference equations have large number of poles. all concepts of nevanlinna theory related to ramification have natural difference analogue. let f be a transcendental entire function and n be a positive integer. hayman [14] and clunie [22] proved that fnf′ assumes every non-zero value a ∈ c infinitely often. let f be a transcendental entire function. as for the value distribution of differential polynomial fn(f −1)f′, fang [15] showed that fn(f −1)f′ assumes every non-zero value a ∈ c infinitely often for n ≥ 4. we recall the following uniqueness theorem due to lin and yi [16, 17]. 2010 mathematics subject classification. 30d35. key words and phrases. meromorphic functions ,difference ,uniqueness. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 124 value distribution and uniqueness theorems 125 2. preliminaries and lemmas. theorem a. let f and g be two non-constant (resp transcendental) entire functions and let n ≥ 7 be an integer. if fn(f − 1)f′ and gn(g − 1)g′ share 1 (resp z) cm then f = g. recently , value distribution in difference analogue has become a subject of great interest. for analogue results in difference laine and yang [4] proved the following result. theorem b. let f be a transcendental entire function of finite order and c be a non-zero complex constant. then for n ≥ 2, f(z)nf(z + c) assumes every non zero value a ∈ c infinitely often. following analogue results in difference are proved by j. zhang [5]. theorem c. let f(z) be a transcendental entire function of finite order and α(z) be a small function with respect to f(z). suppose that c is non-zero complex constant and n ≥ 2 is an integer then f(z)n(f(z)−1)f(z +c) has infinitely many zeros. theorem d. let f(z) and g(z) be a transcendental entire function of finite order and α(z) be a small function with respect to both f(z) and g(z). suppose that c is non-zero complex constant and let n ≥ 7. if f(z)n(f(z) − 1)f(z + c) and g(z)n(g(z) − 1)g(z + c) share α(z) cm, then f(z) ≡ g(z). in this paper we consider the difference analogue of fn(fm − 1)f′ and prove theorem 2 . for m = 1 our result reduces to theorem 4. also proved theorem 3 where we investigate the uniqueness problem when the difference polynomial share α(z) with ignoring multiplicity. for a non-constant meromorphic function f and a set s ⊂ c∪{∞} , we define ef (s) = ⋃ a∈s {z ∈ f(z) −a = 0, counting multipilicities} . we say that two non-constant meromorphic functions f and g share a cm , if ef (s) = eg(s) and s = {a} . in 1976, gross [18] proved that there exists three finite sets sj(j = 1, 2, 3) such that for any two non-constant entire functions f and g , ef (sj) = eg(sj)(j = 1, 2, 3) imply f = g. in the same paper , gross posed the following question: question 1 . can one find two (or possibly even one ) finite sets sj(j = 1, 2) such that any two entire functions f and g satisfying ef (sj) = eg(sj)(j = 1, 2) must be identical? many authors have worked on it and got related results.we recall the following result by li and yang [19]. theorem e. let m ≥ 2, n > 2m+ 6 with n and n−m having no common factors. let a and b be two non-zero constants such that the equation ωn + aωn−m + b = 0 has no multiple roots. let s = {ω|ωn + aωn−m + b = 0} . then for any two non-constant meromorphic functions f and g, the conditions ef (s) = eg(s) and ef {∞} = eg {∞} imply f = g. j zhang [5] considered the difference analogue of this result and proved the following result. theorem f. let m ≥ 2, n ≥ 2m+ 4 with n and n−m having no common factors. let a and b be two non-zero constants such that the equation ωn + aωn−m + b = 0 has no multiple roots. let s = {ω|ωn + aωn−m + b = 0} . suppose that f is a non-constant meromorphic function of finite order. then ef(z)(s) = ef(z+c)(s) 126 bhoosnurmath and kabbur and ef(z) {∞} = ef(z+c) {∞} imply f(z) = f(z + c). in 1998 , frank reinders [20] proved following result. let the polynomial p be defined as, p(ω) = (n− 1)(n− 2) 2 ωn −n(n− 2)ωn−1 + n(n− 1) 2 ωn−2 − c, where n(≥ 3) is an integer and c(6= 0, 1) is a constant. theorem g.let s = {ω|p(ω) = 0} , where p(ω) is as defined above and n ≥ 11 be an integer. then for any two non-constant meromorphic functions f and g the condition ef(z)(s) = eg(z)(s) implies f = g. in this section, we consider the difference analogue of this result and prove theorem 4. the techniques used here greatly improves the condition on n from ′n ≥ 11′ to ′n ≥ 8′. for a non-constant meromorphic function f and a set s ⊂ c∪{∞} , we define ef (s) = ⋃ a∈s {z ∈ f(z) −a = 0, counting multipilicities} . we say that two non-constant meromorphic functions f and g share a cm , if ef (s) = eg(s) and s = {a} . in 1976, gross [18] proved that there exists three finite sets sj(j = 1, 2, 3) such that for any two non-constant entire functions f and g , ef (sj) = eg(sj)(j = 1, 2, 3) imply f = g. in the same paper , gross posed the following question: question 1 . can one find two (or possibly even one ) finite sets sj(j = 1, 2) such that any two entire functions f and g satisfying ef (sj) = eg(sj)(j = 1, 2) must be identical? many authors have worked on it and got related results.we recall the following result by li and yang [19]. theorem h. let m ≥ 2, n > 2m+ 6 with n and n−m having no common factors. let a and b be two non-zero constants such that the equation ωn + aωn−m + b = 0 has no multiple roots. let s = {ω|ωn + aωn−m + b = 0} . then for any two non-constant meromorphic functions f and g, the conditions ef (s) = eg(s) and ef {∞} = eg {∞} imply f = g. j zhang [5] considered the difference analogue of this result and proved the following result. theorem i. let m ≥ 2, n ≥ 2m + 4 with n and n−m having no common factors. let a and b be two non-zero constants such that the equation ωn + aωn−m + b = 0 has no multiple roots. let s = {ω|ωn + aωn−m + b = 0} . suppose that f is a non-constant meromorphic function of finite order. then ef(z)(s) = ef(z+c)(s) and ef(z) {∞} = ef(z+c) {∞} imply f(z) = f(z + c). in 1998 , frank reinders [20] proved following result. let the polynomial p be defined as, p(ω) = (n− 1)(n− 2) 2 ωn −n(n− 2)ωn−1 + n(n− 1) 2 ωn−2 − c, where n(≥ 3) is an integer and c(6= 0, 1) is a constant. theorem j. let s = {ω|p(ω) = 0} , where p(ω) is as defined above and n ≥ 11 be an integer. then for any two non-constant meromorphic functions f and g the condition ef(z)(s) = eg(z)(s) implies f = g. in this section, we consider the difference analogue of this result and value distribution and uniqueness theorems 127 prove theorem 4. the techniques used here greatly improves the condition on n from ′n ≥ 11′ to ′n ≥ 8′. 3. lemmas. in order to prove our results, we need following lemmas. lemma 1. [12]. let f(z) be a meromorphic function of finite order ρ and let c be a fixed non-zero complex constant. then for each � > 0, we have m ( r, f(z + c) f(z) ) + m ( r, f(z) f(z + c) ) = o(rρ−1+�). lemma 2. [6]. let f(z) be a meromorphic function of finite order ρ and let c be a fixed non-zero complex constant . then for each � > 0, we have t(r, f(z + c)) = t(r, f) + o(rρ−1+�). it is evident that s(r, f(z + c)) = s(r, f). lemma 3. [7]. let f(z) be a meromorphic function of finite order ρ and let c be a fixed non-zero complex constant . then n ( r, 1 f(z + c) ) ≤ n ( r, 1 f ) + s(r, f), n(r, f(z + c)) ≤ n(r, f) + s(r, f), n ( r, 1 f(z + c) ) ≤ n ( r, 1 f ) + s(r, f), n(r, f(z + c)) ≤ n(r, f) + s(r, f), outside of possible exceptional set with finite logarithmic measure. lemma 4. let f(z) be a transcendental entire function of finite order ρ. let f = f(z)n(f(z)m − 1)f(z + c). then (1) t(r, f) = (n + m + 1)t(r, f) + s(r, f) from this lemma it is clear that s(r, f) = s(r, f) and similarly s(r, g) = s(r, g) proof. since f is entire function of finite order , we deduce from lemma 1 and the standard valiron mohon’ko theorem that, (n + m + 1)t(r, f(z)) = t(r, f(z)n+1(f(z)m − 1)) + s(r, f) = m(r, f(z)n+1(f(z)m − 1)) + s(r, f) or, (n + m + 1)t(r, f(z)) ≤ m ( r, f(z)n+1(f(z)m − 1) f(z)n(f(z)m − 1)f(z + c) ) + m(r, f) + s(r, f) ≤ m ( r, f(z) f(z + c) ) + m(r, f) + s(r, f) ≤ t(r, f) + s(r, f) therefore, we have (2) (n + m + 1)t(r, f(z)) ≤ t(r, f) + s(r, f) 128 bhoosnurmath and kabbur on the other hand , by lemma 2 and the fact that f is a transcendental entire function of finite order, we get t(r, f ) ≤ t(r, f(z)n(f(z)m − 1)) + t(r, f(z + c)) + s(r, f) = (n + m)t(r, f(z)) + t(r, f(z + c)) + s(r, f) ≤ (n + m + 1)t(r, f(z)) + s(r, f) (3) i.e, t(r, f) ≤ (n + m + 1)t(r, f(z)) + s(r, f) thus (3.1) follows from (3.2) and (3.3). lemma 5. let f(z) and g(z) be a meromorphic function of finite order. if n ≥ m + 6, n, m are positive integers and (4) f(z)n(f(z)m − 1)f(z + c) = g(z)n(g(z)m − 1)g(z + c) then f = tg, where tm = 1. proof. let h(z) = f(z) g(z) . if h(z)n+mh(z + c) 6= 1, then from (3.4) we have g(z)nh(z)n(g(z)mh(z)m − 1)g(z + c)h(z + c) = g(z)n(g(z)m − 1)g(z + c) h(z)n(g(z)mh(z)m − 1)h(z + c) = g(z)m − 1 h(z)n+mh(z + c)g(z)m −h(z)nh(z + c) −g(z)m + 1 = 0 g(z)m(h(z)n+mh(z + c) − 1) = h(z)nh(z + c) − 1 (5) or, g(z)m = h(z)nh(z + c) − 1 h(z)n+mh(z + c) − 1 if 1 is a picard exceptional value of h(z)n+mh(z + c), applying the nevanlinna second main theorem with lemma 2, we get t(r, h(z)n+mh(z + c)) ≤ n(r, h(z)n+mh(z + c)) + n ( r, 1 h(z)n+mh(z + c) ) +n ( r, 1 h(z)n+mh(z + c) − 1 ) + s(r, h) ≤ 2t(r, h(z)) + 2t(r, h(z + c)) + s(r, h) (6) or, t(r, h(z)n+mh(z + c)) ≤ 4t(r, h(z)) + s(r, h) on the other hand, combining the standard valiron mohon’ko theorem with (3.6) and lemma 2, we get (n + m)t(r, h(z)) = t(r, h(z)n+m) + s(r, h) ≤ t(r, h(z)n+mh(z + c)) + t(r, h(z + c)) + s(r, h) ≤ 5t(r, h(z)) + s(r, h) (7) or, (n + m− 5)t(r, h(z)) ≤ s(r, h) which contradicts the hypothesis that n ≥ m + 6. therefore 1 is not a picard exceptional value of h(z)n+mh(z+c). thus there exists z0 such that h(z0) n+mh(z0 + c) = 1 then by (3.5) , we have h(z0) nh(z0 + c) = 1. hence h(z0) m = 1, and (8) n ( r, 1 h(z)n+mh(z + c) − 1 ) ≤ n ( r, 1 h(z)m − 1 ) ≤ mt(r, h) + o(1) value distribution and uniqueness theorems 129 denote, (9) h(z) = h(z)n+mh(z + c) we have t(r, h) ≤ (n + m + 1)t(r, h) + s(r, h). applying the second main theorem to h and using lemma 2 and (3.8), we get t(r, h) ≤ n(r, h(z)) + n ( r, 1 h(z) ) + n ( r, 1 h(z) − 1 ) + s(r, h) ≤ n(r, h(z)) + n ( r, 1 h(z) ) + mt(r, h(z)) + s(r, h) ≤ (m + 4)t(r, h) + s(r, h) therefore, we have (10) t(r, h(z)) ≤ (m + 4)t(r, h) + s(r, h). on the other hand using (3.9) and (3.10) we have (n + m)t(r, h) = t(r, h(z)n+m) + s(r, h) ≤ t(r, h(z)) + t(r, h(z + c)) + s(r, h) ≤ (m + 5)t(r, h) + s(r, h) or, (n−5)t(r, h) ≤ s(r, h). which contradicts our hypothesis , n ≥ m+6. therefore , h(z)n+mh(z + c) ≡ 1, and h(z)nh(z + c) ≡ 1. thus h(z)m ≡ 1. hence we get f(z) = tg(z), where tm = 1. lemma 6. [21] . let f and g be two non-constant meromorphic function. if f and g share 1 cm, then one of the following cases hold: i)max{t(r, f), t(r, g)} ≤ n2 ( r, 1 f ) +n2(r, f)+n2(r, g)+ ( r, 1 g ) +s(r, f )+s(r, g), ii) fg ≡ 1, iii) f ≡ g, where n2 ( r, 1 f ) denoted the counting function of zeros of f such that simple zeros are counted once and multiple zeros are counted twice. lemma 7. [8] . let f and g be two non-constant meromorphic function. let f and g share 1 im. let, (11) h = f ′′ f ′ − 2 f ′ f − 1 − g′′ g′ + 2 g′ g− 1 if h 6= 0, then t(r, f) + t(r, g) ≤ 2 [ n2 ( r, 1 f ) + n2(r, f) + n2(r, g) + n2 ( r, 1 g )] +3 [ n(r, f) + n(r, g) + n ( r, 1 f ) + n ( r, 1 g )] (12) +s(r, f) + s(r, g) lemma 8. [20].let q(ω) = (n− 1)2(ωn − 1)(ωn−2 − 1) −n(n− 2)(ωn−1 − 1)2 be a polynomial of degree 2n− 2 (n ≥ 3). then q(ω) = (ω − 1)4(ω −β1)(ω −β2), . . . , (ω −β2n−6) 130 bhoosnurmath and kabbur where βj ∈ c \{0, 1}. 4. statement and proof of main result. theorem 1. let f(z) be entire function of finite order and α(z) be a small function with respect to f(z). suppose that c is a non-zero complex constant and n is an integer. if n ≥ 2, then f(z)n(f(z)m−1)f(z+c)−α(z) has infinitely many zeros. proof. let f(z) = f(z)n(f(z)m − 1)f(z + c). contrary to the assumption, suppose f(z)n(f(z)m − 1)f(z + c) −α(z) has finitely many zeros. by second fundamental theorem, lemma 3 , we have t(r, f) ≤ n(r, f(z)) + n ( r, 1 f(z) ) + n ( r, 1 f(z) − 1 ) + s(r, f) ≤ n ( r, 1 f(z)n(f(z)m − 1)f(z + c) ) + n ( r, 1 f(z) −α(z) ) + s(r, f) ≤ (m + 2)t(r, f) + s(r, f) using lemma 4 we get (n− 1)t(r, f) ≤ s(r, f), which contradicts our assumption. theorem 2. let f(z) and g(z) be two transcendental entire function of finite order and α(z) be a small function with respect to f(z) and g(z). suppose that c is a non-zero complex constant and let n ≥ m + 6. if f(z)n(f(z)m −1)f(z + c) and g(z)n(g(z)m − 1)g(z + c) share α(z) cm, then f(z) ≡ tg(z), where tm = 1. proof. let (13) f(z) = f(z)n(f(z)m − 1)f(z + c) α(z) and g(z) = g(z)n(g(z)m − 1)g(z + c) α(z) then f(z) and g(z) share 1 cm, except the zeros and poles of α(z). by lemma 4 (14) t(r, f(z)) = (n + m + 1)t(r, f) + s(r, f) (15) and t(r, g(z)) = (n + m + 1)t(r, g) + s(r, g) since f and g are transcendental entire functions, n2(r, f) = s(r, f), and n2(r, g) = s(r, g) by lemma 3, we have n2 ( r, 1 f ) ≤ 2n ( r, 1 fn ) + n ( r, 1 fm − 1 ) + n ( r, 1 f(z + c) ) + s(r, f) ≤ 2n ( r, 1 f ) + mt(r, f) + t(r, f(z + c) + s(r, f) ≤ (m + 3)t(r, f) + s(r, f) therefore (16) n2(r, f) + n2 ( r, 1 f ) ≤ (m + 3)t(r, f) + s(r, f) value distribution and uniqueness theorems 131 similarly, (17) n2(r, g) + n2 ( r, 1 g ) ≤ (m + 3)t(r, g) + s(r, g) by condition of the theorem, suppose case (i) of lemma 6 holds. substituting (4.4) and (4.5) in (i) of lemma 6, we have t(r, f) + t(r, g) ≤ (2m + 6)(t(r, f) + t(r, g)) + s(r, f) + s(r, g) using (4.2) and (4.3) we get (n + m + 1){t(r, f) + t(r, g)} ≤ (2m + 6)(t(r, f) + t(r, g)) + s(r, f) + s(r, g) or, (n−m− 5){t(r, f) + t(r, g)} ≤ s(r, f) + s(r, g) which contradicts our assumption, n ≥ m + 6. hence by lemma 6 f(z)g(z) ≡ α(z)2 or f(z) ≡ g(z). suppose, f(z)g(z) ≡ α(z)2. that is f(z)n (f(z)m − 1) f(z + c) g(z)n (g(z)m − 1) g(z + c) = α(z)2 or, f(z)n (f(z)−1) (f(z)m−1 −···−1) f(z+c) g(z)n (g(z)−1) (g(z)m−1 −···−1) g(z+c) = α(z)2 then, n ( r, 1 f ) = s(r, f) and n ( r, 1 f − 1 ) = s(r, f) from this we have, δ(0, f) + δ(1, f) + δ(∞, f) = 3, which is impossible. from this we conclude that, f(z) ≡ g(z). i, e. f(z)n(f(z)m − 1)f(z + c) ≡ g(z)n(g(z)m − 1)g(z + c) using lemma 5, we conclude that f(z) = tg(z), where tm = 1. theorem 3. let f(z) and g(z) be two transcendental entire function of finite order, and α(z) be a small function with respect to f(z) and g(z). suppose that c is a non-zero complex constant and let n ≥ 4m + 12. if f(z)n(f(z)m − 1)f(z + c) and g(z)n(g(z)m − 1)g(z + c) share α(z) im, then f(z) = tg(z), where tm = 1. proof. let (18) f(z) = f(z)n(f(z)m − 1)f(z + c) α(z) and g(z) = g(z)n(g(z)m − 1)g(z + c) α(z) then f(z) and g(z) share 1 im. let h be as defined in lemma 7. using lemma 3, we have n ( r, 1 f ) ≤ n ( r, 1 fn ) + n ( r, 1 fm − 1 ) + n ( r, 1 f(z + c) ) + s(r, f) (19) ≤ (m + 2)t(r, f) + s(r, f) 132 bhoosnurmath and kabbur using (4.4), (4.5) and (4.7) in (4.10) of lemma 7, we get t(r, f) + t(r, g) ≤ 2 [ n2 ( r, 1 f ) + n2 ( r, 1 g )] + 3 [ n(r, 1 f ) + n(r, 1 g ) ] +s(r, f) + s(r, g) ≤ 2(m + 3) [t(r, f) + t(r, g)] + 3(m + 2) [t(r, f) + t(r, g)] +s(r, f) + s(r, g) ≤ (5m + 12) [t(r, f) + t(r, g)] + s(r, f) + s(r, g) using (4.2) and (4.3) , we have (20) (n + 4m− 11)(t(r, f) + t(r, g)) ≤ s(r, f) + s(r, g) which contradicts the hypothesis that n ≥ 4m + 12. thus we get h ≡ 0. integrating h twice , we obtain (21) f = (b + 1)g + (a− b− 1) bg + (a− b) and g = (a− b− 1) − (a− b)f fb− (b + 1) in the following , we will prove that fg ≡ 1 or f ≡ g. case 1. b 6= 0, −1. if a− b− 1 6= 0, then by (4.9) we have (22) n ( r, 1 f ) = n ( r, 1 g− a−b−1 b+1 ) by the nevanlinna second main theorem and lemma 4, we have t(r, g) ≤ n(r, g) + n ( r, 1 g ) + n ( r, 1 g− a−b−1 b+1 ) + s(r, g) ≤ n ( r, 1 g ) + n ( r, 1 f ) + s(r, g) ≤ n ( r, 1 g(z)n(g(z)m − 1)g(z + c) ) + n ( r, 1 f(z)n(f(z)m − 1)f(z + c) ) + s(r, g) ≤ (m + 2)t(r, g) + (m + 2)t(r, f) + s(r, g) therefore we have, (23) (n + m + 1)t(r, g) ≤ (m + 2)t(r, f) + (m + 2)t(r, g) + s(r, g) (24) (n + m + 1)t(r, f) ≤ (m + 2)t(r, g) + (m + 2)t(r, f) + s(r, f) from (4.11) and (4.12) , we get (n + m + 1)(t(r, f) + t(r, g)) ≤ (m + 2)(t(r, f) + t(r, g)) + s(r, f) + s(r, g) or, (n− 1)(t(r, f) + t(r, g)) ≤ s(r, f) + s(r, g) this contradicts the assumption that n ≥ 4m + 12. thus a− b− 1 = 0, then by (4.9), we have (25) f = (b + 1)g bg + 1 since f is an entire finction, from (4.13) we have n ( r, 1 g + 1 b ) = 0. value distribution and uniqueness theorems 133 proceeding as above we deduce a contradiction. case 2: b = −1, a 6= −1. then by (4.9) we get f = a (a + 1) −g , and n ( r, 1 g− (a + 1) ) = n(r, f) = 0 similarly as in case 1, we get contradiction. hence a = −1, thus we get fg ≡ 1. proceeding as in theorem 2 , we get contradiction. case 3: if b = 0, a 6= 1. from (4.9) we have f = g + (a− 1) a and n ( r, 1 f ) = n ( r, 1 g− (a + 1) ) proceeding as in case 1 , we get a contradiction. thus a = 1, implies f = g. proceeding as in theorem 2 we get f = tg, where tm = 1. theorem 4. let n ≥ 8 be an integer and c(6= 0, 1) is a constant such that the equation p(ω) = 0 has no multiple roots. where (26) p(ω) = (n− 1)(n− 2) 2 ωn −n(n− 2)ωn−1 + n(n− 1) 2 ωn−2 − c let s = {ω|p(ω) = 0} , suppose that f is a non-constant meromorphic function of finite order then ef(z)(s) = ef(z+c)(s) and ef(z)({∞}) = ef(z+c)({∞}) implies f(z) = f(z + c). proof. from the condition of the theorem, we have ef(z)(s) = ef(z+c)(s) then there exists a polynomial q(z), such that (27) p(z + c) p(z) = eq(z) and t(r, eq(z)) = m(r, eq(z)) = s(r, f) rewriting (4.15), we have p(z + c) = eq(z)p(z) or, (n− 1)(n− 2) 2 f(z + c)n −n(n− 2)f(z + c)n−1 + n(n− 1) 2 f(z + c)n−2 − c = eq(z) [ (n− 1)(n− 2) 2 f(z)n −n(n− 2)f(z)n−1 + n(n− 1) 2 f(z)n−2 − c ] or, f(z + c)n−2 [ (n− 1)(n− 2) 2 f(z + c)2 −n(n− 2)f(z + c) + n(n− 1) 2 ] (28) = eq(z) [ (n− 1)(n− 2) 2 f(z)n −n(n− 2)f(z)n−1 + n(n− 1) 2 f(z)n−2 − c + ce−q(z) ] let f(z) = (n− 1)(n− 2) 2 f(z)n −n(n− 2)f(z)n−1 + n(n− 1) 2 f(z)n−2, 134 bhoosnurmath and kabbur or, f(z) = f(z)n−2 [ (n− 1)(n− 2) 2 f(z)2 −n(n− 2)f(z) + n(n− 1) 2 ] , or, (29) f(z) = f(z)n−2(f −α1)(f −α2) where α1, and α2 are roots of the equation, (n− 1)(n− 2) 2 f(z)2 −n(n− 2)f(z) + n(n− 1) 2 = 0. therefore (4.16) can be rewritten as, f(z + c)n−2 [ (n− 1)(n− 2) 2 f(z + c)2 −n(n− 2)f(z + c) + n(n− 1) 2 ] (30) = eq(z) [ f(z) − (c− ce−q(z)) ] using standard valiron moho’nko theorem, we get (31) t(r, f) = nt(r, f) + s(r, f) applying second main theorem to f and using (4.17) and (4.18), we have t(r, f) ≤ n(r, f) + n ( r, 1 f ) + n ( r, 1 f − (c− ce−q(z) ) + s(r, f) ≤ n(r, f) + n ( r, 1 f ) + n ( r, 1 f(z + c)n−2 ) +n ( r, 1 (n−1)(n−2) 2 f(z + c)2 −n(n− 2)f(z + c) + n(n−1) 2 ) + s(r, f) or, t(r,f) ≤ n(r, f) + n ( r, 1 f(z)n−2 ) + n ( r, 1 f −α1 ) +n ( r, 1 f −α2 ) + n ( r, 1 f(z + c)n−2 ) +n ( r, 1 (n−1)(n−2) 2 f(z + c)2 −n(n− 2)f(z + c) + n(n−1) 2 ) + s(r, f) ≤ 4t(r, f) + 3t(r, f(z + c)) + s(r, f) by (4.19), we have n t(r, f) ≤ 4t(r, f) + 3t(r, f(z + c)) + s(r, f) (32) or, (n− 4)t(r, f(z)) ≤ 3t(r, f(z + c)) + s(r, f) (33) similarly (n− 4)t(r, f(z + c)) ≤ 3t(r, f(z)) + s(r, f) using (4.20) and (4.21), we have (n− 4) (t(r, f(z)) + t(r, f(z + c))) ≤ 3 (t(r, f(z)) + t(r, f(z + c))) + s(r, f) or, (n− 7) (t(r, f(z)) + t(r, f(z + c))) ≤ s(r, f), value distribution and uniqueness theorems 135 which contradicts the assumption n ≥ 8. therefore eq(z) = 1. hence from (4.15), we get p(z + c) = p(z). or, (n− 1)(n− 2) 2 f(z + c)n −n(n− 2)f(z + c)n−1 + n(n− 1) 2 f(z + c)n−2 − c = (n− 1)(n− 2) 2 f(z)n −n(n− 2)f(z)n−1 + n(n− 1) 2 f(z)n−2 − c or, (n− 1)(n− 2) 2 (f(z)n −f(z + c)n) −n(n− 2) ( f(z)n−1 −f(z + c)n−1 ) (34) + n(n− 1) 2 ( f(z)n−2 −f(z + c)n−2 ) ≡ 0. taking h(z) = f(z + c) f(z) , we get (35) (n− 1)(n− 2) 2 (hn − 1) f(z)2−n(n−2) ( hn−1 − 1 ) f(z)+ n(n− 1) 2 ( hn−2 − 1 ) = 0 suppose h is not a constant. from (4.23), we have (36) { (n− 1)(n− 2)(hn − 1)f(z) −n(n− 2)(hn−1 − 1) }2 = −n(n− 2)q(h) where q(h) is defined as in lemma 8. using (4.24) and lemma 8, we get{ (n− 1)(n− 2)(hn − 1)f(z) −n(n− 2)(hn−1 − 1) }2 (37) = −n(n− 2)(h− 1)4(h−β1)(h−β2), . . . , (h−β2n−6) from (4.24), all zeros of (h−βj) have order at least 2. applying second fundamental theorem to h, we get (2n− 8)t(r, h) ≤ 2n−6∑ j = 1 n ( r, 1 h−βj ) + s(r, h) ≤ 1 2 2n−6∑ j = 1 n ( r, 1 h−βj ) + s(r, h) ≤ (n− 3)t(r, h) + s(r, h) which contradicts the assumption , n ≥ 8. thus h is a constant, from (5.3.23 ) we obtain hn − 1 = 0. therefore h = 1. hence we conclude that f(z) ≡ f(z + c). theorem 5. let n ≥ 7 be an integer and c(6= 0, 1) is a constant such that the equation p(ω) = 0 has no multiple roots, where p(ω) is as defined in theorem 4 . let s = {ω|p(ω) = 0} , suppose that f is a non-constant entire function of finite order then ef(z)(s) = ef(z+c)(s) implies f(z) = f(z + c). proof. f and g are entire functions. taking n(r,f) = n(r,g) = 0, in the above theorem, we obtain conclusion of this theorem. 6. acknowledgment 136 bhoosnurmath and kabbur this research work is supported by department of science and technology government of india ,ministry of science and technology,technology bhavan,new delhi under the sanction letter no (sr/s4/ms : 520/08) references [1] r.g.halburd and r.j.korhonen,difference analogue of the lemma on the logarithmic derivative with applications to difference equations, ann. math. anal. appl, 314(2006), 477-487. [2] r.g.halburd and r.j.korhonen, nevanlinna theory for the difference operator. ann. acad. sci. fenn. 31, (2006), 463-478. [3] r.g.halburd and r.j.korhonen, finite order meromorphic solutions and the discrete painleive equations, proc. london. math. soc (3) 94 (2) (2007), 443-474. [4] i.laine, c.c.yang, value distribution of difference polynomials, proc. japan acad. sera math. sci. 83(2007), 148-151. [5] jilong zhang , value distribution and shared sets of difference of meromorphic functions , j. math. anal. appl. 367(2010) 401-408. [6] y.m chiang, s.j.feng, on the nevanlinna characteristic of f(z + η) and the difference equations in the complex plane, ramanujan j. 16 (2008) 105-129. [7] xudan luo, wei-chuan lin, value sharing results for shifts of meromorphic functions, . j. math. anal. appl.377(2011), 441 449. [8] j.f.xu and h.x.yi, uniqueness of entire functions and differential polynomials, . bulletin of korean mathematical society, vol.44, no 4. 2007, 623-629. [9] hermann weyl, meromorphic functions and analytic curves. princeton university press, 185 193, p.8. [10] w. k. hayman , meromorphic functions . clarendon press , oxford , 1964. [11] c. c. yang and h. x. yi, uniqueness theory of meromorphic functions , mathematics and its applications , kluwer academic publishers group , dordredt, 2003. [12] w bergweiler and j.k.langley, zeros of differences of meromorphic functions, math. proc. camb. phil. soc. 142, 133 147 (2007). [13] k. ishizaki and n. yanagihara , wiman valiron method for difference equations, . nagoya math.j, 175 , 2004, 75 102. [14] w.k.hayman , picard values of meromorphic functions and their derivatives , ann.math.70(1959), 9-42. [15] m.l.fang , uniqneness and value sharing of entire functions , comput.math.appl.44(2002)823-831. [16] w.c. lin and h. x. yi , uniqueness theorems for meromorphic functions , indian j. pure appl math , vol.35 , no.2 (2004) , 121 132. [17] w. c. lin and h. x. yi , uniqueness theorems for meromorphic functions concerning fixedpoints, complex variables , vol.49 , no.11 (2004) , 793 806. [18] f.gross, factorsiation of meromorphic functions and some open problems, in:complex analysis , proc. conf. univ. kentucky, lexington, ky, 1976, in:lecture notes in maths , vol.599, springer, berlin, 1997, pp . 51-69. [19] p.li, c.c.yang , some further results on the unique range sets of meromorphic functions , kodai.math.j.18(1995), 437-450. [20] g. frank and reinders m, a uniqueness range set for meromorphic functions with 11 elements. complex variables theory appl. 37 (1998), 185 193. [21] c. c. yang and x. h. hua , uniqueness and value sharing of meromorphic functions, ann.acad.aci.fenn.math , vol.22 , (1997) , 395 406. [22] j.clunie , on results of hayman , j.london.math.soc 42(1967), 389-392. department of mathematics, karnatak university, dharwad-580003, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 59-69 http://www.etamaths.com dhage iteration method for generalized quadratic functional integral equations bapurao c. dhage abstract. in this paper we prove the existence as well as approximations of the solutions for a certain nonlinear generalized quadratic functional integral equation. an algorithm for the solutions is developed and it is shown that the sequence of successive approximations starting at a lower or upper solution converges monotonically to the solutions of related quadratic functional integral equation under some suitable mixed hybrid conditions. we rely our main result on dhage iteration method embodied in a recent hybrid fixed point theorem of dhage (2014) in partially ordered normed linear spaces. an example is also provided to illustrate the abstract theory developed in the paper. 1. introduction the quadratic integral equations have been a topic of interest since long time because of their occurrence in the problems of some natural and physical processes of the universe. see argyros [1], deimling [3], chandrasekher [2] and the references therein. the study gained momentum after the formulation of the hybrid fixed point principles in banach algebras due to dhage [4, 5, 6, 7, 8]. the existence results for such quadratic operators equations are generally proved under the mixed lipschitz and compactness type conditions together with a certain growth condition on the nonlinearities involved in the quadratic operator or functional equations. the hybrid fixed point theorems in banach algebras find numerous applications in the theory of nonlinear quadratic differential and integral equations. see dhage [5, 6, 7] and the references therein. the lipschitz and compactness hypotheses are considered to be very strong conditions in the theory of nonlinear differential and integral equations but which still do not yield any algorithm to determine the numerical solutions. therefore, it is of interest to relax or weaken these condition in the existence and approximation theory of quadratic integral equations. this is the main motivation of the present paper. in this paper we prove the existence as well as approximations of the solutions of a certain generalized quadratic integral equation via an algorithm based on successive approximations under partially lipschitz and compactness type conditions. 2010 mathematics subject classification. 45g10, 47h09, 47h10. key words and phrases. quadratic functional integral equation; approximate solution; dhage iteration method; hybrid fixed point theorem. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 59 60 dhage given a closed and bounded interval j = [0,t] of the real line r for some t > 0, we consider the quadratic functional integral equation (in short qfie) (1.1) x(t) = k(t,x(t)) + [ f(t,x(t)) ]( q(t) + ∫ t 0 v(t,s)g(s,x(s)) ds ) , t ∈ j, where q : j → r, v : j ×j → r and f,g,k : j ×r → r are continuous functions. by a solution of the qfie (1.1) we mean a function x ∈ c(j,r) that satisfies the equation (1.1) on j, where c(j,r) is the space of continuous real-valued functions defined on j. the qfie (1.1) is well-known in the literature and studied earlier in the work of dhage [4]. if f(t,x) = 0 for all t ∈ j and x ∈ r the qfie (1.1) reduces to the nonlinear functional equation (1.2) x(t) = k(t,x(t)), t ∈ j, and if k(t,x) = 0 and f(t,x) = 1 for all t ∈ j and x ∈ r, it is reduced to nonlinear usual volterra integral equation (1.3) x(t) = q(t) + ∫ t 0 v(t,s)g(s,x(s)) ds, t ∈ j. therefore, the qfie (1.1) is general and the results of this paper include the existence and approximations results for above nonlinear functional and volterra integral equations as special cases. the paper is organized as follows: in the following section we give the preliminaries and auxiliary results needed in the subsequent part of the paper. the main result is included in section 3. in section 4 some concluding remarks are presented. 2. auxiliary results unless otherwise mentioned, throughout this paper that follows, let e denote a partially ordered real normed linear space with an order relation � and the norm ‖ · ‖. it is known that e is regular if {xn}n∈n is a nondecreasing (resp. nonincreasing) sequence in e such that xn → x∗ as n → ∞, then xn � x∗ (resp. xn � x∗) for all n ∈ n. clearly, the partially ordered banach space c(j,r) is regular and the conditions guaranteeing the regularity of any partially ordered normed linear space e may be found in heikkilä and lakshmikantham [13] and the references therein. we need the following definitions in the sequel. definition 2.1. a mapping t : e → e is called isotone or nondecreasing if it preserves the order relation �, that is, if x � y implies t x �t y for all x,y ∈ e. definition 2.2 (dhage [9]). a mapping t : e → e is called partially continuous at a point a ∈ e if for � > 0 there exists a δ > 0 such that ‖t x−t a‖ < � whenever x is comparable to a and ‖x−a‖ < δ. t called partially continuous on e if it is partially continuous at every point of it. it is clear that if t is partially continuous on e, then it is continuous on every chain c contained in e. definition 2.3. a mapping t : e → e is called partially bounded if t (c) is bounded for every chain c in e. t is called uniformly partially bounded if all generalized quadratic functional integral equations 61 chains t (c) in e are bounded by a unique constant. t is called bounded if t (e) is a bounded subset of e. definition 2.4. a mapping t : e → e is called partially compact if t (c) is a relatively compact subset of e for all totally ordered sets or chains c in e. t is called uniformly partially compact if t (c) is a uniformly partially bounded and partially compact on e. t is called partially totally bounded if for any totally ordered and bounded subset c of e, t (c) is a relatively compact subset of e. if t is partially continuous and partially totally bounded, then it is called partially completely continuous on e. definition 2.5 (dhage [9]). the order relation � and the metric d on a non-empty set e are said to be compatible if {xn}n∈n is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk}n∈n of {xn}n∈n converges to x∗ implies that the whole sequence {xn}n∈n converges to x∗. similarly, given a partially ordered normed linear space (e,�,‖ · ‖), the order relation � and the norm ‖·‖ are said to be compatible if � and the metric d defined through the norm ‖ ·‖ are compatible. clearly, the set r of real numbers with usual order relation ≤ and the norm defined by the absolute value function | · | has this property. similarly, the finite dimensional euclidean space rn with usual componentwise order relation and the standard norm possesses the compatibility property. definition 2.6 (dhage [6]). a upper semi-continuous and nondecreasing function ψ : r+ → r+ is called a d-function provided ψ(0) = 0. let (e,�,‖ · ‖) be a partially ordered normed linear space. a mapping t : e → e is called partially nonlinear d-lipschitz if there exists a d-function ψ : r+ → r+ such that (2.1) ‖t x−t y‖≤ ψ(‖x−y‖) for all comparable elements x,y ∈ e. if ψ(r) = k r, k > 0, then t is called a partially lipschitz with a lipschitz constant k. let (e,�,‖ ·‖) be a partially ordered normed linear algebra. denote e+ = { x ∈ e | x � θ, where θ is the zero element of e } and (2.2) k = {e+ ⊂ e | uv ∈ e+ for all u,v ∈ e+}. the elements of k are called the positive vectors of the normed linear algebra e. the following lemma follows immediately from the definition of the set k and which is often times used in the applications of hybrid fixed point theory in banach algebras. lemma 2.7 (dhage [7]). if u1,u2,v1,v2 ∈ k are such that u1 � v1 and u2 � v2, then u1u2 � v1v2. definition 2.8. an operator t : e → e is said to be positive if the range r(t ) of t is such that r (t ) ⊆k. the dhage iteration principle or method (in short dip or dim) developed in dhage [9, 10, 11] may be formulated as “monotonic convergence of the sequence of successive approximations to the solutions of a nonlinear equation beginning with a lower or an upper solution of the equation as its initial or first approximation” 62 dhage and which is a powerful tool in the existence theory of nonlinear analysis. it is clear that dhage iteration method is different from the usual picard’s successive iteration method and embodied in the following applicable hybrid fixed point theorems proved in dhage [10] which forms a useful key tool for our work contained in this paper. a few other hybrid fixed point theorems involving the dhage iteration method may be found in dhage [9, 10, 11, 12]. theorem 2.9 (dhage [10]). let ( e,�,‖·‖ ) be a regular partially ordered complete normed linear algebra such that the order relation � and the norm ‖ · ‖ in e are compatible in every compact chain of e. let a,b : e →k and b : e → e be three nondecreasing operators such that (a) a and c are partially bounded and partially nonlinear d-lipschitz with dfunctions ψa and ψc respectively, (b) b is partially continuous and uniformly partially compact, and (c) mψa(r)+ψc(r) < r, r > 0, where m = sup{‖b(c)‖ : c is a chain in e}, and (d) there exists an element x0 ∈ x such that x0 � ax0 bx0 + cx0 or x0 � ax0 bx0 + cx0. then the operator equation (2.3) axbx + cx = x has a solution x∗ in e and the sequence {xn} of successive iterations defined by xn+1 = axn bxn + cxn, n = 0, 1, . . . , converges monotonically to x∗. remark 2.10. the compatibility of the order relation � and the norm ‖ · ‖ in every compact chain of e holds if every partially compact subset of e possesses the compatibility property with respect to � and ‖ · ‖. 3. main result the qfie (1.1) is considered in the function space c(j,r) of continuous realvalued functions defined on j. we define a norm ‖ · ‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) for all t ∈ j respectively. clearly, c(j,r) is a banach algebra with respect to above supremum norm and is also partially ordered w.r.t. the above partially order relation ≤. it is known that the partially ordered banach algebra c(j,r) has some nice properties w.r.t. the above order relation in it. the following lemma follows by an application of arzellá-ascolli theorem. lemma 3.1. let ( c(j,r),≤,‖ · ‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2) respectively. then ‖ ·‖ and ≤ are compatible in every partially compact subset of c(j,r). proof. the lemma mentioned in dhage [10], but the proof appears in dhage [11]. since the proof is not well-known, we give the details of the proof. let s be a generalized quadratic functional integral equations 63 partially compact subset of c(j,r) and let {xn}n∈n be a monotone nondecreasing sequence of points in s. then we have (3.3) x1(t) ≤ x2(t) ≤ ···≤ xn(t) ≤ ··· , for each t ∈ r+. suppose that a subsequence {xnk}n∈n of {xn}n∈n is convergent and converges to a point x in s. then the subsequence {xnk (t)}n∈n of the monotone real sequence {xn(t)}n∈n is convergent. by monotone characterization, the whole sequence {xn(t)}n∈n is convergent and converges to a point x(t) in r for each t ∈ r+. this shows that the sequence {xn(t)}n∈n converges point-wise in s. to show the convergence is uniform, it is enough to show that the sequence {xn(t)}n∈n is equicontinuous. since s is partially compact, every chain or totally ordered set and consequently {xn}n∈n is an equicontinuous sequence by arzelá-ascoli theorem. hence {xn}n∈n is convergent and converges uniformly to x. as a result ‖ · ‖ and ≤ are compatible in s. this completes the proof. � we need the following definition in what follows. definition 3.2. a function u ∈ c(j,r) is said to be a lower solution of the qfie (1.1) if it satisfies u(t) ≤ k(t,u(t)) + [ f(t,u(t)) ]( q(t) + ∫ t 0 v(t,s)g(s,u(s)) ds ) (∗) for all t ∈ j. similarly, a function v ∈ c(j,r) is said to be an upper solution of the qfie (1.1) if it satisfies the above inequalities with reverse sign. we consider the following set of assumptions in what follows: (a1) f defines a function f : j ×r → r+. (a2) there exists a constant mf > 0 such that f(t,x) ≤ mf for all t ∈ j and x ∈ r. (a3) there exists a d-function ψf such that 0 ≤ f(t,x) −f(t,y) ≤ ψf (x−y), for all t ∈ j and x,y ∈ r, x ≥ y. (b0) q defines a continuous function q : j → r+. (b1) v defines a continuous and nonnegative function on j ×j. (b2) g defines a function g : j ×r → r+. (b3) there exists a constant mg > 0 such that g(t,x) ≤ mg for all t ∈ j and x ∈ r. (b4) g(t,x) is nondecreasing in x for all t ∈ j. (c1) there exists a constant mk > 0 such that |k(t,x)| ≤ mk for all t ∈ j and x ∈ r. (c2) there exists a d-function ψk, such that 0 ≤ k(t,x) −k(t,y) ≤ ψk(x−y), for all t ∈ j and x,y ∈ r, x ≥ y. (c3) the qfie (1.1) has a lower solution u ∈ c(j,r). theorem 3.3. assume that hypotheses (a1)-(a3), (b0)-(b4) and (c1)-(c3) hold. furthermore, assume that (3.4) ( ‖q‖ + mg t ) ψf (r) + ψk(r) < r, r > 0, 64 dhage then the qfie (1.1) has a solution x∗ defined on j and the sequence {xn}n∈n∪{0} of successive approximations defined by (3.5) xn+1(t) = k(t,xn(t)) + [ f(t,xn(t)) ]( q(t) + ∫ t t0 v(t,s)g(s,xn(s)) ds ) , for all t ∈ j, where x0 = u, converges monotonically to x∗. proof. set e = c(j,r). then, from lemma 3.1 it follows that every compact chain in e possesses the compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in e. define two operators a, b and c on e by (3.6) ax(t) = f(t,x(t)), t ∈ j, (3.7) bx(t) = q(t) + ∫ t t0 v(t,s)g(s,x(s)) ds, t ∈ j, and (3.8) cx(t) = k(t,x(t)), t ∈ j. from the continuity of the integral and the hypotheses (a1) and (b0)-(b2), it follows that a, b and c define the maps a,b : e → k and c : e → e. now by definitions of the operators a, b and c, the qfie (1.1) is equivalent to the quadratic operator equation (3.9) ax(t)bx(t) + cx(t) = x(t), t ∈ j. we shall show that the operators a and b satisfy all the conditions of theorem 2.9. this is achieved in the series of following steps. step i: a, b and b are nondecreasing on e. let x,y ∈ e be such that x ≥ y. then by hypothesis (a3) and (c2), we obtain ax(t) = f(t,x(t)) ≥ f(t,y(t)) = ay(t), and cx(t) = k(t,x(t)) ≥ k(t,y(t)) = cy(t), for all t ∈ j. this shows that a and c are nondecreasing operators on e into e. similarly, using hypothesis (b4), bx(t) = q(t) + ∫ t 0 v(t,s)g(s,x(s)) ds ≤ q(t) + ∫ t 0 v(t,s)g(s,y(s)) ds = by(t) for all t ∈ j. hence, it is follows that the operator b is also nondecreasing on e into itself. thus, a, b and c are nondecreasing operators on e into itself. step ii: a and c are partially bounded and partially d-lipschitz on e. let x ∈ e be arbitrary. then by (a2), |ax(t)| ≤ ∣∣f(t,x(t))∣∣ ≤ mf, generalized quadratic functional integral equations 65 for all t ∈ j. taking supremum over t, we obtain ‖ax‖≤ mf and so, a is bounded. this further implies that a is partially bounded on e. similarly, using hypothesis (c1), it is shown that ‖cx‖≤ mk and consequently c is partially bounded on e. next, let x,y ∈ e be such that x ≥ y. then, by hypothesis (a3), |ax(t) −ay(t)| = ∣∣f(t,x(t)) −f(t,y(t))∣∣ ≤ ψf (|x(t) −y(t)|) ≤ ψf (‖x−y‖), for all t ∈ j. taking supremum over t, we obtain ‖ax−ay‖≤ ψf (‖x−y‖) for all x,y ∈ e with x ≥ y. similarly, by hypothesis (c2), ‖cx−cy‖≤ ψk(‖x−y‖) for all x,y ∈ e with x ≥ y. hence a and c are partially nonlinear d-lipschitz operators on e which further implies that they are also a partially continuous on e into itself. step iii: b is a partially continuous on e. let {xn}n∈n be a sequence in a chain c of e such that xn → x for all n ∈ n. then, by dominated convergence theorem, we have lim n→∞ bxn(t) = lim n→∞ q(t) + lim n→∞ ∫ t 0 v(t,s)g(s,xn(s)) ds = q(t) + ∫ t 0 v(t,s) [ lim n→∞ g(s,xn(s)) ] ds = q(t) + ∫ t 0 v(t,s)g(s,x(s)) ds = bx(t), for all t ∈ j. this shows that bxn converges monotonically to bx pointwise on j. next, we will show that {bxn}n∈n is an equicontinuous sequence of functions in e. let t1, t2 ∈ j be arbitrary with t1 < t2. then |bxn(t2) −bxn(t1)| ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 v(t1,s)g(s,xn(s)) ds− ∫ t1 0 v(t1,s)g(s,xn(s)) ds ∣∣∣∣ ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 v(t2,s)g(s,xn(s)) ds− ∫ t2 0 v(t1,s)g(s,xn(s)) ds ∣∣∣∣ + ∣∣∣∣ ∫ t2 0 v(t1,s)g(s,xn(s)) ds− ∫ t1 0 v(t1,s)g(s,xn(s)) ds ∣∣∣∣ ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 |v(t2,s) −v(t1,s)| |g(s,xn(s))|ds ∣∣∣∣ + ∣∣∣∣ ∫ t2 t1 |v(t1,s)| |g(s,xn(s))|ds ∣∣∣∣ 66 dhage ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣∣ ∫ t 0 |v(t2,s) −v(t1,s)|mg ds ∣∣∣∣∣ + v mg|t2 − t1|.(3.10) since the function q is continuous on compact interval j and v is continuous on compact set j × j, they are uniformly continuous there. therefore, from above inequality (3.10) it follows that |bxn(t2) −bxn(t1)|→ 0 as n →∞ uniformly for all n ∈ n. this shows that the convergence bxn → bx is uniform and hence b is partially continuous on e. step iv: b is a uniformly partially compact operator on e. let c be an arbitrary chain in e. we show that b(c) is a uniformly bounded and equicontinuous set in e. first we show that b(c) is uniformly bounded. let y ∈ b(c) be any element. then there is an element x ∈ c be such that y = bx. now, by hypothesis (b2), |y(t)| ≤ |q(t)| + ∫ t 0 v(t,s)|g(s,x(s))|ds ≤‖q‖ + v mg t ≤ r, for all t ∈ j. taking supremum over t, we obtain ‖y‖ = ‖bx‖≤ r for all y ∈b(c). hence, b(c) is a uniformly bounded subset of e. moreover, ‖b(c)‖ ≤ r for all chains c in e. hence, b is a uniformly partially bounded operator on e. next, we will show that b(c) is an equicontinuous set in e. let t1, t2 ∈ j be arbitrary with t1 < t2. then, for any y ∈b(c), one has |y(t2) −y(t1)| = |bx(t2) −bx(t1)| ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 v(t1,s)g(s,x(s)) ds− ∫ t1 0 v(t1,s)g(s,x(s)) ds ∣∣∣∣ ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 v(t2,s)g(s,x(s)) ds− ∫ t2 0 v(t1,s)g(s,x(s)) ds ∣∣∣∣ + ∣∣∣∣ ∫ t2 0 v(t1,s)g(s,x(s)) ds− ∫ t1 0 v(t1,s)g(s,x(s)) ds ∣∣∣∣ ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣ ∫ t2 0 |v(t2,s) −v(t1,s)| |g(s,x(s))|ds ∣∣∣∣ + ∣∣∣∣ ∫ t2 t1 |v(t1,s)| |g(s,x(s))|ds ∣∣∣∣ ≤ ∣∣q(t1) −q(t2)∣∣ + ∣∣∣∣∣ ∫ t 0 |v(t2,s) −v(t1,s)|mg ds ∣∣∣∣∣ + v mg|t2 − t1| → 0 as n →∞, uniformly for all y ∈ b(c). hence b(c) is an equicontinuous subset of e. now, b(c) is a uniformly bounded and equicontinuous set of functions in e, so it is generalized quadratic functional integral equations 67 compact. consequently, b is a uniformly partially compact operator on e into itself. step v: u satisfies the operator inequality u ≤aubu + cu. by hypothesis (c3), the qfie (1.1) has a lower solution u defined on j. then, we have (3.11) u(t) ≤ k(t,u(t)) + [ f(t,u(t)) ]( q(t) + ∫ t 0 v(t,s)g(s,u(s)) ds ) for all t ∈ j. from definitions of the operators a, b and c it follows that u(t) ≤ au(t)bu(t) + cu(t) for all t ∈ j. hence u ≤aubu + cu. step vi: the d-functions ψa and ψc satisfy the growth condition mψa(r) + ψc(r) < r, r > 0. finally, the d-function φ of the operator a satisfies the inequality given in hypothesis (d) of theorem 2.9, viz., mψa(r) + ψc(r) ≤ (‖q‖ + v mg t) ψf (r) + ψk(r) < r for all r > 0. thus a, b and c satisfy all the conditions of theorem 2.9 and we conclude that the operator equation axbx + cx = x has a solution. consequently the integral equation and the qfie (1.1) has a solution x∗ defined on j. furthermore, the sequence {xn}n∈n of successive approximations defined by (3.5) converges monotonically to x∗. this completes the proof. � the conclusion of theorems 3.3 also remains true if we replace the hypothesis (c3) with the following one: (c′3) the qfie (1.1) has an upper solution v ∈ c(j,r). the proof of theorem 3.3 under this new hypothesis is similar and can be obtained by closely observing the same arguments with appropriate modifications. example 3.4. given a closed and bounded interval j = [0, 1], consider the qfie, x(t) = 1 2 [ 2 + tan−1 x(t) ]( t t + 1 + ∫ t 0 1 t2 + 1 · [1 + tanh x(s)] 4 ds ) + 1 2 tan−1 x(t)(3.12) for t ∈ j. here, q(t) = t t + 1 and v(t,s) = 1 t2 + 1 which are continuous and ‖q‖ = 1 2 and v = 1. similarly, the functions k, f and g are defined by k(t,x) = 1 2 tan−1 x, f(t,x) = 1 2 [ 2 + tan−1 x(t) ] and g(t,x) = 1 + tanh x 4 . the function f satisfies the hypothesis (a3) with ψf (r) = 1 2 · r 1 + ξ2 for each 0 < ξ < r. to see this, we have 0 ≤ f(t,x) −f(t,y) ≤ 1 2 · 1 1 + ξ2 · (x−y) 68 dhage for all x,y ∈ r, x ≥ y and x > ξ > y. moreover, the function f is nonnegative and bounded on j ×r with bound mf = 2 and so the hypothesis (a2) is satisfied. again, since g is nonnegative and bounded on j×r by mg = 1 2 , the hypothesis (b3) holds. furthermore, g(t,x) is nondecreasing in x for all t ∈ j, and thus hypothesis (b4) is satisfied. similarly, the function k satisfies the hypothesis (c2) with ψk(r) = 1 2 · r 1 + ξ2 for every 0 < ξ < r. to see this, we have 0 ≤ k(t,x) −k(t,y) ≤ 1 2 · 1 1 + ξ2 · (x−y) for all x,y ∈ r, x ≥ y and x > ξ > y. moreover, the function k is bounded on j ×r with bound mk = π 4 and so the hypothesis (c1) is satisfied. also we have ( ‖q‖ + mg v t ) ψf (r) + ψk(r) ≤ r 1 + ξ2 < r for every r > 0. thus, condition (3.4) of theorem 3.3 is held. finally, the qfie (3.12) has a lower solution u(t) = 0 on j. thus all the hypotheses of theorem 3.3 are satisfied. hence we apply theorem 3.3 and conclude that the qfie (3.12) has a solution x∗ defined on j and the sequence {xn}n∈n defined by xn+1(t) = 1 2 [ 2 + tan−1 xn(t) ]( t t + 1 + ∫ t 0 1 t2 + 1 · [1 + tanh xn(s)] 4 ds ) + 1 2 tan−1 xn(t),(3.13) for all t ∈ j, where x0 = 0, converges monotonically to x∗. 4. conclusion finally, while concluding this paper we mention that the generalized quadratic integral equation considered here is of very simple nature for which we have illustrated the dhage iteration method to obtain the algorithms for the solutions under weaker partially lipschitz and compactness conditions. however, an analogous study could also be made for other complex quadratic integral equations as well as other different types of quadratic integral equations using similar method with appropriate modifications. some of the results along this line will be reported elsewhere. references [1] i. k. argyros, quadratic equations and applications to chandrasekhar’s and related equations, bull. austral. math. soc. 32 (1985), 275-292. [2] s. chandrasekher, radiative transfer, dover publications, new york, 1960. [3] k. deimling, nonlinear fuctional analysis, springer-verlag, berlin, 1985. [4] b.c. dhage, on α-condensing mappings in banach algebras, the mathematics student 63 (1994), 146-152. [5] b.c. dhage, fixed point theorems in ordered banach algebras and applications, panamer. math. j. 9(4) (1999), 93-102. [6] b.c. dhage, a fixed point theorem in banach algebras with applications to functional integral equations, kyungpook math. j. 44 (2004), 145-155. generalized quadratic functional integral equations 69 [7] b.c. dhage, a nonlinear alternative in banach algebras with applications to functional differential equations, nonlinear funct. anal. & appl. 8 (2004), 563–575. [8] b. c. dhage, periodic boundary value problems of first order carathéodory and discontinuous differential equations, nonlinear funct. anal. & appl. 13(2) (2008), 323-352. [9] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ appl. 5 (2013), 155-184. [10] b.c. dhage, partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45 (4) (2014), 397-426. [11] b.c. dhage, nonlinear d-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, malaya j. mat. 3(1)(2015), 62-85. [12] b.c. dhage, operator theoretic techniques in the theory of nonlinear hybrid differential equations, nonlinear anal. forum 20 (2015), 15-31. [13] s. heikkilä, v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. kasubai, gurukul colony, ahmedpur-413 515, dist: latur maharashtra, india international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 174-191 http://www.etamaths.com parabolic equations in musielak-orlicz-sobolev spaces m.l. ahmed oubeid1, a. benkirane1 and m. sidi el vally2,∗ abstract. we prove in this paper the existence of solutions of nonlinear parabolic problems in musielak-orlicz-sobolev spaces. an approximation and a trace results in inhomogeneous musielak-orlicz-sobolev spaces have also been provided. 1. introduction let ω a bounded open subset of rn and let q be the cylinder ω × (0,t) with some given t > 0. this paper is concerned with the existence of solutions for boundary value problems for quasi-linear parabolic equations of the form   ∂u ∂t + a(u) = f in q u(x,t) = 0 on ∂ω × (0,t) u(x, 0) = u0(x) in ω (1) where a is a leray-lions operator of the form: a(u) = − div (a(x,t,u,∇u)) + a0(x,t,u,∇u), with the coefficients a and a0 satisfying the classical leray-lions conditions. consider first the case where a and a0 have polynomial growth with respect to u and ∇u. therefore a is a bounded operator from lp(0,t,w 1,p0 (ω)), 1 < p < ∞,into its dual. in this setting, it is well known that problems of the form (1) were solved by lions [16], and brzis and browder [9] in the case where p ≥ 2, and by landes [14] and landes and mustonen [15] when 1 < p < 2. see also [6, 7] for related topics. in the case where a and a0 satisfy a more general growth with respect to u and ∇u (for example of exponential or logarithmic type), it is shown in [10] that the adequate space in which (1) can be studied is the inhomogeneous orlicz-sobolev space w 1,xlm (q), where the n-function m is related to the actual growth of a and a0. the solvability of (1) in this setting was proved by donaldson [10] and robert [18] when a is monotone, and by elmahi [11] and elmahi-meskine [12]. 2010 mathematics subject classification. 46e35, 35k15, 35k20, 35k60. key words and phrases. inhomogeneous musielak-orlicz-sobolev spaces; parabolic problems; musielak-orlicz function. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 174 parabolic equations 175 our purpose in this paper is to prove existence theorems for the problem (1) in the setting of inhomogeneous musielak-orlicz-sobolev spaces w 1,xlϕ(q) by applying some new approximation result in inhomogeneous musielak-orlicz-sobolev spaces (see theorem 1), as it is done in the setting of orlicz-sobolev spaces (see [12]), which allows us, on the one hand, to regularize a test function by smooth ones with converging time derivatives (and thus enlarge the set of test functions in order to cover the solution u and then get the energy equality), and, on the other hand, to prove a trace result (see lemma 3) which states that if u ∈ w 1,x0 lϕ(q)∩l 2(q) such that ∂u ∂t ∈ w−1,xlψ(q) + l2(q), then u ∈ c([0,t],l2(ω)), showing that the assumption u0 ∈ l2(ω) cannot be weakened. our result generalizes that of the elmahi-meskine in [12] to the case of inhomogeneous musielak-orlicz-sobolev spaces. let us point out that our result can be applied in the particular case when ϕ(x,t) = tp(x), in this case we use the notations lp(x)(ω) = lϕ(ω), and w m,p(x)(ω) = wmlϕ(ω). these spaces are called variable exponent lebesgue and sobolev spaces. for some classical and recent results on elliptic and parabolic problems in orliczsobolev spaces and a musielak-orlicz-sobolev spaces, we refer to [1, 2, 5, 10, 11, 12]. 2. preliminaries in this section we list briefly some definitions and facts about musielak-orliczsobolev spaces. standard reference is [17]. we also include the definition of inhomogeneous musielak-orlicz-sobolev spaces and some preliminaries lemmas to be used later. musielak-orlicz-sobolev spaces : let ω be an open subset of rn. a musielak-orlicz function ϕ is a real-valued function defined in ω×r+ such that : a): ϕ(x,t) is an n-function i.e. convex, nondecreasing, continuous, ϕ(x, 0) = 0, ϕ(x,t) > 0 for all t > 0 and lim t−→0 sup x∈ω ϕ(x,t) t = 0 lim t−→∞ inf x∈ω ϕ(x,t) t = 0. b): ϕ(., t) is a lebesgue measurable function now, let ϕx(t) = ϕ(x,t) and let ϕ −1 x be the non-negative reciprocal function with respect to t, i.e the function that satisfies ϕ−1x (ϕ(x,t)) = ϕ(x,φ −1 x ) = t. for any two musielak-orlicz functions ϕ and γ we introduce the following ordering : 176 oubeid, benkirane and vally c): if there exists two positives constants c and t such that for almost everywhere x ∈ ω : ϕ(x,t) ≤ γ(x,ct) for t ≥ t we write ϕ ≺ γ and we say that γ dominates ϕ globally if t = 0 and near infinity if t > 0. d): if for every positive constant c and almost everywhere x ∈ ω we have lim t→0 (sup x∈ω ϕ(x,ct) γ(x,t) ) = 0 or lim t→∞ (sup x∈ϕ ϕ(x,ct) γ(x,t) ) = 0 we write ϕ ≺≺ γ at 0 or near ∞ respectively, and we say that ϕ increases essentially more slowly than γ at 0 or near infinity respectively. in the sequel the measurability of a function u : ω 7→ r means the lebesgue measurability. we define the functional %ϕ,ω(u) = ∫ ω ϕ(x, |u(x)|)dx where u : ω 7→ r is a measurable function. the set kϕ(ω) = {u : ω → r mesurable /%ϕ,ω(u) < +∞} . is called the musielak-orlicz class (the generalized orlicz class). the musielak-orlicz space (the generalized orlicz spaces) lϕ(ω) is the vector space generated by kϕ(ω), that is, lϕ(ω) is the smallest linear space containing the set kϕ(ω). equivelently: lϕ(ω) = { u : ω → r mesurable /%ϕ,ω( |u(x)| λ ) < +∞, for some λ > 0 } let ψ(x,s) = sup t≥0 {st−ϕ(x,t)}, ψ is the musielak-orlicz function complementary to ( or conjugate of ) ϕ(x,t) in the sense of young with respect to the variable s. on the space lϕ(ω) we define the luxemburg norm: ||u||ϕ,ω = inf{λ > 0/ ∫ ω ϕ(x, |u(x)| λ )dx,≤ 1}. and the so-called orlicz norm : |||u|||ϕ,ω = sup ||v||ψ≤1 ∫ ω |u(x)v(x)|dx. where ψ is the musielak-orlicz function complementary to ϕ. these two norms are equivalent [17]. the closure in lϕ(ω) of the set of bounded measurable functions with compact support in ω is denoted by eϕ(ω). it is a separable space and eψ(ω) ∗ = lϕ(ω) [17]. parabolic equations 177 the following conditions are equivalent: e): eϕ(ω) = kϕ(ω) f ): kϕ(ω) = lϕ(ω) g): ϕ has the ∆2 property. we recall that ϕ has the ∆2 property if there exists k > 0 independent of x ∈ ω and a nonnegative function h , integrable in ω such that ϕ(x, 2t) ≤ kϕ(x,t) + h(x) for large values of t, or for all values of t, according to whether ω has finite measure or not. let us define the modular convergence: we say that a sequence of functions un ∈ lϕ(ω) is modular convergent to u ∈ lϕ(ω) if there exists a constant k > 0 such that lim n→∞ %ϕ,ω( un −u k ) = 0. for any fixed nonnegative integer m we define wmlϕ(ω) = {u ∈ lϕ(ω) : ∀|α| ≤ m dαu ∈ lϕ(ω)} where α = (α1,α2, ...,αn) with nonnegative integers αi; |α| = |α1|+|α2|+...+|αn| and dαu denote the distributional derivatives. the space wmlϕ(ω) is called the musielak-orlicz-sobolev space. now, the functional %ϕ,ω(u) = ∑ |α|≤m %ϕ,ω(d αu), for u ∈ wmlϕ(ω) is a convex modular. and ||u||mϕ,ω = inf{λ > 0 : %ϕ,ω( u λ ) ≤ 1} is a norm on wmlϕ(ω). the pair 〈wmlϕ(ω), ||u||mϕ,ω〉 is a banach space if ϕ satisfies the following condition : there exist a constant c > 0 such that inf x∈ω ϕ(x, 1) ≥ c, as in [17]. the space wmlϕ(ω) will always be identified to a σ(πlϕ, πeψ) closed subspace of the product ∏ |α|≤m lϕ(ω) = ∏ lϕ. let wm0 lϕ(ω) be the σ(πlϕ, πeψ) closure of d(ω) in w mlϕ(ω). let wmeϕ(ω) be the space of functions u such that u and its distribution derivatives up to order m lie in eϕ(ω), and let w m 0 eϕ(ω) be the (norm) closure of d(ω) in wmlϕ(ω). 178 oubeid, benkirane and vally the following spaces of distributions will also be used: w−mlψ(ω) = {f ∈ d′(ω); f = ∑ |α|≤m (−1)|α|dαfα with fα ∈ lψ(ω)} w−meψ(ω) = {f ∈ d′(ω); f = ∑ |α|≤m (−1)|α|dαfα with fα ∈ eψ(ω)} as we did for lϕ(ω), we say that a sequence of functions un ∈ wmlϕ(ω) is modular convergent to u ∈ wmlϕ(ω) if there exists a constant k > 0 such that lim n→∞ %ϕ,ω( un −u k ) = 0. from [17], for two complementary musielak-orlicz functions ϕ and ψ the following inequalities hold : h) : the young inequality : t.s ≤ ϕ(x,t) + ψ(x,s) for t,s ≥ 0, x ∈ ω i) : the hölder inequality :∣∣∣∣ ∫ ω u(x)v(x) dx ∣∣∣∣ ≤ ||u||ϕ,ω|||v|||ψ,ω. for all u ∈ lϕ(ω) and v ∈ lψ(ω). inhomogeneous musielak-orlicz-sobolev spaces : let ω an bounded open subset of rn and let q = ω×]0,t[ with some given t ¿ 0. let ϕ be a musielak function. for each α ∈ nn , denote by dαx the distributional derivative on q of order α with respect to the variable x ∈ rn. the inhomogeneous musielak-orlicz-sobolev spaces of order 1 are defined as follows. w 1,xlϕ(q) = {u ∈ lϕ(q) : ∀|α| ≤ 1 dαxu ∈ lϕ(q)} and w 1,xeϕ(q) = {u ∈ eϕ(q) : ∀|α| ≤ 1 dαxu ∈ eϕ(q)} the last space is a subspace of the first one, and both are banach spaces under the norm ‖u‖ = ∑ |α|≤m ‖dαxu‖ϕ,q. we can easily show that they form a complementary system when ω is a lipschitz domain [4]. these spaces are considered as subspaces of the product space πlϕ(q) which has (n + 1) copies. we shall also consider the weak topologies σ(πlϕ, πeψ) and σ(πlϕ, πlψ). if u ∈ w 1,xlϕ(q) then the function : t 7−→ u(t) = u(t, .) is defined on (0,t) with values in w 1lϕ(ω). if, further, u ∈ w 1,xeϕ(q) then this function is a w 1eϕ(ω)-valued and is strongly measurable. furthermore the following imbedding holds : w 1,xeϕ(q) ⊂ l1(0,t; w 1eϕ(ω)). the space w 1,xlϕ(q) is not in general separable, if u ∈ w 1,xlϕ(q), we can not conclude that the function u(t) is measurable on (0,t). however, the scalar function t 7→ ‖u(t)‖ϕ,ω is parabolic equations 179 in l1(0,t). the space w 1,x 0 eϕ(q) is defined as the (norm) closure in w 1,xeϕ(q) of d(q). we can easily show as in [4] that when ω a lipschitz domain then each element u of the closure of d(q) with respect of the weak * topology σ(πlϕ, πeψ) is limit, in w 1,xlϕ(q), of some subsequence (ui) ⊂ d(q) for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ 1,∫ q ϕ(x, ( dαxui −dαxu λ )) dxdt → 0 as i →∞, this implies that (ui) converges to u in w 1,xlϕ(q) for the weak topology σ(πlm, πlψ). consequently d(q) σ(πlϕ,πeψ) = d(q) σ(πlϕ,πlψ) , this space will be denoted by w 1,x 0 lψ(q). furthermore, w 1,x 0 eϕ(q) = w 1,x 0 lϕ(q)∩ πeϕ. poincaré’s inequality also holds in w 1,x 0 lϕ(q) i.e. there is a constant c > 0 such that for all u ∈ w 1,x0 lϕ(q) one has∑ |α|≤1 ‖dαxu‖ϕ,q ≤ c ∑ |α|=1 ‖dαxu‖ϕ,q. thus both sides of the last inequality are equivalent norms on w 1,x 0 lϕ(q). we have then the following complementary system( w 1,x 0 lϕ(q) f w 1,x 0 eϕ(q) f0 ) , f being the dual space of w 1,x 0 eϕ(q). it is also, except for an isomorphism, the quotient of πlψ by the polar set w 1,x 0 eϕ(q) ⊥, and will be denoted by f = w−1,xlψ(q) and it is shown that w−1,xlψ(q) = { f = ∑ |α|≤1 dαxfα : fα ∈ lψ(q) } . this space will be equipped with the usual quotient norm ‖f‖ = inf ∑ |α|≤1 ‖fα‖ψ,q where the inf is taken on all possible decompositions f = ∑ |α|≤1 dαxfα, fα ∈ lψ(q). the space f0 is then given by f0 = { f = ∑ |α|≤1 dαxfα : fα ∈ eψ(q) } and is denoted by f0 = w −1,xeψ(q). the following technical lemmas are important for the proof of our main result. lemma 1. if u ∈ w 1,10 (ω), then ||uσ −u||1,ω ≤ σ||∇u||1,ω, where uσ = u∗ρσ and where (ρσ) is a mollifier sequence in rn . 180 oubeid, benkirane and vally lemma 2. let ϕ be an musielak-orlicz function. let (un) be a bounded sequence in w 1,x 0 lϕ(q) ∩ l ∞(0,t; l1(ω)). if un(t) ⇀ u(t) weakly in l 1(ω) for almost every t ∈ [0,t], then un → u strongly in l1(q). proof.for each v ∈ w 1,x0 lϕ(q), denote vσ(x,t) = ∫ rn v(y,t)ρσ(x−y)dy, where v(y,t) = 0 if y /∈ ω and where (ρσ) is a mollifier sequence in rn . since un(t) ⇀ u(t) weakly in l 1(ω), we have unσ(x,t) → uσ(x,t) almost everywhere in q and unσ(t) → uσ(t) strongly in l1(ω) for almost every t ∈ [0,t],we have∫ ω |un(t) −uk(t)|dx ≤ ∫ ω |un(t) −unσ(t)|dx + ∫ ω |unσ(t) −ukσ(t)|dx + ∫ ω |ukσ(t) −uk(t)|dx ≤ σ( ∫ ω |∇un(t)|dx + ∫ ω |∇uk(t)|dx) + ||unσ(t) −ukσ(t)||1,ω integrating this over [0,t] yields ∫ q |un(t) −uk(t)|dxdt ≤ σ( ∫ q |∇un(t)|dxdt + ∫ q |∇uk(t)|dxdt) + ∫ t 0 ||unσ(t) −ukσ(t)||1,ωdt which gives, since lϕ(q) ⊂ l1(q) with continuous imbedding,∫ q |un(t) −uk(t)|dxdt ≤ σc1(||∇un||ϕ,q + ||∇uk||ϕ,q) + ∫ t 0 ||unσ(t) −ukσ(t)||1,ωdt where c1 and c2 are constants which do not depend on n and k such that ||v||1,q ≤ c1||v||ϕ,q for all v ∈ lϕ(q) and ||∇un||ϕ,q ≤ c2 for all n. consequently, we obtain : ∫ q |un(t) −uk(t)|dx ≤ 2c1c2σ + ∫ t 0 ||unσ(t) −ukσ(t)||1,ωdt. since ||unσ(t) −ukσ(t)||1,ω → 0 almost everywhere in [0,t] when n,k →∞ and ||unσ(t)||l1(ω) ≤ ||un(t)||l1(ω) ≤ c uniformly with respect to n and t ∈ [0,t],we deduce by using lebesgue’s theorem that∫ t 0 ||unσ(t) −ukσ(t)||1,ωdt → 0 as n,k →∞ implying, since σ is arbitrary, that ∫ q |un(t)−uk(t)|dxdt → 0 when n and k →∞. hence (un) is a cauchy sequence in l 1(q) and thus un → u strongly in l1(q). 3. approximation and trace results in this section, ω is a bounded lipschitz domain in rn and i is a subinterval of r ( possibly unbounded) and q = ω × i. it is easy to see that q also satisfies lipschitz domain. definition. we say that un → u in w−1,xlψ(q) + l2(q) for the modular convergence if we can write un = ∑ |α|≤1 dαxu α n + u 0 n and u = ∑ |α|≤1 dαxu α + u0 parabolic equations 181 with uαn → uα in lψ(q) for modular convergence for all |α| ≤ 1 and uαn → uα strongly in l2(q). we shall prove the following approximation theorem, which plays a fundamental role in the prove of our main results. theorem 1. if u ∈ w 1,xlϕ(q) ∩l2(q) (respectively w 1,x 0 lϕ(q) ∩l 2(q)) and ∂u ∂t ∈ w−1,xlψ(q) + l2(q), then there exists a sequence (vj) in d(q) (respectively d(i,d(ω)) ) such that vj → u in w 1,xlϕ(q) ∩l2(q) and ∂vj ∂t → ∂u ∂t in w−1,xlψ(q) + l 2(q) for the modular convergence. proof. let u ∈ w 1,xlϕ(q) ∩l2(q) such that ∂u∂t ∈ w −1,xlψ(q) + l 2(q) and let ε > 0 be given. writing ∂u ∂t = ∑ |α|≤1 d α xu α + u0, where uα ∈ lψ(q) for all |α| ≤ 1 and u0 ∈ l2(q), we will show that there exists λ > 0(depending only on u and n) and there exists v ∈ d(q) for which we can write ∂v ∂t = ∑ |α|≤1 d α xv α + v0 with vα,v0 ∈d(q) such that∫ q ϕ(x, dαxv −dαxu λ )dxdt ≤ ε,∀|α| ≤ 1,(2) ||v −u||l2(q) ≤ ε,(3) ||v0 −u0||l2(q) ≤ ε,(4) ∫ q ψ(x, vα −uα λ )dxdt ≤ ε,∀|α| ≤ 1,(5) the equation (2) flows from a slight adaptation of the arguments of [4], (3) and (4) flow also from classical approximation results. regrading the equation (5) it is enough to prove that d(q) is dense in lψ(q), for this end we use the fact that the log-hölder continuity(commutes with the complementarity) i.e :if ϕ is log-hölder the its complementary ψ also it is, and proceed as in [4] (with ϕ and ψ interchanged ) and using of course rn+1 instead of rn and q = ω × (0,t) instead of ω. these facts lead us to prove that ||kεf||ψ,q ≤ c||f||ψ,q,∀f ∈ lψ(q) (with kεf(x,t) = k −1 ε ∫ q kε(x − y)f(kεy,t)dy ,kε(x) = 1εn k( x ε ) and k(x) is a measurable function with support in the ball br = b(0,r) see [4]). and then we deduce that d(q) is dense in lψ(q) for the modular convergence which gives the desired conclusion. the case of w 1,x 0 lϕ(q) ∩l 2(q) is similar to the above arguments as in [4]. remark 1. if, in the statement of theorem 1, one consider ω×r instead of q, we have d(ω×r) is dense in u ∈ w 1,x0 lϕ(ω×r)∩l 2(ω×r) : ∂u ∂t ∈ w 1,x0 lψ(ω× r) + l2(ω × r) for the modular convergence. this follows trivially from the fact that d(r,d(ω)) ≡d(ω ×r). a first application of theorem 1 is the following trace result generalizing a classical result which states that if u belong to l2(a,b; h10 (ω)) and ∂u ∂t belongs to l2(a,b; h−1(ω)), then u is in c([a,b],l2(ω)). 182 oubeid, benkirane and vally lemma 3. let a < b ∈ r and let ω be a bounded lipschitz domain in rn . then {u ∈ w 1,x0 lϕ(ω×(a,b))∩l 2(ω×(a,b)) : ∂u ∂t ∈ w−1,xlψ(ω×(a,b))+l2(ω×(a,b))} is a subset of c([a,b],l2(ω)). proof. let u ∈ w 1,x0 lϕ(ω × (a,b)) ∩ l 2(ω × (a,b)) such that w−1,xlψ(ω × (a,b)) + l2(ω × (a,b)). after two consecutive reflection first with respect to t = b and then with respect to t = a, û(x,t) = u(x,t)χ(a,b) + u(x, 2b− t)χ(b,2b−a) on ω × (a, 2b−a) ũ(x,t) = û(x,t)χ(a,2b−a) + û(x, 2a− t)χ(3a−2b,a) on ω × (3a− 2b, 2b−a), we get a function ũ ∈ w 1,x0 lϕ(ω × (3a− 2b, 2b−a)) ∩l 2(ω × (3a− 2b, 2b−a)) such that ∂ũ ∂t ∈ w−1,xlψ(ω × (3a− 2b, 2b−a)) + l2(ω × (3a− 2b, 2b−a)). now, by letting a function η ∈d(r) with η = 1 on [a,b] and supp η ⊂ (3a−2b, 2b− a), setting u = ηũ, and using standard arguments (see [[8],lemme iv,remarque 10,p.158]), we have u = u on ω × (a,b) ũ ∈ w 1,x0 lϕ(ω × r) ∩ l 2(ω × r) ∂ũ ∂t ∈ w−1,xlψ(ω ×r) + l2(ω ×r). now let vj ∈d(ω ×r) be the sequence given by theorem 1 corresponding to u, that is, vj → u ∈ w 1,x 0 lϕ(ω×r)∩l 2(ω×r) and ∂vj ∂t → ∂u ∂t ∈ w−1,xlψ(ω×r)+l2(ω×r) for the modular convergence. we have∫ ω (vi(τ) −vj(τ))2dx = 2 ∫ ω ∫ τ −∞ (vi −vj)( ∂vi ∂t − ∂vj ∂t )dxdt → 0, as i,j →∞ from which one deduces that vj is a cauchy sequence in c(r,l2(ω)), and since the limit of vj in l 2(ω × r) is u, we have vj → u inc(r,l2(ω)). consequently, u ∈ c([a,b],l2(ω)). 4. existence result let ω be a bounded lipschitz domain in rn (n ≥ 2) , t > 0 and set q = ω × (0,t). throughout this section, we denote qτ = ω × (0,τ) for every τ ∈ [0,t]. let ϕ and γ two musielak-orlicz functions such that γ � ϕ. consider a second-order operator a : d(a) ⊂ w 1,xlϕ(q) → w−1,xlψ(q) of the form a(u) = −div(a(x,t,u,∇u)) + a0(x,t,u,∇u), where a : ω× [0,t]×r×rn → rna0 : ω× [0,t]×r×rn → r are carathodory functions, for almost every(x,t) ∈ ω × [0,t] and all s ∈ r,ξ 6= ξ∗ ∈ rn , parabolic equations 183 |a(x,t,s,ξ)| ≤ β(c(x,t) + ψ−1x γ(x,ϑ|s|) + ψ −1 x ϕ(x,ϑ|ξ|))(6) |a0(x,t,s,ξ)| ≤ β(c(x,t) + ψ−1x γ(x,ϑ|s|) + ψ −1 x ϕ(x,ϑ|ξ|))(7) (a(x,t,s,ξ) −a(x,t,s,ξ∗))(ξ − ξ∗) > 0(8) a(x,t,s,ξ)ξ + a0(x,t,s,ξ)s ≥ αϕ(x, |ξ| λ ) −d(x,t)(9) with c(x,t) ∈ eψ(q),c ≥ 0,d(x,t) ∈ l1(q),α,β,ϑ > 0. furthermore, let f ∈ w−1,xeψ(q)(10) consider then the following parabolic initial-boundary value problem.  ∂u ∂t + a(u) = f in q u(x,t) = 0 on ∂ω × (0,t) u(x, 0) = u0(x) in ω (11) where u0 is a given function in l 2(ω). we shall prove the following existence theorem. theorem 2. assume that (6)-(10) hold true. then there exists at least one distributional solution u ∈ d(a)∩w 1,x0 lϕ(q)∩c(([0,t],l 2(ω)) of (11) satisfying u(x, 0) = u0(x) for almost every x ∈ ω. furthermore, for all τ ∈ [0,t], we have 〈 ∂φ ∂t ,u〉qτ + [ ∫ ω u(t)φ(t)dx]τ0 + ∫ qτ [a(x,t,u,∇u)∇φ + a0(x,t,u,∇u)φ]dxdt = 〈f,φ〉qτ(12) for every φ ∈ w 1,x0 lϕ(q) ∩l 2(q) with ∂φ ∂t ∈ w−1,xlψ(q) + l2(q) and for φ = u, which gives the energy equality 1 2 ∫ ω u2(τ)dx− 1 2 ∫ ω u20dx + ∫ qτ [a(x,t,u,∇u)∇u + a0(x,t,u,∇u)u]dxdt = 〈f,u〉qτ remark 2. note that all the terms in (12) make sense. indeed, it easy to see that the first, third, and fourth terms are well defined. for the second one, we have by the trace result in lemma 3 that φ ∈ c([0,t],l2(ω)), from which we can easily show that the second term of (12) makes sense. note also that taking φ ∈d(q) in (12) shows that u is a distributional solution of (11). remark 3. if a0 ≡ 0 and a(x,t,s,ξ) ≡ a(x,t,ξ) does not depend on s, then the solution u is unique. ineed, let v ∈ w 1,x0 lφ(q) ∩l 2(q) be another solution of (11). using u − v as a test function in both equations corresponding to u and v with τ = t, we get 1 2 [ ∫ ω (u(t) −v(t))2dx]τ0 + ∫ qτ [a(x,t,u,∇u) −a(x,t,u,∇v)][∇u−∇v]dxdt = 0,(13) which implies that, by (8) and the fact that u(0) = v(0),∇u = ∇v. this gives, again by(13), u(t) = v(t) for almost every t ∈ [0,t] and hence u = v. remark 4. note that the trace result in lemma3 shows that the assumption u0 ∈ l2(ω) cannot be weakened in order to get a distributional solution for the cauchy-dirichlet problem (11). 184 oubeid, benkirane and vally remark 5. as in the elliptic case (see, [5]), γ is introduced instead of ϕ in (6) and (7) only to guarantee the boundedness in lψ(q) of ψ −1 x γ(x,ϑ|un|) and ψ−1x γ(x,ϑ|∇un|) whenever un is bounded in w 1,xlϕ(q). in the elliptic case, one usually takes γ = ϕ in the term ψ−1x γ(x,ϑ|un|) since un is bounded in a smaller space lθ(ω) with ϕ � θ; see [5]. however, in the parabolic case, we cannot conclude that there is the boundedness. nevertheless, we can take γ = ϕ if one of the following assertions holds true. (1) ϕ satisfies a 42 condition near infinity. (2) a is monotone, that is 〈a(u)−a(v),u−v〉≥ 0 for all u,v ∈ d(a)∩w 1,x0 lϕ(q). indeed, suppose first that ϕ satisfies a 42 condition.therefore (6) and (7),now with γ = ϕ, imply that, for all ε > 0, |a(x,t,s,ξ)| ≤ βε(cε(x,t) + ψ−1x ϕ(x,ε|s|) + ψ −1 x ϕ(x,ε|ξ|)), |a0(x,t,s,ξ)| ≤ βε(cε(x,t) + ψ−1x ϕ(x,ε|s|) + ψ −1 x ϕ(x,ε|ξ|)) which allows us to deduce the boundedness in lψ(q) of a(x,t,un,∇un) and a(x,t,un,∇un). assume now that a is monotone. we have, for all φ ∈ w 1,x0 eϕ(q),〈a(un) − a(φ),un − φ〉 ≥ 0. this gives 〈a(un),φ〉 ≤ 〈a(un),un〉− 〈a(φ),un − φ〉, which implies that, since un is bounded in w 1,x 0 lϕ(q) and 〈a(un),un〉 is bounded from above, thanks to the a priori estimates, 〈a(un),φ〉≤ cφ for all φ ∈ w 1,x 0 eϕ(q), where cφ is a constant depending on φ but not n. therefore, the banach-steinhauss theorem applies so that we can obtain the boundedness of a(un) in w −1,xlψ(q). proof of theorem 2. we will use a galerkin method due to landes and musten [15]. for the galerkin method, we choose a sequence {w1,w2, .....} in d(ω) such that ⋃∞ n=1 vn with vn = span{w1,w2, ....,wn} is dense in hm0 (ω) with m sufficiently large such that h m 0 (ω) is continuously embedded in c1(ω). for any v ∈ hm0 (ω), there exists a sequence (vk) ⊂ ⋃∞ n=1 vn such that vk → v in hm0 (ω) and c1(ω) too. we denote further vn = c([0,t],vn). it is easy to see that the closure of ⋃∞ n=1 vn with respect to the norm ||v||c1,0(q) = sup|α|≤1{|dαxv(x,t)| : (x,t) ∈ q} contains d(ω). this implies that, for any f ∈ w−1,xeψ(q), there exists a sequence (fk) ⊂ ⋃∞ n=1 vn such that fk → f strongly in w −1,xeψ(q). indeed,let ε > 0 be given . writing f = ∑ |α|≤1 d α xf α for all |α| ≤ 1, there exists gα ∈ d(q) such that ||fα −gα||ψ,q ≤ ε(2n+2) . moreover, by setting g = ∑ |α|≤1 d α xg α, we see that g ∈d(q), and so there exists φ ∈ ⋃∞ n=1 vn such that ||g−φ||∞,q ≤ ε (2meas(q)) . we deduce then that ||f −φ||w−1,xlψ(q) ≤ ∑ |α|≤1 ||fα −gα||ψ,q + ||g −φ||ψ,q for any u0 ∈ l2(ω), there is a sequence u0k ⊂ ⋃∞ n=1 vn such that u0k → u0 in l2(ω). parabolic equations 185 we divide the proof into three steps. step 1( a priori estimates): as in [15], by using [[14],lemma1], we find that there exists a gelerkin solution un of (13) in the following sense. un ∈vn, ∂un ∂t ∈ l1(0,t; vn),un(0) = u0n,(14) and for all φ ∈vn and all τ ∈ [0,t]∫ qτ ∂un ∂t φdxdt + ∫ qτ a(x,t,un,∇un)∇φdxdt + ∫ qτ a0(x,t,un,∇un)φdxdt = ∫ qτ fφdxdt. letting φ = un in (13) with τ = t and using (9) yields ||un||w1,x0 lϕ(q) ≤ c, ||un||l∞(0,t;l2(ω)) ≤ c,∫ q a(x,t,un,∇un)∇undxdt + ∫ q a0(x,t,un,∇un)undxdt ≤ c, where here and below c denotes a constant not depending on n. using (7) and the fact that γ � ϕ, it is easy to see that a0(x,t,un,∇un) is bounded in lψ(q). this implies that ∫ q a(x,t,un,∇un)∇undxdt ≤ c. to prove that a(x,t,un,∇un) is bounded in (lψ(q))n , let φ ∈ (eϕ(q))n , with ||φ||ϕ,q = 1. in view of (8), we have∫ q [a(x,t,un,∇un) −a(x,t,un,φ)][∇un −φ]dxdt ≥ 0, which gives∫ q [a(x,t,un,∇un)φdxdt ≤ ∫ q a(x,t,un,∇un)∇undxdt− ∫ q a(x,t,un,φ)[∇un −φ]dxdt, and since, thinks to (6), a(x,t,un,φ) is uniformly bounded in (lψ(q)) n , we deduce that ∫ q a(x,t,un,∇un)φdxdt ≤ c for all φ ∈ (eϕ(q))n, ||φ||ϕ,q = 1, which implies that, by the use of the dual norm of (lψ(q)) n,a(x,t,un,∇un) is bounded in lψ(q)) n . hence, for a subsequence and some h0 ∈ lψ(q),h ∈ (lψ(q)) n , un ⇀ u in w 1,x 0 lψ(q) for σ(πlϕ, πeψ) and weakly in l 2(q), a0(x,t,un,∇un) ⇀ h0,a(x,t,un,∇un) ⇀ h in lψ(q) for σ(πlψ, πeϕ). as in [15], we get un(t) ⇀ u(t) in l 1(ω) for almost every t ∈ [0,t], and then, by using lemma2, we deduce that un → u strongly in l1(q) and that, for some subsequence still denoted by un, un → u almost everywhere in q. 186 oubeid, benkirane and vally step 2( almost everywhere convergence of the gradients):for every τ ∈ (0,t] and for all φ ∈ c1([0,t],d(ω)), we get from (10)∫ qτ u ∂φ ∂t dxdt + [ ∫ ω u(t)φ(t)dx]τ0 + ∫ qτ h∇φ + ∫ qτ h0φdxdt = 〈f,φ〉qτ ,(15) and then, by choosing τ = t and taking φ to be arbitrary in d(q), we have ∂u ∂t ∈ w−1,xlψ(q). consider now the prolongation of u to ω × r as int the proof of lemma3. we see that there exists a sequence vk in d(ω ×r) such that vk → u in w 1,x 0 lϕ(q) ∩ l 2(q) and ∂vk ∂t → ∂u ∂t in w−1,xlψ(q) + l 2(q) for the modular convergence and so ( see the proof of lemma 3), vk → u in c([0,t],l2(ω)), which implies that, in particular, u ∈ c([0,t],l2(ω)). consequently, lim k→∞ ∫ q ∂vk ∂t (vk −u)dxdt = 0, which gives, by the use of the fact that ∂vk ∂t ∈ eψ(q), lim k→∞ lim n→∞ ∫ q ∂vk ∂t (vk −un)dxdt = 0. this implies that lim sup k→∞ lim sup n→∞ ∫ q ∂un ∂t (vk −un)dxdt ≤ 0. since∫ q ∂un ∂t (vk −un)dxdt = −12 [ ∫ ω (un(t) −vk(t))2dx]t0 + ∫ q ∂vk ∂t (vk −un)dxdt ≤ 1 2 ||u0n −vk(0)||2l2(ω) + ∫ q ∂vk ∂t (vk −un)dxdt and u0n → u0 in l2(ω) and vk(0) → u(0) in l2(ω)( note that u(0) = u0 since un(0) ⇀ u(0) in l 1(ω)). from (14) and (15), we have lim sup n→∞ ( ∫ ω [a(x,t,un∇un)∇un −h∇vk + a0(x,t,un,∇un)un −h0vk]dxdt) ≤ lim sup n→∞ 〈fn,un〉−〈f,vk〉 + lim sup n→∞ (− ∫ q ∂un ∂t undxdt) − ∫ q ∂vk ∂t udxdt + [ ∫ ω u(t)vk(t)dx] t 0 = 〈f,u−vk〉 + lim sup n→∞ ∫ q ∂un ∂t (vk −un)dxdt where we have used the fact that − ∫ q ∂vk ∂t (u)dxdt + [ ∫ ω u(t)vk(t)dx] t 0 = lim n→∞ (− ∫ q ∂vk ∂t (un)dxdt + [ ∫ ω un(t)vk(t)dx] t 0 ) = lim n→∞ ∫ q ∂un ∂t (vk)dxdt. parabolic equations 187 we deduce that lim sup k→∞ lim sup n→∞ ( ∫ ω [a(x,t,un,∇un)∇un −h∇vk + a0(x,t,un∇un)(un −vk)]dxdt) ≤ 0(16) since, as can be easily seen, limn→∞ ∫ ω (a(x,t,un,∇un)∇vk−∇vk+a0(x,t,un∇un)vk)dxdt) = ∫ q (h∇vk+h0vk)dxdt. in the sequel, and for any r > 0 and any k ∈ n, we denote by χrk χ r the characteristic functions of {(x,t) ∈ q : |∇vk| ≤ r} and {(x,t) ∈ q : |∇u| ≤ r}, respectivly. we also denote by ε(n,k,s) all quantities (possibly different) depending on l such that lim s→∞ lim k→∞ lim n→∞ ε(n,k,s) = 0, and this will be the order in which the parametres we use will tend to infinity, that is, first n, then k, and finally s. similarly, we will write onlyε(n), or ε(n,k),...to mean that the limits are only on the specified parametrers. we have, for any l > 0,∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt − ∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ s k]dxdt = ∫ {|un|≤l} a(x,t,un,∇vk.χsk)[∇un −∇vk.χ s k]dxdt + ∫ {|un|≤l} a(x,t,un,∇un)[∇vk.χsk −∇u.χ s]dxdt∫ {|un|≤l} a(x,t,un,∇u.χs)[∇u.χs −∇un]dxdt := i1 + i2 + i3. we shall go to the limit in all integrals ii (for i = 1, 2, 3) as first n, then k and finally s tend to infinity. starting with i1 and letting n →∞ yields i1 = ∫ {|u|≤l} a(x,t,u,∇vk.χsk)[∇u−∇vk.χ s k]dxdt + ε(n) since χ{|un|≤l}a(x,t,un,∇vk.χ s k) → χ{|u|≤l}a(x,t,u,∇vk.χ s k) strongly in (eψ(q)) n by (6) and the lebesgue theorem while ∇un →∇u in (lϕ(q))n . this implies, by letting k →∞ in the integral of last side, that i1 = ∫ {|u|≤l}∩{|∇u|>s} a(x,t,u, 0)∇udxdt + ε(n,k), from which we get i1 = ε(n,k,s), since the first term of the last side goes to 0 as s →∞. for i2, we have, by letting n →∞, i2 = ∫ {|u|≤l}∩{|∇u|>s} h[∇vk.χsk −∇u.χ s]dxdt + ε(n), and so, by letting k → ∞ in the integral of last side and using the fact that ∇vkχsk →∇uχ s strongly in (eϕ(q)) n , we deduce that i2 = ε(n,k) . 188 oubeid, benkirane and vally for the third term i3, we have, by letting n →∞, i3 = − ∫ {|u|≤l}∩{|∇u|>s} a(x,t,u, 0)∇udxdt + ε(n,k), and since the first term of the last side tends to zero as s → ∞ , we obtain i3 = ε(n,k,s). we have then proved that∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt = ∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ s k]dxdt + ε(n,k,s). for all s ≥ r > 0 and all l ≥ δ > 0 , we have 0 ≤ ∫ {|un|≤δ}∩{|∇u|≤r} [a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt(17) ≤ ∫ {|un|≤l}∩{|∇u|≤s} [a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt ≤ ∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇u.χs)][∇un −∇u.χs]dxdt = ∫ {|un|≤l} [a(x,t,un,∇un) −a(x,t,un,∇vk.χsk)][∇un −∇vk.χ s k]dxdt + ε(n,k,s) = − ∫ {|un|≤l} a(x,t,un,∇vk.χsk)[∇un −∇vk.χ s k]dxdt + ∫ q [a(x,t,un,∇un)(∇un −∇vk) + a0(x,t,un,∇un)(un −vk)]dxdt −( ∫ {|un|≤l} a(x,t,un,∇un)[∇un −∇vk]dxdt + ∫ q a0(x,t,un,∇un)(un −vk)]dxdt) + ∫ {|un|≤l}∩{|vk|>s} a(x,t,un,∇un)∇vkdxdt + ε(n,k,s) := j1 + j2 + j3 + j4 + ε(n,k,s). we shall go to the limit sup first over n and next over k and finally over s in all integrals of the last side. first of all, note that j1 = −i1 = ε(n,k,s) and that, thanks to (16), lim sup k→∞ lim sup k→∞ j2 ≤ 0. the third integral reads j3 = − ∫ {|un|>l} [a(x,t,un,∇un)(∇un −∇vk) + a0(x,t,un,∇un)(un −vk)]dxdt − ∫ {|un|≤l} a0(x,t,un,∇un)(un −vk)dxdt, and, by using (9), j3 ≤ ∫ {|un|>l} [a(x,t,un,∇un)∇vk + a0(x,t,un,∇un)vk]dxdt − ∫ {|un|>l} d(x,t)dxdt− ∫ {|un|≤l} a0(x,t,un,∇un)(un −vk)dxdt, parabolic equations 189 which gives lim supn→∞j3 ≤ ∫ {|u|≥l}(h∇vk + h0vk)dxdt− ∫ {|u|≥l}d(x,t)dxdt− ∫ {|u|≤l}h0(u− vk)dxdt, where we have used the strong convergence of χ{|un|>l}|∇vk| and χ{|un|>l}|vk| and χ{|un|≤l}un in eϕ(q) as n →∞. this implies that lim sup k→∞ lim sup n→∞ j3 ≤ ∫ {|u|≥l} (h∇u + h0u)dxdt− ∫ {|u|≥l} d(x,t)dxdt since vk → u in w 1,x 0 lφ(q) for the modular convergence. for j4, we have lim n→∞ j4 = ∫ {|u|≤l}∩{|∇vk|>s} h∇vkdxdt since χ{|un|≤l}∩{|∇vk|>s}∇vk → χ{|u|≤l}∩{|∇vk|>s}∇vk strongly in (eϕ(q)) n as n →∞. this implies that lim k→∞ lim n→∞ j4 = ∫ {|u|≤l}∩{|u|≥s} h∇udxdt ≤ ∫ {|u|≥s} |h∇u|dxdt and thus lim sup s→∞ lim k→∞ lim n →∞j4 ≤ 0. combining these estimates with (17) and passing to the limit sup first over n, then over k, and finally over s, we deduce that 0 ≤ lim sup n→∞ ∫ {|un|≤δ,|∇u|≤r} [a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt ≤ ∫ {|un|≥l} (h∇u + h0u−d(x,t))dxdt, in which we can let l →∞ to get limn→∞ ∫ {|un|≤δ,|∇u|≤r} [a(x,t,un,∇un) −a(x,t,un,∇u)][∇un −∇u]dxdt = 0, and thus, as the elliptic case (see [1]), we deduce that, for a subsequence still denoted by un, ∇un →∇u a.e in q, and so h = a(x,t,u,∇u) and h0 = a0(x,t,u,∇u). therefore, we get for every τ ∈ (0,t] and for all φ ∈ c1([o,τ],d(ω)), − ∫ qτ u ∂φ ∂t dxdt + [ ∫ ω u(t)φ(t)dx]τ0 + ∫ qτ [a(x,t,u,∇u)∇φ + a0(x,t,u,∇u)φ]dxdt = 〈f,φ〉qτ .(18) step 3 ( passage to the limit): let v ∈ w 1,x0 lϕ(q) ∩ l 2(q) such that ∂v ∂t in w−1,xlψ(q) + l 2(q). there exists a prolongation v of v such that (see proof of lemma 3) v = v on q,v ∈ w 1,x0 lϕ(ω ×r) ∩l 2(ω ×r), ∂v ∂t ∈ w−1,xlψ(ω ×r) + l2(ω ×r).(19) by theorem 1 (see also remark 1),there exists a sequence wj ⊂ d(ω × r) such that wj → v ∈ w 1,x 0 lϕ(ω ×r) ∩l 2(ω ×r) and ∂wj ∂t → ∂v ∂t ∈ w−1,xlψ(ω ×r) + l2(ω ×r)(20) 190 oubeid, benkirane and vally for the modular convergence. letting φ = wjχ(0,τ) ( which belongs to c 1([0,τ],d(ω))) as a test function in (18), we get, for every τ ∈ (0,t], − ∫ qτ u ∂wj ∂t dxdt + [ ∫ ω u(t)wj(t)dx] τ 0 + ∫ qτ [a(x,t,u,∇u)∇wj + a0(x,t,u,∇u)wj]dxdt = 〈f,wj〉qτ .(21) we shall now go to the limit as j →∞ in all terms of (21). in view of (20), there is no problem with passing to the limit in the first and last three terms. for the second one, observe that, as in the proof of lemma 3, we have wj → v in c([0,t],l2(ω)), and since, for all t ∈ [0,t],u(t) is in l2(ω), we have, for every t ∈ [0,t],∫ ω u(t)wj(t)dx → ∫ ω u(t)v(t)dx. letting j →∞ in both sides of (21) −〈 ∂v ∂t ,u〉qτ + [ ∫ ω u(t)v(t)dx]τ0 + ∫ qτ [a(x,t,u,∇u)∇v + a0(x,t,u,∇u)v]dxdt = 〈f,v〉qτ . to prove the energy equality, it suffices to take v = u in the above equality (note that this is possible since u ∈ w 1,x0 lϕ(q) ∩l 2(q) and ∂u ∂t ∈ w−1,xlψ(q)). this gives −〈 ∂u ∂t ,u〉qτ + [ ∫ ω u2(t)dx]τ0 + ∫ qτ [a(x,t,u,∇u)∇u + a0(x,t,u,∇u)u]dxdt = 〈f,u〉qτ , and since, as can by easily seen, 〈 ∂u ∂t ,u〉qτ = 1 2 ( ∫ ω u2(τ)dx− ∫ ω u20dx), we get the desired equality. this completes the proof of theorem 2. references [1] m.l.ahmed oubeid,a. benkirane and m. ould mohamedhen val; nonlinear elliptic equations involiving measur data in musielak-orlicz-sobolev spaces, j.a. diff. eq and app volume 4, number 1, pp. 43-57 (2013) [2] a. benkirane and m. ould mohamedhen val; an existence result for nonlinear elliptic equations in musielak-orlicz-sobolev spaces, bull. belg. math. soc. simon stevin volume 20, number 1 (2013), 57-75. [3] a. benkirane, j. douieb, m. ould mohamedhen val; an approximation theorem in musielakorlicz-sobolev spaces, comment. math. (prace matem.) vol. 51, no. 1 (2011), 109-120. [4] a. benkirane and m. ould mohamedhen val; some approximation properties in musielakorlicz-sobolev spaces, thai.j. math. vol. 10, no. 2 (2012), 371-381. [5] a. benkirane and m. ould mohamedhen val; variational inequalities in musielak-orliczsobolev spaces, to appear in bull. belg. math. soc. simon stevin. [6] l. boccardo and f. murat; strongly nonlinear cauchy problems with gradient dependent lower order nonlinearity, pitman research notes in mathematics 208 (pitman, 1989) 247-254. [7] l. boccardo and f. murat; almost everywhere convergence of the gradients, nonlinear anal. 19 (1992) 581-597. [8] h.brzis; analyse fonctionnelle,thorie et applications,3rd end, (masson, paris,1992). [9] h.brzis and f.e.browder; strongly nonlinear parabolic initial boundary value problems, proc.nat.acad.sci.u.s.a.76(1979)38-40. [10] t.donaldson; inhomogeneous orlicz-sobolev spaces and nonlinrar parabolic initial boundary value problems, j.differential equations 16(1974)201-256. [11] a.elmahi; strongly nonlinear parabolic initial-boundary value problems in orlicz spaces,electron. j. differential equations.09 (2002) 203-220. parabolic equations 191 [12] a.elmahi, d.meskine; parabolic equations in orlicz spaces,j.london math.soc.72 (2)(2005) 410-428 [13] j.p. gossez, somme approximation properties in orlicz sobolev spaces, studia math. 74 (1982), 17-24. [14] r.landes and v.mustonen, on the existence of weak solutions for quasilinear parabolic initial-boundary value problems , proc.roy.soc. edinburgh sect. a 89 (1981), 217-237. [15] r.landes and v.mustonen, a strongly nonlinear parabolic initial-boundary value problem, ark. mat. 25 (1987), 29-40. [16] j.l.lions; quelques mthodes de rsolution des problmes aux limites non linaires ;(gauthiersvillars,1969). [17] j. musielak; modular spaces and orlicz spaces ;lecture notes in math. 1034 (1983). [18] j. robert; inequations variationnelles paraboliques fortement non lineaires ;(gauthiersvillars,1969). 1département de mathématiques et informatique, faculté des sciences dhar-mahraz, b. p. 1796 atlas fès, maroc 2department of mathematics, faculty of science, king khalid university,abha 61413, kingdom of saudi arabia ∗corresponding author international journal of analysis and applications volume 19, number 1 (2021), 138-152 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-138 some results of rational contraction mapping via extended cf -simulation function in metric-like space with application habes alsamir∗ finance and banking department, college of business administration, dar aluloom university, riyadh, saudi arabia ∗corresponding author: habes@dau.edu.sa; h.alsamer@gmail.com abstract. in this paper, we introduce a new contraction via cf -simulation function and prove the existence and the uniqueness of our mapping defined on a metric-like space. our work generalizes and extends some theorems in the literature. an example and application of second type of fredholm integral equation are given. 1. introduction many problems in mathematics and other sciences such as physics, chemistry, computer science and engineering resolved by using fixed point theory. the banach contraction mapping principle [1] is one of the essential results in fixed point theory. thus, a huge number of mathematical researchers generalized and extended it in a lot of spaces that appeared after 1922. one of the most spaces introduced in this decade is metric-like space that was presented by amini-harandi [11] in 2012. after that, a lot of researchers proved (common) fixed point results by using different types of contractive conditions in the setting of metric-like spaces, for example see( [2], [3], [6][10]). definition 1.1. [11] let χ is a nonempty set. a function σ : χ × χ → [0,∞) is said to be a metric like space (or dislocated metric) on χ if for any α,ν,ξ ∈ χ, the following conditions hold: received november 13th, 2020; accepted december 10th, 2020; published january 7th, 2021. 2010 mathematics subject classification. 54h25, 47h10. key words and phrases. an extended cf -simulation function, fixed point, metric-like spaces. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 138 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-138 int. j. anal. appl. 19 (1) (2021) 139 (σ1) σ(α,ξ) = 0 ⇒ α = ξ, (σ2) σ(α,ξ) = σ(ξ,α), (σ3) σ(α,ξ) ≤ σ(α,w) + σ(w,ξ). the pair (χ,σ) is called a metric-like space. let (χ,σ) be a metric-like space. a sequence {αn} in χ, if and only if lim n→∞ σ(αn,α) = σ(α,α) a sequence {αn} is called σ-cauchy if the limit limn,m→∞σ(αn,αm) exists and is finite. the metric-like space (χ,σ) is called complete if for each σ-cauchy sequence {αn}, there is some α ∈ χ such that lim n→∞ σ(αn,α) = σ(α,α) = lim n,m→∞ σ(αn,αm). lemma 1.1. [12] let (χ,σ) be a metric-like space. let {αn} be a sequence in χ such that αn → u where α ∈ χ and σ(α,α) = 0. then, for all ξ ∈ χ, we have limn→∞σ(αn,ξ) = σ(α,ξ). definition 1.2. [24] let χ be a nonempty set. a function ß : χ×χ → [0,∞) is a partial metric if for all α,ξ,w ∈ χ, the following conditions are satisfied: (1) α = ξ ⇔ ß(α,α) = ß(α,ξ) = ß(ξ,ξ), (2) ß(α,α) ≤ ß(α,ξ), (3) ß(α,ξ) = ß(ξ,ξ), (4) ß(α,ξ) ≤ ß(α,w) + ß(w,ξ) − ß(w,w). in this case, the pair (χ, ß) is called a partial metric space. it is known that each partial metric is a metric-like, but the converse is not true in general. example 1.1. let χ = {0, 1} and σ : χ×χ → [0,∞) defined by σ(0, 0) = 2, σ(u,v) = 1 if (α,ξ) 6= (0, 0) then, pair (χ,σ) is a metric-like space. note that σ is not a partial metric on χ because σ(0, 0) � σ(1, 0). remark 1.1. let χ = {0, 1}, and σ(α,ξ) = 1 for each α,ξ ∈ χ and αn = 1 for each n ∈ n. then it is easy to see that αn → 0 and αn → 1 and so in metric-like spaces the limit of a convergent sequence is not necessarily unique. definition 1.3. [27] a function ζ : [0,∞) × [0,∞) → r is called an extended simulation function if ζ satisfies the following conditions: (ζ1) ζ(α,ξ) < α− ξ for all α,ξ > 0, int. j. anal. appl. 19 (1) (2021) 140 (ζ2) if {αn} and {ξn} are sequences in (0,∞) such that limn→∞αn = limn→∞ξn = ` ∈ (0,∞) > 0, and αn > l, n ∈ n, then lim n→∞ sup ζ(αn,ξn) < 0. (ζ2)let {αn} be a sequences in (0,∞) such that lim n→∞ αn = ` ∈ [0,∞) > 0, ζ(αn, l) ≥ 0, n ∈ n, then l = 0. many researchers have used the above notation to prove some fixed and common fixed point results, see for example ( [13], [23]). in 2014, ansari [26] introduced the concept of c-class functions as follows: definition 1.4. [26] a mapping f : [0,∞)2 → r is called a c -class function if for any α,ξ ∈ [0,∞), the following conditions hold: (i) f(α,ξ) ≤ α, (ii) f(α,ξ) = α implies that either α = 0 or ξ = 0. as examples of c -class functions, we state: (1) f(α,ξ) = α− ξ for all α,α ∈ [0,∞); (2) f(α,ξ) = lα for all α,ξ ∈ [0,∞) where 0<l<1; definition 1.5. [5] a mapping f : r+ ×r+ → r has the property cf , if there exists cf ≥ 0 such that (fi) f(α,ξ) > cf ⇒ α > ξ, (fii) f(ξ,ξ) ≤ cf for all ξ ∈ [0,∞). the following example of c -class functions that have property cf (1) f1(α,ξ) = α 1+ξ , cf = 1, 2. (2) f2(α,ξ) = α− ξ, cf = r, r ∈ [0,∞). liu [5] linked between a c-class function and cf -simulation function and presented it as the following: definition 1.6. [5] a mapping ζ : r+ × r+ → r is cf -simulation function if satisfying the following conditions: (ζi) ζ(0, 0) = 0 int. j. anal. appl. 19 (1) (2021) 141 (ζii) ζ(α,ξ) < f(α,ξ), where α,ξ > 0, with property cf (ζiii) if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn > 0, and αn < ξn, then lim sup n→∞ ζ(αn,ξn) < cf , example 1.2. [5] let ζ : r+ ×r+ → r be a function defined by ζ(α,ξ) = mf(α,ξ), where α,ξ ∈ [0,∞) and m ∈ r be such that m < 1 and for each α,ξ ∈ [0,∞). considering cf = 1,ζ is a cf -simulation function. choosing f(α,ξ) = α 1+ξ , we get ζ(α,ξ) = mα 1+ξ is also a cf -simulation function with cf = 1. chanda et. al. [25] brought the concept of cf -extended simulation function as the following: definition 1.7. [25] a mapping ζ : r+ ×r+ → r an extended cf -simulation function if satisfying the following conditions: (ζ1) ζ(α,ξ) < f(α,ξ), where α,ξ > 0, with property cf (ζ2) if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn = l, where l ∈ (0,∞) and ξn > l for all n ∈ n, then lim sup n→∞ ζ(αn,ξn) < cf , (ζ3) if {αn} be a sequence (0,∞), such that lim n→∞ αn = l ∈ [0,∞), ζ(αn, l) ≥ cf ⇒ l = 0. example 1.3. [25] let ζ : r+ ×r+ → r be a function defined by ζ(α,ξ) = 3 4 α− ξ, where α,ξ ∈ [0,∞). considering f(α,ξ) = α− ξ with cf = 1, for all α,ξ ∈ [0,∞), we assured that (ζ1)is proved. now if {αn},{ξn} are sequences in (0,∞) such that lim n→∞ αn = lim n→∞ ξn = l > 0 and ξn > l for all n ∈ n, we obtain lim sup n→∞ ζ(αn,ξn) = lim sup n→∞ [ 3 4 αn − ξn] = −l 4 < cf = 1. int. j. anal. appl. 19 (1) (2021) 142 thus ζ(α,ξ) = 3 4 α− ξ meets (ζ2). now, we check for (ζ3). we choose a sequence {ξn} in (0,∞) with lim n→∞ αn = l ≥ 0 for each n ∈ n such that ζ(αn, l) ≥ cf = 1 = 3 4 l−αn ≥ 1 ⇒ αn ≤ 3 4 l− 1. letting n →∞, we have l ≤ 3 4 l− 1 ⇒ 1 4 l ≤ −1 ⇒ l = −4 which is a contradiction to l ≥ 0. hence ζ(α,ξ) = 3 4 α− ξ satisfies all conditions od definition 1.3 and so is an extended cf -simulation function. a functional ϕ : [0,∞) → [0,∞) is lower semicontinuous at a point α0 ∈ χ if (1) ϕ(α0) ≤ lim infα→α0 ϕ(α), (2) ϕ(α) = 0 ⇔ α = 0, lemma 1.2. let (χ,σ) be a metric like space and let {wn} be a sequence in χ such that limn→∞σ(αn,αn+1) = 0. if limn,m→∞σ(αn,αm) 6= 0, then there exist � > 0 and two sequences {nl} and {ml} of positive integers with nl > ml > l such that following three sequences σ(α2nl,α2ml ), σ(α2nl−1,α2ml ), and σ(α2nl,α2ml+1) converge to � + when l →∞. in this article, motivated by the idea of an extended cf -simulation function due to chanda et al 1.3, we prove the existence and the uniqueness of a common fixed point for two mappings satisfying a contraction which involve a lower semicontinuous function is established. an example and application are given to support the obtained work. 2. main result theorem 2.1. assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). suppose that there exist an extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.1) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) ≥ cf int. j. anal. appl. 19 (1) (2021) 143 for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ) + ϕ(α) + ϕ(ξ),σ(α,pα) + ϕ(α) + ϕ(pα),σ(ξ,qξ) + ϕ(ξ) + ϕ(qξ), σ(α,qξ) + ϕ(α) + ϕ(qξ) + σ(pα,ξ) + ϕ(pα) + ϕ(ξ) 4 }.(2.2) then, (p,q) has a common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. proof. let α0 ∈ χ, and define a sequence {αn} by α2n+1 = pα2n and α2n+2 = qα2n+1 for all n ≥ 0. if α2n = α2n+1 for some n, then the proof is done. therefore, if α2n 6= α2n+1 and σ(α2n,α2n+1) = 0, then by (σ1), which is a discrepancy. applying (2.1), we obtain cf ≤ ζ(σ(pα2n,qα2n+1) + ϕ(pα2n) + qϕ(α2n+1),m(α2n,α2n+1)) = ζ(σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2),m(α2n,α2n+1)).(2.3) by applying (ζ2) in (2.3), we obtain cf < f(m(α2n,α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)), which implies (2.4) σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2) < m(α2n,α2n+1) where m(α2n,α2n+1) = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n,pα2n) + ϕ(α2n) + ϕ(pα2n),σ(α2n+1,qα2n+1) +ϕ(α2n+1) + ϕ(qα2n+1), 1 4 (σ(α2n,qα2n+1) + ϕ(α2n) + ϕ(qα2n+1) + σ(pα2n,α2n+1) + ϕ(pα2n) +ϕ(α2n+1))} = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) +ϕ(α2n+1) + ϕ(α2n+2),σ(α2n+1,α2n+2) + ϕ(uα2n+1) + ϕ(α2n+2), 1 4 (σ(α2n,α2n+2) + ϕ(α2n) +ϕ(α2n+2) + σ(α2n+1,α2n+1) + ϕ(α2n+1) + ϕ(α2n+1))} ≤ max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2), 1 4 (σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1) + σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2))} = max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)}.(2.5) int. j. anal. appl. 19 (1) (2021) 144 thus, from (2.4), we get σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2) < max{σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1),σ(α2n+1,α2n+2) + ϕ(α2n+1) + ϕ(α2n+2)}.(2.6) by a similar process, one can also get the following σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1) < max{σ(α2n−1,α2n) + ϕ(α2n−1) + ϕ(α2n),σ(α2n,α2n+1) + ϕ(α2n) + ϕ(α2n+1)}.(2.7) therefore, from (2.6) and (2.7), (2.8) σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1) < max{σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn),σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)}, for all n ∈ n. necessarily, we obtain (2.9) max{σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn),σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)} = σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn), for all n ∈ n. consequently, for all n ∈ n, we have σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1) < σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn) therefore, we find that {σ(αn,αn+1) + +ϕ(αn) + ϕ(αn+1)} is a decreasing sequence. so, there exists l ≥ 0 such that lim n→∞ (σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)) = l. assume that l > 0. then, we deal with {αn} and {ξn} with same limit where αn = σ(pαn,pαn+1) > 0 and αn = σ(qαn,qαn+1) > 0 for all n ∈ n and αn > l for all n ∈ n. lastly we get from condition (ζ2), cf ≤ ζ(σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1),σ(αn−1,αn) + ϕ(αn−1) + ϕ(αn) < cf which is a contradiction. then, we conclude that l = 0 and lim n→∞ (σ(αn,αn+1) + ϕ(αn) + ϕ(αn+1)) = 0, int. j. anal. appl. 19 (1) (2021) 145 which implies (2.10) lim n→∞ σ(αn,αn+1) = 0, and (2.11) lim n→∞ ϕ(αn) = 0. now, we will prove that {αn} is cauchy sequence. after that, we will prove lim n→∞ σ(αn,αm) = 0. assume that lim n→∞ σ(αn,αm) 6= 0. by contradiction.thus, that is l = 0. there exists � > 0 and two sequences {αny} and {αmy} of {αn} with ny > my ≥ l such that for every y with the (smallest number satisfying the condition below) (2.12) σ(αny,αmy ) ≥ �. and (2.13) σ(αny−1,αmy−1) < �. by using (2.12) and (2.13) and the triangular inequality, we get � ≤ σ(αny,αmy ) ≥ σ(αny,αmy−1) + σ(αmy−1,αmy ) < σ(αmy−1,αmy ) + �. by (??) (2.14) lim y→∞ σ(αny,αmy ) = lim y→∞ σ(αny−1,αmy−1) = �. we also have (2.15) σ(αny,αmy−1) −σ(αny,αny−1) −σ(αmy,αmy−1) ≤ σ(αny−1,αmy ), and (2.16) σ(αny−1,αmy ) ≤ σ(αny−1,αny ) + σ(αny,αmy ). letting y →∞ in (2.15) and (2.16) and by using (2.10) and (2.14), we obtain (2.17) lim y→∞ σ(αny−1,αmy ) = �. again, using the triangular inequality, we have (2.18) | σ(αny−1,αmy ) −σ(αny−1,αmy−1) | σ(αmy−1,αmy ). int. j. anal. appl. 19 (1) (2021) 146 letting y →∞ in (2.18)and by using (2.17), we get (2.19) lim y→∞ σ(αny−1,αmy−1) = �. from (2.34), we have m(αny−1,αmy−1) = max{σ(αny−1,αmy−1) + ϕ(αny−1) + ϕ(αmy−1),σ(αny−1,pαny−1) + ϕ(αmy−1) +ϕ(qαmy−1), 1 4 (σ(αny−1,qαmy−1) + ϕ(αny−1) + ϕ(qαmy−1) + σ(pαny−1,αmy−1) +ϕ(pαny−1) + ϕ(αmy−1))} = max{σ(αny−1,αmy−1) + ϕ(αny−1) + ϕ(αmy−1),σ(αny−1,αny ) + ϕ(αn−1) + ϕ(αn), σ(αmy−1,αmy ) + ϕ(αmy−1) + ϕ(αmy ), 1 4 (σ(αny−1,αmy ) + ϕ(αny−1) + ϕ(αmy ) + σ(αny,αmy−1) + ϕ(αny ) + ϕ(αmy−1))}.(2.20) letting y →∞ in (2.20)and by (2.10),(2.11),(2.14),(2.17) and (2.19), it follows that (2.21) lim y→∞ σ(αny,αmy ) = lim y→∞ m(αny−1,αmy−1) = �. applying (theta2), we get cf ≤ ζ(σ(αny,αmy ) + ϕ(αn) + ϕ(αm),m(αny−1,αmy−1)) < cf which is a contradiction. hence αn is a cauchy sequence and hence limn→∞αn = k ∈ χ exists because χ is complete. since ϕ is lower semicontinuous, ϕ(k) ≤ lim inf n→∞ ϕ(αn) ≤ lim n→∞ ϕ(αn), which implies (2.22) ϕ(k) = 0. we claim that k is a common fixed point of p and q. put α = αn and ξ = k in (2.33) for all n, and we obtain (2.23) ζ(σ(pαn,qk) + ϕ(pαn) + ϕ(qk),m(αn,k)) ≥ cf m(αn,k) = max{σ(αn,k) + ϕ(αn) + ϕ(k),σ(αn,pun) + ϕ(αn) + ϕ(pαn),σ(k,qk) + ϕ(k) + ϕ(qk), 1 4 (σ(αn,qk) + ϕ(αn) + ϕ(qk) + σ(pαn,k) + ϕ(pαn) + ϕ(k))} = max{σ(αn,k) + ϕ(αn) + ϕ(k),σ(αn,un+1) + ϕ(αn) + ϕ(αn+1),σ(k,qk) + ϕ(k) + ϕ(qk), 1 4 (σ(αn,qk) + ϕ(αn) + ϕ(qk) + σ(αn+1,k) + ϕ(αn+1) + ϕ(k))}. int. j. anal. appl. 19 (1) (2021) 147 let n →∞ in (2.23) and using (2.22), we have cf ≤ ζ(σ(k,qk) + ϕ(qk),σ(k,qk) + ϕ(qk)) < f(σ(k,qk) + ϕ(qk),σ(k,qk) + ϕ(qk))(2.24) ⇒ (2.25) σ(k,qk) + ϕ(qk) < σ(k,qk) + ϕ(qk) which is absurd. hence σ(k,qk) + ϕ(qk) = 0, and hence (2.26) k = qk and ϕ(qk) = 0. similarly, when we take α = αn and ξ = k in (2.33) for all n we get (2.27) k = pk and ϕ(pk) = 0. equations (2.26) and (2.27) show that k is a common fixed point of p and q. to prove the uniqueness of the common fixed point, we suppose that h is another fixed point of p and q. we argue by contradiction. assume that there exists h 6= k(so σ(h,k) > 0.) such that (2.28) ζ(σ(ph,qk) + ϕ(ph) + ϕ(qk),m(h,k)) ≥ cf , where m(h,k) = max{σ(h,k) + ϕ(h) + ϕ(k),σ(h,ph) + ϕ(h) + ϕ(ph),σ(k,qk) + ϕ(k) + ϕ(qk), σ(h,qk) + ϕ(h) + ϕ(qk) + σ(ph,k) + ϕ(ph) + ϕ(k) 4 } = σ(h,qk).(2.29) hence from (2.30), we obtain cf ≤ ζ(σ(h,k),σ(h,k)) < f(σ(h,k),σ(h,k)) < cf ,(2.30) which is absurd and hence h = k. � we will use the same manner in 2.1 to obtain the following result. theorem 2.2. assume that p,q : χ → χ are two self-maps on a complete partial metric space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.31) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) ≥ cf int. j. anal. appl. 19 (1) (2021) 148 for all u,v ∈ χ, where m(α,ξ) = max{dpar(α,ξ) + ϕ(α) + ϕ(v),dpar(α,pα) + ϕ(α) + ϕ(pα),dpar(ξ,qξ) + ϕ(ξ) + ϕ(qξ), dpar(α,qξ) + ϕ(α) + ϕ(qξ) + dpar(pα,ξ) + ϕ(pα) + ϕ(ξ) 2 }.(2.32) then, (p,q) has a common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. if we put q = p in 2.1, we have the following corollary corollary 2.1. assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.33) ζ(σ(pα,pξ) + ϕ(pα) + ϕ(pξ),m(α,ξ)) ≥ cf for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ) + ϕ(α) + ϕ(ξ),σ(α,pα) + ϕ(α) + ϕ(pα),σ(ξ,pξ) + ϕ(ξ) + ϕ(pξ), σ(α,pξ) + ϕ(α) + ϕ(pξ) + σ(pα,ξ) + ϕ(pα) + ϕ(ξ) 4 }.(2.34) then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. corollary 2.2. assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.35) ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≥ cf for all α,ξ ∈ χ. then, (p,q) has a unique common fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. corollary 2.3. assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.36) ζ(σ(pα,pξ) + ϕ(pα) + ϕ(pξ),σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≥ cf for all α,ξ ∈ χ. then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0 and ϕ(z) = 0. if we take ϕ(t) = 0 in 2.1 and 2.2, we obtain the following two corollaries. corollary 2.4. assume that p,q : χ → χ are two self-maps on a complete metric-like space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.37) ζ(σ(pα,qξ),m(α,ξ)) ≥ cf for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ),σ(α,pα),σ(ξ,qξ), σ(α,qξ) + σ(pα,ξ) 4 }.(2.38) int. j. anal. appl. 19 (1) (2021) 149 then, (p,q) has a unique common fixed point z ∈ χ such that σ(z,z) = 0. corollary 2.5. assume that p : χ → χ be self-map on a complete metric-like space (χ,σ). suppose that there exists a extended cf−simulation function ζ ∈=∗ and ϕ ∈ ∆ such that (2.39) ζ(σ(pα,qξ),m(α,ξ)) ≥ cf for all α,ξ ∈ χ, where m(α,ξ) = max{σ(α,ξ),σ(α,pα),σ(ξ,qξ), σ(α,qξ) + σ(pα,ξ) 4 }.(2.40) then, p has a unique fixed point z ∈ χ such that σ(z,z) = 0. example 2.1. let χ = [0, 1] be equipped with the metric-like mapping σ(α,ξ) = α2 + ξ2 for all α,ξ ∈ χ. let p,q : χ → χ be defined by pα =   α2 α+1 if 0 ∈ [0, 1], α2, otherwise. , and qα =   α3 α+1 if 0 ∈ [0, 1], α3, otherwise. . we also consider ζ(s,t) = 1 3 s− t for all s,t ≥ 0, cf = 0 and ϕ(t) = t for all α ∈ χ. note that (χ,σ) is a complete metric-like space. without loss of generality we assume that α,ξ ∈ χ, σ(pα,qξ) + ϕ(pα) + ϕ(qξ) = σ( α2 α + 1 , ξ3 ξ + 1 + ϕ( α2 α + 1 ) + ϕ( ξ3 ξ + 1 ) = ( α2 α + 1 )2) + ( ξ3 ξ + 1 )2)3 + ϕ( α2 ξ + 1 ) + ϕ( α2 α + 1 ) ≤ 1 6 (α2 + ξ3) + 1 3 (α + ξ) ≤ 1 3 (α2 + ξ3) + α + ξ) = 1 3 (σ(α,ξ) + ϕ(α) + ϕ(ξ)) ≤ 1 3 m(α,ξ). it follows that ζ(σ(pα,qξ) + ϕ(pα) + ϕ(qξ),m(α,ξ)) = 1 3 m(α,ξ) − [σ(pα,qξ) + ϕ(pα) + ϕ(qξ)] ≥ 0. then theorem 2.1 is applicable to (p,q) and ϕ on (χ,σ). moreover, α = 0 is a common fixed point of (p,q). int. j. anal. appl. 19 (1) (2021) 150 3. application in this part, we will apply corollary 2.3 to study the existence and uniqueness of solutions of second type of fredholm integral equation: (3.1) α(ϑ) = ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ α(ϑ) = ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ. for all (ϑ,κ) ∈ [0, ]2. let t = c([0, ],r) is the set of real continuous functions on [0, ] for  > 0, defined by σ(α,ξ) =‖ α− ξ ‖∞= sup t∈ |α(t) − ξ(t)| for all α,ξ ∈ t. then (t,σ) is a complete metric-like space. we consider the operators pα(ϑ) = ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ, qξ(ϑ) = ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ, theorem 3.1. assume that equation (3.1) with the following axioms: (1) π : [0, ] × [0, ] → [0,∞) is a continuous function, (2) $ : [0, ] ×r → r where $(κ,.) is monotone nondecreasing mapping for all κ ∈ [0, ], (3) supϑ,κ∈[0,] ∫  0 π(ϑ,κ)dκ ≤ 1, (4) for every δ ∈ (0, 1) such that for all (ϑ,κ) ∈ [0, ]2 and θ,τ ∈ r, ‖ $(κ,θ(κ)) −$(κ,τ(κ)) ‖≤ δ ‖ α(t) − ξ(t) ‖, then, the system (3.1)has a unique solution. proof. for α,ξ ∈ t and from (3) and (4), for all ϑ and κ, we have σ(pα(ϑ),qξ(ϑ)) = | pα(ϑ) −qξ(ϑ) |(3.2) = | ∫  0 π(ϑ,κ)$(κ,θ(κ))dκ− ∫  0 π(ϑ,κ)$(κ,τ(κ))dκ | ≤ ∫  0 π(ϑ,κ) ‖ $(κ,θ(κ)) −$(κ,τ(κ)) ‖ dκ ≤ ∫  0 π(ϑ,κ)δ ‖ α(ϑ) − ξ(ϑ) ‖∞ dκ ≤ π(ϑ,κ)δ ‖ α(ϑ) − ξ(ϑ) ‖∞ ≤ δσ(α,ξ) ≤ δm(α,ξ).(3.3) int. j. anal. appl. 19 (1) (2021) 151 let (ζ1) and ζ(α,ξ) = δα− ξ for all α,ζ ∈ [0,∞),cg = 0. now (3.4) σ(pα(ϑ),qξ(ϑ)) < δm(α,ξ). then, from (3.2), we obtain ζ(σ(pα,qξ),m(α,ξ)) ≥ cf. applying corollary (3.1), we obtain that (p,q) has a unique common fixed point in c([0, 1]), say x. hence, x is a solution of (3.1). � acknowledgements: the authors extend their appreciation to the deanship of post graduate and scientific research at dar al uloom university for funding this work. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. banach, sur les operations dans les ensembles abstraits et leur applications aux equations integrals, fund. math. 3 (1922), 133-181. [2] h. qawaqneh, m. noorani, w. shatanawi, h. alsamir, common fixed point theorems for generalized geraghty (α,ψ,φ)quasi contraction type mapping in partially ordered metric-like spaces, axioms. 7 (2018), 74. [3] h. alsamir, m. selmi noorani, w. shatanawi, h. aydi, h. akhadkulov, h. qawaqneh, k. alanazi, fixed point results in metric-like spaces via sigma-simulation functions, eur. j. pure appl. math. 12 (2019), 88–100. [4] a.f. roldan-lópez-de-hierro, e. karapinar, c. roldán-lópez-de-hierro, j. mart́ıinez-moreno, coincidence point theorems on metric spaces via simulation functions, j. comput. appl. math. 275 (2015), 345-355. [5] x.l. liu, a.h. ansari, s. chandok, s. radenovic, on some results in metric spaces using auxiliary simulation functions via new functions, j. comput. anal. appl. 24(6) (2018), 1103-1114. [6] h. aydi, a. felhi, best proximity points for cyclic kannan-chatterjeaciric type contractions on metric-like spaces, j. nonlinear sci. appl. 9 (2016), 2458-2466. [7] h. aydi, a. felhi, on best proximity points for various alpha-proximal contractions on metric-like spaces, j. nonlinear sci. appl. 9 (2016), 5202-5218. [8] h. alsamir, m.s. md noorani h. qawagneh, k. alanazi, modified cyclic(α,β)-admissible z-contraction mappings in metric-like spaces, asia-pacific conference on applied mathematics and statistics, 2019. [9] h. alsamir, m. noorani, w. shatanawi, k. abodyah, common fixed point results for generalized (ψ,β)-geraghty contraction type mapping in partially ordered metric-like spaces with application, filomat 31(17) (2017), 5497–5509. [10] h. aydi, a. felhi, h. afshari, new geraghty type contractions on metric-like spaces, j. nonlinear sci. appl. 10 (2017), 780–788. [11] a.a. harandi, metric-like spaces, partial metric spaces and fixed points, fixed point theory appl. 2012 (2012), 204. [12] e. karapinar, p. salimi, dislocated metric space to metric spaces with some fixed point theorems, fixed point theory appl. 2013 (2013), 222. [13] f.yan, y. su, q. feng, a new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. fixed point theory appl. 2012 (2012), 152. int. j. anal. appl. 19 (1) (2021) 152 [14] b. samet, c. vetro, p. vetro, fixed point theorems for a α−ψ-contractive type mappings. nonlinear anal., theory meth. appl. 75(4) (2012), 2154-2165. [15] h. alsamir, m. noorani,w. shatanawi, f. shaddad, generalized berinde-type (η,ξ,ϑ,θ) contractive mappings in b-metric spaces with an application, j. math. anal. 7(6) (2016), 1-12. [16] h. alsamir, m. noorani, w. shatanawi, on fixed points of (η,θ)-quasi contraction mappings in generalized metric spaces. j. nonlinear sci. appl. 9 (2016), 4651-4658. [17] h. alsamir, m. s. m. noorani, w. shatanawi, on new fixed point theorems for three types of (α,β)−(ψ,θ,φ)-multivalued contractive mappings in metric spaces. cogent math. 3(1) (2016), 1257473. [18] w. shatanawi, m. noorani, j. ahmad, h. alsamir, m. kutbi, some common fixed points of multivalued mappings on complex-valued metric spaces with homotopy result. j. nonlinear sci. appl. 10 (2017), 3381-3396. [19] h. akhadkulov, m. s. noorani, a. b. saaban, f. m. alipiah, h. alsamir. notes on multidimensional fixed-point theorems. demonstr. math. 50(1) (2017), 360-374. [20] h. qawagneh, noorani, w. shatanawi, h. alsamir. common fixed points for pairs of triangular α-admissible mappings. j. nonlinear sci. appl 10 (2017), 6192-6204. [21] h. qawagneh, m. s. m. noorani, w. shatanawi, k. abodayeh, h. alsamir. fixed point for mappings under contractive condition based on simulation functions and cyclic (α,β)-admissibility. j. math. anal. 9 (2018), 38-51. [22] h. alsamir, m. noorani, w. shatanawi, fixed point results for new contraction involving c-class functions in partial metric spaces, https://www.researchgate.net/publication/332396635_fixed_point_results_for_new_ contraction_involving_c-class_functions_in_partail_metric_spaces, 2017. [23] h. argoubi, b. samet, c. vetro, nonlinear contractions involving simulation functions in a metric space with a partial order, j. nonlinear sci. appl. 8 (2015), 1082-1094. [24] s. g. matthews, partial metric topology. in proceedings of the 8th summer conference on general topology and applications, ann. n.y. acad. sci. 728 (1994), 183–197. [25] a. chandaa, a. ansari, l. kanta dey, b. damjanovic̀. on non-linear contractions via extended cf-simulation functions. filomat 32(10) (2018), 3731–3750 [26] a.h. ansari. note on φ − ψ-contractive type mappings and related fixed point. in: the 2nd regional conference on mathematics and applications, payame noor university, pp. 377–380, 2014. [27] a.f. roldán-lópez-de-hierro, b. samet. ϕ-admissibility results via extended simulation functions. j. fixed point theory appl. 19(3) (2017), 1997-2015. https://www.researchgate.net/publication/332396635_fixed_point_results_for_new_contraction_involving_c-class_functions_in_partail_metric_spaces https://www.researchgate.net/publication/332396635_fixed_point_results_for_new_contraction_involving_c-class_functions_in_partail_metric_spaces 1. introduction 2. main result 3. application references international journal of analysis and applications volume 19, number 6 (2021), 890-903 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-890 generalized close-to-convexity related with bounded boundary rotation khalida inayat noor1, muhammad aslam noor1,∗ and muhammad uzair awan2 1comsats university islamabad, islamabad, pakistan 2government college university faisalabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. the class pα,m[a,b] consists of functions p, analytic in the open unit disc e with p(0) = 1 and satisfy p(z) = ( m 4 + 1 2 ) p1(z) − ( m 4 − 1 2 ) p2(z), m ≥ 2, and p1, p2 are subordinate to strongly janowski function ( 1+az 1+bz )α , α ∈ (0, 1] and −1 ≤ b < a ≤ 1. the class pα,m[a,b] is used to define vα,m[a,b] and tα,m[a,b; 0; b1], b1 ∈ [−1, 0). these classes generalize the concept of bounded boundary rotation and strongly close-to-convexity, respectively. in this paper, we study coefficient bounds, radius problem and several other interesting properties of these functions. special cases and consequences of main results are also deduced. 1. introduction let a denote the class of analytic functions defined in the open unit disc e = {z : |z| < 1} and be given by (1.1) f(z) = z + ∞∑ n=2 anz n, z ∈ e. received august 6th, 2021; accepted september 23rd, 2021; published october 28th, 2021. 2010 mathematics subject classification. 30c45. key words and phrases. janowski function; subordination bounded boundary rotation; univalent; starlike; close-to-convex; integral operator; coefficient problem. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 890 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-890 int. j. anal. appl. 19 (6) (2021) 891 let s ⊂ a be the class of univalent functions in e and let c, s? and k be the subclasses of s consisting of convex, starlike and close-to-convex functions, respectively. for details, see [3]. for f,g ∈ a, we say f is subordinate to g in e, written as f(z) ≺ g(z), if there exists a schwartz function w(z) such that f(z) = g(w(z)), g(z) = z + ∞∑ n=2 bnz n. furthermore, if the function g is univalent in e, then we have the following equivalence f(z) ≺ g(z) ⇔ f(0) = g(0) and f(e) ⊂ g(e). convolution of f and g is defined as (g ∗f)(z) = z + ∞∑ n=2 anbnz n. the class pα[a,b] of strongly janowski functions is defined as follows. definition 1.1. let p be analytic in e with p(0) = 1. then p ∈ pα[a,b], if p(z) ≺ ( 1+az 1+bz )α , where α ∈ (0, 1] and −1 ≤ b < a ≤ 1. we denote pα[0,b1] as pα[b1], −1 ≤ b1 < 0. the class pα[a,b] is generalized as: definition 1.2. an analytic function p : p(z) = 1 + ∞∑ n=1 cnz n is in the class pα,m[a,b], if and only if, there exist p1,p2 ∈ pα[a,b] such that (1.2) p(z) = ( m 4 + 1 2 ) p1(z) − ( m 4 − 1 2 ) p2(z), m ≥ 2. it is obvious pα,2[a,b] = pα[a,b]. for the class p1[a,b] = p [a,b], we refer to [6]. about the class pα[a,b], we observe the following. (i) p(z) ≺ ( 1+az 1+bz )α implies p ∈ pα[a,b] and it can easily be shown that φα(a,b; z) = ( 1+az 1+bz )α is convex univalent in e. in fact simple calculation yield that reφ′α(a,b; z) ≥ α|a−b| (1 −|a|)α−1 (1 −|b|)α+1 > 0, z ∈ e. this shows φα(a,b; z) is univalent in e. also re { (zφ′α(a,b; z)) ′ φ′α(a,b; z) } ≥ t(r) (1 + ar)(1 + br) , where t(r) = 1 −α(a−b)r −abr2 int. j. anal. appl. 19 (6) (2021) 892 is decreasing on (0, 1) and t(0) = 1. this implies re [ (zφ′α(a,b;z)) ′ φ′α(a,b;z) ] ≥ 0 in e. (ii) for a = 1, b = −1, p ∈ φα(1,−1; z) implies ∣∣ arg p(z)∣∣ ≤ απ 2 , z ∈ e. definition 1.3. let f,g ∈ a, (g∗f) ′(z) z 6= 0, z ∈ e. then f ∈ vα,m[a,b; g], if and only if, (z(g ∗f)′)′ (g ∗f)′ ∈ pα,m[a,b], z ∈ e, with f = zf′, f ∈ rα,m[a,b; g], if and only if, f ∈ vα,m[a,b; g] in e. special cases. (i) v1,m[a,b; z 1−z ] = vm[a,b] ⊂ vm[1,−1] = vm, where vm is the well known class of functions of bounded boundary rotation. see, for example, [2, 10, 12]. (ii) r1,m[a,b; z 1−z ] = rm[a,b] ⊂ rm and rm is the class of functions with bounded radius rotation, see [9]. (iii) vα,m[a,b; z (1−z)2 ] = rα,m[a,b; z 1−z ] = rα,m[a,b]. definition 1.4. let f,g ∈ a with (g ∗ f)(z) 6= 0. then f ∈ tα,m[a,b; 0; b1; g], if there exists ψ ∈ vα,m[a,b; g] such that, for b1 ∈ [−1, 0), (g ∗f)′ (g ∗ψ)′ ∈ pα[b1], z ∈ e. we note that t1[a,b; 0;−1; z1−z ] = tm[a,b]. for certain special cases, see [8, 11, 12]. 2. the class vα,m[a,b; g] theorem 2.1. let f ∈ vα,m[a,b; g] and let g(z) = z + ∞∑ n=2 bnz n. then, with f given by (1.1), an = anbn, an = o(1)n σ, σ = {(m 2 + 1 ) (1 −ρ) − (ρ + 2) } , where ρ = ( 1−a 1−b )α , m ≥ 2(1+ρ) 1−ρ and o(1) denotes a constant. proof. let f = f ∗g. then f ∈ vα,m[a,b]. since p ∈ pα[a,b] implies rep(z) > ρ, ρ = ( 1−a 1−b )α , it follows that vα,m[a,b] ⊂ vm(ρ). now, f ∈ vm(ρ), we can write (2.1) f ′1(z) = (f ′ 1(z)) 1−ρ , f1 ∈ vm, int. j. anal. appl. 19 (6) (2021) 893 see [13]. using a result due to brannan [2], we can write (2.2) zf ′1(z) = (s1(z)) ( m4 + 1 2 )(1−ρ) (s2(z)) ( m4 − 1 2 )(1−ρ) , s1,s2 ∈ s?. therefore, from (2.1), (2.2) and cauchy theorem with z = reιθ, we have n2|an| ≤ 1 2πrn ∫ 2π 0 ∣∣f ′1(z)h(z)∣∣1−ρdθ, h ∈ pα,m[a,b] ⊂ pm(ρ) = 1 2πrn+1 ∫ 2π 0 |s1(z)|( m 4 + 1 2 )(1−ρ) |s2(z)|( m 4 −1 2 )(1−ρ) · |h(z)|1−ρdθ.(2.3) applying distortion result for s2 ∈ s? and holder’s inequality in (2.3), we get n2|an| ≤ 1 rn+1 ( 4 r )( m4 −12 )(1−ρ) ( 1 2π ∫ 2π 0 ∣∣s1(z)∣∣{( m4 + 12 )(1−ρ)} 21+ρ dθ) 1+ρ 2 · ( 1 2π ∫ 2π 0 ∣∣h(z)∣∣2dθ)1−ρ2(2.4) now, for h(z) = 1 + ∞∑ n=1 cnz n, we use parsval identity to have 1 2π ∫ 2π 0 ∣∣h(z)∣∣2dθ = ∞∑ n=0 |cn|2r2n ≤ 1 + m2(1 −ρ)2 ∞∑ n=1 r2n = 1 + [m2(1 −ρ)2 − 1]r2 1 −r2 ,(2.5) where we have used coefficient bounds |cn| ≤ m(1 −ρ), for h ∈ pm(ρ). from (2.5) together with subordination for starlike functions, and a result due to hayman [5] for m ≥ 2(1+ρ) 1−ρ , we have (2.6) n2|an| ≤ c1(m,ρ) ( 1 1 −r ){( m2 +1)(1−ρ)}−ρ , where c1(m,ρ) denotes a constant. taking r = 1 − 1 n in (2.6), we obtain the required result. � special cases. (i) let g(z) = z 1−z , then an = an. take a = 0, and in this case f ∈ vm. this leads us to a known coefficient result that an = o(1)n ( m2 −1). (ii) let f ∈ v1,m [ 0,−1, z (1−z)2 ] = rm ( 1 2 ) . then an = o(1)n m 4 −2, m ≥ 6. int. j. anal. appl. 19 (6) (2021) 894 theorem 2.2. let f ∈ vα,m[a,b; g]. then, for f = f ∗g, z = reιθ, 0 ≤ θ1 < θ2 ≤ 2π, we have (2.7) ∫ θ2 θ1 re { (zf ′(z))′ f ′(z) } dθ > − (m 2 − 1 ) (1 −ρ)π, ρ = ( 1 −a 1 −b )α . proof. proof is straight forward, since vα,m[a,b] ⊂ vm(ρ) and f ∈ vm(ρ) implies there exist f1 ∈ vm with f ′(z) = (f ′1(z)) (1−ρ). now, using essentially the same method given in [2], the required result follows. � remark 2.1. let β ( m 2 − 1 ) (1 − ρ). then, from a result of goodman [4] and from (2.7), it follows that f = f∗g ∈ vα,m[a,b] is univalent for β = ( m 2 − 1 ) (1−ρ) ≤ 1. that is f ∈ s for m ≤ 2(2−ρ) 1−ρ . as a special case, with g(z) = z 1−z , a = 0, b = −1 and α = 1, we have f = f, ρ = 1 2 . then f ∈ v1,m[0,−1] implies ∫ θ2 θ1 re { (zf ′(z))′ f ′(z) } dθ > − ( m 4 − 1 2 ) π for this, we can conclude that v1,m[0,−1] ⊂ s for 2 ≤ m ≤ 6. also, with g(z) = z 1−z , a = 1, b = −1, we have a well known result that f ∈ vm is univalent for 2 ≤ m ≤ 4. theorem 2.3. let f ∈ vα,m[a,b; g], m ≤ 2(2−ρ) 1−ρ and ρ = ( 1−a 1−b )α . then f(e) with f = f ∗g, contains the disc d: d = { w : |w| < 4 8 + αm|a−b| } proof. from theorem 2.2, f is univalent in e. let w0 (w0 6= 0) be any complex number such that f(z) 6= w0 for z ∈ e. then the function f1(z) = w0f(z) w0 −f(z) = z + ( a2 + 1 w0 ) z2 + . . . is analytic and univalent in e. using the well known bieberbach theorem for the best bound for second coefficient of univalent functions, see [3], we have 1 |w0| − |a2| ≤ ∣∣a2 + 1 w0 ∣∣ ≤ 2. this gives us 1 |w0| ≤ 2 + |a2| ≤ 2 + αm|a−b| 4 = 8 + αm|a−b| 4 . this completes the proof. � int. j. anal. appl. 19 (6) (2021) 895 special cases. (i) let a = 1, b = −1, α = 1; (ρ = 0) and so f(e) contains the disc |w| < 2 4+m , m ≤ 4. (ii) with a = 0, b = −1, α = 1 2 , we have ρ = 1 4 , and f(e) contains the disc |w| < 8 16+m , m ≤ 14 3 . the following properties of the class vα,m[a,b; g] can easily be proved with simple computations and well known results and therefore we omit the proof. theorem 2.4. (i) the class vα,m[a,b; g] is preserved under the integral operator l : a → a defined as l(z) = ∫ z 0 (l′1(ξ)) β (l′2(ξ)) γ dξ, where li ∈ vα,m[a,b; g], i = 1, 2 and β,γ are positively real with β + γ = 1. (ii) let f ∈ vα,m [ a,b; z 1−z ] . then, with ρ = ( 1−a 1−b )α , z ∈ e and z = reιθ, we have (1 −br)(1−ρ)( m 4 + 1 2 ) (1 + br)(1−ρ)( m 4 −1 2 ) ≤ |f′(z)| ≤ (1 + br)(1−ρ)( m 4 + 1 2 ) (1 −br)(1−ρ)( m 4 −1 2 ) . for α = 1, f ∈ vm[a,b] and a = 1, b = −1,the result reduces to f ∈ vm studied in [2]. (iii) let f ∈ vα,2 [ a,b; z 1−z ] and define f ∈ a as f(z) = β + 1 zβ ∫ z 0 tβ−1f(t)dt,β > 0. then f is convex of order γ(ρ), ρ = ( 1−a 1−b )α , where γ = γ(ρ) = { (β + 1) 2f1 ( 2(1 −ρ), 1; (β + 2); 1 2 ) −β } , 2f1 represents gauss hypergeometric function. (iv) the set of all points log f′(z) for a fixed z ∈ e and f ranging over the class vα,m[a,b; g] is convex. (v) let f ∈ vα,m [ a,b; z 1−z ] , b 6= 0. then f is close-to-convex for |z| < r1, where r1 = { sin ( π b(γ − 2) ) , b 6= 0, m > 2 γ , γ = 1 − ( 1 −a 1 −b )α (vi) let f ∈ vα,m [a,b; g], and let f = f ∗ g. then f is convex of order ρ = ( 1−a 1−b )α for |z| < rm, where r(m) = m− √ m2 − 4 2 , m ≥ 2. theorem 2.5. let f1,f2 ∈ vα,m [a,b; g], β, δ, c and ν be positively real, c ≥ β ≥ 1, (ν + δ) = β. let f = f1 ∗g, gi = fi ∗g, i = 1, 2 and define (2.8) [f(z)] β = cz(β−c) ∫ z 0 tc−1 (g′1(t)) δ (g′2(t)) ν dt. int. j. anal. appl. 19 (6) (2021) 896 then, for z = reιθ, 0 ≤ θ1 < θ2 ≤ 2π, zf ′ f = p, we have ∫ θ2 θ1 re { p(z) + 1 β zp′(z) p(z) + 1 β (c−β) } dθ > −(1 −ρ) (m 2 − 1 ) π, ρ = ( 1 −a 1 −b )α . proof. first we show that there exists a function f ∈ a satisfying (2.8). we assume f1 ∗g 6= 0, fi ∗g 6= 0, z ∈ e. let q(z) = (g′1(z)) δ (g′2(z)) ν = 1 + d1z + d2z 2 + . . . and choose the branches which equal 1, when z = 0. for k(z) = zc−1 (g′1(z)) δ (g′2(z)) ν = zc−1q(z), we have n(z) = c zc ∫ z 0 k(t)dt = 1 + c c + 1 d1z + . . . hence n is well defined and analytic. now let f(z) = [ zβn(z) ] 1 β = z [n(z)] 1 β , where we choose the branch of [n(z)] 1 β which equal 1 when z = 0. thus f ∈ a and satisfies (2.8). we write (2.9) zf ′(z) f(z) = p(z), f = f1 ∗g. from (2.8) and (2.9) with some calculations βp(z) + βzp′(z) (c−β) + βp(z) = δ [ (zg′1(z)) ′ g′1(z) ] + ν [ (zg′1(z)) ′ g′2(z) ] . that is p(z) + 1 β zp′(z) p(z) + 1 β (c−β) = ] δ β [ (zg′1(z)) ′ g′1(z) ] + ν β [ (zg′1(z)) ′ g′2(z) ] . we now apply theorem 2.2 and obtain the required result. for m ≤ 2(2−ρ) 1−ρ and applying a result proved in [14], it can easily be deduced that∫ θ2 θ1 re{p(z)}dθ > −π, p(z) = z(f1 ∗g)′ f1 ∗g . taking g(z) = z (1−z)2 , it follows that f1 ∈ s in e, see [4]. � 3. the class tα,m[a,b; 0; b1; g] theorem 3.1. let f ∈ tα,m [ a,b; 0; b1; z 1−z ] = tα,m[a,b; 0; b1]. then, for z = re ιθ, 0 ≤ θ1 < θ2 ≤ 2π, ρ1 = ( 1 2 )α , ρ = ( 1−a 1−b )α , ∫ θ2 θ1 re { (zf′(z)) ′ f′(z) } dθ > −βπ, β = [ (1 −ρ1) + (m 2 − 1 ) (1 −ρ) ] . int. j. anal. appl. 19 (6) (2021) 897 proof. for f ∈ tα,m [a,b; 0; b1], we can write f′(z) ψ′(z) = h(z), ψ ∈ vα,m[a,b], h ∈ pα[0,b1]. to prove this result, we shall essentially follow the method due to kaplan [4]. for ψ ∈ vα,m[a,b], it implies that ψ ∈ vm(ρ), where ρ = ( 1−a 1−b )α . also h ∈ pα[0,b1], b1 ∈ [−1, 0) is equivalent to h ≺ ( 1 1+b1z )α . that is, h ∈ p(α1) ⊂ p, α1 = ( 1 2 )α . now, with z = reιθ, write p(z) = arg f′(z) and q(z) = arg ψ′(z). then (3.1) |p(z) −q(z)| < ( 1 − ( 1 2 )α) π 2 let p(r,θ) = p(reιθ) + θ, q(r,θ) = q(reιθ) + θ be defined for 0 ≤ r < 1 and for all θ. this gives us (3.2) |p(r,θ) −q(r,θ)| < ( 1 − ( 1 2 )α) π 2 . from theorem 2.2, for ψ ∈ vα,m[a,b] ⊂ vm(ρ), we have∫ θ2 θ1 re { (zψ′(z)) ′ ψ′(z) } dθ > − (m 2 − 1 )( 1 − ( 1 −a 1 −b )α) π, (z = reιθ). thus (3.3) |q(r,θ1) −q(r,θ2)| < ( 1 − ( 1 −a 1 −b )α)(m 2 − 1 ) π. from (3.2) and (3.3), it follows that |p(r,θ1) −p(r,θ2)| = |{p(r,θ1) −q(r,θ1)}−{p(r,θ2) −q(r,θ2)} + {q(r,θ1) −q(r,θ2)}| < ( 1 − ( 1 2 )α) π 2 + ( 1 − ( 1 2 )α) π 2 + ( 1 − ( 1 −a 1 −b )α)(m 2 − 1 ) π = [( 1 − ( 1 2 )α) + ( 1 − ( 1 −a 1 −b )α)(m 2 − 1 )] π = [ (1 −ρ1) + (m 2 − 1 ) (1 −ρ) ] π = βπ, and this proves our result. � special cases. (i) let α = 1, a = 1 and b = −1. then β = m−1 2 = 1 for m = 3. this implies f ∈ t1,m[1,−1; 0;−1] is univalent for 2 ≤ m ≤ 3. (ii) for a = 0, b = −1, α = 1 we have β = m 4 and, in this case, f is univalent for 2 ≤ m ≤ 4. int. j. anal. appl. 19 (6) (2021) 898 remark 3.1. for f ∈ a, goodman [4] introduced a class k(β) as∫ θ2 θ1 re { (zf ′(z)) ′ f ′(z) } dθ > −βπ, z = reιθ, 0 ≤ θ1 < θ2 ≤ 2π, and β ≥ 0. when 0 ≤ β ≤ 1, k(β) consists of univalent functions (close-to-convex), whilest for β > 1, f need not even be finitely-valent, see [4]. we note that, for ρ1 = ( 1 2 )α , ρ = ( 1−a 1−b )α . tα,m[a,b, 0,b1] ⊂ k (m 2 (1 −ρ) + (ρ−ρ1) ) . this implies f ∈ tα,m[a,b; 0;−1] is univalent for m ≤ 2 [ 1 + ρ1 1−ρ ] . theorem 3.2. for g(z) = z 1−z , let f ∈ tα,2[a,b, 0,b1] and for γ,β > 0, let f1 be defined by (3.4) f1(z) = [ (1 + β)z−β ∫ z 0 tβ−1fγ(t)dt ]1 γ . then f1 ∈ t1,2[a,b; 0; b1] in e. proof. we can write (3.4) as (3.5) f1(z) = [( f(z) z )γ ∗ ( φγ,β(z) z )]1 γ , where (3.6) φγ,β(z) = ∞∑ n=1 ( zn n + γ + β ) is convex in e. since f ∈ tα,2[a,b; 0; b1], there exists ψ1 = zψ′ ∈ rα,2[a,b] such that f ′ ψ′ ∈ pα[0,b1], ψ = vα,2[a,b] in e. let (3.7) g1(z) = [ (β + 1)z−β ∫ z 0 tβ−1ψ γ 1 (t)dt ]1 γ , g1 = zg ′. we first show that g ∈ vα,2[a,b]. from (3.7), it follows that (3.8) { zβg γ 1 (z) }′ = zβ−1 (ψ γ 1 (z)) that is (3.9) (g γ 1 (z)) [β + γh1(z)] = ψ γ 1 (z), h1(z) = zg′1(z) g1(z) logarithmic differentiation of (3.9) and simple computations give us (3.10) h1(z) + zh′1(z) γh1(z) + β = zψ′1 ψ1(z) ≺ ( 1 + az 1 + bz )α ≺ ( 1 + az 1 + bz ) . int. j. anal. appl. 19 (6) (2021) 899 now, using theorem 3.3 of [7, p: 109], it follows from (3.10) that h1 ∈ p [a,b] and g1 = zg′ belongs to r1,2[a,b] = s ?[a,b]. therefore g ∈ v1,2[a,b] = c[a,b]. from (3.4), we have zf ′1(z)f γ−1 1 (z) g γ 1 (z) = φγ,β(z) ∗z ( ψ1(z) z )γ ( zf′(z) · f γ−1(z) ψ γ 1 (z) ) φγ,β(z) ∗z ( ψ1(z) z )γ = φγ,β(z) ∗z ( ψ1(z) z )γ h(z) φγ,β(z) ∗z ( ψ1(z) z )γ , h ∈ pα(b1). since h(z) is analytic in e, h(0) = 1, and φγ,β(z) is convex, ψ1 ∈ s?, we use a result due to ruscheweyh and sheil-small [17] to conclude that ( zf′1f γ−1 g γ 1 ) (e) ⊂ c̄oh(e), where c̄oh(e) denotes convex hull of h(e). this implies f1 ∈ t1,2[a,b; 0; b1] in e. or γ = 1 in (3.4), we obtain the well known bernardi integral operator, see [7]. � theorem 3.3. let f = f ∗g, f ∈ tα,m[a,b; 0; b], b 6= 0. then with ρ = ( 1−a 1−b )α and γ = a−b 3b , (i) (3.11) ( 1 1 + br )α (1 −br)γ(1−ρ)( m 4 + 1 2 ) (1 + br)γ(1−ρ)( m 4 −1 2 ) ≤ |f ′(z)| ≤ (1 + br)γ(1−ρ)( m 4 + 1 2 ) (1 −br)γ(1−ρ)( m 4 −1 2 ) · ( 1 1 −br )α (ii) 2γ(1−ρ) a|b| [ g12(a,b; c;−1) −r−a1 g12(a,b; c;−r1) ] ≤ |f(z)| ≤ 2γ(1−ρ) a|b| · [ g12(a,b; c;−1) −r−a2 g12(a,b; c;−r2) ] , where r1 = −r−12 = 1+br 1−br , m ≤ [ 4(1−α) γ(1−ρ) + 2 ] and a is given in (3.16). proof. we can write for f ∈ tα,m[a,b; 0; b], f ′(z) = g′(z)h(z), h ∈ pα[b1], g = ψ ∗g ∈ vα,m[a,b]. since h ∈ pα[b], it easily follows that (3.12) ( 1 1 + br )α ≤ |h(z)| ≤ ( 1 1 −br )α from theorem 2.4 (ii) and (3.12), the proof of (i) is established. we know proceed to prove (ii). let dr denote the radius of the largest schlicht disc centered at the origin contained in the image of |z| < r under f(z). then there is a point z0, |z0| = r, such that |f(z0)| = dr. the ray from 0 to f(z0) lies entirely int. j. anal. appl. 19 (6) (2021) 900 in the image and the inverse image of this ray is a curve in |z| < r. using (3.11), we have dr = |f(z0)| = ∫ c |f ′(z)||dz|, r = a−b 2b ≥ ∫ |z| 0 [ (1 −bs)γ{(1−ρ)( m 4 + 1 2 )} (1 + bs){γ(1−ρ)( m 4 −1 2 )+α} ] ds = ∫ |z| 0 [( 1 −bs 1 + bs ){γ(1−ρ)(m 4 −1 2 )+α} · (1 −bs)γ(1−ρ−α) ] ds,(3.13) let 1+bs 1−bs = t. then 2b (1−bs)2 = dt, and 1 − bs = 2 1+t . this implies ds = 2 b ( 1 1+t )2 dt. therefore, from (3.13), we have |f(z0)| ≥ ∫ 1+br 1−br 1 t−{(1−ρ)( m 4 −1 2 )−α} · ( 2 1 + t )1−ρ−α · 2 b ( 1 1 + t )2 dt = −2(1−ρ) |b| [∫ 1+br 1−br 0 tγ(1−ρ)( m 4 −1 2 )−α · (1 + t)γ(1−ρ−α)dt − ∫ 1 0 tr(1−ρ)( m 4 −1 2 )+α · (1 + t)r(1−ρ−α)dt ] = 2γ(1−ρ) |b| [i1 + i2].(3.14) now put t = r1u with r1 = 1+br 1−br . then dt = r1du and i1 = ∫ 1 0 (r1u) −[γ(1−ρ)(m4 − 1 2 )−α] · (1 + r1u)1−ρ−αdu = r −{γ(1−ρ)(m 4 −1 2 )+α−1} 1 ∫ 1 0 u−γ(1−ρ)( m 4 −1 2 )−α · (1 + r1u)−{γ(1−ρ)+α}du = r −{γ(1−ρ)(m 4 −1 2 )+α−1} 1 · γ(a)γ(c−a) γ(c) g12(a,b; c;−r1),(3.15) where γ and g12, respectively denote gamma and gauss hypergeometric functions. also, here, b,c are positively real for m ≤ 2 { 1 + 2(1−α) 1−ρ } and are given as a = −γ(1 −ρ) ( m 4 − 1 2 ) −α + 1, γ = a−b 2b , b 6= 0 b = −γ(1 −ρ) + α, c = −γ(1 −ρ) ( m 4 − 1 2 ) −α + 2, (c−a) > 0.(3.16) similarly, we calculate i2 and have i2 = γ(a)γ(c−a) γ(c) g12(a,b; c;−1).(3.17) int. j. anal. appl. 19 (6) (2021) 901 using (3.15), (3.16) and (3.17) in (3.14), we obtain the lower bound of |f(z)|. for the upper bound, we proceed in similar way and have |f(z)| ≤ ∫ |z| 0 (1 + bs)γ(1−ρ)( m 4 + 1 2 ) (1 −bs)γ(1−ρ)( m 4 −1 2 ) · ( 1 1 −bs )α ds = ∫ |z| 0 ( 1 + bs 1 −bs )γ(1−ρ)(m 4 −1 2 )+α · (1 + bs)(1−ρ−α)ds. now similar computations yield the required bound and the proof is complete. � by choosing suitable and permissible values of involved parameters, we obtain several new and also known results. remark 3.2. (i) we use a result of pommerenke [16] together with theorem 3.1 and easily deduce that the class tα,m[a,b; 0;−1], m ≤ 2 { 1 + ρ1 1−ρ } , ρ1 = ( 1 2 )α , ρ = ( 1−a 1−b )α , is a linearly invariant family of order b2 ={ m 2 (1 −ρ) + (ρ−ρ1) + 1 } . with similar argument given in [16], we have the covering result for tα,m[a,b; 0;−1] as: the image of e under f = f ∗g ∈ tα,m[a,b; 0;−1] contains the schlicht disc |z| = 12b2 , where b2 = { m 2 (1 −ρ) + 1 + ρ−ρ1 } , and f(z) = z + ∞∑ n=2 anz n. (ii) let f∗ be defined as f∗(z) = 1 b1 [( 1 + z 1 −z )b2 − 1 ] = z + ∞∑ n=2 a∗nz n, where b1 = {m 2 (1 −ρ) + (ρ−ρ1) + 2 } , b2 = {m 2 (1 −ρ) + (ρ−ρ1) + 1 } . it can be shown, with some computations, that f∗ belongs to the linearly invariant family of tα,m[a,b; 0;−1]. using this concept, together with the same argument of pommerenke [16], we have |an| ≤ |a∗n|, n ≥ 1 and lr(f) ≤ lr(f∗), f ∈ tα,m[a,b; 0;−1] when lr(f) is the length of the image of the circle |z| = r under f , 0 ≤ r < 1. theorem 3.4. let f ∈ tα,m[a,b; 0;−1; g] and let f = f ∗g 6= 0 in e with f(z) = z + ∞∑ n=2 anz n. then, for m > 2, an = o(1) ·nγ1, γ1 = {m 2 (1 −ρ) + [ρ1 − (1 + ρ)] } , where o(1) is a constant depending on m, α, a and b only. int. j. anal. appl. 19 (6) (2021) 902 proof. for f ∈ tα,m[a,b; 0;−1], we can write f ′ = g′h, g ∈ vα,m[a,b], g = ψ ∗g, ψ ∈ vα,m[a,b; g]. since vα,m[a,b] ⊂ vm(ρ), ρ = ( 1−a 1−b )α , and it is well known that there exists gi ∈ vm such that g′(z) = (g′(z)) 1−ρ for z ∈ e. also h ≺ ( 1 1+z )α , which implies |arg h(z)| < ρ1π 2 , ρ1 = ( 1 2 )α . therefore we have f ′ = (g′1) 1−ρ (h1) ρ1, reh1 > 0 in e. for g1 ∈ vm, there exists s ∈ s? such that g′1 = sh (m 2 −1) 2 , m > 2 and reh2 > 0 in e, see [1]. thus, for f ∈ tα,m[a,b; 0;−1], it follows that (3.18) f ′ = (s)1−ρ(h2) (1−ρ)(m 2 −1)(h1) ρ1, hi ∈ p, i = 1, 2 so, by cauchy theorem and (3.18), we have for z = reιθ. n|an| ≤ 1 2πrn ∫ 2π 0 |s|1−ρ|h1|ρ1|h2|(1−ρ)( m 2 −1)dθ ≤ 1 rn ( r (1 −r)2 )(1−ρ) ( 1 2π ∫ 2π 0 |h1|2dθ )ρ1 2 · ( 1 2π ∫ 2π 0 |h2| 2δ 2−ρ1 dθ )2−ρ1 2   , where δ = (1 −ρ) ( m 2 − 1 ) and we have used distortion result for s ∈ s? and holder inequality. now, for m > { 2 + 2−ρ1 1−ρ } , we apply a result due to hayman [5] for hi ∈ p and obtain (3.19) n|an| ≤ c(ρ,ρ1,m) · ( 1 1 −r )1+δ+ρ1−2ρ where c(ρ,ρ1,δ) is a constant. setting r = 1 − 1 n , n →∞ in (3.19), the required result follows as an = o(1) ·n{ m 2 (1−ρ)+[ρ1−(ρ+1)]}, ρ1 = ( 1 2 )α , ρ = ( 1 −a 1 −b )α , and m > { 2 + 2−ρ1 1−ρ } , n ≥ 2. � special cases. (i) a = 1 implies that ρ = 0 and for α = 1, ρ1 = 1 2 . then an = o(1) ·n m 2 −1 2 , m > 7 2 taking m = 4, we have an = o(1) ·n 3 2 . int. j. anal. appl. 19 (6) (2021) 903 (ii) a = 1 2 , b = −1, α = 1 ⇒ ρ = 1 4 . also ρ1 = 1 2 . then m = 5 > 4 implies an = o(1) ·n 9 8 . acknowledgement: this research was supported by hec pakistan under project no: 8081/punjab/nrpu /r&d/hec/2017. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d. a. brannan, j. g. clunie and w. e. kirwan, on the coefficient problem for the functions of bounded boundary rotation, ann. acad. sci. fenn. series ai math. 523(1973), 1-18. [2] d. a. brannan, on functions of bounded boundary rotation, proc. edinburgh math. soc. 16(1969), 339-347. [3] a. w. goodman, univalent functions, vol 1, 11, polygonal publishing house, washington, new jersey, 1983. [4] a. w. goodman, on close-to-convex functions of higher order, ann. univ. sci. budapest, evotous sect. math. 25(1972), 17-30. [5] w. k. hayman, on functions with positive real part, j. lond. math. soc. 36(1961), 34-48. [6] w. janowski, some extremal problems for certain families of analytic functions, ann. polon. math. 28(1973), 297-326. [7] s. s. miller and p. t. mocanu, differential subordination: theory and applications, dekker, new york, 2000. [8] k. i. noor, on a generalization of close-to-convexity, int. j. math. math. sci. 6(1983), 327-334. [9] k. i. noor, some properties of certain classes of functions with bounded radius rotation, honam math. j. 19(1997), 97-105. [10] k. i. noor, b. malik and s. mustafa, a survey on functions of bounded boundary and bounded radius rotation, appl. math. e-notes, 12(2012), 136-152. [11] k. i. noor, on some univalent integral operations, j. math. anal. appl. 128(1987), 586-592. [12] k. i. noor, higher order close-to-convex functions, math. japonica, 37(1992), 1-8. [13] k. padmanabhan and r. parvatham, properties of a class of functions with bounded boundary rotation, ann. polon math. 31(1975), 311-323. [14] r. parvatham and s. radha, on certain classes of analytic functions, ann. polon math. 49(1988), 31-34. [15] b. pinchuk, functions with bounded boundary rotation, isr. j. math. 10(1971), 7-16. [16] ch. pommerenke, linear-invarient families analytischer funktionen 1, math. ann. 155(1964), 108-154. [17] s. ruscheweyh and t. sheil-small, hadamard products of schlicht functions and the polya-shoenberg conjecture, comment. math. helv. 48(1973), 119-135. 1. introduction 2. the class v, m [a, b; g] 3. the class t, m [a, b; 0; b1; g] references international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 22-29 http://www.etamaths.com second hankel determinant for bi-univalent analytic functions associated with hohlov operator g. murugusundaramoorthy∗ and k. vijaya abstract. in the present paper, we consider a subclass of the function class σ of bi-univalent analytic functions in the open unit disk ∆ associated with hohlov operator and we obtain the functional |a2a4−a23| for the function class σ . our result gives corresponding |a2a4 −a23| for the subclasses of σ defined in the literature. 1. introduction let a be the class of functions given by the power series (1.1) f(z) = z + ∞∑ n=2 anz n (z ∈ ∆). and analytic in the open unit disk ∆ := {z : z ∈ c and |z| < 1}. also let ω be the family of functions f ∈a which are univalent in ∆ and satisfying the normalization conditions (see[4]): f(0) = f′(0) − 1 = 0. the well-known koebe one-quarter theorem (see[4]) asserts that the image of ∆ under every univalent function f ∈ ω contains a disk of radius 1 4 . thus, the inverse of f ∈ ω is a univalent analytic function on the disk ∆ρ := {z : z ∈ c and |z| < ρ; ρ ≥ 1 4 }. therefore, for each function f(z) = w ∈ ω, there is an inverse function f−1(w) of f(z) defined by f−1(f(z)) = z (z ∈ ∆) and f(f−1(w)) = w (w ∈ ∆ρ) where (1.2) g(w) = f−1(w) = w −a2w2 + (2a22 −a3)w 3 − (5a32 − 5a2a3 + a4)w 4 + .... a function f ∈ ω is said to be bi-univalent in ∆ if both f and f−1 are univalent in ∆. let σ denote the class of bi-univalent function in ∆ given by (1.1). the concept of bi-univalent analytic functions was introduced by lewin [14] in 1967 and he showed 2010 mathematics subject classification. 30c45. key words and phrases. univalent functions; analytic functions; bi-univalent functions; hohlov operator; coefficient bounds. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 22 bi-univalent analytic functions 23 that |a2| < 1.51. subsequently, brannan and clunie [1] conjectured that |a2| ≤ √ 2. netanyahu [18], on the other hand, showed that maxf∈σ |a2| = 43. the coefficient estimate problem for each of the following taylor-maclaurin coefficients |an| (n ∈ n \ {1, 2})is presumably still an open problem. in [3](see also [2, 7, 20, 22, 23]), certain subclasses of the bi-univalent analytic functions class σ were introduced and non-sharp estimates on the first two coefficients |a2| and |a3| were found. in 1976, noonan and thomas [19] defined the qth hankel determinant of f for q ≥ 1 by hq(n) = ∣∣∣∣∣∣∣∣∣ an an+1 . . . an+q−1 an+1 an+2 . . . an+q ... ... ... ... an+q−1 an+q . . . an+2q−2 ∣∣∣∣∣∣∣∣∣ . further, fekete and szegö [6] considered the hankel determinant of f ∈ a for q = 2 and n = 1, h2(1) = ∣∣∣∣ a1 a2a2 a3 ∣∣∣∣ . they made an early study for the estimates of |a3 −µa22| when a1 = 1 with µ real. the well known result due to them states that if f ∈a, then |a3 −µa22| ≤   4µ− 3 if µ ≥ 1, 1 + 2 exp(−2µ 1−µ) if 0 ≤ µ ≤ 1, 3 − 4µ if µ ≤ 0. furthermore, hummel [9, 10] obtained sharp estimates for |a3 − µa22| when f is convex functions and also keogh and merkes [13] obtained sharp estimates for |a3 − µa22| when f is close-to-convex, starlike and convex in ∆. here we consider the hankel determinant of f ∈a for q = 2 and n = 2, h2(2) = ∣∣∣∣ a2 a3a3 a4 ∣∣∣∣ . for the functions f, g ∈a and given by the series f(z) = ∞∑ n=0 anz n and g(z) = ∞∑ n=0 bnz n (z ∈ ∆), the hadamard product (or convolution) of f and g denoted by f ∗g is defined as (f ∗g)(z) = ∞∑ n=0 anbnz n = (g ∗f)(z) (z ∈ ∆). by using the hadamard product (or convolution ), hohlov (cf.[11]) introduced and studied the linear operator ia,bc : ω → ω defined by ia,bc f(z) = z2f1(a,b; c; z) ∗f(z) (f ∈ ω,z ∈ ∆), where 2f1(z) known as gaussian hypergeometric function is defined by (1.3) 2f1(z) = 2f1(a,b; c; z) = ∞∑ n=0 (a)n(b)n (c)n(1)n zn (a,b ∈ c,c ∈ c\z−0 =: {0,−1,−2, . . .}) 24 murugusundaramoorthy and vijaya and (λ)n is the pochhamer symbol or shifted factorial, written in terms of the gamma function γ, by (λ)n = γ(λ + n) γ(λ) = { 1, n = 0 λ(λ + 1)....(λ + n− 1), n ∈ n := {1, 2, 3, .....}. note that 2f1(z) is symmetric in a and b and that the series (1.3) terminates if at least one of the numerator parameter a and b is zero or a negative integer.observe that for the function f of the form (1.1), we have ia,bc f(z) = z + ∞∑ n=2 (a)n−1(b)n−1 (c)n−1(1)n−1 anz n = z + ∞∑ n=2 φnanz n (z ∈ ∆),(1.4) where φn = (a)n−1(b)n−1 (c)n−1(1)n−1 . making use of hohlov operator we consider a new subclass of σ due to panigarhi and murugusundaramoorthy[20] as given below definition 1.1. [20] a function f ∈ σ and of the form (1.1)is said to be in the class ma,b;cς (β,λ) if the following conditions are satisfied: (1.5) < [ (1 −λ) ia,bc f(z) z + λ ( ia,bc f(z) )′] > β (0 ≤ β < 1,λ ≥ 1,z ∈ ∆) and (1.6) < [ (1 −λ) ia,bc g(w) w + λ ( ia,bc g(w) )′] > β (0 ≤ β < 1,λ ≥ 1,w ∈ ∆) where the function g is the inverse of f given by (1.2). it is of interest to note that by taking a = b and c = 1 we state the following subclass fς(β,λ) due to frasin et al.[7]. example 1.2. [7] a function f ∈ σ and of the form (1.1) is said to be in the class fς(β,λ) if the following conditions are satisfied: (1.7) < [ (1 −λ) f(z) z + λf′(z) ] > β (0 ≤ β < 1,λ ≥ 1,z ∈ ∆) and (1.8) < [ (1 −λ) g(w) w + λg′(w) ] > β (0 ≤ β < 1,λ ≥ 1,w ∈ ∆) where the function g is the inverse of f given by (1.2). it is of interest to note that by taking a = b; c = 1 and λ = 1 we state the following subclass hς(β) due to srivastava et al.[22]. by taking a = b; c = 1 and we state the following : bi-univalent analytic functions 25 example 1.3. [22] a function f ∈ σ and of the form (1.1) is said to be in the class hς(β) if the following conditions are satisfied: < [f′(z)] > β (0 ≤ β < 1,z ∈ ∆) and < [g′(w)] > β (0 ≤ β < 1,w ∈ ∆) where the function g is the inverse of f given by (1.2). the object of the present paper is to determine the functional |a2a4 − a23| for the function f ∈ ma,b;cς (β,λ). our result gives corresponding |a2a4 − a 2 3| for the subclasses of σ defined in the examples 1.2 and 1.3. 2. coefficient bounds for the function class ma,b;cς (β,λ) we need the following lemma for our investigation. lemma 2.1. (see [4], p. 41) let p be the class of all analytic functions p(z) of the form (2.1) p(z) = 1 + ∞∑ n=1 pnz n satisfying <(p(z)) > 0 (z ∈ ∆) and p(0) = 1. then |pn| ≤ 2 (n = 1, 2, 3, ...). this inequality is sharp for each n. in particular, equality holds for all n for the function p(z) = 1 + z 1 −z = 1 + ∞∑ n=1 2zn. lemma 2.2. if the function p ∈p is given by the series (2.2) 2p2 = p 2 1 + x(4 −p 2 1), (2.3) 4p3 = p 3 1 + 2(4 −p 2 1)p1x−p1(4 −p 2 1)x 2 + 2(4 −p21)(1 −|x| 2z), for some x,z with |x| ≤ 1 and |z| ≤ 1. lemma 2.3. [8] the power series for p given in (2.1) converges in ∆ to a function in p if and only if the toeplitz determinants (2.4) dn = ∣∣∣∣∣∣∣∣∣ 2 c1 c2 · · · cn c−1 2 c1 · · · cn−1 ... ... ... ... ... c−n c−n+1 c−n+2 · · · 2 ∣∣∣∣∣∣∣∣∣ , n = 1, 2, 3, . . . and c−k = ck, are all nonnegative. they are strictly positive except for p(z) = m∑ k=1 ρkp0(e itkz), ρk > 0, tk real and tk 6= tj for k 6= j in this case dn > 0 for n < m− 1 and dn = 0 for n ≥ m. in the following theorem we determine the second hankel coefficient results for 26 murugusundaramoorthy and vijaya theorem 2.4. let f ∈ma,b;cς (β,λ) be given by (1.1). then (2.5) |a2a4−a23| ≤   4(1 −β2) [ (1+λ)3φ32+4(1−β) 2(1+3λ)φ4 (1+λ)4(1+3λ)φ42φ4 ] , β ∈ [ 0, 1 − √ (1+λ)3 φ32 8(1+3λ)φ4 ] 9(1+λ)2(1−β)2φ22 2(1+3α)φ4[(1+λ)3φ 3 2−2(1−β)2(1+3λ)φ4] , β ∈ ( 1 − √ (1+λ)3 φ32 8(1+3λ)φ4 , 1 ) . proof. since f ∈ ma,b;cς (β,λ), there exists two functions φ(z) and ψ(z) ∈ p satisfying the conditions of lemma 2.1 such that (2.6) (1 −λ) ia,bc f(z) z + λ ( ia,bc f(z) )′ = β + (1 −β)φ(z) and (2.7) (1 −λ) ia,bc g(w) w + λ ( ia,bc g(w) )′ = β + (1 −β)ψ(z) where (2.8) φ(z) = 1 + c1z + c2z 2 + c3z 3 + ... and (2.9) ψ(w) = 1 + d1w + d2w 2 + d3w 3 + .... . equating the coefficients in (2.6) and (2.7)gives (2.10) (1 + λ)φ2a2 = (1 −β)c1 (2.11) (1 + 2λ)φ3a3 = (1 −β)c2 (2.12) (1 + 3λ)φ4a4 = (1 −β)c3 and (2.13) −(1 + λ)φ2a2 = (1 −β)d1 (2.14) (1 + 2λ)φ3(2a 2 2 −a3) = (1 −β)d2 (2.15) −(1 + 3λ)φ4(5a32 − 5a2a3 + a4) = (1 −β)d3 from (2.10) and (2.13) gives (2.16) a2 = 1 −β (1 + λ)φ2 c1 = − 1 −β (1 + λ)φ2 d1 which implies c1 = −d1 now from(2.11) and (2.14), we obtain (2.17) a3 = (1 −β)2 (1 + λ)2 φ22 c21 + (1 −β) 4(1 + 2λ)φ3 (c1 − c2). on the other hand, subtracting (2.15) from (2.12) and using (2.16), we get (2.18) a4 = 1 2(1 + 3λ)φ4 [ −5(1 + 3λ)(1 −β)3φ4 (1 + λ)3φ32 c31 + 5(1 + 3λ)(1 −β)φ4 (1 + λ)φ2 a3c1 + (1 −β)(c3 −d3) ] . bi-univalent analytic functions 27 thus we establish that (2.19) |a2a4 −a23| = ∣∣∣∣− (1 −β)4(1 + λ)4φ42 c41 + (1 −β) 3c21(c2 −d2) 8(1 + λ)2(1 + 2λ)φ22φ3 + (1 −β)2 2(1 + λ)(1 + 3λ)φ4φ2 c1(c3 −d3) − (1 −β)2(c2 −d2)2 ∣∣∣∣ . according to lemma2.2 we have 2c2 = c 2 1 + x(4 − c 2 1), and 2d2 = d 2 1 + x(4 −d 2 1), hence we have (2.20) c2 = d2 and further 4c3 = c 3 1 + 2(4 − c 2 1)c1x− c1(4 − c 2 1)x 2 + 2(4 − c21)(1 −|x| 2z), 4d3 = d 3 1 + 2(4 −d 2 1)d1x−d1(4 −d 2 1)x 2 + 2(4 −d21)(1 −|x| 2z) (2.21) c3 −d3 = 1 2 c31 + c1(4 − c 2 1)x− 1 2 c1(4 − c21)x 2 (2.22) |a2a4 −a23| = ∣∣∣∣ −(1 −β)4(1 + λ)4φ42 c41 + (1 −β) 2 4(1 + λ)(1 + 3λ)φ4φ2 c41 + (1 −β)2c21(4 − c21)x 2(1 + λ)(1 + 3λ)φ4φ2 − (1 −β)2c21(4 − c21)x2 4(1 + λ)(1 + 3λ)φ4φ2 ∣∣∣∣ letting c1 = c, we may assume without restriction that c ∈ [0, 2] since φ ∈ p so |c1| ≤ 2.thus,applying triangle inequality on (2.19),with µ = |x| ≤ 1, we obtain (2.23) |a2a4 −a23| ≤ (1 −β)4 (1 + λ)4φ42 c4 + (1 −β)2 4(1 + λ)(1 + 3λ)φ4φ2 c4 + (1 −β)2c2(4 − c2)µ 2(1 + λ)(1 + 3λ)φ4φ2 + (1 −β)2c2(4 − c2)µ2 4(1 + λ)(1 + 3λ)φ4φ2 = f(µ) differentiating f(µ), we get f ′(µ) = (1 −β)2c21(4 − c21) 4(1 + λ)(1 + 3λ)φ4φ2 + (1 −β)2c2(4 − c2)µ 2(1 + λ)(1 + 3λ)φ4φ2 by using elementary calculus, one can show that f ′(µ) > 0 for µ > 0 hence f is an increasing function and thus ,the upper bound for f(µ) corresponds to µ = 1,in which case (2.24) f(µ) = f(1) = [ (1 −β)4 (1 + λ)4φ42 + (1 −β)2 4(1 + λ)(1 + 3λ)φ4φ2 ] c4 + 3(1 −β)2c2(4 − c2) 4(1 + λ)(1 + 3λ)φ4φ2 = g(c) assume that g(c) has a maximum value in an interior of c ∈ [0, 2], by elementary calculations we find (2.25) g′(c) = [ 4(1 −β)4 (1 + λ)4φ42 − 2(1 −β)2 (1 + λ)(1 + 3λ)φ4φ2 ] c3+ 6(1 −β)2c (1 + λ)(1 + 3λ)φ4φ2 . 28 murugusundaramoorthy and vijaya then g′(c) = 0 implies the real critical point c01 = 0 or c02 = √ 3(1+λ)3 φ32 (1+λ)3φ32−2(1−β)2(1+3λ)φ4 . after some calculations we concluded following cases: case 1: when β ∈ [ 0, 1 − √ (1+λ)3 φ32 8(1+3λ)φ4 ] , we observe that c02 ≥ 2, that is, c02 is out of the interval (0, 2). therefore the maximum value of g(c) occurs at c01 = 0 or c = c02 which contradicts our assumption of having the maximum value at the interior point of c ∈ [0, 2]. since g is an increasing function in the interval [0, 2], maximum point of g must be on the boundary of c ∈ [0, 2], that is, c = 2. thus, we have max 0≤c≤2 g1(p) = g(2) = 4(1 −β2) [ (1 + λ)3φ32 + 4(1 −β)2(1 + 3λ)φ4 (1 + λ)4(1 + 3λ)φ42φ4 ] case 2: when β ∈ ( 1 − √ (1+λ)3 φ32 8(1+3λ)φ4 , 1 ) , we observe that c02 ≤ 2, that is, c02 is interior of the interval [0, 2]. since g ′′(c02) < 0, the maximum value of g(c) occurs at c = c02. thus, we have max 0≤c≤2 g(c) = g(c02) = g (√ 3(1 + λ)3 φ32 (1 + λ)3φ32 − 2(1 −β)2(1 + 3λ)φ4 ) = 9(1 + λ)2(1 −β)2φ22 2(1 + 3α)φ4[(1 + λ)3φ 3 2 − 2(1 −β)2(1 + 3λ)φ4] . � concluding remarks: suitably specializing the parameter λ one can state the hankel coefficients for various subclasses ofma,b;cς (β,λ). in fact, by choosing a = b and c = 1 we have φ2 = 1; φ3 = 1; φ4 = 1 hence we state the hankel determinant coefficients for the function f ∈fς(β,λ) studied in[7] as given below: (2.26) |a2a4 −a23| ≤   4(1 −β2) [ (1+λ)3+4(1−β)2(1+3λ) (1+λ)4(1+3λ) ] , β ∈ [ 0, 1 − √ (1+λ)3 8(1+3λ) ] 9(1+λ)2(1−β)2 2(1+3α)[(1+λ)3−2(1−β)2(1+3λ)], β ∈ ( 1 − √ (1+λ)3 8(1+3λ) , 1 ) . also by choosing λ = 1 one can easily derive hankel determinant |a2a4 − a23| for the functions f ∈hς studied by srivastava et al.[22]. references [1] d.a. brannan, j.g. clunie(eds.), aspects of contemporary complex analysis (proceeding of the nato advanced study institute held at the university of durham, durham: july 1-20, 1979), academic press, new york and london, 1980. [2] d.a. brannan, j. clunie, w.e. kirwan, coefficient estimates for a class of star-like functions, canad. j. math. 22(1970), 476-485. [3] d.a. brannan, t.s. taha, on some classes of bi-univalent functions, studia univ. babesbolyai math. 31(2)(1986), 70-77. [4] p.l. duren, univalent functions, in: grundlehren der mathematischen wissenschaften, band 259, springer-verlag, new york, berlin, heidelberg and tokyo, 1983. [5] e.deniz, m.ca̧g̀lar, and h. orhan,second hankel determinant for bi-starlike and bi-convex functions of order β,arxiv:1303.2504v2. [6] m.fekete and g.szegö, eine bemerkung über ungerade schlichte funktionen, j. london. math. soc., 8(1933), 85–89. bi-univalent analytic functions 29 [7] b.a. frasin, m.k. aouf, new subclasses of bi-univalent functions, appl. math. lett. 24(2011), 1569–1973. [8] u.grenander and g.szegö, toeplitz forms and their applications, univ. of california press, berkeley and los angeles, 1958. [9] j.hummel, the coefficient regions of starlike functions, pacific. j. math., 7 (1957), 1381– 1389. [10] j.hummel, extremal problems in the class of starlike functions, proc. amer. math. soc., 11 (1960), 741–749. [11] yu.e. hohlov, operators and operations in the class of univalent functions (in russian), izv. vyss̃. uc̃ebn. zaved. matematika 10(1978) 83-89. [12] a.janteng, s.a.halim and m.darus, coefficient inequality for a function whose derivative has a positive real part, j.ineq. pure and appl. math., vol.7, 2 (50) (2006), 1–5. [13] f.r.keogh and e.p.merkes, a coefficient inequality for certain classes of analytic functions, proc. amer. math. soc., 20 (1969), 8–12. [14] m. lewin, on a coefficient problem for bi-univalent functions, proc. amer. math. soc. 18(1967) 63–68. [15] r.j.libera and e.j.zlotkiewicz, early coefficients of the inverse of a regular convex function, proc. amer. math. soc., 85(2) (1982), 225–230. [16] r.j.libera and e.j.zlotkiewicz, coefficient bounds for the inverse of a function with derivative in p, proc. amer. math. soc., 87(2) (1983), 251–289. [17] t.h.macgregor, functions whose derivative has a positive real part, trans. amer. math. soc., 104 (1962), 532–537. [18] e. netanyahu, the minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, arch. rational mech. anal. 32(1969) 100-112. [19] j.w.noonan and d.k.thomas, on the second hankel determinant of areally mean p−valent functions, trans. amer. math. soc., 223 (2) (1976), 337–346 [20] t.panigarhi and g. murugusundaramoorthy, coefficient bounds for biunivalent functions analytic functions associated with hohlov operator, proc. jangjeon math. soc,16 (1) (2013) 91-100. [21] ch. pommerenke, univalent functions, vandenhoeck and ruprecht, göttingen, 1975. [22] h.m. srivastava, a.k. mishra, p. gochhayat, certain subclasses of analytic and bi-univalent functions, appl. math. lett. 23(2010), 1188-1192. [23] t.s. taha, topics in univalent function theory, ph.d thesis, university of london, 1981. school of advanced sciences, vit university, vellore 632014, tamilnadu, india ∗corresponding author int. j. anal. appl. (2022), 20:42 some new types of convergence definitions for random variable sequences saadettin aydın∗ gülhane medical faculty, university of health sciences, ankara, turkey ∗corresponding author: saadettin.aydin@sbu.edu.tr abstract. in this paper, we introduce the concepts of invariant convergence in probability, statistically invariant convergence in probability, invariant convergence almost surely, invariant convergence in distribution and invariant convergence in lpnorm for sequences of random variables. also, we investigate some inclusion relations. 1. introduction in probability theory, we know several different notions of convergence of random variables. the convergence of sequences of random variables to some limit random variable is an essential concept in probability theory and its applications to statistics. the natural density of a set k of positive integers is defined by δ(k) := lim n→∞ 1 n |{k ≤ n : k ∈ k}|, where |{k ≤ n : k ∈ k}| denotes the number of elements of k not exceeding n. the concept of statistical convergence of number sequences was introduced by fast [5]. in [25] schoenberg established some basic features of statistical convergence and studied the notion as a summability method. a continuous linear functional ϕ on `∞, the space of real bounded sequences, is said to be a invariant limit if (a) ϕ(x)≥ 0 when the sequence x =(xn) has xn ≥ 0 for all n, received: jul. 12, 2022. 2010 mathematics subject classification. 40a05, 40d25, 60b10. key words and phrases. random variable; statistical convergence; invariant convergence; strongly invariant convergence; convergence in probability; almost surely convergence. https://doi.org/10.28924/2291-8639-20-2022-42 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-42 2 int. j. anal. appl. (2022), 20:42 (b) ϕ(e)=1 where e =(1,1,1, ...) and (c) ϕ(xσ(m))= ϕ(x) for all x ∈ l∞, where σ : n+ → n+. a sequence x ∈ l∞ is said to be invariant convergent to ` if all of its invariant limits are equal to `. the mappings σ are one-to-one and such that σk(m) 6= m for all k,m ∈ n+ and σk(m) = σ(σk−1(m). thus ϕ extends the limit functional on the space of convergent sequences, in the sense that ϕ(x) = limx for all convergent sequences x. in the case σ is the translation mapping σ(m) = m+1, the σ-mean is often called a banach limit and vσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences. it can be shown that vσ = {(xk)∈ `∞ : lim n→∞ 1 n+1 n∑ k=0 xσk(m) = ` uniformly in m}. in this case, we write σ− limxn = `. the set of all strongly invariant convergent sequences is denoted by [vσ] = {(xk)∈ `∞ : lim n→∞ 1 n+1 n∑ k=0 |xσk(m) − `|=0 uniformly in m}. in [18], the concept of σ-statistically convergent sequence was introduced as follows: a sequence x =(xk) is said to be σ-statistically convergent to ` if for every� > 0, lim n→∞ 1 n |{k ≤ n : |xσk(m) − `| ≥ �}|=0 uniformly in m. throughout the paper, {xn} will denote a sequence of random variables where each xn is defined on the same sample space w with respect to a given class of events s and a given probability function p :s →r. the concept of statistical convergence in probability of a sequence of random variables {xn} was introduced in [8] as follows: a sequence of random variables {xn} is said to be statistically convergent in probability to a random variable x if for any ε,δ > 0 lim n→∞ 1 n |k ≤ n : p(|xk −x| ≥ ε)≥ δ|=0. the reader can also see [9–11] for related works. a sequence of random variables {xn} is said to be strongly cesàro summable in probability to a random variable x if for each ε > 0, lim n→∞ 1 n n∑ k=1 p(|xk −x| ≥ ε)=0. let {xn} be a sequence of random variables. we say {xn} is bounded in probability, if for every ε > 0, there exists mε > 0 such that p(|xn| > ε) < mε for all n. int. j. anal. appl. (2022), 20:42 3 for more information on the concepts covered in this article, see articles [1,2,8–12,27] on random variables, articles [3–7], [13][26] on invariant convergence and statistical convergence. 2. some new types of invariant convergence let denote the set of all σ mapping by m. then⋂ {vσ : σ ∈ m} is equal to the set of all convergent sequences (see [21]). therefore, the definitions and theorems we will give in this section are meaningful. the definitions we will give are more general, and the kinds of convergence previously known for random variable sequences can be derived from the definitions we give. 2.1. invariant convergence in probability. definition 2.1. the sequence {xn} is said to be invariant convergent in probability to a random variable x if for any ε > 0 lim n→∞ 1 n n∑ k=1 p(|xσk(m) −x| ≥ ε)=0 uniformly in m or lim n→∞ 1 n n∑ k=1 p(|xσk(m) −x| < ε)=1 uniformly in m. in this case, we write σ − limp(|xn −x| ≥ ε)= 0 or σ − limp(|xn −x| < ε)= 1. the class of all sequences of random variables which are invariant convergent in probability will be denoted by v pσ . ⋂{ v pσ : σ ∈ m } is equal to the set of all random variable sequences which are convergent in probability. 2.2. statistically invariant convergence in probability. definition 2.2. the sequence {xn} is said to be statistically invariant convergent in probability to a random variable x if for any ε,δ > 0 lim n→∞ 1 n |k ≤ n : p(|xσk(m) −x| ≥ ε)≥ δ|=0 uniformly in m. in this case, we write xn → x(spσ ). the class of all sequences of random variables which are statistically invariant convergent in probability will be denoted by spσ . ⋂ {spσ : σ ∈ m} is equal to the set of all random variable sequences which are statistically convergent in probability. 4 int. j. anal. appl. (2022), 20:42 theorem 2.1. if a sequence of random variables {xn} is strongly invariant convergent in probability to x, then {xn} is statistically invariant convergent in probability to x. proof. suppose that{xn}is strongly invariant convergent in probability to x . for an arbitrary ε > 0,δ, we get 1 n n∑ k=1 p(|xσk(m) −x| ≥ ε) = 1 n   n∑ k=1 p(|x σk(m) −x|≥ε)≥δ p(|xσk(m) −x| ≥ ε)+ n∑ k=1 p(|x σk(m) −x|≥ε)<δ p(|xσk(m) −x| ≥ ε)   ≥ 1 n n∑ k=1 p(|x σk(m) −x|≥ε)≥δ p(|xσk(m) −x| ≥ ε) ≥ δ n |{k ≤ n : p(|xσk(m) −x| ≥ ε)≥ δ}| for each m. hence, we have lim n→∞ 1 n |{k ≤ n : p(|xσk(m) −x| ≥ ε)≥ δ}|=0 uniformly in m, that is, {xn} is statistically invariant convergent in probability to x. � theorem 2.2. if a sequence of random variables {xn} is statistically invariant convergent in probability to x and bounded in probability, then {xn} is strongly invariant convergent in probability to x. proof. suppose that {xn} is statistically invariant convergent in probability to x and {xn} is bounded, say p(|xσk(m) −x| > ε) < mε for all m and k. given ε,δ > 0, we get 1 n n∑ k=1 p(|xσk(m) −x| ≥ ε) = 1 n n∑ k=1 p(|x σk(m) −x|≥ε)≥δ p(|xσk(m) −x| ≥ ε)+ 1 n n∑ k=1 p(|x σk(m) −x|≥ε)<δ p(|xσk(m) −x| ≥ ε) ≤ ε n n∑ k=1 p(|x σk(m) −x|≥ε)≥δ 1+ δ n n∑ k=1 p(|x σk(m) −x|≥ε)<δ 1 ≤ mε n |{k ≤ n : p(|xσk(m) −x| ≥ ε)}|+ δ n |{k ≤ n : p(|xσk(m) −x| ≥ ε)}| for each m, hence we have lim n→∞ 1 n n∑ k=1 p(|xσk(m) −x| ≥ ε)=0 uniformly in m. � int. j. anal. appl. (2022), 20:42 5 theorem 2.3. suppose that xn is invariant convergent in probability to a random variable x and that f is a continuous function. then {f (xn)} is invariant convergent in probability to f (x). proof. if f is continuous, given ε > 0 there exists δ > 0 such that |f (xn)−f (x)| < ε for |xn−x| < δ. since {xn} is invariant convergent in probability to the random variable x, then lim n→∞ 1 n n∑ k=1 p(|xσk(m) −x| < δ)=1 uniformly in m. thus, lim n→∞ 1 n n∑ k=1 p(|f (xσk(m))− f (x)| < ε)=1 uniformly in m. � 2.3. invariant convergence almost surely. we say an event happens “surely” if it always happens. we say an event happens “almost surely” if it happens with probability 1. almost sure convergence is one of the main modes of stochastic convergence. it may be viewed as a notion of convergence for random variables that’s analogous to, but not the same as, the notion of pointwise convergence for real functions. now we will introduce invariant convergent almost surely random variable sequence. definition 2.3. we say that a sequence of random variables {xn} is invariant convergent almost surely to x if p( lim n→∞ 1 n n∑ k=1 xσk(m) = x uniformly in m)=1 that is, p(σ − limxn = x)=1. we will denote the set of all sequences of random variables {xn} which are invariant convergent almost surely by asσ. ⋂ {asσ : σ ∈ m} is equal to the set of all random variable sequences which are convergent almost surely. theorem 2.4. if a sequence of random variables {xn} is invariant convergent almost surely to a random variable x, then {xn} is invariant convergent in probability to x. proof. let {xn} be invariant convergent almost surely to a random variable x and ε > 0. then for every s ≥ n, we can write p(|xn −x| < ε)≥ p(|xs −x| < ε). (2.1) by considering the original sample space, we can write {t : |xs(t)−x(t)| > ε, ∀s ≥ n}⊂{t : |xn(t)−x(t)| > ε}. 6 int. j. anal. appl. (2022), 20:42 since {xn} is invariant convergent almost surely to a random variable x, for every s ≥ n, we have σ − limp(|xs −x| < ε)=1. if we take σ − limits of in (2.1), ∀s ≥ n, we get σ − limp(|xn −x| < ε)≥ σ − limp(|xs −x| < ε)=1. thus, σ − limp(|xn −x| < ε)=1. � 2.4. invariant convergence in distribution. definition 2.4. we say that a sequence of random variables {xn} is invariant convergent in distribution to a random variable x if lim n→∞ 1 n n∑ k=1 fx σk(m) = fx(x) uniformly in m, that is p − limfxn(x)= fx(x), for all x at which fx(x) is continuous. the set of all sequences of random variables {xn} which are invariant convergent in distribution will be denoted idσ. ⋂ {idσ : σ ∈ m} is equal to the set of all random variable sequences which are convergent in distribution. theorem 2.5. if a sequence of random variables {xn} is invariant convergent in distribution to a number a, then {xn} is invariant convergent in probability to a. proof. since {xn} is invariant convergent in distribution to the number a, for any ε > 0, we have σ − limfxn(a−ε)=0, σ− limfxn(a+ ε 2 )=1. we can write for any ε > 0, σ − limp(|xn −a| ≥ ε) = σ − lim[p(xn ≤ a−ε)+p(xn ≥ a+ε)] = σ − limp(xn ≤ a−ε)+σ − limp(xn ≥ a+ε) = σ − limfxn(a−ε)+σ − limp(xn ≥ a+ε) = 0+σ − limp(xn ≥ a+ε) (since σ − limfxn(a−ε)=0) ≤ σ − limp(xn > a+ ε 2 ) = 1−σ − limfxn(a+ ε 2 ) (since σ − limfxn(a+ ε 2 )=1) = 0. int. j. anal. appl. (2022), 20:42 7 since σ− limp(|xn −a| ≥ ε)≥ 0, we conclude that σ− limp(|xn −a| ≥ ε)=0, for all ε > 0, which means {xn} is invariant convergent in probability to a. � 2.5. invariant convergence in pth mean. definition 2.5. let p ≥ 1 be a fixed number. a sequence of random variables {xn} is invariant convergent in the pth mean or in the lp-norm to a random variable x if lim n→∞ 1 n n∑ k=1 e(|xσk(m) −x| p)=0 uniformly in m, that is, σ − lime(|xn −x|p)=0. if p =2, it is called the mean-square invariant convergence. the set of all invariant convergent in the pth mean or in the lp-norm will be denoted by epσ. ⋂ {epσ : σ ∈ m} is equal to the set of all random variable sequences that are convergent in the pth mean or in the lp-norm. theorem 2.6. if a sequence of random variables {xn} is invariant convergent in the lp,(p ≥ 1)norm to x, then {xn} is invariant convergent in probability to x. proof. for any ε > 0 and for each m, we can write p(|xn −x| ≥ ε) = p(|xn −x|p ≥ εp)( since p ≥ 1) ≤ e(|xn −x|p) εp by markov’s inequality and 1 n n∑ k=1 p(|xσk(m) −x| ≥ ε)≤ 1 n n∑ k=1 e(|xσk(m) −x|p) εp since by assumption limn→∞ 1 n ∑n k=1e(|xσk(m) −x| p) uniformly in m, we get σ − limp(|xn −x| ≥ ε)=0, for all ε > 0. � in case σ(m) = m + 1, we have the concepts of almost convergence in probability, statistically almost convergence in probability, almost convergence almost surely, almost convergence in distribution and almost convergence in lp-norm from the definitions 2.1, 2.2, 2.3, 2.4 and 2.5, respectively. in addition, the inclusion relations in the theorems 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 are also valid between these concepts. these definitions and theorems have not been seen elsewhere until now. 8 int. j. anal. appl. (2022), 20:42 conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] l. breiman, probability, siam, 1992. [2] p. brockwell, r. davis, time series: theory and methods, springer, 1991. [3] j.s. connor, the statistical and strong p-cesàro of sequences, analysis, 8 (1988), 47–63. https://doi.org/10. 1524/anly.1988.8.12.47. [4] j.s. connor, on strong matrix summability with respect to a modulus and statistical convergence, can. math. bull. 32 (1989), 194–198. https://doi.org/10.4153/cmb-1989-029-3. [5] h. fast, sur la convergence statistique, colloq. math. 2 (1951), 241–244. http://eudml.org/doc/209960. [6] j.a. fridy, on statistical convergence, analysis, 5 (1985), 301–313. https://doi.org/10.1524/anly.1985.5. 4.301. [7] j.a. fridy, c. orhan, lacunary statistical convergence, pac. j. math. 160 (1993), 43–51. [8] s. ghosal, statistical convergence of a sequence of random variables and limit theorems, appl. math. 58 (2013), 423–437. https://doi.org/10.1007/s10492-013-0021-7. [9] s. ghosal, i-statistical convergence of a sequence of random variables in probability, afr. mat. 25 (2013), 681–692. https://doi.org/10.1007/s13370-013-0142-x. [10] s. ghosal, sλ-convergence of a sequence of random variables, j. egypt. math. soc. 23 (2015), 85–89. https: //doi.org/10.1016/j.joems.2014.03.007. [11] s. ghosal, weighted statistical convergence of order α and its applications, j. egypt. math. soc. 24 (2016), 60–67. https://doi.org/10.1016/j.joems.2014.08.006. [12] p.g. hoel, s.c. port, c.j. stone, introduction to probability theory, houghton mifflin company, 1971. [13] g.g. lorentz, a contribution to the theory of divergent sequences, acta math. 80 (1948), 167–190. https: //doi.org/10.1007/bf02393648. [14] i.j. maddox, a new type of convergence, math. proc. camb. phil. soc. 83 (1978), 61–64. https://doi.org/ 10.1017/s0305004100054281. [15] m. mursaleen, matrix transformation between some new sequence spaces, houston j. math. 9 (1983), 505–509. [16] m. mursaleen, on infinite matrices and invariant means, indian j. pure appl. math. 10 (1979), 457–460. [17] m. mursaleen, o.h.h. edely, on the invariant mean and statistical convergence, appl. math. lett. 22 (2009), 1700–1704. https://doi.org/10.1016/j.aml.2009.06.005. [18] f. nuray, e. savaş, invariant statistical convergence and a-invariant statistical convergence, indian j. pure appl. math. 25 (1994), 267–274. [19] e. savaş, some sequence spaces involving invariant means, indian j. math. 31 (1989), 1–8. [20] e. savaş, strongly σ-convergent sequences, bull. calcutta math. 81 (1989), 295–300. [21] r.a. raimi, invariant means and invariant matrix methods of summability, duke math. j. 30 (1963), 81–94. https://doi.org/10.1215/s0012-7094-63-03009-6. [22] s. ross, a first course in probability, macmillan publishing company, 2nd ed., 1984. [23] p. schaefer, infinite matrices and invariant means, proc. amer. math. soc. 36 (1972), 104–110. https://doi. org/10.1090/s0002-9939-1972-0306763-0. [24] t. šalát, on statistically convergent sequences of real numbers, math. slovaca, 30 (1980), 139–150. http: //dml.cz/dmlcz/136236. [25] i.j. schoenberg, the integrability of certain functions and related summability methods, amer. math. mon. 66 (1959), 361–775. https://doi.org/10.1080/00029890.1959.11989303. https://doi.org/10.1524/anly.1988.8.12.47 https://doi.org/10.1524/anly.1988.8.12.47 https://doi.org/10.4153/cmb-1989-029-3 http://eudml.org/doc/209960 https://doi.org/10.1524/anly.1985.5.4.301 https://doi.org/10.1524/anly.1985.5.4.301 https://doi.org/10.1007/s10492-013-0021-7 https://doi.org/10.1007/s13370-013-0142-x https://doi.org/10.1016/j.joems.2014.03.007 https://doi.org/10.1016/j.joems.2014.03.007 https://doi.org/10.1016/j.joems.2014.08.006 https://doi.org/10.1007/bf02393648 https://doi.org/10.1007/bf02393648 https://doi.org/10.1017/s0305004100054281 https://doi.org/10.1017/s0305004100054281 https://doi.org/10.1016/j.aml.2009.06.005 https://doi.org/10.1215/s0012-7094-63-03009-6 https://doi.org/10.1090/s0002-9939-1972-0306763-0 https://doi.org/10.1090/s0002-9939-1972-0306763-0 http://dml.cz/dmlcz/136236 http://dml.cz/dmlcz/136236 https://doi.org/10.1080/00029890.1959.11989303 int. j. anal. appl. (2022), 20:42 9 [26] h. steinhaus, sur la convergence ordinaire et la convergence asymptotique, colloq. math. 2 (1951), 73–74. [27] d. williams, probability with martingales, cambridge university press, 1991. 1. introduction 2. some new types of invariant convergence 2.1. invariant convergence in probability 2.2. statistically invariant convergence in probability 2.3. invariant convergence almost surely 2.4. invariant convergence in distribution 2.5. invariant convergence in pth mean references int. j. anal. appl. (2022), 20:46 received: jun. 20, 2022. 2010 mathematics subject classification. 65c20. key words and phrases. survival analysis; kaplan-meier; survival curves; statistical power; censoring. https://doi.org/10.28924/2291-8639-20-2022-46 © 2022 the author(s) issn: 2291-8639 1 statistical powers of some tests for checking homogeneity of survival distributions with disjointed ends in the presence of censoring babalola bayowa teniola1,2*, adeleke raphael ayantunji2, halid omobolaji yusuf2, olubiyi adenike olufunmilola2, ogunsakin ropo ebenezer3, adigun kehinde abimbola4, adejuwon samuel oluwaseun4, adarabioyo mumini idowu4, ogunboyo ojo femi5, fadugba sunday emmanuel6, egbon osafu augustine7, akinyemi oluwadare2, ogunwale olukunle daniel2, faweya olanrewaju2, kawiso martin1 1department of mathematics and statistics, kampala international university, uganda 2department of statistics, ekiti state university, ado-ekiti, nigeria 3biostatistics, discipline of public health medicine, howard college, university of kwazulu-natal, south africa 4department of mathematical and physical science, afe babalola university, ado-ekiti, nigeria 5department of epidemiology and biostatistics, university of medical sciences, ondo, nigeria 6department of mathematics, ekiti state university, ado-ekiti, nigeria 7institute of mathematics and computer science, university of sao paulo, sao carlos, brazil *corresponding author: bayowa.babalola@kiu.ac.ug abstract. this paper considered the comparison of some tests for assessing the overall homogeneity of kaplan-meier survival curves under low and high censoring rates when the curves are disjointed towards the end. the performances of these tests were measured by their statistical powers. monte carlo simulation study was conducted to evaluate and numerically compare the relative performances of log-rank,wilcoxon, tarone-ware, peto-peto, modified peto-peto, the fleming-harrington (1,1), and the babalola-adeleke tests. the result obtained shows that the babalola-adeleke and fleming-harrington (1,1) tests have more https://doi.org/10.28924/2291-8639-20-2022-46 2 int. j. anal. appl. (2022), 20:46 robust performances than the other five popular tests with relatively high power in detecting differences when the censoring rates in the groups are both low and high. the highest overall average powers under low and high censoring rates were produced by babalola-adeleke and fleming-harrington (1,1) tests respectively. hence, these two tests are the most suitable tests for diagnosing homogeneity of survival curves under these conditions. 1. introduction the rate at which survival analysis is advancing and gaining popularity in every field of study is pretty impressive. the nature of data obtained in the area of biostatistics has necessitated the growth in the volume of works done in the survival analysis [1-5]. survival analysis is also of massive use in engineering and social sciences fields [6-8]. a very predominant method in survival analysis is kaplan-meier method, which is capable of estimating the survivorship function for different sample sizes. several scholars have established its huge efficiency in capturing necessary survival details in cohort studies and otherwise. the kaplan-meier estimator is a nonparametric method that allows for the incorporation of censoring for the purpose of estimation of probabilities of survival [9-12]. more related and relevant research works have also been reported in the literature. the log-rank test is arguably the most popular test in testing for homogeneity of survival distribution. however, it may fail to recognize some crucial differences that exist among groups whereby the main difference takes place very early in the study or towards the end of the study [13].this is because it was proposed in order to give equal weight to all failures among the follow-up [14]. the shortfall of the log-rank test is in the assumption that the hazard ratio of the groups should be proportional along the follow-up period as that is the only condition that makes the test superior to others [15-17]. when this assumption is not met, that is when the hazard ratio is non-constant, the gehan-wilcoxon and tarone-ware tests can be more powerful than the log-rank test [18,19]. the peto-peto test is also efficient when the proportional hazard assumption is violated [10]. the strength of the fleming-harrington tests (f-h) is in its flexibility. unlike the other tests, it allows for the choice of weights and focuses on crossing the hazard ratios of groups [19]. different combinations of the weight, therefore, yield different tests entirely. [20] compared the statistical powers of some nonparametric tests and concluded that the peto-prentice generalized wilcoxon statistic performed best under the investigated situation. [15,21] examined the properties of the tests based on linear rank statistics and the effect of unequal censoring 3 int. j. anal. appl. (2022), 20:46 by using various combinations of censoring proportions, respectively. in the paper, wilcoxon test had the lowest relative power of all tests examined. [22,23] and [24] were interested in the comparison of the wilcoxon and the log-rank tests under different scenarios. [25] added more tests, which are the tarone-ware, peto-peto, and f-h tests to the comparison of the log-rank and wilcoxon tests when the sample size is quite small. it was concluded in the paper that the choice of weight function has a tremendous impact on the power of the tests under any given situation. the importance of simulations and monte carlo methods in modern research were the focus of [26]. [27] proposed a modified one-sample log-rank test, and a sample size formula was derived based on its exact variance to provide a study design that preserves the type i error. [28] discussed the versatile tests for comparing survival curves based on weighted log-rank statistics. [29] proposed a nonparametric test for the comparison of survival curves using the median. [30] examined the tests for comparing survival curves with right-censored data. in the study, the type i error rate of logrank test was equal or close to the nominal value. [31] developed a new method and demonstrated that this method outclassed some existing methods and relatively performed better under low and high censoring rates when the kaplan-meier survival curves are proportional. it was also ascertained that when there are crossing survival curves, the powers of the tests are relatively low since none of the tests gave statistical power in close of one. other relevant works on censoring and other methodologies are [1],[32-38]. thus, this paper considers a typical situation whereby the survival curves of the two groups are similar at the beginning of the study but gradually diverged towards the end. the censoring rates were categorized into two parts (low and high censoring rates). the censoring times among the groups were carefully chosen to fit into the intended survival pattern. all survival times were simulated from an exponential distribution. the outcome of this study will assist researchers as a further guide for their choice of tests when survival curves are disjointed towards the end. hence, the novelty of this study would be in comparing the relatively new babalola-adeleke test with some of the popular methods for checking homogeneity of kaplan-meier survival curves with disjointed ends under both high and low censoring rates. it is expected that the findings of this study would help the users of survival analysis as it will certainly further expose to them performances of the tests under consideration. it will also guide in decision making when confronted with the choosing of the most appropriate test to detect differences in survival curves 4 int. j. anal. appl. (2022), 20:46 with disjointed ends. to the best of our knowledge, this is the first study that would compare babalola-adeleke test with others under this particular situation. 2. methodology given that there are two groups, that is, groups 1 and 2, where the survival times were observed and recorded as j t . the number of observed failures (death) in group 1 and group 2 being j m 1 and j m 2 respectively, the number not experiencing the event of interest being jj mn 11 − and 2 2j j n m− for group 1 and group 2 respectively, and the number at risk is jjj nnn 21 += table 1. table used for test of equality of the survivorship function in two groups at observed survival time jt event/group 1 2 total number of death j m 1 j m 2 jj mm 21 + number not dying jj mn 11 − 2 2j jn m− 1 2 1 2j j j jn n m m+ − − number at risk j n 1 j n 2 jjj nnn 21 += the various multiple-group versions of the two-group test statistic is obtained by computing a weighted difference between the observed and the expected numbers of events. table 2 presents a k groups pattern for the test of equality. table 2. table used for test of equality of the survivorship function in k groups at observed survival time j t event/group 1 2 … k … k total number of death j m 1 jm2 … kj m … kjm jm number not dying jj mn 11 − jj mn 12 − … kj kj n m− … kjj mn −2 j jn m− number at risk j n 1 jn2 … kj n … kjn jn where, 1 2 ... j j j kj m m m m= + + + 1 2 ... j j j kj n n n n= + + + 5 int. j. anal. appl. (2022), 20:46 1 2 1 2 ... ... j j j j kj j j kj n m n n n m m m− = + + + − − − − based on the argument above, the test hypothesis considered is: 0 1 2 : ( ) ( )h s t s t= 1 1 2 : ( ) ( )h s t s t for the test statistics of the tests, see: [39-42] and [8]. the tests are based on some assumptions namely: censoring is unrelated to prognosis; the survival probabilities are equal for subjects recruited early and late in the study; the events happened at the times specified. simulation study the use of simulation study for the examination of statistical powers of tests under a variety of situations is a popular concept which is well reported in the literature. over the years, monte-carlo simulations have been employed for testing heterogeneity of survival distributions when the proportional hazard assumption is satisfied and when it is not. therefore, a monte carlo simulation to compare the statistical power of the log-rank, wilcoxon, tarone-ware, peto-peto, modified peto-peto, fleming-harrington(1,1), and babalola-adeleke tests was conducted. it is a known fact that due to the flexibility of the fleming-harrington test, there are several options for its weights. hence, for the purpose of placing weights of hazard in the middle, fleming-harrington (1,1) was selected since every other test either places equal weight across the board or places more weight at the beginning or towards the end. figure 1 shows the survival curves of two groups that have a similar pattern for some time but have a disjointed end. therefore, all the simulated datasets followed this pattern. figure 1. figure of the situation for consideration in the simulation study 6 int. j. anal. appl. (2022), 20:46 for each of the combination of the sample sizes, 5000 iterations were simulated in order to obtain statistically viable powers of the aforesaid tests. since the larger the number of iterations, the better the result. the estimated statistical power was obtained as the proportion of 5000 repeated random samples where the hypothesis of no difference in the survival curves (null hypothesis) at the 0.05 significance level is correctly rejected. 3. results considering the sub-situation with low censoring rates in both groups, the survival times in group 1 follow an exponential distribution with a mean of 4 (rate 0.25), and in group 2, the survival times follow an exponential distribution with mean 4(rate 0.25) as well. in order to get disconnected survival curves towards the end, if the survival time in group 2 is greater than or equal to 4, then the survival time is automatically simulated from an exponential distribution with a mean 40(rate 0.025). in order to have low censoring rates in the two groups, if the survival time is greater than the maximum survival time divided by 1.25 into both groups, then the observation was censored. these yielded an overall average censoring rate of 4.50% and 9.99% in groups 1 and 2, respectively. table 3 displays the result of the powers of the seven tests obtained from the simulation conducted for this sub-situation under low censoring rates alongside the censoring rates. the censoring rates in both groups decrease as the sample sizes increase. the same trend is also exhibited in mixed sample sizes. table 3. powers of the tests and censoring rates for the situation (low censoring rates) sample size log rank wilcoxon tarone ware peto peto modified peto-peto fleming harrington babalola adeleke censoring rates (%) 20,20 0.0798 0.0698 0.0732 0.0698 0.0688 0.1016 0.0810 8.3840 11.0760 40,40 0.1824 0.0878 0.1144 0.0884 0.0872 0.1906 0.1884 4.8805 9.7795 50,50 0.2386 0.1082 0.1428 0.1072 0.1062 0.2360 0.2462 4.0228 9.6964 60,60 0.2890 0.1142 0.1554 0.1132 0.1126 0.2706 0.2976 3.4980 9.7923 80,80 0.3914 0.1404 0.2096 0.1390 0.1386 0.3480 0.3998 2.8098 9.5333 100,100 0.4508 0.1610 0.2344 0.1578 0.1578 0.3916 0.4586 2.3252 9.6056 20,50 0.0784 0.0674 0.0668 0.0670 0.0674 0.1082 0.0796 8.5350 9.6608 50,20 0.1880 0.0852 0.1170 0.0836 0.0834 0.1756 0.1966 4.1364 11.0430 50,100 0.2690 0.1104 0.1516 0.1056 0.1054 0.2908 0.2774 4.0152 9.5900 100,50 0.3890 0.1374 0.2016 0.1346 0.1356 0.3218 0.3976 2.3938 9.5516 7 int. j. anal. appl. (2022), 20:46 from table 3, it is evident that the powers of all the tests increase as the sample size increase as the highest powers recorded for all the tests is obtained at sample size (100,100). the babalola-adeleke test has the highest power at the largest equal sample size, with a value of 0.4586. the babalola-adeleke test outperforms all the other tests at all sample sizes except when the sample sizes were (20,40), (40,40) and (20,50) for the fleming-harrington test. the peto-peto and the modified peto-peto produced similar results under this situation with just small differences in the powers of the two tests across all the sample sizes, which is not statistically significant judging by student t-test. however, the peto-peto test still outperforms the modified peto-peto under equal sample sizes. the statistical description of table 3 is given in table 4. table 4. descriptive statistics of the power of the tests for the situation (low censoring rates) log-rank wilcoxon taroneware peto peto modified peto-peto fleming harrington babalola adeleke mean 0.2556 0.1082 0.1466 0.1066 0.1063 0.2435 0.2622 standard error 0.0406 0.0099 0.0178 0.0096 0.0097 0.0312 0.0413 median 0.2538 0.1093 0.1472 0.1064 0.1058 0.2533 0.2618 standard deviation 0.1283 0.0313 0.0563 0.0304 0.0306 0.0988 0.1306 kurtosis -1.0388 -0.9181 -0.9519 -0.9273 -0.9537 -1.1006 -1.0237 skewness 0.0424 0.2763 0.0984 0.2998 0.3179 -0.1156 -0.0025 range 0.3724 0.0936 0.1676 0.0908 0.0904 0.2900 0.3790 minimum 0.0784 0.0674 0.0668 0.0670 0.0674 0.1016 0.0796 maximum 0.4508 0.1610 0.2344 0.1578 0.1578 0.3916 0.4586 table 4 shows that the babalola-adeleke test has the highest mean of 0.2622 as the average power of the method across all the combinations of sample sizes and the standard error of 0.0413. this is followed by the log-rank test with an average statistical power of 0.2556 with a standard error of 0.0406, while the modified peto-peto test resulted in the lowest average statistical power 0.1063 with standard error 0.0097. the descriptive statistics of the modified peto-peto and peto-peto tests are similar. the median powers for the tests arranged in descending order are 0.2618, 0.2538, 0.2533, 0.1472, 0.1093, 0.1064, and 0.1058, which are results of the babalola-adeleke test, log-rank, fleming-harrington, tarone-ware, wilcoxon tests, peto-peto and modified peto-peto, respectively. 8 int. j. anal. appl. (2022), 20:46 for skewness, the result shows that the power of all the tests is positively skewed except for the babalola-adeleke test and fleming-harrington test, which indicates that both the mean and the median are less than the mode of the powers of the tests. the negative values of the kurtosis indicate that the distribution of the powers has lighter tails and a flatter peak than the normal distribution. figure 2. a chart showing the statistical powers of the tests under the situation with low censoring rates 3.1 the situation with high censoring rates in the presence of high censoring rates in both groups, the survival times in group 1 follow an exponential distribution with a mean of 4 (rate 0.25), and in group 2, the survival times follow an exponential distribution with mean 4(rate 0.25) as well. in order to get disconnected survival curves towards the end, if the survival time in group 2 is greater than or equal to 4, then the survival time is automatically simulated from an exponential distribution with a mean 40(rate 0.025). additionally, in order to have high censoring rates in both groups, if the survival time is greater than the minimum survival time plus two. that is, (the minimum survival time in both groups +2), then the observation was censored. these yielded an overall average censoring rate of 59.3096% and 55.6807% in groups 1 and 2, respectively. these censoring rates are quite high since more than half of the cohorts in both groups censored. the result of the powers of the tests when there are high censoring rates is displayed in table 5. unlike the first sub-situation with low censoring rates, the censoring rates in both groups increase with sample size. 9 int. j. anal. appl. (2022), 20:46 table 5. powers of the tests and censoring rates for the situation (high censoring rates) sample size log rank wilcoxon taroneware peto peto modified peto-pet o fleming harringto n babalolaadeleke censoring rates(%) 20,20 0.0546 0.0526 0.0530 0.0520 0.0526 0.0692 0.0554 57.8950 53.9850 40,40 0.0724 0.068 0.0712 0.0664 0.0664 0.0834 0.0722 59.2980 55.7400 50,50 0.0796 0.0802 0.0806 0.0784 0.0786 0.0918 0.0796 59.3356 56.0164 60,60 0.0964 0.0924 0.0916 0.0902 0.0902 0.1004 0.0964 59.7593 56.1180 80,80 0.1118 0.1056 0.1098 0.1038 0.1038 0.1178 0.1118 59.8665 56.3325 100,100 0.1264 0.1230 0.1250 0.1180 0.1180 0.1378 0.1264 59.9832 56.4456 20,50 0.0576 0.0592 0.0584 0.0576 0.0574 0.0680 0.0576 57.9630 55.7360 50,20 0.0786 0.0704 0.0750 0.0698 0.0700 0.1068 0.0788 59.4056 53.8740 50,100 0.0968 0.0934 0.094 0.0890 0.0892 0.0958 0.0966 59.5496 56.5508 100,50 0.1092 0.1022 0.1048 0.1002 0.1000 0.1192 0.1092 60.0398 56.0084 generally, the powers of all the tests are low. even at that, the fleming-harrington still outperforms the other tests. as expected, the powers increase as the sample sizes increase. this could indicate that at much larger sample sizes, the powers of the tests could attain higher values than the ones reported. figure 3. a chart showing the statistical powers of the tests under the situation with high censoring rates 10 int. j. anal. appl. (2022), 20:46 figure 3 above further reiterates the outstanding performance of the fleming-harrington test under this situation and censoring rates. it apparently outclasses all the other tests when the sample sizes are the same in the two groups. the value of its power is only in the range of the other tests when the sample size is 50 in the first group and 100 in the second group. in any other sample size, it outperforms all the other tests. table 6. descriptive statistics of the power of the tests for the situation (high censoring rates) log rank wilcoxon taroneware peto peto modified peto-peto fleming harrington babalola-a deleke mean 0.0883 0.0847 0.0863 0.0825 0.0826 0.0990 0.0884 standard error 0.0075 0.0071 0.0073 0.0068 0.0067 0.0071 0.0075 median 0.0880 0.0863 0.0861 0.0837 0.0839 0.0981 0.0880 standard deviation 0.0238 0.0224 0.0230 0.0214 0.0213 0.0223 0.0236 kurtosis -1.0034 -0.8489 -0.7644 -0.9598 -0.9645 -0.5284 -1.0139 skewness 0.0534 0.1778 0.1554 0.1432 0.1486 0.1630 0.0713 range 0.0718 0.0704 0.0720 0.0660 0.0654 0.0698 0.0710 minimum 0.0546 0.0526 0.0530 0.0520 0.0526 0.0680 0.0554 maximum 0.1264 0.1230 0.1250 0.1180 0.1180 0.1378 0.1264 from table 6, the fleming-harrington test has the highest mean of 0.0990 as the average power of the method across all the combinations of sample sizes and the standard error of 0.0071. this is followed by the babalola-adeleke test with an average statistical power of 0.0884 with a standard error of 0.0075, while the peto-peto test resulted in the lowest average statistical power 0.0825 with standard error 0.0068. as in the case of low censoring rates in this situation, the descriptive statistics of the modified peto-peto and peto-peto tests are similar. however, the modified peto-peto performs better than peto-peto under the condition. the median powers for the tests arranged in descending order are 0.0981, 0.0880, 0.0880, 0.0863, 0.0861, 0.0839, and 0.0837, which are results for fleming-harrington, babalola-adeleke, log-rank, wilcoxon, tarone-ware, modified peto-peto, and peto-peto, respectively. for skewness and kurtosis, the result shows that the power of all the tests is positively skewed with negative kurtosis. 3.2 application of the tests to real-life data survival in patients with acute myelogenous leukemia was studied with the interest of knowing the impact of the standard course of chemotherapy extension [43,44]. the variables in the study were 11 int. j. anal. appl. (2022), 20:46 time, which is the survival or censoring time, and event (recurrence of aml cancer) is indicated by the variable "status" 1 = event (recurrence) and 0 = no event (censored). the treatment group was represented by the variable "x", which indicates if maintenance chemotherapy was given (maintained) or not (non-maintained). this is a popular data set with 8.33% patients censored in group 1(maintained) and 36.36% in the second group (non-maintained). the property of this data set is "slightly" similar to the situation under study as the survival curves have a similar pattern from the beginning of the study till about the week 45(though not exactly the same form from the beginning). then homogeneity of the survival curves can be investigated. this is the closest real-life data we have at our disposal for the situation under study. the test hypothesis is: 0 maintained nonmaintained h : ( ) ( )s t s t= 1 maintained nonmaintained h : ( ) ( )s t s t table 7. comparison of the results of the different tests using the acute myelogenous leukemia method log rank wilcoxon tarone ware peto-peto modified peto-peto fleming harrington babalola adeleke 2  value 3.3964 2.7233 2.9816 3.5880 3.5670 1.4310 3.6236 p-value 0.0654 0.0988 0.0842 0.0582 0.0590 0.2316 0.0570 table 7 clearly shows that all the tests validate that the kaplan-meier survival curves of those who were maintained and those who were not maintained are not significantly different as none of the p-values is less than 0.05. all the tests yielded very low chi-squared values. this result is consistent with the results earlier reported. 4. conclusion generally, the powers of all the tests are low. even at that, the fleming-harrington still outperforms the other tests. the powers increase as the sample sizes increase. this could indicate that at much larger sample sizes, the powers of the tests could attain higher values than the ones reported. a general comment about this situation, that is when the survival curves are separate towards the end is that, the powers of the tests are also low as expected. this means that it is quite difficult for the different tests to correctly diagnose survival curves because of the similarity of the curves for a larger part of the study (not until towards the end of the study). the low values of the 12 int. j. anal. appl. (2022), 20:46 power are expected, and it has been reported by other researches as well. generally, across all the sample sizes, the overall average of the power of the entire tests combined is lower when dealing with high censoring rates (0.0874) than when dealing with lower censoring rate (0.1756). authors’ contributions b.t. conceived the idea presented. b.t., o.a. developed the theory and performed the computations. b.t., o.y., a.o., o.d., o.f, and r.e. verified the analytical methods. b.t., k.a.,o., o., s.o., m.i., m., and s.e. wrote the manuscript with input from all authors. r.a. and o.y. supervised the findings of this paper. all authors discussed the results and contributed to the final manuscript. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] b.t. babalola, r.e. ogunsakin, o.a. egbon, et al. a simulation based comparative study of some tests for checking homogeneity of non-crossing survival curves under high censoring rates, j. appl. probab. stat. 17 (2022), 87-99. [2] j.c. goldsack, a. coravos, j.p. bakker, et al. verification, analytical validation, and clinical validation (v3): the foundation of determining fit-for-purpose for biometric monitoring technologies (biomets), npj digit. med. 3 (2020), 55. https://doi.org/10.1038/s41746-020-0260-4. [3] j. lepš, p. šmilauer, biostatistics with r: an introductory guide for field biologists, cambridge university press, cambridge, (2020). [4] k. sumathi, d. balakrishnan, v. naveen, et al. talent flow employee analysis based turnover prediction on survival analysis, ann. roman. soc. cell biol. 25 (2021), 3844-3857. [5] t. saegusa, z. zhao, h. ke, et al. detecting survival-associated biomarkers from heterogeneous populations, sci rep. 11 (2021), 3203. https://doi.org/10.1038/s41598-021-82332-y. [6] z. cai, y. wang, h. cao, et al. life prediction of self-locking nut for aeroengine based on survival analysis and bayesian network, in: 2020 ieee international conference on industrial engineering and engineering management (ieem), ieee, singapore, singapore, 2020: pp. 414–418. https://doi.org/10.1109/ieem45057.2020.9309737. [7] s. nematolahi, s. nazari, z. shayan, et al. improved kaplan-meier estimator in survival analysis based on partially rank-ordered set samples, comput. math. methods med. 2020 (2020), 7827434. https://doi.org/10.1155/2020/7827434. https://doi.org/10.1038/s41746-020-0260-4 https://doi.org/10.1038/s41598-021-82332-y https://doi.org/10.1109/ieem45057.2020.9309737 https://doi.org/10.1155/2020/7827434 13 int. j. anal. appl. (2022), 20:46 [8] b.t. babalola, w.b. yahya, effects of collinearity on cox proportional hazard model with time dependent coefficients: a simulation study, j. biostat. epidemiol. 5(2020), 172-182. https://doi.org/10.18502/jbe.v5i2.2348. [9] e. ilker, a. sulaiman, a. rukayya, the kaplan meier estimate in survival analysis, biometrics biostat. int. j. 5 (2017), 00128. https://doi.org/10.15406/bbij.2017.05.00128. [10] d.g. kleinbaum, m. klein, survival analysis a self-learning text, springer, new york, 44-66, (2005). [11] c.j. pelz, j.p. klein, analysis of survival data: a comparison of three major statistical packages (sas, spss and bmdp). working paper (medical college of wisconsin, milwaukee). rep.17: 1-6. (1996). https://www.mcw.edu/-/media/mcw/departments/biostatistics/tr017.pdf. [12] e.l. kaplan, p. meier, nonparametric estimation from incomplete observations, j. amer. stat. assoc. 53 (1958), 457-481. [13] j. klein, j. rizzo, m.-j. zhang, n. keiding, statistical methods for the analysis and presentation of the results of bone marrow transplants. part i: unadjusted analysis, bone marrow transplant. 28 (2001), 909–915. https://doi.org/10.1038/sj.bmt.1703260. [14] e.t. lee, j.w. wang, statistical methods for survival data analysis, john wiley & sons inc. new jersey, (2003). [15] t.r. fleming, d.p. harrington, m. o’sullivan, supremum versions of the log-rank and generalized wilcoxon statistics, j. amer. stat. assoc. 82 (1987), 312–320. https://doi.org/10.1080/01621459.1987.10478435. [16] j.w. lee, some versatile tests based on the simultaneous use of weighted log-rank statistics, biometrics. 52 (1996), 721-725. https://doi.org/10.2307/2532911. [17] s. buyske, r. fagerstrom, z. ying, a class of weighted log-rank tests for survival data when the event is rare, j. amer. stat. assoc. 95 (2000), 249–258. https://doi.org/10.1080/01621459.2000.10473918. [18] r.e. tarone, j. ware, on distribution-free tests for equality of survival distributions, biometrika. 64 (1977), 156–160. https://doi.org/10.1093/biomet/64.1.156. [19] m.s. pepe, t.r. fleming, weighted kaplan-meier statistics: a class of distance tests for censored survival data, biometrics. 45 (1989), 497-507. https://doi.org/10.2307/2531492. [20] r.b. latta, a monte carlo study of some two-sample rank tests with censored data, j. amer. stat. assoc. 76 (1981), 713–719. https://doi.org/10.1080/01621459.1981.10477710. [21] m.s. beltangady, r.f. frankowski, effect of unequal censoring on the size and power of the logrank and wilcoxon types of tests for survival data, stat. med. 8 (1989), 937–945. https://doi.org/10.1002/sim.4780080805. [22] e. letón, p. zuluaga, equivalence between score and weighted tests for survival curves, commun. stat. – theory methods. 30 (2001), 591–608. https://doi.org/10.1081/sta-100002138. https://doi.org/10.18502/jbe.v5i2.2348 https://doi.org/10.15406/bbij.2017.05.00128 https://www.mcw.edu/-/media/mcw/departments/biostatistics/tr017.pdf https://doi.org/10.1038/sj.bmt.1703260 https://doi.org/10.1080/01621459.1987.10478435 https://doi.org/10.2307/2532911 https://doi.org/10.1080/01621459.2000.10473918 https://doi.org/10.1093/biomet/64.1.156 https://doi.org/10.2307/2531492 https://doi.org/10.1080/01621459.1981.10477710 https://doi.org/10.1002/sim.4780080805 https://doi.org/10.1081/sta-100002138 14 int. j. anal. appl. (2022), 20:46 [23] e. letón, p. zuluaga, relationships among tests for censored data, biom. j. 47 (2005), 377–387. https://doi.org/10.1002/bimj.200410115. [24] a. akbar, g.r. pasha, properties of kaplan-meier estimator: group comparison of survival curves, eur. j. sci. res. 32 (2009), 391–397. [25] t. jurkiewicz, e. wycinka, significance tests of differences between two crossing survival curves for small samples. acta univ. lodziensis folia oecon. 255 (2011), 114-119. http://hdl.handle.net/11089/690. [26] p.c. austin, generating survival times to simulate cox proportional hazards models with time-varying covariates, stat. med. 31 (2012), 3946–3958. https://doi.org/10.1002/sim.5452. [27] j. wu, a new one-sample log-rank test, j. biometrics biostat. 05 (2014), 1000210. https://doi.org/10.4172/2155-6180.1000210. [28] t.g. karrison, versatile tests for comparing survival curves based on weighted log-rank statistics, stata j. 16 (2016), 678–690. https://doi.org/10.1177/1536867x1601600308. [29] z. chen, g. zhang, comparing survival curves based on medians, bmc med. res. methodol. 16 (2016), 33. https://doi.org/10.1186/s12874-016-0133-3. [30] p.g. karadeniz, i. ercan, examining tests for comparing survival curves with right censored data, stat. transition. new ser. 18 (2017), 311–328. https://doi.org/10.21307/stattrans-2016-072. [31] b.t. babalola, r.a. adeleke, o.y. halid, et al. statistical powers of an alternative test for comparison of survival distributions with crossed survival curves in the presence of censoring: a simulation study, int. j. civil eng. technol. 10 (2019), 366-379. [32] m. stevenson, an introduction to survival analysis, epicentre, ivabs. massey massey university, (2009). http://www.massey.ac.nz/massey/fms/colleges/college%20of%20sciences/epicenter/docs/asvcs/ste venson_survival_analysis_195_721.pdf. [33] x. wang, f. bai, h. pang, et al. bias-adjusted kaplan–meier survival curves for marginal treatment effect in observational studies, j. biopharmaceutical stat. 29 (2019), 592–605. https://doi.org/10.1080/10543406.2019.1633659. [34] r.l.m.c. martinez, j.d. naranjo, a pretest for choosing between logrank and wilcoxon tests in the two-sample problem, metron. 68 (2010), 111–125. https://doi.org/10.1007/bf03263529. [35] j. xie, c. liu, adjusted kaplan–meier estimator and log-rank test with inverse probability of treatment weighting for survival data, stat. med. 24 (2005), 3089–3110. https://doi.org/10.1002/sim.2174. [36] a. winnett, p. sasieni, adjusted nelson–aalen estimates with retrospective matching, j. amer. stat. assoc. 97 (2002), 245–256. https://doi.org/10.1198/016214502753479383. [37] s. galimberti, p. sasieni, m.g. valsecchi, a weighted kaplan-meier estimator for matched data with application to the comparison of chemotherapy and bone-marrow transplant in leukaemia, stat. med. 21 (2002), 3847–3864. https://doi.org/10.1002/sim.1357. https://doi.org/10.1002/bimj.200410115 http://hdl.handle.net/11089/690 https://doi.org/10.1002/sim.5452 https://doi.org/10.4172/2155-6180.1000210 https://doi.org/10.1177/1536867x1601600308 https://doi.org/10.1186/s12874-016-0133-3 https://doi.org/10.21307/stattrans-2016-072 http://www.massey.ac.nz/massey/fms/colleges/college%20of%20sciences/epicenter/docs/asvcs/stevenson_survival_analysis_195_721.pdf http://www.massey.ac.nz/massey/fms/colleges/college%20of%20sciences/epicenter/docs/asvcs/stevenson_survival_analysis_195_721.pdf https://doi.org/10.1080/10543406.2019.1633659 https://doi.org/10.1007/bf03263529 https://doi.org/10.1002/sim.2174 https://doi.org/10.1198/016214502753479383 https://doi.org/10.1002/sim.1357 15 int. j. anal. appl. (2022), 20:46 [38] b.t. babalola, r.a. adeleke, o.y. halid, et al. an alternative test for comparison of survival distributions with proportional hazard functions in the presence of low censoring rates, j. appl. stat. probab. 15 (2020), 61-75. [39] x. lin, q. xu, a new method for the comparison of survival distributions, pharmaceut. stat. 9 (2010), 67–76. https://doi.org/10.1002/pst.376. [40] j. shanahan, a new method for the comparison of survival distributions, master's thesis, university of south carolina, (2013). https://scholarcommons.sc.edu/etd/555. [41] c. dardis, package "survmisc". (2018). https://cran.r-project.org/web/packages/survmisc/survmisc.pdf. [42] h. uno, l. tian, b. claggett, l.j. wei, a versatile test for equality of two survival functions based on weighted differences of kaplan-meier curves, stat. med. 34 (2015), 3680–3695. https://doi.org/10.1002/sim.6591. [43] r.g. miller, survival analysis, john wiley & sons, hoboken, 1981. [44] s.h. embury, l. elias, p.h. heller, et al. remission maintenance therapy in acute myelogenous leukaemia, western j. med. 126 (1977), 267-272. https://doi.org/10.1002/pst.376 https://scholarcommons.sc.edu/etd/555 https://cran.r-project.org/web/packages/survmisc/survmisc.pdf https://doi.org/10.1002/sim.6591 international journal of analysis and applications volume 19, number 2 (2021), 252-263 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-252 fuzzy zagreb indices and some bounds for fuzzy zagreb energy mahesh kale∗, s. minirani department of basic sciences and humanities, mpstme, svkm’s nmims deemed to be university, mumbai, india ∗corresponding author: mnk.maths@gmail.com abstract. topological indices m1/m2 known as first/second zagreb indices are defined as the sum of the sum/product of degrees of pairs of adjacent vertices of a simple graph. these indices and their properties have been studied in detail under chemical graph theory. in this paper we introduce the concepts of first, second and hyper zagreb indices of fuzzy graphs. we also study the zagreb matrices and the associated zagreb energies of fuzzy graphs. some bounds for these energies are also obtained. 1. introduction topological indices are numerical quantities of structural molecular graphs. they are studied and applied in various fields by engineers, pharmacist, graph theorist and mathematicians. i. gutman [1] in 1972, introduced the first zagreb index and randec in [2] introduced randec index, which are oldest among the topological indices. i gutman, eliasi, kulli, kc das and many other experts have contributed in the developments of different zagreb indices, randic indices of simple graphs. in case of classical graphs, both the vertices and edges have membership value one, but in case of fuzzy graphs both vertices and edges are equally important along with their fuzzy membership values. if the description of objects or their relationships or both are vague in nature, then we design a fuzzy graph model. in 1965, zadeh [3] introduced the concept of fuzzy sets and fuzzy relations. further rosenfeld [4], zimmerman [5], thomson [6] and many experts in [7–12] have contributed significantly in the developments of fuzzy graphs. received january 12th, 2021; accepted february 15th, 2021; published march 2nd, 2021. 2010 mathematics subject classification. 05c07, 05c072, 15b15. key words and phrases. fuzzy graphs; fuzzy zagreb indices; fuzzy zagreb matrices; fuzzy zagreb energies. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 252 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-252 int. j. anal. appl. 19 (2) (2021) 253 in [13], anjali and mathew introduced energy of fuzzy graphs and in [14] authors introduced laplacian energy of fuzzy graphs. in [15, 16], authors discussed weiner index of fuzzy graph and found relationships between connectivity index and wiener index of a fuzzy graph. recently, authors in [17, 18], discussed transitive blocks, hamiltonian fuzzy graphs and their applications in fuzzy interconnection networks, human trafficking. the paper is structured as follows: in section 2, we discuss the preliminary definitions required for the development of the content. in section 3, we introduce the important definitions of fuzzy zagreb indices, fuzzy zagreb matrices and corresponding energies. section 4 provides some bounds for the fuzzy zagreb energies. 2. preliminaries in this section, we recall some definitions of zagreb indices and some notions of fuzzy graphs which will play an important role in the subsequent sections of the paper. basics of zagreb indices can be referred in [19, 20]. basics of graphs and fuzzy graphs can be referred in [4, 21]. definition 2.1. let g=(v,e) be a simple graph. degree of vertex u is denoted by du and e = uv ∈ e is an edge in graph g. first zagreb index m1(g) is defined as m1(g) = ∑ uv∈e [du + dv] = ∑ u∈v d2u second zagreb index m2(g) is defined as m2(g) = ∑ uv∈e du.dv and hyper zagreb index hm(g) is defined as hm(g) = ∑ uv∈e [du + dv] 2 definition 2.2. a fuzzy graph g=(v,σ,µ); which also can simply be denoted by g=(σ,µ), is a graph with vertex-membership function σ : v → [0, 1] and edge-strength function µ : v ×v → [0, 1] such that it satisfies the relation µ(x,y) ≤ min{σ(x),σ(y)}, ∀x,y ∈ v . corresponding crisp graph is denoted by g=(σ∗,µ∗). also, we denote strength of vertex u by µ(u), it represents the minimum of strengths of edges incident to the vertex u µ(u) = ∧ uvi∈µ∗ µ(u,vi) 3. fuzzy zagreb indices in this section, we introduce the definitions of fuzzy zagreb first index, fuzzy zagreb second index and fuzzy zagreb hyper index along with associated fuzzy zagreb matrices and the fuzzy zagreb energies. these definitions are required to discuss the main results. int. j. anal. appl. 19 (2) (2021) 254 definition 3.1. the fuzzy zagreb first index of g=(σ,µ) is defined as fm1 (g) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] equivalently the index can also be defined as fm1 (g) = ∑ u∈σ∗ [σ(u)µ(u)du] definition 3.2. the fuzzy zagreb second index of g=(σ,µ) is defined as fm2 (g) = ∑ uv∈µ∗ [σ(u)µ(u).σ(v)µ(v)] definition 3.3. the fuzzy hyper zagreb index of g=(σ,µ) is defined as fhm (g) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] 2 definition 3.4. if g=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un} then first fuzzy zagreb matrix is defined as f z (1) = (fz(1))i,j, where (fz(1))i,j =   σ(ui)µ(ui) + σ(uj)µ(uj) , if i 6= j and ui.uj ∈ µ∗ 0 , if ui.uj /∈ µ∗ 0 , if i = j and second fuzzy zagreb matrix is defined as f z (2) = (fz(2))i,j, where (fz(2))i,j =   σ(ui)µ(ui).σ(uj)µ(uj) , if i 6= j and ui.uj ∈ µ∗ 0 , if ui.uj /∈ µ∗ 0 , if i = j definition 3.5. if g=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, if f z (1) is the first fuzzy zagreb matrix with its eigen values ξ (1) 1 , ξ (1) 2 , ... , ξ (1) n then the first fuzzy zagreb energy is defined as fz e (1) = n∑ i=1 |ξ(1)i | definition 3.6. if g=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, if f z (2) is the second fuzzy zagreb matrix with ξ (2) 1 , ξ (2) 2 , ... , ξ (2) n as its eigen values then the second fuzzy zagreb energy is defined as fz e (2) = n∑ i=1 |ξ(2)i | example 3.1. consider the fuzzy graphs g=(σ,µ) as shown in fig.1 here σ∗ = {u1,u2,u3,u4,u5,u6} with membership values σ(u1) = 0.4, σ(u2) = 0.2, σ(u3) = 0.6, σ(u4) = 0.5, σ(u5) = 0.7, σ(u6) = 0.8 and strengths of edges µ(u1u2) = 0.1, µ(u2u3) = 0.2, µ(u2u4) = 0.1, µ(u3u4) = 0.4, µ(u4u5) = 0.4, µ(u5u6) = 0.5, µ(u5u1) = 0.3, µ(u1u6) = 0.3 then we get fm1 = 1.68, fm2 = 0.1107 and fhm = 0.0468. the first and second fuzzy zagreb matrices are given by int. j. anal. appl. 19 (2) (2021) 255 figure 1. f z (1) = u1 u2 u3 u4 u5 u6    u1 0 0.06 0 0 0.25 0.28 u2 0.06 0 0.14 0.07 0 0 u3 0 0.14 0 0.17 0 0 u4 0 0.07 0.17 0 0.26 0 u5 0.25 0 0 0.26 0 0.45 u6 0.28 0 0 0 0.45 0 and f z (2) = u1 u2 u3 u4 u5 u6    u1 0 0.0008 0 0 0.0084 0.0096 u2 0.0008 0 0.024 0.001 0 0 u3 0 0.024 0 0.006 0 0 u4 0 0.001 0.006 0 0.0105 0 u5 0.0084 0 0 0.0105 0 0.0504 u6 0.0096 0 0 0 0.0504 0 eigen values of fz (1) are given by ξ (1) 1 = 0.7067, ξ (1) 2 = −0.5276, ξ (1) 3 = −0.2595, ξ (1) 4 = 0.2487, ξ (1) 5 = −0.1704, ξ (1) 6 = 0.0021 hence the first fuzzy zagreb energy is fze (1) = 1.915. eigen values of fz (2) are given by ξ (2) 1 = 0.0544, ξ (2) 2 = −0.0515, ξ (2) 3 = 0.0249, ξ (2) 4 = −0.0245, ξ (2) 5 = −0.00404, ξ(2)6 = 0.00069 hence the second fuzzy zagreb energy is fze (2) = 0.16. 4. main results in this paper we will discuss fuzzy zagreb first index and the corresponding first fuzzy zagreb matrix and first fuzzy zagreb energy. the analogous study of second fuzzy zagreb quantities along with first and second fuzzy eztrada zagreb energies will be communicated in forthcoming paper. int. j. anal. appl. 19 (2) (2021) 256 for simplicity of notations, fuzzy zagreb first matrix will be denoted by fz, its (i,j) th element by (fz)i,j and the corresponding fuzzy zagreb energy by fze = n∑ i=1 |ξi| , where ξ1 ≥ ξ2 ≥ . . . ≥ ξn are eigen values of fz. here fz is a real symmetric matrix and hence all its eigen values are real. for non-negative integer k, the kth spectral moment of fz is given by nk = n∑ i=1 (ξi) k = tr(fzk) although elementary, the following result deserves tobe stated as: theorem 4.1. if g=(v,σ,µ) and h=(w,σ′,µ′) are fuzzy isomorphic graphs then fm1 (g) = fm1 (h) and fm2 (g) = fm2 (h) proof. if g=(v,σ,µ) and h=(w,σ′,µ′) are fuzzy isomorphic graphs then there exists a bijective function f : v → w such that for every u ∈ v and uv ∈ µ∗, σ(u) = σ′(f(u)) and µ(uv) = µ′(uv). hence we can easily conclude that fm1 (g) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] = ∑ f(u)f(v)∈µ′∗ [σ′(f(u))µ′(f(u)) + σ′(f(v))µ′(f(v))] = fm1 (h) and also fm2 (g) = ∑ uv∈µ∗ [σ(u)µ(u).σ(v)µ(v)] = ∑ f(u)f(v)∈µ′∗ [σ′(f(u))µ′(f(u)).σ′(f(v))µ′(f(v))] = fm2 (h) � theorem 4.2. if g=(v,σ,µ) with v = {u1,u2, . . . ,un} is a fuzzy graph and fz is its fuzzy zagreb matrix then (1) n1 = tr(fz ) = 0 (2) n2 = tr(fz 2) = 2.fhm int. j. anal. appl. 19 (2) (2021) 257 proof. (1) by definition of fz, all its diagonal elements are zero, hence trace of fz is zero. (2) the diagonal elements of fz2 are given by (fz2)i,i = n∑ j=1 (fz)i,j.(fz)j,i = n∑ j=1 (fz)2i,j = ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 hence n2 = tr(fz 2) = n∑ i=1 ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 = 2. ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 = 2.fhm � theorem 4.3. let g=(v,σ,µ) be a fuzzy graph with |v | = n vertices. let fz be the corresponding fuzzy zagreb matrix with eigen values ξ1, ξ2, . . . , ξn and fze is its fuzzy zagreb energy then√ 2(fhm ) + n(n− 1)|det(fz )| 2 n ≤ fze ≤ √ 2n.(fhm ) proof. upper bound: applying cauchy-schwartz inequality to the n numbers (1, 1, . . . , 1) and (ξ1,ξ2, . . . ,ξn) we get, n∑ i=1 (1. |ξi|) ≤ ( n∑ i=1 1 )1/2 . ( n∑ i=1 |ξi| 2 )1/2 (4.1) n∑ i=1 |ξi| ≤ √ n. √√√√ n∑ i=1 |ξi| 2 also we have, ( n∑ i=1 |ξi| )2 = n∑ i=1 |ξi| 2 + 2. ∑ 1≤i<j≤n |ξi.ξj| = 0 hence (4.2) n∑ i=1 |ξi| 2 = −2. ∑ 1≤i<j≤n |ξi.ξj| int. j. anal. appl. 19 (2) (2021) 258 on comparing the coefficients of n∏ i=1 (ξ − ξi) = |fz − ξ.i|, we get (4.3) ∑ 1≤i<j≤n |ξi.ξj| = − ∑ ui.uj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 using (4.3), eq.(4.2) becomes (4.4) n∑ i=1 |ξi| 2 = 2. ∑ ui.uj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 hence eq.(4.1) becomes n∑ i=1 |ξi| ≤ √ 2n. ∑ ui.uj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 (4.5) fze (g) ≤ √ 2n.(fhm ) lower bound: (fze (g)) 2 = ( n∑ i=1 |ξi| )2 = n∑ i=1 |ξi| 2 + 2. ∑ 1≤i<j≤n |ξi.ξj| (fze (g)) 2 = 2. ∑ ui.uj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 + 2. n(n− 1) 2 .am{|ξi.ξj|} hence (4.6) (fze (g)) 2 ≥ 2.(fhm ) + n(n− 1).gm{|ξi.ξj|} where gm{|ξi.ξj|} =   ∏ 1≤i<j≤n |ξi.ξj|   2 n(n− 1) = ( n∏ i=1 |ξi| n−1 ) 2 n(n− 1) = ( n∏ i=1 |ξi| )2/n = (det fz) 2/n hence eq.(4.6) becomes (fze (g)) 2 ≥ 2.(fhm ) + n(n− 1). (det fz)2/n (4.7) fze (g) ≥ √ 2.(fhm ) + n(n− 1). (det fz)2/n int. j. anal. appl. 19 (2) (2021) 259 � theorem 4.4. let g=(v,σ,µ) be a fuzzy graph and σ∗ = {u1,u2, . . . ,un}. if fz is the corresponding fuzzy zagreb matrix with eigen values ξ1,ξ2, . . . ,ξn and fze is its fuzzy zagreb energy then fze ≤ 2(fhm ) n + √√√√(n− 1) { 2(fhm ) − ( 2 n (fhm ) )2} proof. first we recall some elementary results. if a = (aij)n×n is a symmetric matrix with diagonal elements all zero then λmax ≥ 2n ∑ 1≤i<j≤n aij, where λmax is the maximum eigenvalue of a. hence in case of fuzzy zagreb matrix fz, if ξ1 ≥ ξ2 ≥ ···≥ ξn are its eigen values, then (4.8) |ξ1| ≥ 2 n ∑ uiuj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] = 2 n fm1 using eq.(4.4) of theorem 4.3, (4.9) n∑ i=1 |ξi| 2 = 2. ∑ uiuj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 = 2(fhm ) (4.10) n∑ i=2 |ξi| 2 = 2 (fhm ) −|ξ1| 2 using cauchy-schwartz inequality to the (n− 1) numbers (1, 1, . . . , 1) and (ξ2, ξ3, . . . ,ξn) we get n∑ i=2 (1. |ξi|) ≤ ( n∑ i=2 1 )1/2 . ( n∑ i=2 |ξi| 2 )1/2 (4.11) n∑ i=2 |ξi| ≤ √√√√(n− 1) . ( n∑ i=2 |ξi| 2 ) using eq.(4.10), eq.(4.11) becomes n∑ i=2 |ξi| ≤ √ (n− 1) . { 2(fhm ) −|ξ1| 2 } n∑ i=2 |ξi| + |ξ1| ≤ |ξ1| + √ (n− 1) . { 2(fhm ) −|ξ1| 2 } (4.12) fze ≤ |ξ1| + √ (n− 1) . { 2(fhm ) −|ξ1| 2 } as the function g(x) = x + √ (n− 1) .{2(fhm ) −x2} is decreasing function of x in the interval 2 n (fhm ) < x ≤ √ 2(fhm ) and also |ξ1| ≥ 2n fm1 , we get√ 2 n (fhm ) ≤ 2 n (fhm ) ≤ 2 n fm1 ≤ |ξ1| ≤ √ 2(fhm ) int. j. anal. appl. 19 (2) (2021) 260 hence eq.(4.12) becomes (4.13) fze ≤ 2 n (fhm ) + √√√√(n− 1) . { 2(fhm ) − ( 2 n (fhm ) )2} � theorem 4.5. if g=(v,σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, fz is the corresponding fuzzy zagreb matrix with eigen values ξ1 ≥ ξ2 ≥ ···≥ ξn > 0 and fze is its fuzzy zagreb energy then fze ≥ 2 n fm1 + (n− 1) + ln [ n |det (fz )| 2fm1 ] proof. consider g=(v,σ,µ) is a fuzzy graph and ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 are eigen values of corresponding fuzzy zagreb matrix fz . for simplicity, consider a function f(x) = 1 − x − ln(x), x > 0. elementary calculations shows thatf(x)is decreasing function in (0, 1] and it is increasing function for x ≥ 1. hence f(x) ≥ f(1) = 0 for x > 0 gives, x ≥ 1 + ln(x) for x > 0. fze = n∑ i=1 |ξi| = |ξ1| + n∑ i=2 |ξi| ≥ |ξ1| + n∑ i=2 [1 + ln |ξi|] = |ξ1| + (n− 1) + ln [ n∏ i=2 |ξi| ] = |ξ1| + (n− 1) + ln [ n∏ i=1 |ξi| ] − ln |ξ1| = |ξ1| + (n− 1) + ln |det (fz )|− ln |ξ1| as the function g(x) = x + (n − 1) + ln |det(fz )| − ln x is increasing function in 1 ≤ x ≤ n, hence for |ξ1| ≥ 2n fm1 , fze ≥ 2 n fm1 + (n− 1) + ln |det(fz )|− ln [ 2 n fm1 ] (4.14) fze ≥ 2 n fm1 + (n− 1) + ln [ n |det (fz )| 2fm1 ] � theorem 4.6. if g=(v,σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, fz is the corresponding fuzzy zagreb matrix with eigen values ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 and fze is its fuzzy zagreb energy then fze ≤ 2(fhm ) + 2 n fm1 − ( 2 n fm1 )2 − ln [ n |det (fz )| 2fm1 ] int. j. anal. appl. 19 (2) (2021) 261 proof. consider g=(v,σ,µ) is a fuzzy graph and ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 are the eigen values of fz . recall that, 2fm1 = 2 ∑ uiuj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] ≥ n, hence 2n fm1 ≥ 1. for simplicity, consider a function f(x) = x2 −x− ln(x), x > 0. f(x) is decreasing function in 0 < x ≤ 1 and it is increasing function in x ≥ 1. hence f(x) ≥ f(1) = 0 for x > 0, gives x ≤ x2 − ln(x) for x > 0. fze = n∑ i=1 |ξi| = |ξ1| + n∑ i=2 |ξi| ≤ |ξ1| + n∑ i=2 [ |ξi| 2 − ln |ξi| ] = |ξ1| + n∑ i=1 |ξi| 2 −|ξ1| 2 − ln n∏ i=1 |ξi| + ln |ξ1| = 2(fhm ) + |ξ1|− |ξ1| 2 − ln [ |det(fz )| |ξ1| ] as the function g(x) = 2(fhm ) + x−x2 − ln [|det(fz )|] + ln(x) is increasing function in 0 < x ≤ 1 and it is decreasing function in x ≥ 1, also x ≥ 2 n fm1 ≥ 1 we get, (4.15) fze ≤ 2(fhm ) + 2 n fm1 − ( 2 n fm1 )2 − ln [ n |det(fz )| 2fm1 ] � illustration: consider the fuzzy graph g=(σ,µ) as shown in fig.1. here fze = 1.915, fhm = 0.468 and fm1 = 1.68 fze ≥ √ 2(fhm ) + n(n− 1)|det(fz)| 2 n = √ 2(0.468) + 6(5)|0.00000861| 2 6 = 1.2453 and fze ≤ √ 2n.(fhm ) = √ 2(6).(0.468) = 2.3698 hence it verifies the theorem 4.3. fze ≤ 2(fhm ) n + √√√√(n− 1) { 2(fhm ) − ( 2 n (fhm ) )2} = 2(0.468) 6 + √√√√(5) { 2(0.468) − ( 2 6 (0.468) )2} = 2.291 int. j. anal. appl. 19 (2) (2021) 262 hence it verifies the theorem 4.4. fze ≥ 2 n fm1 + (n− 1) + ln [ n |det(fz )| 2fm1 ] = 2 6 (1.68) + 5 + ln [ 6(0.00000861) 2(1.68) ] = −5.5228 hence the theorem 4.5 is verified. fze ≤ 2(fhm ) + 2 n fm1 − ( 2 n fm1 )2 − ln [ n |det(fz )| 2fm1 ] = 2(0.468) + 2 6 (1.68) − ( 2 6 (1.68) )2 − ln [ 6(0.00000861) 2(1.68) ] = 12.2652 hence the theorem 4.6 is verified. 5. applications in case of human trafficking, objects can be considered as vertices which are reasons for human trafficking while each link between these reasons can be considered as an edge. so each edge has strength of the routes between vertices. concepts of indices can be applied to measure of susceptibility of certain routs which need to be eliminated with respect to human trafficking. similarly, in case of internet routing, fuzzy zagreb indices can be used to identify the nature of particular vertex or strength of the whole system so that it reduces the time consumption in the particular area. 6. conclusion fuzzy zagreb first, second and hyper indices as well as first, second fuzzy zagreb matrices and their energies are studied. some bounds for first fuzzy zagreb energy are studied along with illustration. further study on these fuzzy zagreb indices may reveal more analogous results of these kind and will be discussed in the forthcoming papers. acknowledgements: the authors are highly grateful to the anonymous reviewers for their helpful comments and suggestions for improving the paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (2) (2021) 263 references [1] i. gutman, n. trinajstić, graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons, chem. phys. lett. 17 (4) (1972), 535–538. [2] m. randic, characterization of molecular branching, j. amer. chem. soc. 97 (23) (1975), 6609–6615. [3] l. a. zadeh, fuzzy sets, inform. control. 8 (3) (1965), 338–353. [4] a. rosenfeld, fuzzy graphs, in: fuzzy sets and their applications to cognitive and decision processes, elsevier, 1975, pp. 77–95. [5] h.-j. zimmermann, fuzzy set theory and mathematical programming, in: fuzzy sets theory and applications, springer, 1986, pp. 99–114. [6] m. g. thomason, convergence of powers of a fuzzy matrix, j. math. anal. appl. 57 (2) (1977), 476–480. [7] j. n. mordeson, p. s. nair, fuzzy graphs and fuzzy hypergraphs, vol. 46, physica verlag, heidelberg, 2012. [8] m. sunitha, a. vijayakumar, complement of a fuzzy graph, indian j. pure appl. math. 33 (9) (2002), 1451–1464. [9] s. mathew, m. sunitha, types of arcs in a fuzzy graph, inform. sci. 179 (11) (2009), 1760–1768. [10] a. nagoorgani, v. chandrasekaran, a first look at fuzzy graph theory, allied publication pvt. ltd, chennai, (2010). [11] a. n. gani, k. radha, the degree of a vertex in some fuzzy graphs, int. j. algorithms comput. math 2 (2009), 107–116. [12] a. nagoorgani, k. ponnalagu, a new approach on solving intuitionistic fuzzy linear programming problem, appl. math. sci. 6 (70) (2012), 3467–3474. [13] n. anjali, s. mathew, energy of a fuzzy graph, ann. fuzzy math. inform. 6 (3) (2013), 455–465. [14] s. r. sharbaf, f. fayazi, laplacian energy of a fuzzy graph. iran. j. math. chem. 5 (1) (2014), 1–10 [15] m. binu, s. mathew, j. n. mordeson, wiener index of a fuzzy graph and application to illegal immigration networks, fuzzy sets syst. 384 (2020), 132–147. [16] s. r. islam, s. maity, m. pal, comment on “wiener index of a fuzzy graph and application to illegal immigration networks”, fuzzy sets syst. 384 (2020), 148–151. [17] s. mathew, n. anjali, j. n. mordeson, transitive blocks and their applications in fuzzy interconnection networks, fuzzy sets syst. 352 (2018), 142–160. [18] s. ali, s. mathew, j. mordeson, hamiltonian fuzzy graphs with application to human trafficking, inform. sci. 550 (2021), 268-284. [19] m. kale, s. minirani, on zagreb indices of graphs with a deleted edge, ann. pure appl. math. 21 (1) (2020), 1–14. [20] b. borovicanin, k. c. das, b. furtula, i. gutman, bounds for zagreb indices, match commun. math. comput. chem 78 (1) (2017), 17–100. [21] j. n. mordeson, s. mathew, advanced topics in fuzzy graph theory, springer, 2019. 1. introduction 2. preliminaries 3. fuzzy zagreb indices 4. main results 5. applications 6. conclusion references international journal of analysis and applications volume 18, number 3 (2020), 332-336 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-332 a note on olivier’s theorem and convergence in erdős-ulam density józsef bukor∗ department of mathematics and informatics, j. selye university, 945 01 komárno, slovakia ∗corresponding author: bukorj@ujs.sk abstract. olivier’s theorem says that if ∑ an is a convergent positive series and (an) is monotone decreasing, then nan → 0. šalát and toma [4] proved that the monotonicity condition can be omitted if the convergence of (nan)n is replaced by the statistical convergence. the aim of this note is to give an alternative proof and generalization of this result. 1. introduction a classical olivier’s theorem says that if ∑ an is a convergent positive series and (an) is monotone decreasing, then nan → 0. t. šalát and v. toma proved in 2003 [4] that the monotonicity condition in the above result can be omitted if the convergence of (nan)n is replaced by the statistical convergence. this result was generalized and extended by several authors, see e.g., [3] and [2]. the aim of this note is to give an alternative proof and a generalization of the result of šalát and toma, and extend a result of niculescu and prǎjiturǎ (see [3], theorem 6) which we recall later. from now on, we call a positive function f : n → (0,∞) weight function (or erdős-ulam function) if it satisfies ∞∑ n=1 f(n) = ∞ and lim n→∞ f(n)∑n j=1 f(j) = 0 . received february 20th, 2020; accepted march 19th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 40a30, 40a35. key words and phrases. positive series; weighted density; convergence in density. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 332 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-332 int. j. anal. appl. 18 (3) (2020) 333 with respect to a weight function f the f-weighted densities are defined as follows. for a ⊂ n let f(a,n) = n∑ j=1 f(j) ·χa(j)∑n j=1 f(j) , where χa denotes the characteristic function of a. now we define the lower and upper f-densities of a by df (a) = lim inf n→∞ f(a,n) and df (a) = lim sup n→∞ f(a,n), respectively. in the case when df (a) = df (a) we say that a has the f-density property denoted by df (a). note that the asymptotic density corresponds to f(n) = 1, while the logarithmic density does to f(n) = 1/n. the logarithmic density is related to the asymptotic density via the inequalities 0 ≤ d1(a) ≤ d 1 n (a) ≤ d 1 n (a) ≤ d1(a) ≤ 1 . define the function f∗ by f∗(n) = f(n)∑n j=1 f(j) . (1.1) the logarithmic density can be considered as a density derived from the asymptotic density by (1.1). this method can be extended for an arbitrary weighted density given by the weight function f to provide a new weight function f∗ (and, consequently, a new weighted density). moreover, for arbitrary a ⊂ n we have df (a) ≤ df∗(a) ≤ df∗(a) ≤ df (a) , (1.2) see [1]. the concept of convergence in density is an extension of the concept of statistical convergence. a sequence (an) converges to a number α in density df , which we denote as (df )– limn→∞an = α, provided the set aε = {n ∈ n : |an −α| ≥ ε} has zero f-density, i.e., df (aε) = 0. now, we can rewrite the result of šalát and toma as if ∑ an is a convergent positive series, then (d1)– lim n→∞ nan = 0. (1.3) niculescu and prǎjiturǎ [3] studied an analogous question for the harmonic density. they stated that if ∑ an is a convergent positive series, then (d 1 n )– lim n→∞ (n ln n)an = 0. (1.4) we generalize these results above. int. j. anal. appl. 18 (3) (2020) 334 2. results in the proof of our theorem we will use the following observation. lemma 2.1. let f be an erdős-ulam function and f∗ is defined by (1.1). let a be an infinite set of positive integers such that ∑ k∈a f ∗(k) is convergent. then df (a) = 0. proof. from the assertion of the lemma df∗(a) = 0 follows immediately. but inequality (1.2) does not give any information on the behavior of df (a). taking into account that the upper density of a set does not change by removing finitely many elements. this observation, together with the fact that the tail of a convergent series tends to zero shows df (a) = lim n→∞ ( lim sup m→∞ ∑ k∈a∩[n,m] f(k)∑m k=1 f(k) ) ≤ lim n→∞ ( lim m→∞ ∑ k∈a∩[n,m] f(k)∑k j=1 f(j) ) = lim n→∞ ( lim m→∞ ∑ k∈a∩[n,m] f∗(k) ) ≤ lim n→∞ ∑ k∈a∩[n,∞) f∗(k) = 0. � hence df (a) = 0. theorem 2.1. let f be an erdős-ulam function. if ∑ an is a convergent positive series, then (df )– lim n→∞ ∑n k=1 f(k) f(n) an = 0 . (2.1) proof. fix ε > 0, and consider the set aε = {n ∈ n : ∑n k=1 f(k) f(n) an ≥ ε} . since ε ∑ n∈aε f∗(n) = ε ∑ n∈aε f(n)∑n k=1 f(k) ≤ ∑ n∈aε an ≤ ∑ n∈n an < ∞, applying lemma 2.1 we immediately get that the set aε has zero f-density. then (2.1) holds and the proof is completed. � corollary 2.1. if we consider the asymptotic density in (2.1), then we conclude (1.3). similarly, the logarithmic density (if f(n) = 1/n) leads to (1.4). for f(n) = 1/(n ln n) (the case of loglog-density), we obtain if ∑ an is a convergent positive series, then (d 1 n ln n )– lim n→∞ n(ln n)(ln ln n)an = 0. roughly speaking, if ∑ an is a convergent positive series, then the fast growing of the weight function f guarantees a less speed convergence of (an) to zero in density df . int. j. anal. appl. 18 (3) (2020) 335 for example, let f(n) = e √ n/(2 √ n). in this case ∑n k=1 f(k) ∼ e √ n and we have if ∑ an is a convergent positive series, then (df )– lim n→∞ √ nan = 0. next, we show that (1.3) is best possible in the sense that we cannot replace (d1)– limn→∞nan = 0 with (d1)– limn→∞nωnan = 0, where ωn is an arbitrary sequence tending to infinity. theorem 2.2. let (ωn) be an increasing sequence, tending to infinity. then there exists a sequence (an) of positive terms, such that ∑ an converges and (d1)– limn→∞nωnan 6= 0. proof. the construction of (an) is based on the fact that lim m→∞ 2m∑ k=m 1 kωk ≤ lim m→∞ 1 ωm 2m∑ k=m 1 k = lim m→∞ ln 2 ωm = 0 . (2.2) using (2.2) we are able to define an increasing sequence (mi) for that mi+1 > 2mi and 2mi∑ k=mi 1 kωk < 1 2i , i = 1, 2, . . . . define the sequence (an) as an =   1 n2ωn if n ∈ n r ∞⋃ i=1 [mi, 2mi] 1 nωn if n ∈ ∞⋃ i=1 [mi, 2mi]. then ∑ an converges since ∞∑ n=1 an = ∑ n∈nr∪∞ i=1 [mi,2mi] 1 n2ωn + ∑ n∈∪∞ i=1 [mi,2mi] 1 nωn ≤ ∞∑ n=1 1 n2 + ∞∑ i=1 k=2mi∑ k=mi 1 kωk < π2 6 + ∞∑ i=1 1 2i = π2 6 + 1 . we are going to show that (d1)– limn→∞nωnan = 0 fails. fix ε ∈ (0, 1) and consider the set aε = {n ∈ n : nωnan ≥ ε} . then for any n ∈ [mi, 2mi] we have nωnan = 1 and therefore the set aε does not have zero asymptotic density. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (3) (2020) 336 references [1] j. bukor, f. filip and j. t. tóth, sets with countably infinitely many prescribed weighted densities, rocky mountain j. math.(to appear). [2] a. faisant,g. grekos and l. mǐśık, some generalizations of olivier’s theorem, math. bohem. 141 (4) (2016), 483–494. [3] c. p. niculescu and g. t. prǎjiturǎ, some open problems concerning the convergence of positive series, ann. acad. rom. sci. ser. math. appl. 6 (1) (2014), 92–107. [4] t. šalát and v. toma, a classical olivier’s theorem and statistical convergence, ann. math. blaise pascal, 10 (2) (2003), 305–313. 1. introduction 2. results references international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 40-47 http://www.etamaths.com fejér–hadamard inequlality for convex functions on the coordinates in a rectangle from the plane g. farid∗, m. marwan, and atiq ur rehman abstract. we give fejér–hadamard inequality for convex functions on coordinates in the rectangle from the plane. we define some mappings associated to it and discuss their properties. 1. introduction a real valued function f : i → r, where i is an interval in r, is called convex if f(αx + (1 −α)y) ≤ αf(x) + (1 −α)f(y), where α ∈ [0, 1], for all x,y ∈ i. convex functions play a vital role in the theory of inequalities. a lot of inequalities are established using convex functions, e.g. see for convex functions in [1, 2, 6, 7]. the most classical and fundamental inequality is hermite-hadamard inequality, this is stated as follows: (1) f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 holds for every convex function f : i → r and a, b ∈ i with a < b. this inequality is present in many textbooks and monographs devoted to convex functions and it is also extensively studied by many researchers. with the help of (1) researchers have produced many integral and differential inequalities (see [8, 9]), and operators. very interesting historical remarks concerning the inequality (1) can be found in [13] (see also [14, pp. 62] ). in 1906, fejér (see [16, page 138] and [11]) established the following weighted generalization of the hermite–hadamard inequality for symmetric functions. (2) f ( a + b 2 )∫ b a g(x)dx ≤ ∫ b a f(x)g(x)dx ≤ f(a) + f(b) 2 ∫ b a g(x)dx holds for every convex function f : i → ra,b ∈ i, and g : [a,b] → r+ symmetric about (a + b)/2. in [5] dragomir gave the hermite-hadamard inequality on a rectangle in plane, by defining convex functions on coordinates. definition 1.1. let us consider the two-dimensional interval ∆ := [a,b] × [c,d] in r2 with a < b and c < d. a function f : ∆ → r will be called convex on the coordinates if the partial mappings fy : [a,b] → r, fy(u) := f(u,y), and fx : [a,b] → r, fx(v) := f(x,v), are convex where defined for all y ∈ [c,d] and x ∈ [a,b]. one can note that every convex mapping f : ∆ → r is convex on the coordinates but the converse is not true. for example, f(x,y) = xy is convex on coordinates in r2 but it is not convex. 2010 mathematics subject classification. 26a51, 26d15, 65d30. key words and phrases. convex functions; hadamard inequality; convex functions on coordinates. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 40 fejér–hadamard inequlality 41 theorem 1.2. let f : ∆ → r be convex on co-ordinate in ∆. then we have: f ( a + b 2 , c + d 2 ) ≤ 1 2 [ 1 b−a ∫ b a f(x, c + d 2 )dx + 1 d− c ∫ d c f( a + b 2 ,y)dy ] ≤ 1 (b−a)(d− c) ∫ b a ∫ d c f(x,y)dydx ≤ 1 4 [ 1 (b−a) ∫ b a f(x,c)dx + 1 (b−a) ∫ b a f(x,d)dx + 1 d− c ∫ d c f(a,y)dy + 1 d− c ∫ d c f(b,y)dy ] ≤ 1 4 [f(a,c) + f(a,d) + f(b,c) + f(b,d)] . there in [5] some mappings connected to above inequality are also considered and their properties are discussed. in this paper we are interested to give the fejér–hadamard inequality for a rectangle in plane via convex functions on coordinates. we also study some properties of mappings associated with the fejér–hadamard inequality for convex functions on coordinates. 2. main results theorem 2.1. let ∆ := [a,b] × [c,d] ⊂ r2 and f : ∆ → r be a convex function on coordinates in ∆. also let g1 : [a,b] → r+ and g2 : [c,d] → r+ be two integrable and symmetric functions about (a + b)/2 and (c + d)/2 respectively. then one has the following inequalities (3) f ( a + b 2 , c + d 2 ) ≤ 1 2 [ 1 g1 ∫ b a f ( x, c + d 2 ) g1(x)dx + 1 g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy ] ≤ 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx ≤ 1 4 [ 1 g1 ∫ b a g1(x)f(x,c)dx + 1 g1 ∫ b a g1(x)f(x,d)dx + 1 g2 ∫ d c g2(y)f(a,y)dy + 1 g2 ∫ d c g2(y)f(b,y)dy ] ≤ 1 4 [f(a,c) + f(a,d) + f(b,c) + f(b,d)] , where g1 = ∫ b a g1(x)dx and g2 = ∫ d c g2(y)dy. these inequalities are sharp. proof. since f : ∆ → r is convex on coordinates, it follows that functions fx and fy are convex on [c,d] and [a,b] respectively. thus from (2), we have (4) f ( x, c + d 2 ) ≤ 1 g2 ∫ d c f(x,y)g2(y)dy ≤ f(x,c) + f(x,d) 2 and (5) f ( a + b 2 ,y ) ≤ 1 g1 ∫ b a f(x,y)g1(x)dx ≤ f(a,y) + f(b,y) 2 . 42 farid, marwan, and rehman multiplying (4) by g1(x) g1(x)f ( x, c + d 2 ) ≤ 1 g2 ∫ d c f(x,y)g1(x)g2(y)dy ≤ g1(x) f(x,c) + f(x,d) 2 . now integrating on [a,b], we get (6) ∫ b a g1(x)f ( x, c + d 2 ) dx ≤ 1 g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx ≤ 1 2 [∫ b a g1(x)f(x,c)dx + ∫ b a g1(x)f(x,d)dx ] . now multiplying (5) by g2(y) and integrating on [c,d], we get (7) ∫ d c g2(y)f ( a + b 2 ,y ) dy ≤ 1 g1 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx ≤ 1 2 [∫ d c g2(y)f(a,y)dy + ∫ d c g2(y)f(b,y)dy ] . since g1,g2 > 0, dividing inequalities (6), (7) by g1, g2 respectively and adding we get second and third inequalities in (3). from first part of (4) and (5), we have f ( a + b 2 , c + d 2 ) ≤ 1 g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy and f ( a + b 2 , c + d 2 ) ≤ 1 g1 ∫ b a f ( x, c + d 2 ) g1(x)dx. adding the above two inequalities we get the first inequality in (3). now from second part of (4) and (5), we can get 1 g1 ∫ b a f(x,c)g1(x)dx ≤ f(a,c) + f(b,c) 2 , 1 g1 ∫ b a f(x,d)g1(x)dx ≤ f(a,d) + f(b,d) 2 , 1 g2 ∫ d c f(a,y)g2(y)dy ≤ f(a,c) + f(a,d) 2 , 1 g2 ∫ d c f(b,y)g2(y)dy ≤ f(b,c) + f(b,d) 2 . by adding the above four inequalities, we get last inequality in (3). if in (3) we choose f(x) = xy, then (3) becomes an equality, which shows that inequalities in (3) are sharp. � remark 2.2. if we put g1 ≡ 1 and g2 ≡ 1 in above theorem, then we get theorem 1.2, which is the main theorem of [5]. for a mapping f : ∆ → r, we define the mapping ĥ : [0, 1]2 → r, as follows: (8) ĥ(t,s) = 1 g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 ,sy + (1 −s) c + d 2 ) g1(x)g2(y)dydx. the properties of this mapping are studied in the following theorem. we need a following lemma to give desire results, which is due to levin and stečkin [16, pp. 200]. fejér–hadamard inequlality 43 lemma 2.3. let f be a convex function on [a,b], g be a function symmetric about (a + b)/2 and nonincreasing on [a, (a + b)/2]. then∫ b a f(x)g(x)dx ≥ 1 b−a ∫ b a f(x)dx ∫ b a g(x)dx. theorem 2.4. suppose that f : ∆ → r is convex on the coordinates in ∆. then the mapping ĥ, defined in (8), is convex on the coordinates on [0, 1]2. further if g1 is nonincreasing on [a, (a + b)/2] and g2 is nonincreasing on [c, (c + d)/2], then inf (t,s)∈[0,1]2 ĥ(t,s) = f ( a + b 2 , c + d 2 ) = ĥ(0, 0) and sup (t,s)∈[0,1]2 ĥ(t,s) = 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx = ĥ(1, 1). proof. for convexity, fix s ∈ [0, 1]. then for all α,β ≥ 0 with α + β = 1, and t1, t2 ∈ [0, 1] we have ĥ(αt1 + βt2,s) = 1 g1g2 × ∫ b a ∫ d c f ( (αt1 + βt2)x + (1 − (αt1 + βt2)) a + b 2 ,sy + (1 −s) c + d 2 ) g1(x)g2(y)dydx which gives us ĥ(αt1 + βt2,s) = 1 g1g2 ∫ b a ∫ d c f ( α ( t1x + (1 − t1) a + b 2 ) + β ( t1x + (1 − t1) a + b 2 ) ,sy + (1 −s) c + d 2 ) g1(x)g2(y)dydx ≤ α g1g2 ∫ b a ∫ d c f ( t1x + (1 − t1) a + b 2 ,sy + (1 −s) c + d 2 ) g1(x)g2(y)dydx + β g1g2 ∫ b a ∫ d c f ( t2x + (1 − t2) a + b 2 ,sy + (1 −s) c + d 2 ) g1(x)g2(y)dydx = αĥ(t1,s) + βĥ(t2,s). if t ∈ [0, 1] is fixed, then for all s1,s2 ∈ [0, 1] and α,β ≥ 0 with α + β = 1, we also have: ĥ(t,αs1 + βs2) ≤ αĥ(t,s1) + βĥ(t,s2) and the statement is proved. now to prove the remaining part of the theorem, we take ĥ(t,s) = 1 g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 ,sy + (1 −s) c + d 2 ) g2(y)g1(x)dydx. since f is convex on the coordinates and 1 g2 ∫ d c g2(y)dy = 1, we apply jensen’s inequality for integrals on second coordinate to get ĥ(t,s) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , 1 g2 ∫ d c ( sy + (1 −s) c + d 2 ) g2(y)dy ) g1(x)dx. now it follows from lemma 2.3, that ĥ(t,s) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)dx.(9) 44 farid, marwan, and rehman since 1 g1 ∫ b a g1(x) dx = 1, jensen’s inequality for integrals leads to ĥ(t,s) ≥ f ( 1 g1 ∫ b a ( tx + (1 − t) a + b 2 ) g1(x)dx, c + d 2 ) . now by lemma 2.3, we have ĥ(t,s) ≥ f ( a + b 2 , c + d 2 ) . this gives us the lower bound of ĥ. to get upper bound, we use convexity on second coordinates of f to get ĥ(t,s) ≤ 1 g1g2 ∫ b a [ s ∫ d c f ( tx + (1 − t) a + b 2 ,y ) g2(y)dy + (1 −s)f ( tx + (1 − t) a + b 2 , c + d 2 ) g2(y)dy ] g1(x)dx. this gives ĥ(t,s) ≤ s g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 ,y ) g1(x)g2(y)dydx + (1 −s) g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)g2(y)dydx ≤ s g1g2 ∫ b a ∫ d c [ tf(x,y) + (1 − t)f ( a + b 2 ,y )] g1(x)g2(y)dydx + 1 −s g1g2 ∫ b a ∫ d c [ tf ( x, c + d 2 ) + (1 − t)f ( a + b 2 ,y )] g1(x)g2(y)dydx. on simplification, we have ĥ(t,s) ≤ st g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx + s(1 − t) g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy + (1 −s)t g1 ∫ b a f ( x, c + d 2 ) g1(x)dx + (1 −s)(1 − t)f ( a + b 2 , c + d 2 ) . from inequalities (6) and (7), we have 1 g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy ≤ 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx and 1 g1 ∫ b a f ( x, c + d 2 ) g1(x) ≤ 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx. using above inequalities, we deduce that ĥ(t,s) ≤ [st + s(1 − t) + (1 −s)t + (1 −s)(1 − t)] 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx = 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx, (t,s) ∈ [0, 1]2. now we have to show the monotonicity of the mapping ĥ(t,s). for this firstly, we will show that ĥ(t,s) ≥ ĥ(t, 0) for all (t,s) ∈ [0, 1]2. by (9), we have: ĥ(t,s) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)dx = ĥ(t, 0) fejér–hadamard inequlality 45 for all (t,s) ∈ [0, 1]2. now let 0 ≤ s1 ≤ s2 ≤ 1. by convexity of mapping ĥ(t, .) for all t ∈ [0, 1], we have ĥ(t,s2) − ĥ(t,s1) s2 −s1 ≥ ĥ(t,s1) − ĥ(t, 0) s1 ≥ 0. this completes the proof. � remark 2.5. if we put g1 ≡ 1 and g2 ≡ 1, in theorem 2.4 then we get theorem 2 of [5]. if the function f is convex on ∆, instead of coordinated convex, then we have the following theorem. theorem 2.6. suppose that f : ∆ → r is convex on ∆. (i) the mapping ĥ is convex on ∆. (ii) let ĥ : [0, 1] → r be the mapping defined as ĥ(t) = ĥ(t,t). then ĥ is convex, monotonic nondecreasing on [0, 1] and one has the bounds: inf t∈[0,1] ĥ(t) = f ( a + b 2 , c + d 2 ) = ĥ(0, 0) and sup t∈[0,1] ĥ(t) = 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx = ĥ(1, 1). proof. (i) for convexity, let (t1,s1), (t2,s2) ∈ [0, 1]2 and α,β ≥ 0 with α + β = 1. then ĥ(αt1 + βt2,αs1 + βs2) = 1 g1g2 × ∫ b a ∫ d c f [ α ( t1x + (1 − t1) a + b 2 ,s1y + (1 −s1) c + d 2 ) +β ( t2x + (1 − t2) a + b 2 ,s2y + (1 −s2) c + d 2 )] g1(x)g2(y)dydx ≤ α g1g2 ∫ b a ∫ d c f ( t1 + (1 − t1) a + b 2 ,s1y + (1 −s1) c + d 2 ) g1(x)g2(y)dydx + β g1g2 ∫ b a ∫ d c f ( t2x + (1 − t2) a + b 2 ,s2y + (1 −s2) c + d 2 ) g1(x)g2(y)dydx = αĥ(t1,s1) + βĥ(t2,s2). which shows that h is convex on [0, 1]2. (ii) now we prove the convexity of ĥ on [0, 1]. for this, let t1, t2 ∈ [0, 1] and α,β ≥ 0 with α + β = 1. then ĥ(αt1 + βt2) = ĥ(αt1 + βt2,αt1 + βt2) = ĥ(α(t1, t1) + β(t2, t2)) ≤ αĥ(t1, t1) + βĥ(t2, t2) = αĥ(t1) + βĥ(t2), which shows the convexity of ĥ on [0, 1]. now to prove the remaining part of the theorem, we take ĥ(t) = 1 g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 , ty + (1 − t) c + d 2 ) g2(y)g1(x)dydx. since f is convex on the coordinates and 1 g2 ∫ d c g2(y)dy = 1, we apply jensen’s inequality for integrals on second coordinate to get ĥ(t) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , 1 g2 ∫ d c [ ty + (1 − t) c + d 2 ] g2(y)dy ) g1(x)dx. 46 farid, marwan, and rehman now it follows from lemma 2.3, that ĥ(t) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)dx.(10) since 1 g1 ∫ b a g1(x) dx = 1, jensen’s inequality for integrals leads to ĥ(t) ≥ f ( 1 g1 ∫ b a [ tx + (1 − t) a + b 2 ] g1(x)dx, c + d 2 ) . now by lemma 2.3, we have ĥ(t) ≥ f ( a + b 2 , c + d 2 ) . this gives us the lower bound of ĥ(.). to get upper bound, we use convexity on second coordinates of f to get ĥ(t) ≤ 1 g1g2 ∫ b a [ t ∫ d c f ( tx + (1 − t) a + b 2 ,y ) g2(y)dy + (1 − t)f ( tx + (1 − t) a + b 2 , c + d 2 ) g2(y)dy ] g1(x)dx. this gives ĥ(t) ≤ t g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 ,y ) g1(x)g2(y)dydx + (1 − t) g1g2 ∫ b a ∫ d c f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)g2(y)dydx ≤ t g1g2 ∫ b a ∫ d c [ tf(x,y) + (1 − t)f ( a + b 2 ,y )] g1(x)g2(y)dydx + 1 − t g1g2 ∫ b a ∫ d c [ tf ( x, c + d 2 ) + (1 − t)f ( a + b 2 ,y )] g1(x)g2(y)dydx. on simplification, we have ĥ(t) ≤ t2 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx + t(1 − t) g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy + (1 − t)t g1 ∫ b a f ( x, c + d 2 ) g1(x)dx + (1 − t)2f ( a + b 2 , c + d 2 ) . from inequalities (6) and (7), we have 1 g2 ∫ d c f ( a + b 2 ,y ) g2(y)dy ≤ 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx and 1 g1 ∫ b a f ( x, c + d 2 ) g1(x) ≤ 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx. using above inequalities, we deduce that ĥ(t) ≤ [ t2 + t(1 − t) + (1 − t)t + (1 − t)2 ] 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx = 1 g1g2 ∫ b a ∫ d c f(x,y)g1(x)g2(y)dydx, t ∈ [0, 1]. fejér–hadamard inequlality 47 now we have to show the monotonicity of the mapping ĥ for this firstly, we show that ĥ(t,t) ≥ ĥ(t, 0) for all t ∈ [0, 1]. by (10), we have: ĥ(t) ≥ 1 g1 ∫ b a f ( tx + (1 − t) a + b 2 , c + d 2 ) g1(x)dx = ĥ(t, 0) for all t ∈ [0, 1]. now let 0 ≤ t1 ≤ t2 ≤ 1. by convexity of mapping ĥ(t, .) for all t ∈ [0, 1], we have ĥ(t2) − ĥ(t1) t2 − t1 ≥ ĥ(t1) − ĥ(0) t1 ≥ 0. this completes the proof. � remark 2.7. if we put g1 ≡ 1 and g2 ≡ 1 in theorem 2.6, then we get theorem 3 of [5]. references [1] s. abramovich, g. farid and j. e. pečarić, more about jensens inequality and cauchys means for superquadratic functions, j. math. inequal. 7(1) (2013), 11–24. [2] s. i. butt, j. e. pečarić and atiq ur rehman, non-symmetric stolarsky means, j. math. inequal. 7(2) (2013), 227–237. [3] s. s. dragomir, j. e. pečarić and j. sáandor, a note on the jensen-hadamard inequality, l’ anal. num. theor. l’approx. (romania) 19 (1990), 21–28. [4] s. s. dragomir, d. barbu and c. buşe, a probabilistic argument for the convergence of some sequences associated to hadamard’s inequality, studia univ. babeş-bolgai, mathematica 38 (1993), 29–34. [5] s. s. dragomir, on hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, taiwanese j math. 4 (2001), 775–788. [6] s. s. dragomir, on hadamard’s inequality for convex functions, mat. balkanica 6 (1992), 215–222. [7] s. s. dragomir, refinements of the hermite–hadamard inequality for convex functions, j. inequal. pure appl. math. 6(5) (2005), article 140. [8] s. s. dragomir and n. m. lonescu, some integral inequalities for differentiable convex functions. coll. pap. of the fac. of sci. kragujevac (yugoslavia) 13 (1992), 11–16. [9] s. s. dragomir, some integral inequlities for differentiable convex functions, contributions, macedonian acad. of sci. and arts (scopie) 16 (1992), 77–80. [10] s. s. dragomir, and n. m. ionescu, some remarks in convex functions, l’anal. num. theor. l’approx. (romania), 21 (1992), 31–36. [11] l. fejér, über die fourierreihen, ii. math. naturwiss anz ungar. akad. wiss. 24 (1906), 369–390 [12] d. s. mitrinovísc, j. e. pečarić and a. m. fink, classical and new inequalities in analysis, kluwer academic publishers, 1993. [13] d. s. mitrinović, i. b. lacković, hermite and convexity, aequationes math. 28 (1985). [14] c. p. niculescu, l. e. persson, convex functions and their applications: a contemporary approach, springer, new york, 2006. [15] j. e. pečarić and s. s. dragomir, a generalisation of hadamard’s inequality for isotonic linear functionals, rodovi math. (sarajevo) 7 (1991), 103–107. [16] j. e. pečarić, f. proschan, y. l. tong, convex functions, partial ordering, and stasitcal applications, academic press, inc. 1992. [17] j. e. pečarić and s. s. dragomir, on some integral inequalities for convex functions, bull. mat. inst. pol. iasi 36 (1990), 19–23. comsats institute of information technology, attock campus, pakistan ∗corresponding author: faridphdsms@hotmail.com international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 69-78 http://www.etamaths.com convergence of hybrid fixed point for a pair of nonlinear mappings in banach spaces g. s. saluja abstract. in this paper, we study hybrid fixed point of a modified two-step iteration process with errors for a pair of asymptotically quasi-nonexpansive mapping and asymptotically quasi-nonexpansive mapping in the intermediate sense in the framework of banach spaces. also we establish some strong convergence theorems and a weak convergence theorem for the iteration scheme and mappings. the results presented in this paper extend, improve and generalize some previous work from the existing literature. 1. introduction let k be a nonempty subset of a real banach space e. let t : k → k be a mapping, then we denote the set of all fixed points of t by f(t). the set of common fixed points of two mappings s and t will be denoted by f = f(s) ∩f(t). a mapping t : k → k is said to be: (1) nonexpansive if ‖tx−ty‖ ≤ ‖x−y‖ for all x,y ∈ k; (2) quasi-nonexpansive if f(t) 6= ∅ and ‖tx−p‖ ≤ ‖x−p‖ for all x ∈ k and p ∈ f(t); (3) asymptotically nonexpansive if there exists a sequence {kn} ∈ [1,∞) with limn→∞kn = 1 and ‖tnx−tny‖ ≤ kn‖x−y‖ for all x, y ∈ k and n ≥ 1; 2010 mathematics subject classification. 47h09, 47h10, 47j25. key words and phrases. asymptotically quasi-nonexpansive mapping; asymptotically quasinonexpansive mapping in the intermediate sense; modified two-step iteration process with errors; hybrid fixed point; strong convergence; weak convergence; banach space. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 69 70 saluja (4) asymptotically quasi-nonexpansive if f(t) 6= ∅ and there exists a sequence {kn}∈ [1,∞) such that limn→∞kn = 1 and ‖tnx−p‖ ≤ kn‖x−p‖ for all x ∈ k, p ∈ f(t) and n ≥ 1; (5) uniformly l-lipschitzian if there exists a constant l > 0 such that ‖tnx−tny‖ ≤ l‖x−y‖ for all x, y ∈ k and n ≥ 1; (6) uniformly quasi-lipschitzian if there exists l ∈ [1, +∞) such that ‖tnx−p‖ ≤ l‖x−p‖ for all x ∈ k, p ∈ f(t) and n ≥ 1. remark 1.1. it is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive is uniformly lipschitzian. also, if f(t) 6= ∅, then a nonexpansive mapping is a quasi-nonexpansive mapping, an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive, a uniformly l-lipschitzian mapping must be uniformly quasi-lipschitzian and an asymptotically quasi-nonexpansive mapping must be uniformly quasi-lipschitzian mapping with l = supn≥1{kn}≥ 1 but the converse is not true in general. the class of asymptotically nonexpansive mappings was introduced by goebel and kirk [5] as a generalization of the class of nonexpansive mappings. recall also that a mapping t : k → k is said to be asymptotically quasi-nonexpansive in the intermediate sense [18] provided that t is uniformly continuous and lim sup n→∞ sup x∈k, p∈f(t) ( ‖tnx−p‖−‖x−p‖ ) ≤ 0. from the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense. but the converse does not hold as the following example: example 1.2. let x = r be a normed linear space and k = [0, 1]. for each x ∈ k, we define t(x) = { kx, if x 6= 0, 0, if x = 0, where 0 < k < 1. then |tnx−tny| = kn|x−y| ≤ |x−y| for all x,y ∈ k and n ∈ n. thus t is an asymptotically nonexpansive mapping with constant sequence {1} and lim sup n→∞ {|tnx−tny|− |x−y|} = lim sup n→∞ {kn‖x−y‖−‖x−y‖} ≤ 0, convergence of hybrid fixed point 71 because limn→∞k n = 0 as 0 < k < 1, for all x,y ∈ k, n ∈ n and t is continuous. hence t is an asymptotically nonexpansive mapping in the intermediate sense. example 1.3. let x = r, k = [−1 π , 1 π ] and |k| < 1. for each x ∈ k, define t(x) = { k xsin(1/x), if x 6= 0, 0, if x = 0. then t is an asymptotically nonexpansive mapping in the intermediate sense but it is not asymptotically nonexpansive mapping. since 1972, many authors have studied weak and strong convergence problem of the iterative sequences (with errors) for asymptotically nonexpansive mappings in hilbert spaces and banach spaces (see, e.g., [5, 6, 8, 10, 11, 12, 13, 17] and references therein). in 2007, agarwal et al. [1] introduced the following iteration process: x1 = x ∈ k, xn+1 = (1 −αn)tnxn + αntnyn, yn = (1 −βn)xn + βntnxn, n ≥ 1,(1.1) where {αn} and {βn} are sequences in (0, 1). they showed that this process converge at a rate same as that of picard iteration and faster than mann for contractions. the above process deals with one mapping only. the case of two mappings in iterative processes has also remained under study since das and debate [3] gave and studied a two mappings process. later on, many authors, for example khan and takahashi [8], shahzad and udomene [14] and takahashi and tamura [16] have studied the two mappings case of iterative schemes for different types of mappings. recently, khan et al. [7] studied the modified two-step iteration process for two mappings as follows: x1 = x ∈ k, xn+1 = (1 −αn)tnxn + αnsnyn, yn = (1 −βn)xn + βntnxn, n ≥ 1,(1.2) where {αn} and {βn} are sequences in (0, 1). they established weak and strong convergence theorems in the setting of real banach spaces. inspired and motivated by [1, 7] and many others, in this paper we introduce the following iteration scheme for a pair of asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense. the proposed iteration scheme is as follows. definition 1.4. let s : k → k be an asymptotically quasi-nonexpansive mapping and t : k → k be an asymptotically quasi-nonexpansive mapping in the intermediate sense on a closed convex subset k of a real banach space e with k +k ⊂ k. 72 saluja let {xn} be the sequence defined as follows: x1 = x ∈ k, xn+1 = (1 −αn)tnxn + αnsnyn + un, yn = (1 −βn)xn + βntnxn + vn, n ≥ 1,(1.3) where {αn} and {βn} are sequences in (0, 1) and {un} and {vn} are two sequences in k. the iteration scheme (1.3) is called modified two-step iteration process with errors for a pair of above said mappings. the aim of this paper is to establish some strong convergence theorems and a weak convergence theorem for newly proposed iteration scheme (1.3) in the framework of real banach spaces. the results presented in this paper extend, improve and generalize some previous work from the existing literature. 2. preliminaries for the sake of convenience, we restate the following concepts. a mapping t : k → k is said to be demiclosed at zero, if for any sequence {xn} in k, the condition xn converges weakly to x ∈ k and txn converges strongly to 0 imply tx = 0. a mapping t : k → k is said to be semi-compact [2] if for any bounded sequence {xn} in k such that ‖xn −txn‖ → 0 as n → ∞, then there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ k strongly. we say that a banach space e satisfies the opial’s condition [9] if for each sequence {xn} in e weakly convergent to a point x and for all y 6= x lim inf n→∞ ‖xn −x‖ < lim inf n→∞ ‖xn −y‖. the examples of banach spaces which satisfy the opial’s condition are hilbert spaces and all lp[0, 2π] with 1 < p 6= 2 fail to satisfy opial’s condition [9]. now, we state the following useful lemma to prove our main results. lemma 2.1. (see [15]) let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of nonnegative numbers satisfying the inequality αn+1 ≤ (1 + βn)αn + rn, ∀n ≥ 1. if ∑∞ n=1 βn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞αn exists. 3. main results in this section, we prove some strong convergence theorems and a weak convergence theorem of the iteration scheme (1.3) for a pair of asymptotically quasinonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense in the framework of real banach spaces. convergence of hybrid fixed point 73 theorem 3.1. let e be a real banach space and k be a nonempty closed convex subset of e with k+k ⊂ k. let s : k → k be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn − 1) < ∞ and t : k → k be uniformly l-lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that f = f(s) ∩ f(t) 6= ∅. let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞ and∑∞ n=1 ‖vn‖ < ∞. put dn = max { sup x∈k, q∈f ( ‖tnx−q‖−‖x−q‖ ) ∨ 0 : n ≥ 1 } (3.1) such that ∑∞ n=1 dn < ∞. then {xn} converges to a hybrid fixed point of the mappings s and t if and only if lim infn→∞d(xn,f) = 0, where d(x,f) = infp∈f d(x,p). proof. the necessity is obvious. thus we only prove the sufficiency. let q ∈ f. then from (1.3) and (3.1), we have ‖yn −q‖ = ‖(1 −βn)xn + βntnxn + vn −q‖ ≤ (1 −βn)‖xn −q‖ + βn‖tnxn −q‖ + ‖vn‖ ≤ (1 −βn)‖xn −q‖ + βn[‖xn −q‖ + dn] + ‖vn‖ = (1 −βn)‖xn −q‖ + βn‖xn −q‖ + βndn + ‖vn‖ ≤ ‖xn −q‖ + dn + ‖vn‖.(3.2) again using (1.3), (3.1) and (3.2), we obtain ‖xn+1 −q‖ = ‖(1 −αn)tnxn + αnsnyn + un −q‖ ≤ (1 −αn)‖tnxn −q‖ + αn‖snyn −q‖ + ‖un‖ ≤ (1 −αn)[‖xn −q‖ + dn] + αnkn‖yn −q‖ + ‖un‖ ≤ (1 −αn)‖xn −q‖ + αnkn[‖xn −q‖ + dn + ‖vn‖] +(1 −αn)dn + ‖un‖ ≤ [1 + (kn − 1)αn]‖xn −q‖ + [1 + (kn − 1)αn]dn +‖un‖ + kn‖vn‖ = (1 + tn)‖xn −q‖ + θn(3.3) where tn = (kn − 1)αn and θn = [1 + (kn − 1)αn]dn + ‖un‖ + kn‖vn‖. since by hypothesis of the theorem ∑∞ n=1(kn−1)αn < ∞, ∑∞ n=1 dn < ∞, ∑∞ n=1 ‖un‖ < ∞ and ∑∞ n=1 ‖vn‖ < ∞, it follows that ∑∞ n=1 tn < ∞ and ∑∞ n=1 θn < ∞. hence by lemma 2.1, we know that the limit limn→∞‖xn − q‖ exists. also from (3.3), we obtain d(xn+1,f) ≤ (1 + tn) d(xn,f) + θn(3.4) for all n ≥ 1. from lemma 2.1 and (3.4), we know that limn→∞d(xn,f) exists. since lim infn→∞d(xn,f) = 0, we have that limn→∞d(xn,f) = 0. 74 saluja next, we shall prove that {xn} is a cauchy sequence. since 1 +x ≤ ex for x ≥ 0, therefore, for any m,n ≥ 1 and for given q ∈ f, from (3.3), we have ‖xn+m −q‖ ≤ (1 + tn+m−1)‖xn+m−1 −q‖ + θn+m−1 ≤ etn+m−1‖xn+m−1 −q‖ + θn+m−1 ≤ etn+m−1 [etn+m−2‖xn+m−2 −q‖ + θn+m−2] + θn+m−1 ≤ e(tn+m−1+tn+m−2)‖xn+m−2 −q‖ +e(tn+m−1+tn+m−2)[θn+m−2 + θn+m−1] ≤ . . . ≤ e (∑n+m−1 k=n tk ) ‖xn −q‖ + e (∑n+m−1 k=n tk ) n+m−1∑ k=n θk ≤ e (∑∞ n=1 tk ) ‖xn −q‖ + e (∑∞ n=1 tk ) n+m−1∑ k=n θk = r‖xn −q‖ + r n+m−1∑ k=n θk(3.5) where r = e (∑∞ n=1 tk ) < ∞. since lim n→∞ d(xn,f) = 0, ∞∑ n=1 θn < ∞(3.6) for any given ε > 0, there exists a positive integer n1 such that d(xn,f) < ε 4(r + 1) , ∞∑ k=n θk < ε 2r ∀n ≥ n1.(3.7) hence, there exists q1 ∈ f such that ‖xn −q1‖ < ε 2(r + 1) ∀n ≥ n1.(3.8) consequently, for any n ≥ n1 and m ≥ 1, from (3.5), we have ‖xn+m −xn‖ ≤ ‖xn+m −q1‖ + ‖xn −q1‖ ≤ r‖xn −q1‖ + r n+m−1∑ k=n θk + ‖xn −q1‖ ≤ (r + 1)‖xn −q1‖ + r n+m−1∑ k=n θk < (r + 1). ε 2(r + 1) + r. ε 2r = ε.(3.9) this implies that {xn} is a cauchy sequence in e and so is convergent since e is complete. let limn→∞xn = q ∗. then q∗ ∈ k. it remains to show that q∗ ∈ f. let ε1 > 0 be given. then there exists a natural number n2 such that ‖xn −q∗‖ < ε1 2(l + 1) , ∀n ≥ n2.(3.10) convergence of hybrid fixed point 75 since limn→∞d(xn,f) = 0, there must exists a natural number n3 ≥ n2 such that for all n ≥ n3, we have d(xn,f) < ε1 3(l + 1) (3.11) and in particular, we have d(xn3,f) < ε1 3(l + 1) .(3.12) therefore, there exists z∗ ∈ f such that ‖xn3 −z ∗‖ < ε1 2(l + 1) .(3.13) consequently, we have ‖tq∗ −q∗‖ = ‖tq∗ −z∗ + z∗ −xn3 + xn3 −q ∗‖ ≤ ‖tq∗ −z∗‖ + ‖z∗ −xn3‖ + ‖xn3 −q ∗‖ ≤ l‖q∗ −z∗‖ + ‖z∗ −xn3‖ + ‖xn3 −q ∗‖ ≤ l‖q∗ −xn3 + xn3 −z ∗‖ + ‖z∗ −xn3‖ +‖xn3 −q ∗‖ ≤ l[‖q∗ −xn3‖ + ‖xn3 −z ∗‖] + ‖z∗ −xn3‖ +‖xn3 −q ∗‖ ≤ (l + 1)‖q∗ −xn3‖ + (l + 1)‖z ∗ −xn3‖ < (l + 1). ε1 2(l + 1) + (l + 1). ε1 2(l + 1) < ε1.(3.14) this implies that q∗ ∈ f(t). similarly, we can show that q∗ ∈ f(s). since s is asymptotically quasi-nonexpansive mapping, so it is uniformly quasi-lipschitzian with l = supn≥1{kn} ≥ 1 (by remark 1.1). so, as above we can show that q∗ ∈ f(s). thus q∗ ∈ f = f(s) ∩ f(t), that is, q∗ is a hybrid fixed point of the mappings s and t. this completes the proof. � theorem 3.2. let e be a real banach space and k be a nonempty closed convex subset of e with k+k ⊂ k. let s : k → k be a asymptotically quasi-nonexpansive mapping with sequence {kn}∈ [1,∞) such that ∑∞ n=1(kn−1) < ∞ and t : k → k be uniformly l-lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that f = f(s) ∩ f(t) 6= ∅. let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞,∑∞ n=1 ‖vn‖ < ∞ and dn be taken as in theorem 3.1. suppose that the mappings s and t satisfy the following conditions: (c1) limn→∞‖xn −sxn‖ = 0 and limn→∞‖xn −txn‖ = 0, (c2) there exists a constant a > 0 such that{ a1 ‖xn −sxn‖ + a2 ‖xn −txn‖ } ≥ ad(xn,f) for all n ≥ 1, where a1 and a2 are two nonnegative real numbers such that a1 +a2 = 1. 76 saluja then {xn} converges strongly to a hybrid fixed point of the mappings s and t . proof. from conditions (c1) and (c2), we have limn→∞d(xn,f) = 0, it follows as in the proof of theorem 3.1, that {xn} must converges strongly to a hybrid fixed point of the mappings s and t. this completes the proof. � theorem 3.3. let e be a real banach space and k be a nonempty closed convex subset of e with k+k ⊂ k. let s : k → k be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn − 1) < ∞ and t : k → k be uniformly l-lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that f = f(s)∩f(t) 6= ∅. let {αn} and {βn} be sequences in [δ, 1 − δ] for some δ ∈ (0, 1). let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn − 1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞, ∑∞ n=1 ‖vn‖ < ∞ and dn be taken as in theorem 3.1. suppose limn→∞‖xn−sxn‖ = 0 and limn→∞‖xn−txn‖ = 0. if at least one of the mappings s and t is semi-compact, then the sequence {xn} converges strongly to a hybrid fixed point of the mappings s and t . proof. without loss of generality, we may assume that t is semi-compact. by theorem 3.1, {xn} is bounded and by assumption of the theorem limn→∞ ‖xn − sxn‖ = 0 and limn→∞‖xn − txn‖ = 0. this means that there exists a subsequence {xnk}⊂{xn} such that xnk → x ∗ ∈ k as nk →∞. now again by the hypothesis of the theorem, we find ‖x∗ −tx∗‖ = lim nk→∞ ‖xnk −txnk‖ = 0 and ‖x∗ −sx∗‖ = lim nk→∞ ‖xnk −sxnk‖ = 0. this shows that x∗ ∈ f. according to theorem 3.1, the limit limn→∞‖xn −x∗‖ exists. then lim n→∞ ‖xn −x∗‖ = lim nk→∞ ‖xnk −x ∗‖ = 0, which means that {xn} converges to x∗ ∈ f, that is, the sequence {xn} converges strongly to a hybrid fixed point of the mappings s and t . this completes the proof. � theorem 3.4. let e be a real banach space satisfying opial’s condition and k be a nonempty closed convex subset of e with k + k ⊂ k. let s : k → k be a asymptotically quasi-nonexpansive mapping with sequence {kn} ∈ [1,∞) such that ∑∞ n=1(kn−1) < ∞ and t : k → k be uniformly l-lipschitzian asymptotically quasi-nonexpansive mapping in the intermediate sense such that f = f(s)∩f(t) 6= ∅. let {αn} and {βn} be sequences in [δ, 1−δ] for some δ ∈ (0, 1). let {xn} be the sequence defined by (1.3) with the restrictions ∑∞ n=1(kn−1)αn < ∞, ∑∞ n=1 ‖un‖ < ∞, ∑∞ n=1 ‖vn‖ < ∞ and dn be taken as in theorem 3.1. suppose that s and t have a hybrid fixed point, i − s and i − t are demiclosed at zero and {xn} is an approximating hybrid fixed point sequence for s and t , that is, limn→∞‖xn − sxn‖ = 0 and limn→∞‖xn −txn‖ = 0. then {xn} converges weakly to a hybrid fixed point of the mappings s and t . proof. let p be a hybrid fixed point of s and t. then limn→∞‖xn − p‖ exists as proved in theorem 3.1. we prove that {xn} has a unique weak subsequential limit in f = f(s) ∩ f(t). for, let u and v be weak limits of the subsequences convergence of hybrid fixed point 77 {xni} and {xnj} of {xn}, respectively. by hypothesis of the theorem, we know that limn→∞‖xn −sxn‖ = 0 and i −s is demiclosed at zero, therefore we obtain su = u. similarly, tu = u. thus u ∈ f(s) ∩f(t). again in the same fashion, we can prove that v ∈ f(s) ∩f(t). next, we prove the uniqueness. to this end, if u and v are distinct then by opial’s condition lim n→∞ ‖xn −u‖ = lim ni→∞ ‖xni −u‖ < lim ni→∞ ‖xni −v‖ = lim n→∞ ‖xn −v‖ = lim nj→∞ ‖xnj −v‖ < lim nj→∞ ‖xnj −u‖ = lim n→∞ ‖xn −u‖. this is a contradiction. hence u = v ∈ f . thus {xn} converges weakly to a hybrid fixed point of the mappings s and t . this completes the proof. � remark 3.5. our results extend, improve and generalize many known results from the existing literature (see, e.g., [1], [3], [4], [6]-[8], [10]-[14] and many others). 4. conclusion the results proved in this paper of the iteration scheme (1.3) for a pair of asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mappings in the intermediate sense which is wider than that the class of nonexpansive, quasinonexpansive and asymptotically nonexpansive mappings. thus our results are good improvement and extension of some corresponding previous work from the existing literature. references [1] r.p. agarwal, donal o’regan and d.r. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, nonlinear convex anal. 8(1)(2007), 61–79. [2] c.e. chidume and b. ali, weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 330(2007), 377–387. [3] g. das and j.p. debate, fixed points of quasi-nonexpansive mappings, indian j. pure appl. math. 17(11)(1986), 1263–1269. [4] h. fukhar-ud-din and s.h. khan, convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, j. math. anal. appl. 328(2007), 821–829. [5] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1)(1972), 171–174. [6] s.h. khan and h. fukhar-ud-din, weak and strong convergence of a scheme with errors for two nonexpansive mappings, nonlinear anal. 61(8)(2005), 1295– 1301. 78 saluja [7] s.h. khan, y.j. cho and m. abbas, convergence to common fixed points by a modified iteration process, j. appl. math. comput. doi:10.1007/s12190-0100381-z. [8] s.h. khan and w. takahashi, approximating common fixed points of two asymptotically nonexpansive mappings, sci. math. jpn. 53(1)(2001), 143–148. [9] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc. 73(1967), 591–597. [10] m.o. osilike and s.c. aniagbosor, weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, math. and computer modelling 32(2000), 1181–1191. [11] b.e. rhoades, fixed point iteration for certain nonlinear mappings, j. math. anal. appl. 183(1994), 118–120. [12] j. schu, weak and strong convergence to fixed points of asymptotically nonexpansive mappings, bull. austral. math. soc. 43(1)(1991), 153–159. [13] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, j. math. anal. appl. 158(1991), 407–413. [14] n. shahzad and a. udomene, approximating common fixed points of two asymptotically quasi nonexpansive mappings in banach spaces, fixed point theory and applications 2006(2006), article id 18909, pages 1-10. [15] k.k. tan and h.k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178(1993), 301–308. [16] w. takahashi and t. tamura, convergence theorems for a pair of nonexpansive mappings, j. convex anal. 5(1)(1998), 45–56. [17] b.l. xu and m.a. noor, fixed point iterations for asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 267(2)(2002), 444–453. [18] y. yao and y.c. liou, new iterative schemes for asymptotically quasinonexpansive mappings, journal of inequalities and applications, 2010(2010), article id 934692, 9 pages. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications volume 19, number 4 (2021), 587-603 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-587 received may 14th, 2021; accepted june 7th, 2021; published june 24th, 2021. 2010 mathematics subject classification. 68n30. keywords. mobile application; resource management; performance evaluation; optimizing performance. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 587 analysis of parameters used for measuring performance of mobile applications durga sowjanya kolluru1,*, p. bhaskara reddy2 1department of electronics and communication engineering, mlr institute of technology, hyderabad, india 2director, holy mary institute of technology and science, hyderabad, india *corresponding author: k.durgasowjanya@gmail.com abstract. to make mobile application more reliable, performance is most important parameter. resource management plays crucial role inside development of mobile application. careful attention should be required for number of non-functional requirements to achieve ultimate goal of mobile application development. mobile application and its impact on market are analyzed by performance evaluation process. for getting better performance, efficient utilization of physical resources is required. these resources provide actionable information to application developer for optimizing performance and increases efficiency during development cycle. due to limited processing power and memory resources, keep hardware components always in running states. but slow applications drain batteries of their devices. to maximize utilization of resources, developers should consider hardware limitations too which make an effort to optimize performance and efficiency of application. this paper presents a qualitative performance analysis of mobile applications. this paper describes factors considered for analyzing performance and corrective measures to achieve better performance. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-587 int. j. anal. appl. 19 (4) (2021) 588 1. introduction popularity of mobile applications has fabulous response since last decade. development of mobile applications witnessed change of devices from land line telephones to high-end smart phones. smart phones are driven by powerful operating system that allows users to add and remove applications. business organizations prepare software developers for developing independent application with life cycle which become redundant and expensive [1]. fig.1. performance evaluation proposed paper presents route to analyze performance of mobile application for android operating system. figure 1 shows the evaluation process of performance for mobile application. various parameters are raised by the utilization of resources in mobile application during development time and runtime. then it describes tips for optimizing performance of mobile application. proposed paper is organized into different sections as section 2 for related work, section 3 for overview of parameters affecting performance of mobile application, section 4 describes tips to improve performance and section 4 presents results and discussion and section 5 states conclusion. 2. literature survey existing research focused on the analysis of performance optimization by evaluation and comparison of qualitative properties [2]. state-of-art for mobile application development addresses many challenges and opportunities for developers [3]. it mainly focuses on userperceived application performance. alternative methods for qualitative analysis are presented in [6], [7] and [8]. demo application with graphical user interface is presented in [9]. it describes variious parameters like cpu usage, memory usage and battery consumption. native interface is not used in this method to evaluate behavior of application. it presents study on in-depth parameters for mobile application performance as per quantitative analysis. therefore, int. j. anal. appl. 19 (4) (2021) 589 contribution of existing work for proposed paper is explained in three phases. first phase explains cpuusage, memory and battery usage, multiple response time as they facilitate major impact on user experience. second phase compares java implementations with native implementations for android operating system. third phase analyzes all parameters of development cycle based on user experience. 3. parameters effecting performance performance of mobile application can be determined with many parameters. execution time, memory usage and battery consumption are parameters yields for analyzing performance [9]. application’s execution time and launch time, both are collectively called as latency. analysis of performance is based on various parameters that are measured and evaluated. parameters considered during development time and run times are described in section 3.1 and 3.2. these parameters specify effect on performance of each parameter relevant to overall performance of application. 3.1. parameters effected during development of an application: 3.1.1. threads: managing threads determines stable and consistently high-performing application. 90% of performance issues can be solved by effective working with threads. there are two main types of threads which effects performance of an application: main thread, also called ui thread, and the background thread. figure 2 shows working of ui thread and its effect on performance. any changes that made to the ui block the main thread, and this won’t be assigned to another method or call until the current task is finished and it follows the fifo method. fig.2. working of ui threads int. j. anal. appl. 19 (4) (2021) 590 3.1.2. managing memory leaks: most important problems that appeared in mobile application are memory leaks. a memory leak is the problem occurred when the garbage collector is not able to collect the allocated memory. for example: if the reference is hold in our code to an unused activity, all of activity’s layout and, in turn, all of its views and everything else that activity holds [5]. main causes of memory leaks are as follows: 3.1.2.1. static references: keeping static references in the application degrades the performance. an improper reference causes both activities and fragments with a reduced lifecycle. with static reference, an activity or fragment will not be garbage collected. 3.1.2.2. unregistered events and handlers: it is to be noticed that lot of times developers don’t unregister events and handlers. this causes application could leak. 3.1.3. removing deprecated apis: deprecation essentially references when an api is removed and, it could be that one or two days after a major release, the application might not work on certain devices. to make things even worse, sometimes application is not dependent on an older version of a library, no way is detected to update both apis and tools. the major problem by not using deprecated apis is that after the api is removed from the android system, the developed application won’t be able to find the method and will crash with a runtime exception. similarly, always make sure that the libraries are maintained by the providers otherwise the application get stuck in a pothole without a way out. 3.1.4. compactness: compactness is determined by user experience in downloading application over mobile internet connection. 3.1.5. disk space: there are two methods used to measure disk space. one method is considered by space taken for installing application on device. it is use ful for low end devices with limited resources. another method is verified by space defined for apk size. it is downloadable installer for application in android os. int. j. anal. appl. 19 (4) (2021) 591 3.1.6. without using static final key: it is observed that many developers make the mistake as they declare static constants without the final key. compiler executes initializer method called <clinit> when class is used first time. these values are accessed later with field lookups. so, from the above example, it is understood that when a static field without the final keyword was declared, that variable is considered as a field and not as a constant. 3.2. parameters effected during run time execution of an application: 3.2.1. response time: response time gains importance regarding user experience. figure 3 shows graphical representation of response time during application running. response time is calculated by three different time measures listed as start time, pause time, resume time. start time is time taken by start of application i.e. time taken from tapping application icon to displaying first screen of application [8]. resume time is time taken by application moving from background to foreground and vice versa for pause time. fig.3. graphical representation of response time 3.2.2. memory usage: memory usage defines amount of ram allocated for application. figure 4 shows memory usage of application on profiler. event timeline, which shows activity states, user input events, and screen rotation events as indicated by x-axis. memory usage by each memory category being shon by stacked graph as indicated by y-axis on left and color key at top. dashed line indicates number of allocated objects as indicated by y-axis on right. each garbage collection event is shown by icon. in background, it measures ram memory [9]. int. j. anal. appl. 19 (4) (2021) 592 fig.4. memory usage on profiler 3.2.3. cpu usage: cpu usage measures percentage amount of work done by cpu between two specific time intervals. it depends on operating point of cpu determined by frequency. so, care should be taken to normalize cpu usage for maximum operating frequency which gives comfortable user experience. 3.2.4. battery consumption [18]: for any application, battery consumption is important. figure 5 shows battery consumption of application. if device batteries are drained, no user prefers to use applications. efficient battery usage of application reasons user experience. fig.5.battery usage 3.2.5. application launching time [25]: application launching time is described by one of two states. it is described by time taken for application visible to user. the two states are: • cold state • warm state int. j. anal. appl. 19 (4) (2021) 593 3.2.5.1. cold state: cold state refers to state of application starting from scratch launched for first time after device boot or reload from system. cold state performs three tasks. they are: • loading and launching application. • displaying blank window immediately after launch. • creating application window. 3.2.5.2. warm state: warm state provides lower-overhead than cold state. warm state defines state of activity moves from backgroung to foreground. if all activities are resided in memory, application can avoid object initialisation, layout inflation, and rendering. 3.2.6. latency : latency is the time between moments that request data till the moment that actually start receiving it. 3.2.7. slow rendering: slow rendering is common performance issue for mobile applications. application screen should be updated after every 16ms by attempting redrawn of application activities after every 16ms. if the frame is not completed within 16ms, it is called as dropped frame. figure 6 shows process of slow rendering. if application takes 21.763 ms to draw new picture, system was not ready because time exceeds 16ms. therefore, user can saw new picture with refreshed after 32 ms instead of 16ms. one dropped frame in application causes animation will not be shown with smooth start. rendering is defined as how many times application runs in smooth process at a constant rate of 60 fps without any dropped or delayed frame. fig.6. slow rendering process int. j. anal. appl. 19 (4) (2021) 594 3.2.8. layouts: layouts are fundamental part for any android application, and they directly affect user experience. inadequate implementation of application layout will lead to memory hungry with slow ui. efficient layout structure adds and initializes widget and layout to application. nested instances of linear layout produce deep view hierarchy. nesting linear layout is defined by layout_weight parameter. for list view or grid view, layout is inflated repeatedly. 4. optimizing performance of android mobile application mobile application is developed by giving lot of freedom to developers. after developing application, owner will accesses ever-growing user base. for that developers face many challenges in application development. inventing many versions make developers hard to keep up all updates in application development. it creates big challenge since thousands of devices are running on same os with different versions. application performance can be improved by keeping few things in mind. following are the tips or remedies which optimize application performance. 4.1. never do following things on ui thread: • load images or streams • parse a json • access a local or remote server • make api calls 4.2. steps to remove memory leaks [8]: • avoid using static variables. • always unregister events and listeners. • use tools such as eclipse analyzer or leak canary to find these leaks. • perform code reviews and code review implementations. sometimes peer devices can point out things that might not have noticed at all. usually, this happens when the application have been looking at the same code over and over again. • understand the architecture properly before writing any piece of code. int. j. anal. appl. 19 (4) (2021) 595 4.3. corrective methods for rendering: 4.3.1. hierarchy viewer: hierarchy viewer is a tool which allows developer to inspect properties and layout speed for each view in application layout hierarchy. it measures structure of application view hierarchy and simplifying hierarchy to reduce over draw. 4.3.2. profile gpu rendering: profile gpu rendering presents visual representation of time taken by ui window to render frames relative to 16-ms-per-frame benchmark. rendering can be improved by enable profile gpu rendering. 4.3.3. visual output of gpu profiler: visual output possess graphical representation consists of horizontal axis and vertical axis for elapsing time and frame time respectively in milliseconds. interacting application indicated by growth of vertical bars on screen. each vertical bar determines frame of rendering. green line indicates target at 16 ms. whenever all frames crosses green line, missing frame displays stuttering images on user output. 4.4. resolving application launching time delay: application launching delay was resolved by initialising objects immediately needed rather than creating global static objects. if objects are moved to singleton pattern, application can perform lazily by different threads. it causes delay in application launching. instead of moving objects to singleton pattern, initialise objects for accessing first time. it can be achieved by view hierarchy. in view hierarchy, nested layouts are used for flattening application. it is done by reducing redundant. it updates visual properties depend on bitmaps and other resources. 4.5. battery usage reduction [9]: in android application development, important parameter that ultimately leads to retain user is battery usage. figure 7 shows battery usage reduction by uninstalling unused applications. optimizations of battery draining issue done by reducing uninstall application many times. int. j. anal. appl. 19 (4) (2021) 596 fig.7. battery usage reduction 4.6. improving battery usage [9]: • reducing network calls of user. • avoiding wake lock. • using gps carefully. • using alarm manager carefully. • perform batch scheduling. 4.7. avoid issues with deprecations: • understanding and using of proper apis. • refraction of application dependencies. • avoid using accessing private methods with reflection. • update application dependencies and tools periodically. 4.8. avoid abuse: • avoid calling private apis by reflection. • avoid calling private native methods from ndk or c level. • avoid using adb shell am to communicate with other processes. • avoid runtime.exec to communicate with processes. 4.9. prefer static methods over virtual methods: invocations will be about 15%-20% faster if static methods are accessed than virtual methods. it is a good practice because developer can easily noticed that application method signature which is calling another method cannot alter object’s state. int. j. anal. appl. 19 (4) (2021) 597 4.9.1. using static final for constants: if constants are declared as static final, then the class no longer requires initialization method. constants in static field initializes from dex file. constants declared as integer will use integer value directly. constants accessed as tostring is relatively inexpensive because "string constant" declaration is used instead of field lookup. 4.9.2. using enhanced for-each loop instead of for loop: for-each loop is also known as enhanced-for loop. it is used for iterate through count that implements iterable interface. it is used for arrays. iterator was allocated to interface calls to next (). for iterator, hand-written counted loop was 3 times faster than list array. but for other use cases, enhanced-for loop is exactly same as explicit iterator. so enhanced-for loop is used as default loop but for performance-critical array list iteration, hand-written counted loop is to be considered. 4.9.3. use profilers to profile performance: profiling is one of the most important steps that you should perform on any application that has any kind of performance issues. there are multiple profiler’s available by android that can be used for application. 4.9.4. avoid using float: thumb rule states that floating-point is 2 times slower than integer. for advanced device operations, float and double are same for speed but in terms of space double is 2 times larger than float. for desktop devices space is not an issue, so, developers should prefer double to float [9]. 4.10. corrective methods for layout performance improvements: android sdk uses certain methods to identify problems in layout performance. following corrective methods are used to implement smooth scrolling ui’s with minimal memory trace [10]. 4.10.1. re-using layout: reusing layout allows applications to create reusable complex layout. common elements of application across multiple layouts can be extracted and managed separately. after that those are included in each layout. custom view is used for creating individual ui components by re using layout file and hence it is easy to progress. int. j. anal. appl. 19 (4) (2021) 598 4.10.2. loading views on demand: application layout requires rarely used complex view. complex views include details of item used, progress indicator for itm and undo process of item. as per their need only, views can be loaded in an application to reduce memory usage and to speed up rendering. 4.10.3. optimization of toolbars: prefer toolbar over action bar, and prefer recycler view over list view, especially for animations, and if the application has a bunch of huge images, because it’s optimized for that. it has simpler apis, light-weight, transparent compression, better response caching, and other amazing features [10]. 4.10.4. re-use layout with <include/> and <merge/>: android have specific controls to provide efficient and re-usable interactive elements. developers require a special layout to re-use larger components like custom controls. for efficiently reuse layouts, “include” and “merge” tags are placed to show items for one layout inside another layout [11]. reusable layout allows creating a reusable complicated layout. custom view is used to reuse layout. 4.10.5. delayed view loading: with certain conditions complex views are rarely used in layout. loading view reduces memory usage and speed up layout rendering. delayed loading view of resources is used whenever complex views are needed [10]. it is implemented by using view stub for complex and rarely-used views. 4.10.6. optimized hierarchy: using basic layout structures is not efficient way of consistent and high-performance ui controls for layouts. but it is common misconception amongst community. nesting layout structures possess high performance than basic layout structures. nesting several instances of linear layout uses layout_weight parameter which is time-consuming. each child needs to be measured twice. in list view or grid view, layout is inflated as in view repeatedly. hierarchy viewer and lint are used to optimize application layout performance. hierarchy viewer allows layout to be examined while application is running. it helps to find out bottlenecks in layout performance [12]. int. j. anal. appl. 19 (4) (2021) 599 • use compound drawable: linear layout contains both image view and text view which are handled as compound drawable resource. • merge root frame: replacing root of layout with merge tag for frame layout. then it does not provide background or padding etc. • useless leaf: layout with no children or no background is treated as useless leaf. removing useless leaf provides flatter and efficient layout hierarchy. • useless parent: layout with only children but no siblings are called as parent layout which does not comes under scroll view or root layout. as there is no background, removing parent leaf causes children leafs moved directly into parent for flatter and efficient layout hierarchy. • deep layouts: maximum depth for nesting is 10 for achieving good performance. layouts with more nesting causes bad performance. to enhance performance, instead of nesting, flatter layout like relative layout or grid layout should be used. 4.10.7. working with lint: lint tool provides possible optimizations for view hierarchy. lint replaces layout opt tool and hence achieves greater functionality. lint automatically fixes issues, providing suggestions for others and offending code for review. lint is integrated into android studio and it will automatically run. in android studio, lint has inspection run choices like specific build variant or all build variants [13]. 4.11. know and use libraries: developers always have the knowledge on apis and libraries of application. always prefer to use a library’s code over writing application. system has privilege to replace function calls with library methods using hand-coded assembler. 4.12. using native methods carefully: application developed in native interface with android ndk api is efficient than programming in java. moreover, there is loss associated with java-native transition. if resources like memory on native heap, file descriptors are allotted, it is more complex to arrange resources [4]. https://developer.android.com/reference/android/widget/relativelayout.html int. j. anal. appl. 19 (4) (2021) 600 native code is more useful when native codebase ported to android application. but it is not used for speed up android application written in java. jni tips should be followed for native code in android. 5. results and discussion proposed paper analyzes various parameters that are measured and evaluated for android application. proposed paper discussed that response time and memory usage are to be measured by four different actions, namely start of application, pause application, resume application and navigating from one page to another page on application. response time analyzes user experience. cpu usage provides an interesting benchmark to optimize performance. disk space is preferred for low end devices with limited resources. compactness is relevant to users. battery consumption is essential to reach the application to more users. launch time was measured in phases of launching application and finishing launching. it is recommend from above discussion that limited amount of app-specific code should be added to lifecycle methods. proposed paper discusses corrective methods for each parameter to optimize performance of android application. sample application developed and run on device samsung sm a505f. figure 8 shows experimental results for connected device on profiler. go to profiler will displays screen with parameters cpu usage, memory usage at specific time intervals. fig.8. performance of mobile application on profiler int. j. anal. appl. 19 (4) (2021) 601 6. conclusion proposed paper presents performance analysis of android mobile application by various parameters. the parameters effected during development time and run times are summarized. after that corrective methods used to improve performance or optimizing performance are discussed. finally it is concludes that performance of mobile application depends on architectural requirements and layout hierarcehy. so application developers need to consider essential hardware and software limitations to achieve qualitative, effective and efficient application. acknowledgements i would like to thank department of science and technology (dst), india under wos-a scheme for financial support. i would like to express my very great appreciation to p. bhaskara reddy, my project mentor for their patient guidance, enthusiastic encouragement and useful critiques of this research work. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] v.s. sundara rajan, a. malini, k. sundarakantham, performance evaluation of online mobile application using test my app, in: 2014 ieee international conference on advanced communications, control and computing technologies, ieee, ramanathapuram, india, 2014: pp. 1148–1152. [2] m. willocx, j. vossaert, v. naessens, a quantitative assessment of performance in mobile app development tools, in: 2015 ieee international conference on mobile services, ieee, new york city, ny, usa, 2015: pp. 454–461. [3] d.s. kolluru, p.b. reddy, review on communication technologies in telecommunications from conventional telephones to smart phones, in: international conference on advances in signal processing, vlsi, communications and embedded systems (icsvce-2021), hyderabad, india, april 2021. int. j. anal. appl. 19 (4) (2021) 602 [4] w. gao, l.-y. duan, j. sun, et al. mobile media communication, processing, and analysis: a review of recent advances, in: 2013 ieee international symposium on circuits and systems (iscas2013), ieee, beijing, 2013: pp. 869–872. [5] l. corral, a. sillitti, g. succi, evolution of mobile software development from platform-specific to web-based multiplatform paradigm. in: proceedings of the 10th sigplan symposium on new ideas, new paradigms, and reflections on programming and software, 2011: pp. 181–183 [6] k. suankaewmanee, d.t. hoang, d. niyato, s. sawadsitang, p. wang, z. han, performance analysis and application of mobile blockchain, in: 2018 international conference on computing, networking and communications (icnc), ieee, maui, hi, 2018: pp. 642–646. [7] l. delia, n. galdamez, l. corbalan, p. pesado, p. thomas, approaches to mobile application development: comparative performance analysis, in: 2017 computing conference, ieee, london, 2017: pp. 652–659. [8] durga sowjanya kolluru, p.bhaskara reddy, ip to ip calling through socket programming, in: ieee sponsored asian conference on innovation in technology (asiancon)2021, pune, maharashtra, india, 2021. [9] d. kayande, u. shrawankar, performance analysis for improved ram utilization for android applications, in: 2012 csi sixth international conference on software engineering (conseg), ieee, indore, madhay pradesh, india, 2012: pp. 1–6. [10] s.v.s. prasad, t.s. savithri, i.v.m. krishna, a new technique for color based image segmentation using support vector machines, in: 2014 international conference on medical imaging, m-health and emerging communication systems (medcom), ieee, greater noida, india, 2014: pp. 189–192. [11] m. hort, m. kechagia, f. sarro, m. harman, a survey of performance optimization for mobile applications, ieee trans. software eng. in press, https://doi.org/10.1109/tse.2021.3071193. [12] p. yoosook, p. apirukvorapinit, performance monitoring tool for mobile application, in: 2018 5th international conference on business and industrial research (icbir), ieee, bangkok, 2018: pp. 177– 182. [13] p.k. aggarwal, p.s. grover, l. ahuja, a performance evaluation model for mobile applications, in: 2019 4th international conference on internet of things: smart innovation and usages (iot-siu), ieee, ghaziabad, india, 2019: pp. 1–3. [14] d.s. kolluru, p.b. reddy, development of voice call transfer service between android smart phone and tablet, rev. geint.-gest. inov. tecnol. 11 (2021), 467-480. [15] s.v.s. prasad, t. satya savithri, i.v. murali krishna, performance evaluation of svm kernels on multispectral liss iii data for object classification, int. j. smart sensing intell. syst. 10 (2017) 829–844. int. j. anal. appl. 19 (4) (2021) 603 [16] k.d sowjanya, ch. srinu, instant message transfer between two smart phones using wi-fi, int. j. adv. eng. manage. sci. 2 (2016), 1949-1951. [17] k. kuan, t. adegbija, energy and performance analysis of sttram caches for mobile applications, in: 2019 ieee 13th international symposium on embedded multicore/many-core systems-on-chip (mcsoc), ieee, singapore, singapore, 2019: pp. 257–264. [18] l. zhang, b. tiwana, z. qian, et al. accurate online power estimation and automatic battery behavior based power model generation for smartphones, in: proceedings of the eighth ieee/acm/ifip international conference on hardware/software codesign and system synthesis codes/isss ’10, acm press, scottsdale, arizona, usa, 2010: p. 105. [19] l. pu, performance analysis of short message service, in: 2009 international symposium on intelligent ubiquitous computing and education, ieee, chengdu, china, 2009: pp. 370–373. [20] m.e. joorabchi, a. mesbah, p. kruchten, real challenges in mobile app development, in: 2013 acm / ieee international symposium on empirical software engineering and measurement, ieee, baltimore, maryland, 2013: pp. 15–24. [21] s.v.s. prasad, double block zero padding acquisition algorithm for gps software receiver, j. autom. mobile robot. intell. syst. 12 (2019), 58–63. [22] l. delia, n. galdamez, p. thomas, l. corbalan, p. pesado, multi-platform mobile application development analysis, in: 2015 ieee 9th international conference on research challenges in information science (rcis), ieee, athens, greece, 2015: pp. 181–186. [23] r.r. nimbalkar, mobile application testing and challenges, int. j. sci. res. 2 (2013), 56-58. [24] k.d. sowjanya, v.g. devi, voice call between android devices using wireless sensors, int. j. adv. res. eng. technol. 4 (2016), 10-12. [25] s. blom, m. book, v. gruhn, r. hrushchak, a. k, write once, run anywhere a survey of mobile runtime environments, in: 2008 the 3rd international conference on grid and pervasive computing workshops, ieee, kunming, china, 2008: pp. 132–137. international journal of analysis and applications volume 19, number 1 (2021), 110-122 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-110 strong solutions to 3d-lagrangian averaged boussinesq system ridha selmi1,3,4,∗, leila azem2,4 1department of mathematics, college of sciences, northern border university, p.o. box 1321, arar, 73222, ksa 2department of mathematics, college of sciences and art (turaif), northern border university, ksa 3department of mathematics, faculty of sciences of gabès, university of gabès, gabès, 6072, tunisia 4laboratory of partial differential equations and applications (lr03es04), faculty of sciences of tunis, university of tunis el manar,tunis, 1068, tunisia ∗corresponding author: ridha.selmi@nbu.edu.sa abstract. under suitable assumptions on the initial data, we prove the existence, uniqueness of the strong solutions to a regularized periodic three-dimensional lagrangian averaged boussinesq system, in a sobolev spaces. also, we establish the convergence results of this unique strong solution of this regularized boussinesq system to a strong solution of the three-dimensional boussinesq system, as the regularizing parameter vanishes. 1. introduction and statement of main results we consider the following 3d-lagrangian averaged boussinesq-α system: received april 4th, 2020; accepted april 22nd, 2020; published december 17th, 2020. 2010 mathematics subject classification. primary 35a01, 35a02, 35b40; secondary 35b10. key words and phrases. 3d-lagrangian averaged boussinesq-α system; alpha-regularization; existence; uniqueness; convergence. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 110 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-110 int. j. anal. appl. 19 (1) (2021) 111 (1.1)   ∂v ∂t + (u ·∇)v + ∇ut ·v + ∇p = ν∆v + θe3, (t,x) ∈ r+ ×t3 ∂θ ∂t −κ∆θ + (u ·∇)θ = 0, (t,x) ∈ r+ ×t3 v = (1 −α2∆)u, (t,x) ∈ r+ ×t3 div u = div v = 0, (t,x) ∈ r+ ×t3 (u,θ)|t=0 = (u0,θ0), x ∈ t 3, where by t3 we refer to the 3d-torus, ν > 0 represents the viscosity of the fluid and κ > 0 its thermal conductivity. the unknown vector field u, the scalars p and θ stand respectively for the velocity, the pressure and the temperature of the fluid at the point (t,x) ∈ r+ × t3. the superscript mt denotes the transpose of the matrix m. the data (u0,θ0), are respectively the initial free divergence velocity and the initial temperature. the lagrangian averaged boussinesq-α model (1.1) is the first to use lagrangian averaging to address the turbulence closure problem and also in geophysical modeling [4]. the boussinesq equations were derived in the nineteenth century by joseph boussinesq, despite intense study, there still remain many difficulties and open questions concerning them. namely, the boussinesq system have an incomplete solution theory. it is not known whether global in time strong solutions exist, and although we know weak solutions exist, we are still unable to prove their uniqueness. regularization models are a way to come up with a rather well-posed problem theory. the first attempt to regularize the navier-stokes equations was made in [9] via smoothing the convective velocity by taking the convolution product against a mollifier. here, and among many others methods of regularization presented in the literature, we note that α-regularization models are obtained by applying a smoothing via taking the inverse of the helmholtz operator i − α2∆. the interested reader is referred to [3, 5, 6]. especially, the lans-α model acts by modifying the nonlinearity in the navier-stokes equations without introducing any extra dissipation [6]. it can be seen as a systematic method for modelling the mean circulatory effects of small-scale turbulence, while preserving the mathematical properties that guarantee existence of a unique, regular solution [3, 5]. the inviscid lagrangian averaged euler-α equations were originally derived as euler–poincaré equations in the framework of hamilton’s principle for geometric fluid mechanics [7]. the existence, uniqueness and continuous dependance of solutions to initial date, as well as convergence results of various α-models, as α vanishes can be found in [1, 2, 8, 10, 12] and references therein. before stating our main results, let us introduce some notations that will be used throughout the paper. int. j. anal. appl. 19 (1) (2021) 112 • for n ∈ n, we denote by pn the projection into fourier modes of order up to n, that is pn( ∑ k∈z3 ûk e ikx) = ∑ |k|≤n ûk e ikx. • for s > 0, we define the operator λs acting on hs(t3) as follows. let u ∈ hs(t3) having the fourier series u(x) = ∑ k∈z3 ûk e ikx ∈ hs(t3). then, we define λsu(x) = ∑ k∈z3 |k|sûk eikx ∈ l2(t3). • we denote by ‖ · ‖ḣs the seminorm ‖λ s · ‖l2 . this is compatible with the definition of the sobolev norm. in fact, ‖ · ‖hs is equivalent to ‖ · ‖l2 + ‖ · ‖ḣs . note that in the fourier setting it is more usual to define an equivalent norm on hs by ‖u‖ = ( ∑ k∈z3 (1 + |k|2s)|ûk|2 )1/2 . • we refer to the fractional laplacian √ −∆ by λ. • we denote b(u,v) = [(u ·∇)v], u,v ∈ h1(t3), b̃(u,v) = [(∇×v) ×u], u,v ∈ h1(t3). • we define the space h̃s = {u ∈ hs, div u = 0}. we remark that, for u, v and w ∈ h̃1(t3)( b(u,v),w ) = − ( b(u,w),v ) . • due to the identity (1.2) (u ·∇)v + 3∑ j=1 vj∇uj = −u× (∇×v) + ∇(v ·u), we obtain ( b̃(u,v),w ) = ( b(u,v),w ) − ( b(w,v),u ) . • by the definitions of the operators b(u,v) and b̃(u,v), we deduce the following properties: lemma 1.1. i) let u,v,w ∈ h̃1(t3), then (1.3) ( b(u,v),w ) = − ( b(u,w),v ) which in turn implies that (1.4) ( b(u,v),v ) = 0, u,v ∈ h̃1(t3). int. j. anal. appl. 19 (1) (2021) 113 also, (1.5) ( b̃(u,v),w ) = ( b(u,v),w ) − ( b(w,v),u ) , u,v,w ∈ h̃1(t3) and hence (1.6) ( b̃(u,v),u ) = 0, u,v ∈ h̃1(t3) ii) let u ∈ h̃1(t3), v ∈ h̃2(t3), and w ∈ l2(t3), then (1.7) | ( b(u,v),w ) | ≤ c‖∇u‖l2‖∇v‖ 1/2 l2 ‖4v‖1/2 l2 ‖w‖l2. iii) let u,v,w ∈ h̃1(t3), then (1.8) | 〈 b̃(u,v),w 〉 h̃−1(t3) | ≤ c‖∇u‖l2‖∇v‖l2‖w‖ 1/2 l2 ‖∇w‖l2. here < .,. > denotes the duality pairing of h̃1(t3) and h̃−1(t3). iv) let u ∈ h̃1(t3), v ∈ l2(t3) and w ∈ h̃2(t3), then (1.9) | 〈 b̃(u,v),w 〉 h−2 | ≤ c ( ‖u‖1/2 l2 ‖∇u‖1/2 l2 ‖v‖l2‖∆w‖l2 + ‖v‖l2‖∇u‖l2‖∇w‖ 1/2 l2 ‖∆w‖1/2 l2 ) . proof. see [10] and references therein. � using the above notations and the identity (1.2), we obtain the following equivalent systems of equations: (1.10)   ∂v ∂t + b̃(u,v) −ν4v = θe3, (t,x) ∈ r+ ×t3 ∂θ ∂t −κ4θ + b(u,θ) = 0, (t,x) ∈ r+ ×t3 v = (1 −α24)u, (t,x) ∈ r+ ×t3 div u = div v = 0, (t,x) ∈ r+ ×t3 (u,θ)|t=0 = (u0,θ0), x ∈ t 3, our first result is the following existence and uniqueness theorem. theorem 1.1. let θ0 ∈ ḣ1(t3) and u0 ∈ ḣ2(t3) a divergence free vector field. then, there exists a unique strong solution (uα,θα) of system (1.10), such that uα ∈ c(r+,ḣ2(t3)) ∩l2loc(r+,ḣ 3(t3)) and θα ∈ c(r+,ḣ1(t3)) ∩l2loc(r+,ḣ 2(t3)). int. j. anal. appl. 19 (1) (2021) 114 moreover, ∀ 0 ≤ t ≤ t , this solution satisfies the following energy estimates (1.11) ‖∇θα‖2l2(t3) + ‖∇uα‖ 2 l2(t3) + α 2‖∆uα‖2l2(t3) + κ ∫ t 0 ‖∆θα‖2l2(t3)dτ + ν ∫ t 0 (‖∆uα‖2l2(t3) + α 2‖∇∆uα‖2l2(t3))dτ ≤kα(t), where kα(t) = k(t) 2µ + k3(t) 2µα6 (1 + α2) + ‖∇θ0‖2l2(t3) + ‖∇u 0‖2l2(t3) + α 2‖∆u0‖2l2(t3), and µ = min(ν,κ). here, the function k(t) stands for k(t) = ‖u0‖2l2 + α 2‖∇u0‖2l2 + ‖θ 0‖2l2 + φα(t) and φα is a positive increasing function of time t, defined by (1.12) φα(t) = ( ‖u0‖2 l2 + α2‖∇u0‖2 l2 + ‖θ0‖2 l2 ) (e2t − 1). the proof is based on a galerkin approximation scheme. while trying to close the energy estimates, the buoyancy force presents some difficulties that we overcome by a gronwall’s type technique. after that, we run a compactness method based on aubin-lions lemma [11]. our second result is a convergence theorem, as α → 0: theorem 1.2. let t > 0, u0 ∈ ḣ2(t3) a divergence free vector field, θ0 ∈ ḣ1(t3) and (uα,θα), the solution in [0,t] of system (1.10) and vα = uα −α2∆uα subject of theorem 1.1. then, there exists a time t∗ such that 0 < t∗ ≤ t and subsequences uαk , vαk , θαk , a scalar function θ and a divergence free vector field u belonging both of them to l∞([0,t∗],ḣ1(t3)) ∩l2([0,t∗],ḣ2(t3)) such that as αk → 0+, one has (1) uαk converges to u and θαk converges to θ weakly in l 2([0,t∗],ḣ2(t3)) and strongly in l2([0,t∗],ḣ1(t3)). (2) vαk converges to u weakly in l 2([0,t∗],ḣ1(t3)) and converges strongly in l2([0,t∗],l2(t3)). (3) uαk (t) converges to u(t) and θαk (t) converges to θ(t) weakly in ḣ 1(t3) and uniformly over [0,t∗]. furthermore, (u,θ) is the unique strong solution of the boussinesq system (bq) on [0,t∗] associated to the initial data (u0,θ0) and satisfies, for all t ∈ [0,t∗], the energy inequality ‖u(t)‖2 l2(t3) + ‖θ(t)‖ 2 l2(t3) + 2 ∫ t 0 ν‖∇u(τ)‖2l2(t3) + κ‖∇θ(τ)‖ 2 l2(t3)dτ ≤‖u0‖2 l2(t3) + ‖θ 0‖2 l2(t3) + 2φ(t ∗). here, (bq) and φ denote respectively (1.10) and φα, for α = 0. note that the solution of the regularized lagrangian averaged boussinesq-α system satisfies an energy inequality that depends on the parameter alpha which provide a singularity, as alpha goes to zero. this is a serious impediment that has to be dealt with when taking the limit as αk → 0. we shall use a compactness int. j. anal. appl. 19 (1) (2021) 115 method to obtain strong convergence. the remainder of this paper is divided into two sections; the first is assigned to prove the existence and uniqueness result. the second is concerned by the proofs of convergence results. 2. well-posedness result to study the existence and the regularity of strong solutions. we approximate (1.10) by the following system of ordinary differential equations: (2.1)   ∂vn ∂t + pnb̃(un,vn) −ν4vn −θne3 = 0, (t,x) ∈ r+ ×t3 ∂θn ∂t −κ4θn + pnb(un,θn) = 0, (t,x) ∈ r+ ×t3 vn = (1 −α24)un, (t,x) ∈ r+ ×t3 div un = div vn = 0, (t,x) ∈ r+ ×t3 (un,θn)|t=0 = (pnu0,pnθ0), x ∈ t 3. both the bilinear operators on the left are continuous on l2×l2. then, the above system appears as a system of ordinary differential equations on l2. thus, the usual cauchy–lipschitz theorem yields the existence of a strictly positive maximal time tn such that a unique solution exists which is continuous in time with value in l2. next, we obtain uniform estimates, with respect to the approximating parameter n, on the approximate solutions. to do so, we use conservation laws and product lemmas. taking the l2(t3)-inner product of the equation satisfied by θn in (2.1) against −∆θn and the one satisfied by un against −∆un, to obtain 1 2 d dt ‖∇θn‖2l2(t3) + κ‖δθn‖ 2 l2(t3) = 〈b(un, ,θn), ∆θn〉(2.2) (2.3) 1 2 d dt ( ‖∇un‖2l2(t3) + α 2‖∆un‖2l2(t3) ) + ν ( ‖∆un‖2l2(t3) + α 2‖∇∆un‖2l2(t3) ) = 〈b̃(un,vn), ∆un〉 + 〈θne3,−∆un〉. now, let us first estimate the right hand sides of (2.2) and (2.3). to do so, we recall the following sobolev inequalities [11]: for every ϑ ∈ ḣ1(t3), we have ‖ϑ‖l3 ≤‖ϑ‖ 1/2 l2 ‖ϑ‖1/2 ḣ1 (2.4) and ‖ϑ‖l6 ≤ c‖ϑ‖ḣ1.(2.5) using hölder’s inequality, (2.4) and (2.5), it holds that |〈b(un,θn), ∆θn〉| ≤ ‖un‖l6(t3)‖∇θn‖l3(t3)‖∆θn‖l2(t3) ≤ c‖un‖ḣ1‖∇θn‖ 1/2 l2 ‖∇θn‖ 1/2 ḣ1 ‖∆θn‖l2. int. j. anal. appl. 19 (1) (2021) 116 hence, one obtains |〈b(un,θn), ∆θn〉| ≤ c‖un‖ḣ1‖∇θn‖ 1/2 l2 ‖∆θn‖ 3/2 l2 .(2.6) for every ϑ ∈ ḣ2(t3), the agmon’s inequality [11] reads ‖ϑ‖l∞ ≤‖ϑ‖ 1/2 ḣ1 ‖ϑ‖1/2 ḣ2 .(2.7) the fact that vn = un −α2∆un yields∣∣∣〈b̃(un,vn), ∆un〉∣∣∣ = |〈b(un,vn), ∆un〉−〈b(∆un,vn),un〉| ≤ |〈b(un,un), ∆un〉| + α2 |〈b(un, ∆un), ∆un〉| + |〈b(∆un,un),un〉| + α2 |〈b(∆un, ∆un),un〉| . the first nonlinear term is to be dealt with as follows: |〈b(un,un), ∆un〉| ≤ ‖un‖l∞(t3)‖∇un‖l2(t3)‖∆un‖l2(t3) ≤ c‖un‖ 1/2 ḣ1 ‖un‖ 1/2 ḣ2 ‖∇un‖l2(t3)‖∆un‖l2(t3) ≤ c‖un‖ 1/2 ḣ1 ‖∇un‖l2(t3)‖∆un‖ 3/2 l2(t3) ≤ c‖un‖ḣ1(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 3/2 l2(t3), where we used hölder’s inequality, inequality (2.7) and the facts that ‖∇un‖l2 ≤‖∆un‖l2 and ‖∆un‖l2 ≤ ‖∇∆un‖l2 . similarly, we have |〈b(∆un,un),un〉| ≤ c‖un‖ḣ1(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 3/2 l2(t3). the second nonlinear term is to be dealt with, in the following manner: |〈b(un, ∆un), ∆un〉| ≤ ‖un‖l6(t3)‖∇∆un‖l2(t3)‖∆un‖l3(t3) ≤ c‖un‖ḣ1‖∇∆un‖l2(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 1/2 l2(t3) = c‖un‖ḣ1(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 3/2 l2(t3). similarly, we have |〈b(∆un, ∆un),un〉| ≤ ‖∆un‖l3(t3)‖∇∆un‖l2(t3)‖un‖l6(t3) ≤ c‖un‖ḣ1(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 3/2 l2(t3). it turns out that ∣∣∣〈b̃(un,vn), ∆un〉∣∣∣ ≤ c(1 + α2)‖un‖ḣ1(t3)‖∆un‖1/2l2(t3)‖∇∆un‖3/2l2(t3).(2.8) by cauchy-schwarz and young’s inequalities, we get |〈θne3,−∆un〉| ≤ ‖∇θn‖2l2 + ‖∇un‖ 2 l2.(2.9) int. j. anal. appl. 19 (1) (2021) 117 summing up (2.2) and (2.3) and using (2.6), (2.8) and (2.9), it follows that 1 2 d dt ( ‖∇θn‖2l2(t3) + ‖∇un‖ 2 l2(t3) + α 2‖∆un‖2l2(t3) ) +κ‖∆θn‖2l2(t3) + ν(‖∆un‖ 2 l2(t3) + α 2‖∇∆un‖2l2(t3)) ≤ c‖un‖ḣ1‖∇θn‖ 1/2 l2 ‖∆θn‖ 3/2 l2 + ‖∇θn‖2l2 + ‖∇un‖ 2 l2 c(1 + α2)‖un‖ḣ1(t3)‖∆un‖ 1/2 l2(t3)‖∇∆un‖ 3/2 l2(t3). using young’s inequality, we absorb the remaining diffusion term in the right hand side and we obtain d dt ( ‖∇θn‖2l2 + ‖∇un‖ 2 l2 + α 2‖∆un‖2l2 ) +κ‖∆θn‖2l2 + ν ( ‖∆un‖2l2 + α 2‖∇∆un‖2l2 ) ≤ c‖un‖4ḣ1‖∇θn‖ 2 l2 + ‖∇θn‖2l2 + ‖∇un‖ 2 l2 , where c is a generic constant that may change from line to line. integrating over time, we obtain, for all t ∈ [0,t∗n), ‖∇θn(t)‖2l2(t3) + ‖∇un(t)‖ 2 l2(t3) + α 2‖∆un(t)‖2l2(t3) + κ ∫ t 0 ‖∆θn(τ)‖2l2dτ +ν ∫ t 0 ( ‖∆un(τ)‖2l2(t3) + α 2‖∇∆un(τ)‖2l2(t3) ) dτ ≤ c ∫ t 0 ‖un(τ)‖4ḣ1 (‖∆un(τ)‖ 2 l2 + ‖∇θn(τ)‖ 2 l2 )dτ + ∫ t 0 (‖∇θn(τ)‖2l2 + ‖∇un(τ)‖ 2 l2 )dτ + ‖∇θ 0‖2l2 + ‖∇u 0‖2l2 + α 2‖∆u0‖2l2. at this point, we give the theorem below that states the existence weak solution. the proof follows exactly the lines of the proof given in [12]: theorem 2.1. let θ0 ∈ l2(t3), and u0 ∈ h1(t3) be a divergence-free vector field. then, for any t > 0 there exists a unique weak solution (uα,θα) to (1.1) in the interval [0,t], where uα ∈ c([0,t],h1(t3)) ∩l2([0,t],h2(t3)), and θα ∈ c([0,t],l2(t3)) ∩l2([0,t],h1(t3)). moreover, we have for all t ∈ [0,t] : (‖u(t)‖2l2(t3) + α 2‖∇u(t)‖2l2(t3) + ‖θ(t)‖ 2 l2(t3)) +2µ ∫ t 0 (‖∇u(τ)‖2l2(t3) + α 2‖∆u(τ)‖2l2(t3) + ‖∇θ(τ)‖ 2 l2(t3))dτ(2.10) ≤‖u0‖2l2 + α 2‖∇u0‖2l2 + ‖θ0‖ 2 l2 + φα(t)︸ ︷︷ ︸ k(t) where by φα(t) we refer to the function ( ‖u0‖2l2 + α 2‖∇u0‖2l2 +‖θ0‖ 2 l2 ) (e2t−1). furthermore, this solution is continuously dependent on the initial data (u0,θ0). moreover, this solution is continuously dependent on the initial data (u0,θ0). in particular, it is unique. int. j. anal. appl. 19 (1) (2021) 118 using the energy estimate for weak solution (uα,θα) in theorem above, and the expression of the function ρα given by equation (1.12), we infer that∫ t 0 (‖∇θn(τ)‖2l2 + ‖∇un(τ)‖ 2 l2 )dτ ≤ k(t) 2min(ν,κ) and ∫ t 0 ‖un‖4ḣ1 (‖∆un‖ 2 l2 + ‖∇θn‖ 2 l2 )dτ ≤‖un‖ 4 l∞ t (ḣ1) (‖θn‖2l2 t (ḣ1) + ‖un‖2l2 t (ḣ2) ) ≤ k 2(t) α4 ( k(t) 2κ + k(t) 2να2 ) ≤ k 3(t)(1+α2) 2min(ν,κ)α6 , where k(t) = ‖u0n‖2l2 + α 2‖∇u0n‖2l2 + ‖θ 0 n‖2l2 + φα(t). in conclusion, we obtain the energy estimate (2.11) ‖∇θn(t)‖2l2 + ‖∇un(t)‖ 2 l2 + α2‖∆un(t)‖2l2 + κ ∫ t 0 ‖∆θn(τ)‖2l2dτ +ν ∫ t 0 (‖∆un(τ)‖2l2 + α 2‖∆∇un(τ)‖2l2 )dτ ≤kα(t), where kα(t) = k(t) 2µ + k3(t) 2µα6 (1 + α2) + ‖∇θ0‖2l2(t3) + ‖∇u 0‖2l2(t3) + α 2‖∆u0‖2l2(t3), where by µ = min(ν,κ). notice that the upper bound k(t) in (2.11) is continuous and does not include any singularity with respect to time t. hence, k(t) rules out the finite time blow-up of the solution near t and the solution can be extended to a global in time solution. the estimation (2.11) provides uniform bounds, with respect to n, of the solution un in l ∞ t (ḣ 2(t3)), l2t (ḣ 3(t3)) as well as in l∞t (ḣ 1(t3)) and l2t (ḣ 2(t3)) for θn. this allows to use the aubin-lions lemma so that we can take the limit as n tends to infinity and then obtain existence. as strong solutions are also weak, uniqueness of strong solution simply follows from uniqueness of weak solution. 3. convergence result already, we have proved that an initial data (u0,θ0) gives rise to a global solution (uα,θα). this section is aimed to deal with the convergence result, as the parameter α vanishes. it must be said that the upper bound in the energy estimate (2.11) depends singularly on α and it will fail to control the solution’s norms, as α goes to zero. from (2.10), it is worth mentioning that the dependence of the upper bound in the weak solution’s energy estimate is polynomial, so that such impediment is absent in the weak solution case. to overcome this difficulty, we need to get uniform bounds, with respect to α on ‖∇θα(t)‖2l2(t3) + ‖∇uα(t)‖ 2 l2(t3) + α 2‖∆uα(t)‖2l2(t3), int. j. anal. appl. 19 (1) (2021) 119 and κ ∫ t 0 ‖∆θα(τ)‖2l2dτ + ν ∫ t 0 ( ‖∆uα(τ)‖2l2(t3) + α 2‖∇∆uα(τ)‖2l2(t3) ) dτ. to do so, firstly we mention that ∣∣∣〈b̃(uα,vα), ∆uα〉∣∣∣ ≤ c‖uα‖1/2ḣ1 ‖uα‖1/2ḣ2 ‖∇vα‖l2(t3)‖∆uα‖l2(t3) ≤ c‖uα‖ 1/2 ḣ1 ‖uα‖ 1/2 ḣ2 (‖uα‖ḣ1(t3) + α 2‖∇∆uα‖l2(t3))‖∆uα‖l2(t3) ≤ c(‖uα‖6ḣ1 + α 6‖∆uα‖6l2 ) + ν/2‖∆uα‖ 2 l2(t3) + α 2ν/2‖∇∆uα‖2l2(t3), where we used agmon’s inequality and young’s inequality twice. secondly, using hölder inequality and young’s inequality twice, it turns out that |〈(uα ·∇)θα, ∆θα〉| ≤ c‖uα‖4ḣ1‖∇θα‖ 2 l2 + κ/2‖∆θα‖2l2 ≤ c(‖uα‖6ḣ1 + ‖∇θα‖ 6 l2 ) + κ/2‖∆θα‖2l2. thirdly, the bouancy force can be dealt with via cauchy-schwarz inequality: |〈θαe3,−∆uα〉| ≤ ‖∇θα‖2l2 + ‖∇uα‖ 2 l2 + α 2‖∆uα‖2l2. finally, summing up, one obtains (3.1) 1 2 d dt (‖∇θα‖2l2(t3) + ‖∇uα‖ 2 l2(t3) + α 2‖∆uα‖2l2(t3)) +κ/2‖∆θα‖2l2(t3) + ν/2(‖∆uα‖ 2 l2(t3) + α 2‖∇∆uα‖2l2(t3)) ≤ c(‖∇θα‖6l2 + ‖∇uα‖ 6 l2 + α6‖∆uα‖6l2 +‖∇θα‖2l2 + ‖∇uα‖ 2 l2 + α2‖∆uα‖2l2 ), where c is a constant that does not depend on the parameter α. let g(t) = ‖∇θα(t)‖2l2(t3) +‖∇uα(t)‖ 2 l2(t3) + α2‖∆uα(t)‖2l2(t3). it is clear that g 3 + g ≤ c(g + 1)3. let h(t) = g(t) + 1. the function g is a non-negative function, so h(0) 6= 0. the estimation (3.1) can be written as dh dt ≤ ch3. we integrate this ordinary differential inequality, to obtain for 0 ≤ t ≤ 1 4ch2(0) , h(t) ≤ ch(0). finally, it turns out that for all time t, such that 0 ≤ t ≤ t∗ = min(t, 1 4c(1 + g(0))2 ), we have (3.2) ‖∇θα‖2l2(t3) + ‖∇uα‖ 2 l2(t3) + α 2‖∆uα‖2l2(t3) ≤ c(1 + ‖∇θ0‖2 l2(t3) + ‖∇u 0‖2 l2(t3) + α 2‖∆u0‖2 l2(t3)). int. j. anal. appl. 19 (1) (2021) 120 integrating (3.1) over (0,t∗) and using (3.2), we obtain (3.3) ∫ t∗ 0 ( κ‖∆θα‖2l2) + ν(‖∆uα‖ 2 l2 + α 2‖∇∆uα‖2l2 ) dt ≤ c(1 + ‖∇θ0‖2 l2(t3) + ‖∇u 0‖2 l2(t3) + α 2‖∆u0‖2 l2(t3)). these are non singular bounds with respect to the parameter α. since α is intended to vanish, then there exists some fixed value α0, such that 0 < α ≤ α0. we take α = α0 in (3.2) and (3.3), to obtain a uniform bound with respect to α. namely, the functions θ and u are uniformly bounded in l2([0,t∗],ḣ2(t3)), as for v, it is uniformly bounded in l2([0,t∗],ḣ1(t3)). hence, banach-alaoglu theorem [11] allows to extract subsequences (uαk )k, (vαk )k and (θαk )k (that we relabel (uk), (vk) and (θk)) respectively of uα, vα and θα such that (θk,uk) ⇀ (θ,u) in l 2([0,t∗],ḣ2(t3)) and vk ⇀ u in l2([0,t∗],ḣ1(t3)), as k → +∞. at this step, we proved the two first results of statements 1 and 2 of theorem 1.2. to investigate the two second results of these statements, we establish uniform estimates, independent of α, for d dt θαk and d dt uαk . for a fixed positive time, since θαk is uniformly bounded with respect to α, in l2([0,t∗],ḣ2(t3)), the diffusion ∆θαk belongs to l 2([0,t∗],l2(t3)). using sobolev norm definition and product laws we infer that ∫ t∗ 0 ‖div θkuk‖2l2dτ ≤ ∫ t∗ 0 ‖θk‖2ḣ1‖uk‖ 2 ḣ2 dτ ≤‖θk‖2l∞ t∗(ḣ 1) ‖uk‖2l2 t∗(ḣ 2) . since uk and θk are, respectively, subsequences of uα and θα, then the energy estimates (3.2) and (3.3) apply also for uk and θk and one can control the advection in l 2([0,t∗],l2(t3)). the above temperature diffusion and convection estimates lead to ‖ d dt θk‖l2 t∗(l 2) ≤ k1, where k1 is a real positive constant. to handle the time derivative of the velocity field uk, we apply the helmholtz operator (i − α2∆)−1 on the equation satisfied by u = uk in the system (1.10). so, (3.4) d dt uk = ν∆uk − (i −α2∆)−1b̃(uk,vk) − (i −α2∆)−1∇pk + (i −α2∆)−1θke3. for a fixed positif time, since uk is uniformly bounded with respect to α in the space l 2([0,t∗],ḣ2(t3)), then ∆uk belongs to l 2([0,t∗],l2(t3)). for the remaining terms, we recall that operator (i − α2∆)−1 is bounded from h−2(t3) into l2(t3). moreover, a direct frequencies computation implies that its norm is uniformly bounded and satisfies (3.5) |||(i −α2∆)−1||| ≤ 1. int. j. anal. appl. 19 (1) (2021) 121 since θα ∈ l2t∗(ḣ 2), one infers that ‖(i−α2∆)−1θe3‖l2 t∗(ḣ 4) ≤ k ′ 2, where k ′ 2 is a real positif constant. to estimate the nonlinear term, one has∫ t∗ 0 ‖(i −α2∆)−1b̃(uk,vk)‖2l2 ≤ ∫ t∗ 0 ‖b̃(uk,vk)‖2h−2 ≤ c‖uk‖ 2 l∞ t∗(ḣ 1) ‖vk‖2l2 t∗(ḣ 1) ≤ c‖uk‖2l∞ t∗(ḣ 1) (∫ t∗ 0 ‖∆uk(τ)‖2l2 + α 2‖∇∆uk(τ)‖2l2dτ ) . inequalities (3.2) and (3.3) provide uniform bounds to the non-linearity above. using precedent uniform bounds, thanks to the divergence free-condition, one infers that ‖(i−α2∆)−1∇p‖l2 t∗(ḣ 4) ≤ k ′′ 2 . so, equation (3.4) implies that ‖ d dt uk‖l2 t∗(l 2) ≤ k2. using aubin-lions lemma, we extract two subsequences relabeled uk and θk, that converge strongly in l2([0,t∗],ḣ1) and in l2([0,t∗],l2), respectively. we have ‖vk −uk‖2l2([0,t∗],l2) = α 4 ∫ t 0 ( ∑ k∈z3 |∆̂uk|2) = α4‖uk‖2l2([0,t∗],ḣ2). as uk belongs to l 2([0,t∗],ḣ2), vk converges strongly to u in l 2([0,t∗],l2), we have already proved the statements 1 and 2 of theorem 1.2. now, we turn to the third statement of theorem 1.2. for the first result, since (uk,θk) converges strongly to (u,θ) in (l2([0,t∗],ḣ1))2, then by cauchy-schwarz inequality it converges weakly for almost every t ∈ [0,t∗]. in particular, this holds for the supremum. that is (uk(t),θk(t)) converges to (u(t),θ(t)) weakly in ḣ1(t3) and uniformly over [0,t∗]. to prove the second result, using the precedent bounds of time derivatives, banach-alaoglu theorem in hilbert spaces implies that ( d dt θk, d dt uk) ⇀ ( d dt θ, d dt u) weakly in l2([0,t∗],l2(t3)), ask → +∞, and d dt vk ⇀ d dt u weakly in l2([0,t∗],ḣ−2(t3)), ask → +∞. let λ ∈ ḣ2 be a vector divergence free and ξ ∈ l2 a scaler test functions. taking the inner product and integrating over [0, t], for t ∈ [0,t∗], we obtain 〈θk(t), ξ〉−〈θk(0), ξ〉− ∫ t 0 〈θk, ∆ξ〉dτ + ∫ t 0 〈b(uk,θk), ξ〉dτ = 0, 〈vk(t), λ〉−〈vk(0), λ〉− ∫ t 0 〈vk, ∆λ〉dτ + ∫ t 0 〈b̃(uk,vk), λ〉dτ − ∫ t 0 〈θke3, λ〉dτ = 0. to handle the nonlinear terms, we use a standard compactness argument (thanks to the uniform bounds obtained with respect to αk above) so that b̃(uk,vk) → b(u,u) and b(uk,θk) → b(u,θ). hence, taking the limit, for every t ∈ [0,t∗]\e, to obtain 〈θ(t), ξ〉−〈θ(0), ξ〉− ∫ t 0 〈θ, ∆ξ〉dτ + ∫ t 0 〈b(u,θ), ξ〉dτ = 0, int. j. anal. appl. 19 (1) (2021) 122 〈u(t), λ〉−〈u(0), λ〉− ∫ t 0 〈u, ∆λ〉dτ + ∫ t 0 〈b(u,u), λ〉dτ − ∫ t 0 〈θe3, λ〉dτ = 0. especially, every strong solution fulfills the energy estimates (1.11), so one deduces the energy estimates (1.13) by taking the lower limit as αk → 0+. acknowledgement: the authors gratefully acknowledge the approval and the support of this research study by the grant number sci-2018-3-9-f-8032 from the deanship of scientific research at northern border university, arar, k. s. a. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. cao, e. m. lunasin, e. s. titi, global well-posedness of the three-dimensional viscous and inviscid simplified bardina turbulence models. commun. math. sci. 4 (2006), 823–848. [2] a. chaabani, r. nasfi, r. selmi, m. zaabi, well-posedness and convergence results for strong solution to a 3d-regularized boussinesq system, math. meth. appl. sci. (2016). https://doi.org/10.1002/mma.3950. [3] cheskidov, a., holm, d., olson, e., titi, e.: on a leray-α model of turbulence. proc. r. soc. lond. ser. a 461 (2005), 629–649. [4] b. cushman-roisin, j. m. beckers, introduction to geophysical fluid dynamics, series in international geophysics. academic press, 2nd edition, 2011. [5] c. foias, d. d. holm, e. s. titi, the navier–stokes-alpha model of fluid turbulence. physica d, 152 (2001), 505-519. [6] d. d. holm, c. jeffery, s. kurien, d. livescu, m. a. taylor, b. a. wingate, the lans-α model for computing turbulence. los alamos sci. 29 (2005), 152-171. [7] d. d. holm, j. e. marsden, t. s. ratiu, the euler–poincaré equations and semidirect products with applications to continuum theories. adv. math. 137(1) (1998), 1-81. [8] a. a. ilyin, e. m. lunasin, e. s. titi, a modified leray-alpha subgrid-scale model of turbulence. nonlinearity, 19 (2006), 879–897. [9] j. leray, sur le mouvement d’un liquide visquex emplissant l’espace. acta math. 63 (1934), 193–248. [10] j. s. linshiz, e. s. titi, analytical study of certain magnetohydrodynamic-α models, j. math. phys. 48 (2007), 065504. [11] j. robinson, j. rodrigo, w. sadowski, the three-dimensional navier-stokes equations: classical theory (cambridge studies in advanced mathematics), cambridge university press, cambridge, 2016. [12] r. selmi, global well-posedness and convergence results for the 3d-regularized boussinesq system. canad. j. math. 64(6) (2012), 1415-1435. 1. introduction and statement of main results 2. well-posedness result 3. convergence result references international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 142-150 http://www.etamaths.com characterization of (δ,γ)-dini-lipschitz functions in terms of their jacobi-dunkl transforms a. belkhadir1,∗, a. abouelaz2 and r. daher3 abstract. in this paper, we are going to define a generalized dini-lipschitz class and give a characterization for functions belonging to by means of an asymptotic estimating growth of the norm of their jacobi-dunkl transforms. 1. introduction and preliminaries younis theorem 5.2 [3] characterizes the set of functions in l2(r) satisfying the dini-lipschitz condition by means of an asymptotic estimating growth of the norm of their fourier transforms. i.e. theorem 1.1. [3] let δ ∈ (0, 1) , γ > 0 and f ∈ l2(r) . then the following conditions are equivalents: (1) ‖f(t + h) −f(t)‖l2(r) = o ( hδ(log 1 h )−γ ) , as h → 0 ; (2) ∫ |λ|≥r |f̂(λ)|2dλ = o ( r−2δ(log r)−2γ ) , as r → +∞ . where f̂ is the fourier transform of f . in the following, let α,β and ρ denote 3 reals such that α ≥ β ≥ −1 2 , α 6= −1 2 and ρ = α + β + 1, aα,β(x) = 2 ρ(sinh |x|)2α+1(cosh |x|)2β+1. in [1] we have established a characterization of functions f in l2(r,aα,β(x)dx) satisfying a certain lipschitz condition, namely the equivalence between the two following conditions: (1) ||∆hf|| = ||τhf + τ−hf − 2f||l2(r,aα,β(x)dx) = o(h δ) , as h → 0; (2) ∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = o(r−2δ) , as r →∞ . where fα,β(f) stands for the jacobi-dunkl transform of f, and τh is the related generalized translation operator . this result has been generalized in [2] by using the higher powers: λrα,β and ∆khf = ∆h(∆ k−1 h f), r ∈ n, k ∈ n ∗ . in this way, we are going in section 3 to define a generalized dini-lipschitz class dlip[2, (δ,γ),k,r], δ ∈ (0, 1), γ > 0, and give a characterization for functions belonging to by means of an asymptotic estimating 2010 mathematics subject classification. 47b36, 46e35. key words and phrases. younis theorem; generalized jacobi-dunkl translation; jacobi-dunkl transform; dini-lipschitz class; sobolev spaces. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 142 characterization of (δ,γ)-dini-lipschitz functions 143 growth of the norm of their jacobi-dunkl transforms, i.e. we show the equivalence of the two following conditions: (1) f ∈ dlip[2, (δ,γ),k,r] ; (2) ∫ ∞ s λ2r |fα,β(f)(λ)| 2 dσα,β(λ) = o ( s−2δ(log s)−2γ ) , as s → +∞ . in the following section we recapitulate some results related to the harmonic analysis associated with the jacobi-dunkl operator λα,β (see [4, 5, 6, 7, 9]). 2. notations and preliminaries notations: • dσα,β(λ) = |λ| 8π √ λ2 −ρ2|cα,β( √ λ2 −ρ2)| ir\]−ρ,ρ[(λ)dλ where, cα,β(µ) = 2ρ−iµγ(α + 1)γ(iµ) γ( 1 2 (ρ + iµ))γ( 1 2 (α−β + 1 + iµ)) , µ ∈ c\ (in) . and iω is the characteristic function of ω . • lp(aα,β) (resp. lp(σα,β) ,p ∈]0, +∞[ , the space of measurable functions g on r such that ||g||lp(aα,β) = (∫ r |g(t)|paα,β(t)dt )1/p < +∞ . (resp. ||g||lp(σα,β) = (∫ r |g(λ)|pdσα,β(λ) )1/p < +∞) . • d(r) the space of c∞-functions on r with compact support. • s(r) the usual schwartz space of c∞-functions on r rapidly decreasing together with their derivatives, equipped with the topology of semi-norms lm,n , (m,n) ∈ n2 , where lm,n(f) = sup x∈r,0≤k≤n [ (1 + x2)m ∣∣∣∣ dkdxk f(x) ∣∣∣∣ ] < +∞. • s1(r) = {(cosh t)−2ρf; f ∈s(r)} . the topology of this space is given by the semi-norms l1m,n , (m,n) ∈ n2 , where l1m,n(f) = sup x∈r,0≤k≤n [ (cosh t)−2ρ(1 + x2)m ∣∣∣∣ dkdxk f(x) ∣∣∣∣ ] < +∞. • ( s1(r) )′ the topological dual of s1(r) . now, we introduce the jacobi-dunkl transform and its basic properties: the jacobi-dunkl function with parameters (α,β) , α ≥ β ≥ −1 2 ,α 6= −1 2 , is defined by : (1) ∀x ∈ r, ψ(α,β)λ (x) = { ϕ (α,β) µ (x) − i λ d dx ϕ (α,β) µ (x) , if λ ∈ c\{0}; 1 , if λ = 0. 144 belkhadir, abouelaz and daher with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕ (α,β) µ is the jacobi function given by: (2) ϕ(α,β)µ (x) = f ( ρ + iµ 2 , ρ− iµ 2 ; α + 1,−(sinh x)2 ) , where f is the gaussian hypergeometric function given by f(a,b,c,z) = ∞∑ m=0 (a)m(b)m (c)mm! zm , |z| < 1, a,b,z ∈ c and c /∈−n; (a)0 = 1, (a)m = a(a + 1)...(a + m− 1) . (see [4, 10, 11]). ψ (α,β) λ is the unique c ∞-solution on r of the differentiel-difference equation (3) { λα,βu = iλu , λ ∈ c; u(0) = 1. where λα,β is the jacobi-dunkl operator given by: λα,βu(x) = du dx (x) + a′α,β(x) aα,β(x) × u(x) −u(−x) 2 ; i.e. λα,βu(x) = du dx (x) + [(2α + 1) coth x + (2β + 1) tanh x] × u(x) −u(−x) 2 . the function ψ (α,β) λ can be written in the form below (see [5]), (4) ψ (α,β) λ (x) = ϕ (α,β) µ (x) + i λ 4(α + 1) sinh(2x)ϕ(α+1,β+1)µ (x) , ∀x ∈ r , where λ2 = µ2 + ρ2 , ρ = α + β + 1. the jacobi-dunkl transform of a function f ∈ l1(aα,β) is defined by : (5) fα,β(f)(λ) = ∫ r f(x)ψ (α,β) −λ (x)aα,β(x)dx, ∀λ ∈ r ; the inverse jacobi-dunkl transform of a function h ∈ l1(σα,β) is: (6) f−1α,β(h)(t) = ∫ r h(λ)ψ (α,β) λ (t)dσα,β(λ) . fα,β is a topological isomorphism from s1(r) onto s(r) , and extends uniquely to a unitary isomorphism from l2(aα,β) onto l 2(σα,β) . the plancherel formula is given by (7) ‖f‖l2(aα,β) = ‖fα,β(f)‖l2(σα,β) . for f ∈s1(r) we have the following inversion formula (8) f(x) = ∫ r fα,β(f)(λ)ψ (α,β) λ (x)dσα,β(λ) , ∀x ∈ r , and the relation (9) fα,β(λα,βf)(λ) = iλfα,β(f)(λ) . characterization of (δ,γ)-dini-lipschitz functions 145 let f ∈ l2(aα,β) . for all x ∈ r the operator of jacobi-dunkl translation τx is defined by: (10) τxf(y) = ∫ r f(z)dνα,βx,y (z) , ∀ y ∈ r . where να,βx,y , x,y ∈ r are the signed measures given by (11) dνα,βx,y (z) =   kα,β(x,y,z)aα,β(z)dz , if x,y ∈ r∗; δx , if y = 0; δy , if x = 0. here, δx is the dirac measure at x. and kα,β(x,y,z) = mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αiix,y × ∫π 0 ρθ(x,y,z) ×(gθ(x,y,z)) α−β−1 + sin 2β θdθ. ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x| + |y||, |x| + |y|] , ρθ(x,y,z) = 1 −σθx,y,z + σ θ z,x,y + σ θ z,y,x σθx,y,z =   cosh(x) + cosh(y) − cosh(z) cos(θ) sinh(x) sinh(y) , if xy 6= 0; 0 , if xy = 0. for all x,y,z ∈ r , θ ∈ [0,π]. gθ(x,y,z) = 1 − cosh2 x− cosh2 y − cosh2 z + 2 cosh x cosh y cosh z cos θ . t+ = { t , if t > 0; 0 , if t ≤ 0. and mα,β =   2−2ργ(α + 1) √ πγ(α−β)γ(β + 1 2 ) , if α > β; 0 , if α = β. we have (12) fα,β(τhf)(λ) = ψ α,β λ (h).fα,β(f)(λ) ; h,λ ∈ r . let g ∈ l2(σα,β) . then the distribution tgσα,β defined by (13) 〈tgσα,β,ϕ〉 = ∫ r g(λ)ϕ(λ)dσα,β(λ) , ϕ ∈d(r) , belongs to s′(r) . let f ∈ l2(aα,β) . then the distribution tfaα,β defined by (14) 〈tfaα,β,ϕ〉 = ∫ r f(x)ϕ(x)aα,β(x)dx, ϕ ∈s1(r) , belongs to ( s1(r) )′ . via the correspondance f 7→ tfaα,β , we identify l 2(aα,β) as a subspace of( s1(r) )′ . 146 belkhadir, abouelaz and daher the jacobi-dunkl transform of a distribution t ∈ ( s1(r) )′ is defined by: (15) 〈fα,β(t),ϕ〉 = 〈t,f−1α,β(ϕ̌)〉 , ϕ ∈s(r) , where ϕ̌ is given by ϕ̌(x) = ϕ(−x) . it is clear that fα,β(t) ∈s′(r) . the jacobi-dunkl transform of a distribution defined by f ∈ l2(aα,β) is given by the distribution tfα,β(f)σα,β ; i.e. (16) fα,β(tfaα,β ) = tfα,β(f)σα,β . we identify the tempered distribution given by fα,β(f) and the function fα,β(f) . let t ∈ ( s1(r) )′ and consider the distribution λα,βt defined by (17) 〈λα,β(t),ϕ〉 = −〈t, λα,β(ϕ)〉 , for all ϕ ∈s1(r) . (note that s1(r) is unvariant under λα,β) . by using (9) it is easy to see that (18) fα,β(λα,β(t)) = iλfα,β(t) . for f ∈ l2(aα,β) , we define the finite differences of first and higher order as follows: ∆1hf = ∆hf = τhf + τ−hf − 2f = (τh + τ−h − 2e)f; ∆khf = ∆h(∆ k−1 h )f = (τh + τ−h − 2e) kf , k = 2, 3, ...; where e is the unit operator in l2(aα,β) . lemma 2.1. the following inequalities are valids for jacobi functions ϕα,βµ (h) (1) |ϕ(α,β)µ (h)| ≤ 1 ; (2) |1 −ϕ(α,β)µ (h)| ≤ h2λ2; where λ2 = µ2 + ρ2 . proof. (see [12], lemmas 3.1-3.2) � for α ≥ −1 2 , we introduce the bessel normalized function of the first kind defined by jα(z) = γ(α + 1) ∞∑ n=0 (−1)n(z 2 )2n n!γ(n + α + 1) , z ∈ c. we see that lim z→0 jα(z) − 1 z2 6= 0 , by consequence, there exists c1 > 0 and η > 0 satisfying (19) |z| ≤ η ⇒ |jα(z) − 1| ≥ c1|z|2 . lemma 2.2. let α ≥ β ≥ −1 2 , α 6= −1 2 . then for |υ| ≤ ρ , there exists a positive constant c2 such that |1 −ϕ(α,β)µ+iυ(t)| ≥ c2|1 − jα(µt)| . proof. (see [8], lemma 9) � characterization of (δ,γ)-dini-lipschitz functions 147 3. main results we denote by w 2,k α,β , k ∈ n ∗ , the sobolev space constructed by the operator λα,β ; i.e. (20) w 2,k α,β = { f ∈ l2(aα,β); λ j α,βf ∈ l 2(aα,β), j = 0, 1, 2, ...,k } ; where, λ0α,βf = f, λ 1 α,βf = λα,βf , λ r α,βf = λα,β(λ r−1 α,β f), r = 2, 3, ... lemma 3.1. let f ∈ w 2,kα,β , k ∈ n ∗ . then∥∥∆khλrα,βf∥∥2l2(aα,β) = 22k ∫ r λ2r|1 −ϕµ(h)|2k|fα,β(f)(λ)|2dσα,β(λ) , where r = 0, 1, ...,k. proof. using the eveness of ϕµ and formula (4) we get fα,β(τhf + τ−hf − 2f)(λ = (ψ (α,β) λ (h) + ψ (α,β) λ (−h) − 2).fα,β(f)(λ) = 2(ϕ(α,β)µ (h) − 1).fα,β(f)(λ). and (21) fα,β(∆khf)(λ) = 2 k(ϕ(α,β)µ (h) − 1) k.fα,β(f)(λ). furthermore, we obtain by the formula (18) (22) fα,β(λrα,βf)(λ) = (iλ) rfα,β(f)(λ) . using the formulas (21) and (22) we get fα,β(∆khλ r α,βf)(λ) = 2 k(iλ)r.(ϕ(α,β)µ (h) − 1) k.fα,β(f)(λ). by the plancherel formula (7), we have the result. � definition 3.2. let δ ∈ (0, 1),γ > 0 and k ∈ n∗ . a function f ∈ w 2,kα,β is said to be in the (δ,γ)-dini-lipschitz class, denoted by dlip[2, (δ,γ),k,r] , if∥∥∆khλrα,βf∥∥l2(aα,β) = o ( hδ(log 1 h )−γ ) , as h −→ 0, where r = 0, 1, ...,k. theorem 3.3. let f ∈ w 2,kα,β , k ∈ n ∗ . then the following are equivalents: (1) f ∈ dlip[2, (δ,γ),k,r] ; (2) ∫ ∞ s λ2r |fα,β(f)(λ)| 2 dσα,β(λ) = o ( s−2δ(log s)−2γ ) , as s → +∞ . proof. (1) ⇒ (2): assume that f ∈ dlip[2, (δ,γ),k,r] ; then∥∥∆khλrα,βf∥∥l2(aα,β) = o ( hδ(log 1 h )−γ ) as h −→ 0. by lemma 3.1, we have ∫ r λ2r|1 −ϕµ(h)|2k|fα,β(f)(λ)|2dσα,β(λ) = o ( h2δ(log 1 h )−2γ ) 148 belkhadir, abouelaz and daher if |λ| ∈ [ η 2h , η h ] then |µh| ≤ η (recall that λ2 = µ2 + ρ2). we get by (19): |jα(µh)− 1| ≥ c1µ2h2. from |λ| ≥ η 2h we have, µ2h2 ≥ η2 4 − ρ2h2; then we can find a positive constant c3 = c3(η,α,β) such that µ 2h2 ≥ c3 (take h < 1) ; thus, |jα(µh) − 1| ≥ c1c3. this inequality and lemma 2.2 implys that: |1 −ϕ (α,β) µ (h)| ≥ c1c2c3 = c. hence, 1 ≤ 1 c2 |1 −ϕ(α,β)µ (h)|2. then, ∫ η 2h ≤|λ|≤η h λ2r|fα,β(f)(λ)|2dσα,β(λ) ≤ 1 c2k ∫ η 2h ≤|λ|≤η h λ2r|1 −ϕ(α,β)µ (h)| 2k ×|fα,β(f)(λ)|2dσα,β(λ) ≤ 1 c2k ∫ r λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ) = o ( h2δ(log 1 h )−2γ ) . then we have, ∫ s≤|λ|≤2s λ2r|fα,β(f)(λ)|2dσα,β(λ) = o ( s−2δ(log s)−2γ ) , as s → +∞. or equivalently ∫ s≤|λ|≤2s λ2r|fα,β(f)(λ)|2dσα,β(λ) ≤ k1o ( s−2δ(log s)−2γ ) , as s → +∞, where k1 is some positive constant . it follows that, ∫ |λ|≥s λ2r|fα,β(f)(λ)|2dσα,β(λ) = ∞∑ i=0 ∫ 2is≤|λ|≤2i+1s λ2r|fα,β(f)(λ)|2dσα,β(λ) ≤ k1 ∞∑ i=0 (2is)−2δ(log 2is)−2γ ≤ k1 ( ∞∑ i=0 (2i)−2δ )( s−2δ(log s)−2γ ) ≤ k ( s−2δ(log s)−2γ ) . where k = k1 1 − 2−2δ . this proves that: ∫ |λ|≥s λ2r|fα,β(f)(λ)|2dσα,β(λ) = o ( s−2δ(log s)−2γ ) , as s → +∞. (2) ⇒ (1) : suppose now that∫ |λ|≥s λ2r|fα,β(f)(λ)|2dσα,β(λ) = o ( s−2δ(log s)−2γ ) , as s → +∞. we have to show that: characterization of (δ,γ)-dini-lipschitz functions 149 ∫ r λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ) = o ( h2δ(log 1 h )−2γ ) , as h → 0. write: ∫ r λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ) = i1 + i2, where: i1 = ∫ |λ|≤1 h λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ) ; i2 = ∫ |λ|> 1 h λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ). estimate i1 and i2 . from (1) of lemma 2.1 we can write, i2 ≤ 4k ∫ |λ|> 1 h λ2r|fα,β(f)(λ)|2dσα,β(λ) , (s = 1 h ) = o ( h2δ(log 1 h )−2γ ) . using the inequalities (1) and (2) of lemma 2.1 we get i1 = ∫ |λ|≤1 h λ2r|1 −ϕ(α,β)µ (h)| 2k|fα,β(f)(λ)|2dσα,β(λ) ≤ 22k−1 ∫ |λ|≤1 h λ2r|1 −ϕ(α,β)µ (h)|.|fα,β(f)(λ)| 2dσα,β(λ) ≤ 22k−1h2 ∫ |λ|≤1 h λ2r.λ2|fα,β(f)(λ)|2dσα,β(λ). consider the function ψ(s) = ∫ ∞ s λ2r|fα,β(f)(λ)|2dσα,β(λ). since ψ(s) = o ( s−2δ(log s)−2γ ) , an integration by parts gives: 22k−1h2 ∫ 1 h 0 λ2r.λ2|fα,β(f)(λ)|2dσα,β(λ) = 22k−1h2 ∫ 1 h 0 ( −s2ψ′(s) ) ds = 22k−1h2 ( − 1 h2 ψ( 1 h ) + 2 ∫ 1 h 0 sψ(s)ds ) ≤ 22k−1h2 ∫ 1 h 0 sψ(s)ds ≤ c1.h2 ∫ 1 h 0 s1−2δ(log s)−2γds ≤ c2.h2δ(log 1 h )−2γ. 150 belkhadir, abouelaz and daher hence, i1 = o ( h2δ(log 1 h )−2γ ) . finally we get i1 + i2 = o ( h2δ(log 1 h )−2γ ) + o ( h2δ(log 1 h )−2γ ) = o ( h2δ(log 1 h )−2γ ) which completes the proof of the theorem. � corollary 3.4. let f ∈ w 2,kα,β such that f ∈ dlip[2, (δ,γ),k,r] . then∫ ∞ s |fα,β(f)(λ)| 2 dσα,β(λ) = o ( s−2(δ+r)(log s)−2γ ) , as s → +∞. references [1] a. belkhadir, a. abouelaz, and r. daher, an analog of titchmarsh’s theorem for the jacobidunkl transform in the space l2 α,β (r), international journal of analysis and applications, 8(1) (2015), 15-21. [2] a. belkhadir and a. abouelaz, generalization of titchmarsh’s theorem for the jacobi-dunkl transform, gen. math. notes, 28(2) (2015), 9-20. [3] m. s. younis, fourier transforms of dini-lipschitz functions, international journal of mathematics and mathematical sciences, 9(2) (1986), 301-312. [4] h.b. mohamed and h. mejjaoli, distributional jacobi-dunkl transform and application, afr. diaspora j. math, (2004), 24-46. [5] h.b. mohamed, the jacobi-dunkl transform on r and the convolution product on new spaces of distributions, ramanujan j., 21(2010), 145-175. [6] n.b. salem and a.o.a. salem, convolution structure associated with the jacobi-dunkl operator on r , ramanujan j., 12(3) (2006), 359-378. [7] n.b. salem and a.o.a. salem, sobolev types spaces associated with the jacobi-dunkl operator, fractional calculus and applied analysis, 7(1) (2004), 37-60. [8] w.o. bray and m.a. pinsky, growth properties of fourier transforms via moduli of continuity, journal of functional analysis, 255(2008), 2256-2285. [9] f. chouchane, m. mili and k. trimèche, positivity of the intertwining opertor and harmonic analysis associated with the jacobi-dunkl operator on r, j. anal. appl., 1(4) (2003), 387-412. [10] t.h. koornwinder, jacobi functions and analysis on noncompact semi-simple lie groups, in: r.a. askey, t.h. koornwinder and w. schempp (eds.), special functions: group theoritical aspects and applications, d. reidel, dordrecht, (1984). [11] t.h. koornwinder, a new proof of a paley-wiener type theorems for the jacobi transform, ark. math., 13(1975), 145-159. [12] s.s. platonov, approximation of functions in l2-metric on noncompact rank 1 symetric spaces, algebra analiz., 11(1) (1999), 244-270. [13] e.c. titchmarsh, introduction to the theory of fourier integrals, claredon, oxford, (1948), komkniga, moscow, (2005). department of mathematics and informatics, faculty of science ain chock, university of hassan ii, casablanca, morocco ∗corresponding author international journal of analysis and applications volume 18, number 4 (2020), 550-558 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-550 on q-mocanu type functions associated with q-ruscheweyh derivative operator khalida inayat noor and shujaat ali shah∗ comsats university islamabad, pakistan ∗corresponding author: shaglike@yahoo.com abstract. in this paper, we introduce certain subclasses of analytic functions defined by using the qdifference operator. mainly we give several inclusion results for defined classes. also, certain applications due to q-ruscheweyh derivative operator will be discussed. 1. introduction let a denotes the class of analytic functions f(z) in the open unit disk e = {z : |z| < 1} such that f(z) = z + ∞∑ n=2 anz n. (1.1) subordination of two functions f and g is denoted by f ≺ g and defined as f(z) = g(w(z)), where w(z) is schwartz function in e (see [10]). let s, s∗ and c denote the subclasses of a of univalent functions, starlike functions and convex functions respectively. mocanu [11] introduced the class m (α) of α−convex functions f ∈ s satisfies; ( (1 −α) zf′(z) f(z) + α (zf′(z)) ′ f′(z) ) ≺ 1 + z 1 −z , where α ∈ [0, 1], f(z) z f′(z) 6= 0 and z ∈ e. we see that m0 = s∗ and m1 = c. this class is vastly studied by several authors, see [2, 14]. we recall here some basic definitions and concept details of q-calculus that are used in this paper. received february 4th, 2020; accepted march 31st, 2020; published may 11th, 2020. 2010 mathematics subject classification. 30c45, 30c55. key words and phrases. mocanu functions; q-difference operator; q-ruscheweyh derivative operator. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 550 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-550 int. j. anal. appl. 18 (4) (2020) 551 the q-difference operator, which was introduced by jackson [7], defined by dqf(z) = f(z) −f(qz) (1 −q)z ; q 6= 1, z 6= 0, for q ∈ (0, 1). it is clear that limq→1− dqf(z) = f′(z), where f′(z) is the ordinary derivative of the function. for more properties of dq; see [3–5, 9, 18]. it can easily be seen that, for n ∈ n = {1, 2, 3, ..} and z ∈ e, dq { ∞∑ n=1 anz n } = ∞∑ n=1 [n]q z n−1, where [n]q = 1 −qn 1 −q = 1 + q + q2 + ... . we have the following rules of dq. dq (af (z) ± bg (z)) = adqf (z) ± bdqg (z) . dq (f (z) g (z)) = f (qz) dq (g (z)) + g(z)dq (f (z)) . dq ( f(z) g(z) ) = dq (f(z)) g(z) −f(z)dq (g(z)) g(qz)g(z) , g(qz)g(z) 6= 0. dq (log f(z)) = dq (f(z)) f(z) . some properties related with function theory involving q-theory were first introduced by ismail et al. [6]. moreover, several authors studied in this matter such as [1, 12, 13, 15]. now, by making use of the principle of subordination together with q-difference operator, we have the following classes: let a function p ∈ a with p(0) = 1 is in the class p̃q(β) if and only if p(z) ≺ pq,β(z), where pq,β(z) = ( 1 + z 1 −qz )β , (0 < β ≤ 1) . (1.2) it is very easy to see that pq,β(z) is convex univalent in e for 0 < β ≤ 1. aslo, pq,β(z) is symmetric with respect to the real axis, that is, 0 < <(pq,β(z)) < ( 1 1 −q )β . int. j. anal. appl. 18 (4) (2020) 552 definition 1.1. let function f ∈ a and 0 ≤ α ≤ 1, q ∈ (0, 1). then f ∈ mβq (α) if and only if jq (α,f) ∈ p̃q(β), where jq (α,f) = (1 −α) zdqf f + α dq (zdqf) dqf . moreover, let us denote mβq (0) = s ∗ q (β) , m β q (1) = cq (β) . a function f ∈ a is said to be in s∗q (β) and cq (β) if and only if zdqf(z) f(z) ≺ pq,β(z) and dq (zdqf(z)) dqf(z) ≺ pq,β(z), respectively. special cases: (i) if q → 1−, then the class mβq (α) reduces to the class mβ (α). (ii) if q → 1− and β = 1, then the class mβq (α) reduces to the class m (α) introduced by mocanu [11]. (iii) if q → 1−, α = 0 and β = 1, then the class mβq (α) reduces to the well known class s∗ of starlike functions. (iv) if q → 1−, α = 1 and β = 1, then the class mβq (α) reduces to the well known class c of convex functions. the authors in [8], introduced an operator rλq : a → a defined as: rλqf(z) = zλ+1,q(z) ∗f(z) (1.3) = z + ∞∑ n=2 [n + λ− 1]q! [λ]q! [n− 1]q! anz n, (1.4) where f ∈ a, zλ+1,q(z) = z + ∑∞ n=2 [n+λ−1] q ! [λ] q ![n−1] q ! zn and * denotes convolution. this series (1.4) is absolutely convergent in e. for q → 1−, we have the operator rλ, called ruscheweyh derivative operator introduced in [16]. in this case rλf(z) = lim q→1− rλqf(z) = z + ∞∑ n=2 (n + λ− 1)! λ! (n− 1)! anz n = z (1 −z)λ+1 ∗f(z). we note that r0qf(z) = f(z) and r 1 qf(z) = zdqf(z). also rnq f(z) = zdnq ( zn−1f(z) ) [n]q! ; n ∈ n = {1, 2, 3, ...} . int. j. anal. appl. 18 (4) (2020) 553 the following identity can be easily obtained from (1.4) zdq ( rλqf(z) ) = ( 1 + [λ]q qλ ) rλ+1q f(z) − [λ]q qλ rλqf(z). (1.5) now, we define definition 1.2. let f ∈ a and n ∈ n, 0 ≤ α ≤ 1, q ∈ (0, 1) and β ∈ (0, 1]. then f ∈ mβq (n,α) if and only if r n q f(z) ∈ m β q (α) . moreover, let us denote mβq (n, 0) = s ∗ q (n,β) and m β q (n, 1) = cq (n,β) . note that f ∈ cq (n,β) ⇔ zdqf ∈ s∗q (n,β) . (1.6) 2. main results we need the following basic result to prove our main results: lemma 2.1. [17] let β and γ be complex numbers with β 6= 0 and let h(z) be analytic in e with h(0) = 1 and re{βh(z) + γ} > 0. if p(z) = 1 + p1z + p2z2 + ... is analytic in e, then p(z) + zdqp(z) βp(z) + γ ≺ h(z) implies that p(z) ≺ h(z). theorem 2.1. let 0 ≤ α ≤ 1, β ∈ (0, 1] and q ∈ (0, 1). then mβq (α) ⊂ s ∗ q (β) . proof. let f ∈ mβq (α) and let zdqf(z) f(z) = p(z). (2.1) we note that p(z) is analytic in e with p(0) = 1. the q-logarithmic differentiation of (2.1) yields dq (zdq (f(z))) dqf(z) − dq (f(z)) f(z) = dqp(z) p(z) . equivalently dq (zdq (f(z))) dqf(z) = p(z) + zdqp(z) p(z) . since f ∈ mβq (α), so we get jq (α,f) = p(z) + α zdqp(z) p(z) ≺ pq,β(z). (2.2) int. j. anal. appl. 18 (4) (2020) 554 since re { 1 α pq,β(z) } > 0 in e, so by (2.2) together with lemma 2.1, we obtain p(z) ≺ pq,β(z). consequently f ∈ s∗q (β). � corollary 2.1. for q → 1−, we have mβ(α) ⊂ s∗ (β). furthermore, for β = 1, m(α) ⊂ s∗. corollary 2.2. for q → 1−, α = 1 and β = 1, we have well known fundamental result c ⊂ s∗. theorem 2.2. let α > 1, β ∈ (0, 1] and q ∈ (0, 1). then mβq (α) ⊂ cq (β) . proof. let f ∈ mβq (α). then, by definition 1.1, (1 −α) zdqf(z) f(z) + α dq (zdqf(z)) dqf(z) = p1(z) ∈ p̃q(β). now, α dq (zdqf(z)) dqf(z) = (1 −α) zdqf(z) f(z) + α dq (zdqf(z)) dqf(z) + (α− 1) zdqf(z) f(z) = (α− 1) zdqf(z) f(z) + p1(z). this implies dq (zdqf) dqf = ( 1 α − 1 ) zdqf f + 1 α p1(z) = ( 1 α − 1 ) p2(z) + 1 α p1(z). since p1,p2 ∈ p̃q(β) and is p̃q(β) convex set, so dq(zdqf) dqf ∈ p̃q(β). hence, proof is complete. � theorem 2.3. for 0 ≤ α1 < α2 < 1 mβq (α2) ⊂ m β q (α1) . proof. for α1 = 0, this is obvious from theorem 2.1. let f ∈ mβq (α2). then, by definition 1.1, (1 −α2) zdqf(z) f(z) + α2 dq (zdqf(z)) dqf(z) = q1(z) ∈ p̃q(β). (2.3) now, we can easily write jq (α1,f(z)) = α1 α2 q1(z) + ( 1 − α1 α2 ) q2(z), (2.4) where we have used (2.3) and zdqf(z) f(z) = q2(z) ∈ p̃q(β). since p̃q(β) is convex set, so (2.4) follows our required result. � int. j. anal. appl. 18 (4) (2020) 555 remark 2.1. if α2 = 1 and let f ∈ mβq (1) = cq(β). then, from theorem 2.3, we can write f ∈ mβq (α1) ,for 0 ≤ α1 < 1, now, by making use of theorem 2.1, we obtain f ∈ s∗q (β). thus we have, cq(β) ⊂ s∗q (β). we develop some applications in terms of q-linear operator, which we call q-ruscheweyh derivative operator, given by (1.3). theorem 2.4. let 0 ≤ α ≤ 1, β ∈ (0, 1], n ∈ n0 and q ∈ (0, 1). then mβq (n + 1,α) ⊂ s ∗ q (n + 1,β) . proof. one can easily prove this result by using similar arguments as used in theorem 2.1 and letting zdqfn+1,q(z) fn+1,q(z) = p(z) ( for fn+1,q(z) = r n+1 q f(z) ) , where p(z) is analytic in e with p(0) = 1. � theorem 2.5. let 0 ≤ α ≤ 1, β ∈ (0, 1], n ∈ n0 and q ∈ (0, 1). then s∗q (n + 1,β) ⊂ s ∗ q (n,β) . proof. let f ∈ s∗q (n + 1,β) and let fn+1(z) = rn+1q f(z). then zdqfn+1,q(z) fn+1,q(z) ≺ pq,β(z), where pq,β(z) is given by (1.2). now, let zdqfn,q(z) fn,q(z) = h(z), (2.5) where h(z) is analytic in e with h(0) = 1. using identity (1.5) and (2.5), we get zdq (fn,q(z)) fn,q(z) = (1 + nq) fn+1,q(z) fn,q(z) −nq, equivalently (1 + nq) fn+1,q(z) fn,q(z) = h(z) + nq, ( for nq = [n]q qn ) . the q-logarithmic differentiation yields, zdq (fn+1,q(z)) fn+1,q(z) = p(z) + zdqh(z) h(z) + nq . (2.6) since f ∈ s∗q (n + 1,β), so (2.6) implies p(z) + zdqh(z) h(z) + nq ≺ pq,β(z). (2.7) int. j. anal. appl. 18 (4) (2020) 556 since re{pq,β(z) + nq} > 0 in e, we use lemma 2.1 along with (2.7), to get h(z) ≺ pq,β(z). consequently, f ∈ s∗q (n,β). � theorem 2.6. let 0 ≤ α ≤ 1, β ∈ (0, 1], n ∈ n0 and q ∈ (0, 1). then cq (n + 1,β) ⊂ cq (n,β) . proof. let f ∈ cq (n + 1,β) ⇔ zf′ ∈ s∗q (n + 1,β) (by (1.6)) ⇒ zf′ ∈ s∗q (n,β) (by theorem2.5) ⇔ f ∈ cq (n,β) . (by (1.6)) � remark 2.2. from theorem 2.4 and theorem 2.5, we can extend the inclusions as following mβq (n + 1,α) ⊂ s ∗ q (n + 1,β) ⊂ s ∗ q (n,β) ⊂ ... ⊂ s ∗ q (β) . cq (n + 1,β) ⊂ cq (n,β) ⊂ ... ⊂ cq (β) . theorem 2.7. let f ∈ a. then f ∈ mβq (n + 1,α), α 6= 0, if and only if there exists g ∈ s∗q (n + 1,β) such that f(z) = [ 1 α ] q [∫ t 0 t 1 α −1 ( g(t) t ) 1 α dqt ]α . (2.8) proof. let f ∈ mβq (n + 1,α). then jq (α,f) = (1 −α) zdqf(z) f(z) + α dq (zdqf(z)) dqf(z) ∈ p̃q(β). (2.9) on some simple calculations of (2.8), we get zdqf(z) (f(z)) 1 α −1 = (g(z)) 1 α . (2.10) the q-logarithmic differentiation of (2.10), gives (1 −α) zdqf(z) f(z) + α dq (zdqf(z)) dqf(z) = zdqg(z) g(z) . (2.11) from (2.9) and (2.11), we conclude our required result. � int. j. anal. appl. 18 (4) (2020) 557 theorem 2.8. let f ∈ a and define, for f ∈ mβq (n,α), fc,q(z) = [c + 1]q zc ∫ z 0 tb−1f(t)dqt. (2.12) then fc,q ∈ s∗q (n,β). proof. let f ∈ mβq (n,α). if we set, for fnc,q(z) = rnq (fc,q(z)) zdq ( fnc,q(z) ) fnc,q(z) = q(z), (2.13) where q(z) is analytic in e with q(0) = 1. from (2.12), we can write dq (z cfc,q(z)) [c + 1]q = zc−1f(z). using product rule of the q-difference operator, we get zdqfc,q(z) = ( 1 + [c]q qc ) f(z) − [c]q qc fc,q(z). (2.14) from (2.13), (2.14) and (1.3), we have q(z) = ( 1 + [c]q qc ) z (fn,q(z)) fnc,q(z) − [c]q qc , where fnc,q(z) = r n q (fc,q(z)) and fn,q(z) = r n q (f(z)) on q-logarithmic differentiation, we get zdq (fn,q(z)) fn,q(z) = q(z) + zdqq(z) q(z) + [n]q , ( for nq = [c]q qc ) . (2.15) since f ∈ mβq (n,α) ⊂ s∗q (n,β), so (2.15) implies q(z) + zdqq(z) q(z) + [c]q ≺ pq,β(z). now, by applying lemma 2.1, we conclude q(z) ≺ pq,β(z). consequently, zdq(fnc,q(z)) fnc,q(z) ≺ pq,β(z). hence fc,q ∈ s∗q (n,β). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (4) (2020) 558 references [1] o. altintas, n. mustafa, coefficient bounds and distortion theorems for the certain analytic functions, turk. j. math. 43 (2019), 985–997. [2] j. dziok, classes of functions associated with bounded mocanu variation, j. inequal. appl. (2013), art. id 349. [3] h. exton, q-hypergeometric functions and applications, ellis horwood limited, uk, 1983. [4] g. gasper, m. rahman, basic hypergeometric series, cambridge university press, cambridge, uk, 1990. [5] h.a. ghany, q-derivative of basic hypergeomtric series with respect to parameters, int. j. math. anal. 3 (2009), 1617–1632. [6] m.e.h. ismail, e. merkes, d. styer, a generalization of starlike functions, complex var., theory appl. 14 (1990), 77–84. [7] f.h. jackson, on q-functions and a certain difference operator, trans. r. soc. edin. 46 (1908), 253–281. [8] s. kanas, r. raducanu, some classes of analytic functions related to conic domains, math. slovaca. 64 (2014), 1183–1196. [9] v. koc, p. cheung, quantum calculus, springer, 2001. [10] s.s. miller, p. t. mocanu, differential subordinations theory and applications, marcel dekker, new york, basel, 2000. [11] p.t. mocanu, une propriete de convexite generlise dans la theorie de la representation conforme, math. (cluj). 11 (1969), 127–133. [12] m. naeem, s. hussain, t. mahmood, s. khan, m. darus, a new subclass of analytic functions defined by using salagean q-differential operator, mathematics. 7 (2019), 458. [13] k.i. noor, on generalized q-close-to-convexity, appl. math. inf. sci. 11 (2017), 1383–1388. [14] k.i. noor, s. hussain, on certain analytic functions associated with ruscheweyh derivatives and bounded mocanu variation, j. math. anal. appl. 340 (2008), 1145–1152. [15] k.i. noor, s. riaz, generalized q-starlike functions, stud. sci. math. hungerica. 54 (2017), 509–522. [16] s. ruscheweyh, new criteria for univalent functions. proc. amer. math. soc. 49 (1975), 109–115. [17] h. shamsan, s. latha, on genralized bounded mocanu variation related to q-derivative and conic regions, ann. pure appl. math. 17 (2018), 67–83. [18] h.e.o. ucar, coefficient inequality for q-starlike functions, appl. math. comput. 276 (2016), 122–126. 1. introduction 2. main results references int. j. anal. appl. (2022), 20:7 differential equations models and their applications in metallurgy ridha selmi1,3,4,∗, muflih alhazmi1, abir sboui2,4, hanen louati1, amel touati1, hechmi hattab5,6 1department of mathematics, college of sciences, northern border university, p.o. box 1321, arar, 73222, ksa 2department of mathematics, faculty of sciences and art (turaif), northern border university, ksa 3department of mathematics, faculty of sciences of gabes, 6072, gabès, tunisia 4laboratory of partial differential equations and applications (lr03es04), faculty of sciences of tunis, university of tunis el manar, 1068 tunis, tunisia 5fs sfax, univercité de sfax, route de la soukra km 4 sfax 3038, tunisia 6isim gabès, université de gabès, cité erriadh, zrig, 6072, gabès, tunisia ∗corresponding author: ridha.selmi@nbu.edu.sa, ridhaselmiridhaselmi@gmail.com abstract. this paper focus on the heat recovery from the metallurgical and mining wastes. we propose and study a new and more realistic mathematical model for heat recovery from molten slag. our model is based on time delay differential equations. in the theoretical part, we prove that a unique solution exists to the mathematical problem. in the numerical part, we establish an algorithm based on explicit fourth order runge-kutta method with delay; the new feature is that the delay must be larger enough than the step of integration. compared to the classical model (without time delay), the numerical test proves that our model is more efficient and industrially more profitable. 1. introduction in metallurgy, heat recovery from molten slag is nowadays one of the relevant manners to valorize the huge amount of thermal energy provided by such mining waste. this very high temperature liquid received: oct. 7, 2021. 2010 mathematics subject classification. 34k05; 65l03; 65l20; 97m50. key words and phrases. heat transfer; heat recovery from molten slag; delayed convection; mathematical model; time delay-differential equations; runge-kutta method; numerical solution. https://doi.org/10.28924/2291-8639-20-2022-7 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-7 2 int. j. anal. appl. (2022), 20:7 slag is refrigerated then solidified by quenching procedure via fluids (gas or liquid), in general. after that, the heat recuperated is used for many purposes that range from industrially to domestically use. such huge thermal energy can be used for producing electrical energy, or heating buildings, for example. a review paper about this subject is [13]. the report [11] deals with this matter in australia. the paper [10] presents a global study of the problem, with focus on south africa mining and metallurgical industry. in the literature, from mathematical point of view, models dealing with heat recovery from molten slag are formulated using ordinary differential equations that are taking in account only instantaneous properties of the molten slag. thus, they naturally consider the ordinary time derivative. such classical models were given by the ordinary differential equation of the form: dt dt (t) = f (t,t (t)), where t is the temperature of the molten slag, t is the time variable and f is a given function that describes the process of energy transfer. from physical point of view, as stated in [8], models above are based on the fact that convection and radiation are mainly the two manners of heat dissipation, when liquid slag is quenched. the heat transfer between the quenching fluid and the very hot slag is governed by the energy equality cv ρ dt dt (t) = ah(t (t) −tf ) + aεσ0(t4(t) −t4f ), (1.1) where t is the unknown temperature of the molten slag, it depends only on the time t. the parameters v and a are respectively the volume and the surface of the molten slag; tf is the fixed temperature of the incoming quenching fluid; c and ε state for the heat of the alloy and its integrated radiant remittance, ρ is the melt density, h is the heat transfer coefficient of the interface and finally σ0 refers to the blackbody radiation coefficient. however, we think that models above are neither realistic nor so much efficient. in fact, from physical and industrial point of view, there are two intersecting issues (a property and a phenomenon) that should be considered: • memory effects caused by the viscoelasticity of the molten slag: as for such property, based on physical experiences and chemical analysis the detailed review paper [9] and references therein putted in evidence the viscoelastic response of molten slag, where according to the authors "at high temperatures (which is our case), the time dependent constitutive relations are needed not only for the stress-stress relationship, but also for the heat flux vector"(page 26). also, they reviewed and discussed "the various existing implicit constitutive models for non-linear viscoelastic materials that can be used to model the rheological characteristics of slag" (page 28). int. j. anal. appl. (2022), 20:7 3 • time delay phenomena occurring in the interconnections of different parts of a system: as for such phenomena, authors in [6] used a space-averaging technique and the method of characteristics to propose a time-delay system modelling the flow temperatures of a heat exchanger; they believe that "time delay phenomena naturally occur in the interconnections of different parts of a system, as propagation of matter is not instantaneous. in particular, it occurs in tubular heat exchangers, which are very common devices in industry". tubular heat exchangers are exactly the case of quenching systems used for molten slag. in [12], a dual-phase-lagging model of the micro-scale heat conduction is re-derived analytically from the boltzmann transport equation. then, based on such model, a delay-advanced partial differential equations governing the micro-scale heat conduction are established, as a more realistic model. to take in account facts signaled above, we will opt for a model governed by a time-delay equations. mathematically speaking, this is based on delay differential equations (ddes), where the derivative of the unknown molten slag temperature t at a time t is expressed by the values of the temperature at previous times that is dt dt (t) = f (t,t (t −τ)), (1.2) where t (t−τ) is the delay term that takes in account the memory effects of the system, and τ is the delay time. in the first chapter of the book [1], it was discussed, in details, how delayed differential equations are a reliable mathematical tool to modelize systems with memory. based on (1.2), to make the model of heat recovery from molten slag more realistic, we propose the following new governing equation: cv ρ dt dt (t) = [ah(t (t −τ1) −tf ) + aεσ(t4(t −τ2) −t4f )], (1.3) where we performed a correction via different time delays τ1 and τ2, respectively in the convective term and the radiation term. however, to simplify this model, we recall the following: by one hand, as radiation is an electromagnetic wave travelling, heat transfer by radiation occurs at the speed of light. by the other hand, as convection is a phenomenon related to the physical medium, heat transfer by convection takes place at the speed of convective medium. thus, we are lead to consider that τ2 << τ1, because heat transfer by radiation is faster than heat transfer by convection. mathematically speaking, we can neglect τ2 compared to τ1. thus, model (1.3) reads cv ρ dt dt (t) = [ah(t (t −τ1) −tf ) + aεσ(t4(t) −t4f )]. (1.4) above, we mean by "faster" the way to transfer energy from one medium to another in least amount of time. 4 int. j. anal. appl. (2022), 20:7 from numerical point of view, it is already known, in the literature, that numerical investigation of delayed differential equations is so delicate and some times problematic. among others, authors in the recent publication [4] discussed several problems that arise while implementing numerical methods to solve delay-differential equations, as overlapping, difficulties with error estimation, discontinuities of solution derivatives and their detection, with focus on the runge–kutta methods. let us explicitly say that, one of serious problems in such investigation is the fact that integration in time variable will be perturbed by some shift effects due to delays. in [2, 7], this anomaly was overcome by interpolation. here, we will take the step of time integration so small compared to the delay. so, the time shift will not be in the interval of integration. this allows us to establish an explicit delayed fourth order runge-kutta algorithm (dfor-k) valid for our proposed model. to the best of our knowledge, our idea is new and original. the remainder of this paper will be as follows. in section two, we will prove that our model is mathematically well posed; that is a unique solution exists to the corresponding cauchy problem. section three deals with the numerical investigation, we will establish an explicit forth order rungekutta algorithmi, after which numerical test will be presented to validate the model. this paper will be achieved by a closely conclusion. 2. functional setting and theoretical results in this section, we will prove that a solution to our model exists and it is unique. following mathematical material in [5], we recall the following. • the naturel functional setting used to study retarded differential equations is the set of continuous functions mapping the interval [−r, 0] into rn, r > 0, denoted by c = c([−r, 0],rn), and endowed with the usual norm defined by |φ| = sup −r≤θ≤0 |φ(θ)|, for any given function φ ∈ c. • if we consider σ ∈ r, a ≥ 0 and x ∈ c([σ − r,σ + a],rn), then for any t ∈ [σ − r,σ + a], we let xt ∈ c be defined by xt(θ) = x(t + θ), −r ≤ θ ≤ 0. • if d is a subset of r×c, f : d → rn is a given function and "." represents the derivative, then we say that the relation (dde) ẋ(t) = f (t,xt) is a delayed (retarded or advanced) differential equation (also said functional differential equation) on d and will denote this equation by (dde). equation (dde) is a very general type int. j. anal. appl. (2022), 20:7 5 of equation and includes ordinary differential equations ẋ(t) = f (t,x(t)) whenever r = 0 (and thus θ = 0); differential equations in the form ẋ(t) = f (t,x(t),x(t −τ)), 0 ≤ τ ≤ r, for example which is our case, in this paper. • a delayed initial value problem is simply a delayed differential equation supplemented by an initial value φ at σ: (div p ) { ẋ(t) = f (t,xt) xσ = φ, where σ ∈r, φ ∈ c are given. • a function x is said to be a solution of a delayed differential equation on [σ−r,σ + a) if there are σ ∈ r and a > 0, such that x ∈ c([σ − r,σ + a),rn), (t,xt) ∈ d and x(t) satisfies equation (dde) for t ∈ [σ,σ + a). it follows that x(σ,φ,f ) is a solution to the delayed initial value problem (div p ), that is a solution of (dde) with initial value φ at σ or simply a solution through (σ,φ), if there is an a > 0, such that x(σ,φ,f ) is a solution of equation (dde) on [σ − r,σ + a) and xσ(σ,φ,f ) = φ. for interested readers, [5] and [1] are complete references about this topic. using notation above, we introduce the existence result (theorem 2.3 page 42), in [5]: theorem 2.1. suppose ω is an open subset in r×c and f : ω → rn is continuous, and f (t,φ) is lipschitzian in φ in each compact set in ω. if (σ,φ) ∈ ω, then there is a unique solution of equation (dde) passing through (σ,φ). we use theorem above to deal with existence and uniqueness of solution to our model. (1) existence of solution: the function t : t → t (t) should belong to c(j), so it should be continuous in time t on some time interval j ⊂ r. this interval will be defined in practice by the duration of the quenching process. thus, the operator f : j ×c(j) → r, such that f (t,t (t)) = at (t−τ) + bt4(t) + c is continuous in (t,t ) as a polynomial in the variable t, here we note that there is no explicit dependence of the operator f on the time t and such dependence is implicitly through the temperature t. above, the coefficients a, b and c are given by the parameters of the model after standard computation. so, at this step a solution to our model (1.4) exists. (2) uniqueness of solution: also, as a polynomial, the operator f is lipschitzian in t in each compact set in j × c(j). thus, if we supplement our model by the initial value tσ = tin, theorem above asserts that our model has a unique solution t (σ,tin, f ) passing through (σ,φ = tin). here, we note that tin is the temperature of the molten slag before the 6 int. j. anal. appl. (2022), 20:7 beginning of the quenching process and we recall that tσ is defined by tσ(θ) = t (σ + θ), −r ≤ θ ≤ 0, r > 0, so that tσ ∈ c as it is a constant function of time equal to tin everywhere. the value of σ can be taken equal to zero, without loss of generality. 3. numerical investigation 3.1. the delayed fourth order runge-kutta algorithm. the classical fourth runge-kutta method consists in integrating a differential equation of the form dt dt (t) = f (t,t (t)), using the formula: tn+1 = tn + h, where h is the time step and tn+1 = tn + 1 6 (k1 + 2k2 + 2k3 + k4) , for n = 1, 2, . . . ,m− 1, (3.1) where k1 = hf (tn,tn) k2 = hf (tn + h 2 ,tn + k1 2 ) k3 = hf (tn + h 2 ,tn + k2 2 ) k4 = hf (tn + h,tn + k3) (3.2) and m is the number of integration’s steps. let us consider the following delay differential equation: dt dt (t) = f (t,t (t −τ),t (t)) = 1 cv ρ [ah(t (t −τ) −tf ) + aεσ(t4(t) −t4f )] = αt (t −τ) + βt4(t) + γ, (3.3) where α = (6 ∗ h)/(ρ ∗ c ∗ d), β = (6 ∗ ε ∗ σ)/(ρ ∗ c ∗ d) and γ = (α ∗ tf ) + (β ∗ (t4f )), and we approximate the molten slag grains by spheres of radius d/2. we claim that the diameter d and thus the surface a can be adjustable, by industrial methods to increase or decrease the transfer interface between molten slag and the fluid. in reference ( [8]), the coefficients α, β and γ are calculated based the physical parameters of model (1.1), to obtain α = −5.208 10−6 β = −4.626 10−3 γ = 38381.568. for sake of comparison of our delayed model to the non delayed model introduced in ( [8]), we are using the same coefficients in our numerical study. also, as in ( [8]), we take the initial temperature of the slag to be tin = t (0) = 1773k. thus, in our following numerical investigation, we will consider int. j. anal. appl. (2022), 20:7 7 the delayed initial value problem (s) { dt dt (t) = −5.208 10−6t (t −τ) − 4.626 10−3t4(t) + 38381.568 tin = 1773k. if the delay τ is larger enough than the step of integration, the above fourth runge-kutta method can be extended to a fourth runge kutta algorithm with delay by expressive constants k̃1, k̃2, k̃3 and k̃4 with the delayed form of the f (t,t (t −τ),t (t)). that is k̃1 = hf (t (tn)) = h [ αt (tn −τ) + βt4(tn) + γ ] ; k̃2 = hf (t (tn) + 1 2 k̃1) = h [ α ( t (tn) + 1 2 k̃1 ) (tn−τ) + β(t (tn) + 1 2 k̃1) 4 + γ ] = h [ α ( (t (tn −τ) + 12(k̃1)(tn−τ) ) + β(t (tn) + 1 2 k̃1) 4 + γ ] , (3.4) where (k̃1)(tn−τ) = hf (t (tn −τ)) = h [ αt (tn − 2τ) + βt4(tn −τ) + γ ] ; (3.5) k̃3 = hf (t (tn) + 1 2 k̃2) = h [ α ( t (tn) + 1 2 k̃2 ) (tn−τ) + β ( t (tn) + 1 2 k̃2 )4 + γ ] = h [ α ( (t (tn −τ) + 12(k̃2)(tn−τ) ) + β ( t (tn) + 1 2 k̃2 )4 + γ ] , (3.6) where (k̃2)(tn−τ) = hf ( t (tn −τ) + 12(k̃1)(tn−τ) ) = h [ α ( t (tn − 2τ) + 1 2 (k̃1)(tn−2τ) ) + β ( t (tn −τ) + 1 2 (k̃1)(tn−τ) )4 + γ ] (k̃1)(tn−2τ) = hf (t (tn − 2τ)) = h [ αt (tn − 3τ) + βt4(tn − 2τ) + γ ] (3.7) and k̃4 = hf (t (tn) + k̃3) = h [ α ( t (tn) + k̃3 ) (tn−τ) + β ( t (tn) + k̃3 )4 + γ ] = h [ α ( (t (tn −τ) + (k̃3)(tn−τ) ] + β ( t (tn) + k̃3 )4 + γ ] , (3.8) 8 int. j. anal. appl. (2022), 20:7 where (k̃3)(tn−τ) = hf ( t (tn −τ) + 12(k̃2)(tn−τ) ) = h [ α ( t (tn − 2τ) + 1 2 (k̃2)(tn−2τ) ) + β ( t (tn −τ) + 1 2 (k̃2)(tn−τ) )4 + γ ] (k̃2)(tn−2τ) = hf ( t (tn − 2τ) + 12(k̃1)(tn−2τ) ) = h [ α ( t (tn − 3τ) + 1 2 (k̃1)(tn−3τ) ) + β ( t (tn − 2τ) + 1 2 (k̃1)(tn−2τ) )4 + γ ] (k̃1)(tn−3τ) = hf (t (tn − 3τ)) = h [ αt (tn − 4τ) + βt4(tn − 3τ) + γ ] . (3.9) then, tn+1 = tn + 1 6 ( k̃1 + 2k̃2 + 2k̃3 + k̃4 ) , for n = 1, 2, . . . ,m− 1, (3.10) where tn+1 = tn + h, h is the time step and m is the number of integration’s steps. we will name the algorithm (3.4)-(3.10) the delayed fourth order runge-kutta algorithm that we will denote by (dfor-k). from the above computation, it is clear that since equation (3.3) is with one constant delay τ, the (dfor-k) depends on the delay of time equals to 4τ via (k̃1)(tn−3τ), in the last line of (3.9). this proves the following theorem. theorem 3.1. let τ be the delay in equation (3.3) and h be the integration step in the (dfor-k) (3.4)-(3.10). if τ ≥ h, then the (dfor-k) (3.4)-(3.10) is a valid algorithm. moreover, it depends on delays τ, 2τ, 3τ and 4τ. 3.2. numerical results. we present two numerical results. the first concerns the comparison between the model with delay, that it is physically more realistic, and the one without delay. the second result consists in establishing the variation of the heat transfer as a function of the exchange surface between the molten slag and the quenching fluid. in figure 1 and table 1, we fix the thermal exchange surface of the molten slag with the fluid used for the recovery of the heat to be a = 200 units of the sphere of diameter d modeling the grains of the molten slag and we present the decrease in the temperature of the molten slag as a function of time, with an integration step timestep = 0.5 s, a delay τ = 4∗ timestep, where the number of steps of integration is m = 40. so, the simulation is in a time equal to 20 s. clearly, we can see that, when taking into account the delay, the decrease in temperature is more important, which implies a saving of the time while transferring heat from the molten slag. also, we see that the behaviour of the temperature as a function of time presents some asymptote that seems to be the same for the two models (without and with delay). this leads to think about the optimal time to spend in the heat recovery process, in industry. int. j. anal. appl. (2022), 20:7 9 table 1. table for the delay τ = 4 pastemps = 4 steps of integration. pastemps=0.5 s, a =200 sphere unities, number of integration’s steps m = 40. iteration times (s) delayed temperature not delayed temperature 0 0.0000000000000000 1773.0000000000000 1773.0000000000000 1 0.50000000000000000 1496.3627665694594 1487.6130483245572 2 1.0000000000000000 1228.2200943503881 1260.5919326885185 3 1.5000000000000000 1011.7376907722477 1078.3396255965374 4 2.0000000000000000 843.90934123125589 931.22909171791252 5 2.5000000000000000 715.08690804465118 812.09212122248778 6 3.0000000000000000 616.41586854571949 715.41277990356048 7 3.5000000000000000 540.85202877542747 636.85701598717617 8 4.0000000000000000 482.97008411070431 572.97465271184217 9 4.5000000000000000 438.62089061602052 520.99665252349666 10 5.0000000000000000 404.63356920917545 478.68924422998879 11 5.5000000000000000 378.58332275097030 444.24459090912433 12 6.0000000000000000 358.61453884968427 416.19650804817417 13 6.5000000000000000 343.30637504591556 393.35426511243531 14 7.0000000000000000 331.57045335400164 374.74995043192519 15 7.5000000000000000 322.57283358988946 359.59628591699783 16 8.0000000000000000 315.67440783299958 347.25264075992033 17 8.5000000000000000 310.38531550573572 337.19755665628946 18 9.0000000000000000 306.33005421868648 329.00648616513820 19 9.5000000000000000 303.22076278047979 322.33372710777883 20 10.000000000000000 300.83675480171695 316.89774647706361 21 10.500000000000000 299.00883657838762 312.46924901901673 22 11.000000000000000 297.60728872134200 308.86147204479352 23 11.500000000000000 296.53265440274487 305.92228806723062 24 12.000000000000000 295.70867813729103 303.52777670666762 25 12.500000000000000 295.07689267128490 301.57699142269786 26 13.000000000000000 294.59246909462632 299.98769831306481 27 13.500000000000000 294.22103526397939 298.69290600508037 28 14.000000000000000 293.93623652689627 297.63803951277356 29 14.500000000000000 293.71786551843422 296.77863839112746 30 15.000000000000000 293.55042824457450 296.07848181653878 10 int. j. anal. appl. (2022), 20:7 31 15.500000000000000 293.42204466040243 295.50806134363791 32 16.000000000000000 293.32360570627554 295.04333682354223 33 16.500000000000000 293.24812697431037 294.66472295529155 34 17.000000000000000 293.19025313632625 294.35626369636543 35 17.500000000000000 293.14587796555929 294.10495969758477 36 18.000000000000000 293.11185298858754 293.90022039122590 37 18.500000000000000 293.08576409385415 293.73341762396927 38 19.000000000000000 293.06576024566658 293.59752201192765 39 19.500000000000000 293.05042215001640 293.48680668521655 40 20.000000000000000 293.03866155349442 293.39660593217508 our second numerical test focus on the effect of the variation of the contact surface between the quenching fluid and the molten slag surface on the heat transfer rate from this molten slag. it is clear that increasing this exchange surface has as an effect to increase the speed of the heat recovery from the molten slag, in both cases of modeling with delay (figure 2) and without delay (figure 3). also, it is clear that while passing from surface a = 100 to surface a = 200, temperature decreases faster than while passing from a = 200 to a = 300 an so on, for the same abscise of time, see for example for t = 5s of (figure 2). this leads to think about the optimally profitable surface that should be considered in industrial protocols. moreover, the behavior of the temperature as a function of time presents some horizontal asymptote that should define the stopping time of the heat recovery procedure, in practice. combining the above two statements, as a result, even more heat exchange speed is gained one takes into account the delay and increases the surface of the contact. remark 3.1. the fact when passing from surface a = 100 to surface a = 200, temperature decreases faster than when passing from a = 200 to a = 300 an so on, may be du to the hypothesis we made in the beginning that is τ2 << τ1. in a forthcoming paper, we will deal with the model (1.3) and discuss the situation based on this model that we think to be a compete one. int. j. anal. appl. (2022), 20:7 11 figure 1. decrease in the temperature, τ = 4∗ timestep, timestep= 0.5s and m = 40. figure 2. decrease in the temperature for different surface of heat transfer: a=100, 200, 300, & 400; τ = 4∗ timestep, timestep= 0.5s and m = 40. figure 3. decrease in the temperature for different surface of heat transfer: a=100, 200, 300, & 400, without delay, timestep= 0.5s, m = 40. 12 int. j. anal. appl. (2022), 20:7 4. conclusion heat recovery and re-use of the huge amount of thermal energy offered by the very high temperature of the metallurgical and mining waste, in molten slag allow to built a circular economic model and prevent environmental damages. such industrial procedure should be an efficient process both in the sense of duration of heat recovery and the optimal logistic protocol. in this paper, we proposed a time delayed model that is more realistic then the instantaneous one. our model is mathematically well posed and numerically solvable using a variant of runge-kutta method which is popular. first, we proved that our model is more efficient compared to the classical one, in the sense that the heat recovery is faster; this saves time in industry. also, we proved that there should exist both a stopping time to the quenching process and an optimal surface of the contact between the molten slag and the quenching fluid, so that the heat recovery procedure is maximally profitable, in industry. acknowledgement: the authors extend their appreciation to the deputyship for research & innovation, ministry of education in saudi arabia for funding this research work through the project number ”3400_2020_if ”. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] o. arino, m.l. hbid, e. ait dads, delay differential equations and applications, springer, netherlands, (2006). [2] n. a. ayunni sabri, m. bin mamat, solving delay differential equations (ddes) using nakashima’s 2 stages 4th order pseudo-runge-kutta method, world appl. sci. j. 21 (2013), 181-186. [3] l. cheng, m. xu, l. wang, from boltzmann transport equation to single-phase-lagging heat conduction, int. j. heat mass transfer. 51 (2008), 6018–6023. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.004. [4] a. s. eremin, a. r. humphries, a. a. lobaskin, some issues with the numerical treatment of delay differential equations, aip conf. proc. 2293 (2020), 100003. https://doi.org/10.1063/5.0027149. [5] j. k. hale, s. m. verduyn lunel, introduction to functional differential equations, springer verlag, berlin, 1993. [6] s. hamze, e. witrant, d. bresch-pietri, c. fauvel, estimating heat-transport and time-delays in a heat exchanger, in: 2018 ieee conference on control technology and applications (ccta), ieee, copenhagen, 2018: pp. 1514–1519. https://doi.org/10.1109/ccta.2018.8511359. [7] f. ismail, r. a. al-khasawneh, a. s. lwin, m. b. suleiman, numerical treatment of delay differential equations by runge–kutta method using hermite interpolation, matematika: malaysian j. ind. appl. math. 18 (2002), 79-90 [8] c. liu, h. wu, j. chang, research on a class of ordinary differential equations and application in metallurgy, in: r. zhu, y. zhang, b. liu, c. liu (eds.), information computing and applications, springer berlin heidelberg, berlin, heidelberg, 2010: pp. 391–397. https://doi.org/10.1007/978-3-642-16339-5_52. [9] m. massoudi, p. wang, a brief review of viscosity models for slag in coal gasification, doe/netl-2012/1533, national energy technology laboratory, pittsburgh, pa, 2011. [10] e. matinde, g.s. simate, s. ndlovu, mining and metallurgical wastes: a review of recycling and re-use practices, j. south. afr. inst. min. metall. 118 (2018), 825-844. https://doi.org/10.17159/2411-9717/2018/v118n8a5. https://doi.org/10.1016/j.ijheatmasstransfer.2008.04.004 https://doi.org/10.1063/5.0027149 https://doi.org/10.1109/ccta.2018.8511359 https://doi.org/10.1007/978-3-642-16339-5_52 https://doi.org/10.17159/2411-9717/2018/v118n8a5 int. j. anal. appl. (2022), 20:7 13 [11] d. xie, y. pan, r. flann, b. washington, s. sanetsis, j. donnelley et al., heat recovery from slag through dry granulation, in: 1st csrp annual conference. melbourne (australia), vol. csiro minerals, pp. 29–30, 2007. [12] m. xu, l. wang, dual-phase-lagging heat conduction based on boltzmann transport equation, int. j. heat mass transfer 48 (2005), 5616–5624. https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.040. [13] h. zhang, h. wang, x. zhu, y.-j. qiu, k. li, r. chen and q. liao, a review of waste heat recovery technologies towards molten slag in steel industry, appl. energy 112 (2013), 956-966. https://doi.org/10.1016/j.apenergy. 2013.02.019. https://doi.org/10.1016/j.ijheatmasstransfer.2005.05.040 https://doi.org/10.1016/j.apenergy.2013.02.019 https://doi.org/10.1016/j.apenergy.2013.02.019 1. introduction 2. functional setting and theoretical results 3. numerical investigation 3.1. the delayed fourth order runge-kutta algorithm 3.2. numerical results 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 68-80 http://www.etamaths.com growth analysis of functions analytic in the unit polydisc sanjib kumar datta1,∗, tanmay biswas2, soumen kanti deb3 abstract. in this paper we study some growth properties of composite functions analytic in the unit polydisc. some results related to the generalised n variables based p-th nevanlinna order (generalised n variables based p-th nevanlinna lower order) and the generalised n variables based p-th nevanlinna relative order (generalised n variables based p-th nevanlinna relative lower order) of an analytic function with respect to an entire function are established in this paper where n and p are any two positive integers. in fact in this paper we extend some results of [3] and [4]. 1. introduction, definitions and notations. a function f analytic in the unit disc u = {z : |z| < 1} is said to be of finite nevanlinna order [6] if there exists a number µ such that the nevanlinna characteristic function tf (r) = 1 2π ∫ 2π 0 log+ |f ( reiθ ) |dθ satisfies tf (r) < (1 −r) −µ for all r in 0 < r0 (µ) < r < 1. the greatest lower bound of all such numbers µ is called the nevanlinna order of f. thus the nevanlinna order ρf of f is given by ρf = lim sup r→1 log tf (r) − log (1 −r) . similarly, the nevanlinna lower order λf of f are given by λf = lim inf r→1 log tf (r) − log (1 −r) . l. bernal introduced the relative order between two entire functions of single variables to avoid comparing growth just with the exponential function exp z. in this connection, banerjee and dutta [2] gave the following definition in a unit disc: definition 1. [2] if f be analytic in u and g be entire , then the relative order of f with respect to g denoted by ρg (f) is defined by ρg (f) = inf { µ > 0 : tf (r) < tg [( 1 1 −r )µ] for all 0 < r0 (µ) < r < 1 } . 2010 mathematics subject classification. 30d20, 30d30, 32a15. key words and phrases. growth; analytic function of n complex variables; composite function; generalised n variables based p-th nevanlinna order; generalised n variables based p-th nevanlinna relative order; unit polydisc. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 68 growth analysis of functions analytic in the unit polydisc 69 similarly, one may define λg (f), the relative lower order of f with respect to g. with g (z) = exp z, the definition coincides with the definition of nevanlinna order of f. analogously, λg (f) = lim inf r→1 log t−1g tf (r) − log (1 −r) . extending the notion of single variables to several variables, let f(z1,z2, · · ·,zn) be a non-constant analytic function of n complex variables z1,z2, ··· zn−1 and zn in the unit polydisc u = {(z1,z2, · · ·,zn) : |zj| ≤ 1, j = 1, 2, · · ·,n; r1 > 0,r2 > 0, · · ·rn > 0} . now in the line of nevanlinna order [6], in this paper we introduce the generalised n variables based p-th nevanlinna order and the generalised n variables p-th nevanlinna lower order for functions of n complex variables analytic in a unit polydisc as follows : ρ [p] f (r1,r2, ...,rn) = lim sup r1,r2,...rn→1 log[p] tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] and λ [p] f (r1,r2, ...,rn) = lim inf r1,r2,...rn→1 log[p] tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] where log[k] x = log(log[k−1] x) for k = 1, 2, 3, ... and log[0] x = x. when n = p = 1, the above definition reduces to the definition of juneja and kapoor [6]. likewise, one may introduce the generalised n variables based p-th relative nevanlinna order ( generalised n variables based p-th relative nevanlinna lower order) for functions of n complex variables analytic in a unit polydisc in the following manner : definition 2. let tf (r1,r2, ...,rn) denote the nevanlinna’s characteristic function of f of n variables. the generalised n variables based p-th relative nevanlinna order ρ [p]f g (r1,r2, ...,rn) and generalised n variables based p-th relative nevanlinna lower order λ[p]fg (r1,r2, ...,rn) of an analytic function f in u with respect to another entire function g in n complex variables are defined in the following way : ρ[p]fg (r1,r2, ...,rn) = lim sup r1,r2→∞ log[p] t−1g tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] and λ[p]fg (r1,r2, ...,rn) = lim inf r1,r2→∞ log[p] t−1g tf (r1,r2, ...,rn) − log [(1 −r1) (1 −r2) ... (1 −rn)] where n and p are any two positive integers . if we consider p = n = 1 in definition 2, then it coincides with definition 1. in the paper we establish some results relating to the composition of two nonconstant analytic functions, of n complex variables in the unit polydisc u = {(z1,z2, · · ·,zn) : |zj| ≤ 1, j = 1, 2, · · ·,n; r1 > 0,r2 > 0, · · ·rn > 0} . also we prove a few theorems related to generalised n variables based p-th relative nevanlinna order ρ [p]f g (r1,r2, ...,rn) (generalised n variables based p-th relative 70 datta, biswas and deb nevanlinna lower order λ[p]fg (r1,r2, ...,rn) ) of an analytic function f with respect to an entire function g of n complex variables which are in fact some extensions of earlier results as proved in [3] and [4]. we do not explain the standard definitions and notations in the theory of entire functions of severable variables as those are available in [1], [5] and [7]. 2. theorems. in this section we present the main results of the paper. theorem 1. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u such that 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[q] g (r1,r2, ...,rn) ≤ ρ[q]g (r1,r2, ...,rn) < ∞. then λ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) where p and q are any two positive integers . proof. from the definition of generalised n variables based p-th nevanlinna order and generalised n variables based p-th nevanlinna lower order of analytic functions in the unit polydisc u, we have for arbitrary positive � and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] tf◦g (r1,r2, ...,rn)(1) ≥ ( λ[p] f◦g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] and log[q] tg (r1,r2, ...,rn)(2) ≤ ( ρ[q] g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . now from (1) and (2) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) − � ρ[q] g (r1,r2, ...,rn) + � . as � (> 0) is arbitrary, we obtain that (3) lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] tf◦g (r1,r2, ...,rn)(4) ≤ ( λ[p] f◦g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] growth analysis of functions analytic in the unit polydisc 71 and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[q] tg (r1,r2, ...,rn)(5) ≥ ( λ[q] g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . so combining (4) and (5) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) + � λ[q] g (r1,r2, ...,rn) − � . since � (> 0) is arbitrary, it follows that (6) lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . also for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, we get that log[q] tg (r1,r2, ...,rn)(7) ≤ ( λ[q] g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . now from (1) and (7) , we obtain for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) − � λ[q] g (r1,r2, ...,rn) + � . choosing � → 0, we get that (8) lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . also for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] tf◦g (r1,r2, ...,rn)(9) ≤ ( ρ[p] f◦g (r1,r2, ...,rn) + � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . so from (5) and (9) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) + � λ[q] g (r1,r2, ...,rn) − � . as � (> 0) is arbitrary, we obtain that (10) lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) . 72 datta, biswas and deb thus the theorem follows from (3), (6), (8) and (10). the following theorem can be proved in the line of theorem 1 and so its proof is omitted. theorem 2. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u with 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[l] f (r1,r2, ...,rn) ≤ ρ[l]f (r1,r2, ...,rn) < ∞ where p and l are any two positive integers. then λ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) ≤ λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) . theorem 3. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u such that 0 < ρ[p] f◦g (r1,r2, ...,rn) < ∞ and 0 < ρ[q] g (r1,r2, ...,rn) < ∞. then lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) where p and q are any two positive integers . proof. from the definition of generalised n variables based p-th nevanlinna order, we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[q] tg (r1,r2, ...,rn)(11) ≥ ( ρ[q] g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . now from (9) and (11) , it follows for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) + � ρ[q] g (r1,r2, ...,rn) − � . as � (> 0) is arbitrary, we obtain that (12) lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] tf◦g (r1,r2, ...,rn)(13) ≥ ( ρ[p] f◦g (r1,r2, ...,rn) − � ) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . growth analysis of functions analytic in the unit polydisc 73 so combining (2) and (13) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ ρ[p] f◦g (r1,r2, ...,rn) − � ρ[q] g (r1,r2, ...,rn) + � . since � (> 0) is arbitrary, it follows that (14) lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≥ ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) . thus the theorem follows from (12) and (14) . the following theorem can be carried out in the line of theorem 3 and therefore we omit its proof. theorem 4. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u with 0 < ρ[p] f◦g (r1,r2, ...,rn) < ∞ and 0 < ρ[l] f (r1,r2, ...,rn) < ∞ where p and l are any two positive integers. then lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) ≤ ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) . the following theorem is a natural consequence of theorem 1 and theorem 3: theorem 5. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u such that 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[q] g (r1,r2, ...,rn) ≤ ρ[q]g (r1,r2, ...,rn) < ∞. then lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) ≤ min { λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) } ≤ max { λ[p] f◦g (r1,r2, ...,rn) λ[q] g (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ[q] g (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) where p and q are any two positive integers . analogously one may state the following theorem without its proof. theorem 6. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u with 0 < λ[p] f◦g (r1,r2, ...,rn) ≤ ρ[p]f◦g (r1,r2, ...,rn) < ∞ and 0 < λ[l] f (r1,r2, ...,rn) ≤ ρ[l]f (r1,r2, ...,rn) < ∞ where p and l are any two 74 datta, biswas and deb positive integers .then lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) ≤ min { λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) } ≤ max { λ[p] f◦g (r1,r2, ...,rn) λ[l] f (r1,r2, ...,rn) , ρ[p] f◦g (r1,r2, ...,rn) ρ [l] f (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) . theorem 7. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u such that ρ[l] f (r1,r2, ...,rn) < ∞ and λ[p]f◦g (r1,r2, ...,rn) = ∞. then lim r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[l] tf (r1,r2, ...,rn) = ∞ where p and l are any two positive integers . proof. let us suppose that the conclusion of the theorem do not hold. then we can find a constant β > 0 such that for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, (15) log[p] tf◦g (r1,r2, ...,rn) ≤ β log[l] tf (r1,r2, ...,rn) . again from the definition of ρ[l] f (r1,r2, ...,rn) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[l] tf (r1,r2, ...,rn)(16) ≤ [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . thus from (15) and (16) , we have for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] tf◦g (r1,r2, ...,rn) ≤ β [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] i.e., log[p] tf◦g (r1,r2, ...,rn) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] ≤ β [ ρ[l] f (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] i.e., lim inf r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) [− log [(1 −r1) (1 −r2) ... (1 −rn)]] = λ[p] f◦g (r1,r2, ...,rn) < ∞. this is a contradiction. hence the theorem follows. growth analysis of functions analytic in the unit polydisc 75 remark 1. theorem 7 is also valid with “limit superior” instead of “limit” if λ[p] f◦g (r1,r2, ...,rn) = ∞ is replaced by ρ[p]f◦g (r1,r2, ...,rn) = ∞ and the other conditions remain the same. corollary 8. under the assumptions of theorem 7 and remark 1, lim r1,r2,...rn→1 tf◦g (r1,r2, ...,rn) tf (r1,r2, ...,rn) = ∞ and lim sup r1,r2,...rn→1 tf◦g (r1,r2, ...,rn) tf (r1,r2, ...,rn) = ∞ respectively hold if p = l. the proof is omitted. analogously one may also state the following theorem and corollaries without their proofs as those may be carried out in the line of remark 1, theorem 7 and corollary 8 respectively. theorem 9. let f and g be any two non-constant analytic functions of n complex variables in the unit polydisc u with ρ[q] g (r1,r2, ...,rn) < ∞ and ρ[p]f◦g (r1,r2, ...,rn) = ∞ where p and q are any two positive integers. then lim sup r1,r2,...rn→1 log[p] tf◦g (r1,r2, ...,rn) log[q] tg (r1,r2, ...,rn) = ∞ . corollary 10. theorem 9 is also valid with “limit” instead of “limit superior” if ρ[p] f◦g (r1,r2, ...,rn) = ∞ is replaced by λ[p]f◦g (r1,r2, ...,rn) = ∞ and the other conditions remain the same. corollary 11. under the assumptions of theorem 7 and corollary 10, lim sup r1,r2,...rn→1 tf◦g (r1,r2, ...,rn) tg (r1,r2, ...,rn) = ∞ and lim r1,r2,...rn→1 tf◦g (r1,r2, ...,rn) tg (r1,r2, ...,rn) = ∞ respectively hold if p = q. in the next three theorems we establish some comparative growth properties related to the generalised n variables based p-th relative nevanlinna order (generalised n variables based p-th relative nevanlinna lower order) of an analytic function with respect to an entire function in the unit poly disc u. theorem 12. let f,h be any two analytic functions of n complex variables in u and g be entire in n complex variables such that 0 < λ[p]fg (r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < λ[p]hg (r1,r2, ...,rn) ≤ ρ [p]h g (r1,r2, ...,rn) < ∞. then λ[p]fg (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) ≤ lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) where p is any positive integer. proof. from the definition of generalised n variables based p-th relative nevanlinna order and generalised n variables based p-th relative nevanlinna lower order of an analytic function with respect to an entire function in an unit polydisc u, we have 76 datta, biswas and deb for arbitrary positive � and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] t−1g tf (r1,r2, ...,rn)(17) ≥ [ λ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] and log[p] t−1g th(r1,r2, ...,rn)(18) ≤ [ ρ[p]hg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . now from (17) and (18) , it follows for all sufficiently large values of ( 1 1−r1 ) ,( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) − � ρ [p]h g (r1,r2, ...,rn) + � . as � (> 0) is arbitrary, we obtain that (19) lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . again we have for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that (20) log[p] t−1g tf (r1,r2, ...,rn) ≤ [ λ[p]fg (r1,r2, ...,rn) + � ] [− log (1 −r)] and for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] t−1g th(r1,r2, ...,rn)(21) ≥ [ λ[p]hg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . so combining (20) and (21) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , tending to infinity that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) + � λ[p]hg (r1,r2, ...,rn) − � . since � (> 0) is arbitrary, it follows that (22) lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . also for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] t−1g th(r1,r2, ...,rn)(23) ≤ [ λ[p]hg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . growth analysis of functions analytic in the unit polydisc 77 now from (17) and (23) , we obtain for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , tending to infinity that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) − � λ[p]hg (r1,r2, ...,rn) + � . choosing � (> 0) , we get that (24) lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . also for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) , log[p] t−1g tf (r1,r2, ...,rn)(25) ≤ [ ρ[p]fg (r1,r2, ...,rn) + � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . so from (21) and (25) , it follows for all sufficiently large values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) + � λ[p]hg (r1,r2, ...,rn) − � . as � (> 0) is arbitrary, we obtain from above that (26) lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) . thus the theorem follows from (19) , (22) , (24) and (26) . theorem 13. let f,h be any two analytic functions of n complex variables in u and g be entire in n complex variables with 0 < ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < ρ [p]h g (r1,r2, ...,rn) < ∞ where p is any positive integer. then lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) ≤ lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) . proof. from the definition of generalised n variables based p-th relative nevanlinna order, we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] t−1g th(r1,r2, ...,rn)(27) ≥ [ ρ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . 78 datta, biswas and deb now from (25) and (27) , it follows for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) + � ρ [p]h g (r1,r2, ...,rn) − � . as � (> 0) is arbitrary, we obtain that (28) lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . again for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity, log[p] t−1g tf (r1,r2, ...,rn)(29) ≥ [ ρ[p]fg (r1,r2, ...,rn) − � ] [− log [(1 −r1) (1 −r2) ... (1 −rn)]] . so combining (18) and (29) , we get for a sequence of values of ( 1 1−r1 ) , ( 1 1−r2 ) , ... and ( 1 1−rn ) tending to infinity that log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ ρ [p]f g (r1,r2, ...,rn) − � ρ [p]h g (r1,r2, ...,rn) + � . since � (> 0) is arbitrary, it follows that (30) lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≥ ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) . thus the theorem follows from (28) and (30) . in view of theorem 12 and theorem 13, we may state the following theorem without its proof. theorem 14. let f,h be any two analytic functions of n complex variables in u and g be entire in n complex variables such that 0 < λ[p]fg (r1,r2, ...,rn) ≤ ρ [p]f g (r1,r2, ...,rn) < ∞ and 0 < λ[p]hg (r1,r2, ...,rn) ≤ ρ [p]h g (r1,r2, ...,rn) < ∞. then lim inf r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) ≤ min { λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) , ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) } ≤ max { λ[p]fg (r1,r2, ...,rn) λ[p]hg (r1,r2, ...,rn) , ρ [p]f g (r1,r2, ...,rn) ρ [p]h g (r1,r2, ...,rn) } ≤ lim sup r1,r2,...rn→1 log[p] t−1g tf (r1,r2, ...,rn) log[p] t−1g th(r1,r2, ...,rn) where p is any positive integer. growth analysis of functions analytic in the unit polydisc 79 theorem 15. let f,h be any two analytic functions of n complex variables in u and g be entire in n complex variables such that ρ [p]f h (r1,r2, ...,rn) < ∞ and λ [p]f◦g h (r1,r2, ...,rn) = ∞ where p is any positive integer. then lim r1,r2,...rn→1 log[p] t−1h tf◦g(r1,r2, ...,rn) log[p] t−1h tf (r1,r2, ...,rn) = ∞ . the proof is omitted because it can be carried out using the same technique as involved in theorem 7. remark 2. theorem 15 is also valid with “limit superior” instead of “limit” if λ [p]f◦g h (r1,r2, ...,rn) = ∞ is replaced by ρ [p]f◦g h (r1,r2, ...,rn) = ∞ and the other conditions remain the same. corollary 16. under the assumptions of theorem 15 and remark 2, lim r1,r2,...rn→1 t−1h tf◦g(r1,r2, ...,rn) t−1h tf (r1,r2, ...,rn) = ∞ and lim sup r1,r2,...rn→1 t−1h tf◦g(r1,r2, ...,rn) t−1h tf (r1,r2, ...,rn) = ∞ respectively hold. the proof is omitted. similarly, one may also state the following theorem and corollaries without their proofs as they may be carried out in the line of remark 2, theorem 15 and corollary 16 respectively. theorem 17. let f,h be any two analytic functions of n complex variables in u and g be entire in n complex variables such that ρ [p]g h (r1,r2, ...,rn) < ∞ and ρ [p]f◦g h (r1,r2, ...,rn) = ∞. then lim sup r1,r2,...rn→1 log[p] t−1h tf◦g(r1,r2, ...,rn) log[p] t−1h tg(r1,r2, ...,rn) = ∞ where p is any positive integer. corollary 18. theorem 17 is also valid with “limit” instead of “limit superior” if ρ [p]f◦g h (r1,r2, ...,rn) = ∞ is replaced by λ [p]f◦g h (r1,r2, ...,rn) = ∞ and the other conditions remain the same. corollary 19. under the assumptions of theorem 15 and corollary 18, lim sup r1,r2,...rn→1 t−1h tf◦g(r1,r2, ...,rn) t−1h tg(r1,r2, ...,rn) = ∞ and lim r1,r2,...rn→1 t−1h tf◦g(r1,r2, ...,rn) t−1h tg(r1,r2, ...,rn) = ∞ respectively hold. references [1] agarwal, a. k., on the properties of an entire function of two complex variables, canadian j.math. vol. 20 (1968), pp.51–57. [2] banerjee, d. and dutta, r. k., relative order of functions analytic in the unit disc, bull. cal. math. soc. vol. 101, no. 1 (2009), pp. 95 104. [3] datta, s. k. and deb, s. k. , growth properties of functions analytic in the unit disc, international j. of math. sci & engg. appls (ijmsea), vol. 3, no. iv (2009), pp. 2171-279. [4] datta, s. k. and jerine, e., on the generalised growth properties of functions analytic in the unit disc, wesleyan journal of research, vol.3, no.1 (2010), pp.13-19. 80 datta, biswas and deb [5] fuks, b. a., theory of analytic functions of several complex variables, moscow, 1963. [6] juneja, o. p. and kapoor, g.p., analytic functions-growth aspects, pitman advanced publishing program, 1985. [7] kiselman, c. o., plurisubharmonic functions and potential theory in several complex variables, a contribution to the book project, development of mathematics 1950-2000, edited by jean paul pier. 1department of mathematics,university of kalyani, kalyani, dist-nadia,pin741235, west bengal, india 2rajbari, rabindrapalli, r. n. tagore road, p.o.krishnagar, dist-nadia, pin741101, west bengal, india 3bahin high school, p.o.-bahin,dist.-uttar dinajpur, pin-733157, west bengal, india ∗corresponding author int. j. anal. appl. (2022), 20:3 well-posedness of triequilibrium-like problems misbah iram bloach, muhammad aslam noor∗, khalida inayat noor department of mathematics, comsats university islamabad, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. this work emphasizes in presenting new class of equilibrium-like problems, termed as equilibriumlike problems with trifunction. we establish some metric characterizations for the well-posed triequilibriumlike problems. we give some conditions under which the equilibrium-like problems are strongly wellposed. our results, which give essential and adequate conditions to the well-posedness of triequilibriumlike problems, are acquired by utilizing the assumption of pseudomonotonicity. technique and ideas of this paper inspire further research in this dynamic field. 1. introduction the theory of equilibrium problems is an engrossing and significant offshoot of variational inequalities in practice with a broad variety of industrial, physical, geographical and social applications. across various areas of pure, applied and engineering sciences, equilibrium problem theory has shown incredible potential and great impact. in nearly all disciplines, of mathematics and engineering, this theory has registered its exceptional ever-expanding mark. a new and incisive treatment of a broad list of problems, that occur in ecology, finance, economics, elasticity, network, image reconstruction, optimization and transport, are exhaustively approached by the equilibrium problem theory. in 1994, blum and oettli [3] and noor and oettli [23] rendered equilibrium problems their existing form. the classical equilibrium problems theory revolves around the assumption of convexity of the set and objective function. equilibrium problems cover a diverse set of applications including hemivariational inequalities, variational inequalities, game theory, nash equilibrium, variational-like inequalities and fixed point received: sep. 08, 2021. 2010 mathematics subject classification. 49j40, 90c33. key words and phrases. variational-like inequalities; preinvex functions; equilibrium problems; well-posed; pseudomonotonicity. https://doi.org/10.28924/2291-8639-20-2022-3 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-3 2 int. j. anal. appl. (2022), 20:3 point theory, see [1–7,12–19,21,22,24–29]. recently, the notion of convexity has started expanding to numerous fields showing the capacity for various useful applications. hanson [8] derived invex functions as a special extension of convex functions. different results were presented to make this fact noticeable that what holds for convex functions in mathematical programming also holds for a generalized class of functions known as invex functions. ben-israel and mond [1] and weir and jeyakumar [31] works led to preinvex functions as another generalized class of convex functions. weir and mond [30] has shown in their work the interchangeability of preinvex functions with convex functions in optimization problems. in [15], it is made evident that the minimum of preinvex on the invex set can be disciplined into variational inequalities, widely known as variational-like inequalities. variational-like inequalities and equilibrium-like problems, owing to their specialized nature, cannot allow traditional resolvent method, projection method and their prevalent variant forms to propound any iterative methods. to bridge this gap, we resort to a technique named as auxiliary principle, proposed by glowinski et al. [4]. to solve numerous variational inequalities and equilibrium problems, noor [15, 19, 20] and noor et al. [22, 24, 25, 27] employed the technique of auxiliary principle to propose various iterative methods. research of our paper is devoted to present a new class of equilibrium problems, termed as equilibrium-like problems with trifunction. the notion of well-posedness of variational ineaqualities and equilibrium problems was introduced by lucchetti and patrone [9,10]. we expand the notion of well-posedness to contemplate and establish the well-posedness of triequilibrium-like problems. several interesting and important cases are discussed as applications of the obtained results. 2. basic concepts and formulations let h̃ be a real hilbert space. the inner product and norm on h̃ are denoted by 〈., .〉 and ‖ . ‖ respectively. let k} be a nonempty invex set in h̃. let = : k} → < and }(., .) : h̃ × h̃ → h̃ are continuous functions. first we recall the following well-known results and concepts, before discussing our main results. definition 2.1. ( [1]). a nonempty set k} in h̃ is said to be an invex set, if there exists a bifunction }(., .), such that ξ + τ}(ζ,ξ) ∈k}, ∀ξ,ζ ∈k},τ ∈ [0, 1]. if the set k} is invex at each ξ ∈k}, then k} is also called }−connected set. definition 2.2. ( [1]). a function = : k} → < is said to be a preinvex function, if there exists a bifunction }(., .), such that =(ξ + τ}(ζ,ξ)) ≤ (1 −τ)=(ξ) + τ=(ζ), ∀ξ,ζ ∈k},τ ∈ [0, 1]. int. j. anal. appl. (2022), 20:3 3 the function = : k} →< is said to be preconcave if and only if −= is preinvex. definition 2.3. ( [8]). a differentiable function = : k} → < is said to be an invex function, if there exists a bifunction }(., .), such that =(ζ) −=(ξ) ≥〈=′(ξ) , }(ζ,ξ)〉, ∀ξ,ζ ∈k},τ ∈ [0, 1], where =′(ξ) is the differential of = at ξ. from above definitions, it is clear that the differentiable preinvex functions are the invex functions but the converse is not true, see [21]. we note that, if }(ζ,ξ) = ζ−ξ, the invex set k} reduces to the convex set k and preinvex functions reduce to convex functions. there are some functions which are preinvex but not convex. definition 2.4. . the bifunction }(., .) : h̃ × h̃ →< satisfies the following condition }(ξ + τ1(ζ −ξ),ξ + τ2(ζ −ξ)) = (τ1 −τ2)}(ζ,ξ) , ∀ξ,ζ ∈ h̃. for τ1 = 0, 1 and τ2 = τ, we get condition c of mohan and neogy [11], (i) }(ξ,ξ + τ(ζ −ξ)) = −τ}(ζ,ξ) , (ii) }(ζ,ξ + τ(ζ −ξ)) = (1 −τ)}(ζ,ξ) , ∀ξ,ζ ∈ h̃. mohan and neogy [11] used definition 2.4 to show that an invex function on an invex set k}, is also a preinvex function and the converse also hold. given an operator υ : h̃ → < and a continuous trifunction ψ(., ., .) : h̃ × h̃ × h̃ → <, consider the problem of finding ξ ∈k}, such that ψ(ξ, υ(ξ),}(ζ,ξ)) ≥ 0 , ∀ζ ∈k}. (2.1) the problem (2.1) is called an equilibrium-like problem with trifunction. for ψ(ξ, υ(ξ),}(ζ,ξ)) = 〈ξ, υ(ξ),}(ζ,ξ)〉, problem (2.1) is called variational-like inequality with trifunction of finding ξ ∈k}, such that 〈ξ, υ(ξ),}(ζ,ξ)〉≥ 0 , ∀ζ ∈k}. (2.2) for 〈ξ, υ(ξ),}(ζ,ξ)〉 = 〈υ(ξ),}(ζ,ξ)〉, problem (2.2) is called variational-like inequality of finding ξ ∈ h̃ such that 〈υ(ξ),}(ζ,ξ)〉≥ 0 , ∀ζ ∈k} (2.3) if }(ζ,ξ) = ζ−ξ, then an invex set reduces to a convex set and problem (2.1) is equivalent to finding ξ ∈k, such that ψ(ξ, υ(ξ),ζ −ξ) ≥ 0 , ∀ζ ∈k, (2.4) 4 int. j. anal. appl. (2022), 20:3 which is called an equilibrium problem with trifunction and gives off an impression of being new. variational-like inequality with trifunction (2.2) reduces to variational inequality with trifunction of finding ξ ∈k, such that 〈ξ, υ(ξ),ζ −ξ〉≥ 0 , ∀ζ ∈k. (2.5) also variational-like inequality (2.3) is equivalent to finding ξ ∈k, such that 〈υ(ξ),ζ −ξ〉≥ 0 , ∀ζ ∈k}, (2.6) which is known as variational inequality, proposed and cosidered by stampacchia [20]. in brief, for suitable and appropriate choice of the functions ψ(., ., .),}(., .) and the spaces, one can obtain a number of new and known problems as special cases of the problem (2.1). this shows that problem (2.1) is quite general and unifying. definition 2.5. the operator υ : h̃ →< and the function ψ(., ., .) are said to be: (i) jointly }− pseudomonotone, if ψ(ξ, υ(ξ),}(ζ,ξ)) ≥ 0, =⇒ ψ(ζ, υ(ζ),}(ξ,ζ)) ≤ 0 , ∀ξ,ζ ∈k}. (ii) partially relaxed jointly strong }− monotone, if there exists a constant α > 0 such that ψ(ξ, υ(ξ),}(ζ,z)) + ψ(ζ, υ(ζ),}(z,ζ)) ≤ α ‖ }(z,ξ) ‖2 , ∀ξ,ζ,z ∈k}. (iii) jointly }− monotone, if ψ(ξ, υ(ξ),}(ζ,ξ)) + ψ(ζ, υ(ζ),}(ξ,ζ)) ≤ 0 , ∀ξ,ζ ∈k}. (iv) jointly }− hemicontinuous, if the mapping ψ(ξ + τ}(ζ,ξ), υ(ξ + τ}(ζ,ξ)),}(ζ,ξ)) ∀ξ,ζ ∈k},τ ∈ [0, 1], is continuous. we note that, for z = ξ, partially relaxed jointly strong }− monotonicity reverts to jointly }− monotonicity. lemma 2.1. let the trifunction ψ(., ., .) and operator υ be jointly }−pseudomonotone and jointly }− hemicontinuous. if assumption (2.4) holds, then problem (2.1) is equivalent to finding ξ ∈ k}, such that ψ(ζ, υ(ζ),}(ξ,ζ)) ≤ 0 , ∀ξ,ζ ∈k}. int. j. anal. appl. (2022), 20:3 5 proof: let ξ ∈k} be a solution of equilibrium-like problem (2.1), then ψ(ξ, υ(ξ),}(ζ,ξ)) ≥ 0, implies ψ(ζ, υ(ζ),}(ξ,ζ)) ≤ 0 , ∀ξ,ζ ∈k}, (2.7) since ψ(., ., .) and υ are jointly }− pseudomonotone. conversely, let ∀ξ,ζ ∈k}, we define ζτ = ξ + τ}(ζ,ξ) ∈ h̃. replacing ζ by ζτ in (2.7), we have ψ(ζτ, υ(ζτ ),}(ξ,ζτ )) ≤ 0 , ∀ξ,ζ ∈k}. now by using assumption (2.4) , we have −τψ(ζτ, υ(ζτ ),}(ζ,ξ)) ≤ 0. (2.8) now dividing (2.8) by τ and letting τ −→ 0 and using jointly }− hemicontinuity of ψ(., ., .) and υ, we get ψ(ξ, υ(ξ),}(ζ,ξ)) ≥ 0, which shows that ξ ∈k} is a solution of problem (2.1), the required result. � lemma (2.1) can be considered as the generalized form of minty’s lemma for triequilibrium-like problems. result obtained in above lemma is termed as the dual triequilibrium -like problem. 3. well-posedness we generalize the notion of well-posedness to triequilibrium-like problems. by using the assumption of pseudomonotonicity we obtain some results for well-posed equilibrium-like problems with trifunction. our obtained results could be considered, an extension of the results which were obtained and studied in [6,8,9,16–18] given ε > 0, consider two sets m(ε) = {ξ ∈ h̃ : ψ(ξ , υ(ξ) , }(ζ,ξ)) ≥−ε ‖ }(ζ,ξ) ‖, ∀ζ ∈k}}, (3.1) and n(ε) = {ξ ∈ h̃ : ψ(ζ , υ(ζ) , }(ξ,ζ)) ≤ ε ‖ }(ζ,ξ) ‖, ∀ζ ∈k}. (3.2) for a non-empty set s ⊂ h̃, define the diameter of s, denoted by d(s), as: d(s) = sup[‖ ζ −ξ ‖; ∀ξ,ζ ∈ s]. definition 3.1. [18]. the triequilibrium-like problem (2.1) is well-posed, if (i) for any ε > 0, m(ε) 6= ∅, (ii) d(m(ε)) −→ 0 as ε −→ 0. 6 int. j. anal. appl. (2022), 20:3 theorem 3.1. let the trifunction ψ(., ., .) and operator υ be jointly }− pseudomonotone and jointly }− hemicontinuous. if assumption (2.4) holds, then m(ε) = n(ε). proof: let ξ ∈k} be such that ψ(ξ , υ(ξ) , }(ζ,ξ)) ≥−ε ‖ }(ζ,ξ) ‖, ∀ζ ∈k} implies ψ(ζ , υ(ζ) , }(ξ,ζ)) ≤ ε ‖ }(ζ,ξ) ‖, ∀ζ ∈k}, (3.3) since ψ(., ., .) and υ are jointly }− pseudomonotone. thus m(ε) ⊂ n(ε). (3.4) conversely, for any ξ ∈k}, (3.3) holds. now, let ∀ξ,ζ ∈k} we define ζτ = ξ + τ}(ζ,ξ) ∈k}. replacing ζ by ζτ in (3.3), we have ψ(ζτ, υ(ζτ ),}(ξ,ζτ )) ≤ ε ‖ }(ζτ,ξ) ‖, ∀ξ,ζ ∈k}. now by using assumption (2.4), we have −τψ(ζτ, υ(ζτ ),}(ζ,ξ)) ≤ τε ‖ }(ζ,ξ) ‖ . (3.5) now dividing (3.5) by τ and letting τ −→ 0 and using jointly }− hemicontinuity of ψ(., ., .) and t, we get ψ(ξ, υ(ξ),}(ζ,ξ)) ≥−ε ‖ }(ζ,ξ) ‖, which implies that n(ε) ⊂ m(ε). (3.6) by combining (3.4) and (3.6), we get our required result n(ε) = m(ε). theorem 3.2. let the trifunction ψ(., ., .) and operator υ be jointly }− pseudomonotone and jointly }− hemicontinuous. then for all ε > 0, n(ε) is closed in h̃. proof: let {ξn : n ∈ n} ⊂ n(ε) be a sequence, such that ξn −→ ξ in k} as n −→ ∞. then ξn ∈k} and ψ(ζ , υ(ζ) , }(ξn,ζ)) ≤ ε ‖ }(ζ,ξn) ‖, ∀ζ ∈k}. (3.7) by taking limit n −→∞ in (3.7), we get ψ(ζ , υ(ζ) , }(ξ,ζ)) ≤ ε ‖ }(ζ,ξ) ‖, ∀ζ ∈k}. int. j. anal. appl. (2022), 20:3 7 which implies that ξ ∈ n(ε). thus, n(ε) is closed in k} � theorem 3.3. let the trifunction ψ(., ., .) and operator υ be jointly }− pseudomonotone and jointly }− hemicontinuous. if triequilibrium-like problem (2.1) is well-posed and assumption (2.4) holds, then there exists a unique solution of problem (2.1). proof: let the sequence {ξn : n ∈ n}, defined by ξn ∈ m(1n ). let ε > 0 be very small and let p,q ∈n , such that p ≥ q ≥ 1 ε . then m( 1 p ) ⊂ m( 1 q ) ⊂ m(ε). so ‖ ξp −ξq ‖≤ d(m( 1 p )). thus, the sequence {ξn} is a cauchy sequence and ξn −→ ξ in k}. by using results of theorem (3.1) and theorem (3.2), we get m(ε) is a closed set. thus ξ ∈ ⋂ ε>0 m(ε), so, ξ is solution of problem (2.1). uniqueness of solution ξ follows from second condition of wellposedness. � theorem 3.4. let the trifunction ψ(., ., .) and operator υ be jointly }− pseudomonotone and jointly }− hemicontinuous. if m(ε) 6= 0, for all ε > 0 and m(ε) is bounded for some ε0, then there exists at least one solution of problem (2.1). proof: let ξk ∈ m(1k ), then for large enough k, we have m( 1 k ) ⊂ m(ε). thus for some subsequence ξk −→ ξ ∈k}, we get ψ(ζ , υ(ζ) , }(ξk,ζ)) ≤ 1 k ‖ }(ζ,ξk) ‖ ≤ 1 k {‖ ζ ‖ +c}∀∈ h̃. now by taking limit k −→∞, we get ψ(ζ , υ(ζ) , }(ξ,ζ)) ≤ 0, which implies that ξ ∈ n(0). by theorem (3.1), we get ξ ∈ n(0) = m(0), which implies ξ ∈ m(0). hence triequilibrium-like problem (2.1) has at least one solution. � 8 int. j. anal. appl. (2022), 20:3 remark 3.1. (i) if triequilibrium-like problem has a unique solution, then m(ε) 6= 0,∀ε > 0 and⋂ ε>0 = {ξ0}. (ii) it is well-known, that if there is a unique solution to variational inequality (2.6), then it is not well-posed,see [9]. (iii) theorem (3.3) concludes that, the unique solution of problem (2.1) could be calculated by utilizing ε− equilibrium-like problem with trifunction, that is, find ξε ∈ h̃, such that ψ(ξε , υ(ξε) , }(ζ,ξε)) ≥−ε ‖ }(ζ,ξε) ‖, ∀ζ ∈k}. 4. perspective in this section, we discuss some research perspective of the equilibrium-like problem with trifunction. we show that the results derived in this paper can also be extended for a class of nonconvex equilibrium problem with trifunction. for the given operators υ,g : h̃ →< and a nonlinear continuous trifunction ψ(., ., .) : h̃×h̃×h̃ →<, consider the problem of finding ξ ∈k}, such that ψ(g(ξ), υ(g(ξ)), (g(ζ)) ≥ 0 , ∀ζ ∈k}, (4.1) which is called nonconvex triequilibrium problem. for the formulation and applications of problem (4.1) see, [12,15,18] and references therein. for g ≡ i, where i is the identity operator, then the gconvex set becomes becomes the convex set and the problem (4.1) is called the triequilibrium problem of finding ξ ∈k}, such that ψ(ξ, υ(ξ),ζ) ≥ 0 , ∀ζ ∈k}, (4.2) which was introduced and investigated by noor and oettli [14]. for }(ζ,ξ) = ζ − ξ, the invex set reduces to convex set. thus from (4.1) and (4.2), we obtain that the triequilibrium-like problem is equivalent to the nonconvex equilibrium problem with trifunction (4.1). hence all the above discussed results continue to hold for nonconvex triequilibrium problem (4.1). 5. conclusion in our work we focused on introducing another class of equilibrium-like problems terming it as triequilibrium-like problems. we studied and established the well-posedness of triequilibrium-like problems by using the assumption of pseudomonotonicity. the demonstrated results in this endeavour can rightly be considered as an enhancement and sophistication of already existing work. acknowledgements we wish to express our deepest gratitude to our colleagues, collaborators and friends, who have direct or indirect contributions in the process of this paper. we are also grateful to rector, comsats university islamabad, pakistan for the research facilities and support in our research endeavors. int. j. anal. appl. (2022), 20:3 9 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. ben-israel and b. mond, what is invexity?, j. austral. math. soc. ser. b. 28 (1986), 1-9. https://doi.org/ 10.1017/s0334270000005142. [2] m. i. bloach and m. a. noor, perturbed mixed variational-like inequalities, aims math. 5(3) (2019), 2153-2162. https://doi.org/10.3934/math.2020143. [3] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. student. 63 (1994), 123-145. [4] r. glowinski, j. l. lions and r. tremolieres, numerical analysis of variational inequalities, north-holland, amsterdam, 1981. [5] f. giannessi and a. maugeri, variational inequalities and network equilibrium problems. plenum press, new york, 1995. [6] f. giannessi, a. maugeri and p. m. pardalos, equilibrium problems: nonsmooth optimization and variational inequality models, kluwer academic publishers, dordrecht, holland, 2001. [7] d. goeleven and d. mantaque, well-posed hemivariational inequalities. numer. funct. anal. optim. 16 (1995), 909-921. https://doi.org/10.1080/01630569508816652. [8] m. a. hanson, on sufficiency of the kuhn-tucker conditions, j. math. anal. appl. 80 (1981), 545-550. https: //doi.org/10.1016/0022-247x(81)90123-2. [9] r. lucchetti and f. patrone, a characterization of tykhonov well-posedness for minimum problems with applications to variational inequalities. numer. funct. anal. optim. 3 (1981), 461-476. https://doi.org/10.1080/ 01630568108816100. [10] r. lucchetti and f. patrone, some properties of well-posed variational inequalities governed by linear operators. numer. funct. anal. optim. 5 (1983), 349-361. https://doi.org/10.1080/01630568308816145. [11] s. r. mohan and s. k. neogy, on invex set and preinvex functions. j. math. anal. appl. 189 (1995), 901-908. https://doi.org/10.1006/jmaa.1995.1057. [12] t. v. nghi1 and n. n. tam, general variational inequalities: existence of solutions, tikhonov-type regularization, and well-posedness, acta math. vietnam. (2021). https://doi.org/10.1007/s40306-021-00435-0. [13] b. b. mohsen, m. a. noor, k. i. noor and m. postolache, strongly convex functions of higher order involving bifunction, mathematics, 7(11) (2019), 1028. https://doi.org/10.3390/math7111028. [14] m. a. noor, general variational inequalities. appl. math. lett. 1 (1988), 119-121. https://doi.org/10.1016/ 0893-9659(88)90054-7. [15] m. a. noor, variational-like inequalities. optimization, 30 (1994), 323-330. https://doi.org/10.1080/ 02331939408843995. [16] m. a. noor, new approximation schemes for general variational inequalities. j. math. anal. appl. 251 (2000), 217-229. https://doi.org/10.1006/jmaa.2000.7042. [17] m. a. noor, merit function for variational–like inequalitiets, math. inequal. appl. 1 (2000), 117-128. [18] m. a. noor, well-posed variational inequalities. j. appl. math. comput. 11 (2003), 165-172. https://doi.org/ 10.1007/bf02935729. [19] m. a. noor, fundamentals of mixed quasi variational inequalities. int. j. pure. appl. math. 15 (2004), 137-250. [20] m. a. noor, fundamentals of equilibrium problems. math. inequal. appl. 9 (2006), 529-566. https://doi.org/ 10.7153/mia-09-51. https://doi.org/10.1017/s0334270000005142 https://doi.org/10.1017/s0334270000005142 https://doi.org/10.3934/math.2020143 https://doi.org/10.1080/01630569508816652 https://doi.org/10.1016/0022-247x(81)90123-2 https://doi.org/10.1016/0022-247x(81)90123-2 https://doi.org/10.1080/01630568108816100 https://doi.org/10.1080/01630568108816100 https://doi.org/10.1080/01630568308816145 https://doi.org/10.1006/jmaa.1995.1057 https://doi.org/10.1007/s40306-021-00435-0 https://doi.org/10.3390/math7111028 https://doi.org/10.1016/0893-9659(88)90054-7 https://doi.org/10.1016/0893-9659(88)90054-7 https://doi.org/10.1080/02331939408843995 https://doi.org/10.1080/02331939408843995 https://doi.org/10.1006/jmaa.2000.7042 https://doi.org/10.1007/bf02935729 https://doi.org/10.1007/bf02935729 https://doi.org/10.7153/mia-09-51 https://doi.org/10.7153/mia-09-51 10 int. j. anal. appl. (2022), 20:3 [21] m. a. noor, extended general variational inequalities. appl. math. lett. 22(2) (2009), 182-186. https://doi. org/10.1016/j.aml.2008.03.007. [22] m. a. noor and k. i. noor, some new trends in mixed variational inequalities, j. adv. math. stud. in press. [23] m. a. noor and w. oettli, on general nonlinear complementarity problems and quasi-equilibria. le mathematiche, 49 (1994), 313-331. [24] m. a. noor, k. i. noor and h. m. y. al-bayatti, higher order variational inequalities, inf. sci. lett. 11 (2022), 1-5. [25] m. a. noor, k. i. noor and m. i. baloch, auxiliary principle technique for strongly mixed variational-like inequalities. u.p.b. sci. bull. ser. a, 80 (2018), 93-100. [26] m. a. noor, k. i. noor, a. hamdi and e. h. el-shemas, on difference of two monotone operators, optim. lett. 3 (2009), 329. https://doi.org/10.1007/s11590-008-0112-7. [27] m.a. noor, k.i. noor, m.th. rassias, new trends in general variational inequalities, acta appl. math. 170 (2020), 981–1064. https://doi.org/10.1007/s10440-020-00366-2. [28] m. a. noor, k. i. noor, m. u. awan and a. g. khan, quasi variational inclusions involving three operators, inform. sci. lett. in press. [29] g. stampacchia, formes bilineaires coercitives sur les ensembles convexes. c. r. acad. sci. paris, 258 (1964), 4413-4416. [30] t. weir and b. mond, preinvex functions in multiple objective optimization. j. math. anal. appl. 136 (1988), 29-38. https://doi.org/10.1016/0022-247x(88)90113-8. [31] t. weir and v. jeyakumar, a class of nonconvex functions and mathematical programming. bull. austral. math. soc. 38 (1988), 177-189. https://doi.org/10.1017/s0004972700027441. https://doi.org/10.1016/j.aml.2008.03.007 https://doi.org/10.1016/j.aml.2008.03.007 https://doi.org/10.1007/s11590-008-0112-7 https://doi.org/10.1007/s10440-020-00366-2 https://doi.org/10.1016/0022-247x(88)90113-8 https://doi.org/10.1017/s0004972700027441 1. introduction 2. basic concepts and formulations 3. well-posedness 4. perspective 5. conclusion acknowledgements references international journal of analysis and applications volume 19, number 4 (2021), 518-541 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-518 received march 16th, 2021; accepted april 14th, 2021; published may 11th, 2021. 2010 mathematics subject classification. 49j40. key words and phrases. exponentially preinvex fuzzy mapping; exponentially invex fuzzy mappings; exponentially monotone fuzzy mappings; exponentially fuzzy mixed variational-like inequalities. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 518 exponentially preinvex fuzzy mappings and fuzzy exponentially mixed variational-like inequalities muhammad bilal khan1, muhammad aslam noor1, khalida inayat noor1, hassan almusawa2, kottakkaran sooppy nisar3,* 1department of mathematics, comsats university islamabad, pakistan 2department of mathematics, college of sciences, jazan university, jazan 45142, saudi arabia 3department of mathematics, college of arts and sciences, wadi aldawaser, prince sattam bin abdulaziz university, saudi arabia *corresponding author: n.sooppy@psau.edu.sa; ksnisar1@gmail.com abstract. in this article, our aim is to consider a class of nonconvex fuzzy mapping known as exponentially preinvex fuzzy mapping. with the support of some examples, the notions of exponentially preinvex fuzzy mappings are explored and discussed in some special cases. some properties are also derived and relations among the exponentially preinvex fuzzy mappings (exponentially preinvex-fms), exponentially invex fuzzy mappings (exponentially-ifms), and exponentially monotonicity are established under some mild conditions. in the end, using the fact that fuzzy optimization and fuzzy variational inequalities have close relationships, we have proven that the optimality conditions of exponentially preinvex fuzzy mapping can be distinguished by exponentially fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality. these inequalities render the very interesting outcomes of our main results and appear to be the new ones. presented results in this paper can be considered and the development of previously obtained results. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-518 int. j. anal. appl. 19 (4) (2021) 519 1. introduction in the last few decades, the ideas of convexity and nonconvexity are well-acknowledged in optimization concepts and gifted a vital role in operation research, economics, decision making, and management. hanson [16] initiated to introduce a generalized class of convexity which is known as an invex function. the invex function played a significant role in mathematical programming. a step forward, the invex set invex function and preinvex function were introduced and studied by israel and mond [9], and mohen and neogy [20]. also, noor [23] examined the optimality conditions of differentiable preinvex functions and proved that variational-like inequalities would characterize the minimum. many classical convexity generalizations and extensions have been investigated by several authors, see ([1], [6], [7], [15]) and the references therein. it is well known that logarithmically convex functions have serious importance in convex theory because by using these functions we can derive more accurate inequalities as compare to convex functions. some of the authors discussed different classes of convex and nonconvex functions. these functions have a very close relationship with logarithmically convex functions. bernstain [10], initiated to introduce exponentially convex function. a move forward, the concept of an r-convex function was introduced by avriel [5], while antczak [4] studied the notions of (r, p)-convex functions. for further details on r-convex function, we refer the reader to study the properties of r-convex function, see [34] and the reference therein. exponentially convex functions and their generalizations have many applications in different fields such as information theory, data analysis, statistics, and machine learning; see [2-5]. noor and noor [25-27] have recently considered and characterized another type of exponentially convex functions and characterized its properties, and have demonstrated that convex functions are different from the exponentially convex functions defined by bernstein [10]. furthermore, noor and noor [28] generalized the exponentially convex functions and defined a class of nonconvex functions which is known as exponentially preinvex function and proved that the minimum of a differentiable exponentially convex function is distinguished by exponentially variational-like inequality. similarly, the notions of convexity and nonconvexity play a vital role in optimization under fuzzy domain because, during characterization of the optimality condition of convexity, we obtain fuzzy variational inequalities so variational inequality theory and fuzzy complementary int. j. anal. appl. 19 (4) (2021) 520 problem theory established powerful mechanism of the mathematical problems and they have a friendly relationship. many authors contributed to this fascinating and interesting field. the concept of fuzzy mappings (fms) was proposed by chang and zadeh [11] in 1972. in 1989, nanda and kar [21] were initiated to introduce convex-fms and characterized the notion of convex-fm through the idea of epigraph. a step forward, furukawa [14] and syau [31] proposed and examined fm from space ℝ𝑛 to the set of fuzzy numbers, fuzzy valued lipschitz continuity, logarithmic convex-fms and quasi-convex-fms. besides, chang [12] discussed the idea of convex-fm and find its optimality condition with the support of fuzzy variational inequality. generalization and extension of fuzzy convexity play a vital and significant implementation in diverse directions. so let’s note that, one of the most considered classes of nonconvex-fm is preinvex-fm. noor [22] introduced this idea and proved some results that distinguish the fuzzy optimality condition of differentiable fuzzy preinvex mappings by fuzzy variational-like inequality. fuzzy variational inequality theory and complementary problem theory established a strong relationship with mathematical problems. recently, li and noor [19] established an equivalence condition of preinvex-fm and characterizations about preinvex-fms under some mild conditions. with the support of examples, wu and xu [33] updated the definition of convex-fms and established a new approach regarding the existence of a fuzzy preinvex mapping under the condition of lower or upper semicontinuity. in 2012, rufianlizana et al. [29] reviewed the existing literature and made appropriate modifications to the results obtained by wu and xu [32] regarding invex-fms. for differentiable and twice differentiable preinvex-fms, rufian-lizana et al. [30] gave the required and sufficient conditions. they demonstrated the validity of characterizations with the help of examples and improved the previous results provided by li and noor [19]. we refer to the readers for further analysis of literature on the applications and properties of variational-like inequalities and generalized convex-fms, see ([8], [13], [15], [24], [32]) and the references therein. motivated and inspired by the ongoing research work, we note that convex and generalized convex-fms play an important role in fuzzy optimization. the paper is organized as follows. section 2 recalls some basic and new definitions, preliminary notations, and new results which will be helpful for further study. section 3 introduces and considers a family of classes of nonconvex-fms is called exponentially preinvex-fms and investigates some int. j. anal. appl. 19 (4) (2021) 521 properties. section 4 derives some relations among the exponentially preinvex-fms, exponentially-ifms, and exponentially monotonicity under some mild conditions. section 5 introduces the new classes of fuzzy variational-like inequality which is known as exponentially fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality. several special cases are also discussed. these inequalities give an interesting outcome of our main results. 2. preliminaries let ℝ be the set of real numbers. a fuzzy subset set 𝒜 of ℝ is distinguished by a function 𝜑:ℝ → [0,1] called the membership function. in this study this depiction is approved. moreover, the collection of all fuzzy subsets of ℝ is denoted by �̃�(ℝ). definition 2.1. if 𝜑 be a fuzzy set in ℝ and 𝛾 ∈ [0,1], then 𝛾-level sets of 𝜑 is denoted and is defined as follows 𝜑𝛾 = {𝓊 ∈ ℝ| 𝜑(𝓊) ≥ 𝛾}. (2.1) definition 2.2. a fuzzy number (fn) 𝜑 is a fuzzy set in ℝ with the following properties: (1) 𝜑 is normal i.e. there exists 𝑥 ∈ ℝ such that 𝜑(𝑥) = 1; (2) 𝜑 is upper semi continuous i.e., for given 𝑥 ∈ ℝ, for every 𝑥 ∈ ℝ there exist > 0 there exist 𝛿 > 0 such that 𝜑(𝑥) − 𝜑(𝑦) < for all 𝑦 ∈ ℝ with |𝑥 − 𝑦| < 𝛿. (3) 𝜑 is fuzzy convex i.e., 𝜑((1 − 𝜏)𝑥 + 𝜏𝑦) ≥ 𝑚𝑖𝑛(𝜑(𝑥),𝜑(𝑦)), ∀ 𝑥,𝑦 ∈ ℝ and 𝜏 ∈ [0,1]; (4) 𝜑 is compactly supported i.e., 𝑐𝑙{𝑥 ∈ ℝ| 𝜑(𝑥) > 0} is compact. the lr-fns first introduced by dubois and prade [13] are defined as follows: definition 2.3. let 𝐿,𝑅:[0,1] → [0,1] be two decreasing and upper semicontinuous functions with 𝐿(0) = 𝑅(0) = 1 and 𝐿(1) = 𝑅(1) = 0. then the fn is defined as 𝜑(𝓊) = { 𝐿( 𝑐−𝓊 𝜌 ) 𝑐 − 𝜌 ≤ 𝓊 < 𝑑, 1 𝑐 ≤ 𝓊 < 𝑑, 𝐿( 𝓊−𝑑 𝑟 ) 𝑐 ≤ 𝓊 < 𝑑 + 𝑟, 0 otherwise, where 𝑟,𝜌 > 0 and 𝑑 ≥ 𝑐. int. j. anal. appl. 19 (4) (2021) 522 let 𝔽(ℝ) denotes the set of all fns and let 𝜑 ∈ 𝔽(ℝ) be fn, if and only if, 𝛾-levels 𝜑𝛾 is a nonempty compact convex set of ℝ. this is represented by 𝜑𝛾 = [𝜑∗(𝛾),𝜑 ∗(𝛾)] where 𝜑∗(𝛾) = 𝑖𝑛𝑓{𝓊 ∈ ℝ| 𝜑(𝓊) ≥ 𝛾}, 𝜑 ∗(𝛾) = 𝑠𝑢𝑝{𝓊 ∈ ℝ| 𝜑(𝓊) ≥ 𝛾}. since each 𝜌 ∈ ℝ is also a fn, defined as �̃�(𝓊) = { 1 if 𝓊 = 𝜌 0 if 𝓊 ≠ 𝜌 . it is well known that a fn is a real fuzzy interval and 𝜑 can also be identified by a triplet: {(𝜑∗(𝛾),𝜑 ∗(𝛾),𝛾):𝛾 ∈ [0,1]}. this notion help us to characterize a fn in terms of the two end point functions 𝜑∗(𝛾) and 𝜑∗(𝛾). theorem 2.1 ([13]). suppose that 𝜑∗(𝛾):[0,1] → ℝ and 𝜑 ∗(𝛾):[0,1] → ℝ satisfy the following assertions: • 𝜑∗(𝛾) is a non-decreasing function. • 𝜑∗ (𝛾) is a non-increasing function. • 𝜑∗(1) ≤ 𝜑 ∗(1). • 𝜑∗(𝛾) and 𝜑 ∗(𝛾) are bounded and left continuous on (0,1] and right continuous at 𝛾 = 0. then 𝜑:ℝ → [0,1], defined by 𝜑(𝓊) = 𝑠𝑢𝑝{𝛾:𝜑∗(𝛾) ≤ 𝓊 ≤ 𝜑 ∗(𝛾)}, is a fn, parameterization is given by {(𝜑∗(𝛾),𝜑 ∗(𝛾),𝛾):𝛾 ∈ [0,1]}. moreover, if 𝜑:ℝ → [0,1] is a fn with parametrization given by {(𝜑∗(𝛾),𝜑 ∗(𝛾),𝛾):𝛾 ∈ [0,1]}, then function 𝜑∗(𝛾) and 𝜑 ∗(𝛾) find the conditions (1)-(4). let 𝜑,𝜙 ∈ 𝔽(ℝ) represented parametrically {(𝜑∗(𝛾),𝜑 ∗(𝛾),𝛾):𝛾 ∈ [0,1]} and {(𝜑∗(𝛾),𝜙 ∗(𝛾),𝛾):𝛾 ∈ [0,1]} respectively. we say that 𝜑 ⪯ 𝜙 if for all 𝛾 ∈ (0,1], 𝜑∗(𝛾) ≤ 𝜙∗(𝛾), and 𝜑∗(𝛾) ≤ 𝜙∗(𝛾). if 𝜑 ⪯ 𝜙, then there exist 𝛾 ∈ (0,1] such that 𝜑 ∗(𝛾) < 𝜙∗(𝛾) or 𝜑∗(𝛾) ≤ 𝜙∗(𝛾) . we say comparable if for any 𝜑,𝜙 ∈ 𝔽(ℝ), we have 𝜑 ⪯ 𝜙 or 𝜑 ⪰ 𝜙 otherwise they are non-comparable. some time we may write 𝜑 ⪯ 𝜙 instead of 𝜙 ⪰ 𝜑 and note that, we may say that 𝔽(ℝ) is a partial ordered set under the relation ⪯. int. j. anal. appl. 19 (4) (2021) 523 if 𝜑,𝜙 ∈ 𝔽(ℝ), there exist 𝜓 ∈ 𝔽(ℝ) such that 𝜑 = 𝜙+̃𝜓, then by this result we have existence of hukuhara difference of 𝜑 and 𝜙, and we say that 𝜓 is the h-difference of 𝜑 and 𝜙, and denoted by 𝜑−̃𝜙, see [32]. if h-difference exists, then (𝜓)∗(𝛾) = (𝜑−̃𝜙)∗(𝛾) = 𝜑∗(𝛾) − 𝜙∗(𝛾), (𝜓)∗(𝛾) = (𝜑−̃𝜙)∗(𝛾) = 𝜑∗(𝛾) − 𝜙∗(𝛾). now we discuss some properties of fns under addition and scalar multiplication, if 𝜑,𝜙 ∈ 𝔽(ℝ) and 𝜌 ∈ ℝ then 𝜑+̃𝜙 and 𝜌𝜑 define as 𝜑+̃𝜙 = {(𝜑∗(𝛾) + 𝜙∗(𝛾),𝜑 ∗(𝛾) + 𝜙∗(𝛾),𝛾):𝛾 ∈ [0,1]}, (2.2) 𝜌𝜑 = {(𝜌𝜑∗(𝛾),𝜌𝜑 ∗(𝛾),𝛾):𝛾 ∈ [0,1]}. (2.3) remark 2.1. obviously, 𝔽(ℝ) is closed under addition and nonnegative scaler multiplication and above defined properties on 𝔽(ℝ) are equivalent to those derived from the usual extension principle. furthermore, for each scalar number 𝜌 ∈ ℝ, 𝜑+̃𝜌 = {(𝜑∗(𝛾) + 𝜌,𝜑 ∗(𝛾) + 𝜌,𝛾):𝛾 ∈ [0,1]}. (2.4) definition 2.4. a mapping 𝒯:𝐾 ⊂ ℝ → 𝔽(ℝ) is called fuzzy mapping (fm). for each 𝛾 ∈ [0,1], denote [𝒯(𝓊)]𝛾 = [𝒯∗(𝓊,𝛾),𝒯 ∗(𝓊,𝛾)]. thus a fm 𝒯 can be identified by a parametrized triples 𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯 ∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]}. definition 2.5. let 𝒯:𝐾 ⊂ ℝ → 𝔽(ℝ) be a fm. then 𝒯(𝓊) is said to be continuous at 𝓊 ∈ 𝐾, if for each 𝛾 ∈ [0,1], both end point functions 𝒯∗(𝓊,𝛾) and 𝒯 ∗(𝓊,𝛾) are continuous at 𝓊 ∈ 𝐾. definition 2.6 ([8]). let 𝐿 = (𝑚,𝑛) and 𝓊 ∈ 𝐿 . then fm 𝒯:(𝑚,𝑛) → 𝔽(ℝ) is said to be a generalized differentiable (in short, g-differentiable) at 𝓊 if there exists an element 𝒯 ,(𝓊) ∈ 𝔽(ℝ) such that for all 0 < 𝜏, sufficiently small, there exist 𝒯(𝓊 + 𝜏)−̃𝒯(𝓊), 𝒯(𝓊)−̃𝒯(𝓊 − 𝜏) and the limits lim 𝜏→0+ 𝒯(𝓊+𝜏)−̃𝒯(𝓊) 𝜏 = lim 𝜏→0+ 𝒯(𝓊)−̃𝒯(𝓊−𝜏) 𝜏 = 𝒯 ,(𝓊) or lim 𝜏→0+ 𝒯(𝓊)−̃𝒯(𝓊+𝜏) −𝜏 = lim 𝜏→0+ 𝒯(𝓊−𝜏)−̃𝒯(𝓊) −𝜏 = 𝒯 ,(𝓊) or lim 𝜏→0+ 𝒯(𝓊+𝜏)−̃𝒯(𝓊) 𝜏 = lim 𝜏→0+ 𝒯(𝓊−𝜏)−̃𝒯(𝓊) −𝜏 = 𝒯 ,(𝓊) or lim 𝜏→0+ 𝒯(𝓊)−̃𝒯(𝓊+𝜏) −𝜏 = lim 𝜏→0+ 𝒯(𝓊)−̃𝒯(𝓊−𝜏) 𝜏 = 𝒯 ,(𝓊), where the limits are taken in the metric space (𝔽(ℝ),𝐷), for 𝜑,𝜙 ∈ 𝔽(ℝ) 𝐷(𝜑,𝜙) = sup 0≤𝛾≤1 𝐻(𝜑𝛾,𝜙𝛾), and 𝐻 denote the well-known hausdorff metric on space of intervals. definition 𝟐.𝟕 ([21]). a fm 𝒯:𝐾 → 𝔽(ℝ) is called convex on the convex set 𝐾 if 𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ (1 − 𝜏)𝒯(𝓊)+̃𝜏𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾, 𝜏 ∈ [0,1]. definition 𝟐.𝟖 ([21]). a fm 𝒯:𝐾 → 𝔽(ℝ) is called quasi-convex on the convex set 𝐾 if int. j. anal. appl. 19 (4) (2021) 524 𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ 𝑚𝑎𝑥(𝒯(𝓊),𝒯(𝜗)),∀ 𝓊,𝜗 ∈ 𝐾, 𝜏 ∈ [0,1]. definition 𝟐.𝟗 ([22]). a fm 𝒯:𝐾𝜕 → 𝔽(ℝ) is called preinvex-fm on the invex set 𝐾𝜕 w.r.t. bifunction 𝜕 if 𝒯(𝓊 + 𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝒯(𝓊)+̃𝜏𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1], where 𝜕:𝐾𝜕 × 𝐾𝜕 → ℝ. lemma 2.1 ([19]). let 𝐾𝜕 be an invex set w.r.t. 𝜕 and let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a fm parametrized by 𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯 ∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]},∀ 𝓊 ∈ 𝐾𝜕. then 𝒯 is preinvex-fm on 𝐾𝜕 if and only if, for all 𝛾 ∈ [0,1],𝒯∗(𝓊,𝛾) and 𝒯 ∗(𝓊,𝛾) are preinvex functions w.r.t. 𝜕 on 𝐾𝜕. definition 𝟐.𝟏𝟎 ([22]). a comparable fm 𝒯:𝐾𝜕 → 𝔽(ℝ) is called quasi-preinvex-fm on the invex set 𝐾𝜕 w.r.t. 𝜕 if 𝒯(𝓊 + 𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝒯(𝓊),𝒯(𝜗)), ∀ 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1]. definition 2.11 ([28]). a function 𝒯:𝐾 → ℝ is called exponentially convex on 𝐾, if 𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ (1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. the definition 2.11, can also be written in the following equivalent way, which is due to antczak [4]. definition 2.12. a function 𝒯:𝐾 → ℝ is called exponentially convex on 𝐾, if 𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ 𝑙𝑜𝑔[(1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗)], ∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. strictly exponentially convex function if strict inequality holds for 𝒯(𝓊) ≠ 𝒯(𝜗). 𝒯:𝐾 → 𝔽0 is called exponentially concave function if −𝒯 is exponentially convex on 𝐾. strictly exponentially concave function if strict inequality holds for 𝒯(𝓊) ≠ 𝒯(𝜗). antczak [4] and alirezaei and mathar [3 have discussed the applications of exponentially convex function in the mathematical programing and information theory. similarly, alirezaei and mazhar [3] explore the applications of exponentially concave function with following example in communication and information theory. example. the error function 𝑒𝑟𝑓(𝓊) = 2 √𝛱 ∫ 𝑒−𝑡 2 𝓊 0 𝑑𝓊, becomes an exponentially concave function in the form 𝑒𝑟𝑓(√𝓊), 𝓊 ≥ 0, which defines the bit/symbol error probability of communication system depending on the square root of the int. j. anal. appl. 19 (4) (2021) 525 underlying signal-to-noise ratio. this demonstrates that the exponentially concave functions in the theory communication and information theory may play an important role. definition 2.13 ([28]). a function 𝒯:𝐾 → ℝ is called exponentially quasi convex on 𝐾, if 𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ max (𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. from above definitions we can easily prove that each exponentially convex function is exponentially quasi convex function but the converse is not true such that 𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ (1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗), ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). definition 2.14. let 𝐾𝜕 be an invex set. then fm 𝒯:𝐾𝜕 → 𝔽0 is said to be: • exponentially preinvex-fm (exponentially preinvex-fm) on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ) if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.5) • exponentially preconcave-fm on 𝐾𝜕 if inequality (2.5) is reversed. • strictly exponentially preinvex-fm on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ) if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.6) • strictly exponentially preconcave-fm on 𝐾𝜕 if inequality (2.6) is reversed. remark 2.2. (i) if 𝒯 is exponentially preinvex-fm, then 𝜆𝒯 is also exponentially preinvex-fm for 𝜆 ≥ 0. (ii) if 𝒯(𝓊) and 𝐺(𝓊) both are exponentially preinvex-fms, then max(𝒯(𝓊),𝐺(𝓊)) is also exponentially preinvex-fms. special cases (iii) the exponentially preinvexity of 𝒯 on 𝐾𝜕 is equivalent to preinvexity of 𝑒 𝒯(𝓊). (iv) the exponentially prconcavity of 𝒯 on 𝐾𝜕 is equivalent to preconcavety of 𝑒 𝒯(𝓊). (v) by using remark (iii), if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊, then exponentially preinvex-fm becomes exponentially convex-fm, that is 𝑒𝒯(𝓊+𝜏(𝜗−𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (vi) if 𝜏 = 1 2 , then (2.5) becomes 𝑒 𝒯( 2𝓊+𝜕(𝜗,𝓊) 𝟐 ) ⪯ 𝑒𝒯(𝓊)+̃𝑒𝒯(𝜗) 2 ,∀ 𝓊,𝜗 ∈ 𝐾𝜕. (2.7) int. j. anal. appl. 19 (4) (2021) 526 the inequality (2.7) is known as exponentially jenson preinvex-fms. we can easily discuss the next result with the help of remark 2.2 (iii). theorem 2.2. let k∂ be an invex set w.r.t. ∂ and let 𝒯:k∂ → 𝔽0 be a fm parameterized by 𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯 ∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]},∀ 𝓊 ∈ k∂. then 𝒯 is exponentially preinvex-fm on k∂ if and only if, for all 𝛾 ∈ [0,1], 𝒯∗(𝓊,𝛾) and 𝒯 ∗(𝓊,𝛾) are exponentially preinvex function w.r.t. ∂. proof. the demonstration of proof is similar to lemma 2.1, by remark 2.2 (iii). example 2.1. we consider the fms 𝒯:(−1,1) → 𝔽0 defined by, 𝒯(𝓊)(σ) = { σ − 𝓊2 1 − 𝓊2 σ ∈ [𝓊2,1) 0 otherwise. , then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [𝛾 + (1 − 𝛾)𝓊 2,1 ]. since end point functions 𝒯∗(𝛾), 𝒯∗(𝛾) are exponentially preinvex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially preinvexfm w.r.t. bi-function ∂(ϑ,𝓊) = { ϑ2 − 𝓊2 2𝓊 ϑ2 − 𝓊2 < 0, 0 otherwise. example 2.2. we consider the fms 𝒯:(0,∞) → 𝔽0 defined by, 𝒯(𝓊)(σ) = { σ 𝓊2 σ ∈ [0,𝓊2], 2𝓊2 − σ 𝓊2 σ ∈ (𝓊2,2𝓊2], 0 otherwise. then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [𝛾𝓊 2, (2 − 𝛾)𝓊2]. since end point functions 𝒯∗(𝛾), 𝒯∗(𝛾) are exponentially preinvex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially preinvexfm w.r.t. ∂(ϑ,𝓊) = ϑ − 𝓊. definition 2.15. a comparable fm 𝒯:𝐾𝜕 → 𝔽0 is said to be: • exponentially quasi-preinvex-fm on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.8) • exponentially quasi-preconcave-fm on 𝐾𝜕 if inequality (2.8) is reversed. • strictly exponentially quasi-preinvex-fm on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.9) int. j. anal. appl. 19 (4) (2021) 527 • strictly exponentially quasi-preconcave-fm on 𝐾𝜕 if inequality (2.9) is reversed. proposition 2.1. let comparable fm 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-fm, such that 𝒯(𝜗) ≺ 𝒯(𝓊). then 𝒯 is strictly exponentially quasi-preinvex-fm. proof. let 𝑒𝒯(𝜗) ≺ 𝑒𝒯(𝓊) and 𝒯 be exponentially preinvex-fm. then, for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1] we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), since 𝒯(𝜗) ≺ 𝒯(𝓊), we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ 𝑒𝒯(𝓊) = max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). hence, 𝒯 is strictly exponentially quasi-preinvex-fm. definition 2.16. a fm 𝒯:𝐾𝜕 → 𝔽0 is said to be: • exponentially logarithmic preinvex-fm on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (𝑒𝒯(𝓊)) 1−𝜏 (𝑒𝒯(𝜗)) 𝜏 ,∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.10) • exponentially logarithmic preconcave-fm on 𝐾𝜕 if inequality (2.10) is reversed. • strictly exponentially logarithmic preinvex-fm on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (𝑒𝒯(𝓊)) 1−𝜏 (𝑒𝒯(𝜗)) 𝜏 ,∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. (2.11) • strictly exponentially logarithmic preconcave-fm on 𝐾𝜕 if inequality (2.11) is reversed, where 𝒯(.) ≻ 0̃. for further study, let 𝐾𝜕 be a nonempty closed invex set in ℝ. let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a fm and 𝜕:𝐾𝜕 × 𝐾𝜕 → ℝ be an arbitrary bifunction. we denote 〈. , . 〉 be the inner product. next, we will discuss exponentially preinvex-fms by using remark 2.2, (iii). 3. basic results in this section, we investigate some basic properties of exponentially preinvex-fms and obtain some characterization for exponentially preinvex-fms by using the concepts of level sets and epigraph of fms. definition 3.1. let 𝒯:𝒟 → 𝔽0 and 𝐺:𝒟 → 𝔽0 be two fms on 𝒟 ⊆ ℝ . then 𝒯(𝓊) and 𝐺(𝓊) are said to be fuzzy comonotonic on 𝒟 if, for all 𝓊,𝜗 ∈ 𝒟, [𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗)][𝑒𝐺(𝓊)−̃𝑒𝐺(𝜗)] ⪰ 0̃. int. j. anal. appl. 19 (4) (2021) 528 theorem 3.1. let 𝒯(𝓊) and 𝐺(𝓊) be two exponentially preinvex-fms on 𝐾𝜕 . then 𝒯(𝓊) = 𝒯(𝓊)+̃𝐺(𝓊) is exponentially preinvex-fm on 𝐾𝜕, provide that 𝒯(𝓊) and 𝐺(𝓊) are fuzzy comonotonic on 𝐾𝜕. proof. let 𝒯(𝓊) and 𝐺(𝓊) be two exponentially preinvex-fms on 𝐾𝜕. then, for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1], we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), (3.1) 𝑒𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝐺(𝓊)+̃𝜏𝑒𝐺(𝜗). (3.2) now, 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))+̃𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝑒𝐺(𝓊+𝜏𝜕(𝜗,𝓊)), ⪯ [(1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗)][(1 − 𝜏)𝑒𝐺(𝓊)+̃𝜏𝑒𝐺(𝜗)], = (1 − 𝜏)2𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃(1 − 𝜏)𝜏[𝑒𝒯(𝓊)+̃𝐺(𝜗)+̃𝑒𝒯(𝜗)+̃𝐺(𝓊)] +̃𝜏2𝑒𝒯(𝜗)+̃𝐺(𝜗). (3.3) since 𝒯(𝓊) and 𝐺(𝓊) are fuzzy comonotonic on 𝐾𝜕, such that [𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗)][𝑒𝐺(𝓊)−̃𝑒𝐺(𝜗)] ⪰ 0̃, which implies that 𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝑒𝒯(𝜗)+̃𝐺(𝜗) ⪰ 𝑒𝒯(𝓊)+̃𝐺(𝜗)+̃𝑒𝒯(𝜗)+̃𝐺(𝓊). (3.4) it follows by substituting (3.4) into (3.3) that 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))+̃𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)2𝑒𝒯(𝓊)+̃𝐺(𝓊) + (1 − 𝜏)𝜏[𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝑒𝒯(𝜗)+̃𝐺(𝜗)] +̃𝜏2𝑒𝒯(𝜗)+̃𝐺(𝜗), = (1 − 𝜏)𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝜏𝑒𝒯(𝜗)+̃𝐺(𝜗). hence, 𝒯(𝓊)+̃𝐺(𝓊) is exponentially preinvex-fm on 𝐾𝜕. theorem 3.2. let 𝒯:𝐾𝜕 → 𝔽0 be a fm, with 𝜃 = inf𝓊∈𝐾𝜕(𝑒 𝒯(𝓊)) exists in 𝔽0. (1) if 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-fm on 𝐾𝜕, then set 𝛺 = {𝓊:𝓊 ∈ 𝐾𝜕, 𝑒 𝒯(𝓊) = 𝜃} is an invex set. (2) if 𝒯:𝐾𝜕 → 𝔽0 is a strictly exponentially preinvex-fm on 𝐾𝜕, then 𝛺 is a singleton set or empty. that is, if 𝒯 is strictly exponentially preinvex-fm, then 𝒯 has at least one global minimum. int. j. anal. appl. 19 (4) (2021) 529 proof. (1). let 𝒯 be exponentially preinvex-fm. if 𝛺 is an empty set, then 𝛺 is an invex set. assume that, 𝓊,𝜗 ∈ 𝛺 , that is 𝓊,𝜗 ∈ 𝐾𝜕 and 𝑒 𝒯(𝓊) = 𝜃 = 𝑒𝒯(𝜗) . since 𝒯 is exponentially preinvex-fm, 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), = (1 − 𝜏)𝜃+̃𝜏𝜃 = 𝜃, for all 𝜏 ∈ (0,1). hence, all points of the form 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝛺, 𝜏 ∈ [0,1], which implies that 𝛺 is an invex set. (2). let 𝒯 be a strictly exponentially preinvex-fm on 𝐾𝜕. contrary suppose that there exist distinct points 𝓊,𝜗 ∈ 𝛺 such that 𝑒𝒯(𝓊) = 𝜃 = 𝑒𝒯(𝜗). since 𝒯 is strictly exponentially preinvexfm then, for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ (0,1), we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), = (1 − 𝜏)𝜃+̃𝜏𝜃 = 𝜃, which implies that 𝜃 ≠ inf𝓊∈𝐾𝜕(𝑒 𝒯(𝓊)), so this contradict the fact. hence, the result follows. theorem 3.3. let 𝐾𝜕 be a nonempty set. suppose that 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvexfm on 𝐾𝜕, with 𝜃 = inf𝓊∈𝐾𝜕(𝑒 𝒯(𝓊)) exists in [0,1], and that the set 𝛺 = {𝓊:𝓊 ∈ 𝐾𝜕,𝑒 𝒯(𝓊) = 𝜃} ≠ ∅. if 𝓊 is local minimum of 𝒯, then it is also a global minimum of 𝒯 on 𝐾𝜕. proof. let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-fm and 𝓊 is local minimum of 𝒯. if 𝓊 is not global minimum of 𝒯, then 𝓊 ∉ 𝛺. by hypothesis, 𝛺 ≠ ∅. if 𝜗 ∈ 𝛺, then we must have 𝒯(𝜗) ≺ 𝒯(𝓊). since 𝒯 is exponentially preinvex-fm then, for 𝜏 ∈ (0,1), we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ≺ 𝑒𝒯(𝓊), because 𝒯(𝜗) ≺ 𝒯(𝓊), for any small positive number 𝜏, and this contradiction proves the required result. theorem 3.4. let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially quasi-preinvex-fm on 𝐾𝜕. if 𝓊 ∈ 𝐾𝜕 is a strict local minimum of 𝒯, then it is also a strict global minimum of 𝒯 on 𝐾𝜕. proof. let 𝓊 ∈ 𝐾𝜕 is a strict local minimum of 𝒯. then by definition of strict local minimum, there exist a 𝛿 > 0 such that 𝒯(𝓊) ≺ 𝒯(𝜗) for all 𝜗 ∈ 𝐾𝜕 ∩ 𝐵𝛿(𝓊). assume contrary, that is, for some 𝜗 ∈ 𝐾𝜕, we have 𝒯(𝓊) ⪰ 𝒯(𝜗). since 𝒯 is an exponentially quasi-preinvex-fm then, for 𝜏 ∈ [0,1], we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) = 𝑒𝒯(𝓊), int. j. anal. appl. 19 (4) (2021) 530 since 𝒯(𝓊) ⪰ 𝒯(𝜗), we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑒𝒯(𝓊) this contradicts that 𝓊 ∈ 𝐾𝜕 is a strict local minimum of 𝒯, and hence the results follow. definition 3.2. a set 𝐾𝜕 ⊆ ℝ 2 is said to an invex set w.r.t. 𝜕, if (𝓊 + 𝜏𝜕(𝜗,𝓊), (1 − 𝜏)𝛽1+̃𝜏𝛽2) ∈ 𝐾𝜕, (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐾𝜕, 𝓊,𝜗 ∈ ℝ, 𝜏 ∈ [0,1]. the epigraph of exponentially preinvex-fm 𝒯:𝐾𝜕 → 𝔽0 can be given as 𝐸(𝒯) = {(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒 𝒯(𝓊) ⪯ 𝛽}. we now give a characterization of exponentially preinvex-fms in terms of its epigraph 𝐸(𝒯). theorem 3.5. a fm 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-fm w.r.t. 𝜕, if and only if, its epigraph is an invex set w.r.t. �̃�, where �̃�: 𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺 with �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1), for (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐸(𝒯). proof. let (𝓊,𝛽1) , (𝜗,𝛽2) ∈ 𝐸(𝒯) . then 𝑒 𝒯(𝓊) ⪯ 𝛽1 and 𝑒 𝒯(𝜗) ⪯ 𝛽2 . since fm 𝒯:𝐾𝜕 → 𝔽0 is exponentially preinvex-fm w.r.t. 𝜕, so for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1], we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ⪯ (1 − 𝜏)𝛽1+̃𝜏𝛽2, from which it follows that (𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝛽1+̃𝜏𝛽2) ∈ 𝐸(𝒯), which implies that (𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝛽1+̃𝜏𝛽2) = (𝓊,𝛽1)+ 𝜏(𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1), = (𝓊,𝛽1)+ 𝜏�̃�((𝜗,𝛽2),(𝓊,𝛽1)) ∈ 𝐸(𝒯). hence, 𝐸(𝒯) is an invex set w.r.t. �̃�:𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺, with �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1) for (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐸(𝒯). conversely, 𝐸(𝒯) is an invex set w.r.t. �̃�:𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺, with �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1) for (𝜗,𝛽2),(𝓊,𝛽1) ∈ 𝐸(𝒯). since (𝓊,𝑒 𝒯(𝓊)),(𝜗,𝑒𝒯(𝜗)) ∈ 𝐸(𝒯), we have for 𝜏 ∈ [0,1] (𝓊,𝑒𝒯(𝓊))+ 𝜏�̃�((𝜗,𝑒𝒯(𝜗)),(𝓊,𝑒𝒯(𝓊))) ∈ 𝐸(𝒯), since �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1) 𝛽1), then (𝓊,𝑒𝒯(𝓊)) + 𝜏(𝜕(𝜗,𝓊),𝑒𝒯(𝜗)+̃(−1)𝑒𝒯(𝓊)) ∈ 𝐸(𝒯), int. j. anal. appl. 19 (4) (2021) 531 which implies that (𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗)) ∈ 𝐸(𝒯), so we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗). hence, 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-fm w.r.t. 𝜕. theorem 3.6. let {𝒯𝑗}𝒋∈𝐼 :𝐾𝜕 → 𝔽0 be a family of exponentially preinvex-fms and bounded above on an invex set 𝐾𝜕. then the mapping 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially preinvex-fm on 𝐾𝜕. proof. since each 𝒯𝑗 is an exponentially preinvex-fm on the invex set 𝐾𝜕, its epigraph 𝐸(𝒯) = {(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒 𝒯𝑗(𝓊) ⪯ 𝛽,𝑗 ∈ 𝐼} is an invex set w.r.t. �̃�, by theorem 3.5. then their intersection ⋂ 𝐸(𝒯) = {(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒 𝒯𝑗(𝓊) ⪯ 𝛽,𝑗 ∈ 𝐼}𝒋∈𝐼 , is also an invex set w.r.t. 𝜕. hence, 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially preinvex-fm. definition 3.3. let 𝒯:𝐾𝜕 → 𝔽0 be an fm then the level set of 𝒯 is denoted by 𝒯𝛽 and defined as, 𝒯𝛽 = {𝓊:𝓊 ∈ 𝐾𝜕,𝑒 𝒯(𝓊) ⪯ 𝛽,𝛽 ∈ 𝔽0}. note that the level set is also called 𝛽-cut of 𝒯 and the set 𝒯𝛽 generalizes the standard form of 𝛽level of 𝒯. the set of all level sets of 𝒯 is represented as 𝐿(𝒯𝛽). theorem 3.7. let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-fm. then 𝒯𝛽 is an invex set. proof. let 𝓊,𝜗 ∈ 𝒯𝛽 . then 𝑒 𝒯(𝓊) ⪯ 𝛽 and 𝑒𝒯(𝜗) ⪯ 𝛽 . since 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-fm, 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ⪯ (1 − 𝜏)𝛽 + 𝜏𝛽 = 𝛽, hence 𝒯𝛽 is an invex set. theorem 3.8. the comparable fm 𝒯:𝐾𝜕 → 𝔽0 is said to be exponentially quasi-preinvex-fm if and only if, the level set 𝒯𝛽 is an invex set. proof. let 𝓊,𝜗 ∈ 𝒯𝛽. then by definition of 𝒯𝛽, we have 𝑒 𝒯(𝓊) ⪯ 𝛽 and 𝑒𝒯(𝜗) ⪯ 𝛽. since 𝓊,𝜗 ∈ 𝐾𝜕 and 𝐾𝜕 is invex set then, 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕. since 𝒯 is exponentially quasi-preinvex-fm then, 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊), 𝑒𝒯(𝜗)) ⪯ 𝑚𝑎𝑥(𝛽,𝛽) = 𝛽, ∀ 𝓊,𝜗 ∈ 𝑅, 𝜏 ∈ [0,1], from which we can note that 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝒯𝛽 is an invex set. int. j. anal. appl. 19 (4) (2021) 532 conversely, assume that 𝒯𝛽 be an invex set to prove 𝒯 is exponentially quasi-preinvex-fm. since 𝒯𝛽 is an invex set then, for any 𝓊,𝜗 ∈ 𝒯𝛽 such that 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝒯𝛽 with 𝜏 ∈ [0,1]. now we take 𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) = 𝛽 and 𝑒𝒯(𝓊) ⪰ 𝑒𝒯(𝜗). now by the definition of invex set 𝒯𝛽, we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) ⪯ 𝛽. hence, 𝒯 is exponentially quasi-preinvex-fm. lemma 3.1. for a comparable fm 𝒯:𝐾𝜕 → 𝔽0: exponentially logarithmic preinvexity ⇒ exponentially preinvexity ⇒ exponentially quasipreinvexity. proof. for 𝜏 ∈ [0,1], and 𝓊,𝜗 ∈ 𝐾𝜕, we have 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (𝑒𝒯(𝓊)) 1−𝜏 (𝑒𝒯(𝜗)) 𝜏 ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). theorem 3.9. let fm 𝒯:𝐾𝜕 → 𝔽0 be a exponentially logarithmic preinvex-fm. then its epigraph 𝐸(𝒯) is an invex set. proof. from lemma 3.1, it follows that exponentially logarithmic preinvex-fm 𝒯 is exponentially preinvex-fm. hence by theorem 3.5, epigraph 𝐸(𝒯) is an invex set. theorem 3.10. let {𝒯𝑗}𝒋∈𝐼 :𝐾𝜕 → 𝔽0 be a family of exponentially logarithmic preinvex-fms and bounded above on an invex set 𝐾𝜕. then the mapping 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially logarithmic preinvex-fm on 𝐾𝜕. proof. by using lemma 3.1 and theorem 3.6, the demonstration of proof is analogous to theorem 3.9. 4. exponentially invex fuzzy mappings in this section, we propose the concept of exponentially-ifms and some relations are established between exponentially preinvex-fm, exponentially-ifm and exponentially invex fuzzy monotone mapping. definition 4.1. a fm 𝒯:𝐾𝜕 → 𝔽(ℝ) is called sharply exponentially pseudo invex w.r.t. 𝜕 on 𝐾𝜕, if int. j. anal. appl. 19 (4) (2021) 533 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪰ 0̃ ⟹ 𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑒𝒯(𝜗), ∀ 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1], where 𝒯 ,, is g-differentiable of 𝒯 at 𝓊. theorem 4.1. let 𝒯 be a sharply exponentially pseudo preinvex-fm on 𝐾𝜕. then −̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝜗,𝓊)〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕. proof. let 𝒯 be a sharply exponentially pseudo preinvex-fm on 𝐾𝜕. then for all 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1], we have 𝑒𝒯(𝜗) ⪰ 𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊)), from which we get 𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝜗) 𝜏 ⪯ 0̃. taking limit in the above inequality as 𝜏 → 0, we obtain −̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝜗,𝓊)〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕. hence, the result follows. definition 4.2. a g-differentiable fm 𝒯:𝐾𝜕 → 𝔽(ℝ) is said to be: • invex w.r.t. 𝜕, if 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, (4.1) • strictly incave w.r.t. 𝜕, if inequality (4.1) is reversed. • invex w.r.t. 𝜕, if 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, (4.2) • strictly incave w.r.t. 𝜕, if inequality (4.2) is reversed. ∀ 𝓊,𝜗 ∈ 𝐾𝜕. example 4.1. we consider the fms 𝒯:(0,∞) → 𝔽(ℝ) defined by, 𝒯(𝓊)(σ) = { 4σ − 𝓊 𝓊 σ ∈ [ 1 4 𝓊,𝓊], 3𝓊 − 2σ 𝓊 σ ∈ (𝓊, 3 2 𝓊], 0 otherwise. then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [( 1 4 + 3 4 𝛾)𝓊,( 3 2 − 1 2 𝛾)𝓊]. since end point functions 𝒯∗(𝛾), 𝒯 ∗(𝛾) are exponentially invex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially-ifm w.r.t. ∂(ϑ,𝓊) = ϑ. int. j. anal. appl. 19 (4) (2021) 534 example 4.2. we consider the fms 𝒯:(0,∞) → 𝔽(ℝ) defined by, 𝒯(𝓊)(σ) = { −𝓊 + σ 𝓊 σ ∈ [𝓊,2𝓊], 4𝓊 − σ 2𝓊 σ ∈ (2𝓊,4𝓊], 0 otherwise. then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [(1 + 𝛾)𝓊,2(2 − 𝛾)𝓊]. since end point functions 𝒯∗(𝛾), 𝒯∗(𝛾) are exponentially invex functions for each 𝛾 ∈ [0,1], then 𝒯 is strictly exponentially-ifm w.r.t. ∂(ϑ,𝓊) = ϑ + 𝓊. theorem 4.2. let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a g-differentiable fm and let (i) 𝜕(𝜗,𝓊 + 𝜏𝜕(𝜗,𝓊)) = (1 − 𝜏)𝜕(𝜗,𝓊), (ii) 𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊)) = −𝜏𝜕(𝜗,𝓊). then 𝒯 is an exponentially preinvex-fm, if and only if 𝒯 is an exponentially-ifm. proof. let 𝒯:𝐾𝜕 → 𝔽(ℝ) be g-differentiable exponentially preinvex-fm. since 𝒯 is exponentially preinvex-fm then, for each 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1], therefore 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), = 𝑒𝒯(𝓊)+̃𝜏(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊)), which implies that 𝜏(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊)) ⪰ 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊), 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) 𝜏 . on taking limit in the above inequality as 𝜏 → 0, we obtain 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. conversely, let 𝒯 be an exponentially-ifm. since 𝐾𝜕 is an invex set so we have, 𝜗𝜏 = 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕 for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1]. taking 𝜗=𝜗𝜏 in (4.1), we get 𝑒𝒯(𝜗𝜏)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗𝜏,𝓊)〉. from (i) and (ii), we have 𝑒𝒯(𝜗𝜏)−̃𝑒𝒯(𝓊) ⪰ (1 − 𝜏)〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. (4.3) in a similar way, we get 𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗𝜏) ⪰ 〈𝑒𝒯(𝓊) 𝒯,(𝓊),𝜕(𝓊,𝜗𝜏)〉 = −𝜏〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝜗𝜏)〉, (4.4) multiplying (4.3) by 𝜏 and (4.4) by (1 − 𝜏), and adding the resultant, we have int. j. anal. appl. 19 (4) (2021) 535 𝑒𝒯(𝜗𝜏) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), which implies that 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗). hence, 𝒯 is exponentially preinvex-fm w.r.t. 𝜕. theorem 4.3. let 𝒯 be a g-differentiable exponentially preinvex-fm on the invex set 𝐾𝜕 and let (i) 𝜕(𝜗,𝓊 + 𝜏𝜕(𝜗,𝓊)) = (1 − 𝜏)𝜕(𝜗,𝓊), (ii) 𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊)) = −𝜏𝜕(𝜗,𝓊). then the followings are equivalent (1) 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, (4.5) (2) 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃〈𝑒𝒯(𝜗)𝒯 ,(𝜗),𝜕(𝓊,𝜗)〉 ⪯ 0̃, (4.6) for all 𝓊,𝜗 ∈ 𝐾𝜕. proof. (1) implies (2), let (1) holds. then, by replacing 𝜗 by 𝓊 and 𝓊 by 𝜗 in (4.5), we get 𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗) ⪰ 〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝓊,𝜗)〉, (4.7) adding (4.5) and (4.7), we have 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃〈𝑒𝒯(𝜗)𝒯 ,(𝜗),𝜕(𝓊,𝜗)〉 ⪯ 0̃, (2) implies (1), assume that (2) holds. then 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪯ −̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝓊,𝜗)〉, (4.8) since 𝐾𝜕 is an invex set so we have, 𝜗𝜏 = 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕 for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝑡 ∈ [0,1]. taking 𝜗=𝜗𝜏 in (4.8), we obtain 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊))〉 ⪯ −̃〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝓊 + 𝜏𝜕(𝜗,𝓊),𝓊)〉, from (i) and (ii), we have 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜏𝜕(𝜗,𝓊)〉 ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜏𝜕(𝜗,𝓊)〉, 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝜗,𝓊)〉 ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, (4.9) let ℋ(𝜏) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)). taking g-derivative w.r.t. 𝜏, we get ℋ ,(𝜏) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)).𝜕(𝜗,𝓊) = 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝜗,𝓊)〉. the above inequality along with (4.9) gives ℋ ,(𝜏) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, (4.10) by integrating (4.10) between 0 to 1 w.r.t. 𝜏, we get int. j. anal. appl. 19 (4) (2021) 536 ℋ(1)−̃ℋ(0) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, 𝑒𝒯(𝓊+𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. since 𝒯 is strongly exponentially preinvex-fm, then for 𝜏 = 1, we have 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, for all 𝓊,𝜗 ∈ 𝐾𝜕. note that, if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊 in (4.5), then we obtain the g-differentiable exponentially convexfm. consequently, the exponentially-ifm is more general than g-differentiable convex-fm. the exponentially-ifm type is equivalent to the type of fm whose stationary points are global minima. 5. applications in this section, by using the familiar fact that fuzzy variational inequality problem has deep relationship with the fuzzy optimization problems, we try to explore some applications of exponentially preinvex-fms in fuzzy optimization and prove that the minimum of pfms can be distinguish with variational-like inequality. theorem 5.1. let 𝒯 be an exponentially preinvex-fm on 𝐾𝜕. then 𝓊 ∈ 𝐾𝜕 is the minimum of the fm 𝒯, if and only if, for 𝓊 ∈ 𝐾𝜕 such that 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪰ 0̃, (5.1) for all 𝜗 ∈ 𝐾𝜕. proof. let 𝓊 ∈ 𝐾𝜕 be a minimum of 𝒯. then 𝒯(𝓊) ⪯ 𝒯(𝜗), ∀ 𝜗 ∈ 𝐾𝜕, from which, we have 𝑒𝒯(𝓊) ⪯ 𝑒𝒯(𝜗), ∀ 𝜗 ∈ 𝐾𝜕. (5.2) since 𝐾𝜕 is an invex set, for all 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1], 𝜗𝜏 = 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕. taking 𝜗 = 𝜗𝜏 in (5.2), we obtain 0̃ ⪯ 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) 𝜏 , taking limit in the above inequality as 𝜏 → 0, we get 0̃ ⪯ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. conversely, since 𝒯:𝐾𝜕 → 𝔽(ℝ) is an exponentially preinvex-fm, so 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), int. j. anal. appl. 19 (4) (2021) 537 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) 𝜏 , again taking limit in the above inequality as 𝜏 → 0, we get 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, the above inequality and (5.1) yield 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 0̃. hence, the result follows. remark 5.1. the inequality of the type (5.1) is called exponentially fuzzy variational-like inequality. we emphasize that if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊, the optimality conditions of the exponentially convex-fm can be characterized by the following inequality 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜗 − 𝓊〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕, which is called exponentially fuzzy variational inequality. we consider the functional 𝐼(𝜗), defined as 𝐼(𝜗) = 𝒯(𝜗)+̃𝒥(𝜗), ∀ 𝜗 ∈ ℝ, (5.3) where 𝒯 is a g-differentiable exponentially preinvex-fm and 𝒥 is a non g-differentiable exponentially preinvex-fm. since both 𝒯(𝜗) and 𝒥(𝜗) both are exponentially preinvex-fms, then by remark 2.2 (iii), (5.3) can be written as, 𝐼(𝜗) = 𝑒𝒯(𝜗)+̃𝑒𝒥(𝜗),∀ 𝜗 ∈ ℝ, (5.4) where 𝑒𝒯(𝜗) and 𝑒𝒥(𝜗) both are preinvex-fm w.r.t. same 𝜕. we know that the minimum of the functional 𝐼(𝜗), can be characterized by a class of variational like-inequalities. theorem 5.2. let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a g-differentiable exponentially preinvex-fm and 𝒥:𝐾𝜕 → 𝔽(ℝ) be a non g-differentiable exponentially preinvex-fm. then the functional 𝐼(𝜗) has minimum 𝓊 ∈ 𝐾𝜕, if and only if, 𝓊 ∈ 𝐾𝜕 satisfies 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊) ⪰ 0̃, ∀ 𝜗 ∈ 𝐾𝜕. (5.5) proof. let 𝓊 ∈ 𝐾𝜕 be the minimum of 𝐼 then by definition, for all 𝜗 ∈ 𝐾𝜕 we have 𝐼(𝓊) ⪯ 𝐼(𝜗). (5.6) since 𝐾𝜕 is an invex set so 𝜗𝜏 = 𝓊 + 𝜏𝜕(𝜗,𝓊), for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1]. replacing 𝜗 by 𝜗𝜏 in (5.6), we get int. j. anal. appl. 19 (4) (2021) 538 𝐼(𝓊) ⪯ 𝐼(𝓊 + 𝜏𝜕(𝜗,𝓊)), which implies that, using (5.4), we have 𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊) ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))+̃𝑒𝒥((𝓊+𝜏𝜕(𝜗,𝓊)) . since 𝒥 is exponentially preinvex-fm then, 𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊) ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))+̃(1 − 𝜏)𝑒𝒥(𝓊)+̃𝜏𝑒𝒥(𝜗), that is 0̃ ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)+̃𝜏(𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊)). on dividing by “𝜏” and taking lim 𝜏→0 , we get 0̃ ⪯ lim 𝜏→0 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) 𝜏 +̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊). conversely, let (5.5) be satisfy to prove 𝓊 ∈ 𝐾𝜕 is a minimum of 𝐼. assume that 0̃ ⪯ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊). for all 𝜗 ∈ 𝐾𝜕, we have 𝐼(𝓊)−̃𝐼(𝜗) = 𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊)−̃𝑒𝒯(𝜗)−̃𝑒𝒥(𝜗), = −̃(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊))+̃𝑒𝒥(𝓊)−̃𝑒𝒥(𝜗), using theorem 4.2, we obtain 𝐼(𝓊)−̃𝐼(𝜗) ⪯ −̃[〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉−̃𝑒𝒥(𝜗)+̃𝑒𝒥(𝓊)] ⪯ 0̃, from (5.5), we have 𝐼(𝓊)−̃𝐼(𝜗) ⪯ 0̃, hence, 𝐼(𝓊) ⪯ 𝐼(𝜗). note that the (5.5) is called exponentially fuzzy mixed variational like-inequality. this result shows that the fuzzy optimality conditions of fuzzy functional 𝐼(𝜗) can be characterized by exponentially fuzzy mixed variational-like inequalities. if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊, then from (5.1) and (5.2), we can obtain the family of some new classes. 𝟔. conclusion convex and generalized convex-fms play an important role in fuzzy optimization. therefore, by the importance of nonconvex-fms, we have introduced and considered a class of nonconvex-fms is called exponentially preinvex-fms. it is illustrated that classical convexity int. j. anal. appl. 19 (4) (2021) 539 and nonconvexity are special cases of exponentially preinvex-fms. we have also introduced the notions of quasi-preinvex and log-preinvex-fms and investigated some properties. some relations among the exponentially preinvex-fms, exponentially invex-fms, and exponentially monotonicities are derived under some mild conditions. we have proved that optimality conditions of g-differentiable exponentially preinvex-fms and for the sum of g-differentiable preinvex-fms and non g-differentiable exponentially preinvex-fms can be characterized by exponentially fuzzy variational-like inequalities and exponentially fuzzy mixed variational-like inequalities, respectively. the inequalities (5.1) and (5.2) are the interesting outcome of our main results. it is itself an engaging problem to flourish some well-organized some numerical methods for solving exponentially fuzzy variational-like inequalities and exponentially fuzzy mixed variational-like inequalities together with applications in applied and pure sciences. in the future, we will try to investigate the applications of exponentially fuzzy variational-like inequalities and exponentially fuzzy mixed variational-like inequalities in existence theory. we hope that these concepts and applications will be helpful for other authors to pay their roles in different fields of sciences. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. adamek, on a problem connected with strongly convex functions, math. inequal. appl. 19 (2016), 1287–1293. [2] n. i. akhiezer, the classical moment problem and some related questions in analysis, oliver and boyd, edinburgh, 1965. [3] g. alirezaei, r. mazhar, on exponentially concave functions and their impact in information theory, inform. theory appl. workshop, 9 (2018), 265–274. [4] t. antczak, (p,r)-invex sets and functions, j. optim. theory appl. 263 (2001), 355–379. [5] m. avriel, r-convex functions, math. program. 2 (1972), 309–323. [6] m. u. awan, m. a. noor, e. set, et al. on strongly (p, h)-convex functions, twms j. pure appl. math. 9 (2019), 145-153. int. j. anal. appl. 19 (4) (2021) 540 [7] a. azcar, j. gimnez, k. nikodem, j. l. sánchez, on strongly midconvex functions, opuscula math. 31 (2011), 15–26. [8] b. bede, s. g. gal, generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, fuzzy sets syst. 151(3) (2005), 581-599. [9] a. ben-isreal, b. mond, what is invexity? anziam j. 28 (1986), 1–9. [10] s. n. bernstein, sur les fonctions absolument monotones, acta math. 52 (1929), 1–66. [11] s. s. l. chang, l. a. zadeh, on fuzzy mapping and control, ieee trans. syst., man, cybern. smc-2 (1972), 30–34. [12] s. s. chang, variational inequality and complementarity problems theory and applications. shanghai scientific and technological literature publishing house, shanghai, (1991). [13] d. dubois, h. prade, operations on fuzzy numbers. int. j. syst. sci. 9 (1978), 613-626. [14] n. furukawa, convexity and local lipschitz continuity of fuzzy-valued mappings. fuzzy sets syst. 93 (1998), 113-119. [15] jr, r. goetschel, w. voxman, elementary fuzzy calculus. fuzzy sets syst. 18 (1986), 31-43. [16] m. a. hanson, on sufficiency of the kuhn-tucker conditions, j. math. anal. appl. 80 (1980), 545–550. [17] m. v. jovanovic, a note on strongly convex and strongly quasi convex functions, math. notes, 60 (1966), 584–585. [18] s. karamardian, the nonlinear complementarity problem with applications, part 2. j. optim. theory appl. 4 (1969), 167–181. [19] j. li, m. a. noor, on characterizations of preinvex fuzzy mappings, computers math. appl. 59 (2010), 933-940. [20] m. s. mohan, s. k. neogy, on invex sets and preinvex functions, j. math. anal. appl. 189 (1995), 901– 908. [21] s. nanda, k. kar, convex fuzzy mappings. fuzzy sets syst. 48 (1992), 129-132. [22] m. a. noor, fuzzy preinvex functions. fuzzy sets syst. 64 (1994), 95-104. [23] m. a. noor, variational-like inequalities, optimization, 30 (1994), 323–330. [24] m. a. noor, variational inequalities for fuzzy mappings. (iii), fuzzy sets syst. 110 (2000), 101–108. [25] m. a. noor, k. i. noor, exponentially convex functions, j. orissa math. soc. 39 (2019), 33-51. [26] m. a. noor, k. i. noor, strongly exponentially convex functions, upb sci. bull. ser. a: appl. math. phys. 81 (2019), 75-84. [27] m. a. noor, k. i. noor, strongly exponentially convex functions and their properties, j. adv. math. stud. 12 (2019), 177–185. int. j. anal. appl. 19 (4) (2021) 541 [28] m. a. noor, k. i. noor, some properties of exponentially preinvex functions, facta univ. (nis) ser. math. inform. 34 (2019), 941-955. [29] a. rufián-lizana, y. chalco-cano, r. osuna-gómez, g. ruiz-garzón, on invex fuzzy mappings and fuzzy variational-like inequalities, fuzzy sets syst. 200 (2012), 84-98. [30] a. rufián-lizana, y. chalco-cano, g. ruiz-garzón, h. román-flores, on some characterizations of preinvex fuzzy mappings, top, 22 (2014), 771-783. [31] y. r. syau, on convex and concave fuzzy mappings. fuzzy sets syst. 103 (1999), 163-168. [32] l. stefanini, b. bede, generalized hukuhara differentiability of interval-valued functions and interval differential equations, nonlinear anal., theory meth. appl. 71 (2009), 1311-1328. [33] z. wu, j. xu, generalized convex fuzzy mappings and fuzzy variational-like inequality, fuzzy sets syst. 160 (2009), 1590-1619. [34] y. x. zhao, s. y. wang, l. coladas uria, characterizations of r-convex functions, j. optim. theory appl. 145 (2010), 186–195. international journal of analysis and applications volume 16, number 5 (2018), 733-750 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-733 cubic graphs with application sheikh rashid1, naveed yaqoob2,∗, muhammad akram3, muhammad gulistan1 1department of mathematics, hazara university, mansehra, pakistan 2department of mathematics, college of science al-zulfi, majmaah university, al-zulfi, saudi arabia 3department of mathematics, university of the punjab, new campus, lahore, pakistan ∗corresponding author: nayaqoob@ymail.com abstract. we introduce certain concepts, including cubic graphs, internal cubic graphs, external cubic graphs, and illustrate these concepts by examples. we deal with fundamental operations, cartesian product, composition, union and join of cubic graphs. we discuss some results of internal cubic graphs and external cubic graphs. we also describe an application of cubic graphs. 1. introduction cubic sets are one of the real generalizations of fuzzy sets [27] provided by jun et al. [9–11, 15, 26] during the last five years. they developed cubic set theory in many directions and for more detail about cubic sets one can see [12]. kang and kim [13] studied mappings of cubic sets. muhiuddin et al. [18] presented the idea of stable cubic sets. fuzzy graphs were studied by rosenfeld [23] and give a few theoretical ideas in spite of the fact that the fundamental thought was presented by kauffmann [14] in 1973. bhattacharya [6] gave some remarks on fuzzy graphs. a book written by mordeson and nair [17] is devoted especially to the study of fuzzy graphs received 2018-03-13; accepted 2018-05-22; published 2018-09-05. 2010 mathematics subject classification. 68r10, 05c72. key words and phrases. cubic sets; cubic graphs; internal and external cubic graphs. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 733 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-733 int. j. anal. appl. 16 (5) (2018) 734 and fuzzy hypergraphs. akram et al. gave the idea of interval-valued fuzzy graphs [1, 2], intuitionistic fuzzy graphs [3] and bipolar fuzzy graphs [4, 5]. borzooei and rashmanlou [7] studied cayley interval-valued fuzzy threshold graphs. buckley [8] introduced self-centered graphs. sunitha et al. [25] characterized g-self centered fuzzy graphs. mishra et al. [16] studied coherent category of interval-valued intuitionistic fuzzy graphs. pal et al. [19] and pramanik et al. [21, 22] added some useful results to the theory of interval-valued fuzzy graphs. parvathi et al. [20] provided some different operations on intuitionistic fuzzy graphs and sahoo and pal [24] studied product of intuitionistic fuzzy graphs. in this paper we study some operations on cubic graphs. internal and external cubic graphs are studied with some example. we provided some conditions for union and join of external and internal cubic graphs. 2. preliminaries here we recall some basic helping material from the existing literature. definition 2.1. a graph is denoted by ω∗ = (p,q), where p denotes the set of vertices of ω∗ and q stands for the set of edges of ω∗. definition 2.2. [12] let t be a non-empty set. by a cubic set in t we mean a structure λ = {〈t,$̃λ(t),µλ(t)〉 |t ∈ t} in which $̃λ is an interval-valued fuzzy set in t and µλ is a fuzzy set in t. a cubic set λ = {〈t,$̃λ(t),µλ(t)〉 |t ∈ t} is simply denoted by λ = 〈$̃λ,µλ〉. definition 2.3. [12] let t be a non-empty set. a cubic set λ = 〈$̃λ,µλ〉 in t is said to be an internal cubic (resp., external cubic) set if $−λ (t) ≤ µλ(t) ≤ $ + λ (t) (resp., µλ(t) /∈ ($ − λ (t),$ + λ (t))) for all t ∈ t. definition 2.4. [12] for any λi = {〈t,$̃λi (t),µλi (t)〉 |t ∈ t} where i ∈ i, we define (a) ∪p i∈i λi = {〈 t, ( ∪ i∈i $̃λi ) (t), ( ∨ i∈i µλi ) (t) 〉 |t ∈ t } (p-union) (b) ∪r i∈i λi = {〈 t, ( ∪ i∈i $̃λi ) (t), ( ∧ i∈i µλi ) (t) 〉 |t ∈ t } (r-union) 3. cubic graphs we develop the theory of a cubic graph and some operations on cubic graph. int. j. anal. appl. 16 (5) (2018) 735 definition 3.1. let m ∗ = 〈p,q〉 be a graph. a cubic graph of a graph m ∗ = 〈p,q〉 , is the structure m = 〈α,β〉 , where α = 〈$̃α,µα〉 is the cubic set representation for the vertex p and β = 〈$̃β,µβ〉 denotes the cubic set representation for the edge q, with $̃α : p → d[0, 1], µα : p → [0, 1], and $̃β : q → d[0, 1], µβ : q → [0, 1], such that $̃β(pipj) � r min{$̃α(pi),$̃α(pj)}, µβ(pipj) ≤ max{µα(pi),µα(pj)}, for all (pi,pj) ∈ q ⊆ p ×p. example 3.1. let us consider a graph ω∗ = (p,q) such that p = {p1,p2,p3,p4}, q = {p1p2,p2p3,p3p4,p4p1}. let α be a cubic set of p and let β be a cubic set of q defined by p $̃α µα p1 [0.1, 0.5] 0.7 p2 [0.3, 0.7] 0.2 p3 [0.2, 0.4] 0.2 p4 [0.1, 0.8] 0.7 q $̃β µβ p1p2 [0.1, 0.4] 0.4 p2p3 [0.1, 0.3] 0.1 p3p4 [0.1, 0.4] 0.5 p4p1 [0.1, 0.4] 0.3 figure 1. cubic graph by routine calculations, it can be observed that the graph shown in fig. 1 is a cubic graph. example 3.2. consider a graph ω∗ = (p,q). let α be a cubic set of p and let β be a cubic set of q defined by µα(pi) = $−α (pi) + $ + α (pi) 2 and µβ(ei) = $−β (ei) + $ + β (ei) 2 . then m = 〈α,β〉 is a cubic graph of ω∗. int. j. anal. appl. 16 (5) (2018) 736 remark 3.1. if $̃β(pipj) = [0, 0] and µβ(pipj) = 0, then the cubic graph m = 〈α,β〉 has no edge. definition 3.2. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the cartesian product of m1 and m2 is denoted by m1 × m2 = 〈α1 ×α2,β1 ×β2〉 and is defined as follows: (i)   ($̃α1 × $̃α2 )(p1,p2) = r min{$̃α1 (p1),$̃α2 (p2)}(µα1 ×µα2 )(p1,p2) = max{µα1 (p1),µα2 (p2)} for all (p1,p2) ∈ p = p1 ×p2, (ii)   ($̃β1 × $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)}(µβ1 ×µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)} for all q ∈ p1, and q2p2 ∈ q2, (iii)   ($̃β1 × $̃β2 )((q1,r)(p1,r)) = r min{$̃β1 (q1p1),$̃α2 (r)}(µβ1 ×µβ2 )((q1,r)(p1,r)) = max{µβ1 (q1p1),µα2 (r)} for all r ∈ p2, and q1p1 ∈ q1. example 3.3. consider two cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 2. figure 2. cubic graphs m1 and m2 then, their corresponding cartesian product m1 × m2 is shown in figure 3. figure 3. cubic graph m1 × m2 clearly, m1 × m2 is a cubic graph. proposition 3.1. the cartesian product of two cubic graphs is a cubic graph. int. j. anal. appl. 16 (5) (2018) 737 proof. the conditions for α1 ×α2 are obvious, therefore, we verify only conditions for β1 ×β2. let q ∈ p1, and q2p2 ∈ q2. then ($̃β1 × $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)} � r min{$̃α1 (q),r min{$̃α2 (q2),$̃α2 (p2)}} = r min{r min{$̃α1 (q),$̃α2 (q2)},r min{$̃α1 (q),$̃α2 (p2)}} = r min{($̃α1 × $̃α2 )(q,q2), ($̃α1 × $̃α2 )((q,p2)} (µβ1 ×µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)} ≤ max{µα1 (q), max{µα2 (q2),µα2 (p2)}} = max{max{µα1 (q),µα2 (q2)}, max{µα1 (q),µα2 (p2)}} = max{(µα1 ×µα2 )(q,q2), (µα1 ×µα2 )((q,p2)} similarly, we can prove it for r ∈ p2, and q1p1 ∈ q1. � definition 3.3. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the composition of m1 and m2 is denoted by m1[m2] = 〈α1 ◦α2,β1 ◦β2〉 and is defined as follows: (i)   ($̃α1 ◦ $̃α2 )(p1,p2) = r min{$̃α1 (p1),$̃α2 (p2)}(µα1 ◦µα2 )(p1,p2) = max{µα1 (p1),µα2 (p2)} for all (p1,p2) ∈ p = p1 ×p2, (ii)   ($̃β1 ◦ $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)}(µβ1 ◦µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)} for all q ∈ p1, and q2p2 ∈ q2, (iii)   ($̃β1 ◦ $̃β2 )((q1,r)(p1,r)) = r min{$̃β1 (q1p1),$̃α2 (r)}(µβ1 ◦µβ2 )((q1,r)(p1,r)) = max{µβ1 (q1p1),µα2 (r)} for all r ∈ p2, and q1p1 ∈ q1. (iv)   ($̃β1 ◦ $̃β2 )((q1,q2)(p1,p2)) = r min{$̃α2 (q2),$̃α2 (p2),$̃β1 (q1p1)}(µβ1 ◦µβ2 )((q1,q2)(p1,p2)) = max{µα2 (q2),µα2 (p2),µβ1 (q1p1)} for all q2,p2 ∈ p2, q2 6= p2 and q1p1 ∈ q1. example 3.4. from example 3.3, consider two cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 2. then, their corresponding composition m1[m2] is shown in figure 4. int. j. anal. appl. 16 (5) (2018) 738 figure 4. cubic graph m1[m2] clearly, m1[m2] is a cubic graph. proposition 3.2. the composition of two cubic graphs is a cubic graph. definition 3.4. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the p-union of two cubic graphs m1 and m2 is denoted by m1 ∪p m2 = 〈α1 ∪p α2,β1 ∪p β2〉 and is defined as follows: (i) ($̃α1 ∪p $̃α2 )(p) =   $̃α1 (p) if p ∈ p1 −p2 $̃α2 (p) if p ∈ p2 −p1 r max{$̃α1 (p),$̃α2 (p)} if p ∈ p1 ∩p2 (ii) (µα1 ∪p µα2 )(p) =   µα1 (p) if p ∈ p1 −p2 µα2 (p) if p ∈ p2 −p1 max{µα1 (p),µα2 (p)} if p ∈ p1 ∩p2 (iii) ($̃β1 ∪p $̃β2 )(pipj) =   $̃β1 (pipj) if pipj ∈ q1 −q2 $̃β2 (pipj) if pipj ∈ q2 −q1 r max{$̃β1 (pipj),$̃β2 (pipj)} if pipj ∈ q1 ∩q2 (iv) (µβ1 ∪p µβ2 )(pipj) =   µβ1 (pipj) if pipj ∈ q1 −q2 µβ2 (pipj) if pipj ∈ q2 −q1 max{µβ1 (pipj),µβ2 (pipj)} if pipj ∈ q1 ∩q2 definition 3.5. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the r-union of two cubic graphs m1 and m2 is denoted by m1 ∪r m2 = 〈α1 ∪r α2,β1 ∪r β2〉 and is defined as follows: (i) ($̃α1 ∪r $̃α2 )(p) =   $̃α1 (p) if p ∈ p1 −p2 $̃α2 (p) if p ∈ p2 −p1 r max{$̃α1 (p),$̃α2 (p)} if p ∈ p1 ∩p2 int. j. anal. appl. 16 (5) (2018) 739 (ii) (µα1 ∪r µα2 )(p) =   µα1 (p) if p ∈ p1 −p2 µα2 (p) if p ∈ p2 −p1 min{µα1 (p),µα2 (p)} if p ∈ p1 ∩p2 (iii) ($̃β1 ∪r $̃β2 )(pipj) =   $̃β1 (pipj) if pipj ∈ q1 −q2 $̃β2 (pipj) if pipj ∈ q2 −q1 r max{$̃β1 (pipj),$̃β2 (pipj)} if pipj ∈ q1 ∩q2 (iv) (µβ1 ∪r µβ2 )(pipj) =   µβ1 (pipj) if pipj ∈ q1 −q2 µβ2 (pipj) if pipj ∈ q2 −q1 min{µβ1 (pipj),µβ2 (pipj)} if pipj ∈ q1 ∩q2 example 3.5. consider two cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 5. figure 5. cubic graphs m1 and m2 then, their corresponding p-union m1 ∪p m2 is shown in figure 6. figure 6. cubic graph m1 ∪p m2 also, their corresponding r-union m1 ∪r m2 is shown in figure 7. int. j. anal. appl. 16 (5) (2018) 740 figure 7. cubic graph m1 ∪r m2 clearly, m1 ∪p m2 and m1 ∪r m2 are cubic graphs. proposition 3.3. the p-union and r-union of two cubic graphs is a cubic graph. proof. since all the conditions for α1∪pα2 are automatically satisfied therefore, we verify only conditions for β1 ∪p β2. in the case, when qp ∈ q1 ∩q2. then ($̃β1 ∪p $̃β2 )(qp) = r max{$̃β1 (qp),$̃β2 (qp)} � r max{r min{$̃α1 (q),$̃α1 (p)},r min{$̃α2 (q),$̃α2 (p)}} = r min{r max{$̃α1 (q),$̃α2 (q)},r max{$̃α1 (p),$̃α2 (p)}} = r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)}. (µβ1 ∪p µβ2 )(qp) = max{µβ1 (qp),µβ2 (qp)} ≤ max{max{µα1 (q),µα1 (p)}, max{µα2 (q),µα2 (p)}} = max{max{µα1 (q),µα2 (q)}, max{µα1 (p),µα2 (p)}} = max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}. if qp ∈ q1 and qp /∈ q2, then ($̃β1 ∪p $̃β2 )(qp) � r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)} (µβ1 ∪p µβ2 )(qp) ≤ max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}. if qp ∈ q2 and qp /∈ q1, then ($̃β1 ∪p $̃β2 )(qp) � r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)} (µβ1 ∪p µβ2 )(qp) ≤ max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}. int. j. anal. appl. 16 (5) (2018) 741 hence the p-union of two cubic graphs is a cubic graph. the case for r-union of two cubic graphs can be seen in a similar way. � definition 3.6. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the p-join of two cubic graphs m1 and m2 is denoted by m1 +p m2 = 〈α1 +p α2,β1 +p β2〉 and is defined as follows: (i)   ($̃α1 +p $̃α2 )(p) = ($̃α1 ∪p $̃α2 )(p)(µα1 +p µα2 )(p) = (µα1 ∪p µα2 )(p) for p ∈ p1 ∪p2, (ii)   ($̃β1 +p $̃β2 )(qp) = ($̃β1 ∪p $̃β2 )(qp)(µβ1 +p µβ2 )(qp) = (µβ1 ∪p µβ2 )(qp) for qp ∈ q1 ∩q2, (iii)   ($̃β1 +p $̃β2 )(qp) = r min{$̃α1 (q),$̃α2 (p)}(µβ1 +p µβ2 )(qp) = min{µα1 (q),µα2 (p)} for qp ∈ q∗, where q∗ is the set of all edges joining the vertices of p1 and p2. definition 3.7. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two cubic graphs of the graphs ω∗1 and ω∗2, respectively. the r-join of two cubic graphs m1 and m2 is denoted by m1 +r m2 = 〈α1 +r α2,β1 +r β2〉 and is defined as follows: (i)   ($̃α1 +r $̃α2 )(p) = ($̃α1 ∪r $̃α2 )(p)(µα1 +r µα2 )(p) = (µα1 ∪r µα2 )(p) for p ∈ p1 ∪p2, (ii)   ($̃β1 +r $̃β2 )(qp) = ($̃β1 ∪r $̃β2 )(qp)(µβ1 +r µβ2 )(qp) = (µβ1 ∪r µβ2 )(qp) for qp ∈ q1 ∩q2, (iii)   ($̃β1 +r $̃β2 )(qp) = r min{$̃α1 (q),$̃α2 (p)}(µβ1 +r µβ2 )(qp) = max{µα1 (q),µα2 (p)} for qp ∈ q∗, where q∗ is the set of all edges joining the vertices of p1 and p2. example 3.6. consider two cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 8. figure 8. cubic graphs m1 and m2 int. j. anal. appl. 16 (5) (2018) 742 then, their corresponding p-join m1 +p m2 is shown in figure 9. figure 9. cubic graph m1 +p m2 also, their corresponding r-join m1 +r m2 is shown in figure 10. figure 10. cubic graph m1 +r m2 clearly, m1 +p m2 and m1 +r m2 are cubic graphs. proposition 3.4. the p-join and r-join of two cubic graphs is a cubic graph. 4. internal and external cubic graphs here in this section we discuss some results related with internal and external cubic graphs. definition 4.1. a cubic graph m = 〈α,β〉 is said to be an (i) internal cubic graph (ic-graph) if µα(pi) ∈ [$−α (pi),$ + α (pi)] and µβ(ei) ∈ [$ − β (ei),$ + β (ei)] int. j. anal. appl. 16 (5) (2018) 743 for each pi ∈ p and ei ∈ q. (ii) external cubic graph (ec-graph) if µα(pi) /∈ ($−α (pi),$ + α (pi)) and µβ(ei) /∈ ($ − β (ei),$ + β (ei)) for each pi ∈ p and ei ∈ q. example 4.1. the cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 are internal and external cubic graphs, respectively, as shown in figure 11. figure 11. ic-graph m1 and ec-graph m2 theorem 4.1. let {mi = 〈αi,βi〉 |i ∈ i} be a family of ic-graphs. then ∪p i∈i mi is an ic-graph. proof. since mi is an ic-graph, we have $ − α (p) ≤ µα(p) ≤ $+α (p) and $ − β (e) ≤ µβ(e) ≤ $ + β (e) for i ∈ i. this implies that ( ∪ i∈i $−α ) (p) ≤ ( ∨ i∈i µα ) (p) ≤ ( ∪ i∈i $+α ) (p), and ( ∪ i∈i $−β ) (e) ≤ ( ∨ i∈i µβ ) (e) ≤ ( ∪ i∈i $+β ) (e). hence ∪p i∈i mi is an ic-graph. � the following example shows that the r-union of ic-graphs need not be an ic-graph (ec-graph). example 4.2. consider two ic-graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 12. int. j. anal. appl. 16 (5) (2018) 744 figure 12. ic-graphs m1 and m2 then, their corresponding r-union m1 ∪r m2 is shown in figure 13. figure 13. r-union of ic-graphs m1 and m2 it is easy to see that the cubic graph m1 ∪r m2 is neither ic-graph nor ec-graph. we provide a condition for the r-union of two ic-graphs to be an ic-graph. theorem 4.2. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be ic-graphs such that max{$−α1 (p),$ − α2 (p)}≤ min{µα1 (p),µα2 (p)} and max{$−β1 (e),$ − β2 (e)}≤ min{µβ1 (e),µβ2 (e)} for all p ∈ p and e ∈ q. then the r-union of two ic-graphs m1 and m2 is an ic-graph. proof. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two ic-graphs which satisfy the conditions max{$−α1 (p),$ − α2 (p)}≤ min{µα1 (p),µα2 (p)} and max{$−β1 (e),$ − β2 (e)}≤ min{µβ1 (e),µβ2 (e)} int. j. anal. appl. 16 (5) (2018) 745 for all p ∈ p and e ∈ q. since µα1 (p) ∈ [$−α1 (p),$ + α1 (p)], µβ1 (e) ∈ [$ − β1 (e),$+β1 (e)] and µα2 (p) ∈ [$−α2 (p),$ + α2 (p)], µβ2 (e) ∈ [$ − β2 (e),$+β2 (e)]. this implies that min{µα1 (p),µα2 (p)}≤ ($ + α1 ∪$+α2 )(p) and min{µβ1 (e),µβ2 (e)}≤ ($ + β1 ∪$+β2 )(e) thus from the given condition we get ($−α1 ∪$ − α2 )(p) = max{$−α1 (p),$ − α2 (p)}≤ min{µα1 (p),µα2 (p)}≤ ($ + α1 ∪$+α2 )(p), and ($−β1 ∪$ − β2 )(e) = max{$−β1 (e),$ − β2 (e)}≤ min{µβ1 (e),µβ2 (e)}≤ ($ + β1 ∪$+β2 )(e). this shows that m1 ∪r m2 is an ic-graph. � the following example shows that the p-union and r-union of ec-graphs need not be an ec-graph (ic-graph). example 4.3. consider two ec-graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 14. figure 14. ec-graphs m1 and m2 then, their corresponding p-union m1 ∪p m2 is shown in figure 15. figure 15. p-union of ec-graphs m1 and m2 also, the corresponding r-union m1 ∪r m2 is shown in figure 16. int. j. anal. appl. 16 (5) (2018) 746 figure 16. r-union of ec-graphs m1 and m2 it is easy to see that the cubic graph m1 ∪p m2 and m1 ∪r m2 are neither ec-graph nor ic-graph. we provide a condition for the p-union of two ec-graphs to be an ec-graph. theorem 4.3. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two ec-graphs such that min   max{$ + α1 (p),$−α2 (p)}, max{$−α1 (p),$ + α2 (p)}   > max{µα1 (p),µα2 (p)} ≥ max   min{$ + α1 (p),$−α2 (p)}, min{$−α1 (p),$ + α2 (p)}   and min   max{$ + β1 (e),$−β2 (e)}, max{$+β1 (e),$ − β2 (e)}   > max{µβ1 (e),µβ2 (e)} ≥ max   min{$ + β1 (e),$−β2 (e)}, min{$+β1 (e),$ − β2 (e)}   for all p ∈ p and e ∈ q. then the p-union of two ec-graphs is an ec-graph. we provide a condition for the r-union of two ec-graphs to be an ec-graph. theorem 4.4. let m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 be two ec-graphs such that min   max{$ + α1 (p),$−α2 (p)}, max{$−α1 (p),$ + α2 (p)}   > min{µα1 (p),µα2 (p)} ≥ max   min{$ + α1 (p),$−α2 (p)}, min{$−α1 (p),$ + α2 (p)}   int. j. anal. appl. 16 (5) (2018) 747 and min   max{$ + β1 (e),$−β2 (e)}, max{$+β1 (e),$ − β2 (e)}   > min{µβ1 (e),µβ2 (e)} ≥ max   min{$ + β1 (e),$−β2 (e)}, min{$+β1 (e),$ − β2 (e)}   for all p ∈ p and e ∈ q. then the r-union of two ec-graphs is an ec-graph. theorem 4.5. let m = 〈α,β〉 be a cubic graph which is not an ec-graph. then there exist pi ∈ p and ei ∈ q such that µα(pi) ∈ ($−α (pi),$ + α (pi)) and µβ(ei) ∈ ($ − β (ei),$ + β (ei)). proof. straightforward. � theorem 4.6. let m = 〈α,β〉 be a cubic graph of ω∗. if m = 〈α,β〉 is both an ic-graph and an ec-graph, then µα(pi) ∈ u($̃α) ∪l($̃α) and µβ(ei) ∈ u($̃β) ∪l($̃β) for all pi ∈ p and ei ∈ q ⊆ p ×p. where u($̃α) = {$+α (pi)|pi ∈ p}, l($̃α) = {$ − α (pi)|pi ∈ p} and u($̃β) = {$+β (ei)|ei ∈ q},l($̃β) = {$ − β (ei)|ei ∈ q}. proof. assume that m = 〈α,β〉 is both an ic-graph and an ec-graph. then by definition we have µα(pi) ∈ [$−α (pi),$ + α (pi)], µβ(ei) ∈ [$ − β (ei),$ + β (ei)] and µα(pi) /∈ ($−α (pi),$ + α (pi)), µβ(ei) /∈ ($ − β (ei),$ + β (ei)). thus µα(pi) = $ − α (pi) or µα(pi) = $ + α (pi) and µβ(ei) = $ − β (ei) or µβ(ei) = $ + β (ei). hence µα(pi) ∈ u($̃α) ∪l($̃α) and µβ(ei) ∈ u($̃β) ∪l($̃β) for all pi ∈ p and ei ∈ q ⊆ p ×p. � consider two cubic graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 in ω∗. if we exchange µα1 by µα2 and µβ1 by µβ2 we get the cubic graph as m̂1 = 〈 α̂1, β̂1 〉 and m̂2 = 〈 α̂2, β̂2 〉 , respectively. for any two ic-graphs (or ec-graphs) m1 and m2, two cubic graphs m̂1 and m̂2 may not be ic-graph and ec-graph. int. j. anal. appl. 16 (5) (2018) 748 example 4.4. consider two ic-graphs m1 = 〈α1,β1〉 and m2 = 〈α2,β2〉 as shown in figure 17. figure 17. ic-graphs m1 and m2 then, their corresponding m̂1 and m̂2 are shown in figure 18. figure 18. cubic graphs m̂1 and m̂2 it is easy to see that the cubic graphs m̂1 and m̂2 are neither ic-graph nor ec-graph. similarly, we can provide and example for two ec-graphs that are neither ic-graph nor ec-graph. 5. application fuzzy graph theory is a platform which has wide range of applications in mathematics, computer science etc. cubic graph is a more general approach, which can be used in decision making very effectively. suppose we have a set of three countries like, p = {x, y , z} as a vertex set and the membership of each member of the set denotes the strength of that country over the neighbouring country with respect to future and present time by considering its economic strength. now we want to observe the effect of strength of one country at the another country with respect to economy. let we have a cubic set for each country based on int. j. anal. appl. 16 (5) (2018) 749 certain information and data with respect to economy α =   〈x : [0.6, 0.8], 0.9〉 〈y : [0.5, 0.9], 0.7〉 〈z : [0.3, 0.7], 0.8〉 where interval membership predicts the economy of a certain country for the future and the other membership shows economy of a certain country for the present time based on certain information and data with respect to economy. now on the basis of α, we have the set β of edges as follows β =   〈xy : [0.5, 0.8], 0.9〉 〈y z : [0.3, 0.7], 0.8〉 〈zx : [0.3, 0.7], 0.9〉 where interval membership predicts the effect of economy of a certain country for the future and the other membership shows the effect of economy of a certain country for the present time at the other country. the corresponding cubic graph is shown in figure 19. figure 19. cubic graph so finally we concluded that economy of a certain country effect very much on the economy of the neighboring countries. 6. conclusions graphs are among the most ubiquitous models of both natural and human-made structures. they can be used to model many types of relations and process dynamics in computer science, physical, biological and social systems. we come up here with the idea of cubic graphs and we define different operations of cubic graphs. we also provide a short application of cubic graph. in future we are planning to generalize our notions to (1) cubic line graphs, (2) cubic hypergraphs, and (3) cubic soft graphs. int. j. anal. appl. 16 (5) (2018) 750 references [1] m. akram and w.a. dudek, interval-valued fuzzy graphs, comput. math. appl., 61(2) (2011) 289-299. [2] m. akram, interval-valued fuzzy line graphs, neural comput. appl., 21 (2012) 145-150. [3] m. akram and b. davvaz, strong intuitionistic fuzzy graphs, filomat, 26(1) (2012) 177-196 [4] m. akram, bipolar fuzzy graphs, inf. sci., 181 (2011) 5548-5564. [5] m. akram, bipolar fuzzy graphs with applications, knowl.based syst., 39 (2013) 1-8. [6] p. bhattacharya, some remarks on fuzzy graphs, pattern recognition lett., 6 (1987) 297-302. [7] r.a. borzooei and h. rashmanlou, cayley interval-valued fuzzy threshold graphs, u.p.b. sci. bull., ser. a, 78(3) (2016) 83-94. [8] f. buckley, self-centered graphs, ann. n.y. acad. sci., 576 (1989) 71-78. [9] y.b. jun, c.s. kim and m.s. kang, cubic subalgebras and ideals of bck/bci-algebras, far east j. math. sci., 44 (2010) 239-250. [10] y.b. jun, k.j. lee and m.s. kang, cubic structures applied to ideals of bci-algebras, comput. math. appl., 62(9) (2011) 3334-3342. [11] y.b. jun, g. muhiuddin, m.a. oztürk and e.h. roh, cubic soft ideals in bck/bci-algebras, j. comput. anal. appl., 22(5) (2017) 929-940. [12] y.b. jun, c.s. kim and k.o. yang, cubic sets, ann. fuzzy math. inf., 4(1) (2012) 83-98. [13] j.g. kang and c.s. kim, mappings of cubic sets, commun. korean math. soc., 31(3) (2016) 423-431. [14] a. kauffman, introduction a la theorie des sous-emsembles flous, masson et cie, 1 (1973). [15] m. khan, y.b. jun, m. gulistan, n. yaqoob, the generalized version of jun’s cubic sets in semigroups, j. intell. fuzzy syst., 28(2) (2015) 947-960. [16] s.n. mishra, h. rashmanlou and a. pal, coherent category of interval-valued intuitionistic fuzzy graphs, j. mult.-val. log. soft comput., 29(3-4) (2017) 355-372. [17] j.n. mordeson and p.s. nair, fuzzy graphs and fuzzy hypergraphs, physica verlag, heidelberg (2001). [18] g. muhiuddin, s.s. ahn, c.s. kim and y.b. jun, stable cubic sets, j. comput. anal. appl., 23(5) (2017) 802-819. [19] m. pal, s. samanta and h. rashmanlou, some results on interval-valued fuzzy graphs, int. j. comput. sci. electr. eng., 3(3) (2015) 2320-4028. [20] r. parvathi, m. g. karunambigai and k. atanassov, operations on intuitionistic fuzzy graphs, proc. ieee int. conf. fuzzy syst., (2009) 1396-1401. [21] t. pramanik, m. pal and s. mondal, interval-valued fuzzy threshold graph, pac. sci. rev. a: nat. sci. eng., 18(1) (2016) 66-71. [22] t. pramanik, s. samanta and m. pal, interval-valued fuzzy planar graphs, international j. mach. learn. cybern., 7(4) (2016) 653-664. [23] a. rosenfeld, fuzzy graphs, fuzzy sets and their applications, academic press, new york, (1975) 77-95. [24] s. sahoo and m. pal, product of intuitionistic fuzzy graphs and degree, j. intell. fuzzy syst., 32(1) (2017) 1059-1067. [25] m.s. sunitha and k. sameena, characterization of g-self centered fuzzy graphs, j. fuzzy math., 16 (2008) 787-791. [26] s. vijayabalaji and s. sivaramakrishnan, a cubic set theoretical approach to linear space, abstr. appl. anal., 2015 (2015) article id 523129 8 pages. [27] l.a. zadeh, fuzzy sets, inf. control, 8 (1965) 338-353. 1. introduction 2. preliminaries 3. cubic graphs 4. internal and external cubic graphs 5. application 6. conclusions references international journal of analysis and applications volume 19, number 5 (2021), 760-772 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-760 fixed point theorems for contractive mappings in probabilistic modular spaces shahnaz jafari1,∗, maryam shams1 and asier ibeas2 1department of pure mathematics, shahrekord university, iran 2department of telecommunications and system engineering, faculty of engineering, universitat autonoma de barcelona, 08193 bellaterra, cerdanyola del valles, barcelona, spain ∗corresponding author: jafari.shahnaz@yahoo. com abstract. in this paper we introduce the concept of contractive maps and prove some related fixed point theorems in probabilistic modular spaces. in addition, we investigate the existence of common fixed points for a finite linear combination of contractive mappings. finally, some results concerned with the convergence properties of sequences defined by contractive maps in probabilistic modular spaces are also given. 1. introduction in recent times, fixed point theory has become an important tool in pure and applied sciences, such as biology [1], chemistry [2], engineering and physics , to cite just a few. the banach’s fixed point theory, widely known as the contraction principle, is an important tool in the theory of metric spaces [3], [4]. moreover, since the location of the fixed point can be obtained by means of an iterative process it can be implemented on a computer to find the fixed point of contraction mappings easily. the fixed point theory has been widely developed and extended to very general classes of spaces such as [5], [7], [16]. the concept of modular space was firstly introduced by nakano [8] and it was later generalized by musielak and orlicz [9]. many authors have worked ever since on the fixed point theory in modular spaces, see [10], [11], [12]. in 2009, nourouzi introduced in [13] the notion of probabilistic modular space according to menger’s probabilistic approach [6]. received june 20th, 2021; accepted july 13th, 2021; published august 12th, 2021. 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. probabilistic modular space; contrative map; fixed point; linear combination; converging sequences. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 760 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-760 int. j. anal. appl. 19 (5) (2021) 761 in this paper we introduce the concept of ϕ-contractive maps in probabilistic modular spaces and prove some related fixed theorems. there are no such results in probabilistic modular spaces and this paper contributes to fill in this gap. moreover, the existence of fixed points for a finite linear combination of ϕ-contractive mappings in a probabilistic modular space is also investigated. finally, we will provide some results concerned with the convergence properties of iterative sequences defined by ϕ-contractive maps in probabilistic modular spaces. 2. preliminaries we denote the function min by ∧, z+ = {z ∈ z : z > 0}, z0+ = z+ ∪{0}, r+ = {z ∈ r : z > 0}, r0+ = r+ ∪ {0}.we also will denote the set of all non-decreasing functions f : r −→ r+0 satisfying inft∈r f(t) = 0, and supt∈r f(t) = 1 by ∆. the latter properties imply that limt→∞f(t) = 1. the set of those distribution functions such that f(0) = 0 is denoted by ∆+. the space ∆+ is partially ordered by the usual pointwise ordering of functions, and has a maximal element �0, defined by �0(t) =   0 t 6 0,1 t > 0. definition 2.1. let x be an arbitrary vector space. a functional ρ : x → [0,∞] is called modular if for any arbitrary x,y ∈ x: (ii) ρ(x) = 0, iff x = 0, (ii) ρ(αx) = ρ(x), for every scalar α with |α| = 1, (iii) ρ(αx + βy) 6 ρ(x) + ρ(y) for all x,y ∈ x, and α,β ∈ r+0 , α + β = 1. definition 2.2. [13] a probabilistic modular space (briefly, pm-space) is a pair (x,µ) in which x is a real vector space and µ is a mapping from x into ∆ (for x ∈ x, the function µ(x) is denoted by µx, and µx(t) is the value of µx at t ∈ r) satisfying the following conditions: (i) µx(0) = 0, (ii) µx(t) = 1, for all t > 0 iff x = 0, (iii) µ−x(t) = µx(t), for all x ∈ x, (iv) µαx+βy(s + t) ≥ µx(s) ∧µy(t) for all x,y ∈ x, and α,β,s,t ∈ r+0 , α + β = 1. definition 2.3. [13] we say that (x,µ) is β-homogeneous, where β ∈ (0, 1] if, µαx(t) = µx( t |α|β ), for every x ∈ x,t > 0 and α ∈ r\{0}. int. j. anal. appl. 19 (5) (2021) 762 example 2.1. let ρ : x → x be defined by ρ(x) = 1 α |x|, for every α ∈ r\{0}. it is easy to see that ρ is a modular on x. define µx(t) =   0 t 6 0,t t+ρ(x) t > 0 for all x ∈ x. then (x,µ) is a β–homogeneous pm-space, for β = 1. example 2.2. consider the real vector space x with µx defined as: µx(t) =   0 t 6 ρ(x),1 t > ρ(x), where ρ is a modular on x. then (x,µ) is a pm-space. definition 2.4. [13] let (x,µ) be a pm-space. 1) a sequence {xn} in x is said to be µ-convergent to a point x ∈ x and denoted by xn −→ x, if for every t > 0 and r ∈ (0, 1), there exists a positive integer k such that µxn−x(t) > 1 − r for all n ≥ k. in this case, the point x ∈ x is said to be the µ-limit of the µ-converging sequence {xn}. 2) a sequence {xn} in x is said to be µ-cauchy sequence, if for every t > 0 and r ∈ (0, 1), there exists a positive integer k such that µxn−xm(t) > 1 −r for all n,m ≥ k. 3) the modular space (x,µ) is said to be µ-complete if each µ-cauchy sequence in x is µ-convergent to a point of x. lemma 2.1. [13] let (x,µ) be a pm-space. then the µ-limit of any µ-convergent sequence is unique. lemma 2.2. [13] the operations +, . in the β-homogeneous p-modular space (x,µ) are continuous. definition 2.5. [13] let (x,µ) and (y,ν) be two pm-spaces. a mapping t from (x,µ) to (y,ν) is said to be sequentially pm-continuous (probabilistic modular continuous) at x ∈ x if t(xn) ν−→ t(x) for every sequence {xn} of points in x that converges to x ∈ x, xn µ−→ x. the definition below is introced by sherstnev in 1963, [14]. definition 2.6. a random normed space (briefly, rn-space) is a triple (x,ν,t) where x is a vector space, t is a continuous t-norm, and ν is a mapping from x into ∆+ (for x ∈ x, if νx denotes the value of x ∈ x, the following conditions hold: (i) νx(t) = ε0(t), for all t > 0 iff x = 0, (ii) ναx(t) = νx( t |α|), for every x ∈ x,t > 0 and α ∈ r\{0}. (iii) νx+y(s + t) ≥ t(νx(s),νy(t)) for all x,y ∈ x and s,t ≥ 0. theorem 2.1. let (x,ν,t) be a rn-space with t-norm t(a,b) = min(a,b) for all a,b ∈ x. then (x,ν) is a pm-space. int. j. anal. appl. 19 (5) (2021) 763 proof. (1) ν−x(t) = ν(−1)x(t) = νx( t |−1|) = νx(t), for all x ∈ x. (2) let x,y ∈ x, α,β,s,t ∈ r+0 and α + β = 1. hence ναx+βy(t) = ναx+βy((α + β)t) ≥ t(ναx(αt),νβy(βt)) = t(νx( αt α ),νy( βt β )) = t(νx(t),νy(t)) ≥ t(νx( t 2 ),νy( t 2 )) = νx( t 2 ) ∧νy( t 2 ).(2.1) � 3. fixed point theorems for ϕ-contractive mappings in this section we define the notion of ϕ-contractive mapping in probabilistic modular spaces and prove some fixed point theorems related to this concept. let us introduce the following definition: definition 3.1. a function ϕ : [0,∞) → [0,∞) is said to be a φ-function if it satisfies the following conditions: (i) ϕ(t) is continuous, (ii) ϕ(t) is strictly monotone increasing and ϕ(t) →∞ as t →∞, (iii) ϕ(αt) 6 αφ(t), for all α ∈ (0, 1) and t ≥ 0. it is easy to see that the condition (iii) of definition 3.1 is equivalent to the following one: ϕ(0) = 0. example 3.1. ϕ(t) = k tr, is a simple example of φ-function for k > 0 and r > 1. lemma 3.1. a direct consequence of condition (iv) of definition 2.2 is: µσn i=1 αixi(t) ≥ µx1 ( t n ) ∧µx2 ( t n ) · · ·∧µxn( t n )︸ ︷︷ ︸ n ,(3.1) for all x1,x2, ...,xn ∈ x and αi, t ∈ r+0 with σ n i=1αi = 1. int. j. anal. appl. 19 (5) (2021) 764 proof. it is obtained by induction as follows: µσn i=1 αixi(t) = µσn−1 i=1 αixi+αnxn ( (n− 1)t n + t n ) = µ (σ n−1 i=1 αi σ n−1 i=1 αixi σ n−1 i=1 αi +αnxn) ( (n− 1)t n + t n ) ≥ µ σ n−1 i=1 αixi σ n−1 i=1 αi ( (n− 1)t n ) ∧µxn( t n ) ≥ µx1 ( t n ) ∧µx2 ( t n ) · · ·∧µxn( t n )︸ ︷︷ ︸ n .(3.2) � definition 3.2. let (x,µ) be a probabilistic modular space (pm-space). a mapping t : x −→ x is said to be ϕ-contractive if µtx−ty(ϕ(t)) ≥ µl(x−y)(ϕ( t c )),(3.3) where l,c ∈ (0, 1) and ϕ ∈ φ. it is easy to see that every ϕ-contractive map is sequentially pm-continuous. in fact, if xn → x, hence, for every t > 0 and r ∈ (0, 1), there exists n such that µxn−x(t) > 1 − r for all n ≥ n. therefore we get µtxn−tx(ϕ(t)) ≥ µxn−x(ϕ( t c )) > 1 −r. remark 3.1. we can see that the definition 3.2 generalizes the previous ones introduced in [15]. theorem 3.1. let (x,µ) be a β-homogeneous µ-complete pm-space and t : x −→ x be a ϕ-contractive map. then t has a unique fixed point x∗ ∈ x and the iterative sequence {tn(x0)}, generated by the initial element x0 ∈ x, converges to the fixed point x∗ ∈ x as n →∞ . proof. choose x ∈ x arbitrarily. we first prove that {tn(x)} is a µ-cauchy sequence. let s > 0 be given. since ϕ is continuous at 0 and ϕ(0) = 0, we can find t > 0 such that ϕ(t) < s. hence, we have: µtnx−tn+px(s) ≥ µtnx−tn+px(ϕ(t)) ≥ µl(tn−1x−tn+p−1x)(ϕ( t c )) = µtn−1x−tn+p−1x(l −βϕ( t c )) ≥ µtn−1x−tn+p−1x(ϕ( t c )) ... ≥ µx−tpx(l−βϕ( t cn )) ≥ µx−tpx(ϕ( t cn )).(3.4) int. j. anal. appl. 19 (5) (2021) 765 on the other hand, we have: µx−tpx(ϕ( t cn )) = µ(x−tx)+(tx−tpx)(ϕ( t cn )) ≥ µ2(x−tx)( 1 2 ϕ( t cn )) ∧µ2(tx−tpx)( 1 2 ϕ( t cn )) ≥ µx−tx( 1 2β+1 ϕ( t cn )) ∧µtx−tpx( 1 2β+1 ϕ( t cn )) ≥ µx−tx(ϕ( t 2β+1cn )) ∧µtx−tpx(ϕ( t 2β+1cn )) ≥ µx−tx(ϕ( t 2β+1cn )) ∧µl(x−tp−1x)(ϕ( t 2β+1cn+1 )) = µx−tx(ϕ( t 2β+1cn )) ∧µx−tp−1x(l−βϕ( t 2β+1cn+1 )) ≥ µx−tx(ϕ( t 2β+1cn )) ∧µx−tp−1x(ϕ( t 2β+1cn+1 )).(3.5) by induction we get: µx−tpx(ϕ( t cn )) ≥ µx−tx(ϕ( t 2β+1cn )) ∧µx−tx(ϕ( t 22(β+1)cn+1 )) ∧·· · ∧µx−tx(ϕ( t 2p(β+1)cn+p−1 )).(3.6) according to property (ii) of φ-function and since µ(∞) = 1, from (3.4) and (3.6) we get limn→∞µtnx−tn+px(s) = 1. since x is µ-complete, there exists x∗ ∈ x with limn→∞tnx∗ = x∗. we will prove now that x∗ is a fixed point of t . the ϕ-contractivity of t yields sequentialy pm-continuity. therefore, x∗ = limn→∞t n+1x∗ = limn→∞t(t nx∗) = tx∗; i.e x∗ is a fixed point of t . in order to prove that the fixed point if unique, assume that there exists another fixed point y∗ ∈ x such that y∗ = ty∗. hence, tnx∗ = x∗ and tny∗ = y∗, and there exists t1 > 0 such that µy∗−x∗ (t1) = a < 1. then, a = µy∗−x∗ (t1) ≥ µy∗−x∗ (ϕ(t)) = µtny∗−tnx∗ (ϕ(t)) ≥ µy∗−x∗ (l−nβϕ( t cn )).(3.7) letting n →∞ in (3.7), according the property (ii) of φ-function and since µ(∞) = 1, we get a ≥ 1, that is contradiction. therefore y∗ = x∗. � the following theorem shows that a linear combination of a family of ϕ-contrative mappings possesing a common fixed point has a fixed point and it can be calculated by using an interative process. theorem 3.2. let (x,µ) be a β-homogeneous µ-complete pm-space and fi : x −→ x (i = 1, 2, · · · ,m) be a finite family of ϕ-contractive maps for ϕ ∈ φ and c ∈ (0, 1 m ). define f = ∑m i=1 λifi, where λi ∈ [0, 1], σmi=1λi = 1. then f has fixed point x ∗ ∈ x, which is common to each linear operator’s one and the iterative sequence {fn(x)} defined by the initial element x0 ∈ x, converges to x∗ ∈ x. int. j. anal. appl. 19 (5) (2021) 766 proof. since fi have a common fixed point x ∗ ∈ x, then: f(x∗) = λ1f1(x ∗) + λ2f2(x ∗) + ... + λmfm(x ∗) = (λ1 + λ2 + ... + λm)x ∗ = x∗,(3.8) this means that x∗ is a fixed of f (and common to ech operator’s fixed point). now we prove that f is a ϕ-contractive map. we have: µfx−fy(ϕ(t)) = µ ∑ m i=1 λifixi− ∑ m i=1 λifiy(ϕ(t)) ≥ µf1x−f1y( 1 m ϕ(t)) ∧µf2x−f2y( 1 m ϕ(t)) ∧·· ·∧µfnx−fny( 1 m ϕ(t))︸ ︷︷ ︸ m ≥ µf1x−f1y(ϕ( t m )) ∧µf2x−f2y(ϕ( t m )) ∧·· ·∧µfnx−fny(ϕ( t m ))︸ ︷︷ ︸ m ≥ µl(x−y)(ϕ( t mc )) ∧µl(x−y)(ϕ( t mc )) ∧·· ·∧µl(x−y)(ϕ( t mc )) ≥ µl(x−y)(ϕ( t mc )) = µl(x−y)(ϕ( t k )),(3.9) where k ∈ (0, 1). hence, f is ϕ-contractive and according to theorem 3.1, the sequence {fn(x0)} converges to the fixed point x∗ ∈ x for any arbitrary initial element x0. � the subsequent results are concerned with the convergence properties of ϕ-contractive maps. pm-space. lemma 3.2. the following property hold: if tn : x → x,∀n ∈ z+ are continuous and {tn} uniformly converges to {t}, then {tmn } uniformly converge to {tm}, ∀m ∈ z+. proof. we prove these properties with induction. assume that {tjn} converge to {tj}, as n → ∞, for all 1 ≤ j ≤ m and for any given m ∈ z+. we have: µ t j n(tnx)−tj(tx) (t) ≥ µ 2(t j n(tnx)−tj(tnx)) ( t 2 ) ∧µ2(tj(tnx)−tj(tx))( t 2 )(3.10) since tn : x → x is continuous and {tjn} converge to {tj}, there exists a big enough n such that µtj(tnx)−tj(tx)( t 2 ) > 1−λ and µ t j n(tnx)−tj(tnx) ( t 2 ) > 1−λ, for any given λ ∈ (0, 1). thus, from (3.10), we have µ t j+1 n x−tj+1x (t) > (1 −λ) ∧ (1 −λ) = 1 −λ. thus, tj+1n converge to t j+1 as n →∞, for all 1 6 j 6 m. � theorem 3.3. let (x,µ) be a β-homogeneous µ-complete pm-space and {tn} be a sequence of sequentially pm-continuous operators with fix(tn) = {x∗n}, such that: int. j. anal. appl. 19 (5) (2021) 767 (i) {tn} uniformly converge to t for some t : x −→ x. (ii) t is ϕ-contractive, with t(x∗) = x∗. then {x∗n}→ x∗. proof. according to the definition of convergence in pm-space, we show that limn→∞µx∗n−x∗ (t) = 1, for every t > 0. in this way, we have: µx∗n−x∗ (ϕ(t)) = µtmn x∗n−tmx∗ (ϕ(t)) ≥ µ2(tmn x∗n−tmx∗n)(ϕ( t 2 )) ∧µ2(tmx∗n−tmx∗)(ϕ( t 2 )) ≥ µ2(tmn x∗n−tmx∗n)(ϕ( t 2 )) ∧µx∗n−x∗ (ϕ( t 2β+1cm )), ∀n,m ∈ z0+,(3.11) if we take the limit m →∞ in (3.11) we get limn→∞µx∗n−x∗ (ϕ(t)) = 1. thus, {x ∗ n}→{x∗}. � theorem 3.4. let (x,µ) be a β-homogeneous µ-complete pm-space and {tn} be a sequence of ϕ-contractive operators tn : x → x for some l,c ∈ (0, 1), ϕ ∈ φ with fix(tn) = {x∗n}. moreover, let t : x −→ x be a ϕ-contractive mapping with fix(t) = {x∗}. assume the following properties hold: (a) {tn} converge to t , (b) there exists a subsequence {x∗nm} of {x ∗ n}, converging to a point z ∈ x. then z = x∗ and the iterated sequence generated by xn+1 = tnxn converges to the fixed point x ∗, for any given x0 ∈ x and n ∈ z+ proof. we first prove that {x∗nm} converge to x ∗. proceed by assuming, since {x∗nm}→{z} and {tn}→ t, for any given δ ∈ (0, 1) and t > 0 there exists n1(∈ z0+) = n1(δ,t) such that for n,m ≥ n1, µx∗nm−z(ϕ(t)) > 1 − δ and µtnmz−tz(ϕ(t)) > 1 − δ , where ϕ ∈ φ. therefore, µx∗nm−tz(ϕ(t)) = µtnmx ∗ nm −tz(ϕ(t)) ≥ µ2(tnmx∗nm−tnmz)(ϕ(t)) ∧µ2(tnmz−tz)(ϕ(t)) ≥ µx∗nm−z(ϕ( t 2βc )) ∧µtnmz−tz(2 −βϕ(t)) ≥ (1 − δ) ∧ (1 − δ) = (1 − δ).(3.12) int. j. anal. appl. 19 (5) (2021) 768 this result means that {x∗nm} converges to tz. hence, by the uniqueness of the limit, we get tz = z implying that z = x∗. also for α ∈ (0, 2c) we have: µxn+1−x∗ (ϕ(t)) = µtnxn−tx∗ (ϕ(t)) ≥ µ2(tnxn−tnx∗)(αϕ(t)) ∧µ2(tnx∗−tx∗)((1 −α)ϕ(t)) ≥ µxn−x∗ (ϕ( αt 2βc )) ∧µ2(tnx∗−tx∗)((1 −α)ϕ(t)).(3.13) on the other hand: µxn−x∗ (ϕ( αt 2βc )) ≥ µxn−1−x∗ (ϕ( α2t 22(β+1)c2 )) ∧µ2(tnx∗−tx∗)((1 −α)ϕ( αt 2βc )).(3.14) by induction from (3.13) and (3.14), we have µxn+1−x∗ (ϕ(t)) ≥ µx0−x∗ (ϕ( αnt 2n(β+1)cn )) ∧µ2(tnx∗−tx∗)((1 −α)ϕ(t)) ∧µ2(tnx∗−tx∗)((1 −α)ϕ( αt 2βc )) ∧·· · ∧µ2(tnx∗−tx∗)((1 −α)ϕ( αnt 2n(β+1)cn )).(3.15) letting n →∞ in (3.15) we get lim µxn+1−x∗ (ϕ(t)) ≥ 1, i.e {xn}→ x∗. � theorem 3.5. let (x,µ) be a β-homogeneous µ-complete pm-space, and {tn} be a sequence of operators such that {tn} are ϕ-contractive for some l,c ∈ (0, 1) and ϕ ∈ φ. assume that {tn} converge to t for some t : x −→ x. then the following properties hold: (a) t is ϕ-contractive for some c ∈ (0, 1 4 ), (b) {x∗n}→ x∗, where fix(tn) = {x∗n}, ∀n ∈ z+, and fix(t) = {x∗}, (c) the iterative sequence generated by xn+1 = tnxn converges to x ∗, for any given x0 ∈ x arbitrary and n ∈ z+. proof. first we prove that t is ϕ-contractive. for any x,y ∈ x we have: µtnx−tny(ϕ(t)) ≥ µl(x−y)(ϕ( t c )).(3.16) additionally, µtx−ty(ϕ(t)) = µtx−tnx+tnx−ty(ϕ(t)) ≥ µ2(tx−tnx)((1 −α)ϕ(t)) ∧µ2(tnx−ty)(αϕ(t)) ≥ µtx−tnx( ϕ((1 −α)t) 2β ) ∧µtnx−ty( ϕ(αt) 2β ).(3.17) int. j. anal. appl. 19 (5) (2021) 769 on the other hand µtnx−ty( ϕ(αt) 2β ) ≥ µ2(tnx−tny)( ϕ(αt) 2β+1 ) ∧µ2(tny−ty)( ϕ(αt) 2β+1 ) ≥ µtnx−tny(ϕ( αt 22(β+1) )) ∧µtny−ty( ϕ(αt) 22(β+1) ) ≥ µl(x−y)(ϕ( c−1αt 22(β+1) )) ∧µtny−ty( ϕ(αt) 22(β+1) ).(3.18) by using equation (3.17) and (3.18) we have µtx−ty(ϕ(t)) ≥ µtx−tnx( ϕ((1 −α)t) 2β ) ∧µl(x−y)(ϕ( c−1αt 22(β+1) )) ∧µtny−ty( ϕ(αt) 22(β+1) ).(3.19) letting n →∞ in (3.19), we get µtx−ty(ϕ(t)) ≥ 1 ∧µl(x−y)(ϕ( c−1αt 22(β+1) )) ∧ 1 = µl(x−y)(ϕ( c−1t 22(β+1) )) ≥ µl(x−y)(ϕ( t k )),(3.20) where k ∈ (0, 1 4 ). hence t is ϕ-contractive. finally, according to theorem 3.3, {x∗n} converges to x∗ and by the same method of proof of theorem 3.4, {xn} converges to x∗. � remark 3.2. every probabilistic modular space (x,µ) induces a probabilistic metric space (x,f,∧) with f : x ×x → ∆ via fx,y = µx−y for all x,y ∈ x. 4. numerical examples in this section we present some numerical examples in order to illustrate the main results discussed in the previous sections. example 4.1. let x = r and µx(t) = tt+ρ(x) , x,y ∈ x, t > 0, where ρ(x) = |x| is a modular functional on x. define a mapping f : r → r by f(x) = x 8 for all x ∈ r. let ϕ(t) = 2 t2. then f is ϕ-contractive with the constants l = 1 2 and c ≥ 1 2 . indeed, for x,y ∈ r, we have µfx−fy(ϕ(t)) = 2t2 2t2 + 1 8 |x−y| , µl(x−y)(ϕ( t c )) = 2t2 c2 2t2 c2 + 1 2 |x−y| . it is easy to see that 2t2 2t2 + 1 8 |x−y| ≥ 2t2 c2 2t2 c2 + 1 2 |x−y| , for all c ∈ [ 1 2 , 1). accordingly, f is ϕ-contractive and it has a unique fixed point, as predicted by theorem 3.1. in addition, it is easy to check that x = 0 is the fixed point of f. int. j. anal. appl. 19 (5) (2021) 770 example 4.2. let x = r and µx(t) = tt+ρ(x) , x,y ∈ x, t > 0, where ρ(x) = |x| is a modular functional on x. define the mappings fi : r → r by f1(x) = x4 and f2(x) = x 2 for all x ∈ r. let ϕ(t) = t, l = 2 3 and c = 3 4 . define f = 1 2 f1 + 1 2 f2. we can see that f1 and f2 are ϕ-contractive maps. we prove that f(x) = 1 2 f1x + 1 2 f2x = 3x 8 is ϕ-contractive. µfx−fy(ϕ(t)) = t t + 3 8 |x−y| , µl(x−y)(ϕ( t c )) = 4 3 t 4 3 t + 2 3 |x−y| , thus, t t + 3 8 |x−y| ≥ 4 3 t 4 3 t + 2 3 |x−y| , consequently, f is a ϕ-contractive map and it has a unique fixed point, as predict by theorem 3.2. it is easy to check that x = 0 is the fixed point of f. example 4.3. let x = r and µx(t) = tt+ρ(x) , x,y ∈ x, t > 0, where ρ(x) = |x| is a modular functional on x. let ϕ(t) = t and define tnxn = (n+1)xn (2n+3)(1+x2n) . we show that tn is ϕ-contractive. we have: µtnx−tny(ϕ(t)) = t t + (n+1)|x−y| (2n+3)(1+x2)(1+y) = (2n + 3)(1 + x2)(1 + y)t (2n + 3)(1 + x2)(1 + y)t + (n + 1)|x−y| , and µl(x−y)(ϕ( t c )) = t c t c + l|x−y| = t t + lc|x−y| . so the condition (3.3) becomes: (2n + 3)(1 + x2)(1 + y)t (2n + 3)(1 + x2)(1 + y)t + (n + 1)|x−y| ≥ t t + lc|x−y| .(4.1) eq. (4.1) leads to 2n+3 n+1 (1 + x2)(1 + y) ≥ 1 lc , that holds for every l,c ∈ (0, 1). hence tn is ϕ-contractive. on the other hand we have: t = lim n→∞ tn = lim n→∞ (n + 1)x (2n + 3)(1 + x2) = x 2(1 + x2) . similarly to the above method, we can see that t is also ϕ-contractive. therefore, theorem 3.5 holds and the iterative scheme: xn+1 = (n + 1)xn (2n + 3)(1 + x2n) (4.2) converges to the unique fixed point of t . it is easy to check out that x∗ = 0 is the fixed point of t . figure 1 shows the evolution of the iterative scheme (4.2) for different initial conditions. in figure 1, we can observe that the sequence {xn} converges to zero as predicted by theorem 3.5. int. j. anal. appl. 19 (5) (2021) 771 0 1 2 3 4 5 6 7 iteration (n) -3 -2 -1 0 1 2 3 x n figure 1. evolution of the sequence of iterates for different initial conditions 5. conclusions this paper has introduced the concept of ϕ-contractive maps in probabilistic modular spaces. furthermore, the existence of fixed points for these operators in probabilistic modular spaces is investigated as well. afterwards, the results are extended to a finite linear combination of ϕ-contractive mappings. finally, we also investigate some convergence properties of sequences constructed by these operators which are either convergent to either a ϕ-contractive map. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. b. amar, a. jeribi, m. mnif, some fixed point theorems and application to biological model, numer. funct. anal. optim. 29 (1-2) (2008), 1–23. [2] l. marks, fixed-point optimization of atoms and density in dft, j. chem. theory comput. 9 (6) (2013), 2786–2800. [3] m. a. khamsi, w. a. kirk, an introduction to metric spaces and fixed point theory, vol. 53, wiley, new york, 2011. [4] l. b. ćirić, b. fisher, fixed point theory: contraction mapping principle, faculty of mechanical engineering, university of belgrade, serbia, 2003. [5] z. zuo, some fixed point property for multivalued nonexpansive mappings in banach spaces, math inequal appl. 7 (1) (2013), 129–137. [6] k. menger, statistical metrics, proc. nat. acad. sci. 28 (12) (1942), 535–537. [7] m. abtahi, common fixed point theorems of meir-keeler type in metric spaces, fixed point theory. 1 (2017), 47–56. int. j. anal. appl. 19 (5) (2021) 772 [8] h. nakano, modular semi-ordered spaces, maruzen co. ltd., tokyo, japan, 1950. [9] j. musielak, w. orlicz, on modular spaces, stud. math. 18 (1) (1959) 49–65. [10] p. kumam, fixed point theorems for nonexpansive mappings in modular spaces, arch. math. 40 (4) (2004), 345–353. [11] k. kuaket, p. kumam, fixed points of asymptotic pointwise contractions in modular spaces, appl. math. lett. 24 (11) (2011), 1795–1798. [12] m. de la sen, d. o’regan, r. saadati, characterization of modular spaces, j. comput. anal. appl. 22 (2017), 558–572. [13] k. fallahi, k. nourouzi, probabilistic modular spaces and linear operators, acta appl. math. 105 (2) (2009), 123–140. [14] a. n. sherstnev, on the notion of a random normed space, dokl. akad. nauk sssr. 149 (1963), 280–283. [15] f. lael, k. nourouzi, fixed points of mappings defined on probabilistic modular spaces, bull. math. anal. appl. 4 (3) (2012), 23–28. [16] x. wang, c. zhang, y. cui, some sufficient conditions for points of multivalued nonexpansive mappings in banach spaces, math. inequal. appl. 11 (1) (2017), 113–120. 1. introduction 2. preliminaries 3. fixed point theorems for -contractive mappings 4. numerical examples 5. conclusions references int. j. anal. appl. (2022), 20:1 controlled k −g−fusion frames in hilbert c∗−modules mohamed rossafi1,∗, fakhr-dine nhari2 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco 2laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, kenitra, morocco ∗corresponding author: rossafimohamed@gmail.com abstract. controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. in this paper, we introduce the concepts of controlled g−fusion frame and controlled k−g−fusion frame in hilbert c∗−modules and we give some properties. also, we study the perturbation problem of controlled k − g−fusion frame. moreover, an illustrative example is presented to support the obtained results. 1. introduction frames for hilbert spaces were introduced by duffin and schaefer [4] in 1952 to study some deep problems in nonharmonic fourier series by abstracting the fundamental notion of gabor [6] for signal processing. many generalizations of the concept of frame have been defined in hilbert c∗-modules [5,7,9,13– 17]. controlled frames in hilbert spaces have been introduced by p. balazs [3] to improve the numerical efficiency of iterative algorithms for inverting the frame operator. rashidi and rahimi [10] are introduced the concept of controlled frames in hilbert c∗−modules. received: nov. 15, 2021. 2010 mathematics subject classification. 42c15. key words and phrases. fusion frame; g-fusion frame; k−g−fusion frame; controlled k−g−fusion frames; hilbert c∗−modules. https://doi.org/10.28924/2291-8639-20-2022-1 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-1 2 int. j. anal. appl. (2022), 20:1 the paper is organized in the following manner. in section 3, we introduced the notion of g−fusion frames and controlled g−fusion frames in hilbert c∗−modules and estabilish some properties. section 4 is devoted to introduce the concept of controlled k −g−fusion frames in hilbert c∗−modules and gives some results, finally in section 5 we study the perturbation of controlled k −g−fusion frames. 2. preliminaires let a be a unital c∗−algebra, let j be countable index set. throughout this paper h and l are countably generated hilbert a−modules and {hj}j∈j is a sequence of submodules of l. for each j ∈ j, end∗a(h,hj) is the collection of all adjointable a−linear maps from h to hj, and end ∗ a(h,h) is denoted by end∗a(h). also let gl +(h) be the set of all positive bounded linear invertible operators on h with bounded inverse. definition 2.1. [8] let a be a unital c∗-algebra and h be a left a-module, such that the linear structures of a and h are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued inner product 〈., .〉 : h ×h → a, such that is sesquilinear, positive definite and respects the module action. in the other words, (i) 〈f , f 〉≥ 0 for all f ∈ h and 〈f , f 〉 = 0 if and only if f = 0. (ii) 〈af + g,h〉 = a〈f ,h〉 + 〈g,h〉 for all a ∈a and f ,g,h ∈ h. (iii) 〈f ,g〉 = 〈g,f 〉∗ for all f ,g ∈ h. for f ∈ h, we define ||f || = ||〈f , f 〉|| 1 2 . if h is complete with ||.||, it is called a hilbert a-module or a hilbert c∗-module over a. for every a in a c∗-algebra a, we have |a| = (a∗a) 1 2 and the a-valued norm on h is defined by |f | = 〈f , f 〉 1 2 for f ∈ h. define l2({hj}j∈j) by l2({hj}j∈j) = {{fj}j∈j : fj ∈ hj, || ∑ j∈j 〈fj, fj〉|| < ∞}. with a−valued inner product is given by 〈{fj}j∈j,{gj}j∈j〉 = ∑ j∈j 〈fj,gj〉, l2({hj}j∈j) is a hilbert a−module. the following lemmas was used to proof our results: lemma 2.1. [1] if φ : a → b is a ∗−homomorphism between c∗−algebras, then φ is increasing, that is, if a ≤ b, then φ(a) ≤ φ(b). lemma 2.2. [2] let t ∈ end∗a(h,l) and h,l are hilberts a−modules.the following statemnts are multually equivalent: (i) t is surjective. int. j. anal. appl. (2022), 20:1 3 (ii) t∗ is bounded below with respect to the norm, i.e., there is m > 0 such that ||t∗f || ≥ m||f || for all f ∈ l. (iii) t∗ is bounded below with respect to the inner product, i.e, there is m ′ > 0 such that 〈t∗f ,t∗f 〉≥ m ′ 〈f , f 〉 for all f ∈ l. lemma 2.3. [1] let h and l are two hilbert a-modules and t ∈ end∗a(h,l). then: (i) if t is injective and t has closed range, then the adjointable map t∗t is invertible and ‖(t∗t )−1‖−1 ≤ t∗t ≤‖t‖2. (ii) if t is surjective, then the adjointable map tt∗ is invertible and ‖(tt∗)−1‖−1 ≤ tt∗ ≤‖t‖2. lemma 2.4. [2] let h be a hilbert a-module over a c∗-algebra a, and t ∈ end∗a(h) such that t∗ = t . the following statements are equivalent: (i) t is surjective. (ii) there are m,m > 0 such that m‖f‖≤‖tf‖≤ m‖f‖, for all f ∈ h. (iii) there are m′,m′ > 0 such that m′〈f , f 〉≤ 〈tf,tf 〉≤ m′〈f , f 〉 for all f ∈ h. lemma 2.5. [12] let h be a hilbert a-module. if t ∈ end∗a(h), then 〈tf,tf 〉≤ ‖t‖2〈f , f 〉, ∀f ∈ h. lemma 2.6. [18] let e,h and l be hilbert a−modules, t ∈ end∗a(e,l) and t ′ ∈ end∗a(h,l). then the following two statements are equivalent: (1) t ′ (t ′ )∗ ≤ λtt∗ for some λ > 0; (2) there exists µ > 0 such that ‖(t ′ )∗z‖≤ µ‖t∗z‖ for all z ∈ l. lemma 2.7. [11] let {wj}j∈j be a sequence of orthogonally complemented closed submodules of h and t ∈ end∗a(h) invertible, if t ∗twj ⊂ wj for each j ∈ j, then {twj}j∈j is a sequence of orthogonally complemented closed submodules and pwjt ∗ = pwjt ∗ptwj. 3. controlled g−fusion frame in hilbert c∗−modules firstly we give the definition of g−fusion frame in hilbert c∗−modules. definition 3.1. [11] let {wj}j∈j be a sequence of closed submodules orthogonally complemented of h, {vj}j∈j be a family of weights in a, ie., each vj is positive invertible element frome the center of a and λj ∈ end∗a(h,hj) for each j ∈ j. we say that λ = {wj, λj,vj}j∈j is a g−fusion frame for h if there exists 0 < a ≤ b < ∞ such that a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉≤ b〈f , f 〉, ∀f ∈ h. (3.1) 4 int. j. anal. appl. (2022), 20:1 the constants a and b are called the lower and upper bounds of the g−fusion frame, respectively. if a = b then λ is called tight g−fusion frame and if a = b = 1 then we say λ is a parseval g−fusion frame. the operator s : h → h defined by sf = ∑ j∈j v2j pwj λ ∗ j λjpwjf , ∀f ∈ h. is called g−fusion frame operator. now we define the notion of (c,c ′ )−controlled g−fusion frame in hilbert c∗−modules. definition 3.2. let c, c ′ ∈ gl+(h), {wj}j∈j be a sequence of closed submodules orthogonally complemented of h, {vj}j∈j be a family of weights in a, i.e., each vj is a positive invertible element frome the center of a and λj ∈ end∗a(h,hj) for each j ∈ j. we say that λcc′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h if there exists 0 < a ≤ b < ∞ such that a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ b〈f , f 〉, ∀f ∈ h. (3.2) the constants a and b are called the lower and upper bounds of the (c,c ′ )−controlled g−fusion frame, respectively. when a = b, the sequence λ cc ′ = {wj, λj,vj}j∈j is called (c,c ′ )−controlled tight g−fusion frame, and when a = b = 1, it is called a (c,c ′ )−controlled parseval g−fusion frame. if only upper inequality of (3.2) hold, then λ cc ′ is called an (c,c ′ )−controlled g−fusion bessel sequence for h. example 3.1. let l∞ be the set of all bounded complex-valued sequences. for any u = {uj}j∈n, v = {vj}j∈n ∈ l∞, we have uv = {ujvj}j∈n,u∗ = {uj}j∈n, ||u|| = sup j∈n |uj|. then a = {l∞, ||.||} is a c∗−algebra. let h = c0 be the set of all sequences converging to zero. for any u, v ∈ h we define 〈u,v〉 = uv∗ = {ujvj}j∈n. then h is a hilbert a−module. now let {ej}j∈n be the standard orthonormal basis of h. we construct hj = span{e1,e2, ...,ej} and wj = span{ej} for each j ∈n. define λj : h → hj by λj(f ) = ∑j k=1〈f , ej√ j 〉ek. the adjoint operator λ∗j : hj → h define by λ ∗ j (g) = ∑j k=1〈g, ek√ j 〉ej. and the projection orthogonal pwj define by pwj (f ) = 〈f ,ej〉ej. int. j. anal. appl. (2022), 20:1 5 let us define cf = 2f and c ′ f = 1 2 f . then for any f ∈ h, we have 〈λjpwjcf, λjpwjc ′ f 〉 = 〈 2 √ j 〈f ,ej〉 j∑ k=1 ek, 1 2 √ j 〈f ,ej〉 j∑ k=1 ek〉 = 1 j 〈f ,ej〉〈ej, f 〉〈 j∑ k=1 ek, j∑ k=1 ek〉 = 1 j 〈f ,ej〉〈ej, f 〉 j∑ k=1 ||ek||2 = 1 j 〈f ,ej〉〈ej, f 〉j = 〈f ,ej〉〈ej, f 〉. therefore, for each f ∈ h,∑ j∈n 〈λjpwjcf, λjpwjc ′ f 〉 = ∑ j∈n 〈f ,ej〉〈ej, f 〉 = 〈f , f 〉. hence {wj, λj, 1}j∈n is a (c,c ′ )−controlled parseval g−fusion frame for h. suppose that λ cc ′ be a (c,c ′ )−controlled g−fusion bessel sequence for h. the bounded linear operator t (c,c ′ ) : l2({hj}j∈j) → h define by t (c,c ′ ) ({fj}j∈j) = ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j fj, ∀{fj}j∈j ∈ l 2({hj}j∈j). (3.3) is called the synthesis operator for the (c,c ′ )−controlled g−fusion frame λ cc ′. the adjoint operator t∗ (c,c ′ ) : h → l2({hj}j∈j) given by t∗ (c,c ′ ) (g) = {vjλjpwj (c ′ c) 1 2 g}j∈j (3.4) is called the analysis operator for the (c,c ′ )−controlled g−fusion frame λ cc ′. when c and c ′ commute with each other, and commute with the operator pwj λ ∗ j λjpwj, for each j ∈ j, then the (c,c ′ )−controlled g−fusion frame operator s (c,c ′ ) : h → h is defined as s (c,c ′ ) (f ) = t (c,c ′ ) t∗ (c,c ′ ) (f ) = ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjcf, ∀f ∈ h. (3.5) and we have 〈s (c,c ′ ) (f ), f 〉 = ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉, ∀f ∈ h. (3.6) from now we assume that c and c ′ commute with each other, and commute with the operator pwj λ ∗ j λjpwj, for each j ∈ j lemma 3.1. let λ cc ′ be a (c,c ′ )−controlled g−fusion frame for h. then the (c,c ′ )−controlled g−fusion frame operator s (c,c ′ ) is positive, self-adjoint and invertible. 6 int. j. anal. appl. (2022), 20:1 proof. for each f ∈ h we have s (c,c ′ ) (f ) = ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjcf then ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉 = 〈 ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjcf,f 〉 = 〈s(c,c′)(f ), f 〉. since λ cc ′ is a (c,c ′ )−controlled g−fusion frame for h, then a〈f , f 〉≤ 〈s (c,c ′ ) (f ), f 〉≤ b〈f , f 〉, ∀f ∈ h (3.7) it is clear that s (c,c ′ ) is positive, bounded and linear operator. on the other hand for each f , g ∈ h 〈s (c,c ′ ) (f ),g〉 = 〈 ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjcf,g〉 = 〈f , ∑ j∈j v2j cpwj λ ∗ j λjpwjc ′ g〉 = 〈f ,s (c ′ ,c) (g)〉. that implies s∗ (c,c ′ ) = s (c ′ ,c) . also as c and c ′ commute with each other, and commute with the operator pwj λ ∗ j λjpwj, for each j ∈ j, we have s(c,c′) = s(c′,c). so the (c,c ′ )−controlled g−fusion frame operator s (c,c ′ ) is self-adjoint. and from inequality (3.7) we have aih ≤ s(c,c′) ≤ bih. (3.8) therefore, the (c,c ′ )−controlled g−fusion frame operator s (c,c ′ ) is invertible. � we estabilish an equivalent definition of (c,c ′ )−controlled g−fusion frame. theorem 3.1. λ cc ′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h. if and only if there exists two constants 0 < a ≤ b < ∞ such that a||f ||2 ≤ || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| ≤ b||f ||2, ∀f ∈ h. (3.9) proof. if λ cc ′ be a (c,c ′ )−controlled g−fusion frame for h, then we have inequality (3.9). converselly, assume that (3.9) holds. from (3.4), the (c,c ′ )−controlled g−fusion frame operator s (c,c ′ ) is positive, self-adjoint and invertible. then we have for all f ∈ h 〈(s (c,c ′ ) ) 1 2 f , (s (c,c ′ ) ) 1 2 f 〉 = 〈s (c,c ′ ) f , f 〉 = ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉. (3.10) using (3.9) and (3.10), we conclude that √ a||f || ≤ ||s 1 2 (c,c ′ ) f || ≤ √ b||f ||, ∀f ∈ h. so by lemma 2.4, λ cc ′ is a (c,c ′ )−controlled g−fusion frame for h. � int. j. anal. appl. (2022), 20:1 7 theorem 3.2. let {wj, λj,vj}j∈j be a g−fusion frame for h with frame operator s and let c, c ′ ∈ gl+(h). then {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h. proof. let {wj, λj,vj}j∈j be a g−fusion frame for h with frame bounds a and b. then for each f ∈ h a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉≤ b〈f , f 〉 (3.11) we have || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| = ||〈s (c,c ′ ) f , f 〉|| = ||c||.||c ′ ||.||〈sf,f 〉||, (3.12) using (3.11) and (3.12), we conclude a||c||.||c ′ ||||〈f , f 〉|| ≤ || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| ≤ b||c||.||c ′ ||||〈f , f 〉||, ∀f ∈ h. therefore, {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h with bounds a||c||.||c ′ || and b||c||.||c ′ ||. � remark 3.1. when c = c ′ we say that the sequence {wj, λj,vj}j∈j is a c2−controlled g−fusion frame for h. theorem 3.3. let c ∈ gl+(h). the sequence {wj, λj,vj}j∈j is a g−fusion frame for h if and only if {wj, λj,vj}j∈j is a c2−controlled g−fusion frame for h. proof. suppose that {wj, λj,vj}j∈j is a g−fusion frame for h. with bounds a and b. then a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉≤ b〈f , f 〉, ∀f ∈ h. we have for each f ∈ h,∑ j∈j v2j 〈λjpwjcf, λjpwjcf 〉≤ b〈cf,cf 〉≤ b||c|| 2〈f , f 〉. (3.13) on the other hand for each f ∈ h a〈f , f 〉 = a〈c−1cf,c−1cf 〉≤ a||c−1||2〈cf,cf 〉 ≤ ||c−1||2 ∑ j∈j v2j 〈λjpwjcf, λjpwjcf 〉. (3.14) so from (3.13) and (4.1), we have a||c−1||−2〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjcf 〉≤ b||c|| 2〈f , f 〉, ∀f ∈ h. 8 int. j. anal. appl. (2022), 20:1 we conclude that {wj, λj,vj}j∈j is a c2−controlled g−fusion frame for h. converselly, let {wj, λj,vj}j∈j be a c2−controlled g−fusion frame for h with bounds a ′ and b ′ . then for all f ∈ h, a ′ 〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjcf 〉≤ b ′ 〈f , f 〉 we have for each f ∈ h, ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉 = ∑ j∈j v2j 〈λjpwjcc −1f , λjpwjcc −1f 〉 ≤ b ′ 〈c−1f ,c−1f 〉 ≤ b ′ ||c−1||2〈f , f 〉. (3.15) also for each f ∈ h, a ′ 〈c−1f ,c−1f 〉≤ ∑ j∈j v2j 〈λjpwjcc −1f , λjpwjcc −1f 〉 = ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉 and a ′ ||(c−1c−1)−1||−1〈f , f 〉≤ a ′ 〈c−1f ,c−1f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉 (3.16) from (3.15) and (3.16), we have a ′ ||(c−2)−1||−1〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉≤ b ′ ||c−1||2〈f , f 〉, ∀f ∈ h. hence {wj, λj,vj}j∈j is a g−fusion frame for h. � theorem 3.4. let c, c ′ ∈ gl+(h), and c, c ′ commute with each other and commute with pwj λ ∗ j λjpwj for all j ∈ j. then λcc′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion bessel sequence for h with bound b if and only if the operator t (c,c ′ ) : l2({hj}j∈j) → h given by t (c,c ′ ) ({gj}j∈j) = ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j gj, ∀{gj}j∈j ∈ l 2({hj}j∈j). is well defined and bounded operator with, ||t (c,c ′ ) || ≤ √ b. proof. let λ cc ′ is a (c,c ′ )−controlled g−fusion bessel sequence with bound b for h. as a result of theorem 3.1, || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| ≤ b||f ||2, ∀f ∈ h. (3.17) int. j. anal. appl. (2022), 20:1 9 for any {gj}j∈j ∈ l2({hj}j∈j), ||t (c,c ′ ) ({gj}j∈j)|| = sup ||f ||=1 ||〈t (c,c ′ ) ({gj}j∈j), f 〉|| = sup ||f ||=1 ||〈 ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j gj, f 〉|| = sup ||f ||=1 || ∑ j∈j 〈vj(cc ′ ) 1 2 pwj λ ∗ j gj, f 〉|| = sup ||f ||=1 || ∑ j∈j 〈gj,vjλjpwj (cc ′ ) 1 2 f 〉|| ≤ sup ||f ||=1 || ∑ j∈j 〈gj,gj〉|| 1 2 || ∑ j∈j v2j 〈λjpwj (cc ′ ) 1 2 f , λjpwj (cc ′ ) 1 2 f 〉|| 1 2 = sup ||f ||=1 || ∑ j∈j 〈gj,gj〉|| 1 2 || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| 1 2 ≤ sup ||f ||=1 || ∑ j∈j 〈gj,gj〉|| 1 2 √ b||f || = √ b||{gj}j∈j||. therefore, the sum ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j gj is convergent, and we have ||t (c,c ′ ) ({gj}j∈j)|| ≤ √ b||{gj}j∈j|| hence the operator t (c,c ′ ) is well defined, bounded and ||t (c,c ′ ) || ≤ √ b. for the converse, suppose that the operator t (c,c ′ ) is well defined, bounded and ||t (c,c ′ ) || ≤ √ b. for all f ∈ h, we have || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| = || ∑ j∈j v2j 〈c ′ pwj λ ∗ j λjpwjcf,f 〉|| = || ∑ j∈j v2j 〈(cc ′ ) 1 2 pwj λ ∗ j λjpwj (cc ′ ) 1 2 f , f 〉|| = ||〈t (c,c ′ ) ({gj}j∈j), f 〉|| ≤ ||t (c,c ′ ) ||||{gj}j∈j||||f || = ||t (c,c ′ ) |||| ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| 1 2 ||f || where gj = vjλjpwj (cc ′ ) 1 2 f . hence || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| 1 2 ≤ √ b||f || then || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| ≤ b||f ||2 (3.18) 10 int. j. anal. appl. (2022), 20:1 the adjoint operator of t (c,c ′ ) is given by t∗ (c,c ′ ) (g) = {vjλjpwj (cc ′ ) 1 2 g}j∈j, ∀g ∈ h. and we have for each f ∈ h || ∑ j∈j 〈λjpwjcf, λjpwjc ′ f 〉|| = || ∑ j∈j v2j 〈λjpwj (cc ′ ) 1 2 f , λjpwj (cc ′ ) 1 2 f 〉|| = ||〈t∗ (c,c ′ ) (f ),t∗ (c,c ′ ) (f )〉|| = ||t∗ (c,c ′ ) (f )||2 frome (3.18), we have ||t∗ (c,c ′ ) (f )|| ≤ √ b||f ||, ∀f ∈ h. so, t∗ (c,c ′ ) is bounded a−linear operator, then there exist a constant m > 0 such that 〈t∗ (c,c ′ ) f ,t∗ (c,c ′ ) f 〉≤ m〈f , f 〉, ∀f ∈ h. hence ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ m〈f , f 〉. ∀f ∈ h. this give that λ cc ′ is a (c,c ′ )−controlled g−fusion bessel sequence for h. � theorem 3.5. let {wj, λj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h with bounds a and b, with operator frame s (c,c ′ ) . let θ ∈ end∗a(h) be injective and has a closed range. suppose that θ commute with c, c ′ and pwj for all j ∈ j. then {wj, λjθ,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h. proof. let {wj, λj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h with bounds a and b, then a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ b〈f , f 〉, ∀f ∈ h. for each f ∈ h, we have∑ j∈j v2j 〈λjθpwjcf, λjθpwjc ′ f 〉 = ∑ j∈j v2j 〈λjpwjcθf, λjpwjc ′ θf 〉 ≤ b〈θf,θf 〉 ≤ b||θ||2〈f , f 〉 (3.19) and a〈θf,θf 〉≤ ∑ j∈j v2j 〈λjθpwjcf, λjθpwjc ′ f 〉, by lemma 2.3, we have a||(θ∗θ)−1||−1〈f , f 〉≤ a〈θf,θf 〉 int. j. anal. appl. (2022), 20:1 11 so a||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈j v2j 〈λjθpwjcf, λjθpwjc ′ f 〉 (3.20) using (3.19) and (3.20) we conclude that a||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈j v2j 〈λjθpwjcf, λjθpwjc ′ f 〉≤ b||θ||2〈f , f 〉, ∀f ∈ h. therefore {wj, λjθ,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h. � theorem 3.6. let {wj, λj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h. with bounds a and b. let θ ∈ end∗a(l,h) be injective and has a closed range. suppose that θ commute with λjpwjc and λjpwjc ′ for all j ∈ j. then {wj,θλj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h. proof. let {wj, λj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h with bounds a and b, then a〈f , f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ b〈f , f 〉, ∀f ∈ h. we have for each f ∈ h∑ j∈j v2j 〈θλjpwjcf,θλjpwjc ′ f 〉≤ ||θ||2 ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉 ≤ b||θ||2〈f , f 〉 (3.21) on the other hand, a〈θf,θf 〉≤ ∑ j∈j v2j 〈θλjpwjcf,θλjpwjc ′ f 〉 = ∑ j∈j v2j 〈λjpwjcθf, λjpwjc ′ θf 〉 by lemma 2.3, we have a||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈j v2j 〈θλjpwjcf,θλjpwjc ′ f 〉 (3.22) using (3.21) and (3.22) , we conclude that a||(θ∗θ)−1||−1〈f , f 〉≤ ∑ j∈j v2j 〈θλjpwjcf,θλjpwjc ′ f 〉≤ b||θ||2〈f , f 〉, ∀f ∈ h. hence, {wj,θλj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for h. � under wich conditions a (c,c ′ )−controlled g−fusion frame for h with h a c∗−module over a unital c∗−algebras a is also a (c,c ′ )−controlled g−fusion frame for h with h a c∗−module over a unital c∗−algebras b. the following theorem answer this questions. we teak in next theorem hj ⊂ h, ∀j ∈ j. 12 int. j. anal. appl. (2022), 20:1 theorem 3.7. let (h,a,〈., .〉a) and (h,b,〈., .〉b) be two hilbert c∗−modules and let φ : a → b be a ∗−homomorphisme and θ be a map on h such that 〈θf,θg〉b = φ(〈f ,g〉a) for all f , g ∈ h. suppose that λ cc ′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled g−fusion frame for (h,a,〈., .〉a) with frame operator sa and lower and upper bounds a and b respectively. if θ is surjective such that θλjpwj = λjpwjθ for each j ∈ j and θc = cθ and θc ′ = c ′ θ. then {wj, λj,φ(vj)}j∈j is a (c,c ′ )−controlled g−fusion frame for (h,b,〈., .〉b) with frame operator sb and lower and upper bounds a and b respectively and 〈sbθf,θg〉b = φ(〈saf ,g〉a). proof. since θ is surjective, then for every g ∈ h there exists f ∈ h such that θf = g. using the definition of (c,c ′ )−controlled g−fusion frame for (h,a,〈., .〉a) we have a〈f , f 〉a ≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉a ≤ b〈f , f 〉a by lemma 2.1 we have φ ( a〈f , f 〉a ) ≤ φ (∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉a ) ≤ φ ( b〈f , f 〉a ) frome the definition of ∗−homomorphisme we have aφ ( 〈f , f 〉a ) ≤ ∑ j∈j φ(v2j )φ ( 〈λjpwjcf, λjpwjc ′ f 〉a ) ≤ bφ ( 〈f , f 〉a ) using the relation betwen θ and φ we get a〈θf,θf 〉b ≤ ∑ j∈j φ(vj) 2〈θλjpwjcf,θλjpwjc ′ f 〉b ≤ b〈θf,θf 〉b since θλjpwj = λjpwjθ for each j ∈ j and θc = cθ and θc ′ = c ′ θ we have a〈θf,θf 〉b ≤ ∑ j∈j φ(vj) 2〈λjpwjcθf, λjpwjc ′ θf 〉b ≤ b〈θf,θf 〉b therefore, a〈g,g〉b ≤ ∑ j∈j φ(vj) 2〈λjpwjcg, λjpwjc ′ g〉b ≤ b〈g,g〉b, ∀g ∈ h. this implies that {wj, λj,φ(vj)}j∈j is a (c,c ′ )−controlled g−fusion frame for (h,b,〈., .〉b) with bounds a and b. moreover we have φ ( 〈saf ,g〉a ) = φ ( 〈 ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjcf,g〉a ) = ∑ j∈j φ(vj) 2φ ( 〈λjpwjcf, λjpwjc ′ g〉a ) int. j. anal. appl. (2022), 20:1 13 = ∑ j∈j φ(vj) 2〈θλjpwjcf,θλjpwjc ′ g〉b = 〈 ∑ j∈j φ(vj) 2c ′ pwj λ ∗ j λjpwjcθf,θg〉b = 〈sbθf,θg〉b. � 4. (c,c ′ )−controlled k −g−fusion frames in hilbert c∗−modules firstly we give the definition of k −g−fusion frame in hilbert c∗−modules. definition 4.1. [11] let a be a unital c∗−algebra and h be a countably generated hilbert a−module. let ( vj ) j∈j be a family of weights in a,i.e.,each vj is a positive invertible element frome the center of a, let ( wj ) j∈j be a collection of orthogonally complemented closed submodules of h. let ( kj ) j∈j a sequence of closed submodules of k and λj ∈ end∗a(h,kj) for each j ∈ j and k ∈ end ∗ a(h). we say λ = (wj, λj,vj)j∈j is k − g−fusion frame for h with respect to (kj)j∈j if there exist real constants 0 < a ≤ b < ∞ such that a〈k∗f ,k∗f 〉≤ ∑ j∈j v2j 〈λjpwjf , λjpwjf 〉≤ b〈f , f 〉, ∀f ∈h. (4.1) the constants a and b are called a lower and upper bounds of k −g−fusion frame, respectively. if the left-hand inequality of (4.1) is an equality, we say that λ is a tight k−g−fusion frame. if k = ih then λ is a g−fusion frame and if k = ih and λj = pwj for any j ∈ j, then λ is a fusion frame for h definition 4.2. let c, c ′ ∈ gl+(h) and k ∈ end∗a(h). {wj}j∈j be a sequence of closed submodules orthogonally complemented of h, {vj}j∈j be a family of weights in a, i.e., each vj is a positive invertible element frome the center of a and λj ∈ end∗a(h,hj) for each j ∈ j. we say λcc′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled k −g−fusion frame for h if there exists 0 < a ≤ b < ∞ such that a〈k∗f ,k∗f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ b〈f , f 〉, ∀f ∈ h. (4.2) the constants a and b are called a lower and upper bounds of (c,c ′ )−controlled k − g−fusion frame, respectively. if the left-hand inequality of (4.2) is an equality, we say that λ cc ′ is a tight (c,c ′ )−controlled k −g−fusion frame for h. remark 4.1. if λ cc ′ is a (c,c ′ )−controlled k−g−fusion frame for h with bounds a and b we have akk∗ ≤ s (c,c ′ ) ≤ bih. (4.3) from equality (3.6) and inequality (4.3) we have 14 int. j. anal. appl. (2022), 20:1 proposition 4.1. let k ∈ end∗a(h), and λcc′ be a (c,c ′ )−controlled g−fusion bessel sequence for h. then λ cc ′ is a (c,c ′ )−controlled k −g−fusion frame for h if and only if there exist a constant a > 0 such that akk∗ ≤ s (c,c ′ ) where s (c,c ′ ) is the frame operator for λ cc ′. theorem 4.1. let λ cc ′ = {wj, λj,vj}j∈j and γcc′ = {vj, γj,uj}j∈j be two (c,c ′ )−controlled g−fusion bessel sequences for h with bounds b1 and b2, respectively. suppose that tλ cc ′ and tγ cc ′ be their synthesis operators such that tγ cc ′t ∗ λ cc ′ = k∗ for some k ∈ end∗a(h). then, both λ cc ′ and γ cc ′ are (c,c ′ )−controlled k and k∗ −g−fusion frames for h, respectively. proof. for each f ∈ h, we have 〈k∗f ,k∗f 〉 = 〈tγ cc ′t ∗ λ cc ′ f ,tγcc′ t∗λ cc ′ f 〉≤ ||tγcc′ || 2〈t∗λ cc ′ ,t ∗ λ cc ′ f 〉 ≤ b2 ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉, hence b−12 〈k ∗f ,k∗f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉. this means that λ cc ′ is a (c,c ′ )−controlled k − g−fusion frame for h. similarly, γ cc ′ is a (c,c ′ )−controlled k∗ −g−fusion frame for h with the lower bound b−11 . � theorem 4.2. let u ∈ end∗a(h) be an invertible operator on h and λcc′ = {wj, λj,vj}j∈j be a (c,c ′ )−controlled k −g−fusion frame for h for some k ∈ end∗a(h). suppose that u ∗uwj ⊂ wj, ∀j ∈ j and c, c ′ commute with u. then γ cc ′ = {uwj, λjpwju ∗,vj}j∈j is a (c,c ′ )−controlled uku∗ −g−fusion frame for h. proof. since λ cc ′ is a (c,c ′ )−controlled k −g−fusion frame for h, ∃ a,b > 0 such that a〈k∗f ,k∗f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ b〈f , f 〉, ∀f ∈ h. also, u is an invertible linear operator on h, so for any j ∈ j, uwj is closed in h. now, for each f ∈ h, using lemma 2.7, we obtain ∑ j∈j v2j 〈λjpwju ∗puwjcf, λjpwju ∗puwjc ′ f 〉 = ∑ j∈j v2j 〈λjpwju ∗cf, λjpwju ∗c ′ f 〉 = ∑ j∈j v2j 〈λjpwjcu ∗f , λjpwjc ′ u∗f 〉 ≤ b〈u∗f ,u∗f 〉 ≤ b||u||2〈f , f 〉. int. j. anal. appl. (2022), 20:1 15 on the other hand, for each f ∈ h a〈(uku∗)∗f , (uku∗)∗f 〉 = a〈uk∗u∗f ,uk∗u∗f 〉 ≤ a||u||2〈k∗u∗f ,k∗u∗f 〉 ≤ ||u||2 ∑ j∈j v2j 〈λjpwjc(u ∗f ), λjpwjc ′ (u∗f )〉 = ||u||2 ∑ j∈j v2j 〈λjpwju ∗cf, λjpwju ∗c ′ f 〉 ≤ ||u||2 ∑ j∈j v2j 〈λjpwju ∗puwjcf, λjpwju ∗puwjc ′ f 〉, then a ||u||2 〈(uku∗)∗f , (uku∗)∗f 〉≤ ∑ j∈j v2j 〈λjpwju ∗puwjcf, λjpwju ∗puwjc ′ f 〉 therefore, γ cc ′ is a (c,c ′ )−controlled uku∗ −g−fusion frame for h. � theorem 4.3. let u ∈ end∗a(h) be an invertible operator on h and γcc′ = {uwj, λjpwju ∗,vj}j∈j be a (c,c ′ )−controlled k−g−fusion frame for h for some k ∈ end∗a(h). suppose that u ∗uwj ⊂ wj, ∀j ∈ j and c, c ′ commute with u. then λ cc ′ = {wj, λj,vj}j∈j is a (c,c ′ )−controlled u−1ku − g−fusion frame for h. proof. since γ cc ′ = {uwj, λjpwju ∗,vj}j∈j is a (c,c ′ )−controlled k −g−fusion frame for h, ∃ a, b > 0 such that a〈k∗f ,k∗f 〉≤ ∑ j∈j v2j 〈λjpwju ∗puwjcf, λjpwju ∗puwjc ′ f 〉≤ b〈f , f 〉. ∀f ∈ h. let f ∈ h, we have a〈(u−1ku)∗f , (u−1ku)∗f 〉 = a〈u∗k∗(u−1)∗f ,u∗k∗(u−1)∗f 〉 ≤ a||u||2〈k∗(u−1)∗f ,k∗(u−1)∗f 〉 ≤ ||u||2 ∑ j∈j v2j 〈λjpwju ∗puwjc(u −1)∗f , λjpwju ∗puwjc ′ (u−1)∗f 〉 ≤ ||u||2 ∑ j∈j v2j 〈λjpwju ∗c(u−1)∗f , λjpwju ∗c ′ (u−1)∗f 〉 = ||u||2 ∑ j∈j v2j 〈λjpwju ∗(u−1)∗cf, λjpwju ∗(u−1)∗c ′ f 〉 = ||u||2 ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉. then, for each f ∈ h, we have a ||u||2 〈(u−1ku)∗f , (u−1ku)∗f 〉≤ ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉. 16 int. j. anal. appl. (2022), 20:1 also, for each f ∈ h, we have ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉 = ∑ j∈j v2j 〈λjpwjcu ∗(u−1)∗f , λjpwjc ′ u∗(u−1)∗f 〉 = ∑ j∈j v2j 〈λjpwju ∗c(u−1)∗f , λjpwju ∗c ′ (u−1)∗f 〉 = ∑ j∈j v2j 〈λjpwju ∗puwjc(u −1)∗f , λjpwju ∗puwjc ′ (u−1)∗f 〉 ≤ b〈(u−1)∗f , (u−1)∗f 〉 ≤ b||u−1||2〈f , f 〉. thus, λ cc ′ is a (c,c ′ )−controlled u−1ku −g−fusion frame for h. � theorem 4.4. let k ∈ end∗a(h) be an invertible operator on h and λcc′ = {wj, λj,vj}j∈j be a (c,c ′ )−controlled g−fusion frame for h with frame bounds a, b and s (c,c ′ ) be the associated (c,c ′ )−controlled g−fusion frame operator. suppose that for all j ∈ j, t∗twj ⊂ wj, where t = ks−1 (c,c ′ ) and c, c ′ commute with t . then {ks−1 (c,c ′ ) wj, λjpwjs −1 (c,c ′ ) k∗,vj}j∈j is a (c,c ′ )−controlled k − g−fusion frame for h with the corresponding (c,c ′ )−controlled g−fusion frame operator ks−1 (c,c ′ ) k∗. proof. we now t = ks−1 (c,c ′ ) is invertible on h and t∗ = (ks−1 (c,c ′ ) )∗ = s−1 (c,c ′ ) k∗. for each f ∈ h, we have 〈k∗f ,k∗f 〉 = 〈s (c,c ′ ) s−1 (c,c ′ ) k∗f ,s (c,c ′ ) s−1 (c,c ′ ) k∗f 〉 ≤ ||s (c,c ′ ) ||2〈s−1 (c,c ′ ) k∗f ,s−1 (c,c ′ ) k∗f 〉 ≤ b2〈s−1 (c,c ′ ) k∗f ,s−1 (c,c ′ ) k∗f 〉. now for each f ∈ h, we get ∑ j∈j v2j 〈λjpwjt ∗ptwjc(f ), λjpwjt ∗ptwjc ′ (f )〉 = ∑ j∈j v2j 〈λjpwjt ∗c(f ), λjpwjt ∗c ′ (f )〉 = ∑ j∈j v2j 〈λjpwjct ∗(f ), λjpwjc ′ t∗(f )〉 ≤ b〈t∗f ,t∗f 〉 ≤ b||t ||2〈f , f 〉 ≤ b||s−1 (c,c ′ ) ||2||k||2〈f , f 〉 ≤ b a2 ||k||2〈f , f 〉. int. j. anal. appl. (2022), 20:1 17 on the other hand, for each f ∈ h, we have∑ j∈j v2j 〈λjpwjt ∗ptwjc(f ), λjpwjt ∗ptwjc ′ (f )〉 = ∑ j∈j v2j 〈λjpwjt ∗c(f ), λjpwjt ∗c ′ (f )〉 = ∑ j∈j v2j 〈λjpwjct ∗(f ), λjpwjc ′ t∗(f )〉 ≥ a〈t∗f ,t∗f 〉 = a〈s−1 (c,c ′ ) k∗f ,s−1 (c,c ′ ) k∗f 〉 ≥ a b2 〈k∗f ,k∗f 〉. thus {ks−1 (c,c ′ ) wj, λjpwjs −1 (c,c ′ ) k∗,vj}j∈j is a (c,c ′ )−controlled k −g−fusion frame for h. for each f ∈ h, we have∑ j∈j v2j c ′ ptwj (λjpwjt ∗)∗(λjpwjt ∗)ptwjcf = ∑ j∈j v2j c ′ ptwjtpwj λ ∗ j (λjpwjt ∗)ptwjcf = ∑ j∈j v2j c ′ (pwjt ∗ptwj ) ∗λ∗j λj(pwjt ∗ptwj)cf = ∑ j∈j v2j c ′ tpwj λ ∗ j λjpwjt ∗cf = ∑ j∈j v2j tc ′ pwj λ ∗ j λjpwjct ∗f = t ( ∑ j∈j v2j c ′ pwj λ ∗ j λjpwjct ∗f ) = ts (c,c ′ ) t∗(f ) = ks−1 (c,c ′ ) k∗(f ). this implies that ks−1 (c,c ′ ) k∗ is the associated (c,c ′ )−controlled g−fusion frame operator. � the next theorem we give an equivqlent definition of (c,c ′ )−controlled k −g−fusion frame. theorem 4.5. let k ∈ end∗a(h). then λcc′ is a (c,c ′ )−controlled k −g−fusion frame for h if and only if there exist constants a, b > 0 such that a||k∗f ||2 ≤ || ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉|| ≤ b||f ||2, ∀f ∈ h. (4.4) proof. evidently, every (c,c ′ )−controlled k −g−fusion frame for h satisfies (4.4). for the converse, we suppose that (4.4) holds. for any {fj}j∈j ∈ l2({hj}j∈j), || ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j fj|| = sup ||g||=1 ||〈 ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j fj,g〉|| = sup ||g||=1 || ∑ j∈j 〈vj(cc ′ ) 1 2 pwj λ ∗ j fj,g〉|| 18 int. j. anal. appl. (2022), 20:1 = sup ||g||=1 || ∑ j∈j 〈fj,vjλjpwj (cc ′ ) 1 2 g〉|| ≤ sup ||g||=1 || ∑ j∈j 〈fj, fj〉|| 1 2 || ∑ j∈j v2j 〈λjpwj (cc ′ ) 1 2 g, λjpwj (cc ′ ) 1 2 g〉|| 1 2 = sup ||g||=1 || ∑ j∈j 〈fj, fj〉|| 1 2 || ∑ j∈j v2j 〈λjpwjcg, λjpwjc ′ g〉|| 1 2 ≤ sup ||g||=1 || ∑ j∈j 〈fj, fj〉|| 1 2 √ b||g|| = √ b||{fj}j∈j||. thus the series ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j fj converges in h unconditionally. since 〈tf,{fj}j∈j〉 = ∑ j∈j 〈vjλjpwj (cc ′ ) 1 2 f , fj〉 = 〈f , ∑ j∈j vj(cc ′ ) 1 2 pwj λ ∗ j fj〉. t is adjointable. now for each f ∈ h we have 〈tf,tf 〉 = ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉≤ ||t ||2〈f , f 〉. on the other hand the left-hand inequality of (4.4) gives ||k∗f ||2 ≤ 1 a ||tf ||2, ∀f ∈ h. then the lemma 2.6 implies that there exist a constant µ > 0 such that kk∗ ≤ µt∗t, and hence 1 µ 〈k∗f ,k∗f 〉≤ 〈tf,tf 〉 = ∑ j∈j v2j 〈λjpwjcf, λjpwjc ′ f 〉, ∀f ∈ h. consequently, λ cc ′ is a (c,c ′ )−controlled k −g−fusion frame for h. � 5. perturbation of (c,c ′ )−controlled k −g−fusion frame in hilbert c∗−modules theorem 5.1. let λ cc ′ = {wj, λj,vj}j∈j be a (c,c ′ )−controlled k − g−fusion frame for h with frame bounds a, b and γj ∈ end∗a(h,hj). suppose that for each f ∈ h, ||((vjλjpwj −ujγjpvj )(cc ′ ) 1 2 f )j∈j|| ≤λ1||(vjλjpwj (cc ′ ) 1 2 f )j∈j||+ λ2||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| + �||k∗f ||. where 0 < λ1,λ2 < 1 and � > 0 such that � < (1 −λ1) √ a. then {wj, γj,uj}j∈j is a (c,c ′ )−controlled k −g−fusion frame for h. int. j. anal. appl. (2022), 20:1 19 proof. we have for each f ∈h || ∑ j∈j u2j 〈γjpvjcf, γjpvjc ′ f 〉|| 1 2 = ||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| = ||(ujγjpvj (cc ′ ) 1 2 f )j∈j + (vjλjpwj (cc ′ ) 1 2 f )j∈j − (vjλjpwj (cc ′ ) 1 2 f )j∈j|| ≤ ||((ujγjpvj −vjλjpwj )(cc ′ ) 1 2 f )j∈j|| + ||(vjλjpwj (cc ′ ) 1 2 f )j∈j|| ≤ (λ1 + 1)||(vjλjpwj (cc ′ ) 1 2 f )j∈j|| + λ2||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| + �||k∗f ||. so (1 −λ2)||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| ≤ (λ1 + 1) √ b||f || + �||k∗f ||. then ||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| ≤ (λ1 + 1) √ b||f || + �||k∗f || 1 −λ2 ≤ ( (λ1 + 1) √ b + �||k|| 1 −λ2 )||f ||. hence || ∑ j∈j u2j 〈γjpvjcf, γjpvjc ′ f 〉|| ≤ ( (λ1 + 1) √ b + �||k|| 1 −λ2 )2||f ||2. on the other hand for each f ∈h || ∑ j∈j u2j 〈γjpvjcf, γjpvjc ′ f 〉|| 1 2 = ||(ujγjpvj (cc ′ ) 1 2 f )j∈j|| = ||((ujγjpvj −vjλjpwj )(cc ′ ) 1 2 f )j∈j + (vjλjpwj (cc ′ ) 1 2 f )j∈j|| ≥ ||(vjλjpwj (cc ′ ) 1 2 f )j∈j|| − ||((ujγjpvj −vjλjpwj )(cc ′ ) 1 2 f )j∈j|| ≥ (1 −λ1)||(vjλjpwj (cc ′ ) 1 2 f )j∈j|| −λ2||(ujγjpvj (cc ′ ) 1 2 f )j∈j||− �||k∗f ||. hence || ∑ j∈j u2j 〈γjpvjcf, γjpvjc ′ f 〉|| ≥ ( (1 −λ1) √ a− � 1 + λ2 )2||k∗f ||2. by theorem 4.5, we conclude that {vj, γj,uj}j∈j is a (c,c ′ )−controlled k−g−fusion frame for h. � 20 int. j. anal. appl. (2022), 20:1 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. alijani, m. dehghan, ∗-frames in hilbert c∗-modules, u.p.b. sci. bull. ser. a, 73 (4) (2011), 89–106. [2] l. arambasic, on frames for countably generated hilbert c∗-modules, proc. amer. math. soc. 135 (2006), 469–478. https://doi.org/10.1090/s0002-9939-06-08498-x. [3] p. balazs, j.-p. antoine, a. grybos, weighted and controlled frames: mutual relationship and first numerical properties, int. j. wavelets multiresolut. inf. process. 08 (2010), 109–132. https://doi.org/10.1142/ s0219691310003377. [4] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341–366. https://doi.org/10.1090/s0002-9947-1952-0047179-6. [5] m. frank, d. r. larson, a-module frame concept for hilbert c∗-modules, functional and harmonic analysis of wavelets, contempt. math. 247 (2000), 207-233. [6] d. gabor, theory of communication, j. inst. electric. eng. part iii, 93 (1946), 429–457. [7] s. kabbaj, m. rossafi, ∗-operator frame for end∗a(h), wavelet linear algebra, 5 (2018), 1-13. [8] i. kaplansky, modules over operator algebras, amer. j. math. 75 (1953), 839. https://doi.org/10.2307/2372552. [9] a. khosravi, b. khosravi, fusion frames and g-frames in hilbert c∗-modules, int. j. wavelets multiresolut. inf. process. 06 (2008), 433–446. https://doi.org/10.1142/s0219691308002458. [10] m. r. kouchi and a. rahimi, on controlled frames in hilbert c∗−modules, int. j. wavelets multiresolut. inf. process. 15 (2017), 1750038. https://doi.org/10.1142/s0219691317500382. [11] f. d. nhari, r. echarghaoui, m. rossafi, k−g−fusion frames in hilbert c∗−modules, int. j. anal. appl. 19 (2021), 836-857. https://doi.org/10.28924/2291-8639-19-2021-836. [12] w. paschke, inner product modules over b∗-algebras, trans. amer. math. soc. 182 (1973), 443-468. https: //doi.org/10.1090/s0002-9947-1973-0355613-0. [13] m. rossafi, s. kabbaj, ∗-k-operator frame for end∗a(h), asian-eur. j. math. 13 (2020), 2050060. https://doi. org/10.1142/s1793557120500606. [14] m. rossafi, s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra, 8 (2019), 85-95. [15] m. rossafi, s. kabbaj, ∗-k-g-frames in hilbert a-modules, j. linear topol. algebra, 7 (2018), 63-71. [16] m. rossafi, s. kabbaj, ∗-g-frames in tensor products of hilbert c∗-modules, ann. univ. paedagog. crac. stud. math. 17 (2018), 17-25. [17] m. rossafi, s. kabbaj, generalized frames for b(h,k), iran. j. math. sci. inf. accepted. [18] x. fang, m.s. moslehian, q. xu, on majorization and range inclusion of operators on hilbert c∗-modules, linear multilinear algebra. 66 (2018), 2493–2500. https://doi.org/10.1080/03081087.2017.1402859. https://doi.org/10.1090/s0002-9939-06-08498-x https://doi.org/10.1142/s0219691310003377 https://doi.org/10.1142/s0219691310003377 https://doi.org/10.1090/s0002-9947-1952-0047179-6 https://doi.org/10.2307/2372552 https://doi.org/10.1142/s0219691308002458 https://doi.org/10.1142/s0219691317500382 https://doi.org/10.28924/2291-8639-19-2021-836 https://doi.org/10.1090/s0002-9947-1973-0355613-0 https://doi.org/10.1090/s0002-9947-1973-0355613-0 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1080/03081087.2017.1402859 1. introduction 2. preliminaires 3. controlled g-fusion frame in hilbert c-modules 4. (c,c')-controlled k-g-fusion frames in hilbert c-modules 5. perturbation of (c,c')-controlled k-g-fusion frame in hilbert c-modules references international journal of analysis and applications volume 18, number 6 (2020), 939-956 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-939 a polynomial linear regression approach to estimate sensitive parameters in the novel double diabetes model rashida hussain1, asma arbab1, salahuddin2, mohammad munir3, nasreen kausar4,∗ and asghar ali1 1department of mathematics, mirpur university of science and technology, mirpur-ajk, pakistan 2department of mathematics, jazan university, jazan, kingdom of saudi arabia 3department of mathematics, government postgraduate college, abbottabad, pakistan 4department of mathematics, university of agriculture, faisalabad, pakistan ∗corresponding author: kausar.nasreen57@gmail.com abstract. sensitivity analysis characterizes the changes in the model outputs due to the changes in the model parameters. in this article, we estimate the most sensitive parameters in the novel double diabetes model (nddm) through the polynomial linear regression approach; this way we develop a direct relation between the sensitivity analysis and the paramter estimation. the nddm has more than seventeen parameters, and estimating them simultanously is difficult. we select the most commonly used five parameters in the glucose-insulin dynamics for the sensitivity analysis. the model outputs-glucose concentrations in the plasma and the subcutaneous compartments are sensitive to the selected parameters whereas the insulin concentrations in the plasma and the subcutaneous compartment are sensitive only to the insulin transfer rate from the subcutaneous to the plasma compartment. system sensitivity of the model for the selected parameters is also in agreement with the individual sensitivities of the parameters. consequently, we estimate the parameters which are more sensitive by the polynomial linear regression approach. received july 19th, 2020; accepted august 10th, 2020; published september 3rd, 2020. 2010 mathematics subject classification. 92c45, 34a34. key words and phrases. glucose-insulin dynamics; compartmental model; sensitivity analysis; linear regression. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 939 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-939 int. j. anal. appl. 18 (6) (2020) 940 1. introduction mathematical modeling possesses a pivotal position in understanding the different physiological processes [12]. one of the most important biological processes is the glucose-insulin regulatory system [5], [1]. in order to understand the physiological behavior of this system, various mathematical models have been proposed by different scientists, and are available in the literature. some of these models are very helpful for the development of a closed-loop algorithm in order to control the bloodglucose concentration in the body [18]. bergman et al. proposed the four compartmental mathematical model of glucose-insulin kinetic on the basis of the intravenous glucose tolerance test (ivgtt) [3] in the late seventies. this model is commonly used in the mathematical theory and in the clinical research to understand the glucose-insulin system. a detailed study and analysis of this model is found in [13]. in literature, more than 500 studies based on minimal model have been found [13]. hovarka et al. [6], sorenson [16] and puckett [15] modified minimal model and focused on type-1 diabetic patients [18]. topp et al., roy et al., introduced mathematical models [4], which deal with type-2 diabetes, however, these models ignore the effect of delays. inclusion of the delay factors are useful for the better understanding of the oscillatory behavior of the insulin and glucose regulatory system. researchers suggested two delays, hepatic glucose production(hgp) delay and insulin secretion(is) delay to be relevant to the oscillations of glucose and insulin. hgp delay was considered as the time taken for “remote insulin” to stimulate hgp to secrete glucose. is delay is defined as the time for the elevated plasma glucose to the change the insulin secretion. sturis et al., [17] proposed compartmentel model similar to bergman minimal model and introduced hgp delay to study oscillating behavior of glucose and insulin. li et al., in [9], and [10] proposed one-compartment models, considering is and hgp delays explicitly. the combined effect of the two delays was considered by using the sum of the two delays and proposed that the combined effect of the two delays influenced the dynamics of the glucose-insulin feedback mechanism, but not each individual delay. the novel double diabetes model studied in [5] considered the effect of two delays with subcutaneously injected insulin and glucose measurements. insulin administered via subcutaneous has the advantage to make the management of diabetes more effectively than intravenous. through sensitivity analysis of the novel double diabetes model, it is possible to investigate the behavior of the system which provides rich information for the diabetes control. in the present study, we present a novel diabetes model in section 2. the sensitivity analysis is briefly described in section 3 and the sensitivity studies of the nddm model is made in subsection 3.1. the detail of numerical implementation for the model is given in section 4 and the results are discussed in section 5. the general solution of the model is presented in section 6 and the selected parameters are estimated in section 7. the conclusion of the whole analysis is given in section 8. int. j. anal. appl. 18 (6) (2020) 941 2. novel double diabetes model novel double diabetes model consists of a set of six ordinary differential equations and seven subfunctions [18]. this model has seventeen parameters. the description of all state variables, sub-functions and parameters of the model are described in table( 1), table( 2), and table( 3) respectively [18]. ġp = gin + hgp −uii −e −k1gp + k2gi ġi = k1gp −k2gi −uid ˙q1a = pu−ka1q1a −lda q̇1b = u(1 −p) −ka2q1b −ldb q̇2 = ka1q1a −ka1q2 i̇p = αs + ka1q2 + ka2q1b −keip (2.1) the seven sub-functions used in the model are described below: (1) uid = β ×f3(gi) ×f4(q1a,q1b,q2), where f3(gi) = 0.01gi/vg, and f4(q1a,q1b,q2) = 4 + 90 [1 + exp(z)] , z = −1.772log[(q1a + q1b + q2) × (1/vii + 0.03/e)] + 7.76. (2) hgp(ip ) = 160 1 + exp ( 0.29 ( ip vip − 7.5 )), (3) s(gp) = 210 1 + exp(5.21 − 0.03gp vgp ) , (4) uii = −72 ( 1 − exp ( −8.267 × 10−4gp )) , (5) e(gp) =   0.0005[gp − 339bw ] if gp > 339bw, 0 if gp < 339bw, int. j. anal. appl. 18 (6) (2020) 942 variable definition unit gp amount of glucose in plasma compartment mg gi amount of glucose in subcutaneous compartment mg q2 amount of insulin in subcutaneous compartment (slow channel) mu q1a amount of insulin in subcutaneous compartment (slow channel) mu q1b amount of insulin in subcutaneous compartment (fast channel) mu ip amount of insulin in plasma compartment mu table 1. state variable of the novel model variable definition unit gin glucose intake rate pmg min −1 uii insulin independent glucose utilization mg min −1 uid insulin dependent glucose utilization mg min −1 e renal excretion mg min−1 s insulin secreted by pancreas mu min−1 lda local insulin degradation mu min −1 ldb local insulin degradation mu min −1 table 2. sub-functions of the novel model (6) lda(q1a) = vmax-ld ×q1a/(km-ld + q1a), (7) ldb(q1b) = vmax-ld ×q1b/(km-ld + q1b). the mechanism of delays plays a vital role in physiological system [18]. in the model, two delay parameters τ1 and τ2 are used which enhance the accuracy of the glucose-insulin system under examination. glucose moves in arteries and peripherals. brain and central nervous system(cns) use glucose without insulin dependency, this mechanism is known as insulin independent glucose utilization uii. liver, muscles and adipose tissue use glucose depending upon insulin; this mechanism is known as insulin dependent glucose utilization uid. int. j. anal. appl. 18 (6) (2020) 943 parameter definition values ke insulin clearance rate of plasma by kidney & liver 0.37 min −1 k1 transfer rate from plasma to subcutaneous glucose compartment 0.032 min−1 k2 transfer rate from subcutaneous to plasma glucose compartment 0.02 min−1 ka1 transfer rate from insulin subcutaneous (slow channel) to plasma insulin compartment 0.14 min−1 ka2 transfer rate from insulin subcutaneous (fast channel) to plasma insulin compartment 0.13 min−1 p proportion of insulin flux passing through slow channel 0.55 vmax-ld saturation rate for continuous infusion and bolus 235 mumin −1 insulin mass at which insulin degradation is equal to half km-ld maximal value 65 mu e insulin exchange rate between plasma and subcutaneous 0.2 min−1 τ1 hgp delay 37 mgmin −1 τ2 insulin secretion delay(is) 45 mgmin −1 vii insulin distribution volume in subcutaneous compartment 7 l/kg vip insulin distribution volume in plasma 3.15 l/kg vgp glucose distribution volume in plasma 8.4 l/kg vgi glucose distribution volume in subcutaneous compartment 7 l/kg table 3. parameters of the novel model 3. sensitivity analysis since novel model is a compartmental model with six outputs gp, gi, q1a, q1b and q2, we consider a multiple-output system with the measurable outputs [8], [14] modeled by (3.1) f(t,θ) = col(f1(t,θ), · · · ,f`(t,θ), , 0 ≤ t ≤ t, where t > 0 is fixed. the open set, θ ∈ u ⊂ rp, is the set of admissible parameters. we assume that the output model given by equation (3.1) is a valid description of the real system for all t ∈ [0,t] and θ ∈ u and the component outputs fk, k = 1, .....,`, are sufficiently smooth. θ is the vector of parameters which is to be int. j. anal. appl. 18 (6) (2020) 944 estimated. the sensitivity of the model output fk with respect to the parameter component θn, represented by sfk θn , is defined by. sfk θn (t) = lim ∆θn→∞ ∆fk(t,θ)/fk(t,θ) ∆θn/θn , n = 1, · · · ,r, k = 1, · · · ,`. the above result simplifies to (3.2) sfk θn (t) = ∂fk(t,θ) ∂θn · θn fk(t,θ) , n = 1, · · · ,r, k = 1, · · · ,`. here assume that fk and θn are non-zero k = 1, · · · ,`, and n = 1, · · · ,r. the sensitivity function describes the behavior of model function output by changing the parameters’ values [11]. it quantifies to which parameter is the model output is the most or less sensitive. the sensitivity functions are used in the parameter identification problems [7]. the system sensitivity with respect to a model parameter θn over time interval [0,t] describes the sensitivity of all model outputs fk,k = 1, · · · ,` with respect to θn, and is defined by the following relation [13]. sθn = (∑̀ k=1 ( sfk θn (t) )2)1/2 , n = 1, · · · ,r.(3.3) 3.1 sensitivity analysis of the nddm. in order to check the sensitivity of the parameters (k1,k2,ka1,ka2,ke), we use all parametric values except (k1,k2,ka1,ka2,ke) in the model. then, the model equations become ġp = −k1gp + k2gi + 160 1 + exp (0.0921ip − 2.175) − 72 ( 1 − exp ( −8.267 × 10−4gp )) ġi = k1gp −k2gi − 2.97024 × 10−3gi − 0.0668304gi 1 + 20658.4 (q1a + q1b + q2) −1.772 ˙q1a = 5.5 −ka1q1a − 235q1a 65 + q1a q̇1b = 4.5 −ka2q1b − 235q1b 65 + q1b q̇2 = ka1q1a −ka1q2 i̇p = 88.2 1 + exp (5.21 − 3.5714 × 10−3gp) + ka1q2 + ka2q1b −keip. (3.4) 4. numerical implementation we take θ = (k1,k2,ka1,ka2,ke), then a system of sensitivity equations is obtained using equation (3.2) with repsect to θ. taking the initial condition as gp(0) = 15, gi(0) = 14, q1a(0) = 2, q1b(0) = 1.5, q2(0) = 1.5, ip(0) = 1, we solve the system of sensitivity equations by using the matlab ode solver ode45 over the time interval [0, 10]. int. j. anal. appl. 18 (6) (2020) 945 5. results in first subsection, we find the sensitivities of the model outputs with respect to the model parameters, and in the second subsection we find the system senstivity of the whole model output. 5.1 sensitivites. sensitivity analysis identifies the parameters to which the model output is the most or the least sensitive [2]. the sensitivities of the model outputs with respect to the selected parameters are given in the following parts. (1) sensitivity of gp with respect to parameters k1, k2, ka1, ka2, ke: sensitivity of the plasma glucose, gp, with respect to the parameters k1, k2, ka1, ka2 and ke is presented in figure (1). it shows that the plasma glucose, gp, is sensitive to all five parameter upto 10 minutes. this means the changes in the true value of these parameters change gp output. the parameter k1, k2 are the glucose transfer rates modeling the material exchange between the plasma and the subcutaneous region. since the output, gp, is sensitive to k1 and k2, these parameters affect the evaluation of the glucose concentration in the plasma glucose region. if k1 is greater in amount, then the glucose transfers from the plasma to the subcutaneous compartment and extra glucose moves back from subcutaneous to the plasma compartment through k2, then the glucose level tends to normal range. if k1 is smaller in amount, then less amount of glucose transfers from the plasma to the subcutaneous compartment through k2 and less glucose moves from the subcutaneous to the plasma compartment, then the glucose level increases from normal range that leads to diabetes. the model output, gp, is sensitive to the paramters ka1, ka2. the glucose-insulin regulatory system shows large variation due to small changes in these parameters. if the transfer rates, ka1, ka2, are greater, than more insulin moves from the subcutaneous tissues to the plasma, and maintain the blood glucose level. plasma glucose, gp, is also sensitive to the insulin clearance rate, namely ke. when ke decreases, its results is shown in higher plasma insulin that maintains the blood glucose level. when it increases, the results is shown in terms of low plasma insulin that leads to diabetes. the combined sensitivity of the output, gp, for the selected parameter is presented in figure (3) (upper panel) which shows that gp is the most sensitive to the parameters k1, ke, ka1 and less sensitive to the parameters k2, and ka2. (2) sensitivity of gi with respect to parameters k1, k2, ka1, ka2, ke : sensitivity of the subcutaneous glucose, gi, with respect to the selected parameters is presented in figure (2). the combined sensitivity of the output, gi, for the selected parameters is presented in figure (3) (lower panel) which shows that gi is the most sensitive to the parameter k1 than the other four parameters. int. j. anal. appl. 18 (6) (2020) 946 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.18 -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 sensitivities of g p with respect to parameters k 1 , k 1 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 sensitivities of g p with respect to parameters k 2 , k 2 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 sensitivities of g p with respect to parameters k a1 , k a1 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 sensitivities of g p with respect to parameters k a2 , k a2 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 sensitivities of g p with respect to parameters k e k e fig. 1. sensitivity of gp w.r.t the selected parameters. (3) sensitivity of q1a, q1b, q2 and ip with respect to the parameters k1, k2, ka1, ka2, ke: the combined sensitivities of the outputs q1a, q1b, q2 and ip w.r.t the parameters k1, k2, ka1, ka2, ke are presented in figure (4). it is observed that q1a, q2 and ip outputs are not sensitive to the four parameters k1, k2, ka2 and ke upto 10 minutes. this means the changes in the true value of the parameters k1, k2, ka2 and ke do not bring changes in q1a, q2 and ip output. these output are only sensitive to ka1 for the 1 minutes. this implies that changes in parameter, ka1, will change in the outputs q1a, q2 and ip in the beginning and not afterwards. moreover, q1b is not sensitive to all the five parameters k1, k2, ka1, ka2 and ke upto 10 minutes. this implies the changes in the true value of the parameters k1, k2, ka1, ka2 and ke do not bring change in the output, q1b. 5.2 system sensitivities. system sensitivities of the parameters combine the individual affects of the sensitivities of both glucose and insulin with respect to all parameters. using equation (3.3), the system sensitivities of the model is evaluated, and is presented in figure (5). the information given by the time int. j. anal. appl. 18 (6) (2020) 947 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 sensitivities of g i with respect to parameters k 1 , k 1 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 sensitivities of g i with respect to parameters k 2 , k 2 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s ×10 -3 0 1 2 3 4 5 6 sensitivities of g i with respect to parameters k a1 , k a1 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s ×10 -3 0 1 2 3 4 5 6 7 sensitivities of g i with respect to parameters k a2 , k a2 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.05 -0.04 -0.03 -0.02 -0.01 0 sensitivities of g i with respect to parameters k e k e fig. 2. sensitivity of gi w.r.t the selected parameters. time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.2 -0.15 -0.1 -0.05 0 0.05 sensitivities of g p with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.2 0 0.2 0.4 0.6 0.8 1 sensitivities of g i with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e fig. 3. sensitivity of gp and gi w.r.t the all selected parameters. courses of the system sensitivities are more or less similar to the information given by their individual sensitivities. the system sensitivities indicate that the model output of the whole glucose-insulin regularuty system is the most sensitive with respect to the parameter ka1 than all the other parameters. from the system sensitivity, we quantify that the parameters in the descending order of their sensitivities are ka1, k1, ke, ka2, k2. from these results, we observe that the novel double diabetes model is majorly affected by the int. j. anal. appl. 18 (6) (2020) 948 time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 sensitivities of q 1a with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -1 -0.5 0 0.5 1 sensitivities of q 1b with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 sensitivities of q 2 with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e time (min) 0 1 2 3 4 5 6 7 8 9 10 s e n s it iv it ie s -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 sensitivities of i p with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e fig. 4. sensitivity of the insulin variables w.r.t the selected parameters. time (min) 0 1 2 3 4 5 6 7 8 9 10 s y s te m s e n s it iv it ie s 0 1 2 3 4 5 6 7 system sensitivities with respect to parameters k 1 ,k 2 ,k a1 ,k a2 ,k e k 1 k 2 k a1 k a2 k e fig. 5. system sensitivity of model rate at which insulin is injected through subcutaneous tissues. model is also affected by the rate at which glucose move from plasma to subcutaneous tissues and insulin clearance rate through the kidney and liver at which insulin deactivates. ka2 and k2 are the least sensitive, this means the changes in the true value of the parameters ka2 and k2 do not bring any major change in the model output. the sensitivity analysis signifies that the parameters are identified by the availability of data for any compartment gp, gi, q1a, q1b, q2, ip. in order to estimate model parameter, we need to find the general solution. int. j. anal. appl. 18 (6) (2020) 949 6. linearization and general solution nddm is nonlinear system. we linearize it about the equilibrium point. the equilibrium point is obtained by putting the rate of change of each state to zero, and then assigning values to parameters [18], as given in table 3. we get the feasible equilibrium point as (6.1) (gp,gi,q1a,q1b,q2,ip) = (0.7235, 1.0066, 1.5, 1.2, 1.5, 2.2982). to estimate the parameters, (k1,k2,ka1,ka2,ke), we use the model equation (3.4) and linearize it using the jacobin matrix given by equation (6.2) about the equilibrium point given in (6.1). (6.2) a =   ∂gp ∂gp ∂gp ∂gi ∂gp ∂q1a ∂gp ∂q1b ∂gp ∂q2 ∂gp ∂ip ∂gi ∂gp ∂gi ∂gi ∂gi ∂q1a ∂gi ∂q1b ∂gi ∂q2 ∂gi ∂ip ∂q1a ∂gp ∂q1a ∂gi ∂q1a ∂q1a ∂q1a ∂q1b ∂q1a ∂q2 ∂q1a ∂ip ∂q1b ∂gp ∂q1b ∂gi ∂q1b ∂q1a ∂q1b ∂q1b ∂q1b ∂q2 ∂q1b ∂ip ∂q2 ∂gp ∂q2 ∂gi ∂q2 ∂q1a ∂q2 ∂q1b ∂q2 ∂q2 ∂q2 ∂ip ∂ip ∂gp ∂ip ∂gi ∂ip ∂q1a ∂ip ∂q1b ∂ip ∂q2 ∂ip ∂ip   putting the valuse of the corresponding entries in the jacobian matrix a, we get the matrices a, a2, a3, · · · , as follow. a =   −k1 − 0.05946 k2 0 0 0 1.59065 k1 −k2 − 0.00301 −0.00002 −0.00002 −0.00002 0 0 0 −ka1 − 3.451 0 0 0 0 0 0 −ka2 − 3.4855 0 0 0 0 ka1 0 −ka1 0 3.5506 ∗ 10−3 0 0 ka2 ka1 −ke   , a2 =   k21 + 0.11892k1 +k1k2 + 0.00919 k2(−k1 − k2 − 0.06247) −0.00002k2 −0.00002k2 + 1.59065ka2 −0.00002k2 + 1.59065ka1 1.59065(−k1 − ke − 0.05946) k1(−k1 −k2 − 0.06247) k1k2 + k 2 2 +0.00602k2 +0.00001 −0.00002(−k2 −3.454) −0.00002(−k2 − ka1 − 3.48851) −0.00002(−k2 − ka2 − 3.4885) 1.59065k1 0 0 (−ka1 − 3.451)2 0 0 0 0 0 0 (−ka2 − 3.4855)2 0 0 0 0 −2k2a1 − 3.451ka1 0 (ka1) 2 0 3.5506 ∗ 10−3(−k1 −ke − 0.05946) 3.5506 ∗ 10−3k2 k2a1 ka2 (−ka2 − ke − 3.4855) −k 2 a1 − keka1 0.00565 + k 2 e   · · · . int. j. anal. appl. 18 (6) (2020) 950 general solution is x (t) = exp (at) x (0) + ∫ τ 0 exp (a(t− τ))bu (τ) dτ,(6.3) where exp(at) = i + at + a2t2/2! + a3t3/3! + · · · + aktk/k! + · · · , b =   1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1   , u = [ 0 0 1 1 1 0 ]t . when we neglect a3, and the higher power of a, we get, exp(at) = i + at + a2t2/2!. the general solution then becomes (6.4) x (t) = [ gp(t) gi(t) q1a(t) q1b(t) q2(t) ip(t) ]t , gp(t) = 150 + t[−150k1 + 132k2 + 13.69095] + t2[75(k1)2 + 6.533(k1) + 9(k1)(k2) − 66(k2)2 − 4.12(k2) + 3.5789(ka2) + 3.49(ka1) − 2.386ke − 0.133] + t3[−9.9 × 10−6k2 + 0.265ka2 + 0.265ka1] + t 4[(−k1 − 0.05946) × (−3.3 × 10−6(k2) + 0.265(ka1) + 0.265(ka2) − 1.67 × 10−5) + 9.9 × 10−6(k2)2 + 1.735(k2) − 1.7 × 10−6(ka1)(k2) − 1.7 × 10−6(ka2)(k2) + 0.1319(ka2) 2 + 1.4096(ka2) + 0.13266(0.37)(ka2) (6.5) int. j. anal. appl. 18 (6) (2020) 951 + 0.132(0.37)(ka1) + 0.39766(k1)(ka2) + 0.397(k1)(ka1) + 0.0236(ka1) + 0.0017] + t 5[−2.5 × 10−3k31 − 2.5 × 10−3k21 + 0.021k 1 + 0.534ka2k 2 1 + 0.515k1ka2 + 28.38ka2 + 0.13k1k 2 a2 + 1 × 10 −3k1k 2 e + 2.65 × 10−3k2e + 0.1325k1k 2 a2 + 0.265kek1ka2 + 0.1325kek 2 a2 + 0.469keka2 + 7.88 × 10 −3k2a2 + 0.027ka2 + 5.9 × 10−5k1ke + 1.57 × 10−4ke − 2.34 × 10−4], gi(t) = 132 + t(150k1 − 132k2 − 0.397608) + t2(−149.6k2 − 146.58k1 + 66k1k2 + 66k22 + 4.4 × 10 −5ka2 − 9.36903) + t3(6.96992 × 10−5 + 3.3 × 10−6(ka1 + k2 + ka2) + t 4(0.26511ka2k1 − 0.1326ka1k1 − 0.39771k2k1 + (1.5 × 10−5) × (k1k2 + k22 + 0.00602k2 + 0.00001) + ((−k2 − 0.00301) × (3.3 × 10 −6(k2 + ka1 + ka2)1.16259 × 10−5) + ((−3.3 × 10−6) × (ka2 + 3.4855)2 + ((−5 × 10−6) × (−k2 −ka1 − 3.4885) × (ka2 + 3.4855) + (−3.333 × 10−6) × (ka1 + 3.451)2 + ((−5 × 10−6) × (−k2 − 3.454) × (ka1 + 3.451) − 0.3977k1k2 − 0.3977k1ka1 + 3.3 × 10−6k2a1 + 1.138 × 10 −5) + t5((0.5)k1 × (−k1 −k2 − 0.06247) × (−9.9 × 10−6k2) + 0.265ka1 + 0.265ka2 + 0.5 × (k1k2 + k22 + 0.00602k2 + 0.00001) × (1.32 × 10 −5k2 + 3.3 × 10−6(ka1 + ka2) + 2.30241 × 10−5) + (−1.66 × 10−6) × (−k2 −ka1 − 3.4885) × (ka2 + 3.4855)2 + (1.666 × 10−6) × (k2 + 3.454) × (ka1 + 3.451)2 + 0.13255k1k 2 a1 + (0.13255) × (−k1k2k 2 a2 −kek1k2 − 3.4855k1ka2) − 1.3255(k1k2a1) − 1.3255k1keka1 − 1.6 × 10 −6(k2 + ka2 + 3.4885) − 5.52 × 10−6ka1(k2 + ka2 + 3.4885)), q1a(t) = t + t 2(−0.5(ka1 + 3.451)) + t3(0.167(ka1 + 3.451)2) + t4(0.25(ka1 + 3.451) 3) + t5(0.083(ka1 + 3.451) 4) − (ka1 + 3.451)3), q1b(t) = t + t 2(−0.5ka2 − 1.7443) + t3(0.167(ka2 + 3.48851)2) + t4(0.083(ka2 + 3.4885) 3) + t5(0.083(ka2 + 3.4885) 4), int. j. anal. appl. 18 (6) (2020) 952 q2(t) = t + t 3(−0.667k2a1 − 2.251ka1) + t 4(0.167(2k2a1 + 3.451ka1)ka1 − 0.167(ka1)3 + 0.167(ka1(ka1 + 3.451)2) + 0.25(−2k2a1 − 3.451ka1)(ka1 + 3.451)) + t 5(0.083(ka1) 2)(−2k2a1 − 3.451ka1) + 0.083(k4a1) + 0.083(ka1 + 3.451) 2(−2k2a1 − 3.451ka1) and ip(t) = 4.3 + t(0.5325 + 4.5ka2 + 4.4ka1 − 1.3ke) + t2(−0.26625k1 − 0.26625ke − 1.5831 × 10−3 + 0.2343k2 + 0.551k2a1 − 2.25ka2(−ka1 −ke − 3.4885) − 2.2keka1 + 2.15k2e − 2.2keka1 + 0.5ka2 + 0.5ka1) + t 3(0.5k2a1 − 0.167k2a2 − 0.5keka1 − 0.167keka2) + t 4(8.87 × 10−8k2 − 4.69 × 10−4ka2 − 4.69 × 10−4ka1 − 0.083k2eka1 − 0.25k 2 eka2 + 0.167ka2(ka2 + 3.4885) 2) + 0.25ka2(−ka2 −ke − 3.4885)(ka2 + 3.4885) + 0.167(−k3a1 − 3.451k 2 a1) + 0.167keka2(ka2 + ke + 3.4885)) + t5((1.7755 × 10−8k2) × (−k1 −ke − 0.005946) + (4.7 × 10−4ka2) × (−k1 −ke − 0.05946) + (4.71 × 10−4ka1) × (−k1 −ke − 0.05946) + (5.85 × 10−9k2)(−3k2 −ka1 −ka2 − 10.431) + (0.08k2a1) × (ka1 + 3.451)2 + 0.08ka2(−ka2 −ke − 3.4885)(ka2 + 3.4885)2 + 0.08ka2(0.00565 + k 2 e)(−ka2 −ke − 3.4885) − 0.08keka1(0.00565 + k 2 e)). general solution of the novel double diabetes does not exist in the closed form. different methods of parameter estimation exist in literature. for example, the maximum likelihood method, methods of moments, cramer-von mises method, and the least square method etc. since general solution of model is in the complex form, we are unable to convert solution into probability density function. so, estimation method like maximum likelihood method, methods of moments, cramer-von mises method are not used for estimation purposes in this case. however, model differential equation yields solution in the form of a polynomial regression. so, polynomial regression least square method is the appropriate method for the parameter estimation of the model in this case. 7. polynomial regression least square approach polynomial regression least square method is used to estimate the rate parameters k1,k2,ka1,ka2 by the collection of data of the model output, gp. through sensitive analysis of gp with respect to parameters, it int. j. anal. appl. 18 (6) (2020) 953 t 0 120 90 150 120 300 gp(t) 150 210 240 225 250 215 table 4. numerical measurements of gp has been observed that gp is more sensitive with respect to k1, ka1, ke rather than k2,ka2. so, we assign parametric values from table 3 to k2 and ka2, we only estimate the sensitive parameters k1, ka1, and ke are estimated. from equation (6.5), taking: β0 = 150, β1 = −150k1 + 132k2 + 13.69095, β2 = 75k 2 1 + 6.533k1 + 9k1k2 − 66k 2 2 − 4.12k2 + 3.5789ka2 + 3.49ka1 − 2.386ke − 0.133, β3 = −9.9 × 10−6k2 + 0.265ka2 + 0.265ka1. (7.1) and similarly adjusting β4 and β5, then gp(t) becomes: gp(t) = β0 + β1t + β2t 2 + β3t 3 + β4t 4 + β5t 5.(7.2) the equation (7.2) yields polynomial regression model gp(t) = qβ.(7.3) we consider the measurement scheme of gp(t) against the values of time t as given in table( 4) [18]. then the least square estimates β̂ are determined by β̂ = (q ′ q)−1q ′ gp(t).(7.4) by using data, equation (7.4) becomes gp(t) =   1 0 0 0 0 0 1 120 14400 1728000 207360000 2.48832 × 1010 1 90 8100 729000 65610000 5904900000 1 150 22500 3375000 506250000 7.59375 × 1010 1 120 14400 1728000 207360000 2.48832 × 1010 1 300 90000 27000000 8100000000 2.43 × 1012   ×   β0 β1 β2 β3 β4 β5   here int. j. anal. appl. 18 (6) (2020) 954 q =   1 0 0 0 0 0 1 120 14400 1728000 207360000 2.48832 × 1010 1 90 8100 729000 65610000 5904900000 1 150 22500 3375000 506250000 7.59375 × 1010 1 120 14400 1728000 207360000 2.48832 × 1010 1 300 90000 27000000 8100000000 2.43 × 1012   , β =   β0 β1 β2 β3 β4 β5   and, (q ′ q)−1q ′ =   1 0 0 0 0 0 −0.0293 −0.0753 0.1330 0.0493 −0.0753 0.0005 0.0002 −0.0011 −0.0034 −0.0035 −0.0011 −0.0055 0 0 0 −0.0001 0 0.0001 0 0 0 0 0 0 0 0 0 0 0 0   now equation (7.4) becomes β̂ = (q ′ q)−1q ′ gp(t) =   150 4.0964 −3.2647 −0.0219 −0.0017 0   , which gives β0 = 150, β1 = 4.0964, β2 = −3.2647, β3 = −0.0219, β4 = −0.0017 and β5 = 0. to solve the system of linear equations (7.1) after substituting the values of regression coefficients β0, · · · ,β5, we use mathematica software. we have gotten back the values of the sensitive parameters called the estmated values of them. these estmates are given in table( 5). int. j. anal. appl. 18 (6) (2020) 955 parameters estimated values k1 0.0816 ka1 −0.2126 ke 1.5895 table 5. estimated values of sensitive parameter 8. conclusions (1) plasma glucose gp and subcutaneous glucose gi are sensitive with respect to all selected parameters k1,k2,ka1, ka2, ke. (2) two insulin subcutaneous compartments q1a, q2 and one plasma insulin compartment ip are sensitive with respect to ka1. (3) insulin subcutaneous compartments q1b is not sensitive with respect to all parameters. (4) the system sensitivity of the model output is also in agreement with the individual sensitivity of the parameters. (5) the sensitivity analysis signifies that parameters are identified by availability of data for any compartment gp, gi, q1a, q1b, q2, ip. (6) finally, we concluded that model output gp is more sensitive with respect to k1,ka1, ke rather than k2, ka2. therefore, sensitive parameters k1,ka1, ke are estimated using polynomial regression least square method and found closed to the literature value. (7) in a future work, we want to implement this method to a more precise and studied model so that the results are more close to the real values. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] e. ackerman, l.c. gatewood, j.w. rosevear, g.d. molnar, model studies of blood-glucose regulation. j. bull math biophys. 27 (1) (1965), 21-37. [2] a. ali, m. munir, r. hussain, s. aziz, mathematical analysis of tb model, j. sci. arts. 20 (1) (2020), 119-128. [3] r.n. bergman, y.z. ider, c.r. bowden, c. cobelli, quantitative estimation of insulin sensitivity., amer. j. physiol., endocrinol. metab. 236 (1979), e667. [4] w. boutayeb, m. lamlili, a. boutayeb, m. derouich, the impact of obesity on predisposed people to type 2 diabetes: mathematical model. in: ortuño f., rojas i. (eds) bioinformatics and biomedical engineering. iwbbio 2015. lecture notes in computer science, vol 9043. springer, cham. 2015. https://doi.org/10.1007/978-3-319-16483-0 59 int. j. anal. appl. 18 (6) (2020) 956 [5] g. eigner, b. kurtan, i.j. rudas, c.c. kong, l.a. kovacs, examination of a novel double diabetes model, in: 2015 ieee 13th international symposium on applied machine intelligence and informatics (sami), ieee, herl’any, slovakia, 2015: pp. 41–46. [6] r. hovorka, f. shojaee-moradie, p.v. carroll, l.j. chassin, i.j. gowrie, n. c. jackson, r.s. tudor, a. m. umpleby, r.h. jones, partitioning glucose distribution/transport, disposal, and endogenous production during ivgtt. amer. j. physiol., endocrinol. metab. 282 (5) (2002), 992-1007. [7] f. kappel, m. munir, a new approach to optimal design problems, proc. int. conf. nonlinear anal. optim. october 6 – 10, 2008, budva (montenegro). [8] f. kappel, m. munir, generalized sensitivity functions for multiple output systems, j. inverse ill-posed probl. 25 (4) (2017), 499-519. [9] j. li, y. kuang, c. c. mason, modeling the glucose–insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays, j. theor. biol. 242(3) (2006), 722-735. [10] j. li, y. kuang, analysis of a model of the glucose-insulin regulatory system with two delays. siam j. appl. math. 67 (3) (2007), 757-776. [11] m. munir, sensitivity and generalized sensitivity studies of the sir and seir models of computer virus. proc. pakistan acad. sci. a. phys. comput. sci. 54 (2) (2017), 167-178. [12] m. munir, a. ali, and r. hussain, an improved mathematical model of solute kinetic during hemodialysis, punjab univ. j. math. 50 (1) (2018), 55-66. [13] m. munir, generalized sensitivity analysis of the minimal model of the intravenous glucose tolerance test, math. biosci. 300 (2018), 14-26. [14] m. munir, on the concept of off-diagonal generalized sensitivity functions and their relations to the parameter estimates and correlation. punjab univ. j. math. 51 (1) (2019), 61-77. [15] w.r. puckett, dynamic modeling of diabetes mellitus, phd diss., department of chemical engineering, the university of wisconsin-madision usa, 1992. [16] j. t. sorensen, a physiologic model of glucose metabolism in man and its use to design and assess improved insulin therapies for diabetes phd diss., massachusetts institute of technology usa, 1985. [17] j. sturis, k.s. polonsky, e. mosekild, e. van cauter. computer model for mechanisms underlying ultradian oscillations of insulin and glucose, amer. j. physiol., endocrinol. metab., 260 (5) (1991) e801-9. [18] w.u. zimei, mathematical models with delays for glucose-insulin regulation and applications in artificial pancreas, phd diss., national university of singapore, 2013. 1. introduction 2. novel double diabetes model 3. sensitivity analysis 3.1. sensitivity analysis of the nddm 4. numerical implementation 5. results 5.1. sensitivites 5.2. system sensitivities 6. linearization and general solution 7. polynomial regression least square approach 8. conclusions references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 171-178 http://www.etamaths.com the s-transform on hardy spaces and its duals sunil kumar singh∗ and baby kalita abstract. in this paper, continuity and boundedness results for the continuous s-transform in bmo and hardy spaces are obtained. furthermore, the continuous s-transform is also studied on the weighted bmok and weighted hardy spaces associated with a tempered weight function which was proposed by l. hörmander in the study of the theory of partial differential equations. 1. introduction the s-transform is a time-frequency localization technique that has characteristics superior to both of the fourier transform and the wavelet transform[12]. the n-dimensional continuous s-transform of a function f with respect to the window function ω is defined as [13] (1.1) (sωf)(τ,ξ) = ∫ rn f(t) ω(τ − t,ξ) e−i2π〈ξ,t〉 dt, for τ,ξ ∈ rn, provided the integral exists. in signal analysis, at least in dimension n = 1, r2n is called the time-frequency plane, and in physics r2n is called the phase space[11]. equation(1.1) can be rewritten as a convolution (1.2) (sωf)(τ,ξ) = ( f(·)e−i2π〈ξ,·〉 ∗ω(·,ξ) ) (τ). applying the convolution property for the fourier transform in (1.2), we obtain (1.3) (sωf)(τ,ξ) = f −1 { f̂(· + ξ) ω̂(·,ξ) } (τ), where f̂(η) = (ff)(η) = ∫ rn f(t) e −i2π〈η,t〉dt, is the fourier transform of f. 2. the s-transform on bmo spaces the bounded mean oscillation space bmo(rn) was first introduced by f. john and l. nirenberg in 1961 [3]. it is the dual space of the real hardy space h1 and serves in many ways as a substitute space for l∞. the bmo(rn) space has become extremely important in various areas of analysis including harmonic analysis, pdes and function theory. 2010 mathematics subject classification. 65r10, 32a37, 30h10. key words and phrases. s-transform; bmo space; hardy space. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 171 172 singh and kalita definition 2.1. the bounded mean oscillation space bmo(rn) is defined as the space of all locally lebesgue integrable functions defined on rn such that ‖ f ‖bmo= sup b⊂rn 1 |b| ∫ b |f(x) −fb|dx < ∞, here the supremum is taken over the ball b in rn of measure |b| and fb stands for the mean of f on b, namely (2.1) fb := 1 |b| ∫ b f(x)dx ≤ 1 |b| ∫ b |f(x)|dx ≤ m < ∞. lemma 2.1. let f ∈ l1(rn), then e−i2π<ξ,·>f(·) ∈ l1(rn) and ‖ e−i2π<ξ,·>f(·) ‖bmo ≤ ‖ f ‖bmo +2m where m is a constant given in equation (2.1). proof. ‖ e−i2π<ξ,·>f(·) ‖bmo = sup b⊂rn 1 |b| ∫ b ∣∣∣∣e−i2π<ξ,x>f(x) − 1|b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣dx = sup b⊂rn 1 |b| ∫ b ∣∣∣∣e−i2π<ξ,x>f(x) − e−i2π<ξ,x>|b| ∫ b f(t)dt + e−i2π<ξ,x> |b| ∫ b f(t)dt− 1 |b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣dx ≤ sup b⊂rn 1 |b| ∫ b (∣∣∣∣e−i2π<ξ,x> ( f(x) − 1 |b| ∫ b f(t)dt )∣∣∣∣ + ∣∣∣∣ 1|b| ∫ b f(t)dt ∣∣∣∣ + ∣∣∣∣ 1|b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣ ) dx ≤ sup b⊂rn 1 |b| ∫ b |f(x) −fb|dx + sup b⊂rn 1 |b| ∫ b |fb|dx + sup b⊂rn 1 |b| ∫ b ( 1 |b| ∫ b |f(t)|dt ) dx ≤ ‖ f ‖bmo + 1 |b| m|b| + 1 |b| m|b| = ‖ f ‖bmo + 2m. � theorem 2.2. suppose ω(·,ξ) ∈ l1(rn) ⋂ l2(rn), then, for any fixed ξ ∈ rn0 = rn \{0}, the operator sω : bmo(rn) → bmo(rn) is continuous. furthermore, we have ‖ (sωf)(·,ξ) ‖bmo ≤ ‖ ω(·,ξ) ‖l1 (‖ f ‖bmo +2m) . proof. for any arbitrary ball b in rn, we have (sωf)b(τ,ξ) = 1 |b| ∫ b (sωf)(τ,ξ)dτ = 1 |b| ∫ b ∫ rn e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dxdτ, the s-transform on hardy spaces and bmo 173 and hence |(sωf)(τ,ξ) − (sωf)b(τ,ξ)| = ∣∣∣∣ ∫ rn e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dx − 1 |b| ∫ b ∫ rn e−i2π<ξ,α−x>f(α−x)ω(x,ξ)dxdα ∣∣∣∣ = ∣∣∣∣ ∫ rn e−i2π<ξ,τ−x>f(τ −x)ω(x,ξ)dx − ∫ rn ω(x,ξ) ( 1 |b| ∫ b e−i2π<ξ,α−x>f(α−x)dα ) dx ∣∣∣∣ = ∣∣∣∣ ∫ rn ω(x,ξ) ( e−i2π<ξ,τ−x>f(τ −x) − 1 |b| ∫ b e−i2π<ξ,α−x>f(α−x)dα ) dx ∣∣∣∣ ≤ ∫ rn |ω(x,ξ)| ∣∣∣∣e−i2π<ξ,τ−x>f(τ −x) − 1 |b| ∫ b e−i2π<ξ,α−x>f(α−x)dα ∣∣∣∣dx. therefore, ‖ (sωf)(·,ξ) ‖bmo = sup b⊂rn 1 |b| ∫ b |(sωf)(τ,ξ) − (sωf)b(τ,ξ)|dτ ≤ sup b⊂rn 1 |b| ∫ b (∫ rn |ω(x,ξ)| ∣∣∣∣e−i2π<ξ,τ−x>f(τ −x) − 1 |b| ∫ b e−i2π<ξ,α−x>f(α−x)dα ∣∣∣∣dx ) dτ = ∫ rn |ω(x,ξ)| ( sup k⊂rn 1 |k| ∫ k ∣∣∣∣e−i2π<ξ,y>f(y) − 1 |k| ∫ k e−i2π<ξ,t>f(t)dt ∣∣∣∣dy ) dx ≤ ‖ ω(·,ξ) ‖l1‖ e−i2π<ξ,·>f(·) ‖bmo, here k = b −x for x ∈ rn. by using above lemma we get, ‖ (sωf)(·,ξ) ‖bmo≤‖ ω(·,ξ) ‖l1 (‖ f ‖bmo +2m) . � 3. the s-transform on weighted bmo spaces. definition 3.1. a positive function k defined on rn is called a tempered weight function[2] if there exists positive constants c and n such that (3.1) k(ξ + η) ≤ (1 + c|ξ|)nk(η) for all ξ,η ∈ rn. 174 singh and kalita definition 3.2. for 1≤ p ≤∞, the weighted lebesgue space lpk(r n) is defined as the space of all measurable functions f on rn such that ‖ f ‖lp k = (∫ rn |f(x)|pk(x)dx )1 p < ∞. definition 3.3. the weighted bounded mean oscillation space bmok(rn) is defined as the space of all weighted lebesgue integrable (locally) functions defined on rn such that ‖ f ‖bmok = sup b⊂rn 1 |b|k ∫ b |f(x) −fb|k(x)dx < ∞, where the supremum is taken over the ball b in rn and |b|k = ∫ b k(x)dx. lemma 3.1. let f ∈ l1k(r n), then e−i2π<ξ,·>f(·) ∈ l1k(r n) and ‖ e−i2π<ξ,·>f(·) ‖bmok≤‖ f ‖bmok + 2m, where m is a constant defined in equation (2.1). proof. ‖ e−i2π<ξ,·>f(·) ‖bmok = sup b⊂rn 1 |b|k ∫ b ∣∣∣∣e−i2π<ξ,x>f(x) − 1|b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣k(x)dx = sup b⊂rn 1 |b|k ∫ b ∣∣∣∣e−i2π<ξ,x>f(x) − e−i2π<ξ,x>|b| ∫ b f(t)dt + e−i2π<ξ,x> |b| ∫ b f(t)dt− 1 |b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣k(x)dx ≤ sup b⊂rn 1 |b|k ∫ b (∣∣∣∣e−i2π<ξ,x> ( f(x) − 1 |b| ∫ b f(t)dt )∣∣∣∣ + ∣∣∣∣ 1|b| ∫ b f(t)dt ∣∣∣∣ + ∣∣∣∣ 1|b| ∫ b e−i2π<ξ,t>f(t)dt ∣∣∣∣ ) k(x)dx ≤ sup b⊂rn 1 |b|k ∫ b |f(x) −fb|k(x)dx + sup b⊂rn 1 |b|k ∫ b |fb|k(x)dx + sup b⊂rn 1 |b|k ∫ b ( 1 |b| ∫ b |f(t)|dt ) k(x)dx ≤ ‖ f ‖bmok + 1 |b|k m ∫ b k(x)dx + 1 |b|k m ∫ b k(x)dx = ‖ f ‖bmok + 1 |b|k m|b|k + 1 |b|k m|b|k = ‖ f ‖bmok +2m. � theorem 3.2. suppose ω is a window function such that for any fixed ξ ∈ rn0 (3.2) ∫ rn |ω(x,ξ)|(1 + c|x|)ndx ≤ a < ∞, the s-transform on hardy spaces and bmo 175 where a,c and n are positive constants. then the operator sω : bmok(rn) → bmok(rn) is continuous. furthermore, we have ‖ (sωf)(·,ξ) ‖bmok≤ a (‖ f ‖bmok +2m) where m is a constant given in equation (2.1). proof. by using the techniques of theorem 2.2, for any arbitrary ball b in rn, we have ‖ (sωf)(·,ξ) ‖bmok = sup b⊂rn 1 |b|k ∫ b |(sωf)(τ,ξ) − (sωf)b(τ,ξ)|k(τ)dτ ≤ sup b⊂rn 1 |b|k ∫ b (∫ rn |ω(x,ξ)| ∣∣∣∣e−i2π<ξ,τ−x>f(τ −x) − 1 |b| ∫ b e−i2π<ξ,α−x>f(α−x)dα ∣∣∣∣dx ) k(τ)dτ ≤ sup k⊂rn 1 |k|k ∫ k (∫ rn |ω(x,ξ)| ∣∣∣∣e−i2π<ξ,y>f(y) − 1 |k| ∫ k e−i2π<ξ,t>f(t)dt ∣∣∣∣dx ) (1 + c|x|)nk(y)dy = ∫ rn |ω(x,ξ)|(1 + c|x|)n ( sup k⊂rn 1 |k|k ∫ k ∣∣∣∣e−i2π<ξ,y>f(y) − 1 |k| ∫ k e−i2π<ξ,t>f(t)dt ∣∣∣∣k(y)dy ) dx ≤ a ‖ e−i2π<ξ,·>f(·) ‖bmok, here k = b −x for x ∈ rn. by using above lemma we get ‖ (sωf)(·,ξ) ‖bmok≤ a (‖ f ‖bmok +2m) . � 4. the s-transform on hardy spaces. definition 4.1. the hardy space is defined as the space of all functions f ∈ l1(rn) such that ‖ f ‖h1 = ∫ rn sup t>0 |(f ∗φt) (x)|dx < ∞, where φ is any test function with ∫ φ 6= 0 and φt(x) = t−nφ(x/t); t > 0,x ∈ rn. theorem 4.1. let f ∈ l1(rn) such that (4.1) sup t>0 ∣∣∣∣ ∫ rn f(x−y)φt(y)dy ∣∣∣∣ = sup t>0 ∫ rn |f(x−y)φt(y)|dy < ∞. then for any fixed ξ ∈ rn0 , the operator sω : h1(rn) → h1(rn) is continuous. furthermore, we have ‖ (sωf)(·,ξ) ‖h1≤ 3 ‖ ω(·,ξ) ‖l1‖ f ‖h1 . 176 singh and kalita proof. since ((sωf)(·,ξ) ∗φt) (τ) = ((∫ rn e−i2π<ξ, ·−x>f(·−x)ω(x,ξ)dx ) ∗φt ) (τ) = ∫ rn (∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)ω(x,ξ) dx ) φt(y) dy = ∫ rn ω(x,ξ) (∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy ) dx. thus ‖ (sωf)(·,ξ) ‖h1 = ∫ rn sup t>0 |((sωf)(·,ξ) ∗φt) (τ)|dτ = ∫ rn sup t>0 ∣∣∣∣ ∫ rn ω(x,ξ) (∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy ) dx ∣∣∣∣dτ ≤ ∫ rn |ω(x,ξ)| (∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy ∣∣∣∣dτ ) dx = ∫ rn |ω(x,ξ)| (∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy ∣∣∣∣dη ) dx. also, ∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy ∣∣∣∣dη = ∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy − ∫ rn f(η −y)φt(y)dy + ∫ rn f(η −y)φt(y)dy ∣∣∣∣dη = ∫ rn sup t>0 ∣∣∣∣ ∫ rn (e−i2π<ξ,η−y> − 1)f(η −y)φt(y)dy + ∫ rn f(η −y)φt(y)dy ∣∣∣∣dη ≤ ∫ rn sup t>0 ∫ rn ∣∣(e−i2π<ξ,η−y> − 1)∣∣ |f(η −y)φt(y)|dydη + ∫ rn sup t>0 ∣∣∣∣ ∫ rn f(η −y)φt(y)dy ∣∣∣∣dη ≤ 2 ∫ rn sup t>0 ∫ rn |f(η −y)φt(y)|dydη+ ‖ f ‖h1 = 2 ‖ f ‖h1 + ‖ f ‖h1 = 3 ‖ f ‖h1 . therefore, ‖ (sωf)(·,ξ) ‖h1≤ ∫ rn |ω(x,ξ)|3 ‖ f ‖h1 dx = 3 ‖ ω(·,ξ) ‖l1‖ f ‖h1 . � the s-transform on hardy spaces and bmo 177 5. the s-transform on weighted hardy spaces. definition 5.1. the weighted hardy space is defined as the space of all functions f ∈ l1k(r n) such that ‖ f ‖h1 k = ∫ rn sup t>0 |(f ∗φt) (x)|k(x)dx < ∞. theorem 5.1. suppose ω is a window function and satisfies the condition (3.2). let f ∈ l1(rn) and satisfies the condition (4.1). then, for any fixed ξ ∈ rn0 , the operator sω : h 1 k(r n) → h1k(r n) is continuous. furthermore, we have ‖ (sωf)(·,ξ) ‖h1 k ≤ 3a ‖ f ‖h1 k . proof. since ‖ (sωf)(·,ξ) ‖h1 k = ∫ rn sup t>0 |((sωf)(·,ξ) ∗φt) (τ)|k(τ)dτ = ∫ rn sup t>0 ∣∣∣∣ ∫ rn ω(x,ξ) (∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy ) dx ∣∣∣∣k(τ)dτ ≤ ∫ rn |ω(x,ξ)| (∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,τ−x−y>f(τ −x−y)φt(y)dy ∣∣∣∣k(τ)dτ ) dx ≤ ∫ rn |ω(x,ξ)| (∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy ∣∣∣∣ (1 + c|x|)nk(η)dη ) dx = ∫ rn |ω(x,ξ)|(1 + c|x|)n (∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy ∣∣∣∣k(η)dη ) dx. and ∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy ∣∣∣∣k(η)dη = ∫ rn sup t>0 ∣∣∣∣ ∫ rn e−i2π<ξ,η−y>f(η −y)φt(y)dy − ∫ rn f(η −y)φt(y)dy + ∫ rn f(η −y)φt(y)dy ∣∣∣∣k(η)dη ≤ ∫ rn sup t>0 ∣∣∣∣ ∫ rn (e−i2π<ξ,η−y> − 1)f(η −y)φt(y)dy ∣∣∣∣k(η)dη + ∫ rn sup t>0 ∣∣∣∣ ∫ rn f(η −y)φt(y)dy ∣∣∣∣k(η)dη ≤ 2 ∫ rn sup t>0 ∫ rn |f(η −y)φt(y)|dy k(η) dη + ∫ rn sup t>0 ∣∣∣∣ ∫ rn f(η −y)φt(y)dy ∣∣∣∣k(η)dη = 2 ‖ f ‖h1 k + ‖ f ‖h1 k = 3 ‖ f ‖h1 k . therefore, using equation (3.2), we have ‖ (sωf)(·,ξ) ‖h1 k ≤ ∫ rn |ω(x,ξ)|(1 + c|x|)n 3 ‖ f ‖h1 k dx ≤ 3a ‖ f ‖h1 k . 178 singh and kalita this completes the proof. � references [1] chuong. n. m. and duong. d. v., boundedness of the wavelet integral operator on weighted function spaces, russian journal of mathematical physics, 20(3) (2013), 268-275. [2] hörmander. l., the analysis of linear partial differential operators ii, springer-verlag, berlin heidelberg new york, 1983. [3] john. f. and nirenberg. l., on functions of bounded mean oscillation , communications on pure and applied mathematics, 14 (1961), 415-426. [4] pathak. r. s. and singh. s. k., boundedness of the wavelet transform in certain function spaces, journal of inequalities in pure and applied mathematics, 8(1) (2007), article 23, 8 pages. [5] singh. s. k., the s-transform on spaces of type s, integral transforms and special functions, 23(7) (2012), 481-494. [6] singh. s. k., the s-transform on spaces of type w, integral transforms and special functions, 23(12) (2012), 891-899. [7] singh. s. k., the fractional s-transform of tempered ultradistibutions, investigations in mathematical sciences, 2(2) (2012), 315-325. [8] singh. s. k., the fractional s-transform on spaces of type s, journal of mathematics, 2013 (2013), article id 105848. [9] singh. s. k., the fractional s-transform on spaces of type w, journal of pseudo-differential operators and applications, 4(2) (2013), 251-265. [10] singh. s. k., a new integral transform: theory part, investigations in mathematical sciences, 3(1) (2013), 135-139. [11] singh. s. k., the s-transform of distributions, the scientific world journal, 2014 (2014), article id 623294. [12] stockwell. r. g., mansinha. l. and lowe. r. p., localization of the complex spectrum: the s transform, ieee trans. signal process., 44(4) (1996), 998-1001. [13] ventosa. s., simon. c., schimmel. m. , dañobeitia. j. and mànuel. a., the s-transform from a wavelet point of view, ieee trans. signal process., 56(07) (2008), 2771-2780. department of mathematics, rajiv gandhi university, doimukh-791112, arunachal pradesh, india ∗corresponding author international journal of analysis and applications volume 18, number 6 (2020), 900-919 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-900 ruled surfaces with constant slope ruling according to darboux frame in minkowski space ayşe yavuz1,∗, yusuf yayli2 1department of mathematics and science education, necmettin erbakan university, konya, turkey 2department of mathematics, faculty of science, ankara university, ankara, turkey ∗corresponding author: ayasar@erbakan.edu.tr abstract. in this study, three different types of ruled surfaces are defined. the generating lines of these ruled surfaces are given by points on a curve x in minkowski space, while the position vector of x have constant slope with respect to the planes (t, y) , (t, n) , (n, y). it is observed that the lorentzian casual characters of the ruled surfaces with constant slope can be timelike or spacelike. furthermore, striction lines of these surfaces are obtained and investigated under various special cases. finally, new investigations are obtained on the base curve of these types of ruled surfaces. 1. introduction a ruled surface is a special surface which is formed by moving a line along a given curve in 3-dimensional minkowski space. the line is called the generating line and the curve is called the direction curve of the surface. thus, a ruled surface has a parametrization m(u,v) = α(u) + vx(u) where α and x are curves. the curve α is called the directrix or base curve and x is called the director curve of the ruled surface. thus, the ruled surfaces in minkowski space can be classified according to the lorentzian character of their ruling and surface normal. developable ruled surfaces are surfaces which can received june 6th, 2020; accepted july 8th, 2020; published september 3rd, 2020. 2010 mathematics subject classification. 53a04, 53a05. key words and phrases. ruled surface; surface with constant slope; darboux frame; minkowski space. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 900 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-900 int. j. anal. appl. 18 (6) (2020) 901 be made isometric to part of the plane. the necessary and sufficient conditions for these surfaces to become developable are characterized by vanishing gaussian curvature. in this study “developable” and “torsal” are used as synonyms, since a surface is developable if and only if it is a torsal ruled surface. cylindrical, conical, torse surfaces, a plane and surfaces of polyhedrons are examples of torsal surfaces. these surfaces can be developed on a plane without any lap break. on the ground that their isometrics with planes are becoming interesting to discover more ways to use these surfaces in different applications in [5] . a regular curve in e31 , whose position vector is obtained by frenet frame vectors on another regular curve, is called smarandache curve [1]. in this study, it is shown that if developable surface’s generating line is a smarandache curve and asymptotic or geodesic curve, then the basic curve is a general helix. k. malecek and others defined the surfaces with a constant slope with respect to the given surface in euclidean space in [7]. at the same time in [10], yavuz, ateş and yaylı investigated surface with a constant slope ruling with respect to osculating plane by using frenet frame according to casual characters in minkowski space. by the definition of surfaces with a constant slope ruling with respect to the given surface in study [7] , in this study new surface definitions are obtained. these surfaces in three types were studied according to the darboux frame in minkowski space. furthermore, necessary and sufficient conditions are given for these surfaces to become developable in minkowski 3-space. striction lines of the surfaces are obtained and investigated under various special cases. finally, the ruled surfaces with constant slope ruling visualized of given curves as examples, separately. 2. preliminaries a tangent vector v on a semi-riemanian manifold m is spacelike if g (v,v) > 0 or v = 0, is null if g (v,v) = 0 and v 6= 0, timelike if g (v,v) < 0. the norm of a tangent vector v is given by |v| = √ |g (v,v)|. a curve in a manifold m is a smooth mapping α : i → m, where i is an open interval in the real line r. a curve α in a semi-riemannian manifold m is spacelike if all of its velocity vectors α ′ (s) are spacelike, is null if all of its velocity vector α ′ (s) are null, timelike if all of its velocity vectors α ′ (s) are timelike [8]. in this study, the darboux frames and formulas in the minkowski space e31 are given with metric g = −dx21 + dx 2 2 + dx 2 3. let s be an oriented surface in e31 and let consider a non-null curve α (s) lying fully on s . since the curve α (s) lies on the surface s there exists a frame along the curve α (s). this frame is called darboux frame and denoted by {t,y,n} which gives us an opportunity to investigate the properties of the curve according to the surface. in this frame t is the unit tangent of the curve, n is the unit normal of the surface s along curve α (s) and y is a unit vector given by y = ∓n × t. according to the lorentzian casual characters of the surface and the curve lying on surface, the derivative formulae of the darboux frame can be changed as int. j. anal. appl. 18 (6) (2020) 902 follows: i) if the surface is timelike, then the curve α (s) lying on surface can be spacelike or timelike. thus, the derivative formulae of the darboux frame is given by  t′ y′ n′   =   0 kg −εkn kg 0 εtr kn tr 0     t y n   〈t,t〉 = ε = ∓1,〈y,y〉 = −ε,〈n,n〉 = 1 ii) if the surface is spacelike, then the curve α (s) lying on surface is spacelike. thus, the derivative formulae of the darboux frame is given by   t′ y′ n′   =   0 kg kn -kg 0 tr kn tr 0     t y n   where 〈t,t〉 = 1,〈y,y〉 = 1,〈n,n〉 = −1 here, kg (s) = 〈 α ′′ (s) ,y (s) 〉 is the geodesic curve, kn is the normal curvature defined by equality kn (s) = 〈 α ′′ (s) ,n (s) 〉 and tr is the geodesic torsion of α(s) defined by tr (s) = − 〈 n ′ (s) ,y (s) 〉 [2], [3], [4]. if u,v ∈ e31, lorentzian vector product of u and v is to the unique vector by u×v that satisfies 〈u×v,w〉 = det (u,v,w) where u×v vector product is defined as follows u×v = ∣∣∣∣∣∣∣∣∣ −i j k u1 u2 u3 v1 v2 v3 ∣∣∣∣∣∣∣∣∣ int. j. anal. appl. 18 (6) (2020) 903 [6]. the relations between geodesic curve, normal curvature, geodesic torsion and κ and τ are given, if both surface and curve are timelike or spacelike, then kg = κ cos θ kn = κ sin θ if surface is timelike and curve is spacelike, then kg = κ cosh θ kn = κ sinh θ [2], [3], [4]. let −→x and −→y be future pointing (or past pointing) timelike vectors in r31. then there is a unique real number θ > 0 such that 〈−→x ,−→y 〉 = −‖−→x‖‖−→y ‖cos hθ. let −→x and −→y be spacelike vectors in r31 that span a timelike vector subspace. then there is a unique real number θ > 0 such that 〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖cos hθ. let −→x and −→y be spacelike vectors in r31 that span a spacelike vector subspace. then there is a unique real number θ > 0 such that 〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖cos θ. let −→x be a spacelike vector and −→y be a timelike vector in r31. then there is a unique real number θ > 0, such that 〈−→x ,−→y 〉 = ‖−→x‖‖−→y ‖sinh θ. [9]. 3. ruled surfaces with constant slope ruling according to darboux frame in minkowski 3-space let m be a ruled surface whose generating lines are given by points on the curve x in minkowski space , while in all points they have the constant slope with respect to the tangent planes to the given surface .these surfaces will be called ruled surfaces with constant slope ruling with respect to the given surface in minkowski space. in this section, the developable conditions are investigated for the ruled surfaces with constant slope ruling with respect to the given surface with darboux frame {t,n,y} and the striction lines of the surface are obtained. at the same time, various relations and special cases about developable conditions int. j. anal. appl. 18 (6) (2020) 904 and striction line of the surfaces are given. 3.1. ruled surfaces with constant slope ruling with respect to the (t, y) planes. generating lines of the surface m are given by points on the curve x(s) and they have the constant slope σ with respect to the (t,y) planes to the curve at every point on the curve x(s). the surface will be called the ruled surfaces with a constant slope ruling with respect to the (t,y) planes. we give the definition of a ruled surface with a constant slope ruling with respect to the (t,y) planes according to casual characters in following three cases in minkowski space where 〈x(s),n(s)〉 = σ. case 3.1. if α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surface is spacelike where 1 + σ2 > 0. direction vector of generating line of the spacelike surface is given by x(s) = sin w(s).t(s) + cos w(s).y(s) + σn(s) and the surface is parametrized by m1 (s,v) = α (s) + v (x(s)) . case 3.2. if α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is spacelike where σ2 − 1 > 0 and the surface is timelike where σ2 − 1 < 0. direction vector of generating line of the surface is given by u(s) = cosh w(s)t(s) + sinh w(s)y(s) + σn(s) and the surface is parametrized by m2 (s,v) = α (s) + v (u(s)) . case 3.3. if α(s) is a timelike curve with the principal spacelike normal vector field n(s), then surface is timelike where σ2 + 1 > 0 and direction vector is given as follows δ(s) = sinh w(s)t(s) + cosh w(s)y(s) + σn(s) so the timelike surface is parametrized by m3 (s,v) = α (s) + v (δ(s)) where the vector t is the direction vector of a tangent to the curve x, n is the direction vector of a normal to the surface and y = n× t is the direction vector of intersection line of a tangent plane to the surface and the normal plane curve x at the point. int. j. anal. appl. 18 (6) (2020) 905 theorem 3.1. the spacelike surface m1(s,v) is developable if and only if cos w(s) (sin w(s).kn + cos w(s).tr) = σ (sin w(s) (kg −w′(s)) + σtr) . proof. the surface m1 is developable if and only if det(t,x,x ′) = 0. thus derivative of the direction vector of generating lines of the surface is obtained as follows x′(s) = (( w ′ (s) −kg ) cos w(s) + σkn ) t(s) + ( − ( w ′ (s) + kg ) sin w + σtr ) y(s) + (kn sin w(s) + tr cos w(s)) .n(s) det(t,x,x′) = 〈tλx, x′〉 . and 〈tλx, x′〉 = cos w(s) (sin w(s).kn + cos w(s).tr) −σ (sin w(s) (kg −w′(s)) + σtr) thus developable condition for surface m1 (s,v) is given by cos w(s) (sin w(s).kn + cos w(s).tr) = σ (sin w(s) (kg −w′(s)) + σtr) . � corollary 3.1. developable surface’s generating line x(s) is a asymptotic and smarandache curve if and only if α(s) is a general helix with τ κ = σx1 x22 −σ2 where sinw(s) = x1 = const.and cosw(s) = x2 = const. proof. if x(s) is asymptotic curve, then kn = 0 kn = 0 ⇒ cos2 w(s).tr = σ (sin w(s) (kg −w′(s)) + σtr) and if surface and curve are the same character, then κ2 = k2n + k 2 g, so kg = κ, tr = τ. if we replace the values in the last equation, we get the following equality τ κ = σx1 x22 −σ2 . m1(s,v) developable surface’s generating line x(s) is a geodesic and smarandache curve if and only if α(s) is a general helix, so that τ κ = x1x2 σ2 −x22 int. j. anal. appl. 18 (6) (2020) 906 where sinw(s) = x1 = const.and cosw(s) = x2 = const. � corollary 3.2. m1(s,v) spacelike developable surface’s generating line x(s) is a line curvature kn kg = σ cos w(s) where w(s) = const. theorem 3.2. the striction line on spacelike surface m1(s,v) is given by β = α(s) − −σkn − (w′ (s)−kg) cos w (s) (w′ (s)−kg)   (w′ (s)−kg) (sin2 w(s)−cos2 w(s)) −2σ (kn cos w (s) +tr sin w (s))   + (kn sin w (s) +tr cos w (s)) 2 + σ2 ( −k2n+t 2 r ) x (s) . remark 3.1. if x(s) is a geodesic curve and w (s) = −σ ∫ κ sec w (s) d (s) , then striction line of spacelike surface is equal to base curve. remark 3.2. if x(s) is a asymptotic curve and w (s) = ∫ κd (s) where cos w (s) 6= 0, then striction line of spacelike surface is equal to base curve. theorem 3.3. the spacelike surface m2(s,v) is developable if and only if sinh w(s) (cosh w(s).kn + sinh w(s)tr) −σ (cosh w(s) (kg + w′(s)) + σtr) = 0 where σ2 − 1 > 0. if σ2 − 1 < 0, then the surface is timelike. so the timelike surface is developable if and only if ε sinh w(s) (cosh w(s).kn + sinh w(s)tr) −σ (cosh w(s) (kg + w′(s)) + σtr) = 0 where ε = 〈t,t〉 = ∓1. corollary 3.3. if sinhw(s) = x3 = const.and coshw(s) = x4 = const, then generating lines of the surface u(s) is a smarandache curve. let m(s,v) be a spacelike developable surface and if u(s) be a smarandache and asymptotic curve, then α(s) is a general helix, so that τ κ = σx4 x23 −σ2 int. j. anal. appl. 18 (6) (2020) 907 if also u(s) is a geodesic, then base curve is a general helix, so that τ κ = x3x4 σ2 −x23 corollary 3.4. let m2(s,v) be a spacelike developable surface and if u(s) is a line and smarandache curve, kn kg = σ x3 where sinhw(s) = x3 = const.and coshw(s) = x4 = const. corollary 3.5. if sinhw(s) = x3 = const.and coshw(s) = x4 = const , generating lines of the surface u(s) be a smarandache curve. let m2(s,v) be a timelike developable surface and u(s) be a smarandache curve , if at the same time u(s) be a asymptotic, then base curve is a general helix, so that τ κ = σx4 εx23 −σ2 if also u(s) is a geodesic, then base curve is a general helix, so that τ κ = εx3x4 εx23 −σ2 . proof. if m2(s,v) is a timelike developable surface and the base curve is spacelike, then κ 2 = k2g −k2n . so we replace values kn = 0,kg = κ,tr = τ in condition of developable equation ε sinh w(s) (sinh w(s).tr) = σ (cosh w(s) (kg) + σtr) εx3 (x3.τ) = σ (x4.κ + στ) τ κ = σx4 εx23 −σ2 . if u(s) is a geodesic, then we write values kg = 0,kn = −κ,tr = τ in condition of developable equation for the timelike surface, we obtained as follows εx3 (x4. (−κ) + x3.τ) = σ. (σ.τ) τ κ = εx3x4 εx23 −σ2 . � remark 3.3. let m2(s,v) be a timelike developable surface and if u(s) be a line and smarandache curve, kn kg = σ εx3 where sinhw(s) = x3 = const.and coshw(s) = x4 = const. int. j. anal. appl. 18 (6) (2020) 908 theorem 3.4. the striction line on spacelike surface m2(s,v) is given by β = α(s) − −σkn − (w′ (s) −kg) sinh w (s) (w′ (s)−kg) 2 + σ2 ( −k2n+t 2 r ) + (2σ (w′ (s) −kg) (−kn sinh w (s) +tr cosh w (s))) −(kn cosh w (s) +tr sinh w (s)) 2 .x (s) where σ2 − 1 > 0, and striction line on timelike surface is given by β = α(s) − −σkn − (w′ (s) −kg) sinh w (s) (w′ (s)−kg) 2 + σ2 ( −k2n+t 2 r ) + (2σ (w′ (s) −kg) (−kn sinh w (s) +tr cosh w (s))) +ε2 (kn cosh w (s) +tr sinh w (s)) 2 .x (s) where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1. remark 3.4. if u(s) is a geodesic curve and w (s) = −σ ∫ κ sinh w (s) d (s) where σ2 − 1 > 0, then striction line of spacelike surface is equal to base curve. remark 3.5. if u(s) is a asymptotic curve and w (s) = ∫ κd (s) where σ2 − 1 > 0 and w (s) 6= 0, then striction line of spacelike surface is equal to base curve. theorem 3.5. the timelike surface m3 (s,v) is developable if and only if ε cosh w(s) (−sinh w(s)kn + cosh w(s)tr) −σ (sinh w(s) (kg + w′(s)) + σtr) = 0. remark 3.6. if sinhw(s) = x3 = const.and coshw(s) = x4 = const ,generating lines of the surface δ(s) be a smarandache curve. let m3(s,v) be a developable timelike surface and δ(s) be a smarandache curve , if at the same time δ(s) be a asymptotic, then the base curve is a general helix, so that τ κ = σx3 εx24 −σ2 , if also δ(s) be a geodesic, then the base curve is a general helix, so that τ κ = −εx3x4 σ2 −εx24 . int. j. anal. appl. 18 (6) (2020) 909 corollary 3.6. if u(s) be a line and smarandache curve, then kn kg = σ −εx4 where sinhw(s) = x3 = const.and coshw(s) = x4 = const. theorem 3.6. the striction line on timelike surface m3 (s,v) is obtained as follows β = α− −σkn − (w′ (s) −kg) . cosh w (s) −(w′ (s)−kg) 2 + σ2 ( −k2n+t 2 r ) + (2σ (w′ (s) −kg) (−kn cosh w (s) +tr sinh w (s))) +ε2 (−kn sinh w (s) +tr cosh w (s)) 2 x (s) . remark 3.7. if δ(s) is a geodesic curve and w (s) = −σ ∫ κ cosh w (s) d (s) , then striction line of surface is equal to the base curve. remark 3.8. if δ(s) is a asymptotic curve and w (s) = ∫ κd (s) , then striction line of timelike surface is equal to the base curve. 3.2. ruled surfaces with constant slope ruling with respect to the (t, n) planes. generating lines of the surface m̃ are given by points on the curve x̃(s) and they have the constant slope σ with respect to the (t,n) planes to the curve at every point on the curve x̃(s).the surface will be called the ruled surfaces with constant slope ruling with respect to the (t,n) planes where 〈 x̃(s),y(s) 〉 = σ. case 3.4. if α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surfaces is spacelike character where 1 + σ2 > 0. direction vector of generating line of the spacelike surface is given by x̃(s) = sin w(s).t(s) + cos w(s).n(s) + σ.y(s) the surface with constant slope ruling m̃1 parametrization obtained by m̃1 (s,v) = α (s) + v ( x̃(s) ) . int. j. anal. appl. 18 (6) (2020) 910 case 3.5. if α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is spacelike character where σ2−1 > 0 and the surfaces is timelike character where σ2−1 < 0. direction vector of generating line of the surface is given by ũ(s) = cosh w(s).t(s) + sinh w(s).n(s) + σ.y(s) and the surface is obtained by m̃2 (s,v) = α (s) + v (ũ(s)) . case 3.6. if α(s) is a timelike curve with the principal spacelike normal vector field n(s), then the surface is timelike where σ2 + 1 > 0 and direction vector is given as follows δ̃(s) = sinh w(s).t(s) + cosh w(s).n(s) + σ.y(s). the surface is parametrized by m̃3 (s,v) = α (s) + v ( δ̃(s) ) . theorem 3.7. the spacelike surface m̃1 (s,v) is developable if and only if cos w(s) (sin w(s)kg + cos w(s)tr) −σ ((kn −w′(s)) sin w(s) + σtr) = 0. corollary 3.7. developable surface’s m̃1 (s,v) generating line x̃(s) is an asymptotic and smarandache curve if and only if α̃(s) is a general helix with τ κ = x1x2 σ2 −x22 where sinw(s) = x1 = const.and cosw(s) = x2 = const. corollary 3.8. m̃1(s,v) developable surface’s generating line x̃(s) is a geodesic and smarandache curve if and only if α̃(s) is a general helix, so that τ κ = σx1 x22 −σ2 where sinw(s) = x1 = const.and cosw(s) = x2 = const. corollary 3.9. x̃(s) is a line and smarandache curve if and only if kn kg = σ x2 where cosw(s) = x2 = const. int. j. anal. appl. 18 (6) (2020) 911 theorem 3.8. the striction line on spacelike surface m̃1 (s,v) is given by β = α− σkg − (w′ (s) + kn) . cos w (s) −w′ (s)2 −2w′ (s) .kn+k2n ( sin2 w(s)−cos2 w(s) ) + σ2 ( −k2g+t 2 r ) −2σ (kg. (w′ (s) + kn) . cos w(s) − tr. sin w (s) . (−w ′ (s) + kn)) + ( sin w(s).kg+ cos w (s) .tr )2 .x (s) . corollary 3.10. if x̃(s) is an asymptotic curve and w (s) = σ ∫ κ cos w (s) d (s) , then striction line of surface is equal to the base curve. corollary 3.11. if x̃(s) is a geodesic curve and w (s) = − ∫ κd (s) where cos w (s) 6= 0, then striction line of the surface is equal to the base curve. theorem 3.9. the spacelike surface m̃2 (s,v) is developable if and only if sinh w(s) (cosh w(s).kg + sinh w(s)tr) −σ (cosh w(s) (kn + w′(s)) + σtr) = 0 where σ2 − 1 > 0. if σ2 − 1 < 0, then the surface is timelike ruled surface. so the timelike surface is developable if and only if sinh w(s). (cosh w(s).kg + sinh w(s)tr) −σ (cosh w(s) (−εkn + w′(s)) + σεtr) = 0 where ε = 〈t,t〉 = ∓1. remark 3.9. if sinhw(s) = x3 = const.and coshw(s) = x4 = const , generating line of the surface ũ(s) be a smarandache curve. let m̃2(s,v) be a developable spacelike surface and ũ(s) be a smarandache curve , if at the same time ũ(s) be an asymptotic, then the base curve is a general helix with τ κ = x3x4 σ2 −x23 , if also ũ(s) be a geodesic, then the base curve is a general helix, so that τ κ = σ x3 . int. j. anal. appl. 18 (6) (2020) 912 remark 3.10. if sinhw(s) = x3 = const.and coshw(s) = x4 = const,generating lines of the surface ũ(s) be a smarandache curve. let m̃2(s,v) be a developable timelike surface and ũ(s) be a smarandache curve , if at the same time ũ(s) be an asymptotic, then the base curve is a general helix with τ κ = x3x4 x23 −εσ2 if also ũ(s) be a geodesic, then the base curve is a general helix with τ κ = εσx4 εσ2 −x23 if ũ(s) be a line curvature, then kn kg = −εσ x3 . theorem 3.10. the striction line on spacelike surface m̃3(s,v) is given by β = α− σkg − (w′ (s) + kn) sinh w (s) (w′ (s) +kn) 2 + σ2 ( −k2g+t 2 r ) + ((2σ (w′ (s) + kn) kg sinh w (s) +tr cosh w (s))) + (kg cosh w (s) +tr sinh w (s)) 2 x (s) where σ2 − 1 > 0, and striction line on timelike surface is given by β = α− −σkg − (w′ (s) + kn) sinh w (s) w′ (s) 2 +k2n−2.kn.w′ (s) ( sinh2 w(s) −ε cosh2 w(s) ) +σ2 ( k2g+t 2 r ) +2σ (kg sinh w (s) . (w ′ (s) + kn) +ε.tr cosh w (s) . (w ′ (s) −εkn)) + (kg cosh w (s) +tr sinh w (s)) 2 x (s) where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1. remark 3.11. if ũ(s) is an asymptotic curve and w (s) = σ ∫ κ sinh w (s) d (s) where σ2 − 1 > 0, then striction line of spacelike surface is equal to the base curve and w (s) = −σ ∫ kg sinh w (s) d (s) where σ2 − 1 < 0, then striction line of timelike surface is equal to the base curve. remark 3.12. if ũ(s) is a geodesic curve and w (s) = − ∫ κd (s) int. j. anal. appl. 18 (6) (2020) 913 where σ2 − 1 > 0 then striction line of spacelike surface is equal to base curve and w (s) = ∫ knd (s) where σ2 − 1 < 0 and sinh w (s) 6= 0, then striction line of timelike surface is equal to base curve. corollary 3.12. the surface m̃3 (s,v) is torsal if and only if cosh w(s) (sinh w(s)kg + cosh w(s)tr) −σ (sinh w(s) (−εkn + w′(s)) + εσtr) = 0 where ε = 〈t,t〉 = ∓1. corollary 3.13. if sinhw(s) = x3 = const.and coshw(s) = x4 = const, generating lines of the surface δ̃(s) is a smarandache curve. let m̃3(s,v) be a developable timelike surface and δ̃(s) is a smarandache curve , if at the same time δ̃(s) be a asymptotic, then basic curve is a general helix, so that τ κ = x3x4 σ2 −x24 if also δ̃(s) be a geodesic, then base curve is a general helix, so that τ κ = x3 σx4 if δ̃(s) be a line curvature, then kn kg = −εσ x4 such that ε = 〈t,t〉 = ∓1. theorem 3.11. the striction line on timelike surface m̃3(s,v) is given by β = α− −σkg−(w′ (s) + kn) . cosh w (s) − ( w′ (s) 2 +k2n ) −2w′ (s) .kn. ( cosh2 w(s) + ε sinh2 w(s) ) +σ2 ( k2g+t 2 r ) + (sinh w(s).kg + cosh w (s) .tr) 2 x (s) . remark 3.13. if generating line is a asymptotic curve on timelike surface m̃3(s,v) and w (s) = −σ ∫ κ cosh w (s) d (s) then striction line of surface is equal to base curve. int. j. anal. appl. 18 (6) (2020) 914 remark 3.14. if generating line is a geodesic curve on timelike surface m̃3(s,v) and w (s) = − ∫ κd (s) then striction line of the surface is equal to the base curve. 3.3. ruled surfaces with constant slope ruling with respect to the (n,y) planes. surface m is given by points on the curve x(s) and they have the constant slope σ with respect to the (n,y) planes.the surface will be defined the ruled surfaces with constant slope ruling with respect to the (n,y) planes to the curve where 〈 x(s), t(s) 〉 = σ. case 3.7. if α(s) is a spacelike curve with the principal spacelike normal vector field n(s), then surface is spacelike where 1 + σ2 > 0. direction vector of generating line of the spacelike surface is given by x(s) = sin w(s).n(s) + cos w(s).y(s) + σt(s). the surface with constant slope m1 is parametrized by m1 (s,v) = α(s) + v ( x(s) ) . case 3.8. if α(s) is a spacelike curve with the principal timelike normal vector field n(s), then surfaces is spacelike where σ2 − 1 > 0 and the surface is timelike where σ2 − 1 < 0. direction vector of generating line of the surface is given by u(s) = cosh w(s).n(s) + sinh w(s).y(s) + σt(s) and the surface is obtained as follows m2 (s,v) = α(s) + v (u(s)) . case 3.9. if α(s) is a timelike curve with the principal spacelike normal vector field n(s), then the surface is timelike where σ2 + 1 > 0 and direction vector is defined as follows δ(s) = sinh w(s).n(s) + cosh w(s).y(s) + σt(s). the surface parametrization is given as follows m3 (s,v) = α(s) + v ( δ(s) ) . theorem 3.12. the spacelike surface m1 (s,v) is developable if and only if tr(sin 2 w(s) − cos2 w(s)) + σ (cos w(s)kn + sin w(s)kg) −w′(s) = 0. int. j. anal. appl. 18 (6) (2020) 915 corollary 3.14. developable surface’s generating line x(s) is a asymptotic and smarandache curve if and only if α(s) is a general helix, so that τ κ = σx1 x22 −x21 where sinw(s) = x1 = const.and cosw(s) = x2 = const. corollary 3.15. m1(s,v) developable surface’s generating line x(s) is a geodesic and smarandache curve if and only if α(s) is a general helix, so that τ κ = σx2 x22 −x21 where sinw(s) = x1 = const.and cosw(s) = x2 = const. corollary 3.16. x(s) is a line and smarandache curve kn kg = − x2 x1 where sinw(s) = x1 = const.and cosw(s) = x2 = const. theorem 3.13. the striction line on the spacelike surface m1(s,v) is given by β = α− kn. sin w(s) −kg. cos w(s)( w′(s) − t2r )( sin2 w(s) − cos2 w(s) ) − 2tr.w′(s) + σ2. ( k2n + k 2 g ) + (kn. sin w(s) −kg. cos w(s)) 2 .x (s) . remark 3.15. if kn kg = −cot w(s), then the striction line on the surface m1(s,v) equal to the base curve. theorem 3.14. the spacelike ruled surface with constant slope m2 (s,v) is developable if and only if sinh w(s) (cosh w(s).kn + ε sinh w(s)kg) −σ (cosh w(s) (tr + w′(s)) + σkg) = 0 ε = ±1. if σ2 −1 > 0, the surface is spacelike and ε = −1. if σ2 −1 < 0, the surface is timelike and ε = +1. int. j. anal. appl. 18 (6) (2020) 916 corollary 3.17. if sinhw(s) = x3 = const.and coshw(s) = x4 = const,generating line of the surface u(s) is a smarandache curve. let m2(s,v) be a developable surface and u(s) be a smarandache curve , if at the same time u(s) is an asymptotic, then the base curve is a general helix with τ κ = εx23 −σ2 σx4 , if also u(s) is a geodesic , then the base curve is a general helix with τ κ = ε ( σ2 −x3x4 ) σx4 ,σ2 − 1 > 0 if u(s) is a line curvature, then kn kg = x3x4 σ2 − �x24 . remark 3.16. the striction line on the spacelike surface m2(s,v) is given by β = α (s) + kn. cosh w (s) −kg. sinh w (s) (w′ (s) + tr) 2 . ( sinh2 w (s) + cosh2 w (s) ) + σ2 ( k2g+k 2 n ) + (2σ (w′ (s) + tr) (kn. sinh w (s) +kg. cosh w (s))) −(kn. cosh w (s)−kg. sinh w (s)) 2 x (s) where σ2 − 1 > 0, and striction line on the timelike surface is given by β = α (s) − kn. cosh w (s) + kg. sinh w (s) (w′ (s) + tr) 2 +σ2 ( k2g+k 2 n ) + (kn. cosh w (s) + kg. sinh w (s)) 2 + (2σ (w′ (s) + tr) (ε.kn. sinh w (s) +kg. cosh w (s))) .x (s) where σ2 − 1 < 0, ε = 〈t,t〉 = ∓1. theorem 3.15. the spacelike surface m3 (s,v) is developable if and only if sinh w(s) (sinh w(s) (tr + w ′(s)) + σkg) − cosh w(s) (cosh w(s) (εtr + w′(s)) −εσkn) where ε = 〈t,t〉 = ∓1. remark 3.17. if sinhw(s) = x3 = const.and coshw(s) = x4 = const, generating line of the surface δ(s) is a smarandache curve. let m3(s,v) be a developable timelike surface and δ(s) be a smarandache curve , if at the same time δ(s) be an asymptotic, then the base curve is a general helix, so that τ κ = σx3 εx24 −x23 int. j. anal. appl. 18 (6) (2020) 917 for ε = 1, τ κ = σx3 if also δ(s) is a geodesic, then base curve is a general helix, so that τ κ = −εσ x23 −εx24 for ε = 1, τ κ = σ if δ(s) is a line curvature, then kn kg = −εx4 x3 where ε = 〈t,t〉 = ∓1. theorem 3.16. the striction line on timelike surface m3(s,v) is given by β = α (s) − kn. sinh w(s) + kg. cosh w(s) −(cosh w(s). (ε.tr + w′(s))−εσ.kn) 2 + (sinh w(s). (tr + w ′(s)) +σ.kg) 2 + ( kn. sinh w(s) + kg. cosh w(s) )2 .x (s) . 4. some numerical examples in this section, we give examples of the surfaces with a constant slope ruling according to darboux frame in minkowski space with respect to the given planes. example 4.1. the curve α(s) given by α(s) = ( r sin s r ,r cos s r , s r ) and ruled surfaces with constant slope ruling is parametrized by m1(s,v) = α(s) + v(sin w(s)t(s) + cos w(s)y(s) + σn(s)). surface is visualized in following figure for w(s) = π 2 ,r = 10,σ = 2. int. j. anal. appl. 18 (6) (2020) 918 figure 1. example 4.2. ruled surfaces with constant slope ruling with respect to the {t,n} planes m̃1(s,v) = α(s) + v(sin w(s)t(s) + cos w(s)n(s) + σy(s)). is shown following figure for w(s) = s π ,r = 2,σ = √ 3. figure 2. example 4.3. the curve β(s) is given by β(s) = ( r cosh s r ,r sinh s r , s r ) and the timelike surface with a constant slope ruling is shown following figure for w(s) = 3π 2 , r = 10,σ = 1 10 , s� ( −5, π 2 4 ) , u� (−1, 1). int. j. anal. appl. 18 (6) (2020) 919 figure 3. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.t. ali, special smarandache curves in the euclidean space. int. j. math. comb. 2 (2010), 30-36. [2] h.h. ugurlu, h. kocayigit, the frenet and darboux instantaneous rotain vectors of curves on timelike surfaces, math. comput. appl. 1 (2) (1996), 133-141. [3] s. kızıltuğ, a. çakmak, developable ruled surfaces with darboux frame in minkowski 3-space. life science journal (2013), 10(4). [4] s. kızıltuğ, y. yaylı, timelike curves on timelike parallel surfaces in minkowski 3-space e31, math. aeterna, 2 (2012), 689 700. [5] s. n. krivoshapko, s. shambina, design of developable surfaces and the application of thin-walled developable structures, serbian architect. j. 4 (3) (2012), 298-317. [6] r. lópez, differential geometry of curves and surfaces in lorentz-minkowski space. int. electron. j. geom. 7 (1) (2014), 44-107. [7] k. malecek, j. szarka, d. szarkova, surfaces with constant slope with their generalisation. j. polish soc. geom. eng. graph. 19 (2009), 67-77. [8] b. o’neill, elementary differential geometry, academic press, new york, 1966. [9] m. önder, h.h. uğurlu, frenet frames and invariants of timelike ruled surfaces, ain shams eng. j. 4 (3) (2013), 507-513. [10] a. yavuz, f. ateş, y. yaylı, non-null surfaces with constant slope ruling with respect to osculating plane. adıyaman univ. j. sci. 10 (2020), 240-255. 1. introduction 2. preliminaries 3. ruled surfaces with constant slope ruling according to darboux frame in minkowski 3-space 3.1. ruled surfaces with constant slope ruling with respect to the ( t,y) planes 3.2. ruled surfaces with constant slope ruling with respect to the ( t,n) planes 3.3. ruled surfaces with constant slope ruling with respect to the ( n,y) planes 4. some numerical examples references international journal of analysis and applications volume 18, number 3 (2020), 482-492 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-482 received january 18th, 2020; accepted february 21st, 2020; published may 1st, 2020. 2010 mathematics subject classification. 62p20, 91g70. key words and phrases. potential exports; stochastic frontier gravity model; agriculture; iran. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 482 an application of stochastic frontier gravity approach (the case of iran's potential agricultural exports) shokrollah hajivand, reza moghaddasi*, yaaghoob zeraatkish, amir mohammadinejad department of agricultural economics, education and extension, science and research branch, islamic azad university, tehran, iran *corresponding author: r.moghaddasi@srbiau.ac.ir abstract. agriculture plays a crucial role in iranian economy in terms of food supply, job creation, food security, and foreign earnings. the main purpose of this study is to provide an estimate of the country's agricultural exports potential and to determine how efficient iran is in realizing this capacity. using data for 38 destination countries for the period spanning from 1982 to 2017, proper stochastic frontier gravity model was estimated. main findings revealed direct and significant impact of trade partners' gdp and population on iran's agricultural exports, while distance and border barriers imposed by destination countries show significant reverse effect. furthermore, on average, 69 percent of the country's agricultural export potential has been realized through the study period. measures to promote competitive exports along with pursuing free bilateral and regional trade agreements for removing border barriers are recommended. 1. introduction export potential is the maximum amount of exports which is realized in free (no impediment or restriction) trade between nations. such an amount can be determined for any level of export drivers (miankhel et al. in [1]). due to heterogeneity of countries in terms of economic development, trade relationships and border policies, different factors might be responsible for inefficiency in realizing trade potential. some exist within the exporting country (internal factors) while others relate to destination countries (external factors). https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-482 int. j. anal. appl. 18 (3) (2020) 483 moreover, due to more severe rigidities, it seems viable assuming that 'external factors' exert stronger prohibitive impact on export potential in developing countries than developed nations. from the policy-making point of view knowing trade potential for different products with corresponding trading partners is of great importance as it provides insights on public trade policies' efficiencies as well as a guide for necessary reforms and trade negotiations to get export potential fully exploited. for almost six decades, the gravity model has been the most common tool among economists interested in analysis of trade flows between countries. (baier and bergstrand in [2]; anderson and wincoop in [3]). the first application of this model in economics dates back to 1960's inspiring from the newton's physical gravity model which has been introduced in 1687 (linder in [4]; tinbergen in [5]). lack of theoretical background on one hand and their purely empirical nature on the other hand, have put early papers under serious criticism. anderson in [6] made the first attempt to prove that theoretical justification for the gravity model exists. further the same assumption was used by bergstrand in [7], who aimed at developing further the microeconomic foundations of the gravity equation. helpman and krugman in [8] used the same assumptions and the same constant elasticity of substitution (ces) utility function to again prove that the gravity equation is consistent with theory of international trade. deardoff in [9] derived the gravity equation from the well-known two-factor, twocommodity heckscher-ohlin model. all of that allowed economists to see that the gravity approach to estimating trade flows is not only a useful technique, but can also be transformed in many ways to analyze other issues of international trade. agriculture is recognized as a key sector in iranian economy. it constitutes about 9.5 percent of gross domestic product, while its share in total workforce, non-oil exports and food supply is reported at 23, 21, and 80 percent, respectively (cbi1 in [10]). as a source of non-oil foreign earning, there is high priority on agricultural exports promotion in national five-year development plans. iran has exported, on average, usd 1.3 billion agricultural products (raw and processed) to the world each year during 1982-2017 rising from usd 250 million in 1982 to more than usd 2.7 billion in 2017. despite of this remarkable growth, a common belief is that the country's capacity to touch higher figures is great (roosta et al. in 1 central bank of iran int. j. anal. appl. 18 (3) (2020) 484 [11]). so, this paper tries to empirically quantify iran's agricultural export potential for different trading partners. 2. materials and methods the difference between real (observed) exports and the predicted (fitted) values is commonly defined as potential exports. estimation of the gravity equation with ordinary least squares (ols) to find trade potential between a pair of countries leads to estimates that represent the centered values of the data set. however, potential trade refers to open and frictionless trade between countries. thus, for policy purposes, it is sensible to define potential trade as the maximum trade that can occur between any two countries which have bilaterally liberalized trade regimes given the conventional determinants of trade (size of the trading countries, the geographical distance, etc.). this means that the estimation of potential trade requires a procedure that represents the upper limits of the data instead of its centered values (kalirajan in [12]). in addition, in the conventional gravity model, it is also arguable that trade costs are dependent not only on geographical distance between countries but also on other factors emanating from the existing infrastructural, institutional, socio-economic, and political rigidities in both trade partners. these latter costs are defined as ‘economic distance’ in the literature (anderson in [6]). thus, the conventional gravity model given above has omitted this potentially important explanatory variable. furthermore, this inherent omitted variable bias is overlooked by ols estimation. in simpler language, omission of the economic distance term leads to heteroskedastic errors which results in bias in the estimation of the model parameters. the log-linearization of the empirical model in the presence of heteroskedasticity leads to inconsistent estimates because the expected value of the logarithm of a random variable depends on higher-order moments of its distribution (silva and tenreyro in [13]). also, it affects the normality assumption of the error term (matyas in [14]). as a result, an ordinary least squares estimation will lead to biased results (kalirajan in [12]). following kalirajan in [12] and miankhel et al. in [1], the gravity equation for exports can be estimated alternatively as: 𝐿𝑛𝑋𝑖𝑗 = 𝐿𝑛 𝑓(𝑍𝑗 , 𝛽)𝑒𝑥𝑝 (𝑣𝑖−𝑢𝑖) (1) where the term 𝑋𝑖𝑗 represents the actual exports from country i (iran in this study) to country j. the term 𝑓(𝑍𝑗 , 𝛽) denotes a function of the drivers of potential bilateral trade 𝑍𝑗 , int. j. anal. appl. 18 (3) (2020) 485 which include distance, gdp, and population to represent supply and demand conditions, and 𝛽 is a vector of unknown parameters. the inclusion of the composite error term in the above gravity equation, which accounts for the impact of other unobservable variables influencing exports costs, is to remove the bias that is inherent in the conventional gravity model. to estimate equation (1), the stochastic frontier framework is used. the sfgm1 recognizes two separate groups of constraints on export including behind and beyond border constraints. the latter can be divided into explicit beyond the border constraints, which are observable, and implicit beyond the border constraints, which are not observable. explicit beyond the border constraints, for example, can be measured from the applied tariffs of importing countries (kalirajan and singh in [15]; miankhel et al. in [1]). implicit beyond the border constraints, which emanate from institutional weaknesses and policy rigidities existing in the importing countries, are difficult to measure and are commonly considered as given. however, miankhel et al. in [1] address this issue and highlight that implicit beyond the border constraints affect the exporting countries uniformly. through the trade balance relationship equation in anderson’s theoretical framework for the gravity model, the implicit beyond the border constraints would affect, and may probably reduce, planned expenditures in exporting countries if the exporting countries are not taking measures to overcome these constraints through conforming to, or initiating, certain measures for becoming more efficient (anderson in [6]). in order to overcome implicit beyond the border constraints, and to maintain their market shares or realize their export potential, exporting countries need to become more efficient by removing behind the border constraints. these put additional transaction costs on the trade flows. these costs include institutional costs stemming from the inefficient prevalent practices in the institutions, regulatory and legislative costs, equipment and training costs, and political costs due to the inability to take on trade facilitation measures due to geostrategic interests. specific behind the border measures could range from product standards and conformity assessment measures, business facilitation, and trade financing, to hard (physical) and soft (regulatory) infrastructure including efficient transport links and logistics and poor governance in the regulatory institutions. behind the border constraints could also be due to the retention of imperfect institutions, caused by rent seeking agents 1 stochastic frontier gravity model int. j. anal. appl. 18 (3) (2020) 486 through lobbying, and resistance from the elite towards introducing institutional innovations. in addition, these costs could come from the stance of certain institutions aimed at achieving policy objectives. following the methodology given above, behind the border constraints and explicit beyond the border constraints are included within the gravity equation in the form of μi and trade weighted effective applied tariffs, respectively. 𝐿𝑛 𝑋𝑖𝑗 = 𝐿𝑛 𝑓 (𝑍𝑗 , 𝛽 )𝑒𝑥𝑝 (𝑡𝑎𝑟𝑖𝑓𝑓+ 𝑣𝑖−𝑢𝑖), (2) the single sided error term, μi is the exporting country’s share of the economic distance bias, referred to by anderson in [6], which is due to the influence of the behind the border constraints. this bias, which is country specific to the exporting country for each importer, creates the difference between actual and potential trade between the exporting and importing countries concerned. it is difficult to get full information on all behind the border constraints that exist within the exporting country. nevertheless, drawing on kalirajan and singh in [15], the combined effect of these constraints can be modeled by the random variable μi that takes values between 0 and 1 and is usually assumed to follow a truncated (at 0) distribution, 𝑁(0, 𝜎𝜇 2). when μi is 0, this indicates that the constraints are not important, and the actual exports and potential exports are the same (assuming there are no statistical errors). when μi takes a value other than 0 (but less than or equal to 1), this indicates that the constraints are important, and they constrain actual exports from reaching potential exports. thus, the term μi, which is bilateral observation-specific, represents the bias that is a function of the behind the border constraints within the exporting county’s control. unlike the conventional approach, this method of estimating the gravity model does not exclude the influence of economic distance bias on trade flows between two countries. the error term 𝑣𝑖 captures the influence of omitted variables on trade flows and implicit beyond the border factors, in addition to measurement errors that are randomly distributed across observations in the sample. implicit beyond the border constraints are not controlled by exporting countries, and it is assumed that these are randomly distributed, affecting the exporting countries uniformly. the random distribution of 𝑣𝑖 also implies efficient, conforming exporting countries could gain market share at the expense of less efficient countries in specific product markets in the importing country. the model formulation supports the assumption that 𝑣𝑖 is a double sided and is usually assumed to be 𝑁(0, 𝜎𝑣 2). int. j. anal. appl. 18 (3) (2020) 487 with the stochastic framework followed in this analysis, in some cases when 𝑣𝑖 > 0 due to favorable external developments, it is possible that actual exports exceed estimated potential exports. for such situations, the results need to be interpreted as the realization of country's export potential. moreover, measurement errors could also lead to these situations. actual exports may also be different from potential exports due to measurement errors emanating from alternative trade institutions that have evolved over time due to the weakness of formal contracting institutions. nunn and trefler in [16] argue that these may deal with hold-up problems and could take the form of repeat relationships, ethnic networks, culture, and vertical integration. for example, repeated interactions could lead to the creation of non-kin-based networks that act as a substitute for legal contract enforcement and also help in sharing risk and pooling information. in this case, it would have an export-enhancing effect. conversely, these alternative institutions are not without costs, as they may create barriers to entry and, when old partnerships become less productive, may result in inefficiencies. gould in [17], while explaining trade with the usa, finds positive a correlation between the presence of immigrant populations from a particular country and trade with that particular country. nunn and trefler in [16] further state that if there are underinvestment problems due to hold-up, and for example, if both parties underinvest, then this problem could be alleviated by allocating control to one party or the other. therefore, vertical integration provides an additional tool to alleviate underinvestment. for example, in multinationals this decision could involve whether to incentivize the headquarters or supplier, with the final decision affecting the pattern of trade from a particular country. to estimate a sfgm, maximum likelihood methods can be applied to either crosssectional or panel data to verify how important behind the border constraints are in limiting exports from their potential. in addition, estimating with this methodology also demonstrates whether total variations from the mean in the potential exports, given as 𝜎2 = 𝜎𝑣 2 + 𝜎𝜇 2, are due to random factors 𝜎𝑣 2 or country-specific behind the border constraints 𝜎𝜇 2. the gamma coefficient (𝛾) captures the total variation in the model due to the influence of country-specific institutional, socio-economic, and political factors that constitute the behind the border constraints to exports. this is given as 𝛾 = 𝜎𝜇 2/𝜎2. a large size and significance of gamma imply that country-specific behind the border constraints are responsible for a large proportion of the mean total variation in the model. int. j. anal. appl. 18 (3) (2020) 488 3. data the bilateral data for iran's agricultural exports to 36 destination countries for the period of 1982-2017 were retrieved from the united nations commodity trade statistics (un comtrade) database. the real gdp (2010 constant prices) and population data were obtained from the world bank’s world development indicators (wdi). the trade-weighted, effective applied tariff rates have been downloaded from the trade analysis and information system (trains) using world integrated trade solutions (wits world bank). the bilateral population weighted distances are in kilometers and have been downloaded from geodist. all the variables in the model are in natural logs except tariffs. computer software frontier 4.1 was used to estimate the sfgm (coelli in [18]). 4. results and discussion table (1) portrays estimation results for equation (2). table1. estimation results of sfgm coefficient std error zstatistic prob constant 7.14 2.24 3.19 0.01 gdp 0.54 0.12 4.50 0.008 population 0.02 0.01 2.00 0.03 distance -1.14 0.08 -14.25 0.001 tariff -0.01 0.005 -2.00 0.03 𝝈𝒗 0.25 0.001 𝝈𝒖 1.04 0.12 𝝈𝟐 1.25 0.002 𝜸 0. 79 0.001 log likelihood -123.14 wald 𝒄𝒉𝒊𝟐 126.57 observations 36×36 int. j. anal. appl. 18 (3) (2020) 489 the coefficients of gdp and population of the trade partners, and distance from iran, have the expected signs, though the impact of distance is much greater than gdp. the estimated 𝜎2 is significant implying that variation of export potentials in the period was considerable but this variation is due to random factors 𝜎𝑣 2 only. the significance and level of 𝛾 suggests that almost three-fourths of the estimated variations in iran’s potential agricultural exports with its trading partners were due to behind the border constraints. moreover, tariffs imposed by trade partners are estimated to be another significant factor on iran's agricultural exports. a similar result was found by roosta et al. in [11] and kalirajan in [12]. based on the results provided in table 1, potential agricultural exports has been calculated for each importer and is presented in tables 2. the comparison between actual and potential values for each country provides an estimate of the export capacity utilization with the respective countries and the extent to which trade is limited by behind the border constraints. the highest rate of exports potential realization relates to syria which has close political and economic ties with iran. some major european partners such as france, italy, and the netherlands show same result. countries with great unrealized exports capacity are saudi arabia, hong kong, south korea, china, and sri lanka. it means that iranian government, especially ministry of agriculture, should pay special attention to expansion of markets in this group of countries. int. j. anal. appl. 18 (3) (2020) 490 table2. iran's actual and potential agricultural exports by importer (average over 1982-2017) importer actual exports (1000usd) (1) potential exports (1000usd) (2) (𝟏) (𝟐) ⁄ germany 897832.7 1282618.1 0.70 italy 454955.9 541614.1 0.84 turkey 342475.6 482360.0 0.71 iraq 280167.8 466946.3 0.60 united arab emirates 229245.2 301638.4 0.76 pakistan 179078.2 284251.1 0.63 china 163732.2 314869.6 0.52 india 157652.5 222045.7 0.71 azerbaijan 122295.2 160914.7 0.76 afghanistan 89671.8 137956.6 0.65 turkmenistan 57461.9 79808.1 0.72 poland 50452.2 62286.6 0.81 belgium 39313.5 57813.9 0.68 vietnam 39309.3 71471.4 0.55 switzerland 36471.7 52857.5 0.69 ukraine 28606.6 40866.5 0.70 united kingdom 27533.8 33577.8 0.82 france 22073.2 25968.4 0.85 hong kong 22042.4 44984.4 0.49 uzbekistan 21084.3 39781.6 0.53 russian federation 20532.6 29757.3 0.69 bangladesh 17490.8 22715.3 0.77 spain 17287.7 26193.4 0.66 sri lanka 17196.2 32445.6 0.53 netherlands 14211.4 16334.9 0.87 qatar 10397.6 15068.9 0.69 armenia 10228.7 14011.9 0.73 georgia 9959.2 13458.3 0.74 kuwait 9256.5 11867.3 0.78 south korea 8684.2 17027.8 0.51 saudi arabia 6999.9 16666.4 0.42 philippines 6839.1 11591.6 0.59 egypt 6464.7 11544.1 0.56 japan 6406.7 7718.9 0.83 indonesia 5779.6 6643.2 0.87 syrian arab republic 5379.2 6044.0 0.89 average 95404.7 137881.2 0.69 int. j. anal. appl. 18 (3) (2020) 491 5. conclusions empirical estimation was performed to investigate the presence of institutional, socioeconomic and political behind the border impediments to iran's agricultural exports during 1982-2017 using stochastic frontier gravity model. in the sfgm framework, which considers both the demand and supply side effects, the results showed significant direct impact of partners' gdp and population on iran's agricultural exports. opposite relationship was found for partners' import tariff and geographical distance between countries. the empirical results also demonstrate that iran is not realizing its full agricultural exports potential with its main partners including neighboring countries. iran needs more regional focus in order to smooth consumption across borders and insulate the region from future shocks. besides, signing bilateral and regional agreements to set preferential tariff regime, can lead to more realization of exports capacity. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.k. miankhel, a.k. kaliappa, and s.m. thangavelu, australia's export potential: an exploratory analysis, j. asia. pac. econ. 19 (2) (2014), 230-246. [2] s.l. baier, and j. h. bergstrand, bonus vetus ols: a simple method for approximating international trade-cost effects using gravity equation, j. int. econ. 77 (2009), 77-85. [3] j. e. anderson, and e. van wincoop, gravity with gravitas: a solution to the border puzzle, amer. econ. rev. 93 (1) (2003), 170–192. [4] s. linder, an essay in trade and transformation, new york: wiley, (1961). [5] j. tinbergen, shaping the world economy: suggestions for an international economic policy, new york: twentieth century fund, (1962). [6] j. e. anderson, a theoretical foundation for the gravity equation, amer. econ. rev. 69 (1) (1979), 106–116. [7] j. h. bergstrand, jeffrey, the gravity equation in international trade: some microeconomic foundations and empirical evidence, rev. econ. stat. 67 (3) (1985), 474–481. [8] e. helpman, and p.krugman, market structure and foreign trade: increasing returns, imperfect competition, and the international economy. cambridge, ma: mit press, (1985). int. j. anal. appl. 18 (3) (2020) 492 [9] a. v. deardoff, determinants of bilateral trade: does gravity work in the neoclassical world? nber working papers 5377, national bureau of economic research, inc, (1995). [10] central bank of iran, statistical yearbook, (2017). [11] r. a. roosta, r. moghaddasi, and s. s. hosseini, export target markets of medicinal and aromatic plants, j. app. res. med. arom. plant. 7 (1) (2017), 84-88. [12] k. kalirajan, regional cooperation and bilateral trade flows: an empirical measurement of resistance model, intl. trade. j. 21 (2) (2007), 195-209. [13] j. m. c. s. silva, and s. tenreyro, gravity-defying trade, working paper 03, federal reserve bank of boston, boston, mass (2003). [14] l. mátyás, l, proper econometric specification of the gravity model, world. econ. 20 (1997), 363– 368. [15] k. kalirajan, and k. singh, a comparative analysis of recent export performances of china and india, asian. econ. papers. (2007). [16] n. nunn, and d. trefler, domestic institutions as a source of comparative advantage, working paper 18851, national bureau of economic research (nber), cambridge ma, (2013). [17] d. m. gould, immigrants link to the home country: empirical implications for u.s. bilateral trade flows, rev. econ. stat. 76 (2) (1994), 1-25. [18] t. coelli, a guide to frontier version 4.1: a computer program for stochastic frontier production and cost function estimation, center for efficiency and productivity analysis working paper 96/07, university of new england, armidale, australia, (1996). international journal of analysis and applications volume 16, number 3 (2018), 437-444 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-437 the eoq model: a differential cyclic system for calculating economic order quantity rabha w. ibrahim1,∗, samir b. hadid2 1faculty of computer science and information technology, university malaya, 50603, malaysia 2ajman university of science and technology, college of education and basic sciences, uae ∗corresponding author: rabhaibrahim@yahoo.com abstract. the eoq model determines the quantity that minimizes the total sum of all cost functions. we suggest a common structure for economic order quantity type non-linear differential models with costs functions with respect to time in a cyclic period. for this model, we analyze the related optimization problem and develop a relaxed method for determining a bounded interval containing the optimal cycle length. also for a special class of transportation functions, we study these consequences and introduce algorithms to calculate the optimal size and the corresponding optimal order stage. 1. introduction the general economy challenged by governments in recent years has been described by almost a linear model. for such a model, it must satisfy some kind of stability. therefore, it would be problematic to consider it likely that decisions occupied on the basis of past actions could cause to accurate future consequences. in view of the uncertainty and complexity usual of this setting, the question arises of the need to seek out new methods to deal with it. one of the most dynamic features in managing any economic units is the economic order quantity (eoq). it might be summed up, among other features, as confirming both good customer service and efficient production while keeping records as low as possible. received 2018-01-27; accepted 2018-03-14; published 2018-05-02. 2010 mathematics subject classification. 44a45. key words and phrases. differential system; economic order quantity; cost function. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 437 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-437 int. j. anal. appl. 16 (3) (2018) 438 the behavior of the cost function needs not be linear. furthermore, to be able to cover all possible conditions, it would be sensible to signify them as variables with respect to time, position and controller. these models usually are driven analytically by suggesting bounded intervals containing an optimal cycle length (reorder interval) to arrive at concave functions. this method brings a minimal value of eoq. for the other methods of cycle optimal lengths that evaluate the upper bound by an algorithm or by utilizing some concepts of the convex optimization problem. the convexity concept does not change in each bounded interval. in this case the procedure calculates the interval, such that the global minimum point resides. another structure is that the cost function imposes in bounded intervals. hence, the cost function is convex in all intervals; such an algorithm introduces a local solution in each interval (see [1]). several generalizations and modifications can be suggested to the eoq model, counting backordering costs and multiple units. moreover, the economic order interval can be computed from the eoq and the economic production quantity typical (which controls the optimal production quantity) can be calculated in a similar technique. the baumol-tobin model, has also been utilized to compute the money demand function, where a person’s holdings of money balances can be realized in an approach parallel to a firm’s holdings of inventory [2]. malakooti presented a model with conditions that could be minimized the total cost, order quantity and shortages [3]. a historical overview can be found in [4]. recently, lagodimoset et al. introduced a discrete-time eoq model [5]. methods for minimization can be found in [6]. in this paper, we introduce a new mathematical modeling for eoq. our method is based on the fixed point theorems in some of compact sets. the minimization of eoq is described by considering a geometric frame for the total cost function during a time. our study is related to the minimum norm solution. examples of linear and non-linear systems illustrated in the sequel. some of the systems have equilibrium points, which are given approximately. this technique allows eoq problems to be under unfixed credit period, where the classical models are satisfied fixed credit period. 2. processing the model of eoq was established by ford w. harris [7] under the formula: q = √ 2ab/c, (2.1) where a is the annual demand quantity, b is the fixed cost per item and c is the annual holding cost per item. in this note we suggest that (2.1) changes over time taking the formula q(τ) = √ 2a(τ)b(τ)/c(τ). (2.2) int. j. anal. appl. 16 (3) (2018) 439 to minimize (2.2), we shall prepare a differential equation as follows: by squaring (2.2) and differentiating the result, we have d dτ q(τ) = q(τ) 2 ( a′(τ) a(τ) + b′(τ) b(τ) − c′(τ) c(τ) ) := q(τ) (f1(τ) + f2(τ) + f3(τ)) := q(τ)f(τ,a,b,c). (2.3) eq. (2.3) is a general case of the suggested model (differential eoq) given in [8]. our aim is to find a minimum solution for (2.3). for this purpose, we introduce the following operator π : c → c, where c is any closed bounded and convex subset of a banach space b. the map π is called non-expensive if the following inequality holds: ‖πξ − πη‖ < ‖ξ −η‖; and it is called a contraction mapping if ‖πξ − πη‖≤ `‖ξ −η‖, ` ∈ (0, 1). we need the following outcomes: lemma 2.1. ( leray-schauder theorem) let π be a continuous and compact mapping of a banach space b into itself, such that the set {ξ ∈ b : ξ = λπξ for some 0 ≤ λ ≤ 1} is bounded. then π has a fixed point. lemma 2.2. (zorn’s lemma) for any weakly compact convex subset c and any non-expansive map π : c → c, c has a minimal (π -invariant) subset. lemma 2.3. [9] let b be a strictly convex normed space with norm ‖.‖. the set of fixed points of a non-expansive mapping π : b → b is either empty or closed and convex. 3. results in this section, we introduce two cases of the function f in (2.3). let the function f,q be continuous with respect to τ ∈ j : [0,t] and f be lipschitz with respect to (a,b,c) then there is a positive constant ℘ ≥ 0 such that int. j. anal. appl. 16 (3) (2018) 440 ‖f(τ,a,b,c) −f(τ,a,b,c)‖≤ ℘ (‖a−a‖ + ‖b − b‖ + ‖c − c‖) . moreover, we let f(τ, 0, 0, 0) = 0. define the following operator: (π)(q)(τ) := ∫ τ 0 q(ς)f(ς,a,b,c)dς, q(0) = 0. in our study, we involve some type of specification of how eoq moves from step to step; that is q(τ + 1) = πq(τ). the stationary step can be defined as a case for which πq(τ) = q(τ). this can be viewed as an equilibrium point of the system. that is the fixed points of π are the states at which the process of eoq is clear (stable). we have the following result: theorem 3.1. consider the eq (2.3). if ℘t (‖a‖ + ‖b‖ + ‖c‖) ≥ 1; then it admits at least one solution. if ℘t (‖a‖ + ‖b‖ + ‖c‖) < 1 then it admits a unique solution which minimizes the problem (2.2). proof. obviously, |(π)(q)(τ)| = | ∫ τ 0 q(ς)f(ς,a,b,c)dς| ≤ ∫ τ 0 |q(ς)||f(ς,a,b,c)|dς = ∫ τ 0 |q(ς)||f(ς,a,b,c) −f(τ, 0, 0, 0)|dς ≤ ℘t (‖a‖ + ‖b‖ + ‖c‖)‖q‖; consequently, we obtain 1 ℘t (‖a‖ + ‖b‖ + ‖c‖) |(π)(q)(τ)| := λ|(π)(q)(τ)| ≤ ‖q‖. by taking the maximum value, we get λ‖(π)(q)‖ = ‖q‖. int. j. anal. appl. 16 (3) (2018) 441 the operator π is continuous, bounded and compact in a subset of banach space b. hence, in view of lemma 2.1, the (2.3) has a solution. let p ∈ b such that (π)(p)(τ) = ∫ τ 0 p(ς)f(ς,a,b,c)dς, p(0) = 0. a calculation implies that |(π)(q)(τ) − (π)(p)(τ)| = ∣∣∣∫ τ 0 q(ς)f(ς,a,b,c)dς − ∫ τ 0 p(ς)f(ς,a,b,c)dς ∣∣∣ ≤ ∫ τ 0 |q(ς) −p(ς)||f(ς,a,b,c)|dς ≤ t |f(ς,a,b,c) −f(ς, 0, 0, 0)|‖q−p‖ ≤ ℘t (‖a‖ + ‖b‖ + ‖c‖)‖q−p‖; if ℘t (‖a‖ + ‖b‖ + ‖c‖) < 1 then by banach fixed point theorem, we conclude that (2.3) has a unique solution. it is clear that ‖(π)(q) − (π)(p)‖≤‖q−p‖. thus operator π is not-expensive map. therefore, in view of lemma 2.2 and lemma 2.3, the set of fixed point is a minimal close convex set. note that the fixed point corresponds to equilibrium of eoq. � 4. examples in this section, we shall consider some applications. example 1. assume that a company’ s production in two seasons during a year satisfying the following eoq system: q(1) = a(1)b(1) −c(1)q(2) q(2) = a(2)b(2) −c(2)q(1). (4.1) define the company’ s reaction function ρ : r2+ → r2+ by ρ(q(1),q(2)) := ( a(1)b(1) −c(1)q(2),a(2)b(2) −c(2)q(1) ) . int. j. anal. appl. 16 (3) (2018) 442 if max{c(1),c(2)} < 1 then a computation with respect to city block metric, gives |ρ(q) −ρ(q′)| = ∣∣∣a(1)b(1) −c(1)q(2) −(a(1)b(1) −c(1)q′(2))∣∣∣ + ∣∣∣a(2)b(2) −c(2)q(1) −(a(2)b(2) −c(2)q′(1))∣∣∣ = c(1)|q′(2) −q(2)| + c(2)|q′(1) −q(1)| ≤ max{c(1),c(2)} ( |q′(2) −q(2)| + |q′(1) −q(1)| ) = max{c(1),c(2)}|q−q′|. thus, the existence and uniqueness outcome for the equilibrium, where ρ has a unique fixed point q∗ if c(i) < 1, i = 1, 2. we have the following facts: • the condition c(i) < 1, i = 1, 2 is sufficient for equilibrium but not necessary. • this condition implies the convexity of the set of the fixed point ( lemma 2.3). • we may replace the metric with the euclidean metric to obtain the same result. • in view of the condition c(i) < 1, i = 1, 2 the system (4.1) can be considered as a fuzzy system (see fig. 1). −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 q(1) q(2) equilibrium point figure 1: example 1; the functions q(1) in the horizontal axis and q(2) in the vertical axis. here, we suppose c(i) = 0.5, i = 1, 2. and a(i) = 0.1,b(i) = 0.3. example 2. assume that a recent state of market is q(0) = (q1(0),q2(0)). a company j is imposing qj(0) units. at time τ = 1 each company selects its outcome level qj(1) by replaying to the quantity. consequently, we have the system (see fig.2) q1(τ + 1) = a(1)b(1) −c(1)q2(τ) q2(τ + 1) = a(2)b(2) −c(2)q1(τ). (4.2) int. j. anal. appl. 16 (3) (2018) 443 the sequence < q(τ) > converges the equilibrium point q∗ whenever max{c(1),c(2)} < 1. 0 2 4 6 8 10 −4 −3 −2 −1 0 equilibrium point τ q (τ ) 2.5 ∗ 0.3 − 0.5 ∗ τ 1.5 ∗ 0.3 − 0.2 ∗ τ figure 2: example 2; the functions q(τ) with respect to τ. we suppose c(1) = 0.5,c(2) = 0.2. example 3. suppose the following non-linear system q1(τ) = 1 2(1 + q2(τ)) q2(τ) = exp(−q1(τ)) 2 . (4.3) it is clear that the function % : r2+ → r2+, where %(q1,q2) = ( 1 2(1 + q2(τ)) , exp(−q1(τ)) 2 ) is a contraction , so an equilibrium point exists and unique. 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 equilibrium point τ q (τ ) q1 q2 figure 3: example 3: the functions q(τ) with respect to τ. there is a unique equilibrium point for the system corresponding to a unique solution. int. j. anal. appl. 16 (3) (2018) 444 example 4. suppose the following non-linear equation q : r+ → r+ q(τ) = 1 2 ( τ + c 1 τ ) , τ 6= 0. (4.4) it is clear that limτ−→0 q(τ) = ∞. to minimize (2.2), we shall use the an approximation technique. the approximation form of (4.4) is as follows: τm+1 = 1 2 ( τm + c 1 τm ) . (4.5) it is clear that this sequence converges to √ c. therefore, since q(τ) is continuous, then √ c is a fixed point of q. also, in view of theorem 3.1, for 0 < √ c < 1, eq. (4.4) has a unique fixed point corresponding to the solution of it. the reaction of (4.5) showed that it has a limit and this limit converges to the fixed point of (4.4). therefore, no need to show that the function q is a contraction mapping. 5. discussion in general, economic can be performed by a set of fixed points; thus fixed point theorems can provide the equilibria of economic. the frame of this work was to consider a new formula of eoq model. we introduced a technique of minimizing it by using the concept of fixed point theory. we developed this method to be suitable to the model (see theorem 3.1). applications are given including linear and non-linear systems. these systems are based on changing the solution during time in a fixed interval. the equilibrium points of each system was established, where it represented to the stationary state at which the markets clear. also, this stationary state acted when the company wants to change its strategy. references [1] r. w. grubbstrom, modelling production opportunities an historical overview, int. j. product. econ. 41 (1995), 1-14. [2] a caplin, j. leahy, john, economic theory and the world of practice: a celebration of the (s, s) model, j. econ. persp. 24 (1)(2010), 183-201. [3] b. malakooti, operations and production systems with multiple objectives (2013). john wiley & sons. [4] m.holmbom, a. segerstedt, economic order quantities in production: from harris to economic lot scheduling problems, int. j. product. econ. 155 (2014) , 82-90. [5] a. g. lagodimos, et al., the discrete-time eoq model: solution and implications, eur. j. oper. res. 266 (2018) ,112-121. [6] r. w. ibrahim, maximize minimum utility function of fractional cloud computing system based on search algorithm utilizing the mittag-leffler sum, int. j. anal. appl. 16(1) (2018), 125-136. [7] r. w. harris, how many parts to make at once, operat. res. 38 (6)(1990), 947. [8] g. mahata, p mahata, analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain, math. comput. model. 53 (2011), 1621-1636. [9] s.g. ferreira, the existence and uniqueness of the minimum norm solution to certain linear and nonlinear problems, signal proc. 55 (1996), 137-139. 1. introduction 2. processing 3. results 4. examples 5. discussion references international journal of analysis and applications volume 18, number 5 (2020), 724-737 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-724 he’s variational iteration method for solving multi-dimensional of navier stokes equation mohamed zellal1,2,∗, kacem belghaba2 1department of common core in exact sciences and informatics, faculty of exact sciences and informatics, hassiba benbouali university of chlef, ouled fares p.o. box 78, 02180, chlef, algeria 2laboratory of mathematics and its applications (lamap), university of oran1, oran, algeria ∗corresponding author: m.zellal@yahoo.fr abstract. in this paper, he’s variational iteration method (vim), established by he in (1999), is adopted to solve two and three dimensional of navier-stokes equation in cartesian coordinates. this method is a powerful tool to handle linear and nonlinear models. the main property of the method is its softness and ability to solve nonlinear equations, accurately and easily. using variational iteration method, it is possible to find the exact solution or a closed approximate solution of a problem. to illustrate the capacity and reliability of this method, some examples and numerical results are provided. 1. introduction the main aim of this work, is to solve the model of the navier stokes equation for an incompressible fluid flow is given as follows [5, 8, 17].  ut + (u.∇) u = ρ0∇2u − 1ρ∇p, on ω × (0,t), ∇.u = 0, on ω × (0,t), u = 0, on ∂ω × (0,t), (1.1) received may 18th, 2020; accepted june 8th, 2020; published june 24th, 2020. 2010 mathematics subject classification. 35n05, 35q30, 65a05, 65n25, 65q20. key words and phrases. navier–stokes equation; he’s variational iteration method; lagrange multiplier; correction functional; exact solution; approximate solution. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 724 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-724 int. j. anal. appl. 18 (5) (2020) 725 where u = (u,v,w), t, p, denote the fluid vector at the point (x; y; z), time and the pressure, respectively, ω ⊂ r3 and ∂ω its boundary, ρ is the density, ρ0 denotes the kinematic viscosity of the flow. in cartesian coordinates, the (3d) navier-stokes equation becomes:  ut + uux + vuy + wuz = ρ0 (uxx + uyy + uzz) − 1ρpx, vt + uvx + vvy + wvz = ρ0 (vxx + vyy + vzz) − 1ρpy, wt + uwx + vwy + wwz = ρ0 (wxx + wyy + wzz) − 1ρpz. (1.2) various kinds of analytical methods and numerical methods were used to solve navier-stokes equation, as adomian decomposition method [2, 17], fractional reduced differential transformation method [5], modified laplace decomposition method [15], the meshless local petrov-galerkin method [22], homotopy perturbation method [23]. the organization of this paper is as follows: in section 2, we review the procedure of he’s variational iteration method. in section 3, we show in some examples and numerical results that the present method gives the exact solution or a very good approximation of the exact solution, even for a small n, to illustrate the method, its accuracy, effectiveness and simplicity. a conclusion is given in section 4. 2. he’s variational iteration method variational iteration method was first proposed by the chinese mathematician he ( [9]– [14]). this method has been employed to solve a large variety of linear and nonlinear problems which has various applications in science and engineering, with approximations converging rapidly to accurate solutions. this approach is successfully and effectively applied to various equations such as burger’s and coupled burger’s equations [1, 4]. this technique is also employed to solve multispecies lotka–volterra equations in [3], the cauchy reaction-diffusion problem and several test examples are given in [6], in [7] the applications of the present method to solve the fokker–planck equation is provided. author of [9] investigated (vim) for solving delay differential equations. he’s variational iteration method is used in [10] to give the solution of blasius equation, autonomous ordinary differential equations [11], in [12] the variational iteration method is used to construct solitary solutions and compacton-like solutions for nonlinear dispersive equations. also this procedure is investigated in [16] for solving helmholtz equation, for solving integro-differential equations [19]. authors of [20] employed the variational iteration method for solving a parabolic inverse problem. wazwaz has used this method for handling linear and nonlinear diffusion equations [24], for solving linear and nonlinear systems of pdes [25], for rational solutions for kdv, k(2, 2) , burger, and cubic boussinesq equations which has various applications in science and engineering [26]. in [27], for analytic treatment for linear and nonlinear odes, the (vim) modified using he’s polynomials to solve biological population model [28]. the convergence of (vim) is studied in [18, 21]. the (vim) gives rapidly convergent successive approximations of the exact solution if such a solution exists. int. j. anal. appl. 18 (5) (2020) 726 to illustrate the procedure of this approach, we write a system as shown below [3, 25]  ltu + r1(u,v,w) + n1(u,v,w) = f1, ltv + r2(u,v,w) + n2(u,v,w) = f2, ltw + r3(u,v,w) + n3(u,v,w) = f3, (2.1) subject to initial conditions,   u(x, 0) = g1(x), v(x, 0) = g2(x), w(x, 0) = g3(x), (2.2) where lt is considered a first-order partial differential operator, rk and nk, 1 ≤ k ≤ 3 are linear and nonlinear operators respectively, and f1, f2, f3 source terms. the correction functionals for equations of (2.1) can be written as  un+1(x,t) = un(x,t) + ∫ t 0 λ1(τ)[lun(τ) + r1(ũn, ṽn, w̃n) + n1(ũn, ṽn, w̃n) −f1(τ)]dτ, vn+1(x,t) = vn(x,t) + ∫ t 0 λ2(τ)[lvn(τ) + r2(ũn, ṽn, w̃n) + n2(ũn, ṽn, w̃n) −f2(τ)]dτ, wn+1(x,t) = wn(x,t) + ∫ t 0 λ3(τ)[lwn(τ) + r3(ũn, ṽn, w̃n) + n3(ũn, ṽn, w̃n) −f3(τ)]dτ, (2.3) where λk, 1 ≤ k ≤ 3, are general lagrange multipliers, which can be identified optimally via the variational theory, and ũn , ṽn , and w̃n are restricted variations which means δũn = 0 , δṽn = 0 and δw̃n = 0. it is required first to determine the lagrange multipliers λk that will be identified optimally via integration by parts. the successive approximations un+1(x,t) , vn+1(x,t) , wn+1(x,t) , n ≥ 0, of the solutions u(x,t) , v(x,t), and w(x,t) will follow immediately upon using the lagrange multipliers obtained and by using selected functions u0 , v0 and w0. the initial values are usually used for the selected zeroth approximations. with the lagrange multipliers λk determined, then several approximations un(x,t), vn(x,t), wn(x,t), n ≥ 1, can be determined. finally, the solutions are given   u(x,t) = lim n→∞ un (x, t ), v(x,t) = lim n→∞ vn (x, t ), w(x,t) = lim n→∞ wn (x, t ). (2.4) int. j. anal. appl. 18 (5) (2020) 727 3. test examples example 3.1. consider two-dimensional (2d) navier-stokes equations: [2, 5].   ut + uux + vuy = ρ0 (uxx + uyy) ,vt + uvx + vvy = ρ0 (vxx + vyy) , (3.1) with the initial conditions:   u(x,y, 0) = −sin(x + y),v(x,y, 0) = sin(x + y). (3.2) the correction functional for (3.1) reads   un+1(x,y,t) = un(x,y,t) + ∫ t 0 λ1(τ) [ ∂un(x,y,τ) ∂τ + ũn ∂ũn ∂x + ṽn ∂ũn ∂y −ρ0 ( ∂2ũn ∂x2 + ∂2ũn ∂y2 )] dτ, vn+1(x,y,t) = vn(x,y,t) + ∫ t 0 λ2(τ) [ ∂vn(x,y,τ) ∂τ + ũn ∂ṽn ∂x + ṽn ∂ṽn ∂y −ρ0 ( ∂2ṽn ∂x2 + ∂2ṽn ∂y2 )] dτ. (3.3) this yields the stationary conditions   1 + λ1 = 0, λ ′ 1(τ = t) = 0, 1 + λ2 = 0, λ ′ 2(τ = t) = 0. (3.4) as a result we find λ1 = λ2 = −1. (3.5) substituting these values of the lagrange multipliers into the functionals (3.3) gives the iteration formulas   un+1(x,y,t) = un(x,y,t) − ∫ t 0 [ ∂un(x,y,τ) ∂τ + un ∂un ∂x + vn ∂un ∂y −ρ0 ( ∂2un ∂x2 + ∂2un ∂y2 )] dτ, vn+1(x,y,t) = vn(x,y,t) − ∫ t 0 [ ∂vn(x,y,τ) ∂τ + un ∂vn ∂x + vn ∂vn ∂y −ρ0 ( ∂2vn ∂x2 + ∂2vn ∂y2 )] dτ. (3.6) we can select u0(x,y,t) = −sin(x+y), v0(x,y,t) = sin(x+y), by using the given initial values. accordingly, we obtain the following successive approximations: int. j. anal. appl. 18 (5) (2020) 728 u1(x,y,t) = −sin(x + y)(1 − 2ρ0t), v1(x,y,t) = sin(x + y)(1 − 2ρ0t), u2(x,y,t) = −sin(x + y)(1 − 2ρ0t + 2ρ20t 2), v2(x,y,t) = sin(x + y)(1 − 2ρ0t + 2ρ20t 2), u3(x,y,t) = −sin(x + y)(1 − 2ρ0t + 2ρ20t 2 − 4 3 ρ30t 3), v3(x,y,t) = sin(x + y)(1 − 2ρ0t + 2ρ20t 2 − 4 3 ρ30t 3), ... un(x,y,t) = −sin(x + y)(1 − 2ρ0t + 2ρ20t 2 + · · ·), vn(x,y,t) = sin(x + y)(1 − 2ρ0t + 2ρ20t 2 + · · ·). the final form of the solution will be as follows  u(x,y,t) = −sin(x + y) ( 1 + (−2ρ0t) + (−2ρ0t) 2 2! + (−2ρ0t) 3 3! + · · · ) = −sin(x + y)e−2ρ0t, v(x,y,t) = sin(x + y) ( 1 + (−2ρ0t) + (−2ρ0t) 2 2! + (−2ρ0t) 3 3! + · · · ) = sin(x + y)e−2ρ0t. which is an exact solution and is same as obtained by [5]. table 1: numerical results when ρ0 = 0.5, x = 0.1 and y = 0.5 in example 3.1. t u u3 |u−u3| v v3 |v −v3| 0 -0.56464247 -0.56464247 0 0.56464247 0.56464247 0 0.05 -0.53710453 -0.53710453 0 0.53710453 0.53710453 0 0.1 -0.51090964 -0.51090968 0.4628e-7 0.51090964 0.51090968 0.4628e-7 0.15 -0.48599228 -0.48599262 0.9719 0.48599228 0.48599262 0.9719 0.2 -0.46229015 -0.46229162 0.9246 0.46229015 0.46229162 0.9246 0.25 -0.43974400 -0.43974841 0.8795 0.43974400 0.43974841 0.8795 0.30 -0.41829743 -0.41830832 0.1089e-4 0.41829743 0.41830832 0.1089e-4 0.35 -0.39789682 -0.39792017 0.2335e-4 0.39789682 0.39792017 0.2335e-4 0.40 -0.37849117 -0.37853631 0.4514e-4 0.37849117 0.37853631 0.4514e-4 int. j. anal. appl. 18 (5) (2020) 729 figure 1. the behavior of u,u3 and v,v3 in example 3.1 at t = 1 with the parameters ρ0 = 0, 5. example 3.2. consider two-dimensional (2d) navier-stokes equations:  ut + uux + vuy = ρ0 (uxx + uyy) ,vt + uvx + vvy = ρ0 (vxx + vyy) , (3.7) with the initial conditions:   u(x,y, 0) = −e x+y, v(x,y, 0) = ex+y. (3.8) we follow the same procedure discussed in example (3.1), we can select u0(x,y,t) = −ex+y, v0(x,y,t) = ex+y in the iteration formulas(3.6), by using the given initial values. accordingly, we obtain the following successive approximations: int. j. anal. appl. 18 (5) (2020) 730 u1(x,y,t) = −ex+y(1 + 2ρ0t), v1(x,y,t) = e x+y(1 + 2ρ0t), u2(x,y,t) = −ex+y(1 + 2ρ0t + 2ρ20t 2), v2(x,y,t) = e x+y(1 + 2ρ0t + 2ρ 2 0t 2), u3(x,y,t) = −ex+y(1 + 2ρ0t + 2ρ20t 2 + 4 3 ρ30t 3), v3(x,y,t) = e x+y(1 + 2ρ0t + 2ρ 2 0t 2 + 4 3 ρ30t 3), u4(x,y,t) = −ex+y(1 + 2ρ0t + 2ρ20t 2 + 4 3 ρ30t 3 + 2 3 ρ40t 4), v4(x,y,t) = e x+y(1 + 2ρ0t + 2ρ 2 0t 2 + 4 3 ρ30t 3 + 2 3 ρ40t 4), ... un(x,y,t) = −ex+y(1 + (2ρ0t) + (2ρ0t) 2 2! + · · ·), vn(x,y,t) = e x+y(1 + (2ρ0t) + (2ρ0t) 2 2! + · · ·). finally, the exact solution may be obtained as follows  u(x,y,t) = −e x+ye2ρ0t = −ex+y+2ρ0t, v(x,y,t) = ex+ye2ρ0t = ex+y+2ρ0t, which are exact solutions [5]. consider the following tables 2 and 3 with the observation that v = −u. table 2: numerical results when ρ0 = 1,x = 0.1 and y = 0.5 in example 3.2. t u u4 |u−u4| = |v −v4| 0 -1.8221188 -1.8221188 0 0.05 -2.0137527 -2.0137526 1.5441055e-7 0.1 -2.2255409 -2.2255359 5.0256955e-6 0.15 -2.4596031 -2.4595643 3.8824935e-5 0.2 -2.7182818 -2.7181155 1.6647662e-4 0.25 -3.0041660 -3.0036490 5.1706393e-4 0.30 -3.3201169 -3.3188072 1.3097397e-3 0.35 -3.6692967 -3.6664144 8.8236261e-3 0.40 -4.0552000 -4.0494768 5.7231449e-3 int. j. anal. appl. 18 (5) (2020) 731 table 3: absolute error= |u−u4| when ρ0 = 1 in example 3.2. t t = 0.1 t = 0.3 t = 0.5 (x,y) |u−u4| |u−u4| |u−u4| (0.1,0.1) 3.36882444 e-6 8.7794477954 e-4 1.2151119386 e-2 0.3,0.1) 4.11469146 e-6 1.0723241752 e-3 1.4841410733 e-2 (0.5,0.2) 5.55425251 e-6 1.4474862325 e-3 2.0033808995 e-2 (0.7,0.2) 6.78397933 e-6 1.7679636768 e-3 2.4469349563 e-2 (0.3,0.3) 5.02569550 e-6 1.3097397053 e-3 1.8127340004 e-2 (0.5,0.3) 6.13839835 e-6 1.5997196885 e-3 2.2140783079 e-2 (0.7,0.5) 9.15741425 e-6 2.3865013406 e-3 3.3030167023 e-2 (0.9,0.5) 1.11848910 e-5 2.9148793198 e-3 4.0343137104 e-2 example 3.3. consider three-dimensional (3d) navier-stokes equations:  ut + uux + vuy + wuz = ρ0 (uxx + uyy + uzz) , vt + uvx + vvy + wvz = ρ0 (vxx + vyy + vzz) , wt + uwx + vwy + wwz = ρ0 (wxx + wyy + wzz) , (3.9) with the initial condition:   u(x,y,z, 0) = −1 2 x + y + z, v(x,y,z, 0) = x− 1 2 y + z, w(x,y,z, 0) = x + y − 1 2 z. (3.10) the correction functionals for (3.9) read  un+1(x,y,z,t) = un + ∫ t 0 λ1(τ) [ ∂un ∂τ + ũn ∂ũn ∂x + ṽn ∂ũn ∂y + w̃n ∂ũn ∂z −ρ0 ( ∂2ũn ∂x2 + ∂2ũn ∂y2 + ∂2ũn ∂z2 )] dτ, vn+1(x,y,z,t) = vn + ∫ t 0 λ2(τ) [ ∂vn ∂τ + ũn ∂ṽn ∂x + ṽn ∂ṽn ∂y + w̃n ∂ṽn ∂z −ρ0 ( ∂2ṽn ∂x2 + ∂2ṽn ∂y2 + ∂2ṽn ∂z2 )] dτ, wn+1(x,y,z,t) = wn + ∫ t 0 λ3(τ) [ ∂wn ∂τ + ũn ∂w̃n ∂x + ṽn ∂w̃n ∂y + w̃n ∂w̃n ∂z −ρ0 ( ∂2w̃n ∂x2 + ∂2w̃n ∂y2 + ∂2w̃n ∂z2 )] dτ. (3.11) the stationary conditions are thus given by  1 + λ1 = 0, λ ′ 1(τ = t) = 0, 1 + λ2 = 0, λ ′ 2(τ = t) = 0, 1 + λ3 = 0, λ ′ 3(τ = t) = 0, (3.12) so that, the lagrange multipliers can be identified as follows: λ1 = λ2 = λ3 = −1. (3.13) int. j. anal. appl. 18 (5) (2020) 732 he’s variational iteration method consists of the following scheme:  un+1(x,y,z,t) = un − ∫ t 0 [ ∂un ∂τ + un ∂un ∂x + vn ∂un ∂y + wn ∂un ∂z −ρ0 ( ∂2un ∂x2 + ∂2un ∂y2 + ∂2un ∂z2 )] dτ, vn+1(x,y,z,t) = vn − ∫ t 0 [ ∂vn ∂τ + un ∂vn ∂x + vn ∂vn ∂y + wn ∂vn ∂z −ρ0 ( ∂2vn ∂x2 + ∂2vn ∂y2 + ∂2vn ∂z2 )] dτ, wn+1(x,y,z,t) = wn − ∫ t 0 [ ∂wn ∂τ + un ∂wn ∂x + vn ∂wn ∂y + wn ∂wn ∂z −ρ0 ( ∂2wn ∂x2 + ∂2wn ∂y2 + ∂2wn ∂z2 )] dτ. (3.14) starting with initial approximations: u0(x,y,z,t) = −12x+y+z,v0(x,y,z,t) = x− 1 2 y+z, and w0(x,y,z,t) = x + y − 1 2 z, from (3.14), other terms of the sequence are computed as follows:  u1(x,y,z,t) = − 1 2 x + y + z − 9 4 xt, v1(x,y,z,t) = x− 1 2 y + z − 9 4 yt, w1(x,y,z,t) = x + y − 1 2 z − 9 4 zt,   u2(x,y,z,t) = − 1 2 x + y + z + (− 1 2 x + y + z) 9 4 t2 − 9 4 xt− 27 16 xt3, v2(x,y,z,t) = x− 1 2 y + z + (x− 1 2 y + z) 9 4 t2 − 9 4 yt− 27 16 yt3, w2(x,y,z,t) = x + y − 1 2 z + (x + y − 1 2 z) 9 4 t2 − 9 4 zt− 27 16 zt3,   u3(x,y,z,t) = (− 1 2 x + y + z) + (− 1 2 x + y + z) 9 4 t2 + (− 1 2 x + y + z) 27 8 t4 + (− 1 2 x + y + z) 81 64 t6 − 9 4 xt− 81 16 xt3 − 243 64 xt5 − 729 1792 xt7, v3(x,y,z,t) = (x− 1 2 y + z) + (x− 1 2 y + z) 9 4 t2 + (x− 1 2 y + z) 27 8 t4 + (x− 1 2 y + z) 81 64 t6 − 9 4 yt− 81 16 yt3 − 243 64 yt5 − 729 1792 yt7, w3(x,y,z,t) = (x + y − 1 2 z) + (x + y − 1 2 z) 9 4 t2 + (x + y − 1 2 z) 27 8 t4 + (x + y − 1 2 z) 81 64 t6 − 9 4 zt− 81 16 zt3 − 243 64 zt5 − 729 1792 zt7, int. j. anal. appl. 18 (5) (2020) 733   u4(x,y,z,t) = (− 1 2 x + y + z) + (− 1 2 + y + z) 9 4 t2 + (− 1 2 x + y + z) 81 16 t4 + (− 1 2 + y + z) 243 32 t6 + (− 1 2 x + y + z) 51759 7168 t8 + (− 1 2 x + y + z) 72171 17920 t10 + (− 1 2 x + y + z) 59049 57344 t12 + (− 1 2 x + y + z) 59049 802816 t14 − 9 4 xt− 81 16 xt3 − 3159 320 xt5 − 21141 1792 xt7 − 31347 3584 xt9 − 98415 28672 xt11 − 59049 114688 xt13 − 177147 16056320 xt15, v4(x,y,z,t) = (x− 1 2 y + z) + (x− 1 2 y + z) 9 4 t2 + (x− 1 2 y + z) 81 16 t4 + (x− 1 2 y + z) 243 32 t6 + (x− 1 2 y + z) 51759 7168 t8 + (x− 1 2 y + z) 72171 17920 t10 + (x− 1 2 y + z) 59049 57344 t12 + (x− 1 2 y + z) 59049 802816 t14 − 9 4 yt− 81 16 yt3 − 3159 320 yt5 − 21141 1792 yt7 − 31347 3584 yt9 − 98415 28672 yt11 − 59049 114688 yt13 − 177147 16056320 yt15, w4(x,y,z,t) = (x + y − 1 2 z) + (x + y − 1 2 z) 9 4 t2 + (x + y − 1 2 z) 81 16 t4 + (x + y − 1 2 z) 243 32 t6 + (x + y − 1 2 z) 51759 7168 t8 + (x + y − 1 2 z) 72171 17920 t10 + (x + y − 1 2 z) 59049 57344 t12 + (x + y − 1 2 z) 59049 802816 t14 − 9 4 zt− 81 16 zt3 − 3159 320 zt5 − 21141 1792 zt7 − 31347 3584 zt9 − 98415 28672 zt11 − 59049 114688 zt13 − 177147 16056320 zt15. therefore,  u(x,y,z,t) = ( − 1 2 x + y + z )( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) − 9 4 xt ( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) = −1 2 x + y + z − 9 4 xt 1 − 9 4 t2 , v(x,y,z,t) = ( x− 1 2 y + z )( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) − 9 4 yt ( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) = x− 1 2 y + z − 9 4 yt 1 − 9 4 t2 , w(x,y,z,t) = ( x + y − 1 2 z )( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) − 9 4 zt ( 1 + ( 9 4 ) t2 + ( 9 4 )2 t4 + · · · ) = x + y − 1 2 z − 9 4 zt 1 − 9 4 t2 . the same result as [5]. int. j. anal. appl. 18 (5) (2020) 734 table 4: the (vim) results for u and u4 approximations in example 3.3 t t = 0.01 t = 0.3 t = 0.5 (x,y,z) u4 |u−u4| u4 |u−u4| u4 |u−u4| (0.1,0.1,0.2) 0.2478057563 1.44e-11 0.2285415892 2.99e-4 0.2897586783 2.45e-2 (0.3,0.1,0.2) 0.1432822385 4.50e-11 -0.0640993439 1.73e-3 -0.3486963088 7.99e-2 (0.5,0.2,0.3) 0.2388037309 7.51e-11 -0.1068322397 2.89e-3 -0.5811605148 1.33e-1 (0.7,0.2,0.3) 0.1342802131 1.06e-10 -0.3994731729 4.92e-3 -1.2196155019 2.36e-1 (0.9,0.3,0.4) 0.2298017055 1.36e-10 -0.4422060687 6.07e-3 -1.4520797078 2.91e-1 (0.6,0.3,0.4) 0.3865869822 8.97e-11 -0.0032446693 3.03e-3 -0.4943972270 1.34e-1 (0.8,0.4,0.5) 0.4821084745 1.20e-10 -0.0459775653 4.18e-3 -0.7268614329 1.87e-1 (0.9,0.4,0.5) 0.4298467157 1.35e-10 -0.1922980316 5.19e-3 -1.0460889265 2.40e-1 table 5: the (vim) results for v and v4 approximations in example 3.3. t t = 0.01 t = 0.3 t = 0.5 (x,y,z) v4 |v −v4| v4 |v −v4| v4 |v −v4| (0.1,0.1,0.2) 0.2478057563 1.43e-11 0.2285415892 2.99e-4 0.2897586783 2.45e-2 (0.3,0.1,0.2) 0.4478507665 1.35e-11 0.4784496263 1.17e-3 0.6957494596 7.57e-2 (0.5,0.2,0.3) 0.6956565227 2.77e-11 0.7069912153 1.47e-3 0.9855081381 1.00e-1 (0.7,0.2,0.3) 0.8957015329 2.70e-11 0.9568992524 2.35e-3 1.3914989194 1.51e-1 (0.9,0.3,0.4) 1.1435072892 4.12e-11 1.1854408415 2.65e-3 1.6812575977 1.76e-1 (0.6,0.3,0.4) 0.8434397739 4.24e-11 0.8105787860 1.33e-3 1.0722714258 9.92e-2 (0.8,0.4,0.5) 1.0912455303 5.66e-11 1.0391203750 1.63e-3 1.3620301041 1.24e-1 (0.9,0.4,0.5) 1.1912680354 5.63e-11 1.1640743935 2.07e-3 1.5650254948 1.49e-1 int. j. anal. appl. 18 (5) (2020) 735 table 6: the (vim) results for w and w4 approximations in example 3.3. t t = 0.01 t = 0.3 t = 0.5 (x,y,z) w4 |w −w4| w4 |w −w4| w4 |w −w4| (0.1,0.1,0.2) 0.0955214924 3.00e-11 -0.0427328960 1.15e-3 -0.2324642059 5.33e-2 (0.3,0.1,0.2) 0.2955665025 2.93e-11 0.2071751411 2.79e-4 0.1735265755 2.09e-3 (0.5,0.2,0.3) 0.5433722588 4.35e-11 0.4357167302 1.95e-5 0.4632852538 2.24e-2 (0.7,0.2,0.3) 0.7434172689 4.28e-11 0.6856247672 8.96e-4 0.8692760351 7.36e-2 (0.9,0.3,0.4) 0.9912230253 5.70e-11 0.9141663564 1.19e-3 1.1590347135 9.81e-2 (0.6,0.3,0.4) 0.6911555100 5.82e-11 0.5393043008 1.19e-4 0.5500485415 2.19e-2 (0.8,0.4,0.5) 0.9389612664 7.24e-11 0.7678458899 1.79e-4 0.8398072198 4.59e-2 (0.9,0.4,0.5) 1.0389837714 7.20e-11 0.8927999085 6.17e-4 1.0428026105 7.15e-2 it can be observed through tables [1-6] that this method is efficient and accurate for different values of time and place. conclusion. there are two main objectives for this work. the first presents an alternative approach to variation iteration method to handle non-linear problems. the second is to use this method to solve the twodimensional (2d) and three-dimensional (3d) navier-stokes equations in cartesian coordinates. it is obvious that the method gives several successive approximations through determining the lagrange multipliers and using the iteration. (vim) reduces the size of calculations and facilitates the computational work when compared with (adm) or (hpm) techniques. he’s variational iteration method is suitable as an alternative approach to current techniques being employed to a wide variety of problems in physics. the navier-stokes equations were examined. the desired solutions were obtained rapidly and in a direct way. the two goals were achieved. acknowledgements: the two authors, sincerely thank the editor and the referees for their guidance and suggestions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.a. abdou, a.a. soliman, variational iteration method for solving burger’s and coupled burger’s equations, j. comput. appl. math. 181 (2005), 245–251. [2] a. alsaif, analytical approximate solutions for two-dimensional incompressible navier-stokes equations, adv. phys. theor. appl. 49 (2015), 69–86. [3] b. batiha, m.s.m. noorani, i. hashim, variational iteration method for solving multispecies lotka–volterra equations, comput. math. appl. 54 (2007), 903–909. int. j. anal. appl. 18 (5) (2020) 736 [4] j. biazar, h. aminikhah, exact and numerical solutions for non-linear burger’s equation by vim, math. computer model. 49 (2009), 1394–1400. [5] b.k. singh, p. kumar, frdtm for numerical simulation of multi-dimensional, time-fractional model of navier–stokes equation, ain shams eng. j. 9 (2018), 827–834. [6] m. dehghan, f. shakeri, application of he’s variational iteration method for solving the cauchy reaction–diffusion problem, j. comput. appl. math. 214 (2008), 435–446. [7] m. dehghan, m. tatari, the use of he’s variational iteration method for solving the fokker-planck equation, phys. scr. 74 (2006), 310–316. [8] m. el-shahed, a. salem, on the generalized navier–stokes equations, appl. math. comput. 156 (2005), 287–293. [9] j.h. he, variational iteration method for delay differential equations, commun. nonlinear sci. numer. simul. 2 (4) (1997), 235–236. [10] j.h. he, variational iteration method–a kind of non-linear analytical technique: some examples, internat. j. non-linear mech. 34 (1999), 699–708. [11] j.h. he, variational iteration method for autonomous ordinary differential systems, appl. math. comput. 114 (2/3) (2000), 115–123. [12] j.h. he, x.h. wu, construction of solitary solution and compacton-like solution by variational iteration method, chaos solitons fractals. 29 (2006), 108–113. [13] j. h. he, variational iteration method-some recent results and new interpretations, j. comput. appl. math. 207 (2007), 3–17. [14] j. h. he,, x. h. wu, variational iteration method: new development and applications, comput. math. appl. 54 (7-8) (2007), 881–894. [15] s. kumar, d. kumar, s. abbasbandy, and m. m. rashidi, analytical solution of fractional navier-stokes equation by using modified laplace decomposition method, ain shams eng. j. 5 (2) (2014), 569–574. [16] s. momani, s. abuasad, application of he’s variational iteration method to helmholtz equation, choas solitons fractals. 27 (2006), 1119–1123. [17] s. momani, z. odibat, analytical solution of a time-fractional navier–stokes equation by adomian decomposition method, appl. math. comput. 177 (2) (2006), 488–494. [18] z. m. odibat, a study on the convergence of variational iteration method, math. computer model. 51 (2010), 1181–1192. [19] x. shang, d. han, application of the variational iteration method for solving nth-order integro-differential equations, j. comput. appl. math. 234 (2010), 1442–1447. [20] m. tatari, m. dehghan, he’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, chaos solitons fractals. 33 (2007), 671–677. [21] m. tatari, m. dehghan, on the convergence of he’s variational iteration method, j. comput. appl. math. 207 (2007), 121–128. [22] n. thamareerat, a. luadsong, n. aschariyaphotha, the meshless local petrov–galerkin method based on moving kriging interpolation for solving the time fractional navier–stokes equations, springerplus. 5 (2016), 417. [23] h. a. wahab, j. anwar, b. saira, n. muhammad, s. muhammad and h. sajjad, application of homotopy perturbation method to the navier-stokes equations in cylindrical coordinates, comput. ecol. softw. 5 (2) (2015), 139–151. [24] a. m. wazwaz, the variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations, comput. math. appl. 54 (2007), 933–939. int. j. anal. appl. 18 (5) (2020) 737 [25] a. m. wazwaz, the variational iteration method for solving linear and nonlinear systems of pdes, comput. math. appl. 54 (2007), 895–902. [26] a. m. wazwaz, the variational iteration method for rational solutions for kdv, k(2,2), burgers, and cubic boussinesq equations, j. comput. appl. math. 207 (2007), 18–23. [27] a. m. wazwaz, the variational iteration method for analytic treatment for linear and nonlinear odes, appl. math. comput. 212 (2009), 120–134. [28] m. zellal, k. belghaba, an accurate algorithm for solving biological population model by the variational iteration method using he’s polynomials, arab j. basic appl. sci. 25 (3) (2018), 142–149. 1. introduction 2. he's variational iteration method 3. test examples conclusion references int. j. anal. appl. (2022), 20:18 on magnetic curves according to killing vector fields in euclidean 3-space m. khalifa saad1,∗, h. s. abdel-aziz2 and haytham a. ali2 1department of mathematics, faculty of science, islamic university of madinah, 170 al-madinah, ksa 2department of mathematics, faculty of science, sohag university, 82524 sohag, egypt ∗corresponding author: mohammed.khalifa@iu.edu.sa abstract. in the geometric theory of space curves, a magnetic field generates magnetic flow. the trajectories of magnetic flow are called magnetic curves. in the present paper, we obtain magnetic curves corresponding to killing magnetic fields in euclidean 3-space e3. the magnetic curves of the spherical indicatrices of the tangent, principal normal and binormal for a regular space curve are said to be meant curves. also, we investigate the magnetic curves of the tangent indicatrix and obtain the trajectories of the magnetic fields called tt-magnetic, nt-magnetic and bt-magnetic curves. finally, some computational examples in support of our main results are given and plotted. 1. introduction the magnetic curves on three dimensional riemannian manifold (m3,g) are trajectories of charged particles moving on m3 under the action of a magnetic field f. each trajectory γ may be found by solving the lorentz equation ∇γ′γ′ = φ(γ′), where φ is the lorentz force corresponding to f and ∇ is the levi civita connection of g. in particular, the trajectories of (charged) particles moving without the action of a magnetic field are geodesics, which satisfy ∇γ′γ′ = 0 (see for more details [1,2]). in a three-dimensional space, when a charged particle moves along a regular curve, the tangent, normal and binormal vectors describe the kinematic and geometric properties of this particle. these vectors and the time dimension affect the trajectory of the charged particle during the motion in a magnetic field [3, 4]. moreover, the study of magnetic curves was extended to other ambient spaces, such as complex space forms [5,6], sasakian 3-manifold [7,8]. recently, results of classification for the killing received: jan. 22, 2022. 2010 mathematics subject classification. 53a04, 53a17, 53b20. key words and phrases. magnetic curves; killing vector field; lorentz force; spherical indicatrices. https://doi.org/10.28924/2291-8639-20-2022-18 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-18 2 int. j. anal. appl. (2022), 20:18 magnetic trajectories on two special 3-dimensional manifolds, namely e3 and s2 ×r, were obtained in [9] and [10], respectively. barros and romero proved that if (m3,g) has constant curvature, then the magnetic curves corresponding to a killing magnetic field are center lines of kirchhoff elastic rods [11]. the curves and their frames play an important role in differential geometry and in many branches of science such as mechanics and physics. so, we are interested here in studying some of these curves called magnetic curves, which have many applications in modern physics. in this work, we investigate the trajectories of the magnetic fields called as tt-magnetic, nt-magnetic and btmagnetic curves and obtain some solutions of the lorentz force equation. we are looking forward to see that our results will be helpful to researchers who are specialized in mathematical modeling, mechanics and modern physics. 2. basic concepts in this section, we list some notions, formulae and conclusions for curves in three-dimensional euclidean space which can be found in the text books on differential geometry (see for instance [1, 12, 13]). let e3 denotes the real vector space with its usual vector structure. we denote by (x1,x2,x3) the coordinates of a vector with respect to the canonical basis of e3. for any two vectors x = (x1,x2,x3) and y = (y1,y2,y3), the metric g on e3 is defined by g(x,y) = x1y1 + x2y2 + x3y3. the norm of x is given by ‖x‖ = √ g(x,x), and the vector product is denoted by x×y = ((x2y3 −x3y2), (x3y1−x1y3), (x1y2 −x2y1)). the sphere of radius r > 0 with center at the origin is given by s2 = {x = (x1,x2,x3) ∈e3 : g(x,x) = r2}. let γ = γ(s) : i ⊂ r →e3 be an arbitrary curve in e3, s be the arclength parameter of γ. it is well known that each unit speed curve with at least four continuous derivatives, one can associate three mutually orthogonal unit vector fields t , n and b called the tangent, the principal normal and the binormal vector fields, respectively [14]. let {t (s),n(s),b(s)} be the moving frame along γ, where these vectors are mutually orthogonal vectors satisfying 〈t (s),t (s)〉 = 〈n(s),n(s)〉 = 〈b(s),b(s)〉 = 1. int. j. anal. appl. (2022), 20:18 3 the frenet equations for γ are given by [15]  t ′(s) n′(s) b′(s)   =   0 κ(s) 0 −κ(s) 0 τ(s) 0 −τ(s) 0     t (s) n(s) b(s)   , (2.1) where κ(s) and τ(s) are called the curvatures of γ. for spherical images of a regular curve in euclidean 3-space, we present the following definition: definition 2.1. [16,17] let γ be a curve in euclidean 3-space with frenet vectors t , n and b. the unit tangent vectors along the curve γ(s) generate a curve γt = t on the sphere of radius 1 about the origin. the curve γt is called the spherical indicatrix of t or more commonly, γt is called tangent indicatrix of the curve γ. if γ = γ(s) is a natural representations of the curve γ, then γt = t (s) will be a representation of γt . similarly, one can consider the principal normal indicatrix γn = n(s) and binormal indicatrix γb = b(s). let γ be a curve in e3 and consider γt = t (s) as the tangent indicatrix of γ with {tt ,nt ,bt} as its frenet vectors. then we have the frenet formula as follows:  t ′t (st ) n′t (st ) b′t (st )   =   0 κt 0 −κt 0 τt 0 −τt 0     tt (st ) nt (st ) bt (st )   , where   tt = n, nt = −1√ 1+f 2 t + f√ 1+f 2 b, bt = f√ 1+f 2 t + 1√ 1+f 2 b, and st = ∫ κ(s)ds, κt = √ 1 + f 2, τt = σ √ 1 + f 2, f = τ(s) κ(s) , (2.2) taking into consideration that σ = f ′(s) κ(s) (1 + f 2(s)) 3/2 , is the geodesic curvature of the principal image of the principal normal indicatrix of the curve γ, st is a natural representation parameter of the tangent indicatrix of γ and also it is the total curvature of the curve γ and κt , τt are the curvature and torsion of γt . therefore, we can see that τt κt = σ. let us introduce the following notions [6,11,18]. definition 2.2. a magnetic field on a three-dimensional oriented riemannian manifold (m3,g) is defined as a closed 2-form f on m3, related to a skew-symmetric (1, 1)−tensor field φ called the lorentz force of f, and we have g(φ(x),y ) = f (x,y ), ∀ x,y ∈ χ(m). 4 int. j. anal. appl. (2022), 20:18 the magnetic trajectories of f are curves γ on m3 which satisfy the lorentz equation: ∇γ′γ′ = φ(γ′). let v be a killing vector field on m3, then the lorentz force can be written as φ(x) = v ×x, (2.3) in this case, the lorentz force equation is given by ∇γ′γ′ = v ×γ′. note that, for a trivial magnetic field; f = 0, the lorentz equation becomes ∇γ′γ′ = 0 and then the solutions are geodesics. proposition 2.1. let γ : i ⊂ r → m3 be a curve in the three-dimensional oriented riemannian manifold (m3,g) and v be a vector field along the curve γ. then, one can take a variation of γ in the direction of v , say, a map π : i × (−�,�) → m3 which satisfies π(s, 0) = γ(s), ( ∂π ∂s (s,t) ) = v (s). in this setting, we have the following functions: 1. the speed function v(s,t) = ∥∥∂π ∂s (s,t) ∥∥; t is the time dimension, 2. the curvature κ(s,t) and the torsion τ(s,t) are functions of γ(s). the variations of these functions at t = 0 are given as follows: v (v) = ( ∂v ∂t (s,t) )∣∣∣∣ t=0 = g(∇tv,t ), v (κ) = ( ∂κ ∂t (s,t) )∣∣∣∣ t=0 = g(∇2tv,n) − 2κ g(∇tv,t ) + g(r(v,t )t,n), v (τ) = ( ∂τ ∂t (s,t) )∣∣∣∣ t=0 = [ 1 κ g(∇2tv + r(v,t )t,b) ]′ +g(r(v,t )n,b)+τg(∇tv,t )+2κ g(∇tv,b), where r is the curvature tensor of m3. corollary 2.1. let v (s) be a restriction to γ(s) of a killing vector field v of m3, then v (v) = v (κ) = v (τ) = 0. 3. magnetic curves of the tangent indicatrix definition 3.1. [11, 18] let γt : i → s2 ⊂ e3 be a tangent indicatrix of a regular curve γ in three-dimensional euclidean space e3, and f be a magnetic field on m3, then the curve γt is (i) tt−magnetic curve if tt satisfies the lorentz force equation, ∇tt tt = φ(tt ) = v ×tt , (ii) nt−magnetic curve if nt satisfies the lorentz force equation, ∇tt nt = φ(nt ) = v ×nt , (iii) bt−magnetic curve if bt satisfies the lorentz force equation, ∇tt bt = φ(bt ) = v ×bt . in the light of this definition, we can investigate the following. int. j. anal. appl. (2022), 20:18 5 3.1. tt-magnetic curve. proposition 3.1. let γt be a tt−magnetic curve in e3, with the frenet apparatus {tt ,nt ,bt ,κt ,τt}. then, we have the frenet formula:  t ′t (st ) n′t (st ) b′t (st )   =   0 √ 1 + f 2 0 − √ 1 + f 2 0 σ √ 1 + f 2 0 −σ √ 1 + f 2 0     tt (st ) nt (st ) bt (st )   , and the lorentz force in the frenet frame can be written as  φ(tt ) φ(nt ) φ(bt )   =   0 √ 1 + f 2 0 − √ 1 + f 2 0 ψ1 0 −ψ1 0     tt nt bt   . (3.1) where ψ1 is a certain function defined by ψ1 = g(φ(nt ),bt ). proof. from definition 3.1, one can write φ(tt ) = √ 1 + f 2 nt . (3.2) since φ(nt ) ∈ span{tt ,nt ,bt}, we have φ(nt ) = λ1tt + λ2nt + λ3bt . use the following equalities: g(φ(nt ),tt ) = −g(φ(tt ),nt ) = − √ 1 + f 2, g(φ(nt ),nt ) = 0, g(φ(nt ),bt ) = ψ1, to get λ1 = − √ 1 + f 2, λ2 = 0, λ3 = ψ1. hence, φ(nt ) = − √ 1 + f 2tt + ψ1bt . (3.3) similarly, we can easily obtain φ(bt ) = −ψ1nt . (3.4) from eqs. (3.2), (3.3) and (3.4), we get the required result. � proposition 3.2. the curve γt is a tt-magnetic trajectory of a magnetic field f if and only if the vector field v is given by v = ψ1tt + √ 1 + f 2 bt . (3.5) 6 int. j. anal. appl. (2022), 20:18 proof. let γt be a tt-magnetic trajectory of a magnetic field f. then, by using proposition 3.1 and eq. (2.3), we can easily have v = ψ1tt + √ 1 + f 2 bt . conversely, we assume that eq. (3.5) holds, then we get v ×tt = φ(tt ) and so the curve γt is a tt-magnetic curve. � theorem 3.1. let γt be a tt−magnetic curve and v be a killing vector field on a space form (m3(k),g). if γt is one of the tt−magnetic trajectories of (m3(k),g,v ), then its curvatures satisfying the following relations: ψ1 = const., (1 + f 2) ( ψ1 2 −σ √ 1 + f 2 ) = a1, (√ 1 + f 2 )′′ + σ(1 + f 2)ψ1 −σ2(1 + f 2)3/2 + k √ 1 + f 2 + 1 2 (1 + f 2)3/2 = a2 √ 1 + f 2, where k is the curvature of riemannian space m3 and a1, a2 are constants. proof. let v be a vector field in riemannian manifold m3, then v satisfies eq. (3.5). so, differentiating eq. (3.5) with respect to s, we get ∇tv = ψ′1tt + √ 1 + f 2(ψ1 −σ √ 1 + f 2)nt + (√ 1 + f 2 )′ bt . (3.6) since v is a killing vector then from corollary 2.1, v (v) = 0 and ∇tv has no tangential component, i.e., ψ1 = const. also, the differentiation of eq. (3.6) and using eq. (2.3) lead to ∇2tv = (1 + f 2)(σ √ 1 + f 2 − ψ1)tt + ((√ 1 + f 2 )′′ + σ(1 + f 2)ψ1 −σ2(1 + f 2)3/2 ) bt + ((√ 1 + f 2 )′( ψ1 − 2σ √ 1 + f 2 ) − √ 1 + f 2 ( σ √ 1 + f 2 )′) nt . (3.7) thus, from eqs. (3.6) and (3.7) and corollary 2.1, we have (v ( √ 1 + f 2) = 0). so, we get (1 + f 2) ( ψ1 2 −σ √ 1 + f 2 ) + a1 = 0. (3.8) similarly, according to proposition 2.2, when eqs. (3.6) and (3.7) are considered with the condition v (σ √ 1 + f 2) = 0, we can easily obtain[ 1 √ 1 + f 2 ((√ 1 + f 2 )′′ + σ(1 + f 2)ψ1 −σ2(1 + f 2)3/2 + g(r(v,tt )tt ,bt ) )]′ + √ 1 + f 2 (√ 1 + f 2 )′ = 0. if m3 has constant curvature k, then g(r(v,tt )tt ,bt ) = kg(v,bt ) = k √ 1 + f 2, int. j. anal. appl. (2022), 20:18 7 therefore,(√ 1 + f 2 )′′ + σ(1 + f 2)ψ1 −σ2(1 + f 2)3/2 + k √ 1 + f 2 + 1 2 (1 + f 2)3/2 = a2 √ 1 + f 2. (3.9) hence, the proof is completed. � 3.2. nt-magnetic curve. proposition 3.3. let γt be a nt-magnetic curve in e3 with frenet apparatus {tt ,nt ,bt ,κt ,τt}. then, the lorentz force in the frenet frame can be written as  φ(tt ) φ(nt ) φ(bt )   =   0 √ 1 + f 2 ψ2 − √ 1 + f 2 0 σ √ 1 + f 2 −ψ2 −σ √ 1 + f 2 0     tt nt bt   , (3.10) where ψ2 is a function defined by ψ2 = g(φ(tt ),bt ). proof. from definition 3.1, one can write φ(nt ) = − √ 1 + f 2tt + σ √ 1 + f 2bt . (3.11) since φ(tt ) ∈ span{tt ,nt ,bt}, then we have φ(tt ) = µ1tt + µ2nt + µ3bt . using the following equalities: g(φ(tt ),tt ) = 0, g(φ(tt ),bt ) = ψ2, g(φ(tt ),nt ) = −g(φ(nt ),tt ) = √ 1 + f 2, we get µ1 = 0, µ2 = √ 1 + f 2, µ3 = ψ2, and therefore, φ(tt ) = √ 1 + f 2nt + ψ2bt . (3.12) similarly, we can easily obtain that φ(bt ) = −ψ2tt −σ √ 1 + f 2nt . (3.13) hence, from eqs. (3.11), (3.12) and (3.13), the proof is completed. � corollary 3.1. let γt be a curve in e3. then, the curve γt is a nt-magnetic trajectory of a magnetic field f if and only if the vector field v along γ is written as v = σ √ 1 + f 2tt − ψ2nt + √ 1 + f 2bt . (3.14) proof. the proof is similar to that we have considered in proposition 3.2. � 8 int. j. anal. appl. (2022), 20:18 theorem 3.2. let γt be a nt−magnetic curve and v be a killing vector field on a space form (m3(k),g). if the curve γt is one of the nt−magnetic trajectories of (m3(k),g,v ), then its curvatures satisfying the following relations: ψ2 = ( σ √ 1 + f 2 )′ √ 1 + f 2 , ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − ψ′′2 = kψ2,(√ 1 + f 2 )′′ − 2ψ′2σ √ 1 + f 2 − ψ2 ( σ √ 1 + f 2 )′ + k √ 1 + f 2 + (1 + f 2)3/2(1 + σ) 2 = a3 √ 1 + f 2, where a3 is a constant. proof. differentiating eq. (3.14) with respect to s, we get ∇tv = ( ψ2 √ 1 + f 2 + ( σ √ 1 + f 2 )′) tt − ψ′2nt + ((√ 1 + f 2 )′ − ψ2σ √ 1 + f 2 ) bt . (3.15) since v is a killing vector, then from proposition 3.2 (v (v) = 0), we have ψ2 = ( σ √ 1 + f 2 )′ √ 1 + f 2 . also, differentiation of eq. (3.15) together with eq. (2.2), gives ∇2tv = ψ ′ 2 √ 1 + f 2tt + ( ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − ψ′′2 ) nt + ((√ 1 + f 2 )′′ − 2ψ′2σ √ 1 + f 2 − ψ2 ( σ √ 1 + f 2 )′) bt . (3.16) thus, from eqs. (3.15) and (3.16) together with proposition 2.2 (v ( √ 1 + f 2) = 0), we get ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − ψ′′2 + g(r(v,tt )tt ,nt ) = 0. if m3 has a constant curvature k, then g(r(v,tt )tt ,nt ) = kg(v,nt ) = −kψ2, and therefore ψ2σ 2(1 + f 2) −σ √ 1 + f 2 (√ 1 + f 2 )′ − ψ′′2 −kψ2 = 0. (3.17) using the condition v (σ √ 1 + f 2) = 0 in eqs. (3.15) and (3.16), we obtain[ 1 √ 1 + f 2 ((√ 1 + f 2 )′′ − 2ψ′2σ √ 1 + f 2 − ψ2 ( σ √ 1 + f 2 )′ + k √ 1 + f 2 )]′ + √ 1 + f 2 (√ 1 + f 2 )′ + σ √ 1 + f 2 ( σ √ 1 + f 2 )′ = 0. (3.18) integrating eq. (3.18) leads to(√ 1 + f 2 )′′ − 2ψ′2σ √ 1 + f 2 − ψ2 ( σ √ 1 + f 2 )′ + k √ 1 + f 2 + (1 + f 2)3/2(1 + σ) 2 = a3 √ 1 + f 2. (3.19) int. j. anal. appl. (2022), 20:18 9 thus, this completes the proof. � corollary 3.2. let γt be a nt−magnetic curve in euclidean 3-space with ψ2 is zero, then γt is a circular helix. moreover, the axis of the circular helix is the vector field. proof. it is clear from theorem 3.2. � 3.3. bt-magnetic curve. proposition 3.4. let γt be a bt-magnetic curve in e3 with frenet apparatus {tt ,nt ,bt ,κt ,τt}. then, the lorentz force in the frenet frame can be written as  φ(tt ) φ(nt ) φ(bt )   =   0 ψ3 0 −ψ3 0 σ √ 1 + f 2 0 −σ √ 1 + f 2 0     tt nt bt   . (3.20) where ψ3 is given by ψ3 = g(φ(tt ),nt ). proof. as we mentioned the above, we can write φ(bt ) = −σ √ 1 + f 2nt , (3.21) φ(tt ) = υ1tt + υ2nt + υ3bt . using the following conditions: g(φ(tt ),tt ) = 0, g(φ(tt ),nt ) = ψ3, g(φ(tt ),bt ) = −g(φ(bt ),tt ) = 0, we can obtain µ1 = 0, µ2 = ψ3, µ3 = 0. from this, we get φ(tt ) = ψ3nt . (3.22) also, we obtain φ(nt ) = −ψ3tt + σ √ 1 + f 2bt . (3.23) therefore, the proof is completed. � corollary 3.3. let γt be a curve in e3. the curve γt is a bt-magnetic trajectory of a magnetic field f if and only if the vector field v along γ is written as v = σ √ 1 + f 2tt + ψ3bt . (3.24) 10 int. j. anal. appl. (2022), 20:18 theorem 3.3. let γt be a bt−magnetic curve and v be a killing vector field on a space form (m3(k),g). if the curve γt is one of the bt−magnetic trajectories of (m3(k),g,v ), then its curvatures satisfying the following relations: σ √ 1 + f 2 = const., ψ′3 = 1 2 (√ 1 + f 2 )′ , ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − ψ3 ) + kψ3 + (1 + f 2)3/2 4 = a4 √ 1 + f 2 ; a4 is constant. proof. since v is a vector field, differentiating eq. (3.24) with respect to s, we get ∇tv = ( σ √ 1 + f 2 )′ tt + σ √ 1 + f 2 (√ 1 + f 2 − ψ3 ) nt + ψ ′ 3bt . (3.25) since v is a killing vector, then we have σ √ 1 + f 2 = const. (3.26) again, differentiating eq. (3.25) and using eq. (2.2), we get ∇2tv = −σ(1 + f 2) (√ 1 + f 2 − ψ3 ) tt + σ √ 1 + f 2 ((√ 1 + f 2 )′ − 2ψ′3 ) nt + ( ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − ψ3 )) bt , (3.27) which leads to ψ′3 = 1 2 (√ 1 + f 2 )′ . (3.28) similarly, using the condition v (σ √ 1 + f 2) = 0 in eqs. (3.25) and (3.27), we obtain[ 1 √ 1 + f 2 ( ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − ψ3 ) + g(r(v,tt )tt ,bt ) )]′ +ψ′3 √ 1 + f 2 = 0. (3.29) if k = const., then we have g(r(v,tt )tt ,bt ) = kg(v,bt ) = kψ ′ 3, and therefore ψ′′3 + σ 2(1 + f 2) (√ 1 + f 2 − ψ3 ) + kψ′3 + (1 + f 2)3/2 4 = a4 √ 1 + f 2, (3.30) thus, this completes the proof. � corollary 3.4. let γt be a bt−magnetic curve in euclidean 3-space with ψ3 constant, then γt is a circular helix. moreover, the axis of the circular helix is the vector field. proof. it is obvious from eq. (3.26) and eq. (3.28). � int. j. anal. appl. (2022), 20:18 11 using eq. (3.30), we obtain the following second-order nonlinear ordinary differential equation u′′(s)+σ2(1+f 2)u(s)+ku′(s)+2u3(s)−2a4u(s) = 0, u(s) = 1 2 √ 1 + f 2; k and σ √ 1 + f 2 = const. now, we can consider the above differential equation in euclidean 3space e3, in 3sphere s3 and in hyperbolic 3space h3, respectively. case 3.1. euclidean 3space e3 (k = 0, σ √ 1 + f 2 = 3) : u′′(s) + 2u3(s) + 7u(s) = 0, a sample of individual solutions for this equation is given in the following figures: figure 1 sample solution family: figure 2. trajectories of the curvature κt of b-magnetic curve in euclidean 3-space. 12 int. j. anal. appl. (2022), 20:18 case 3.2. 3-sphere s3 (k = 1, σ √ 1 + f 2 = 3): u′′(s) + u′(s) + 2u3(s) + 7u(s) = 0, a sample of individual solutions for this equation is given in the following figures: figure 3. sample solution family: figure 4. trajectories of the curvature κt of b-magnetic curve in 3-sphere. case 3.3. 3hyperbolic space h3(k = −1, σ √ 1 + f 2 = 3): u′′(s) −u′(s) + 2u3(s) + 7u(s) = 0, k = −1, σ √ 1 + f 2 = 3, a sample of individual solutions for this equation is given in the following figures: int. j. anal. appl. (2022), 20:18 13 figure 5. sample solution family: figure 6. trajectories of the curvature κt of b-magnetic curve in hyperbolic 3-space. remark 3.1. according to the study that we have considered in the case of magnetic curves of the tangent indicatrix of γ, we can do similar study for the other spherical indicatrices, the principal normal indicatrix and the binormal indicatrix. 4. applications in what follows, we give two computational examples to illustrate our main results. example 4.1. let α : i →e3 be a regular curve in the three-dimensional euclidean space e3, can be written as α = ( s 2 cos[ln[ s 2 ]], s 2 sin[ln[ s 2 ]], s √ 2 ) , 14 int. j. anal. appl. (2022), 20:18 taking the first derivative of the curve α we get t (s) = ( 1 2 ( cos[ln[ s 2 ]] − sin[ln[ s 2 ]] ) , 1 2 ( cos[ln[ s 2 ]] + sin[ln[ s 2 ]] ) , 1 √ 2 ) . also, we can get the principal normal and binormal vectors of α respectively, n(s) = ( − cos[ln[ s 2 ]] + sin[ln[ s 2 ]] √ 2 , cos[ln[ s 2 ]] − sin[ln[ s 2 ]] √ 2 , 0 ) , b(s) = ( 1 2 ( sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 2 ( −sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 √ 2 ) , the curvatures of α are κ(s) = τ(s) = 1 √ 2s . it is clear that α is a general helix. the tangent indicatrix of α is obtained as follows αt = ( 1 2 ( cos[ln[ s 2 ]] − sin[ln[ s 2 ]] ) , 1 2 ( cos[ln[ s 2 ]] + sin[ln[ s 2 ]] ) , 1 √ 2 ) , from direct calculations, we can get the frenet vectors of αt tt (st ) = ( − cos[ln[ s 2 ]] + sin[ln[ s 2 ]] √ 2 , cos[ln[ s 2 ]] − sin[ln[ s 2 ]] √ 2 , 0 ) , nt (st ) = ( 1 √ 2 ( sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 1 √ 2 ( −sin[ln[ s 2 ]] − cos[ln[ s 2 ]] ) , 0 ) , bt (st ) = (0, 0, 1) . the natural representation and the curvatures of αt are respectively, st = 1 √ 2 ln[s], f = 1, σ = 0, κt = √ 2, τt = 0, in addition, the certain function of αt is ψ1 = const., it means that αt is a tt-magnetic curve. example 4.2. we consider the circular helix γ in euclidean 3− space defined by γ(s) = ( cos [ s √ 2 ] , sin [ s √ 2 ] , s √ 2 ) . differentiating this equation, we get the tangent vector t as follows: t (s) = ( −1 √ 2 sin [ s √ 2 ] , 1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) . it follows that, the principal normal and binormal vectors of γ respectively, are given by n(s) = ( −cos [ s √ 2 ] ,−sin [ s √ 2 ] , 0 ) , b(s) = ( 1 √ 2 sin [ s √ 2 ] , −1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) , and so, the curvatures of γ are obtained κ(s) = τ(s) = 1 2 . int. j. anal. appl. (2022), 20:18 15 from the above calculations, the tangent indicatrix of γ is given as follows γt (st ) = ( −1 √ 2 sin [ s √ 2 ] , 1 √ 2 cos [ s √ 2 ] , 1 √ 2 ) . the frenet vectors of γt are given as follows tt (st ) = ( −cos [ s √ 2 ] ,−sin [ s √ 2 ] , 0 ) , nt (st ) = ( sin [ s √ 2 ] ,−cos [ s √ 2 ] , 0 ) , bt (st ) = (0, 0, 1) . moreover, the natural representation and the curvature of γt are respectively, st = 1 2 s, f = 1, σ = 0, κt = √ 2, in addition, the torsion and the certain function of γt are respectively, τt = 0 and ψ2 = 0, it means that γt is nt-magnetic as well as bt-magnetic curve. (a) (b) figure 7. the circular helix γ and its spherical image γt . 5. conclusion the value of this paper is due to the important and prominent role of the theory of curves in differential geometry as well as magnetic fields that generate magnetic flow whose trajectories give so-called magnetic curves. in this sense, the idea of this work is devoted to examine some conditions to construct special magnetic curves of spherical images for a regular curve γ in euclidean 3-space. some characterizations of magnetic curves of the tangent indicatrix of γ are obtained. an application to confirm our main results is given and plotted. 16 int. j. anal. appl. (2022), 20:18 acknowledgment the researchers wish to extend their sincere gratitude to the deanship of scientific research at the islamic university of madinah for the support provided to the post-publishing program 1. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.p. do carmo, differential geometry of curves and surfaces, prentice hall, englewood cliffs, 1976. [2] r. talman, geometric mechanics: toward a unification of classical physics, second ed., wiley-vch, new york, 2007. [3] a. comtet, on the landau levels on the hyperbolic plane, ann. phys. 173 (1987), 185–209. https://doi.org/ 10.1016/0003-4916(87)90098-4. [4] t. sunada, magnetic flows on a riemann surface, in: proceedings of kaist mathematics workshop, (1993), 93-108. [5] t. adachi, kähler magnetic fields on a complex projective space, proc. japan acad. ser. a math. sci. 70 (1994). https://doi.org/10.3792/pjaa.70.12. [6] t. adachi, kähler magnetic flows for a manifold of constant holomorphic sectional curvature, tokyo j. math. 18 (1995). https://doi.org/10.3836/tjm/1270043477. [7] j.l. cabrerizo, m. fernández, j.s. gómez, the contact magnetic flow in 3d sasakian manifolds, j. phys. a: math. theor. 42 (2009), 195201. https://doi.org/10.1088/1751-8113/42/19/195201. [8] j.l. cabrerizo, m. fernández, j.s. gómez, on the existence of almost contact structure and the contact magnetic field, acta math. hung. 125 (2009), 191–199. https://doi.org/10.1007/s10474-009-9005-1. [9] s.l. druţă-romaniuc, m.i. munteanu, magnetic curves corresponding to killing magnetic fields in e3, j. math. phys. 52 (2011), 113506. https://doi.org/10.1063/1.3659498. [10] m.i. munteanu, a.i. nistor, the classification of killing magnetic curves in s2 ×r, j. geom. phys. 62 (2012), 170-182. https://doi.org/10.1016/j.geomphys.2011.10.002. [11] m. barros, a. romero, magnetic vortices, europhys. lett. 77 (2007), 34002. https://doi.org/10.1209/ 0295-5075/77/34002. [12] j. koenderink, solid shape, mit press, cambridge, ma, 1990. [13] b. o’neil, semi-riemannian geometry with applications to relativity, academic press, inc., new york, 1983. [14] h.h. hacisalihoglu, differential geometry, ankara university, faculty of science press, 2000. [15] d.j. struik, lectures in classical differential geometry, addison wesley, reading, ma, 1961. [16] i̇. arslan and h. h. hacısalihoğlu, on the spherical representatives of a curve, int. j. contemp. math. sciences, 4 (2009), 1665-1670. [17] s. yılmaz, e. özyılmaz, m. turgut, new spherical indicatrices and their characterizations, an. şt. univ. ovidius constanta, 18 (2010), 337-354. [18] m. barros, j.l. cabrerizo, m. fernández, a. romero, magnetic vortex filament flows, j. math. phys. 48 (2007), 082904. https://doi.org/10.1063/1.2767535. https://doi.org/10.1016/0003-4916(87)90098-4 https://doi.org/10.1016/0003-4916(87)90098-4 https://doi.org/10.3792/pjaa.70.12 https://doi.org/10.3836/tjm/1270043477 https://doi.org/10.1088/1751-8113/42/19/195201 https://doi.org/10.1007/s10474-009-9005-1 https://doi.org/10.1063/1.3659498 https://doi.org/10.1016/j.geomphys.2011.10.002 https://doi.org/10.1209/0295-5075/77/34002 https://doi.org/10.1209/0295-5075/77/34002 https://doi.org/10.1063/1.2767535 1. introduction 2. basic concepts 3. magnetic curves of the tangent indicatrix 3.1. tt-magnetic curve 3.2. nt-magnetic curve 3.3. bt-magnetic curve 4. applications 5. conclusion acknowledgment references international journal of analysis and applications volume 18, number 4 (2020), 531-549 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-531 continuous and discrete wavelet transforms associated with hermite transform c. p. pandey∗ and pranami phukan department of mathematics, north eastern regional institute of science and technology, nirjuli, 791109, arunachal pradesh, india ∗corresponding author: drcppandey@gmail.com abstract. in this paper, we accomplished the concept of continuous and discrete hermite wavelet transforms. we also discussed some basic properties of hermite wavelet transform. inversion formula and parsevals formula for continuous hermite wavelet transform is established. moreover the discrete version of wavelet transform is discussed. 1. introduction many authors have defined wavelet transforms associated with different integral transforms. in ( [6], [5]) pathak and dixit, pathak and pandey defined the wavelet transform which are associated with the hankel and laguree transform respectively. in [7] upadhyay and tripathi defined continuous wavelet transform corresponding to watson transform. in 2017 prasad and mandal [4] studied the kontorovich-lebedev wavelet transform and derived many important properties related to the kl-wavelet transform. in [1] pathak and abhishek studied the continuous and discrete wavelet transform associated with index whittaker transform. hans-jurgen glaeske [3] defined the translation and convolution operator associated with hermite transform and proved so many important results related to these operator. now, however to best our knowledge wavelet received march 24th, 2020; accepted april 22nd, 2020; published may 11th, 2020. 2010 mathematics subject classification. 42c40, 65r10, 44a35. key words and phrases. hermite transforms; continuous hermite wavelet transform; discrete hermite wavelet transform; hermite convolution. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 531 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-531 int. j. anal. appl. 18 (4) (2020) 532 associated with the hermite transform is not defined. so we are interested to define the wavelet associated to hermite transform and study the continuous as well as discrete wavelet transforms associated with this. the wavelet transform [8] of the function f ∈ l2(r) with respect to the wavelet φ ∈ l2(r) is defined by (wφf) (ρ,σ) = ∫ ∞ −∞ f(t)φρ,σ(t)dt,ρ ∈ r,σ > 0, (1.1) where φρ,σ(t) = σ −1 2 φ ( t−ρ σ ) . (1.2) in terms of translation τρ defined by τρφ(t) = φ(t−ρ),ρ ∈ r, and dilation dσ is defined by dσφ(t) = σ −1 2 φ ( t σ ) ,σ > 0, we can write φρ,σ(t) = τρdσφ(t). (1.3) from equation 1.1 and 1.3 it is clear that wavelet transform of the function f on r is an integral transform for which the kernel is the dilated translate of φ. we can also express equation 1.1 as the convolution (wφf) (ρ,σ) = (f ∗g0,σ) (ρ), (1.4) where g(t) = φ(−t). since associated with each integral transform there exists a special kind of convolution, one can construct wavelet transform corresponding to an integral transform using the associated convolution. we construct wavelet and wavelet transform on the interval (−∞,∞) by using the theory of hermite transforms [2] and associated convolution involving the function h(µ)n (x) = exp ( −x2 2 ) h̃(µ)n (x),x ∈ r, where h̃ (µ) n (x) is the normalized hermite polynomial, where α > −1, is given by h̃(µ)n (x) =   h (µ) 2k (x) h (µ) 2k (0) = r (µ−12 ) k (x 2),n = 2k h (µ) 2k+1 (x)( h (µ) 2k+1 (x) ) ′(0) = xr (µ+ 12 ) k (x 2),n = 2k + 1 , int. j. anal. appl. 18 (4) (2020) 533 and h(µ)n (x) =   (−1)k22kk!l( µ−1 2 ) k (x 2),n = 2k (−1)k22k+1k!xl( µ+ 1 2 ) k (x 2),n = 2k + 1 . set dµ(x) = e−x 2 |x|2µdx. (1.5) let us consider the measurable function f(x) on the interval (−∞,∞). then the hermite transform is defined by h[f](n) = f̂(n) = ∫ ∞ −∞ f(x)h̃n(x)dµ(x),n ∈ n. (1.6) the inverse hermite transform defined by f(x) = ∞∑ n=0 f̂(n)h̃(µ)n (x) [ h̃(µ)n ]−1 . (1.7) where h(µ)n = 2 2nγ ([n 2 ] + 1 ) γ ([ (n + 1) 2 ] + µ + 1 2 ) . let the space of those real measurable functions f on (−∞,∞) be lp,µ(−∞,∞), 1 ≤ p < ∞, for which ‖f‖p,µ = { ∫ ∞ −∞ | f(x) |p dµ(x)} 1 p ,p < ∞. (1.8) ‖f‖p,µ = esssupx∈r | f(x) |,p = ∞. (1.9) an inner product on l2,µ, is defined by 〈f,g〉 = ∫ ∞ −∞ f(x)g(x)dµ(x). (1.10) 2. hermite translation and convolution in this section, hermite translation and associated convolution will be discussed. to define the hermite convolution ’*’ we have to introduce hermite translation. for this purpose we need the basic function k (µ) h (x,y,z) ∼ ∞∑ n=0 [ h̃(µ)n ]−1 h̃(µ)n (x)h̃ (µ) n (y)h̃ (µ) n (z). (2.1) hence by equation 1.6 and 1.7, we have∫ ∞ −∞ k (µ) h (x,y,z)h̃ (µ) n (z)dµ(z) = h̃ (µ) n (x)h̃ (µ) n (y). (2.2) clearly k (µ) h (x,y,z) is symmetric in x,y and z. setting n = 0 in equation 2.2, we have∫ ∞ −∞ k (µ) h (x,y,z)dµ(z) = 1. (2.3) int. j. anal. appl. 18 (4) (2020) 534 the hermite translation τy of f ∈ lp,µ(−∞,∞), 1 ≤ p < ∞ is defined τyf(x) = f(x,y) = ∫ ∞ −∞ f(z)k (µ) h (x,y,z)dµ(z), 1 ≤ p < ∞. (2.4) lemma 2.1. for f ∈ lp,µ and 1 ≤ p < ∞, ‖τ(µ)y f‖p,µ ≤‖f‖p,µ, (2.5) and the map: f → τyf is continuous and linear in lp,µ. proof. proof is referred from [3]. � let p,q,r ∈ (−∞,∞) and 1 r = 1 p + 1 q − 1. then the hermite convolution [3] of f ∈ lp,µ(−∞,∞) and g ∈ lq,µ(−∞,∞) is defined by following equation (f ∗g) (y) = ∫ ∞ −∞ τ(µ)y (f; x) g(x)dµ(x). (2.6) by using the relation defined in equation 2.4, convolution (f ∗g) can be defined as (f ∗g) (x) = ∫ ∞ −∞ ∫ ∞ −∞ f(z)g(x)dm(µ)x,y(z) = ∫ ∞ −∞ ∫ ∞ −∞ f(z)g(x)k (µ) h (x,y,z)dµ(x)dµ(z). (2.7) also recall the following lemma from [3]. lemma 2.2. let p,q,r ∈ (−∞,∞) and 1 r = 1 p + 1 q − 1,f ∈ lp,µ(−∞,∞) and g ∈ lq,µ(−∞,∞). then the convolution (f ∗g) defined by equation 2.7 satisfies the following norm inequality: (i)‖f ∗g‖r,µ ≤‖f‖p,µ‖g‖q,µ. (2.8) moreover f,g ∈ l2,µ, we get (ii) (f ∗g)∧ (n) = f̂(n)ĝ(n). (2.9) lemma 2.3. for any f ∈ l2,µ the following parseval identity holds for hermite transform:∑ n [ h̃(µ)n ]−1 | f̂(n) |2= ‖f‖22,µ. (2.10) proof. proof is referred from theorem 1 in ref. [2]. � for any f1,f2 ∈ l2,µ(−∞,∞) the below parseval identity holds for hermite transform. see ref. [2].∑ n [ h̃(µ)n ]−1 f1(n)f2(n) = ∫ ∞ −∞ f1(x)f2(x)dµ(x) and ∑ n [ h̃(µ)n ]−1 f1(n)f2(n) = ∫ ∞ −∞ h−1[f1(n)][f2(n)]dµ(x). int. j. anal. appl. 18 (4) (2020) 535 in this paper, following the technique of pathak and dixit [6] and trimeche [9], hermite wavelet transform is defined. the continuity and boundedness properties of hermite wavelet transform is derived. a semi discrete hermite wavelet transform is defined. furthermore discrete hermite wavelet transform is investigated. using discrete hermite wavelet, frame and riesz basis [8] are also studied. 3. continuous hermite wavelet transform for a function φ ∈ lp,µ(−∞,∞), defined the dilation dσ by dσφ(t) = φ(σt),σ > 0. (3.1) using the hermite translation 2.4 and above dilation, the hermite wavelet φρ,σ(t) is defined as follows: φρ,σ(t) = τρdσφ(t) = τρφ(σt) (3.2) = ∫ ∞ 0 φ(σz)k (µ) h (ρ,t,z)dµ(z). (3.3) where ρ ≥ 0 and σ > 0. the integral is convergent by virtue of inequality 2.5. definition 3.1. admissible hermite wavelet the function φ(·) ∈ lp,µ(−∞,∞) is said to be admissible hermite wavelet if φ(·) satisfies the following admissibility condition cφ = ∞∑ n=0 | φ̂(n) |2 | n | < ∞, where φ̂(n) is the hermite transform of φ. continuous hermite wavelet transform using the wavelet φρ,σ we now define the continuous hermite wavelet transform. ( h̃ (µ) φ f ) (ρ,σ) = 〈f(t),φρ,σ(t)〉 = ∫ ∞ −∞ f(t)φρ,σ(t)dµ(t) (3.4) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)k (µ) h (ρ,t,z)dµ(z)dµ(t) (3.5) provided the integral is convergent. since by inequality 2.5 and definition φρ,σ ∈ lp,µ whenever φ ∈ lp,µ. by virtue of lemma 2.2, the integral 3.5 is convergent for f ∈ lq,µ, 1p + 1 q = 1. the hermite wavelet transform can be expressed in the form of hermite transform as follows. h [( h̃ (µ) φ f ) (ρ,σ) ] = f̂(n)φ̂(σ,n). int. j. anal. appl. 18 (4) (2020) 536 also the hermite wavelet transform can be written as ( h̃ (µ) φ f ) (ρ,σ) = (f ∗φ(σ, ·)) (ρ). the continuity and boundedness results follow from the following theorem. theorem 3.1. let f(·) ∈ lp,µ and φ(·) ∈ lq,µ,σ > 0 with 1 ≤ p,q < ∞ and 1p + 1 q = 1. and ( h̃ (µ) φ f ) (ρ,σ) be continuous hermite wavelet transform 3.5. then (i) ‖ ( h̃ (µ) φ f ) (ρ,σ)‖r,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ, 1r = 1 p + 1 q − 1, 1 ≤ p,q,r < ∞. (ii) ‖ ( h̃ (µ) φ f ) (ρ,σ)‖∞,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ, 1p + 1 q = 1. proof. (ii) using representation 3.5, we have ( h̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)k (µ) h (ρ,t,z)dµ(z)dµ(t) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)k (µ) 1 p h (ρ,t,z)k (µ) 1 q h (ρ,t,z)dµ(z)dµ(t) using holder’s inequality, we get | ( h̃ (µ) φ f ) (ρ,σ)| ≤ (∫ ∞ −∞ ∫ ∞ −∞ | f(t) |p k(µ)h (ρ,t,z)dµ(z)dµ(t) )1 p × (∫ ∞ −∞ ∫ ∞ −∞ | φ(σz) |q k(µ)h (ρ,t,z)dµ(z)dµ(t) )1 q = (∫ ∞ −∞ | f(t) |p dµ(t) ∫ ∞ −∞ k (µ) h (ρ,t,z)dµ(z) )1 p × (∫ ∞ −∞ | φ(σz) |q dµ(z) ∫ ∞ −∞ k (µ) h (ρ,t,z)dµ(t) )1 q by using equation 2.3, it follows that | ( h̃ (µ) φ f ) (ρ,σ)| ≤ ‖f‖p,µ‖φ(σ, ·)‖q,µ; so that ‖ ( h̃ (µ) φ f ) (ρ,σ)‖∞,µ ≤‖f‖p,µ‖φ(σ, ·)‖q,µ. the inequality 1.1 follows from inequality 2.8. � theorem 3.2. if φ is a basic hermite wavelet and ψ is any bounded function, then (φ∗ψ) is also a hermite wavelet. int. j. anal. appl. 18 (4) (2020) 537 proof. c(φ∗ψ) = ∞∑ n=0 |(φ∗ ψ)∧(n)|2 n = ∞∑ n=0 |(φ)∧(n)(ψ)∧(n)|2 n ≤ |(ψ)∧(n)| ∞∑ n=0 |(φ)∧(n)|2 n < ∞ hence (φ∗ ψ) is a hermite wavelet. � basic properties of continuous hermite wavelet transform theorem 3.3. let φ and ψ be two wavelets and f,g be two functions belong to lp,µ(−∞,∞), then (i) linearity property: h (µ) φ (ηf + ζg)(σ,ρ) = ηh (µ) φ (f)(σ,ρ) + ζh (µ) φ (g)(σ,ρ) where η and ζ are any two scalars. (ii) shift property ( h (µ) φ f ) (x− τ)(σ,ρ) = ( h (µ) φ f ) (σ,ρ− τ) where τ is any scalar. (iii) scaling property if c 6= 0 is any scalar, then the hermite wavelet transform of the scaled function fc(x) = 1 c f ( 1 2 ) is ( h (µ) φ fc ) (ρ,σ) = h (µ) φ f (σ c , ρ c ) (iv) symmetry property: ( h (µ) φ f ) (σ,ρ) = ( h (µ) φ f ) (φ) ( 1 σ , −1 ρ ) (v) parity property ( h (µ) pφ pf ) (σ,ρ) = ( h (µ) φ f ) (σ,−ρ) where p is the parity operator defined by pf(x) = f(−x). proof. the proof is the straight forward application of hermite transform. � plancharel and persevals relation for continuous hermite wavelet transform let f,g ∈ l2,µ(−∞,∞) and φ1,φ2 ∈ l2,µ(−∞,∞) are two hermite wavelets. then we have 〈 ( h (µ) φ1 f ) (σ,ρ), ( h (µ) φ2 g ) (σ,ρ)〉l2,µ((−∞,∞)×(−∞,∞)) = cφ1,φ2〈f,g〉l2,µ(−∞,∞), (3.6) int. j. anal. appl. 18 (4) (2020) 538 where cφ1,φ2 = ∫ ∞ 0 φ1(σ,n)φ2(ρ,n)dµ(σ). proof. let f,g ∈ l2,µ(−∞,∞) then from 3.4, we have∫ ∞ 0 ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ) ( h (µ) φ g ) (σ,ρ)dµ(σ)dµ(ρ) = ∫ ∞ 0 ∫ ∞ −∞ h (µ)−1 φ [ f̂(n)φ1(σ,n) ] (ρ) h (µ)−1 φ [ĝ(n)φ2(σ,n)] (ρ)dµ(σ)dµ(ρ) now by using 2.10 we get ∫ ∞ 0 ∫ ∞ −∞ f(x)φ1(σ,x)(ρ)g(x)φ2(σ,x)(ρ)dµ(σ)dµ(ρ) = ∑ n [ h̃(µ)n ]−1 f̂(n)ĝ(n) ∫ ∞ −∞ φ1(σ,n)φ2(σ,n)dµ(σ) (3.7) = cφ1,φ2 ∑ n [ h̃(µ)n ]−1 f̂(n)ĝ(n). hence by using the parseval formula for hermite transform, we get∫ ∞ 0 ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ) ( h (µ) φ g ) (σ,ρ)dµ(σ)dµ(ρ) = cφ1,φ2f̂(n)ĝ(n) = cφ1,φ2〈f,g〉l2,µ(−∞,∞). (3.8) � theorem 3.4. (inversion formula) let f ∈ l2,µ(−∞,∞) and φ is hermite wavelet defines continuous hermite wavelet transform. then, f(x) = 1 cφ ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ), where cφ is the admissible hermite wavelet. proof. let h(x) ∈ l2,µ(−∞,∞) be any function, then by applying previous theorem, we have cφ〈f,h〉l2,µ(−∞,∞) = ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ) ( h (µ) φ h ) (σ,ρ)dµ(σ)dµ(ρ) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ) ∫ ∞ −∞ h(t)φρ,σ(t)dtdµ(σ)dµ(ρ) = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ)φρ,σ(t)h(t)dtdµ(σ)dµ(ρ) = ∫ ∞ −∞ g(t)h(t)dt = 〈g,h〉, (3.9) int. j. anal. appl. 18 (4) (2020) 539 where, g = 1 2 ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ). then, cφ〈f,h〉 = 〈g,h〉 f = 1 cφ g = 1 2cφ ∫ ∞ −∞ ∫ ∞ −∞ ( h (µ) φ f ) (σ,ρ)φρ,σ(t)dµ(σ)dµ(ρ). if f = h, ‖f‖2l2,µ(−∞,∞) = ∫ ∞ −∞ ∫ ∞ −∞ | ( h (µ) φ f ) (σ,ρ) |2 dµ(σ)dµ(ρ). moreover the hermite wavelet transform is isometry from l2,µ(−∞,∞) to l2,µ(−∞,∞)×l2,µ(−∞,∞). � a general reconstruction formula in this section, we show that the function f can be recovered from its hermite wavelet transform. in derived the reconstruction formula, we need the following lemma. lemma 3.1. let f ∈ l2,µ and φ ∈ l2,µ be a basic wavelet, which defines hermite wavelet transform 3.5. then ( h̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n), (3.10) where φ̂(σ,n) = ∫ ∞ −∞ φ(σz)h̃(µ)n (z)dµ(z). (3.11) proof. using representation 3.5, we have( h̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)k (µ) h (ρ,t,z)dµ(z)dµ(t) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)dµ(z)dµ(t) ( ∞∑ n=0 [ h̃(µ)n ]−1 h̃(µ)n (ρ)h̃ (µ) n (t)h̃ (µ) n (z) ) = ∞∑ n=0 [ h̃(µ)n ]−1 h̃(µ)n (ρ) (∫ ∞ −∞ f(t)h̃(µ)n (t)dµ(t) ∫ ∞ −∞ φ(σz)h̃(µ)n (z)dµ(z) ) = ∞∑ n=0 [ h̃(µ)n ]−1 h̃(µ)n (ρ)f̂(n)φ̂(σ,n) = ( f̂(n)φ̂(σ,n) )∨ (ρ). ∴ ( h̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n). this completes the proof. � int. j. anal. appl. 18 (4) (2020) 540 theorem 3.5. let f ∈ l2,µ and φ be a basic wavelet which defines hermite wavelet transform by equation 3.5. let q(σ) > 0 be a weight function such that q(n) = ∫ ∞ 0 q(σ) | φ̂(σ,n) |2 dµ(σ) > 0. (3.12) set φ̂ρ,σ(n) = φ̂ρ,σ(n) q(n) . (3.13) then f(t) = ∫ ∞ 0 ∫ ∞ −∞ q(σ) ( h̃ (µ) φ f ) (ρ,σ)φb,a(t)dµ(σ)dµ(ρ). (3.14) proof. from equation 3.10, we have ( h̃ (µ) φ f )∧ (ρ,σ) = f̂(n)φ̂(σ,n) ⇒ ∫ ∞ −∞ ( h̃ (µ) φ f ) (ρ,σ)h̃ (µ) φ (b)dµ(b) = f̂(n)φ̂(σ,n). multiplying both sides by φ̂(σ,n) and weight function q(σ) and integrating with respect to dµ(σ), we have∫ ∞ 0 q(σ)φ̂(σ,n) (∫ ∞ −∞ ( h̃ (µ) φ f ) (ρ,σ)h̃(µ)n (ρ)dµ(ρ) ) dµ(σ) = ∫ ∞ 0 q(σ)f̂(n)φ̂(σ,n)φ̂(σ,n)dµ(σ) ⇒ ∫ ∞ 0 q(σ)φ̂(σ,n) (∫ ∞ −∞ ( h̃ (µ) φ f ) (ρ,σ)h̃(µ)n (ρ)dµ(ρ) ) dµ(σ) = ∫ ∞ 0 q(σ)f̂(n) | φ(σ,n) |2 dµ(σ). (3.15) equation 3.11 and 3.15 gives f̂(n)q(n) = ∫ ∞ 0 q(σ)φ̂(σ,n)dµ(σ) ∫ ∞ −∞ ( h̃ (µ) φ f ) (ρ,σ)h̃(µ)n (ρ)dµ(ρ) f̂(n) = 1 q(n) ∫ ∞ 0 q(σ) ∫ ∞ −∞ ( h̃ (µ) φ f ) (ρ,σ)φ̂(σ,n)h̃(µ)n (ρ)dµ(σ)dµ(ρ) (3.16) we also have from equation 3.3, φρ,σ(t) = ∫ ∞ −∞ φ(σz) ∞∑ n=0 [ h(µ)n ]−1 h̃(µ)n (ρ)h̃ (µ) n (t)h̃ (µ) n (z)dµ(z) = ∞∑ n=0 [ h(µ)n ]−1 h̃(µ)n (ρ)h̃ (µ) n (t) ∫ ∞ −∞ φ(σz)h̃(µ)n (z)dµ(z) = ∞∑ n=0 [ h(µ)n ]−1 h̃(µ)n (ρ)h̃ (µ) n (t)φ̂(σ,n) = ( φ̂(σ,n)h̃(µ)n (ρ) )∨ (t). int. j. anal. appl. 18 (4) (2020) 541 therefore φρ,σ(t) = ( φ̂(σ,n)h̃(µ)n (ρ) )∨ (t). φ̂ρ,σ(t) = φ̂(σ,n)h̃ (µ) n (ρ). (3.17) using equation 3.17 in 3.16, we have f̂(n) = 1 q(n) ∫ ∞ 0 q(σ) ∫ ∞ −∞ φ̂ρ,σ(n) ( h̃ (µ) φ f ) (ρ,σ)dµ(σ)dµ(ρ). (3.18) from equation 3.13 it follows that f̂(n) = ∫ ∞ 0 q(σ) ∫ ∞ −∞ φ̂ρ,σ(n) ( h̃ (µ) φ f ) (ρ,σ)dµ(σ)dµ(ρ). (3.19) from equation 1.7 and 3.19, we have f(t) = ∞∑ n=0 [ h(µ)n ]−1 h̃(µ)n (t) ∫ ∞ −∞ ∫ ∞ −∞ q(σ) ( h̃ (µ) φ f ) (ρ,σ)φ̂ρ,σ(n)dµ(σ)dµ(ρ) = ∞∑ n=0 q(σ) ( h̃ (µ) φ f ) (ρ,σ) ∞∑ n=0 [ h(µ)n ]−1 φ̂ρ,σ(n)h̃(µ)n (t)dµ(σ)dµ(ρ) = ∫ ∞ 0 ∫ ∞ −∞ q(σ) ( h̃ (µ) φ f ) (ρ,σ)φρ,σ(n)dµ(σ)dµ(ρ). this completes the proof of theorem 3.5. a characterization of φρ,σ is given below. � theorem 3.6. assume that there exist positive constant a and b such that, 0 < a ≤ q(n) ≤ b < ∞ (3.20) let φσ(t) = ∞∑ n=0 [ h (µ) n ]−1 q(n) φ̂(σ,n)h̃(µ)n (t). (3.21) then (i)φρ,σ(t) = τρφ σ(t); (3.22) (ii)‖φρ,σ‖2,µ ≤ a−1‖φρ,σ‖2,µ. (3.23) int. j. anal. appl. 18 (4) (2020) 542 proof. (i) using equations 1.7, 2.2, 3.13 and 3.16, we have φρ,σ(t) = φρ,σ(n) q(n) = φ(σ,n)h̃ (µ) n (ρ) q(n) = ∑∞ n=0 φ̂(σ,n)h̃ (µ) n (t) [ h̃ (µ) n ]−1 h̃ (µ) n (ρ) q(n) = ∑∞ n=0 φ̂ρ,σ(n) h̃ (µ) n (ρ) h̃ (µ) n (t) [ h̃ (µ) n ]−1 h̃ (µ) n (ρ) q(n) = ∑∞ n=0 φ̂ ρ,σ(n)q(n)h̃ (µ) n (t) [ h̃ (µ) n ]−1 q(n) = ∞∑ n=0 φ̂ρ,σ(n)h̃ (µ) n (t) [ h̃(µ)n ]−1 = ∞∑ n=0 [ h̃(µ)n ]−1 φ̂ρ,σ(n) q(n) h̃(µ)n (t) = ∞∑ n=0 [ h̃ (µ) n ]−1 q(n) φ̂(σ,n)h̃(µ)n (ρ)h̃ (µ) n (t) = ∞∑ n=0 [ h̃ (µ) n ]−1 q(n) φ̂(σ,n) (∫ ∞ −∞ k (µ) h (x,y,z)h̃ (µ) n (z)dµ(z) ) = ∫ ∞ −∞ k (µ) h (x,y,z)   ∞∑ n=0 [ h̃ (µ) n ]−1 q(n) φ̂(σ,n)h̃(µ)n (z)  dµ(z) = ∫ ∞ −∞ φσk (µ) h (x,y,z)dµ(z) = τρφ σ(z), where φσ(t) is given in equation 3.21. (ii) from equation 3.13, we have | φ̂ρ,σ |≤ a−1 | φρ,σ(n) |; (3.24) so that ∞∑ n=0 [ h(µ)n ]−1 | φ̂ρ,σ(n) |2≤ a−2 ∞∑ n=0 [ h(µ)n ]−1 | φρ,σ(n) |2 . using equation 2.10, we get ‖φρ,σ‖2,µ ≤‖φρ,σ‖2,µ. � int. j. anal. appl. 18 (4) (2020) 543 4. the discrete hermite wavelet transform the continuous hermite wavelet transform of the function f in terms of two continuous parameters σ and ρ can be converted into a semi-discrete hermite wavelet transform by assuming that σ = 2−j,j ∈ z and ρ ∈ r+. in what follows we assume that φ ∈ l1,µ ∩l2,µ satisfies the so called ’stability condition’. a ≤ ∞∑ j=−∞ | φ̂(2−jn) |2≤ b a.e. (4.1) for certain positive constants a and b, −∞ < a ≤ b < ∞. the function φ ∈ l1,µ ∩ l2,µ satisfying condition 4.1 is called dyadic wavelet. using definition 3.4, we define the semi discrete hermite wavelet transform of any f ∈ l1,µ ∩l2,µ by ( h φ j f ) (ρ) = ( h φ j f ) (ρ, 2−j) (4.2) = 〈f(t),φρ,2−j (t)〉 = ∫ ∞ −∞ f(t)φρ,2−j (t)dµ(t) = ∫ ∞ −∞ f(t)τρφ(2−jt)dµ(t) = ( f ∗φj ) (ρ), where φj(z) = φ(2 −jz),j ∈ z. theorem 4.1. assume that the semi discrete hermite wavelet transform of any f ∈ l1,µ ∩l2,µ is defined by the equation 4.2. let us consider another wavelet φ∗ defined by means of its hermite transform φ̂∗(n) = φ̂(n)∑∞ l→−∞ | φ̂(2−ln) |2 . (4.3) then f(t) = ∞∑ j=−∞ ∫ ∞ −∞ ( h φ j f ) (ρ) ( φ̂∗(2−jn)h̃(µ)n (t) )∨ (ρ)dµ(ρ). (4.4) proof. in view of relations 1.7, 3.23 and 2.9, int. j. anal. appl. 18 (4) (2020) 544 ∞∑ j=−∞ ∫ ∞ −∞ ( h φ j f ) (ρ) ( φ̂∗(2−jn)h̃(µ)n (t) )∨ (ρ)dµ(ρ) = ∞∑ j=−∞ ∫ ∞ −∞ ( h φ j f ) (ρ) [ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)h̃(µ)n (t)h̃ (µ) n (ρ) ] dµ(ρ) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)h̃(µ)n (t) ∫ ∞ −∞ ( h φ j f ) (ρ)h̃(µ)n (ρ)dµ(ρ) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)h̃(µ)n (t) ( h φ j f )∧ (n) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)h̃(µ)n (t) ( f ∗φj )∧ (n) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 φ̂∗(2−jn)h̃(µ)n (t)f̂(n)φ̂(2 −jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)h̃(µ)n (t)φ̂ ∗(2−jn)φ̂(2−jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)h̃(µ)n (t) φ̂(2−jn)∑ l | φ̂(2−j2−ln) |2 φ̂(2−jn) = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)h̃(µ)n (t) | φ̂(2−jn) |2∑ l | φ̂(2−j2−ln) |2 = ∞∑ j=−∞ ∞∑ n=0 [ h̃(µ)n ]−1 f̂(n)h̃(µ)n (t) = f(t). the above theorem leads to the following definition of dyadic dual. � definition 4.1. a function φ̃ ∈ l1,µ ∩ l2,µ, is called a dyadic dual of a dyadic wavelet φ, if every f ∈ l1,µ ∩l2,µ can be expressed as f(t) = ∑ j ∫ ∞ −∞ ( h φ j f ) (ρ) ( φ̃(2−jn)h̃(µ)n (t) )∨ (ρ)dµ(ρ). (4.5) so far we have considered semi-discrete hermite wavelet transform of any f ∈ l1,µ ∩ l2,µ discretizing only variable a. now, we discretize the translation parameter b also by restricting it to the discrete set of points: ρj,k = k 2j ρ0; j ∈ z,k ∈ n, (4.6) where ρ0 > 0 is a fixed constant. we write, φρ0;j,k(t) = φρ,j,k;σj (t) = φ(2 −jt, 2−jkρ0). (4.7) int. j. anal. appl. 18 (4) (2020) 545 then the discrete hermite wavelet transform of any f ∈ l2,µ can be expressed as( h (µ) φ f ) (ρj,k,σj) = 〈f,φρ0;j,k〉,j ∈ z,k,n ∈ n. (4.8) the ’stability condition’ for this reconstruction takes the form a‖f‖22,µ ≤ ∑ j∈z |〈f,φρ0;j,k〉| 2 ≤ b‖f‖22,µ,k ∈ n, (4.9) where a and b are positive constant such that 0 ≤ a ≤ b < ∞. theorem 4.2. assume that the discrete hermite wavelet transform of any f ∈ l2,µ is defined by 4.8 and stability condition 4.9 holds. let t be a linear operator on l2,µ defined by tf = ∑ j∈z,k∈n0 〈f,φρ0;j,k〉µφρ0;j,k. (4.10) then f = ∑ 〈f,φρ0;j,k〉µφ j,k ρ0 , where, φj,kρ0 = t −1φρ0;j,k; j ∈ z,k ∈ n0. proof. from the stability condition 4.9, it follows that the operator defined by equation 4.10 is a one-one bounded linear operator. set g = tf,f ∈ l2,µ. then from equation 4.10, we have 〈tf,f〉 = ∑ j∈z,k∈n0 | f,φρ0;j,k | 2 . therefore, from condition 4.9, a‖t−1g‖22,µ = a‖f‖t 2 2 ≤ ∑ | 〈f,ψρ0;j,k〉 | 2= 〈tf,f〉µ = 〈gt−1g〉µ ≤‖g‖2,µ‖t−1g‖2,µ by schwartz equality. therefore, ‖t−1g‖2,µ ≤ 1 a ‖g‖2,µ. hence, every f ∈ l2,µ can be constructed from its discrete hermite wavelet transform given by 4.8. thus f = t−1tf = ∑ j∈z,k∈n0 〈f,φρ0;j,k〉t −1φρ0;j,k. (4.11) int. j. anal. appl. 18 (4) (2020) 546 finally, set φj,kρ0 = t −1φρ0;j,k; j ∈ z,n ∈ n0. then the construction 4.11 can be expressed as f = ∑ j∈z,k∈n0 〈f,φρ0;j,k〉φ j,k ρ0 , which completes the proof of theorem 4.2. � frames and riesz basis in l2,µ in this section, using φρ0;j,k a frame is defined and riesz basis of l2,µ is studied. definition 4.2. a function f ∈ l2,µ is said to generate a frame {φρ0;j,k} of l2,µ with sampling rate ρ0 if condition 4.8 holds for some positive constant a and b. if a = b, then the frame is called a tight frame. definition 4.3. a function f ∈ l2,µ is said to generate a riesz basis of {φρ0;j,k} with sampling rate ρ0 if the following two properties are satisfied. (i) the linear span 〈φρ0;j,k; j ∈ z〉 is dense in l2,µ. (ii) there exist positive constants a and b with 0 < a ≤ b < ∞ such that a‖cj,k‖2l2 ≤‖ ∑ j,k∈z cj,kφρ0;j,k‖ 2 2,µ ≤ b‖cj,k‖ 2 l2 (4.12) for all {cj,k}∈ l2(n2). here a and b are called the riesz bounds of {φρ0;j,k}. theorem 4.3. let φ ∈ l2,µ, then the following statements are equivalent. (i) {φρ0;j,k} is a riesz basis of l2,µ. (ii) {φρ0;j,k} is a frame of l2,µ and is also an l2 linearly independent family in the sense that if∑ j,k cj,kφρ0;j,k = 0 and {cj,k} ∈ l 2, then cj,k = 0. furthermore, the riesz bounds and frame bounds agree. proof. it follows from property 4.12 that any riesz basis is l2-linearly independent. let {φρ0;j,k} be a riesz basis with riesz bounds a and b, and consider the matrix operator: m = [γl,m;j,k](l,m)(j,k)∈n×n , where the entries are defined by γl,m;j,k = 〈φρ0;l,m,φρ0;j,k〉µ. (4.13) then from property 4.12, we have a‖{cj,k}‖2l2 ≤ ∑ l,m,j,k cl,mγl,m;j,kcj,k ≤ b‖{cj,k}‖2l2 ; int. j. anal. appl. 18 (4) (2020) 547 so that m is positive definite. we denote the inverse of m by m−1 = [µl,m;j,k](l,m)(j,k)∈n×n , (4.14) which means that both ∑ r,s µl,m;r,sγr,s;j,k = δl,jδm,k; l,m,j,k ∈ n, (4.15) and b−1‖{cj,k}‖2l2 ≤ ∑ l,m,j,k cl,mµl,m;j,kcj,k ≤ a−1‖{cj,k}‖2l2, (4.16) are satisfied. this allows us to introduce φl,m(x) = ∑ j,k µl,m;j,kφρ0;j,k(x). (4.17) clearly, φl,m ∈ l2,µ and it follows from equation 4.15 and 4.13 that 〈φl,m,φρ0;j,k〉µ = δl,jδm,k;l,m,j,k ∈ n, which means that {φl,m} is the basis of l2,µ, which is dual to {φρ0;j,k}. furthermore from equation 4.15 and 4.17, we conclude that 〈φl,m,φj,k〉µ = µl,m;j,k and the reisz bounds of {φl,m} are b−1 and a−1. in particular, for any f ∈ l2,µ we may write f(x) = ∑ j,k 〈f,φρ0;j,k〉µφ j,k(x) and b−1 ∑ j,k | 〈f,φρ0;j,k〉µ | 2≤‖f‖22,µ ≤ a −1 ∑ j,k | 〈f,φρ0;j,k〉µ | 2 . (4.18) since condition 4.18 is equivalent to condition 4.8, therefore statement (i) implies statement (ii). to prove the converse part, we recall theorem 3.5 and we have any g ∈ l2,µ and f = t−1g, g(x) = ∑ m∈z,n∈n 〈f,φρ0;j,k〉µφρ0;j,k. also by the l2-linear independence of {φρ0;j,k}, this representation is unique. from the banach-steinhaus and open mapping theorem it follows that {φρ0;j,k}, is a riesz basis of l2,µ. � int. j. anal. appl. 18 (4) (2020) 548 example 4.1. let the mother wavelet be φ(t) =   1, 0 ≤ t ≤ 1 2 −1, −1 2 ≤ t < 1 0, otherwise . (4.19) this mother wavelet is called haar wavelet. this is piecewise continuous. using this wavelet we have following expression for φ(σt). φ(σt) =   1, 0 ≤ t ≤ 1 2σ −1, 1 2σ ≤ t < 1 σ 0, otherwise (4.20) let f(t) = t−2µe−2t. then hermite transform of f(t) is given by h [ t−2µe−2t ] = ∫ ∞ −∞ f(t)h̃(µ)n (t)dµ(t) = ∫ ∞ −∞ f(t)e−t 2 | t |2µ h̃(µ)n (t)dt = ∫ ∞ −∞ t−2µe−2te t2 2 h(µ)n (t) | t | 2µ e−t 2 dt = e − ( t2+2t−t 2 2 ) h(µ)n (t)dt = √ π(2σ)neα 2 (4.21) now, ∫ ∞ −∞ φ(σz)h̃(µ)n (z)dµ(z) = ∫ 1 2σ 0 h̃(µ)n (z)dµ(z) − ∫ 1 σ 1 2σ h̃(µ)n (z)dµ(z) = 2 ∫ 1 σ 0 h̃(µ)n (z)dµ(z) − ∫ 1 2σ 0 h̃(µ)n (z)dµ(z) = 2φ1(n,µ) −φ2(n,µ), (4.22) where φ1(n,µ) = ∫ 1 σ 0 h̃ (µ) n (z)dµ(z) and φ2(n,µ) = ∫ 1 2σ 0 h̃ (µ) n (z)dµ(z). using representation 3.5 and 2.1, we have( h̃ (µ) φ f ) (ρ,σ) = ∫ ∞ −∞ ∫ ∞ −∞ f(t)φ(σz)k (µ) h (ρ,t,z)dµ(z)dµ(t) = ∞∑ n=0 [ h̃(µ)n ]−1 h̃(µ)n (ρ) (∫ ∞ −∞ f(t)h̃(µ)n (z)dµ(z) )(∫ ∞ −∞ φ(σz)h̃(µ)n (z)dµ(z) ) . from equations 4.21 and 4.22, it follows that( h̃ (µ) φ f ) (ρ,σ) = ∞∑ n=0 22nγ ([n 2 ] + 1 ) γ ([ n + 1 2 ] + µ + 1 ) h̃(µ)n (ρ) √ π(2σ)neα 2 (2φ1(n,µ) −φ2(n,µ)). int. j. anal. appl. 18 (4) (2020) 549 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. pathak, abhishek, wavelet transforms associated with the index whittaker transform, arxiv:1908.03766 [math]. 2019. [2] c. markett, the product formula and convolution structure associated with the generalized hermite polynomials, j. approx. theory, 73 (1993) 199–217. [3] h. glaeske, convolution structure of (generalized) hermite transforms, algebra analysis and related topics, banach center publications, vol 53 (1), 113-120 (2000). [4] a. prasad, u.k. mandal, wavelet transforms associated with the kontorovich–lebedev transform, int. j. wavelets multiresolut inf. process. 15 (2017) 1750011. [5] r.s. pathak, c.p. pandey, laguerre wavelet transforms, integral transforms spec. funct. 20 (2009), 505–518. [6] r.s. pathak, m.m. dixit, continuous and discrete bessel wavelet transforms, j. comput. appl. math. 160 (12) (2003), 241-250. [7] s.k. upadhyay, a. tripathi, continuous watson wavelet transform, integral transforms spec. funct. 23 (2012), 639-647. [8] c.k. chui, an introduction to wavelets, academic press, new york, 1992. [9] e.c. titchmarch, introduction to theory of fourier integrals, 2nd edition, oxford university press, oxford, u.k. 1948. 1. introduction 2. hermite translation and convolution 3. continuous hermite wavelet transform 4. the discrete hermite wavelet transform references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 110-122 http://www.etamaths.com strong metrizability for closed operators and the semi-fredholm operators between two hilbert spaces mohammed benharrat1 and bekkai messirdi2,∗ abstract. to be able to refine the completion of c(h1, h2), the set of all closed densely defined linear operators between two hilbert spaces h1 and h2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset l(h1, h2) of bounded elements of c(h1, h2). in addition, several operator norm inequalities concerning the equivalence of some metrics on l(h1, h2) are presented. we also establish the semi-fredholmness and fredholmness of unbounded operators in terms of bounded pure contractions. 1. introduction let h, h1, h2 be a complex hilbert spaces endowed with the appropriate scalar product and the associated norm. the inner product in h1 × h2 is defined by < (x,y); (x′,y′) >=< x; x′ > + < y; y′ >. for t linear operator from h1 to h2, the symbols d(t) ⊂ h1, n(t) ⊂ h1 and r(t) ⊂ h2 will denote the domain, null space and the range space of t , respectively. the set g(t) = {(x,tx) : x ∈ d(t)} ⊂ h1 × h2 is called the graph of t . the operator t is closed if and only if g(t) is a closed subset of h1 ×h2, and is densely defined if d(t) = h1, where d(t) denote the closure of d(t) in h1. the set of all closed and densely defined linear operators from h1 to h2 will be denoted by c(h1,h2). denote by l(h1,h2) the banach space of all bounded linear operators from h1 to h2. if h1 = h2, write c(h1,h2) = c(h1) and l(h1,h2) = l(h1). if t ∈c(h1,h2), the adjoint t∗ of t exists, is unique and t∗ ∈c(h2,h1). an operator a ∈l(h1,h2) is a pure contraction if ‖ax‖2 < ‖x‖1 for all nonzero x in h1. we denote by l0(h1,h2) the set of all pure contractions. in [9] w. e. kaufman showed that if t ∈ c(h) then t is represented as t = γ(a) = a(i −a∗a)−1/2 using a unique pure contraction a defined in h, where i denote the identity in h. since the publication of kaufman [9] in 1978 and its follows papers, this kaufman’s representation is used to reformulate questions about unbounded operators in terms of bounded ones: • in [9] [11], kaufman proved that the map γ preserves many properties of operators: self-adjontness, nonnegative conditions, normality and quasinormality. in [12] he also defined by the use of γ−1 a metric in the space of closed densely defined hilbert space which is stronger than the gap metric and sharing many of its properties. on the the bounded operators it is equivalent to the metric generated by the usual operator-norm. • in [5] hirasawa showed that a pure contraction a is hyponormal if and only if γ(a) is formally hyponormal, and if a is quasinormal then tn = an(i − a∗a)−n/2 is quasinormal for all integers n ≥ 2. • in [2], cordes and labrousse proved that if a closed and densely defined operator t is semi fredholm then so is the bounded operator γ−1(t) = t(i + t∗t)−1/2. 2010 mathematics subject classification. 47a10, 47a30 and 47a53. key words and phrases. pure contractions, closed densely defined linear operators, the gap metric, the gap topology, semi-fredholm operators. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 110 strong metrizability for closed operators 111 • in [1], benharrat and messirdi prove that if a pure contraction a on a hlibert space h is semi fredholm, then the closed densely defined linear operator λi −t = λi −a(i − a∗a)−1/2 is semi fredholm operator for all λ ∈ c such that |λ| < γ(a) 1+γ(a) . recently, j. j. koliha in [14] extend kaufman’s results to operators between two hilbert spaces and showed that the mapping γ maps the set l0(h1,h2) one-to-one onto the set c(h1,h2). more precisely, we have the following result. theorem 1.1. [14, theorem 5.] let l0(h1,h2) be the set of all pure contractions from h1 to h2, c(h1,h2) the set of all closed and densely defined linear operators from h1 to h2, and g ∈l+(h1) a positive bijection. the mapping γg defined by γg(a) = ag 1/2(g1/2(i −a∗a)g1/2)−1/2g1/2, a ∈l0(h1,h2), is a bijection of l0(h1,h2) onto c(h1,h2) with the inverse γ−1g (t) = t(g + t ∗t)−1/2, t ∈c(h1,h2). in this paper, by the use of the generalized kaufman’s representation, we discuss some metrics in the space c(h1,h2) endowed with the gap metric. more precisely, in section 2, we define some metrics on c(h1,h2) equivalent to the gap metric. in section 3, we define on c(h1,h2) a metric in term of γ−1, strictly stronger than the gap metric and is equivalent to the metric associated to the operator-norm in l(h1,h2). we characterize essentially the closure of l(h1,h2) in c(h1,h2) for this metric. in section 4, we prove some operator norm inequalities for bounded operators between two hilbert spaces. in the last section, we establish some characterizations of fredholm unbounded operators in terms of bounded pure contractions by treating unbounded operators between two hilbert spaces rather than restricting the investigation to operators on a single space. 2. a strong metric for closed operators between two hilbert spaces recall that, if t ∈ c(h1,h2), then the operator rt = (i + t∗t)−1 is self-adjoint positive operator defined on all h1, and has a unique positive definite self-adjoint square root, which we denoted by st . the fundamental properties of rt and st are, see [15]: rt ,st ∈l(h1,h2), ‖rt‖≤ 1, ‖st‖≤ 1, ‖trt‖≤ 1, ‖tst‖≤ 1, and (2.1) (trt ) ∗ = t∗rt∗, (tst ) ∗ = t∗st∗, t ∗st∗tst = i −rt . in the sequel l+(h1) denotes the set of all positive bijective operators in l(h1). let g be an element of l+(h1). by theorem 1.1, for a given pure contraction a ∈l0(h1,h2), there exists a unique operator t ∈c(h1,h2) such that the equation (2.2) xgx = i −a∗a, admits a unique solution given by (2.3) x = (g + t∗t)−1/2 = g−1/2(g1/2(i −a∗a)g1/2)1/2g−1/2, with t = γg(a). further, the operator tg −1/2 has domain g1/2d(t), which is clearly dense in h1. therefore rtg−1/2 is defined on all h1 and (2.4) (g + t∗t)−1 = g−1/2rtg−1/2g −1/2. hence (g + t∗t)−1 ∈l(h1). let pg(t) denote the correspondence which assigns to each t ∈ c(h1,h2) the orthogonal projection from h1 ×h2 onto the graph g(t) of t . 112 benharrat and messirdi lemma 2.1. let t ∈c(h1,h2) and g ∈l+(h1). then (2.5) pg(tg−1/2) = ( g1/2(g + t∗t)−1g1/2 g1/2(g + t∗t)−1t∗ t(g + t∗t)−1g1/2 i −t(g + t∗t)−1t∗ ) . proof. let v ∈d(g1/2t∗) and x = (u,v) ∈ h1 ×h2 such that pg(tg−1/2)x = (x,tg−1/2x), for x ∈ g1/2d(t). since g(tg−1/2)⊥ = v (g(g−1/2t∗)) with the isomorphism v from h1×h2 to h2 ×h1 defined by v (x1,x2) = (−x2,x1), we get pg(tg−1/2)x from the decomposition x = (u,v) = (x,tg−1/2x) + (−g1/2t∗y,y), y ∈d(g1/2t∗), x ∈ g1/2d(t), where x and y are the solutions of the system:{ u = x−g1/2t∗y v = tg−1/2x + y. by solving this system we get{ x = g1/2(g + t∗t)−1g1/2u + g1/2(g + t∗t)−1t∗v y = t(g + t∗t)−1g1/2u + (i −t(g + t∗t)−1t∗)v. � by (2.4), we can rewrite (2.5) as follows (2.6) pg(tg−1/2) = ( rtg−1/2 rtg−1/2g −1/2t∗ tg−1/2rtg−1/2 i −tg−1/2rtg−1/2g−1/2t∗ ) . if h1 = h2, we also have (2.7) pg(tg−1/2) = ( rtg−1/2 g −1/2t∗rg−1/2t∗ tg−1/2rtg−1/2 i −rg−1/2t∗ ) . in the case of g = ih1 lemma 2.1 reduces to the following well-known statement, see [15]. corollary 2.2. let t ∈c(h1,h2). then the orthogonal projection pg(t) in h1 ⊕h2 onto the graph g(t) of t , is given by (2.8) pg(t) = ( rt t ∗rt∗ trt i −rt∗ ) . definition 2.3. let g ∈ l+(h1) and t,s ∈ c(h1,h2). the gap metric between t and s associated to g is defined by (2.9) gg(t,s) = ∥∥pg(tg−1/2) −pg(sg−1/2)∥∥ . note that if g = ih1 , we have the usual gap metric (cf. [8, p. 201]) for t,s ∈c(h1,h2), (2.10) g(t,s) = ∥∥pg(t) −pg(s)∥∥ for all t,s ∈c(h1,h2). thus, for an infinite sequence (tn) of c(h1,h2), g(tn,t) → 0 if and only if each the following conditions hold (i) ‖rtn −rt‖→ 0, (ii) ‖tnrtn −trt‖→ 0, (iii) ∥∥rt∗n −rt∗∥∥ → 0, (iv) ∥∥t∗nrt∗n −t∗rt∗∥∥ → 0. similarly, we can express the convergence with respect to the metric gg as follows : proposition 2.4. for an infinite sequence (tn) of c(h1,h2), g(tn,t) → 0 in the sens of definition 2.3 if and only if each the following conditions hold (i) ∥∥rtng−1/2 −rtg−1/2∥∥ → 0, (ii) ∥∥tng−1/2rtng−1/2 −tg−1/2rtg−1/2∥∥ → 0, (iii) ∥∥tng−1/2rtng−1/2g−1/2t∗n −tg−1/2rtg−1/2g−1/2t∗∥∥ → 0, (iv) ∥∥rtng−1/2g−1/2t∗n −rtg−1/2g−1/2t∗∥∥ → 0. strong metrizability for closed operators 113 with g ∈l+(h1). by (2.9) and (2.10) we can deduce that gg(t,s) = g(tg −1/2,sg−1/2). let m,n be two closed linear subspaces of the hilbert space h. denote by pm and pn the orthogonal projection onto m and n respectively. set δ(m,n) = ‖(i −pn )pm‖ , δ is a pseudo-distance, for its properties we can see also [2]. we define another metric on c(h1,h2) as follows, dg(t,s) = ∥∥(i −pg(tg−1/2))pg(sg−1/2)∥∥ + ∥∥∥(i −pg(s)g−1/2)pg(tg−1/2)∥∥∥ , for all t,s ∈c(h1,h2) with g ∈l+(h1). we notice that gg(t,s) = max{δ(g(tg −1/2),g(sg−1/2)),δ(g(sg−1/2),g(tg−1/2))} and dg(t,s) = δ(g(tg−1/2),g(sg−1/2)) + δ(g(sg−1/2),g(tg−1/2)). the following result is immediately obtained. corollary 2.5. if g ∈ l+(h1); then dg and gg are equivalent metrics on c(h1,h2), in particular we have gg(t,s) ≤ dg(t,s) ≤ 2gg(t,s). put p = (i −pg(tg−1/2))pg(sg−1/2), let us remark that (2.11) p = ut [ 0 0 tg−1/2stg−1/2ssg−1/2 −s∗tg−1/2sg −1/2ssg−1/2 0 ] us. with ut = [ stg−1/2 ttg−1/2g −1/2t∗ tg−1/2stg−1/2 s ∗ tg−1/2 ] . then we can deduce that corollary 2.6. if g ∈l+(h1), then for t,s ∈c(h1,h2) we have dg(t,s) = ∥∥∥tg−1/2stg−1/2ssg−1/2 −s∗tg−1/2sg−1/2ssg−1/2∥∥∥ + ∥∥∥sg−1/2ssg−1/2stg−1/2 −s∗sg−1/2tg−1/2stg−1/2∥∥∥ . furthermore, if t,s are bounded, then dg(t,s) = ∥∥∥g−1/2s∗tg−1/2 (t −s)ssg−1/2∥∥∥ + ∥∥∥g−1/2s∗sg−1/2 (s −t)stg−1/2∥∥∥ . for an operator g ∈l+(h1), we define a third metric on c(h1,h2) by pg(t,s) = [ ‖rtg−1/2 −rsg−1/2‖ 2 + ∥∥∥tg−1/2rtg−1/2 −sg−1/2rsg−1/2∥∥∥2]1/2 . it easy to see that pg(t,s) ≤ gg(t,s). hence theorem 2.7. the topology induced from the gap metric gg on c(h1,h2) is strictly stronger than that induced from pg. the following example exclude the possibility that the metrics pg and gg generate the same topology even in the case of g = ih1 . 114 benharrat and messirdi example 2.8. let h1 and h2 two separable hilbert spaces and {φn}, {ψn} an orthonormal basis in h1, h2 respectively. put for n ∈ n∗, tnφk = { kψk, if k < n −kψk+1 if k ≥ n. then, t∗nψk =   kφk, if k < n 0 if k = n −kφk if k > n and thus, rtnφk =   1 1+k2 φk, if k < n φn if k = n 1 1+(k−1)2 φk if k > n define the operator t by tφk = kψk, k ∈ n∗. then, rt = rt∗ = rt∗n , ∥∥rt∗ −rt∗n∥∥ = 0, and t∗rt∗ −t∗nrt∗n ψk =   0 if k < n n 1+n2 φn if k = n k 1+k2 φk if k > n thus ‖trt −tnrtn‖ = ∥∥t∗rt∗ −t∗nrt∗n∥∥ ≤ 2√ 1 + n2 → 0. on the other hand, (rt −rtn )φk =   0 if k < n ( n 1+n2 − 1)φn if k = n 1−2k (1+k2)(1+(k−1)2 φk if k > n then ‖rt −rtn‖≥ n2 √ 1 + n2 → 1. finally, if we put sn = t ∗ n and s = t ∗, we get pg(s,sn) → 0 and gg(s,sn) → 1. 3. a new strong metric than the gap metric let g ∈l+(h1) and t,s ∈c(h1,h2). we define another metric in terms of γ−1g , given in theorem 1.1, as follows, qg(t,s) = ∥∥γ−1g (t) − γ−1g (s)∥∥ . clearly c(h1,h2) is isometric to the subset l0(h1,h2) of the unit ball in l(h1,h2) under the operator-norm, so that qg(t,s) ≤ 2 for all t,s ∈ c(h1,h2). the related convergence in the space c(h1,h2), called quotient-convergence associated to g. the purpose of the following theorem is to prove that the metric qg is stronger than dg. theorem 3.1. let g ∈ l+(h1). the metric topology induced by qg is stronger than that induced by the gap metric gg in c(h1,h2). proof. let t ∈c(h1,h2) and (tn) an infinite sequence of c(h1,h2), such that qg(tn,t) → 0. by theorem 1.1 then we can write t = γg(a) (resp tn = γg(an)) with a unique positive contraction a ∈l0(h1,h2) (resp. an ∈l0(h1,h2) for all n). thus, (g + t∗t)−1/2g(g + t∗t)−1/2 = i −a∗a, and (g + t∗ntn) −1/2g(g + t∗ntn) −1/2 = i −a∗nan. therefore, by (2.3) the orthogonal projections pg(tg−1/2) and pg(tng −1/2 n ) are easily computed from (2.3), and we obtain respectively, (3.1) pg(tg−1/2) = ( ug−1u ug−1ug−1/2(γg(a)) ∗ ag−1/2u i −ag−1/2ug−1/2(γg(a))∗ ) , strong metrizability for closed operators 115 and (3.2) pg(tng−1/2) = ( ung −1un ung −1ung −1/2(γg(an)) ∗ ang −1/2un i −ang−1/2ung−1/2(γg(an))∗ ) , where u = (g1/2(i −a∗a)g1/2)1/2 and un = (g1/2(i −a∗nan)g1/2)1/2 for all n ∈ n. consequently, if an converges to a in l0(h1,h2), then un converges to u and this assures the convergence pg(tng−1/2) −→ pg(tg−1/2) as n −→∞, hence gg(tn,t) → 0. � in the following example, we show that is not possible that the metrics qg and gg generate the same topology even for g = ih1 . example 3.2. let h1 and h2 be two separable hilbert spaces and {φn}, {ψn} an orthonormal basis in h1, h2 respectively. put for n ∈ n∗, tnφk = { kψk, if k < n −kψk if k ≥ n. then, t∗nψk = { kφk, if k < n −kφk if k ≥ n and thus, rtnφk = 1 1 + k2 φk, rt∗n ψk = 1 1 + k2 ψk. if we define the operator t by tφk = kψk, k ∈ n∗. then, t = γ(a) where aφk = k√1+k2 ψk, a ∈l0(h1,h2), we see that the conditions (i)-(iv) of proposition 2.4 holds. thus gg(tn,t) → 0. on the other hand, as n −→∞, qg(tn,t) = ‖an −a‖ = 2n √ 1 + n2 → 2. we have also the following result: corollary 3.3. the topology induced on c(h1,h2) by the metric qg is strictly stronger than the topology induced by the metric dg. lemma 3.4. let g ∈l+(h1). an operator t ∈c(h1,h2) is bounded if and only if ∥∥γ−1g (t)∥∥ < 1. in this case, (3.3) ‖t‖ = ∥∥γ−1g (t)∥∥∥∥g1/2∥∥2√ 1 − ∥∥γ−1g (t)∥∥2 . proof. let t ∈c(h1,h2) a bounded operator, then for all x ∈ h1, we have ‖x‖2 = ∥∥∥g1/2(g + t∗t)−1/2x∥∥∥2 + ∥∥∥t(g + t∗t)−1/2x∥∥∥2 ≤ [ 1 + ∥∥∥g−1/2∥∥∥2 ‖t‖2]∥∥∥g1/2(g + t∗t)−1/2x∥∥∥2 . thus, ∥∥∥g1/2(g + t∗t)−1/2x∥∥∥2 ≥ 1 1 + ∥∥g−1/2∥∥2 ‖t‖2 ‖x‖2 . consequently∥∥∥t(g + t∗t)−1/2x∥∥∥2 = ‖x‖2 −∥∥∥g1/2(g + t∗t)−1/2x∥∥∥2 ≤ ‖t‖2∥∥g1/2∥∥2 + ‖t‖2 ‖x‖2 . hence (3.4) ∥∥∥t(g + t∗t)−1/2∥∥∥ ≤ ‖t‖√∥∥g1/2∥∥2 + ‖t‖2 < 1. 116 benharrat and messirdi conversely, assume that ∥∥γ−1g (t)∥∥ < 1. then i − (γ−1g (t))∗γ−1g (t) is invertible and for all x ∈ h1, we have〈 g1/2(i − (γ−1g (t)) ∗γ−1g (t))g 1/2x,x 〉 = ∥∥∥g1/2x∥∥∥2 −∥∥∥γ−1g (t)g1/2x∥∥∥2 ≥ [ 1 − ∥∥γ−1g (t)∥∥2]∥∥∥g1/2x∥∥∥2 . hence ∥∥∥∥[g1/2(i − (γ−1g (t))∗γ−1g (t))g1/2]−1/2 ∥∥∥∥ ≤ 1√ 1 − ∥∥γ−1g (t)∥∥2 . since t = γ−1g (t)g 1/2 [ g1/2(i − (γ−1g (t)) ∗γ−1g (t))g 1/2 ]−1/2 g1/2, we obtain (3.5) ‖t‖≤ ∥∥γ−1g (t)∥∥∥∥g1/2∥∥2√ 1 − ∥∥γ−1g (t)∥∥2 . this implies the boundedness of t . furthermore, by (3.4) and (3.5) we obtain (3.3). � theorem 3.5. l(h1,h2) is dense open subset of c(h1,h2) endowed with the metric qg. proof. l(h1,h2) is an open subset of c(h1,h2) with respect to the metric qg follows immediately from lemma 3.4. now suppose that t ∈ c(h1,h2), then γ−1g (t) is in the unit closed ball of l(h1,h2), relative to the operator-norm. hence { nn+1 γ −1 g (t)} is a sequence {an} of operators such that for each n, ‖an‖ < 1 and ∥∥an − γ−1g (t)∥∥ −→ 0. for each n, we put tn = γg(an), by theorem 1.1 each tn is in l(h1,h2) and, clearly, qg(tn,t) =∥∥γ−1g (tn) − γ−1g (t)∥∥ −→ 0. this complete the proof. � definition 3.6. let t1,t2 ∈c(h1,h2). we put σg(t1,t2) = [ 2qg(t1,t2) 2 + ∥∥st1g−1/2 −st2g−1/2∥∥2 + ∥∥∥sg−1/2t∗1 −sg−1/2t∗2 ∥∥∥2 ]1/2 . σg is a metric on c(h1,h2) and note that the sequence defined in the example 2.8 converges on c(h1,h2) for the metric qg but is not convergent for the metric σg. theorem 3.7. the topology induced on c(h1,h2) by the metric σg is strictly stronger than the topology induced from the metric qg. by theorem 3.7 and theorem 3.3 we obtain the following results. corollary 3.8. the topology induced on c(h1,h2) by the metric σg is strictly stronger than the topology induced from the gap metric gg. theorem 3.9. l(h1,h2) is dense open subset of c(h1,h2) endowed with the metric σg. for the proof of this theorem we need the following lemma lemma 3.10. if t ∈c(h1,h2) and b ∈l(h1,h2) such that σg(t,b) < 1√ 1 + ∥∥bg−1/2∥∥2 , then t ∈l(h1,h2). proof. let x ∈ g1/2d(t), then for all y ∈ h2,〈 tg−1/2x,y 〉 − 〈 x,g−1/2b∗y 〉 = 〈 (x,tg−1/2x), (−g−1/2b∗y,y) 〉 = 〈 pg(tg−1/2)(x,tg −1/2x), (i −pg(bg−1/2))(−g −1/2b∗y,y) 〉 , strong metrizability for closed operators 117 by using schwarz inequality, (3.6) ∣∣∣〈(t −b)g−1/2x,y〉∣∣∣ ≤ gg(t,b) ∥∥∥(x,tg−1/2x)∥∥∥∥∥∥(−g−1/2b∗y,y)∥∥∥ . setting y = (t −b)g−1/2x in (3.6) it follows that ∥∥∥(t −b)g−1/2x∥∥∥ ≤ σg(t,b)√‖x‖2 + ∥∥tg−1/2x∥∥2√1 + ∥∥bg−1/2∥∥2. let us put σg(t,b) √ 1 + ∥∥bg−1/2∥∥2 = 1 − �, � > 0. thus, ∥∥∥tg−1/2x∥∥∥ ≤ ∥∥∥bg−1/2x∥∥∥ + (1 − �)[‖x‖ + ∥∥∥tg−1/2x∥∥∥], finally ∥∥∥tg−1/2x∥∥∥ ≤ 1 � [ ∥∥∥g1/2∥∥∥ + ∥∥∥bg−1/2∥∥∥] ∥∥∥g−1/2x∥∥∥ , what shows that t is bounded from h1 to h2. � proof of theorem 3.9. from lemma 3.10 l(h1,h2) is an open subset of c(h1,h2). now, we show the density. let t ∈c(h1,h2), then there exists an unique pure contraction a such that a = γ−1g (t). we put tn = n n+1 a as in the proof of theorem 3.5. then tn ∈ l(h1,h2) and qg(tn,t) −→ 0. on the other hand, let rgt = (g + t ∗t)−1 and sgt = (g + t ∗t)−1/2, then∥∥rgtn −rgt ∥∥ = ∥∥sgtnt∗ntnsgtn −sgt t∗tsgt ∥∥ = ∥∥sgtnt∗ntnsgtn + sgtnt∗ntsgt −sgtnt∗ntsgt −sgt t∗tsgt ∥∥ ≤ ∥∥sgtnt∗n∥∥∥∥tnsgtn −tsgt ∥∥ + ∥∥sgtnt∗n −sgt t∗∥∥∥∥tsgt ∥∥ ≤ [∥∥sgtnt∗n∥∥ + ∥∥tsgt ∥∥]qg(tn,t). thus, limn→+∞ ∥∥rgtn −rgt ∥∥ = 0, hence limn→+∞∥∥sgtn −sgt ∥∥ = 0. by (2.4), we observe that stg−1/2 = (g 1/2rgt g 1/2)1/2, then we conclude that lim n→+∞ ∥∥stng−1/2 −stg−1/2∥∥ = ∥∥∥sg−1/2t∗n −sg−1/2t∗∥∥∥ = 0. thus σg(tn,t) −→ 0, this shows the density of l(h1,h2) in c(h1,h2). � 4. some equivalent metrics for bounded operators between two hilbert spaces in this section we present several operator norm inequalities to compare the metric qg, the gap metric gg, and the usual operator norm metric. more presicily, we show that these three metrics are equivalent in l(h1,h2). our results extend those obtained in [12] and [13] to the bounded operators between two hilbert spaces. lemma 4.1. if t1,t2 ∈l(h1,h2), then ‖t1 −t2‖≤ 1 2 ∥∥(sgt1 )−1 + (sgt2 )−1∥∥qg(t1,t2) + 1 2 ∥∥(rgt1 )−1∥∥∥∥(rgt2 )−1∥∥∥∥γ−1g (t1) + γ−1g (t2)∥∥2 qg(t1,t2). 118 benharrat and messirdi proof. let t1,t2 ∈l(h1,h2), we have ‖t1 −t2‖ = ∥∥γ−1g (t1)(sgt1 )−1 − γ−1g (t2)(sgt2 )−1∥∥ ≤ 1 2 qg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 12 ∥∥t1sgt1 + t2sgt2∥∥∥∥(sgt1 )−1 − (sgt2 )−1∥∥ ≤ 1 2 qg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 14 ∥∥t1sgt1 + t2sgt2∥∥∥∥(rgt1 )−1 − (rgt2 )−1∥∥ ≤ 1 2 qg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 1 4 ∥∥(rgt1 )−1∥∥∥∥(rgt2 )−1∥∥∥∥t1sgt1 + t2sgt2∥∥∥∥rgt1 −rgt2∥∥ . since ∥∥rgt1 −rgt2∥∥ = ∥∥(γ−1g (t1))∗γ−1g (t1) − (γ−1g (t2))∗γ−1g (t2)∥∥ ≤ qg(t1,t2) ∥∥γ−1g (t1) + γ−1g (t2)∥∥ ,(4.1) it follows the desired inequality. � lemma 4.2. if t1,t2 ∈l(h1,h2), then qg(t1,t2) ≤ (1 + 1 4 ‖t1 + t2‖ 2 )‖t1 −t2‖ . proof. let t1,t2 ∈l(h1,h2), we have qg(t1,t2) = ∥∥γ−1g (t1) − γ−1g (t2)∥∥ = ∥∥∥∥12 (t1 −t2)(sgt1 + sgt2 ) + 12 (t1 + t2)(sgt1 −sgt2 ) ∥∥∥∥ ≤‖t1 −t2‖ + 1 2 ‖t1 + t2‖ ∥∥sgt1 −sgt2∥∥ ≤‖t1 −t2‖ + 1 2 ‖t1 + t2‖ ∥∥(sgt1 )−1 − (sgt2 )−1∥∥ ≤‖t1 −t2‖ + 1 4 ‖t1 + t2‖ ∥∥(rgt1 )−1 − (rgt2 )−1∥∥ = ‖t1 −t2‖ + 1 4 ‖t1 + t2‖‖t∗1 t1 −t ∗ 2 t2‖ ≤‖t1 −t2‖(1 + 1 4 ‖t1 + t2‖ 2 ). � combining lemma 4.1 and lemma 4.2, we obtain the following result: theorem 4.3. let g ∈ l+(h1). the restriction of the metric qg to l(h1,h2) is equivalent to the operator-norm. note that this theorem is extended to the unbounded operators between two hilbert spaces, the result was shown by w. e. kaufman in [12, theorem 2] in the case of unbounded operators defined on a single hilbert space and when g = ih1 . lemma 4.4. if t1,t2 ∈l(h1,h2), then qg(t1,t2) ≤ 1 2 [∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 12 ‖g + t∗1 t1‖‖g + t∗2 t2‖ ] gg(t1,t2). strong metrizability for closed operators 119 proof. let t1,t2 ∈l(h1,h2), we have qg(t1,t2) = ∥∥t1rgt1 (sgt1 )−1 −t2rgt2 (sgt2 )−1∥∥ ≤ 1 2 ∥∥t1rgt1 −t2rgt2∥∥∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 1 2 ∥∥t1rgt1 + t2rgt2∥∥∥∥(sgt1 )−1 − (sgt2 )−1∥∥ ≤ 1 2 gg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 12 ∥∥(sgt1 )−1 − (sgt2 )−1∥∥ ≤ 1 2 gg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 14 ∥∥(rgt1 )−1 − (rgt2 )−1∥∥ ≤ 1 2 gg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 14 ∥∥(rgt1 )−1∥∥∥∥(rgt2 )−1∥∥∥∥(rgt1 ) − (rgt2∥∥ ≤ 1 2 gg(t1,t2) ∥∥(sgt1 )−1 + (sgt2 )−1∥∥ + 14 ∥∥(rgt1 )−1∥∥∥∥(rgt2 )−1∥∥gg(t1,t2). � lemma 4.5. if t1,t2 ∈l(h1,h2), then g2g(t1,t2) ≤ [ ( ∥∥∥g1/2∥∥∥4 + 1) ∥∥γ−1g (t1) + γ−1g (t2)∥∥2 + ∥∥∥g1/2∥∥∥2 ] q2g(t1,t2) + 2 ∥∥∥g1/2∥∥∥2 ∥∥γ−1g (t1) + γ−1g (t2)∥∥3/2 q3/2g (t1,t2) + 1 2 ∥∥∥g1/2∥∥∥2 ∥∥γ−1g (t1) + γ−1g (t2)∥∥3 qg(t1,t2). proof. by using the representation (2.5), we get g2g(t1,t2) ≤ ∥∥∥g1/2∥∥∥4 ∥∥rgt1 −rgt2∥∥2 + 2 ∥∥∥g1/2∥∥∥2 ∥∥tg1 rgt1 −t2rgt2∥∥2 + ∥∥t1rgt1t∗1 −t2rgt2t∗2 ∥∥2 . we have∥∥t1rgt1 −t2rgt2∥∥ ≤ 12 qg(t1,t2) ∥∥sgt1 + sgt2∥∥ + 12 ∥∥γ−1g (t1) + γ−1g (t2)∥∥∥∥sgt1 −sgt2∥∥ ≤ qg(t1,t2) + 1 2 ∥∥γ−1g (t1) + γ−1g (t2)∥∥∥∥sgt1 −sgt2∥∥ ≤ qg(t1,t2) + 1 2 ∥∥γ−1g (t1) + γ−1g (t2)∥∥∥∥rgt1 −rgt2∥∥1/2 ≤ qg(t1,t2) + 1 2 ∥∥γ−1g (t1) + γ−1g (t2)∥∥3/2 q1/2g (t1,t2). in view of these estimations and the fact that, by (4.1), both ∥∥t1rgt1t∗1 −t2rgt2t∗2 ∥∥ and∥∥rgt1 −rgt2∥∥ are majorized by qg(t1,t2) ∥∥γ−1g (t1) + γ−1g (t2)∥∥ we get the required inequality. � combining lemma 4.4 and lemma 4.5 we obtain: theorem 4.6. in l(h1,h2) the metric qg is equivalent to the gap metric gg. combining theorem 4.3, theorem 4.6 and corollary 2.5 we deduce: corollary 4.7. the metrics qg, gg, pg and the operator-norm metric are equivalent on l(h1,h2). 120 benharrat and messirdi 5. pure contractions and semi-fredholm operators in this section, by the use of the generalized kaufman’s representation, we present some results concerning the characterization of unbounded semi fredholm operators in terms of bounded ones. we begin by introduce now some important classes of operators in fredholm theory. in the sequel, for every t ∈ c(h1,h2), let α(t) and β(t) be the nullity and the deficiency of t defined as α(t) := dim n(t), and β(t) := codimr(t). if the range r(t) of t is closed and α(t) < ∞ (resp. β(t) < ∞), then t is called an upper (resp. a lower) semi-fredholm operator. if t is either upper or lower semi-fredholm, then t is called a semi-fredholm operator, and the index of t is defined by ind(t) := α(t) − β(t). if both α(t) and β(t) are finite, then t is a called a fredholm operator. in the following, a denotes a pure contraction from h1 to h2, and t the closed and densely-defined operator γg(a) = ag 1/2b−1g1/2 from h1 to h2, with b = (g 1/2(i − a∗a)g1/2)1/2 such that g−1/2bg−1/2 is the unique solution of the equation (2.2) with g ∈gl+(h1). note that since a is a pure contraction, b is a positive and injective element of l(h1). recall that the reduced minimum modulus of a non-zero operator t is defined by γ(t) = inf x∈n(t)⊥ ‖tx‖ ‖x‖ if t = 0 then we take γ(t) = ∞. note that (see [8]): γ(t) > 0 ⇔ r(t) is closed. lemma 5.1 ([8]). (1) if δ(m,n) < 1 then dim m ≤ dim n. (2) δ(m,n) = δ(n⊥,m⊥). the main results of this section is: theorem 5.2. let a ∈ l0(h1,h2). if a is upper semi-fredholm operator then λc − γg(a) is upper semi-fredholm operator for all c ∈l(h1,h2) and |λ| < γ(a) 1+γ(a) ‖g1/2‖ ‖c‖ . proof. let a ∈ l0(h1,h2), c ∈ l(h1,h2) and b denote the positive member (g1/2(i − a∗a)g1/2)1/2 of l0(h). since a is a pure contraction, b is one-to-one with dense range in h1, and the fact that λc − γg(a) = (λcg−1/2bg−1/2 −a)g1/2b−1g1/2, it follows that to prove λc − γg(a) is upper semi-fredholm operator it suffices to prove that (λcg−1/2bg−1/2 −a) is upper semi-fredholm one. for each nonzero x in h1, ‖x‖ 2 −‖ax‖2 = ∥∥bg−1/2x∥∥2; thus∥∥∥bg−1/2x∥∥∥ ≤‖x‖ + ‖ax‖ . hence ∥∥∥cg−1/2bg−1/2x∥∥∥ ≤ ∥∥∥g−1/2∥∥∥∥∥∥bg−1/2x∥∥∥ ≤‖c‖ ∥∥∥g−1/2∥∥∥ (‖x‖ + ‖ax‖).(5.1) let λ in c. we prove that if |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) then 0 < γ(λcg −1/2bg−1/2 − a) < ∞ and hence r(λcg−1/2bg−1/2 − a) is closed. first if we use (5.1) with λx instead of x and by [7, theorem 1a], we obtain that γ(λcg−1/2bg−1/2 − a) > 0 for |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) . now to prove that γ(λcg−1/2bg−1/2 − a) < ∞, we proceed by contraposition. in fact γ(λcg−1/2bg−1/2 −a) = ∞ implies that (λcg−1/2bg−1/2 −a)x = 0 for all x ∈ h1. hence ‖ax‖ = |λ| ∥∥∥cg−1/2bg−1/2x∥∥∥ ≤‖c‖|λ|∥∥∥g−1/2∥∥∥ (‖x‖ + ‖ax‖), and so (5.2) γ(a)‖x‖≤‖ax‖≤ |λ|‖c‖ ∥∥g−1/2∥∥ 1 −|λ|‖c‖ ∥∥g−1/2∥∥ ‖x‖ strong metrizability for closed operators 121 for x ∈ n(a)⊥ with x 6= 0. it follows that |λ| ≥ ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) . we next prove that (5.3) δ(n(λcg−1/2bg−1/2 −a),n(a)) ≤ |λ|‖c‖ ∥∥g−1/2∥∥ (1 −|λ|‖c‖ ∥∥g−1/2∥∥)γ(a). let x ∈ h1, γ(a) ∥∥(i −pn(a))pn(λcg−1/2bg−1/2−a)x∥∥ ≤ ∥∥apn(λcg−1/2bg−1/2−a)x∥∥ . since pn(λcg−1/2bg−1/2−a)x ∈ n(λcg−1/2bg−1/2 −a) by the same calculation given before we have γ(a) ∥∥(i −pn(a))pn(λcg−1/2bg−1/2−a)x∥∥ ≤ |λ|‖c‖ ∥∥g−1/2∥∥ (1 −|λ|‖c‖ ∥∥g−1/2∥∥) ‖x‖ . recalling the definition of δ(n,m), this proves (5.3). the right side of (5.3) is smaller than one if |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) , thus lemma 5.1 shows that (5.4) α(λcg−1/2bg−1/2 −a) ≤ α(a) for |λ| < ∥∥g1/2∥∥γ(a) ‖c‖(1 + γ(a)) . we then conclude that λcg−1/2bg−1/2−a is upper semi-fredholm operator for |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) . this complete the proof of the theorem. � theorem 5.3. let a ∈ l0(h1,h2) is a lower semi-fredholm operator. then λc − γg(a) is a lower semi-fredholm operator for all c ∈l(h1,h2) and λ such that |λ| < ‖g1/2‖γ(a) ‖c‖(1+γ(a)) . proof. since r(a) is closed, by the first part of the proof of theorem 5.2, r(λcg−1/2bg−1/2− a) is closed and r(λcg−1/2bg−1/2 − a) = n(λg−1/2b∗g−1/2c∗ − a∗)⊥ for all |λ| < ‖g1/2‖γ(a) ‖c‖(1+γ(a)) . from (5.3) we deduce that δ(r(λcg−1/2bg−1/2 −a)⊥,r(a)⊥) = δ(n(λg−1/2b∗g−1/2c∗ −a∗),n(a∗)) ≤ |λ|‖c‖ ∥∥g−1/2∥∥ (1 −|λ|‖c‖ ∥∥g−1/2∥∥)γ(a), because γ(a) = γ(a∗). now by lemma 5.1 we have β(λcg−1/2bg−1/2 −a) ≤ β(a) for |λ| < ∥∥g1/2∥∥γ(a) ‖c‖(1 + γ(a) . consequently, λcg−1/2bg−1/2 −a is lower semi-fredholm one for all |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) and hence λc − γg(a) is lower semi-fredholm operator for all λ such that |λ| < ‖g1/2‖γ(a) ‖c‖(1+γ(a)) . � corollary 5.4. if a ∈ l0(h1,h2) is a semi-fredholm operator (resp. fredholm operator), then λc−γg(a) is a semi-fredholm operator (resp. fredholm operator) for all c ∈l(h1,h2) and λ such that |λ| < ‖ g1/2‖γ(a) ‖c‖(1+γ(a)) . we proceed as in the proof of [2, lemma 5.2, p. 708], by taking in count that the operator t is defined between two hilbert spaces, we can easily check the following result. proposition 5.5. if t ∈ c(h1,h2) is a semi-fredholm operator (resp. fredholm operator), then a = t(g + t∗t)−1/2 is a semi-fredholm operator (resp. fredholm operator) from h1 to h2, and n(a) = n(t),n(a ∗) = n(t∗). 122 benharrat and messirdi proof. it is easy to see that n(t) = {x ∈ h1 : (g + t∗t)−1gx = x}. since g ∈ gl+(h1), (g+t∗t)−1 is bounded self-adjoint and (g+t∗t)−1g leaves n(t) as well as n(t)⊥ invariant, so this two subspaces are invariant by (g + t∗t)−1 and its square root. accordingly n(t) = n(a). it is also clear that y ∈ n(a∗) if and only if 〈 t(g + t∗t)−1/2x,y 〉 = 0 for all x ∈ h1 i.e. 〈 t(g + t∗t)−1z,y 〉 = 0 for all z ∈ h1 i.e if y ∈ n((t(g + t∗t)−1)∗) = n(t∗). thus n((t(g + t∗t)−1/2)∗) = n(t∗). � by proposition 5.5 and corollary 5.4 we obtain the following results theorem 5.6. let a ∈ l0(h1,h2) . then a is a semi-fredholm operator (resp. fredholm operator) if and only if γg(a) is a semi-fredholm operator (resp. fredholm operator). in this case ind(a) = ind(γg(a)). remark 5.7. theorems 5.2, 5.3 and 5.6 generalize [1, theorem 1], [1, theorem 2] and [1, theorem 3] respectively, by taking h1 = h2 = h and g = c = i. references [1] m. benharrat, b. messirdi. semi-fredholm operators and pure contractions in hilbert space. rend. circ. mat. palermo, 62 (2013), 267–272. [2] h. o. cordes and j. p. labrousse, the invariance of the index in the metric space of closed operators. j. math. mech. 12 (1963), 693–719. [3] g. djellouli, s. messirdi and b. messirdi, some stronger topologies for closed operators in hilbert space. int. j. contemp. math. sciences, 5(25) (2010), 1223–1232. [4] s. goldberg, unbounded linear operators. mcgraw-hill, new-york, (1966). [5] g. hirasawa, quotient of bounded operators and kaufman’s theorem. math. j. toyama univ. 18 (1995), 215-224. [6] s. izumino, quotients of bounded operators, proc. amer. math. soc. 106 (1989), 427–435. [7] t. kato, perturbation theory for nullity, deficiency and other quantities of linear operators. j. anal. math. 6 (1958), 261–322. [8] t. kato, perturbation theory for linear operators, springer-verlag, new york, (1966). [9] w. e. kaufman, representing a closed operator as a quotient of continuous operators, proc. amer. math. soc. 72 (1978), 531–534. [10] w. e. kaufman, semiclosed operators in hilbert space, proc. amer. math. soc. 76 (1979), 67–73. [11] w. e. kaufman, closed operators and pure contractions in hilbert space, proc. amer. math. soc. 87 (1983), 83–87. [12] w. e. kaufman, a strong metric for closed operators in hilbert space, proc. amer. math. soc. 90 (1984), 83–87. [13] f. kittaneh, on some equivalent metrics for bounded operators on hilbert space proc. amer. math. soc. 110 (1990), 789–798. [14] j. j. koliha, on kaufman’s theorem j. math. anal. appl. 411(2014), 688–692. [15] j.p. labrousse, quelques topologies sur des espaces d’opérateurs dans des espaces de hilbert et leurs application. i, faculté des sciences de nice (math.), 1970. [16] j. weidmann, linear operators in hilbert spaces, springer (1980). 1laboratory of fundamental and applicable mathematics of oran, department of mathematics and informatics, national polytechnic school of oran, bp 1523 oran-el m’naouar, oran, algeria 2laboratory of fundamental and applicable mathematics of oran, department of mathematics, university of oran 1, bp 1524 oran-el m’naouar, oran, algeria ∗corresponding author international journal of analysis and applications volume 18, number 4 (2020), 624-632 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-624 on the normality of the product of tow operators in hilbert space mohammed meziane1,∗, abdelkader benali2 and mohammed hichem mortad3 1higher school of economics, b.p 65 ch2 ptt achaba hnifi, bir el djir, oran,laboratoire d’analyse mathematiques et applications (lama), algeria 2faculty of the exact sciences and computer, mathematics department, university of hassiba benbouali, chlef algeria. b.p. 151 hay essalem, chlef 02000, algeria 3department of mathematics, university of oran, b.p. 1524, el menouar, oran 31000, algeria ∗corresponding author: benali4848@gmail.com abstract. in this paper we present the results of the maximality of operators not necessarily bounded. for that, we will see the results obtained by operators in situation of extension. regarding the normal product of normal operators we seem to be the key to maximality. 1. introduction first, we assume that all operators operators are non necessarily bounded on a complex hilbert space h, let us, however, recall some notations that will be met below. if a and b are two operators with dense domains d(a) and d(b) respectively, then b is called an extension of a, and we write a ⊂ b, if d(a) ⊂ d(b) and if a and b coincide on d(a). the product ab of two operators is definded by ab(x) = a(bx) for x ∈ d(ab) received february 20th, 2020; accepted april 1st, 2020; published may 19th, 2020. 2010 mathematics subject classification. primary 47a05, secondary 47a10, 47b20, 47b25. key words and phrases. normal; self-adjoint; symmetric operators; commutativity; maximality of operators. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 624 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-624 int. j. anal. appl. 18 (4) (2020) 625 wehere d(ab) = {x ∈ d(b) : bx ∈ d(a)}. recall too that the unbounded operator a, defined on d(a), is said to be invertible if there exists an everywhere defined (i.e. on the whole of h) bounded operator b, which then will be designated by a−1, such that a−1a ⊂ aa−1 = i where i is the indentity operator on h. an operator a is said to be closed if its graph is closed in h ⊕h. the closing of the domain d(a) of a implies the closing of a if a is bounded on d(a). it is known that the product operators ab is closed if for instance a is closed and b ∈ b(h), or a−1 ∈ b(h) and b is closed. we also recall that an operator s is said to be densely defined if its domain d(s) is dense in h. it is known that in such case its adjoint s∗ exists and is unique. if t ⊂ s, then s∗ ⊂ t∗. notice that if s, t and st are all densely defined, then we are only sure of t∗s∗ ⊂ (st)∗, and a full equality occurring if e.g. t−1 ∈ b(h) or s ∈ b(h). the bounded operator t ∈ b(h) is said to be unitary if tt∗ = t∗t = i. a densely defined operator s is said to be symmetric if s ⊂ s∗. it is called self-adjoint if s = s∗. s is called essential self-adjoint if the closure of s is self-adjoint (i.e. (s)∗ = s). we say that s is normal if s is densely defined, and ‖sx‖ = ‖s∗x‖ for all x ∈ d(s) = d(s∗) (hence from known facts normal operators are automatically closed). recall that the previous is equivalent to s is closed and ss∗ = s∗s. other classes of operators are defined in the usual fashion. let us also agree that any operator is linear and non necessarily bounded unless we specify that it belongs to b(h). we also assume the basic theory of operators (see e.g. [1] or [20]). we do recall the celebrated fuglede-putnam theorem though: theorem 1.1. (for a proof, see e.g. [1]) let t ∈ b(h) and let m,n be two normal non necessarily bounded operators. then tn ⊂ mt =⇒ tn∗ ⊂ m∗t. one of the main objectives of this work is to impose conditions to obtain other results, starting from an extension. the following theorem and corollary result are a powerful tool to prove results on unbounded operators. for instance, statement (3) of the next theorem is used in the proof of the ”unbounded” version of the spectral theorem of normal operators (see e.g. [15]). for other uses, see e.g. [6] or [10]. let us now list some known (see e.g. [15] or [16]) maximality results: theorem 1.2. let s,t be two operators with (dense when necessary) domains d(s) and d(t) respectively such that s ⊂ t . then s = t when one of the following occurs: int. j. anal. appl. 18 (4) (2020) 626 (1) s is surjective and t is injective. (2) t is symmetric and s is self-adjoint (resp. normal). we then say that self-adjoint (resp. normal) operators are maximally symmetric. (3) t and s are normal (we say that normal operators are maximally normal). hence, self-adjoint (resp. normal) operators are maximally normal (resp. self-adjoint). commutativity of operators must be handled with care. first, recall the definition of two strongly commuting (normal) operators (see e.g. [16]): definition 1.1. let a and b be two normal operators. we say that a and b strongly commute if all the projections in their associated projection-valued measures commute. now, let us recall results obtained by devinatz-nussbaum (and von neumann) on strong commutativity: theorem 1.3. (devinatz-nussbaum-von neumann, [2] and cf. [13]). if there exists a self-adjoint operator a such that a ⊆ bc, where b and c are self-adjoint, then b and c strongly commute. corollary 1.1. let a, b and c be self-adjoint operators. then a ⊆ bc =⇒ a = bc 2. main results the normality of unbounded products of normal operators has been studied recently. see e.g. [5] and the references therein. we recall theorem 2.1. (for a proof, see e.g. [11]) let a,b be normal operators with b ∈ b(h). if ba ⊂ ab, then ab and ba are both normal (and so ab = ba). theorem 2.2. let t,a,b be non necessarily bounded operators such that t and b are self-adjoint with b ∈ b(h) and a is normal. assume further that ba ⊂ t . then ba = t. proof. we have: ba ⊂ t =⇒ ba ⊂ t ⊂ a∗b =⇒ ba∗ ⊂ ab (by fuglede-putnam theorem). it is clear that ba is closable and densely defined. let’s show now that ba is normal. indeed (ba)∗ba = (ba)∗(ba)∗∗ = a∗b(a∗b)∗ ⊃ a∗bba ⊃ baba ⊃ b2a∗a. int. j. anal. appl. 18 (4) (2020) 627 since the operators b2,a∗a, (ba)∗ba are self-adjoint with b2 ∈ b(h), then (ba)∗ba ⊂ a∗ab2, by corollary 1.1, we obtain (ba)∗ba = a∗ab2. similarly, we obtain ba(ba)∗ = a∗ab2, establishing the normality of ba. theorem 1.2 gives us ba = t. � corollary 2.1. let t,b,a be non necessarily bounded operators such that t is normal and b symmetric and invertible (hence b is self-adjoint) and that a is self-adjoint, then t ⊂ ba =⇒ a = b−1t. proof. clearly, t ⊂ ba =⇒ b−1t ⊂ a, by theorem 2.2, we obtain a = b−1t. � proposition 2.1. let a,b and t be operators where b ∈ b(h). assume that t∗ is symmetric, b is self-adjoint and a is normal. if t ⊂ ab, then ba is essential self-adjoint. proof. since ab is closed, we have t ⊂ ab =⇒ t ⊂ ab, and t ⊂ ab =⇒ ba∗ ⊂ t∗ ⊂ t∗∗ = t ⊂ ab =⇒ ba ⊂ a∗b (by fuglede-putnam theorem) =⇒ ba ⊂ a∗b (because a∗b is closed ). we can show the normality of ba. we have (ba)∗ba = a∗b(a∗b)∗ ⊃ a∗bba ⊃ b2a∗a. int. j. anal. appl. 18 (4) (2020) 628 since b2, a∗a are self-adjoint with b2 ∈ b(h), then (ba)∗ba ⊂ a∗ab2, by corollary 1.1, we obtain (ba)∗ba = a∗ab2. similarly, ba(ba)∗ = a∗ab2, i.e. ba is normal. since (ba)∗ too is normal and normal operators are maximally normal, we get ba = (ba)∗ = a∗b, i.e. ba is essentially self-adjoint. � proposition 2.2. let t,b,a be non necessarily bounded operators such that t is sels-adjoint and b symmetric and invertible (hence b is self-adjoint) and that a∗ is symmetric. then ab ⊂ t =⇒ tb−1 = b−1t = a. proof. clearly, ab ⊂ t =⇒ a ⊂ tb−1 =⇒ b−1t ⊂ a∗ ⊂ a∗∗ = a ⊂ tb−1 = tb−1. from theorem 2.1, we have tb−1 and b−1t are normal. hence tb−1 = b−1t = a. � proposition 2.3. let t,b,a be non necessarily bounded operators such that t and b are self-adjoint with b ∈ b(h) and invertible and a∗ is symmetric. if ba ⊂ t , then ba = ab = t . proof. we have: ba ⊂ t =⇒ a ⊂ b−1t =⇒ tb−1 ⊂ a∗ ⊂ a∗∗ = a ⊂ b−1t = b−1t. left and right multiplying by b give bt ⊂ bab ⊂ tb. by theorem 2.1, we obtain tb,bt are normal and tb = bt = bab, int. j. anal. appl. 18 (4) (2020) 629 hence ba = ab = t. � theorem 2.3. let t,a,b be non necessarily bounded operators such that a and b are normal with b ∈ b(h) and t is unitary. assume further that ta est normal. if ba ⊂ tab and , then ab and ba are normal. also, if a and t commute, then tab is normal. proof. obviously, ba ⊂ tab =⇒ b∗a∗t∗ ⊂ a∗b∗ =⇒ b∗ta ⊂ ab∗ (by fuglede-putnam theorem). =⇒ ba∗ ⊂ a∗t∗b. it is clear that ab is closed and we have: (ab)∗ab ⊃ b∗a∗ab ⊃ b∗a∗ba ⊃ b∗ba∗a. since (ab)∗ab,b∗b,a∗a are self-adjoint with b∗b ∈ b(h), then (ab)∗ab ⊂ a∗ab∗b, and by corollary 1.1, we obtain (ab)∗ab = a∗ab∗b. we also have, ab(ab)∗ ⊃ abb∗a∗ ⊃ ab∗ba∗ ⊃ b∗taba∗ ⊃ b∗tt∗baa∗ = b∗baa∗. similarly, we obtain ab(ab)∗ = a∗ab∗b (because a and b are normal). and this marks the end of the proof of the normality of ab. let’s show now that ba is normal. indeed (ba)∗ba = a∗b∗(a∗b∗)∗ ⊃ a∗b∗ba ⊃ b∗a∗t∗ba ⊃ b∗ba∗a, i.e. (ba)∗ba = a∗ab∗b similarly, ba(ba)∗ = a∗ab∗b, int. j. anal. appl. 18 (4) (2020) 630 that is, ba is normal. let’s show now that tab is normal. we have tab is closed because t is invertible and ab is closed. we also have ta ⊂ at =⇒ ta∗ ⊂ a∗t (by fuglede-putnam theorem). =⇒ t∗a ⊂ at∗ =⇒ t∗a∗ ⊂ a∗t∗ (by fuglede-putnam theorem). indeed tab(tab)∗ ⊃ b∗a∗t∗tab = b∗a∗ab ⊃ b∗a∗t∗ba ⊃ b∗ba∗a, since tab(tab)∗,b∗b,a∗a are self-adjoint with b∗b ∈ b(h), we get tab(tab)∗ ⊂ a∗ab∗b. by corollary 1.1, we obtain tab(tab)∗ = a∗ab∗b. similarly, (tab)∗tab = tab(tab)∗ = a∗ab∗b, and this marks the end of the proof of the normality of tab. � the folowing result is already seen in ( [12]), we can consider it as a consequence of the prceding theorem. also for t = i (where i is the indentity operator on h) we will get the theorem 2.1. corollary 2.2. let a,b be normal operators with b ∈ b(h). assume that ba ⊂ λab where λ ∈ c. then ab and ba are both normal if |λ| = 1 (and so ab = λba). proof. for t = λi where i is the indentity operator on h, we obtain t∗ = λi, i.e. t is unitary (because |λ| = 1). theorem 2.3 yiels the normality of ab, ba and λab. since ab is closed, we may also write ba ⊂ λab =⇒ ba ⊂ λab but, normal operators are maximally normal, therefore, we finally infer that ba = λab. � closely related to the previous results, we have another proof for the closure of bounded operators on a domain. proposition 2.4. let t is a bounded operator on d(t). then t is closed if d(t) is closed on h. int. j. anal. appl. 18 (4) (2020) 631 the proof requires the following lemma whose proof is very akin to the one in [15]. lemma 2.1. let f : x −→ y is continous such that y is hausdorff space. then the graph of f is closed on x ×y . now we prove proposition 2.4 proof. we denote the graphe of t by gr(t). by lemma 2.1, we obtain gr(t) is closed on d(t) ×h, i.e. (gr(t)) c is open. we may write (gr(t)) c = ∪ (i,j)∈i×j u1i ×u 2 j where i, j are arbitrary and u1i , u 2 j are open on d(t), h respectively. hence gr(t) = ∩ (i,j)∈i×j (f1i ∩d(t)) ×f 2 j , with f1i , f 2 j closed sets on h. therefore gr(t) becommes closed on h ×h when d(t) is closed on h (it’s the induced topology). � remark 2.1. in the previous proof, we did not use the linearity of t , we used only topological notions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. b. conway, a course in functional analysis, (2nd edition), springer, 1990 . [2] a. devinatz, a. e. nussbaum, j. von neumann, on the permutability of self-adjoint operators, ann. math. 62 (2) (1955), 199-203. [3] a. devinatz, a. e. nussbaum, on the permutability of normal operators, ann. math. 65 (2) (1957), 144-152. [4] b. abdelkader and h. mortad mohammed, generalizations of kaplansky´s theorem involving unbounded linear operators. bull. polish acad. sci. math. 62 (2) (2014), 181–186. [5] k. gustafson, m. h. mortad, unbounded products of operators and connections to dirac-type operators, bull. sci. math., 138 (5) (2014), 626-642. [6] k. gustafson, m. h. mortad, conditions implying commutativity of unbounded self-adjoint operators and related topics, j. oper. theory, 76 (1) (2016), 159-169. [7] il bong jung, m. h. mortad, j. stochel, on normal products of selfadjoint operators, kyungpook math. j. 57 (2017), 457-471. [8] m. h. mortad, an all-unbounded-operator version of the fuglede-putnam theorem, complex anal. oper. theory, 6 (6) (2012), 1269-1273. [9] m. h. mortad, commutativity of unbounded normal and self-adjoint operators and applications, oper. matrices, 8 (2) (2014), 563-571. [10] m.h. mortad, a criterion for the normality of unbounded operators and applications to self-adjointness, rend. circ. mat. palermo, 64 (2015), 149-156. int. j. anal. appl. 18 (4) (2020) 632 [11] m. meziane, m.h. mortad, maximality of linear operators, rend. circ. mat. palmero, ser. 2, 68 (2019), 441–451 [12] c. chellali, m.h. mortad, commutativity up to a factor of bounded operators and applications, j. math. anal. appl. 419 (2014), 114-122. [13] a. e. nussbaum, a commutativity theorem for unbounded operators in hilbert space, trans. amer. math. soc. 140 (1969), 485-491. [14] f. c. paliogiannis, a generalization of the fuglede-putnam theorem to unbounded operators, j. oper. 2015 (2015). art. id 804353. [15] w. rudin, functional analysis, mcgraw-hill book co., second edition, international series in pure and applied mathematics, mcgraw-hill, inc., new york, 1991. [16] k. schmüdgen, unbounded self-adjoint operators on hilbert space, springer gtm 265 (2012). [17] z. sebestyén, j. stochel, on suboperators with codimension one domains, j. math. anal. appl. 360 (2009), 391-397. [18] j. stochel, an asymmetric putnam-fuglede theorem for unbounded operators, proc. amer. math. soc. 129 (2001), 22612271. [19] j. stochel, f. h. szafraniec, domination of unbounded operators and commutativity, j. math. soc. japan, 55 (2003), 405-437. [20] j. weidmann, linear operators in hilbert spaces (translated from the german by j. szücs), srpinger-verlag, gtm 68 (1980). 1. introduction 2. main results references international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 58-63 http://www.etamaths.com fixed point results of altman integral type mappings in s-metric spaces mujeeb ur rahman1,∗, muhammad sarwar1 and muhib ur rahman2 abstract. in this article, we introduce the concept of ϕ-weakly commuting self-mappings pairs in s-metric space. using this idea we establish a common fixed point theorem of altman integral type for four self-mappings in the context of s-metric space. example is constructed to support our result. 1. introduction and preliminaries fixed point theory is one of the most dynamic research subject in nonlinear analysis. in the field of metric fixed point theory the first important and significant result was proved by banach in 1922 for contraction mapping in complete metric space. the well known banach contraction theorem may be stated as follows: ”every contraction mapping of a complete metric space x into itself has a unique fixed point”(bonsall 1962). in [1] altman proved a fixed point theorem for a single self-mapping in compact metric space satisfying the following contractive condition: d(tx,ty) ≤ q(d(x,y)) ∀ x,y ∈ x where q : [0,∞) → [0,∞) is an increasing function satisfies the following conditions: (1) 0 < q(t) < t, t ∈ (0,∞); (2) ρ(t) = t t−q(t) is a decreasing function; (3) t1∫ 0 ρ(t)dt < +∞ for some positive number t1. remark 1.1. by (1) and that q is increasing we have q(0) = 0 also q(t) = t ⇐⇒ t = 0. common fixed point for altman type mapping has been discussed by garbone and singh [2] and li and gu [3] in metric spaces. in 2006, mustafa and sims [4] introduced a new structure of generalized metric space called g-metric space. gu and ye [5] obtained a common fixed point theorem for altman integral type mapping in complete g-metric space. recently, sedghi et al. [6] initiated the idea of s-metric space as a generalization of g-metric space. while in [7] sedghi proved fixed point theorems for implicit relation in complete s-metric space. in this paper, we derive a common fixed point altman integral type mapping for two pairs of ϕ-weakly commuting self-mappings in complete s-metric space. we begin with the following definitions and results in the framework of s-metric space which can be found in [6, 7]. definition 1.2. let x be a non-empty set. an s-metric is a function s : x × x × x → [0,∞) satisfying the following conditions for all x,y,z,a ∈ x s1) s(x,y,z) = 0 if and only if x = y = z; s2) s(x,y,z) ≤ s(x,a,a) + s(y,a,a) + s(z,a,a). the pair (x,s) is called s-metric space. example 1.3. let x=(−∞, +∞) the distance function s : x ×x ×x → [0,∞) is defined by s(x,y,z) = |x−z| + |y −z| for all x,y,z ∈ x. 2010 mathematics subject classification. primary 39b82; secondary 44b20, 46c05. key words and phrases. altman type mapping; common fixed point; self-mapping; ϕ-weakly commuting selfmappings. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 58 altman integral type mappings 59 then (x,s) is a complete s-metric space. definition 1.4. let (x,s) be an s-metric space. a sequence {xn} in x converges to x ∈ x if s(xn,xn,x) → 0 as n →∞. we write xn → x for brevity. definition 1.5. let (x,s) be an s-metric space. a sequence {xn} in x is called cauchy sequence if for � > 0, there exists n0 ∈ n such that for all n,m ≥ n0 we have s(xn,xn,xm) < �. definition 1.6. an s-metric space (x,s) is said to be complete if every cauchy sequence in x converges in x. lemma 1.7. limit of the convergent sequence in s-metric space is unique. lemma 1.8. s-metric is jointly continuous on all three variables. lemma 1.9. in an s-metric space, we have s(x,x,y) = s(y,y,x) for all x,y ∈ x. now we introduce the concept of ϕ-weakly commuting pairs of self-mappings in s-metric space as follows: definition 1.10. a pair of self-mappings (s,t) on s-metric space is called ϕ-weakly commuting. if there exist a continuous function ϕ : [0,∞) → [0,∞), ϕ(0) = 0 such that s(stx,stx,tsx) ≤ ϕ(s(sx,sx,tx)) ∀ x ∈ x. example 1.11. let x = [0,∞), s(x,y,z) = |x− z| + |y − z| for all x,y,z ∈ x. let s,t : x → x are defined by sx = x 8 and tx = x 2 then s(stx,stx,tsx) = s( x 16 , x 16 , x 16 ) ≤ 1 2 3 4 x = 1 2 s(sx,sx,tx) s(stx,stx,tsx) ≤ ϕ(s(sx,sx,tx)). lemma 1.12. [5]. let ρt ba a lebesgue integrable function and ρ(t) > 0 for all t > 0. let f(x) = x∫ 0 ρ(t)dt, then f(x) is an increasing function in [0, +∞). definition 1.13. [8]. let s and t be two self-mappings on a set x. any point x ∈ x is called coincidence point of s and t if sx = tx for some x ∈ x and we called u = sx = tx is a point of coincidence of s and t . definition 1.14. a function φ : [0,∞) → [0,∞) is called contractive modulus if it satisfy φ(t) ≤ t for all t ≥ 0. 2. main results theorem 2.1. let (x,s) be a complete s-metric space and p,t,f,g : x → x be self-mappings. if there exists an increasing function q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) also the following conditions holds: (4) p(x) ⊆ g(x) and t(x) ⊆ f(x); (5) s(px,px,ty)∫ 0 ρ(t)dt ≤ φ( q(s(fx,fx,gy))∫ 0 ρ(t)dt) for all x,y ∈ x and φ is contractive modulus where ρ(t) is a lebesgue integrable function which is summable nonnegative and such that δ∫ 0 ρ(t)dt > 0 ∀ δ > 0. (6) if (p,f) and (t,g) are two pairs of continuous ϕ-weakly commuting mappings. then p,t,f and g have a unique common fixed point in x. 60 rahman, sarwar and rahman proof. since p(x) ⊆ g(x) and t(x) ⊆ f(x) so we define two sequences {xn} and {yn} in x by the rule y2n+1 = px2n = gx2n+1 and y2n+2 = tx2n+1 = fx2n+2 n = 0, 1, 2, ..... now consider s(y2n+1,y2n+1,y2n+2)∫ 0 ρ(t)dt = s(px2n,px2n,tx2n+1)∫ 0 ρ(t)dt. using (5) we have ≤ φ ( q(s(fx2n,fx2n,gx2n+1))∫ 0 ρ(t)dt ) = φ ( q(s(y2n,y2n,y2n+1))∫ 0 ρ(t)dt ) . using the property of φ we have ≤ q(s(y2n,y2n,y2n+1))∫ 0 ρ(t)dt. let tn = s(yn,yn+1) then the above inequality take the form t2n+1∫ 0 ρ(t)dt ≤ q(t2n)∫ 0 ρ(t)dt. now by the property of q and lemma 1.12 we have t2n+1 ≤ q(t2n) < t2n. similarly we can show that t2n ≤ q(t2n−1) < t2n−1. hence {tn} is a nonnegative strictly decreasing sequence and hence convergent. thus tn+1 ≤ q(tn) < tn for all n = 0, 1, 2, 3, ..... now to prove that {yn} is a cauchy sequence consider for m ≥ n and by triangle inequality we have s(yn,yn,ym) ≤ 2 m−1∑ i=n s(yi,yi,yi+1) = 2 m−1∑ i=n ti = 2 m−1∑ i=n ti(ti − ti+1) (ti − ti+1) ≤ 2 m−1∑ i=n ti(ti − ti+1) (ti −q(ti)) ≤ 2 m−1∑ i=n ti∫ ti+1 t (t−q(t)) dt = 2 tn∫ tm t (t−q(t)) dt = 2 tn∫ tm p(t)dt. it follows from the convergence of the sequence {tn} and by condition (3) we have lim n→∞ tn∫ tm p(t)dt = 0. thus {yn} is a cauchy sequence in x. since x is complete so there must exists u ∈ x such that lim n→∞ yn = u. also the subsequences {y2n+1} and {y2n+2} converges to u. therefore lim n→∞ y2n+1 = lim n→∞ px2n = lim n→∞ gx2n+1 = u lim n→∞ y2n+2 = lim n→∞ tx2n+1 = lim n→∞ fx2n+2 = u. since (p,f) are continuous ϕ-weakly commuting pair so s(pfx2n,pfx2n,fpx2n) ≤ ϕ(s(px2n,px2n,fx2n)). altman integral type mappings 61 taking limit n →∞ and since (p,f) is continuous pair of mappings thus s(pu,pu,fu) ≤ ϕ(s(u,u,u)) = ϕ(0) = 0. which implies that pu = fu. similarly from continuous ϕ-weakly commuting pair (t,g) we can show that tu = gu. now by condition (5) and using other given information we have s(pu,pu,tu)∫ 0 ρ(t)dt ≤ φ ( q(s(fu,fu,gu))∫ 0 ρ(t)dt ) ≤ q(s(fu,fu,gu))∫ 0 ρ(t)dt s(pu,pu,tu) ≤ q(s(fu,fu,gu)) ≤ s(fu,fu,gu) ≤ s(fu,fu,pu) + s(fu,fu,pu) + s(gu,gu,pu) ≤ s(gu,gu,tu) + s(gu,gu,tu) + s(pu,pu,tu) = s(pu,pu,tu). which implies that fu = gu. thus fu = gu = pu = tu and let z = fu = gu = pu = tu. therefore u is the common coincidence point of mappings p,t,f and g. again since (p,f) are ϕ-weakly commuting pair so s(pz,pz,fz) = s(pfu,pfu,fpu) ≤ ϕ(s(pu,pu,fu)) = ϕ(0) = 0. implies that pz = fz. similarly we can show that tz = gz. thus pfu = fpu and tgu = gtu. again by condition (5) we have s(pz,pz,z)∫ 0 ρ(t)dt = s(ppu,ppu,tu)∫ 0 ρ(t)dt ≤ φ ( q(s(fpu,fpu,gu))∫ 0 ρ(t)dt ) ≤ q(s(fpu,fpu,gu))∫ 0 ρ(t)dt. by lemma 1.12 and using the property of q we have s(pz,pz,z) ≤ q(s(fpu,fpu,gu)) ≤ s(fpu,fpu,gu) = s(pfu,pfu,gu) = s(pz,pz,z). which implies pz = z but pz = fz therefore pz = fz = z. similarly we can prove that tz = gz = z. hence pz = fz = gz = tz = z. thus z is a common fixed point of mappings p,t,f and g. uniqueness. assume that common fixed point of p,t,f and g is not unique i.e z 6= w be two distinct fixed points of p,t,f and g. then using condition (5) we have s(z,z,w)∫ 0 ρ(t)dt = s(pz,pz,tw)∫ 0 ρ(t)dt ≤ φ ( q(s(fz,fz,gw))∫ 0 ρ(t)dt ) ≤ q(s(fz,fz,gw))∫ 0 ρ(t)dt. by lemma 1.12 and using the property of q we have s(z,z,w) ≤ q(s(fz,fz,gw)) ≤ s(z,z,w). which is a contradiction hence z = w. therefore, fixed point of p,t,f and g is unique. � remark 2.2. if we take (1) p = t (2) f = g (3) p = t and f = g = i (4) φ = i in theorem 2.1. then we obtain several new results in the setting of s-metric space. 62 rahman, sarwar and rahman corollary 2.3. let (x,s) be a complete s-metric space and p,t,f,g : x → x be self-mappings. if there exists an increasing function q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) also the following conditions holds: (4) p(x) ⊆ g(x) and t(x) ⊆ f(x); (5) s(px,px,ty) ≤ φ(q(s(fx,fx,gy))) for all x,y ∈ x where φ is contractive modulus; (6) if (p,f) and (t,g) are two pairs of continuous ϕ-weakly commuting mappings. then p,t,f and g have a unique common fixed point in x. proof. putting ρ(t) = i in theorem 2.1, one can easily obtain the proof of corollary 2.3 from theorem 2.1. � now we construct an example to support corollary 2.3. example 2.4. let x = [0,∞)) and s(x,y,z) = |x − y| + |y − z| for all x,y,z ∈ x with selfmappings defined on x is given by px = x 8 ,fx = x,tx = x 16 and gx = x 2 . clearly p(x) ⊆ g(x) and t(x) ⊆ f(x). also we have s(px,px,ty) = s ( x 8 , x 8 , y 16 ) = | x 8 − y 16 | + | x 8 − y 16 | = 1 8 ( |x− y 2 | + |x− y 2 | ) = 1 8 s(fx,fx,gy) s(px,px,ty) = 1 8 s(fx,fx,gy). let φ(t) = 3 4 t and q(t) = 1 2 t. then φ(t) ≤ t and q(t) satisfies conditions (1)-(3). then we have s(px,px,ty) = 1 8 s(fx,fx,gy) ≤ 3 4 · 1 2 s(fx,fx,gy) = 3 4 q(s(fx,fx,gy)). on the other side if ϕ(t) = 1 2 t for all t ∈ [0,∞). then one can easily show that (p,f) and (t,g) are two pairs of continuous ϕ-weakly commuting mappings in x. so that all the conditions of corollary 2.3 are satisfied. therefore, 0 is the unique common fixed point of p,t,f and g. corollary 2.5. let (x,s) be a complete s-metric space and p,t : x → x be self-mappings. if there exists an increasing function q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) and φ is a contractive modulus also the following condition holds: s(px,px,ty) ≤ φ(q(s(x,x,y))) for all x,y ∈ x. then p and t have a unique common fixed point in x. corollary 2.6. let (x,s) be a complete s-metric space and p : x → x be self-mappings. if there exists an increasing function q : [0,∞) → [0,∞) satisfying conditions from (1)-(3) and φ is a contractive modulus also the following condition holds: s(px,px,py) ≤ φ(q(s(x,x,y))) for all x,y ∈ x. then p has a unique fixed point in x. references [1] m. altman, a fixed point theorem in compact metric spaces, american mathematical monthly, 82(1975), 827-829. [2] a. garbone and s.p. singh, common fixed point theorem for altman type mapping, indian journal of pure and applied mathematics, 18(1987), 1082-1087. [3] y. li and f. gu, common fixed point theorem of altman integral type mapping, the journal of n0nlinear sciences and applications, 2(2009), 214-218. [4] z. mustafa and b. sims, a new approach to generalized metric spaces, journal of non-linear convex analysis, 7(2006), 289-297. [5] f. gu and h. ye, common fixed point theorem of altman integral type mapping in g-metric spaces, abstract and applied analysis, 2012(2012) article id 630457. [6] s. sedghi, n. shobe, a. aliouche, a generalization of fixed point theorem in s-metric spaces, matema. bech., 64(2012), 258-266. altman integral type mappings 63 [7] s. sedghi and v.n. dung, fixed point theorems on s-metric spaces, matema. bech., 66(2014), 113-124. [8] g. jungck, compatible mappings and common fixed points, international journal of mathematics and mathematical sciences, 9(1986), 771-779. [9] g. jungck, common fixed points for non-continuous non-self mappings on non-metric spaces, far east journal of mathematical sciences, 4(1996), 199-212. [10] m. sarwar and m.u. rahman, six maps version for hardy-rogers type mapping in dislocated metric space, proceeding of a. razmadze mathematical institute, 166(2014), 121-132. [11] m.u. rahman and m. sarwar, a fixed point theorem for three pairs of mappings satisfying contractive condition of integral type in dislocated metric space, journal of operetors, 2014 (2014), article id 750427. [12] a. branciari, a fixed point theorem for mappings satisfying general contractive condition of integral type, international journal of mathematics and mathematical sciences, 29(2002), 531-536. 1department of mathematics, university of malakand,chakdara dir(l), khyber pukhtunkhwa, pakistan 2department of electrical (telecom), mcs, nust, rawalpindi, pakistan ∗corresponding author: mujeeb846@yahoo.com international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 17-23 http://www.etamaths.com harmonic analysis associated with the generalized q-bessel operator ahmed abouelaz, radouan daher, el mehdi loualid∗ abstract. in this article, we give a new harmonic analysis associated with the generalized q-bessel operator. we introduce the generalized q-bessel transform, the generalized q-bessel translation and the generalized q-bessel convolution product. 1. introduction in this paper we consider a generalized q-bessel operator ∆q,α,n defined by (1) ∆q,α,nf(x) = 1 x2 [ q2nf(q−1x) − (1 + q2α+4n)f(x) + q2α+2nf(qx) ] where n = 0, 1, ... . for n = 0, we regain the q-bessel operator (2) ∆q,αf(x) = 1 x2 [ f(q−1x) − (1 + q2α)f(x) + q2αf(qx) ] through this paper, we provide a new harmonic analysis corresponding to the generalized q-bessel operator ∆d,α,n. the structure of the paper is as follows: in section 2, we set some notations and collect some basic results about q-harmonnic analysis. in section 3, we give some facts about harmonic analysis related to the generalized q-bessel operator ∆q,α,n, we define the generalized q-bessel transform and we give some proprieties. in section 4, we define the generalized q-bessel translation tαq,x,n and the generalized q-bessel convolution product related to tαq,x,n. 2. element of q-harmonnic analysis in this section, we recapitulate some facts about harmonic analysis related to the bessel operator ∆d,α,n. we cite here, as briefly as possible, some properties. for more details we refer to [5, 6, 1, 2]. throught this paper, we assume that 0 < q < 1 and α > −1. let a ∈ c, the q-shifted factorial are defined by (a; q)0 = 1, (a; q)n = n−1∏ k=0 (1 −aqk), (a; q)∞ = ∞∏ k=0 (1 −aqk) the q-derivative of a function f is given by dqf(x) = f(x) −f(qx) (1 −q)x if x 6= 0 the q-jackson integrals from 0 to a and from 0 to ∞ are defined by [3, 4]∫ a 0 f(x)dqx = (1 −q)a ∞∑ 0 f(aqn)qn, 2010 mathematics subject classification. 05a30, 44a15. key words and phrases. generalized q-bessel operator; generalized q-bessel transform; generalized q-bessel translation; genaralized q-bessel translation. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 17 18 abouelaz, daher and loualid ∫ ∞ 0 f(x)dqx = (1 −q)a ∞∑ n=−∞ f(qn)qn. we have dq ∫ a x h(t)dqt = −h(x). the q-analogue of the integration theorem by a change of variable can be stated as follows∫ b a h( λ r )λ2α+1dqλ = r 2α+2 ∫ b r a r h(t)t2α+1dqt, ∀r ∈ r+q the q-integration by parts formula is given by∫ b a g(x)dqf(x)dqx = [f(b)g(b) −f(a)g(a)] − ∫ b a f(qx)dqg(x)dq. the third jackson q-bessel function jα (also called hahn-exton q-bessel functions) is defined by the power series [7] jα(x; q) = (qα+1; q)∞ (q; q)∞ xα ∞∑ n=0 (−1)n q n(n+1) 2 (qα+1; q)n(q; q)n x2n, and has the normalized form jα(x; q) = ∞∑ n=0 (−1)n q n(n+1) 2 (qα+1; q)n(q; q)n x2n. it satisfies the following estimate [5] |jα(qn,q2)| ≤ (−q2; q2)∞(−q2α+2; q2)∞ (q2α+2; q2)∞ { 1 if n ≥ 0 qn 2−(2α+1)n if n < 0 if x ∈ c∗ \r then we have the following asymptotic expansion as |x|→∞ jα(x; q 2) ∼ (x2q2; q2)∞ (q2α+2; q2)∞ also the normalized q-bessel functions satisfies an orthogonality relation c2q,α ∫ ∞ 0 jα(xt,q 2)jα(yt,q 2)t2α+1dqt = δq(x,y) where δq(x,y) = { 0, if y 6= x; 1 (1−q)x2(α+1) , if x = y. (3) cq,α = 1 1 −q (q2α+2; q2)∞ (q2; q2)∞ . the function x 7→ jα(λx,q2) is a solution of the following q-differential equation (4) ∆q,αf(x) = −λ2f(x), where ∆q,α is the q-bessel operator given by (2). for 1 ≤ p < ∞ we denote by lpq,α the set of all real functions on r+q for which ‖f‖q,p,α = (∫ ∞ 0 |f(x)|px2α+1dqx ) 1 p < ∞. proposition 2.1. let f,g ∈l2q,α such that ∆q,αf, ∆q,αg ∈l2q,α then (5) ∫ ∞ 0 ∆q,αf(x)g(x)x 2α+1dqx = ∫ ∞ 0 f(x)∆q,αg(x)x 2α+1dqx. generalized q-bessel operator 19 the q-bessel fourier transform fq,α was introduced and studies in [5, 6] (6) fq,αf(x) = cq,α ∫ ∞ 0 f(t)jα(xt,q 2)t2α+1dqt, the q-bessel translation operator is defined as follows [5, 6] (7) tαq,xf(y) = cq,α ∫ ∞ 0 fq,α(f)(t)jα(xt,q2)jα(yt,q2)t2α+1dqt. proposition 2.2. we have for all x,y ∈ r+q tαq,xf(y) = t α q,yf(x) the q-translation operator is positive if tαq,xf ≥ 0, ∀f ≥ 0, ∀x ∈ r + q . the domaine of positivity of the q-translation operator is qα = {q ∈]0, 1[, tαq,x is positive for all x ∈ r + q }. in [1] it was proved that if −1 < α < α′ then qα ⊂ qα′. as a consequence: • if 0 ≤ α then qα =]0, 1[. • if −1 2 ≤ α < 0 then ]0,q0[⊂ q−1 2 ⊂ qα (]0, 1[, q0 ' 0.43. • if −1 ≤ α < −1 2 then qα ⊂ q−1 2 . in the rest of this section we always assume that the q-translation operator is positive. the q-convolution product of two functions is given by [5, 6] (8) f ∗q,α g(x) = cq,α ∫ ∞ 0 tαq,xf(y)g(y)y 2α+1dqy. the following theorem summarize some result about q-bessel fourier transform [6] theorem 2.3. the q-bessel fourier transform satisfies (1) for all functions f ∈lpq,α, f2q,αf(x) = f(x), ∀x ∈ r + q . (2) for all functions f ∈l2q,α, (9) ‖f2q,αf ‖q,α,2=‖ f ‖q,α,2 . (3) for all functions f ∈lpq,α, where p ≥ 1 then fq,αf ∈lpq,α. if 1 ≤ p ≤ 2 then (10) ‖fq,αf‖q,α,p ≤ b 2 p −1 q,α ‖f‖q,α,p. where (11) bq,α = 1 1 −q (−q2; q2)∞(−q2α+2; q2)∞ (q2; q2)∞ (4) let f ∈lpq,α and g ∈lrq,α the f ∗q g ∈lsq,α and (12) fq,α(f ∗q,α g)(x) = fq,αf(x) ×fq,αf(x), ∀x ∈ r+q . where 1 ≤ p,r,s such that 1 p + 1 r − 1 = 1 s . proposition 2.4. [2] for all x,y ∈ r+q , we have (13) tαq,xjα(λy,q 2) = jα(λx,q 2)jα(λy,q 2). proposition 2.5. [2] for any function f ∈l2q,α we have (14) fq,α(tαq,xf)(λ) = jα(λx,q 2)fq,α(f)(λ), ∀λ,x ∈ r+q . 20 abouelaz, daher and loualid 3. generalized q-bessel transform let • m the map defined by (15) mf(x) = x2nf(x). • lpq,α,n the class of measurable functions f on r+q for which ‖f‖q,α,p,n = ‖m−1f‖q,α+2n,p < ∞. ∀x ∈ r+q , put (16) ψα,n(λx,q2) = x2njα+2n(λx,q2). proposition 3.1. (i): the map m is a topological isomorphism from lpq,α onto lpq,α,n (ii): we have (17) ∆q,α,n ◦m = m◦ ∆q,α+2n. (iii): ψα,n(λ.,q2) satisfies the differential equation (18) ∆q,α,nψα,n(λ.,q2) = −λ2ψα,n(λ.,q2) proof. assertion (i) is easily checked. (ii) easy combination of (1), (2) and (16). using (4) and (18), we have ∆q,α,nψα,n(λ.,q 2) = m◦ ∆q,α+2n ◦m−1ψα,n(λ.,q2), = m◦ ∆q,α+2njα+2n(λ.,q2), = −λ2mjα+2n(λ.,q2), = −λ2ψα,n(λ.,q2) which prove (iii). definition 3.2. the generalized q-bessel transform of a function f ∈l1q,α,nis defined by (19) fq,α,n(f)(x) = cq,α+2n ∫ ∞ 0 f(t)ψα,n(λt,q 2)t2α+1dqt where cq,α+2n is given by (3). proposition 3.3. (i): for all f ∈l1q,α,n we have (20) fq,α,n(f)(λ) = fq,α+2n ◦m−1f(λ). (ii): for all f ∈l1q,α,n (21) fq,α,n(∆q,α,nf)(λ) = −λ2fq,α,n(f)(λ). proof. let f ∈l1q,α,n. from (6), (17) and (20), we have fq,α,n(f)(λ) = cq,α+2n ∫ ∞ 0 f(t)ψα,n(λt,q 2)t2α+1dqt, = cq,α+2n ∫ ∞ 0 f(t)x2njα+2n(λt,q 2)t2α+1dqt, = cq,α+2n ∫ ∞ 0 m−1f(t)jα+2n(λt,q2)t2α+4n+1dqt, = fq,α+2n ◦m−1f(λ). generalized q-bessel operator 21 which prove (i). let f ∈l1q,α,n. from (5), (19) and (20), we have fq,α,n(∆q,α,nf)(λ) = cq,α+2n ∫ ∞ 0 ∆q,α,nf(t)ψα,n(λt,q 2)t2α+1dqt, = cq,α+2n ∫ ∞ 0 f(t)∆q,α,nψα,n(λt,q 2)t2α+1dqt, = cq,α+2n ∫ ∞ 0 (−λ2)f(t)ψα,n(λt,q2)t2α+1dqt, = −λ2fq,α,n(f)(λ). theorem 3.4. (1) for f ∈lpq,α,n, we have (22) ‖fq,α,nf‖q,α,n,∞ ≤ bq,α+2n‖f‖q,α,n,1. where bq,α+2n is given by (11) (2) let f ∈l1q,α,n, then (23) ‖fq,α,nf‖q,α,n,2 = ‖f‖q,α,n,2. proof. let f ∈l1q,α,n, from (10), (11) and (21) we have ‖fq,α,nf‖q,α,n,∞ = ‖fq,α+2n ◦m−1f‖q,α+2n,∞, ≤ bq,α+2n‖m−1f‖q,α+2n,1, ≤ bq,α+2n‖f‖q,α,n,1. which prove 1). let f ∈l1q,α,n. using (9) and (21), we have ‖fq,α,nf‖q,α,n,2 = ‖fq,α+2n ◦m−1f‖q,α+2n,2, = ‖m−1f‖q,α+2n,2, = ‖f‖q,α,n,2. 4. generalized convolution product associated with ∆q,α,n definition 4.1. the generalized q-bessel translation operators tαq,x,n associated with ∆q,α,n are defined by (24) tαq,x,n = x 2nm◦tα+2nq,x ◦m −1 where tα+2nq,x is given by (7). the generalized q-bessel translation operator is positive if tαq,x,nf ≥ 0, ∀f ≥ 0, ∀x ∈ r + q . the domaine of positivity of the generalized q-bessel translation operator is qα,n = {q ∈]0, 1[, tαq,x,n is positive for all x ∈ r + q }. in the rest of this paper we always assume that the generalized q-bessel translation operator is positive. proposition 4.2. (i): let f ∈l1q,α,n, we have (25) tαq,x,nf(y) = t α q,y,nf(x) and tαq,x,nf(0) = f(x). (ii): ∀x,y ∈ r+q , we have (26) tαq,x,nψα,n(λy,q 2) = ψα,n(λx,q 2)ψα,n(λy,q 2) 22 abouelaz, daher and loualid (iii): for any function f ∈l2q,α,n, we have (27) fq,α,n(tαq,x,nf)(λ) = ψα,n(λx,q 2)fq,α,n(f)(λ). proof. let f ∈l1q,α,n, from porosition 2.2 and (25), we have tαq,x,nf(y) = x 2nm◦tα+2nq,x ◦m −1f(y), = x2ny2ntα+2nq,y ◦m −1f(x), = y2nm◦tα+2nq,y ◦m −1f(x), = tαq,y,nf(x). which prove (i). let x,y ∈ r+q . from (13) and (25), we have tαq,x,nψα,n(λy,q 2) = x2nm◦tα+2nq,x ◦m −1(y2njα+2n(λy,q 2)), = x2ny2ntα+2nq,x jα+2n(λy,q 2)), = x2njα+2n(λx,q 2))y2njα+2n(λy,q 2)), = ψα,n(λx,q 2)ψα,n(λy,q 2). which prove (ii). let f ∈l2q,α,n. from (14), (17), (21) and (25), we have fq,α,n(tαq,x,nf)(λ) = fq,α,n(x 2nm◦tα+2nq,x ◦m −1f)(λ), = x2nfq,α+2n(tα+2nq,x ◦m −1f)(λ), = x2njα+2n(λx,q 2)fq,α+2n(m−1f)(λ), = ψα,n(λx,q 2)fq,α,n(f)(λ). which prove (iii). definition 4.3. the generalized q-convolution product of both function f,g ∈l1q,α,n is defined by (28) f ∗q,α,n g(x) = cq,α+2n ∫ ∞ 0 tαq,x,nf(y)g(y)y 2α+1dqy. where cq,α+2n is given by (3). proposition 4.4. for f,g ∈l1q,α,n (29) f ∗q,α,n g = m [ (m−1f) ∗q,α+2n (m−1f) ] . proof. let f,g ∈l1q,α,n. from (8), (25) and (29), we have f ∗q,α,n g(x) = cq,α+2n ∫ ∞ 0 tαq,x,nf(y)g(y)y 2α+1dqy, = cq,α+2nx 2n ∫ ∞ 0 tα+2nq,x m −1f(y)g(y)y2α+2n+1dqy, = cq,α+2nx 2n ∫ ∞ 0 tα+2nq,x m −1f(y)m−1g(y)y2α+4n+1dqy, = x2n [ (m−1f) ∗q,α+2n (m−1g) ] (x), = m [ (m−1f) ∗q,α+2n (m−1g) ] (x). proposition 4.5. for f,g ∈l1q,α,n, then f ∗q,α,n g ∈l1q,α,n and (30) fq,α,n(f ∗q,n g)(λ) = fq,α,n(f)(λ)fq,α,n(g)(λ). generalized q-bessel operator 23 proof. let f,g ∈l1q,α,n, we have ‖f ∗q,n g‖q,α,n,1 = ‖m−1(f ∗q,n g)‖q,α+2n,1, ≤ ‖m−1f‖q,α+2n,1‖m−1g‖q,α+2n,1, = ‖f‖q,α,n,1‖g‖q,α,n,1. on the other hand, from (12), (21) and (30), we have fq,α,n(f ∗q,α,n g)(λ) = fq,α,n ( m [ (m−1f) ∗q,α+2n (m−1f) ]) (λ), = fq,α+2n ◦m−1 ( m [ (m−1f) ∗q,α+2n (m−1f) ]) (λ), = fq,α+2n ( m−1f ) (λ) ×fq,α+2n ( m−1g ) (λ), = fq,α,n(f)(λ)fq,α,n(g)(λ). proposition 4.6. let f ∈l1q,α,n, we have (31) tαq,x,nf(y) = ∫ ∞ 0 f(z)dα,n(x,y,z)z 2α+1dqz, where dα,n(x,y,z) = c2q,α+2n ∫∞ 0 ψα,n(xt,q 2)ψα,n(yt,q 2)ψα,n(zt,q 2)t2α+4n+1dqt proof. let f ∈l1q,α,n, from (7), (19) and (24) we have tαq,x,nf(y) = x 2n ( m◦tα+2nq,x ◦m −1) (f)(y), = x2ny2ntα+2nq,x ◦m −1f(y), = x2ny2ncq,α+2n ∫ ∞ 0 fq,α+2n ◦m−1f(t)jα+2n(xt,q2)jα+2n(yt,q2)t2α+4n+1dqt, = cq,α+2n ∫ ∞ 0 fq,α,n(f)(t)ψα,n(xt,q2)ψα,n(yt,q2)t2α+4n+1dqt, = ∫ ∞ 0 f(z) [ c2q,α+2n ∫ ∞ 0 ψα,n(xt,q 2)ψα,n(yt,q 2)ψα,n(zt,q 2)t2α+4n+1dqt ] z2α+1dqz, = ∫ ∞ 0 f(z)dα,n(x,y,z)z 2α+1dqz. references [1] a. fitouhi and l. dhaouadi, positivity of the generalized translation associated with the q-hankel transform, constructive approximation, 34(2011), 435-472. [2] ahmed fitouhi and m. moncef hamza, the q-jαbessel function, journal of approximation theory 115(2002), 144-166. [3] f.h. jackson, on a q-defnite integrals, quarterly journal of pure and application mathematics, 41(1910), 193-203. [4] g.gasper and m.rahman, basic hypergeometric series, encycopedia of mathematics and its applications, volume 35, cambridge university press (1990). [5] l.dhaouadi, a.fitouhi and j.el kamel, inequalities in q-fourier analysis, journal of inequalities in pure and applied mathematics, 7(2006), article id 171. [6] l.dhaoudi, on the q-bessel fourier transform, bulletin of mathematical analysis and applications, 5(2013), 42-60. [7] r. f. swarttouw, the hahn-exton q-bessel functions, ph. d. thesis, delft technical university (1992). department of mathematics, faculty of sciences aïn chock,, university of hassan ii, casablanca, morocco ∗corresponding author: mehdi.loualid@gmail.com international journal of analysis and applications volume 19, number 6 (2021), 904-914 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-904 received august 15th, 2021; accepted october 4th, 2021; published november 3rd, 2021. 2010 mathematics subject classification. 54h25, 47h10. key words and phrases. self-mappings; occasionally weakly compatible mappings; probabilistic 2-metric space; clr’s-property. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 904 some results by using clr’s-property in probabilistic 2-metric space v. srinivas1, k. satyanna2,* 1department of mathematics, university college of science, osmania university, hyderabad, telangana state, india 2department of mathematics, m.a.l.d. government degree college, gadwal, palamoor university, mahaboobnagar, telangana state, india *corresponding author: satgjls@gmail.com abstract. the aim of this paper is to generate two fixed point theorems in probabilistic 2-metric space by applying clr’s-property and occasionally weakly compatible mappings (owc), these two results generalize the theorem proved by v. k. gupta, arihant jain and rajesh kumar. further these results are justified with suitable examples. 1. introduction menger [1] pioneered the statistical metric(sm) space theory. one of the major achievements was the translation of probabilistic concepts into geometry. menger used the notation of new distance distribution function from p to q by a fpq. b. schweizer, and a. sklar [2] introduced a new notion of a probabilistic-norm. this norm naturally generates topology, convergence ,continuity and completeness in sm-space. mishra [3] used compatible mappings and generated some fixed points in menger space. altumn turkoglu [4] proved some more results of sm-space by utilizing the implicit relation in multivalued mappings. zhang, xiaohong, huacan he, and yang xu [5] employed the schweizer-sklar t-norm established fuzzy https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-904 int. j. anal. appl. 19 (6) (2021) 905 logic system to contribute in development of sm-space. sehgal, v. m., and a. t. bharucha-reid [6] used classical banach contraction to establish the first result of menger space for coincidence points. weakly compatible mappings were generalized by al-thagafi and shahzad [7], by introducing occasionally weakly compatible mappings. futher chauhan, sunny, wutiphol sintunavarat, and poom kumam[9] proved some more theorems by using clr’s-property in fuzzy metric space. further some more results can be witnessed by using the concepts of sub sequentially continuous and semi compatible mappings in menger space [10]. 2. preliminaries definition 2.1 [8] a continuously t-norm is mapping t: [0, 1] × [0, 1] → [0, 1] and it satisfies the following properties (𝑡1) 𝑡 𝑖𝑠 𝑎𝑏𝑒𝑙𝑖𝑎𝑛 & 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒 (𝑡2) 𝑡(𝛾, 1) = 𝛾, ∀ 𝛾 ∈ [0, 1] (𝑡3) 𝑡 (𝛾, 𝜔) ≤ 𝑡 (𝛼, 𝜗) 𝑓𝑜𝑟 𝛾 ≤ 𝛼 𝑎𝑛𝑑 𝜔 ≤ 𝜗 ∀ 𝛾, 𝛼, 𝜗, 𝜔 ∈ [0, 1]. definition 2.2 [8] the pair (𝑋, 𝐹) named as probabilistic 2-metric space (2-pm space) where 𝑋 ≠ ∅ and 𝐹 ∶ 𝑋 × 𝑋 × 𝑋 → 𝐿 here l is the set of all distribution functions and the f value at (𝑢, 𝑣, 𝑤) ∈ 𝑋 × 𝑋 × 𝑋 is represented by 𝐹𝑢,𝑣,𝑤 and obeys properties as under a) 𝐹𝑢,𝑣,𝑤 (0) = 0, b) for all distinct u, v in x ∃ w ∈ x with 𝐹𝑢,𝑣,𝑤 (t) < 1 for some t > 0, c) 𝐹𝑢,𝑣,𝑤 (t) = 1 ∀ t > 0, if any two of the three points have to be the same, d) 𝐹𝑢,𝑣,𝑤 ( t) = 𝐹𝑣,𝑤,𝑢 (t) = 𝐹𝑤,𝑢,𝑣 ( t), e) 𝐹𝑢,𝑣,𝑤 ( 𝑡𝑎) = 𝐹𝑣,𝑤,𝑢 (𝑡𝑏) = 𝐹𝑤,𝑢,𝑣 ( 𝑡𝑐) = 1 then 𝐹𝑢,𝑣,𝑤 (𝑡𝑎 + 𝑡𝑏 + 𝑡𝑐 ) = 1. definition 2.3 [8] a sequence 〈𝑥𝑛 〉 in 2-menger space ( 𝑋, 𝐹, 𝑡) is (i) converges to 𝛽 if for each 𝜖 ∗ > 0, 𝑡 > 0, ∃ n(𝜖 ∗) ∈ n implies 𝐹𝑥𝑛,𝛽,𝑎 (ϵ *) > 1 –t, ∀ 𝑎𝜖 𝑋 𝑎𝑛𝑑 𝑛 ≥ 𝑁 (𝜖 ∗). (ii) cauchy if for each 𝜖 > 0, 𝑡 > 0, ∃ n(𝜖) ∈ n implies 𝐹𝑥𝑛,𝑥𝑚,𝑎 (𝜖) > 1 – t, ∀ 𝑎 ∈ 𝑋 𝑎𝑛𝑑 𝑛, 𝑚 ≥ 𝑁(𝜖). (iii) if the cauchy sequence converges in x then it is referred as a complete 2-menger space. int. j. anal. appl. 19 (6) (2021) 906 definition 2.4 [8] self-mappings p, s in 2-menger space ( 𝑋, 𝐹, 𝑡) are called (i) compatible if 𝐹𝑃𝑆𝑥𝑛,𝑆𝑃𝑥𝑛, 𝑎(δ) → 1, ∀ a ∈ x and δ > 0 whenever a sequence 〈𝑥𝑛〉 in x ∋ 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑧 where z is an element of x as n → ∞. (ii) weakly compatible if the mappings commute at their coincidence points. (iii) occasionally weakly compatible (owc) if ∃ x in x ∋ px = sx ⇒ psx = spx. remark 2.5 two weakly compatible mappings are obviously owc mappings, but the converse does not have to be the case. example 2.6. by treating 𝑋 = [0, 1] and d be the usual metric on x and for all 𝑡1 ∈ [0, 1], define 𝐹𝑢, 𝑣, 𝑎(𝑡1) = { 𝑡1 𝑡1+⎸𝛼−𝛽⎹ , 𝑡1 > 0 0 , 𝑡1 = 0 ∀ 𝛼, 𝛽 in x and fixed a, 𝑡1 > 0. define mappings 𝑃, 𝑆 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃(𝑥) = 𝑥2 2 , x ∈ [0, 1] and 𝑆(𝑥) = 𝑥 3 , x ∈ [0, 1]. we notice that the pair (p, s) has two coincidence points 0, 2 3 . if x = 2 3 then p ( 2 3 ) = s ( 2 3 ) = 2 9 (2.6.1) 𝑃𝑆 ( 2 3 ) = 𝑃( 2 9 ) = 2 81 , (2.6.2) 𝑆𝑃( 2 3 ) = s( 2 9 ) = 2 27 . (2.6.3) from (2.6.2) and (2.6.3) ps( 2 3 ) ≠ sp( 2 3 ). at x = 0, p ( 0 ) = s (0 ) and 𝑃𝑆( 0 ) = 𝑆𝑃( 0 ). this shows the mappings p, s are owc but not weakly compatible. definition 2.7 [9] “self maps p and s of a 2-menger space (x, f, t) are said to satisfy clr’s – property (common limit range property) if there exists a sequence 〈𝑥𝑛 〉 ∈ 𝑋 ∋ 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑆𝑧, for some element z ∈ 𝑋 as n → ∞. this example shows that mappings p, s satisfy clr’sproperty but they do not have closed ranges. example 2.8. take x = (0, 1] and 𝑡 ∈ [0, 1], define int. j. anal. appl. 19 (6) (2021) 907 𝐹𝑢,𝑣(t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 ∀ 𝛼, 𝛽 in x and 𝑡 > 0. 𝐷𝑒𝑓𝑖𝑛𝑒 𝑃, 𝑆 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃(𝑥 ) = { 1 − 𝑥, 𝑥 ∈ (0, 𝟐 𝟑 ) 𝑥, 𝑥 ∈ [ 𝟐 𝟑 , 1] (2.8.1) and 𝑆(𝑥) = { 2𝑥, 𝑥 ∈ (0, 2 3 ] 1, 𝑥 ∈ ( 2 3 , 1] . (2.8.2) consider a sequence 𝑥𝑛 = 1 3 − 1 3𝑛 for n = 1, 2, 3… then (2.8.3) p𝑥𝑛 = 1 − ( 𝟏 𝟑 − 1 3𝑛 ) = 𝟐 𝟑 + 1 3𝑛 → 𝟐 𝟑 (2.8.4) s𝑥𝑛 = 2( 𝟏 𝟑 − 1 3𝑛 ) = 𝟐 𝟑 − 2 9𝑛 → 𝟐 𝟑 as n → ∞. (2.8.5) thus 𝑃𝑥𝑛,𝑆𝑥𝑛 → 𝑆( 𝟏 𝟑 ) = 𝟐 𝟑 as n → ∞. (2.8.6) where p(x) = ( 1 3 , 1], s(x) = (0, 4 3 ] u{1} this shows that p, s are satisfy clr’sproperty but they do not have closed ranges. now we give the statement of theorem (a). it is proved by v. k. gupta et al. theorem (a) [8] “ let a, b, s and t be self -mappings on a complete probabilistic 2-metric space (x̃, f, t) satisfying: (a1) a(x̃) ⊆ t(x̃), b(x̃) ⊆ s( x̃) (𝐴2) one of a(x̃), b(x̃), t(x̃) or s(x̃) is complete, (𝐴3) pairs (a, s) and (b, t) are weakly compatible, (𝐴4) 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (t) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (t) for all x, y and t > 0, where r: [0, 1] → [0, 1] is some continuous function such that r(t) > t for each o < t < 1, then a,b,s and t have unique common fixed point in x̃ ’’. we now generalize theorem(a) as under. 3. main result theorem 3.1 let a, b, s and t be self -mappings on a complete probabilistic 2-metric space (x̃, f, 𝑡 ∗) satisfying : (3.1.1) a(x̃) ⊆ t(x̃) , b(x̃) ⊆ s(x̃), (3.1.2) the pairs (a, s), (b, t) share the clr’s property with owc, int. j. anal. appl. 19 (6) (2021) 908 (3.1.3) 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (𝑡 ∗ ) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (𝑡 ∗ ) for all x, y and 𝑡 ∗ > 0, where r: [0, 1] → [0, 1] is some continuous function such that r(𝑡 ∗ ) > 𝑡 ∗ for each o < 𝑡 ∗ < 1 then a, b, s and t have unique common fixed point in x̃. proof: iteratively the sequences 〈𝑦𝑛 〉 and 〈𝑥𝑛 〉 can be constructed as x0 ∈ x̃ ⇒ ax0 ∈ a(x̃) ⊆ t( x̃), ∃ x1 ∈ x̃ in such a way that ax0 = tx1, bx1 ∈ b( x̃) ⊆ s( x̃) then we have x2 ∈ x̃ with bx1 = sx2 〈𝑦2𝑛 〉 = a𝑥2𝑛= t𝑥2𝑛+1and 〈𝑦2𝑛+1〉 = b𝑥2𝑛+1 = s𝑥2𝑛+ 2. (3.1.4) now our claim is to show 〈𝑦𝑛 〉 is cauchy sequence. for this take 𝑥 = 𝑥2𝑛, 𝑦 = 𝑥2𝑛+1 in (3.1.3) we get 𝐹𝐴𝑥2𝑛, 𝐵 𝑥2𝑛+1, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹𝑆 𝑥2𝑛 . 𝑇 𝑥2𝑛+1 (𝑡 ∗), (3.1.5) ⇒𝐹𝑦2𝑛, 𝑦2𝑛+1, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹 𝑦2𝑛−1 . 𝑦2𝑛, 𝑎 (𝑡 ∗) > 𝐹 𝑦2𝑛−1 . 𝑦2𝑛, 𝑎 (𝑡 ∗). (3.1.6) similarly 𝐹𝑦2𝑛+1, 𝑦2𝑛+2, 𝑎 (𝑡 ∗) > 𝐹 𝑦2𝑛 . 𝑦2𝑛+1, 𝑎 (𝑡 ∗). (3.1.7) in general we have 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) > 𝐹𝑦𝑛, 𝑦𝑛−1, 𝑎 (𝑡 ∗) for all values of n. then < 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) > is an increasing sequence bounded above by 1 therefore it must converge to l, where l ≤ 1. if l < 1 then 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (𝑡 ∗) = l > r(1) > 1 as a result of the contradiction, l= 1. hence 𝐹𝑦𝑛+1. 𝑦𝑛, 𝑎 (t) = 1 for all n and p. as a result, because cauchy sequence exists in complete space x, it has a limit z in x̃ and consequently each sub sequence has the same limit z. that is ax2n , sx2n → z and bx2n+1, tx2n+1 → z as n → ∞. (3.1.8) on using clrs-property of (a, s) , (b, t) implies there are sequences (an ) as well as (bn ) in order for aan , san , b bn , t bn → sµ as n → ∞ for some µ in x̃. (3.1.9) to prove z = sµ put x = a2n , y = x2n+5 in (3.1.3) we get 𝐹a𝑎2𝑛. 𝐵x2𝑛+5, 𝑎 (𝑡 ∗) ≥ r (𝐹s𝑎2𝑛. 𝑇x2𝑛+5, 𝑎 (𝑡 ∗) ) as n → ∞ (3.1.10) ⇒ 𝐹sµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹sµ. z, 𝑎 (𝑡 ∗)).> 𝐹sµ. z, 𝑎 (𝑡 ∗). (3.1.11) resulting 𝐹sµ. z, 𝑎 (𝑡 ∗). > 𝐹sµ. z, 𝑎 (𝑡 ∗) (3.1.12) which is a contradiction. hence sµ =z. (3.1.13) int. j. anal. appl. 19 (6) (2021) 909 claim a µ = s µ. put x = µ, y = x2n+3 in (3.1.3) we get 𝐹aµ. 𝐵x2𝑛+3, 𝑎 (𝑡 ∗) ≥ r (𝐹sµ. 𝑇x2𝑛+3, 𝑎 (𝑡 ∗) ) as n → ∞ (3.1.14) ⇒ 𝐹aµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹sµ. z, 𝑎 (𝑡 ∗)) using (3.1.13) (3.1.15) ⇒ 𝐹aµ. z, 𝑎 (𝑡 ∗) ≥ r (𝐹z. z, 𝑎 (𝑡 ∗)) = r(1) = 1. (3.1.16) this results a µ = s µ = z. (3.1.17) since the pair (a, s) obeys owc resulting a µ = s µ ⇒ sa µ = as µ. that is az = sz. (3.1.18) claim az = z. substitute y = x2n+3, x = z in (3.1.3) we have 𝐹az. 𝐵x2𝑛+3, 𝑎 (𝑡 ∗) ≥ r 𝐹sz. 𝑇x2𝑛+3, 𝑎 (𝑡 ∗ ) letting n → ∞. (3.1.19) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗ ) ≥ 𝑟𝐹𝑆𝑧 . 𝑧, 𝑎 ( 𝑡 ∗) using (3.1.18) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗), (3.1.20) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗ ) > 𝐹𝐴𝑧 . 𝑧, 𝑎 ( 𝑡 ∗). (3.1.21) this is a contradiction. thus z = az. resulting az = sz = z. (3.1.22) since az ∈ a(x̃) ⊆ t(x̃) then ∃ ρ ∈ x̃ such that az = t 𝜌. (3.1.23) claim z = b𝜌. by employing x = x4n, y = 𝜌 of (3.1.3) we obtain 𝐹𝐴𝑥2𝑛, 𝐵𝜌 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑥 2𝑛 . 𝑇𝜌 , 𝑎 ( 𝑡 ∗) as n → ∞ . (3.1.24) from (3.1.22) & (3.1.23) ⇒ 𝐹𝑧, 𝐵𝜌 (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑇𝜌 , 𝑎 ( 𝑡 ∗ ) = r(1) = 1. (3.1.25) thus z = b𝜌 = t𝜌. since the pair of mappings (b, t) obeys owc, this results b𝜌 = t𝜌 ⇒ bt𝜌 = tb𝜌. that is bz = tz. (3.1.26) claim z = bz. by substituting y = z, x = z in (3.1.3) results 𝐹𝐴𝑧, 𝐵𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧, 𝑎 ( 𝑡 ∗) using (3.1.22) &(3.1.26) (3.1.27) 𝐹𝑧, b𝑧, a (𝑡 ∗ ) ≥ 𝑟𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗). (3.1.28) resulting 𝐹𝑧, b𝑧, a (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 ( 𝑡 ∗). it is impossible. therefore bz =z. combining all we get az = bz = z = sz = tz. int. j. anal. appl. 19 (6) (2021) 910 thus 𝑧 is the required common fixed point for these mappings a, b, s and t. uniqueness: assume z1 is second common fixed point. now assume z ≠ z1. by considering y = z1 , x = z in (3.1.4) we obtain 𝐹𝐴𝑧, 𝐵𝑧1 , a (𝑡∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧1, 𝑎 (𝑡 ∗) 𝐹𝑧, 𝑧1, a (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) > 𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) 𝐹𝑧, 𝑧1, a (𝑡 ∗) > 𝐹𝑧 . 𝑧1, 𝑎 (𝑡 ∗) which is absurd. hence z = z1. as a result, four selfmappings a, b, s, and t have the only one common fixed point. now we justify our theorem as under. 3.2 example let us take x = [0, 𝜋] and each 𝑡 ∈ [0, 1], define 𝐹𝑢,𝑣( t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼, 𝛽 𝑖𝑛 𝑋 , 𝑡 > 0. define mappings 𝑃, 𝑆, 𝑇 & 𝑄 ∶ 𝑋 → 𝑋 𝑎𝑠 𝐴 (𝑥) = 𝐵(𝑥) = { 2𝑒 −𝜋𝑥 , 𝑥 ∈ [0, 𝜋 2 ) 𝜋 − 𝑥, 𝑥 ∈ [ 𝜋 2 , 𝜋] (3.2.1) 𝑎𝑛𝑑 𝑆(𝑥) = 𝑇(𝑥) = { 2𝑒 −𝜋𝑥 2 , 𝑥 ∈ [0, 𝜋 2 ) 𝑥, 𝑥 ∈ [ 𝜋 2 , 𝜋] (3.2.2) now a(x) = b(x) = [0, 2] and s(x) = t(x) = [0, 𝜋] implies 𝐴(𝑋 ) ⊆ 𝑇(𝑋) 𝑎𝑛𝑑 𝐵(𝑋 ) ⊆ 𝑆(𝑋). clearly 𝜋 2 and 1 are coincidence points for the mappings b, t. at x = 𝜋 2 , b( 𝜋 2 ) = t ( 𝜋 2 ) and 𝐵𝑇 ( 𝜋 2 ) = 𝐵 ( 𝜋 2 ) = 𝜋 2 , 𝑇𝐵 ( 𝜋 2 ) = t( 𝜋 2 ) = 𝜋 2 implies 𝐵𝑇 ( 𝜋 2 ) = 𝑇𝐵 ( 𝜋 2 ). at x = 1, b(1) = t(1) and bt(1) ≠ tb(1). thus the pairs (a, s), (b, t) satisfy owc but are not weakly compatible. if 𝑥𝑛 = 𝜋 2 − 1 𝑛 for all n ≥ 1. then (3.2.3) s𝑥𝑛= t𝑥𝑛= s ( 𝜋 2 − 1 𝑛 ) = 𝜋 2 − 1 𝑛 → 𝜋 2 . (3.2.4) a𝑥𝑛= b𝑥𝑛= a ( 𝜋 2 − 1 𝑛 ) = 𝜋 – ( 𝜋 2 − 1 𝑛 ) = 𝜋 2 + 1 𝑛 → 𝜋 2 as n → ∞. (3.2.5) int. j. anal. appl. 19 (6) (2021) 911 ⇒ a𝑥𝑛, s𝑥𝑛, t𝑥𝑛, b𝑥𝑛 → s( π 2 ) as n → ∞. (3.2.6) this gives the pairs of maps (a, s), (b, t) sharing the clr’s property with owc. thus a, b, s and t satisfy all the norms of theorem and having the unique commonly fixed point at 𝜋 2 as a ( 𝜋 2 )= s( 𝜋 2 ) = b( 𝜋 2 ) = t( 𝜋 2 ) = 𝜋 2 . now we present another generalization of theorem (a) as under. theorem 3.3 let a, b, s and t be self -mappings on a complete probabilistic 2-metric space (x̃, f, 𝑡∗) satisfying : (3.3.1) a(x̃) ⊆ t(x̃), b(x̃) ⊆ s( x̃) (3.3.2) the pair (a, s) satisfies clr’s property with owc and (b, t) satisfies owc. (3.3.3) further 𝐹𝐴𝑥. 𝐵𝑦, 𝑎 (𝑡 ∗) ≥ r 𝐹𝑆𝑥. 𝑇𝑦, 𝑎 (𝑡 ∗) for all elements x, y in x̃ and 𝑡 ∗> 0 r is continuous self-map on [0, 1] such that r(𝑡 ∗) > 𝑡∗ for each o < 𝑡∗< 1. then a, b, s and t have unique common fixed point in x̃. proof: take the constructed sequences < xn> , < yn> in theorem (3.1) as 〈𝑦2𝑛 〉 = a𝑥2𝑛= t𝑥2𝑛+1and 〈𝑦2𝑛+1〉 = b𝑥2𝑛+1 = s𝑥2𝑛+ 2 . (3.3.4) it is already shown that 〈𝑦𝑛 〉 as cauchy sequence. as a result each sub sequence has the same limit point z in complete space x̃. that is ax2n , sx2n → z and bx2n+1, tx2n+1 → z. the pair (a, s) obeys clrs-property this implies there is a sequence 〈𝑧𝑛 〉 such that 𝐴𝑧𝑛 , 𝑆𝑧𝑛 → 𝑆𝑣 for some v in x̃. claim z = sv. by putting y = x2n+1 , x = zn, in (3.3.3), that results 𝐹𝐴𝑧𝑛, 𝐵𝑥2𝑛+1, a (𝑡∗) ≥ 𝑟𝐹𝑆𝑧𝑛 . 𝑇𝑥2𝑛+1 , 𝑎 (𝑡 ∗) as n → ∞ (3.3.5) ⇒𝐹𝑆𝑣, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) > 𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗), (3.3.6) ⇒𝐹𝑆𝑣, 𝑧, a (𝑡 ∗) > 𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) . (3.3.7) this is absurd. as a result sv =z. (3.3.8) claim av = sv. (3.3.9) by inserting x = v, y = x2n+3 in (3.3.3), that results 𝐹𝐴𝑣, 𝐵𝑥 2𝑛+3, 𝑎 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑇𝑥 2𝑛+3 , 𝑎 (𝑡 ∗) letting as n → ∞ (3.3.10) ⇒ 𝐹𝐴𝑣, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑧, 𝑎 (𝑡 ∗) using (3.3.8) (3.3.11) ⇒ 𝐹𝐴𝑣, 𝑆𝑣, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑣 . 𝑆𝑣, 𝑎 (𝑡 ∗) = r(1) = 1. (3.3.12) int. j. anal. appl. 19 (6) (2021) 912 ⇒ av = sv = z. (3.3.13) since the pair (a, s) satisfies owc property, that results av = sv ⇒ sav = asv. this gives az = sz. (3.3.14) claim az = z. by replacing y = x2n+1, x = z in (3.3.3), as a result 𝐹𝐴𝑧, 𝐵𝑥2𝑛+1, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑥2𝑛+1 , 𝑎 (𝑡 ∗). as n → ∞ (3.3.15) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑧, 𝑎 (𝑡 ∗), using ( 3.3.14) (3.3.16) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗) , (3.3.17) ⇒ 𝐹𝐴𝑧, 𝑧, a (𝑡 ∗) > 𝐹𝐴𝑧 . 𝑧, 𝑎 (𝑡 ∗). this is a contradiction. consequently az = z. (3.3.18) by combining (3.3.14) and (3.3.18) gives z = sz = az. (3.3.19) since az ∈ a( x̃) ⊆ t( x̃) then ∃ 𝜔 ∈ x̃ such that az = t 𝜔. (3.3.20) claim z = b 𝜔. by using x = x4n, y = 𝜔 of (3.3.4), we obtain 𝐹𝐴𝑥 2𝑛,, 𝐵𝜔 (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑥 2𝑛, . 𝑇𝜔 , 𝑎 (𝑡 ∗). (3.3.21) taking limit as n → ∞ and from (3.3.19) and (3.3.20) we get ⇒ 𝐹𝑧, 𝐵𝜔 (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝑧, 𝑎 (𝑡 ∗) = r(1) = 1. (3.3.22) thus z = b 𝜔 = t 𝜔. (3.3.23) since the pair (b, t) obeys owc property gives b 𝜔 = t 𝜔 ⇒ bt 𝜔 = tb 𝜔 implying bz = tz. (3.3.24) claim z = bz. applying x = y = z in (3.3.3), this resulting 𝐹𝐴𝑧, 𝐵𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑆𝑧 . 𝑇𝑧, 𝑎 (𝑡 ∗), (3.3.25) using (3.3.19) and (3.3.24) ⇒ 𝐹𝑧, b𝑧, a (𝑡 ∗) ≥ 𝑟𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗), (3.3.26) ⇒ 𝐹𝑧, b𝑧, a (𝑡 ∗) > 𝐹𝑧 . 𝐵𝑧, 𝑎 (𝑡 ∗). (3.3.27) contradicting the fact implies bz =z. as a result az = bz = sz = tz = z. as a consequence four self-mappings a, b, s, and t, there is a fixed point commonly. uniqueness can be easily proved as in the theorem (3.1). int. j. anal. appl. 19 (6) (2021) 913 now the theorem (3.3) can be supported by discussing with suitable example. 3.4 example we choose x = [0, 1], d be usual metric on x and each 𝑡 ∈ [0, 1], define 𝐹𝑢,𝑣 ( t) = { 𝑡 𝑡+⎸𝛼−𝛽⎹ , 𝑡 > 0 0 , 𝑡 = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝛼, 𝛽 𝑖𝑛 𝑋 , 𝑡 > 0. choose mappings 𝑃, 𝑆, 𝑇 & 𝑄 ∶ 𝑋 → 𝑋 𝑎𝑠 𝑃 (𝑥) = 𝑄(𝑥) = { 1 − 2𝑥, 𝑥 ∈ [0, 0.2] 𝑥2, 𝑥 ∈ (0.2, 1] (3.4.1) 𝑎𝑛𝑑 𝑆(𝑥) = 𝑇(𝑥) = { 3𝑥, 𝑥 ∈ [0, 0.2] 𝑥3, 𝑥 ∈ (0.2, 1] . (3.4.2) now p(x) = q(x) = (0.04, 1] and s(x) = t(x) = [0, 1] so that 𝑃(𝑋 ) ⊆ 𝑇(𝑋) 𝑎𝑛𝑑 𝑄(𝑋 ) ⊆ 𝑆(𝑋). clearly 0.2 and 1 are coincedence points of the graphs q, t. at x = 0.2, q (0.2) = t(0.2) = 0.6 but qt(0.2) = q(0.6) = 0.36, tq(0.2) = t(0.6) = 0.216 at x = 1, q (1) = t(1) and 𝑄𝑇(1) = 𝑄(1) = 1 = 𝑇(1) = 𝑇𝑄(1). this demonstrates that the pairs (p, s), (q, t) are owc mappings, although they are not weakly compatible. if we choose 𝑥𝑛 = 1 − 4 3𝑛 for all n ≥ 1. then (3.4.3) p𝑥𝑛= p ( 1 − 4 3𝑛 ) = ( 1 − 4 3𝑛 )2 → 1 (3.4.4) s𝑥𝑛= s ( 1 − 4 3𝑛 ) = ( 1 − 4 3𝑛 )3 → 1 as n → ∞. (3.4.5) this implies p𝑥𝑛, s𝑥𝑛 → s(1) as n → ∞. (3.4.6) this gives the pair (p, s) satisfies clr’s-property with owc and the pair (q, t) is owc. thus the mappings p, q, s and t satisfy all the norms of the theorem (3.3), containing unique common fixed point at 1 as 1=p(1) = q(1) = s(1) = t(1). 4. conclusion in this paper theorem (a) is generalized in two ways. (a) theorem (3.1) is formulated by employing clr’s-property and applying owc for both the pairs instead of assuming weakly compatible mappings. int. j. anal. appl. 19 (6) (2021) 914 (b) theorem (3.2) is formulated by employing clr’s-property and owc for one pair and owc for the other pair instead of assuming weakly compatible mappings. further these two results are justified with suitable examples. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] k. menger, statistical metrics, proc. natl. acad. sci. usa. 28(12) (1942) ,535-537. [2] c. alsina, b. schweizer, and a. sklar, on the definition of a probabilistic normed space, aequationes math. 46(1-2) (1993), 91-98. [3] s. n. mishra, n. sharma and s. l. singh, common fixed points of maps on fuzzy metric spaces, int. j. math. math. sci. 17(1994), 253-258. [4] altun, ishak, and duran turkoglu, some fixed point theorems for weakly compatible mappings satisfying an implicit relation, taiwan. j. math. 13 (2009), 1291-1304. [5] x. zhang, h. he, y. xu, a fuzzy logic system based on schweizer-sklar t-norm, sci. china ser. f: inform. sci. 49(2) (2006), 175-188. [6] v. m. sehgal, a. t. bharucha-reid, fixed points of contraction mappings on probabilistic metric spaces, math. syst. theory 6(1-2) (1972), 97-102. [7] m. a. al-thagafi, n. shahzad, a note on occasionally weakly compatible maps, int. j. math. anal 3.2 (2009), 55-58. [8] v.k. gupta, a. jain and r. kumar, common fixed point theorem in probabilistic 2-metric space by weak compatibility, int. j. theor. appl. sci. 11(1) (2019), 09-12. [9] s. chauhan, w. sintunavarat, and p. kumam, common fixed point theorems for weakly compatible mappings in fuzzy metric spaces using (jclr) property, appl. math. 3 (2012), 976-982. [10] k. satyanna, v. srinvas, fixed point theorem using semi compatible and sub sequentially continuous mappings in menger space, j. math. comput. sci. 10(6) (2020), 2503-2515. international journal of analysis and applications volume 19, number 3 (2021), 389-404 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-389 on a new approach by modified (p; q)-szász-mirakyan operators vishnu narayan mishra1,∗, ankita r. devdhara2, khursheed j. ansari3, seda karateke4 1department of mathematics, indira gandhi national tribal university, lalpur, amarkantak 484 887, madhya pradesh, india 2applied mathematics and humanities department, svnit, surat-395007, india 3 department of mathematics, college of science, king khalid university, 61413, abha, saudi arabia 4department of mathematics and computer science, faculty of science and letters, istanbul arel university, istanbul-34537, turkey ∗corresponding author: vishnunarayanmishra@gmail.com abstract. in this paper, we introduce a new type of (p; q) exponential function with some properties and a modified (p; q)-szász-mirakyan operators by virtue of this function by investigating approximation properties. we obtain moments of generalized (p; q)-szász-mirakyan operators. furthermore, we derive direct results, rate of convergence, weighted approximation result, statistical convergence and voronovskaya type result of these operators with numerical examples. graphical representations reveal that modified (p; q)-szász-mirakyan operators have a better approximation to continuous functions than pioneer one. 1. introduction approximation theory is one of the oldest branches of mathematics. to approximate continuous functions with q-analogue of linear positive operators is significant application of q-calculus in approximation theory. cieśliński [1] established alternative definition of q-exponential function. he defined q-exponential function received november 29th, 2020; accepted january 4th, 2021; published april 9th, 2021. 2010 mathematics subject classification. 46b28. key words and phrases. (p; q)-szász-mirakyan operators; uniform convergence; voronovskaya type result; statistical convergence. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 389 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-389 int. j. anal. appl. 19 (3) (2021) 390 using cayley transformation. the main advantages of the new q-exponential function consist of better qualitative properties i.e., its properties are more similar to properties of ez,z ∈ c [1]. over the years, many research papers were developed on q-analogue of various linear positive operators and their approximation properties. recently, in [2] research of bernstein-stancu operators on (p; q)integers were performed and discussed uniform convergence and direct result of the operators. eventually, in [3] (p; q)-analogue of bernstein operators was investigated and developed the same convergence. acar [5] and mursaleen et.al [4], [12] proposed (p; q)-generalization of szász-mirakyan operators and discussed uniform convergence, rate of convergence, voronovskaya result in those papers. the motivation of recent work is developing a new type of (p; q) exponential function and utilizing this new exponential function to modify (p; q)-szász-mirakyan operators. we studied uniform convergence and statistical convergence of modified (p; q)-szász-mirakyan operators. in the first section, we discussed some sequences and rate of convergence of operators. we also proved voronovskaya type result. in the last section, we present some graphical representations. consider, 0 < q < p ≤ 1. the definition of (p; q)-integer is, {m}p,q = pm −qm p−q , m ∈ n,(1.1) {0}p,q = 0, and (p; q)−factorial is {m}p,q! = m∏ k=1 {k}p;q, m ∈ n {0}p;q! = 1.(1.2) (p; q)-exponential function is defined as [6] ep,q(z) = ∞∑ j=0 p j(j−1) 2 zj {j}p,q! ,(1.3) ep,q(z) = ∞∑ j=0 q j(j−1) 2 zj {j}p,q! .(1.4) the (p; q)-exponential functions have following property: ep,q(z)ep,q(−z) = ep,q(z)ep,q(−z) = 1. int. j. anal. appl. 19 (3) (2021) 391 another way of defining two (p; q)-exponentials as infinite products is ep,q(z) = ∞∏ j=0 1 (pj −qj(p−q)z) ,(1.5) ep,q(z) = ∞∏ j=0 (pj + qj(p−q)z).(1.6) 2. new type of (p; q)-exponential function new (p; q)-exponential function is determined as (2.1) ep,q(z) = ep,q(z/2)ep,q(z/2) = ∞∏ j=0 pj + qj(p−q)z 2 pj −qj(p−q)z 2 , ep,q(z),ep,q(z) are usual (p; q)-exponential functions. theorem 2.1. (p; q)-exponential function ep,q(z) is analytic in |z| < rp,q (2.2) ep,q(z) = ∞∑ j=0 zj [j]p,q! , |z| < rp,q where, rp,q =   2 p−q , 0 < q < p < 1. 2q q−p, q > p. ∞, p = q = 1. (2.3) [j]p,q = pj −qj p−q . 2 pj−1 + qj−1 = {j}p,q. 2 pj−1 + qj−1 ,(2.4) and [j]p,q! = j∏ m=1 [m]p,q = j∏ m=1 {m}p,q 2 pj−1 + qj−1 = {j}p,q! 2j∏j−1 m=0(p m + qm) .(2.5) proof. since (1.5) and (1.6) are absolutely convergent in |z| < 1, multiplying (1.5) and (1.6), we obtain ep,q(z/2)ep,q(z/2) = ∞∑ n=0 ∞∑ k=0 p n(n−1) 2 q k(k−1) 2 (z 2 )n+k {n}p,q!{k}p,q! = ∞∑ j=0 (z 2 )j {j}p,q! ∞∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! .(2.6) using formula for the (p; q)-binomial coefficients [7], we have (2.7) j−1∏ r=0 (pr + xqr) = j∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! xk. in particular, (2.8) j∑ k=0 p (j−k)(j−k−1) 2 q k(k−1) 2 {j}p,q! {j −k}p,q!{k}p,q! = (1 + 1)(p + q)...(pj−1 + qj−1). int. j. anal. appl. 19 (3) (2021) 392 substituting (2.8) into (2.6), we obtain (2.2), where [j]p,q defined as in (2.4). to get the radius of convergence, lim n→∞ ∣∣∣∣ zj+1[j + 1]p,q! ∣∣∣∣ ∣∣∣∣[j]p,q!zj ∣∣∣∣ = lim n→∞ ∣∣∣∣ z[j + 1]p,q ∣∣∣∣ =   (p−q)|z| 2 , for q < p, (q−p)|z| 2q , for q > p. (2.9) applying d’alembert’s test on (2.9), we obtain (p,q 6= 1) the radius of convergence (2.3). for p = q = 1, ep,q(z) is ez, thus r1 = ∞. � theorem 2.2. the ep,q(z) satisfies the following properties: (2.10) 1. ep,q(−z) = (ep,q(z))−1, 2. |ep,q(ix)| = 1. proof. the first part of above equation (2.10) directly comes from the definition of ep,q(z). that implies, ep,q(z) = ep,q(z̄). then, |ep,q(ix)|2 = ep,q(ix) ep,q(ix) = 1. � the above (p; q)-exponential function (2.1) has more improved properties similar to function ez. the definition of (p,q)-szász-mirakyan operators in [5] is (2.11) am,p,q(f; x) = ∞∑ j=0 1 ep,q({m}p,qx) q j(j−1) 2 {m}jp,qxj {j}p,q! f ( {j}p,q qj−2{m}p,q ) . acar obtained moments, uniform convergence and voronovskaya result of the above operators. we define a different sort of modified (p,q)-szász-mirakyan operators via new (p; q)-exponential function for f ∈ c[0,∞] in (2.1) is (2.12) sn,p,q(f; x) = 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! f ( [k]p,q [n]p,q ) , where 0 < q < p ≤ 1, n ∈ n, 0 ≤ x < 2 (p−q)[n]p,q = p n−1+qn−1 pn−qn . remark 2.1. we choose an x between 0 and p n−1+qn−1 pn−qn because we want ep,q([n]p,qx) to be convergent. remark 2.2. from calculations for every k ∈ n; [k]p,q [n]p,q = (pk−qk)(pn−1+qn−1) (pn−qn)(pk−1+qk−1) , 0 ≤ [k]p,q [n]p,q < p n−1+qn−1 pn−qn , int. j. anal. appl. 19 (3) (2021) 393 then we consider (2.13) sn(p,q; x) = 1 ep,q([n]p,qx) . [n]kp,qx k [k]p,q! . clearly, sn(p,q; x) is positive for 0 < q < p ≤ 1, n ∈ n and every 0 ≤ x < 2(p−q)[n]p,q . the operator sn,p,q is linear and positive. 3. moments of sn,p,q here, we determine approximation moments of operators (2.12). lemma 3.1. for n ∈ n and 0 < q < p ≤ 1. below equalities are verified: sn,p,q(1; x) = 1,(3.1) sn,p,q(t; x) = x,(3.2) sn,p,q(t 2; x) = x2 + x [n]p,q ,(3.3) sn,p,q(t 3; x) = x3 + 3x2 [n]p,q + x [n]2p,q .(3.4) proof. the result is obvious for sn,p,q(1; x). now sn,p,q(t; x) = 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q ) = 1 ep,q([n]p,qx) ∞∑ k=1 [n]k−1p,q x k [k − 1]p,q! = x. and sn,p,q(t 2; x) = 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q )2 = 1 ep,q([n]p,qx) ∞∑ k=1 [n]k−1p,q x k [k − 1]p,q! ( 1 [n]p,q ) + 1 ep,q([n]p,qx) ∞∑ k=2 [n]k−2p,q x k [k − 2]p,q! = x [n]p,q + x2. also sn,p,q(t 3; x) = 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]p,q [n]p,q )3 = 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ( [k]3p,q − 3[k]2p,q + 2[k]p,q + 3[k]2p,q − 2[k]p,q [n]3p,q ) = x3 + 3x2 [n]p,q + x [n]2p,q . int. j. anal. appl. 19 (3) (2021) 394 � central moments are: sn,p,q(t−x; x) = 0,(3.5) sn,p,q((t−x)2; x) = x [n]p,q .(3.6) remark 3.1. from our choice of p and q, we know that limn→∞[n]p,q = 1 p−q . but, to get the uniform convergence and other results of approximation for sn,p,q we suppose that sequences qn ∈ (0,pn); pn ∈ (qn, 1] such that qn, pn → 1 and pnn → a, qn ′ n → b as n tending to infinity, i.e., limn→∞ 1/[n]p,q = 0. now, we have uniform convergence of new kind of operators for all f ∈ cϑ[0,∞) where cϑ[0,∞) = {f ∈ c[0,∞) : |f(t)| ≤ a(1 + t)ϑ} for a > 0, ϑ > 0 and ‖f‖ = supx≥0 |f(x)| 1+x2 . theorem 3.1. let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity then for each f ∈ cϑ[0,∞) (3.7) lim n→∞ ‖sn,p,q(f) −f‖ϑ = 0. proof. from korovkin’s result, we put evidence that lim n→∞ ‖sn,p,q(ti) −xi‖ϑ = 0, i = 0, 1, 2. since sn,p,q(1; x) = 1, the result is clear for i = 0. for i = 1 lim n→∞ ‖sn,p,q(t) −x‖ϑ = lim n→∞ ‖x−x‖ϑ = 0. and for i = 2 lim n→∞ ‖sn,p,q(t2) −x2‖ϑ = lim n→∞ ‖x2 + x [n]p,q −x2‖ϑ = 0. hence sn,p,q(f; x) is uniformly convergent to f ∈ cϑ[0,∞]. � example 3.1. for p = 0.99 and q = 0.96, sequences of sn,p,q defined by (2.12) is convergent to f(x) = x2 −5x + 10 (fig. 1) and g(x) = x3 −x + 1 (fig. 2) with increasing values of n (n = 10, 20, 30) respectively. int. j. anal. appl. 19 (3) (2021) 395 figure 1. approximation to f by sn,p,q for n = 10, 20, 30. figure 2. approximation to g by sn,p,q for n = 10, 20, 30. int. j. anal. appl. 19 (3) (2021) 396 example 3.2. for different choices of p and q, the sequence of operators sn,p,q defined by (2.12) is convergent to f(x) = x2 − 5x + 10 (fig. 3) and g(x) = x3 −x + 1 (fig. 4) with n = 50 respectively. figure 3. approximation to f by sn,p,q for n = 50. figure 4. approximation to g by sn,p,q for n = 50. 4. some consequences for this section, we provide several results on local approximation for sn,p,q(f; x). here, cb[0,∞) is the set of bounded, continuous functions f on [0,∞). attached norm on cb[0,∞) is defined by‖f‖ = sup x∈[0,∞) |f(x)|. peetre’s k-functional is given by k2(f,δ) = inf h∈w2 {‖f −h‖ + δ‖h′′‖} where w2 = {h ∈ cb[0,∞) : h′,h′′ ∈ cb[0,∞)}. from ( [9], p.177), there exists a > 0 such that k2(f,δ) ≤ aω2(f,δ 1/2),δ > 0, where ω2(f,δ 1/2) = sup 0<η<δ1/2,x∈[0,∞) |f(x + 2η) − 2f(x + η) + f(x)| is the second order modulus of continuity of functions f in cb[0,∞). the first order modulus of continuity of function f ∈ cb[0,∞) is defined by ω(f,δ1/2) = sup 0<η<δ1/2,x∈[0,∞) |f(x + η) −f(x)|. int. j. anal. appl. 19 (3) (2021) 397 theorem 4.1. let 0 < q < 1 and p ∈ (q, 1]. the operators sn,p,q map from cb into cb. also, the following inequality is satisfied. (4.1) ‖sn,p,q(f; x)‖cb ≤‖f‖cb. proof. from the definition of sn,p,q(f; x), |sn,p,q(f; x)| ≤ 1 ep,q([n]p,qx) ∞∑ k=0 [n]kp,qx k [k]p,q! ∣∣∣∣f ( [k]p,q [n]p,q )∣∣∣∣. applying supremum to both sides here sup x≥0 |sn,p,q(f; x)| ≤ sup x≥0 |f(x)| 1 ep,q sn,p,q(1; x). one has ‖sn,p,q(f; x)‖cb ≤‖f‖cb. � theorem 4.2. let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity. then for f ∈ cb[0,∞), there exists a > 0 such that (4.2) |sn,p,q(f; x) −f(x)| ≤ aω2 ( f, √ x [n]p,q ) . proof. for h ∈ w2, using taylor’s expansion h(t) = h(x) + (t−x)h′(x) + ∫ t x (t−u)h′′(u)du. now |sn,p,qh(t) −h(x)| ≤ 12‖h ′′‖sn,p,q((t−x)2; x). also |sn,p,q(f; x)| ≤ ‖f‖. hence |sn,p,q(f; x) −f(x)| ≤ |sn,p,q((f −h)(x); x) − (f −h)(x)| + |sn,p,q(h; x) −h(x)| ≤ 2‖f −h‖ + 1 2 ‖h′′‖sn,p,q((t−x)2; x). taking infimum of the right hand side of above inequality for all h ∈ w2, |sn,p,q(f; x) −f(x)| ≤ 2k2 ( f; 1 4 x [n]p,q ) . since ω2(f,λδ) ≤ (λ + 1)2ω2(f; δ), |sn,p,q(f; x) −f(x)| ≤ aω2 ( f, √ x [n]p,q ) . int. j. anal. appl. 19 (3) (2021) 398 � 5. rate of convergence suppose that c[0,∞) is set of all continuous functions on [0,∞) and consider following sets: cϑ[0,∞) = {f ∈ c[0,∞) : |f(t)| ≤ a(1 + t)ϑ} for a > 0, ϑ > 0 and c∗ϑ[0,∞) = {f ∈ cϑ[0,∞) : limx→∞ |f(x)| 1+x2 < ∞}. the first order modulus of continuity on [0,a] is defined as ωa(f,δ) = sup |t−x|≤δ sup 0≤x≤a |f(t) −f(x)|. theorem 5.1. suppose that f ∈ c∗ϑ[0,∞). let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity and ωa+1(f,δ) be the modulus of continuity on [0,a + 1] ⊂ [0,∞). then (5.1) ‖sn,p,q(f; x) −f(x)‖ϑ ≤ 6af (1 + a2) a [n]p,q + 2ωa+1 ( f; √ a [n]p,q ) . proof. for 0 ≤ x ≤ a; 0 ≤ t,∞; from [8] |f(t) −f(x)| ≤ 6af (1 + a2)(t−x)2 + ωa+1(f; δ) ( |t−x| δ + 1 ) . combining above inequality and cauchy-schwarz inequality, ‖sn,p,q(f; x) −f(x)‖ϑ ≤ sn,p,q(|(f; x) −f(x)|; x) ≤ 6af (1 + a2)sn,p,q((t−x)2; x) + ωa+1(f; δ) ( 1 + 1 δ2 sn,p,q((t−x)2; x) ) . from the central moments of operators and for 0 ≤ x ≤ a sn,p,q((t−x)2; x) = x[n]p,q ≤ a [n]p,q = ζn. taking δ = √ ζ ‖sn,p,q(f; x) −f(x)‖ϑ ≤ 6af (1 + a2) a[n]p,q + 2ωa+1 ( f; √ a [n]p,q ) . � 6. weighted approximation result here, we discuss weighted approximation of sn,p,q through polynomial weight over the space cm defined below. consider that w0(x) = 1, wm (x) = (1 + x m )−1, (x ≥ 0,m ∈ n), int. j. anal. appl. 19 (3) (2021) 399 cm = {f ∈ c[0,∞) : wm (f) is continuous, bounded and uniformly convergent.}. the norm is defined by ‖f‖m = sup x≥0 wm (x)|f(x)|. also, we refer some results associated to steklov means. for h > 0 it is defined in [11] fh(x) = ( 2 h )2 ∫ h/2 0 ∫ h/2 0 [2f(x + s + t) −f(x + 2(s + t))]dsdt. we have f(x) −fh(x) = ( 2 h )2 ∫ h/2 0 ∫ h/2 0 ∆2s+tf(x)dsdt, f′′h (x) = h −2[8∆2 h/2 f(x) − ∆2hf(x)], and hence (6.1) ‖f −fh‖m ≤ w2m (f,h), ‖f ′′ h‖m ≤ 9h −2w2m (f,h) theorem 6.1. suppose that sn,p,q(f; x) is defined as in (2.12), where (pn) and (qn) are the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity. let m ∈ n∗, then for f ∈ cm , (6.2) wm|sn,p,q(f; x) −f(x)| ≤ nmw2m ( f; √ x [n]p,q ) ,nm > 0. proof. for m = 0, the result comes from theorem 4.2. for f ∈ cm, m ∈ n, wm|sn,p,q(f; x) −f(x)| ≤ wm|sn,p,q(|f −fh|; x)| + wm|sn,p,q(fh; x) −fh(x)| + wm|fh(x) −f(x)|. from theorem 4.1 and the first property of steklov means, wm|sn,p,q(|f −fh|; x)| ≤ ‖sn,p,q(f −fh)‖m ≤‖f −fh‖m ≤ w2m (f,h).(6.3) also, by taylor’s expansion wm|sn,p,q(fh; x) −fh(x)| ≤ ‖f′h‖msn,p,q((t−x); x) + 1 2 ‖f′′h‖msn,p,q((t−x) 2; x). from moments of operators sn,p,q and the second property of steklov means wm|sn,p,q(fh; x) −fh(x)| ≤ 9 2h2 w2m (f,h)sn,p,q((t−x) 2; x) ≤ 9 2h2 w2m (f,h)sn,p,q((t−x) 2; x) ≤ 9 2h2 w2m (f,h) x [n]p,q .(6.4) int. j. anal. appl. 19 (3) (2021) 400 setting h = √ x [n]p,q and from (6.1), (6.3), (6.4), we get wm|sn,p,q(f; x) −f(x)| ≤ nmw2m ( f; √ x [n]p,q ) . � 7. statistical convergence in this section, we obtain statistical convergence for new modified (p; q)-szász-mirakyan operators. we need the following theorem [10] to prove statistical convergence of the operators on h′ and we set all real valued functions on real-valued functions on [0,∞) with condition |f(x) −f(y)| ≤ ω(|x−y|). theorem 7.1. let mn be the sequence of positive linear operators from h ′ into cb[0,∞) with three conditions st− lim n→∞ ‖mn(tj; x) −xj‖cb = 0, j=0,1,2. then st− lim n→∞ ‖mn(f; x) −f‖cb = 0. now, the result on statistical convergence of the operators defined in (2.12). theorem 7.2. suppose that sn,p,q(f; x) is defined as in (2.12), where (pn) and (qn) are the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity. then (7.1) st− lim n→∞ ‖sn,p,q(f; x) −f‖cb = 0. proof. from above theorem, we only have to prove that st− lim n→∞ ‖sn,p,q(tj; x) −xj‖cb = 0, j = 0, 1, 2. from the moments of sn,p,q(f; x), it is obvious that the result is true for j = 0, 1. for j = 2, ‖sn,p,q(t2; x) −x2‖cb ≤ 1 [n]p,q . but st− lim n→∞ 1 [n]p,q = 0. we define u = {n : ‖sn,p,q(t2; x) −x2‖cb ≥ �} u1 = {n : 1 [n]p,q ≥ �}. int. j. anal. appl. 19 (3) (2021) 401 clearly, u ⊆ u1. then δ{k ≤ n : ‖sn,p,q(t2; x) −x2‖cb ≥ �}≤ δ{k ≤ n : 1 [n]p,q ≥ �} but the right hand side of the above inequality is zero because st− lim n→∞ 1 [n]p,q = 0. hence, st− lim n→∞ ‖sn,p,q(t2; x) −x2‖cb = 0. the theorem is proved. � 8. voronovskaya type result theorem 8.1. let (pn) and (qn) be the sequences such that pn → 1, qn → 1 and pnn → a, qn ′ n → b as n tending to infinity then for each function f, f′, f′′ ∈ c∗ϑ[0,∞) (8.1) lim n→∞ [n]p,q[sn,p,q(f; x) −f(x)] = x 2 f′′(x), is uniformly convergent on [0,a], a > 0. proof. consider taylor’s formula on f ∈ c∗ϑ[0,∞) f(t) = f(x) + (t−x)f′x + 1 2 (t−x)2f′′(x) + p(t,x)(t−x)2, where p(t,x) is peano’s remainder, p(t,x) → 0 as t → x. now [n]p,q[sn,p,q(f; x) −f(x)] = [n]p,qf′(x)sn,p,q((t−x); x) + [n]p,q f′′(x) 2 sn,p,q((t−x)2; x) + [n]p,qsn,p,q(p(t,x)(t−x)2; x). from cauchy-schwarz inequality, sn,p,q(p(t,x)(t−x)2; x) ≤ √ sn,p,q((p2(t,x); x) √ sn,p,q((t−x)4; x) is satisfied. since p(t,x) ∈ c∗ϑ[0,∞) and p(x,x) = 0, lim n→∞ sn,p,q((p 2(t,x); x) = p2(x,x) = 0, uniformly convergent for x ∈ [0,a]. so lim n→∞ [n]p,qsn,p,q(p(t,x)(t−x)2; x) = 0 is obtained. also lim n→∞ [n]p,qsn,p,q((t−x); x) = 0. int. j. anal. appl. 19 (3) (2021) 402 and lim n→∞ [n]p,qsn,p,q((t−x)2; x) = x. hence, lim n→∞ [n]p,q[sn,p,q(f; x) −f(x)] = x 2 f′′(x). � 9. numerical examples example 9.1. we compute the absolute error (a.e.) of sn,p,q with the function f(x) = x 2 − 5x + 10 and g(x) = x3 −x + 1 for the different values of n taking x = 1 and x = 2 in table 1 and table 2, respectively. also, the absolute error can be seen graphically in the figure 5 and 6. table 1. a.e. of operators and function at x = 1 n |sn,p,qf −f| |sn,p,qg −g| 10 0.1252 0.3914 20 0.0798 0.2458 30 0.0673 0.2064 40 0.0633 0.1940 50 0.0631 0.1934 table 2. a.e. of operators and function at x = 2 n |sn,p,qf −f| |sn,p,qg −g| 10 0.2505 1.5342 20 0.1596 0.9704 30 0.1346 0.8165 40 0.1267 0.7682 50 0.1263 0.7657 int. j. anal. appl. 19 (3) (2021) 403 figure 5. absolute error of sn,p,q at x = 1. figure 6. absolute error of sn,p,q at x = 2. 10. acknowledgements the second author would like to express his gratitude to king khalid university, abha, saudi arabia, for providing administrative and technical support. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.l. cieśliński, improved q-exponential and q-trigonometric functions, appl. math. lett. 24 (2011), 2110–2114. [2] m. mursaleen, k.j. ansari and a. khan, on (p; q)-analogue of bernstein operators, appl. math. comput. 266 (2015), 874-882. [erratum: appl. math. comput. 278 (2016) 70-71]. [3] m. mursaleen, k.j. ansari and a. khan, some approximation results by (p; q)-analogue of bernstein-stancu operators, appl. math. comput. 264 (2015), 392-402. [corrigendum: appl. math. comput 269 (2015) 744-746]. [4] m. mursaleen, a. alotaibi and k.j. ansari, on a kantorovich variant of (p; q)-szász-mirakjan operators, j. funct. spaces, 2016 (2016), 1035253. [5] t. acar, (p; q)-generalization of szász-mirakyan operators, math. meth. appl. sci. 39 (10) (2016), 2685-2695. [6] r. jagannathan, k.s. rao, two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, proc. int. conf. numb. theory math. phys. kumbakonam, india, 2005. [7] r.b. corcino, on p,q-binomial coefficients, integers: electron. j. comb. numb. theory. 8 (2008), #a29. [8] v. gupta, a. aral, convergence of the q-analogue of szász-beta operators. appl. math. comput. 216 (2010), 374–380. int. j. anal. appl. 19 (3) (2021) 404 [9] r.a. devore, g.g. lorent, constructive approximation, springer, berlin, 1993. [10] a.d. gadjev, c. orham, some approximation theorems via statistical convergence, rocky mt. j. math. 32(1) (2002) 129-138. [11] r.b. gandhi, deepmala, v.n. mishra, local and global results for modified szász-mirakjan operators, math. method. appl. sci., 40(7) (2017), 2491-2504. [12] m. mursaleen, aah. al-abied and a. alotaibi, on (p; q) szász-mirakyan operators and their approximation properties. j. inequal. appl. 2017 (2017), 196. 1. introduction 2. new type of (p;q)-exponential function 3. moments of sn,p,q 4. some consequences 5. rate of convergence 6. weighted approximation result 7. statistical convergence 8. voronovskaya type result 9. numerical examples 10. acknowledgements references international journal of analysis and applications volume 16, number 4 (2018), 556-568 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-556 on g-β-irresolute functions on generalized topological spaces m k ghosh∗ department of mathematics, kalyani mahavidyalaya, kalyani-741235, nadia, west bengal, india ∗corresponding author: manabghosh@gmail.com abstract. in this paper, we introduce and investigate a new kind of function namely g-β-irresolute function along with its two weak and strong forms in generalized topological spaces. several characterizations and interesting properties of these functions are discussed. 1. introduction concepts of generalized topological spaces (gts), generalized open sets and generalized continuity (= (g,g′)-continuous functions) were introduced by a. császár [8, 11, 14]. since then, several research works devoted to generalize the existing notions of topological spaces to generalized topological spaces have appeared. in [22], [23], w. k. min introduced the notions of weak (g,g′)-continuity and almost (g,g′)-continuity on generalized topological spaces. the concept of g-α-irresolute functions on generalized topological spaces was introduced by bai and zuo [4]. in 2013, bayhan et al. [7] investigated some functions between generalized topological spaces. recently, acikgoz et al. [2] also studied some functions between gts’s. on the other hand, abd el-monsef et al. [1] introduced the notions of β-open sets and β-continuity in topological spaces early in 1983. andrijevic [3] introduced the notion of semi-preopen sets which are equivalent to β-open sets. since then, β-open sets [1] played a significant role in the theory of generalized open sets in topological spaces. in [21], mahmoud and el-monsef defined and studied β-irresolute functions. received 2018-02-08; accepted 2018-04-06; published 2018-07-02. 2010 mathematics subject classification. 54a05, 54c05, 54c08. key words and phrases. generalized topology; g-β-open; g-β-regular; g-β-θ-open; g-β-irresolute; weakly g-β-irresolute; strongly g-β-irresolute. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 556 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-556 int. j. anal. appl. 16 (4) (2018) 557 t. noiri [26] studied some weak and strong forms of β-irresolute functions in 2003. this work is concerned with the extension various forms of β-irresolute functions to generalized topological spaces. 2. preliminaries a collection g of subsets of x is called a generalized topology (briefly gt) on x [11] if and only if ∅∈ g and gi ∈ g for i ∈ i 6= ∅ implies g = ⋃ i∈i gi ∈ g. a set x with a gt g on x is called a generalized topological space (gts) and is denoted by (x,g). by a space x or (x,g), we will always mean a gts. a gt g on x is called a strong gt [13] if x ∈ g. for a space (x,g), the elements of g are called g-open sets and the complements of g-open sets are called g-closed sets. for a ⊂ x, the g-closure of a, denoted by ca is the intersection of all g-closed sets containing a and the g-interior of a, denoted by ia is the union of all g-open sets contained in a. it was pointed out in [14] that each of the operations ia and ca are monotonic i.e. if a ⊂ b ⊂ x, then ia ⊂ ib and ca ⊂ cb, idempotent [16], i.e. if a ⊂ x, then i(ia) = ia and c(ca) = ca, ia is restricting [16], i.e. if a ⊂ x, then ia ⊂ a, ca is enlarging [16], i.e., if a ⊂ x, then a ⊂ ca. in a space (x,g), for a ⊂ x, x ∈ ia if and only if there exists an g-open set v containing x such that v ⊂ a and x ∈ ca if and only if v ∩a 6= ∅ for every g-open set v containing x [9]. in a space (x,g), a ⊂ x is g-open if and only if a = ia and is g-closed if and only if a = ca [8] and ca = x \ i(x \a). a subset a of a topological space is called β-open [1] if a ⊂ cl(int(cl(a))). the complement of a β-open set is called β-closed. for a subset a of a topological space (x,τ), the β-closure of a, denoted by βcl(a) is the intersection of all β-open sets containing a and the β-interior of a, denoted by βint(a) is the union of all β-open sets contained in a. in a gts (x,g), a subset a of x is said to be g-β-open (resp. g-α-open, g-preopen, g-semiopen) [14] if a ⊂ cica (resp. a ⊂ icia, a ⊂ ica, a ⊂ cia). we denote by β(gx) (resp. α(gx), π(gx), σ(gx)) the class of all g-β-open (resp. resp. g-α-open, g-preopen, g-semiopen) sets of (x,g). from [14], it is clear that, g ⊂ α(gx) ⊂ σ(gx) ⊂ β(gx), α(gx) ⊂ π(gx) ⊂ β(gx) and each of β(gx) (resp. α(gx), π(gx), σ(gx)) forms a gt on x. the complements of a g-β-open sets (resp. g-α-open, g-preopen, g-semiopen) is called g-β-closed (resp. g-α-closed, g-preclosed, g-semiclosed) set. we denote by β(gx,x), the set of all g-β-open sets of (x,g) containing x ∈ x and by βc(gx) the class of all g-β-closed sets of (x,g). for a ⊂ x, we denote by βca the intersection of all g-β-closed sets containing a and by βia the union of all g-β-open sets contained in a. 3. g-β-regular sets and g-β-θ-open sets we first state a lemma which will be used in the sequel. proofs can be checked easily and therefore omitted. int. j. anal. appl. 16 (4) (2018) 558 lemma 3.1. the following hold for a subset a of gts (x,g): (i) arbitrary union of g-β-open sets is g-β-open. (ii) arbitrary intersection of g-β-closed sets is g-β-closed. (iii) βia = a∩ cica. (iv) βca = a∪ icia. (v) x ∈ βca if a∩u 6= ∅ for every g-β-open set u of x containing x. (vi) βc(x \a) = x \βia. (vii) a is g-β-closed if and only if a = βca. (viii) βia is g-β-open and βca is g-β-closed. lemma 3.2. [22] in a gts (x,g), x is both g-semiopen and g-β-open. definition 3.1. a subset a of a space x is said to be g-β-regular if it is both g-β-open and g-β-closed. the family of all g-β-regular sets of a space x is denoted by βr(x) and those of containing a point x of x by βr(x,x). theorem 3.1. for a subset a of a gts (x,g), (i) a ∈ β(gx) if and only if βca ∈ βr(x). (ii) a ∈ βc(gx) if and only if βia ∈ βr(x). proof: (i) first suppose, a ∈ β(gx). then a ⊂ cica and therefore, βc(ca) ⊂ βc(cica) = cica ⊂ cic(βca) i.e. βca is g-β-open. since βca is g-β-open and g-β-closed, βca ∈ βr(x). next suppose, βca ∈ βr(x). then a ⊂ βca ⊂ cic(βca) ⊂ cic(ca) = cica. hence a ∈ β(gx). (ii) follows from (i) and lemma 3.1 (vi). theorem 3.2. the following are equivalent for a subset a of a gts (x,g). (i) a ∈ βr(x); (ii) a = βiβca; (iii) a = βcβia. proof: proofs of (i) ⇒ (ii) and (i) ⇒ (iii) are obvious. proofs of (ii) ⇒ (i) and (iii) ⇒ (i) follow from lemma 3.1 and theorem 3.1. definition 3.2. in a gts (x,g), a point x ∈ x is said to be in the g-β-θ-closure of a, denoted by β-θ-ca, if a∩βcv 6= ∅ for every g-β-open set v of x containing x. if β-θ-ca = a, then a is said to be g-β-θ-closed. the complement of a g-β-θ-closed set is said to be g-β-θ-open. for a subset a of x, union of all g-β-θ-open sets contained in a is said to be g-β-θ-interior of a, denoted by β-θ-ia. int. j. anal. appl. 16 (4) (2018) 559 lemma 3.3. for a subset a of a space (x,g), β-θ-ca = ∩{v : a ⊂ v and v is g-β-θ-closed } = ∩{v : a ⊂ v and v ∈ βr(x)} proof: we give a proof of the first equality, because that of the other is quite similar. suppose that, x 6∈ βθ-ca. then there exists, g-β-open set v containing x such that βcv ∩ a = ∅. therefore by theorem 3.1, x\βcv is g-β-regular and so g-β-θ-closed set containing a such that x 6∈ x\βcv . hence, x 6∈ ∩{v : a ⊂ v and v is g-β-θ-closed set}. conversely, suppose that, x 6∈ ∩{v : a ⊂ v and v is g-β-θ-closed set}. then there exist, a g-β-θ-closed set v containing a and x 6∈ v . also, there exists a u ∈ β(gx) such that x ∈ u ⊂ βcu ⊂ x \v . then we have, βcu ∩a ⊂ βcu ∩v = ∅ and so x 6∈ β-θ-ca. lemma 3.4. let a and b be any subset of a gts (x,g). then the following properties hold: (i) x ∈ β-θ-ca if and only if a∩v 6= ∅ for every v ∈ βr(x,x). (ii) if a ⊂ b then β-θ-ca ⊂ β-θ-cb. (iii) β-θ-c(β-θ-ca) = β-θ-ca. (iv) intersection of an arbitrary family of g-β-θ-closed sets in x is g-β-θ-closed in x. (v) a is g-β-θ-open if and only if for each x ∈ a, there exists v ∈ βr(x,x), such that x ∈ v ⊂ a. (vi) if a ∈ β(g) then βca = β-θ-ca. (vii) if a ∈ βr(x) then a is g-β-θ-closed. (viii) a ∈ βr(x) if and only if a is g-β-θ-open and g-β-θ-closed. proof: we give only the proofs of (iii) and (iv). others proofs are obvious. (iii) we have β-θ-ca ⊂ β-θ-c(β-θ-ca). now, if x /∈ β-θ-ca, there exits v ∈ βr(x,x) such that a∩v = ∅. since v ∈ βr(x,x), we have β-θ-ca∩v = ∅. this implies x 6∈ β-θc(β-θca) and so β-θ-c(β-θ-ca) ⊂ β-θ-ca. (iv) let aα be a g-β-θ-closed for each α ∈ ∆. then for each α ∈ ∆, we have aα = β-θ-caα. therefore βθ-c(∩α∈∆aα) ⊂ ∩α∈∆β-θ-caα=∩α∈∆aα ⊂ β-θ-c(∩α∈∆aα). hence, β-θ-c(∩α∈∆aα) = ∩α∈∆aα. therefore, ∩α∈∆aα is g-β-θ-closed. corollary 3.1. for a subset a of a gts (x,g), the following properties hold: (i) a is g-β-θ-open in x if and only if for each x ∈ a there exists v ∈ βr(x,x) such that x ∈ v ⊂ a. (ii) β-θ-ca is g-β-θ-closed and β-θ-ia is g-β-θ-open. (iii) arbitrary union of g-β-θ-open sets is g-β-θ-open. theorem 3.3. for a subset a of a gts (x,g), the following properties hold: (i) if a ∈ β(gx) then βca = β-θ-ca. (ii) a ∈ βr(x) if and only if a is g-β-θ-open and g-β-θ-closed. int. j. anal. appl. 16 (4) (2018) 560 proof: (i) let a ∈ β(gx) and x /∈ βca. then, there exists v ∈ β(gx,x) such that a∩v = ∅. now, since a ∈ β(gx) we have a∩βcv = ∅. this implies x /∈ β-θ-ca and so β-θ-ca ⊂ β-ca. also, for every subset a of x, we have β-ca ⊂ β-θ-ca. hence, βca = β-θ-ca. (ii) suppose a ∈ βr(x). then a = βca=β-θ-ca. hence a is g-β-θ-closed. again, since x \a ∈ βr(x), we get x \a is g-β-θ-closed and so a is β-θ-open. the converse part is obvious. remark 3.1. it is clear that in a gts (x,g), g-β-regular ⇒ g-β-θ-open ⇒ g-β-open. but the converses are not necessarily true. example 3.1. let x = {a,b,c,d} and g = {∅,{a},{a,b},{b,c},{a,b,c}} be a gt on x. then the subsets {a,b}, {a,b,c} and {a,b,d} of x are g-β-θ-open but not g-β-regular. example 3.2. let x = {a,b,c,d} and g = {∅,{a,c},{b,c},{a,b,c}} be a gt on x. then the subset {b,c} of x is g-β-open but not g-β-θ-open. 4. g-β-irresolute functions definition 4.1. let gx and gy be generalized topologies on x and y respectively. then a function f : (x,gx) → (y,gy ) is defined to be generalized continuous or more properly (gx,gy )-continuous [11] if f−1(v ) ∈ gx for each v ∈ gy . definition 4.2. a function f : (x,gx) → (y,gy ) is called (β,gy )-continuous [24] if f−1(v ) ∈ β(gx) for each v ∈ gy . definition 4.3. [4] a function f : (x,gx) → (y,gy ) is called g-α-irresolute if f−1(v ) is g-α-open in x for every g-α-open set v of y . definition 4.4. a function f : (x,gx) → (y,gy ) is called g-β-irresolute if the inverse image of each g-β-open set of y is g-β-open in x. definition 4.5. a function f : (x,gx) → (y,gy ) is said to be g-β-irresolute at x ∈ x if for each v ∈ β(gy ,f(x)), there exists u ∈ β(gx,x) such that f(u) ⊂ v . definition 4.6. a function f : (x,gx) → (y,gy ) is called weakly g-β-irresolute (resp. strongly g-βirresolute) if for each point x ∈ x and each g-β-open set v of y containing f(x), there exists a g-β-open set u of x containing x such that f(u) ⊂ βcv (resp. f(βcu) ⊂ v ). remark 4.1. from the above definitions we have the following implications: strongly g-β-irresolute ⇒ g-β-irresolute ⇒ weakly g-β-irresolute and g-α-irresolute ⇒ g-β-irresolute ⇒ (β,gy )-continuous. int. j. anal. appl. 16 (4) (2018) 561 we now state basic properties of a g-β-irresolute function. some results of the following theorem may be analogous to theorem 3.18 of [24] in terms of other terminologies. theorem 4.1. let f : (x,gx) → (y,gy ) be a function. then the following are equivalent: (i) f is g-β-irresolute; (ii) f−1(f) is g-β-closed in x for every g-β-closed subset f of y ; (iii) f(βca) ⊂ βc(f(a)) for every subset a of x; (iv) βc(f−1(b)) ⊂ f−1(βcb) for every subset b of y ; (v) f−1(βiv ) ⊂ βi(f−1(v )) for every subset v of y ; (vi) for every x ∈ x and for every g-β-open set v containing f(x), there exists a g-β-open set u of x containing x such that f(u) ⊂ v ; proof: (i) ⇒ (ii): obvious. (ii) ⇒ (iii): let a be any subset of x. then since f−1(βc(f(a))) is a g-β-closed set we get βca ⊂ βc(f−1(f(a))) ⊂ βc(f−1(βc(f(a)))) = f−1(βc(f(a))). hence f(βca) ⊂ βc(f(a)). (iii) ⇒ (iv): for any subset v of y , using (iii) we get f(βc(f−1(v ))) ⊂ βc(ff−1(v )) ⊂ βcv . therefore, βc(f−1(v )) ⊂ f−1f(βc(f−1(v )) ⊂ f−1(βcv ). (iv) ⇒ (v): for any subset v of y , using (iv) we get, f−1(βc(y\v )) ⊃ βc(f−1(y\v )) = βc(x\f−1(v )). now by lemma 3.1, f−1(βiv ) = f−1(y \βc(y \v )) = x\f−1(βc(y \v )) ⊂ x\βc(x\f−1(v )) = βi(f−1(v )). (v) ⇒ (i): let v be any g-β-open subset of y . then f−1(v ) = f−1(βiv ) ⊂ βi(f−1(v )) ⊂ f−1(v ). this implies f−1(v ) = βi(f−1(v )) i.e. f−1(v ) is g-β-open set of x. hence f is g-β-irresolute. (i) ⇒ (vi): let f be g-β-irresolute. also let x ∈ x and v ∈ β(gy ,f(x)). then x ∈ f−1(v ) = βi(f−1(v )). if we set u = f−1(v ), then u ∈ β(gx) and f(u) ⊂ v . hence f is g-β-irresolute for each x ∈ x. (vi) ⇒ (i): let v ∈ β(gy ) and x ∈ f−1(v ). then f(x) ∈ v . so, there exists u ∈ β(gx,x) such that f(u) ⊂ v . then x ∈ u ⊂ f−1f(u) ⊂ f−1(v ) and x ∈ u = βiu ⊂ βi(f−1(v )). therefore f−1(v ) ⊂ βi(f−1(v )) and so f−1(v ) = βi(f−1(v )). hence f is g-β-irresolute. theorem 4.2. let f : (x,gx) → (y,gy ) be a bijective function. then f is g-β-irresolute if and only if βi(f(u)) ⊂ f(βiu) for every subset u of x. proof: let v be any subset of x. then by above theorem 4.1, f−1(βi(f(v ))) ⊂ βi(f−1f(v )) = βiv . therefore, ff−1(βi(f(v ))) ⊂ f(βiv ) so βi(f(v )) ⊂ f(βiv ). conversely, let v be any g-β-open set of y . then v = βiv = βi(ff−1(v )) ⊂ f(βi(f−1(v ))) i.e. f−1(v ) ⊂ f−1f(βi(f−1(v )). since f is bijective, this implies f−1(v ) ⊂ f−1f(βi(f−1(v ))) = βi(f−1(v )) i.e f−1(v ) = βi(f−1(v )). therefore f−1(v ) is g-β-open set of x and so f is g-β-irresolute. int. j. anal. appl. 16 (4) (2018) 562 definition 4.7. [10] a gts (x,g) is called β-compact if each cover of x by g-β-open sets of x, has a finite subcover. theorem 4.3. let f : (x,gx) → (y,gy ) be a g-β-irresolute function. if (x,gx) is β-compact then so is (y,gy ). proof: let {uα : α ∈ λ} be a g-β-open cover of y . then since f is g-β-irresolute, {f−1(uα) : α ∈ λ} is a g-β-open cover of x. now, since (x,gx) is β-compact, there exists a finite subcover, say {f−1(uα1 ),f−1(uα2 ), ...,f−1(uαn )} such that {ff−1(uα1 ),ff−1(uα2 ), ...,ff−1(uαn )}⊂{uα1,uα2, ...,uαn} is a finite subcover of (y,gy ). hence (y,gy ) is β-compact. 5. properties of weakly g-β-irresolute functions theorem 5.1. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is weakly g-β-irresolute; (ii) f−1(v ) ⊂ βi(f−1(βcv )) for every g-β-open set v of y ; (iii) βc(f−1(v )) ⊂ f−1(βcv ) for every g-β-open set v of y . proof: (i) ⇒ (ii): let v be any g-β-open set of y and x ∈ f−1(v ). then f(u) ⊂ βcv for some u ∈ β(gx,x). this implies u ⊂ f−1(βcv ) and x ∈ u ⊂ βi(f−1(βcv )). hence f−1(v ) ⊂ βi(f−1(βcv )). (ii) ⇒ (iii): let v ∈ β(gy ) and x 6∈ f−1(βcv ). then f(x) /∈ βcv . so, there exists w ∈ β(gy ,f(x)) such that v ∩w = ∅. now, since v is g-β-open, we have v ∩βcw = ∅ and so f−1(v ) ∩βi(f−1(βcw)) = ∅. as x ∈ f−1(w) ⊂ βi(f−1(βcw)) ∈ β(gx), we have x /∈ βc(f−1(v )). hence, βc(f−1(v )) ⊂ f−1(βcv ). (iii) ⇒ (i): for x ∈ x, suppose v ∈ β(gy ,f(x)). then by lemma 3.1, βcv ∈ βr(y ) and x /∈ f−1(βc(y\βcv ). since y \βcv is a g-β-open set of y , we get x /∈ βc(f−1(y \βcv )). so there exists u ∈ β(gx,x) such that f−1(y \βcv )∩u = ∅. this implies f(u)∩(y \βcv ) = ∅ and so f(u) ⊂ βcv i.e. f is weakly g-β-irresolute. theorem 5.2. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is weakly g-β-irresolute; (ii) βc(f−1(v )) ⊂ f−1(β-θ-cv ) for every subset v of y ; (iii) f(βcu) ⊂ β-θ-c(f(u)) for every subset u of x; (iv) f−1(f) ∈ βc(gx) for every g-β-θ-closed set f of y ; (v) f−1(g) ∈ β(gx) for every g-β-θ-open set g of y . proof: (i) ⇒ (ii): let v be any subset of y and x /∈ f−1(β-θ-cv ). then f(x) /∈ β-θ-cv and so there exists q ∈ β(gy ,f(x)) such that v ∩βcq = ∅. since f is weakly g-β-irresolute, there exists p ∈ β(gx,x) such that f(p) ⊂ βcq. hence f(p) ∩ v = ∅ and so p ∩ f−1(v ) = ∅. therefore, x /∈ βc(f−1(v )) and consequently βc(f−1(v )) ⊂ f−1(β-θ-cv ). int. j. anal. appl. 16 (4) (2018) 563 (ii) ⇒ (iii) : for any subset u of x, we have βcu ⊂ βc(f−1(f(u))) ⊂ f−1(β-θ-f(u)) and so f(βcu) ⊂ β-θc(f(u)). (iii) ⇒ (iv): for any g-β-θ-closed set f of y , f(βc(f−1(f))) ⊂ β-θ-c(f(f−1(f))) ⊂ β-θ-cf = f. this implies βc(f−1(f)) ⊂ f−1(f) and hence βc(f−1(f)) = f−1(f). therefore f−1(f) ∈ βc(gx). (iv) ⇒ (v): obvious. (v) ⇒ (i): for any x ∈ x, let q ∈ β(gy ,f(x)). then by theorem 3.1 (i) and theorem 3.3 (ii), we get βcq is g-β-θ-open in y . if we set p = f−1(βcq), then p ∈ β(gx,x) and f(p) ⊂ βcq. hence f is weakly g-β-irresolute. theorem 5.3. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is weakly g-β-irresolute; (ii) for each x ∈ x and v ∈ β(gy ,f(x)), there exists u ∈ β(gx,x) such that f(βcu) ⊂ βcv ; (iii) f−1(r) ∈ βr(x) for every r ∈ βr(y ). proof: (i) ⇒ (ii): for any x ∈ x, let v ∈ β(gy ,f(x)). then by theorem 3.1 and 3.3, βcv is g-β-θ-open and g-β-θ-closed in y . if we set u = f−1(βcv ), then by theorem 5.2, we have u ∈ βr(x) and so u ∈ β(gx,x). also we have f(βcu) ⊂ βcv . (ii) ⇒ (iii): let r ∈ βr(y ) and x ∈ f−1(r). then we have f(x) ∈ r and there exists u ∈ β(gx,x) such that f(βcu) ⊂ r. this implies x ∈ u ⊂ βcu ⊂ f−1(r) and so f−1(r) ∈ β(gx). again since y \r ∈ βr(y ) f−1(y \r) = x \f−1(r) ∈ β(gx). thus f−1(r) ∈ βc(gx) and consequently f−1(r) ∈ βr(x). (iii) ⇒ (i): for any x ∈ x, suppose v ∈ β(gy ,f(x)). then by theorem 3.1, we get βcv ∈ βr(y,f(x)) and f−1(βcv ) ∈ βr(x,x). if we take, u = f−1(βcv ), then u ∈ β(gx,x) and f(u) ⊂ βcv . hence f is weakly g-β-irresolute. theorem 5.4. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is weakly g-β-irresolute; (ii) f−1(v ) ⊂ β-θ-i(f−1(β-θ-cv )) for every g-β-open set v of y ; (iii) β-θ-c(f−1(v )) ⊂ f−1(β-θ-cv ) for every g-β-open set v of y . proof: the proof is quite similar to the proof of theorem 5.1, if we observe that every g-β-closed set is g-β-θ-closed. theorem 5.5. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is weakly g-β-irresolute; (ii) β-θ-c(f−1(v )) ⊂ f−1(β-θ-cv ) for every subset v of y ; (iii) f(β-θ-cu) ⊂ β-θ-c(f(u)) for every subset u of x; int. j. anal. appl. 16 (4) (2018) 564 (iv) f−1(f) is g-β-θ-closed in x for every g-β-θ-closed set f of y ; (v) f−1(g) is g-β-θ-open set in x for every g-β-θ-open set g of y . proof: the proof is quite similar to proof of theorem 5.2 and hence omitted. definition 5.1. a gts (x,g) is said to be g-β-regular if for each f ∈ βc(gx) and each x /∈ f , there exist disjoint g-β-open sets u and v such that x ∈ u and f ⊂ v . lemma 5.1. the following properties are equivalent in a gts (x,g): (i) x is g-β-regular; (ii) for each u ∈ β(gx) and each x ∈ u, there exists v ∈ β(gx) such that x ∈ v ⊂ βcv ⊂ u; (iii) for each u ∈ β(gx) and each x ∈ u, there exists v ∈ βr(x) such that x ∈ v ⊂ u proof: follows from theorem 3.1. theorem 5.6. a function f : (x,gx) → (y,gy ) is g-β-irresolute if and only if it is weakly g-β-irresolute and (y,gy ) is g-β-regular. proof: suppose that f is weakly g-β-irresolute. let v be any g-β-open set of y and x ∈ f−1(v ), then f(x) ∈ v . now, since y is g-β-regular, by above lemma 5.1, there exists w ∈ β(gx) such that f(x) ∈ w ⊂ βcw ⊂ v . again since f is weakly g-β-irresolute, there exists u ∈ β(gx,x) such that f(u) ⊂ βcw. this implies x ∈ u ⊂ f−1(v ) and f−1(v ) ∈ β(gx). hence f is g-β-irresolute. the converse part is obvious. proposition 5.1. [28] let (x,gx) and (y,gy ) be generalized topological spaces and let u = {u × v : u ∈ gx,v ∈ gy}. then u generates a generalized topology gx×y on x ×y , called the generalized product topology on x ×y , i.e. gx×y = {all possible union of members of u} proposition 5.2. [28] let (x,gx) and (y,gy ) be generalized topological spaces, gx×y be the generalized topology on x ×y , a ⊂ x, b ⊂ y and k ⊂ x ×y . then the following hold: (i) k is gx×y -open if and only if for each (x,y) ∈ k, there exist ux ∈ gx and vy ∈ gy such that (x,y) ∈ ux ×vy ⊂ k. (ii) c(a×b) = ca× cb. (iii) i(a×b) = ia× ib. proposition 5.3. let (x,gx) and (y,gy ) be generalized topological spaces, gx×y be the generalized topology on x ×y , a ⊂ x, b ⊂ y and k ⊂ x ×y . then the following hold: (ii) βc(a×b) = βca×βcb. (iii) βi(a×b) = βia×βib. int. j. anal. appl. 16 (4) (2018) 565 theorem 5.7. a function f : (x,gx) → (y,gy ) is weakly g-β-irresolute if the graph function, defined by g(f) = (x,f(x)) for each x ∈ x, is weakly g-β-irresolute. proof: let x ∈ x and v ∈ β(gx). then using lemma 3.2, we get x×v is g-β-open set of x×y containing g(f). since g is weakly g-β-irresolute, there exists u ∈ β(gx,x) such that g(u) ⊂ βc(x ×v ) ⊂ x ×βcv . hence f(u) ⊂ βcv i.e. f is weakly g-β-irresolute. definition 5.2. [27] a gts (x,g) is said to be g-β-t2 if and only if for each pair of distinct points x,y ∈ x, there exits disjoint g-β-open sets containing x and y respectively. lemma 5.2. a gts (x,g) is g-β-t2 if and only if for each pair of distinct points x,y ∈ x, there exist u ∈ β(gx,x) and v ∈ β(gx,y) such that βcu ∩βcv = ∅. proof: follows from theorem 3.1. theorem 5.8. if y is a g-β-t2 space and f : (x,gx) → (y,gy ) is a weakly g-β-irresolute injection, then x is g-β-t2. proof: let x,y be any two distinct points of x, then since f is an injection, we have f(x) 6= f(y). now y being g-β-t2, by lemma 5.1, there exists u ∈ β(gy ,f(x)) and v ∈ β(gy ,f(y)) such that βcu ∩βcv = ∅. again since f is weakly g-β-irresolute, there exist p ∈ β(gx,x) and q ∈ β(gx,y) such that f(p) ⊂ βcu and f(q) ⊂ βcv . this implies p ∩q = ∅. therefore x is g-β-t2. definition 5.3. a function f : (x,gx) → (y,gy ) is said to have strongly g-β-closed graph if for each (x,y) ∈ (x ×y ) \g(f), there exist u ∈ β(gx,x) and v ∈ β(gy ,y) such that (βcu ×βcv ) ∩g(f) = ∅. theorem 5.9. if a function f : (x,gx) → (y,gy ) is weakly g-β-irresolute, where y is g-β-t2, then g(f) is strongly g-β-closed. proof: let (x×y) ∈ (x×y )\g(f). then since y 6= f(x), by lemma 5.1, there exists, u ∈ β(gx,f(x)) and v ∈ β(gy ,y) such that βcu ∩βcv = ∅. again since f is weakly g-β-irresolute, by theorem 5.3, there exists w ∈ β(gx,x) such that f(βcw) ⊂ βcu. this implies f(βcw) ∩βcv = ∅ and so (βcw ×βcv ) ∩g(f) = ∅. hence g(f) is strongly g-β-closed in x ×y . theorem 5.10. if a function f : (x,gx) → (y,gy ) is weakly g-β-irresolute injection and g(f) is strongly g-β-closed, then x is g-β-t2. proof: let x,y ∈ x and x 6= y. since f is an injection f(x) 6= f(y) and (x,f(y)) /∈ g(f). again since g(f) is strongly g-β-closed, there exists u ∈ β(gx,x) and v ∈ β(gy ,f(y)) such that f(βcu) ∩βcv = ∅. also, f being weakly g-β-irresolute, there exists w ∈ β(gx,y) such that f(w) ⊂ βcv . hence f(βcu) ∩f(w) = ∅ and so u ∩w = ∅. therefore x is g-β-t2. int. j. anal. appl. 16 (4) (2018) 566 definition 5.4. a gts (x,g) is said to be connected [29] if there are no nonempty disjoint sets a,b ∈ g such that a∪b = x. a gts (x,g) is said to be β-connected [29] if (x,β(gx)) is connected. theorem 5.11. if a function f : (x,gx) → (y,gy ) is a weakly g-β-irresolute surjection and x is βconnected, then y is β-connected. proof: if possible, suppose that y is not β-connected. then there exists nonempty disjoint sets a,b ∈ β(gy ) such that y = a ∪ b. this implies a,b ∈ βr(y ) by lemma 3.2. now, since f is weakly g-β-irresolute, by lemma 3.2 and theorem 5.3, we get f−1(a),f−1(b) ∈ βr(x). moreover f being a surjection, x = f−1(a) ∪f−1(b) where f−1(a) and f−1(b) are disjoint nonempty sets. therefore x is not β-connected. 6. properties of strongly g-β-irresolute functions theorem 6.1. for a function f : (x,gx) → (y,gy ), the following are equivalent: (i) f is strongly g-β-irresolute; (ii) for each x ∈ x, and each v ∈ β(gy ,f(x)), there exists u ∈ β(gx,x) such that f(β-θ-cu) ⊂ v ; (iii) for each x ∈ x and each v ∈ β(gy ,f(x)), there exists u ∈ βr(x,x) such that f(u) ⊂ v ; (iv) for each x ∈ x and for each v ∈ β(gy ,f(x)), there exist an g-β-open set u in x containing x such that f(u) ⊂ v ; (v) f−1(g) is g-β-θ-open in x for every g ∈ β(gy ); (vi) f−1(f) is g-β-θ-closed in x for every f ∈ βc(gy ); (vii) f(β-θ-ca) ⊂ βc(f(a)) for every subset a of x; (viii) β-θ-c(f−1(b)) ⊂ f−1(βcb) for every subset b of y . proof: we first observe that (i) to (iv) are equivalent from theorem 3.1 and theorem 3.3. (iv) ⇒ (v): let g ∈ β(gy ) and x ∈ f−1(g). then we have f(x) ∈ g and there exists a g-β-θ-open set u in x containing x such that f(u) ⊂ g. therefore, x ∈ u ⊂ f−1(g). hence by using corollary 3.1, f−1(g) is g-β-θ-open in x. (v) ⇒ (vi): obvious. (vi) ⇒ (vii): let a be any subset of x. then f−1(βc(f(a))) is g-β-θ-closed in x and so we get β-θ-ca ⊂ βθ-c(f−1(f(a))) ⊂ β-θ-c(f−1(βc(f(a)))) = f−1(βc(f(a))). hence f(β-θ-ca) ⊂ β-c(f(a)). (vii) ⇒ (viii): let b be any subset of y . then we have f(β-θ-c(f−1(b))) ⊂ βc(f(f−1(b))) ⊂ βcb. hence β-θ-c(f−1(b)) ⊂ f−1(βcb). (viii) ⇒ (i): let x ∈ x and v ∈ β(gy ,f(x)). since y \ v ∈ βc(gy ), we have β-θ-c(f−1(y \ v )) ⊂ f−1(βc(y \v )) = f−1(y \v ). this implies f−1(y \v ) is g-β-θ-closed in x and so f−1(v ) is a β-θ-open set containing x. then there exists u ∈ β(gx,x) such that βcu ⊂ f−1(v ) i.e. f(βcu) ⊂ v . therefore f is strongly g-β-irresolute. int. j. anal. appl. 16 (4) (2018) 567 theorem 6.2. a g-β-irresolute function f : (x,gx) → (y,gy ) is strongly g-β-irresolute if and only if x is a g-β-regular space. proof: first let, every g-β-irresolute function be strongly g-β-irresolute. the identity function id : (x,gx) → (x,gx) is g-β-irresolute and therefore strongly g-β-irresolute. therefore, for any p ∈ β(gx) and any point x = id(x) ∈ p , there exists q ∈ β(gx,x) such that id(βcq) ⊂ p . this implies x ∈ q ⊂ βcq ⊂ p . hence by lemma 5.1, x is g-β-regular. conversely, let f : (x,gx) → (y,gy ) be g-β-irresolute and x be g-β-regular. then for any x ∈ x and any q ∈ β(gy ,f(x)), we get f−1(q) ∈ β(gx,x). now, since x is g-β-regular, there exists p ∈ β(gx,x) such that x ∈ p ⊂ βcp ⊂ f−1(q) i.e. f(βcp) ⊂ q. hence f is strongly g-β-irresolute. corollary 6.1. let x be a g-β-regular space. then a function f : (x,gx) → (y,gy ) is strongly g-βirresolute if and only if it is g-β-irresolute. theorem 6.3. let f : (x,gx) → (y,gy ) be a function and g(f) : x → x ×y be the graph of f. if g(f) is strongly g-β-irresolute, then f is strongly g-β-irresolute and x is g-β-regular. proof: suppose g(f) is strongly g-β-irresolute. to show, f is strongly g-β-irresolute, let x ∈ x and q ∈ β(gy ,f(x)). now by lemma 3.2, we have x × q is a g-β-open set of x × y containing g(f). since g(f) is strongly g-β-irresolute, there exists p ∈ β(gx,x) such that g(βcp) ⊂ x × q. this implies f(βcp) ⊂ q and so f is strongly g-β-irresolute. to show x is g-β-regular, let p ∈ β(gx,x). then since g(f) ∈ p ×y , using lemma 3.2 we get, p ×y is g-β-open set in x ×y . hence there exists s ∈ β(gx,x) such that g(βcs) ⊂ p ×y . therefore we obtain, x ∈ s ⊂ βcs ⊂ p . so by lemma 5.1, x is g-β-regular. references [1] m. e. abd el monsef, s. n. el-deeb and r. a. mahmoud, β-open sets and β-continuous mappings, bull. fac. sci. assiut univ., 12 (1) (1983), 77-90. [2] a. acikgoz, n. a. tas and m. s. sarsak, contra g-αand g-β-preirressloute functions on gts’s, math. sci., 9 (2015), 79-86. [3] d. andrijević, semi-preopen sets, mat. vesnik, 38 (1986), 24-32. [4] s. z. bai and y. p. zuo, on g-α-irresolute functions, acta math. hungar., 130 (4) (2011), 382-389. [5] c. k. basu and m. k. ghosh, β-closed spaces and β-θ-subclosed graphs, european jour. pure appl. math., 1 (2008), 40-50. [6] c. k. basu and m. k. ghosh, locally β-closed spaces, european jour. pure appl. math., 2 (1)(2009), 85-96. [7] s. bayhan, a. kanibir and i. l. reilly, on functions between generalized topological spaces, appl. gen. topology, 14 (2)(2013), 195-203. [8] á. császár, generalized open sets, acta math. hungar., 75 (1-2) (1997), 65-87. [9] á. császár, on the γ-interior and γ-closure of set, acta math. hungar., 80 (1-2) (1998), 89-93. [10] á. császár, γ-compact spaces, acta math. hungar., 87 (2000), 99-107. int. j. anal. appl. 16 (4) (2018) 568 [11] á. császár, generalized topology, generalized continuity, acta math. hungar., 96 (4) (2002), 351-357. [12] á. császár, separation axioms for generalized topologies, acta math. hungar., 104 (1-2) (2004), 63-69. [13] á. császár, extremally disconnected generalized topologies, annales univ. budapest, section math, 17 (2004), 151-161. [14] á. császár, generalized open sets in generalized topologies, acta math. hungar., 106 (1-2) (2005), 53-66. [15] á. császár, further remarks on the formula for γ-interior, acta math. hungar., 113(4) (2006), 325-328. [16] á. császár, remarks on quasi topologies, acta math. hungar., 119 (2008), 197-200. [17] á. császár, δand θ-modifications of generalized topologies, acta math. hungar., 120 (3) (2008), 275-279. [18] j. dugundji, topology, allyn and bacon, boston, mass, (1966). [19] r. engelking, general topology, second edition, sigma series in pure mathematics, 6, heldermann verlag, berlin, (1989). [20] m. k. ghosh and c. k. basu, generalized connectedness on generalized topologies, jour. adv. research in appl. math., 6(3) (2014), 23-34. [21] r. a. mahmoud and m. e. abd el-monsef, β-irresolute and β-topological invariant, proc. pakistan acad. sci., 27 (1990), 285-296. [22] w. k. min, weak continuity on generalized toological spaces, acta math. hungar., 124 (1-2) (2009), 73-81. [23] w. k. min, almost continuity on generalized toological spaces, acta math. hungar., 125 (1-2) (2009), 121-125. [24] w. k. min, generalized continuous functions defined by generalized open sets on generalized toological spaces, acta math. hungar., 128 (4) (2010), 299-306. [25] t. noiri, unified characterizations for modifications of r0 and r1 topological spaces, rend. circ. mat. palermo, 55 (2) (2006), 29-42. [26] t. noiri, weak and strong forms of β-irresolute functions, acta math. hungar., 99(4) (2003), 315–328. [27] m. s. sarsak, weak separation axioms in generalized topological spaces, acta math. hungar., 131 (1-2) (2011), 110-121. [28] m. s. sarsak, on µ-compact sets in µ-spaces, quest. answers gen. topology, 31 (2013), 49-57. [29] r. x. shen, a note on generalized connectedness, acta. math. hungar., 122 (3) (2009), 231-235. 1. introduction 2. preliminaries 3. g–regular sets and g—open sets 4. g–irresolute functions 5. properties of weakly g–irresolute functions 6. properties of strongly g–irresolute functions references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 10-19 http://www.etamaths.com fuzzy stability of generalized square root functional equation in several variables: a fixed point approach k. ravi1,∗ and b.v. senthil kumar2 abstract. in this paper, we investigate the generalized hyers-ulam stability of the generalized square root functional equation in several variables in fuzzy banach spaces, by applying the fixed point method. 1. introduction in the last forty years, fuzzy theory has gained paramount importance and validty on the mathematical scenario by facilitating to focus an ardent attention on multifarious avenues of development in the theory of fuzzy sets to explore the fuzzy analogues of the classical set theory. in the effulgent light of the authentic investigations executed in this branch, the fuzzy sets are being tapped to augment a wide range of applications in science and engineering with platonic dimensions. various mathematical visions, viewed in different perspectives, have triggered scores of scholars to come out with different definitions of fuzzy norms on a vector space. for example, a.k. katsaras [26] had accomplished a detailed survey to define a fuzzy norm on a vector space to help to construct a fuzzy vector topological structure. in 1991, r. biswas [6] defined and studied fuzzy inner product spaces in linear space. in 1992, c. felbin [18] introduced an alternative definition of a fuzzy norm on a linear topological structures of a fuzzy normed linear spaces. similarly, t. bag and s.k. samanta [4], gliding along the mathematical track of s.c. cheng and j.m. mordeson [12], proved that the corresponding fuzzy metric of a fuzzy norm would be the same as that of the metric executed by i. kramosil and j. michalek [28]. they had initiated a decomposition theorem of a fuzzy norm into a family of crisp norms by undertaking an analytical investigation of some of the properties of fuzzy normed spaces. an inquisitive question that was given a serious thought by s.m. ulam [45] concerning the stability of group homomorphisms gave rise to the stability problem of functional equations. the laborious intellectual strivings of d.h. hyers [23] did not go in vain because he was the first to come out with a partial answer to solve the question posed by ulam on banach spaces. in course of time, the theorem formulated by hyers was generalized by t. aoki [2] for additive mappings and by th.m. rassias [43] for linear mappings by taking into consideration an unbounded 2010 mathematics subject classification. 46s40, 39b72, 39b52, 46s50, 26e50. key words and phrases. fuzzy normed space, fixed point, generalized square root functional equation, generalized hyers-ulam stability. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 10 fuzzy stability of generalized square root functional equation 11 cauchy difference. the findings of th.m. rassias have exercised a delectable influence on the development of what is addressed as the generalized hyers-ulam stability of hyersulam-rassias stability of functional equations. a generalized and modified form of the theorem evolved by th.m. rassias was advocated by p. gavruta [21] who replaced the unbounded cauchy difference by driving into study a general control function within the viable approach designed by th.m. rassias. a further research materialized by f. skof [44] found solution to hyers-ulam-rassias stability problem for quadratic functional equation (1) f(x + y) + f(x−y) = 2f(x) + 2f(y) for a class of functions f : a → b, where a is a normed space and b is a banach space. the stability problems of several functional equations have been extensively investigated by a number of scholars, possed with creative thinking and critial dissent who have arrived at interesting results (see [3], [10], [11], [17], [22], [24], [27], [42]). in 1996, g. isac and th.m. rassias [25] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. by using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [8], [9], [35], [36], [39]). functional equations find a lot of application in information theory, information science, measure of information, coding theory, fuzzy system models, economics, social sciences and physics. the paper presenters have made use of some basic concept concerning fuzzy normed spaces and some fundamental results in fixed point theory. let x be a real linear space. a function n : x × r → [0, 1] is said to be a fuzzy norm on x if for all x,y ∈ x and all u,v ∈ r : (n1) n(x,u) = 0 for u ≤ 0 (n2) x = 0 if and only if n(x,u) = 1 for all u > 0 (n3) n(ux,v) = n ( x, v|u| ) if u 6= 0 (n4) n(x + y,u + v) ≥ min{n(x,u),n(y,v)} (n5) n(x,.) is non-decreasing function on r and lim t→∞ n(x,u) = 1 (n6) for x 6= 0, n(x,.) is (uppersemi) continuous on r. the pair (x,n) is called a fuzzy normed linear space. one may regard n(x,u) as the truth value of the statement the norm of x is less than or equal to the real number u. definition 1.1. let (x,n) be a fuzzy normed linear space. let {xn} be a sequence in x. then {xn} is said to be convergent if there exists x ∈ x such that lim n→∞ n(xn−x,u) = 1 for all u > 0. in this case, x is called the limit of the sequence {xn} and we denote it by nlim n→∞ xn = x. definition 1.2. a sequence {xn} in x is called cauchy if for each � > 0 and each u > 0, there exists n0 ∈ n such that for all n ≥ n0 and all p > 0, we have n(xn+p −xn,u) > 1 − �. it is known that every convergent sequence in fuzzy normed space is cauchy. if each cauchy sequence is convergent, then the fuzzy norm is said to be complete 12 ravi and kumar and the fuzzy normed space is called a fuzzy banach space. we say that a mapping f : x → y between fuzzy normed linear spaces x and y is continuous at a point x0 ∈ x if for each sequence {xn} converging to x0 in x, then the sequence {f(xn)} converges to f(x0). if f : x → y is continuous at each x ∈ x, then f : x → y is said to be continuous on x. let x be a set. a function d : x ×x → [0,∞] is called a generalized metric on x if d satisfies (1) d(x,y) = 0 if and only if x = y; (2) d(x,y) = d(y,x) for all x,y ∈ x; (3) d(x,z) ≤ d(x,y) + d(y,z) for all x,y,z ∈ x. theorem 1.3. let (x,d) be a complete generalized metric space and let σ : x → x be a strictly contractive mapping with lipschitz constant l < 1. then for each given element x ∈ x, either d(σnx,σn+1x) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that (1) d(σnx,σn+1x) < ∞ for all n ≥ n0; (2) the sequence {σnx} converges to a fixed point y∗ of σ; (3) y∗ is the unique fixed point of σ in the set y = {y ∈ x/d(σn0x,y) < ∞}; (4) d(y,y∗) ≤ 1 1−ld(y,σy) for all y ∈ y . k. ravi and b.v. senthil kumar[41] introduced the generalized square root functional equation (or gsrf equation) in several variables of the form (2) s   p∑ i=1 ρixi + 2 p−1∑ i=1 p∑ j=i+1 √ ρiρjxixj   = p∑ i=1 √ ρis(xi) for arbitrary but fixed real numbers (ρi,ρ2, . . . ,ρp) 6= (0, 0, . . . , 0), so that 0 < ρ = √ ρ1 + √ ρ2 + · · · + √ ρp = ∑p i=1 √ ρi 6= 1 and s : x → r with x as space of non-negative real numbers and investigated generalized hyers-ulam stability of equation (2). it is easy to verify that the function f : x → r such that f(x) = √ x is a solution of the functional equation (2). in this paper, we will show the generalized hyers-ulam stability of the equation (2) on fuzzy normed spaces using fixed point method. throughout this paper, let us assume that x be space of non-negative real numbers and y be a fuzzy normed linear space. for the sake of convenience, let us define dρs(x1,x2, . . . ,xp) = s   p∑ i=1 ρixi + 2 p−1∑ i=1 p∑ j=i+1 √ ρiρjxixj  − p∑ i=1 √ ρis(xi) for all x1,x2, . . . ,xp ∈ x and p ∈ n−{1}. 2. generalized hyers-ulam stability of the functional eqution (2) in fuzzy normed spaces theorem 2.1. let ϕ : xp → r be a function such that there exists an l < 1 with ϕ(ρ2x1,ρ 2x2, . . . ,ρ 2xp) ≤ ρlϕ(x1,x2, . . . ,xp) fuzzy stability of generalized square root functional equation 13 for all x1,x2, . . . ,xp ∈ x. let f : x → r be a mapping satisfying (1) n (dρf(x1,x2, . . . ,xp), t) ≥ t t + ϕ(x1,x2, . . . ,xp) for all x1,x2, . . . ,xp ∈ x and all t > 0, where 0 < ρ = ∑p i=1 √ ρi < 1. then s(x) = nlim n→∞ ρ−nf ( ρ2nx ) exists for each x ∈ x and defines a square root mapping s : x → y such that (2) n(f(x) −s(x), t) ≥ (1 −l)t (1 −l)t + lϕ(x,x,. . . ,x) for all x ∈ x and all t > 0. proof. taking xi as x for 1 ≤ i ≤ p in (1), we get (3) n ( f(ρ2x) −ρf(x) ) ≥ t t + ϕ(x,x,. . . ,x) for all x ∈ x. consider the set s = {g : x → y/g is a function} and introduce the generalized metric d on s as follows: d(g,h) = inf{c ∈ r+ : n(g(x) −h(x),ct) ≥ t t + ϕ(x,x,. . . ,x) ,∀x ∈ x,∀t > 0}, where, as usual, inf φ = +∞. it is easy to show that (s,d) is complete. (see the proof of lemma 2.1 of [30]). define a mapping σ : s → s by σh(x) = 1 ρ h ( ρ2x ) (x ∈ x) let g,h ∈ s be given such that d(g,h) = �. then n (g(x) −h(x),�t) ≥ t t + ϕ(x,x,. . . ,x) for all x ∈ x and all t > 0. hence n(σg(x) −σh(x),l�t) = n ( 1 ρ g(ρ2x) − 1 ρ h(ρ2x),l�t ) = n ( g(ρ2x) −h(ρ2x),ρl�t ) ≥ ρlt ρlt + ϕ(ρ2x,ρ2x,. . . ,ρ2x) ≥ ρlt ρlt + ρlϕ(x,x,. . . ,x) ≥ t t + ϕ(x,x,. . . ,x) 14 ravi and kumar for all x ∈ x and all t > 0. so d(g,h) = � implies that d(σg,σh) ≤ l�. this means that d(σg,σh) ≤ ld(g,h) for all g,h ∈ s. it follows from (3) that n ( 1 ρ f(ρ2x) −f(x), lt ρ ) ≥ t t + ϕ(x,x,. . . ,x) for all x ∈ x and all t > 0. so d(σf,f) ≤ l ρ . by theorem 1.3, there exists a mapping s : x → y satisfying the following: (1) s is a fixed point of σ, ie., (4) s(ρ2x) = ρs(x) for all x ∈ x. the mapping s is a unique fixed point of σ in the set µ = {g ∈ s : d(f,g) < ∞}. this implies that s is a unique mapping satisfying (4) such that there exists a c ∈ (0,∞) satisfying n(f(x) −s(x),ct) ≥ t t + ϕ(x,x,. . . ,x) for all x ∈ x. (2) d(σnf,s) → 0 as n →∞. this implies the equality nlim n→∞ 1 ρn f(ρ2nx) = s(x) for all x ∈ x. (3) d(f,s) ≤ 1 1−ld(σf,f), which implies the inequality d(f,s) ≤ l ρ−ρl . this implies that the inequality (2) holds. by (1), n ( 1 ρn dρf(ρ 2nx1,ρ 2nx2, . . . ,ρ 2nxp), t ρn ) ≥ t t + ϕ(ρ2nx1,ρ2nx2, . . . ,ρ2nxp) for all x1,x2, . . . ,xp ∈ x, all t > 0 and all n ∈ n. so n ( 1 ρn dρf(ρ 2nx1,ρ 2nx2, . . . ,ρ 2nxp), t ) ≥ ρnt ρnt + lnρnϕ(x1,x2, . . . ,xp) for all x1,x2, . . . ,xp ∈ x, all t > 0 and all n ∈ n. since lim n→∞ ρnt ρnt + lnρnϕ(x1,x2, . . . ,xp) = 1 fuzzy stability of generalized square root functional equation 15 for all x1,x2, . . . ,xp ∈ x, all t > 0, n (dρs(x1,x2, . . . ,xp), t) = 1 for all x1,x2, . . . ,xp ∈ x, all t > 0. thus the mapping s : xp → y is square root as desired. � theorem 2.2. let ϕ : xp → y be a function such that there exists an l < 1 with ϕ ( x1 ρ2 , x2 ρ2 , . . . , xp ρ2 ) ≤ l ρ ϕ(x1,x2, . . . ,xp) for all x1,x2, . . . ,xp ∈ x. let f : x → y be a mapping satisfying (5) n (dρf(x1,x2, . . . ,xp), t) ≥ t t + ϕ(x1,x2, . . . ,xp) for all x1,x2, . . . ,xp ∈ x and all t > 0, where 0 < ρ = ∑p i=1 √ ρi > 1. then s(x) = nlim n→∞ ρnf ( ρ−2nx ) exists for each x ∈ x and defines a square root mapping s : x → y such that (6) n(f(x) −s(x), t) ≥ (1 −l)t (1 −l)t + lϕ(x,x,. . . ,x) for all x ∈ x and all t > 0. proof. taking xi as x ρ2 for 1 ≤ i ≤ p in (5) and proceeding further using similar arguments as in theorem 2.1, the proof is complete. � corollary 2.3. let c1 ≥ 0 and α be real numbers with α > 12 or α < 1 2 . let f : xp → y be a mapping satisfying n (dρf(x1,x2, . . . ,xp), t) ≥ t t + c1 ( ∑p i=1 |xi|α) for all x1,x2, . . . ,xp ∈ x and all t > 0. then there exists a unique square mapping s : x → y such that n(f(x) −s(x), t) ≥   (ρα−ρ 1 2 )t (ρα−ρ 1 2 )t+ρ 1 2 pc1|x|α for α > 1 2 and 0 < ρ = ∑p i=1 √ ρi < 1 (ρ 1 2 −ρα)t (ρ 1 2 −ρα)t+ραpc1|x|α for α < 1 2 and 0 < ρ = ∑p i=1 √ ρi > 1 for all x ∈ x and all t > 0. proof. by taking ϕ(x1,x2, . . . ,xp) = c1 ( ∑p i=1 |xi| α) for all x1,x2, . . . ,xp ∈ x in theorem 2.1 and theorem 2.2, and choosing respectively l = ρ 1 2 −α and l = ρα− 1 2 , we get the deisred result. � 16 ravi and kumar corollary 2.4. let c2 ≥ 0 and α be real numbers with α > 12 or α < 1 2 . let f : xp → y be a mapping satisfying (7) n (dρf(x1,x2, . . . ,xp), t) ≥ t t + c2 (∏p i=1 |xi| α p ) for all x1,x2, . . . ,xp ∈ x and all t > 0. then there exists a unique square root mapping s : x → y such that n(f(x) −s(x), t) ≥   (ρα−ρ 1 2 )t (ρα−ρ 1 2 )t+ρ 1 2 c2|x|α for α > 1 2 and 0 < ρ = ∑p i=1 √ ρi < 1 (ρ 1 2 −ρα)t (ρ 1 2 −ρα)t+ραc2|x|α for α < 1 2 and 0 < ρ = ∑p i=1 √ ρi > 1 for all x ∈ x and all t > 0. proof. by taking ϕ(x1,x2, . . . ,xp) = c2 (∏p i=1 |xi| α p ) for all x1,x2, . . . ,xp ∈ x in theorem 2.1 and theorem 2.2, and choosing respectively l = ρ 1 2 −α and l = ρα− 1 2 , we get the desired result. � corollary 2.5. let c3 ≥ 0 and α be real numbers with α > 12 or α < 1 2 . let f : xp → y be a mapping satisfying n (dρf(x1,x2, . . . ,xp), t) ≥ t t + c3 [∑p i=1 (∏p j=1,j 6=i |xj| α p−1 )] for all x1,x2, . . . ,xp ∈ x and all t > 0. then there exists a unique square root mapping s : x → y such that n(f(x) −s(x), t) ≥   (ρα−ρ 1 2 )t (ρα−ρ 1 2 )t+ρ 1 2 pc3|x|α for α > 1 2 and 0 < ρ = ∑p i=1 √ ρi < 1 (ρ 1 2 −ρα)t (ρ 1 2 −ρα)t+ραpc3|x|α for α < 1 2 and 0 < ρ = ∑p i=1 √ ρi > 1 for all x ∈ x and all t > 0. proof. by taking ϕ(x1,x2, . . . ,xp) = c3 [∑p i=1 (∏p j=1,j 6=i |xj| α p−1 )] for all x1,x2, . . . ,xp ∈ x in theorem 2.1 and theorem 2.2, and choosing respectively l = ρ 1 2 −α and l = ρα− 1 2 , we get the desired result. � corollary 2.6. let c4 ≥ 0 and α be real numbers with α > 12 or α > 1 2 . let f : xp → y be a mapping satisfying n (dρf(x1,x2, . . . ,xp), t) ≥ t t + c4 [∏p i=1 |xi| α p + ( ∑p i=1 |xi|α) ] fuzzy stability of generalized square root functional equation 17 for all x1,x2, . . . ,xp ∈ x and all t > 0. then there exists a unique square root mapping s : x → y such that n(f(x)−s(x), t) ≥   (ρα−ρ 1 2 )t (ρα−ρ 1 2 )t+ρ 1 2 (p+1)c4|x|α for α > 1 2 and 0 < ρ = ∑p i=1 √ ρi < 1 (ρ 1 2 −ρα)t (ρ 1 2 −ρα)t+ρα(p+1)c4|x|α for α < 1 2 and 0 < ρ = ∑p i=1 √ ρi > 1 for all x ∈ x and all t > 0. proof. by taking ϕ(x1,x2, . . . ,xp) = c4 [∏p i=1 |xi| α p + ( ∑p i=1 |xi| α) ] for all x1,x2, . . . ,xp ∈ x in theorem 2.1 and theorem 2.2, and choosing respectively l = ρ 1 2 −α and l = ρα− 1 2 , we get the desired result. � references [1] j. aczel and j. dhombres, functional equations in several variables, cambridge univ. press, 1989. [2] t. aoki, on the stability of the linear transformation in banach spaces, j. math. soc. japan, 2 (1950), 64-66. [3] c. baak and m.s. moslehian, on the stability of j∗-homomorphisms, nonlinear analysistma 63 (2005), 42-48. [4] t. bag and s.k. samanta, finite dimensional fuzzy normed linear spaces, j. fuzzy math., 11(3) (2003), 687-705. [5] t. bag and s.k. samanta, fuzzy bounded linear operators,, fuzzy sets and syst.,151 (2005), 513-547. [6] r. biswas, fuzzy inner product spaces and fuzzy norm functions, inform. sci., 53 (1991), 185-190. [7] l. cadariu and v. radu, fixed points and the stability of jensen’s functional equation, j. inequal. pure appl. math., 4(1), art. id 4 (2003). [8] l. cadariu and v. radu, on the stability of the cauchy functional equation: a fixed point approach, grazer math. ber., 346 (2004), 43-52. [9] l. cadariu and v. radu, fixed point methods for the generalized stability of functional equations in a single variable, fixed point theory and appl., 2008, art. id 749392 (2008). [10] i.s. chang and h.m. kim, on the hyers-ulam stability of quadratic functional equations, j. ineq. appl. math., 33 (2002), 1-12. [11] i.s. chang and y.s. jung, stability of functional equations deriving from cubic and quadratic functions, j. math. anal. appl., 283 (2003), 491-500. [12] s.c. cheng and j.m. mordeson, fuzzy linear operators and fuzzy normed linear spaces, bull. calcutta math. soc., 86 (1994), 429-436. [13] p.w. cholewa, remarks on the stability of functional equations, aequationes math., 27 (1984), 76-86. [14] s. czerwik, on the stability of the quadratic mappings in normed spaces, abh. math. sem. univ. hamburg, 62 (1992), 59-64. 18 ravi and kumar [15] p. czerwik, functional equations and inequalities in several variables, world scientific publishing co. new jersey, hong kong, singapore and london, 2002. [16] j. diaz and b. margolis, a fixed point theorem of the alternative for contractions on a generalized complete metric space, bull. amer. math. soc., 74 (1968), 305-309. [17] m. eshaghi gordji, s. zolfaghari, j.m. rassias and m.b. savadkouhi, solution and stability of a mixed type cubic and quartic functional equation in quasi-banach spaces, abstract and applied analysis, volume 2009, article id 417473 (2009), 1-14. [18] c. felbin, finite dimensional fuzzy normed linear space, fuzzy sets and syst., 48 (1992), 239-248. [19] g.l. forti, hyers-ulam stability of functional equations in several variables, aequationes math., 50 (1995), 143-190. [20] z. gajda, on stability of additive mappings, int. j. math. math. sci., 14 (1991), 431-434. [21] p. gavruta, a generalization of the hyers-ulam-rassias stability of approximately additive mappings, j. math. anal. appl., 184 (1994), 431-436. [22] n. ghobadipour and c. park, cubic-quartic functional equations in fuzzy normed spaces, int. j. nonlinear anal. appl., 1 (2010), 12-21. [23] d.h. hyers, on the stability of the linear functional equation, proc. nat. acad. sci. u.s.a., 27 (1941), 222-224. [24] d.h. hyers, g. isac and th.m. rassias, stability of functional equations in several variables, birkhauser, basel, 1998. [25] g. isac and th.m. rassias, stability of ψ-additive mappings: applications to nonlinear analysis, int. j. math. math. sci., 19 (1996), 219-228. [26] a.k. katsaras, fuzzy topological vector spaces ii, fuzzy sets and syst., 12 (1984), 143-154. [27] h. khodaei and th.m. rassias, approximately generalized additive functions in several variables, int. j. nonlinear anal. appl., 1 (2010), 22-41. [28] i. kramosil and j. michalek, fuzzy metric and statistical metric spaces, kybernetica, 11 (1975), 326-334. [29] s.v. krishna and k.k.m. sarma, separation of fuzzy normed linear spaces, fuzzy sets and syst., 63 (1994), 207-217. [30] d. mihet and v. radu on the stability of the additive cauchy functional equation in random normed spaces, j. math. anal. appl., 343 (2008), 567-572. [31] a.k. mirmostafaee, m. mirzavaziri and m.s. moslehian, fuzzy stability of the jensen functional equation, fuzzy sets and syst., 159 (2008), 730-738. [32] a.k. mirmostafaee and m.s. moslehian, fuzzy versions of hyers-ulam-rassias theorem, fuzzy sets and syst., 159 (2008), 720-729. [33] a.k. mirmostafaee and m.s. moslehian, fuzzy approximately cubic mappings, information sci., 178 (2008), 3791-3798. [34] m. mirzavaziri and m.s. moslehian, a fixed point approach to stability of a quadratic equation, bull. braz. math. soc., 37 (2006), 361-376. [35] c. park, fixed points and hyers-ulam-rassias stability of cauchy-jensen functional equations in banach algebras, fixed point theory and appl., 2007, art. id 50175 (2007). fuzzy stability of generalized square root functional equation 19 [36] c. park, generalized hyers-ulam-rassias stability of quadratic functional equations: a fixed point approach, fixed point theory and appl., 2008, art. id 493751 (2008). [37] c. park, fuzzy stability of a functional equation associated with inner product spaces, fuzzy sets and syst., 160 (2009), 1632-1642. [38] c. park, s. lee and s. lee, fuzzy stability of a cubic-quadratic functional equation: a fixed point approach, korean j. math., 17(3) (2009), 315-330. [39] v. radu, the fixed point alternative and the stability of functional equations, fixed point theory, 4 (2003), 91-96. [40] j.m. rassias, on approximation of approximately linear mappings by linear mappings, j. funct. anal., 46 (1982), 126-130. [41] k. ravi and b.v. senthil kumar, rassias stability of generalized square root functional equation, int. j. math. sci. engg. appl., 3(iii) (2009), 35-42. [42] k. ravi, j.m. rassias m. arunkumar and r. kodandan, stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, j.of ineq.pure & appl. math., 10 (4) (2009), 1-29. [43] th.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72 (1978), 297-300. [44] f. skof, proprieta locali e approssimazione di operatori, rend. sem. mat. fis. milano 53 (1983), 113-129. [45] s.m. ulam, problems in modern mathematics, chapter vi, wiley-interscience, new york, 1964. [46] j.z. xiao and x.h. zhu, fuzzy normed spaces of operators and its completeness, fuzzy sets and systems, 133 (2003), 389-399. 1pg & research department of mathematics, sacred heart college, tirupattur 635 601, tamilnadu, india 2department of mathematics, c. abdul hakeem college of engg. and tech., melvisharam 632 509, tamilnadu, india ∗corresponding author international journal of analysis and applications volume 19, number 3 (2021), 319-340 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-319 the downside and upside beta valuation in the variance-gamma model roman v. ivanov∗ laboratory of control under incomplete information, v.a. trapeznikov institute of control sciences of ras, moscow, russian federation ∗corresponding author: roivanov@yahoo.com abstract. the paper is aimed to assess the risks and gains of investment portfolio which relate to the impact of a particular asset. we consider the investment portfolios which consist of assets with variancegamma, gamma distributed and deterministic returns. the returns are assumed to be dependent. we derive analytical formulas for the downside and upside betas in the discussed framework. the established formulas depend on the values of a number of special mathematical functions including the values of the generalized hypergeometric ones. 1. introduction the basic monetary risk measures value at risk (see, for example, berkowitz et al. [6], chen and tang [8], ivanov [20], stoyanov et al. [42]) and conditional value at risk (kalinchenko et al. [22], mafusalov and uryasev [29], rockafellar and uryasev [37]) serve to assess the downward risk of the investment portfolio. but if we want to rate the influence of a specific asset on the return of the portfolio, we exploit the market beta. when we form the investment portfolio, it is necessary to estimate how the share increase or decrease for a particular asset impacts the risks and the expected profit of the portfolio. the downside beta serves to evaluate the risk size, the upside beta is used to outlay the profit. received february 22nd, 2021; accepted march 17th, 2021; published april 1st, 2021. 2010 mathematics subject classification. 60e07, 60e08, 65c20, 91g10, 91g60, 33c20, 33c90. key words and phrases. downside and upside betas; variance-gamma distribution; investment portfolio; dependence; appell function. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 319 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-319 int. j. anal. appl. 19 (3) (2021) 320 the ideas of use the downside and upside betas go back to the paper by roy [39] and the monograph by markowitz [30], where it was argued that investors more care about downside losses and upside gains. in this context, markowitz [30] suggested to use the semivariance as the basic risk measure. the semivariance beta was introduced in hogan and warren [19]. the advantage of downside and upside betas over the traditional ones is proposed in ang et al. [2] and tehir et al. [43]. the work by estrada [13] suggests a capital asset pricing model based on the downside beta. guy [18] presents a portfolio construction based on the assessment of the values of the upside and downside betas. rutkowska-ziarko and pyke [38] introduce the downside accounting beta suggesting to use it for the measurement of the systemic risk. altigan et al. [1] claim that the downside beta valuation is not sufficient for the asset pricing on international markets contrary to the results for the us equity market. and therefore it is required to take into account the upside one also if we want to create a general model. advantages of downside beta-based capital asset pricing model over the traditional one are presented in ayub et al. [4] and post and van vliet [35]. in the context of the general theory, the downside beta relates to the class of loss-based risk measures which is considered in cont et al. [9]. the variance-gamma distribution was proposed as a model for market stock returns in madan and seneta [28]. an utility-based option pricing theory which exploits the variance-gamma distribution was suggested in madan and milne [27]. the price of european call option in the variance-gamma model was derived analytically in the paper by madan et al. [26]. there is a number of modern research papers which confirms statistically the idea of use the variancegamma distribution for the financial index modeling. daal and madan [12] and finlay and seneta [14] approve the variance-gamma model for the exchange rate simulation. linders and stassen [23], moosbrucker [31] and rathgeber et al. [36] model with the variance-gamma distribution the dow jones index returns. mozumder et al. [32] consider the s&p500 index options in the variance-gamma model. luciano and schoutens [25] model the s&p500, the nikkei225 and the eurostoxx50 financial indexes by the variancegamma process. luciano et al. [24] and wallmeier and diethelm [44] confirm the use of the variance-gamma distribution for the modeling of the us and the swiss stock markets, respectively. groups of various financial indices are modeled by the multivariate variance-gamma distribution in nitithumbundit and chan [34]. flora and vargiolu [15] find that the variance-gamma process is the best fit for the carbon price dynamics. göncü et al. [16] show that the variance-gamma model fits well with the financial data of developed markets. this paper is set to compute the downside and upside betas for the investment portfolio with the variancegamma, gamma distributed and deterministic asset returns. the gamma and deterministic returns relate to the modeling of credit risk, see ivanov [21] and my [33]. as usually, the variance-gamma random variables are modeled as the normal mean-variance mixtures and it is supposed that the normal distributions are correlated. the paper develops the direction of research of madan et al. [26], ano and ivanov [3] and int. j. anal. appl. 19 (3) (2021) 321 ivanov [20], where closed form expressions in the variance-gamma framework are derived for various targets of mathematical finance. 2. main notations we denote by γ = γ(a,b) the gamma random variable with parameters a,b > 0. the gamma distribution has the probability density function f(γ,x) = baxa−1e−bx γ (a) , x > 0,(2.1) where γ(χ) is the gamma function. it has the characteristic function ψ(γ,u) = ( 1 − iu b )−a ,(2.2) the mean and the variance a b and a b2 , respectively. by definition, the variance-gamma distribution is the mean-variance normal mixture, where the mixing density is the gamma distribution. that is, the variance-gamma random variable h is defined as h = r + θγ + σ √ γn,(2.3) where r,θ ∈ r, σ > 0, n is the standard normally distributed random variable and the gamma random variable γ is independent with n. throughout this work, we do not assume that γ has necessary the mean 1, that is the identity a = b is not required. next, we set sg(χ) :=   1 if χ > 0, 0 if χ = 0, −1 if χ < 0, and use notations n(χ),χ ∈ r, b(χ1,χ2),χ1 > 0,χ2 > 0, kχ1 (χ2),χ1 ∈ r,χ2 > 0 for the normal distribution function, the beta function and the macdonald function (the modified bessel function of the second kind), respectively. the hypergeometric gauss function is denoted as f(χ1,χ2,χ3; χ4), χ1,χ2,χ3 ∈ r,χ4 < 1. int. j. anal. appl. 19 (3) (2021) 322 also, we discuss one of the degenerate appell functions (or the humbert series) which is the double sum φ(χ1,χ2,χ3; χ4,χ5) = ∞∑ m=0 ∞∑ n=0 (χ1)m+n(χ2)m m!n!(χ3)m+n χm4 χ n 5 with χ1,χ2,χ3,χ5 ∈ r and |χ4| < 1, where (χ)l, l ∈ n ∪ {0}, is the pochhammer’s symbol. for more information on the special mathematical functions above and relations between them, see the monographs by bateman and erdélyi [5] and srivastava and karlsson [40], the handbook by gradshteyn and ryzhik [17] and the papers by chaudhry et al. [7] and srivastava et al. [41]. 3. setup and results let aj,t, j = 1, 2, ...,n, be the values of n assets at time moments t = 0, 1. it is assumed that aj,0 are constant but aj,1 are random with law (aj,1 −aj,0) = hj, where hj = rj + θjγj + σj √ γjnj(3.1) are constant, gamma or variance-gamma random variables in dependence with the values of the parameters rj,θj ∈ r, σj ≥ 0. we suggest that σj > 0 for at least one j ∈ {1, 2, ...,n}. it is supposed that the normal random variables nj and nl are correlated with coefficients ρjl, j, l ≤ n. all the gamma random variables are assumed to be independent with the normal ones. we suggest that γj = κjγ, where γ = γ(a,b) is gamma distributed, κj ≥ 0, j = 1, 2, ...,n. also, we set for the simplicity of formulas below that ρlm = 0 if σl = 0 or σm = 0. this model is a particular case of a more general model which is discussed in ivanov [20]. briefly, we assume here that the asset returns are highly dependent with each other. it agrees with the last investigations on the financial market structure, see cont and sirignano [10]. together with it, the strong dependence between stocks originates to the decisions of a large investor (cont and wagalath [11]). the value it at time moments t = 0, 1 of the investment portfolio x = (x1,x2, ...,xn) ∈ rn is defined as it = n∑ j=1 ij,t, where ij,t = xjxj,t, j = 1, 2, ...,n. int. j. anal. appl. 19 (3) (2021) 323 since all xj,0 are constant, it is enough for the aim to evaluate the portfolio risks to discuss the random increment x = i1 − i0 = n∑ j=1 xjhj(3.2) of the portfolio and the random increments of the investments xj = ij,1 − ij,0 = xj (xj,1 −xj,0) = xjhj, j = 1, 2, ...,n.(3.3) throughout this work we consider the downside beta β−, the investment portfolio characteristics which is defined as β− = e [ (xj − exj)(x − ex)i{x≤u} ] e [ (x − ex)2i{x≤u} ] , u ∈ r. taking into account the value of β−, one could analyze is it expedient to put the size xj into the asset j or not. for example, if β− � 0, it means that the investment xjxj,0 accelerates the downside risk of the portfolio substantially. together with the downside beta, we discuss the upside beta β+ = e [ (xj − exj)(x − ex)i{x≥u} ] e [ (x − ex)2i{x≥u} ] , u ∈ r. the upside beta shows the impact of the investment xj on the potential earnings of the investment portfolio. to introduce the results, now we suggest some auxiliary abbreviations. let hj = xj ( rj + θjaj bj ) , h = n∑ l=1 hl, ŝ = n∑ l=1 xlrl û = u− ŝ, s1 = n∑ l=1 xlθlκl, s2 = n∑ l=1 ρjlxlσl √ κl, s3 = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ κmκl, s = û √ s21 + 2bs 2 3 s3|s3| , q = − sg(s3)s1√ s21 + 2bs 2 3 . next, we set yl = xlσl √ γlnl, xl,σl 6= 0. the lemma below computes the value of the expectation f1 = e ( yj n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) . lemma 3.1. if xj,σj 6= 0, the expectation f1 = xjσjs2b a√κj s3γ(a) ( s3λ(û) − exp ( ûs1 s23 ) √ 2π ( ûθ(û, 0) −s1θ(û, 1) )) ,(3.4) int. j. anal. appl. 19 (3) (2021) 324 where λ(û) = γ ( a + 3 2 ) ba+1 √ 2π ( b ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b f ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )) i{û=0}+ + |s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π (( |s|ka+ 3 2 (|s|) + ska+ 1 2 (|s|) ) φ̂(0)− − (1 + q)ska+ 1 2 (|s|) φ̂(1) ) i{û6=0} and θ(û,j) = γ ( a + 1 2 + j )( 2s23 s21 + 2bs 2 3 )a+ 1 2 +j i{û=0}+ + 2 ( û2 s21 + 2bs 2 3 )a+j 2 + 1 4 ka+j+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) i{û 6=0} with φ̂(j) = b(a + 1 + j, 1)φ ( a + 1 + j,−a,a + 2 + j; 1 + q 2 ,−s(1 + q) ) . next, we discuss the expectation f2(ζ,α) = e ( γ ζ j ( n∑ l=1 xlθlγl )α yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) . for ζ,α ∈ n∪{0}. lemma 3.2. when xj,σj 6= 0, we have that f2(ζ,α) = − s2xjσjs α 1 b aκ ζ+ 1 2 j γ(a) √ π ( 2ζ+α+as 2(ζ+α+a) 3 γ ( α + a + 1 2 ) (s21 + 2bs 2 3) ζ+α+a+ 1 2 i{û=0}+(3.5) + exp ( ûs1 s23 )√ 2 s3 ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) i{û6=0} ) . also, we calculate the function f3(ζ,α) = e ( γ ζ j ( n∑ l=1 xlθlγl )α i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) . int. j. anal. appl. 19 (3) (2021) 325 lemma 3.3. let ζ,α ∈ n∪{0}. then f3(ζ,α) = κ ζ js α 1 γ(ζ + α + a + 1 2 ) γ(a)bζ+α √ 2π i{û=0} ( b ( 1 2 ,ζ + α + a ) √ 2 − s1 s3 √ b × × f ( ζ + α + a + 1 2 , 1 2 , 3 2 ;− s21 2bs23 )) + κ ζ j|s| ζ+α+a−1 2 (1 + q)ζ+α+ai{û 6=0} γ(a)bζ+αe−ss−α1 √ 2π × × ( b(ζ + α + a, 1) ( |s|kζ+α+a+ 1 2 (|s|) + skζ+α+a−1 2 (|s|) ) φ̃(0)− − (1 + q)sb(ζ + α + a + 1, 1)kζ+α+a−1 2 (|s|) φ̃(1) ) ,(3.6) where φ̃(j) = =φ ( ζ + α + a + j, 1 − ζ −α−a,ζ + α + a + 1 + j; 1 + q 2 ,−s(1 + q) ) . set f4(ζ,α) = e ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) for ζ,α ∈ n∪{0}. lemma 3.4. the expectation f4(ζ,α) = − s3b asα1 κ ζ j γ(a) √ 2π ( 2e ûs1 s2 3 kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) i{û=0}×(3.7) ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 + γ ( ζ + α + a + 1 2 )( 2s23 s21 + 2bs 2 3 )ζ+α+a+ 1 2 i{û6=0} ) . finally, set f5 = e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) . lemma 3.5. let the functions θ(û,j) and λ(û) be defined in lemma 3.1. then f5 = s3b a γ(a) ( s1√ 2π θ(0, 1) + s3λ(0) ) i{û=0}+(3.8) + s3b a γ(a) ( s3λ(û) + exp ( ûs1 s23 ) √ 2π (s1θ(û, 1) − ûθ(û, 0)) ) i{û6=0}. the proofs of lemmas 3.1–3.5 are placed in section 4. now can introduce the main results of the paper. the theorem below gives us an analytical expression for the value of the downside beta. int. j. anal. appl. 19 (3) (2021) 326 theorem 3.1. the downside beta β− = β−n β−d , with β−n = xj [ rj ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) +θj ( ŝf3(1, 0)+(3.9) + f3(1, 1) + f4(1, 0) )] +ŝf2(0, 0) + f2(0, 1) + f1 −hj ( ŝf3(0, 0)+ + f3(0, 1) + f4(0, 0) −hf3(0, 0) ) −h [ xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0) ] and β−d = ŝ 2f3(0, 0) + 2ŝf3(0, 1) + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1)+(3.10) + f5 − 2h ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) +h2f3(0, 0), where the expectations f1,f2,f3,f4,f5 are computed in lemmas 3.1–3.5, respectively. let ĥj = xj n∑ l=1 xl ( rj ( rl + θlal bl ) +θj ( rlaj bj + + θlκlκj(a + 1)a b2 ) +σj ( rl + σlρlja √ κlκj b )) and ĥ = n∑ l,m=1 xlxm ( rl ( rm + θmam bm ) +θl ( rmal bl + + θmκlκma(a + 1) b2 ) + σlσmρlma √ κlκm b ) . the next theorem derives the size of the upside beta. theorem 3.2. the upside beta β+ = ĥj −hjh−β−n ĥ−h2 −β−d , where β−n and β − d are defined in (3.9) and (3.10). the proofs of theorems 3.1 and 3.2 are given in section 4. the following example considers the case of the investment portfolio which consists of three assets and two of them are risk-free and low risk ones. int. j. anal. appl. 19 (3) (2021) 327 example 3.1. assume that n = 3, h1 = r1, h2 = r2 + θ2γ, h3 = r3 + θ3γ + σ3 √ γn3 and j = 2. then h2 = x2 ( r2 + θ2a2 b2 ) , h = r1 + ∑3 l=2 xl ( rl + θlal bl ) , ŝ = ∑3 l=1 xlrl, û = u − ŝ, s1 = ∑3 l=2 xlθl, s2 = 0, s3 = σ3|x3|, ĥ2 = =x2 ( x1 ( r2r1 + θ2r1a b ) + 3∑ l=2 xl ( r2 ( rl + θla b ) +θ2 (rla b + θl(a + 1)a b2 ))) , ĥ = 3∑ l,m=1 xlxm ( rl ( rm + θmi{m6=1}a b ) + +θli{l 6=1} ( rma b + θmi{m 6=1}a(a + 1) b2 )) + σ23a b , f1 ≡ 0, f2(ζ,α) ≡ 0, β−n = xj ( rj ( ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ) + +θj ( ŝf3(1, 0) + f3(1, 1) + f4(1, 0) )) −hj ( ŝf3(0, 0) + f3(0, 1)+ +f4(0, 0) −hf3(0, 0) ) −h ( xj ( rjf3(0, 0) + θjf3(1, 0) )) , where f3, f4 are determined by lemma 3.3, lemma 3.4 and β − d is calculated in (3.10). example 3.2 discusses the case when there are n assets in the portfolio and they are the medium dependent between each other. example 3.2. let γ1 ≡ γ2 ≡ ... ≡ γn ≡ γ and ρlm = 0, l 6= m. we have that hj = xj ( rj + θja b ) , h = ∑n l=1 hl, ŝ = ∑n l=1 xlrl, û = u− ŝ, s1 = ∑n l=1 xlθl, s2 = xjσj, s3 = √∑n l=1 x 2 l σ 2 l , ĥj = xj ( xjσ 2 ja b + n∑ l=1 xl ( rj ( rl + θla b ) +θj (rla b + θl(a + 1)a b2 ) +σjrl )) , ĥ = n∑ l,m=1 xlxm ( rl ( rm + θma b ) +θl ( rma b + θma(a + 1) b2 )) + a b n∑ l=1 x2l σ 2 l and β−n , β − d are computed with respect to (3.9), (3.10). 4. proofs proof of lemma 3.1. it is easy to notice that ( yj, ∑n l=1 yl ∣∣∣γ1,γ2, ...,γn) is a gaussian vector with the covariance matrix   (xjσj)2γj ∑nl=1 ρjlxjσjxlσl√γjγl∑n l=1 ρjlxjσjxlσl √ γjγl ∑n m,l=1 ρmlxmσmxlσl √ γmγl   . int. j. anal. appl. 19 (3) (2021) 328 hence law ( yj, n∑ l=1 yl ∣∣∣γ1,γ2, ...,γn)= law(yj, ỹ ),(4.1) where ỹ = σỹ ñ with (4.2) σỹ = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ γmγl and the standard normal random variables nj and ñ are correlated with the coefficient (4.3) ρjỹ = ∑n l=1 ρjlxjσjxlσl √ γjγl xjσj √ γj √∑n m,l=1 ρmlxmσmxlσl √ γmγl . set σ̂j = xjσj √ γj(4.4) and (4.5) ũ = u− n∑ l=1 xl(rl + θlγl). then e ( yj n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= =e ( yjỹ i{ỹ≤ũ} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ ∫ ∞ −∞ xy 2πσỹ σ̂j √ 1 −ρ2 jỹ × ×exp  − 1 2 ( 1 −ρ2 jỹ ) [ x2 σ2 ỹ − 2ρjỹ xy σỹ σ̂j + y2 σ̂2j ]dydx int. j. anal. appl. 19 (3) (2021) 329 and since ∫ ∞ −∞ y exp ( − 1 2 ( 1 −ρ2 jỹ ) [y2 σ̂2j − 2ρjỹ xy σỹ σ̂j ]) dy = = exp ( ( xρjỹ )2 2σ2 ỹ ( 1 −ρ2 jỹ ))∫ ∞ −∞ y exp ( − 1 2 ( 1 −ρ2 jỹ ) [ y σ̂j − xρjỹ σỹ ]2 ) dy = = σ̂2j exp ( ( xρjỹ )2 2σ2 ỹ ( 1 −ρ2 jỹ ))(∫ ∞ −∞ ( y − xρjỹ σỹ ) × × exp ( − 1 2 ( 1 −ρ2 jỹ ) [y − xρjỹ σỹ ]2 ) dy + xρjỹ σỹ × × ∫ ∞ −∞ exp ( − 1 2 ( 1 −ρ2 jỹ ) [y − xρjỹ σỹ ]2 ) dy ) = = exp ( ( xρjỹ )2 2σ2 ỹ ( 1 −ρ2 jỹ ))xρjỹ σ̂2j σỹ √ 2π ( 1 −ρ2 jỹ ) , we have that e ( yj n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ x2ρjỹ σ̂j σ2 ỹ √ 2π × × exp ( − x2 2σ2 ỹ ) dx = − ρjỹ σ̂j√ 2π ∫ ũ −∞ xd exp ( − x2 2σ2 ỹ ) = = − ρjỹ σ̂j√ 2π ũ exp ( − ũ2 2σ2 ỹ ) + ρjỹ σ̂j√ 2π ∫ ũ −∞ exp ( − x2 2σ2 ỹ ) dx = = − ρjỹ σ̂j√ 2π ũ exp ( − ũ2 2σ2 ỹ ) +ρjỹ σỹ σ̂jn ( ũ σỹ ) .(4.6) next, one can observe that σỹ = √√√√γ n∑ m,l=1 ρmlxmσmxlσl √ κmκl and ũ = u− n∑ l=1 xlrl −γ n∑ l=1 xlθlκl. moreover, we have that ρjỹ = γ ∑n l=1 ρjlxjσjxlσl √ κjκl xjσj √ κjγ √ γ ∑n m,l=1 ρmlxmσmxlσl √ κmκl int. j. anal. appl. 19 (3) (2021) 330 and σ̂j = xjσj √ κjγ. set û = u− n∑ l=1 xlrl, s1 = n∑ l=1 xlθlκl, s2 = n∑ l=1 ρjlxlσl √ κl, s3 = √√√√ n∑ m,l=1 ρmlxmσmxlσl √ κmκl. then σỹ = s3 √ γ, ũ = û−s1γ, ρjỹ = s2 s3 and we get from (4.6) that e ( yj n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.7) =xjσj s2 s3 √ κj ( s3 ∫ ∞ 0 gn ( û−s1g s3 √ g ) f(γ,g)dg− − 1 √ 2π ∫ ∞ 0 (û−s1g) √ g exp ( − (û−s1g)2 2s23g ) f(γ,g)dg ) = = xjσjs2b a√κj s3γ(a) ( s3 ∫ ∞ 0 gan ( û−s1g s3 √ g ) exp(−bg)dg− − exp ( ûs1 s23 ) √ 2π ∫ ∞ 0 (û−s1g)ga− 1 2 exp ( − û2 + (s1g) 2 2s23g − bg ) dg ) . first, ∫ ∞ 0 ga± 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg = =2 ( û2 s21 + 2bs 2 3 )a+1 2 ±1 4 ka+1±1 2 ( |û| √ s21 + 2bs 2 3 s23 ) (4.8) with respect to the formula 3.471.9 from gradshteyn and ryzhik [17] if û 6= 0. when û = 0,∫ ∞ 0 ga± 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg = = γ ( a + 1 ± 1 2 )( 2s23 s21 + 2bs 2 3 )a+1±1 2 .(4.9) next, the integral i = ∫ ∞ 0 gan ( û−s1g s3 √ g ) exp(−bg)dg int. j. anal. appl. 19 (3) (2021) 331 is quite similar to the one at the bottom of p.207 of ivanov and ano [3]. if û = 0, i = γ ( a + 3 2 ) ba+1 √ 2π ( b ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b f ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )) (4.10) due to case 2.2, p.208 of ivanov and ano [3]. when û 6= 0, i = |s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π ( b(a + 1, 1) ( |s|ka+ 3 2 (|s|) +(4.11) ska+ 1 2 (|s|) ) φ ( a + 1,−a,a + 2; 1 + q 2 ,−s(1 + q) ) − (1 + q)sb(a + 2, 1)ka+ 1 2 (|s|) φ ( a + 2,−a,a + 3; 1 + q 2 ,−s(1 + q) )) , where s = û √ s21+2bs 2 3 s3|s3| and q = − sg(s3)s1√ s21+2bs 2 3 , with respect to case 3.2, p.210 of ivanov and ano [3]. hence we get (3.4) from (4.7)–(4.11). proof of lemma 3.2. keeping in mind (4.1), we get that e ( yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = e ( yji{ỹ≤ũ} ∣∣∣γ1,γ2, ...,γn)= ∫ ũ −∞ ∫ ∞ −∞ y 2πσỹ σ̂j √ 1 −ρ2 jỹ × × exp  − 1 2 ( 1 −ρ2 jỹ ) [ x2 σ2 ỹ − 2ρjỹ xy σỹ σ̂j + y2 σ̂2j ]dydx = = ∫ ũ −∞ xρjỹ σ̂j σ2 ỹ √ 2π exp ( − x2 2σ2 ỹ ) dx = − ρjỹ σ̂j√ 2π exp ( − ũ2 2σ2 ỹ ) , where σỹ , ρjỹ , σ̂j, ũ are defined in (4.2), (4.3), (4.4), (4.5), respectively. hence e ( γ ζ j ( n∑ l=1 xlθlγl )α yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) = = − s2xjσj( ∑n l=1 κlxlθl) ακ ζ+ 1 2 j s3 √ 2π e ( γζ+α+ 1 2 exp ( − (û−s1γ)2 2s23γ )) = = − s2xjσj( ∑n l=1 κlxlθl) α exp ( ûs1 s23 ) baκ ζ+ 1 2 j s3γ(a) √ 2π × × ∫ ∞ 0 gζ+α+a− 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg = = − s2xjσj( ∑n l=1 κlxlθl) α exp ( ûs1 s23 ) baκ ζ+ 1 2 j √ 2 s3γ(a) √ π × × ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) (4.12) int. j. anal. appl. 19 (3) (2021) 332 due to the formula 3.471.9 from gradshteyn and ryzhik [17] when û 6= 0. if û = 0, e ( γ ζ j ( n∑ l=1 xlθlγl )α yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) = = − s2xjσj( ∑n l=1 κlxlθl) αbaκ ζ+ 1 2 j s3γ(a) √ 2π ∫ ∞ 0 gζ+α+a− 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg = = − 2ζ+α+as2s 2(ζ+α+a) 3 xjσj( ∑n l=1 κlxlθl) αbaγ ( α + a + 1 2 ) κ ζ+ 1 2 j γ(a) (s21 + 2bs 2 3) ζ+α+a+ 1 2 √ π .(4.13) thus, we get (3.5) from (4.13) and (4.12). proof of lemma 3.3. since p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)= = p ( ỹ ≤ ũ ∣∣γ1,γ2, ...,γn)= n( ũ σỹ ) , we have that e ( γ ζ j ( n∑ l=1 xlθlγl )α i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) = = κ ζ j ( n∑ l=1 xlθlκl )α e ( γζ+αn ( ũ σỹ )) = = baκ ζ j (∑n l=1 xlθlκl )α γ(a) ∫ ∞ 0 gζ+α+a−1 exp(−bg)n ( û−s1g s3 √ g ) dg. hence we get similarly to the proof of lemma 3.1 that e ( γ ζ j ( n∑ l=1 xlθlγl )α i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.14) = baκ ζ j (∑n l=1 xlθlκl )α γ(ζ + α + a + 1 2 ) γ(a)bζ+α+a √ 2π ×( b ( 1 2 ,ζ + α + a ) √ 2 − s1 s3 √ b f ( ζ + α + a + 1 2 , 1 2 , 3 2 ;− s21 2bs23 )) int. j. anal. appl. 19 (3) (2021) 333 if û = 0 and e ( γ ζ j ( n∑ l=1 xlθlγl )α i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.15) = baκ ζ j (∑n l=1 xlθlκl )α |s|ζ+α+a− 1 2 es(1 + q)ζ+α+a γ(a)bζ+α+a √ 2π ×( b(ζ + α + a, 1) ( |s|kζ+α+a+ 1 2 (|s|) + skζ+α+a−1 2 (|s|) ) × φ ( ζ + α + a, 1 − ζ −α−a,ζ + α + a + 1; 1 + q 2 ,−s(1 + q) ) − − (1 + q)sb(ζ + α + a + 1, 1)kζ+α+a−1 2 (|s|)× φ ( ζ + α + a + 1, 1 − ζ −α−a,ζ + α + a + 2; 1 + q 2 ,−s(1 + q) )) when û 6= 0. we have (3.6) from (4.14) and (4.15). proof of lemma 3.4. we have with respect to (4.1) that e ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = γ ζ j ( n∑ l=1 xlθlγl )α e ( ỹ i{ỹ≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = γ ζ j ( n∑ l=1 xlθlγl )α∫ ũ −∞ x σỹ √ 2π exp ( − x2 2σ2 ỹ ) dx = = − σỹ γ ζ j (∑n l=1 xlθlγl )α √ 2π exp ( − ũ2 2σ2 ỹ ) . hence e ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.16) = − s3b asα1 κ ζ j γ(a) √ 2π ∫ ∞ 0 gζ+α+a− 1 2 exp ( −bg − (û−s1g)2 2s23g ) dg = = − s3b asα1 κ ζ j exp ( ûs1 s23 )√ 2 γ(a) √ π ( û2 s21 + 2bs 2 3 )ζ+α+a 2 + 1 4 × × kζ+α+a+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 ) int. j. anal. appl. 19 (3) (2021) 334 if û 6= 0 and e ( γ ζ j ( n∑ l=1 xlθlγl )α n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.17) = − s3b asα1 κ ζ j γ(a) √ 2π γ ( ζ + α + a + 1 2 )( 2s23 s21 + 2bs 2 3 )ζ+α+a+ 1 2 when û = 0 similarly to (4.8) and (4.9), respectively. we get (3.7) from (4.16) and (4.17). proof of lemma 3.5. conditional expectation e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)= = e ( ỹ 2i{ỹ≤u−∑nl=1 xl(rl+θlγl)}) ∣∣∣γ1,γ2, ...,γn)= = ∫ ũ −∞ x2 σỹ √ 2π exp ( − x2 2σ2 ỹ ) dx = − σỹ√ 2π ( ũ exp ( − ũ2 2σ2 ỹ ) − − ∫ ũ −∞ exp ( − x2 2σ2 ỹ ) dx ) = − ũσỹ√ 2π exp ( − ũ2 2σ2 ỹ ) +σ2 ỹ n ( ũ σỹ ) . therefore e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) = =s3 ( s1√ 2π e ( γ 3 2 exp ( − s21 2s23 γ )) +s3e ( γn ( − s1 s3 √ γ ))) = = s3b a γ(a) ( s1√ 2π ∫ ∞ 0 ga+ 1 2 exp ( − s21 + 2bs 2 3 2s23 g ) dg+ +s3 ∫ ∞ 0 gan ( − s1 s3 √ g ) exp(−bg)dg ) =(4.18) = s3b a γ(a) ( s1√ 2π γ ( a + 3 2 )( 2s23 s21 + 2bs 2 3 )a+ 3 2 + + s3γ ( a + 3 2 ) ba+1 √ 2π [ b ( 1 2 ,a + 1 ) √ 2 − s1 s3 √ b f ( a + 3 2 , 1 2 , 3 2 ;− s21 2bs23 )]) int. j. anal. appl. 19 (3) (2021) 335 if û = 0 as in (4.9) and (4.10). when û 6= 0, e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) = s3 ( s3e ( γn (û−s1γ s3 √ γ )) + + 1 √ 2π [ s1e ( γ 3 2 exp ( − (û−s1γ)2 2s23γ )) −ûe ( γ 1 2 exp ( − (û−s1γ)2 2s23γ ))]) = = s3b a γ(a) ( s3 ∫ ∞ 0 gan (û−s1g s3 √ g ) exp(−bg)dg+ + exp ( ûs1 s23 ) √ 2π [ s1 ∫ ∞ 0 ga+ 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg− − û ∫ ∞ 0 ga− 1 2 exp ( − û2 2s23g − s21 + 2bs 2 3 2s23 g ) dg ]) and hence e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ) =(4.19) = s3b a γ(a) ( s3|s|a+ 1 2 es(1 + q)a+1 ba+1 √ 2π [ b(a + 1, 1) ( |s|ka+ 3 2 (|s|) + + ska+ 1 2 (|s|) ) φ ( a + 1,−a,a + 2; 1 + q 2 ,−s(1 + q) ) − (1 + q)sb(a + 2, 1)ka+ 1 2 (|s|) φ ( a + 2,−a,a + 3; 1 + q 2 ,−s(1 + q) )] + exp ( ûs1 s23 )√ 2 √ π [ s1 ( û2 s21 + 2bs 2 3 )a 2 + 3 4 ka+ 3 2 ( |û| √ s21 + 2bs 2 3 s23 ) − − û ( û2 s21 + 2bs 2 3 )a 2 + 1 4 ka+ 1 2 ( |û| √ s21 + 2bs 2 3 s23 )]) in this case similarly to (4.8) and (4.11). we establish (3.8) from (4.18) and (4.19). proof of theorem 3.1. we have that β− = e [ (xj − exj)(x − ex)i{x≤u} ] e [ (x − ex)2i{x≤u} ] = = e ( xjxi{x≤u} ) − exje ( xi{x≤u} ) e ( x2i{x≤u} ) − 2exe ( xi{x≤u} ) + (ex) 2 p(x ≤ u) + = exjexp(x ≤ u) − exe ( xji{x≤u} ) e ( x2i{x≤u} ) − 2exe ( xi{x≤u} ) + (ex) 2 p(x ≤ u) (4.20) and hence it is needed to compute consequently e ( xjxi{x≤u} ) , e ( xi{x≤u} ) , e ( xji{x≤u} ) , p(x ≤ u) and e ( x2i{x≤u} ) . int. j. anal. appl. 19 (3) (2021) 336 one can see that e ( xjxi{x≤u}|γ1,γ2, ...,γn ) = xj(rj + θjγj) n∑ l=1 xl(rl + θlγl)× ×p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+xj(rj + θjγj)× ×e ( n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ n∑ l=1 xl(rl + θlγl)× ×e ( yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ +e ( yj n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn). hence we have that e ( xjxi{x≤u} ) = xj ( rj [ ŝf3(0, 0) + f3(0, 1) ] + + θj [ ŝf3(1, 0) + f3(1, 1) ] +rjf4(0, 0) + θjf4(1, 0) ) + + ŝf2(0, 0) + f2(0, 1) + f1.(4.21) next, e ( xi{x≤u}|γ1,γ2, ...,γn ) = n∑ l=1 xl(rl + θlγl)× ×p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ +e ( n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and therefore e ( xi{x≤u} ) = ŝf3(0, 0) + f3(0, 1) + f4(0, 0).(4.22) further, e ( xji{x≤u}|γ1,γ2, ...,γn ) = xj(rj + θjγj)× ×p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ +e ( yji{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and then e ( xji{x≤u} ) = xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0).(4.23) int. j. anal. appl. 19 (3) (2021) 337 also, p(x ≤ u|γ1,γ2, ...,γn) = p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn) and hence p(x ≤ u) = f3(0, 0).(4.24) moreover, e ( x2i{x≤u}|γ1,γ2, ...,γn ) = ( n∑ l=1 xl(rl + θlγl) )2 × × p ( n∑ l=1 yl ≤ u− n∑ l=1 xl(rl + θlγl) ∣∣∣γ1,γ2, ...,γn)+ + 2 n∑ l=1 xl(rl + θlγl)e ( n∑ l=1 yli{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn)+ + e (( n∑ l=1 yl )2 i{∑nl=1 yl≤u−∑nl=1 xl(rl+θlγl)} ∣∣∣γ1,γ2, ...,γn) and e ( x2i{x≤u} ) = ŝ2f3(0, 0) + 2ŝf3(0, 1)+ + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1) + f5(0, 0).(4.25) keeping in mind the identities exj = hj and ex = h, we get exploiting (4.20)–(4.25) that β− = β−n β−d , where β−n = xj ( rj [ ŝf3(0, 0) + f3(0, 1) ] + + θj [ ŝf3(1, 0) + f3(1, 1) ] +rjf4(0, 0) + θjf4(1, 0) ) + + ŝf2(0, 0) + f2(0, 1) + f1 −hj [ ŝf3(0, 0)+ f3(0, 1) + f4(0, 0) ] +hjhf3(0, 0) −h [ xj ( rjf3(0, 0) + θjf3(1, 0) ) +f2(0, 0) ] int. j. anal. appl. 19 (3) (2021) 338 and β−d = ŝ 2f3(0, 0) + 2ŝf3(0, 1)+ + f3(0, 2) + 2ŝf4(0, 0) + 2f4(0, 1) + f5(0, 0)− − 2h [ ŝf3(0, 0) + f3(0, 1) + f4(0, 0) ] +h2f3(0, 0). proof of theorem 3.2. one can observe that β+ = e [ (xj − exj)(x − ex)i{x≥u} ] e [ (x − ex)2i{x≥u} ] = = e [(xj − exj)(x − ex)] − e [ (xj − exj)(x − ex)i{x≤u} ] e [(x − ex)2] − e [ (x − ex)2i{x≤u} ] = = exjx − exjex − e [ (xj − exj)(x − ex)i{x≤u} ] ex2 − (ex)2 − e [ (x − ex)2i{x≤u} ] . since exjx = e [ xj(rj + θjγj + σj √ γjnj) n∑ l=1 xl(rl + θlγl + σl √ γlnl) ] = = xj [ rj n∑ l=1 xl ( rl + θlal bl ) + θj n∑ l=1 xl ( rlaj bj + θleγjγl ) + + σj n∑ l=1 xl ( rl + σle √ γlγjnlnj )] = xj [ rj n∑ l=1 xl ( rl + θlal bl ) + + θj n∑ l=1 xl ( rlaj bj + θlκlκj(a + 1)a b2 ) + σj n∑ l=1 xl ( rl + σlρlja √ κlκj b )] = ĥj and ex2 = e ( n∑ l=1 xl(rl + θlγl + σl √ γlnl) )2 = =e   n∑ l,m=1 xlxm(rl + θlγl + σl √ γlnl)(rm + θmγm + σm √ γmnm)   = = n∑ l,m=1 xlxm ( rl ( rm + θmam bm ) +θl (rmal bl + θmκlκma(a + 1) b2 ) + + σlσmρlma √ κlκm b ) = ĥ, we get that β+ = ĥj −hjh−β−n ĥ−h2 −β−d . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (3) (2021) 339 references [1] y. altigan, t.g. bali, k.o. demirtas and a.d. gunaydin, downside beta and equity returns around the world, j. portfolio manage. 44 (7) (2018), 39–54. [2] a. ang, j. chen and y. xing, downside risk, rev. financ. stud. 19 (4) (2006), 1191–1239. [3] k. ano and r.v. ivanov, on exact pricing of fx options in multivariate time-changed lévy models, rev. deriv. res. 19(3) (2016), 201–216. [4] u. ayub, s. kausar, u. noreen, m. zakaria and i. abbas jadoon, downside risk-based six-factor capital asset pricing model (capm): a new paradigm in asset pricing, sustainability 12 (2020), 6756. [5] h. bateman and a. erdélyi, higher transcendental functions, mcgraw-hill, new york, 1953. [6] j. berkowitz, m. pritsker, m. gibson and h. zhou, how accurate are value-at-risk models at commercial banks, j. finance, 57 (2002), 1093–1111. [7] a.m. chaudhry, a. qadir, h.m. srivastava and r.b. paris, extended hypergeometric and confluent hypergeometric functions, appl. math. comput. 159(2) (2004), 589–602. [8] s.x. chen and c.y. tang, nonparametric inference of value-at-risk for dependent financial returns, j. financ. econ. 3(2) (2005), 227–255. [9] r. cont, r. deguest and x.d. he, loss-based risk measures, stat. risk model. 30(2) (2013), 133–167. [10] r. cont and j. sirignano, universal features of price formation in financial markets: perspectives from deep learning, quant. finance, 19(9) (2019), 1449–1459. [11] r. cont and l. wagalath, institutional investors and the dependence structure of asset returns, int. j. theor. appl. finance, 19(2) (2016), 1650010. [12] e.a. daal and d.b. madan, an empirical examination of the variance-gamma model for foreign currency options, j. bus. 78(6) (2005), 2121–2152. [13] j. estrada, mean-semivariance behavior: downside risk and capital asset pricing, int. rev. econ. finance, 16(2) (2007), 169–185. [14] r. finlay and e. seneta, stationary-increment student and variance-gamma processes, j. appl. probab. 43 (2006), 441–453. [15] m. flora and t. vargiolu, price dynamics in the european union emissions trading system and evaluation of its ability to boost emission-related investment decisions, eur. j. oper. res. 280 (2020), 383–394. [16] a. göncü, m.o. karahan and t.u. kuzubas, a comparative goodness-of-fit analysis of distributions of some lévy processes and heston model to stock index returns, north amer. j. econ. finance, 36 (2016), 69–83. [17] i.s. gradshteyn and i.m. ryzhik, table of integrals, series and products, academic press, new york, 1980. [18] a. guy, upside and downside beta portfolio construction: a different approach to risk measurement and portfolio construction, risk gov. control: financ. mark. inst. 5(4) (2015), 243–251. [19] w. hogan and j. warren, computation of the efficient boundary in the es portfolio selection model, j. financ. quant. anal. 7(4) (1972), 1881–1896. [20] r.v. ivanov, on risk measuring in the variance-gamma model, stat. risk model. 35(1-2) (2018), 23–33. [21] r.v. ivanov, a credit-risk valuation under the variance-gamma asset return, risks, 6(2) (2018), 58. [22] k. kalinchenko, s. uryasev and r.t. rockafellar, calibrating risk preferences with generalized capm based on mixed cvar deviation, j. risk, 15(1) (2012), 45–70. [23] d. linders and b. stassen, the multivariate variance gamma model: basket option pricing and calibration, quant. finance, 16(4) (2016), 555–572. int. j. anal. appl. 19 (3) (2021) 340 [24] e. luciano, m. marena and p. semeraro, dependence calibration and portfolio fit with factor-based subordinators, quant. finance, 16(7) (2016), 1037-1052. [25] e. luciano and w. schoutens, a multivariate jump-driven financial asset model, quant. finance, 6(5) (2016), 385–402. [26] d. madan, p. carr and e. chang, the variance gamma process and option pricing. eur. finance rev. 2 (1998), 79–105. [27] d. madan and f. milne, option pricing with vg martingale components, math. finance, 1(4) (1991), 39–55. [28] d. madan and e. seneta, the variance gamma (v.g.) model for share market returns, j. bus. 63 (1990), 511–524. [29] a. mafusalov and s. uryasev, cvar (superquantile) norm: stochastic case, europ. j. oper. res. 249 (2016), 200–208. [30] h. markowitz, portfolio selection: efficient diversification of investments, yale university press, yale, 1959. [31] t. moosbrucker, explaining the correlation smile using variance gamma distributions, j. fixed income, 16(1) (2006), 71–87. [32] s. mozumder, g. sorwar and k. dowd, revisiting variance gamma pricing: an application to s&p500 index options, int. j. financ. eng. 2(2) (2015), 1550022. [33] s.t. my, credit risk and bank stability of vietnam commercial bank: a bk approach, int. j. anal. appl. 18(6) (2020), 1066–1082. [34] t. nitithumbundit and j.s.k. chan, ecm algorithm for auto-regressive multivariate skewed variance gamma model with unbounded density, methodol. comput. appl. probab. 22 (2020), 1169–1191. [35] t. post and p. van vliet, downside risk and asset pricing, j. bank. finance, 30(3) (2006), 823–849. [36] a. rathgeber, j. stadler and s. stöck, modeling share returns – an empirical study on the variance gamma model, j. econ. finance, 40(4) (2016), 653–682. [37] r.t. rockafellar and s. uryasev, optimization of conditional value-at-risk, j. risk, 2 (2000), 21–41. [38] a. rutkowska-ziarko and ch. pyke, the development of downside accounting beta as a measure of risk, econ. bus. rev. 3(4) (2017), 55–65. [39] a.d. roy, safety first and the holding of assets, econometrica, 20(3) (1952), 431–449. [40] h.m. srivastava and p.w. karlsson, multiple gaussian hypergeometric series, wiley, new york, 1985. [41] h.m. srivastava, m.i. qureshi, k.a. quraishi and r. singh, applications of some hypergeometric summation theorems involving double series, j. appl. math. stat. inform. 8(2) (2012), 37–48. [42] s.v. stoyanov, s.t. rachev and f.g. fabozzi, sensitivity of portfolio var and cvar to portfolio return characteristics, ann. oper. res. 205 (2013), 169–187. [43] m. tahir, q. abbas, s. sargana, u. ayub and s. saeed, an investigation of beta and downside beta based capm-case study of karachi stock exchange, amer. j. sci. res. 85 (2013), 118–135. [44] m. wallmeier and m. diethelm, multivariate downside risk: normal versus variance gamma, j. futures mark. 32 (2012), 431–458. 1. introduction 2. main notations 3. setup and results 4. proofs references international journal of analysis and applications issn 2291-8639 volume 8, number 1 (2015), 30-38 http://www.etamaths.com on the wallis formula bai-ni guo1,∗, feng qi2,3 abstract. by virtue of complex methods and tools, the authors express the famous wallis formula as a sum involving binomial coefficients, establish the expansions for sink x and cosk x in terms of cos(mx), find the general formulas for the derivatives of sink x and cosk x, and recover the general multiple-angle formulas for sin(kx) and cos(kx), where k ∈ n and m ∈ z. 1. introduction it is well known [8, 9, 16, 18, 23] that (1.1) in = ∫ π/2 0 cosn x d x = ∫ π/2 0 sinn x d x = (n− 1)!! n!! × {π 2 for n even 1 for n odd for n ∈ n, where n!! denotes a double factorial. usually we call (1.1) the wallis cosine or sine formula, or simply say, the wallis formula, in the literature. in mathematical analysis, the wallis formula (1.1) is derived generally by integrating by parts and mathematical induction. the formula (1.1) may also be represented by in = √ π γ((n + 1)/2) nγ(n/2) = √ π 2 γ((n + 1)/2) γ((n + 2)/2) , where γ(x) stands for the classical euler gamma function which may defined by γ(z) = ∫ ∞ 0 tz−1e−t d t, <(z) > 0. the wallis ratio is defined [42] as wn = (2n− 1)!! (2n)!! = (2n)! 22n(n!)2 = 1 √ π γ ( n + 1 2 ) γ(n + 1) , n ∈ n. it is clear that for n ∈ n (1.2) wn = 2 π i2n = 1 22n ( 2n n ) and i2n−1i2n = π 4n . there have existed plenty of literature about bounding the wallis ratio. see, for example, [4, 5, 6, 7, 9, 16, 17, 19, 20, 22, 42, 43, 47]. 2010 mathematics subject classification. primary 33b10; secondary 26a06, 26a09, 33b15. key words and phrases. wallis formula; sine; cosine; derivative; multiple-angle formula. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 30 on the wallis formula 31 in [18], the wallis formula (1.1) was generalized as i(t) = ∫ π/2 0 cost x d x = ∫ π/2 0 sint x d x = √ π 2 γ((t + 1)/2) γ((t + 2)/2) , t ≥ 0. see also [27, section 2.3] and [48, 49]. in [2, p. 123], it was claimed that if im,n is a primitive of sin m x cosn x for m,n ∈ r, then im+2,n = − sinm+1 x cosn+1 x m + n + 2 + m + 1 m + n + 2 im,n is a primitive of sinm+2 x cosn x if m + n + 2 6= 0. with the aid of this formula the formula (1.1) may be recovered. in [3, 10], by establishing double inequalities for i2n−1 and i2n, the double inequality √ π√ 1 + (9π/16 − 1)/n ≤ ∫ √n − √ n e−x 2 d x < √ π√ 1 − 3/(4n) was obtained for n ∈ n. as a result, the probability integral∫ ∞ 0 e−x 2 d x = √ π 2 was recovered. for more information, please refer to [2, p. 123], [22, 34] and related references therein. in [13, 44], among other things, the sequence ni2n for n ∈ n, which originates from computation of the probability of intersecting between a plane couple and a convex body, was proved to be increasing. for recent developments on the gamma function and the ratios of two gamma functions, please refer to the papers [11, 12, 14, 15, 21, 24, 25, 26, 29, 30, 32, 33, 35, 36, 37, 40, 41, 45, 46], the expository and survey articles [27, 28, 38, 39] and closely related references therein. the aims of this paper are, by virtue of complex methods and tools, to express the sequence i2n−1 as a sum involving binomial coefficients and to recover the identity (1.2). as by-products, the expansions for sink x and cosk x in terms of cos(mx) for m ∈ z, the derivatives for sink x and cosk x, and the general multipleangle formulas for sin(kx) and cos(kx) are established and recovered. 2. main results now we are in a position to establish and recover our main results and byproducts. theorem 2.1. for n ∈ n, we have (2.1) i2n−1 = (−1)n+1 22n−1 2n−1∑ k=0 (−1)k 2n− 2k − 1 ( 2n− 1 k ) . first proof. let i = √ −1 be the imaginary unit. then for n ∈ n we have i2n−1 = ∫ π/2 0 ( eix + e−ix 2 )2n−1 d x 32 guo and qi = 1 22n−1 ∫ π/2 0 2n−1∑ `=0 ( 2n− 1 ` ) ei`xe−i(2n−1−`)x d x = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` )∫ π/2 0 ei(2`−2n+1)x d x = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 i(2`− 2n + 1) [ ei(2`−2n+1)π/2 − 1 ] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 i [ 1 −ei(2`−2n+1)π/2 ] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 sin (2`− 2n + 1)π 2 = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) 1 2`− 2n + 1 cos[(`−n)π] = 1 22n−1 2n−1∑ `=0 ( 2n− 1 ` ) (−1)`−n 2`− 2n + 1 . the formula (2.1) follows. � second proof. for n ∈ n, we have in = ∫ π/2 0 ( eix −e−ix 2i )n d x = 1 2n ∫ π/2 0 [ ei(x−π/2) −e−i(x+π/2) ]n d x = 1 2n ∫ π/2 0 n∑ `=0 (−1)n−` ( n ` ) ei`(x−π/2)e−i(n−`)(x+π/2) d x = 1 2n n∑ `=0 (−1)n−` ( n ` )∫ π/2 0 ei[(2`−n)x−nπ/2] d x = 1 2n n∑ `=0 (−1)n−` ( n ` )∫ π/2 0 cos [ (2`−n)x−n π 2 ] d x. therefore, it follows that i2n−1 = −1 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` )∫ π/2 0 cos [ (2`− 2n + 1)x− (2n− 1) π 2 ] d x = (−1)n 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` )∫ π/2 0 sin[(2`− 2n + 1)x] d x = (−1)n+1 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` ) 1 2`− 2n + 1 [ cos (2`− 2n + 1)π 2 − 1 ] on the wallis formula 33 = (−1)n 22n−1 2n−1∑ `=0 (−1)` ( 2n− 1 ` ) 1 2`− 2n + 1 . the proof is completed. � corollary 2.1. for ` ∈ n, we have cos` x = 1 2` ∑̀ q=0 ( ` q ) cos[(2q − `)x],(2.2) sin` x = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ) cos [ (2q − `)x− ` 2 π ] ,(2.3) and ∑̀ q=0 ( ` q ) sin[(2q − `)x] = 0,(2.4) ∑̀ q=0 (−1)q ( ` q ) sin [ (2q − `)x− ` 2 π ] = 0.(2.5) proof. from the second proof of theorem 2.1, we conclude that cos` x = 1 2` (eix + e−ix)` = 1 2` ∑̀ q=0 ( ` q ) eqixe−(`−q)ix = 1 2` ∑̀ q=0 ( ` q ) e(2q−`)ix = 1 2` ∑̀ q=0 ( ` q ) {cos[(2q − `)x] + i sin[(2q − `)x]}. equating the real and imaginary parts in the above equality gives equalities (2.2) and (2.4). similarly, we have sin` x = 1 (2i)` ∑̀ q=0 (−1)`−q ( ` q ) eqixe−(`−q)ix = (−1)` (2i)` ∑̀ q=0 (−1)q ( ` q ) e(2q−`)ix = (−1)` 2` e−πi`/2 ∑̀ q=0 (−1)q ( ` q ) e(2q−`)ix = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ) e[(2q−`)x−π`/2]i = (−1)` 2` ∑̀ q=0 (−1)q ( ` q ){ cos [ (2q − `)x− ` 2 π ] + i sin [ (2q − `)x− ` 2 π ]} . hence, we obtain equalities (2.3) and (2.5). � corollary 2.2. for m,k ∈ n, we have dm cosk x d xm = 1 2k k∑ q=0 ( k q ) (2q −k)m cos [ π 2 m + (2q −k)x ] ,(2.6) dm sink x d xm = (−1)k 2k k∑ q=0 (−1)q ( k q ) (2q −k)m cos [ (m−k) π 2 + (2q −k)x ] ,(2.7) 34 guo and qi and k∑ q=0 ( k q ) (2q −k)m sin [ π 2 m + (2q −k)x ] = 0, k∑ q=0 (−1)q ( k q ) (2q −k)m sin [ (m−k) π 2 + (2q −k)x ] = 0. proof. these identities follow from directly differentiating on all the sides of the identities in corollary 2.1. � remark 2.1. the formulas (2.6) and (2.7) were established and applied in the paper [31]. theorem 2.2. for n ∈ n, we have (2.8) i2n = π 22n+1 ( 2n n ) . first proof. a direct calculation reveals that i2n = ∫ π/2 0 ( eix + e−ix 2 )2n d x = 1 22n ∫ π/2 0 2n∑ `=0 ( 2n ` ) ei`xe−i(2n−`)x d x = 1 22n 2n∑ `=0 ( 2n ` )∫ π/2 0 ei(2`−2n)x d x = 1 22n [( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` )∫ π/2 0 ei(2`−2n)x d x + π 2 ( 2n n )] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) 1 i(2`− 2n) [ ei(2`−2n)π/2 − 1 ] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) i 2`− 2n [ 1 −ei(2`−2n)π/2 ] = π 22n+1 ( 2n n ) + 1 22n ( n−1∑ `=0 + 2n∑ `=n+1 )( 2n ` ) 1 2(`−n) sin 2(`−n)π 2 = π 22n+1 ( 2n n ) . consequently, the formula (2.8) is proved. � second proof. by virtue of (2.3), it follows that i2n = 1 22n 2n∑ `=0 (−1)` ( 2n ` )∫ π/2 0 cos[(2`− 2n)x−nπ] d x = (−1)n 22n 2n∑ `=0 (−1)` ( 2n ` )∫ π/2 0 cos[(2`− 2n)x] d x on the wallis formula 35 = (−1)n 22n [ (−1)n ( 2n n ) π 2 + ( n−1∑ `=0 + 2n∑ `=n+1 ) (−1)` ( 2n ` ) 1 2`− 2n sin (2`− 2n)π 2 ] = π 22n+1 ( 2n n ) . as a result, the formula (2.8) is proved. � third proof. letting ` = 2n and integrating from 0 to π 2 on both sides of (2.2) arrive at the formula (2.8). � remark 2.2. in [2, p. 100], the formula (2.8) was proved alternatively. 3. general multiple-angle formulas for sine and cosine let i = √ −1 be the imaginary unit. then ik =   i, k = 1 + 4`, −1, k = 2 + 4`, −i, k = 3 + 4`, 1, k = 4 + 4`, where k ∈ n and ` ≥ 0. the quantity ik may also be computed by ik = (−1) 1 2 [ k−1−(−1) k 2 ] i 1−(−1)k 2 and ik = ekπi/2 = cos kπ 2 + i sin kπ 2 . it is well known [1, p. 72] that the first few multiple-angle formulas are sin(2x) = 2 sin x cos x, cos(2x) = cos2 x− sin2 x = 2 cos2 x− 1 = 1 − 2 sin2 x, sin(3x) = 3 sin x− 4 sin3 x = 4 sin x sin ( π 3 + x ) sin ( π 3 −x ) , cos(3x) = 4 cos3 x− 3 cos x = 4 cos x cos ( π 3 + x ) cos ( π 3 −x ) , sin(4x) = 8 cos3 x sin x− 4 cos x sin x, cos(4x) = 8 cos4 x− 8 cos2 x + 1. theorem 3.1. for k ≥ 2, the general multiple-angle formulas for the sine and cosine functions are sin(kx) = k∑ `=0 ( k ` ) sin `π 2 sin` x cosk−` x and cos(kx) = k∑ `=0 ( k ` ) cos `π 2 sin` x cosk−` x. proof. by the formula ekxi = cos(kx) + i sin(kx), we have ekxi = ( exi )k = (cos x + i sin x)k 36 guo and qi = k∑ `=0 ( k ` ) i` sin` x cosk−` x = k∑ `=0 ( k ` )[ cos `π 2 + i sin `π 2 ] sin` x cosk−` x = k∑ `=0 ( k ` ) cos `π 2 sin` x cosk−` x + i k∑ `=0 ( k ` ) sin `π 2 sin` x cosk−` x. further equating the real and imaginary parts yields the required general multipleangle formulas for the sine and cosine functions. the proof of theorem 3.1 is complete. � corollary 3.1. for k ≥ 2, we have sin(kx) = b(k−1)/2c∑ `=0 ( k 2` + 1 ) sin (2` + 1)π 2 sin2`+1 x cosk−2`−1 x = b(k−1)/2c∑ `=0 ( k 2` + 1 ) (−1)` sin2`+1 x cosk−2`−1 x and cos(kx) = bk/2c∑ `=0 ( k 2` ) cos(`π) sin2` x cosk−2` x = bk/2c∑ `=0 ( k 2` ) (−1)` sin2` x cosk−2` x, where bxc is called as the floor function which expresses the biggest integer not more than x. references [1] m. abramowitz and i. a. stegun (eds), handbook of mathematical functions with formulas, graphs, and mathematical tables, national bureau of standards, applied mathematics series 55, 10th printing, dover publications, new york and washington, 1972. [2] n. bourbaki, functions of a real variable, elementary theory, translated from the 1976 french original by philip spain. elements of mathematics (berlin). springer-verlag, berlin, 2004. [3] j. cao, d.-w. niu, and f. qi, a wallis type inequality and a double inequality for probability integral, aust. j. math. anal. appl. 4 (2007), no. 1, art. 3. [4] c.-p. chen and f. qi, best upper and lower bounds in wallis’ inequality, j. indones. math. soc. (mihmi) 11 (2005), no. 2, 137–141. [5] c.-p. chen and f. qi, completely monotonic function associated with the gamma function and proof of wallis’ inequality, tamkang j. math. 36 (2005), no. 4, 303–307. [6] c.-p. chen and f. qi, the best bounds in wallis’ inequality, proc. amer. math. soc. 133 (2005), no. 2, 397–401. [7] c.-p. chen and f. qi, the best bounds to (2n)! 22n(n!)2 , math. gaz. 88 (2004), 540–542. [8] j. t. chu, a modified wallis product and some applications, amer. math. monthly 69 (1962), no. 5, 402–404. [9] t. dana-picard and d. g. zeitoun, parametric improper integrals, wallis formula and catalan numbers, internat. j. math. ed. sci. tech. 43 (2012), no. 4, 515–520. on the wallis formula 37 [10] b.-n. guo and f. qi, a class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, j. korean math. soc. 48 (2011), no. 3, 655–667. [11] b.-n. guo and f. qi, a class of completely monotonic functions involving the gamma and polygamma functions, cogent math. 1 (2014), 1:982896, 8 pages. [12] b.-n. guo and f. qi, logarithmically complete monotonicity of a power-exponential function involving the logarithmic and psi functions, glob. j. math. anal. 3 (2015), no. 2, 77–80. [13] b.-n. guo and f. qi, on the increasing monotonicity of a sequence originating from computation of the probability of intersecting between a plane couple and a convex body, turkish j. anal. number theory 3 (2015), no. 1, 21–23. [14] b.-n. guo and f. qi, sharp inequalities for the psi function and harmonic numbers, analysis (berlin) 34 (2014), no. 2, 201–208. [15] b.-n. guo, f. qi, j.-l. zhao, and q.-m. luo, sharp inequalities for polygamma functions, math. slovaca 65 (2015), no. 1, 103–120. [16] s. guo, j.-g. xu, and f. qi, some exact constants for the approximation of the quantity in the wallis’ formula, j. inequal. appl. 2013, 2013:67, 7 pages. [17] t. hyde, a wallis product on clovers, amer. math. monthly 121 (2014), no. 3, 237–243. [18] d. k. kazarinoff, on wallis’ formula, edinburgh math. notes no. 40 (1956), 19–21. [19] s. koumandos, remarks on a paper by chao-ping chen and feng qi, proc. amer. math. soc. 134 (2006), 1365–1367. [20] m. kovalyov, removing magic from the normal distribution and the stirling and wallis formulas, math. intelligencer 33 (2011), no. 4, 32–36. [21] v. krasniqi and f. qi, complete monotonicity of a function involving the p-psi function and alternative proofs, glob. j. math. anal. 2 (2014), no. 3, 204–208. [22] p. levrie and w. daems, evaluating the probability integral using wallis’s product formula for π, amer. math. monthly 116 (2009), no. 6, 538–541. [23] d. s. mitrinović, analytic inequalities, springer, berlin, 1970. [24] c. mortici and f. qi, asymptotic formulas and inequalities for the gamma function in terms of the tri-gamma function, results math. 66 (2015), in press. [25] f. qi, a completely monotonic function involving the gamma and tri-gamma functions, available online at http://arxiv.org/abs/1307.5407. [26] f. qi, a completely monotonic function related to the q-trigamma function, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 76 (2014), no. 1, 107–114. [27] f. qi, bounds for the ratio of two gamma functions, j. inequal. appl. 2010 (2010), article id 493058, 84 pages. [28] f. qi, bounds for the ratio of two gamma functions: from gautschi’s and kershaw’s inequalities to complete monotonicity, turkish j. anal. number theory 2 (2014), no. 5, 152–164. [29] f. qi, complete monotonicity of a function involving the triand tetra-gamma functions, proc. jangjeon math. soc. 18 (2015), no. 2, 253–264. [30] f. qi, complete monotonicity of functions involving the q-trigamma and q-tetragamma functions, rev. r. acad. cienc. exactas f́ıs. nat. ser. a math. racsam. 109 (2015), in press. [31] f. qi, derivatives of tangent function and tangent numbers, available online at http://arxiv.org/abs/1202.1205. [32] f. qi, integral representations and complete monotonicity related to the remainder of burnside’s formula for the gamma function, j. comput. appl. math. 268 (2014), 155–167. [33] f. qi, properties of modified bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, math. inequal. appl. 18 (2015), no. 2, 493–518. [34] f. qi, l.-h. cui, and s.-l. xu, some inequalities constructed by tchebysheff ’s integral inequality, math. inequal. appl. 2 (1999), no. 4, 517–528. [35] f. qi and b.-n. guo, necessary and sufficient conditions for a function involving divided differences of the diand tri-gamma functions to be completely monotonic, available online at http://arxiv.org/abs/0903.3071. [36] f. qi and b.-n. guo, integral representations and complete monotonicity of remainders of the binet and stirling formulas for the gamma function, researchgate technical report, available online at http://dx.doi.org/10.13140/2.1.2733.3928. [37] f. qi and w.-h. li, a logarithmically completely monotonic function involving the ratio of gamma functions, available online at http://arxiv.org/abs/1303.1877. 38 guo and qi [38] f. qi and q.-m. luo, bounds for the ratio of two gamma functions: from wendel’s asymptotic relation to elezović-giordano-pečarić’s theorem, j. inequal. appl. 2013, 2013:542, 20 pages. [39] f. qi and q.-m. luo, bounds for the ratio of two gamma functions—from wendel’s and related inequalities to logarithmically completely monotonic functions, banach j. math. anal. 6 (2012), no. 2, 132–158. [40] f. qi and q.-m. luo, complete monotonicity of a function involving the gamma function and applications, period. math. hungar. 69 (2014), no. 2, 159–169. [41] f. qi and b.-n. guo, a note on additivity of polygamma functions, available online at http://arxiv.org/abs/0903.0888. [42] f. qi and c. mortici, some best approximation formulas and inequalities for the wallis ratio, appl. math. comput. 253 (2015), 363–368. [43] f. qi and c. mortici, some inequalities for the trigamma function in terms of the digamma function, available online at http://arxiv.org/abs/1503.03020. [44] f. qi, c. mortici, and b.-n. guo, some properties of a sequence arising from computation of the intersecting probability between a plane couple and a convex body, researchgate research, available online at http://dx.doi.org/10.13140/rg.2.1.1176.0165. [45] f. qi and s.-h. wang, complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified bessel functions, glob. j. math. anal. 2 (2014), no. 3, 91–97. [46] f. qi and x.-j. zhang, complete monotonicity of a difference between the exponential and trigamma functions, j. korea soc. math. educ. ser. b pure appl. math. 21 (2014), no. 2, 141–145. [47] j. wästlund, an elementary proof of the wallis product formula for pi, amer. math. monthly 114 (2007), no. 10, 914–917. [48] g. n. watson, a note on gamma functions, proc. edinburgh math. soc. 11 (1958/1959), no. 2, edinburgh math notes no. 42 (misprinted 41) (1959), 7–9. [49] y.-q. zhao and q.-b. wu, wallis inequality with a parameter, j. inequal. pure appl. math. 7 (2006), no. 2, art. 56. 1school of mathematics and informatics, henan polytechnic university, jiaozuo city, henan province, 454010, china 2college of mathematics, inner mongolia university for nationalities, tongliao city, inner mongolia autonomous region, 028043, china 3department of mathematics, college of science, tianjin polytechnic university, tianjin city, 300387, china ∗corresponding author international journal of analysis and applications volume 18, number 5 (2020), 774-783 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-774 class of (n, m)-power-d-hyponormal operators in hilbert space cherifa chellali1,∗, abdelkader benali2 1higher school of economics oran, algeria 2faculty of the exact sciences and computer, mathematics department, university of hassiba benbouali, chlef algeria. b.p. 151 hay essalem, chlef 02000, algeria ∗corresponding author: benali4848@gmail.com abstract. in this paper, we introduce a new classes of operators acting on a complex hilbert space h, denoted by [(n,m)dh], called (n,m)-power-d-hyponormal associated with a drazin inversible operator using its drazin inverse. some proprieties of (n,m)-power-d-hyponormal, are investigated with some examples. 1. introduction let h be a complex hilbert space. let b(h) be the algebra of all bounded linear operators defined in h. let t be an operator in b(h). the operator t is called normal if it satisfies the following condition t ∗t = tt ∗ , i.e.,t commutes with t ∗. the class of quasi-normal operators was first introduced and studied by a. brown in [5] in 1953. the operator t is quasi-normal if t commutes with t ∗t , i.e. t (t ∗t ) = (t ∗t )t and it is denoted by [qn]. a.a.s. jibril [6, 7], in 2008 introduced the class of n power normal operators as a generalization of normal operators. the operator t is called n power normal if t n commutes with t ∗ , i.e., t nt ∗ = t ∗t n and is denoted by [nn]. in the year 2011, o.a. mahmoud sid ahmed introduced n power quasi normal operators [14], as a generalization of quasi normal operators. the operator t is called n power quasi normal if t n commutes with t ∗t , i.e., t n(t ∗t ) = (t ∗t )t n and it is denoted by [nqn]. recently in [13], the authors introduced and studied the operator [(n, m)dn] and [(n, m)dq].in this search, received april 11th, 2020; accepted may 26th, 2020; published july 14th, 2020. 2010 mathematics subject classification. primary 47a05, secondary 47a10, 47b20, 47b25. key words and phrases. hilbert space; (n,m)-power-d-hyponormal operators; d-idempotent; power-d-hyponormal operators; (2,2)-power-d-normal operators; drazin inversible operator. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 774 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-774 int. j. anal. appl. 18 (5) (2020) 775 we introduce a new class of operators t namely (n, m)-power-d-hyponormal operator for a positive integer n, m if t ∗m(t d)n ≥ (t d)nt ∗m, m = n = 1, 2, ... denoted by [(n, m)dh]. and we in this work, we will try to apply the same results obtained in [8] for this new classes. definition 1.1. an operator t ∈ b(h) be drazin inversible operator. we said that t is (n, m)-power-dhyponormal operator for a positive integer n, m if t ∗m(t d)n ≥ (t d)nt ∗m, m = n = 1, 2, ... we denote the set of all (n, m)-power-d-hyponormal operators by [(n, m)dh] remark 1.1. clearly n = m = 1, then (1, 1)-power-d-hyponormal operator is precisely power-dhyponormal operator. definition 1.2. an operator t ∈b(h)d is said to be (n, m)-power-d-hyponormal if t ∗m(t d)n−(t d)nt ∗m is positive i.e: t ∗m(t d)n − (t d)nt ∗m ≥ 0 or equivalently 〈 ( t ∗m(t d)n − (t d)nt ∗m ) u | u〉≥ 0 for all u ∈h. example 1.1. let t =   3 −2 0 −3   , s =   1 1 −1 0   ∈b(r2). a simple computation shows that t d = 1 9   3 −2 0 −3   , sd =   0 −1 1 1   , s∗ =   1 −1 1 0   , t ∗ =   3 0 −2 −3   . then t ∈ [(2, 2)dh], but t /∈ [(3, 3)dh] and s ∈ [(3, 2)dh], but s /∈ [(2, 2)dh] proposition 1.1. if s, t ∈b(h)d are unitarily equivalent and if t is (n, m)-power-d-hyponormal operators then so is s proof. let t be an (n, m)-power-d-hyponormal operator and s be unitary equivalent of t. then there exists unitary operator u such that s = utu∗ so sn = ut nu∗ int. j. anal. appl. 18 (5) (2020) 776 we have s∗m(sd)n = (ut mu∗) ∗ u(t d)nu∗ = ut ∗mu∗u(t d)nu∗ = ut ∗m(t d)nu∗ ≥ u(t d)nt ∗mu∗ ≥ u(t d)nu∗ut ∗mu∗ = (sd)ns∗m hence, s∗m(sd)n − (sd)ns∗m ≥ 0 � proposition 1.2. let t ∈ b(h)d be an (n, m)-power-d-hyponormal operator. then t ∗ is (n, m)-powerd-co-hyponormal operator proof. since t is (n, m)-power-d-hyponormal operator. we have t ∗m(t d)n ≥ (t d)nt ∗m ⇒ ( t ∗m(t d)n)∗ ≥ ( (t d)nt ∗m )∗ ⇒ (t d)∗nt m ≥ t m(t d)∗n. hence, t ∗ is (n, m)-power-d-co-hyponormal operator. � theorem 1.1. if t, t ∗ are two (n, m)-power-d-hyponormal operator, then t is an (n, m)-power-d-normal operator. proposition 1.3. if t is (2, 2)-power-d-hyponormal operator and t dt ∗ = −t ∗t d. tthen t is (2, 2)power-d-normal operator. proof. since (t d)2t ∗2 = t dt dt ∗t ∗ = −t dt ∗t dt ∗ = t dt ∗t ∗t d = −t ∗t dt ∗t d = t ∗2(t d)2 and t ∗2(t d)2 = t ∗t ∗t dt d = −t ∗t dt ∗t d = t dt ∗t ∗t d = −t dt ∗t dt ∗ = (t d)2t ∗2 so int. j. anal. appl. 18 (5) (2020) 777 t is (2, 2)-power-d-hyponormal, then (t d)2t ∗2 ≤ t ∗2(t d)2 ⇒ t dt dt ∗t ∗ ≤ t ∗t ∗t dt d ⇒ −t dt ∗t dt ∗ ≤−t ∗t dt ∗t d ⇒ t dt ∗t dt ∗ ≥ t ∗t dt ∗t d ⇒ t dt ∗t ∗t d ≥ t dt ∗t ∗t d ⇒ −t ∗t dt ∗t d ≥−t dt ∗t dt ∗ ⇒ t ∗t dt ∗t d ≤ t dt ∗t dt ∗ ⇒ t ∗2(t d)2 ≥ (t d)2t ∗2. hence t ∗2(t d)2 = (t d)2t ∗2. � example 1.2. let t =   1 0 −1 0 0 0 1 0 −1   ∈ b(c3). a simple computation, shows that ; t∗ =   1 0 1 0 0 0 −1 0 −1   ,td =   0 0 0 0 0 0 0 0 0   . then power-d-hyponormal operator, but t∗t 6= tt ∗and ‖tu‖ 6≥ ‖t ∗u‖. lemma 1.1. let tk, sk ∈b(h)d, k = 1, 2 such that t1 ≥ t2 ≥ 0 and s1 ≥ s2 ≥ 0, then ( t1 ⊗s1 ) ≥ ( t2 ⊗s2 ) ≥ 0. theorem 1.2. . let t, s ∈b(h)d, such that (sd)ns∗ ≥ 0 and (t d)nt ∗ ≥ 0, then . t ⊗s is (n, 1)-power-d-hyponormal if and only if t and s are (n, 1)-power-d-hyponormal operators proof. assume that t, s are (n, 1)-power-d-hyponormal operators. then ( ( t ⊗s )d )n ( t ⊗s )∗ = ( t d ⊗sd )n( t ∗ ⊗s∗ ) = (t d)nt ∗ ⊗ (sd)ns∗ ≤ t ∗(t d)n ⊗s∗(sd)n = ( t ⊗s )∗ ( ( t ⊗s )d )n. which implies that t ⊗s is (n, 1)-power-d-hyponormal operator. int. j. anal. appl. 18 (5) (2020) 778 conversely, assume that t ⊗s is (n, 1)-power-d-hyponormal operator.we aim to show that t, s are (n, 1)power-d-hyponormal. since t ⊗s is a (n, 1)-power-d-hyponormal operator, we obtain (t ⊗s) is (n, 1)-power-d-hyponormal ⇐⇒ (( ( t ⊗s )d )n ( t ⊗s )∗ ≤ (t ⊗s)∗((t ⊗s)d)n ⇐⇒ (t d)nt ∗ ⊗ (sd)ns∗ ≤ t ∗(t d)n ⊗s∗(sd)n. then, there exists d > 0 such that   d (t d)nt ∗ ≤ t ∗(t d)n. and d−1(sd)ns∗ ≤ s∗(sd)n a simple computation shows that d = 1 and hence (t d)nt ∗ ≤ t ∗(t d)n and (sd)ns∗ ≤ s∗(sd)n. therefore, t, s are (n, 1)-power-d-hyponormal. � proposition 1.4. if t, s ∈ b(h)d are (n, 1)-d-power-hyponormal operators commuting, such that such that s∗(sd)nt ∗(t d)n ≥ (sd)ns∗(t d)nt ∗ ≥ 0 and (t d)nt ∗ ≥ 0, then ts ⊗ t, ts ⊗ s, st ⊗ t and st ⊗s ∈b(h⊗h)d are (n, 1)-power-d-power-d-hyponormal if the following assertions hold: (1) s∗(t d)n = (t d)ns∗. (2) t ∗(sd)n = (sd)nt ∗. proof. assume that the conditions (1) and (2) are hold. since t and s are (n, 1)-power-d-hyponormal, we have ( ( ts ⊗t )d )n ( ts ⊗t )∗ = ( (ts)d ⊗t d )n( (ts)∗ ⊗t ∗ ) = ( ((ts)d)n(ts)∗ ⊗ (t d)nt ∗ ) = ( ((sd)n(t d)n)s∗t ∗ ⊗ (t d)nt ∗ ) = ( (sd)ns∗(t d)nt ∗ ⊗ (t d)nt ∗ ) int. j. anal. appl. 18 (5) (2020) 779 ≤ ( s∗(sd)nt ∗(t d)n ⊗t ∗(t d)n ) = ( s∗t ∗(sd)n(t d)n ⊗t ∗(t d)n ) = ( (ts)∗((ts)d)n ⊗t ∗(t d)n ) = ( (ts)∗ ⊗t ∗ )( ((ts)d)n ⊗ (t d)n ) = ( ts ⊗t )∗ ( ( ts ⊗t )d )n then ts ⊗s is (n, 1)-power-d-hyponormal operator. in the same way, we may deduce the (n, 1)-power-d-hyponormal operator of ts⊗s, st ⊗t and st ⊗s. � theorem 1.3. if t, s ∈b(h)d two operators commuting. then :( i ⊗s ) , ( t ⊗ i ) are (n, 1)-power-d-hyponormal then t � s is (n, 1)-power-d-hyponormal. proof. firstly, observe that if ( i ⊗ s ) , ( t ⊗ i ) are (n, 1)-power-d-hyponormal, then we have following inequalities ( (t ⊗ i )d )n ( t ⊗ i )∗ ≤ (t ⊗ i)∗((t ⊗ i)d)n and ( (s ⊗ i )d )n ( s ⊗ i )∗ ≤ (s ⊗ i)∗((s ⊗ i)d)n. then ((t � s)d)n(t � s)∗ = ( (t ⊗ i + i ⊗s)d )n( t ⊗ i + i ⊗s )∗ = ( (t ⊗ i)d + (i ⊗s)d )n( (t ⊗ i)∗ + (i ⊗s )∗ ≤ ((t ⊗ i)d)n ( t ⊗ i)∗ + ((t ⊗ i)d)n(i ⊗s )∗ + ((i ⊗s)d)n(t ⊗ i)∗ + ((i ⊗s)d)n(i ⊗s )∗ ≤ ( t ⊗ i)∗((t ⊗ i)d)n + (i ⊗s )∗ ((t ⊗ i)d)n + (t ⊗ i)∗((i ⊗s)d)n + (i ⊗s )∗ ((i ⊗s)d)n = (t � s)∗((t � s)d)n. then t � s is (n, 1)-power-d-hyponormal. � theorem 1.4. let t1, t2, ...., tm are (n, 1)-power-d-hyponormal operator in b(h)d, such that (t dk ) nt ∗k ≥ 0, ∀k ∈ {1, 2...m} . then (t1 ⊕t2 ⊕ ....⊕tm) is (n, 1)-power-d-hyponormal operators and (t1 ⊗t2 ⊗ ....⊗tm) is (n, 1)-power-d-hyponormal operators. int. j. anal. appl. 18 (5) (2020) 780 proof. since ( (t1 ⊕t2 ⊕ ...⊕tm)d )n (t1 ⊕t2 ⊕ ...⊕tm) ∗ = ( (t d1 ) n ⊕ (t d2 ) n ⊕ ...⊕ (t dm ) n ) (t ∗1 ⊕t ∗ 2 ⊕ ...⊕t ∗ m) = ((t d1 ) nt ∗1 ⊕ (t d 2 ) nt ∗2 ⊕ ...⊕ (t d m ) nt ∗m) ≤ (t ∗1 (t d 1 ) n ⊕t ∗(t d2 ) n 2 ⊕ ...⊕t ∗ m(t d m ) n) = (t ∗1 ⊕t ∗ 2 ⊕ ...⊕t ∗ m) ( (t d1 ) n ⊕ (t d2 ) n ⊕ ...⊕ (t dm ) n ) = (t1 ⊕t2 ⊕ ...⊕tm) ∗ ( (t1 ⊕t2 ⊕ ...⊕tm)d )n . then (t1 ⊕t2 ⊕ ....⊕tm) is (n, 1) -power-d-hyponormal operators. now, ( (t1 ⊗t2 ⊗ ...⊗tm)d )n (t1 ⊗t2 ⊗ ...⊗tm) ∗ = ( (t d1 ) n ⊗ (t d2 ) n ⊗ ...⊗ (t dm ) n ) (t ∗1 ⊗t ∗ 2 ⊗ ...⊗t ∗ m) = ((t d1 ) nt ∗1 ⊗ (t d 2 ) nt ∗2 ⊗ ...⊗ (t d m ) nt ∗m) ≤ (t ∗1 (t d 1 ) n ⊗t ∗(t d2 ) n 2 ⊗ ...⊗t ∗ m(t d m ) n) = (t ∗1 ⊗t ∗ 2 ⊗ ...⊗t ∗ m) ( (t d1 ) n ⊗ (t d2 ) n ⊗ ...⊗ (t dm ) n ) = (t1 ⊗t2 ⊗ ...⊗tm) ∗ ( (t1 ⊗t2 ⊗ ...⊗tm)d )n . then (t1 ⊗t2 ⊗ ....⊗tm) is (n, 1) -power-d-hyponormal operators. � proposition 1.5. if t is (2, 1)-power-d-hyponormal and t is d-idempotent. then t is power-dhyponormal operator proof. since t is (2, 1)-power-d-hyponormal operator, then (t d)2t ∗ ≤ t ∗(t d)2 since t is d-idempotent (t d)2 = t d, wich implies t dt ∗ ≤ t ∗t d thus t is is power-d-hyponormal operator � proposition 1.6. if t is (3, 1)-power-d-hyponormal and t is d-idempotent. then t is power-dhyponormal operator proof. since t is (3, 1)-power-d-hyponormal operator, then (t d)3t ∗ ≤ t ∗(t d)3 since t is d-idempotent (t d)2 = t d, wich implies (t d)t ∗ ≤ t ∗t d then t is power-d-hyponormal operator � int. j. anal. appl. 18 (5) (2020) 781 proposition 1.7. if t, s are (2, 1)-power-d-hyponormal operators commuting, such that t ds∗ = s∗t d and t ds −st d = 0, then s + t is (2, 1)-power-d-hyponormal operator. proof. since t ds −st d = 0, hence (t d)2s2 + s2(t d)2 = 0, so ( sd + t d )2 = (sd)2 + (t d)2. ( (t + s)d )2 (s + t ) ∗ = ( (sd)2 + (t d)2 ) (s∗ + t ∗) = (sd)2s∗ + (sd)2t ∗ + (t d)2s∗ + (t d)2t ∗ = (sd)2s∗ + t ∗(sd)2 + s∗(t d)2 + (t d)2t ∗ ≤ s∗(sd)2 + t ∗(sd)2 + s∗(t d)2 + t ∗(t d)2 = (s + t ) ∗ ( (t + s)d )2 then s + t is (2, 1)-power-d-hyponormal operator. � proposition 1.8. if t, s are (2, 1)-power-d-hyponormal operators commuting, such that t ds∗ = s∗t d and t ds −st d = 0, ts = st = s + t then st is (2, 1)-power-d-hyponormal operator. proof. since t ds −st d = 0, hence (t d)2s2 + s2(t d)2 = 0, so ( sd + t d )2 = (sd)2 + (t d)2. since, ( (st )d )2 (st ) ∗ = ( (t + s)d )2 (s + t ) ∗ = (sd)2s∗ + (sd)2t ∗ + (t d)2s∗ + (t d)2t ∗ = (sd)2s∗ + t ∗(sd)2 + s∗(t d)2 + (t d)2t ∗ ≤ s∗(sd)2 + t ∗(sd)2 + s∗(t d)2 + t ∗(t d)2 = (s + t ) ∗ ( (t + s)d )2 = (st ) ∗ ( (ts)d )2 hence( (st )d )2 (st ) ∗ ≥ (st )∗ ( (st )d )2 . then st is (2, 1)-power-d-hyponormal operator. � example 1.3. let t =   1 1 1 −1   , s =   −1 1 1 1   ∈b(c2). a simple computation shows that t ∗ =   1 1 1 −1   , s∗ =   −1 1 1 1   , t d = 1 2   1 1 1 −1   , sd = 1 2   −1 1 1 1   . int. j. anal. appl. 18 (5) (2020) 782 then t is (2, 1)-power-d-hyponormal operator, but〈( (t d)2t ∗ −t ∗(t d)2 ) u v   |   u v   〉 = 0. for all (u, v) ∈ (c2) and s is (2, 1)-power-d-hyponormal operator, but〈( (sd)2s∗ −s∗(sd)2 ) u v   |   u v   〉 = 0. for all (u, v) ∈ (c2) such that ts + st = 0 and t ds∗ 6= s∗t d but s + t and st are (2, 1)-power-d-hyponormal operator the following example shows that proposition (1.7) is not necessarily true if t ds∗ 6= s∗t d proposition 1.9. let t, s ∈ b(h)d are commuting and are (n, 1)-power-d-hyponormal operators, such that t ds∗ = s∗t d and (t + s) ∗ is commutes with ∑ 1≤p≤n−1 ( n p )( (t d)p(sd)n−p ) . then (t + s) is an (n, 1)-power-d-hyponormal operator. proof. since ( (t + s)d )n( t + s )∗ =   ∑ 0≤p≤n ( n p )( (t d)p(sd)n−p )(t + s)∗ = (sd)ns∗ + ∑ 1≤p≤n−1 ( n p )( (t d)p(sd)n−p )( t + s )∗ + (t d)ns∗ + (sd)nt ∗ + (t d)nt ∗ = (sd)ns∗ + ∑ 1≤p≤n−1 ( n p )( (t d)p(sd)n−p )( t + s )∗ + s∗(t d)n + t ∗(sd)n + (t d)nt ∗ ≤ s∗(sd)n + ( t + s )∗ ∑ 1≤p≤n−1 ( n p )( (t d)p(sd)n−p ) + s∗(t d)n + t ∗(sd)n + t ∗(t d)n ≤ ( t + s )∗  ∑ 0≤p≤n ( n p )( (t d)p(sd)n−p ) = ( t + s )∗( (t + s)d )n . then (t + s) is an (n, 1)-power-d-hyponormal operator. � int. j. anal. appl. 18 (5) (2020) 783 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.benali, on the class of n-real power positive operators on a hilbert space. funct. anal. approx. comput. 10(2)(2018), 23-31. [2] a. aluthge, onp-hyponormal operators for 0 < p < 1, integr. equ. oper. theory. 13 (1990), 307–315. [3] a. benali, on the class of (a,n)-real power positive operators in semi-hilbertian space, glob. j. pure appl. sci. 25 (2019), 161-166. [4] a. benali, s.a. ould ahmed mahmoud, (α,β)-a-normal operators in semi-hilbertian spaces, afr. mat. 30 (2019), 903–920. [5] a. brown, on a class of operators, proc. amer. math. soc. 4 (1953), 723-728. [6] a.a.s. jibril, on n-power normal operators, arab. j. sci. eng. 33 (2a) (2008), 247-251. [7] a.a.s. jibril, on subprojection sperators, int. math. j. 4 (3) (2003), 229–238. [8] c. cherifa, a. benali, class of (a,n)-power-hyponormal and quasihyponormal operators in semi-hilbertian space, funct. anal. approx. comput. 11 (2) (2019), 13-21. [9] t. veluchamy, k.m. manikandan, n-power-quasi normal operators on the hilbert space, iosr j. math. 12 (1) (2016), 6-9. [10] m.h. mortad, on the normality of the sum of two normal operators, complex anal. oper. theory. 6 (2012), 105–112. [11] m. guesba, m. nadir, on operators for wich t2 ≥ −t∗2. aust. j. math. anal. appl. 13 (1) (2016), article 6. [12] s.a. mohammady, s.a.o. beinane, s.a.o.a. mahmoud, on (n,m)-a-normal and (n, m)-a-quasinormal semihilbert space operators, arxiv:1912.03304 [math]. (2019). [13] s.a.o.a. mahmoud, o.b.s. ahmed, on the classes of (n,m)-power d-normal and (n,m)-power d-quasi-normal operators, oper. matrices. (2019), 705–732. [14] o.a.m.s. ahmed, on the class of n-power quasi-normal operators on hilbert spaces, bull. math. anal. appl. 3 (2) (2011), 213–228. [15] s.a. ould ahmed mahmoud, on some normality-like properties and bishop’s property (β) for a class of operators on hilbert spaces, int. j. math. math. sci. 2012 (2012), 975745. [16] o.a.m.s. ahmed, a. benali, hyponormal and k-quasi-hyponormal operators on semi-hilbertian spaces, aust. j. math. anal. appl. 13 (1) (2016), article 7. [17] r.r. al-mosawi and h.a. hassan a note on normal and n-normal operators, j. univ. thi-qar 9 (1) 2014, 1-4. [18] s.h. jah, class of (a, n)-power-quasi-hyponormal operators in semi-hilbertian spaces, int. j. pure appl. math. 93 (1) (2014), 61-83. [19] s. panayappan, n. sivamani, on n-binormal operators gen. math. notes, 10 (2) (2012), 1-8. 1. introduction references international journal of analysis and applications volume 19, number 4 (2021), 576-586 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-576 a comparative study of the impact of dummy variables on regression coefficients and canonical correlation indices: an empirical perspective nsisong ekong1,∗, imoh moffat2, anthony usoro1, iseh matthew1 1department of statistics, akwa ibom state university, ikot akpaden, akwa ibom state, nigeria 2department of statistics, university of uyo, uyo, akwa ibom state, nigeria ∗corresponding author: nsisongekong@aksu.edu.ng abstract. in this paper, the impact of dummy variables on regression coefficients and canonical correlation indices from an empirical perspective is investigated. to do this, a regression analysis of crude oil prices on us dollars naira exchange rates is performed, and the extent of the significance of the relationship is noted. secondly, dummy variables (coded with respect to various economic regimes of interest) is introduced into the regression of the two variables and the impact of such introduction is also noted. and also, a canonical correlation analysis (cca) of inflation rate, the dummy variables and crude oil prices and the dummy variables is conducted. finally, we compare the significant role of the introduction of the dummy variables on the coefficients of the regression and the canonical correlation indices. the results showed that the introduction of dummy variables impact more on the canonical correlation indices than it does on the regression coefficients. 1. introduction there are numerous approaches to unravelling the relationships between two or more entities; regression and correlation analyses, amongst others, are some of these approaches. with these myriads of techniques, researchers are often faced with the problems of which to use, when, where and how to use them. although received january 13th, 2021; accepted february 3rd, 2021; published june 17th, 2021. 2010 mathematics subject classification. 62p10. key words and phrases. regression analysis; canonical correlation analysis (cca); dummy variables; crude oil prices; exchange rates; nigeria. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 576 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-576 int. j. anal. appl. 19 (4) (2021) 577 misuses of statistics are inadvertent, it is necessary to put things in check and raise the necessary alarm on its consequences such as erroneous relationships and flawed conclusions. in this paper, focus is on canonical correlation and regression analysis. some of these consequences are timebound while others span through ages as they are printed and relied upon by others in the field. therefore, getting to know the proper usability of these techniques is of essence. in the case of analysing the relationships between two or more variables, we, more often than not, see insignificant relationships whereas there may exist a significant one in actual sense, and vice versa. this issue may be attributed to the approach we use in the analysis. the question now begging for answer is can we checkmate the problem of false results (significance or nonsignificance)? putting it differently, can a relationship be significant in one approach and insignificant in another? in attempting to answer this, we need to put the methods of interest side by side while comparing their pros and cons. also, we sometimes encounter the problem of interrelationship between different approaches. for instance, in ascertaining the relationship between quantitative and qualitative variables, we could be faced with questions like: would an introduction of dummy variables to a regression line be similar or dissimilar to an introduction of dummy variables to a canonical correlation analysis of these variables? this paper is the at the intersection of the two issues raised above. in the literature, a handful of works have been done on the relationship between regression and canonical correlation analyses. in studying the relationship between canonical correlation analysis and multivariate multiple regression, lutz and eckert (1994) posited that the similarities and dissimilarities between multivariate multiple regression and canonical correlation analysis has been inconsistently acknowledged in the literature. in trying to put this in the right perspective, the authors stated that although the two analyses seems to have different objectives, the underlying aspects of the approaches are mathematically equivalent. in their postulation, they put it that a multivariate multiple regression analysis that incorporates discriminant analysis as part of its post hoc investigation will produce identical result as a canonical correlation analysis in terms of omnibus significance testing, variable weighting schemes, and dimension reduction analysis. similarly, sun, et al (2008) showed that for a high dimensional data, cca in multi-label classification can be formulated as least squares problems. the authors did this by proposing several canonical correlation extensions including sparse cca using what they call 1-norm regularization base on equivalence relationship. also, kakade and foster(2007) in investigating the canonical correlation analysis through multi-view regression approach, provided a semi-supervised algorithm in the multi-view regression problem where the input variable can be partitioned into two different views. this algorithm, according to them, first uses unlabeled data to learn a norm and the uses labeled data in ridge regression algorithm to provide the predictor. by doing this, the authors were able to characterized the intrinsic dimensionality of the subsequent ridge regression problem. in a more succint way, foucart (1998) says that the results obtained by the selection of canonical variables are better than those given by calssical int. j. anal. appl. 19 (4) (2021) 578 regession and principal component regression. apart from these, other pieces of research are carried out on this topic and they could be found in the literature (everitt (2005); george (1975); and others). different from these works, we move to ascertain the impact of dummy variables on the regresion coefficients compared to its impact on canonical correlation indices via an empirical perspective. to do this, we first of all regress the dependent variable (in this case, the exchange rates) with the independent variable (crude oil price in this case). secondly, we introduce dummy variables into the regression model by taking cognizance of pre-covid 19 era, covid 19 era, and post covid 19 era. thirdly, we also increase the dimensionality of both the dependent and independent variables with a dummy variable and use same to conduct a canonical correlation analysis. we compare the various statistics from both methods to see the signicance induced by the introduction of the dummy variables. this paper is organised as follows; in section 1, a general introduction is given, section 2 gives an overview of the methodology employed in the data analysis, and section 3 presents the results of the analyses. final;y, section 4 concludes with a summary. 2. methodology in the analysis of data related to this research, we will employ the regression analysis, dummy regression analysis, and canonical correlation analysis. 2.1. linear regression analysis. regression analysis have become an integral component of any data analysis when it comes to deciphering the relationship between a response variable and explanatory variable(s) given that the dataset are discrete in nature. this is a procedure for fitting a linear model that relates one or more independent variables, x′s to a dependent variable y . the independent variables are also referred to as explanatory variables while the dependent variable is also called response variable. walpole and myers (1989), linear regression is a concept of arriving at the best estimate of the relationship between a particular set of variables linearly. here, the term linearity implies linear in coefficients. the technique typically uncovers the variability by the explanatory variable(s) in the response variable. when the response variable is regress against one explanatory variable, the method is called a simple linear regression. on the other hand, if the number of explanatory variables exceed one, the approach employed in the analysis is called a multiple regression analysis. the general form of the model with n-independent variables is given as (2.1) y = β0 + β1x1 + β2x2 + ... + βnxn + ε where y is the response variable, x1, x2, ...,xn are the predictor variables with associated parameters β1 β2, ..., βn respectively. ε is the error term associated with the model. int. j. anal. appl. 19 (4) (2021) 579 for the variables under consideration, that is us dollars naira exchange rates and crude oil prices (in dollars), we have a simple linear regression of the form, (2.2) y = β0 + β1x1 + ε where y is the us dollars naira exchange rates, x1 is crude oil prices (in dollars) with associated parameters β1, that is, the slope of the regression line. β0 is the regression constant, that is the intercept of the regression line. ε is the error term associated with the model. 2.2. dummy regression analysis. a dummy variable is a numerical variable used in regression analysis to represent sub-groups of the sample in the study. it is used in research design to distinguish different treatment groups. according to draper & smith (1981), an assignment of some levels to variables in order to account for the fact that the variable may have seperate deterministic effects on the response; variables that result from this sort of assignment are called dummy variables. the variables are limited to two specific values, 0 or 1. typically, 1 represents the presence of a qualitative attribute, and 0 represents the absence. in applying the regression technique, we are often presented with explanatory variables which are categorical in nature and/or are in levels. the presence of these levels makes place for such a variable to be coded with dummies. a regression analysis where these dummy variables are used in coding the different levels in the explanatory variables which are categorical in nature is called dummy regression. of course, we should know that if the response variable is also categorical, we make use of the logistic regression technique. the dummy regression model is equivalent to the regular regression model but for the fact that the there is a dummy variable incorporated into the model to make sense of the levels in the explanatory variables. for every k levels, we need k − 1 dummy variables. for the model under consideration, that is, us dollars naira exchange rates and crude oil prices (in dollars) regression model, we will consider 3 levels. these are, the pre-covid-19 era and the covid-19 era. this means we will need 2 dummy variables. the resulting dummy regression model is given as (2.3) y = β0 + β1x1 + α1z1 + ε where y is the us dollars naira exchange rates, x1 is crude oil prices (in dollars) with associated parameters β1, that is, the slope of the regression line. β0 is the regression constant, that is the intercept of the regression line. z1 is the dummy variable coded with 1’s for the pre-covid-19 era and 0’s otherwise. its associated parameter is α1. ε is the error term associated with the model. covid-19 era is used in this case as the reference subgroup, hence not coded with dummies. 2.3. canonical correlation analysis. canonical correlation analysis , introduced by harold hotelling in 1936, is a way of making sense of cross-covariance matrices. canonical correlation analysis (cca) is a well-known technique for finding the correlations between two sets of multi-dimensional variables. it int. j. anal. appl. 19 (4) (2021) 580 projects both sets of variables into a lower-dimensional space in which they are maximally correlated. this nimplies that the canonical correlation indices only provides a measure of linear association between the two variables; when the two are uncorrelated, i.e. where their correlation is zero, this means that there is no linear function that can describe their relationship. thus, some non-linear relationship could suffice. in partitioning the correlation matrix, we consider the two sets of data y1,y2, ...,yp and x1,x2, ...,xq such that we have p dependent variables and q independent variables. the dimensions of the correlation matrix r are (p + q)(p + q). the matrix can be partitioned so that r =   r11 r12 r21 r22   where r11 is the matrix of intercorrelations among the p dependent variables, r22 is the matrix of intercorrelations among the q independent variables, r12 is the matrix of intercorrelations between the p dependent and the q independent variables, and then r21 is the transpose of r12 the partitioned intercorrelation matrix reveals the basic pattern of association within and between the two sets of variables. to find the pattern, or canonical correlations, we first have to find the latent roots of the canonical equation given as (2.4) (r−122 r −1 21 r −1 11 r12 −λi) = 0 where λ is the latent root and i is the identity matrix. eq (2.4) can be written as (2.5) (m −λi) = 0 where m = r−122 r −1 21 r −1 11 r12 we have to note that the matrix m represents the ways in which the two sets covary with one another and the latent roots of this matrix indicates the size of the common patterns. the canonical roots are the square of the latent roots and is given as ω̄i = √ λi,∀i = 1, 2, ...,min{p,q} in order to test the significance of the roots, we use the test statistic known as the wilk’s lambda (λ), which is described from the canonical correlation roots, and is given as (2.6) λ = min{p,q}∏ i=1 (i −λi) to test the hypothesis h00 : the variables are not correlated against int. j. anal. appl. 19 (4) (2021) 581 h10 : the variables are correlated given that we have n independent observations in a sample and λi is the estimated correlation for i=1,2,...,min{p,q}, each row can be tested for significance with the following methods. for the ith row, the test statistic is (2.7) χ2 = −(n− 1 − 1 2 (p + q + 1))logeλ which is asymptotically distributed as a chi-square with pq degrees of freedom. we reject h00 , the hypothesis of no relationship if χ2 ≤ χ2pq(α) in this study, the two sets of variable of interest are the crude oil price with the dummy variable (z1), and the exchange rates with a dummy variable (z1). the over hypothesis with respect to this study is, h01 : the impact of dummy variables on the two approaches are the same against h11 : the impact of dummy variables on the two approaches differs. the statistic of interest will be the coefficient of determination (the square of the correlation coefficient) r2 which measures the strength of the relationship between the variables under investiagtion. we compare this statistic from the two approaches and then either accept the h01 if their differences is negligible, or otherwise reject. 3. data analysis and results r software is used in the analysis of this work. 3.1. data collection. data used for this research work are secondary data obtained from the central bank of nigeria (cbn) website. the exchange rates data can be found on https://www.cbn.gov.ng/rates/exrate.asp while crude oil prices can be found on https://www.cbn.gov.ng/rates/crudeoil.asp. the monthly data are collected from the year 2010 to 2020. 3.2. analyses and results. the data used in this study were dummy coded to reflect the pre-covid 19 and covid 19 era as reflected on table 1 of the appendix. firstly, a regression analysis of crude oil prices on exchange rates was carried out. the output is as given in table 2 of the appendix, and the fitted regression line is given as; (3.1) y = 426.8999 − 2.5217x1 its associated statistics are as follows; residual standard error: 89.29 on 177 degrees of freedom multiple r-squared: 0.3663, adjusted r-squared: 0.3627 f-statistic: 102.3 on 1 and 177 df, p-value: < 2.2e-16. secondly, a multiple regression analysis of exchange rates against crude oil prices and the dummy variable was done. the output is as given in table 3 of the appendix, and the fitted regression line is given below; int. j. anal. appl. 19 (4) (2021) 582 (3.2) y = 513.2538 − 2.0981x1 − 128.4192z1 its associated statistics are s follows; residual standard error: 83.78 on 176 degrees of freedom multiple r-squared: 0.4452, adjusted r-squared: 0.4389 f-statistic: 70.62 on 2 and 176 df, p-value: < 2.2e-16 finally, a canonical correlation analysis was carried out on exchange rates, the dummy variable and crude oil prices with the dummy variable. the output is as presented in table 4 of the appendix canonical correlation analysis of:exchange rates, the dummy variable and crude oil prices with the dummy variable. the correlations are high and the statistic of interest, r2 is given as 1.0000 and 0.2878 for the two roots respectively. 4. discussion of results from the results we obtain from the first analysis given in table 2 of the appendix, although the r2 is relatively small in value, we found that there is a significant relationship linear between the exchange rates (x) and the crude oil prices (y), as the p-value is very small < 2.2e-16. the result also shows that only about 37% of the variation in exchange rates is explained by crude oil prices. in order to ascertain the performance of the exchange rates in the two era (pre-covid 19 and covid 19), we incorporated the dummy variables, coded with 1’s for the pre-covid 19 era and 0’s otherwise. this introduction of dummy variables into the model will also give an opportunity to discover the improvability of the model, as the dummy variable is seen to be statistically significant in the model as shown in table 3 of the appendix. also, the r2 has been noticed to improve from 37% to about 45%. furthermore, as noticed in table 4 of the appendix, the canonical correlation between exchange rates, the dummy variable and crude oil prices with the dummy variable revealed a strong correlation between the two sets of variables. the first canonical r2 has been seen to be a perfect correlation of 100%, far better than that of the multile regression of exchange rates with crude oil prices and the dummy variable. the second canonical r2 of about 29% is also relatively strong. comparing the r2 from the multiple regression analysis and the canonical correlation analysis, we therefore reject the h01 and conclude that the introduction of dummy variables to the approaches impacts more on the canonical correlation analysis than it does on the multiple regression analysis. 5. conclusion in this paper, the impact of dummy variables on canonical correlation indices and regression coefficients was compared empirically. to achieve this, we considered exchange rates and crude oil prices as dependent and independent variable respectively. first of al, a simple linear regression analysis of exchange rates and crude oil prices was carried out. at this point, a degree of linear relationship between the two variables was investigated. having established a significant relationship between the duo, a dummy variable was int. j. anal. appl. 19 (4) (2021) 583 introduced to the model to examine the improvability of the relationship given a regime-based (pre-covid 19 and covid 19 era) approach to studying the relationship. it was uncovered that there is a need to analyse the relationship based on the two era, as the coefficient of the dummy variable was highly significant and the model obviously improved. finally, the canonical correlation indices from the exchange rates and the dummy variable versus the crude oil prices and the dummy variable were observed to outperformed the multiple regression of exchange rates versus the crude oil prices and the dummy variable. the implication of this finding shows that the dummy variable impacts more on the canonical correlation indices than it does on the regression coefficient. 6. appendix table 1. data on exchange rates (x), crude oil prices (y), and the dummy variable (z), this is a monthly dataset from january 2006 to november 2020. the dummy variable is coded to reflect the pre-covid 19 (1’s) and covid 19 (0’s) era. month no. of pregnant women no. of positive enrolled not enrolled january 291 23 16 7 february 48 15 12 3 march 193 12 9 3 april 200 13 11 2 may 89 20 13 7 june 36 12 10 2 july 132 33 16 17 august 96 12 10 2 september 143 9 7 2 october 92 5 0 5 november 250 12 11 1 total 1570 166 115 51 int. j. anal. appl. 19 (4) (2021) 584 table 2. model summary for exchange rates and crude oil prices coefficients: estimate std. error t value pr(< |t|) (intercept) 426.8999 20.3925 20.93 < 2e-16 *** crude.oil.price -2.5217 0.2493 -10.11 < 2e-16 *** — signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 residual standard error: 89.29 on 177 degrees of freedom multiple r-squared: 0.3663, adjusted r-squared: 0.3627 f-statistic: 102.3 on 1 and 177 df, p-value: ¡ 2.2e-16 table 3. model summary for exchange rates and crude oil prices, dummy variable coefficients: estimate std. error t value pr(< |t|) (intercept) 513.2538 25.7657 19.920 ¡ 2e-16 *** crude.oil.price -2.0981 0.2488 -8.433 1.18e-14 *** dummy -128.4192 25.6611 -5.004 1.35e-06 *** — signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 residual standard error: 83.78 on 176 degrees of freedom multiple r-squared: 0.4452, adjusted r-squared: 0.4389 f-statistic: 70.62 on 2 and 176 df, p-value: ¡ 2.2e-16 table 4. canonical correlation analysis of: exchange rates, the dummy variable and crude oil prices with the dummy variable canr canrsq eigen percent cum scree 1 1.0000 1.0000 -2.252e+15 1.000e+02 100********************** 2 0.5365 0.2878 4.041e-01 -1.795e-14 100 canr lr test stat approx f numdf dendf pr(> f) 1 1.00000 0.0000 4 350 2 0.53647 0.7122 71.121 1 176 1.181e-14 *** signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 int. j. anal. appl. 19 (4) (2021) 585 abbreviations. cca, canonical correlation analysis; cbn, central bank of nigeria. consent for publication. not applicable ethics approval and consent to participate. not applicable funding. not applicable availability of data and materials. all data used for supporting the conclusions of this article are available from the public data repository at the website https://zenodo.org/record/4419710. authors’ contributions. nsisong ekong designed, coordinated the research, and conducted the analysis and drafted the manuscript, imoh moffat supervised the research processes, anthony usoro oversaw the technical aspect of the research and ensured appropriate research language usage, and matthew iseh sourced for data and contributed in the analysis. recommendations for further research. there are a number of aspects around the role of dummy variables in canonical correlation analysis (cca) that follow from our findings which will be of essense for further investigation, including a more robust evaluation of the hypothetical test and approaches respectfully constructed and employeded here: (1) factor loading: another approach, though similar to cca, which could be used to tackle this problem is the use of factor loadings for the comparisons. factor loadings are part of the outcome from factor analysis, which serves as a data reduction method designed to explain the correlations between observed variables using a smaller number of factors. this, acording to our projection, may give a deeper and more robust insight into the role of dummy variable in canonical correlation analysis. (2) hypothesis test: setting up hypothesis to be ejected based on r value may either be confusing or misleading. a more appropriate and reliable hypothesis testing approach should be developed for questions like this. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.g. lutz, t.l. eckert, the relationship between canonical correlation analysis and multivariate multiple regression, educ. psychol. measure. 54 (1994), 666–675. [2] n.r. draper, h. smith, applied regression analysis. 2nd ed. wiley, new york, (1981). [3] b.s. everitt, multiple regression and canonical correlation, in: an r and s-plus® companion to multivariate analysis, springer london, london, 2005: pp. 157–170. [4] t. foucart, paper on multiple linear regression on canonical correlation variable.biometrical j. 41 (1998), 559-572. int. j. anal. appl. 19 (4) (2021) 586 [5] l.k. george, a comparison of canonical correlation and multiple regression in the analysis of change. in: proceedings of the 1975 joint statistical association, (1975), 262-269. [6] s.m. kakade, d.p. foster, multi-view regression via canonical correlation analysis. in: international conference on computational learning theory, (2007), 82-96. [7] l. sun, s. ji, j. ye, least squares formulations for canonical correlation analysis. in: international conference on machine learning, (2008), 1024-1031. [8] r.e. walpole, r.h. myers,probability and statistics for engineers and scientists. 4th ed. macmillan publishing company, new york, (1989). 1. introduction 2. methodology 2.1. linear regression analysis 2.2. dummy regression analysis 2.3. canonical correlation analysis 3. data analysis and results 3.1. data collection 3.2. analyses and results 4. discussion of results 5. conclusion 6. appendix abbreviations consent for publication ethics approval and consent to participate funding availability of data and materials authors' contributions recommendations for further research references int. j. anal. appl. (2022), 20:35 on the solutions of the second kind nonlinear volterra-fredholm integral equations via homotopy analysis method a. s. rahby1,∗, m. a. abdou2, g. a. mosa1 1department of mathematics, faculty of science, benha university, 13518, egypt 2department of mathematics, faculty of education, alexandria university, 21544, egypt ∗corresponding author: a.s.rahby@fsc.bu.edu.eg abstract. in this paper, we discuss the existence and uniqueness of the solution of the second kind nonlinear volterra-fredholm integral equations (nv-fies) which appear in mathematical modeling of many phenomena, using picard’s method. in addition, we use banach fixed point theorem to show the solvability of the first kind nv-fies. moreover, we utilize the homotopy analysis method (ham) to approximate the solution and the convergence of the method is investigated. finally, some examples are presented and the numerical results are discussed to show the validity of the theoretical results. 1. introduction in the application of physical mathematics and engineering, the second kind of nv-fies are often arisen [1–8]. therefore, there exist great efforts to approximate the solution of this kind of nv-fies. yousefi and razzaghi [9] presented a numerical method based upon legendre wavelet approximations for solving the nv-fies. cui and du [10] obtained the representation of the exact solution for the nv-fies in the reproducing kernel space and the exact solution was given by the form of series. the approximate solutions of the nv-fies were pesented using modified decomposition method by bildik and inc [11]. ghasemi et al. [12] presented homotopy perturbation method for solving nv-fies. in addition, rationalized haar functions are developed to approximate the solution of the nv-f-hammerstein ies by ordokhani and razzaghi [13]. he’s variational iteration method was used by yousefi [14] to approximate the solution of a type of nv-fies. hashemizadeh et al. [15] introduced an approximation received: oct. 30, 2021. 2010 mathematics subject classification. 45g10, 46b07, 65r20. key words and phrases. nonlinear fredholm-volterra integral equations; picard’s method; homotopy analysis method and error analysis. https://doi.org/10.28924/2291-8639-20-2022-35 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-35 2 int. j. anal. appl. (2022), 20:35 method based on hybrid legendre and block-pulse functions for solving the nv-fies. a computational technique based on the composite collocation method was presented by marzban et al. [16] for the solution of the nv-f-hammerstein ies. moreover, maleknejad et al. [17] utilized a method to solve nv-f-hammerstein ies in terms of bernstein polynomials. parand and rad [18] proposed the collocation method based on radial basis functions to approximate the solution of nv-f-hammerstein ies. a numerical method based on hybrid of block-pulse functions and taylor series is proposed by mirzaee and hoseini [19] to approximate the solution of nv-fies. chen and jiang [20] developed a simple and effective method for solving nv-fies based on lagrange interpolation functions. the approximate solution of the nv-f-hammerstein ies is obtained by gouyandeh et al. [21] using the tau-collocation method. the present paper shall utilize ham for solving the nfvies of the first and second kind. foremost, in section 2, we discuss the solvability of the second kind nf-vies using picard’s method. moreover, in section 3, banach fixed point theorem is used to discuss the existence and uniqueness of the solution of the first kind nf-vies. in addition, the basic idea of ham and how to utilize ham for the nf-vies of the second and first kind are presented in section 4. finally, we present the numerical results in section 5. 2. existence and uniqueness of the second kind nv-fies consider the following second kind nv-fies of the form µu(t) = f (t) + λ ∫ b a k1(t,s)n1(u(s))ds + λ ∫ t 0 k2(t,s)n2(u(s))ds. (2.1) now, we shall discuss the solvability of eq.(2.1) under the following assumptions (1) the function f (t) is continuous in the space c[0,t ], such that ‖f (t)‖c[0,t] = max t∈[0,t] |f (t)| ≤ p1 and µ ∈r−{0}. (2) the kernels k1(t,s) and k2(t,s) are continuous in c[0,t ] and satisfy |k1(t,s)| ≤ p2 and |k2(t,τ)| ≤ p3, ∀t,τ ∈ [0,t ], and 0 ≤ τ ≤ t ≤ t < 1. (3) the nonlinear functions ni (u(t)), i = 1, 2 satisfy i. the lipschitz condition |ni (u2(t)) −ni (u1(t))| ≤li |u2(t) −u1(t)|, ii. the following inequality ‖ni (u(t))‖≤ σi ‖u(t)‖, where p1 : p3,l1,l2,σ1 and σ2 are positive constants. theorem 2.1. if assumptions (1), (2) and (3.i) are satisfied and |λ| < |µ| p2l1 + p3l2t , (2.2) then eq.(2.1) has a unique solution u(t) in the space c[0,t ]. int. j. anal. appl. (2022), 20:35 3 proof. foremost, using picard’s method, the solution of eq.(2.1) can be expressed as a sequence of functions {un(t)} as n →∞ based on um(t) = f (t) + λ ∫ b a k1(t,s)n1(um−1(s))ds + λ ∫ t 0 k2(t,s)n2(um−1(s))ds, (2.3) with u0(t) = f (t). let vm(t) = um(t) −um−1(t), and v0(t) = f (t). (2.4) with un(t) = n∑ i=0 vi (t), n = 1, 2, 3, ... , (2.5) where vm(t),m = 1, 2, ..., are continuous functions. now we shall prove that the series ∞∑ i=0 vi (t) is uniformly convergent. using eqs.(2.3), (2.4) and norm properties yield |µ|‖um(t) −um−1(t)‖≤|λ| ∥∥∥∥ ∫ b a k1(t,s) [n1(um−1(s)) −n1(um−2(s))] ds ∥∥∥∥ + |λ| ∥∥∥∥ ∫ t 0 k2(t,s) [n2(um−1(s)) −n2(um−2(s))] ds ∥∥∥∥ , (2.6) at n = 1, from (2.6) and using the given assumptions, we get |µ|‖v1(t)‖≤ |λ| ∥∥∥∥ ∫ b a |k1(t,s)|l1|v0|ds ∥∥∥∥ + |λ| ∥∥∥∥ ∫ t 0 |k2(t,s)|l2|v0|ds ∥∥∥∥ . (2.7) hence, we obtain ‖v1(t)‖≤ |λ| |µ| (p2l1 + p3l2t )p1, (2.8) where max t∈[0,t] |t| = t. in addition, at n = 2, we have |µ|‖v2(t)‖≤ |λ|‖ ∫ b a |k1(t,s)|l1|v1|ds + |λ|‖ ∫ t 0 |k2(t,s)|l2|v1|ds‖, (2.9) which leads to ‖v2(t)‖≤ ( |λ| |µ| (p2l1 + p3l2t ) )2 p1. (2.10) subsequently, the mathematical induction is applied to obtain ‖vm(t)‖≤ γm1 p1, γ1 = (p2l1 + p3l2t )|λ| |µ| . (2.11) note that, the series ∞∑ i=0 vi (t) is uniformly convergent if and only if the series ∞∑ m=0 γm1 p3 is convergent. therefore, since |λ| < |µ| p2l1+p3l2t , we get γ1 < 1 and this implies that the series ∞∑ m=0 γm1 p3 is convergent. thus, for n →∞, we get u(t) = ∞∑ i=0 vi (t) (2.12) 4 int. j. anal. appl. (2022), 20:35 represents a solution of eq (2.1). now, to show the solution is unique, we assume that there exists another continuous solution ũ(t) of eq.(2.1). so, we get ‖u(t) − ũ(t)‖≤ ∥∥∥∥λ ∫ b a k1(t,s)(n1(u(s) −n1(ũ(s)))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t a k2(t,s)(n2(u(s) −n1(ũ(s)))ds ∥∥∥∥ . (2.13) note that under the given conditions, inequality (2.13) yields ‖u(t) − ũ(t)‖≤ γ1‖u(t) − ũ(t)‖. (2.14) if ‖u(t)− ũ(t)‖ 6= 0, then (2.14) yields γ1 ≥ 1 which is a contradiction. therefore, ‖u(t)− ũ(t)‖ = 0 and it is implied that u(t) = ũ(t) which means the solution is unique. � 3. existence and uniqueness of the first kind nv-fies if we have µ = 0 in eq (2.1), we get the first kind nv-fies f (t) + λ ∫ b a k1(t,s)n1(u(s))ds + λ ∫ t 0 k2(t,s)n2(u(s))ds = 0. (3.1) now, we shall use banach fixed point theorem which is used in case of failure of picard’s method at µ = 0. so, eq.(3.1) will be first expressed in its integral operator form uu = f + uu, (3.2) where uu = u1u + u2u, u1u = λ ∫ b a k1(t,s)n1(u(s))ds, and u2u = λ ∫ t 0 k2(t,s)n2(u(s))ds. for the normality of the integral operator, we use (3.2) with the help of the given assumptions and norm properties to obtain ‖uu‖≤ ∥∥∥∥λ ∫ b a k1(t,s)n1(u(s))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t 0 k2(t,s)n2(u(s))ds ∥∥∥∥ ≤ |λ|p2σ1‖u(s)‖ + |λ|p3σ2t‖u(s)‖ ≤ γ2‖u(s)‖, γ2 = |λ|(p2σ1 + p3σ2t ). (3.3) if |λ| < 1 p2σ1‖+p3σ2t , we get γ2 < 1 which means u is a contraction operator and this implies that the integral operator u has a normality which leads directly after using the condition (1) to the normality of the operator u. int. j. anal. appl. (2022), 20:35 5 for the continuity of the integral operator, if we assume that the two functions u1(t) and u2(t) ∈ c[0,t ] with the help of the norm properties under the given conditions, then we get ‖uu1 −uu2‖ = ‖uu1 −uu2‖ ≤ ∥∥∥∥λ ∫ b a k1(t,s)(n1(u1(s)) −n1(u2(s)))ds ∥∥∥∥ + ∥∥∥∥λ ∫ t 0 k2(t,s)(n2(u1(s)) −n2(u2(s)))ds ∥∥∥∥ ≤ |λ|(p2l1 + p3l2t )‖u1(s) −u2(s)‖ ≤ γ3‖u1(s) −u2(s)‖, γ3 = (p2l1 + p3l2t )|λ|. (3.4) if |λ| < 1 p2l1+p3l2t , we get γ2 < 1 which means u is a contraction operator and leads to the continuity of the integral operator u in the space c[0,t ]. using banach fixed point theorem, u has a unique fixed point that means the nv-fies (3.1) of the first kind has a unique solution. 4. homotopy analysis method for nv-fies we shall introduce the basic idea of ham [22,23] for solving the operator equation n(u(t)) = 0, t ∈ [0,t ] (4.1) where n denotes the nonlinear operator, and u(t) is an unknown function. foremost, we define the homotopy operator h, h(φ,p) = (1 −p)(φ(t; p) −u0(t)) −phn(φ(t; p)), (4.2) where p ∈ [0, 1] is the embedding parameter, h 6= 0 denotes the convergence control parameter, u0(t) describes the initial approximation of the solution of (4.1). considering h(φ,p) = 0, we get the so-called zero-order deformation equation (1 −p)(φ(t; p) −u0(t)) = phn(φ(t; p)). (4.3) for p = 0, we have φ(t; 0) −u0(t) = 0 which implies that φ(t; 0) = u0(t), whereas for p = 1, we have n(φ(t; 1)) = 0 that means φ(t; 1) = u(t), where u(t) is the solution of (4.1). in this way, the variation of parameter p : 0 → 1 corresponds with the change of problem from the trivial problem to the original one (and with the change of solution from u0(t) → u(t)). expanding φ(x; p) into the maclaurin series with respect to p, we get φ(t; p) = φ(t; 0) + ∞∑ m=1 1 m! ∂mφ(t; p) ∂pm ∣∣∣∣∣ p=0 pm, (4.4) by distinguishing vm(t) = ∞∑ m=1 1 m! ∂mφ(t; p) ∂pm ∣∣∣∣∣ p=0 , m = 1, 2, 3, ... . (4.5) 6 int. j. anal. appl. (2022), 20:35 eq.(4.4) becomes φ(t; p) = v0(t) + ∞∑ m=1 vm(t)p m. (4.6) if the above series is convergent at p = 1, we obtain u(t) = ∞∑ m=0 vm(t). (4.7) to determine function vm(t), we differentiate the both sides of eq.(4.3) m times with respect to p, next we divide the received result by m! and we substitute p = 0. herein, we get the so-called m th-order deformation equation (m > 0) vm(t) −χmvm−1(t) = hrm (v̄m−1,t) (4.8) where v̄m−1 = {v0(t),v1(t), . . . ,vm−1(t)}, χm = { 0 m ≤ 1 1 m > 1 (4.9) and rm (v̄m−1,t) = 1 (m− 1)! ( ∂m−1 ∂pm−1 n ( ∞∑ i=0 vi (t)p i ))∣∣∣∣∣ p=0 . (4.10) since, we can not determine the sum of series in (4.7), we shall accept the partial sum of this series u(t) ≈ un(t) = n∑ m=0 vm(t) (4.11) as the approximate solution of considered equation. secondly, we introduce ham for nf-vie (2.1) and operator n can be defined as n(v(t)) =µv(t) − f (t) −λ ∫ b a k1(t,s)n1(v(s))ds −λ ∫ t 0 k2(t,s)n2(v(s))ds. (4.12) applying the ham, we get the following formula for functions vm(t) vm(t) = χmvm−1(t) + hrm (v̄m−1,t) , (4.13) where χm and rm are defined by (4.9) and (4.10), respectively. using definitions of the respective operators, we obtain v1(t) = hr1 (v̄0,t) = h ( µv0(t) − f (t) −λ ∫ b a k1(t,s)n1(v0(s))ds −λ ∫ t 0 k2(t,s)n2(v0(s))ds ) (4.14) int. j. anal. appl. (2022), 20:35 7 and for m ≥ 2, we get vm(t) =(1 + hµ)vm−1(t) − λ h (m− 1)! ∫ b a k1(t,s) [ ∂m−1 ∂pm−1 n2 ( ∞∑ i=0 vi (s)p i )] p=0 ds − λ h (m− 1)! ∫ t a k2(t,s) [ ∂m−1 ∂pm−1 n2 ( ∞∑ i=0 vi (s)p i )] p=0 ds. (4.15) in case of µ = 0, eq.(4.14) becomes v1(t) = hr1 (v̄0,t) = h ( −f (t) −λ ∫ b a k1(t,s)n1(v0(s))ds −λ ∫ t 0 k2(t,s)n2(v0(s))ds ) (4.16) and eq.(4.15) gives vm(t) =vm−1(t) − λ h (m− 1)! ∫ b a k1(t,s) [ ∂m−1 ∂pm−1 n2 ( ∞∑ i=0 vi (s)p i )] p=0 ds − λ h (m− 1)! ∫ t a k2(t,s) [ ∂m−1 ∂pm−1 n2 ( ∞∑ i=0 vi (s)p i )] p=0 ds. (4.17) theorem 4.1. suppose that the nonlinear operators n1 and n2 satisfies lipschitz condition (3.i). if the series ∑+∞ m=0vm(t) converges to u(t), where vm(t) is governed by eq.(4.8) under the definitions (4.9) and (4.10), then u(t) will be the exact solution of the nf-vie (2.1). proof. firstly, we define him (t) = 1 m!   ∂m ∂pm ni   ∞∑ j=0 vj(t)p j     ∣∣∣∣∣∣ p=0 , i = 1, 2. (4.18) from (4.9), we have n∑ m=1 [vm(t) −χmvm−1(t)] =v1(t) + [v2(t) −v1(t)] + [v3(t) −v2(t)] + · · · + [vn(t) −vn−1(t)] = vn(t). (4.19) from the convergence of ∑+∞ m=0vm(t), lim m→∞ vm(t) = 0 , t ∈ [0,t ]. (4.20) using eq.(4.20), eq.(4.19) becomes ∞∑ m=1 [vm(t) −χmvm−1(t)] = lim n→∞ vn(t) = 0. (4.21) 8 int. j. anal. appl. (2022), 20:35 eq.(4.21) and eq.(4.13) yield h ∞∑ m=1 rm (v̄m−1,t) = ∞∑ m=1 vm(t) −χmvm−1(t) = 0. (4.22) since h 6= 0, eq.(4.22) gives ∞∑ m=1 rm (v̄m−1,t) = 0. (4.23) now, eq.(4.12) and definitions (4.5) and (4.10) give 0 = +∞∑ m=1 [rm (~vm−1,t)] = +∞∑ m=1  µvm−1(t)−(1−χm)f (t)−λ∫ b a k1(t,s) ∂m−1 (m−1)!∂pm−1 n1 [ k=+∞∑ k=0 vk(s)p k ]∣∣∣∣∣ p=0 ds −λ ∫ t a k2(t,s) ∂m−1 (m− 1)!∂pm−1 n2 [ k=+∞∑ k=0 vk(s)p k ]∣∣∣∣∣ p=0 ds   = +∞∑ m=1 [ µvm−1(t)−(1−χm)f (t)−λ ∫ b a k1(t,s)h1m−1(s)ds−λ ∫ t a k2(t,s)h2m−1(s)ds ] = +∞∑ m=1 µvm−1(t) −f (t) −λ ∫ b a k1(t,s) +∞∑ m=1 h1m−1(s)ds −λ ∫ t a k2(t,s) +∞∑ m=1 h2m−1(t)ds (4.24) since the nonlinear operators n1 and n2 are contraction; therefore, if the series ∑+∞ m=0 vm(t) converges to u(t), then the series ∑+∞ m=0 h1m−1(t) and ∑+∞ m=0 h2m−1(t) will converge to n1(u(t)) and n2(u(t)), respectively [24]. so, eq.(4.24) becomes µu(t) = f (t) + λ ∫ b a k1(t,s)n1(u(s))ds + λ ∫ t 0 k2(t,s)n2(u(s))ds. (4.25) hence, u(t) is the exact solution of nf-vie (2.1). in case of µ = 0, eq.(4.24) gives f (t) + λ ∫ b a k1(t,s)n1(u(s))ds + λ ∫ t 0 k2(t,s)n2(u(s))ds = 0. (4.26) which indicates that u(t) is the exact solution of nf-vie (3.1). � theorem 4.2. (convergence theorem) assume that h is properly chosen for which there exists 0 < α < 1 so that ‖vk+1‖6 α‖vk‖ ,∀k > k0, for some k0 ∈n, then un (t,h) in (4.11) converges as n −→ +∞. proof. define the sequence um as u0 = v0 u1 = v0 + v1 ... un = v0 + v1 + · · · + vm (4.27) int. j. anal. appl. (2022), 20:35 9 now, we show that um is a cauchy sequence in the space c[0,t ]. consider ‖um+1 −um‖ = ‖vm+1‖6 α‖vm‖6 α2‖vm−1‖6 · · ·6 αm−k0+1‖vk0‖ (4.28) for every l,m ∈n,m > l > k0, we have ‖um−ul‖ = ‖(um −um−1)+(um−1 −um−2) + · · · + (ul+1 −ul)‖ 6 ‖(um −um−1)‖ + ‖(um−1 −um−2)‖ + · · · + ‖(ul+1 −ul)‖ 6 αm−k0 ‖vk0‖ + α m−k0−1‖vk0‖ + · · · + α m−k0+1‖vk0‖ = 1 −αm−l 1 −α αl−k0+1‖vk0‖ . (4.29) since 0 < α < 1, we get lim m,l→∞ ‖um −ul‖ = 0. (4.30) therefore, um is a cauchy sequence in the space c[0,t ] and un = un(t,h) converges as n →∞ and the proof is complete. � theorem 4.3. if assumptions of theorem 4.2 are satisfied, n ∈ n and n ≥ k0, then we obtain the estimation of error of the approximate solution defined by ‖u(t) −un(t)‖≤ αn+1−k0 1 −α ‖vk0‖ . (4.31) proof. . let n ∈ n and n ≥ k0, we get ‖u(t) −un(t)‖ = sup t∈[0,t] ∣∣∣∣∣u(t) − n∑ m=0 vm(x,t) ∣∣∣∣∣ ≤ sup t∈[0,t] ( ∞∑ m=n+1 |vm(t)| ) ≤ ∞∑ m=n+1 sup t∈[0,t] (|vm(x,t)|) ≤ ∞∑ m=n+1 αm−k0 ‖vk0‖ = αn+1−k0 1 −α ‖vk0‖ . (4.32) � 5. illustrating examples in this section, some examples will be given to investigate the efficiency and accuracy of the proposed method 10 int. j. anal. appl. (2022), 20:35 example 5.1. consider the following strongly nv-fie u(t) = f (t) + ∫ 1 0 (t2 − s)u2(s)ds + ∫ t 0 ts2u3(s)ds, (5.1) where f (t) = e−tt2− 6(e2−7)t2+109 8e2 + e −3tt 2187 (3t(3t(t(3t(3t(t(3t(3t + 8) + 56) + 112) + 560) + 2240) + 2240) + 4480) + 4480) − 4480t 2187 + 15 8 and (the exact solution u(t) = t2e−t). in this example, fig.(1) shows the behaviour of error using ham at n = 10 and h = −1.9, t ∈ [0, 0.8]. also, fig.(2) presents the valid region of h at n = 10. 0.2 0.4 0.6 0.8 t 1.×10-16 2.×10-16 3.×10-16 4.×10-16 en(t) figure 1. the error behaviour at n =10,h =−0.1 using ham un(0.8,h) un ′ (0.8,h) u n ′′(0.8,h) -40 -20 20 40 h -4 -2 2 figure 2. the h-curves at n =10 int. j. anal. appl. (2022), 20:35 11 example 5.2. consider the following strongly nv-fie [20] u(t) = f (t) + ∫ 1 0 (t − s)u5(s)ds + ∫ t 0 (t + s)u4(s)ds, (5.2) where f (t) = 77107623−70t(792t5+324787) 151200 + 5t−6 25e5 + 1 16 e−4t(8t + 1) + 5(10t−13) 32e4 + 10(17t−26) 27e3 + 5(19t−42) 4e2 + 5(65t−326) e + 4 27 e−3t(9t(2t+1)+2)+ 1 4 e−2t(6t(t(4t+5)+4)+9)+e−t(4t(t(t(2t+7)+18)+30)+97) and (the exact solution u(t) = t + et). in this example, fig.(3) displays the behaviour of error using ham at n = 10 and h = −1.9. in addition, fig.(4) explains the valid region of h at n = 10. moreover, table 1 presents the maximum error emax = max i |u(ti ) −un(ti )| ∀ti ∈ [0, 0.9] for every n = 2, 5, 8, 12, 16, 20 and a comparison with results in ref. [20]. 0.2 0.4 0.6 0.8 t 2.×10-14 4.×10-14 6.×10-14 8.×10-14 1.×10-13 en(t) figure 3. the error behaviour at n =10,h =−1.9 using ham un(0.9,h) un ′ (0.9,h) u n ′′(0.9,h) -30 -20 -10 10 20 30 h -4 -2 2 4 figure 4. the h-curves at n =10 12 int. j. anal. appl. (2022), 20:35 table 1. the maximum error emax for different values of n with corresponding h for example (5.2) n emax in [20] emax of ham at h = −0.1 2 2.94 × 10−2 4.3876 × 10−12 5 5.95 × 10−4 9.45667 × 10−13 8 1.27e × 10−4 1.3152 × 10−13 12 3.93 × 10−5 1.65706 × 10−13 16 1.60 × 10−5 1.96266 × 10−13 20 7.66 × 10−6 2.02851 × 10−13 example 5.3. consider the following strongly nv-fie f (t) = ∫ 1 0 s2tu2(s)ds + ∫ t 0 st2u3(s)ds, (5.3) where f (t) = −1 4 3t3 cosh(t)+ 1 12 t3 cosh(3t)+ 3 4 t2 sinh(t)− 1 36 t2 sinh(3t)− t 6 + 3 8 t sinh(2)−1 4 t cosh(2) and (the exact solution u(t) = sinh(t)). in this example, fig.(5) shows the behaviour of error using ham at at n = 20, h = 0.1 and t ∈ [0, 0.8]. in addition, fig. (6) investigates the valid region of h at n = 20. 0.2 0.4 0.6 0.8 t 5.×10-17 1.×10-16 1.5×10-16 2.×10-16 en(t) figure 5. the error behaviour at n =20,h =0.1 using ham int. j. anal. appl. (2022), 20:35 13 un(0.8,h) un ′ (0.8,h) u n ′′(0.8,h) -40 -20 20 40 h -0.5 0.5 1.0 1.5 figure 6. the h-curves at n =20 6. conclusion in this paper, we used picard’s method to prove the existence and uniqueness of the solution of the second kind nv-fies which has many application in mathematical physics. moreover, we utilized banach fixed point theorem to discuss the solvability of the first kind nv-fies. in addition, we applied the ham to approximate the solution and discussed the convergence analysis. furthermore, we investigated illustrative examples to indicate the validity and accurately of the presented method showing the error behaviour. based on the results, we observed that that ham is an effective method for solving the first and second kind nf-vies. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] g.a. mosa, m.a. abdou, a.s. rahby, numerical solutions for nonlinear volterra-fredholm integral equations of the second kind with a phase lag, aims math. 6 (2021), 8525–8543. https://doi.org/10.3934/math.2021495. [2] m.a. abdou, m.n. elhamaky, a.a. soliman, g.a. mosa, the behaviour of the maximum and minimum error for fredholm-volterra integral equations in two-dimensional space, j. interdiscip. math. 24 (2021), 2049–2070. https://doi.org/10.1080/09720502.2020.1814497. [3] c. constanda, m.e. pérez, integral methods in science and engineering, volume 1. birkhäuser, boston, 2010. https://doi.org/10.1007/978-0-8176-4899-2. [4] k. e. atkinson, the numerical solution of integral equations of the second kind, cambridge university press, cambridge, 1997. https://doi.org/10.1017/cbo9780511626340. [5] a. jerri, introduction to integral equations with applications, john wiley & sons, 1999. [6] m.m. el-borai, m.a. abdou, m.m. el-kojok, on a discussion of nonlinear integral equation of type volterrafredholm, j. korean soc. ind. appl. math. 10 (2006), 59–83. [7] a.m. wazwaz, linear and nonlinear integral equations: methods and applications, springer berlin, heidelberg, 2011. https://doi.org/10.1007/978-3-642-21449-3. https://doi.org/10.3934/math.2021495 https://doi.org/10.1080/09720502.2020.1814497 https://doi.org/10.1007/978-0-8176-4899-2 https://doi.org/10.1017/cbo9780511626340 https://doi.org/10.1007/978-3-642-21449-3 14 int. j. anal. appl. (2022), 20:35 [8] s.m. zemyan, classical theory of integral equations: a concise treatment, birkhäuser boston, ma, 2012. https: //doi.org/10.1007/978-0-8176-8349-8. [9] s. yousefi, m. razzaghi, legendre wavelets method for the nonlinear volterra-fredholm integral equations, math. computers simul. 70 (2005), 1–8. https://doi.org/10.1016/j.matcom.2005.02.035. [10] m. cui, h. du, representation of exact solution for the nonlinear volterra-fredholm integral equations, appl. math. comput. 182 (2006), 1795–1802. https://doi.org/10.1016/j.amc.2006.06.016. [11] n. bildik, m. inc, modified decomposition method for nonlinear volterra-fredholm integral equations, chaos solitons fractals. 33 (2007), 308–313. https://doi.org/10.1016/j.chaos.2005.12.058. [12] m. ghasemi, m.t. kajani, e. babolian, numerical solutions of the nonlinear volterra-fredholm integral equations by using homotopy perturbation method, appl. math. comput. 188 (2007), 446–449. https://doi.org/10. 1016/j.amc.2006.10.015. [13] y. ordokhani, m. razzaghi, solution of nonlinear volterra-fredholm-hammerstein integral equations via a collocation method and rationalized haar functions, appl. math. lett. 21 (2008), 4–9. https://doi.org/10.1016/ j.aml.2007.02.007. [14] s.a. yousefi, a. lotfi, m. dehghan, he’s variational iteration method for solving nonlinear mixed volterrafredholm integral equations, computers math. appl. 58 (2009), 2172–2176. https://doi.org/10.1016/j. camwa.2009.03.083. [15] e. hashemizadeh, k. maleknejad, b. basirat, hybrid functions approach for the nonlinear volterra-fredholm integral equations, procedia computer sci. 3 (2011), 1189–1194. https://doi.org/10.1016/j.procs.2010.12. 192. [16] h.r. marzban, h.r. tabrizidooz, m. razzaghi, a composite collocation method for the nonlinear mixed volterrafredholm-hammerstein integral equations, commun. nonlinear sci. numer. simul. 16 (2011), 1186–1194. https: //doi.org/10.1016/j.cnsns.2010.06.013. [17] k. maleknejad, e. hashemizadeh, b. basirat, computational method based on bernstein operational matrices for nonlinear volterra-fredholm-hammerstein integral equations, commun. nonlinear sci. numer. simul. 17 (2012), 52–61. https://doi.org/10.1016/j.cnsns.2011.04.023. [18] k. parand, j.a. rad, numerical solution of nonlinear volterra-fredholm-hammerstein integral equations via collocation method based on radial basis functions, appl. math. comput. 218 (2012), 5292–5309. https: //doi.org/10.1016/j.amc.2011.11.013. [19] f. mirzaee, a.a. hoseini, numerical solution of nonlinear volterra-fredholm integral equations using hybrid of block-pulse functions and taylor series, alexandria eng. j. 52 (2013), 551–555. https://doi.org/10.1016/j. aej.2013.02.004. [20] z. chen, w. jiang, an efficient algorithm for solving nonlinear volterra-fredholm integral equations, appl. math. comput. 259 (2015), 614–619. https://doi.org/10.1016/j.amc.2015.02.079. [21] z. gouyandeh, t. allahviranloo, a. armand, numerical solution of nonlinear volterra-fredholm-hammerstein integral equations via tau-collocation method with convergence analysis, j. comput. appl. math. 308 (2016), 435–446. https://doi.org/10.1016/j.cam.2016.06.028. [22] s. liao, homotopy analysis method in nonlinear differential equations, springer berlin, heidelberg, 2012. https: //doi.org/10.1007/978-3-642-25132-0. [23] l. shijun, homotopy analysis method: a new analytic method for nonlinear problems, appl. math. mech. 19 (1998), 957–962. https://doi.org/10.1007/bf02457955. [24] y. cherruault, convergence of adomian’s method, kybernetes. 18 (1989), 31–38. https://doi.org/10.1108/ eb005812. https://doi.org/10.1007/978-0-8176-8349-8 https://doi.org/10.1007/978-0-8176-8349-8 https://doi.org/10.1016/j.matcom.2005.02.035 https://doi.org/10.1016/j.amc.2006.06.016 https://doi.org/10.1016/j.chaos.2005.12.058 https://doi.org/10.1016/j.amc.2006.10.015 https://doi.org/10.1016/j.amc.2006.10.015 https://doi.org/10.1016/j.aml.2007.02.007 https://doi.org/10.1016/j.aml.2007.02.007 https://doi.org/10.1016/j.camwa.2009.03.083 https://doi.org/10.1016/j.camwa.2009.03.083 https://doi.org/10.1016/j.procs.2010.12.192 https://doi.org/10.1016/j.procs.2010.12.192 https://doi.org/10.1016/j.cnsns.2010.06.013 https://doi.org/10.1016/j.cnsns.2010.06.013 https://doi.org/10.1016/j.cnsns.2011.04.023 https://doi.org/10.1016/j.amc.2011.11.013 https://doi.org/10.1016/j.amc.2011.11.013 https://doi.org/10.1016/j.aej.2013.02.004 https://doi.org/10.1016/j.aej.2013.02.004 https://doi.org/10.1016/j.amc.2015.02.079 https://doi.org/10.1016/j.cam.2016.06.028 https://doi.org/10.1007/978-3-642-25132-0 https://doi.org/10.1007/978-3-642-25132-0 https://doi.org/10.1007/bf02457955 https://doi.org/10.1108/eb005812 https://doi.org/10.1108/eb005812 1. introduction 2. existence and uniqueness of the second kind nv-fies 3. existence and uniqueness of the first kind nv-fies 4. homotopy analysis method for nv-fies 5. illustrating examples 6. conclusion references 22 2 2 0 2 2 2 2 ( )2 0 ( c o s h 2 c o s 2 ) d c z                       22 bac          a b1 tan     0 , , ,z f z     ba   hzh      1 , , , , 0 z h z z k z                   2 , , , , 0 z h z z k z                 , , ( , ) b z g z              , , , , ( ) h m h z p z d z          2 2 2 2 2 2 2 ( c o s h 2 c o s 2 ) m d a c                   ma         1 1 1 2 2 2 1 1 c o s s i n ( ) c o s s i n ( ) c o s s i n ( ) c o s s i n ( ) a a b a b a b a a b a a b a b a a b a a b                                0, , ( , )f       n n f    b    2 2 2 2 2 ( c o s h 2 c o s 2 ) m q a                  2 22 4 d ca q m       2 2 , 2, , ( , )nn n n nc f c e q c e q                   2 2 , 2, , ( , )nn n n nf c f c e q c e q             2 2 0 ( , )n n c e q d               2 0 2 ),( dqceff nn , 2 n c  qec n  ,2   2    2 0 2 2 ),( dqce n ),(),(),( 2 2 0 2   nn fdfqce           0 2 2 2 , 1 n n n n qbec f c             2 2 , 2 0 2 1 , , ( , ) , n nn n n n n f f c e q c e q c e b q                               2 2 , 2 0 0 2 ( )1 , , ( , ) , n nn m n n m nm n f f t p z z c e q c e q c e b q                           ),( )(1 , 2 0 2 0 qectff zp zg n n nnn m m m                   ,)(),,(,, 2    h h mn dzzpzfzf  ,)( 2    h h mm dzzp   )sin()cos( zawzaqzp mmmmm      )sin()cos( 2121 hahaaq mmmm       )sin()cos( 1221 hahaw mmm   1 21   ., 2211 kk     , 22 2 2 0 2 2 2 2 ( )2 1 ( c o s h 2 c o s 2 ) d t c z k t                         22 bac                a b1 tan  , , , 0 0z        0 , , , , ,z t f z t     ba   hzh      1 , , , , , , 0 z h z t z t k z                   2 , , , , , , 0 z h z t z t k z                 , , , ( , , ) b z t g z t           1 k 2 k    , , , , , , ( ) h m h t z t p z d z          2 2 2 2 2 2 2 1 ( c o s h 2 c o s 2 ) m d t a c k t                      m a         1 1 1 2 2 2 1 1 c o s s i n ( ) c o s s i n ( ) c o s s i n ( ) c o s s i n ( ) a a b a b a b a a b a a b a b a a b a a b                                0, , , ( , , )t f t       n n f    b    2 2 2 2 2 ( c o s h 2 c o s 2 ) *q                      2 22 4 d c k ska q m                   2 2 , 2, , , ( , )nn n n ns c f c e q c e q                qce n , 2   qce n , 2       2 2 , 2, , , ( , )nn n n nf s c f c e q c e q              2 2 0 ( , )n n c e q d               2 0 2 ),( dqceff nn , 2 n c  qec n  ,2   2    2 0 2 2 ),( dqce n ),(),(),( 2 2 0 2   nn fdfqce           0 2 2 2 , 1 n n n n qbec f c             2 2 , 2 0 2 1 , , , ( , ) , n nn n n n n f f t c e q c e q c e b q                              2 , 2 , 2 2 , 0 0 2 2 , 2 2 ,2 0 ( , )( )1 , , , , e x p ( ) ( ) n n m n n mm mn m n n m t nn n mn c e q c e qp z z t d c e b q d p f f t k t t d t                                         2 , 2 , 2 2 , 0 0 2 2 , 2 2 ,2 0 ( , )( )1 , , , e x p ( ) ( ) n n m n n mm mn m n n m t nn n mn c e q c e qp z g z t d c e b q d p f f k t t d t                                  , 4 2 2 222 ,2 kh d kphq mn    ,)(),,,(,, 2    h h mn dzzptzfzf  ,)( 2    h h mm dzzp   )sin()cos( zawzaqzp mmmmm      )sin()cos( 2121 hahaaq mmmm       )sin()cos( 1221 hahaw mmm   1 21   ., 2211 kk   international journal of analysis and applications volume 18, number 6 (2020), 957-964 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-957 generalized absolute riesz summability of infinite series and fourier series bağdagül kartal∗ department of mathematics, erciyes university, 38039 kayseri, turkey ∗corresponding author: bagdagulkartal@erciyes.edu.tr abstract. in this paper, two known theorems dealing with |n̄,pn|k summability of infinite series and fourier series have been generalized to ϕ−|n̄,pn; β|k summability. 1. introduction a sequence (an) is said to be δ-quasi-monotone if an → 0, an > 0 ultimately and ∆an ≥ −δn, where ∆an=an −an+1 and δ = (δn) is a sequence of positive numbers (see [1]). a sequence (gn) is said to be of bounded variation, denoted by (gn) ∈bv, if ∑∞ n=1 |∆gn| < ∞. let ∑ an be a given infinite series with the partial sums (sn). let (ϕn) be a sequence of positive real numbers. the series ∑ an is said to be summable ϕ−|n̄,pn; β|k, k ≥ 1 and β ≥ 0, if (see [22]) ∞∑ n=1 ϕβk+k−1n |un −un−1| k < ∞ where (pn) is a sequence of positive numbers such that pn = n∑ v=0 pv →∞ as n →∞ (p−i = p−i = 0, i ≥ 1) , received august 13th, 2020; accepted september 1st, 2020; published september 14th, 2020. 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g99. key words and phrases. absolute summability; fourier series; hölder’s inequality; infinite series; minkowski’s inequality; riesz mean; summability factor. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 957 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-957 int. j. anal. appl. 18 (6) (2020) 958 and un = 1 pn n∑ v=0 pvsv. for ϕn = pn pn and β = 0, ϕ −|n̄,pn; β|k summability reduces to |n̄,pn|k summability (see [2]). taking ϕn = n, β = 0 and pn = 1 for all values of n, ϕ −|n̄,pn; β|k summability reduces to |c, 1|k summability (see [8]). if we write xn = n∑ v=1 pv/pv, then (xn) is a positive increasing sequence tending to infinity with n. in [3], the following theorem on δ-quasi-monotone sequences has been proved. theorem 1.1. let (λn) → 0 as n →∞ and (pn) be a sequence of positive numbers such that pn = o(npn) as n →∞. suppose that there exists a sequence of numbers (an) which is δ-quasi-monotone with∑ nxnδn < ∞, ∑ anxn is convergent, and |∆λn| ≤ |an| for all n. if the condition m∑ n=1 pn pn |tn|k = o(xm) as m →∞(1.1) is satisfied, where (tn) is the n-th (c, 1) mean of the sequence (nan), then the series ∑ anλn is summable |n̄,pn|k, k ≥ 1. lemma 1.1. [3] under the conditions of theorem 1.1, we have that |λn|xn = o (1) as n →∞,(1.2) nxnan = o(1) as n →∞,(1.3) ∞∑ n=1 nxn|∆an| < ∞.(1.4) 2. main result there are some papers on absolute summability (see [4–6,9–12,16–18,23–25]). now we generalize theorem 1.1 as in the following form. theorem 2.1. let (ϕn) be a sequence of positive real numbers such that ϕnpn = o(pn),(2.1) m+1∑ n=v+1 ϕβk−1n 1 pn−1 = o ( ϕβkv 1 pv ) as m →∞.(2.2) int. j. anal. appl. 18 (6) (2020) 959 if all conditions of theorem 1.1 are satisfied with the condition (1.1) replaced by m∑ n=1 ϕβk−1n |tn| k = o(xm) as m →∞,(2.3) then the series ∑ anλn is summable ϕ−|n̄,pn; β|k, k ≥ 1 and 0 ≤ β < 1/k. 3. proof of theorem 2.1 let (in) indicates (n̄,pn) mean of the series ∑ anλn. then, for n ≥ 1, we obtain ∆̄in = in − in−1 = pn pnpn−1 n∑ v=1 pv−1avλv = pn pnpn−1 n∑ v=1 pv−1λv v vav. applying abel’s transformation, we get ∆̄in = pn pnpn−1 n−1∑ v=1 λv+1 v pvtv − pn pnpn−1 n−1∑ v=1 v + 1 v pvλvtv + pn pnpn−1 n−1∑ v=1 v + 1 v pvtv∆λv + (n + 1) npn pnλntn = in,1 + in,2 + in,3 + in,4. for the proof of theorem 2.1, it is sufficient to show that ∞∑ n=1 ϕβk+k−1n | in,r | k< ∞, for r = 1, 2, 3, 4. first, m+1∑ n=2 ϕβk+k−1n | in,1 | k ≤ m+1∑ n=2 ϕβk+k−1n ( pn pnpn−1 )k (n−1∑ v=1 pv |tv| |λv+1| v )k = m+1∑ n=2 ϕβk−1n ( ϕnpn pn )k 1 pkn−1 ( n−1∑ v=1 pv |tv| |λv+1| v )k . here (2.1) gives ( ϕnpn pn )k = o(1), also using hölder’s inequality, we obtain m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 ( n−1∑ v=1 pv |tv| k |λv+1|k v )( 1 pn−1 n−1∑ v=1 pv v )k−1 . now using the fact that pv = o(vpv), m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 ( n−1∑ v=1 pv |tv| k |λv+1|k )( 1 pn−1 n−1∑ v=1 pv )k−1 . int. j. anal. appl. 18 (6) (2020) 960 then, we have m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 n−1∑ v=1 pv |tv| k |λv+1|k = o(1) m∑ v=1 pv|λv+1|k−1|λv+1| |tv| k m+1∑ n=v+1 ϕβk−1n 1 pn−1 . here, by using (2.2) and (1.2), m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m∑ v=1 ϕβkv pv pv |λv+1| |tv| k = o(1) m∑ v=1 ϕβk−1v ( ϕvpv pv ) |λv+1| |tv| k . again, from (2.1), we obtain m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m∑ v=1 ϕβk−1v |λv+1| |tv| k . hence, we get m+1∑ n=2 ϕβk+k−1n | in,1 | k = o(1) m−1∑ v=1 ∆|λv+1| v∑ r=1 ϕβk−1r |tr| k + o(1)|λm+1| m∑ v=1 ϕβk−1v |tv| k = o(1) m−1∑ v=1 |av+1|xv+1 + o(1)|λm+1|xm+1 = o(1) as m →∞, by using abel’s transformation, hypotheses of theorem 2.1, and lemma 1.1. now, we have m+1∑ n=2 ϕβk+k−1n | in,2 | k = o(1) m+1∑ n=2 ϕβk+k−1n ( pn pnpn−1 )k (n−1∑ v=1 pv |λv| |tv| )k = o(1) m+1∑ n=2 ϕβk−1n ( ϕnpn pn )k 1 pkn−1 ( n−1∑ v=1 pv |λv| |tv| )k = o(1) m+1∑ n=2 ϕβk−1n 1 pkn−1 ( n−1∑ v=1 pv |λv| |tv| )k . using hölder’s inequality, we get m+1∑ n=2 ϕβk+k−1n | in,2 | k = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 ( n−1∑ v=1 pv |λv| k |t v |k )( 1 pn−1 n−1∑ v=1 pv )k−1 = o(1) m∑ v=1 pv|λv|k|tv|k m+1∑ n=v+1 ϕβk−1n 1 pn−1 . int. j. anal. appl. 18 (6) (2020) 961 by (2.2), (2.1) and (1.2), we get m+1∑ n=2 ϕβk+k−1n | in,2 | k = o(1) m∑ v=1 ϕβk−1v |λv||tv| k. here, using abel’s transformation as in in,1, we have m+1∑ n=2 ϕβk+k−1n | in,2 | k = o(1) as m →∞. again, using hölder’s inequality, we have m+1∑ n=2 ϕβk+k−1n | in,3 | k = o(1) m+1∑ n=2 ϕβk+k−1n ( pn pnpn−1 )k (n−1∑ v=1 pv |tv| |∆λv| )k = o(1) m+1∑ n=2 ϕβk−1n ( ϕnpn pn )k 1 pkn−1 ( n−1∑ v=1 pv |tv| |av| )k = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 ( n−1∑ v=1 pv |tv| k (v |av|)k )( 1 pn−1 n−1∑ v=1 pv )k−1 = o(1) m+1∑ n=2 ϕβk−1n 1 pn−1 n−1∑ v=1 pv |tv| k (v |av|)k−1(v |av|). using (1.3), we get (v |av|)k−1 = o(1), then m+1∑ n=2 ϕβk+k−1n | in,3 | k = o(1) m∑ v=1 pv|tv|kv |av| m+1∑ n=v+1 ϕβk−1n 1 pn−1 . now using the conditions (2.2) and (2.1), we get m+1∑ n=2 ϕβk+k−1n | in,3 | k = o(1) m∑ v=1 ϕβk−1v |tv| kv |av| . then, we have m+1∑ n=2 ϕβk+k−1n | in,3 | k = o(1) m−1∑ v=1 ∆(v |av|) v∑ r=1 ϕβk−1r |tr| k + o(1)m |am| m∑ v=1 ϕβk−1v |tv| k = o(1) m−1∑ v=1 ∆(v |av|)xv + o(1)m |am|xm = o(1) m−1∑ v=1 vxv|∆av| + o(1) m−1∑ v=1 |av+1|xv+1 + o(1)m |am|xm = o(1) as m →∞, by using abel’s transformation, hypotheses of theorem 2.1, and lemma 1.1. int. j. anal. appl. 18 (6) (2020) 962 finally, we get m∑ n=1 ϕβk+k−1n | in,4 | k = o(1) m∑ n=1 ϕβk+k−1n ( pn pn )k |λn|k−1|λn||tn|k = o(1) m∑ n=1 ϕβk−1n |λn||tn| k. here, as in in,1, we get m∑ n=1 ϕβk+k−1n | in,4 | k = o(1) as m →∞. hence, the proof of theorem 2.1 is completed. 4. applications there are some different papers dealing with applications of fourier series (see [14, 15, 19–21]). let f be a periodic function with period 2π and lebesgue integrable over (−π,π). the trigonometric fourier series of f is defined as f(x) ∼ 1 2 a0 + ∞∑ n=1 (an cos nx + bn sin nx) = ∞∑ n=0 cn(x) where a0 = 1 π ∫ π −π f(x)dx, an = 1 π ∫ π −π f(x) cos(nx)dx and bn = 1 π ∫ π −π f(x) sin(nx)dx. write φ(t) = 1 2 {f(x + t) + f(x− t)} and φ1(t) = 1 t ∫ t 0 φ(u)du. if φ1(t) ∈ bv(0,π), then tn(x) = o(1), where tn(x) is the n-th (c, 1) mean of the sequence (ncn(x)) (see [7]). by using this, the following theorem has been obtained in [3]. theorem 4.1. if φ1(t) ∈bv(0,π), and the sequences (pn), (λn) and (xn) satisfy the conditions of theorem 1.1, then the series ∑ cn(x)λn is summable | n̄,pn |k, k ≥ 1. the following theorem gives a generalization of theorem 4.1 for ϕ−|n̄,pn; β|k summability. theorem 4.2. if φ1(t) ∈bv(0,π), and the sequences (pn), (λn), (an), (ϕn) and (xn) satisfy the conditions of theorem 2.1, then the series ∑ cn(x)λn is summable ϕ−|n̄,pn; β|k, k ≥ 1 and 0 ≤ β < 1/k. int. j. anal. appl. 18 (6) (2020) 963 5. conclusions if we take ϕn = pn pn and β = 0 in theorem 2.1, then the condition (2.3) reduces to the condition (1.1), and the conditions (2.1) and (2.2) are provided. thus, theorem 2.1 reduces to theorem 1.1. if we take ϕn = n, β = 0 and pn = 1 for all values of n, then we have a result for |c, 1|k summability of an infinite series (see [13]). also, if we take ϕn = pn pn and β = 0 in theorem 4.2, then we get theorem 4.1. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.p. boas, quasi-positive sequences and trigonometric series, proc. lond. math. soc. s3-14a (1965), 38–46. [2] h. bor, on two summability methods, math. proc. cambridge philos. soc. 97 (1) (1985), 147–149. [3] h. bor, on quasi-monotone sequences and their applications, bull. austral. math. soc. 43 (2) (1991), 187-192. [4] h. bor, h. s. özarslan, on absolute riesz summability factors, j. math. anal. appl. 246 (2) (2000), 657-663. [5] h. bor, h. s. özarslan, a note on absolute summability factors, adv. stud. contemp. math. (kyungshang) 6 (1) (2003), 1-11. [6] h. bor, h. seyhan, on almost increasing sequences and its applications, indian j. pure appl. math. 30 (10) (1999), 1041-1046. [7] k.k. chen, functions of bounded variation and the cesàro means of a fourier series, acad. sinica science record 1 (1945), 283-289. [8] t.m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. lond. math. soc. s3-7 (1957), 113-141. [9] a. karakaş, a note on absolute summability method involving almost increasing and δ-quasi-monotone sequences, int. j. math. comput. sci. 13 (1) (2018), 73-81. [10] a. karakaş, a new factor theorem for generalized absolute riesz summability, carpathian math. publ. 11 (2) (2019), 345-349. [11] b. kartal, on generalized absolute riesz summability method, commun. math. appl. 8 (3) (2017), 359-364. [12] b. kartal, new results for almost increasing sequences, ann. univ. paedagog. crac. stud. math. 18 (2019), 85-91. [13] s.m. mazhar, on generalized quasi-convex sequence and its applications, indian j. pure appl. math. 8 (7) (1977), 784-790. [14] h.s. özarslan, a note on |n̄,pn; δ|k summability factors, erc. üni. fen bil. enst. derg., cilt. 16 (2000), 95-100. [15] h.s. özarslan, a note on |n̄,pαn|k summability factors, soochow j. math. 27 (1) (2001), 45-51. [16] h.s. özarslan, on almost increasing sequences and its applications, int. j. math. math. sci. 25 (5) (2001), 293-298. [17] h.s. özarslan, a note on ∣∣n̄,pn; δ∣∣k summability factors, indian j. pure appl. math. 33 (3) (2002), 361–366. [18] h.s. özarslan, on |n̄,pn; δ|k summability factors, kyungpook math. j. 43 (1) (2003), 107–112. [19] h.s. özarslan, a note on |n̄,pn|k summability factors, int. j. pure appl. math. 13 (4) (2004), 485–490. [20] h.s. özarslan, on the local properties of factored fourier series, proc. jangjeon math. soc. 9 (2) (2006), 103-108. [21] h.s. özarslan, local properties of factored fourier series, int. j. comp. appl. math. 1 (1) (2006), 93-96. [22] h. seyhan, on the local property of ϕ− ∣∣n̄,pn; δ∣∣k summability of factored fourier series, bull. inst. math. acad. sin. 25 (4) (1997), 311-316. int. j. anal. appl. 18 (6) (2020) 964 [23] h. seyhan, a note on absolute summability factors, far east j. math. sci. 6 (1) (1998), 157-162. [24] h. seyhan, on the absolute summability factors of type (a,b), tamkang j. math. 30 (1) (1999), 59-62. [25] h. seyhan, a. sönmez, on ϕ− ∣∣n̄,pn; δ∣∣k summability factors, portugal. math. 54 (4) (1997), 393–398. 1. introduction 2. main result 3. proof of theorem 2.1 4. applications 5. conclusions references international journal of analysis and applications volume 19, number 4 (2021), 494-502 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-494 pricing options in a delayed market driven by le’vy noise ismail hamed elsanousi∗ department of mathematics, faculty of sciences al-baha university,p.o.box-1988, alaqiq, al-baha-65431, saudi arabia ∗corresponding author: i elsanousi@hotmail.com abstract. in this paper we studied stochastic delayed differential equations driven by le’vy noise. the analogue of itô formula is considered. the black-scholes formula analogue for vanilla call option price formula is derived. 1. introduction in this paper we studied the stochastic delay differential equations driven by le’vy noise which arise in many applications of stochastic analysis in finance specifically in pricing of options security markets. as known such systems are quite hard to study due to their lack of markovianity which is a key property for the study of option prices. basically, the difficulties arises from the fact that delay systems have, in general, an infinite dimensional nature. the model for the stock price ζ(t) that we consider satisfies a stochastic delay differential equation driven by le’vy noise with volatility σ depending on time t and the path ζt = {ζ(t + θ),θ ∈ [−τ, 0]} called a level and past-dependent volatility. an analogue of itô’s formula for such a stochastic systems is obtained. received march 29th, 2021; accepted april 23rd, 2021; published may 11th, 2021. 2010 mathematics subject classification. 60g40, 34k50. key words and phrases. stochastic delay equations; black-scholes formula; (b,s)-securities market. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 494 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-494 int. j. anal. appl. 19 (4) (2021) 495 an option price value of the form g(t,ζt) = ∫ 0 −τ erθf(ζ(t + θ),ζ(t), t)dθ is studied when f is in c0,1,2(r×r×r3). a special case of g(t,ζt) of the form g(t,ζt) = g1(ζ(t), t) + ∫ 0 −τ erθg2(ζ(t + θ), t)dθ when g1(ζ(t), t) is a classical black-scholes call option is studied. the partial differential equation of blackscholes type is derived for such option. 2. notations and preliminaries let us consider a probability space (ω,z,ρ) on which is defined ((b(t))t≥0, (η(t))t≥0) where ((b(t))t≥0 and (η(t))t≥0) are independent stochastic processes -((b(t))t≥0 is a standard brownian motion with respect to its natural filtration, -(η(t))t≥0) is a pure jump le’vy process. the poisson random measure n(t) of the process η is defined by n(t,a) = ∑ ζ∈[0,t] ia(η(s) −η(s−)), a ⊂ r. the le’vy measure ν of the process η is supposed to be ∫ r z 2ν(dz) < +∞, since (η(t))t≥0) is a pure jump ν({0}) = 0. define the measure-valued process (ñ(t))t≥0 by ñ(t,dz) := n(t,dz) −ν(dz) so that the compensated poisson random measure is ñ(dt,dz) := n(dt,dz) −ν(dz)dt let (zt)t≥0 be the filtration generated by the process b(t) and η(t) as defined above. since b and η are assumed independent, then b(.) and ñ(.,a) are still brownian motion and square integrable martingale with respect to the filtration (zt)t≥0. int. j. anal. appl. 19 (4) (2021) 496 2.1. stochastic delay differential equations driven by le’vy noise consider the following sdde: dx(t) =µ(t,xt)dt + σ(t,xt)dw(t) + ∫ r γ(x−t ,z)ñ(dt,dz), t ∈ [0,t] x0 = x(θ), θ ∈ [−r, 0] (2.1) where µ : [0,t] ×d → rn,σ : [0,t] ×d → rnd,γ : d ×d → r are predictable processes. for the unknown process (x(t))t∈[−r,t] in r,t < ∞ is a fixed finite time, x(t) is the value of x at t ∈ [0,t] and xt its segment, i.e. its value in the past time interval [t − r,t] i.e. xt(.) : [−r, 0] → r defined by xt(θ) : x(t + θ) for all θ ∈ [−r, 0]. the initial data x(θ) is assumed to be in the space d := d([−r, 0],r) of càdlàg random variables from [−r, 0] to r. hypotheses 2.1 (i). there exists constant l > 0 such that for all t ∈ [0,t] and for all x1,x2 ∈ d, |µ(t,x1) −µ(t,x2)| + |σ(t,x1) −σ(t,x2) + ∫ r |γ(x1,z) −γ(x2,z)|ν(dz) ≤ l|x1 −x2|. (ii). the functions µ,σ,γ satisfy the linear growth condition, i.e. there exists a constant k > 0 such that for all x ∈ d, |µ(t,x)| + |σ(t,x)| + ∫ r |γ(x,z)|ν(dz) ≤ k(1 + |x|). theorem 2.1. suppose that hypotheses (i) and (ii) hold. then there exists a unique ca′dla′g adapted solution to equation (2.1). for proof we refer to [7]. the infinitesimal operator of the solution of equation (2.1) in reformulation of equation (2.1) in infinite dimension, the following linear stochastic evolution equation in the space h dx(t) = ax(t)dt+ < σ,x(t) > −n̂dw(t) + ∫ r < r(z),x(t−) > −n̂ñ(dt,dz) (2.1)′ will represents equation (2.1) in the sense that x(t) = (x0(t),x1(t)) = (x(t),x(t + θ) : θ ∈ [−t, 0],∀t ≥ 0 int. j. anal. appl. 19 (4) (2021) 497 where a is defined on d(a = {y = (y0,y1(.)) ∈ h; y1(.) ∈ w 1,2([−1, 0],r),y0 = y1(0)} by ay = (µ0y0 + ∫ 0 −t µ1(θ)y1(θ)dθ + µ2y1(−t),y ′ 1(.)) is the generator of a strongly continuous semigroup (s(t))t≥0 on h. here n̂ = (1, 0) ∈ h and x(t−) := lims↑t x(s) = lims↑t(x(s),s(s+.)) with the limit taken in h. theorem 2.1. let y ∈ c and let x(.; y),x(.; y) representing the solutions of (2.1) and (2.1)′ respectively. then x(.; y) represents s(.; y) in the sense that x(t; y) = (x(t; y),x(t + θ; y) : θ ∈ [−t, 0]),∀t ≥ 0. for proof see [11]. the infinitesimal operator of the process x(., 0) is formally defined as [lφ](y) :=< ay,ϕy(y) > + 1 2 < σ,y > ϕy0y0 (y) +∫ r [ϕ(y+ < r(z),y > n̂) −ϕ(y) −ϕy0 (y) < r(z),y >]ν(dz), y ∈ d(a),ϕ ∈ c 2(h; r). 3. option price formula assume that the stock price satisfy the following stochastic delay differential equation of the form ds(t) =rs(t)dt + σ(t,st)dw(t) + ∫ r γ(s−t ,z)ñ(dt,dz), t ∈ [0,t] s(0) = y0,s(θ) = y1(θ), θ ∈ [−r, 0] (3.1) where y := (y0,y1(.)) ∈ c is positive. here c is the subspace of the hilbert space h := r × l2−r := r×l2([−r, 0],r) whose inner product < .,. > is < .,. >=< .,. >r + < .,.,>l2r c := y ∈ h : y1(.) admits a càdla′g representative. in (3.1) r ∈ r,σ := (σ0,σ1(.) ∈ h,γ(.) := (γ0,γ1(.)(.)) ∈ l2(r,ν; h) are functional parameters. analogue of black-scholes formula for vanilla call option price: suppose that the financial market under consideration as follows: (i). a risk free asset given by ds0(t) = r(s0)(t)dt; t ∈ [0.t]. (ii). a risky asset given by equation (3.1)-(3.2). int. j. anal. appl. 19 (4) (2021) 498 a portfolio in such market is an ft predictable process π(t) representing the number of units held at time t of the assets number 0, 1, . . . ,n respectively, then the wealth process x(t) = xπ(t) associated to the portfolio π is defined to be: xπ(t) = π(t)s(t) = n∑ i=1 πi(t)si(t). absence of arbitrage in order to have no arbitrage opportunities in the considered market, the return from the portfolio must be risk-free with interest rate r. in what follows we adopt that π(t) is riskless during [t,t + dt] and instantaneously earn the same rate of return as other short-term risk-free assets. these assumptions on π(t) gives dπ(t) = rπ(t)dt. let the option price value has the form (3.2) g(t,st) = ∫ 0 −t e−rθf(s(t + θ),s(t), t)dθ where f ∈ c0,2,1(r×r2 ×r+). lemma 3.1. (itô formula) suppose s(t) is given by (3.1) and a functional g : r+ ×c → r has the form (3.3) g(t,st) = ∫ 0 −τ g(θ)f(st(θ),st(0), t)dθ, where f ∈ c0,2,1(r×r×r+) and g ∈ c1([−τ, 0],r). hence in view of the classical itô formula, we have g(t,st) =g(0,ζ) + ∫ t 0 ag(s,ss)ds + ∫ t 0 σ(s,ss)s(s)bg(s,ss)dw(s) + ∫ t 0 ∫ r {f(t,s(t−) + γ(s,z)) −f(t,s(t−)) −f(t,s(t)−)}n(dt,dz) (3.4) where for (t,x) ∈ r+ ×c. ag(t,y) =g(0)f(x0,x0, t) −g(−τ)f(x(−τ),x0, t)−∫ 0 −τ g′(θ)f(x(θ),x0, t)dθ + ∫ 0 −τ g(θ)lf(x(θ),x0, t)dθ + ∫ r {f(t,x(t−) + γ(x,z)) −f(t,x(t−))}n(dt,dz), int. j. anal. appl. 19 (4) (2021) 499 with lf(x(θ,x(0), t) =rx(0)f ′2(x(θ,x(0), t)+ σ2(t,x)x2(0) 2 f ′′22(x(θ,x(0), t) + f ′ 3(x(θ,x(0), t) + ∫ r {f(x + γ(x,z)) −f(x) −f ′2.γ(x,z)}ν(dz), where f ′i, i = 1, 2, 3 represents the derivative of f with respect to the i th argument. portfolio concepts for financial markets driven by le’vy process: theorem 3.1. the option price value given by (3.2) satisfies the equation 0 = f |θ=0 −erθf|θ=−τ + ∫ 0 −τ e−rθ{(f ′3 + rs(t)f ′ 2 + 1 2 σ2(t,s(t))s2(t)f ′′22)dθ +∫ r {f(s + γ(s,z)) −f(s)) −f ′2γ(s,z)}ν(dz)}dθ. proof. we sketch the proof as in [12] as follows: by considering a portfolio consists of -1 derivative and bg(t,st) shares. then if π(t) represents the portfolio value we have π(t) = −g(t,st) + bg(t,st)s(t) = −dg + d(bgs) = d(bg)s + bgds −df. hence dπ(t) = −dg + bgds. since we assume bg is held constant during the time-step dt yields d(bg) equal zero. substituting for dg and ds from equation (3.4) and (3.1) we get (3.5) dπ = −agdt−σsbgdw + bg(rsdt + σsdw) where a = a− ∫ t r {f(t,s(t−) + γ(s,z)) −f(t,s(t−)) −f(t,s(t)−)}n(dt,dz). by considering risk-free during the time dt gives (3.6) dπ = rπdt. int. j. anal. appl. 19 (4) (2021) 500 by equating the last equations we get ag(t,st) = rg(t,st). which gives an equation for f(s(t + θ),s(t), t) in the form 0 = f|θ=0 −erθf|θ=−τ + ∫ 0 −τ e−rθ{(f ′3 + rs(t)f ′ 2 + 1 2 σ2(t,s(t))s2(t)f ′′22)dθ +∫ r {f(s + γ(s,z)) −f(s)) −f ′2γ(s,z)}ν(dz)}dθ. 4. price formula for european call option: theorem 4.1.(black-scholes pde type) in view of (3.3) the option price value g(t,st ) will has the form (4.1) g(t,st ) = max(s(t) −k, 0). for simplification we assume that g(t,st) takes the form (4.2) g(t,st) = g1(s(t), t) + ∫ 0 −τ e−rθg2(s(t + θ), t)dθ, where g1(s(t), t) is a classical black-scholes call option price with variance assumed equal to a long-run variance rate v, then (4.3) g1(s(t), t) = s(t)n(d1) −ke−r(t−t)n(d2), where n(x) = 1 √ 2π ∫ x −∞ e− x2 2 dx d1 = ln( s(t) k ) + (r + v 2 )(t − t) d2 = d1 − √ v (t − t). then g(t,st) satisfies the following equation: ∂g ∂t + rs(t) ∂g1 ∂s + 1 2 [σ2(t,st)]s 2(t) ∂2g1 ∂s2 +∫ r {f(s + γ(s,z)) −f(s)) −f ′2γ(s,z)}ν(dz) = rg. int. j. anal. appl. 19 (4) (2021) 501 proof. by substituting (4.1) into (3.5) we obtain the equation for g2 as follows g2(s(t), t) −erτg2(s(t− τ), t)+ ∫ 0 tau e−rθ ∂g2 ∂t dθ = 1 2 (v −σ2(t,s(t)))s2(t) ∂2g1 ∂s2 +∫ r {f(s + γ(s,z)) −f(s)) −f ′2γ(s,z)}ν(dz). since g1 satisfies the classical black-scholes pde we have: ∂g1 ∂t + rs(t) ∂g1 ∂s + 1 2 [v ]s2(t) ∂2g1 ∂s2 = rg1. by combining the last equation we get the following equation g(t,st) of the form : ∂g ∂t + rs(t) ∂g1 ∂s + 1 2 [σ2(t,st)]s 2(t) ∂2g1 ∂s2 +∫ r {f(s + γ(s,z)) −f(s)) −f ′2γ(s,z)}ν(dz) = rg, which is an integro-differential equation. remark. for pricing american put option we give the following discussion : suppose the functional (3.3) has the form s(t)− ∫ 0 −τ s(t+ξ)dξ then the pricing formula for the european put option is to minimize the following functional overall ft stoping time, τ < ∞ a.s. for all t ≥ 0: f(τ,sτ ) = e[(k − (s(τ) − ∫ 0 −t s(t + ξ)dξ))+] where k > 0. the dynamics of the stock prices are driven by the following stochastic differential equation: ds(t) =µ(s(t) −s(t−t))dt + α(s(t) −µ ∫ 0 −t s(t + ξ))dw(t)+ β ∫ r (s(t−) −µ ∫ 0 −t s(t− + ξ)dξ)zñdtdz with the initial conditions s(0) = y0; s(θ) = y1(θ),θ ∈ [−t, 0), where µ,α,β are constants and y := (y0,y1(.) ∈ c. the assumptions y0 −µ ∫ 0 −t y1(ξ)dξ > 0 and ν ≡ 0 on (−∞, 0] are imposed to ensure that positivity of the solution and which it is exist and the allowance only for positive jumps. int. j. anal. appl. 19 (4) (2021) 502 the problem of finding the optimal exercise time of f(τ,sτ ) = e[(k − (s(τ) − ∫ 0 −t s(t + ξ)dξ))+] is strictly connected to the corresponding optimal stoping problem of such stochastic system. this problem was studied by [13] where the solution is obtained by rewriting the problem in infinite dimension from which they found the condition under which the problem is reduced to one dimensional case which yields explicit solution. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. chojnowska-michalik, representation theorem for general stochastic delay equations, bull. acad. polon. sci. ser. sci. math. astronom. phys. 26 (1978), 635-642. [2] g. da prato, j. zabczyk, stochastic equations in infinite dimensions, encyclopedia of mathematics and its applications, cambridge university press, cambridge (uk), 1992. [3] i. elsanousi, b. pksendal, a. sulem, some solvable stochastic control problems with delay, stoch. stoch. rep. 71 (2000), 69-89. [4] j.c. hull, a. white, the pricing of options on assets with stochastic valatilities, j. finance, 42 (1987), 281-300. [5] s.-e.a. mohammed, stochastic differential systems with memory: theory, examples and applications, in: l. decreusefond, b. øksendal, j. gjerde, a.s. üstünel (eds.), stochastic analysis and related topics vi, birkhäuser boston, boston, ma, 1998: pp. 1–77. [6] f. black, m. scholes, the pricing of options and corporate liabilities, j. polit. econ. 81 (1973), 637-654. [7] b. oksendal, a. sulem, applied stochastic control of jump diffusions, springer, berlin, 2005. [8] m. arriojas, y. hu, s.-e. mohammed, g. pap, a delayed black and scholes formula, stoch. anal. appl. 25 (2007), 471–492. [9] d. applebanm, le’vy process and stochastic calculus, cambridge studies in advanced mathematics vol. 116, cambridge university press, cambridge, 2009. [10] p.e. protter, stochastic integration and differential equations, springer, berlin, 2005. [11] m. reiß, m. riedle, o. van gaans, delay differential equations driven by lévy processes: stationarity and feller properties, stoch. proc. appl. 116 (2006), 1409–1432. [12] y. kazmerchuk, a. swishchuk, j. wu, the pricing of options for securities markets with delayed response, math. computers simul. 75 (2007), 69–79. [13] s. federico, b.k. øksendal, optimal stopping of stochastic differential equations with delay driven by lévy noise, potential anal. 34 (2011), 181–198. 1. introduction 2. notations and preliminaries 3. option price formula 4. price formula for european call option: references international journal of analysis and applications volume 16, number 4 (2018), 472-483 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-472 a new fixed point theorem in modular metric spaces ali̇ mutlu1, kübra özkan1,∗ and utku gürdal2 1manisa celal bayar university, faculty of science and arts, department of mathematics, 45140, manisa/turkey 2mehmet akif ersoy university, faculty of science and arts, department of mathematics, burdur/turkey ∗corresponding author: kubra.ozkan@hotmail.com abstract. in this article, we first give a new fixed point theorem which is main theorem of our study in modular metric spaces. after that, by using this theorem, we express some interesting results. moreover, we characterize completeness in modular metric spaces via this theorem. finally, we use our main result to show the existence of solution for a specific problem in dynamic programming. 1. introduction the fixed point theory is used in many different fields of mathematics such as topology, analysis, nonlinear analysis and operator theory. moreover, it can be applied to different disciplines such as statistics, economy, engineering, etc. in literature, studies of fixed point theory cover a wide range. the most basic and famous fixed point theorem is banach fixed point theorem which was introduced in 1922 [6]. it guarantees the existence and uniqueness of solution of a functional equation. besides banach, many different fixed point theorems were introduced such as kannan, caristi, coupled, suzuki, etc [1, 2, 7, 8, 13–16, 19, 23, 24]. in 1950, nakano introduced modular spaces [21]. then chistyakov introduced the concept of modular metric spaces, which have a physical interpretation, via f-modulars [9] in 2008 and he further developed received 2018-03-20; accepted 2018-05-09; published 2018-07-02. 2010 mathematics subject classification. 46a80, 47h10, 54h25. key words and phrases. modular metric space; fixed point theorem; complete modular metric. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 472 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-472 int. j. anal. appl. 16 (4) (2018) 473 the theory of these spaces in 2010 [10]. then many authors made various studies on this structures, e.g. [3–5, 11, 12, 17, 18, 20]. in this paper, we first give a new fixed point theorem which is main theorem of our study. after that, by using this theorem, we express some interesting results. moreover, we characterize completeness in modular metric spaces via this theorem. finally, we use our main theorem to show the existence of solution for a specific problem in dynamic programming. 2. modular metric spaces here, we express a series of definitions of some basic concepts related to modular metric spaces. definition 2.1. [22] let x be a linear space on r. if a functional ρ : x → [0,∞] satisfies the following conditions, we call that ρ is a modular on x: (1) ρ(0) = 0; (2) if x ∈ x and ρ(αx) = 0 for all numbers α > 0, then x = 0; (3) ρ(−x) = ρ(x), for all x ∈ x; (4) ρ(αx + βy) ≤ ρ(x) + ρ(y) for all α,β ≥ 0 with α + β = 1 and x,y ∈ x. let x 6= ∅ and λ ∈ (0,∞). generally, a function ω : (0,∞) ×x ×x → [0,∞] is denoted as ωλ(x,y) = ω(λ,x,y) for all x,y ∈ x and λ > 0. definition 2.2. [10] let x 6= ∅. a function ω : (0,∞) × x × x → [0,∞], which satisfies the following conditions for all x,y,z ∈ x, is called a metric modular on x: (m1) ωλ(x,y) = 0 for all λ > 0 ⇔ x = y; (m2) ωλ(x,y) = ωλ(y,x) for all λ > 0; (m3) ωλ+µ(x,y) ≤ ωλ(x,z) + ωµ(z,y) for all λ,µ > 0. if 0 < µ < λ, from properties of metric modular, we obtain that ωλ(x,y) ≤ ωλ−µ(x,x) + ωµ(x,y) = ωµ(x,y) for all x,y ∈ x. from [10, 11], we know that for a fixed x0 ∈ x, the two sets xω = xω(x0) = {x ∈ x : ωλ(x,x0) → 0 as λ →∞} and x∗ω = x ∗ ω(x0) = {x ∈ x : ∃λ = λ(x) > 0 such that ωλ(x,x0) < ∞} are said to be modular spaces. int. j. anal. appl. 16 (4) (2018) 474 it is known [10, 11] that if ω is a metric modular on a nonempty set x, then the modular space xω can be equipped with a metric, generated by ω and given by dω(x,y) = inf{λ > 0 : ωλ(x,y) ≤ λ} for all x,y ∈ xω. the pair (xω,dω) is called a modular metric space. definition 2.3. [18] let xω be a modular metric space, {xn}n∈n be a sequence in xω and c ⊆ xω. then (1) {xn}n∈n is called a modular convergent sequence such that xn → x, x ∈ xω, if for λ > 0 ωλ(xn,x) → 0 as n →∞. (2) {xn}n∈n is called a modular cauchy sequence, if for λ > 0 ωλ(xn,xm) → 0 as m,n →∞. (3) c is called closed, if the limit of a modular convergent sequence in c always belongs to c. (4) c is called complete modular, if every modular cauchy sequence {xn} in c is modular convergent in c. (5) c is called bounded, if δω(c) = sup{ωλ(x,y) : x,y ∈ c, λ > 0} < ∞. 3. main results let ω : (0,∞)×x ×x → [0,∞] be a metric modular on x, xω be a modular metric space, c ⊆ xω and ψ : c → r+ be a function on c. ψ is called lower semi-continuous (l.s.c.) on c if lim n→∞ ωλ(xn,x) = 0 ⇒ ψ(x) ≤ lim n→∞ inf(ψ(xn)) for all {xn}⊆ c and λ > 0. theorem 3.1. let ω be a metric modular on x, xω be a complete modular metric space, ψ : xω → r+ be a lower semi-continuous function on xω and t : xω → xω be a mapping satisfying the inequality ωλ(x,tx) ≤ ψ(x) −ψ(tx) (3.1) for all x ∈ xω and λ > 0. then t has a fixed point in xω. proof. for any x ∈ xω denote, p(x) = {y ∈ xω : ωλ(x,y) ≤ ψ(x) −ψ(y) for all λ > 0}, α(x) = inf{ψ(y) : y ∈ p(x)}. int. j. anal. appl. 16 (4) (2018) 475 as x ∈ p(x), p(x) is not empty and 0 ≤ α(x) ≤ ψ(x). let x ∈ xω be an arbitrary point. now, we construct a sequence {xn} in xω as follows: let x1 = x and when x1,x2, ...,xn have been chosen, choose xn+1 ∈ p(xn) such that ψ(xn+1) ≤ α(xn) + 1n, for all n ∈ n. by doing so, we get a sequence {xn} satisfying the conditions ωλ(xn,xn+1) ≤ ψ(xn) −ψ(xn+1), α(xn) ≤ ψ(xn+1) ≤ α(xn) + 1n (3.2) for all n ∈ n and λ > 0. then {ψ(xn)} is a nonincreasing sequence in r and it is bounded from below by zero. so, the sequence {ψ(xn)} is convergent to a number m ≥ 0. by virtue of (3.2), we get m = lim n→∞ ψ(xn) = lim n→∞ α(xn). (3.3) now, let k ∈ n be arbitrary. from (3.2) and (3.3), there exists at least a number nk such that ψ(xn) < m + 1k for all n ≥ nk. since ψ(xn) is monotone, we get m ≤ ψ(xm) ≤ ψ(xn) < m + 1 k for m ≥ n ≥ nk. it follows that ψ(xn) −ψ(xm) < 1 k for all m ≥ n ≥ nk. (3.4) preserving the generality, suppose that m > n and m,n ∈ n. from (3.2), we get ω λ m−n (xn,xn+1) ≤ ψ(xn) −ψ(xn+1) for λ m−n > 0. now, we obtain that ωλ(xn,xm) ≤ ω λ m−n (xn,xn+1) + ω λ m−n (xn+1,xn+2) + · · · + ω λ m−n (xm−1,xm) ≤ ψ(xn) −ψ(xn+1) + ψ(xn+1) −ψ(xn+2) + · · · + ψ(xm−1) −ψ(xm) = ψ(xn) −ψ(xm) (3.5) for all m,n ≥ nk. then by (3.4), ωλ(xn,xm) < 1 k for all m ≥ n ≥ nk. (3.6) letting k or m and n tend to infinity in (3.6), we conclude that lim m,n→∞ ωλ(xn,xm) = 0. then {xn}n∈n is a modular cauchy sequence. since xω is complete modular, there exists a point u ∈ xω such that xn → u as n →∞. since ψ is lower semi-continuous, using (3.5), we have ψ(u) ≤ lim m→∞ inf ψ(xm) ≤ lim m→∞ inf(ψ(xn) −ωλ(xn,xm)) = ψ(xn) −ωλ(xn,u) int. j. anal. appl. 16 (4) (2018) 476 and hence ωλ(xn,u) ≤ ψ(xn) −ψ(u). thus, u ∈ p(xn) for all n ∈ n and hence α(xn) ≤ ψ(u). then by (3.3), we get m ≤ ψ(u). moreover, using lower semi-continuity of ψ and (3.3), we have ψ(u) ≤ lim n→∞ inf ψ(xn) = m. so, ψ(u) = m. from (3.1), we know that tu ∈ p(u). since u ∈ p(u) for n ∈ n, we have ωλ(xn,tu) ≤ ωλ 2 (xn,u) + ωλ 2 (u,tu) ≤ ψ(xn) −ψ(u) + ψ(u) −ψ(tu) = ψ(xn) −ψ(tu). then tu ∈ p(xn) and this implies α(xn) ≤ ψ(tu). hence, we obtain m ≤ ψ(tu). from (3.1), we get ψ(tu) ≤ ψ(u). as ψ(u) = m, we have ψ(u) = m ≤ ψ(tu) ≤ ψ(u). therefore, ψ(tu) = ψ(u). then from (3.1), we get ωλ(u,tu) ≤ ψ(u) −ψ(tu) = 0. thus, tu = u. � theorem 3.2. let ω be a metric modular on x and xω be a complete modular metric space and ψ : xω → r be a lower semi-continuous function on xω. if ψ is bounded below, then there exists a point u ∈ xω such that ψ(u) < ψ(x) + ωλ(u,x) for each x ∈ xω, x 6= u and λ > 0. proof. following the proof theorem 3.1, we obtain a sequence {xn} that converges to some u ∈ xω. under the same notations, for any u ∈ xω, define p(u) = {x ∈ xω : ωλ(u,x) ≤ ψ(u) −ψ(x) for all λ > 0} α(u) = inf{ψ(x) : x ∈ p(u)}. int. j. anal. appl. 16 (4) (2018) 477 we will show that u /∈ p(u) as x 6= u. suppose the contrary, that is, we get v ∈ p(u) for some v 6= u. then 0 < ωλ(u,v) ≤ ψ(u) −ψ(v) implies ψ(v) < ψ(u) = m. since ωλ(xn,v) ≤ ωλ 2 (xn,u) + ωλ 2 (u,v) ≤ ψ(xn) −ψ(u) + ψ(u) −ψ(v) = ψ(xn) −ψ(v) for all λ > 0, v ∈ p(xn). so, α(xn) ≤ ψ(v) for all n ∈ n. letting n tends to infinity, we get m ≤ ψ(v). this equation contradicts with ψ(v) < m = ψ(u). therefore, for each x ∈ xω, x 6= u implies x /∈ p(u), that is x 6= u ⇒ ωλ(u,x) > ψ(u) −ψ(x). � theorem 3.3. let xω and yω be complete modular metric spaces and the mapping t : xω → xω be arbitrary. assume that there exists a closed mapping s : xω → yω, a lower semi-continuous mapping ψ : s(xω) → r+ and a constant c > 0 such that for any x ∈ xω and λ > 0 ωλ(x,tx) ≤ ψ(sx) −ψ(stx) and c ·ωλ(sx,stx)} ≤ ψ(sx) −ψ(stx). (3.7) then the mapping t has a fixed point. proof. for any x ∈ xω, we set p(x) = {z ∈ xω : ωλ(x,z) ≤ ψ(sx) −ψ(sz) and c ·ωλ(sx,sz)}≤ ψ(sx) −ψ(sz) for all λ > 0} α(x) = inf{ψ(sz) : z ∈ p(x)}. as x ∈ p(x), it is clear that p(x) 6= ∅ and 0 ≤ α(x) ≤ ψ(sx). similar to the proof of theorem 3.1, choose a sequence {xn} in xω: x1 = x, xn+1 ∈ p(xn) such that ψ(sxn+1) ≤ α(xn) + 1 n for all n ≥ 1. thus we obtain that ωλ(xn,xn+1) ≤ ψ(sxn) −ψ(sxn+1), c ·ωλ(sxn,sxn+1)} ≤ ψ(sxn) −ψ(sxn+1) (3.8) int. j. anal. appl. 16 (4) (2018) 478 and ψ(sxn+1) − 1 n ≤ α(xn) ≤ ψ(sxn+1). (3.9) from (3.8), {ψ(sxn)} is a nonincreasing and bounded sequence on r. so, {ψ(sxn)} is a modular convergent sequence. therefore, by (3.9) there is a number m ≥ 0 such that m = lim n→∞ α(xn) = lim n→∞ ψ(sxn). (3.10) now, let k ∈ n be an arbitrary point. from (3.10), there exists some nk such that ψ(sxn) < m + 1k for all n ≥ nk. thus, by monotonicity of {ψ(sxn)}, for all m ≥ n ≥ nk we have m ≤ ψ(sxm) ≤ ψ(sxn) ≤ m + 1 k . so, ψ(sxn) −ψ(sxm) ≤ 1 k . (3.11) preserving the generality, suppose that m > n and m,n ∈ n. from (3.8), we easily obtain that ω λ m−n (xn,xn+1) ≤ ψ(sxn) −ψ(sxn+1) and c ·ω λ m−n (sxn,stxn+1)} ≤ ψ(sxn) −ψ(sxn+1) (3.12) for λ m−n > 0. from (3.8), (3.12) and condition (m3) of modular metric, we have ωλ(xn,xm) ≤ ω λ m−n (xn,xn+1) + ω λ m−n (xn+1,xn+2) + · · · + ω λ m−n (xm−1,xm) ≤ ψ(sxn) −ψ(sxn+1) + ψ(sxn+1) −ψ(sxn+2) + · · · + ψ(sxm−1) −ψ(sxm) = ψ(sxn) −ψ(sxm) c ·ωλ(sxn,sxm) ≤ c ·ω λ m−n (sxn,sxn+1) + c ·ω λ m−n (sxn+1,sxn+2) + · · · + cω λ m−n (sxm−1,sxm) ≤ ψ(sxn) −ψ(sxn+1) + ψ(sxn+1) −ψ(sxn+2) + · · · + ψ(sxm−1) −ψ(sxm) = ψ(sxn) −ψ(sxm). (3.13) from (3.11), we get ωλ(xn,xm) < 1 k and c ·ωλ(sxn,sxm) < 1 k for all m ≤ n ≤ nk and k ∈ n. therefore, {xn}n∈n is a modular cauchy sequence in xω and {sxn}n∈n is a modular cauchy sequence in yω. by completeness of xω and yω, there exist p ∈ xω and q ∈ yω such that int. j. anal. appl. 16 (4) (2018) 479 xn → p and sxn → q. the fact that, s is a closed mapping implies sp = q. since ψ is lower semi-continuous, using equation (3.13), we have ψ(q) = ψ(sp) ≤ lim m→∞ inf ψ(sxm) ≤ lim m→∞ inf(ψ(sxn) −ωλ(xn,xm)) = ψ(sxn) −ωλ(xn,p). then we obtain ωλ(xn,p) ≤ ψ(sxn) −ψ(sp) for λ > 0. similarly, we get c ·ωλ(xn,p) ≤ ψ(sxn) −ψ(sp). thus, p ∈ p(xn) for all n ∈ n. then α(xn) ≤ ψ(sp). so, by (3.10), we get m ≤ ψ(sp). on the other hand, using lower semi-continuity of ψ and (3.10), we have ψ(q) = ψ(sp) = lim m→∞ α(xn) = m. therefore, ψ(sp) = m. by benefiting from the proof of theorem 3.2, we obtain that x 6= p implies x /∈ p(p). from (3.7), it’s clear that tp ∈ p(p), then we have tp = p. � corollary 3.1. theorem 3.3 holds with inequality max{ωλ(x,tx),c ·ωλ(sx,stx)}≤ ψ(sx) −ψ(stx) in the place of inequality (3.7). example 3.1. let x = r. we define the mapping ω : (0,∞) × r × r → [0,∞] by ωλ(x,y) = |x−y| 1+λ for all x,y ∈ r and λ > 0. then it is obvious that rω is a complete modular metric space. define t : rω → rω by tx = x 4 and ψ : rω → [0,∞] by ψ(x) = 3|x|. then for all x,y ∈ r and λ > 0, we have ωλ(x,tx) = |x−tx| 1 + λ = |x− x 4 | 1 + λ = 3|x| 4(1 + λ) ≤ 3 4 |x| and ψ(x) −ψ(tx) = 3|x|− 3|x| 4 = 9 4 |x|. hence, ωλ(x,tx) ≤ ψ(x) − ψ(tx). from theorem 3.1, the mapping t has a fixed point. moreover, it is 0 ∈ rω. int. j. anal. appl. 16 (4) (2018) 480 4. characterization of completeness we now prove a new theorem, which together with theorem 3.1 characterizes completeness in modular metric spaces. theorem 4.1. let xω be a modular metric space which is not complete modular. then there exists a fixed point free function t : xω → xω and a lower semi-continuous mapping ψ : xω → r+ such that ωλ(x,tx) ≤ ψ(x) −ψ(tx) for all x ∈ xω and λ > 0. proof. let {xn}⊂ xω be a modular cauchy sequence, which has no limit. we define a function φ : xω → r+ by φ(u) = lim m→∞ ωλ(u,xm), u ∈ xω for all λ > 0. given x ∈ xω and let n denote the smallest positive integer such that 0 < 1 2 ωλ(x,xn) ≤ φ(x) −φ(xn) for all λ > 0. (4.1) note that φ(xn) → 0 as φ(x) > 0. with n so determined, we define function t : xω → xω as tx = xn. let ψ(x) = 2φ(x). then from (4.1), we obtain that ωλ(x,tx) ≤ ψ(x) −ψ(tx). � 5. application let xω be a complete modular metric space, y be a banach space, m ⊆ xω, s ⊆ y and θ : m ×s → m, h : m ×s ×r → r be two functions. using theorem 3.1, we show the existence of a bounded solution for the following problem in dynamic programming: we take a g ∈ b(m) such that g(t) = sup s∈s {h(t,s,g(θ(t,s)))} (5.1) where t ∈ m and b(m) is a banach space which consists of all bounded real functionals on m with the norm ‖g‖ = supt∈m |g(t)|. we define a complete modular metric on b(m) with ωλ(g,k) = sup t∈z {∣∣∣∣g(t) −k(t)1 + λ ∣∣∣∣ } (5.2) for all g,k ∈ b(m) and λ > 0. if we take a cauchy sequence {gn}n∈n in b(m), then from completeness of xω, there exists a function u ∈ b(m) such that the sequence {gn}n∈n is convergent to u. int. j. anal. appl. 16 (4) (2018) 481 theorem 5.1. let θ : m ×s → m, h : m ×s × r → r be bounded and ψ : b(m) → r+ be lower semi continuous on xω and define by ψ(g) = ‖g‖. we define a operator t : b(m) → b(m) by t(g)(t) = sup s∈s {h(t,s,g(θ(t,s)))} for all g ∈ b(m) and t ∈ m. if sup t∈m ∣∣∣∣g(t) −h(t,s,g(θ(t,s)))λ ∣∣∣∣ ≤ ψ(g) −ψ(t(g)) (5.3) for all λ > 0, g,k ∈ b(m) and s ∈ s, then the functional equation (5.1) has a bounded solution. proof. let t ∈ m and g ∈ b(m). then there exists s ∈ s and � > 0 such that t(g)(x) < h(t,s,g(θ(t,s))) + � (5.4) and t(g)(x) > h(t,s,g(θ(t,s))). (5.5) on the other hand, it is obvious that g(t) < g(t) + � (5.6) and g(t) > g(t) − �. (5.7) for all � > 0. by using the inequalities (5.5) and (5.6) we obtain that g(t) −t(g)(t) < g(t) −h(t,s,g(θ(t,s))) + � ≤ |g(t) −h(t,s,g(θ(t,s)))| + �. (5.8) similarly, by using the inequalities (5.4) and (5.7) we obtain that t(g)(t) −g(t) < h(t,s,g(θ(t,s))) −g(t) + 2� ≤ |h(t,s,g(θ(t,s))) −g(t)| + 2�. (5.9) therefore, from the inequalities (5.8) and (5.9), we get |g(t) −t(g)(t)| < |g(t) −h(t,s,g(θ(t,s)))| + 2�. (5.10) if we divide both sides of the inequality (5.10) by 1 + λ, we get∣∣∣∣g(t) −t(g)(t)1 + λ ∣∣∣∣ < ∣∣∣∣g(t) −h(t,s,g(θ(t,s)))1 + λ ∣∣∣∣ + 2�1 + λ (5.11) for all λ > 0. since 2� 1+λ > 0 in the inequality (5.11), we can ignore the contrary 2� 1+λ . then we have∣∣∣∣g(t) −t(g)(t)1 + λ ∣∣∣∣ < ∣∣∣∣g(t) −h(t,s,g(θ(t,s)))1 + λ ∣∣∣∣ int. j. anal. appl. 16 (4) (2018) 482 for all λ > 0. then from property of supremum, we get sup t∈z ∣∣∣∣g(t) −t(g)(t)1 + λ ∣∣∣∣ < sup t∈z ∣∣∣∣g(t) −h(t,s,g(θ(t,s)))1 + λ ∣∣∣∣. then from inequalities (5.2) and (5.3) we obtain that ωλ(g,t(g)) < ψ(g) −ψ(t(g)). therefore, from theorem 3.1, t has a fixed point u ∈ b(z). then the functional equation (5.1) has a bounded solution. � references [1] ö. acar and i. altun, some generalizations of caristi type fixed point theorem on partial metric spaces, filomat, 26(4) (2012), 833-837. [2] h. aydi and m. abbas, tripled coincidence and fixed point results in partial metric spaces, appl. gen. topol., 13(2) (2012), 193-206. [3] a.a.n. abdou and m.a. khamsi, fixed point results of pointwise contractions in modular metric spaces, fixed point theory appl., 2013 (2013), article id 163. [4] a.a.n. abdou and m.a. khamsi, fixed points of multivalued contraction mappings in modular metric spaces, fixed point theory appl., 2014 (2014), article id 249. [5] c. alaca, m.e. ege and c. park, fixed point results for modular ultrametric spaces, j. comput. anal. appl., 20(1) (2016), 1259–1267. [6] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund. math., 3 (1922), 133–181. [7] t.g. bhaskar and v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal., 65 (2006), 1379–1393. [8] j. caristi, fixed point theorems for mappings satisfying inwardness conditions, trans. am. math. soc., 215 (1976), 241–251. [9] v.v. chistyakov, modular metric spaces generated by f-modulars, folia math., 14 (2008), 3–25. [10] v.v. chistyakov, modular metric spaces, i: basic concepts, nonlinear anal., 72 (2010), 1–14. [11] v.v. chistyakov, fixed points of modular contractive maps, doklady math., 86(1) (2012), 515–518. [12] y.j. cho, r. saadati and g. sadeghi, quasi-contractive mappings in modular metric spaces, j. appl. math., 2012 (2012), 907–951. [13] i. erhan, e. karapinar and d. turkoglu, different types meir-keeler contractions on partial metric, j. comput. anal. appl, 14 (2012), 1000-1005. [14] r. kannan, some results on fixed points, bull. calcutta math. soc., 60 (1968), 71–76. [15] m.a. khamsi, remarks on caristi’s fixed point theorem, nonlinear anal., 71(1–2) (2009), 227–231. [16] w.a. kirk, caristi’s fixed point theorem and metric convexity, colloq. math., 36 (1976), 81–86. [17] p. kumam, fixed point theorems for nonexpansive mapping in modular spaces, arch. math., 40 (2004), 345–353. [18] c. mongkolkeha, w. sintunavarat and p. kumam, fixed point theorems for contraction mappings in modular metric spaces, fixed point theory appl., 2011 (2011), article id 93. [19] a. mutlu, k. özkan and u. gürdal, coupled fixed point theorems on bipolar metric spaces, eur. j. pure appl. math., 10(4) (2017), 655–667. int. j. anal. appl. 16 (4) (2018) 483 [20] a. mutlu, k. özkan and u. gürdal, coupled fixed point theorem in partially ordered modular metric spaces and its an application, j. comput. anal. appl., 25(2) (2018), 1–10. [21] h. nakano, modulared semi-ordered linear spaces, in tokyo math. book ser., 1, maruzen co. tokyo, 1950. [22] j. musielak and w. orlicz, on modular spaces, studia math., 18 (1959), 49–65. [23] w. shatanawi, b. samet and m. abbas, coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, math. comput. model., 55(3-4) (2012), 680-687. [24] t. suzuki, a new type of fixed point theorem in metric spaces, nonlinear anal., 71 (2009), 5313–5317. 1. introduction 2. modular metric spaces 3. main results 4. characterization of completeness 5. application references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 91-101 http://www.etamaths.com fixed points under ψ-α-β conditions in ordered partial metric spaces zoran kadelburg1,∗ and stojan radenović2 abstract. recently, e. karapinar and p. salimi [fixed point theorems via auxiliary functions, j. appl. math. 2012, article id 792174] have obtained fixed point results for increasing mappings in a partially ordered metric space using three auxiliary functions in the contractive condition. in this paper, these results are extended to 0-complete ordered partial metric spaces with a more general contractive condition. examples are given showing that these extensions are proper. 1. introduction matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. he showed that the banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. subsequently, several authors (see, e.g., [2, 3, 4, 5, 6]) proved variuos more general fixed point results in partial metric spaces. the notion of weakly contractive conditions in metric spaces was first used by rhoades [7] who proved the following theorem 1.1. [7] let (x,d) be a metric space. if t : x → x satisfies the condition d(tx,ty) ≤ d(x,y) −ϕ(d(x,y)), ∀x,y ∈ x, where ϕ : [0, +∞) → [0, +∞) is a continuous and nondecreasing function such that ϕ(t) = 0 iff t = 0, then t has a unique fixed point. subsequently, several authors (see, e.g., [8, 9]) proved various generalizations and refinements of this result. fixed point theory has developed rapidly in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. the first result in this direction was given by ran and reurings [10] who presented its applications to matrix equations. further, a lot of results appeared, we mention just those contained in [11, 12, 13, 14, 15, 16]. these results use weaker contractive conditions (mostly just for comparable elements of the given space), but at the expense of some additional restrictions to the mappings involved. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. common fixed point, partial metric space, weakly contractive condition. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 91 92 kadelburg and radenović consider the following classes of functions from [0, +∞) into itself: ψ = {ψ : ψ is nondecreasing and lower semicontinuous }, φ1 = {α : α is upper semicontinuous }, φ2 = {β : β is lower semicontinuous }. very recently, karapinar and salimi [16] proved the following theorem 1.2. let (x,�,d) be a complete ordered metric space and let t : x → x be a nondecreasing selfmap. assume that there exist ψ ∈ ψ, α ∈ φ1 and β ∈ φ2 such that for all s,t ≥ 0, (1.1) t > 0 and (s = t or s = 0) implies ψ(t) −α(s) + β(s) > 0, and (1.2) ψ(d(tx,ty)) ≤ α(d(x,y)) −β(d(x,y)) for all comparable x,y ∈ x. suppose that, either, t is continuous, or x is regular (see further definition 2.5). if there exists x0 ∈ x such that x0 � tx0, then t has a fixed point. we shall prove in the present paper that this result can be extended to 0-complete ordered partial metric spaces with a contractive condition more general than (1.2). examples will be given showing that this extension is proper. remark 1.3. it was shown very recently (see [17]) that in some cases fixed point results in partial metric spaces can be directly reduced to their standard metric counterparts. we note that the results of the present paper do not fall into this category. moreover, using the method from [17], it is not possible to conclude that if x is a fixed point of the mapping under consideration, then p(x,x) = 0, which is an important conclusion for applications in computer science. 2. preliminaries and auxiliary results the following definitions and details can be seen in [1, 2, 3, 4, 18]. definition 2.1. a partial metric on a nonempty set x is a function p : x×x → r+ such that for all x,y,z ∈ x: (p1) x = y ⇐⇒ p(x,x) = p(x,y) = p(y,y), (p2) p(x,x) ≤ p(x,y), (p3) p(x,y) = p(y,x), (p4) p(x,y) ≤ p(x,z) + p(z,y) −p(z,z). a partial metric space is a pair (x,p) of a nonempty set x and a partial metric p on x. if, moreover, � is a partial order on x, then the triple (x,�,p) is called an ordered partial metric space. it is clear that, if p(x,y) = 0, then from (p1) and (p2), x = y. but p(x,x) may not be 0. a sequence {xn} in (x,p) converges to a point x ∈ x if limn→∞p(x,xn) = p(x,x). this will be denoted as xn → x (n → ∞) or limn→∞xn = x. clearly, a limit of a sequence in a partial metric space need not be unique. moreover, the function p(·, ·) need not be continuous in the sense that xn → x and yn → y imply p(xn,yn) → p(x,y). fixed points in partial metric spaces 93 example 2.2. (1) a paradigmatic example of a partial metric space is the pair (r+,p), where p(x,y) = max{x,y} for all x,y ∈ r+. (2) [1] let x = { [a,b] : a,b ∈ r, a ≤ b} and let p([a,b], [c,d]) = max{b,d}− min{a,c}. then (x,p) is a partial metric space. definition 2.3. let (x,p) be a partial metric space. then: (1) a sequence {xn} in (x,p) is called a cauchy sequence if limn,m→∞p(xn,xm) exists (and is finite). (2) the space (x,p) is said to be complete if every cauchy sequence {xn} in x converges to a point x ∈ x such that p(x,x) = limn,m→∞p(xn,xm). (3) [2] a sequence {xn} in (x,p) is called 0-cauchy if limn,m→∞p(xn,xm) = 0. the space (x,p) is said to be 0-complete if every 0-cauchy sequence in x converges (in τp) to a point x ∈ x such that p(x,x) = 0. lemma 2.4. let (x,p) be a partial metric space. (a) [3] if p(xn,z) → p(z,z) = 0 as n → ∞, then p(xn,y) → p(z,y) as n → ∞ for each y ∈ x. (b) [2] if (x,p) is complete, then it is 0-complete. the converse assertion of (b) does not hold as an easy example in [2] shows. definition 2.5. let (x,�,p) be an ordered partial metric space. we say that x is regular if the following holds: if {zn} is a nondecreasing sequence in x with respect to � such that zn → z ∈ x as n →∞ (in (x,p)), then zn � z for all n ∈ n and if {zn} is a nonincreasing sequence in x with respect to � such that zn → z ∈ x as n →∞ (in (x,p)), then zn � z for all n ∈ n. we will also need the following lemma which was proved in the metric case, e.g., in [19]. the proof is similar in the partial metric case, and so we omit it. lemma 2.6. let (x,p) be a partial metric space and let {xn} be a sequence in x such that (2.1) lim n→∞ p(xn+1,xn) = 0. if {xn} is not a 0-cauchy sequence in (x,p), then there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that nk > mk > k and the following four sequences tend to ε+ when k →∞: (2.2) p(xmk,xnk ), p(xmk,xnk+1), p(xmk−1,xnk ), p(xmk−1,xnk+1). 3. main results our first result is the following theorem 3.1. let (x,�,p) be a 0-complete ordered partial metric space and let t : x → x be a nondecreasing selfmap. assume that there exist functions ψ ∈ ψ, α ∈ φ1 and β ∈ φ2 such that for all t,s ≥ 0, (3.1) t > 0 and (s = t or s = 0) implies ψ(t) −α(s) + β(s) > 0, and (3.2) ψ(p(tx,ty)) ≤ α(m(x,y)) −β(m(x,y)) 94 kadelburg and radenović for all comparable x,y ∈ x, where m(x,y) = max { p(x,y),p(x,tx),p(y,ty), 1 2 [p(x,ty) + p(y,tx)] } . suppose that, either t is continuous, or x is regular. if there exists x0 ∈ x such that x0 � tx0, then t has a fixed point z ∈ x satisfying that p(z,z) = 0. proof. starting with the given x0, construct the picard sequence {xn} by xn+1 = txn, n ∈ n0. using the assumption x0 � tx0 and that t is nondecreasing, we conclude that x0 � x1 � ···� xn � xn+1 � ··· . if xn+1 = xn for some n ∈ n0, a fixed point of t is found. suppose that xn+1 6= xn for all n ∈ n0. apply assumption (3.2) for x = xn and y = xn+1 to obtain ψ(p(xn+1,xn+2)) = ψ(p(txn,txn+1))(3.3) ≤ α(m(xn,xn+1)) −β(m(xn,xn+1)), where m(xn,xn+1) = max{p(xn,xn+1),p(xn,xn+1),p(xn+1,xn+2), 1 2 (p(xn,xn+2) + p(x2n+1,x2n+1))} = max{p(xn,xn+1),p(xn+1,xn+2)} (condition (p4) of partial metric was used). suppose that p(xn+1,xn+2) > p(xn,xn+1) for some n ∈ n0. then (3.3) implies that ψ(p(xn+1,xn+2)) ≤ α(p(xn+1,xn+2)) −β(p(xn+1,xn+2)). by the assumption (3.1) it follows that p(xn+1,xn+2) = 0 and, hence, xn+1 = xn+2, which is already excluded. hence, p(xn+1,xn+2) ≤ p(xn,xn+1), and m(xn,xn+1) = p(xn,xn+1) for all n ∈ n0. thus, the sequence {p(xn,xn+1)} is nonincreasing. since it is bounded from below, there exists r ≥ 0 such that limn→∞p(xn,xn+1) = r. it follows from (3.3) that ψ(p(xn+1,xn+2) ≤ α(p(xn,xn+1)) −β(p(xn,xn+1)), and using the properties of functions ψ,α,β we get that ψ(r) ≤ lim inf ψ(p(xn+1,xn+2)) ≤ lim sup ψ(p(xn+1,xn+2)) ≤ lim sup[α(p(xn,xn+1)) −β(p(xn,xn+1))] = lim sup α(p(xn,xn+1)) − lim inf β(p(xn,xn+1))] ≤ α(r) −β(r). using again the condition (3.1), we get that it is only possible if r = 0. next, we claim that {xn} is a 0-cauchy sequence in the partial metric space (x,p). suppose that this is not the case. then, using lemma 2.6 we get that there exist ε > 0 and two sequences {mk} and {nk} of positive integers such that nk > mk > k and sequences (2.2) tend to ε when k →∞. applying condition (3.2) to elements x = xnk−1 and y = xmk we get that (3.4) ψ(p(xnk,xmk+1)) ≤ α(m(xnk−1,xmk )) −β(m(xnk−1,xmk )), fixed points in partial metric spaces 95 where m(xnk−1,xmk ) = max{p(xnk−1,xmk ),p(xnk−1,xnk ),p(xmk,xmk+1), 1 2 [p(xnk−1,xmk+1) + p(xnk,xmk )]}. using lemma 2.6 we get that limk→∞m(xnk−1,xmk ) = ε. passing to the upper limit in (3.4) and using properties of the functions ψ,α,β, we get that ψ(ε) ≤ lim inf ψ(p(xnk,xmk+1)) ≤ lim sup ψ(p(xnk,xmk+1)) ≤ lim sup[α(m(xnk−1,xmk )) −β(m(xnk−1,xmk ))] = lim sup α(m(xnk−1,xmk )) − lim inf β(m(xnk−1,xmk )) ≤ α(ε) −β(ε). this is (because of ε > 0) a contradiction with (3.1). we conclude that {xn} is a 0-cauchy sequence in (x,p). now, since (x,p) is a 0-complete partial metric space, it follows that there exists x ∈ x such that xn → x as n →∞, i.e., p(xn,x) → p(x,x) = 0, as n →∞. assume first that the mapping t is continuous. then limn→∞p(txn,tx) = p(tx,tx). it follows that p(tx,x) ≤ p(tx,txn) + p(xn+1,x) −p(xn+1,xn+1) ≤ p(tx,txn) + p(xn+1,x) → p(tx,tx) as n →∞. it follows that p(tx,x) = p(tx,tx). since x � x, we can apply (3.2) to obtain ψ(p(tx,tx)) ≤ α(m(x,x)) −β(m(x,x)) = α(p(x,tx)) −β(p(x,tx)) = α(p(tx,tx)) −β(p(tx,tx)). by (3.1), this is possible only if p(tx,tx) = 0, i.e., p(x,tx) = 0. hence tx = x. assume now that the space (x,�,p) is regular. since the sequence {xn} is increasing and xn → x as n → ∞, we get that xn � x for n ∈ n. hence, we can apply (3.2) to get (3.5) ψ(p(txn,tx)) ≤ α(m(xn,x)) −β(m(xn,x)), where m(xn,x) = max{p(xn,x),p(xn,xn+1),p(x,tx), 12 [p(xn,tx) + p(xn+1,x)]} → p(x,tx) as n →∞. on the other hand, since p(xn,x) → p(x,x) = 0, lemma 2.4.(1) implies that p(txn,tx) = p(xn+1,tx) → p(x,tx). hence, passing to the upper limit in (3.5), similarly as in the previous case we get that it is only possible that p(x,tx) = 0, i.e., tx = x. in all possible cases we have obtained that x is a fixed point of the mapping t satisfying p(x,x) = 0 and the theorem is proved. � remark 3.2. taking p to be a standard metric in the contractive condition (3.2), and assuming ψ = α to be continuous, we obtain [13, corollary 3.3]. in a similar way one can prove the following two assertions. 96 kadelburg and radenović theorem 3.3. let all the conditions of theorem 3.1 be fulfilled, except that condition (3.2) is replaced by (3.6) ψ(p(tx,ty)) ≤ α(p(x,y)) −β(p(x,y)). then t has a fixed point in x. theorem 3.4. let all the conditions of theorem 3.1 be fulfilled, except that condition (3.2) is replaced by (3.7) ψ(p(tx,ty)) ≤ α(m1(x,y)) −β(m1(x,y)), where m1(x,y) = max { p(x,y), 1 2 [p(x,tx) + p(y,ty)], 1 2 [p(x,ty) + p(y,tx)] } . then t has a fixed point in x. the following simple example shows that conditions of theorem 3.1 are not sufficient for the uniqueness of fixed points. example 3.5. let x = {(1, 0), (0, 1)}, let (a,b) � (c,d) if and only if a ≤ c and b ≤ d, and let p be the euclidean metric. the function t((a,b)) = (a,b) is continuous. the only comparable pairs of points in x are x � x for x ∈ x and then m(x,x) = 0 and p(tx,tx) = 0, hence the condition ψ(p(fx,fy)) ≤ α(m(x,y))−β(m(x,y)) is fulfilled, e.g., for the functions ψ ∈ ψ, α ∈ φ1, β ∈ φ2 given as ψ(t) = α(t) = t, β(t) = kt, 0 < k < 1. however, t has two fixed points (1, 0) and (0, 1). using the same example we can show that there exist situations where conditions of theorem 3.1, taken in the case without order may not be sufficient. example 3.6. if the previous example is considered without order, then one has also to take into account the case when x 6= y. but then p(x,y) = √ 2 and m(tx,ty) = √ 2, and so the condition (3.2) reduces to ψ( √ 2) ≤ α( √ 2) −β( √ 2) and cannot be valid for any functions ψ,α,β satisfying (3.1). now we give a sufficient condition for the uniqueness of fixed point. theorem 3.7. let all the conditions of theorem 3.1 be fulfilled and, moreover, the space (x,�,p) satisfy the following condition: for all x,y ∈ x there exists z ∈ x, z � tz, satisfying both x � z and y � z or there exists z ∈ x, z � tz, satisfying both x � z and y � z. then the fixed point of t is unique. proof. let x and y be two fixed points of t, i.e., tx = x and ty = y. consider the following two possible cases. 1. x and y are comparable. then we can apply condition (3.2) and obtain that ψ(p(x,y)) = ψ(p(tx,ty)) ≤ α(m(x,y)) −β(m(x,y)), where m(x,y) = max{p(x,y),p(x,tx),p(y,ty), 1 2 [p(x,ty) + p(y,tx)]} = p(x,y) and hence ψ(p(x,y)) ≤ α(p(x,y)) −β(p(x,y)) which is possible only if p(x,y) = 0 and hence x = y. fixed points in partial metric spaces 97 2. suppose now that x and y are not comparable. choose an element z ∈ x, z � tz comparable with both of them. then also x = tnx is comparable with tnz for each n (since t is nondecreasing). applying (3.1) one obtains that ψ(p(x,tnz)) = ψ(p(ttn−1x,ttn−1z)) ≤ α(m(tn−1x,tn−1z)) −β(m(tn−1x,tn−1z)), where m(tn−1x,tn−1z) = max{p(tn−1x,tn−1z),p(tn−1x,tnx),p(tn−1z,tnz), 1 2 [p(tn−1x,tnz) + p(tnx,tn−1z)]} = max{p(x,tn−1z),p(tn−1z,tnz), 1 2 [p(x,tnz) + p(x,tn−1z))} ≤ max{p(x,tn−1z),p(x,tnz)}, for n sufficiently large, because p(tn−1z,tnz) → 0 when n →∞ (the last assertion can be proved, starting from the assumption z � tz, in the same way as a similar conclusion in the proof of theorem 3.1). similarly as in the proof of theorem 3.1, it can be shown that p(x,tnz) ≤ m(x,tn−1z) ≤ p(x,tn−1z). it follows that the sequence p(x,tnz) is nonincreasing and it has a limit l ≥ 0. assuming that l > 0 and passing to the limit in the relation ψ(p(x,tnz)) ≤ α(m(x,tn−1z)) −β(m(x,tn−1z)) one obtains that l = 0, a contradiction. in the same way it can be deduced that p(y,tnz) → 0 as n → ∞. now, passing to the limit in p(x,y) ≤ p(x,tnz) + p(tnz,y), it follows that p(x,y) = 0. hence, x = y and the uniqueness of the fixed point is proved. � remark 3.8. if two selfmaps t,s : x → x are given, then in a similar way a common fixed point result can be obtained, under an appropriate contractive condition and assuming, e.g., that t and s are weakly increasing (for the details see, e.g., [13]). the proof uses the standard method of jungck sequences. 4. examples our first example (inspired by [5]) shows that it may happen that the contractive condition (3.7) is not satisfied, hence the existence of a fixed point cannot be obtained using theorem 3.3. however, the condition (3.2) is fulfilled and theorem 3.1 can be used to obtain the conclusion. example 4.1. consider the set x = {a,b,c} and the function p : x × x → r given by p(a,b) = p(b,c) = 1, p(a,c) = 3 2 , p(x,y) = p(y,x), p(a,a) = p(c,c) = 1 2 and p(b,b) = 0. obviously, p is a partial metric on x, not being a metric (since p(x,x) 6= 0 for x = a and x = c). define an order-relation � on x by a � b � c. then, (x,�,p) is a 0-complete ordered partial metric space. define a selfmap t on x by t : ( a b c b b a ) . then t is not a (banach)-contraction since p(fc,fc) = p(a,a) = 1 2 = p(c,c) 98 kadelburg and radenović and there is no λ ∈ [0, 1) such that p(fc,fc) ≤ λp(c,c). moreover, condition (3.7) (i.e., condition (1.2) of theorem 1.2, with d replaced by p) cannot hold for functions ψ ∈ ψ, α ∈ φ1, β ∈ φ2 satisfying (1.1) because for otherwise x = y = c would imply ψ( 1 2 ) −α( 1 2 ) + β( 1 2 ) ≤ 0 which cannot hold. we will check that t satisfies the condition (3.2) of theorem 3.1 with functions ψ,α,β given as ψ(t) = t, α(t) = t, β(t) = 1 3 t (obviously belonging to the respective classes). note that in this case α(t) −β(t) = 2 3 t. if x,y ∈{a,b}, then p(tx,ty) = p(b,b) = 0 and (3.2) trivially holds. let, e.g., y = c; then we have the following three cases: ψ(p(ta,tc)) = p(b,a) = 1 ≤ 2 3 · 3 2 = 2 3 max{p(a,c),p(a,ta),p(c,tc), 1 2 [p(a,tc) + p(c,ta)]}, ψ(p(tb,tc)) = p(b,a) = 1 ≤ 2 3 · 3 2 = 2 3 max{p(b,c),p(b,tb),p(c,tc), 1 2 [p(b,tc) + p(c,tb)]}, ψ(p(tc,tc)) = p(a,a) = 1 2 < 2 3 · 3 2 = 2 3 max{p(c,c),p(c,tc)}. thus, conditions of theorem 3.1 are satisfied and the existence of a fixed point of t follows. using an example of romaguera [2], we present another example showing how theorem 3.1 can be used. it also shows that there are situations when standard metric arguments cannot be used to obtain the existence of a fixed point. example 4.2. let x = [0, 1]∩q be equipped with the partial metric p defined by p(x,y) = max{x,y} for x,y ∈ x and the standard order. let t : x → x be given by tx = x2 1 + x . it is easy to see that the space (x,p) is 0-complete. take the functions ψ,α,β given by ψ(t) = α(t) = t, β(t) = 1 2 t. the contractive condition (3.2) for (say) x ≥ y takes the form p(tx,ty) = max { x2 1 + x , y2 1 + y } = x2 1 + x ≤ 1 2 max{p(x,y),p(x, x 2 1+x ),p(y, y 2 1+y ), 1 2 [p(x, y 2 1+y ) + p(y, x 2 1+x )]} = 1 2 max{x,x,y, 1 2 [x + max{y, x 2 1+x }]} = 1 2 x, and it is satisfied for all x,y ∈ x, since 0 ≤ x ≤ 1. hence, all the conditions of theorem 3.1 and theorem 3.7 are satisfied and t has a unique fixed point (z = 0). however, if we consider the same ordered set x = [0, 1] ∩ q equipped with the standard metric d(x,y) = |x−y|, we obtain a non-complete ordered metric space. hence, the existence of a fixed point of t cannot be deduced using theorem 1.2. we present now an example (inspired by [16]) showing the usage of theorem 3.1 with (at least some of) functions ψ,α,β not being continuous. fixed points in partial metric spaces 99 example 4.3. consider the set x = [0, 1] equipped with the standard order and the partial metric given as p(x,y) = max{x,y}. it is easy to show that (x,�,p) is a regular, 0-complete ordered partial metric space. let t : x → x be given by tx = 1 2 x− 1 4 x2 and take the functions ψ,α,β defined by ψ(t) = { t + 3 2 , t > 0 1, t = 0, , α(t) = t + 5 2 , β(t) = 1 2 t + 1. note that ψ is lower semicontinuous and the condition (3.1) is satisfied. both conditions (3.2) and (3.7) are satisfied. we shall check, e.g., condition (3.2). let x,y ∈ x and, for example, x ≤ y. then ψ(p(tx,ty)) = ψ(ty) = { 1 2 y − 1 4 y2 + 3 2 , y > 0 1, y = 0, and m(x,y) = max{max{x,y}, max{x,tx}, max{y,ty}, 1 2 [max{x,ty} + max{y,tx}]} = max{y,x,y, 1 2 [max{x,ty} + y]} = y, since max{x,ty} ≤ y. the condition (3.2) reduces to 1 2 y − 1 4 y2 + 3 2 ≤ 1 2 y + 3 2 if y > 0 and to 1 ≤ 3 2 if y = 0 and it is fulfilled in both cases. by theorems 3.1 and 3.7, t has a unique fixed point (which is z = 0). finally, the following example demonstrates the situation when contractive conditions using partial metric can guarantee the existence of a fixed point, while the respective conditions with the standard metric cannot. example 4.4. consider the set x = [0, +∞) endowed with the partial metric p(x,y) = max{x,y} and the order given by x � y =⇒ x = y ∨ (x,y ∈ [0, 1] ∧x ≤ y). (x,�,p) is a regular, 0-complete ordered partial metric space. let t : x → x be given by tx =   x2 1 + x , x ∈ [0, 1] x 2 , x > 1 and consider the functions ψ ∈ ψ, α ∈ φ1, β ∈ φ2 defined as ψ(t) = α(t) = t, β(t) =   t 1 + t , t ∈ [0, 1] t 2 , t > 1. clearly, ψ,α,β satisfy condition (3.1). we shall show that also condition (3.2) is satisfied. let x,y ∈ x and suppose that, e.g., y � x. then, there are two possibilities: 1. x ∈ [0, 1] (and hence also y ∈ [0, 1] and y ≤ x). then p(tx,ty) = max { x2 1 + x , y2 y + x } = x2 1 + x , 100 kadelburg and radenović and m(x,y) = max { x,x,y, 1 2 [ x + max { y, x2 1 + x }]} = x, since max{y, x 2 1+x }≤ x. condition (3.2) reduces to x2 1 + x ≤ x− x 1 + x and obviously holds true. 2. x > 1 (and hence y = x). then p(tx,ty) = x 2 and m(x,y) = x. hence, (3.2) reduces to x 2 ≤ x− x 2 and is also satisfied. all the conditions of theorems 3.1 and 3.7 are fulfilled and t has a unique fixed point (which is z = 0). consider now the same problem but in the case that instead od the partial metric p, the standard metric d(x,y) = |x − y| is used (note that this is the socalled associated metric of p, see, e.g., [1]). take x = 1 and y = 1 2 . then ψ(d(tx,ty)) = ∣∣∣∣12 − 16 ∣∣∣∣ = 13, m(x,y) = max { 1 2 , 1 2 , 1 3 , 1 2 ( 5 6 + 0 )} = 1 2 , α(m(x,y)) −β(m(x,y)) = 1 2 − 1 2 1 + 1 2 = 1 6 < 1 3 , and the contractive condition (3.2) with p = d is not satisfied (neither is condition (1.2) of theorem 1.2). acknowledgements the authors are thankful to the ministry of education, science and technological development of serbia. references [1] s.g. matthews, partial metric topology, proc. 8th summer conference on general topology and applications, ann. new york acad. sci. 728 (1994), 183–197. [2] s. romaguera, a kirk type characterization of completeness for partial metric spaces, fixed point theory appl. (2010) article id 493298, 6 pages. [3] t. abdeljawad, e. karapinar, k. taş, existence and uniqueness of a common fixed point on partial metric spaces, appl. math. lett. 24 (2011), 1900–1904. [4] e. karapinar, i.m. erhan, fixed point theorems for operators on partial metric spaces, appl. math. lett. 24 (2011), 1894–1899. [5] c. di bari, z. kadelburg, h.k. nashine, s. radenović, common fixed points of g-quasicontractions and related mappings in 0-complete partial metric spaces, fixed point theory appl. 2012:113 (2012), doi:10.1186/1687-1812-2012-113. [6] n. hussain, z. kadelburg, s. radenović, f.r. al-solamy, comparison functions and fixed point results in partial metric spaces, abstract appl. anal. 2012, article id 605781, 15 pages, doi:10.1155/2012/605781. [7] b.e. rhoades, some theorems on weakly contractive maps, nonlinear anal. 47 (2001), 2683– 2693. [8] p.n. dutta, b.s. choudhury, a generalization of contraction principle in metric spaces, fixed point theory appl. (2008), article id 406368, doi:10.1155/2008/406368. [9] b.s. choudhury, a. kundu, (ψ,α,β)-weak contractions in partially ordered metric spaces, appl. math. lett. 25 (2012), 6–10. fixed points in partial metric spaces 101 [10] a.c.m. ran, m.c.b. reurings, a fixed point theorem in partially ordered sets and some application to matrix equations, proc. amer. math. soc. 132 (2004), 1435–1443. [11] j. harjani, k. sadarangani, fixed point theorems for weakly contractive mappings in partially ordered sets, nonlinear anal. 71 (2009), 3403–3410. [12] j. harjani, k. sadarangani, generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, nonlinear anal. 72 (3-4) (2010), 1188–1197. [13] s. radenović, z. kadelburg, generalized weak contractions in partially ordered metric spaces, comput. math. appl. 60 (2010), 1776–1783. [14] z. golubović, z. kadelburg, s. radenović, common fixed points of ordered gquasicontractions and weak contractions in ordered metric spaces, fixed point theory appl. 2012:20 (2012), doi:10.1186/1687-1812-2012-20. [15] h. aydi, e. karapinar, b. samet, remarks on some recent fixed point theorems, fixed point theory appl. 2012:76 (2012), doi:10.1186/1687-1812-2012-76. [16] e. karapinar, p. salimi, fixed point theorems via auxiliary functions, j. appl. math. 2012, article id 792174, 9 pages, doi:10.1155/2012/792174. [17] r.h. haghi, sh. rezapour, n. shahzad, be careful on partial metric fixed point results, topology appl. 160 (2013), 450–454. [18] h.k. nashine, z. kadelburg, s. radenović, common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, math. comput. modelling 57 (2013), 2355–2365. [19] s. radenović, z. kadelburg, d. jandrlić and a. jandrlić, some results on weak contraction maps, bull. iranian math. soc. 38(3) (2012), 625–645. 1faculty of mathematics, university of belgrade, studentski trg 16, 11000 beograd, serbia 2faculty of mechanical engineering, university of belgrade, kraljice marije 16, 11120 beograd, serbia ∗corresponding author international journal of analysis and applications volume 18, number 4 (2020), 672-688 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-672 geometric singularities of the poisson’s equation in a non-smooth domain with applications of weighted sobolev spaces yasir nadeem anjam1,2,∗ 1school of mathematical sciences, shanghai jiao tong university, shanghai 200240, p.r.china 2department of applied sciences, national textile university, faisalabad 37610, pakistan ∗corresponding author: ynanjam@ntu.edu.pk abstract. the solution fields of the elliptic boundary value problems may exhibit singularities near the corners, edges, crack tips, and so forth of the physical domain. this paper deals with the boundary singularities of weak solutions of boundary value problems governed by the poisson equation in a two-dimensional non-smooth domain with singular points on the boundary. the presence of these points on the boundary, generally, generates local singularities in the solution. the applications of fourier transform and weighted sobolev spaces make it possible to describe the qualitative properties of the solution including its regularity. the general theory of v. a. kondratiev is followed to obtain these results. 1. introduction let d be a 2-dimensional bounded plane polygonal domain d ⊂ r2, (see figure-1) whose boundary ∂d comprises the corner points (ω 6= π) and the points where the type of boundary conditions changes (ω = π). let n denote the set of these boundary points which consists of { p1, ...,pn } ⊂ ∂d. note that a point p ∈ ∂d is said to be a corner point if there exists a neighborhood η(p) of the point p such that d∩η(p) is diffeomorphic to an angle κ intersected with the unit circle. received april 24th, 2020; accepted may 14th, 2020; published june 3rd, 2020. 2010 mathematics subject classification. primary 35q99; secondary 35j05, 35b65, 35j25. key words and phrases. poisson equation; mixed boundary conditions; non-smooth domain; regularity; weighted sobolev spaces. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 672 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-672 int. j. anal. appl. 18 (4) (2020) 673 n p 1 p 2 p 3 p 1  n  2  3  i  1  2  3  figure 1. a polygonal domain. let pi denote the vertices of the polygon and the open edges γi connecting the vertices pi+1 and pi, 1 ≤ i ≤ n. let pn+1 = p1, γn+1 = γ1, γ0 = γn , and the interior angles are ωi = γi − γi−1. suppose that the boundary ∂d = γ0 ∪ γ1 and γ0 ∩ γ1 = ∅ with meas(γ0) > 0 (lebesgue measure). further, we assume that j1 and j2 be the disjoint subsets of { 1, 2, ...,n } , where we can set γ0 = ⋃ i∈j1 γ̄i and γ1 = ⋃ i∈j2 γ̄i, respectively, denote the union of the boundary parts, where the dirichlet boundary conditions and the neumann boundary conditions are given. as well, the combinations of the boundary points with different boundary conditions are considered. to characterize them, we denote the dirichlet-dirichlet boundary conditions by (dd), i.e., pi ∈ j1, the neumann-neumann boundary conditions by (nn), pi ∈ j2 and the dirichlet-neumann (mixed) boundary conditions by (dn), pi ∈j1j2. note that n = j1 ∪j2 ∪j1 j2. to describe the problem mathematically, let us consider the mixed boundary value problem for the poisson equation   −∆v = f in d, v = 0 on γ0, ∂v ∂n = 0 on γ1, (1.1) where n = (n1,n2) is known as the unit outward normal vector to the boundary, f ∈ l2(d) and ∆ = ( ∂ 2 ∂x2 + ∂ 2 ∂y2 ) is the laplacian operator. it is known from the theory of elliptic boundary value problems in domains with boundary irregularities, like corners, conic vertices, edges, and cracks, etc., the solution may exhibit singularities. numerous interesting results about the regularity of the solution cannot be extended if one of the following situations appears: the domain has corners, edges or angular points on the boundary, the change of the boundary conditions at some points, the discontinuities of the solution and the singularities of the coefficients. principally, the theory for smooth domains cannot be applied directly to non-smooth domains having corners or edges on the boundary, and the points where the type of boundary conditions changes. the asymptotic expansion of the solution near the conical or angular points plays an important role to describe the int. j. anal. appl. 18 (4) (2020) 674 regularity behavior of the solution accurately. moreover, the information of the singularity functions in nonsmooth domains can help to improve the rate of convergence of the numerical methods for approximations, for instance, the finite element approximation, singular function method or the dual singular function method, and the graded mesh refinement [13, 19, 20]. presently, there exists a wide-ranging theory for parabolic, hyperbolic, and elliptic boundary value problems having a smooth boundary. generally, the results of this theory conclude that if the boundary of the domain, the boundary operators, the coefficients of the equations, and the right-hand sides are sufficiently smooth, then the solution of the considered problem is itself sufficiently smooth [12, 13, 18, 29]. generally, three types of singularities arise in elliptic type problems: the angular type singularities, the interface, and the infinity type singularities in unbounded solution domains. this paper deals with the angular type singularities and several approaches to find these singularities are discussed in [5, 10, 27, 30]. in recent, [4] has comprehensively discussed the methods to find the singular behavior of the solution structure of the elliptic boundary value problems in a polygonal domain with convex and non-convex vertices. further, it is noted from the general theory on h2-regularity for linear elliptic boundary value problems [14, 15, 25], the general solution u for two or three-dimensional domain d with corner or edge singularities and any right-hand side function f ∈ l2(d) can be broken down as a sum of a singular and a regular part u = n∑ m=1 cmsm + ur, (1.2) where ur ∈ h2(d). the second part is the locally acting singular part that is a combination of explicit model singular solutions sm and the unknown coefficients cm. the special singular functions sm rely on the geometry of the model problem, the differential operator, and the characteristic boundary conditions. the unknown coefficients cm relating to singularity functions are some real numbers or unique scalar constants which are stated as the stress intensity factors. the rigorous formulas for their derivations are of constant interest and a challenging task [7, 13, 14, 25]. the mathematical analysis like well-posedness and regularity results of such type of elliptic boundary value problems in non-smooth domains have attracted many mathematicians and scientists to examine the singular behavior of the solution structure near the singular points [9, 15, 16, 21, 23]. the main purpose of this paper is the derivation and the computation of the singular terms of the solutions of the generalized boundary eigenvalue problem for the poisson equation in a bounded plane polygonal domain with singular points on its boundary. the theory developed by kondratiev [22, 23] and further extended by [28] for scalar problems is used in the context of weighted sobolev spaces. generally, the sobolev spaces are not suitable to define the regularity results of the boundary value problems in non-smooth domains. so, [22, 28] have introduced weighted sobolev spaces with kondratiev type weights for parabolic and elliptic problems in polygonal domains. in [25], where the method of special ansatzes and spherical coordinates are used to calculate the singular terms for the dirichlet problem of the poisson equation. int. j. anal. appl. 18 (4) (2020) 675 analogous to [6], where the mellin transform and the method of the special ansatzes is used to obtain the asymptotic singular representations of the solution of the biharmonic operator on a bounded domain with angular corners. the technique of fourier transform is used here to obtain the generalized form of the boundary eigenvalue problem for the poisson equation with the mixed boundary conditions. the achieved eigenvalues and eigensolutions generate singular terms. the information about the singular terms allows us to evaluate the optimal regularity of the corresponding weak solution of the considered boundary value problem. the rest of this paper is organized as follows: section 2 is dedicated to present the weak formulation of the problem and introduce some function spaces. in section 3, determine a parametric boundary eigenvalue problem with a complex parameter ξ, the poisson equation is considered in an infinite cone with various combinations of the boundary conditions. furthermore, the distribution of the eigenvalues and the eigenfunctions are discussed. in section 4, the regularity and expansion results for the corresponding problem with various conditions are investigated. some concluding remarks are given in the last section 5. 2. analytical preliminaries besides the strong formulation, let us consider the weak formulation of the mixed boundary value problem (1.1) which reads: find v ∈ u(d) = {v ∈ h1(d) : v = 0 on γ0} such that a(v,u) = f(v) ∀u ∈ u(d), (2.1) where a(v,u) = ∫ d ∇v ·∇udx and f(v) = ∫ d f ·udx. the lax-milgram theorem [12–14] deduces that the variational problem (2.1) has a unique solution. hence, we have to analyze the smoothness of the weak solution v and see how it depends on the size of the angle ωi, i = 1, ...,n. 2.1. weighted sobolev spaces. to analyze the regularity results of the weak solution of the corresponding boundary value problem in a non-smooth domain with singular points, firstly, we introduce some function spaces in line with [1, 11, 22, 28]. let n be the set of singular points on the boundary, i.e., n ⊂ ∂d. denote c∞n = { v ∈ c∞(d), supp v ∩n = ∅ } , where the supp v is bounded. we assume that dβv be the multi-index notation for higher-order derivatives and in cartesian coordinates is defined by dβv = ∂|β|v ∂x β1 1 ∂x β2 2 , β = (β1, β2), |β| = β1 + β2. int. j. anal. appl. 18 (4) (2020) 676 let α = ( α1, ...,αn ) be an n−tuple of real numbers which satisfying 0 < αi < 1 for 1 ≤ i ≤ n. therefore, the weight function is characterized by φα+m(x) = n∏ i=1 ( ri(x) )αi+m , where m is an any integer and ( ri(x) ) = dist (x,pi). let wm,pα (d) be the weighted sobolev spaces and is the closure of c∞n (d) equipped with the norm ‖v‖wm,pα (d) = ( ∑ |β|≤m ∫ d |x|p(α−m+|β|) ∣∣dβv∣∣pdx)1p . (2.2) let q = { (τ,θ) : −∞ < τ < ∞, 0 < θ < ω0 } denote the infinite strip with positive width ω0. for any real h > 0, and for an integer m ≥ 0, the spaces are defined as wmh (q) = { u ∈ l2(q) : ∑ |β|≤m ∫ q e2hτ ∣∣dβu∣∣2 dτdθ < ∞}, where ‖u‖wm h (q) = ( ∑ |β|≤m ∫ q e2hτ ∣∣dβu∣∣2 dτ dθ)12 . 3. the boundary value problem in an infinite cone in this section, we will see the occurrence of the singular terms near the singular points and the structure which they have. so, to analyze these results, the following steps are followed. (1) we localize the model problem in the neighborhood of the corner point or a point where the boundary conditions changes (known as a singular point), and then the model problem is considered in an infinite cone. (2) the model problem is transformed in the form of local polar coordinates (r, θ) and then the variable transformation r = eτ is used. afterward, the complex fourier transform respecting the variable τ is applied to attain a boundary value problem which depends on the complex parameter ξ. moreover, the operator v̂(ξ) is used to represent the generalized form of this parametric boundary eigenvalue problem. (3) the eigenvalues and the generalized eigensolutions of this parametric boundary eigenvalue problem with various kinds of boundary conditions are obtained. they exhibit the asymptotic development of the solution of the model problem near the singular points. finally, the regularity results can be followed by the general theory of ellipticity. int. j. anal. appl. 18 (4) (2020) 677 3.1. localization of the model problem. the regularity analysis of the solutions of mixed boundary value problems in a bounded plane polygonal domain is a local problem. if we presume that d is a polygonal domain, then the regularity principles work well in the interior of the domain and also on ∂d\ ⋃n i=1 η(pi), where η(pi) is the neighborhood of the corner points or the points where the type of boundary conditions changes. usually, these points are called singular points. to show that the weak solution v is regular, we have to investigate its behavior near the corner points pi, i = 1, 2, ...,n. let us consider one corner point pn as an origin with an angle ω0, and an appropriate infinite differentiable cut-off function χ(|x|) = χ(r) is defined as χ(r) =   1 for 0 ≤ r ≤ �, 0 for r ≥ 2�, (3.1) and it depends on the distance from the point pn . the number � is so small that pn is the only corner point of the domain d that lies inside the circle {x : |x| ≤ 2�}. afterward, multiplying the smooth cut-off function χ on both sides of (1.1), then substituting u = χv in (1.1). the derivatives are considered in the distribution sense. thus, the boundary value problem is transformed into an infinite cone s = { (r, θ) : 0 < r < ∞, 0 < θ < ω0 } , which coincides with the original problem near the corner point pn . then the system (1.1) become s np 0 ,s w ,0s 0  figure 2. infinite cone s with opening angle ω0.   −∆u = f in s, u = 0 on γs, 0, γs,ω0 if γs, 0, γs,ω0 ⊂ γ0, ∂u ∂n = g on γs, 0, γs,ω0 if γs, 0, γs,ω0 ⊂ γ1, (3.2) where f = χf − 2∇χ ·∇v−v ∆χ and g(x) = 0 for r < � and r > 2�. the behavior of u near the point pn illustrate the regularity of the solution v in the neighborhood of pn . if we suppose that the right-hand side in (1.1) is f ∈ l2(d), then f ∈ l2(s). the following boundary conditions are prescribed on the subsequent int. j. anal. appl. 18 (4) (2020) 678 edges γs, 0 (θ = 0) and γs,ω0 (θ = ω0) of the cone (see figure-2). just one condition is considered per edge to differentiate between the mixed boundary conditions. to analyze the regularity results of the boundary value problem (3.2), we rewrite the operators in the structure of polar coordinates (r,θ). hence, the transformed form is − (∂2ǔ ∂r2 + 1 r ∂ǔ ∂r + 1 r2 ∂2ŭ ∂θ2 ) =f̌(r,θ) in ŝ, ǔ(r,θ) ∣∣ θ=0,ω0 =0, ∂ǔ ∂θ (r,θ) ∣∣ θ=0,ω0 =ǧ(r,θ) ∣∣ θ=0,ω0 , (3.3) where ŝ is the infinite half-strip in the (r,θ)-plane and ǔ(r,θ) = u(x,y), f̌(r,θ) = f(x,y) and ǧ(r,θ) = g(x,y). now, a variable τ with the relation r = eτ is introduced, then (3.3) is transformed to the infinite strip with the width ω0 as − (∂2ũ ∂τ2 + ∂2ũ ∂θ2 ) =f̃(τ,θ) in s̄, ũ(τ,θ) ∣∣ θ=0,ω0 =0, ∂ũ ∂θ (τ,θ) ∣∣ θ=0,ω0 =g̃(τ,θ) ∣∣ θ=0,ω0 . (3.4) here, s̄ = { (τ, θ) : −∞ < τ < ∞, 0 < θ < ω0 } and ũ = ǔ(eτ,θ), f̃ = e2τ f̌(eτ,θ) and g̃ = eτ ǧ(eτ,θ). to obtain the boundary eigenvalue value problem, some basic properties of the complex fourier transform respecting variable τ in line with [16, 23, 28] are described as f[u](ξ) = û(ξ) = (2π)− 1 2 ∫ ∞ −∞ e−iξτ u(τ)dτ, ξ ∈ c, (3.5) and the inverse fourier transform is f−1[u](ξ) = u(τ) = (2π)− 1 2 ∫ ∞+ih −∞+ih eiξτû(ξ) dξ. (3.6) it defines an isomorphic mapping, i.e., f[u](ξ) = { u(τ) : ∫ ∞ −∞ e2hτ|u(τ)|2dτ < ∞ } → l2(r + ih), (3.7) for ξ = s + ih, where h = constant, r + ih = { ξ ∈ c : im ξ = h } . therefore, the subsequent parseval identity holds ∫ ∞ −∞ e2hτ |u(τ)|2 dτ = ∫ ∞+ih −∞+ih |û(ξ)|2 dξ. (3.8) we have f ( dm dτm u(τ) ) (ξ) = (iξ)mf ( u(τ) ) (ξ). (3.9) int. j. anal. appl. 18 (4) (2020) 679 moreover, it is noted that if h1 < h2 and the following properties are satisfied∫ +∞ −∞ e2h1τ |u(τ)|2 dτ < ∞, ∫ +∞ −∞ e2h2τ |u(τ)|2 dτ < ∞, (3.10) then û(ξ) is holomorphic in the strip h1 < im ξ < h2. now, by applying (3.5) to (3.4) with respect to τ, the parametric boundary value problem for the unknown function û is obtained that depend on the complex parameter ξ and holds in the interval i = (0,ω0). consequently, the transformed form of (3.4) is ξ2û− ∂2û ∂θ2 =f̂(ξ,θ), û(ξ,θ) ∣∣ θ=0,ω0 =0, ∂û ∂θ (ξ,θ) ∣∣ θ=0,ω0 =ĝ(ξ,θ) ∣∣ θ=0,ω0 . (3.11) let v̂(ξ) represent the operator of (3.11) and it maps from v̂(ξ) : w 2, 2(0,ω0) into l2(0,ω0) ×c×c. note that the operator v̂(ξ) can be defined for every boundary point in the sense of [2, 3]. so, the operator v̂(ξ)(ξ,θ) = 0 is used to describe a generalized eigenvalue problem and the solvability of these type of problems is discussed in [24]. the operator v̂(ξ) realizes an isomorphism for all ξ ∈ c apart from some isolated points (known as the eigenvalues of v̂(ξ)). so, the resolvent r(ξ) = [ v̂(ξ) ]−1 is an operator-valued, meromorphic function of ξ has poles of finite multiplicity. to compute the eigenvalues ξµ (generally referred for multiple eigenvalues) and the corresponding eigenfunctions, we proceed as. definition 3.1. a complex number ξ = ξ0 is known as the eigenvalue of v̂(ξ) if there exists a nontrivial solution which is holomorphic at ξ0, i.e., û(.,ξ0) 6= 0, and v̂(ξ0) û(θ,ξ0) = 0, where û(θ,ξ0) is an eigenfunction of v̂(ξ0) corresponding to the eigenvalue ξ0. the set of fields { û0(θ,ξ0), û0,1(θ,ξ0), ..., û0,s(θ,ξ0) } with û0,0 = û0 is said to be a jordan chain corresponding to the eigenvalue ξ0, if the equation σ∑ q=0 1 q! ( ∂ ∂ξ )q v̂(ξ) û0,m−q(θ,ξ)∣∣∣ ξ=ξ0 = 0 for m = 1, 2, ...,s, is satisfied. the number s + 1 is called the length of the jordan chain. remark 3.1. it is noted from [22–24] that if the complex number ξ is not an eigenvalue of the operator v̂(ξ), then v̂(ξ) is an isomorphism among the spaces v̂(ξ) : w 2, 2(0,ω0) and l2(0,ω0) ×c×c. theorem 3.1. let lh = {ξ ∈ c : im ξ = h}. if no eigenvalues of v̂(ξ) lies on the line lh, then the system (3.11) admits a unique solution û ∈ w 2,2(0,ω0) provided (f̂, 0,ĝ) ∈ l2(0,ω0) × c × c, and it holds for all int. j. anal. appl. 18 (4) (2020) 680 ξ ∈ lh: ‖û‖2w2,2(0,ω0) ≤ c { ‖f̂‖2l2(0,ω0) + |ξ||ĝ| 2 } , (3.12) with the constant c is independent of ξ. proof. a similar theorem is proved in ( [17], theorem 4.9). so, we omit its proof. � taking note from the above-mentioned results, and [22,23,28], we can derive a fundamental regularity and expansion theorem, based on the fourier transform, for the mixed boundary value problem for the poisson equation in a two-dimensional bounded domain with singular points on the boundary. by considering the substitution re α = −im ξ − 2 for α ∈ c, it improves theorems ( [24], theorem 8.2.1 and theorem 8.2.2.) which are based on the mellin transform technique and used for the solvability of the elliptic systems. theorem 3.2. (regularity and expansion theorem). let α1 and α2 be real numbers and satisfying α1 − 1 < α2 < α1. let v ∈ wm, 2α1 (d) be a solution of the mixed boundary value problem (1.1) and f ∈ wm1−2,pα2 (d) ∩w m−2, 2 α1 (d), where 1 ≤ p < ∞, m1 ≥ m ≥ 2 and α1 ≥ α2 ≥ 0. subsequently, the following implications holds: (1) if the strip α2 + 2 p −m1 ≤ im ξ ≤ α1 + 1 −m, is free of eigenvalues of the operator v̂(ξ), then the solution v ∈wm1,pα2 (d) and holds the following estimate ‖u‖wm1,pα2 (d) ≤ c(d)‖f‖wm1−2,pα2 (d) . (2) let ξ1,ξ2, ...,ξm be the eigenvalues of the operator v̂(ξ) and suppose that no eigenvalue lie on the lines im ξ = α2 + 2 p −m1 and im ξ = α1 + 1 −m. if the eigenvalues ξ1,ξ2, ...,ξm are situated in the strip α2 + 2 p −m1 < im ξ < α1 + 1 −m, then the solution v admits the following expansion in the neighborhood pδ of the corner point p , i.e., v = χ(r) [ m∑ µ=1 iµ∑ σ=1 κµσ−1∑ κ=0 cµ,σ,κ φµ,σ,κ(r, θ) ] + vr(r, θ), (3.13) where vr(r, θ) ∈ wm1,pα2 (pδ). here, we set m be the number of eigenvalues of the operator v̂(ξ) in the strip, the constants cµ,σ,κ depends on the data and the singular functions, iµ = dim ker v̂(ξµ) represents the geometrical multiplicity of ξµ, κµσ is the length of the jordan chains of v̂(ξµ), and the corresponding singular function is described as φµ,σ,κ(r, θ) = r iξµ κ∑ j=0 (i log r)j j! ψσ,κ−jµ (θ), (3.14) where ψσ,κ−jµ (θ) is a canonical system of jordan chains of v̂(ξ) respecting ξµ. it is noted from (3.13) and (3.14) that the eigenvalues ξµ = 0 does not yield singularities in the development of the solution in the neighborhood pδ. int. j. anal. appl. 18 (4) (2020) 681 it is recognized for elliptic boundary value problems that the eigenvalues of the operator v̂(ξ) which lies in the strip have a significant role in the regularity results. the assertions 1 and 2 of theorem 3.2 represent the regularity and the expansion of the solution of the system (1.1) near the singular points. 3.2. the calculation of the eigenvalues. in this section, the eigenvalues and eigenfunctions of the boundary value problem (3.11) with relationships of various boundary conditions are evaluated. generally, no unique solution exists for different boundary conditions, and the multiple solutions are marked with an index µ or l. the computed eigenvalues ξl and corresponding eigenfunctions φl(θ) of the considered problem with various conditions are defined as follows. dirichlet boundary conditions (dd) for dirichlet boundary conditions, the eigenvalues of the operator v̂(ξ) are ξl = i lπω0 , l = ±1,±2, ..., and the corresponding eigenfunctions are φl(θ) = sin lπ ω0 θ. neumann boundary conditions (nn) for neumann boundary conditions, the eigenvalues of the operator v̂(ξ) are ξl = i lπω0 , l = 0,±1,±2, ..., and the corresponding eigenfunctions are φl(θ) = cos lπ ω0 θ. mixed boundary conditions (nd) similar to the latter cases of the boundary conditions, the eigenvalues of the corresponding operator v̂(ξ) are ξl = i(l + 1 2 ) π ω0 , l = 0,±1,±2, ..., and the eigenfunctions are φl(θ) = cos(l + 12 ) π ω0 θ. remark 3.2. it is noted that if the versed boundary conditions are used which means that the dirichlet condition is at θ = 0 and the neumann condition is at θ = ω0, then the same eigenvalues of v̂(ξ) like the mixed boundary conditions (nd) are obtained but the corresponding eigenfunctions are φl(θ) = sin(l+ 1 2 ) π ω0 θ. 4. the regularity results in this section, the regularity results and the expansion of the solution u or v of the boundary value problem (3.2) or (1.1) are defined. to analyze the regularity results of the boundary value problem (3.11), the combinations of the boundary points with different boundary conditions are considered. first of all, it is to be determined that the righthand sides functions in (3.4) are fourier transform in the sense of (3.7). we know from (3.2) that f ∈ l2(s), and further note that for all α ≥ 0, f ∈w0, 2α (s). since, f ∈w0, 2α (s), we have∫ s |f(x)|2 |x|2 α dx = ∫ s̄ e2(τα+τ)|f̃(τ, θ)|2dτ dθ < ∞, (4.1) where h = α− 1 for all α ≥ 0 and it is meaningful according to (3.7). consequently, the fourier transform of f̃(τ,θ) is meaningful in the half plane h = im ξ ≥−1 for almost all θ ∈ (0, ω0). the following regularity results of the boundary value problem (3.2) for various combinations of the boundary conditions are achieved as a direct consequence of theorem 3.2 and the contemplations in section 3. int. j. anal. appl. 18 (4) (2020) 682 dirichlet boundary conditions (dd) let v̂(ξ) denote the operator of the problem (3.11) for the dirichlet-dirichlet boundary conditions (dd) and v̂(ξ) : w 2, 2(0,ω0) → l2(0,ω0) × c × c. if ξ is no eigenvalue of v̂(ξ), then for any f̂ ∈ l2(0,ω0) a unique weak solution û of (3.11) exists. we write û = v̂−1(ξ)[f̂, 0, 0], (4.2) where v̂−1(ξ) represent the inverse (or resolvent) operator and v̂−1(ξ) : l2(0,ω0) × c × c → w 2, 2(0,ω0). moreover, the inverse fourier transform of û produces the solution ũ(τ,θ) = u(x,y) of (3.2) and the subsequent regularity result holds. if no eigenvalues of v̂(ξ) are lie on the line h = im ξ = α− 1, α ≥ 0, then the inverse fourier transform which can be read as follows in formula (3.6) exists and ũh(τ, θ) = uh(x) ∈ w2, 2α (s). further, uh(x) be the unique solution of (3.2) from w2, 2α (s). it follows from the theory of kondratiev in [22, 23], a regularity result yields that u ∈w2, 21 (s). therefore, we have u(x) = u0(x). let us derive an expansion of the solution u(x) in s, the main question is the inverse fourier transformation of the right-hand sides of (3.11) which can be read as follows ũh(τ, θ) = (2π) −1 2 ∫ ∞+ih −∞+ih eiξτ v̂−1(ξ)[f̂, 0, 0] dξ. (4.3) using the cauchy theorem, yields ũh(τ, θ) = (2π) −1 2 lim n→∞ {∫ −n+iδ −n+ih eiξτ v̂−1(ξ)[f̂, 0, 0] dξ + ∫ n+iδ −n+iδ eiξτ v̂−1(ξ)[f̂, 0, 0] dξ + ∫ n+ih n+iδ eiξτ v̂−1(ξ)[f̂, 0, 0] dξ } + 1 √ 2π 2πi ∑ −1<im ξ<0 −n<re<n res ( eiξτ v̂−1(ξ)[f̂, 0, 0] ) . (4.4) as n →∞, the first integral and the third integral tend to zero by [22] and the second integral yields wh(x) ∈w2, 2α (s), the calculation of the residue gives the singular terms. if the eigenvalues of the operator v̂(ξ) does not lie on the line im ξ = α − 1, ∀α ≥ 0, then the residue vanishes and the inverse fourier transform exist. now, we ought to calculate the residue of eiξτ v̂−1(ξ)[f̂, 0, 0] and therefore to look at the singularities of (4.2). we know that v̂−1(ξ) is an operator-valued, meromorphic function of ξ with poles at ξl = i lπω0 , l = ±1,±2, .... so as to calculate the residuum, the only poles ξl that lies in −1 < im ξl < 0 plays a significant role in the regularity results. hence, if ω0 > π, then ξ−1 = −i πω0 lies in this strip. further from [26], the operator v̂−1(ξ) has the following expansion in the neighborhood of ξ−1, i.e., v̂−1(ξ) = q1 (ξ − ξ−1) + γ(ξ), (4.5) int. j. anal. appl. 18 (4) (2020) 683 where q1 and γ(ξ) map l 2(0,ω0) × c × c into w 2, 2(0,ω0). the operator q1 behaves as the space of eigenfunctions of v̂(ξ) corresponding to ξ−1 and γ(ξ) is holomorphic. moreover, (3.10) and (4.1) imply that f̂(ξ,θ) is holomorphic respecting ξ in the strip −1 < im ξ < 0. hence, we can write [f̂, 0, 0] = ∞∑ m=0 bm(θ) (ξ − ξ−1)m, (4.6) in a neighborhood of ξ−1, where the coefficients bm(θ) are elements of l 2(0,ω0) ×c×c. finally, we have eiξτ = eiξ−1 [ 1 + i(ξ − ξ−1)τ + ... + [i(ξ − ξ−1)τ]m m! + ... ] . (4.7) from (4.5)-(4.7), it follows that eiξτ v̂−1(ξ) [ f̂, 0, 0 ] = eiξ−1τ [ 1 + ... ] ∞∑ m=0 [ q1bm(θ) (ξ − ξ−1)m (ξ − ξ−1) + γ(ξ)bm(θ) (ξ − ξ−1)m ] . (4.8) we conclude that res [ eiξτ v̂−1(ξ) [ f̂, 0, 0 ]]∣∣∣ ξ=ξ−1 = eiξ−1τq1b0(θ), = e πτ ω0 c1 sin π ω0 θ, (4.9) where c1 is a complex constant and the formulas for its precise computation can be found in [7, 8, 20]. for ω0 > π, (4.4) yields u(x) = ũ0(τ, θ) = e πτ ω0 c1 sin π ω0 θ + w(x), (4.10) where w(x) ∈ w2, 20 (s) and u(x) ∈ w 1, 2 0 (s) is the solution of the boundary value (3.2). now, substituting r = eτ , we get u(x) = ũ0(τ, θ) = c1 r π ω0 sin π ω0 θ + w(x). (4.11) if ω0 ≤ π, then u(x) = w(x) ∈w 2, 2 0 (s). neumann boundary conditions (nn) let pi be a boundary point at which the neumann-neumann (nn) conditions appear. using the same approach which is used for the dirichlet-dirichlet conditions, the following fourier transformed form of (3.2) is obtained as ξ2û− ∂2û ∂θ2 =f̂(ξ,θ), ∂û ∂θ (ξ, 0) =ĝ(ξ, 0), ∂û ∂θ (ξ,ω0) =ĝ(ξ,ω0). (4.12) int. j. anal. appl. 18 (4) (2020) 684 let v̂(ξ) denote the operator of the problem (4.12) for the neumann-neumann conditions (nn) and v̂(ξ) : w 2, 2(0,ω0) → l2(0,ω0) × c × c. if ξ is no eigenvalue of v̂(ξ), then for any f̂ ∈ l2(0,ω0) a unique weak solution û of (4.12) exists. we write û = v̂−1(ξ)[f̂,ĝ(0),ĝ(ω0)], (4.13) where v̂−1(ξ) represent the inverse (or resolvent) operator and v̂−1(ξ) : l2(0,ω0) × c × c → w 2, 2(0,ω0). besides, the inverse fourier transform of û yields the solution ũ(τ,θ) = u(x,y) of (3.2) and the subsequent regularity result holds. if no eigenvalues of v̂(ξ) are lie on the line h = im ξ = α− 1, α ≥ 0, then the inverse fourier transform which can be read as follows in formula (3.6) exists and ũh(τ, θ) = uh(x) is the unique solution of (4.12) from w2, 2α (s). it follows from the theory of kondratiev in [22, 23], a regularity result yields that u ∈ w 2, 2 γ+1(s) where γ is a small positive real number. to derive an expansion of the solution u(x) in s, where v ∈ w 1,2(d) is the unique weak solution of the boundary value problem (1.1). the main question is the inverse fourier transformation of the right-hand sides of (4.12) which can be read as follows u(x) = uγ(x) = (2π) −1 2 ∫ ∞+iγ −∞+iγ eiξτ v̂−1(ξ)[f̂,ĝ(0),ĝ(ω0)] dξ. (4.14) the integral (4.14) can be calculated in the same way by considering the cauchy theorem and the approach used for calculating the regularity results of dirichlet boundary conditions. hence, we conclude that for ω0 > π, the following expansion of the solution of the boundary value (3.2) is obtained u(x) = c1 + c2 e πτ ω0 cos π ω0 θ + w(x), (4.15) where w(x) ∈ w2, 20 (s) and u(x) ∈ w 1, 2 0 (s) is the solution of the boundary value (3.2). now, substituting r = eτ , we get u(x) = c1 + c2 r π ω0 cos π ω0 θ + w(x). (4.16) mixed boundary conditions (nd) let pi be a boundary point at which the neumann-dirichlet conditions (nd) appear. using the same approach which is used for the latter cases, i.e., (dirichlet and neumann) conditions, the following fourier transformed form of (3.2) is obtained ξ2û− ∂2û ∂θ2 =f̂(ξ,θ), ∂û ∂θ (ξ, 0) =ĝ(ξ, 0), û(ξ,ω0) =0. (4.17) int. j. anal. appl. 18 (4) (2020) 685 let v̂(ξ) denote the operator of the problem (4.17) and v̂(ξ) : w 2, 2(0,ω0) → l2(0,ω0) × c × c. if ξ is no eigenvalue of the operator v̂(ξ), then for any f̂ ∈ l2(0,ω0) a unique weak solution û of (4.17) exists. we can write û = v̂−1(ξ)[f̂,ĝ(0), 0)], (4.18) where v̂−1(ξ) represent the resolvent operator and v̂−1(ξ) : l2(0,ω0) × c × c → w 2, 2(0,ω0). further, the inverse fourier transform of û yields the solution ũ(τ,θ) = u(x,y) of (3.2) and the subsequent regularity result holds. again we have, if no eigenvalues of v̂(ξ) lie on the line h = im ξ = α− 1, α ≥ 0, then the inverse fourier transform which can be read as follows ũh(τ, θ) = (2π) −1 2 ∫ ∞+ih −∞+ih eiξτ v̂−1(ξ)[f̂,ĝ(0), 0] dξ = uh(x) ∈w2, 2α (s), (4.19) exists and uh(x) is the uniquely determined solution from w2, 2α (s) of (3.2) for mixed conditions (nd). from [22, 23], a regularity result yields that u ∈ w2, 2γ+1(s) where γ is a sufficiently small positive real number. to derive an expansion of the solution u(x) in s, the main question is the inverse fourier transformation of the right-hand sides of (4.17) which can be read as u(x) = uγ(x) = (2π) −1 2 ∫ ∞+iγ −∞+iγ eiξτ v̂−1(ξ)[f̂,ĝ(0), 0] dξ. (4.20) the integral (4.20) can be calculated using the cauchy theorem as same in (4.4) and the approach used for calculating the regularity results of the latter conditions. moreover, the rectangle choosing here have the corner points −n+iγ, −n−i, n−i, n+iγ. since, we have the eigenvalues ξl = i(l + 12 ) π ω0 , l = 0,±1,±2, ...,. for ω0 ∈ (π2 , 3π 2 ), we have ξ−1 = ( −i 2 ) π ω0 and ω0 > 3π 2 yield ξ−2 = ( −3i 2 ) π ω0 . let these eigenvalues lie in the rectangle and the following expansion of the solution of the boundary value (3.2) is obtained u(x) = c1 e πτ 2ω0 cos π 2ω0 θ + w(x), for ω0 ∈ ( π 2 , 3π 2 ), u(x) = c1 e πτ 2ω0 cos π 2ω0 θ + c2 e 3πτ 2ω0 cos 3π 2ω0 θ + w(x), for ω0 > 3π 2 , (4.21) where w(x) ∈ w2, 20 (s) and u(x) ∈ w 1, 2 0 (s) is the solution of the boundary value (3.2). now, substituting r = eτ , we get u(x) = c1 r π 2ω0 cos π 2ω0 θ + w(x), for ω0 ∈ ( π 2 , 3π 2 ), u(x) = c1 r π 2ω0 cos π 2ω0 θ + c2 r 3π 2ω0 cos 3π 2ω0 θ + w(x), for ω0 > 3π 2 . (4.22) remark 4.1. it is observed that if the versed boundary conditions are used which means that the dirichlet condition is at θ = 0 and the neumann condition is at θ = ω0, then the similar regularity results can be int. j. anal. appl. 18 (4) (2020) 686 obtained like the mixed conditions (nd) but the eigenvalues and the corresponding eigenfunctions discussed in remark 3.2 are used. 4.1. the regularity of the boundary value problem in a polygonal domain. in section 1, we have described that d is a polygonal domain and n denote the set of the boundary points which consists of{ p1, ...,pn } ⊂ ∂d. to investigate the regularity of the solution v of the boundary value problem (1.1) in d, the following set of boundary points of n are considered. let we denote (1) j1 be the corresponding index set for the boundary points with the dirichlet-dirichlet boundary conditions (dd), where ωi > π, (2) j2 be the corresponding index set for the boundary points with the neumann-neumann boundary conditions (nn), where ωi > π, (3) j1j2 be the corresponding index set for the boundary points with the neumann-dirichlet boundary conditions (nd), where ωi ∈ (π2 , 3π 2 ), (4) j2j1 be the corresponding index set for the boundary points with the neumann-dirichlet boundary conditions (nd), where ωi > 3π 2 . let v ∈ w 1,2(d) be the uniquely determined weak solution of (1.1). we investigate its regularity which can be written in the form v = ∑ i∈j1 χ2iv + ∑ i∈j2 χ2iv + ∑ i∈j1j2 χ2iv + ∑ i∈j2j1 χ2iv + ( 1 − ∑ i∈j1∪j2∪j1j2∪j2j1 χ2iv ) , (4.23) where the function χ is defined in (3.1) and u = χv. using the expansions (4.11), (4.16) and (4.22) into (4.23), we get v = ∑ i∈j1 χici r π ωi i sin π ωi θi + ∑ i∈j2 χici r π ωi i cos π ωi θi + ∑ i∈j1j2 χici r π 2ωi i cos π 2ωi θi + ∑ i∈j2j1 ( χici r π 2ωi i cos π 2ωi θi + ćiχi r 3π 2ωi i cos 3π 2ωi θi ) + w(x), (4.24) where w(x) ∈ w 2,2(d). let θi represents the locally variable angle, i.e., 0 < θi < ωi, whereas ci, ći are the singularity coefficients and their computations can be found in [7, 8, 20]. finally, (4.24) completely describe the regularity of the solution v of the boundary value problem (1.1). 5. conclusion it is well-known from the theory of elliptic boundary value problems in domains with boundary irregularities, like corners, conic vertices, edges, and cracks, etc., the solution may exhibit singularities. generally, the flows over corners usually change their behaviors and properties as a result of a rapid geometrical change in the shape. in this article, we have studied the boundary singularities and regularity of the weak solution int. j. anal. appl. 18 (4) (2020) 687 of the mixed boundary value problem for the poisson equation in a non-smooth domain with singular points on the boundary. the singular structure of the solution of the considered problem near the corner points is investigated through the fourier transform and the suitable weighted sobolev spaces that best characterize the singular behavior of the solution are presented. it is observed for dirichlet and neumann boundary conditions that if d has reentrant corners (ωi > π : i = 1, 2, ...n), then the weak solution v ∈ w 1,2 0 (d) of the considered problem has the form (4.11) and (4.16). if the domain d is a convex polygonal domain, then the solution v ∈ w 2,2(d). for the mixed boundary conditions, the general solution is presented in the form of (4.22). moreover, it is shown that the solution of the given problem can be decomposed into the singular and regular parts near the corner points for the values of ωi ∈ (π2 , 3π 2 ) and ωi > 3π 2 and does not belong locally to space h2. finally, (4.24) completely describe the regularity result of the original boundary value problem in a domain d with singular points on the boundary. the results to be achieved here can be further extended to three-dimensional domains, for instance, polyhedral domain, etc. with straight edges to analyze the edge singularities and the regularity expansion of the solutions. additionally, the technique to be presented here can be modified for investigating and treating numerous linear boundary value problems in two-dimensional domains with corners, such as lame’s equations, stokes equations and so forth. conflict of interest: the author declares that no conflict of interest regarding the publication of this paper. references [1] r. a. adams and j. j. fournier, sobolev spaces, academic press, amsterdam, 2003. [2] s. agmon, a. douglis, and l. nirenberg, estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. i, commun. pure. appl. math. 12(4) (1959), 623–727. [3] s. agmon, a. douglis, and l. nirenberg, estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. ii, commun. pure. appl. math. 17(1) (1964), 35–92. [4] y. n. anjam, singularities and regularity of stationary stokes and navier-stokes equations on polygonal domains and their treatments, aims math. 5(1) (2020), 440–466. [5] h. blum, m. dobrowolski, on finite element methods for elliptic equations on domains with corners, computing. 28 (1982), 53–63. [6] h. blum and r. rannacher, on the boundary value problem of the biharmonic operator on domains with angular corners, math. methods appl. sci. 2(4) (1980), 556–581. [7] z. cai, s. kim, and h.c. lee, error estimate of a finite element method using stress intensity factor, comput. math. appl. 76(10) (2018), 2402–2408. [8] h. j. choi, w. choi, and y. koh, a finite element method for elliptic optimal control problem on a non-convex polygon with corner singularities, comput. math. appl. 75(1) (2018), 45–58. [9] m. dauge, stationary stokes and navier-stokes systems on two or three-dimensional domains with corners. part i. linearized equations, siam j. math. anal. 20 (1989), 74–97. int. j. anal. appl. 18 (4) (2020) 688 [10] m. elliotis, g. georgiou, and c. xenophontos, the solution of laplacian problems over l-shaped domains with a singular function boundary integral method, commun. numer. methods eng. 18 (2002), 213–222. [11] a. kufner, o. john, s. fučik, function spaces, noordhoff, leyden, 1977. [12] d. gilbarg and n. s. trudinger, elliptic partial differential equations of second order, springer, 2015. [13] v. girault and p. a. raviart, finite element methods for navier-stokes equations: theory and algorithms, springer, 2012. [14] p. grisvard, singularities in boundary value problems, springer, 1992. [15] p. grisvard, behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, in: numerical solution of partial differential equations-iii, elsevier, 1976, 207–274. [16] p. grisvard, elliptic problems in nonsmooth domains, volume 69 of classics in applied mathematics. siam, philadelphia, pa, 1985. [17] b. guo and c. schwab, analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces, j. comput. appl. math. 190(1–2) (2006), 487–519. [18] w. hackbusch, elliptic differential equations theory and numerical treatment: the poisson equation, springer, 2010. [19] s. kim and h. c. lee, a finite element method for computing accurate solutions for poisson equations with corner singularities using the stress intensity factor, comput. math. appl. 71(11) (2016), 2330–2337. [20] s. kim and h. c. lee, finite element method to control the domain singularities of poisson equation using the stress intensity factor: mixed boundary condition, int. j. numer. anal. model. 14(4–5) (2017), 500–510. [21] t. kinoshita, y. watanabe, n. yamamoto, and m. t. nakao, some remarks on a priori estimates of highly regular solutions for the poisson equation in polygonal domains, japan j. ind. appl. math. 33(3) (2016), 629–636. [22] v. a. kondrat́iev, boundary value problems for elliptic equations in domains with conical or angular points, tr. mosk. mat. obshch. 16 (1967), 209–292. [23] v. a. kondrat́iev and o. a. oleinik, boundary value problems for partial differential equations in non-smooth domains, russ. math. surv. 38(2) (1983), 1–86. [24] v. a. kozlov, v. g. maźya, and j. rossmann, elliptic boundary value problems in domains with point singularities, american mathematical society, 1997. [25] v. a. kozlov, v. g. maźya, and j. rossmann, spectral problems associated with corner singularities of solutions to elliptic equations, american mathematical society, 2001. [26] s. g. krejn and v. p. trofimov, holomorphic operator-valued functions of several complex variables, funct. anal. i priložen. 3(4) (1969), 85–86. [27] z. c. li and t. t. lu, singularities and treatments of elliptic boundary value problems, math. comput. model. 31(8–9) (2000), 97–145. [28] a. m. sändig, some applications of weighted sobolev spaces, vieweg+teubner verlag, 1987. [29] r. temam, navier-stokes equations: theory and numerical analysis, north-holland: elsevier, 1979. [30] j. r. whiteman and n. papamichael, treatment of harmonic mixed boundary problems by conformal transformation methods, z. angew. math. phys. 23(4) (1972), 655–664. 1. introduction 2. analytical preliminaries 2.1. weighted sobolev spaces 3. the boundary value problem in an infinite cone 3.1. localization of the model problem 3.2. the calculation of the eigenvalues 4. the regularity results 4.1. the regularity of the boundary value problem in a polygonal domain 5. conclusion references int. j. anal. appl. (2022), 20:8 n-convexity via delta-integral representation of divided difference on time scales hira ashraf baig∗, naveed ahmad school of mathematics and computer sciences, institute of business administration, karachi, pakistan ∗corresponding author: habaig@iba.edu.pk abstract. we introduce the delta-integral representation of divided difference on arbitrary time scales and utilize it to set criteria for n-convex functions involving delta-derivative on time scales. consequences of the theory appear in terms of estimates which generalize and extend some important facts in mathematical analysis. 1. introduction time scale calculus is a well known and rapidly growing theory in mathematical analysis which unifies two distinct well-known mathematical areas named as continuous and discrete analysis. for supplementary details and basics of time scale calculus, we invoke [1–3]. the notion of convexity with its various types have a noteworthy presence in literature, see [4–7] and the references therein. the notion is firstly generalized on an arbitrary time scale in 2008 by cristian dinu [8], subsequently a large number of estimation and inequalities for the functions that are convex on time scales are in the continuous state of development, some of them are present in [9,10]. here we consult with an exclusive variety of these functions, that is n-convex functions. the n-convexity or higher order convexity firstly investigated by eberhard hopf [11] in his scholarly thesis. further it was discussed in different narrations by popoviciu [12,13]. a comprehensive review of this family of functions is elaborated in [5, 14]. in [15] m. rozarija, and j. pečarić discussed some "jensen-type inequalities on time scales" involving real-valued n-convex functions. higher order convex functions has been discussed on time scales with constant graininess function by h. a. baig and n. ahmad in [16], so there is a need to explore this class of functions on arbitrary time scales. received: nov. 18, 2021. 2010 mathematics subject classification. 26d20, 39b62, 36a51, 34n05. key words and phrases. n-convex functions; delta integrals; time scales; integral inequalities. https://doi.org/10.28924/2291-8639-20-2022-8 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-8 2 int. j. anal. appl. (2022), 20:8 this article is structured as follows. in section 2 we furnish few preliminaries, utilizing in the main results. section 3 is dedicated to construct a relationship between nth delta derivatives and nth-order divided difference on arbitrary time scales. afterward, we presented some mathematical inequalities as consequences of our main results in the last section. 2. preliminaries a time scale t is defined to be an arbitrary closed subset of the real numbers r, with the standard inherited topology. the forward jump operator and the backward jump operator are defined by σ(t) := inf{s ∈ t : s > t}, and ρ(t) := sup{s ∈ t : s < t}, where infφ = supt and supφ = inft. let u : t→r, u∆(t) is representing the first delta derivative of function u at t ∈tκ. the second-order delta derivative of u at t is defined as, provided it exists u∆ 2 (t) = u∆∆(t) = (u∆(t))∆ : tκ 2 →r similarly higher-order derivatives are defined as u∆ n (t) : tκ n → r. the definition for rd-continuous functions can be seen in [2]. the set of rd-continuous functions u : t→r is denoted by crd = crd(t,r) = crd(t). the set consisting of first-order delta differentiable functions u and whose derivative is rd-continuous is denoted by c1rd = c 1 rd(t,r) = c 1 rd(t). the substitution rule and first mean value theorem for delta-integrals in time scales are presented in [1–3]. theorem 2.1. assume ν : t→r is strictly increasing and t̃ := ν(t) is a time scale. if u ∈ crd and ν ∈ c1rd, then for a,b ∈t ∫ b a u(t)ν∆(t)∆t = ∫ ν(b) ν(a) ( u ◦ν−1 ) (s)∆̃s. (2.1) theorem 2.2. let ν and u be bounded and integrable functions on [a,b], and let ν be nonnegative (or nonpositive) on [a,b]. let us set m = sup{u(t) : t ∈ [a,b)} m = inf{u(t) : t ∈ [a,b)}. then there exists a real number λ satisfying the inequalities m < λ < m such that∫ b a u(t)ν(t)∆t = λ ∫ b a ν(t)∆t. the time scale monomials have been defined in [1,3,17] recursively as g0(t,s) = h0(t,s) = 1 for s,t ∈t, int. j. anal. appl. (2022), 20:8 3 gk+1(t,s) = ∫ t s gk (σ(γ),s) ∆γ, hk+1(t,s) = ∫ t s hk (γ,s) ∆γ, k ∈n0. (2.2) these monomials satisfy the following relation for t ∈t and s ∈tκ: gn (t,s) = (−1)nhn (s,t) . (2.3) remark 2.1. [17] the functions hn and gn satisfy gn(t,s) ≥ 0 and hn(t,s) ≥ 0 for all t ≥ s. let us recall the taylor’s formula defined on time scales from [17]. theorem 2.3. let u be n-times delta-differentiable on tκ n , t ∈t and tα ∈tκ n−1 . we have u(t) − n−1∑ k=0 hk(t,tα)u ∆k (tα) = ∫ ρn−1(t) tα hn−1(t,σ(γ))u ∆n (γ)∆γ, (2.4) similarly, u(t) − n−1∑ k=0 (−1)ngk(tα,t)u∆ k (tα) = ∫ ρn−1(t) tα (−1)ngn−1(σ(γ),t)u∆ n (γ)∆γ, (2.5) where k ∈n0. higher order convex functions defined on r as well as on z through nth-order divided difference, in which we randomly select n + 1 points {a0,a1, . . . ,an} from r or from z, respectively and compute the nth-order divided difference by the formula [a0,a1, · · · ,an; u] = [a1,a2, · · · ,an; u] − [a0,a1, · · · ,an−1; u] an −a0 . (2.6) if (2.6) is non-negative we say that u is an n-convex function. here (2.6) remains same for every permutation of n + 1 points. to construct the criteria for n-convexity we need to introduce the forward operator σ in the definition of higher order convexity. so we adopt the same strategy as we did in [16]. assume n + 1 distinct points t0, · · · ,tn ∈t and arrange them in an increasing order. relabel these points in the time scale t̃ in terms of forward operator, that is t̃ = {t0,σ(t0), · · · ,σn(t0)}. consequently we can define the nth-order divided difference for n + 1 points as [t0,σ(t0), · · · ,σn(t0); u] = [σ(t0),σ 2(t0), · · · ,σn(t0); u] − [t0,σ(t0), · · · ,σn−1(t0); u] σn(t0) − t0 . (2.7) so a function u : t→r, is said to be n-convex if [t0,σ(t0), · · · ,σn(t0); u] ≥ 0, (2.8) where σ : t ⋂ t̃→t ⋂ t̃. 4 int. j. anal. appl. (2022), 20:8 3. main results here we want to establish a criteria for n-convex function on arbitrary time scales which is stated as u ∈ cnrd is n-convex iff u ∆n ≥ 0. it is sufficient to prove this on t̃. firstly we introduce a new representation of divided difference in terms of delta-integral, that can be seen in the next theorem. theorem 3.1. suppose u ∈ cnrd(t,r). let t0,t1, · · · ,tn be n + 1 distinct points in t, then [t0,σ(t0), · · · ,σn(t0); u] = ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−1 0 ∆sn ×u∆ n (sn[σ n(t0) −σn−1(t0)] + · · · + s1[σ(t0) − t0] + t0), (3.1) where n ≥ 1 and si ∈ [0, 1]. proof. consider t0,t1, · · · ,tn, n + 1 distinct points and the corresponding time scale t̃ = {t0,σ(t0), · · · ,σn(t0)}. we prove (4.3) by induction method. for this we first show that [t0,σ(t0); u] = ∫ 1 0 u∆(s1[σ(t0) − t0] + t0)∆s1. (3.2) let us use the time scales substitution rule for integration (2.1), let the new variable of integration β in the following manner (since σ(t0) 6= t0) β = v−1(s1) = s1[σ(t0) − t0] + t0 ⇒ v(s1) = s1 − t0 σ(t0) − t0 , here v−1 : [0, 1] → t̃. by calculating delta derivative of v(s1) with respect to s1 we get v∆(s1) = 1 σ(t0)−t0 therefore, s1 ∈ [0, 1] and v(s1) is strictly increasing such that v[t0,σ(t0)] = [0, 1]. hence the corresponding limits are (s1 = 0) → (β = t0); (s1 = 1) → (β = σ(t0)). since σ(t0) 6= t0, thus (3.2) can be written as∫ 1 0 u∆(s1[σ(t0) − t0] + t0)∆s1 = ∫ v(σ(t0)) v(t0) u∆(v−1(s1))∆s1 = ∫ σ(t0) t0 u∆(β) σ(t0) − t0 ∆β = 1 σ(t0) − t0 ( u(β) ∣∣∣∣σ(t0) t0 ) = u(σ(t0)) −u(t0) σ(t0) − t0 . int. j. anal. appl. (2022), 20:8 5 now we make the inductive hypothesis that [t0,σ(t0), · · · ,σn−1(t0); u] = ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−2 0 ∆sn−1 ×u∆ n−1 (sn−1[σ n−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0). in the integral in (3.1) we apply substitution rule of integration of time scales (2.1) by replacing the variable of integration sn with β. β =v−1(sn) = sn[σ n(t0) −σn−1(t0)] + · · · + s1[σ(t0) − t0] + t0 ⇒ v(sn) = sn − (sn−1[σn−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0) σn(t0) −σn−1(t0) . so that the delta derivative of v(sn) with respect to sn gives us v∆(sn) = 1 σn(t0) −σn−1(t0) . the corresponding limits are (sn = 0) → ( β = β0 ≡ sn−1[σn−1(t0) −σn−2(t0)] + · · · + s1[σ(t0) − t0] + t0 ) (sn = sn−1) → (β = β1 ≡ sn−1[σn(t0) −σn−2(t0)] + sn−2[σn−2(t0) −σn−3(t0)]+ · · · + s1[σ(t0) − t0] + t0). thus the innermost integral of (4.3) can transform in the following manner, since σn(t0) 6= σn−1(t0) ∫ sn−1 0 u∆ n (sn[σ n(t0) −σn−1(t0)]) + · · · + s1[σ(t0) − t0] + t0)∆sn = ∫ β1 β0 u∆ n (β) σn(t0) −σn−1(t0) ∆β = 1 σn(t0) −σn−1(t0) ( u∆ n−1 (β) ∣∣∣∣β1 β0 ) = u∆ n−1 (β1) −u∆ n−1 (β0) σn(t0) −σn−1(t0) . however, by applying the inductive hypothesis we have ∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−2 0 ∆sn−1 ( u∆ n−1 (β1) −u∆ n−1 (β0) σn(t0) −σn−1(t0) ) = u[t0,σ(t0), · · · ,σn−2(t0),σn(t0)] −u[t0,σ(t0), · · · ,σn−2(t0),σn−1(t0)] σn(t0) −σn−1(t0) = [t0,σ(t0), · · · ,σn(t0); u]. 6 int. j. anal. appl. (2022), 20:8 � in the next theorem we establish a relation between nth-order divided difference and nth-delta derivative on arbitrary time scales, since in this result the points ti ∈t need not to be distinct. theorem 3.2. let u ∈ cnrd(t,r), then for n + 1 points form t we have [t0,σ(t0), · · · ,σn(t0); u] = u∆ n (ξ) (hi (sn−i, 0)) , (3.3) where s0 = 1, 0 ≤ i ≤ n, and ξ ∈ [t0,σn(t0)]t. proof. by using the time scale monomials (2.2) we can write a general notation for the integral∫ 1 0 ∆s1 ∫ s1 0 ∆s2 · · · ∫ sn−1 0 ∆sn, that is hi (sn−i, 0) = ∫ sn−i 0 hi−1(sn−i+1, 0)∆sn−i+1. (3.4) by the remark 2.1 we can conclude that hn(si, 0) > 0 in (3.4) because all si > 0. now by applying theorem 2.2, (3.1) yields x (hi (sn−i, 0)) ≤ [t0,σ(t0), · · · ,σn(t0); u] ≤ x (hi (sn−i, 0)) , or x ≤ [t0,σ(t0), · · · ,σn(t0); u] (hi (sn−i, 0)) ≤ x, where x ≡ min u∆ n (t) and x ≡ max u∆ n (t) for t ∈ [t0,σn(t0)]t. then by the rd-continuity of u∆ n there exists a λ in this interval that is u∆ n (ξ) = λ, such that [t0,σ(t0), · · · ,σn(t0); u] (hi (sn−i, 0)) = u∆ n (ξ). � here, we can directly achieve the next result. corollary 3.1. let u : t→r is n-convex function iff u∆ n ≥ 0, given that u∆ n exists. another useful property of n-convex function is represented in the next result. theorem 3.3. let u(t) ∈ cnrd(t,r) is n-convex function, then for every r ∈ n, 1 ≤ r ≤ n − 1, u ∆r is (n− r)-convex. proof. by corollary 3.1 u∆ n ≥ 0. since u∆ r exists for every 1 ≤ r ≤ n− 1. let us choose (n− r + 1) points from [ta,tb]t such that t̃ = {t0,σ(t0), · · ·σn−r (t0)}, then by using (3.3) we can write [t0,σ(t0), · · · ,σn−r (t0); u∆ r ] = ( u∆ r (ξ) )∆n−r (hn−r (sr, 0)) = (u(ξ)) ∆n (hn−r (sr, 0)) ≥ 0, (3.5) where ξ ∈ [t0,σn−r (t0)]t. thus (3.5) shows that u∆ r is (n− r)-convex for every 1 ≤ r ≤ n− 1. � int. j. anal. appl. (2022), 20:8 7 4. applications: inequalities for n-convex functions let us present levinson’s type inequality for higher-order convex functions on time scales for this we require the next result. let ti ∈ [ta,tb]t, for i = 1, · · · ,z. let bi > 0 such that ∑z i=1 bi = 1 therefore t ∈ [ta,tb]t denoted by ∑z i=1 biti. theorem 4.1. let u is (n + 2)-convex on t. then for every t ∈t the function u(t) = [t,σ(t), · · · ,σn(t); u], (4.1) is a convex function. proof. by using (3.1), (4.1) can be expressed as u(t) = [t,σ(t), · · · ,σn(t); u] = ∫ 1 0 ∫ s1 0 · · · ∫ sn−1 0 u∆ n (sn[σ n(t) −σn−1(t)] + · · · + s1[σ(t) − t] + t)∆sn · · ·∆s1. therefore u∆ n is convex by theorem 3.3, thus for fixed sj, σj(t) for j = 1, · · · ,n we can write u∆ n   n∑ j=1 sj[σ j(t) −σj−1(t)] + z∑ i=1 biti   ≤ z∑ i=1 biu ∆n   n∑ j=1 sj[σ j(t) −σj−1(t)] + ti   , which concludes the proof. � theorem 4.2. if u is (n + 2)-convex on t, then the given inequality is true u[t,σ(t), · · · ,σn(t)] ≤ z∑ i=1 bi [ti,σ(ti ), · · · ,σn(ti ); u]. (4.2) proof. the proof is the direct consequence of theorem 4.1. � remark 4.1. let t = r in theorem 4.2, inequality (4.2) coincides with inequality (4) in [18], this levinson’s type inequality itself having a great importance in literature which is used to develop further divided difference estimates for n-convex functions in [19]. further, we present certain useful inequalities involving n-convex functions on time scales by using the criteria for n-convexity, that is u∆ n ≥ 0. theorem 4.3. let tα,tβ ∈tκ n , suppose u ∈ cn+1 rd (t,r) be (n+ 1)-convex function on [tα,tβ]. then for each t ∈ (tα,tβ), the following inequalities hold n−1∑ k=0 hk(t,tα)u ∆k (tα) + u ∆n (tα) ∫ ρn−1(t) tα hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tα)u ∆k (tα) + u ∆n (tβ) ∫ ρn−1(t) tα hn−1(t,σ(γ))∆γ, (4.3) 8 int. j. anal. appl. (2022), 20:8 where tα < ρn−1(tβ). if n is odd, then n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tα) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ, (4.4) and if n is even, the given inequality holds n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tα) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) ≤ n−1∑ k=0 hk(t,tβ)u ∆k (tβ) + u ∆n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ. (4.5) proof. if u is (n + 1)−convex on tκ n which implies that u∆ n+1 ≥ 0, then u∆ n is increasing on tκ n , i.e u∆ n (tα) ≤ u∆ n (γ) ≤ u∆ n (tβ) for each γ ∈ [tα,tβ], let σ(γ) ≤ t so that hn−1(t,σ(γ)) is non-negative, then from (2.4) we get ∫ ρn−1(t) tα hn−1(t,σ(γ))u(tα)∆γ ≤ u(t) − n−1∑ k=0 hk(t,tα)u ∆k (tα) ≤ ∫ ρn−1(t) tα hn−1(t,σ(γ))u ∆n (tβ)∆γ, which executes the proof for (4.3). let n is odd and t ≤ σ(γ) so that gn−1(σ(γ),t) ≥ 0, thus we can write∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ),t)u∆ n (tα)∆γ ≤ ∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ,t))u∆ n (γ)∆γ ≤ ∫ tβ ρn−1(t) (−1)n−1gn−1(σ(γ),t)u∆ n (tβ)∆γ, ⇒ u∆ n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ ∫ ρn−1(t) tβ hn−1(t,σ(γ))u ∆n (γ)∆γ ≤ u∆ n (tα) ∫ ρn−1(t) tβ hn−1(σ(γ),t)∆γ, int. j. anal. appl. (2022), 20:8 9 which gets the form u∆ n (tβ) ∫ ρn−1(t) tβ hn−1(t,σ(γ))∆γ ≤ u(t) − n−1∑ k=0 hk(t,tβ)u ∆k (tβ) ≤ u∆ n (tα) ∫ ρn−1(t) tβ hn−1(σ(γ),t)∆γ, which executes the proof for (4.4). let n is even then we have (−1)n−1u∆ n (tβ) ≤ (−1)n−1u∆ n (γ) ≤ (−1)n−1u∆ n (tα), then by adopting the same steps we can prove (4.5). � therefore, we can extract the particular cases of theorem 4.3 by considering different time scales. first by taking t = r we obtained the following result which agrees theorem 1 in [20]. theorem 4.4. let u(t) be (n+1)−convex on [tα,tβ]. then for all t ∈ (tα,tβ), the following inequality holds n∑ k=0 u(k)(tα) k! (t − tα)k ≤ u(t) ≤ n−1∑ k=0 u(k)(tα) k! (t − tα)k + u(n)(tβ) n! (t − tα)n. (4.6) for odd n the following inequality is true n∑ k=0 u(k)(tβ) k! (t − tβ)k ≤ u(t) ≤ n−1∑ k=0 u(k)(tβ) k! (t − tβ)k + u(n)(tα) n! (t − tβ)n, (4.7) and for even n the following inequality holds n−1∑ k=0 u(k)(tβ) k! (t − tβ)k + u(n)(tα) n! (t − tβ)n ≤ u(t) ≤ n∑ k=0 u(k)(tβ) k! (t − tβ)k. (4.8) now by considering t = z in theorem 4.3 we get the discrete analogues of the inequalities (4.6), (4.7) and (4.8). therefore, σ(t) = t + 1, σn(t) = t + n, ρ(t) = t − 1 and ρn(t) = t −n. theorem 4.5. let ut : [tα,tβ] → r be an (n + 1)−convex sequence. then for all t ∈ (tα,tβ), the following inequality holds n−1∑ k=0 ∆kutα k! (t − tα)k + ∆nutα t−n∑ γ=tα (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.9) ≤ n−1∑ k=0 ∆kutα k! (t − tα)k + ∆nutβ t−n∑ γ=tα (t −γ − 1)(n−1) (n− 1)! . (4.10) 10 int. j. anal. appl. (2022), 20:8 for odd n the following inequality is true n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutβ t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.11) ≤ n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutα t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! , (4.12) and for even n the following inequality holds n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutα t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! ≤ ut (4.13) ≤ n−1∑ k=0 ∆kutβ k! (t − tβ)k + ∆nutβ t−n∑ γ=tβ (t −γ − 1)(n−1) (n− 1)! . (4.14) the next result is obtained by considering n = 1 in (4.3) and (4.4). corollary 4.1. let tα,tβ ∈ tκ, if u is convex on [tα,tβ], then the given inequalities hold for all t ∈ [tα,tβ] max{u(tα) + u∆(tα)(t − tα),u(tβ) + u∆(tβ)(t − tβ)}≤ u(t) ≤ min{u(tα) + u∆(tβ)(t − tα),u(tβ) + u∆(tα)(t − tβ)}. (4.15) the next result is obtained by considering n = 2 in (4.3) and (4.5). corollary 4.2. let tα,tβ ∈ tκ 2 , if u is 3−convex on [tα,tβ], then the given inequalities hold for all t ∈ [tα,tβ]t max { u(tα) + u ∆(tα)(t − tα) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tα)∆γ,u(tβ) + u∆(tβ)(t − tβ) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tβ)∆γ } ≤ u(t) ≤ min { u(tα) + u ∆(tβ)(t − tα) + u∆ 2 (tα) ∫ ρ(t) tα (γ − tβ)∆γ,u(tβ) + u∆(tα)(t − tβ) + u∆ 2 (tβ) ∫ ρ(t) tα (γ − tβ)∆γ } . remark 4.2. when we take t = r in corollaries 4.1 and 4.2 we get the results which coincide with corollary 1 and corollary 2 in [20] respectively. moreover corollary 4.1 for t = r is used to derive more useful result in [21]. 5. conclusion the notion of n-convexity has been discussed in [16], on specific time scales that are r or hz. here we extend the theory on arbitrary time scale and developed the relationship between the delta derivatives of order n and the nth-order divided difference using integral representation of nth-order divided difference on time scales, see [5, 22]. further we utilized this relationship to derive some int. j. anal. appl. (2022), 20:8 11 dynamic inequalities from which we are able to extract some difference inequalities that are equally important in the study of difference equations and their applications. authors contribution: both authors have equivalent contribution in this research. both authors have inspect the manuscript and certified the final version. funding information: the authors acknowledge the moral and financial support by the higher education commission (hec), pakistan, through the funding of indigenous scholarship phase i, batch v. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. bohner, a. peterson, dynamic equations on time scales: an introduction with applications, springer science & business media, 2001. [2] m. bohner, a. peterson, advances in dynamic equations on time scales, springer science & business media, 2002. [3] m. bohner, s.g. georgiev, multivariable dynamic calculus on time scales, springer, 2016. [4] l. ciurdariu, a note concerning several hermite-hadamard inequalities for different types of convex functions, int. j. math. anal. 6 (2012), 33–36. [5] j. e. pečarić, y. l. tong, convex functions, partial orderings, and statistical applications, academic press, 1992. [6] s. zaheer ullah, m. adil khan, y.m. chu, a note on generalized convex functions, j inequal appl. 2019 (2019), 291. https://doi.org/10.1186/s13660-019-2242-0. [7] d.e. varberg, a. w. roberts, convex functions, academic press, new york-london, 1973. [8] c. dinu, convex functions on time scales, ann. univ. craiova, math. comp. sci. ser. 35 (2008), 87–96. [9] r.p. agarwal, d. o’regan, s.h. saker, hardy type inequalities on time scales, springer, 2016. [10] r.p. agarwal, d. o’regan, s. saker, dynamic inequalities on time scales, springer, 2014. [11] e. hopf, über die zusammenhänge zwischen gewissen höheren differenzen-quotienten reeller funktionen einer reellen variablen und deren differenzierbarkeitseigenschaften, ph.d. thesis, norddeutsche buchdr. u. verlagsanst. (1926). [12] t. popoviciu, on some properties of functions of one or two variables réthem, ph.d. thesis, institutul de arte grafice "ardealul (1933). [13] t. popoviciu, les fonctions convexes, actualites sci. ind. 992 (1945). [14] p. bullen, a criterion for n-convexity, pac. j. math. 36 (1971), 81–98. https://doi.org/10.2140/pjm.1971.36. 81. [15] r. mikic, j. pečarić, jensen-type inequalities on time scales for n-convex functions, commun. math. anal. 21 (2018), 46–67. [16] h.a. baig, n. ahmad, the weighted discrete dynamic inequalities for 4-convex functions, and its generalization on time scales with constant graininess function, j. inequal. appl. 2020 (2020), 168. https://doi.org/10.1186/ s13660-020-02435-4. [17] r.p. agarwal, m. bohner, basic calculus on time scales and some of its applications, results. math. 35 (1999), 3–22. https://doi.org/10.1007/bf03322019. [18] d. zwick, a divided difference inequality for n-convex functions, j. math. anal. appl. 104 (1984), 435–436. https://doi.org/10.1016/0022-247x(84)90008-8. [19] r. farwig, d. zwick, some divided difference inequalities for n-convex functions, j. math. anal. appl. 108 (1985), 430–437. https://doi.org/10.1016/0022-247x(85)90036-8. https://doi.org/10.1186/s13660-019-2242-0 https://doi.org/10.2140/pjm.1971.36.81 https://doi.org/10.2140/pjm.1971.36.81 https://doi.org/10.1186/s13660-020-02435-4 https://doi.org/10.1186/s13660-020-02435-4 https://doi.org/10.1007/bf03322019 https://doi.org/10.1016/0022-247x(84)90008-8 https://doi.org/10.1016/0022-247x(85)90036-8 12 int. j. anal. appl. (2022), 20:8 [20] i. brnetić, inequalities for n-convex functions, j. math. inequal. 5 (2011), 193–197. [21] i. brnetić, inequalities for convex and 3-log convex functions, rad hazu (515) (2013), 189–194. [22] e. isaacson, h.b. keller, analysis of numerical methods, courier corporation, 2012. 1. introduction 2. preliminaries 3. main results 4. applications: inequalities for n-convex functions 5. conclusion references international journal of analysis and applications volume 19, number 6 (2021), 929-948 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-929 received may 15th, 2021; accepted june 24th, 2021; published november 16th, 2021. 2010 mathematics subject classification. 34a45. key words and phrases. stiff systems; variable step; variable order; adams family; milne’s estimate. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 929 a computational strategy of variable step, variable order for solving stiff systems of ordinary differential equations j. g. oghonyon*, p. o. ogunniyi, i. f. ogbu department of mathematics, covenant university, ota, nigeria *corresponding author: godwin.oghonyon@covenantuniversity.edu.ng abstract: this research study focuses on a computational strategy of variable step, variable order (csvsvo) for solving stiff systems of ordinary differential equations. the idea of newton’s interpolation formula combine with divided difference as the basis function approximation will be very useful to design the method. analysis of the performance strategy of variable step, variable order of the method will be justified. some examples of stiff systems of ordinary differential equations will be solved to demonstrate the efficiency and accuracy. nomenclature csvsvo: errors in csvsvo for solving test application problem 1, 2 and 3. memployed: approach employed. maxerrors: the magnitude of the maximum errors of csvsvo. convcriteria: convergence criteria source of application problem i: see [5] for more info source of application problem ii: see [28] for more info. source of application problem iii: see [18] for mnore info. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-929 int. j. anal. appl. 19 (6) (2021) 930 1. introduction in diverse applied sciences, like chemical kinetics, mass-spring-damper systems, and control system analysis, we find systems of differential equations whose analytical solutions comprise terms with magnitudes that change at rates that are substantially unlike. for instance, whenever the analytical solution includes the terms 𝑒−𝑎𝑡 and 𝑒−𝑏𝑡, with 𝑎,𝑏 > 0, where the magnitude of 𝑎 is majorly greater than 𝑏, then 𝑒−𝑎𝑡 decays to zero at extremely quicker rate than 𝑒−𝑏𝑡 does. in cases of a quickly decaying transient analytical solution, a sure computational technique turns unstable except the step length is immoderately small. explicit techniques universally are submitted to this stability control, which necessitates the usage of very small step length not only necessarily improve the amount of functions to find the analytic solution, and as such stimulates round-off error to spring up, hence, having bounded accuracy and efficiency. implicit techniques, then again, are release of stability limitations and are thus favourable for computing stiff systems differential equations [27]. the conception of stiff initial value problems can be best valued by studying the succeeding general one-dimensional systems with changeless constants: 𝑦′ = 𝐴𝑦 + 𝐵(𝑥), 𝑦(𝑎) = 𝑦0, (1) where 𝐴 is an 𝑚 × 𝑚 matrix with actual entries and 𝐵(𝑥),𝑦,𝑦′ are 𝑚 − 𝑣𝑒𝑐𝑡𝑜𝑟𝑠. the theoretic solutions to (1) is seen as 𝑦(𝑥) = ∑ 𝛼𝑖𝑒 𝜆𝑖𝑥𝑐𝑖 𝑚 𝑖=1 + 𝑦𝑝(𝑥), (2) where 𝜆𝑖, 𝑖 = 1(1)𝑚 are the eigenvalues of 𝐴 , with 𝑐𝑖, 𝑖 = 1(1)𝑚 the matching eigenvectors. 𝑦𝑝(𝑥) is a special solution to (1), and 𝛼𝑖, 𝑖 = 1(1)𝑚 are actual constants that are unambiguously determined by the related initial conditions 𝑦(𝑎) = 𝑦0 [14, 18-19]. definition 1: a solution vector (or solution) of the system (1) on the interval 𝐼 is an 𝑚 × 1 matrix (or vector) of the form 𝑦(𝑥) = ( 𝑦1(𝑥) 𝑦2(𝑥) ⋮ 𝑦𝑚(𝑥) ), where the 𝑦𝑖(𝑥) are differentiable functions that gratifies (1) on 𝐼[1] for details. definition 2: any set {𝛼1}𝑖=1 𝑚 = {( 𝛼1𝑖 𝛼2𝑖 ⋮ 𝛼𝑚𝑖 )} 𝑖=1 𝑚 of 𝑚 linearly independent solution vectors of 𝑦′ = 𝐴𝑦 on an interval 𝐼 is called a fundamental set of solutions of 𝑦′ = 𝐴𝑦 on 𝐼. see [1] for more info. int. j. anal. appl. 19 (6) (2021) 931 definition 3: the stiffness ratio s of the system (1) is established as 𝑆 = { |𝜆𝑖|(𝑏−𝑎) in(tol) }, (3) where in(tol) is the exponential logarithm of tol. the stiffness ratio as established by (3) is a standard of the dispersion of the fourth dimension constants for (1), and in actual problems, may be of the order of 108. see [8] for more info. definition 4: the initial value problem (1) is stated to be stiff if it gratifies (4) and (5); i.e., whenever (i) 𝑟𝑒(𝜆𝑖) < 0,𝑖 = 1(1)𝑚, and (ii) the stiffness ratio 𝑠 > 1 . nevertheless, it should be observed that this is a quite general resolution with respect to mathematics. stiffness takes place whenever the step length is limited by stability, rather than order, conditions. see [8] for more info. definition 5: the initial value problem 𝑦′ = 𝑓(𝑥,𝑦), 𝑦(𝑎) = 𝑦0, 𝑦 = (𝑦1,𝑦2,… ,𝑦𝑚) 𝑇,𝑦0 = (𝜂1,𝜂2,… ,𝜂𝑚) 𝑇 is stated to stiff oscillatory whenever the eigenvalues 𝜆𝑖 = 𝑢𝑖 + 𝑗𝑣𝑖, 𝑖 = 1(1)𝑚 of the jacobian 𝐽 = ( 𝜕𝑓 𝜕𝑑𝑦 ) have the succeeding attributes: 𝑢𝑖 < 0,, 𝑖 = 1(1)𝑚, max 1≤𝑖≤𝑚 |𝑢𝑖| > min 1≤𝑖≤𝑚 |𝑢𝑖|, or whenever the stiffness ratio gratifies max 1≤𝑖≤𝑚 | 𝑢𝑖 𝑢𝑗 | > 1 and |𝑢𝑖| < |𝑣𝑖| for at least single pair of 𝑖 ∈ 1 ≤ 𝑖 ≤ 𝑚. see [8] for more info. definition 6: stiffness occurs when stability requirements, rather than those of accuracy constrain the step length. see [18-19] for details. definition 7: stiffness occurs when some components of the solution decay much more rapidly than others. see [18-19] for details. authors have contributed immensely towards solving stiff systems of ordinary differential equations using diverse strategies. [9] implemented the extensions of the predictorcorrector method for the solution of systems of ordinary differential equations. [11] formulated int. j. anal. appl. 19 (6) (2021) 932 the resultant of variable mesh size on the constancy of multi-step methods. [12] constructed the constancy and convergency of variable order multi-step methods. [14] developed the diagonally implicit block backward differentiation formula with optimal stability properties for stiff ordinary differential equations. [15] launched a varying-step, varying-order multistep method for the numerical solution of ordinary differential equations. [16] designed the algorithms for changing the step size. [17] worked on changing step-size in the integration of differential equations using modified divided differences. [21-25] developed and implemented a variable step, variable step size with same order for solving ordinary differential equations. [26] derived the constancy, consistence and convergency of varying k-step methods for numerical integration of large-systems of ordinary differential equations. this research is designed to extend the idea of [21-25] by inventing variable step, variable order together with variable step size for solving stiff systems of ordinary differential equations. [18-19] specifies that this idea leads to better efficiency and accuracy and as well bypass theorem 4. nevertheless, nonstiff algorithmic program possess a limited region of absolute stability (ras), whilst stiff algorithmic program possess unlimited ras. this describes why stiff algorithmic program adapt the usage of large step length beyond the transient (nonstiff) phase. in the transient phase (boundary layer), automatic codification seeks to name the optimal mesh size that holds the local truncation error inside the bound of the observed accuracy. again, for the transient phase, accuracy necessities oblige a numeric integrator to accept a mesh size of the order of the littlest time constant. beyond the transient phase (i.e., in the stiff phase) the increase of disseminated errors (instability) checks the selection of mesh size whenever a nonstiff technique is followed, since stability limitations are autonomous of the accuracy necessities. for the transient phase, there is ever the need to accept fairly large mesh size, but this is frequently bounded by stability conditions. see [8] for more info. theorem 1: the set {𝛼1}𝑖=1 𝑚 = {( 𝛼1𝑖 𝛼2𝑖 ⋮ 𝛼𝑚𝑖 )} 𝑖=1 𝑚 is linearly independent if and only if the wronskian 𝑊({𝛼1}𝑖=1 𝑚 ) = |𝛼1𝛼2 …𝛼𝑚| = | 𝛼11𝛼12 …𝛼1𝑚 𝛼21𝛼22 …𝛼2𝑚 ⋮ ⋮ ⋯ ⋮ 𝛼𝑚1𝛼𝑚2 …𝛼𝑚𝑚 | ≠ 0 see [1] for more info. int. j. anal. appl. 19 (6) (2021) 933 theorem 2: let 𝑆 = {𝛼1}𝑖=1 𝑚 = {( 𝛼1𝑖 𝛼2𝑖 ⋮ 𝛼𝑚𝑖 )} 𝑖=1 𝑚 be a set of 𝑚 linearly independent solutions of 𝑦′ = 𝐴𝑦. then every solution of 𝑦′ = 𝐴𝑦 is a linear combination of these solutions. see [1] for more info. theorem 3 (existence and uniqueness): assume that each of the functions 𝑓1(𝑥,𝑦1,𝑦2,…,𝑦𝑚), 𝑓2(𝑥,𝑦1,𝑦2,…,𝑦𝑚),…,𝑓𝑚(𝑥,𝑦1,𝑦2,…,𝑦𝑚) and the partial derivatives 𝜕𝑓1 𝜕𝑥1 , 𝜕𝑓2 𝜕𝑥2 ,…, 𝜕𝑓𝑚 𝜕𝑥𝑚 are continuous in a region 𝑅 containing the point (𝑥,𝑦1,𝑦2,…,𝑦𝑚). then, the initial-value problem { 𝑦1 ′ = 𝑓1(𝑥,𝑦1,𝑦2,…,𝑦𝑚) 𝑦2 ′ = 𝑓2(𝑥,𝑦1,𝑦2,…,𝑦𝑚) ⋮ 𝑦𝑚 ′ = 𝑓𝑚(𝑥,𝑦1,𝑦2,…,𝑦𝑚) 𝑦1(𝑥0) = 𝑡1,… ,𝑦𝑚(𝑥0) = 𝑡𝑚 (4) has a unique solution { 𝑦1 = ∅1(𝑥) 𝑦2 = ∅2(𝑥) ⋮ 𝑦𝑚 = ∅𝑚(𝑥) (5) on the interval 𝐼 contanining 𝑥 = 𝑥0. see [1, 3] for more info. theorem 4 (dahlquist barrier theorem): an a − stable linear multistep method • must be implicit and • the most accurate a-stable linear multistep method is the trapezoidal scheme of order 𝑝 = 2 and error constant 𝑐3 = − 1 12 . see [8] for more info. the dahlquist barrier theorem 4 can be outwitted by accepting unconventional numeric integrators, some of which are • nonlinear multistep schemes, • multiderivative multistep schemes, • exponentially fitting, and • extrapolation process. the variable step, variable order of the predictor-corrector algorithmic program came forth as result of the broad computational experience that holds throughout the years. this vsvo-pint. j. anal. appl. 19 (6) (2021) 934 cap is fundamental to high level efficiency and accuracy with the potential to change automatically not exclusively the step length but as well the order (and thus the step number) of the techniques utilized. algorithmic program with such a potential are recognized as variable step, variable order or vsvo, algorithmic program. apart from been built to handle nonstiff initial value problems, though various subsisting vsvo codification admit alternative for stiff systems. the necessary elements of vsvo algorithmic programs are: • a family of methods, • a starting procedure, • a local error estimator, • a strategy for determining when to vary step length and/ or order, and • a technique for varying step length and/ or order • a written softcode in any mathematical packages is required if manual computation is very tedious. • a special basis function approximation for estimating stiff systems is necessary if the desired accuracy and efficiency is not achieved. in addition, the convergence attributes of predictor-corrector methods antecedently proved on the presumption of changeless step length and changeless order still maintain in a vsvo algorithmic program conceptualizations. the results of [11-12] indicate that a vsvo algorithmic program established on adams-bashforth-moulton pair in 𝑃𝐸𝐶𝐸 mode (abm) with stepvarying attained by a variable coefficient technique is ever convergent ( as the maximal step length used in the time interval of integration inclines to zero). whenever an interpolatory technique is utilized then convergence is ensured whenever the step/order-varying technique is such that there subsists a changeless 𝑁 such that in any 𝑁 sequential steps there are ever 𝑘 𝑠𝑡𝑒𝑝𝑠 of changeless length considered by the same 𝑘𝑡ℎ − 𝑜𝑟𝑑𝑒𝑟 abm method, for some value of 𝑘. these results stress so far once again that variable coefficient technique, although in general is more costly and cumbersome to carry out, are essentially more effective than interpolatory techniques. see [18-19] for more details. the succeeding sections will demonstrate the usefulness of these strategies. int. j. anal. appl. 19 (6) (2021) 935 ii. materials and methods [4] was evidently the first author to propose a “placid” conversion of a step-size ℎ to novel step-size 𝑤ℎ. [9, 15] unfolded his ideas: we study an arbitrary grid (𝑥𝑛) and announce the step sizes by ℎ𝑛 = 𝑥𝑛+1 − 𝑥𝑛. we presume that estimations 𝑦𝑖 to 𝑦(𝑥𝑖) are recognized for 𝑖 = 𝑛 − 𝑘 + 1,…,𝑛 and insert 𝑓𝑖(𝑥𝑖,𝑦𝑖) and announce 𝑝(𝑥) as the multinomial which interpolates the values (𝑥𝑖,𝑓𝑖) for 𝑖 = 𝑛 − 𝑘 + 1,…,𝑛. utilizing newton’s interpolation formula we get 𝑝(𝑥) = ∑ ∏ (𝑥 − 𝑥𝑛−𝑖)𝛿 𝑖𝑓[𝑥𝑛,𝑥𝑛−1,…,𝑥𝑛−𝑖] 𝑖−1 𝑖=0 𝑘−1 𝑖=0 (6) where the divided differences 𝛿𝑖𝑓[𝑥𝑛,𝑥𝑛−1,…,𝑥𝑛−𝑖] are specified algorithmic by 𝛿𝑖𝑓[𝑥𝑛] = 𝑓𝑛 𝛿𝑖𝑓[𝑥𝑛,𝑥𝑛−1,…,𝑥𝑛−𝑖] = 𝛿𝑖−1𝑓[𝑥𝑛,𝑥𝑛−1,…,𝑥𝑛−𝑖+1]−𝛿 𝑖−1𝑓[𝑥𝑛−1,…,𝑥𝑛−𝑖] 𝑥𝑛−𝑥𝑛−𝑖 (7) it is virtual to rescript (6) as 𝑝(𝑥) = ∑ ∏ 𝑥−𝑥𝑛−𝑖 𝑥𝑛+1−𝑥𝑛−𝑖 ∙ φ𝑖 ∗(𝑛)𝑖−1𝑖=0 𝑘−1 𝑖=0 , (8) where φ𝑖 ∗(𝑛) = ∏ (𝑥𝑛+1 − 𝑥𝑛−𝑖) ∙ 𝛿 𝑖𝑓[𝑥𝑛,𝑥𝑛−1,…,𝑥𝑛−𝑖] 𝑖−1 𝑖=0 (9) we immediately specify the estimations to 𝑦(𝑥𝑛+1) by 𝑦𝑛+1 = 𝑦𝑛 + ∫ 𝑝(𝑥)𝑑𝑥 𝑥𝑛+1 𝑥𝑛 (10) replacing equation (6) into (10) we have 𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑛 ∑ 𝑔𝑖(𝑛)φ𝑖 ∗(𝑛)𝑘−1𝑖=0 (11) with 𝑔𝑖(𝑛) = 1 ℎ𝑛 ∫ ∏ 𝑥−𝑥𝑛−𝑖 𝑥𝑛+1−𝑥𝑛−𝑖 𝑑𝑥𝑖−1𝑖=0 𝑥𝑛+1 𝑥𝑛 . (12) equation (11) is the elongation of the explicit adams method (1) to variable step sizes [13]. the variable step size implicit adams methods can be inferred likewise. we assume 𝑝∗(𝑥) be the multinomial of degree 𝑘 that interpolates (𝑥𝑖,𝑓𝑖) for 𝑖 = 𝑛 − 𝑘 + 1 (the value 𝑓𝑛+1 = 𝑓(𝑥𝑛+1,𝑦𝑛+1) comprises the unknown physical solution 𝑦𝑛+1). once more, employing newton’s interpolation formula, we get 𝑝∗(𝑥) = 𝑝(𝑥) + ∏ (𝑥 − 𝑥𝑛−𝑖) ∙ 𝛿 𝑘𝑓[𝑥𝑛+1,𝑥𝑛,… ,𝑥𝑛−𝑘+1] 𝑖−1 𝑖=0 . the numeric solution specified by int. j. anal. appl. 19 (6) (2021) 936 𝑦𝑛+1 = 𝑦𝑛 + ∫ 𝑝 ∗(𝑥)𝑑𝑥 𝑥𝑛+1 𝑥𝑛 , is established immediately by 𝑦𝑛+1 = 𝑝𝑛+1 + ℎ𝑛𝑔𝑘(𝑛)φ𝑘(𝑛 + 1) (13) where 𝑝𝑛+1 is the numeric estimation got by the explicit adams method 𝑝𝑛+1 = 𝑦𝑛 + ℎ𝑛 ∑ 𝑔𝑖(𝑛)φ𝑖 ∗(𝑛) 𝑘−1 𝑖=0 and where φ𝑘(𝑛 + 1) = ∏ (𝑥𝑛+1 − 𝑥𝑛−𝑖) ∙ 𝛿 𝑘𝑓[𝑥𝑛+1,𝑥𝑛,… ,𝑥𝑛−𝑘+1] 𝑖−1 𝑖=0 . (14) defining the reoccurrence relations for 𝑔𝑖(𝑛),φ𝑖(𝑛) and φ𝑖 ∗(𝑛). the price of calculating consolidation coefficients is the greatest weakness to allowing arbitrary fluctuations in the step size [16]. the values φ𝑖 ∗(𝑛)(𝑖 = 0,…,𝑘 − 1) and φ𝑘(𝑛 + 1) can be calculated expeditiously with reoccurrence relations φ0(𝑛) = φ𝑖 ∗(𝑛) = f𝑛 φ𝑖+1(𝑛) = φ𝑖(𝑛)− φ𝑖 ∗(𝑛 − 1) (15) φ𝑖 ∗(𝑛) = 𝛽𝑖(𝑛)φ𝑖(𝑛), which are an instant effect of (9) and (14). the coefficients 𝛽𝑖(𝑛) = ∏ 𝑥𝑛+1 − 𝑥𝑛−𝑖 𝑥𝑛 − 𝑥𝑛−𝑖−1 𝑑𝑥 𝑖−1 𝑖=0 can be computed by 𝛽0(𝑛) = 1, 𝛽𝑖(𝑛) = 𝛽𝑖−1(𝑛) 𝑥𝑛+1−𝑥𝑛−𝑖+1 𝑥𝑛−𝑥𝑛−1 . the computation of the coefficients 𝑔𝑖(𝑛) is wilier [17]. we bring in the 𝑞 − 𝑓𝑜𝑙𝑑 integral 𝑐𝑖𝑞(𝑥) = (𝑞−1)! ℎ𝑛 𝑞 ∫ … 𝑥 𝑥𝑛 ∫ ∏ 𝜖0−𝑥𝑛−𝑖 𝑥𝑛+1−𝑥𝑛−𝑖 𝑑𝜖0 … 𝑖−1 𝑖=0 𝜖𝑞−1 𝑥𝑛 𝑑𝜖𝑞−1 (16) and remark that 𝑔𝑖(𝑛) = 𝑐𝑖𝑞(𝑥𝑛+1) [13]. a. theoretical analysis of the method definition 8: the order of the operator 𝐿ℎ is the highest 𝑟 such that whenever 𝑦(𝑥) possesses a continuous (𝑟 + 1)𝑡ℎ the derivative, then int. j. anal. appl. 19 (6) (2021) 937 𝐿ℎ(𝑦(𝑥)) = 0(ℎ 𝑟+1). (17) whenever we presume a continuous (𝑟 + 2)𝑡ℎ derivative for 𝑦, then we can replace the taylor’s series for 𝑦 and 𝑦′ with 0(ℎ𝑟+1) remains. whenever the terms in ℎ0,ℎ2,…,ℎ𝑟+1 are assembled unitedly, we will arrive at 𝐿ℎ(𝑦(𝑥)) = ∑ 𝐶𝑞 𝑟+1 𝑞=0 ℎ𝑞𝑦(𝑞)(𝑥) + 0(ℎ𝑟+1) where 𝐶𝑞 = { ∑ 𝛼𝑖 𝑘 𝑖=0 𝑞 = 0 ∑ [ (−𝑖)𝑞 𝑞! 𝛼𝑖 − (−𝑖)𝑞−1 (𝑞−1)! 𝛽𝑖] 𝑞 > 0 𝑘 𝑖=0 (18) the linear equations 𝐶𝑞 = 0,𝑞 ≤ 𝑟, are the equations which decides an 𝑟𝑡ℎ order method. given the number of truncation error put in for each one step is 𝐶𝑟+1 ∑ 𝛽𝑖 𝑘 𝑖=0 ℎ𝑟+1𝑦(𝑟+1) + 0(ℎ𝑟+2). therefore the natural standization is to take ∑ 𝛽𝑖 𝑘 𝑖=0 = 1 [10]. (19) theorem 5: whenever the multi-step method (29) is unchanging and of order 1 so it is convergent. whenever the method (29) is unchanging and of order 𝑝 so it is convergent of order 𝑝 [13]. theorem 6: the multi-step method (29) is of order 𝑝, whenever one and only of the following tantamount precondition is true conditions is met: (i) ∑ 𝛼𝑖 𝑘 𝑖=0 = 0 and ∑ 𝛼𝑖 𝑘 𝑖=0 𝑖 𝑞 = 𝑞∑ 𝛽𝑖𝑖 𝑞−1𝑘 𝑖=0 for 𝑞 = 1,…,𝑝; (ii) 𝜚(𝑒ℎ)− ℎ𝜎(𝑒ℎ) = 𝑂(ℎ𝑝+1) for ℎ → 0; (iii) 𝜚(𝜍) 𝑙𝑜𝑔𝜍 − 𝜎(𝜍) = 𝑂((𝜍 − 1)𝑝) for 𝜍 → 1. where the linear difference operator 𝐿 specified by 𝐿(𝑦,𝑥,ℎ) = ∑ (𝛼𝑖𝑦(𝑥 + 𝑖ℎ) − h𝛽𝑖𝑦 ′(𝑥 + 𝑖ℎ))𝑘𝑖=0 [13]. (20) proof enlarging 𝑦(𝑥 + 𝑖ℎ) and 𝑦′(𝑥 + 𝑖ℎ) using taylor series and inserting the truncated series. put the taylor series expansion into (20) gives 𝐿(𝑦,𝑥,ℎ) = ∑ (∑ 𝑖𝑞 𝑞!𝑞≥0 ℎ𝑞𝑦(𝑞)(𝑥) − ℎ𝛽𝑖 ∑ 𝑖𝑟 𝑟!𝑟≥0 ℎ𝑟𝑦(𝑟+1)(𝑥))𝑘𝑖=0 (21) 𝑦(𝑥)∑ 𝛼𝑖 𝑘 𝑖=0 + ∑ ℎ𝑞 𝑞!𝑞≥1 𝑦(𝑞)(𝑥)(∑ 𝛼𝑖𝑖 𝑞 − 𝑞∑ 𝛽𝑖𝑖 𝑞−1𝑘 𝑖=0 𝑘 𝑖=0 ). int. j. anal. appl. 19 (6) (2021) 938 this means the par of precondition (i) with 𝐿(𝑦,𝑥,ℎ) = 𝑂(ℎ𝑝+1) for entirely sufficient degree of regular functions 𝑦(𝑥). we will continue to express that the three precondition of proposition 1 are tantamount. the individuality operator 𝐿(𝑒𝑥𝑝,0,ℎ) = 𝜚(𝑒ℎ) − ℎ𝜎(𝑒ℎ) where 𝑒𝑥𝑝 announces the exponential function, unitedly with 𝐿(𝑒𝑥𝑝,0,ℎ) = ∑ 𝛼𝑖 𝑘 𝑖=0 + ∑ ℎ𝑞 𝑞!𝑞≥1 (∑ 𝛼𝑖𝑖 𝑞 − 𝑞∑ 𝛽𝑖𝑖 𝑞−1𝑘 𝑖=0 𝑘 𝑖=0 ), which succeeds from (21), proves the par of the preconditions (i) and (ii). by use of the translation 𝜍 = 𝑒ℎ (𝑜𝑟 ℎ = 𝑙𝑜𝑔𝜍) precondition (ii) can be spelt in the pattern 𝜚(𝜍) − 𝑙𝑜𝑔𝜍.𝜎(𝜍) = 𝑂((𝑙𝑜𝑔𝜍)𝑝+1) for 𝜍 → 1. but this precondition is par to (iii), since 𝑙𝑜𝑔𝜍 = (𝜍 − 1) + 𝑂((𝜍 − 1)2) for 𝜍 → 1. see [13] for more info. b. convergence convergence for variable step size adams method was first considered by [26]. in order to show convergence for the general case we present the vector 𝑌𝑛 = (𝑦𝑛+𝑘−1,… ,𝑦𝑛+1,𝑦𝑛) 𝑇. the method 𝑦𝑛+𝑘 + ∑ 𝛼𝑖𝑦𝑛+𝑖 = ℎ𝑛+𝑘−1 𝑘−1 𝑖=0 ∑ 𝛽𝑖𝑛𝑓𝑛+𝑖 𝑘 𝑖=0 (22) then turns tantamount to 𝑌𝑛+1 = (𝐴𝑛 ⊗ 𝐼)𝑌𝑛 + ℎ𝑛+𝑘−1φ𝑛(𝑥𝑛,𝑌𝑛,ℎ𝑛) , (23) where 𝐴𝑛 is established by                 = −−− − 01 001 00...01 ...... ,0,1,1 a   nnnk n (24) the comrade matrix and φ𝑛(𝑥𝑛,𝑌𝑛,ℎ𝑛) = (𝑒1 ⊗ 𝐼)ψ𝑛(𝑥𝑛,𝑌𝑛,ℎ𝑛). the value ψ = ψ𝑛(𝑥𝑛,𝑌𝑛,ℎ𝑛) is specified without any doubt by ψ = 𝑦𝑛+𝑘 + ∑ 𝛽𝑖𝑛𝑓(𝑥𝑛+𝑖,𝑦𝑛+𝑖) + 𝛽𝑘𝑛𝑓(𝑥𝑛+𝑘,ℎψ − 𝑘−1 𝑖=0 ∑ 𝛼𝑖𝑛𝑦𝑛+𝑖 𝑘 𝑖=0 . we advance by announcing int. j. anal. appl. 19 (6) (2021) 939 𝑌(𝑥𝑛) = (𝑦(𝑥𝑛+𝑘−1),…,𝑦(𝑥𝑛+1),𝑦(𝑥𝑛)) 𝑇 the precise values to be estimated by 𝑌𝑛 . the convergence theorem can immediately be developed as succeeds. see [13] for more info. theorem 7: assume that • the method (27) is stable of order 𝑝, and has bounded coefficients 𝛼𝑖𝑛 and 𝛽𝑖𝑛; • the starting values satisfy ‖𝑌(𝑥𝑛) − 𝑌0‖ = 𝑂(ℎ0 𝑝 ); • the step size ratios are bounded ( ℎ𝑛 ℎ𝑛−1 ≤ ω). then the method is convergent of order 𝑝, i.e., for each differential equation 𝑦′ = 𝑓(𝑥,𝑦),𝑦(𝑥0) = 𝑦0 with 𝑓 sufficiently differentiable the global error satisfies ‖𝑌(𝑥𝑛) − 𝑦𝑛‖ ≤ 𝐶ℎ 𝑝 for 𝑥𝑛 ≤ 𝑥, where ℎ = 𝑚𝑎𝑥ℎ𝑖. see [13] for more info. proof because the approach has order p and the physical coefficients and step-size ratios are restricted, the expression becomes 𝑦(𝑥𝑛+𝑘)+ ∑ 𝛼𝑖𝑛𝑦(𝑥𝑛+𝑖) − ℎ𝑛+𝑘−1 𝑘−1 𝑖=0 ∑ 𝛽𝑖𝑛𝑦 ′(𝑥𝑛+𝑖) = 𝑘 𝑖=0 𝑂(ℎ𝑛 𝑝+1 ), we justify that the truncated-local error 𝛿𝑛+1 = 𝑌(𝑥𝑛+1)− (𝐴𝑛 ⊗ 𝐼)𝑌(𝑥𝑛) − ℎ𝑛+𝑘−1φ𝑛(𝑥𝑛,𝑌(𝑥𝑛),ℎ𝑛) (25) gratifies 𝛿𝑛+1 = 𝑂(ℎ𝑛 𝑝+1 ). (26) deducting (23) from (25) we get 𝑌(𝑥𝑛+1)− 𝑌𝑛+1 = (𝐴𝑛 ⊗ 𝐼)𝑌(𝑥𝑛) − 𝑌𝑛) +ℎ𝑛+𝑘−1(φ𝑛(𝑥𝑛,𝑌(𝑥𝑛),ℎ𝑛)− φ𝑛(𝑥𝑛,𝑌𝑛,ℎ𝑛)) + 𝛿𝑛+1 and by induction it succeeds that 𝑌(𝑥𝑛+1) − 𝑌𝑛+1 = (𝐴𝑛 ….𝐴0) ⊗ 𝐼)𝑌(𝑥0) − 𝑌0) + ∑ ℎ𝑖+𝑘−1 𝑛 𝑖=0 (𝐴𝑛 ….𝐴𝑖+1)⊗ 𝐼)(φ𝑖(𝑥𝑖,𝑌(𝑥𝑖),ℎ𝑖) − φ𝑖(𝑥𝑖,𝑌𝑖,ℎ𝑖)) + ∑ (𝐴𝑛 ….𝐴𝑖+1) ⊗ 𝐼) 𝑛 𝑖=0 𝛿𝑖+1. as in the proof of theorem 1, we derive that the φ𝑛 gratifies a uniform lipschitz precondition with respect to 𝑌𝑛. this unitedly with stability and (26), means that ‖𝑌(𝑥𝑛+1) − 𝑌𝑛+1‖ ≤ ∑ ℎ𝑖+𝑘−1 𝑛 𝑖=0 𝐿‖𝑌(𝑥𝑖) − 𝑌𝑖‖ + 𝐶1ℎ 𝑝. int. j. anal. appl. 19 (6) (2021) 940 to figure out this difference, we bring in the succession { 𝑛} specified as 0 = ‖𝑌(𝑥0) − 𝑌0‖, 𝑛+1 = ∑ ℎ𝑖+𝑘−1 𝑛 𝑖=0 𝐿 𝑖 + 𝐶1ℎ 𝑝. (27) a simple induction statement proves that ‖𝑌(𝑥𝑛) − 𝑌𝑛‖ ≤ 𝑛. (28) from (27) we get for 𝑛 ≥ 1 𝑛+1 = 𝑛 + ℎ𝑛+𝑘−1𝐿 𝑛 ≤ exp (𝐿ℎ𝑛+𝑘−1) 𝑛 so that in addition 𝑛 ≤ exp(𝑥 − 𝑥0)𝐿) 1 = exp(𝑥 − 𝑥0)𝐿) ∙ (ℎ𝑘−1𝐿‖𝑌(𝑥0) − 𝑌0‖ + 𝐶1ℎ 𝑝). the inequality unitedly with (28) completes the proof of theorem 7. see [13] for more info. c. implementing the convergence criteria of variable step, variable order the use of milne’s estimate for the principal local truncation error necessitate that predictor-corrector method possess like order. this is attained by accepting the predictor to be a 𝑘 − 𝑠𝑡𝑒𝑝 adams bashforth method and the corrector to be a (𝑘 − 1) − 𝑠𝑡𝑒𝑝 adams-moulton, both then possess order 𝑝 = 𝑘. the 𝑘 − 𝑠𝑡𝑒𝑝 𝑘𝑡ℎ order abm pair is therefore 𝑦𝑛+1 = 𝑦𝑛 + ℎ∑ 𝛾𝑖 ∗∇𝑖𝑓𝑛, 𝑘−1 𝑖=0 𝑝 ∗ = 𝑘, 𝐶𝑘+1 ∗ = 𝛾𝑘 ∗ 𝑦𝑛+1 = 𝑦𝑛 + ℎ∑ 𝛾𝑖∇ 𝑖𝑓𝑛+1, 𝑘−1 𝑖=0 𝑝 = 𝑘, 𝐶𝑘+1 = 𝛾𝑘 }𝑘 = 1,2,… (29) whenever we imagine (29) being employed in 𝑃(𝐸𝐶)𝜇𝐸1−𝑡 mode then, in the second of (29), 𝑦𝑛+1 will be substituted by 𝑦𝑛+1 [𝑣+1] , and the one value 𝑓𝑛+1 on the right side by 𝑓𝑛+1 [𝑣] , the leftover values 𝑓𝑛−𝑗 being substituted by 𝑓𝑛−𝑗 [𝜇−𝑡] , 𝑗 = 0,1,…,𝑘 − 1. we can surmount this trouble by defining ∇𝑣 𝑖 𝑓𝑛+1 [𝜇] to be ∇𝑖𝑓𝑛+1 [𝜇] with the one value 𝑓𝑛+1 [𝜇] substituted by 𝑓𝑛+1 [𝑣] . that is, ∇𝑣 𝑖 𝑓𝑛+1 [𝜇] = ∇𝑖𝑓𝑛+1 [𝜇] + 𝑓𝑛+1 [𝑣] − 𝑓𝑛+1 [𝜇] (30) we may rewrite (30) in the form ∑ (𝛾𝑖∇ 𝑖𝑓𝑛+1 [𝜇] − 𝛾𝑖 ∗∇𝑖𝑓𝑛 [𝜇] ) = 𝛾𝑘−1 ∗ 𝑓𝑛+1 [𝜇]𝑘−1 𝑖=0 (31) where the notational system is specified by (29). we immediately employ the pair (29) in 𝑃(𝐸𝐶)𝜇𝐸1−𝑡 mode, and utilize the structure of the adams methods to formulate a form of abm method which is computationally commodious and frugal. the mode is specified by 𝑃: 𝑦𝑛+1 [0] = 𝑦𝑛 [𝜇] + ℎ∑ 𝛾𝑖 ∗∇𝑖𝑓𝑛 [𝜇−1]𝑘−1 𝑖=0 (32) int. j. anal. appl. 19 (6) (2021) 941 (𝐸𝐶)𝜇 𝑓𝑛+1 [𝑣] = 𝑓(𝑥𝑛+1,𝑦𝑛+1 [𝑣] ) 𝑦𝑛+1 [𝑣+1] = 𝑦𝑛 [𝜇] + ℎ∑ 𝛾𝑖∇𝑣 𝑖 𝑓𝑛+1 [𝜇−1]𝑘−1 𝑖=0 } 𝑣 = 0,1,…,𝜇 − 1 (33) (𝐸1−𝑡 𝑓𝑛+1 [𝜇] = 𝑓(𝑥𝑛+1,𝑦𝑛+1 [𝜇] ) whenever 𝑡 = 0. to employ the milne estimate, we require calculating 𝑦𝑛+1 [𝜇] − 𝑦𝑛+1 [0] . deducting (32) from (33) with 𝑣 = 𝜇 − 1 establishes 𝑦𝑛+1 [𝜇] − 𝑦𝑛+1 [0] = ℎ∑ (𝛾𝑖∇𝜇−1 𝑖 𝑓𝑛+1 [𝜇−𝑡] − 𝛾𝑖 ∗∇𝑖𝑓𝑛 [𝜇−𝑡] ) 𝑘−1 𝑖=0 = ℎ𝛾𝑖 ∗∇𝜇−1 𝑘 𝑓𝑛+1 [𝜇−𝑡] . since 𝐶𝑘+1 ∗ = 𝛾𝑘 ∗ and 𝐶𝑘+1 = 𝛾𝑘, the milne estimate 𝑊 = 𝐶𝑝+1 𝐶𝑝+1 ∗ − 𝐶𝑝+1 for the principal local truncation error at 𝑥𝑛+1 (which we shall denote by 𝑇𝑛+1) is established by 𝑇𝑛+1 = 𝐶𝑝+1 𝐶𝑝+1 ∗ − 𝐶𝑝+1 ( 𝑦𝑛+1 [𝜇] − 𝑦𝑛+1 [0] ) = 𝛾𝑘 𝛾𝑘 ∗−𝛾𝑘 = ℎ𝛾𝑖 ∗∇𝜇−1 𝑘 𝑓𝑛+1 [𝜇−𝑡] . whenever 𝛾𝑘 ∗ − 𝛾𝑘 = 𝛾𝑘−1 ∗ , wherefrom 𝑇𝑛+1 == ℎ𝛾𝑖 ∗∇𝜇−1 𝑘 𝑓𝑛+1 [𝜇−𝑡] . see [2, 6-7, 18-19, 21-25] for more info. iii. practical examples of stiff systems of first order odes we are interested with the computation interpretation of these attributes. a problem is stiff whenever the analytical solution being looked for changes tardily, but there are close solutions that change speedily, so the numerical approach must consider small steps size to get acceptable results. see [20] for more info. application problem 1 an engineering example: in chemical engineering, a complicated production activity may involve several reactors connected with inflow and outflow pipes. if there are n reactors, the whole process is governed by a system of 𝑛 differential equations of the form [ 𝑥′(𝑡) 𝑦′(𝑡) 𝑧′(𝑡) ] = [ − 8 3 − 4 3 1 − 17 3 − 4 3 1 − 35 3 14 3 −2] [ 𝑥1 𝑥2 𝑥3 ] + [ 12 29 48 ] analytical solution int. j. anal. appl. 19 (6) (2021) 942 𝑥(𝑡) = 1 6 𝑒−3𝑡(6 − 50𝑒𝑡 + 10𝑒2𝑡 + 34𝑒3𝑡), 𝑦(𝑡) = 1 6 𝑒−3𝑡(12 − 125𝑒𝑡 + 40𝑒2𝑡 + 73𝑒3𝑡), 𝑧(𝑡) = 1 6 𝑒−3𝑡(14 − 200𝑒𝑡 + 70𝑒2𝑡 + 116𝑒3𝑡). see [5] for more info. application problem 2 we examine the coefficient matrix a with the initial conditions 𝑦(0) and the forcing function 𝑔(𝑡) given by 𝐴 = [ 0 −1 1 0 2 0 −2 −1 3 ], 𝑦(0) = [ 0 0 1 ], 𝑔(𝑡) = [ 1 𝑡 𝑒−𝑡 ]. analytical solution 𝑦(𝑡) = [ 7 12 𝑒2𝑡 + 3 2 𝑒𝑡 + 𝑒−𝑡 6 − 𝑡 2 − 9 4 𝑒2𝑡 4 − 𝑡 2 − 1 4 17 12 𝑒2𝑡 + 3 2 − 𝑒−𝑡 6 − 𝑡 2 − 7 4 ] . see [28] for more info. application problem 3 we study the initial value problem 𝑦′(𝑡) = 𝐴𝑦(𝑡),𝑦(0) = [1,0,−1]𝑇, where 𝐴 = [ −21 19 −20 19 −21 20 40 −40 −40 ]. analytical solution 𝑦(𝑡) = [ 1 2 𝑒−2𝑡 + 1 2 𝑒−40𝑡(𝑐𝑜𝑠(40𝑡) + sin(40𝑡))) 1 2 𝑒−2𝑡 − 1 2 𝑒−40𝑡(𝑐𝑜𝑠(40𝑡) + sin(40𝑡))) −𝑒−40𝑡 + (𝑐𝑜𝑠(40𝑡) − sin(40𝑡))) ] . see [18] for more info. int. j. anal. appl. 19 (6) (2021) 943 iv. results this aspect implements the computational strategy of variable step, variable order for solving stiff systems of ordinary differential equations. the different application problems of stiff system of ordinary differential equations were implemented on diverse convergence criteria of the following: 10−3,10−3,10−4,10−5,10−6,10−7,10−8,10−9,10−10 𝑎𝑛𝑑 10−11 . the results of the application problems 1, 2 and 3 are displayed on tables 1, 2 and 3. table i. csvsvo implementation 𝑀𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑 𝑀𝑎𝑥𝑒𝑟𝑟𝑜𝑟𝑠 𝐶𝑜𝑛𝑣𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 csvsvo 6.73296e-05 10−2 csvsvo 1.50195e-01 10−2 csvsvo 2.30889e-01 10−2 csvsvo 7.17852e-09 10−4 csvsvo 1.81011e-03 10−4 csvsvo 3.07211e-03 10−4 csvsvo 7.17852e-13 10−6 csvsvo 1.84418e-05 10−6 csvsvo 3.16097e-05 10−6 csvsvo 6.24284e-16 10−8 csvsvo 1.84762e-07 10−8 csvsvo 3.17e-07 10−8 csvsvo 1.96457e-16 10−10 csvsvo 1.84796e-09 10−10 csvsvo 3.17e-09 10−10 int. j. anal. appl. 19 (6) (2021) 944 table ii. csvsvo implementation 𝑀𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑 𝑀𝑎𝑥𝑒𝑟𝑟𝑜𝑟𝑠 𝐶𝑜𝑛𝑣𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 csvsvo 1.33456e-04 10−3 csvsvo 1.10578e-02 10−3 csvsvo 3.81005e-02 10−3 csvsvo 1.43869e-08 10−5 csvsvo 2.10822e-04 10−5 csvsvo 4.47082e-04 10−5 csvsvo 1.43996e-12 10−7 csvsvo 2.22913e-06 10−7 csvsvo 4.47991e-06 10−7 csvsvo 1.11022e-16 10−9 csvsvo 2.24143e-08 10−9 csvsvo 4.48e-08 10−9 csvsvo 1.11022e-16 10−11 csvsvo 2.24226e-10 10−11 csvsvo 4.48e-10 10−11 int. j. anal. appl. 19 (6) (2021) 945 table iii. csvsvo implementation 𝑀𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑 𝑀𝑎𝑥𝑒𝑟𝑟𝑜𝑟𝑠 𝐶𝑜𝑛𝑣𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 csvsvo 6.73296e-05 10−2 csvsvo 1.50195e-01 10−2 csvsvo 2.30889e-01 10−2 csvsvo 7.17852e-09 10−4 csvsvo 1.81011e-03 10−4 csvsvo 3.07211e-03 10−4 csvsvo 7.17852e-13 10−6 csvsvo 1.84418e-05 10−6 csvsvo 3.16097e-05 10−6 csvsvo 6.24284e-16 10−8 csvsvo 1.84762e-07 10−8 csvsvo 3.17e-07 10−8 csvsvo 1.96457e-16 10−10 csvsvo 1.84796e-09 10−10 csvsvo 3.17e-09 10−10 v. conclusion applications problem 1, 2 and 3 represents stiff systems of ordinary differential equations which seems to generates an unstable system behavior, and as such requires a technical approach like csvsvo to guarantee an improve efficiency and better accuracy. the stiff systems of ordinary differential equations are carried out employing the csvsvo implementation. the csvsvo has the capacity to introduce the convergence criteria in order to engender the desired result is achieved. these convergence criteria decide whether the result is accepted or rejected. the csvsvo implementation is done utilizing a multiprocessor approach executed under the mathematica software platform. tables 1, 2 and 3 displays the int. j. anal. appl. 19 (6) (2021) 946 computational results established that the csvsvo is reached via the convergence criteria of the following: 10−3,10−3,10−4,10−5,10−6,10−7,10−8,10−9 ,10−10 𝑎𝑛𝑑 10−11. in addition, with the trend of the maximum errors achieved via the different convergence criteria, we can conclude that the csvsvo is capable of resolving stiff systems of ordinary differential equations with better efficiency and accuracy as exhibited in tables 1, 2 and 3. see [5, 18, 21—25, 28] for details. acknowledgements: the authors would like to appreciate covenant university for providing sponsorship throughout the study period of time. many thanks to the anonymous reviewers for their continuous contribution. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. l. abell, j. p. braselton, differential equations with mathematica, elsevier academic press, usa, 2004 [2] u. m. ascher, l. r. petzoid, computer methods for ordinary differential equations and differential-algebraic equations, siam, usa, 1998. [3] k. atkinson, w. han, d. stewart, numerical solution of ordinary differential equations, john wiley & sons, inc., new jersey, 2009. [4] f. ceschino, modification de la longueur du pas dans l’ integration numerique par les methods a pas lies, chifres, vol. 2, 101-106, 1961. [5] w. cheney, d. kingaid, numerical mathematics and computing, thomson brooks/cole, usa, 2008. [6] j. r. dormand, numerical methods for differential equations, crc press, new york, 1996. [7] j. d. faires, r. l. burden, initial-value problems for odes, dublin city university, 2012. [8] s. o. fatunla, numerical methods for initial value problems in ordinary differential equations, academic press, inc., new york, 1988. [9] c. v. d. forrington, extensions of the predictor-corrector method for the solution of systems of ordinary differential equations, comput. j. 4 (1961), 80-84. int. j. anal. appl. 19 (6) (2021) 947 [10] c. w. gear, numerical value problems in odes, prentice-hall, inc., new jersey, usa, 1971. [11] c. w. gear, k. w. tu, the effect of variable mesh size on the stability of multistep methods, siam j. numer. anal. 11 (1974), 1025-1043. [12] c. w. gear, d. s. watanabe, stability and convergence of variable order multistep methods, siam j. numer. anal. 11 (1974), 1044-1058. [13] e. hairer, s. p. norsett, g. wanner, solving ordinary differential equations i, springer heidelberg dordrecht, new york, 2009. [14] m. i. hazizah, b. i. zarina, diagonally implicit block backward differentiation formula with optimal stability properties for stiff ordinary differential equations, symmetry, 11 (2019), 1342. [15] f. t. krogh, a variable step variable order multistep method for the numerical solution of ordinary differential equations, information processing 68 north-holland, amsterdam, 194-199, 1969. [16] f. t. krogh, algorithm for changing the step size, siam j. num. anal. 10 (1973), 949-965. [17] f. t. krogh, changing step size inthe integration of differential equations using modified divided differences, proceedings of the conference on the num. sol. of ode, lecture notes in math vol. 362, 22-71, 1974. [18] j. d. lambert, computational methods in ordinary differential equations, john wiley & sons, inc., new york, 1973. [19] j. d. lambert, numerical methods for ordinary differential systems, john wiley & sons, inc., new york, 1991. [20] c. b. moler, numerical computing with matlab, siam, usa, 2004. [21] j. g. oghonyon, s. a. okunuga, n. a. omoregbe, o. o. agboola, a computational approach in estimating the amount of pond and determining the long time behavioural representation of pond pollution”, global journal of pure and applied mathematics, vol. 11, 2773-2786, 2015. [22] j. g. oghonyon, j. ehigie, s. k. eke, investigating the convergence of some selected properties on block predictor-corrector methods and its applications, j. eng. appl. sci. 11 (2017), 2402-2408. int. j. anal. appl. 19 (6) (2021) 948 [23] j. g. oghonyon, o. a. adesanya, h. akewe, h. i. okagbue, softcode of multi-processing milne’s device for estimating first-order ordinary differential equations, asian j. sci. res. 11 (2018), 553-559. [24] j. g. oghonyon, o. f. imaga, p. o. ogunniyi, the reversed estimation of variable step size implementation for solving nonstiff ordinary differential equations, int. j. civil eng. technol. 9 (2018), 332-340. [25] j. g. oghonyon, s. a. okunuga, h. i. okagbue, expanded trigonometrically matched block variable-step-size technics for computing oscillating vibrations, lecture notes in engineering and computer science, vol. 2239, 552-557, 2019. [26] p. piotrowsky, stability, consistency and convergence of variable k-step methods for numerical integration of large systems of ordinary differential equations, lecture notes in math., vol. 109, 221—227, 1969. [27] s. e. ramin, numerical methods for engineers and scientists using matlab®, crc press, london, 2017. [28] p. m. shankar, differential equations: a problem solving approach based on matlab®, crc press, usa, 2018. international journal of analysis and applications volume 19, number 2 (2021), 239-251 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-239 boundary value problems for fractional differential equation in special banach space beddani moustafa1,∗, hedia benaouda2 1department of mathematics, djillali liabes university of sidi bel-abbés, po box 89 22000 bel-abbés, algeria 2laboratory of mathematics, university of tiaret, po box 78 14000 tiaret, algeria ∗corresponding author: beddani2004@yahoo.fr abstract. this paper studies the existence of solutions of boundary value problem for fractional differential equations on the half-line in a special banach space. the main result is based on mönch fixed point theorem combining with a suitable measure of non-compactness, an example is given to illustrate our approach. 1. introduction fractional differential equations play a very important role in describing some real world problems, for example, in the description of hereditary properties of various materials and processes. they are also widely applied in the mathematical modeling of processes in physics, chemistry, aerodynamics, electro-dynamics of complex medium, polymer rheology, etc. consequently, the fractional calculus and its applications in various fields of science and engineering have received much attention and have developed very rapidly (cf. [18,20,23] for instance). very recently, many research papers have appeared concerning the fractional differential equations in banach spaces, some of them investigated the existence results of solutions on finite intervals by classical tools from received october 16th, 2020; accepted november 11th, 2020; published february 24th, 2021. 2010 mathematics subject classification. 34b15, 34b40, 26a33. key words and phrases. boundary value problem, measure of non-compactness, unbounded domain, special banach space, mönch fixed point theorem, riemann-liouville fractional derivative. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 239 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-239 int. j. anal. appl. 19 (2) (2021) 240 functional analysis; see, for example references. [1, 3, 6–8, 19, 21, 22]. in [4]. a. arara and m. benchohra studied the following problem  cdα 0+ y(t) = f(t,y(t)), t ∈ j = (0, +∞), 1 < α ≤ 2, y(0) = y0, y is bounded on j, where cdα 0+ is the caputo fractional derivative of order α, f : j × r → r is a is a continuous function and y0 ∈ r. the main approach is based on schauder’s fixed point theorem. my results is to generalize the previous work. this article concerns the existence of solutions of boundary value problem for fractional differential equations on unbounded interval. we consider the boundary value problem (1.1) dα0+y(t) = f(t,y(t)), t ∈ j = (0, +∞), (1.2) i2−α 0+ y(0+) = y0, (1.3) dα−1 0+ y(∞) = y∞. dα 0+ denote riemann-liouville fractional derivative introduced, 1 < α ≤ 2. the operator i2−α 0+ denotes the left-sided riemann-liouville fractional integral, the state y(·) takes value in a banach space e, f : (0,∞) ×e → e will be specified in later sections and (y0,y∞) ∈ e ×e. this paper is organized in the following way. in section 2, we give some general results and preliminaries and in section 3, we present existence results for the problem (1.1)-(1.3), using the mönch’s fixed point theorem combined with the technique of measure of noncompactness. finally an illustrative example will be presented in section 4. 2. preliminary results in this section, we introduce some notation and technical results which are used throughout this paper. let i ⊂ j be a compact interval and denote by c(i,e) the banach space of continuous functions y : i → e with the usual norm ‖y‖∞ = sup{‖y(t)‖, t ∈ i}. l1(j,e) the space of e-valued bochner integrable functions on j with the norm ‖f‖l1 = ∫ +∞ 0 ‖f(t)‖dt. int. j. anal. appl. 19 (2) (2021) 241 we consider the space of functions cα([0,∞),e) = {y ∈ c(j,e) : lim t→0+ t2−αy(t) exists and finite}. for y ∈ cα((0,∞),e), we define yα by yα(t) =   t 2−αy(t), t ∈ (0,∞), limt−→0 t 2−αy(t), t = 0. it is clear that yα ∈ c([0,∞),e). we consider the following banach space xα([0,∞),e) = {y ∈ cα([0,∞),e) : lim t→∞ t2−αy(t) 1 + tα exists and finite}. a norm in this space is given by ‖y‖α = sup t∈j t2−α‖y(t)‖ 1 + tα . we begin with some definitions from the theory of fractional calculus. let α > 0, n = dαe + 1 (the least integer greater than or equal to α) and h ∈ c(j,e). definition 2.1. [18]. (1) the riemann-liouville fractional integral of the function h of order α is defined by iα0+h(t) = gα(t) ∗h(t) = ∫ t 0 gα(t−s)h(s)ds, t > 0, where ∗ denotes convolution and gα(t) = tα−1/γ(α). (2) the riemann-liouville fractional derivative of the function h of order α is defined by dα0+h(t) = dn dtn (gn−α(t) ∗h(t)), for all t > 0. where γ is the gamma function. for the existence of solutions for the problem (1.1)-(1.3), we need the following auxiliary lemmas. lemma 2.1. [26] let α > 0, then the differential equation dα0+h(t) = 0, has solutions h(t) = c1t α−1 + c2t α−2 + . . . + cnt α−n, for some ci ∈ r, i = 1 . . .n, where n = [α] + 1. lemma 2.2. [26] let α > 0, then iα0+d α 0+h(t) = h(t) + c1t α−1 + c2t α−2 + . . . + cnt α−n, for some ci ∈ r, i = 0, . . . ,n, where n = [α] + 1. int. j. anal. appl. 19 (2) (2021) 242 remark 2.1. for α > 0, k > −1, we have iα0+t k = γ(k + 1) γ(α + k + 1) tα+k and dα0+t k = γ(k + 1) γ(k −α + 1) tk−α, t > 0, giving in particular dα0+t α−m = 0, m = 1, . . . ,n, where n is the smallest integer greater than or equal to α. remark 2.2. if h is suitabe function (see for instance [18, 20, 23] ), we have the composition relations dα0+i α 0+h(t) = h(t), α > 0 and d α 0+i k 0+h(t) = i k−α 0+ h(t), k > α > 0, t > 0. we note that γ,γc and γxα the kuratowski noncompactness measure of bounded sets in the spaces e,c(i,e) and xα, respectively. as for the definition of the kuratowski noncompactness measure, we refer to references [5, 17]. the following properties of the kuratowski measure of noncompactness and mönch fixed point theorem are needed for our discussion. lemma 2.3. if h ⊂ c(i,e) is bounded and equicontinuous, then γ(h(t)) is continuous on i and γc(h) = max t∈i γ(h(t)), γ ( { ∫ i x(t)dt : x ∈ h} ) ≤ ∫ i γ(h(t))dt, where h(t) = {x(t) : x ∈ h}. theorem 2.1. [2, 24] let d be a bounded, closed and convex subset of a banach space e such that 0 ∈ d , and let n be a continuous mapping of d into itself. if the implication v = convn(v ) or v = n(v ) ∪{0} =⇒ γ(v ) = 0, holds for every subset v of d, then n has a fixed point. 3. main result we will need to introduce the following hypotheses which are assumed here after. (h1) there exists a nonnegative functions a,b ∈ c(j,r+) such that ‖f(t,u)‖≤ a(t) + t2−αb(t)‖u‖ for all t ∈ j and u ∈ e, where ∫ ∞ 0 (1 + tα)b(t)dt < γ(α), ∫ ∞ 0 a(t)dt < ∞. (h2) ∀t ∈ (0,l],∀x,y ∈ e : ‖f(t,x) −f(t,y)‖≤ t 2−α 1+tα ‖x−y‖. (h3) there exists nonnegative function ` ∈ l1(j,r+) such that for each nonempty, bounded set ω ⊂ xα(j,e) γ(f(t, ω(t))) ≤ t2−α`(t)γ(ω(t)), for all t ∈ j, where ∫ ∞ 0 (1 + tα)`(t)dt ≤ γ(α). int. j. anal. appl. 19 (2) (2021) 243 definition 3.1. a function y ∈ xα([0, +∞)) is said to be solution of the problem (1.1)-(1.3) if y satisfies the equation dα 0+ y(t) = f(t,y(t)) and the conditions (1.2 − 1.3). lemma 3.1. let 1 < α < 2 and let h : j → e be continuous. a function y is a solution of the fractional integral equation (3.1) y(t) = 1 γ(α) [y∞ − ∫ ∞ 0 h(t)dt]tα−1 + y0 γ(α− 1) tα−2 + 1 γ(α) ∫ t 0 (t−s)α−1h(s)ds if and only if y is solution of the problem (3.2) dα0+y(t) = h(t), t ∈ j = (0, +∞), (3.3) i2−α 0+ y(0+) = y0, (3.4) dα−1 0+ y(∞) = y∞. proof. assume that y satisfies the problem (3.2)-(3.4). we may apply lemma 2.2 to reduce equation (3.2) to an equivalent integral equation (3.5) y(t) = c1t α−1 + c2t α−2 + iα0+h(t), for some c1, c2 ∈ r. applying i2−α0+ to both side of (3.5), we have i2−α 0+ y(t) = c1i 2−α 0+ tα−1 + c2i 2−α 0+ tα−2 + i2−α 0+ iα0+h(t). from remark 2.1, we then get i2−α 0+ y(t) = c1γ(α) γ(2) t + c2γ(α− 1) + 1 γ(2) ∫ t 0 (t−s)h(s)ds. as t −→ 0, we obtain c2 = y0 γ(α− 1) . applying dα−1 0+ to both side of (3.5), we have dα−1 0+ y(t) = c1d α−1 0+ tα−1 + c2d α−1 0+ tα−2 + dα−1 0+ iα0+h(t). from remark 2.1 and remark 2.2, we then get dα−1 0+ y(t) = c1γ(α) + 1 γ(1) ∫ t 0 h(s)ds. int. j. anal. appl. 19 (2) (2021) 244 hence c1 = 1 γ(α) (y∞ − ∫ ∞ 0 h(t)dt). thus, we have y(t) = 1 γ(α) [y∞ − ∫ ∞ 0 h(t)dt]tα−1 + y0 γ(α− 1) tα−2 + 1 γ(α) ∫ t 0 (t−s)α−1h(s)ds. conversely. the proof is simple. � consider the operator n : xα([0,∞),e) → xα([0,∞),e) defined by n(y)(t) = y∞ γ(α) tα−1 + y0 γ(α− 1) tα−2 − 1 γ(α) ∫ t 0 [tα−1 − (t−s)α−1]f(s,y(s))ds − 1 γ(α) ∫ ∞ t tα−1f(s,y(s))ds. the following several lemmas present some properties of the operator n, which are necessary for the proof of our main result. lemma 3.2. suppose that conditions (h1) and (h2) are valid. then the operator n is bounded and continuous. proof. for y ∈ xα([0,∞),e), it is easy to deduce from (h1), and that ny ∈ xα(j,e). furthermore, (h1) guarantees that t2−α‖n(y)(t)‖ 1 + tα ≤ ‖y∞‖ γ(α) + ‖y0‖ γ(α− 1) + 1 γ(α) ∫ ∞ 0 ‖f(s,y(s))‖ds ≤ ‖y∞‖ γ(α) + ‖y0‖ γ(α− 1) + ‖y‖α γ(α) ∫ ∞ 0 (1 + tα)b(t)dt + 1 γ(α) ∫ ∞ 0 a(t)dt. hence, n : xα(j,e) → xα(j,e) is bounded. next we prove that n is continuous. let {yn}∞n=1 ⊂ xα(j,e) and y ∈ xα(j,e) such that yn → y as n →∞. then, {yn}∞n=1 is a bounded set of xα(j,e), i.e. there exists m > 0 such that ‖yn‖α ≤ m for n > 1. we also have by taking limit that ‖y‖α ≤ m. in view of condition (h1), for any ε > 0, there exists l > 0 such that∫ ∞ l a(t)dt < γ(α) 6 ε, ∫ ∞ l (1 + tα)b(t)dt < γ(α)ε 6m ε, and there exists ñ ∈ n such that, for all n ≥ ñ, we have ‖f(s,yn(s)) − (s,y(s))‖ < γ(α) 3l ε. therefore, for all t ∈ j and n > ñ, we can obtain from t2−α 1 + tα ‖n(yn)(t) −n(y)(t)‖≤ 1 γ(α) ∫ t 0 ‖f(s,yn(s)) −f(s,y(s))‖ds + 1 γ(α) ∫ ∞ t ‖f(s,yn(s)) −f(s,y(s))‖ds. int. j. anal. appl. 19 (2) (2021) 245 if t ≤ l and n > ñ, we can obtain from t2−α 1 + tα ‖n(yn)(t) −n(y)(t)‖≤ 1 γ(α) ∫ t 0 ‖f(s,yn(s)) −f(s,y(s))‖ds + 1 γ(α) ∫ ∞ t ‖f(s,yn(s)) −f(s,y(s))‖ds ≤ 1 γ(α) ∫ t 0 ‖f(s,yn(s)) −f(s,y(s))‖ds + 1 γ(α) [∫ l t ‖f(s,yn(s)) −f(s,y(s))‖ds + ∫ ∞ l ‖f(s,yn(s)) −f(s,y(s))‖ds ] ≤ 2 γ(α) ∫ l 0 ‖f(s,yn(s)) −f(s,y(s))‖ds + 2m γ(α) ∫ ∞ l (1 + sα)b(s)ds + 2 γ(α) ∫ ∞ l a(s)ds ≤ ε 3 + ε 3 + ε 3 = ε. the case when t > l and n > ñ is treated similarly. thus we conclude that ‖yn −y‖α → 0 as n →∞, namely, n is continuous and the conclusion of the lemma follows. � lemma 3.3. let condition (h1) be satisfied and b be a bounded subset of xα(j,e). then (i) t2−αn(b)(t) 1+tα is equicontinuous on any compact interval of j. (ii) for given ε > 0, there exists a constant n1 > 0 such that ‖ t 2−α 1 n(y)(t1) 1+tα1 − t 2−α 2 n(y)(t2) 1+tα2 ‖ < ε for any t1, t2 ≥ n1 and y(.) ∈ b. proof. we have ny(t) = y∞ − ∫∞ 0 f(t,y(t))dt γ(α) tα−1 + y0 γ(α− 1) tα−2 + 1 γ(α) ∫ t 0 (t−s)α−1f(s,y(s))ds. in view of condition (h1) and the boundedness of b, there exists m > 0 such that (3.6) ∫ ∞ 0 ‖f(t,y(t))‖dt ≤ m for any y ∈ b. int. j. anal. appl. 19 (2) (2021) 246 in order to prove (i), let the constant r be such that ‖y‖α ≤ r for any y ∈ b, and without loss of generality, let [a,b] ⊂ j be a compact interval and t1, t2 ∈ [a,b] with t1 < t2. then ‖ t2−α1 n(y)(t1) 1 + tα1 − t2−α2 n(y)(t2) 1 + tα2 ‖≤ ‖y∞‖ + m γ(α) | t1 1 + tα1 − t2 1 + tα2 | + ‖y0‖ γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + 1 γ(α) ∥∥∥∥ ∫ t1 0 (t1 −s)α−1f(s,y(s))ds− ∫ t2 0 (t2 −s)α−1f(s,y(s))ds ∥∥∥∥ ≤ ‖y∞‖ + m γ(α) | t1 1 + tα1 − t2 1 + tα2 | + ‖y0‖ γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + 1 γ(α) ∫ t1 0 |(t2 −s)α−1 − (t1 −s)α−1|‖f(s,y(s))‖ds + 1 γ(α) ∫ t2 t1 (t2 −s)α−1‖f(s,y(s))‖ds ≤ ‖y∞‖ + m γ(α) | t1 1 + tα1 − t2 1 + tα2 | + ‖y0‖ γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + 1 γ(α) ∫ t1 0 |(t2 −s)α−1 − (t1 −s)α−1|a(s)ds + r γ(α) ∫ t1 0 |(t2 −s)α−1 − (t1 −s)α−1|(1 + sα)b(s)ds + 1 γ(α) ∫ t2 t1 (t2 −s)α−1a(s)ds + r γ(α) ∫ t2 t1 (t2 −s)α−1(1 + sα)b(s)ds ≤ ‖y∞‖ + m γ(α) | t1 1 + tα1 − t2 1 + tα2 | + ‖y0‖ γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + a∗ + b∗r γ(α) (∫ t1 0 (t2 −s)α−1 − (t1 −s)α−1ds ) + a∗ + b∗r γ(α) ∫ t2 t1 (t2 −s)α−1ds + 2b∗r γ(α) (∫ t2 0 (t2 −s)α−1sαds− ∫ t1 0 (t1 −s)α−1sαds ) ≤ ‖y∞‖ + m γ(α) | t1 1 + tα1 − t2 1 + tα2 | + ‖y0‖ γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + a∗ + b∗r γ(1 + α) (tα2 − t α 1 − (t2 − t1) α) + a∗ + b∗r γ(1 + α) (t2 − t1)α + 2b∗rb(α,α + 1) γ(α) (t2α2 − t 2α 1 ), where a∗ = maxt∈[a,b] a(t) and b ∗ = maxt∈[a,b] b(t). as t2 → t1 the right-hand side of the above inequality tends to zero. then t2−αn(b)(t) 1+tα is equicontinuous on [a,b]. next we verify assertion (ii). let ε > 0, we heve ‖ t2−α1 n(y)(t1) 1 + tα1 − t2−α2 n(y)(t2) 1 + tα2 ‖≤ ‖y∞‖ + m γ(α) ∣∣∣∣ t11 + tα1 − t21 + tα2 ∣∣∣∣ + ‖y0‖γ(α− 1) ∣∣∣∣ 11 + tα1 − 11 + tα2 ∣∣∣∣ + 1 γ(α) ∥∥∥∥ ∫ t1 0 t2−α1 (t1 −s) α−1 1 + tα1 f(s,y(s))ds− ∫ t2 0 t2−α2 (t2 −s) α−1 1 + tα2 f(s,y(s))ds ∥∥∥∥ . int. j. anal. appl. 19 (2) (2021) 247 it is sufficient to prove that∥∥∥∥ ∫ t1 0 t2−α1 (t1 −s) α−1 1 + tα1 f(s,y(s))ds− ∫ t2 0 t2−α2 (t2 −s) α−1 1 + tα2 f(s,y(s))ds ∥∥∥∥ ≤ ε. relation (3.6) yields that there exits n0 > 0 such that (3.7) ∫ ∞ n0 ‖f(t,y(t))‖dt ≤ ε 3 for any y ∈ b. on the other hand, since limt→∞ t2−α(t−n0)α−1 1+tα = 0, there exists n1 > n0 such that, for any t1, t2 ≥ n1 and s ∈ [0,n0], we have (3.8) ∣∣∣∣t2−α2 (t2 −s)α−11 + tα2 − t 2−α 1 (t1 −s) α−1 1 + tα1 ∣∣∣∣ < ε3m . now taking t1, t2 ≥ n1, from (3.7), (3.8) we can arrive at∥∥∥∥ ∫ t1 0 t2−α1 (t1 −s) α−1 1 + tα1 f(s,y(s))ds− ∫ t2 0 t2−α2 (t2 −s) α−1 1 + tα2 f(s,y(s))ds ∥∥∥∥ ≤ ∫ n1 0 ∣∣∣∣t2−α2 (t2 −s)α−11 + tα2 − t 2−α 1 (t1 −s) α−1 1 + tα1 ∣∣∣∣‖f(s,y(s))‖ds + ∫ t1 n1 t2−α1 (t1 −s) α−1 1 + tα1 ‖f(s,y(s))‖ds + ∫ t2 n1 t2−α2 (t2 −s) α−1 1 + tα2 ‖f(s,y(s))‖ds < ε 3m ∫ ∞ 0 ‖f(s,y(s))‖ds + 2 ∫ ∞ n1 ‖f(s,y(s))‖ds < ε. therefore, we complete the proof of lemma 3.3. � lemma 3.4. [25] suppose that condition (h1) holds and b is a bounded subset of xα(j,e). then γxα(n(b)) = supt∈j γ ( t2−αn(b)(t) 1+tα ) . now we are in a position to give the main result of this work. let b = {y ∈ xα([0,∞),e) : ‖y‖α ≤ r}. theorem 3.1. suppose that conditions (h1), (h2) and (h3) are valid. if (h4) r > ‖y∞‖ + (α− 1)‖y0‖ + ∫∞ 0 a(t)dt γ(α) − ∫∞ 0 (1 + tα)b(t)dt . then the problem (1.1)-(1.3) has at least one solution. proof. first we transform problem (1.1)-(1.3) into a fixed point problem. consider the operator n : xα([0,∞),e) → xα([0,∞),e) defined by n(y)(t) = y∞ γ(α) tα−1 + y0 γ(α− 1) tα−2 − 1 γ(α) ∫ t 0 [tα−1 − (t−s)α−1]f(s,y(s))ds − 1 γ(α) ∫ ∞ t tα−1f(s,y(s))ds. int. j. anal. appl. 19 (2) (2021) 248 from lemma 3.1, the fixed points of n are solutions to (1.1)-(1.3). we shall show that n satisfies the assumptions of mönch fixed point theorem (theorem 2.1). then we can derive that n : b → b. indeed, for any y ∈ b, by condition (h1) we get ‖ t2−αn(y)(t) 1 + tα ‖≤ ‖y∞‖ γ(α) + ‖y0‖ γ(α− 1) + 1 γ(α) ∫ ∞ 0 ‖f(t,y(t))‖dt ≤ (1 γ(α) ( ‖y∞‖ + (α− 1)‖y0‖ + ∫ ∞ 0 a(t)dt + r ∫ ∞ 0 (1 + tα))b(t)dt ) < r. hence, from (h4) we have ‖ny‖α ≤ r, and we conclude that n : b → b. clearly b is a bounded, convex and closed subset of xα([0,∞),e), together with lemma 3.2 we know that n : b → b is continuous. finally we need to prove the following implication v ⊂ conv{n(v ) ∪{0}} =⇒ γxα(v ) = 0, for any v ⊂ b. let v ⊂ b such that v ⊂ conv{n(v ) ∪{0}} and t ∈ j, we choose ξ > 0 and n > 0 such that ξ < t < n. for each y ∈ v , we consider nξ,n(y)(t) = y∞ γ(α) tα−1 + y0 γ(α− 1) tα−2 + 1 γ(α) ∫ t ξ [tα−1 − (t−s)α−1]f(s,y(s))ds + 1 γ(α) ∫ n t (t−s)α−1f(s,y(s))ds. then from (h1), we obtain that t2−α 1 + tα ‖nξ,n(y)(t) −n(y)(t)‖≤ 1 γ(α) ∫ ξ 0 ‖f(t,y(t))‖dt + 1 γ(α) ∫ ∞ n ‖f(t,y(t))‖dt ≤ 1 γ(α) (∫ ξ 0 a(t)dt + r ∫ ξ 0 (1 + tα))b(t)dt + ∫ ∞ n a(t)dt + r ∫ ∞ n (1 + tα))b(t)dt ) , this shows that hd ( t2−αnξ,n(v )(t) 1+tα , t2−αn(v )(t) 1+tα ) → 0 as ξ → 0 and n → ∞, t ∈ j. where hd denotes the hausdorff metric in space e. by the prorerty of noncompactness mearure we get (3.9) lim ξ→0, n→∞ γ ( t2−αnξ,n(v )(t) 1 + tα ) = γ ( t2−αn(v )(t) 1 + tα ) . from lemma 3.3, the set {t 2−αn(v )(t) 1+tα }⊂ xα([0,∞),e) is equicontinuous on any compact of j. by (h1), il is easy to know that {f(.,y(.)) : y ∈ v} is equicontinuous on [ξ,n]. moreover {f(.,y(.)) : y ∈ v} is bounded int. j. anal. appl. 19 (2) (2021) 249 on [ξ,n], by (h1). using lemma 2.3, lemma 3.4 and (h3), we arrive at γ ( t2−αnξ,nv (t) 1 + tα ) ≤ 1 γ(α) ∫ n ξ (1 + tα)`(t)γ ( t2−αv (t) 1 + tα ) dt ≤ 1 γ(α) ∫ n ξ (1 + tα)`(t)γ ( t2−αn(v )(t) 1 + tα ) dt ≤ 1 γ(α) ∫ n ξ (1 + tα)`(t)γxα(n(v ))dt. from (3.9), we know that γ ( t2−αn(v )(t) 1 + tα ) ≤ 1 γ(α) ∫ ∞ 0 (1 + tα)`(t)γxα(n(v ))dt. thus γxα(n(v )) ≤ 1 γ(α) ∫ ∞ 0 (1 + tα)`(t)γxα(n(v ))dt. consequently, by condition (h3). we get γxα(n(v )) = 0; that is γxα(v ) = 0. from the theorem 2.1, we conclude that n has a fixed point y ∈ b which is a solution of problem (1.1)-(1.3). � 4. example we consider the following problem. (4.1) d 3 2 y(t) = ( √ tyn(t) (1 + t 3 2 )e10t + 2t) (1 + t2)2 )∞ n=1 , t ∈ j = (0, +∞), (4.2) i 1 2 0+ y(t) = y0, (4.3) d 1 2 0+ y(∞) = y∞. let e = {(y1, . . . ,yn, . . .) : sup |yn| < ∞}, with the norm ‖y‖ = supn |yn|, then e is a banach space and problem (4.1)-(4.3) can be regarded as a problem of the form (1.1)-(1.3), with α = 3 2 and f(t,y(t)) = (f(t,y1(t)), . . . ,f(t,yn(t)), . . .), where f(t,yn(t)) = √ tyn(t) (1 + t 3 2 )e10t + 2t (1 + t2)2 ,n ∈ n∗. int. j. anal. appl. 19 (2) (2021) 250 we shall verify the conditions (h1) − (h3). evidently, f is continuous in j ×e and ‖f(t,y(t))‖≤ √ t (1 + t 3 2 )e10t ‖y(t)‖ + 2t (1 + t2)2 . with the aid of simple computation we find that∫ ∞ 0 e−5tdt = 1 10 < γ( 3 2 ) and ∫ ∞ 0 2t (1 + t2)2 dt = 1 < ∞. finally, we verify condition (h3). for any bounded set b ⊂ e, we have f(t,b(t)) = √ t (1 + t 3 2 )e5t b(t) + { 2t (1 + t2)2 }. then γ(f(t,b(t)) ≤ √ t (1 + t 3 2 )e5t γ(b(t)). since ∫∞ 0 e−10tdt = 0.1 < γ( 3 2 ), we conclude that condition (h3) is satisfied. therefore, theorem 3.1 ensures that problem (4.1)-(4.3) has a solution. acknowledgments: the authors would like to express their thanks to the editor and anonymous referees for his/her comments that improved the quality of the paper. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. p. agarwal, b. hedia and m. beddani, structure of solution sets for implulsive fractional differential equations,fract. calc. appl. anal. 9 (2018), 15-36. [2] r. p. agarwal, m. meehan and d. o’regan, fixed point theory and applications , cambridge tracts in mathematics, 141 , cambridge university press, cambridge, 2001. [3] b. ahmad, existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, appl. math. lett. 23 (2010), 390-394. [4] a. arara, m. benchohra, n. hamidi and j.j. nieto, fractional order differential equations on an unbounded domain, nonlinear anal., theory methods appl. 72 (2010), 580-586. [5] z.b. bai and h.s. lü, positive solutions for boundary value problem of nonlinear fractional differential equation, j. math. anal. appl. 311 (2005), 495-505. [6] k. balachandran and j.y. park, nonlocal cauchy problem for abstract fractional semilinear evolution equations, nonlinear anal., theory methods appl. 71 (2009), 4471-4475. [7] k. balachandran and j.j. trujillo, the nonlocal cauchy problem for nonlinear fractional integrodifferential equations in banach spaces, nonlinear anal., theory methods appl. 72 (2010), 4587-4593. [8] k. balachandran, s. kiruthika and j.j. trujillo, existence results for fractional impulsive integrodifferential equations in banach spaces, commun. nonlinear sci. numer. simul. 16 (2011), 1970-1977. [9] m. benchohra, s. hamani and s.k. ntouyas, boundary value problems for differential equations with fractional order and nonlocal conditions, nonlinear anal., theory methods appl. 71 (2009), 2391-2396. int. j. anal. appl. 19 (2) (2021) 251 [10] l. byszewski, existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal cauchy problem, selected problems of mathematics, 50th anniv. cracow univ. technol. anniv. issue 6, cracow univ. technol. krakow, (1995), 25-33. [11] l. byszewski and v.lakshmikantham, theorem about the existence and uniqueness of a solution of a nonlocal abstract cauchy problem in a banach space, appl. anal. 40 (1991), 11-19. [12] g. christopher, existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, computers math. appl. 61(2011), 191-202. [13] w. cheung, j. ren, p.j.y. wong and d. zhao, multiple positive solutions for discrete nonlocal boundary value problems, j. math. anal. appl. 330 (2007), 900-915. [14] k. deng, exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, j. math. anal. appl. 179 (1993), 630-637. [15] a. m. a. el-sayed, nonlinear functional differential equations of arbitrary orders, nonlinear anal., theory methods appl. 33 (1998), 181-186. [16] a. m. a. el-sayed and a. g. ibrahim, multivalued fractional differential equations, appl. math. comput. 68 (1995), 15-25. [17] d.j. guo, v. lakshmikantham and x. liu, nonlinear integral equations in abstract spaces, kluwer academic publishers, dordrecht, 1996. [18] a.a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier b.v. amsterdam, 2006. [19] g. m. n’guérékata, a cauchy problem for some fractional abstract differential equation with non local conditions, nonlinear anal., theory methods appl. 70 (2009), 1873-1876. [20] i. podlubny, fractional differential equations, in: mathematics in science and engineering, vol. 198, academic press, new york, london, toronto, 1999. [21] h.a.h. salem, on the fractional calculus in abstract spaces and their applications to the dirichlet-type problem of fractional order, comput. math. appl. 59 (2010), 1278-1293. [22] h.a.h. salem, multi-term fractional differential equation in reflexive banach space, math. comput. model. 49 (2009), 829-834. [23] s.g. samko, a.a. kilbas and o.i. marichev, fractional integrals and derivatives: theory and applications, gordon and breach, yverdon, 1993. [24] s. szufla, on the application of measure of noncompactness to existence theorems, rend. sem. mat. univ. padova. 75 (1986), 1-14. [25] x. su, solutions to boundary value problem of fractional order on unbounded domains in a banach space, nonlinear anal., theory methods appl. 74 (2011), 2844-2852. [26] s. zhang, positive solutions for boundary-value problems of nonlinear fractional differential equations, electron. j. differ. equations, 2006 (2006), 36. 1. introduction 2. preliminary results 3. main result 4. example references international journal of analysis and applications volume 18, number 5 (2020), 718-723 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-718 spherical-radial multipliers on the heisenberg group m.e. egwe∗ department of mathematics, university of ibadan, ibadan, nigeria ∗corresponding author: murphy.egwe@ui.edu.ng abstract. let ihn be the (2n + 1)-dimensional heisenberg group. we consider a radial fourier multiplier which is a spherical function on ihn and show that it is a herz-schur multiplier. 1. introduction the theory of multipliers has grown over the years to yield several results and applications in virtually all aspects of analysis and mathematics in general. its use in harmonic analysis has assumed an enormous dimension. the theory was introduced on the heisenberg group by g. mauceri [13] and several other authors. recently, bagchi [2] revisited fourier multipiers on the hesisenberg group showing some variance of the results of [14] and [15]. a transference result of fourier multipliers from su(2) to the heisenberg group was considered by f.ricci [15]. the spherical functions form a large subject matter on this group [3], [1]. a construction of spherical radial functions on the heisenberg group was given in [5], [6] and [7]. the concept of schur multipliers or completely bounded functions has attained an exciting peak in harmonic analysis. however, the version of the result we shall consider in this work can be seen in [4]. received march 25th, 2020; accepted april 16th, 2020; published june 17th, 2020. 2010 mathematics subject classification. 43a90, 42a45, 22e46. key words and phrases. spherical-radial multipliers; herz-schur; heisenberg group. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 718 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-718 int. j. anal. appl. 18 (5) (2020) 719 2. main result definition 2.0.1 [2]: given a bounded measurable function m(η) on irn, we can define a transformation tm by setting (t̂mf) = m(η)f̂(η), f ∈ l2(irn). (1) by placherel theorem, tm is a bounded operator on l 2(irn). definition 2.0.1: let p ∈ [0,∞), if m is a continuous function on irn such that ∀ � > 0, the operators (m̂�f) = m(� −1n)f̂(n) (2) are uniformly bounded multiplier operator on lp(ttn), then m defines a bounded multiplier operator on lp(irn). when tm extends to l p(irn) as a bounded operator, we say that m (or equivalently tm) is a fourier multiplier for lp(irn). theorem 2.0.3 (hormander’s multiplier theorem): let k = [n 2 ] + 1 and m be of class ck away from the origin. if for any β ∈ inn satisfying |β| < k, we have sup r r|β|− n 2 (∫ irn |dβm(η)|2χ{r<|η|<2r}(η)dη )1/2 < ∞, (3) then m is a fourier multiplier for lp(irn) for 1 < p < ∞. in particular, if |dβm(η)| ≤ c|η|−|β|, then m is an lp-multiplier, 1 < p < ∞. 2.1 the heisenberg group (ihn) define the heisenberg group of dimension (2n + 1) by ihn = ic × ir equipped with the group law (z,t)(z′, t′) = (z + z′, t + t′ + 1 2 =z.z′), z.z′ = n∑ j=1 zj.z̄′j t ∈ ir, z ∈ ic. (4) this gives a two-step nilpotent lie group with centre given by z = {(0, t) : t ∈ ir}. (5) full details on the ubiquity of this group can be found in [12] [9], [17], [14]. for each µ > 0, we have two non-equivalent irreducible representations of ihn on the fock space f µ consisting of the entire functions f on icn such that ‖f‖2fµ = µ π ∫ ic |f(w)|2e−µ|w| 2 dw < ∞. (6) int. j. anal. appl. 18 (5) (2020) 720 these representations have the form [15] (ρµ(γ,t)f)(w) = eµ(it+γw+ 1 2 |γ|2)f(w + γ) (ρµ(γ,t)f)(w) = eµ(−it−γw+ 1 2 |γ|2)f(w − γ̄).   (7) the monomials ηλj (w) = ( µj j! )1/2 wj form a orthonormal basis for fµ and the matrix entries corresponding to the representations with respect to the monomials is given by τ ±µ ij (µ,t) = 〈ρ ±µ(γ,t)η µ i ,η µ j 〉. (8) now, let du = ( 1 2π2 ) dzdt denote the normalized haar measure on ihn. then, given an integrable function f on ihn and a nonzero real number µ, we have a countably infinite matrix with (i,j) entry given by f̂(µ,i,j) = ∫ ihn f(u)τ µ ij(u)du (9) with this normalisation and matrix entries, we obtain the plancherel formula given by∫ ihn |f(u)|2du = ∫ ∞ ∞ ∞∑ i,j=0 |f̂(µ,i,j)|2|µ|dµ. (10) we now give the following definition following hormander’s theorem. definition 2.1.1: let µ 6= 0 and m(µ) a countably infinite matrix with entries m(µ,i,j) that are measurable in µ for each i,j. we say that this induces a bounded multiplier on lp(ihn) if ‖mf‖p ≤ c‖f‖, (11) where (̂mf)(µ) = f̂(µ)m(µ) for some f in some dense subspace of lp(ihn). in what follows, we shall construct the spherical radial multipliers following [5], [6]. let ϕkλ be a k-spherical function on ihn. that is the distinguished spherical function restricted to l1(k \g/k) where (k,g) is a gelfand pair, k a compact subgroup of aut(ihn). in this case, g may be taken as a semi-direct product of k and ihn denoted as g := k o ihn [1]. now, recall that the heisenberg group heat equation defined on ihn × ir+ is given by ∂tu(u,t) = 4u(u,t), u(u,t) ∈ ihn × ir+. (12) the fundamental solution of (12) is given in [16] as kt(x,u,ξ) = cn ∫ ir eλee−tλ 2 ( λ sinh λt ) e 1 4 λ(coth tλ)(x.x+u.u)dλ, (13) int. j. anal. appl. 18 (5) (2020) 721 where cn = (4π) −n, λ ∈ ir∗ := ir\{0}. by a unique transformation of kt(x,u,ξ) given explicitly in [6], we obtain that kt(u) = cnt −n/2ϕkλ (u)δ −2 r (u)e |u|2 4t . (14) this gives a representations in (8) of ihn with respect to the dilations on the group. thus, (9) becomes f̂(µ,i,j) = ∫ ihn f(u)kλij(u)du, (15) where (from (14) we have) kλij = 〈ϕ k λ (ξ,t)η λ i ,η λ j 〉. (16) the spherical transforms of a function on ihn are then obtained and given as [1], [5]: f̃(λ,t) = ∫ ihn f(z,t)ϕkλ (z,t)dzdt (17) and f̃(0,ρ) = ∫ ihn f(z,t)jρ0 (z)dzdt, (18) where ϕkλ = e 2πiλte−2π|λ||z| 2 n∏ j=1 l0k(4π|λ||zj| 2), λ ∈ ir∗,k ∈ (z+)n (19) and jρ0 = n∏ j=1 j0(ρj · |zj|), ρ ∈ (ir+)n. (20) here, l0k is the laguerre polynomial of degree k and j0 is the bessel function (of first kind) of index 0. definition 2.1.2: let m = {m(λ) ∈ b(l2(irn)) : λ ∈ ir∗} be a family of operators. suppose that tm is the corresponding group fourier multiplier. also, let ϕkm (λ)be the spherical fourier multiplier associated with the parameter and operator m(λ). then, it becomes clear that tmf(z,t) = ∫ ir e−iλtϕk m(λ) (λ)fλ(t)dλ, for all f ∈ l1 ∩l2(ihn). this implies that tmf(z,t) = ∫ ir kλjkf(t)e −iλtdλ, f ∈ s (ihn) [1]. (21) definition 2.1.3: we shall say a matrix-valued function m(µ) = (m(µ,i,j))i,j∈n is a bounded fourier multiplier for ihn if m(., i,j) ∈ l∞(ir) for every i,j ∈ n, and if ‖m‖∞ = ess. sup‖m(µ)‖l(`2) < ∞. this definition together with 2.1.1 yield the following theorem as seen in [14]. theorem 2.1.4: if m is a bounded fourier multiplier on ihn, the requirement that t̂mf(µ) = f̂(µ)m(µ) int. j. anal. appl. 18 (5) (2020) 722 defines a bounded left invariant operator tm on l 2(ihn), with ‖tm‖l(l2(ihn)) = ‖m‖∞. conversely, for any bounded left-invariant operator t on l2(ihn), there is a bounded fourier multiplier m such that t = tm. definition 2.1.5: a function f : irn −→ ir is said to be radial if there is a function φ defined on [0,∞) such that f(x) = φ(|x|) for almost every x ∈ irn. simple and classical examples of radial functions and their properties can be seen in for example [10], [3], [6] and [11]. thus, given a k-spherical function, ϕkλ restricted to l 1(k \g/k) where (k,ihn) is a gelfand pair, k a compact subgroup of aut(ihn), then ϕ k λ is a unique radial function since it is a radial eigenfunction of 4ihn [6], [7]. thus, ϕkλ (u) = ψ(|u|) this forces forces (15) to become ϕkλ (u) = cnψ(e −iθ|u|, t) = cnk λ ij(|u|, t). this establishes (21). in fact, we have the following result [14]. prposition 2.1.6: kλjk is a unique radial function in span{ϕ k λ : |λ| ∈ ∑ }, where ∑ is the heisenberg fan, up to scalar multiples. definition 2.1.7: let ihn be the 2n + 1-dimensional heisenberg group. then f on ihn is said to be herz-schur, f ∈ b2(ihn) if there exist u,ν ∈ ihn such that f(u−1ν) = 〈ρµ1 (u), ρ µ 2 (ν)〉, (22) where ρ µ 1 and ρ µ 2 (ν) are irreducible unitary representations of ihn on l 2(ir). here, we assume that sup u∈ihn ‖ρµ1 (u)‖ < ∞ and sup ν∈ihn ‖ρµ2 (ν)‖ < ∞, where ‖ · ‖ is the fourier multiplier norm equivalent to the koranyi norm [8]. theorem 2.1.8 let tm be the group fourier multiplier on ihn acting on a k-bounded spherical function, f ∈ s (ihn). then, tm is a herz-schur multiplier on ihn. proof: following [4], any f can be expressed in the form given in (22). thus, if we consider (21) above, we readily see that up to scalar multiples, k µ ij is the unique radial function with ‖k µ ij‖≤ 1. thus, |tmf(z,t)| = ∣∣∣∣ ∫ ir k µ ijf(t)e −iµt dµ ∣∣∣∣ ≤ ∫ ir ∣∣∣〈ϕkλ (ξ,t)ηµi ,ηµj 〉f(t)e−iµtdµ∣∣∣ ≤ sup |ϕkλ (ξ,t)| ∫ ∣∣∣f(t)e−iµtdµ∣∣∣ = mk,λ‖f‖s (ihn). int. j. anal. appl. 18 (5) (2020) 723 since the representations of ihn are uniformly bounded on l 2 and tm is acting on k-bounded spherical functions, then the last expression shows that tm ∈ b2(ihn) and therefore herz-schur. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. astengo, d.b. blasio, and f. ricci. gelfand pairs on the heisenberg group and schwartz functions. j. funct. anal. 256 (5) (2009), 1565-1587. [2] s. bagchi. fourier multipliers on the heisenberg groups revisited arxiv:1710.02822v2 [math.ca], 2017. [3] c. benson, j. jenkins. bounded k-spherical functions on the heisenberg groups.j. funct. anal. 105 (1992), 409-443. [4] m. bozejko, g. fendler. herz-schur multipliers and uniformly bounded representations of discrete groups, arch. math. 57 (1991), 290-298. [5] m.e. egwe. aspects of harmonic analysis on the heisenberg group. ph.d. thesis, university of ibadan, ibadan, nigeria, 2010. [6] m.e. egwe, u.n. bassey. on isomorphism between certain group algebras on the heisenberg group, j. math. phys. anal. geom. 9 (2) (2013), 150-164. [7] m.e. egwe. a k-spherical-type solution for invariant differential operators on the heisenberg group. int. j. math. anal. 8 (30) 2014, 1475-1486. [8] m.e. egwe. the equivalence of certain norms on the heisenber group. adv. pure math. 3 (6) (2013), 576-578. [9] g.b. folland, harmonic analysis in phase space, princeton university press, princeton, n.j, 1989. [10] r. gangolli. spherical functions on semisimple lie groups. in: symmetric spaces, w. boothy and g. weiss (eds.). marcel dekker, inc. new york, 1972. [11] s. helgason, groups and geometric analysis: integral geometry, invariant differential operators, and spherical functions, academic press, orlando, 1984. [12] r. howe. on the role of the heisenberg group in harmonic analysis. bull. amer. math. soc. 3 (1980), 821-843. [13] g. mauceri. lp-multipliers on the heisenberg group. michigan math. j. 26 (1979), 361-371. [14] f. ricci. fourier and spectral multipliers in rn and in the heisenberg group. http://homepage.sns.it/fricci/papers/ multipliers.pdf [15] f. ricci, r.l. rubin. transferring fourier multipliers from su(2) to the heisenberg group. amer. j. math. 108 (3) (1986), 571-588. [16] s. thangavelu. spherical means on the heisenberg group and a restriction theorem for the symplectic fourier transform. revista math. iberoamericana 7 (2) (1991), 135-165. [17] s. thangavelu, harmonic analysis on the heisenberg group, birkhäuser boston, boston, ma, 1998. http://homepage.sns.it/fricci/papers/multipliers.pdf http://homepage.sns.it/fricci/papers/multipliers.pdf 1. introduction 2. main result 2.1 the heisenberg group (ihn) references international journal of analysis and applications volume 18, number 3 (2020), 421-438 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-421 study of the blow up of the maximal solution to the three-dimensional magnetohydrodynamic system in lei-lin-gevrey spaces ridha selmi1,2,3,∗, jamel benameur4 1department of mathematics, college of sciences, northern border university, p.o. box 1321, arar, 73222, ksa 2faculty of sciences of gabes, university of gabes, 6072, gabès, tunisia 3laboratory of partial differential equations and applications (lr03es04), faculty of sciences of tunis, university of tunis el manar, 1068 tunis, tunisia 4department of mathematics, issat gabès, university of gabès, tunisia ∗corresponding author: ridha.selmi@nbu.edu.sa abstract. in this paper, we investigate existence, uniqueness and blowup in finite time of the local solution to the three dimensional magnetohydrodynamic system, in gevrey-lei-lin spaces. to prove the blowup results and give the blow profile as a function of time, two key points are used. the first is a frequency decomposition of the spectrum of the initial data. this allows to use leray theory. the second is a technical lemma we proved to state that the lei-lin space is an interpolation space between the gevrey-lei-lin and the lebesgue square integrable functions spaces. to prove uniqueness, we use a penalization procedure and energy methods. about existence, we split the initial condition into low frequencies part and high frequencies part. the former are considered as initial data to the linear part of the system. the latter will be taken as small as needed, so that smallness theory applies and allows to run a fixed point argument. received february 8th, 2020; accepted february 28th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 35a01, 35a02, 35b40, 35b44, 35b45. key words and phrases. magnetohydrodynamic system; critical spaces; existence; uniqueness; blowup; gevrey-lein-lin space; frequency decomposition. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 421 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-421 int. j. anal. appl. 18 (3) (2020) 422 let us consider the following three dimensional incompressible magnetohydrodynamic system, (mhd)   ∂tu− ∆u + u ·∇u− b ·∇b + ∇(p + 12|b| 2) = 0, (t,x) ∈ r+ ×r3 ∂tb− ∆b + u ·∇b− b ·∇u = 0, (t,x) ∈ r+ ×r3 div u = 0, (t,x) ∈ r+ ×r3 div b = 0, (t,x) ∈ r+ ×r3 (u,b)(0) = (u0,b0), x ∈ r3, where u, b and p denote respectively the unknown velocity, the unknown magnetic field and the unknown pressure at the point (t,x). if the initial data u0 and b0 are quite regular, the divergence free conditions determine the pressure p. we aim to study the existence, uniqueness and blowup in finite time of local solution to the mhd system, in the framework of gevrey-lei-lin spaces. these spaces are defined for the real numbers a > 0, σ > 1 and ρ, by xρa,σ(r 3) = {f ∈s′(r3); f̂ ∈ l1loc(r 3) , ∫ r3 |ξ|ρea|ξ| 1/σ |f̂(ξ)|dξ < ∞} and endowed with its naturel norm ‖f‖xρa,σ(r3) = ∫ r3 |ξ|ρea|ξ| 1/σ |f̂(ξ)|dξ, where f̂ denotes the fourier transform of f. in [15], the authors defined the lei-lin space by x−1(r3) = {f ∈s′(r3), f̂ ∈ l1loc(r 3), ∫ r3 |f̂(ξ)| |ξ| dξ < ∞} and endowed with its natural norm ‖f‖x−1(r3) = ∫ r3 |f̂(ξ)| |ξ| dξ here, l1loc(r 3) states for the set of locally r3-lebesgue integrable distributions. in this critical space, the distinguishable fact was that to obtain global well-posedness to the navier-stokes equations, the norm of the initial data have to be exactly less than the viscosity of the fluid. however, in the wide fluid mechanic literature, it was always assumed that the initial data must be very small, especially smaller than the viscosity multiplied by a tiny positive constant. such assumption is mandatory to run the smallness argument and to obtain global well-posedness; see for example [11–14] and a complete survey in [10]. for many fluid mechanics equations, well-posedness and asymptotic behavior, as time goes to infinity or as small parameter goes to zero, were investigated by the authors, in various spaces; see for example [6–9, 18–20]. about blowup, it is worthwhile to emphasize that several authors studied this phenomena to the navier-stokes equations; see for example [1–3, 17] and references therein. the author observed, in [4], that in the case of the navier-stokes equations, the blowup phenomena depends on the chosen space not on the nonlinear part. to do so, he used fourier analysis in sobolev-gevrey spaces, for int. j. anal. appl. 18 (3) (2020) 423 sobolev index s > 3/2. his blowup result was improved later on, in [5], where the authors gave precisely an exponential type of the blowup profile, in sobolev-gevrey spaces but with less regularity on the initial data since they dealt with s = 1. in [16], authors studied the cauchy problem for a two-components high-order camassa-holm system. first, they proved the local well-posedness of the system in besov spaces. then, using littlewood-paley theory, they derived a blowup criterion for the strong solution. finally, they studied gevrey regularity and analyticity of the solutions to the camassa-holm system in the gevrey-sobolev spaces. in this paper, we begin by addressing the problem of local well-posedness. our result is summarized in the following existence and uniqueness theorem. theorem 0.1. let (u0,b0) ∈ (x−1a,σ(r3))2. then, there exist a time t > 0 and a unique solution (u,b) ∈ c([0,t], (x−1a,σ(r3))2) of (mhd), such that (u,b) ∈ l1([0,t], (x1a,σ(r3))2). to prove uniqueness of solution, we use a penalisation method. this allows to put the problem in a form where gronwall inequality can be applied. to establish existence of solution, we split the initial condition into low and high frequencies. the former will be considered as initial data to the linear part of the mhd system. the latter, taken as small as needed, will be the initial data to the remaining nonlinear part, for which smallness theory applies and allows to run a fixed point argument, in (x−1a,σ(r3))2. then, we turn to the blowup result that we state in the following theorem. theorem 0.2. let (u,b) ∈ c([0,t∗[, (x−1a,σ(r3))2) be the maximal solution of mhd system, where t∗ < ∞. then, there exists c0 > 0, such that lim inf t↗t∗ (t∗ − t)1/3e−c0(t ∗−t)− 1 3σ ‖(u,b)(t)‖x−1a,σ ≥ 1 2 . the proof is somewhat technical. our idea is to use a suitable frequency decomposition and to impose ”the problematic” large frequencies part to be a square integrable function, so that we fully profit from leray theory. some technical lemmas are specially derived to handle technical difficulties, mainly lemma 1.1, where we proved that x0 is an interpolation space between the lei-lin spaces xs and the space of lebesgue square integrable functions. the structure of our proof is as follows. starting with the energy estimates, we prove that the x−1a,σ norm of the solution blows up, in finite time. gronwall type inequality allows to infer that blowup holds also in x0a′,σ, for any a ′ ∈ (0,a). using a particular choice of parameter a′, we deduce that our solution blows up in the x0 norm, as a limit of x0a′,σ spaces. we split the initial data into two parts, a large frequencies one that belongs to x−1a,σ ∩l2 and a small frequencies remainder that leis in x−1a,σ. the smallness theory applies an leads to a global and continuous in time solution that belongs to x−1a,σ. this continuity plays an important role. about the large frequencies part, the l2 theory applies and allows to derive a leray type energy estimates. we use the above two estimates to dominate the x0 norm int. j. anal. appl. 18 (3) (2020) 424 of the solution. finally, lemma 1.1 with a judicious choice of the index s, finish the proof of the blowup result, in x−1a,σ and determine its profile as a function of time. this paper is organized as follows. in section 2, we give some notations and useful preliminary results. section 3 is devoted to prove existence and uniqueness of local in time solution. in section 4, we establish the blowup result. 1. technical lemmas in this section, we prove some technical lemmas that will be used later on. lemma 1.1. let s ≥ 3/2, there exists c > 0 such that, ∀f ∈ l2(r3) ∩xs(r3), ‖f‖x0 ≤ c‖f‖ s s+ 3 2 l2 ‖f‖ 3 2 s+ 3 2 xs . proof. for λ > 0, we consider the decomposition ‖f‖x0 = ∫ |ξ|<λ |f̂(ξ)|dξ︸ ︷︷ ︸ i(λ) + ∫ |ξ|>λ |f̂(ξ)|dξ︸ ︷︷ ︸ j(λ) . by cauchy-schwarz inequality for i(λ) and direct computation for j(λ), we get i(λ) ≤ √ 4πλ3/2‖f‖l2, j(λ) ≤ λ−s‖f‖xs. for a = √ 4π‖f‖l2 and b = ‖f‖xs, let ϕ(λ) = aλ3/2 + bλ−s. clearly the value λ0 = ( 2sb 3a ) 1 s+ 3 2 is a minimum of ϕ and for all s ≥ 3/2, ϕ(λ0) ≤ ca s s+ 3 2 b 3/2 s+ 3 2 . � lemma 1.2. let σ > 1, then there is a constant c > 0, such that for all f,g ∈x−1a,σ(r3), we have ‖fg‖x0a,σ ≤ c‖f‖x0a,σ‖g‖x0a σ ,σ + c‖g‖x0a,σ‖f‖x0a σ ,σ . proof. in a first step, using the triangular inequality and the fact that |ξ| ≤ max(|ξ −η|, |η|)(1 + min(|ξ −η|, |η|) max(|ξ −η|, |η|) ), we obtain ‖fg‖x0a,σ ≤ ∫ ξ ∫ η e a max(|ξ−η|,|η|)1/σ(1+ min(|ξ−η|,|η|) max(|ξ−η|,|η|) ) 1/σ |f̂(ξ −η)|.|ĝ(η)|dηdξ. in a second step, the inequality (1 + r)θ ≤ 1 + θrθ, ∀r ∈ [0, 1], ∀θ ∈]0, 1] leads to a max(|ξ −η|, |η|)1/σ(1 + min(|ξ−η|,|η|) max(|ξ−η|,|η|) ) 1/σ ≤ a max(|ξ −η|, |η|)1/σ + a σ min(|ξ −η|, |η|)1/σ. finally, distinguishing the two cases |ξ −η| < |η| and |ξ −η| > |η|, we obtain the desired result. � int. j. anal. appl. 18 (3) (2020) 425 by cauchy-schwarz inequality, we prove the following lemma. lemma 1.3. for all f ∈xs−1a,σ (r3) ∩xs+1a,σ (r3), with σ > 1 and a > 0, we have ‖f‖xsa,σ ≤‖f‖ 1/2 xs−1a,σ ‖f‖1/2 xs+1a,σ . using the fact that |ξ|1/σ ≤ (|ξ −η| + |η|)1/σ ≤ |ξ −η|1/σ + |η|1/σ, we have the lemma below. lemma 1.4. for all f ∈x0a,σ(r3), g ∈x1a,σ(r3), with σ > 1 and a > 0, we have ‖f.∇g‖x0a,σ ≤‖f‖x0a,σ‖g‖x1a,σ. lemma 1.5. let u,v ∈ l∞([0,t],x−1a,σ(r3)) ∩l1([0,t],x1a,σ(r3)). then ‖ ∫ t 0 e(t−τ)∆div(uv)dτ‖x−1a,σ ≤‖u‖ 1/2 l∞ t (x−1a,σ) ‖u‖1/2 l1 t (x1a,σ) ‖v‖1/2 l∞ t (x−1a,σ) ‖v‖1/2 l1 t (x1a,σ) . proof. first of all, let us prove that for f,g ∈x−1a,σ(r3) ∩x1a,σ(r3), we have ‖fg‖x0a,σ ≤‖f‖ 1/2 x−1a,σ ‖f‖1/2x1a,σ‖g‖ 1/2 x−1a,σ ‖g‖1/2x1a,σ. (1.1) to do so, we recall that ‖fg‖x0a,σ ≤ ∫ ξ ea|ξ| 1/σ (∫ η |f̂(ξ −η)||ĝ(η)|dη ) dξ. using the inequality ea|ξ| 1/σ ≤ ea|ξ−η| 1/σ ea|η| 1/σ , we obtain ‖fg‖x0a,σ ≤ ∫ ξ ( ∫ η ea|ξ−η| 1/σ |f̂(ξ −η)|.ea|η| 1/σ .|ĝ(η)|dη)dξ. put f(ξ) = ea|ξ| 1/σ |f̂(ξ)| and g(ξ) = e a|ξ|1/σ |ξ| |f̂(ξ)|. it holds that, ‖fg‖x0a,σ ≤ ‖f ∗g‖l1 ≤ ‖f‖l1‖g‖l1 ≤ ‖f‖x0a,σ‖g‖x0a,σ. by cauchy-schwarz inequality, we get (1.1). to continue proving lemma above, we have∫ t 0 ‖e(t−τ)∆div(uv)dτ‖x−1a,σ ≤‖ ∫ t 0 e(t−τ)∆div(uv)‖x−1a,σdτ ≤ ∫ t 0 ‖uv)‖x0a,σdτ ≤ ∫ t 0 ‖u‖x0a,σ‖v‖x0a,σdτ ≤ ∫ t 0 ‖u‖1/2 x−1a,σ ‖u‖1/2x1a,σ‖v‖ 1/2 x−1a,σ ‖v‖1/2x1a,σdτ ≤‖u‖1/2 l∞ t (x−1a,σ) ‖v‖1/2 l∞ t (x−1a,σ) ∫ t 0 ‖u‖1/2x1a,σ‖v‖ 1/2 x1a,σ dτ ≤‖u‖1/2 l∞ t (x−1a,σ) ‖v‖1/2 l∞ t (x−1a,σ) ‖u‖1/2 l1 t (x1a,σ) ‖v‖1/2 l1 t (x1a,σ) . � int. j. anal. appl. 18 (3) (2020) 426 lemma 1.6. let u,v ∈ l∞t (x −1 a,σ(r 3)) ∩l1t (x 1 a,σ(r 3)). then ‖ ∫ t 0 e(t−τ)∆div(uv)dτ‖l1 t (x1a,σ) ≤‖u‖ 1/2 l∞ t (x−1a,σ) ‖v‖1/2 l∞ t (x−1a,σ) ‖u‖1/2 l1 t (x1a,σ) ‖v‖1/2 l1 t (x1a,σ) . proof. using the definition of x1a,σ norm and integrating the function e−(t−τ)|ξ| 2 twice with respect to τ ∈ [0, t] and t ∈ [0,t], it follows that ∫ t 0 ‖ ∫ t 0 e(t−τ)∆div(uv)dτ‖x1a,σdt ≤ ∫ t 0 ∫ t 0 ∫ r3 e−(t−τ)|ξ| 2 |ξ|2ea|ξ| 1/σ |ûv(τ,ξ)|dτdtdξ ≤ ∫ r3 |ξ|2ea|ξ| 1/σ (∫ t 0 ∫ t 0 e−(t−τ)|ξ| 2 |ûv(τ,ξ)|dτdt ) dξ ≤ ∫ r3 |ξ|2ea|ξ| 1/σ  ∫ t 0 |ûv(τ,ξ)|  [−e−(t−τ)|ξ|2 |ξ|2 ]t τ  dτ  dξ ≤ ∫ r3 |ξ|2ea|ξ| 1/σ (∫ t 0 |ûv(τ,ξ)|( 1 −e−(t−τ)|ξ| 2 |ξ|2 )dτ ) dξ ≤ ∫ t 0 ‖uv‖x0a,σdτ. using equation (1.1) finishes the proof. � 2. well-posedness results to prove uniqueness of solution to the (mhd), we consider two solutions (u1,b1) and (u2,b2) that belong to (c([0,t],x−1a,σ(r3)))2 ∩ (l1([0,t],x1a,σ(r3)))2 and have the same initial data. let δ = u1 − u2 and η = b1 − b2, it follows that  ∂tδ − ∆δ + δ ·∇u2 + u1 ·∇δ −η ·∇b2 − b1 ·∇η + ∇(p1 −p2 + 12 (|b1| 2 −|b2|2)) = 0 ∂tη − ∆η + u2 ·∇η −η ·∇u1 + δ ·∇b1 − b2 ·∇δ = 0 div δ = 0 div η = 0 (δ,η) = (0, 0). (2.1) taking the fourier transform and using divergence free condition, one infers that ∂tδ̂ + |ξ|2δ̂ + ̂(δ ·∇u2) + ̂(u1 ·∇δ) − ̂(η.∇b2) − ̂(b1 ·∇η) = 0 (2.2) ∂tη̂ + |ξ|2η̂ + ̂(u2 ·∇η) − ̂(η ·∇u1) + ̂(δ ·∇b1) − ̂(b2 ·∇δ) = 0. (2.3) multiplying (2.2) (respectively (2.3)) by δ̂ (respectively η̂) and its conjugate by δ̂ (respectively η̂), summing up the four resulting equations, and dominating the real part of any complex quantity by its modulus, we int. j. anal. appl. 18 (3) (2020) 427 obtain 1 2 ∂t(|δ̂|2 + |η̂|2) + |ξ|2(|δ̂|2 + |η̂|2) ≤ | ̂(δ.∇u2)||δ̂| + | ̂(u1.∇δ)||δ̂| + | ̂(η.∇b2)||δ̂| + | ̂(b1.∇η)||δ̂| + | ̂(u2.∇η)||η̂| + | ̂(η.∇u1)||η̂| + 2| ̂(δ.∇b1)||η̂| + | ̂(b2.∇δ)||η̂|. (2.4) let ε > 0 be a penalizing parameter. one has ∂t(|δ̂|2 + |η̂|2) = ∂t(|δ̂|2 + |η̂|2 + ε2) = 2 √ |δ̂|2 + |η̂|2 + ε2.∂t √ |δ̂|2 + |η̂|2 + ε2. (2.5) substituting (2.5) in (2.4), dividing by √ |δ̂|2 + |η̂|2 + ε2, integrating with respect to time, letting ε → 0 and using that |δ̂|+|η̂|√ 2 ≤ √ |δ̂|2 + |η̂|2, we infer that |δ̂| + |η̂| + ∫ t 0 |ξ|2(|δ̂| + |η̂|)dτ ≤ √ 2( ∫ t 0 (| ̂(δ.∇u2)||δ̂| + | ̂(u1.∇δ)||δ̂| + | ̂(η.∇b2)||δ̂| + | ̂(b1.∇η)||δ̂|)dτ + ∫ t 0 (| ̂(u2.∇η)||η̂| + | ̂(η.∇u1)||η̂| + 2| ̂(δ.∇b1)||η̂| + | ̂(b2.∇δ)||η̂|dτ). multiplying by e a|ξ| 1 σ |ξ| and integrating with respect to ξ. by divergence free, we get ‖δ‖x−1a,σ + ‖η‖x−1a,σ + ∫ t 0 (‖∆δ‖x−1a,σ + ‖∆η‖x−1a,σ )dτ ≤ √ 2( ∫ t 0 ‖δu2‖x0a,σ + ‖u1δ‖x0a,σ + ‖ηb2‖x0a,σ + ‖b1η‖x0a,σdτ + ∫ t 0 ‖u2η‖x0a,σ + ‖ηu1‖x0a,σ + ‖δb1‖x0a,σ + ‖b2δ‖x0a,σdτ). using the product young inequality, it follows that ‖δu2‖x0a,σ ≤ ‖δ‖x0a,σ‖u2‖x0a,σ ≤ ‖δ‖ 1 2 x−1a,σ ‖∆δ‖ 1 2 x−1a,σ ‖u2‖ 1 2 x−1a,σ ‖∆u2‖ 1 2 x−1a,σ ≤ 2‖δ‖x−1a,σ‖u2‖x−1a,σ‖∆u2‖x−1a,σ + 1 2 ‖∆δ‖x−1a,σ, and so on for the other terms. thus, ‖δ‖x−1a,σ + ‖η‖x−1a,σ ≤ 2 √ 2 ∫ t 0 (‖δ‖x−1a,σ + ‖η‖x−1a,σ ) ∑ 1≤i≤2 ‖ui‖x−1a,σ‖∆ui‖x−1a,σ + ‖bi‖x−1a,σ‖∆bi‖x−1a,σdτ. since the function t 7→ ∑ 1≤i≤2 ‖ui‖x−1a,σ‖∆ui‖x−1a,σ + ‖bi‖x−1a,σ‖∆bi‖x−1a,σ belongs to l 1([0,t]), gronwall inequality implies that δ = 0 on [0,t]. thus, uniqueness holds. we turn to the existence result. to do so, let r ∈ (0, 1 10 ), such that (‖(u0,b0)‖x−1a,σ + r) 1/2r1/2 < 1 32 , int. j. anal. appl. 18 (3) (2020) 428 and n ∈ n, such that ∫ |ξ|>n ea|ξ| 1/σ |ξ| |û0(ξ)|dξ + ∫ |ξ|>n ea|ξ| 1/σ |ξ| |b̂0(ξ)|dξ < r 5 . let v0 = f−1(1{|ξ|<n}û0(ξ)) (respectively c0 = f−1(1{|ξ|<n}b̂0(ξ))) the low frequencies part of u0 (respectively b0) and w0 (respectively d0) its high frequencies part corresponding to (|ξ| > n). clearly, one has ‖(d0,w0)‖x−1a,σ < r 5 . (2.6) by a standard fourier computation, one infers that (v,c) = eνt∆(v0,c0) is the unique solution to the following linear system (mhdl)   ∂tv − ∆v = 0, (t,x) ∈ r+ ×r3 ∂tc− ∆c = 0, (t,x) ∈ r+ ×r3 div v = 0, (t,x) ∈ r+ ×r3 div c = 0, (t,x) ∈ r+ ×r3 (v,c)(0) = (v0,c0), x ∈ r3 and that for all t ≥ 0, ‖(v,c)‖x−1a,σ ≤‖(u 0,b0)‖x−1a,σ. (2.7) by definition of x1a,σ norm, expression of (v,c) and tonelli’s theorem, we infers that ‖(v,c)‖l1 t (x1a,σ) ≤ ∫ r3 (1 −e−νt|ξ| 2 )|ξ|−1ea|ξ| 1/σ (|û0(ξ)| + |b̂0(ξ)|)dξ. using the dominated convergence theorem, we get lim t→0+ ‖(v,c)‖l1 t (x1a,σ) = 0. (2.8) as it will be seen below, for instance, the stability condition of the fixed point argument requires a choice of ε > 0 such that ε1/2‖u0‖1/2 x−1a,σ < 1 18 . moreover, for the operator ψ to be a contraction mapping, ε has to fulfill the supplementary condition (‖(u0,b0)‖x−1a,σ + r) 1/2ε1/2 < 1 32 . for this choice of ε, by (2.8), there exists a time t = t(ε) > 0 such that ‖v‖l1 t (x1a,σ) < ε. (2.9) int. j. anal. appl. 18 (3) (2020) 429 put w = u−v and d = b−c, the 3+3 components vector (w,d) satisfies, for all (t,x), the following nonlinear system denoted (mhdnl),   ∂tw − ∆w + (w + v) ·∇(w + v) − (d + c) ·∇(d + c) + ∇(p + 12|d + c| 2) = 0 ∂td− ∆d + (w + v) ·∇(d + c) − (d + c) ·∇(w + v) = 0 div w = 0 div d = 0 (u,b) = (u0,b0). to run a fixed point argument, we introduce the following operator ψ defined for all (w,d)t by the right hand side of the following integral equation  w d   = eνt∆   w0 d0  −∫ t 0 eν(t−τ)∆   (w + v) ·∇(w + v) − (d + c) ·∇(d + c) (w + v) ·∇(d + c) − (d + c) ·∇(w + v)  dτ, and we consider the space xt := c([0,t], (x−1a,σ(r3))2) ∩ l1([0,t], (x1a,σ(r3))2), endowed with its naturel norm ‖f‖xt := ‖f‖l∞ t ((x−1a,σ)2) + ‖f‖l1t ((x1a,σ)2). in a first step, let us prove that xt is stable under the operator ψ. to do so, we denote by br the subset of xt defined by br = {(u,b) ∈xt ;‖(u,b)‖l∞ t (x−1a,σ) ≤ r;‖(u,b)‖l1t (x1a,σ) ≤ r}. for (w,d) ∈ br, we have ψ((w,d)) ∈ br. in fact, it holds that ‖ψ(w,d)(t)‖x−1a,σ ≤ iww + iwv + ivw + ivv + idd + idc + icd + icc + iwd + iwc + ivd + ivc + idw + idv + icw + icv, (2.10) where we denoted, for any divergence free vector field υ and ω, iυω = ∫ t 0 ‖e(t−τ)∆υ ·∇ω‖x−1a,σdτ. to estimate ‖ψ(w,d)(t)‖x−1a,σ , we recall that according to the choice of n, we have ‖w0‖x−1a,σ + ‖d 0‖x−1a,σ < r 9 . using divergence free condition and lemma 1.5, we obtain that ivv ≤‖v‖l∞ t (x−1a,σ)‖v‖l1t (x1a,σ) ≤ ε‖u 0‖x−1a,σ < r 18 . the same holds for icc, ivc and icv. moreover, iww ≤‖w‖l∞ t (x−1a,σ)‖w‖l1t (x1a,σ) ≤ r 2 < r 18 , and the same holds for idd, iwd and idw. furthermore, ivw ≤‖v‖ 1/2 l∞ t (x−1a,σ) ‖v‖1/2 l1 t (x1a,σ) ‖w‖1/2 l∞ t (x−1a,σ) ‖w‖1/2 l1 t (x1a,σ) ≤ rε1/2‖u0‖1/2 x−1a,σ < r 18 , int. j. anal. appl. 18 (3) (2020) 430 and the same holds for the seven remaining integrals. finally, we obtain ‖ψ(w,d)(t)‖x−1a,σ ≤ r. (2.11) let us estimate ‖ψ(w)(t)‖l1(x1a,σ). as above, we have ‖ψ(w)(t)‖l1(x1a,σ) ≤ jww + jwv + jvw + jvv + jdd + jdc + jcd + jcc + jwd + jwc + jvd + jvc + jdw + jdv + jcw + jcv, (2.12) where we denoted, for any divergence free vector field υ and ω, jυω = ∫ t 0 ‖ ∫ t 0 e(t−τ)∆υ ·∇ωdτ‖x1a,σdt. by the facts that ‖v‖l1 t (x1a,σ),‖c‖l1t (x1a,σ) < ε, we can take ‖v‖l1 t (x1a,σ) + ‖c‖l1t (x1a,σ) < r 9 . using lemma 1.6 and the fact that w ∈ br, we get jvv ≤‖v‖l∞ t (x−1a,σ)‖v‖l1t (x1a,σ) ≤ ε‖u 0‖x−1a,σ < r 18 and so on for jcc, jvc and jcv. also, we get jww ≤‖w‖l∞ t (x−1a,σ)‖w‖l1t (x1a,σ) ≤ r 2 < r 18 and so on for jdd, jwd and jdw. moreover, jvw ≤‖v‖ 1/2 l∞ t (x−1a,σ) ‖v‖1/2 l1 t (x1a,σ) ‖w‖1/2 l∞ t (x−1a,σ) ‖w‖1/2 l1 t (x1a,σ) ≤ rε1/2‖u0‖1/2 x−1a,σ < r 18 and so on for the seven remaining integrals. thus, ‖ψ(w,d)(t)‖l1(x1a,σ) ≤ r. (2.13) combining (2.11) and (2.13), we deduce that ψ(br) ⊂ br. in a second step, to prove that ψ is a contraction mapping on br. one has ψ(w2,d2) − ψ(w1,d1) = − ∫ t 0 e(t−τ)∆   αww + αdd αwd + αdw  dτ, where αww = (w2 + v) ·∇(w2 −w1) − (w2 −w1) ·∇(w1 + v) αdd = −(d2 + c) ·∇(d2 + c) + (d1 + c) ·∇(d1 + c) αwd = (w2 + v) ·∇(d2 + c) − (w1 + v) ·∇(d1 + c) αdw = −(d2 + c) ·∇(w2 + v) + (d1 + c) ·∇(w1 + v). int. j. anal. appl. 18 (3) (2020) 431 or equivalently, in an adequate form to be estimated, one has αww = (w2 + v) ·∇(w2 −w1) − (w2 −w1) ·∇(w1 + v) αdd = −(d2 + c) ·∇(d2 −d1) + (d2 −d1) ·∇(d1 + c) αwd = (w2 −w1) ·∇(d2 + c) + (w1 −v) ·∇(d2 −d1) αdw = −(d2 −d1) ·∇(w2 + v) − (d1 + c) ·∇(w2 −w1). it follows that ‖ψ((w2,d2)) − ψ((w1,d1))‖x−1a,σ ≤ 2∑ i=1 k(i)ww + k (i) dd + k (i) wd + k (i) dd , where k(1)ww = ‖ ∫ t 0 e(t−τ)∆(w2 + v)∇(w2 −w1)dτ‖x−1a,σ, k(2)ww = ‖ ∫ t 0 e(t−τ)∆(w2 −w1)∇(w1 + v)dτ‖x−1a,σ, and so on for the other integrals. using lemma 1.5, triangle inequality, the fact that wi belongs to br, inequalities (2.7) and (2.9), we infer that k (1) ww, k (2) ww ≤‖v + w2‖ 1/2 l∞ t (x−1a,σ) ‖v + w2‖ 1/2 l1 t (x1a,σ) ‖w2 −w1‖ 1/2 l∞ t (x−1a,σ) ‖w2 −w1‖ 1/2 l1 t (x1a,σ) ≤ (‖u0‖x−1a,σ + r) 1/2(ε + r)1/2‖w2 −w1‖xt ≤ (‖(u0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2‖w2 −w1‖xt . similarly, k (1) dd , k (2) dd ≤ (‖(u 0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2‖d2 −d1‖xt k (1) wd, k (2) dw ≤ (‖(u 0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2‖w2 −w1‖xt k (2) wd, k (1) dw ≤ (‖(u 0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2‖d2 −d1‖xt . thus, ‖ψ(w2,d2) − ψ(w1,d1)‖l∞ t (x−1a,σ) ≤ 4(‖(u0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2(‖w2 −w1‖xt + ‖d2 −d1‖xt ). (2.14) to estimate the l1t (x 1 a,σ) norm, we proceed as above; ‖ψ((w2,d2)) − ψ((w1,d1))‖l1(x1a,σ) ≤ 2∑ i=1 l(i)ww + l (i) dd + l (i) wd + l (i) dd, where l(1)ww = ∫ t 0 ‖ ∫ t 0 e(t−τ)∆(w2 + v)∇(w2 −w1)dτ‖x1a,σdt l(2)ww = ∫ t 0 ‖ ∫ t 0 e(t−τ)∆(w2 −w1)∇(w1 + v)dτ‖x1a,σdt, int. j. anal. appl. 18 (3) (2020) 432 and so on for the other integrals. using lemma 1.6, triangle inequality, the fact that wi belongs to br, inequalities (2.7) and (2.9), we infer that l(1)ww, l (2) ww ≤ (‖(u 0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2‖w2 −w1‖|xt . the same holds for l (i) dd,l (i) wd and l (i) dw, and one obtains ‖ψ(w2,d2) − ψ(w1,d1)‖l1 t (x1a,σ) ≤ 4(‖(u0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2(‖w2 −w1‖xt + ‖d2 −d1‖xt ). (2.15) by (2.14) and (2.15), we infer that ‖ψ(w2,d2) − ψ(w1,d1)‖xt ≤ 8(‖(u0,b0)‖x−1a,σ + r) 1/2(ε + r)1/2(‖w2 −w1‖xt + ‖d2 −d1‖xt ). this implies that ‖ψ(w2,d2) − ψ(w1,d1)‖xt ≤ 1 2 (‖w2 −w1‖xt + ‖d2 −d1‖xt ) (2.16) and ψ is a contraction mapping. the fixed point theorem implies that there is a unique (w,d) ∈ br, such that (u,b) = (v + w,c + d) is the solution of (mhd) with (u,b) ∈xt . 3. blowup results in this section we prove theorem 0.2. first of all, the following energy estimates holds in x−1a,σ, ‖(u,b)(t)‖x−1a,σ + ∫ t 0 ‖(u,b)(t)‖x1a,σ ≤ ‖(u0,b0)‖x−1a,σ + ∫ t 0 luu(τ) + lbb(τ) + lub(τ) + lbu(τ)dτ, where lυω = ‖div (υω)‖x−1a,σ . by lemma 1.2, we have lυω ≤ ‖υ‖x0a,σ‖ω‖x0a σ ,σ + ‖ω‖x0a,σ‖υ‖x0a σ ,σ ≤ 2‖(υ,ω)‖x0a,σ‖(u,b)‖x0a σ ,σ . it follows that ‖(u,b)(t)‖x−1a,σ + ∫ t 0 ‖(u,b)(t)‖x1a,σ ≤‖(u0,b0)‖x−1a,σ + 8 ∫ t 0 ‖(u,b)‖x0a σ ,σ ‖(u,b)‖x0a,σdτ. using lemma 1.3 and product young inequality, we obtain ‖(u,b)(t)‖x−1a,σ + 1 2 ∫ t 0 ‖(u,b)(z)‖x1a,σdz ≤‖(u0,b0)‖x−1a,σ + 32 ∫ t 0 ‖(u,b)‖2x0a σ ,σ ‖(u,b)‖x−1a,σdτ. (3.1) int. j. anal. appl. 18 (3) (2020) 433 however, a direct computation implies that ‖(u,b)‖x0a σ ,σ ≤ m0‖(u,b)‖x−1a,σ, where the constant m = m0(a,σ) = supr≥0 re −a( 1 σ −1)r1/σ . then, estimation (3.1) becomes ‖(u,b)(t)‖x−1a,σ + 1 2 ∫ t 0 ‖(u,b)(z)‖x1a,σdz ≤‖(u0,b0)‖x−1a,σ + 32m 2 0 ∫ t 0 ‖(u,b)‖3x−1a,σdz. let t = sup{t ∈ [0,t∗[; supz∈[0,t] ‖(u,b)(z)‖x−1a,σ ≤ 2‖(u0,b0)‖x−1a,σ}. by continuity of (u,b), we have t ∈ ]0,t∗] and 2‖(u0,b0)‖x−1a,σ ≤‖(u0,b0)‖x−1a,σ + 128m 2 0 t‖(u0,b0)‖ 3 x−1a,σ . we infer that (128m20 ) −1 ‖(u0,b0)‖2x−1a,σ ≤ t ≤ t∗. if we consider the magnetohydrodynamic system starting at initial time t, with the data (u,b)(t) , we get (128m20 ) −1 ‖(u,b)(t)‖2 x−1a,σ ≤ t∗ − t, or equivalently (128m20 ) −1 t∗ − t ≤‖(u,b)(t)‖2x−1a,σ. therefore, we infers that lim t→t∗ ‖(u,b)(t)‖x−1a,σ = +∞. (3.2) applying gronwall inequality to inequality (3.1), we get ‖(u,b)(t)‖x−1a,σ + 1 2 ∫ t 0 ‖(u,b)(t)‖x1a,σ ≤‖(u0,b0)‖x−1a,σ exp(32 ∫ t 0 ‖(u,b)‖2x0a σ ,σ dz). by equation (3.2), we obtain ∫ t∗ 0 ‖(u,b)‖2x0a σ ,σ dz = ∞. (3.3) this implies that lim t→t∗ ‖(u,b)(t)‖x0a σ ,σ = +∞. (3.4) at this point, we proved that x0a σ ,σ norm of the solution blows up, in finite time. let a′ = a σ ∈ (0,a), using the same method, we obtain ‖(u,b)(t)‖x0 a′,σ + ∫ t 0 ‖(u,b)(t)‖x2 a′,σ ≤ ‖(u0,b0)‖x0 a′,σ + ∫ t 0 ruu(z) + rbb(z) + rub(z) + rbu(z)dz, where rxy(t) = ‖x.∇y‖x0 a′,σ . using lemma 1.4, we get rxy(t) ≤ ‖x‖x0 a′,σ ‖y‖x1 a′,σ ≤ ‖(x,y)‖x0 a′,σ ‖(x,y)‖x1 a′,σ . int. j. anal. appl. 18 (3) (2020) 434 it follows that ‖(u,b)(t)‖x0 a′,σ + ∫ t 0 ‖(u,b)(t)‖x2 a′,σ ≤‖(u0,b0)‖x0 a′,σ + 4 ∫ t 0 ‖(u,b)‖x0 a′,σ ‖(u,b)‖x1 a′,σ dz. by lemma 1.3, we obtain ‖(u,b)(t)‖x0 a′,σ + ∫ t 0 ‖(u,b)‖x2 a′,σ ≤‖(u0,b0)‖x0 a′,σ + 4 ∫ t 0 ‖(u,b)‖3/2x0 a′,σ ‖(u,b)‖1/2x2 a′,σ . using product young inequality, we get ‖(u,b)(t)‖x0 a′,σ + 1 2 ∫ t 0 ‖(u,b)‖x2 a′,σ ≤‖(u0,b0)‖x0 a′,σ + 4 ∫ t 0 ‖(u,b)‖3x0 a′,σ . gronwall lemma gives ‖(u,b)(t)‖x0 a′,σ ≤‖(u0,b0)‖x0 a′,σ exp ( 4 ∫ t 0 ‖(u,b)‖2x0 a′,σ dz ) . or equivalently, 8‖(u,b)(t)‖2x0 a′,σ exp ( − 8 ∫ t 0 ‖(u,b)‖2x0 a′,σ dz ) ≤ 8‖(u0,b0)‖2x0 a′,σ . integrating over [0,t∗) and using (3.4), we infer that 1 ≤ 8‖(u0,b0)‖2x0 a′,σ t∗. since a σn < ... < a σ < a, by the same method we used for a′ ∈ (0,a), we prove that 1 ≤ 8‖(u0,b0)‖2x0a σn ,σ t∗, ∀n ∈ n∗. by dominate convergence theorem, we obtain 1 2 √ 2 √ t∗ ≤‖(u0,b0)‖x0. consider the (mhd) system starting at t ∈ [0,t∗), by time translation, we have 1 2 √ 2 √ t∗ − t ≤‖(u,b)(t)‖x0. (3.5) at this point, we proved that the x0 norm of the solution blows up, in finite time. let k ∈ n∗, we consider the subset ak defined by ak = {ξ ∈ r3; |ξ| ≤ k and |û0(ξ)| ≤ k} and v0 and c0 in l2(r3)∩x−1a,σ(r3), such that (v0k,c 0 k) = f −1 ( 1ak(ξ)((û 0, b̂0) ) . let (w0k,d 0 k) = (u 0 −v0k,b 0 − c0k), one has limk→∞‖(w 0 k,d 0 k)‖x−1a,σ = 0. so, there exists k ∈ n, such that ‖(w 0 k,d 0 k)‖x−1a,σ < 1 16 . using int. j. anal. appl. 18 (3) (2020) 435 smallness theory, we prove that a unique and global in time solution (wk,dk) to the system (mhd1)   ∂tw − ∆w + w ·∇w −d ·∇d = −∇π1, in r+ ×r3 ∂td− ∆d + w ·∇d−d ·∇w = 0, in r+ ×r3 div w = div d = 0, in r+ ×r3 (w,d)(0,x) = (w0k,d 0 k)(x), in r 3, exists in cb(r+,x−1a,σ(r3)) ∩l1(r+,x1a,σ(r3)) and satisfies for t ≥ 0, ‖(wk,dk)(t)‖x−1a,σ + 1 2 ∫ t 0 ‖(wk,dk)(z)‖x1a,σdz ≤‖(w 0 k,d 0 k)‖x−1a,σ. (3.6) consider (vk,ck) = (u−wk,b− ck), it belongs to c([0,t∗),x−1a,σ(r3)) and satisfies, for all (t,x) ∈ r × r3, the following (mhd2) system,  ∂tvk − ∆vk + vk ·∇vk + vk ·∇wk + wk ·∇vk − ck ·∇ck − ck ·∇dk −dk ·∇ck = −∇π2 ∂tck − ∆ck + vk ·∇ck + vk ·∇dk + wk ·∇ck − ck ·∇vk − ck ·∇wk −dk ·∇vk = 0 div w = div d = 0 (vk,ck)(0,x) = (v 0 k,c 0 k)(x). having that (v0k,c 0 k) ∈ l 2(r3), we take the scalar product and use l2 theory. under divergence free condition, we infers that 1 2 d dt ‖(vk,ck)(t)‖2l2 + ‖(∇vk,∇ck)(t)‖ 2 l2 ≤ c‖(dk,wk)‖ 2 l∞‖(vk,ck)(t)‖ 2 l2. the gronwall lemma implies that ‖(vk,ck)(t)‖2l2 + ∫ t 0 ‖(∇vk,∇ck)(z)‖2l2dz ≤‖(v 0 k,c 0 k)‖ 2 l2 exp(c ∫ t 0 ‖(wk,dk)‖2l∞dz). using that ‖f‖l∞ ≤‖f̂‖l1 = ‖f‖x0 ≤‖f‖ 1/2 x−1‖f‖ 1/2 x1 , we obtain ‖(vk,ck)(t)‖2l2 + ∫ t 0 ‖(∇vk,∇ck)(z)‖2l2dz ≤ ‖(v0k,c 0 k)‖ 2 l2 e c ∫ t 0 ‖(wk,dk)‖x−1‖(wk,dk)‖x1dz ≤ ‖(v0k,c 0 k)‖ 2 l2 e c ∫ t 0 ‖(wk,dk)‖x−1a,σ‖(wk,dk)‖x1a,σdz . by the energy estimates (3.6), we obtain the following l2 energy estimates ‖(vk,ck)(t)‖2l2 + ∫ t 0 ‖(∇vk,∇ck)(z)‖2l2dz ≤ α0, (3.7) where α0 = ‖(v0k,c 0 k)‖ 2 l2 exp(2c‖(w0k,d 0 k)‖ 2 x−1a,σ ). at this point, thanks to the properties of small frequencies part in x−1a,σ, we closed the l2 energy estimate of the large frequencies part of the solution. using x−1a,σ int. j. anal. appl. 18 (3) (2020) 436 energy estimates (3.6), we obtain ‖(u,b)‖x0 ≤ ‖(vk,ck)‖x0 + ‖(wk,dk)‖x0 ≤ ‖(vk,ck)‖x0 + m1‖(wk,dk)‖x−1a,σ ≤ ‖(vk,ck)‖x0 + m1‖(w0k,d 0 k)‖x−1a,σ, where m1 = m1(a,σ) = supr≥0 re −ar1/σ. inequality (3.5) implies that the function t → ‖(u,b)(t)‖x0 is continuous on [0,t∗) and tends to infinity when t approaches t∗. thus, there is t0 ∈ [0,t∗), such that 1/4 (t∗ − t)1/2 ≤‖(vk,ck)(t)‖x0, ∀t ∈ [t0,t∗). (3.8) at this point, we proved that the high frequencies part considered above blows up in the x0 norm. thus, according to equation (3.5), one can infers that only high frequencies are responsible for this phenomena. now, using lemma 1.1 and (3.8), we can involve xs. mainly, for s ≥ 3/2 and t ∈ [t0,t∗), 1/4 (t∗ − t)1/2 ≤‖(vk,ck)(t)‖ s s+ 3 2 l2 ‖(vk,ck)(t)‖ 3 2 s+ 3 2 xs . using inequality (3.7), we can omit the l2 norm and obtain 1/4 (t∗ − t)1/2 ≤ α s s+ 3 2 0 ‖(vk,ck)(t)‖ 3 2 s+ 3 2 xs . this implies for n ∈ n, such that n σ − 1 ≥ 3/2 or n ≥ n0 = [ 52σ] + 1, 4−1/3 (t∗ − t)1/3 an n! ( 4− 23σ α− 23σ0 (t∗ − t) 1 3σ )n ≤ an n! ‖(vk,ck)(t)‖x nσ−1. summing up for n ≥ n0, we get 4−1/3 (t∗−t)1/3 ∑ n≥n0 an n! ( 4− 23σ α− 23σ0 (t∗ − t) 1 3σ )n ≤ ∑ n≥n0 an n! ‖(vk,ck)(t)‖x nσ−1 ≤ ‖(vk,ck)(t)‖x−1a,σ. using triangular inequality and x−1a,σ energy estimates (3.6), we have ‖(vk,ck)(t)‖x−1a,σ ≤‖(u,b)(t)‖x−1a,σ + ‖(w 0 k,d 0 k)‖x−1a,σ. dividing both sides of the resulting inequality by exp ( a 4 − 2 3σ α − 2 3σ 0 (t∗−t) 1 3σ ) , we infer that lim inf t→t∗ (t∗ − t)1/3 exp ( − 4− 2 3σ α − 2 3σ 0 (t∗ − t) 1 3σ ) ‖(u,b)(t)‖x−1a,σ ≥ 4 −1/3. this gives the blowup profile and finishes the proof. int. j. anal. appl. 18 (3) (2020) 437 remark 3.1. let u0 = (u0,b0) ∈x−1a,σ(r3), where a > 0, σ ≥ 1. let u = (u,b) be the maximal solution of (mhd) system. using the fact x−1a,σ(r 3) ↪→x−1a′,σ(r 3) ↪→x−1(r3), ∀0 < a′ < a, one infers that u ∈ c([0,t∗a,σ),x−1a,σ(r3)), u ∈ c([0,t∗a′,σ),x −1 a′,σ(r 3)) and u ∈ c([0,t∗),x−1(r3)), where the maximal times of existence t∗a,σ, t ∗ a′,σ and t ∗ belong all of them to (0, +∞]. these times satisfy t∗a,σ ≤ t∗a′,σ ≤ t ∗. by the method we used in the proof of the blowup result (a′ = a σn ), we proved that t∗a,σ = t ∗, if σ > 1. however, we note that if σ = 1 our technics failes. so, this critical case seems to need an other approach different from us. remark 3.2. using the same technics we can prove existence, uniqueness and blowup results in finite time of solution to the periodic mhd, the three dimensional navier-stokes system, the two dimensional quasi-geostrophic equations with subcritical dissipation, in the lei-lin-gevrey space. acknowledgement: the authors gratefully acknowledge the approval and the support of this research study by the grant number sci-2017-1-7-f-7081 from the deanship of scientific research at northern border university, arar, k. s. a. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. t. beale, t. kato and a. majda, remarks on the breakdown of smooth solutions for the 3-d euler equations, commun. math. phys. 94 (1984), 61—66. [2] j. benameur, on the blow-up criterion of 3d navier-stokes equations, j. math. anal. appl. 371 (2010), 719-–727. [3] j. benameur, on the blow-up criterion of the periodic incompressible fluids, math. methods appl. sci. 36 (2) (2013), 143—153. [4] j. benameur, on the exponential type explosion of navier-stokes equations, nonlinear anal., theory methods appl. 103 (2014), 87–97. [5] j. benameur and l. jlali, on the blow-up criterion of 3d-nse in sobolev-gevrey spaces, j. math. fluid mech. 18 (4) (2016), 805-822. [6] j. benameur and r. selmi, anisotropic rotating mhd system in critical anisotropic spaces, mem. differential equations math. phys. 44 (2008), 23–44. [7] j. benameur and r. selmi, study of anisotropic mhd system in anisotropic sobolev spaces, ann. fac. sci. toulouse math., 17 (1) (2008), 1–22. [8] j. benameur and r. selmi, long time decay to the leray solution of the two-dimensional navier-stokes equations, bull. london math. soc. 44 (5) (2012), 1001-1019. [9] j. benameur and r. selmi, time decay and exponential stability of solutions to the periodic 3d navier-stokes equation in critical spaces, math. methods appl. sci. 37 (17) (2014), 2817-2828. int. j. anal. appl. 18 (3) (2020) 438 [10] m. cannone, harmonic analysis tools for solving the incompressible navier-stokes equations, handbook of mathematical fluid dynamics, vol 3, eds. s. friedlander and d. serre, elsevier, 2003. [11] j-y. chemin, remarques sur l’existence global pour le systeme de navier-stokes incompressible, siam j. math. anal. 23 (1) (1992), 27–50. [12] h. fujita and t. kato, on the navier-stokes initial value problem i, arch. ration. mech. anal. 16 (1964), 269–315. [13] t. kato, strong lp solutions of the navier-stokes equations in rm with applications to weak solutions, math. z. 187 (4)(1984), 471-480. [14] h. koch and d. tataru, well-posedness for the navier-stokes equations, adv. math. 157 (1) (2001), 22–35. [15] z. lei and f. lin, global mild solutions of navier-stokes equations, commun. pure appl. math. 64 (2011), 1297–1304. [16] z. lei and l. xiuting, the local well-posedness, blow-up criteria and gevrey regularity of solutions for a two-component high-order camassa-holm system, nonlinear anal., real. world appl. 35 (2017), 414-440. [17] j. leray, essai sur le mouvement d’un liquide visqueux emplissant l’espace, acta math. 63 (1933), 22–25. [18] r. selmi, convergence results for mhd system, internat. j. math. math. sci. 2006 (2006), article id 28704. [19] r. selmi, asymptotic study of mixed rotating mhd system, bull. korean math. soc. 47 (2) (2010), 231–250. [20] r. selmi, global well-posedness and convergence results for 3d-regularized boussinesq system, canad. j. math. 64 (6) (2012), 1415–1435. 1. technical lemmas 2. well-posedness results 3. blowup results references international journal of analysis and applications volume 16, number 4 (2018), 542-555 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-542 generalized (h,r)-harmonic convex functions and inequalities muhammad aslam noor∗, khalida inayat noor, sabah iftikhar, farhat safdar department of mathematics, comsats institute of information technology, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. the main aim of this paper is to introduce a new class of harmonic convex functions with respect to non-negative function h, which is called generalized (h,r)-harmonic convex functions. we derive some new fejer-hermite-hadamard type inequalities for generalized harmonic convex functions. some special cases are also discussed. the ideas and techniques of this paper may stimulate further research. 1. introduction convexity theory has become a rich source of inspiration in pure and applied sciences. this theory had not only stimulated new and deep results in many branches of mathematical and engineering sciences, but also provided us a unified and general framework for studying a wide class of unrelated problems. for recent applications, generalizations and other aspects of convex functions, see [1, 2, 4, 6, 8, 10, 11, 13–16, 18–23, 25, 27–29, 32, 35] and the references therein. varosanec [31] introduced the class of h-convex functions with respect to an arbitrary non-negative function h, which is quite flexible and unifying one. pearce et. al [18] generalized the hermite-hadamard inequality to a r -convex positive functions. gordji et al. [3, 4] considered a new class of convex functions, which is called the generalized convex( ϕ-convex) functions. for some properties of the generalized convex functions, see [3–5]. anderson et al. [1] and iscan [9] introduced and studied the harmonic convex functions, which can be viewed as an important and siginificant generaliztion received 2018-01-29; accepted 2018-04-07; published 2018-07-02. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. convex functions; general preinvex functions; differentiability; hermite-hadamard inequality. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 542 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-542 int. j. anal. appl. 16 (4) (2018) 543 of convex functions. noor et. al. [24] introduced and investigated new class of convex functions, which is called relative harmonic (s,η)-convex functions. they discussed some basic results of harmonic (s,η)-convex functions and also derived the hermite-hadamard and fejer type inequalities for this class of functions. noor et al. [17–23, 25, 27–29] have derived various error estimates for different classes of generalized convex functions. inspired and motivated by the ongoing research, we introduce the concept of (h,r)-harmonic convex functions with respect to an arbitrary nonnegative function h and r ≥ 0. this class is more general and contains several new classes of harmonic r-convex functions functions as special cases. we discuss some properties of generalized harmonic r-convex function. we establish several hermite-hadamard inequalities for generalized harmonic r-convex function. our results represent a significant refinement of the known results. 2. preliminaries in this section, we recall some basic concepts. let η(·, ·) : r×r −→ r be a continuous bifunction. definition 2.1. [13] a set i = [a,b] ⊂ r is said to be a convex set, if (1 − t)x + ty ∈ i, ∀x,y ∈ i,t ∈ [0, 1]. definition 2.2. [13] a function f : i = [a,b] → r is said to be a convex function, if f((1 − t)x + ty) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ i,t ∈ [0, 1]. definition 2.3. [31] let h : j = [0, 1] → r be a nonnegative function. a function f : i = [a,b] ⊂ r → r is said to be an h-convex function, if f((1 − t)x + ty) ≤ h(1 − t)f(x) + h(t)f(y), ∀x,y ∈ i,t ∈ [0, 1]. definition 2.4. [18] a function f : i = [a,b] ⊂ r → r is r-convex, if f is positive and for all x,y ∈ i and t ∈ [0, 1], we have f((1 − t)x + ty) =   ((1 − t)[f(x)] r + t[f(y)]r) 1 r ,r 6= 0 (f(x))1−t(f(y))t ,r = 0   . it is clear that 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions. ngoc et. al [12] obtained the hermite-hadamard inequality for r-convex function. hap and vinh [7] established a hermite-hadamard inequality for (h,r)-convex functions. gordi et al [4] introduced another class of convex functions, which is called the generalized convex functions. int. j. anal. appl. 16 (4) (2018) 544 definition 2.5. [3,4] a function f : i = [a,b] ⊂ r → r is said to be generalized convex (φ-convex) function, if and only if, f((1 − t)x + ty) ≤ f(x) + tη(f(y),f(x)), ∀x,y ∈ i,t ∈ [0, 1]. noor [14] introuced and studied the genralized r-convex functions. definition 2.6. [14] a function f : i = [a,b] ⊂ r → r is said to be generalized r-convex, if f is positive and for all x,y ∈ i and t ∈ [0, 1], we have f((1 − t)x + ty) =   ((1 − t)[f(x)] r + t[f(x) + η(f(y),f(x))]r) 1 r ,r 6= 0 (f(x))1−t(f(x) + η(f(y),f(x)))t ,r = 0   . it is clear that generalized 0-convex functions are simply generalized log-convex functions [29] and generalized 1-convex functions are generalized convex (φ-convex) functions, see [3]. definition 2.7. [33]. a set i = [a,b] ⊂ r\{0} is said to be a harmonic convex set, if xy tx + (1 − t)y ∈ i, ∀x,y ∈ i,t ∈ [0, 1]. definition 2.8. [9]. a function f : i = [a,b] ⊂ r\{0}→ r is said to be harmonic convex function, if f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ i,t ∈ [0, 1]. in particular, it has been shown that f is a harmonic convex function, if and only if, f ( 2ab a + b ) ≤ ab b−a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 , x ∈ [a,b], which is called hermite-hadamard inequality for harmonic convex function. definition 2.9. [26] let r 6= 0 be a real number and h : j → r be a nonnegative function. we say that f : i = [a,b] ⊆ r\{0}→ r is harmonic (h,r)-convex function, if f ( xy tx + (1 − t)y ) ≤ [h(1 − t)[f(x)]r + h(t)[f(y)]r] 1 r , ∀x,y ∈ i,t ∈ [0, 1]. it is clear that harmonic (h, 0)-convex functions are simply harmonic logarithmic h-convex functions and harmonic (h, 1)-convex functions are harmonic h-convex functions [17]. we now introduce some new concepts. throughout this paper, we take r 6= 0, a real number, unless otherwise specified. definition 2.10. let h : j = [0, 1] → r be a nonnegative function. a function f : i = [a,b] ⊂ r\{0}→ r is said to be generalized (h,r)-harmonic convex function, or f belongs to the class hr(h,r), if and only if, f ( xy tx + (1 − t)y ) =   [h(1 − t)[f(x)] r + h(t)[f(x) + η(f(y),f(x))]r] 1 r ,r 6= 0 (f(x))h(1−t)(f(x) + η(f(y),f(x)))h(t) ,r = 0   . (2.1) int. j. anal. appl. 16 (4) (2018) 545 the function f is said to be generalized (h,r)-harmonic concave function, if and only if, -f is generalized (h,r)-harmonic convex function. for t = 1 2 , we have f ( 2xy x + y ) =   [ h ( 1 2 )]1 r ( [f(x)]r + [f(x) + η(f(y),f(x))]r )1 r ,r 6= 0√ f(x)(f(x) + η(f(y),f(x))) ,r = 0   . (2.2) the function f is called jensen generalized (h,r)-harmonic convex function. we now discuss some special cases of generalized (h,r)-harmonic convex function, which appear to be new ones. i. if h(t) = t in definition 2.10, then it reduces to the definition of generalized r-harmonic convex functions. definition 2.11. a function f : i = [a,b] ⊂ r\{0}→ r is r-harmonic convex function, if and only if, f ( xy tx + (1 − t)y ) ≤ [(1 − t)[f(x)]r + t[f(x) + η(f(y),f(x))]r] 1 r , ∀x,y ∈ i,t ∈ [0, 1]. ii. if r = 1 in definition 2.10, then it reduces to the definition of generalized hharmonic convex functions. iii. if h(t) = ts in definition 2.10, then it reduces to the definition of breckner type of generalized (s,r)-harmonic convex functions. definition 2.12. a function f : i = [a,b] ⊂ r\{0}→ r is (s,r)-harmonic convex function, where s ∈ (0, 1), if f ( xy tx + (1 − t)y ) ≤ [(1 − t)s[f(x)]r + ts[f(x) + η(f(y),f(x))]r] 1 r , ∀x,y ∈ i,t ∈ [0, 1]. iv. if h(t) = t−s in definition 2.10, then it reduces to the definition of godunova-levin type of generalized (s,r)-harmonic convex functions. definition 2.13. a function f : i = [a,b] ⊂ r \ {0} → r is godunova-levin type of generalized (s,r)harmonic convex functions where s ∈ (0, 1), if f ( xy tx + (1 − t)y ) ≤ [(1 − t)−s[f(x)]r + t−s[f(x) + η(f(y),f(x))]r] 1 r , ∀x,y ∈ i,t ∈ (0, 1). if s = 1, then godunova-levin type of generalized r-harmonic convex functions reduces to godunovalevin type of generalized (1,r)-harmonic convex functions. lemma 2.1. suppose that a,b,c ∈ r. then (1) min{a,b}≤ a+b 2 . int. j. anal. appl. 16 (4) (2018) 546 (2) if c ≥ 0, c. min{a,b} = min{ca,cb}. remark 2.1. if i = [a,b] ⊂ r\{0} and if we consider the function g : [ 1 b , 1 a ] → r defined by g(t) = f ( 1 t ) , then f is generalized r-harmonic convex on [a,b], if and only if, g is generalized r-convex in the usual sense on [ 1 b , 1 a ] . generalized logarithmic means of order r of positive numbers x,y is defined by: lr(x,y) =   r r+1 ( xr+1−yr+1 xr−yr ) , r 6= {−1, 0},x 6= y x−y ln x−ln y , r = 0,x 6= y xy ln x−ln y x−y , r = −1,x 6= y x, x = y. minkowskis inequality is stated as follows: let r ≥ 1, 0 < ∫ b a [f(x)]rdx < ∞, 0 < ∫ b a [g(x)]rdx < ∞. then (∫ b a [f(x) + g(x)]rdx )1 r ≤ (∫ b a [f(x)]rdx )1 r + (∫ b a [g(x)]rdx )1 r . 3. main results in this section, we obtain several new hermite-hadamard type inequalities for generalized (h,r)-harmonic convex functions. theorem 3.1. let f : i = [a,b] ⊂ r \ {0} −→ r be a generalized (h,r)-harmonic convex function. if f ∈ l[a,b], then ab b−a ∫ b a f(x) x2 dx ≤ min { [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r (∫ 1 0 [h(t)] 1 r dt ) , [[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r (∫ 1 0 [h(t)] 1 r dt )} ≤ 1 2 { [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r +[[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r }(∫ 1 0 [h(t)] 1 r dt ) . proof. let f be a generalized (h,r)-harmonic convex function. then f ( ab ta + (1 − t)b ) ≤ [h(1 − t)[f(a)]r + h(t)[f(a) + η(f(b),f(a))]r] 1 r , and f ( ab (1 − t)a + tb ) ≤ [h(1 − t)[f(b)]r + h(t)[f(b) + η(f(a),f(b))]r] 1 r , ∀x,y ∈ i,t ∈ [0, 1]. int. j. anal. appl. 16 (4) (2018) 547 thus, we have f ( ab ta + (1 − t)b ) + f ( ab (1 − t)a + tb ) ≤ [h(1 − t)[f(a)]r + h(t)[f(a) + η(f(b),f(a))]r] 1 r +[h(1 − t)[f(b)]r + h(t)[f(b) + η(f(a),f(b))]r] 1 r , (3.1) integrating (3.1) over the interval [0, 1] and using minkowskis inequality, we have ∫ 1 0 f ( ab ta + (1 − t)b ) dt + ∫ 1 0 f ( ab (1 − t)a + tb ) dt ≤ [(∫ 1 0 [h(1 − t)] 1 r f(a)dt )r + (∫ 1 0 [h(t)] 1 r [f(a) + η(f(b),f(a))]dt )r]1 r + [(∫ 1 0 [h(1 − t)] 1 r f(b)dt )r + (∫ 1 0 [h(t)] 1 r [f(b) + η(f(a),f(b))]dt )r]1 r = { [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r +[[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r }(∫ 1 0 [h(t)] 1 r dt ) . this implies ab b−a ∫ b a f(x) x2 dx ≤ 1 2 { [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r +[[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r }(∫ 1 0 [h(t)] 1 r dt ) , which is the required result. � corollary 3.1. under the assumptions of theorem 3.1 with r = 1, we have ab b−a ∫ b a f(x) x2 dx ≤ min { f(a) ∫ 1 0 [h[(1 − t)] + h(t)]dt +η(f(b),f(a)) ∫ 1 0 h(t)dt,f(b) ∫ 1 0 [h[(1 − t)] + h(t)]dt +η(f(a),f(b)) ∫ 1 0 h(t)dt } ≤ [f(a) + f(b)] ∫ 1 0 h(t)dt + η(f(b),f(a)) + η(f(a),f(b)) 2 ∫ 1 0 h(t)dt. int. j. anal. appl. 16 (4) (2018) 548 theorem 3.2. let f : i = [a,b] ⊂ r \ {0} −→ r be a generalized (h,r)-harmonic convex function. if f ∈ l[a,b], then f ( 2ab a + b ) ≤ min {[ h (1 2 )]1r ([ f ( ab ta + (1 − t)b )]r + [ f ( ab ta + (1 − t)b ) + η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b ))]r)1r , [ h (1 2 )]1r ([ f ( ab (1 − t)a + tb )]r + [ f ( ab (1 − t)a + tb ) + η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb ))]r)1r } . (3.2) proof. let f be a generalized (h,r)-harmonic convex function. then, taking x = ab ta+(1−t)b and y = ab (1−t)a+tb in (2.2), we have f ( 2ab a + b ) ≤ [ h (1 2 )]1r ([ f ( ab ta + (1 − t)b )]r + [ f ( ab ta + (1 − t)b ) + η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b ))]r)1r , and f ( 2ab a + b ) ≤ [ h (1 2 )]1r ([ f ( ab (1 − t)a + tb )]r + [ f ( ab (1 − t)a + tb ) + η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb ))]r)1r . thus, f ( 2ab a + b ) ≤ min {[ h (1 2 )]1r ([ f ( ab ta + (1 − t)b )]r + [ f ( ab ta + (1 − t)b ) + η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b ))]r)1r , [ h (1 2 )]1r ([ f ( ab (1 − t)a + tb )]r + [ f ( ab (1 − t)a + tb ) + η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb ))]r)1r } . the required result. � int. j. anal. appl. 16 (4) (2018) 549 corollary 3.2. under the assumptions of theorem 3.3 with r = 1, we have f ( 2ab a + b ) ≤ min { h (1 2 )[ 2f ( ab ta + (1 − t)b ) +η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b ))] , h (1 2 )[ 2f ( ab (1 − t)a + tb ) +η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb ))]} . theorem 3.3. let f : i = [a,b] ⊂ r \ {0} −→ r be a generalized (h,r)-harmonic convex function. if f ∈ l[a,b], then 2 1−r r h ( 1 2 )(f( 2ab a + b ))r − ( ab b−a ∫ b a f(x) + η(f( abx (a+b)x−ab),f(x)) x2 dx )r ≤ ( ab b−a ∫ b a f(x) x2 dx )r ≤ 1 2r ( [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r +[[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r )r(∫ 1 0 [h(t)] 1 r dt )r . proof. let f be a generalized (h,r)-harmonic convex function. from inequality (3.2) and lemma 2.1, we have 2 [h ( 1 2 ) ] 1 r f ( 2ab a + b ) ≤ [(∫ 1 0 f ( ab ta + (1 − t)b ) dt )r + (∫ 1 0 f ( ab (1 − t)a + tb ) dt )r + (∫ 1 0 f ( ab ta + (1 − t)b ) dt + ∫ 1 0 η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b )) dt )r + (∫ 1 0 f ( ab (1 − t)a + tb ) dt + ∫ 1 0 η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb )) dt )r]1 r = [ 2 ( ab b−a ∫ b a f(x) x2 dx )r +2 ( ab b−a ∫ b a f(x) + η(f( abx (a+b)x−ab),f(x)) x2 dx )r]1 r . int. j. anal. appl. 16 (4) (2018) 550 this implies 2 1−r r h ( 1 2 )(f( 2ab a + b ))r − ( ab b−a ∫ b a f(x) + η(f( abx (a+b)x−ab),f(x)) x2 dx )r ≤ ( ab b−a ∫ b a f(x) x2 dx )r ≤ 1 2r ( [[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r +[[f(b)]r + [f(b) + η(f(a),f(b))]r] 1 r )r(∫ 1 0 [h(t)] 1 r dt )r , which is the required result. � corollary 3.3. under the assumptions of theorem 3.3 with r = 1, we have 1 2h ( 1 2 )f( 2ab a + b ) − ab 2(b−a) ∫ b a η(f( abx (a+b)x−ab),f(x)) x2 dx ≤ ab b−a ∫ b a f(x) x2 dx ≤ [f(a) + f(b)] ∫ 1 0 h(t)dt + η(f(b),f(a)) + η(f(a),f(b)) 2 ∫ 1 0 h(t)dt. one can also obtain the hermite-hadamard inequality for generalized (h,r)-harmonic convex functions as: 1 h ( 1 2 )fr( 2ab a + b ) − ab b−a ∫ b a [f(x) + η(f( abx (a+b)x−ab),f(x))] r x2 dx ≤ ab b−a ∫ b a fr(x) x2 dx ≤ [[f(a)]r + [f(a) + η(f(b),f(a))]r] (∫ 1 0 [h(t)] 1 r dt )r . we now obtain some fejer type integral inequalities for generalized (h,r)-harmonic convex functions. theorem 3.4. let f,g : i = [a,b] ⊂ r \ {0} −→ r be generalized (h,r)-harmonic convex functions. if fg ∈ l[a,b], the ∫ b a f(x)g(x) x2 dx ≤ ab 2(b−a) ∫ b a [h ( a(b−x) x(b−a) ) [f(a)] r + h ( b(x−a) x(b−a) ) [f(a) + η(f(b),f(a))] r ] 1 r ] g(x) x2 dx + ab 2(b−a) ∫ b a [h ( a(b−x) x(b−a) ) [f(b)] r + h ( b(x−a) x(b−a) ) [f(b) + η(f(a),f(b))] r ] 1 r ] g(x) x2 dx, where g : [a,b] ⊂ r\{0} is symmetric, nonnegative, integrable and satisfies g(x) = g ( abx [a + b]x−ab ) , ∀x ∈ [a,b]. int. j. anal. appl. 16 (4) (2018) 551 proof. let f be a generalized harmonic r-convex function. then, multiplying inequality (3.1) with g ( ab ta+(1−t)b ) and integrating over t, we have∫ 1 0 [ f ( ab ta + (1 − t)b ) + f ( ab (1 − t)a + tb )] g ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 [h(1 − t)[f(a)]r + h(t)[f(a) + η(f(b),f(a))]r] 1 r ]g ( ab ta + (1 − t)b ) dt + ∫ 1 0 [h(1 − t)[f(b)]r + h(t)[f(b) + η(f(a),f(b))]r] 1 r ]g ( ab ta + (1 − t)b ) dt. since g is symmetric, we have∫ b a f(x)g(x) x2 dx ≤ 1 2 ∫ b a [h ( a(b−x) x(b−a) ) [f(a)]r + h ( b(x−a) x(b−a) ) [f(a) + η(f(b),f(a))]r] 1 r ] g(x) x2 dx + 1 2 ∫ b a [h ( a(b−x) x(b−a) ) [f(b)]r + h ( b(x−a) x(b−a) ) [f(b) + η(f(a),f(b))]r] 1 r ] g(x) x2 dx, the required result. � corollary 3.4. under the assumptions of theorem 3.4 with r = 1, we have∫ b a f(x)g(x) x2 dx ≤ f(a) + f(b) 2 ∫ b a [h ( a(b−x) x(b−a) ) + h ( b(x−a) x(b−a) ) ] g(x) x2 dx + η(f(b),f(a)) + η(f(a),f(b)) 2 ∫ b a h ( b(x−a) x(b−a) ) g(x) x2 dx. theorem 3.5. let f,g : i = [a,b] ⊂ r \ {0} −→ r be generalized (h,r)-harmonic convex functions. if fg ∈ l[a,b], then f ( 2ab a + b )∫ b a g(x) x2 dx ≤ ∫ b a g(x) x2 min {[ h (1 2 )]1r ( [f(x) ]r + [ f(x) + η(f ( abx (a + b)x−ab ) ,f(x))]r )1 r , [ h (1 2 )]1r ([ f ( abx (a + b)x−ab )]r +[f ( abx (a + b)x−ab ) + η ( f(x),f ( abx (a + b)x−ab )) ]r )1 r } dx. where g : [a,b] ⊂ r\{0} is symmetric, nonnegative, integrable and satisfies g(x) = g ( abx [a + b]x−ab ) , ∀x ∈ [a,b]. int. j. anal. appl. 16 (4) (2018) 552 proof. let f,g be generalized (h,r)-harmonic convex functions. then multiplying (3.2) with g ( ab ta+(1−t)b ) and integrating over t, we have f ( 2ab a + b )∫ 1 0 g ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 g ( ab ta + (1 − t)b ) min {[ h (1 2 )]1r ([ f ( ab ta + (1 − t)b )]r + [ f ( ab ta + (1 − t)b ) + η ( f ( ab (1 − t)a + tb ) ,f ( ab ta + (1 − t)b ))]r)1r , [ h (1 2 )]1r ([ f ( ab (1 − t)a + tb )]r + [ f ( ab (1 − t)a + tb ) + η ( f ( ab ta + (1 − t)b ) ,f ( ab (1 − t)a + tb ))]r)1r } dt. by the symmetry of g on [a,b], we have f ( 2ab a + b )∫ b a g(x) x2 dx ≤ ∫ b a g(x) x2 min {[ h (1 2 )]1r ( [f(x) ]r + [ f(x) + η(f ( abx (a + b)x−ab ) ,f(x))] r )1 r , [ h (1 2 )]1r ([ f ( abx (a + b)x−ab )]r +[f ( abx (a + b)x−ab ) + η ( f(x),f ( abx (a + b)x−ab )) ] r )1 r } dx, which is the required result. � corollary 3.5. under the assumptions of theorem 3.5 with r = 1, we have 1 2h ( 1 2 )f( 2ab a + b )∫ b a g(x) x2 dx ≤ ∫ b a g(x) x2 min { f(x) + 1 2 η(f ( abx (a + b)x−ab ) ,f(x)), f ( abx (a + b)x−ab ) + 1 2 η ( f(x),f ( abx (a + b)x−ab ))} dx ≤ ∫ b a f(x)g(x) x2 dx + 1 2 ∫ b a g(x) x2 [ η ( f(x),f ( abx (a + b)x−ab ))] dx. theorem 3.6. let f : i = [a,b] ⊂ r \{0}−→ r be generalized r-harmonic convex function. if fg ∈ l[a,b], then ab b−a ∫ b a f(x) x2 dx ≤   r r+1 ( [f(a)]r+1−[f(a)+η(f(b),f(a))]r+1 [f(a)]r−[f(a)+η(f(b),f(a))]r ) , r 6= {−1,0},f(a) 6= f(b) η(f(b),f(a)) ln[f(a)+η(f(b),f(a))]−ln f(a), r = 0,f(a) 6= f(b) f(a)[f(a) + η(f(b),f(a))] ln[f(a)+η(f(b),f(a))]−ln f(a) η(f(b),f(a)) , r = −1,f(a) 6= f(b) f(a), f(a) = f(b). int. j. anal. appl. 16 (4) (2018) 553 proof. let f be a harmonic r-convex functions. then f ( ab ta + (1 − t)b ) ≤ [(1 − t)[f(a)]r + [f(a) + η(f(b),f(a))]r] 1 r , i. the case r 6= {−1, 0},f(a) 6= f(b). ab b−a ∫ b a f(x) x2 dx = ∫ 1 0 f ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 [(1 − t)[f(a)]r + t[f(a) + η(f(b),f(a))]r] 1 r dt = r r + 1 ( [f(a)]r+1 − [f(a) + η(f(b),f(a))]r+1 [f(a)]r − [f(a) + η(f(b),f(a))]r ) . ii. the case r = 0,f(a) 6= f(b). ab b−a ∫ b a f(x) x2 dx = ∫ 1 0 f ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 [f(a)]1−t + [f(a) + η(f(b),f(a))]tdt = η(f(b),f(a)) ln[f(a) + η(f(b),f(a))] − ln f(a) . iii. the case r = −1,f(a) 6= f(b). ab b−a ∫ b a f(x) x2 dx = ∫ 1 0 f ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 [(1 − t)[f(a)]−1 + t[f(a) + η(f(b),f(a))]−1]−1dt = f(a)[f(a) + η(f(b),f(a))] η(f(b),f(a)) ∫ f(a)+η(f(b),f(a)) f(a) 1 u du = f(a)[f(a) + η(f(b),f(a))] ln[f(a) + η(f(b),f(a))] − ln f(a) η(f(b),f(a)) . iv. the case f(a) = f(b) is obvious. this completes the proof. � acknowledgements the authors would like to thank the rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. int. j. anal. appl. 16 (4) (2018) 554 references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen. generalized convexity and inequalities. j. math. anal. appl., 335(2007), 1294-1308. [2] g. cristescu and l. lupsa. non-connected convexities and applications. kluwer academic publisher, dordrechet, holland, (2002). [3] m. r. delavar and s. s. dragomir, on η-convexity, math. inequal. appl., 20(1), 203-216. [4] m. e. gordji, m. r. delavar and m. de la sen, on ϕ-convex functions, j. math. inequal., 10(1)(2016), 173-183. [5] m. e. gordji, s. s. dragomir and m. r. delavar, an inequality related to η-convex functions (ii), int. j. nonlinear anal. appl., 6(2)(2015), 26-32. [6] j. hadamard. etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann. j. math. pure appl., 58(1893), 171-215. [7] l. v. hap and n. v. vinh, on some hadamard-type inequalities for (h,r)−convex functions, int. j. math. anal., 7(42)(2013), 2067 2075. [8] c. hermite, sur deux limites dune integrale definie. mathesis, 3(1883), 82. [9] i, iscan. hermite-hadamard type inequalities for harmonically convex functions. hacettepe, j. math. stats., 43(6)(2014), 935-942. [10] m. a. latif, s.s. dragomir and e. momoniat. some fejer type inequalities for harmonically convex functions with applications to special means, int. j. anal. appl., 13(1)(2017), 1-14 [11] m. v. mihai, m. a. noor, k. i. noor and m. u. awan. some integral inequalities for harmonically h-convex functions involving hypergeometric functions. appl. math. comput., 252(2015), 257-262. [12] n. p. n. ngoc, n. v. vinh and p. t. t. hien, integral inequalities of hadamard type for r-convex functions, int. mathe. forum, 4(35)(2009), 1723 1728. [13] c. p. niculescu and l. e. persson. convex functions and their applications. springer-verlag, new york, (2006). [14] m. a. noor, advanced convex analysis and optimization, lecture notes, mathematics department, comsats institute of information technology, islamabad, pakistan. (20015-2018). [15] m. a. noor, some deveolpments in general variational inequalities, appl. mathh. comput. 251(2004), 199-277. [16] m. a. noor, b. b. mohsen, k. i, noor and s. iftikhar, relative strongly harmonic convex functions and their characterizations, j. nonlinear sci. appl. in press. [17] m. a. noor, k. i. noor, m. u. awan and s. costache. some integral inequalities for harmonically h-convex functions. u.p.b. sci. bull. serai a, 77(1)(2015), 5-16. [18] m. a. noor, k. i. noor and u. awan, some new estimates of hermite-hadamard inequalities via harmonically r-convex functions, le matematiche, 71(2)(2016), 117-127. [19] m. a. noor, k. i. noor and s. iftikhar, some newton’s type inequalities for harmonic convex functions, j. adv. mathe. stud., 9(1)(2016), 07-16. [20] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic nonconvex functions, magnt res. rep., 4(1)(2016), 24-40. [21] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions(survey), twms j. pure appl. math., 7(1)(2016), 3-19. [22] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inform. sci., 10(5)(2016), 1811-1814. int. j. anal. appl. 16 (4) (2018) 555 [23] m. a. noor, k. i. noor, s. iftikhar and c. ionescu, hermite-hadamard inequalities for co-ordinated harmonic convex functions, u.p.b. sci. bull., ser: a, 79(1)(2017), 25-34. [24] m. a. noor, k. i. noor, s. iftikhar and f. safdar, integral inequalities for relative harmonic (s,η)-convex functions, appl. math. comput. sci., 1 (1) (2016), 27-34. [25] m. a. noor, k. i. noor and s. iftikhar, some characterizations of harmonic convex functions, int. j. anal. appl., 15(2)(2017), 179-187. [26] m. a. noor, k. i. noor and s. iftikhar, on harmonic (h,r)-convex functions, proc. jangjeon math. soc., 21(2)(2018), 239-251. [27] m. a. noor, k. i. noor and s. iftikhar, inequalities via (p,r)-convex functions, rad hazu, matematicke znanosti, in press. [28] m. a. noor, k. i. noor and f. safdar, integral inequalities via generalized (α,m)-convex functions, j. nonlinear funct. anal., 2017(2017), article id 32. [29] m. a. noor, k. i. noor, f. safdar, m. u. awan and s. ullah, inequalities via generalized log m-convex functions, j. nonlinear sci. appl., 10 (2017), 5789-5802. [30] j. park, on the hermite-hadamard-like type inequalities for co-ordinated (s,r)-convex mappings, int. j. pure appl. math., 74(2)(2012), 251-263. [31] c.e.m. pearce, j. pecaric and v. simic, on weighted generalized logarithmic means, houston j. math., 24(3)(1998), 459. [32] m. z. sarikaya, h. yaldiz and h. bozkurt, on the hadamard type integral inequalities involving several differentiable φ − r-convex functions, (2012), arxiv:1203.2278 [math.ca]. [33] h. n. shi and zhang. some new judgement theorems of schur geometric and schur harmonic convexities for a class of symmetric functions. j. inequal. appl., 2013(2013), art. id 527. [34] s. varosanec, on h-convexity, j. math. anal. appl., 326(2007), 303-311. [35] g. s. yang, refinements of hadamards inequality for r -convex functions, indian j. pure appl. math , 32(10)(2001), 1571-1579. 1. introduction 2. preliminaries 3. main results acknowledgements references international journal of analysis and applications volume 18, number 6 (2020), 1066-1082 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1066 received september 17th, 2020; accepted october 8th, 2020; published november 3rd, 2020. 2010 mathematics subject classification. 91g40. key words and phrases. bank stability; credit bank; bank performance. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1066 credit risk and bank stability of vietnam commercial bank: a bk approach sang tang my* ho chi minh city university of economics and finance (uef). 141-145 dien bien phu street, ward 15, binh thanh district, ho chi minh city, vietnam *corresponding author: sangtm@uef.edu.vn abstract: through the secondary data collected from 2005 to 2019, we used the bk approach in which, pooled ols, fem, rem, and gmm methods and sobel’s test are used to check the relationship between bank credit risk and bank stability of vietnam commercial bank system. the results show that bank credit risk, profitability, and bank stability have a direct relationship and have a partial indirect relationship. the size and profitability of the previous period have a positively correlated with bank profitability, nonperforming loan, loan loss provision, non-interest income, efficiency, and bank credit growth have a negative impact on bank profitability, bank profitability has no impact on bank credit risk. profitability and bank stability of the previous period have an impact on current bank stability. nonperforming loans, non-interest income hurt bank stability, loan loss provision and bank stability of the previous period have a positively correlated with current bank stability. 1. introduction financial stability is the foundation for sustained economic growth in every country. so, this attaches the attention of governments, international organizations such as the world bank, international monetary organization, and all regions of the world. countries have identified financial stability as a condition for macroeconomic stability due to the close link between these two goals. money and banking is an extremely sensitive field, affected by many different factors, https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1066 int. j. anal. appl. 18 (6) (2020) 1067 therefore, bank stability, monetary stability are considered as an important factor in the goal of financial stability. in the financial system, a commercial bank is one of the largest financial institutions, supply the capital for corporations, households, individuals, and other organizations. because commercial bank role as the main intermediary financial institution of the economy, the operation of the commercial banking system has a great impact on other entities, especially corporations, and then affects economic growth. since then, the bank's activities have received much attention from the objects of the economy, especially about bank efficiency. the bank's performance depends on various factors such as profitability, innovation, service quality [22] of which the most important factor considered is bank credit risk. there’s a positive relationship between profitability and liquidity of 12 banks in europe, north america, and australia [12]. the relationship between profitability and capital ratio of american banks is positive in the 1980s [10]. the relationship between bank credit risk and the profitability of commercial banks is positive in switzerland [28]. one of the most important factors that have a direct impact on the business performance of banks is non-performing loans ([7], [38], [27]). theoretically, credit risk is the most important risk, stemming from the operational nature of commercial banks, credit risk is the result of the adverse selection and moral hazard. this risk hurts the performance of commercial banks. frederic s. the commercial banks want to earn high profits, they must recover all loans to customers, only need a small percentage of non-performing loan can lead to commercial bank failures [30]. because of this importance, the operation of the banking system is subject to a lot of different rules, one of the principles applied by the global commercial banking system in the governance process is basel accord, developed by the basel committee. up to now, basel has undergone 3 revisions, helping banks to maintain a stable position in the complicated fluctuations of the financial market. one of the contents identified by the basel committee is credit risk management because effective credit risk management will help banks increase profitability and then help the bank system more stability, contribute to stabilizing the financial system, and allocate capital in the economy more effectively. after the recent financial crisis, the number of failed banks increased, leading to a lot of complicated and difficult problems. these issues are clearly expressed in the financial statements of banks, from which the world financial market has focused more on the business performance int. j. anal. appl. 18 (6) (2020) 1068 of the commercial banking system and especially credit risk because there is evidence that credit banks have an impact on financial stability [37]. as the main source of capital in the economy, the profitability of vietnam's commercial bank system has a great impact on the operation of corporations and the stability of the financial system. in the period from 2015 to 2018, profitability improved markedly, increasing from 6.42% in 2015 to 9.06% in 2018. the improvement in the profitability of vietnam commercial banks in 2018 is considered to be the average level of southeast asia since the global crisis in 2008. according to the national financial supervisory commission, the operation of the bank system is still facing many challenges such as non-performing loans and rising costs. the total value of non-performing loans of 22 banks to march 2019 was about vnd 84 trillion, an increase more than vnd 4.6 trillion, about 5.9% compared to the end of 2018 while the growth of outstanding loans was only at 3.46%, of which 15 banks have increased npls. among the non-performing loan groups, the owe losses-loss loans account for a high proportion, affect to credit growth, the credit growth of this period was lower than in 2017 and 2018 [6]. bank credit growth fall and high non-performing loan greatly affect the income and efficiency of vietnam commercial banking system. for example, bank for investment and development of vietnam (bidv), the largest bank in vietnam in 2019, due to high non-performing loan, they have to make provision and profit after provisioning has fallen by nearly 80% compared to the original. in the banking sector development strategy, the state bank has identified the period from 2005 to 2020 as a stage of building a safe, healthy, competitive, and sustainable banking system. therefore, bank stability is also identified as a very important target of the bank system. to strengthen bank stability, improve efficiency and reduce bank credit risk in the coming period, vietnam's banking system needs to have effective and appropriate management measures, as well as to identify the overall factors including internal and external elements, to make appropriate forecasts and strategies for hedging the increasingly complex developments of the macro environment. 2. literature review credit risk is defined as a potential change in net income and market value of capital resulting from a client's failure to pay or to make a late repayment [38]. credit risk is the biggest risk in an accounting book of a bank and if not properly managed, it can lead to a bank financial weakening int. j. anal. appl. 18 (6) (2020) 1069 [25]. this is a potential risk in which the customer can not meet its obligations under the agreed terms [9]. in 2001, the basle committee identified additional credit risk as to the main risk in the operation of commercial banks, the risk associated with the bank's key activities, related to capital mobilization and lending. so, credit risk is the risk arising in the process of credit extension of banks, in which customers cannot repay debts or repay debts on time to banks. the impact of credit on bank performance received great interest from researchers. internal factors such as loan loss provision, the ratio of equity to total assets, and operating costs have an impact on bank profits with high significance [7]. bank performance is defined as the ability to minimize costs or maximize the profit of banks [10]. the performance of commercial banks is reflected in the relationship between output and input of commercial banks. specifically, commercial banks generate the largest output with the smallest input [10]. bank performance is interested in managers and investors because high performance will help banks preserve capital, increase market share, and attract investment. bank performance is the net income after tax of commercial banks [30], which is the net profit of the bank [33], involving the return on the initial investment ([9], which is an increase in profit compared to operating costs [39]. this ratio is used to measure the profit made by the bank based on revenue, capital, assets, and earnings per share. bank performance is one of the indicators showing growth and success in banking operations [29], which is the final goal of investment, demonstrating the efficiency of resource management [27], expressed in two ways including accounting profit and economic profit [5]. corporations often have more benefits when performance increases. to measure the bank performance of a bank, various methods can be used but financial indicators are the most common method [33]. theoretically, roe is favored more than other indicators ([21], [15]). in an increasingly competitive environment, bank performance is one of the key factors that help the bank operate smoothly, continuously, and have a direct impact on its development process [38]. studying bank performance is very important because this is not only the most important factor in the existence of a bank but also provides information about the health of the bank system. bank stability is the status of effective implementation of important economic functions such as resource allocation, dispersion, and risk handling, which is the ability to fully absorb the shock the system faces [16], assess changes in financial risks, and effective allocation of resources [23], shows the flexibility of all financial-related activities and sectors to minimize losses and bank int. j. anal. appl. 18 (6) (2020) 1070 crises [5]. bank instability comes from inefficient banks leading to liquidity risk which leads to shocks [37] and economic efficiency is likely to be reduced due to financial fluctuations [15]. z-score is considered as an indicator of bank stability. a higher z-score indicates a more stable bank ([16], [17], [10], [20]). because of this importance, some studies have been carried out in various aspects to test and identify signs of instability to take countermeasures. the research study on the agricultural banking system in ghana shows that the risk of capital mobilization and the stability of the bank also tend to fluctuate in the same direction [2]. determinants of bank stability is the differences between banks and the type of banks [18]. to study the relationship between bank credit risk and bank profitability, the research used the theories including efficiency theory [3], the agency cost theory [26]. efficiency theory proposed that good governance and effective banks are more profitable, efficiency management not only increases profits but also helps banks increase market share and improve market concentration [7]. for the banking industry, the efficient theory said that large banks have better governance and management experience, so they can reduce operating costs, and profit will be higher than small banks [36]. besides, the efficiency theory also suggests that efficiency operating effects to bank profitability by reducing operating costs [10]. the agency cost theory assumes that the financial structure of a firm is used by managers as a means of dealing with cash flow problems. in the organization structure of joint-stock enterprises, managers and owners are two different objects in which managers represent for the owner to manage the company and they may take actions that are inconsistent with the owner's goal resulting in agency costs. this cost is divided into three sub-groups including management supervision cost, bond cost, and the remaining costs [26]. 3. research objectives: the research was conducted to study the relationship between credit risk and bank stability of the vietnam commercial bank system, thereby propose solutions to increase bank stability of the bank system. the specific objective is to check the relationship between credit risk and bank performance, the relationship between credit risk and bank stability, the relationship between bank performance and bank stability of vietnam commercial bank. suggest solutions to decrease bank credit risk and improve bank stability for vietnam commercial banks. 4. data and methodology: 4.1 research data and estimation methodology int. j. anal. appl. 18 (6) (2020) 1071 the research uses a panel data of 286 observations collected from the financial statements from 2005 to 2019, in which some data collected from bankscope and some microeconomic data from the adb indicators. because credit risk and bank stability is mediation relationship, so bk approach is used to solve the data, this method used to demonstrate that using the regression technique suffers from a serious drawback, this produces larger standard errors for the path coefficients, so the results are inaccurate and not comprehensive [24]. according to the bk approach, to measure all relationships between variables in the model. the first step will be to measure the direct relationship between variables in the model. if the measurement result has a relationship that is not statistically significant, there is no mediation and the researcher should stop. conversely, if all models are statistically significant, continue to the second step. in the second step, the result obtained from sobel's coefficient will be used to conclude the intermediate relationship of the model. to check the direct relationship, the research uses time-series data, the model estimation method chosen is pooled ols, fem, rem, gls, and gmm method [24]. 4.2 model and hypotheses: empirical studies on credit risk and bank performance with three main research trends including checking the relationship between profitability and bank stability, the relationship between credit risk and bank performance, and the relationship between credit risk and bank stability, this is the basis for selecting relevant research models. most models from empirical research use multivariate regression models with panel data for commercial banks of each country or many countries. this study approach towards research on the choice of the model comes from two reasons. firstly, choose a multivariate regression model, using panel data of vietnam commercial banks from 2005 to 2019. secondly, to ensure the estimation of the regression model is accurate, choosing the appropriate variables, the authors have conducted a multi-collinear test, variance change, and autocorrelation. besides, pooled ols, fem, rem, gls, and gmm methods are used to test the hypotheses proposed, this approach is consistent with research trends. the selection of basic variables and making research hypotheses are mainly based on empirical evidence, mainly from the research of athanasoglou [7] and diaconu [18]. the research proposes three models for three phases, the first phase will determine the factors that affect to bank's credit risk, the second phase will study the impact of profitability on bank performance, the third phase will study the impact of credit risk on bank stability. model in the first phase as follows: int. j. anal. appl. 18 (6) (2020) 1072 𝑅𝑂𝐸 = 𝛼0 + ∑ 𝛽𝑗 𝑋𝑖𝑡 𝐽 + ∑ 𝛽𝑙 𝑋𝑖𝑡 𝑙 + 𝑖𝑡 (1) 𝐿 𝑙=1 𝐽 𝑗=1 where roe is the bank performance of bank i. 𝛼0 is constant. x are independent variables, group j includes internal factors of the bank (liquidity, non-performing loan, loan loss provision, bank size, leverage, non-interest income, efficiency, bank credit growth). 𝛽𝑙 is the impact of the independent variable on bank performance. 𝛽𝑗 is the effect of the lag variable on bank performance. 𝑖𝑡 is the disturbance. based on the research of athanasoglou [7] and diaconu [18], variables of the model are shown in table 1. table 1. variables of the model (1) variables description expected effect authors non-performing loan (npl) non-performing loan/ total loans berger (1997) loan loss provision (llr) loan loss provision/ total loans chaibi (2003) leverage (lev) liabilities / total asset + abreu (2000) non-interest income (nii) non-interest income/ total income + alexiou (2009) dietrich (2011) bank size (size) natural log total assets + pasiouras (2007) guillen (2014) efficiency (eff) operation cost /operation income memmel (2010) hoffmann (2011) pasiouras (2007) liquidity (eta) equity/ total asset radic (2012) berger (1997) bank credit growth (crg) (total loans at year t – total loans at year t-1)/ total loans at year t-1 baron (2015) rashid (2014) source: summary of previous studies in the second phase, to examine the relationship between bank stability and bank performance, the research use the model based on research by tan [39] 𝑦𝑖𝑡 = 𝛼0 + 𝛼1𝑦𝑖,𝑡−1 + 𝛼2𝑦𝑖,𝑡−2 + 𝛽1𝑥𝑖,𝑡−1 + 𝛽2𝑥𝑖,𝑡−2 + 𝜃𝑡 + 𝛿𝑖 + 𝜇𝑖𝑡 (2) int. j. anal. appl. 18 (6) (2020) 1073 where y is bank performance (roe), 𝑥 is bank stability (z-score), i and t represent for bank i in year t, 𝛼0 𝑖𝑠 the intercept, 𝛼1, 𝛼2, 𝛽1, 𝛽2 are the coefficients to be estimated, 𝜃𝑡 is the time effect, 𝛿𝑖 stands for individual bank effect, and 𝜇𝑖𝑡 is the error term, 𝜇𝑖𝑡 is the error term. z-score = [e(roaa) + ebq/abq]/ σ(roaa) where roaa is the return on average of total assets. ebq/abq is the ratio of average equity to an average of total assets. σi(roaa) is the standard deviation of roaa model in the final phase used to study the impact of profitability on bank stability 𝑍 − 𝑠𝑐𝑜𝑟𝑒 = 𝛼0 + ∑ 𝛽𝑗 𝑋𝑖𝑡 𝐽 + ∑ 𝛽𝑙 𝑋𝑖𝑡 𝑙 + 𝑖𝑡 𝐿 𝑙=1 𝐽 𝑗=1 (3) where roe is bank performance i. 𝛼0 is constant. x are independent variables, group j includes internal factors of the bank (liquidity, credit risk, capital ratio, bank size, operating expenses) and group l includes macro factors (inflation, economic growth). 𝛽𝑙 is the impact of the independent variable on profitability. 𝛽𝑗 is the effect of the lag variable on the profitability of the bank. 𝑖𝑡 is the disturbance. based on the research of athanasoglou [7] and diaconu [18], independent variables of the model are shown in table 2. table 2. variables of the model (3) variables description expected effect authors liquidity (eta) total loans/ total assets + ongore (2014) non-performing loans (llp) non-performing loans / total loans) hasan (2003) loan loss provision (llp) loan loss provision/ total loans athanasoglou (2006) diamond (2011) dang (2011) efficiency (eff) operation cost / operation income + obamuyi (2013) memmel (2010) bank size (size) natural log total assets + bowa (2015) hoffmann (2011) non-interest income (nii) non-interest income/ total income + alexiou (2009) bank credit growth (crg) (total loans at year t – total loans at year t-1)/ total loans at year t-1 carlson (2019) source: summary of previous studies int. j. anal. appl. 18 (6) (2020) 1074 5. empirical analysis the descriptive statistics of the research are shown in table 3. this indicates that the dispersion between observations in the sample is expressed as mean, maximum, minimum, and standard deviation. the values of variables are unevenly distributed, through mean and standard deviation. the data is unbalanced. table 3. descriptive statistics variables obs mean standard deviation min max npl 349 .0198539 .0135546 .0002 .1032 roe 384 .0986214 .0829285 -.82 .3714 size 384 17.95054 1.550093 11.8835 20.9956 nii 383 .2199755 .6205942 -2.1087 11.6503 lev 384 .8897294 .134121 -.422 1.7112 eff 375 .5024067 .1916247 .1741 2.2091 crg 380 .4343045 1.020566 -1 11.3268 eta 380 .1023092 .0616704 .0041 .4624 llr 386 .3474319 4.39156 -.0071 86.3019 z_score 371 2.262678 1.804082 .0528 12.5477 source: calculating from stata 12. the correlation coefficient matrix in three models shows the results that the correlation coefficients are relatively small, without serious multi-collinear phenomena due to low correlation coefficients value, the comparison standard is 0.8. checking the multicollinearity phenomenon with the vif coefficient, the result shows that the vif of all independent variables is less than 10. firstly, the study uses the regular panel data regression model with the pool ols method to estimate the regression equations and test some hypotheses of the ols model. after that, the research estimated all three models: pooled, fem, and rem, but due to the variance change, white test results with prob> chi2 = 0.0000, less than 1%, so the research will eventually regress according to gmm method. the results of the regression of model 1 shown in table 4 table 4. summary of regression results of model 1 int. j. anal. appl. 18 (6) (2020) 1075 pooled ols fem rem gmm roe roe roe roe size 0.0123*** 0.0102*** 0.0113*** 0.0153*** [4.39] [2.70] [3.42] [4.81] nii -0.0750*** -0.0770*** -0.0765*** -0.0827*** [-17.88] [-19.55] [-19.83] [-28.98] lev 0.0840*** 0.0832*** 0.0830*** 0.0552 [2.74] [2.81] [2.88] [1.45] llr -1.456*** -1.909*** -1.818*** -1.980*** [-4.17] [-5.29] [-5.24] [-7.88] eff -0.318*** -0.306*** -0.312*** -0.283*** [-15.63] [-14.02] [-15.17] [-19.06] crg -0.00177 -0.00125 -0.00127 -0.00307*** [-1.09] [-0.84] [-0.87] [-3.26] npl -0.425 -0.326 -0.322 -0.756* [-1.51] [-1.17] [-1.19] [-1.67] eta -0.0879 -0.122* -0.111 -0.0700 [-1.18] [-1.71] [-1.59] [-0.99] l.roe 0.197*** [2.59] _cons 0.00518 0.0440 0.0232 -0.0468 [0.08] [0.58] [0.34] [-0.57] n 317 317 317 274 r-sq 0.672 0.696 t statistics in brackets * p<0.1, ** p<0.05, *** p<0.01 source: calculating from stata 12 in model 2, four methods including pooled ols, fem, rem, and gls used to process data, but due to the variance change, the white test results prob> chi2 = 0.0000, less than 1% so the research will regress according to the gls method. the result is shown in table 5. int. j. anal. appl. 18 (6) (2020) 1076 table 5. summary of regression results of model 2 pooled ols fem rem gls z_score z_score z_score z_score z_score-1 0.336*** 0.212*** 0.336*** 0.185*** [5.67] [3.48] [5.67] [3.19] z_score-2 -0.0188 -0.107* -0.0188 0.0122 [-0.32] [-1.78] [-0.32] [0.21] roe-1 -2.535* -3.034** -2.535* -2.674** [-1.93] [-2.20] [-1.93] [-2.06] roe-2 -0.752 -1.638 -0.752 -1.115 [-0.57] [-1.19] [-0.57] [-0.87] _cons 2.071*** 2.697*** 2.071*** 2.409*** [8.46] [9.81] [8.46] [8.92] n 317 317 317 317 r-sq 0.145 0.078 t statistics in brackets *p<0.1, ** p<0.05, *** p<0.01 source: calculating from stata 12. to test model 3, the research also uses four methods including pool ols, fem, rem, and gmm to estimate, autocorrelation phenomena, variance change, multicollinearity do not occur and the estimation results in table 6 show that the gmm method is most suitable. int. j. anal. appl. 18 (6) (2020) 1077 table 6. summary of regression results of model 3 pool ols fem rem gmm z_score z_score z_score z_score crg 9.534*** 15.37*** 12.75*** -14.48 [3.32] [5.31] [4.48] [-0.74] eff 2.572*** 0.890 2.032*** -0.505 [3.85] [1.21] [2.98] [-0.69] llr 30.43** 28.11** 30.81** 26.83* [2.39] [2.02] [2.33] [1.87] npl -19.26*** -19.85*** -20.33*** -20.97*** [-2.64] [-2.81] [-2.88] [-3.54] lev 0.00749 0.156 0.0240 -33.51 [0.01] [0.13] [0.02] [-1.56] nii -0.964 -1.231* -1.208* -2.613*** [-1.55] [-1.85] [-1.89] [-4.36] size 0.266*** 0.853*** 0.502*** 0.0269 [2.60] [6.24] [4.38] [0.05] l.z_score 0.121*** [3.24] _cons -4.479* -14.95*** -8.734*** 34.07 [-1.84] [-5.27] [-3.44] [1.60] n 326 326 326 270 r-sq 0.100 0.207 t statistics in brackets *p<0.1, ** p<0.05, *** p<0.01 source: calculating from stata 12. in the final step, sobel's test is used to check the indirect relationship between bank stability, credit risk, and bank stability and the estimation results in table 7 show that there is a partial mediation relationship between the three variables. int. j. anal. appl. 18 (6) (2020) 1078 table 7. the result of sobel’s test estimates delta sobel monte carlo indirect effect 9.065 9.065 9.110 std. err. 2.680 2.680 2.790 z-value 3.383 3.383 3.265 p-value 0.001 0.001 0.001 conf. interval 3.814 , 14.317 3.814 , 14.317 3.807 , 14.606 baron and kenny approach to testing mediation step 1 roe:npl (x -> m) with b=-1.5e+03 and p=0.000 step 2 zscore:roe (m -> y) with b=-0.006 and p=0.000 step 3 zscore:npl (x -> y) with b=-17.602 and p=0.008 as step 1, step 2 and step 3 as well as the sobel's test above are significant the mediation is partial! source: calculating from stata 12 the research results of model 1 show that the bank size and profitability of the previous year have positively correlated with bank performance. non-interest income, loan loss provision, bank performance, bank credit growth, and non-performance have negatively correlated with bank performance. these results are consistent with the study of berger [10], chaibi [41], pasiouras [33], abreu [1]. research results of the impact of business efficiency on bank stability, the impact of credit risk on bank stability show that non – performing loans, loan loss provision, non-interest income, and bank stability in the previous period have an impact on bank stability. the research results after 3 steps and sobel’s test show that the mediation relationship is a partial, this means that credit risk affects bank stability through bank performance. bank stability of the vietnam commercial bank system depends on both credit risk and bank performance. npl impact on roe, roe has no impact on npl, non-performing loans impact on the bank performance but bank performance has no impact on non-performing loans ; roe impact on z-score, the bank performance impact on bank stability with one year lag; zscore-1 impact on z-score, bank stability of previous period impact on current bank stability. npl impact on z-score, non-performing loan impact on bank stability; npl, roe, and z-score are partial relationships, bank stability is explained by two variables including non-performing loans and bank performance. int. j. anal. appl. 18 (6) (2020) 1079 6. conclusion and policy implications 6.1 conclusion the results obtained from the research models show that, in terms of direct impact, npl has an impact on roe and z-score, roe has a one-way impact on z-score. because the first model does not show the roe's adverse effect on the npl, the study of the opposite effect will end immediately [24]. in terms of indirect effects, the relationship between npl, roe, and z-score is a partial mediation relationship. combining these relationships, it can be said that bank stability is explained by both the two variables including credit risk and bank performance. credit risk has a direct impact on bank stability, when credit risk increases, bank stability decreases. bank performance increases, bank stability tends to decrease. when credit risk increases, combined with bank performance increased, banks' volatility will certainly increase. from this judgment, recommendations on reducing credit risk and improving bank stability are proposed. the groups of proposed solutions include the group of solutions for credit risk management, the solution group to reduce operating costs, the solution group to increase non-interest income, and the solution of capital structure management. 6.2 policy implications from the research results, the research proposed implications for the management of vietnam commercial bank's focus on credit risk management. firstly, banks need to enhance credit risk governance to reduce the loan loss provision ratio. commercial banks need to pay more attention to lending to reduce nonperforming loans, not lending to customers who do not have a healthy financial capacity. secondly, banks need to classify debts and make provisions for credit losses based on combining with internal credit ratings according to current regulations, strengthening inspection and supervision in the debt classification process and the bank should accept a decrease in profits due to the provision, as these will be the money used to handle loans when non-performance loans are incurred, avoiding the bank's crisis based on bad debt as well as excess liquidity. finally, banks need to build and perfect the risk management system to comply with basel ii standards. identification of risk signs is one of the most important steps in the credit risk management process. identification of the warning signals for a loan with a risk probability of exceeding the permitted standard is the most important and least expensive for the bank. identifying and forecasting credit risks is one of the areas that need to be improved and carried out regularly and continuously. int. j. anal. appl. 18 (6) (2020) 1080 6.3 limitations the study was conducted for the case of vietnam but there was no comparison with other countries. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] m. abreu, v. mendes. commercial bank interest margins and profitability: evidence for some eu countries. in pan-european conference jointly organised by the iefs-uk & university of macedonia economic & social sciences, thessaloniki, greece, vol. 34, no. 2, pp. 17-20. (2001). [2] s. o. adeusi, f. t. kolapo, o. a. aluko. determinants of commercial banks’ profitability: panel evidence from nigeria. int. j. econ. commerce manage. 2(12) (2014), 1–18. [3] a. a. alchian, h. demsetz, the property right paradigm, j. eco. history. 33 (1973), 16–27. [4] c. alexiou, v. sofoklis. determinants of bank profitability: evidence from the greek banking sector. econ. ann. 54(182) (2009), 93-118. [5] f. n. alshubiri, determinants of financial stability: an empirical study of commercial banks listed in muscat security market. j. bus. retail manage. res. 11(4) (2017), 192-200. [6] p. t. k. anh, handling bad debts and credit growth at vietnamese commercial banks. ministry of finance website, institute of strategy and policy. retrieved oct. 23rd 2019 from http://tapchitaichinh.vn/ngan-hang/xu-ly-no-xau-va-tang-truong-tin-dung-tai-cac-ngan-hangthuong-mai-viet-nam-309460.html [7] p. p. athanasoglou, s. n. brissimis, m. d. delis. bank-specific, industry-specific and macroeconomic determinants of bank profitability. j. int. financ. mark. inst. money. 18(2) (2008), 121-136. [8] m. baron, countercyclical bank equity issuance, rev. financ. stud. 33 (2020), 4186–4230. [9] basle committee. credit risk modelling: current practices and applications basle committee on banking supervision, bank of international settlements, basle, switzerland. (1999). [10] a. n. berger, r. deyoung. problem loans and cost efficiency in commercial banks. working paper, office of the comptroller of the currency, (1995). [11] m. bofondi, g. gobbi. bad loans and entry in local credit markets. bank of italy, mimeo, 2003. [12] p. bourke. concentration and other determinants of bank profitability in europe, north america and australia. j.bank. finance, 13(1) (1989), 65-79. [13] m. carlson, d. c. wheelock. interbank markets and banking crises: new evidence on the establishment and impact of the federal reserve. amer. econ. rev. 106(5) (2016), 533-537. int. j. anal. appl. 18 (6) (2020) 1081 [14] v. castro. macroeconomic determinants of the credit risk in the banking system: the case of the gipsi. econ. model. 31 (2013), 672-683. [15] j. chant. financial stability as a policy goal. essays financ. stab. 1 (2003), 57-58. [16] m. m. cihák, h. hesse. islamic banks and financial stability: an empirical analysis. imf working paper no. 08/16. (2008). [17] g. d. nicolo, m. g. zephirin, p. f. bartholomew, j. zaman. bank consolidation, internationalization and conglomeration: trends and implications for financial risk. imf working paper no. 03/158. (2003). [18] r. i. diaconu, d. c. oanea, the main determinants of bank’s stability. evidence from romanian banking sector, procedia econ. finance. 16 (2014), 329–335. [19] a. dietrich, g. wanzenried, determinants of bank profitability before and during the crisis: evidence from switzerland, j. int. financ. mark. inst. money. 21 (2011), 307–327. [20] gamaginta, rokhim r (2015). the stability comparison between islamic banks and conventional banks: evidence in indonesia. in h a el-karanshawy et al. (eds.), financial stability and risk management in islamic financial institutions. doha, qatar: bloomsbury qatar foundation. 2015. [21] j. goddard, p. molyneux, j.o.s. wilson, the profitability of european banks: a cross-sectional and dynamic panel analysis, manchester school. 72 (2004), 363–381. [22] j. a. haslem. a statistical analysis of the relative profitability of commercial banks. j. finance, 23(1) (1968), 167-176. [23] j. kakes, g. j. schinasi. toward a framework for safeguarding financial stability (june 2004). imf working paper no. 04/101. [24] d. iacobucci, n. saldanha, x. deng, a meditation on mediation: evidence that structural equations models perform better than regressions, j. consumer psychol. 17 (2007), 139–153. [25] p. jackson, w. perraudin. the nature of credit risk: the effect of maturity, type of obligator, and country of domicile. financial stability review, november, bank of england: london. pp. 122–150. 1999. [26] m. c. jensen, w. h. meckling. agency costs and the theory of the firm. j. financ. econ. 3(4) (1976), 305-360. [27] s. h. kargi. credit risk and the performance of nigerian banks. thesis, department of accouting, ahmadu bello university, zaira nigeria. 2011. [28] t. f. kolapo, r. k. ayeni, m. o. oke. credit risk and commercial banks' performance in nigeria: a panel model approach. aust. j. bus. manage. res. 2(2) (2012), 31-38. [29] c. memmel, p. raupach. how do banks adjust their capital ratios? j. financ. intermediation, 19(4) (2010), 509-528. int. j. anal. appl. 18 (6) (2020) 1082 [30] f. s. mishkin. the economics of money, banking, and financial markets. 7th ed., usa: columbia university, pp. 411-434. (2004). [31] f. s. mishkin. how big a problem is too big to fail? a review of gary stern and ron feldman's too big to fail: the hazards of bank bailouts. j. econ. literature, 44(4) (2006), 988-1004. [32] a. h. m. noman, s. pervin, m. m. chowdhury, h. banna. the effect of credit risk on the banking profitability: a case on bangladesh. glob. j. manage. bus. res. c finance. 15(3) (2015), 41-48. [33] a. hoffmann, g. gunkel, bank filtration in the sandy littoral zone of lake tegel (berlin): structure and dynamics of the biological active filter zone and clogging processes, limnologica. 41 (2011), 10– 19. [34] f. pasiouras, a. liadaki, c. zopounidis. bank efficiency and share performance: evidence from greece. appl. financ. econ. 18(14) (2008), 1121-1130. [35] e. richard, m. chijoriga, e. kaijage, c. peterson, h. bohman. credit risk management system of a commercial bank in tanzania. int. j. emerg. mark. 3(3) (2008), 323-332. [36] m. s. saeed, n. zahid. the impact of credit risk on profitability of the commercial banks. j. bus. financ. affairs, 5(2) (2016), 2167-0234. [37] m. segoviano, c. goodhart, banking stability measures, imf working paper 09/4. 2009. [38] f. a. takang, c. t. ntui, bank performance and credit risk management, dissertation, institutionen för teknik och samhälle, skövde, 2008. [39] y. tan. the impacts of risk and competition on bank profitability in china. j. int. financ. mark. inst. money, 40 (2016), 85-110. [40] t. w. koch, s. s. macdonald. bank management. south-western cengage learning, 7th edition, chapter 13. 1995. [41] m. ulici, a. chaibi, c. rault, shock and volatility transmissions between bank stock returns in romania: evidence from a var-garch approach, j. appl. bus. res. 30 (2014) 689-700. [42] u. uwuigbe, o. r. uwuigbe, b. oyewo. credit management and bank performance of listed banks in nigeria, j. econ. sustain. develop. 6(2) (2015), 27-32. international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 71-76 http://www.etamaths.com additive units of product system of hilbert modules biljana vujošević∗ abstract. in this paper we consider the notion of additive units and roots of a central unital unit in a spatial product system of two-sided hilbert c∗-modules. this is a generalization of the notion of additive units and roots of a unit in a spatial product system of hilbert spaces introduced in [b. v. r. bhat, m. lindsay, m. mukherjee, additive units of product system, arxiv:1501.07675v1 [math.fa] 30 jan 2015]. we introduce the notion of continuous additive unit and continuous root of a central unital unit ω in a spatial product system over c∗-algebra b and prove that the set of all continuous additive units of ω can be endowed with a structure of two-sided hilbert b−b module wherein the set of all continuous roots of ω is a hilbert b−b submodule. 1. introduction the notion of additive units and roots of a unit in a spatial product system of hilbert spaces is introduced and studied in [1, section 3]. in more details, an additive unit of a unit u = (ut)t>0 in a spatial product system e is a measurable section a = (at)t>0, at ∈ et, that satisfies as+t = asut + usat for all s,t > 0, i.e. a is ”additive with respect to the given unit u”. an additive unit a = (at)t>0 of a unit u = (ut)t>0 is a root if for all t > 0 〈at,ut〉 = 0. in the same paper it is, also, proved that the set of all additive units of a unit u is a hilbert space wherein the set of all roots of u is a hilbert subspace. the goal of this paper is to generalize the notion of additive units and roots of a unit in a spatial product system of hilbert spaces (from [1, section 3]) and to obtain some similar results as therein but in a more general context. to this purpose, we observe a spatial product system of two-sided hilbert modules over unital c∗-algebra b (it presents a product system that contains a central unital unit). we introduce the notion of continuous additive unit and continuous root of a central unital unit. also, we show that the set of all continuous additive units of a central unital unit is continuous in a certain sense. finally, we prove that the set of all continuous additive units of a central unital unit ω can be provided with a structure of two-sided hilbert b−b module wherein the set of all continuous roots of ω is a hilbert b−b submodule. throughout the whole paper, b denotes a unital c∗-algebra and 1 denotes its unit. also, we use ⊗ for tensor product, although � is in common use. the rest of this section is devoted to basic definitions. definition 1.1. a) a hilbert b-module f is a right b-module with a map 〈 , 〉 : f ×f →b which satisfies the following properties: • 〈x,λy + µz〉 = λ〈x,y〉 + µ〈x,z〉 for x,y,z ∈ f and λ,µ ∈ c; • 〈x,yβ〉 = 〈x,y〉β for x,y ∈ f and β ∈b; • 〈x,y〉 = 〈y,x〉∗ for x,y ∈ f ; • 〈x,x〉≥ 0 and 〈x,x〉 = 0 ⇔ x = 0 for x ∈ f ; 2010 mathematics subject classification. 46h25. key words and phrases. additive unit; hilbert module; c∗-algebra. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 71 72 vujošević and f is complete with respect to the norm ‖ ·‖ = ‖〈·, ·〉‖ 1 2 . b) a hilbert b−b module is a hilbert b-module with a non-degenerate ∗−representation of b by elements in the c∗-algebra ba(f) of adjointable (and, therefore, bounded and right linear) mappings on f . the homomorphism j : b → ba(f) is contractive. in particular, since c∗-algebra b is unital, the unit of b acts as the unit of ba(f). also, for x,y ∈ f and β ∈ b there holds 〈x,βy〉 = 〈β∗x,y〉 where βy = j(β)(y). for basic facts about hilbert c∗-modules we refer the reader to [5] and [6]. definition 1.2. a) a product system over c∗-algebra b is a family (et)t≥0 of hilbert b−b modules, with e0 ∼= b, and a family of (unitary) isomorphisms ϕt,s : et ⊗es → et+s, where ⊗ stands for the so-called inner tensor product obtained by identifications ub ⊗ v ∼ u ⊗ bv, u⊗vb ∼ (u⊗v)b, bu⊗v ∼ b(u⊗v), (u ∈ et, v ∈ es, b ∈b) and then completing in the inner product 〈u⊗v,u1 ⊗v1〉 = 〈v,〈u,u1〉v1〉; b) unit on e is a family u = (ut)t≥0, ut ∈ et, so that u0 = 1 and ϕt,s(ut ⊗us) = ut+s, which we shall abbreviate to ut⊗us = ut+s. a unit u = (ut) is unital if 〈ut,ut〉 = 1. it is central if for all β ∈b and all t ≥ 0 there holds βut = utβ. definition 1.3. the spatial product system is a product system that contains a central unital unit. for a more detailed approach to this topic, we refer the reader to [2], [8], [9], [4]. 2. additive units in this section we define all notions and prove auxiliary statements that are necessary for the proof of main result that we present in section 3. throughout the whole paper, ω = (ωt)t≥0 is a central unital unit in a spatial product system e = (et)t≥0 over unital c ∗-algebra b. definition 2.1. a family a = (at), at ∈ et, is said to be an additive unit of ω if a0 = 0 and as+t = as ⊗ωt + ωs ⊗at, s,t ≥ 0. definition 2.2. an additive unit a = (at) of a unit ω = (ωt) is said to be a root if 〈at,ωt〉 = 0 for all t ≥ 0. the previous definitions do not include any technical condition, such as measurability or continuity. it occurs that it is sometimes more convenient to pose the continuity condition directly on units. definition 2.3. for β ∈b, let fa,bβ : [0,∞) →b be the map defined by (1) f a,b β (s) = 〈as,βbs〉, s ≥ 0, where a,b are additive units of ω in e. we say that the set of additive units of ω s is continuous if the map fa,bβ is continuous for all a,b ∈ s, β ∈ b. we say that a is a continuous additive unit of ω if the set {a} is continuous, i.e. if the map f a,a β is continuous for each β ∈b. denote the set of all continuous additive units of ω by aω and the set of all continuous roots of ω by rω. remark 2.4. we should tell the difference between the continuous set of additive units of ω and the set of continuous additive units of ω. in the second case only f a,a β should be continuous for all a ∈s, β ∈b, whereas in the first case all fa,bβ should be continuous. the following example assures us that the set of all continuous additive units of a central unital unit ω in a spatial product system is not empty. additive units of product system of hilbert modules 73 example 2.5. for γ ∈ b, the family (as)s≥0, where as = sγωs = sωsγ, is an additive unit of ω since for s,t ≥ 0 there holds as+t = (s + t)γωs ⊗ωt = sγωs ⊗ωt + tωsγ ⊗ωt = sγωs ⊗ωt + tωs ⊗γωt = = sγωs ⊗ωt + ωs ⊗ tγωt = as ⊗ωt + ωs ⊗at and a0 = 0. since f a,a β : s 7→ 〈sωsγ,β(sωsγ)〉 = s 2γ∗βγ is a continuous mapping for all β ∈ b, the additive unit a belongs to aω. the properties of additive units of ω are given in the following lemma: lemma 2.6. 1. if a is a continuous additive unit of ω, then (2) 〈ωs,as〉 = s〈ω1,a1〉, s ≥ 0. 2. if a,b are continuous roots of ω and β ∈b, then (3) f a,b β (s) = sf a,b β (1), s ≥ 0. 3. if a is a continuous additive unit of ω, then a family (a′s)s≥0, where (4) a′s = as −〈ωs,as〉ωs, is a continuous root of ω. proof. 1. let ga : [0,∞) →b be the map defined by ga(s) = 〈ωs,as〉, s ≥ 0. for s,t ≥ 0 we obtain ga(s + t) = 〈ωs+t,as ⊗ωt + ωs ⊗at〉 = 〈ωs ⊗ωt,as ⊗ωt〉 + 〈ωs ⊗ωt,ωs ⊗at〉 = = 〈ωt,〈ωs,as〉ωt〉 + 〈ωt,〈ωs,ωs〉at〉 = 〈ωs,as〉 + 〈ωt,at〉 = ga(s) + ga(t) and ‖ga(s) −ga(0)‖2 = ‖〈ωs,as〉‖2 ≤‖ωs‖2‖as‖2 = = ‖〈as,as〉‖ = ‖f a,a 1 (s)‖→‖f a,a 1 (0)‖ = 0, s → 0. hence, the map ga is continuous. therefore, ga(s) = sga(1), i.e. 〈ωs,as〉 = s〈ω1,a1〉. 2. let s,t ≥ 0. since a,b ∈rω, we see that f a,b β (s + t) = 〈as ⊗ωt + ωs ⊗at,β(bs ⊗ωt + ωs ⊗ bt)〉 = = 〈ωt,〈as,βbs〉ωt〉 + 〈at,〈ωs,βωs〉bt〉 = 〈as,βbs〉 + 〈at,βbt〉 = f a,b β (s) + f a,b β (t) and ‖fa,bβ (s) −f a,b β (0)‖ 2 = ‖〈as,βbs〉‖2 ≤‖〈as,as〉‖‖β‖2‖〈bs,bs〉‖→ 0, s → 0. hence, the map f a,b β is continuous and, therefore, f a,b β (s) = sf a,b β (1). 3. for s,t ≥ 0, we obtain that a′s+t = as ⊗ωt + ωs ⊗at −〈ωs ⊗ωt,as ⊗ωt + ωs ⊗at〉ωs ⊗ωt = = as ⊗ωt + ωs ⊗at − (〈ωt,〈ωs,as〉ωt〉 + 〈ωt,〈ωs,ωs〉at〉)ωs ⊗ωt = = as ⊗ωt + ωs ⊗at −〈ωs,as〉ωs ⊗ωt −〈ωt,at〉ωs ⊗ωt = = (as −〈ωs,as〉ωs) ⊗ωt + ωs ⊗ (at −〈ωt,at〉ωt) = a′s ⊗ωt + ωs ⊗a ′ t and 〈a′s,ωs〉 = 0. therefore, a′ is a root of ω. let β ∈b. by (4) and (2), it follows that f a′,a′ β (s) = f a,a β (s) −s 2〈a1,ω1〉β〈ω1,a1〉, s ≥ 0. hence, the map f a′,a′ β is continuous which implies that a ′ ∈rω. � remark 2.7. let a be a continuous additive unit of ω. by (2) and (4), it can be decomposed as as = s〈ω1,a1〉ωs + a′s, s ≥ 0, where a′ is a continuous root of ω. 74 vujošević let a,b be two continuous additive units of ω. by remark 2.7, we can decompose them as (5) as = s〈ω1,a1〉ωs + a′s, bs = s〈ω1,b1〉ωs + b ′ s, s ≥ 0, where a′,b′ ∈rω. therefore, f a′,b′ β (1) = 〈a1 −〈ω1,a1〉ω1,β(b1 −〈ω1,b1〉ω1)〉 = = f a,b β (1) −〈a1,ω1〉β〈ω1,b1〉, β ∈b. let s ≥ 0 and β ∈b. since, by (3), there holds f a′,b′ β (s) = sf a′,b′ β (1), it follows that (6) f a′,b′ β (s) = sf a,b β (1) −s〈a1,ω1〉β〈ω1,b1〉. now, by (5) and (6), we obtain that (7) f a,b β (s) = sf a,b β (1) + (s 2 −s)〈a1,ω1〉β〈ω1,b1〉. it follows that the map f a,b β is continuous. therefore, we conclude that the set of all continuous additive units of ω is continuous in the sense of definition 2.3. 3. the result in this section we prove the main result. throughout the whole section, ω = (ωt)t≥0 is a central unital unit in a spatial product system e = (et)t≥0 over unital c ∗-algebra b. theorem 3.1. the set aω (the set of all continuous additive units of ω) is a b−b module under the point-wise addition and point-wise scalar multiplication. the set rω (the set of all continuous roots of ω) is a b−b submodule in aω. proof. let a = (as), b = (bs) ∈ aω and β ∈ b. for s ≥ 0, (a + b)s = as + bs, (aβ)s = asβ and (βa)s = βas. let s,t ≥ 0. since (a+b)s+t = (a+b)s⊗ωt +ωs⊗(a+b)t and f a+b,a+b β = f a,a β +f b,a β +f a,b β +f b,b β , it follows that a + b ∈aω. let γ ∈ b. since the unit ω is central, we obtain that (aγ)s+t = (aγ)s ⊗ ωt + ωs ⊗ (aγ)t. also, f aγ,aγ β (s) = γ ∗f a,a β (s)γ which implies that the map f aγ,aγ β is continuous. therefore, aγ ∈aω. similarly, (γa)s+t = (γa)s ⊗ωt + ωs ⊗ (γa)t. by remark 2.7, as = s〈ω1,a1〉ωs + a′s, a′ ∈rω, and we obtain that f γa,γa β (s) = s 2〈a1,ω1〉γ∗βγ〈ω1,a1〉 + f a′,a′ γ∗βγ(s). by (3), the map f γa,γa β is continuous. therefore, γa ∈aω. the associativity and the commutativity follow directly. the neutral element is 0 = (0s) and the inverse of a is −a = (−as). the other axioms of two-sided b−b module (aβ)γ = a(βγ), β(γa) = (βγ)a, β(a + b) = βa + βb, (a + b)β = aβ + bβ, (β + γ)a = βa + γa, a(β + γ) = aβ + aγ, 1a = a1 = a follow directly. if a,b ∈ rω, then 〈as + bs,ωs〉 = 0, 〈asβ,ωs〉 = β∗〈as,ωs〉 = 0 and 〈βas,ωs〉 = 〈as,β∗ωs〉 = 〈as,ωsβ∗〉 = 〈as,ωs〉β∗ = 0. hence, a + b,aβ,βa ∈rω. since also 0 = (0s) and −a = (−as) ∈rω, we see that rω is a b−b submodule in aω. � for every b 3 β ≥ 0 there is a map 〈 , 〉β : aω ×aω →b given by (8) 〈a,b〉β = 〈a1,βb1〉. proposition 3.2. the pairing (8) satisfies the following properties: 1. 〈a,λb + µc〉β = λ〈a,b〉β + µ〈a,c〉β for all a,b,c ∈aω and λ,µ ∈ c; 2. 〈a,bγ〉β = 〈a,b〉βγ for all a,b ∈aω and γ ∈b; 3. 〈a,b〉β = 〈b,a〉∗β for all a,b ∈aω; 4. 〈a,a〉β ≥ 0 for all a ∈aω; additive units of product system of hilbert modules 75 5. 〈a,a〉1 = 0 ⇔ a = 0 for all a ∈aω; 6. 〈a,γb〉1 = 〈γ∗a,b〉1 for all a,b ∈aω and γ ∈b. proof. 1, 2, 3 straightforward calculation. 4 since β ≥ 0, it follows that β = γ∗γ for some γ ∈b. thus, 〈a,a〉β = 〈a1,γ∗γa1〉 = 〈γa1,γa1〉≥ 0. 5 if 〈a,a〉1 = 0, then a1 = 0 by (8). by remark 2.7, as = s〈ω1,a1〉ωs + a′s, a′ ∈ rω and s ≥ 0, implying that as = a ′ s. therefore, 〈as,as〉 = s〈a′1,a′1〉 by (3). now, it follows that 〈as,as〉 = 0, i.e. as = 0 for all s ≥ 0. 6 straightforward calculation. � theorem 3.3. the set aω (the set of all continuous additive units of ω) is a hilbert b−b module under the inner product 〈 , 〉 : aω ×aω →b defined by (9) 〈a,b〉 = 〈a1,b1〉, a,b ∈aω. the set rω (the set of all continuous roots of ω) is a hilbert b−b submodule in aω. proof. we notice that the mapping 〈 , 〉 in (9) is equal to the mapping 〈 , 〉1 in (8). therefore, by theorem 3.1 and proposition 3.2, we obtain that 〈 , 〉 is a b-valued inner product on b−b module aω. therefore, aω is a pre-hilbert b−b module. now, we need to prove that aω is complete with respect to the inner product (9). let (an) be a cauchy sequence in aω and s ≥ 0. if β = 1 and a = b = am −an in (7), it follows that ‖ams −a n s‖ 2 ≤ (s2 + 2s)‖am1 −a n 1‖ 2 = (s2 + 2s)‖am −an‖2. (the last equality follows by (9).) thus, (ans ) is a cauchy sequence in es and denote (10) as = lim n→∞ ans . let ε > 0 and s,t ≥ 0. there is n0 ∈ n so that ‖ans −as‖≤ ε 3 , ‖ant −at‖≤ ε 3 and ‖ans+t−as+t‖≤ ε 3 for n > n0. then, ‖as+t −as ⊗ωt −ωs ⊗at‖≤‖as+t −ans+t‖ + ‖a n s+t −as ⊗ωt −ωs ⊗at‖≤ ≤‖as+t −ans+t‖ + ‖(a n s −as) ⊗ωt‖ + ‖ωs ⊗ (a n t −at)‖≤ ε. hence, a is an additive unit of ω. let β ∈b. by (1), (10) and (7), f a,a β (s) = limn→∞ f an,an β (s) = limn→∞ [sf an,an β (1) + (s 2 −s)〈an1 ,ω1〉β〈ω1,a n 1〉] = = sf a,a β (1) + (s 2 −s)〈a1,ω1〉β〈ω1,a1〉. hence, the map f a,a β is continuous, i.e. a ∈aω. by (9) and (10), ‖a n −a‖ = ‖an1 −a1‖→ 0, n →∞. therefore, aω is complete with respect to the inner product (9). let (an) be a sequence in rω satisfying lim n→∞ an = a. the only question is whether the continuous additive unit a belongs to rω. however, this immediately follows from (10) since 〈as,ωs〉 = lim n→∞ 〈ans ,ωs〉 = 0 for all s ≥ 0. � references [1] b. v. r. bhat, martin lindsay and mithun mukherjee, additive units of product system, arxiv:1501.07675v1 [math.fa] 30 jan 2015. [2] s. d. barreto, b. v. r. bhat, v. liebscher and m. skeide, type i product systems of hilbert modules, j. funct. anal. 212 (2004), 121–181. [3] b. v. r. bhat and m. skeide, tensor product systems of hilbert modules and dilations of completely positive semigroups, infin. dimens. anal. quantum probab. relat. top. 3 (2000), 519–575. [4] d. j. kečkić, b. vujošević, on the index of product systems of hilbert modules, filomat, 29 (2015), 1093-1111. [5] e. c. lance, hilbert c∗-modules: a toolkit for operator algebraists, cambridge university press, (1995). [6] v. m. manuilov and e. v. troitsky, hilbert c∗-modules, american mathematical society (2005). [7] m. skeide, dilation theory and continuous tensor product systems of hilbert modules, pqqp: quantum probability and white noise analysis xv (2003), world scientific. [8] m. skeide, hilbert modules and application in quantum probability, habilitationsschrift, cottbus (2001). [9] m. skeide, the index of (white) noises and their product systems, infin. dimens. anal. quantum probab. relat. top. 9 (2006), 617–655. 76 vujošević faculty of mathematics, university of belgrade, studentski trg 16-18, 11000 beograd, serbia ∗corresponding author: bvujosevic@matf.bg.ac.rs international journal of analysis and applications volume 19, number 3 (2021), 405-439 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-405 received september 1st, 2020; accepted september 16th, 2020; published april 12th, 2021. 2010 mathematics subject classification. 62e17, 05a15. key words and phrases. libby-novick kumaraswamy distribution; moment generating function; mean deviation; reliability measures; maximum likelihood estimation. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 405 libby-novick kumaraswamy distribution with its properties and applications abdul saboor1,*, zafar iqbal2, muhammad hanif1, munir ahmad1 1school of social sciences, national college of business administration & economics, lahore, pakistan 2department of statistics, government college satellite town gujranwala, pakistan *corresponding author: saboorfsd@hotmail.com abstract: the kumaraswamy distribution is one of the most popular probability distributions with applications to real life data. in this paper, an extension of this distribution called the libby-novick kumaraswamy (lnk) distribution is presented which is believed to provide greater flexibility to model scenarios involving skew-normal data than original one. analytical expressions for various mathematical properties including its cdf, quantile function, moments, factorial moments, conditional momennts, moment generating function, characteristic function, vitality function, information generating function, reliability measures, mean deviations, mean residual function, bonferroni and lorenz curves are derived.the parameters’ estimation of lnk distribution is undertaken using the method of maximum likelihood estimation. a simulation study for different values of sample sizes, to assess the performance of the parameters of lnk distribution is provided. for illustration and performance evaluation of lnk distribution three real-life data sets from the field of engineering and science adapted from earlier studies are used. on comparing the results to previously used methods, lnk distribution shows that it can give consistently better fit than other existing important lifetime models. it is found that the lnk distribution is more suitable and useful to study lifetime data. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-405 int. j. anal. appl. 19 (3) (2021) 406 1. introduction iqbal et al., (2017) [1] defined the libby-novick kumaraswamy (lnk) distribution. the probability density function (pdf) and cumulative distribution function (cdf) are respectively defined as ( ) ( ) ( ) 1 1 1 1 ; 0 1, , , 0 1 1 b a a b a x x f x abc x a b c c x − − + − =     − −  (1) ( ) ( ) 1 11 1 b a a x f x c x  − =   − −  (2) the libby-novick kumaraswamy (lnk) distribution is a continuous probability distribution with double-bounded support. it is very similar, in many respects, to the libby-novick beta (lnb, 1982) [2] distribution and kumaraswamy (kum, 1980) [3] distribution. the kum distribution is a two parameter distribution like beta distribution where as m.c. jones (2009) [4] find out numerous benefits of the kum distribution over classical beta distribution.the kum distribution is a special case of mcdonald’s (1984) [5] generalized beta of the first kind distribution. one key difference between the kum and beta distributions is the availability for the former, but not for the latter, of an invertible closed form cumulative distribution function presented by mitnick (2013) [6]. the two distributions lnk and lnb are very flexible and can take approximately the same shapes and the former distribution is also a generalized distribution of kum distribution. there are, however, important realistic differences between lnb and lnk distributions. on the one hand, the availability for the lnk , but not for the lnb distribution, of an invertible closedform cumulative distribution function makes the lnk distribution much better suited than the lnb for activities that require the generation of random varieties, in particular simulation modeling and simulation-based model estimation. in contrast, the lack of tractable-enough expressions for the mean and variance of the lnk distribution has stalled its utilization for modeling purposes; in spite of the advantages that the availability of an invertible closed-form cumulative distribution function entails, the lnk distribution may use rather sparingly in the modeling of stochastic phenomena and processes. libby-novick (1982) [2] derived two new multivariate probability density functions, which are the generalized forms of beta distribution int. j. anal. appl. 19 (3) (2021) 407 and f distribution. they proved that these distributions that seem to be suitable in utility modeling. they reproduced the same distributional form in the cases of marginal and conditional distributions. chen and novick (1984) [7] used the libby-novick (1982) [2] univariate generalized beta distribution as a prior and estimated the parameters of bernoulli and binomial distributions. these expressions are the generalized forms of the standard beta class. ristic et al., (2013) [8] derived new family of skewed distributions such as libby and novick’s generalized beta exponential distribution and found some useful properties of this family of distributions. cordeiro et al., (2014) [9] defined a family of distributions, named the libby-novick beta family of distributions, which includes the classical beta generalized and exponentiated generators. this extended family gave reasonable parametric fits to real data in several areas because the additional shape parameters controlled the skewness and kurtosis simultaneously. ali (2019) [10] worked on new form of libby-novick (nln) distribution and explored some properties of nln distribution. this model was compared with other distributions by fitting them to a real data set. rashid et al., (2020) [11] derived different entropy measures and characterized libby-novick generalized beta (lngb) distribution through various methods. iqbal et al., (2021) [12] derived mathematical properties of lngb distribution and applied it to modeling on three real data sets. in this paper we derive basic and advanced properties of lnk distribution and find applications of lnk distribution to three real data sets. in section 2, a detailed remarks about the graphs of pdf, hazard rate function, reverse hazarz rate function and survival function are provided. section 2 also contains the derivation and results of some important mathematical properties of lnk distribution. maximum likelihood estimators of three parameters of lnk distribution are derived in section 3. in section 4, a simulation of the parameters is carried out for different sample sizes. a number of deduced models are shown in section 5. in section 6, lnk distribution is compared with some other models through three data sets. some concluding remarks are presented in section 7. 2. mathematical properties 1.1 shape properties of the pdf the lnk density function defined in (1) has real flexibility and it is shown through graphs w.r.t. some different combinations of values of the parameters. int. j. anal. appl. 19 (3) (2021) 408 a 0.5 ,b 0.5 ,c 1 a 5.0 ,b 1.0 ,c 1 a 1.0 ,b 3.0 ,c 1 a 2.0 ,b 5.0 ,c 1 a 2.0 ,b 2.0 ,c 1 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 f x a 0.5 ,b 0.5 ,c 1.5 a 5.0 ,b 1.0 ,c 1.5 a 1.0 ,b 3.0 ,c 1.5 a 2.0 ,b 5.0 ,c 1.5 a 2.0 ,b 2.0 ,c 1.5 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 f x a 0.5 ,b 0.5 ,c 0.5 a 5.0 ,b 1.0 ,c 0.5 a 1.0 ,b 3.0 ,c 0.5 a 2.0 ,b 5.0 ,c 0.5 a 2.0 ,b 2.0 ,c 0.5 0.2 0.4 0.6 0.8 1.0 x 0.5 1.0 1.5 2.0 2.5 3.0 3.5 f x int. j. anal. appl. 19 (3) (2021) 409 figure 2.1 pdf graph of lnk distribution shapes of pdf (i) for a > 0, c > 0 and 0 < b < 1 then lnk distribution is u-shaped. (ii) for 0 < a < 1, b > 1 and 0 < c < 1 then lnk distribution is s-shaped. (iii) for a = b = 2, c ≥ 1 (c < 1) and if c →  (c → 0) then lnk distribution increases its positively skewed (negatively skewed) from the symmetric with decreasing mode (increasing mode). (iv) for a = b = c = 1, the lnk distribution is uniform distribution. (v) for a > 2, b > 2 and c ≥ 1 the lnk distribution is unimodel positively skewed with decreasing mode when c→ . (vi) for a > 2, b = 1, 0 < c < 1, the form of lnk distribution is an increasing. a 0.5 ,b 0.5 ,c 0.1 a 5.0 ,b 1.0 ,c 0.1 a 1.0 ,b 3.0 ,c 0.1 a 2.0 ,b 5.0 ,c 0.1 a 2.0 ,b 2.0 ,c 0.1 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 f x a 0.5 ,b 0.5 ,c 0.5 a 5.0 ,b 1.0 ,c 0.5 a 1.0 ,b 3.0 ,c 2.0 a 2.0 ,b 2.0 ,c 3.0 a 2.0 ,b 5.0 ,c 0.1 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 f x int. j. anal. appl. 19 (3) (2021) 410 (vii) for a > 2, b = 1, 1< c < 3.5, the lnk distribution increase but it increases slowly when c increases in the interval and for c ≥ 3.5 the lnk distribution again turns to unimodel. (viii) for b = c = 1, the lnk distribution is power distribution. (ix) for a = c = 1, the lnk distribution is lnk is a special case of kum-distribution or reflected exponentiated distribution. 1.2 distribution function the cumulative distribution function (cdf) of lnk distribution is ( ) ( ) ( ) 1 1 1 0 1 1 1 b a ax b a t t f x abc dt c t − − + − =  − −   (3) by making substitutions 1 + cz a c z t z and u z = = − (4) and after some simplification we have the expression of cdf as shown in (2) , , figure 2.2 cdf graph of lnk distribution 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 f x a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 f x a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 int. j. anal. appl. 19 (3) (2021) 411 2.3 quantile function the quantile function of lnk distribution is given by: ( )f x p= where 0 < p < 1 and it 1 ( )x f p − = 1 1 1 (1 ) b a a x p c x  − − =  − −  after simplification we find the quantile function of the lnk distribution as under; ( ) 1/ 1/ 1/ 1 (1 ) ; , , (0,1) 1 (1 )(1 ) a b b p q p a b c p c p  − − =   − − −  where ( ); , ,q p a b c is the inverse of lnk function or quantile of lnk function at p. clearly, the function has explicit form and it can be numerically solved through software for different set of parameters’ values.the graph can also be used to illustrate the behaviour of quantile function of lnk distribution. i. for a > 1, b > 1 by increasing c, the quantile value of the lnk distribution decreses comparatively. ii. for 1,a  c > 1, and for any value of b, the quantile value of the lnk distribution increases slowly for 0.5p  and for 0.5p  , the quantile value sharply increases. iii. for c → 0, the quantile value of the lnk distribution increases sharply. 2.4 hazard rate function the hazard or instantaneous rate function is denoted by h(x). the hazard function of x can be interpreted as instantaneous rate or the conditional probability density of failure at time x, given that the unit has survived until x. the hazard rate function of lnk distribution as ( ) 1 ( ; , , ) 1 (1 ) 1 a a a abcx h x a b c c x x − =  − − −  int. j. anal. appl. 19 (3) (2021) 412 figure 2.3 hazard rate function graph of lnk distribution a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 25 h x a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 25 h x a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 h x int. j. anal. appl. 19 (3) (2021) 413 the hazard rate function of lnk distribution is of bath-tub shape, increasing shape and decreasing shape for specific sets of values of parameters. 2.5 reverse hazard rate function the reverse hazard can be interpreted as an approximate probability of failure in  , + dx x , given that the failure has occurred in 0, x . the reverse hazard function ( ; , , )r x a b c of lnk distribution is defined as ( ) ( ) ( )( ) 1 1 (1 ) ( ; , , ) 1 (1 ) 1 (1 ) 1 a a b b b a a a abcx x r x a b c c x c x x − − − = − − − − − − figure 2.4 reverse hazard rate graph of lnk distribution a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 25 30 r x a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 r x int. j. anal. appl. 19 (3) (2021) 414 2.6 survival function the survival function or reliability function of lnk distribution is defined as ( ) 1 1 (1 ) b a a x s x c x  − =   − −  figure 2.5 survival function graph of lnk distribution a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 s x a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 s x int. j. anal. appl. 19 (3) (2021) 415 2.7 the rth moment the rth moments about origion of the lnk distribution is defined as 1 0 ( ) ( ) r r e x x f x dx=  by substitution (4) becomes   1 1 1 0 (1 ) 1 (1 ) r ba b bcy y dy c y − + − = − −  1 1 0 (1 ) r a bu b u du c u uc −  = −  + −   ( ) ( ) 1 1 00 / 1 (1 ) 1 / ! kr r b ka a k r a k bc u u u du r a k c −  − =  +   = − −      ( ) ( ) 1 1 0 0 / 1 1 (1 ) / ! kr r k ba a k r a k bc u u du r a k c −  + − =  +   = − −       ( ) ( ) ( ) / 0 / 1 ( ) 1 / 1, / ! k r r a k r a kb e x b r a k b c r a k c  =  +   = − + +      corollary ( ) ( ) ( ) 1/ 0 1 / 1 ( ) 1 1 / 1, 1 / ! k a k a kb e x b a k b c a k c  =  +   = − + +      ( ) ( ) ( )2 2/ 0 2 / 1 ( ) 1 2 / 1, 2 / ! k a k a kb e x b a k b c a k c  =  +   = − + +      2.8 moment generating function the moment generating function of the lnk distribution about zero is ( ) ( )tm t = e e xx ( ) = 0 0 t1 = 1 1, ! ! r rk a r k bc r r k b k b r a c a k r a −   =       + − + +                  0 e ! r x r x r  = =  int. j. anal. appl. 19 (3) (2021) 416 2.9 factorial moments this section devotes to inceasing and decreasing factorial moments of lnk distribution as under: 2.9.1 decreasing factorial moments of lnk distribution the decreasing factorial moments of the lnk distribution is defined as: ( ) ( ) n r r = 0 = s n, r e x ( )n r = 0 0 n, r 1 = 1 1, ! kr a k s r r bc k b k b r a c a k a − =       + − + +                 where ( )s n, r are the stirling’s numbers of first kind. 2.9.2 increasing factorial moments of lnbd the increasing factorial moments of the lnk distribution is defined as: ( ) ( ) n r r = 0 = n, r e xd ( )n r = 0 0 n, r 1 = 1 1, ! kr a k d r r bc k b k b r a c a k a − =       + − + +                 where ( )n, rd can be deduced from the relation 2.10 negative moments the negative moments of the lnk distribution is defined as: ( ) ( ) 1 r r 0 e x = ; a, b dx f x x − −  by applying the substitution (4), we have ( )r 0 1 e x = ( 1) 1 1, kr ka k n r bc b k b k c a  − =       − − − + +            int. j. anal. appl. 19 (3) (2021) 417 2.11 incomplete moments the incomplete moments of the lnk distribution is defined as: 0 ( ) ( ) x r r x x f x dx =  1 1 1 0 (1 ) 1 (1 ) x r a a b b a x abcx x dx c x − − + − =  − −   0 1 1 ( ; 1, ) 1 ! k a r a a ka r k b cx ra i k b r c x cx a kc a  =    +     = − + +  − +         2.12 scaled total time for aging properties the scaled total time of the lnk distribution is defined as: ( )( ) ( ) ( ) 1 0 0 1 1 x x f y s f x s y dy f t dt dy   = =  ( )( ) ( ) 0 1 x f s f x x f y dy    = −     ( )( ) 0 1 1 1 (1 ) bx a f a y s f x dy c y  − =   − −   2.13 conditional moments the conditional moments of the lnk distribution is defined as: ( ) 1 r r 1 m = t t dt f x f  by applying the substitution (4) and after some simplification, we have ( )( ) ( ) r 0 1 1 1 m = 1 ( ; 1, ) 11 ! b a k a rb r a a a ka r k c x b cx ra i k b r c x cx ax kc a   =     +  − −     − − + +  − +    −       2.14 mean residual function the mean residual function of the lnk distribution is defined as: r r m = e x x > x    int. j. anal. appl. 19 (3) (2021) 418 ( )m = e x x > x x x −  ( )m = e x x > x x x  −  ( )( ) ( ) 1 1 0 1 1 1 1 1 = 1 ( ; 1, ) 1 11 ! b a k a b a a a ka k c x b cxa i k b x c x cx ax kc a   =     +  − −    − − + + −  − +    −       2.15 vitality function the vitality function of the lnk distribution is defined as: ( )v = e x x > x x   ( ) ( )( ) ( ) 1 1 0 1 1 1 1 1 v = = 1 ( ; 1, ) 1 11 ! b a k a b a a a ka k c x b cxa x i k b c x cx ax kc a   =     +  − −    − − + +  − +    −       2.16 geometric vitality function the geometric vitality function of the lnk distribution is defined as: ( )logg = e log x x > x x   ( ) ( ) ( ) 1 1 logg = log t t dt 1 f x x x f −  ( ) ( ) ( ) 1 0 0 1 = log t t dtlog t t dt 1 f x x f f     −     ( ) ( ) ( )( ) ( ) ( ) 1 0 1 1 ln c 1 1 1 1 + 1 b 1, log t t dt 1 f x c i x i b b i b f a a i    =     = − − + − + −    −        ( ) ( ) ( )( ) ( ) ( ) ( ) 1 1 1 1 0 1 1 1 0 1 ln c 1 1 1 1 + 1 b 1, (1 ) (ln ) c1 1 f x (1 ) ln( ) a a a a cx c xi b i cx c x b b b b i b u u du a a i a b u c u uc du a   − −  − = − − −       − − + − + − −         =   −     + − + −       int. j. anal. appl. 19 (3) (2021) 419 2.17 characteristics function the characteristic function of the lnk distribution is defined as: ( ) ( )tt = e ei xx ( ) 0 0 1 1 1, ! ! r rk a r k itbc r r k b k b r a c a k r a −   = =      =  + − + +                  0 e ! r x r x r  = =  2.18 information generating function the information generating function of the lnk distribution is defined by p(s) and found from ( ) ( )1sp s e f −= ( ) ( ) 1 11 b+1 0 1 abc dx 1 1 s b a a a x x c x − − −  =   − −    by applying the substitution (4), we have ( ) ( ) 11 ( 1)/ ( 1)(s 1)/ ( 1) 0 0 ( 1)(1 ) / 1 1 (1 ) (( 1)(1 ) / ) ! i s s s a a a s b i i a s a i p s a b c u u u du a s a i c  − − − − − =  + − +   = − −   + −     ( ) 11 ( 1)/ ( 1)( 1)/ ( 1) 0 0 ( 1)(1 ) / 1 1 (1 ) (( 1)(1 ) / ) ! i s s s a a s a i s b i a s a i a b c u u du a s a i c  − − − − + − =  + − +   = − −   + −     ( ) ( )1 ( 1)/ 0 ( 1)(1 ) / 1 1 ( 1)( 1) / 1, ( 1) 1 (( 1)(1 ) / ) ! i s s s a i a s a i a b c b a s a i s b a s a i c  − − =  + − +   = − − − + + − +   + −    2.19 some other measures of averages 2.19.1 harmonic mean this section contains harmonic mean of x for the lnk distribution the harmonic mean of x is as under: 1 1 x h e x =       consider 1 0 1 1 ( )e f x dx x x   =     int. j. anal. appl. 19 (3) (2021) 420 1 2 1 1 0 (1 ) [1 (1 ) ] a a b a b x x abc dx c x − − + − = − −  by substituting a x y= , we have 1 1 1 1 0 (1 ) [1 (1 ) ] ba b bcy y dy c y − − + − = − −  after substitution (4), and simplification, we have 1 1 1 1 0 (1 ) [ ] ba a bu u du c u uc − − − − = + −  1 1 1 1 0 (1 ) ( ) ba abu u c u uc du − − = − + − 11 1 1 00 1 / 1 (1 ) (1 ) i ba a i a bc u u u du i c −  − =    = − − −       1 0 1 /1 1 1 (1 ) ( 1, ) i a i a e bc b i b ix c a  =   −    = − − + +           1 0 1 1. 1 / 1 1 (1 ) ( 1, ) ix a i h for a a bc b i b c ai  = =    −  − − + +       corollary if c=1, then 1 1 1, x h bb b a = −  +    is the harmonic mean of kum distribution 2.19.2 geometric mean the logarithm of the geometric x g of a distribution with random variable x is the arithmetic mean of( )ln x , or, equivalently, its expected value ( )ln lnxg e x= . 1 0 (ln ) (ln ) ( )e x x f x dx=  int. j. anal. appl. 19 (3) (2021) 421 1 1 1 1 0 (1 ) (ln ) (ln ) 1 (1 ) a a b b a abcx x e x x dx c x − − + − =  − −   by substitution a x y= and (4), after simplification, we have ( ) ( )( ) ( ) 1 1 ln c 1 1 = exp 1 1 + 1 b 1, c i x i g b b i b a a i    =    − − + − +        2.19.3 mode mode is obtained by solving ( ) d = 0 d f x x ( ) ( ) ( ) ( ) b 1a a 1 a + b c 1 = b a, b 1 1 c −− − − −   x x f x x mode of lnk distribution is obtained by solving ( ) 0 d f x dx = 1 1 1 ( 1) (1 )( 1) '( ) { } ( ) (1 ) [1 (1 ) ] a a a a a a b x a b c x f x f x x x c x − − − − + − = − − − − − 1 1 1 ( 1) (1 )( 1) 0 (1 ) [1 (1 ) ] a a a a a a b x a b c x x x c x − − − − + − − − = − − − multiplying both sides by x ( 1) (1 )( 1) 1 0 (1 ) [1 (1 ) ] a a a a a b x a b c x a x c x − + − − − − = − − − put a x y= ( 1) y (1 )( 1) 1 0 1 1 (1 ) a b a b c a y c y − + − − − − = − − − ( 1)(1 y)(1 y cy) {a(b 1) y(1 y cy)} {a(1 b)(c 1) y(1 y)} 0a − − − + − − − + − + − − = ( )( ) ( )22 (1 ) 2 1 1 0a c y c ab y a− − + − + + − = 2 0ay by c+ + = 2 b b 4ac y = 2a −  − int. j. anal. appl. 19 (3) (2021) 422 1/ 2 b b 4ac x = 2a a  −  −       where a y x= 2 (1 )a a c= − − 2b c abc= − − ( 1)c a= − 2.20 points of inflection the points of the inflection of the lnk distribution are found from '' ( ) 0f x = under ''' ( ) 0f x  or equavalentally as 2 2 ln ( ) 0 d f x dx = and 3 3 ln ( ) 0 d f x dx  where as 2 2 2 2 2 2 2 ln ( ) ( 1) ( 1) [( 1) ] (1 x ) (1 )(1 ) [( 1) (1 ) ] [1 (1 ) ] a a a a a a d f x a a b x a x dx x a b c x a c x c x − − − − − − + = − + − + − − + − − − so 2 2 2 ( 1) ( 1) [( 1) ] (1 x ) a a a a a b x a x x − − − − − + − − + 2 2 (1 )(1 ) [( 1) (1 ) ] [1 (1 ) ] a a a a b c x a c x c x − + − − + − − − =0 after some simplification 4 3 2 0ay by cy dy e+ + + + = where a y x= 2 ( 1)(1 )a a c= − + − 2 (1 )[(2 )( 2 ) (1 )]b c c a a abc a= − − + − − + 2 2 (1 )(4 6) ( 2)c c a abc c c= − − − − − ( 1)[( 2)( 2) ]d a c a abc= − − + + ( 1)e a= − int. j. anal. appl. 19 (3) (2021) 423 table 2.1 points of inflection of lnk distribution a b c point (s) of inflection 1 2 0.1 0.8482 1 2 0.2 0.6585 1 2 0.3 0.4146 1 2 0.4 0.0894 1 3 0.1 0.7318 1 3 0.2 0.3965 2 3 0.1 0.4432 , 0.8678 2 3 0.2 0.5681 , 0.6607 2 4 0.1 0.4625, 0.8089 2 5 0.1 0.4888, 0.7454 2 6 0.1 0.5342, 0.6644 3 4 0.1 0.5492, 0.8741 3 5 0.1 0.5665, 0.8335 3 6 0.1 0.5897, 0.7881 3 7 0.1 0.6294, 0.7273 2.21 bonferroni curve the bonferroni curve of the lnk distribution is as: ( )( ) ( ) ( ) 0 1 x f b f x y f y dy f x =  ( )( ) ( ) 0 0 1 1 1 1 ( ; 1, ) 1 1 ! = 1 1 1 1 11 1, 11 1 ! k a a a k f b ka a k k cxa i k b c x cx a k a b f x k x a b k b c x c a k a  =  =    +     − + +  − +           +   −       − + +       − −                2.22 lorenz curve the lorenz curve of the lnk distribution is as: ( )( ) ( ) ( )( )l f x f x b f x= int. j. anal. appl. 19 (3) (2021) 424 ( )( ) 0 0 1 1 1 1 ( ; 1, ) 1 1 ! 1 1 1 1 1, 1 ! k a a a k k k k cxa i k b c x cx a k a l f x k a b k b c a k a  =  =    +     − + +  − +        =    +       − + +                2.23 gini coefficient the gini coefficient of the lnk distribution is explained as: ( )( ) 1 21 0 1 1g f x dx − = − − ( ) 2 1 0 1 1 1 1 1 b a a x dx c x  − = −   − −   2.24 mean deviations the mean deviation of the lnk distribution from arithmetic mean and median are denoted by ( )1δ y and ( )2δ y respectively and are found from ( ) ( ) 1 1 0 δ = y y f y dy− ( ) ( ) ( ) ( ) 1 1 0 0 = + f y dy y f y dy y f y dy f y dy      − −    ( ) ( ) = 2 2 f j  − ( ) 1 0 1 1 1 1 = 2 1 2 1 ( ; 1, ) 11 1 1 ! b ka a a a a ka k b ca i k b c c c a kc a        =    +   −    − − − + +     − − − +            the mean deviation about median ( ) ( ) 1 2 0 δ = y y m f y dy− ( ) ( ) ( ) ( ) 1 1 0 0 = + m m m m m f y dy y f y dy y f y dy m f y dy− −    int. j. anal. appl. 19 (3) (2021) 425 ( ) = 2 j m − 1 0 1 1 1 = 2 1 ( ; 1, ) 1 1 ! k a a a ka k b cma i k b c m cm a kc a   =    +     − − + +  − +         3. estimation here, we consider 3.1 maximum-likelihood estimation the likelihood function of random sample 1 2, ,..., nx x x of observation is given by 1 2 1 ( , ,..., ;a, b,c) ( , , , ) n n i i f x x x f x a b c = = 1 ( , , , ) ( , , , ) n i i i l x a b c f x a b c = = ( ) ( ) 1 1 1 1 1 1 abc (1 ) ( , , , ) 1 (1 ) a b n n n a i i i i i b n a i i x x l x a b c c x − − = = + =     −        =   − −       taking ln both sides ( ) 1 1 1 ln ln ( 1) ln (b 1) ln(1 ) (1 ) ln 1 (1 c) n n n a a i i i i i i l n abc a x x b x = = = = + − + − − − + − −   it follows that the maximum-likelihood estimates ( )a, b, c , say ( )ˆ垐a, b, c , are the simultaneous solutions of the equations: ( )1 1 1 ln (1 ) ln ln ( 1) (1 ) (1 ) 1 (1 c) x a an n n i i i i i a a i i ii i l n x x c x x x b b a a x= = =  − = + − − + +  − − −    ( ) 1 1 ln(1 ) ln 1 (1 c) x n n a a i i i i l n x b b = =  = + − − − −    int. j. anal. appl. 19 (3) (2021) 426 1 (1 ) 1 (1 c) x an i a i i l n x b c c =  = − +  − −  for the ml estimators, we solve the following three equations simultaneously we have 0, 0, 0 l l l a b c    = = =    since the equations are of implicit and have the complicated forms. therefore, for numerically solution of the equations we can use differenent softwares like the r mathematica 10.2 etc. 4. simulation study table 1 the bias, mse values for the lnk model when a=b=c=2.0 n=10 n=20 n=30 n=50 n=70 n=100 bias(a) 0.6789 0.2966 0.1591 0.0732 0.0398 0.0134 bias(b) 148.4399 98.1754 80.9118 56.4929 42.0678 37.0448 bias(c) 519.4312 63.2427 5.1913 0.3873 -0.1171 -0.6091 mse(a) 2.3026 0.7537 0.3908 0.1723 0.1132 0.0843 mse(b) 45131.8214 22559.3813 17215.8524 9705.1161 5973.0134 5221.2081 mse(c) 4729852.1234 592121.2123 17854.2713 33.3845 15.21264 4.6655 n=150 n=200 n=300 n=500 n=700 n=1000 bias(a) 0.0165 0.0013 0.0025 -0.0071 -0.0157 -0.0134 bias(b) 25.9659 17.9847 12.0603 6.9705 5.6428 4.7767 bias(c) -0.7730 -0.9022 -0.9606 -0.9643 -0.9604 -0.9707 mse(a) 0.0578 0.0445 0.0286 0.0179 0.0123 0.0081 mse(b) 2688.8471 1429.0832 685.7248 176.8713 80.2722 36.4003 mse(c) 2.2229 1.8662 1.5139 1.3256 1.2007 1.1071 int. j. anal. appl. 19 (3) (2021) 427 table 2 the bias, mse values for the lnk model when a=b=c=2.5 n=10 n=20 n=30 n=50 n=70 n=100 bias(a) 0.7101 0.1423 0.0226 -0.0336 -0.0764 -0.0892 bias(b) 148.6944 83.5053 70.3768 49.9135 40.9383 28.8218 bias(c) 908.6814 240.0717 101.7437 6.0567 3.3570 1.6298 mse(a) 2.5889 0.6457 0.4488 0.2476 0.1627 0.1103 mse(b) 46025.2324 19184.2612 13459.1232 7983.5764 5993.6893 3566.5871 mse(c) 8932954 2172753 858508 1049.3921 181.7757 45.5042 n=150 n=200 n=300 n=500 n=700 n=1000 bias(a) -0.1165 -0.1040 -0.1170 -0.1267 -0.1286 -0.1309 bias(b) 16.4083 11.5937 8.2968 4.7545 3.6887 3.1612 bias(c) 0.9505 1.0242 0.6185 0.4136 0.4096 0.2923 mse(a) 0.0804 0.0619 0.0491 0.0373 0.0310 0.0282 mse(b) 1369.2381 791.9238 402.1093 108.3226 41.8772 16.9204 mse(c) 15.8173 13.1791 6.4383 3.0282 2.2463 1.4012 table 3 the bias, mse values for the lnk model when a=b=c=3.0 n=10 n=20 n=30 n=50 n=70 n=100 bias(a) 0.4603 0.0535 -0.0688 -0.1884 -0.2438 -0.2725 bias(b) 139.5197 81.7559 59.6459 39.1355 28.2413 16.5283 bias(c) 1318.3864 540.5148 167.8884 61.5224 12.9443 9.0277 mse(a) 2.1234 0.9155 0.6307 0.3236 0.2394 0.2077 mse(b) 43208.6434 18732.8834 11124.5823 5996.8751 3686.9514 2041.9723 mse(c) 12800727 4713924 1058158 285476.4 1760.2651 455.9306 n=150 n=200 n=300 n=500 n=700 n=1000 bias(a) -0.2895 -0.3107 -0.3037 -0.3063 -0.3198 -0.3133 bias(b) 10.2456 6.4328 3.8052 2.2502 1.5573 1.3494 bias(c) 7.5233 7.4066 5.9516 5.0419 5.2533 4.9514 mse(a) 0.1698 0.1596 0.1384 0.1194 0.1204 0.1101 mse(b) 845.9625 368.9467 139.7221 35.0732 8.4393 3.1515 mse(c) 238.0884 154.0213 83.8912 46.6563 42.8393 34.4719 int. j. anal. appl. 19 (3) (2021) 428 5. some transformed distributions 5.1 following table show the dffierent transformation of lnk distribution: sr. no. a b c transformation resulting distribution 1 a b 1 x kumaraswamy distribution (1980) 2 a b 1 1 – x corderio (2012) distribution ( ) ( )( ) = 11 1 ; 0 < < 1, a,b > 0 b a f x x x − − 3 1 b 1 x b(1,b) and kum(1,b) power distribution 4 a 1 1 x b(a,1) and kum (a,1) power distribution 5 a 1 1 -logx exp(a) 6 1 b 1 -log(1-x) exp(b) 7 a b 1 x kum(a,b) or g 1b (a,1,1,b) 8 1 b 1 1 x x+ ( ) ( ) 1 1 1 b f x x = − + 0 < < , b > 0x  9 1 b c 1 cx x cx− + ( ) ( ) = 1 1 ; 0 < < 1, b > 0 b f x x x − − 10 1 1 1 x uniform (0,1) 5.2 asymptotes and shapes the asymptotes of (1.1), (1.2) and (3.1) as 0,1x → are given by ( ) pdf f x ( ) hrf h x 0x → a 1 , , 1a b c x where a b c −  a 1 , , 1a b c x where a b c −  1x → ( ) b 1 1 , , 1 a a b c x where a b c − −  ( )1 a abc x− , , 1where a b c  int. j. anal. appl. 19 (3) (2021) 429 6. application in order to prove that lnk distribution can be a better model than the power distribution, beta distribution with (a = 1), beta distribution, kumaraswamy distribution, let us use three real data sets. the following tables show the numerical values with mles and their corresponding standard errors (in parentheses) of the model parameters including loglikelihood, kolmogorovsmirnov test (ks), akaike information criterion (aic) and consistent akaike information criterion (caic) for comparing lnk distribution with the power distribution, beta distribution with (a =1), beta distribution, kumaraswamy distribution. it is quite evident from the reports that lnk distribution is better. the plots of the fitted distributions to real datasets are shown in figures. data set 1: the following right to skewed dataset presented by cordeiro and brito (2012) [13] is obtained from the measurements on petroleum rock samples. the data consists of 48 rock samples from a petroleum reservoir. the dataset corresponds to twelve core samples from petroleum reservoirs that were sampled by four cross-sections. each core sample was measured for permeability and each cross-section has the following variables: the total area of pores, the total perimeter of pores and shape. we analyze the shape perimeter by squared (area) variable and the observations are: 0.0903296, 0.2036540, 0.2043140, 0.2808870, 0.1976530, 0.3286410, 0.1486220, 0.1623940, 0.2627270, 0.1794550, 0.3266350, 0.2300810, 0.1833120, 0.1509440, 0.2000710, 0.1918020, 0.1541920, 0.4641250, 0.1170630, 0.1481410, 0.1448100, 0.1330830, 0.2760160, 0.4204770, 0.1224170, 0.2285950, 0.1138520, 0.2252140, 0.1769690, 0.2007440, 0.1670450, 0.2316230, 0.2910290, 0.3412730, 0.4387120, 0.2626510, 0.1896510, 0.1725670, 0.2400770, 0.3116460, 0.1635860, 0.1824530, 0.1641270, 0.1534810, 0.1618650, 0.2760160, 0.2538320, 0.2004470. data set 2 the symmetric behavior of the following dataset, discussed by dasgupta (2011) [14], consists of 50 observations relates to holes operation on jobs made of iron sheet. this dataset is as follows: 0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16. int. j. anal. appl. 19 (3) (2021) 430 data set 3: the following second data set presented by cordeiro and brito (2012) [13], displays the skewed symmetric trend of data. this data discusses the total milk production in the first birth of 107 cows from sindi race. these cows are property of the carnaúba farm which belongs to the agropecuária manoel dantas ltda (amda), located in taperoá city, paraíba (brazil). the data of proportion of total milk is as under: 0.4365, 0.4260, 0.5140, 0.6907, 0.7471, 0.2605, 0.6196, 0.8781, 0.4990, 0.6058, 0.6891, 0.5770, 0.5394, 0.1479, 0.2356, 0.6012, 0.1525, 0.5483, 0.6927, 0.7261, 0.3323, 0.0671, 0.2361, 0.4800, 0.5707, 0.7131, 0.5853, 0.6768, 0.5350, 0.4151, 0.6789, 0.4576, 0.3259, 0.2303, 0.7687, 0.4371, 0.3383, 0.6114, 0.3480, 0.4564, 0.7804, 0.3406, 0.4823, 0.5912, 0.5744, 0.5481, 0.1131, 0.7290, 0.0168, 0.5529, 0.4530, 0.3891, 0.4752, 0.3134, 0.3175, 0.1167, 0.6750, 0.5113, 0.5447, 0.4143, 0.5627, 0.5150, 0.0776, 0.3945, 0.4553, 0.4470, 0.5285, 0.5232, 0.6465, 0.0650, 0.8492, 0.8147, 0.3627, 0.3906, 0.4438, 0.4612, 0.3188, 0.2160, 0.6707, 0.6220, 0.5629, 0.4675, 0.6844, 0.3413, 0.4332, 0.0854, 0.3821, 0.4694, 0.3635, 0.4111, 0.5349, 0.3751, 0.1546, 0.4517, 0.2681, 0.4049, 0.5553, 0.5878, 0.4741, 0.3598, 0.7629, 0.5941, 0.6174, 0.6860, 0.0609, 0.6488, 0.2747. estimated parameters by mle with their s.e. and goodness of fit data set 1 model ̂ ̂ ̂ ln( )l ks aic caic power 0.63 (0.0909) -6.0118 0.4295 14.0237 14.1107 beta ( =1) 3.9643 (0.5722) 30.2205 0.9156 62.4412 62.5281 kumaraswamy 44.6597 (17.5894) 2.7187 (0.2937) 52.4915 0.1533 108.9831 109.2497 beta 5.9417 (1.1825) 21.2057 (4.3513) 55.6002 0.1427 115.2004 115.4671 libby novick kumaraswamy 5.5040 (0.6078) 0.7469 (0.2042) 10.4007 (130.7012) 57.7939 0.0852 121.588 122.1334 int. j. anal. appl. 19 (3) (2021) 431 figure 6.1: pdf, cdf and q-q graphs of the densities for data set 1 int. j. anal. appl. 19 (3) (2021) 432 estimated parameters by mle with their s.e. and goodness of fit data set 2 model ̂ ̂ ̂ ln( )l ks aic caic power 0.5033 (0.0711) -15.0139 0.4364 32.0279 32.1113 beta ( =1) 5.4693 (0.7735) -44.0998 0.8786 90.1997 90.2831 kumaraswamy 33.1367 (13.9342) 2.0773 (0.2551) -56.0686 0.0902 116.1374 116.3927 beta 2.6825 (0.5074) 13.8656 (2.8295) -54.6066 0.1214 113.2133 113.4686 libby novick kumaraswamy 2.0376 (0.2687) 226.0664 (980.0583) 0.1342 (0.5945) -56.2244 0.0900 118.4489 118.9707 figure 6.2: pdf and cdf graphs of the compitators distributions for data set 2 int. j. anal. appl. 19 (3) (2021) 433 figure 6.3: q-q plots of the densities for data set 2 estimated parameters by mle with their s.e. and goodness of fit data3 model ̂ ̂ ̂ ln( )l− ks aic caic power 1.1123 (0.1075) -0.58545 0.2418 3.17090 3.209 beta ( =1) 1.4181 (0.1371) -5.8304 0.9669 13.6608 13.6989 kumaraswamy 3.4363 (0.5822) 2.1948 (0.2224) -25.3946 0.0769 54.7893 54.9047 beta 2.4125 (0.3145) 2.8296 (0.3745) -23.7772 0.0816 51.5544 51.6698 libby novick kumaraswamy 1.7588 (1.0411) 18.3624 (201.0970) 0.1044 (1.2976) -27.2446 0.0698 60.4892 60.7222 int. j. anal. appl. 19 (3) (2021) 434 figure 6.4: pdf and cdf graphs of the compitators distributions for data set 3 figure 6.5: q-q plots of the densities for data set 3 int. j. anal. appl. 19 (3) (2021) 435 6 conclusion the lnk distribution has some properties like the lnb distribution but, there are numerous benefits of the lnk distribution over lnb distribution: the lnk distribution has simple closed form of both its cdf and quantile function and that’s why it is easy to use in simulation studies.the distribution and quantile function of lnk do not involve any special functions. to compare the proposed model with other models, we apply these models to three sets of real data from different fields of science and engineering and it is examined by using well-known statistics. we conclude that the lnk distribution is better than the power, beta with (a=1), beta distribution and kumaraswamy distribution. int. j. anal. appl. 19 (3) (2021) 436 appendix 1: quantile values of lnk distribution a b c 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.999 0.5 0.5 0.1 0.49155 0.72089 0.83237 0.89632 0.93652 0.96296 0.98050 0.99171 0.99798 0.99998 1 0.5 0.5 0.2 0.29135 0.54420 0.70361 0.80797 0.87890 0.92795 0.96158 0.98353 0.99597 0.99996 1 0.5 0.5 0.6 0.07899 0.23413 0.40237 0.55900 0.69444 0.80539 0.89110 0.95181 0.98798 0.99988 0.99999 0.5 0.5 1.0 0.03610 0.12960 0.26010 0.40960 0.56250 0.70560 0.82810 0.92160 0.98010 0.99980 0.99999 0.5 0.5 1.5 0.01828 0.07438 0.16780 0.29416 0.44444 0.60493 0.75831 0.88581 0.97037 0.99970 0.99999 0.5 0.5 3.0 0.00525 0.02493 0.06634 0.13845 0.25000 0.40495 0.59472 0.79012 0.94204 0.99940 0.99999 0.5 0.5 6.0 0.00141 0.00734 0.02185 0.05224 0.11111 0.21777 0.39386 0.64000 0.88898 0.99880 0.99998 0.5 0.5 10 0.00052 0.00283 0.00888 0.02278 0.05325 0.11851 0.25277 0.49827 0.82493 0.99800 0.99998 1.0 1.0 0.1 0.52631 0.71428 0.81081 0.86956 0.90909 0.93750 0.95890 0.97561 0.98901 0.99899 0.99990 1.0 1.0 0.2 0.35714 0.55555 0.68181 0.76923 0.83333 0.88235 0.92105 0.95238 0.97826 0.99798 0.99980 1.0 1.0 0.6 0.15625 0.29411 0.41666 0.52631 0.62500 0.71428 0.79545 0.86956 0.93750 0.99397 0.99940 1.0 1.0 1.0 0.10000 0.20000 0.30000 0.40000 0.50000 0.60000 0.70000 0.80000 0.90000 0.99000 0.99900 1.0 1.0 1.5 0.06896 0.14285 0.22222 0.30769 0.40000 0.50000 0.60869 0.72727 0.85714 0.98507 0.99850 1.0 1.0 3.0 0.03571 0.07692 0.12500 0.18181 0.25000 0.33333 0.43750 0.57142 0.75000 0.97058 0.99700 1.0 1.0 6.0 0.01818 0.04000 0.06666 0.10000 0.14285 0.20000 0.28000 0.40000 0.60000 0.94285 0.99403 1.0 1.0 10.0 0.01098 0.02439 0.04109 0.06250 0.0909 0.13043 0.18918 0.28571 0.47368 0.90825 0.99008 2.0 2.0 0.1 0.59248 0.73576 0.81319 0.86269 0.89751 0.92368 0.94444 0.96184 0.97764 0.99449 0.998371 2.0 2.0 0.2 0.46139 0.60921 0.70282 0.76984 0.82120 0.86253 0.89722 0.92775 0.95673 0.98907 0.99675 2.0 2.0 0.6 0.28757 0.40544 0.49548 0.57148 0.63906 0.70143 0.76103 0.82049 0.88475 0.96824 0.99034 int. j. anal. appl. 19 (3) (2021) 437 2.0 2.0 1.0 0.22653 0.32492 0.40415 0.47476 0.54119 0.60625 0.67251 0.74349 0.82690 0.94868 0.98406 2.0 2.0 1.5 0.18656 0.27009 0.33935 0.40308 0.46517 0.52843 0.59585 0.67213 0.76838 0.92582 0.97637 2.0 2.0 3.0 0.13308 0.19456 0.24718 0.29735 0.34831 0.40283 0.46458 0.54018 0.64719 0.86602 0.95434 2.0 2.0 6.0 0.09452 0.13889 0.17751 0.21507 0.25412 0.29715 0.34781 0.41330 0.51469 0.77459 0.91442 2.0 2.0 10.0 0.07334 0.10800 0.13838 0.16815 0.19943 0.23435 0.27618 0.33167 0.42164 0.68824 0.86823 2.5 2.5 1.0 0.27941 0.37373 0.44615 0.50896 0.56705 0.62340 0.68063 0.74227 0.81622 0.93330 0.97426 3.0 3.0 0.1 0.64095 0.75811 0.82327 0.86619 0.89717 0.9210 0.94038 0.95702 0.97268 0.99101 0.99632 3.0 3.0 0.2 0.53323 0.65307 0.72871 0.78371 0.82677 0.86225 0.89283 0.92060 0.94813 0.98233 0.99270 3.0 3.0 0.6 0.38309 0.48491 0.55810 0.61822 0.67111 0.71995 0.76713 0.81531 0.86978 0.95043 0.97871 3.0 3.0 1.0 0.32557 0.41540 0.48216 0.53897 0.59088 0.64085 0.69143 0.74602 0.81222 0.92230 0.965489 3.0 3.0 1.5 0.28551 0.36582 0.42659 0.47932 0.52858 0.57723 0.62799 0.68488 0.75764 0.89138 0.94991 3.0 3.0 3.0 0.22749 0.29276 0.34308 0.38769 0.43040 0.47385 0.52089 0.57626 0.65255 0.81847 0.90856 3.0 3.0 6.0 0.18092 0.23334 0.27416 0.31075 0.34627 0.38301 0.42366 0.47297 0.54441 0.72284 0.84343 3.0 3.0 10.0 0.15271 0.19714 0.23187 0.26316 0.29369 0.32550 0.36102 0.40471 0.46950 0.64388 0.77952 int. j. anal. appl. 19 (3) (2021) 438 appendix 2: variance, skewness and kurtosis of lnk distribution following table shows the skewness and kutosis alongwith variance of lnk distribution a b c variance skewness kurtosis 0.5 0.5 0.1 0.05296 −1.90608 6.01753 0.5 0.5 0.2 0.07621 −1.27182 3.53694 0.5 0.5 0.6 0.11146 −0.46476 1.80245 0.5 0.5 1 0.12190 −0.13530 1.5386 0.5 0.5 1.5 0.12589 0.11638 1.50603 0.5 0.5 3.0 0.12346 0.54039 1.77633 0.5 0.5 6.0 0.11145 0.97311 2.46619 0.5 0.5 10.0 0.09890 1.30763 3.28243 1.0 1.0 0.1 0.04264 -1.82849 5.97880 1.0 1.0 0.2 0.05954 −1.19142 3.56560 1.0 1.0 0.6 0.08049 −0.35626 1.95722 1.0 1.0 1 0.08333 0.00000 1.80000 1.0 1.0 1.5 0.08152 0.28209 1.89855 1.0 1.0 3.0 0.07109 0.78491 2.56442 1.0 1.0 6.0 0.05508 1.34776 4.06188 1.0 1.0 10.0 0.04264 1.82849 5.97880 1.5 1.5 0.1 0.03501 -87.1461 386.365 1.5 1.5 0.2 0.04780 -40.6365 139.708 1.5 1.5 0.6 0.06162 -13.6714 32.6900 1.5 1.5 1 0.06205 −0.04546 2.00878 1.5 1.5 1.5 0.05922 0.21997 2.1023 1.5 1.5 3.0 0.04919 0.68276 2.73117 1.5 1.5 6.0 0.03603 1.18148 4.11004 1.5 1.5 10.0 0.02662 1.58988 5.82322 2.0 2.0 0.1 0.02935 −1.80078 6.22199 2.0 2.0 0.2 0.03939 −1.20368 3.91541 2.0 2.0 0.6 0.04923 −0.43784 2.35643 2.0 2.0 1 0.04888 −0.12530 2.18005 2.0 2.0 3.0 0.03789 0.50497 2.69308 2.0 2.0 6.0 0.02764 0.89112 3.64976 2.0 2.0 10.0 0.02052 1.17422 4.71545 2.5 2.5 0.1 0.02502 -694.098 3782.96 2.5 2.5 0.2 0.03311 -273.261 1177.61 2.5 2.5 0.6 0.04050 -278.916 868.895 2.5 2.5 1.5 0.03761 0.00355 2.3443 2.5 2.5 3.0 0.03094 0.33237 2.64818 2.5 2.5 6.0 0.02295 0.62511 3.25548 2.5 2.5 10.0 0.01744 0.81623 3.85719 int. j. anal. appl. 19 (3) (2021) 439 conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] z. iqbal, a. saboor, m. ahmad, a note on libby-novick kumaraswamy distribution. presented in 15th issos conference (2017). [2] d. l. libby, m. r. novick, multivariate generalized beta distributions with applications to utility assessment. j. educ. stat. 7(4) (1982), 271-294. [3] p. kumaraswamy, a generalized probability density function for double-bounded random processes. j. hydrol. 46(1-2) (1980), 79-88. [4] m. c. jones, kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. stat. methodol. 6(1) (2009), 70-81. [5] j. mcdonald, some generalized functions for the size distribution of income. econometrica. 52(3) (1984), 647–665. [6] p. a. mitnik, new properties of the kumaraswamy distribution. commun. stat., theory meth. 42(5) (2013), 741-755. [7] j. j. chen, m. r. novick, bayesian analysis for binomial models with generalized beta prior distributions. j. educ. stat. 9(2) (1984), 163-175. [8] m. m. ristić, b. v. popović, s. nadarajah, libby and novick's generalized beta exponential distribution. j. stat. comput. simul. 85(4) (2013), 740-761. [9] g. m. cordeiro, l. h. de santana, e. m. ortega, r. r. pescim, a new family of distributions: libbynovick beta. int. j. stat. probab. 3(2) (2014), 63. [10] m. ali ahmed, the new form libby-novick distribution, communications in statistics theory and methods. 50 (2021), 540. [11] m. rashid, z. iqbal, m. hanif, characterizations and entropy measures of the libby-novick generalized beta distribution. adv. appl. stat. 63(2) (2020), 235-259 [12] z. iqbal, m. rashid, m. hanif, properties of the libby-novick generalized beta distribution with application. int. j. anal. appl. 19 (2021), 360-388. [13] g. m. cordeiro, r. dos santos brito, the beta power distribution. brazil. j. probab. stat. 26(1) (2012), 88112. [14] r. dasgupta, on the distribution of burr with applications. sankhya b, 73 (2011), 1-19. international journal of analysis and applications volume 19, number 5 (2021), 725-742 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-725 application of successive linearisation method on the boundary layer flow problem of heat and mass transfer with radiation effect ahmed a. khidir, salihah l. alsharari∗ department of mathematics, faculty of science, university of tabuk, p.o. box 741, tabuk 71491, saudi arabia ∗corresponding author: salihah1972@yahoo.com abstract. in this paper, we applied the successive linearization method (slm) in solving highly system of nonlinear boundary value problem. the method is presented in detail by solving the problem of free convective heat and mass transfer of an incompressible fluid past a moving vertical plate in the presence of radiation effect. the governing partial differential equations are converted into system of non linear ordinary differential equations by a similarity transformation, which are converted into system of linear ordinary differential equations using slm. the linear system is solved using the chebyshev spectral method to find solutions that are accurate and converge rapidly to the full numerical solution. comparison with previously published works are performed to test the validity of the obtained results with focus on the accuracy and convergence of the solution. the effects of selected fluid parameters on the velocity as well as the temperature and concentration distribution are determined and discussed. 1. introduction most problems that arise in engineering are nonlinear with no analytic solutions and developing new methods that give rapid convergence, are robust and easy to use is a core function of numerical analysis. in the last few decades, a great deal of interest has been generated in the area of heat and mass transfer on a continuously stretching surface with a given temperature or heat flux distribution and they have received june 20th, 2021; accepted july 26th, 2021; published august 12th, 2021. 2010 mathematics subject classification. 80a20. key words and phrases. successive linearisation method; boundary layer flow; heat and mass transfer. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 725 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-725 int. j. anal. appl. 19 (5) (2021) 726 included several different physical models. transport of heat is being extensively studied, as understanding the associated transport processes becomes increasingly important. this interest stems from the variety of cases which can be modeled or approximated as transport through porous media, such as packed sphere beds, high performance insulation for buildings, chemical catalytic reactors, grain storage, and many others. literature concerning convective flow in porous media is abundant, and many published studies such as [3, 6, 7, 11, 14, 16, 17]. finding exact solutions to steady hydromagnetic flow and heat transfer could be of great benefit to polymer technology. in particular, there are many applications involve the cooling of continuous strips or filaments by drawing them though a quiescent fluid. by drawing such strips in an electrically conducting fluid subjected to a magnetic field, the rate of cooling can be controlled and final products of desired characteristics are obtainable [4]. numerical study for the effect of thermal-diffusion and diffusion-thermo on combined heat and mass transfer of flows induced a rotating disk has been investigated by emmanuel et al. [13]. the effect of mhd coupled heat and mass transfer of free convection from a moving permeable vertical was studied by surface was investigated by yih [18]. a similar problem, including natural convection about a vertical impermeable flat plate, was investigated by sparrow and cess [15]. the magneto-hydrodynamics (mhd) convection in porous medium have been studied by makinde [9, 10]. alan and rahman [2] examined dufour and soret effects on mixed hydrogenair convective flow past a vertical porous flat plate embedded in a porous medium. numerical study of natural convection of water in a partially heated enclosure with soret and dufour effects were discussed by nithyadevi and yang [12]. recently, ibrahim and makinde [5] studied the combined effects of wall suction and magnetic fields on boundary layer flow with heat and mass transfer over an accelerating vertical plate. in this paper, we aims to extend the work of olanrewaju et. all [1] to include the radiation effect of mhd boundary layer flow of heat and mass transfer. the successive linearisation method (slm) has been used to convert the governing non linear equations into a system of linear differential equations. the chebyshev pseudospectral method willbe used to solve the higher order deformation on linear differential equations. the auxiliary linear operator is defined in terms of the chebyshev spectral collocation differentiation matrix described in [21]. the slm has been used in a number of recent studies (see [21–25]. they showed that successive linearisation method is accurate and converges rapidly to the numerical results when compared to other recent semi-analytical methods such as the adomian decomposition method. the slm method can further be used in place of traditional numerical methods such as finite differences and runge-kutta methods to solve highly nonlinear systems of boundary value problems. int. j. anal. appl. 19 (5) (2021) 727 2. problem formulation we consider the steady free convective heat and mass transfer flow of a viscous, incompressible, electrically conducting fluid past a moving vertical plate in the presence of thermal diffusion (soret) and diffusion-thermo (dufour) effects. the non uniform transverse magnetic field b0 is imposed along the y-axis. the induced magnetic field is neglected as the magnetic reynolds number of the flow is assumed very small. it is further assumed that the external electric field is zero and that the electric field due to charge polarization is negligible. the temperature and the concentration of the ambient fluid are t∞ and c∞ respectively, while those at the surface are, respectively, tw(x) and cw(x). the pressure gradient, viscous and electrical dissipation are also neglected. the fluid properties are assumed constant, apart from the density in the buoyancy terms of the linear momentum equation, which is estimated using boussinesq’s approximation. under the above assumptions, the boundary layer form of the governing equation can be written as [8]: ∂u ∂x + ∂v ∂y = 0 (2.1) u ∂u ∂x + v ∂u ∂y = v ∂2u ∂2y + gβt (t −t∞) + gβc(c −c∞) − σb20 ρ u (2.2) u ∂t ∂x + v ∂t ∂y = 1 cp ( k + 16σ∗t3∞ 3k∗ ) ∂2t ∂y2 + α ∂2t ∂y2 + dmkt cscp ∂2c ∂y2 (2.3) u ∂c ∂x + v ∂c ∂y = dm ∂2c ∂y2 + dmkt tm ∂2t ∂y2 (2.4) the boundary conditions for equations (2.1)–(2.4) are expressed as v = v,u = bx,t = tw = t∞ + ax,c = cw = c∞ + bx at y = 0 u → 0, t → t∞,c → c∞ as y →∞ where b is a constant, a and b denote the stratification rate of the gradient of ambient temperature and concentration profiles and (u,v) are the fluid velocity components in the x and y directions, respectively regarding the plate, t is the temperature,βt is the volumetric coefficient of thermal expansion, α is the thermal diffusivity and g is the acceleration due to gravity. fluid parameters are ν, the kinematic viscosity, dm the coefficient of diffusion in the mixture, c the species concentration, σ the electrical conductivity, kt is the themal diffusion ratio, cs is the concentration susceptibility, cp is the specific heat at constant pressure and tm the mean fluid temperature, k ∗ is the mean absorption coefficient and σ∗ is the stefan–boltzmann constant. b0 is the externally imposed magnetic field in the y direction. we introduce the following nondimensional variables: η = √ b v y f(η) = ψ x √ bv , θ(η) = t −t∞ tw −t∞ , φ(η) = c −c∞ cw −c∞ (2.5) where f(η) is a dimensionless stream function, θ(η) is a dimensionless temperature of the fluid in the boundary layer region, φ(η) is a dimensionless species concentration of the fluid in the boundary layer region int. j. anal. appl. 19 (5) (2021) 728 and η is the similarity variable. the velocity components u and v are respectively obtained as follows u = ∂ψ ∂y = xbf ′, v = − ∂ψ ∂x = − √ bvf, (2.6) where fw = v √ bv is the dimensionless suction velocity. following equation (2.6), the partial differential equations (2.2)–(2.4) are transformed into local similarity equations as follows: f ′′′(η) + f(η)f ′′(η) − (f ′(η) + m)f ′(η) + grθ(η) + gcφ(η) = 0 (2.7)( 1 + 4 3 rd ) θ′′(η) + prfθ ′(η) −prf ′(η)θ(η) + prduφ′′(η) = 0 (2.8) φ′′(η) + scfφ ′(η) −scf ′(η)φ(η) + scsrθ′′(η) = 0 (2.9) the boundary conditions are also transformed into the form f ′ = 1 f = −fw,θ = 1,φ = 1 at η = 0 (2.10) f ′ = 0 θ = 0 φ = 0 as η →∞ (2.11) where m = σb20 ρb is the magnetic parameter, pr = v α is the prandtl number, sc = v dm is the schmidt number, gr = gβt (tw −t∞) xb2 is the local temperature grashof number, gc = gβc(cw −c∞) xb2 is the local concentration grashof number, du = dmkt (cw−c∞) cscp(tw−t∞) is the dufour number, sr = dmkt (tw−t∞) tm(cw−c∞)v is the soret number and rd = 4σ ∗t3∞/k ∗k is the radiation number. 3. successive linearisation method (slm) the slm procedure linearises the governing non linear equation. in order to fully describe of the slm algorithm, let us consider the following boundary value problem of order n in the form: l[u(x),u′(x),u′′(x), ...,un + n[u(x),u′(x),u′′(x), ...,un] = g(x), (3.1) where l and n are linear and non linear operators ,u(x) is an unknown function to be determined, g(x) is a known function. we assume that equation (3.1) is to be solve for x ∈ [a,b] subject to the boundary conditions u(a) = a0, u(b) = b0 (3.2) we represent the vertical difference between the function u(x) and the initial guess u0(x) by a function u1(x) as (see figure 1) u1(x) = u(x) −u0(x), (3.3) where u1(x) is an unknown functions and u0(x) is the initial guess which is chosen to satisfy boundary conditions (3.2). it is reasonable to assume, for example, that the initial approximation u0(x) is a linear int. j. anal. appl. 19 (5) (2021) 729 figure 1. geometric representation of the function ui(x) function in case of second order problems defined on a finite domain and an exponential function for problems defined on an infinite or a semi-infinite domain. substituting (3.3) into (3.1), yields l[u1(x),u ′ 1(x),u ′′ 1 (x), ....,u (n) 1 ] + l[u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ] + n[u1(x) + u0(x),u ′ 1(x) + u ′ 0(x),u ′′ 1 (x) + u ′′ 0 (x), ...,u (n) 1 (x) + u (n) 0 (x)] = g(x), (3.4) this equation is non-linear in u1(x), so it may not be possible to find an exact solution. we therefore look for a solution which is obtained by solving the linear part of the equation and neglecting the non-linear terms containing u1(x) and its derivatives. we further assume that u1(x) and its derivatives are very small and denote the solution of the linearised equation (3.4) by u1(x), that is u1(x) ≈ u1(x). equation (3.4) can be written as l[u1(x),u ′ 1(x),u ′′ 1 (x), ....,u (n) 1 ] + f0,0u1(x) + f1,0u ′ 1(x) + f2,0u ′′ 1 (x) + .... + fn,0u (n) 1 (x) = r1(x) (3.5) where f0,0 = ∂n ∂u1(x) (u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ), f1,0 = ∂n ∂u′1(x) (u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ), f2,0 = ∂n ∂u′′1 (x) (u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ), fn,0 = ∂n ∂u (n) 1 (x) (u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ), r1(x) = g(x) −l[u0(x),u′0(x),u ′′ 0 (x), ....,u (n) 0 ] −n[u0(x),u ′ 0(x),u ′′ 0 (x), ....,u (n) 0 ]. int. j. anal. appl. 19 (5) (2021) 730 since the left hand side of equation (3.5) is linear and the right hand side is known, the equation can be solved for u1(x) subject to the boundary conditions u(a) = 0,u(b) = 0 (3.6) assuming that the solution of the linear equation (3.5) is close to the solution of the non linear equation (3.4), then the first approximation of the solution (order 1) is u(x) ≈ u0(x) + u1(x) (3.7) to improve this solution we define the vertical difference between the function u1(x) and u1(x) by the function u2(x) as u2(x) = u1(x) −u1(x) (3.8) substitute (3.8) into equation (3.1) to give l[u2(x),u ′ 2(x),u ′′ 2 (x), ....,u (n) 2 ] + l[u0(x) + u1(x),u ′ 0(x) + u ′ 1(x),u ′′ 0 (x) + u ′′ 1 (x), ....,u (n) 0 + u (n) 1 (x)] + n[u2(x) + u0(x) + u1(x),u ′ 2(x) + u ′ 0(x) + u ′ 1(x) ′u′′2 (x) + u ′′ 0 (x) + u ′′ 1 (x), ...,u (n) 2 (x) + u (n) 0 (x) + u (n) 1 (x)] = g(x) (3.9) since u0(x) and u1(x) are known and this equation is non-linear in u2(x), we solve the linearized equation after neglecting the non-linear terms containing u2(x) and its derivatives. we further assume that u2(x) and its derivatives are very small that is u2(x) ≈ u2(x). equation(3.9)can be written as l[u2(x),u ′ 2(x),u ′′ 2 (x), ....,u (n) 2 (x)] + f0,0u2(x) + f1,0u ′ 2(x) + f2,0u ′′ 2 (x) + ..... + fn,0u (n) 2 (x) = r2(x), (3.10) where f0,0 = ∂n ∂u2(x) (u0(x) + u1(x),u ′ 0(x) + u ′ 1(x),u ′′ 0 (x) + u ′′ 1 (x), ....,u (n) 0 (x) + u n 1 (x)), f1,0 = ∂n ∂u′2(x) (u0(x) + u1(x),u ′ 0(x) + u ′ 1(x),u ′′ 0 (x) + u ′′ 1 (x), ....,u (n) 0 (x) + u n 1 (x)), f2,0 = ∂n ∂u′′1 (x) (u0(x) + u1(x),u ′ 0(x) + u ′ 1(x),u ′′ 0 (x) + u ′′ 1 (x), ....,u (n) 0 (x) + u n 1 (x)), fn,0 = ∂n ∂u (n) 1 (x) (u0(x) + u1(x),u ′ 0(x) + u ′ 1(x),u ′′ 0 (x) + u ′′ 1 (x), ....,u (n) 0 (x) + u n 1 (x)), r2(x) = g(x) −l[u0(x) + u1(x),u′0(x) + u ′ 1(x), ....,u (n) 0 (x) + u (n) 1 (x)] −n[u0(x) + u1(x),u′0(x) + u ′ 1(x), ....,u (n) 0 (x)]. after solving the equation (3.10), the second order approximation of u(x) is given by u(x) ≈ u0(x) + u1(x) + u2(x). (3.11) int. j. anal. appl. 19 (5) (2021) 731 this process is repeated for m = 2, 3, ..., i. thus, u(x) is given by u(x) = u1(x) + u0(x), = u2(x) + u0(x) + u1(x), = u3(x) + u0(x) + u1(x) + u2(x), ... = ui+1(x) + u0(x) + u1(x) + u2(x) + ... + ui(x), = ui+1(x) + i∑ m=0 um(x). thus, for large i, we can approximate the ith order solution u(x) by u(x) = i∑ m=0 um(x). (3.12) the solution ui(x) can be determined from the linearized original equation (3.1) starting from the initial guess u0(x) and solving the linear equations for ui(x). in general, the form of the linearized equation for ui(x) is given by l[ui(x),u ′ i(x),u ′′ i (x), ....,u (n) i ] + f0,i−1ui(x) + f1,i−1u ′ i(x) + f2,i−1u ′′ i (x) + ..... + fn,i−1u (n) i (x) = ri−1(x), i = 1, 2, ...,m, (3.13) subject to the boundary conditions ui(a) = 0,ui(b) = 0, (3.14) where m is termed the order of the slm. f0,i−1 = ∂n ∂ui(x) ( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)), f1,i−1 = ∂n ∂u′i(x) ( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)), f2,i−1 = ∂n ∂u′′i (x) ( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)), int. j. anal. appl. 19 (5) (2021) 732 ... fn,i−1 = ∂n ∂u (n) i (x) ( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)), ri−1(x) = g(x) −l( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)) −n( i−1∑ m=0 um(x), i−1∑ m=0 u′m(x), i−1∑ m=0 u′′m(x), ...., i−1∑ m=0 u(n)m (x)). then ordinary differential equation is linear and can easily be solved using any analytical or numerical method. 4. method of solution the system of non linear equation to be solved usin slm are f ′′′ + ff ′′ − (f ′ + m)f ′ + grθ + gcφ = 0 (4.1) λθ′′ + fθ′ −f ′θ + duφ′′ = 0 (4.2) φ′′ + scfφ ′ −scf ′φ + scsrθ′′ = 0 (4.3) subject to the boundary conditions f ′ = 1 f = −fw,θ = 1,φ = 1 at η = 0 (4.4) f ′ = 0 θ = 0 φ = 0 as η →∞ (4.5) where λ = 1 pr ( 1 + 4 3 rd ) . the slm algorithm starts with the assumption that the independent variables f(η),θ(η) and φ(η) can be expressed as f(η) = fi(η) + i−1∑ m=0 fm(η), θ(η) = θi(η) + i−1∑ m=0 θm(η), φ(η) = φ(η) + i−1∑ m=0 φm(η) (4.6) where fi,θi,φi (i = 1, 2, 3, ...) are unknown functions and fm,θm and φm(m < i) are known functions which are obtained by recursively solving the linear part of the equation system that results from substituting in the governing equations. the main assumption of the slm is that fi,θi,φi become increasingly smaller when i becomes large that is lim i→∞ fi = lim i→∞ θi = lim i→∞ φ = 0. (4.7) thus, starting from the initial guesses f0(η) = 1 − fw − e−η,θ0(η) = e−η,φ0(η) = e−η, which are chosen to satisfy boundary conditions (4.4) and (4.5), the subsequent solutions for fi,φi,θi, i ≥ 1 are obtained by successively solving the linearised form of the equations which are obtained by substituting equation (1)in the governing equations and considering only the linear terms. the linearised equations to be solved are int. j. anal. appl. 19 (5) (2021) 733 given by substituting the assumptions (4.6) in the system (4.1) (4.3) and neglecting the non linear terms, gives f ′′′i + i−1∑ m=0 fmf ′′ i + i−1∑ m=0 f ′′mfi − 2 i−1∑ m=0 f ′mf ′ i −mf ′ i + crθi + gcrφi = r1 (4.8) λθ′′i + i−1∑ m=0 θ′m(η)fi + i−1∑ m=0 fm(η)θ ′ i − i−1∑ m=0 θmf ′ i − i−1∑ m=0 f ′mθi + duφ ′′ i = r2 (4.9) φ′′i + sc i−1∑ m=0 φ′mfi + sc i−1∑ m=0 fmφ ′ i −sc i−1∑ m=0 φmf ′ i + sc i−1∑ m=0 f ′mφi + scθ ′′ i = r3 (4.10) subject to the boundary conditions fi(0) = f ′ i (0) = f ′ i (∞) = θi(0) = θi(∞) = φ(0) = φi(∞) = 0 (4.11) where r1 = − ( i−1∑ m=0 f ′′′m + i−1∑ m=0 fm i−1∑ m=0 f ′′m − i−1∑ m=0 f ′m i−1∑ m=0 f ′m −m i−1∑ m=0 f ′m + gr i−1∑ m=0 θm + gc i−1∑ m=0 φm ) r2 = − ( λ i−1∑ m=0 θ′′m + i−1∑ m=0 fm(η) i−1∑ m=0 θ′m − i−1∑ m=0 f ′m t−1∑ m=0 θm + du i−1∑ m=0 φ′′m ) r3 = − ( i−1∑ m=0 φ′′m + sc i−1∑ m=0 fm i−1∑ m=0 φ′m −sc i−1∑ m=0 f ′m i−1∑ m=0 φm + scsr i−1∑ m=0 θ′′m ) once each solution for fi,θi,φi(i ≥ 1) has been found from iteratively solving equations (36 ), the approximate solutions for f(η),θ(η) and φ(η) are obtained as f(η) ≈ m∑ m=0 fm(η),θ(η) ≈ m∑ m=0 θm(η),φ(η) ≈ m∑ m=0 φm(η), (4.12) where m is the order of slm approximation. since the coefficient parameters and the right hand side of equations (4.8) (4.10), for i = 1, 2, 3, ... are known (from previous iterations). the equation system can easily be solved analytically (wherever possible) or using any numerical method such as finite differences, finite elements, runge-kutta based shooting methods or collocation methods’ in this work, we used the chebyshev spectral collocation method. this method is based on approximating the unknown functions by the chebyshev interpolating polynomials in such a way that they are collocated at the gauss-lobatto points defined as (see [19, 20]) ξj = cos πj n , j = 0, 1, . . . ,n where n + 1 is the number of collocation points used. in order to implement the method, the physical region [ 0,∞) is transformed into the region [−1, 1] using the any domain truncation technique in which the problem is solved on the interval instead of [0,∞). the unknown functions fi,θi and φi are approximated int. j. anal. appl. 19 (5) (2021) 734 at the collocation points by fi(ξ) ≈ n∑ k=0 fi (ξ z k) tk (ξj) ,θi(ξ) ≈ n∑ k=0 θi (ξk) tk (ξj) ,φ(ξ) ≈ n∑ k=0 φ (ξk) tk (ξj) ,j = 0, 1, . . .n (4.13) where tkis the kth chebyshev polynomial defined as tk(ξ) = cos[k cos −1 (ξ)] (4.14) the derivatives of the variables at the collocation points are represented as drfi dηr = n∑ k=0 drkj ·fi (ξk) , drθi dηr = n∑ k=0 drkjθi (ξk) , drφi dηr = n∑ k=0 drkjφi (ξk) ,j = 0, 1, . . . ,n (4.15) where r is the order of differentiation and d being the chebyshev spectral differentiation matrix hose entries are defined as [19, 20] d00 = 2 n 2+1 6 djk = cj(−1)j+k ckξj−ξk j = /k; j,k = 0, 1, . . . ,n dkk = − ξk 2(1−ξ2k) ,k = 1, 2, . . . ,n − 1, dnn = −2 n 2+1 6   (4.16) substituting equations (4.15) into the system (4.8) (4.10) gives the following system of algebraic equations a11fi + a12θi + a13φi = r1,i−1 a21fi + a22θi + a23φi = r2,i−1 a31fi + a32θi + a33φi = r2,i−1 which leads to the matrix equation given as ai−1xi = ri−1, (4.17) in which ai−1 is a (3n + 3)×(3n + 3) square matrix and xi and ri−1 are (3n + 1)1 column vectors defined by ai−1 =   a11 a12 a31 a21 a22 a32 a31 a32 a33   , xi =   fi θi φi   , ri−1 =   r1,i−1 r2,i−1 r3,i−1   (4.18) where fi = [ fi (η0) ,fi (η1) , . . . ,fi (ηn−1) ,fi (ηn )] t θi = [ θi (η0) ,θi (η1) , . . . ,θi (ηn−1) ,θi (ηn )] t φi = [ φi (η0) ,φi (η1) , . . . ,φi (ηn−1) ,φi (ηn )] t r1,i−1 = [r1,i−1 (η0) ,r1,i−1 (η1) , . . . ,r1,i−1 (ηn−1) ,r1,i−1(ηn )] t int. j. anal. appl. 19 (5) (2021) 735 r2,i−1 = [r2,i−1 (η0) ,r2,i−1 (η1) , . . . ,r2,i−1 (ηn−1) ,r2,i−1(ηn )] t r3,i−1 = [r3,i−1 (η0) ,r3,i−1 (η1) , . . . ,r3,i−1 (ηn−1) ,r3,i−1(ηn )] t a11 = d 3 + i−1∑ m=0 fmd 2 − ( 2 i−1∑ m=0 f ′m + m ) d + [ i−1∑ m=0 f ′′m ] , a12 = [cr], a13 = [gr], a21 = − i−1∑ m=0 θmd + [ i−1∑ m=0 θ′m(η) ] ,a22 = λd 2 + i−1∑ m=0 fmd − [ i−1∑ m=0 f ′m ] ,a23 = dud 2, a31 = −sc i−1∑ m=0 φmd + sc [ i−1∑ m=0 φ′m(η) ] ,a32 = srscd 2,a33 = d 2 and [..] stands to a diagonal matrix of size (3n + 3)×(3n + 3). the boundary conditions (4.15) are imposed to the system (4.17) as displayed in figure 2. after modifying the matrix system (4.17) to incorporate boundary conditions, the solution is obtained as xi = a −1 i−1ri−1 (4.19) figure 2. imposing the boundary conditions (4.15) into the system (4.19). int. j. anal. appl. 19 (5) (2021) 736 5. results and discussion the system of non-linear ordinary differential equations (2.7) (2.9) together with the boundary conditions (2.10) and (2.11) are solved numerically using successive linearization method. in this chapter we give the obtained results for the main parameters affecting the flow. we use matlab software in all obtained results in this research. to check the accuracy of the proposed successive linearisation method (slm), comparison was made with those obtained in literature. the graphs and tables presented in this work, unless otherwise specified, were generated using n = 30 and η∞ ' l = 15 gave sufficient accuracy for the slm results. 6. convergence of the solution in this section, comparisons with previously published works are performed to test the validity and the convergence of the obtained results. also, the effects of some physical parameters on the velocity, temperature and concentration profiles are obtained and discussed. table 1 showed the slm results of f ′′(0), −θ′(0) and −φ′(0) at various values of rd for different iterations. it is cleared form the table that the results obtained by slm are in excellent agreement with a few order of slm series starting from the third iteration and giving accuracy up to eight decimal places.. in table 2, the slm results were compared for −f ′′(0), −θ′(0) and −φ′(0) for different values of fw with those obtained by olanrewaju et. all [1] in the absence of radiation effect. they used the shooting iteration technique together with a sixth order runge–kutta integration scheme. it can be seen from the table that they are in good agreement between the results.. 7. velocity profiles figure 3 illustrates the effect of magnetic parameter m parameter on velocity as a function of the similarity variable η. it is observed from the figure that an increase in the magnetic parameter m leads to decreases in the velocity profile. the same result was obtained by olanrewaju et. all [1]. figure 4 represents the the effect of the radiation parameter rd on dimensionless velocity distributions. it shows that the velocity enhanced as the radiation parameter increased. 8. temperature profiles the effect of magnetic filed parameter m on temperature distribution profile θ(η) as a function of variable η in displayed on figure 5. from the figure, we see that an increase in the magnetic parameter m leads to increases in the temperature distribution. figure 6 shows the effect of increasing the radiation parameter rd on dimensionless temperature distributions. we observe that increasing the radiation parameter leads to enhanced the temperature distributions θ(η) int. j. anal. appl. 19 (5) (2021) 737 table 1. the slm results for f ′′(0), −θ′(0) and −φ′(0) to different values of the radiation number rd when fw = 1,m = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,sc = 0.62,sr = 0.1,du = 0.03 rd first iteration second iteration third iteration fourth iteration f ′′(0) 0.1 0.56946888 0.56947404 0.56947405 0.56947405 0.2 0.56880563 0.56881060 0.56881061 0.56881061 0.5 0.56717266 0.56717699 0.56717699 0.56717699 1.0 0.56523532 0.56523914 0.56523914 0.56523914 −θ′(0) 0.1 0.52976296 0.52974957 0.52974955 0.52974955 0.2 0.50638146 0.50636189 0.50636185 0.50636185 0.5 0.45023485 0.45019930 0.45019924 0.45019924 1.0 0.38601330 0.38596450 0.38596442 0.38596442 −φ′(0) 0.1 0.51484762 0.51483002 0.51482998 0.51482998 0.2 0.51622774 0.51621125 0.51621122 0.51621122 0.5 0.51958417 0.51957245 0.51957243 0.51957243 1.0 0.52347723 0.52347206 0.52347205 0.52347205 9. concentration profiles figure 7 showed the effect of magnetic parameter m parameter on concentration profile as a function of the similarity variable η. it is observed from the figure that an increase in the magnetic parameter m leads to increases in the concentration profile φ(η). the effect of radiation parameter rd is decreased concentration distribution via increasing rd . int. j. anal. appl. 19 (5) (2021) 738 table 2. comparison between the slm, and ref. [1] for −f ′′(0), −θ′(0) and −φ′(0) for different values of fw when m = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,rd = 0,sc = 0.62,sr = 0.1,du = 0.03 ref. [1] slm fw −f ′′(0) −θ′(0) −φ′(0) −f ′′(0) −θ′(0) −φ′(0) 0.5 0.724431 0.673004 0.605696 0.724428 0.673003 0.605696 0.3 0.801102 0.728095 0.648537 0.801097 0.728093 0.648536 0.0 0.934398 0.820892 0.720212 0.934389 0.820888 0.720212 −0.3 1.090854 0.926778 0.801866 1.090838 0.926772 0.801864 −0.5 1.208097 1.004942 0.862315 1.208074 1.004934 0.862312 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η f ′ ( η ) m = 0 m = 0.3 m = 0.5 m = 1 figure 3. effect of m on the velocity profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,rd = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . 10. conclusion in this paper, we applied the successive linearization method in solving highly system of nonlinear boundary value problem. the method is applied on the mhd free convective heat and mass transfer with radiation effect. the set of governing equations and the boundary conditions are reduced to ordinary differential equations with appropriate boundary conditions. the results were compared with other methods in the literature int. j. anal. appl. 19 (5) (2021) 739 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η f ′ ( η ) rd = 0.0 rd = 0.5 rd = 1.0 rd = 1.5 figure 4. effect of m on the concentration profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,m = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η θ (η ) m = 0 m = 0.3 m = 0.5 m = 1 figure 5. effect of m on the temperature profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,rd = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . such as the shooting iteration technique together with a sixth order runge–kutta integration scheme with focus on the accuracy and convergence of the results. graphs were presented showing the effects of various physical parameters on the fluid properties. the main conclusions emerging from this paper are as follows: • the slm suggested a standard method for choosing the linear operators and initial approximations by using any form of initial guess as long as it satisfies the boundary conditions, while the initial int. j. anal. appl. 19 (5) (2021) 740 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η θ (η ) rd = 0.0 rd = 0.5 rd = 1.0 rd = 1.5 figure 6. effect of m on the concentration profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,m = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η φ (η ) m = 0 m = 0.3 m = 0.5 m = 1 figure 7. effect of m on the concentration profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,rd = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . guess in the other method such as hpm and ham can be selected that will make the integration of the higher order deformation equations possible. • the slm is highly accurate, efficient and converges rapidly with a few iterations required to achieve the accuracy of the numerical results, in this study it was found that for a few iterations of slm was sufficient to give good agreement with the exact solution. int. j. anal. appl. 19 (5) (2021) 741 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η φ (η ) rd = 0.0 rd = 0.5 rd = 1.0 rd = 1.5 figure 8. effect of m on the concentration profile when fw = 0.1,gr = 0.1,gc = 0.1,pr = 0.72,m = 0.1,sc = 0.62,sr = 0.1,du = 0.03 . • when the magnetic field increases, the velocity profile is decreased while the temperature and concentration components are enhanced. • when the radiation parameter increases, the velocity profile temperature distribution are increased while the concentration component is reduced. finally, the successive linearisation method has high accuracy and simple for solving nonlinear boundary value problems compared with the runge kutta, finite difference, finite element and keller-box methods. because of its efficiency and easy of use. the extension to systems of nonlinear bvps allows the method to be used as alternative to the traditional of those methods. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. o. olanrewaju, o. d. makinde, effects of thermal diffusion and diffusion thermo on chemically reacting mhd boundary layer flow of heat and mass transfer past a moving vertical plate with suction/injection, arab. j. sci. eng. 36 (2011), 1607-1619. [2] m. s. alam, m. m rahman, dufour and soret effects on mixed convection flow past a vertical porous flat plate with variablesuction, nonlinear anal.: model. control, l1 (1) (2006), 3–12. [3] a. bejan, i. dincer, s. lorente, a. f. miguel, a. h. reis, porous and complex flow structures in modern technologies. springer, new york, (2004). [4] a. chakrabarti, a. s. gupta, hydromagnetic flow and heat transfer over a stretching sheet. quart. appl. math. 37 (1979), 73–78. int. j. anal. appl. 19 (5) (2021) 742 [5] s. y. ibrahim, o. d. makinde, chemically reacting mhd boundary layer flow of heat and mass transfer past a moving vertical plate with suction. sci. res. essay. 5 (19) (2010), 2875–2882. [6] d. b. ingham, a. bejan, e. mamut, i. pop, emerging technologies and techniques in porous media, kluwer, dordrecht, (2004). [7] d. b. ingham, i. pop, transport phenomena in porous media, vol. iii. pergamon, oxford, (2005). [8] n. g. kafoussias, e. w. williams, thermal-diffusion and diffusion-thermo effects on mixed free convective and mass transfer boundary layer flow with temperature dependent viscosity. int. j. eng. sci. 33 (1995), 1369–1384. [9] o. m. makinde, mhd steady flow and heat transfer on the sliding plate. amse model. meas. control b. 70(1) (2001), 61–70. [10] o. d. makinde, on mhd boundary-layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux. int. j. numer. meth. heat fluid flow, 19 (3/4) (2009), 546–554. [11] d. a. nield, a. bejan, convection in porous media, 3rd edn. springer, new york, (2006). [12] n. nithyadevi, r. j. yang, double diffusive natural convection in a partially heated enclosure with soret and dufour effects. int. j. heat fluid flow, 30 (5) (2009), 902-910. [13] e. osalusi, j. side, r. harris, thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady mhd convective and slip flow due to a rotating disk with viscous dissipation and ohmic heating. int. commun. heat mass transfer. 35 (2008), 908–915. [14] i. pop, d. b. ingham, convective heat transfer: mathematical and computational modeling of viscous fluids and porous media. pergamon, oxford, (2001). [15] e. m. sparrow, r. d. cess, the effect of a magnetic field on free convection heat transfer, int. j. heat mass transfer. 3 (1961), 267–274. [16] p. vadasz, emerging topics in heat and mass transfer in porous media. springer, new york (2008). [17] k. vafai, handbook of porous media. taylor & francis, new york (2005). [18] k. a. yih, free convection effect on mhd coupled heat and mass transfer of a moving permeable vertical surface. int. commun. heat mass transf. 26(1), 95–104 (1999). [19] c. canuto, m. y. hussaini, a. quarteroni, t. a. zang, spectral methods in fluid dynamics, springer-verlag, berlin (1998). [20] l. n. trefethen, spectral methods in matlab, siam, 2000. [21] z. g. makukula, p. sibanda and s. s. motsa, a note on the solution of the von karman equations using series and chebyshev spectral methods, bound. value probl. 2010 (2010), 471793. [22] z. makukula and s. s. motsa, on new solutions for heat transfer in a viscolastic fluid between parallel plates, int. j. math. models meth. appl. sci. 4 (2010), 221-230. [23] s. s. motsa, p. sibanda and s. shateyi, on a new quasi-linearization method for systems of nonlinear boundary value problems. math. meth. appl. sci. 34 (2011), 1406–1413. [24] f. g. awad, p. sibanda, s. s. motsa and o. d. makinde, convection from an inverted cone in a porous medium with cross-diffusion effects, computers math. appl. 61 (2011), 1431-1441. [25] f. g. awad, p. sibanda, m. narayana and s. s. motsa, convection from a semi-finite plate in a fluid saturated porous medium with cross-diffusion and radiative heat transfer, int. j. phys. sci. 6 (21) (2011), 4910–4923. 1. introduction 2. problem formulation 3. successive linearisation method (slm) 4. method of solution 5. results and discussion 6. convergence of the solution 7. velocity profiles 8. temperature profiles 9. concentration profiles 10. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 98-107 http://www.etamaths.com some generalized steffensen’s inequalities via a new identity for local fractional integrals tuba tunç1,∗, mehmet zeki sarikaya1 and h. m. srivastava2,3 abstract. in this study, we first give an identity for local fractional integrals. we then make use of this identity in order to derive several generalizations of the celebrated steffensen’s inequality associated with local fractional integrals. relevant connections of the results presented in this paper with those that were proven in earlier works are also pointed out. 1. introduction as long ago as 1919, steffensen [25] established the following result which is known in the literature as steffensen’s inequality. theorem 1.1. let a and b be real numbers such that a < b. also let f,g : [a,b] → r be integrable functions such that f is nonincreasing and, for every x ∈ [a,b] , 0 5 g(x) 5 1. then∫ b b−λ f(x)dx 5 ∫ b a f(x)g(x)dx 5 ∫ a+λ a f(x)dx, (1.1) where λ = ∫ b a g(x)dx. steffensen’s inequality (1.1) happens to be the most basic inequality which deals with the comparison between integrals over a whole interval [a,b] and integrals over a subset of [a,b]. in fact, the inequality (1.1) has attracted considerable attention and interest from mathematicians and researchers. in this connection, the interested reader is referred to a number of works (see, for example, [3], [4], [7][10], [13], [14], [17] and [26]) for various related integral inequalities. recently, wu and srivastava [27] proved the following inequality which is a weighted version of the inequality (1.1). theorem 1.2. let f, g and h be integrable functions defined on [a,b] with f nonincreasing. also let 0 5 g(x) 5 h(x) for all x ∈ [a,b] . then the following inequalities hold true:∫ b b−λ f(x)h(x)dx 5 ∫ b b−λ (f(x)h(x) − [f(x) −f(b−λ)] [h(x) −g(x)]) dx 5 ∫ b a f(x)g(x)dx 5 ∫ a+λ a ( f(x)h(x) − [f(x) −f(a + λ)] [h(x) −g(x)] ) dx 5 ∫ a+λ a f(x)h(x)dx, where λ is given by ∫ a+λ a h(x)dx = ∫ b a g(x)dx = ∫ b b−λ h(x)dx. received 12th august, 2016; accepted 7th october, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d15, 26d99, 26a33. key words and phrases. steffensen’s inequality; local fractional integrals; fractal space; fink’s identity. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 98 some generalized steffensen’s inequalities 99 we recall the following identity given by fink [1]: 1 n ( f(x) + n−1∑ k=1 fk(x) ) − 1 b−a ∫ b a f(t)dt = 1 n!(b−a) ∫ b a (x− t)n−1 κ(t.x)f(n)(t)dt, (1.2) where fk(x) = ( n−k k! )( f(k−1)(a)(x−a)k −f(k−1)(b)(x− b)k b−a ) (1.3) and κ(t,x) =   t−a (a 5 t 5 x 5 b) t− b (a 5 x < t 5 b) pečarić et al. [15] gave generalizations of steffensen’s inequality (1.1) via fink’s identity (1.2). subsequently, pečarić et al. [16] derived several new identities related to various generalizations of steffensen’s inequality (1.1). 2. definitions, notations and preliminaries the concepts of fractional calculus [8] and local fractional calculus (also called fractal calculus) (see, for details, [28] and [32]) are becoming increasingly useful in a wide variety of problems in mathematical, physical and engineering sciences (see, for example, the recent works [29] to [39]). with a view to introducing the definition of the local fractional derivative and the local fractional integral, we need the following notations and preliminaries (see [28] and [32]). for 0 < α 5 1, we have the following α-type sets of elements: zα : the α-type set of integers defined by zα := {0α,±1α,±2α, · · · ,±nα, · · · } . qα : the α-type set of the rational numbers defined by qα := { mα : mα = ( p q )α (p,q ∈ z; q 6= 0) } . jα : the α-type set of the irrational numbers defined by jα := { mα : mα 6= ( p q )α (p,q ∈ z; q 6= 0) } . rα : the α-type set of the real line numbers defined by rα := qα ∪jα. proposition. let aα,bα and cα belong to the set rα of real line numbers. then (1) aα + bα and aαbα belong to the set rα; (2) aα + bα = bα + aα = (a + b) α = (b + a) α ; (3) aα + (bα + cα) = (a + b) α + cα; (4) aαbα = bαaα = (ab) α = (ba) α ; (5) aα (bαcα) = (aαbα) cα; (6) aα (bα + cα) = aαbα + aαcα; (7) aα + 0α = 0α + aα = aα and aα1α = 1αaα = aα. the definitions of the local fractional derivative and the local fractional integral can now be given as follows. definition 2.1. (see [28] and [32]) a non-differentiable function f : r → rα ( x → f(x) ) is said to be local fractional continuous at x = x0 if, for any ε > 0, there exists δ > 0 such that |f(x) −f(x0)| < εα holds true for |x−x0| < δ (ε,δ ∈ r). 100 tunç,sarikaya and srivastava if the function f(x) is local continuous on the interval (a,b) , we denote this property as follows: f(x) ∈ cα(a,b). definition 2.2. (see [28] and [32]) the local fractional derivative of f(x) of order α (0 < α 5 1) at x = x0 is defined by f(α)(x0) = dαf(x) dxα ∣∣∣∣ x=x0 = lim x→x0 ∆α ( f(x) −f(x0) ) (x−x0) α , (2.1) where ∆α ( f(x) −f(x0) ) =̃ γ(α + 1) ( f(x) −f(x0) ) . if there exists f(k+1)α(x) = k+1 times︷ ︸︸ ︷ dαx · · ·d α xf(x) (x ∈ i ⊆ r), then we write f ∈ d(k+1)α(i) (k ∈ n0 := {0, 1, 2, · · ·} = n∪{0}). definition 2.3. (see [28] and [32]) let f(x) ∈ cα [a,b] . then the local fractional integral of f(x) of order α (0 < α 5 1) is defined by ai α b f(x) = 1 γ(1 + α) ∫ b a f(t)(dt)α = 1 γ(1 + α) lim ∆t→0 n−1∑ j=0 f(tj)(∆tj) α (2.2) with ∆tj = tj+1 − tj and ∆t = max{∆t1, ∆t2, · · · , ∆tn−1} , where [tj, tj+1] (j = 0, 1, · · · ,n − 1) and a = t0 < t1 < · · · < tn−1 < tn = b is a partition of the interval [a,b] . clearly, we find from definition 2.3 that ai α b f(x) =   0 (a = b) − biαa f(x) (a < b). (2.3) if, for any x ∈ [a,b] , there exists aiαx f(x), then we denote it simply as follows: f(x) ∈ iαx [a,b] . lemma 2.1. (see [28] and [32]) it is asserted that (i) dαxkα dxα = γ(1 + kα) γ(1 + (k − 1) α) x(k−1)α; (ii) 1 γ(1 + α) ∫ b a xkα(dx)α = γ(1 + kα) γ(1 + (k + 1) α) ( b(k+1)α −a(k+1)α ) (k ∈ r). lemma 2.2. (see [28] and [32]) each of the following assertions holds true. (1) local fractional integration is anti-differentiation: suppose that f(x) = g(α)(x) ∈ cα [a,b] . then ai α b f(x) = g(b) −g(a). (2) local fractional integration by parts: suppose that f(x),g(x) ∈ dα [a,b] and f(α)(x), g(α)(x) ∈ cα [a,b] . then ai α b f(x)g (α)(x) = f(x)g(x)|ba −a i α b f (α)(x)g(x). we next recall that sarikaya et al. [23] proved the following generalized steffensen inequality for local fractional integrals. some generalized steffensen’s inequalities 101 theorem 2.1. let f(x),g(x) ∈ iαx [a,b] such that the function f(x) is non-increasing and 0 5 g(x) 5 1 on [a,b] (a < b). then b−λi α b f(x) 5 ai α b f(x)g(x) 5 ai α a+λf(x), (2.4) where λα = γ(1 + α) ai α b g(x). sarikaya et al. [23] also stated the following identities which we shall use in order to prove our main results in this paper. 1 γ(α + 1) ∫ a+λ a f(x)(dx)α − aiαb f(x)g(x) = 1 γ(α + 1) ∫ a+λ a [f(x) −f (a + λ)] [1 −g(x)] (dx)α + 1 γ(α + 1) ∫ b a+λ [f (a + λ) −f(x)] g(x)(dx)α (2.5) and ai α b f(x)g(x) − 1 γ(α + 1) ∫ b b−λ f(x)(dx)α = 1 γ(α + 1) ∫ b−λ a [f(x) −f (b−λ)] g(x)(dx)α + 1 γ(α + 1) ∫ b b−λ [f (b−λ) −f(x)] [1 −g(x)] (dx)α. (2.6) the interested reader is referred to several other related works including (for example) [2], [5], [6], [11], [12], [18] to [24] and [28] to [39] for the theory and applications of local fractional calculus. in section 3 of this paper, we give several inequalities which provide generalizations steffensen’s inequality (1.1) for local fractional integrals. 3. main results we start with the following important identity for our work. throughout this paper, tk(x) is defined by tk(x) = (n− 1 −k)α γ(1 + kα) ( f(kα)(a)(x−a)kα −f(kα)(b)(x− b)α (b−a)α ) . (3.1) lemma 3.1. let f(n−1)α(t) be absolutely continuous on [a,b] with f(n)α ∈ iαx [a,b]. then 1 nα ( f(x) γ(1 + α) + n−1∑ k=1 fk ) − 1 (b−a)αγ(1 + α) ∫ b a f(y)(dy)α = 1 nα (b−a)αγ ( 1 + (n− 1)α ) [γ(1 + α)]2 × ∫ b a (x− t)(n−1)α κα(t,x)f(nα)(t)dt, where fk(x) = (n−k)α γ(1 + kα) ( f(k−1)α(a)(x−a)kα −f(k−1)α(b)(x− b)kα (b−a)α ) and κα(t,x) =   (t−a)α (a 5 t 5 x 5 b) (t− b)α (a 5 x < t 5 b). 102 tunç,sarikaya and srivastava proof. we begin by recalling the following local fractional taylor’s formula: f(x) = f(y) + n−1∑ k=1 f(kα)(y)(x−y)kα γ(1 + kα) + 1 γ(1 + α) ∫ b a f(nα)(t)(x− t)(n−1)α γ ( 1 + (n− 1)α ) (dt)α, which, upon integration with respect to y from y = a to y = b, yields∫ b a (dy)α ∫ x y (dt)α = ∫ x a (dy)α ∫ x y (dt)α + ∫ b x (dy)α ∫ x y (dt)α = ∫ x a (dt)α ∫ t a (dy)α − ∫ b x (dy)α ∫ y x (dt)α = ∫ x a (dt)α ∫ t a (dy)α − ∫ b x (dt)α ∫ b t (dy)α. thus, by evaluating the last integral, we have f(x)(b−a)α γ(1 + α) = 1 γ(1 + α) ∫ b a f(y)(dy)α + n−1∑ k=1 ik + 1 γ ( 1 + (n− 1)α ) [γ(1 + α)]2 × ∫ b a f(nα)(t)(x− t)(n−1)ακα(t,x)(dt)α, (3.2) where ik(x) = 1 γ(1 + α) ∫ b a f(kα)(y)(x−y)kα γ(1 + kα) (dy)α. by local fractional integration by parts in the above expression for ik(x), we get ik(x) = ik−1(x) − (b−a)αfk(x)(n−k)−α (1 5 k 5 n− 1). hence (n−k)α [ik(x) − ik−1(x)] = −(b−a)αfk(x). (3.3) the sum from k = 1 to k = n− 1 in (3.3) is given by n−1∑ k=1 ik = −(b−a)α n−1∑ k=1 fk(x) + (n− 1)αi0. (3.4) substituting from (3.4) into (3.2) and rearranging the resulting equation, we have desired inequality asserted by lemma 3.1. � theorem 3.1. let f : [a,b] 7→ rα be such that f(n−1)α is absolutely continuous for some n ∈ n\{1}. suppose that the functions g,h ∈ iαx [a,b] are such that h is positive and 0α 5 g 5 1α on [a,b]. also let aia+λh(t) = aibg(t)h(t) and the function s1 be defined by s1(x) =   1 γ(1 + α) ∫x a [1 −g(t)] h(t)(dt)α (x ∈ [a,a + λ]) 1 γ(1 + α) ∫ b x g(t)h(t)(dt)α (x ∈ [a + λ,b]). (3.5) some generalized steffensen’s inequalities 103 then 1 γ(1 + α) ∫ a+λ a f(t)h(t)(dt)α − 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α − γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s1(x)tk(x)(dx) α = − 1 (b−a)αγ ( 1 + (n− 2)α ) [γ(1 + α)]2 × ∫ b a (∫ b a s1(x)(x− t)(n−2)α κα(t,x)(dx)α ) f(nα)(t)(dt)α. (3.6) proof. it is easily seen that 1 γ(1 + α) ∫ a+λ a f(t)h(t)(dt)α − 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α = 1 γ(1 + α) ∫ a+λ a [f(t) −f (a + λ)] [1 −g(t)] h(t)(dt)α + 1 γ(1 + α) ∫ b a+λ [f (a + λ) −f(t)] g(t)h(t)(dt)α = ( 1 γ(1 + α) ∫ t a [1 −g(x)] h(x)(dx)α ) [f(t) −f (a + λ)] ∣∣∣∣a+λ a − 1 γ(1 + α) ∫ a+λ a ( 1 γ(1 + α) ∫ t a [1 −g(x)] h(x)(dx)α ) (df(t))α + ( 1 γ(1 + α) ∫ t b g(x)h(x)(dx)α ) [f (a + λ) −f(t)] ∣∣∣∣b a+λ − 1 γ(1 + α) ∫ b a+λ ( 1 γ(1 + α) ∫ b t g(x)h(x)(dx)α ) (df(t))α = − 1 γ(1 + α) ∫ a+λ a ( 1 γ(1 + α) ∫ t a [1 −g(x)] h(x)(dx)α ) (df(t))α − 1 γ(1 + α) ∫ b a+λ ( 1 γ(1 + α) ∫ b t g(x)h(x)(dx)α ) (df(t))α = − 1 γ(1 + α) ∫ b a s1(t)d(f(t)) α = − 1 γ(1 + α) ∫ b a s1(x)f (α)(x)(dx)α. by applying lemma 3.1 with f(α) and replacing n by n− 1 (n ∈ n\{1}), we obtain f(α)(x) = −γ(1 + α) n−2∑ k=0 tk(x) + 1 (b−a)αγ ( 1 + (n− 2)α ) γ(1 + α) × ∫ b a (x− t)(n−2)α κα(t,x)f(nα)(t)(dt)α. (3.7) moreover, by using the equation (3.7), we get 104 tunç,sarikaya and srivastava 1 γ(1 + α) ∫ b a s1(x)f (α)(x)(dx)α = −γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s1(x)tk(x)(dx) α + 1 (b−a)αγ ( 1 + (n− 2)α ) [γ(1 + α)]2 × ∫ b a s1(x) (∫ b a (x− t)(n−2)α κα(t,x)f(nα)(t)(dt)α ) (dx)α. (3.8) finally, by applying fubini’s theorem for local fractional double integrals in the last term in (3.8), we arrive at the assertion (3.6) of theorem 3.1. � theorem 3.2. let f : [a,b] 7→ rα be such that f(n−1)α is absolutely continuous for some n ∈ n\{1}. suppose that the functions g,h ∈ iαx [a,b] are such that h is positive and 0α 5 g 5 1α on [a,b]. also let b−λibh(t) = aibg(t)h(t) and the function s2 be defined by s2(x) =   1 γ(1 + α) ∫x a g(t)h(t) (dt)α (x ∈ [a,b−λ]) 1 γ(1 + α) ∫ b x [1 −g(t)] h(t)(dt)α (x ∈ [b−λ,b]). (3.9) then 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α − 1 γ(1 + α) ∫ b b−λ f(t)h(t)(dt)α − γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s2(x)tk(x)(dx) α = − 1 (b−a)αγ ( 1 + (n− 2)α ) [γ(1 + α)]2 × ∫ b a (∫ b a s2(x)(x− t)(n−2)α κα(t,x)(dx)α ) f(nα)(t)(dt)α. (3.10) some generalized steffensen’s inequalities 105 proof. we observe that 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α − 1 γ(1 + α) ∫ b b−λ f(t)h(t)(dt)α = 1 γ(1 + α) ∫ b−λ a [f(t) −f (b−λ)] g(t)h(t)(dt)α + 1 γ(1 + α) ∫ b b−λ [f (b−λ) −f(t)] [1 −g(t)] h(t)(dt)α = ( 1 γ(1 + α) ∫ t a g(x)h(x)(dx)α ) [f(t) −f (b−λ)] ∣∣∣∣b−λ a − 1 γ(1 + α) ∫ b−λ a ( 1 γ(1 + α) ∫ t a g(x)h(x)(dx)α ) (df(t))α + ( 1 γ(1 + α) ∫ t b [1 −g(x)] h(x)(dx)α ) [f (b−λ) −f(t)] ∣∣∣∣b b−λ − 1 γ(1 + α) ∫ b b−λ ( 1 γ(1 + α) ∫ b t [1 −g(x)] h(x)(dx)α )( df(t) )α = − 1 γ(1 + α) ∫ b−λ a ( 1 γ(1 + α) ∫ t a g(x)h(x)(dx)α )( df(t) )α − 1 γ(1 + α) ∫ b b−λ ( 1 γ(1 + α) ∫ b t [1 −g(x)] h(x)(dx)α )( df(t) )α = − 1 γ(1 + α) ∫ b a s2(t)d ( f(t) )α = − 1 γ(1 + α) ∫ b a s2(x)f (α)(x)(dx)α. by making use of lemma 3.1 with f(α) and replacing n by n− 1 (n ∈ n\{1}), we obtain f(α)(x) = −γ(1 + α) n−2∑ k=0 tk(x) + 1 (b−a)αγ ( 1 + (n− 2)α ) γ(1 + α) × ∫ b a (x− t)(n−2)α κα(t,x)f(nα)(t)(dt)α. (3.11) thus, by using this last equation (3.11), we get 1 γ(1 + α) ∫ b a s2(x)f (α)(x)(dx)α = −γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s2(x)tk(x)(dx) α + 1 (b−a)αγ ( 1 + (n− 2)α ) [γ(1 + α)]2 × ∫ b a s2(x) (∫ b a (x− t)(n−2)α κα(t,x)f(nα)(t)(dt)α ) (dx)α. (3.12) finally, by applying fubini’s theorem of local fractional double integrals in the last term in (3.12), we deduce the result asserted by theorem 3.2. � theorem 3.3. let f : [a,b] 7→ rα be such that f(n−1)α is absolutely continuous for some n ∈ n\{1}. suppose that the functions g,h ∈ iαx [a,b] are such that h is positive and 0α 5 g 5 1α on [a,b]. also let aia+λh(t) = aibg(t)h(t) 106 tunç,sarikaya and srivastava and the function s1 be defined by (3.5). if the function f is n-convex for local fractional calculus and∫ b a s1(x)(x− t)(n−2)α κα(t,x)(dx)α 5 0 (t ∈ [a,b]), then the following inequality holds true: 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α 5 1 γ(1 + α) ∫ a+λ a f(t)h(t)(dt)α − γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s1(x)tk(x)(dx) α. proof. since the function f is n-convex, we can suppose that f is n times differentiable and f(nα) = 0. using this property and the assumption (3.6) of theorem 3.1, we get the required inequality asserted by theorem 3.3. � theorem 3.4. let f : [a,b] 7→ rα be such that f(n−1)α is absolutely continuous for some n ∈ n\{1}. suppose that the functions g,h ∈ iαx [a,b] are such that h is positive and 0α 5 g 5 1α on [a,b]. also let b−λibh(t) = aibg(t)h(t) and the function s2 be defined by (3.9). if the function f is n-convex for local fractional calculus and∫ b a s2(x)(x− t)(n−2)α κα(t,x)(dx)α 5 0 (t ∈ [a,b]), then the following inequality holds true: 1 γ(1 + α) ∫ b a f(t)g(t)h(t)(dt)α = 1 γ(1 + α) ∫ b b−λ f(t)h(t)(dt)α + γ(1 + α) n−2∑ k=0 1 γ(1 + α) ∫ b a s2(x)tk(x)(dx) α. proof. since the function f is n-convex, we can suppose that f is n times differentiable and f(nα) = 0. using this property and the assumption (3.10) of theorem 3.2, we obtain the required inequality asserted by theorem 3.4. � 4. concluding remarks and observations the present investigation is motivated essentially by widespread applications of fractional calculus and local fractional calculus (also called fractal calculus) in a large variety of problems in mathematical, physical and engineering sciences. here, in this paper, we have first derived an identity for local fractional integrals. we then make use of this identity in order to derive several generalizations of the celebrated steffensen’s inequality associated with local fractional integrals. we have also briefly considered relevant connections of the results presented in this paper with the results which were proven in earlier works. references [1] a. m. fink, bounds of the deviation of a function from its averages, czechoslovak math. j. 42 (1992), 289–310. [2] h. budak, m. z. sarikaya and h. yildirim, new inequalities for local fractional integrals, iranian j. sci. tech. trans. a sci. (in press). [3] s. abramovich, m. k. bakula, m. matić and j.e. pečarić, a variant of jensen-steffensen’s inequality and quasiarithmetic means, j. math. anal. appl. 307 (2005), 370–386. [4] j. bergh, a generalization of steffensen’s inequality, j. math. anal. appl. 41 (1973), 187–191. [5] g.-s. chen, generalizations of hölder’s and some related integral inequalities on fractal space, j. funct. spaces appl. 2013 (2013), article id 198405. [6] s. erden and m. z. sarikaya, generalized pompeiu type inequalities for local fractional integrals and its applications, appl. math. comput. 274 (2016), 282–291. [7] h. gauchman, on a further generalization of steffensen’s inequality, j. inequal. appl. 5 (2000), 505–513. [8] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractinal differential equations, north-holland mathematical studies, vol. 204, elsevier (north-holland) science publishers, amsterdam, 2006. some generalized steffensen’s inequalities 107 [9] d. s. mitrinović, analytic inequalities, springer-verlag, new-york, heidelberg and berlin, 1970. [10] d. s. mitrinović, j. e. pečarić and a. m. fink, classical and new inequalities in analysis, east european series on mathematical analysis, vol. 61, kluwer academic publishers, dordrecht, boston and new york, 1993. [11] h. mo, generalized hermite-hadamard inequalities involving local fractional integral [arxiv:1410.1062]. [12] h. mo, x. sui and d. yu, generalized convex functions on fractal sets and two related inequalities, abstr. appl. anal. 2014 (2014), article id 636751. [13] j. e. pečarić, on the bellman generalization of steffensen’s inequality, j. math. anal. appl. 88 (1982), 505–507. [14] j. e. pečarić, on the bellman generalization of steffensen’s inequality. ii, j. math. anal. appl. 104 (1984), 432–434. [15] j. e. pečarić, a. perušić and a. vukelić, generalizations of steffensen’s inequality via fink identity and related results, adv. inequal. appl. 2014 (2014), article id 9. [16] j. e. pečarić, a. perušić and a. vukelić, generalizations of steffensen’s inequality via fink identity and related results. ii, rend. ist. mat. univ. trieste 47 (2015), 1–19. [17] f. qi and b.-n. guo, on steffensen pairs, j. math. anal. appl. 271 (2002), 534–541. [18] m. z. sarikaya and h. budak, generalized ostrowski type inequalities for local fractional integrals,proc. amer. math. soc. (in press). [19] m. z. sarikaya and h. budak, on generalized hermite-hadamard inequality for generalized convex function, rgmia res. rep. collect. 18 (2015), article id 64. [20] m. z. sarikaya, h. budak and s. erden, on new inequalities of simpson’s type for generalized convex functions, rgmia res. rep. collect. 18 (2015), article id 66. [21] m. z. sarikaya, s. erden and h. budak, some integral inequalities for local fractional integrals, rgmia res. rep. collect. 18 (2015), article id 65. [22] m. z. sarikaya, s. erden and h. budak, some generalized ostrowski type inequalities involving local fractional integrals and applications, adv. inequal. appl. 2016 (2016), article id 6. [23] m. z. sarikaya, t. tunç and s. erden, generalized steffensen inequalities for local fractional integrals, rgmia res. rep. collect. 18 (2015), article id 85. [24] m. z. sarikaya, t. tunç and h. budak, on generalized some integral inequalities for local fractional integrals, appl. math. comput. 276 (2016), 316–323. [25] j. f. steffensen, on certain inequalities and methods of approximation, j. inst. actuar. 51 (1919), 274–297. [26] u. m. ozkan and h. yildirim, steffensen’s integral inequality on time scales, j. inequal. appl. 2007 (2007), article id 46524. [27] s.-h. wu and h. m. srivastava, some improvements and generalizations of steffensen’s integral inequality, appl. math. comput. 192 (2007), 422–428. [28] x.-j. yang, advanced local fractional calculus and its applications, world science publisher, new york, london and hong kong, 2012. [29] x.-j. yang, local fractional integral equations and their applications, adv. comput. sci. appl. 1 (2012), 234–239. [30] x.-j. yang, generalized local fractional taylor’s formula with local fractional derivative, j. expert syst. 1 (2012), 26–30. [31] x.-j. yang, local fractional fourier analysis, adv. mech. engrg. appl. 1 (2012), 12–16. [32] x.-j. yang, d. baleanu and h. m. srivastava, local fractional integral transforms and their applications, academic press (elsevier science publishers), amsterdam, heidelberg, london and new york, 2016. [33] y.-j. yang, d. baleanu and x.-j. yang, analysis of fractal wave equations by local fractional fourier series method, adv. math. phys. 2013 (2013), article id 632309. [34] x.-j. yang, d. baleanu and h. m. srivastava, local fractional similarity solution for the diffusion equation defined on cantor sets, appl. math. lett. 47 (2015), 54–60. [35] x.-j. yang, j. a. t. machado and h. m. srivastava, a new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, appl. math. comput. 274 (2016), 143–151. [36] x.-j. yang and h. m. srivastava, an asymptotic perturbation solution for a linear operator of free damped vibrations in fractal medium described by local fractional derivatives, commun. nonlinear sci. numer. simulat. 29 (2015), 499–504. [37] x.-j. yang, h. m. srivastava and d. baleanu, initial-boundary value problems for local fractional laplace equation arising in fractal electrostatics, j. appl. nonlinear dyn. 4 (2015), 349–356. [38] x.-j. yang, h. m. srivastava and c. cattani, local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, romanian rep. phys. 67 (2015), 752–761. [39] y. zhang, h. m. srivastava and m.-c. baleanu, local fractional variational iteration algorithm ii for nonhomgeneous model associated with the non-differentiable heat flow, adv. mech. engrg. 7 (10) (2015), 1–5. 1department of mathematics, faculty of science and arts, düzce university, düzce-turkey 2department of mathematics and statistics, university of victoria, victoria, british columbia v8w 3r4, canada 3center for general education (department of science and technology), china medical university, taichung 40402, taiwan, republic of china ∗corresponding author: tubatunc@duzce.edu.tr 1. introduction 2. definitions, notations and preliminaries 3. main results 4. concluding remarks and observations references analytic solutions of special functional equations © ,fgg  ,,)( dxxxf  d .f g g ,f g ,g .f f xdu  x x )( h .h r, vu vu  [,] vu r[,:] vug ,r)(lim)(lim   wxgxg vxux [,] u [.,[ v [,][,:] vuvuf  [,])),(()( vuxxfgxg  [,] vu ;)(lim,)(lim uxfvxf vxux    ff   1 [;,] vu [;,] vu ,1},{n}),{\[,](  nnvucg n  });{\[,]( vucf n  },{\[,] vu [),,(] 2 vucg  0)( g r  )(lim: 1 xf x })\{[,(])[,(] 21  vucvucf  ;1)(  f [),,(] 3 vucg  0)( g r  )(lim: 1 xf x ,)(lim: 2 rxf x     })[\{,(][),(] 32  vucvucf  ; )( )( 3 2 )( 2    g g f    g , f    ;1 f ,|: [,] ul gg  ;|: [,] vr gg   ],])}()(;[,[sup{))(()( 000 1 0  uxxgxgvxxggxf lrlr    [,[)}()(;],]inf{))(()( 000 1 0 vxxgxguxxggxf rlrl     .f  ,\[,] vux    ., xgx ,g g       .,,, 1111 xxxgxxgx    .1xxf             .:,\[,],1   fvuxxfgxgxg x  xfu ],,]  [,,[ v v . x [,[ v  xf ],,] u  .u [,0]  g   ,0[,,0][,0:]  gg ],0]  [,,[      .limlim 0   xgxg xx ,, ghg  hg , , hg ff  hg ff , ,g h [.,0],,  ghg hgghg  ,,  g ; hg ff  ;,|:,|:, [,[],0] 11 ggggggghgh rlllrr      [,0[[,0[:    ,00   .gh  x )( xizom  xxt :  x x  x ,x ld };;{   xxxd l r d };;{   xxxd r xdg ll : }.{\,))(()(  ll dxxizomxg    xdg rr : }.{\,))(()(  rr dxxizomxg    )()(  rl gg  ),()( rl grgr  )( gr ,g )( xg g x         ., ,: rrll rl dxxgxgdxxgxg xdddg    ddf : dxxfgxg  )),(()( f d  1 f ff   1 ;d ,)}()(;sup{))(()( 000 1 0 llrrlr dxxgxgdxxggxf   .)}()(;inf{))(()( 000 1 0 rrllrl dxxgxgdxxggxf   , ~ g g .       ,0~~,  wgzgwzh         .lim 0 0 ~ ~ lim,0, zw zwg zg w h zz          ,\[,] vu f . g ~ g g ~ [.,] vu f ~ , ~~~ fgg  f ~ f [.,] vu f ~ .                                           .0 !12 ~ !2 ~ ~~ ,0 !12 ~ !2 ~ ~~ 122 2 122 2                               zfzf k g k g zfgzfg zz k g k g zgzg kk k kk k km 2 m g ~   .g , ~~~ fgg      .lim  fzfz                           , ~~ !2/ ~ !2/ ~~~ 2/1 2 2 k k k zfzfkg zzkg z fzf                   .  f  ~ k2 f ~     .1 ~   ff □ v         .0im,re;,0im,re; ,0im,re;,0im,re;     zzvzvzzvzv zzvzvzzvzv rr ll   v   vfvf ~ : ~                      . ~ , ~ , ~ , ~ , ~~   lrlrrlrl vfvfvfvfvfvfvfvfidff  f ~ f f ~ ,0 x f □ w [,] vu f ~ .w       zf z 1 1 z   2 zzf                2 2 ~ 11       z f z z zf z .1 z ,1 1 2   n nan 1, na n ,11 a 2, na n     .2,   zzfzzf       zf z 1 . □    zzf           1 2 1 .1, n n n n n anzaz      ,2,1,0,1, 2 2 1 1       nmpapazazh nn n nm nn n n n  nn mp , ,       . ~ ~ 2 ~ zg zg zzf                      .0 2 ~ ~~~ 0 22 zzfzzf zg zzfzgzgzfg      0 zzf   ,zzf  □ g ~ u ,h  u , f ~                         ,~~2 ~ ,0im ~ 0im , ~~ , ~~~ 1    uguguuf zufzu uuffugufg  .u □         .0,0,0,expexp   axxfxfxx aa        .0/:,0,expexp  abxxbfxfbxx  [,0][,0:] f     ;0,0  ff b/1 ;f ff   1 [;,0]   xf               [;,/1[,expexp];/1,0]inf ],/1,0],expexp[;,/1[sup 0000 0000   bxbxxbxxbxxf bxbxxbxxbxxf f f ~ [,,0]    , ~~ ,1/1 ~ idffbf   f ~ ,/1 b   . 2 12~ bz bz b zzf       bxxxg  exp 0   11 /1   bebg  ,,0 1b  1bg 0 [.,[ 1   b g , ~ g            .,1,,exp1~,0~ 11 cznnbznbzbzgbg nnn     ,0~ 1  bg □ g [,,] vu , 1 b g    .[,,] vgugvu      fbzbzzzg ~ ,,0re,exp ~ 1   f [,0]     haaxx  , .h         .0,; ,,;,; 111 hhhvhxvx xuuvvuxvaxxauuahux    x           ,,],/1[; ,[/1,[; 1 1 rlr l dddibvbvxvd ibbuvxvd          v .v ,: ddf         .,expexp duubfufbuu  ib 1 ;f f ff   1 ;d f ld rd rd ;ld f               ;,expexp;inf ,,expexp;sup 0000 0000 rl lr dubuubuuduuf dubuubuuduuf   ,au  , 1 b      .2)2( 11   buibuibuuf    buuug  exp ,, rl dd g [,] vu ,, 21 lduu  .2,1,  jdu rj 21 , uu  1,0t                              .1 1 11 21 21 21 221 21 1 21 21 2121 utgugt ddxgtddxgt ddtxxtgtuutg ee uu ee uu ee uu          rl dd ,  ,\ 1 ibdu l        0exp  ibubuug l    .\,0 1 ibduug rr   rl gg ,      ,:, 121 ufugrug ll  f f ,f           .,/1 2112 rduvbufufu           ,/1,, 1ubuttfgtg rl  ,1u                  .21 1 1 1 1 1 rrrue u rue u ll grugufgdtfgdtgug       .lr grgr  □ f        ,0,11   xeeee xfxfxx     .0,1ln   xexf x     .0re,1ln ~   zezf z     [,0],1   xeexg xx .2ln ,f          .22 xxfxxfxxfxxf eeeeeeee     ,2ln,0   xee xxf □ f ~        2 2 ~ ,1, ~~~         z z zgzzfgzg   . 1 ~   z z zf international journal of analysis and applications volume 19, number 2 (2021), 288-295 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-288 fixed point theorem for monotone non-expansive mappings joseph frank gordon∗ department of mathematics, nanjing university of aeronautics and astronautics, nanjing 211106, china ∗corresponding author: jgordon@aims.edu.gh abstract. in this paper, we study the fixed point theorem for monotone nonexpansive mappings in the setting of a uniformly smooth and uniformly convex smooth banach space. 1. introduction given a complete metric space (x ,d), the most well-studied types of self-maps are referred to as lipschitz mappings (or lipschitz maps, for short), which are given by the metric inequality d(tx,ty) ≤ kd(x,y),(1.1) for all x,y ∈x , where k > 0 is a real number, usually referred to as the lipschitz constant of t . the metric inequality (1.1) can be classified into three categories, thus contraction mappings for the case where k < 1, non-expansive mappings for the case where k = 1 and expansive mappings for the case where k > 1. the most important property of (1.1) is that they are uniformly continuous. thus, for any sequence {xn}n≥1 converging to x in x , we have d(txn,tx) = 0 as n → ∞. it is well known that when x is complete and t is a contraction mapping, then t has unique fixed point and the sequence of picard iteration tn(x) converges to the fixed point of t as n → ∞. fixed points problems of contraction mappings always exist received january 28th, 2021; accepted february 22nd, 2021; published march 17th, 2021. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. monotone nonexpansive mappings; normalised duality mappings; uniformly convex spaces; uniformly smooth spaces; fixed points. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 288 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-288 int. j. anal. appl. 19 (2) (2021) 289 and it’s unique due to banach [1]. edelstein [2] also showed when t is a contractive mapping (that is, d(tx,ty) < d(x,y)) on a compact metric space x , then t has a unique fixed point and the fixed point can be iteratively approximated by the picard iteration xn+1 = txn. in metric spaces, the only non-trivial thing one can say about nonexpansive mappings is that the picard iteration is a bounded sequence, which is as a result of the inequality d(tnx,tny) ≤ d(x,y),∀n ≥ 0, x,y ∈x , where t is a nonexpansive mapping. even in compact metric spaces, with the exception of the contractive mappings described above, generally one cannot find a fixed point (if it exists) by the picard iteration. it is therefore imperative that one considers the more specialised complete metric spaces: that is, the banach spaces, where linearity and homogeneity affords more structure to the nonexpansive mappings and their fixed points. we use fix(t) to denote the set of fixed points of the mapping t (that is, fix(t) = {x ∈c : tx = x}). an approximate fixed point sequence of a nonexpansive self-map t on a closed convex subset c of banach space x is any sequence {xn}n≥1 ⊂c such that lim n→∞ ‖xn −txn‖ = 0. when c is bounded or fix(t) 6= ∅, then such a sequence always exists. one of the ways to construct an approximate fixed point sequence for nonexpansive mappings is to use the banach contraction mapping theorem [1] to obtain a sequence {xn} in c such that xn = αnx0 + (1 −αn)txn, n ≥ 1 where the initial guess x0 is taken arbitrarily in c and {αn} is a sequence in the interval (0,1) such that αn → 0 as n →∞. by assuming that fix(t) 6= ∅, this sequence {xn} is bounded (indeed, ‖xn−p‖≤‖x0−p‖ for all p ∈ fix(t)). hence ‖xn −txn‖ = αn‖x0 −txn‖→ 0, and {xn} is an approximate fixed point sequence for t. the immediate conclusion from the above deduction is the following result on compact star-convex sets. theorem 1.1. let t be a nonexpansive self-mapping on a compact star-convex subset of a banach space. then t has a fixed point. int. j. anal. appl. 19 (2) (2021) 290 theorem 1.1 is proved by means of the banach contraction mapping theorem [1], and it is in this spirit that we employ the monotone contraction mapping theorem [8] to prove the following weaker but generalized version of theorem 1.1. theorem 1.2 (main theorem). let x be a uniformly smooth, uniformly convex smooth banach space with a sequentially weakly continuous normalized duality mapping, c ⊂x be a weakly-compact star-domain such that 0 ∈ kerc. then every monotone nonexpansive mapping, t : c →c has a fixed point. it is not clear to the author if the ‘sequentially weakly continuity’ condition can be removed, which would be desirable; however, all attempts to do so presently has not been successful and we hope that we may be able to remove it in subsequent work. throughout this paper, < denotes the real part of a complex number. we also use ker(c) to denote the kernel of a star convex subset c (equivalently, star-domain) of a normed linear space, that is {x ∈c : ax + (1 −a)y ∈c,∀∈ [0, 1],y ∈c}. definition 1.1 (normalised duality mapping, see lunner [7],1961). let x be a banach space with the norm ‖·‖ and let x∗ be the dual space of x. denote 〈·, ·〉 as the duality product. the normalised duality mapping j from x to x∗ is defined by jx := {f ∈x∗ : ‖f‖2∗ = ‖x‖ 2 = 〈x,f〉 = fx}, for all x ∈ x . hahn banach theorem guarantees that jx 6= ∅ for every x ∈ x . for our purposes in this work, our interest mostly lies on the case when jx is single-valued for all x ∈x , which is equivalent to the statement that x is a smooth banach space. we say that the normalized duality map j of a banach space x is sequentially weakly continuous if a sequence {xn}n≥1 in x is weakly convergent to x, then the sequence {jxn}n≥1 in x∗ is weak-star convergent to jx. that is, given that xn ⇀ x ∈x , then {jxn}n≥1 ∗ ⇀ jx ∈x∗. remark 1.1. by virtue of the riesz-representation theorem, it follows that jx = x (j is the identity map) when we are in a hilbert space. definition 1.2 (monotone contraction mapping, see gordon [8], 2020). let x be a smooth banach space and let c be a closed subset of x. then the mapping t : c →c is said to be a monotone contraction mapping if there exists 0 ≤ c < 1 such that for all x,y ∈c, the following two conditions are satisfied: 1. <〈tx−ty,jtx−jty〉≤ c<〈x−y,jx−jy〉, 2. <〈tm+1x−tmy,jtn+1x−jtny〉≤ 0, where j is the normalised duality mapping and for all m,n ≥ 0 with m 6= n. int. j. anal. appl. 19 (2) (2021) 291 in this paper, we consider the case where c = 1 in the above definition and introduce the following set of new mappings. definition 1.3 (monotone nonexpansive mapping). let x be a smooth banach space and let c be a closed subset of x. then the mapping t : c → c is said to be a monotone nonexpansive mapping if the following two conditions are satisfied: 1. <〈tx−ty,jtx−jty〉≤<〈x−y,jx−jy〉, 2. <〈tm+1x−tmy,jtn+1x−jtny〉≤ 0, where j is the normalised duality mapping and for all m,n ≥ 0 with m 6= n. we should note here that, monotone nonexpansive mappings reduce to the nonexpansive type of mappings in (1.1) when in hilbert spaces because in hilbert spaces j is the identity mapping. these references browder [3], göhde [4], alpach [5] and kirk [6] can be consulted for fixed point problems on nonexpansive mappings. 2. preliminaries we introduce the following theorem, proposition and lemmas that will be used in the proof of our main result. as before, all notations employed remain as defined. theorem 2.1 (monotone contraction mapping theorem, see gordon [8], 2020). let c be a closed subset of a uniformly convex smooth banach space x and let t : c → c be a monotone contraction mapping. then t has a unique fixed point, that is, fix(t) = {p} and that the picard iteration associated to t , that is, the sequence defined by xn = t(xn−1) = t n(x0) for all n ≥ 1 converges to p for any initial guess x0 ∈x. proposition 2.1 (see for instance ezearn, [9]). let x be a normed linear space. then for any jx ∈ jx,jy ∈ jy (‖x‖−‖y‖)2 ≤<〈x−y,jx− jy〉≤ ‖x−y‖(‖x‖ + ‖y‖). thus, <〈x−y,jx− jy〉≥ 0. moreover if <〈x−y,jx− jy〉 = 0, int. j. anal. appl. 19 (2) (2021) 292 then jx ∈ jy and jy ∈ jx; in particular, when x is smooth (resp. strictly convex) then equality occurs if and only if jx = jy (resp. x=y). proposition 2.2 (see for instance ezearn, [9]). let x be a banach space and let x∗ be the dual space of x. denote 〈·, ·〉 the duality product. now for {xn}n≥1 ⊂ x and {fn}n≥1 ⊂ x∗, suppose either of the following conditions hold • {xn} ⇀ x and {fn}→ f • {xn}→ x and {fn} ∗ ⇀ f then limn→∞〈xn,fn〉 = 〈x,f〉. lemma 2.1 (uniform continuity in uniformly smooth spaces). let x be a uniformly smooth banach space. then the normalised duality map j : x →x∗ is norm-to-norm uniformly continuous. 3. main results in this section, we first give the proof of theorem 1.1 following the proof of our main result, theorem 1.2. proof of theorem 1.1. let c be a compact star convex subset of a banach space x with a distinguished point ‘p’. let t : c →c be a non-expansive mapping on c. for n ≥ 1, define tn : c →c by, tnx = ( n n + 1 ) tx + ( 1 n + 1 ) p,∀x ∈c. obviously, tn is a contraction mapping on c. therefore, by the banach contraction mapping theorem [1], tn has a unique fixed point xn in c. now consider, ‖txn −xn‖ = ‖txn −tnxn‖, = ∥∥∥txn −( n n + 1 ) txn − ( 1 n + 1 ) p ∥∥∥, = ( 1 n + 1 )∥∥∥txn −p∥∥∥,∀n ≥ 1. hence ‖txn − xn‖ → 0 as n → ∞ since c is bounded. since c is compact, the sequence {xn}n≥1 has a convergence subsequence {xnk}k≥1 which converges to some x ∗ ∈ c and by continuity of t , txnk → tx ∗. then consider, txnkxnk = xnk = ( nk nk + 1 ) txnk + ( 1 nk + 1 ) p. by passing k →∞, we have tx∗ = x∗ and hence x∗ is a fixed point of t in c and that completes the proof. int. j. anal. appl. 19 (2) (2021) 293 the proof of our theorem uses the ideas of the proof of theorem 1.1 by creating an internal contraction in order to obtain an approximate fixed point sequence for these new mappings. the proof of our main result is as follows. proof of theorem 1.2. now for every natural number n ≥ 1, define a new mapping tn : c →c as tn(x) = ( 1 − 1 n ) tx. clearly, tn is a self-mapping since 0 ∈ kerc. now we have the following: tx = 1( 1 − 1 n )tnx and ty = 1( 1 − 1 n )tny. by substituting tx and ty into definition 1.3, we have the following: <〈 ( 1 − 1 n )−1 tnx− ( 1 − 1 n )−1 tny, ( 1 − 1 n )−1 j(tnx) − ( 1 − 1 n )−1 j(tny)〉≤<〈x−y,jx−jy〉,( 1 − 1 n )−2 <〈tnx−tny,jtnx−jtny〉≤<〈x−y,jx−jy〉. multiply the last inequality by ( 1 − 1 n )2 to obtain <〈tnx−tny,jtnx−jtny〉≤ ( 1 − 1 n )2 <〈x−y,jx−jy〉. since 0 ≤ ( 1 − 1 n )2 < 1, then by theorem 2.1, tn has a unique fixed point say xn, that is, xn = txn =( 1 − 1 n ) txn and therefore ‖xn −txn‖ = 1n‖txn‖. since c is bounded, then supn≥1 ‖txn‖ = d < ∞ where d is constant. hence lim n→∞ ‖xn −txn‖ = 0,(3.1) where {xn}n≥1 is an approximate fixed point sequence for the monotone nonexpansive mapping t. clearly, equation (3.1) implies xn − txn → 0 as n → ∞. since c is weakly-compact, then the sequence {xn}n≥1 has a weakly converging subsequence. without loss of generality, let {xn}n≥1 be the weakly converging subsequence and x ∈ c be the weak limit of this subsequence, that is, xn ⇀ x as n → ∞. given that xn −txn ⇀ 0 (strong convergence implies weak convergence) and xn ⇀ x, then txn ⇀ x as n →∞. we can clearly see that definition 1.3 is equivalent to the following evaluation: <〈x−y + tx−ty,jx−jy −jtx + jty〉≥<〈tx−ty,jx−jy〉−<〈x−y,jtx−jty〉(3.2) int. j. anal. appl. 19 (2) (2021) 294 for all x,y ∈c. since {xn}n≥1 and it weak limit are both contained in c, then by replacing y with = xn in equation (3.2), we obtain the following <〈x−xn + tx−txn,jx−jxn −jtx + jtxn〉≥<〈tx−txn,jx−jxn〉 −<〈x−xn,jtx−jtxn〉 (3.3) taking limit as n →∞, the left hand side of equation (3.3) becomes lim n→∞ <〈x−xn + tx−txn,jx−jxn −jtx + jtxn〉.(3.4) since xn − txn → 0, then by lemma 2.1, we have that jxn − jtxn → 0. now with txn ⇀ x and jxn −jtxn → 0, then by proposition 2.2, as n →∞, equation (3.4) becomes (3.5) <〈tx−x,jx−jtx〉 = −<〈x−tx,jx−jtx〉. again, taking limit of the right hand side of equation (3.3) as n →∞, we have (3.6) lim n→∞ [<〈tx−txn,jx−jxn〉−<〈x−xn,jtx−jtxn〉]. given that xn = ( 1 − 1 n ) txn for all n ≥ 1, then substituting this sequence into equation (3.6), we obtain lim n→∞ [<〈tx−txn,jx− (1 − 1 n )jtxn〉−<〈x− (1 − 1 n )txn,jtx−jtxn〉], which by expansion gives the following: lim n→∞ [<〈tx,jx〉− (1 − 1 n )<〈tx,jtxn〉−<〈txn,jx〉 −<〈x,jtx〉 + <〈x,jtxn〉 + (1 − 1 n )<〈txn,jtx〉]. (3.7) by the sequentially weakly continuity of x , if txn ⇀ x then jtxn ∗ ⇀ jx so that by proposition 2.2, we have <〈tx,jtxn〉 → <〈tx,jx〉 and <〈txn,jx〉 → <〈x,jx〉 as n → ∞. hence as n → ∞, equation (3.7) reduces to: (3.8) <〈tx,jx〉−<〈tx,jx〉−<〈x,jx〉−<〈x,jtx〉 + <〈x,jx〉 + <〈x,jtx〉 = 0. by equation (3.5) and equation (3.8), equation (3.3) reduces to <〈x−tx,jx−jtx〉≤ 0, which by proposition 2.1 leads to <〈x−tx,jx−jtx〉 = 0. since x is strictly convex, then by proposition 2.1, we have x−tx = 0 which implies x ∈ fix(t) and that completes the proof. int. j. anal. appl. 19 (2) (2021) 295 availability of data and material: no data and material were used for this research. acknowledgements: the author thanks colleagues for their proof reading and other helpful suggestions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. banach, h. steinhaus, sur le principe de la condensation de singularités. fundam. math. 3(9) (1927), 50-61. [2] m. edelstein, an extension of banach’s contraction principle. proc. amer. math. soc. 12(1) (1961), 7-10. [3] f.e. browder, nonexpansive nonlinear operators in a banach space. proc. natl. acad. sci. usa. 54(4) (1965), 1041-1044. [4] d. göhde, zum prinzip der kontraktiven abbildung. math. nachr. 30(3-4) (1965), 251-258. [5] d.e. alspach, a fixed point free nonexpansive map. proc. amer. math. soc. 82(3) (1981), 423-424. [6] w.a. kirk, m.a. khamsi, introduction to metric spaces and fixed point theory. john willy & sons, inc., new york, chichester, singapore, toronto, 2001. [7] l. günter, semi-inner-product spaces. trans. amer. math. soc. 100(1) (1961), 29-43. [8] j.f. gordon, the monotone contraction mapping theorem. j. math. 2020 (2020), 2879283. [9] j. ezearn, fixed point theory of some generalisations of lipschitz mappings with applications to linear and non-linear problems, phd. thesis, kwame nkrumah university of science and technology, kumasi, ghana, 2017. 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 7, number 2 (2015), 104-128 http://www.etamaths.com a spectral analysis of linear operator pencils on banach spaces with application to quotient of bounded operators bekkai messirdi1,∗, abdellah gherbi2 and mohamed amouch3 abstract. let x and y two complex banach spaces and (a,b) a pair of bounded linear operators acting on x with value on y. this paper is concerned with spectral analysis of the pair (a,b). we establish some properties concerning the spectrum of the linear operator pencils a−λb when b is not necessarily invertible and λ ∈ c. also, we use the functional calculus for the pair (a,b) to prove the corresponding spectral mapping theorem for (a,b). in addition, we define the generalized kato essential spectrum and the closed range spectra of the pair (a,b) and we give some relationships between this spectrums. as application, we describe a spectral analysis of quotient operators. 1. introduction let l(x,y ) be the banach algebra of all bounded linear operators from one complex banach space x to another y. if x = y, then l(x,x) = l(x). for a ∈ l(x,y ) we denote by r(a) its range, n(a) its null space and σ(a) its spectrum. if a ∈ l(x), we denote by %(a) the resolvent set of a. let ix (respectively iy ) denotes the identity operator in x (respectively in y ). recall that an operator a ∈ l(x) is called nilpotent if ap = 0 for some p ∈ n∗ and a is said to be quasinilpotent if σ(a) = {0} . for a set m, let ∂m, m denote the boundary and the closure of m, respectively. let a−λb be a linear operator pencil, where a and b are in l(x,y ) and λ ∈ c. the operator b is not considered injective or surjective. for the study of spectral properties of the quotient operators a/b : bx −→ ax, defined by a/b(bx) = ax, where n(b) ⊂ n(a), we need to consider the spectrum of the operator pencil a−λb where λ ∈ c. furthermore, many authors consider the generalized eigenvalue problems ax = λbx and discussed the spectra of quadratic operator pencils, see [2, 13, 22]. note that, in the finite dimensional case the generalized eigenvalue problems is one of the basic problems in the control theory of linear systems with finite dimensional state space. the solution of this problem is well-known as rosenbrok’s theorem [29]. however, in the infinite dimensional case, a complete description of the spectra of operators a − λb is known when the pair (a,b) is exactly controllable, that is the matrix operator [b,ab,...,ap−1b] ∈ l(xp,y ) is right invertible for some integer p. if 2010 mathematics subject classification. 47a05, 47b33, 47a10. key words and phrases. operator pencils; functional calculus; spectral mapping theorem; browder spectrum; generalized kato type spectrum; quotient of bounded operators. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 104 a spectral analysis of linear operator pencils 105 b is self-adjoint, positive and invertible then the eigenvalue problem ax = λbx is equivalent to b−1ax = λx or to b−1/2ab−1/2y = λy with y = b1/2x, and the problem is also equivalent to a standard one for a self-adjoint operator where the spectrum is real. thus, the interesting case is when both a and b are not sign-definite, the pencil spectrum can be non-real. in particular, if neither a nor b is invertible, then the problem poses major difficulties. typical problems include: characterization of the spectrum of a − λb, localization of non-real eigenvalues, asymptotic of real eigenvalues, dependence on parameters and often the use of complex analysis. similar problems, as well as some other related questions, have been studied in a variety of situations in mathematical literature, see [9, 16, 32]. in physical literature, our problem appears in the study of electron waveguides in graphene, see [18, 31] and many references there. the objective of this paper is to investigate the spectrum of linear operator pencils of type a−λb. our work generalizes initially some results of [28] to the case of operators defined on a banach space x with values in another banach space y which is not necessarily equal to x. thereafter, we extend our study to some different essential spectra. we state some basic results for linear operator pencils with non-empty resolvent set. we present particularly a simple demonstration than that obtained by ditkin in [14] on the spectrum of (a − λb) when b is assumed to be compact. the originality of our technique allows us to operate a functional calculus on linear operator pencils. we also got a spectral characterization on quotient operators through that we have established on linear operator pencils. the obtained results bring quite information for the investigation of joint spectra and in particular the spectra of quotients operators. the present work is organized as follow: after the second section where several basic definitions and facts will be recalled, in section 3, we study some basic spectral properties for linear operator pencils. the fourth section is consecrated to the functional calculus of a pair of bounded operators. in section five, we investigate the isolated points of the spectrum of a pair of bounded linear operators. we define various essential spectra of linear operator pencils on a banach space. we define the generalized kato essential spectrum of a pair of bounded operators, and we also give some relationships between this spectrums and the closed range spectra. the obtained results are finally used in the last section to describe a spectral analysis of quotient operators. 2. preliminaries we begin this section by the following definitions. definition 2.1. for a pair (a,b) of operators in l(x,y ), the spectrum σ(a,b) of the linear operator pencil (a−λb), or of the pair (a,b), is defined by: σ(a,b) = {λ ∈ c such that (a−λb) is not invertible } = {λ ∈ c such that 0 ∈ σ(a−λb)} . the resolvent set %(a,b) of the pair (a,b) is the complement of the set σ(a,b) in c. (2.1) %(a,b) = c\σ(a,b) = { λ ∈ c such that rλ(a,b) = (a−λb)−1 exists in l(x,y ) } rλ(a,b) is called the resolvent of (a−λb). so σ(a,b) = {λ ∈ c such that n(a−λb) 6= {0} or r(a−λb) 6= y} . 106 messirdi, gherbi and amouch thus, the spectrum σ(a,b) of (a−λb) is the set of all scalars λ in c for which the operator (a−λb) fails to be an invertible element of the banach algebra l(x,y ). from [33, theorem 3.2.4], σ(a−λb) can be an unbounded set. besides that the spectrum σ(a,b) can be an empty set. according to the nature of such a failure, σ(a,b) can be split into many disjoint parts. a classical partition comprises three parts. the point spectrum of (a,b) defined by (2.2) σp(a,b) = {λ ∈ c : (a−λb) is not injective} . a complex number λ ∈ c is an eigenvalue of (a−λb) if there exists a nonzero vector x in x such that ax = λbx, then n(a − λb) 6= {0} . the algebraic multiplicity of an eigenvalue λ is the dimension of the respective eigenspace n(a − λb). the second parts is the set σc(a,b) of those λ for which (a−λb) has a densely defined but unbounded inverse on its range; (2.3) σc(a,b) = { λ ∈ c : n(a−λb) = {0} ,r(a−λb) = y and r(a−λb) 6= y } which is referred to as the continuous spectrum of (a−λb). the third parts is the residual spectrum of (a−λb) is the set σr(a,b) of all scalars λ such that (a−λb) has an inverse on its range that is not densely defined; (2.4) σr(a,b) = { λ ∈ c : n(a−λb) = {0} and r(a−λb) ( y } . the collection {σp(a,b),σc(a,b),σr(a,b)} forms a partition of σ(a,b), which means that they are pairwise disjoint and σ(a,b) = σp(a,b)∪σc(a,b)∪σr(a,b). remark 2.2. 1) if x = y and b = ix, the spectrum of the linear operator pencil a−λix is the spectrum of a, ie σ(a,ix) = σ(a). %(a,ix) = %(a) and rλ(a,ix) = rλ(a) = (a−λix)−1 if λ ∈ %(a). 2) if x = y is a finite dimensional vector space, dim x < ∞, the spectrum σ(a− λb) coincides with the complex plane or it contains no more than n points. example 2.3. let x = y = l2([0, 1]) and define the multiplication operators a and b in l2([0, 1]) by af(x) = (x + 1)f(x) and bf(x) = xf(x). then a and b are bounded with ‖a‖ = 2, ‖b‖ = 1. if (a − λb)f(x) = [(1 −λ)x + 1] f(x) = 0, then f = 0 in l2([0, 1]) when λ ∈ c\{1} . thus, (a − λb) has no eigenvalues if λ ∈ c\{1} . however, if λ = 1, (a − b)f(x) = f(x), thus 1 ∈ σp(a,b). consequently, σp(a,b) = {1} . if xλ = 1 λ−1 or else λ ∈ c\ [2, +∞[ , then [(1 −λ)x + 1] −1 f(x) ∈ l2([0, 1]) for any f ∈ l2([0, 1]) because [(1 −λ)x + 1]−1 is bounded on [0, 1]. thus, c\({1}∪ [2, +∞[) is in %(a,b). if λ ∈ [2, +∞[ , then (a−λb) is not onto, because c [(1 −λ)x + 1]−1 /∈ l2([0, 1]) for c 6= 0, so the nonzero constant functions c do not belong to the range of (a−λb). however, the range of (a−λb) is dense. indeed, for any f ∈ l2([0, 1]), let fn(x) = { f(x) if |x−xλ| ≥ 1n 0 if |x−xλ| < 1n. then, lim n→+∞ fn = f in l 2([0, 1]) and fn ∈ r(a−λb), since [(1 −λ)x + 1] −1 fn ∈ l2([0, 1]), then it follows that σp(a,b) = {1}, σc(a,b) = [2, +∞[, σr(a,b) = ∅ and σ(a,b) = {1}∪ [2, +∞[. a spectral analysis of linear operator pencils 107 remark 2.4. if x∗ and y ∗ are respectively the dual spaces of x and y, and a∗,b∗ : y ∗ −→ x∗ are the adjoint of a and b respectively, then σ(a,b) = σ(a∗,b∗). the spectra and sub-spectra of the pair (a,b) and its adjoint (a∗,b∗) are related by the following relations: theorem 2.5. let (a,b) a pair of operators in l(x,y ), then the following hold: (1) σr(a,b) ⊂ σp(a∗,b∗) ⊂ σr(a,b)∪ σp(a,b). (2) σp(a,b) ⊂ σr(a∗,b∗)∪ σp(a∗,b∗). (3) σc(a,b) ⊂ σr(a∗,b∗) ∪σc(a∗,b∗). (4) σc(a ∗,b∗) ⊂ σc(a,b). (5) σr(a ∗,b∗) ⊂ σp(a,b) ∪σc(a,b). proof. (1) let λ ∈ σr(a,b), then r(a−λb) is not dense in y. by the hahn– banach theorem, there exists a non-zero y∗ ∈ y ∗ that vanishes on r(a − λb). thus, for all x ∈ x, 〈(a−λb)x,y∗〉 = 〈x, (a∗ −λb∗)y∗〉 = 0. therefore (a∗ −λb∗)y∗ = 0 and λ ∈ σp(a∗,b∗). next suppose that (a∗ −λb∗)z∗ = 0 where z∗ 6= 0, that is 〈x, (a∗ −λb∗)z∗〉 = 〈(a−λb)x,z∗〉 = 0 for all x ∈ x. if r(a − λb) is dense, then z∗ must be the zero functional, which is a contradiction. the claim is proved. in particular, when x and y are reflexive banach spaces, we have σr(a ∗,b∗) ⊂ σp(a∗∗,b∗∗) = σp(a,b). one shows (2) to (5) by the same argument. there are some overlapping parts of the spectrum of linear operator pencils which are commonly used. for instance, the compression spectrum σcp (a,b) and the approximate point spectrum σap (a,b), which are defined respectively by: σcp (a,b) = {λ ∈ c : r(a−λb) is not dense in y}(2.5) σap (a,b) = {λ ∈ c : (a−λb) is not bounded below} . let (a,b) a pair of operators in l(x,y ), we list below some classical results concerning σcp (a,b) and σap (a,b). theorem 2.6. (1) the following assertions are pairwise equivalent. (i) for every ε > 0, there is a unit vector xε ∈ x such that ‖(a−λb)xε‖y < ε. (ii) there is a sequence (xn)n∈n of unit vectors in x such that lim n→+∞ ‖(a−λb)xn‖y = 0. (iii) λ ∈ σap (a,b). (2) the approximate point spectrum σap (a,b) is a closed subset of c and that includes the boundary ∂σ(a,b) of the spectrum σ(a,b). (3) if x and y are reflexive banach spaces, we have σcp (a,b) = σp(a ∗,b∗) and σr(a,b) = σcp (a,b)\σp(a,b). proof. (1) clearly (i) implies (ii). if (ii) holds, then there is no constant δ > 0 such that δ = δ‖xn‖x ≤ ‖(a−λb)xn‖y for all n ∈ n. thus, (a − λb) is not bounded below, and so (ii) implies (iii). conversely, if (a − λb) is not bounded below, then there is no constant δ > 0 such that δ‖x‖x ≤ ‖(a−λb)x‖y for all x ∈ x or, equivalently, for every ε > 0 there exists a nonzero tε in x such that 108 messirdi, gherbi and amouch ‖(a−λb)tε‖y ≤ ε‖tε‖x . set xε = tε ‖tε‖x , hence (iii) implies (ii). (2) the quantity j(a−λb) = inf ‖x‖x ‖(a−λb)x‖y is called the injective modulus of the pair (a,b) at λ, and obviously by virtue of 1) we have j(a−λb) = 0 if and only if λ ∈ σap (a,b). moreover, it is easy to show that (2.6) { |j(a−λb) − j(a−µb)| ≤ |λ−µ|‖b‖ j(a−λb) = ‖rλ(a,b)‖ −1 ; for all λ ∈ %(a,b). since the function j(a − λb) is continuous at λ and σap (a,b) is the inverse image by j of 0, it follows that σap (a,b) is closed. now, let λ ∈ %(a,b), then j(a−λb) = ‖rλ(a,b)‖ −1 > 0 and λ /∈ σap (a,b). hence σap (a,b) ⊂ σ(a,b). the case ∂σ(a,b) = ∅ is obvious. if λ ∈ ∂σ(a,b) = σ(a,b) ∩ %(a,b), then there exists a sequence (λn)n∈n in %(a,b) such that lim n→+∞ λn = λ. since (a−λb) is not bounded invertible, then there exists a subsequence of (λn)n∈n for which lim n→+∞ ‖rλn(a,b)‖ = +∞. thus, lim n→+∞ ‖rλn(a,b)‖ −1 = lim n→+∞ j(a−λnb) = 0. by continuity of j(.), we deduce that j(a−λb) = 0 and then λ ∈ σap (a,b). (3) the proof of (3) is similar to the proof of (1) in the previous theorem. the condition 0 ∈ %(a,b) is understood as the continuous reversibility of the operator a, furthermore it is quite simple to see that σ(a,b)\{0} = { 1 λ : λ ∈ σ(b,a) } . this result can be extended to 0 and ∞ by introducing the concept of extended spectrum of a pair (a,b) of bounded operators from x to y . let c̃ = c∪{∞} denote the riemann sphere. c̃ is equipped with the following topology: u ⊆ c̃ is open if and only if u ⊆ c and u is open in c or if u = v ∪{∞} where v ⊆ c such that c\v, the complement of v in c, is compact in c. then c̃ is a compact hausdorff space. definition 2.7. the extended spectrum σ̃(a,b) of a pair (a,b) of bounded operators from x to y is a subset of c̃ which coincides with σ(a,b) if both functions %(a,b) 3 λ −→ brλ(a,b) : c̃ −→ l(y ) and %(a,b) 3 λ −→ rλ(a,b)b : c̃ −→ l(x) are holomorphic at the point ∞ and coincide with σ(a,b) ∪ {∞} otherwise. the set %̃(a,b) = c̃\σ̃(a,b) is called the extended resolvent set of the pair (a,b). we set (a−∞b)−1 = 0. for λ ∈ %(a,b) the two operators rλ,l(a,b) = brλ(a,b) ∈l(y ) and rλ,r(a,b) = rλ(a,b)b ∈ l(x) are called the left and the right resolvent of the pair (a,b), respectively. note that they are also called pseudo resolvent (see [34]). through this definition we have then immediately σ̃(a,b) and σ̃(b,a) are compact subsets of c̃ and (2.7) σ̃(a,b) = { 1 λ : λ ∈ σ̃(b,a) } . a spectral analysis of linear operator pencils 109 for more details on the spectrum σ̃(a,b), let λ0 be a fixed point of %(a,b) and define φ0 : c̃ −→ c̃ by: (2.8) φ0(λ) =   1 λ−λ0 if λ 6= λ0, λ 6= ∞ ∞ if λ = λ0 0 if λ = ∞. then, φ0 is an homeomorphism, its inverse mapping is given by (2.9) λ = φ−10 (µ) =   1 µ + λ0 if µ 6= λ0, µ 6= ∞ ∞ if µ = 0 0 if µ = ∞. however, a−λb = −µ−1(a−λ0b)[rλ0,r(a,b) −µix] = −µ−1[rλ0,l(a,b) −µiy ](a−λ0b) where µ = (λ−λ0)−1 6= 0. so, λ ∈ %(a,b) if and only if µ = φ0(λ) ∈ %(rλ0,j(a,b)), j = r, l, then φ0(σ̃(a,b)) = σ(rλ0,r(a,b)) = σ(rλ0,l(a,b))(2.10) σ̃(a,b) = φ−10 (σ(rλ0,j(a,b))), j = r, l. we can also directly deduce that if λ ∈ %(a,b) then (2.11) dist(λ,σ̃(a,b)) ≥ 1 ‖rλ,j(a,b)‖ ; j = l,r. 3. some basic spectral properties of linear operator pencils in this section we give some spectral properties of the operator pencils (a−λb). we begin by the following theorem. theorem 3.1. let a,b ∈l(x,y ). then the following assertions hold: (1) σ(a,b) is a closed set in c. (2) if %(a,b) 6= ∅, then a(n(b)) is closed in y. (3) if a is invertible, then (a−λb) is equivalent to the linear pencil ix −λa−1b and hence %(a,b) = %(ix − λa−1b), σ(a,b) = σ(ix − λa−1b), σi(a,b) = σi(ix −λa−1b), i = p,c,r, and λ ∈ %(a,b) for sufficiently small |λ| . (4) the resolvent operator rλ(a,b) for λ ∈ %(a,b) is holomorphic function on %(a,b) with values in l(y,x) and dn dλn rλ(a,b) = n!rλ(a,b)b n(rλ(a,b)) n(3.1) = n!(rλ(a,b)) nbnrλ(a,b). (5) if λ,µ ∈ %(a,b), then we have the equalities rλ(a,b) −rµ(a,b) = (λ−µ)rλ(a,b)brµ(a,b)(3.2) rλ,j(a,b) −rµ,j(a,b) = (λ−µ)rλ,j(a,b)rµ,j(a,b) ; j = l,r. (6) if λ,µ ∈ %(a,b), then (3.3) rλ(a,b)brµ(a,b) = rµ(a,b)brλ(a,b). we say that the operators rλ(a,b) and rµ(a,b) commute modulo b. (7) for all λ ∈ %(a,b), rλ,l(a,b) and rλ,r(a,b) have the same spectrum that is, σ(rλ,l(a,b)) = σ(rλ,r(a,b)). 110 messirdi, gherbi and amouch (8) σ(a,b) = ∅ if and only if a is continuously invertible and a−1b is quasinilpotent on x. proof. (1) let (λn)n∈n be a sequence of elements in σ(a,b) such that (λn)n∈n converges to α ∈ c. then, (a−λnb)n∈n is a sequence of non-invertible operators in l(x,y ) which converges strongly to the operator (a − αb). we deduce that (a−αb) can not be invertible in l(x,y ) since the set of all non-invertible operators in l(x,y ) is closed. consequently, α ∈ σ(a,b) and this shows that σ(a,b) is closed set in c. (2) observe that for any λ ∈ c, a(n(b)) = (a−λb)(n(b)). now, if λ ∈ %(a,b), the operator (a − λb) has a continuous inverse, so it maps closed subspaces to closed subspaces. the claim is now proved since n(b) is closed in x. the statements in (3) follow from the equality, (a−λb) = a(ix −λa−1b) and the fact that the set of all invertible operators in l(x,y ) is open. indeed, select λ sufficiently small so that |λ| ∥∥a−1b∥∥ < 1. then (ix −λa−1b) is invertible and (ix −λa−1b)−1 = ∑∞ n=0 λ na−nbn ∈l(x). therefore (3.4) rλ(a,b) = ∞∑ n=0 λna−nbna−1. (4) observe that rλ(a,b) = [(a−λ0b) − (λ−λ0)b] −1 = rλ0 (a,b) [iy − (λ−λ0)brλ0 (a,b)] −1 . if |λ−λ0| < 1‖rλ0,l(a,b)‖ , then the second inverse above is given by a convergent neumann series: (3.5) rλ(a,b) = rλ0 (a,b) ∞∑ n=0 bn(rλ0 (a,b)) n(λ−λ0)n. thus, rλ(a,b) is given by a convergent power series about any point λ0 ∈ %(a,b) that is, the resolvent set %(a,b) is open, so rλ(a,b) defines an l(y,x) -valued holomorphic function on the resolvent set %(a,b) of (a−λb). note that from the series one obtains that dn dλn rλ(a,b) |λ=λ0 = n!(rλ0 (a,b)b n(rλ0 (a,b)) n = n!(rλ0 (a,b)) nbnrλ0 (a,b). hence, we obtain (3.1) for any λ ∈ %(a,b). (5) (a−λb)−1 − (a−µb)−1 = (a−λb)−1[(a−µb) − (a−λb)](a−µb)−1 = (λ−µ)rλ(a,b)brµ(a,b). a spectral analysis of linear operator pencils 111 by the same method and (2.11) we obtain the identities for the left and the right resolvent of the pair (a,b). indeed if for example j = l, rλ,l(a,b) −rµ,l(a,b) = b[rλ(a,b) −rµ(a,b)] = (λ−µ)brλ(a,b)brµ(a,b) = (λ−µ)rλ,l(a,b)rµ,l(a,b). (6) by using (2.11) we have (λ−µ) [rλ(a,b)brµ(a,b) −rµ(a,b)brλ(a,b)] = 0, this proves the result. (7) for all λ ∈ %(a,b), rλ,l(a,b) and rλ,r(a,b) are similar : rλ(a,b)rλ,l(a,b) = rλ,r(a,b)rλ(a,b). it is clear, that similar operators rλ,l(a,b) and rλ,r(a,b) have the same spectral properties and, particularly, σ(rλ,l(a,b)) = σ(rλ,r(a,b)). (8) if σ(a,b) = ∅, then rλ(a,b) ∈ l(y,x) for all λ ∈ c. in particular, a is invertible with bounded inverse a−1 ∈ l(y,x) and hence σ(a,b) = σ(a(ix −λa−1b)) = σ((ix −λa−1b)). if λ0 ∈ σ(a−1b)) and λ0 6= 0, then (λ0ix − a−1b) and (ix − 1λ0 a −1b) are not invertible in l(x), this gives 1 λ0 ∈ σ(a,b), this is a contradiction. since a−1b ∈ l(x), then σ(a−1b) 6= ∅, so λ0 = 0 and hence σ(a−1b) = {0} . conversely, suppose that λ0 ∈ σ(a,b) and a is invertible in l(x,y ). thus λ0 6= 0 and (a− λ0b) = λ0a( 1 λ0 ix −a−1b). as (a−λ0b) is not invertible in l(x,y ), we deduce that ( 1 λ0 ix −a−1b) is not invertible in l(x). hence, 1λ0 ∈ σ(a −1b). note that σ(a,b) is not necessarily bounded, see [33, theorem 3.2.4]. the following examples shows that σ(a,b) can be the whole complex plane and it can be discrete. moreover, it may be empty. example 3.2. 1) let a = ( 2 2 0 3 ) and b = ( 1 0 0 0 ) . then det(a−λb) = 3(2 −λ) and σ(a,b) = {2} . 2) if a = ( 1 2 0 3 ) andb = ( 0 1 0 0 ) , then det (a−λb) = 3,σ(a,b) = ∅ and %(a,b) = c. 3) if a = ( 1 2 0 0 ) and b = ( 1 0 0 0 ) , then det (a−λb) = 0,σ(a,b) = c and %(a,b) = ∅. 4) let x = h2 (]0, 1[) ∩h10 (]0, 1[) be the hilbert space of complex measurable functions f on ]0, 1[ such that∫ 1 0 (|f(t)|2 + |f ′ (t)|2 + |f ′′ (t)|2)dt < ∞, < f,g >x= ∫ 1 0 (f(t)g(t) + f ′ (t)g ′ (t) + f ′′ (t)g ′′ (t))dt ; f,g ∈ x and f(0) = f(1) = 0, where the derivatives are taken in the distribution sense. let y = l2 (]0, 1[) be the hilbert space of complex measurable functions on ]0, 1[ such 112 messirdi, gherbi and amouch that < f,g >y = ∫ b a f(t)g(t)dt and ∫ 1 0 |f(t)|2 dt < ∞ . define a ∈l(x,y ) and b ∈l(x,y ) by setting{ af(t) = f ′′ (t) + π2f(t) bf(t) = f(t). for each g ∈ y and λ ∈ c we wish to find f ∈ x to solve the differential equation f ′′ (t) + (π2 −λ)f(t) = g(t). with the above notations this equation can be written in the form (a−λb)f(t) = g(t) and hence the solution is given, if λ /∈ σ(a,b), by f(t) = (a−λb)−1g(t). set ek(t) = √ 2 sin(kπt), k ∈ n∗, t ∈]0, 1[. it is well known that each f ∈ x can be written as f = ∞∑ k=1 fkek such that ∞∑ k=1 ( 1 + k2π2 )2 |fk| 2 < ∞ and each g ∈ y can be written as g = ∑∞ k=1 gkek such that ∑∞ k=1 |gk| 2 < ∞. it is easy to see that (a − λb) is invertible in l(x,y ). indeed, by the equation of coefficients in the respective fourier series we obtain the solution of the differential equation: f1 = − g1 λ and fk = − gk λ + (k2 − 1)π2 , k = 2, 3, ... thus, f = − g1 λ e1 − ∞∑ k=2 gkek (k2 − 1)π2 ( 1 − λ (k2 − 1)π2 + ... ) the expansion is a laurent series of f with a pole of order 1 at 0. hence, σ(a,b) = σp(a,b) = {0}. even if a and b are self-adjoint operators on banach spaces, the spectrum of the pencil (a−λb) is often complex. for the finite-dimensional example consider the hermitian matrices a = ( 1 i √ 2 −i √ 2 1 ) and b = ( 1 1 1 0 ) . then det(a−λb) = −λ2 −λ− 1 and σ(a,b) = { −1 − i √ 3 2 , −1 + i √ 3 2 } . for the infinite-dimensional case, this follows from the fact that the operator t = ( b ix −a 0 ) is not self-adjoint on l(x) ⊕l(x), knowing that under certain conditions the linear operator pencils (a−λb) is equivalent to the quadratic operator pencils mλ = a−λb + λ2ix (see e.g. [12], [4]). theorem 3.3. let a,b ∈l(x). if %(a,b) 6= ∅ and b is invertible, then %(a,b) = %(ab−1) = %(b−1a),(3.6) σ(a,b) = σ(ab−1) = σ(b−1a). a spectral analysis of linear operator pencils 113 proof. let λ ∈ %(a,b), then for all y ∈ y, b(a−λb)−1)(ab−1 −λiy )y = b(a−λb)−1)(a−λb)b−1y = y. thus, (ab−1 − λiy )rλ,l(a,b) = rλ,l(a,b)(ab−1 − λiy ), λ ∈ %(ab−1) and rλ,l(a,b) = (ab −1 −λiy )−1. by the same argument we have also λ ∈ %(b−1a) and rλ,r(a,b) = (b −1a−λix)−1. this imply that %(a,b) ⊂ %(ab−1)∩%(b−1a). conversely, since b is invertible λ ∈ %(a,b) once λ ∈ %(b−1a) or λ ∈ %(ab−1). thus, %(a,b) = %(ab−1) ∩%(b−1a). equality (3.6) follows from σ(rλ,l(a,b)) = σ(rλ,r(a,b)). corollary 3.4. if b is invertible, then rλ,l(a,b) and rλ,r(a,b) are resolvent operators at λ respectively of ab−1 and b−1a. now it can be shown that under certain conditions any closed complex subspace is the spectrum of a linear operator pencils (see [28]). theorem 3.5. there exists a pair of bounded operators (a,b) on a separable banach space x such that for every closed subspace m of c we have σ̃(a,b) = m. proof. let (zj)j∈n be a dense subset of m. if 0 /∈ m, then there exists δ > 0 such that |zj| ≥ δ or each j ∈ n. consider an arbitrary bounded invertible operator c on h where h is assumed to be a separable banach space, then σ̃(c − ( 1 zj )λc) = {zj}. denote x = ⊕∞ j=0 h, a = ⊕∞ j=0 c and b = ⊕∞ j=0( 1 zj )c. thus, σ̃(a,b) = ∞⋃ j=0 σ̃(c − ( 1 zj )λc) = (zj)j∈n = m. if 0 ∈ m, there exists γ > 0, such that (zj)j∈n = (uk)k∈k∪(vl)l∈l with k∪l = n, k ∩ l = ∅, (uk)k ∩ (vl)l = ∅, |uk| ≤ γ for all k ∈ k and |vl| > γ for all l ∈ l. take now with the same previous considerations, ã1 = ⊕∞ j=0 c, b̃1 = ⊕ k∈k ukc, ã2 = ⊕ l∈l ( 1 vl ) c and b̃2 = ⊕∞ j=0 c. then a = ã1 ⊕ ã2, b = b̃1 ⊕ b̃2 are bounded on x and σ̃(a,b) = σ̃(ã1, b̃1) ∪ σ̃(ã2, b̃2) = (uk)k∈k ∪ (vl)l∈l = (zj)j∈n = m. 4. functional calculus on a pair of bounded operators the functional calculus under consideration in this article is of riesz-dunford type, but extended to unbounded spectra. since σ(a,b) can be unbounded, it is necessary to make some assumptions on a and b ∈ l(x,y ). the first such functional calculus was defined by bade [5] for operators with spectrum in a strip. but there are now several other classes of operators with similar functional calculus. let a,b ∈ l(x,y ), where b is not necessarily invertible and let ω an open set of the extended complex plane c̃ containing the extended spectrum σ̃(a,b) of the pair (a,b). denoted by symbol h(ω) the set of holomorphic functions on ω with topology of uniform convergent on compact subsets from ω. h(ω) is a commutative 114 messirdi, gherbi and amouch algebra. more precisely, let h̃(ω) be the set of pairs (f,d̃), where d̃ is an open subset of c̃ containing ω and f is an analytic function on d̃. we introduce the relation (f1,d̃1) ∼ (f2,d̃2) if and only if f1 = f2 in a neighborhood of ω contained in d̃1∩d̃2. we set h(ω) = h̃(ω)� ∼ . let γ be a contour in a domain of f ∈h(ω) that encircles σ̃(a,b) and consists of a finite number of rectifiable jordan curves with a positive orientation. then similar to dunford’s operator calculus we define an operator-function f(a,b) of the pair of bounded operators (a,b) as follows: (4.1) f(a,b) = 1 2πi ∫ γ f(λ)rλ(a,b)dλ where f(a,b) is a well-defined continuous linear operator from h(ω) to l(y,x). note that f(a,b) is a bounded operator, by the de notion of the spectrum and the properties of integration. it is also useful for our study to introduce the two following operators fl(a,b) from h(ω) to l(y ) and fr(a,b) from h(ω) to l(x) by the formula: (4.2) fj(a,b) = 1 2πi ∫ γ f(λ)rλ,j(a,b)dλ, j = r, l. note that if σ̃(a,b) is the whole riemann sphere, then the functional calculus is trivial, since h(c̃) coincides with constant functions. the first main result of this section is the following: theorem 4.1. let a,b ∈ l(x,y ). for f,g ∈ h(ω) and λ0 ∈ %(a,b) we have the following assertions : (1) if f∗(λ) = 1 (λ−λ0) , then rλ0 (a,b) = f ∗(a,b). (2) fl(a,b) and fr(a,b) are continuous homomorphisms of algebra h(ω) and we have the following properties: (i) fl(a,b) = bf(a,b) and fr(a,b) = f(a,b)b. (ii) fr(a,b)g(a,b) = g(a,b)fl(a,b). (iii) fl(a,b)(a−µb) = (a−µb)fr(a,b). (iv) f∗l (a,b) = brλ0 (a,b) and f ∗ r (a,b) = rλ0 (a,b)b. (3) if x = y, cf(a,b)c−1 = f(cac−1,cbc−1) holds for any bounded invertible operator c in l(x). proof. (1) f∗ ∈h(ω) since f∗ is holomorphic on c\{λ0}. thus, (a−λ0b)f∗(a,b) = 1 2πi ∫ γ (a−λ0b) (λ−λ0) rλ(a,b)dλ = 1 2πi ∫ γ 1 (λ−λ0) [iy + brλ(a,b)]dλ = 1 2πi ∫ γλ0 dλ (λ−λ0) iy + 1 2πi intγbrλ(a,b)dλ = iy where γλ0 is a closed curve having λ0 in its interior. similarly we obtain the equality f∗(a,b)(a − λ0b) = ix. thus, (a − λ0b) is invertible in l(x,y ) and (a − λ0b)−1 = rλ0 (a,b) = f∗(a,b). (2) it is clear that the maps f(a,b), fl(a,b) and fr(a,b) are linear on h(ω). let us show that fl(a,b) and fr(a,b) a spectral analysis of linear operator pencils 115 are multiplicative. let f,g ∈h(ω). choose a contour γ2 around γ1 both in ω. − 1 4π2 ∫ γ1 f(λ)rλ,l(a,b)dλ ∫ γ2 g(µ)rµ,l(a,b)dµ = − 1 4π2 ∫ γ1 ∫ γ2 f(λ)g(µ) (λ−µ) [rλ,l(a,b) −rµ,l(a,b)]dλdµ = − 1 4π2 ∫ γ1 f(λ)rλ,l(a,b)dλ ∫ γ2 g(µ) (λ−µ) dµ + 1 4π2 ∫ γ2 g(µ)rµ,l(a,b)dµ ∫ γ1 f(λ) (λ−µ) dλ. but ∫ γ1 f(λ) (λ−µ) dλ = 0 and ∫ γ2 g(µ) (λ−µ) dµ = −2πig(λ). thus fl(a,b)gl(a,b) = (fg)l(a,b). by a similar calculation we obtain fr(a,b)gr(a,b) = (fg)r(a,b). consequently, fl(a,b) and gl(a,b) (resp. fr(a,b) and gr(a,b)) commute. equalities in (i) follow directly from commutation of bounded operators with integration. for (ii), fr(a,b)g(a,b) = f(a,b)bg(a,b) = f(a,b)gl(a,b) = − 1 4π2 ∫ γ1 ∫ γ2 f(λ)g(µ)rλ(a,b)brµ(a,b)dλdµ = − 1 4π2 ∫ γ1 ∫ γ2 f(λ)g(µ)rµ(a,b)brλ(a,b)dλdµ = g(a,b)fl(a,b). this, since rλ(a,b) and rµ(a,b) commute modulo b (see formula (3.3)). (iii) (a−µb)fr(a,b) = 1 2πi ∫ γ f(λ)(a−µb)rλ(a,b)bdλ = 1 2πi ∫ γ (λ−µ)f(λ)brλ(a,b)bdλ = 1 2πi ∫ γ f(λ)brλ(a,b)adλ−µfl(a,b)b = fl(a,b)a−µfl(a,b)b = fl(a,b)(a−µb). (iv) by virtue of (i), we obtain: f∗l (a,b) = 1 2πi ∫ γ 1 (λ−λ0) brλ(a,b)dλ = bf ∗(a,b) = brλ0 (a,b), f∗r (a,b) = 1 2πi ∫ γ 1 (λ−λ0) rλ(a,b)bdλ = f ∗(a,b)b = rλ0 (a,b)b. (3) let x = y, cf(a,b)c−1 = 1 2πi ∫ γ f(λ)[c(a−λb)c−1]−1dλ = 1 2πi ∫ γ f(λ)[(cac−1 −λcbc−1)]−1dλ = f(cac−1,cbc−1). 116 messirdi, gherbi and amouch sahin and ragimov gave in [30] a result on the absence of the point ∞ in the extended spectrum σ̃(a,b) of a pair (a,b) of bounded linear operators in different banach spaces by considering a reducing decomposition of the pair (a,b). precisely, they showed that ∞ /∈ σ̃(a,b) if and only if (a,b) has the following reducing decomposition : x = x1 ⊕ x2, y = y1 ⊕ y2 in direct sums of their respectively closed subspaces xj, yj such that axj ⊂ yj, bxj ⊂ yj, aj = a|xj, bj = b|xj , j = 1, 2, (4.3) (a,b) = (a1 ⊕a2,b1 ⊕b2) = (a1,b1) ⊕ (a2,b2), where the operators a2 and b1 are continuously invertible, (a−12 b2) 2 = 0 and σ̃(a,b) = σ(a,b) = σ(a1,b1). now we will prove the second main result of this section: theorem 4.2. let a,b ∈l(x,y ), f ∈h(ω) and λ0 ∈ %(a,b). then, (1) (4.4) σ(rλ0,l(a,b)) = σ(rλ0,r(a,b)) = { 1 λ−λ0 : λ ∈ σ̃(a,b)}. (2) spectral mapping theorem of a pair of bounded linear operators: (4.5) σ(fr(a,b)) = σ(fl(a,b)) = f(σ̃(a,b)) = {f(λ) : λ ∈ σ̃(a,b)}. proof. (1) if ∞ /∈ σ̃(a,b), by virtue of the reduction (4.3) we can consider b = b1 invertible in l(x,y ) and a = a1. by theorem 3.1, we also have σ(rλ0,l(a,b)) = σ(rλ0,r(a,b)). therefore, a−λb = (a−λ0b)[ix − (λ−λ0)rλ0,r(a,b)] = [iy − (λ−λ0)rλ0,l(a,b)](a−λ0b). thus, λ0 6= λ ∈ %(a,b) if and only if 1(λ−λ0) ∈ %(rλ0,r(a,b)) (or 1 (λ−λ0) ∈ %(rλ0,l(a,b))) which gives the equality (4.3). ∞∈ σ̃(a,b) means that rλ0,l(a,b) and rλ0,r(a,b) are not invertible. (2) here we take the same constructs used by sahin and ragimov given through the gelfand representation theory developed in [19, theorem 5.8.4]. let lr (resp. ll) be the closed subalgebra of l(x) (resp. l(y )) containing the set {rλ,r(a,b) : λ ∈ %(a,b)} and ix (resp. {rλ,l(a,b) : λ ∈ %(a,b)} and iy ) and mr and ml are their spaces of maximal ideals respectively. then there exists a continuous c̃-valued function αj on mj such that for all λ ∈ %(a,b) and m ∈mj, rλ,j(a,b)(m) = 1 (λ−αj(m)) mj and αj(mj) = {αj(m) : m ∈mj} are holomorphic, j = r, l. particularly, as λ0 ∈ %(a,b), then according to (4.4), αj(mj) = σ̃(a,b) and the space of maximal ideals of algebras lj are homeomorphic, j = r, l. thus, for all m ∈mj, j = r, l, fj(a,b)(m) = 1 2πi ∫ γ f(λ)rλ,j(a,b)(m)dλ = 1 2πi ∫ γ f(λ) 1 (λ−αj(m)) dλ = f(αj(m)). a spectral analysis of linear operator pencils 117 consequently, σ(fj(a,b)) is the range of the function f on αr(mr) = αl(ml) = σ̃(a,b) and σ(fj(a,b)) = f(σ̃(a,b)), j = r, l. the following result is well-known, it concerns linear operator pencils having a discrete spectrum. it has been proved by ditkin [14], we get here this result directly as a consequence of the previous theorem. indeed if λ0 ∈ %(a,b) and b is compact it follows from (3.6) that σ̃(a,b) = φ−10 (σ(rλ0,j(a,b))). since rλ0,j(a,b) is compact, σ(rλ0,j(a,b))\{0} consists of eigenvalues (µk)k with finite-dimensional eigenspaces. the only possible point of accumulation of σ(rλ0,j(a,b)) is 0, if (µk)k is infinite, lim k→+∞ µk = 0. thus, σ̃(a,b) = {φ−10 (µk)}k ∪{∞}. theorem 4.3. let a,b ∈l(x,y ). suppose that %(a,b) 6= ∅. then, (1) if b is of finite rank, σ(a,b) is of finite cardinal. (2) if b is compact, then σ̃(a,b) is at most countable and consists only of eigenvalues of finite algebraic multiplicity which accumulate at most at infinity. proof. (1) let λ0 ∈ %(a,b). σ̃(a,b) is homeomorphic to σ(rλ,l(a,b)). or, rλl(a,b) is of finite rank, thus σ̃(a,b) = σ(a,b) is a finite set. (2) b is a limit of finite-rank operators bn and σ̃(a,b) ⊆ ⋃ n σ(a,bn) is at most countable as a countable union of finite sets. if σ̃(a,b) = (λn)n is infinite and lim n→+∞ λn = λ, then there exists n0 ∈ n such that λ = λn0 since σ̃(a,b) is closed, which necessarily implies that λ = λn0 = ∞. 5. isolated points of linear operator pencils let λ0 be an isolated point of σ̃(a,b), thus, λ0 6= ∞ and there exists δ0 > 0 such that {λ ∈ c : |λ−λ0| < δ0}∩ σ̃(a,b) = {λ0} and γ0 ∩ σ(a,b) = ∅ if γ0 = {λ ∈ c : |λ−λ0| = δ0} with clockwise orientation. the left and right riesz projectors corresponding to λ0 and the pair (a,b) are respectively defined in l(y ) and l(x) by: (5.1) pλ0,j(a,b) = − 1 2πi ∫ γ0 rλ,j(a,b)dλ ; j = l,r which corresponds to f(λ) = −1 in the functional calculus formula 4.2. in this section we investigate the isolated points of the spectrum of a pair of bounded linear operators a,b ∈l(x,y ). theorem 5.1. let a,b ∈l(x,y ) and λ0 an isolated point of σ̃(a,b). then the following hold: (1) pλ0,j(a,b), j = l(respectively, j = r) are projections operators in l(y ) (respectively, l(x)). (2) the spaces x and y can be written as a direct sum x = r(pλ0,r(a,b)) ⊕n(pλ0,r(a,b)) and y = r(pλ0,l(a,b)) ⊕n(pλ0,l(a,b)). (3) apλ0,r(a,b) = pλ0,l(a,b)a. (4) let x = y, then we have n(a − λ0b) ⊂ r(pλ0,r(a,b)) and if b commutes with rλ(a,b), then n(a−λ0b) ⊂ r(pλ0,l(a,b)). (5) if x = y is a hilbert space, λ0 ∈ r, a and b are self-adjoint, b is invertible 118 messirdi, gherbi and amouch and ab−1 = b−1a, then pλ0,j(a,b) where j = l,r, are the orthogonal projections onto n(a−λ0b). in particular, r(pλ0,j(a,b)) = n(a−λ0b) where j = l,r. proof. (1) let γ1 = {λ ∈ c : |λ−λ0| = δ1} such that δ0 < δ1, {λ ∈ c : |λ−λ0| < δ1} ∩ σ̃(a,b) = {λ0} and γ1 ⊂ %(a,b). in view of the resolvent identity (3.2) we obtain: p 2λ0,j(a,b) = − 1 4π2 ∫ γ0 ∫ γ1 [rλ,j(a,b) −rµ,j(a,b)] (λ−µ) dλdµ = 1 4π2 ∫ γ0 rλ,j(a,b)[ ∫ γ1 dµ (µ−λ) ]dλ − 1 4π2 ∫ γ1 rµ,j(a,b)[ ∫ γ0 dλ (µ−λ) ]dµ = pλ0,j(a,b), j = l,r because ∫ γ1 dµ (µ−λ) = 2πi and ∫ γ0 dλ (µ−λ) = 0. (2) follows directly from (1). (3) apλ0,r(a,b) = − 1 2πi ∫ γ0 arλ,r(a,b)dλ = − 1 2πi ∫ γ0 λbrλ(a,b)bdλ = − 1 2πi ∫ γ0 brλ(a,b)((λb −a) + a)dλ = − 1 2πi ∫ γ0 rλ,l(a,b)adλ = pλ0,l(a,b)a. (4) let x ∈ n(a − λ0b). then for all λ ∈ γ0, (a − λb)x = (λ0 − λ)bx or else x = (λ0 −λ)(a−λb)−1bx = (λ0 −λ)rλ,r(a,b)x. thus, pλ0,r(a,b)x = − 1 2πi ∫ γ0 dλ (λ0 −λ) x = x. so x ∈ r(pλ0,r(a,b)). now if b commutes with rλ(a,b), pλ0,r(a,b) = pλ0,l(a,b). (5) we use now the parametrization λ = λ0 + δ0e it, −π ≤ t ≤ π, of all point λ of γ0, then pλ0,l(a,b) = − δ0 2π ∫ π −π b[a− (λ0 + δ0eit)b]−1eitdt, p∗λ0,l(a,b) = − δ0 2π ∫ π −π [a− (λ0 + δ0e−it)b]−1be−itdt. by the change s = −t, we obtain since λ0 is real p∗λ0,l(a,b) = − δ0 2π ∫ π −π b[a− (λ0 + δ0eis)b]−1eisds = pλ0,l(a,b). a spectral analysis of linear operator pencils 119 similarly we obtain p∗λ0,r(a,b) = pλ0,r(a,b). it remains now to show that r(pλ0,j(a,b)) ⊂ n(a−λ0b), j = l,r. indeed, (a−λ0b)pλ0,j(a,b) = − 1 2πi ∫ γ0 (a−λ0b)(a−λb)−1bdλ = − 1 2πi ∫ γ0 (λ−λ0)b(a−λb)−1bdλ = − 1 2πi ∫ γ0 (λ−λ0)(ab−1 −λix)−1bdλ. now we can choose δ0 such that (λ − λ0)(ab−1 − λix)−1 extends to analytic function on {λ ∈ c : |λ−λ0| < δ0}. hence by cauchy’s theorem, the last integral is identically zero which gives (a−λ0b)pλ0,j(a,b) = 0. note also that if λ0 be an isolated point of σ̃(a,b), the laurent series for the resolvent (λb−a)−1 in a small neighborhood of the isolated singularity λ0 is given by (5.2) (λb −a)−1 = +∞∑ n=−∞ (λ−λ0)nsn, where sn = − 1 2πi ∫ γ0 1 (λ−λ0)n+1 rλ(a,b)dλ , n ∈ z. the coefficients sn are bounded operators and satisfies the following properties: (i) snbsm = (1−τn−τm)sn+m, where τn = 1 if n ≥ 0 and τn = 0 if n < 0. indeed, assume that λ0 = 0, since 0 is an isolated point of σ̃(a,b), then there exists δ > 0 such that {λ ∈ c : |λ| < δ}∩ σ̃(a,b) = {0}. denote γr = {λ ∈ c : |λ| = r} for 0 < r < δ. let r < r1, we have by using the resolvent identities that snbsm = 1 (2πi)2 ∫ γr ∫ γr1 λ−n−1µ−m−1rλ(a,b)brµ(a,b)dλdµ = 1 (2πi)2 ∫ γr ∫ γr1 λ−n−1µ−m−1 (µ−λ) [(λb −a)−1 − (µb −a)−1]dλdµ. by computing the double integral on the right in any order and the fact that 1 2πi ∫ γr λ−n−1 (λ−µ) dλ = −τnµ−n−1 1 2πi ∫ γr1 µ−m−1 (λ−µ) dµ = (τm − 1)λ−m−1. we obtain snbsm = (1 − τn − τm) 2πi ∫ γr λ−n−m−2(λb −a)−1dλ(5.3) = (1 − τn − τm)sn+m+1. (ii) multiplying (5.3) on the left and the right by (λb −a), we obtain (λb −a) +∞∑ n=−∞ (λ−λ0)nsn = iy and +∞∑ n=−∞ (λ−λ0)nsn(λb −a) = ix. 120 messirdi, gherbi and amouch thus, iy = +∞∑ n=−∞ (λ−λ0)n[bsn−1 + (λ0b −a)sn], ix = +∞∑ n=−∞ (λ−λ0)n[sn−1b + sn(λ0b −a)]. the uniqueness of the laurent series expansion yields ix = s−1b + s0(λ0b −a), iy = bs−1 + (λ0b−a)s0 and sn−1b + sn(λ0b−a) = 0, bsn−1 + (λ0b−a)sn for all n 6= 0. then,   s−1b = ix −s0(λ0b −a) bs−1 = iy − (λ0b −a)s0 sn−1b = sn(a−λ0b) , n 6= 0 bsn−1 = (a−λ0b)sn , n 6= 0. from the standard terminology of the complex theory, we call the operator s−1 in the laurent series (5.2) the residue operator at λ0. by taking n = m = −1 in (5.3), bs−1 and s−1b are projections which coincide respectively with the left and right riesz projectors pλ0,l(a,b) and pλ0,r(a,b) at λ0. definition 5.2. let x = y and a,b ∈ l(x) are with non empty resolvent set %(a,b). we say that a and b commute in the sense of resolvent if for all λ ∈ %(a,b), rλ,l(a,b) = rλ,r(a,b). remark 5.3. if a and b commute in the sense of resolvent then for all λ ∈ %(a,b), we deduce that arλ(a,b) = rλ(a,b)a brλ(a,b)b = b 2rλ(a,b) = rλ(a,b)b 2 abrλ(a,b) = rλ(a,b)ab = rλ(a,b)ba = barλ(a,b). then, pλ0,l(a,b) = pλ0,r(a,b) = pλ0 (a,b), apλ0 (a,b) = pλ0 (a,b)a, bpλ0 (a,b) = pλ0 (a,b)b and snb = bsn for all n ∈ z, if a and b commute in the sense of resolvent. by setting rλ,b(a,b) = −rλ,l(a,b) = −rλ,r(a,b), d = s−2b = bs−2 and e = −bs0 = −s0b, the relation (snbsm) gives bs−k = d k−1 for k ≥ 2 bsk = −ek+1 for k ≥ 0. the laurent series (5.2) around λ0 is equivalent to (5.4) rλ,b(a,b) = +∞∑ n=1 dn (λ−λ0)n+1 + pλ0 (a,b) (λ−λ0) − +∞∑ n=1 (λ−λ0)nen+1. thus,   rλ,b(a,b)pλ0 (a,b) = +∞∑ n=1 dn (λ−λ0)n+1 + pλ0 (a,b) (λ−λ0) rλ,b(a,b)(ix −pλ0 (a,b)) = − +∞∑ n=1 (λ−λ0)nen+1, a spectral analysis of linear operator pencils 121 where (a−λ0b)pλ0 (a,b) = (apλ0 (a,b) −λ0b)pλ0 (a,b) = d and (a−λ0b)e = ix −pλ0 (a,b). hence (5.5)   e = (a−λ0b)−1|r(ix−pλ0 (a,b))(ix −pλ0 (a,b)) epλ0 (a,b) = pλ0 (a,b)e = 0 de = ed = 0 d = dpλ0 (a,b) = pλ0 (a,b)d. now suppose that λ0 is a pole of the resolvent (a−λb) of order m, then s−m 6= 0 and sn = 0 for all n > m. since bs−1 = s−1b = pλ0 (a,b), it follows that (a−λb)m−1pλ0 (a,b) = s−m 6= 0 and (a−λb)mpλ0 (a,b) = s−m−1 = 0, then the operator d = (a−λ0b)pλ0 (a,b) is nilpotent of order m. now, we give the following fundamental results: theorem 5.4. let a,b ∈ l(x) such that a and b commute in the sense of resolvent. if λ0 is an isolated point in the spectrum σ̃(a,b), then λ0 is a pole of the resolvent of order m ∈ n∗. laurent series around λ0 is given by (5.2) with the residue operator bs−1 = s−1b coincides with the riesz projection pλ0 (a,b) associated to λ0 and the relations (5.5) are satisfied. on the other hand, the operator d = (a−λ0b)pλ0 (a,b) is nilpotent of order m. the discrete spectrum of the pair (a,b) denoted σd(a,b) is the set of isolated points λ ∈ c of the spectrum σ(a,b) such that the corresponding riesz projectors pλ,j(a,b) are finite dimensional. thus, σd(a,b) ⊂ σp(a,b). define also the essential spectra of the pair (a,b) by: (5.6) σess(a,b) = σ̃(a,b)\σd(a,b). the largest open set of c̃ on which the resolvent rλ(a,b) is finitely meromorphic is precisely %ess(a,b) = σd(a,b) ∪ %(a,b) = c̃\σess(a,b). let x = y, λ ∈ %ess(a,b) and let pλ,j(a,b) be the corresponding finite rank riesz projector, j = l,r. since r(pλ,j(a,b)) and n(pλ,j(a,b)) are pλ,j(a,b)-invariant, j = l,r, we may define the operators: (5.7) qλ,j(a,b) = (a−λb)(i −pλ,j(a,b)) + pλ,j(a,b) ; j = l,r. with respect to the decomposition x = r(pλ,j(a,b)) ⊕ n(pλ,j(a,b)), j = l,r, we can write: (5.8) qλ,j(a,b) = (a−λb)|n(pλ,j(a,b)) ⊕ ix ; j = l,r. since σ((a − λb)|n(pλ,j(a,b))) = σ̃(a,b)\{0}, qλ,j(a,b) has bounded inverse denoted by rλ,j(a,b), j = l,r. rλ,l(a,b) and rλ,r(a,b) are called respectively the left browder and the right browder resolvent operator of the pair (a,b), that is, rλ,j(a,b) = ((a−λb)|n(pλ,j(a,b))) −1(i −pλ,j(a,b)) + pλ,j(a,b)(5.9) j = l,r, λ ∈ %ess(a,b). 122 messirdi, gherbi and amouch this clearly extends the resolvent rλ,j(a,b) from %(a,b) to %ess(a,b) and admits the following properties for j = l,r : pλ,j(a,b)rλ,j(a,b) = rλ,j(a,b)pλ,j(a,b),(5.10) pλ,j(a,b)qλ,j(a,b) = qλ,j(a,b)pλ,j(a,b) = pλ,j(a,b). proposition 5.5. let a,b ∈l(x) and λ,µ ∈ %ess(a,b), then for j = l,r, rλ,j(a,b) −rµ,j(a,b) = (λ−µ)rλ,j(a,b)brµ,j(a,b) +rλ,j(a,b)mj(λ,µ)rµ,j(a,b) where mj(λ,µ) is a finite rank operator defined on x by: mj(λ,µ) = [(a− (λ + 1)b)pλ,j(a,b) − (a− (µ + 1)b)pµ,j(a,b)] proof. by computing rλ,j(a,b) −rµ,j(a,b), we directly obtain: m(λ,µ) = [(a−λb − ix)pλ,j(a,b) − (a−µb − ix)pµ,j(a,b)] . the browder resolvent, through its properties mentioned above, can be used to study the question of existence of solutions of boundary value problems with singularities defined by a given boundary condition:{ ax = λbx + f γx = ϕ. where f ∈ x, γ is a boundary operator and λ is a spectral parameter such that λ−1 ∈ %ess(a,b). for more details one can consult [23]. remark 5.6. as the matter of fact, this decomposition is not “the simplest”; there are many different definitions of σess(a,b) for a,b ∈l(x,y ) : 1) λ ∈ σess,1(a,b) if (a−λb) is not semi-fredholm (t ∈l(x,y ) is semi-fredolm if r(t) is closed in y and n(t) or the quotient space y/r(t) are finite-dimensional); 2) λ ∈ σess,2(a,b) if r(a − λb) is not closed in y or n(a − λb) is infinitedimensional in x; 3) λ ∈ σess,3(a,b) if (a−λb) is not fredholm (t ∈ l(x,y ) is fredolm if r(t) is closed in y and n(t) and y/r(t) are finite-dimensional); 4) λ ∈ σess,4(a,b) if (a − λb) is not fredholm with index zero (recall that index(t) = dim n(t) − dim y/r(t) = dim n(t) − co dim r(t)); 5) σess,5(a,b) is the union of σess,1(a,b) with all components of c\σess,1(a,b) that do not intersect with the resolvent set %(a,b). note that, (5.11) σess,1(a,b) ⊂ σess,2(a,b) ⊂ σess,3(a,b) ⊂ σess,4(a,b) ⊂ σess,5(a,b) ⊂ σ̃(a,b) and that the essential spectrum σess,i(a,b) is invariant under compact perturbations for i = 1, 2, 3, 4, but not for i = 5. the case i = 4 gives the part of the spectrum that is independent of compact perturbations, that is, σess,4(a,b) = ⋂ lim k∈∈k(x,y ) σ(a + k,b) where k(x,y ) denotes the set of compact operators from x to y. as a generalization of the usual notion of wolf essential spectrum, the essential spectrum of linear operator pencils was introduced by faierman, mennicken and moller in [15]. a spectral analysis of linear operator pencils 123 note that if b = ix, we recover the usual definition of the essential spectra of a bounded linear operator, that is, σess,i(a,ix) = σess,i(a), i = 1, ..., 4. we denote the dimension of the null space or nullity of an operator t ∈l(x,y ) by n(t) and the codimension of the range or defect of t by d(t). the ascent of t , α(t), is the smallest integer p such that n(tp) = n(tp+1), and the descent of t , β(t), is the smallest integer q such that r(tq) = r(tq+1). (it may happen that α(t) = ∞ or β(t) = ∞). one of the central questions in the study of the essential spectra of bounded linear operators consists in showing when different notions of essential spectrum coincide and studying the invariance of σess,i(a,b) by some class of perturbations. for a detailed study, see [1]. the following result is given in [20]. proposition 5.7. let a,b ∈l(x,y ). then the following hold: (1) σess,3(a,b) is closed subset of c. (2) index(a−λb) is constant on any component of c\σess,3(a,b). (3) n(a − λb) and d(a − λb) are constant on any component of c\σess,3(a,b) except on a discreet set of points at which they have larger values. (4) if c\σess,3(a,b) is connected and %(a,b) is not empty, then σess,3(a,b) = σess,4(a,b). the following result is a generalization of [24, theorem 1]. theorem 5.8. let a,b ∈ l(x) such that a and b commute in the sense of resolvent and λ ∈ σ(a,b). the following statements are equivalent: (1) λ ∈ σess,2(a,b); (2) λ is a pole of the resolvent rλ(a,b) of finite rank; (3) α(a−λb) = β(a−λb) < ∞ and n(a−λb) < ∞. proof. the equivalence of (1) and (2) can be obtained in the same manner as in the proof of [11, lemma 17]. (2) =⇒ (3). if λ is a pole of (a−λb)−1 of order m, then n((a−λb)m) = n((a−λb)m+1). indeed, as n((a−λb)m) ⊂ n((a−λb)m+1) it suffices to prove the inverse inclusion. we proceed by contradiction. let x ∈ n((a−λb)m+1) and x /∈ n((a−λb)m), that is, the vector y = (a−λb)mx 6= 0, it follows that (a−λb)y = 0. this implies, by (4) of theorem 5.1, that pλ,l(a,b)y = pλ,r(a,b)y = pλ(a,b)y = y. consequently, 0 = (a−λb)mpλ(a,b)x = pλ(a,b)(a−λb)mx = pλ(a,b)y = y, which is a contradiction. hence, n((a − λb)m) = n((a − λb)m+1) and α(a − λb) ≤ m. now, notice that (a − λb)m−1pλ(a,b) 6= 0 which guarantees the existence of some vector x ∈ r(pλ(a,b)) such that (a − λb)m−1x = (a − λb)m−1pλ(a,b)x 6= 0. from (a − λb)mx = (a − λb)mpλ(a,b)x = 0, it follows that (5.12) n((a−λb)m) 6= n((a−λb)m−1). this shows α(a − λb) ≥ m. thus, α(a − λb) = m. now if we consider the decomposition σ̃(a,b) = {λ}∪ (σ̃(a,b)\{λ}), then (a − λb)n is invertible on n(pλ(a,b)) for all n ∈ n and (a−λb)mpλ(a,b) = 0 implies that (a−λb)m = 0 on r(pλ(a,b)). consequently, r((a−λb)m) = n(pλ(a,b)) = r((a−λb)m+1). thus, (a − λb) has finite descent β(a − λb) = m. (3) =⇒ (1) assume that α(a − λb) = β(a − λb) = m < ∞. then n(pλ(a,b)) = r((a − λb)m) = r((a−λb)n) and r(pλ(a,b)) = n((a−λb)m) = n((a−λb)n) for all n ≥ m. 124 messirdi, gherbi and amouch it follows that dn = (a−λb)npλ(a,b) = 0 for all n ≥ m, and so λ is a pole of the resolvent of order k with k ≤ m. but from (5.12), necessarily k = m. remark 5.9. if a,b ∈ l(x) commute in the sense of resolvent and λ is a pole of order m of the resolvent, then λ ∈ σp(a,b) and x = n((a−λb)n) ⊕r((a−λb)n) for all n ≥ m. now, we introduce an important class of bounded operators which involves the concept of semi-regularity see e.g. muller [26] and rakocevic [27] , mbekhta and ouahab [25]. definition 5.10. let a ∈l(x). the algebraic core c(a) of a is defined to be the greatest subspace m of x for which a(m) = m. the reduced minimum modulus of a is defined by γ(a) = { inf x/∈n(a) ‖ax‖ dist(x,n(a)) if a 6= 0 ∞ if a = 0. a is said to be semi-regular if r(a) is closed and n(an) ⊆ r(a) for all n ∈ n. a is said to admit a generalized kato decomposition or a is of generalized kato type, if there exists a pair of closed subspaces (m,n) of x such that: (i) x = m ⊕n. (ii) a(m) ⊂ m and a|m is semi-regular. (iii) a(n) ⊂ n and a|n is quasi-nilpotent. note that if a ∈ l(x), r(a) is closed in x if and only if γ(a) > 0 and γ(a) = γ(a∗). if a is semi-regular, then γ(an) ≥ (γ(a))n and an is semi-regular for all n ∈ n, c(a) is closed and c(a) = ⋂ n∈n r(an). a is semi-regular if and only if a∗ is semi-regular. on the other hand if a pair of closed subspaces (m,n) of x reduces a (x = m ⊕n, a(m) ⊂ m and a(n) ⊂ n), then a is semi-regular if and only if a|m and a|n are semi-regular. if a|n is nilpotent, a is said to be of kato type [22]. semi-regular operators are of kato type with m = x and n = {0} . if 0 is an isolated point in σ(a), or equivalently 0 is a pole of the resolvent of a, then a is of generalized kato type [8]. using rather direct technique different from [3], we extend the results to semi-regular operators and those who admit a generalized kato decomposition. indeed, an immediate and direct generalization of [3, theorem 1.31] we provided the following result: theorem 5.11. let a,b ∈ l(x), a be semi-regular and bc(a) = c(a). then (a−λb) is semi-regular for all |λ| < γ(a)‖b‖ . for a,b ∈l(x), let us define the generalized kato spectrum for the pair (a,b) as follows: (5.13) σgk(a,b) = {λ ∈ c : (a−λb) is not of generalized kato type} σgk(a,b) is not necessarily non-empty. for example, each pair of quasi-nilpotent (resp. nilpotent) operator a and b = ix has empty generalized kato spectrum. the next theorem is a generalization of theorem 2.2 of [21]. theorem 5.12. let a,b ∈ l(x), a be of generalized kato type and bc(a) = c(a). then there exists an open disc d(0,ε) for which (a − λb) is semi-regular for all λ ∈ d(0,ε)\{0} . a spectral analysis of linear operator pencils 125 proof. b = ix corresponds to the theorem 2.2 of [21]. if b 6= ix, note that x = m ⊕ n, a(m) ⊂ m, a|m is semi-regular, a(n) ⊂ n and a|n is quasinilpotent. if m = {0} , a is quasi-nilpotent and thus (a−λb) = (a−λix) [ ix −λ(a−λix)−1(b − ix) ] is invertible if |λ| < 1‖(a−λix)−1‖‖b−ix‖ = η. this shows that (a − λb) is semiregular in d(0,η)\{0} . if m 6= {0} , a = ( a|m 0 0 a|n ) and (a−λb) = ( (a−λb)|m 0 0 (a−λb)|n ) . since a|n is quasi-nilpotent, (a − λix) is invertible in l(x) for all λ non-zero complex number. then, (a−λb)|n is invertible and semi-regular for |λ| < η. as a|m is semi-regular operator, then γ(a|m ) > 0 and by theorem 29, (a − λb)|m is semi-regular for all |λ| < γ(a|m )‖b‖ . consequently, (a−λb) is semi-regualar for all |λ| < ε, where ε = min(η, γ(a|m )‖b‖ ). we deduce in particular from this theorem that the generalized kato spectrum of a pair of bounded operators is a closed subset. the following result gives the relation between the closed range spectrum σess,2(a,b) and the generalized kato spectrum σgk(a,b) of a pair (a,b) of bounded operators which extend some results of [6, 7, 21]. theorem 5.13. let a,b ∈l(x) such that bc(a) = c(a). (1) if λ ∈ σess,2(a,b) is non-isolated point then λ ∈ σgk(a,b). (2) the symmetric difference σgk(a,b)∆σess,2(a,b) is at most countable. proof. (1) let λ ∈ σess,2(a,b) be a non-isolated point and assume that (a−λb) is of generalized kato type. then by theorem 29 there exists an open disc d(λ,�) such that (a−µb) is semi-regular in d(λ,�)\{λ} , so that r(a−µb) is closed in x for all µ ∈ d(λ,�)\{λ} . this contradicts our assumption that λ is a non-isolated point. (2) we have σgk(a,b)∆σess,2(a,b) is equal to (σgk(a,b) ∩ (c\σess,2(a,b))) ∪ (σess,2(a,b) ∩ (c\σgk(a,b))) . hence, from (1), the set (σess,2(a,b)\σgk(a,b)) is at most countable, we have c\σess,2(a,b) = ∞⋃ m=1 { λ ∈ c : γ(a−λb) ≥ 1 m } and σgk(a,b) ∩ (c\σess,2(a,b)) = ∞⋃ m=1 σgk(a,b) ∩ { λ ∈ c : γ(a−λb) ≥ 1 m } . the set am = σgk(a,b)∩ { λ ∈ c : γ(a−λb) ≥ 1 m } is necessarily at most countable for all m ≥ 1. indeed, let ζ be a non-isolated point of am, then there exists a sequence (λk)k∈n in am such that lim k→+∞ λk = ζ. thus, γ(a − ζb) ≥ 1m, since 126 messirdi, gherbi and amouch{ λ ∈ c : γ(a−λb) ≥ 1 m } is closed in c (see e.g. [6]), and ζ /∈ σgk(a,b) which contradicts the closedness of σgk(a,b). 6. spectrum of the quotient of two bounded operators let here x = y be an infinite dimensional complex hilbert space equipped with the inner product 〈.; .〉 and the associated norm ‖.‖ . the quotient a/b of bounded operators a and b on x, b 6= 0, is defined by the mapping bx −→ ax, x ∈ x when n(b) ⊂ n(a) and a 6= b. if a = b, take a/b = ix. we note that the quotient of two bounded operators is not necessarily bounded whose domain is r(b) and its rang is r(a). the question of boundedness, compactness and invertibility of quotient operators is very important and for the reader’s convenience, let us summarize all what has been obtained in [17]. theorem 6.1. [17] let a,b ∈l(x) such that n(b) ⊂ n(a). then the following hold: (1) a/b is bounded if and only if r(a∗) ⊂ r(b∗). (2) if r(b) is closed in x then a/b is bounded. (3) if r(b) is closed in x and b is invertible, then a/b = ab−1. (4) if a/b is compact then a is compact. conversely, if r(b) is closed in x and a is compact then a/b is also compact. (5) if n(a) = n(b), then a/b is invertible and (a/b)−1 = b/a. (6) if n(a) = n(b) and r(a) is closed in x, then a/b has a bounded inverse b/a. (7) a/b has an everywhere defined and bounded inverse if and only if the operator a is invertible in l(x) and (a/b)−1 = b/a = ba−1. the aim of this section is to give some fundamental characterizations of the spectrum of quotient operators using the basic spectral properties of linear operator pencils. note here that this is the first time where the notion of the spectrum of a quotient of two operators is studied by using the theory of linear operator pencils. remark 6.2. if n(b) ⊂ n(a) then n(b) ⊂ n(a−λb) and [(a/b) −λix] is well defined by (a−λb)/b for all λ ∈ c, then by property (7) of theorem 6.1, we can write % (a/b) = {λ ∈ c : (a−λb)/b is invertible in l(x)}(6.1) = {λ ∈ c : (a−λb) is invertible in l(x)} = % (a,b) and (6.2) σ (a/b) = σ̃ (a,b) thus, if λ ∈ % (a/b) , then (6.3) [(a/b) −λix] −1 = b(a−λb)−1 = rλ,l(a,b). using the results of the previous sections, we obtain the following properties on the spectra of quotient operators through those previously established on a pair of bounded linear operators. theorem 6.3. let a,b ∈l(x) such that n(b) ⊂ n(a). then (1) if 0 ∈ % (a) then 0 ∈ % (a,b) . (2) if n(a) = n(b) and r(a) is closed in x, then 0 ∈ % (a,b) . (3) if r(b) is closed in x and a is compact, then a spectral analysis of linear operator pencils 127 a) σ̃ (a/b) = {0}∪{λj : j ∈ j} , where either j = ∅, or j = n, or j = {1, ...,n} for some n ∈ n. b) σ̃ (a/b)\{0} = σp (a,b) . each λj is an eigenvalue having a finite multiplicity. c) if j = n, then (λj) −→ 0 as j → ∞. this means that for all ε > 0, the set σ (a/b)\d(0,ε) is finite where d(0,ε) = {λ ∈ c : |λ| < ε} . (4) if b is compact and % (a/b) 6= ∅, then σ̃ (a/b) is at most countable and consists only of eigenvalues of finite algebraic multiplicity which accumulate at most at infinity. remark 6.4. this is a first attempt to establish the link between the spectral theory of quotient operators and linear operator pencils. our results give rise to other interesting perspectives on the study of quotients operators. references [1] f. abdmouleh, a. ammar and a. jeribi, stability of the s-essential spectra on a banach space, math. slovaca, 63, 2 (2013), 1-22. [2] f. aguirre and c. conca, eigenfrequencies of a tube bundle immersed in a fluid, appl. math. optim., 18, (1988), 1-38. [3] p. aiena, fredholm and local spectral theory, with applications to multipliers, kluwer academic pub., 2004. [4] r. i. andrushkiw, on the spectral theory of operator pencils in a hilbert space, nonlinear math. phys., 2, 3/4 (1995), 356-366. [5] w.g. bade, an operational calculus for operators with spectrum in a strip, pacific j. math. 3, (1953), 257-290. [6] m. benharrat and b. messirdi, on the generalized kato spectrum, serdica math. j., 37, (2011), 283-294. [7] m. benharrat and b. messirdi, relationship between the kato essential spectrum and a variant of essential spectrum, gen. math. rev., 20, 4 (2012), 71-88. [8] m. benharrat and b. messirdi, essential spectrum a brief survey of concepts and applications, azerb. j. math., 2, 1 (2012), 35-61. [9] m. sh. birman, a. laptev, discrete spectrum of the perturbed dirac operator, mathematical results in quantum mechanics (blossin, 1993), 55-59. oper. theory adv. appl., 70, birkhauser, basel, 1994. [10] m. sh. birman and m. z. solomyak, asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, vestn. leningr. univ., 1, (1977), 13-21. [11] f. e. browder, on the spectral theory of elliptic differential operators, i, math. ann. 142,(1961), 22-130. [12] j. bronski, m. johnson and t. kapitula, an instability index theory for quadratic pencils and applications, comm. math. physics 327, 2 (2014), 521-550. [13] d. chu and g. h. golub, on a generalized eigenvalue problem for nonsquare pencils, siam j. matrix anal. appl., 28, 3 (2006), 770-787. [14] v. v. ditkin, certain spectral properties of a pencil of linear bounded operators, mathematical notes of the academy of sciences of the ussr, 31, 1 (1982), 39-41. [15] m. faierman, r. mennicken and m. moller, a boundary eigenvalue problem for a system of partial differential operators occuring in magnetohydrodynamics, math. nachr., 173, (1995), 141–167. [16] gestztesy g., gurarie d., h. holden, m. klaus, l. sadun, b. simon and p. vogl, trapping and cascading of eigenvalues in the large coupling limit, commun. math. phys. 118, (1988), 597-634. [17] a. gherbi, b. messirdi and m. benharrat, on the quotient of two bounded operators, submitted, june 2014. [18] r.r.hartmann, n.j.robinson and m.e.portnoi smooth electron waveguides in graphene, phys. rev. b, 81, 24 (2010), 245-431. [19] e. hille and r. c. phillips, functional analysis and semi-groups, russian translation, il, moscow (1962). 128 messirdi, gherbi and amouch [20] a. jeribi, n. moalla and s. yengui, s-essential spectra and application to an example of transport operators, math. methods appl. sci., 37 (2012), 2341-2353. [21] q. jiang and h. zhong, generalized kato decomposition, single-valued extension property and approximate point spectrum, j. math. anal. appl. 356, (2009) 322-327. [22] t. kato, perturbation theory for linear operators, springer-verlag, new york, (1995). [23] n. khaldi, m. benharrat and b. messirdi, on the spectral boundary value problems and boundary approximate controllability of linear systems, rend. circ. mat. palermo, 63, (2014),141– 153. [24] d. lay, characterizations of the essential spectrum of f. e. browder, bull. amer. math. soc., 74, (1968), 246–248. [25] m. mbekhta and a. ouahab, opérateur s-régulier dans un espace de banach et théorie spectrale, pub. irma. lille. vol. 22, (1990), no xii. [26] v. muller, on the regular spectrum, j. operator theory, 31, (1994), 363-380. [27] v. rakočević, generalized spectrum and commuting compact perturbations, proc. ed-inb. math. soc. 36, (1993), 197-209. [28] f. g. ren, x. m. yang, some properties on the spectrum of linear operator pencils, basic sci. j. of textile universities, 25, 2 (2012), 127-131. [29] h. h. rosenbrok, state space and multivariable theory, thomas nelson, london (1970). [30] m. sahin, m. b. ragimov, spectral theory of ordered pairs of the linear operators acting in different banach spaces and applications, internat. math. forum, 2, 5 (2007), 223 236. [31] d.a.stone, c.a.downing, and m.e.portnoi searching for confined modes in graphene channels: the variable phase method, phys. rev. b, 86, 7 (2012), 075464. [32] c. tretter, linear oerator pencils (a−λb) with discrete spectrum, integral equations oper. theory, 317, (2000) 127-141. [33] c. tretter, spectral theory of block operator matrices and applications, imperial college press, london, (2008). [34] k. yosida, functional analysis, springer-verlag berlin, (1965). 1department of mathematics, university of oran 1, b.p. 1524 el m’naouar, oran, algeria 2laboratory of fundamental and applicable mathematics (lmfao), department of mathematics, university of oran 1, algeria 3department of mathematics, university chouaib doukkali, faculty of sciences, b.p 20 eljadida, marocco ∗corresponding author international journal of analysis and applications volume 18, number 3 (2020), 461-481 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-461 received january 24th, 2020; accepted february 18th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 90b50. key words and phrases. meeting effectiveness; leadership; substantive conflict; internal communication; agenda. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 461 determinants to gain more effective meetings in the context of vietnamese organizations ly dan thanh1,2, le van chon1,2 , bui quang thong1,2, nhu-ty nguyen1,2,* 1school of business, international university (iu), vietnam 2vietnam national university, hcm city, vietnam *corresponding author: nhutynguyen@hcmiu.edu.vn; nhutynguyen@gmail.com abstract. meetings are the primary communicative practice in every organization in order to fulfill the vital consensus, make changes and exchange ideas. much time and effort are devoted to meetings aiming at information sharing, decision making, and problem solving. therefore, finding out how voice and leadership power affect meeting effectiveness becomes essential, especially in vietnamese organizations. first, the paper reviews factors affecting meeting effectiveness including leadership, agenda, substantive conflicts and internal communication. next, a structured questionnaire was completed by a sample of 157 participants who are working at 31 vietnamese organizations from a variety of sectors such as tax, banking, health service, airlines, education and business. finally, the results reveal two antecedents affecting meeting effectiveness: leadership and substantive conflict. leaders play the vital role in formulating an organization vision, making effective plans for vision implementation in reality as well as creating a healthy environment and organizational culture to grow ethical behaviors inside the organization. their subordinates surely become more committed to the organization when they are working with inspirational leaders who willingly instruct them in uncertainty and encourage their abilities and talents. in addition, it is obvious that during the process of interaction, conflicts may exist and therefore how to resolve conflicts needs to be concerned. at any circumstances, most authors from previous studies believe that when conflicts occur in the meeting, if they are resolved in a constructive way, they will surely bring more benefits for the organizations. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-461 int. j. anal. appl. 18 (3) (2020) 462 1introduction meetings are the common activities in every organization for several purposes such as fulfilling vital goals, making changes and exchanging ideas [57][65]. obviously, all meetings are unlike. they vary in several ways, depending on the way people involved, group’s size, tools used, management styles, and overall design of the meeting[62][70]. moreover, much time and effort is devoted to work meetings with the aims of information sharing, decision making, and problem solving [2]. moreover, meetings offer an exciting gateway to dynamic social processes in organizations [29]. during their meeting interactions, employees exchange information, build common ground, create new ideas, manage relationships, and make or break team climate [54]. everyday experience makes it evident that, not all meetings are effective [23]. to most working adults, meetings are often viewed as time-wasters but better or worse, it becomes a common workplace activity, occurring everyday around the world. they play the central role of the work environment that can affect many different aspects of one's job, such as job satisfaction with several purposes which may include decision making, information sharing, product design and development. according to the previous reviews and surveys of managers and staff, nicholas [36] also states that meetings are an important part of one’s working life [36]. above all, meetings need to be held to accomplish several tasks such as reaching important consensus, making changes, coming up with new ideas and the forth. according to previous researches, they reveal that as many as half of these meetings are considered poor in quality [69][66][62][61][68]. meeting effectiveness, more or less, becomes crucial in vietnamese organizations under more intense competition. due to the difference from people in low-context culture in which people tend to be direct, verbal, explicit, and individualistic (us, most of western europe, etc.), vietnamese people belong to high-context culture in which people are considered to be nonverbal, indirect, implicit and collectivistic (vietnam, greece, etc.) [25]. in most meetings, subordinates rarely or never raise their ideas, even though they disagree with ideas from their superiors. they are considered to be obedient and passive. in other meetings, some subordinates suggest solutions and receive an approval from their boss but it still doesn’t work because the boss did promise but don’t keep it. vietnamese superiors seem to be so conservative and high-power distance [57][59]. they direct the meeting without agenda and lack of internal and problem-focused communication. that’s the reason why most meetings in int. j. anal. appl. 18 (3) (2020) 463 vietnamese organizations have poor quality, leading to diminish staff’s job enthusiasm and in turn weakening the organizational commitment. effective and efficient meetings will motivate subordinates make more contributions and increase commitment to their workplace. thus, what makes meetings more effective are conducted [61] [58]. the paper aims to build a model of determinants to gain more effective meeting in vietnamese organizations and through which meeting organizers can direct their meeting’s quality more effectively and efficiently, later on facilitate and inspire their subordinates to have more engagement in organizational commitment. the authors design a survey based on the two research questions: what makes subordinates look forward to their work meetings? and what makes subordinate threatened by their work meetings? 2literature review 2.1 meeting effectiveness in general, meetings are considered as the strategic role in the social practice that brings consequential strategic outcomes to the organization [59][55][52][67]. furthermore, they can be recognized as the focal points for organizational members’ essential activities [17]. there are several types of meeting such as board meetings, committee meetings, departmental meetings and so forth [6]. rogelberg [40] points out that if the meetings are effective in facilitating organizations and employees to reach their goals, their benefits as an organizational tool is obvious[40]. thus, meeting effectiveness needs to be improved in order to get higher performance of organizational members. it was closely related to goal attainment and decision satisfaction. the research suggests that effective meetings need to be open in communicating, task-focused, impartial and strict to the use of agenda [3][37]. according to nixon [37], employees’ goals and an organization’s goals will lead to meeting effectiveness which is a timed process as well. it should bring benefits to the entire organization. the effective meeting shouldn’t be lack of the clear purpose and specific agenda, date, duration and materials [5]. besides that, bagire [5] emphasized that the central role of the chairperson who conducts the meeting decides the meeting effectiveness. put it another way, some authors state several factors affecting meeting productivity such as irrelevant topics or issues, excessive length of time and poor or inadequate preparation int. j. anal. appl. 18 (3) (2020) 464 [36]. volkema [47] emphasized that not only the use of agenda and meeting minutes but also the role of group leaders/facilitators controlling the meeting affect the meeting effectiveness [47]. researchers of ethnography have more explanations in the differentiation of behaviors and attitudes of organizational members, known as organizational culture and they also state that cultural behaviors to some extent enforce the rules, laws and norms. for instances, the meanings of communication are implied by the culture and the context of an organization [42]. sharing activities among organizational members are shaped by organizational values. the way members share their insights will be supported by behaviors from organizational culture [1]. undoubtedly, in order to make meeting effective, several factors need to be discussed. actually, an organization is mostly influenced by the top leader who has the strongest power in final decision-making. this most powerful person will get involved either directly or indirectly in the meeting decision. a middle manager who hosts the meeting is still there but unable to conclude or give any solutions. as a result, the leader’s style and role become a decisive factor in setting organizational culture. it is known as leadership. 2.2 leadership from the meeting literature perspective, the role of the meeting leader is vital [37]. in a highly diverse workforce, leadership becomes too complicated and needs to be more skillful. it is considered as the key factor in determining whether the organization succeeds [30]. the style of leading should be “simpatico” or “diversity-friendly”. a diversity leader from ceo to the first line supervisor is considered as corporate manager who leads subordinates in a fair, effective and respectful way. nine characteristics that a diversity leader must possess are sensitive, impartial, mediators, patient, amiable, teachers, involved, communicators, and optimistic [15]. also, in term of leadership, simola [43] recommends transformational leadership in which leaders aim to transform, motivate and enhance their subordinates’ actions and ethical aspirations. it contains four dimensions which are idealized influence, inspirational motivation, intellectual stimulation and individualized consideration [19] [43]. furthermore, this type of leadership brings more benefits for leading present workgroups because today’s followers turn more challenged and empowered. followers are in the need of an inspirational leader to guide them in uncertainty and intellectually stimulate and encourage their abilities and talents [7]. put it another way, kirkpatrick [20] emphasizes leader’s traits which include int. j. anal. appl. 18 (3) (2020) 465 achievement, motivation, ambition, energy, tenacity and initiative. leaders should be provided essential skills such as formulating an organization vision, making effective plans for vision implementation in reality [20]. from most previous studies about leadership, the type of charisma becomes emerging. partly like ethical one, emotionality is the main dimension in charismatic leadership, the nature of which is not very rational. problem-solving is not mostly based on authority but rather on personal characteristics [26]. leadership cannot be fulfilled without groups who have the common goals. surely, it is hard for leaders or managers effectively achieving organization’s goals and that the leader can only achieve goals through followers’ efforts and actions [4]. fry [12] highly appreciates type of servant leadership which consists of four elements such as being a servant first, making sure that other people’s needs are served; serving through listening; serving through people building and serving through leadership creation [12]. similarly, another type of leadership is transformational leadership by which leaders motivates followers by appealing to their higher-order needs and induce employees to transcend self-interest for the sake of the group or the organization [31]. for the emphasis, wallis [48] strengthens that followers are mainly influenced by leadership’s inspiration in which values and beliefs are shared by both leaders and followers. zhu [51] believes in ethical leaders who behave morally and always tend to create a healthy environment and organizational culture to grow ethical behaviors inside the organization [51]. above all, researchers in this field point out several definitions of leadership, but to what extent, leadership is defined or limited by its cultural context [48]. in reality, the meeting will be more effective if it is led by the transitional or charismatic leadership. therefore, the authors propose: proposition 1: leadership significantly affects meeting effectiveness. besides leadership, internal communication assists to transform information more specifically and effectively. 2.3 internal communication internal communication is an essential process by which people exchange information, create relationship and build organizational culture and values as well. it is somehow called employee communication [10][30]. moreover, martic [27]emphasizes “through internal communication, executives "pilots" the organization, as well as assure and guide employees to int. j. anal. appl. 18 (3) (2020) 466 follow the mission and goals, encourage loyalty, enhance employees to identify with the organization, increase their motivation and satisfaction with their work, develop mutual positive relationships between employees and the impact on the socialization of employees and organizational culture.” [27]. above all, the best method for facilitating employees to gain specific goals is face-to-face communication [38]. even though, several blocks in communication happen such as age, gender, previous history of organization, distrust in management, regional differences and so far [44]. if it is symmetrical, it has the positive effect on the relationship between employees and their organization which in turn leads to employee advocacy. men [30] also claims that there is a linkage among leadership, communication and employee outcomes which positively cultivates the quality of this relationship [31][32]. if communication is effective, it plays as an useful weapon for an organization [41][50]. communication behaviors have an indirect contribution to the success of the company through employee attitudes[28] . furthermore, effective communication will foster the closer relationship between senior managers and employees[50]. especially, in the change process, along with commitment, social and cultural values, it plays a key role in which employees share information, build relationship and make things meaningful [24] [33]. from the same view point, daly [19] strengthens that internal communication is also a key issue with regard to how successful change management programmers are performed [19]. in the process of constructing a culture of transparency in an organization between management and employees, face-to-face communication is one of the important means of internal communication [34]. mishra [34] and vercic [46] strongly state that the executives choose communication strategies in the aim of building trust and engagement with employees and actually, they consider internal communication as a management function in charge of intra-organizational communication [34][46]. and therefore, this is the proposition of the relationship between international communication and meeting effectiveness. proposition 2: internal communication significantly affects meeting effectiveness. it is unavoidable that internal communication may cause conflicts. how to manage conflicts is considered as art and science. from the perspective of conflict literature, substantive conflict is highly recommended. int. j. anal. appl. 18 (3) (2020) 467 2.4 substantive conflict one of the strategic problems occurring in the workplace is conflict. organizational members, everyday face with resolving conflicts with subordinates, supervisors, peers and stakeholders [39]. conflict [11][35] normallyrelates to a negative connotation which should be undesirable and avoided. it may originate from an individual, a team or an organization and often results in disagreement and frustration but not all conflicts are harmful. previous studies reveal that groups in conflict would terminate or reach a consensus in decision-making [13][22] meetings. esquive [11] also finds out the positive effect of conflict on the process of making decision. conflict consists of two different types which are called c-type conflict and a-type conflict. while the former is substantive, issue-related differences of opinion that tend to improve team effectiveness and originated from the agenda’s content, the latter depends on personal feelings, someone’s own agenda or interpersonal struggle related to the group’s agenda problems[11][13]. guetzkow [13] named these two types: subjective conflict and affective conflict. conflict is caused by 3 main ingredients which are individual characteristics, interpersonal factors and issues. the three most prominent categories of conflict management style are avoidance, distributive, and integrative [21]. while avoidance style tends to ignore or shift a conversation to a different issue, distributive is a confrontive approach. among the three styles, integrative brings more effective decision, implying an effort to come to the best or at least agreeable solution for all concerned members. from another perspective, conflict is also classified by the two dimensions of assertiveness and cooperativeness, expressed by five conflict-handling modes including competing, collaborating, compromising, avoiding vand accommodating) [45]. moderators that can influence in conflict [18] are “amplifiers (those variables that amplify the conflict-outcome relationship, strengthening both the positive and negative effects), suppressors (those variables that weaken both the positive and negative effects on outcomes), ameliorators (those variables that decrease negative effects and increase positive effects), and execrators (those variables that increase negative effects of conflict and decrease positive effects)” [18]. importantly, effective managers select a range of different strategies in different contexts, aiming at achieving a desirable outcome[14]. ultimately, substantive conflict is considered to have much positive effects on meeting effectiveness. therefore, the proposition is suggested as: int. j. anal. appl. 18 (3) (2020) 468 proposition 3: substantive conflict significantly affects meeting effectiveness. even though, several prominent factors affecting meeting effectiveness are abovementioned, it would be inadequate without agenda of the meeting in advance. 2.5agenda agenda is another meeting issue that need to be concerned because it affects member preparation, time-use effectiveness and finally, meeting effectiveness [37]. depending on agenda-based meeting management, an agenda enables meeting leaders to manage one or more meetings for locally-located participants, remote participants or both [8]. basically, an agenda makes teamwork more task-focused and issue-focused. it is viewed as the “purchase point” decision for team members [16]. a formal meeting agenda brings meeting participants or members involved specific information about the structure of a meeting time, place, topics related, or other preparatory work [49]. moreover, it keeps the meeting happening in the correct sequence and covering the right topics. there are a couple of benefits for either the chair of the meeting to make sure the agenda is correct or participants to prepare for a meeting [6]. above all, an agenda in advance is indispensable to meeting effectiveness. as a result, the proposition is suggested as: proposition 4: agenda significantly affects meeting effectiveness. to sum up, from previous studies of the meeting literature, it seems that there are four dominant factors affecting meeting effectiveness in the context of vietnamese organizations as the authors’ suggestion in the following conceptual model. leadership substantive conflict internal communication meeting effectiveness agenda the conceptual model int. j. anal. appl. 18 (3) (2020) 469 3method and results data collection the data for the research is based on the survey of one hundred and fifty-seven participants who are working at about 31 vietnamese organizations from a variety of sectors such as tax, banking, health service, airlines, education and business. specifically, they all are subordinates with various titles from middle managers to staffs, but not in the top management board. in other words, participants are those who lead a meeting, but still are led by other meeting organizers. the questionnaires included five variables: meeting effectiveness, agenda, leadership, substantive conflict and internal communication and were distributed as hard copies that required handwritten responses. these questions contained 30 items using fivepoint likert scale: totally disagree, disagree, neutral, agree, and totally agree. a total of completed 157 questionnaires performed within five months in hochiminh city and kien giang province in southern vietnam were returned and valid. quantitative research is conducted by non-probability sampling. data analysis and results to ensure the items in the questionnaire and the model to be valid and reliable, a part of the questionnaires is conducted as a pilot test for testing the clarity of contents and misspelling. then, one hundred and fifty-seven participants are surveyed. the result is applied spss software with the following steps: statistic analysis; evaluation of cronbach alpha for each factor; efa, then used amos to analyze sem model based on the efa’s result. the result of descriptive statistics shows that it ranged with mean from 3.55 to 4.17 (table 1). int. j. anal. appl. 18 (3) (2020) 470 table 1. descriptive statistics n minimum maximum mean std. deviation agen1.meetings start on time. 157 1 5 4.13 .899 agen2.meetings end when you expect them to end. 157 1 5 3.66 1.010 agen3.a written agenda is provided before the meetings. 157 1 5 4.09 .929 agen4.overall, i am satisfied with the meeting process. 157 1 5 3.81 .761 agen5.the meeting was time well spent. 157 1 5 3.80 .845 agen6.a verbal agenda is provided at the meetings. 157 1 5 3.92 .874 lds1.in the meeting, the leader will express the objective opinion with followers. 157 1 5 3.95 .830 lds2.in the meeting, the leader will remain impartial rather than speaking out and expressing his/her views. 157 1 5 3.90 .846 lds3.in the meeting, the leader will express the nonconservative opinion with followers. 157 1 5 3.85 .856 lds4.in the meeting, the leader will interact with followerssocial distance is low. 157 1 5 3.90 .826 lds5.in the meeting, the leader will support and encourage followers to express their ideas. 157 1 5 4.03 .812 lds6.in the meeting, the leader will foster group goals. 157 1 5 4.17 .741 lds7.in the meeting, the leader will communicate a high degree of confidence in the followes' ability to meet expectations. 157 1 5 3.83 .831 lds8.in the meeting, the leader will express high performance expectations for followers. 157 1 5 4.06 .727 lds9.in the meeting,the leader provides recognition/rewards when others reach their goals. 157 1 5 3.87 .830 lds10.in the meeting, the leader empowers his/her followers to make the final decision. 157 1 5 3.55 .957 cft1.when conflicts happen in the meeting, your leader and the group search for the real causes of the problem and find out suitable solutions. 157 1 5 3.94 .778 cft2.when conflicts happen in the meeting, your leader provides the accurate information and solves together with flollowers. 157 1 5 3.93 .743 cft3.when conflicts happen in the meeting, your leader combines his/her opinion with the group’s opinion for making the final decision. 157 1 5 3.84 .820 ic1.this company encourages differences of opinions. 157 1 5 3.89 .725 ic2.most communication between management and other employees in this organization can be said to be two-way communication. 157 1 5 3.80 .838 ic3.your leader makes you feel comfortable working with him/her. 157 1 5 3.85 .778 int. j. anal. appl. 18 (3) (2020) 471 ic4.you would feel comfortable working with your leader. 157 1 5 3.73 .859 met1.when the meeting is finally over, you feel satisfied with the results. 157 1 5 3.80 .766 met2.the meeting states each problem with a clear solution. 157 1 5 3.83 .839 met3.most of conflicts raising in the meeting are solved satisfactorily. 157 1 5 3.55 .865 met4.after the meeting, you achive your work goals. 157 1 5 4.01 .789 met5.after the meeting, you get your leader’s understanding about your difficulties. 157 1 5 3.72 .861 met6.after the meeting, you receive your leader’s instruction and sympathy with what you are fulfilling. 157 1 5 3.80 .822 met7.the meeting provides you with an opportunity to acquire useful information. 157 1 5 3.98 .755 valid n (listwise) 157 efa factor analysis is classified into 2 steps. while the first step is for independent variables, the second step is for the dependent variable. the first step, 4 independent variables are included in efa factor analysis with principal components method and rotation varimax. kmo and bartlett’s test is significant (p<.001)and kaiser-meyer-olkin measure of sampling adequacy equal to 0.920 (>0.5) (table 2) and the evaluation of cronbach alpha is .953. table 2. kmo and bartlett’s test kaiser-meyer-olkin measure of sampling adequacy. .920 bartlett's test of sphericity approx. chi-square 2593.761 df 253 sig. .000 after rotation method varimax with kaiser normalization, 22 items of independent variables are grouped into 4 groups. however, factor 4 contains only 1 item which should be eliminated. therefore, there actually exits 3 groups with 21 items which are named as leadership for group 1, agenda for group 2 and conflicts for group 3. meeting effective ness contains 7 items and is also named meeting effectiveness. the evaluation of cronbach alpha after efa analysis for 3 factors: leadership, agenda and conflict are simultaneously at .944; .814; and .817 (table 3). they all are accepted. int. j. anal. appl. 18 (3) (2020) 472 table 3. efa result rotated component matrixa component 1 2 3 4 agen1 .782 agen2 .806 agen3 .731 agen4 .661 agen5 .543 .512 agen6 .742 lds1 .598 lds2 .584 lds3 .649 lds4 .767 lds5 .722 lds6 .674 lds7 .604 lds8 .523 lds9 lds10 .876 cft1 .572 .538 cft2 .619 .546 cft3 .572 ic1 .587 ic2 .775 ic3 .826 ic4 .775 eigenvalue 7.829 2.568 2.637 cumulative 60.222 64.294 65.917 cronbach alpha .944 .814 .817 next, the depedent variable “meeting effectiveness”is evaluated by kmo and barlett’s test and efa analysis. the result is that the evaluation of cronbach alpha for dependent variable “meeting effectiveness” is .909 which is also accepted. furthermore, kmo and bartlett’s test is significant (p<.001) and kaiser-meyer-olkin measure of sampling adequacy equals to 0.891 (>0.5) and factor loadings are all more than .50. table 4. kmo and bartlett’s test kmo and bartlett's test kaiser-meyer-olkin measure of sampling adequacy. .891 bartlett's test of sphericity approx. chi-square 644.649 df 21 sig. .000 int. j. anal. appl. 18 (3) (2020) 473 table 5. component analysis communalities initial extraction met1 1.000 .683 met2 1.000 .715 met3 1.000 .645 met4 1.000 .693 met5 1.000 .598 met6 1.000 .628 met7 1.000 .579 cfa factor analysis figure 1. results of cfa concepts of research model (standardized) p=.000; cfi = .871; tli = .858; gfi = .743; rmsea = .089. int. j. anal. appl. 18 (3) (2020) 474 table 6. standardized regression weights estimate s.e. c.r. p label meetef <--ledship .683 .255 2.679 .007 meetef <--agenda -.023 .185 -.124 .901 meetef <--conflict .408 .122 3.353 *** lds2 <--ledship 1.216 .136 8.912 *** lds3 <--ledship 1.261 .138 9.128 *** lds4 <--ledship 1.215 .133 9.115 *** lds5 <--ledship 1.250 .131 9.525 *** lds6 <--ledship 1.135 .120 9.476 *** lds7 <--ledship 1.214 .134 9.055 *** lds8 <--ledship 1.000 agen5 <--agenda 1.260 .176 7.172 *** agen4 <--agenda 1.012 .150 6.736 *** agen3 <--agenda 1.000 agen1 <--conflict 1.000 agen2 <--conflict .864 .108 8.003 *** cft1 <--conflict 1.175 .077 15.185 *** cft2 <--conflict 1.000 met04 <--meetef 1.000 met03 <--meetef 1.001 .101 9.885 *** met02 <--meetef 1.101 .094 11.677 *** met01 <--meetef .984 .087 11.347 *** cft3 <--ledship 1.090 .132 8.244 *** ic01 <--ledship 1.091 .117 9.329 *** ic02 <--ledship 1.230 .135 9.099 *** ic03 <--ledship 1.110 .125 8.851 *** ic04 <--ledship 1.218 .139 8.787 *** met05 <--meetef .914 .104 8.827 *** met06 <--meetef .970 .096 10.140 *** met07 <--meetef .898 .088 10.239 *** int. j. anal. appl. 18 (3) (2020) 475 the results of cfa factor analysis of the research model are presented in figure 1. while processing data, the authors eliminate two items which are lds1 and agen6 because they are insignificant in the model in order to produce the valid results. these results show that the conditions are stated as follow: p < 0.05; cfi, gfi ≥ 0.8 and rmsea is approximately0.08. they all meet the requirements. considering the above conditions, the model is consistent with market data. based on the results in table 6, the parameters (standardized) are statically significant (p<0.05). according to the regression weight between factors shown, two factors that are leadership and substantive conflict have significant effects on meeting effectiveness with weight of 0.683 and 0.408 and p-value <0.05respectively, while agenda with weight of -0.023 and p-value 0.901does not. in other words, leadership affects positively meeting effectiveness and when leadership goes up by 1 standard deviation, meeting effectiveness goes up by 0.683 standard deviation. similarly, when substantive conflict increases by 1 standard deviation, meeting effectiveness increases by 0.408 standard deviation. 4discussion meetings become frequent activities in every organization for such purposes as fulfilling vital goals, making changes and exchanging ideas. it is evident that meeting effectiveness is closely related to goal attainment and decision satisfaction. therefore, meetings need be improved in an effective and efficient way so that subordinates make more contributions and increase commitment to their workplace. it is found that meeting effectiveness is significantly influenced by the two dominant factors consisting of leadership and substantive conflict. from previous study, kirkpatrich [20] confirms that leader’s styles such as achievement, motivation, ambition, energy, tenacity and initiative are highly appreciated. they should be trained essential skills: formulating an organization vision, making effective plans for vision implementation in reality [20]. besides, both leaders and subordinates should have the common goals [4]. servant leadership in which leaders need to make sure that other people’s needs are served by listening and observing is strongly recommended by [12]. furthermore, wallis [48]and zhu [51] also emphasize that leadership’s inspiration should be shared with followers and the leaders should behave morally int. j. anal. appl. 18 (3) (2020) 476 and always expect to create a healthy environment and organizational culture to grow ethical behaviors inside the organization. actually, whether the meeting is effective or not depends on the meeting leaders’ guide. actually, leadership plays a very important role in transforming, motivating and enhancing subordinates ‘actions and ethical aspirations. subordinates surely become more committed to the organization when they are working with inspirational leaders who willingly instruct them in uncertainty and encourage their abilities and talents [7]. that’s why leadership strongly affects meeting effectiveness in reality. during the process of interaction, conflicts may exist and therefore, how to resolve conflicts needs to be concerned. conflicts are double-faced. while affective conflict may improve and bring benefits to team effectiveness, subjective one may destroy the relationship and reduce members’ job performance [11][13]. transparently, from previous studies, substantive conflicts which are issue-related differences of opinion are proved to aim at improving team effectiveness. it also confirms that substantive conflicts positively influence meeting effectiveness. in short, empirically, in order to host a meeting effectively, meeting organizers should control their leadership in a proper way and solve thoroughly any conflicts raising in a constructive way. 5-implications and conclusion implications for future research, based on the literature of meeting effectiveness, it also has the great impact on organizational commitment. therefore, what we should do next is to find out the relationship between meeting effectiveness and organizational commitment which motivates and inspires subordinates to engage more closely in their organization. int. j. anal. appl. 18 (3) (2020) 477 conclusion the findings show practical meaning of meeting effectiveness in the context of vietnamese organizations. empirically, the two significant factors that mainly affect meeting effectiveness are leadership and substantive conflict. based on the previous studies in the world [16] agenda plays an essential role in the meeting, but the result shows that it’s statistically insignificant with p-value equals to 0.90 > 0.05 which is showed in table 6 standardized regression weights. regarding to national values differences across the worldwide subsidiaries, while according to western cultures, people belong to polychromic culture, they tend to be on time, and vietnamese people almost belonging to asian culture tend to be influenced by polychromic culture [25]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] alavi, m., kayworth, t. r., & leidner, d. e. an empirical examination of the influence of organizational culture on knowledge management practices. j. manage. inform. syst. 22(3)(2005-6), 191–224. [2] allen, j., sands, s., mueller, s., frear, k., mudd, m. and rogelberg, s. employees' feelings about more meetings: an overt analysis and recommendations for improving meetings, manage. res. rev. 35(5) (2012), 405-418. [3] allen, j. a., willenbrock, n. l., & landowski, n. linking pre-meeting communication to meeting effectiveness. j. manager. psychol. 29(2014), 1064-1081. [4] andersen, j. a. leadership, personality and effectiveness, j. soc. econ. 35(6) (2006), 1078-1091. [5] bagire, v., byarugaba, j., & kyogabiirwe, j. organizational meetings: management and benefits, j. manage. develop. 34(8)(2015), 960 972. [6] baker, h. writing meeting minutes and agenda. lancashire, uk: universe of learning ltd.(2010). [7] bass, b. m., & riggio, r. e. transformational leadership. london: lawrence erlbaum associates, inc.(2006). [8] butt, d. agenda based meeting management system, interface and method. google patents. (2006). [9] daly, p. j. k. f. internal communication during change management. corp. commun. 17(2002), 46-53. int. j. anal. appl. 18 (3) (2020) 478 [10] deetz, s. conceptual foundations in the new handbook of organizational communication: advances in theory, research and methods (pp. 3-46): thousand oaks, ca: sage.(2001). [11] esquivel, m. a., & kleiner, b. h. the importance of conflict in work team effectiveness, team perform. manage. 2(3)(1996), 42 48. [12] fry, l. w., matherly, l. l., whittington, j. l., & winston, b. e. spiritual leadership as an integrating paradigm for servant leadership.integrat. spirit. organiz. leadership. (2007), 70-82. [13] guetzkow, h., & gyr, j. an analysis of conflict in decision-making groups. human relations, 7(3) (1954), 367-382. [14] holmes, j. m.,marra, m. leadership and managing conflict in meetings. int. pragmat. assoc. 14(4)(2004), 439-462. [15] hopkins, w. e., & hopkins, s. a. diversity leadership: a mandate for the 21st century workforce. j. leadership stud. 5(3)(1998), 129-140. [16] inglis, s., & weaver, l. designing agendas to reflect board roles and responsibilities: results of a study, nonprofit manage. leadership, 11(1)(2000), 65-77. [17] jarzabkowski, p., & seidl, d. the role of meetings in the social practice of strategy. organ. stud. 29(11)(2008), 1391-1426. [18] jehn, k. a., & bendersky, c. intragroup conflict in organizations: a contingency perspective on the conflict-outcome relationship. res. organ. behavior, 25(2003), 187–242. [19] judge, t. a., & bono, j. e. five-factor model of personality and transformational leadership. j. appl. psychol. 85(5)(2000), 751-765. [20] kirkpatrick, s. a., & locke, e. a. leadership: do traits matters? acad. manage. execut. 5(2)(1991), 4860. [21] kuhn, t., & poole, m. l. do conflict management styles affect group decision making?. human commun. res. 26(4)(2000), 558–590 [22] laursen, b., & collins, w. a. interpersonal conflict during adolescence. psychol. bull. 115(2)(1994), 197-209. [23] leach, d. j., rogelberg, s. g., warr, p. b., & burnfield, s. g. perceived meeting effectivenss: the role of design characteristics. j. bus. psychol. 24(2009), 65-76. [24] linke, a., & zerfass, a. internal communication and innovation culture: developing a change framework, j. commun. manage. 15(4)(2011), 332-348. [25] locker, k., & keinzler, d. business and administrative communication (9th ed.), mc graw, (2009). int. j. anal. appl. 18 (3) (2020) 479 [26] marjosola, i. a., & takala, t. charismatic leadership, manipulation and the complexity of organizational life. j. workplace learning,12 (4) (2000),146-158. [27] martic, m. communication between employees. paper presented at the symorg 2014, serbira, 2014. [28] mazzei, a. promoting active communication behaviours through internal communication,corp. commun., int. j. 15(3)(2010), 221-234 [29] meinecke, a. l., & lehmann-willenbrock, n. k. social dynamics at work: meetings as a gateway, in j. a. allen, n. lehmann-willenbrock, & s. g. rogelberg (eds.), cambridge handbooks in psychology. the cambridge handbook of meeting science (p. 325–356). cambridge university press, (2015). [30] men, l. r. strategic internal communication: transformational leadership, communication channels, and employee satisfaction. manage. commun. q. 28(2)(2014), 264 –284 [31] men, l. r.why leadership matters to internal communication: linking transformational leadership, symmetrical communication, and employee outcomes. j. public relat. res. 26(2014), 256–279. [32] men, l. r., & jiang, h. cultivating quality employee-organization relationship. int.j.strat.commun. 10(5)(2016), 462–479. [33] men, l. r., & stacks, d. the effects of authentic leadership on strategic internal communication and employee-organization relationships, j. public relat. res. 26(2014), 301-324. [34] mishra, k., lois boynton, l., & mishra, a. driving employee engagement: the expanded role of internal communications. int. j. bus. commun. 51(2)(2014), 183 –202. [35] morton deutsch, m., coleman, p. t., & marcus, e. c. culture and conflict. in the handbook of conflict resolution (pp. 641). san francisco, ca 94103-1741 jossey-bass a wiley imprint, (2006). [36] nicholas, c. r., & jay, f. n. meeting analysis: findings from research and practice, proceedings of the 34th annual hawaii international conference on system sciences, maui, hi, usa.(2001). [37] nixon, c. t., & littlepage, g. l. impact of meeting procedures on meeting effectiveness. j. bus. psychol. 6(2014), 361-369. [38] okanovic, m., stefanovic, t., & suznjevic, m. new media in internal communication. paper presented at the symorg 2014, serbira, (2014). [39] putnam, l. l. communication and interpersonal conflict in organizations, manage. commun. q. 1(1988), 293-301. [40] rogelberg, s. g., leach, d. j., warr, p. b., & burnfield, j. l. “not another meeting!” are meeting time demands related to employee well-being? j. appl. psychol. 91(1)(2006), 86-96. int. j. anal. appl. 18 (3) (2020) 480 [41] ruck, k., & welch, m.valuing internal communication; management and employee perspectives. public relat. rev. 38(2012), 223-230. [42] safriadi, hamdat, s., lampe, m., & munizu, m. (2006). organizational culture in perspective anthropology, int. j. sci. res. publ. 6(6) (2016), 773-776. [43] simola, s., barling, j., & turner, n. (2012). transformational leadership and leader's mode of care reasoning,j. bus. ethics, 108(2012), 229–237. [44] smith, l., & mounter, p. effective internal communication. usa: replika press pvt ltd, (2008). [45] thomas, k. w.conflict and conflict management: reflections and update. j. organ. behavior, 13(1992), 265-274. [46] vercic, a. t., vercic, d., & sriramesh, k. internal communication: definition, parameters, and the future. public relat. rev. 38(2012), 223-230. [47] volkema, r. j., & fred niederman, f. planning and managing organizational meetings: an empirical analysis of written and oral communications, j.bus. commun. 33(1996), 275-296. [48] wallis, j. drawing on revisionist economics to explain the inspirational dimension of leadership. j. soc. econ. 31(2002), 59-74. [49] welch, d. d. conflicting agendas. eugene, oregon: published by the pilgrim press, (2008). [50] welch, m. (2011). appropriateness and acceptability: employee perspectives of internal communication. public relat. rev.38(2) (2012), 246-254. [51] zhu, w., may, d. r., & avolio, b. j. the impact of ethical leadership behavior on employee outcomes: the roles of psychological empowerment and authenticity. j. leadershiporgan. stud. 11 (1)(2004), 16-26. [52] pham, l. h. t., nguyen, n. t., & tran, t. t. on the factors affecting start-up intention of millennials in vietnam. in. j. adv. appl. sci. 6(1)(2019), 1-8. [53] nguyen n.t., nguyen b.p.u., and tran t.t. application of grey system theory and arima model to forecast factors of tourism: a case of binh thuan province in vietnam. int. j. adv. appl. sci. 7(1)(2020), 87-99. [54] nguyen n.t., nguyen t.t.t., and tran t.t. forecasting vietnamese tourists’ accommodation demand using grey forecasting and arima models. int. j. adv. appl. sci. 6(11)(2019), 42-54. [55] nguyen, n. t. optimizing factors for accuracy of forecasting models in food processing industry: a context of cacao manufacturers in vietnam. ind. eng. manage. syst. 18(4)(2019), 808-824. [56] nguyen, n. t., & nguyen, l. x. t. applying dea model to measure the efficiency of hospitality sector: the case of vietnam. int. j. anal. appl. 17(6) (2019), 994-1018. int. j. anal. appl. 18 (3) (2020) 481 [57] nguyen, n. t., & tran, t. t. mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dyn. nat. soc. 2015(2015), article id 858157. [58] nguyen, n. t., & tran, t. t. facilitating an advanced product layout to prioritize hot lots in 450 mm wafer foundry in the semiconductor industry. int. j. adv. appl. sci. 3(6) (2016), 14-23. [59] nguyen, n. t., & tran, t. t. a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9) (2018), 73-81. [60] nguyen, n. t., & tran, t. t. a study of the strategic alliance for vietnam domestic pharmaceutical industry: a dynamic integration of a hybrid dea and gm (1, 1) approach. j. grey syst. 30(4)(2018), 134-151. [61] nguyen, n. t., & tran, t. t. a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9)(2018), 73-81. [62] nguyen, n. t., & tran, t. t. raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl. 31(11)(2019), 7963-7974. [63] nguyen, n. t., tran, t. t., wang, c. n., & nguyen, n. t. optimization of strategic alliances by integrating dea and grey model. journal of grey system, 27(1)(2015), 38-56. [64] nguyen, n.t. performance evaluation in strategic alliances: a case of vietnamese construction industry. glob. j. flex. syst. manage. 21 (2020), 85–99. [65] tran, t. t. evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci, 3(12)(2016), 5-20. [66] tran, t. t. an empirical research on selecting the targeted suppliers and purchasing process of supermarket. int. j. adv. appl. sci. 4(4)(2017), 96-109. [67] tran, t. t. forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci. 4(9)(2017), 86-95. [68] tran, t. t. factors affecting the purchase and repurchase intention smart-phones of vietnamese staff. int. j. adv. appl. sci. 5(3)(2018), 107-119. [69] wang, c. n., le, t. m., nguyen, h. k., & ngoc-nguyen, h. using the optimization algorithm to evaluate the energetic industry: a case study in thailand. processes, 7(2) (2019), 87. [70] wang, c. n., nguyen, n. t., & tran, t. t. integrated dea models and grey system theory to evaluate past-to-future performance: a case of indian electricity industry. sci. world j. 2015 (2015), article id 638710. international journal of analysis and applications volume 19, number 3 (2021), 440-454 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-440 numerical study of rayleigh-benard problem under the effect of magnetic field abdelfatah abasher1,∗, mohammed gubara2 , suliman sheen2, ibrahim bashir2 1mathematics department, faculty of science, jazan university, jazan, ksa 2mathematics department, faculty of mathematical science and statistic, al-neelain university, khartoum, sudan ∗corresponding author: amoaf84@gmail.com abstract. in this paper, a linear stability analysis is studied for rayliegh-benard problem with the effect of magnetic field, a perturbation equations is solved numerically by using spectral chebyshev tau method, the boundaries are considered both are free, both are rigid, the lower is free and the upper is rigid, the results were illustrated graphically and compared with previous studies. 1. introduction the aim of this work is to solve the perturbation equations that represent rayleigh-benard problem under the effect of magnetic field ( [1], page (160, 161)) as follows (1.1) ∂u ∂t = −∇ ( δp ρ0 + µ h � h 4πρ0 ) + gαθk + v∇2u + µ 4πρ0 (h �∇) h, (1.2) ∂h ∂t = (h �∇) u + η∇2h, received february 24th, 2021; accepted march 26th, 2021; published april 28th, 2021. 2010 mathematics subject classification. 00a69. key words and phrases. rayleigh-benard problem; linear stability analysis; chebyshev tau method. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 440 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-440 int. j. anal. appl. 19 (3) (2021) 441 (1.3) ∇ � h = 0, (1.4) ∂θ ∂t = βw + κ∇2θ, (1.5) ∇ � u = 0, where, u, δp, θ and h are the perturbation in the velocity, pressure, temperature and magnetic field respectively, ρ0 is the density at mean temperature t0, v is the kinematic viscosity, κ is the thermal diffusivity, α the coefficient of volume expansion, µ is the magnetic permeability, η is the resistivity and β is the temperature gradient defined as (1.6) β = t0 −td d , the problem is studied analytically by various authors ( [1][4], [12][14]). also many numerical methods were used, one of these method is galerkin method which discussed in [14]. in this paper we used the numerical method namely spectral chebyshev tau method to determine the conditions of instability for various cases of the boundary conditions. the paper outlined as follows. in section 1 we formulate the governing mathematical perturbation equations. in section 2 a linear stability analysis for the perturbation equations. in section 3 we describe the method of solution spectral chebyshev tau method for the three cases of the boundary conditions. finally, we present our numerical results in which are computed using matlab. 2. linear stability analysis by taking the curl operator of equation (1.1), we get (2.1) ∂ω ∂t = gα ( ∂θ ∂y i− ∂θ ∂x j ) + v∇2ω + µ 4πρ0 (h �∇) v, where ω = ∇× u is the vorticity vector and v = ∇× h is the current density induced by the perturbation. taking the curl operator of (2.1) again, we get (2.2) ∂∇2u ∂t = −gα ( ∂2θ ∂z∂x i + ∂2θ ∂z∂y j − ( ∂2θ ∂x2 + ∂2θ ∂y2 ) k ) + v∇4u + µ 4πρ0 (h �∇)∇2h. take the curl of equation (1.2), we get (2.3) ∂v ∂t = (h �∇) ω + η∇2v. int. j. anal. appl. 19 (3) (2021) 442 now by equating the z-component of equation (1.2), (2.1), (2.2) and (2.3) respectively, we get (2.4) ∂hz ∂t = η∇2hz + (h �∇) w, (2.5) ∂ζ ∂t = v∇2ζ + µ 4πρ0 (h �∇) ξ, (2.6) ∂∇2w ∂t = gα ( ∂2θ ∂x2 + ∂2θ ∂y2 ) + v∇4w + µ 4πρ0 (h �∇) hz, (2.7) ∂ξ ∂t = η∇2ξ + (h �∇) ζ, where w, ζ and ξ and hz are the z-components of the velocity, the vorticity, the current density and the magnetic field respectively. when the direction of the magnetic field coincides with the vertical direction h = (0, 0,h), the required perturbation equations become (2.8) ∂ ∂t ∇2w = gα ( ∂2θ ∂x2 + ∂2θ ∂y2 ) + v∇4w + µh 4πρ0 ∂ ∂z ∇2hz, (2.9) ∂hz ∂t = η∇2hz + h ∂w ∂z , (2.10) ∂ξ ∂t = η∇2ξ + h ∂ζ ∂z , (2.11) ∂ζ ∂t = v∇2ζ + µh 4πρ0 ∂ξ ∂z . in addition to equation (1.4). the boundary conditions of θ, w and ζ for the three cases free-free, rigid-rigid and rigid-free are (2.12)   θ = 0,w = 0 at z = 0 and z = d, dζ dz = 0, d 2w dz2 = 0 at free boundary, ζ = 0, dw dz = 0 at rigid boundary. int. j. anal. appl. 19 (3) (2021) 443 3. normal mode analysis let w,θ,ζ,hz and ξ be defined as (3.1)   zθ ζ w ξ hz   =   θ(z) z(z) w(z) x(z) k(z)   exp (i(kxx + kyy) + γt), where k = √ k2x + k 2 y is the wave number and γ is a constant. substitute (3.1) into the system (1.4), (2.8)-(2.11), the system become (3.2) γθ = βw + κ ( d2 dz2 −k2 ) θ, (3.3) γk = η ( d2 dz2 −k2 ) k + h dw dz , (3.4) γ ( d2 dz2 −k2 ) w = −gαk2θ + v ( d2 dz2 −k2 )2 w + µh 4πρ0 d dz ( d2 dz2 −k2 ) k, (3.5) γx = η ( d2 dz2 −k2 ) x + h dz dz , (3.6) γz = v ( d2 dz2 −k2 ) z + µh 4πρ0 dx dz . and the boundary conditions (2.12) become (3.7)   θ = 0,w = 0 at z = 0 and z = d, dz dz = 0, d 2w dz2 = 0 at free boundary, z = 0, dw dz = 0 at rigid boundary. define the following non-dimensional variables, (3.8) a = kd, σ = γd2 v , z∗ = z d , p1 = v κ , and p2 = v η , the operators d dz = 1 d d dz∗ and d 2 dz2 = 1 d2 d2 dz∗2 , assume d = d dz∗ , then by substituting (3.8) into the system (3.2)-(3.6), we get (3.9) ( d2 −a2 −p1σ ) θ = − ( βd2 κ ) w, int. j. anal. appl. 19 (3) (2021) 444 (3.10) ( d2 −a2 −p2σ ) k = − ( hd η ) dw, (3.11) ( d2 −a2 )( d2 −a2 −σ ) w + ( µhd 4πρ0v ) d ( d2 −a2 ) k = ( gαd2 v ) a2θ, (3.12) ( d2 −a2 −p2σ ) x = − ( hd η ) dz, (3.13) ( d2 −a2 −σ ) z = − ( µhd 4πρ0v ) dx, at the marginal state (σ = 0), then equations (3.9)-(3.13) become (3.14) ( d2 −a2 ) θ = − ( βd2 κ ) w, (3.15) ( d2 −a2 ) k = − ( hd η ) dw, (3.16) ( d2 −a2 )2 w + ( µhd 4πρ0v ) d ( d2 −a2 ) k = ( gαd2 v ) a2θ (3.17) ( d2 −a2 ) x = − ( hd η ) dz, (3.18) ( d2 −a2 ) z = − ( µhd 4πρ0v ) dx, and the boundary conditions (3.7) become (3.19)   θ = 0, w = 0 at z = 0 and z = 1, dz = 0, d2w = 0 at free boundary, z = 0, dw = 0 at rigid boundary. by taking the operator d for equation (3.15) and substituting in (3.16), we get (3.20) ( d2 −a2 )2 w −qd2w = ( gαd2 v ) a2θ, int. j. anal. appl. 19 (3) (2021) 445 where, (3.21) q = µh2d2 4πρ0vη , is chandrasekhar number [1]. taking the operator (d2 −a2) for equation (3.20), and using equation (3.14), we get (3.22) (d2 −a2) [( d2 −a2 )2 −qd2 ] w = −ra2w, where, (3.23) r = gαβd4 κv , is rayleigh number. by taking the operator (d2 −a2) for equation (3.18) and using (3.17), we get (3.24) [( d2 −a2 )2 −qd2 ] z = 0, we must seek the solution of (3.22) and (3.24) subject to the boundary conditions (3.19). 4. spectral chebyshev tau method spectral chebyshev tau method is a numerical method to solve the differential equations and the eigen values problems, (see [6], [7], [8] and [10]). to solve equations (3.22) and (3.24) subject to the boundary conditions (3.19), first convert the domain to chebyshev polynomials domain [−1, 1], use the relation x = 2z − 1, if z ∈ [0, 1] implies x ∈ [−1, 1] and the derivative d dz = 2 d dx , d 2 dz2 = 4 d 2 dx2 , then (3.22), (3.24) and (3.19) become (4.1) (4d2 −a2) [( 4d2 −a2 )2 − 4qd2 ] w = −ra2w, (4.2) [( 4d2 −a2 )2 − 4qd2 ] z = 0, (4.3)   w = 0 at x = −1 and x = 1, dz = 0, d2w = 0 at free boundary, z = 0, dw = 0 at rigid boundary, where d = d dx . let s = [( 4d2 −a2 )2 − 4qd2]w then we can write (4.1) (4.3) as (4.4) [( 4d2 −a2 )2 − 4qd2 ] w −s = 0, int. j. anal. appl. 19 (3) (2021) 446 (4.5) ( 4d2 −a2 ) s = −ra2w, (4.6) [( 4d2 −a2 )2 − 4qd2 ] z = 0, (4.7)   w = 0,s = 0 at x = −1 and x = 1, dz = 0, d2w = 0 at free boundary, z = 0, dw = 0 at rigid boundary. now, expand w,s and z as chebyshev polynomials w = n∑ n=0 wntn(x) = [ w0 · · · wn ]   t0 ... tn   = wφ,(4.8) s = n∑ n=0 sntn(x) = [ s0 · · · sn ]   t0 ... tn   = sφ,(4.9) z = n∑ n=0 zntn(x) = [ z0 · · · zn ]   t0 ... tn   = zφ,(4.10) where w, s and z are row vectors represent the coefficients of w,s and z respectively, and φ is the vector of chebyshev polynomials t0 up to tn . furthermore we can expand the derivatives dw, d 2w and d4w as chebyshev polynomials as (4.11) dw = n∑ n=0 w(1)n tn(x) = wdφ, (4.12) d2w = n∑ n=0 w(2)n tn(x) = wd 2φ, (4.13) d4w = n∑ n=0 w(4)n tn(x) = wd 4φ, int. j. anal. appl. 19 (3) (2021) 447 similarly for the derivatives of s and z (4.14) d2s = n∑ n=0 s(2)n tn(x) = sd 2φ, (4.15) d2z = n∑ n=0 z(2)n tn(x) = zd 2φ, where d and d2 are (n + 1)×(n + 1) matrices represent the coefficients of the first and second chebyshev derivatives, (see [6] and [10] for more details of formulations of d and d2). d =   0 0 0 0 0 · · · 0 1 0 0 0 0 · · · 0 0 4 0 0 0 · · · 0 3 0 6 0 0 · · · 0 0 8 0 8 0 · · · 0 ... ... ... ... ... . . . 0 n 0 2n 0 2n · · · 0   ,(4.16) d2 =   0 0 0 0 0 0 · · · 0 0 0 0 0 0 0 · · · 0 4 0 0 0 0 0 · · · 0 0 24 0 0 0 0 · · · 0 32 0 48 0 0 0 · · · 0 0 120 0 80 0 0 · · · 0 ... ... ... ... ... ... . . . 0 0 n(n2 − 1) 0 n(n2 − 9) 0 n(n2 − 25) · · · 0   ,(4.17) and d4 = (d)4 = d2d2. by substitute (4.8)–(4.15) into equations (4.4)-(4.6) we get, (4.18) w [ 16d4 − (8a2 + 4q)d2 + a4i ] φ− sφ = 0, (4.19) s [ 4d2 −a2i ] φ = −ra2wφ, (4.20) z [ 16d4 − (8a2 + 4q)d2 + a4i ] φ = 0, int. j. anal. appl. 19 (3) (2021) 448 where i is the identity matrix of order (n + 1). by taking the inner product with tn,n = 0, 1, ..,n, for each equation in the system (4.18) – (4.20) and using the property of orthogonality of chebyshev polynomials, we obtain 3(n + 1) equations as follows (4.21) [ 16w(4)n − ( 8a2 + 4q ) w(2)n + a 4wn ] −sn = 0, n = 0, 1, . . . ,n (4.22) 4s(2)n −a 2sn = −ra2wn, n = 0, 1, . . . ,n (4.23) 16z(4)n − (8a 2 + 4q)z(2)n + a 4zn = 0. n = 0, 1, . . . ,n rewriting these equations in matrices form as (4.24) w [ 16d4 − (8a2 + 4q)d2 + a4i ] − s = 0, (4.25) s [ 4d2 −a2i ] = −ra2w, (4.26) z [ 16d4 − (8a2 + 4q)d2 + a4i ] = 0, or, (4.27) ax = rbx, where a and b are square matrices of order 3(n + 1) given as a =   l1 −i 0 0 l2 0 0 0 l1   , b =   0 0 0 −a2i 0 0 0 0 0   , and x =   wt st zt  (4.28) where l1 = [ 16d4 − (8a2 + 4q)d2 + a4i ]t and l2 = [ 4d2 −a2i ]t . now returning to the boundary conditions (4.7), the p-derivative of chebyshev polynomials at x = ±1 is given by the formula (4.29) dptn dxp ∣∣∣∣ x=±1 = (±1)n+p p−1∏ k=0 n2 −k2 2k + 1 , for free-free boundaries, we have eight boundary conditions as w = 0, d2w = 0, s = 0 and dz = 0 for x = ±1. int. j. anal. appl. 19 (3) (2021) 449 bc1 : w(−1) = 0 ⇒ n∑ n=0 wntn(−1) = 0 ⇒ n∑ n=0 (−1)n wn = 0, bc2 : w(1) = 0 ⇒ n∑ n=0 wntn(1) = 0 ⇒ n∑ n=0 wn = 0, bc3 : d2w(−1) = 0 ⇒ n∑ n=0 wnt ′′ n (−1) = 0 ⇒ n∑ n=0 (−1)n+2 n2(n2 − 1) 3 wn = 0, bc4 : d2w(1) = 0 ⇒ n∑ n=0 wnt ′′ n (1) = 0 ⇒ n∑ n=0 n2(n2 − 1) 3 wn = 0, bc5 : s(−1) = 0 ⇒ n∑ n=0 sntn(−1) = 0 ⇒ n∑ n=0 (−1)n sn = 0, bc6 : s(1) = 0 ⇒ n∑ n=0 sntn(1) = 0 ⇒ n∑ n=0 sn = 0, bc7 : dz(−1) = 0 ⇒ n∑ n=0 znt ′ n(−1) = 0 ⇒ n∑ n=0 (−1)n+1 n2zn = 0, bc8 : dz(1) = 0 ⇒ n∑ n=0 znt ′ n(1) = 0 ⇒ n∑ n=0 n2zn = 0, similarly, for rigid-rigid boundaries, w = 0, dw = 0, s = 0 and z = 0 for x = ±1. and for rigid-free boundaries, w = 0, dw = 0, s = 0 and z = 0 for x = −1 rigid. w = 0, d2w = 0, s = 0 and dz = 0 for x = 1 free. for each case of the boundary conditions, insert bc1 up to bc4 into the rows (n −2)th up to (n + 1)th of the first column in the matrix a in (4.27), bc5 and bc6 into the rows (2n + 1)th and (2n + 2)th of the second column in a, bc7 and bc8 into the rows (3n + 2)th and (3n + 3)th of the third column in a. the int. j. anal. appl. 19 (3) (2021) 450 corresponding rows in the matrix b are zeros, then we can write the system (4.27) as follows  l1 −i 0 bc1 0 . . . 0 0 . . . 0 bc2 0 . . . 0 0 . . . 0 bc3 0 . . . 0 0 . . . 0 bc4 0 . . . 0 0 . . . 0 0 l2 0 0 . . . 0 bc5 0 . . . 0 0 . . . 0 bc6 0 . . . 0 0 0 l1 0 . . . 0 0 . . . 0 bc7 0 . . . 0 0 . . . 0 bc8     w0 ... wn s0 ... sn z0 ... zn   = r   0 0 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 −a2i 0 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 0 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0 0 . . . 0     w0 ... wn s0 ... sn z0 ... zn   (4.30) where, l1, l2, 0 and i are the sub matrices of l1, l2, 0 and i respectively. now we have generalized eigen value (4.30), we can find the minimum eigen values r(a), then the critical rayleigh number values rc and the corresponding wave number ac for various values of q using matlab software, the results are illustrated in the next section. 5. results and conclusion • spectral chebyshev tau method give results in full agreement with the results that obtained by the analytical solution given by chandrasekhar [1]. • for the three cases of the boundaries free-free, rigid-rigid and rigid-free the critical rayleigh number determine the stability. if r < rc the motion is stable and no convection, when r = rc it is stationary convection, and unstable motion when r > rc. • it is also observed the effect of the magnetic field, as q increases, the value of the critical rayleigh number and the critical wave number also increases. int. j. anal. appl. 19 (3) (2021) 451 q chandrasekhar [1] present study ac rc ac rc 0 2.233 657.511 2.22 657.51 10 2.590 923.070 2.59 923.07 50 3.270 1762.04 3.27 1762.04 100 3.702 2653.71 3.7 2653.71 200 4.210 4258.49 4.21 4258.49 500 4.998 8578.88 5.00 8578.89 1000 5.684 15207.0 5.68 15207 2000 6.453 27699.9 6.46 27699.79 5000 7.585 63135.9 7.61 63135.48 10000 8.588 119832 8.61 119831.71 20000 9.706 230038 9.72 230038.48 40000 10.95 445507 10.96 445506.86 table 1. the relation between between rc and ac when the boundaries are free-free and q = 0, 10, 50, . . . , 40000. wave number a 0 5 10 15 r a y li e g h n u m b e r r ×10 4 0 1 2 3 4 5 6 7 8 9 10 free free boundaries figure 1. the relation between between rc and ac when the boundaries are free-free and q = 0, 10, 50, . . . , 40000. int. j. anal. appl. 19 (3) (2021) 452 q chandrasekhar [1] present study ac rc ac rc 0 3.13 1707.8 3.12 1707.77 10 3.25 1945.9 3.27 1945.75 50 3.68 2803.1 3.68 2802.01 100 4.00 3757.4 4.01 3757.23 200 4.45 5488.6 4.45 5488.54 500 5.16 10110 5.16 10109.78 1000 5.80 17103 5.81 17102.85 2000 6.55 30125 6.56 30124.81 4000 7.40 54697 7.39 54700.75 6000 7.94 78391 7.93 78441.25 8000 8.34 101606 8.30 101930.17 10000 8.66 124509 8.52 125856.50 table 2. the relation between between rc and ac when the boundaries are rigid-rigid and q = 0, 10, 50, . . . , 10000. wave number a 0 5 10 15 r a y li e g h n u m b e r r ×10 4 0 1 2 3 4 5 6 7 8 9 10 rigid rigid boundaries figure 2. the relation between between rc and ac when the boundaries are rigid-rigid and q = 0, 10, . . . , 10000. int. j. anal. appl. 19 (3) (2021) 453 q chandrasekhar [1] present study ac rc ac rc 0 2.68 1100.75 2.7 1100.81 2.5 2.75 1167.2 2.71 1167.45 12.5 2.97 1415.5 3.01 1415.81 25 3.17 1699.4 3.21 1699.79 50 3.45 2217.6 3.51 2217.98 125 4.00 3586.1 4.01 3585.85 250 4.50 5613.3 4.51 5612.93 500 5.10 9304.5 5.11 9303.95 1000 5.75 16119 5.71 16118.68 1500 6.20 22592 6.21 22591.30 2000 6.50 28879 6.51 28877.86 2500 6.75 35044 6.81 35043.57 5000 7.65 64847 7.61 64836.70 10000 8.65 122140 8.71 121512.72 table 3. the relation between between rc and ac when the boundaries are rigid-free and q = 0, 2.5, 12.5, . . . , 10000. wave number a 0 5 10 15 r a y li e g h n u m b e r r ×10 4 0 1 2 3 4 5 6 7 8 9 10 rigid free boundaries figure 3. the relation between between rc and ac when the boundaries are rigid-free and q = 0, 2.5, 12.5. . . , 10000. int. j. anal. appl. 19 (3) (2021) 454 conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. chandrasekhar, hydrodynamic and hydromagnetic stability, oxford university press, oxford, 1961. [2] s. chandrasekhar, the instability of a layer of fluid heated below and subject to coriolis forces, proc. r. soc. lond. a. 217 (1953), 306–327. [3] s. chandrasekhar, the instability of a layer of fluid heated below and subject to the simultaneous action of a magnetic field and rotation, proc. r. soc. lond. a. 225 (1954), 173–184. [4] s. chandrasekhar, d.d. elbert, the instability of a layer of fluid heated below and subject to coriolis forces. ii, proc. r. soc. lond. a. 231 (1955), 198–210. [5] a.j. chorin, a numerical method for solving incompressible viscous flow problems, j. comput. phys. 2 (1967), 12–26. [6] j.j. dongarra, b. straughan, d.w. walker, chebyshev tau-qz algorithm methods for calculating spectra of hydrodynamic stability problems, appl. numer. math. 22 (1996), 399–434. [7] l. fox, chebyshev methods for ordinary differential equations, computer j. 4 (1962), 318–331. [8] m.a. hernandez, chebyshev’s approximation algorithms and applications, comput. math. appl. 41 (2001), 433–445. [9] a.a. hill, convection in porous media and legendre, chebyshev galerkin methods, ph.d. thesis, durham university, 2005. [10] d. johnson, chebyshev polynomials in the spectral tau method and applications to eigenvalue problems, nasa contractor report 198451, 1996. [11] s. talukdar, a study of some steady and unsteady mhd flow problems with heat and mass transfer, ph.d. thesis, gauhati university, 2013. [12] d.y. tzou, instability of nanofluids in natural convection, j. heat transfer, 130 (2008), 072401. [13] b.-f. wang, d.-j. ma, z.-w. guo, d.-j. sun, linear instability analysis of rayleigh–bénard convection in a cylinder with traveling magnetic field, j. crystal growth, 400 (2014), 49–53. [14] d. yadav, j. wang, r. bhargava, j. lee, h.h. cho, numerical investigation of the effect of magnetic field on the onset of nanofluid convection, appl. therm. eng. 103 (2016), 1441–1449. 1. introduction 2. linear stability analysis 3. normal mode analysis 4. spectral chebyshev tau method 5. results and conclusion references international journal of analysis and applications volume 18, number 5 (2020), 849-858 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-849 on some subclasses of strongly starlike analytic functions el moctar ould beiba∗ el moctar ould beiba, department of mathematics and computer sciences, faculty of sciences and techniques, university of nouakchott al aasriya, p.o. box 5026, nouakchott, mauritania ∗corresponding author: elbeiba@yahoo.fr abstract. the aim of the present article is to investigate a family of univalent analytic functions on the unit disc d defined for m ≥ 1 by < (zf′(z) f(z) ) > 0, ∣∣∣∣ ( zf′(z) f(z) )2 −m ∣∣∣∣ < m, z ∈ d. some proprieties, radius of convexity and coefficient bounds are obtained for classes in this family. 1. introduction let a be the set of analytic function on the unit disc d with the normalization f(0) = f′(0) − 1 = 0. f ∈a if f is of the form (1.1) f(z) = z + +∞∑ n=2 anz n, z ∈ d. s denotes the subclass of a of univalent functions. a function f ∈s is said to be strongly starlike of order α, 0 < α ≤ 1, if it satisfies the condition ∣∣argzf′(z) f(z) ∣∣ < απ 2 , ∀z ∈ d. received april 12th, 2020; accepted may 8th, 2020; published july 28th, 2020. 2010 mathematics subject classification. 30c45. key words and phrases. strongly starlike functions; subordination; radius of convexity; coefficient bounds. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 849 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-849 int. j. anal. appl. 18 (5) (2020) 850 this class is denoted by ss∗(α) and was first introduced by d. a. brannan and w. e. kirwan [1] and independently by j. stankiewicz [9]. ss∗(1) is the well known class s∗ of starlike functions. recall that a function f ∈s belongs to s∗ if the image of d under f is a starlike set with respect to the origin or, equivalently, if < (zf′(z) f(z) ) > 0, z ∈ d. a function f ∈s belongs to ss∗(α) if the image of d under zf ′(z) f(z) lies in the angular sector ωα = { z ∈ c, ∣∣argz∣∣ < απ 2 } . let b denotes the set of schwarz functions, i.e. ω ∈ b if and only ω is analytic in d, ω(0) = 0 and∣∣ω(z)∣∣ < 1 for z ∈ d. given two functions f and g analytic in d, we say that f is subordinate to g and we write f ≺ g if there exists ω ∈b such that f = g ◦ω in d. if g is univalent on d, f ≺ g is equivalent to f(0) = g(0) and f(d) ⊂ g(d). we obtain from the schwarz lemma that if f ≺ g then ∣∣f′(0)∣∣ ≤ ∣∣g′(0)∣∣. as a consequence of this statement, we have (1.2) f, g ∈a , f(z) z ≺ g(z) z =⇒ ∣∣a2∣∣ ≤ ∣∣b2∣∣, where a2 and b2 are respectively the second coefficients of f and g. w. janowski [2] investigated the subclass s∗(m) = { f ∈s, zf′(z) f(z) ∈dm,∀z ∈ d } , where dm = { w ∈ c, ∣∣w −m∣∣ < m}, m ≥ 1 j. sókol and j. stankiewicz [8] introduced a subclass of ss∗( 1 2 ), namely, the class s∗l defined by s∗l = { f ∈s, zf′(z) f(z) ∈l1,∀z ∈ d } , where l1 = { w ∈ c,<w > 0, ∣∣w2 − 1∣∣ < 1}. l1 is the interior of the right half of the bernoulli’s lemniscate ∣∣w2 − 1∣∣ = 1. in the present paper we are interested to the family of subclass of s (1.3) s∗l(m) = { f ∈s, zf′(z) f(z) ∈lm,∀z ∈ d } , m ≥ 1, int. j. anal. appl. 18 (5) (2020) 851 where (1.4) lm = { w ∈ c,<w > 0, ∣∣w2 −m∣∣ < m}. is the interior of the right half of the cassini’s oval ∣∣w2 −m∣∣ = m. for the particular case m = 1, s∗l(1) stands for the class s∗l introduced by j. sókol and j. stankiewicz [8]. since lm ⊂ ω( 1 2 ), all functions in s∗l(m) are strongly starlike of order 1 2 . note that all classes above correspond to particular cases of the classes of s∗(ϕ) introduced by w. ma and d. minda [3], s∗(ϕ) = { f ∈a, zf′(z) f(z) ≺ ϕ } . where ϕ is analytic univalent function with real positive part in the unit disc d, ϕ ( d ) is symmetric with respect to the real axis and starlike with respect to ϕ(0) = 1 and ϕ′(0) > 0. let m = 1 − 1 m and ϕm be the function ϕm(z) = √ 1 + z 1 −mz , z ∈ d where the branch of the square root is chosen so that ϕm(0) = 1. we have (1.5) s∗l(m) = s ∗(ϕm) = { f ∈a, zf′(z) f(z) ≺ ϕm } . observe that s∗l corresponds to m = 0 so that s ∗ l = s ∗( √ 1 + z). 2. some properties of the class s∗l(m) let p the class of analytic functions p in d with p(0) = 1 and <p(z) > 0 in d. for m ≥ 1, let pl(m) = { p ∈ p, ∣∣p2(z) −m∣∣ < m, z ∈ d}. it is easy to see that pl(m1) ⊂ pl(m2) for m1 ≤ m2. remark 2.1. a function f ∈a belongs to s∗l(m) if and only if there exists p ∈ pl(m) such that zf′(z) f(z) = p(z), z ∈ d. theorem 2.1. a function f belongs to s∗l(m) if and only if there exists p ∈ pl(m) such that (2.1) f(z) = z exp ∫ z 0 p(ξ) − 1 ξ dξ. proof. (2.1) is an immediate consequence of the remark 2.1 � int. j. anal. appl. 18 (5) (2020) 852 let fm ∈a be the unique function such that (2.2) zf ′ m(z) fm(z) = ϕm(z), z ∈ d with m = 1 − 1 m . fm belongs to s∗l(m) and we have (2.3) fm(z) = z exp ∫ z 0 ϕm(ξ) − 1 ξ dξ. evaluating the integral in (2.3), we get (2.4) fm(z) = 4z exp ∫ϕm(z) 1 hm(t)dt( ϕm(z) + 1 )2 , z ∈ d, where hm(t) = 2mt + 2 mt2 + 1 , m = 1 − 1 m for m = 1, h0 is the constant function h(t) = 2 and we have f0(z) = 4z exp ( 2 √ 1 + z − 2 ) (√ 1 + z + 1 )2 for z ∈ d. f0 is extremal function for problems in the class s∗l (see [8]). it is easy to see that (2.5) fm(z) = z + m + 1 2 z2 + (m + 1)(5m + 1) 16 z3 + (m + 1)(21m2 + 6m + 1) 96 z4 + . . . we need the following result by st. ruscheweyh [5] lemma 2.1. [ [5], theorem 1] let g be a convex conformal mapping of d, g(0) = 1, and let f(z) = z exp ∫ z 0 g(ξ) − 1 ξ dξ. let f ∈a. then we have zf′(z) f(z) ≺ g if and only if for all ∣∣s∣∣ ≤ 1, ∣∣t∣∣ ≤ 1 tf(sz) sf(tz) ≺ tf(sz) sf(tz) . theorem 2.2. if f belongs to s∗l(m) then (2.6) f(z) z ≺ fm(z) z . proof. from (1.5), we obtain by applying lemma 2.1 to the convex univalent function g = ϕm, tf(z) f(tz) ≺ tfm(z) fm(tz) . letting t −→ 0, we obtain the desired conclusion. � int. j. anal. appl. 18 (5) (2020) 853 corollary 2.1. let f belongs to s∗l(m) and |z| = r < 1, then (2.7) −fm(−r) ≤ |f(z)| ≤ fm(r); (2.8) f ′ m(−r) ≤ |f ′ (z)| ≤ f ′ m(r). proof. (2.7) follows from (2.6). now if m ≥ 1 we have 0 ≤ m < 1. thus for 0 ≤ r < 1 (2.9) min |z|=r |ϕm(z)| = ϕm(−r), max |z|=r |ϕm(z)| = ϕm(r) from (2.6) and (2.9) we get (2.8) by applying theorem 2 ( [3], p. 162). � 3. radius of convexity for the class s∗l(m) in the sequel m = 1 − 1 m . for m ≥ 1, let p(m) be the family of analytic functions p in d satisfying (3.1) p(0) = 1, |p(z) − m| < m, for z ∈ d. we have (3.2) f ∈s∗l(m) ⇐⇒ ∃p ∈p(m) / zf′(z) f(z) = √ p. we need the two following lemmas by janowski [2]: lemma 3.1. [ [2] , theorem 1] for every p(z) ∈p(m) and |z| = r, 0 < r < 1, we have (3.3) inf p∈p(m) <p(z) = 1 −r 1 + mr . the infimum is attained by (3.4) p(z) = 1 − �z 1 + �mz , |�| = 1. lemma 3.2. (theorem 2, [2]) for every p(z) ∈p(m) and |z| = r, 0 < r < 1, we have (3.5) inf p∈p(m) < zp ′ (z) p(z) = − ( 1 + m ) r( 1 −r )( 1 + mr ). the infimum is attained by (3.6) p(z) = 1 − �z 1 + �mz , |�| = 1. theorem 3.1. the radius of convexity of the class s∗l(m) is is the unique root in (0, 1) of the equation (3.7) 4 ( 1 + mr )( 1 −r )3 −(1 + m)2r2 = 0. int. j. anal. appl. 18 (5) (2020) 854 proof. let f ∈s∗l(m). from (3.2), there exists p ∈p(m) such that (3.8) zf ′ (z) f(z) = √ p(z), z ∈ d. (3.8) can be written zf ′ (z) = f(z) √ p(z) which gives 1 + zf ′′ (z) f ′ (z) = z ( zf ′)′ (z) zf ′ (z) = √ p(z) + 1 2 zp ′ (z) p(z) . this yields for |z| = r, 0 < r <, (3.9) < ( 1 + zf ′′ (z) f ′ (z) ) ≥ inf p∈p(m) < √ p(z) + 1 2 inf p∈p(m) < zp ′ (z) p(z) . replacing (3.3) and(3.5 in (3.9), we obtain (3.10) < ( 1 + zf ′′ (z) f ′ (z) ) ≥ √ 1 −r 1 + mr − 1 2 ( 1 + m ) r( 1 −r )( 1 + mr ) let h m be defined by h m = √ 1 −r 1 + mr − 1 2 ( 1 + m ) r( 1 −r )( 1 + mr ). h m is decreasing in the interval (0, 1) , h m (0) = 1 and the limit of h m in 1− is −∞. let r m−1 be the unique solution of h m (r) = 0 in (0, 1), then f is convex on the disc |z| < r m−1 . on the other hand, 1 + zf ′′ m(z) f ′ m(z) = √ 1 + z 1 −mz + 1 2 ( 1 + m ) z( 1 −mz )( 1 + z ) vanishes in z = −r m−1 . thus rm−1 is the best value. to finish, we observe that the equation h m (r) = 0 is equivalent in the interval (0, 1) to the equation 4 ( 1 + mr )( 1 −r )3 −(1 + m)2r2 = 0. � remark 3.1. as a consequence of theorem 3.1 applying for m = 1, we find theorem 4 [8] which gives r0 the radius of convexity of the class s∗l. r0 = 1 12 ( 11 + 3 √√ 44928 − 181 − 3 √√ 44928 + 181 ) ≈ 0.5679591 remark 3.2. as observed above, s∗l(m) increases with m. therefore rm−1 decreases when m increases. let r∞ = lim m→+∞ r m−1. substituting in (3.7), we obtain int. j. anal. appl. 18 (5) (2020) 855 ( 1 + r∞ )( 1 −r∞ )3 −r2∞ = 0. solving this equation in (0, 1), we get r∞ = 1 2 ( 1 − √ 2 + √√ 8 − 1 ) ≈ 0.46899. we have r∞ ≤ rm−1 ≤ r0. 4. coefficient bounds for s∗l(m) theorem 4.1. let f(z) = ∑∞ n=0 anz n be a function in s∗l(m) . then for 1 ≤ m ≤ 2 we have (4.1) ∞∑ n=2 ( (1 −m)n2 − 2 ) |an|2 ≤ 1 + m and for m > 2 we have (4.2) ∑ n≥ √ 2 1−m ( (1 −m)n2 − 2 ) |an|2 ≤ 1 + m− ∑ 2≤k< √ 2 1−m ( (1 −m)k2 − 2 ) |ak|2. with m = m−1 m . proof. if f ∈s∗l(m) there exists ω ∈b such that (4.3) ( 1 −mω(z) )( zf ′ (z) )2 −f(z)2 = ω(z)f(z)2, z ∈ d. for 0 < r < 1 we have 2π ∞∑ n=1 |an|2r2 = ∫ 2π 0 |f ( reiθ ) |2dθ ≥ ∫ 2π 0 |ω ( reiθ ) ||f ( reiθ ) |2dθ(4.4) replacing (4.3) in the right side of (4.5) we obtain 2π ∞∑ n=1 |an|2r2 ≥ ∫ 2π 0 ∣∣(1 −mω(reiθ))(reiθf′(reiθ))2 −f(reiθ)2∣∣dθ ≥ ∫ 2π 0 ∣∣(1 −mω(reiθ))(reiθf′(reiθ))2∣∣dθ −∫ 2π 0 ∣∣f(reiθ)2∣∣dθ ≥ (1 −m) ∫ 2π 0 ∣∣(reiθf′(reiθ))2∣∣dθ −∫ 2π 0 ∣∣f(reiθ)2∣∣dθ = 2π ∞∑ n=1 (1 −m)n2|an|2r2 − 2π ∞∑ n=1 |an|2r2. int. j. anal. appl. 18 (5) (2020) 856 thus 2 ∞∑ n=1 |an|2r2 ≥ ∞∑ n=1 (1 −m)n2|an|2r2. if we let r → 1−, we obtain from le last inequality 2 ∞∑ n=1 |an|2 ≥ ∞∑ n=1 (1 −m)n2|an|2 which gives, (4.5) 1 + m ≥ ∞∑ n=2 ( (1 −m)n2 − 2 ) |an|2. since (1 −m)n2 − 2 ≥ 0 for all n ≥ 2 if and only if 1 ≤ m ≤ 2 then (4.5) yields (4.1) and (4.2) according to the case 1 ≤ m ≤ 2 or m > 2. � the following corollary is an immediate consequence of (4.2). corollary 4.1. let f(z) = ∑∞ n=0 anz n be a function in s∗l(m) .then for 1 ≤ m ≤ 2 we have (4.6) |an| ≤ √ 1 + m (1 −m)n2 − 2 , for n ≥ 2 and for m > 2 we have (4.7) |an| ≤ √√√√1 + m−∑2≤k<√ 21−m ((1 −m)k2 − 2)|ak|2( 1 −m ) n2 − 2 ; for n ≥ √ 2 1 −m . with m = m−1 m . remark 4.1. for m = 1, (4.1) and (4.6) give respectivly theorem 1 and corollary 1 [6]. theorem 4.2. let f(z) = ∑∞ n=0 anz n be a function in s∗l(m) .then (i) |a2| ≤ m+12 , for 0 ≤ m ≤ 1; (ii) |a3| ≤ m+14 , for 0 ≤ m ≤ 3 5 ; (iii) |a4| ≤ m+16 , for 0 ≤ m ≤ √ 3−1 7 . this estimations are sharp. proof. if f ∈s∗l(m) there exists ω(z) = ∑∞ n=1 cnz n ∈b such that (4.8) ( zf ′ (z) )2 −f(z)2 = ω(z)(m(zf′(z))2 + f(z)2), z ∈ d. let f(z)2 = ∑∞ n=2 anz n, ( zf ′ (z) )2 = ∑∞ n=2 bnz n. (4.8) becomes (4.9) ∞∑ n=2 ( bn −an ) zn = ( ∞∑ n=2 ( mbn + an ) zn )( ∞∑ n=1 cnz n ) int. j. anal. appl. 18 (5) (2020) 857 equating coefficients for n = 2, n = 3 in both sides of (4.9), we obtain (sm)   b3 −a3 = ( mb2 + a2 ) c1 b4 −a4 = ( mb2 + a2)c2 + ( mb3 + a3 ) c1 b5 −a5 = ( mb2 + a2)c3 + ( mb3 + a3 ) c2 + ( mb4 + a4 ) c1 a little calculation yields a2 = a1 = 1, a3 = 2a2, a4 = 2a3 + a 2 2, a5 = 2a4 + 2a2a3 and b2 = a1 = 1, b3 = 4a2, b4 = 6a3 + 4a 2 2, b5 = 8a4 + 12a2a3. replacing in (sm), we obtain  (1) 2a2 = ( m + 1 ) c1 (2) 4a3 + 3a 2 2 = ( m + 1)c2 + ( 4m + 2)a2c1 (3) 6a4 + 10a2a3 = ( m + 1 ) c3 + ( 2m + 1 )( m + 1 ) c1c2 + ( (6m + 2)a3 + (4m + 1)a 2 2 ) c1 since |c1| ≤ 1 then (1) implies that |a2| ≤ 1+m2 . this proves the assertion (i). on the other hand we have from (1) and (2) a3 = 1 + m 4 c2 + (5m + 1)(m + 1) 16 c21. thus |a3| ≤ 1 + m 4 ( |c2| + 5m + 1 4 |c1| ) . it is well known that |c2| ≤ 1 −|c1|2. therefore we obtain |a3| ≤ 1 + m 4 ( 1 −|c1|2 + 5m + 1 4 |c1| ) = 1 + m 4 ( 1 + 5m− 3 4 |c1| ) .(4.10) since 5m− 3 ≤ 0 if and only if m ≤ 3 5 then (4.10) yields the assertion (ii). replacing the values of a2 and a3 in the equation (3), we obtain a4 = ( m + 1 ) 6 c3 + ( m + 1 )( 9m + 1 ) 24 c1c2 + ( m + 1 )( 21m2 + 6m + 1 ) 96 c31 = m + 1 6 ( c3 + 9m + 1 4 c1c2 + 21m2 + 6m + 1 16 c31 ) .(4.11) let µ = 9m+1 4 and ν = 21m 2+6m+1 16 . under the assumption 0 ≤ m ≤ √ 3−1 7 , we have (µ,ν) ∈ d1 (see [4], p. 127). therefore by lemma 2 [4] we obtain∣∣∣∣c3 + 9m + 14 c1c2 + 21m 2 + 6m + 1 16 c31 ∣∣∣∣ ≤ 1 int. j. anal. appl. 18 (5) (2020) 858 which yields from (4.11) the assertion (iii). the sharpness of (i) is given by the function fm. if we take in (4.8) ω(z) = z 2 and ω(z) = z3 successively, we obtain two functions in s∗l(m): f1,m(z) = z + m + 1 4 z3 + . . . and f2,m(z) = z + m + 1 6 z4 + . . . which give respectively the sharpness of estimations (ii) and (iii). � remark 4.2. the estimation (i) can be obtained directly from (2.6). remark 4.3. if we take m = 0 in theorem 4.2, we obtain as particular case theorem 2 [6]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.a. brannan, w.e. kirwan, on some classes of bounded univalent functions, journal of the london mathematical society. s2-1 (1969), 431–443. [2] w. janowski, extremal problems for a family of functions with positive real part and for some related families, ann. polon. math. 23 (1970), 159-177. [3] wancang ma, david minda, a unified treatment of some classes of univalent functions, proceeding of the international conference on complex analysis at the nankai institute of mathematics, 1992, 157-169. [4] dimitri v. prokhorov, jan szynal, inverse coefficients for (α,β)-convex functions, ann. univ. mariae curie-sklodowska, xxxv (15) (1981), 125-143. [5] s. ruscheweyh, a subordination theorem for φ-like functions, j. lond. math. soc. s2-13 (1976), 275–280. [6] janusz sókol, coefficient estimates in a class of strongly starlike functions, kyungpook math. j. 49 (2009), 349-353. [7] janusz sókol, on some subclass of strongly starlike functions, demonstr. math. xxxi (1) (1988), 81-86. [8] j. sókol, j. stankiewicz, radius of convexity of some subclass of strongly starlike functions, folia sci. univ. tech. resovienis, math. 19 (1996), 101-105. [9] j. stankiewicz, quelques problèmes extrémaux dans les classes des fonctions α-angulairement étoilées, ann. univ. mariae curie-sklodowska sect. a, 20 (1966), 59-75. 1. introduction 2. some properties of the class s*l(m) 3. radius of convexity for the class s*l(m) 4. coefficient bounds for s*l(m) references international journal of analysis and applications volume 18, number 6 (2020), 989-997 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-989 extension of vikor method for mcdm under bipolar fuzzy set bashayer alsolame∗ and noura omair alshehri department of mathematics, faculty of science, university of jeddah ∗corresponding author: baalsolame@uj.edu.sa abstract. in this paper, we extend the vikor method (serbian name: vlsekriterijumska optimizacija i kompromisno resenje, means multi-criteria optimization and compromise solution) to solve multi-criteria decision making (mcdm) with bipolar fuzzy environment. firstly, the bipolar fuzzy set concept is described. then, proposed the vikor strategy in bipolar set environment to handle mcdm problems. finally, two numerical examples illustrate an application of bf-vikor method, and analysis the results of different values of the decision making weights of criteria on ranking order of the alternatives. 1. introduction multi-criteria decision making (mcdm) has seen an incredible amount of use over last several decades. its role in different applications areas has increased significantly, especially as new methods develop and as old methods improve. in classical mcdm methods, the goal is to designate a preferred alternative, classify alternatives in a small number of categories and rank alternatives in a subjective preference order. however, most of the real world decisions are made in an environment in which goals and constraints cannot be precisely expressed due to their complexity. the fuzzy sets theory was first proposed by zadeh in 1965 ( [1], [2]). afterward, bellman and zadeh described a decision-making method in a fuzzy environment [3]. multi-criteria optimization and compromise solution (vikor in serbian) method, developed by opricovic (1998) [4], opricovic and tzeng (2004) [5], is one of the most outranking mcdm method. the first paper that proposes to use fuzzy inputs with vikor method was published in 2002 [6]. later, opricovic [7] proposes a received august 15th, 2020; accepted september 7th, 2020; published september 29th, 2020. 2010 mathematics subject classification. 03e72. key words and phrases. bipolar fuzzy set; vikor method. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 989 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-989 int. j. anal. appl. 18 (6) (2020) 990 fuzzy extension of vikor to find a fuzzy compromise solution.wang and chang [8] introduced fuzzy vikor for solving multi-criteria group decision-making problem. since that, many researchers have been dealing with decision-making problems by applying vikor method in fuzzy environment ( [9], [10], [11], [12], [13]), interval-valued intuitionistic fuzzy environment [14], intuitionistic fuzzy vikor [15] and hesitant fuzzy vikor [16]. in 1994, zhang [17] proposed the concept of bipolar fuzzy sets, which is a generalization of fuzzy set allowing the membership degree having two values. one locates in the interval [0,1] represents the satisfaction degree of a certain property associated to the fuzzy set and the other locates in the interval [-1,0] and represents the satisfaction degree of a counter-property to the concerned fuzzy set. bipolarity is important to distinguish between positive information, which represents what is guaranteed to be possible, for example what has already been observed or experienced and negative information, which represents what is impossible or forbidden, or surely false. thus providing an efficient approach to solving a widely larger number of complex decision making problems. in recent years, many decision-making problems have been applied in bipolar fuzzy environment. in this paper, we present bipolar fuzzy vikor (bf-vikor) method solving mcdm problems that are equipped with bipolar fuzzy information. we illustrate our proposed method with numerical examples. 2. preliminaries in this section, some preliminaries from the fuzzy set theory and bipolar fuzzy sets are induced. definition 2.1. a fuzzy set of the universe x is a mathematical object a described by its characteristic function (membership function) µa : x → [0, 1] defined as [1]: (2.1) a = {(x,µa(x)) | x ∈ x} definition 2.2. a bipolar fuzzy set a on the universe x is defined as [17]: (2.2) a = {(x,µa (x) ,νa (x)) | x ∈ x} where µa : x → [0, 1] is a positive membership degree denotes the satisfaction degree of an element to the property corresponding to a, and νa : x → [−1, 0] is the negative membership degree denotes the satisfaction degree of an element to some implicit counter-property corresponding to a. 3. vikor method the vikor method focuses on ranking and selecting from a set of alternatives with conflicting criteria, and determines compromise solution based on particular measure. assuming that each alternative is int. j. anal. appl. 18 (6) (2020) 991 evaluated according to each criterion function, the compromise ranking can be presented by comparing the degree of closeness to the ideal alternative. the compromise solution, whose foundation was established by yu [18] and zeleny [19], is a feasible solution, which is the closest to the ideal, and here “compromise” means an agreement established by mutual concessions. the multi-criteria measure for compromise ranking is developed from the lp− metric used as an aggregating function in a compromise programming method defined as: (3.1) lp,i =   n∑ j=1 [ wj ( f∗j −fij )( f∗j −f − j )]p   1 p , 1 ≤ p ≤∞ where l1,i is defined as the maximum group utility, and l∞,i is defined as the minimum individual regret of the opponent. for a mcdm problem involving m alternatives and n criteria, whereby the performances of the alternatives are expressed by using bipolar fuzzy set, let ψ = {ψ1,ψ2, . . . ,ψm} = {ψi : i = 1, 2, . . . ,m} and c = {c1,c2, . . . ,cn} = {cj : j = 1, 2, . . . ,n} be the sets of alternatives and criteria that are determined by a decision maker, respectively. formally, a bipolar fuzzy mcdm problem can be express in matrices format as follows: (3.2) d =   (µ11,ν11) (µ12,ν12) . . . (µ1n,ν1n) (µ21,ν21) (µ22,ν22) . . . (µ2n,ν2n) . . . . (µm1,νm1) (µm2,νm2) . . . (µmn,νmn)   m×n the ratings of any alternative ψi ∈ ψ for each criterion cj are given by a bipolar fuzzy set fi = {(ψi,fij) | j = 1, 2, . . . ,n} where fij = (µij,νij) represent respectively, the satisfaction degree (µij ∈ [0, 1]) and the dissatisfaction degree (νij ∈ [−1, 0]) that are determined for ψi with respect to cj. now, determine the best f∗j and the worst f-j value of all criterion functions, j = 1, 2, . . . ,n as: (3.3) f∗j = { (max µij, min νij) , for the benefit criterion (min µij, max νij) , for the cost criterion } , i = 1, 2, . . . ,m. (3.4) f−j = { (min µij, max νij) , for the benefit criterion (max µij, min νij) , for the cost criterion } , i = 1, 2, . . . ,m in the vikor development methodology si (as l1,i in equation 3.1) and ri (as l∞,i in equation 3.1) are used to formulate ranking measure [4], [5]. in light of the distance measure for each alternatives functions int. j. anal. appl. 18 (6) (2020) 992 fij = (µij,νij) and the best f ∗ j and the worst f j value of all criterion functions. (3.5) si = n∑ j=1 wj d ( f∗j ,fij ) d ( f∗j ,f − j ), (3.6) ri = max j [ wj d ( f∗j ,fij ) d ( f∗j ,f − j )] where wj are the weights of criteria, expressing the decision-makers preference of the criteria, such that wj ∈ [0, 1] and n∑ j=1 wj = 1. furthermore, the relative closeness degree of each alternative ψi (i = 1, 2, . . . ,m) with respect to si and ri is given as: (3.7) qi = v (si −s∗) (s− −s∗) + (1 −v) (ri −r∗) (r− −r∗) where, s∗ = min i si , s − = max i si r∗ = min i ri , r − = max i ri v and (1 −v) are the weights for the strategy of maximum group utility and individual regret, respectively. usually, the value of v can be assumed to be ( v = 0.5 ).the qirepresent the distance of ψi alternative from the best solution ”compromise solution”. so the alternative, which has the minimum value in qi would be the compromise solution if the following two conditions are satisfied: c1) acceptable advantage: (3.8) q(ψ(2)) −q(ψ(1)) ≥ 1 m− 1 where, ψ(1) and ψ(2) are the top two alternatives in qi [such that qi sort in ascending order], m is the number of the alternatives. c2) acceptable stability: the compromise solution should be the best rank by si and/or ri. if one of the conditions is not satisfied, then a set of compromise solutions is proposed, which consists of: • ψ(1) and ψ(2) if only the condition c2 is not satisfied, or • ψ(1),ψ(2), . . . ,ψ(m) if the condition c1 is not satisfied (equation 3.8); ψ(m) is determined by the relation: q(ψ(m)) −q(ψ(1)) < 1 m− 1 ; maximum m. int. j. anal. appl. 18 (6) (2020) 993 4. vikor algorithm according to the above discussion, the procedures of bf-vikor method can be summarized in figure 1. figure 1. the flow chart of bf-vikor algorithm 5. numerical examples 5.1. example 1. assume that we have a decision maker who is confused in choosing the a new location to start his sales business. after the initial consideration of the available alternatives, four alternatives has been identified as suitable. suppose that he is evaluating suitable alternatives based on the following criteria: c1 :rental space quality; c2 :rental space adequacy; c3 :location quality; c4 :location distance from the city center, and c5 :rental price. so, these locations represent the alternatives {ψ1,ψ2,ψ3,ψ4} and the mentioned features represent the criteria {c1,c2,c3,c4,c5} in our mcdm problem. ratings of the alternatives, in table 1, and weights of the criteria are given by the decision maker in matrices format with bipolar fuzzy and fuzzy values, respectively, as follows: w = [ 0.18 0.24 0.21 0.16 0.21 ]t clearly, the given weights satisfy the normalized condition. from equation 3.3, equation 3.4 the best and worst values of all criterion ratings are determined as follows in table 2. further, the equations 3.5 3.7 calculated the values of si, ri and qi in table 3 were (v = 0.5) and rank the alternatives by ranking si,ri and qi in decreasing order as shown in table 4. thus, the compromise solution is ψ4 and ψ1. int. j. anal. appl. 18 (6) (2020) 994 table 1. ratings of the alternatives c1 c2 c3 c4 c5 ψ1 (0.67,−0.16) (0.79,−0.32) (1, 0) (1, 0) (0.81,−0.10) ψ2 (0.47,−0.18) (0.43,−0.42) (1,−.26) (1, 0) (0.77,−0.10) ψ3 (0.64,−0.10) (0.61,−0.16) (0.49, 0) (0.41,−0.10) (0.70,−0.13) ψ4 (0.81,−0.10) (1,−0.15) (0.53, 0) (0.53,−0.10) (0.60,−0.10) table 2. the best and worst value of all criterion functions f∗j f − j c1 (0.81,−0.18) (0.47,−0.10) c2 (1,−0.42) (0.43,−0.15) c3 (1,−0.26) (0.49, 0) c4 (1,−0.10) (0.41, 0) c5 (0.60,−0.10) (0.80,−0.13) table 3. the values of si, ri and qi for all alternatives ψ1 ψ2 ψ3 ψ4 si 0.491 0.595 0.751 0.467 ri 0.208 0.217 0.21 0.197 qi 0.311 0.726 0.826 0 table 4. final ranking of alternatives ψ1 ψ2 ψ3 ψ4 ranking si 0.491 0.595 0.751 0.467 ψ4,ψ1,ψ2,ψ3 ri 0.208 0.217 0.21 0.197 ψ4,ψ1,ψ3,ψ2 qi 0.311 0.726 0.826 0 ψ4,ψ1,ψ2,ψ3 5.2. example 2. assume that we have a decision maker who is confused in choosing the best airline. after the initial consideration of the available alternatives, five alternatives has been identified as suitable. suppose that he is evaluating suitable alternatives based on the following criteria: c1 :empathy, represents how airline deal with the customer complaints and provide thoughtful services; c2 :assurance, represents the certainty that airline provides for customers; c3 :tangibility, means the physical service presentation such as quality of the food, and c4 :cost, represents the tickets and services prices. int. j. anal. appl. 18 (6) (2020) 995 so, these airlines represent the alternatives {ψ1,ψ2,ψ3,ψ4,ψ5} and the mentioned features represent the criteria {c1,c2,c3,c4} in our mcdm problem. ratings of the alternatives, in table 5, and weights of the criteria are given by the decision maker in matrices format with bipolar fuzzy and fuzzy values, respectively, as follows: w = [ 0.25 0.3 0.25 0.2 ]t clearly, the given weights satisfy the normalized condition. using the equations 3.5 3.7 the values of si, ri and qi are calculated as in table 6. table 5. rating of the alternatives c1 c2 c3 c4 ψ1 (0.5,−0.25) (0.8,−0.7) (0.3,−0.1) (0.6,−0.6) ψ2 (0.2,−0.8) (0.9,−0.4) (0.6,−0.3) (0.55,−0.5) ψ3 (0.33,−0.25) (0.75,−0.4) (0.25,−0.7) (0.3,−0.1) ψ4 (0.65,−0.6) (0.3,−0.75) (0.8,−0.35) (0.65,−0.7) ψ5 (1,−0.5) (0.4,−0.35) (0.2,−0.6) (0.25,−0.65) table 6. the values ofsi,ri and qi for all alternatives ψ1 ψ2 ψ3 ψ4 ψ5 ranking compromise solutions si 0.637 0.622 0.558 0.657 0.675 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 ri 0.23 0.206 0.223 0.25 0.266 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 qi (v = 0.5) 0.538 0.274 0.143 0.781 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 after rank the alternatives by sorting the values si, ri and qi in decreasing order. the results are three ranking lists, which is depicted in table 6. so, the compromise solution are ψ3 and ψ2. the parameter v in vikor technique normally considered as (v = 0.5). therefore, by changing value of v in the interval [0, 1] is performed for the obtained results. the ranking for five alternatives under different v values are illustrated in table 7. as can be seen, when v is changed, there are some deviations in ranking of alternatives. ψ3 is the best ranked alternative for v ≥ 0.7; also, ψ2 has the best rank for v = 0. moreover, ψ5 is the worst ranked alternative for different values of v. 6. conclusion in mcdm methods the alternatives are compared against each other based on how they perform relative to each criterion. one the other hand, vikor method require comparison of the criteria to determine the relative importance of each criterion. the alternative with the highest rank is selected as the best compromise int. j. anal. appl. 18 (6) (2020) 996 table 7. ranking orders of alternatives under different v values. ψ1 ψ2 ψ3 ψ4 ψ5 ranking compromise solutions si 0.637 0.622 0.558 0.657 0.675 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 ri 0.23 0.206 0.223 0.25 0.266 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 qi(v) 0 0.399 0 0.285 0.722 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2 0.1 0.427 0.055 0.257 0.734 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.2 0.455 0.109 0.228 0.746 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.3 0.482 0.164 0.2 0.758 1 ψ2,ψ3,ψ1,ψ4,ψ5 ψ2,ψ3 0.4 0.51 0.219 0.171 0.769 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.5 0.538 0.274 0.143 0.781 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.6 0.566 0.328 0.114 0.793 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3,ψ2 0.7 0.594 0.383 0.086 0.805 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 0.8 0.621 0.438 0.057 0.816 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 0.9 0.649 0.493 0.029 0.828 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 1 0.677 0.547 0 0.84 1 ψ3,ψ2,ψ1,ψ4,ψ5 ψ3 solution. in this paper, we proposed bf-vikor method since the vikor method is an effective mcdm method to reach a compromise solution, particularly in a situation where the decision maker is not able to express his preference at the beginning. furthermore, bfss are an effective tool to depict fuzziness and bipolarity in assessment information. the obtained solution is compromised by a maximum group utility of the majority, and a minimum of the individual regret. in future study, we may apply vikor method to problems that are under two fuzzy concept(such that bipolar hesitant fuzzy). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] l. zadeh, fuzzy sets, information and control. 8 (3) (1965), 338-353. [2] l. zadeh, the concept of a linguistic variable and its application to approximate reasoning, inform. sci. 8 (3) (1975), 199-249. [3] r. e. bellman, l. a. zadeh, decision making in a fuzzy environment, manage. sci. 17 (4) (1970), 141-164. [4] s. opricovic, g.h. tzeng, compromise solution by mcdm methods: a comparative analysis of vikor and topsis, eur. j. oper. res. 156 (2) (2004), 445-455. int. j. anal. appl. 18 (6) (2020) 997 [5] s. opricovic, g.h. tzeng, extended vikor method in comparison with outranking methods, eur. j. oper. res. 178 (2) (2004), 514-529. [6] s. opricovic, g.h. tzeng, multi-criteria planning of post-earthquake sustainable reconstruction, comput.-aided civil infr. eng. 17 (3) (2002), 211-220. [7] s. opricovic, a fuzzy compromise solution for multicriteria problems, int. j. uncertain. fuzziness knowl.-based syst. 15 (3) (2007), 363-380. [8] t.c. wang, t.h. chang, fuzzy vikor as a resolution for multi-criteria group decision-making, in the 11th international conference on industrial engineering and engineering management. 2005, 352-356. [9] a. shemshadi, h. shirazi, m. toreihi, m.j. tarokh, a fuzzy vikor method for supplier selection based on entropy measure for objective weighting. expert syst. appl. 38 (10) (2011), 12160-12167. [10] h.-c. liu, l. liu, j. wu, material selection using an interval 2-tuple linguistic vikor method considering subjective and objective weights, mater. des. (1980-2015). 52 (2013), 158–167. [11] m.s. kuo, g.s. liang, a soft computing method of performance evaluation with mcdm based on interval-valued fuzzy numbers, appl. soft comput. 12 (1) (2012), 476-485. [12] a. sanayei, s. f. mousavi, a. yazdankhah, group decision making process for supplier selection with vikor under fuzzy environment, expert syst. appl. 37 (1) (2010), 24-30. [13] n. yalcin, a. bayrakdaroglu, c. kahraman, application of fuzzy multi-criteria decision making methods for financial performance evaluation of turkish manufacturing industries, expert syst. appl. 39 (1) (2012), 350-364. [14] x. y. zhao, s. tang, s. l. yang, k.d huang, extended vikor method based on cross-entropy for interval-valued intuitionistic fuzzy multiple criteria group decision making, j. intell. fuzzy syst. 25 (4) (2013), 1053-1066. [15] k. devi, extension of vikor method in intuitionistic fuzzy environment for robot selection, expert syst. appl. 38 (11) (2011), 14163-14168. [16] n. zhang, g.w wei, extension of vikor method for decision making problem based on hesitant fuzzy set, appl. math. model. 37 (7) (2013), 4938-4947. [17] w. r. zhang, yin-yang bipolar fuzzy sets, in 1998 ieee international conference on fuzzy systems proceedings. 1998, 835-840. [18] p.l. yu, a class of solutions for group decision problems. manage. sci. 19 (8) (1973), 936-946. [19] m. zeleny, multiple criteria decision making, mcgraw-hill, new york, 1982. 1. introduction 2. preliminaries 3. vikor method 4. vikor algorithm 5. numerical examples 5.1. example 1 5.2. example 2 6. conclusion references int. j. anal. appl. (2022), 20:60 better uniform approximation by new bivariate bernstein operators asha ram gairola1, suruchi maindola1, laxmi rathour2,∗, lakshmi narayan mishra3, vishnu narayan mishra4 1department of mathematics, doon university, dehradun 248 001 uttarakhand, india 2ward no-16, bhagatbandh, anuppur 484 224, madhya pradesh, india 3department of mathematics, school of advanced sciences, vellore institute of technology, vellore 632 014, tamil nadu, india 4department of mathematics, indira gandhi national tribal university, amarkantak, anuppur 484 887, madhya pradesh, india ∗corresponding author: laxmirathour817@gmail.com, rathourlaxmi562@gmail.com abstract. in this paper we introduce new bivariate bernstein type operators bm,in (f ;x,y), i = 1,2,3. the rates of approximation by these operators are calculated and it is shown that the errors are significantly smaller than those of ordinary bivariate bernstein operators for sufficiently smooth functions. 1. introduction in approximation theory the bernstein operators are the most studied linear positive operators. let, b[0,1] denote the class of bounded functions and f ∈ b[0,1], then the bernstein operator bn : b[0,1]→ c∞[0,1] of degree n with respect to f is defined by bnf (x)= n∑ k=0 pn,k(x)f ( k n ) where pn,k(x)=   ( n k ) xk(1−x)n−k if k =0,1, ...,n, 0 if k < 0ork > n. here c∞[0,1] is the class of infinitely differentiable functions on [0,1]. received: jun. 23, 2022. 2010 mathematics subject classification. 41a35, 41a10, 41a25. key words and phrases. linear operators; approximation by polynomials; rate of convergence. https://doi.org/10.28924/2291-8639-20-2022-60 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-60 2 int. j. anal. appl. (2022), 20:60 the operators bnf (x) exhibit interesting approximation properties. we mention a few of them in the following. (1) bnf (0)= f (0) and bnf (1)= f (1) i.e. bnf (x) has end point interpolation property, (2) the sequence bnf (x) converges uniformly to f (x) whenever f is continuous on [0,1], (3) bnf (x)≥ 0 for a positive function f (x), (4) the polynomials bnf (x) are convex for a convex function f (x), (5) the bernstein operators demonstrate simultaneous approximation property i.e. the derivative dr dtr ∣∣∣ t=x bnf (t) converges to the corresponding derivative f (r)(x) whenever the later derivative exists. (6) the polynomials bnf (x) provide a quadrature rule(see [1]) i.e. for f ∈ c[0,1],n ∈n ∫ 1 0 f (x)dx = lim n→∞ 1 n+1 n∑ k=0 f ( k n ) . for other interesting properties we refer to the monograph [2]. inspite of extensive research work of more than one century, new and interesting results have been continuously appearing on the properties and applications of these polynomials, see [2] and the references therein. although the operators bnf (x) are easy to work with and show remarkable properties, they lack rapid convergence. in fact, the identity bn(t2;x)= x2+ x(1−x) n implies that the order of approximation for the function, f (t)= t2 is not less than o(n−1) while the function is sufficiently smooth. in fact, it is known that the order of approximation is o(n−1/2) for the operators bn(f ;x) and can not be improved as such. this fact is indicated by the following result due to voronovskaya(see, [3]) theorem 1.1. let f (x) be bounded in [0,1] and suppose that the second derivative f ′′(x) exists at a certain point x of [0,1], then bnf (x)− f (x)= 1 n ( x(1−x) 2 f ′′(x) ) +o ( 1 n ) . for the details see [4], pp. 102. in order to improve the degree of approximation by the bernstein operators arab et al. [5] have introduced kernel decomposition method recently. this method, then have been applied by maria et al. in [6] and gairola et al. in [7] for other sequence of operators. the aim of this paper is to extend this technique to modify the bivariate bernstein operators and apply the modified operators for surface plotting. we show that the new bivariate operator approximate a surface better than the classical bivariate bernstein operator (1.1). let d denote the open square (0,1)× (0,1), and the closed square d := {(x,y) : 0≤ x ≤ 1,0≤ y ≤ 1}. int. j. anal. appl. (2022), 20:60 3 the class b(d) consists of the functions f (x,y) defined and bounded on d. then the bivariate bernstein operators of degrees n,m with respect to f ∈ b(d) are defined by (cf. [8]) bn,m(f ;x,y)= n∑ k=0 m∑ j=0 pn,k(x)pm,j(y)f ( k n , j m ) , (1.1) where pn,k(x) = ( n k ) xk(1− x)n−k,k ∈ 0,n and pm,j(y) = ( m j ) y j(1− y)m−j, j ∈ 0,m. we will need the class cs(d),s ∈ n0 in order to establish estimates for the smooth functions. cs(d) := { f ∈ c(d) : ∂rf ∂i∂r−i ∈ c(d), i ∈ 0, r, r ∈ 0,s } . here, n0 :=n∪{0}. approximation properties of multivariate bernstein operators were studied by a number of researchers(see [8][16]) and they have established noteworthy results on degree of approximation, convergence properties etc. it is worth to mention the relevant work in [17][22]. in 1953 p.l. butzer [23] discussed the simultaneous approximation properties of the operator 1.1. moreover, pop [24] obtained the rate of convergence in terms of voronovskaja type asymptotic theorem and modulus of continuity for bivariate bernstein operator bm,n(f ;x,y) on the square d. in 1960 d.d. stancu [25] defined another bivariate bernstein operators on the triangle and derived the rate of convergence in terms of complete modulus of continuity. subsequently, in 1992 zhou [26] defined the two dimensional bernstein-durrmeyer operators. in [28], n. deo introduced a certain modification of bernstein operator and obtained direct results. the bivariate bernstein operators have been also studied on a simplex by several authors(see [26][30]). by straightforward calculations it follows that bn,m(x2+y2;x,y)= x2+y2+ x(1−x) n + y(1−y) m , so that as in the case of the univariate berntsein operator bn(f ;x), the degree of approximation is at most o ( 1 n , 1 m ) . thus, this estimate can not be improved as such even for the sufficiently smooth function x2 + y2 on d. although, these operators are easy to compute and demonstrate nice approximation properties, the slow rate of convergence has been a matter of concern. in order to achieve higher order of approximation we propose three bivariate bernstein operators bm,rn,m(f ;x,y), r =1,2,3 as follows. 2. the first order operator bm,1n,m(f ;x,y) let b(d) and c(d) denote the space of bounded functions and space of continuous functions f (x,y) on d with the defining norm ‖f‖ := sup (x,y)∈d |f (x,y)|. for f ∈ b(d), the operator bm,1nm (f ;x,y) is defined by bm,1n,m(f ;x,y)= n∑ k=0 m∑ j=0 pm,1 n,m;k,j (x,y)f ( k n , j m ) , (2.1) where pm,1 n,m;k,j (x,y)= pm,1 n,k (x)pm,1 m,j (y), pm,1 n,k (x)= a(x,n)pn−1,k(x)+a(1−x,n)pn−1,k−1(x), 4 int. j. anal. appl. (2022), 20:60 pm,1 m,j (y)= b(y,m)pm−1,j(y)+b(1−y,m)pm−1,j−1(y) and a(x,n) = a1(n)x + a0(n), b(x,n) = b1(m)y + b0(m). here ai(n),bi(m) n,m = 0,1,2, are unknown sequences which are to be determined by imposing suitable convergence conditions. for a1(n) = b1(m) = −1, a0(n) = b0(m) = 1 the operator (2.1) reduces to the ordinary bivariate operator (1.1). the total modulus of continuity ω(f ;δ1,δ2) of f ∈ c(d) is defined as ω(f ;δ1,δ2) := sup { |f (s,t)− f (x,y)| : s,t ∈ d, |s −x| ≤ δ1, |t −y| ≤ δ2 } , where δ1,δ2 > 0. the modulus ω(f ;δ1,δ2) satisfies the following properties. (i) ω(f ;δ1,δ2)→ 0, if δ1 → 0, δ2 → 0, (ii) |f (s,t)− f (x,y)| ≤ ω(f ;δ1,δ2) ( 1+ |s−x| δ1 )( 1+ |t−y| δ2 ) . the full modulus of continuity, ω(f ;δ) for f ∈ c(d) is defined as ω(f ;δ) := max√ (s−x)2+(t−y)2≤δ (x,y)∈d |f (s,t)− f (x,y)| . other tools used to measure smoothness of a functions in c(d) are the partial modulus of continuity with respect to x and y and these are defined as ω(1)(f ;δ)= sup{|f (x1,y)− f (x2,y)| : y ∈ [0,1], |x1 −x2| ≤ δ} , ω(2)(f ;δ)= sup{|f (x,y1)− f (x,y2)| : x ∈ [0,1], |y1 −y2| ≤ δ} . 3. preliminaries let ei,j be the function ei,j(x,y) = x iy j. for a sequence ln,m(f ;x,y) of bivariate linear positive operators we have the following extension of the korovkin theorem in the square d. theorem 3.1. [?] if {ln,m} is a sequence of linear positive operators satisfying the conditions lim n,m→∞ ‖ln,m(e00;x,y)−1‖c(d) =0, lim n,m→∞ ‖ln,m(e10;x,y)−x‖c(d) =0, lim n,m→∞ ‖ln,m(e01;x,y)−y‖c(d) =0, lim n,m→∞ ∥∥ln,m(e201 +e210;x,y)− (x2 +y2)∥∥c(d) =0 then for any f ∈ c(d), lim n,m→∞ ‖ln,m(f ;x,y)− f (x,y)‖c(d) =0. lemma 3.1. the operator bm,1n,m verifies following identities: (1) bm,1n,m(e00,x,y)= (2a0(n)+a1(n))(2b0(m)+b1(m)), (2) bm,1n,m(e10,x,y)= [ x(2a0(n)+a1(n))+ (1−2x)(a0(n)+a1(n)) n ] (2b0(n)+b1(m)), int. j. anal. appl. (2022), 20:60 5 (3) bm,1n,m(e01,x,y)= (2a0(n)+a1(n)) [ y(2b0(m)+b1(m))+ (1−2y)(b0(m)+b1(m)) m ] , (4) bm,1n,m(e11,x,y)= [ x(2a0(n)+a1(n))+ (1−2x)(a0(n)+a1(n)) n ] × [ y(2b0(m)+b1(m))+ (1−2y)(b0(m)+b1(m)) m ] , (5) bm,1n,m(e20,x,y)= [ x2(2a0(n)+a1(n))+ (4x−6x2)a0(n)+(3x−5x2)a1(n) n + (1−4x+4x2)(a1(n)+a0(n)) n2 ] (2b0(n)+b1), (6) bm,1n,m(e02,x,y)= (2a0(n)+a1(n)) [ y2(2b0(m)+b1(m))+ (4y−6y2)b0(m)+(3y−5y2)b1(m) m + (1−4y+4y2)(b1(m)+b0(m)) m2 ] , (7) bm,1n,m(e22,x,y)= [ x2(2a0(n)+a1(n))+ (4x−6x2)a0(n)+(3x−5x2)a1(n) n + (1−4x+4x2)(a1(n)+a0(n)) n2 ] [ y2(2b0(m)+b1(m))+ (4y−6y2)b0(m)+(3y−5y2)b1(m) m + (1−4y+4y2)(b1(m)+b0(m)) m2 ] , (8) bm,1n,m(e12,x,y)= [ x(2a0(n)+a1(n))+ (1−2x)(a0(n)+a1(n)) n ] [ y2(2b0(m)+b1(m))+ (4y−6y2)b0(m)+(3y−5y2)b1(m) m + (1−4y+4y2)(b1(m)+b0(m)) m2 ] , (9) bm,1n,m(e21,x,y)= [ x2(2a0(n)+a1(n))+ (4x−6x2)a0(n)+(3x−5x2)a1(n) n + (1−4x+4x2)(a1(n)+a0(n)) n2 ] × [ x(2a0(n)+a1(n))+ (1−2x)(a0(n)+a1(n)) n ] . proof. the proof follows by straightforward calculations. � 4. rate of approximation in the next theorem we show that the linear positive operator sequence bm,1n,m defined by equation (2.1) converges uniformly to f with the help theorem3.1 given by volkov in [31]. theorem 4.1. let f ∈ b(d) and the sequences a0(n),a1(n),b0(m),b1(m) be convergent. if 2a0(n)+ a1(n)=2b0(m)+b1(m)=1 and the operator (2.1) is positive, then lim n,m→∞ bm,1n,m(f ;x,y)= f (x,y). further, the convergence is uniform if f ∈ c(d). proof. by lemma 3.1, for each i, j with 0≤ i ≤ 2,0≤ j ≤ 2 limn,m→∞bm,1n,m(eij,x,y)= eij, uniformly on d. hence, lim n,m→∞ ∥∥bm,1n,m(e00;x,y)−1∥∥c(d) =0 lim n,m→∞ ∥∥bm,1n,m(e10;x,y)−x∥∥c(d) =0 lim n,m→∞ ∥∥bm,1n,m(e01;x,y)−y∥∥c(d) =0 lim n,m→∞ ∥∥bn,m(e201 +e210;x,y)− (x2 +y2)∥∥c(d) =0 lim n,m→∞ ‖bn,m(f ;x,y)− f (x,y)‖c(d) =0 now the proof follows by the theorem 3.1. � 6 int. j. anal. appl. (2022), 20:60 theorem 4.2. (voronovskaja theorem)let f ∈ c(d), fx, fy, fxx and fyy exist at a certain point (x,y) ∈ d and operator bm,1n,m(f ;x,y) be positive. if a1(n)+a0(n) = b1(m)+b0(m) = 0 and then we have, lim n,m→∞ 2n ( bm,1n,m(f ,x,y)− f (x,y) ) = ( (4x −6x2) lim n→∞ a0(n)+(3x −5x2) lim n→∞ a1(n)fxx ) + ( (4y −6y2) lim m→∞ b0(m)+(3y −5y2) lim m→∞ b1(m)fyy ) , where fxx and fyy are second order partial derivatives of f with respect to x and y respectively. moreover, the above relation holds uniformly on d if fxx, fyy ∈ c(d). proof. by taylor’s formula, f ( k n , j m ) = f (x,y)+ (( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y ) f (x,y) + 1 2! (( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y )2 f (x,y) +ε ( k n , j m )(( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y )2 f (x,y). the function ε(s,t) is bounded on d and lim(s,t)→(x,y))ε(s,t) = 0. multiplying the above equation by pm,1 n,k (x)pm,1 n,j (y) and taking sum over i, j it follows that bm,1n,m(f ;x,y)= f (x,y)+b m,1 n,m((e10 −x);x,y)fx +b m,1 n,m((e01 −y);x,y)fy + 1 2 (bm,1n,m((e10 −x) 2;x,y)fxx +b m,1 n,m((e01 −y) 2;x,y)fyy +bm,1n,m(((e10 −x);x,y)((e01 −y);x,y)fxfy)) +bm,1n,m ( ε(e10,e01)(((e10,x,y)−x) fx +((e01,x,y)−y) fy)2 ;x,y ) . in view of the relation 2a0(n)+a1(n)=2b0(m)+b1(m)=1 and lemma 1, it follows that bm,1n,m(f ;x,y)= f (x,y)+ ( 1 2 (4a0(n)+3a1(n))x − (6a0(n)+5a1(n))x2 n ) fxx + ( 1 2 (4b0(m)+3b1(m))y − (6b0(m)+5b1(m))y2 m ) fyy +bm,1n,n ( ε(e10,e01)(((e10,x,y)−x) fx +((e01,x,y)−y) fy)2 ;x,y ) . (4.1) since ε ( k n , j m ) is bounded on d and lim(s,t)→(x,y))ε(s,t)=0, proceeding in the standard manner, it follows that lim n,m→∞ bm,1n,m ( ε(e10,e01)(((e10,x,y)−x) fx +((e01,x,y)−y) fy)2 ;x,y ) =o ( 1 n , 1 m ) . (4.2) now from equation (4.1) and (4.2) we obtain the desired result. � int. j. anal. appl. (2022), 20:60 7 theorem 4.3. let bm,1n,m(f ;x,y) be the operator (2.1). if f ∈ c(d) and a0(n),a1(n),b0(m) and b1(m) are convergent sequences such that for each n and m, 2a0(n)+a1(n)=2b0(m)+b1(m)=1 then, ∣∣bm,1n,m(f ;x,y)− f (x,y)∣∣ ≤ 2 ( (1+3|a1(n)|)ω(1) ( f , n−1 2n √ n−2 ) +(1+3|b1(m)|)ω(2) ( f , m−1 2m √ m−2 )) . proof. we have∣∣bm,1n,m(f ;x,y)− f (x,y)∣∣ ≤ ∣∣∣ n∑ k=0 m∑ j=0 ((a0(n)+a1(n))pn−1,k(x)+(a0(n)+a1(n)(1−x))pn−1,k−1(x)) ((b0(m)+b1(m))pm−1,j(y)+(b0(m)+b1(m)(1−y))pm−1,j−1(y))× × ( f ( k n , j m ) − f ( x, j m ))∣∣∣ + ∣∣∣ n∑ k=0 m∑ j=0 ((a0(n)+a1(n))pn−1,k(x)+(a0(n)+a1(n)(1−x))pn−1,k−1(x)) ((b0(m)+b1(m))pm−1,j(y)+(b0(m)+b1(m)(1−y))pm−1,j−1(y))× × ( f ( k n , j m ) − f ( k n ,y ))∣∣∣ ≤ ( |a0(n)+a1(n)| n∑ k=0 pn−1,k(x)+ |(a0(n)+a1(n)(1−x))| n∑ k=0 pn−1,k−1(x) ) × × ∣∣∣∣f ( k n , j m ) − f ( x, j m )∣∣∣∣ +  |b0(m)+b1(m)| m∑ j=0 pm−1,j(y)+ |(b0(m)+b1(m)(1−y))| m∑ j=0 pm−1,j−1(y)  × × ∣∣∣∣f ( k n , j m ) − f ( k n ,y )∣∣∣∣ ≤ |a0(n)+a1(n)| n∑ k=0 pn−1,k(x)ω (1) (∣∣∣∣kn −x ∣∣∣∣ ) + |a0(n)+a1(n)(1−x)| n∑ k=0 pn−1,k−1(x)ω (1) (∣∣∣∣kn −x ∣∣∣∣ ) + |b0(m)+b1(m)| m∑ j=0 pm−1,j(y)ω (2) (∣∣∣∣ jm −y ∣∣∣∣ ) + |b0(m)+b1(m)(1−y)| m∑ j=0 pm−1,j−1(y)ω (2) (∣∣∣∣ jm −y ∣∣∣∣ ) . now using the property ω(f ,λδ)≤ (λ+1)ω(f ,δ), λ ≥ 0 8 int. j. anal. appl. (2022), 20:60 of modulus of continuity, we get, ω(1) ( f ; ∣∣∣∣kn −x ∣∣∣∣ ) ≤ ( 1+ 2n √ n−2 n−1 ∣∣∣∣kn −x ∣∣∣∣ ) ω(1) ( f , n−1 2n √ n−2 ) . similarly, ω(2) ( f ; ∣∣∣∣ jm −y ∣∣∣∣ ) ≤ ( 1+ 2m √ m−2 m−1 ∣∣∣∣ jm −y ∣∣∣∣ ) ω(2) ( f , m−1 2m √ m−2 ) . therefore,∣∣bm,1n,m(f ;x,y)− f (x,y)∣∣ |a0(n)+a1(n)|ω(1) ( f , n−1 2n √ n−2 )( 1+ 2n √ n−2 n−1 n∑ k=0 pn−1,k(x) ∣∣∣∣kn −x ∣∣∣∣ ) + |a0(n)+a1(n)(1−x)|ω(1) ( f , n−1 2n √ n−2 )( 1+ 2n √ n−2 n−1 n∑ k=0 pn−1,k−1(x) ∣∣∣∣kn −x ∣∣∣∣ ) + |b0(m)+b1(m)|ω(2) ( f , m−1 2m √ m−2 )1+ 2m√m−2 m−1 m∑ j=0 pm−1,j(y) ∣∣∣∣ jm −x ∣∣∣∣   + |b0(m)+b1(m)(1−y)|ω(2) ( f , m−1 2m √ m−2 )1+ 2m√m−2 m−1 m∑ j=0 pm−1,j−1(y) ∣∣∣∣ jm −y ∣∣∣∣   . now an application of schwarz inequality and lemma 1 yields, n∑ k=0 pn−1,k(x) ∣∣∣∣kn −x ∣∣∣∣ ≤ ( n∑ k=0 pn−1,k(x) ( k n −x )2)12 ( n∑ k=0 pn−1,k(x) )1 2 = ( (2−n)x2 n2 + (n−1)x n2 )1 2 , and n∑ k=0 pn−1,k−1(x) ∣∣∣∣kn −x ∣∣∣∣ ≤ ( n∑ k=0 pn−1,k−1(x) ( k n −x )2)12 ( n∑ k=0 pn−1,k−1(x) )1 2 = ( (2−n)x2 n2 + (n−3)x n2 + 1 n2 )1 2 . since, max x∈[0,1],n>2 ( (2−n)x2 n2 + (n−1)x n2 ) = max x∈[0,1],n>2 ( (2−n)x2 n2 + (n−3)x n2 + 1 n2 ) = (n−1)2 4n2(n−2) therefore, n∑ k=0 pn−1,k(x) ∣∣∣∣kn −x ∣∣∣∣ ≤ (n−1)2n√n−2, and similarly n∑ k=0 pn−1,k−1(x) ∣∣∣∣kn −x ∣∣∣∣ ≤ (n−1)2n√n−2. int. j. anal. appl. (2022), 20:60 9 in the same manner, m∑ j=0 pm−1,j(y) ∣∣∣∣ jm −y ∣∣∣∣ ≤ (m−1)2m√m−2, m∑ j=0 pm−1,j−1(y) ∣∣∣∣ jm −y ∣∣∣∣ ≤ (m−1)2m√m−2. finally, using 2a0(n)+a1(n)=2b0(m)+b1(m)=1, we get∣∣bm,1n,m(f ;x,y)− f (x,y)∣∣ ≤ 4ω(1) ( f , n−1 2n √ n−2 )(∣∣∣∣a1(n)x + 1−a1(n)2 ∣∣∣∣+ ∣∣∣∣a1(n)(1−x)+ 1−a1(n)2 ∣∣∣∣ ) +4ω(2) ( f , m−1 2m √ m−2 )(∣∣∣∣b1(m)y + 1−b1(m)2 ∣∣∣∣+ ∣∣∣∣b1(m)(1−y)+ 1−b1(m)2 ∣∣∣∣ ) ≤ 2 [ (1+3|a1(n)|)ω(1) ( f , n−1 2n √ n−2 ) +(1+3|b1(m)|)ω(2) ( f , m−1 2m √ m−2 )] . the proof is now completed. � remark 4.1. for a lipschitz class we can established the degree of approximation by bivariate operator bm,1n,m(f ;x,y). for 0 < θ1 ≤ 1 and 0 < θ2 ≤ 1 and f ∈ b(d) the lipschitz class lipm(θ1,θ2) is defined as follows lipm(θ1,θ2) := { f : |f (x1,x2)− f (y1,y2)| ≤ m |x1 −y1|θ1 |x2 −y2|θ2 } , where m > 0. so, if f (x,y) belongs to the class lipm(θ1,θ2) then, there holds the estimate∣∣bm,1n,m(f ;x,y)− f (x,y)∣∣ ≤ m(n,m)n θ12 m θ22 in view of the upper bounds ω(1) ( f , n−1 2n √ n−2 ) ≤ cnθ1/2 and ω(2) ( f , m−1 2m √ m−2 ) ≤ cmθ2/2. here, the constant m(n,m)=2max{(1+3 |a1(n)|) ,(1+3 |b1(n)|)} . 5. the second order operator bm,2n,m(f ;x,y) we extend our results to achieve 2nd order approximation on two variable. the proposed operator bm,2n (f ;x,y) is defined as bm,2n,m(f ;x,y)= n∑ k=0 m∑ j=0 pm,2 n,k (x)pm,2 m,j (y)f ( k n , j m ) , (5.1) where pm,2 n,k (x)= c(x,n)pn−2,k(x)+d(x,n)pn−2,k−1(x)+c(1−x,n)pn−2,k−2(x), pm,2 m,j (y)= e(y,m)pm−2(j)(y)+ f (y,m)pm−2,j−1(y)+e(1−y,m)pm−2(j −2)(y) and c(x,n)= c2(n)x 2 +c1(n)x +c0(n), d(x,n)= d0(n)x(1−x), e(y,m)= e2(m)y 2 +e1(m)y +e0, f (x,n)= f0(m)y(1−y). 10 int. j. anal. appl. (2022), 20:60 as in the first order case, ci(n),ei(m), i =0,1,2, d0(n), f0(m) are the unknown sequences. remark 5.1. if we put c2(n)= e2(m)=1,c1(n)= e1(m)=−2,c0(n)= e0(m)=1 and d0 = f0 =2 then the operator (5.1) becomes ordinary bivariate bernstein operator (1.1). by straightforward calculation we obtain values of unknown sequences as follows. c2 = n 2 , c1 = −2−n 2 , c0 =1, d0 = n (5.2) e2 = m 2 , e1 = −2−m 2 , e0 =1, f0 = m. (5.3) using these sequences we prove the following lemma. lemma 5.1. for the operator bm,2n,m(f ,x,y) with the sequences chosen in (5.2)-(5.3), there holds the estimates (1) bm,2n,m(e00,x,y)=1, (2) bm,2n,m(e10,x,y)= x, (3) bm,2n,m(e01,x,y)= y, (4) bm,2n,m(e20,x,y)= x 2 + 2x(1−x) n2 , (5) bm,2n,m(e02,x,y)= y 2 + 2y(1−y) m2 , (6) bm,2n,m(e30,x,y)= x 3 + 2x(1−x)(5x−1) n2) − 6x(1−x)(2x−1) n3 , (7) bm,2n,m(e03,x,y)= y 3 + 2y(1−y)(5y−1) m2 − 6y(1−y)(2y−1) m3 . proof. the proof follows by straightforward calculations. � remark 5.2. by the lemma 5.1 we get improved rate of approximation for the functions eij, i, j =0(2) i.e. bm,2n,m((e01 − y),x,y) = 0,bm,2n,m((e10 − x),x,y) = 0,bm,2n,m((e20 − x2),x,y) = 2x(1−x) n2 and bm,2n,m((e02 −y2),x,y)= 2y(1−y) m2 . for twice differentiable function we obtain the following theorem. theorem 5.1. let bm,2n,m be the operator, f ∈ c3(d),n = m and let the sequences c2,c1,c0, e2,e1,e0,d0, f0 be bounded. then, lim n→∞ n2(bm,2n,n (f ,x,y)− f (x,y))= x(1−x) ( fxx + (2x −1) 3 fxxx ) +y(1−y) ( fyy + (2y −1) 3 fyyy ) . proof. we prove theorem for the cases when the operator bm,2n,n (f ,x,y) is positive. since f ∈ c3(d), we can write int. j. anal. appl. (2022), 20:60 11 f ( k n , j m ) = f (x,y)+ (( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y ) f (x,y) + 1 2! (( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y )2 f (x,y) + 1 3! (( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y )3 f (x,y) + � ( k n , j m )(( k n −x ) ∂ ∂x + ( j m −y ) ∂ ∂y )3 f (x,y). (5.4) multiplying equation (5.4) by pm,1 n,k (x)pm,1 n,j (y) and taking the sum over i, j we obtain bm,2n,n (f ;x,y)= f (x,y)+b m,2 n,n ((e10 −x);x,y)fx +b m,2 n,n ((e01 −y);x,y)fy + 1 2 (bm,2n,n ((e10 −x) 2;x,y)fxx +b m,2 n,n ((e01 −y) 2;x,y)fyy +bm,2n,n (((e10 −x);x,y)((e01 −y);x,y)fxfy) + 1 6 ( bm,2n,n ((e10 −x);x,y) 3fxxx +b m,2 n,n ((e01 −y) 3;x,y)fyyy ) +3bm,2n,n ((e10 −x);x,y)(e01 −y) 2;x,y)fxfyy +3bm,2n,n ((e10 −x) 2;x,y)(e01 −y);x,y)fxxfy) +bm,2n,n ( ε(e10,e01)(((e10,x,y)−x) fx +((e01,x,y)−y) fy)3 ;x,y ) here, bm,2n,n ( ε(e10,e01)(((e10,x,y)−x) fx +((e01,x,y)−y) fy)3 ;x,y ) ≤ � � is the error term which we have proved earlier for bm,1n,m, now by lemma 5.1 we have, bm,2n,n (f ;x,y)− f (x,y) = 1 2 ( 2x(1−x) n2 fxx + 2y(1−y) n2 fyy ) + 1 6 ( 2x(1−x)(2x −1) n2 fxxx + 2y(1−y)(2y −1) n2 fyyy ) + � finally, by standard computations, we have lim n→∞ n2(bm,2n,n (f ,x,y)− f (x,y))= x(1−x) ( fxx + (2x −1) 3 fxxx ) +y(1−y) ( fyy + (2x −1) 3 fyyy ) . � remark 5.3. by theorem 5.1, for thrice differentiable functions, the degree of approximation is proved to be o(n−2) which is higher than the degree o(n−1) obtained by the ordinary operator bn,n(f ;x,y). 12 int. j. anal. appl. (2022), 20:60 6. the third order operator bm,3n,m(f ;x,y) the third order operator bm,3n,m(f ;x,y) on d is defined by bm,3n,m(f ;x,y)= n∑ k=0 m∑ j=0 f ( k n , j m ) pm,3 n,k (x)pm,3 m,j (y) (6.1) where, pm,3 n,k (x) = g(x,n)pn−4,k(x)+ r(x,n)pn−4,k−1(x)+ s(x,n)pn−4,k−2(x) +r(1−x,n)pn−4,k−3(x)+g(1−x,n)pn−4,k−4(x), pm,3 m,j (y) = h(y,m)pm−4,j(y)+ t(y,m)pm−4,j−1(y)+w(y,m)pm−4,j−2(y) +t(1−y,m)pm−4,j−3(y)+g(1−y,n)pm−4,j−4(y). and g(x,n)= g4(n)x 4 +g3(n)x 3 +g2(n)x 2 +g1(n)x +g0(n), r(x,n)= r4(n)x 4 + r3(n)x 3 + r2(n)x 2 + r1(n)x + r0(n), s(x,n)= s0(n)(x(1−x)2)2, h(y,m)= h4(m)y 4 +h3(m)y 3 +h2(m)y 2 +h1(m)y +h0(m), t(y,m)= t4(m)y4 + t3(m)y 3 + t2(m)y 2 + t1(m)y + t0(m), w(y,m)= w0(m)(y(1−y)2)2. note 1. if we put g4(n) = h4(m) = 1,g3(n) = h3(m) = −4,g2(n) = h2(m) = 6,g1(n) = h1(m) = −4,g0(n) = h0(m) = 0, r4(n) = t4(m) = −4, r3(n) = t3(m) = 12, r2(n) = t2(m) = −12, r0(n) = t0(m) = 0 and s0(n) = w0(m) = 6,then the operator become the bernstein operator (1.1) of two variable. by straightforward calculations we have the following values of the unknown sequences gi(n),hi(m), ri(n),ti(m), i =0,1,2,3,4. s0(n) and w0(m) for which the operator bm,3n,m(f ;x,y) attains the order o ( 1 n2 , 1 m2 ) . g4(n)=1+ 23 12 n+ 1 8 n2, g3(n)=−4− 143 n− 1 4 n2, g2(n)=6+ 10 3 n+ 1 8 n2, g1(n)=−4− 712n, g0(n)=1 r4(n)=−4− 233 n− 1 2 n2, r3(n)=12+17n+n 2, r2(n)=−12− 313 n− 1 2 n2 r1(n)=4+n, r0(n)=0, s0(n)=6+ 23 2 n+ 3 4 n2 h4(m)=1+ 23 12 m+ 1 8 m2, h3(m)=−4− 143 m− 1 4 m2, h2(m)=6+ 10 3 m+ 1 8 m2, h1(m)=−4− 712m, h0(m)=1 t4(m)=−4− 233 m− 1 2 m2, t3(m)=12+17m+m 2, t2(m)=−12− 313 m− 1 2 m2 t1(m)=4+m, t0(m)=0, w0(m)=6+ 23 2 m+ 3 4 m2. int. j. anal. appl. (2022), 20:60 13 lemma 6.1. the operator bm,3n,m(f ,x,y) together with the above values gi(n),hi(m), ri(n),ti(m), s0(n) and w0(m), i =0(4) verifies (1) bm,3n,m(e00,x,y)=1 (2) bm,3n,m(e10,x,y)= x (3) bm,3n,m(e01,x,y)= y (4) bm,3n,m(e20,x,y)= x 2 (5) bm,3n,m(e02,x,y)= y 2 (6) bm,3n,m(e30,x,y)= x 3 (7) bm,3n,m(e03,x,y)= y 3 (8) bm,3n,m(e 2 10 +e 2 01;x,y)= x 2 +y2. proof. the proof follows by straightforward computations. � proceeding along the lines of theorem 5.1, we obtain following theorem. theorem 6.1. let f ∈ c4(d) and together the values of gi,hi, ri,ti,si,wi, we have lim n,m→∞ (bm,2n,n (f ,x,y)− f (x,y))=o ( 1 n3 , 1 m3 ) . 7. numerical verification to verify the order of approximation by operator bm,3n,m(f ;x,y) we choose the function sinπx cosπy and for its images under the operator bm,3n,m for n =5,10 and m =5,10. figure 1. comparison of the function sinπx cosπy with bm,3n,m(f ,x,y) for n = m = 5,n = m =10. 14 int. j. anal. appl. (2022), 20:60 table 1 provides values of the function sinπx cosπy at equidistant nodes (x,y) with step size 0.2 and tables 2, 3 provide values of the functions bm,35,5 (f ;x,y) and b m,3 10,10(f ;x,y) at the corresponding nodes. table 1. f (x,y)= sinπx cosπy, (starred ∗ rows are of order 10−17) 0 0.2 0.4 0.6 0.8 1 0. 0. 0. 0. 0. 0. 0. 0.2 0.587785 0.475528 0.181636 -0.181636 -0.475528 -0.587785 0.4 0.951057 0.769421 0.293893 -0.293893 -0.769421 -0.951057 0.6 0.951057 0.769421 0.293893 -0.293893 -0.769421 -0.951057 0.8 0.587785 0.475528 0.181636 -0.181636 -0.475528 -0.587785 ∗1 12.2465 9.9076 3.78437 -3.78437 -9.9076 -12.2465 table 2. bm,35,5 (f ;x,y) 0 0.2 0.4 0.6 0.8 1 0. 0. 0. 0. 0. 0. 0. 0.2 0.643288 0.527466 0.200931 -0.200931 -0.527466 -0.643288 0.4 1.03431 0.848086 0.323067 -0.323067 -0.848086 -1.03431 0.6 1.03431 0.848086 0.323067 -0.323067 -0.848086 -1.03431 0.8 0.643288 0.527466 0.200931 -0.200931 -0.527466 -0.643288 1. 0. 0. 0. 0. 0. 0. table 3. bm,310,10(f ;x,y), (starred ∗ rows are of order 10−12) 0 0.2 0.4 0.6 0.8 1 0. 0. 0. 0. 0. 0. 0. 0.2 0.594148 0.482834 0.184553 -0.184553 -0.482834 -0.594148 0.4 0.961538 0.781392 0.298671 -0.298671 -0.781392 -0.961538 0.6 0.961538 0.781392 0.298671 -0.298671 -0.781392 -0.961538 0.8 0.594148 0.482834 0.184553 -0.184553 -0.482834 -0.594148 ∗1 5.38325 4.37469 1.67213 -1.67213 -4.37469 -5.38325 it follows that the maximum absolute errors by bm,3n,n (f ;x,y) for n =5,10 are found to be 0.0943346 at (0.5,0.919599) and 0.0134438 at (0.5,0.879425) respectively. hence, in the figure 2 we compare f (x,y) with bm,3n,n (f ;x,y) for n =5 and n =10 in the neighbourhood of the point (0.5,0.8). int. j. anal. appl. (2022), 20:60 15 figure 2. comparison of the function sinπx cosπy with bm,3n,m(f ,x,y) for n = m = 5,n = m =10. the next example compares bm,2n,n (f ;x,y) with b m,3 n,n (f ;x,y) for n =5 (see fig. 3.) corresponding to the sufficiently smooth function e−x−y. it is observed that the accuracy increases as we move from bm,2n,n (f ;x,y) to b m,3 n,n (f ;x,y) for a fixed number of nodes. figure 3. comparison of the function e−x−y with bm,2n,m(f ,x,y), b m,3 n,m(f ,x,y) for n = 5,m =5. 16 int. j. anal. appl. (2022), 20:60 figure 4. comparison of the function e−x−y with bm,2n,m(f ,x,y), b m,3 n,m(f ,x,y) for n = 5,m =5 in small rectangle the following tables show the values of f (x,y) and corresponding values under the operator bm,3n,m for n =5,10,15,20m =5,10,15,20 at different nodes with steps size 0.1. table 4. f (x,y)= e−(x+y) 0 0.2 0.4 0.6 0.8 1 0 1. 0.818731 0.67032 0.548812 0.449329 0.367879 0.2 0.818731 0.67032 0.548812 0.449329 0.367879 0.301194 0.4 0.67032 0.548812 0.449329 0.367879 0.301194 0.246597 0.6 0.548812 0.449329 0.367879 0.301194 0.246597 0.201897 0.8 0.449329 0.367879 0.301194 0.246597 0.201897 0.165299 1 0.367879 0.301194 0.246597 0.201897 0.165299 0.135335 table 5. bm,35,5 (f ;x,y) 0 0.2 0.4 0.6 0.8 1 0 1. 0.819147 0.670916 0.549393 0.449698 0.367879 0.2 0.819147 0.671002 0.549579 0.450034 0.368369 0.301347 0.4 0.670916 0.549579 0.450129 0.368597 0.30171 0.246816 0.6 0.549393 0.450034 0.368597 0.301833 0.247061 0.202111 0.8 0.449698 0.368369 0.30171 0.247061 0.202228 0.165435 1 0.367879 0.301347 0.246816 0.202111 0.165435 0.135335 int. j. anal. appl. (2022), 20:60 17 table 6. bm,310,10(f ;x,y) 0 0.2 0.4 0.6 0.8 1 0 1. 0.818788 0.670397 0.548882 0.44937 0.367879 0.2 0.818788 0.670414 0.548913 0.449418 0.367939 0.301215 0.4 0.670397 0.548913 0.449432 0.367969 0.301257 0.246625 0.6 0.548882 0.449418 0.367969 0.301271 0.246651 0.201922 0.8 0.44937 0.367939 0.301257 0.246651 0.201934 0.165314 1 0.367879 0.301215 0.246625 0.201922 0.165314 0.135335 table 7. bm,320,20(f ;x,y) 0 0.2 0.4 0.6 0.8 1 0 1. 0.818738 0.67033 0.54882 0.449334 0.367879 0.2 0.818738 0.670332 0.548825 0.44934 0.367887 0.301197 0.4 0.67033 0.548825 0.449342 0.367891 0.301202 0.246601 0.6 0.54882 0.44934 0.367891 0.301204 0.246604 0.2019 0.8 0.449334 0.367887 0.301202 0.246604 0.201901 0.165301 1 0.367879 0.301197 0.246601 0.2019 0.165301 0.135335 the global errors of approximation of the function by bm,25,5 (f ,x,y)and b m,3 5,5 (f ,x,y)are0.00975603 and 0.000803997 which are obtained at (0.278459,0.278459) and (0.365884,0.365884) respectively. here, again we choose small intervals for comparison in fig 4 to show the three surfaces sufficiently separated. 8. conclusion and future scope the surface plotting of a smooth function by a bivariate linear operators is constructed using the m,n nodes k/n, j/m k = 0,n, j = 0,m. the operator bm,jn,m(f ;x,y), j = 1,2,3 approximate such surfaces with the degree of convergence o(n−j), j = 1,2,3 which is significantly high in comparison of approximation by regular bivariate operator bn,m(f ;x,y). we observe that the global absolute error decreases with the order o(n−1) from bm,2n,n (f ;x,y) to b m,3 n,n (f ;x,y) while keeping the degree of the polynomials same. on the other hand choosing large number of nodes can also improve the degree of approximation however, for a given fixed number of values it is better to apply the methods b m,j n,m(f ;x,y) for j = 2,3, ... it is of interest to determine exact class of the functions for which the methods bm,jn,m(f ;x,y) provide optimal degree of approximation. this can be considered as an open problem. authors’ contributions: all authors contributed equally and significantly in writing this paper. all authors read and approved the final manuscript. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 18 int. j. anal. appl. (2022), 20:60 references [1] j. reinkenhof, differentiation and integration using bernstein’s polynomials, int. j. numer. meth. eng. 11 (1977), 1627–1630. https://doi.org/10.1002/nme.1620111012. [2] j. bustamante, bernstein operators and their properties, springer, cham, 2017. https://doi.org/10.1007/ 978-3-319-55402-0. [3] e. voronovskaja, détermination de la forme asymptotique d’approximation des fonctions par les polynômes de m. bernstein, c.r. acad. sci. urss, 4 (1932), 79–85. [4] g.g. lorentz, approximation of functions, athena series, holt, rinehart and winston, new york, (1966). [5] h. khosravian-arab, m. dehghan, m.r. eslahchi, a new approach to improve the order of approximation of the bernstein operators: theory and applications, numer. algorithms, 77 (2017), 111–150. https://doi.org/10. 1007/s11075-017-0307-z. [6] a.m. acu, v. gupta, g. tachev, better numerical approximation by durrmeyer type operators, results math. 74 (2019), 90. https://doi.org/10.1007/s00025-019-1019-6. [7] a.r. gairola, k.k. singh, l.n. mishra, degree of approximation by certain durrmeyer type operators, discontin. nonlinear. complex. 11 (2022), 253–273. https://doi.org/10.5890/dnc.2022.06.006. [8] e.h. kingsley, bernstein polynomials for functions of two variables of class c(k), proc. amer. math. soc. 2 (1951), 64–71. https://doi.org/10.2307/2032622. [9] t. acar, a. kajla, degree of approximation for bivariate generalized bernstein type operators, results math. 73 (2018), 79. https://doi.org/10.1007/s00025-018-0838-1. [10] c. ding, f. cao, k-functionals and multivariate bernstein polynomials, j. approx. theory. 155 (2008), 125–135. https://doi.org/10.1016/j.jat.2008.03.011. [11] a. fellhauer, approximation of smooth functions using bernstein polynomials in multiple variables, arxiv:1609.01940, (2016). https://doi.org/10.48550/arxiv.1609.01940. [12] c. munoz, a. narkawicz, formalization of bernstein polynomials and applications to global optimization, j. autom. reason. 51 (2012), 151–196. https://doi.org/10.1007/s10817-012-9256-3. [13] a. pallini, bernstein-type approximations of smooth functions, statistica. 65 (2005), 169–191. https://doi. org/10.6092/issn.1973-2201/84. [14] t. sauer, multivariate bernstein polynomials and convexity, computer aided geom. design. 8 (1991), 465–478. https://doi.org/10.1016/0167-8396(91)90031-6. [15] a.yu. veretennikov, e.v. veretennikova, on partial derivatives of multivariate bernstein polynomials, sib. adv. math. 26 (2016), 294–305. https://doi.org/10.3103/s1055134416040039. [16] j. wang, z. peng, s. duan, j. jing, derivatives of multivariate bernstein operators and smoothness with jacobi weights, j. appl. math. 2012 (2012), 346132. https://doi.org/10.1155/2012/346132. [17] v.n. mishra, k. khatri, l.n. mishra, deepmala, inverse result in simultaneous approximation by baskakov-durrmeyer-stancu operators, j. inequal. appl. 2013 (2013), 586. https://doi.org/10.1186/ 1029-242x-2013-586. [18] r.b. gandhi, deepmala, v.n. mishra, local and global results for modified szász–mirakjan operators, math. methods appl. sci. 40 (2017), 2491–2504. https://doi.org/10.1002/mma.4171. [19] v.n. mishra, k. khatri, l.n. mishra, statistical approximation by kantorovich type discrete q−beta operators, adv. differ. equ. 2013 (2013), 345. https://doi.org/10.1186/1687-1847-2013-345. [20] v.n. mishra, k. khatri, l.n. mishra, on simultaneous approximation for baskakov-durrmeyer-stancu type operators, j. ultra sci. phys. sci. 24 (2012), 567–577. [21] a.r. gairola, deepmala, l.n. mishra, rate of approximation by finite iterates of q-durrmeyer operators, proc. natl. acad. sci., india, sect. a phys. sci. 86 (2016), 229–234. https://doi.org/10.1007/s40010-016-0267-z. https://doi.org/10.1002/nme.1620111012 https://doi.org/10.1007/978-3-319-55402-0 https://doi.org/10.1007/978-3-319-55402-0 https://doi.org/10.1007/s11075-017-0307-z https://doi.org/10.1007/s11075-017-0307-z https://doi.org/10.1007/s00025-019-1019-6 https://doi.org/10.5890/dnc.2022.06.006 https://doi.org/10.2307/2032622 https://doi.org/10.1007/s00025-018-0838-1 https://doi.org/10.1016/j.jat.2008.03.011 https://doi.org/10.48550/arxiv.1609.01940 https://doi.org/10.1007/s10817-012-9256-3 https://doi.org/10.6092/issn.1973-2201/84 https://doi.org/10.6092/issn.1973-2201/84 https://doi.org/10.1016/0167-8396(91)90031-6 https://doi.org/10.3103/s1055134416040039 https://doi.org/10.1155/2012/346132 https://doi.org/10.1186/1029-242x-2013-586 https://doi.org/10.1186/1029-242x-2013-586 https://doi.org/10.1002/mma.4171 https://doi.org/10.1186/1687-1847-2013-345 https://doi.org/10.1007/s40010-016-0267-z int. j. anal. appl. (2022), 20:60 19 [22] a.r. gairola, deepmala, l.n. mishra, on the q-derivatives of a certain linear positive operators, iran. j. sci. technol. trans. sci. 42 (2017), 1409–1417. https://doi.org/10.1007/s40995-017-0227-8. [23] p.l. butzer, on two-dimensional bernstein polynomials, can. j. math. 5 (1953), 107–113. https://doi.org/ 10.4153/cjm-1953-014-2. [24] o. t. pop, about the generalization of voronovskaja’s theorem for bernstein polynomials of two variables, int. j. pure appl. math. 38 (2007), 297-308. [25] d.d. stancu, on some polynomials of bernstein type, stud. cerc. st. mat. iaşi, 11 (1960), 221-233. [26] x. zhou, approximation by multivariate bernstein operators, results. math. 25 (1994), 166–191. https://doi. org/10.1007/bf03323150. [27] a. bayad, t. kim, s.h. rim, bernstein polynomials on simplex, arxiv:1106.2482, (2011). https://doi.org/10. 48550/arxiv.1106.2482. [28] n. deo, n. bhardwaj, some approximation theorems for multivariate bernstein operators, southeast asian bull. math. 34 (2010), 1023–1034. [29] t.d. do, s. waldron, multivariate bernstein operators and redundant systems, j. approx. theory. 192 (2015), 215–233. https://doi.org/10.1016/j.jat.2014.12.001. [30] r.t. farouki, t.n.t. goodman, t. sauer, construction of orthogonal bases for polynomials in bernstein form on triangular and simplex domains, computer aided geom. design. 20 (2003), 209–230. https://doi.org/10. 1016/s0167-8396(03)00025-6. [31] v. i. volkov, on the convergence of sequences of linear positive operators in the space of continuous functions of two variables, dokl. akad. nauk sssr, 115 (1957), 17–19. https://doi.org/10.1007/s40995-017-0227-8 https://doi.org/10.4153/cjm-1953-014-2 https://doi.org/10.4153/cjm-1953-014-2 https://doi.org/10.1007/bf03323150 https://doi.org/10.1007/bf03323150 https://doi.org/10.48550/arxiv.1106.2482 https://doi.org/10.48550/arxiv.1106.2482 https://doi.org/10.1016/j.jat.2014.12.001 https://doi.org/10.1016/s0167-8396(03)00025-6 https://doi.org/10.1016/s0167-8396(03)00025-6 1. introduction 2. the first order operator bm,1n,m(f;x,y) 3. preliminaries 4. rate of approximation 5. the second order operator bn,mm,2(f;x,y) 6. the third order operator bn,mm,3(f;x,y) 7. numerical verification 8. conclusion and future scope references international journal of analysis and applications volume 19, number 6 (2021), 915-928 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-915 common fixed point of four maps in sm-metric space v. srinivas1,∗, k. mallaiah2 1department of mathematics, ucs, saifabad, ou,hyderabad, telangana, india 2lecturer in mathematics, jngp, ramanthapur, hyderabad, telangana, india ∗corresponding author: srinivasmaths4141@gmail.com abstract. in this paper, first, we deal with new metric space sm-metric space that combines multiplicative metric space and s-metric space. we generate a common fixed point theorem in a sm-metric space using the notions of reciprocally continuous mappings, faintly compatible mappings and occasionally weakly compatible mappings (owc). we are also studying the well-posedness of sm metric space. further, some examples are presented to support our outcome. 1. introduction the idea of multiplicative metric space(mms for short ) was first introduced by bashirove [1] in 2008.ozaksar and cevical [2] investigated and proved the properties of mms. following that, several theorems like [3] and [4] in this area of mms were developed. sedhi.s et al. [5] introduced a new structure of s-metric space and developed some fixed point theorems. pant et al. [6] used the concept of reciprocally continuous mappings which is weaker than continuous mappings. in this article, we use the multiplicative metric space and s metric space and generated a new sm-metric space [7]. we used the concept of occasionally weakly compatible (owc for shot) [9]mappings, reciprocally continuous and faintly compatible mappings [10] to generate a common fixed point theorem in sm-metric space. we also discuss the well-posedness property [11] in sm-metric space. furthermore, some examples are provided to support our new findings. received august 23rd, 2021; accepted october 13th, 2021; published november 5th, 2021. 2010 mathematics subject classification. 54h25. key words and phrases. multiplicative metric space; s-metric space; sm-metric space; occasionally weakly compatible; reciprocally continuous; faintly compatible mappings; well-posed property. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 915 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-915 int. j. anal. appl. 19 (6) (2021) 916 2. mathematical preliminaries: definition 2.1. [1] ” let x 6= φ. an operator δ : x2 → r+ be a multiplicative metric space (mms) holding in the conditions below: (m1) δ(α,β) ≥ 1, and δ(α,β) = 1 ⇐⇒ α = β (m2) δ(α,β) = δ(β,α) (m3) δ(α,β) ≤ δ(α,γ)δ(γ,β),∀α,β,γ ∈ x. mapping δ together with x, (x,δ) is called a mms”. a three-dimensional metric space was proposed by sedghi et al . [5], and it is called s-metric space. definition 2.2. [5] ” let x 6= φ defined on a function s : x3 → [0,∞) satisfying: (s1) s(α,β,γ) ≥ 0; (s2) s(α,β,γ) = 0; ⇐⇒ α = β = γ, (s3) s(α,β,γ) ≤ s(α,α,ω) + s(β,β,ω) + s(γ,γ,ω),∀α,β,γ,ω ∈ x. the pair (x,s) is known as s-metric space on x ”. we now present the concept of sm-metric space which is consolidation of multiplicative metric space defined by bashirov [1] and s-metric space defined by sedgi [5] by as follows definition 2.3. [7] ” let x 6= φ . a function sm : x3 → r+ holding the conditions below: (ms1) sm(α,β,γ) ≥ 1 (ms2) sm(α,β,γ) = 1 ⇐⇒ α = β = γ (ms3) sm(α,β,γ) ≤ sm(α,α,ω)sm(β,β,ω)sm(γ,γ,ω),∀α,β,γ,ω ∈ x. mapping sm together with x, (x,sm) is known as sm-metric space.” example 2.1. ” let x 6= φ,sm : x3 → [0,∞) by sm(α,β,γ) = a |α−γ|+|β−γ|, where α,β,γ,a ∈ x, then (x,sm) is a sm -metric space on x.” example 2.2. let x=r+ ,define sm : x3 → [0,∞) by sm(α,β,γ) = a |β+γ−2α|+|β−γ|, where α,β,γ ∈ x, then (x,sm) is a sm -metric space on x. now we present some definitions in sm -metric space. definition 2.4. [7] suppose (x,sm) is a sm-metric space, a sequence {αk}∈ x is called ( 2.4.1) cauchy sequence ⇐⇒ sm(αk,αk,αl) → 1, for all k,l →∞; int. j. anal. appl. 19 (6) (2021) 917 ( 2.4.2) convergent ⇐⇒ ∃α ∈ x such that sm(αk,αk,α) → 1 as k →∞; ( 2.4.3) is complete if every cauchy sequence is convergent. definition 2.5. [8] ” the mappings g and i be compatible mappings in sm-metric space if sm(giαk,giαk,igαk) = 1, whenever a sequence {αk} in x such that limk→∞gαk = limk→∞iαk = η for some η ∈ x. ” definition 2.6. [8] ”let g and i be weakly compatible mappings in sm-metric space if for all η ∈ x, gη = iη =⇒ giη = igη”. definition 2.7. [9] ” suppose g and i are mappings in sm-metric are said to be occasionally weakly compatible (owc for shot) iff ∃η ∈ x such that gη = iη =⇒ giη = igη. ” example 2.3. let x = [0,∞) is a sm-metric space on x , sm(α,β,γ) = a |α−β|+|β−γ|+|γ−α|, for every α,β,γ ∈ x. construct two self maps g and i as g(α) = 3α− 2 and i(α) = α2. consider a sequence {αk} given by αk = 2 + 1k for k ≥ 0. g(αk) = 3(2 + 1 k ) − 2 = 4 and i(αk) = (2 + 1k ) 2 = 4 as k →∞ therefore gαk = iαk = 4 6= φ. moreover, gi(βk) = gi(2 + 1 k ) = g(2 + 1 k )2 = g(4 + 4 1 k + 1 k2 ) = 3(4 + 4 1 k + 1 k2 ) − 2 =10 and ig(βk) = ig(2 + 1 k ) = i(3(2 + 1 k ) − 2) = i(4 + 3 k ) = (4 + 3 k )2 = 16 as k →∞. this gives sm(giαk,giαk,igαk) = sm(10, 10, 16) 6= 1. hence, (g,i) is not compatible. now g(1)= i(1)=1 also gi(1)= ig(1)=1 =⇒ gi(1)= ig(1). g(2)=4, i(2)=4 also gi(2)=10 , ig(2)=16 =⇒ gi(2) 6= ig(2). as a result g and i have owc, but not weakly compatible. definition 2.8. [10] ” two self maps g and i in sm-metric space as conditionallycompatible if there exists a sequence {αk}∈ x such that gαk = iαk 6= φ,∃ a sequence {βk}∈ x such that gβk = iβk → η for some η ∈ x and sm(giβk,giβk,igη) = 1 as k →∞. ” definition 2.9. [10] ” two self maps g and i in sm-metric space are called as faintly compatible iff (g , i )is conditionallycompatible and g and i commute on a non -empty subset of the set of coincidence points if the collection of coincidence points is nonempty.” int. j. anal. appl. 19 (6) (2021) 918 definition 2.10. [6] ”a reciprocally continuous mappings g and i of a sm-metric space is defined as sm(giαk,giαk,iη) = 1andsm(igαk,igαk,gη) = 1 letting k → ∞ if there exists a sequence {αk} ∈ x such that limk→∞gαk = limk→∞iαk = η as η ∈ x. ” example 2.4. let x = [0,∞) is a sm-metric space on x , sm(α,β,γ) = a |α−β|+|β−γ|+|γ−α|, for every α,β,γ ∈ x. construct two self maps g and i as g(α) = α2 − 3α + 2 and i(α) = 3α2 − 7α + 2. consider a sequence {αk} given by αk = 2 + 1k for k ≥ 0. then g(αk) = (2 + 1 k )2 − 3(2 + 1 k ) + 2 = 0 and i(αk) = 3(2 + 1 k )2 − 7(2 + 1 k ) + 2 = 0 as k → ∞ therefore limk→∞gαk = limk→∞iαk = 0 6= φ. moreover, gi(αk) = gi(2 + 1 k ) = g[3(2 + 1 k )2 −7(2 + 1 k ) + 2] = g( 3 k2 + 5 k ) = ( 3 k2 + 5 k )2 −3( 3 k2 + 5 k ) + 2 = 2 and ig(αk) = ig(2 + 1 k ) = i[(2 + 1 k )2−3(2 + 1 k ) + 2] = i( 1 k2 + 1 k ) = 3( 1 k2 + 1 k )2−7( 1 k2 + 1 k ) + 2 = 2 as k →∞. =⇒ sm(giαk,giαk,igαk) = sm(2, 2, 2) = 1. hence, (g,i) is compatible. consider another sequence {βk} given by βk = 1k for k ≥ 0. g(βk) = ( 1 k2 − 3 k + 2) = 2 and i(βk) = ( 3 k2 − 7 k + 2) = 2 as k →∞ therefore limk→∞gβk = limk→∞iβk = 2. further gi(βk) = gi( 1 k ) = g( 3 k2 − 7 k + 2) = ( 3 k2 − 7 k + 2)2 − 3( 3 k2 − 7 k + 2) + 2 = 0 and ig(βk) = ig( 1 k ) = i( 1 k2 − 3 k + 2) = 3( 1 k2 − 3 k + 2) − 7( 1 k2 − 3 k + 2) + 2 = −6 as k →∞. this gives sm(giαk,igαk,η) = sm(0, 0,−6) 6= 1. the pair (g,i) is not compatible hence (g,i) is conditionally compatible. compatibility is distinct from the concept of conditional compatibility, now g(2)=0,i(2)=0 and gi(2)=2,ig(2)=2. also g(0)=i(0)=2 and gi(0)=ig(0)=0. hence the pair (g,i) is faintly compatible. as a result, the mappings g and i have faintly compatible, but they are not compatible. definition 2.11. [11] ”the mappings g and i of a sm-metric space are called well-posed if • g and i have a unique common fixed point η in x • if αk ∈ x such that sm(gαk,gαk,αk) = 1 and sm(iαk,iαk,αk) = 1 as k → ∞ we have sm(αk,αk,η) = 1 as k →∞.” int. j. anal. appl. 19 (6) (2021) 919 3. main theorem theorem: suppose g, h, i and j are self-mapping in a complete sm-metric space x, suppose that there exist λ ∈ (0, 12 ) such that the conditions (3.1.1) g(x) ⊆ j(x) and h(x) ⊆ i(x) (3.1.2) sm(gα,gα,hβ) ≤ { max[sm(gα,gα,iα)sm(hβ,hβ,jβ),sm(gα,gα,jβ)sm(iα,iα,hβ), sm(gα,gα,jβ)sm(hβ,hβ,jβ),sm(gα,gα,iα)sm(hβ,hβ,iα)] }λ (3.1.3) the pair (h,j) is owc (3.1.4) and the pair (g,i) is reciprocally continuous and faintly compatible. then the common fixed point problem of g, h, i and j is well-posed. proof: we begin with using (3.1.1), then there is a point α0 ∈ x, such that gα0 = jα1 = β0. for this point α1then there ∃α2 ∈ x such that hα1 = iα2 = β1. in general, by induction choose αk+1 so that β2k = gα2k = jα2k+1 and β2k+1 = hα2k+1 = iα2k+2 for k ≥ 0. we show that {βk} is a cauchy sequence in sm metric space . indeed, it follows that sm(β2k,β2k,β2k+1) = sm(gα2k,gα2k,hα2k+1) ≤ max { sm(gα2k,gα2k,iα2k)sm(hα2k+1,hα2k+1,jα2k+1), sm(gα2k,gα2k,jα2k+1)sm(hα2k+1,hα2k+1,iα2k), sm(gα2k,gα2k,jα2k+1)sm(hα2k+1,hα2k+1,jα2k+1), sm(gα2k,gα2k,iα2k)sm(hα2k+1,hα2k+1,iα2k) }λ int. j. anal. appl. 19 (6) (2021) 920 sm(β2k,β2k,β2k+1) ≤ max { sm(β2k,β2k,β2k−1)sm(β2k+1,β2k+1,β2k), sm(β2k,β2k,β2k)sm(β2k+1,β2k+1,β2k−1), sm(β2k,β2k,β2k)sm(β2k+1,β2k+1,β2k), sm(β2k,β2k,β2k−1)sm(β2k+1,β2k+1,β2k−1) }λ on simplification sm(β2k,β2k,β2k+1) ≤ sm(β2k−1,β2k−1,β2k+1)λ. sm(β2k,β2k,β2k+1) ≤{sm(β2k−1,β2k−1,β2k)sm(β2k,β2k,β2k+1)}λ. s1−λm (β2k,β2k,β2k+1) ≤ s λ m(β2k−1,β2k−1,β2k). sm(β2k,β2k,β2k+1) ≤ s λ 1−λ m (β2k−1,β2k−1,β2k). sm(β2k,β2k,β2k+1) ≤ spm(β2k−1,β2k−1,β2k). where p = λ 1 −λ now this gives sm(βk,βk,βk+1) ≤ spm(βk−1,βk−1,βk) ≤ s p2 m (βk−2,βk−2,βk−1) ≤ ···s pn m (β0,β0,βn) by using triangular inequality, sm(βk,βk,βn) ≤ sp k m (β0,β0,βl) ≤ s pk+1 m (β0,β0,βn) ≤ ···s pn−1 m (β0,β0,βn) hence {βk} is a cauchy sequence in sm-metric space. now x being complete in sm-metric space ∃η ∈ x such that limk→∞βk → η . consequently, the sub sequences {gα2k}, {iα2k}, {jα2k+1} and {hα2k+1} of {βk} also converges to the point η ∈ x. since the pair (g,i) is faintly compatible mappings, so that ∃ another sequence νk ∈ x such that limk→∞gνk = limk→∞iνk = ω for ω ∈ x satisfying limk→∞s(giνk,giνk,igνk) = 1 and the pair (g,i) is reciprocally continuous sm(giνk,giνk,iω) = 1, and sm(igνk,igνk,gω) = 1. as k →∞. (3.1) gω = iω int. j. anal. appl. 19 (6) (2021) 921 on putting α = ω and β = α2k+1 in (3.1.2) we get sm(gω,gω,hα2k+1) ≤ { max[sm(gω,gω,iω)sm(hα2k+1,hα2k+1,jα2k+1), sm(gω,gω,jα2k+1)sm(iω,iη,hα2k+1), sm(gω,gω,jα2k+1)sm(hα2k+1,hα2k+1,jα2k+1), sm(gω,gω,iω)sm(hα2k+1,hα2k+1,iω)] }λ and sm(gω,gω,η) ≤ { max[sm(gω,gω,iω)sm(η,η,η),sm(gω,gω,η)sm(iω,iω,η), sm(gω,gω,η)s ∗(η,η,η),sm(gω,gω,iω)sm(η,η,iω)] }λ which gives sm(gω,gω,η) ≤ { max[sm(gω,gω,gω)sm(η,η,η),sm(gω,gω,η)sm(gω,gω,η), sm(gω,gω,η)sm(η,η,η),sm(gω,gω,gω)sm(η,η,gω)] }λ implies sm(gω,gω,η) ≤ { max[1,s2m(gω,gω,η),sm(gω,gω,η),sm(gω,gω,η)] }λ this gives sm(gω,gω,η) ≤ { s2λm (gω,gω,η) } this implies gω = η. therefore gω = iω = η.(3.2) since the pair (g,i) is faintly compatible, so that gω = iω this gives giω = igω this implies gη = iη. by using the inequality (3.1.2) on putting α = η and β = α2k+1 we get sm(gη,gη,hα2k+1) ≤ { max[sm(gη,gη,iη)sm(hα2k+1,hα2k+1,jα2k+1), sm(gη,gη,jα2k+1)sm(iη,iη,hα2k+1), sm(gη,gη,jα2k+1)sm(hα2k+1,hα2k+1,jα2k+1), sm(gη,gη,iη)sm(hα2k+1,hα2k+1,iη)] }λ int. j. anal. appl. 19 (6) (2021) 922 and sm(gη,gη,η) ≤ { max[sm(gη,gη,iη)sm(η,η,η),sm(gη,gη,η)sm(iη,iη,η), sm(gη,gη,η)sm(η,η,η),sm(gη,gη,iη)smη,η,iη)] }λ which gives sm(gη,gη,η) ≤ { max[sm(gη,gη,gη)sm(η,η,η),sm(gη,gη,η)sm(gη,gη,η), sm(gη,gη,η)sm(η,η,η),sm(gη,gη,gη)sm(η,η,gη)] }λ implies sm(gη,gη,η) ≤ { max[1,sm 2(gη,gη,η),sm(gη,gη,η),sm(gη,gη,η)] }λ which implies sm(gη,gη,η) ≤ { sm 2λ(gη,gη,η) } =⇒ gη = η.(3.3) gη = iη = η.(3.4) =⇒ η = gη ∈ g(x) ⊆ j(x) =⇒ gη = jv for some v ∈ x. gη = iη = jv = η.(3.5) using the inequality (3.1.2) on putting α = η and β = v we have sm(gη,gη,hv) ≤ { max[sm(gη,gη,iη)sm(hv,hv,jv),sm(gη,gη,jv)sm(iη,iη,hv), sm(gη,gη,jv)sm(hv,hv,jv),sm(gη,gη,iη)sm(hv,hv,iη)] }λ this implies sm(η,η,hv) ≤ { max[sm(η,η,η)sm(hv,hv,η),sm(η,η,η)sm(η,η,hv), sm(η,η,η)sm(hv,hv,η),sm(η,η,η)sm(hv,hv,η)] }λ which implies sm(η,η,hv) ≤ { max[sm(hv,hv,η),sm(η,η,hv), sm(hv,hv,η),sm(hv,hv,η)] }λ int. j. anal. appl. 19 (6) (2021) 923 this gives sm(η,η,hv) ≤ { sm(hv,hv,η) }λ which gives hv = η. gη = iη = jv = hv = η.(3.6) again (h,j) is owc with v ∈ x so that hv = jv =⇒ hjv = jhv which implies that hη = jη. using the inequality (3.1.2) and take α = η and β = η we get sm(gη,gη,hη) ≤ { max[sm(gη,gη,iη)sm(hη,hη,jη),sm(gη,gη,jη)sm(iη,iη,hη), sm(gη,gη,jη)sm(hη,hη,jη),sm(gη,gη,iη)sm(hη,hη,iη)] }λ this implies sm(η,η,hη) ≤ { max[sm(η,η,η)sm(hη,hη,η),sm(η,η,η)sm(η,η,hη), sm(η,η,η)sm(hη,hη,η),sm(η,η,η)sm(hη,hη,η)] }λ where sm(η,η,hη) ≤ { max[sm(hη,hη,η),sm(η,η,hη),sm(hη,hη,η),sm(hη,hη,η)] }λ this gives sm(η,η,hη) ≤ { sm(hη,hη,η) }λ this gives hη = η. hη = jη = η.(3.7) from( 3.4) and ( 3.7) gη = iη = jη = hη = η.(3.8) =⇒ η is a common fixed point for the mappings g,h,i and j. for the proof of well-posed property suppose ρ(ρ 6= η) is one more fixed point of g,i,h and j int. j. anal. appl. 19 (6) (2021) 924 i.e gρ = iρ = hρ = jρ = ρ. using the inequality (3.1.2) take α = ρ and β = η we have sm(gρ,gρ,hη) ≤ { max[sm(gρ,gρ,iη)sm(hη,hη,jη),sm(gρ,gρ,jη)sm(iρ,iρ,hη), sm(gρ,gρ,jη)sm(hη,hη,jη),sm(gρ,gρ,iρ)sm(hη,hη,iρ)] }λ this gives sm(ρ,ρ,η) ≤ { max[sm(ρ,ρ,η)sm(η,η,η),sm(ρ,ρ,η)sm(ρ,ρ,η), sm(ρ,ρ,η)sm(η,η,η),sm(ρ,ρ,ρ)sm(η,η,ρ)] }λ this gives sm(ρ,ρ,η) ≤ { max[1,sm(ρ,ρ,η), 1, 1] }λ which gives ∴ sm(ρ,ρ,η) ≤ sm(ρ,ρ,η)λ this gives ρ = η. hence η is the unique common fixed point of g,h,i and j suppose {αk} be a sequence in x such that sm(gαk,gαk,αk) = sm(iαk,iαk,αk) = 1 and sm(hαk,hαk,αk) = sm(jαk,jαk,αk) = 1 as k →∞. we have to show that sm(αk,αk,η) = 1, sm(αk,αk,η) ≤ sm(gαk,gαk,η)sm(gαk,gαk,αk) sm(αk,αk,η) ≤{ max[sm(gαk,gαk,iαk)sm(hη,hη,jη),sm(gαk,gαk,jη)sm(iαk,iαk,hη), sm(gαk,gαk,jη)sm(hη,hη,jη),sm(gαk,gαk,iαk)sm(hη,hη,iαk)] }λ sm(gαk,gαk,αk) this gives sm(αk,αk,η) ≤{ max[sm(gαk,gαk,gη)sm(iαk,iαk,iη)sm(hη,hη,jη),sm(gαk,gαk,jη)sm(iαk,iαk,hη), sm(gαk,gαk,jη)sm(hη,hη,jη),sm(gαk,gαk,iαk)sm(hη,hη,iαk)] }λ sm(gαk,gαk,αk) int. j. anal. appl. 19 (6) (2021) 925 which gives sm(αk,αk,η) ≤{ max[sm(αk,αk,η)sm(αk,αk,η))sm(η,η,η),sm(αk,αk,η)sm(αk,αk,η), sm(αk,αk,η)sm(η,η,η),sm(αk,αk,αk)sm(η,η,αk)] }λ sm(gαk,gαk,αk) therefore sm(αk,αk,η) ≤ { sm(αk,αk,η) }λ sm(gαk,gαk,αk) sm(αk,αk,η) 1−λ ≤ sm(gαk,gαk,αk) sm(αk,αk,η) ≤ s 1 1−λ m (gαk,gαk,αk) sm(αk,αk,η) = 1 as k →∞. thus g,h,i and j is well-posed. 4. example suppose x = [0, 1] ,smmetric space by sm(α,β,γ) = e |α−β|+|β−γ|+|γ−α|, when α,β,γ ∈ x. define g ,i ,h j:xxx → x as follows g(α) =   1−α 2 if 0 ≤ α ≤ 1 3 ; 3−2α 4 if 1 3 < α ≤ 1. j(α) =   1+4α 7 if 0 ≤ α ≤ 1 3 ; 2α+1 4 if 1 3 < α ≤ 1. and h(α) =   2−α 5 if 0 ≤ α ≤ 1 3 ; 3−2α 4 if 1 3 < α ≤ 1. i(α) =   2α+1 5 if 0 ≤ α ≤ 1 3 ; α+5 11 if 1 3 < α ≤ 1. then g(x)=( 7 12 , 1 4 ] and j(x) =[ 1 7 , 1 3 ] ∪ ( 5 12 , 3 4 ] . and also h(x)=( 7 21 , 1 4 ] and i(x)=[ 1 5 , 1 3 ] ∪ ( 16 33 , 6 11 ] thisimpliesimpliesg(x) ⊆ j(x) and h(x) ⊆ i(x) hence the inequality (3.1.1) holds. take a sequence {αk} as αk = 12 + 1 k as k ≥ 0. now g(αk) = g( 1 2 + 1 k )= 3−2( 1 2 + 1 k ) 4 = 1 2 and j(αk) = j( 1 2 + 1 k )= 2( 1 2 + 1 k )+1 4 = 1 2 . and also h(αk) = h( 1 2 + 1 k )= 3−2( 1 2 + 1 k ) 4 = 1 2 and i(αk) = i( 1 2 + 1 k )= 5+( 1 2 + 1 k ) 11 = 1 2 . ∴ gαk=jαk= 1 2 and hαk=jαk= 1 2 as k →∞. then gi(αk) = gi( 1 2 + 1 k ) = g( 1 2 + 1 11k ) = 3−2( 1 2 + 1 11k ) 4 = 1 2 and ig(αk) = ig( 1 2 + 1 k ) = i( 1 2 − 1 2k ) = 5+( 1 2 − 1 2k ) 11 = 1 2 as k →∞. hj(αk) = hj( 1 2 + 1 k ) = h( 5+( 1 2 + 1 k ) 11 ) = h( 1 2 + 1 11k ) = 3−2( 1 2 + 1 11k ) 4 = 1 2 int. j. anal. appl. 19 (6) (2021) 926 and jh(αk) = jh( 1 2 + 1 k ) = j( 3−2( 1 2 + 1 k ) 4 ) = i( 1 2 − 1 2k ) = 1+2( 1 2 − 1 2k ) 4 = 1 2 as k →∞. limk→∞sm(giαk,giαk,igαk) = sm( 1 2 , 1 2 , 1 2 ) = 1 and limk→∞sm(hjαk,hjαk,jhαk) = sm( 1 2 , 1 2 , 1 2 ) = 1 hence the pairs (g,i) and (h,j) are satisfies compatible property . take another sequence {βk} as βk = 13 − 1 k as k ≥ 0. now g(βk) = g( 1 3 − 1 k )= 1−( 1 3 −1 k ) 2 = 1 3 = and i(βk) = i( 1 3 − 1 k )= 2+( 1 3 −1 k ) 5 = 1 3 . and also h(βk) = h( 1 3 − 1 k )= 2−( 1 3 −1 k ) 5 = 1 3 and j(βk) = j( 1 3 − 1 k )= 4( 1 3 −1 k )+1 7 = 1 3 as k →∞. ∴ gβk=iβk= 1 3 = η similarly hβk=jβk= 1 3 = η as k →∞. further gi(βk) = gi( 1 3 − 1 k )=g( 2( 1 3 −1 k )+1 5 )=g( 1 3 − 2 5k )= 1−( 1 3 − 2 5k ) 2 = 1 3 and ig((βk) = ig( 1 3 − 1 k )=i( 1−( 1 3 −1 k 2 )=i( 1 3 + 1 2k )= 5+( 1 3 + 1 2k ) 11 = 16 33 hj(βk) = hj( 1 3 − 1 k ) = h( 4( 1 3 −1 k )+1 7 = h( 1 3 − 4 7k ) = 2−( 1 3 + 4 7k ) 11 = 1 3 and jh(βk) = jh( 1 3 − 1 k ) = j( 2−( 1 3 −1 k 5 ) = j( 1 3 + 1 5k ) = 2( 1 3 + 1 5k )+1 4 = 5 12 as k →∞. this implies limk→∞sm(giβk,giβk,igβk) = sm( 1 3 , 1 3 , 66 33 ) 6= 1 which shows that the pairs (g,i) is faintly compatible mappings. moreover i( 1 2 ) = 1 2 and g( 1 2 ) = 1 2 and also limk→∞sm(giαk,giαk,iw) = sm( 1 2 , 1 2 , 1 2 ) = 1 limk→∞sm(igαk,igαk,gw) = sm( 1 2 , 1 2 , 1 2 ) = 1 this shows that the pairs (g,i) is reciprocally continuous. the inequity (3.1.4) holds. further h( 1 2 ) = 5+ 1 2 11 = 1 2 and j( 1 2 ) = 2 1 2 +1 4 = 1 2 . ∴ h( 1 2 ) = j( 1 2 ) = ( 1 2 ) where 1 2 ∈ x. and also hj( 1 2 ) = 1 2 and jh( 1 2 ) = 1 2 , =⇒ hj( 1 2 ) = jh( 1 2 ) = 1 2 . moreover, h( 1 3 ) = 2−1 3 5 = 1 3 and j( 1 3 ) = 4 1 3 +1 7 = 1 3 . ∴ h( 1 3 ) = j( 1 3 ) = ( 1 3 ) where 1 3 ∈ x. and also hj( 1 3 ) = 1 3 and jh( 1 3 ) = 1 3 , =⇒ hj( 1 3 ) = jh( 1 3 ) = 1 3 . which shows that the pair(h,j) satisfies owc. so that the inequity (3.1.3) holds. further more sm(gβk,gβk,βk) = sm( 1 3 , 1 3 , 1 3 ) = 1,sm(iβk,iβk,βk) = sm( 1 3 , 1 3 , 1 3 ) = 1, when k → ∞. which implies limk→∞sm(βk,βk,η) = limk→∞sm( 1 3 , 1 3 , 1 3 ) = 1, case-i let α,β ∈ [0, 1 3 ],while we have sm(α,β,γ) = e |α−γ|+|β−γ| in the inequality(3.1.2) putting α = 1 3 and β = 1 4 implies sm(0.375, 0.375, 0.36) ≤{ max[sm(0.375, 0.375, 0.28)sm(0.36, 0.36, 0.286)sm(0.375, 0.375, 0.286)sm(0.36, 0.36, 0.28), sm(0.375, 0.375, 0.286)sm(0.36, 0.36, 0.286),sm(0.375, 0.375, 0.28)sm(0.36, 0.36, 0.28)] }λ =⇒ e0.03 ≤ { max[e0.19e0.15,e0.18e0.16,e0.18e0.15,e0.19e0.16] }λ int. j. anal. appl. 19 (6) (2021) 927 e0.03 ≤{ max[e0.34,e0.34,e0.33,e0.35]}λ =⇒ e0.03 ≤ e0.35λ which gives λ = 0.08, where λ ∈ (0, 1 2 ). case-ii let α,β ∈ [ 1 2 , 1], then sm(α,β,γ) = e |α−γ|+|β−γ| in the inequality(3.1.2) putting α = 2 3 and β = 3 4 implies sm(0.42, 0.42, 0.375) ≤{ max[sm(0.42, 0.42, 0.52)sm(0.375, 0.375, 0.47)sm(0.42, 0.42, 0.47)sm(0.375, 0.375, 0.52), sm(0.42, 0.42, 0.47)sm(0.375, 0.375, 0.47),sm(0.42, 0.42, 0.52)sm(0.375, 0.375, 0.52)] }λ =⇒ e0.09 ≤ { max[e0.2e0.19,e0.1e0.29,e0.1e0.19,e0.2e0.29] }λ e0.09 ≤{ max[e0.39,e0.39,e0.29,e0.49]}λ =⇒ e0.09 ≤ e0.49λ which gives λ = 0.18, where λ ∈ (0, 1 2 ). hence the inequality(3.1.2) holds. the verification in the remaining intervals is also simple. it can be observed that 1 2 is a unique common fixed point of g,h,i and j. 5. conclusion: this article, aimed to prove a common fixed point theorem in sm-metric space using conditions owc, reciprocally continuous and faintly compatible mappings. also proved the wellposed property. further our result is supported with a suitable example. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.e. bashirov, e.m. kurpınar, a. özyapıcı, multiplicative calculus and its applications, j. math. anal. appl. 337 (2008), 36–48. [2] m. özavşar, fixed points of multiplicative contraction mappings on multiplicative metric spaces, j. eng. technol. appl. sci. 2 (2017), 65–79. [3] v. srinivas, k. mallaiah, some results on weaker class of compatible mappings in s-metric space, malaya j. mat. 8 (2020), 1132-1137. [4] v. srinivas, k. satyanna. some results in mnrger space by using sub compatible, faintly compatible mappings, malaya j, mat. 9 (2021), 725-730. int. j. anal. appl. 19 (6) (2021) 928 [5] s. sedghi, n. shobkolaei, m. shahraki, t. došenović, common fixed point of four maps in s-metric spaces, math. sci. 12 (2018), 137–143. [6] r.p. pant, common fixed points of four mappings, bull. cal. math. soc. 90 (1998), 281–286. [7] v. naga raju some properties of multiplicative s-metric spaces, adv. math.: sci. j. 10 (2021), 105–109. [8] g. jungck, b.e rhoades, fixed points for set valued functions without continuity, indian j. pure appl. math. 29 (1998), 227-238. [9] m.a. al-thagafi, n. shahzad generalized -non expansive self maps and invariant approximations acta math. sin. (engl. ser.), 24 (2008), 867-876. [10] r.k. bisht, n. shahzad, faintly compatible mappings and common fixed points, fixed point theory appl. 2013 (2013), 156. [11] f.s. de blasi, j. myjak, sur la porosité de l’ensemble des contractions sans point fixe. c. r. acad. sci. paris 308 (1989), 51–54. 1. introduction 2. mathematical preliminaries: 3. main theorem 4. example 5. conclusion: references international journal of analysis and applications volume 18, number 4 (2020), 572-585 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-572 micro separation axioms hariwan z. ibrahim∗ department of mathematics, faculty of education, university of zakho, zakho, kurdistan region, iraq ∗corresponding author: hariwan.ibrahim@uoz.edu.krd abstract. in this paper, some new types of spaces are defined and studied in micro topological spaces namely, micro t0, micro t1, micro t2, micro r0 and micro r1 spaces. properties and the relationships of these spaces are introduced. finally, the relationships between these spaces and the related concepts are investigated. 1. introduction topology and its branches have become hot topics not only for almost all fields of mathematics, but also for many areas of science such as chemistry [21], and information systems [23]. the notion of rough sets was introduced by pawlak [22]. rough set theory is an important tool for data mining. lower and upper approximation operators are two important basic concepts in the rough set theory. the classical pawlak rough approximation operators are based on equivalence relations and have been extended to relation-based generalized rough approximation operators. the notation of nano topology was introduced by thivagar et al [25, 26] which was defined in terms of approximations and boundary region of a subset of an universe using an equivalence relation on it. the concept of micro topology was introduced and investigated by chandrasekar [1]. the notion of micro t1 2 space was introduced by ibrahim [16]. in the past few years, different forms of separation axioms have been studied [2–14, 17–20]. received february 21st, 2020; accepted march 19th, 2020; published may 11th, 2020. 2010 mathematics subject classification. primary 22a05, 22a10; secondary 54c05. key words and phrases. micro topology; micro open set; micro t0; micro t1; micro r0; micro r1. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 572 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-572 int. j. anal. appl. 18 (4) (2020) 573 2. preliminaries the following recalls requisite ideas and preliminaries necessitated in the sequel of this work. definition 2.1. [24] let u be a nonempty finite set of objects called the universe and r be an equivalence relation on u named as the indiscernibility relation. elements belonging to the same equivalence class are said to be indiscernible with one another. the pair (u,r) is said to be the approximation space. let x ⊆ u. (1) the lower approximation of u with respect to r is the set of all objects, which can be for certain classified as x with respect to r and its is denoted by lr(x). that is, lr(x) = ⋃ x∈u{r(x) : r(x) ⊆ x}, where r(x) denotes the equivalence class determined by x. (2) the upper approximation of u with respect to r is the set of all objects, which can be possibly classified as x with respect to r and it is denoted by lr(x). that is, lr(x) = ⋃ x∈u{r(x) : r(x)∩x 6= φ}. (3) the boundary region of u with respect to r is the set of all objects, which can be classified neither as x nor as not-x with respect to r and it is denoted by br(x). that is, br(x) = lr(x)−lr(x). definition 2.2. [25, 26] let u be the universe, r be an equivalence relation on u and τr(x) = {u,φ,lr(x),lr(x),br(x)}, where x ⊆ u. then, τr(x) satisfies the following axioms: (1) u and φ ∈ τr(x). (2) the union of the elements of any subcollection of τr(x) is in τr(x). (3) the intersection of the elements of any finite subcollection of τr(x) is in τr(x). that is, τr(x) is a topology on u called the nano topology on u with respect to x. we call (u,τr(x)) as the nano topological space. the elements of τr(x) are called as nano open sets. a subset f of u is nano closed if its complement is nano open. definition 2.3. [1] let (u,τr(x)) be a nano topological space. then, µr(x) = {n ∪ (n ′ ∩µ) : n,n ′ ∈ τr(x) and µ /∈ τr(x)} is called the micro topology on u with respect to x. the triplet (u,τr(x),µr(x)) is called micro topological space and the elements of µr(x) are called micro open sets and the complement of a micro open set is called a micro closed set. definition 2.4. [1] the micro closure of a set a is denoted by mic-cl(a) and is defined as mic-cl(a) = ∩{b : b is micro closed and a ⊆ b}. definition 2.5. [1] let (u,τr(x),µr(x)) be a micro topological space. let a and b be any two subsets of u. then: (1) a is a micro closed set if and only if mic-cl(a) = a. (2) if a ⊆ b, then mic-cl(a) ⊆ mic-cl(b). (3) mic-cl(mic-cl(a)) = mic-cl(a). int. j. anal. appl. 18 (4) (2020) 574 remark 2.1. [15] let (u,τr(x),µr(x)) be a micro topological space and a be any subset of u. then: (1) mic-cl(a) is micro closed. (2) a ⊆ mic-cl(a). (3) x ∈ mic-cl(a) if and only if for every micro open subset l of u containing x, a∩l 6= φ. definition 2.6. [16] let (u,τr(x),µr(x)) be a micro topological space. a subset a of u is said to be a micro generalized closed (briefly, micro g.closed) if mic-cl(a) ⊆ l whenever a ⊆ l and l is a micro open set in u. definition 2.7. [16] let (u,τr(x),µr(x)) be a micro topological space. then, a subset a of u is called a micro difference set (briefly, md-set) if there are l,k ∈ µr(x) such that l 6= u and a = l\k. definition 2.8. [16] let (u,τr(x),µr(x)) be a micro topological space and a be a subset of u. then, the micro kernel of a denoted by mker(a) is defined to be the set mker(a) = ∩{l ∈ µr(x): a ⊆ l}. definition 2.9. [16] let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be micro t1 2 if every micro g.closed in u is micro closed. definition 2.10. [16] let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be micro symmetric if for x and y in u such that x ∈ mic-cl({y}) implies y ∈ mic-cl({x}). definition 2.11. [16] let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be: (1) micro d0 if for any pair of distinct points x and y of u there exists a md-set of u containing x but not y or a md-set of u containing y but not x. (2) micro d1 if for any pair of distinct points x and y of u there exists a md-set of u containing x but not y and a md-set of u containing y but not x. (3) micro d2 if for any pair of distinct points x and y of u there exist disjoint md-set g and e of u containing x and y, respectively. theorem 2.1. [16] let (u,τr(x),µr(x)) be a micro topological space. then, u is a micro t1 2 if and only if {x} is micro closed or micro open, for each x ∈ u. remark 2.2. [16] let (u,τr(x),µr(x)) be a micro topological space. if u is micro dk, then it is micro dk−1, for k = 1, 2. theorem 2.2. [16] let (u,τr(x),µr(x)) be a micro topological space. then, then the following statements are equivalent: int. j. anal. appl. 18 (4) (2020) 575 (1) u is a micro symmetric. (2) {x} is micro g.closed, for each x ∈ u. theorem 2.3. [16] let (u,τr(x),µr(x)) be a micro topological space and x ∈ u. then, y ∈ mker({x}) if and only if x ∈ mic-cl({y}). theorem 2.4. [16] let (u,τr(x),µr(x)) be a micro topological space and a be a subset of u. then, mker(a) = {x ∈ u: mic-cl({x}) ∩a 6= φ}. theorem 2.5. [16] let (u,τr(x),µr(x)) be a micro topological space. then, for any points x and y in u the following statements are equivalent: (1) mker({x}) 6= mker({y}). (2) mic-cl({x}) 6= mic-cl({y}). definition 2.12. [1] let (u,τr(x),µr(x)) and (v,τr(y ),µr(y )) be two micro topological spaces. then, a function f : u → v is said to be: micro-continuous if f−1(k) is micro open in u, for every micro open set k in v . 3. micro tk (k = 0, 1, 2) the following definitions are introduced via micro open sets. definition 3.1. let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be: (1) micro t0 if for each pair of distinct points x,y in u, there exists a micro open set l such that either x ∈ l and y /∈ l or x /∈ l and y ∈ l. (2) micro t1 if for each pair of distinct points x,y in u, there exist two micro open sets l and k such that x ∈ l but y /∈ l and y ∈ k but x /∈ k. (3) micro t2 if for each distinct points x,y in u, there exist two disjoint micro open sets l and k containing x and y respectively. theorem 3.1. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro t0 if and only if for each pair of distinct points x,y of u, mic-cl({x}) 6= mic-cl({y}). proof. necessity. let u be micro t0 and x,y be any two distinct points of u. then, there exists a micro open set l containing x or y, say x but not y. then, u \l is a micro closed set which does not contain x but contains y. since mic-cl({y}) is the smallest micro closed set containing y, then mic-cl({y}) ⊆ u \l and therefore x /∈ mic-cl({y}). consequently mic-cl({x}) 6= mic-cl({y}). sufficiency. suppose that x,y ∈ u, x 6= y and mic-cl({x}) 6= mic-cl({y}). let z be a point of u such that z ∈ mic-cl({x}) but z /∈ mic-cl({y}). we claim that x /∈ mic-cl({y}). for, if x ∈ mic-cl({y}) then int. j. anal. appl. 18 (4) (2020) 576 mic-cl({x}) ⊆ mic-cl({y}). this contradicts the fact that z /∈ mic-cl({y}). consequently x belongs to the micro open set u \mic-cl({y}) to which y does not belong. � theorem 3.2. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro t1 if and only if the singletons are micro closed sets. proof. let u be micro t1 and x any point of u. suppose y ∈ u\{x}, then x 6= y and so there exists a micro open set l such that y ∈ l but x /∈ l. consequently y ∈ l ⊆ u \{x}, that is u \{x} = ∪{l : y ∈ u \{x}} which is micro open. conversely, suppose {p} is micro closed for every p ∈ u. let x,y ∈ u with x 6= y. now, x 6= y implies y ∈ u \{x}. hence, u \{x} is a micro open set contains y but not x. similarly u \{y} is a micro open set contains x but not y. accordingly u is micro t1. � theorem 3.3. let (u,τr(x),µr(x)) be a micro topological space. then, the following statements are equivalent: (1) u is micro t2. (2) let x ∈ u. for each y 6= x, there exists a micro open set l containing x such that y /∈ mic-cl(l). (3) for each x ∈ u, ∩{mic-cl(l) : l ∈ µr(x) and x ∈ l} = {x}. proof. (1) ⇒ (2): since u is micro t2, then there exist disjoint micro open sets l and k containing x and y respectively. so, l ⊆ u \k. therefore, mic-cl(l) ⊆ u \k. so, y /∈ mic-cl(l). (2) ⇒ (3): if possible for some y 6= x, we have y ∈ mic-cl(l) for every micro open set l containing x, which then contradicts (2). (3) ⇒ (1): let x,y ∈ u and x 6= y. then, there exists a micro open set l containing x such that y /∈ miccl(l). let k = u \mic-cl(l), then y ∈ k, x ∈ l and l∩k = φ. thus, u is micro t2. � theorem 3.4. let (u,τr(x),µr(x)) be a micro topological space. then, then the following statements are hold: (1) every micro t2 space is micro t1. (2) every micro t1 space is micro t1 2 . (3) every micro t1 2 space is micro t0. proof. (1) the proof is straightforward from the definitions. (2) the proof is obvious by theorem 3.2. (3) let x and y be any two distinct points of u. by theorem 2.1, the singleton set {x} is micro closed or micro open. (a) if {x} is micro closed, then u \{x} is micro open. so y ∈ u \{x} and x /∈ u \{x}. therefore, we have u is micro t0. int. j. anal. appl. 18 (4) (2020) 577 (b) if {x} is micro open, then x ∈{x} and y /∈{x}. therefore, we have u is micro t0. � remark 3.1. let (u,τr(x),µr(x)) be a micro topological space. then, (1) if u is micro t1, then µr(x) is discrete micro topology on u. (2) u is micro t1 if and only if it is micro t2. example 3.1. consider u = {a,b,c} with u/r = {{a},{b,c}} and x = {a}. then, τr(x) = {u,φ,{a}}. if µ = {a,c}, then µr(x) = {u,φ,{a},{a,c}}. then, u is micro t0 but not micro t1 2 . example 3.2. consider u = {a,b,c} with u/r = {{c},{a,b}} and x = {a,b}. then, τr(x) = {u,φ,{a,b}}. if µ = {a,c}, then µr(x) = {u,φ,{a},{a,b},{a,c}}. then, u is micro t1 2 but not micro t1. remark 3.2. let (u,τr(x),µr(x)) be a micro topological space. if u is micro tk, then it is micro dk, for k = 0, 1, 2. proof. obvious. � theorem 3.5. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro d0 if and only if it is micro t0. proof. suppose that u is micro d0. then, for each distinct pair x,y ∈ u, at least one of x,y, say x, belongs to a md-set g but y /∈ g. let g = l1 \ l2 where l1 6= u and l1,l2 ∈ µr(x). then, x ∈ l1, and for y /∈ g we have two cases: (a) y /∈ l1 (b) y ∈ l1 and y ∈ l2. in case (a), x ∈ l1 but y /∈ l1. in case (b), y ∈ l2 but x /∈ l2. thus in both the cases, we obtain that u is micro t0. conversely, if u is micro t0, by remark 3.2, u is micro d0. � corollary 3.1. if u is micro d1, then it is micro t0. proof. follows from remark 2.2 and theorem 3.5. � here is an example which shows that the converse of corollary 3.1 is not true in general. example 3.3. consider u = {a,b,c} with u/r = {{a},{b,c}} and x = {a}. then, τr(x) = {u,φ,{a}}. if µ = {a,b}, then µr(x) = {u,φ,{a},{a,b}}. then, u is micro t0 but not micro d1 because there is no md-set containing c but not b. int. j. anal. appl. 18 (4) (2020) 578 corollary 3.2. let (u,τr(x),µr(x)) be a micro topological space. if u is micro t1, then it is micro symmetric. proof. in micro t1, every singleton is micro closed and therefore is micro g.closed. then, by theorem 2.2, u is micro symmetric. � corollary 3.3. let (u,τr(x),µr(x)) be a micro topological space. then, the following statements are equivalent: (1) u is micro symmetric and micro t0. (2) u is micro t1. proof. by corollary 3.2 and theorem 3.4, it suffices to prove only (1) ⇒ (2). let x 6= y and as u is micro t0, we may assume that x ∈ l ⊆ u \ {y} for some l ∈ µr(x). then, x /∈ mic-cl({y}) and hence y /∈ mic-cl({x}). there exists a micro open set k such that y ∈ k ⊆ u \{x} and thus u is a micro t1 space. � theorem 3.6. let (u,τr(x),µr(x)) be a micro topological space. if u is micro symmetric, then the following statements are equivalent: (1) u is micro t0. (2) u is micro t1 2 . (3) u is micro t1. proof. (1) ⇔ (3): obvious from corollary 3.3. (3) ⇒ (2) and (2) ⇒ (1): directly from theorem 3.4. � corollary 3.4. let (u,τr(x),µr(x)) be a micro topological space. if u is micro symmetric, then the following statements are equivalent: (1) u is micro t0. (2) u is micro d1. (3) u is micro t1. proof. (1) ⇒ (3). follows from corollary 3.3. (3) ⇒ (2) ⇒ (1). follows from remark 3.2 and corollary 3.1. � definition 3.2. a function f : u → v is called micro-open if the image of every micro open set in u is a micro open set in v . theorem 3.7. suppose that f : u → v is micro-open and surjective. then: int. j. anal. appl. 18 (4) (2020) 579 (1) if u is micro t0, then v is micro t0. (2) if u is micro t1, then v is micro t1. (3) if u is micro t2, then v is micro t2. proof. we prove only the case for micro t1 the others are similarly. let u be micro t1 and y1,y2 ∈ v with y1 6= y2. since f is surjective, so there exist distinct points x1,x2 of u such that f(x1) = y1 and f(x2) = y2. since u is micro t1, then there exist micro open sets g and h such that x1 ∈ g but x2 /∈ g and x2 ∈ h but x1 /∈ h. since f is micro-open, then f(g) and f(h) are micro open sets of v such that y1 = f(x1) ∈ f(g) but y2 = f(x2) /∈ f(g), and y2 = f(x2) ∈ f(h) but y1 = f(x1) /∈ f(h). hence, v is micro t1. � theorem 3.8. if f : u → v is a micro-continuous injective function and v is micro t2, then u is micro t2. proof. let x and y in u be any pair of distinct points, then there exist disjoint micro open sets a and b in v such that f(x) ∈ a and f(y) ∈ b. since f is micro-continuous, then f−1(a) and f−1(b) are micro open in u containing x and y respectively, we have f−1(a) ∩f−1(b) = φ. thus, u is micro t2. � 4. micro rk (k = 0, 1) definition 4.1. let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be micro r0 if l is a micro open set and x ∈ l, then mic-cl({x}) ⊆ l. theorem 4.1. let (u,τr(x),µr(x)) be a micro topological space. then, the following properties are equivalent: (1) u is micro r0. (2) for any f ∈ µcr(x), x /∈ f implies f ⊆ l and x /∈ l for some l ∈ µr(x). where µ c r(x) is the family of all micro closed sets. (3) for any f ∈ µcr(x), x /∈ f implies f ∩mic-cl({x}) = φ. (4) for any distinct points x and y of u, either mic-cl({x}) = mic-cl({y}) or mic-cl({x}) ∩ miccl({y}) = φ. proof. (1) ⇒ (2): let f ∈ µcr(x) and x /∈ f . then by (1), mic-cl({x}) ⊆ u \f. set l = u \mic-cl({x}), then l is a micro open set such that f ⊆ l and x /∈ l. (2) ⇒ (3): let f ∈ µcr(x) and x /∈ f . then, there exists l ∈ µr(x) such that f ⊆ l and x /∈ l. since l ∈ µr(x), then l∩mic-cl({x}) = φ and f ∩mic-cl({x}) = φ. (3) ⇒ (4): suppose that mic-cl({x}) 6= mic-cl({y}) for distinct points x,y ∈ u. then, there exists z ∈ miccl({x}) such that z /∈ mic-cl({y}) (or z ∈ mic-cl({y}) such that z /∈ mic-cl({x})). there exists k ∈ µr(x) int. j. anal. appl. 18 (4) (2020) 580 such that y /∈ k and z ∈ k; hence x ∈ k. therefore, we have x /∈ mic-cl({y}). by (3), we obtain miccl({x}) ∩mic-cl({y}) = φ. (4) ⇒ (1): let k ∈ µr(x) and x ∈ k. for each y /∈ k, x 6= y and x /∈ mic-cl({y}). this shows that mic-cl({x}) 6= mic-cl({y}). by (4), mic-cl({x}) ∩ mic-cl({y}) = φ for each y ∈ u \ k and hence mic-cl({x}) ∩ ( ⋃ y∈u\k mic-cl({y})) = φ. on other hand, since k ∈ µr(x) and y ∈ u \k, we have miccl({y}) ⊆ u \k and hence u \k = ⋃ y∈u\k mic-cl({y}). therefore, we obtain (u \k) ∩mic-cl({x}) = φ and mic-cl({x}) ⊆ k. this shows that u is micro r0. � theorem 4.2. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro t1 if and only if u is both micro t0 and micro r0. proof. necessity. let l be any micro open subset of u and x ∈ l. then by theorem 3.2, we have miccl({x}) ⊆ l and so by theorem 3.4, it is clear that u is micro t0 and micro r0. sufficiency. let x and y be any distinct points of u. since u is micro t0, then there exists a micro open set l such that x ∈ l and y /∈ l. as x ∈ l implies that mic-cl({x}) ⊆ l. since y /∈ l, so y /∈ mic-cl({x}). hence, y ∈ k = u \mic-cl({x}) and it is clear that x /∈ k. thus, it follows that there exist micro open sets l and k containing x and y respectively, such that y /∈ l and x /∈ k. this implies that u is micro t1. � theorem 4.3. let (u,τr(x),µr(x)) be a micro topological space. then, the following properties are equivalent: (1) u is micro r0. (2) x ∈ mic-cl({y}) if and only if y ∈ mic-cl({x}), for any points x and y in u. proof. (1) ⇒ (2): assume that u is micro r0. let x ∈ mic-cl({y}) and k be any micro open set such that y ∈ k. now by hypothesis, x ∈ k. therefore, every micro open set which contain y contains x. hence, y ∈ mic-cl({x}). (2) ⇒ (1): let l be a micro open set and x ∈ l. if y /∈ l, then x /∈ mic-cl({y}) and hence y /∈ mic-cl({x}). this implies that mic-cl({x}) ⊆ l. hence, u is micro r0. � remark 4.1. from definition 2.10 and theorem 4.3, the notions of micro symmetric and micro r0 are equivalent. theorem 4.4. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro r0 if and only if for every x and y in u, mic-cl({x}) 6= mic-cl({y}) implies mic-cl({x}) ∩mic-cl({y}) = φ. proof. necessity. suppose that u is micro r0 and x,y ∈ u such that mic-cl({x}) 6= mic-cl({y}). then, there exists z ∈ mic-cl({x}) such that z /∈ mic-cl({y}) (or z ∈ mic-cl({y}) such that z /∈ mic-cl({x})) and there exists k ∈ µr(x) such that y /∈ k and z ∈ k, hence x ∈ k. therefore, we have x /∈ mic-cl({y}). int. j. anal. appl. 18 (4) (2020) 581 thus, x ∈ u \mic-cl({y}) ∈ µr(x), which implies mic-cl({x}) ⊆ u \mic-cl({y}) and mic-cl({x}) ∩miccl({y}) = φ. sufficiency. let k ∈ µr(x) and x ∈ k. we still show that mic-cl({x}) ⊆ k. let y /∈ k, that is y ∈ u \k. then, x 6= y and x /∈ mic-cl({y}). this shows that mic-cl({x}) 6= mic-cl({y}). by assumption, miccl({x}) ∩ mic-cl({y}) = φ. hence, y /∈ mic-cl({x}) and therefore mic-cl({x}) ⊆ k. thus, u is micro r0 � theorem 4.5. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro r0 if and only if for any points x and y in u, mker({x}) 6= mker({y}) implies mker({x}) ∩mker({y}) = φ. proof. suppose that u is micro r0. thus, by theorem 2.5, for any points x and y in u if mker({x}) 6= mker({y}) then mic-cl({x}) 6= mic-cl({y}). now we prove that mker({x}) ∩ mker({y}) = φ. assume that z ∈ mker({x}) ∩ mker({y}). by z ∈ mker({x}) and theorem 2.3, it follows that x ∈ mic-cl({z}). since x ∈ mic-cl({x}), by theorem 4.1, mic-cl({x}) = mic-cl({z}). similarly, we have mic-cl({y}) = miccl({z}) = mic-cl({x}). this is a contradiction. therefore, we have mker({x}) ∩mker({y}) = φ. conversely, suppose that for any points x and y in u, mker({x}) 6= mker({y}) implies mker({x}) ∩ mker({y}) = φ. if mic-cl({x}) 6= mic-cl({y}), then by theorem 2.5, mker({x}) 6= mker({y}). hence, mker({x})∩mker({y}) = φ which implies mic-cl({x})∩mic-cl({y}) = φ. because z ∈ mic-cl({x}) implies that x ∈ mker({z}) and therefore mker({x}) ∩ mker({z}) 6= φ. by hypothesis, we have mker({x}) = mker({z}). then z ∈ mic-cl({x})∩mic-cl({y}) implies that mker({x}) = mker({z}) = mker({y}). this is a contradiction. therefore, mic-cl({x}) ∩mic-cl({y}) = φ and by theorem 4.1, u is micro r0. � theorem 4.6. let (u,τr(x),µr(x)) be a micro topological space. then, the following properties are equivalent: (1) u is micro r0. (2) for any non-empty set a and g ∈ µr(x) such that a∩g 6= φ, there exists f ∈ µcr(x) such that a∩f 6= φ and f ⊆ g. (3) for any g ∈ µr(x), we have g = ∪{f ∈ µcr(x): f ⊆ g}. (4) for any f ∈ µcr(x), we have f = ∩{g ∈ µr(x): f ⊆ g}. (5) for every x ∈ u, mic-cl({x}) ⊆ mker({x}). proof. (1) ⇒ (2): let a be a non-empty subset of u and g ∈ µr(x) such that a ∩ g 6= φ. then, there exists x ∈ a ∩ g. since x ∈ g ∈ µr(x), so mic-cl({x}) ⊆ g. set f = mic-cl({x}), then f ∈ µcr(x), f ⊆ g and a∩f 6= φ. (2) ⇒ (3): let g ∈ µr(x), then g ⊇∪{f ∈ µcr(x): f ⊆ g}. let x be any point of g. then, there exists f ∈ µcr(x) such that x ∈ f and f ⊆ g. therefore, we have x ∈ f ⊆ ∪{f ∈ µ c r(x): f ⊆ g} and hence int. j. anal. appl. 18 (4) (2020) 582 g = ∪{f ∈ µcr(x): f ⊆ g}. (3) ⇒ (4): obvious. (4) ⇒ (5): let x be any point of u and y /∈ mker({x}). then, there exists k ∈ µr(x) such that x ∈ k and y /∈ k, hence mic-cl({y}) ∩k = φ. by (4), (∩{g ∈ µr(x): mic-cl({y}) ⊆ g}) ∩k = φ and there exists g ∈ µr(x) such that x /∈ g and mic-cl({y}) ⊆ g. therefore, mic-cl({x}) ∩g = φ and y /∈ mic-cl({x}). consequently, we obtain mic-cl({x}) ⊆ mker({x}). (5) ⇒ (1): let g ∈ µr(x) and x ∈ g. let y ∈ mker({x}), then x ∈ mic-cl({y}) and y ∈ g. this implies that mker({x}) ⊆ g. therefore, we obtain x ∈ mic-cl({x}) ⊆ mker({x}) ⊆ g. this shows that u is micro r0. � corollary 4.1. let (u,τr(x),µr(x)) be a micro topological space. then, the following properties are equivalent: (1) u is micro r0. (2) mic-cl({x}) = mker({x}) for all x ∈ u. proof. (1) ⇒ (2): suppose that u is micro r0. by theorem 4.6, mic-cl({x}) ⊆ mker({x}) for each x ∈ u. let y ∈ mker({x}), then x ∈ mic-cl({y}) and by theorem 4.1, mic-cl({x}) = mic-cl({y}). therefore, y ∈ mic-cl({x}) and hence mker({x}) ⊆ mic-cl({x}). this shows that mic-cl({x}) = mker({x}). (2) ⇒ (1): follows from theorem 4.6. � theorem 4.7. let (u,τr(x),µr(x)) be a micro topological space. then, the following properties are equivalent: (1) u is micro r0. (2) if f is micro closed, then f = mker(f). (3) if f is micro closed and x ∈ f , then mker({x}) ⊆ f . (4) if x ∈ u, then mker({x}) ⊆ mic-cl({x}). proof. (1) ⇒ (2): let f be micro closed and x /∈ f. thus, (u \f) is a micro open set containing x. since u is micro r0, mic-cl({x}) ⊆ (u \ f). thus, mic-cl({x}) ∩ f = φ and by theorem 2.4, x /∈ mker(f). therefore, mker(f) = f. (2) ⇒ (3): in general, a ⊆ b implies mker(a) ⊆ mker(b). therefore, it follows from (2), that mker({x}) ⊆ mker(f) = f. (3) ⇒ (4): since x ∈ mic-cl({x}) and mic-cl({x}) is micro closed, by (3), mker({x}) ⊆ mic-cl({x}). (4) ⇒ (1): we show the implication by using theorem 4.3. let x ∈ mic-cl({y}). then by theorem 2.3, y ∈ mker({x}) and by (4), we obtain y ∈ mker({x}) ⊆ mic-cl({x}). therefore, x ∈ mic-cl({y}) implies y ∈ mic-cl({x}). the converse is obvious and u hence is micro r0. � int. j. anal. appl. 18 (4) (2020) 583 definition 4.2. let (u,τr(x),µr(x)) be a micro topological space. then, u is said to be micro r1 if for x,y in u with mic-cl({x}) 6= mic-cl({y}), there exist disjoint micro open sets l and k such that mic-cl({x}) ⊆ l and mic-cl({y}) ⊆ k. theorem 4.8. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro r1 if it is micro t2. proof. let x and y be any points of u such that mic-cl({x}) 6= mic-cl({y}). by theorem 3.4 (1), u is micro t1. therefore, by theorem 3.2, mic-cl({x}) = {x}, mic-cl({y}) = {y} and hence {x} 6= {y}. since u is micro t2, then there exist disjoint micro open sets l and k such that mic-cl({x}) = {x} ⊆ l and mic-cl({y}) = {y}⊆ k. this shows that u is micro r1. � theorem 4.9. let (u,τr(x),µr(x)) be a micro topological space. then, the following statements are equivalent: (1) u is micro r1. (2) if x,y ∈ u such that mic-cl({x}) 6= mic-cl({y}), then there exist micro closed sets f1 and f2 such that x ∈ f1, y /∈ f1, y ∈ f2, x /∈ f2 and u = f1 ∪f2. proof. obvious. � theorem 4.10. if u is micro r1, then u is micro r0. proof. let l be micro open such that x ∈ l. if y /∈ l, then x /∈ mic-cl({y}) and mic-cl({x}) 6= mic-cl({y}). so, there exists a micro open set k such that mic-cl({y}) ⊆ k and x /∈ k, which implies y /∈ mic-cl({x}). hence, mic-cl({x}) ⊆ l. therefore, u is micro r0. � corollary 4.2. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro r1 if and only if for x,y ∈ u, mker({x}) 6= mker({y}), there exist disjoint micro open sets l and k such that mic-cl({x}) ⊆ l and mic-cl({y}) ⊆ k. proof. follows from theorem 2.5. � theorem 4.11. let (u,τr(x),µr(x)) be a micro topological space. then, u is micro r1 if and only if x ∈ u \mic-cl({y}) implies that x and y have disjoint micro open neighbourhoods. proof. necessity. let x ∈ u \mic-cl({y}). then, mic-cl({x}) 6= mic-cl({y}). thus, x and y have disjoint micro open neighbourhoods. sufficiency. first, we show that u is micro r0. let l be a micro open set and x ∈ l. suppose that y /∈ l. then, mic-cl({y})∩l = φ and x /∈ mic-cl({y}). so, there exist micro open sets lx and ly such that x ∈ lx, y ∈ ly and lx ∩ly = φ. hence, mic-cl({x}) ⊆ mic-cl(lx) and mic-cl({x}) ∩ly ⊆ mic-cl(lx) ∩ly = φ. int. j. anal. appl. 18 (4) (2020) 584 therefore, y /∈ mic-cl({x}). consequently, mic-cl({x}) ⊆ l and hence u is micro r0. next, we show that u is micro r1. suppose that mic-cl({x}) 6= mic-cl({y}). then, we can assume that there exists z ∈ miccl({x}) such that z /∈ mic-cl({y}). then, there exist micro open sets kz and ky such that z ∈ kz, y ∈ ky and kz ∩ky = φ. since z ∈ mic-cl({x}), then x ∈ kz. since u is micro r0, we obtain mic-cl({x}) ⊆ kz, mic-cl({y}) ⊆ ky and kz ∩ky = φ. this shows that u is micro r1. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. chandrasekar, on micro topological spaces, j. new theory, 26 (2019) 23-31. [2] h. z. ibrahim, on new separation axioms via γ-open sets, int. j. adv. res. technol. 1 (1) (2012), 1-3. [3] h. z. ibrahim, operation on regular spaces, j. adv. stud. topol. 4 (1) (2013), 138-149. [4] h. z. ibrahim, weak forms of γ-open sets and new separation axioms, int. j. sci. eng. res. 3 (4) (2012), 1-4. [5] h. z. ibrahim, b-γ-continuous and b-γ-irresolute, int. electron. j. pure appl. math. 5 (4) (2012), 145-156. [6] h. z. ibrahim, α-γ-g.closed sets and α-γ-g.closed graph, int. j. pure appl. math. 83 (4) (2013), 575-588. [7] h. z. ibrahim, pre-γ-t1 2 and pre-γ-continuous, j. adv. stud. topol. 4 (2) (2013), 1-9. [8] h. z. ibrahim, β-γ-irresolute and β-γ-closed graph, gen. math. notes, 15 (2) (2013), 32-44. [9] h. z. ibrahim, on some separation axioms via β-γ-open sets, gen. math. notes, 15 (2) (2013), 14-31. [10] h. z. ibrahim, on a class of αγ-open sets in a topological space, acta sci. technol. 35 (3) (2013), 539-545. [11] h. z. ibrahim, on a class of γ-b-open sets in a topological space, gen. math. notes, 16 (2) (2013), 66-82. [12] h. z. ibrahim, bc-separation axioms in topological spaces, gen. math. notes, 17 (1) (2013), 45-62. [13] h. z. ibrahim, αγ-open sets, αγ-functions and some new separation axioms, acta sci. technol. 35 (4) (2013), 725-731. [14] h. z. ibrahim, on α (γ,γ ′ ) -separation axioms, int. j. anal. appl. 16 (5) (2018), 775-782. [15] h. z. ibrahim, on micro b-open sets, communicated. [16] h. z. ibrahim, on micro t1 2 space, int. j. appl. math. in press. [17] a. b. khalaf and and h. z. ibrahim, some applications of γ-p-open sets in topological spaces, int. j. pure appl. math. sci. 5 (1-2) (2011), 81-96. [18] a. b. khalaf and h. z. ibrahim, pγ-open sets and pγ,β-continuous mappings in topological spaces, j. adv. stud. topol. 3 (4) (2012), 102-110. [19] a. b. khalaf, h. z. ibrahim and a. k. kaymakci, operation-separation axioms via α-open sets, acta univ. apulensis, (47) (2016), 99-115. [20] a. b. khalaf, s. jafari and h. z. ibrahim, bioperations on α-separations axioms in topological spaces, sci. math. jpn. 81 (2018), 1-15. [21] a. m. kozae, s. a. saleh , m. a. elsafty and m. m. salama, entropy measures for topological approximations of uncertain concepts, jokull j. 65 (1) (2015), 192-206. [22] z. pawlak, rough sets, int. j. inform. computer sci. 11 (1982), 341-356. [23] z. pawlak, granularity of knowledge, indiscernibility and rough sets, proc. ieee int. conf. fuzzy syst. (1998), 106-110. [24] i. l. reilly and m. k. vamanamurthy, on α-sets in topological spaces, tamkang j. math. 16 (1985), 7-11. [25] m. l. thivagar and c. richard, note on nano topological spaces, communicated. int. j. anal. appl. 18 (4) (2020) 585 [26] m. l. thivagar, c. richard and n. r. paul, mathematical innovations of a modern topology in medical events, int. j. inform. sci. 2 (4) (2012), 33-36. 1. introduction 2. preliminaries 3. micro tk (k=0, 1, 2) 4. micro rk (k=0, 1) references int. j. anal. appl. (2022), 20:9 solutions of linear and nonlinear fractional fredholm integro-differential equations abdelhalim ebaid1,∗, hind k. al-jeaid2 1department of mathematics, faculty of science, university of tabuk, p.o. box 741, tabuk 71491, saudi arabia 2department of mathematical sciences, umm al-qura university, makkah, saudi arabia ∗corresponding author: aebaid@ut.edu.sa, halimgamil@yahoo.com abstract. the present paper analyzes a class of first-order fractional fredholm integro differential equations in terms of caputo fractional derivative. in the literature, such kind of fractional integrodifferential equations have been solved using several numerical methods, while the exact solutions were not obtained. however, the exact solutions are obtained in this paper for various linear and nonlinear examples. it is shown that the exact solution of the linear problems is unique, while multiple exact solutions exist for the nonlinear ones. moreover, the obtained results reduce to the classical ones in the relevant literature as the fractional order becomes unity. the obtained exact solutions can be further invested by other researchers to validate their numerical/approximation methods. 1. introduction the fractional calculus (fc) has gained observable interest in recent years due to its applications several fields [1-14]. the fc has been also extended to integro-differential equations (fides) as observed in the literature [15-28], where various numerical and analytical methods were applied to solve for approximate solutions. we are concerned here with fractional fredholm integro-differential equations (ffides) of first-order. although important results were reported [15-28] for fides, obtaining the exact solution of ffides is not an easy task, even for simple equations as will be shown later. so, received: nov. 3, 2021. 2010 mathematics subject classification. 34k37. key words and phrases. fractional calculus; analytic solution; fredholm integro-differential equations. https://doi.org/10.28924/2291-8639-20-2022-9 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-9 2 int. j. anal. appl. (2022), 20:9 we consider in this paper the following class of ffides: c 0 d α x u(x) = f (x) + λ ∫ b2 b1 k(x,τ,u(τ)) dτ, 0 < α ≤ 1, (1.1) u(0) = h, (1.2) where h, λ, b1 and b2 are given constants, f (x) is a given continuous function on [b1,b2]. the objective of this paper is to introduce a direct analytic approach for obtaining exact solutions for the class (1-2). it will be shown that the solution is unique when k(x,τ,u(τ)) is a linear function in the unknown function u(τ). in addition, it will be declared that multiple exact solutions exists when k(x,τ,u(τ)) is a nonlinear function in u(τ). the caputo definition is chosen as a fractional derivative in eq. (1) and the structure of the paper is as follows. in section 2, we give the main aspects of the fc. in addition, a basic lemma will be provided for the formal exact solution of the class (1-2). sections 3 investigates the application of the present approach on several linear and nonlinear problems. besides, the way of obtaining exact dual solution for the nonlinear case will be demonstrated in section 3. moreover, it will be shown that the present exact solutions reduce to the classical ones as α → 1. finally, section 5 outlines the conclusions. 2. main aspects of fc the riemann-liouville fractional integral of order α is defined as [1]: jα0 u(x) = 1 γ(α) ∫ x 0 (x −τ)α−1 u(τ)dτ α > 0. (2.1) the caputo’s fd of order α of a function u(x) is defined by c 0 d α x u(x) = 1 γ(n−α) ∫ x 0 (x −τ)n−α−1u(n)(τ)dτ, n− 1 < α ≤ n. (2.2) the jα0 and c 0 d α x are related by: jα0 ( c 0 d α x u(x) ) = u(x) − n−1∑ m=0 u(m)(0) m! xm(0), (2.3) which is useful when solving fdes/fies. a basic property of the jα0 is jα0 (x r ) = γ(r + 1) γ(α + r + 1) xα+r, r > −1. (2.4) int. j. anal. appl. (2022), 20:9 3 the mittag-leffler function (mlf) of one-parameter is defined as eα(z) = ∞∑ m=0 zm γ(αm + 1) , z ∈ c, (2.5) while the two-parameter mlf is given as eα,β(z) = ∞∑ m=0 zm γ(αm + β) , α > 0, β > 0. (2.6) the following properties are also hold: e1,2(z) = (e z − 1) / (z) , (2.7) e2,1(−z2) = cos(z), e2,2(−z2) = sin(z) z . (2.8) lemma 1. the analytic solution of the first-order ffide (1-2) is given by u(x) = h + aλ ( xα γ(α + 1) ) + jα0 (f (x)) , (2.9) provided that the fractional integral of f (x), i.e., jα0 (f (x)), exists and a is the constant given by a = ∫b2 b1 k(x,τ,u(τ)) dτ. proof: the bounded integral involved in eq. (1) can be assumed as a constant. besides, we assume that such integral is given by the constant a as a = ∫ b2 b1 k(x,τ,u(τ)) dτ. (2.10) operating with jα0 on eq. (1) and implementing (2), (5), and (14), it then follows u(x) −u(0) = jα0 (aλ) + j α 0 (f (x)) , (2.11) or u(x) = h + aλjα0 (1) + j α 0 (f (x)) . (2.12) calculating jα0 (1) from eq. (6) at r = 0, we have j α 0 (1) = xα γ(α+1) . substituting this last result into eq. (14) we obtain eq. (11) which completes the proofs. � 4 int. j. anal. appl. (2022), 20:9 3. examples example 1: consider the ffide [29] c 0 d α x u(x) = 2 ( 1 − ∫ 1 0 u(τ) dτ ) , u(0) = 0. (3.1) let a1 = ∫ 1 0 u(τ) dτ, (3.2) where a1 is an unknown constant. accordingly, eq. (15) becomes c 0 d α x u(x) = 2 (1 −a1) . (3.3) applying the integral operator jα0 on eq. (17) and making use of eq. (5), we have u(x) = u(0) + 2 (1 −a1) jα0 (1), (3.4) or u(x) = 2 (1 −a1) xα γ(α + 1) . (3.5) the constant a1 is evaluated by inserting (19) into (16), this yields a1 = 2 (1 −a1) ∫ 1 0 τα γ(α + 1) dτ = 2 (1 −a1) γ(α + 2) . (3.6) solving eq. (20) for a1, we obtain a1 = 2 2 + γ(α + 2) , (3.7) and hence, eq. (19) becomes u(x) = ( 2γ(α + 2) γ(α + 1) (2 + γ(α + 2)) ) xα. (3.8) as α → 1, eq. (22) reduces to the exact solution u(x) = x for the classical form of eq. (15), given by u′(x) = 2 ( 1 − ∫ 1 0 u(τ) dτ ) . example 2: consider the ffide [29] c 0 d α x u(x) = 3 + 6x + x ∫ 1 0 τu(τ) dτ, u(0) = 0. (3.9) suppose that a2 = ∫ 1 0 τu(τ) dτ, (3.10) int. j. anal. appl. (2022), 20:9 5 where a2 is a constant to be determined, then eq. (23) becomes c 0 d α x u(x) = 3 + (6 + a2) x. (3.11) operating with jα0 on eq. (25), it then follows u(x) = 3xα γ(α + 1) + (6 + a2)x α+1 γ(α + 2) . (3.12) from eq. (24), we have a2 = ∫ 1 0 ( 3τα+1 γ(α + 1) + (6 + a2)τ α+2 γ(α + 2) ) dτ, = 3 (α + 2)γ(α + 1) + (6 + a2) (α + 3)γ(α + 2) , (3.13) which gives a2 = 3α2 + 18α + 21 (α + 2) [(α + 3)γ(α + 2) − 1] . (3.14) therefore, u(x) = 3xα γ(α + 1) + ( 6 + 3α2 + 18α + 21 (α + 2) [(α + 3)γ(α + 2) − 1] ) xα+1 γ(α + 2) , (3.15) and reduces, as α → 1, to u(x) = 3x + 4x2 which is the same solution in ref. [29] for the classical form: u′(x) = 3 + 6x + x ∫ 1 0 τu(τ) dτ. example 3: this example considers the ffide [29] c 0 d α x u(x) = −1 + cos x + ∫ π/2 0 τu(τ) dτ, u(0) = 0, (3.16) which takes the form: c 0 d α x u(x) = (a3 − 1) + cos x, (3.17) where a3 is a constant defined by a3 = ∫ π/2 0 τu(τ) dτ. (3.18) expressing cos x as maclaurin series and then applying jα0 on both sides of eq. (31), gives u(x) = (a3 − 1) jα0 (1) + j α 0 ( ∞∑ m=0 (−1)mx2m (2m)! ) , = (a3 − 1) xα γ(α + 1) + ∞∑ m=0 (−1)m (2m)! × γ(2m + 1)xα+2m γ(α + 2m + 1) , (3.19) 6 int. j. anal. appl. (2022), 20:9 which is simplified as u(x) = (a3 − 1) xα γ(α + 1) + ∞∑ m=0 (−1)mxα+2m γ(α + 2m + 1) , (3.20) or in terms of the mlf e2,α+1(−x2) as u(x) = (a3 − 1) xα γ(α + 1) + xαe2,α+1(−x2). (3.21) from eq. (32) and eq. (35), we have a3 = ∫ π/2 0 ( (a3 − 1) τα+1 γ(α + 1) + τα+1e2,α+1(−τ2) ) dτ, = (a3 − 1) (π/2)α+2 (α + 2)γ(α + 1) + i, (3.22) where the integral i is defined by i = ∫ π/2 0 τα+1e2,α+1(−τ2) dτ. (3.23) solving eq. (36) for a3, we obtain a3 = − (π/2)α+2 (α + 2)γ(α + 1) − (π/2)α+2 + (α + 2)γ(α + 1) (α + 2)γ(α + 1) − (π/2)α+2 i. (3.24) therefore, u(x) is finally given by u(x) = ( − (π/2)α+2 (α + 2)γ(α + 1) − (π/2)α+2 + (α + 2)γ(α + 1) (α + 2)γ(α + 1) − (π/2)α+2 i − 1 ) × xα γ(α + 1) + xαe2,α+1(−x2), (3.25) and i is already defined by eq. (37). the solution given by eq. (39) reduces, as α → 1, to u(x) = ( − (π/2)3 3 − (π/2)3 + 3 3 − (π/2)3 ∫ π/2 0 τ2e2,2(−τ2) dτ − 1 ) x + xe2,2(−x2), = ( − (π/2)3 3 − (π/2)3 + 3 3 − (π/2)3 ∫ π/2 0 τ sin τ dτ − 1 ) x + x ( sin x x ) , = ( − (π/2)3 3 − (π/2)3 + (π/2)3 3 − (π/2)3 ) x + sin x, where ∫ π/2 0 τ sin τ dτ = 1, = sin x, (3.26) which is the corresponding solution for the classical form: u′(x) = −1 + cos x + ∫π/2 0 τu(τ) dτ. int. j. anal. appl. (2022), 20:9 7 example 4: in this example, we considers the ffide [29]: c 0 d α x u(x) = −10x + ∫ 1 −1 (x −τ) u(τ) dτ, u(0) = 1. (3.27) assuming that a4 = ∫ 1 −1 u(τ) dτ, a5 = ∫ 1 −1 τu(τ) dτ, (3.28) then eq. (41) becomes c 0 d α x u(x) = (a4 − 10) x −a5, (3.29) where a4 and a5 are constants. following the same analysis of the previous examples, we obtain u(x) = 1 + (a4 − 10) xα+1 γ(α + 2) − a5x α γ(α + 1) . (3.30) substituting eq. (44) into eqs. (42) and performing the associated integrals, we get the following system: ( 1 − ( 1 − (−1)α γ(α + 3) )) a4 + ( 1 + (−1)α γ(α + 2) ) a5 = 2 − 10 ( 1 − (−1)α γ(α + 3) ) , (3.31)( 1 + (−1)α (α + 3)γ(α + 2) ) a4 − ( 1 + 1 − (−1)α (α + 2)γ(α + 1) ) a5 = 10 ( 1 + (−1)α (α + 3)γ(α + 2) ) . (3.32) the solution of the system (45-46) can be obtained as a4 = ∆1 ∆ , a5 = ∆2 ∆ , (3.33) where ∆, ∆1, and ∆2 are given by the determinants: ∆ = ∣∣∣∣∣∣∣∣∣∣ 1 − 1 − (−1)α γ(α + 3) 1 + (−1)α γ(α + 2) 1 + (−1)α (α + 3)γ(α + 2) −1 − 1 − (−1)α (α + 2)γ(α + 1) ∣∣∣∣∣∣∣∣∣∣ , (3.34) and ∆1 = ∣∣∣∣∣∣∣∣∣∣ 2 − 10 ( 1−(−1)α γ(α+3) ) 1 + (−1)α γ(α + 2) 10 ( 1+(−1)α (α+3)γ(α+2) ) −1 − 1 − (−1)α (α + 2)γ(α + 1) ∣∣∣∣∣∣∣∣∣∣ , (3.35) ∆2 = ∣∣∣∣∣∣∣∣∣∣ 1 − 1 − (−1)α γ(α + 3) 2 − 10 ( 1−(−1)α γ(α+3) ) 1 + (−1)α (α + 3)γ(α + 2) 10 ( 1+(−1)α (α+3)γ(α+2) ) ∣∣∣∣∣∣∣∣∣∣ . (3.36) 8 int. j. anal. appl. (2022), 20:9 hence, the solution of eq. (41) is finally given by u(x) = 1 + ( ∆1 ∆ − 10 ) xα+1 γ(α + 2) − ( ∆2 ∆ ) xα γ(α + 1) . (3.37) as α → 1, u(x) given by eq. (44) implies that u(x) = 1 + 1 2 ([ ∆1 ∆ ] α→1 − 10 ) x2 − [ ∆2 ∆ ] α→1 x. (3.38) calculating ∆, ∆1, and ∆2 when α → 1, we obtain ∆ = − 10 9 , ∆1 = 20 9 , ∆2 = 0. (3.39) thus, eq. (52) becomes u(x) = 1 − 6x2, (3.40) which is the corresponding solution of the classical form: u′(x) = −10x + ∫ 1 −1 (x −τ) u(τ) dτ. example 5: in order to show how to apply the present direct approach to solving nonlinear ffides, we consider here a simple example, given by the nonlinear ffide: c 0 d α x u(x) = ∫ 1 0 u2(τ) dτ, u(0) = 0, (3.41) which can be written as c 0 d α x u(x) = a6, (3.42) where a6 is a constant defined by a6 = ∫ 1 0 u2(τ) dτ. (3.43) on solving eq. (56), we have u(x) = a6x α γ(α + 1) . (3.44) evaluating a6 from eq. (57), we obtain a6 = a 2 6 ∫ 1 0 τ2α (γ(α + 1)) 2 dτ, (3.45) which leads to a6 ( a6 (2α + 1) (γ(α + 1)) 2 − 1 ) = 0. (3.46) solving this equation for a6, we obtain a6 = 0, a6 = (2α + 1) (γ(α + 1)) 2 . (3.47) int. j. anal. appl. (2022), 20:9 9 the first value a6 = 0 leads u(x) = 0, which is a trivial solution. while the value a6 = (2α + 1) (γ(α + 1)) 2 gives u(x) = (2α + 1)γ(α + 1)xα, (3.48) as a second solution. as α → 1, we obtain the corresponding solution u(x) = 3x for the classical form u′(x) = ∫ 1 0 u2(τ) dτ. example 6: we consider an additional nonlinear example: c 0 d α x u(x) = 10x − 5 + ∫ 1 0 u2(τ) dτ, u(0) = 0, (3.49) following the above analysis, we can obtain u(x) = (a7 − 5)xα γ(α + 1) + 10xα+1 γ(α + 2) , (3.50) and a7 is given as a7 = ∫ 1 0 u2(τ) dτ. (3.51) substituting eq. (64) into eq. (65), yields a7 = ∫ 1 0 [ (a7 − 5)2τ2α (γ(α + 1))2 + 20(a7 − 5)τ2α+1 γ(α + 1)γ(α + 2) + 100τ2α+2 (γ(α + 2))2 ] dτ. (3.52) performing this integral, we find that a7 is governed by the equation: a7 = (a7 − 5)2 (2α + 1)(γ(α + 1))2 + 10(a7 − 5) (γ(α + 2))2 + 100 (2α + 3)(γ(α + 2))2 . (3.53) eq. (67) can be rewritten as a(a7 − 5)2 + b(a7 − 5) + c = 0, (3.54) where a = (a7 − 5)2 (2α + 1)(γ(α + 1))2 , b = 10 (γ(α + 2))2 − 1, c = 100 (2α + 3)(γ(α + 2))2 − 5. (3.55) solving eq. (68) for the constant a7, we get a7 = 5 + 1 2a ( −b ± √ b2 − 4ac ) . (3.56) from eq. (64) and eq. (70), we obtain u(x) = 1 2a ( −b ± √ b2 − 4ac )( xα γ(α + 1) ) + 10xα+1 γ(α + 2) . (3.57) 10 int. j. anal. appl. (2022), 20:9 it can be seen from eq. (71) that there are two different solutions for the present nonlinear example, the first one is given by u1(x) = 1 2a ( −b + √ b2 − 4ac )( xα γ(α + 1) ) + 10xα+1 γ(α + 2) , (3.58) while the second solution is u2(x) = − 1 2a ( b + √ b2 − 4ac )( xα γ(α + 1) ) + 10xα+1 γ(α + 2) . (3.59) in order to check these two solutions, we evaluate them as α → 1. in this case, we have from eqs. (69) that a = 1 3 , b = 3 2 , c = 0. (3.60) hence, (u1(x))α→1 = 5x 2, (3.61) and (u2(x))α→1 = − 9 2 x + 5x2, (3.62) the solutions (75) and (76) are the same obtained one in ref. [29] for the classical nonlinear version u′(x) = 10x − 5 + ∫ 1 0 u2(τ) dτ. 4. conclusion a class of first-order ffides was investigated in terms of caputo definition in fc. the analytic solutions of several linear and nonlinear examples were obtained. for the linear ffides, a unique solution was obtained, while multiple solutions were obtained for the nonlinear ffides. it was shown that the linear problems posses unique solution of, while the nonlinear ones posses multiple solutions. furthermore, as the fractional order is unity, the results agree with to the corresponding classical problems. this study may deserve extensions to further ffides of higher-order. funding: no funding availability of data and materials: not applicable. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2022), 20:9 11 references [1] r. hilfer, applications of fractional calculus in physics, world scientific publishing company, singapore, 2000. [2] j.h. he, approximate analytical solution for seepage flow with fractional derivatives in porous media, comput. meth. appl. mech. eng. 167 (1998), 57-68. https://doi.org/10.1016/s0045-7825(98)00108-x. [3] o.p. agrawal, a new lagrangian and a new lagrange equation of motion for fractionally damped systems, j. appl. mech. 68 (2001), 339–341. https://doi.org/10.1115/1.1352017. [4] v. kiryakova, generalized fractional calculus and applications, pitman research notes in mathematics, longman, harlow, 1994. [5] yu. f. luchko, h.m srivastava, the exact solution of certain differential equations of fractional order by using operational calculus, computers math. appl. 29(8) (1995), 73-85. https://doi.org/10.1016/0898-1221(95) 00031-s. [6] n. sebaa, z.e.a. fellah, w. lauriks, c. depollier, application of fractional calculus to ultrasonic wave propagation in human cancellous bone, signal process. 86 (2006), 2668-2677. https://doi.org/10.1016/j.sigpro.2006. 02.015. [7] yongsheng ding, haiping yea, a fractional-order differential equation model of hiv infection of cd4+t-cells, math. comput. model. 50 (2009), 386-392. https://doi.org/10.1016/j.mcm.2009.04.019. [8] a. ebaid, d.m.m. el-sayed, m.d. aljoufi, fractional calculus model for damped mathieu equation: approximate analytical solution, appl. math. sci. 6 (2012), 4075–4080. [9] lei song, shiyun xu, jianying yang, dynamical models of happiness with fractional order, commun. nonlinear sci. numer. simulat. 15 (2010), 616-628. https://doi.org/10.1016/j.cnsns.2009.04.029. [10] a. ebaid, analysis of projectile motion in view of the fractional calculus, appl. math. model. 35 (2011), 1231-1239. https://doi.org/10.1016/j.apm.2010.08.010. [11] a. ebaid, e.r. el-zahar, a.f. aljohani, bashir salah, mohammed krid, j. tenreiro machado, analysis of the twodimensional fractional projectile motion in view of the experimental data, nonlinear dyn. 97 (2019), 1711-1720. https://doi.org/10.1007/s11071-019-05099-y. [12] e.r. elzahar, a.a. gaber, a.f. aljohani, j.t. machado, a. ebaid, generalized newtonian fractional model for the vertical motion of a particle, appl. math. model. 88 (2020), 652-660. https://doi.org/10.1016/j.apm.2020. 06.054. [13] e.r. el-zahar, a.m. alotaibi, a. ebaid, a.f. aljohani, j.f. gómez aguilar, the riemann-liouville fractional derivative for ambartsumian equation, results phys. 19 (2020), 103551. https://doi.org/10.1016/j.rinp.2020. 103551. [14] a. ebaid, c. cattani, a.s. al juhani1, e.r. el-zahar, a novel exact solution for the fractional ambartsumian equation, adv. differ. equ. 2021 (2021), 88. https://doi.org/10.1186/s13662-021-03235-w. [15] a. peedas, e. tamme, spline collocation method for integro-differential equations with weakly singular kernels, j. comput. appl. math. 197 (2006), 253-269. https://doi.org/10.1016/j.cam.2005.07.035. [16] h. jafari, v. daftardar-gejji, solving a system of nonlinear fractional differential equations using adomian decomposition, j. comput. appl. math. 196(2) (2006), 644-651. https://doi.org/10.1016/j.cam.2005.10.017. [17] d. nazari, s. shahmorad, application of the fractional differential transform method to fractional-order integrodifferential equations with nonlocal boundary conditions, j. comput. appl. math. 234 (2010), 883-891. https: //doi.org/10.1016/j.cam.2010.01.053. [18] b. ghazanfari, a.g. ghazanfari, f. veisi, homotopy perturbation method for nonlinear fractional integro-differential equations, aust. j. basic appl. sci. 12(4) (2010), 5823-5829. [19] s. karimi vanani, a. aminataei, operational tau approximation for a general class of fractional integro-differential equations, comput. appl. math. 3(30) (2011), 655-674. https://doi.org/10.1590/s1807-03022011000300010. https://doi.org/10.1016/s0045-7825(98)00108-x https://doi.org/10.1115/1.1352017 https://doi.org/10.1016/0898-1221(95)00031-s https://doi.org/10.1016/0898-1221(95)00031-s https://doi.org/10.1016/j.sigpro.2006.02.015 https://doi.org/10.1016/j.sigpro.2006.02.015 https://doi.org/10.1016/j.mcm.2009.04.019 https://doi.org/10.1016/j.cnsns.2009.04.029 https://doi.org/10.1016/j.apm.2010.08.010 https://doi.org/10.1007/s11071-019-05099-y https://doi.org/10.1016/j.apm.2020.06.054 https://doi.org/10.1016/j.apm.2020.06.054 https://doi.org/10.1016/j.rinp.2020.103551 https://doi.org/10.1016/j.rinp.2020.103551 https://doi.org/10.1186/s13662-021-03235-w https://doi.org/10.1016/j.cam.2005.07.035 https://doi.org/10.1016/j.cam.2005.10.017 https://doi.org/10.1016/j.cam.2010.01.053 https://doi.org/10.1016/j.cam.2010.01.053 https://doi.org/10.1590/s1807-03022011000300010 12 int. j. anal. appl. (2022), 20:9 [20] y. ordokhani, n. rahimi, numerical solution of fractional volterra integro-differential equations via the rationalized haar functions, j. sci. kharazmi univ. 14(3) (2014), 211-224. [21] s. bushnaq, b. maayah, s. momani, a. alsaedi, a reproducing kernel hilbert space method for solving systems of fractional integrodifferential equations, abstr. appl. anal. 2014 (2014), article id 103016. https://doi.org/10. 1155/2014/103016. [22] m.r. eslahchi, m. dehghanb, m. parvizi, application of the collocation method for solving nonlinear fractional integro-differential equations, j. comput. appl. math. 257 (2014), 105-128. https://doi.org/10.1016/j.cam. 2013.07.044. [23] m.h. heydari, m.r. hooshmandasl, f. mohammadi, c. cattani, wavelets method for solving systems of nonlinear singular fractional volterra integro-differential equations, commun. nonlinear sci. 19 (2014), 37-48. https://doi. org/10.1016/j.cnsns.2013.04.026. [24] w. jiang, t. tian, numerical solution of nonlinear volterra integro-differential equations of fractional order by the reproducing kernel method, appl. math. model. 39 (2015), 4871-4876. https://doi.org/10.1016/j.apm.2015. 03.053. [25] d. nazari susahab, m. jahanshahi, numerical solution of nonlinear fractional volterra fredholm integro-differential equations with mixed boundary conditions, int. j. ind. math. 7 (2017), 63-69. [26] a.a. al-marashi, approximate solution of the system of linear fractional integro-differential equations of volterra using b-spline method, amer. res. math. stat. 3(2) (2015), 39-47. https://doi.org/10.15640/arms.v3n2a6. [27] r.l. jian, p. chang, a. isah, new operational matrix via genocchi polynomials for solving fredholm volterra fractional integro-differential equations, adv. math. phys. 2017 (2017), 3821870. https://doi.org/10.1155/ 2017/3821870. [28] h. khan, m. arif, s.t. mohyud-din, s. bushnaq, numerical solutions to systems of fractional voltera integro differential equations, using chebyshev wavelet method, j. taibah univ. sci. 12 (2018), 584–591. https://doi. org/10.1080/16583655.2018.1510149. [29] a.m. wazwaz, linear and nonlinear integral equations, methods and applications, higher education press, beijing and springer-verlag, berlin heidelberg (2011). https://doi.org/10.1155/2014/103016 https://doi.org/10.1155/2014/103016 https://doi.org/10.1016/j.cam.2013.07.044 https://doi.org/10.1016/j.cam.2013.07.044 https://doi.org/10.1016/j.cnsns.2013.04.026 https://doi.org/10.1016/j.cnsns.2013.04.026 https://doi.org/10.1016/j.apm.2015.03.053 https://doi.org/10.1016/j.apm.2015.03.053 https://doi.org/10.15640/arms.v3n2a6 https://doi.org/10.1155/2017/3821870 https://doi.org/10.1155/2017/3821870 https://doi.org/10.1080/16583655.2018.1510149 https://doi.org/10.1080/16583655.2018.1510149 1. introduction 2. main aspects of fc 3. examples 4. conclusion references international journal of analysis and applications volume 18, number 3 (2020), 439-447 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-439 some fixed point results for multivalued mappings in b−multiplicative and b−metric space mazhar mehmood∗, abdullah shoaib and hamza khalid department of mathematics and statistics,riphah international university, islamabad 44000, pakistan ∗corresponding author: mazharm53@gmail.com abstract. the main outcome of this paper is to introduce the notion of hausdorff b-multiplicative metric space and to present some fixed point results for multivalued mappings in this space. moreover, we obtain some fixed point results satisfying rational type contractive condition on closed ball for multivalued mappings in b-metric space.the proven results are original in nature. 1. introduction bourbaki and bakhtin [6], were the first ones who gave the idea of b-metric. after that, czerwik [7] gave an axiom and formally defined a b-metric space. for further results on b-metric space, see [11–13]. ozaksar and cevical [10] investigated multiplicative metric space and proved its topological properties. mongkolkeha et al. [9] described the concept of multiplicative proximal contraction mapping and proved best proximity point theorems for such mappings. recently, abbas et al. [1] proved some common fixed points results of quasi weak commutative mappings on a closed ball in the setting of multiplicative metric spaces. they also describe the main conditions for the existence of common solution of multiplicative boundary value problem. for further results on multiplicative metric space, see [2, 3, 8]. in 2017, ali et al. [4] introduced the notion of b-multiplicative and proved some fixed point result. as an application, they established an existence theorem for the solution of a system of fredholm multiplicative integral equations. shoaib et received january 7th, 2020; accepted january 27th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. fixed point; closed ball; b-multiplicative metric space; b-metric space; multivalued mapping; contractive condition. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 439 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-439 int. j. anal. appl. 18 (3) (2020) 440 al. [13], discussed the result for fuzzy mappings on a closed ball in a b-metric space. for further results on closed ball, see [5, 12, 14, 15]. in this paper, we proved some fixed point results for multivalued mappings in b-multiplicative and b-metric space. 2. preliminaries and basic definitions in this section we include some basic definitions and theorems which are useful to understand the results presented in this paper. first we give definition of b-metric space and its relevant results. definition 2.1. [11] let w be a non-empty set and s ≥ 1 be a real number. a mapping d : w × w → <+ ∪{0} is said to be b-metric with coefficient ”s”, if for all w,y,z ∈ w , the following conditions hold i. d(w,y) = 0 if and only if w = y. ii. d(w,y) = d(y,w). iii. d(w,z) ≤ s[d(w,y) + d(y,z)]. the pair (w,d) is called b-metric space. example 2.1. [11] let (w,d) be a metric space. then for a real number k > 1, we define a function d1(a,b) = (d(a,b)) k, then d1 is a b-metric with b = 2 k−1. definition 2.2. [11] let (w,b) be a b-metric space. i. a sequence {wn} in (w,b) is called convergent if and only if there exists w ∈ w such that b(wn,w) → 0, as n → +∞. ii. a sequence {wn} in (w,b) is a cauchy sequence, if and only if b(wn,wm) → 0, as n,m → +∞. iii. a b-metric space (w,b) is said to be complete if every cauchy sequence in w converge to a point of w. definition 2.3. let (w,d) be a b-metric space. a be a non empty subset of w with some w0 ∈ w . an element a ∈ a is called a best approximation in a if d(wo,a) = d(wo,a), where d(w0,a) = infw∈a d(wo,w), if each wo ∈ w has at least one best approximation in a, then a is called a proximinal set. we denote p(w), the set of all closed proximinal subsets of w . definition 2.4. let (w,d) be b-metric space. the function h : p(w) ×p(w) →<+,defined by h(a,b) = max{supw∈a d(w,b), supy∈b d(a,y)}, is called hausdorff b-metric on p(w). lemma 2.1. let (w,d) be b-metric space. let h be a hausdorff b-metric on p(w). then for all a,b ∈ p(w) and for each w ∈ a there exist y ∈ b satisfying d(w,b) = d(w,y) then h(a,b) ≥ d(w,y). now, we include the definition of b-multiplicative metric space and its relevant results. int. j. anal. appl. 18 (3) (2020) 441 definition 2.5. [4] let w be a non-empty set and let s ≥ 1 be a given real number. a mapping m : w ×w → [1,∞) is called a b-multiplicative metric with coefficient s, if the following conditions hold: i. m(w,y) > 1 for all w,y ∈ w with w 6= y and m(w,y) = 1 if and only if w = y. ii. m(w,y) = m(y,w) for all w,y ∈ w . iii. m(w,z) ≤ [m(w,y).m(y,z)]s for all w,y,z ∈ w . the triplet (w,m,s) is called b-multiplicative metric space. example 2.2. [4] let w = [ 0,∞) . define a mapping ma : w ×w → [ 1,∞) , ma(w,y) = a(w−y) 2 , where a > 1 is any fixed real number. then for each a, ma is b-multiplicative metric on w with s = 2. note that ma is not a multiplicative metric on w . definition 2.6. [4] let (w,m) be a b-multiplicative metric space. i. a sequence {wn} is convergent iff there exists w ∈ w , such that m(wn,w) → 1, as n → +∞. ii. a sequence {wn} is called b-multiplicative cauchy, iff m(wm,wn) → 1, as m,n → +∞. iii. a b-multiplicative metric space (w,m) is said to be complete if every multiplicative cauchy sequence in w is convergent to some y ∈ w . definition 2.7. let (w,d) be a b-multiplicative metric space. a be a non empty subset of w with some wo ∈ w . an element a ∈ a is called a best approximation in a if d(wo,a) = d(wo,a), where d(wo,a) = infw∈a d(wo,w), if each wo ∈ w has at least one best approximation in a, then a is called a proximinal set. we denote p(w), the set of all closed proximinal subsets of w . definition 2.8. let (w,d) be a b-multiplicative metric space. the function h : p(w) × p(w) → <+, defined by h(a,b) = max{supw∈a d(w,b), supy∈b d(a,y)}, is called hausdorff b-multiplicative metric on p(w). lemma 2.2. let (w,d) be a b-multiplicative metric. let h be a hausdorff b-multiplicative metric on p(w). then for all a,b ∈ p(w) and for each w ∈ a there exist y ∈ b satisfying d(w,b) = d(w,y) then h(a,b) ≥ d(w,y). definition 2.9. let (w,d) be a complete b-multiplicative metric and w0 ∈ w and s : w → p(w) be the multivalued mapping on w , then there exist w1 ∈ sw0 be an element such that d(w0,sw0) = d(w0,w1). let w2 ∈ sw1 be such that d(w1,sw1) = d(w1,w2). let w3 ∈ sw2 be such that d(w2,sw2) = d(w2,w3). continuing this process, we construct a sequence {wn} of points in w such that wn+1 ∈ swn, d(wn,swn) = d(wn,wn+1). we denote this iterative sequence by {ws(wn)}. we say that {ws(wn)} is a sequence in w generated by w0. int. j. anal. appl. 18 (3) (2020) 442 3. results for b−multiplicative metric space definition 3.1. let (w,d) be a b-multiplicative metric space and s : w → p(w) be the multivalued mapping. let w0 ∈ w and {ws(wn)} be a sequence in w generated by w0. we define the family m(s) of all functions a : w ×w → [ 0, 1) which satisfy the following property a( wn,wn+1) ≤ a( w0,w1) , for all n ∈ n ∪{0}. also, if {ws(wn)}→ h, then a(wn,h) ≤ a(w0,h). theorem 3.1. let (w,d) be a complete b-multiplicative metric space with coefficient s, s : w → p(w) be a multivalued mapping on w and β,σ,ψ ∈ m(s). if the following relations hold: h(swn,swn+1) ≤ [d(wn,wn+1)]β(wn,wn+1).[d(wn,swn+1).d(wn+1,swn)]σ(wn,wn+1) .[d(wn,swn).d(wn+1,swn+1)] ψ(wn,wn+1), (3.1) for all wn,wn+1 ∈{ws(wn)}, n ∈ n ∪{0}, a,b ≥ 0, and sβ(w0,w1) + (s 2 + s)σ(w0,w1) + (s + 1)ψ(w0,w1) < 1 for w0,w1 ∈{ws(wn)}, (3.2) then {ws(wn)}→ w∗ ∈ w . also, if inequalities 3.1 and 3.2 hold for h, then s has a fixed point w∗. proof. considering a sequence {ws(wn)} in w generated by w0, then we have wn+1 ∈ swn, where n = 0, 1, 2, · · · now by using lemma 2.2, we can write d(wn,wn+1) ≤ h(swn−1,swn) ≤ [d(wn−1,wn)]β(wn−1,wn).[d(wn−1,swn).d(wn,swn−1)]σ(wn−1,wn) .[d(wn−1,swn−1).d(wn,swn)] ψ(wn−1,wn) ≤ [d(wn−1,wn)]β(wn−1,wn).[d(wn−1,wn+1).d(wn,wn)]σ(wn−1,wn) .[d(wn−1,wn).d(wn,wn+1)] ψ(wn−1,wn) by using the definition 3.1 and triangle inequality, we can write d(wn,wn+1) ≤ [d(wn−1,wn)]β(w0,w1).[d(wn−1,wn).d(wn,wn+1)]sσ(w0,w1) .[d(wn−1,wn).d(wn,wn+1)] ψ(w0,w1).[d(wn,wn+1)] 1−sσ(w0,w1)−ψ(w0,w1) ≤ [d(wn−1,wn)]β(w0,w1)+sσ(w0,w1)+ψ(w0,w1) d(wn,wn+1) ≤ [d(wn−1,wn)] β(w0,w1) + sσ(w0,w1) + ψ(w0,w1) 1 −sσ(w0,w1) −ψ(w0,w1) = [d(wn−1,wn)]k. (3.3) now, d(wn−1,wn) ≤ h(swn−2,swn−1) ≤ [d(wn−2,wn−1)]β(wn−2,wn−1).[d(wn−2,swn−1).d(wn−1,swn−2)]σ(wn−2,wn−1) .[d(wn−2,swn−2).d(wn−1,swn−1)] ψ(wn−2,wn−1) ≤ [d(wn−2,wn−1)]β(w0,w1).[d(wn−2,wn−1).d(wn−1,wn)]sσ(w0,w1).[d(wn−2,wn−1).d(wn−1,wn)]ψ(w0,w1) int. j. anal. appl. 18 (3) (2020) 443 [d(wn−1,wn)] 1−sσ(w0,w1)−ψ(w0,w1) ≤ [d(wn−2,wn−1)]β(w0,w1)+sσ(w0,w1)+ψ(w0,w1) d(wn−1,wn) ≤ [d(wn−2,wn−1)] β(w0,w1) + sσ(w0,w1) + ψ(w0,w1) 1 −sσ(w0,w1) −ψ(w0,w1) = [d(wn−2,wn−1)]k. (3.4) from 3.3 and 3.4, we can write d(wn,wn+1) ≤ [d(wn−1,wn)]k ≤ [d(wn−2,wn−1)k]k = [d(wn−2,wn−1)]k 2 ≤ [d(wn−3,wn−2)]k 3 ≤ ···≤ [d(w0,w1)]k n (3.5) now, for m,n ∈ n, with m > n, we have d(wn,wm) ≤ d(wn,wn+1)s.d(wn+1,wn+2)s 2 · · · .d(wm−2,wm−1)s m−n−1 .d(wm−1,wm) sm−n by using the inequality 3.5, we have d(wn,wm) ≤ d(wn,wn+1)s.d(wn+1,wn+2)s 2 · · · .d(wm−2,wm−1)s m−n−1 .d(wm−1,wm) sm−n ≤ [d(w0,w1)]sk n .[d(w0,w1)] s2kn+1 · · · .[d(w0,w1)]s m−n−1km−2.[d(w0,w1)] sm−nkm−1 ≤ [d(w0,w1)]sk n(1+sk+(sk)2+···+sm−n−2km−n−2+sm−n−1km−n−1) ≤ [d(w0,w1)]sk n(1+sk+(sk)2+···+(sk)m−n−2+(sk)m−n−1) < [d(w0,w1)] skn(1+sk+(sk)2+···) = [d(w0,w1)] skn( 1 1−sk ). taking limm,n→∞, we get d(wn,wm) → 1. hence {ws(wn)} is a b-multiplicative cauchy sequence. by completeness of (w,d), we have wn → w∗ ∈ w . also lim n→∞ d(wn,w ∗) = 1. (3.6) now, d(w∗,sw∗) ≤ [d(w∗,wn+1).d(wn+1,sw∗)]s ≤ d(w∗,wn+1)s.h(swn,sw∗)s ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(wn,w ∗).[d(wn,sw ∗).d(w∗,swn)] sσ(wn,w ∗).[d(wn,swn).d(w ∗,sw∗)]sψ(wn,w ∗) ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(wn,w ∗).[d(wn,w ∗).d(w∗,sw∗)]s 2σ(wn,w ∗) .d(w∗,wn+1) sσ(wn,w ∗).[d(wn,wn+1).d(w ∗,sw∗)]sψ(wn,w ∗). by using definition 2.8, we can write d(w∗,sw∗) ≤ d(w∗,wn+1)s.[d(wn,w∗)]sβ(w0,w ∗).[d(wn,w ∗).d(w∗,sw∗)]s 2σ(w0,w ∗).d(w∗,wn+1) sσ(w0,w ∗) .[d(wn,wn+1).d(w ∗,sw∗)]sψ(w0,w ∗). on taking limn→∞ and by using inequality 3.6, we get d(w∗,sw∗) ≤ [d(w∗,sw∗)]s 2σ(w0,w ∗).[d(w∗,sw∗)]sψ(w0,w ∗) [d(w∗,sw∗)]1−s 2σ(w0,w ∗)−sψ(w0,w∗) ≤ 1 d(w∗,sw∗) ≤ (1) 1 1 −s2σ(w0,w∗) −sψ(w0,w∗) ≤ 1. this implies that d(w∗,sw∗) = 1 and hence w∗ is a fixed point of mapping s. � int. j. anal. appl. 18 (3) (2020) 444 theorem 3.2. let (w,d) be a complete b-multiplicative metric space with coefficient s, s : w → p(w) be the multivalued mapping on w and β ∈ m(s). if the following relations hold: h(swn,swn+1) ≤ [d(wn,wn+1)]β(wn,wn+1), (3.7) for all wn,wn+1 ∈{ws(wn)}, n ∈ n ∪{0},a,b ≥ 0 and sβ(w0,w1) < 1 for w0,w1 ∈{ws(wn)}, then {ws(wn)}→ w∗ ∈ w. (3.8) also if inequalities 3.7 and 3.8 hold for h, then s has a fixed point w∗. theorem 3.3. let (w,d) be a complete b-multiplicative metric space with coefficient s, s : w → p(w) be the multivalued mapping on w and σ ∈ m(s). if the following relations hold: h(swn,swn+1) ≤ [d(wn,swn+1).d(wn+1,swn)]σ(wn,wn+1), (3.9) for all wn,wn+1 ∈{ws(wn)}, n ∈ n ∪{0}, a,b ≥ 0 and (s2 + s)σ(w0,w1) < 1 for w0,w1 ∈{ws(wn)}, then {ws(wn)}→ w∗ ∈ w. (3.10) also if inequalities 3.9 and 3.10 hold for h, then s has a fixed point w∗. theorem 3.4. let (w,d) be a complete b-multiplicative metric space with coefficient s, s : w → p(w) be the multivalued mapping on w and ψ ∈ m(s). if the following relations hold: h(swn,swn+1) ≤ [d(wn,swn).d(wn+1,swn+1)]ψ(wn,wn+1), (3.11) for all wn,wn+1 ∈{ws(wn)}, n ∈ n ∪{0}, a,b ≥ 0 (s + 1)ψ(w0,w1) < 1 for w0,w1 ∈{ws(wn)}, then {ws(wn)}→ w∗ ∈ w. (3.12) also if inequalities 3.11 and 3.12 hold for h, then s has a fixed point w∗ 4. results for b−metric space theorem 4.1. let (w,d) be a complete b-metric space with coefficient s and w0 be any point in w . let the mapping s : w → p(w) satisfy the following relations: h(swn,swn+1) ≤ a1d(wn,wn+1) + a2[ a + d(wn,swn) b + d(wn,wn+1) ]d(wn,swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,swn) ]d(wn+1,swn+1) + a4[d(wn,swn+1) + d(wn+1,swn)], (4.1) for all wn,wn+1 ∈ b(w0; r) ∩{ws(w0)} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with a ≤ b, c ≤ d ′ . also d(w0,sw0) ≤ β(1 −sβ)r, where β = a1 + a2 + sa4 1 −a3 −sa4 , r > 0 and sβ < 1. (4.2) int. j. anal. appl. 18 (3) (2020) 445 then {ws(wn)} is a sequence in b(w0; r) and {ws(wn)}→ h ∈ b(w0; r). also if inequality 4.1 holds for h, then s has a fixed point h in b(w0; r). proof. considering a sequence {ws(wn)} in w generated by w0, then, we have wn+1 ∈ swn, where n=0,1,2,... from 4.2, we have d(w0,w1) ≤ β(1 − sβ)r ≤ r. this implies w1 ∈ b(w0; r). now by using lemma 2.1 and inequality 4.1, we can write d(w1,w2) = d(w1,sw1) ≤ h(sw0,sw1) ≤ a1d(w0,w1) + a2[ a + d(w0,sw0) b + d(w0,w1) ]d(w0,sw0) + a3[ c+d(w0,w1) d ′ +d(w0,sw0) ]d(w1,sw1) +a4[d(w0,sw1) + d(w1,sw0)] ≤ a1d(w0,w1) + a2[ a + d(w0,w1) b + d(w0,w1) ]d(w0,w1) + a3[ c+d(w0,w1) d ′ +d(w0,w1) ]d(w1,w2) + sa4d(w0,w1) +sa4d(w1,w2). as a ≤ b and c ≤ d ′ , we have [1−a3−sa4]d(w1,w2) ≤ [a1+a2+sa4]d(w0,w1)d(w1,w2) ≤ [a1+a2+sa41−a3−sa4 ]d(w0,w1) ≤ βd(w0,w1) ≤ β 2(1−sβ)r. now, d(w0,w2) ≤ s[d(w0,w1)+d(w1,w2)] ≤ s[β(1−sβ)r+β2(1−sβ)r] ≤ βs(1−sβ)(1+β)r ≤ βs(1−sβ)(1+sβ)r ≤ βs[1 − (sβ)2]r ≤ r. this implies w2 ∈ b(w0; r). considering w3,w4,w5, · · · ,wj ∈ b(w0; r). now, d(wj,wj+1) ≤ h(swj−1,swj) ≤ a1d(wj−1,wj) + a2[ a + d(wj−1,swj−1) b + d(wj−1,wj) ]d(wj−1,swj−1) + + a3[ c + d(wj−1,wj) d ′ + d(wj−1,swj−1) ]d(wj,swj) + a4[d(wj−1,swj) + d(wj,swj−1)] ≤ a1d(wj−1,wj) + a2[ a + d(wj−1,wj) b + d(wj−1,wj) ]d(wj−1,wj) + a3[ c + d(wj−1,wj) d ′ + d(wj−1,wj) ]d(wj,wj+1) + sa4[d(wj−1,wj) + d(wj,swj+1)]. as a ≤ b and c ≤ d ′ , we have (1 −a3 −sa4)d(wj,wj+1) ≤ (a1 + a2 + sa4)d(wj−1,wj)d(wj,wj+1) ≤ [ a1 + a2 + sa4 1 −a3 −sa4 ]d(wj−1,wj) = βd(wj−1,wj) ≤ β[βd(wj−2,wj−1)] ≤ β2[βd(wj−3,wj−2)]. (1 −a3 −sa4)d(wj,wj+1) ≤ βjd(w0,w1) ∀ j ∈ n (4.3) now, d(w0,wj+1) ≤ sd(w0,w1) + s2d(w1,w2) + · · · + sjd(wj,wj+1) ≤ sd(w0,w1) + s2βd(w′,w1) + · · · + sj+1βjd(w0,w1) ≤ s(1 + sβ + (sβ)2 + · · · + (sβ)jd(w0,w1)) ≤ s[ 1 − (sβ)j 1 −sβ ]β(1 −sβ)r ≤ sβ[1 − (sβ)j]r ≤ r. thus wj+1 ∈ b(w0; r). hence wn ∈ b(w0; r) for all n ∈ n∪{0}, therefore {ws(wn)} is a sequence in b(w0; r). now, the inequality 4.3 can be written as d(wn,wn+1) ≤ βnd(w0,w1) for all n ∈ n. (4.4) hence for any m > n d(wn,wm) ≤ sd(wn,wn+1) + s2d(wn+1,wn+2) + · · · + sm−1d(wm−1,wm) ≤ [sβn + s2βn+1 + · · · + int. j. anal. appl. 18 (3) (2020) 446 sm−nβm−1]d(w0,w1) by using 4.4 ≤ sβn[1 + sβ + (sβ)2 + · · · + sm−n−1βm−n−1]d(w0,w1) ≤ sβn[1 + sβ + (sβ)2 + · · · ]d(w0,w1) ≤ [ (sβ)n 1−sβ ]d(w0,w1) → 0 as m, n → ∞. thus, we prove that {ws(wn)} is a cauchy sequence in b(w0; r). as every closed ball in complete b-metric space is complete, so there exists h ∈ b(w0; r) such that {ws(wn)} → h or limn→∞d(wn,h) = 0. if h ∈ sh, the desired result is obvious and straight forward. if h /∈ sh then d(h,sh) = z > 0, that is d(h,sh) ≤ s[d(h,wn+1)+d(wn+1,sh)] ≤ s[d(h,wn+1)+h(swn,sh)] ≤ sd(h,wn+1)+sa1d(wn,h)+a2[ a+d(wn,swn) b+d(wn,h) ]d(wn,swn)+a3[ c+d(wn,h) d ′ +d(wn,swn) ]d(h,sh)+a4[d(wn,sh)+d(h,swn)] ≤ sd(h,wn+1) + sa1d(wn,h) + a2[ a+d(wn,wn+1) b+d(wn,h) ]d(wn,wn+1) + a3[ c+d(wn,h) d ′ +d(wn,wn+1) ]d(h,sh) + sa4[d(wn,h) + d(h,sh)]. taking n →∞, it follows that d(h,sh) ≤ a3d(h,sh) + sa4d(h,sh) (1 −a3 −sa4)d(h,sh) ≤ 0. this implies d(h,sh) = z ≤ 0, a contradiction, so h ∈ sh. hence proved � corollary 4.1. let (w,d) be a complete metric space and w0 be any point in w . let s : w → w be a mapping and wn = swn−1 be a picard sequence. if d(swn,swn+1) ≤ a1d(wn,wn+1) + a2[ a + d(wn,swn) b + d(wn,wn+1) ]d(wn,swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,swn) ]d(wn+1,swn+1) + a4[d(wn,swn+1) + d(wn+1,swn)], (4.5) for all wn,wn+1 ∈ b(w0; r) ∪{wn} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with a ≤ b, c ≤ d ′ . also d(w0,sw0) ≤ β(1 −β)r, where β = a1+a2+a4 1−a3−a4 , β < 1. then {wn} is a sequence in b(w0; r) and wn → h ∈ b(w0; r). also if inequality 4.5 holds for h, then s has a fixed point h in b(w0; r). corollary 4.2. let (w,d) be a complete b-metric space with coefficient s and w0 be any point in w . let the mapping s : w → p(w) satisfy the following: h(swn,swn+1) ≤ a1d(wn,wn+1) + a2[( a + d(wn,swn) b + d(wn,wn+1) ]d(wn,swn) + a3[ c + d(wn,wn+1) d ′ + d(wn,swn) ]d(wn+1,swn+1) + a4[d(wn,swn+1) + d(wn+1,swn)], (4.6) for all wn,wn+1 ∈{ws(wn)} and a,b,c,d ′ ,a1,a2,a3,a4 > 0 with sa1 + sa2 + a3 + (s 2 + s)a4 < 1, and a ≤ b, c ≤ d ′ . then {ws(wn)}→ h ∈ w . also if inequality 4.6 holds for h, then s has a fixed point h in w . authors contribution: all authors contributed equally and signifcantly in writing this paper. all authors read and approved the origenal manuscript. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (3) (2020) 447 references [1] m. abbas, b. ali, and yi. suleiman, common fixed points of locally contractive mappings in multiplicative metric spaces with application, int. j. math. math. sci. 2015 (2015), article id 218683. [2] m. abbas, m. d. sen, and t. nazir, common fixed points of generalized rational type cocyclic mapping in multiplicative metric spaces, discrete dynamics in nature and society, 2015 (2015), id 532725. [3] a. e. al-mazrooei, d. lateef, and j. ahmad, common fixed point theorems for generalized contractions, j. math. anal. 8 (3) (2017), 157-166. [4] m. u. ali, t. kamran, and a. kurdi, fixed point in b-multiplicative metric spaces, u.p.b. sci. bull. ser. a, 79 (3) (2017), 107-116. [5] m. arshad, a. shoaib, m. abbas and a. azam, fixed points of a pair of kannan type mappings on a closed ball in ordered partial metric spaces, miskolic math. notes, 14 (3) (2013), 769-784. [6] i. a. bakhtin, the contraction mapping principle in almost metric spaces, funct. anal. 30 (1989), 26-30. [7] s. czerwick, contraction mapping in b-metric spaces, acta math. inform. univ. ostrav. 1 (1993), 5-11. [8] c. mongkolkeha, and w. sintunaravat, optimal approximate solutions for multiplicative proximal contraction mappings in multiplicative metric spaces, proc. nat. acad. sci. 86 (1) (2016), 15-20. [9] c. mongkolkeha, and w. sintunavarat, best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces, j. nonlinear sci. appl. 8 (6) (2015), 1134-1140. [10] m. ozavsar and a. c. cervikel, fixed points of multiplicative contraction mappings on multiplicative metric spaces, arxiv:1205.5131v1 [math.gm], 2012. [11] m. sarwar, and m. rahman, fixed point theorems for ciric’s and generalized contractions in b-metric spaces, int. j. anal. appl. 7 (1) (2015), 70-78. [12] a. shoaib, a. azam, m. arshad, and e. ameer, fixed point results for multivalued mappings on a sequence in a closed ball with application, j. math. computer sci. 17 (2017), 308-316. [13] a shoaib, p kumam, a shahzad, s phiangsungnoen, q mahmood, fixed point results for fuzzy mappings in a b-metric space. fixed point theory appl. 2018 (2018), 2. [14] a. shoaib, m. arshad, and a. azam, fixed points of a pair of locally contractive mappings in ordered partial metric spaces. mat. vesnik, 67 (1) (2015), 26-38. [15] a. shoaib, m. arshad and m. a. kutbi, common fixed points of a pair of hardy rogers type mappings on a closed ball in ordered partial metric spaces. j. comput. anal. appl. 17 (2) (2014), 266-264. 1. introduction 2. preliminaries and basic definitions 3. results for b-multiplicative metric space 4. results for b-metric space references international journal of analysis and applications volume 16, number 4 (2018), 503-517 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-503 a perturbed version of general weighted ostrowski type inequality and applications waseem ghazi alshanti∗ department of general studies, jubail university college, saudi arabia ∗corresponding author: shantiw@ucj.edu.sa abstract. the main purpose of this paper is to derive some new generalizations of weighted ostrowski type inequalities. the new established inequalities are carried out for a twice differentiable mapping in different lp spaces. applications throught considering grüss type inequality and numerical integration are also provided. 1. introduction the ostrowski’s inequality [1] can be considered as a very powerful tool for enhancement of numerical integration rules. it provides convenient potintial window for establishing bounds for the well known newtoncotes rules. to illustrate this point, consider f : [a,b] → r to be a bounded function such that b − a is small, then i = b∫ a f (x) dx, can be, simply, approximated by sampling at one point as i∗(x) = (b−a) f(x) for some x ∈ [a,b] . now, if f ′ exists and is bounded, the inequality of ostrowski may be stated as follows | i∗(x) − i| ≤  1 4 + ( x− a+b 2 b−a )2‖f′‖∞ , (1.1) received 2018-02-12; accepted 2018-04-27; published 2018-07-02. 2010 mathematics subject classification. 26d15. key words and phrases. ostrowski’s inequality; weight function; grüss inequality; numerical integration. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 503 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-503 int. j. anal. appl. 16 (4) (2018) 504 where ‖f′‖∞ = sup x∈[a,b] |f′ (x)| . consequently, over the past few decades, there have been many studies on obtaining sharp bounds of (1.1) by considering the mappings and their derivatives in various lebesgue spaces. further, the new bounds have been carried out by implementing weighted and non-weighted peano kernel. several weighted and nonweighted versions of (1.1) have been derived. applications in both numerical integration and probability are also presented in this regards. for instance, roumeliotis et. al [2] proved a weighted integral inequality of ostrowski’s type for mappings whose second derivatives are bounded. cerone [3] obtained bounds for the deviation of a function from a combination of integral means over the end intervals covering the entire interval. qayyum et. al [4] established a new ostrowski’s type inequality using weight function which generalizes the inequality in [3]. barnett [5] reported a companion of (1.1) and the generalized trapezoid inequalites for various classes of functions, including functions of bounded variation, lipschitzian, convex and absolutely continuous functions. recently, budak et. al [6] presented a new generalization of weighted ostrowski’s type inequality for mappings of bounded variation. several further generalizations of (1.1) are provided in [7] [16]. in [12], qayyum et. al proved the following non-weighted generalization of ostrowski’s type integral inequality. theorem 1.1. let f : [a,b] → r be a twice differentiable mapping. then∣∣∣∣ 12 (α + β) [α(x−a)−β (b−x)]f ′(x)−f(x) + 1 α + β   α x−a x∫ a f(t)dt + β b−x b∫ x f(t)dt   ∣∣∣∣∣∣ , ≤   [ α(x−a)2 + β (b−x)2 ] ‖f ′′‖∞ 6(α+β) , f ′′ ∈ l∞ [a,b] , 1 (2q+1) 1 q [ αq (x−a)q+1 + βq (b−x)q+1 ]1 q ‖f ′′‖ p 2(α+β) , f ′′ ∈ lp [a,b] , p > 1, 1 p + 1 q = 1, [α(x−a) + β (b−x) + |α(x−a)−β (b−x)|] ‖f ′′‖ 1 4(α+β) , f ′′ ∈ l1 [a,b] . (1.2) where α and β are non-negative real numbers such that not both zero. in this paper, motivated by the non-weighted case in [12], new general weighted peano kernel has been defined. to obtain new general weighted inequality of ostrowski’s type that is more generalized and extended as compare to [12]. we consider a twice differentiable mapping f where, respectively, f ′′ ∈ l∞, f ′′ ∈ lp int. j. anal. appl. 16 (4) (2018) 505 and f ′′ ∈ l1. moreover, we utilize grüss type inequality to present the perturbed verion of our result. finally, we investigate the new general weighted inequality in numerical integration. before we introduce our main result for a general weighted inequality of ostrowski’s type, we commence with the following definition and lemma. definition: let ω : (a,b) → (0,∞) be a non-negative weighted function (density) such that b∫ a ω (t) dt < ∞. the domain of ω may be finite or infinite and may vanish at the boundary points. we denote the moments m(a,b) = 1 b−a b∫ a ω (t) dt, m(a,b) = 1 b−a b∫ a tω (t) dt, n(a,b) = 1 b−a b∫ a t2ω (t) dt, µ(a,b) = m(a,b) m(a,b) , σ2(a,b) = n(a,b) m(a,b) −µ2(a,b). (1.3) furthermore, for a function f : [a,b] → r, we define the functional s(f; a,b) = 1 b−a b∫ a f(t)ω (t) dt. (1.4) lemma 1.1. let f : [a,b] → r be a twice differentiable mapping. denote by pω : [a,b] 2 → r the weighted peano kernel function that is given by pω(x,t) =   α α+β 1 x−a t∫ a (t−u) ω(u)du, t ∈ [a,x] β α+β 1 b−x t∫ b (t−u) ω(u)du, t ∈ (x,b] , (1.5) where α,β ≥ 0 and not both zero. then the following identity b∫ a pω(x,t)f ′′ (t) dt = z(f; α,β), (1.6) where z(f; α,β) = 1 α + β {[αm(a,x)(x−µ(a,x)) + βm(x,b)(x−µ(x,b))] f ′(x) − [αm(a,x) + βm(x,b)] f(x) + αs(f; a,x) + βs(f; x,b)} , (1.7) holds. int. j. anal. appl. 16 (4) (2018) 506 proof: from (1.5), we have b∫ a pω(x,t)f ′′ (t) dt = α α + β 1 x−a x∫ a   t∫ a (t−u) ω(u)du  f ′′ (t) dt + β α + β 1 b−x b∫ x   t∫ b (t−u) ω(u)du  f ′′ (t) dt = α α + β 1 x−a  f ′(x) x∫ a (x−u) ω(u)du −f(x) x∫ a ω(u)du + x∫ a f(t)ω(t)dt   + β α + β 1 b−x  f ′(x) b∫ x (x−u) ω(u)du − f(x) x∫ b ω(u)du + b∫ x f(t)ω(t)dt   . after further simplification, the identity (1.6) can be obtained. 2. main results theorem 2.1. let f : [a,b] → r be a twice differentiable mapping in (a,b) . then ∀x ∈ [a,b], we have |z(f; α,β)| ≤   { αm(a,x) [ (x−µ(a,x))2 + σ2 (a,x) ] + βm(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]} ×‖ f ′′‖∞ 2(α+β) , f ′′ ∈ l∞ [a,b] , [ αq (x−a)q x∫ a (t−a)2q mq(a,t)dt + β q (b−x)q b∫ x (b− t)2q mq(t,b)dt ]1 q ‖f ′′‖ p 2(α+β) , f ′′ ∈ lp [a,b] , p > 1, 1 p + 1 q = 1, max{αm(a,x) (x−µ(a,x)) , βm(x,b) (µ(x,b) −x)} ‖ f ′′‖ 1 α+β , f ′′ ∈ l1 [a,b] . (2.1) where the functional z(f; α,β) is defined in (1.6) . int. j. anal. appl. 16 (4) (2018) 507 proof: taking the modulus of the right hand side of (1.6) , yields∣∣∣∣∣∣ b∫ a pω(x,t)f ′′ (t) dt ∣∣∣∣∣∣ ≤ b∫ a |pω(x,t)| |f ′′ (t)|dt. (2.2) hence, for f ′′ ∈ l∞ [a,b] |z(f; α,β)| ≤ ‖f ′′‖∞ b∫ a |pω(x,t)|dt. (2.3) now, using (1.5) provids, b∫ a |pω(x,t)|dt = α α + β 1 x−a  x2 2 x∫ a ω (u) du − x x∫ a uω (u) du + 1 2 x∫ a t2ω (t) dt   + β α + β 1 b−x  x2 2 b∫ x ω (u) du − x b∫ x uω (u) du + 1 2 b∫ x t2ω (t) dt   . (2.4) thus, by combining (2.3) and (2.4) , the first inequality of (2.1) results. further, from (2.2) and by using hölder’s integral inequality for f ′′ ∈ lp [a,b] , we have |z(f; α,β)| ≤ ‖f ′′‖p   b∫ a |pω(x,t)| q dt   1 q , (2.5) where 1 p + 1 q = 1 with p > 1. now, by (1.5) and utilizing the weighted mean value theorem for integrals, we have b∫ a |pω(x,t)| q dt = ( α α + β )q 1 (x−a)q x∫ a   t∫ a (t−u) ω(u)du  q dt int. j. anal. appl. 16 (4) (2018) 508 + ( β α + β )q 1 (b−x)q b∫ x   t∫ b (t−u) ω(u)du  q dt = ( α α + β )q 1 (x−a)q x∫ a  t−a 2 t∫ a ω(u)du  q dt + ( β α + β )q 1 (b−x)q b∫ x  b− t 2 b∫ t ω(u)du   q dt, (2.6) and so, by considering (2.5) and (2.6) the second inequality of (2.1) is obtained. finally, for f ′′ ∈ l1 [a,b] , we have from (1.6) z(f; α,β) ≤ sup t∈[a,b] |pω(x,t)|‖f ′′‖1 , (2.7) where, sup t∈[a,b] |pω(x,t)| = max   αα + β 1x−a x∫ a (x−u) ω(u)du , β α + β 1 b−x b∫ x (u−x) ω(u)du   = 1 α + β max{αm(a,x)(x−µ(a,x) , βm(x,b)(µ(x,b) −x]} . (2.8) therefore, combining (2.7) and (2.8) gives the third inequality of (2.1) , and so, the theorem is now completely proven. remark 2.1. setting ω(u) = 1 in theorem (2.1) provids the corresponding non-weighted result (1.2) in [12]. for different weights, a variety of results can be obtained. int. j. anal. appl. 16 (4) (2018) 509 corollary 2.1. let the condition of theorem (2.1) holds. then |z(f; α,β)| ≤ { αm(a,x) [ (x−µ(a,x))2 + σ2 (a,x) ] + βm(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]} × ‖f ′′‖∞ 2(α + β) ≤ [ α (x−a)2 m(a,x) + β (x− b)2 m(x,b) ] × ‖f ′′‖∞ 2(α + β) . (2.9) proof: from (1.5), we have b∫ a |pω(x,t)|dt = 1 2 α α + β 1 x−a x∫ a (x− t)2 ω (t) dt + 1 2 β α + β 1 b−x b∫ x (x− t)2 ω (t) dt. now, by noting that x∫ a (x− t)2 ω (t) dt ≤ sup t∈[a,x] (x− t)2 [(x−a) m(a,x)] = (x−a)3 m(a,x), and b∫ x (x− t)2 ω (t) dt ≤ sup t∈(x,b] (x− t)2 [(b−x) m(x,b)] = (b−x)3 m(x,b). the desired second inequality of (2.9) can be obtained. corollary 2.2. setting α = β in theorem (2.1) gives∣∣∣∣12 {[m(a,x)(x−µ(a,x)) + m(x,b)(x−µ(x,b))] f ′(x) − (m(a,x) + m(x,b)) f(x) + s(f; a,x) + s(f; x,b)}| int. j. anal. appl. 16 (4) (2018) 510 ≤   { m(a,x) [ (x−µ(a,x))2 + σ2 (a,x) ] + m(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]} × ‖f ′′‖∞ 4 , f ′′ ∈ l∞ [a,b] , [ 1 (x−a)q x∫ a (t−a)2q mq(a,t)dt + 1 (b−x)q b∫ x (b− t)2q mq(t,b)dt ]1 q ‖f ′′‖ p 4 , f ′′ ∈ lp [a,b] , p > 1, 1 p + 1 q = 1, max{m(a,x) (x−µ(a,x)) , m(x,b) (µ(x,b)−x)} ‖f ′′‖ 1 2 , f ′′ ∈ l1 [a,b] . (2.10) remark 2.2. setting ω(u) = 1 in corollary (2.2) gives the corresponding non-weighted result obtained by [12] for the case α = β. corollary 2.3. setting x = (a + b) /2 in theorem (2.1) gives∣∣∣∣ 1α + β {[ αm ( a, a + b 2 )( a + b 2 −µ ( a, a + b 2 )) + βm ( a + b 2 ,b )( a + b 2 −µ ( a + b 2 ,b ))] f ′ ( a + b 2 ) − [ αm ( a, a + b 2 ) + βm ( a + b 2 ,b )] f ( a + b 2 ) + αs ( f; a, a + b 2 ) + βs ( f; a + b 2 ,b )}∣∣∣∣ ≤   { αm(a, a+b 2 ) [ (a+b 2 −µ(a, a+b 2 ))2 + σ2 ( a, a+b 2 )] + βm(a+b 2 ,b) [ (a+b 2 −µ(a+b 2 ,b))2 + σ2 ( a+b 2 ,b )]} ×‖f ′′‖∞ , f ′′ ∈ l∞ [a,b] , [ αq a+b 2∫ a (t−a)2q mq(a,t)dt + βq b∫ a+b 2 (b− t)2q mq(t,b)dt   1 q ‖f ′′‖ p (b−a)(α+β), f ′′ ∈ lp [a,b] , p > 1, 1 p + 1 q = 1, max { αm(a, a+b 2 ) ( a+b 2 −µ(a, a+b 2 ) ) , β m(a+b 2 ,b) ( µ(a+b 2 ,b) − a+b 2 )} ‖f ′′‖ 1 α+β , f ′′ ∈ l1 [a,b] . (2.11) int. j. anal. appl. 16 (4) (2018) 511 remark 2.3. setting ω(u) = 1 in corollary (2.3) , gives the corresponding non-weighted result obtained by [12] when x = (a + b) /2. corollary 2.4. if (2.10) evaluated at x = (a + b) /2, then∣∣∣∣12 {[ m(a, a + b 2 )( a + b 2 −µ(a, a + b 2 )) + m( a + b 2 ,b)(x−µ( a + b 2 ,b)) ] f ′( a + b 2 ) − ( m(a, a + b 2 ) + m( a + b 2 ,b) ) f( a + b 2 ) + s(f; a, a + b 2 ) + s(f; a + b 2 ,b) }∣∣∣∣ ≤   { m(a, a+b 2 ) [ (a+b 2 −µ(a, a+b 2 ))2 + σ2 ( a, a+b 2 )] + m(a+b 2 ,b) [ (a+b 2 −µ(a+b 2 ,b))2 + σ2 ( a+b 2 ,b )]} ×‖ f ′′‖∞ 4 , f ′′ ∈ l∞ [a,b] , [ a+b 2∫ a (t−a)2q mq(a,t)dt + b∫ a+b 2 (b− t)2q mq(t,b)dt   1 q ‖f ′′‖ p 2(b−a) , f ′′ ∈ lp [a,b] , p > 1, 1 p + 1 q = 1, max { m(a, a+b 2 ) ( a+b 2 −µ(a, a+b 2 ) ) , m(a+b 2 ,b) ( µ(a+b 2 ,b) − a+b 2 )} ‖f ′′‖ 1 2 , f ′′ ∈ l1 [a,b] . (2.12) 3. perturbed results the grüss inequality is as follows [13]. theorem 3.1. let f,g : [a,b] → r be integrable functions on [a,b] such that ϕ ≤ f(x) ≤ φ and γ ≤ g(x) ≤ γ ∀x ∈ [a,b] , where ϕ, φ, γ, γ are constants. then∣∣∣∣∣∣ 1b−a b∫ a f(t)g(t)dt− 1 b−a b∫ a f(t)dt · 1 b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ 1 4 (φ −ϕ) (γ −γ) , (3.1) where the constant 1 4 is sharp. now, the perturbed verions of the results in the pervious section may be obtained by using grüss type inequalities involving the čebyŝev functional [14], t (f,g) = m(fg; a,b) −m(f; a,b)m(g; a,b), (3.2) int. j. anal. appl. 16 (4) (2018) 512 where m(f; a,b) = 1 b−a b∫ a f(t)dt, is the integral mean of f over [a,b] . theorem 3.2. let f : [a,b] → r be a twice differentiable mapping such that γ ≤ f ′′(x) ≤ γ ∀x ∈ [a,b] and α ≥ 0,β ≥ 0, α + β 6= 0. then∣∣∣∣z(f; α,β) − k2(α + β) [αm(a,x) [(x−µ(a,x))2 + σ2 (a,x)] + βm(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]]∣∣ ≤ (b−a) n(x) [ ‖f ′′‖22 b−a −k2 ]1 2 , ≤ (b−a) n(x) γ −γ 2 , ≤ (b−a) (γ −γ) 4 (α + β) max{αm(a,x) (x−µ(a,x)) , βm(x,b) (µ(x,b) −x)} , (3.3) where z(f; α,β) is given by (1.7) , k = (f ′(b) −f ′(a)) / (b−a) , and n(x) =    ( α x−a )2 x∫ a (t−a)4 m2(a,t)dt + ( β b−x )2 b∫ x (b− t)4 m2(t,b)dt   1 4(α + β)2 (b−a) (3.4) − {[ αm(a,x) [ (x−µ(a,x))2 + σ2 (a,x) ] + βm(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]] 1 2(α + β) (b−a) }2}12 . proof: replacing f(t) by pω(x,t) and g(t) by f ′′(x) in (3.2) yields, t (pω(x,t),f ′′(x)) = m(pω(x,t)f ′′(x); a,b) (3.5) −m(pω(x,t); a,b)m(f ′′(x); a,b). now, by using both (1.6) and (2.4) , we have t (pω(x,t),f ′′(x)) = 1 b−a {z(f; α,β) − k 2(α + β) [ αm(a,x) [ (x−µ(a,x))2 + σ2 (a,x) ] + βm(x,b) [ (x−µ(x,b))2 + σ2 (x,b) ]]} , (3.6) int. j. anal. appl. 16 (4) (2018) 513 where k is the secant slope of f ′ over [a,b] as given in (3.4) . moreover, by [3], we have t (pω(x,t),f ′′(x)) ≤ t 1 2 (pω(x,t),pω(x,t)) t 1 2 (f ′′(x),f ′′(x)) , (pω(x,t), f ′′(x) ∈ l2 [a,b] ) ≤ γ −γ 2 t 1 2 (pω(x,t),pω(x,t)) , (γ ≤ f ′′(x) ≤ γ, ∀x ∈ [a,b]) ≤ 1 4 (φ −ϕ) (γ −γ) , (3.7)( ϕ ≤ pω(x,t) ≤ φ, ∀(x,t) ∈ [a,b] 2 ) . but, 0 ≤ t 1 2 (f ′′(x),f ′′(x)) = [ m((f ′′(x)) 2 ; a,b) −m2(f ′′(x); a,b) ]1 2 =   1b−a b∫ a (f ′′(x)) 2 dx−   b∫ a f ′′(x)dx b−a   2 1 2 = [ 1 b−a ‖f ′′‖22 −k 2 ]1 2 ≤ γ −γ 2 , (3.8) where γ ≤ f ′′(x) ≤ γ, ∀x ∈ [a,b] . now, for t 1 2 (pω(x,t),pω(x,t)) , we consider (1.5) as follows 0 ≤ t 1 2 (pω(x,t),pω(x,t)) = [ m((pω(x,t)) 2 ; a,b) −m2(pω(x,t); a,b) ]1 2 =   1b−a b∫ a (pω(x,t)) 2 dx−   b∫ a pω(x,t)dx b−a   2 1 2 = n(x), (3.9) where n(x) is given in (3.4) . therefore, by combining (3.6), (3.7), (3.8), and (3.9) gives the first and the second inequalities of (3.3) . further, to determine the values of ϕ and φ for which ϕ ≤ pω(x,t) ≤ φ, ∀(x,t) ∈ [a,b] 2 , it may be noticed int. j. anal. appl. 16 (4) (2018) 514 from the definition of pω(x,t) in (1.5) that for α,β ≥ 0, α + β 6= 0, we have φ = sup t∈[a,b] pω(x,t) = 1 α + β max{αm(a,x) (x−µ(a,x)) ,βm(x,b) (µ(x,b) −x)} , ϕ = inf t∈[a,b] pω(x,t) = 0. (3.10) hence, from (3.6), (3.8), (3.10) and the last inequality in (3.7) , we obtain the third inequality in (3.3) and the theorem is now completely proved. 4. application in numerical integration let a = ζ◦ < ζ1 < · · · < ζn−1 < ζn = x = η◦ < η1 < · · · < ηn−1 < ηn = b be a partition of the interval [a,b], with xi ∈ [ζi,ζi+1] for i = 0, 1, .....,n − 1, x∗j ∈ [ηj,ηj+1] for j = 0, 1, .....,n − 1, δ = ζi+1 − ζi, and ∆ = ηj+1 −ηj. consider the following general quadrature rule. a (f,ζ,η,x) = n−1 α ∑ i=0 mi [f (xi) − (xi −µi) f ′ (xi)] + n−1 β ∑ j=0 m∗j [ f (x∗j ) − ( x∗j −µ ∗ j ) f ′ ( x∗j )] . (4.1) theorem 4.1. let the conditions of theorem (2.1) hold. the following weighted quadrature rule for weighted integral holds α δ x∫ a f(t)ω (t) dt + β ∆ b∫ x f(t)ω (t) dt = a (f,ζ,η,x) + r (f,ζ,η,x) , (4.2) where a (f,ζ,η,x) is defined by (4.1), the remainder r (f,ζ,η,x) satisfies the estimate r (f,ζ,η,x) ≤ { n−1 α ∑ i=0 mi [ (xi −µi) 2 + σ2i ] + n−1 β ∑ j=0 m∗j [( x∗j −µ ∗ j )2 + σ∗2j ]  ‖f ′′‖∞ 2 , (4.3) and the parameters mi, µi, σ 2 i , m ∗ j , µ ∗ j , and σ ∗2 j are given by mi = m(ζi,ζi+1), µi = µ (ζi,ζi+1) , σ 2 i = σ 2 (ζi,ζi+1) , m∗j = m (ηj,ηj+1) , µ ∗ j = µ (ηj,ηj+1) , and σ ∗2 j = σ 2 (ηj,ηj+1) . (4.4) int. j. anal. appl. 16 (4) (2018) 515 proof: applying the first inequality of (2.1) over the interval [ζi,ζi+1] with x = xi ∈ [ζi,ζi+1] and over the interval [ηj,ηj+1] with x = x ∗ j ∈ [ηj,ηj+1] gives∣∣αmi(xi −µi)f ′(xi) + βm∗j (x∗j − µ∗j )f ′(x∗j ) −αmif(xi) + βm∗jf(x∗j ) + α δ ζi+1∫ ζi f(t)ω (t) dt + β ∆ ηj+1∫ ηj f(t)ω (t) dt ∣∣∣∣∣∣∣ ≤ { αmi [ (xi −µi)2 + σ2i ] + βm∗j [ (x∗j −µ ∗ j ) 2 + σ∗2j ]} ‖f ′′‖∞ 2 , for all i,j = 0, 1, .....,n− 1. summing over i,j from 0 to n− 1 and using the triangle inequality produces the desired result (4.2). theorem 4.2. let the conditions of theorem (2.1) hold. the following weighted quadrature rule for weighted integral holds α δ x∫ a f(t)ω (t) dt + β ∆ b∫ x f(t)ω (t) dt = a (f,ζ,η,x) + r (f,ζ,η,x) , where a (f,ζ,η,x) is defined by (4.1), the remainder r (f,ζ,η,x) satisfies the estimate r (f,ζ,η,x) ≤  αq δq ζi+1∫ ζi (t− ζi) 2q mq(ζi, t)dt + βq ∆q ηj+1∫ ηj (ηj+1 − t) 2q mq(t,ηj+1)dt   1 q ‖f ′′‖p 2 . (4.5) proof: applying the second inequality of (2.1) over the interval [ζi,ζi+1] with x = xi ∈ [ζi,ζi+1] and over the interval [ηj,ηj+1] with x = x ∗ j ∈ [ηj,ηj+1] gives∣∣αmi(xi −µi)f ′(xi) + βm∗j (x∗j − µ∗j )f ′(x∗j ) −αmif(xi) + βm∗jf(x∗j ) + α δ ζi+1∫ ζi f(t)ω (t) dt + β ∆ ηj+1∫ ηj f(t)ω (t) dt ∣∣∣∣∣∣∣ ≤  αq δq ζi+1∫ ζi (t− ζi) 2q mq(ζi, t)dt + βq ∆q ηj+1∫ ηj (ηj+1 − t) 2q mq(t,ηj+1)dt   1 q ‖f ′′‖p 2 . int. j. anal. appl. 16 (4) (2018) 516 for all i,j = 0, 1, .....,n− 1. summing over i,j from 0 to n− 1 and using the triangle inequality produces the desired result (4.5). theorem 4.3. let the conditions of theorem (2.1) hold. the following weighted quadrature rule for weighted integral holds α δ x∫ a f(t)ω (t) dt + β ∆ b∫ x f(t)ω (t) dt = a (f,ζ,η,x) + r (f,ζ,η,x) , where a (f,ζ,η,x) is defined by (4.1), the remainder r (f,ζ,η,x) satisfies the estimate r (f,ζ,η,x) ≤ max [ αmi (xi −µi) ,βm∗j ( µ∗j −x ∗ j )] ‖f ′′‖1 . (4.6) proof: applying the third inequality of (2.1) over the interval [ζi,ζi+1] with x = xi ∈ [ζi,ζi+1] and over the interval [ηj,ηj+1] with x = x ∗ j ∈ [ηj,ηj+1] gives ∣∣αmi(xi −µi)f ′(xi) + βm∗j (x∗j − µ∗j )f ′(x∗j ) −αmif(xi) + βm∗jf(x∗j ) + α δ ζi+1∫ ζi f(t)ω (t) dt + β ∆ ηj+1∫ ηj f(t)ω (t) dt ∣∣∣∣∣∣∣ ≤ max { αmi (xi −µi) ,βm∗j ( µ∗j −x ∗ j )} ‖f ′′‖1 . for all i,j = 0, 1, .....,n− 1. summing over i,j from 0 to n− 1 and using the triangle inequality produces the desired result (4.6). references [1] a. ostrowski, über die absolutabweichung einer differentienbaren funktionen von ihren integralimittelwert, comment. math. hel. 10 (1938), 226-227. [2] j. roumeliotis, p. cerone, and s. s. dragomir, an ostrowski type inequality for weighted mappings with bounded second derivatives, rgmia res. rep. collect. 1 (1998), 101-111. [3] p. cerone, a new ostrowski type inequality involving integral means over end intervals, tamkang j. math. 33(2) (2002), 109-118. [4] a. qayyum, m. shoaib, and i. faye, on new weighted ostrowski type inequalities involving integral means over end intervals and application, turkish j. anal. numb. theory. 3(2) (2015), 61-67. [5] n. barnett, s. s. dragomir, and i. gomm, a companion for the ostrowski and the generalised trapezoid inequalities, math. comput. model. 50(1) (2009), 179-187. [6] h. budak, s. erden, and m. sarikaya, new weighted ostrowski type inequalities for mappings whose nth derivatives are of bounded variation, int. j. anal. appl. 12(1) (2016), 71-79. [7] m. alomari, a companion of ostrowski’s inequality for mappings whose first derivatives are bounded and applications in numerical integration, transylvanian j. math. mech. 4(2) (2012), 103–109. int. j. anal. appl. 16 (4) (2018) 517 [8] m. alomari, a generalization of companion inequality of ostrowski’s type for mappings whose first derivatives are bounded and applications in numerical integration, kragujevac j. math. 36(1) (2012), 77–82. [9] w. liu, new bounds for the companion of ostrowski’s inequality and applications, filomat. 28 (2014), 167-178. [10] r. agarwal, p. ravi, m. luo, and r. raina, on ostrowski type inequalities, fasciculi math. 56(1) (2016), 5-27. [11] h. budak, m. sarikaya, a new companion of ostrowski type inequalities for functions of two variables with bounded variation, facta universitatis, series: mathematics and informatics. 31(2) (2016), 447-463. [12] a. qayyum, m. shoaib, a. matouk, and m. latif, on new generalized ostrowski type integral inequalities, abstr. appli. anal. 2014 (2014), art. id 275806. [13] g. grüss, uber das maximum des absoluten betrages von 1/(b − a) b∫ a f(x)g(x)dx − 1/(b − a)2 b∫ a f(x)dx b∫ a g(x)dx, mathematische zeitschrift. vol. 39(1) (1935), 215–226. [14] p. l. čebyŝev, sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, proc. math. soc. kharkov. 2 (1882), 93–98. [15] w.g. alshanti and a. qayyum, a note on new ostrowski type inequalities using a generalized kernel. bull. math. anal. appl. 9(1) (2017), 74-91. [16] w.g. alshanti, a. qayyum and m.a. majid, ostrowski type inequalities by using general quadratic kernel. j. inequal. spec. funct. 8(4) (2017), 111-135. 1. introduction 2. main results 3. perturbed results 4. application in numerical integration references international journal of analysis and applications volume 19, number 4 (2021), 561-575 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-561 application and graphical interpretation of a new two-dimensional quaternion fractional fourier transform khinal parmar∗, v. r. lakshmi gorty nmims, mpstme, vile parle west, mumbai 400056, india ∗corresponding author: khinal.parmar@nmims.edu abstract. in this paper, a new two-dimensional quaternion fractional fourier transform is developed. the properties such as linearity, shifting and derivatives of the quaternion-valued function are studied. the convolution theorem and inversion formula are also established. an example with graphical representation is solved. an application related to two-dimensional quaternion fourier transform is also demonstrated. 1. introduction in 1853, quaternions were developed by w. r. hamilton [10]. the necessity of enlarging the operations on three-dimensional vectors to include multiplication and division led hamilton to introduce the fourdimensional algebra of quaternions. in 1993, ell [6] introduced quaternion fourier transform for application to two-dimensional linear time-invariant systems of partial differential equations. in 2001 [3], authors defined non-commutative hypercomplex fourier transforms of multidimensional signals. in 2007 [9], author introduced right side quaternion fourier transform. in 2008 [8], the concept of fractional quaternion fourier transform was presented. in [11], the author studied the uncertainty principle for the quaternion fourier transform. authors in [1] developed quaternion domain fourier transforms and its application in mathematical statistics. in [4], plancherel theorem and quaternion fourier transform for square-integrable functions were studied. received april 7th, 2021; accepted may 13th, 2021; published may 25th, 2021. 2010 mathematics subject classification. 44a35, 44a05, 46s10. key words and phrases. quaternion fractional fourier transform; convolution; operational calculus; graphical representation. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 561 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-561 int. j. anal. appl. 19 (4) (2021) 562 quaternion fourier transform transfers signals from the real-valued time domain to quaternion-valued frequency domain. but the proposed two-dimensional quaternion fractional fourier transform will transfer the signal to unified time-frequency domains. hence, it has a wide range of applications in the field of optics and signal processing. the organization of the paper is as follows: in section 2, some basic facts of quaternions and quaternionvalued functions are illustrated. in section 3, the two-dimensional quaternion fractional fourier transform is defined and its inversion formula and operational properties are developed. graphical interpretation of two-dimensional quaternion fractional fourier transform is also illustrated. in section 4, the application of the two-dimensional quaternion fractional fourier transform is shown. 2. preliminary results in quaternions, every element is a linear combination of a real scalar and three imaginary units i, j and k with real coefficients. let q be a quaternion defined in (2.1) h = {q = x0 + ix1 + jx2 + kx3 : x0,x1,x2,x3 ∈ r} be the division ring of quaternions, where i, j, k satisfy hamilton’s multiplication rules (see, e.g. [9]) (2.2) ij = −ji = k, jk = −kj = i, ki = −ik = j, i2 = j2 = k2 = ijk = −1. the quaternion conjugate of q is defined by (2.3) q̄ = x0 − ix1 − jx2 − kx3; x0,x1,x2,x3 ∈ r. the norm of q ∈ h is defined as (2.4) |q| = √ qq̄ = √ x20 + x 2 1 + x 2 2 + x 2 3. alternatively, in [13] the quaternions are defined as (2.5) h = {q = q1 + jq2 : q1,q2 ∈ c} where j is the imaginary number satisfying following conditions: j2 = −1, jr = rj, ∀ r ∈ r, ji = −ij, where i is the imaginary number. from [13] f ∈ l2(r2; h), then the function is expressed as (2.6) f(u,v) = f0(u,v) + if1(u,v) + jf2(u,v) + kf3(u,v). for some applications the quaternions can be rewritten by replacing k with ij as given in [9], q = x0 + ix1 + jx2 + ix3j. int. j. anal. appl. 19 (4) (2021) 563 another way of rewritting quaternion is q = x+ + x−; x± = 1 2 (q ± iqj) . x± can also be expressed as x± = {x0 ±x3 + i(x1 ∓x2)} 1 ± k 2 = 1 ± k 2 {x0 ±x3 + j(x2 ∓x1)} . the real scalar part of the quaternion can be written as [9], (2.7) x0 = 〈q〉0 . we can also rewrite the function f ∈ l2(r2,h) as [9], f = f0 + if1 + jf2 + if3j. we can also split the function as [9], f = f+ + f−; f+ = 1 2 (f + ifj) , f− = 1 2 (f − ifj) . for f,g ∈ l2(r2,h) and u = (u,v) = ue1 + ve2 ∈ r2 with {e1,e2} as the basis of r2, the quaternion-valued inner product is defined in [9] as (2.8) (f,g) = ∫ r2 f(u)ḡ(u)d2u, with real symmetric part (2.9) 〈f,g〉 = 1 2 [(f,g) + (g,f)] = ∫ r2 〈f(u)ḡ(u)〉0 d 2u. the norm of f ∈ l2(r2,h) is defined as (2.10) ||f|| = √ (f,f) = √ 〈f,f〉 = ∫ r2 |f(u)|2d2u. 3. main results definition 3.1. let f ∈ l2(r2,h), then two-dimensional quaternion fractional fourier transform (2dqfrft) of particular order α,β using [9, 12] is defined as (3.1) f̂α,β (w1,w2) = fα,β [f (u,v) ; w1,w2] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv where 0 < α,β ≤ 1. analogous to [5, page 112],the integral will converge for values of w1 and w2 in the strips −s1 < im(w1) < s1 and −s2 < im(w2) < s2 respectively; where s1 < re(p1), s2 < re(p2), for p1 = iw1,p2 = jw2. int. j. anal. appl. 19 (4) (2021) 564 the sufficient condition for f(u,v) to have 2d-qfrft is that ∞∫ −∞ ∞∫ −∞ |f(u,v)|dudv exists. inversion formula: consider the inverse formula of quaternion fourier transform as defined in [9] f(u,v) = 1 (2π) 2 ∞∫ −∞ ∞∫ −∞ eixuf̂(x,y)ejyvdxdy. substituting x = w 1 α 1 and y = w 1 β 2 . then, f(u,v) = 1 (2π) 2 ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1 α −1 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1 β −1 2 dw1 α dw2 β f(u,v) = 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1−β β 2 dw1dw2. hence, the inversion formula is defined as (3.2) f−1α,β [ f̂α,β (w1,w2) ] = f(u,v) = 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) e jw 1 β 2 vw 1−β β 2 dw1dw2. property 3.1 (left linearity). for f1,f2 ∈ l2(r2,h) and k1,k2 ∈{q|q = x0 + ix1,x0,x1 ∈ r}; (3.3) fα,β [k1f1(u,v) + k2f2(u,v)] = k1fα,β [f1(u,v)] + k2fα,β [f2(u,v)] . proof. for f1,f2 ∈ l2(r2,h); k1,k2 ∈ r and using (3.1), we get fα,β [k1f1(u,v) + k2f2(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [k1f1(u,v) + k2f2(u,v)] e −jw 1 β 2 vdudv = k1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [f1(u,v)] e −jw 1 β 2 vdudv + k2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u [f2(u,v)] e −jw 1 β 2 vdudv = k1fα,β [f1(u,v)] + k2fα,β [f2(u,v)] . � property 3.2 (right linearity). for f1,f2 ∈ l2(r2,h) and k′1,k ′ 2 ∈{q|q = x0 + jx2,x0,x2 ∈ r}; (3.4) fα,β [f1(u,v)k ′ 1 + f2(u,v)k ′ 2] = fα,β [f1(u,v)] k ′ 1 + fα,β [f2(u,v)] k ′ 2. int. j. anal. appl. 19 (4) (2021) 565 the proof is similar to property 3.1. property 3.3 (shifting). for f ∈ l2(r2,h) and a,b ∈ r; (3.5) fα,β [f(u−a,v − b)] = e−iw 1 α 1 afα,β [f(u,v)] e −jw 1 β 2 b proof. for f ∈ l2(r2,h); a,b ∈ r and using (3.1), we get fα,β [f(u−a,v − b)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u−a,v − b)e−jw 1 β 2 vdudv. substituting u−a = s and v − b = t gives = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 (s+a)f(s,t)e−jw 1 β 2 (t+b)dsdt = e−iw 1 α 1 afα,β [f(s,t)] e −jw 1 β 2 b. � property 3.4 (2d-qfrft of derivatives). for f ∈ l2(r2,h), the two-dimensional quaternion fractional fourier transform with derivatives of f(u,v) are as follows: (3.6) i) fα,β [ ∂ ∂u f(u,v) ] = ( iw 1 α 1 ) fα,β [f(u,v)] . (3.7) ii) fα,β [ ∂ ∂v f(u,v) ] = fα,β [f(u,v)] ( jw 1 β 2 ) . (3.8) iii) fα,β [ ∂2 ∂u∂v f(u,v) ] = ( iw 1 α 1 ) fα,β [f(u,v)] ( jw 1 β 2 ) . in general (3.9) iv) fα,β [ ∂n ∂un f(u,v) ] = ( iw 1 α 1 )n fα,β [f(u,v)] . (3.10) v) fα,β [ ∂n ∂vn f(u,v) ] = fα,β [f(u,v)] ( jw 1 β 2 )n . (3.11) vi) fα,β [ ∂n ∂un ∂m ∂vm f(u,v) ] = ( iw 1 α 1 )n fα,β [f(u,v)] ( jw 1 β 2 )m . int. j. anal. appl. 19 (4) (2021) 566 proof. i) for f ∈ l2(r2,h), the first order derivative over f(u,v) w.r.t. u is given by fα,β [ ∂ ∂u f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂u f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 uf(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 uf(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv = ( iw 1 α 1 ) fα,β [f(u,v)] . ii) for f ∈ l2(r2,h), the first order derivative over f(u,v) w.r.t. v is given by fα,β [ ∂ ∂v f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂v f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞ e−iw 1 α 1 u   [ f(u,v)e−jw 1 β 2 v ] − ∞∫ −∞ f(u,v)e−jw 1 β 2 v ( −jw 1 β 2 ) dv  du = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv ( jw 1 β 2 ) = fα,β [f(u,v)] ( jw 1 β 2 ) . iii) for f ∈ l2(r2,h), the second order derivative over f(u,v) w.r.t. u,v is given by fα,β [ ∂2 ∂u∂v f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂2 ∂u∂v f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂ ∂v f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂ ∂v f(u,v)du  e−jw 1 β 2 vdv = ( iw 1 α 1 ) ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂v f(u,v)e−jw 1 β 2 vdudv int. j. anal. appl. 19 (4) (2021) 567 = ( iw 1 α 1 ) ∞∫ −∞ e−iw 1 α 1 u   [ f(u,v)e−jw 1 β 2 v ] − ∞∫ −∞ f(u,v)e−jw 1 β 2 v ( −jw 1 β 2 ) dv  du = ( iw 1 α 1 ) ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv ( jw 1 β 2 ) = ( iw 1 α 1 ) fα,β [f(u,v)] ( jw 1 β 2 ) . iv) by using mathematical induction for n = 1 by (3.6), we get fα,β [ ∂ ∂u f(u,v) ] = iw 1 α 1 fα,β [f(u,v)] . for n = 2, the result holds true. fα,β [ ∂2 ∂u2 f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂2 ∂u2 f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂ ∂u f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂ ∂u f(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂ ∂u f(u,v)e−jw 1 β 2 vdudv = iw 1 α 1 ∞∫ −∞   [ e−iw 1 α 1 uf(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 uf(u,v)du  e−jw 1 β 2 vdv = ( iw 1 α 1 )2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudv = ( iw 1 α 1 )2 fα,β [f(u,v)] . for n = k − 1, fα,β [ ∂k−1 ∂uk−1 f(u,v) ] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂k−1 ∂uk−1 f(u,v)e−jw 1 β 2 vdudv = ∞∫ −∞   [ e−iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v) ] − ∞∫ −∞ −iw 1 α 1 e −iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v)du  e−jw 1 β 2 vdv = iw 1 α 1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 u ∂k−2 ∂uk−2 f(u,v)e−jw 1 β 2 vdudv. int. j. anal. appl. 19 (4) (2021) 568 on repeating the integration by parts, we get fα,β [ ∂k−1 ∂uk−1 f(u,v) ] = ( iw 1 α 1 )k−1 fα,β [f(u,v)] . by method of mathematical induction, the result is true for all n = k. fα,β [ ∂k ∂uk f(u,v) ] = ( iw 1 α 1 )k fα,β [f(u,v)] . thus, it is true for all n. similarly, v) and vi) can be proved. � property 3.5 (power of u,v). for f ∈ l2(r2,h) (3.12) i) fα,β [uf(u,v)] = ( i α w 1−α α 1 ) ∂ ∂w1 fα,β [f(u,v)] . (3.13) ii) fα,β [vf(u,v)] = ∂ ∂w2 fα,β [f(u,v)]  j β w 1−β β 2   . proof. for f ∈ l2(r2,h) and using (3.1), we get fα,β [uf(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uuf (u,v) e−jw 1 β 2 vdudv = ∞∫ −∞ ∞∫ −∞ ( i α w 1−α α 1 ) ∂ ∂w1 e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv = ( i α w 1−α α 1 ) ∂ ∂w1 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv = ( i α w 1−α α 1 ) ∂ ∂w1 fα,β [f(u,v)] . fα,β [vf(u,v)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uvf (u,v) e−jw 1 β 2 vdudv = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) ∂ ∂w2 e−jw 1 β 2 v  j β w 1−β β 2  dudv = ∂ ∂w2 ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf (u,v) e−jw 1 β 2 vdudv  j β w 1−β β 2   = ∂ ∂w2 fα,β [f(u,v)]  j β w 1−β β 2   . hence the proof. � int. j. anal. appl. 19 (4) (2021) 569 property 3.6 (power of i, j). for f ∈ l2(r2,h); m,n ∈ n (3.14) fα,β [i mf(u,v)jn] = imfα,β [f(u,v)] j n. proof. for f ∈ l2(r2,h) and using (3.1), we get fα,β [i mf(u,v)jn] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uimf(u,v)jne−jw 1 β 2 vdudv = im ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 uf(u,v)e−jw 1 β 2 vdudvjn = imfα,β [f(u,v)] j n. hence the proof. � definition 3.2. the convolution for the quaternion valued functions f,g ∈ l2(r2,h) is defined [14] by (3.15) f ∗g = ∞∫ −∞ ∞∫ −∞ f(u,v)g(x−u,y −v)dudv. theorem 3.7 (convolution theorem). for f,g ∈ l2(r2,h); (3.16) fα,β [f ∗g] = fα,β [f] fα,β [g] . proof. for f,g ∈ l2(r2,h); x = (x1,x2), y = (y1,y2) and z = (z1,z2); fα,β [f ∗g] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1 (f ∗g) (x) e−jw 1 β 2 x2dx = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1   ∞∫ −∞ ∞∫ −∞ f(y )g(x −y )dy  e−jw 1β2 x2dx = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 x1f(y )   ∞∫ −∞ ∞∫ −∞ g(x −y )e−jw 1 β 2 x2dx  dy . substituting z = x −y , we get fα,β [f ∗g] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 (y1+z1)f(y )   ∞∫ −∞ ∞∫ −∞ g(z)e−jw 1 β 2 (y2+z2)dz  dy =   ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 y1f(y )e−jw 1 β 2 y2dy     ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 z1g(z)e−jw 1 β 2 z2dz   = fα,β [f] fα,β [g] . � int. j. anal. appl. 19 (4) (2021) 570 theorem 3.8. the scalar product of two quaternion-valued functions f,g ∈ l2(r2,h) is given by the scalar product of the corresponding 2d-qfrfts f̂ and ĝ: (3.17) 〈f,g〉 = 1 (2π) 2 αβ 〈fα,β(w1,w2),gα,β(w1,w2)〉 . proof. for f, g ∈ l2(r2,h) and using (2.8), we get 〈f, g〉 = ∫ ∞ ∞ ∫ ∞ ∞ 〈 f(u,v)g(u,v) 〉 dudv = ∫ ∞ ∞ ∫ ∞ ∞ 〈 1 (2π) 2 αβ ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 f̂α,β (w1,w2) × ejw 1 β 2 vw 1−β β 2 dw1dw2g(u,v) 〉 dudv = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2) ∞∫ −∞ ∞∫ −∞ eiw 1 α 1 uw 1−α α 1 e jw 1 β 2 vw 1−β β 2 × g(u,v)dudv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2)w 1−α α 1 w 1−β β 2 × ∞∫ −∞ ∞∫ −∞ ejw 1 β 2 vg(u,v)eiw 1 α 1 ududv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 f̂α,β(w1,w2)w 1−α α 1 w 1−β β 2 × ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 vg(u,v)e−jw 1 β 2 ududv 〉 dw1dw2 = 1 (2π) 2 αβ ∫ ∞ ∞ ∫ ∞ ∞ 〈 w 1−α α 1 f̂α,β(w1,w2)w 1−β β 2 ĝα,β(w1,w2) 〉 dw1dw2 = 1 (2π) 2 αβ 〈fα,β(w1,w2),gα,β(w1,w2)〉 where fα,β(w1,w2) = w 1−α α 1 f̂α,β(w1,w2); gα,β(w1,w2) = w 1−β β 2 ĝα,β(w1,w2). thus, the theorem holds true. � int. j. anal. appl. 19 (4) (2021) 571 figure 1. kernel of 2d-qfrft at α = 1 and β = 1. figure 2. kernel of 2d-qfrft at α = 1/2 and β = 1/2. figure 1 shows the kernel of 2d-qfrft for various values of w1,w2 at order α = 1 and β = 1 which is a particular case of the study developed in this paper. figure 2 shows the kernel of 2d-qfrft for various values of w1,w2 at order α = 1/2 and β = 1/2. for both the figures the range of x and y is between −3 and 3. the 2d-qfrft is superior in disparity estimation and analyzing genuine 2d texture as compared to other fractional fourier transforms and [7]. int. j. anal. appl. 19 (4) (2021) 572 figure 3. 2d-qfrft kernel. left: top row: (1 + ij)/2 and (i−j)/2 components. bottom row: (1 − ij)/2 and (i + j)/2 components at α = 1, β = 1 ; right: top row: (1 + ij)/2 and (i−j)/2 components. bottom row: (1−ij)/2 and (i + j)/2 components at α = 1/2, β = 1/2 the components (1 + ij)/2, (i − j)/2, (1 − ij)/2 and (i + j)/2 are shown in figure 3 at α = 1, β = 1 and α = 1/2, β = 1/2 which represents 2d-qft extended to 2d-qfrft. we can also observe the scale-invariant feature of 2d-qfrft. example 3.1. find the quaternion fractional fourier transform of the function: (3.18) f(x,y) =   1; |x| < 1, |y| < 1 0; otherwise. by using (3.1), we get fα,β[f(x,y)] = ∞∫ −∞ ∞∫ −∞ e−iw 1 α 1 xf (x,y) e−jw 1 β 2 ydxdy. fα,β[f(x,y)] = 1∫ −1 1∫ −1 e−iw 1 α 1 xe−jw 1 β 2 ydxdy fα,β[f(x,y)] = 4 sin w 1 α 1 w 1 α 1 · sin w 1 β 2 w 1 β 2 .(3.19) the graphical representation of the quaternion fractional fourier transform of the function (3.18) obtained using α = 1 and β = 1 in (3.19), is now a particular case of (3.1) which is represented in the following figure: int. j. anal. appl. 19 (4) (2021) 573 figure 4. graph of fα,β[f(x,y)] with α = 1 and β = 1. the graphical representation of the quaternion fractional fourier transform of the function (3.18) obtained using α = 1/2 and β = 1/2 in (3.19) is represented in the following figure: figure 5. graph of fα,β[f(x,y)] with α = 1/2 and β = 1/2. the graphical representation of the quaternion fractional fourier transform of the function (3.18) obtained using α = 1/2 and β = 1 in (3.19) is represented in the following figure: figure 6. graph of fα,β[f(x,y)] with α = 1/2 and β = 1. int. j. anal. appl. 19 (4) (2021) 574 figures 1-3 are plotted using online freeware version of wolframalpha. figures 4-6 are plotted using online freeware version 3d surface plotter of academo. 4. application let us consider the initial value problem from [2]: (4.1) ∂h ∂t −o2h = 0, on r0, 2 × (0,∞), and (4.2) h(u,v) = f(u,v), f ∈s(r0, 2; h) at t = 0, where s(r0, 2; h) is the quaternion schwartz space and o2 = ∂2 ∂2u + ∂2 ∂2v . applying the definition of 2d-qfrft to both sides of (4.1), we get fα,β [ ∂h ∂t ] = ( iw 1 α 1 )2 fα,β[h] + fα,β[h] ( jw 1 β 2 )2 (4.3) ∂ ∂t fα,β [h] = − ( w 2 α 1 + w 2 β 2 ) fα,β[h]. the general solution of (4.3) is given by (4.4) fα,β [h] = ce − ( w 2 α 1 +w 2 β 2 ) t , where c is a quaternion constant. by using the initial value condition, we get (4.5) fα,β [h] = e − ( w 2 α 1 +w 2 β 2 ) t fα,β [f] . analogous to [2, equation 6.6], we have (4.6) 1 4πt fα,β [ e − ( u 2 α +v 2 β ) /4t ] = e − ( w 2 α 1 +w 2 β 2 ) t . applying the inversion formula of 2d-qfrft to (4.5), we get h = f−1α,β  e− ( w 2 α 1 +w 2 β 2 ) t fα,β [f]   = f−1α,β [ 1 4πt fα,β [ e − ( u 2 α +v 2 β ) /4t ] fα,β [f] ] . using convolution theorem, we have (4.7) h = kt ∗f int. j. anal. appl. 19 (4) (2021) 575 where kt = 1 4πt e − ( u 2 α +v 2 β ) /4t . 5. conclusion the authors developed a new two-dimensional quaternion fractional fourier transform in this study. the properties such as linearity, shifting and derivatives of the quaternion-valued function are demonstrated. the convolution theorem and inversion formula are also established. an example is illustrated with graphical representation. in the concluding section, an application related to the two-dimensional quaternion fourier transform is also demonstrated. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. bahri, a.k. amir, c. lande, the quaternion domain fourier transform and its application in mathematical statistics, iaeng int. j. appl. math. 48 (2018), 1-7. [2] m. bahri, r. ashino, r. vaillancourt, continuous quaternion fourier and wavelet transforms, int. j. wavelets multiresolut. inf. process. 12 (2014), 1460003. [3] t. bülow, m. felsberg, g. sommer, non-commutative hyper complex fourier transforms of multidimensional signals, in: g. sommer (eds) geometric computing with clifford algebras, springer, berlin, heidelberg, 2001. [4] d. cheng, k.i. kou, plancherel theorem and quaternion fourier transform for square integrable functions, complex var. elliptic equ. 64 (2019), 223–242. [5] b. davies, integral transforms and their applications, springer, new york, 1978. [6] t.a. ell, quaternion-fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: proceedings of 32nd ieee conference on decision and control, ieee, san antonio, tx, usa, 1993: pp. 1830–1841. [7] t.a. ell, s.j. sangwine, hypercomplex fourier transforms of color images, ieee trans. image process. 16 (2007), 22–35. [8] x. guanlei, w. xiaotong, x. xiaogang, fractional quaternion fourier transform, convolution and correlation, signal processing. 88 (2008), 2511–2517. [9] e.m.s. hitzer, quaternion fourier transform on quaternion fields and generalizations, adv. appl. clifford alg. 17 (2007), 497–517. [10] a.c. lewis, william rown hamilton, lectures on quaternions (1853), in: landmark writings in western mathematics 1640-1940, elsevier, 2005: pp. 460–469. [11] p. lian, uncertainty principle for the quaternion fourier transform, j. math. anal. appl. 467 (2018), 1258–1269. [12] l. romero, r. cerutti, l. luque, a new fractional fourier transform and convolutions products, int. j. pure appl. math. 66 (2011), 397-408. [13] r. roopkumar, quaternionic one-dimensional fractional fourier transform, optik, 127 (2016), 11657-11661. [14] a.h. zemanian, generalized integral transformations, john wiley sons inc., new york, 1968. 1. introduction 2. preliminary results 3. main results 4. application 5. conclusion references international journal of analysis and applications volume 19, number 3 (2021), 360-388 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-360 received may 22nd, 2020; accepted june 15th, 2020; published april 9th, 2021. 2010 mathematics subject classification. 62e10. key words and phrases. libby-novick beta distribution (lnbd); moment generating function; mean deviation; bonferroni curve; lorenze curve; gini index truncated moment. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 360 properties of the libby-novick beta distribution with application zafar iqbal1, muhammad rashad2,3,*, muhammad hanif3 1department of statistics, government college, gujranwala, pakistan 2superior university, lahore, pakistan 3national college of business administration & economics, lahore, pakistan *corresponding author: rashid.geostat@gmail.com abstract. the beta distribution is one of the most popular probability distributions with applications to real life data. in this paper, an extension of this distribution called the libby-novick beta distribution (lnbd) which is believed to provide greater flexibility to model scenarios involving skew-normal data than original one. analytical expressions for various mathematical properties along with characterization based on one truncated moment. the estimation of lnbd’s parameters is undertaken using the method of maximum likelihood estimation. for illustration and performance evaluation of lnbd two real-life data sets adapted from earlier studies are used. 1. introduction in statistical distributions, a vast activity has been observed in generalizing classical distributions by adding more parameters to make them more flexible in analyzing empirical data. in the past few decades, a major research effort has been devoted to the study of skewsymmetric distributions. such distributions have been constructed by adding a new parameter https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-360 int. j. anal. appl. 19 (3) (2021) 361 to probability density function (pdf) of the cumulative distribution function (cdf) of symmetric distribution, resulting in families of asymmetric more flexible distributions that are analytically more flexible. a motivation for the [15] beta distribution is a threeparameter generalization of the usual two-parameter beta distribution. it is perhaps the simplest generalization of the twoparameter beta distribution that allows for significant additional flexibility. 2. mathematical properties the pdf of the libby-novick beta distribution (lnbd) is as under: ( ) ( ) ( ) ( ) 11 + 1 = ; 0 < < 1, , , > 0 b , 1 1 ba a a b c x x f x x a b c a b c x −− − − −   (2.1) 2.1 shape properties of the pdf the lnbd defined in (2.1) has real flexibility and it is shown through graphs w.r.t. some different combinations of values of the parameters. figure 2.1 shapes of pdf (i) for a > 0, c > 0 and 0 < b < 1 then lnbd is u-shaped. (ii) for 0 < a < 1, b > 1 and 0 < c < 1 then lnbd is s-shaped. (iii) for a = b = 2, c ≥ 1 (c < 1) and if c →  (c → 0) then lnbd increases its positively skewed (negatively skewed) from the symmetric with decreasing mode (increasing mode). (iv) for a = b = c = 1, the lnbd is uniform distribution. (v) for a > 2, b > 2 and c ≥ 1 the lnbd is unimodel positively skewed with decreasing mode when c → . (vi) for a > 2, b = 1, 0 < c < 1, the form of lnbd is an increasing. a 0.5 ,b 0.5 ,c 3.0 a 0.5 ,b 1.5 ,c 0.1 a 2.0 ,b 2.0 ,c 0.5 a 2.0 ,b 5.0 ,c 2.0 a 3.0 ,b 2.0 ,c 0.3 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 f x a 0.5 ,b 0.5 ,c 0.5 a 5.0 ,b 1.0 ,c 0.5 a 1.0 ,b 3.0 ,c 2.0 a 2.0 ,b 2.0 ,c 3.0 a 2.0 ,b 5.0 ,c 0.1 0.2 0.4 0.6 0.8 1.0 x 1 2 3 4 5 f x int. j. anal. appl. 19 (3) (2021) 362 (vii) for a > 2, b = 1, 1 < c < 3.5, the lnbd increase but it increases slowly when c increases in the interval and for c ≥ 3.5 the lnbd again turns to unimodel. (viii) for b = c = 1, the lnbd is power distribution. (ix) for a = c = 1, the lnbd is a special case of kum-distribution or reflected exponentiated distribution. 2.2 distribution function the cumulative distribution function (cdf) of lnbd is ( ) ( ) ( ) 1 1 f = i , cx c x x a b + −   (2.2) where ( ) ( ) ( ) 11 0 1 i , = y 1 y dy b , x ba x a b a b −− − figure 2.2 2.3 quantile function the quantile function is given by: ( )f = where 0 < < 1x p p ( ) ( ) 1 1 i , = cx c x a b p + −   and is describe as an inverse of cdf, as follows ( ) ( ) ( ) 1 1 = + 1 − − − , , , , i p a b x c c i p a b ( ) ( ) ( ) = + 1 − ; , ; , q p a b c c q p a b (2.3) 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 f x 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 f x a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 int. j. anal. appl. 19 (3) (2021) 363 where ( ) ; ,q p a b is the inverse of beta function or quantile of beta function at p, and it can be numerically solved through software for different set of parameters’ values. the graph can also be used to illustrate the behaviour of quantile function of lnbd. i. for a > 1, b > 1 by increasing c, the quantile value of the lnbd decreases comparatively. ii. for 1,a  c > 1, and for any value of b, the quantile value of the lnbd increases slowly for 0.5p  and for 0.5p  , the quantile value sharply increases. iii. for c → 0, the quantile value of the lnbd increases sharply. 2.4 averages 2.4.1 harmonic mean the harmonic mean is of the pdf (2.1) is 1 1 − = + − x a h a bc (2.4) the following are the limiting cases of hx: 0→ lim x a h is not defined. 1 0 → → = =lim limx x a b h h 0 1 → → = =lim limx x a b h h corollary (i) if c = 1 then 1 1 − = + − x a h a b is the harmonic mean of beta distribution. (ii) if a = b & c = 1 then 1 2 1 − = − x a h a and 1 1 + 1 − − = − x b h ca b (2.5) the following are the limiting cases of h1-x : 1 0 − → lim x b h is not defined. int. j. anal. appl. 19 (3) (2021) 364 1 1 1 0 − − → → = =lim limx x b a h h 1 1 0 1 − − → → = =lim limx x a b h h corollary (iii) if c = 1 then 1 1 1 − − = + − x b h a b (iv) if a = b & c = 1 then 1 1 2 1 − − = − x a h a a relation between 1−x xh and h is as: ( )( ) ( )( ) 1 1 1 1 − − + − = − + − + − x x c a b a b h h a bc ac b (2.6) when the c = 1 then 1 1 − − = = + − x x a b h h a bc 2.4.2 geometric mean the geometric mean xg of the pdf (2.1) is ( ) ( ) ( ) ( ) 1 , 1 1 = + 1 , i x i b a i b g exp a a b ln c i c b a b    =  +  − − + −       (2.7) and ( ) ( ) 1 0 , 1 1 = 1 1 , i x i b a i b g exp b a b i c  − =   +    − −           (2.8) 2.4.3 mode mode is obtained by solving ( ) d f x = 0 dx ( ) ( ) ( ) ( ) 11 1 = 1 1 ba a a + b c x x f x b a, b c x −− − − −   ( ) ( ) ( ) ( ) ( ) ( ) = + 1 + 1 1 1 1lnf x ln c a lnx b ln x a + b ln c x− − − − − −   where ( ) ac c = b a, b int. j. anal. appl. 19 (3) (2021) 365 ( ) ( )( ) ( ) 11 1 = 1 1 1 a + b cd a b lnf x dx x x c x −− − − + − − − ( ) = 0 d lnf x dx ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 + 1 1 = 0 1 1 1 a x c x b x c x a b c x x x x c x − − − − − − − − + − −       − − −   ( ) ( ) ( )22 1 3 1 = 0c x a c bc x a− + − − − + − 2 4 = 2 −  −b b ac x a x d= when the quadratic equation cuts to x – axis. 2 4 0b ac−  ( ) ( )( ) 2 3 8 1 1 0a c bc a c− − − − − −  ( ) ( ) ( ) 22 2 + 1 6 2 1a b c c a + b a c+  + + (2.9) table 2.1 the mode for the lnbd (for different values) a b c mode 1 1.1 0.1 0.99444 1 1.2 0.2 0.97500 1 1.3 0.3 0.93557 1 1.4 0.4 0.86667 1 1.5 0.5 0.75000 1 1.6 0.6 0.55000 1 1.7 0.7 0.18333 1 1.8 0.8 0 in table 2.1, if a = 1, 1 < b < 2 and 0 < c < 1 then the mode moves from 1 to 0. int. j. anal. appl. 19 (3) (2021) 366 table 2.2 the mode for the lnbd (for different values) a b c mode 0.1 0.1 0.1 0.45779 0.2 0.1 0.2 0.41890 0.3 0.1 0.3 0.38120 0.4 0.1 0.4 0.34322 0.5 0.1 0.5 0.30371 0.6 0.1 0.6 0.26127 0.7 0.1 0.7 0.21405 0.8 0.1 0.8 0.15920 0.9 0.1 0.9 0.09160 in table 2.2, if 0 < a < 1, b = 0.1 and 0 < c < 1 then the mode moves from 0.45 to 0.09. 2.5 mean deviations the mean deviation of the lnbd from arithmetic mean and median are denoted by ( )1 y and ( )2 y respectively and are found from 2.5.1 from mean ( ) ( ) 1 1 0 = y y f y dy − ( ) ( ) = 2 2 f j  − ( ) ( ) ( ) ( ) ( ) 1 1 1 1 0 + 2 1 = 2 1 1 ( ) ( ) i c c c c i n i i a, b b a + i + , b c b a, b n c      + − + −       =    − −      (2.10) 2.5.2 from median ( ) ( ) 1 2 0 = y y m f y dy − ( )= 2 j m − ( ) ( ) ( ) 1 1 0 + 2 1 = 1 + + 1, ( ) ( ) i cm c m i n i b a i b c b a, b n c   + −   =    − −      (2.11) int. j. anal. appl. 19 (3) (2021) 367 2.6 rth moment the rth moments about origin of the pdf (2.1) is ( ) ( ) ( ) 0 + 1 1 = 1 + , ( ) ! ( ) i r r i r i e x b a + r i b c b a, b i r c  =    −      (2.12) corollary  if c = 1, then ( ) a e x = a + b , ( ) ( ) ( )( ) 2 = a a + 1 e x a + b a + b + 1 , ( ) ( ) ( ) 2 = ab v x a + b a + b + 1  if a = b & c = 1 then ( ) 1 = 2 e x 2.7 moment generating function the moment generating function of the pdf (2.1) about zero is ( ) ( ) ( ) ( ) = 0 0 + 1 1 = 1 + , ( ) ( ) ! ri x r r i tn i m t b a + r i b b a, b c n c r   =    −      (2.13) 2.8 factorial moments 2.8.1 decreasing factorial moments of lnbd the decreasing factorial moments of the lnbd is defined as: ( ) 1 = 1 r r i e x e x i =     − +      ( ) ( ) r = 0 = n rs n, r e x ( ) ( ) ( ) r = 0 0 + 1 1 = 1 + , ( ) ( ) in r i s n, r n i b a + r i b b a, b c n c  =    −      (2.14) where ( )s n, r are the stirling numbers of first kind. 2.8.2 increasing factorial moments of lnbd the increasing factorial moments of the lnbd is defined as: ( ) 1 = e 1 r r i e x x i =     + −      ( ) ( ) = 0 = n r r d n, r e x ( ) ( ) ( ) r = 0 0 + 1 1 = 1 + , ( ) ( ) in r i d n, r n i b a + r i b b a, b c n c  =    −      (2.15) int. j. anal. appl. 19 (3) (2021) 368 where ( )d n, r can be deduced from the relation ( ) ( ) ( ) = 1 n r s n, r d n, r − − 2.9 negative moments the negative moments of the pdf (2.1) is: ( ) ( ) ( ) ( ) 0 + 1 = 1 1 + , ( ) ( ) ir ir i n ic e x b a r i b b a, b n c  − =    − − −      (2.16) 2.10 incomplete moments the incomplete moments of the pdf (2.1) is defined as: ( ) ( ) ( ) ( ) 1 1 0 + 1 φ = 1 + , ( ) ( ) i s s cx c x i n i1 t b a + s i b c b a, b n c  + −   =    −      (2.17) 2.11 scaled total time for aging properties the scaled total time of the pdf (2.1) is defined as: ( )( ) ( ) ( ) 1 0 0 1 1 x x f y s f t s y dy f t dt dy   = =  ( )( ) ( ) ( ) ( )cy 1 c 1 11 0 0 1 1 1 , yx ba fs f t x t t dt dy b a b + −   −−    = − −       ( )( ) ( ) ( ) ( ) cy 1 c 1 0 1 1 , , y x fs f x x i a b dy b a b + −     = −      (2.18) 2.12 conditional moments the conditional moments of the pdf (2.1) is defined as: = > r rm e x x x    ( ) 1 r r x 1 m = t f t dt f   by applying the substitution (2.1) and (2.2) and after some simplification, we have ( ) ( ) ( ) ( ) ( ) 1 1 1 1 i 1 a r cx c x r cx c x c a + r, b m = b a, b i a, b   + −    + −     −    −   (2.19) 2.13 mean residual function the mean residual function of the pdf (2.1) is defined as: int. j. anal. appl. 19 (3) (2021) 369 ( ) = m x e x x x > x −  ( ) = > m x e x x x x  −  ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 + 1 = 1 a cx c x cx c x c i a , b m x x b a, b i a, b   + −    + −    −   −  −   (2.20) 2.14 vitality function the vitality function of the pdf (2.1) is defined as: ( ) = > v x e x x x   ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 + 1, = 1 a cx c x cx c x c i a b v x b a, b i a, b   + −    + −    −    −   (2.21) 2.15 gometric vitality function the geometric vitality function of the pdf (2.1) is defined as: ( ) = > logg x e logx x x   ( ) ( ) ( ) 1 1 = 1 x logg x log t f t dt f t−  by using (2.1), (2.2) and (2.3), we have ( ) ( ) ( )  ( ) ( )( ) ( ) ( ) ( ) 1 1 1 1 1 11 1 1 1 1 1 1 + 1 1 = 1 cx c x cx c x b ba a a cx c x c lny y y dy ln c c y y y dy logg x b a, b i a, b  + −   + −     − −− −  + −      − − − −      −     (2.22) 2.16 characteristics function the characteristic function of the pdf (2.1) is defined as: ( ) ( ) = itxx t e e int. j. anal. appl. 19 (3) (2021) 370 ( ) ( ) ( ) = 0 0 + 1 1 = 1 + , ( ) ( ) ! rj r r j jtn j b a + r j b b a, b n c c r   =    −      (2.23) 2.17 information generating function the information generating function of the (2.1) is defined by p(s) and found from ( ) ( )1sp s e f −= ( )( ) ( ) ( ) ( ) 1 0 1 1 1 sb ssa ss.a s a b t t tc dx b a,b c t −− + − = − −    by applying the substitution (2.1) and (2.2), we have ( ) ( ) ( ) ( ) ( ) ( )( ) 1 0 2 + 2 1 = 1 1 + + 1 s 1 + 1 2 2 , is s i s ic p s b s a i b s cb a, b −  =  −   − − −   −      (2.24) 2.18 points of inflection the points of inflection of the pdf (2.1) are found from ( ) = 0f x under ( ) 0f x  or equivalentally as ( ) 2 2 = 0 d lnf x dx & ( ) 3 3 0 d lnf x dx  ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) 2 2 2 2 2 2 22 2 2 1 11 1 1 1 = = 0 1 11 1 1 1 a + b c a + b cd a b a b lnf x dx x xx xc x c x − −− − − − − − + − − + − −− − − −       (2.25) after some simplification 4 3 2 = 0a x bx c x d x e+ + + + where ( ) ( )( ) ( )( ) 2 2 1 2 1 + 3 1 3 2 + 3, ,a c b c a bc c c c a b bc c= − = − − + − = − − + − ( )( ) ( )2 1 2 1d a c and e a= − − − = − − it is four degree equation; it can either be solved analytically or through software for specific values of parameters. int. j. anal. appl. 19 (3) (2021) 371 table 2.3 the points of inflection for the lnbd (for fixed values) a b c points of inflection 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2 2.5 3.0 3.5 4.0 4.5 5.0 5.5 2 2.5 3.0 3.5 4.0 4.5 5.0 5.5 0.47248 0.03203 0.04577, 0.31463 0.05223, 0.27525 0.05522, 0.24560 0.05637, 0.22252 0.05651, 0.20302 0.05605, 0.18703 table 2.4 the points of inflection for the lnbd (for different values) a b c points of inflection 2.0 2.0 2.0 3.0 3.0 3.0 4.0 4.0 4.0 5.0 5.0 5.0 5.0 5.0 5.0 6.5 6.5 6.5 8.0 8.0 8.0 9.5 9.5 9.5 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3 0.77523 0.54776, 0.89745 0.31849, 0.83656 0.78934, 0.94129 0.32553, 0.60080 0.46964, 0.44414 0.79880, 0.93391 0.62896, 0.86694 0.49366, 0.80073 0.80560, 0.92840 0.64689, 0.85817 0.52260, 0.79087 int. j. anal. appl. 19 (3) (2021) 372 2.19 reliability measures the sf of lnbd is denoted by r(x) and is defined as under with graphs for various values of the parameters: ( ) ( ) = 1 r x f x− ( ) ( ) ( ) 1 1 = 1 cx c x r x i a, b + −   − (2.26) figure 2.3 2.21 hazard function (hrf) the hazard or instantaneous rate function is denoted by h(x). the hazard function of x can be interpreted as instantaneous rate or the conditional probability density of failure at time x, given that the unit has survived until x. the hazard function is defined to be ( ) ( ) ( )( ) 1 = h x f x f x − ( ) ( ) ( ) ( ) ( ) ( ) 11 1 1 1 = 1 1 1 i ba a a + b cx c x c x x h x b a, b c x a, b −− + −   −  − − −     (2.27) figure 2.4 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 r x 0.2 0.4 0.6 0.8 1.0 x 0.2 0.4 0.6 0.8 1.0 r x 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 25 f x 0.2 0.4 0.6 0.8 1.0 x 5 10 15 20 25 30 f x a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 a 0.5,b 0.5,c 3.0 a 0.5,b 1.5,c 0.1 a 2.0,b 2.0,c 0.5 a 2.0,b 5.0,c 2.0 a 3.0,b 2.0,c 0.3 a 0.5,b 0.5,c 0.5 a 5.0,b 1.0,c 0.5 a 1.0,b 3.0,c 2.0 a 2.0,b 2.0,c 3.0 a 2.0,b 5.0,c 0.1 int. j. anal. appl. 19 (3) (2021) 373 2.22 reverse hazard function the reverse hrf of lnbd is defined as. ( ) ( ) ( ) 1 = r x f x f x −    ( ) ( ) ( ) ( ) ( ) ( ) 11 1 1 1 = 1 1 i ba a a + b cx c x c x x r x b a, b c x a, b −− + −   − − −   (2.28) 2.23 bonferroni curve the bonferroni curve of the pdf (2.1) is defined as: ( )( ) ( ) ( ) 0 1 x fb f x y f y dy f x =  ( )( ) ( ) ( ) ( ) ( ) ( ) 1 1 0 1 1 0 + 1 1 + 1 + , = + 1 1 + + , ( ) ( ) ( ) ( ) i cx c x i f i cx c x i n i b a i b n c b f x n i i a, b b a r i b n c  + −   =  + −   =    −        −       (2.29) 2.24 lorenz curve the lorenz curve of the pdf (2.1) is defined as: ( )( ) ( ) ( )( )l f x f x b f x= ( )( ) ( ) ( ) ( ) 1 1 0 0 + 1 1 + 1 + , + 1 1 + , ( ) ( ) ( ) ( ) i cx c x i i i n i b a i b n c l f x n i b a + r i b n c  + −   =  =    −     =    −       (2.30) 2.25 gini coefficient the gini coefficient of the pdf (2.1) is explained as: ( )( ) 1 2 0 1 1 1g f x dx  = − − ( ) ( ) ( ) 2 1 11 11 0 0 1 1 1 1 1 , cx c x bat t dt dx b a b + −   −−    = − − −      (2.31) int. j. anal. appl. 19 (3) (2021) 374 table 2.6 the gini coefficient for the lnbd a b c gini coefficient 2 2 2 0 2.5 2.5 2.5 0.13001 3 3 3 0.11912 4 4 4 0.10052 4.5 4.5 4.5 0.09282 5 5 5 0.08602 5.5 5.5 5.5 0.08003 table 2.7 the gini coefficient for the lnbd a b c gini coefficient 0.5 0.5 3.0 0.37516 0.5 1.0 2.5 0.39541 0.5 1.5 2.0 0.22469 0.5 1.0 0.5 0.16378 1.0 2.0 3.0 0.28570 2.0 3.0 4.0 0.03762 3.0 4.0 5.0 0.10899 4.0 5.0 6.0 0.11417 5.0 6.0 0.7 0.10840 6.0 0.7 0.8 1.10059 2.26 asymptotes and shapes the asymptotes of (1.1), (1.2) and (3.1) as 0, 1x → are as explained below: int. j. anal. appl. 19 (3) (2021) 375 table 2.8 ( ) pdf f x ( ) cdf f x ( ) hrf h x 0x → ( ) 1 a ac x b a, b − ( ) a ac x a b a, b ( ) 1 a ac x b a, b − 1x → ( ) ( ) 1 1 b b x c b a, b − − ( ) ( ) 1 1 ba b c x bc b a, b − − 1 b x− 2.19 transformation table 2.8 different transformation of lnbd sr. no. a b c transformation resulting distribution 1 a b 1 x beta type – i b(a, b) 2 a b 1 1 – x miror of beta b(a, b) 3 a b 1 1 x x− beta 2nd kind or ( ) b a, b 4 a b c 1 cx x− ( )b a, b 5 2 n 2 m c ( )1 mcx n x− ( )f n, m 6 1 b 1 x  ( ) kum , b 7 1 b 1 x ( )1 kum , b 8 a 1 1 lnx− exp(a) 9 a b c ( )1 1 cx c x− − beta type – i b(a, b) 10 1 b c ( )1 1 cx c x   − − libby-novick (b, c, ) 11 1 1 1 x uniform u(0, 1) 12 3 2 3 2 1 2rx r− wigner sanecar distribution 13 1 2 1 2 1 x arcsine distribution int. j. anal. appl. 19 (3) (2021) 376 3 characterization 3.1 based on one truncated moment from [32], [33] the following two lemmas to characterize different univariate continuous distributions. here, we discuss characterizations of lnbg class distributions through one truncated moment. assumption 3.1 suppose the random variable x is absolutely continuous with cumulative distribution function (cdf) f(x) and probability density function (pdf) f(x). we assume that ( )  = > 0sup x f x and ( )  = < 1inf x f x . we further assume that e(x) exists. lemma 3.1 under the assumption 3.1 if ( ) ( ) ( ) = e x x x g x x , where ( ) ( ) ( ) = f x x f x  and g(x) is a continuous differentiable function of x with the condition that ( ) ( ) x u g u du g u  −  is finite for all x, x   , and then ( ) ( ) ( ) = x u g u du g u f x ce −  , where c is determined by the condition ( ) = 1f x dx    . proof. it follows from ahsanullah et al., (2016) and ahsanullah (2017). lemma 3.2 under the assumption 3.1, if ( ) ( ) ( ) = e x x x h x r x , where ( ) ( ) ( ) = 1 f x r x f x− and ( )h x is a continuous differentiable function of x with the condition that ( ) ( ) x u h u du h u  +  is finite for all x, x   , then ( ) ( ) ( ) = x u h u du h u f x ce  + − , where c is determined by the condition ( ) = 1f x dx    . proof. it follows from [32], [33]. int. j. anal. appl. 19 (3) (2021) 377 theorem 3.2 under the assumption 3.1 with = 0 and = 1 , if ( ) ( ) ( ) = e x x x g x x , where ( ) ( ) ( ) ( ) 1 1 1 1 1 = 1 a + b ba a b a, b c x g x m c x x −− − −    − and ( ) ( ) ( ) = f x x f x  , if and only if ( ) ( ) ( ) ( ) 1 1 1 = 1 1 ba a a + b c x x f x b a, b c x −− − − −   proof. if ( ) ( ) ( ) ( ) 1 1 1 = 1 1 ba a a + b c x x f x b a, b c x −− − − −   , then ( ) ( ) ( ) 1 0 = = x f x g x uf u du m thus ( ) ( ) ( ) ( ) 1 1 1 1 1 = 1 a + b ba a b a, b c x g x m c x x −− − −    − suppose ( ) ( ) ( ) ( ) 1 1 1 1 1 = 1 a + b ba a b a, b c x g x m c x x −− − −    − , then ( ) ( ) ( ) ( ) ( )( ) ( ) 1 1 1 1 1 1 1 1 = + 1 1 11 a + b ba a b a, b c x c a + b a b g x x m c x x xc x x −− − −   − − −   − − +  − − −−   ( ) ( ) ( )( ) ( ) 1 1 1 = + 1 1 1 c a + b a b g x x g x c x x x  − − −  − − +  − − −  thus ( ) ( ) ( )( ) ( ) 1 1 1 = 1 1 1 x g x c a + b a b g x c x x x − − − − + − − − − by lemma 3.1 and after some simplification, we obtain the following result ( ) ( ) ( ) ( ) 11 = 1 1 1 a + b baf x k c x x x − −− − − −   , where k is a constant. int. j. anal. appl. 19 (3) (2021) 378 using the condition ( ) 1 0 = 1f x dx , we obtain ( ) ( ) ( ) ( ) 1 1 1 = 1 1 baa a + b x xc f x b a, b c x −− − − −   theorem 3.3 under the assumption 3.1 with = 0 and = 1 , if ( ) ( ) ( )e x x = h rx x x , where ( ) ( ) ( ) ( ) ( )( )1 1 1 1 1 = 1 a + b ba a b a, b c x h x e x m c x x −− − −   − − and ( ) ( ) ( ) = 1 f x r x f x− , if and only if ( ) ( ) ( ) ( ) 1 1 1 = 1 1 baa a + b x xc f x b a, b c x −− − − −   proof. if ( ) ( ) ( ) ( ) 1 1 1 = 1 1 baa a + b x xc f x b a, b c x −− − − −   , then ( ) ( ) ( ) ( ) 1 1 x = = f x h x u f u du e x m− thus ( ) ( ) ( ) ( ) ( )( )111 1 1 = 1 a + b ba a b a, b c x h x e x m c x x −− − −   − − suppose ( ) ( ) ( ) ( ) ( )( )111 1 1 = 1 a + b ba a b a, b c x h x e x m c x x −− − −   − − , then ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) 1 1 1 1 1 1 1 1 = + 1 1 11 a + b ba a b a, b c x c a + b a b h x x e x m c x x xc x x −− − −   − − −  − − − − +  − − −−   ( ) ( ) ( )( ) ( ) 1 1 1 = + 1 1 1 c a + b a b h x x h x c x x x  − − −  − − − +  − − −  int. j. anal. appl. 19 (3) (2021) 379 ( ) ( ) ( )( ) ( ) 1 1 1 = 1 1 1 x + h x c a + b a b h x c x x x − − − − + − − − − by lemma 3.2 ( ) ( ) ( )( ) ( ) 1 1 1 = 1 1 1 f x c a + b a b f x c x x x  − − − + − − − − integrating the both sides of the above equation with respect to x, we obtain ( ) ( ) ( ) 11 1 = 1 1 ba a + b x x f x k c x −− − − −   , where k is a constant. using the condition ( ) 1 0 = 1f x dx , we obtain ( ) ( ) ( ) ( ) 1 1 1 = 1 1 baa a + b x xc f x b a, b c x −− − − −   4 estimation 4.1 maximum-likelihood estimation let 1 2 3 . . , , , . , nx x x x be a random sample having probability density function ( ) ( ) 1 = n i l x ; a, b, c f x =  ( ) ( ) ( ) ( )  1 1 1 1 1 1 = 1 1 ; a b n n ina i i n a + b n i i x x c l x a, b, c b a, b c x − − = = =     −             − −       the log-likelihood function is ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 = + 1 + 1 1 + 1 1 n n i i i=1 i n i i lnl x ; a, b, c nalnc nlnb a, b a lnx b ln x a b ln c x = = − − − − − − −      it follows that the maximum-likelihood estimates( )a, b, c , say ( )ˆ垐a, b, c , are the simultaneous solutions of the equations: int. j. anal. appl. 19 (3) (2021) 380 ( ) ( ) 1 1 0 = = ψ( ) ψ + + 1 1 a n n i i i i lnl nlnc n a a b lnx ln c x = =  − − − − −          ( ) ( ) ( ) 1 1 0 = = ψ( ) ψ + 1 1 1 n n i i i i lnl n b a + b ln x ln c x b = =  − − − − − −          ( ) ( )1 0 = = + 1 1 n i i lnl an c a b c c c x=  −  − −  these equations can be easily solved by using the r statistical package. 4.2 a simulation study the study was based on 1000 generated data sets from the lnbd with different values of the parameters for n = 10, 20, 30, 50, 70, 100, 150, 200, 300, 500, 700 and 1000. we calculate of the bias and mse for different values of the parameters, these results suggest that the mse decrease when sample size increases. table 4.2.1 the bias, mse values for the lnb model when a = b = c = 2.5 n = 10 n = 20 n = 30 n = 50 n = 70 n = 100 bias (a) bias (b) bias (c) -2.05042 -1.50497 -2.40205 -2.49065 -2.01256 -2.071787 -0.7122865 -2.171628 -2.466312 -0.86810 15.99567 0.177780 -0.58077 17.03842 -1.71462 0.57136 27.10133 -1.41986 mse (a) mse (b) mse (c) 67.98561 258.9330 65.35506 64.23299 237.71409 56.42674 55.0937 195.15059 47.268447 41.68756 189.3684 36.1156 9.82927 149.6461 30.09384 1.43641 111.326 19.86549 n = 150 n = 200 n = 300 n = 500 n = 700 n = 1000 bias (a) bias (b) bias (c) 0.45077 21.20987 -1.67034 0.38837 15.69702 -1.714691 0.36992 13.10631 -1.76891 -1.64893 9.31559 -1.76459 0.30763 8.41728 -1.79250 0.30824 7.78502 0.30603 mse (a) mse (b) mse (c) 0.91711 1411.456 3.49029 0.51585 728.7420 3.31447 0.34143 459.8135 3.30728 4.45167 164.5743 3.233171 0.17203 107.269 3.30078 0.15320 81.3317 0.14827 5 application in order to prove that lnbd can be a better model than the power distribution, beta distribution with (a = 1), beta distribution, kumaraswamy distribution, let us use three real data sets. the following tables show the numerical values with mles and their corresponding standard errors (in parentheses) of the model parameters including loglikelihood, kolmogorovint. j. anal. appl. 19 (3) (2021) 381 smirnov test (ks), akaike information criterion (aic) and consistent akaike information criterion (caic) for comparing lnbd with the power distribution, beta distribution with (a = 1), beta distribution, kumaraswamy distribution. it is quite evident from the reports that lnbd is better. the plots of the fitted distributions to real datasets are shown in figures. data set 1: the following dataset which is skewed to right, present the sar image modeling on oil slick visibility in ocean. the values are: 0.6244, 0.1868, 0.5444, 0.3399, 0.4864, 0.4825, 0.2528, 0.2612, 0.2086, 0.3303, 0.5453, 0.2025, 0.4231, 0.2310, 0.7167, 0.2706, 0.1922, 0.5390, 0.5550, 0.2282, 0.5434, 0.4799, 0.4570, 0.3448, 0.2271, 0.4731, 0.1875, 0.3188, 0.1824, 0.3229, 0.1962, 0.3743, 0.1614, 0.1543, 0.4985, 0.1515, 0.2553, 0.1734, 0.1617, 0.2271, 0.2253, 0.2635, 0.5441, 0.1281, 0.4764, 0.3443, 0.3770, 0.5101, 0.3143, 0.1645, 0.1211, 0.2900, 0.4265, 0.2084, 0.5753, 0.2526, 0.2469, 0.2301, 0.5180, 0.4176, 0.1776, 0.6351, 0.3362, 0.2355, 0.3916, 0.4615, 0.6178, 0.3272, 0.3876, 0.2010, 0.3614, 0.1480, 0.3105, 0.1710, 0.2771, 0.4655, 0.1468, 0.2113, 0.3071, 0.4291, 0.2777, 0.2101, 0.4991, 0.2567, 0.3065, 0.5470, 0.3353, 0.1948, 0.2686, 0.2061, 0.5123, 0.1567, 0.3749, 0.3714, 0.3618, 0.5189, 0.3500, 0.2633, 0.1928, 0.4022, 0.1120, 0.3621, 0.4664, 0.3106, 0.2465, 0.2388, 0.4497, 0.2979, 0.1524, 0.1822, 0.3955, 0.1744, 0.3800, 0.4578, 0.1872, 0.2587, 0.4699, 0.2329, 0.3943, 0.3613. table 5.1: estimated parameters by mle with their s.e. and goodness of fit model â b̂ ĉ ln (l) ks aic caic power 5.7155×10-3 0.07560 -2.2764 0.3442 6.5527 6.5867 beta1 (a = 1) 0.04690 (0.2166) -34.2384 70.4768 70.5107 beta 0.2118 (0.4602) 0.9438 (0.9715) -73.6054 0.08153 151.2108 151.3134 kumaraswamy 4.1328 (2.3293) 0.03734 (0.1932) -71.5770 0.08800 147.1539 147.2565 libby-novick beta 324.8720 (18.1072) 1. 6564 (1.2870) 338. 7979 (18.4065) -75.00942 0.06152 156.0188 156.2257 int. j. anal. appl. 19 (3) (2021) 382 figure 5.1: pdf and cdf graphs of the densities figure 5.2: q-q plots of the densities data set 2: the following right to skewed dataset presented by cordeiro and brito (2012) is obtained from the measurements on petroleum rock samples. the data consists of 48 rock samples from a petroleum reservoir. the dataset corresponds to twelve core samples from petroleum reservoirs that were sampled by four cross-sections. each core sample was measured for permeability and int. j. anal. appl. 19 (3) (2021) 383 each cross-section has the following variables: the total area of pores, the total perimeter of pores and shape. we analyze the shape perimeter by squared (area) variable and the observations are: 0.090330, 0.203654, 0.204314, 0.280887, 0.197653, 0.328641, 0.448622, 0.562394, 0.462727, 0.279455, 0.520635, 0.830081, 0.183312, 0.350944, 0.201071, 0.191802, 0.054192, 0.564125, 0.217063, 0.048141, 0.544810, 0.433083, 0.176016, 0.420477, 0.422417, 0.228595, 0.113852, 0.025214, 0.376969, 0.200744, 0.156705, 0.031623, 0.791029, 0.041273, 0.238712, 0.362651, 0.389051, 0.472567, 0.340077, 0.711646, 0.463586, 0.282453, 0.264127, 0.151181, 0.461865, 0.306016, 0.553832, 0.105447 table 5.2: estimated parameters by mle with their s.e. and goodness of fit model â b̂ ĉ ln (l) ks aic caic power 0.01068 (0.1033) -3.008295 0.2737 8.01659 8.1035 beta1 (a = 1) 0.1086 (0.32948) -12.6421 0.9825 27.2843 27.3712 beta 0.08118 (0.2849) 0.4367 (0.6608) -14.8539 0.06485 33.7077 33.9744 kumaraswamy 0.6939 (0.8330) 0.04087 (0.2022) -14.8116 0.06492 33.6232 33.8899 libby-novick beta 0.2447 (0.4947) 2.5122 (1.5850) 1.2131 (1.1014) -14.8696 0.06488 35.7393 36.2847 figure 5.3: pdf and cdf graphs of the densities int. j. anal. appl. 19 (3) (2021) 384 figure 5.4: q-q plots of the densities 6 conclusion this particular research work consists of basic mathematical properties of lnbd i.e. definition of probability density function (pdf) with its shape discussion, cumulative distribution function (cdf), quantiles, skewness, kurtosis, variance, mode and points of inflection along with their numerical behavior given in the form of tables due to various values of the parameters, harmonic mean, geometric mean, rth moment about origin, moment generating function (mgf), factorial moments (increasing and decreasing), negative moments, conditional moments, mean deviations from mean and median etc. we have deduced the reliability measures in the shape of survival function, hazard rate function, reverse hazard rate function with graphical presentation. we developed transformations of lnbd to other density function, asymptotes and shapes of pdf, cdf and hrf. we cultivated a mathematical treatment bonferroni and lorenz curves, gini index, entropy measures, characterizations for lnbd with one truncated moment. int. j. anal. appl. 19 (3) (2021) 385 we derive mles of the parameters, simulation study of lnbd with the comparison of biases and mses due to different values of the parameters. by means of application of lnbd to two real data sets and comparing it with the power distribution, beta distribution with (a = 1), beta distribution, kumaraswamy distribution model, we come to the conclusion that lnbd is better. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. azzalini, a class of distributions which includes the normal ones, scand. j. stat. 12(2) (1985), 171-178. [2] a. k. sheikh, m. ahmad, z. ali, some remarks on the hazard functions of the inverted distributions, reliab. eng. 19 (1987), 255–261. [3] a. p. prudnikov, y. a. brychkov, and o.i. marichev, integrals and series. vol. 1. gordon and breach science publishers, amsterdam, (1986). [4] a. réyni, probability theory, dover publications, new york, 1970. [5] a. telcs, w. glanzel, and a. schubert, characterization and statistical test using truncated expectations for a class of skew distributions. math. soc. sci. 10 (1985), 169-178. [6] a. w marshall, and i. olkin, a new method for adding a parameter to a family of distributions with application to the exponential and weibull families. biometrika. 84 (1997), 641-652. [7] b.d. sharma, i.j. taneja, entropy of type (α, β) and other generalized measures in information theory, metrika. 22 (1975), 205–215. [8] b. d. sharma, p. mittal, new non-additive measure of relative information. j. comb. inform. syst. sci. 2 (1977), 122-133. [9] c. alexander, g. m. cordeiro, e. m. m. ortega, j. m. sarabia, generalized beta-generated distributions, comput. stati. data anal. 56 (2012), 1880–1897. [10] c.-d. lai, g. jones, beta hazard rate distribution and applications, ieee trans. rel. 64 (2015), 44– 50. [11] c. e. shannon, a mathematical theory of communication, sigmobile mob. comput. commun. rev. 5 (2001), 3–55. [12] c. tsallis, possible generalization of boltzmann-gibbs statistics, j. stat. phys. 52 (1988), 479–487. [13] d. chotikapanich, modeling income distributions and lorenz curves, springer, ny, 2008. int. j. anal. appl. 19 (3) (2021) 386 [14] d. desai, v. mariappan, and m. sakhardanda, nature of reverse hazard rate: an investigation. int. j. perform. eng. 7(2) (2011), 165-171. [15] d. l. libby, m. r. novick, multivariate generalized beta distributions with applications to utility assessment, j. educ. stat. 7 (1982), 271–294. [16] d. salomon, data compression, springer, new york, 1998. [17] d. n. shanbagh, the characterization for exponential and geometric distributions. j. amer. stat. assoc. 65(331) (1970), 1256-1259. [18] s. t. dara, recent advances in moment distributions and their hazard rate. ph.d. thesis, national college of business administration and economics, lahore, pakistan, (2012). [19] e. boekee, and a. c. j. van der lubbe, the r-norm information measure. inform. control. 45 (1980), 136-155. [20] e. r. barlow, w. a. marshall, and f. proschan, properties of probability distribution with monotone hazard rate. ann. math. stat. 34(2) (1963), 375-389. [21] e. m. ghitany, the monotonicity of the reliability measures of the beta distribution. appl. math. lett. 17 (2004), 1277-1283. [22] f. farmoye, c. lee and o. olumolade, the beta-weibull distribution. j. stat. theory appl. 4 (2005), 121-136. [23] g. m. cordeiro, l. h. de santana, e. m. m. ortega, r. r. pescim, a new family of distributions: libby-novick beta, int. j. stat. probab. 3 (2014), 63-80. [24] g. g. hamedani, on certain generalized gamma convolution distributions ii. technical report, no. 484, mscs, marquette university, 2013. [25] g. m. cordeiro, m. de castro, a new family of generalized distributions. j. stat. comput. simul. 81(7) (2011), 883-898. [26] a. k. gupta, s. nadarajah, handbook of beta distribution and its applications. marcel dekker, new york, 2004. [27] i. s. gradshteyn, i. m. ryzhik, table of integrals, series and products. 7th ed. academic press, diego, 2004. [28] j. havrda, f. s. charvat, quantification method of classification processes: concept of structuralentropy, khbernetika. 3(1967), 30-35. [29] j. navarro, a. guillamon, m.c. ruiz, generalized mixture in reliability modeling: applications to the construction of bathtub shaped hazard model and the study of systems. appl. stoch. models bus. ind. 25(3) (2009), 323-337. int. j. anal. appl. 19 (3) (2021) 387 [30] j. n. kapur, generalized entropy of order α and type β. math. seminar, 4 (1967), 78-94. [31] m. ahsanullah, g. g. hamedani, characterizations of certain continuous univariate distributions based on the conditional distribution of generalized order statistics. pak. j. stat. 28 (2012), 253-258. [32] m. ahsanullah, characterizations of univariate continuous distributions. atlantis press, paris, france 2017. [33] m. ahsanullah, m. shakil, m. b. golam kibria, characterizations of continuous distributions by truncated moment. j. mod. appl. stat. meth. 1(15) (2016), 316-331. [34] m. a. awad, j. a. alawneh, application of entropy to life-time model. ima j. math. control inform. 4(1987), 143-147. [35] m. c. jones, families of distributions arising from distributions of order statistics. test, 13(2004), 143. [36] g.m. giorgi, s. nadarajah, bonferroni and gini indices for various parametric families of distributions, metron. 68 (2010), 23–46. [37] m. h. barakat, h. y. abdelkader, computing the moments of order statistics from non-identical random variables. stat. meth. appl. 13 (2004), 15-26. [38] m. i. kamien, n.l. schwartz, a generalized hazard rate, econ. lett. 5(3) (1980), 245-249. [39] m. mitra, s. k. basu, on some properties of the bathtub failure rate family of life distributions. microelectron. reliab. 36(5) (1996), 679-684. [40] m. r. fazlollah, an introduction to information theory. mcgraw-hill, new york, 1961. [41] m. m. ristić, b. v. popović, s. nadarajah, libby and novick’s generalized beta exponential distribution, j. stat. comput. simul. 85 (2015), 740–761. [42] n. eugene, c. lee, felix famoye, beta-normal distribution and its application. commun. stat. – theory meth. 31 (2002), 497–512. [43] n. pushkarna, j. saran, and r. tiwari, bonferroni and gini indices and recurrence relation for moments progressive type-ii right censored order statistics from marshall-olkin exponential distribution. j. stat. theory appl. 12(3) (2013), 306-320. [44] p. t. gia, n. turkkan, determine of the beta distribution form its lorenz curve. math. comput. model. 16(2) (1992), 73-84. [45] r core team, r a language and environment for statistical computing. austria, vienna: r foundation for statistical computing, 2013. [46] r. d. gupta, d. kundu, generalized exponential distributions, aust nz j stat. 41 (1999), 173–188. [47] r. s. varma, generalizations of renyi’s entropy of order α. j. math. sci. 1 (1966), 34-48. https://www.tandfonline.com/author/eugene%2c+nicholas https://www.tandfonline.com/author/lee%2c+carl int. j. anal. appl. 19 (3) (2021) 388 [48] s. a. hasnain, z. iqbal, m. ahmad, one exponentiated moment exponential distribution. pak. j. stat. 31(2) (2015), 267-280. [49] s. arimoto, information theoretical considerations on estimation theory. inform. control, 73 (1971), 181-190. [50] s. kullback, information theory and statistics, wiley, ny, 1959. [51] s.m. sunoj, p.g. sankaran, s.s. maya, characterizations of life distributions using conditional expectations of doubly (interval) truncated random variables, commun. stat. – theory meth. 38 (2009) 1441–1452. [52] s. nadarajah, a. k. gupta, the beta fréchet distribution. far east theor. stat. 14 (2004),15-24. [53] s. nadarajah, s. kotz, the beta exponential distribution. reliab. eng. syst. safe. 91 (2006), 689-697. [54] s. nadarajah, s. kotz, the beta gumbel distribution. math probl. eng. 10 (2004), 323-332. [55] s. pundir, s. arora, k. jain, bonferroni curve and the related statistical inference. stat. probab. lett. 75(2005), 140-150. [56] t. m. cover, j. a. thomas, elements of information theory, john wiley, new york, 1991. [57] u. n. nair, g. p. sankaran, properties of a mean residual life function arising from renewal theory. naval res. logist. 57 (2010), 373-379. [58] v. mameli, m. musio, a new generalization of the skew-normal distribution: the beta skew-normal. commun. stat. – theory meth. 42 (2013) 2229-2244. [59] v. mameli, the kumaraswamy skew-normal distribution, stat. probab. lett. 104 (2015), 75–81. [60] v. mameli, two generalizations of the skew-normal distribution and two variants of mccarthys theorem. doctoral dissertation, cagliari university, italy, 2012. [61] v. dardanoni, a. forcina, inference for lorenz curve orderings. econ. j. 2 (1999), 49-75. [62] w. f. sharpe, investments, prentice hall, englewood cliffs, 1985. international journal of analysis and applications volume 18, number 4 (2020), 663-671 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-663 invariant summability and unconditionally cauchy series nimet pancaroḡlu akin∗ afyon kocatepe university department of mathematics and science education ∗corresponding author: npancaroglu@aku.edu.tr abstract. in this study, we will give new characterizations of weakly unconditionally cauchy series and unconditionally convergent series through summability obtained by the invariant convergence. 1. introduction let σ be a mapping of the positive integers into itself. a continuous linear functional ϕ on m, the space of real bounded sequences, is said to be an invariant mean or a σ mean, if and only if, (1) φ(x) ≥ 0, when the sequence x = (xj) is such that xj ≥ 0 for all j, (2) φ(e) = 1,where e = (1, 1, 1....), (3) φ(xσ(j)) = φ(x) for all x ∈ m. the mappings φ are assumed to be one-to-one and such that σi(j) 6= j for all positive integers j and i, where σi(j) denotes the ith iterate of the mapping σ at j. thus φ extends the limit functional on c, the space of convergent sequences, in the sense that φ(x) = lim x for all x ∈ c. in case σ is translation mappings σ(j) = j + 1, the σ mean is often called a banach limit and vσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences. received march 27th, 2020; accepted may 14th, 2020; published june 1st, 2020. 2010 mathematics subject classification. 40a05. key words and phrases. unconditionally cauchy series; invariant convergence; invariant convergent series. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 663 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-663 int. j. anal. appl. 18 (4) (2020) 664 it can be shown that vσ = {x = (xj) : lim i tij(x) = ` uniformly in j,` = σ − lim x} where, tij(x) = xσ(j) + xσ2(j) + · · · + xσi(j) i + 1 several authors including raimi [19], schaefer [20], mursaleen and edely [10], mursaleen [12], savaş [22, 23], nuray and savaş [14], pancaroǧlu and nuray [16, 17] and some authors have studied invariant convergent sequences. the concept of strongly σ-convergence was defined by mursaleen [11]. savaş and nuray [24] introduced the concepts of σ-statistical convergence and lacunary σ-statistical convergence and gave some inclusion relations. now, we recall the basic concepts and some definitions and notations (see [1, 3–5, 7–9, 13, 15, 21]). let x be a normed space. for any given series ∑ i xi in x, let us consider the sets s( ∑ i xi) = {(ai) ∈ `∞ : ∑ i aixi convergent} sw( ∑ i xi) = {(ai) ∈ `∞ : ∑ i aixi convergent for the weak topology}. the above sets endowed with the sup norm and they will be called the space of convergence and the space of weak convergence associated to the series ∑ i xi. definition 1.1. a series ∑ i xi in a normed space x is said to be a weakly unconditionally cauchy(wuc) if for each ε > 0 and f ∈ x∗, an n0 ∈ n can be found such that for each finite subset f ⊂ n with f ∩{1, . . . ,n0} 6= ∅ is ∑ i∈f |f(xi)| < ε. as a consequence, ∑ i xi is a wuc series in x if and only if each functional f ∈ x ∗ satisfies that∑∞ i=1 |f(xi)| < ∞. in [18] it is proved that a normed space x is complete if and only if for every weakly unconditionally cauchy (wuc) series ∑ i xi, the space s( ∑ i xi) is also complete. diestel [6] proved the following characterization that will be used throughout the paper. theorem 1.1. let ∑ i xi be a series in a normed space x. then, the series ∑ i xi is wuc if and only if there exists h > 0 such that int. j. anal. appl. 18 (4) (2020) 665 h = sup{‖ n∑ i=1 aixi‖ : n ∈ n, |ai| ≤ 1, i ∈{1, . . . ,n}} = sup{‖ n∑ i=1 εixi‖ : n ∈ n,εi ∈{−1, 1}, i ∈{1, . . . ,n}} = sup{ n∑ i=1 |f(xi)| : f ∈ bx∗} where bx∗ is denotes the closed unit ball in x ∗ 2. main results proposition 2.1. let x be a normed space and (xn) an invariant convergent sequence in x. then (xn) ∈ `∞(x). proof. let (xn) be a sequence in x such that σ − limn xn = x0 for some x0 ∈ x. we can fix ε > 0 and i0 ∈ n satisfying that ∥∥∥∥ 1i + 1 i∑ k=0 xσk(j) ∥∥∥∥ ≤‖x0‖ + ε for every i ≥ i0 and j ∈ n. then we have that for every j ∈ n is ‖xj‖ = ‖xσ0(j)‖ = ∥∥∥∥i0 + 2i0 + 1 i0+1∑ k=0 xσk(j) i0 + 2 − i0+1∑ k=1 xσk(j) i0 + 1 ∥∥∥∥ ≤ ( i0 + 2 i0 + 1 + 1 ) (‖x0‖ + ε) where the last term is a fixed constant, what concludes the proof. � definition 2.1. a series ∑ i xi in x is said to be invariant convergent to x0 ∈ x if σ − limn sn = x0, where sn = ∑n i=1 xi is sequence of partial sums, and we will denote it by vσ − ∑ i xi = x0. therefore, vσ − ∑ i xi = x0 if and only if lim i→∞ ( j∑ k=1 xk + 1 i + 1 i∑ k=1 [ (i−k + 1)xσk(j) ]) = x0 uniformly in j ∈ n. definition 2.2. x0 is said to be weak invariant limit of a sequence (xn) if each function f ∈ x∗ verifies that σ − lim f(xn) = f(x0) and we will write wσ − lim xn = x0. let x be a normed space and ∑ i xi a series in x. we define following sets: sσ( ∑ i xi) = {(ai) ∈ `∞ : vσ − ∑ i aixi exists} swσ( ∑ i xi) = {(ai) ∈ `∞ : wvσ − ∑ i aixi exists}. these spaces are the vector subspaces of `∞ and we consider them endowed with the sup norm. int. j. anal. appl. 18 (4) (2020) 666 theorem 2.1. let x be a banach space and ∑ i xi a series in x. then ∑ i xi is wuc(weakly unconditionally cauchy) if and only if sσ( ∑ i xi) is complete. proof. let ∑ i xi be a wuc series. we will prove that sσ( ∑ i xi) is closed in `∞. let (a n) be a sequence in sσ( ∑ i xi), a n = (ani ) for each n ∈ n and let also be a0 ∈ `∞ such that limn‖a n −a0‖ = 0. we will show that a0 ∈ sσ( ∑ i xi). let h > 0 be such that h ≥ sup{‖ n∑ i=1 aixi‖ : n ∈ n, |ai| ≤ 1, i ∈{1, . . . ,n}}. for each natural n there exists yn ∈ x such that yn = vσ − ∑ i a n i xi. we will see that (yn) is a cauchy sequence. if ε > 0 is given, there exists an n0 such that if p,q ≥ n0 ,then ‖ap −aq‖ < ε 3h . if p,q ≥ n0 are fixed, there exists i ∈ n verifying ∥∥∥∥yp − ( j∑ k=1 a p kxk + 1 i + 1 i∑ k=1 [ (i−k + 1)ap σk(j) xσk(j) ])∥∥∥∥ < ε3 (2.1) ∥∥∥∥yq − ( j∑ k=1 a q kxk + 1 i + 1 i∑ k=1 [ (i−k + 1)aq σk(j) xσk(j) ])∥∥∥∥ < ε3 (2.2) for each j ∈ n. then, if p,q ≥ n0 we have that ‖yp −yq‖≤ (2.1) + (2.2) + ∥∥∥∥ j∑ k=1 (a p k −a q k)xk + i∑ k=1 [ i−k + 1 i + 1 (a p σk(j) −aq σk(j) )xσk(j) ]∥∥∥∥, (2.3) where (2.3) ≤ ε 3 . therefore, since x is banach space, there exists y0 ∈ x such that limn‖yn −y0‖ = 0. we will check that σ ∑ i a 0 ixi = y0, that is, lim i→∞ ( j∑ k=1 a0kxk + 1 i + 1 i∑ k=1 [ (i−k + 1)a0σk(j)xσk(j) ]) = y0, uniformly in j ∈ n. if ε > 0 is given, we can fix a natural n such that ‖an −a0‖ < ε 3h and ‖yn −y0‖ < ε 3 . now, we can also fix i0 such that for every i ≥ i0 is ∥∥∥∥yn − ( j∑ k=1 ankxk + 1 i + 1 i∑ k=1 [ (i−k + 1)anσk(j)xσk(j) ])∥∥∥∥ < ε3 int. j. anal. appl. 18 (4) (2020) 667 for every j ∈ n. then, if i ≥ i0 it is satisfied that∥∥∥∥y0 − ( j∑ k=1 a0kxk + 1 i + 1 i∑ k=1 [ (i−k + 1)a0σk(j)xσk(j) ])∥∥∥∥ ≤‖y0 −yn‖ + ∥∥∥∥yn − ( j∑ k=1 ankxk + 1 i + 1 i∑ k=1 [ (i−k + 1)anσk(j)xσk(j) ])∥∥∥∥ + ∥∥∥∥ j∑ k=1 (an −a0)xk + 1 i + 1 i∑ k=1 [ (i−k + 1)(anσk(j) −a 0 σk(j))xσk(j) ]∥∥∥∥ ≤ 2ε3 + ‖an −a0‖ (σ(j)∑ k=1 (ank −a 0 k) ‖an −a0‖ xk + i∑ k=1 [ (i−k + 1) i + 1 (an σk(j) −a0 σk(j) ) ‖an −a0‖ xσk(j) ]) ≤ 2ε 3 + ε 3h h ≤ ε for every j ∈ n. thus (a0n) ∈ sσ( ∑ i xi). conversely, if sσ( ∑ i xi) is closed, since c00 ⊂ sσ( ∑ i xi), we deduce that c0 ⊂ sσ( ∑ i xi). suppose that∑ i xi is not wuc series. then there exists f ∈ x ∗ verifying ∑∞ i=1 |f(xi)| = +∞. we can choose a natural n1 such that ∑n1 i=1 |f(xi)| > 2.2 and for i ∈ {1, . . . ,n1} we define ai = 1 2 if f(xi) ≥ 0 or ai = −1 2 if f(xi) < 0. there exists n2 > n1 such that ∑n2 i=n1+1 |f(xi)| > 3.3 and for i ∈ {n1 + 1, . . . ,n2} we define ai = 1 3 if f(xi) ≥ 0 or ai = −1 3 if f(xi) < 0. in this manner we obtain an increasing sequence (nk)k in n and a sequence a = (ai)i in c0 such that∑∞ i=1 aif(xi) = +∞. since (ai)i ∈ sσ( ∑ i xi), it follows that σ ∑ i aixi exists and therefore (∑n i=1 aif(xi) ) n is bounded sequence, which is a contradiction. � then we have the following result. corollary 2.1. let x be a banach space and ∑ i xi a series in x. then ∑ i xi is a wuc(weakly unconditionally cauchy) series if and only if for each sequence (ai)i ∈ c0 it is satisfied that vσ − ∑ i aixi exists. proof. let ∑ i xi be a wuc series in x. then, we have that sσ( ∑ i xi) is complete. since c00 ⊂ sσ( ∑ i xi), we deduce that c0 ⊂ sσ( ∑ i xi), that is, vσ − ∑ i aixi exists for every sequence (ai) ∈ c0. the converse is proved similar to the end of the previous demonstration. � remark 2.1. let x be a normed space and ∑ i xi a series in x. we consider the linear map t : sσ( ∑ i xi) → x defined by t(a) = vσ − ∑ i aixi. int. j. anal. appl. 18 (4) (2020) 668 suppose that ∑ i xi is a wuc series and consider h = sup{‖ ∑n i=1 aixi‖ : n ∈ n, |ai| ≤ 1, i ∈ {1, . . . ,n}}. then, it is easy to check that if a ∈ sσ( ∑ i xi) then ‖t(a)‖ = ‖vσ − ∑ i aixi‖ ≤ h‖a‖ and therefore t is continuous. conversely if t is continuous and {a1, . . . ,aj} ⊂ [−1, 1], it is satisfied that ‖ ∑j i=1 aixi‖ = ‖vσ −∑∞ i=1 aixi‖≤‖t‖ (considering ai = 0 if i > j), which implies that ∑ i xi is a wuc series. in the next theorem we study the completeness of space swσ( ∑ i xi). theorem 2.2. let x be a banach space and ∑ i xi a series in x. then ∑ i xi is a wuc series if and only if swσ( ∑ i xi) is complete. proof. consider ∑ i xi to be a wuc series. it will be enough to prove that swσ( ∑ i xi) is closed in `∞. let (a n) be sequence in swσ( ∑ i xi), a n = (ani )i for each n ∈ n and let also be a 0 ∈ `∞ such that limn‖an −a0‖ = 0. we will show that a0 ∈ swσ( ∑ i xi). let h > 0 be such that h ≥ sup{‖ n∑ i=1 aixi‖ : n ∈ n, |ai| ≤ 1, i ∈{1, . . . ,n}} for each natural n there exists yn ∈ x such that yn = wvσ− ∑ i a n i xi. we will check that (yn)n is cauchy sequence. if ε > 0 is given, there exists an n0 such that if p,q ≥ n0 ,then ‖ap − aq‖ < ε 3h . we fix p,q ≥ n0 and we have that there exists f ∈ sx∗ (unit sphere in x∗)verifying ‖yp − yq‖ = |f(yp − yq)|. since vσ − ∑ i a p i f(xi) = f(yp) and vσ − ∑ i a q if(xi) = f(yq), there exists i ∈ n such that∣∣∣∣f(yp) − ( j∑ k=1 a p kf(xk) + 1 i + 1 i∑ k=1 [ (i−k + 1)ap σk(j) f(xσk(j)) ])∣∣∣∣ < ε3 (2.4) ∣∣∣∣f(yq) − ( j∑ k=1 a q kf(xk) + 1 i + 1 n∑ k=1 [ (i−k + 1)aq σk(j) f(xσk(j)) ])∣∣∣∣ < ε3 (2.5) for each j ∈ n. then, if p,q ≥ n0 we have that ‖yp −yq‖ =|f(yp) −f(yq)| ≤ (2.4) + (2.5) (2.6) + ∣∣∣∣ j∑ k=1 (a p k −a q k)f(xk) + i∑ k=1 [ i−k + 1 i + 1 (a p σk(j) −aq σk(j) )f(xσk(j)) ]∣∣∣∣, (2.7) where (2.6) ≤ ε 3 . therefore, since x is banach space, there exists y0 ∈ x such that limn‖yn −y0‖ = 0. we will check that wvσ − ∑ i a 0 ixi = y0. if ε > 0 is given, we can fix a natural n such that ‖an −a0‖ < ε 3h and ‖yn −y0‖ < ε 3 . consider a functional f ∈ bx∗ . we have that there exists i0 ∈ n such that if i ≥ i0 is int. j. anal. appl. 18 (4) (2020) 669 ∣∣∣∣f(yn) − ( j∑ k=1 ankf(xk) + 1 i + 1 i∑ k=1 [ (i−k + 1)anσk(j)f(xσk(j)) ])∣∣∣∣ < ε3 for every j ∈ n. then, if i ≥ i0 and j ∈ n, we have that∣∣∣∣f(y0) − ( j∑ k=1 a0kf(xk) + 1 i + 1 i∑ k=1 [ (i−k + 1)a0σk(j)f(xσk(j)) ])∣∣∣∣ ≤ |f(y0 −yn)| + ∣∣∣∣f(yn) − ( j∑ k=1 ankf(xk) + 1 i + 1 i∑ k=1 [ (i−k + 1)anσk(j)f(xσk(j)) ])∣∣∣∣ + ∣∣∣∣ j∑ k=1 (an −a0)f(xk) + 1 i + 1 i∑ k=1 [ (i−k + 1)(anσk(j) −a 0 σk(j))f(xσk(j)) ]∣∣∣∣ ≤ ε that is, wvσ − ∑ i a 0 ixi = y0 and a 0 ∈ swσ( ∑ i xi). conversely, if swσ( ∑ i xi) is complete, which implies that c0 ⊂ swσ( ∑ i xi). suppose that there exists f ∈ x∗ verifying ∑∞ i=1 |f(xi)| = +∞. then, as we did in theorem 2.1, a sequence a = (ai) in c0 can be obtained such that ∑ i aif(xi) = +∞ since a ∈ swσ( ∑ i xi), there will exists x0 ∈ x such that wvσ− ∑ i aixi = x0 and it will be vσ− ∑ i aif(xi) = x0. but this implies that the sequence (∑n i=1 aif(xi) ) n is bounded which is a contradiction. � remark 2.2. let x be a banach space ∑ i xi a series in x. we consider the linear map t:swσ( ∑ i xi) → x defined by t(a) = wvσ − ∑ i aixi. we will show that ∑ i xi is wuc series if and only if t is continuous. we define h = sup{‖ ∑n i=1 aixi‖ : n ∈ n, |ai| ≤ 1, i ∈ {1, . . . ,n}} and take a ∈ swσ( ∑ i xi). then wvσ− ∑ i aixi = x0 exists and we can take f ∈ sx∗ such that |t(a)| = |f(t(a))| = |vσ− ∑ i aif(xi)| ≤ h‖a‖. conversely if t is continuous.then if {a1, . . . ,an}⊂ [−1, 1], we have that ‖ ∑n i=1 aixi‖ = ‖wvσ − ∑∞ i=1 aixi‖ ≤ ‖t‖ (considering ai = 0 if i > n), which implies that ∑ i xi is a wuc series. from the previous theorem and its proof the following corallary can be easily proved. corollary 2.2. let x be a banach space ∑ i xi a series in x. then the following are equivalent: (1) ∑ i xi is a wuc series. (2) swσ( ∑ i xi) is complete. (3) c0 ⊂ swσ( ∑ i xi) (wvσ − ∑ i aixi exists for each a = (ai) ∈ c0). let us see that the hypothesis of completeness in the two previous theorems is completely necessary. let x be a non-complete normed space. then it is easy to prove that there exists a sequence ∑ i xi in x such that ‖xi‖ < 1 i2i and ∑ i xi = x ∗∗ ∈ x∗∗\x. then we have that vσ − ∑ i xi = x ∗∗. if we consider int. j. anal. appl. 18 (4) (2020) 670 the series ∑ i zi defined by zi = ixi for each n ∈ n, we have that ∑ i zi is wuc series. consider the sequence a = (ai) ∈ c0 given by ai = 1 i . it is satisfied that vσ − ∑ i aizi ∈ x ∗∗\x and therefore a /∈ sσ( ∑ i zi) and a /∈ swσ( ∑ i zi). let x be a normed space and x∗ its dual space. let also ∑ i fi be a series in x ∗. it is known that [6],∑ i fi is wuc if and only if ∑ i |fi(x)| < ∞ for each x ∈ x. now we consider the vector space s∗wσ( ∑ k fi) = {a = (ai) ∈ `∞ : ∗wvσ − ∑ i aifi exists} , where ∗wvσ − ∑ i aifi = f0 if and only if vσ − ∑ i aifi(x) = f0(x) for each x ∈ x. theorem 2.3. let x be a normed space. it is satisfied that 1 ⇒ 2 ⇒ 3, where: (1) ∑ i f(i) is a wuc series. (2) s∗wσ( ∑ i fi) = `∞. (3) if x ∈ x and m ⊂ n, then vσ − ∑ i∈m fi(x) exists. besides, if x is a barelled normed space, the three items are equivalent. proof. from the ∗ weak compacity of bx∗ we deduce that 1 ⇒ 2. it is clear that 2 ⇒ 3. we suppose now that x is barelled and we will prove that 3 ⇒ 1. effectively, our goal is to prove that e = { ∑n i=1 aifi : n ∈ n, |ai| ≤ 1, i ∈ {1, . . . ,n}} is pointwise bounded for each x ∈ x and therefore e is bounded, which implies that ∑ i fi is wuc series. suppose that e is not pointwise bounded, that is, there exists x0 ∈ x such that ∑ i |fi(x0)| = +∞. then, we can choose a subset m ⊂ n such that∑ i∈m fi(x0) = + −∞. but, by hypothesis, vσ − ∑ i∈m fi(x0) exists, which is a contradiction. � when σ(j) = j + 1, we have the almost all definitions and theorems in [2] concerning almost summability. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. aizpuru, a. gutierrez-davila, a. sala, unconditionally cauchy series and cesaro summability, j. math. anal. appl. 324 (2006), 39–48. [2] a. aizpuru, r. armario, f.j. perez-fernandez, almost summability and unconditionally cauchy series, bull. belg. math. soc. simon stevin. 15 (2008), 635–644. [3] s. banach, théorie des opérations linéaires. chelsea publishing company, new york,(1978). [4] c. bessaga, a. pelczynski, on bases and unconditional convergence of series in banach spaces. stud. math. 17 (1958), 151–164. [5] j. boos, p. cass, classical and modern methods in summability, oxford university press, oxford; new york, 2000. int. j. anal. appl. 18 (4) (2020) 671 [6] j. diestel, sequences and series in banach spaces, springer-verlag, new york, 1984. [7] h. fast, sur la convergence statistique, colloq. math. 2 (1951), 241–244. [8] g. lorentz, a contribution to the theory of divergent sequences, acta math. 80 (1948), 167–190. [9] c.w. mcarthur.on relationships amongst certain spaces of sequences in an arbitrary banach space, can. j. math. 8 (1956), 192–197. [10] m. mursaleen, o.h.h. edely, on the invariant mean and statistical convergence, appl. math. lett. 22 (2009), 1700–1704. [11] m. mursaleen, matrix transformation between some new sequence spaces, houston j. math. 9 (1983), 505–509. [12] m. mursaleen, on finite matrices and invariant means, indian j. pure appl. math. 10 (1979), 457–460. [13] f. nuray, w.h. ruckle, generalized statistical convergence and convergence free spaces, j. math. anal. appl. 245 (2000), 513–527. [14] f. nuray, e. savaş, invariant statistical convergence and a-invariant statistical convergence, indian j. pure appl. math. 10 (1994), 267–274. [15] f. nuray, h. gök, u. ulusu, iσ-convergence, math. commun. 16 (2011), 531–538. [16] n. pancaroǧlu, f. nuray, statistical lacunary invariant summability, theor. math. appl. 3 (2) (2013), 71–78. [17] n. pancaroǧlu, f. nuray, on invariant statistically convergence and lacunary invariant statistically convergence of sequences of sets, prog. appl. math. 5 (2) (2013), 23–29. [18] f.j. perez-fernandez, f. benitez-trujillo, a. aizpuru, characterizations of completeness of normed spaces through weakly unconditionally cauchy series, czechoslovak math. j. 50 (2000), 889?896. [19] r.a. raimi, invariant means and invariant matrix methods of summability, duke math. j. 30 (1963), 81–94. [20] p. schaefer, infinite matrices and invariant means, proc. amer. math. soc. 36 (1972), 104–110. [21] i. j. schoenberg, the integrability of certain functions and related summability methods, amer. math. monthly, 66 (1959), 361–375. [22] e. savaş, some sequence spaces involving invariant means, indian j. math. 31 (1989), 1–8. [23] e. savaş, strong σ-convergent sequences, bull. calcutta math. 81 (1989), 295–300. [24] e. savaş, f. nuray, on σ-statistically convergence and lacunary σ-statistically convergence, math. slovaca, 43 (3) (1993), 309–315. 1. introduction 2. main results references international journal of analysis and applications volume 18, number 5 (2020), 876-889 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-876 received may 3rd, 2020; accepted may 27th, 2020; published august 10th, 2020. 2010 mathematics subject classification. 91b02. key words and phrases. measurement model; pls-sem; structural equation modelling; formative; reflective. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 876 formative vs. reflective measurement model: guidelines for structural equation modeling research mohd hafiz hanafiah* faculty of hotel and tourism management, universiti teknologi mara, selangor, malaysia *corresponding author: hafizhanafiah@uitm.edu.my abstract. various social sciences researchers have always debated the operationalisation of formative or a reflective measurement in partial least squares structural equation modeling (pls-sem). this paper aims to offer guidance on formative and reflective measurement model assessment in pls-sem. first, this paper explores and discuss the similarities and differences between the formative and reflective measurement model. next, this paper reviews the practice of measurement model assessment for formative and reflective construct based on the latest methodological background. finally, this paper proposes a set of guidelines in classifying the formative and reflective constructs and the steps in assessing the formative and reflective measurement model. this paper addresses the issue of measurement misspecification in pls-sem assessment by providing logical guidelines for researchers. this paper also helps to explain and suggest appropriate pls-sem assessment for researchers as they plan future research projects. 1. introduction partial least squares structural equation modeling (pls-sem) is a second-generation data analysis technique in the family of structural equation modelling ([1]; [2]). different from the sem covariance-based groups, pls-sem is a prediction-oriented approach to sem, usually used for exploratory research and also appropriate for confirmatory research ([3]; [4]). lauro and vinzi ([5]) suggested that pls-sem is particularly useful for causal-predictive analysis in situations of high complexity and low theoretical information availability. meanwhile, other researchers used https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-876 int. j. anal. appl. 18 (5) (2020) 877 the pls-sem approach because of its advantages over the covariance approach ([2]; [3]). the benefits of this soft-modelling approach include its ability to account the theoretical conditions, measurement conditions, distributional considerations, and practical considerations ([3]). besides, pls-sem is also an exploratory statistical tool that is able to process primary or secondary data ([6]). meanwhile, other researchers claimed that the pls-sem approach is suitable with prediction-oriented objective, abnormal data distribution and accommodates small sample sizes ([7]; [8]; [9]; [10]). table 1.1 illustrates how the pls-sem approach is compared to the covariance-based structural equation modeling (cb-sem) approach. table 1.1. comparing partial least square (pls) to covariance-based (cb) approaches of sem criterion pls-sem cb-sem research objective prediction oriented parameter oriented approach variance covariance assumption non-parametric parametric implication optimal for prediction optimal for parameter estimation model complexity large complexity small to moderate complexity sample size minimum of 30-100 based on power analysis software smartpls, warppls, pls-graph amos, lisrel, mplus as shown above, the approach of pls-sem is to explain the variance, similar to basic regression analysis and therefore, it is essential to note that pls-sem also provides coefficient of determination (r2) values besides indicating the significant relationships that lie among the construct which are able to denote on the performance of the model such as how far the model is performing. among the advantages possessed by pls-sem if to be compared with the basic regression is its ability in handling various independent variable at one time even when it displays multicollinearity ([3]; [6]). besides, some of the assumptions on regression are also shared by pls-sem, i.e. the ones which concern on the outliers as well as nonlinear data relationships. lastly, the pls-sem’s characteristics which include the minimal demands when it comes to measurement scales, sample size and also residual distributions allow it to be utilised int. j. anal. appl. 18 (5) (2020) 878 in the circumstances either the relationships exist or not, and it can also be utilised in suggesting the propositions for the later testing ([6]; [7]). pls-sem involves a two-step approach which revolves around the estimation of the measurement model right before an analysis is done for the structural model. it is also known as an iterative algorithm which has the ability in separately solving out the blocks of the measurement model and later estimates the path coefficients in the structural model. this paper focuses on the differences and assessment of the formative and reflective measurement models. 2. measurement model a measurement model is a component of the general model where latent constructs are prescribed. measurement models, as discussed in the psychological, sociological and management literature identify various instances where reflective and formative measures differ. the most common distinction between reflective and formative measures has to do with the relationship that is present when it comes to the construct and its measurement items ([11]; [12]; [13]; [14]; [15]). commonly, the reflective modes act as the indicator of causality from constructs to measurement items, and it is the other way around for formative modes. figure 1 below exhibits the differences between formative and reflective construct. figure 2.1. differences between formative and reflective construct in term of formative construct, the latent variable is considered a consequence of its respective indicators ([16]) and because the latent variable is defined by its indicators, int. j. anal. appl. 18 (5) (2020) 879 changing/replacing a formative indicator will alter the meaning of the latent variable ([17]). alternatively, in reflective construct, indicators are deemed as the consequences of the latent variable to which they belong, which means items are manifested by the construct ([2]; [18]; [19]). the use of reflective indicators is interchangeable, and to a certain extent, it can even be removed. another critical differentiation between the two models is whether the measurement items possess any correlation. in reference to the formative model, all measurement items are not necessary to appear having a high correlation, while the reflective model stipulates that there is a need for all measurement items to be highly correlated. 3. construct classification as the research data was collected prior to the model specification, the next step involved classifying the constructs as either formative or reflective. while some scholars argue that no construct is inherently reflective or formative, others suggest that a construct must be either reflective or formative based on its theoretical considerations ([14]; [15]; [20]). the rationale of these theoretical considerations is to develop items that measure the actual construct. the choice of measurement would affect the content, parsimony and criterion validity of the measurement model ([13]). other researchers suggested that usage of incorrect measurement model will undermine the content validity of constructs, misrepresents the structural relationships between them, and ultimately lowers the usefulness of the research findings ([14]; [15]; [20]). after reviewing the works of literature, this study found that three criteria: (i) the nature of the construct, (ii) the direction of causality between the indicators and the latent construct; and (iii) indicators characteristics that are used to measure the construct, is applicable in classifying the research constructs into formative or reflective measurement model. 3.1. nature of the construct. based on a reflective model, the latent construct is present (in an absolute sense) independently of the measures ([21]). this aspect is in line with many businesses and related methodological studies that use reflective measurement ([2]; [22]). alternatively, for the formative model, the latent construct is dependent based on constructive, operational or int. j. anal. appl. 18 (5) (2020) 880 instrumental interpretation ([23], [24]). it is vital to highlight that due to the fact that formative indicators define the latent variable, they are not interchangeable. however, it was found that only there were only limited examples of formative models included in the literature of social science, specifically with regards to secondary data ([14]; [25]; [26]). they argued that secondary data tends to be very descriptive, may be challenging to obtain, and most of the time, it may not measure all the variables that are important to the research construct. despite the limitations, they also supported that secondary data allows the researchers in testing complex hypotheses which involve multiple variables as well as large samples that act in facilitating the use of statistical techniques (e.g., structural equation modelling). 3.2. the direction of causality. the direction of causality between the construct and the indicators is the second consideration in deciding whether the measurement model is reflective or formative ([13]; [16]). reflective models assume that the flow of causality flows is from construct to the indicators. hence when there is a change in the construct, there will be a change in the indicators as well. meanwhile, the reverse is true for formative models, where causality flows from the indicators to the particular construct. when there is a change in the indicators, it will result in a change in the construct under study. also, it is essential to note that different causal direction can contribute towards significant implications in terms of the measurement error as well as the model estimation ([13]). formative and reflective models were also found to have the main difference of which is basically on the treatment of measurement error, which then may affect the model estimation result. 3.3. characteristics of indicators. finally, in confirming whether the measurement model is reflective or formative, the differences with regards to specific indicator characteristics need to be analysed. for a reflective model, the content validity of the construct is not triggered by the inclusion or exclusion of one or even more indicators outside a domain. the indicators are interchangeable as they shared a common theme ([4]; [27]; [28]). however, in the case of formative models, types of indicators representing the construct as well as the number of constructs affect int. j. anal. appl. 18 (5) (2020) 881 the constructs itself, and thus, the conceptual meaning of the construct can change there is an addition or removal of an indicator. in this case, if the indicators represent the model conceptually, they are still considered adequate in the viewpoint of the empirical prediction. based on the above measurement and theoretical considerations, research constructs can be classified to either formative or reflective measurement model ([13]; [14]; [15]). table 3.1 describes the justification process used to determine which constructs were reflective and which ones were formative. table 3.1. formative and reflective construct assessments the nature of the construct direction of causality characteristics of indicators verdict latent construct is dependent upon a constructive, operational or instrumental interpretation causality flows from the indicators to the construct; a change in the indicators results in a change in the construct understudy construct is sensitive to the number and types of indicators representing it formative latent construct exists (in an absolute sense) independently of the measures causality flows from the construct to the indicators; a change in the construct causes a change in the indicators construct is not sensitive; does not materially alter the content validity of the construct reflective 4. reflective measurement model assessment reflective measurement specifies that a latent or unobservable concept causes variation in a set of observable indicators, which therefore can be used to gain an indirect measurement of the concept. in order to examine the reflective measurement models, four parameters were examined: (i) internal consistency reliability, (ii) indicator reliability, (iii) convergent validity and (iv) discriminant validity ([3]; [9]; [29]). the criteria for the reflective measurement model fitting are presented below in table 4.1. int. j. anal. appl. 18 (5) (2020) 882 table 4.1. reflective outer model assessments criterion recommendations/rules of thumb / thresholds sources internal consistency reliability do not use cronbach’s alpha; composite reliability > 0.70 bagozzi and yi (1988) [18] indicator reliability standardized indicator loadings > 0.70; in exploratory studies, loadings of 0.40 are acceptable hulland (1999) [28] convergent validity average variance extracted (ave) > 0.50 bagozzi and yi (1988) [18] discriminant validity fornell-larcker criterion each construct’s ave should be higher than its squared correlation with any other construct fornell and larcker (1981) [30] cross loadings each indicator should load highest on the construct it is intended to measure chin (1999) [7] heterotrait-monotrait ratio of correlations (htmt) no discriminant validity problems (htmt>0.85 criterions) henseler et al. (2009) [8] a threshold value of 0.70 was applied in assessing the internal consistency of the model specifically in the effort to determine the item’s minimum factor loadings (18). measurements with loadings lesser than 0.70 were removed in cases where failure to eliminate them may contribute towards the increase of composite reliability that is greater than the threshold value ([3]). meanwhile, the convergent validity was determined using the widely accepted method ‘average variance extracted (ave)’ ([3]). the ave value indicates that; on average, each construct can explain more than half of the variance of its measuring items and must be more than 0.50 ([14]; [18]). fornell and larcker ([30]) criteria were used in examining the discriminant validity at the construct-level, whereas the discriminant validity at the item level was examined using chin’s criteria ([7]). implementing this two-fold technique in testing for the discriminant validity is supported by various researchers, as they suggested that the variance extracted estimates should be greater than the squared correlation estimate ([8]; [9]; [13]; [18]). lately, many researchers proposed the heterotrait-monotrait ratio of correlations (htmt) to assess discriminant validity in pls-sem ([31]). the htmt can achieve higher specificity and sensitivity rates compared to the int. j. anal. appl. 18 (5) (2020) 883 cross-loadings and fornell-lacker criterion. from the htmt results, if the htmt values are less than 0.85, it indicated no discriminant validity problems and implied that the htmt criterion did not detect the collinearity problems among the latent constructs ([31]). 5. formative measurement model assessment formative measurement specifies that the observable indicators are considered to cause the latent construct. thus, formative constructs should be assessed based on the statistical significance and size of the indicator weights as well as collinearity among indicators ([6]). for the evaluation of the formative measurement model, this study adopted the guidelines outlined by ([6]; [20]; [32]). specifically, three parameters should be examined: (i) multicollinearity; (ii) construct validity; and (iii) indicator reliability. the criteria for the formative measurement model fitting are presented below in table 5.1. table 5.1. formative outer model assessments criterion recommendations/rules of thumb /thresholds sources multicollinearity variance inflation factor (vif) is used to determine whether there was high correlation between the formative indicators hair et al. (2017) [6] construct validity estimate the indicator weights to measure the contribution of each formative indicator to the variance of the latent variable. petter et al. (2007) [33] indicator reliability calculates the outer loadings of the formative construct; if the item loadings are relatively high (>.50), the indicator should be retained hair et al. (2012) [9] different to the reflective indicator by which moderate multicollinearity between construct indicators is desirable, the balance of the model can be triggered by the high multicollinearity present in the formative construct ([15]; [32]; [33]). multicollinearity occurs when two or more predictors in the model are correlated and provide redundant information about the response. in determining whether there was a too high correlation between the formative indicators, many studies used the variance inflation factor (vif). vif is the reciprocal of the tolerance value; small vif values indicate low correlation among variables and vice-versa. int. j. anal. appl. 18 (5) (2020) 884 the higher the value, the greater the correlation of the variable with other variables ([34]). for formative measures, there is a rule of thumb that clarifies; if vif values are greater than 5, thus it represents high multicollinearity ([20]). recently, other researchers recommended that multicollinearity exists if the vif value is higher than 10 ([6]; [31]). next, it is important to note that the cronbach’s alpha and the composite reliability will not be estimated, as much as formative indicators are not internally consistent ([7]; [35]). moreover, the aves were not calculated, given the assumption that formative indicators demonstrate convergent validity ([35]). therefore, in order to test for the construct validity, the estimation of the indicator weights in measuring the contribution of every each of the formative indicators to the variance of the latent variable should be applied. the item weights indicate whether or not an indicator can explain a significant portion of the variance of a formative construct ([36]; [37]). this step is in line with other researchers who suggested that indicator weights can be used to test the construct validity ([12]; [13]; [15]). lastly, researchers should also look at indicator reliability. the outer loadings of a formative construct should be tested to confirm the indicator reliability. when an indicator’s weight is not significant, but the corresponding item loadings are relatively high (>.70), the indicator should be retained, as been proposed by researchers ([9]; [29]). this will ensure that measurements are prioritised according to their reliability with regard to making estimations ([15]; [38]). 6. proposition in any study, it is vital to acknowledge the different types of measurement models and understand the criteria involved when it comes to determining the measurement models’ mode. the formulation of the measurement model depends on the direction of the relationships that are present in reference to latent variables as well as the corresponding manifest variables. in general, there are distinctive types of measurement model that are available namely, (i) the reflective model or also called outward-directed model; and (ii) the formative model or also called the directed model. the general distinction between measurement model for reflective and formative measures must be distinguished. it is important to note that reflective measures generally represent the causality from the constructs to the specified measurement items, and meanwhile, int. j. anal. appl. 18 (5) (2020) 885 the formative measures consider the opposite. this study found that each measurement model must be tested by assessing the validity and reliability of the items and constructs used in each (reflective and formative) model. these specific steps must be taken to ensure that only reliable and valid constructs and measures are used, prior to assessing the nature of the relationships proposed by the research hypotheses. this study, therefore, suggests the following guidelines for researchers in assessing the differences between reflective and formative constructs as per table 6.1 below. table 6.1. criteria used to determine the mode of measurement models reflective model formative model using existing latent construct and involves a realist interpretation of a latent construct latent construct is formed, constructivist, operationalist, or instrumentalist interpretations causality from constructs to indicators; causality from indicators to constructs indicators are manifested by the construct; they are interchangeable and share a common theme; dropping an indicator does not alter the meaning of the construct indicators define the construct; they need not share a common theme; they are not interchangeable; dropping an indicator may alter the meaning of the construct measures have a high correlation, as they are all dependent on the same unobservable variable measures have positive, negative, low or zero correlation with one another measures have similar sign and significance of relationships with the antecedents/ consequences as the construct measures may not have the similar significance of relationships with the antecedents/consequences as the construct taking measurement error into account at the measurement level; error terms in indicators can be identified taking measurement error into account at the construct level; error term cannot be identified if the formative measurement model is estimated in isolation assessments: internal consistency reliability, indicator reliability, convergent validity, discriminant validity fornell-larcker criterion, cross loadings, and heterotrait-monotrait ratio of correlations (htmt) assessments: multicollinearity, construct validity, and indicator reliability int. j. anal. appl. 18 (5) (2020) 886 as per table 6.1, the most typical distinction between reflective and formative measures has to do with the relationship between the construct and its measurement items. the reflective modes indicate causality from constructs to measurement items, whereas formative modes reflect the opposite. in formative measurement models, the latent variable is considered a consequence of its respective indicators and because the latent variable is defined by its indicators, changing/replacing a formative indicator will alter the meaning of the latent variable. alternatively, in reflective measurement models, indicators are regarded as the consequences of the latent variable to which they belong, which means items are manifested by the construct. the reflective indicators can be used interchangeably and can, to a certain extent, even be discarded. another critical differentiation between the two models has to do with whether or not the measurement items are correlated. in the formative model, it is not essential for all measurement items to be highly correlated, while the reflective model stipulates that all measurement items need to have a high level of correlation. 7. conclusion this paper proposes a set of guidelines in classifying the formative and reflective constructs and the steps in assessing the formative and reflective measurement model. in addition, this paper confirms that there are apparent differences between reflective and formative constructs and the construct identification and validation depends on the type of construct specified by the researcher. this paper proposes that quantitative researchers that the decision whether to use a formative or reflective indicator should be based on the theoretical grounds. misspecification of measurement models may affect research outcome or mislead future research. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. acknowledgement: this research work was supported by the universiti teknologi mara malaysia. int. j. anal. appl. 18 (5) (2020) 887 references [1] j.f. hair, ed., a primer on partial least squares structural equations modeling (pls-sem), sage, los angeles, 2014. [2] r.b. kline, principles and practice of structural equation modeling, guilford press, new york, 2015. [3] j.f. hair, c.m. ringle, sarstedt m. pls-sem: indeed a silver bullet. j. market. theory practice. 19(2)(2011), 139-152. [4] x. wang, l.m. jessup, p.f. clay. measurement model in entrepreneurship and small business research: a ten year review. int. entrepren. manage. j. 11(1)( 2015), 183-212. [5] v.e. vinzi, c.n. lauro, s. amato, pls typological regression: algorithmic, classification and validation issues, in: h.-h. bock, et al. (eds.), new developments in classification and data analysis, springer-verlag, berlin/heidelberg, 2005: pp. 133–140. [6] j.f. hair jr., l.m. matthews, r.l. matthews, m. sarstedt, pls-sem or cb-sem: updated guidelines on which method to use. int. j. multivar. data anal. 1(2)( 2017), 107-123. [7] w.w. chin, p.r. newsted, structural equation modeling analysis with small samples using partial least squares. stat. strat. small sample res. 1(1)( 1999), 307-341. [8] r.r. sinkovics, ed., new challenges to international marketing, emarald, london, 2009. [9] j.f. hair, m. sarstedt, c.m. ringle, j.a. mena. an assessment of the use of partial least squares structural equation modeling in marketing research. j. acad. market. sci. 40(3)( 2012), 414-433. [10] qureshi i, compeau d. assessing between-group differences in information systems research: a comparison of covariance-and component-based sem. mis quart. 33(2009), 197-214. [11] a. diamantopoulos, p. riefler, k.p. roth. advancing formative measurement models. j. bus. res. 61(12)(2008), 1203-1218. [12] a. diamantopoulos, incorporating formative measures into covariance-based structural equation models. mis quart. 35(2011), 335-358. [13] k.a. bollen, a. diamantopoulos, notes on measurement theory for causal-formative indicators: a reply to hardin. psychol. meth. 22(2017), 605–608. [14] t. coltman, t.m. devinney, d.f. midgley, s. venaik. formative versus reflective measurement models: two applications of formative measurement. j. bus. res. 61(12)( 2008), 1250-62. [15] e.a. khan, m.n.a. dewan, m.m.h. chowdhury. reflective or formative measurement model of sustainability factor? a three industry comparison. corp. owner. control. 13(2)( 2016), 83-92. [16] k. bollen, r. lennox. conventional wisdom on measurement: a structural equation perspective. psychol. bull. 110(2)( 1991), 305–314. int. j. anal. appl. 18 (5) (2020) 888 [17] a. diamantopoulos, h.m. winklhofer. index construction with formative indicators: an alternative to scale development. j. market. res. 38(2)(2001), 269-277. [18] r.p. bagozzi, y. yi. on the evaluation of structural equation models. j. acad. market. sci. 16(1)(1988), 74-94. [19] c.b. jarvis, s.b. mackenzie, p.m. podsakoff. a critical review of construct indicators and measurement model misspecification in marketing and consumer research. j. consumer res. 30(2)(2003), 199-218. [20] a. diamantopoulos, j.a. siguaw. formative versus reflective indicators in organisational measure development: a comparison and empirical illustration. br. j. manage. 17(4)(2006), 263-282. [21] d. borsboom, a.o.j. cramer, r.a. kievit, a.z. scholten, s. franić. the end of construct validity. in r. w. lissitz (ed.), the concept of validity: revisions, new directions, and applications (p. 135–170). iap information age publishing, 2009. [22] r.g. netemeyer, w.o. bearden, s. sharma, scaling procedures: issues and applications, sage publications, thousand oaks, calif, 2003. [23] j.b. wilcox, r.d. howell, e. breivik, questions about formative measurement. j. bus. res. 61(12)(2008), 1219-1228. [24] d. borsboom, g.j. mellenbergh, j. van heerden, the theoretical status of latent variables., psychol. rev. 110(2003), 203–219. [25] d.r. allen, t. finlayson, a. abdul-quader, a. lansky. the role of formative research in the national hiv behavioral surveillance system. public health rep. 124(1)(2009), 26-33. [26] j.r. macnamara. research in public relations: a review of the use of evaluation and formative research. asia-pac. public relat. j. 1(1992), 2-11. [27] j.c. nunnally, i.h. bernstein. psychological theory. mcgraw-hill, new york, 1994. [28] j. hulland, use of partial least squares (pls) in strategic management research: a review of four recent studies. strat. manage. j. 20(2)(1999), 195-204. [29] j.f. hair, j.j. risher, m. sarstedt, c.m. ringle. when to use and how to report the results of pls-sem. eur. bus. rev. 31(1)(2019), 2-24. [30] c. fornell, d.f. larcker. structural equation models with unobservable variables and measurement error: algebra and statistics. sage publications sage ca: los angeles, ca; 1981. [31] j. henseler, c.m. ringle, m. sarstedt. a new criterion for assessing discriminant validity in variancebased structural equation modeling. j. acad. market. sci. 43(1)(2015), 115-135. [32] r.t. cenfetelli, g. bassellier. interpretation of formative measurement in information systems research. mis quart. 33(2009), 689-707. int. j. anal. appl. 18 (5) (2020) 889 [33] s. petter, d. straub, a. rai, specifying formative constructs in information systems research. mis quart. 31(2007), 623-656. [34] a.h. westlund, m. källström, j. parmler. sem-based customer satisfaction measurement: on multicollinearity and robust pls estimation. total qual. manage. 19(7-8)(2008), 855-869. [35] w.w. chin. the partial least squares approach to structural equation modeling. mod. meth. bus. res. 295(2)(1998), 295-336. [36] g.r. franke, k.j. preacher, e.e. rigdon. proportional structural effects of formative indicators. j. bus. res. 61(12)(2008), 1229-1237. [37] n. roberts, j. thatcher, conceptualizing and testing formative constructs: tutorial and annotated example, sigmis database. 40(2009), 9–39. [38] k.a. bollen, a. diamantopoulos. in defense of causal-formative indicators: a minority report. psychol. meth. 22(3)(2017), 581-596. int. j. anal. appl. (2022), 20:10 well-posedness and stability for a system of klein-gordon equations naaima latioui1,∗, amar guesmia1, amar ouaoua2 1laboratory of applied mathematics and history and didactics of mathematics "lamahis", department of mathematics, university 20 août 1955 skikda, algeria 2laboratory of applied mathematics and history and didactics of mathematics "lamahis", department of technologic, university 20 août 1955 skikda, algeria ∗corresponding author: loubnalatioui@gmail.com, n.latioui@univ-skikda.dz abstract. in this paper, we study the weak existence of solution for a non-linear hyperbolic coupled system of klein-gordon equations with memory and source terms using the faedo-galerkin method techniques and compactness results, we have demonstrated the uniqueness of the solution by using the classical technique. in addition, we show that the solution remains stable over time. the reaction of the proper lyapunov function is the primary tool of the proof. 1. introduction in this paper, we consider a non-linear hyperbolic system of klein-gordon equations, defined as the following  utt − ∆u −αut + k ∗ ∆u −div(|v| 2∇u) + u|∇v|2 = 0 in ω × (0,t ), vtt − ∆v −βvt + l ∗ ∆v −div(|u|2∇v) + v|∇u|2 = 0 in ω × (0,t ), (1.1) with boundary conditions u(x,t) = v(x,t) = 0 on γ × (0,t ), (1.2) ut(x,t) = vt(x,t) = 0 on γ × (0,t ), (1.3) received: dec. 3, 2021. 2010 mathematics subject classification. 81q05, 37l45, 93d20. key words and phrases. klein-gordon system; faedo-galerkin method; lyapunov function. https://doi.org/10.28924/2291-8639-20-2022-10 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-10 2 int. j. anal. appl. (2022), 20:10 and initial conditions u(x, 0) = u0(x), v(x, 0) = v0(x) on ω, (1.4) ut(x, 0) = u1(x), vt(x, 0) = v1(x) on ω. (1.5) where ω is a bounded domain of rn (n ≥ 1) with smooth boundary γ and let t > 0, α and β are non-positive constants, and (n∗w)(t) = ∫ t 0 n(t − s)w(s)ds. (1.6) several authors have studied the klein-gordon non-linear system among them medeiros & m. miranda [8] considered the non-linear system  ∂2u ∂t2 − ∆u + u −|v|ρ|u|ρu = f1, ∂2v ∂t2 − ∆v + v −|u|ρ|v|ρv = f2, (1.7) they prove the existence and uniqueness of weak global solutions in ω×[0,t ], where ρ is a real number meeting a specific condition and ω is any domain of rn. d. andrade & a. mognon [2], considered the non-linear system with memory term  utt − ∆u + f (u,v) + k ∗ ∆u = 0, vtt − ∆v + g(u,v) + l ∗ ∆v = 0, (1.8) for x ∈ ω and t > 0 where f (u,v) = |u|ρ−2u|v|ρ, and g(u,v) = |v|ρ−2v|u|ρ, with ρ > 0 if n = 1, 2 and 1 < ρ ≤ n− 1 n− 2 if n ≥ 3, they use the argument from komornik and zuazua [6] to prove the existence of weak and strong solutions in ω × (0,t ) given initial and boundary conditions. a. t. louredo & m. m. miranda [7], considered the non-linear system  u ′′ − ∆u + αv2u = 0, v ′′ − ∆v + αu2v = 0, (1.9) with the nonlinear boundary conditions, ∂u ∂ν + h1(.,u ′) = 0 on γ1 × (0,∞), ∂v ∂ν + h2(.,v ′) = 0 on γ1 × (0,∞), and boundary conditions u = v = 0 on (γ/γ1)×(0,∞), where ω is a bounded open set of rn (n ≤ 3), α > 0 a real number, γ1 is a subset of the border γ of ω and hi a real function defined on γ1×(0,∞). they use the galerkin approach to demonstrate the existence of global solutions. int. j. anal. appl. (2022), 20:10 3 k. zennir & a. guesmia [10], considered the non-linear κth-order with non-linear sources and memory terms   u′′1 + (−1) κ∆κu1 + m 2 1u1 + α1(t) ∫ t 0 g1(t − s)∆κu1(x,s)ds + |u′1| r−2|u′1| = |u1|p−2u1|u2|p, u′′2 + (−1) κ∆κu2 + m 2 2u2 + α2(t) ∫ t 0 g2(t − s)∆κu2(x,s)ds + |u′2| r−2|u′2| = |u2|p−2u2|u1|p, (1.10) using the potential well method, they verify the existence of global solutions in the a bounded domain ω of rn, where mi = 1, 2 are non-negative constants, r,p ≥ 2,κ ≥ 1. c. l. frota & a. vicente [5], studied the non-linear system of klien-gordon with acoustic boundary conditions   u ′′ − ∆u + |v|ρ+2|uρ|u = f1 in ω × (0,t ), v ′′ − ∆v + |u|ρ+2|vρ|v = f1 in ω × (0,t ), (1.11) they demonstrate the existence of both global and weak solutions, as well as their uniqueness. our objective is to prove that the problem (1.1)-(1.5) has a weak and unique solution such that the kernel terms k, l have some hypothesis as well as using some ideas from articles ( [2]) and ( [9]). 2. preliminaries let ω be a domain in rn with smooth boundary γ let t > 0. the inner product and norm in l2(ω) are denoted by 〈u,v〉 = ∫ ω u(x)v(x)dx, |u|2 = (∫ ω |u(x)|2dx )1 2 . (2.1) the norm in h10 (ω) is denoted by ‖u‖h10 (ω) = (∫ ω |∇u|2dx )1 2 . (2.2) we assume that k, l: r+ −→r+ are non-increasing differentiable functions satisfying : l1 = ( 1 − ∫ t 0 l(s)ds ) > 0 and k1 = ( 1 − ∫ t 0 k(s)ds ) > 0, (2.3) and k′(t) ≤−k(t), l′(t) ≤−l(t). (2.4) if w = w(t,x) is a function in l2(0.t ; h10 (ω)) and k is continuous we put: (k ◦w)(t) = ∫ t 0 k(t − s)|∇w(t) −∇w(s)|22ds. 4 int. j. anal. appl. (2022), 20:10 lemma 2.1. [2] w ∈ c1((0,t ); h10 (ω)), k ∈ c 1(0,∞) ∫ t 0 k(t − s) 〈 ∇w(s)∇w ′(t) 〉 ds = − 1 2 d dt (k ◦w)(t) + 1 2 d dt (∫ t 0 k(s)ds ) |∇w(t)|22 (2.5) +(k′ ◦w)(t) −k(t)|∇w(t)|22. lemma 2.2. [9] (young’s inequality) let a, b ≥ 0 and 1 q + 1 p = 1 for 1 < p,q < +∞, then one has the inequality ab ≤ δaq + c(δ)bp, where δ > 0 is an arbitrary constant, and c(δ) is a positive constant depending on δ. lemma 2.3. [1] (sobolev-poincaré inequality) let s be a number with 2 ≤ s < +∞ if n ≤ 2 and 2 ≤ s ≤ 2n n−2 if n > 2. then there is a constant c depending on ω and s such that ‖u‖s ≤ c‖∇u‖2, u ∈ h10. theorem 2.1. let u0, v0 ∈ l2(ω) and u1, v1 ∈ l1(ω). then, under assumptions on two functions k and l, the problem (1.1)-(1.5) has a local solution (u(x,t),v(x,t)) such that u,v ∈ l∞(0,t ; h10 (ω)) ∩l ∞(0,t ; l2(ω)), (2.6) ut,vt ∈ l∞(0,t ; l2(ω)). (2.7) theorem 2.2. let u,v :→ l2(ω) be functions in the class (2.6) and (2.7) satisfying from (1.1) to (1.5), with u,v ∈ h2(ω). then the solution (u,v) obtained in theorem (2.1) is unique. 3. global existence step 1: approximate solution. using the faedo-galerkin process, we will determine the existence of a local solution to the problem (1.1)-(1.5) in this section. let {wi} be a basis for both h2(ω)∩h10 (ω) and l2(ω) for each positive integer m we put v = span{w1,w2, . . . ,wm}. we look for an approximate solution in the form um(t) = m∑ i=1 uimwi and vm(t) = m∑ i=1 vimwi, int. j. anal. appl. (2022), 20:10 5 satisfying the approximate problem∫ ω {umtt − ∆u m −αumt }widx − ∫ t 0 k(t − s)〈∇um(s),∇wi〉ds (3.1) + ∫ ω |vm|2∇um∇widx + ∫ ω um|∇vm|widx = 0,∫ ω {vmtt − ∆v m −βvmt }widx − ∫ t 0 l(t − s)〈∇vm(s),∇wi〉ds (3.2) + ∫ ω vm|∇um|widx + ∫ ω |um|2∇vm∇wi = 0, with initial conditions satisfying u m(0) = um0 , ∑m i=1 aimwi = u m 0 → u0, v m(0) = vm0 , ∑m i=1 bimwi = v m 0 → v0 in l 2(ω), umt (0) = u m 1 , ∑m i=1 a 1 imwi = u m 1 → u1, v m t (0) = v m 1 , ∑m i=1 b 1 imwi = v m 1 → v1 in l 1(ω). (3.3) since the vectors {wi} are linearly independent, this means det(wi,wj) 6= 0, the latter ensuring that the problem admits a local solution (um(t),vm(t)) in the interval [0,tm]. step 2: a priori estimate. our system’s energy functional e(t) is given by 2e(t) = |umt | 2 l2(ω) + |v m t | 2 l2(ω) + (k ◦u m)(t) + (l ◦vm)(t) (3.4) + ( 1 − ∫ t 0 k(s)ds ) |∇um|22 + ( 1 − ∫ t 0 l(s)ds ) |∇vm|22 + |v m∇um|22 + |u m∇vm|22. after that, we multiply (3.1) by ut, (3.2) by vt, and use identity (2.5) to get d dt e(t) = (k′ ◦um)(t) + (l′ ◦vm)(t) −k(t)|∇um|22 − l(t)|∇v m|22 + α|u m t | 2 2 + β|v m t | 2 2 ≤ 0. (3.5) we found that d dt e(t) is a non-positive function, this last indicates that e(t) is a non-increasing function, meaning there exists a positive constant c1, independent of t and m such that |umt | 2 2 + |v m t | 2 2 + |∇u m|22 + |∇v m|22 + |u m∇vm|22 + |v m∇um|22 ≤ c1. (3.6) from this estimation, deduce that tm = t. in addition, we get  um,vm is bounded in l ∞ (0,t ; h10 (ω)), um,vm is bounded in l ∞ (0,t ; l2(ω)), umt ,v m t is bounded in l ∞ (0,t ; l2(ω)). (3.7) by the holder inequality, the embedding h10 (ω) ↪→ l 6(ω) and (3.7), we obtain |um|∇vm|2|22 ≤‖u m‖2l6(ω)‖∇v m‖4l6(ω) ≤ c1, (3.8) ||vm|2∇um|22 ≤‖v m‖4l6(ω)‖∇u m‖2l6(ω) ≤ c2 ∀(u m,vm) in h2(ω). (3.9) 6 int. j. anal. appl. (2022), 20:10 therefore (um|∇vm|2) is bounded in l∞(0,t ; l2(ω)), (3.10) (|vm|2∇um) is bounded in l∞(0,t ; l2(ω)). (3.11) analogously (vm|∇um|2) is bounded in l∞(0,t ; l2(ω)), (3.12) (|um|2∇vm) is bounded in l∞(0,t ; l2(ω)). (3.13) step 3: passage to the limit. from (3.7), (3.10), (3.11), (3.12) and (3.13) there exists a subsequence of (um) and a subsequence of (vm), denoted by same symbols, such that  um → u and vm → v weak star in l∞(0,t ; h10 (ω)), um → u and vm → v weak star in l∞(0,t ; l2(ω)), umt → ut and vmt → vt weak star in l∞(0,t ; l2(ω)), um|∇vm|2 → χ1 weak star in l∞(0,t ; l2(ω)), vm|∇um|2 → χ2 weak star in l∞(0,t ; l2(ω)), |vm|2∇um → χ3 weak star in l∞(0,t ; l2(ω)), |um|2∇vm → χ4 weak star in l∞(0,t ; l2(ω)). (3.14) from (3.14) and aubin-lions compactness lemma in ( [3]), we obtain um → u, vm → v strongly in l∞(0,t ; l 2 (ω)), (3.15) since ∇um and ∇vm are bounded, then we have  um|∇vm|2 → u|∇v|2 strongly in l2(0,t ; l2(ω)), vm|∇um|2 → v|∇u|2 strongly in l2(0,t ; l2(ω)), |um|2∇vm →|u|2∇v strongly in l2(0,t ; l2(ω)), |vm|2∇um →|v|2∇u strongly in l2(0,t ; l2(ω)). (3.16) then, there exists a subsequences of um and vm, which we will denote by um,vm respectively, such that   um|∇vm|2 → u|∇v|2 almoust everywhere in (0,t ) × ω, vm|∇um|2 → v|∇u|2 almoust everywhere in (0,t ) × ω, |um|2∇vm →|u|2∇v almoust everywhere in (0,t ) × ω, |vm|2∇um →|v|2∇u almoust everywhere in (0,t ) × ω. (3.17) int. j. anal. appl. (2022), 20:10 7 from lemma (3.15) in ( [11]) and (3.17) we deduce  um|∇vm|2 → u|∇v|2 weakly in l∞(0,t ; l2(ω)), vm|∇um|2 → v|∇u|2 weakly in l∞(0,t ; l2(ω)), |um|2∇vm →|u|2∇v weakly in l∞(0,t ; l2(ω)), |vm|2∇um →|v|2∇u weakly in l∞(0,t ; l2(ω)). (3.18) by the last formula (3.18) and (3.14) we get χ1 = u|∇v|2, χ2 = v|∇u|2, (3.19) χ3 = |v|2∇u, χ4 = |u|2∇v. taking wi = 1 in (3.1) become (umtt, 1) −α(u m t , 1) + (u m|∇vm|2, 1) = 0 (3.20) |(umtt, 1)| = ∣∣α(umt , 1) − (u|∇vm|2, 1)∣∣ . using the cauchy schwartz inequality, we have ‖umtt‖l1(ω) ≤ |α||u m t |2m 1 2 (ω) + |um|∇vm|2|2m 1 2 (ω), such that, m(ω) is a measure of ω. since, the measure of ω is finite, and (3.14), we obtain ‖umtt‖l1(ω) ≤ c1. (3.21) analogously ‖vmtt‖l1(ω) ≤ c2. (3.22) then  u m tt is bounded in l ∞(0,t ; l1(ω)), vmtt is bounded in l ∞(0,t ; l1(ω)). (3.23) similarly we have  u m tt → utt weakly star in l∞(0,t ; l1(ω)), vmtt → vtt weakly star in l∞(0,t ; l1(ω)). (3.24) from (3.14), (3.24) and lemma (3.1.7) in ( [12]) with b = l2(ω) and b = l1(ω) we get u m 0 → u(0), v m 0 → v(0) weakly star in l 2(ω), um1 → u1(0), v m 1 → v1(0) weakly star in l 1(ω). (3.25) 8 int. j. anal. appl. (2022), 20:10 from (3.25) and (3.3) we get u(0) = u0, v(0) = v0, (3.26) u1(0) = u1, v1(0) = v1. (3.27) setting up m →∞ and passing to the limit in (3.1), (3.2), we obtained ∫ ω {utt − ∆u −αut}widx − ∫ t 0 k(t − s)〈∇u(s),∇wi〉ds (3.28) + ∫ ω |v|2∇u∇widx + ∫ ω u|∇v|widx = 0,∫ ω {vtt − ∆v −βvt}widx − ∫ t 0 l(t − s)〈∇v(s),∇wi〉ds (3.29) + ∫ ω v|∇u|widx + ∫ ω |u|2∇v∇wi = 0. i = 1...,m. since (wi)∞i=1 is a base of h 1 0 (ω), we deduce that (u,v) satisfies (1.1). the proof is complete. lemma 3.1. let u0,v0 ∈ h10 (ω) and u1,v1 ∈ l 2(ω) be given. assume that (2.3) and (2.4) are true. then the problem’s local solution (1.1)-(1.5) is global in time. proof. since the map t → e(t) is a non-increasing function, i.e there exists a positive constant c1, independent of t, such that delete m from these equations c1 ≥ 2e(t) = |ut|2l2(ω) + |vt| 2 l2(ω) + (k ◦u)(t) + (l ◦v)(t) (3.30) + ( 1 − ∫ t 0 k(s)ds ) |∇u|22 + ( 1 − ∫ t 0 l(s)ds ) |∇v|22 + |v∇u| 2 2 + |u∇v| 2 2 > 0, which give c1 ≥ 2e(t) ≥ |ut|2l2(ω) + |vt| 2 l2(ω) + ( 1 − ∫ t 0 k(s)ds ) |∇u|22 + ( 1 − ∫ t 0 l(s)ds ) |∇v|22 (3.31) + |v∇u|22 + |u∇v| 2 2 > 0, consequently, ∀t ∈ [0,t ], we have |ut|2l2(ω) + |vt| 2 l2(ω) + |∇u|22 + |∇v| 2 2 + |v∇u| 2 2 + |u∇v| 2 2 ≤ c1. this deduces that the solution. int. j. anal. appl. (2022), 20:10 9 4. uniqueness let (u,v) and (u1,v1) two solutions of (1.1), we assume that u = u −u1 and v = v −v1 satisfy utt − ∆u −αut + k ∗ ∆u −div(|v|2∇u −|v1|2∇u1) + (u|∇v|2 −u1|∇v1|2) = 0 in ω × (0,t ), (4.1) vtt − ∆v −βvt + k ∗ ∆v −div(|u|2∇v −|u1|2∇v1) + (v|∇u|2 −v1|∇u1|2) = 0 in ω × (0,t ), (4.2) with u(0) = v (0) = 0 ut(0) = vt(0) = 0. (4.3) let as put 2e2(t) = |ut|22 + |vt| 2 2 + (k ◦u)(t) + (l ◦v )(t) (4.4) + ( 1 − ∫ t 0 k(s)ds ) |∇u|22 + ( 1 − ∫ t 0 l(s)ds ) |∇v |22. multiplying (4.1) by ut(t) and (4.2) by (vt(t)) and summing up the product result we have d dt e2(t) ≤ ∫ ω div(|v|2∇u −|v1|2∇u1)ut − (u|∇v|2 −u1|∇v1|2)utdx (4.5) + ∫ ω div(|u|2∇v −|u1|2∇v1)vt − (v|∇u|2 −v1|∇u1|2)vtdx, d dt e2(t) ≤ ∫ ω ∣∣|v|2∇u −|v1|2∇u1∣∣|∇ut| + ∣∣u|∇v|2 −u1|∇v1|2∣∣|ut|dx (4.6) + ∫ ω ∣∣|u|2∇v −|u1|2∇v1∣∣|∇vt| + ∣∣v|∇u|2 −v1|∇u1|2∣∣|vt|dx, ∫ ω (|v|2∇u −|v1|2∇u1)∇ut + (|u|∇v|2 −u1|∇v1|2)utdx (4.7) = ∫ ω |v|2∇u∇ut + ∇u1 [ |v|2 −|v1|2 ] ∇ut + ut|∇v|2u + u1 [ |∇v|2 −|∇v1|2 ] dx. from the mean value theorem, it follows that∫ ω ∣∣|v|2∇u −|v1|2∇u1∣∣|∇ut| + ∣∣u|∇v|2 −u1|∇v1|2∣∣|ut|dx ≤ ∫ ω |v|2|∇u||∇ut| + 2|∇u1|2 [|v| + |v1|] |∇ut||v | + |∇v|2|ut||u|. working in the same way as in argument of lemma (2.2) in ( [2]) there exists c > 0 such that∫ ω ∣∣|v|2∇u −|v1|2∇u1∣∣|∇ut| + ∣∣|u|∇v|2 −u1|∇v1|2∣∣|ut|dx ≤ c {|∇u|22 + |∇v |22 + |ut|22} . analogously we have∫ ω ∣∣|u|2∇v −|u1|2∇v1∣∣|∇vt| + ∣∣|v|∇u|2 −v1|∇u1|2∣∣|vt|dx ≤ c {|∇v |22 + |∇u|22 + |vt|22} , 10 int. j. anal. appl. (2022), 20:10 and from (4.6) we have d dt e2(t) ≤ c { |∇u|22 + |∇v | 2 2 + |ut| 2 2 + |vt| 2 2 } (4.8) d dt e2(t) ≤ ce2(t). (4.9) then, by using gronwall’s lemma (1.3) in ( [4]) we get |∇u|22 = |∇v | 2 2 = |ut| 2 2 = |vt| 2 2 = 0. (4.10) this proves the uniqueness of the solution. 5. stability theorem 5.1. let u0,v0 ∈ h10 (ω) and u1,v1 ∈ l 2(ω) be given. assume that (2.3) and (2.4) hold. then there exists two positive constants µ1 and µ2 independent of t such that 0 < e(t) ≤ µ1e −µ2t,∀t ≥ 0. proof. we define the function of laypunov, for � > 0 as follows l(t) = e(t) + � ∫ ω utu + vtvdx. (5.1) we prove that l(t) and e(t) are equivalent, meaning that there exist two positive constants n and m depending on � such that for t ≥ 0 ne(t) ≤ l(t) ≤ me(t). (5.2) from the lemma (2.2), we have l(t) ≤ e(t) + � [ 1 2δ |ut|22 + δ|u| 2 2 ] + � [ 1 2δ |vt|22 + δ|v| 2 2 ] . by using the poincaré inequality, we get l(t) ≤ e(t) + � [ 1 2δ |ut|22 + δc1|∇u| 2 2 ] + � [ 1 2δ |vt|22 + δc2|∇v| 2 2 ] . from (3.31) we have l(t) ≤ e(t) + � [ 1 δ e(t) + 2δ c1 k1 e(t) ] + � [ 1 δ e(t) + 2δ c2 l1 e(t) ] l(t) ≤ e(t) + 2� 1 δ e(t) + 2�δ c1 k1 e(t) + 2�δ c2 l1 e(t) l(t) ≤ me(t) such that m = 1 + 2� 1 δ + 2�δ c1 k1 + 2�δ c2 l1 . int. j. anal. appl. (2022), 20:10 11 on the other hand, we have l(t) ≥ e(t) − � [ 1 2δ |ut|22 + δ|u| 2 2 ] − � [ 1 2δ |vt|22 + δ|v| 2 2 ] ≥ e(t) − � [ 1 2δ |ut|22 + δc1|∇u| 2 2 ] − � [ 1 2δ |vt|22 + δc2|∇v| 2 2 ] ≥ e(t) − � [ 1 δ e(t) + 2δ c1 k1 e(t) ] − � [ 1 δ e(t) + 2δ c2 l1 e(t) ] , l(t) ≥ ne(t) such that n = 1 − 2� 1 δ − 2�δ c1 k1 − 2�δ c2 l1 . now we have d dt l(t) = d dt e(t) + � ∫ ω [ u2t + uttu + v 2 t + vttv ] dx (5.3) � ∫ ω uttudx = � ∫ ω [ u.∆u + αuut −u.k ∗ ∆u + u.div(|v|2∇u) −u|∇v|2.u ] dx ≤ � [ −|∇u|22 + α 1 2δ |ut|22 + αδ|u| 2 2 −|v∇u| 2 2 −|u∇v| 2 2 + ∫ ω ∇u ∫ t 0 k(t − s)∇u(s)dsdx ] (5.4) ≤ � [ −|∇u|22 + α 1 2δ |ut|22 + αc1δ|∇u| 2 2 −|v∇u| 2 2 −|u∇v| 2 2 + ∫ ω ∇u ∫ t 0 k(t − s)∇u(s)dsdx ] . analogous � ∫ ω vttvdx = � ∫ ω [ v.∆v + βvvt −v.k ∗ ∆v + v.div(|u|2∇v) −v|∇u|2.v ] dx (5.5) ≤ � [ −|∇v|22 + β 1 2δ |vt|22 + βc1δ|∇v| 2 2 −|u∇v| 2 2 −|v∇u| 2 2 + ∫ ω ∇v ∫ t 0 l(t − s)∇v(s)dsdx ] . d dt l(t) ≤ d dt e(t) + �|ut|22 + �|vt| 2 2 − �|∇u|22 + �α 1 2δ |ut|22 + �αc1δ|∇u| 2 2 − �|v∇u| 2 2 − �|u∇v| 2 2 + � ∫ ω ∇u ∫ t 0 k(t − s)∇u(s)dsdx − �|∇v|22 + �β 1 2δ |vt|22 + �βc1δ|∇v| 2 2 − �|u∇v| 2 2 − �|v∇u| 2 2 + � ∫ ω ∇v ∫ t 0 l(t − s)∇v(s)ds.dx. (5.6) the last term of relation (5.6) can be estimated as follow.∣∣∣∣ ∫ ω ∇u ∫ t 0 k(t − s)∇u(s)dsdx ∣∣∣∣ ≤ ∫ ω (∫ t 0 k(t − s)|∇u(s) −∇u(t)|ds ) dx + ∫ t 0 k(s)ds|∇u|22 (5.7) ≤ (1 + η)(1 −k1)|∇u|22 + 1 4η (k ◦∇u)(t) for η > 0. 12 int. j. anal. appl. (2022), 20:10 analogously ∣∣∣∣ ∫ ω ∇v ∫ t 0 l(t − s)∇v(s)dsdx ∣∣∣∣ ≤ (1 + η)(1 − l1)|∇v|22 + 1 4η (l ◦∇v)(t) for η > 0. so d dt l(t) ≤ (k′ ◦u)(t) + (l′ ◦v)(t) −k(t)|∇u|22 − l(t)|∇v| 2 2 + α|ut| 2 2 + β|vt| 2 2 + �|ut| 2 2 + �|vt| 2 2 − �|∇u|22 + �α 1 2δ |ut|22 + �αc1δ|∇u| 2 2 − �|v∇u| 2 2 − �|u∇v| 2 2 + �(1 + η)(1 −k1)|∇u| 2 2 + � 1 4η (k ◦∇u)(t) − �|∇v|22 + �β 1 2δ |vt|22 + �βc1δ|∇v| 2 2 − �|u∇v| 2 2 − �|u∇u| 2 2 + �(1 + η)(1 − l1)|∇v|22 + � 1 4η (l ◦∇v)(t), (5.8) so d dt l(t) ≤ ( α + � + �α 1 2δ ) |ut|22 + ( β + � + �β 1 2δ ) |vt|22 + (−k(t) − � + �αc1δ + �(1 + η)(1 −k1)) |∇u|22 + (−l(t) − � + �βc1δ + �(1 + η)(1 − l1))|∇v|22 + (−2�− 1)|v∇u| 2 2 + (−2�− 1)|v∇u| 2 2 − (k ◦u)(t) − (l ◦v)(t) + |u∇v|22 + |v∇u| 2 2 + � 1 4η (k ◦∇u)(t) + � 1 4η (l ◦∇v)(t), (5.9) so d dt l(t) ≤ γe(t) + λ, (5.10) we choosing � small enough, such that γ =min(α + � + �α 1 2δ ; β + � + �β 1 2δ ;−k(t) − � + �αc1δ + �(1 + η)(1 −k1) (5.11) − l(t) − � + �βc1δ + �(1 + η)(1 − l1); (−2�− 1);−1) < 0, and λ = |u∇v|22 + |v∇u| 2 2 + � 1 4η (k ◦∇u)(t) + � 1 4η (l ◦∇v)(t). (5.12) from (5.2), we have d dt l(t) ≤ γ m l(t) + λ, (5.13) by integrating the previous differential inequality (5.13) between 0 and t, we obtain the following estimate for the function l l(t) ≤ ce γ m t − λm γ , ∀t ≥ 0, (5.14) by using (5.2), we conclude e(t) ≤ c1e γ m t − λm γn , ∀t ≥ 0. (5.15) conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2022), 20:10 13 references [1] r.a. adams, j.j.f. fournier, sobolev spaces, academic press, new york (2003). [2] d. andrade, a mognon, global solutions for a system of klein-gordon equations with memory, bol. soc. paran. mat. ser. 3, 21 (2003), 127–136. [3] s. brrimi, s.a. messaoudi, exponential decay of solutions to a viscoelastic equation with nonlinear localized damping, electron. j. differ. equ. 2004 (2004), no. 88, pp. 1–10. [4] y. boukhatem and b. benabdrrahmane, a. rahmoune, mèthode de faedo-galerkin pour un problème aux limites non linéaire. anale universităţii oradea, fasc. math. tom xvi (2009), 167-181. [5] c.l. frota, a. vicente, a hyperbolic system of klein-gordon type with acoustic boundary conditions, int. j. pure appl. math. 47 (2008), 185-198. [6] v. komornik, e. a zuazua, a direct method for boundary stabilization of the wave equation, j. math. pure appl. 69 (1990), 33-54. [7] a.t. louredo, m.m. miranda, nonlinear boundary dissipation for a coupled system of klein-gordon equations, electron. j. differ. equ. 2010 (2010), no. 120, pp. 1-19. [8] m.m. miranda, l.a. medeiros, on existence of global solutions of a coupled nonlinear klein-gordon equation, funkcialaj ekvacioj 30 (1987), 147-161. [9] a. ouaoua, m. maouni, a. khaldi, exponential decay of solutions with lp norm for class to semilinear wave equation with damping and source terms, open j. math anal. 4 (2020), 123-131. https://doi.org/10.30538/ psrp-oma2020.0071. [10] k. zennir, a. guesmia, existence of solutions to nonlinear κth-order coupled klein-gordon equations with nonlinear sources and memory terms, appl. math. e-notes, 15 (2015), 121-136. [11] s. zheng, nonlinear evolution equations, chapman & hall/ crc monographs and surveys in pure and applied mathematics 133, chapman & hall/crc, boca raton, (2004). [12] y. zhijian, initial boundary value problem for a class of non-linear strongly damped wave equations, math. meth. appl. sci. 26 (2003), 1047–1066. https://doi.org/10.1002/mma.412. https://doi.org/10.30538/psrp-oma2020.0071 https://doi.org/10.30538/psrp-oma2020.0071 https://doi.org/10.1002/mma.412 1. introduction 2. preliminaries 3. global existence step 1: approximate solution step 2: a priori estimate step 3: passage to the limit 4. uniqueness 5. stability references bibliographie international journal of analysis and applications issn 2291-8639 volume 2, number 1 (2013), 1-18 http://www.etamaths.com families of meromorphic multivalent functions associated with the dziok-raina operator g. murugusundaramoorthy1,∗ and m. k. aouf2 abstract. making use a linear operator, which is defined here by means of the hadamard product (or convolution), involving the wright’s generalized hypergeometric function , we introduce two novel subclasses ∑ p (q,s,α1; a,b,λ) and ∑+ p (q,s,α1;a,b,λ) of meromorphically multivalent functions of order λ (0 ≤ λ < p) in the punctured disc u∗. in this paper we investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. we extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions . we also derive many interesting results for the hadamard products of functions belonging to the class ∑+ p (q,s,α1; a,b,λ). 1. . introduction let ∑ p denote the class of functions of the form : (1.1) f(z) = z−p + ∞∑ k=1 akz k−p (p ∈ n = {1, 2, ...}) , which are analytic and p-valent in the punctured disc u∗ = {z : z ∈ c and 0 < |z| < 1} = u\{0}. for functions f(z) ∈ ∑ p given by (1.1), and g(z) ∈ ∑ p given by (1.2) g(z) = z−p + ∞∑ k=1 bkz k−p (p ∈ n) , we define the hadamard product (or convolution) of f(z) and g(z) by (1.3) (f ∗g)(z) = z−p + ∞∑ k=1 akbkz k−p = (g ∗f)(z). if f(z) and g(z) are analytic in u, we say that f(z) is subordinate to g(z) ; written symbolically as follows : f ≺ g or f(z) ≺ g(z) (z ∈ u), if there exists a schwarz function w(z) in u such thatf(z) = g(w(z)) (z ∈ u). let α1,a1, ...,αq,aq and β1,b1, ...,βs,bs (q,s ∈ n) be positive real parameters such that 2010 mathematics subject classification. 30c45. key words and phrases. wright’s generalized hypergeometric function, hadamard product, meromorphic functions, neighborhoods. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 murugusundaramoorthy and aouf (1.4) 1 + s∑ n=1 bn − q∑ n=1 an ≥ 0. the wright generalized hypergeometric function [36] ( see also [33] ) qψs[(α1,a1), ..., (αq,aq); (β1,b1), ...., (βs,bs); z] = qψs[(αn,an)1,q ; (βn,bn)1,s ; z] is defined by qψs[(αn,an)1,q ; (βn,bn)1,s ; z] = (1.5) ∞∑ k=0 { q∏ n=1 γ(αn + kan) }{ s∏ n=1 γ(βn + kbn) }−1 zk k! (z ∈ u). if an = 1(n = 1, ...,q) and bn = 1(n = 1, ..,s), we have the relationship : (1.6) ω qψs[(αn,1)1,q ; (βn,1)1,s ; z] = qfs(α1, ...,αq; β1, ...,βs; z), where qfs(α1, ...,αq ; β1, ...,βs ; z) is the generalized hypergeometric function ( see for details [9] , [10] , [11] , and [18] ) and (1.7) ω = ( q∏ n=1 γ(αn)) −1( s∏ n=1 γ(βn)). the wright generalized hypergeometric functions were invoked in the geometric function theory ( see [8] , [25] , [26], [27] and [28] ). we define a function qφs[(αn,an)1,q ; (βn,bn)1,s; z ] = ωz −p qψs[(αn,an)1,q ; (βn,bn)1,s ; z] and consider the following linear operator whp[(αn,an)1,q ; (βn,bn)1,s] : ∑ p → ∑ p , defined by the convolution (1.8) whp[(αn,an)1,q; (βn,bn) 1,s]f(z) = qφs[(αn,an)1,q ; (βn,bn)1,s ; z] ∗f(z), so that , for a function f(z) of the form (1.1) , we have (1.9) whp[(αn,an)1,q; (βn,bn) 1,s]f(z) = z −p + ∞∑ k=1 ωσk(α1)akz k−p, where (1.10) σk(α1) = γ(α1 + ka1).....γ(αq + kaq) γ(β1 + kb1).....γ(βs + kbs)k! . if , for convenience , we write ∆p,q,s[α1]f(z) = whp[(α1,a1), ....., (αq,aq); (β1,b1), ...., (βs,bs)]f(z), then one can easily verify from the definition (1.9) that (1.11) za1(∆p,q,s[α1]f(z)) ′ = α1∆p,q.s[α1 + 1]f(z) − (α1 + pa1)∆p,q,s[α1]f(z). for an = 1(n = 1, ...,q) and bn = 1(n = 1, ...,s), the operator ∆p,q,s[α1]f(z) = hp,q,s(α1)f(z) was introduced and studied by liu and srivastava [21]. families of meromorphic multivalent functions 3 for fixed parameters a , b and λ(−1 ≤ b < a ≤ 1 ; 0 ≤ λ < p ; p ∈ n), we say that a function f(z) ∈ σp is in the class ∑ p(q,s,α1; a,b,λ) of meromorphically pvalent functions in u if it also satisfies the following subordination condition : (1.12) α1 (∆p,q,s[α1 + 1]f(z)) ′ (∆p,q,s[α1]f(z)) ′ − (α1 + pa1) ≺−a1 p + [pb + (a−b)(p−λ)]z 1 + bz , or , by using (1.11) , if it satisfies the following subordination condition : (1.13) 1 (p−λ) (1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + λ) ≺− 1 + az 1 + bz (z ∈ u) or , equivalently , if the following inequality holds true : (1.14) ∣∣∣∣∣∣∣∣∣ 1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + p b(1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + p) + (a−b)(p−λ) ∣∣∣∣∣∣∣∣∣ < 1 (z ∈ u). furthermore, we say that a function f(z) ∈ ∑+ p (q,s,α1; a,b,λ) wherever f(z) is of the form (cf. equation (1.1)] : (1.15) f(z) = z−p + ∞∑ k=p |ak|zk (p ∈ n) . we note that for an = 1(n = 1, ...,q) and bn = 1(n = 1, ...,s) the classes ∑ p(q,s,α1; a,b, 0) = ωp,q,s(α1; a,b) and ∑+ p (q,s,α1; a,b, 0) = ω + p,q,s(α1; a,b) are studied by liu and srivastava [21]. also we note that ∑+ p (q,s,α1; β,−β,λ) = ∑+ p (q,s,α1; λ,β) =  f(z) ∈ σp and ∣∣∣∣∣∣∣∣∣ 1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + p 1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ −p + 2λ ∣∣∣∣∣∣∣∣∣ < β (1.16) (z ∈ u; 0 ≤ λ < p; 0 < β ≤ 1; p ∈ n) } . meromorphic multivalent functions have been extensively studied by (for example) (mogra [22] and [23]), uralegaddi and ganigi [34], uralegaddi and somanatha [35], aouf ([4] and [5]), aouf and hossen [6], srivastava et al. [32], owa et al. [24], joshi and aouf [14], joshi and srivastava [15], aouf et al. [7], raina and srivastava [29], yang ([37] and [38]), kulkarni et al. [16], liu [17] and liu and srivastava ([19] and [20]). in this paper we investigate the various important properties and characteristics of the classes ∑ p(q,s,α1; a,b,λ) and ∑+ p (q,s,α1; a,b,λ). following the recent investigations by altintas et al. [3, p. 1668], we extend the concept of neighborhoods of analytic functions, which was considered earlier by (for example) goodman [12] and ruscheweyh [30], to meromorphically multivalent functions, belonging to the classes∑ p(q,s,α1; a,b,λ) and ∑+ p (q,s,α1; a,b,λ). we also derive many interesting results for the hadamard products of functions belonging to the p-valently meromorphic function class ∑+ p (q,s,α1; a,b,λ). 4 murugusundaramoorthy and aouf 2. . inclusion properties of the class ∑ p(q,s,α1; a,b,λ) for proving our first inclusion result , we shall make use of the following lemma. lemma 1. ( see jack [13] ). let the (nonconstant) function w(z) be analytic in u with w(0) = 0. if |w(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 ∈ u, then (2.1) z0w ′ (z0) = γw(z0) , where γ is a real number and γ ≥ 1. theorem 1. . let α1 ∈ r\{0}. if (2.2) α1 ≥ a1(a−b)(p−λ) 1 + b (−1 < b < a ≤ 1 ; 0 ≤ λ < p; p ∈ n; a1 > 0) , then ∑ p (q,s,α1 + 1; a,b,λ) ⊂ ∑ p (q,s,α1; a,b,λ) . proof. let f(z) ∈ ∑ p(q,s,α1 + 1; a,b,λ) and suppose that (2.3) (∆p,q,s[α1 + 1]f(z)) ′ (∆p,q,s[α1]f(z)) ′ = 1 + a1(b −a)(p−λ) w(z) α1(1 + bw(z)) , where the function w(z) is either analytic or meromorphic in u, with w(0) = 0. by differeniating (2.3) with respect to z logarithmically and using (1.11), we have 1 + (∆p,q,s[α1 + 1]f(z)) ′′ (∆p,q,s[α1 + 1]f(z)) ′ + p = ( α1 + 1 a1 )( (∆p,q,s[α1 + 2]f(z)) ′ (∆p,q,s[α1 + 1]f(z)) ′ − 1) (2.4) = (b −a)(p−λ) w(z) 1 + bw(z) + (b −a)(p−λ)z w ′ (z) (1 + bw(z)){α1 + [bα1 + a1(b −a)(p−λ)] w(z)} . if we suppose now that (2.5) max |z|≤|z0| |w(z)| = |w(z0)| = 1 (z0 ∈ u) , and apply jack’s lemma, we find that (2.6) z0w ′ (z0) = γw(z0) (γ ≥ 1) . writing w(z0) = e iθ (0 ≤ θ ≤ 2π) and putting z = z0 in (2.4), we get after some computations that∣∣∣∣∣∣∣∣∣ 1 + z(∆p,q,s[α1 + 1]f(z)) ′′ (∆p,q,s[α1 + 1]f(z)) ′ + p b(1 + z(∆p,q,s[α1 + 1]f(z)) ′′ (∆p,q,s[α1 + 1]f(z)) ′ + p) + (a−b)(p−λ) ∣∣∣∣∣∣∣∣∣ 2 − 1 = ∣∣∣∣(α1 + γa1) + [α1b + a1(b −a)(p−λ)]eiθα1 + [b(α1 −γa1) + a1(b −a)(p−λ)]eiθ ∣∣∣∣2 − 1 (2.7) = γ2a21(1−b 2)+2γa1[α1(1+b 2)+a1b(b−a)(p−λ)]+2γa1[2α1b+a1(b−a)(p−λ)] cos θ |α1+[b(α1−γa1)+a1(b−a)(p−λ)]eiθ|2 . families of meromorphic multivalent functions 5 set g(θ) = γ2a21(1 −b 2) + 2γa1[α1(1 + b 2) + a1b(b −a)(p−λ)] (2.8) +2γa1[2α1b + a1(b −a)(p−λ)] cos θ (0 ≤ θ ≤ 2π) (−1 < b < a ≤ 1; 0 ≤ λ < p ; p ∈ n ; α1 ∈ r\{0}; a1 > 0; γ ≥ 1; 0 ≤ θ ≤ 2π). then, by hypothesis, we have g(0) = γ2a21(1 −b 2) + 2γa1(1 + b)[α1(1 + b) + a1(b −a)(p−λ)] ≥ 0 and g(π) = γ2a21(1 −b 2) + 2γa1(1 −b)[α1(1 −b) −a1(b −a)(p−λ)] ≥ 0 which, together, show that (2.9) g(θ) ≥ 0 (0 ≤ θ ≤ 2π) . in view of (2.9) , (2.7) would obviously contradict our hypothesis that f(z) ∈ ∑ p(q,s,α1 + 1; a,b,λ). hence, we must have (2.10) |w(z)| < 1 (z ∈ u), and we conclude from (2.3) that f(z) ∈ ∑ p (q,s,α1; a,b,λ). the proof of theorem 1 is thus completed. � next we prove an inclusion property associated with a certain integral transform introduced below. theorem 2. . let µ be a complex number such that <(µ) > (a−b)(p−λ) 1 + b (−1 < b < a ≤ 1 ; 0 ≤ λ < p ; p ∈ n). if f(z) ∈ ∑ p(q,s,α1; a,b,λ) , then the function f(z) defined by (2.11) f(z) = µ zµ+p z∫ 0 tµ+p−1f(t) dt also belongs to the class ∑ p(q,s,α1; a,b,λ) . proof. from (2.11), we readily have (2.12) z(∆p,q,s[α1])f(z)) ′ = µ ∆p,q,s[α1]f(z) − (µ + p)∆p,q,s[α1]f(z). suppose that f(z) ∈ ∑ p(q,s,α1; a,b,λ) and put (2.13) α1 a1 ( (∆p,q,s[(α1 + 1)]f(z)) ′ (∆p,q,s[α1]f(z)) ′ − 1) = (b −a)(p−λ) w(z) 1 + bw(z) , where the function w(z) is either analytic or meromorphic in u, with w(0) = 0. then, by using (2.12), (2.13) and the identity (1.11), we find after some calculations that 6 murugusundaramoorthy and aouf α1 a1 ( (∆p,q,s[α1 + 1]f(z)) ′ (∆p,q,s[α1]f(z)) ′ − 1) (2.14) = (b −a)(p−λ)w(z) 1 + bw(z) + (b −a)(p−λ)z w ′ (z) (1 + bw(z)){µ + [µb + (b −a)(p−λ)]w(z)} . the remaining part of the proof of theorem 2 is similar to that theorem 1.we choose to omit the details involved . � 3. . properties of the class ∑+ p (q,s,α1; a,b,λ) in this section we assume further that : αn, an > 0(n = 1, ...,q), βn, bn > 0 (n = 1, ...,s), 0 ≤ b < 1, 0 ≤ λ < pand p ∈ n. we first determine a necessary and sufficient condition for a function f(z) ∈ σp of the form (1.15) to be in the class ∑+ p (q,s,α1; a,b,λ) of meromorphically p-valent functions with positive coefficients. theorem 3. . let f(z) ∈ σp be given by (1.15). then f(z) ∈ ∑+ p (q,s,α1; a,b,λ) if and only if (3.1) ∞∑ k=p ωkσk+p(α1)[(k + p)(1 + b) + (a−b)(p−λ)]|ak| ≤ p(a−b)(p−λ) , where, for convenience, (3.2) σm(α1) = γ(α1 + ma1).....γ(αq + maq) γ(β1 + mb1).....γ(βs + mbs)m! . proof. let f(z) ∈ ∑+ p (q,s,α1; a,b,λ) is given by (1.15 ). then from (1.14 ) and (1.15 ), we have∣∣∣∣∣ z(∆p,q,s[α1]f(z)) ′′ + (1 + p)(∆p,q,s[α1]f(z)) ′ b (z(∆p,q,s[α1]f(z)) ′′ + (1 + p)(∆p,q,s[α1]f(z)) ′ ) + (a−b)(p−λ)(∆p,q,s[α1]f(z)) ′ ∣∣∣∣∣ = ∣∣∣∣∣∣∣∣ ∞∑ k=p ωk(k + p)σk+p(α1)|ak|zk+p p(a−b)(p−λ) − ∞∑ k=p [b(k + p) + (a−b)(p−λ)]ωkγk+p(α1)|ak|zk+p ∣∣∣∣∣∣∣∣ < 1 (z ∈ u). since |<(z)| ≤ |z|(z ∈ c), we have <   ∞∑ k=p ωk(k + p)σk+p(α1)|ak|zk+p p(a−b)(p−λ) − ∞∑ k=p [b(k + p) + (a−b)(p−λ)]ωkσk+p(α1)|ak|zk+p   (3.3) < 1 (z ∈ u). we consider real values of z and take z = r with 0 ≤ r < 1 . then , for r = 0, the denominator of (3.3) is positive and so is positive for all r(0 < r < 1). letting z = r → 1−, (3.3) yields ∞∑ k=p ωk(k + p)σk+p(α1)|ak| ≤ p(a−b)(p−λ) families of meromorphic multivalent functions 7 − ∞∑ k=p [b(k + p) + (a−b)(p−λ)]ωkσk+p(α1)|ak|, which leads us at once to (3.1). in order to prove the converse, we assume that the inequality (3.1) holds true. then we get∣∣∣∣∣∣∣∣∣ 1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + p b(1 + z(∆p,q,s[α1]f(z)) ′′ (∆p,q,s[α1]f(z)) ′ + p) + (a−b)(p−λ) ∣∣∣∣∣∣∣∣∣ ≤ ∞∑ k=p ωk(k + p)σk+p(α1)|ak| p(a−b)(p−λ) − ∞∑ k=p [b(k + p) + (a−b)(p−λ)]ωkσk+p(α1)|ak| < 1 ( z ∈ u). hence, by the maximum modulus theorem, we have f(z) ∈ ∑+ p (q,s,α1; a,b,λ). this completes the proof of theorem 3. corollary 1. . let f(z) ∈ σp be given by ( 1.15 ). if f(z) ∈ ∑+ p (q,s,α1; a,b,λ), then (3.4) |ak| ≤ p(a−b)(p−λ) ωkσk+p(α1)[(k + p)(1 + b) + (a−b)(p−λ)] (k ≥ p; p ∈ n) . the result is sharp for the function f(z) given by (3.5) f(z) = z−p + p(a−b)(p−λ) ωkσk+p(α1)[(k + p)(1 + b) + (a−b)(p−λ)] zk (k ≥ p; p ∈ n). putting λ = 0 , an = 1(n = 1, ...,q) and bn = 1(n = 1, ...,s) in theorem 3 , we obtain corollary 2. .let f(z) ∈ σp be given by (1.15). then f(z) ∈ ω+p,q,s(α1; a,b), if and only if (3.6) ∞∑ k=p kγk+p(α1)[(k + p)(1 + b) + p(a−b)] |ak| ≤ p2(a−b), where (3.7) γm(α1) = (α1)m.......(αq)m (β1)m.......(βs)mm! (m ∈ n), and (θ)γ is the pochhammer symbol defined, in terms of the gamma function γ by (θ)γ = γ(θ + ν) γ(θ) = { 1 , (ν = 0 ; θ ∈ c/{0}) θ(θ + 1)....(θ + ν − 1) , (ν ∈ n ; θ ∈ c). remark 1. .we note the result obtained by liu and srivastava [20 , theorem 3 ] is not correct . the correct result is given by corollary 2 . next we prove the following growth and distortion properties for the class ∑+ p (q,s,α1; a,b,λ). theorem 4. . let the function f(z) of the form (1.15) belong to the class ∑+ p (q,s,α1; a,b,λ) . if the sequence {δk} is nondecreasing, then 8 murugusundaramoorthy and aouf (3.8) r−p − p(a−b)(p−λ) ωδp rp ≤ |f(z)| ≤ r−p + p(a−b)(p−λ) ωδp rp (0 < |z| = r < 1), where (3.9) δk = kσk+p (α1) [(k + p)(1 + b) + (a−b)(p−λ)] (k ≥ p ; p ∈ n) and σk+p(α1) is given by (3.2). if the sequence {δkk } is nondecreasing, then pr−p−1 − p2(a−b)(p−λ) ωδp rp−1 ≤ ∣∣∣f′ (z)∣∣∣ ≤ pr−p−1 + p2(a−b)(p−λ) ωδp rp−1 (3.10) (0 < |z| = r < 1). each of these results is sharp with the extremal function f(z) given by (3.11) f(z) = z−p + (a−b)(p−λ) ωσ2p (α1)[2p(1 + b) + (a−b)(p−λ)] zp (p ∈ n). proof. let the function f(z) , given by (1.15), be in the class ∑+ p (q,s,α1; a,b,λ). if the sequence {δk} is nondecreasing and positive, then , by theorem 3 , we have (3.12) ∞∑ k=p |ak| ≤ p(a−b)(p−λ) ωδp and if the sequence {δk k } is nondecreasing and positive , theorem 3 also yields (3.13) ∞∑ k=p k |ak| ≤ p2(a−b)(p−λ) ωδp . making use of the conditions (3.12) and (3.13), in conjunction with the definition (1.15), we readily obtain the assertions (3.8) and (3.10) of theorem 4. finally , it is easy to see that the bounds in (3.8) and (3.10) are attained for the function f(z) given by (3.11). next we determine the radii of meromorphically p-valent starlikeness of order ϕ(0 ≤ ϕ < p) for functions in the class ∑+ p (q,s,α1; a,b,λ). theorem 5. .let the function f(z) defined by (1.15) be in the class ∑+ p (q,s,α1; a,b,λ). then , f(z) is meromorphically p-valent starlike of order ϕ(0 ≤ ϕ < p) in the disc |z| < r1 , that is , (3.14) <{− zf ′ (z) f(z) } > ϕ (|z| < r1; 0 ≤ ϕ < p ; p ∈ n), where (3.15) r1 = inf k≥p { ωkσk+p(α1)(p−ϕ)[(k + p)(1 + b) + (a−b)(p−λ)]) p(a−b)(p−λ)(k + ϕ) } 1 k + p . and σk+p(α1) is given by (3.2). the result is sharp for the function f(z) given by (3.5). families of meromorphic multivalent functions 9 proof. from the definition (1.15), we easily get (3.16) ∣∣∣∣∣∣∣∣∣ zf ′ (z) f(z) + p zf ′ (z) f(z) −p + 2ϕ ∣∣∣∣∣∣∣∣∣ ≤ ∞∑ k=p (k + p)|ak||z|k+p 2(p−ϕ) − ∞∑ k=p (k −p + 2ϕ)|ak||z|k+p . thus, we have the desired inequality : (3.17) ∣∣∣∣∣∣∣∣∣ zf ′ (z) f(z) + p zf ′ (z) f(z) −p + 2φ ∣∣∣∣∣∣∣∣∣ ≤ 1 (0 ≤ ϕ < p; p ∈ n) if (3.18) ∞∑ k=p ( k + ϕ p−ϕ ) |ak||z|k+p ≤ 1 . hence, by theorem 3, (3.18) will be true if (3.19) ( k + ϕ p−ϕ ) |z|k+p ≤ ωkσk+p(α1)[(k + p)(1 + b) + (a−b)(p−λ)] p(a−b)(p−λ) (k ≥ p; p ∈ n) . the last inequality (3.19) leads us immediately to the disc |z| < r1, where r1 is given by (3.15). 4. . neighborhoods in this section , we also assume that αn,an > 0 (n = 1, ...,q) and βn,bn > 0 (n = 1, ...,s) and 1 + s∑ n=1 bn − s∑ n=1 an ≥ 0. following the earlier works (based upon the familiar concept of neighorhoods of analytic functions) by goodman [12] and ruscheweyh [30], and (more recenlty) by altintas et al. ([1],[2] and [3]), liu [17], and liu and srivastava ([19], [20] and [21]), we begin by introducing here the δ-neighborhood of a function f(z) ∈ σp of the form (1.1) by means of the definition given below : nδ(f) = { g : g ∈ σp , g(z) = z−p + ∞∑ k=1 bkz k−p and ∞∑ k=1 ω(k + p)σk(α1)[(a−b)(p−λ) + k(1 + |b|)] p(a−b)(p−λ) |ak − bk| ≤ δ (4.1) (−1 ≤ b < a ≤ 1; δ > 0 ; 0 ≤ λ < p; p ∈ n)} . making use of the definition (4.1), we now prove theorem 6 below. theorem 6. . let the function f(z) defined by (1.1) be in the class ∑ p(q,s,α1; a,b,λ). if f(z) satisfies the following condition : 10 murugusundaramoorthy and aouf (4.2) f(z) + �z−p 1 + � ∈ ∑ p (q,s,α1; a,b,λ) (� ∈ c, |�| < δ,δ > 0) , then (4.3) nδ(f) ⊂ ∑ p (q,s,α1; a,b,λ) . proof. it is easily seen from (1.14) that g(z) ∈ ∑ p(q,s,α1; a,b,λ) if and only if, for any complex ζ with |ζ| = 1, (4.4) 1 + z(∆p,q,s[α1]g(z)) ′′ (∆p,q,s[α1]g(z)) ′ + p b(1 + z(∆p,q,s[α1]g(z)) ′′ (∆p,q,s[α1]g(z)) ′ ) + [pb + (a−b)(p−λ)] 6= ζ (z ∈ u; ζ ∈ c; |ζ| = 1), which is equivalent to (4.5) (g ∗h)(z) z−p 6= 0 (z ∈ u) , where, for convenience, h(z) = z−p + ∞∑ k=1 ckz k−p (4.6) = z−p + ∞∑ k=1 ω(k −p)σk(α1)[(a−b)(p−λ)ζ + k(bζ − 1)] pζ(b −a)(p−λ) zk−p . from (4.6), we have |ck| = ∣∣∣∣ω(k −p)σk(α1)[(a−b)(p−λ)ζ + k(bζ − 1)]pζ(b −a)(p−λ) ∣∣∣∣ (4.7) ≤ ω(k + p)σk(α1)[(a−b)(p−λ) + k(1 + |b|)] p(a−b)(p−λ) (k,p ∈ n) . now , if f(z) = zp + ∞∑ k=1 akz k−p ∈ σp satisfies the condition (4.2), then (4.5) yields (4.8) ∣∣∣∣(f ∗h)(z)z−p ∣∣∣∣ ≥ δ (z ∈ u; δ > 0) . by letting (4.9) g(z) = z−p + ∞∑ k=1 bkz k−p ∈ nδ(f) , so that ∣∣∣∣[f(z) −g(z)] ∗h(z)z−p ∣∣∣∣ = ∣∣∣∣∣ ∞∑ k=1 (ak − bk)ckzk ∣∣∣∣∣ families of meromorphic multivalent functions 11 ≤ |z| ∞∑ k=1 ω(k + p)σk(α1)[(a−b)(p−λ) + k(1 + |b|)] p(a−b)(p−λ) |ak − bk| (4.10) < δ (z ∈ u; δ > 0), which leads us to (4.5), and hence also (4.4) for any ζ ∈ c such that |ζ| = 1. this implies that g(z) ∈∑ p(q,s,α1; a,b,λ), which evidenlty completes the proof of the assertion (4.3) of theorem 6. we now define the δ-neighorhood of a function f(z) ∈ σp of the form (1.15) as follows n+δ (f) =  g : g ∈ σp , g(z) = z−p + ∞∑ k=p |bk|zk and ∞∑ k=p ωkσk+p(α1)[(a−b)(p−λ) + (k + p)(1 + b)] p(a−b)(p−λ) ||ak|− |bk|| ≤ δ (4.11) (0 ≤ b < a ≤ 1 ; δ > 0 ; 0 ≤ λ < p; p ∈ n)} . theorem 7. . let the function f(z) defined by (1.13) be in the class ∑+ p (q,s,α1 + 1; a,b,λ)(0 ≤ b < a ≤ 1 ; 0 ≤ λ < p; p ∈ n). then (4.12) n+δ (f) ⊂ ∑+ p (q,s,α1; a,b,λ) (δ = 2p α1 + 2p ) . the result is sharp in the sense that δ cannot be increased . proof. making use of the same method as in the proof of theorem 6, we can show that [cf. equation (4.6)] h(z) = z−p + ∞∑ k=p ckz k (4.13) = z−p + ∞∑ k=p ωkσk+p(α1)[(a−b)(p−λ)ζ + (k + p)(bζ − 1)] pζ(b −a)(p−λ) zk . if f(z) ∈ ∑+ p (q,s,α1 + 1; a,b,λ) is given by (1.15), we obtain ∣∣∣∣(f ∗h)(z)z−p ∣∣∣∣ = ∣∣∣∣∣∣1 + ∞∑ k=p ck|ak|zk+p ∣∣∣∣∣∣ ≥ 1 − α1 α1 + 2p ∞∑ k=p kσk+p(α1 + 1)[(a−b)(p−λ) + (k + p)(1 + b)] p(a−b)(p−λ) |ak| ≥ 1 − α1 α1 + 2p = 2p α1 + 2p = δ, by appealing to assertion (3.1) of theorem 3. the remaining part of our proof of theorem 7 is similar to that of theorem 6, and we skip the details involed. to show sharpness of the assertion theorem 7 , we consider the functions f(z) and g(z) given by (4.14) f(z) = z−p + (a−b)(p−λ) ωσ2p(α1 + 1)[(a−b)(p−λ) + 2p(1 + b)] zp ∈ ∑+ p (q,s,α1 + 1; a,b,λ) and 12 murugusundaramoorthy and aouf g(z) = z−p + [ (a−b)(p−λ) ωσ2p(α1 + 1)[(a−b)(p−λ) + 2p(1 + b)] + (4.15) (a−b)(p−λ)δ ′ ωσ2p(α1)[(a−b)(p−λ) + 2p(1 + b)] ] zp, where δ ′ > δ = 2p α1 + 2p . clearly, the function g(z) belongs to n+ δ ′ (f). on the other hand, we find from theorem 3 that g(z) is not in the class ∑+ p (q,s,α1; a,b,λ). thus the proof of theorem 7 is completed. finally , we prove the following theorem. theorem 8. . let f(z) ∈ σp be given by (1.1) and define the partial sums s1(z) and sn(z) as follows : (4.16) s1(z) = z −p and sn(z) = z −p + n−1∑ k=1 akz k−p (n ∈ n) , it being understood that an empty sum is (as usual) nil. suppose also that (4.17) ∞∑ k=1 dk|ak| ≤ 1 (dk = ω(k + p)σk(α1)[(a−b)(p−λ) + k(1 + |b|)] p(a−b)(p−λ) ) . then (i) f(z) ∈ ∑ p(q,s,α1; a,b,λ), (ii) if {σk(α1)}(k ∈ n) is nondecreasing and (4.18) σ1(α1) > p(a−b)(p−λ) ω(1 + p)[(a−b)(p−λ) + (1 + |b|)] , then (4.19) < { f(z) sn(z) } > 1 − 1 dn (z ∈ u ; n ∈ n) , and (4.20) < { sn(z) f(z) } > dn 1 + dn (z ∈ u ; n ∈ n) . each of the bounds in (4.19) and (4.20) is the best possible for each n ∈ n. proof. (i) it is not difficult to see that z−p ∈ ∑ p(q,s,α1; a,b,λ)(p ∈ n). thus, from theorem 6 and the hypothesis (4.17) of theorem 8, we have (4.21) n1(z −p) ⊂ ∑ p (q,s,α1; a,b,λ) (0 ≤ λ < p; p ∈ n), which shows that f(z) ∈ ∑ p(q,s,α1; a,b,λ) . (ii) under the hypothesis in part (ii) of theorem 8, we can see from (4.17) that (4.22) dk+1 > dk > 1 (k ∈ n) . families of meromorphic multivalent functions 13 therefore, we have (4.23) n−1∑ k=1 |ak| + dn ∞∑ k=n |ak| ≤ ∞∑ k=1 dk|ak| ≤ 1 , where we have used the hypothesis (4.17) again. by setting (4.24) g1(z) = dn [ f(z) sn(z) − (1 − 1 dn ) ] = 1 + dn ∞∑ k=n akz k 1 + n−1∑ k=1 akzk , and applying (4.23), we find that (4.25) ∣∣∣∣g1(z) − 1g1(z) + 1 ∣∣∣∣ ≤ dn ∞∑ k=n |ak| 2 − 2 n−1∑ k=1 |ak|−dn ∞∑ k=n |ak| ≤ 1 (z ∈ u), which readily yields the assertion (4.19) of theorem 8. if we take (4.26) f(z) = z−p − zn−p dn , then f(z) sn(z) = 1 − zn dn → 1 − 1 dn ( z → 1−) , which shows that the bound in (4.19) is the best possible for each n ∈ n. similarly, if we put (4.27) g2(z) = (1 + dn)( sn(z) f(z) − dn 1 + dn ) = 1 − (1 + dn) ∞∑ k=n akz k 1 + ∞∑ k=1 akzk and make use of (4.23), we can deduce that (4.28) ∣∣∣∣g2(z) − 1g2(z) + 1 ∣∣∣∣ ≤ (1 + dn) ∞∑ k=n |ak| 2 − 2 n−1∑ k=1 |ak| + (1 −dn) ∞∑ k=n |ak| ≤ 1 (z ∈ u), which leads us immediately to the assertion (4.20) of theorem 8. the bound in (4.20) is sharp for each n ∈ n, with the extremal function f(z) given by (4.26). the proof of theorem 8 is thus completed. 5. . convolution properties for the functions (5.1) fj(z) = z −p + ∞∑ k=p |ak,j|zk (j = 1, 2; p ∈ n) , 14 murugusundaramoorthy and aouf we denote by (f1 ∗f2)(z) the hadamard product (or convolution) of the functions f1(z) and f2(z), that is, (5.2) (f1 ∗f2)(z) = z−p + ∞∑ k=p |ak,1||ak,2|zk . throughout this section, we assume further that the sequence {σm(α1)}(m ∈ n) is nondecreasing , where σm(α1) is given by (3.2), (5.3) c(p,λ,a,b,k) = (k + p)(1 + b) + (a−b)(p−λ) (k ≥ p) and (5.4) d(p,λ,a,b) = p(a−b)(p−λ) . theorem 9. . let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class ∑+ p (q,s,α1; a,b,λ). then (f1 ∗f2)(z) ∈ ∑+ p (q,s,α1; a,b,γ), where (5.5) γ = p(1 − 2(1 + b)(a−b)(p−λ)2 ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)]2 − (a−b)2(p−λ)2 ). the result is sharp for the functions fj(z)(j = 1, 2) given by (5.6) fj(z) = z −p + (a−b)(p−λ) ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)] zp (j = 1, 2; p ∈ n) . proof. employing the technique used earlier by schild and silverman [31], we need to find the largest γ such that (5.7) ∞∑ k=p ωkσk+p(α1)c(p,γ,a,b,k) d(p,γ,a,b) |ak,1||ak,2| ≤ 1 for fj(z) ∈ ∑+ p (q,s,α1; a,b,λ)(j = 1, 2). since fj(z) ∈ ∑+ p (q,s,α1; a,b,λ)(j = 1, 2), we readily see that (5.8) ∞∑ k=p ωkσk+p(α1)c(p,λ,a,b,k) d(p,λ,a,b) |ak,j| ≤ 1 (j = 1, 2). therefore, by the cauchy-schwarz inequality, we obtain (5.9) ∞∑ k=p ωkσk+p(α1)c(p,λ,a,b,k) d(p,λ,a,b) √ |ak,1||ak,2| ≤ 1 . this implies that we only need to show that (5.10) c(p,γ,a,b,k) (p−γ) |ak,1||ak,2| ≤ c(p,λ,a,b,k) (p−λ) √ |ak,1||ak,2| (k ≥ p) or, equivalently, that (5.11) √ |ak,1||ak,2| ≤ (p−γ)c(p,λ,a,b,k) (p−λ)c(p,γ,a,b,k) (k ≥ p). families of meromorphic multivalent functions 15 hence, by the inequality (5.9), it is sufficient to prove that (5.12) d(p,λ,a,b) ωkσk+p(α1)c(p,λ,a,b,k) ≤ (p−γ)c(p,λ,a,b,k) (p−λ)c(p,γ,a,b,k) (k ≥ p) . it follows from (5.12) that (5.13) γ ≤ p− p(k + p)(1 + b)(a−b)(p−λ)2 ωkσp+k(α1)[c(p,λ,a,b,k)]2 −p(a−b)2(p−λ)2 (k ≥ p). now, defining the function φ(k) by (5.14) φ(k) = p− p(k + p)(1 + b)(a−b)(p−λ)2 ωkσp+k(α1)[c(p,λ,a,b,k)]2 −p(a−b)2(p−λ)2 (k ≥ p) , we see that φ(k) is an increasing function of k. therefore, we conclude that (5.15) γ ≤ φ(p) = p(1 − 2(1 + b)(a−b)(p−λ)2 ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)]2 − (a−b)2(p−λ)2 ), which evidently completes the proof of theorem 9. putting a = β and b = −β (0 < β ≤ 1) in theorem 9, we obtain the following consequence. corollary 3. . let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class ∑+ p (q,s,α1; λ,β). then (f1 ∗f2)(z) ∈ ∑+ p (q,s,α1; γ,β), where (5.16) γ = p(1 − β(1 −β)(p−λ)2 ωσ2p(α1)(p−λβ)2 −β2(p−λ)2 ) . the result is sharp for the functions fj(z)(j = 1, 2) given by (5.17) fj(z) = z −p + β(p−λ) ωσ2p(α1)(p−λβ) zp (j = 1, 2; p ∈ n) . using arguments similar to those in the proof of theorem 9, we obtain the following result . theorem 10. . let the function f1(z) defined by (5.1) be in the class ∑+ p (q,s,α1; a,b,λ). suppose also that the function f2(z) defined by (5.1) be in the class ∑+ p (q,s,α1; a,b,γ). then (f1 ∗ f2)(z) ∈∑+ p (q,s,α1; a,b,ξ), where ξ = p(1 − 2(1 + b)(a−b)(p−λ)(p−γ) ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)][2p(1 + b) + (a−b)(p−γ)] −m ) (5.18) (m = (a−b)2(p−λ)(p−γ)) . the result is sharp for the functions fj(z)(j = 1, 2) given by (5.19) f1(z) = z −p + (a−b)(p−λ) ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)] zp (p ∈ n) and (5.20) f2(z) = z −p + (a−b)(p−γ) ωσ2p(α1)[2p(1 + b) + (a−b)(p−γ)] zp (p ∈ n) . 16 murugusundaramoorthy and aouf putting a = β and b = −β (0 < β ≤ 1) in theorem 10, we obtain corollary 4 below. corollary 4. . let the function f1(z) defined by (5.1) be in the class ∑+ p (q,s,α1; λ,β). suppose also that the function f2(z) defined by (5.1) be in the class ∑+ p (q,s,α1; γ,β). then (f1∗f2)(z) ∈ ∑+ p (q,s,α1; η,β), where (5.21) η = p(1 − β(1 −β)(p−λ)(p−γ) ωσ2p(α1)(p−λβ)(p−γβ) −β2(p−λ)(p−γ) ) . the result is the best possible for the functions fj(z)(j = 1, 2) given by (5.22) f1(z) = z −p + β(p−λ) ωσ2p(α1)(p−λβ) zp (p ∈ n) and (5.23) f2(z) = z −p + β(p−γ) ωσ2p(α1)(p−γβ) zp (p ∈ n). theorem 11. . let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class ∑+ p (q,s,α1; a,b,λ). then the function h(z) defined by (5.24) h(z) = z−p + ∞∑ k=p (|ak,1|2 + |ak,2|2)zk belongs to the class ∑+ p (q,s,α1; a,b,ζ), where (5.25) ζ = p(1 − 4(1 + b)(a−b)(p−λ)2 ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)]2 − 2(a−b)2(p−λ)2 ) . this result is sharp for the functions fj(z)(j = 1, 2) given already by (5.6). proof. noting that ∞∑ k=p [ωkσk+p(α1)c(p,λ,a,b,k)] 2 [d(p,λ,a,b)]2 |ak,j|2 (5.26) ≤ ( ∞∑ k=p ωkσk+p(α1)c(p,λ,a,b,k) d(p,λ,a,b) |ak,j|)2 ≤ 1 (j = 1, 2) , for fj(z) ∈ ∑+ p (q,s,α1; a,b,λ)(j = 1, 2), we have (5.27) ∞∑ k=p [ωkσk+p(α1)c(p,λ,a,b,k)] 2 2[d(p,λ,a,b)]2 (|ak,1|2 + |ak,2|2) ≤ 1 . therefore, we have to find the largest ζ such that (5.28) c(p,ζ,a,b,k) (p− ζ) ≤ ωkσk+p(α1)[c(p,λ,a,b,k)] 2 2p(a−b)(p−λ)2 (k ≥ p) , that is, that families of meromorphic multivalent functions 17 (5.29) ζ ≤ p− 2p(k + p)(1 + b)(a−b)(p−λ)2 ωkσk+p(α1)[c(p,λ,a,b,k)]2 − 2p(a−b)2(p−λ)2 (k ≥ p) . now, defining a function ψ(k) by (5.30) ψ(k) = p− 2p(k + p)(1 + b)(a−b)(p−λ)2 ωkσk+p(α1)[c(p,λ,a,b,k)]2 − 2p(a−b)2(p−λ)2 (k ≥ p) , we observe that ψ(k) is an increasing function of k. we thus conclude that (5.31) ζ ≤ ψ(p) = p(1 − 4(1 + b)(a−b)(p−λ)2 ωσ2p(α1)[2p(1 + b) + (a−b)(p−λ)]2 − 2(a−b)2(p−λ)2 ) , which completes the proof of theorem 11. putting a = β and b = −β (0 < β ≤ 1) in theorem 11, we obtain the following corollary. corollary 5. . let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class ∑+ p (q,s,α1 ; λ,β). then the function h(z) defined by (5.24) belongs to the class ∑+ p (q,s,α1 ; τ,β), where (5.32) τ = p(1 − 2β(1 −β)(p−λ)2 ωσ2p(α1)(p−λβ)2 − 2β2(p−λ)2 ) . the result is sharp for the functions fj(z)(j = 1, 2) given already by (5.17). remark 2. .we note the results obtained by liu and srivastava [ 21 , theorems 4,5 and 7 ] are not correct . the correct results are given by theorems 4,5 and 7 , respectively , after putting λ = 0 ,an = 1(n = 1, ...,q) and bn = 1(n = 1, ...,s). references [1] o. altintas and s. owa, neighborhoods of certain analytic functions with negative coefficients, internat. j. math. math. sci. 19(1996), 797-800. [2] o. altintas, o. ozkan and h. m. srivastava, neighborhoods of a class of analytic functions with negative coefficients, appl. math. lett. 13(2000), no. 3, 63-67. [3] o. altintas, o. ozkan and h. m. srivastava, neighborhoods of a certain family of multivalent functions with negative coefficients, comput. math. appl. 47(2004), 1667-1672. [4] m. k. aouf, on a class of meromorphic multivalent functions with positive coefficients, math. japon. 35(1990), 603-608. [5] m. k. aouf, a generalization of meromorphic multivalent functions with positive coefficients, math. japon. 35(1990), 609-614. [6] m. k. aouf and h. m. hossen, new criteria for meromorphic p-valent starlike functions, tsukuba j. math. 17(1993), 481-486. [7] m. k. aouf, h. m. hossen and h. e. elattar, a certain class of meromorphic multivalent functions with positive and fixed second coefficients, punjab univ. j. math. 33(2000), 115-124. [8] j. dziok and r. k. raina, families of analytic functions associated with the wright generalized hypergemetrric function , demonstratio math. 37 (2004), no.3, 533-542. [9] j. dziok and h. m. srivastava, classes of analytic functions associated with the generalized hypergeometric function, appl. math. comput. 103(1999), 1-13. [10] j. dziok and h. m. srivastava, certain subclasses of analytic functions associated with the generalized hypergeometric function, integral transform. spec. funct. 14(2003), 7-18. [11] a. gangadharan, t. n. shanmugam and h. m. srivastava, generalized hypergeometric functions associated with k-uniformly convex functions, comput. math. appl. 44(2002), no. 12, 1515-1526. [12] a. w. goodman, univalent functions and nonanalytic curves, proc. amer. math. soc. 8(1957), 598-601. [13] i. s. jack, functions starlike and convex functions of order α, j. london math. soc. (ser. 2) 2(1971), no. 3, 469-474. [14] s. b. joshi and m. k. aouf, meromorphic multivalent functions with positive and fixed second coefficients, kyungpook math. j. 35(1995), 163-169. [15] s. b. joshi and h. m. srivastava, a certain family of meromorphically multivalent functions, comput. math. appl. 38(1999), no. 3-4, 201-211. 18 murugusundaramoorthy and aouf [16] s. r. kulkarni, u. h. naik and h. m. srivastava, a certain class of meromorphically p-valent quasi-convex functions, panamer. math. j. 8(1998), no. 1, 57-64. [17] j. -l. liu, properties of some families of meromorphic p-valent functions, math. japon. 52(2000), no. 3, 425-434. [18] j. -l. liu, strongly starlike functions associated with the dziok-srivastava operator, tamkang j. math. 35(2004), no. 1, 37-42. [19] j. -l. liu and h. m. srivastava, a linear operator and associated families of meromorphically multivalent functions, j. math. anal. appl. 259(2001), 566-581. [20] j. -l. liu and h. m. srivastava, subclasses of meromorphically multivalent functions associated with a certain linear operator, math. comput. modelling 39(2004), 35-44. [21] j. -l. liu and h. m. srivastava, classes of meromorphically multivalent functions associated with the generalized hypergeometric function, math. comput. modelling 39(2004), 21-34. [22] m. l. mogra, meromorphic multivalent functions with positive coefficients. i, math. japon. 35(1990), no. 1, 1-11. [23] m. l. morga, meromorphic multivalent functions with positive coefficients. ii, math. japon. 35(1990), no. 6, 1089-1098. [24] s. owa, h. e. darwish and m. k. aouf, meromorphic multivalent functions with positive and fixed second coefficients, math. japon. 46(1997), no. 2, 231-236. [25] r. k. raina, on certain classes of analytic functions and applications to fractional calculus operators , integral tranform. spec. funct. 5(1997), 247-260.. [26] r. k. raina and t. s. nahar, a note on boundedness properties of wright’s generalized hypergeometric function, ann. math. blaise pascal 4 (1997), 83-95. [27] r. k. raina and t. s. nahar, on characterization of certain wright’s generalized hypergeometric functions involving certain subclasses of analytic functions, informatica 10 (1999), 219-230. [28] r. k. raina and t. s. nahar, on univalent and starlike wright’s hypergeometric functions, rend. sem. math. univ. padava 95 (1996) , 11-22. [29] r. k. raina and h. m. srivastava, a new class of meromorphically multivalent functions with applications to generalized hypergeometric functions, math. comput. modelling 43(2006), 350-356. [30] st. ruscheweyh, neighborhoods of univalent functions, proc. amer. math. soc. 81(1981), 521-527. [31] a. schild and h. silverman, convolution of univalent functions with negative coefficients, ann. univ. mariae curiesklodowska sect. a, 29(1975), 99-107. [32] h. m. srivastava h. m. hossen and m. k. aouf, a unified presentation of some classes of meromorphically multivalent functions, comput. math. appl. 38(1999), 63-70. [33] h. m. srivastava and p. w. karlsson, multiple gaussian hypergeometric series, halsted press ( ellis horwood ltd., chichester ), john wiley and sons, new york , chichester, brisbane and london 1985. [34] b. a. uralegaddi and c. somanatha, certain classes of meromorphic multivalent functions, tamkang j. math. 23(1992), 223-231. [35] b. a. uralegaddi and c. somanatha, certain classes of meromorphic multivalent functions , tamkang j. math. 23 (1992), 223-231. [36] e. m. wright, the asymptotic expansion of the generalized hypergeometric function, proc. london math. soc. 46 (1946), 389-408. [37] d. -g. yang, on new subclasses of meromorphic p-valent functions, j. math. res. exposition 15(1995), 7-13. [38] d. -g. yang, subclasses of meromorphic p-valent convex functions, j. math. res. exposition 20(2000), 215-219. 1school of advanced sciences, vit university, vellore-632014, india 2department of mathematics, mansoura university, mansoura 35516, egypt ∗corresponding author international journal of analysis and applications volume 18, number 6 (2020), 920-938 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-920 some multistep iterative methods for nonlinear equation using quadrature rule gul sana∗, muhammad aslam noor, khalida inayat noor department of mathematics, comsats university islamabad, park road, islamabad, pakistan ∗corresponding author: gulsana123@yahoo.com abstract. we introduce a sequence of third and fourth order iterative schemes to determine the roots of nonlinear equations by applying quadrature formula and decomposition approach. we also examined the convergence of suggested iterative methods under varied constraint. various numerical test examples are presented to exhibit the validity, efficiency and implementaion of our algorithms. 1. introduction an extensive range of problems that appeared in diverse directions of pure and applied sciences can be considered by mean of nonlinear equations in the framework of novel and inventive techniques. many authors proposed, analyzed and modified diversity of numerical methods using various approaches such as taylor series, decomposition methods, quadrature formulas, modified homotopy perturbation method and variational iterative method, and. a vast literature is available that highlight different methods used to solve nonlinear equations for example see, [1, 3–33] and references therein. in adomian decomposition method solution is considered in terms of an infinite series, which converges to towards exact solution. chun [6] and abbasbandy [1] constructed and investigated different higher order iterative methods by applying the decomposition technique of adomian [2]. darvishi and barati [9] also applied the adomian decomposition received june 22nd, 2020; accepted july 21st, 2020; published september 3rd, 2020. 2010 mathematics subject classification. 41a55. key words and phrases. convergence analysis; quadrature formula; iterative method; decomposition technique. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 920 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-920 int. j. anal. appl. 18 (6) (2020) 921 technique to develop newton type methods that are cubically convergent for the solution of system of nonlinear equation. implementation of this adomian decomposition technique required higher order derivatives evaluation, which is major pitfall of this method. to overcome this drawback, several new techniques have been suggested and analyzed by many researchers. daftardar-gejji and jafari [10] used different modifications of adomian decomposition method and introduced simple technique which do not need derivative evaluation of the adomian polynomial. numerous authors used the decomposition technique [10] and derived different higher order iterative methods. numerical computation of univariate integral is one of the most notable problem in numerical analysis. weerakon and fernando cite32 improved the convergence of newton method using quadrature rule. later on, ozban [27] investigated some new variant forms of newton method by using the concept of harmonic mean and mid-point rule. frontini and sormani [11] modified the p-dimensional case using quadrature formula and developed new iterative methods for system of nonlinear equations that are better than classical methods. noor [25] developed fifth order convergent iterative method using gaussian quadrature formula and investigated its efficacy as compared to the methods already existed in literature. ali et al.[3] introduced a family of iterative method using the quadrature formula as well as the fundamental theorem of calculus and decomposition technique and checked the validity and performance of these methods by examining two mathematical models. also in [21, 29, 30], decomposition technique [10] is merged sophistically with coupled system of equation to investigate various order iterative methods. motivated and inspired by the on-going research activities in this direction in this work we consider the well-known fixed point iterative method in which we rewrite the nonlinear equation f(u) = 0 as u = g(u). in section2, we use fundamental law of calculus, along with the functional equation u = g(u) as coupled system of equation and applying the decomposition technique [10]. we present and introduce some new iterative methods. section 3 consists of convergence analysis of proposed methods. some numerical examples are presented to make a comparative study of newly constructed methods with known third and fourth order convergent iterative algorithms. 2. creation of iterative methods this section comprises of some new multistep third and fourth order convergent iterative methods in view of mid-point, trapezoidal rule and decomposition technique [10]. 2.1. mid-point rule. consider the nonlinear equation f(u) = 0 (2.1) which is equivalent to u = g(u). (2.2) int. j. anal. appl. 18 (6) (2020) 922 assume that α is simple root of nonlinear equation (2.1) and γ is initial guess sufficiently close to root. using fundamental theorem of calculus and mid-point quadrature formula, we have u = g(γ) + (u−γ)g′( u + γ 2 ). (2.3) now using the technique of he [16], the nonlinear equation (2.1) can be written as an equivalent coupled system of equations u = g(γ) + (u−γ)g′( u + γ 2 ) + h(u) h(u) =g(u) −g(γ) − (u−γ)g′( u + γ 2 ) =u ( 1 −g′( u + γ) 2 ) + γg′( u + γ 2 ) −g(γ), (2.4) from which, it follows that u = h(u) 1 −g′ ( u+γ 2 ) + g(γ) −γg′(u+γ2 ) 1 −g′ ( u+γ 2 ) =c + m(u), (2.5) where c =γ (2.6) m(u) = h(u) 1 −g′(u+γ 2 ) + g(γ) −γ 1 −g′(u+γ 2 ) . (2.7) it is clear that m(u) is nonlinear operator.now we establish sequence of higher order iterative methods implementing decomposition technique presented by daftardar-gejji and jafari [10]. in this technique, the solution of (2.1) can be represented as in terms of infinite series. u = ∞∑ i=0 ui. (2.8) here the operator m(u) can be decomposed as m(u) = m(uo) + ∞∑ i=1 {m( i∑ j=0 uj) −m( i−1∑ j=0 uj)}. (2.9) thus from equation (2.5),(2.8) and (2.9), we have ∞∑ i=1 ui = c + m(uo) = ∞∑ i=1 m( i∑ j=0 uj) −m( i−1∑ j=0 uj), (2.10) int. j. anal. appl. 18 (6) (2020) 923 which generates the following iterative scheme uo = c u1 = m(uo) u2 = m(uo + u1) −m(uo) . . . un+1 = m( n∑ j=0 uj) −m( n−1∑ j=0 uj) n = 1, 2, ... (2.11) consequently, it follows that u1 + u2 + ... + un+1 = m(uo + u1 + u2 + ... + un), and u = c + ∞∑ i=1 ui. (2.12) it is noted that u is approximated by un = (uo + u1 + u2 + ... + un), and lim x→∞ un = u. for n=0, u ≈ uo = uo = c = γ. (2.13) from (2.4), it can easily be computed as h(uo) = 0. using(2.7), we get u1 = m(uo) = h(u0) 1 −g′(uo+γ 2 ) + g(γ) −γ 1 −g′(uo+γ 2 ) = g(γ) −γ 1 −g′(uo+γ 2 ) for n=1, u ≈ u1 = uo + u1 = uo + m(uo), int. j. anal. appl. 18 (6) (2020) 924 u ≈ u1 = uo + u1 = γ + g(γ) −γ 1 −g′(uo+γ 2 ) . using(2.13), we have u = g(γ) −γg′(γ) 1 −g′(γ) . (2.14) this fixed point formulation is used to suggest the following algorithm. algorithm2a. for a given u0 (initial guess), approximate solution un+1 is computed by the following iterative scheme un+1 = g(un) −ung′(un) 1 −g′(un) . (2.15) kang et al. [28] developed this algorithm and proved that algorithm 2a has quadratic convergence. from(2.4) and (2.7),we have h(uo + u1) = g(uo + u1) −g(γ) − (uo + u1 −γ)g′( uo + u1 + γ 2 ). thus u1 + u2 = m(uo + u1) = h(uo + u1) 1 −g′(uo+u1+γ 2 ) + g(γ) −γ 1 −g′(uo+u1+γ 2 ) = g(uo + u1) −g(γ) − (uo + u1 −γ)g′(uo+u1+γ2 ) 1 −g′(uo+u1+γ 2 ) + g(γ) −γ 1 −g′(uo+u1+γ 2 ) = g(uo + u1) 1 −g′(uo+u1+γ 2 ) − (uo + u1 −γ)g′(uo+u1+γ2 ) 1 −g′(uo+u1+γ 2 ) − γ 1 −g′(uo+u1+γ 2 ) for n=2, u ≈ u2 = uo + u1 + u2 = c + m(uo + u1). = γ − γ 1 −g′(uo+u1+γ 2 ) + g(uo + u1) 1 −g′(uo+u1+γ 2 ) − (uo + u1 −γ)g′(uo+u1+γ2 ) 1 −g′(uo+u1+γ 2 ) = g(uo + u1) 1 −g′(uo+u1+γ 2 ) − (uo + u1)g ′(uo+u1+γ 2 ) 1 −g′(uo+u1+γ 2 ) take uo + u1 = v = g(γ) −γg′(γ) 1 −g′(γ) = g(v) 1 −g′(v+γ 2 ) − vg′(v+γ 2 ) 1 −g′(v+γ 2 ) . int. j. anal. appl. 18 (6) (2020) 925 this relation yields the following two step method for solving nonlinear equation algorithm 2b. for a given initial guess u0, un+1 the approximated solution un+1 can be computed by the following iterative schemes. vn = g(un) −ung′(un) 1 −g′(un) (2.16) un+1 = g(vn) −vng′( un+vn) 2 1 −g′(un+vn) 2 ) , n = 0, 1, 2, .... (2.17) it is noted that u0 + u1 + u2 = w = g(uo + u1) 1 −g′(uo+u1+γ 2 ) − (uo + u1)g ′( (uo+u1+γ 2 ) 1 −g′(uo+u1+γ 2 ) . (2.18) from (2.4) and (2.7), we can write h(uo + u1 + u2) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)g′( uo + u1 + u2 + γ 2 ). and u1 + u2 + u3 = m(uo + u1 + u2) = h(uo + u1 + u2) 1 −g′(uo+u1+u2+γ 2 ) + g(γ) −γ 1 −g′(uo+u1+u2+γ 2 ) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 ) 1 −g′(uo+u1+u2+γ) 2 + g(γ) −γ 1 −g′(uo+u1+u2+γ 2 ) = g(uo + u1 + u2) 1 −g′(uo+u1+u2+γ 2 ) − (uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 ) 1 −g′(uo+u1+u2+γ 2 ) − γ 1 −g′(uo+u1+u2+γ 2 ) . for n=3, u ≈ u3 = uo + u1 + u2 + u3 = uo + m(uo + u1 + u2). = γ − γ 1 −g′(uo+u1+u2+γ 2 ) + g(uo + u1 + u2) 1 −g′(uo+u1+u2+γ 2 ) − (uo + u1 + u2 −γ)g′(uo+u1+u2+γ2 ) 1 −g′(uo+u1+u2+γ 2 ) . using (2.18), we have = γ + g(w) 1 −g′(w+γ 2 ) − (w −γ)g′(w+γ 2 ) 1 −g′(w+γ 2 ) − γ 1 −g′(w+γ 2 ) = g(w) 1 −g′(w+γ 2 ) − wg′(w+γ 2 ) 1 −g′(w+γ 2 ) . using this relation, we suggest the following three step method for solving nonlinear equation (2.1). int. j. anal. appl. 18 (6) (2020) 926 algorithm 2c. for a given initial guessu0, compute the approximate solution un+1 by the following iterative scheme vn = g(un) −ung′(un) 1 −g′(un) wn = g(vn) −vng′( un+vn) 2 1 −g′(un+vn) 2 ) (2.19) un+1 = g(wn) −wng′( un+wn) 2 1 −g′(un+wn) 2 ) , n = 0, 1, 2, ... (2.20) 2.2. trapezoidal rule. again using the technique of he [16] and fundamental law of calculus along with trapezoidal rule, we can obtain . u = g(γ) + (u−γ) (g′(u) + g′(γ) 2 ) + h(u) h(u) =g(u) −g(γ) − (u−γ) (g′(u) + g′(γ) 2 ) =u [ 1 − (g′(u) + g′(γ) 2 )] + γ (g′(u) + g′(γ) 2 ) −g(γ), (2.21) from which it follows that u = h(u) 1 − ( g′(u)+g′(γ) 2 ) + g(γ) −γ ( g′(u)+g′(γ) 2 ) 1 − ( (g′(u)+g′(γ) 2 ) =c + m(u), (2.22) where c =γ (2.23) m(u) = h(u) 1 − ( g′(u)+g′(γ) 2 ) + g(γ) −γ 1 − ( g′(u)+g′(γ) 2 ). (2.24) now applying decomposition technique of daftardar-gejji and jafari [10], we have for n=0, u ≈ uo = uo = c = γ. (2.25) from (2.21), it can easily be computed as h(uo) = 0. using(2.24), we get u1 = m(uo) = h(uo) 1 − ( g′(uo)+g′(γ) 2 ) + g(γ) −γ 1 − ( g′(uo)+g′(γ) 2 ) = g(γ) −γ 1 − ( g′(uo)+g′(γ) 2 ). int. j. anal. appl. 18 (6) (2020) 927 for n=1, u ≈ u1 = uo + u1 = uo + m(uo) (2.26) u ≈ u1 = uo + u1 = γ + g(γ) −γ 1 − ( (g′(uo)+g′(γ) 2 ). using(2.26),we have u = g(γ) −γg′(γ) 1 −g′(γ) . (2.27) this formulation determines the algorithm 2a. from(2.21) and (2.24),we have h(uo + u1) = g(uo + u1) −g(γ) − (uo + u1 −γ) (g′(uo + u1) + g′(γ) 2 ) . and u1 + u2 = m(uo + u1) = h(uo + u1) 1 − ( g′(uo+u1)+g′(γ) 2 ) + g(γ) −γ 1 − ( g′(uo+u1)+g′(γ) 2 ) = g(uo + u1) −g(γ) − (uo + u1 −γ) ( g′(uo+u1)+g ′(γ) 2 ) 1 − ( (g′(uo+u1)+g′(γ) 2 ) + g(γ) −γ 1 − ( g′(uo+u1)+g′(γ) 2 ) = g(uo + u1) 1 − ( g′(uo+u1)+g′(γ) 2 ) − (uo + u1 −γ) ( g′(uo+u1)+g ′(γ) 2 ) 1 − ( g′(uo+u1)+g′(γ) 2 ) − γ 1 − ( g′(uo+u1)+g′(γ) 2 ). for n=2, u ≈ u2 = uo + u1 + u2 = c + m(uo + u1). =γ − γ 1 − ( g′(uo+u1)+g′(γ) 2 ) + g(uo + u1) 1 − ( g′(uo+u1)+g′(γ) 2 ) − (uo + u1 −γ) ( (g′(uo+u1)+g ′(γ) 2 ) 1 − ( g′(uo+u1)+g′(γ) 2 ) . take uo + u1 = v = g(γ) −γg′(γ) 1 −g′(γ) . =γ − γ 1 − ( g′(v)+g′(γ) 2 ) − g(v) 1 − ( g′(v)+g′(γ) 2 ) − (v −γ) ( (g′(v)+g′(γ) 2 ) 1 − ( g′(v)+g′(γ) 2 ) = g(v) 1 − ( g′(v)+g′(γ) 2 ) − v ( g′(v)+g′(γ) 2 ) 1 − ( g′(v)+g′(γ) 2 ). int. j. anal. appl. 18 (6) (2020) 928 this relation yields the following two step method for solving nonlinear equation algorithm 2d. for a given initial guessu0, the approximate solution un+1 can be computed by the following iterative schemes vn = g(un) − (un)g′(un) 1 −g′(un) (2.28) un+1 = g(vn) −vn ( g′(vn)+g ′(un) 2 ) 1 − ( g′(vn)+g′(un) 2 ) n = 0, 1, 2, ... (2.29) it is noted that uo + u1 + u2 = w = g(uo + u1) 1 − ( g′(uo+u1)+g′(γ) 2 ) − (uo + u1) ( g′(uo+u1)+g ′(γ) 2 ) 1 − ( g′(uo+u1)+g′(γ) 2 ) . (2.30) from (2.21) and (2.24),we can write h(uo + u1 + u2) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ) (g′(uo + u1 + u2) + g′(γ) 2 ) and u1 + u2 + u3 =m(uo + u1 + u2) = h(uo + u1 + u2) 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) + g(γ) −γ 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) = g(uo + u1 + u2) −g(γ) − (uo + u1 + u2 −γ) ( g′(uo+u1+u2)+g ′(γ) 2 ) 1 − ( (g′(uo+u1+u2)+g′(γ) 2 ) + g(γ) − (γ)( 1−g′(uo+u1+u2)+g′(γ) 2 ) = g(uo + u1 + u2) 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) − (uo + u1 + u2 −γ) ( (g′(uo+u1+u2)+g ′(γ) 2 ) 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) − γ 1 − ( g′(uo+u1+u2)+g′(γ) 2 ). for n=3, u ≈ u3 = uo + u1 + u2 + u3 = uo + m(uo + u1 + u2) = γ − γ 1 − ( g(uo+u1+u2)+g′(γ) 2 ) + g(uo + u1 + u2) 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) − (uo + u1 + u2 −γ) ( (g′(uo+u1+u2)+g ′(γ) 2 ) 1 − ( g′(uo+u1+u2)+g′(γ) 2 ) . int. j. anal. appl. 18 (6) (2020) 929 using (2.29), we obtain =γ − γ 1 − ( g′(w)+g′(γ) 2 ) − g(w) 1 − ( g′(w)+g′(γ) 2 ) − (w −γ) ( g′(w)+g′(γ) 2 ) 1 − ( g′(w)+g′(γ) 2 ) = g(w) 1 − ( g′(w)+g′(γ) 2 ) − w ( g′(w)+g′(γ) 2 ) 1 − ( g′(w)+g′(γ) 2 ). this formulation yeilds the following three step method for solving nonlinear equation(2.1). algorithm 2e. for a given initial guessu0, the approximated solution un+1 is computed by the following iterative schemes. vn = g(un) −ung′(un) 1 −g′(un) wn = g(vn) −vn ( g′(vn)+g ′(un) 2 ) 1 − ( g′(vn)+g′(un) 2 ) (2.31) un+1 = g(wn) −wn ( g′(wn)+g ′(un) 2 ) 1 − ( g′(wn)+g′(un) 2 ) , n = 0, 1, 2, ... (2.32) 3. convergence analysis of proposed iterative methods this section comprises of convergence analysis of proposed algorithms 2b, 2d, 2c and 2e and it shown that these methods are third and fourth order convergent respectively. theorem 3.1.let i ⊂ r be an open interval and f : i → r is differential function.if β ∈ i be root of f(u) = 0 and u0 is sufficiently close to β then multistep method defined by algorithm 2b and algorithm 2c has at least third and fourth order of convergence and satisfies the error equation. poof. let β be the root of nonlinear equation f(u) = 0 or equivalently u = g(u). let en and en+1 be the errors at nth and (n+1) iterations respectively. now expanding g(u) and g′(u) by using taylor series about α, we have g(u) = β + eng ′(β) + e2n 2 g′′(β) + e3n 6 g′′′(β) + o(e4n) g′(u) = g′(β) + eng ′′(β) + e2n 2 g′′′(β) + e3n 6 giv(β) + o(e4n) g(un) −ung(un) = β −βg′(β) −βg′′(β)en − 1 2 ( g′′(β) + βg′′′(β) ) e2n + 1 6 ( 2g′′′(β) + βgiv(β) ) + o(e4n) 1 −g′(un) = 1 −g′(β) −g′′(β)en − e2n 2 g′′′(β) − e3n 6 giv(β) + o(e4n) g(un) −ung(un) 1 −g′(un) = β + g′′(β) 2(−1 + g′(β)) e2n − 2g′′′(β) − 2g′′′(β)g′(β) + 3g′′2(β) 6(−1 + g′(β))2 e3n + o(e 4 n). int. j. anal. appl. 18 (6) (2020) 930 from (2.16), we obtain vn = β + g′′(β) 2(−1 + g′(β)) e2n + (−2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β) 6(−1 + g′(β))2 ) e3n + 1 12(−1 + g′(β))3 ( 2giv(β) − 4giv(β)g′(β) + 2giv(β)g′2(β) + 7g′′(β)g′′′(β) − 7g′′(β)g′′′(β)g′(β) + 6g′′3(β) ) e4n + o(e 5 n). g(vn) = β+ g′(β)g′′(β) 2(−1 + g′(β)) e2n + ( g′(β) ( − 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β) )) 6(−1 + g′(β))2 e3n + 1 24(−1 + g′(β))3 ( 4giv(β)g′(β) − 8giv(β)g′2(β) + 4giv(β)g′3(β) − 14g′′(β)g′′′(β)g′2(β) +14g′′(β)g′′′(β)g′(β) + 15g′′3(β)g′(β) − 3g′′3(β) ) e4n + o(e 5 n). (3.1) expanding g′ ( vn+un 2 ) in terms of taylors series about α, we have g′ (vn + un 2 ) = g′(β) + g′′(β) 2 en + ( 2g′′2(β) + g′′′(β)g′(β) −g′′′(β) ) 8(−1 + g′(β)) e2n + (−14g′′(β)g′′′(β) + 14g′′(β)g′′′(β)g′(β) + giv(β) − 12g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β) 48(−1 + g′(β))2 ) e3n + 1 96(−1 + g′(β))3 ( 11giv(β)g′′(β) − 22giv(β)g′(β)g′′(β) + 11giv(β)g′2(β)g′′(β) + 37g′′2(β)g′′′(β) − 37g′′2(β)g′′′(β)g′(β) + 24g′′4(β) + 8g′′′2(β) − 16g′′′3(β)g′(β) + 8g′2(β)g′′′2(β) ) + o(e5n). (3.2) g(vn) −vng′ (vn + un 2 ) = −β(−1 + g′(β)) − 1 2 g′′(β)en − 1 8(−1 + g′(β) ( β ( 2g′′2(β) −g′′′(β) −g′′′(β)g′(β) ) e2n − 1 48(−1 + g′(β))2 ( − 14βg′′(β)g′′′(β) + 14βg′(β)g′′(β)g′′′(β) − 12βg′′′2(β) +βgiv(β) − 2(β)giv(β)g′(β) + βgiv(β)g′2(β) − 12g′′2(β) + 12g′(β)g′′2(β) ) e3n + 1 96(−1 + g′(β))3 ( − 44g′′2(β)g′′′(β)g′(β) + 22g′′′(β)g′′(β)g′2(β) + 8βg′′′2(β) + 24g′′3(β) −24g′(β)g′′3(β) + 22g′′′(β)g′′(β) + 11βg′′(β)giv(β) + 37βg′′2(β)g′′′(β) − 16βg′′′2(β)g′(β) +8g′′′2(β)g′2(β) + 24βg′′4(β) − 22βg′(β)g′′(β)giv(β) + 11βg′2(β)g′′(β)giv(β) − 37βg′(β)g′′′(β)g′′2(β) ) e4n + o(e 5 n). (3.3) int. j. anal. appl. 18 (6) (2020) 931 1 −g′ (vn + un 2 ) = 1 −g′(β) − g′′(β) 2 en + ( 2g′′2(β) + g′′′(β)g′(β) −g′′′(β)) ) 8(−1 + g′(β)) e2n − (−14g′′(β)g′′′(β) + 14g′′(β)g′′′(β)g′(β) + giv(β) − 12g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β) 48(−1 + g′(β))2 ) e3n − 1 96(−1 + g′(β))3 ( 11giv(β)g′′(β) − 22giv(β)g′(β)g′′(β) + 11giv(β)g′2(β)g′′(β) + 37g′′2(β)g′′′(β) −37g′′2(β)g′′′(β)g′(β) + 24g′′4(β) + 8g′′′2(β) − 16g′′′3(β)g′(β) + 8g′2(β)g′′′2(β) ) e4n + o(e 5 n) (3.4) dividing (3.3),(3.4) and from (2.19), we have wn = g(vn) −vng′ ( vn+un 2 ) 1 −g′ ( vn+un 2 ) = β + ( g′′2(β) 4(−1 + g′(β))2 ) e3n + 1 48(−1 + g′(β))3 ( g′′(β) ( 11g′′′(β)g′(β) − 11g′′′(β) − 18g′′2(β) )) e4n + o(e5n). expanding g(wn) in terms of taylor series about α, we obtain g(wn) = β + ( g′(β)g′′2(β) 4(−1 + g′(β))2 ) e3n − 1 48(−1 + g′(β))3 ( g′(β)g′′(β) ( − 11g′′′(β)g′(β) + 11g′′′(β) + 18g′′2(β) )) e4n + o(e 5 n). (3.5) expanding g ( wn+un 2 ) in terms of taylor series about α, we get g′ (wn + un 2 ) = g′(β) + g′′(β) 2 en + g′′′(β) 8 e2n + 1 48(−1 + g′(β))2 ( 6g′′3(β) + giv(β) − 2giv(β)g′(β) +giv(β)g′2(β) )) e3n − 1 96(−1 + g′(β))3 ( g′′2(β) − 17g′′′(β)g′(β) + 17g′′′(β) + 18g′′2(β) ) e4n + o(e 5 n). 1 −g ( wn + un 2 ) = 1 −g′(β) − g′′(β) 2 en − g′′′(β) 8 e 2 n − ( 6g′′3(β) + giv(β) − 2giv(β)g′(β) + giv(β)g′2(β)) ) 48(−1 + g′(β))2 e 3 n − ( g ′′2 (β) (−17g′′(β)g′(β) − 17g′′′(β) + 18g′′2(β) 96(−1 + g′(β))3 )) e 4 n + o(e 5 n). (3.6) now using (2.20)and dividing (3.5),(3.6), simplifying un+1 = g(wn) −wng′( wn+un2 ) 1 −g′( wn+un 2 ) =β + ( g′′3(β) 8(−1 + g′(β))3 ) e 4 n + o(e 5 n) en+1 =β + ( g′′3(β) 8(−1 + g′(β))3 ) e 4 n + o(e 5 n). int. j. anal. appl. 18 (6) (2020) 932 theorem 3.2.let f : i → r is differential function and i ⊂ r be an open interval . if β ∈ i be root of f(u) = 0 and u0 is sufficiently close to β then multistep method defined by algorithm 2d and algorithm 2e has at least third and fourth order of convergence and satisfies the error equation. proof. from equation (3.1), we have g(vn) = β + g′(β)g′′(β) 2(−1 + g′(β)) e 2 n + ( g′(β) ( − 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β) )) 6(−1 + g′(β))2 e 3 n + 1 24(−1 + g′(β))3 ( 4g iv (β)g ′ (β) − 8giv(β)g′2(β) + 4giv(β)g′3(β) − 14g′′(β)g′′′(β)g′2(β) +14g ′′ (β)g ′′′ (β)g ′ (β) + 15g ′′3 (β)g ′ (β) − 3g′′3(β) ) e 4 n + o(e 5 n). g ′ (vn) = g ′ (β) + g′′2(β) 2(−1 + g′(β)) e 2 n + ( − 2g′′′(β) + 2g′′′(β)g′(β) − 3g′′2(β) ) 6(−1 + g′(β))2 e 3 n + 1 24(−1 + g′(β)3) ( g ′′ (β) ( 4g iv (β) − 8giv(β)g′(β) + 4giv(β)g′2(β) + 11g′′(β)g′′′(β) −11g′′(β)g′′′(β)g′(β) + 12g′′3(β) )) e 4 n + o(e 5 n). expanding ( g′(vn)+g ′(un) 2 ) in terms of taylors series about α, we have ( g′(vn) + g ′(un) 2 ) = g ′ (β) + g′′(β) 2 en + ( g′′2(β) + g′′′(β)g′(β) −g′′′(β) ) 4(−1 + g′(β)) e 2 n + 1 12(−1 + g′(β))2 ( − 2g′′(β)g′′′(β) + 2g′′(β)g′′′(β)g′(β) + giv(β) − 3g′′3(β) − 2giv(β)g′(β) +g iv (β)g ′2 (β) ) e 3 n + 1 48(−1 + g′(β))3 ( g ′′ (β) ( 4g iv (β) − 8giv(β)g′(β) + 4giv(β)g′2(β) +11g ′′ (β)g ′′′ (β) − 11g′′(β)g′′′(β)g′(β) + 12g′′3(β) )) + o(e 5 n). g(vn) −vn ( g′(vn) + g ′(un) 2 ) = −β(−1 + g′(β)) − 1 2 βg ′′ (β)en − 1 4(−1 + g′(β) ( β ( −g′′′(β) −g′′2(β) +g ′′′ (β)g ′ (β) ) e 2 n − 1 12(−1 + g′(β))2 ( − 2βg′′(β)g′′′(β) + 2βg′(β)g′′(β)g′′′(β) + βgiv(β) − 2βgiv(β)g′(β) +βg iv (β)g ′2 (β) − 3(β)g′′2(β) − 3g′′2(β) + 3g′(β)g′′2(β) ) e 3 n − 1 48(−1 + g′(β))3 ( g ′′ (β) ( 4βg iv (β) + 11βg ′′′ (β)g ′′ (β) − 8βgiv(β)g′(β) + 4βgiv(β)g′2(β) −11βg′(β)g′′(β)g′′′(β) − 28g′(β)g′′′(β) + 14g′′(β)g′2(β) + 12g′′2(β) − 12g′(β)g′′2(β) +14g ′′′ (β) + 12g ′′3 (β) ) e 4 n + o(e 5 n). (3.7) 1 − ( g′(vn) + g ′(un) 2 ) = 1 −g′(β) − g′′(β) 2 en − ( g′′2(β) + g′′′(β)g′(β) −g′′′(β)) ) 4(−1 + g′(β)) e 2 n − (−2g′′(β)g′′′(β) + 2g′′(β)g′′′(β)g′(β) + giv(β) − 3g′′3(β) − 2giv(β)g′(β) + giv(β)g′2(β) 12(−1 + g′(β))2 ) e 3 n − 1 48(−1 + g′(β))3 ( g ′′ (β) ( 4g iv (β) − 8giv(β)g′(β) + 4giv(β)g′2(β) + 11g′′(β)g′′′(β) −11g′′(β)g′′′(β)g′(β) + 12g′′3(β) )) + o(e 5 n). (3.8) int. j. anal. appl. 18 (6) (2020) 933 subsituting the values (3.7),(3.8) in (2.30) and simplifying, we obtain wn = g(vn) −vn ( g′(vn)+g ′(un) 2 ) 1 − ( g′(vn)+g′(un) 2 ) =β + ( g′′2(β) 4(−1 + g′(β))2 ) e 3 n + 1 24(−1 + g′(β))3 ( g ′′ (β) ( 7g ′′′ (β) − 7g′′′(β) − 9g′′2(β) )) e 4 n + o(e 5 n). expanding g(wn) in terms of taylor series about α, we obtain g(wn) = β + ( g′(β)g′′2(β) 4(−1 + g′(β))2 ) e 3 n + 1 24(−1 + g′(β))3 ( g ′′ (β) ( 7g ′′′ (β)g ′ (β) − 7g′′′(β) − 9g′′2(β) )) e 4 n + o(e 5 n). (3.9) expanding ( g′(wn)+g ′(un) 2 ) in terms of taylor series about α, we have ( g′(wn) + g ′(un) 2 ) = g ′ (β) + g′′(β) 2 en + g′′′(β) 4 e 2 n + 1 24(−1 + g′(β))2 ( 2g iv (β) − 4giv(β)g′(β) + 2g iv (β)g ′2 (β) + 3g ′′3 (β)) ) e 3 n + (g′′2(β)(7g′′′(β)g′(β) − 7g′′′(β) − 9g′′2(β) 48(−1 + g′(β))3 ) e 4 n + o(e 5 n). (3.10) g(wn) −wn ( g′(wn) + g ′(un) 2 ) = −β(−1 + g′(β)) − 1 2 βg ′′ (β)en − 1 4 βg ′′′ (β)e 2 n − 1 24(−1 + g′(β))2 ( β ( 2g iv (β) − 4giv(β)g′(β) + 2giv(β)g′2(β) + 3g′′3(β) )) e 3 n − 1 48(−1 + g′(β))3 ( g ′′2 (β) ( 7βg ′′′ (β)g ′ (β) − 9βg′′2(β) − 7βg′′′(β) − 6g′′(β) + 6g′(β)g′′(β) )) e 4 n +o(e 5 n). (3.11) 1 − ( g′(wn) + g ′(un) 2 ) = 1 −g′(β) − g′′(β) 2 en − g′′′(β) 4 e 2 n − 1 24(−1 + g′(β))2 ( 2g iv (β) −4giv(β)g′(β) + 2giv(β)g′2(β) + 3g′′3(β) ) e 3 n − 1 48(−1 + g′(β))3 ( g ′′2 (β) ( 7g ′′′ (β)g ′ (β) − 7g′′′(β) − 9g′′2(β) )) e 4 n +o(e 5 n). (3.12) substituting the values from (3.11) and (3.12) in (2.31) and simplifying un+1 = g(wn) −wng′( g ′(wn)+g ′(un) 2 ) 1 − ( g ′(wn)+g′(un) 2 ) =β + ( g′′2(β) 8(−1 + g′(β))3 ) e 4 n + o(e 5 n) en+1 =β + ( g′′2(β) 8(−1 + g′(β))3 ) e 4 n + o(e 5 n). this shows that algoithm 2e has fourth order convergence. � int. j. anal. appl. 18 (6) (2020) 934 4. numerical results and applications this section elaborated the efficacy of algorithms introduced in this paper with the support of examples. we obtain estimated simple root rather than the exact based on the exactness � of the computer. we utilize �=10−15,for computational work we use the following stopping criteria (i). |un+1 −un| < � and (ii). |f(un+1| < � we make a comparative representation of the chun method (cm)[6], newton method (nm), the abbasbandy method (am)[1],halley method (hm) [12,13], weerakoon and fernando(wf) [30], with the algorithm 2b (ag1), algorithm 2c (ag2) and algorithm 2d (ag3) and algorithm 2e (ag4).as for the convergence criteria, it was required that the distance of two successive estimation for the zero was less than 10−15. in table 1 we also displayed, the number of iterations (it), the approximate root un and the valuef(un) the examples are the same as in chun [4] f1(u) = u 3 − 10, g(u) = √ 10 u f2(u) = cosu−u, g(u) =cosu f3(u) = (u− 1)3 − 1, g(u) =1 + √ 1 u− 1 f4(u) = e u2+7u−30 − 1, g(u) = 1 7 (30 −u2) f5(u) = sin 2 u−u2 + 1, g(u) =sinu + 1 sinu + u f6(u) = u 2 −eu − 3u, g(u) = u2 −eu + 2 3 table1.numerical examples f1(u) = u 3 − 10, g(u) = √ 10 u , u0=1.5 methods it un f(un) δ = |un −un−1| nm 6 2.1544346900318837 7.857615e-27 3.486728e-14 hm 6 2.1544346900318837 6.573851e-30 1.008516e-15 am 4 2.1544346900318837 3.071853e-26 1.972938e-09 cm 9 2.1544346900318837 1.399647e-29 1.040560e-15 wf 4 2.1544346900318837 7.066989e-31 5.866631e-11 ag1 3 2.1544346900318837 8.936935e-27 3.625733e-09 ag2 3 2.1544346900318837 9.833722e-66 1.458067e-16 ag3 3 2.1544346900318837 4.856920e-26 6.374604e-09 ag4 3 2.1544346900318837 7.081450e-63 7.553158e-16 f2(u) = cosu−u, g(u) = cosu , u0=1.7 int. j. anal. appl. 18 (6) (2020) 935 nm 4 0.7390851332151609 3.924473e-16 3.258805e-08 hm 5 0.7390851332151606 2.373589e-27 8.014391e-14 am 4 0.7390851332151606 8.935133e-33 2.845706e-11 cm 5 0.7390851332151606 7.035845e-25 9.756878e-13 wf 3 0.7390851332151606 1.735664e-21 4.084953e-07 ag1 3 0.7390851332151606 5.258162e-26 8.637471e-09 ag2 3 0.7390851332151606 3.458981e-60 3.722340e-15 ag3 3 0.7390851332151606 1.117701e-27 2.392676e-09 ag4 3 0.7390851332151606 2.655872e-66 1.101871e-16 f3(u) = (u− 1)3 − 1, g(u) = 1 + √ 1 u−1 , u0=3.5 nm 7 2 2.484117e-21 2.877567e-11 hm 8 2 8.422670e-24 1.675577e-12 am 6 2 2.626229e-20 6.615927e-11 cm 5 2 7.505295e-35 2.657272e-12 wf 5 2 9.839450e-37 6.550899e-13 ag1 3 2 1.956948e-17 4.708259e-06 ag2 3 2 1.845148e-41 1.408551e-10 ag3 3 2 2.746096e-20 5.271145e-07 ag4 3 2 9.745968e-50 1.200801e-12 f4(u) = e u2+7u−30 − 1, g(u) = 1 7 (30 −u2) , u0=3.5 nm 12 3 5.475744e-24 2.530687e-13 hm diverge am 7 3 4.775960e-18 2.353603e-07 cm 8 3 9.665693e-21 7.518279e-12 wf 8 3 5.290016e-24 1.916154e-09 ag1 4 3 1.938304e-114 2.931715e-38 ag2 3 3 2.118685e-21 2.446184e-05 ag3 3 3 3.811229e-37 1.704784e-12 ag4 3 3 4.174513e-90 1.629758e-22 int. j. anal. appl. 18 (6) (2020) 936 f5(u) = sin 2u−u2 + 1, g(u) = sinu + 1 sinu+u , u0= -1 nm 6 1.4044916482153412 1.819126e-25 3.058088e-13 hm 6 1.4044916482153412 9.561689e-28 2.217103e-14 cm 6 1.4044916482153412 1.669602e-16 6.551037e-09 am 4 1.4044916482153412 6.138132e-21 1.329320e-07 wf 4 1.4044916482153412 9.413532e-30 1.793023e-10 ag1 4 1.4044916482153412 2.097175e-88 9.878333e-30 ag2 3 1.4044916482153412 7.391875e-68 3.273071e-17 ag3 3 1.4044916482153412 1.895915e-31 9.551650e-11 ag4 3 1.4044916482153412 3.473304e-75 4.818950e-19 f6(u) = u 2 −eu − 3u, g(u) = u 2−eu+2 3 , u0=2 nm 5 1.4044916482153412 3.439576e-27 9.869210e-14 hm 8 1.4044916482153412 1.163256e-29 5.739414e-15 cm 4 1.4044916482153412 7.664653e-31 1.280280e-10 am 5 1.4044916482153412 9.348485e-20 3.638191e-10 wf 4 1.4044916482153412 6.103947e-34 1.630507e-11 ag1 3 1.4044916482153412 4.442861e-18 5.125169e-06 ag2 3 1.4044916482153412 7.862110e-56 7.105574e-14 ag3 4 1.4044916482153412 6.398407e-50 6.398407e-501 ag4 3 1.4044916482153412 5.810512e-38 2.083378e-09 5. conclusion we have developed the algorithms 2a,2b,2c,2d,2e by approximating the definite integral ∫u γ g(u)du = g(u)−g(γ) by mean of quadrature rule along with writing the nonlinear equation as coupled system of equation and using the decomposition method. we have determined the convergence analysis of the newly suggested iterative schemes. with the help of test examples, computational comparison has been made with well known third and fourth order convergent iterative methods. moreover it is observed that multistep algorithms suggested in this article requires only first derivative evaluation of function and our results can be considered as an alternative to already known third and fourth order convergent iterative method. int. j. anal. appl. 18 (6) (2020) 937 acknowledgements the authors would like to thank the rector, comsats university islamabad,islamabad pakistan, for providing excellent research and academic environments. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. abbasbandy, improving newton–raphson method for nonlinear equations by modified adomian decomposition method., appl. math. comput. 145 (2003), 887–893. [2] g. adomian, nonlinear stochastic systems and applications to physics, kluwer academic publishers, dordrecht, 1989. [3] f. ali, w. aslam, a. rafiq, some new iterative techniques for the problems involving nonlinear equations, int. j. comput. math. 16 (2019),1-18. [4] e. babolian, j. biazar, a.r. vahidi, solution of a system of nonlinear equations by adomian decomposition method, appl. math. comput. 150 (3) (2004), 847–854. [5] f.a. shah, m. darus, i. faisal, m.a. shafiq, decomposition technique and family of efficient schemes for nonlinear equations, discrete dyn. nat. soc. 2017 (2017), 3794357. [6] c. chun, iterative methods improving newton’s method by the decomposition method, comput. math. appl. 50 (2005), 1559–1568. [7] c. chun, y. ham, some fourth-order modifications of newton’s method, appl. math. comput. 197 (2) (2008), 654–658. [8] a. cordero, j.r. torregrosa, variants of newton’s method using fifth-order quadrature formulas, appl. math. comput. 190 (1) (2007), 686–698. [9] m.t. darvishi, a. barati, a third-order newton-type method to solve systems of nonlinear equations, appl. math. comput. 187 (2007) ,630–635. [10] v. daftardar-gejji and h. jafari, an iterative method for solving nonlinear functional equations, j. math. anal.and appl. 316 (2) (2006), 753–763. [11] m. frontini, e. sormani, third-order methods from quadrature formulae for solving systems of nonlinear equations, appl. math. comput. 149 (3) (2004), 771– 782. [12] e. halley, a new, exact and easy method for finding the roots of equations generally, and that without any previous reduction, phil. r. soc. lond. 18 (1964), 136–147. [13] a. melman, geometry and convergence of halley’s method, siam rev. 39 (4) (1997), 728–735. [14] j.h. he, variational iteration method some new results and new interpretations, j. comput., appl. math. 207 (1) (2007), 3-17. [15] j.h. he, homotpy perturbation technique, comp. math. appl. mech. eng. 178 (3-4) (1999), 257-262. [16] j.h. he, a new iteration method for solving algebraic equations, appl. math. comput. 135 (2003), 81–84. [17] h.h. homeier, on newton-type methods with cubic convergence, j. comput. appl. math. 176 (2) (2005), 425–432. [18] v.i. hasanov, i.g. ivanov, and g. nedzhibov, a new modification of newton method, appl. math. eng. 27 (2002), 278–286. [19] m.a. noor, k.i. noor, s.t. mohyud-din, a. shabbir, an iterative method with cubic convergence for nonlinear equations, appl. math. comput. 183 (2006), 1249–1255. [20] m.a. noor, new iterative schemes for nonlinear equations, appl. math. comput. 187 (2007), 937–943. int. j. anal. appl. 18 (6) (2020) 938 [21] m.a. noor, k.i. noor, e. al-said, m. waseem, some new iterative methods for nonlinear equations, math. probl. eng. 2010, (2010), 198943. [22] m.a. noor, some iterative methods for solving nonlinear equations using homotopy perturbation method, int. j. comput. math. 87 (2010), 141–149. [23] m.a. noor, k.i. noor, e. al-said, m. waseem, higher-order iterative algorithms for solving nonlinear equations, world appl. sci. j., 16 (2012), 1657–1663. [24] m.a. noor, m. waseem, k.i. noor, m.a. ali, new iterative technique for solving nonlinear equations, appl. math. comput. 265 (2015), 1115–1129. [25] m.a. noor, fifth order convergent iterative method for solving nonlinear equation using quadrature formula, j. math. control sci. appl. 4 (1) (2018), 95-104. [26] o. ogbereyivwe, k.o. muka, on the efficiency of family of quadrature based methods for solving nonlinear equations, glob. sci. j. 6 (9) (2018), 149-159. [27] a.y. ozban, some new variants of newton’s method, appl. math. lett. 17 (6) (2004), 677–682. [28] s.m. kang, a. rafiq, y.c. kwun, a new second-order iteration method for solving nonlinear equations, abstr. appl. anal. 2013 (2013), 48706. [29] s.m. kang, m. saqib, m. fahad, w. nazeer, two new third and fourth order algorithms for resolution of nonlinear scalar equations based on decomposition technique, far east j. math. sci. 101 (3) (2017), 457-471. [30] m. saqib, m. iqbal, some multi-step iterative methods for solving nonlinear equations, open j. math. sci. 1 (2017), 25-33. [31] m. saqib, m. iqbal, s. ali, t. ismail, new fourth and fifth order iterative methods for solving nonlinear equations, appl. math. 6 (2015), 1220-1227. [32] s. weerakoon, t.g.i. fernando, a variant of newton’s method with accelerated third-order convergence, appl. math. letter., 13(8) (2000), 87-93. [33] m. waseem, m.a. noor, k.i. noor, f.a. shah, k.i. noor, an efficient technique to solve nonlinear equations using multiplicative calculus, turk. j. math. 42 (2018), 679-691. 1. introduction 2. creation of iterative methods 2.1. mid-point rule 2.2. trapezoidal rule 3. convergence analysis of proposed iterative methods 4. numerical results and applications 5. conclusion acknowledgements references international journal of analysis and applications volume 19, number 4 (2021), 619-632 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-619 on ω-interpolative berinde weak contraction in quasi-partial b-metric space pragati gautam1,∗, swapnil verma1, manuel de la sen2, prachi rakesh marwaha1 1department 0f mathematics, kamala nehru college, university of delhi, august kranti marg, new delhi 110049, india 2institute of research and development of processes, university of basque country, campus of leioa(bizkaia)-aptdo, 644-bilbao, bilbao, 48080, spain ∗corresponding author: pgautam@knc.du.ac.in abstract. the aim of this paper is to introduce interpolative weak contraction in the notion of berinde weak operator in quasi-partial b-metric space and to extend and generalize fixed point results by adopting the condition of ω-admissibility. we also discussed convex contraction mapping and obtained a fixed point result in the setting of berinde weak operator in quasi-partial b-metric space. consequently, we present some examples to show the applicability of the concept. 1. introduction and preliminaries in 1922, banach [1] introduced the highly recognized banach’s contraction principle in the field of nonlinear analysis. this result is used to prove the uniqueness of fixed point theorems as well as in picard theorems. the banach’s contraction in metric space is stated as follow : theorem 1.1. [1] let us consider (m,d) to be a complete metric space and t : m → m is the given self mapping. let ζ ∈ (0, 1) such that d(tτ,tυ) ≤ ζd(τ,υ) received may 27th, 2021; accepted june 21st, 2021; published july 8th, 2021. 2010 mathematics subject classification. 47h10, 49t99, 54h25. key words and phrases. quasi-partial b-metric space; ω-admissibility; interpolation; berinde weak contractions; fixed point. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 619 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-619 int. j. anal. appl. 19 (4) (2021) 620 for all τ,υ ∈ m. then t has a unique fixed point in m. in 1992, matthews [2] brought up the concept of partial metric space. motivated by which in 1993, czerwik [3] introduced a more generalized form of banach fixed point theorem in b-metric space. many authors including miculescu et al. [4], oltra et al [5], and valero [6] introduced some fixed point results and its topological properties. later, karapinar [7] and shukla [8] introduced quasi-partial metric-space and partial b-metric space respectively. in 1968, kannan [9] removed the continuity condition from the banach contraction principle. theorem 1.2. [9] suppose (m,d) is a complete metric space and t : m → m is called the kannan contraction mapping. let ζ ∈ [ 0, 1 2 ) such that d(tτ,tυ) ≤ ζ[d(τ,tτ) + d(υ,tυ)] for all τ,υ ∈ m. then t has a unique fixed point in m. in the year 2004, berinde [10] brought up the concept of berinde contraction also known as almost contractions and stated that : theorem 1.3. [10] let (m,d) is a complete metric space and t : m → m is almost contraction if there exists ζ ∈ [0, 1) and some r ≥ 0 such that d(tτ,tυ) ≤ ζd(τ,υ) + rd(υ,tτ) for all τ,υ ∈ m. hence, t has a unique fixed point in m and is called a weak contraction. additionally, berinde obtained some fixed point theorems that serves as the most important results in literature. some results are the generalization of banach, kannan, chatterjea, and c̀iric̀ etc. others can be found in [11–13]. recently, turkoglu [14] formulated a fixed point theorem consisting of four mappings using the concept of berinde contraction in partial metric spaces. in 2018, the idea of interpolative kannan-type contraction was introduced by karapinar [15] which is proved to be a very notable outcome for solving mathematical analysis. theorem 1.4. [15] suppose a self mapping t : m → m in complete metric space (m,d) is an interpolative kannan-type contraction, i.e. there exist ζ ∈ [0, 1) and α ∈ (0, 1) such that d(tτ,tυ) ≤ ζ[d(τ,tυ)]α[d(υ,tυ)]1−α for all τ,υ ∈ m with τ 6= tτ. then t has a fixed point in m. taking this approach forward ampadu [16] introduced the concept of interpolative berinde weak contraction in metric space and expanded its approach in cone metric and partial metric space as well. int. j. anal. appl. 19 (4) (2021) 621 definition 1.1. [16] let us consider (m,d) is a metric space. let t : m → m is an interpolative berinde weak operator if it satisfies, (1.1) d(tτ,tυ) ≤ ζ[d(τ,υ)]α.[d(τ,tτ)]1−α for all τ,υ ∈ m and τ,υ /∈ fix(t), where, ζ ∈ [0, 1) and α ∈ (0, 1) . remark 1.1. the fix(t) denotes the set of all fixed points of t . the idea of ω-orbital admissible maps was introduced by popescu [17] as a refining of α-admissible maps of samet et al [18]. further, karapinar proposed that definition 1.2. [19] let ω : m×m → [0,∞) and m is non-empty subset and t : m×m be a self mapping is called ω-orbital admissible if for all v ∈ m, we have (1.2) ω(v,tv) ≥ 1 implies ω(tv,tv2) ≥ 1. recently in 2015, gupta and gautam [20, 21] brought up the concept of quasi-partial b-metric space and its related topological properties. several authors [22–29] have made valuable contributions in this research field. definition 1.3. [20] a quasi-partial b-metric on a non-empty set m is a function qpb : m ×m → r+ such that for some real number s ≥ 1 and for all τ,υ,χ ∈ m following condition hold: (qpb1) qpb(τ,τ) = qpb(τ,υ) = qpb(υ,υ) implies τ = υ, (qpb2) qpb(τ,τ) ≤ qpb(τ,υ), (qpb3) qpb(τ,τ) ≤ qpb(υ,τ), (qpb4) qpb(τ,υ) ≤ s[qpb(τ,χ) + qpb(χ,υ)] −qpb(χ,χ). a quasi-partial b-metric space is a pair (m,qpb) such that m is a nonempty set and qpb is a quasi-partial b-metric on m. the number s is called the coefficient of (m,qpb). example 1.1. let m = [0,∞). define qpb = ln(k) |τ −υ| + τ, where k ≥ 1 here qpb(τ,τ) = qpb(τ,υ) = qpb(υ,υ) =⇒ ln(k) |τ − τ| + τ = ln(k) |τ −υ| + τ = ln(k) |υ −υ| + υ. therefore, (qpb1) holds for any (τ,υ ∈ m ×m). now as ln(k) |τ −υ| ≥ |τ −υ| ≥ 0 then qpb(τ,τ) = τ ≤ qpb(τ,υ) int. j. anal. appl. 19 (4) (2021) 622 and qpb(τ,τ) = τ = |τ −υ + υ| ≤ |τ −υ| + υ ≤ ln(k) |τ −υ| + υ ≤ ln(k) |υ − τ| + υ ≤ qpb(υ,τ). therefore, qpb2 and qpb3 both holds. now, qpb(τ,υ) + qpb(χ,χ) = ln(k) |τ −υ| + τ + χ. since |τ −υ| ≥ 0 ln(k) |τ −υ| ≥ |τ −υ| k(|τ −χ| + |χ−υ|) ≥ 0. since, ln(k) is an increasing function. therefore, qpb(τ,υ) + qpb(χ,χ) = ln(k) |τ −υ| + τ + χ ≤ ln(k)(|τ −χ| + |χ−υ|) + τ + χ ≤ k(|τ −χ| + |χ−υ|) + τ + χ ≤ (k)ln(k) |τ −χ| + (k)ln(k) |χ−υ| + τ + χ ≤ (k)(ln(k) |τ −χ| + |χ−υ|) + τ + χ ≤ s(qpb(τ,χ) + qpb(υ,χ)). for all τ,υ,χ ∈ m and s ≥ k. so (m,qpb) is a quasi-partial b-metric space with s ≥ 1. remark 1.2. note that a metric space is included in the class of quasi-partial b-metric space. in fact, the notions of convergent sequence, cauchy sequence and complete space are defined as in metric spaces. miculescu and mihail [4] (lemma 2. 2.) obtain the following result for b-metric spaces. int. j. anal. appl. 19 (4) (2021) 623 lemma 1.1. every sequence {xn} of elements from a b-metric space (x,d,b), having the property that there exists λ ∈ [0, 1) such that (1.3) d(xn+2,xn+1) ≤ λd(xn+1,xn), for any n ∈ n, is cauchy. remark 1.3. note that lemma 1.1 holds in quasi-partial b-metric space (see proof of lemma 2. 2 in [4]). 2. main results in this section, we introduce interpolative berinde weak contractions in quasi-partial b-metric space and adopted the condition of ω-admissibility to obtain a fixed point. definition 2.1. let (m,qpb) be a complete quasi-partial b-metric space. we say that self-mapping t : m → m is an interpolative berinde weak operator if there exist ζ ∈ [ 0, 1 s ) and α ∈ (0, 1) such that (2.1) qpb(tτ,tυ) ≤ ζ[qpb(τ,υ)]α [ 1 s2 qpb(τ,tτ) ]1−α , for all τ,υ ∈ m\fix(t). theorem 2.1. let (m,qpb) be a complete quasi-partial b-metric space and t be an interpolative berinde weak operator. then t has a fixed point in m. proof. let τ0 ∈ (m,qpb). consider a constructive sequence τn by τn+1 = tn(τ0) for all n ∈ n ∪{0}. we assume that τn = τn+1. indeed if there exist n0 such that τn0 = τn0+1 = tτn0 , then, τn0 forms a fixed point. thus, we have qpb(τn,tτn) = qpb(τn,τn+1) > 0, for all n ∈ n∪{0}. let τ = τn+1,υ = τn+2 qpb(τn+1,τn+2) = qpb(tτn,tτn+1) ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 qpb(τn,τn+1) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 qpb(τn−1,τn+1) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 [sqpb(τn−1,τn) + qpb(τn,τn+1) −qpb(τn,τn) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s qpb(τn−1,τn) + qpb(τn,τn+1) ]1−α (2.2) by induction, for all n ∈ n∪{0} we get qpb(τn,τn+1) ≤ ζnqpb(τ0,τ1). int. j. anal. appl. 19 (4) (2021) 624 on generalising the inequality, qpb(τn,τn+1) = sn−1τn (1 −sτ)n and qpb(τn+1,τn) = sn−1τn (1 −sτ)n . now, we shall show that τn is cauchy sequence. let n,k ∈ n qpb(τn,τn+k) ≤ sqpb(τn,τn+1) + s2qpb(τn+1,τn+2) + . . . + skqpb(τn+k−1,τn+k) ≤ s.sn−1.ζn.τn (1 −sτ)n + s2.sn.ζn+1.τn+1 (1 −sτ)n+1 + . . . + sn−k.sn−2.ζn−1.τn−1 (1 −sτ)n−1 ≤ sn.τn (1 −sτ)n + sn+2.τn+1 (1 −sτ)n+1 + . . . + s2n−k−2.τn−1 (1 −sτ)n−1 ≤ sn.τn (1 −sτ)n [ 1 + s2.τ (1 −sτ) + . . . + s2n−k−2.τn−k−1 (1 −sτ)n−k−1 ] the inequality, 0 ≤ τ ≤ 1 s2(s+1) then s 2.τ (1−sτ) ≤ 1. qpb(τn,τn+k) ≤ ( sτ 1−sτ )n { 1 − ( s2.τ 1−sτ )n−k} ( 1 − s2.τ 1−sτ ) ≤ ( sτ a−sτ )n (1 −sτ) 1 −sτ −s2τ .(2.3) therefore, we claim that τn is a cauchy sequence in (m,qpb). let m,n ∈ n. by triangle inequality in (2.2), we deduce that qpb(τn+m,τn+m+k) ≤ ( sτ a−sτ )n (1 −sτ) 1 −sτ −s2τ since, sτ 1−sτ ≤ 1 and taking n →∞ in (2.3) and using limn→∞τ(t n) = 0 for t ≥ 0, we get qpb(τn+m,τn+m+k) ≤ ( sτ a−sτ )n (1 −sτ) 1 −sτ −s2τ therefore, lim n→∞ qpb(τn,τn+k) = lim m→∞,n→∞ qpb(τn+m,τn+m+k) = 0(2.4) since m is complete, so there exist χ ∈ m such that lim n→∞ τn = χ. suppose τn 6= tτn for each n ≥ 0 qpb(τn+1,tχ) = qpb(tτn,tχ) ≤ ζ[qpb(τn,χ)]α. [ 1 s2 qpb(τn,tχ) ]1−α int. j. anal. appl. 19 (4) (2021) 625 since 1 s ≤ 1, this implies 1 s2 ≤ 1 therefore ≤ ζ[qpb(τn,χ)]α. [ 1 s2 qpb(τn,tχ) ]1−α ≤ ζ[qpb(τn,χ)]α.[qpb(τn,tχ)]1−α.(2.5) letting n →∞ in (2.5), we get qpb(χ,tχ) = 0 which is a contradiction. thus tχ = χ. � definition 2.2. [7] let t : m → m be a map and α: m × m → r be a function. then t is said to be α-admissible if (2.6) α(τ,υ) ≥ 1 implies α(tτ,tυ) ≥ 1. we introduce ω-admissible interpolative berinde weak contraction in quasi-partial b-metric space engrossed by gupta and gautam [20]. definition 2.3. let (m,qpb) be a complete quasi-partial b-metric space. the map t : m → m is said to be an ω-interpolative berinde weak contraction if there exists ζ, ω : m ×m → [0,∞) and α ∈ (0, 1) such that (2.7) ω(τ,υ)qpb(tτ,tυ) ≤ ζ[qpb (τ,υ)]α [ 1 s2 qpb (τ,tτ) ]1−α , for all τ,υ ∈ m\fix(t). theorem 2.2. let us consider (m,qpb) to be a complete quasi-partial b-metric space with self mapping t : m → m is ω-orbital admissibile and forms an ω-interpolative berinde weak contraction on a complete quasi-partial b-metric space (m,qpb). if there exists τ0 ∈ m such that ω(τ0,tτ1) ≥ 1, then t posses a fixed point in m. proof. let τ ∈ m be a point such that τn+1 = tn(τ0) for all n ∈ n ∪{0}. if we have, τn0 = τn0+1 then τn is a fixed point of t which ends the proof otherwise τn 6= τn+1 for all n ∈ n ∪{0}. we have ω(τ0,τ1) ≥ 1. since t is ω orbital admissible, ω(τ1,τ2) = ω(tτ0,tτ1) ≥ 1(2.8) int. j. anal. appl. 19 (4) (2021) 626 continuing ω(τn,τn+1) ≥ 1. let τ = τn+1, υ = τn+2, we have qpb(τn,τn+1) ≤ ω(τn+1,τn+2)qpb(tτn+1,tτn+2) qpb(τn+1,τn+2) = qpb(tτn,tτn+1) ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 qpb(τn,τn+1) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 qpb(τn−1,τn+1) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s2 [sqpb(τn−1,τn) + qpb(τn,τn+1) −qpb(τn,τn) ]1−α ≤ ζ[qpb(τn,τn+1)]α. [ 1 s qpb(τn−1,τn) + qpb(τn,τn+1) ]1−α (2.9) by induction, for all n ∈ n∪{0} we get ω(τn+1,τn+2)qpb(τn,τn+1) ≤ ζnqpb(τ0,τ1). on generalising the inequality, qpb(τn,τn+1) = sn−1τn (1 −sτ)n and qpb(τn+1,τn) = sn−1τn (1 −sτ)n . now we shall show that τn is cauchy sequence. let n,k ∈ n qpb(τn,τn+k) ≤ sqpb(τn,τn+1) + s2qpb(τn+1,τn+2) + . . . + sn−kqpb(τn+k−1,τn+k) ≤ s.sn−1.ζn.τn (1 −sτ)n + s2.sn.ζn+1.τn+1 (1 −sτ)n+1 + . . . + sn−k.sn−2.ζn−1.τn−1 (1 −sτ)n−1 ≤ sn.τn (1 −sτ)n + sn+2.τn+1 (1 −sτ)n+1 + . . . + s2n−k−2.τn−1 (1 −sτ)n−1 ≤ sn.τn (1 −sτ)n [ 1 + s2.τ (1 −sτ) + . . . + s2n−k−2.τn−k−1 (1 −sτ)n−k−1 ] the inequality, 0 ≤ τ ≤ 1 s2(s+1) then s 2.τ (1−sτ) ≤ 1 qpb(τn,τn+k) ≤ ( sτ 1−sτ )n { 1 − ( s2.τ 1−sτ )n−k} ( 1 − s2.τ 1−sτ ) ≤ ( sτ a−sτ )n (1 −sτ) 1 −sτ −s2τ .(2.10) therefore, we claim that τn is a cauchy sequence in (m,qpb). let m,n ∈ n. on account of the triangle inequality in (2.9), we deduce that qpb(τn+m,τn+m+k) ≤ ( sτ a−sτ )n (1 −sτ) 1 −sτ −s2τ int. j. anal. appl. 19 (4) (2021) 627 since, sτ 1−sτ ≤ 1, and taking n →∞ in (2.10) and using limn→∞τ(t n) = 0 for t ≥ 0, we get lim n→∞ qpb(τn,τn+k) = lim m→∞,n→∞ qpb(τn+m,τn+m+k) = 0 since m is complete, so there exists χ ∈ m such that lim n→∞ τn = χ. suppose τn 6= tτn for each n ≥ 0 qpb(τn+1,tχ) = qpb(tτn,tχ) ≤ ζ[qpb(τn,χ)]α.[ 1 s2 qpb(τn,tχ)] 1−α. since 1 s ≤ 1, this implies 1 s2 ≤ 1, therefore ≤ ζ[qpb(τn,χ)]α.[qpb(τn,tχ)]1−α.(2.11) letting n →∞ in (2.11), we get qpb(χ,tχ) = 0 which is a contradiction. thus tχ = χ. � corollary 2.1. let (m,qpb) be quasi-partial b-metric space. let t : m → m be the mapping, such that ω(τ,υ)qpb(tτ,tυ) ≤ ζ[qpb(τ,tυ)]α. [ 1 s2 qpb(τ,tτ) ]1−α for all τ,υ ∈ m\fix(t) with τ � υ,s ≥ 1 and α ∈ (0, 1). assume that : (1) t is non-decreasing with respect to �, (2) there exists τ0 ∈ x such that τ0 � tτ0, (3) t is continuous. then, t has a fixed point in m. proof. it suffices to take, in theorem 2.1, ω(τ,υ) =   1 if (τ � υ) or (υ � τ), 0 otherwise. � corollary 2.2. assume that the subsets b1 and b2 of a quasi-partial b-metric space (m,qpb) are closed. suppose that t : b1 ∩b2 → b1 ∩b2 satisfies, ω(τ,υ)qpb(tτ,tυ) ≤ ζ[qpb(τ,tυ)]α. [ 1 s2 qpb(τ,tτ) ]1−α int. j. anal. appl. 19 (4) (2021) 628 for all τ ∈ b1 and υ ∈ b2, such that τ,υ /∈ fix(t), where α ∈ (0, 1),s ≥ 1, if t(b1) ⊆ b2 and t(b2) ⊆ b1, then there exists a fixed point of t in b1 ∩b2. proof. it suffices to take, in theorem 2.1, ω(τ,υ) =   1 if (τ � υ) or (υ � τ), 0 otherwise. � example 2.1. let a set m = [0, 4] with qpb(τ,υ) = log(k)|τ −υ|+ τ. let t be a self-mapping on m shown as tτ =   10 3 , if τ ∈ [3, 4], 3 4 , if τ ∈ [0, 3]. we illustrate the self-mappings of t in the fig1. figure 1. 10 3 and 3 4 are the fixed points of t. take, ω(τ,υ) =   1, if τ,υ ∈ [3, 4], 0 otherwise. let τ,υ ∈ m such that τ 6= tτ,υ 6= tυ and ω(τ,υ) ≥ 1. then τ,υ ∈ [3, 4] and τ,υ /∈ 10 3 . we have tτ = tυ = 10 3 . for τ0 = 4, we have ω(4,t4) = ω(4, 10 3 ) = 1 now, let τ,υ ∈ m be such that ω(τ,υ) ≥ 1. this shows that τ,υ ∈ [3, 4], so tτ = tυ in [3, 4]. hence, ω(tτ,tυ) ≥ 1, that is, t is ω-orbital admissible. suppose τn to be a sequence in m such that ω(τn,τn+1) ≥ 1 int. j. anal. appl. 19 (4) (2021) 629 for each n ∈ n. then, τn ⊂ [3, 4]. if τn → w as n →∞, we have |τn −w|→ 0 as n →∞. hence, w ∈ [3, 4] and so, ω(τn,w) = 1. therefore, theorem 2.1 holds. so, 10 3 and 3 4 are the two fixed points of t . in 2016, berinde and fukhar-ud-din [30] modified the concept of convex metric space and applied it to obtain fixed point results of quasi-contractive operators. motivated by this, ampadu [16] introduced convex interpolative berinde weak operator in metric spaces. forging this approach, we introduce the following result in quasi-partial b-metric space. definition 2.4. let (m,qpb) be a complete quasi-partial b-metric space with continuous mapping t : m → m is convex interpolative berinde weak operator if the following is true for all τ,υ ∈ x,τ,υ /∈ fix(t),fix(t2). (2.12) qpb(t 2τ,t2υ) ≤ ζ1qpb(τ,υ) 1 2 + ζ2qpb(tτ,tυ) 1 2 qpb(tτ,t 2τ) 1 2 , where ζ1,ζ2 ∈ [ 0, 1 s ) with ζ1 + ζ2 ≤ 1s for s ≥ 1. theorem 2.3. let (m,qpb) be a complete quasi-partial b-metric space with self mapping t : m → m be a convex interpolative berinde weak operator. if (m,qpb) is complete, then the fixed point exists. proof. let τn be a sequence in m such that τn+1 = tτn = t 2τn−1, for all positive integers n. now, we observe that qpb(τn+1,τn+2) = qpb(t 2τn−1,t 2τn) ≤ ζ1qpb(τn,τn−1) 1 2 qpb(τn−1,tτn−1) 1 2 + ζ2qpb(tτn,tτn−1) 1 2 qpb(tτn−1,t 2τn−1) 1 2 = ζ1qpb(τn,τn−1) 1 2 qpb(τn−1,τn) 1 2 + ζ2qpb(τn,τn+1) 1 2 qpb(τn,τn+1) 1 2 = ζ1qpb(τn,τn−1) + ζ2qpb(τn,τn+1) ≤ (ζ1 + ζ2)max{qpb(τn,τn−1),qpb(τn,τn−1)} = (ζ1 + ζ2)qpb(τn,τn+1). from the above, we deduce that qpb(τn+1,τn+2) ≤ hqpb(τn,τn+1) =⇒ qpb(τn+1,τn+2) ≤ 1 s qpb(τn,τn+1). where h := ζ1 + ζ2 ≤ 1s . by induction, the following is clear for all n ∈ n ⋃ 0 qpb(τn,τn+1) ≤ hnqpb(τ0,τ1). int. j. anal. appl. 19 (4) (2021) 630 now, we shall show that τn is a cauchy sequence. for this, let n,m ∈ n with m ≥ n, and we have qpb(τm,τn) ≤ qpb(τm,τm−1) + qpb(τm−1,τm−2) + . . . + qpb(τn+1,τn) ≤ sqpb(τm,τm−1) + s2qpb(τm−1,τm−2) + . . . + smqpb(τn+1,τn) ≤ [ s(ζ1 + ζ2) n + s2(ζ1 + ζ2) n+1 + . . . + sm(ζ1 + ζ2) m+n−1]qpb(τ0,τ1) ≤ [ s(h)n + s2(h)n+1 + . . . + sm(h)m+n−1 ] qpb(τ0,τ1) ≤ sm m+n−1∑ i=n hiqpb(τ0,τ1) ≤ sm ∞∑ i=n hiqpb(τ0,τ1). now, letting m,n → ∞ in the above inequality it follows that τn is a cauchy sequence and since m is complete χ ∈ m such that lim n→∞ τn = χ. suppose qpb(χ,tχ) = 0, but qpb(χ,tχ) ≥ 0. therefore we observe that 0 ≥ qpb(χ,t2χ) ≥ qpb(χ,τn+1) + qpb(τn+1,t2χ) = qpb(χ,τn+1) + qpb(t 2τn−1,t 2χ) ≥ qpb(χ,τn+1) + ζ1qpb(τn−1,χ) 1 2 qpb(τn−1,tτn−1) 1 2 + ζ2qpb(tτn−1,tχ) 1 2 qpb(tτn−1,t 2τn−1) 1 2 . taking n → ∞ in the above inequality we find that qpb(χ,tχ) = 0, which is a contradiction. thus tχ = χ. � 3. conclusion the main contribution of this paper is to prove the existence of fixed points via interpolative berinde weak contraction. the interpolative weak contraction is extended to adapt various nonlinear self mappings leading to achieve best topological and geometrical results. one of the major applications of berinde weak contraction is that it is used to solve multivalued mappings, that is, it can be used to get more than one fixed point. also, it is used to solve initial value problems in ordinary differential equations and integral equations. weak contraction merged large amount of contractive operators and formulated fixed points by the means of picard iteration. int. j. anal. appl. 19 (4) (2021) 631 acknowledgements: all authors are grateful to the spanish government for grant rti2018-094366b-i00 (mciu/aei/feder, ue) and to the basque government for grant it1207-19. funding: this manuscript has received the funding from spanish government grant rti2018-094366-b-i00 (mciu/aei/feder, ue) in collaboration with basque government grant it1207-19. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. banach, sur les operations dans les ensembles abstraits et leur application aux equations integrales, fund. math. 3 (1922), 133–181. [2] s. g. matthews, partial metric topology, ann. n. y. acad. sci. 728 (1994), 183-197. [3] s. czerwik, contraction mappings in b-metric spaces, acta math. inform. univ. ostrav. 1 (1993), 5-11. [4] r. miculescu, a. mihail, new fixed point theorems for set-valued contractions in b-metric spaces, j. fixed point theory appl. 19 (2017), 2153-2163. [5] s. oltra, o. valero, banach’s fixed point theorem for partial metric spaces, rend. istit. mat. univ. trieste, 36 (2004), 17-26. [6] o. valero, on banach fixed point theorems for partial metric spaces, appl. gen. topol. 6(2) (2005), 229-240. [7] e. karapınar, i.m. erhan, a. öztürk, fixed point theorems on quasi-partial metric spaces, math. comput. model. 57 (2013), 2442-2448. [8] s. shukla, partial b-metric spaces and fixed point theorems, mediterr. j. math. 11 (2014), 703-711. [9] r. kannan, some results on fixed points, bull. calcutta math. soc. 60 (1968), 71–76. [10] v. berinde, approximating fixed points of weak contractions using the picard iteration, nonlinear anal. forum. 9(1) (2004), 43–53. [11] m. abbas, d. ilic, common fixed points of generalized almost non expansive mappings, filomat. 24(3) (2010), 11–18. [12] v. berinde, some remarks on a fixed point theorem for c̀iric̀ type almost contractions, carpathian j. math. 25(2) (2009), 157–162. [13] i. altun, o. acar, fixed point theorems for weak contractions in the sense of berinde on partial metric spaces, topol. appl. 159(10-11) (2012), 2642–2648. [14] a. d. t. turkoglu, v. ozturk, common fixed point results for four mappings on partial metric spaces, abstr. appl. anal. 2012 (2012), 190862. [15] e. karapinar, revisiting the kannan type contractions via interpolation, adv. theory nonlinear anal. appl. 2 (2018), 85-87. [16] c. boateng ampadu, some fixed point theory results for the interpolative berinde weak operator, earthline j. math. sci. 4(2) (2020), 253-271. [17] o. popescu, some new fixed point theorems for α-geraghty contractive type maps in metric spaces, fixed point theory appl. 2014 (2014), 190. [18] b. samet, c. vetro, p. vetro, fixed point theorems for α − ψ-contractive type mappings, nonlinear anal., theory meth. appl. 75 (2012), 2154–2165. int. j. anal. appl. 19 (4) (2021) 632 [19] h. aydi, e. karapinar, a. f. rold́an lópez de hierro, ω-interpolative ćirić-reich-rus-type contractions, mathematics. 7 (2019), 57. [20] a. gupta, p. gautam, quasi-partial b-metric spaces and some related fixed point theorems, fixed point theory appl. 2015 (2015), 18. [21] a. gupta, p. gautam, topological structure of quasi-partial b-metric space, int. j. pure math sci. 17 (2016), 8-18. [22] p. gautam, v. n. mishra, k. negi, common fixed point theorems for cyclic reich-rus-ćirić contraction mappings in quasi-partial b-metric space, ann. fuzzy math. inf. 12(1) (2020), 47-56. [23] p. gautam, v. n. mishra, r. ali, s. verma, interpolative chatterjea and cyclic chatterjea contraction on quasi-partial b-metric space, aims math. 6(2) (2021), 1727-1742. [24] v. n. mishra, l. m. sánchez ruiz., p. gautam, s. verma, interpolative reich-rus-ćirić and hardy-rogers contraction on quasi-partial b-metric space and related fixed point results, mathematics. 8 (2020), 1598. [25] l. m. sánchez ruiz., p. gautam, s. verma, fixed point of interpolative reich-rus-ćirić and contraction mapping on rectangular quasi-partial b-metric space, symmetry. 13(1) (2021), art. id 32. [26] p. gautam, s. verma, fixed point via implicit contraction mapping on quasi-partial b-metric space, j. anal. (2021). https://doi.org/10.1007/s41478-021-00309-6. [27] p. gautam, s. verma, m. de la sen, s. sundriyal, fixed point results for ω-interpolative chatterjea type contraction in quasi-partial b-metric space, int. j. anal. appl. 19 (2021), 280-287. [28] p. gautam, l. m. sánchez ruiz, s. verma, g. gupta, common fixed point results on generalized weak compatible mapping in quasi-partial b-metric space, j. math. 2021 (2021), art. id 5526801. [29] p. gautam, s. verma, s. gulati, omega-interpolative ćirić-reich-rus type contraction on quasi-partial b-metric space, filomat. accepted. [30] h. fukhar-ud-din, v. berinde, iterative methods for the class of quasi-contractive type operators and comparsion of their rate of convergence in convex metric spaces, filomat. 30(1) (2016), 223-230. 1. introduction and preliminaries 2. main results 3. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 185-197 http://www.etamaths.com generalized meir-keeler type ψ-contractive mappings and applications to common solution of integral equations hüseyi̇n işik1,∗, mohammad imdad2, duran turkoglu3 and nawab hussain4 abstract. the goal of the present article to introduce the notion of generalized meir-keeler type ψ-contractions and prove some coupled common fixed point results for such type of contractions. the theorems proved herein extend, generalize and improve some results of the existing literature. several examples and an application to integral equations are also given in order to illustrate the genuineness of our results. 1. introduction and preliminaries the meir-keeler contraction defined in 1969 by meir and keeler [13] is one of the most significant generalizations of banach contraction principle [1]. owing to it’s utility, generality and effectiveness, the result of meir and keeler [13] remains a novel result in metric fixed point theory. in recent years, many authors extended and generalized this result in different ways and by now there exists extensive literature on this theme. to mention a few, we recall [3, 7, 15, 16] and references cited therein. guo and lakshmikantham [8] established some coupled fixed point theorems which has attracted the attention of many researchers (e.g. [5–7,9–12] and references therein). bhaskar and lakshmikantham [2] introduced the notion of mixed monotone mapping to prove results on coupled fixed points. as an application, they also proved the existence and uniqueness of solution for a periodic boundary value problem associated to a first order ordinary differential equation. recently, ding et al. [6] introduced the notion of weakly increasing mappings with two variables and established several coupled common fixed point theorems for these mappings in ordered metric spaces. in this paper, we introduce the notion of generalized meir-keeler type ψ-contractions and prove some coupled common fixed point theorems for such contractions using weakly increasing property instead of mixed monotone property. our results extend, generalize and improve several results of the existing literature. several interesting consequences of our theorems are derived besides furnishing an example. as an application of the results presented herein, we discuss the existence of the common solution for a system of integral equations. we start by recalling some definitions and notions. in the sequel, the letters r,r+and n will denote the set of all real numbers, the set of all non negative real numbers and the set of all natural numbers, respectively. definition 1.1 ( [2]). an element (x,y) ∈ x2 is said to be a coupled fixed point of the mapping f : x2 → x if x = f(x,y) and y = f(y,x). definition 1.2 ( [6]). an element (x,y) ∈ x2 is called a coupled common fixed point of mappings f,g : x2 → x if f(x,y) = g(x,y) = x and f(y,x) = g(y,x) = y. we denote the set of all coupled common points of f and g by f (f,g) . definition 1.3 ( [2]). let (x,�) be a partially ordered set and f : x2 → x. we say that f has the mixed monotone property if f(x,y) is monotone nondecreasing in x and monotone nonincreasing in y, that is, for any x,y ∈ x, x1,x2 ∈ x, x1 � x2 ⇒ f(x1,y) � f(x2,y) 2010 mathematics subject classification. 47h10. key words and phrases. coupled common fixed point; meir-keeler contraction; weakly increasing mapping; integral equation. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 185 186 işik, imdad, turkoglu and hussain and y1,y2 ∈ x, y1 � y2 ⇒ f(x,y2) � f(x,y1). definition 1.4 ( [6]). let (x,�) be a partially ordered set. two mappings f,g : x2 → x are said to be weakly increasing if f(x,y) � g(f(x,y),f(y,x)) and g(x,y) � f(g(x,y),g(y,x)), hold for all (x,y) ∈ x2. example 1.1. let x = [1, +∞) be endowed with the usual ordering ≤ and f,g : x2 → x be given by f(x,y) = x2 + 2y and g(x,y) = 2x + y2. then, for all (x,y) ∈ x2, f(x,y) = x2 + 2y ≤ g(f(x,y),f(y,x)) = g(x2 + 2y,y2 + 2x) = 2(x2 + 2y) + (y2 + 2x)2 and g(x,y) = 2x + y2 ≤ f(g(x,y),g(y,x)) = f(2x + y2, 2y + x2) = ( 2x + y2 )2 + 2 ( 2y + x2 ) . thus, f and g are two weakly increasing mappings with respect to ≤. on the lines of [4], we denote by ψ the family of all functions ψ : r+ 4 → r+ such that (ψ1) ψ is nondecreasing and continuous in each coordinate; (ψ2) ψ (t,t,t, t) ≤ t for all t > 0; (ψ3) ψ (t1, t2, t3, t4) = 0 iff t1 = t2 = t3 = t4 = 0. in the rest of the paper, we denote by (x,�,d) an ordered metric space wherein � is a partial order on the set x while d is a metric on x. in addition, we say that (x,y) ∈ x2 is comparable to (u,v) ∈ x2 if x � u and y � v or u � x and v � y. for brevity, we show this with (x,y) � (u,v) or (x,y) � (u,v) . (x2,δ) is a metric space under the following metric: δ ((x,y) , (u,v)) := d (x,u) + d (y,v) , for all (x,y) , (u,v) ∈ x2. by the definition of δ, it is obvious that δ ((x,y) , (u,v)) = δ ((y,x) , (v,u)) . definition 1.5. let (x,�,d) be an ordered metric space and f,g : x2 → x be two mappings. we say that (f,g) is a generalized meir-keeler type ψ-contraction pair if, for all ε > 0, there exists η > 0 such that, for all comparable pairs (x,y) , (u,v) ∈ x2, ε ≤ 1 2 m (x,y,u,v) ≤ ε + η (ε) ⇒ d (f (x,y) ,g (u,v)) < ε, (1.1) where m (x,y,u,v) = ψ   δ ((x,y) , (u,v)) ,δ ((x,y) , (f (x,y) ,f (y,x))) , δ ((u,v) , (g (u,v) ,g (v,u))) , 1 2 [ δ ((x,y) , (g (u,v) ,g (v,u))) +δ ((u,v) , (f (x,y) ,f (y,x))) ]   with ψ ∈ ψ. if we take f = g in the above definition, then we have: definition 1.6. let (x,�,d) be an ordered metric space and f : x2 → x be a mapping. we say that f is a generalized meir-keeler type ψ-contraction if, for all ε > 0, there exists η > 0 such that, for all comparable pairs (x,y) , (u,v) ∈ x2, ε ≤ 1 2 mf (x,y,u,v) ≤ ε + η (ε) ⇒ d (f (x,y) ,f (u,v)) < ε, generalized meir-keeler contractions 187 where mf (x,y,u,v) = ψ   δ ((x,y) , (u,v)) ,δ ((x,y) , (f (x,y) ,f (y,x))) , δ ((u,v) , (f (u,v) ,f (v,u))) , 1 2 [ δ ((x,y) , (f (u,v) ,f (v,u))) +δ ((u,v) , (f (x,y) ,f (y,x))) ]   with ψ ∈ ψ. remark 1.1. let (x,�,d) be an ordered metric space and let f,g : x2 → x be two given mappings. if (f,g) is a generalized meir-keeler type ψ-contraction pair, then d (f (x,y) ,g (u,v)) < 1 2 m (x,y,u,v) , for all comparable pairs (x,y) , (u,v) ∈ x2 when m (x,y,u,v) > 0. also, if m (x,y,u,v) = 0, then d (f (x,y) ,g (u,v)) = 0, that is, d (f (x,y) ,g (u,v)) ≤ 1 2 m (x,y,u,v) , for all comparable pairs (x,y) , (u,v) ∈ x2. 2. main results our main result is stated as follows: theorem 2.1. let (x,�,d) be a complete ordered metric space, f,g : x2 → x be two weakly increasing mappings with respect to � and (f,g) be a generalized meir-keeler type ψ-contraction pair. if either f or g is continuous, then f and g have a coupled common fixed point. proof. let x0,y0 ∈ x. define two sequences {xn} and {yn} in x as follows: x2n+1 = f(x2n,y2n), x2n+2 = g(x2n+1,y2n+1), and y2n+1 = f(y2n,x2n), y2n+2 = g(y2n+1,x2n+1), for all n ∈ n0 where n0 = n∪{0} . since f and g are weakly increasing, we have x1 = f (x0,y0) � g (f (x0,y0) ,f (y0,x0)) = g (x1,y1) = x2 � f (g (x1,y1) ,g (y1,x1)) = f (x2,y2) = x3 � ··· , and y1 = f (y0,x0) � g (f (y0,x0) ,f (x0,y0)) = g (y1,x1) = y2 � f (g (y1,x1) ,g (x1,y1)) = f (y2,x2) = y3 � ··· , so that the sequences {xn} and {yn} are nondecreasing. step 1. now, we show that {xn} and {yn} are cauchy sequences in (x,�,d). case 1. if for some n ∈ n0, xn = xn+1 and yn = yn+1, then xn+1 = xn+2 and yn+1 = yn+2. if not, then for n = 2m where m ∈ n, by remark 1.1, we have d (xn+1,xn+2) = d (x2m+1,x2m+2) = d (f(x2m,y2m),g (x2m+1,y2m+1)) < 1 2 m (x2m,y2m,x2m+1,y2m+1) , 188 işik, imdad, turkoglu and hussain where m (x2m,y2m,x2m+1,y2m+1) = ψ   δ ((x2m,y2m) , (x2m+1,y2m+1)) , δ ((x2m,y2m) , (f (x2m,y2m) ,f (y2m,x2m))) , δ ((x2m+1,y2m+1) , (g (x2m+1,y2m+1) ,g (y2m+1,x2m+1))) , 1 2 [ δ ((x2m,y2m) , (g (x2m+1,y2m+1) ,g (y2m+1,x2m+1))) +δ ((x2m+1,y2m+1) , (f (x2m,y2m) ,f (y2m,x2m))) ]   = ψ   δ ((x2m,y2m) , (x2m+1,y2m+1)) ,δ ((x2m,y2m) , (x2m+1,y2m+1)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [ δ ((x2m,y2m) , (x2m+2,y2m+2)) +δ ((x2m+1,y2m+1) , (x2m+1,y2m+1)) ]   = ψ { 0, 0,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [δ ((x2m,y2m) , (x2m+2,y2m+2))] } . since ψ is nondecreasing, we deduce m (x2m,y2m,x2m+1,y2m+1) ≤ ψ   δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) ,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [δ ((x2m,y2m) , (x2m+1,y2m+1)) + δ ((x2m+1,y2m+1) , (x2m+2,y2m+2))]   ≤ψ { δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) ,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) ,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) } ≤ δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) = δ ((xn+1,yn+1) , (xn+2,yn+2)) . hence, it follows that d (xn+1,xn+2) < 1 2 δ ((xn+1,yn+1) , (xn+2,yn+2)) . (2.1) similarly, we can also show that d (yn+1,yn+2) < 1 2 δ ((xn+1,yn+1) , (xn+2,yn+2)) . (2.2) thus, from (2.1) and (2.2) δ ((xn+1,yn+1) , (xn+2,yn+2)) < δ ((xn+1,yn+1) , (xn+2,yn+2)) , which is a contradiction. hence we must have xn+1 = xn+2 and yn+1 = yn+2 when n is even. by similar arguments we can show that this equality holds also when n is odd. therefore, in any case for all those n for which xn = xn+1 and yn = yn+1 holds, we always obtain xn+1 = xn+2 and yn+1 = yn+2. repeating above process inductively, one obtains xn = xn+k and yn = yn+k for all k ∈ n. hence {xn} and {yn} are constant sequences so that {xn} and {yn} are cauchy sequences in (x,�,d). case 2. suppose that xn 6= xn+1 and yn 6= yn+1 for all n ∈ n0. then, for n = 2m + 1, using remark 1.1, we have d (xn,xn+1) = d (x2m+1,x2m+2) = d (f(x2m,y2m),g (x2m+1,y2m+1)) < 1 2 m (x2m,y2m,x2m+1,y2m+1) , (2.3) where m (x2m,y2m,x2m+1,y2m+1) = ψ   δ ((x2m,y2m) , (x2m+1,y2m+1)) ,δ ((x2m,y2m) , (x2m+1,y2m+1)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [δ ((x2m,y2m) , (x2m+2,y2m+2)) + δ ((x2m+1,y2m+1) , (x2m+1,y2m+1))]   . since, δ ((x2m+1,y2m+1) , (x2m+1,y2m+1)) = d (x2m+1,x2m+1) + d (y2m+1,y2m+1) = 0, generalized meir-keeler contractions 189 and δ ((x2m,y2m) , (x2m+2,y2m+2)) = d (x2m,x2m+2) + d (y2m,y2m+2) ≤ d (x2m,x2m+1) + d (x2m+1,x2m+2) + d (y2m,y2m+1) + d (y2m+1,y2m+2) = δ ((x2m,y2m) , (x2m+1,y2m+1)) + δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , so we get m (x2m,y2m,x2m+1,y2m+1) ≤ ψ   δ ((x2m,y2m) , (x2m+1,y2m+1)) ,δ ((x2m,y2m) , (x2m+1,y2m+1)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [δ ((x2m,y2m) , (x2m+1,y2m+1)) + δ ((x2m+1,y2m+1) , (x2m+2,y2m+2))]   . (2.4) if δ ((x2m,y2m) , (x2m+1,y2m+1)) ≤ δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) for some m, since ψ is nondecreasing, we obtain m (xn−1,yn−1,xn,yn) = m (x2m,y2m,x2m+1,y2m+1) ≤ ψ   δ ((x2m,y2m) , (x2m+1,y2m+1)) ,δ ((x2m,y2m) , (x2m+1,y2m+1)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , 1 2 [δ ((x2m,y2m) , (x2m+1,y2m+1)) + δ ((x2m+1,y2m+1) , (x2m+2,y2m+2))]   ≤ ψ { δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) ,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) , δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) ,δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) } ≤ δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) = δ ((xn,yn) , (xn+1,yn+1)) , and so from (2.3) d (xn,xn+1) < 1 2 δ ((xn,yn) , (xn+1,yn+1)) . (2.5) similarly, we can show that d (yn,yn+1) < 1 2 δ ((xn,yn) , (xn+1,yn+1)) . (2.6) thus, from (2.5) and (2.6), we get δ ((xn,yn) , (xn+1,yn+1)) < δ ((xn,yn) , (xn+1,yn+1)) , which is a contradiction. hence, it must be δ ((x2m+1,y2m+1) , (x2m+2,y2m+2)) = δ ((xn,yn) , (xn+1,yn+1)) < δ ((xn−1,yn−1) , (xn,yn)) = δ ((x2m,y2m) , (x2m+1,y2m+1)) , for all n ∈ n0. set δn := {δ ((xn,yn) , (xn+1,yn+1))} , then the sequence {δn} is decreasing and bounded below. thus, there exists r ≥ 0 such that limn→∞δn = r. notice that r = inf {δn : n ∈ n0} . (2.7) we now prove that r = 0. if not, then by (2.4), we deduce that lim n→∞ m (xn−1,yn−1,xn,yn) = r. then there exists a positive integer p such that ε ≤ 1 2 m (xp−1,yp−1,xp,yp) < ε + η (ε) , where ε = r/2. owing to the condition (1.1), we have d (f(xp−1,yp−1),g (xp,yp)) < ε, which implies d (xp,xp+1) < ε. 190 işik, imdad, turkoglu and hussain similarly, we can obtain that d (yp,yp+1) < ε. summing the two foregoing inequalities, we get δ ((xp,yp) , (xp+1,yp+1)) < 2ε = r, which contradicts (2.7) for n = p. thus, ε = r/2 = 0, that is, lim n→∞ δn = lim n→∞ [d (xn,xn+1) + d (yn,yn+1)] = 0. (2.8) now, we prove that {xn} and {yn} are cauchy sequences. it is sufficient to show that {x2n} and {y2n} are cauchy sequences in (x,d). suppose, to the contrary, that at least one of {x2n} or {y2n} is not cauchy sequence. then, there exists an ε > 0 for which we can find subsequences {x2mk},{x2nk} of {x2n} and {y2mk},{y2nk} of {y2n} such that nk is the smallest index for which nk > mk > k and d(x2nk,x2mk ) + d(y2nk,y2mk ) ≥ ε and d(x2nk−1,x2mk ) + d(y2nk−1,y2mk ) < ε. (2.9) using the triangular inequality and (2.9), we get ε ≤ d(x2nk,x2mk ) + d(y2nk,y2mk ) ≤ d(x2mk,x2nk−1) + d(x2nk−1,x2nk ) + d(y2mk,y2nk−1) + d(y2nk−1,y2nk ) < ε + δ2nk−1. taking k →∞ in the above inequality and using (2.8), we deduce lim k→∞ [d(x2nk,x2mk ) + d(y2nk,y2mk )] = ε. (2.10) again, by the triangle inequality, we get d(x2nk,x2mk ) + d(y2nk,y2mk ) ≤ d(x2nk,x2nk+1) + d(x2nk+1,x2mk−1) + d(x2mk−1,x2mk ) + d(y2nk,y2nk+1) + d(y2nk+1,y2mk−1) + d(y2mk−1,y2mk ) ≤ δ2nk + δ2mk−1 + d(x2nk+1,x2mk ) + d(x2mk,x2mk−1) + d(y2nk+1,y2mk ) + d(y2mk,y2mk−1) ≤ δ2nk + 2δ2mk−1 + d(x2nk+1,x2nk ) + d(x2nk,x2mk ) + d(y2nk+1,y2nk ) + d(y2nk,y2mk ) = 2δ2nk + 2δ2mk−1 + d(x2nk,x2mk ) + d(y2nk,y2mk ). letting k →∞ in the above inequality besides using (2.8) and (2.10), we have limk→∞ [d(x2nk+1,x2mk−1) + d(y2nk+1,y2mk−1)] = ε, and limk→∞ [d(x2nk+1,x2mk ) + d(y2nk+1,y2mk )] = ε. (2.11) on the other hand, we also obtain d(x2nk,x2mk−1) + d(y2nk,y2mk−1) ≤ d(x2nk,x2mk ) + d(x2mk,x2mk−1) +d(y2nk,y2mk ) + d(y2mk,y2mk−1) = d(x2nk,x2mk ) + d(y2nk,y2mk ) + δ2mk−1, which implies lim k→∞ [d(x2nk,x2mk−1) + d(y2nk,y2mk−1)] ≤ ε. (2.12) since (x2nk,y2nk ) � (x2mk−1,y2mk−1) for nk > mk, using remark 1.1, we have d (x2nk+1,x2mk ) = d (f(x2nk,y2nk ),g (x2mk−1,y2mk−1)) < 1 2 m (x2nk,y2nk,x2mk−1,y2mk−1) , (2.13) generalized meir-keeler contractions 191 where m (x2nk,y2nk,x2mk−1,y2mk−1) = ψ   δ ((x2nk,y2nk ) , (x2mk−1,y2mk−1)) , δ ((x2nk,y2nk ) , (f(x2nk,y2nk ),f(y2nk,x2nk ))) , δ ((x2mk−1,y2mk−1) , (g (x2mk−1,y2mk−1) ,g (y2mk−1,x2mk−1))) , 1 2 [ δ ((x2nk,y2nk ) , (g (x2mk−1,y2mk−1) ,g (y2mk−1,x2mk−1))) +δ ((x2mk−1,y2mk−1) , (f(x2nk,y2nk ),f(y2nk,x2nk ))) ]   = ψ   δ ((x2nk,y2nk ) , (x2mk−1,y2mk−1)) ,δ ((x2nk,y2nk ) , (x2nk+1,y2nk+1)) , δ ((x2mk−1,y2mk−1) , (x2mk,y2mk )) , 1 2 [δ ((x2nk,y2nk ) , (x2mk,y2mk )) + δ ((x2mk−1,y2mk−1) , (x2nk+1,y2nk+1))]   . by a similar method, we can also show that d (y2nk+1,y2mk ) < 1 2 m (x2nk,y2nk,x2mk−1,y2mk−1) . (2.14) summing the inequalities (2.13) and (2.14), we get d (x2nk+1,x2mk ) + d (y2nk+1,y2mk ) < m (x2nk,y2nk,x2mk−1,y2mk−1) . now, using (2.8), (2.10), (2.11) and (2.12) as k →∞ in the above inequality, we deduce ε < lim k→∞ m (x2nk,y2nk,x2mk−1,y2mk−1) = ψ{ lim k→∞ δ ((x2nk,y2nk ) , (x2mk−1,y2mk−1)) , 0, 0,ε} ≤ ψ{ε, 0, 0,ε}≤ ε, which is a contradiction. therefore, {xn} and {yn} are cauchy sequences in x. step 2. now, we prove the existence of coupled common fixed point of f and g. owing to the completeness of (x,d) , there exist x,y ∈ x such that lim n→∞ xn = x and lim n→∞ yn = y. (2.15) without loss of generality, we assume that f is continuous. now, we have x = lim n→∞ x2n+1 = lim n→∞ f(x2n,y2n) = f ( lim n→∞ x2n, lim n→∞ y2n ) = f (x,y) . and y = lim n→∞ y2n+1 = lim n→∞ f(y2n,x2n) = f ( lim n→∞ y2n, lim n→∞ x2n ) = f (y,x) . we now assert that d (x,g (x,y)) = d (y,g (y,x)) = 0. to establish the claim, assume that d (x,g (x,y)) > 0 and d (y,g (y,x)) > 0. since (x,y) ∈ x2 is comparable to its own, making use of remark 1.1, we obtain d (x,g (x,y)) = d (f (x,y) ,g (x,y)) < 1 2 m (x,y,x,y) , (2.16) where m (x,y,x,y) = ψ   δ ((x,y) , (x,y)) ,δ ((x,y) , (f(x,y),f(y,x))) , δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 [δ ((x,y) , (g (x,y) ,g (y,x))) + δ ((x,y) , (f(x,y),f(y,x)))]   = ψ { 0, 0,δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 δ ((x,y) , (g (x,y) ,g (y,x))) } ≤ δ ((x,y) , (g (x,y) ,g (y,x))) . similarly, we have d (y,g (y,x)) < 1 2 m (x,y,x,y) . (2.17) 192 işik, imdad, turkoglu and hussain thus, it follows from (2.16) and (2.17) that d (x,g (x,y)) + d (y,g (y,x)) < m (x,y,x,y) ≤ δ ((x,y) , (g (x,y) ,g (y,x))) = d (x,g (x,y)) + d (y,g (y,x)) , which implies d (x,g (x,y)) = d (y,g (y,x)) = 0. hence, x = f (x,y) = g (x,y) and y = f (y,x) = g (y,x) . � now, we furnish the following example which illustrates the results of theorem 2.1. example 2.1. let x = [0, 1] be equipped with the usual metric and the partial order defined by x � y ⇐⇒ y ≤ x. define the function ψ : r+ 4 → r+ by ψ (t1, t2, t3, t4) = max{t1, t2, t3, t4} and the mappings f,g : x2 → x by f (x,y) = x+y 7 and g (x,y) = x+y 6 . then, it is easy to see that f and g are weakly increasing with respect to � . also, (f,g) is a generalized meir-keeler type ψ-contraction. indeed, for all comparable (x,y) , (u,v) ∈ x2 d (f (x,y) ,g (u,v)) = ∣∣∣∣x + y7 − u + v6 ∣∣∣∣ ≤ 1 7 (|x−u| + |y −v|) = 1 7 δ ((x,y) , (u,v)) ≤ 1 7 m (x,y,u,v) < 1 7 · 2 (ε + η (ε)) < ε, which holds if we choose η (ε) < 5 2 ε. thus, it can easily see that all the hypotheses of theorem 2.1 are fulfilled. therefore, f and g have a coupled common fixed point which is (0, 0) . definition 2.1. let (x,�,d) be an ordered metric space. we say that (x,�,d) is regular if for non-decreasing sequence {xn} with d(xn,x) → 0 implies that xn � x for all n. in our next theorem, we replace the continuity of f or g in theorem 2.1 with the regularity of (x,�,d). theorem 2.2. let (x,�,d) be a complete ordered metric space, f,g : x2 → x be two weakly increasing mappings with respect to � and (f,g) be a generalized meir-keeler type ψ-contraction pair. if (x,�,d) is regular, then f and g have a coupled common fixed point. proof. we define sequences {xn} and {yn} as in theorem 2.1. proceeding on the lines of the proof of theorem 2.1, we can show that the sequences {xn} and {yn} are non-decreasing and limn→∞xn = x and limn→∞yn = y. then, since (x,�,d) is regular, we conclude that (xn,yn) is comparable to (x,y) for all n ∈ n0. now, by remark 1.1, we obtain d (x2n+1,g (x,y)) = d (f(x2n,y2n),g (x,y)) < 1 2 m (x2n,y2n,x,y) , (2.18) where m (x2n,y2n,x,y) = ψ   δ ((x2n,y2n) , (x,y)) ,δ ((x2n,y2n) , (f(x2n,y2n),f(y2n,x2n))) , δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 [ δ ((x2n,y2n) , (g (x,y) ,g (y,x))) +δ((x,y) , (f(x2n,y2n),f(y2n,x2n))) ]   = ψ   δ ((x2n,y2n) , (x,y)) ,δ ((x2n,y2n) , (x2n+1,y2n+1)) , δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 [δ ((x2n,y2n) , (g (x,y) ,g (y,x))) + δ((x,y) , (x2n+1,y2n+1))]   . generalized meir-keeler contractions 193 similarly, we have d (y2n+1,g (y,x)) = d (f(y2n,x2n),g (y,x)) < 1 2 m (x2n,y2n,x,y) . (2.19) thus it follows from (2.18) and (2.19) that d (x2n+1,g (x,y)) + d (y2n+1,g (y,x)) < m (x2n,y2n,x,y) . letting n →∞ in the above inequality, we get d (x,g (x,y)) + d (y,g (y,x)) < ψ { 0, 0,δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 δ ((x,y) , (g (x,y) ,g (y,x))) } ≤ δ ((x,y) , (g (x,y) ,g (y,x))) = d (x,g (x,y)) + d (y,g (y,x)) , which implies d(x,g (x,y)) = 0 and d (y,g (y,x)) = 0 that is, x = g (x,y) and y = g (y,x) . since (x,y) ∈ x2 is comparable to its own, using of remark 1.1, we obtain d (f (x,y) ,x) = d (f (x,y) ,g (x,y)) < 1 2 m (x,y,x,y) , (2.20) where m (x,y,x,y) = ψ   δ ((x,y) , (x,y)) ,δ ((x,y) , (f(x,y),f(y,x))) , δ ((x,y) , (g (x,y) ,g (y,x))) , 1 2 [δ ((x,y) , (g (x,y) ,g (y,x))) + δ ((x,y) , (f(x,y),f(y,x)))]   = ψ { 0,δ ((x,y) , (f(x,y),f(y,x))) , 0, 1 2 δ ((x,y) , (f(x,y),f(y,x))) } ≤ δ ((x,y) , (f(x,y),f(y,x))) . similarly, we have d (f (y,x) ,y) < 1 2 m (x,y,x,y) . (2.21) thus, it follows from (2.20) and (2.21) that d (f (x,y) ,x) + d (f (y,x) ,y) < m (x,y,x,y) ≤ δ ((x,y) , (f(x,y),f(y,x))) = d (x,f (x,y)) + d (y,f (y,x)) , which implies d (x,f (x,y)) = d (y,f (y,x)) = 0. therefore, x = f (x,y) = g (x,y) and y = f (y,x) = g (y,x) . � definition 2.2. let (x,�,d) be an ordered metric space and f,g : x2 → x be two mappings. we say that (f,g) is a generalized meir-keeler type contraction if, for all ε > 0, there exists η > 0 such that, for all comparable (x,y) , (u,v) ∈ x2, ε ≤ 1 2 mmax (x,y,u,v)≤ ε + η (ε)⇒ d (f (x,y) ,g (u,v)) < ε, (2.22) where mmax (x,y,u,v) = max   δ ((x,y) , (u,v)) ,δ ((x,y) , (f (x,y) ,f (y,x))) , δ((u,v) , (g (u,v) ,g (v,u))), 1 2 [ δ ((x,y) , (g (u,v) ,g (v,u))) +δ (u,v) , (f (x,y) ,f (y,x)) ]   . if we take ψ (t1, t2, t3, t4) = max{t1, t2, t3, t4} in theorem 2.1, we have the following result. corollary 2.1. let (x,�,d) be a complete ordered metric space, f,g : x2 → x be weakly increasing mappings with respect to � and (f,g) be a generalized meir-keeler type contraction. assume that the following conditions are satisfied: (a) f (or g) is continuous or (b) (x,�,d) is regular. then f and g have a coupled common fixed point. 194 işik, imdad, turkoglu and hussain proposition 2.1. let (x,�,d) be an ordered metric space and let f,g : x2 → x be two given mappings. if the following contraction is satisfied, then (f,g) is a generalized meir-keeler type contraction: d (f (x,y) ,g (u,v)) ≤ k 2 mmax (x,y,u,v) , k ∈ [0, 1). proof. assume that the above inequality holds. then, for all ε > 0, we can easily show that (2.22) is satisfied with η (ε) = ( 1 k − 1 ) ε. � definition 2.3. let (x,�) be an ordered set and f : x2 → x. we say that f is nondecreasing if, for any x,y ∈ x, x1,x2 ∈ x, x1 � x2 ⇒ f(x1,y) � f(x2,y) and y1,y2 ∈ x, y1 � y2 ⇒ f(x,y1) � f(x,y2). if we choose f = g in theorem 2.1, we have the following corollary. corollary 2.2. let (x,�,d) be a complete ordered metric space, f : x2 → x be a nondecreasing mapping and f be a generalized meir-keeler type ψ-contraction. assume that the following conditions hold: (a) f is continuous or (b) (x,�,d) is regular. then f has a coupled fixed point. we denote by φ the family of all functions φ : r+ → r+ such that (φ1) φ is nondecreasing and right continuous; (φ2) φ (0) = 0 and φ (t) > 0 for any t > 0. theorem 2.3. let (x,�,d) be a complete ordered metric space and f,g : x2 → x be two weakly increasing mappings with respect to �. assume that, for all ε > 0, there exists η > 0 such that, for all comparable (x,y) , (u,v) ∈ x2, ε ≤ φ ( 1 2 m (x,y,u,v) ) ≤ ε + η (ε)⇒ φ (d (f (x,y) ,g (u,v))) < ε, (2.23) where φ ∈ φ. then (f,g) is a generalized meir-keeler type ψ-contraction. proof. fix ε > 0. since φ (ε) > 0, there exists γ > 0 such that, for all comparable (u,v) , (w,z) ∈ x2, φ (ε) ≤ φ ( 1 2 m (u,v,w,z) ) ≤ φ (ε) + γ ⇒ φ (d (f (x,y) ,g (u,v))) < φ (ε) . (2.24) due to the right continuity of φ, there exists η > 0 such that φ (ε + η) < φ (ε) +γ. for any comparable (x,y) , (u,v) ∈ x2, such that ε ≤ 1 2 m (x,y,u,v) ≤ ε + η. since φ is nondecreasing, we have: φ (ε) ≤ φ ( 1 2 m (x,y,u,v) ) < φ (ε + η) < φ (ε) + γ, by (2.24), we obtain φ (d (f (x,y) ,g (u,v))) < φ (ε) which implies d (f (x,y) ,g (u,v)) < ε, as φ is nondecreasing. this completes the proof. � corollary 2.3. let (x,�,d) be a complete ordered metric space and f,g : x2 → x be two weakly increasing mappings with respect to �. assume that, for all ε > 0, there exists η > 0 such that, for every comparable pair (x,y) , (u,v) ∈ x2, ε ≤ ∫ 1 2 m(x,y,u,v) 0 θ (t) dt ≤ ε + η (ε)⇒ ∫ d(f (x,y),g(u,v)) 0 θ (t) dt < ε, where θ is a locally integrable function from r+ into itself satisfying ∫ s 0 θ (t) dt > 0 for all s > 0. suppose also that the following conditions are satisfied: generalized meir-keeler contractions 195 (a) f (or g) is continuous or (b) (x,�,d) is regular. then f and g have a coupled common fixed point. corollary 2.4. let (x,�,d) be a complete ordered metric space and f,g : x2 → x be two weakly increasing mappings with respect to �. assume that, for all comparable (x,y) , (u,v) ∈ x2,∫ d(f (x,y),g(u,v)) 0 θ (t) dt ≤ k ∫ 1 2 m(x,y,u,v) 0 θ (t) dt, where k ∈ (0, 1) and the function θ is defined as in corollary 2.3. suppose also that the following conditions are satisfied: (a) f (or g) is continuous or (b) (x,�,d) is regular. then f and g have a coupled common fixed point. proof. for any ε > 0, choosing η (ε) = ( 1 k − 1 ) ε and applying corollary 2.3, the desired result is obtained. � 3. an application consider the following integral equations: x (s) = ∫ b a h1 (s,r,x (r) ,y (r)) dr, y (s) = ∫ b a h1 (s,r,y (r) ,x (r)) dr, (3.1) and x (s) = ∫ b a h2 (s,r,x (r) ,y (r)) dr, y (s) = ∫ b a h2 (s,r,y (r) ,x (r)) dr, (3.2) where s ∈ i = [a,b] , h1,h2 : i × i ×r×r → r and b > a ≥ 0. in this section, we present an existence theorem for a common solution to (3.1) and (3.2) that belongs to x := c(i,r) (the set of continuous functions defined on i) by using the obtained result in corollary 2.1. we consider the operators f,g : x2 → x given by f (x,y) (s) = ∫ b a h1 (s,r,x (r) ,y (r)) dr, x,y ∈ x, s ∈ i, and g (x,y) (s) = ∫ b a h2 (s,r,x (r) ,y (r)) dr, x,y ∈ x, s ∈ i. then the existence of a common solution to (3.1) and (3.2) is equivalent to the existence of a coupled common fixed point of f and g. it is well known that x endowed with the metric d defined by d(x,y) = sups∈i|x (s) −y (s) | for all x,y ∈ x, forms a complete metric space. also, equip x with the partial order � given by x,y ∈ x, x � y ⇔ x(s) ≤ y(s), ∀s ∈ i. recall that in [14], it is proved that (x,�,d) is regular. suppose that the following conditions hold. (a) h1,h2 : i × i ×r×r → r are continuous; (b) for all s,r ∈ i and x,y ∈ x, we have h1 (s,r,x (r) ,y (r))≤h2 ( s,r, ∫ b a h1 (r,τ,x (τ) ,y (τ)) dτ, ∫ b a h1 (r,τ,y (τ) ,x (τ)) dτ ) and h2 (s,r,x (r) ,y (r))≤h1 ( s,r, ∫ b a h2 (r,τ,x (τ) ,y (τ)) dτ, ∫ b a h2 (r,τ,y (τ) ,x (τ)) dτ ) (c) for all comparable (x,y) , (u,v) ∈ x2 and for every s,r ∈ i, we have 196 işik, imdad, turkoglu and hussain |h1 (s,r,x (r) ,y (r)) −h2 (s,r,u (r) ,v (r))| 2 ≤ k2 4 γ (s,r) [|x (r) −u (r)| + |y (r) −v (r)|]2 , where k ∈ [0, 1) and γ : i2 → r+ is a continuous function satisfying sups∈i ∫ b a γ (s,r) ≤ 1/ (b−a) . theorem 3.1. assume that conditions (a)-(c) are satisfied. then, integral equations (3.1) and (3.2) have a common solution in x. proof. from condition (b), the mappings f and g are weakly increasing with respect to �. let (x,y) is comparable to (u,v) . then, by (c), for all s ∈ i, we deduce |f (x,y) (s) −g (u,v) (s)|2 ≤ (∫ b a |h1 (s,r,x (r) ,y (r)) −h2 (s,r,u (r) ,v (r))|dr )2 ≤ ∫ b a 12dr ∫ b a |h1 (s,r,x (r) ,y (r)) −h2 (s,r,u (r) ,v (r))| 2 dr ≤ (b−a) ∫ b a k2 4 γ (s,r) [|x (r) −u (r)| + |y (r) −v (r)|]2 dr ≤ k2 4 (b−a) ∫ b a γ (s,r) [d (x,u) + d (y,v)] 2 dr ≤ k2 4 (b−a) sup s∈i (∫ b a γ (s,r) dr ) [δ ((x,y) , (u,v))] 2 ≤ [ k 2 δ ((x,y) , (u,v)) ]2 ≤ [ k 2 mmax (x,y,u,v) ]2 , where mmax (x,y,u,v) = max   δ ((x,y) , (u,v)) ,δ ((x,y) , (f (x,y) ,f (y,x))) , δ((u,v) , (g (u,v) ,g (v,u))), 1 2 [ δ ((x,y) , (g (u,v) ,g (v,u))) +δ (u,v) , (f (x,y) ,f (y,x)) ]   . therefore, we obtain ( sup s∈i |f (x,y) (s) −g (u,v) (s)| )2 ≤ [ k 2 mmax (x,y,u,v) ]2 , and so d (f (x,y) ,g (u,v)) ≤ k 2 mmax (x,y,u,v) . (3.3) hence, by proposition 2.1, (f,g) is a generalized meir-keeler type contraction. therefore, from corollary 2.1, f and g have a coupled common fixed point, that is, integral equations (3.1) and (3.2) have a common solution in x. � references [1] s. banach, sur les operations dans les ensembles abstraits et leur applications aux equations integrales, fund. math. 3 (1922), 133-181. [2] t. g. bhaskar and v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal. 65 (2006), 1379–1393. [3] c. chen and i. j. lin, common fixed points of generalized cyclic meir-keeler-type contractions in partially ordered metric spaces, abstr. appl. anal. 2013 (2013), art. id 787342. [4] s. h. cho and j. s. bae, fixed points of weak α-contraction type maps, fixed point theory appl. 2014 (2014), art. id 175. [5] b. s. choudhury and p.maity, cyclic coupled fixed point result using kannan type contractions, journal of operators 2014 (2014), article id 876749. [6] h. s. ding, l. li and w. long, coupled common fixed point theorems for weakly increasing mappings with two variables, j. comput. anal. appl. 15(8) (2013), 1381-1390. [7] m. e. gordji, y. j. cho, s. ghods, m. ghods and m. h. dehkordi, coupled fixed point theorems for contractions in partially ordered metric spaces and applications, math. probl. eng. 2012 (2012), art. id 150363. [8] d. guo and v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal. 11 (1987), 623-632. generalized meir-keeler contractions 197 [9] n. hussain, m. abbas, a. azam and j. ahmad, coupled coincidence point results for a generalized compatible pair with applications, fixed point theory appl. 2014 (2014), art. id 62. [10] h. işık and d. turkoglu, coupled fixed point theorems for new contractive mixed monotone mappings and applications to integral equations, filomat 28(6) (2014), 1253-1264. [11] v. lakshmikantham and l. ciric, coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, nonlinear anal. 70 (2009), 4341–4349. [12] n. v. luong and n. x. thuan, coupled fixed points in partially ordered metric spaces and application, nonlinear anal. 74 (2011), 983-992. [13] a. meir and e. keeler, a theorem on contraction mappings, j. math. anal. appl. 28 (1969), 326–329. [14] j. j. nieto and r. rodriguez-lopez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22 (2005), 223–239. [15] b. samet, coupled fixed point theorems for a generalized meir-keeler contraction in partially ordered metric spaces, nonlinear anal. 72(12) (2010), 4508–4517. [16] t. suzuki, fixed point theorem for asymptotic contractions of meir-keeler type in complete metric spaces, nonlinear anal. 64 (2006), 971-978. 1department of mathematics, faculty of science and arts, muş alparslan university, muş 49100, turkey 2department of mathematics, aligarh muslim university, aligarh, uttar pradesh, 202002, india 3department of mathematics, faculty of science, university of gazi, 06500-teknikokullar, ankara, turkey 4department of mathematics, king abdulaziz university, p.o. box 80203, jeddah, 21589, saudi arabia ∗corresponding author: isikhuseyin76@gmail.com 1. introduction and preliminaries 2. main results 3. an application references int. j. anal. appl. (2023), 21:54 on the exponential stability of the implicit differential systems in hilbert spaces nor el-houda beghersa1,∗, mehdi benabdallah2, mohamed hariri3 1faculty of mathematics and computer sciences, department of mathematics, university of sciences and technology of oran mohamed boudiaf usto-mb, b.p 1505 el-m’nouar, bir el djir 31000, oran, algeria 2faculty of mathematics and computer sciences, department of mathematics, university of sciences and technology of oran mohamed boudiaf usto-mb, b.p 1505 el-m’nouar, bir el djir 31000, oran, algeria 3mohamed hariri, department of mathematics, ain temouchent university, 46000, ain temouchent, algeria ∗corresponding author: norelhouda.beghersa@univ-usto.dz abstract. the aim of this research is to study the exponential stability of the stationary implicit system: ax ′(t)+bx(t)=0, where a and b are bounded operators in hilbert spaces. the achieved results are the generalization of liapounov theorem for the spectrum of the operator pencil λa + b. we also establish the exponential stability conditions for the corresponding perturbed and quasi-linear implicit systems. 1. introduction consider the general implicit differential system described by the following form: ax′(t)+bx(t)= θ(t,x(t)), t ≥ t0, t0 ≥ 0, (1.1) where a and b are bounded operators acting from the hilbert space x into another hilbert space y, θ is an operator, usually non-linear from [t0,+∞[×x into y . next, we assume that the system 1.1 has solutions. the system 1.1 has been considered in various forms by many authors as a. favini and a. yagi [6], received: apr. 10, 2023. 2020 mathematics subject classification. 34l05, 15a22, 93d05, 34d20. key words and phrases. spectral theory; operator pencil; stability; implicit system. https://doi.org/10.28924/2291-8639-21-2023-54 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-54 2 int. j. anal. appl. (2023), 21:54 a.g. rutkas [7], l.a. vlasenko [8] and others. in the present paper, we study not only the stationary implicit system ax′(t)+bx(t)=0, t ≥ t0, (1.2) but also the quasi-linear system 1.1, with the initial condition x(t0)= x0. in [2] the authors obtained results concerning the stability of the degenerate difference systems that is similar to 1.1. we can find some practical examples for the above systems in [6–8]. we introduce at first, the basic bellow definition 1.1 of the exponential system. in section 2, we extend the famous general theorem of liapounov [4] which plays an important role in our paper. in section 3, we establish some conditions of the exponential stability for the perturbed implicit systems and finally, we present our main results about the exponential stability for the solution of the quasi-linear implicit system 1.1. definition 1.1. the system 1.1 is said to be exponential if there exist the constants m and α such that for all solutions x(t), t ≥ t0, we have: ‖x(t)‖≤ meα(t−t0)‖x0‖. (1.3) if α < 0, then the system 1.1 is said to be exponentially stable. 2. stationary systems consider the implicit stationary system 1.2 above where a and b are bounded operators acting from x into y . definition 2.1. [7] the system 1.2 is well-posed, if it is determined (i.e, if x(t0) = x0 = 0 then, x(t)=0,∀t ≥ t0), and its evolution operator s(t) : x0 7→ x(t) is bounded for all t ≥ t0. in this work, we use the spectral theory of the operator pencil λa+b. definition 2.2. [3,4] the complex number λ ∈c is said to be a regular point of the pencil λa+b, if the resolvant operator r(λ) = (λa+b)−1 exists and it is bounded. the set of all regular points is denoted by ρ(a,b), and its complement σ(a,b)=c\ρ(a,b) is called the spectrum of the pencil λa+b. the set of all eigen-values of the pencil λa+b is denoted by σp(a,b)= {λ ∈c\∃v 6=0, (λa+b)v =0}. (2.1) proposition 2.1. the system 1.2 is exponential if and only if it is well-posed. int. j. anal. appl. (2023), 21:54 3 proof. if the system 1.2 is exponential then, it admits a unique solution. in fact, if x0 =0, according to 1.3, we have ‖x(t)‖≤ 0. hence, x(t)≡ 0 for all t ≥ t0, in one hand. on the other hand, we have ‖x(t)‖= ‖s(t)x0‖≤ meα(t−t0)‖x0‖, so, the operator s(t) is bounded, therefore, the system 1.2 is well-posed. conversely, if the system 1.2 is well-posed then, according to [7], we have lim t→∞ ln‖s(t)‖ t = ω < ∞. hence, ∀ε > 0, ∃nε/‖s(t)‖≤ e(ω+ε)t, ∀t > nε. thus, ‖x(t)‖≤‖s(t)x0‖≤ e(ω+ε)t‖x0‖, ∀t > nε. if we put m1 = sup t∈[t0,nε] ‖s(t)‖, we obtain ‖x(t)‖= ‖s(t)x0‖≤ me(ω+ε)(t−t0)‖x0‖, with m = sup t∈[t0,nε] { m1 e(ω+ε)(t−t0) ,e(ω+ε)t}. therefore, the system 1.2 is exponential. � we can find some necessary and sufficient conditions for the system 1.2 to be well-posed in [7]. proposition 2.2. if the system 1.2 is exponential then, all the eigen-values of the pencil λa+b are in the half plane (re(λ)≤ α), where α is the constant appearing in 1.3. in particular, if the system 1.2 is exponentially stable then, all the eigen-values of the pencil λa+b belong to the left half plane, i.e: σp(a,b)⊂{λ : re(λ) < 0}. proof. suppose that there exists an eigen-value λ0 ∈ σp(a,b) with re(λ0) > α. then, (λ0a+b)v = 0 and v is the corresponding eigen-vector. therefore, y(t) = eλ0(t−t0)v for t ≥ t0 is a solution of the system 1.2 such that v = y(t0) = y0. moreover, we have: ‖y(t)‖= ‖eλ0(t−t0)y0‖= ere(λ0)(t−t0)‖y0‖ > eα(t−t0)‖y0‖. so, the solution y(t) does not satisfy the condition 1.3 hence, the system 1.2 is not exponential. � the general liapounov theorem (see theorem 2.4, [3]) can be also extended to the operator pencil λa+b for an arbitrary constant α as follows: 4 int. j. anal. appl. (2023), 21:54 theorem 2.1. a necessary condition for the sepctrum σ(a,b) of the pencil λa+b to lie inside the half-plane re(λ) < α, is that for any uniformly positive operator g � 01, there exists an operator w � 0 such that: b∗wa+a∗wb +2αa∗wa = g β (β 6=0), (2.2) and a sufficent condition is that α∓ iβ ∈ ρ(a,b) and there exists an operator w � 0 such that: b∗wa+a∗wb +2αa∗wa � 0. (2.3) proposition 2.3. if a and b are bounded operators in hilbert spaces x, y and there exists an operator w � 0 such that: f = b∗wa+a∗wb +2αa∗wa � 0, then, there exists a real number β 6=0 satisfies the property: α+ iβ /∈ σp(a,b). proof. suppose that, for all β 6= 0, we have α+ iβ ∈ σp(a,b), there exists v 6= 0 an eigen-vector verifies [(α+ iβ)a+b]v =0. now, we compute the inner product then, we get < fv,v > = < b∗wav +a∗wbv +2αa∗wav,v >, = < wav,bv > + < wbv,av > +2α < wav,av >, = −(α− iβ) < wav,av > −(α+ iβ) < wav,av > +2α < wav,av >, = 0. so, we obtain a contradiction with our hypothesis < fv,v >≥ c‖v‖2 > 0, which proves the proposition. � proposition 2.4. in finite dimentional spaces (i.e, dim(x)= dim(y ) < ∞), if σ(a,b)= σp(a,b)⊂{λ : re(λ) < ω}, then, the system 1.2 is exponential. moreover, we have α ≤ ω, where α is the constant appearing in 1.3. proof. suppose that, the system 1.2 is not exponential. using the method of elementary divisors (see for example f.r gantmacher [5]) and noting that the pencil of matrices λa+b is regular (i.e, det(λa+b) 6=0) to prove our proposition. so, λa+b ∼ λã+ b̃ = {nµ1,nµ2, ...,nµs;=+λi}; 1it means that g = g∗ and that < gx,x >> c‖x‖, ∀c ∈ r and for all x with ‖x‖=1. int. j. anal. appl. (2023), 21:54 5 where the first diagonal blocks correspond to the infinite elementary divisors. now, we put x(t)= qz(t) with det(q) 6=0. so, the system 1.2 is equivalent to the following system:   az ′(t)+bz(t)=0, ã = aq, b̃ = bq, λã+ b̃ =(λa+b)q. (2.4) in accordance with the diagonal blocks, the system 1.2 can be written as follows:  nµk dzk dt =0, k =1,2, ...,s. dz̃k dt +=z̃ =0, where z =(z1,z2, ...,zs, z̃)t. (2.5) since, σ(a,b)= σ(ã,b̃)= σ(i,=)= σ(−=)⊂{λ : re(λ) < ω}, then: ‖e−=t‖≤ mωeωt, and ‖z̃(t)‖= ‖e−=(t−t0)z̃(t0)‖≤ mωeω(t−t0)‖z̃(t0)‖. so, ‖x(t)‖= ‖qz(t)‖≤‖q‖mωeω(t−t0)‖z(t)‖. therefore, the system 1.2 is exponential for α ≤ ω. we obtain a contradiction with our hypothesis (< fv,v >≥ c‖v‖2), which proves the proposition. � corollary 2.1. if dimx = dimy < ∞, then the following conditions are equivalents: (1) the system 1.2 is exponential. (2) σ(a,b)= σp(a,b)⊂{λ : re(λ) < α}. (3) ∃w � 0 such that b∗wa+a∗wb +2αa∗wa � 0. according to the proposition 2.2 and 2.3, we have (1) ⇐⇒ (2) also, from theorem 2.1 and proposition 2.4, we obtain (2)⇐⇒ (3). in particular, if α =0 then, we obtain the next result: corollary 2.2. if the spaces x and y have the same finite dimension then, the following assertions are equivalents: (1) the system 1.2 is exponentially stable. (2) σ(a,b)= σp(a,b)⊂{λ : re(λ) < 0}. (3) ∃w � 0such that b∗wa+a∗wb � 0. 6 int. j. anal. appl. (2023), 21:54 3. perturbed sytems we can use the method of variation of constants [7] to prove the following lemma: lemma 3.1. suppose that in the system 1.2, the operator a0 = a/d0 is invertible 2 with d0 = {x0}. if θ(s,x(s))∈ ad0,∀s ≥ t0 and the function s(t − s)a−10 θ(s,x(s)) is integrable(with respect to s), where s(t) is the evolution operator of the system 1.2. then, for all x0 ∈ ad0, the system 1.1 is equivalent to x(t)= s(t − s)x0 + ∫ t t0 s(t − s)a−10 θ(s,x(s))ds. (3.1) in the following, we use the lemma of gronwall-bellman: lemma 3.2. [1] if: g(t)≤ c + ∫ t t0 g(s)h(s), ∀t ≥ t0, (3.2) where h is a continuous positive real function and c > 0 is an arbitrary constant. then, g(t)≤ c.exp[ ∫ t t0 h(s)ds]. (3.3) for the non stationary perturbation of the system 1.2 with: θ(t,x(t))=−(b +b(t)), we have: theorem 3.1. suppose that: (1) the system 1.2 is well-posed. (2) the operator a0 is invertible. (3) the linear operators b(t), t ≥ t0 which transforme d0 into ad0 such that∫ ∞ t0 ‖a−10 b(t)‖dt < ∞. (3.4) then, the perturbed system: ax′(t)+(b +b(t))x(t)=0, t ≥ t0 (3.5) is exponential with the same constant α as in 1.3. proof. according to the lemma 3.1 with ψ(t,x(t))≡−(b+b(t))x(t), the system 3.5 is equivalent to: x(t)= s(t − t0)x0 − ∫ t t0 s(t − t0)a−10 b(s)x(s)ds, (3.6) 2in particular, if the system 1.2 is well posed then, the operator a0 (i.e the restriction of a in d0) is invertible. int. j. anal. appl. (2023), 21:54 7 where s(t) is the evolution operator of the system 1.2. using the hypothesis (1) and the proposition 2.1, we obtain: ‖s(t − t0)x0‖) ≤ meα(t−t0)‖x0‖, (3.7) ‖s(t − t0)a−10 b(s)x(s)‖ ≤ me α(t−s)‖a−10 b(s)x(s)‖, (3.8) from (2) and (3), we have a−10 b(s)x(s)∈ d0. according to 3.6, we obtain: ‖x(t)‖≤ meα(t−t0)‖x0‖+m ∫ t t0 eα(t−s)‖a−10 b(s)‖‖x(s)‖ds, (3.9) or e−α(t−t0)‖x(t)‖≤ m‖x0‖+m ∫ t t0 eα(t0−s)‖a−10 b(s)‖‖x(s)‖ds. (3.10) applying lemma 3.2, where g(t)= e−α(t−t0)‖x(t)‖, h(t)= m‖a−10 b(t)‖, c = m‖x0‖, (3.11) then, e−α(t−t0)‖x(t)‖ ≤ m‖x0)‖.exp[m ∫ t t0 ‖a−10 b(s)‖ds], (3.12) ≤ m‖x0‖.exp[m ∫ ∞ t0 ‖a−10 b(s)‖ds]. (3.13) thus, ‖x(t)‖≤ m1eα(t−t0)‖x0‖, where m1 = m.exp[m ∫ ∞ t0 ‖a−10 b(s)‖ds] < ∞. � corollary 3.1. under the condition of theorem 3.1, if the unperturbed system 1.2 is exponentially stable then, the perturbed system 3.5 is also exponentially stable. remark 3.1. the theorem 3.1 is a generalization of dini-hukuhara’s theorem [1], (a ≡ i, b(t)≡ −t(t), α =0). example 3.1. consider the system 3.5 in finite dimensional spaces (dimx = dimy = 2), with the matrices: a = ( 1 0 1 0 ) , b = ( 1 0 1 1 ) , b(t)= e−t ( 1 1 1 0 ) , t ≥ t0. in our case we have: d0 = {(a,b)/b =0}, ad0 = {(a,b)/b = a}, λa+b = ( λ+1 0 λ+1 1 ) , (λa+b)−1 = 1 λ+1 ( 1 0 −λ−1 λ+1 ) it’s clear that b(t) : d0 7→ ad0, t ≥ t0 and a0 is invertible. since, σ(a,b)= σp(a,b)= {−1}. 8 int. j. anal. appl. (2023), 21:54 then, the system 1.2 is exponentially stable (see corollary 2.2). according to corollory 3.1, we conclude that the perturbed system 3.5 is also exponentially stable because:∫ ∞ t0 ‖a−10 b(t)‖dt ≤‖a −1 0 ‖ ∫ ∞ t0 ‖b(t)‖dt = ‖a−10 ‖ ∫ ∞ t0 e−tdt = e−t0‖a−10 ‖ < ∞. 4. quasi-linear systems using the same way for demonstration of theorem 3.1, we can prove the following theorem: theorem 4.1. suppose that: (1) the system 1.2 is exponential. (2) the operator a0 is invertible. (3) the non-linear operator θ transforms [t0,∞[×d0 into ad0 such that ‖a−10 θ(t,v)‖≤ ϕ(t).‖v‖, ∀v ∈ d0, where ϕ is real positive function satisfies ∫ ∞ t0 ϕ(t)dt < ∞, then, the quasi-linear system 1.1 is exponential with the same constant α as in 1.3. corollary 4.1. if the linear system 1.2 is exponentially stable then, under the conditions of theorem 4.1, the quasi-linear system 1.1 is also exponentially stable. theorem 4.2. suppose that: (1) the system 1.2 is exponential. (2) the operator a0 is invertible. (3) the non-linear operator θ(t, .) transforms d0 into ad0 such that ‖a−10 θ(t,v)‖≤ γ‖av‖, ∀v ∈ d0, ∀t ≥ t0. (4.1) then, the quasi-linear system 1.1 is also exponential with the constants: α1 = α+γm, m1 = m. (4.2) this result represents the generalization of the famous liapounov theorem on the stability by the first approximation for quasi-linear systems. corollary 4.2. if under the conditions of theorem 4.2, the constant γ is small enough (γ < − α m ) then, the exponential stability of the linear system 1.2 implies the exponential stability of the corresponding quasi-linear system 1.1. the obtained results on the exponential stability can be used to obtain some conditions on the exponential stabilization of implicit controlled systems. finally, we note that similar results for the discrete implicit systems are obtained in the paper [2]. int. j. anal. appl. (2023), 21:54 9 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. bellman, k.l cooke, differential-difference equations, academic press, london, 1963. [2] m. benabdallakh, a.g. rutkas, a.a. solov’ev, on the stability of degenerate difference systems in banach spaces, j. soviet math. 57 (1991), 3435-3439. https://doi.org/10.1007/bf01880215. [3] m. benabdallah, m. hariri, on the stability of the quasi-linear implicit equations in hilbert spaces, khayyam j. math. 5 (2019), 105-112. https://doi.org/10.22034/kjm.2019.81222. [4] j.l. daleckii, m.g. krien, stability of solutions of differential equations in banach spaces, translations of mathematical monographs, volume 43, american mathematical society, providence, ri, 1974. [5] f.r. gantmakher, applications of the theory of matrices, dover publications, mineola, 2005. [6] a. favini, a. yagi, degenerate differential equations in banach spaces, marcel dekkar inc, new york, 1999. [7] a.g. rutkas, spectral methods for studying degenerate differential-operator equations. i, j. math. sci. 144 (2007), 4246-4263. https://doi.org/10.1007/s10958-007-0267-2. [8] l.a. vlasenko, evolutionary models with implicit and degenerate differential equations, sistemnye tekhnologii, dnepropetrovsk, 2006. https://doi.org/10.1007/bf01880215 https://doi.org/10.22034/kjm.2019.81222 https://doi.org/10.1007/s10958-007-0267-2 1. introduction 2. stationary systems 3. perturbed sytems 4. quasi-linear systems references international journal of analysis and applications volume 19, number 1 (2021), 65-76 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-65 on a new constraint reduction heuristic using improved bisection method for mixed integer linear programming hande günay akdemir∗ giresun university, faculty of arts and sciences, department of mathematics, giresun, turkey ∗corresponding author: hande.akdemir@giresun.edu.tr abstract. in this study, we develop a surrogate relaxation-based procedure to reduce mixed-integer linear programming (milp) problem sizes. this technique starts with one surrogate constraint which is a nonnegative linear combination of multiple constraints of the problem. at this initial step, we calculate optimal lagrangian multipliers from lp relaxation of the problem and use them as initial surrogate multipliers. we incorporate the improved bisection method (ibm) (b. gavish, f. glover, and h. pirkul, surrogate constraints in integer programming, j. inform. optim. sci. 12(2) (1991), 219–228.) into our algorithm. this simple heuristic algorithm is designed to iteratively generate a new surrogate cut that is to guarantee to satisfy the most violated two constraints of the corresponding iteration. the performance of the heuristic is tested using both some problems from the or libraries and randomly generated ones. 1. introduction the objective here is to attempt to reduce the number of constraints of milp type of problems. in the surrogate relaxation, a subset of constraints is substituted by a linear combination of these constraints, that is, we multiply these constraints with non-negative multipliers, and then aggregate them together to generate one combined constraint. with the optimal surrogate multipliers, this combined constraint serves as a proxy for the others and captures useful information. by doing that, we may obtain infeasible but near-optimal solutions while reducing processing time and complexity relative to the original model. received october 26th, 2020; accepted november 19th, 2020; published december 4th, 2020. 2010 mathematics subject classification. 90c10, 90c27, 90c46, 49n15. key words and phrases. mixed integer linear programming; surrogate relaxation; surrogate constraints; lagrangian multipliers; improved bisection method; heuristics; reduction. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 65 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-65 int. j. anal. appl. 19 (1) (2021) 66 besides, the use of surrogate constraints is known to provide better bounds relative to lagrangian bounds. in the proposed heuristic, as initial surrogate multipliers, we simply use the optimal dual values (lagrangian multipliers) of the lp relaxation to get the strongest initial surrogate constraint. we incorporate the ibm [12], which is an updated version of the bisection algorithm [13], into our heuristic. at each iteration, the procedure evaluates two infeasible constraints at a time, and uses the ibm to find the optimal surrogate multipliers while upper (or lower) bound decreases (or increases). furthermore, we generate not one but several surrogate constraints and only relax inequality constraints. so, we solve a sequence of computationally easy relaxed and reduced problems to reach the optimal solution in a small amount of time. our proposed heuristic also enables us to examine which ratio of the problem constraints are redundant and which ones are critical. the heuristic is implemented in the matlab environment. in order to test the algorithm, we use problems available for download from beasley’s or library [4] and the mixed integer programming library [23]. we also conduct a set of testings using randomly generated instances. all of the computational experiments are done on a pc with an intel core i5-7400 cpu (3.00 ghz) and 4 gb of ram. consider the following milp problem: (1.1) (p) min ct x s.t. ax ≤ b x ∈ x where c,x ∈ rn,b ∈ rm,a ∈ rm×n and x is a discrete set that may be defined by some linear equalities and bound constraints of the decision variables. by moving the constraints into the objective function, we generate lagrange relaxation: (1.2) min ct x + λt (ax− b) s.t. x ∈ x where lagrange multipliers vector λ ≥ 0. analogously, by assembling multiple constraints into a single new surrogate constraint, we generate surrogate relaxation: (1.3) min ct x s.t. µt (ax− b) ≤ 0 x ∈ x where surrogate multipliers vector µ ≥ 0. both techniques enlarge the feasible region and provide a lower bound on the optimal objective value of problem (p). but, surrogate lower bound is tighter than the lagrangian lower bound [14, 17]. int. j. anal. appl. 19 (1) (2021) 67 relaxation-based search or dual algorithms, both exact and heuristics, have been extensively used in finding bounds for integer programming (ip). let us now survey the related literature which investigates surrogate relaxation-based heuristics to solve combinatorial optimization problems. in [15], the author proposed a class of surrogate constraint heuristics that provide a variety of supplementing new alternatives and independent solution strategies. lorena and narciso [31, 39] proposed six heuristics based on both the surrogate and lagrangian relaxations and a subgrandient search algorithm for large scale generalized assignment problems. the authors showed that the use of procedures based on surrogate constraint analysis is effective for satisfiability problem in [30]. applications of a classical combinatorial optimization problem called the set covering were given in [1,11]. the former used the surrogate constraint normalization technique to create appropriate weights for surrogate constraint relaxations, and the latter compared heuristics based on lagrangian and surrogate relaxations for the maximal covering location problem. karabati et al. [22] handled a class of discrete problems with min-max-sum objectives by using line search and surrogate procedures to obtain optimal surrogate multipliers. the combined use of surrogate and lagrangian relaxation was considered for traveling salesman problem in [26, 32, 36]. the reduction is a preprocessing technique that is fundamental for developing efficient integer programming methods and based on dynamic programming and upper bounds. for a comprehensive literature review of reduction techniques, see [3, 10, 40]. riberio and lorena [46] examined an ip problem called cartographic label placement by using lagrangian/surrogate heuristics. for a detailed review on solution techniques based on surrogate relaxations for p−median and facility location problems, see [43]. for a review of the approaches which combine metaheuristics with exact ip techniques, see [42]. in [27], a critical event tabu search heuristic was presented to solve the multidimensional knapsack problems (mkps) with generalized upper bound constraints. osorio et al. [41] considered a combined cutting and surrogate constraint analysis strategy for mkps, see also [19]. the authors introduced problem-size reduction heuristics for mkps in [21]. for heuristics and metaheuristics for mkps and their variants, see recent works of [2, 5, 6, 8, 18, 20, 24, 25, 28, 29, 33, 35, 47–50], and references therein. for graph theory applications, such as graph coloring, weighted maximum clique, and shortest path problems see [9, 16, 37, 38]. choi and choi [7] proposed a redundancy identification method that is based on surrogate constraints. the relaxation adaptive memory programming approach based on surrogate constraints was proposed for combinatorial optimization problems in [44] and for capacitated minimum spanning tree problems in [45]. to review the scatter search method which was conceived as an extension of surrogate constraint relaxation, see [34]. the rest of this paper is organized as follows. section 2 gives a brief version of the ibm, followed by the proposed heuristic is introduced in section 3. section 4 dercribes computational experiments. section 5 concludes the paper. int. j. anal. appl. 19 (1) (2021) 68 2. computing the optimal surrogate multipliers by ibm now, we consider problems with only two constraints. we start by determining which constraint is tighter, namely, it produces a lower (greater) objective value for maximization (minimization) problem considering only one constraint and ignoring the other one. renumarate the tighter constraint as constraint 1, and the other as constraint 2. the surrogate multiplier of constraint 1 is constant and equals to 1. other multiplier is µ which is going to be updated. the algorithm can be summarized as follows: step 1. initial values of µl and µh are as following. the solution of the problem with multipliers (1,µl) and (1,µh ) satisfies and does not satisfy constraint 1, respectively. step 2. let µ = (µl+ µh )/2. solve the problem with the multipliers (1,µ). if (i) both constraints are satisfied stop, (1,µ) is the optimal multipliers, and the optimal solution of the problem is obtained. (ii) only constraint 1 is satisfied let µl = f�g where f is the amount of oversatisfaction of constraint 1 and g is the amount of undersatisfaction of constraint 2. (iii) only constraint 2 is satisfied let µh = f�g where f is the amount of undersatisfaction of constraint 1 and g is the amount of oversatisfaction of constraint 2. step 3. if µl < µh go to step 2, otherwise stop, (1,µ) is the optimal multipliers. note that, we can also stop if the objective function does not improve, or if the number of iterations reached an upper limit. 3. constraint reduction heuristic to determine the surrogate multipliers of the most violated two constraints of the current step, we apply the following iterative process which finds appropriate surrogate constraints. initial solution: we start by calculating optimal lagrangian multipliers from lp relaxation of the problem and use them as initial surrogate multipliers. this surrogate constraint is our first surrogate constraint. then, we solve our problem with this constraint and integrality restrictions. adding a new surrogate constraint (generating a cut): we first determine the most violated two constraints. let gi(x) = n∑ j=1 aijxj − bi, i ∈ i = {1, 2, . . . ,m} , and max i∈i gi(x) = gt(x), max i∈i−{t} gi(x) = gk(x). add a new surrogate constraint as µtgt(x) + µkgk(x) ≤ 0 where µt = µk = 1. solve the problem adding this new surrogate constraint to prior surrogate constraint(s). if gt(x) and gk(x) are non-positive while µt = µk = 1, add another surrogate constraint. otherwise, apply the ibm supposing the positive one as int. j. anal. appl. 19 (1) (2021) 69 constraint 1. if both constraints are satisfied then add a new surrogate constraint. repeat these steps until all constraints are satisfied (within a tolerance value). pseudo-code is given in algorithm 3.1. 4. computational results this section presents the experimental results obtained with the proposed simple heuristic. both test problems from the or libraries [4, 23] and randomly generated ones are used to conduct the computational experiment. we only consider milp problem cases which have non-negative lagrange multipliers and at least one of them is non-zero. if this is not the case, we can assume that all of the initial surrogate multipliers are 1. we denote the number of constraints by m, and the number of variables by n. for randomly generated 0-1 mkps, we choose m ∈{3000, 4000, 5000} and n ∈{150, 200} . the constraint coefficients aij and objective function coefficients cj are integers and randomly generated from the discrete uniform distributions u {1, 2, . . . , 500} and u {1, 2, . . . , 100}, respectively. it is assumed that there are no correlations between the objective function and constraint coefficients. the right hand side values bi are set to equal to si n∑ j=1 aij, where si is a slackness ratio and drawn from uniform distribution between 0.65 and 0.95. for each combination of (m,n), we generate 10 problems. table 1 gives, for each combination of m×n, minimum, average (rounded off) and maximum computing times for both the intlinprog solver and the proposed heuristic. the heuristic performs better when the methods are compared by their average execution times which are less except for only one case. table 1 indicates average execution times are reduced for almost every uncorrelated instance in which expected slackness ratio is 0.80. in addition to the mkp instances, we test the proposed heuristic on a few instances from or libraries [4, 23]. refer to table 2 for the results. note that, in all our calculations, the tolerance value are fixed to 10−6, the algorithm terminates if the upper bound does not improve 30 times, and for the ibm, the iteration upper limit is set to 10. 5. conclusions in this paper, we try to provide equivalent formulations with a fewer number of constraints for combinatorial optimization problems. to do that, we present a surrogate cut generation procedure based on the ibm. experiments performed on problems with lots of redundant constraints have shown that the proposed heuristic gives reduced models which can be solved significantly faster than original models. int. j. anal. appl. 19 (1) (2021) 70 algorithm 3.1 constraint reduction heuristic input: coefficient matrices and vectors of the milp problem (p) (a,b,c, and the set x) output: optimal integer solution x; number of constraints k after reduction; constraint coefficient matrix sur after reduction; right hand side vector surrhs after reduction; optimal surrogate weights matrix µ 1: solve the lp relaxation of problem (p) and calculate lagrange multipliers, assign the corresponding lagrange multipliers into the row vector µ (µ(1, i) = λ(i) for i ∈ i = {1, 2, . . . ,m}), 2: constnum ← the number of inequality constraints (m), tol ← 1e − 6, iteruplim ← 10, lowbound1 ← 0, lowbound2 ← 0, w ← 0 and k ← 1. 3: function solvesur(a,b,c,x,µ) 4: solve min ct x s.t. µax ≤ µb x ∈ x 5: e ← ax− b 6: return x,e 7: end function 8: function lowbound(k,c,x) 9: if k is odd then 10: lowbound1 = ct x 11: else 12: lowbound2 = ct x 13: end if 14: return lowbound1, lowbound2 15: end function 16: solvesur(a,b,c,x,µ) 17: lowbound(k,c,x) 18: while max(e) > tol and k <= constnum and w <= 30 do 19: if lowbound1 == lowbound2 then 20: w + + 21: end if 22: r ← 0 23: for i ← 1,constnum do 24: f(i, 1) = e(i) 25: f(i, 2) = i 26: end for 27: sort the rows of f in ascending order based on the elements in the first column and assign it to d 28: ind1 = d(constnum, 2) 29: ind2 = d(constnum− 1, 2) 30: mlow = 0 31: mhigh = 1 32: k + + 33: if e(ind1) > tol and e(ind2) > tol then 34: µ(k,ind1) = 1 35: µ(k,ind2) = 1 36: solvesur(a,b,c,x,µ) int. j. anal. appl. 19 (1) (2021) 71 37: if e(ind1) > 0 and e(ind2) < 0 then 38: if e(ind1) > tol then 39: µ(k,ind2) = 0.5 40: solvesur(a,b,c,x,µ) 41: while (e(ind1) > tol or e(ind2) > tol) and r <= iteruplim do 42: if e(ind1) > 0 and e(ind2) 6= 0 then 43: mhigh = |e(ind1)/e(ind2)| 44: if mlow <= mhigh then 45: µ(k,ind2) = (mlow + mhigh)/2 46: solvesur(a,b,c,x,µ) 47: r + + 48: if mlow == mhigh then 49: break 50: end if 51: else 52: break 53: end if 54: else if e(ind2) > 0 and e(ind2) 6= 0 then 55: mlow = |e(ind1)/e(ind2)| 56: if mlow <= mhigh then 57: µ(k,ind2) = (mlow + mhigh)/2 58: solvesur(a,b,c,x,µ) 59: r + + 60: if mlow == mhigh then 61: break 62: end if 63: else 64: break 65: end if 66: end if 67: lowbound(k,c,x) 68: end while 69: end if 70: else if e(ind2) > 0 and e(ind1) < 0 then 71: if e(ind2) > tol then 72: µ(k,ind1) = 0.5 73: solvesur(a,b,c,x,µ) 74: lowbound(k,c,x) 75: while (e(ind1) > tol or e(ind2) > tol) and r <= iteruplim do 76: if e(ind1) > 0 and e(ind1) 6= 0 then 77: mlow = |e(ind2)/e(ind1)| 78: if mlow <= mhigh then 79: µ(k,ind1) = (mlow + mhigh)/2 80: solvesur(a,b,c,x,µ) 81: r + + 82: if mlow == mhigh then 83: break 84: end if int. j. anal. appl. 19 (1) (2021) 72 85: else 86: break 87: end if 88: else if e(ind2) > 0 and e(ind1) 6= 0 then 89: mhigh = |e(ind2)/e(ind1)| 90: if mlow <= mhigh then 91: µ(k,ind1) = (mlow + mhigh)/2 92: solvesur(a,b,c,x,µ) 93: r + + 94: if mlow == mhigh then 95: break 96: end if 97: else 98: break 99: end if 100: end if 101: lowbound(k,c,x) 102: end while 103: end if 104: end if 105: else if e(ind1) > tol and e(ind2) < tol then 106: µ(k,ind1) = 1 107: solvesur(a,b,c,x,µ) 108: lowbound(k,c,x) 109: end if 110: end while 111: sur = µa 112: surrhs = µb int. j. anal. appl. 19 (1) (2021) 73 table 1. computational results of randomly generated 0-1 mkps. cpu time (seconds) intlinprog heuristic number of const. m×n min ave max min ave max min ave max 3000 × 150 85 232 520 34 145 538 17 33 62 3000 × 200 425 1050 3203 168 960 3860 31 40 60 4000 × 150 59 257 626 31 207 805 18 36 76 4000 × 200 66 528 1391 30 443 2043 20 28 41 5000 × 150 149 861 2051 21 942 2761 26 45 83 5000 × 200 670 2000 2994 209 1390 4296 27 42 58 table 2. computational results for instances from or libraries [4, 23]. number of problem inequality equality variables ineq. const. key constraints constraints after reduction st1 30 0 60 13 st2 30 0 60 14 pb6 30 0 40 13 pb7 30 0 37 15 f2gap40400 40 0 400 21 f2gap401600 40 0 1600 27 f2gap801600 80 0 1600 47 app2-1 1038 0 3283 2 app2-2 335 0 1226 1 supportcase14 127 107 304 4 misc07 177 35 206 136 blend2 185 89 353 64 bppc8-02 39 20 232 33 beasleyc1 1250 500 2500 628 int. j. anal. appl. 19 (1) (2021) 74 data availability: the data that support the findings of this study are available from the corresponding author upon reasonable request. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] j.h. ablanedo-rosas and c. rego, surrogate constraint normalization for the set covering problem, eur. j. oper. res. 205(3) (2010), 540–551. [2] b. alidaee, v.p. ramalingam, h. wang, and b. kethley, computational experiment of critical event tabu search for the general integer multidimensional knapsack problem, ann. oper. res. 269(1-2) (2018), 3–19. [3] s. balev, n. yanev, a. fréville, and r. andonov, a dynamic programming based reduction procedure for the multidimensional 0—1 knapsack problem, eur. j. oper. res. 186(1) (2008), 63–76. [4] j. e. beasley, or-library: distributing test problems by electronic mail, j. oper. res. soc. 41(11) (1990), 1069–1072. [5] v. boyer, m. elkihel, and d. el baz, heuristics for the 0–1 multidimensional knapsack problem, eur. j. oper. res. 199(3) (2009), 658–664. [6] m. chih, three pseudo-utility ratio-inspired particle swarm optimization with local search for multidimensional knapsack problem, swarm. evol. comput. 39 (2018), 279–296. [7] j. choi and i.c. choi, identifying redundancy in multi-dimensional knapsack constraints based on surrogate constraints, int. j. comput. math. 91(12) (2014), 2470–2482. [8] c. d’ambrosio, s. martello, and l. mencarelli, relaxations and heuristics for the multiple non-linear separable knapsack problem, comput. oper. res. 93 (2018), 79–89. [9] s.m. douiri, m.b.o. medeni, s. elbernoussi, and e.m. souidi, a new steganographic method for grayscale image using graph coloring problem, appl. math. inform. sci. 7(2) (2013), 521–527. [10] a. freville, the multidimensional 0−1 knapsack problem: an overview, eur. j. oper. res. 155(1) (2004), 1–21. [11] r.d. galvao, l.g.a. espejo, and b. boffey, a comparison of lagrangean and surrogate relaxations for the maximal covering location problem, eur. j. oper. res. 124(2) (2000), 377–389. [12] b. gavish, f. glover, and h. pirkul, surrogate constraints in integer programming, j. inform. optim. sci. 12(2) (1991), 219–228. [13] b. gavish and h. pirkul, efficient algorithms for solving multiconstraint zero-one knapsack problems to optimality, math. program. 31(1) (1985), 78–105. [14] f. glover, surrogate constraint duality in mathematical programming, oper. res. 23(3) (1975), 434–451. [15] f. glover, heuristics for integer programming using surrogate constraints, decision sci. 8(1) (1977), 156–166. [16] f. glover, tutorial on surrogate constraint approaches for optimization in graphs, j. heuristics. 9(3) (2003), 175–227. [17] h.j. greenberg and w.p. pierskalla, surrogate mathematical programming, oper. res. 18 (1970), 924–939. [18] b. haddar, m. khemakhem, s. hanafi, and c. wilbaut, a hybrid quantum particle swarm optimization for the multidimensional knapsack problem, eng. appl. artif. intel. 55 (2016), 1–13. [19] s. hanafi and f. glover, exploiting nested inequalities and surrogate constraints, eur. j. oper. res. 179(1) (2007), 50–63. [20] s. hanafi and c. wilbaut, improved convergent heuristics for the 0-1 multidimensional knapsack problem, ann. oper. res. 183(1) (2011), 125–142. int. j. anal. appl. 19 (1) (2021) 75 [21] r.r. hill, y. k. cho, and j. t. moore, problem reduction heuristic for the 0–1 multidimensional knapsack problem, comput. oper. res. 39(1) (2012), 19–26. [22] s. karabati, p. kouvelis, and g. yu, a min-max-sum resource allocation problem and its aapplications, oper. res. 49(6) (2001), 913–922. [23] t. koch, t. achterberg, e. andersen, et al., miplib 2010, math. program. comput. 3 (2) (2011), 103–163. [24] x. kong, l. gao, h. ouyang, and s. li, solving large-scale multidimensional knapsack problems with a new binary harmony search algorithm, comput. oper. res. 63 (2015), 7–22. [25] s. laabadi, m. naimi, h. el amri, and b. achchab, the 0/1 multidimensional knapsack problem and its variants: a survey of practical models and heuristic approaches, amer. j. oper. res. 8(05) (2018), 395–439. [26] m. lama and d. pinto, a preprocessing that combines heuristic and surrogate constraint analysis to fix variables in tsp, in mex int. conf. artif. i. (727–734). springer, berlin, heidelberg, 2004, april. [27] v.c. li and g.l. curry, solving multidimensional knapsack problems with generalized upper bound constraints using critical event tabu search, comput. oper. res. 32(4) (2005), 825–848. [28] y. li, o. ergun, and g.l. nemhauser, a dual heuristic for mixed integer programming, oper. res. lett. 43(4) (2015), 411–417. [29] c.j. lin, m.s. chern, and m. chih, a binary particle swarm optimization based on the surrogate information with proportional acceleration coefficients for the 0-1 multidimensional knapsack problem, j. ind. product. eng. 33(2) (2016), 77–102. [30] a. lokketangen, and f. glover, surrogate constraint analysis-new heuristics and learning schemes for satisfiability problems, in dimacs series in discrete mathematics and theoretical computer science, 35 (1997), 537–572. [31] l.a.n. lorena and m.g. narciso, relaxation heuristics for a generalized assignment problem, eur. j. oper. res. 91(3) (1996), 600–610. [32] l.a.n. lorena and m.g. narciso, using logical surrogate information in lagrangean relaxation: an application to symmetric traveling salesman problems, eur. j. oper. res. 138(3) (2002), 473–483. [33] s. martello and m. monaci, algorithmic approaches to the multiple knapsack assignment problem, omega 90 (2020), art. id 102004. [34] r. marti, a., corberan, and j. peiro, scatter search, in handbook of heuristics, 1–24 (2016). [35] t. meng and q.k. pan, an improved fruit fly optimization algorithm for solving the multidimensional knapsack problem, appl. soft comput. 50 (2017), 79–93. [36] f. molina, m.o.d. santos, f. toledo, and s.a.d. araujo, an approach using lagrangian/surrogate relaxation for lotsizing with transportation costs, pesquisa oper. 29(2) (2009), 269–288. [37] a. moumen, m. bouye, and h. sissaoui, new secure partial encryption method for medical images using graph coloring problem, nonlinear dynam. 82(3) (2015), 1475–1482. [38] a. nagih and f. soumis, nodal aggregation of resource constraints in a shortest path problem, eur. j. oper. res. 172(2) (2006), 500–514. [39] m.g. narciso and l.a.n. lorena, lagrangean/surrogate relaxation for generalized assignment problems, eur. j. oper. res. 114(1) (1999), 165–177. [40] s.m. neiro and j.m. pinto, decomposition techniques for the long-range production planning of oil complexes, in comput. aided chem. eng. 18 (2004), 967–972. [41] m.a. osorio, f. glover, and p. hammer, cutting and surrogate constraint analysis for improved multidimensional knapsack solutions, ann. oper. res. 117(1-4) (2002), 71–93. int. j. anal. appl. 19 (1) (2021) 76 [42] g.r. raidl and j. puchinger, combining (integer) linear programming techniques and metaheuristics for combinatorial optimization, in hybrid metaheuristics (31–62), springer, berlin, heidelberg, 2008. [43] j. reese, solution methods for the p–median problem: an annotated bibliography, networks. 48(3) (2006), 125–142. [44] c. rego, ramp: a new metaheuristic framework for combinatorial optimizatio, in metaheuristic optimization via memory and evolution (441-460), springer, boston, ma, 2005. [45] c. rego, f. mathew, and f. glover, ramp for the capacitated minimum spanning tree problem, ann. oper. res. 181(1) (2010), 661–681. [46] g.m. ribeiro and l.a.n. lorena, heuristics for cartographic label placement problems, comput. geosci. 32(6) (2006), 739–748. [47] a. sakallioglu, surrogate constraint applications to network models in operations research, master thesis, department of mathematics, giresun university, 2019. [48] f. taniguchi, t. yamada, and s. kataoka, heuristic and exact algorithms for the max–min optimization of the multiscenario knapsack problem, comput. oper. res. 35(6) (2008), 2034–2048. [49] j. wang, t. liu, k. liu, b. kim, j. xie, z. han, computation offloading over fog and cloud using multi-dimensional multiple knapsack problem, in: 2018 ieee global communications conference (globecom), ieee, abu dhabi, united arab emirates, 2018: pp. 1–7. [50] b. zhang, q.k. pan, x.l. zhang, and p.y. duan, an effective hybrid harmony search-based algorithm for solving multidimensional knapsack problems, appl. soft. comput. 29 (2015), 288–297. 1. introduction 2. computing the optimal surrogate multipliers by ibm 3. constraint reduction heuristic 4. computational results 5. conclusions references international journal of analysis and applications volume 18, number 4 (2020), 633-643 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-633 pair (f,h) upper class on some fixed point results in probabilistic menger space sh. jafari1,∗, m. shams1, a. h. ansari2 and m. de la sen3 1department of pure mathematics, university of shahrekord, shahrekord 88186-34141, iran 2department of mathematics, karaj branch, islamic azad university, karaj, iran 3institute of research and development of processes iidp, faculty of science and technology, university of the basque country, po box 644, de bilbao, barrio sarriena, 48940 leioa, bizkaia, spain ∗corresponding author: jafari.shahnaz@yahoo.com abstract. in this paper, we define the concept of (f,h,α,β,ψ)contractive mappings in a probabilistic menger space, which generalizes some previous related concepts. also, we investigate the existence of fixed points for such mappings. some examples are given to support the obtained results. 1. introduction and mathematical preliminaries the study of fixed points of mappings in a menger pm-space satisfying certain contractive conditions has been at the center of vigorous research activity. menger pm-space were introduced in 1942 by menger [9]. afterwards the study of these spaces was performed by schweizer and sklar [8] and many others, [3–6]. in 1984 khan et al. introduced the concept of altering distance function [15]. a ϕ-function is the extension of altering distance function and has been worked by many authors, [16], [17].the concepts of α−ψ−type contractive and α− admissible mappings were introduced by gopal et. al. [7], who also established some fixed point theorems for these mappings in complete menger spaces. after that, shams and jafari generalized this concept to (α,β,ψ)-contractive and α−β−admissible mappings and proved some fixed point theorems for such maps [14]. received february 10th, 2020; accepted march 5th, 2020; published may 26th, 2020. 2010 mathematics subject classification. 47h10, 47h09. key words and phrases. fixed point; contractive mapping; probabilistic menger space; t-norm. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 633 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-633 int. j. anal. appl. 18 (4) (2020) 634 in this paper, we give a generalization of the concepts discussed in [1, 14]. thus we introduce the concept of (f,h,α,β,ψ)contractive mapping in menger pm-space, and establish corresponding fixed point theorems for such contractive mappings which are based on the mentioned generalized notion of such a contractive mapping. in particular, the presented theorems extend, generalize and improve the results in [1, 14]. also, some examples are given to support the obtained results. we first bring notion, definitions and known results, which are related to our work. for more details, we refer the reader to [2]. we denote by r the set of real numbers, r+ the set of non-negative real numbers and n the set of positive integers definition 1.1. a distribution function is a function f : (−∞,∞) → [0, 1], that is non-decreasing and left continuous on r. moreover, inft∈r f(t) = 0 and supt∈r f(t) = 1. the set of all the distribution functions is denoted by d, and the set of those distribution functions such that f(0) = 0 is denoted by d+. we will denote the specific heaviside distribution function by: h(t) =   1 t > 0 0 t ≤ 0. definition 1.2. a binary operation t : [0, 1]×[0, 1] → [0, 1] is a continuous t-norm if the following conditions hold: (a) t is commutative and associative, (b) t is continuous, (c) t(a, 1) = a for all a ∈ [0, 1], (d) t(a,b) ≤ t(c,d) whenever a ≤ c and b ≤ d, for a,b,c,d ∈ [0, 1]. the following are three basic continuous t-norms. (i) the minimum t-norm, say tm , defined by tm (a,b) = min{a,b}. (ii) the product t-norm, say tp, defined by tp(a,b) = a.b. (iii)the lukasiewicz t-norm, say tl, defined by tl(a,b) = max{a + b− 1, 0}. these tnorms are related in the following way: tl ≤ tp ≤ tm . definition 1.3. a menger pm-space is a triple (x,f,t), where x is a nonempty set, t is a continuous t-norm, and f is a mapping from x ×x into d+ such that the following conditions hold: (pm1) fx,y(t) = h(t) if and only if x = y, (pm2) fx,y(t) = fy,x(t) (pm3) fx,y(t + s) ≥ t(fx,z(t),fz,y(s)) for all x,y,z ∈ x and s,t ≥ 0 int. j. anal. appl. 18 (4) (2020) 635 definition 1.4. let (x,f,t) be a menger pm-space. then (i) a sequence xn in x is said to be convergent to x if, for every � > 0 and 0 < λ < 1, there exists a positive integer n such that fxnx(�) > 1 −λ, whenever n ≥ n. (ii) a sequence xn in x is called cauchy sequence if, for every � > 0 and λ > 0, there exists a positive integer n scuh that fxnxm (�) > 1 −λ whenever n,m ≥ n. (iii) a menger pm-space is said to be complete if and only if every cauchy sequence in x is convergent to a point in x. (iv) a sequence xn is called g-cauchy if lim n→∞ fxnxn+m (t) = 1, for each m ∈ n and t > 0. (v) the space (x,f,t) is called g-complete if every g-cauchy sequence in x is convergent. it follows immediately that a cauchy sequence is a g-cauchy sequence. the converse is not always true. this has been established by an example in [11]. according to [8], the (�,λ)-topology in menger pm-space (x,f,t) is introduced by the family of neighborhoods nx of a point x ∈ x given by nx = nx(�,λ) : � > 0,λ ∈ (0, 1), where nx(�,λ) = {y ∈ x : fx,y(�) > 1 −λ}. the (�,λ) -topology is a hausdorff topology. in this topology, a function f is continuous in x0 ∈ x if and only if f(xn) → f(x0), for every sequence xn → x0, as n →∞. definition 1.5. [10] a function φ : [0,∞) → [0,∞) is said to be a φ-function if it satisfies the following conditions: (i) φ(t) = 0 if and only if t = 0, (ii) φ(t) is strictly monotone increasing and φ(t) →∞ as t →∞, (iii) φ is left continuous in (0,∞), (iv) φ is continuous at 0. in the sequel, the class of all φ-functions will be denoted by φ. also we denote by ψ the class of all continuous functions ψ : [0,∞) → [0,∞) such that ψ(0) = 0 and ψn(an) → 0, whenever an → 0 as n →∞. theorem 1.1. [1] let (x,f,t) be a g-complete menger pm-space and f : x → x be a mapping satisfying the following inequality 1 ffx,fy(ϕ(ct)) − 1 ≤ ψ ( 1 fx,y(ϕ(t)) − 1 ) (1.1) where x,y ∈ x,c ∈ (0, 1), ϕ ∈ φ,ψ ∈ ψ and t > 0 such that fx,y(ϕ(t)) > 0. then f has a unique fixed point. int. j. anal. appl. 18 (4) (2020) 636 definition 1.6. [14] let (x,f,t) be a menger pm-space and f : x → x be a given mapping and α,β : x ×x × (0,∞) → [0,∞), be two functions, we say that f is α−β-admissible if (i) for all x,y ∈ x and for all t > 0, α(x,y,t) ≥ 1 ⇒ α(fx,fy,t) ≥ 1, (ii) for all x,y ∈ x and for all t > 0, β(x,y,t) ≤ 1 ⇒ β(fx,fy,t) ≤ 1. 2. fixed point theorems for (f,h,α,β,ψ)-contractive mappings in this section, we state some allied definitions and results which are needed for the development of the present topic. also we introduce the notion of (f,h,α,β,ψ)-contractive mappings in menger pm-spaces and prove some fixed point theorems for such a contractive mappings. definition 2.1. [12, 13] we say that the function h: r+ × r+ → r is a function of subclass of type i, if x ≥ 1 =⇒ h(1,y) ≤ h(x,y) for all y ∈ r+. example 2.1. [12, 13] define h: r+ ×r+ → r by: (a) h(x,y) = (y + l)x, l > 1; (b) h(x,y) = (x + l)y, l > 1; (c) h(x,y) = xny, n ∈ n; (d) h(x,y) = y; (e) h(x,y) = 1 n+1 (∑n i=0 x i ) y, n ∈ n; (f) h(x,y) = [ 1 n+1 (∑n i=0 x i ) + l ]y , l > 1, n ∈ n for all x,y ∈ r+. then h is a function of subclass of type i. definition 2.2. [12, 13] let h,f : r+ ×r+ → r, then we say that the pair (f,h) is an upper class of type i, if h is a function of subclass of type i and: (i) 0 ≤ s ≤ 1 =⇒f(s,t) ≤f(1, t), (ii) h(1,y) ≤f(1, t) =⇒ y ≤ t for all t,y ∈ r+. example 2.2. [12, 13] define h,f : r+ ×r+ → r by: (a) h(x,y) = (y + l)x, l > 1 and f(s,t) = st + l; (b) h(x,y) = (x + l)y, l > 1 and f(s,t) = (1 + l)st; (c) h(x,y) = xmy, m ∈ n and f(s,t) = st; (d) h(x,y) = y and f(s,t) = t; (d) h(x,y) = 1 n+1 (∑n i=0 x i ) y, n ∈ n and f(s,t) = st; (e) h(x,y) = [ 1 n+1 (∑n i=0 x i ) + l ]y , l > 1, n ∈ n and f(s,t) = (1 + l)st for all x,y,s,t ∈ r+. then the pair (f,h) is an upper class of type i. int. j. anal. appl. 18 (4) (2020) 637 definition 2.3. [12, 13] we say that the function h : r+ ×r+ ×r+ → r is a function of subclass of type ii, if x,y ≥ 1 =⇒ h(1, 1,z) ≤ h(x,y,z) for all z ∈ r+. example 2.3. [12, 13] define h: r+ ×r+ ×r+ → r by: (a) h(x,y,z) = (z + l)xy, l > 1; (b) h(x,y,z) = (xy + l)z, l > 1; (c) h(x,y,z) = z; (d) h(x,y,z) = xmynzp,m,n,p ∈ n; (e) h(x,y,z) = x m+xnyp+yq 3 zk,m,n,p,q,k ∈ n for all x,y,z ∈ r+. then h is a function of subclass of type ii. definition 2.4. [12, 13] let h: r+×r+×r+ → r and f : r+×r+ → r, then we say that the pair (f,h) is an upper class of type ii, if h is a subclass of type ii and: (i) 0 ≤ s ≤ 1 =⇒f(s,t) ≤f(1, t), (ii) h(1, 1,z) ≤f(s,t) =⇒ z ≤ st for all s,t,z ∈ r+. example 2.4. [12, 13] define h: r+ ×r+ ×r+ → r and f : r+ ×r+ → r by: (a) h(x,y,z) = (z + l)xy, l > 1,f(s,t) = st + l; (b) h(x,y,z) = (xy + l)z, l > 1,f(s,t) = (1 + l)st; (c) h(x,y,z) = z,f(s,t) = st; (d) h(x,y,z) = xmynzp,m,n,p ∈ n,f(s,t) = sptp (e) h(x,y,z) = x m+xnyp+yq 3 zk,m,n,p,q,k ∈ n,f(s,t) = sktk for all x,y,z,s,t ∈ r+. then the pair (f,h) is an upper class of type ii. now we introduce the following definition that is a generalization of inequality (1.1) introduced in [1], see example 2.6. definition 2.5. let (x,f,t) be a pm-space and f : x → x be a given mapping. we say that f is an (f,h,α,β,ψ)contractive mapping if there exist two functions α,β : x ×x × (0,∞) → [0,∞) and ψ ∈ ψ satisfying the following inequality f ( β(x,y,t),ψ( 1 fx,y(ϕ(t)) − 1) ) ≥ h ( α(x,y,t), ( 1 ffx,fy(ϕ(ct)) − 1) )) (2.1) for all x,y ∈ x and for all t > 0 such that fx,y(ϕ(t)) > 0, where c ∈ (0, 1) and ϕ ∈ φ. remark 1 if α(x,y,t) = 1 and β(x,y,t) = 1 for all x,y ∈ x and for all t > 0, the condition (2.1) reduce to condition (1.1), but the converse is not necessarily true, (see example 2.6 ). int. j. anal. appl. 18 (4) (2020) 638 the following theorem shows that a (f,h,α,β,ψ) contractive mapping under which conditions has fixed points. also, examples 2.5 and 2.6 show that this theorem extends the previous results in [1, 14]. theorem 2.1. let (x,f,t) be a g-complete menger pm-space and f : x → x be a (f,h,α,β,ψ) contractive mapping satisfying the following conditions: (i) f is α−β-admissible. (ii) there exists x0 ∈ x such that α(x0,fx0, t) ≥ 1 and β(x0,fx0, t) ≤ 1, for all t > 0. (iii) if {xn} is a sequence in x such that α(xn,xn+1, t) ≥ 1 and β(xn,xn+1, t) ≤ 1 for all n ∈ n and for all t > 0, and xn → x as n →∞, then α(xn,x,t) ≥ 1 and β(xn,x,t) ≤ 1 for all n ∈ n and for all t > 0. then f has a fixed point, i.e, there exists a point u ∈ x such that fu = u. proof. let x0 ∈ x be such that α(x0,fx0, t) ≥ 1 and β(x0,fx0, t) ≤ 1 for all t > 0. define a sequence {xn} in x so that xn+1 = fxn, for all n ∈ n. assume that xn+1 6= xn for all n ∈ n. then, by using the fact that f is α−β-admissible, we write α(x0,fx0, t) = α(x0,x1, t) ≥ 1 ⇒ α(x1,x2, t) = α(fx0,fx1, t) ≥ 1. similarly we write β(x0,fx0, t) = β(x0,x1, t) ≤ 1 ⇒ β(x1,x2, t) = β(fx0,fx1, t) ≤ 1. by induction, it follows that α(xn,xn+1, t) ≥ 1 and β(xn,xn+1, t) ≤ 1, for all t > 0. from the properties of function ϕ, we can find t > 0 such that fx0x1 (ϕ(t)) > 0. thuse by applying (2.1), we have f ( 1,ψ( 1 fx0,x1 (ϕ(t)) − 1) ) ≥ f ( β(x0,x1, t),ψ( 1 fx0,x1 (ϕ(t)) − 1) ) ≥ h ( α(x0,x1, t), ( 1 ffx0,fx1 (ϕ(ct)) − 1 ) ≥ h ( 1, ( 1 ffx0,fx1 (ϕ(ct)) − 1) ) =⇒ 1 ffx0,fx1 (ϕ(ct)) − 1 ≤ ψ( 1 fx0,x1 (ϕ(t)) − 1) (2.2) repeating the above procedure successively n times, we obtain f ( 1,ψn( 1 fx0,x1 (ϕ( ct cn−1 )) − 1) ) ≥ h ( 1, ( 1 fxn,xn+1 (ϕ(ct)) − 1) ) . hence we have 1 fxn,xn+1 (ϕ(ct)) − 1 ≤ ψn−1 ( 1 fx1,x2 (ϕ( ct cn−1 )) − 1 ) . int. j. anal. appl. 18 (4) (2020) 639 rewrite this sentence for n > r, 1 fxn,xn+1 (ϕ(c rt)) − 1 ≤ ψn−r ( 1 fxn,xn+1 (ϕ( crt cn−r )) − 1 ) . (2.3) since ψn(an) → 0 whenever an → 0, we have from (2.3), for all r > 0 fxn,xn+1 (ϕ(c rt)) → 1. (2.4) now let � > 0 be given, then by virtue of the properties of ϕ, we can find r > 0 such that ϕ(crt) < �. then it follows from (2.4) that fxn,xn+1 (�) → 1, as n →∞ for every � > 0. on the other hand, we know that fxn,xn+p (�) ≥ t(fxn,xn+1 ( � p ),t(fxn+1,xn+2 ( � p ), ..., (fxn+p−1,xn+p ( � p ))...). thus, letting n →∞, we have for any integer p, fxn,xn+p (�) → 1, as n →∞ for every � > 0. hence {xn} is a g-cauchy sequence as (x,f,t) is g-complete, {xn} is convergent and hence xn → u as n →∞ for some u ∈ x. again ffu,u(�) ≥ t ( ffu,xn+1 ( � 2 ),fxn+1,u( � 2 ) ) . (2.5) using the properties of ϕ-function, we can find a t2 > 0 such that ϕ(t2) < � 2 . hence there exists n0 ∈ n such that for all n > n0, fxn,u(ϕ(t2)) > 0. then, for n > n0, we write f ( 1,ψ( 1 fxn,u(ϕ( t2 c )) − 1) ) ≥ f ( β(xn,u,ϕ(t2)),ψ( 1 fxn,u(ϕ( t2 c )) − 1) ) ≥ h ( α(xn,u,ϕ(t2)), ( 1 ffxn,fu(ϕ(t2)) − 1) ) ≥ h ( 1, 1 ffxn,fu(ϕ(t2)) − 1 ) =⇒ 1 fxn+1,fu( � 2 ) − 1 ≤ 1 ffxn,fu(ϕ(t2)) − 1 ≤ ψ( 1 fxn,u(ϕ( t2 c )) − 1). making n →∞, utilizing ψ(0) = 0 and the continuity of ψ, we obtain fxn+1,fu( � 2 ) → 1 as n →∞. (2.6) passing to the limit for n →∞ in (2.5), from (2.6), the continuity of t and the fact that xn → u as n →∞, we have ffu,u(�) = 1 for every � > 0. hence u = fu. � the uniqueness of the fixed point is proved in the next theorem. theorem 2.2. with the same hypotheses of theorem 2.1, if for all x ∈ x and for all t > 0, there exists z ∈ x such that α(x,z,t) ≥ 1 and β(x,z,t) ≤ 1, then f has a unique fixed point. int. j. anal. appl. 18 (4) (2020) 640 proof. let u,v ∈ x be such that fu = u and fv = v. from hypotheses there exists z ∈ y such that α(u,z,t) ≥ 1 and α(v,z,t) ≥ 1, β(u,z,t) ≤ 1 and β(v,z,t) ≤ 1. since f is α−β-admissible, we get α(u,fnz,t) ≥ 1, α(v,fnz,t) ≥ 1, β(u,fnz,t) ≤ 1, β(v,fnz,t) ≤ 1, (2.7) for all t > 0 and n ∈ n. so by using (2.1) and (2.7), we obtain f ( 1,ψ( 1 fu,fn−1z(ϕ(t)) − 1 ) ≥ f ( β(u,fn−1z,t),ψ( 1 fu,fn−1z(ϕ(t)) − 1) ) ≥ h ( α(u,fn−1z,t), 1 ffu,f(fn−1z)(ϕ(ct)) − 1 ) ≥ h ( 1, 1 ffu,f(fn−1z)(ϕ(ct)) − 1 ) this implies that ( 1 fu,fnz(ϕ(ct)) − 1) ≤ ψn( 1 fu,z(ϕ( t cn )) − 1). finally, making n → ∞, we obtain fnz → u. a similar argument shows that for all n ∈ n, fnz → v as n →∞. now, the uniqueness of the limit gives us u = v and hence the proof is complete. � corollary 2.1. [14] let (x,f,t) be a g-complete menger pm-space and f : x → x be a mapping satisfying the following conditions: (i) for all x,y ∈ x and for all t > 0 such that fx,y(ϕ(t)) > 0, where c ∈ (0, 1) and ϕ ∈ φ α(x,y,t)( 1 ffx,fy(ϕ(ct)) − 1) ≤ β(x,y,t)ψ( 1 fx,y(ϕ(t)) − 1) (2.8) (ii) f is α−β-admissible. (iii) there exists x0 ∈ x such that α(x0,fx0, t) ≥ 1 and β(x0,fx0, t) ≤ 1, for all t > 0. (iv) if {xn} is a sequence in x such that α(xn,xn+1, t) ≥ 1 and β(xn,xn+1, t) ≤ 1 for all n ∈ n and for all t > 0, and xn → x as n →∞, then α(xn,x,t) ≥ 1 and β(xn,x,t) ≤ 1 for all n ∈ n and for all t > 0. then f has a fixed point, i.e, there exists a point u ∈ x such that fu = u the following examples show the usefulness of definition 2.5 proposed in this paper to extend previous results in [1, 14]. int. j. anal. appl. 18 (4) (2020) 641 example 2.5. let x = r,t(a,b) = min{a,b} for all a,b ∈ [0, 1] and fx,y(t) = tt+|x−y| for all x,y ∈ x and for all t > 0. clearly (x,f,t) is a g-complete menger pm-space. define the mapping f : x → x by fx =   1 2 x ∈ [0, 1) 1 x = 1 2 otherwise and the functions α,β : x ×x × (0,∞) → [0,∞) by α(x,y,t) =   1 x,y ∈ [0, 1]1 2 otherwise, β(x,y,t) =   4 x,y ∈ [0, 1) 1 x = y = 1 1 3 otherwise. we define h: r+ × r+ → r, f : r+ × r+ → r by h(x,y) = (y + l)x and f(s,t) = st + l. let c = 1 2 . if we define ϕ,ψ : [0,∞) → [0,∞) by ϕ(t) = ψ(t) = t, then the mapping f satisfies the hypotheses of theorem 2.1. let x,y ∈ x be such that α(x,y,t) ≥ 1 and β(x,y,t) ≤ 1 for all t > 0, then by definition of f and α and β, we have α(fx,fy,t) = 1 and β(fx,fy,t) ≤ 1, that is, f is a α−β-admissible. also for x0 = 1 we have α(1,f(1), t) = 1 and β(1,f(1), t) = 1. suppose both x,y are in [0, 1), then α(x,y,t) = 1 and β(x,y,t) = 4 and so by definition of f and h we have 4t + l ≥ l, so inequality (2.1) holds. if x = y = 1, then t + l ≥ l, so inequality (2.1) holds. also in other cases inequality (2.1) hold. next, let {xn} is a sequence such that α(xn,xn+1, t) ≥ 1 and β(xn,xn+1, t) ≤ 1, for all n ∈ n and for all t > 0, and xn → x as n → ∞, this implies xn = 1 and so α(xn,x,t) = 1 and β(xn,x,t) = 1, for all n ∈ n and for all t > 0. hence we conclude that all the conditions of theorem 2.1 hold and so, f has three fixed points x = 1, x = 1 2 and x = 2. we show that corollary 2.1 is not applicable in this case. consider x = 1 and y = 2, then by applying inequality (2.8) we have 9 ≤ c, which gives a contradiction to the fact that c ∈ (0, 1). example 2.6. let x = r,t(a,b) = min{a,b} for all a,b ∈ [0, 1] and fx,y(t) = tt+|x−y| for all x,y ∈ x and for all t > 0. clearly (x,f,t) is a g-complete menger pm-space. define the mapping f : x → x by fx =   1 5 x ∈ [0, 1 4 ] x2 3 otherwise and the functions α,β : x ×x × (0,∞) → [0,∞) by int. j. anal. appl. 18 (4) (2020) 642 α(x,y,t) =   1 x,y ∈ [0, 1 4 ] 2 3 otherwise, β(x,y,t) =   1 2 x,y ∈ [0, 1 4 ] 2 otherwise. we define h: r+ × r+ → r, f : r+ × r+ → r by h(x,y) = xy and f(s,t) = st. let c = 1 2 . if we define ϕ,ψ : [0,∞) → [0,∞) by ϕ(t) = ψ(t) = t, then the mapping f satisfies the hypotheses of theorem 2.1. it is easy to show that f is α−β-admissible. also for x0 = 14 , we have α( 1 4 ,f( 1 4 ), t) = 1 and β( 1 4 ,f( 1 4 ), t) = 1 2 . next, let {xn} is a sequence such that α(xn,xn+1, t) ≥ 1 and β(xn,xn+1, t) ≤ 1 for all n ∈ n, t > 0 and xn → x as n →∞, this implies that {xn},x ∈ [0, 14 ]. so by definition of α,β this implies that β(xn,x,t) ≤ 1 and α(xn,x,t) ≥ 1 for all n ∈ n and for all t > 0. the other conditions are the same as example 2.5. hence we conclude that all the conditions of theorem 2.1 hold and so, f has two fixed points x = 3 and x = 1 5 . we show that theorem 1.1 is not applicable in this case. consider x = 1, and y = 2, then by applying inequality (1.1) we givea a contradiction to the fact that c ∈ (0, 1). acknowledgements: the fourth author thanks the basque government for grant it1207-19. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. n. dutta, b. s. choudhury, k. p. das, some fixed point results in menger spaces using a control function, surv. math. appl. 4 (2009 ), 41-52. [2] o. hadžić, e. pap, fixed point theory in probabilistic metric spaces, kluwer academic publishers, dordrecht, 2001. [3] y. liu, zh. li, coincidence point theorems in probabilistic and fuzzy metric spaces, fuzzy sets syst. 158 (2007), 58–70. [4] d. mihet, a banach contraction theorem in fuzzy metric spaces, fuzzy sets syst. 144 (2004), 431–439. [5] d. mihet, fixed point theorems in probabilistic metric spaces, chaos soliton fract. 41(2009), 1014–1019. [6] j. jachymski, on probabilistic ϕ-contractions on menger spaces, nonlinear anal. theory methods appl. 73 (2010), 2199–2203. [7] d. gopal, m. abbas, c. vetro, some new fixed point theorems in menger pm-spaces with application to volterra type integral equation, appl. math. comput. 232 (2014), 955-967. [8] b. schweizer, a. sklar, probabilistic metric space, elsevier, north-holland, new york, 1983. [9] k. menger, statistical metric, proc. natl. sci. 28 (1942), 535-538. [10] b. s. choudhury, k. p. das, a new contraction principle in menger spaces, acta math. sin. 24 (2008), 1379-1386. [11] r. vasuki, p. veeramani, fixed point theorems and cauchy sequences in fuzzy metric spaces, fuzzy sets syst. 135 (2003), 415-417. [12] a. h. ansari, note on α-admissible mappings and related fixed point theorems,the 2nd regional conference onmathematics and applications, pnu, september 2014, 373-376. int. j. anal. appl. 18 (4) (2020) 643 [13] a. h. ansari, s. shukla, some fixed point theorems for ordered f-(f,h)-contraction and subcontractions in 0-f-orbitally complete partial metric spaces, j. adv. math. stud. 9 (2016), 37-53. [14] m. shams, sh. jafari, some fixed point theorem for cyclic (α,β,ψ)-contractive mapping in probabilistic menger space, int. j. pure appl. math. 3 (2016), 727–740. [15] m. s. khan, m. swaleh, s. sessa, fixed point theorems by altering distances between the points, bull. aust. math. soc. 30 (1984), 1–9. [16] v. la rosa, p. vetro, common fixed points for α−ψ−φ-contractions in generalized metric spaces, nonlinear anal., model. control, 19 (2014), 43–54. [17] d. mihet, altering distances in probabilistic menger spaces, nonlinear anal., theory methods appl. 71 (2009), 2734–2738. 1. introduction and mathematical preliminaries 2. fixed point theorems for (f,h,,,)-contractive mappings references international journal of analysis and applications volume 16, number 4 (2018), 462-471 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-462 convexity of integral operators involving dini functions saddaf noreen, muhey u din and mohsan raza∗ department of mathematics, government college university faisalabad, pakistan ∗corresponding author: mohsan976@yahoo.com abstract. in this article, we are mainly interested to find some covexity properties for certain families of integral operators involving dini functions in the open unit disc. the main tool in the proofs of our results are some functional inequalities of dini functions. some particular cases involving dini functions are also a part of our investigations. 1. introduction let a denote the class of functions f of the form f(z) = z + ∞∑ n=2 anz n, (1.1) which are analytic in the open unit disc u = {z : |z| < 1} and s denote the class of all functions which are univalent in u. let s∗ (α) , c (α) and k(α) denote the classes of starlike, convex and close-to-convex functions of order α and are defined as: s∗ (α) = { f : f ∈a and re ( zf′ (z) f (z) ) > α, z ∈u, α ∈ [0, 1) } , c (α) = { f : f ∈a and re ( 1 + zf′′ (z) f′ (z) ) > α, z ∈u, α ∈ [0, 1) } and k(α) = { f : f ∈a and re ( zf′ (z) g (z) ) > α, z ∈u, α ∈ [0, 1) , g ∈s∗ } . received 2018-01-31; accepted 2018-04-06; published 2018-07-02. 2010 mathematics subject classification. 30c45, 33c10, 30c20, 30c75. key words and phrases. analytic functions; convex functions; integral operators; bessel functions; dini functions. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 462 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-462 int. j. anal. appl. 16 (4) (2018) 463 it is clear that s∗ (0) = s∗, c (0) = c and k(0) = k. special functions have great importance in pure and applied mathematics. the widely use of these functions have attracted many researchers to work on the different directions. geometric properties of special functions such as hypergeometric functions, bessel functions, struve functions, mittage-lefller functions, wright functions and some other related functions is an ongoing part of research in geometric function theory. we refer for some geometric properties of these functions [1, 2, 5, 6, 11] and references therein. the bessel function of the first kind jv is defined by jv(z) = ∞∑ n=1 (−1)n n!γ(v + n + 1) (z 2 )2n+v , (1.2) where γ stands for euler gamma function. it is a particular solution of the second order linear homogeneous differential equation z2w′′(z) + zw′(z) + (z2 −v2)w(z) = 0, where v ∈ c. for some details see [2, 13]. bessel functions are indispensable in many branches of pure and applied mathematics. thus, it is important to study their properties in many aspects. recently baricz et al [3] studied the close-to-convexity of dini functions and some monotonicity properties and functional inequalities for the modified dini function are discussed in [4]. further some geometric properties of dini functions are studied in [3, 4, 7]. now, we consider the normalized dini functions qυ : u → c defined as qυ (z) = 2 υγ (υ + 1) z1− υ 2 ( (1 −υ) jυ (√ z ) + √ zj′υ (√ z )) = z + ∞∑ n=1 (−1)n (2n + 1) γ (υ + 1) 4nn!γ (υ + n + 1) zn+1, z ∈u. (1.3) the pochhammer (or appell) symbol, defined in terms of euler’s gamma functions is given as (x)n = γ(x + n)/γ(x) = x(x + 1)...(x + n− 1). recently, deniz et al. [8], din et al. [9], din et al. [10] and srivastava et al. [12] have obtained sufficient conditions for the univalence of certain families of integral operators defined by bessel, dini, struve and mittage-leffler functions respectively. the families of integral operators are defined below: fα1,...,αn,ζ (z) =  ζ z∫ 0 tζ−1 n∏ i=1 ( fi(t) t ) 1 αi dt   1/ζ , (1.4) gξ,n (z) =  (nξ + 1) z∫ 0 n∏ i=1 (fi (t)) ξ dt   1/(nξ+1) , (1.5) hδ1,...,δn,µ (z) =  µ z∫ 0 tµ−1 n∏ i=1 (f′i(t)) δi dt   1/µ (1.6) int. j. anal. appl. 16 (4) (2018) 464 and qλ (z) =  λ z∫ 0 tλ−1 ( ef(t) )λ dt   1/λ . (1.7) in this paper, we are mainly interested in the convexity of the integral operators involving dini function qv. these integral operators are defined as fv1,...,vn,α1,...,αn (z) = z∫ 0 n∏ i=1 ( qvi(t) t )αi dt (1.8) and hv1,...,vn,δ1,...,δn (z) = z∫ 0 n∏ i=1 ( q′vi(t) )δi dt. (1.9) now we prove some functional inequalities which are useful in establishing our main results. lemma 1.1. let v ∈ r and consider the normalized dini function qυ : u → c, defined by qυ (z) = 2 υγ (υ + 1) z1− υ 2 ( (1 −υ) jυ (√ z ) + √ zj′υ (√ z )) , where jυ is the bessel function of first kind. then the following inequalities hold for all z ∈u. (i) ∣∣∣zq′υ(z)qυ(z) − 1∣∣∣ ≤ 3v+64v2+5v−2, v > −5+√578 , (ii) ∣∣∣zq′′υ(z)q′υ(z) ∣∣∣ ≤ 3v+62v2+v−4, v > −1+√334 . proof. (i) by using the well known triangle inequality with the equality γ (v + 1) γ (v + n + 1) = 1 (v + 1) (v + 2) · · ·(v + n) = 1 (v + 1)n , n ∈ n and the inequality 4nn! (v + 2)n−1 ≥ 4 3 n (2n + 1) (v + 2) n−1 , n ∈ n, we obtain ∣∣∣∣q′υ (z) − qυ (z)z ∣∣∣∣ ≤ 34 (v + 1) ∞∑ n=1 ( 1 v + 2 )n−1 = 3 (v + 2) 4 (v + 1) 2 . furthermore, if we use the reverse triangle inequality and the inequality 4nn! (v + 2)n−1 ≥ 1 3 (2n + 1) (v + 2) n−1 , n ∈ n, int. j. anal. appl. 16 (4) (2018) 465 then we get ∣∣∣∣qυ (z)z ∣∣∣∣ = ∣∣∣∣∣1 + ∞∑ n=1 (−1)n (2n + 1) γ (υ + 1) 4nn!γ (υ + n + 1) ∣∣∣∣∣ ≥ 1 − 3 4 (v + 1) ∞∑ n=0 ( 1 v + 2 )n−1 = 4 (v + 1) 2 − 3 (v + 2) 4 (v + 1) 2 . by combining the above inequalities, we get∣∣∣∣zq′υ (z)qυ (z) − 1 ∣∣∣∣ ≤ 3v + 64v2 + 5v − 2, v > −5 + √ 57 8 . (ii) by using the well known triangle inequality with equality γ (v + 1) γ (v + n + 1) = 1 (v + 1) (v + 2) · · ·(v + n) = 1 (v + 1)n , n ∈ n and the inequality 4nn! (v + 2)n−1 ≥ 4 3 (2n + 1) (v + 2) n−1 , n ∈ n, we have |zq′′υ (z)| ≤ ∣∣∣∣∣ ∞∑ n=1 n (n + 1) (2n + 1) γ (υ + 1) 4nn!γ (υ + n + 1) ∣∣∣∣∣ ≤ 3 2 (v + 1) ∞∑ n=1 ( 1 v + 2 )n−1 = 3 (v + 2) 2 (v + 1) 2 . furthermore, if we use the reverse triangle inequality and the inequality 4nn! (v + 2)n−1 ≥ 2 3 ( 2n2 + 3n + 1 ) (v + 2) n−1 , n ∈ n, then we get |q′υ (z)| ∣∣∣∣∣≥ 1 − ∞∑ n=1 (2n + 1) (n + 1) γ (υ + 1) 4nn!γ (υ + n + 1) ∣∣∣∣∣ ≥ 1 − 3 2 (v + 1) ∞∑ n=1 ( 1 v + 2 )n−1 = 2v2 + v − 4 4 (v + 1) 2 by combining the above inequalities, it can be easily obtained∣∣∣∣zq′′υ (z)q′υ (z) ∣∣∣∣ ≤ 3v + 62v2 + v − 4, v > −1 + √ 33 4 . � int. j. anal. appl. 16 (4) (2018) 466 2. convexity of integral operators defined by generalized dini functions the main objective of this paper is to give convexity properties of integral operators involving dini function. the main results are given as follows. theorem 2.1. let v1, . . . ,vn > −5+ √ 57 8 , where n ∈ n. let qυi : u → c be defined as qυi (z) = 2 υiγ (υ + 1) z1− υi 2 ( (1 −υi) jυi (√ z ) + √ zj′υi (√ z )) . (2.1) suppose that v = min{v1, . . . ,vn} and α1, . . . ,αn be positive real numbers such that these numbers satisfy the following inequality 0 ≤ 1 − 3v + 6 4v2 + 5v − 2 n∑ i=1 αi < 1. then, the function fv1,...,vn,α1,...,αn defined by (1.8), is in the class c (β), where β = 1 − 3v + 6 4v2 + 5v − 2 n∑ i=1 αi. proof. we observe that qυi,∀i = 1, 2, · · ·n are such that qυi (0) = q′υi (0) − 1 = 0. it is also clear that fv1,...,vn,α1,...,αn ∈ a. that is fv1,...,vn,α1,...,αn (0) = f ′v1,...,vn,α1,...,αn (0) − 1 = 0. on the other hand, it is easy to see that f ′v1,...,vn,α1,...,αn (z) = n∏ i=1 ( qvi(z) z )αi . (2.2) differentiating logarithmically, we get zf ′′v1,...,vn,α1,...,αn (z) f ′v1,...,vn,α1,...,αn (z) = n∑ i=1 αi ( zq′vi (z) qvi (z) − 1 ) . (2.3) this implies that re { 1 + zf ′′v1,...,vn,α1,...,αn (z) f ′v1,...,vn,α1,...,αn (z) } = n∑ i=1 αire ( zq′vi (z) qvi (z) ) + ( 1 − n∑ i=1 αi ) . now, by using the assertion (i) of lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtain re { 1 + zf ′′v1,...,vn,α1,...,αn (z) f ′v1,...,vn,α1,...,αn (z) } ≥ n∑ i=1 αi ( 1 − 3vi + 6 4v2i + 5vi − 2 ) + ( 1 − n∑ i=1 αi ) = 1 − n∑ i=1 αi 3vi + 6 4v2i + 5vi − 2 . consider the function φ : ( −5+ √ 57 8 ,∞ ) → r defined as φ (x) = 3x + 6 4x2 + 5x− 2 is decreasing function such that 3vi + 6 4v2i + 5vi − 2 ≤ 3v + 6 4v2 + 5v − 2 , ∀i = 1, 2, · · ·n. int. j. anal. appl. 16 (4) (2018) 467 therefore re { 1 + zf ′′v1,...,vn,α1,...,αn (z) f ′v1,...,vn,α1,...,αn (z) } > 1 − 3v + 6 4v2 + 5v − 2 n∑ i=1 αi. since 0 ≤ 1 − 3v+6 4v2+5v−2 n∑ i=1 αi < 1, therefore fv1,...,vn,α1,...,αn ∈c (β), where β = 1 − 3v + 6 4v2 + 5v − 2 n∑ i=1 αi, which completes the proof. � by setting α1 = α2 = · · · = αn = α in theorem 2.1, we obtain the result given below. corollary 2.1. let v1, . . . ,vn > −5+ √ 57 8 , where n ∈ n. let qυi : u → c be defined as qυi (z) = 2 υiγ (υ + 1) z1− υi 2 ( (1 −υi) jυi (√ z ) + √ zj′υi (√ z )) . (2.4) suppose that v = min{v1, . . . ,vn} and α be positive real numbers such that these numbers satisfy the following inequality 0 ≤ 1 − nα (3v + 6) 4v2 + 5v − 2 < 1. then, the function fv1,...,vn,α defined by fv1,...,vn,α (z) = z∫ 0 n∏ i=1 ( qvi(t) t )α dt is in the class c (β1), where β = 1 − nα (3v + 6) 4v2 + 5v − 2 . the next theorem gives convexity properties of the integral operator defined in (1.9). the key tool in the proof is inequality (ii) of lemma 1.1. theorem 2.2. let v1, . . . ,vn > −1+ √ 33 4 , where n ∈ n. let qυi : u → c be defined as qυi (z) = 2 υiγ (υ + 1) z1− υi 2 ( (1 −υi) jυi (√ z ) + √ zj′υi (√ z )) . (2.5) suppose that v = min{v1, . . . ,vn} and δ1, . . . ,δn be positive real numbers such that these numbers satisfy the following inequality 0 ≤ 1 − 3v + 6 2v2 + v − 4 n∑ i=1 δi < 1. then, the function hv1,...,vn,δ1,...,δn defined by (1.9), is in the class c (γ), where γ = 1 − 3v + 6 2v2 + v − 4 n∑ i=1 δi. int. j. anal. appl. 16 (4) (2018) 468 proof. it can easily be observed that, the operator defined in (1.9) belongs to class a, that is hv1,...,vn,δ1,...,δn (0) = h′v1,...,vn,δ1,...,δn (0) − 1 = 0. differentiating (1.9), we have h′v1,...,vn,δ1,...,δn (z) = n∏ i=1 ( q′vi(z) )δi . differentiating logarithmically, we obtain zh′′v1,...,vn,δ1,...,δn (z) h′v1,...,vn,δ1,...,δn (z) = n∑ i=1 δi ( zq′′vi (z) q′vi (z) ) . this implies that re { 1 + zh′′v1,...,vn,δ1,...,δn (z) h′v1,...,vn,δ1,...,δn (z) } = 1 + n∑ i=1 δire ( zq′′vi (z) q′vi (z) ) . now, by using the assertion (ii) of lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtain re { 1 + zh′′v1,...,vn,δ1,...,δn (z) h′v1,...,vn,δ1,...,δn (z) } > 1 − n∑ i=1 δi ( 3vi + 6 2v2i + vi − 4 ) . consider the function ϕ : ( −1+ √ 33 4 ,∞ ) → r defined as ϕ (x) = 3x + 6 2x2 + x− 4 is decreasing function such that 3vi + 6 2v2i + vi − 4 ≤ 3v + 6 2v2 + v − 4 , ∀i = 1, 2, · · ·n. it follows that re { 1 + zh′′v1,...,vn,δ1,...,δn (z) h′v1,...,vn,δ1,...,δn (z) } > 1 − 3v + 6 2v2 + v − 4 n∑ i=1 δi. since 0 ≤ 1 − 3v + 6 2v2 + v − 4 n∑ i=1 δi < 1, therefore hv1,...,vn,δ1,...,δn ∈ c (γ), where γ = 1 − 3v + 6 2v2 + v − 4 n∑ i=1 δi, which completes the proof. � by setting δ1 = δ2 = δn = δ in theorem 2.2, we obtain the result given below. corollary 2.2. let v1, . . . ,vn > −1+ √ 33 4 , where n ∈ n. let qυi : u → c be defined as qυi (z) = 2 υiγ (υ + 1) z1− υi 2 ( (1 −υi) jυi (√ z ) + √ zj′υi (√ z )) . (2.6) suppose that v = min{v1, . . . ,vn} and δ be positive real numbers such that these numbers satisfy the following inequality 0 ≤ 1 − nδ (3v + 6) 2v2 + v − 4 < 1. int. j. anal. appl. 16 (4) (2018) 469 then, the function hv1,...,vn,δ defined as hv1,...,vn,δ (z) = z∫ 0 n∏ i=1 ( q′vi(t) )δ dt is in the class c (γ1), where γ1 = 1 − nδ (3v + 6) 2v2 + v − 4 . 3. some particular cases of dini function by choosing v = 1 2 and v = 3 2 in (1.2), we get the following forms of the normalized dini function q1 2 (z) = 3 2 √ z ( sin √ z + √ z cos √ z ) , q3 2 (z) = 3 2 √ z ( (z − 1) sin √ z + √ z cos √ z ) . in particular, the results of the above mentioned theorems are given below. corollary 3.1. let v > −5+ √ 57 8 and qυ : u → c be defined as qυ (z) = 2 υγ (υ + 1) z1− υ 2 ( (1 −υ) jυ (√ z ) + √ zj′υ (√ z )) . suppose that α be positive real number such that these numbers satisfy the following inequality 0 ≤ 1 − α (3v + 6) 4v2 + 5v − 2 < 1. then, the function fv,α defined by fv,α (z) = z∫ 0 ( qv(t) t )α dt is in the class c (β2), where β2 = 1 − α (3v + 6) 4v2 + 5v − 2 . in particular, (i) if α ≤ 15, then the function f1 2 ,α : u → c defined by f1 2 ,α (z) = z∫ 0 ( 3 2 ( sin √ t + √ t cos √ t ) √ t )α dt is in the class c (β3), where β3 = 1 − α15. (ii)if α ≤ 29 21 , then the function f3 2 ,α : u → c defined by f3 2 ,α (z) = z∫ 0 ( 3 2 ( (t− 1) sin √ t + √ t cos √ t ) t )α dt is in the class c (β4), where β4 = 1 − 21α29 . int. j. anal. appl. 16 (4) (2018) 470 corollary 3.2. let v > −1+ √ 33 4 and qυ : u → c be defined as qυ (z) = 2 υγ (υ + 1) z1− υ 2 ( (1 −υ) jυ (√ z ) + √ zj′υ (√ z )) . suppose that δ be positive real numbers such that these numbers satisfy the following inequality 0 ≤ 1 − δ (3v + 6) 2v2 + v − 4 < 1. then, the function hv,δ defined as hv,δ (z) = z∫ 0 (q′v(t)) δ dt is in the class c (γ2), where γ2 = 1 − nδ (3v + 6) 2v2 + v − 4 . in particular, (i) if δ ≤ 4 21 , then the function h3 2 ,α : u → c defined by h3 2 ,α (z) = z∫ 0 ( q′3 2 (t )δ dt is in the class c (γ3), where γ3 = 1 − 214 δ. references [1] á. baricz, bessel transforms and hardy space of generalized bessel functions, mathematica, 48(71) (2006), 127-136. [2] á. baricz, generalized bessel functions of the first kind, lecture notes in mathematics vol. 1994, springerverlag, berlin, 2010. [3] á. baricz, e. deniz and n. yagmur, close-to-convexity of normalized dini functions, math. nech., 289 (2016), 1721-1726. [4] á. baricz, s. ponnusamy and s. singh, modified dini functions: monotonicity patterns and functional inequalities, acta math. hungrica, 149(2016), 120-142. [5] á. baricz and r. szász, the radius of convexity of normalized bessel functions, anal. math., 41(3) (2015) 141-151. [6] d. bansal and j. k. prajapat. certain geometric properties of the mittag-leffler functions, complex var. ellipt. equ., 61(3) (2016), 338-350. [7] e. deniz, ş. gören and m. çağlar, starlikeness and convexity of the generalized dini functions, aip conference proceedings 1833, 020004 (2017); doi: 10.1063/1.4981652. [8] e. deniz, h. orhan and h. m. srivastava, some sufficient conditions for univalence of certain families of integral operators involving generalized bessel functions, taiwanese j. math. 15(2) (2011), 883-917. [9] m. din, m. raza and e. deniz, sufficient conditions for univalence of integral operators involving dini functions, (submitted). [10] m. din, h. m. srivastava, m. raza, univalence of integral operators involving generalized struve functions, hacettepe j. math. stat., doi: 10.15672/hjms.2017.485. [11] h. orhan and n. yagmur, geometric properties of generalized struve functions, in the international congress in honour of professor hari m. srivastava, (2012), 23-26. int. j. anal. appl. 16 (4) (2018) 471 [12] h. m. srivastava, b. a. frasin and virgil pescar, univalence of integral operators involving mittag-leffler functions, appl. math. inf. sci. 11(3) (2017), 635-641. [13] g. n. watson. a treatise on the theory of bessel functions, second edition, cambridge university press, cambridge, london and new york, 1944. 1. introduction 2. convexity of integral operators defined by generalized dini functions 3. some particular cases of dini function references int. j. anal. appl. (2023), 21:67 fuzzy initial and final segments in adl’s g. srikanya1, g. prakasam babu1, k. ramanuja rao2,∗, ch. santhi sundar raj1 1department of mathematics, andhra university, visakhapatnam-530003, andhra pradesh, india 2department of mathematics, fiji national university, lautoka, fiji ∗corresponding author: ramanuja.kotti@fnu.ac.fj abstract. in this paper, we define the concepts of fuzzy initial and final segments in an almost distributive lattice (adl) and certain properties of these are discussed. it is proved that the set of fuzzy initial segments forms a complete lattice and that the set of fuzzy final segments of an adl a forms a complete lattice if and only if a is a bounded distributive lattice. 1. introduction a fuzzy subset a of a non empty set x is a mapping of x into [0, 1]. the notion of fuzzy subsets originally introduced by zadeh in his pioneering work [17]. since rosenfield [7] applied this concept to the theory of groups, many researchers are turned and engaged in fuzzyfying various concepts of abstract algebra. kuroki [2], malik and mordeson [4] studied fuzzy ideals and bi-ideals in semi groups. wang-jin liu [3] studied fuzzy ideals of a ring followed by mukherjee and sen [5]. swamy and swamy [15] have introduced the concept of a fuzzy prime ideal of a ring and developed theory of fuzzy ideals. further, swamy and dv raju [13] have introduced the concepts of fuzzy ideals and congruences of lattices. further swamy, sundar raj and natnael [8, 9, 10] have applied some concepts from theory of adls and introduced the notion of fuzzy ideals (filters) and prime (maximal) ideals (filters). natnael, srikanya and sundar raj [6] constructed an l-fuzzy prime spectrum of adls. george boole’s attempt to formalize propositional logic led to the concept of boolean algebra which is a complemented distributive lattice. m.h. stone [10] has proved that any boolean algebra can be made into a boolean ring (a ring with unity, in which every element is idempotent) and viceversa and established a strong duality between boolean algebras (rings) and boolean spaces (compact received: may 3, 2023. 2020 mathematics subject classification. 55-03. key words and phrases. adl; fuzzy initial segment; fuzzy final segment; frame. https://doi.org/10.28924/2291-8639-21-2023-67 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-67 2 int. j. anal. appl. (2023), 21:67 hausdorff totally disconnected spaces). swamy and rao [14] have introduced a common abstraction of these ring theoretic and lattice theoretic generalizations of boolean rings and boolean algebras in the form of an almost distributive lattice (abbreviated as adl) as an algebra (a,∧,∨, 0) of type (2, 2, 0) satisfying the following identities: (1) 0 ∧a = 0 (2) a∨ 0 = a (3) a∧ (b∨c) = (a∧b) ∨ (a∧c) (4) (a∨b) ∧c = (a∧c) ∨ (b∧c) (5) a∨ (b∧c) = (a∨b) ∧ (a∨c) (6) (a∨b) ∧b = b. we first recall certain elementary definitions and results concerning almost distributive lattices. these are collected from [14]. example 1.1. let r be a commutative regular ring with identity (i.e. r is a commutative ring with identity in which, for each a ∈ r, there exists an (unique) idempotent a0 in r such that ar = a0r ). for any a,b ∈ r, define a∧b = a0b and a∨b = a + b−a0b. then (r,∧,∨, 0) is an adl. example 1.2. let x be a nonempty set and fix an arbitrarily chosen element 0 ∈ x. for any a and b ∈ x, define a∧b =  0 if a = 0 b if a 6= 0 and a∨b =  b if a = 0 a if a 6= 0. then (x,∧,∨, 0) is an adl and is called a discrete adl. definition 1.1. let a = (a,∧,∨, 0) be an adl. for any a and b ∈ a, define a ≤ b if and only if a = a∧b (⇔ a∨b = b). it is known that ≤ is a partial order in a. theorem 1.1. the following hold for any elements a,b and c in an adl a. (1) a∧ 0 = 0 = 0 ∧a and a∨ 0 = a = 0 ∨a (2) a∧a = a = a∨a (3) a∧b ≤ b ≤ b∨a (4) a∧b = a ⇔ a∨b = b and a∧b = b ⇔ a∨b = a (5) (a∧b) ∧c = a∧ (b∧c) (i.e., ∧ is associative) (6) a∨ (b∨a) = a∨b (7) a ≤ b ⇒ a∧b = a = b∧a and a∨b = b = b∨a (8) a∧b∧c = b∧a∧c int. j. anal. appl. (2023), 21:67 3 (9) (a∨b) ∧c = (b∨a) ∧c (10) a∨b = b∨a and a∧b = b∧a whenever a∧b = 0 (11) a ∨b = b ∨a and a ∧b = b ∧a (and hence a is a distributive lattice), whenever a ≤ x and b ≤ x for some x ∈ a. (12) a∧b = b∧a ⇔ a∨b = b∨a (13) a∧b = inf{a,b}⇔ a∨b = sup{a,b}. (14) the set {y ∈ a : y ≤ a} is a bounded distributive lattice under the induced operations ∧ and ∨ with 0 is the least element and a is the largest element. (15) m is maximal in (a,≤) ⇔ m∧a = a (⇔ m∨a = m) for all a ∈ a. in this paper, we introduce the concepts of fuzzy initial and final segments in an adl a and discuss certain properties of these. throughout this paper l stands for a non-trivial complete lattice (l,∧,∨, 0, 1) satisfying the infinite meet distributive law: a∧ ( ∨ s∈s s ) = ∨ s∈s ( a∧ s ) for all a ∈ l and s ⊆ l such a lattice is called a frame. infact, the interval [0, 1] is a frame under usual ordering. also a stands for an adl (a,∧,∨, 0) with a maximal element unless otherwise stated. as usual by l-fuzzy subset (simply, fuzzy subset) of an adl a, we mean a mapping λ : a → l. 2. fuzzy initial segments first we recall from [16] that a non empty subset i of an adl a is called an (crisp) initial segment of a if a∧x ∈ i for all x ∈ i and a ∈ a (equivalently, a ∈a and a ≤ x ∈ i implies a ∈ i) definition 2.1. a fuzzy subset λ of a is said to be a fuzzy initial segment of a if, λ(x0) = 1 for some x0 ∈ a and x ≤ y ⇒ λ(y) ≤ λ(x) for all x,y ∈ a. note that if λ is a fuzzy initial segment of an adl (a,∧,∨, 0) then λ(0) = 1. it can be easily verified that a fuzzy subset λ of a is a fuzzy initial segment if and only if, for any α ∈ l, λα is an initial segment of a, where λα = {x ∈ a : α ≤ λ(x)}, called the α−cut of λ. an initial segment i of an adl a need not satisfy the condition that, x ∧a ∈ i for any x ∈ i and a ∈ a (∗) for example, let d be a 3-element discrete adl, say {0,x,y}. then i = {0,x} is an initial segment of d; but x ∧y = y /∈ i. however for any non empty subset i of an adl a satisfying ∗ and, for any x ∈ i and a ∈ i, we have a∧x = a∧x ∧x = (x ∧a) ∧x (by 1.4(8)) and, since x ∧a ∈ i, we get (x ∧a) ∧x ∈ i. (by ∗) so that i is an initial segment of a. this can be extended to fuzzy sets in the following. 4 int. j. anal. appl. (2023), 21:67 theorem 2.1. let λ be a fuzzy subset of a satisfying λ(x0) = 1 for some x0 ∈ a and λ(x ∨y) ≤ λ(x) ∧λ(y) for all x,y ∈ a. then λ is a fuzzy initial segment of a. proof. let x,y ∈ a with x ≤ y. then, λ(y) = λ(x ∨y) ≤ λ(x) ∧λ(y) ≤ λ(x) therefore λ is a fuzzy initial segment of a. � the converse of above theorem is not true. for, consider the example given in the following. example 2.1. let a = {0,x,y} be a dicrete adl and l = {0,s, 1} be a chain with 0 < s < 1. define λ : d −→ l by λ(0) = 1, λ(x) = 1 and λ(y) = s. then λ is a fuzzy initial segment of d; but, λ(x ∨y) = λ(x) = 1 ≮ s = 1 ∧ s = λ(x) ∧λ(y). for any fuzzy subset λ and µ of a, we write λ ≤ µ to mean λ(x) ≤ µ(x) in the ordering of l, for all x ∈ a. it can be easily verified that ≤ is a partial order on the set of all fuzzy subsets of a and is called the point wise ordering. in [16], it is proved that the set intersection of any class of initial segments of an adl a is again an initial segment, so that the set of all initial segments of a is a complete lattice under the usual set inclusion ordering. theorem 2.2. the set of all fuzzy initial segments of a is a complete lattice with respect to the point wise ordering in which, for any set {λi}i∈∆ of fuzzy initial segments of a, g.l.b {λi}i∈∆ = ∧ ß∈∆ λi, the point wise infimum l.u.b {λi}i∈∆ = ∨ i∈∆ λi = ∧{ λ : λis a fuzzy initial segment of a and λi ≤ λ for all i ∈ ∆ } . in fact, if λ = ∧ i∈∆ λi and α ∈ l, then λα = ⋂ i∈∆ λiα. proof. let {λi}i∈∆ be a class of fuzzy initial segments of a and let λ = ∧ ß∈∆ λi, the point wise infimum of {λi}i∈∆; that is, λ(x) = ∧ ß∈∆ λi (x) = g.l.b { λi (x) : i ∈ ∆} in l, for any x ∈ a. since λi (0) = 1 for all i ∈ ∆, it follows that λ(0) = 1. let x,y ∈ a such that x ≤ y. then λi (y) ≤ λi (x), since each λi is fuzzy initial segment. so that ∧ i∈∆ λi (y) ≤ ∧ i∈∆ λi (x) and hence λ(y) ≤ λ(x). therefore λ is a fuzzy initial segment of a. also, λ is the g.l.b {λi}i∈∆ under the point wise ordering. thus the set of all fuzzy initial segments of a is a complete lattice under point wise ordering. � in the following, we describe the smallest fuzzy initial segment containing a given fuzzy subset. definition 2.2. for any fuzzy subset λ of a, define λ̄ : a −→ l by λ̄(x) = ∧{ µ(x) : µ is a fuzzy initial segment of a and λ ≤ µ } . int. j. anal. appl. (2023), 21:67 5 by theorem 2.4, λ̄ is a fuzzy initial segment of a and, with respect to the point wise ordering λ̄ is the smallest fuzzy initial segment of a such that λ ≤ λ̄ ; in the sense that for any fuzzy initial segment µ of a, λ ≤ µ ⇔ λ̄ ≤ µ. λ̄ is called the fuzzy initial segment of a generated by λ. in the following, we give a point wise description of λ̄ for any given fuzzy subset λ of a. theorem 2.3. let λ be a fuzzy subset of a. then the fuzzy initial segment generated by λ is given by λ̄(0) = 1 and λ̄(x) = ∨ {λ(a) : x ≤ a for some a ∈ a} for any 0 6= x ∈ a. proof. clearly λ ≤ λ̄. if x ≤ y in a.then for any a ∈ a, y ≤ a ⇒ x ≤ a ⇒ λ(a) ≤ λ̄(x). which implies that λ̄(y) ≤ λ̄(x). therefore λ̄ is a fuzzy initial segment of a. finally, if µ is a fuzzy initial segment of a such that λ ≤ µ, then for any x ∈ a with x ≤ a, a ∈ a, we have λ(a) ≤ µ(a) ≤ µ(x). therefore λ̄(x) ≤ µ(x) for all x ∈ a so that λ̄ ≤ µ. � 3. fuzzy final segments recall that a non empty subset f of an adl a is called a (crisp) final segment of a if a ∈ a,x ∈ f and x ≤ a ⇒ a ∈ f (equivalently, x ∈ f and a ∈ a ⇒ x ∨a ∈ f ). definition 3.1. a fuzzy subset λ of a is said to be a fuzzy final segment of a if λ(x0) = 1 for some x0 ∈ a and x ≤ y ⇒ λ(x) ≤ λ(y) for all x,y ∈ a. it can be easily verified that a fuzzy subset λ of a is a fuzzy final segment of a if and only if, for any α ∈ l, λα is a (crisp) final segment of a. a final segment f of an adl a need not satisfies that properly that, x ∈ f and a ∈ a =⇒ a ∨ x ∈ f. for example, in any discrete adl d, for any x,y ∈ d,{x} is a final segment; but y ∨x = y /∈{x}. however, a non empty subset f of an adl a satisfying the property that, x ∈ f and a ∈ a =⇒ a∨x ∈ f is a final segment of a (refer theorem 3.2 in [16]). this result can be seen by fuzzyfying in the following. theorem 3.1. let λ be a fuzzy subset of a satisfying λ(x0) = 1 for some x0 ∈ a and λ(x ∧y) ≤ λ(x) ∧λ(y) for all x,y ∈ a. then λ is a fuzzy final segment of a. proof. let x,y ∈ a such that x ≤ y. then λ(x) = λ(x ∧y) ≤ λ(x) ∧λ(y) ≤ λ(y). therefore λ is a fuzzy final segment of a. � the converse of above theorem is not true. for, consider the following example. example 3.1. let a = {0,x,y} be a discrete adl and l = {0,s, 1} be a chain with 0 < s < 1. if we define λ(x) = s, λ(y) = 1 and λ(0) = 0, then λ is a fuzzy final segment of a; but λ(x ∧ y) = λ(y) = 1 ≮ s = λ(x) ∧λ(y). 6 int. j. anal. appl. (2023), 21:67 it can be easily verified that, for any non empty subset f of a, f is a final(initial)segment of a if and only if χf is a fuzzy final(initial) segment of a, where χf (x) =  1 if x ∈ f 0 if x /∈ f. note that for any 0 6= α ∈ l, the α-cut of χf ; (χf )α = {x ∈ a : α ≤ χf (x)} = f in contrast to the case of fuzzy initial segments, the point wise infimum of any class of fuzzy final segments may not be a fuzzy final segment. however, in the following we establish a set of equivalent conditions for the point wise infimum of any family of fuzzy final segments to be fuzzy final segment and inturn it is equivalent to saying that the set of all fuzzy final segments form a complete lattice. theorem 3.2. let a be an adl. then the following statements are equivalent to each other. (1) for any non empty family {λi : i ∈ ∆} of fuzzy final segments of a, ∧ i∈∆ λi (the point wise infimum of λi’s) is a fuzzy final segment of a. (2) a has a largest element. (3) a has a unique maximal element. (4) there exists smallest fuzzy final segment. (5) a is a bounded distributive lattice. (6) the set of all final segments of a is a complete lattice under ⊆. (7) the set of all fuzzy final segments of a is a complete lattice under point wise ordering ≤. proof. (1) ⇒ (2) : for any a ∈ a, let [a) = {x ∈ a : a ≤ x}. then [a) is a final segment containing a and hence χ[a) is a fuzzy final segment of a. then, by (1), λ= ∧ a∈a χ[a); the point wise infimum of χ[a)’s is a fuzzy final segment of a. let 0 6= α ∈ l. then, for any a ∈ a, the α-cut of χ[a) = ( χ[a) ) α = [a). now λα = ( ∧ a∈a χ[a) ) α = ⋂ a∈a (χ[a) ) α = ⋂ a∈a [a). since λα is non empty, there exist m ∈ a such that a ≤ m for all a ∈ a. then m is the largest element of a. (2) ⇒ (3) : it is trivial, since the largest element will be the unique maximal element. (3) ⇒ (4) : if m is the unique maximal element in a, then x ∨m is maximal, and hence x ∨m = m, so that x ≤ m for all x ∈ a. now [m) is a final segment of a and hence χ[m) is a fuzzy final segment of a. let λ be any fuzzy final segment of a such that λ(x0) = 1 for some x0 ∈ a. since x0 ≤ m, we get 1 = λ(x0) ≤ λ(m), so that λ(m) = 1. this implies that χ[m) ≤ λ, for all fuzzy final segments λ of a. thus χ[m) is the smallest fuzzy final segment of a. int. j. anal. appl. (2023), 21:67 7 (4) ⇒ (5) : suppose µ is the smallest fuzzy final segment of a such that µ(m) = 1 for some m ∈ a. now for each x ∈ a, [x) is a final segment of a and hence χ[x) is a fuzzy final segment of a. since µ ≤ χ[x), we get 1 = µ(m) ≤ χ[x)(m) and hence χ[x)(m) = 1, so that m ∈ [x) and hence x ≤ m for all x ∈ a. therefore m is the largest element in a. thus a is a bounded distributive lattice ( by theorem 1.4 (11) ) . (5) ⇒ (6) : let {fα : α ∈ ∆} be a class of final segments of a. put f = ⋂ α∈∆ fα. since the largest element of a is necessarily an element in every final segment of a, it implies that f is non empty and hence f is a final segment of a. also, f is the g.l.b {fα : α ∈ ∆}. thus the set of all final segments form a complete lattice under ⊆. (6) ⇒ (7) : it is obvious. � the point wise infimum of any two fuzzy final segments of an adl need not be a fuzzy final segment. however, we prove the following. theorem 3.3. let a be an adl. then the following are equivalent to each other. (1) the point wise meet of any two fuzzy final segments of a is again a fuzzy final segment. (2) (a,≤) is directed above. (3) a is a distributive lattice. (4) the set of all fuzzy final segments of a is a lattice. (5) the set of all final segments of a is a lattice. (6) the set of all final segments of a is closed under finite intersections. proof. (1) ⇒ (2) : let a and b ∈ a. then χ[a) and χ[b) are fuzzy final segment of a, and by (1), χ[a) ∧ χ[b) is also a fuzzy final segment of a, in perticular, there exists x0 ∈ a such that( χ[a)∧χ[b) ) (x0) = χ[a)(x0)∧χ[b)(x0) = 1. then x0 ∈ [a)∩[b), so that a ≤ x0 and b ≤ x0. therefore (a,≤) is directed above. (2) ⇒ (3) : it is consequence of theorem 1.4 (11) (3) ⇒ (1) : let λ and µ be fuzzy final segments of a. choose x and y ∈ a such that λ(x) = 1 and λ(y) = 1. by (3), x ≤ x ∨y and y ≤ x ∨y and hence λ(x) ≤ λ(x ∨y) and µ(y) ≤ µ(x ∨y). ∴ λ(x ∨y) = 1 and µ(x ∨y) = 1 for any a,b ∈ a with a ≤ b, we have λ(a) ∧µ(a) ≤ λ(b) ∧µ(b) and hence (λ∧µ)(a) = (λ∧µ)(b). therefore λ∧µ is a fuzzy final segment of a. (1) ⇒ (4) : let λ and µ be fuzzy final segments of a. then, by (1), λ∧µ is a fuzzy final segment of a and is the g.l.b {λ,µ} with respect to the point wise ordering ≤. also, by (1), clearly λ∨µ ; the point wise supremum of λ and µ is a fuzzy final segment of a and which is the l.u.b {λ,µ} with 8 int. j. anal. appl. (2023), 21:67 respect to ≤. thus the set of all fuzzy final segments of a is lattice under ≤. (4) ⇒ (1) : it is clear. (3) ⇒ (5) ⇒ (6) ⇒ (3) is well known result ( by theorem 3.4 in [16] ). � conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] j.a. goguen, l-fuzzy sets, j. math. anal. appl. 18 (1967), 145-174. https://doi.org/10.1016/0022-247x(67) 90189-8. [2] n. kuroki, on fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy sets syst. 5 (1981), 203-215. https: //doi.org/10.1016/0165-0114(81)90018-x. [3] w. liu, fuzzy invariant subgroups and fuzzy ideals, fuzzy sets syst. 8 (1982), 133-139. https://doi.org/10. 1016/0165-0114(82)90003-3. [4] d.s. malik, j.n. mordeson, extensions of fuzzy subrings and fuzzy ideals, fuzzy sets syst. 45 (1992), 245-251. https://doi.org/10.1016/0165-0114(92)90125-n. [5] t.k. mukherjee, m.k. sen, on fuzzy ideals of a ring i, fuzzy sets syst. 21 (1987), 99-104. https://doi.org/ 10.1016/0165-0114(87)90155-2. [6] n. teshale amare, s. gonnabhaktula, ch. santhi sundar raj, l-fuzzy prime spectrums of adls, adv. fuzzy syst. 2021 (2021), 5520736. https://doi.org/10.1155/2021/5520736. [7] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 512-517. https://doi.org/10.1016/0022-247x(71) 90199-5. [8] ch.s.s. raj, n.t. amare, u.m. swamy, fuzzy prime ideals of adl’s, int. j. comput. sci. appl. math. 4 (2018), 32-36. https://doi.org/10.12962/j24775401.v4i2.3187. [9] ch.s.s. raj, n.t. amare, u.m. swamy, prime and maximal fuzzy filters of adls, palestine j. math. 9 (2020), 730-739. [10] m.h. stone, the theory of representation for boolean algebras, trans. amer. math. soc. 40 (1936), 37-111. https://doi.org/10.2307/1989664. [11] u.m. swamy, ch.s.s. raj, a.n. teshale, fuzzy ideals of almost distributive lattices, ann. fuzzy math. inf. 14 (2017), 371-379. [12] u.m. swamy, d.v. raju, algebraic fuzzy systems, fuzzy sets syst. 41 (1991), 187-194. https://doi.org/10. 1016/0165-0114(91)90222-c. [13] u.m. swamy, d.v. raju, fuzzy ideals and congruences of lattices, fuzzy sets syst. 95 (1998), 249-253. https: //doi.org/10.1016/s0165-0114(96)00310-7. [14] u.m. swamy, g.c. rao, almost distributive lattices, j. aust. math. soc. a. 31 (1981), 77-91. https://doi. org/10.1017/s1446788700018498. [15] u.m. swamy, k.l.n. swamy, fuzzy prime ideals of rings, j. math. anal. appl. 134 (1988), 94-103. https: //doi.org/10.1016/0022-247x(88)90009-1. [16] r.v. babu, b. venkateswarlu, initial and final segments in adl’s, southeast asian bull. math. 41 (2017), 127–131. [17] l.a. zadeh, fuzzy sets, inf. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65)90241-x. https://doi.org/10.1016/0022-247x(67)90189-8 https://doi.org/10.1016/0022-247x(67)90189-8 https://doi.org/10.1016/0165-0114(81)90018-x https://doi.org/10.1016/0165-0114(81)90018-x https://doi.org/10.1016/0165-0114(82)90003-3 https://doi.org/10.1016/0165-0114(82)90003-3 https://doi.org/10.1016/0165-0114(92)90125-n https://doi.org/10.1016/0165-0114(87)90155-2 https://doi.org/10.1016/0165-0114(87)90155-2 https://doi.org/10.1155/2021/5520736 https://doi.org/10.1016/0022-247x(71)90199-5 https://doi.org/10.1016/0022-247x(71)90199-5 https://doi.org/10.12962/j24775401.v4i2.3187 https://doi.org/10.2307/1989664 https://doi.org/10.1016/0165-0114(91)90222-c https://doi.org/10.1016/0165-0114(91)90222-c https://doi.org/10.1016/s0165-0114(96)00310-7 https://doi.org/10.1016/s0165-0114(96)00310-7 https://doi.org/10.1017/s1446788700018498 https://doi.org/10.1017/s1446788700018498 https://doi.org/10.1016/0022-247x(88)90009-1 https://doi.org/10.1016/0022-247x(88)90009-1 https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. fuzzy initial segments 3. fuzzy final segments references international journal of analysis and applications volume 19, number 6 (2021), 836-857 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-836 k −g−fusion frames in hilbert c∗−modules fakhr-dine nhari1,∗, rachid echarghaoui1 and mohamed rossafi2 1laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, kenitra, morocco 2lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco ∗corresponding author: nharidoc@gmail.com abstract. in this paper, we introduce the concepts of g−fusion frame and k −g−fusion frame in hilbert c∗−modules and we give some properties. also, we study the stability problem of g−fusion frame. the presented results extend, generalize and improve many existing results in the literature. 1. introduction and preliminaries frame theory is recently an active research area in mathematics, computer science, and engineering with many exciting applications in a variety of different fields. a frame is a set of vectors in a hilbert space that can be used to reconstruct each vector in the space from its inner products with the frame vectors. these inner products are generally called the frame coefficients of the vector. but unlike an orthonormal basis each vector may have infinitely many different representations in terms of its frame coefficients. frames for hilbert spaces were introduced by duffin and schaefer [1] in 1952 to study some deep problems in nonharmonic fourier series by abstracting the fundamental notion of gabor [2] for signal processing. received september 29th, 2021; accepted october 21st, 2021; published october 28th, 2021. 2010 mathematics subject classification. 42c15. key words and phrases. fusion frame; g-fusion frame; k − g−fusion; hilbert c∗−module. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 836 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-836 int. j. anal. appl. 19 (6) (2021) 837 hilbert c∗-modules is a generalization of hilbert spaces by allowing the inner product to take values in a c∗-algebra rather than in the field of complex numbers. many generalizations of the concept of frame have been defined in hilbert c∗-modules [3, 5, 9–13]. in the following, we recall some definitions and results that will be used to prove our mains results. let a be a unital c∗−algebra, let j be countable index set. throughout this paper h and k are countably generated hilbert a−modules and (kj)j∈j is a sequence of closed hilbert submodules of k. for each j ∈ j, end∗a(h,kj) is the collection of all adjointable a−linear maps from h to kj, and end∗a(h,h) is denoted by end ∗ a(h). definition 1.1. [6] let a be a unital c∗-algebra and h be a left a-module, such that the linear structures of a and h are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued inner product 〈., .〉 : h×h→a, such that is sesquilinear, positive definite and respects the module action. in the other words, (i) 〈f,f〉≥ 0 for all f ∈h and 〈f,f〉 = 0 if and only if f = 0. (ii) 〈af + g,h〉 = a〈f,h〉 + 〈g,h〉 for all a ∈a and f,g,h ∈h. (iii) 〈f,g〉 = 〈g,f〉∗ for all f,g ∈h. for f ∈h, we define ||f|| = ||〈f,f〉|| 1 2 . if h is complete with ||.||, it is called a hilbert a-module or a hilbert c∗-module over a. for every a in a c∗-algebra a, we have |a| = (a∗a) 1 2 and the a-valued norm on h is defined by |f| = 〈f,f〉 1 2 for f ∈h. define l2((kj)j∈j) by l2((kj)j∈j) = {(fj)j∈j : fj ∈kj, || ∑ j∈j 〈fj,fj〉|| < ∞}. with a−valued inner product is given by 〈(fj)j∈j, (gj)j∈j〉 = ∑ j∈j 〈fj,gj〉, l2((kj)j∈j) is a hilbert a−module. lemma 1.1. [7] let t ∈ end∗a(h,k) and h,k are hilberts a−modules.the following statemnts are multually equivalent: (i) t is surjective. (ii) t∗ is bounded below with respect to the norm, i.e., there is m > 0 such that ||t∗f|| ≥ m||f|| for all f ∈k. (iii) t∗ is bounded below with respect to the inner product, i.e, there is m ′ > 0 such that 〈t∗f,t∗f〉 ≥ m ′ 〈f,f〉 for all f ∈k. int. j. anal. appl. 19 (6) (2021) 838 lemma 1.2. [7] let h be a hilbert a-module over a c∗-algebra a, and t ∈ end∗a(h) such that t ∗ = t . the following statements are equivalent: (i) t is surjective. (ii) there are m,m > 0 such that m‖f‖≤‖tf‖≤ m‖f‖, for all f ∈h. (iii) there are m′,m′ > 0 such that m′〈f,f〉≤ 〈tf,tf〉≤ m′〈f,f〉 for all f ∈h. lemma 1.3. [4] let h and k are two hilbert a-modules and t ∈ end∗(h,k). then: (i) if t is injective and t has closed range, then the adjointable map t∗t is invertible and ‖(t∗t)−1‖−1 ≤ t∗t ≤‖t‖2. (ii) if t is surjective, then the adjointable map tt∗ is invertible and ‖(tt∗)−1‖−1 ≤ tt∗ ≤‖t‖2. lemma 1.4. [8] let h be a hilbert a-module. if t ∈ end∗a(h), then 〈tf,tf〉≤ ‖t‖2〈f,f〉, ∀f ∈h. lemma 1.5. [14] let a be a c∗−algebra, e, h, k be hilbert a-modules. let t ∈ end∗a(e,k) and t ′ ∈ end∗a(h,k). if r(t∗) is orthogonally complemented, then the following statements are equivalent: (i) r(t ′ ) ⊆r(t). (ii) t ′ (t ′ )∗ ≤ λtt∗ for some λ > 0. (iii) there exists a positive real number µ > 0 such that ‖(t ′ )∗f‖≤ µ‖t∗f‖, for all f ∈k. (iv) there exists a solution x ∈ end∗a(h,e) of the so-called douglas equation t ′ = tx. 2. k −g−fusion frames in hilbert c∗−modules we begin this section with the following lemma. lemma 2.1. let (wj)j∈j be a sequence of orthogonally complemented closed submodules of h and t ∈ end∗a(h) invertible, if t ∗twj ⊂ wj for each j ∈ j, then (twj)j∈j is a sequence of orthogonally complemented closed submodules and pwjt ∗ = pwjt ∗pt wj . proof. firstly for each j ∈ j, t : wj → twj is invertible, so each twj is a closed submodule of h. we show that h = twj ⊕t(w⊥j ). since h = th, then for each f ∈h, there exists g ∈h sutch that f = tg. on the other hand g = u + v, for some u ∈ wj and v ∈ w⊥j . hence f = tu + tv, where tu ∈ twj and tv ∈ t(w⊥j ) plainly twj ∩t(w ⊥ j ) = (0), therefore h = twj ⊕t(w ⊥ j ). hence for every g ∈ wj, h ∈ w ⊥ j we have t∗tg ∈ wj and therefore 〈tg,th〉 = 〈t∗tg,h〉 = 0, so t(w⊥j ) ⊂ (twj) ⊥ and consequentely t(w⊥j ) = (twj) ⊥ witch implies that twj is orthogonally complemented. int. j. anal. appl. 19 (6) (2021) 839 let f ∈ h we have f = pt wjf + g, for some g ∈ (twj)⊥, then t∗f = t∗pt wjf + t∗g. let v ∈ wj then 〈t∗g,v〉 = 〈g,tv〉 = 0 then t∗g ∈ w⊥j and we have pwjt ∗f = pwjt ∗pt wjf + pwjt ∗g, then pwjt ∗f = pwjt ∗pt wjf thus implies that for each j ∈ j we have pwjt∗ = pwjt∗pt wj . � now we define the notion of k −g−fusion frame in hilbert c∗−modules. definition 2.1. let a be a unital c∗−algebra and h be a countably generated hilbert a−module. let (vj)j∈j be a family of weights in a,i.e.,each vj is a positive invertible element frome the center of a. let (wj)j∈j be a collection of orthogonally complemented closed submodules of h. let (kj)j∈j a sequence of closed submodules of k and λj ∈ end∗a(h,kj) for each j ∈ j. we say λ = (wj, λj,vj)j∈j is g−fusion frame for h with respect to (kj)j∈j if there exist real constants 0 < a ≤ b < ∞ such that a〈f,f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈f,f〉, ∀f ∈h. the counstants a and b are called the lower and upper bounds of g−fusion frame, respectively. if a = b then λ is called tight g−fusion frame and if a = b = 1 then we say λ is a parseval g−fusion frame. if the family λ satisfies ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈f,f〉, ∀f ∈h. then it is called a g−fusion bessel sequence for h with bound b. lemma 2.2. let λ = (wj, λj,vj)j∈j be a g−fusion bessel sequence for h with bound b. then for each sequence (fj)j∈j ∈ l2((kj)j∈j), the series ∑ j∈j vjpwj λ ∗ jfj is converge unconditionally. proof. let i be a finite subset of j, then || ∑ j∈i vjpwj λ ∗ jfj|| = sup ||g||=1 ||〈 ∑ j∈i vjpwj λ ∗ jfj,g〉|| ≤ || ∑ j∈i 〈fj,fj〉|| 1 2 sup ||g||=1 || ∑ j∈i v2j〈λjpwjg, λjpwjg〉|| 1 2 ≤ √ b|| ∑ j∈i 〈fj,fj〉|| 1 2 . and it follows that ∑ j∈j vjpwj λ ∗ jfj is unconditionally convergent in h. � now, we can define the synthesis operator by lemma 2.2 definition 2.2. let λ = (wj, λj,vj)j∈j be a g−fusion bessel sequence for h. then the operator tλ : l2((kj)j∈j)) →h defined by tλ((fj)j∈j) = ∑ j∈j vjpwj λ ∗ jfj, ∀(fj)j∈j ∈ l 2((kj)j∈j). int. j. anal. appl. 19 (6) (2021) 840 is called synthesis operator. we say the adjoint t∗λ of the synthesis the analysis operator and it is defined by t∗λ : h→ l 2((kj)j∈j) such that t∗λ(f) = (vjλjpwj (f))j∈j, ∀f ∈h. the operator sλ : h→h defined by sλf = tλt∗λf = ∑ j∈j v2j pwj λ ∗ j λjpwj (f), ∀f ∈h. is called g−fusion frame operator. it can be easily verify that (2.1) 〈sλf,f〉 = ∑ j∈j v2j〈λjpwj (f), λjpwj (f)〉, ∀f ∈h. furthermore, if λ is a g−fusion frame with bounds a and b, then a〈f,f〉≤ 〈sλf,f〉≤ b〈f,f〉, ∀f ∈h. it easy to see that the operator sλ is bounded, self-adjoint, positive, now we proof the inversibility of sλ. let f ∈h we have ||t∗λ(f)|| = ||(vjλjpwj (f))j∈j|| = || ∑ j∈j v2j〈λjpwj (f), λjpwj (f)〉|| 1 2 . since λ is g−fusion frame then √ a||〈f,f〉|| 1 2 ≤ ||t∗λf||. then √ a||f|| ≤ ||t∗λf||. frome lemma1.1, tλ is surjective and by lemma1.3, tλt ∗ λ = sλ is invertible. we now, aih ≤ sλ ≤ bih and this gives b−1ih ≤s−1λ ≤ a −1ih definition 2.3. let a be a unital c∗−algebra and h be a countably generated hilbert a−module. let (vj)j∈j be a family of weights in a,i.e.,each vj is a positive invertible element frome the center of a, let (wj)j∈j be a collection of orthogonally complemented closed submodules of h. let (kj)j∈j a sequence of closed submodules of k and λj ∈ end∗a(h,kj) for each j ∈ j and k ∈ end ∗ a(h). we say λ = (wj, λj,vj)j∈j is k−g−fusion frame for h with respect to (kj)j∈j if there exist real constants 0 < a ≤ b < ∞ such that (2.2) a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈f,f〉, ∀f ∈h. the constants a and b are called a lower and upper bounds of k − g−fusion frame, respectively. if the left-hand inequality of (2.2) is an equality, we say that λ is a tight k −g−fusion frame. if k = ih then λ is a g−fusion frame and if k = ih and λj = pwj for any j ∈ j, then λ is a fusion frame for h int. j. anal. appl. 19 (6) (2021) 841 example 2.1. let h be a hilbert c∗−module with dimensional 3 and let {e1,e2,e3} be standard basis. we define the operator k on h by ke1 = e2, ke2 = e3, ke3 = e3; suppose that wj = kj = span{ej} where j = 1, 2, 3. let λjx = 〈x,ej〉ej, it is clear that (wj, λj, 1)j∈j is a k −g−fusion frame for h. remark 2.1. if λ = (wj, λj,vj)j∈j is k −g−fusion frame for h with bounds a and b, then we have (2.3) akk∗ ≤sλ ≤ bih from inequalities (2.1) and (2.3), we have lemma 2.3. let k ∈ end∗a(h) and λ = (wj, λj,vj)j∈j be a g−fusion bessel sequence for h. then λ is k −g−fusion frame for h if and only if there exist a constant a > 0 such that akk∗ ≤ sλ, where sλ is the frame operator for λ. theorem 2.1. let k ∈ end∗a(h), and λ = (wj,pwj,vj)j∈j be a g−fusion bessel sequence for h with frame operator sλ such that r(s 1 2 λ ) is orthogonally complemented. then λ is k −g−fusion frame for h if and only if k = s 1 2 λ m for some m ∈ end ∗ a(h). proof. suppose λ is a k −g−fusion frame for h, then there exist a > 0 such that akk∗ ≤sλ, and sλ is self-adjoint and positive thus s 1 2 λ is self-adjoint and positive, so we have kk∗ ≤ 1 a s 1 2 λs 1 2 λ . by lemma1.5, there exists some m ∈ end∗a(h) such that k = s 1 2 λ m. suppose that there exists an operator m ∈ end∗a(h) so that k = s 1 2 λ m. from lemma1.5 , we know that akk∗ ≤sλ for some constant a > 0, from lemma2.3, λ is a k −g−fusion frame. � theorem 2.2. if u ∈ end∗a(h) and λ is a k−g−fusion frame for h, and r(u) ⊂r(k) such that r(k∗) is orthogonaly complemented. then λ is u −g−fusion frame for h. proof. by lemma1.5, ∃λ > 0: uu∗ ≤ λkk∗, then for each f ∈h we have 〈u∗f,u∗f〉 = 〈uu∗f,f〉≤ 〈λkk∗f,f〉≤ λ〈k∗f,k∗f〉. it follows that, a λ 〈u∗f,u∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉, ∀f ∈h. so λ is a u −g−fusion frame for h. � int. j. anal. appl. 19 (6) (2021) 842 theorem 2.3. let λ = (wj, λj,vj)j∈j and γ = (vj, γj,uj)j∈j be two g−fusion bessel sequence for h with bounds b1 and b2, respectively. suppose that tλ and tγ are their synthesis operators such that tγt ∗ λ = k ∗ where k ∈ end∗a(h). then, both λ and γ are k and k ∗ −g−fusion frames, respectively. proof. let f ∈h, we have 〈k∗f,k∗f〉 = 〈tγt∗λf,tγt ∗ λf〉≤ ||tγ|| 2〈t∗λf,t ∗ λf〉≤ b2 ∑ j∈j v2j〈λjpwjf, λjpwjf〉. so, b−12 〈k ∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉. thus λ is k − g− fusion frame for h. similarly, γ is k∗ − g−fusion frame for h with the lower bound b−11 . � theorem 2.4. let k ∈ end∗a(h) and λ = (wj, λj,vj)j∈j be a g−fusion bessel sequence for h, with synthesis operator tλ of λ. suppose that r(t∗λ) and r(k∗) are orthogonaly complemented, the following statements hold: (1) if λ is a tight k −g−fusion frame for h, then r(k) = r(tλ). (2) r(k) = r(tλ) if and only if there exist two constants 0 < a ≤ b < ∞ such that: a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈k ∗f,k∗f〉, ∀f ∈h. proof. (1) suppose that λ is a tight k −g−fusion frame for h, then there exixt a > 0, such that for each f ∈h a〈k∗f,k∗f〉 = ∑ j∈j v2j〈λjpwjf, λjpwjf〉 = 〈(vjλjpwjf)j∈j, (vjλjpwjf)j∈j〉 = 〈t∗λf,t ∗ λf〉. then 〈akk∗f,f〉 = 〈tλt∗λf,f〉. so akk∗ = tλt ∗ λ. then by lemma1.5 , r(tλ) = r(k) (2) suppose that r(k) = r(tλ), by lemma1.5 there exist two constants a, b > 0 such that akk∗ ≤ tλt∗λ ≤ bkk ∗. int. j. anal. appl. 19 (6) (2021) 843 which implies that for each f ∈h 〈akk∗f,f〉≤ 〈tλt∗λf,f〉≤ 〈bkk ∗f,f〉. a〈kk∗f,f〉≤ 〈tλt∗λf,f〉≤ b〈kk ∗f,f〉. therefore a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈k ∗f,k∗f〉. suppose that there exist two constants a, b > 0 such that for each f ∈h a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b〈k ∗f,k∗f〉. then a〈kk∗f,f〉≤ 〈tλt∗λf,f〉≤ b〈kk ∗f,f〉. so akk∗ ≤ tλt∗λ ≤ bkk ∗. since by lemma1.5 , r(tλ) = r(k). � theorem 2.5. let u ∈ end∗a(h) be an invertible operator on h and λ = (wj, λj,vj)j∈j be a k−g−fusion frame for h for some k ∈ end∗a(h). suppose that u ∗uwj ⊂ wj, ∀j ∈ j. then γ = (uwj, λjpwju∗,vj)j∈j is a uku∗ −g−fusion frame for h. proof. since λ is a k −g−fusion frame for h, ∃ a,b > 0 such that a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwj (f), λjpwj (f)〉≤ b〈f,f〉, ∀f ∈h. also, u is an invertible linear operator on h, so for any j ∈ j, uwj is closed in h. now, for each f ∈ h, using lemma2.1, we obtain ∑ j∈j v2j〈λjpwju ∗puwj (f), λjpwju ∗puwj (f)〉 = ∑ j∈j v2j〈λjpwju ∗(f), λjpwju ∗(f)〉 ≤ b〈u∗f,u∗f〉 ≤ b||u||2〈f,f〉. int. j. anal. appl. 19 (6) (2021) 844 on the other hand, for each f ∈h a〈(uku∗)∗f, (uku∗)∗f〉 = a〈uk∗u∗f,uk∗u∗f〉 ≤ a||u||2〈k∗u∗f,k∗u∗f〉 ≤ ||u||2 ∑ j∈j v2j〈λjpwju ∗(f), λjpwju ∗(f)〉 ≤ ||u||2 ∑ j∈j v2j〈λjpwju ∗puwj (f), λjpwju ∗puwj (f)〉, then a ||u||2 〈(uku∗)∗f, (uku∗)∗f〉≤ ∑ j∈j v2j〈λjpwju ∗puwj (f), λjpwju ∗puwj (f)〉 therefore, γ is uku∗ −g−fusion frame for h. � theorem 2.6. let u ∈ end∗a(h) be an invertible operator on h and γ = (uwj, λjpwju ∗,vj)j∈j be a k − g−fusion frame for h for some k ∈ end∗a(h). suppose that u ∗uwj ⊂ wj, ∀j ∈ j. then λ = (wj, λj,vj)j∈j is a u −1ku −g−fusion frame for h. proof. since γ = (uwj, λjpwj,vj)j∈j is k −g−fusion frame for h, for all f ∈h, ∃ a, b > 0 such that a〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwju ∗puwj, λjpwju ∗puwj〉≤ b〈f,f〉. let f ∈h, we have a〈(u−1ku)∗f, (u−1ku)∗f〉 = a〈u∗k∗(u−1)∗f,u∗k∗(u−1)∗f〉 ≤ a||u∗||2〈k∗(u−1)∗f,k∗(u−1)∗f〉 ≤ ||u||2 ∑ j∈j v2j〈λjpwju ∗puwj (u −1)∗f, λjpwju ∗puwj (u −1)∗f〉 ≤ ||u||2 ∑ j∈j v2j〈λjpwju ∗(u−1)∗f, λjpwju ∗(u−1)∗f〉 = ||u||2 ∑ j∈j v2j〈λjpwjf, λjpwjf〉. then, for each f ∈h, we have a ||u||2 〈(u−1ku)∗f, (u−1ku)∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉. int. j. anal. appl. 19 (6) (2021) 845 also, for each f ∈h, we have ∑ j∈j v2j〈λjpwjf, λjpwjf〉 = ∑ j∈j v2j〈λjpwju ∗(u−1)∗f, λjpwju ∗(u−1)∗f〉 = ∑ j∈j v2j〈λjpwju ∗puwj (u −1)∗f, λjpwju ∗puwj (u −1)∗f〉 ≤ b〈(u−1)∗f, (u−1)∗f〉 ≤ b||u−1||2〈f,f〉. thus, λ is a u−1ku −g−fusion frame for h. � theorem 2.7. let k ∈ end∗a(h) be an invertible operator on h and λ = (wj, λj,vj)j∈j be a g−fusion frame for h with frame bounds a, b and sλ be the associated g−fusion frame operator. suppose that for all j ∈ j, t∗twj ⊂ wj, where t = ks−1λ . then (ks −1 λ wj, λjpwjs −1 λ k ∗,vj)j∈j is a k −g−fusion frame for h with the corresponding g−fusion frame operator ks−1λ k ∗. proof. we now t = ks−1λ is invertible on h and t ∗ = (ks−1λ ) ∗ = s−1λ k ∗. for each f ∈h, we have 〈k∗f,k∗f〉 = 〈sλs−1λ k ∗f,sλs−1λ k ∗f〉 ≤ ||sλ||2〈s−1λ k ∗f,s−1λ k ∗f〉 ≤ b2〈s−1λ k ∗f,s−1λ k ∗f〉. now for each f ∈h, we get ∑ j∈j v2j〈λjpwjt ∗pt wj (f), λjpwjt ∗pt wj (f)〉 = ∑ j∈j v2j〈λjpwjt ∗(f), λjpwjt ∗(f)〉 ≤ b〈t∗f,t∗f〉 ≤ b||t ||2〈f,f〉 ≤ b||s−1λ || 2||k||2〈f,f〉 ≤ b a2 ||k||2〈f,f〉. on the other hand, for each f ∈h, we have ∑ j∈j v2j〈λjpwjt ∗pt wj (f), λjpwjt ∗pt wj (f)〉 = ∑ j∈j v2j〈λjpwjt ∗(f), λjpwjt ∗(f)〉 ≥ a〈t∗f,t∗f〉 = a〈s−1λ k ∗f,s−1λ k ∗f〉 ≥ a b2 〈k∗f,k∗f〉. int. j. anal. appl. 19 (6) (2021) 846 thus (ks−1λ wj, λjpwjs −1 λ k ∗,vj)j∈j is a k −g−fusion frame for h. for each f ∈h, we have ∑ j∈j v2j pt wj (λjpwjt ∗)∗(λjpwjt ∗)pt wjf = ∑ j∈j v2j pt wjtpwj λ ∗ j (λjpwjt ∗)pt wjf = ∑ j∈j v2j (pwjt ∗pt wj ) ∗λ∗j λj(pwjt ∗pt w j)f = ∑ j∈j v2j tpwj λ ∗ j λjpwjt ∗f = t( ∑ j∈j v2j pwj λ ∗ j λjpwjt ∗f) = tsλt∗(f) = ks−1λ k ∗(f). this implies that ks−1λ k ∗ is the associated g−fusion frame operator. � theorem 2.8. let λ = (wj, λj,vj)j∈j be a k − g−fusion frame for h with bounds a, b and for each j ∈ j, tj ∈ end∗a(kj) be invertible operator. suppose 0 < m = inf j∈j 1 ||t−1j || ≤ sup j∈j ||tj|| = m. if t ∈ end∗a(h) is an invertible operator on h with kt = tk and t ∗twj ⊂ wj, ∀j ∈ j then γ = (twj,tjλjpwjt ∗,vj)j∈j is a k −g−fusion frame for h. proof. since t and tj(for each j ∈ j) are invertible, so 〈k∗f,k∗f〉 = 〈(t−1)∗t∗k∗f, (t−1)∗t∗k∗f〉 ≤ ||(t−1)||2〈t∗k∗f,t∗k∗f〉, 〈f,f〉 = 〈t−1j tjf,t −1 j tjf〉 ≤ ||t−1j || 2〈tjf,tjf〉. for each f ∈h, we have ∑ j∈j v2j〈tjλjpwjt ∗pt wjf,tjλjpwjt ∗pt wjf〉 = ∑ j∈j v2j〈tjλjpwjt ∗f,tjλjpwjt ∗f〉 ≤ ∑ j∈j ||tj||2v2j〈λjpwjt ∗f, λjpwjt ∗f〉 ≤ m2 ∑ j∈j v2j〈λjpwjt ∗f, λjpwjt ∗f〉 ≤ m2b〈t∗f,t∗f〉 ≤ m2b||t ||2〈f,f〉. int. j. anal. appl. 19 (6) (2021) 847 on the other hand, for each f ∈h, we have ∑ j∈j v2j〈tjλjpwjt ∗pt wjf,tjλjpwjt ∗pt wjf〉 = ∑ j∈j v2j〈tjλjpwjt ∗f,tjλjpwjt ∗f〉 ≥ ∑ j∈j 1 ||t−1j ||2 v2j〈λjpwjt ∗f, λjpwjt ∗f〉 ≥ m2 ∑ j∈j v2j〈λjpwjt ∗f, λjpwjt ∗f〉 ≥ m2a〈k∗t∗f,k∗t∗f〉 ≥ m2a ||(t−1)||2 〈k∗f,k∗f〉. thus, γ is a k −g−fusion frame for h. � in this theorem we give a necessary and sufficient condition for a quotient operator to be bounded. theorem 2.9. let k ∈ end∗a(h), and λ = (wj, λj,vj)j∈j be a k − g−fusion frame for h with frame operator sλ and frame bounds a and b. let u ∈ end∗a(h) be an invertible operator on h, and u ∗uwj ⊂ wj,∀j ∈ j. then the following statements are equivalent: (1) γ = (uwj, λjpwju ∗,vj)j∈j is a uk −g−fusion frame. (2) the quotient operator [(uk)∗/s 1 2 λ u ∗] is bounded. (3) the quotient operator [(uk)∗/(usλu∗) 1 2 ] is bounded. proof. (1) =⇒ (2) since γ is k −g−fusion frame then there exist a, b > 0 such that for each f ∈h a〈(uk)∗f, (uk)∗〉≤ ∑ j∈j v2j〈λjpwju ∗puwjf, λjpwju ∗puwjf〉≤ b〈f,f〉 for each f ∈h we have ∑ j∈j v2j〈λjpwju ∗puwjf, λjpwju ∗puwjf〉 = ∑ j∈j v2j〈λjpwju ∗f, λjpwju ∗f〉 = 〈sλu∗f,u∗f〉 = 〈s 1 2 λ u ∗f,s 1 2 λ u ∗f〉. then a〈(uk)∗f, (uk)∗f〉≤ 〈s 1 2 λ (u ∗f),s 1 2 λ (u ∗f)〉. we define the operator: t : r(s 1 2 λ u ∗) →r((uk)∗) by t(s 1 2 λ u ∗f) = (uk)∗f, ∀f ∈h. int. j. anal. appl. 19 (6) (2021) 848 t is linear operator and ker(s 1 2 λ u ∗) ⊂ ker((uk)∗). thus t is well-defined quotient operator. therefore for each f ∈h ||t(s 1 2 λ u ∗f)|| = ||(uk)∗f|| ≤ 1 √ a ||〈s 1 2 λ u ∗f,s 1 2 λ u ∗f〉|| 1 2 ≤ 1 √ a ||s 1 2 λ u ∗f||. so t is bounded. (2) =⇒ (3) suppose that the quotion operator [(uk)∗/s 1 2 λ u ∗] is bounded. thus for all f ∈h, ∃c > 0 such that ||(uk)∗f|| ≤ c||s 1 2 λ u ∗f|| ≤ c||〈s 1 2 λ u ∗f,s 1 2 λ u ∗f〉|| 1 2 ≤ c||〈usλu∗f,f〉|| 1 2 ≤ c||〈(uku∗) 1 2 f, (uku∗) 1 2 f〉|| 1 2 ≤ c||(usλu∗) 1 2 f||. hence the quotient operator [(uk)∗/(usλu∗) 1 2 ] is bounded. (3) =⇒ (1) for each f ∈h, we have ∑ j∈j v2j〈λjpwju ∗puwjf, λjpwju ∗puwjf〉 = ∑ j∈j v2j〈λjpwju ∗f, λjpwju ∗f〉 ≥ a〈k∗(u∗f),k∗(u∗f)〉 = a〈(uk)∗f, (uk)∗f〉. on the other hand for each f ∈h ∑ j∈j v2j〈λjpwju ∗puwjf, λjpwju ∗puwjf〉 = ∑ j∈j v2j〈λjpwju ∗f, λjpwju ∗f〉 ≤ b〈u∗f,u∗f〉 ≤ b||u||2〈f,f〉. hence γ is a uk −g−fusion frame for h. � studying g−fusion frame in hilbert c∗-modules with different c∗-algebras is interesting and important. in the following, we study this situation. in the next theorem we take kj ⊂h for each j ∈ j. int. j. anal. appl. 19 (6) (2021) 849 theorem 2.10. let (h,a,〈., .〉a) and (h,b,〈., .〉b) be two hilbert c∗−modules and φ : a → b be a ∗−homomorphism and θ be a map on h such that 〈θf,θg〉b = φ(〈f,g〉a) for all f,g ∈h. also, suppose that λ = (wj, λj,vj)j∈j is a g−fusion frame for (h,a,〈., .〉a) with g−fusion frame operator sa and lower and upper g−fusion frame bounds a, b respectively. if θ is surjective and θλjpwj = λjpwjθ for each j ∈ j, then λ = (wj, λj,φ(vj))j∈j is a g−fusion frame for (h,b,〈., .〉b) with g−fusion frame operator sb and lower and upper g−fusion frame bounds φ(a), φ(b) respectively, and 〈sbθf,θg〉b = φ(〈saf,g〉a). proof. let g ∈h then there exists f ∈h such that θf = g. by the definition of g−fusion frame we have a〈f,f〉a ≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉a ≤ b〈f,f〉a. then φ(a〈f,f〉a) ≤ ∑ j∈j φ(v2j〈λjpwjf, λjpwjf〉a) ≤ φ(b〈f,f〉a). by the definition of ∗−homomorphism we have aφ(〈f,f〉a) ≤ ∑ j∈j φ(v2j )φ(〈λjpwjf, λjpwjf〉a) ≤ bφ(〈f,f〉a). by the relation betwen θ and φ we get a〈θf,θf〉b ≤ ∑ j∈j φ(vj) 2〈θλjpwjf,θλjpwjf〉b ≤ b〈θf,θf〉b. then a〈θf,θf〉b ≤ ∑ j∈j φ(vj) 2〈λjpwjθf, λjpwjθf〉b ≤ b〈θf,θf〉b. so, we have a〈g,g〉b ≤ ∑ j∈j φ(vj) 2〈λjpwjg, λjpwjg〉b ≤ b〈g,g〉b, ∀g ∈h. int. j. anal. appl. 19 (6) (2021) 850 on the other hand we have φ(〈saf,g〉a) = φ(〈 ∑ j∈j v2j pwj λ ∗ j λjpwjf,g〉a) = ∑ j∈j φ(v2j〈λjpwjf, λjpwjg〉a) = ∑ j∈j φ(vj) 2〈θλjpwjf,θλjpwjg〉b = ∑ j∈j φ(vj) 2〈λjpwjθf, λjpwjθg〉b = ∑ j∈j φ(vj) 2〈pwj λ ∗ j λjpwjθf,θg〉b = 〈 ∑ j∈j φ(vj) 2pwj λ ∗ j λjpwjpwjθf,θg〉b = 〈sbθf,θg〉b. � 3. stability of g-fusion frames in hilbert c∗−modules frome theorem 2.7 if λ = (wj, λj,vj)j∈j is a g−fusion frame for h with associated frame operator sλ, such that s−2λ wj ⊂ wj, for all j ∈ j then λ̃ = (s −1 λ wj, λjpwjs −1 λ ,vj)j∈j is called the canonical dual g−fusion frame of λ. the frame operator sλ̃ of λ̃ is described by, for each f ∈h sλ̃(f) = ∑ j∈j v2j ps−1 λ wj (λjpwjs −1 λ ) ∗(λjpwjs −1 λ )ps−1 λ wj (f) = ∑ j∈j v2j ps−1 λ wj s−1λ pwj λ ∗ j λj(pwjs −1 λ ps−1 λ wj )(f) = ∑ j∈j v2j (pwjs −1 λ ps−1 λ wj )∗λ∗j λj(pwjs −1 λ ps−1 λ wj )(f) = ∑ j∈j v2j (pwjs −1 λ ) ∗λ∗j λjpwjs −1 λ (f) = ∑ j∈j v2js −1 λ pwj λ ∗ j λjpwjs −1 λ (f) = s−1λ ∑ j∈j v2j pwj λ ∗ j λjpwjs −1 λ (f) = s−1λ (sλ(s −1 λ f)) = s −1 λ (f). theorem 3.1. let λ = (wj, λj,vj)j∈j and γ = (vj, γj,vj)j∈j be two g−fusion frames for h with lower frame bounds a and c, respectively. suppose that s−2λ wj ⊂ wj and s −2 γ vj ⊂ vj, ∀j ∈ j. if there exist real int. j. anal. appl. 19 (6) (2021) 851 constant d > 0 such that for all f ∈h || ∑ j∈j v2j〈λjpwjf, λjpwjf〉− ∑ j∈j v2j〈γjpvjf, γjpvjf〉|| ≤ d||〈f,f〉||. then for all f ∈h || ∑ j∈j v2j〈λ̃jpw̃j (f), λ̃jpw̃j (f)〉− ∑ j∈j v2j〈γ̃jpṽj (f), γ̃jpṽj (f)〉|| ≤ d ac ||〈f,f〉||. such that w̃j = s−1λ wj, λ̃j = λjpwjs −1 λ , ṽj = s −1 γ vj, γ̃j = γjpvjs −1 γ . proof. we have for every f ∈h ||sλ −sγ|| = sup ||f||=1 ||〈(sλ −sγ)f,f〉|| = sup ||f||=1 ||〈sλf,f〉−〈sγf,f〉|| = sup ||f||=1 || ∑ j∈j v2j〈λjpwjf, λjpwjf〉− ∑ j∈j v2j〈γjpvjf, γjpvjf〉|| ≤ d. therefore, ||s−1λ −s −1 γ || ≤ ||s −1 λ ||||sλ −sγ||||s −1 γ || ≤ d ac . and for all f ∈h ∑ j∈j v2j〈λjpwjs −1 λ ps−1 λ wj f, λjpwjs −1 λ ps−1 λ wj f〉 = ∑ j∈j v2j〈λjpwjs −1 λ f, λjpwjs −1 λ f〉 = ∑ j∈j v2j〈pwj λ ∗ j λjpwjs −1 λ f,s −1 λ f〉 = 〈 ∑ j∈j v2j pwj λ ∗ j λjpwj (s −1 λ f),s −1 λ f〉 = 〈sλ(s−1λ ),s −1f〉 = 〈f,s−1λ f〉. similarly we have for all f ∈h ∑ j∈j v2j〈γjpvjs −1 γ ps−1 γ vj f, γjpvjs −1 γ ps−1 γ vj f〉 = 〈f,s−1γ f〉. then || ∑ j∈j v2j〈λjpwjs −1 λ ps−1 λ wj f, λjpwjs −1 λ ps−1 λ wj f〉− ∑ j∈j 〈γjpvjs −1 γ ps−1 γ vj f, γjpvjs −1 γ ps−1 γ vj f〉|| int. j. anal. appl. 19 (6) (2021) 852 = ||〈f,s−1λ f〉−〈f,s −1 γ f〉|| = ||〈f, (s−1λ −s −1 γ )f〉|| ≤ ||s−1λ −s −1 γ ||||f|| 2 ≤ d ac ||〈f,f〉||. � now we give a characterazation of g−fusion frames for hilbert a−modules. theorem 3.2. let h be a hilbert a−module over c∗−algebra. then λ = (wj, λj,vj)j∈j is a g−fusion frame for h if and only if there exist two constants 0 < a ≤ b < ∞ such that for all f ∈h a||〈f,f〉|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| ≤ b||〈f,f〉||. proof. suppose λ is g−fusion frame for h, since there is 〈f,f〉≥ 0 then for all f ∈h, a||〈f,f〉|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| ≤ b||〈f,f〉|| conversely for each f ∈h we have || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| = || ∑ j∈j 〈vjλjpwjf,vjλjpwjf〉|| = ||〈(vjλjpwjf)j∈j, (vjλjpwjf)j∈j〉|| = ||(vjλjpwjf)j∈j|| 2. we define the operator l : h→ l2((kj)j∈j) by l(f) = (vjλjpwjf)j∈j, then ||l(f)||2 = ||(vjλjpwjf)j∈j|| 2 ≤ b||f||2. l is a−linear bounded operator, then there exist c > 0 sutch that 〈l(f),l(f)〉≤ c〈f,f〉, ∀f ∈h. so ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ c〈f,f〉, ∀f ∈h. therefore λ = (wj, λj,vj)j∈j is g−fusion bessel sequence for h. now we cant define the g−fusion frame operator sλ on h. so ∑ j∈j v2j〈λjpwjf, λjpwjf〉 = 〈sλf,f〉, ∀f ∈h. since sλ is self-adjoint and positive, then 〈s 1 2 λ f,s 1 2 λ f〉 = 〈sλf,f〉, ∀f ∈h. int. j. anal. appl. 19 (6) (2021) 853 that implies a||〈f,f〉|| ≤ ||〈s 1 2 λ f,s 1 2 λ f〉|| ≤ b||〈f,f〉||, ∀f ∈h. frome lemma1.2 there exist two canstants a ′ ,b ′ > 0 such that a ′ 〈f,f〉≤ 〈s 1 2 λ f,s 1 2 λ f〉≤ b ′ 〈f,f〉, ∀f ∈h. so a ′ 〈f,f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ b ′ 〈f,f〉, ∀f ∈h. hence λ is a g−fusion frame for h. � theorem 3.3. let λ = (wj, λj,vj)j∈j be a g−fusion frame for h with frame bounds a and b. if γ = (wj, γj,vj)j∈j is g−fusion bessel sequence with bound m < a, then (wj, λj + γj,vj)j∈j is g−fusion frame for h. proof. let f ∈h, we have || ∑ j∈j v2j〈(λj + γj)pwjf, (λj + γj)pwjf〉|| 1 2 = ||(vj(λj + γj)pwjf)j∈j||. ≤ ||(vjλjpwjf)j∈j|| + ||(vjγjpwjf)j∈j|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 + || ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 ≤ √ b||f|| + √ m||f|| ≤ ( √ b + √ m)||f||. on the other hand, for each f ∈h || ∑ j∈j v2j〈(λj + γj)pwjf, (λj + γj)pwjf〉|| 1 2 = ||(vj(λj + γj)pwjf)j∈j||. ≥ ||(vjλjpwjf)j∈j||− ||(vjγjpwjf)j∈j|| ≥ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 −|| ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 ≥ √ a||f||− √ m||f|| ≥ ( √ a− √ m)||f||. so, ( √ a− √ m)2||f||2 ≤ || ∑ j∈j v2j〈(λj + γj)pwjf, (λj + γj)pwjf〉|| ≤ ( √ b + √ m)2||f||2, ∀f ∈h. hence (wj, λj + γj,vj)j∈j is g−fusion frame for h. � int. j. anal. appl. 19 (6) (2021) 854 theorem 3.4. let (wj, λj,vj)j∈j be a g−fusion frame for h with frame bounds a and b. and γj ∈ end∗a(h,kj), ∀j ∈ j. then the following statements are equivalent: (1) (wj, γj,vj)j∈j is g−fusion frame for h. (2) there exist a canstant m such that ∀f ∈h we have: || ∑ j∈j v2j〈(λj − γj)pwjf, (λj − γj)pwjf〉|| ≤ m min(|| ∑ j∈j v2j〈λjpwjf, λjpwjf〉||, || ∑ j∈j v2j〈γjpwjf, γjpwjf〉||). proof. (1) =⇒ (2) let (wj, γj,vj)j∈j be a g−fusion frame for h, with frame bounds c and d, then for any f ∈h, we have || ∑ j∈j v2j〈(λj − γj)pwjf,(λj − γj)pwjf〉|| 1 2 = ||(vj(λj − γj)pwjf)j∈j|| ≤ ||(vjλjpwjf)j∈j|| + ||(vjγjpwjf)j∈j|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 + || ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 + √ d||f|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 + √ d √ a || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 ≤ (1 + √ d a )|| ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 . similary, for each f ∈h, we can obtain || ∑ j∈j v2j〈(λj − γj)pwjf, (λj − γj)pwjf〉|| 1 2 ≤ (1 + √ b c )|| ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 . we put m = min((1 + √ b c )2, (1 + √ d a )2). (2) =⇒ (1) we have for each f ∈h √ a||f|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 ≤ ||(vjλjpwjf)j∈j|| ≤ ||(vj(λj − γj)pwjf)j∈j|| + ||(vjγjpwjf)j∈j|| ≤ √ m|| ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 + || ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 ≤ ( √ m + 1)|| ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 . int. j. anal. appl. 19 (6) (2021) 855 and we have ∀f ∈h || ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| 1 2 = ||(vjγjpwjf)j∈j|| ≤ ||(vjλjpwjf)j∈j|| + ||(vj(λj − γj)f)j∈j|| ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 + √ m|| ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 ≤ ( √ m + 1)|| ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| 1 2 ≤ ( √ m + 1) √ b||f||. so a (1 + √ m)2 ||f||2 ≤ || ∑ j∈j v2j〈γjpwjf, γjpwjf〉|| ≤ ( √ m + 1)2b||f||2, ∀f ∈h. hence (wj, γj,vj)j∈j is g−fusion frame for h. � 4. perturbation of k-g-fusion frames perturbation of frames has been discussed by casazza and christensen. in this section, we present a perturbation of k − g−fusion frames. first we give a characterazation of k − g−fusion frame for hilbert a−modules. theorem 4.1. let k ∈ end∗a(h). suppose that the operator t : h → l 2((kj)j∈j) is given by t(f) = (vjλjpwjf)j∈j, ∀f ∈h, and r(t) is orthogonally complemented. then λ = (wj, λj,vj)j∈j is k−g−fusion frame for h if and only if there exist constants 0 < a ≤ b < ∞ such that (4.1) a||k∗f||2 ≤ || ∑ j∈j v2j〈λjpwjf, λjpwjf〉|| ≤ b||f|| 2, ∀f ∈h. proof. if λ is k − g−fusion frame for h, then the equation (4.1) is satisfies. conversly, we have for each (fj)j∈j ∈ l2((kj)j∈j) and any finite i ⊂ j || ∑ j∈i vjpwj λ ∗ j (fj)|| = sup ||g||=1 ||〈 ∑ j∈i vjpwj λ ∗ jfj,g〉|| = sup ||g||=1 || ∑ j∈i 〈fj,vjλjpwjg〉|| ≤ sup ||g||=1 || ∑ j∈i 〈fj,fj〉|| 1 2 || ∑ j∈i v2j〈λjpwjg〉|| 1 2 ≤ √ b|| ∑ j∈i 〈fj,fj〉|| 1 2 . then ∑ j∈j vjpwj λ ∗ jfj converge unconditionally in h, and we have ∀f ∈h, ∀(fj)j∈j ∈ l 2((kj)j∈j) 〈tf, (fj)j∈j〉 = 〈(vjλjpwjf)j∈j, (fj)j∈j〉 = ∑ j∈j 〈vjλjpwjf,fj〉 = 〈f, ∑ j∈j vjpwj λ ∗ jfj〉 int. j. anal. appl. 19 (6) (2021) 856 so t is adjointable and t∗((fj)j∈j) = ∑ j∈j vjpwj λ ∗ jfj also frome (4.1) we have ||k∗f||2 ≤ 1 a ||tf||2, ∀f ∈h. by lemma1.5, there exist ν > 0 such that kk∗ ≤ νt∗t , then 〈kk∗f,f〉≤ ν〈tf,tf〉, ∀f ∈h. therefore 1 ν 〈k∗f,k∗f〉≤ ∑ j∈j v2j〈λjpwjf, λjpwjf〉, ∀f ∈h. and we have for each f ∈h, 〈tf,tf〉≤ ||t ||2〈f,f〉, then ∑ j∈j v2j〈λjpwjf, λjpwjf〉≤ ||t || 2〈f,f〉, ∀f ∈h. and the proof is completed. � theorem 4.2. let λ = (wj, λj,vj)j∈j be a k − g−fusion frame for h with frame bounds a, b and let γj ∈ end∗a(h,kj), for all j ∈ j. suppose that t : h → l 2((kj)j∈j) define by t(f) = (ujγjpvjf)j∈j, ∀f ∈h. and r(t) is orthogonally complemented, such that for each f ∈h ||((vjλjpwj −ujγjpvj )f)j∈j|| ≤ λ1||(vjλjpwjf)j∈j|| + λ2||(ujγjpvjf)j∈j|| + �||k ∗f||. where 0 < λ1,λ2 < 1 and � > 0 such that � < (1 −λ1) √ a. then γ = (vj, γj,uj)j∈j is a k −g−fusion frame for h. proof. we have for each f ∈h || ∑ j∈j u2j〈γjpvjf, γjpvjf〉|| 1 2 = ||(ujγjpvjf)j|| = ||(ujγjpvjf)j + (vjλjpwjf)j − (vjλjpwjf)j|| ≤ ||((ujγjpvj −vjλjpwj )f)j|| + ||(vjλjpwjf)j|| ≤ (λ1 + 1)||(vjλjpwjf)j|| + λ2||(ujγjpvjf)j|| + �||k ∗f||. so (1 −λ2)||(ujγjpvjf)j|| ≤ (λ1 + 1) √ b||f|| + �||k∗f||. then ||(ujγjpvjf)j|| ≤ (λ1 + 1) √ b||f|| + �||k∗f|| 1 −λ2 ≤ ( (λ1 + 1) √ b + �||k|| 1 −λ2 )||f||. int. j. anal. appl. 19 (6) (2021) 857 hence || ∑ j∈j u2j〈γjpvjf, γjpvjf〉|| ≤ ( (λ1 + 1) √ b + �||k∗|| 1 −λ2 )2||f||2. on the other hand for each f ∈h || ∑ j∈j u2j〈γjpvjf, γjpvjf〉|| 1 2 = ||(ujγjpvjf)j|| = ||((ujγjpvj −vjλjpwj )f)j + (vjλjpwjf)j|| ≥ ||(vjλjpwjf)j||− ||((ujγjpvj −vjλjpwj )f)j|| ≥ (1 −λ1)||(vjλjpwjf)j||−λ2||(ujγjpvjf)j||− �||k ∗f||. hence || ∑ j∈j u2j〈γjpvjf, γjpvjf〉|| ≥ ( (1 −λ1) √ a− � 1 + λ2 )2||k∗f||2. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. j. duffin, a. c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341-366. [2] d. gabor, theory of communications, j. elec. eng. 93 (1946), 429-457. [3] a. khorsavi, b. khorsavi, fusion frames and g-frames in hilbert c∗-modules, int. j. wavelet multiresolution inform. process. 6 (2008), 433-446. [4] a. alijani, m. dehghan, ∗-frames in hilbert c∗modules, u. p. b. sci. bull. ser. a. 2011. [5] m. frank, d. r. larson, a-module frame concept for hilbert c∗-modules, funct. harmonic anal. wavelets contempt. math. 247 (2000), 207-233. [6] i. kaplansky, modules over operator algebras, amer. j. math. 75 (1953), 839-858. [7] lj. arambašić, on frames for countably generated hilbert c∗-modules, proc. amer. math. soc. 135 (2007), 469-478. [8] w. paschke, inner product modules over b∗-algebras, trans. amer. math. soc. (182) (1973), 443-468. [9] m. rossafi, s. kabbaj, ∗-g-frames in tensor products of hilbert c∗-modules, ann. univ. paedagog. crac. stud. math. 17 (2018), 17-25. [10] m. rossafi, s. kabbaj, ∗-k-g-frames in hilbert a-modules, j. linear topol. algebra, 7 (2018), 63-71. [11] m. rossafi, s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra, 8 (2019), 85-95. [12] s. kabbaj, m. rossafi, ∗-operator frame for end∗a(h), wavelet linear algebra, 5 (2) (2018), 1-13. [13] m. rossafi, s. kabbaj, ∗-k-operator frame for end∗a(h), asian-eur. j. math. 13(3) (2020), 2050060. [14] x. fang, j. yu, h. yao, solutions to operator equations on hilbert c∗−modules, linear algebra appl. 431 (2009), 21422153. 1. introduction and preliminaries 2. k-g-fusion frames in hilbert c-modules 3. stability of g-fusion frames in hilbert c-modules 4. perturbation of k-g-fusion frames references int. j. anal. appl. (2022), 20:40 some hermite-hadamard type inequalities via katugampola fractional for pq-convex on the interval-valued coordinates jen chieh lo∗ general education center, national taipei university of technology, taipei, taiwan ∗corresponding author: jclo@mail.ntut.edu.tw abstract. in this paper, we established the hermite-hadamard inequalities via katugampola fractional. meanwhile, interval analysis is a particular case of set-interval analysis. we established the fractional inequalities and these results are an extension of a previous research. 1. introduction the classical hermite-hadamard inequalities such that f ( a + b 2 ) ≤ 1 b−a b∫ a f (t) dt ≤ f (a) + f (b) 2 where f : i →r is a convex function on the closed bounded interval i of r,and a,b ∈ i with a < b. since then, some improved and generalized results of hermite-hadamard inequality on convex function have been explored and study by many authors(e.g. [2], [7], [10–12], [20], [23,24], [27], [29]). on the other hand, interval analysis is a particular case of set-valued analysis which is the study of sets in the spirit of mathematical analysis and general topology. it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. an old example of interval enclosure is archimede’s method which is related to the computation of the circumference of circle. in 1966, the first book related to interval analysis was given by moore who is known as the first user of intervals in computational mathematics. after this book, several scientists started to investigate theory and application of interval arithmetic. received: jun. 22, 2022. 2010 mathematics subject classification. 34a40. key words and phrases. hermite-hadamard inequalities; katugampola fractional; pq-convex; interval-valued. https://doi.org/10.28924/2291-8639-20-2022-40 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-40 2 int. j. anal. appl. (2022), 20:40 nowadays, because of its application, interval analysis is a useful tool in various areas related to uncertain data. we can see applications in computer graphics, experimental and computational physics, error analysis, robotics and many others. 2. interval calculus a real valued interval x is bounded, closed subset of r and is defined by x = [ x,x ] = { t ∈r : x ≤ t ≤ x } where x,x ∈ r and x ≤ x. the number x and x are called the left and right endpoints of interval x, respectively. when x = x = a, the interval x is said to be degenerate and we use the form x = a = [a,a] . also we call x positive if x> 0 or negative if x < 0. the set of all closed intervals of r, the sets of all closed positive intervals of r and closed negative intervals of r is denoted by ri,r+i and r − i , respectively. the pompeiu-hausdorff distance between the intervals x and y is defined by d (x,y ) = d ([ x,x ] , [ y ,y ]) = max { |x −y | , ∣∣x −y ∣∣} . it is known that (ri,d) is a complete metric space. now, we give the definitions of basic interval arithmetic operations for the intervals x and y as follows: x + y = [ x + y ,x + y ] , x −y = [ x −y ,x −y ] , x ·y = [min s, max s] where s = { xy ,xy ,xy ,xy } , x/y = [min t, max t ] where t = { x/y ,x/y ,x/y ,x/y } and 0 /∈ y. scalar multiplication of the interval x is defined by λx = λ [ x,x ] =   [ λx,λx ] , λ > 0, 0, λ = 0,[ λx,λx ] , λ < 0, where λ ∈r. the opposits of the interval x is −x := (−1) x = [ −x,−x ] , where λ = −1. the subtraction is given by int. j. anal. appl. (2022), 20:40 3 x −y = x + (−y ) = [ x −y ,x −y ] . in general, −x is not additive inverse for x, i.e. x −x 6= 0. use of monotonic functions f (x) = [ f (x) ,f ( x )] . the definitons of operations lead to a number of algebraic properties which allows ri to be quasilinear space. they can be listed as follows (1)(associativity of addition) (x + y ) + z = x + (y + z) for all x,y,z ∈ri, (2)(additivity elemant) x + 0 = 0 + x = x for all x ∈ri, (3)(commutativity of addition) x + y = y + x for all x,y ∈ri, (4)(cancellation law) x + z = y + z =⇒ x = y for all x,y,z ∈ri, (5)(associativity of multiplication) (x ·y ) ·z = x · (y ·z) for all x,y,z ∈ri, (6)(commutativity of multiplication) x ·y = y ·x for all x,y ∈ri, (7)(unity element) x · 1 = 1 ·x for all x ∈ri, (8)(associativity law) λ (µx) = (λµ) x for all x ∈ri, and for all λ,µ ∈r, (9)(first distributiviyu law) λ (x + y ) = λx + λy for all x,y ∈ri, and for all λ ∈r, (10)(second distributiviyu law) (λ + µ) x = λx + µx for all x ∈ri, and for all λ,µ ∈r. but, this law holds in certain cases. if y ·z > 0, then x ·y + z = x ·y + x ·z. what’s more, one of the set property is the inclusion ⊆ that is given by x ⊆ y ⇐⇒ y ≤ x and x ≤ y . considering together with arthmetic operations and inclusion, one has the following property which is called inclusion isotone of interval operations: let � be the addition, multiplication, subtraction or division. if x,y,z and t areintervals such that x ⊆ y and z ⊆ t, then the following relation is valid x �z ⊆ y �t. 3. intgral of interval-valued functions in this section, the notion of integral is mentioned for interval-valued functions. before the definition of integral, the necessary concepts will be given as the following: a function f is said to be an interval-valued function of t on [a,b] , if it assigns a nonempty interval to each t ∈ [a,b] , 4 int. j. anal. appl. (2022), 20:40 f (t) = [ f (t) ,f (t) ] . a partition of [a,b] is any finite ordered subset p having the form: p : a = t0 < t1 < ... < tn = b. the mesh of a partition p defined by mesh (p ) = max{ti − ti−1 : i = 1, 2, ...,n} . we denoted by p ([a,b]) the set of all partition of [a,b] . let p (δ, [a,b]) be the set of all p ∈ p ([a,b]) such that mesh (p ) < δ. choose an arbitrary point ξi in interval [ti−1,ti ] , (i = 1, 2, ...,n) and let us define the sum s (f,p,δ) = n∑ i=1 f (ξi ) [ti − ti−1] , where f : [a,b] →ri. we call s (f,p,δ) a riemann sum of f corresponding to p ∈ p (δ, [a,b]) . definition 3.1 a function f : [a,b] →ri is called interval riemann intrgrable ( (ir)-integrable) on [a,b] , if there exists a ∈ri such that, for each � > 0, there exists δ > 0 such that d (s (f,p,δ) ,a) < � for every riemann sum s of f corresponding to each p ∈ p (δ, [a,b]) and independent from choice of ξi ∈ [ti−1,ti ] for all 1 ≤ i ≤ n. in this case, a is called the (ir)-integral of f on [a,b] and is denoted by a = (ir) b∫ a f (t) dt. the collection of all functions that are (ir)-integrable on [a,b] will be denoted by ir([a,b]). the following theorem gives relation between (ir)−integrable and riemann integrable (r)integrable. theorem 3.2 let f : [a,b] → ri be an interval-valued function such that f (t) = [ f (t) ,f (t) ] . f ∈ ir([a,b]) if and only if f (t) ,f (t) ∈ r([a,b]) and (ir) b∫ a f (t) dt =  (r) b∫ a f (t) dt, (r) b∫ a f (t) dt   , where r([a,b]) denoted the all r-integrable functions. int. j. anal. appl. (2022), 20:40 5 it is seen easily that, if f (t) ⊆ g (t) for all t ∈ [a,b] , then (ir) b∫ a f (t) dt ⊆ (ir) b∫ a g (t) dt. furthermore, if {ti−1,ti} m i=1 is a δ-fine p1 of [a,b] and if { sj−1,sj }n j=1 is a δ-fine p2 of [c,d] , then retangles 4i,j = [ti−1,ti ] × [ sj−1,sj ] are the partition of retangle 4 = [a,b] × [c,d] and the point ( ξi,ηj ) are inside the retangles [ti−1,ti ] × [ sj−1,sj ] . and we denote the set of all δ-fine partition p of 4 with p1 ×p2, where p1 ∈ p (δ, [a,b]) and p2 ∈ p (δ, [c,d]) . let 4ai,j be the area retangle 4i,j, where 1 ≤ i ≤ m, 1 ≤ j ≤ n, choose arbitrary ( ξi,ηj ) and get s (f,p,δ,4) = m∑ i=1 n∑ j=1 f ( ξi,ηj ) 4ai,j. definition 3.3 a function f : 4 → ri is called interval double riemann integrable ( (id)-integrable) on 4 = [a,b] × [c,d] with the id-integral i = (id) ∫∫ 4 f (t,s) da, if there exists i ∈ri such that, for each � > 0, there exists δ > 0 such that d (s (f,p,δ,4) , i) < � for each p ∈ p (δ,4) . we denote by ir(4) the set of all id-integrable function on 4, and by r([a,b]), ir([a,b]), the set of all r-integrable and ir-integrable functions on [a,b] ,respectively. theorem 3.4 let 4 = [a,b] × [c,d] . if f : 4→ri is id-integrable on 4, then we have (id) ∫∫ 4 f (t,s) da = (ir) ∫ b a (ir) ∫ d c f (s,t) dsdt. in [21], sadowska obtained the following hermite-hadamard inequality for interval-valued functions: theorem 3.5 let f : [a,b] → r+ i be an interval-valued function such that f (t) = [ f (t) ,f (t) ] and f ∈ ir([a,b]). then f (a) + f (b) 2 ⊆ 1 b−a (ir) b∫ a f (t) dt ⊆ f ( a + b 2 ) . 6 int. j. anal. appl. (2022), 20:40 4. fractional integrals in [2], katugampola introduced a new fractional which generalizes the riemann-liouville and the hadamard fractional integrals into a single form as follow. definition 4.1 let [a,b] ⊂r be a finite interval. then, the leftand right-side katugampola fractional integralsod order α > 0 of f ∈ xpc (a,b) are defined by ρiαa+f (x) = ρ1−α γ (α) ∫ x a tρ−1 (xρ − tρ)1−α f (t) dt and ρiαb−f (x) = ρ1−α γ (α) ∫ b x tρ−1 (tρ −xρ)1−α f (t) dt where a < x < b and ρ > 0, if the integral exists. theorem 4.2 let α > 0 and ρ > 0. then for x > a, 1. lim ρ→1 ρiα a+ f (x) = jα a+ f (x) , 2. lim ρ→0+ ρiα a+ f (x) = hα a+ f (x) . similar results also hold for right-sided operators. theorem 4.3 let α > 0 and ρ > 0. let f : [aρ,bρ] →r be a positive function with 0 ≤ a ≤ b and f ∈ xpc (a,b) . if f is also a convex function on [a,b] , then the following inequalities hold: f ( aρ + bρ 2 , cρ + dρ 2 ) ≤ ραγ (α + 1) 2 (bρ −aρ)α [ ρiαa+f (b ρ) +ρ iαb−f (a ρ) ] ≤ f (aρ) + f (bρ) 2 where the fractional integral are considered for the function f (xρ) and evaluated at a and b, respectively. in [28], yaldiz established the new definitions and theorem related katugampola fractional integrals for two variables functions: definition 4.4 let f ∈ l1 ([aρ,bρ] × [cρ,dρ]) . the katugampola fractional integrals ρi α,β a+,c+ f (x,y) , ρi α,β a+,d− f (x,y) ,ρ i α,β b−,c+ f (x,y) and ρiα,β b−,d− f (x,y) of α,β > 0 are defined by ρi α,β a+,c+ f (x,y) : = ρ1−α γ (α) ρ1−β γ (β) ∫ x a ∫ y c tρ−1sρ−1 (xρ − tρ)1−α (yρ − sρ)1−β f (t,s) dsdt, x > a,y > c, ρi α,β a+,d− f (x,y) : = ρ1−α γ (α) ρ1−β γ (β) ∫ x a ∫ d y tρ−1sρ−1 (xρ − tρ)1−α (sρ −yρ)1−β f (t,s) dsdt, x > a,y < d, int. j. anal. appl. (2022), 20:40 7 ρi α,β b−,c+ f (x,y) : = ρ1−α γ (α) ρ1−β γ (β) ∫ b x ∫ y c tρ−1sρ−1 (tρ −xρ)1−α (yρ − sρ)1−β f (t,s) dsdt, x < b,y > c, ρi α,β b−,d− f (x,y) : = ρ1−α γ (α) ρ1−β γ (β) ∫ b x ∫ d y tρ−1sρ−1 (tρ −xρ)1−α (sρ −yρ)1−β f (t,s) dsdt, x < b,y < d, with a < x < b and c < y < d with ρ > 0. similarly, we introduce the following integrals: ρiαa+f ( x, c + d 2 ) := ρ1−α γ (α) ∫ x a tρ−1 (xρ − tρ)1−α f ( t, c + d 2 ) dt, x > a, ρiαb−f ( x, c + d 2 ) := ρ1−α γ (α) ∫ b x tρ−1 (tρ −xρ)1−α f ( t, c + d 2 ) dt, x < b, ρi β c+ f ( a + b 2 ,y ) := ρ1−β γ (β) ∫ y c sρ−1 (yρ − sρ)1−β f ( a + b 2 ,s ) ds, y > c, and ρiαd−f ( a + b 2 ,y ) := ρ1−β γ (β) ∫ x a sρ−1 (sρ −yρ)1−β f ( a + b 2 ,s ) ds, y < d. theorem 4.5 let α,β > 0 and ρ > 0. let f : 4ρ ⊂r2 →r be a coordinated convex on 4ρ := [aρ,bρ]×[cρ,dρ] in r2 with 0 ≤ a ≤ b, 0 ≤ c ≤ d and f ∈ l1 (4ρ) . then the following inequalities hold: f ( aρ + bρ 2 , cρ + dρ 2 ) ≤ ρα+βγ (α + 1) γ (β + 1) 4 (bρ −aρ)α (dρ −cρ)β × [ ρi α,β a+,c+ f (b,d) +ρ i α,β a+,d− f (b,c) +ρ i α,β b−,c+ f (a,d) +ρ i α,β b−,d− f (a,c) ] ≤ f (aρ,cρ) + f (aρ,dρ) + f (bρ,cρ) + f (bρ,dρ) 4 with a < x < b and c < y < d. definition 4.6 let i ⊂ (0,∞) be a real interval and p ∈ r\{0} . a function f : i → r is said to be a p-convex function, if f ( [txp + (1 − t) yp] 1 p ) ≤ tf (x) + (1 − t) f (y) for all x,y ∈ i and t ∈ [0, 1] . if the inequality is reserved, then f is said to be p-concave. in [5], fang and shi established the following inequlaity theorem 4.7 8 int. j. anal. appl. (2022), 20:40 let f : i →r be a p-convex function and a,b ∈ i with a < b. if f ∈ l [a,b] , then we have f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f (x) x1−p dx ≤ f (a) + f (b) 2 . in [27], toplu et al. established the following inequality theorem 4.8 let f : i → r be a p-convex function, p > 0, α > 0 and a,b ∈ i with a < b. if f ∈ l [a,b] , then the following inequality for fractional integrals holds: f ([ ap + bp 2 ]1 p ) ≤ pαγ (α + 1) bp −ap [ piαa+f (b) + p iαb−f (a) ] ≤ f (a) + f (b) 2 . in this paper, we can give a different version of the definition of the pq-convex function as below. definition 4.9 let i ⊂ (0,∞) ×(0,∞)be a real interval and p,q ∈ r\{0} . a function f : i → r is said to be a pq-convex function, if f ( [txp + (1 − t) yp] 1 p , [λzq + (1 −λ) wq] 1 q ) ≤ tλf (x,z) + t (1 −λ) f (x,w) + (1 − t) λf (y,z) + (1 − t) (1 −λ) f (y,w) for all (x,z) , (x,w) , (y,z) , (y,w) ∈ i and t,λ ∈ [0, 1] . if the inequality is reserved, then f is said to be pq-concave. we recall the following special functions and inequalities. (1)the gamma function: the gamma γ function is defined by γ (z) = γ (α) = ∫ ∞ 0 e−ttα−1dt for all complex numbers z with re(z) > 0, respectively. the gamma function is a natural extension of the factorial from integers n to real (and complex) numbers z. (2)the beta function: β (x,y) = γ (x) γ (y) γ (x + y) = ∫ 1 0 tx−1 (1 − t)y−1dt, x,y > 0. (3)the hypergeometric function: 2f1 (a,b; c,z) = 1 β (b,c −b) ∫ 1 0 tb−1 (1 − t)c−b−1 (1 −zt)−a dt, c > b > 0, |z| < 1. int. j. anal. appl. (2022), 20:40 9 5. main result theorem 5.1 let f : i × i → r be an interval-valued pq-convex function such that f (t) = [ f (t) , f (t) ] and the order p,q > 0, α,β > 0 and a,b,c,d ∈ i with a < b and c < d. if f ∈ id([a,b]×[c,d]), then the following inequality for fractional integrals holds: f ([ ap + bp 2 ]1 p , [ cq + dq 2 ]1 q ) ⊇ pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β × [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] ⊇ f (a,c) + f (a,d) + f (b,c) + f (b,d) 4 . proof : since f is pq-convex function on [a,b] × [c,d] , we have for all (x,z) , (x,w) , (y,z) , (y,w) ∈ [a,b] × [c,d] (with t,λ = 1 2 ) f ([ xp + yp 2 ]1 p , [ zq + wq 2 ]1 q ) ⊇ f (x,z) + f (x,w) + f (y,z) + f (y,w) 4 . by choosing x = [tap + (1 − t) bp] 1 p ,y = [(1 − t) ap + tbp] 1 p ,z = [λcq + (1 −λ) dq] 1 q and w = [(1 −λ) cq + λdq] 1 q , then we get f ([ ap + bp 2 ]1 p , [ cq + dq 2 ]1 q ) ⊇ 1 4 [ f ( [tap + (1 − t) bp] 1 p , [λcq + (1 −λ) dq] 1 q ) + f ( [tap + (1 − t) bp] 1 p , [(1 −λ) cq + λdq] 1 q ) +f ( [(1 − t) ap + tbp] 1 p , [λcq + (1 −λ) dq] 1 q ) + f ( [(1 − t) ap + tbp] 1 p , [(1 −λ) cq + λdq] 1 q )] . multiplying both sides of the inequality by tα−1λβ−1 and then integrating the resulting inequality with respect to t over [0, 1] and with respect to λ over [0, 1] , then we obtain, 4 αβ f ([ ap + bp 2 ]1 p , [ cq + dq 2 ]1 q ) ⊇ (id) ∫ 1 0 ∫ 1 0 [ f ( [tap + (1 − t) bp] 1 p , [λcq + (1 −λ) dq] 1 q ) + f ( [tap + (1 − t) bp] 1 p , [(1 −λ) cq + λdq] 1 q ) +f ( [(1 − t) ap + tbp] 1 p , [λcq + (1 −λ) dq] 1 q ) + f ( [(1 − t) ap + tbp] 1 p , [(1 −λ) cq + λdq] 1 q )] dtdλ 10 int. j. anal. appl. (2022), 20:40 ⊇ (id) ∫ d c ∫ b a ( bp −xp bp −ap )α−1( dq −yq dq −cq )β−1 f (x,y) pxp−1 bp −ap qyq−1 dq −cq dxdy + (id) ∫ d c ∫ b a ( bp −xp bp −ap )α−1( yq −cq dq −cq )β−1 f (x,y) pxp−1 bp −ap qyq−1 dq −cq dxdy + (id) ∫ d c ∫ b a ( xp −ap bp −ap )α−1( dq −yq dq −cq )β−1 f (x,y) pxp−1 bp −ap qyq−1 dq −cq dxdy + (id) ∫ d c ∫ b a ( xp −ap bp −ap )α−1( yq −cq dq −cq )β−1 f (x,y) pxp−1 bp −ap qyq−1 dq −cq dxdy = pq (bp −ap)α (dq −cq)β [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] . thus we have f ([ ap + bp 2 ]1 p , [ cq + dq 2 ]1 q ) ⊇ pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β × [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] which completes the proof of the first inequality. for the second inequalitiy, br using pq-convexity of f , we have f ( [tap + (1 − t) bp] 1 p , [λcq + (1 −λ) dq] 1 q ) ⊇ tλf (a,c) + t (1 −λ) f (a,d) + (1 − t) λf (b,c) + (1 − t) (1 −λ) f (b,d) , f ( [(1 − t) ap + tbp] 1 p , [λcq + (1 −λ) dq] 1 q ) ⊇ (1 − t) λf (a,c) + (1 − t) (1 −λ) f (a,d) + tλf (b,c) + t (1 −λ) f (b,d) , f ( [tap + (1 − t) bp] 1 p , [(1 −λ) cq + λdq] 1 q ) ⊇ t (1 −λ) f (a,c) + tλf (a,d) + (1 − t) (1 −λ) f (b,c) + (1 − t) λf (b,d) , and f ( [(1 − t) ap + tbp] 1 p , [(1 −λ) cq + λdq] 1 q ) ⊇ (1 − t) (1 −λ) f (a,c) + (1 − t) λf (a,d) + t (1 −λ) f (b,c) + tλf (b,d) . by adding these inequalities, then, we have int. j. anal. appl. (2022), 20:40 11 f ( [tap + (1 − t) bp] 1 p , [λcq + (1 −λ) dq] 1 q ) + f ( [(1 − t) ap + tbp] 1 p , [λcq + (1 −λ) dq] 1 q ) +f ( [tap + (1 − t) bp] 1 p , [(1 −λ) cq + λdq] 1 q ) + f ( [(1 − t) ap + tbp] 1 p , [(1 −λ) cq + λdq] 1 q ) ⊇ f (a,c) + f (a,d) + f (b,c) + f (b,d) . multiplying both sides of the inequality by tα−1λβ−1 ,α > 0,β > 0 and then integrating the resulting inequality with respect to t over [0, 1] and with respect to λ over [0, 1] , then we similarly obtain, pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β × [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] ⊇ f (a,c) + f (a,d) + f (b,c) + f (b,d) 4 . lemma 5.2 let f : i × i → r be a partial differentiable function with 0 ≤ a < b and 0 ≤ c < d. then the equality holds. kf (α,β,a,b,c,d) = (bp −ap) (dq −cq) 4pq   ∫ 1 0 ∫ 1 0 [(1 − t)α − tα] [ (1 −λ)β −λβ ] ∂2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ   = 1 4 {f (a,c) + f (a,d) + f (b,c) + f (b,d) + pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] − pαγ (α + 1) (bp −ap)α [piαa+f (x,c) + p iαb−f (x,c) + p iαa+f (x,d) + p iαb−f (x,d)] − qβγ (β + 1) (dq −cq)β [ qi β c+ f (a,y) +q i β d− f (a,y) +q i β c+ f (b,y) +q i β d− f (b,y) ]} where mp (a,b,t) = [tap + (1 − t) bp] 1 p and mq (c,d,λ) = [λcq + (1 −λ) dq] 1 q . proof : let i1 = (bp −ap) (dq −cq) 4pq {∫ 1 0 ∫ 1 0 (1 − t)α (1 −λ)β ∂ 2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ } i2 = (bp −ap) (dq −cq) 4pq {∫ 1 0 ∫ 1 0 (1 − t)α λβ ∂ 2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ } 12 int. j. anal. appl. (2022), 20:40 i3 = (bp −ap) (dq −cq) 4pq {∫ 1 0 ∫ 1 0 tα (1 −λ)β ∂ 2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ } and i4 = (bp −ap) (dq −cq) 4pq {∫ 1 0 ∫ 1 0 tαλβ ∂ 2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ } . by using integrating by part, we have, i1 = 1 4 { f (b,d) − βq (dq −cq) ∫ d c yq−1 (yq −cq)1−β f (b,y) dy − αp (bp −ap) ∫ b a xp−1 (xp −ap)1−α f (x,d) dx + αp (bp −ap) βq (dq −cq) ∫ d c ∫ b a xp−1 (xp −ap)1−α yq−1 (yq −cq)1−β f (x,y) dxdy } and similarly we get i2 = −1 4 { f (b,c) − βq (dq −cq) ∫ d c yq−1 (dq −yq)1−β f (b,y) dy − αp (bp −ap) ∫ b a xp−1 (xp −ap)1−α f (x,c) dx + αp (bp −ap) βq (dq −cq) ∫ d c ∫ b a xp−1 (xp −ap)1−α yq−1 (dq −yq)1−β f (x,y) dxdy } i3 = −1 4 { f (a,d) − βq (dq −cq) ∫ d c yq−1 (yq −cq)1−β f (b,y) dy − αp (bp −ap) ∫ b a xp−1 (bp −xp)1−α f (x,d) dx + αp (bp −ap) βq (dq −cq) ∫ d c ∫ b a xp−1 (bp −xp)1−α yq−1 (yq −cq)1−β f (x,y) dxdy } and i4 = 1 4 { f (a,c) − βq (dq −cq) ∫ d c yq−1 (dq −yq)1−β f (b,y) dy − αp (bp −ap) ∫ b a xp−1 (bp −xp)1−α f (x,d) dx + αp (bp −ap) βq (dq −cq) ∫ d c ∫ b a xp−1 (bp −xp)1−α yq−1 (dq −yq)1−β f (x,y) dxdy } . so that we combine i1 − i2 − i3 + i4, we will get the equality. theorem 5.3 let f : i × i →r be an interval-valued pq-convex function such that f (t) = [ f (t) , f (t) ] and the order p,q > 0, α,β > 0 and a,b,c,d ∈ i with a < b and c < d. if f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂λf ∣∣∣m is pq −convex on [a,b] × [c,d] for m ≥ 1, then the following inequality for fractional integrals holds: int. j. anal. appl. (2022), 20:40 13 ∣∣∣∣14 {f (a,c) + f (a,d) + f (b,c) + f (b,d) + pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] − pαγ (α + 1) (bp −ap)α [piαa+f (x,c) + p iαb−f (x,c) + p iαa+f (x,d) + p iαb−f (x,d)] − qβγ (β + 1) (dq −cq)β [ qi β c+ f (a,y) +q i β d− f (a,y) +q i β c+ f (b,y) +q i β d− f (b,y) ]∣∣∣∣ ⊇ (bp −ap) (dq −cq) 4pq m 1− 1 m 1 (α,β) × { m2 (α,β) ∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m + m3 (α,β) ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m +m4 (α,β) ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m + m5 (α,β) ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m } 1 m . proof : from lemma by using the property of the modulus, the power mean inequality and the pq-convexity of ∣∣∣ ∂2∂t∂λf ∣∣∣m , then we have∣∣∣∣∣∣(bp −ap)(dq −cq)4pq  (id) ∫ 1 0 ∫ 1 0 [(1− t)α − tα] [ (1−λ)β −λβ ] ∂2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   ∣∣∣∣∣∣ ⊇ (bp −ap)(dq −cq) 4pq ×  (id) ∫ 1 0 ∫ 1 0 ∣∣∣[(1− t)α − tα][(1−λ)β −λβ]∣∣∣ · ∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ))∣∣∣ [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   ⊇ (bp −ap)(dq −cq) 4pq  (id) ∫ 1 0 ∫ 1 0 ∣∣∣[(1− t)α − tα][(1−λ)β −λβ]∣∣∣ [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   1− 1 m ×  (id) ∫ 1 0 ∫ 1 0 ∣∣∣[(1− t)α − tα][(1−λ)β −λβ]∣∣∣ · ∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ))∣∣∣m [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   1 m ⊇ (bp −ap)(dq −cq) 4pq  (id) ∫ 1 0 ∫ 1 0 ∣∣∣[(1− t)α − tα][(1−λ)β −λβ]∣∣∣ [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   1− 1 m ×  (id) ∫ 1 0 ∫ 1 0 ∣∣∣[(1− t)α − tα][(1−λ)β −λβ]∣∣∣ [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q × [ tλ ∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m + t (1−λ) ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m +(1− t)λ ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m +(1− t)(1−λ) ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m dtdλ } 1 m = (bp −ap)(dq −cq) 4pq m 1−1 q 1 (α,β)× { m2 (α,β) ∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m +m3 (α,β) ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m +m4 (α,β) ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m +m5 (α,β) ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m } 1 m , 14 int. j. anal. appl. (2022), 20:40 where by simple computation, we obtain, m1 (α,β) = ∫ 1 0 ∫ 1 0 [(1 − t)α − tα] [ (1 −λ)β −λβ ] [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ = ∫ 1 0 [(1 − t)α − tα] [tap + (1 − t) bp]1− 1 p dt ∫ 1 0 [ (1 −λ)β −λβ ] [λcq + (1 −λ) dq]1− 1 q dλ = { b1−p α + 1 [ 2f1 ( 1 − 1 p , 1; α + 2, 1 − ap bp ) +2 f1 ( 1 − 1 p ,α + 1; α + 2, 1 − ap bp )]} × { d1−q β + 1 [ 2f1 ( 1 − 1 q , 1; β + 2, 1 − cq dq ) +2 f1 ( 1 − 1 q ,β + 1; β + 2, 1 − cq dq )]} m2 (α,β) = ∫ 1 0 ∫ 1 0 [(1 − t)α − tα] [ (1 −λ)β −λβ ] [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q tλdtdλ = ∫ 1 0 [(1 − t)α − tα] [tap + (1 − t) bp]1− 1 p tdt ∫ 1 0 [ (1 −λ)β −λβ ] [λcq + (1 −λ) dq]1− 1 q λdλ = { b1−p α + 2 [ 1 α + 12 f1 ( 1 − 1 p , 2; α + 3, 1 − ap bp ) +2 f1 ( 1 − 1 p ,α + 2; α + 3, 1 − ap bp )]} × { d1−q β + 2 [ 1 β + 12 f1 ( 1 − 1 q , 2; β + 3, 1 − cq dq ) +2 f1 ( 1 − 1 q ,β + 2; β + 3, 1 − cq dq )]} m3 (α,β) = ∫ 1 0 ∫ 1 0 [(1 − t)α − tα] [ (1 −λ)β −λβ ] [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q t (1 −λ) dtdλ = ∫ 1 0 [(1 − t)α − tα] [tap + (1 − t) bp]1− 1 p tdt ∫ 1 0 [ (1 −λ)β −λβ ] [λcq + (1 −λ) dq]1− 1 q (1 −λ) dλ = { b1−p α + 2 [ 1 α + 12 f1 ( 1 − 1 p , 2; α + 3, 1 − ap bp ) +2 f1 ( 1 − 1 p ,α + 2; α + 3, 1 − ap bp )]} × { d1−q β + 1 [ 2f1 ( 1 − 1 q , 1; β + 3, 1 − cq dq ) + 1 β + 12 f1 ( 1 − 1 q ,β + 1; β + 3, 1 − cq dq )]} m4 (α,β) = ∫ 1 0 ∫ 1 0 [(1 − t)α − tα] [ (1 −λ)β −λβ ] [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q (1 − t) λdtdλ = ∫ 1 0 [(1 − t)α − tα] [tap + (1 − t) bp]1− 1 p (1 − t) dt ∫ 1 0 [ (1 −λ)β −λβ ] [λcq + (1 −λ) dq]1− 1 q λdλ = { b1−p α + 1 [ 2f1 ( 1 − 1 p , 1; α + 3, 1 − ap bp ) + 1 α + 12 f1 ( 1 − 1 p ,α + 1; α + 3, 1 − ap bp )]} × { d1−q β + 2 [ 1 β + 12 f1 ( 1 − 1 q , 2; β + 3, 1 − cq dq ) +2 f1 ( 1 − 1 q ,β + 2; β + 3, 1 − cq dq )]} int. j. anal. appl. (2022), 20:40 15 m5 (α,β) = ∫ 1 0 ∫ 1 0 [ (1 − t)α − tα ][ (1 −λ)β −λβ ] [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q (1 − t) (1 −λ) dtdλ = ∫ 1 0 [ (1 − t)α − tα ] [tap + (1 − t) bp]1− 1 p (1 − t) dt ∫ 1 0 [ (1 −λ)β −λβ ] [λcq + (1 −λ) dq]1− 1 q (1 −λ) dλ = { b1−p α + 1 [ 2f1 ( 1 − 1 p , 1; α + 3, 1 − ap bp ) + 1 α + 12 f1 ( 1 − 1 p ,α + 1; α + 3, 1 − ap bp )]} × { d1−q β + 1 [ 2f1 ( 1 − 1 q , 1; β + 3, 1 − cq dq ) + 1 β + 12 f1 ( 1 − 1 q ,β + 1; β + 3, 1 − cq dq )]} . theorem 5.4 let f : i × i →r be an interval-valued pq-convex function such that f (t) = [ f (t) , f (t) ] and the order p,q > 0, α,β > 0 and a,b,c,d ∈ i with a < b and c < d. if f ∈ id([a,b]×[c,d]) and ∣∣∣ ∂2∂t∂λf ∣∣∣m is pq −convex on [a,b] × [c,d] for m ≥ 1, then the following inequality for fractional integrals holds:∣∣∣∣14 {f (a,c) + f (a,d) + f (b,c) + f (b,d) + pαqβγ (α + 1) γ (β + 1) (bp −ap)α (dq −cq)β [ p,qi α,β a+,c+ f (b,d) +p,q i α,β a+,d− f (b,c) +p,q i α,β b−,c+ f (a,d) +p,q i α,β b−,d− f (a,c) ] − pαγ (α + 1) (bp −ap)α [piαa+f (x,c) + p iαb−f (x,c) + p iαa+f (x,d) + p iαb−f (x,d)] − qβγ (β + 1) (dq −cq)β [ qi β c+ f (a,y) +q i β d− f (a,y) +q i β c+ f (b,y) +q i β d− f (b,y) ]∣∣∣∣ ⊇ (bp −ap) (dq −cq) 4pq { m 1 n 6 (α,β) + m 1 n 7 (α,β) + m 1 n 8 (α,β) + m 1 n 9 (α,β) } × { 1 4 [∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m ]} 1 m and 1 m + 1 n = 1. proof : from lemma by using the property of the modulus, the hölder inequality and the pq-convexity of∣∣∣ ∂2∂t∂λf ∣∣∣m , then we have∣∣∣∣∣∣(bp −ap)(dq −cq)4pq  (id) ∫ 1 0 ∫ 1 0 [(1− t)α − tα] [ (1−λ)β −λβ ] ∂2 ∂t∂λ f (mp (a,b,t) ,mq (c,d,λ)) [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ   ∣∣∣∣∣∣ ⊇ (bp −ap)(dq −cq) 4pq ×   ( (id) ∫ 1 0 ∫ 1 0 (1− t)αn (1−λ)βn [tap +(1− t)bp]1− 1 p [λcq +(1−λ)dq]1− 1 q dtdλ )1 n × ( (id) ∫ 1 0 ∫ 1 0 ∣∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ)) ∣∣∣∣m dtdλ ) 1 m 16 int. j. anal. appl. (2022), 20:40 + ( (id) ∫ 1 0 ∫ 1 0 tαn (1 −λ)βn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ )1 n × ( (id) ∫ 1 0 ∫ 1 0 ∣∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ)) ∣∣∣∣m dtdλ ) 1 m + ( (id) ∫ 1 0 ∫ 1 0 (1 − t)αn λβn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ )1 n × ( (id) ∫ 1 0 ∫ 1 0 ∣∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ)) ∣∣∣∣m dtdλ ) 1 m + ( (id) ∫ 1 0 ∫ 1 0 tαnλβn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ )1 n × ( (id) ∫ 1 0 ∫ 1 0 ∣∣∣∣ ∂2∂t∂λf (mp (a,b,t) ,mq (c,d,λ)) ∣∣∣∣m dtdλ ) 1 m } ⊇ (bp −ap) (dq −cq) 4pq { m 1 n 6 (α,β) + m 1 n 7 (α,β) + m 1 n 8 (α,β) + m 1 n 9 (α,β) } × { (id) ∫ 1 0 ∫ 1 0 tλ ∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m + t (1 −λ) ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m + (1 − t) λ ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m + (1 − t) (1 −λ) ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m dtdλ } 1 m = (bp −ap) (dq −cq) 4pq { m 1 n 6 (α,β) + m 1 n 7 (α,β) + m 1 n 8 (α,β) + m 1 n 9 (α,β) } × { 1 4 [∣∣∣∣ ∂2∂t∂λf (a,c) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (a,d) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (b,c) ∣∣∣∣m + ∣∣∣∣ ∂2∂t∂λf (b,d) ∣∣∣∣m ]} 1 m , and by calculations of integrals, we obtain, m6 (α,β) = ∫ 1 0 ∫ 1 0 (1 − t)αn (1 −λ)βn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ = (∫ 1 0 (1 − t)αn [tap + (1 − t) bp]1− 1 p dt )(∫ 1 0 (1 −λ)βn [λcq + (1 −λ) dq]1− 1 q dλ ) = b(1−p)n αp + 1 2f1 ( n− n p , 1; αn + 2, 1 − ap bp ) · d(1−q)n βq + 1 2f1 ( n− n q , 1; βn + 2, 1 − cq dq ) m7 (α,β) = ∫ 1 0 ∫ 1 0 tαn (1 −λ)βn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ = (∫ 1 0 tαn [tap + (1 − t) bp]1− 1 p dt )(∫ 1 0 (1 −λ)βn [λcq + (1 −λ) dq]1− 1 q dλ ) = b(1−p)n αp + 1 2f1 ( n− n p ,αn + 1; αn + 2, 1 − ap bp ) · d(1−q)n βq + 1 2f1 ( n− n q , 1; βn + 2, 1 − cq dq ) int. j. anal. appl. (2022), 20:40 17 m8 (α,β) = ∫ 1 0 ∫ 1 0 (1 − t)αn λβn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ = (∫ 1 0 (1 − t)αn [tap + (1 − t) bp]1− 1 p dt )(∫ 1 0 λβn [λcq + (1 −λ) dq]1− 1 q dλ ) = b(1−p)n αp + 1 2f1 ( n− n p ,αn + 1; αn + 2, 1 − ap bp ) · d(1−q)n βq + 1 2f1 ( n− n q ,βn + 1; βn + 2, 1 − cq dq ) m9 (α,β) = ∫ 1 0 ∫ 1 0 tαnλβn [tap + (1 − t) bp]1− 1 p [λcq + (1 −λ) dq]1− 1 q dtdλ = (∫ 1 0 tαn [tap + (1 − t) bp]1− 1 p dt )(∫ 1 0 λβn [λcq + (1 −λ) dq]1− 1 q dλ ) = b(1−p)n αp + 1 2f1 ( n− n p ,αn + 1; αn + 2, 1 − ap bp ) · d(1−q)n βq + 1 2f1 ( n− n q ,βn + 1; βn + 2, 1 − cq dq ) conclusion in this work, the author established hermite-hadamard type inequalities via katugampola fractional integral. furthermore, the author extend the ineqalities on interval-valued coordinated. it is an interesting issue, and many researchers work to generalize the ostrowski’ inequalities, chebyshev type inequalities and opial-type inequalities on fuzzy interval-valued set. we hope to establish the general fractional integrals in their future research. author contributions: the author contributed has read and agreed to the published version of the manuscript. acknowledgment: the author would like to express their sincere to the editor and the anonmous reviewers for their helpful comments and suggestions. funding: the work was supportes by the ministry of science and technology of taiwan (most1102115-m-027-003-my2). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] h.y. budak, c.c. bilişik, a. kashuri and m.a. ali, hermite-hadamard type inequalities for the interval-valued harmonically h-convex functions via fractional integrals, appl. math. e-notes, 21 (2021), 12-32. https://www. emis.de/journals/amen/2021/amen-200121.pdf. https://www.emis.de/journals/amen/2021/amen-200121.pdf https://www.emis.de/journals/amen/2021/amen-200121.pdf 18 int. j. anal. appl. (2022), 20:40 [2] h. chen, u.n. katugampola, hermite–hadamard and hermite–hadamard–fejér type inequalities for generalized fractional integrals, j. math. anal. appl. 446 (2017), 1274–1291. https://doi.org/10.1016/j.jmaa.2016.09. 018. [3] t. m. costa, jensen’s inequality type integral for fuzzy-interval-valued functions, fuzzy sets syst. 327 (2017), 31–47. https://doi.org/10.1016/j.fss.2017.02.001. [4] t.m. costa, h. román-flores, some integral inequalities for fuzzy-interval-valued functions, inform. sci. 420 (2017), 110–125. https://doi.org/10.1016/j.ins.2017.08.055. [5] z.b.fang, r. shi, on the (p, h)-convex function and some integral inequalities, j. inequal. appl. 2014 (2014), 45. https://doi.org/10.1186/1029-242x-2014-45. [6] y. guo, g. ye, d. zhao, w. liu, some integral inequalities for log-h-convex interval-valued functions, ieee access. 7 (2019), 86739–86745. https://doi.org/10.1109/access.2019.2925153. [7] g. hu, h. lei, t. du, some parameterized integral inequalities for p-convex mappings via the right katugampola fractional integrals, aims math. 5 (2020), 1425–1445. https://doi.org/10.3934/math.2020098. [8] m.b. khan, m.a. noor, k.i. noor, y.m. chu, new hermite-hadamard-type inequalities for (h1,h2)-convex fuzzyinterval-valued functions, adv. differ. equ. 2021 (2021), 149. https://doi.org/10.1186/s13662-021-03245-8. [9] h. kara, m.a. ali, h. budak, hermite-hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals, math. meth. appl. sci. 44 (2020), 104–123. https://doi.org/10. 1002/mma.6712. [10] u.n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218 (2011), 860–865. https://doi.org/10.1016/j.amc.2011.03.062. [11] u.n. katugampola, new approach to a generalized fractional derivatives, bull. math. anal. appl. 6 (2014), 1-15. [12] u.n. katugampola, mellin transforms of generalized fractional integrals and derivatives, appl. math. comput. 257 (2015), 566–580. https://doi.org/10.1016/j.amc.2014.12.067. [13] z. li, kamran, m.s. zahoor, h. akhtar, hermite–hadamard and fractional integral inequalities for interval-valued generalized p-convex function, j. math. 2020 (2020), 4606439. https://doi.org/10.1155/2020/4606439. [14] x. liu, g. ye, d. zhao, w. liu, fractional hermite–hadamard type inequalities for interval-valued functions, j inequal appl. 2019 (2019), 266. https://doi.org/10.1186/s13660-019-2217-1. [15] f.-c. mitroi, k. nikodem, s. wąsowicz, hermite–hadamard inequalities for convex set-valued functions, demonstr. math. 46 (2013), 655-662. https://doi.org/10.1515/dema-2013-0483. [16] r.e. moore, interval analysis, prentice-hall, englewood cliff, 1966. [17] r.e. moore, r.b. kearfott, m.j. cloud, introduction to interval analysis, siam, philadelphia, 2009. [18] m. noor aslam, m. awan uzair, k. noor inayat, integral inequalities for two-dimensional pq-convex functions, filomat. 30 (2016), 343–351. https://doi.org/10.2298/fil1602343n. [19] m.a. noor, k.i. noor, s. iftikhar, nonconvex functions and integral inequalities, punjab univ. j. math. 47 (2015), 19-27. [20] m.a. noor, k.i. noor, m.v. mihai, m.u. awan, hermite-hadamard inequalities for differentiable p-convex functions using hypergeometric functions, researchgate. (2015), https://doi.org/10.13140/rg.2.1.2485.0648. [21] r. osuna-gómez, m.d. jiménez-gamero, y. chalco-cano, m.a. rojas-medar, hadamard and jensen inequalities for s-convex fuzzy processes, in: soft methodology and random information systems, springer berlin heidelberg, berlin, heidelberg, 2004: pp. 645–652. https://doi.org/10.1007/978-3-540-44465-7_80. [22] e. sadowska, hadamard inequality and a refinement of jensen inequality for set—valued functions, results. math. 32 (1997), 332–337. https://doi.org/10.1007/bf03322144. [23] e. set, i̇. mumcu, hermite–hadamard-type inequalities for f-convex functions via katugampola fractional integral, math. probl. eng. 2021 (2021), 5549258. https://doi.org/10.1155/2021/5549258. https://doi.org/10.1016/j.jmaa.2016.09.018 https://doi.org/10.1016/j.jmaa.2016.09.018 https://doi.org/10.1016/j.fss.2017.02.001 https://doi.org/10.1016/j.ins.2017.08.055 https://doi.org/10.1186/1029-242x-2014-45 https://doi.org/10.1109/access.2019.2925153 https://doi.org/10.3934/math.2020098 https://doi.org/10.1186/s13662-021-03245-8 https://doi.org/10.1002/mma.6712 https://doi.org/10.1002/mma.6712 https://doi.org/10.1016/j.amc.2011.03.062 https://doi.org/10.1016/j.amc.2014.12.067 https://doi.org/10.1155/2020/4606439 https://doi.org/10.1186/s13660-019-2217-1 https://doi.org/10.1515/dema-2013-0483 https://doi.org/10.2298/fil1602343n https://doi.org/10.13140/rg.2.1.2485.0648 https://doi.org/10.1007/978-3-540-44465-7_80 https://doi.org/10.1007/bf03322144 https://doi.org/10.1155/2021/5549258 int. j. anal. appl. (2022), 20:40 19 [24] e. set and a. karaoğlan, hermite-hadamard and hermite-hadamard-fejér type inequalities for (k,h)-convex function via katugampola fractional integrals, konuralp j. math. 5 (2017), 181-191. https://dergipark.org. tr/en/download/article-file/351071. [25] f. shi, g. ye, d. zhao, w. liu, some fractional hermite–hadamard type inequalities for interval-valued functions, mathematics. 8 (2020), 534. https://doi.org/10.3390/math8040534. [26] f. shi, g. ye, d. zhao, w. liu, some fractional hermite–hadamard-type inequalities for interval-valued coordinated functions, adv. differ. equ. 2021 (2021), 32. https://doi.org/10.1186/s13662-020-03200-z. [27] t. toplu, e. set, i̇. i̇şcan, s. maden, hermite-hadamard type inequalities for p-convex functions via katugampola fractional integrals, facta univ., ser.: math. inform. 34 (2019) 149-164. https://doi.org/10.22190/ fumi1901149t. [28] h. yaldiz, a.o. akdemir, katugampola fractional integrals within the class of convex functions, turk. j. sci. iii (2018), 40-50. [29] y. yu, h. lei, g. hu, t. du, estimates of upper bound for differentiable mappings related to katugampola fractional integrals and p-convex mappings, aims math. 6 (2021), 3525–3545. https://doi.org/10.3934/math.2021210. [30] d. zhao, m.a. ali, g. murtaza, z. zhang, on the hermite–hadamard inequalities for interval-valued coordinated convex functions, adv. differ. equ. 2020 (2020), 570. https://doi.org/10.1186/s13662-020-03028-7. [31] d. zhao, t. an, g. ye, w. liu, new jensen and hermite–hadamard type inequalities for h-convex interval-valued functions, j. inequal. appl. 2018 (2018), 302. https://doi.org/10.1186/s13660-018-1896-3. https://dergipark.org.tr/en/download/article-file/351071 https://dergipark.org.tr/en/download/article-file/351071 https://doi.org/10.3390/math8040534 https://doi.org/10.1186/s13662-020-03200-z https://doi.org/10.22190/fumi1901149t https://doi.org/10.22190/fumi1901149t https://doi.org/10.3934/math.2021210 https://doi.org/10.1186/s13662-020-03028-7 https://doi.org/10.1186/s13660-018-1896-3 1. introduction 2. interval calculus 3. intgral of interval-valued functions 4. fractional integrals 5. main result conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 82-92 http://www.etamaths.com on the banach space techniques in the existence and uniqueness of the fuzzy fractional klein-gordon equation’s solution a. ebadian1, m. shams yousefi2, f. farahrooz1,∗ and m. najand foumani2 abstract. in this paper, we study the existence and uniqueness of the solution of all fuzzy fractional differential equations, which are equivalent to the fuzzy integral equation. we use the banach space techniques in this study. also we will show that the fuzzy fractional klein-gordon equation (ffkge) is equivalent to a fuzzy integral equation. we use parametric form of ffkge with respect to definition and give new homotopy analysis method to obtain the approximate solution of this equation. 1. introduction in many cases of the modeling of real world phenomena, fuzzy initial value problems appear naturally, because information about the behavior of a dynamical system is uncertain. in order to obtain a more adequate model, we have to take into account these uncertainties. on the other hand, fractional calculus found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control, relaxation, processes and so on [1]. the klein-gordon equation, which is denoted by kge in this paper, is nowadays regarded as the relativistic form of the schrödinger equation. it affords appropriate description for spin zero particles. since the solution of the kge is often a complicated problem, use of pure mathematical methods is required. ebaid, applied exp-function method for solving kge [2]. raicher et.al used a novel solution to the kge in the presence of a strong rotating electric field [3]. but discussion on the fuzzy fractional klein-gordon equation (ffkge) has not been done. we consider the ffkge with boundary conditions as follows ∂2γũ(x,t) ∂t2γ = ∂2ũ(x,t) ∂x2 + ũ(x,t) 0 < x < 1 , 0 < t < 1 , 0 < γ ≤ 1 where γ is a parameter describing the order of fractional time derivative and , ũ(x, 0) = k̃(1 + sin x),k̃ = (0.25β,−0.25β), 0 ≤ β ≤ 1 and ũt(x, 0) = 0, 0 < x < 1, where ũ(x,t) : (0, 1) × [0, 1) −→ rf is fuzzy number-valued function and rf is the set of all fuzzy numbers. in this paper we give a new homotopy analysis transform method for solving ffkge. we use the parametric form of the above equation and find the approximate solution of this equation. the existence and uniqueness of the solution and convergence of the proposed method are proved in details. for this purpose we show that the ffkge is equivalent to fuzzy integral equation. the concept of conformable fuzzy fractional derivative will define in this paper. also, we define the fuzzy banach space. since the fixed point theorems in banach spaces are powerful tools to prove existence and uniqueness of solution for integral equations, so in this study, we use fixed point theorem and introduce a contraction operator on a suitable banach space. the paper is organized as follows: in sect. 2.2 we present some concepts and results about the fuzzy number. we explain the fractional transform and it is applied for ffkge, then the equivalency to the received 13th august, 2016; accepted 7th october, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 34a12; 26a33. key words and phrases. fractional calculus; convergence; existence; homotopy analysis method; uniqueness; kleingordon equation. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 82 on the banach space techniques 83 fuzzy integral equation is also proved in sect. 3. the existence and uniqueness of the solution for fuzzy integral equation is discussed in sect. 4, where we define the cf ([0, 1]) and its properties and use the functional analysis methods. in sect. 5 the homotopy analysis transform method is applied to solve this equation. 2. preliminaries we now recall some definitions and symbols needed through the paper. we follow [4] in definitions and notations. definition 2.1. a fuzzy number is a function u : r → [0, 1] satisfying the following properties: a. u is upper semicontinuous on r, b. u(x) = 0 outside of some interval [c,d], c. there are the real numbers a and b with c ≤ a ≤ b ≤ d, such that u is increasing on [c,a], decreasing on [b,d] and u(x) = 1 for each x ∈ [a,b], d. u is fuzzy convex set (that is u(λx + (1 −λ)y) ≥ min{u(x),u(y)}, ∀x,y ∈ r,λ ∈ [0, 1]) the set of all fuzzy numbers is denoted by rf . in the following we introduce a concept that will be very efficient and useful to use and identification of fuzzy numbers. definition 2.2. for any u ∈ rf the α−cut set of u is denoted by [u]α and defined by [u]α = {x ∈ r | u(x) ≥ α}, where 0 ≤ α ≤ 1. the notation, [u]α = [uα,uα]; α ∈ [0, 1] refers to the lower and upper branches on u, in other words uα = min{x | x ∈ uα}, uα = max{x | x ∈ uα} an arbitrary fuzzy number u is represented, in parametric form, by an ordered pair of functions u = (u,u), which define the end points of the α−cuts, satisfying the three conditions: a. u is a bounded non-decreasing left continuous function on (0, 1], and right continuous at 0, b. u is a bounded non-increasing left continuous function on (0, 1], and right continuous at 0, c. u(r) ≤ u(r), 0 ≤ r ≤ 1. for arbitrary u = (u,u), v = (v, v̄) and k ≥ 0, addition (u + v) and multiplication by k as (u + v)(r) = u(r) + v(r), (u + v)(r) = u(r) + v̄(r), ku(r) = ku(r), ku(r) = ku(r),k ≥ 0, and ku(r) = kū(r),ku(r) = ku(r), k < 0 are defined. it is well-known that the addition and multiplication operations of real numbers can be extended to rf . in other words, for any u,v ∈ rf and λ ∈ r, we define uniquely the sum u⊕v and the product λ�u by [u⊕v]α = [u]α ⊕ [v]α, [λ�u]α = λ[u]α, ∀α ∈ [0, 1]. is the hukuhara difference (h-difference), it means that w v = u if and only if u ⊕ v = w for all u,v,w ∈ rf . definition 2.3. for arbitrary fuzzy number u = (u(r),u(r)), v = (v(r),v(r)) the hausdorff distance between these fuzzy numbers given by d : rf ×rf → r+ ∪{0}, d(u,v) = sup r∈[0,1] max{|u(r) −v(r)|, |u(r) − v̄(r)|}, where d is a metric on rf and has the following properties (see [1]). a. d(u⊕w,v ⊕w) = d(u,v),∀u,v,w ∈ rf , b. d(k �u,k �v) = |k|d(u,v),∀k ∈ r,u,v ∈ rf , c. d(u⊕v,w ⊕e) ≤ d(u,w) + d(v,e),∀u,v,w,e ∈ rf , d. (rf ,d) is a complete metric space. 84 a. ebadian, m. shams yousefi, f. farahrooz and m. najand foumani definition 2.4. the function f : t −→ rf is called a fuzzy function, and the α−cut set of f is represented by f(t; α) = [f(t; α),f(t; α)]; ∀α ∈ [0, 1], where f(t; α) = f(t) α , f(t; α) = f(t) α . a fuzzy function may have fuzzy domain and fuzzy range. so the function f : rf −→ rf is also a fuzzy function. definition 2.5. let f : r → rf be a fuzzy function. if for an arbitrary fixed number t0 ∈ r and ε > 0, exists δ > 0 such that |t− t0| < δ =⇒ d(f(t),f(t0)) < ε t ∈ rf , then f is said to be continuous at t0. definition 2.6. the fuzzy function f : r −→ rf is called to be fuzzy bounded if there exists m > 0 such that ‖ f ‖f.u:=sup d(f(u), 0̂) ≤ m, (u ∈ r). proposition 2.1. let f : [a,b] ⊆ r −→ rf be a fuzzy continuous function. then it is fuzzy bounded. in the following we consider the concept of integral of a fuzzy function. the integration on the α-cut of fuzzy function is also defined. definition 2.7. let f : [a,b] −→ rf be a fuzzy function. for each partition p = {x1,x2, · · · ,xm} of [a,b] and for arbitrary xi−1 ≤ ξi ≤ xi, 2 ≤ i ≤ m, let rp = m∑ i=2 f(ξi)(xi −xi−1). the define integral of f(x) over [a,b] is ,∫ b a f(x; y) = lim rp, max |xi −xi−1| −→ 0 provided that this limit exists in metric d. if the function f is continuous in the metric d, its definite integral exists [5]. furthermore: ∫ b a f(x; α) = ∫ b a f(x; α), and ∫ b a f(x; α) = ∫ b a f̄(x; α) more details about the properties of the fuzzy integral are given in [5]. now, we want to introduce a new definition of fuzzy fractional derivative as [8]: definition 2.8. let f : [a,b] −→ rf . the fuzzy γ-fractional integral of fuzzy-valued function f is defined as follows: (iγf)(x) = ∫ x a f(t) t1−γ dt, x > a , 0 < γ < 1. let us consider the α−cut representation of fuzzy-valued function f is f(x; α) = [f(x; α), f̄(x; α)], for 0 ≤ α ≤ 1, then we indicate the fuzzy γ − fractional integral of fuzzy-valued function f based on its lower and upper functions as follows: theorem 2.1. let f : [a,b] −→ rf . the fuzzy γ−fractional integral of fuzzy-valued function f can be expressed as follows: (iγf)(x; α) = [(iγf)(x; α), (iγf̄)(x; α)], 0 < α < 1 where (iγf)(x; α) = ∫ x a f(t; α) t1−γ dt, (iγf̄)(x; α) = ∫ x a f̄(t; α) t1−γ dt. on the banach space techniques 85 theorem 2.2. for γ ∈ [0, 1] and f : [a,b] −→ rf d γ t f(t) = lim ε→0 f(t + εt1−γ) f(t) ε . for t > 0, γ ∈ (0, 1); dγt f(t) is called the conformable fuzzy fractional derivative of f of order γ [9, 10]. using this kind of fractional derivative and some useful formulas can convert differential equations into integer-order differential equations. some properties for the suggested conformable fuzzy fractional derivative given in [8] are as follows: d γ t (t η) = ηtη−γ,η ∈ r, (2.1) d γ t (f(t)g(t)) = g(t)d γ t f(t) ⊕f(t)d γ t g(t). (2.2) d γ t f[g(t)] = f ′ g[g(t)]d γ t g(t) = d γ gf[g(t)](g ′(t))γ. (2.3) 3. the fractional transform in this section, we reduce ffkge to an ordinary differential equation, then we show that this equation is equivalent to fuzzy integral equation. we consider the ffkge with boundary conditions as follows ∂2γũ(x,t) ∂t2γ = ∂2ũ(x,t) ∂x2 + ũ(x,t) 0 < x < 1 , 0 < t < 1 , 0 < γ ≤ 1. (3.1) now, we introduce the following transformations : ū(x,t) = ū(ξ), ξ = ax + btγ γ a,b > 0, a + b γ < 1 so, we can say that 0< ξ <1 by using (1) and (3) and by substituting into equation (3.1) it is derived that b2u ′′ −a2u ′′ −u = 0. (3.2) now, we show that this fuzzy equation is equivalent to fuzzy integral equation as form: u ′′ (ξ) = f̄(ξ) =⇒ u ′ (ξ) = u ′ (0) + ∫ ξ 0 f̄(z)dz =⇒ u(ξ) = u(0) + u ′ (0)ξ + ∫ ξ 0 ∫ ξ 0 f̄(z)dzdξ, on the other hand, ∫ ξ 0 · · · ∫ ξ 0 f(ξ)(dξ)n = 1 (n− 1)! ∫ ξ 0 (ξ −z)n−1f(z)dz, therefore, u(ξ) = u(0) + u ′ (0)ξ + ∫ ξ 0 (ξ −z)f̄(z)dz. by substituting into equation (8) we have (b2 −a2)f̄(ξ) −u(0) −u ′ (0)ξ − ∫ ξ 0 (ξ −z)f̄(z)dz = 0, f̄(ξ) = [ −(u(0) + u ′ (0)ξ) b2 −a2 ] ︸ ︷︷ ︸ ḡ(ξ) + ∫ ξ 0 (ξ −z) b2 −a2︸ ︷︷ ︸ k(ξ,z) f̄(z)dz = 0, f̄(ξ) = ḡ(ξ) + ∫ ξ 0 k(ξ,z)f̄(z)dz. similarly, f(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)f(z)dz. 86 a. ebadian, m. shams yousefi, f. farahrooz and m. najand foumani 4. existence and convergence analysis in this section, we prove the existence and uniqueness of the solution and convergence of the method by using the following assumptions. we consider fuzzy integral equation as follow: f(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)f(z)dz, where k is an arbitrary positive kernel on [0, 1] × [0,ξ] and functions f,g : [0, 1] −→ rf are continuous fuzzy number-valued functions. we assume that k is continuous and therefore it is uniformly bounded so there exists m1 > 0 such that |k(ξ,z)| ≤ m1 0 ≤ ξ ≤ 1, 0 ≤ z ≤ ξ. now consider the set, cf ([0, 1]) = { f : [0, 1] −→ rf ; f is continuous } , which is the space of fuzzy continuous function because for f,g ∈ cf ([0, 1]) and α ∈ r,αf + g is continuous. regarding to def. 2.6 we define the fuzzy uniform norm as form ‖f‖f.u := sup ξ∈[0,1] d(f(ξ), 0̂). in the next theorem we show that cf ([0, 1]) is a banach space. theorem 4.1. (cf ([0, 1]),‖.‖f.u) is a banach space. proof. let {fn}∞n=1 be a cauchy sequence in cf ([0, 1]). then for each ε > 0 there exists m ∈ n such that d(fn(ξ),fm(ξ)) < ε for all n,m ≥ m, and for all ξ ∈ [0, 1]. that is, sup d(fn(ξ),fm(ξ)) =‖ fn −fm ‖f.u< ε for all n,m ≥ m. this implies that for each ξ ∈ [0, 1], {fn(ξ)} is a cauchy sequence in the complete metric space rf . so there exists a function f such that fn(ξ) −→ f(ξ) for all ξ ∈ [0, 1]. it means that the pointwise limit function f(ξ) = limn→∞fn(ξ) exists. at first, we want to prove that {fn} also converges uniformly to f, that is ‖fn −f‖f.u −→ 0 (n →∞). in other words for each ε > 0 we need to find m such that ‖fn −f‖f.u ≤ ε for n > m. for this, let ε > 0 and then fix m such that ‖fn −fm‖f.u < ε2 for all n,m ≥ m. we can do this since {fn} is a cauchy sequence. using the triangle inequality, we have ‖fn −f‖f.u ≤‖fn −fm‖f.u + ‖f −fm‖f.u. as we know that for n ≥ m, we have ‖fn −fm‖f.u < ε2 . therefore ‖f −fm‖ = lim n→∞ ‖fn −fm‖f.u < ε 2 . so ‖fn −f‖f.u ≤ ε2 + ε 2 = ε for n > m which means ‖fn −f‖f.u −→ 0 (n →∞). now, we have to show that f is continuous. so let ε > 0 and ξ1 ∈ [0, 1], we want to find δ > 0 such that for an arbitrary fixed number ξ2; d(f(ξ1),f(ξ2)) < ε when |ξ1 − ξ2|, with using the triangle inequality d(f(ξ1),f(ξ2)) ≤ d(f(ξ1),fn(ξ1)) + d(fn(ξ1),fn(ξ2)) + d(fn(ξ2),f(ξ2)) for some n. now, we pick n such that d(f(ξ1),fn(ξ1)) < ε 3 and d(fn(ξ2),f(ξ2)) < ε 3 , on the other hand fn is continuous. so, d(fn(ξ1),fn(ξ2)) < ε 3 and we have d(f(ξ1),f(ξ2)) ≤ ε3 + ε 3 + ε 3 = ε. hence f ∈ cf ([0, 1]), so cf ([0, 1]) is a banach space. � now we define the operator t as t(f)(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)f(z) dz, ∀ξ ∈ [0, 1], ∀f ∈ cf ([0, 1]), g : [0, 1] → rf . on the banach space techniques 87 t(f)(ξ) can be represented as form t(f)(ξ) = (t(f)(ξ),t(f)(ξ)) where, t(f)(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)f(z) dz, t(f)(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)f(z) dz. sufficient conditions for the existence of a unique solution for the above integral equation will be given in the following. theorem 4.2. let k = k(ξ,z) be continuous and positive for 0 ≤ ξ ≤ 1 , 0 ≤ z ≤ ξ and f,g : [0, 1] −→ rf be fuzzy continuous functions on [0, 1]. if m1 ξ < 1, than the homotopy analysis method f0(ξ) = g(ξ), fm(ξ) = g(ξ) + ∫ ξ 0 k(ξ,z)fm−1(z)dz m ≥ 1. convergence to the unique solution f. proof. first we show that t(cf ([0, 1])) ⊆ cf ([0, 1])). since g is continuous on the compact set [0, 1], so it is uniformly continuous. therefore ∀ε1 > 0 ∃ρ1 > 0 s.t. |ξ1 − ξ2| < ρ1 =⇒ d(g(ξ1),g(ξ2)) < ε1. it means; sup max{|g(ξ1) −g(ξ2)|, |g(ξ1) −g(ξ2)|} < ε1 0 ≤ ξ1,ξ2 ≤ 1, consequently, |g(ξ1) −g(ξ2)| < ε1 , |g(ξ1) −g(ξ2)| < ε1. as mentioned above f is continuous thus f is bounded. it means ∃ m2 > 0 s.t. |f| ≤ m2, we must show that, ∀ε > 0 ∃ ρ > 0 s.t. |ξ1 − ξ2| < ρ =⇒‖t(f)(ξ1) −t(f)(ξ2)‖f.u < ε. since ‖t(f)(ξ1) −t(f)(ξ2)‖f.u = sup ξ1,ξ2∈[0,1] max {|t(f)(ξ1) −t(f)(ξ2)|, |t(f)(ξ1) −t(f)(ξ2)|} it is enough to show that, |t(f)(ξ1) −t(f)(ξ2)| < ε , |t(f)(ξ1) −t(f)(ξ2)| < ε |t(f)(ξ1) −t(f)(ξ2)| ≤ |g(ξ1) −g(ξ2)| + | ∫ ξ1 0 k(ξ1,z)f(z)dz − ∫ ξ2 0 k(ξ2,z)f(z)dz| ≤ ε1 + | ∫ ξ1 0 k(ξ1,z)f(z)dz + ∫ 0 ξ2 k(ξ2,z)f(z)dz| ≤ ε1 + ∫ ξ1 0 |k(ξ1,z)||f(z)|dz + ∫ 0 ξ2 |k(ξ2,z)||f(z)|dz ≤ ε1 + ∫ ξ1 0 m1|f(z)|dz + ∫ 0 ξ2 m1|f(z)|dz = ε1 + ∫ ξ1 ξ2 m1|f(z)|dz ≤ ε1 + m1.m2 ∫ ξ1 ξ2 dz = ε1 + m1m2(ξ1 − ξ2). choosing ε1 = ε 2 , ξ1 − ξ2 = ξ 2m1m2 , we have, |t(f)(ξ1) −t(f)(ξ2)| < ε. similarly; |t(f)(ξ1) −t(f)(ξ2)| < ε. 88 a. ebadian, m. shams yousefi, f. farahrooz and m. najand foumani so t(cf ([0, 1])) ⊆ cf ([0, 1]). now, we show that the operator t is a contraction. so for f,h ∈ cf ([0, 1]) and ξ ∈ [0, 1] d(t(f)(ξ),t(h)(ξ)) ≤ d(g(ξ),g(ξ)) + d( ∫ ξ 0 k(ξ,z)f(z)dz, ∫ ξ 0 k(ξ,z)h(z)dz) = ∫ ξ 0 |k(ξ,z)|d(f(z),h(z))dz ≤ m1 ∫ ξ 0 d(f(z),h(z))dz = m1ξd(f,h), therefore, d(t(f)(ξ),t(h)(ξ)) ≤ m1ξ d(f,h). after taking supremum we have ‖t(f)(ξ) −t(h)(ξ)‖f.u ≤ m1ξ ‖f −h‖f.u, since m1ξ < 1 the operator t is a contraction on banach space (cf ([0, 1]),‖ . ‖f.u) consequently, the banach’s fixed point theorem implies that this integral equation has a unique solution f in cf ([0, 1]). � the existence and uniqueness of solution for integral equation was proved. we conclude that the ffkge also has a unique solution. corollary 4.1. the ffkge has a unique solution. proof. combine the sect. 3 (equivalency the ffkge and fuzzy integral equation) and theorem. 4.2. � 5. the homotopy analysis transform method the application of homotopy analysis method in linear and nonlinear problems has been devoted by scientists and engineers. the fundamental work was done by liao and he [6]. he’s technique in particular, eliminated some of the traditional limitations of methods and was successfully applied to solve many problems in various, fields including fluid mechanics, heat transfer and so on [7]. this method has a significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear problems in applied science. to illustrate the basic idea of this method to solve the ffkge. we consider the parametric from of this equation as follows : ∂2γu(x,t) ∂t2γ = ∂2u(x,t) ∂x2 + u(x,t), (5.1) ∂2γu(x,t) ∂t2γ = ∂2u(x,t) ∂x2 + u(x,t). equation (5.1) can be written as ∂u(x,t) ∂t = ∂1−2γ∂2u(x,t) ∂t1−2γ∂x2 + ∂1−2γ ∂t1−2γ u(x,t). now the methodology consists of applying the laplace transform first on both sides of above equation, we get l[u(x,t)] − u(x, 0) s − ut(x, 0) s2 = 1 s2γ l [ ∂1−2γ ∂t1−2γ ∂2u(x,t) ∂x2 ] + 1 s2γ l [ ∂1−2γ ∂t1−2γ u(x,t) ] . (5.2) equation (5.2) can be written as a nonlinear operator form as follow: n[u(x,t)] = 0, where n is nonlinear operator, u(x,t) is unknown function and x,t are independent variables, u(x, 0) is auxiliary parameter. on the banach space techniques 89 using q ∈ [0, 1] as an embedding parameter, then we have n[φ̄(x,t; q)] = l[φ̄(x,t; q)] − u(x, 0) s − ut(x, 0) s2 = 1 s2γ l [ ∂1−2γ ∂t1−2γ ∂2φ̄(x,t; q) ∂x2 ] + 1 s2γ l [ ∂1−2γ ∂t1−2γ φ̄(x,t; q) ] , where φ̄(x,t; q) is the real function of x,t and q. by means of generalizing the traditional homotopy methods construct the zero-order deformation equation. (1 −q)l[φ̄(x,t; q) −u0(x,t)] = qhn[φ(x,t; q)], (5.3) where h is a nonzero auxiliary parameter. when q = 0 and q = 1, it holds φ̄(x,t; 0) = u0(x,t), φ̄(x,t; 1) = u(x,t). expanding φ̄i(x,t; q) in taylor’s series with respect to q we have φ̄(x,t; q) = u0(x,t) + σ ∞ m=1um(x,t)q m, where um(x,t) = 1 m! ∂mφ̄(x,t; q) ∂qm ∣∣∣∣ q=0 . differentiating deformation (5.2), m times with respect to q, dividing by m! and setting q = 1, we have the mth-order deformation l[um(x,t) −xmum−1(x,t)] = hrm(um−1(x,t)), where rm(um−1(x,t)) = 1 (m− 1)! ∂m−1n[φ̄(x,t; q)] ∂qm−1 ∣∣∣∣ q=0 , and xm = {1 m>1, 0 m61. therefore for this equation, rm(um−1(x,t)) = l[um−1(x,t)] − (1 −xm) k(β)(1 + sin x) s = 1 s2γ l[ ∂1−2γ ∂t1−2γ ∂2um−1(x,t) ∂x2 ] + 1 s2γ l[ ∂1−2γ ∂t1−2γ um−1(x,t)], u0(x,t; β) = k(β)(1 + sin x), m = 1 =⇒ r1(u0(x,t; β) = −k(β) s2γ+1 , if h = −1 then u1(x,t; β) = k(β)t2γ γ(2γ + 1) . by using this method, u2(x,t; β) = k(β)t4γ γ(4γ + 1) , u3(x,t; β) = k(β)t6γ γ(6γ + 1) , u4(x,t; β) = k(β)t8γ γ(8γ + 1) , u5(x,t; β) = k(β)t10γ γ(10γ + 1) . proceeding in this manner, the rest of the components ūn(x,t; β) for n ≥ 5 can be completely obtained. we get the approximated solution of fuzzy fractional differential equation as follow u(x,t; β) = [(1 + sin x) + t2γ γ(2γ + 1) + t4γ γ(4γ + 1) + t6γ γ(6γ + 1) + t8γ γ(8γ + 1) + t10γ γ(10γ + 1) ]k(β), 90 a. ebadian, m. shams yousefi, f. farahrooz and m. najand foumani u(x,t; β) = [(1 + sin x) + t2γ γ(2γ + 1) + t4γ γ(4γ + 1) + t6γ γ(6γ + 1) + t8γ γ(8γ + 1) + t10γ γ(10γ + 1) ]k(β). let k(β) = 0.25β and k(β) = −0.25β for 0 ≤ β ≤ 1 so the exact solution is given by u(x,t; β) = (1 + sin x)k(β) + σ∞r=1 t2rγ γ(2rγ + 1) k(β), u(x,t; β) = (1 + sin x)k(β) + σ∞r=1 t2rγ γ(2rγ + 1) k(β), and u(x,t; β) = (u(x,t; β),u(x,t; β)). in the following tables and figures comparison between the exact solution and the different terms of approximation solution for u,u is given by the homotopy analysis transform method at γ = 1 8 . if we increase the computational process, the approximation solution will be closer to the exact solution. table. 1. (x,t,β) uapprox[5] uapprox[15] uapprox[25] uexact (0.2,0.2,0.1) 0.07711367095 0.08075077885 0.08075381528 0.08075381592 (0.2,0.4,0.1) 0.1004436229 0.1133482422 0.1134055766 0.1134055435 (0.2,0.6,0.1) 0.1498635015 0.1495343794 0.1498624662 0.1498635015 (0.2,0.8,0.1) 0.1409872942 0.1914962138 0.1926482315 0.1926555602 (0.4,0.2,0.1) 0.08188239622 0.08551950412 0.08552254055 0.08552254120 (0.6,0.2,0.1) 0.08626299950 0.08990010740 0.08990314382 0.08990314448 (0.8,0.2,0.1) 0.09008083995 0.09371794785 0.09372098428 0.09372098492 (0.2,0.2,0.2) 0.1542273419 0.1615015577 0.1615076306 0.1615076318 (0.2,0.4,0.2) 0.2008872458 0.2266964845 0.2268109531 0.2268110870 (0.2,0.6,0.2) 0.2425774142 0.2990687587 0.2997249324 0.2997270030 (0.2,0.8,0.2) 0.2819745885 0.3829924276 0.3852964630 0.3853111204 (0.4,0.2,0.2) 0.1637647924 0.1710390082 0.1710450811 0.1710450824 (0.6,0.2,0.2) 0.1725259990 0.1798002148 0.1798062876 0.1798062890 (0.8,0.2,0.2) 0.1801616799 0.1874358957 0.1874419686 0.1874419698 table. 2. (x,t,β) uapprox[5] uapprox[15] uapprox[25] uexact (0.2,0.2,-0.1) -0.07711367095 -0.08075077885 -0.08075381528 -0.08075381592 (0.2,0.4,-0.1) -0.1004436229 -0.1133482422 -0.1134055766 -0.1134055435 (0.2,0.6,-0.1) -0.1498635015 -0.1495343794 -0.1498624662 -0.1498635015 (0.2,0.8,-0.1) -0.1409872942 -0.1914962138 -0.1926482315 -0.1926555602 (0.4,0.2,-0.1) -0.08188239622 -0.08551950412 -0.08552254055 -0.08552254120 (0.6,0.2,-0.1) -0.08626299950 -0.08990010740 -0.08990314382 -0.08990314448 (0.8,0.2,-0.1) -0.09008083995 -0.09371794785 -0.09372098428 -0.09372098492 (0.2,0.2,-0.2) -0.1542273419 -0.1615015577 -0.1615076306 -0.1615076318 (0.2,0.4,-0.2) -0.2008872458 -0.2266964845 -0.2268109531 -0.2268110870 (0.2,0.6,-0.2) -0.2425774142 -0.2990687587 -0.2997249324 -0.2997270030 (0.2,0.8,-0.2) -0.2819745885 -0.3829924276 -0.3852964630 -0.3853111204 (0.4,0.2,-0.2) -0.1637647924 -0.1710390082 -0.1710450811 -0.1710450824 (0.6,0.2,-0.2) -0.1725259990 -0.1798002148 -0.1798062876 -0.1798062890 (0.8,0.2,-0.2) -0.1801616799 -0.1874358957 -0.1874419686 -0.1874419698 on the banach space techniques 91 figure 1. comparison between the exact solution and the 5th-order of approximation solution given by the homotopy analysis transform method(uapprox[5](x,t,β)) figure 2. comparison between the exact solution and the 5th-order of approximation solution given by the homotopy analysis transform method(uapprox[5](x,t,β)) figure 3. comparison between the exact solution and the 10th-order of approximation solution given by the homotopy analysis transform method(uapprox[10](x,t,β)) figure 4. comparison between the exact solution and the 10th-order of approximation solution given by the homotopy analysis transform method(uapprox[10](x,t,β)) 6. conclusion the homotopy analysis method is applied for solving the ffkge. this equation is equivalent to fuzzy integral equation. the existence, uniqueness of the solution and convergence of this method are proved. 92 a. ebadian, m. shams yousefi, f. farahrooz and m. najand foumani references [1] a. a. kilbas, h. m. srivastava, j. j. trujilo, theory and applications of fractional differential equations, elsevier, the netherlands, 2006. [2] a. ebaid, exact solutions for the generalized klein-gordon equation via atransformation and exp-function method and comparison with adomian’s method. journal of computational an applied mathematics. 223 (2009), 278-290. [3] e. raicher, s. eliezer, a. zigler, a novel solution to the klein-gordon equation in the presence of astrong rotating electric field. physics letters b. 75 (2015), 76-81. [4] l. a. zadeh, k. tanaka, m. shimura, fuzzy sets and their applications to congnitive and decision processes, the university of california, berkeley, california, 1974. [5] r. goetschel, w. voxman, elementary calculus. fuzzy sets and system. 18(1986), 31-43. [6] sj. liao , introduction to the homotopy analysis method. boca raton: chapman and hall / crc, 2003. [7] jy park , yc kwan , jv jeong , existence of solutions of fuzzy integral equations in banach spaces. fuzzy sets and systems. 72 (3) (1995), 373-378. [8] r. khalili , m.al horani , a. yousef , m. sababbeh , a new definition of fractional derivatives. j. comput. appl. math. 264 (2014), 65-70. [9] t. abdeljawad , m.al horani , r. khalili , conformable fractional semiqroup operators. j. semiqroup theory appl. 2015 (2015), article id 7. [10] t. abdeljawad , on conformable fractional calculus. j. comput. appl. math. 279(1) (2015), 57-66. 1department of mathematics, payame noor university, po. box 19395-3697 tehran, iran 2department of mathematis, faculty of science university of guilan, po box 1914 rasht, iran ∗corresponding author: f.farahrooz@yahoo.com 1. introduction 2. preliminaries 3. the fractional transform 4. existence and convergence analysis 5. the homotopy analysis transform method 6. conclusion references international journal of analysis and applications volume 19, number 5 (2021), 743-759 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-743 iterative schemes for triequilibrium-like problems misbah iram bloach, muhammad aslam noor∗ and khalida inayat noor department of mathematics, comsats university islamabad, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this work, we present a new class of equilibrium problems, termed as triequilibrium-like problems with trifunction in invexity settings. classical varilunational-like inequalities and equilibrium-like problems can be obtained as specific variants of triequilibrium-like problems. certain new iterative methods are proposed and examined for the solution of triequilibrium-like problems by using auxiliary principle technique. convergence analysis of these proposed methods is examined under some mild conditions. 1. introduction equilibrium problems theory has shown tremendous potential and great influence in various branches of pure and applied sciences. this theory has shown continuously expanding growth in almost all areas of engineering and mathematical sciences. equilibrium problems cover a diverse set of applications including hemicariational inequalities, variational inequalities, game theory, nash equilibrium, variational-like inequalities as special use cases, see [16–26]. blum and oettli [5] and noor and oettli [12] introduced the present form of equilibrium problems. the classical equilibrium problems theory revolves around the assumption of convexity of the set and objective function . recently, the notion of convexity has started expanding to numerous fields showing the capacity for various useful applications. hanson [8] derived invex functions as a special extension of convex functions. different received june 26th, 2021; accepted july 20th, 2021; published august 12th, 2021. 2010 mathematics subject classification. 49j40, 90c33. key words and phrases. auxiliary principle technique; equilibrium problems; proximal point methods; convergence; variational-like inequalities. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 743 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-743 int. j. anal. appl. 19 (5) (2021) 744 results were presented to make this fact noticeable that what holds for convex functions in mathematical programming also holds for a generalized class of functions known as invex functions. ben-israel and mond [3] and weir and jeyakumar [29] works led to preinvex functions as another generalized class of convex functions. weir and mond [30] has shown in their work the interchangeability of preinvex functions with convex functions in optimization problems. noor [13] proved that the minimum of preinvex on the invex set can be disciplined into variational inequalities, widely known as variational-like inequalities. variational-like inequalities and equilibrium-like problems, owing to their specialized nature, cannot allow traditional resolvent method, projection method and their prevalent variant forms to propound any iterative methods. to bridge this gap, we resort to a technique named as auxiliary principle, proposed by glowinski et al. [6]. to solve numerous variational inequalities and equilibrium problems, noor [14–20, 26] employed the technique of auxiliary principle to propose various iterative methods. motivated by the recent research in this field, we introduce another class of equilibrium problems in invexity settings, termed triequilibrium-like problems. we analyze the proximal point methods for triequilibriumlike problems and prove convergence of the generated sequences to a solution of the problem, assuming the existence of solutions and rather mild jointly pseudo monotonicity property of the trifunction ψ(., ., .) and operator υ. in the end, in combination with the bregman function, we use the auxiliary principle technique again to propose and evaluate some three-step iterative methods for solving triequilibrium-like problems. the convergence of iterative methods is considered by using the partially relaxed stronglyη monotonicity and by assuming that the nonlinear term φ(., .) of the triequilibrium-like problems is a skew-symmetric function. since variational inequalities and equilibrium problems are special variants of triequilibrium-like problems, so our results will continue to hold o true for these problems. the techniques and ideas of this paper may be a strating point for further research activities in these dynamic fields. 2. preliminaries let h̃ be a real hilbert space. the inner product and norm on h̃ are denoted by 〈., .〉 and ‖ . ‖ respectively. let k} be a nonempty invex set in h̃. let = : k} →< and }(., .) : h̃ × h̃ → h̃ are continuous functions. before going on to our main results, first, we recall the following well-known results and concepts. definition 2.1. a nonempty set k} in h̃ is said to be an invex set, if there exists a bifunction }(., .), such that ξ + τ}(ζ,ξ) ∈ k}, ∀ξ,ζ ∈ k},τ ∈ [0, 1]. if the set k} is invex at each ξ ∈ k}, then k} is also called }−connected set. int. j. anal. appl. 19 (5) (2021) 745 definition 2.2. a function = : k} →< is said to be a preinvex function, if there exists a bifunction }(., .), such that =(ξ + τ}(ζ,ξ)) ≤ (1 − τ)=(ξ) + τ=(ζ), ∀ξ,ζ ∈ k},τ ∈ [0, 1]. the function = : k} →< is said to be preconcave if and only if −= is preinvex. definition 2.3. a differentiable function = : k} → < is said to be an invex function, if there exists a bifunction }(., .), such that =(ζ) −=(ξ) ≥〈=′(ξ) , }(ζ,ξ)〉, ∀ξ,ζ ∈ k},τ ∈ [0, 1], where =′(ξ) is the differential of = at ξ. from above definitions, it is clear that the differentiable preinvex functions are the invex functions but the converse is not true, see [30]. definition 2.4. a function ψ : k} →< is said to be a strongly preinvex function with respect to a bifunction }(., .), if there exists a constant υ > 0, such that ψ(ξ + τ}(ζ,ξ)) ≤ (1 − τ)ψ(ξ) + τψ(ζ) −υτ(1 − τ) ‖ }(ζ,ξ) ‖2, ∀ξ,ζ ∈ k},τ ∈ [0, 1]. we note that, if }(ζ,ξ) = ζ −ξ, the invex set k} reduces to the convex set k and preinvex functions and strongly preinvex functions reduce to convex functions and strongly convex functions. assumption 2.1. the bifunction }(., .) : h̃ × h̃ →< satisfies the following condition }(ξ + τ1(ζ − ξ),ξ + τ2(ζ − ξ)) = (τ1 − τ2)}(ζ,ξ) , ∀ξ,ζ ∈ h̃. for τ1 = 0, 1 and τ2 = τ, we get condition c of mohan and neogy [10], (i) }(ξ,ξ + τ(ζ − ξ)) = −τ}(ζ,ξ) , (ii) }(ζ,ξ + τ(ζ − ξ)) = (1 − τ)}(ζ,ξ) , ∀ξ,ζ ∈ h̃. assumption 2.2. the bifunction }(., .) : h̃ × h̃ →< satisfies the following condition }(ξ,ζ) = }(ξ,z) + }(z,ζ) , ∀ξ,ζ,z ∈ h̃. from assumption (2.2), we get (i) }(ξ,ξ) = 0, ∀ξ ∈ h̃, (ii) }(ξ,ζ) = −}(ζ,ξ), ∀ξ,ζ ∈ h̃. assumption (2.2) has a significant role in studying the existence of unique solution of variational-like inequalities and it is also used to suggest and analyze some iterative methods for various classes of variational-like inequalities and equilibrium-like problems. int. j. anal. appl. 19 (5) (2021) 746 definition 2.5. the bifunction }(., .) : h̃ × h̃ → h̃ is said to be: (a) strongly monotone, if there exists a constant σ > 0, such that 〈}(ζ,ξ),ζ − ξ〉≥ σ ‖ ζ − ξ ‖2 , ∀ξ,ζ ∈ h̃. (b) lipschitz continuous, if there exists a constant δ > 0, such that ‖ }(ζ,ξ) ‖≤ δ ‖ ζ − ξ ‖ , ∀ξ,ζ ∈ h̃. from (a) and (b), it is observed that σ ≤ δ. remark 2.1. if }(ζ,ξ) = υ(ζ) − υ(ξ), then strong monotonicity and lipschitz continuity of bifunction }(., .) reduces to the strong monotonicity and lipschitz continuity of the nonlinear operator υ. lemma 2.1. for all ξ,ζ ∈ h̃, we have (i) 2〈ξ,ζ〉 =‖ ξ + ζ ‖2 −‖ ξ ‖2 −‖ ζ ‖2 (ii) 2〈ξ,ζ〉≤‖ ξ ‖2 + ‖ ζ ‖2 . given an operator υ : h̃ → <, a continuous trifunction ψ(., ., .) : h̃ × h̃ × h̃ → < and a continuous bifunction φ(., .) : h̃ × h̃ →<∪{+∞}, consider the problem of finding ξ ∈ h̃, such that ψ(ξ, υ(ξ),}(ζ,ξ)) + φ(ζ,ξ) − φ(ξ,ξ) + υ ‖ }(ζ,ξ) ‖2≥ 0 , ∀ζ ∈ h̃.(2.1) the problem (2.1) is called a triequilibrium-like problem. for the physical and mathematical formulation of the problem (2.1), see [1–19, 21–24, 27–31] and the references therein. now we discuss some special variants of problem (2.1). for ψ(ξ, υ(ξ),}(ζ,ξ)) = 〈ξ, υ(ξ),}(ζ,ξ)〉, problem (2.1) reduces to a trivariational-like inequality: 〈ξ, υ(ξ),}(ζ,ξ)〉 + φ(ζ,ξ) − φ(ξ,ξ) + υ ‖ }(ζ,ξ) ‖2≥ 0 , ∀ζ ∈ h̃.(2.2) for }(ζ,ξ) = ζ − ξ, the invex set reduces to the convex set and the problem (2.1) reduces to: ψ(ξ, υ(ξ),ζ − ξ) + φ(ζ,ξ) − φ(ξ,ξ) + υ ‖ ζ − ξ ‖2≥ 0 , ∀ζ ∈ h̃,(2.3) which is called a triequilibrium problem and appears to be new. also the trivariational-like inequality (2.2) reduces to: 〈ξ, υ(ξ),ζ − ξ〉 + φ(ζ,ξ) − φ(ξ,ξ) + υ ‖ ζ − ξ ‖2≥ 0 , ∀ζ ∈ h̃.(2.4) the problem (2.4) is called a trivariational inequality. briefly, choosing proper ψ(., ., .), }(., .) and spaces, one can get many known and unknown equilibrium problems and special variants of (2.1), see [17–19, 21–24]. int. j. anal. appl. 19 (5) (2021) 747 definition 2.6. the operator υ : h̃ →< and the trifunction ψ(., ., .) are said to be: (i) jointly pseudo }− monotone, if ψ(ξ, υ(ξ),}(ζ,ξ)) + φ(ζ,ξ) − φ(ξ,ξ) + υ ‖ }(ζ,ξ) ‖2≥ 0, =⇒ ψ(ζ, υ(ζ),}(ζ,ξ)) + φ(ζ,ξ) − φ(ξ,ξ) −υ ‖ }(ζ,ξ) ‖2≥ 0 , ∀ξ,ζ ∈ h̃, (ii) partially relaxed strongly jointly }− monotone, if a constant α > 0 exists, such that ψ(ξ, υ(ξ),}(ζ,z)) + ψ(ζ, υ(ζ),}(z,ζ)) ≤ α ‖ }(z,ξ) ‖2 , ∀ξ,ζ,z ∈ h̃, (iii) jointly }− monotone, if ψ(ξ, υ(ξ),}(ζ,ξ)) + ψ(ζ, υ(ζ),}(ξ,ζ)) ≤ 0 , ∀ξ,ζ ∈ h̃, (iv) jointly hemicontinuous, if the mapping ψ(ξ + τ}(ξ,ζ), υ(ξ + τ}(ξ,ζ)),}(ζ,ξ)) ∀ξ,ζ ∈ h̃,τ ∈ [0, 1], is continuous. we noted that, for z = ξ, partially relaxed strongly jointly }− monotonicity reduces to jointly }− monotonicity. definition 2.7. the bifunction φ(., .) : h̃ × h̃ →<∪{+∞} is said to be skew symmetric if and only if, φ(ξ,ξ) − φ(ξ,ζ) − φ(ζ,ξ) + φ(ζ,ζ) ≥ 0 , ∀ξ,ζ ∈ h̃. if the bifunction φ(., .) is skew symmetric and bilinear then, φ(ξ,ξ) − φ(ξ,ζ) − φ(ζ,ξ) + φ(ζ,ζ) = φ(ξ − ζ,ξ − ζ) ≥ 0 , ∀ξ,ζ ∈ h̃, which shows that the bifunction φ(., .) is nonnegative. 3. iterative methods in this section, we derive new iterative methods for a triequilibrium-like problem (2.1) via auxiliary principle technique. given ξ ∈ h̃ satisfying (2.1), consider an auxiliary problem of finding ς ∈ h̃, such that ρψ(ς, υ(ς),}(ζ,ς)) + 〈ς − ξ,ζ − ς〉 + ρ φ(ζ,ς) −ρ φ(ς,ς) + ρ υ ‖ }(ζ,ς) ‖2≥ 0 , ∀ζ ∈ h̃,(3.1) which is called an auxiliary triequilibrium-like problem. if ς = ξ, then ς is the solution of (2.1). due to this perception, we consider and evaluate the present iterative methods for solving problem (2.1). int. j. anal. appl. 19 (5) (2021) 748 algorithm 3.1. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn+1, υ(ξn+1),}(ζ,ξn+1)) + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ζ,ξn+1) ‖2≥ 0 , ∀ζ ∈ h̃,(3.2) which is termed as proximal point method for solving triequilibrium-like problems (2.1). for ψ(ξ, υ(ξ),}(ζ,ξ)) = 〈ξ, υ(ξ),}(ζ,ξ)〉, algorithm (3.1) reduces to: algorithm 3.2. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρ〈ξn+1, υ(ξn+1),}(ζ,ξn+1)〉 + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ζ,ξn+1) ‖2≥ 0 , ∀ζ ∈ h̃.(3.3) for }(ζ,ξ) = ζ − ξ, algorithm (3.1) reduces to: algorithm 3.3. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn+1, υ(ξn+1),ζ − ξn+1) + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ ζ − ξn+1 ‖2≥ 0 , ∀ζ ∈ h̃.(3.4) for }(ζ,ξ) = ζ − ξ, algorithm (3.2) reduces to: algorithm 3.4. for the given ξ0 ∈ h̃, find the approximate solution ξn+1 by the iterative scheme ρ〈ξn+1, υ(ξn+1),ζ − ξn+1〉 + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ ζ − ξn+1 ‖2≥ 0 , ∀ζ ∈ h̃.(3.5) convergence analysis of algorithm (3.1) is considered, by adopting the technique of noor [22]. theorem 3.1. let ξ̄ ∈ h̃ be the solution of (2.1) and let ξn+1 be the approximate solution attained by algorithm (3.1). if the trifunction ψ(., ., .) and υ are jointly pseudo }−monotone and φ(., .) is skew symmetric, then ‖ ξn+1 − ξ̄ ‖2 ≤‖ ξn − ξ̄ ‖2 −‖ ξn+1 − ξn ‖2 .(3.6) proof: let ξ̄ ∈ h̃ be the solution of (2.1). then, using the jointly pseudo }− monotonicity of ψ(., ., .), we get ψ(ζ, υ(ζ),}(ζ, ξ̄)) + φ(ζ, ξ̄) − φ(ξ̄, ξ̄) −υ ‖ }(ζ, ξ̄) ‖2 ≥ 0 , ∀ζ ∈ h̃.(3.7) int. j. anal. appl. 19 (5) (2021) 749 now taking ζ = ξn+1 in (3.7), we have ρψ(ξn+1, υ(ξn+1),}(ξn+1, ξ̄)) + ρ φ(ξn+1, ξ̄) −ρ φ(ξ̄, ξ̄) −ρυ ‖ }(ξn+1, ξ̄) ‖2 ≥ 0.(3.8) now taking ζ = ξ̄ in (3.2) , we get ρψ(ξn+1, υ(ξn+1),}(ξ̄,ξn+1)) + 〈ξn+1 − ξn, ξ̄ − ξn+1〉 + ρ φ(ξ̄,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ξ̄,ξn+1) ‖2≥ 0,(3.9) which can be written as 〈ξn+1 − ξn, ξ̄ − ξn+1〉≥−ρψ(ξn+1, υ(ξn+1),}(ξ̄,ξn+1)) −ρυ ‖ }(ξ̄,ξn+1) ‖2 + ρ φ(ξn+1,ξn+1) −ρ φ(ξ̄,ξn+1) ≥ ρ{φ(ξ̄, ξ̄) − φ(ξ̄,ξn+1) − φ(ξn+1, ξ̄) + φ(ξn+1,ξn+1)} ≥ 0,(3.10) where we used (3.8) and skew symmetry of φ(., .). now by using lemma (2.1), we get ‖ ξn+1 − ξ̄ ‖2 ≤‖ ξn − ξ̄ ‖2 −‖ ξn+1 − ξn ‖2, the desired result. � theorem 3.2. let a finite-dimensional space h̃, and an approximate solution {ξn+1}, given by algorithm (3.1). if ξ̄ ∈ h̃ be the solution of (2.1), then limn−→∞ξn = ξ̄. proof : let ξ̄ ∈ h̃ be the solution of (2.1). from (3.6), it follows that ‖ ξ̄ − ξn ‖ is monotonically decreasing and hence {ξn} is bounded. furthermore from (3.6), we have ∞∑ n=0 ‖ ξn+1 − ξn ‖2≤‖ ξ0 − ξ̄ ‖2 . from above, we get lim n−→∞ ‖ ξn+1 − ξn ‖= 0.(3.11) a subsequence {ξnj} of bounded sequence {ξn} is convergent to the cluster point ξ̂ ∈ h̃ of {ξn}. replacing ξn by ξnj and letting nj −→∞ in (3.2) and using (3.11), we have ψ(ξ̂, υ(ξ̂) , }(ζ, ξ̂)) + φ(ζ, ξ̂) − φ(ξ̂, ξ̂) + υ ‖ }(ζ, ξ̂) ‖2≥ 0 , ∀ζ ∈ h̃, int. j. anal. appl. 19 (5) (2021) 750 which implies that ξ̂ solves (2.1) and ‖ ξn+1 − ξ̂ ‖2 ≤‖ ξn − ξ̂ ‖2 . from above it follows that, the sequence {ξn} has a unique cluster point ξ̂ and lim n−→∞ ξn = ξ̂, which is our required result. � for the implementation of the proximal methods, to determine the approximate solution implicitly is a tricky problem. to deal with this issue, we propose another iterative method for solving (2.1). for a given ξ ∈ h̃ satisfying (2.1), we consider the auxiliary problem of finding ς ∈ h̃, such that ρψ(ξ, υ(ξ),}(ζ,ς)) + 〈ς − ξ,ζ − ς〉 + ρ φ(ζ,ς) −ρ φ(ς,ς) +ρυ ‖ }(ζ,ς) ‖2≥ 0, ∀ζ ∈ h̃,(3.12) where constant ρ > 0. notably the problem (3.1) is not decomposable and not the interpretation of an optimization problem whereas problem (3.12) is almost the optimization problem. for ς = ξ, ς is the solution of (2.1). by using this observation we suggest and analyze the present iterative methods for solving problem (2.1). algorithm 3.5. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn, υ(ξn),}(ζ,ξn+1)) + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρυ ‖ }(ζ,ξn+1) ‖2≥ 0, ∀ζ ∈ h̃.(3.13) for ψ(ξ, υ(ξ),}(ζ,ξ)) = 〈ξ, υ(ξ),}(ζ,ξ)〉, algorithm (3.5) reduces to: algorithm 3.6. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρ〈ξn, υ(ξn),}(ζ,ξn+1)〉 + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ζ,ξn+1) ‖2≥ 0 , ∀ζ ∈ h̃.(3.14) for }(ζ,ξ) = ζ − ξ, algorithm (3.5) reduces to: algorithm 3.7. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn, υ(ξn),ζ − ξn+1) + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ ζ − ξn+1 ‖2≥ 0 , ∀ζ ∈ h̃.(3.15) for }(ζ,ξ) = ζ − ξ, algorithm (3.6) reduces to: int. j. anal. appl. 19 (5) (2021) 751 algorithm 3.8. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρ〈ξn, υ(ξn),ζ − ξn+1〉 + 〈ξn+1 − ξn,ζ − ξn+1〉 + ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ ζ − ξn+1 ‖2≥ 0 , ∀ζ ∈ h̃.(3.16) equivalently, many unique and familiar algorithms for solution of a triequilibrium-like problems can be obtained for best fitting operators and spaces. we consider the convergence analysis of algorithm (3.5) by using the condition of partially relaxed strongly }− monotonicity. theorem 3.3. let ξ̄ ∈ h̃ be the solution of (2.1) and let ξn+1 be the approximate solution attained by algorithm (3.5). if the trifunction ψ(., ., .) and υ be partially relaxed strongly jointly }− monotone with constant α > 0 and }(., .) : h̃ × h̃ → h̃ be lipchitz-continuous with constant δ > 0, then {1 − 4ρυδ2} ‖ ξn+1 − ξ̄ ‖2 ≤‖ ξn − ξ̄ ‖2 −{1 − 2ραδ2} ‖ ξn+1 − ξn ‖2 .(3.17) proof: let ξ̄ ∈ h̃ be the solution of (2.1), then ψ(ξ̄, υ(ξ̄),}(ζ, ξ̄)) + φ(ζ, ξ̄) − φ(ξ̄, ξ̄) + υ ‖ }(ζ, ξ̄) ‖2≥ 0 , ∀ζ ∈ h̃.(3.18) now taking ζ = ξn+1 in (3.18), we have ψ(ξ̄, υ(ξ̄),}(ξn+1, ξ̄)) + φ(ξn+1, ξ̄) − φ(ξ̄, ξ̄) + υ ‖ }(ξn+1, ξ̄) ‖2 ≥ 0.(3.19) now taking ζ = ξ̄ in (3.13) , we get ρψ(ξn, υ(ξn),}(ξ̄,ξn+1)) + 〈ξn+1 − ξn, ξ̄ − ξn+1〉 + ρ φ(ξ̄,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ξ̄,ξn+1) ‖2≥ 0 , ∀ζ ∈ h̃.(3.20) by adding (3.19) and (3.20), we get 〈ξn+1 − ξn, ξ̄ − ξn+1〉≥−ρ{ψ(ξn, υ(ξn),}(ξ̄,ξn+1)) + ψ(ξ̄, υ(ξ̄),}(ξn+1, ξ̄))} − 2ρυ ‖ }(ξn+1, ξ̄) ‖2 +ρ {φ(ξ̄, ξ̄) − φ(ξ̄,ξn+1) − φ(ξn+1, ξ̄) + φ(ξn+1,ξn+1)} ≥−ρα ‖ }(ξn,ξn+1) ‖2 −2ρυ ‖ }(ξn+1, ξ̄) ‖2 ≥−ραδ2 ‖ ξn+1 − ξn ‖2 −2ρυδ2 ‖ ξn+1 − ξ̄ ‖2,(3.21) where we used the partially relaxed strongly jointly }− monotonicity of ψ(., ., .) with constant α > 0, the skew symmetry of φ(., .) and the lipschitz continuity of }(., .). now by using lemma (2.1), we get 2〈ξn+1 − ξn, ξ̄ − ξn+1〉 = ‖ ξn − ξ̄ ‖2 −‖ ξn+1 − ξ̄ ‖2 −‖ ξn+1 − ξn ‖2 .(3.22) int. j. anal. appl. 19 (5) (2021) 752 by combining (3.21) and (3.22), we have {1 − 4ρυδ2} ‖ ξn+1 − ξ̄ ‖2 ≤‖ ξn − ξ̄ ‖2 −{1 − 2ραδ2} ‖ ξn+1 − ξn ‖2, which is the required (3.17). to inquire the convergence of algorithm (3.5), we can employ the similar procedure of theorem (3.2). now we consider the inertial proximal method for the solution of problem (2.1) by using auxiliary principle technique. for a given ξ ∈ h̃ satisfying (2.1), we consider the auxiliary problem of finding ς ∈ h̃, such that ρψ(ς, υ(ς),}(ζ,ς)) + 〈ς − ξ −γ(ξ − ξ),ζ − ς〉 + ρ φ(ζ,ς) −ρ φ(ς,ς) + ρ υ ‖ }(ζ,ς) ‖2≥ 0 , ∀ζ ∈ h̃,(3.23) where ρ > 0 and γ > 0 are constants. the problem (3.23) is called the auxiliary triequilibrium-like problem. if ς = ξ, then ς is the solution of (2.1). this observation motivates us to suggest and analyze the below-mentioned inertial proximal method for (2.1). algorithm 3.9. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn+1, υ(ξn+1),}(ζ,ξn+1)) + 〈ξn+1 − ξn −γn(ξn − ξn−1),ζ − ξn+1〉 +ρ φ(ζ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ζ,ξn+1) ‖2≥ 0 , ∀ζ ∈ h̃,(3.24) for γn = 0, algorithm (3.9) reduces to the algorithm (3.1). this shows that the proximal methods are included in inertial proximal methods as special variants, see [2, 14]. we now consider the convergence criteria of algorithm (3.9) by adopting the technique of alvarez [1] and noor [15]. theorem 3.4. let ξ̄ ∈ h̃ be the solution of (2.1) and ξn+1 be the approximate solution attained by algorithm (3.9). if the trifunction ψ(., ., .) and υ are jointly pseudo }− monotone and the bifunction }(., .) is lipschitz continuous with constant δ > 0, then ‖ ξn+1 − ξ̄ ‖2 ≤‖ ξn − ξ̄ ‖2 −‖ ξn+1 − ξn −γn(ξn − ξn−1) ‖2 + γn{‖ ξn − ξ̄ ‖2 −‖ ξ̄ − ξn−1 ‖2 +2 ‖ ξn − ξn−1 ‖2}(3.25) proof: let ξ̄ ∈ h̃ be a solution of (2.1), then by using the jointly pseudo }− monotonicity of ψ(., ., .) and υ, we get ψ(ζ, υ(ζ),}(ζ, ξ̄)) + φ(ζ, ξ̄) − φ(ξ̄, ξ̄) −υ ‖ }(ζ, ξ̄) ‖2 ≥ 0 , ∀ζ ∈ h̃.(3.26) int. j. anal. appl. 19 (5) (2021) 753 now taking ζ = ξn+1 in (3.26), we have ρψ(ξn+1, υ(ξn+1),}(ξn+1, ξ̄)) + ρφ(ξn+1, ξ̄) −ρφ(ξ̄, ξ̄) −ρυ ‖ }(ξn+1, ξ̄) ‖2 ≥ 0.(3.27) now taking ζ = ξ̄ in (3.24) , we get ρψ(ξn+1, υ(ξn+1),}(ξ̄,ξn+1)) + 〈ξn+1 − ξn −γn(ξn − ξn−1), ξ̄ − ξn+1〉 +ρ φ(ξ̄,ξn+1) −ρ φ(ξn+1,ξn+1) + ρυ ‖ }(ξ̄,ξn+1) ‖2≥ 0,(3.28) by combining (3.27) and (3.28) and by using skew symmetry of φ(., .) , we get 〈ξn+1 − ξn −γn(ξn − ξn−1), ξ̄ − ξn+1〉≥−ρψ(ξn+1, υ(ξn+1),}(ξ̄,ξn+1)) + ρφ(ξn+1,ξn+1) −ρφ(ξ̄,ξn+1) −ρυ ‖ }(ξ̄,ξn+1) ‖2 ≥ ρ{φ(ξ̄, ξ̄) − φ(ξ̄,ξn+1) − φ(ξn+1, ξ̄) + φ(ξn+1,ξn+1)} ≥ 0.(3.29) we can write (3.29) in the form 〈ξn+1 − ξn, ξ̄ − ξn+1〉≥ γn{〈ξn − ξn−1, ξ̄ − ξn〉 + 〈ξn − ξn−1,ξn − ξn+1〉}.(3.30) now by using lemma (2.1), we get 2〈ξn+1 − ξn, ξ̄ − ξn+1〉 =‖ ξn − ξ̄ ‖2 −‖ ξn+1 − ξ̄ ‖2 −‖ ξn+1 − ξn ‖2(3.31) 2γn〈ξn − ξn−1, ξ̄ − ξn〉 = γn{‖ ξ̄ − ξn−1 ‖2 −‖ ξn − ξ̄ ‖2 −‖ ξn − ξn−1 ‖2}.(3.32) by substituting (3.31) and (3.32) in (3.30) and rearranging terms, we get (3.25), the desired result . � theorem 3.5. let a finite-dimensional space h̃, and an approximate solution {ξn+1}, given by algorithm (3.9). if ξ̄ ∈ h be the solution of (2.1), then there exists γ ∈ (0, 1), such that 0 ≤ γn ≤ γ, ∀n ∈ n and the following condition holds ∞∑ n=1 γn ‖ ξn − ξn−1 ‖2≤∞, then limn−→∞ξn = ξ̄. int. j. anal. appl. 19 (5) (2021) 754 proof : let ξ̄ ∈ h̃ be a solution of (2.1). we first examine the case where γn = 0, which corresponds to the standard proximal method. in this case from (3.25), as a result the sequence {‖ ξ̄−ξn ‖} is monotonically decreasing and {ξn} is bounded. furthermore from (3.25), we get ∞∑ n=0 ‖ ξn+1 − ξn ‖2≤‖ ξ0 − ξ̄ ‖2, from above, we get lim n−→∞ ‖ ξn+1 − ξn ‖= 0.(3.33) a subsequence {ξnj} of bounded sequence {ξn} is convergent to the cluster point ξ̂ ∈ h̃ of {ξn}. replacing ξn by ξnj and letting nj −→∞ in (3.24) and using (3.33), we have ψ(ξ̂, υ(ξ̂) , }(ζ, ξ̂)) + φ(ζ) − φ(ξ̂) + υ ‖ }(ζ, ξ̂) ‖2≥ 0 , ∀ζ ∈ h̃, implies that, ξ̂ solves the triequilibrium-like problem (2.1) and ‖ ξn+1 − ξ̂ ‖2 ≤‖ ξn − ξ̂ ‖2 . hence, the sequence {ξn} has exactly one cluster point ξ̂ and lim n−→∞ ξn = ξ̂, which is our required result. now we consider the case γn > 0. from (3.25) and using the technique of alvarez [1, 2] and noor [15], we have ∞∑ n=1 ‖ ξn+1 − ξn −γn(ξn − ξn−1) ‖2 ≤‖ ξ0 − ξ̄ ‖2 + ∞∑ n=1 γ(‖ ξn − ξ̄ ‖2 +2 ‖ ξn − ξn−1 ‖2) ≤∞, which implies that lim n−→∞ ‖ ξn+1 − ξn −γn(ξn − ξn−1) ‖2= 0. by repeating the above arguments, rest of the proof runs as in the case γn = 0, one can easily find the required result, that is limn−→∞ξn = ξ̂. � for a given ξ ∈ h̃ satisfying problem (2.1), we consider the auxiliary problem of finding ς ∈ h̃, such that ρψ(ξ , υ(ξ) , }(ζ,ς)) + 〈e′(ς) −e′(ξ) , }(ζ,ς)〉 + ρ φ(ζ,ς) −ρ φ(ς,ς) + ρ υ ‖ }(ζ,ς) ‖2≥ 0, ∀ζ ∈ h̃,(3.34) int. j. anal. appl. 19 (5) (2021) 755 which is called an auxiliary triequilibrium-like problem associated with the problem (2.1). problem (3.34) has a unique solution because e′(.) is the differential of strongly preinvex function. noted that, if ς = ξ, then ς is the solution of problem (2.1). this observation leads us to suggest and examine the following methods for problem (2.1). algorithm 3.10. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ςn , υ(ςn) , }(ζ,ξn+1)) + 〈e′(ξn+1) −e′(ςn) , }(ζ,ξn+1)〉+ ρφ(ζ,ξn+1) −ρφ(ξn+1,ξn+1) + ρυ ‖ }(ζ,ξn+1) ‖2≥ 0, ∀ζ ∈ h̃,(3.35) λψ(yn , υ(yn) , }(ζ,ςn)) + 〈e′(ςn) −e′(yn) , }(ζ,ςn)〉 + λφ(ζ,ςn) −λ φ(ςn, ςn) + λ υ ‖ }(ζ,ςn) ‖2≥ 0, ∀ζ ∈ h̃,(3.36) µψ(ξn , υ(ξn) , }(ζ,yn)) + 〈e′(yn) −e′(ξn) , }(ζ,yn)〉 + µ φ(ζ,yn) −µ φ(yn,yn) + υ µ ‖ }(ζ,yn) ‖2≥ 0, ∀ζ ∈ h̃,(3.37) which is the three-step iterative algorithm for solving the triequilibrium like problem (2.1), where ρ > 0,λ > 0 and µ > 0 are constants. for µ = 0, algorithm (3.10) reduces to the two-step iterative algorithm for (2.1). algorithm 3.11. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ςn , υ(ςn) , }(ζ,ξn+1)) + 〈e′(ξn+1) −e′(ςn) , }(ζ,ξn+1)〉+ ρφ(ζ,ξn+1) −ρφ(ξn+1,ξn+1) + ρυ ‖ }(ζ,ξn+1) ‖2≥ 0, ∀ζ ∈ h̃, λψ(ξn , υ(ξn) , }(ζ,ςn)) + 〈e′(ςn) −e′(ξn) , }(ζ,ςn)〉 + λφ(ζ,ςn) −λ φ(ςn, ςn) + λ υ ‖ }(ζ,ςn) ‖2≥ 0, ∀ζ ∈ h̃, for µ = 0,λ = 0, algorithm (3.10) as follows. algorithm 3.12. given ξ0 ∈ h̃, find the approximate solution ξn+1, by the iterative scheme ρψ(ξn , υ(ξn) , }(ζ,ξn+1)) + 〈e′(ξn+1) −e′(ξn) , }(ζ,ξn+1)〉+ ρφ(ζ,ξn+1) −ρφ(ξn+1,ξn+1) + ρυ ‖ }(ζ,ξn+1) ‖2≥ 0, ∀ζ ∈ h̃. now we consider the convergence criteria of algorithm (3.10) by using the technique of noor [16]. int. j. anal. appl. 19 (5) (2021) 756 theorem 3.6. let e(ξ) be a differentiable strongly preinvex function with modulus β > 0. let the trifunction ψ(., ., .) and operator υ : h̃ → < be partially relaxed strongly-jointly }− monotone with constant α > 0. if assumption (2.2) holds and there exists the constants 0 < ρ < β α , 0 < λ < β α , and 0 < µ < β α , then an approximate solution ξn+1 given by algorithm (3.10) converges to solution ξ ∈ h̃ of (2.1). proof: let ξ ∈ h̃ be a solution of (2.1), then ρψ(ξ, υ(ξ), }(ζ,ξ)) + ρφ(ζ,ξ) −ρφ(ξ,ξ) +ρυ ‖ }(ζ,ξ) ‖2≥ 0, ∀ζ ∈ h̃,(3.38) λψ(ξ, υ(ξ), }(ζ,ξ)) + λφ(ζ,ξ) −λφ(ξ,ξ) +λυ ‖ }(ζ,ξ) ‖2≥ 0, ∀ζ ∈ h̃,(3.39) µψ(ξ, υ(ξ), }(ζ,ξ)) + µφ(ζ,ξ) −µφ(ξ,ξ) +µυ ‖ }(ζ,ξ) ‖2≥ 0, ∀ζ ∈ h̃.(3.40) taking ζ = ξn+1 in (3.38), we get ρψ(ξ, υ(ξ), }(ξn+1,ξ)) + ρφ(ξn+1,ξ) −ρφ(ξ,ξ) + ρυ ‖ }(ξn+1,ξ) ‖2≥ 0.(3.41) now taking ζ = ξ in (3.35), we get ρψ(ςn, υ(ςn), }(ξ,ξn+1)) + 〈e′(ξn+1) −e′(ςn) , }(ξ,ξn+1)〉 +ρ φ(ξ,ξn+1) −ρ φ(ξn+1,ξn+1) + ρ υ ‖ }(ξ,ξn+1) ‖2≥ 0.(3.42) consider the function b(z) = e(ξ) −e(z) −〈e′(z) , }(ξ,z)〉 ≥ β ‖ }(ξ,z) ‖2, ∀ξ,z ∈ h̃.(3.43) int. j. anal. appl. 19 (5) (2021) 757 since e(.) is strongly preinvex function. by combining (3.41), (3.42) and (3.43) and using assumption (2.2), we get b(ςn) −b(ξn+1) = e(ξn+1) −e(ςn) −〈e′(ςn),}(ξ,ςn)〉 + 〈e′(ξn+1),}(ξ,ξn+1)〉 = e(ξn+1) −e(ςn) + 〈e′(ξn+1) −e′(ςn),}(ξ,ξn+1)〉 −〈e′(ςn),}(ξn+1, ςn)〉 ≥ β ‖ }(ξn+1, ςn) ‖2 +〈e′(ξn+1) −e′(ςn),}(ξ,ξn+1)〉 ≥ β ‖ }(ξn+1, ςn) ‖2 −ρ[ψ(ςn , υ(ςn) , }(ξ,ξn+1)) + ψ(ξ , υ(ξ) , }(ξn+1,ξ))] + ρ[φ(ξn+1,ξn+1) − φ(ξ,ξn+1) − φ(ξn+1,ξ) + φ(ξ,ξ)] + ρυ1 ‖ }(ξ,ξn+1) ‖2 ≥ (β −ρα) ‖ }(ξn+1, ςn) ‖2 +ρυ1 ‖ }(ξ,ξn+1) ‖2 ≥ (β −ρα) ‖ }(ξn+1, ςn) ‖2, where we used skew symmetry of bifunction φ(., .) and partially relaxed strongly joint }− monotonicity of trifunction ψ(., ., .) and operator υ with constant α > 0. similarly by using (3.36), (3.37), (3.39) and (3.40) and repeating the same arguments, we get b(yn) −b(ςn) ≥ (β −λα) ‖ }(ςn,yn) ‖2, b(ξn) −b(yn) ≥ (β −υα) ‖ }(yn,ξn) ‖2 . if ξn+1 = ςn = yn = ξn, then obviously ξn is the solution of problem(2.1). otherwise for 0 < ρ < β α , 0 < λ < β α and 0 < µ < β α the sequences b(ςn) − b(ξn+1), b(yn) − b(ςn) and b(ξn) −b(yn) are non negative, and we must have lim n−→∞ ‖ }(ξn+1, ςn) ‖= 0, lim n−→∞ ‖ }(ςn,yn) ‖= 0, lim n−→∞ ‖ }(yn,ξn) ‖= 0. so, lim n−→∞ ‖ }(ξn+1,ξn) ‖= lim n−→∞ ‖ }(ξn+1, ςn) ‖ + lim n−→∞ ‖ }(ςn,yn) ‖ + lim n−→∞ ‖ }(yn,ξn) ‖= 0. zhu and marcotte [31] technique is employed to show that the entire sequence {ξn} is convergent to the cluster point ξ̄, which satisfies the triequilibrium-like problem(2.1). � int. j. anal. appl. 19 (5) (2021) 758 4. conclusion we have investigated another class of equilibrium problems, termed as triequlibrium-like problems. several new iterative methods for the solution of this newly class have been proposed. convergence of these iterative methods was tested by using only partially relaxed strongly jointly }− monotonicity or jointly pseudo }− monotonicity of the operator υ and the trifunction ψ. further exploration is needed to actualize these algorithms which will be very fascinating applications for new specialists. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] f. alvarez, on the minimization property of a second order dissipative system in hilbert space, siam j. control optim. 38(2000), 1102-1119. [2] f. alvarez and h. attouch, an inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator damping, set-valued anal. 9(2001), 3-11. [3] a. ben-israel and b. mond, what is invexity? j. austral. math. soc. ser. b. 28 (1986), 1-9. [4] m.i. bloach and m.a. noor, perturbed mixed variational-like inequalities, aims math. 5(3)(2019), 2153-2162. [5] e. blum and w. oettli, from optimization and variational inequalities to equilibrium problems, math. student. 63(1994), 123-145. [6] r. glowinski, j. l. lions and r. tremolieres, numerical analysis of variational inequalities, north-holland, amsterdam, 1981. [7] f. giannessi, a. maugeri and p. m. pardalos, equilibrium problems: nonsmooth optimization and variational inequality models, kluwer academic publishers, dordrecht, holland, 2001. [8] m. a. hanson, on sufficiency of the kuhn-tucker conditions, j. math. anal. appl. 80(1981), 545-550. [9] j. l. lions and g. stampacchia, variational inequalities, comm. pure appl. math. 20(1967), 493-512. [10] s.r. mohan and s.k. neogy, on invex set and preinvex functions, j. math. anal. appl. 189 (1995), 901-908. [11] b.b. mohsen, m.a. noor, k.i. noor, and m. postolache, strongly convex functions of higher order involving bifunction, mathematics, 7(11)(2019), 1028. [12] m. a. noor and w. oettli, on general nonlinear complementarity problems and quasi-equilibria, le mathematiche, 49 (1994), 313-331. [13] m. a. noor, variational-like inequalities, optimization, 30(1994), 323-330. [14] m. a. noor, proximal methods for mixed quasi variational inequalities, j. optim. theory appl. 115(2002), 447-452. [15] m. a. noor, m. akhter and k. i. noor, inertial proximal method for mixed quasi variational inequalities, nonlinear funct. anal. appl. 8(2003), 489-496. [16] m. a. noor, fundamentals of mixed quasi variational inequalities, int. j. pure. appl. math. 15(2004), 137-250. [17] m. a. noor and k. i. noor, on equilibrium problems, appl. math. e-notes. 4(2004), 125-132. [18] m. a. noor, mixed quasi invex equilibrium problems, int. j. math. sci. 57(2004), 3057-3067. [19] m. a. noor, auxiliary principle technique for equilibrium problems, j. optim. theory. appl. 122(2004), 371-386. [20] m. a. noor, some developments in general variational inequalities, appl. math. comput. 251(2004), 199-277. int. j. anal. appl. 19 (5) (2021) 759 [21] m. a. noor, invex equilibrium problems, j. math. anal. appl. 302(2005), 463-475 [22] m. a. noor, fundamentals of equilibrium problems, math. inequal. appl. 9 (2006), 529-566. [23] m. a. noor, k. i. noor and v. gupta, on equilibriumlike problems, appl. anal. 86(2007), 807-818. [24] m. a. noor, k. i. noor and m.i baloch, auxiliary principle technique for strongly mixed variational-like inequalities, u.p.b. sci. bull. ser. a. 80(2018), 93–100. [25] m. a. noor, k. i. noor, a. hamdi and e. h. el-shemas, on difference of two monotone operators, optim. lett. 3(2009), 329. [26] m. a. noor, k. i. noor and m. th. rassias, new trends in general variational inequalitis, acta appl. math. 170(1)(2020), 981-1046. [27] j. parida and a. sen, a variational-like inequality for multifunctions with applications, j. math. anal. appl., 124(1987), 73-81. [28] g. stampacchia, formes bilineaires coercitives sur les ensembles convexes, c. r. acad. sci. paris, 258(1964), 4413-4416. [29] t. weir and v. jeyakumar, a class of nonconvex functions and mathematical programming, bull. austral. math. soc. 38(1988), 177-189. [30] t. weir and b. mond, preinvex functions in multiple objective optimization, j. math. anal. appl. 136(1988), 29–28. [31] d.l. zhu and p. marcotte, cocoercivity and its role in the convergence of iterative schemes for solving variational inequalities, siam j. optim. 6(3)(1996), 714-726. 1. introduction 2. preliminaries 3. iterative methods 4. conclusion references international journal of analysis and applications volume 19, number 1 (2021), 47-64 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-47 on domination topological indices of graphs a.m. hanan ahmed1,∗, anwar alwardi2, m. ruby salestina1 1department of mathematics, yuvaraja’s college, university of mysore, mysuru, india 2department of mathematics, university of aden, yemen ∗corresponding author: hananahmed1a@gmail.com abstract. topological indices and domination in graphs are the essential topics in the theory of graphs. a set of vertices d ⊆ v (g) is said to be a dominating set for g if any vertex v ∈ v −d is adjacent to some vertex u ∈ d. in this research work, we define a new degree of each vertex v ∈ v (g), called the domination degree of v and denoted by dd(v), along with this new degree some domination indices based on domination degree are introduced. we study some basic properties of the domination degree function. exact values and bounds for domination zagreb indices of some families of graphs including the join and corona product are obtained. finally, we generalize the domination degree of the vertex and new general indices are defined. 1. introduction in this research article, we assume that g = (v,e) is a connected simple graph. in the field of chemistry, graph theory has provided many useful tools, such as topological indices. chem-informatics is one of the latest concepts which is a join of chemistry, mathematics, and information science. topological indices are numerical parameters of the graph, such that these parameters are the same for the received september 21st, 2020; accepted october 20th, 2020; published november 24th, 2020. 2010 mathematics subject classification. 05c69, 05c90, 05c35. key words and phrases. domination zagreb indices; domination degree; minimal dominating set; total number of minimal domination sets. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 47 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-47 int. j. anal. appl. 19 (1) (2021) 48 graph which they are isomorphism. some of the major classes of topological indices are distance basedtopological indices (see [2], [3]) and degreebased-topological indices (see [5], [20], [23]). degree basedtopological indices are of great significance. the wiener index w(g) is the old index and the first distancebased, introduced by chemist wiener [24] in 1974. after the introduction of the wiener index, many another distance-based topological indices, have been proposition and take into consideration in chemical and mathematical chemical literature. for example, harary index [17] and eccentric connectivity index [22]. presently a great number of vertex-degree-based graph invariants are being studied in mathematical and mathematical chemical literature ( [8], [9]). among them, the zagreb indices m1(g) and m2(g) are the most widely investigated. those have been inserted more than forty years ago ( [11], [12]), which are defined as follows: m1(g) = ∑ v∈v (g) d2(v) = ∑ uv∈e(g) d(u) + d(v) m2(g) = ∑ uv∈e(g) d(u)d(v). for properties of the two zagreb indices see [10] and the papers cited therein. the zagreb co-indices defined in [21], and are given by : m1(g) = ∑ d(u) + d(v) and m2(g) = ∑ d(u)d(v), where uv is not an edge in e(g). the degree of a vertex u in g, d(u) is the number of edges that are incident to u in g. the maximum and minimum degrees of vertices of a graph g are denoted by ∆(g) and δ(g) respectively. g is the complement of a graph g, having the same vertex set of g so that two vertices of g are neighboring if and only if they are not neighboring in g. if for every two vertices u,v ∈ v, there exists a (u,v)-path in g, then g is connected, otherwise, g is said to be disconnected. a set d ⊆ v is said to be a dominating set of g, if for any vertex v ∈ v −d there exists a vertex u ∈ d such that u and v are adjacent. the domination number γ(g) of g is the minimum cardinality of a minimal dominating set in g. the upper domination number γ(g) of g is the maximum cardinality of a minimal dominating set in g [4]. for a survey of domination in graphs, refer to ( [14], [15]). a dominating-set d = {v1,v2, ...,vr} is minimal if d−vi is not a dominating set. we use tm to denote the number of minimal dominating sets. in [7] a graph has at most o(1.7159n) minimal dominating-sets and there exist graphs with at leasto(1.5705n) minimal dominating-sets. for more definitions or properties, we refer to ( [7], [18], [19]). 2. domination degree in graphs in this partition, we sitting the definition of domination degree of the vertex v. we consider the lower and upper bound for this degree, and we study some basic properties of the domination degree. definition 2.1. int. j. anal. appl. 19 (1) (2021) 49 for each vertex v ∈ v (g), the domination degree denoted by dd(v) and defined as the number of minimal dominating sets of g which contains v. the minimum domination and maximum domination degree of g are denoted by δd(g) = δd and ∆d(g) = ∆d respectively, where δd = min{dd(v) : v ∈ v (g)} and ∆d = max{dd(v) : v ∈ v (g)}. let v ∈ v (g) and v ′ ∈ g ′ . then dd g(v) = dd g′ (v ′ ) if g ∼= g ′ . the domination degree function is obviously invariant under isomorphism. observation 2.1. 1 ≤ dd(v) ≤ tm(g), where tm(g) denotes the total number of minimal dominating sets. observation 2.2. suppose g is a graph of n ≥ 2 vertices having ∆(g) = n − 1. then γ(g) = 1 and dd(v) ≥ 1, for any v ∈ v (g). also, dd(v) = 1 if and only if d(v) = n− 1. corollary 2.1. if g ∼= kn and g is the complement of g, then dd g(v) = dd g(v), and tm(g) = 1. observation 2.3. let g(v,e) be a graph with minimal dominating sets s1,s2, ...,st. then tγ(g) ≤ ∑ v∈v (g) dd(v) ≤ tγ(g). we use the notion ρ(g) = ∑ v∈v (g) dd(v). proposition 2.1. let g be the complete bipartite graph kr,s. then dd g(v) = dg(v). proof. let the bipartite sets of kr,s be a and b, where a contains the vertices of degree s and b contains the vertices of degree r. also tm(g) = rs, such that if v ∈ a ⇒ dd g(v) = s. similarly if v ∈ b ⇒ dd g(v) = r. � observation 2.4. let g = ⋃t i=1 gi be the disjoint union of graphs g1, g2,...,gt. then γ(g) = ∑t i=1 γ(gi) and tm(g) =∏t i=1 tm(gi). for v ∈ v (gi), dd g(v) = dd gi (v) ∏t j=1 tm(gj), j 6= i. proposition 2.2. given h is a spanning subgraph of g with v (h) is the same as v (g). if the domination number of h is the same as the domination number of g, then tm(h) ≤ tm(g). int. j. anal. appl. 19 (1) (2021) 50 proof. use the first presumption, every dominating set for h is also a dominating set of g. as γ(h) = γ(g), its ensured that every minimal dominating set of minimum cardinality for h is also a minimal dominating set of minimum cardinality for g. � definition 2.2. the graph g is called k−domination regular graph if and only if dd(v) = k for all v ∈ v (g). example 2.1. sr and kn are 1−domination regular graph. proposition 2.3. let g be the double star graph sr,s . then tm(g) = 4, and any double star graph is 2−domination regular graph. proof. let {v,v1, ...,vr−1,w,w1, ...,ws−1} be the set of all vertices of g with {v,w} be the center vertices. there are four type of minimal dominating sets as following: {v,w}, {v1,v2, ...,vr−1,w1,w2, ...,ws−1}, {v,w1,w2, ...,ws−1} and {w,v1,v2, ...,vr−1}. ⇒ tm(g) = 4. from this we get, dd(v) = 2 for all v ∈ v (g). � 3. domination zagreb indices of a graph definition 3.1. let g be a simple connected graph, the first domination, second domination zagreb and modified first zagreb indices are define as : dm1(g) = ∑ v∈v (g) d2d(v) , dm2(g) = ∑ uv∈e(g) dd(u)dd(v) , dm∗1 = ∑ uv∈e(g) [dd(u) + dd(v)] . lemma 3.1. tm(sr) = 2 and tm(kn) = n. for all v ∈ v (sr) or v ∈ v (kn), we get dd(v) = 1. proposition 3.1. (1) for the star graph sr, with r + 1 vertices. dm1(sr) = r + 1 , dm2(sr) = r and dm ∗ 1 (sr) = 2r. (2) for the complete graph kn dm1(kn) = n , dm2(kn) = n(n− 1) 2 and dm∗1 (kn) = n(n− 1). int. j. anal. appl. 19 (1) (2021) 51 (3) for the double star graph sr,s dm1(sr,s) = 4(r + s + 2) dm2(sr,s) = 4(r + s + 1) and dm ∗ 1 (sr,s) = 4(r + s + 1). lemma 3.2. tm(kr,s) = rs + 2 and dd(v) =   r + 1;s + 1. for all v ∈ v (kr,s). theorem 3.1. if g ∼= kr,s, then dm1(g) = r(r + 1) 2 + s(s + 1)2 , dm2(g) = (r + 1)(s + 1)rs, dm∗1 (g) = rs(r + s + 2). proof. by using the definition of domination zagreb indices and lemma 3.2, we get the results. � corollary 3.1. let g be the complete bipartite kr,s. then (1) dm1(g) = m1(g) + 4rs + (r + s). (2) dm2(g) = m2(g) + m1(g) + rs. (3) dm∗1 (g) = m1(g) + 2rs. proof. in the complete bipartite kr,s, we can see that dd(vi) = d(vi) + 1 for all i = 1, 2, ...,r + s dm1(g) = ∑ v∈v (g) d2d(v) = ∑ v∈v (g) (d(v) + 1)2 = m1(g) + 4rs + (r + s) and, dm2(g) = ∑ uv∈e(g) dd(u)dd(v) = ∑ uv∈e(g) (d(u) + 1)(d(v) + 1) = m2(g) + m1(g) + rs, dm∗1 (g) = ∑ uv∈e(g) [dd(u) + dd(v)] = ∑ uv∈e(g) [(d(u) + 1) + (d(v) + 1)] = m1(g) + 2rs. � proposition 3.2. int. j. anal. appl. 19 (1) (2021) 52 if g ∼= kr,s, then (1) dm1(g) = dm ∗ 1 (g) = m1(g). (2) dm2(g) = m2(g). proof. see proposition 2.1. � lemma 3.3. let g be the windmill graph wdsr. then tm(wd s r) = (r − 1)s + 1. and dd(v) =   1, if v is the center vertex;(r − 1)s−1, otherwise. theorem 3.2. if g ∼= wdsr, then dm1(g) = 1 + s(r − 1)2s−1, dm2(g) = s((r − 1)s + (r − 1)2s−1( r − 2 2 )), dm∗1 (g) = s(r − 1)(1 + (r − 1) s). proof. dm1(g) = ∑ v∈v (g) d2d(v) = 1 + ∑ v∈v (g)−1 d2d(v) = 1 + (r − 1)2(s−1)(|v (g)|− 1) = 1 + s(r − 1)2s−1. let e1 denote the set of all edges which are incident with the center vertex and e2 be the set of all edges of the complete graph, then dm2(g) = ∑ uv∈e(g) dd(u)dd(v) = ∑ uv∈e1 dd(u)dd(v) + ∑ uv∈e2 dd(u)dd(v) = (r − 1)s−1|e1| + s(r − 1)2s−2|e2| = s((r − 1)s + (r − 1)2s−1( r − 2 2 )), dm∗1 (g) = ∑ uv∈e(g) [dd(u) + dd(v)] = ∑ uv∈e1 [dd(d) + dd(v)] + ∑ uv∈e2 [dd(u) + dd(v)] = (1 + (r − 1)s−1)|e1| + 2s(r − 1)s−1|e2| = s(r − 1)(1 + (r − 1)s). � int. j. anal. appl. 19 (1) (2021) 53 proposition 3.3. if g is k−domination regular graph with n vertices, and m edges, then dm1(g) = nk 2, dm2(g) = mk 2 and dm∗1 (g) = 2mk. definition 3.2. let p3 be the 3 vertex tree, is rooted in one of its terminal vertices . for k = 2, 3, 4, ... build the rooted tree bk by identifying the roots of k-copies of p3. the vertex obtained by identifying the roots of p3−trees is the root of bk [16]. definition 3.3. let d ≥ 2 be an integer. let β1,β2, ...,βd be as specified in definition 3.2 i.e., β1,β2, ...,βd ∈ {b2,b3, ...}. a kragujevac tree t is a tree has a vertex of degree d, neighboring to the roots of β1,β2, ...,βd. this vertex be the centralvertex of t , where d is the degree of the tree t . the subgraphs β1,β2, ...,βd are the branches of t . recall that some (or all) branches of t may be mutually isomorphic [16]. the branch bk has 2k + 1 vertices. therefore, if in the kragujevac tree t, specified in definition 3.3, βi ∼= βki , i = 1, 2, ...,d then its order is n(t) = 1 + ∑d i=1(2ki + 1). proposition 3.4. let t be the kragujevac tree of order n(t) = 1 + ∑d i=1(2ki + 1) and size m. then dm1(t) = 4[1 + d∑ i=1 (2ki + 1)], dm2(t) = dm ∗ 1 (t) = 4m. proof. it is easy to see that, in the kragujevac tree of order n(t) = 1 + ∑d i=1(2ki + 1) and size m there are four types of minimal dominating sets. the set which contains the center vertex and all pendent vertices, the set which contains the center vertex and all vertices adjacent to the pendent vertices, the set which contains the roots of β1,β2, ...,βd and all pendent vertices, and the set which contains the roots of β1,β2, ...,βd and all vertices adjacent to the pendent vertices. hence, dd(v) = 2 for all v ∈ v (t). so by using the definition of domination zagreb indices we get the result. � definition 3.4. let g1 and g2 be any two graphs. the cartesian product g1 ×g2 is defined as [6] the graph has vertex set (v (g1) ×v (g2)) such that any two vertices u = (u1,u2) and v = (v1,v2) are adjacent if and only if either ([u1 = v1 and {u2,v2}∈ e(g2)]) or ([u2 = v2 and {u1,v1}∈ e(g1)]). definition 3.5. int. j. anal. appl. 19 (1) (2021) 54 book graph br is a cartesian product of a star and single edge sr+1 × p2. the generalization of the book graph to n “stacked“ is the (r,s)−stacked book graph [13]. lemma 3.4. if g ∼= br, then tm(g) = 2r + 3. further, for any vertex v ∈ v (br) dd(v) =   3, if v is the center vertex;2r−1 + 1, otherwise. proof. let uv be the center edge in book graph such that {u,v} is the set of center vertices. let {v1,v2, ...,vr} be the set of all vertices which are adjacent with the center vertex v. similarly {u1,u2, ...,ur} be the set of all vertices which are adjacent with the center vertex u. there are four types of minimal dominating sets. first type is{u,v}. second type is {v,u1,u2, ...,ur} and {u,v1,v2, ...,vr}. third type is {u,u1,u2, ...,ur} and {v,v1,v2, ...,vr}. fourth type is only those minimal dominating sets which are formed by taking one vertex from each section other than u and v. so there are 2r − 2 minimal dominating sets of fourth type. hence, tm(br) = 2 r + 3 and for all v ∈ v (br) we get dd(v) =   3, if v is the center vertex;2r−1 + 1, otherwise. � theorem 3.3. let g be a book graph br where r ≥ 3. then dm1(br) = 2r(2 r−1 + 1)2 + 18 , dm2(br) = r(2 r−1 + 1)[2r−1 + 7] + 9 , dm∗1 (br) = 2 r+1r + 2r(4 + 2r−1) + 6 . proof. dm1(br) = ∑ w∈v (br) d2d(w) = ∑ w∈v (br−{u,v}) (2r−1 + 1)2 + ∑ w∈{u,v} 32 = 2r(2r−1 + 1)2 + 18. there are three type of edges in the book graph. let e1 denote the set of r edges (uivi) with initial and terminal vertices of the same domination degree 2r−1 + 1, e2 denote the set containing only one edge (uv) with initial and terminal vertices of the same domination degree 3, and e3 denote the set of 2r edges of initial vertices of the domination degree 3 and terminal vertices of domination degree 2r−1 + 1. hence, int. j. anal. appl. 19 (1) (2021) 55 dm2(br) = ∑ uv∈e(br) dd(u)dd(v) = ∑ uv∈e1 (2r−1 + 1)2 + ∑ un∈e2 9 + ∑ uv∈e3 3(2r−1 + 1) = r(2r−1 + 1)[2r−1 + 7] + 9 , dm∗1 (br) = ∑ uv∈e(br) [dd(u) + dd(v)] = ∑ uv∈e1 [(2r−1 + 1) + (2r−1 + 1)] + ∑ uv∈e2 [3 + 3] + ∑ uv∈e3 [3 + 2r−1 + 1] = 2r+1 + 2r(4 + 2r−1) + 6. � lemma 3.5. let g ∼= kn1,n2,...,nk . then tm(g) = k∑ i=2 n1ni + k∑ i=3 n2ni + ... + nk−1nk + k . theorem 3.4. if g ∼= kn1,n2,...,nk , then dm1(g) = m1(g) + 4(tm(g) −k) + k∑ i=1 ni , dm2(g) = m2(g) + m1(g) + tm(g) −k , dm∗1 (g) = m1(g) + 2(tm(g) −k) . proof. suppose g ∼= kn1,n2,...,nk , note that for any vertex v ∈ g we have dd(v) = d(v) + 1, and |e(g)| = tm(g) −k. so, by the definition of domination zagreb indices we get the result. � lemma 3.6. for any connected graph g with n1 vertices and m1 edges. let h ∼= g◦kn2 , where kn2 is the complete graph of n2 vertices and m2 edges. there are (n2 + 1) n1 minimal domination sets in h, and dd(v) = (n2 + 1) n1−1. theorem 3.5. for any connected graph g of n1 vertices and m1 edges, we have dm1(g◦kn2 ) = (n1 + n1n2)(n2 + 1) 2(n1−1) , dm2(g◦kn2 ) = (n2 + 1) 2(n1−1)[2m1 + n2(n2 + 2n1 − 1)] , dm∗1 (g◦kn2 ) = 4(n2 + 1) n1−1[2m1 + n2(n2 + 2n1 − 1)] . int. j. anal. appl. 19 (1) (2021) 56 proof. note that |v (g◦kn2 )| = n1 + n1n2. hence, by the definition of first domination zagreb indices and lemma 3.6, we get dm1(g◦kn2 ) = (n1 + n1n2)(n2 + 1) 2(n1−1). there are three types of edges in g◦kn2 . all edges of g, all edges of kn2 and let e1 denote the set of all edges that connect vertex from g and vertex from kn2 . so, we have dm2(g◦kn2 ) = ∑ uv∈e(gokn2 ) dd(u)dd(v) = ∑ uv∈e(g) dd(u)dd(v) + ∑ uv∈e(kn2 ) dd(u)dd(v) + ∑ uv∈e1 dd(u)dd(v) = m1(n2 + 1) 2(n1−1) + (n2 + 1) 2(n1−1)|e(kn2 )| + n1n2(n2 + 1) 2(n1−1) = (n2 + 1) 2n1−2[m1 + n2(n2 − 1) 2 + n1n2] = (n2 + 1) 2(n1−1)[2m1 + n2(n2 + 2n1 − 1)], dm∗1 (g◦kn2 ) = ∑ uv∈e(g◦kn2 ) [dd(u) + dd(v)] = ∑ uv∈e(g) 2(n2 + 1) n1−1 + ∑ uv∈e(kn2 ) 2(n2 + 1) n1−1 + ∑ uv∈e1 2(n2 + 1) n1−1 = 4(n2 + 1) n1−1[2m1 + n2(n2 + 2n1 − 1)]. � lemma 3.7. let h ∼= g◦kn2 , where g be any connected graph of order n1. then tm(h) = n1∑ i=0 ( n1 i ) . theorem 3.6. if g be a graph of order n1 and size m1. let h ∼= g◦kn2 then dm1(h) = (tm(h) − 2n1−1)2(n1 + n1n2) , dm2(h) = (tm(h) − 2n1−1)2(m1 + n1n2) , dm∗1 (h) = (2tm(h) − 2n1 )(m1 + n1n2) . proof. for any vertex v ∈ v (h), it is not easy to see that h ∼= g◦kn2 is domination regular graph. every v ∈ v (h) is contained in every minimal dominating sets of h except int. j. anal. appl. 19 (1) (2021) 57 ( n1−1 0 ) + ( n1−1 1 ) + ... + ( n1−1 n1−2 ) + ( n1−1 n1−1 ) = 2n1−1 minimal dominating sets. hence, dd h (v) = tm(h) − 2n1−1 and dm1(h) = (tm(h) − 2n1−1)2(n1 + n1n2) , dm2(h) = (tm(h) − 2n1−1)2(m1 + n1n2) , dm∗1 (h) = (2tm(h) − 2n1 )(m1 + n1n2) . � a join of two graphs g1 and g2 is denoted by g1 + g2, with disjoint vertex sets v1 and v2 is the graph on the vertex set v1 ∪v2 and the edge set e1 ∪e2 ∪{u1u2 : u1 ∈ v1,u2 ∈ v2} [1]. lemma 3.8. let g1 and g2 be any non complete graphs of n1, n2 vertices respectively. then tm(g1 + g2) = tm(g1) + tm(g2) + n1n2, and dd g1+g2 (v) =   dd g1 (v) + n2, if v ∈ v (g1);dd g2 (v) + n1, if v ∈ v (g2). proof. there are three types of minimal dominating sets in g1 + g2 graph: the minimal-dominating sets of g1, all the minimal dominating sets of g2 and the sets of size two of all minimal dominating sets containing one vertex from g1 and another vertex from g2. hence, tm(g1 + g2) = tm(g1) + tm(g2) + n1n2, and dd g1+g2 (v) =   dd g1 (v) + n2, if v ∈ v (g1);dd g2 (v) + n1, if v ∈ v (g2). � theorem 3.7. let g1 and g2 be any non complete graphs having n1, n2 vertices and m1, m2 edges respectively. then (1) dm1(g1 + g2) = dm1(g1) + dm2(g2) + 2n2ρ(g1) + 2n1ρ(g2) + n1(n 2 2 + n2n1), (2) dm2(g1 + g2) = dm2(g1)(1 + n2) + m1n 2 2 + dm2(g2)(1 + n1) + m2n 2 1 + [n1n2 + ρ(g1)][n1n2 + ρ(g2)], int. j. anal. appl. 19 (1) (2021) 58 (3) dm∗1 (g1 + g2) = dm ∗ 1 (g1) + dm ∗ 1 (g2) + 2m1n2 + 2n1m2 + (ρ(g2) + n2(n1 + 1))(ρ(g1) + n1n2). proof. dm1(g1 + g2) = ∑ v∈v (g1+g2) d2d g1+g2 (v) = ∑ v∈v (g1) (dd g1 (v) + n2) 2 + ∑ v∈v (g2) (dd g2 (v) + n1) 2 = ∑ v∈v (g1) dd g1 (v) 2 + 2n2 ∑ v∈v (g1) dd g1 (v) + n 2 2 ∑ v∈v (g1) 1 + ∑ v∈v (g2) dd g2 (v) 2 + 2n1 ∑ v∈v (g2) dd g2 (v) + n 2 1 ∑ v∈v (g1) 1 = dm1(g1) + dm2(g2) + 2n2 ∑ v∈v (g1) dd g1 (v) + 2n1 ∑ v∈v (g2) dd g2 (v) + n1(n 2 2 + n2n1) = dm1(g1) + dm2(g2) + 2n2ρ(g1) + 2n1ρ(g2) + n1(n 2 2 + n2n1). and, dm2(g1 + g2) = ∑ uv∈e(g1+g2) dd g1+g2 (u)dd g1+g2 (v) = ∑ uv∈e(g1) dd g1+g2 (u)dd g1+g2 (v) + ∑ uv∈e(g2) dd g1+g2 (u)dd g1+g2 (v) + ∑ u∈v (g1),v∈v (g2) dd g1+g2 (u)dd g1+g2 (v) we will find every part independently (1) ∑ uv∈e(g1) dd g1+g2 (u)dd g1+g2 (v) = ∑ uv∈e(g1) (dd g1 (u) + n2)(dd g1 (v) + n2) = dm2(g1)(1 + n2) + m1n 2 2 int. j. anal. appl. 19 (1) (2021) 59 (2) ∑ uv∈e(g2) dd g1+g2 (u)dd g1+g2 (v) = ∑ uv∈e(g2) (dd g2 (u) + n1)(dd g2 (v) + n1) = dm2(g2)(1 + n1) + m2n 2 1 (3) ∑ u∈v (g1),v∈v (g2) dd g1+g2 (u)dd g1+g2 (v) = (dd g1 (u1) + n2)(dd g2 (v1) + n1) + ... + (dd g1 (u1) + n2)(dd g2 (vn2 ) + n1) + (dd g1 (u2) + n2)(dd g2 (v1) + n1) + ... + (dd g1 (u2) + n2)(dd g2 (vn2 ) + n1) + ... + (dd g1 (un1 ) + n2)(dd g2 (v1) + n1) + ... + (dd g1 (un1 ) + n2)(dd g2 (vn2 ) + n1) = (dd g1 (u1) + n2)[ ∑ v∈v (g2) (dd g2 (v) + n1)] + ... + (dd g1 (un1 ) + n2)[ ∑ v∈v (g2) (dd g2 (v) + n1)] = [ ∑ u∈v (g1) (dd g1 (u) + n2)][ ∑ v∈v (g2) (dd g2 (v) + n1)] = [n1n2 + ∑ u∈v (g1) dd g1 (u)][n1n2 + ∑ v∈v (g2) dd g2 (v)] = [n1n2 + ρ(g1)][n1n2 + ρ(g2)] hence, dm2(g1 + g2) = dm2(g1)(1 + n2) + m1n 2 2 + dm2(g2)(1 + n1) + m2n 2 1 + [n1n2 + ρ(g1)][n1n2 + ρ(g2)]. and, dm∗1 (g1 + g2) = ∑ uv∈e(g1+g2) [ddg1+g2 (u) + ddg1+g2 (v)] = ∑ uv∈e(g1) [dd g1+g2 (u) + dd g1+g2 (v)] + ∑ uv∈e(g2) [dd g1+g2 (u) + dd g1+g2 (v)] + ∑ u∈v (g1),v∈v (g2) [dd g1+g2 (u) + dd g1+g2 (v)]. int. j. anal. appl. 19 (1) (2021) 60 we will find every part independently (1) ∑ uv∈e(g1) [dd g1+g2 (u) + dd g1+g2 (v)] = ∑ uv∈e(g1) (dd g1 (u) + n2) + (dd g1 (v) + n2) = dm∗1 (g1) + 2m1n2, (2) ∑ uv∈e(g2) [dd g1+g2 (u) + dd g1+g2 (v)] = ∑ uv∈e(g2) (dd g2 (u) + n1) + (dd g2 (v) + n1) = dm∗1 (g2) + 2n1m2, (3) ∑ u∈v (g1),v∈v (g2) [dd g1+g2 (u) + dd g1+g2 (v)] = (dd g1 (u1) + n2) + (dd g2 (v1) + n1) + ... + (dd g1 (u1) + n2) + (dd g2 (vn2 ) + n1) + (dd g1 (u2) + n2) + (dd g2 (v1) + n1) + ... + (dd g1 (u2) + n2) + (dd g2 (vn2 ) + n1) + ... + (dd g1 (un1 ) + n2) + (dd g2 (v1) + n1) + ... + (dd g1 (un1 ) + n2) + (dd g2 (vn2 ) + n1) = (dd g1 (u1) + n2)[ ∑ v∈v (g2) (dd g2 (v) + n1) + n2] + ... + (dd g1 (un1 ) + n2)[ ∑ v∈v (g2) (dd g2 (v) + n1) + n2] = [ ∑ v∈v (g2) (dd g2 (v)) + n2(n1 + 1)][ ∑ u∈v (g1) (dd g1 (u)) + n1n2)] = (ρ(g2) + n2(n1 + 1))(ρ(g1) + n1n2). hence, dm∗1 (g1 + g2) = dm ∗ 1 (g1) + dm ∗ 1 (g2) + 2m1n2 + 2n1m2 + (ρ(g2) + n2(n1 + 1))(ρ(g1) + n1n2). � int. j. anal. appl. 19 (1) (2021) 61 corollary 3.2. dm1(g1 + g2) ≤ dm1(g1) + dm2(g2) + 2n1γ(g1)tm(g1) + 2n1γ(g2)tm(g2) + n1(n 2 2 + n2n1). corollary 3.3. dm2(g1 + g2) ≤ dm2(g1)(1 + n2) + m1n22 + dm2(g2)(1 + n1) + m2n 2 1 + [n1n2 + γ(g1)tm(g1)][n1n2 + γ(g2)tm(g2)]. corollary 3.4. dm∗1 (g1 + g2) ≤ dm ∗ 1 (g1) + dm ∗ 1 (g2) + 2m1n2 + 2n1m2 + (γ(g2)tm(g2) + n2(n1 + 1))(γ(g1)tm(g1) + n1n2). 4. some bounds of domination zagreb indices theorem 4.1. let g be a graph of order n. then dm1(g) ≥ 1n (ρ(g)) 2. equality hold if and only if g is one-domination regular graph. proof. we have dm1(g) = ∑ v∈v (g) d 2 d(v) = d 2 d(v1) + d 2 d(v2) + ... + d 2 d(vn) we use cauchy-schwartz inequality on vectors (dd(v1),dd(v2), ...,dd(vn)) and (1, 1, ..., 1) to get dm1(g).n = (d 2 d(v1),d 2 d(v2), ...,d 2 d(vn))(1 2, 12, ..., 12) ≥ (dd(v1).1 + dd(v2).1 + ... + dd(vn).1)2 = ( n∑ i=1 dd(vi)) 2 = (ρ(g))2. to prove the equality, suppose g is one domination regular graph ⇒ dd(vi) = 1 for all 1 ≤ i ≤ n. and dm1(g) = n. conversely if dm1(g) = 1 n ( ∑n i=1 dd(vi)) 2 ⇒ dm1(g).n = ( ∑n i=1 d(vi)) 2, hence g is one domination regular graph. � theorem 4.2. let g be a graph with n vertices. then dm1(g) ≤ ( ∑ v∈v (g) √ dd(v)) 2. int. j. anal. appl. 19 (1) (2021) 62 proof. dm1(g) = ∑ v∈v (g) d 2 d(v) = d 2 d(v1) + d 2 d(v2) + ... + d 2 d(vn) as dd(v1),dd(v2), ...,dd(vn) is positive integers so, we get dm1(g) ≤ ( ∑ v∈v (g) √ dd(v)) 2. � proposition 4.1. if g be any graph such that |v (g)| = n, then n ≤ dm1(g) ≤ n(tm(g))2 , ρ(g) + n ≤ dm∗1 (g) ≤ ntm(g) + ρ(g). proof. see observation 2.1. � theorem 4.3. let g be a graph such that g 6∼= k2. then dm2(g) ≥ γ(g)tm(g) equality hold if and only if g ∼= p3. proof. note that dd(v) ≥ 1, so, ∑ uv∈e(g) dd(u)dd(v) ≥ ∑ u∈v (g) dd(u). from observation 2.3, we get dm2(g) ≥ γ(g)tm(g). � theorem 4.4. suppose g is a connected simple graph. then dm2(g) ≤ γ(g)(tm(g))2. proof. we have dd(v) ≤ tm(g), so ∑ uv∈e(g) dd(u)dd(v) ≤ tm(g) ∑ u∈v (g) dd(u). by observation 2.3, dm2(g) ≤ γ(g)(tm(g))2. � finally, we can generalize the definition of the domination degree of the vertex by using any subset of vertices with property p like an independent set, independent dominating set, total dominating set, hup set, edge dominating set, different distance set,... so on. definition 4.1. let g = (v,e) be a graph and let s be any subset of vertices with property p . then for any vertex v, the p set degree of the vertex v denoted by dp (v) = |{s ⊆ v (g) : s has property p and v ∈ s}|. int. j. anal. appl. 19 (1) (2021) 63 and we can define the zagreb and forgotten indices as following pm1(g) = ∑ v∈v (g) d2p (v) , pm∗1 (g) = ∑ uv∈e(g) dp (u) + dp (v) , pm2(g) = ∑ uv∈e(g) dp (u)dp (v) , pf(g) = ∑ v∈v (g) d3p (v) , pf∗(g) = ∑ uv∈e(g) d2p (u) + d 2 p (v) . 5. conclusion in this research work, we define new topological indices based on the minimal dominating sets. the authors are working now in some of the other types of topological indices by replacing the standers degree of the vertex by the domination degree of the vertex. finally we have the following open problems for research: (1) is there any relation between the normal degree and domination degree of the graph. (2) what is the necessary and sufficient conditions in a graph to become domination regular graph. (3) what is the natural relation between the domination indices. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.r. ashrafi, t. doslic, a. hamzeha, the zagreb coindices of graph operations, discrete appl. math. 158 (2010), 1571– 1578. [2] t. al-fozan, p. manuel, i. rajasingh, r. sundara rajan, computing szeged index of certain nanosheets using partition technique, match commun. math. comput. chem. 72 (2014), 339–353. [3] m. azari, a. iranmanesh, harary index of some nano-structures, match commun. math. comput. chem. 71 (2014), 373–382. [4] b. basavanagoud, on minimal and vertex minimal dominating graph, j. inform. math. sci. 1 (2009), 139—146. [5] d. dimitrov, on structural properties of trees with minimal atom-bond connectivity index ii: bounds on b 1 and b 2 -branches, discrete appl. math. 204 (2016), 90–116. [6] t. došlić, m. hosseinzadeh, eccentric connectivity polynomial of some graph operations, utilitas math. 84 (2011), 297–309. [7] f.v. fomin, f. grandoni, a.v. pyatkin, a.a. stepanov. combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. acm trans. algorithms 5(1) (2008), 9. [8] b. furtula, i. gutman, and m. dehmer, on structure-sensitivity of degree-based topological indices, appl. math. comput. 219 (2013), 8973—8978. [9] i. gutman, degree-based topological indices, croat. chem. acta, 86 (2013), 351—361. int. j. anal. appl. 19 (1) (2021) 64 [10] i. gutman, k.c. das, the first zagreb index 30 years after, match commun. math. comput. chem. 50 (2004) 83—92. [11] i. gutman, b.ruščić, n.trinajstić, c.f. wilcox, graph theory and molecular orbitals. xii. acyclic polyenes, j. chem. phys. 62 (1975), 3399–3405. [12] i. gutman, n. trinajstić, graph theory and molecular orbitals. total π-electron energy of alternant hydrocarbons, chem. phys. lett. 17 (1972), 535—538. [13] f. harary, graph theory, narosa publishing house, new delhi, 2001. [14] t.w. haynes, s.t. hedetniemi, p.j.slater, fundamentals of domination in graphs, marcel dekker, new york. (1998). [15] t.w. haynes, s.t. hedetniemi, p.j. slater, (eds.) domination in graphs: advanced topics. marcel dekker, inc. new york. (1998). [16] ivan gutman, kragujevac trees and their energy, appl. math. inform. mech. 2 (2014), 71-79. [17] o. ivanciuc, t.s. balaban, and a.t. balaban, reciprocal distance matrix, related local vertex invariants and topological indices, j. math. chem. 12 (1993), 309–318. [18] f.c. jear, r. letourneur, m. liedloff, on the number of minimal dominating sets on some graph classes, theor. comput. sci. 562 (2015), 634–642. [19] b.n. kavitha, i.p. kelkar, k.r. rajanna, perfect domination in book graph and stacked book graph, int. j. math. trends technol. 56 (2018), 511-514. [20] m.f. nadeem, s. zafar, z. zahid, on certain topological indices of the line graph of subdivision graphs, appl. math. comput. 271 (2015), 790–794. [21] p.s. ranjini, v. lokesha, the smarandache-zagreb indices on the three graph operators, int. j. math. comb. 3 (2010), 1–10. [22] v. sharma, r. goswami, a.k. madan, eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, j. chem. inf. comput. sci. 37 (1997), 273–282. [23] m. teresa de bustos muñoz, j.l.g. guirao, j. vigo-aguiar, decomposition of pseudo-radioactive chemical products with a mathematical approach, j. math. chem. 52 (4) (2014), 1059–1065. [24] h. wiener, structural determination of the paraffin boiling points, j. amer. chem. soc. 69 (1947), 17-20. 1. introduction 2. domination degree in graphs 3. domination zagreb indices of a graph 4. some bounds of domination zagreb indices 5. conclusion references international journal of analysis and applications volume 19, number 2 (2021), 165-179 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-165 ∗−conformal η−ricci solitons on α−cosymplectic manifolds abdul haseeb1, d. g. prakasha2,∗, h. harish3 1department of mathematics, faculty of science, jazan university, jazan-2097, kingdom of saudi arabia 2department of mathematics, davangere university, davangere 577 007, india 3department of mathematics, sri mahaveera college, kodangallu post, mudbidri 574 197, india ∗corresponding author: prakashadg@gmail.com abstract. the object of this paper is to study ∗−conformal η−ricci solitons on α−cosymplectic manifolds. first, α−cosymplectic manifolds admitting ∗−conformal η−ricci solitons satisfying the conditions r(ξ, .) ·s and s(ξ, .) ·r = 0 are being studied. further, α−cosymplectic manifolds admitting ∗−conformal η−ricci solitons satisfying certain conditions on the m−projective curvature tensor are being considered and obtained several interesting results. among others it is proved that a φ−m−projeectively semisymmetric α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton is an einstein manifold. finally, the existence of ∗−conformal η−ricci soliton in an α−cosymplectic manifolds has been proved by a concrete example. 1. introduction in recent years, ricci solitons and their generalizations are enjoying rapid growth by providing new techniques in understanding the geometry and topology of arbitrary riemannian manifolds. ricci soliton is a natural generalization of einstein metric, and is also a self-similar solution to hamilton’s ricci flow [20,21]. it plays a specific role in the study of singularities of the ricci flow. a solution g(t) of the non-linear evolution pde: ∂ ∂t g(t) = −2s(g(t)), t ∈ [0,i] is called the ricci flow [30], where s is the ricci tensor field associated received september 3rd, 2020; accepted september 30th, 2020; published february 1st, 2021. 2010 mathematics subject classification. 53c21, 53c25, 53d15. key words and phrases. α−cosymplectic manifold; ∗−conformal η−ricci soliton; η−einstein manifold; einstein manifold; m−projective curvature tensor. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 165 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-165 int. j. anal. appl. 19 (2) (2021) 166 to the metric g. a riemannian manifold (m,g) is called a ricci soliton (g,v,λ) if there are a smooth vector field v and a scalar λ ∈ r such that (1.1) s + £v g = λg on m, where s is the ricci tensor and £v g is the lie derivative of the metric g. if the potential vector field v vanishes identically, then the ricci soliton becomes trivial, and in this case manifold is an einstein one. as a generalization of ricci soliton, the notion of η-ricci soliton was introduced by cho and kimura [10]. an η-ricci soliton is a tuple (g,v,λ,µ), where v is a vector field on m, λ and µ are constants, and g is a riemannian metric satisfying the equation (1.2) £v g + 2s + 2λg + 2µη ⊗η = 0, where s is the ricci tensor associated to g. the notion of ∗-ricci soliton has been studied by kaimakamis and panagiotidou [22] within the framework of real hypersurfaces of complex space forms. they essentially modified the definition of ricci soliton by replacing the ricci tensor s in (1.1) with the ∗-ricci tensor s∗. here, it is mentioned that the notion of ∗-ricci tensor was first introduced by tachibana [39] on almost hermitian manifolds and further studied by hamada [19] on real hypersurfaces of non-flat complex space forms. a riemannian metric g on a smooth manifold m is called a ∗-ricci soliton if there exists a smooth vector field v and a real number λ, such that (£v g)(x,y ) + 2s ∗(x,y ) + 2λg(x,y ) = 0(1.3) where s∗(x,y ) = g(q∗x,y ) = trace{φ◦r(x,φy )}(1.4) for all vector fields x,y on m. here, φ is a tensor field of type (0, 2). in this connection, we recommend the papers [5, 6, 11, 15, 24, 31, 32, 36, 40] and the references therein for more details about the study of ricci solitons, η-ricci solitons and ∗-ricci solitons in the context of contact riemannian geometry. in 2004, fischer [14] introduced a variation of the classical ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. the resulting equations are named the conformal ricci flow equations and are given by ∂g ∂t + 2(s + g n ) = −pg and r = −1, for a dynamically evolving metric g, ricci tensor s, scalar curvature r and a scalar non-dynamical field p. since, these equations are the vector field sum of a conformal flow equation and a ricci flow equation, they play an important role in conformal geometry. in the riemannian setting, the notion of conformal ricci int. j. anal. appl. 19 (2) (2021) 167 soliton was introduced by basu and bhattacharyya [3] on a kenmotsu manifold of dimension n as (1.5) £v g + 2s = (2λ− (p + 2 n ))g, where λ is a constant and £v is the lie derivative along the vector field v . this notion was also studied by several authors on various kinds of almost contact metric manifolds (see, [13, 25, 37]). further, siddiqi [38] introduced the notion of conformal η-ricci soliton as (1.6) £v g + 2s + (2λ− (p + 2 n ))g + 2µη ⊗η = 0, where λ and µ are constants. recently, roy et al. [37] introduced and studied the notion of ∗−conformal η−ricci soliton on an n-dimensional sasakian manifold. a riemannian metric g on m is called ∗−conformal η−ricci soliton, if £ξg + 2s ∗ + (2λ− (p + 2 n ))g + 2µη ⊗η = 0,(1.7) where £ξ is the lie derivative along the vector field ξ, s ∗ is the ∗−ricci tensor and λ, µ are constants. on the other hand, the geometry of contact riemannian manifolds and related topics have also drawn a great deal of interest in the last years. an important class of almost contact manifolds is given by cosymplectic manifolds. they were introduced by goldberg and yano [16] in 1969. a cosymplectic manifold is a (2n+1)-dimensional smooth manifold equipped with closed 1-form η and closed 2-form ω such that η∧ωn is a volume form. the products of almost kaehlerian manifolds and the real line r or the s1 circle are the simplest examples of almost cosymplectic manifolds [28]. we refer to [9] for a nice overview on cosymplectic geometry and its connection with other areas of mathematics (especially, geometric mechanics) as well as with physics. in this paper we undertake the study of α−cosymplectic manifolds admitting ∗-conformal η-ricci solitons. the present paper is organized as follows: section 2 is concerned about preliminaries on α−cosymplectic manifolds. in section 3, α−cosymplectic manifolds admitting a ∗−conformal η−ricci solitons is being studied. section 4 is devoted to the study of m−projective curvature tensor on α−cosymplectic manifolds admitting ∗-conformal η-ricci solitons. in the last section, we construct an example of a 5-dimensional manifold which verifies existence of ∗-conformal η-ricci soliton on a α−cosymplectic manifold. 2. preliminaries let m be an n-dimensional differentiable manifold equipped with a triple (φ,ξ,η), where φ is a (1, 1)-tensor field, ξ is a vector field, η is a 1-form on m such that [7] (2.1) φ2 = −i + η ⊗ ξ, η(ξ) = 1, int. j. anal. appl. 19 (2) (2021) 168 which implies (2.2) φξ = 0, η ·φ = 0, rank(φ) = n− 1. if m admits a riemannian metric g, such that (2.3) g(φx,φy ) = g(x,y ) −η(x)η(y ), g(x,ξ) = η(ξ) then m is said to admit almost contact structure (φ,ξ,η,g) [7]. on such a manifold the 2-form φ of m is defined as (2.4) φ(x,y ) = g(φx,y ) for all x,y ∈ χ(m); where χ(m) denotes the collection of all smooth vector fields of m. an almost contact metric manifold (m,φ,ξ,η,g) is said to be almost cosymplectic [16] if dη = 0 and dφ = 0, where d is the exterior differential operator. an almost contact manifold (m,φ,ξ,η,g) is said to be normal if the nijenhuis torsion nφ(x,y ) = [φx,φy ] −φ[φx,y ] −φ[x,φy ] + φ2[x,y ] + 2dη(x,y )ξ vanishes for any vector fields x and y . a normal almost cosymplectic manifolds is called a cosymplectic manifolds. an almost contact metric manifold m is said to be almost α-kenmotsu if dη = 0 and dφ = 2αη ∧ φ, α being a non-zero real constant. kim and pak [23] combined almost α-kenmotsu and almost cosymplectic manifolds into a new class called almost α-cosymplectic manifolds, where α is a scalar. if we join these two classes, we obtain a new notion of an almost α-cosymplectic manifold, which is defined by the following formula dη = 0, dφ = 2αη ∧ φ, for any real number α. a normal almost α-cosymplectic manifold is called an α-cosymplectic manifold. an α-cosymplectic manifold is either cosymplectic under the condition α = 0 or α-kenmotsu under the condition α 6= 0, for α ∈ r. for detailed study of α−cosymplectic manifolds we refer to the readers ( [1, 2, 4, 8, 17, 29]) and many others. in an α−cosymplectic manifold, we have [18] (2.5) (∇xφ)y = α(g(φx,y )ξ −η(y )φx). let m be a n-dimensional α-cosymplectic manifold. from eq. (2.5), it is easy to see that (2.6) ∇xξ = −αφ2x = α[x −η(x)ξ]. int. j. anal. appl. 19 (2) (2021) 169 where ∇ denotes the riemannian connection. on an α-cosymplectic manifold m, the following relations are hold: (2.7) η(r(x,y )z) = α2(η(y )g(x,z) −η(x)g(y,z)), (2.8) r(ξ,x)y = α2(η(y )x −g(x,y )ξ), (2.9) r(x,y )ξ = α2(η(x)y −η(y )x), (2.10) r(ξ,x)ξ = α2(x −η(x)ξ), (2.11) s(x,ξ) = −α2(n− 1)η(x) for all vector fields x,y,z ∈ χ(m). definition 2.1. an α−cosymplectic manifold is said to be an η-einstein manifold if the ricci tensor s is of the form [41] s(x,y )y = ag(x,y ) + bη(x)η(y ), where a and b are smooth functions on the manifold. if b = 0, then the manifold is said to be an einstein manifold. lemma 2.1. in an α−cosymplectic manifold (m,φ,ξ,η,g), we have r̄(x,y,φz,φw) = r̄(x,y,z,w)(2.12) +α2[φ(x,z)φ(y,w) − φ(y,z)φ(x,w) +g(y,z)g(x,w) −g(x,z)g(y,w)] for any x,y,z,w on m, where r̄(x,y,z,w) = g(r(x,y )z,w) and φ is the fundamental 2-form of m defined by φ(x,y ) = g(φx,y ). proof. in view of (2.13) r(x,y )z = ∇x∇y z −∇y∇xz −∇[x,y ] int. j. anal. appl. 19 (2) (2021) 170 we can write r̄(x,y,φz,φw) = g(∇x∇y φz,φw) −g(∇y∇xφz,φw) −g(∇[x,y ]φz,φw).(2.14) by making use of (2.1), (2.3), (2.6) and (2.5), the eq.(2.14) takes the form r̄(x,y,φz,φw) = g(∇x∇y z −∇y∇xz −∇[x,y ]z,w) + α2[g(φy,z)g(x,φw) −g(φx,z)g(y,φw) − g(x,z)g(y,w) + g(y,z)g(x,w) + g(x,z)η(y )η(w) −g(y,z)η(x)η(w)] − η(∇x∇y z −∇y∇xz −∇[x,y ]z)η(w) which in view of (2.7) and (2.13) turns to r̄(x,y,φz,φw) = r̄(x,y,z,w) +α2[g(φy,z)g(x,φw) −g(φx,z)g(y,φw)(2.15) −g(x,z)g(y,w) + g(y,z)g(x,w)]. this completes the proof. � lemma 2.2. in an n−dimensional α−cosymplectic manifold (m,φ,ξ,η,g), the ∗−ricci tensor is given by s∗(y,z) = s(y,z) + α2(n− 2)g(y,z) + α2η(y )η(z)(2.16) for any y,z ∈ χ(m), where s and s∗ are the ricci tensor and the ∗−ricci tensor of type (0, 2), respectively on m. proof. let {ei} , i = 1, 2, 3.....n be an orthonormal basis of the tangent space at each point of the manifold. from the equations (2.12) and (1.4), we have s∗(y,z) = n∑ i=1 r̄(ei,y,φz,φei) = n∑ i=1 r̄(ei,y,z,ei) + α 2 n∑ i=1 [φ(ei,z)φ(y,ei) − φ(y,z)φ(ei,ei) + g(y,z)g(ei,ei) −g(ei,z)g(y,ei)]. by using (2.3) and φ(x,y ) = g(φx,y ) in the above equation, lemma 2.2 follows. � int. j. anal. appl. 19 (2) (2021) 171 3. α−cosymplectic manifolds admitting ∗−conformal η−ricci solitons in this section, first let us consider an n-dimensional α−cosymplectic manifold m admitting a ∗−conformal η−ricci soliton. then, from (1.7) we have (3.1) (£ξg)(y,z) + 2s ∗(y,z) + (2λ− (p + 2 n ))g(y,z) + 2µη(y )η(z) = 0. in an α−cosymplectic manifold, from (2.6)we write (3.2) (£ξg)(y,z) = g(∇xξ,y ) + g(x,∇y ξ) = 2α(g(y,z) −η(y )η(z)). by making use of (3.2) in (3.1), we find (3.3) s∗(y,z) = −(λ + α− 1 2 (p + 2 n ))g(y,z) + (α−µ)η(y )η(z). further, by plugging (3.3) in (2.16) we get s(y,z) = ag(y,z) + bη(y )η(z).(3.4) where a = −{λ + α + α2(n − 2) − 1 2 (p + 2 n )} and b = −{µ + α2 − α}. therefore, the manifold m is an η-einstein manifold. next, taking z = ξ in (3.4), we find (3.5) s(y,ξ) = (a + b)η(y ). it yields (3.6) qξ = (a + b)ξ. in view of (2.11) we obtain from (3.5) that (3.7) λ + µ = 1 2n (np + 2). hence, we are able to state the following: proposition 3.1. let m be an n−dimensional α−cosymplectic manifold. if the manifold m admits a ∗−conformal η−ricci soliton, then m is an η−einstein manifold and the soliton constants λ and µ are related by λ + µ = 1 2n (np + 2). next, assume that an n−dimensional α−cosymplectic manifold m admitting ∗−conformal η−ricci soliton satisfies the condition (3.8) r(ξ,x) ·s = 0. the condition (3.8) implies that (3.9) s(r(ξ,x)y,z) + s(y,r(ξ,x)z) = 0 int. j. anal. appl. 19 (2) (2021) 172 for any vector fields x,y,z ∈ χ(m). by using (3.4) in (3.9), we find b[η(r(ξ,x)y )η(z) + η(r(ξ,x)z)η(y )] = 0, which in view of (2.7) takes the form (3.10) α2b[g(x,y )η(z) + g(x,z)η(y ) − 2η(x)η(y )η(z)] = 0. contracting (3.10) over x and y we get (3.11) (n− 1)α2bη(z) = 0. in general η(z) 6= 0, therefore, we have either α2 = 0 or b = 0. hence, for α2 = 0, i.e., α = 0 the manifold m reduces to cosymplectic manifiold. and, for the latter case, from (3.7) we find λ = 1 2 (p + 2 n ) + α2 −α. thus, we state the following: theorem 3.1. let m be an n−dimensional α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton. if the manifold m satisfies the condition r(ξ,x)·s = 0, then the manifold m is either a cosymplectic or λ = 1 2 (p + 2 n ) + α2 −α and µ = α−α2. further, consider that the manifold m admitting a ∗−conformal η−ricci soliton satisfies the condition (3.12) s(ξ,y ) ·r = 0. the condition (3.12) implies that s(y,r(u,v )w)ξ −s(ξ,r(u,v )w)y + s(y,u)r(ξ,v )w − s(ξ,u)r(y,v )w + s(y,v )r(u,ξ)w −s(ξ,v )r(u,y )w + s(y,w)r(u,v )ξ −s(ξ,w)r(u,v )y = 0(3.13) for any vector fields y,u,v,w ∈ χ(m). taking the inner product of (3.13) with ξ, we have s(y,r(u,v )w) −s(ξ,r(u,v )w)η(y ) + s(y,u)η(r(ξ,v )w) − s(ξ,u)η(r(y,v )w) + s(y,v )η(r(u,ξ)w) −s(ξ,v )η(r(u,y )w) + s(y,w)η(r(u,v )ξ) −s(ξ,w)η(r(u,v )y ) = 0.(3.14) by putting u = w = ξ in (3.14) and then using (2.7), (2.8) and (3.5) we have (3.15) α2[s(y,v ) − (n− 1)α2{g(y,v ) − 2η(y )η(v )}] = 0. thus, we have either α = 0, or s(y,v ) = (n− 1)α2g(y,v ) − 2(n− 1)α2η(y )η(v ). this can be stated in the following form: int. j. anal. appl. 19 (2) (2021) 173 theorem 3.2. let m be an n−dimensional α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton. if the manifold m satisfies the condition s(ξ,y )·r = 0, then the manifold m is either a cosymplectic or an η-einstein manifold. 4. m−projective curvature tensor on α−cosymplectic manifolds admitting ∗−conformal η−ricci solitons in riemannian geometry, one of the basic interest is curvature properties and to what extend these determine the manifold itself. the m-projective curvature tensor differs from the riemannian curvature tensor and is an important tensor field from the differential geometric point of view because it bridges the gap between the conformal curvature tensor, conharmonic curvature tensor and concircular curvature tensor on one side and the h-projective curvature tensor on the other. it is known that, m-projectively flat riemannian manifold is an einstein manifold. the m−projective curvature tensor in an n−dimensional α−cosymplectic manifold is defined by [26] m(x,y )z = r(x,y )z − 1 2(n− 1) [s(y,z)x −s(x,z)y + g(y,z)qx −g(x,z)qy ],(4.1) where r(x,y )z and s(x,y ) are the curvature tensor and the ricci tensor of m, respectively; and q is the ricci operator defined as s(x,y ) = g(qx,y ). in 1985, ojha showed some properties of m-projective curvature tensor in a sasakian manifold [27]. subsequently, many geometers have studied this curvature tensor and obtained important properties of various kinds of riemannian and pseudo-riemannian manifolds (see, for instance, [2, 33, 34, 42]). in this section, we study α−cosymplectic manifolds admitting ∗−conformal η−ricci solitons satisfying certain conditions on the m−projective curvature tensor. first, let us consider an n−dimensional α−cosymplectic manifold m admitting ∗−conformal η−ricci soliton, which is ξ −m−projectively flat, i.e., m(x,y )ξ = 0. then, from (4.1) it follows that (4.2) r(x,y )ξ = 1 2(n− 1) [s(y,ξ)x −s(x,ξ)y + g(y,ξ)qx −g(x,ξ)qy ]. making use of (2.9) and (3.5) in the above equation, we have (4.3) η(y )qx −η(x)qy = {(a + b) + 2(n− 1)α2}[η(x)y −η(y )x]. again, taking y = ξ in (4.3) and then using (3.5) we obtain qx = −{(a + b) + 2(n− 1)α2}x + 2{(a + b) + (n− 1)α2}η(x)ξ. int. j. anal. appl. 19 (2) (2021) 174 taking the inner product of the above equation with w, we obtain s(x,w) = −{(a + b) + 2(n− 1)α2}g(x,w) + 2{(a + b) + (n− 1)α2}η(x)η(w).(4.4) in m, by virtue of (3.7) and with the values of a and b, it follows that (4.5) a + b = −(n− 1)α2. by using (4.5), the eq. (4.4) reduce to (4.6) s(x,w) = −(n− 1)α2g(x,w). thus, the manifold m is an einstein. hence, we have the following: theorem 4.1. let m be an n−dimensional α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton. if the manifold m is ξ −m−projectively flat, then the manifold m is an einstein manifold. definition 4.1. an α−cosymplectic manifold is said to be φ−m−projectively semisymmetric if [12, 35] m(x,y ) ·φ = 0 for all x,y on m. next, let us consider an n−dimensional α−cosymplectic manifold m admitting ∗−conformal η−ricci soliton, which is φ−m−projectively semisymmetric, i.e., m(x,y ) ·φ = 0. then, it follows that (4.7) (m(x,y ) ·φ)w = m(x,y )φw −φm(x,y )w = 0. by virtue of (4.1), (4.7) takes the form r(x,y )φw −φr(x,y )w + 1 2(n− 1) [s(y,w)φx −s(x,w)φy −s(y,φwz)x + s(x,φw)y + g(y,w)φqx −g(x,w )φqy − g(y,φw)qx + g(x,φw)qy ] = 0.(4.8) putting y = ξ in (4.8) and using (2.1), (2.2), (2.8) and (3.5), we find [a + b + 2(n− 1)α2](g(x,φw)ξ + η(w)φx) + s(x,φwz)ξ + η(w)φqx = 0 whcih by taking the inner product with ξ reduces to (4.9) s(x,φw) = −[a + b + 2(n− 1)α2]g(x,φw). int. j. anal. appl. 19 (2) (2021) 175 replacing w by φw in (4.9) and using (2.1) and (3.5), we obtain s(x,w ) = −[a + b + 2(n− 1)α2)]g(x,w) + 2[a + b + (n− 1)α2]η(x)η(w).(4.10) by using (4.5), the eq. (4.10) turns to (4.11) s(x,w) = −(n− 1)α2g(x,w). thus, we have the following: theorem 4.2. let m be an n−dimensional α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton. if the manifold m is φ − m−projectively semisymmetric, then the manifold m is an einstein manifold. finally, consider an n−dimensional α−cosymplectic manifold m admitting ∗−conformal η−ricci soliton, which satisfies the condition (4.12) r(ξ,x) ·m = 0. from (4.12) it follows that r(ξ,x)m(u,v )w −m(r(ξ,x)u,v )w − m(u,r(ξ,x)v )w −m(u,v )r(ξ,x)w = 0.(4.13) in view of (2.8), (4.13) takes the form α2[η(m(u,v )w)x −g(x,m(u,v )w)ξ −η(u)m(x,v )w + g(x,u)m(ξ,v )w −η(v )m(u,x)w + g(x,v )m(u,ξ)w − η(w)m(u,v )x + g(x,w)m(u,v )ξ] = 0.(4.14) taking the inner product of (4.14) with ξ, we have α2[η(m(u,v )w)η(x) −g(x,m(u,v )w) −η(u)η(m(x,v )w) + g(x,u)η(m(ξ,v )w) −η(v )η(m(u,x)w) + g(x,v )η(m(u,ξ)w) − η(w)η(m(u,v )x) + g(x,w)η(m(u,v )ξ)] = 0.(4.15) int. j. anal. appl. 19 (2) (2021) 176 from (4.1), we find η(m(u,v )w) = (α2 + 2a + b 2(n− 1) )(g(u,w)η(v ) −g(v,w)η(u)),(4.16) η(m(ξ,v )w) = (α2 + 2a + b 2(n− 1) )(η(v )η(w) −g(v,w)),(4.17) η(m(u,v )ξ) = 0.(4.18) in view of (4.16)-(4.18), (4.15) reduces to (4.19) α2[g(x,m(u,v )w) − (α2 + 2a + b 2(n− 1) )(g(u,w)g(x,v ) −g(x,u)g(v,w))] = 0. thus we have either α = 0, or (4.20) g(x,m(u,v )w) = (α2 + 2a + b 2(n− 1) )(g(u,w)g(x,v ) −g(x,u)g(v,w)). by using (4.1), (4.20) takes the form g(r(u,v )w,x) = 1 2(n− 1) [s(v,w)g(x,u) −s(u,w)g(x,v ) + s(x,u)g(v,w) −s(x,v )g(u,w )] = [α2 + 2a + b 2(n− 1) ](g(u,w)g(x,v ) −g(x,u)g(v,w)).(4.21) let {ei}, i = 1, 2, 3....n be an orthonormal basis of the tangent space at any point of the manifold. if we put x = u = ei in (4.21) and taking summation with respect to i(1 ≤ i ≤ n), then we get (4.22) s(v,w) = 1 2n {r − (n− 1)(2(n− 1)α2 − (2a + b))}g(v,w), where a and b are given in (3.4). thus we can state the following: theorem 4.3. let m be an n−dimensional α−cosymplectic manifold admitting a ∗−conformal η−ricci soliton. if the manifold m satisfies the condition r(ξ,x) ·m = 0, then the manifold m is an einstein manifold. 5. example we consider the 5-dimensional manifold m = {(x1,x2,y1,y2,z) ∈ r5}, where (x1,x2,y1,y2,z) are the standard coordinates in r5. let e1, e2, e3, e4 and e5 be the vector fields on m given by e1 = e αz ∂ ∂x1 , e2 = e αz ∂ ∂x2 , e3 = e αz ∂ ∂y1 , e4 = e αz ∂ ∂y2 , e5 = − ∂ ∂z = ξ. int. j. anal. appl. 19 (2) (2021) 177 let g be the riemannian metric defined by g(ei,ej) = 0, i 6= j, i,j = 1, 2, 3, 4, 5 and g(e1,e1) = g(e2,e2) = g(e3,e3) = g(e4,e4) = g(e5,e5) = 1. let η be the 1-form on m defined by η(x) = g(x,e5) = g(x,ξ) for all x ∈ χ(m). let φ be the (1, 1) tensor field on m defined by φe1 = −e2, φe2 = e1, φe3 = −e4, φe4 = e3, φe5 = 0. by applying the linearity of φ and g, we have η(ξ) = 1, φ2x = −x + η(x)ξ, η(φx) = 0, g(x,ξ) = η(x), g(φx,φy ) = g(x,y ) −η(x)η(y ) for all x,y ∈ χ(m). then we have [e1,e2] = [e1,e3] = [e1,e4] = [e2,e3] = [e2,e4] = [e3,e4] = 0, [e1,e5] = αe1, [e2,e5] = αe2, [e3,e5] = αe3, [e4,e5] = αe4. the riemannian connection ∇ of the metric g is given by 2g(∇xy,z) = xg(y,z) + y g(z,x) −zg(x,y ) −g(x, [y,z]) + g(y, [z,x]) + g(z, [x,y ]), which is known as koszul’s formula. using koszul’s formula, we can easily calculate ∇e1e1 = −αe5, ∇e1e2 = 0, ∇e1e3 = 0, ∇e1e4 = 0, ∇e1e5 = αe1, ∇e2e1 = 0, ∇e2e2 = −αe5, ∇e2e3 = 0, ∇e2e4 = 0, ∇e2e5 = αe2, ∇e3e1 = 0, ∇e3e2 = 0, ∇e3e3 = −αe5, ∇e3e4 = 0, ∇e3e5 = αe3, ∇e4e1 = 0, ∇e4e2 = 0, ∇e4e3 = 0, ∇e4e4 = −αe5, ∇e4e5 = αe4, ∇e5e1 = 0, ∇e5e2 = 0, ∇e5e3 = 0, ∇e5e4 = 0, ∇e5e5 = 0. it can be easily verified that the manifold satisfy ∇xξ = α[x −η(x)ξ] and (∇xφ)y = α[g(φx,y )ξ −η(y )φx] for ξ = e5. thus the manifold m is an α−cosymplectic manifold. by using (2.13) we can easily obtain the non-vanishing components of the curvature tensors as follows: r(e1,e2)e2 = r(e1,e3)e3 = r(e1,e4)e4 = r(e1,e5)e5 = −α2e1, r(e1,e2)e1 = α 2e2, r(e1,e3)e1 = r(e2,e3)e2 = r(e5,e3)e5 = α 2e3, int. j. anal. appl. 19 (2) (2021) 178 r(e2,e3)e3 = r(e2,e4)e4 = r(e2,e5)e5 = −α2e2,r(e3,e4)e4 = −α2e3, r(e1,e5)e2 = r(e1,e5)e1 = r(e4,e5)e4 = r(e3,e5)e3 = α 2e5, r(e1,e4)e1 = r(e2,e4)e2 = r(e3,e4)e3 = r(e5,e4)e5 = α 2e4. with the help of the above results we get the components of the ricci tensor as follows: (5.1) s(e1,e1) = s(e2,e2) = s(e3,e3) = s(e4,e4) = s(e5,e5) = −4α2. from (3.4), we have s(e5,e5) = −4α2−λ−µ+ 12 (p+ 2 5 ). by equating both the values of s(e5,e5), we obtain λ + µ = 1 2 (p + 2 5 ). hence, λ and µ satisfies the equation (3.7) for n = 5 and, so g defines a ∗−conformal η−ricci soliton on a 5-dimensional α−cosymplectic manifold. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. a. akyol, conformal anti-invariant submersions from cosymplectic manifolds, hacettepe j. math. stat. 46 (2017), 177-192. [2] g. ayar, s. k. chaubey, m−projective curvature tensor over cosymplectic manifolds, differ. geom.-dyn. syst. 21 (2019), 23-33. [3] n. basu, a. bhattacharyya, conformal ricci soliton in kenmotsu manifold, glob. j. adv. res. class. mod. geom. 4 (2015), 15-21. [4] s. beyendi, g. ayar, g. n. atkan, on a type of α−cosymplectic manifolds, commun. fac. sci. univ. ankara, ser. a1, math. stat. 68 (2019), 852-861. [5] a. m. blaga, η-ricci solitons on para-kenmotsu manifolds. balkan j. geom. appl. 20(1) (2015), 1–13. [6] a. m. blaga, η-ricci solitons on lorentzian para-sasakian manifolds. filomat, 30(2) (2016), 489–496. [7] d. e. blair, contact manifolds in riemannian geometry, lecture notes in math. 509, springer-verlag, berlin, 1976. [8] d. e. blair, riemannian geometry of contact and symplectic manifolds, second edition, progress in mathematics, vol. 203, birkhauser boston, inc., boston, ma, 2010. [9] b. cappelletti-montano, a. de nicola, i. yudin. a survey on cosymplectic geometry, rev. math. phys. 25(10) (2013), 1343002. [10] j. t. cho, m. kimura, ricci solitons and real hypersurfaces in a complex space form. tohoku math. j. 61(2) (2009), 205–212. [11] k. de, c. dey, ∗−ricci solitons on (�)−para sasakian manifolds, bull. transilvania univ. brasov, ser. iii: math. inform. phys. 12(61) (2019), 265-274. [12] u. c. de, p. majhi, φ−semisymmetric generalized sasakian space-forms, arab j. math. sci. 21 (2015), 170-178. [13] d. dey and p. majhi, almost kenmotsu metric as conformal ricci soliton, conform. geom. dyn. 23(2019), 105-116. [14] a. e. fischer, an introduction to conformal ricci flow, class. quantum grav. 21 (2004), 171-218. int. j. anal. appl. 19 (2) (2021) 179 [15] a. ghosh, d. s. patra, ∗−ricci soliton within the frame-work of sasakian and (k,µ)−contact manifold, int. j. geom. meth. mod. phys. 15(7) (2018), 1850120. [16] s. i. goldberg and k. yano: integrability of almost cosymplectic structure, pac. j. math. 31 (1969), 373–381. [17] hakan, ozturk, some curvature conditions on α−cosymplectic manifolds, math. stat. 4 (2013), 188-195. [18] hakan, ozturk, c. murathan, n. aktan, a. t. vanli, almost α-cosymplectic f−manifolds, ann. alexandru ioan cuza univ.-math. 60 (1) (2014), 211-226. [19] t. hamada, real hypersurfaces of complex space forms in terms of ricci ∗−tensor, tokyo j. math. 25 (2002), 473-483. [20] r.s. hamilton, the formation of singularities in the ricci flow, surveys in differntial geometry vol. ii, international press, cambridge ma (1993), 7-136. [21] r. hamilton, the ricci flow on surfaces, contemp. math. 71 (1988), 237–261. [22] g. kaimakamis, k. panagiotidou, ∗−ricci solitons of real hypersurfaces in non-flat complex space forms, j. geom. phys. 86 (2014), 408-413. [23] t. w. kim, h. k. pak, canonical foliations of certain classes of almost contact metric structures, acta math. sin. 21(4) (2005), 841-846. [24] p. majhi, u. c. de, y. j. suh, ∗−ricci solitons on sasakian 3-manifolds. publ. math. debrecen, 93 (2018), 241-252. [25] h. g. nagaraja, k. venu, f-kenmotsu metric as conformal ricci soliton, ann. west univ. timisoara-math. computer sci. 55 (2017), 119-127. [26] r. h. ojha, a note on the m−projective curvature tensor, indian j. pure appl. math. 8(12) (1975), 1531-1534. [27] r. h. ojha, m-projectively flat sasakian manifolds, indian j. pure appl. math. 17(4) (1986), 481-484. [28] z. olszak, on almost cosymplectic manifolds. kodai math. 4(2) (1981), 239–250. [29] z. olszak, locally conformal almost cosymplectic manifolds, colloq. math. 57 (1989) 73-87. [30] g. perelman, ricci flow with surgery on three-manifolds, preprint, arxiv:math/0303109 [math.dg], 2003. [31] d. g. prakasha, b. s. hadimani, η-ricci solitons on para-sasakian manifolds, j. geom. 108 (2017), 383–392. [32] d. g. prakasha, p. veeresha, para-sasakian manifolds and ∗−ricci solitons, afr. mat. 30 (2018), 989-998. [33] d. g. prakasha, l. m. fernandez, k. mirji, the m-projective curvature tensor field on generalized (κ,µ)-paracontact metric manifolds, georgian math. j. 27 (1) (2020), 141-147. [34] d. g. prakasha, f. o. zengin, v. chavan, on m-projectively semisymmetric lorentzian α-sasakian manifolds, afr. mat. 28(5-6) (2017), 899-908. [35] r. prasad, a. haseeb, u. k. gautam, on φ−semisymmetric lp-kenmotsu manifolds with a qsnm connection, kragujevac j. math. 45 (2021), 815-827. [36] a. haseeb, r. prasad, η-ricci solitons in lorentzian α−sasakian manifolds, facta univ. ser., math. inform. 35(3) (2020), 713-725. [37] s. roy, s. dey, a. bhattacharyya, s. k. hui, ∗−conformal η-ricci soliton on sasakian manifold, preprint, arxiv:1909.01318 [math.dg], 2019. [38] m. d. siddiqi, conformal η-ricci solitons in δlorentzian trans sasakian manifolds, int. j. maps math. 1(1) (2018), 15-34. [39] s. tachibana, on almost-analytic vectors in almost-kahlerian manifolds, tohoku math. j. 11(2) (1959), 247-265. [40] venkatesha, d. m. naik, h. aruna kumara, ∗-ricci solitons and gradient almost ∗−ricci solitons on kenmotsu manifolds, math. slovaca, 69 (2019), 1447-1458. [41] k. yano, m. kon, structures on manifolds, world scientific, singapore, 1984. [42] f.o. zengin, m-projectively flat spacetimes, math. rep. 14(64) (2012), 363–370. 1. introduction 2. preliminaries 3. 0.2cm-cosymplectic manifolds admitting *-conformal -ricci solitons 4. 0.2cmm-projective curvature tensor on -cosymplectic manifolds admitting *-conformal -ricci solitons 5. 0.2cmexample references international journal of analysis and applications volume 18, number 4 (2020), 614-623 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-614 on janowski close-to-convex functions associated with conic regions afis saliu∗, khalida inayat noor department of mathematics, comsats university islamabad, park road, tarlai kalan, islamabad 45550, pakistan ∗corresponding author: saliugsu@gmail.com abstract. in this work, we introduce and investigate a class of analytic functions which is a subclass of close-to-convex functions of janowski type and related to conic regions. length of the image curve |z| = r < 1 under the generalized janowski close-to-convex function is derived. furthermore, rate of growth of coefficients and hankel determinant for this class are obtained. relevant connections of our results with the earlier known results are also pointed out. 1. introduction let e = {z : |z | < 1} and h be the class of functions f(z) defined as f(z) = z + ∞∑ n=2 anz n (1.1) which are analytic in e. a function f(z) is subordinate to another function g(z) (written as f(z) ≺ g(z)) if there exists an analytic function w(z) in e with w(0) = 1 and |w(z)| < 1 for z ∈ e such that f(z) = g(w(z)). let pm(α) be the class of analytic functions p(z) in e satisfying the condition p(0) = 1 and∫ 2π 0 ∣∣∣re p(z) −α 1 −α ∣∣∣dθ ≤ mπ, (1.2) received march 5th, 2020; accepted april 13th, 2020; published may 14th, 2020. 2010 mathematics subject classification. 30c45, 30c50, 30c55. key words and phrases. analytic functions; open unit disk; janowski functions; subordination; univalent functions; conic domains. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 614 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-614 int. j. anal. appl. 18 (4) (2020) 615 where m ≥ 2,z = reiθ, 0 ≤ α < 1, see [12]. the case α = 0 gives the class pm introduced by pinchuk [13]. for α = 0, m = 2, we obtain the well-known class p of carathéodory functions and for m = 2, p2(α) ≡ p(α) is the class of functions whose real parts are greater than α. it is known in [12] that p ∈ pm(α) has the integral representation p(z) = 1 2π ∫ 2π 0 1 + (1 − 2α)ze−it 1 −ze−it dv(t), (1.3) where v(t) is a function of bounded variation on [0, 2π] such that∫ 2π 0 dv(t) = 2π and ∫ 2π 0 |dv(t)| ≤ mπ. (1.4) it is seen from (1.3) and (1.4) that p ∈ pm(α) has a representation p(z) = m + 2 4 p1(z) − m− 2 4 p2(z), (1.5) where pi ∈ p(α) for i = 1, 2. denote by cv ,s∗,k,cv (α),s∗(α), k(α), are the subclasses of s (the class of univalent functions in e) which consist of functions that are convex, starlike, close-to-convex, convex of order α, starlike of order α and close-to-convex of order α (0 ≤ α < 1) respectively. we have the following class of analytic functions in e : vm(α) = { f ∈ h : (zf′)′ f′ ∈ pm(α), z ∈ e, m ≥ 2, 0 ≤ α < 1 } , see [12] (1.6) and note that v2(α) ≡ cv (α) and v2(0) ≡ cv . recently, noor [11] extended the conic domain ωk,k ≥ 0 introduced by kanas and wisniowska [2, 3] to that of janowski type, ωk[a,b], −1 ≤ b < a ≤ 1 and defined it as ωk[a,b] = { u + iv = [(b2 − 1)(u2 + v2) − 2(ab − 1)u + (a2 − 1)]2 >k2[(−2(b + 1)(u2 + v2) + 2(a + b + 2)u− 2(a + 1))2 + 4(a−b)2v2] } . (1.7) denoted by k − p(a,b), the class of functions p(z) that map e onto ωk[a,b]. equivalently, we say p ∈ k −p(a,b) if and only if p(z) ≺ (a + 1)pk(z) − (a− 1) (b + 1)pk(z) − (b − 1) , k ≥ 0, −1 ≤ b < a ≤ 1, (1.8) where the definition of pk is given in [2]. also, it is worthy mentioning that p ∈ k−p(a,b) ⊂ p(γ1) which implies that p(z) = (1 −γ1)h1(z) + γ1, (see [11]) where h1 ∈ p and γ1 is given by γ1 = 2k + 1 −a 2k + 1 −b . (1.9) if in (1.5), p1,p2 ∈ k−p(a,b), we say p ∈ k−pm(a,b) and if pm(α) in (1.6) is replaced with k−pm(a,b), we say f belongs to the class k −uvm(a,b). we note that k −pm(a,b) ⊂ pm(γ1), where γ1 is given by (1.9). thus, k −uvm(a,b) ⊂ vm(γ1). int. j. anal. appl. 18 (4) (2020) 616 we introduce the following class of functions. definition 1.1. let f ∈ h,−1 ≤ b < a ≤ 1, −1 ≤ d < c ≤ 1, k ≥ 0 and m1,m2 ≥ 2. then f ∈ k −hm1m2 (a,b,c,d) if there exists g ∈ k −uvm2 (c,d) such that f′(z) g′(z) ∈ k −pm1 (a,b). in particular, (i) for k = 0, m1 = m = m2,a = 1,b = −1,c = 1 − 2α,d = −1, k − hm1m2 (1,−1, 1 − 2α,−1) ≡ hmm(α) is the class of analytic functions studied by noor [9], (ii) for k = 0, m1 = m = m2,a = 1,b = −1,c = 1,d = −1, k −hm1m2 (1,−1, 1,−1) ≡ kmm is the class of analytic functions investigated by noor [8], (iii) for k = 0, m1 = 2 = m2,a = 1,b = −1,c = 1,d = −1, k−h22(1,−1, 1,−1) ≡ k is the class of close to convex functions first introduced and examined by kaplan [4] (iv) for k = 0, m1 = 2 = m2,h22(a,b,c,d) ≡ k − uk(a,b,c,d)) is the class of analytic functions examined by mahmood et al [5]. we note that k −hm1m2 (a,b, 1,−1) ≡ hm1m2 (γ1,σ), where σ = k k+1 . 2. some preliminary lemmas we need the following lemmas to investigate our results. lemma 2.1. [10] let p ∈ pm(γ), 0 ≤ γ < 1,m ≥ 2. then for z = reiθ, (i) 1 2π ∫ 2π 0 |p(z) |2 dθ ≤ 1 + ( m2(1 −γ)2 − 1 ) r2 1 −r2 , (2.1) (ii) 1 2π ∫ 2π 0 |p ′(z) | dθ ≤ m(1 −γ) 1 −r2 . (2.2) lemma 2.2. [12] (i) f ∈ vm(α) if and only if there exist f1,f2 ∈ s∗ such that f′(z) = ( f1(z) z )( m+2 4 )(1−α) ( f2(z) z )( m−2 4 )(1−α) . (2.3) (ii) let f ∈ vm(α). then r ( (1 −r)( m−2 4 ) (1 + r)( m+2 4 ) )(1−α) ≤ |zf′(z) | ≤ r ( (1 + r)( m−2 4 ) (1 −r)( m+2 4 ) )(1−α) . (2.4) int. j. anal. appl. 18 (4) (2020) 617 we will need the hypergeometric function γ(a)γ(c−a) γ(c) g(a,b,c; z) = ∫ 1 0 ua−1(1 −u)c−a−1(1 −zu)−b du. (2.5) unless otherwise stated, we assume, m1,m2 ≥ 2,k ≥ 0 − 1 ≤ b < a ≤ 1, and − 1 ≤ d < c ≤ 1. 3. main results theorem 3.1. let f ∈ k −hm1m2 (a,b,c,d). then for 0 < r < 1, l(r,f) ≤ π { c(m2,k,γ2,c,d)m(r) log 1 1 −r + 2b+1γ1 a [ [g(a,b,c,−1) − 2g(a, 1 + b,c− 1)] (3.1) +ra1 [2g(a, 1 + b,c,−r1) −g(a,b,c,−r1)] ]} , (3.2) where m(r) = max θ |f(reiθ)|, c(m2,k,γ2,c,d) is a constant depending on m2,k,γ2,c and d, a = (m2 2 − 1 ) (1 −γ2), b = 2(γ2 − 1), c = a + 1 and r1 = 1 −r 1 + r , where γ1 = 2k + 1 −a 2k + 1 −b , γ2 = 2k + 1 −c 2k + 1 −d . (3.3) proof. let z = reiθ. then l(r,f) = ∫ 2π 0 |zf′(z) |dθ = ∫ 2π 0 |zg′(z)p(z) |dθ, where g ∈ k −vm2 (c,d) and p ∈ k −pm1 (a,b) ≤ r∫ 0 2π∫ 0 (zg′(z))′p(z) |dθdρ + r∫ 0 2π∫ 0 |zg′(z)p ′(z) |dθdρ =j1(r) + j2(r). (3.4) let (zg′)′(z) g′(z) = h(z) = 1 + ∞∑ n=1 dnz n ∈ k −pm2 (c,d). then by schwarz inequality and perseval’s theorem, we have j1(r) ≤2π   r∫ 0 2π∫ 0 |f′(z) |2 dθdρ   1 2   r∫ 0 2π∫ 0 |h(z) |2 dθdρ   1 2 =2π (∫ r 0 ∞∑ n=1 n2|an |2ρ2n−2dρ )1 2 (∫ r 0 ∞∑ n=0 |dn |2ρ2ndρ )1 2 . int. j. anal. appl. 18 (4) (2020) 618 it is easy to see that |dn | ≤ m2(c −d)|δk | 4 , where δk has its definition given in [11]. therefore, j1(r) ≤ √ 2πm2(c −d)|δk | 4 ( 1 r ∞∑ n=1 n2 2n− 1 |an |2r2n )1 2 ( log 1 + r 1 −r )1 2 ≤ √ 2πm2(c −d)|δk | 4 m(r) (1 r log 1 + r 1 −r )1 2 , (3.5) where we used the fact that a(r) = π ∞∑ n=1 n|an |2r2n is the area of the image of |z | < r bounded by w = f(z) and a(r) ≤ πm2(r). next, we estimate j2(r). since p ∈ k −pm1 (a,b) ⊂ pm1 (γ1), then using (1.3), we get p ′(z) = (1 −γ1) π ∫ 2π 0 eit (1 −zeit)2 dv(t) and ∫ 2π 0 1 −ρ2 |1 −zeit |2 dv(t) = 2π(rep(z) −γ1) 1 −γ1 . therefore, j2(r) ≤ (1 −γ1) π r∫ 0 2π∫ 0 2π∫ 0 |zg′(z) | |1 −ze−it |2 dv(t)dθdρ =2 r∫ 0 2π∫ 0 |zg′(z) |(rep(z) −γ1) 1 1 −ρ2 dθdρ =2 r∫ 0 2π∫ 0 re ( zg′(z)e−i arg zg ′(z)p(z) ) 1 1 −ρ2 dθdρ− 2γ1 r∫ 0 2π∫ 0 |zg′(z) |dθdρ. integration by parts, application of (1.2) and lemma 2.2(ii) give j2(r) ≤2π(m2(1 −γ2) + 2γ2) ∫ r 0 m(ρ) 1 −ρ2 dρ− 4πγ1 ∫ r 0 ρ (1 −ρ)( m−2 4 )(1−γ2)−1 (1 + ρ)( m+2 4 )(1−γ2)+1 dρ ≤π(m2(1 −γ2) + 2γ2)m(r) log 1 + r 1 −r + 4πγ1(l1(r) −l2(r)), (3.6) where l1(r) = ∫ r 0 (1 −ρ)( m−2 4 )(1−γ2)−1 (1 + ρ)( m+2 4 )(1−γ2)+1 dρ and l2(r) = ∫ r 0 (1 −ρ)( m−2 4 )(1−γ2)−1 (1 + ρ)( m+2 4 )(1−γ2) dρ. let u = 1−ρ 1+ρ , so that dρ = − 2 (1+u)2 . then l1(r) = ( 1 2 )2(2−γ2)−1 [∫ 1 0 u( m2 2 −1)(1−γ2)−1(1 + u)2(1−γ2)du− ∫ r1 0 u( m2 2 −1)(1−γ2)−1(1 + u)2(1−γ2)du ] = 1 a g(a,b,c,−1) − ∫ r1 0 u( m2 2 −1)(1−γ2)−1(1 + u)2(1−γ2)du, (3.7) where a,b,c and r1 are given in theorem 3.1. for the second integral in (3.7), we let u = r1v. then l1(r) = 2b−1 a [g(a,b,c,−1) −ra1g(a,b,c,−r1)] (3.8) int. j. anal. appl. 18 (4) (2020) 619 in a similar way, we obtain l2(r) = 2b a [g(a, 1 + b,c,−1) −ra1g(a, 1 + b,c,−r1)] . (3.9) using (3.8), (3.9) in (3.6), we get j2(r) ≤π(m2(1 −γ2) + 2γ2)m(r) log 1 + r 1 −r + π2b+1γ1 a { [g(a,b,c,−1) − 2g(a, 1 + b,c− 1)] + ra1 [2g(a, 1 + b,c,−r1) −g(a,b,c,−r1)] } . (3.10) the estimates for j1(r) and j2(r) yield the required result. � corollary 3.1. let f ∈ km1m2, then for 0 ≤ r < 1, l(r,f) ≤ c(m2)m(r) log 1 1 −r , where m(r) = max θ |f(reiθ)|, c(m2) is a constant depending on m2. corollary 3.2. let f ∈ hmm(α), then for 0 ≤ r < 1, l(r,f) ≤ c(m,α)m(r) log 1 1 −r , where m(r) = max θ |f(reiθ)|, c(m,α) is a constant depending on m and α. corollary 3.3. let f ∈ k. then for 0 ≤ r < 1, l(r,f) ≤ cm(r) log 1 1 −r , where m(r) = max θ |f(reiθ)|, c is a constant. theorem 3.2. let f(z) be of the form (1.1) and f ∈ k −hm1m2 (a,b,c,d). then |an | ≤ π n ( c1(m2,k,γ2,c,d)m (n− 1 n ) log n + 2b+1γ1 a { [g(a,b,c,−1) − 2g(a, 1 + b,c− 1)] + ra1 [ 2g ( a, 1 + b,c,− 1 2n− 1 ) −g ( a,b,c,− 1 2n− 1 )]}) , where γ1,γ2,a,b and c are given as in theorem 3.1. noonan and thomas [6] define for q ≥ 1,n ≥ 1, the qth hankel determinant of f(z) ∈ h as follows : hq(n) = ∣∣∣∣∣∣∣∣∣∣∣∣∣ an an+1 . . . an+q−1 an+1 an+2 . . . an+q−2 ... ... ... ... an+q−1 an+q−2 . . . an+2q−2 ∣∣∣∣∣∣∣∣∣∣∣∣∣ . (3.11) int. j. anal. appl. 18 (4) (2020) 620 to estimate the growth rate of hankel determinant for f ∈ k − hm1m2 (a,b,c,d), we need the following results due to noonan and thomas [6]. lemma 3.1. let f ∈ h and suppose the qth hankel determinant of f(z) for q ≥ 1,n ≥ 1 is given by (3.11). then writing ∆j(n) = ∆j(n,z1,f), we have hq(n) = ∣∣∣∣∣∣∣∣∣∣∣∣∣ ∆2q−2(n) ∆2q−3(n + 1) . . . ∆q−1(n + q − 1) ∆2q−3(n + 1) ∆2q−4(n + 2) . . . ∆q−2(n + q − 2) ... ... ... ... ∆q−1(n + q − 1) ∆q−2(n + q − 2) . . . ∆q(n + 2q − 2) ∣∣∣∣∣∣∣∣∣∣∣∣∣ , (3.12) where with ∆0(n,z1,f) = an, we define for j ≥ 1, ∆j(n,z1,f) = ∆j−1(n,z1,f) −z1∆j−1(n + 1,z1,f). (3.13) lemma 3.2. with x = ( n n+1 )y,u ≥ 0 an integer, ∆j(n + u,u,x,zf ′(z)) = j∑ i=0 ( j i ) yi(u− (i− 1)n) (n + 1)i · ∆j−i(n + u + i,y,f). (3.14) remark 3.1. consider any determinant of the form d = ∣∣∣∣∣∣∣∣∣∣∣∣∣ y2q−2 y2q−3 . . . yq−1 y2q−3 y2q−4 . . . yq−2 ... ... ... ... yq−1 yq−2 . . . y0 ∣∣∣∣∣∣∣∣∣∣∣∣∣ . (3.15) with 1 ≤ i, j ≤ q and αij = y2q−(i+j), d = det(αij). thus d = ∑ v1∈sq (sgn v1) q∏ j=1 (y2q − (v1(j) + j) , where sq is the symmetric group on q elements and sgn v1 is either +1 or −1. thus, in the expansion of d, each summand has q factor and the sum of the subscripts of the factor of each summand is q2 −q. now let n be given and hq(n) is as lemma 3.1, then each summand in the expression of hq(n) is of the form q∏ i=1 ∆v1(i) (n + 2q − 2 −v1i) , where v1 ∈ sq and q∑ i=1 v1(i) = q 2 −q; 0 ≤ v1(i) ≤ 2q − 2. int. j. anal. appl. 18 (4) (2020) 621 theorem 3.3. let f ∈ hm1m2 (γ1,σ) and (m2 + 2)(1−γ2) ≥ 4j. if the qth hankel determinant of f(z) for q ≥ 1,n ≥ 1 is given by (3.11), then hq(n) = o(1)   n ( m2 2 +1)( 1k+1 )−1, q = 1, n( m2 2 +1)( 1k+1 )q−q 2 , q ≥ 2, m2 ≥ 8(k + 1)(q − 1) − 2 , (3.16) where o(1) is a constant that depends on m1,m2,j,γ1,k only, with γ1 given by (3.3). proof. since f ∈ hm1m2 (γ1,σ), then f′(z) = p(z)g′(z), where g′(z) ∈ k −vm2 (1,−1) ⊂ vm2 (σ) and p(z) ∈ k −pm1 (a,b) ⊂ pm1 (γ1). setting f(z) = (zf′(z))′, and (zg′(z))′ g′(z) = h(z), then f(z) = g′(z)(h(z)p(z) + zp′(z)). now, for j ≥ 0,z1 any nonzero complex number, consider ∆j(n,z1,f(z)) as defined by (3.13). then ∆j(n,z1,f(z)) ≤ 1 2πrn+j ∣∣∣∣∣ ∫ 2π 0 (z −z1)jf(z)e−i(n+j)θ dθ ∣∣∣∣∣ ≤ 1 2πrn+j ∫ 2π 0 |z −z1 |j|g′(z) ||h(z)p(z) + zp ′(z) |dθ. using lemma 2.2(i) and the distortion theorems for starlike function, then for (m2 + 2)(1 − σ) ≥ 4j, we obtain ∆j(n,z1,f(z)) ≤ 1 2πrn+j ∫ 2π 0 |(z −z1)f1(z)|j |f1(z)|( m2+2 4 )(1−σ)−j |f2(z)|( m2−2 4 )(1−σ) |h(z)p(z) + zp′(z)|dθ ≤ 1 2πrn+j−σ 2π∫ 0 |(z −z1)f1(z)|j ( r (1 −r)2 )( m2+2 4 )(1−σ)−j ( (1 + r)2 r )( m2−2 4 )(1−σ) ×|h(z)p(z) + zp ′(z) |dθ. using the result of golusin [1] and schwarz inequality, we arrive at |∆j(n,z1,f(z)) | ≤ 2( m2−2 2 )(1−σ)+j rn−1 ( 1 1 −r )( m2−22 )(1−σ)−j × {( 1 2π ∫ 2π 0 |h(z) |2 )1 2 ( 1 2π ∫ 2π 0 |p(z) |2 )1 2 + 1 2π ∫ 2π 0 |zp ′(z) | } dθ. int. j. anal. appl. 18 (4) (2020) 622 in view of lemma 2.1, we get |∆j(n,z1,f(z)) | ≤ 2( m2−2 2 )(1−σ)+j rn−1 ( 1 1 −r )( m2+22 )(1−σ)−j {[1 + (m22(1 −σ)2 − 1)r2 1 −r2 ]1 2 × [ 1 + ( m21(1 −γ1)2 − 1 ) r2 1 −r2 ]1 2 + rm1(1 −γ1) 1 −r2 } ≤ 2( m2−2 2 )(1−γ2)+j ( m1(1 −γ1) + (m21(1 −γ1)2 + 1) 1 2 (m22(1 −σ)2 + 1) 1 2 ) rn−1 × ( 1 1 −r )( m2−22 )(1−σ)−j+1 . applying lemma 3.2 with z1 = ( n 1+n )2 eiθn, (n →∞), r = 1 − 1 n , we have for (m2 + 2)(1 −γ2) ≥ 4j, ∆j(n,e iθn,f(z)) = o(1)n( m2+2 2 )(1−σ)−j+1, where o(1) is a constant that depends on m1,m2,γ1 and σ. we estimate the rate of growth of hq(n) for f ∈ hm1m2 (γ1,σ). then, for q=1, h1(n) = an = ∆0(n) and h1(n) = o(1)n( m2+2 2 )( 1 1+k )−1. for q ≥ 2, we use similar arguments from noonan and thomas [6] along with lemma 3.1 and remark 3.1 to arrive at hq(n) = o(1)n( m2+2 2 )( 1 1+k )q−q2, m2 ≥ 8(k + 1)(q − 1) − 2. � corollary 3.4. [8] if f ∈ kmm, then hq(n) = o(1)   n m2 2 , q = 1, n( m2 2 +1)q−q2, q ≥ 2, m2 ≥ 8(q − 1) − 2 , where o(1) is a constant that depends on m and j, only. corollary 3.5. if f ∈ 1 −hm1m2(a,b, 1,−1), then hq(n) = o(1)   n m2 4 −1 2 , q = 1, n( m2 4 + 1 2 )q−q 2 , q ≥ 2, m2 ≥ 16(q − 1) − 2 , where o(1) is a constant that depends on γ1,m1 and j, only. int. j. anal. appl. 18 (4) (2020) 623 4. conclusion arc length and rate of growth of hankel determinant problems have always been the main interests of many researchers in geometric function theory. many studies associated to these problems revolved around classes of normalized analytic univalent functions. in this particular work, length of the image curve |z| = r < 1 under the generalized janowski close-to-convex function was proved; rate of growth of coefficients and hankel determinant for this class were also obtained. acknowledgements the authors would like to thank prof. dr. raheel qamar, rector, comsats university islamabad, pakistan, for providing excellent research and academic environments. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] g. golusin, on distortion theorems and coefficients of univalent functions, mat. sb. 19(1946), 183–203. [2] s. kanas, a. wisniowska, conic regions and k-uniform convexity, j. comput. appl. math. 105(1-2)(1999), 327-336. [3] s. kanas, a. wisniowska, conic domains and starlike functions, rev. roumaine math. pures appl. 45(4)(2000), 647-658. [4] w. kaplan, close-to-convex schlicht functions, michigan math. j. 1(2)(1952), 169-185. [5] s. mahmood, m. arif, s. n. malik, janowski type close-to-convex functions associated with conic regions, j. inequal. appl. 2017(1)(2017), 259. [6] j. w. noonan, d. k.thomas, on the hankel determinants of areally mean p-valent functions, proc. lond. math. soc. 3(3)(1972), 503-524. [7] k. i. noor, on a generalization of close-to-convexity, int. j. math. math. sci. 6(2)(1983), 327-333. [8] k. i. noor, on analytic functions related with functions of bounded boundary rotation, comment. math. univ. st. pauli, 30(2)(1981), 113-118. [9] k. i. noor, m. a. noor, higher order close-to-convex functions, math. japon. 1992. [10] k. i. noor, on subclasses of close-to-convex functions of higher order, int. j. math. math. sci. 15(2)(1992), 279-289. [11] k. i. noor, s. n. malik, on coefficient inequalities of functions associated with conic domains, comput. math. appl. 62(5) (2011), 2209-2217. [12] k. s. padmanabhan, r. parvatham, properties of a class of functions with bounded boundary rotation, ann. polon. math. 3(31)(1976), 311-323. [13] b. pinchuk, functions of bounded boundary rotation, israel j. math. 10(1)(1971), 6-16. [14] c. pommerenke, uber nahezu konvexe analytische funktionenber nahezu konvexe analytische funktionen, arch. math. (basel), 16(1)(1965), 344-347. [15] d. k. thomas, on starlike and close-to-convex univalent functions, j. lond. math. soc. 1(1)(1967), 427-435. 1. introduction 2. some preliminary lemmas 3. main results 4. conclusion acknowledgements references international journal of analysis and applications volume 16, number 5 (2018), 751-762 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-751 a comparison and error analysis of error bounds a. r. kashif1, t. s. khan2, a. qayyum3,∗, and i. faye4 1department of mathematics, capital university of science & technology islamabad, pakistan 2department of mathematics, university of peshawar, peshawar, pakistan 3department of mathematics, university of hail, 2440, saudi arabia 4fundamental and applied sciences department, universiti teknologi petronas, malaysia ∗corresponding author: atherqayyum@gmail.com abstract. in this paper, we present an error analysis with the help of ostrowski type inequalities for n-times differentiable mappings by using n-times peano kernel. a comparison is also presented which shows that obtained error bounds are better than the previous error bounds. 1. introduction integral inequalities have many potential applications in many practical problems of the real world. theory of integral inequalities is rapidly growing with the help of some basic tools of functional analysis, topology and fixed point theory. ostrowski inequality is one of them, that can be defined as: estimate the deviation of functional value from its average value and the estimation of approximating area under the curve. in the last few years, many researchers tried to obtain better bounds of ostrowski inequality in the form of different lebesgue spaces. (see for instance [16][19]). in many practical problems, it is important to bound one quantity by another quantity. the classical inequalities such as ostrowski are very useful for this purpose. received 2018-02-13; accepted 2018-04-27; published 2018-09-05. 2010 mathematics subject classification. 00a00. key words and phrases. ostrowski inequality; čebyŝev-grüss inequality and čebyŝev functional. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 751 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-751 int. j. anal. appl. 16 (5) (2018) 752 to make new ostrowski type inequalities, peano kernel is the most important tool. with the help of different peano kernels, different types of ostrowski type inequalities can be obtained. efficiency of quadrature rules also depend on the selection of kernel. an analysis with the help of some graphs are also shown. at the end, a comparison with the previous results is also presented. in 1938, ostrowski [15] discovered the following useful integral inequality. theorem 1.1. let f : [a,b] → r be continuous on [a,b] and differentiable on (a,b) , whose derivative f′ : (a,b) → r is bounded on (a,b) , i.e. ‖f′‖∞ = sup t∈[a,b] |f′ (t)| < ∞ then for all x ∈ [a,b] ∣∣∣∣∣∣ f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤  1 4 + ( x− a+b 2 b−a )2 (b−a)‖f′‖∞ . (1.1) we mention another inequality called grüss inequality [13] which is stated as the integral inequality that establishes a connection between the integral of the product of two functions and the product of the integrals, which is given below. ∣∣∣∣∣∣ 1b−a b∫ a f(x)g(x)dx− 1 b−a b∫ a f(x)dx. 1 b−a b∫ a g(x)dx ∣∣∣∣∣∣ (1.2) ≤ 1 4 (φ −ϕ)(γ −γ), where ϕ ≤ f (x) ≤ φ and γ ≤ g (x) ≤ γ, for all x ∈ [a,b] . the constant 1 4 is sharp in (1.2) . in [8], dragomir and wang combined ostrowski and grüss inequality to give a new inequality which they named ostrowski-grüss type inequalities. in [9], guessab and schmeisser proved the following ostrowski’s inequality: theorem 1.2. let f: [a,b] → r satisfy the lipschitz condition i.e., |f(t) −f(s)| ≤ m |t−s| . then for all x ∈ [ a, a+b 2 ] , we have∣∣∣∣∣∣f(x) + f(a + b−x)2 − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤  1 8 + 2 ( x− 3a+b 4 b−a )2 (b−a) m. (1.3) in (1.3), the point x = 3a+b 4 yields the following trapezoid type inequality.∣∣∣∣∣∣ f ( 3a+b 4 ) + f ( a+3b 4 ) 2 − 1 b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ b−a8 m. (1.4) the constant 1 8 is sharp in (1.4). int. j. anal. appl. 16 (5) (2018) 753 in [4], barnett et.al proved some ostrowski and generalized trapezoid inequalities. dragomir [7] and liu [10] established some companions of ostrowski type integral inequalities. alomari [1] proved the following inequality: let f: [a,b] → r be a differentiable mapping on (a,b). if f ′ ∈ l1 [a,b] and γ ≤ f ′(t) ≤ γ, for all t ∈ [a,b] , then ∣∣∣∣∣∣f (x) + f (a + b−x)2 − 1b−a b∫ a f (t) dt ∣∣∣∣∣∣ ≤ 18 (b−a) (γ −γ) . (1.5) recently, liu [11] and liu et.al [12] proved some ostrowski type inequalities. in all references mentioned above, authors proved their results by using kernels with two or three steps. qayyum et. al [18] presented a refinements of ostrowski inequality for n-th differentiable functions as: define n-times peano kernel p (x,.) : [a,b] → r by pn(x,t) =   1 n! (t−a)n , t ∈ ( a, a+x 2 ] 1 n! ( t− 3a+b 4 )n , t ∈ ( a+x 2 ,x ] 1 n! ( t− a+b 2 )n , t ∈ (x,a + b−x] 1 n! ( t− a+3b 4 )n , t ∈ ( a + b−x, a+2b−x 2 ] 1 n! (t− b)n , t ∈ ( a+2b−x 2 ,b ] , (1.6) for all x ∈ [ a, a+b 2 ] ,then following lemma holds: lemma 1.1. let f : [a,b] → r be an n-times differentiable function such that f(n−1)(x) for n ∈ n is absolutely continuous on [a,b] then 1 b−a b∫ a pn(x,t)f (n)(t)dt int. j. anal. appl. 16 (5) (2018) 754 = n−1∑ k=0 (−1)n+k+1 (k + 1)! × [ 1 2k+1 { (x−a)k+1 − ( x− a + b 2 )k+1} f(k) ( a + x 2 ) + {( x− 3a + b 4 )k+1 − ( x− a + b 2 )k+1} f(k) (x) + (−1)k+1 {( x− a + b 2 )k+1 − ( x− 3a + b 4 )k+1} f(k) (a + b−x) + ( −1 2 )k+1 {( x− a + b 2 )k+1 − (x−a)k+1 } f(k) ( a + 2b−x 2 )] + (−1)n b−a b∫ a f(t)dt for all x ∈ [ a, a+b 2 ] . 2. main results 2.1. integral inequalities for ∥∥f(n)∥∥∞. theorem 2.1. let f : [a; b] → r be an n-times differentiable function such that f(n−1) (x) for n ∈ n on (a,b) is absolutely continuous on [a,b], then ∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) n−1∑ k=0 (−1)k (k + 1)! [ 1 2k+1 { (x−a)k+1 − ( x− a + b 2 )k+1} (2.1) ×f(k) ( a + x 2 ) + {( x− 3a + b 4 )k+1 − ( x− a + b 2 )k+1} f(k) (x) + (−1)k+1 {( x− a + b 2 )k+1 − ( x− 3a + b 4 )k+1} f(k) (a + b−x) + ( −1 2 )k+1 {( x− a + b 2 )k+1 − (x−a)k+1 } f(k) ( a + 2b−x 2 )] ≤ ∥∥f(n)∥∥∞ (n + 1)! [ 1 2n (x−a)n+1 + (1 + (−1)n) ( x− 3a + b 4 )n+1 + ( −1 + (−1)n+1 2n+1 − (1 + (−1)n) )( x− a + b 2 )n+1∣∣∣∣∣ for all x ∈ [ a, a+b 2 ] . the following new quadrature rules can be obtained while investigating error bounds using above theorem. int. j. anal. appl. 16 (5) (2018) 755 qn,1 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 (b−a)k+2 2k+1 (k + 1)! [ f(k) (a) + (−1)k f(k) (b) ] + [ f(n−1)(b) −f(n−1)(a) ] (b−a)n 2n+1 (n + 1)! (1 + (−1)n) qn,2 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 (b−a)k+2 (−1)k 4k+1 (k + 1)! [ f(k) ( 3a + b 4 ) + { 1 + (−1)k } f(k) ( a + b 2 ) + (−1)k f(k) ( a + 3b 4 )] + ( f(n−1)(b) −f(n−1)(a) ) × 2 4n+1 (b−a)n (n + 1)! ((−1)n + 1) qn,3 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 (−1)k (k + 1)! (b−a)k+2 4k+1 [ 1 + (−1)k 2k+1 ( f(k) ( 7a + b 8 ) + f(k) ( a + 7b 8 )) + { (−1)k f(k) ( 3a + b 4 ) + f(k) ( a + 3b 4 )}] + [ f(n−1)(b) −f(n−1)(a) ] (b−a)n 4n+1 (n + 1)! ((−1)n + 1) ( 1 2n + 1 ) . from [12], using theorem 3.1, cerone’s quadrature rules qn,1 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 (b−a)k+1 (k + 1)! f(k) (a) qn,2 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 (b−a)k+1 (k + 1)! ( 1 + (−1)k ) f(k) ( a + b 2 ) int. j. anal. appl. 16 (5) (2018) 756 qn,3 (f) := b∫ a f(t)dt ≈ n−1∑ k=0 1 (k + 1)! (b−a)k+1 4k+1 ( (1) k+1 + (−1)k ) f(k) ( 3a + b 4 ) now, we present a comparison between cerone’s error bouds with obtained error bounds. 3. a comparison and error analysis of error bounds from [6], cerone’s error bounds are given: ec,1 = (b−a)n+1 (n + 1)! , ec,2 = (b−a)n+1 2n (n + 1)! , ec,3 = (b−a)n+1 ( 1 + 3n+1 ) 4n+1 (n + 1)! . our error bounds obtained from above theorem, are given below: ea,1 = (b−a)n+1 (n + 1)! [ 1 2n+1 + (−1)n + 1 2n+1 ] , ea,2 = (b−a)n+1 2n (n + 1)! [ 1 2n+3 + (−1)n + 1 23n+4 ] , ea,3 = (b−a)n+1 ( 1 + 3n+1 ) 4n+1 (n + 1)! × [ (−1)n + 1 1 + 33n+1 + (−1)n + 1 2n+1 (1 + 33n+1) + 3 2n+1 (1 + 33n+1) ] . to show that our error bounds are less than cerone’s error bound, we have to show that 1 2n+1 + (−1)n + 1 2n+1 < 1 1 2n+3 + (−1)n + 1 23n+4 < 1 (−1)n + 1 1 + 33n+1 + (−1)n + 1 2n+1 (1 + 33n+1) + 3 2n+1 (1 + 33n+1) < 1 case 1 first we will prove that 1 2n+1 + (−1)n + 1 2n+1 < 1 since 2n > 1 ∀ n ∈{1, 2, 3, ....} int. j. anal. appl. 16 (5) (2018) 757 also 2n+1 > 2 ∀ n ∈{1, 2, 3, ....} =⇒ 1 2n+1 < 1 2 ∀ n ∈{1, 2, 3, ....} now, since 1 + (−1)n 2n+1 =   0, if n is odd1 2n , if n is even and 1 2n+1 + 1 + (−1)n 2n+1 < 1 2 < 1 ∀ odd n now 2n+1 > 4 1 2n+1 < 1 4 ∀ even n therefore, 1 2n+1 + 1 + (−1)n 2n+1 < 1 4 + 1 2n < 1 4 + 1 2 < 1 hence proved that 1 2n+1 + (−1)n + 1 2n+1 < 1 ∀ n ∈{1, 2, 3, ....} . case 2 now, we will prove that 1 2n+3 + (−1)n + 1 23n+4 < 1 since 2n > 1 ∀ n ∈{1, 2, 3, ....} also 2n+3 > 23 1 2n+3 < 1 8 < 1 ∀ n ∈{1, 2, 3, ....} int. j. anal. appl. 16 (5) (2018) 758 since 1 + (−1)n 23n+4 =   0, if n is odd1 23n+3 , if n is even and 1 2n+3 + 1 + (−1)n 23n+4 < 1 ∀ odd n now 2n+3 > 29 ∀ even n 1 23n+4 < 1 29 ∀ even n therefore 1 2n+3 + 1 + (−1)n 23n+4 < 1 29 + 1 8 < 1 2 + 1 8 < 1 ∀ even n hence proved that 1 2n+3 + (−1)n + 1 23n+4 < 1 ∀ n ∈{1, 2, 3, ....} . case 3 we will prove that (−1)n + 1 1 + 33n+1 + (−1)n + 1 2n+1 (1 + 33n+1) + 3 2n+1 (1 + 33n+1) < 1 since 1 + (−1)n 1 + 33n+1 =   0, if n is odd2 1+33n+1 , if n is even, 1 + (−1)n 2n+1 (1 + 33n+1) =   0, if n is odd1 2n(1+33n+1) , if n is even, now 2n+1 ( 1 + 33n+1 ) > 2n+1.33n+1 > 22 · 34 > 2 · 3 ∀ odd n or 1 2n+1 (1 + 33n+1) < 1 6 3 2n+1 (1 + 33n+1) < 1 2 ∀ odd n int. j. anal. appl. 16 (5) (2018) 759 therefore 1 + (−1)n 1 + 33n+1 + 1 + (−1)n 2n+1 (1 + 33n+1) + 3 2n+1 (1 + 33n+1) < 1 2 < 1. now 1 + (−1)n 1 + 33n+1 = 2 1 + 33n+1 ∀ even n and 1 + (−1)n 2n+1 (1 + 33n+1) = 1 + (−1)n 2n (1 + 33n+1) ∀ even n. as we know that 1 + 33n+1 > 33n+1 ∀ even n or 1 1 + 33n+1 < 1 33n+1 < 1 3 ∀ even n. again, we know that 2n ( 1 + 33n+1 ) > 2n · 33n+1 ∀ even n or 1 2n (1 + 33n+1) < 1 2n · 33n+1 < 1 12 ∀ even n. now 2n+1 ( 1 + 33n+1 ) > 2n+1 · 33n+1 or 1 2n+1 (1 + 33n+1) < 1 2n+1 · 33n+1 < 1 24 < 1 8 ∀ even n. hence proved that 1 + (−1)n 1 + 33n+1 + 1 + (−1)n 2n+1 (1 + 33n+1) + 3 2n+1 (1 + 33n+1) < 1. int. j. anal. appl. 16 (5) (2018) 760 0 5 10 15 20 25 30 10 −60 10 −40 10 −20 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 1 for odd number of intervals present work’s error bound cerone’s error bound 0 5 10 15 20 25 30 10 −60 10 −40 10 −20 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 2 for odd number of intervals present work’s error bound cerone’s error bound 0 5 10 15 20 25 30 10 −60 10 −40 10 −20 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 3 for odd number of intervals present work’s error bound cerone’s error bound figure 1. comparison of error bounds for odd intervals 4. discussion cerone et al. [6] developed error bounds using 2-step kernel. but in our case, we developed error bounds with the help of 5-step kernel. in fig. 1 & 2, we established a comparison between error bounds e1, e2 and e3 for odd and even number of intervals. these figures show that our error bounds are smaller than the error bounds of cerone et al. [6] in both cases i.e. for even and odd number of intervals. also, it can be seen from the graphs that the error decreases with the increase in number of intervals. actually, use of 5-step kernel and a proper choice of scheme play a major role in minimizing the errors. it can be concluded that we have developed a very efficient new integral inequality which gives us better approximations for the quadrature. int. j. anal. appl. 16 (5) (2018) 761 0 5 10 15 20 25 30 10 −50 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 1 for even number of intervals present work’s error bound cerone’s error bound 0 5 10 15 20 25 30 10 −60 10 −40 10 −20 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 2 for even number of intervals present work’s error bound cerone’s error bound 0 5 10 15 20 25 30 10 −60 10 −40 10 −20 10 0 number of intervals e rr o r b o u n d s ( lo g a x is ) comparison between error bounds of the quadrature rule q 3 for even number of intervals present work’s error bound cerone’s error bound figure 2. comparison of error bounds for even intervals references [1] m. w. alomari, a companion of ostrowski’s inequality for mappings whose first derivatives are bounded and applications numerical integration, kragujevac j. math. 36 (2012), 77 82. [2] w. g. alshanti, a. qayyum and m. a. majid, ostrowski type inequalities by using a generalized quadratic kernel, j. inequal. spec. funct. 8 (4) (2017), 111-135. [3] w. g. alshanti and a. qayyum, a note on new ostrowski type inequalities using a generalized kernel, bull. math. anal. appl. 9 (1) (2017), 74-91. [4] n. s. barnett, s. s. dragomir and i. gomma, a companion for the ostrowski and the generalized trapezoid inequalities, j. math. comput. model. 50 (2009), 179-187. int. j. anal. appl. 16 (5) (2018) 762 [5] h. budak, m. z. sarikaya and a. qayyum, improvement in companion of ostrowski type inequalities for mappings whose first derivatives are of bounded variation and applications, filomat 31 (2017), 5305–5314. [6] p. cerone, s. s. dragomir, j. roumeliotis and j. sunde, a new generalization of the trapezoid formula for n-time differentiable mappings and applications, demonstr. math. 33(4) (2000), 719–736. [7] s. s. dragomir, some companions of ostrowski’s inequality for absolutely continuous functions and applications, bull. korean math. soc. 40 (2) (2005), 213-230. [8] s. s. dragomir and s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, comput. math. appl. 33 (11) (1997), 15-20. [9] a. guessab and g. schmeisser, sharp integral inequalities of the hermite-hadamard type, j. approx. theory, 115 (2) (2002), 260-288. [10] z. liu, some companions of an ostrowski type inequality and applications, j. inequal. pure appl. math. 10 (2) (2009), 10-12. [11] w. liu, new bounds for the companion of ostrowski’s inequality and applications, filomat, 28 (2014), 167-178. [12] w. liu, y. zhu and j. park, some companions of perturbed ostrowski-type inequalities based on the quadratic kernel function with three sections and applications, j. inequal. appl. 2013 (2013), article id 226. [13] d. s. mitrinvić, j. e. pecarić and a. m. fink, classical and new inequalities in analysis, kluwer academic publishers, dordrecht, 1993. [14] d. s. mitrinović, j. e. pecarić and a. m. fink, inequalities involving functions and their integrals and derivatives, mathematics and its applications. (east european series), kluwer acadamic publications dordrecht, 1991. [15] a. ostrowski, über die absolutabweichung einer differentienbaren funktionen von ihren integralimittelwert, comment. math. hel. 10 (1938), 226-227. [16] a. qayyum and s. hussain, a new generalized ostrowski grüss type inequality and applications, appl. math. lett. 25 (2012), 1875-1880. [17] a. qayyum, m. shoaib and i. faye, companion of ostrowski-type inequality based on 5-step quadratic kernel and applications, j. nonlinear sci. appl. 9 (2016), 537-552. [18] a. qayyum, m. shoaib and i. faye, on new refinements and applications of efficient quadrature rules using n-times differentiable mappings, j. comput. anal. appl. 23 (4) (2017), 723-739. [19] a. qayyum, a weighted ostrowski grüss type inequality for twice differentiable mappings belongs to lp [a, b]and applications, the proceedings of international conference of engineering mathematics, london, u.k., 1-3 july (2009). [20] n. ujević, new bounds for the first inequality of ostrowski-grüss type and applications, comput. math. appl. 46 (2003), 421-427. 1. introduction 2. main results 2.1. integral inequalities for "026b30d f( n) "026b30d 3. a comparison and error analysis of error bounds 4. discussion references international journal of analysis and applications volume 19, number 6 (2021), 984-996 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-984 optimal control of a partially known coupled system of bod and do laouar chafia1,2,∗, ayadi abdelhamid1 and hafdallah abdelhak3 1laboratoire des systèmes dynamiques et contrôle, larbi ben m’hidi university, oum el bouaghi, algeria 2labbes laghrour university, bp 1252 route of batna khenchela, 40004 khenchela, algeria 3laboratory of mathematics, informatics and systems (lamis), larbi tebessi university, tébessa, algeria ∗corresponding author: laouar chafia@univ-khenchela.dz abstract. the work presented in this paper is concerned with the organic pollution problem and water quality valuation. biochemical oxygen demand has been used to evaluate the quality of water. if organic matter is present the dissolved oxygen is consumed. this article considers an optimal control problem of coupled system with missing initial conditions, which presents the relation between the biochemical oxygen demand and the dissolved oxygen. the main objective is to control the concentration of dissolved oxygen using the information given in the biochemical oxygen demand equation. the main tool used to characterize the optimal control of the investigate system under the pareto control formulation. 1. introduction the environmental pollution problem is one most serious problems faced by the world. it is always linked to some terrible problems, which are unable to find a solution and causing irreparable nature damage. the presence of a sufficient concentration of dissolved oxygen (do) is all-important and necessary to preserve water life. if more oxygen is consumed than is produced, dissolved oxygen levels decline and some sensitive animals may move away, weaken, or die. oxygen is gained from the atmosphere and plants as a result of photosynthesis. running water, because of its churning, dissolves more oxygen than still water. respiration received august 23rd, 2021; accepted october 7th, 2021; published november 25th, 2021. 2010 mathematics subject classification. 49j21, 93c20, 93c41. key words and phrases. optimal control; problem with missing data; pareto control; no regret control; least regret controls; optimality system. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 984 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-984 int. j. anal. appl. 19 (6) (2021) 985 by aquatic animals, decomposition, and various chemical reactions consume oxygen. the required quantity of dissolved oxygen by aerobic biological organisms which is used for decomposing organic material under aerobic conditions at a specified temperature is called biochemical oxygen demand (bod). the decrease of bod is the way for judging the effectiveness of water purification [1–3]. many studies have been published in the context of improving the quality of the method and procedure can use to reduce the bod level, we refer to works by d.m. reynolds, s.r. ahmad in [12], salguero, jazmin and valverde, jhonny in [13], magdalena zajda and urszula aleksander-kwaterczak in [14], etc. this article provides the main insights into the debate on optimal control choice of an evolution coupled system that presents the relation between biochemical oxygen demand and dissolved oxygen. because the concentration of dissolved oxygen is of prime importance in considering the quality of water, we try to control its level by giving an assessment of the biochemical oxygen demand and for studying too its physicochemical characteristics. elsewhere, the posed coupled systems are given with unknown initial conditions that present some barriers. the main aim of our work is to characterize the optimal control. for finding the characterization of this optimal control, we dispose of the incomplete data by introducing the concepts of no-regret control and the sequence of least regret controls. the optimality coupled systems of the no regret control are formed by passing to the limit. 2. setting the problem in this section, we present a mathematical model that is used for studying the pollution problem. this considered model is not standard because it contains some missing initial conditions. we consider a fixed final time t > 0, and ω a bounded open subset of rn, n ∈ 1, 2, 3 of smooth boundary γ. we denote by q = ω×]0,t[ the space-time cylinder and by σ = γ×]0,t[ her boundary. we are interested in an evolutionary organic pollution problem in surface waters for example lakes or estuaries which is reduced to this reaction-dispersion/diffusion problem with uncertainly (2.1)   ∂y ∂t − div(d(x)oy) + r(x)y = 0 ∂z ∂t − div(d(x)oz) + r̃(x)z + r(x)y = ωχo in q, y(x, 0) = g1, z(x, 0) = g2 in ω, z = 0, ∂z ∂ν = 0 on σ, where y,z are bod and do in a given water sample at a certain temperature over a specific time period. the control function ω presents the sources of dissolved oxygen from the atmosphere and photosynthesis of plants on the control region o, and χo is the characteristic function of o. we suppose that for all ω ∈uad, we have (2.2) uad = {ω ∈ l2(q) : ωmin ≤ ω ≤ ωmax} is non-empty closed, convex, int. j. anal. appl. 19 (6) (2021) 986 were ωmin and ωmax present the minimum and the maximum concentrations of dissolved oxygen that would be present in water at a specific temperature, in the absence of other factors. the initial conditions (g1,g2) ∈ g ⊂ h− 1 2 (ω) × h10 (ω) assumed to be unknown. the boundary conditions (z, ∂z ∂ν ) ∈ h− 1 2 (σ) × h− 1 2 (σ). the functions r, r̃ and d are reaction coefficients. the coupled systems (2.1) has a unique pair solution (y,g) = (y(ω,g),z(ω,g)) where (y,z) ∈ l2(q) ∩c∞(0,t; h− 1 2 (ω)) ×l2(0,t; h10 (ω) ∩h 2(ω)) ∩c∞(0,t; h10 (ω)). there are many factors that can be reduced the level of dissolved oxygen like the respiration of the plant life and the animal life, decomposition of organic matter, reduction due to other gases, temperature increase, and others. for these reasons, the main goal of this work is to control the evolutionary organic pollution problem. exactly, we control the level of dissolved oxygen, where we point out here that we did not insert any control function in the biochemical oxygen demand equations. for fixed pair (yd,zd) ∈ (l2(ω))2 and for n > 0,g = (g1,g2) ∈ g, we define the quadratic cost function associated to (2.1) (2.3) j(ω,g) = ‖y(ω,g) −yd‖2l2(q) + ‖z(ω,g) −zd‖ 2 l2(q) + ∫ t 0 ∫ o nω2dtdx. let us minimize the following optimal control with incomplete data: (2.4) inf ω∈uad j(ω,g) ∀g ∈g. the problem (2.4) has no sense in the case of the space g is infinite. then, if g is finite we try to solve the inf – sup problem (2.5) inf ω∈uad sup g∈g j(ω,g). however, in this situation, it is very difficult to ensure that supg∈g j(ω,g) is bounded. in (1992), j.l.lions has done a good idea by adding an additional concept which is called ”no regret control. the concept of no-regret control (or, equivalently, pareto control ) of distributed systems with missing data is used by j.l. lions in [4, pareto control of distributed systems, page 90]. in [4–7], j.l. lions applied the pareto control and he associated it with a sequence of low-regret controls defined by a quadratic perturbation for deterministic distributed systems with incomplete data. in [10], o. nakolima, r. dorville, and a. omrane studied how the no regret control can be extended to the hyperbolic case. they also generalized these concepts in the case of ill-posed deterministic problems, without assuming slater’s condition [8, 9]. in [11], hafdallah a, and ayadi a applied no regret and low regret concepts to control a thermoelastic body with missing initial conditions. a. hafdallah, a. ayadi, and c. laouar applied the no-regret control notion to control an ill-posed wave equation, see [15]. int. j. anal. appl. 19 (6) (2021) 987 the principle of this idea is based on looking for controls such that (2.6) j(ω,g) ≤ j(0,g) ∀g ∈g. condition (2.6) implies sup g∈g [ j(ω,g) −j(0,g) ] is bounded. in that case, we solve the following problem (2.7) inf ω∈uad sup g∈g [ j(ω,g) −j(0,g) ] . in the following, we are defining the no regret control for the partially known problem (2.1). 3. defining the no-regret control we say that ω̂ ∈uad defines a no-regret control for (2.1) if it is the optimal solution of (2.7). lemma 3.1. for every ω ∈uad the problem (2.7) is equivalent to (3.1) inf ω∈uad ( j(ω, 0) −j(0, 0) + 2 sup g∈g ∫ ω [ g1.ζ(ω)(x, 0) + g2.ξ(ω)(x, 0) ] dx, g = (g1,g2) ∈g, where (ζ,ξ) = (ζ(ω, 0)(x,t),ξ(ω, 0)(x,t)) satisfies the following backward coupled equations (3.2)   − ∂ζ ∂t − div(d(x)oζ) + r(x)ζ + r(x)ξ = y(ω, 0) − ∂ξ ∂t − div(d(x)oξ) + r̃(x)ξ = z(ω, 0) in q, ζ(x,t) = 0, ξ(x,t) = 0 in ω, ζ = 0, ∂ζ ∂ν = 0 on σ. proof. by linearity, we can write the solution to (2.7) in the form y(ω,g) = y(ω, 0) + y(0,g), z(ω,g) = z(ω, 0) + z(0,g). then, the functional j(ω,g) can be written j(ω,g) = j(ω, 0) −j(0, 0) + 2 ∫∫ q [ y(ω, 0)y(0,g) + z(ω, 0)z(0,g) ] dtdx. we introduce (ζ(ω),ξ(ω)) the solution of (3.2). then, we use integration by parts, we obtain (3.3) ∫∫ q −( ∂ζ ∂t + div(d(x)oζ) + r(x)ζ + r(x)ξ)y(0,g)dtdx = ∫ ω g1ζ(ω)(x, 0)dx + ∫∫ q ξr(x)y(0,g)dtdx, and (3.4) ∫∫ q (− ∂ξ ∂t − div(d(x)oξ) + r̃(x)ξ)z(0,g)dtdx = ∫ ω g2.ξ(ω)(x, 0)dx + ∫∫ q ξ( ∂z ∂t (0,g) − div(d(x)oz(0,g)) + r̃(x)z(0,g))dtdx. int. j. anal. appl. 19 (6) (2021) 988 adding (3.3) to (3.4), we get∫∫ q [ y(ω, 0)y(0,g) + z(ω, 0)z(0,g) ] dtdx = ∫ ω [ g1.ζ(ω)(x, 0) + g2.ξ(ω)(x, 0) ] dx. � remark 3.1. the no regret control exist only if g1 and ζ(ω)(x, 0) (respectively g2 and ζ(ω)(x, 0)) are perpendicular to each other in h− 1 2 (ω) (respectively in h10 (ω)). for this reason, we consider the following set of admissible controls ûad = {ω ∈uad : 〈g1,ζ(ω)(x, 0)〉 h −1 2 (ω) = 0, 〈g2,ξ(ω)(x, 0)〉h10 (ω) = 0}. in [4–7], j.l. lions applied the no control and he associated it with a sequence of low-regret controls defined by a quadratic perturbation for deterministic distributed systems with incomplete data. the sequence of low-regret controls is expected to converge to the no regret control. 4. defining the sequence of low-regret controls (least regret controls) for every γ > 0, we relax the problem (3.1) by introducing a quadratic perturbation such that j(ω,g) −j(0,g) ≤ γ‖g‖2g, ∀g ∈g. we say that ω̂γ ∈uad is the sequence of low-regret controls for (2.1) if ω̂γ is the solution to (4.1) inf ω∈uad sup g∈g [ j(ω,g) −j(0,g) −γ ( ‖g1‖2 h −1 2 (ω) + ‖g2‖2h10 (ω) )] . lemma 4.1. problem (4.1) can be written as (4.2) inf ω∈uad jγ(ω), where (4.3) jγ(ω) = j(ω, 0) −j(0, 0) + 1 γ ‖ζ(ω)(x, 0)‖2 h −1 2 (ω) + 1 γ ‖ξ(ω)(x, 0)‖2h10 (ω). proof. from (3.1) and (3.2), the problem (4.1) is written as inf ω∈uad ( j(ω, 0) − j(0, 0) + sup g∈g ∫ ω [ (2g1ζ(ω)(x, 0) − γg21 ) + (2g2ξ(ω)(x, 0) − γg 2 2 ) ] dx ) . the functions f : g1 7→ (2g1ζ(ω)(x, 0) − γg21 ) and f̃ : g2 7→ (2g2ξ(ω)(x, 0) − γg22 ) are concave. then, it’s absolutely clear that sup g1∈h −1 2 (ω) f(g1) = 1 γ ‖ζ(ω)(x, 0)‖2 h −1 2 (ω) , sup g2∈h10 (ω) f̃(g2) = 1 γ ‖ξ(ω)(x, 0)‖2h10 (ω). � lemma 4.2. the problem (4.2)-(4.3) has a unique solution ω̂γ, which called sequence of least regret controls. furthermore, when γ → 0, the control ω̂γ converges weakly to the unique no regret control ω̂. int. j. anal. appl. 19 (6) (2021) 989 proof. since the set of admissible controls uad is non-empty closed and bounded, we have jγ(ω) ≥−j(0, 0) = −‖yd‖2l2(ω) −‖zd‖ 2 l2(ω), ∀ω ∈uad. then there exists dγ := inf ω∈uad jγ(ω) ≥ 0. let (ωγn) ∈uad be a minimizing sequence such that dγ = limn→∞jγ(ωγn) = jγ(ωγ). then, we get dγ ≤jγ(ωγn) < d γ + 1 n < dγ + 1. so, we deduce the bounds (4.4) ‖ωγn‖l2(0,t;o) ≤ c γ, ‖y(ωγn, 0)‖l2(q) ≤ c γ, ‖z(ωγn, 0)‖l2(q) ≤ c γ, 1 √ γ ‖ζ(ωγn)(x, 0)‖h−12 (ω) ≤ c γ, 1 √ γ ‖ξ(ωγn)(x, 0)‖h10 (ω) ≤ c γ. where cγ is a positive constant and (yγn,z γ n) = (y(ω γ n, 0),z(ω γ n, 0)) solves the coupled systems (4.5)   ∂yγn ∂t − div(d(x)oyγn) + r(x)yγn = 0 ∂zγn ∂t − div(d(x)ozγn) + r̃(x)zγn + r(x)yγn = ωγnχo in q, yγn(x, 0) = 0, z γ n(x, 0) = 0, in ω, zγn = 0, ∂zγn ∂ν = 0 on σ. multiplying the first equality of (4.5) by yγn and the second equality by z γ n. we integrate over ω, we find 1 2 d dt ∫ ω |yγn(t)| 2dx + ∫ ω r(x)|yγn(t)| 2dx− ∫ ω div (d(x)yγn(t))y γ n(t)dx = 0, and 1 2 d dt ∫ ω |zγn(t)| 2dx + ∫ ω r̃(x)|zγn(t)| 2 − div( d(x).zγn(t))z γ n(t) + r(x)y γ n(t)z γ n(t)dx = ∫ o ωγn(t)z γ n(t)dx. by integrating over [0,t] and by applying the gronwall lemma we obtain ‖yγn‖l∞(0,t;y) ≤ c γ, ‖zγn‖l∞(0,t;h10 (ω)) ≤ c γ, where cγ is a positive constant. from (4.4), we deduce ‖ ∂zγn ∂t − div(d(x)ozγn) + r̃(x)z γ n + r(x)y γ n‖l2(0,t;o) ≤ c γ. int. j. anal. appl. 19 (6) (2021) 990 then, there exists a subsequence of (ωγn), that we denote with the same indices such that, when n goes to +∞, ωγn ⇀ ω̂ γ weakly in l2(0,t;o), yγn ⇀ ŷ γ weakly in l∞(0,t;y), zγn ⇀ ẑ γ weakly in l∞(0,t; h10 (ω)), ∂zγn ∂t − div(d(x)ozγn) + r̃(x)z γ n + r(x)y γ n ⇀ f weakly in l 2(0,t;o). the space l∞(q) (respectively l∞(0,t; h10 (ω))) is continuously embedded in l 2(q) (respectively l2(0,t; h10 (ω))). clearly, we have (4.6) yγn ⇀ ŷ γ weakly in l2(q), zγn ⇀ ẑ γ weakly in l2(0,t; h10 (ω)). multiplying two equalities in (4.5) by two test functions ϕ,ψ ∈ d(q), we obtain 〈yγn,− ∂ϕ ∂t − div(d(x)oϕ) + r(x)ϕ〉l2(q) = 0, 〈zγn,− ∂ψ ∂t − div(d(x)oψ) + r̃(x)ψ〉l2(q) + 〈yγn,r(x)ψ〉l2(q) = 〈ω γ n,ψ〉l2(q). by adding the last two equalities and passing to the limit, we get (4.7)   ∂ŷγ ∂t − div(d(x)oŷγ) + r(x)ŷγ = 0 ∂ẑγ ∂t − div(d(x)oẑγ) + r̃(x)ẑγ + r(x)ŷγ = ω̂γχo in l2(0,t,o). from (4.6) and (4.7), we get ŷγ(x, 0) = 0, ẑγ(x, 0) = 0. now, we have to prove that (ζγn,ξ γ n) converges to (ζ̂ γ, ξ̂γ). let ζγn = ζ(ω γ n) and ξ γ n = ξ(ω γ n). reverse time variable by taking ζ̃γn(x,t) = ζ γ n(x,t − t), ξ̃γn(x,t) = ξγn(x,t − t), ỹγn(x,t) = yγn(x,t − t) and z̃γn(x,t) = zγn(x,t − t). then, we have  − ∂ζ̃γn ∂t − div(d(x)oζ̃γn) + r(x)ζ̃γn + r(x)ξ̃γn = ỹγn − ∂ξ̃γn ∂t − div(d(x)oξ̃γn) + r̃(x)ξ̃γn = z̃γn in q, ζ̃γn(x, 0) = 0, ξ̃ γ n(x, 0) = 0 in ω, ζ̃γn = 0, ∂ζ̃γn ∂ν = 0 on σ. then, we deduce that ζ̃γn ⇀ ζ̂ γ weakly in l2(q), ξ̃γn ⇀ ξ̂ γ weakly in l2(0,t; h10 (ω)). hence, ζ̃γn(x, 0) ⇀ ζ̂ γ(x, 0) weakly in h− 1 2 (ω), ξ̃γn(x, 0) ⇀ ξ̂ γ(x, 0) weakly in h10 (ω). int. j. anal. appl. 19 (6) (2021) 991 at last, we have lim n→∞ jγ(ωγn) = j γ(ωγ) = inf ω∈uad jγ(ω). the functional jγ is quadratic coercive, thus ω̂γ is unique. � the characterization of the sequence of least regret controls is given in the following proposition 4.1. proposition 4.1. the unique sequence of least regret controls ω̂γ is characterized by the following coupled system (4.8)   ∂ŷγ ∂t − div(d(x)oŷγ) + r(x)ŷγ = 0 ∂ẑγ ∂t − div(d(x)oẑγ) + r̃(x)ẑγ + r(x)ŷγ = ω̂γχo in q, ŷγ(x, 0) = 0, ẑγ(x, 0) = 0, in ω, ẑγ = 0, ∂ẑγ ∂ν = 0 on σ, (4.9)   − ∂ζ̂γ ∂t − div(d(x)oζ̂γ) + r(x)ζ̂γ + r(x)ξ̂γ = y(ω − ω̂γ) − ∂ξ̂γ ∂t − div(d(x)oξ̂γ) + r̃(x)ξ̂γ = z(ω − ω̂γ) in q, ζ̂γ(x,t) = 0, ξ̂γ(x,t) = 0 in ω, ζ̂γ = 0, ∂ζ̂γ ∂ν = 0 on σ, (4.10)   ∂ρ̂γ ∂t − div(d(x)oρ̂γ) + r(x)ρ̂γ = 0 ∂σ̂γ ∂t − div(d(x)oσ̂γ) + r̃(x)σ̂γ + r(x)ρ̂γ = 0 in q, ρ̂γ(x, 0) = − 1 γ ζ(ω − ω̂γ)(x, 0), σ̂γ(x, 0) = − 1 γ ξ(ω − ω̂γ)(x, 0) in ω, ẑγ = 0, ∂ẑγ ∂ν = 0 on σ, and (4.11)   − ∂p̂γ ∂t − div(d(x)op̂γ) + r(x)p̂γ + r(x)q̂γ = ŷγ −yd + ρ̂γ − ∂q̂γ ∂t − div(d(x)oq̂γ) + r̃(x)q̂γ = ẑγ −zd + σ̂γ in q, p̂γ(x,t) = 0, q̂γ(x,t) = 0 in ω, p̂γ = 0, ∂q̂γ ∂ν = 0 on σ. furthermore, for all ω ∈uad, we have (4.12) ∫ t 0 ∫ o (q̂γ + nω̂γ)(ω − ω̂γ)dxdt ≥ 0. proof. the functional jγ is quadratic coercive, thus it possesses a unique minimum ω̂γ. this minimum is a solution to the euler equation, thus for all ω ∈uad, we have lim h→0 jγ ( ω̂γ + h(ω − ω̂γ) ) −jγ(ω̂γ) h ≥ 0. int. j. anal. appl. 19 (6) (2021) 992 so, we have (4.13) ∫∫ q [ y(ω − ω̂γ)(ŷγ −yd) + z(ω − ω̂γ)(ẑγ −zd) ] dtdx + ∫ t 0 ∫ o nω̂γ(ω − ω̂γ)dtdx + 1 γ ∫ ω [ ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0) + ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0) ] dx ≥ 0. we introduce (ρ̂γ, σ̂γ) = (ρ(ω̂γ, 0)(x,t),σ(ω̂γ, 0)(x,t)) solution to (4.10). by integration by parts, we get (4.14) ∫∫ q ζ̂γ( ∂ρ̂γ ∂t − div(d(x)oρ̂γ) + r(x)ρ̂γ)dtdx = ∫∫ q ρ̂γ(− ∂ζ̂γ ∂t − div(d(x)oζ̂γ) + r(x)ζ̂γ)dtdx + 1 γ ∫ ω ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0)dx = 0, and (4.15) ∫∫ q ξ̂γ( ∂σ̂γ ∂t − div(d(x)oσ̂γ) + r̃(x)σ̂γ + r(x)ρ̂γ)dtdx = ∫∫ q σ̂γz(ω − ω̂γ)dtdx + ∫∫ q ρ̂γr̃(x)ξ̂γdtdx + 1 γ ∫ ω ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0)dx = 0. adding (4.14) to (4.15) amounts to (4.16) 1 γ ∫ ω [ ζ̂γ(x, 0)ζ(ω − ω̂γ)(x, 0) + ξ̂γ(x, 0)ξ(ω − ω̂γ)(x, 0) ] dx = ∫∫ q [ρ̂γy(ω − ω̂γ) + σ̂γz(ω − ω̂γ)]dtdx. replacing (4.16) in (4.13), we find (4.17) ∫∫ q [ y(ω − ω̂γ)(ŷγ −yd + ρ̂γ) + z(ω − ω̂γ)(ẑγ −zd + σ̂γ) ] dtdx + ∫ t 0 ∫ o nω̂γ(ω − ω̂γ)dtdx ≥ 0. we introduce now the coupled adjoint state (p̂γ, q̂γ) = (p(ω̂γ, 0)(x,t),q(ω̂γ, 0)(x,t)) solution to (4.11). finally, we obtain (4.12) by replacing (4.11) in (4.17) and integration by parts. � we need some a priori estimations, which we make in the following lemma. lemma 4.3. there exist some positive constants c independent of γ satisfy the following estimations: (4.18) ‖ω̂γ‖l2(0,t,o) ≤c, ‖ŷγ‖l2(q) ≤c, ‖ẑγ‖l2(q) ≤c, 1 √ γ ‖ζ̂γ(x, 0)‖ h −1 2 (ω) ≤c, 1 √ γ ‖ξ̂γ(x, 0)‖h10 (ω) ≤c, int. j. anal. appl. 19 (6) (2021) 993 and, ‖ŷγ‖l2(q) ≤c, ‖ ∂ŷγ ∂t ‖l2(q) ≤c, ‖ẑγ‖l2(0,t;h10 (ω)) ≤c, ‖ ∂ẑγ ∂t ‖l2(q) ≤c,(4.19) ‖ζ̂γ‖l2(q) ≤c, ‖ ∂ζ̂γ ∂t ‖l2(q) ≤c, ‖ξ̂γ‖l2(0,t;h10 (ω)) ≤c, ‖ ∂ξ̂γ ∂t ‖l2(q) ≤c,(4.20) ‖ρ̂γ‖l2(q) ≤c, ‖ ∂ρ̂γ ∂t ‖l2(q) ≤c, ‖σ̂γ‖l2(0,t;h10 (ω)) ≤c, ‖ ∂σ̂γ ∂t ‖l2(q) ≤c,(4.21) ‖p̂γ‖l2(q) ≤c, ‖ ∂p̂γ ∂t ‖l2(q) ≤c, ‖q̂γ‖l2(0,t;h10 (ω)) ≤c, ‖ ∂q̂γ ∂t ‖l2(q) ≤c.(4.22) proof. since ω̂γ is the sequence of least regret controls, we have jγ(ω̂γ) ≤jγ(ω) ∀ω ∈uad. in particular case when ω = 0, we get j(ω̂γ, 0) −j(0, 0) + 1 γ ‖ζ̂γ(x, 0)‖2 h −1 2 (ω) + 1 γ ‖ξ̂γ(x, 0)‖2h10 (ω) ≤ 0. thus, we have (4.23) ‖ŷγ −yd‖2l2(q) + ‖ẑ γ −zd‖2l2(q) + n‖|ω̂ γ‖2l2(q) + 1 γ ‖ζ̂γ(x, 0)‖2 h −1 2 (ω) + 1 γ ‖ξ̂γ(x, 0)‖2h10 (ω) ≤‖yd‖2l2(ω) + ‖zd‖ 2 l2(ω) = constant. so (4.18) holds. multiplying the first equality of (4.8) by ŷγ and the second equality by ẑγ and we integrate over ω, we find ∫ ω ŷγ(t) (∂ŷγ(t) ∂t − div(d(x)oŷγ(t)) + r(x)ŷγ(t) ) dx = 1 2 d dt ∫ ω |ŷγ(t)|2dx + ∫ ω r(x)|ŷγ(t)|2dx− ∫ ω div (d(x)ŷγ(t))ŷγ(t)dx = 0, and ∫ ω ẑγ(t) (∂ẑγ(t) ∂t − div(d(x)oẑγ(t)) + r̃(x)ẑγ(t) + r(x)ŷγ(t) ) dx = 1 2 d dt ∫ ω |ẑγ(t)|2dx + ∫ ω r̃(x)|ẑγ(t)|2dx− ∫ ω div( d(x).ẑγ(t))ẑγ(t) + ∫ ω r(x)ŷγ(t)ẑγ(t)dx = ∫ o ω̂γ(t)ẑγ(t)dx. by integrating over [0,t] and by applying the gronwall lemma we obtain ‖ŷγ‖l2(q) ≤cγ, ‖ ∂ŷγ ∂t ‖l2(q) ≤cγ, ‖ẑγ‖l2(0,t;h10 (ω)) ≤cγ, ‖ ∂ẑγ ∂t ‖l2(q) ≤cγ, where cγ is a positive constant. from the last estimations we get (4.19). we follow a similar method to demonstrate (4.19) for finding (4.20). � int. j. anal. appl. 19 (6) (2021) 994 when we pass to the limit when γ → 0, the sequence of least regret controls converges ω̂γ to the no regret control ω̂. theorem 4.1. the no regret control ω̂ = limγ→0 ω̂ γ is characterized by the unique set {(ŷ, ẑ), (ζ̂, ξ̂), (ρ̂, σ̂), (p̂, q̂)} solution to the following coupled optimality system (so) (4.24)   ∂ŷ ∂t − div(d(x)oŷ) + r(x)ŷ = 0 ∂ẑ ∂t − div(d(x)oẑ) + r̃(x)ẑ + r(x)ŷ = ω̂χo in q, ŷ(x, 0) = 0, ẑ(x, 0) = 0, in ω, ẑ = 0, ∂ẑ ∂ν = 0 on σ, (4.25)   − ∂ζ̂ ∂t − div(d(x)oζ̂) + r(x)ζ̂ + r(x)ξ̂ = y(ω − ω̂) − ∂ξ̂ ∂t − div(d(x)oξ̂) + r̃(x)ξ̂ = z(ω − ω̂) in q, ζ̂(x,t) = 0, ξ̂(x,t) = 0 in ω, ζ̂ = 0, ∂ζ̂ ∂ν = 0 on σ, (4.26)   ∂ρ̂ ∂t − div(d(x)oρ̂) + r(x)ρ̂ = 0 ∂σ̂ ∂t − div(d(x)oσ̂) + r̃(x)σ̂ + r(x)ρ̂ = 0 in q, ρ̂(x, 0) = ρ0, σ̂(x, 0) = σ0 in ω, σ̂ = 0, ∂σ̂ ∂ν = 0 on σ, and (4.27)   − ∂p̂ ∂t − div(d(x)op̂) + r(x)p̂ + r(x)q̂ = ŷ −yd + ρ̂ − ∂q̂ ∂t − div(d(x)oq̂) + r̃(x)q̂ = ẑ −zd + σ̂ in q, p̂(x,t) = 0, q̂(x,t) = 0 in ω, p̂ = 0, ∂p̂ ∂ν = 0 on σ, with the variational inequality (4.28) ∫ t 0 ∫ o (q̂ + nω̂)(ω − ω̂)dxdt ≥ 0, with the following limits: ρ0 = − lim γ→0 1 γ ζ(ω − ω̂γ)(x, 0), σ0 = − lim γ→0 1 γ ξ(ω − ω̂γ)(x, 0). proof. from inequality (4.23), we may extract some some subsequence of (ω̂γ, ŷγ, ẑγ)γ that we denote with the same indices such that, when γ goes to 0, we have ω̂γ ⇀ ω̂ weakly in l2(0,t;o), (ŷγ, ẑγ) ⇀ (ŷ, ẑ) weakly in l2(q) × l2(0,t; h10 (ω)). int. j. anal. appl. 19 (6) (2021) 995 on the other hand, from the optimality coupled systems in proposition 4.1, the sequences ( ∂ŷγ ∂t −div(d(x)oŷγ)+ r(x)ŷγ)γ and ( ∂ẑγ ∂t − div(d(x)oẑγ) + r̃(x)ẑγ + r(x)ŷγ)γ are bounded in l2(q). so, we have ∂ŷγ ∂t − div(d(x)oŷγ) + r(x)ŷγ ⇀ ∂ŷ ∂t − div(d(x)oŷ) + r(x)ŷ weakly in l2(q), ∂ẑγ ∂t − div(d(x)oẑγ) + r̃(x)ẑγ + r(x)ŷγ ⇀ ∂ẑ ∂t − div(d(x)oẑ) + r̃(x)ẑ + r(x)ŷ in l2(q). taking to the limit γ → 0, we get (4.24). also, from to a priori estimates of lemma 4.3, and by using the same method we found (4.25)-(4.27). from (4.18), we have − 1 γ ( ζ(ω − ω̂γ)(x, 0),ξ(ω − ω̂γ)(x, 0) ) ⇀ (ρ0,σ0) weakly in h −1 2 (ω) ×h10 (ω). in close, the inequality (4.28) can be deduced using the weak convergence of p̂γ, q̂γ and ω̂γ. � 5. conclusion this work examined an evolution coupled system, with missing initial conditions that presented the relation between biochemical oxygen demand (bod) and dissolved oxygen (do). since the decrease of bod is a good way for judging the effectiveness of water purification, our main objective was to control the level of dissolved oxygen, for giving more information about it. we gave the characterization of the optimal control through applied the idea of no regret, where we solved an optimal control problem with uncertainly. the obtained method optimization problem is to be transformed into classical optimal control via the notion of low regret control. finally, the coupled optimality system for the least regret control converges weakly to the coupled optimality systems for no regret control or the optimal control. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. j. mocuba, dissolved oxygen and biochemical oxygen demand in the waters close to the quelimane sewage discharge. ms thesis. the university of bergen (2010). [2] j. liu, g. olsson, b. mattiasson, short-term bod (bodst) as a parameter for on-line monitoring of biological treatment process, biosensors bioelectron. 20 (2004), 562–570. [3] j. liu, b. mattiasson, microbial bod sensors for wastewater analysis, water res. 36 (2002), 3786–3802. [4] j. l. lions contrôle de pareto de systèmes distribués. le cas stationnaire c. r. acad. sci. paris sér. i, 302 (6) (1986), 223-227. [5] j. l. lions, control of distributed systems with incomplete data. ams colloquium, berkeley, 1983. [6] j. l. lions contrôle à moindres regrets des systèmes distribuès. c. r. acad. sci. paris sér. i, 315(12) (1992), 1253–1257. [7] j. l. lions, no-regret and low-regret control, environment, economics and their mathematical models, masson, paris, (1994). int. j. anal. appl. 19 (6) (2021) 996 [8] r. dorville, o. nakoulima, a. omrane, on the control of ill-posed distributed parametersystems, esaim: proc. 17 (2007), 50–66. [9] r. dorville, o. nakoulima, a. omrane, contrôle optimal pour les problèmes de contrôlabilité des systèmes distribués à données manquantes, c. r. math. acad. sci. paris, ser. i, 338(12) (2004), 921–924. [10] o. nakoulima, a. omrane, j. velin, no-regret control for nonlinear distributed systems with incomplete data, j. math. pures appl. 81 (2002), 1161–1189. [11] a. hafdallah, a. ayadi, optimal control of a thermoelastic body with missing initial conditions, int. j. control. 93 (2020), 1570–1576. [12] d. m. reynolds, s. r. ahmad, rapid and direct determination of wastewater bod values using a fluorescence technique, water res. 31 (1997), 2012–2018. [13] j. salguero, j. valverde flores, reduction of the biochemical oxygen demand of the water samples from the lower basin of the chillon river by means of air-ozone micronanobubbles, ventanilla callao, j. nanotechnol. (lima). 1 (2017), 25-35. [14] m. zajda, u. aleksander-kwaterczak, wastewater treatment methods for effluents from the confectionery industry – an overview, j. ecol. eng. 20 (2019), 293–304. [15] a. hafdallah, a. ayadi, c. laouar, no-regret optimal control characterization for an ill-posed wave equation, int. j. math. trends techno. 41 (2017), 1-6. 1. introduction 2. setting the problem 3. defining the no-regret control 4. defining the sequence of low-regret controls (least regret controls) 5. conclusion references int. j. anal. appl. (2022), 20:36 mϕa-h-convexity and hermite-hadamard type inequalities sanja varošanec∗ department of mathematics, faculty of science, university of zagreb, bijenička 30, 10000 zagreb, croatia ∗corresponding author: varosans@math.hr abstract. we investigate a family of mϕa-h-convex functions, give some properties of it and several inequalities which are counterparts to the classical inequalities such as the jensen inequality and the schur inequality. we give the weighted hermite-hadamard inequalities for an mϕa-h-convex function and several estimations for the product of two functions. 1. preliminaries it is known that the classical convexity can be generalized to an mn-convexity, where m and n are means which is described in [8]. the other direction of generalization leads to the concept of h-convexity, [13]. it is interesting to see properties of a function which definition combines some elements of mn-convexity and of h-convexity. let m and n be two means in two variables. we say that a function f : i →r is mn-convex if f (m(x,y))≤ n(f (x), f (y)) for every x,y ∈ i. in this paper we will focus on a somewhat special type of means. let ϕ be a continuous, strictly monotone function defined on the interval i. by mϕ we denote a quasi-arithmetic mean: mϕ(x,y;t,1− t) := ϕ−1(tϕ(x)+(1− t)ϕ(y)), x,y ∈ i,t ∈ [0,1]. it is obvious that the power mean mp corresponds to ϕ(x)= xp if p 6=0 and to ϕ(x)= logx if p =0. received: jun. 26, 2022. 2010 mathematics subject classification. 26a51, 26d15. key words and phrases. the hermite-hadamard inequality; the jensen inequality; mϕa-h-convex function; quasiarithmetic mean; the schur inequality. https://doi.org/10.28924/2291-8639-20-2022-36 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-36 2 int. j. anal. appl. (2022), 20:36 let ϕ and ψ be two continuous, strictly monotone functions defined on intervals i and k respectively. let h : j → r be a non-negative function, (0,1) ⊆ j and let f : i → k such that h(t)ψ(f (x))+ h(1− t)ψ(f (y))∈ ψ(k) for all x,y ∈ i,t ∈ (0,1). we say that a function f is mϕmψ-h-convex if f (mϕ(x,y;t,1− t))≤ mψ(f (x), f (y);h(t),h(1− t)) for all x,y ∈ i and all t ∈ (0,1). especially, a function f : i →r is called mϕa-h-convex if f (mϕ(x,y;t,1− t))≤ h(t)f (x)+h(1− t)f (y) (1.1) for all x,y ∈ i and t ∈ (0,1). the mϕmψ-h-concavity is defined on a natural way. some particular cases of mϕmψ-h-convex functions are recently investigated. if h(t) = t, then the mϕmψ-h-convexity collapses to the mϕmψ-convexity which is described in [8]. if mϕ, mψ are an arithmetic mean (a), a geometric mean (g) or a harmonic mean (h), then we can find several results. for example, aa-h-convexity or simply h-convexity firstly appeared in [13]. an ha-h-convexity or harmonic-h-convexity is described in [2] and [10]. hg-h-convexity investigated in [10] and ag-hconvexity or log-h-convexity in [9]. amp-h-convexity or (h,p)-convexity is described in [6] while some properties of mpa-h-convex functions are given in [4]. also, we have to mention article [1] devoted to the mn-h-convexity where m,n ∈{a,g,h}. the aim of this paper is to give several statements about mϕa-h-convex functions primarly related to the hermite-hadamard inequality and the jensen inequality. the following section is devoted to the properties of mϕa-h-convex functions. also in that section we give counterparts to the jensen and the schur inequality and some related results. in the third section we prove several inequalities of hermite-hadamard type. 2. properties of mϕa-h-convex functions and jensen-type inequalities proposition 2.1. let ϕ be a continuous, strictly monotone function defined on the interval i. let h be a non-negative function defined on the interval j, (0,1) ⊆ j. a function f is mϕa-h-convex (concave) on i if and only if the function f ◦ϕ−1 is h-convex (concave) on ϕ(i). proof. let us suppose thatf is mϕa-h-convex on i and let u,v ∈ ϕ(i), t ∈ (0,1). since ϕ is continuous and strictly monotone on i, there exist x,y ∈ i such that u = ϕ(x),v = ϕ(y). then (f ◦ϕ−1)(tu +(1− t)v) = (f ◦ϕ−1)(tϕ(x)+(1− t)ϕ(y))) = f (mϕ(x,y;t,1− t))≤ h(t)f (x)+h(1− t)f (y) = h(t)f (ϕ−1(u))+h(1− t)f (ϕ−1(v)) = h(t)(f ◦ϕ−1)(u)+h(1− t)(f ◦ϕ−1)(v) which means that f ◦ϕ−1 is h-convex. the second case is proved similarly. � int. j. anal. appl. (2022), 20:36 3 proposition 2.2. let ϕ be a continuous, strictly monotone function defined on the interval i. let h,h1,h2 be non-negative functions defined on the interval j, (0,1)⊆ j. (i) let h1 and h2 have a property h2(t)≤ h1(t), t ∈ (0,1). if f : i → [0,∞) is mϕa-h2-convex, then f is an mϕa-h1-convex function. (ii) if f ,g are mϕa-h-convex functions, λ > 0, then f +g and λf are mϕa-h-convex. (iii) let f , : i → [0,∞) be similarly ordered functions on i, i.e. (f (x)− f (y))(g(x)−g(y))≥ 0, x,y ∈ i and h(t)+h(1− t)≤ c for all t ∈ (0,1), where h =max{h1,h2} and c is a fixed positive number. if f is mϕa-h1-convex and g is mϕa-h2-convex, then the product f g is mϕa-h-convex. proof. the proof is based on the known results for h-convex functions and characterization given in proposition 2.1. let us prove part (i). if f is mϕa-h2-convex, then f ◦ϕ−1 is h2-convex. then, using proposition 8 from [13], we get that f ◦ϕ−1 is h1-convex, i.e. f is mϕa-h1-convex. other parts are proved similarly by applying propositions 9 and 10 from [13]. � the following theorem gives a counterpart of the schur inequality. theorem 2.1. let h be a non-negative supermultiplicative function defined on the interval j, (0,1)⊆ j. let ϕ be a continuous, strictly monotone function defined on the interval i. let f : i → [0,∞) be mϕa-h-convex. if ϕ is increasing, then for any x1,x2,x3 ∈ i such that x1 < x2 < x3 and ϕ(x3)−ϕ(x2), ϕ(x3)−ϕ(x1), ϕ(x2)−ϕ(x1)∈ j the following holds h(ϕ(x3)−ϕ(x2))f (x1)−h(ϕ(x3)−ϕ(x1))f (x2)+h(ϕ(x2)−ϕ(x1))f (x3)≥ 0. (2.1) if ϕ is decreasing, then for any x1,x2,x3 ∈ i such that x1 < x2 < x3 and ϕ(x2)−ϕ(x3), ϕ(x1)−ϕ(x3), ϕ(x1)−ϕ(x2)∈ j the following holds h(ϕ(x2)−ϕ(x3))f (x1)−h(ϕ(x1)−ϕ(x3))f (x2)+h(ϕ(x1)−ϕ(x2))f (x3)≥ 0. (2.2) proof. let assume that ϕ is increasing. for x1,x2,x3 ∈ i such that x1 < x2 < x3 we have u1 := ϕ(xi) < u2 := ϕ(x2) < u3 := ϕ(x3). since a function g := f ◦ϕ−1 is h-convex, using proposition 16 from [13], we get h(u3 −u2)g(u1)−h(u3 −u1)g(u2)+h(u2 −u1)g(u3)≥ 0 and after appropriate substitutions we obtain inequality (2.1). inequality (2.2) is proved in a similar way. � 4 int. j. anal. appl. (2022), 20:36 the following theorem is a counterpart of the discrete jensen inequality and its converse for an mϕa-h-convex function. theorem 2.2. let h : j → r be a non-negative supermultiplicative function, (0,1) ⊆ j. let ϕ be a continuous, strictly monotone function defined on the interval i. let f : i → [0,∞) be a mϕah-convex function. let w1, . . . ,wn be non-negative real numbers such that wn = ∑n i=1wi 6= 0 and wi wn ∈ j, i =1, . . . ,n. (i) then for all x1, . . . ,xn ∈ i the following holds f ( ϕ−1 ( 1 wn n∑ i=1 wiϕ(xi) )) ≤ n∑ i=1 h ( wi wn ) f (xi). (ii) then for all x1, . . . ,xn ∈ (a,b)⊆ i the following holds n∑ i=1 h ( wi wn ) f (xi) ≤ f (a) n∑ i=1 h ( wi wn ) h ( ϕ(b)−ϕ(xi) ϕ(b)−ϕ(a) ) +f (b) n∑ i=1 h ( wi wn ) h ( ϕ(xi)−ϕ(a) ϕ(b)−ϕ(a) ) . proof. since f is a mϕa-h-convex function, then f ◦ϕ−1 is h-convex on ϕ(i) and using the jensen inequality for h-convex functions and its converse ( [13, theorems 19 and 21]), we get the above results. � the following result is a property of subadditivity for an index set function. let k be a finite non-empty set of positive integers. let us define the index set function f by f(k)= h(wk)f ( ϕ−1 ( 1 wk ∑ i∈k wiϕ(xi) )) − ∑ i∈k h(wi)f (xi), where wi ∈ j, wk := ∑ i∈k wi ∈ j, xi ∈ i. theorem 2.3. let h : j → r be a non-negative supermultiplicative function and let m and k be finite non-empty sets of positive integers with m ∩ k = ∅. let wi > 0, (i ∈ m ∪ k) be such that wk,wm,wm∪k ∈ j. let ϕ be a continuous, strictly monotone function defined on the interval i. if f : i → [0,∞) is mϕa-h-convex, then the following inequality holds f(m ∪k)≤ f(m)+f(k). furthermore, if mk := {1, . . . ,k}, k =2, . . . ,n and wmk ∈ j, then f(mn)≤ f(mn−1)≤ . . . ≤ f(m2)≤ 0 int. j. anal. appl. (2022), 20:36 5 and f(mn) ≤ min 1≤i<j≤n { h(wi +wj)f ( ϕ−1 ( wiϕ(xi)+wjϕ(xj) wi +wj )) −h(wi)f (xi)−h(wj)f (xj) } . proof. let us consider the following difference f(m)+f(k)−f(m ∪k) = h(wm)f ( ϕ−1 ( 1 wm ∑ i∈m wiϕ(xi) )) +h(wk)f ( ϕ−1 ( 1 wk ∑ i∈k wiϕ(xi) )) −h(wm∪k)f ( ϕ−1 ( 1 wm∪k ∑ i∈m∪k wiϕ(xi) )) . since numbers u := 1 wm ∑ i∈m wiϕ(xi) and v := 1 wk ∑ i∈k wiϕ(xi) belong to ϕ(i), there exist numbers x,y ∈ i such that ϕ(x) = u, ϕ(y) = v. using a definition of the mϕa-h-convexity for t = wmwm∪k , 1− t = wk wm∪k , and x,y and supermultiplicativity of h, we get f ( mϕ(x,y; wm wm∪k , wk wm∪k ) ) ≤ h(wm) h(wm∪k) f (x)+ h(wk) h(wm∪k) f (y) (2.3) and inequality f(m)+f(k)−f(m ∪k)≥ 0 follows from (2.3) immediately. � remark 2.1. if mϕ = a, then the above results related to the jensen inequality, its converse and to the index set function for an h-function were proved in [13]. if mϕ = h, then the jensen type inequality for ha-h-convex function is given in [2]. if mϕ = mp and h(t)= t, then the jensen inequality for mpa-convex was proved in [4]. if mϕ ∈{a,g,h}, then results from this section are given in [1]. 3. hermite-hadamard type inequality and related results counterparts of the hermite-hadamard inequality appear in the study of every kind of convexity. namely, in the classical convexity, the left-hand side or the right-hand side of the hermite-hadamard inequality are equivalent to the definition of convexity. the hermite-hadamard inequality for an hconvex function was proved in [3] and [11] and has the following form. if h is an integrable function, h(1 2 ) 6= 0, then for an integrable h-convex function f : [a,b] → r, the following sequence of inequalities hold: 1 2h(1 2 ) f ( a+b 2 ) ≤ 1 b−a ∫ b a f (x)dx ≤ [f (a)+ f (b)] ∫ 1 0 h(x)dx. (3.1) this section begins with the weighted hermite-hadamard inequality for an mϕa-h-convex function. this result is usually called the hermite-hadamard-féjer inequality. 6 int. j. anal. appl. (2022), 20:36 theorem 3.1. let h be a non-negative function defined on the interval j, (0,1) ⊆ j, h(1 2 ) 6= 0 and ϕ be a differentiable, strictly monotone function defined on [a,b]. let w : [a,b]→ [0,∞) be a function such that wϕ′ ∈ l([a,b]) and w ( ϕ−1(tϕ(a)+(1− t)ϕ(b)) ) = w ( ϕ−1((1− t)ϕ(a)+ tϕ(b)) ) (3.2) for all t ∈ (0,1). if f is mϕa-h-convex, f wϕ′ ∈ l([a,b]), then 1 2h(1 2 ) f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 ))∫ b a w(x)ϕ′(x)dx ≤ ∫ b a f (x)w(x)ϕ′(x)dx (3.3) ≤ [f (a)+ f (b)] ∫ b a h ( ϕ(b)−ϕ(x) ϕ(b)−ϕ(a) ) w(x)ϕ′(x)dx, provided that all integrals exist. moreover, 1 2h(1 2 ) f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) ≤ 1 ϕ(b)−ϕ(a) ∫ b a f (x)ϕ′(x)dx ≤ [f (a)+ f (b)] ∫ 1 0 h(x)dx, (3.4) provided that all integrals exist. proof. let us prove the first inequality in (3.3). since ϕ is continuous, strictly monotone, then for fixed t ∈ (0,1) there exist u,v ∈ [a,b] such that ϕ(u)= tϕ(a)+(1−t)ϕ(b)and ϕ(v)= (1−t)ϕ(a)+tϕ(b). then, we get f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) = f ( ϕ−1 ( 1 2 [tϕ(a)+(1− t)ϕ(b)]+ 1 2 [(1− t)ϕ(a)+ tϕ(b)] )) ≤ h ( 1 2 ) f (u)+h ( 1 2 ) f (v). multiplying the above inequality with w ( ϕ−1(tϕ(a)+(1− t)ϕ(b)) ) , integrating over [0,1] and using condition (3.2), we get f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 ))∫ 1 0 w ( ϕ−1(tϕ(a)+(1− t)ϕ(b)) ) dt ≤ h ( 1 2 )∫ 1 0 f (ϕ−1(tϕ(a)+(1− t)ϕ(b)))w ( ϕ−1(tϕ(a)+(1− t)ϕ(b)) ) dt +h ( 1 2 )∫ 1 0 f (ϕ−1((1− t)ϕ(a)+ tϕ(b)))w ( ϕ−1((1− t)ϕ(a)+ tϕ(b)) ) dt = 2h(1 2 ) ϕ(b)−ϕ(a) ∫ b a f (x)w(x)ϕ′(x)dx and the first inequality in (3.3) is proved. int. j. anal. appl. (2022), 20:36 7 multiplying inequality (1.1) with w ( ϕ−1(tϕ(a)+(1− t)ϕ(b)) ) and integrating, we get∫ 1 0 f (ϕ−1(tϕ(a)+(1− t)ϕ(b)))wϕ−1(tϕ(a)+(1− t)ϕ(b))dt ≤ ∫ 1 0 [h(t)f (a)+h(1− t)f (b)]wϕ−1(tϕ(a)+(1− t)ϕ(b))dt. using condition (3.2) and substitution tϕ(a)+(1− t)ϕ(b) = ϕ(x), we get the second inequality in (3.3). the last inequality follows from (3.3) for the particular weight w(t)=1. � remark 3.1. some particular results related to the non-weighted hermite-hadamard inequality are know. if h(t) = t, then the counterpart of the hermite-hadamard inequality (3.4) is given in [7]. the hermite-hadamard inequality (3.4) for a ha-h-convex function is given in [10], see also [15]. inequality (3.4) for an mϕa-convex function is given in [14] and for mpa-h-convex is given in [5]. corollary 3.1. let h be a non-negative function defined on the interval j, (0,1)⊆ j. let w : [a,b]→ [0,∞), [a,b]⊂ (0,∞) be a function such that w(atb1−t)= w(a1−tbt) for all t ∈ (0,1). if f is ga-h-convex, then 1 2h(1 2 ) f ( √ ab) ∫ b a w(x) x dx ≤ ∫ b a f (x) w(x) x dx ≤ [f (a)+ f (b)] ∫ b a h ( logb/x logb/a ) w(x) x dx, provided that all integrals exist. furthermore, 1 2h(1 2 ) f ( √ ab)≤ ∫ b a f (x) x dx ≤ [f (a)+ f (b)] ∫ 1 0 h(t)dt, provided that all integrals exist. proof. putting in inequalities (3.3) and (3.4) ϕ(x)= logx, we get the required results. � the following theorem contains estimations for the integral mean of the product of two mϕa-hconvex functions. theorem 3.2. let ϕ be a differentiable, strictly monotone function defined on the interval [a,b]. let hi, i = 1,2 be non-negative functions defined on the interval ji, (0,1) ⊆ ji, and let f ,g : [a,b] → [0,∞). if f is mϕa-h1-convex and g is mϕa-h2-convex, then the following hold: 8 int. j. anal. appl. (2022), 20:36 (i) 1 ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx ≤ m(a,b) ∫ 1 0 h1(t)h2(t)dt +n(a,b) ∫ 1 0 h1(t)h2(1− t)dt (3.5) (ii) 1 2h1( 1 2 )h2( 1 2 ) f ( mϕ(a,b; 1 2 , 1 2 ) ) g ( mϕ(a,b; 1 2 , 1 2 ) ) − 1 ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx ≤ m(a,b) ∫ 1 0 h1(t)h2(1− t)dt +n(a,b) ∫ 1 0 h1(t)h2(t)dt, (3.6) where h1( 1 2 )h2( 1 2 ) 6=0. (iii) 1 2(ϕ(b)−ϕ(a))2 ∫ b a ∫ b a ∫ 1 0 ϕ′(x)ϕ′(y)f (mϕ(x,y;t,1− t))g(mϕ(x,y;t,1− t))dt dy dx ≤ 1 ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx ∫ 1 0 h1(t)h2(t)dt +[m(a,b)+n(a,b)] ∫ 1 0 h1(t)dt ∫ 1 0 h2(t)dt ∫ 1 0 h1(t)h2(1− t)dt (3.7) (iv) 1 ϕ(b)−ϕ(a) ∫ b a ∫ 1 0 ϕ′(x)f (mϕ(x,ϕ −1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))× ×g(mϕ(x,ϕ−1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))dt dx ≤ 1 ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx ∫ 1 0 h1(t)h2(t)dt +[m(a,b)+n(a,b)] { h1 ( 1 2 ) h2 ( 1 2 )∫ 1 0 h1(t)h2(t)dt + [ h1 ( 1 2 )∫ 1 0 h2(t)dt +h2 ( 1 2 )∫ 1 0 h1(t)dt ]∫ 1 0 h1(t)h2(1− t)dt } , (3.8) where m(a,b)= f (a)g(a)+ f (b)g(b), n(a,b)= f (a)g(b)+ f (b)g(a) and provided that all integrals exist. proof. (i) since f is mϕa-h1-convex and g is mϕa-h2-convex, we get f (mϕ((a,b;t,1−t))≤ h1(t)f (a)+h1(1−t)f (b) and g(mϕ(a,b;t,1−t))≤ h2(t)g(a)+h2(1−t)g(b). int. j. anal. appl. (2022), 20:36 9 multiplying these two inequalities and integrating it over [0,1], we obtain∫ 1 0 f (mϕ(a,b;t,1− t))g(mϕ(a,b;t,1− t))dt ≤ m(a,b) ∫ 1 0 h1(t)h2(t)dt +n(a,b) ∫ 1 0 h1(t)h2(1− t)dt and after a substitution mϕ(a,b;t,1− t)= x, we get inequality (3.5). (ii) since ϕ(a)+ϕ(b) 2 = 1 2 (tϕ(a)+(1− t)ϕ(b))+ 1 2 ((1− t)ϕ(a)+ tϕ(b)) for t ∈ (0,1) and since f is mϕa-h1-convex and g is mϕa-h2-convex, we get f ( mϕ(u,v; 1 2 , 1 2 ) ) ≤ h1 ( 1 2 ) [f (u)+ f (v)] and g ( mϕ(u,v; 1 2 , 1 2 ) ) ≤ h2 ( 1 2 ) [g(u)+g(v)], where ϕ(u) = tϕ(a)+(1− t)ϕ(b) and ϕ(v) = (1− t)ϕ(a)+ tϕ(b). multiplying these inequalities, we obtain f ( mϕ(a,b; 1 2 , 1 2 ) ) g ( mϕ(a,b; 1 2 , 1 2 ) ) = f ( mϕ(u,v; 1 2 , 1 2 ) ) g ( mϕ(u,v; 1 2 , 1 2 ) ) ≤ h1 ( 1 2 ) h2 ( 1 2 ){ f (u)g(u)+ f (v)g(v)+ f (u)g(v)+ f (v)g(u) } ≤ h1 ( 1 2 ) h2 ( 1 2 ){ [f (u)g(u)+ f (v)g(v)]+ f (a)g(a)[h1(t)h2(1− t)+h1(1− t)h2(t)] +f (a)g(b)[h1(t)h2(t)+h1(1− t)h2(1− t)]+ f (b)g(a)[h1(1− t)h2(1− t)+h1(t)h2(t)] +f (b)g(b)[h1(1− t)h2(t)+h1(t)h2(1− t)] } , where in the last inequality we used the mϕa-h-convexity again. integrating the above inequality and using into account that∫ 1 0 f (u)g(u)dt = ∫ 1 0 f (v)g(v)dt = 1 ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx ∫ 1 0 h1(t)h2(1− t)dt = ∫ 1 0 h1(1− t)h2(t)dt, ∫ 1 0 h1(t)h2(t)dt = ∫ 1 0 h1(1− t)h2(1− t)dt we obtain inequality (3.6). (iii) since f is mϕa-h1-convex and g is mϕa-h2-convex, we get f (mϕ(x,y;t,1− t))≤ h1(t)f (x)+h1(1− t)f (y) and g(mϕ(x,y;t,1− t))≤ h2(t)g(x)+h2(1− t)g(y). 10 int. j. anal. appl. (2022), 20:36 multiplying these two inequalities, then multiplying with ϕ′(x)ϕ′(y) and integrating it over [a,b] with respect to x and y and over [0,1] with respect to t, we obtain∫ b a ∫ b a ∫ 1 0 ϕ′(x)ϕ′(y)f (mϕ(x,y;t,1− t))g(mϕ(x,y;t,1− t))dt dy dx ≤ 2(ϕ(b)−ϕ(a)) ∫ 1 0 h1(t)h2(t)dt ∫ b a f (x)g(x)ϕ′(x)dx +2 ∫ 1 0 h1(1− t)h2(t)dt ∫ b a f (x)ϕ′(x)dx ∫ b a g(x)ϕ′(x)dx. using the right-hand side of inequality (3.4) to estimate ∫b a f (x)ϕ′(x)dx and ∫b a g(x)ϕ′(x)dx and some simple transformations, we get (3.7). (iv) in this case we begin with inequalities f (mϕ(x,ϕ −1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))≤ h1(t)f (x)+h1(1− t)f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) g(mϕ(x,ϕ −1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))≤ h2(t)g(x)+h2(1− t)g ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) and proceed in the similar way, i.e. multiply them mutually, then multiply with ϕ′(x) and integrate with respect to x and t. we get∫ b a ∫ 1 0 ϕ′(x)f (mϕ(x,ϕ −1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))× ×g(mϕ(x,ϕ−1 ( ϕ(a)+ϕ(b) 2 ) ;t,1− t))dt dx ≤ ∫ b a f (x)g(x)ϕ′(x)dx ∫ 1 0 h1(t)h2(t)dt +g ( ϕ−1 ( ϕ(a)+ϕ(b) 2 ))∫ b a f (x)ϕ′(x)dx ∫ 1 0 h1(t)h2(1− t)dt +f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 ))∫ b a g(x)ϕ′(x)dx ∫ 1 0 h1(1− t)h2(t)dt +f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) g ( ϕ−1 ( ϕ(a)+ϕ(b) 2 ))∫ 1 0 h1(1− t)h2(1− t)dt. in the next step we use the right-hand side of (3.4) to estimate ∫b a f (x)ϕ′(x)dx and∫b a g(x)ϕ′(x)dx and definition of mϕa-h-convexity to estimate f ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) and g ( ϕ−1 ( ϕ(a)+ϕ(b) 2 )) . after a short calculation we get the required inequality (3.8). � remark 3.2. particular cases of the above results are already known. if ϕ(x) = x, then (3.5) and (3.6) for aa-h-convex functions are given in [11]. the above results for mϕa-convex functions, i.e. with h(t) = t are given in [14]. if ϕ(x) = xp, p 6= 0, then (3.5) -(3.8) for mpa-h-convex functions are given in [5]. int. j. anal. appl. (2022), 20:36 11 the case p =0 is not considered in [5], so, in the following corollary we give this, complementary result. corollary 3.2. let hi, i = 1,2 be non-negative functions defined on the interval ji, (0,1) ⊆ ji, and let f ,g : [a,b]→ [0,∞), [a,b]⊂ (0,∞). if f is ga-h1-convex and g is ga-h2-convex, then (i) 1 logb/a ∫ b a f (x)g(x) dx x ≤ m(a,b) ∫ 1 0 h1(t)h2(t)dt +n(a,b) ∫ 1 0 h1(t)h2(1− t)dt (ii) 1 2h1( 1 2 )h2( 1 2 ) f ( √ ab)g( √ ab)− 1 logb/a ∫ b a f (x)g(x) dx x ≤ m(a,b) ∫ 1 0 h1(t)h2(1− t)dt +n(a,b) ∫ 1 0 h1(t)h2(t)dt, where h1( 1 2 )h2( 1 2 ) 6=0 (iii) 1 2log2b/a ∫ b a ∫ b a ∫ 1 0 1 xy f (xty1−t)g(xty1−t)dt dy dx ≤ 1 logb/a ∫ b a f (x)g(x) dx x ∫ 1 0 h1(t)h2(t)dt +[m(a,b)+n(a,b)] ∫ 1 0 h1(t)dt ∫ 1 0 h2(t)dt ∫ 1 0 h1(t)h2(1− t)dt. proof. applying the function ϕ(x)= logx in theorem 3.2, we get results of this corollary. � the following theorem also contains some estimations for the integral mean of the product of two functions, but the proofs of these inequalities are based on the following inequality: if a ≤ b and c ≤ d, then ad +cb ≤ bd +ac. (3.9) theorem 3.3. let the assumptions of theorem 3.2 be satisfied. then (i) ∫ b a [ g(a)h2 ( ϕ(b)−ϕ(x) ϕ(b)−ϕ(a) ) +g(b)h2 ( ϕ(x)−ϕ(a) ϕ(b)−ϕ(a) )] f (x)ϕ′(x)dx + ∫ b a [ f (a)h1 ( ϕ(b)−ϕ(x) ϕ(b)−ϕ(a) ) + f (b)h1 ( ϕ(x)−ϕ(a) ϕ(b)−ϕ(a) )] g(x)ϕ′(x)dx ≤ (ϕ(b)−ϕ(a)) [ m(a,b) ∫ 1 0 h1(t)h2(t)dt +n(a,b) ∫ 1 0 h1(t)h2(1− t)dt ] + ∫ b a f (x)g(x)ϕ′(x)dx (3.10) 12 int. j. anal. appl. (2022), 20:36 (ii) 1 ϕ(b)−ϕ(a) ∫ b a [ h2 ( 1 2 ) f ( mϕ(a,b; 1 2 , 1 2 ) ) g(x) + h1 ( 1 2 ) g ( mϕ(a,b; 1 2 , 1 2 ) ) f (x) ] ϕ′(x)dx ≤ h1 ( 1 2 ) h2 ( 1 2 ) ϕ(b)−ϕ(a) ∫ b a f (x)g(x)ϕ′(x)dx +h1 ( 1 2 ) h2 ( 1 2 )[ m(a,b) ∫ 1 0 h1(t)h2(1− t)dt +n(a,b) ∫ 1 0 h1(t)h2(t)dt ] + 1 2 f ( mϕ(a,b; 1 2 , 1 2 ) ) g ( mϕ(a,b; 1 2 , 1 2 ) ) , (3.11) where m(a,b)= f (a)g(a)+ f (b)g(b), n(a,b)= f (a)g(b)+ f (b)g(a). proof. (i) putting in (3.9) a = f (mϕ(a,b;t,1− t)) , b = h1(t)f (a)+h1(1− t)f (b) c = g (mϕ(a,b;t,1− t)) , d = h2(t)g(a)+h2(1− t)g(b) and integrating obtained inequality with respect to t, we get∫ 1 0 [g(a)h2(t)+g(b)h2(1− t)] f (mϕ(a,b;t,1− t)) dt + ∫ 1 0 [f (a)h1(t)+ f (b)h1(1− t)]g (mϕ(a,b;t,1− t)) dt ≤ m(a,b) ∫ 1 0 h1(t)h2(t)dt +n(a,b) ∫ 1 0 h1(t)h2(1− t)dt + ∫ 1 0 f (mϕ(a,b;t,1− t))g (mϕ(a,b;t,1− t)) dt. after substitution u = mϕ(a,b;t,1−t) in integrals the above inequality collapses to inequality (3.10). (ii) from mϕa-hi-convexity we get f ( mϕ(a,b; 1 2 , 1 2 ) ) = f ( mϕ(u,v; 1 2 , 1 2 ) ) ≤ h1 ( 1 2 ) f (u)+h1 ( 1 2 ) f (v) and g ( mϕ(a,b; 1 2 , 1 2 ) ) = g ( mϕ(u,v; 1 2 , 1 2 ) ) ≤ h2 ( 1 2 ) g(u)+h2 ( 1 2 ) g(v), where ϕ(u)= (1− t)ϕ(a)+ tϕ(b) and ϕ(v)= tϕ(a)+(1− t)ϕ(b). putting in (3.9) a = f ( mϕ(a,b; 1 2 , 1 2 ) ) , b = h1 ( 1 2 ) [f (u)+ f (v)] c = g ( mϕ(a,b; 1 2 , 1 2 ) ) , d = h2 ( 1 2 ) [g(u)+g(v)] int. j. anal. appl. (2022), 20:36 13 and integrating with respect to t, we get h2 ( 1 2 ) f ( mϕ(a,b; 1 2 , 1 2 ) )∫ 1 0 [g(u)+g(v)]dt +h1 ( 1 2 ) g ( mϕ(a,b; 1 2 , 1 2 ) )∫ 1 0 [f (u)+ f (v)]dt ≤ h1 ( 1 2 ) h2 ( 1 2 )[∫ 1 0 f (u)g(u)dt + ∫ 1 0 f (v)g(v)dt ] +2h1 ( 1 2 ) h2 ( 1 2 )[ m(a,b) ∫ 1 0 h1(t)h2(1− t)dt +n(a,b) ∫ 1 0 h1(t)h2(t)dt ] +f ( mϕ(a,b; 1 2 , 1 2 ) ) g ( mϕ(a,b; 1 2 , 1 2 ) ) . after substitution ϕ(x)= (1−t)ϕ(a)+tϕ(b) in integrals ∫1 0 f (u)g(u)dt, ∫1 0 f (u)dt and ∫1 0 g(u)dt, and substitution ϕ(x)= tϕ(a)+(1− t)ϕ(b) in integrals ∫1 0 f (v)g(v)dt, ∫1 0 f (v)dt and ∫1 0 g(v)dt, we obtain inequality (3.11). � remark 3.3. if h(t)= t, results (3.10) and (3.11) are given in [14]. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] m.w. alomari, some properties of h − mn-convexity and jensen’s type inequalities, j. interdiscip. math. 22 (2019), 1349–1395. https://doi.org/10.1080/09720502.2019.1698402. [2] i.a. baloch, m. de la sen, i. iscan, characterizations of classes of harmonic convex functions and applications, int. j. anal. appl. 17 (2019), 722–733. https://doi.org/10.28924/2291-8639-17-2019-722. [3] m. bombardelli, s. varošanec, properties of h-convex functions related to the hermite-hadamard-féjer inequalities, computers math. appl. 58 (2009), 1869–1877. https://doi.org/10.1016/j.camwa.2009.07.073. [4] t.h. dinh, k.t.b. vo, some inequalities for operator (p,h)-convex functions, linear multilinear algebra, 66 (2018), 580–592. https://doi.org/10.1080/03081087.2017.1307914. [5] z.b. fang, r. shi, on the (p,h)-convex function and some integral inequalities, j. inequal. appl. 2014 (2014), 45. https://doi.org/10.1186/1029-242x-2014-45. [6] l.v. hap, n.v. vinh, on some hadamard-type inequalities for (h,r)-convex functions, int. j. math. anal. 7 (2013), 2067–2075. https://doi.org/10.12988/ijma.2013.28236. [7] f.c. mitroi, c.i. spiridon, hermite-hadamard type inequalities of convex functions with respect to a pair of quasi-arithmetic means, math. rep. 14 (2012), 291–295. [8] c.p. niculescu, l.e. persson, convex functions and their applications: a contemporary approach, springer new york, new york, ny, 2006. https://doi.org/10.1007/0-387-31077-0. [9] m.a. noor, f. qi, m.u. awan, some hermite–hadamard type inequalities for log-h-convex functions, analysis. 33 (2013), 367–375. https://doi.org/10.1524/anly.2013.1223. [10] m.a. noor, k.i. noor, m.u. awan, et al. some integral inequalities for harmonically h-convex functions, u.p.b. sci. bull. ser. a. 77 (2015), 5–16. https://doi.org/10.1080/09720502.2019.1698402 https://doi.org/10.28924/2291-8639-17-2019-722 https://doi.org/10.1016/j.camwa.2009.07.073 https://doi.org/10.1080/03081087.2017.1307914 https://doi.org/10.1186/1029-242x-2014-45 https://doi.org/10.12988/ijma.2013.28236 https://doi.org/10.1007/0-387-31077-0 https://doi.org/10.1524/anly.2013.1223 14 int. j. anal. appl. (2022), 20:36 [11] m.z. sarikaya, a. saglam, h. yildirim, on some hadamard-type inequalities for h-convex functions, j. math. inequal. 2 (2008), 335–341. [12] s. turhan, m. kunt, i. iscan, hermite-hadamard type inequalities for mϕa-convex functions, int. j. math. model. comput. 10 (2020), 57–75. [13] s. varošanec, on h-convexity, j. math. anal. appl. 326 (2007), 303–311. https://doi.org/10.1016/j.jmaa. 2006.02.086. [14] s. wu, m.u. awan, m.a. noor, et al. on a new class of convex functions and integral inequalities, j. inequal. appl. 2019 (2019), 131. https://doi.org/10.1186/s13660-019-2074-y. [15] d. zhao, t. an, g. ye, et al. on hermite-hadamard type inequalities for harmonical h-convex interval-valued functions, math. inequal. appl. 23 (2020), 95–105. https://doi.org/10.7153/mia-2020-23-08. https://doi.org/10.1016/j.jmaa.2006.02.086 https://doi.org/10.1016/j.jmaa.2006.02.086 https://doi.org/10.1186/s13660-019-2074-y https://doi.org/10.7153/mia-2020-23-08 1. preliminaries 2. properties of ma-h-convex functions and jensen-type inequalities 3. hermite-hadamard type inequality and related results references 1 international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 11-20 http://www.etamaths.com _________ 2010 mathematics subject classification. 49m15. key words and phrases. hod, approximation, operators, convexity. © 2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 11 applications octav olteanu abstract. this review work presents the general statemen t of a its applications. we consider the estimation of the absolu te error too. on e makes the connec tion to the contracti on principle. one of the applications is approximating ,1, /1 pa p where a a positive s elfadjoint operator is acting on a hilbert space. one tone operators. for the local approach, we mention appropriate references. 1. introduction ([1], [4]). the evaluat a version for convex monotone operators. this statement and its proof are important due to the generality of the obtained result. one applies this method to equations involving functions of matrices or of operators, as well to the successive approximation method from the contraction principle. in particular, one 12 olteanu approximates arppa p ,,1, /1  being a selfadjoint operator with the spectrum contained in the positive semi axis. this work is related to the papers [2], [5], [6]. for some other aspects on the same subject (involving analyticity), see [1]. the background is contained in [3] and [4]. the rest of the paper is organized as follows. section 2 contains the detailed proof of the in section 3, some direct applications to concrete scalar and operatorial equations are considered. section 4 makes the connection to the contraction principle, by approximating apa p ,1, /1  being a positive selfadjoint operator acting on a hilbert space. 2. general statements let x be a  complete vector lattice, endowed with a solid  yxyx  and o continuous norm  .0 xxxx norderinn let y be a normed vector space, endowed with an order relation defined by a closed convex cone. for ,,, baxba  we denote    .,, bxaxxba  pet   .,, 1 ybacp  in most of our applications, we have ,yx  where y is an order complete banach lattice of selfadjoint operators, that is also a commutative algebra (see [3], p. 303-305). theorem 2.1 assume additionally that for each       xylxpbax ,,, 1    and that      .bpxpapbxa  if     ,0,0  bpap then there exists a unique solution  x of the equation   ,0xp where       .,,:,liminf: 1 10 nkxpxpxxbxxxx kkkkkk     (1) moreover, we have 13      .0, 1   kk xpapxxbxa (2) proof. using induction upon ,k we prove that   .,,0 1 nkxxxp kkk   (3) the last relations (1) and the convexity of ,p yield                 0,0 110   kkkkkkk xpxpxxxpxpxpbpxp hence   .0 nkxp k  this relations lead to       .0 1 1 1 nkxxxpxpxx kkkkkk     we derive the following useful relations                     ., 00,0 1 1 nkaxxaxaxpxp xpapxpxpap kkkkk kkk     using the hypothesis on the space ,x there exists  x defined by the first relations (1), and from (3) we infer that the sequence   kk x is decreasing. passing through the limit in the recurrence relations (1) one obtains         .00 1     xpxpxp from the assumptions on the positivity of    ,, apbp  and from the definition of ,  x we infer that .bxa   in order to prove (2), one uses the convexity once more:                         .,0 11 nkxpapxxxxxpap xxapxxxpxpxpxp kkkk kkkk     the uniqueness of the solution follows quite easily:               .00 21212 1 221221     xxxxxpxpxxxpxpxp similarly, we can write: ,0 12   xx hence .21   xx □ the corresponding statement for convex decreasing operators holds. 14 olteanu theorem 2.2 assume that for any  bax , there exists    1  xp such that            ., 111     bpxpbxaxyxp if     ,0,0  bpap then there exists an unique solution [,] bax   of the equation   ,0xp  x being given by:       .,,,limsup 1 10 nkxpxpxxaxxxx kkkkkk     moreover, the sequence   kk x is increasing and the convergence rate is given by the inequalities      ., 1 nkxpbpxx kk   3. direct applications during this section we mention some applications of the general theorems of section 2. the difficulties consists only in technical details concerning verifying conditions from general theorems. that is why we do not prove all the statements. theorem 3.1 (see theorem 2.4 [6]). let h be a hilbert space and x the commutative algebra defined in [3], p. 303-305. let   0,0,,...,1,0, 0   nj bbnjxb be such that    n j j bb 1 0 and 1 21 2   n n unbubb  is invertible for any  .,0 iu  then there exists a unique ,0, iuu  such that ,0 01 1 1    bububub n n n n  and this solution verifies in particular the relation    n j j bbui 1 0 15 proof. the space x is an order-complete banach lattice and a commutative algebra of selfadjoint operators. let   xxip   ,0: be defined by   . 01 bububup n n   one can show that the operator   n n uup  is convex on  x (see [6] for details). now it follows easily that p is also convex on ,x since all the coefficients   xb k and all the operators in x are permutable. on the other hand, we have:              .,0, ~ , ~ 2 ~ 2 1 12 11 12 1 iuxvvbubunbvup vbubunbvup n n n n        we also have                 .,00 00,0 11 1 1 1 iuipupp vbnbvbunb vbvpiuup n n n       because of the hypothesis, one has     ,0,00 0 1 0    bbipbp n k k so that all requirements of theorem 2.1 are accomplished. it follows that there exists a unique solution [,0] iu  of the equation   ,0up that verifies relation (9) for :0k      .0 01 1 1 1 bbbbippui n    16 olteanu now the proof is complete. □ theorem 3.2 let xh , be as in the preceding theorem, xb  ,1 such that the spectrum     [.,ln]  bs there is a unique solution xiu  [,0] of the equation   0exp  ibu  and this solution verifies in particular the relation .exp 1 ibbui   the next result is an application of the scalar version of theorem 2.2, when .ryx  proposition 3.1 let 0,,  be such that   .1exp1   then there exists a unique solution [1,0]  x of the equation   0exp   xx and we have   .1,0 exp 1 0         x theorem 3.3 let h be a finite dimensional hilbert space, and x the space defined in [3], p. 303-305. let xcba ,, ~ be such the spectrums of a ~ and b are contained in [.,0]  assume also that   . ~ exp icba  then there exists a unique solution [,0] iu  of the equation   0 ~ exp  cbuua and the following estimation holds:    iccibaau   ,0 ~ exp ~~ 1 17 proposition 3.2 there is a unique solution [2,2/3]  x of the equation ,0142 23  xx and this solution verifies .2 152 27 2   x theorem 3.4 let a be a selfadjoint operator acting on a hilbert space, with the spectrum    .2,2/3as let  axyx  defined in [3], p. 303-305. then there exists a unique operator xu  such that: 042 23  iuu and this operator verifies   [2, 152 27 2] us 4. approximating ;1, /1 pa p connection to contraction principle let h be a hilbert space, a a selfadjoint operator acting on ,h with the spectrum   )([,,0] axxas  the associated commutative algebra according to [3], p. 303-305. we denote   hahhah haha ,sup,,inf 11  theorem 4.1 (see theorem 2.1 [2]). let a be as above,   .,1, rppispa  there exists a unique operator [,] /1/1 iiu p a p ap   such that ,0 au p p and this solution verifies the relations 18 olteanu     ., 1 11 /1 /1 /1 /1       ai p iuai p ui a pp ap apapp a p p a   corollary 4.1 with the above notations and assumptions, we have . 1 lnln 11   aiai aaa a aa    remark 4.1          xpxpxxxx kk 1 1 :,     the mapping  is a contraction, the rate of co nvergence of the sequence   kk x is given by contraction principle. next, we show that this is the case of the operator   ,auup p  which leads to the positive solution . /1 p p au  theorem 4.2 (see [2]). let xap ,, be as above. then the newton recurrence for the equation   0 auup p is   ., 11 , 1 1 /1 0 nkau p u p p uuiu p kkkk p a       the convergence rate for p k au /1  is given by ., 1 1/1/1 nkai p p au a p a k p k             proof. p is:       ., 1111 , 111 1 1 1 /1 0 nkuau p u p p au p u p u aupuuuibu k p kk p k pp kk p k p kkk p a          19 let  ,;: /1 pauxum  where the root is obtained by the aid of functional calculus for .a clearly, m is closed in ,x hence it is complete. let xm : be defined by   ., 11 1 muau p u p p u p      a straightforward computation shows that   ./1/1 pp aa  first we show that     ;[,0] muus   (in particular this proves that   mm  ). one can show that  is convex on the subset of all operators in x having the spectrum contained in the positive semiaxis. in particular,  is convex on .m direct computations yield        ./1/1/1/1 pppp aauaau   thus   mu  for all u with spectrum   [.,0] us now we prove that mm : is a contraction, with contraction constant . 1 p p q   precisely we prove that   ., 1 mu p p u    we have:     ., 1 1 0; 1 mu p p uiaui auiaumuaui p p u p ppp           20 olteanu now the conclusion on  being a contraction follows by a standard differential calculus argument. now application of contraction theorem and an elementary computation shows that   ., 1 1 1/1 00 /1 nkai p p uu q q au a p a kk p k                the proof is complete. □ references [1] argyros, i. k., on the convergence and applications of newton -like methods for analytic operators, j. appl. math. & computing, 10, 1-2 (2002), 41-50. [2] balan, v., olteanu, a. & olteanu, o., some applications, romanian journal of pures and applied math ematics, 51, 3 (2006), 277-290. [3] cristescu, r ., ord ered vector spaces and linear op erators, academiei , bucharest, and abacus press, tunbridge wells, kent, 1976. [4] kantorovich, l. v. & akilov, g. p., functioanal analysis, scien tific and encyclop edic publishing house, bucharest, 1986 (in romanian). [5] j. m., olteanu, a. & olteanu, o., applications of newton convex monotone operators, mathematical reports, 7(57), 3 (2005), 219-231. [6] olteanu, o. & si mion, gh., a new geometric aspect of the i mplicit function principle and method for operators. mathematical reports 5 (55), 1 (2003), 61-84. depart ment of mathe mat ics -in for mat ics, un iversity p ol itehn ica of , 060042 bucharest, romania international journal of analysis and applications volume 19, number 4 (2021), 542-560 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-542 methods for designing experiments to study the actual causes of the housing crisis alanazi talal abdulrahman∗ department of mathamtics, college of science, university of ha’il, saudi arabia ∗corresponding author: t.shyman@uoh.edu.sa abstract. in this paper, i use the edge design analysis, contrast method analysis and regression analysis to study the actual reasons that led to the housing crisis. also, applications of the housing crisis in ksa have been obtained, for example, the extent of this phenomenon with two levels (+, -). 1. introduction having an appropriate house is perhaps the main issues for all individuals. the universal declaration of human rights stressed the privilege of each human to have an appropriate house. subsequently, numerous investigations have since featured the significance of having reasonable homes that serve the necessities, everything being equal. this would help guarantee social and financial soundness for everybody and empower people to be more dynamic local area individuals. because of a few significant socioeconomics, social, social and financial changes, numerous nations have confronted lodging issues and issues around the arrangement of appropriate houses over the most recent couple of many years. as of late, oil incomes have empowered the gulf cooperation council nations to observe quick improvement that has changed them from helpless nations into cutting edge ones. numerous advancements have additionally been seen. because of such turn of events, the legislatures in gulf cooperation council nations have progressed the urban areas and increased the living expectations through making upgrades to the monetary, social and metropolitan climate. saudi received april 11th, 2021; accepted may 4th, 2021; published may 25th, 2021. 2010 mathematics subject classification. 91b25. key words and phrases. the housing crisis; edge designs; contrast method; linear model; screening design; data analysis. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 542 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-542 int. j. anal. appl. 19 (4) (2021) 543 arabia, as the biggest nation of the gulf cooperation council nations with complete land space of the kingdom is around 1,960,582 sq. km. likewise, as perhaps the most populated ones arriving at the quantity of 34710 million, as per ministry of municipal , rural affairs(see figure 1). this fast increment has caused lodging issues and lodging shortages( [1]). figure 1. ksa urbanization from 2015-2035 notwithstanding, the nation has accomplished observable advancement in numerous areas and fields inside a brief timeframe. this is credited primarily to the expanded incomes delivered by oil trades. the lodging area is one of the areas that has seen huge improvement throughout the most recent couple of many years, both the lodging approaches of saudi arabia are portrayed by the giving of enormous help to this area to make the securing of houses simpler for the residents. all things considered, because of fast metropolitan turn of events, close by monetary and social changes, including movement to urban communities, diverse region of saudi arabia have encountered major and continuous changes to the metropolitan climate and the occupants’ ways of life. these progressions have accordingly influenced the requirements, wants and goals of the occupants and additionally, the interest for lodging. presently, the lodging areas in saudi arabia are confronting various difficulties, including, in addition to other things: a prerequisite to give an immense number of houses. there has been a reduction in the quantity of houses offered on reasonable installment plans. expanded houses and property costs. proprietorship rates waiting be raised as they have diminished throughout the most recent decade. more interest being created in the primary locale and urban areas. as inhabitants are the principle partners and major parts as far as the lodging issue, the investigation of their perspectives, lodging necessities and the variables influencing their current and future lodging choices would be useful in outlining int. j. anal. appl. 19 (4) (2021) 544 future insights and plans. in light of the study’s aftereffects of general authority for statistics,2018, the absolute number of homes busy with saudi families made up (3591098) in 2018, contrasted with (3504690) in 2017. the quantity of people living in such abodes recorded (21420372), with a normal size of saudi families coming to (5,96), contrasted with (5,97) in 2017( [2]). nonetheless, the normal size can be gotten by partitioning the quantity of people by the complete number of families. it merits referencing here that the lodging issue in saudi arabia has pulled in the consideration of specialists from various fields and foundations ( [3]). nonetheless, during this examination, it was noticed that there has been a lessening throughout the most recent couple of years in the quantity of scholastic investigations being attempted that attention on the examination of this issue according to the perspective of the inhabitants and their longings. along these lines, this is the primary inspiration for the determination of the subject of this exploration. a goal of this examination is to find the actual causes of the housing crisis in saudi society. to accomplish this objective, this examination used subjective and quantitative information to decide the genuine purposes behind of the housing crisiss. this paper starts by the method using in different examples. 2. method target population and study sample the main technique used in this survey is that it is based on only statistics and statistical analysis that included 16 questions about housing crisis. the number of people who participated to answer these questions has reached 410 individual. data collection by then learn the achievement levels for the concentrated on understudy using the sensible model. this is displayed in table 1, which clarifies the information gathered. i have collected data using the techniques of application of edge design and supersaturated designs to study the housing crisis. also, to analyse the data from both techniques and applying this data in the app of spss. int. j. anal. appl. 19 (4) (2021) 545 table 1. the reasons of the housing crisis in data reasonsr signal yes/no the banking sector + yes − no the lack of the citizen’s culture of the importance of house owning + yes − no the religion contributed in the housing crisis by prohibiting the works of the banks + yes − no earth association and family extended roots + yes − no the increasing prices of the real estate compared wit individual’s income + yes − no the unemployment + yes − no the natural factors and climate (earth quick) + yes − no landowners monopolization of the lands and not investing in them + yes − no the poor system of the current housing plans + yes − no int. j. anal. appl. 19 (4) (2021) 546 table 1. (cont.) the limited availability of the residential lands and their usability + yes − no the lack of the governmental service facilities + yes − no the rural exodus that has conquered the cities + yes − no the lack of the housing units offered + yes − no the absence of the supervision over offices and real estate companies + yes − no the increasing of building materials and manpower prices + yes − no real estate developer’s’ exploitation of residential lands + yes − no y(response) number the individual’s income int. j. anal. appl. 19 (4) (2021) 547 how to select a sample a pre-plan comprising of six components and twelve preliminaries is chosen by edge design method. then, a pre-plan comprising of 16 components and 14 preliminaries is chosen by supersaturated design method . these plans are considered taking all things together the information that has been placed in the past advances and if there should arise an occurrence of acquiring the entire plan is put the estimation of the reaction and the quantity of the individual who addressed the survey. consolidating and adjusting distinctive examination technique the plan picked in the past advance is dissected utilizing the edges , supersaturated design and regression analysis methods . on the off chance that the elements that we get from the edge plan technique are equivalent to supersaturated designs , these elements are the genuine reasons for housing crisis. 3. application of data analysis 3.1. use of the edge plans in screening tests. statistical significance was analyzed using edge design analysis using the spss computer software to determine the actual causes that led to of the housing crisis in saudi society. [4] have presented the edge plans. the edge relies upon a model-autonomous test that can be utilized for dynamic factors. to know the dynamic factors, the estimations are organizes into a gathering of e sets. in this methodology the estimations contrasts in just a single part. it is normal in screening tests, and suggests that practically all distinctions zi,j := yi −yj, (i,j) ∈ e, more details about this method can be found in [5]. in order to address these ethical concerns, the following example are taken. example 3.1. let n = 6 and the lack of the governmental service facilitie x1, the rural exodus that has conquered the cities x2, the lack of the housing units offered x3, the absence of the supervision over offices and real estate companies x4, the increasing of building materials and manpower prices x5 and real estate developer’s’ exploitation of residential lands x6. int. j. anal. appl. 19 (4) (2021) 548 table 2. one replicate for example 3.1. run x1 x2 x3 x4 x5 x6 y 1 + + + + + 840 2 + + + + + 10000 3 + + + + + 2000 4 + + 900 5 + + 2000 6 + + 10000 7 + + + + 9000 8 + + + + 15000 9 + + + + 5000 10 + 0 11 + 3000 12 + 3000 presently an investigation information in table 2 (utilizing liner regreesion) with the program bundle spss. according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 2) is under 0.05, which is there is dynamic factor the poor system of the current housing plans x3 and allow an evaluated liner show y = 14900 − 6550x3 + �, with r − sg = 87.9%� of mean 8450 and standard deviation σ = 6325.77. also from the result, the residual is normal because p-value (0.130) is more than 0.05(see figure 3). int. j. anal. appl. 19 (4) (2021) 549 figure 2. final model for example 3.1 figure 3. test of normal for example 3.1 example 3.2. let n = 6 and the poor system of the current housing plans x1, the limited availability of the residential lands and their usabilityx2 , the lack of the governmental service facilitie x3, the rural exodus that has conquered the cities x4, the lack of the housing units offered x5 and the absence of the supervision over offices and real estate companies x6. int. j. anal. appl. 19 (4) (2021) 550 table 3. one replicate for example 3.2. run x1 x2 x3 x4 x5 x6 y 1 + + + + + 1000 2 + + + + + 4000 3 + + + + + 6800 4 + + 4000 5 + + 7000 6 + + 55000 7 + + + + 2400 8 + + + + 7000 9 + + + + 4855 10 + 2000 11 + 3000 12 + 4000 now an analysis data in table 3 (using liner regression) with the software package spss. according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 4) is under 0.05 that revealed there are active variables the poor system of the current housing plans x1, the limited availability of the residential lands and their usabilityx2 the rural exodus that has conquered the cities x4, and the lack of the housing units offered x5 and give an estimated liner model y = 18075 − 764.5x1 − 2078.25x2 + 885.5x4 + 151.5x5 + �, with r − sg = 94.6%� of mean 4296.25 and standard deviation σ = 2007.43. also from the result, the residual is normal because p-value (0.965) is more than 0.05( see figure 5). int. j. anal. appl. 19 (4) (2021) 551 figure 4. final model for example 3.2 figure 5. test of normal for example 3.2 example 3.3. let n = 6 and the natural factors and climate (earth quick) x1, landowners monopolization of the lands and not investing in themx2 , the poor system of the current housing plans x3, the limited availability of the residential lands and their usability x4, the lack of the governmental service facilities x5 and the rural exodus that has conquered the cities x6. int. j. anal. appl. 19 (4) (2021) 552 table 4. one replicate for example 3.3. run x1 x2 x3 x4 x5 x6 y 1 + + + + + 3000 2 + + + + + 2500 3 + + + + + 10000 4 + + 5000 5 + + 800 6 + + 6000 7 + + + + 5000 8 + + + + 14000 9 + + + + 7000 10 + 10000 11 + 5600 12 + 10000 now an analysis data in table 4 (using liner regression) with the software package spss. according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 6) is under 0.05 that revealed there is active variable x1 the lack of the governmental service facilities and give an estimated liner model y = −703.33 − 4022x1 + �, with r− sg = 86.6%� of mean 5061.66 and standard deviation σ = 4777.55. also from the result, the residual is normal because p-value (0.983) is more than 0.05(see figure 7). int. j. anal. appl. 19 (4) (2021) 553 figure 6. final model for example 3.3 figure 7. test of normal for example 3.3 moreover, from the model and research, it can be concluded that the examination with the edge plans and regression analysis given results that are associated and just like the investigation that is performed with the whole data. in this manner, the poor system of the current housing plans, the poor system of the current housing plans , the limited availability of the residential lands and their usability, the rural exodus that has conquered the cities , and the lack of the housing units offered and the lack of the governmental service facilities are the factor that driven to the housing crisis in saudi society. int. j. anal. appl. 19 (4) (2021) 554 3.2. application of the supersaturated designs in screening experiments. statistical significance was analyzed using supersaturated designs with the spss computer software to determine the actual causes that led to of the housing crisis in saudi society. [6]who presented a technique for examining ssds that utilizing another agreements based strategy. supersaturated plans are partial factorial plans in which the run size (n) is too little to even consider assessing every one of the fundamental impacts. under the impact sparsity supposition, the utilization of supersaturated plan can give the minimal effort distinguishing proof of the trivial few, potentially ruling components (screening). in order to address these ethical concerns, the following example are taken. example 3.4. in this example, we assume n = 16 that shown in table 5. examination of these information (forward table 5. supersaturated design(ssd) for example 3.4. run x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 y 1 + + + + + + + 0 2 + + + + + + + + + + + 9400 3 + + + + + + + + + + + + 7000 4 + + + + + + + + + + + + 1500 5 + + + + + + + + + + 0 6 + + + + + + + 0 7 + + + + + 12000 8 + + + + + 12000 9 + + + + 8700 10 + + + + + 0 11 + + + + + + + 9000 12 + + + + + 6000 13 + + + + + + + + + + + 0 14 + + + 34000 determination) with program bundle spss. according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 8) is under 0.05 that uncovers as dynamic factors x3,x16,x7,x14,x15 and x8, an assessed direct demonstrate is: y = 15113.78 + 8046.65x3 − 6461.90x16 + 4094.22x7 − 2459.401x14 + 2116.295x15 + 2003.552x8 + ε with � of mean 7114.28 and standard deviation σ = 9062.67. also from the result, the residual is normal because p-value (0.384) is more than 0.05(see figure 9). int. j. anal. appl. 19 (4) (2021) 555 figure 8. final model for example 3.4 figure 9. test of normal for example 3.4 int. j. anal. appl. 19 (4) (2021) 556 example 3.5. table 6. supersaturated design(ssd) for example 3.5. run x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 y 1 + + + + + 22000 2 + + + 15000 3 + 9000 4 + + + + + + 3000 5 + + + + + + + + + + + + + 3200 6 + + + + + 1000 7 + + + + + + + + + + + 1000 8 + + + + + 6000 9 + + + + + + + + + + 990 10 + + + + + + + + 5600 11 + + + + + + + + + + + + + + + 12344 12 + + + + + + + + + + + + + + + + 2000 13 + + + + + + + + + + + + + + 8700 14 + + + + + + + + + + + + + 2600 analysis of these data based on forward selection on table 6, according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 10) is under 0.05 that there are two active variables x2andx15. the estimated linear model: y = 8348.688 − 4875.313x15 + 3325.75x2 + ε. with � of mean 6602.42 and standard deviation σ = 6251.82. also from the result, the residual is normal because p-value (0.590) is more than 0.05(see figure 11). int. j. anal. appl. 19 (4) (2021) 557 figure 10. final model for example 3.5 figure 11. test of normal for example 3.5 example 3.6. table 7. supersaturated design(ssd) for example 3.6. run x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 y 1 + + 1000 2 + + 3000 3 + + + 0 4 + + + + 0 5 + + + 0 6 + + + + + + + + + + + 0 7 + + + + + + + + + + + 12670 8 + + + + + + + + + + + 0 9 + + + + + + + + + + + 10000 10 + + + + + + + + + + 0 11 + + + + + + 5500 12 + + + + + + + + 0 13 + + + + + + + + + + + + 10000 14 + + + + + + + + + + + + 10000 int. j. anal. appl. 19 (4) (2021) 558 analysis of these data based on forward selection on table 7, according to the findings of the final model it tends to be seen that the p-value from the analysis of variances table (figure 12) is under 0.05 that there is active variable x11. the estimated linear model: with � of mean 3762.42 and standard deviation σ = 4854.58. also from the result, the residual is normal because p-value (0.197) is more than 0.05(see figure 13). figure 12. final model for example 3.6 figure 13. test of normal for example 3.6 moreover, from the model and research, it can be concluded that the examination with the supersaturated design and regression analysis given results that are associated and just like the investigation that is performed with the whole data. in this manner the lack of the citizen’s culture of the importance of house owning, the religion contributed in the housing crisis by prohibiting the works of the banks, the natural factors and climate (earth quick), landowners monopolization of the lands and not investing in them, the absence of the supervision over offices and real estate companies, the increasing of building materials and manpower prices and real estate developer’s’ exploitation of residential lands and the lack of the governmental service facilities are the factor that driven to the housing crisis in saudi society. int. j. anal. appl. 19 (4) (2021) 559 4. discussion during this examination, it was noticed that there has been a lessening in the course of the most recent couple of years in the quantity of scholarly investigations being embraced that emphasis on the examination of this issue according to the perspective of the inhabitants and their cravings. one of the points of this consider was to consider the genuine causes that driven to the spread of the actual causes of the housing crisis .statistical signi cance was dissected utilizing edge design, supersaturated design with the regression examination utilizing the spss computer computer program to decide the actual causes that driven to the actual causes of the housing crisis in saudi society. i can say that the examination with the edge plans and supersaturated design given results that are associated and just like the investigation that is performed with the whole data. in this manner, tthe lack of the governmental service facilities is the factor that driven to the housing crisis in saudi society. on this premise, it is suggested that saudi wellbeing experts create an operational arrange to consider the maladies and crises that the housing crisis. on this basis, it is recommended that saudi goverment develop an operational plan to study these causes. in future investigations, it might be possible for data to use supersaturated designs, where many factors are investigated using only a few experimental runs. furthermore, there is a requirement for more research thinks about that address different parts of diabetes in the saudi setting, for example, the extent of this phenomenon with three levels (+,0,-). acknowledgment. i might want to thank the university of ha’il in saudi arabia for monetarily supporting my exploration project. ethics endorsement and agree to participate. not relevant availability of information and material. the information upholds the finding of this investigation are accessible inside the article and its advantageous material. funding. not relevant conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] c. susilawati, m. al surf, challenges facing sustainable housing in saudi arabia: a current study showing the level of public awareness, in: proceedings of the 17th annual pacific rim real estate society conference, pacific rim real estate society, 2011, pp. 1–12. [2] ministry of municipal & rural affairs, saudi cities report 2019, https://unhabitat.org/saudi-cities-report-2019, (2019). [3] m.h. aldebasi, n.a. alsobaie, a.y. aldayel, k.m. alwusaidi, t. alasbali, public awareness regarding the differences between ophthalmologists and optometrists among saudi adults living in riyadh: a quantitative study, j. ophthalmol. 2018 (2018), 7418269. int. j. anal. appl. 19 (4) (2021) 560 [4] c. elster, a. neumaier, screening by conference designs, biometrika, 82(3) (1995), 589–602. [5] t. alanazi, s. georgiou, s. stylianou, construction and analysis of edge designs from skew-symmetric supplementary difference sets, commun. stat.-theory meth. 47(20) (2018), 5064–5076. [6] c. koukouvinos, s. stylianou, a method for analyzing supersaturated designs, commun. stat.-simul. comput. 34 (2005), 929–937. 1. introduction 2. method 3. application of data analysis 3.1. use of the edge plans in screening tests 3.2. application of the supersaturated designs in screening experiments 4. discussion acknowledgment ethics endorsement and agree to participate availability of information and material funding references international journal of analysis and applications volume 18, number 5 (2020), 838-848 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-838 surfaces as graphs of finite type in h2 ×r ahmed azzi1,∗, zoubir hanifi2, mohammed bekkar1 1department of mathematics, faculty of sciences, university of oran 1, ahmed benbella algeria 2ecole nationale polytechnique d’oran, département de mathématiques, et informatique b.p 1523 el m’naour, oran, algérie ∗corresponding author: azzi.mat@hotmail.fr abstract. in this paper, we prove that ∆x = 2h where ∆ is the laplacian operator, r = (x,y,z) the position vector field and h is the mean curvature vector field of a surface s in h2 ×r and we study surfaces as graphs in h2 × r which has finite type immersion. 1. introduction the h2 ×r geometry is one of eight homogeneous thurston 3-geometries e3, s3, h3, s2 ×r, h2 ×r, ˜sl(2,r), nil, sol. the riemannian manifold (m,g) is called homogeneous if for any x, y ∈ m there exists an isometry φ : m → m such that y = φ(x). the two and three-dimensional homogeneous geometries are discussed in detail in [6] . a euclidean submanifold is said to be of finite chen-type if its coordinate functions are a finite sum of eigenfunctions of its laplacian [3]. b. y. chen posed the problem of classifying the finite type surfaces in the received may 24th, 2020; accepted july 1st, 2020; published july 27th, 2020. 2010 mathematics subject classification. 53b05, 53b21, 53c30. key words and phrases. laplacian operator; h2 × r geometry; surfaces of coordinate finite type; minimal surfaces. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 838 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-838 int. j. anal. appl. 18 (5) (2020) 839 3-dimensional euclidean space e3. further, the notion of finite type can be extended to any smooth function on a submanifold of a euclidean space or a pseudo-euclidean space. let s be a 2-dimensional surface of the euclidean 3-space e3. if we denote by r, h and ∆ the position vector field, the mean curvature vector field and the laplace operator of s respectively, then it is well-known that [3] (1.1) ∆r = −2h. a well-known result due to takahashi states that minimal surfaces and spheres are the only surfaces in e3 satisfying the condition ∆r = λr for a real constant λ. from (1.1), we know that minimal surfaces and spheres also verify the condition (1.2) ∆h = λh, λ ∈ r. equation (1.1) shows that s is a minimal surface of e3 if and only if its coordinate functions are harmonic. in [9], d. w. yoon studied surfaces invariant under the 1-parameter subgroup in sol3. in 2012, m. bekkar and b. senoussi [1] studied the translation surfaces in the 3-dimensional euclidean and lorentz-minkowski space under the condition ∆iiiri = µiri, µi ∈ r, where ∆iii denotes the laplacian of the surface with respect to the third fundamental form iii. a surface s in the euclidean 3-space e3 is called minimal when locally each point on the surface has a neighborhood which is the surface of least area with respect to its boundary [5]. in 1775, j. b. meusnier showed that the condition of minimality of a surface in e3 is equivalent with the vanishing of its mean curvature function, h = 0. let z = f(x,y) define a graph s in the euclidean 3-space e3. if s is minimal, the function f satisfies (1 + (fy) 2)fxx − 2fxyfxfy + (1 + (fx)2)fyy = 0, which was obtained by j. l. lagrange in 1760. in 1835, h. f. scherk studied translation surfaces in e3 and proved that, besides the planes, the only minimal translation surfaces are given by z(x, y) = 1 λ log |cos(λx)|− 1 λ log |cos(λy)| , where λ is a non-zero constant. in 1991, f. dillen, l. verstraelen and g. zafindratafa. [4] generalized this result to higher-dimensional euclidean space. int. j. anal. appl. 18 (5) (2020) 840 in 2015, d. w. yoon [8] studied translation surfaces in the product space h2×r and classified translation surfaces with zero gaussian curvature in h2 ×r. in 2019, b. senoussi, m. bekkar [7] studied translation surfaces of finite type in h3 and sol3 and the authors gived some theorems. a surface s(γ1, γ2) in h2 ×r is a surface parametrized by s : ω ⊆ r2 → h2 ×r, x(s, t) = γ1(s) ∗γ2(t) or x(s, t) = γ2(t) ∗γ1(s), where γ1 and γ2 are any generating curves in r3. since the multiplication ∗ is not commutative. in this work we study the surfaces as graphs of functions ϕ = f(s, t)) in h2 ×r satisfy the condition (1.3) ∆xi = λixi, λi ∈ r. 2. preliminaries let h2 be represented by the upper half-plane model {(x, y) ∈ r | y > 0} equipped with the metric gh = (dx2 + dy2) y2 . the space h2, with the group structure derived by the composition of proper affine maps, is a lie group and the metric gh is left invariant. therefore, the product space h2 ×r is a lie group with the left invariant product metric g = (dx2 + dy2) y2 + dz2, we can define the multiplication law on h2 ×r as follows (x,y,z) ∗ (x̄, ȳ, z̄) = (yx̄ + x,yȳ,z + z̄). the left identity is (0, 1, 0) and the inverse of (x,y,z) is (− x y , 1 y ,−z), on h2 ×r a left-invariant metric ds2 = (ω1)2 + (ω2)2 + (ω3)2, where ω1 = dx y , ω2 = dy y , ω3 = dz, is the orthonormal coframe associated with the orthonormal frame e1 = y ∂ ∂x , e2 = y ∂ ∂y , e3 = ∂ ∂z , the corresponding lie brackets are [e1, e2] = −e1, [ei, ei] = [e3, e1] = [e2, e3] = 0,∀i = 1, 2, 3. int. j. anal. appl. 18 (5) (2020) 841 the levi-civita connection ∇ of h2 ×r is given by  ∇e1e1 ∇e1e2 ∇e1e3   =   0 1 0 −1 0 0 0 0 0     e1 e2 e3   , ∇e2ei = ∇e3ei = 0, ∀i = 1, 2, 3. let s be an immersed surface in h2×r given as the graph of the function z = f(x,y). hence, the position vector is described by r(x,y) = (x,y,f(x,y)) and the tangent vectors rx = ∂r ∂x and ry = ∂r ∂y in terms of the orthonormal frame (e1,e2,e3) are described by rx = ∂ ∂x + fr ∂ ∂z = 1 y e1 + fxe3,(2.1) ry = ∂ ∂y + fy ∂ ∂z = 1 y e2 + fye3.(2.2) definition 2.1. [3] the immersion (s,r) is said to be of finite chen-type k if the position vector x admits the following spectral decomposition r = r0 + k∑ i=1 ri, where ri are e3-valued eigenfunctions of the laplacian of (s,r) : ∆ri = λiri, λi ∈ r, i = 1, 2, ..,k. if λi are different, then s is said to be of k-type. for the matrix g = (gij) consisting of the components of the induced metric on s, we denote by g−1 = (gij) (resp. d = det(gij)) the inverse matrix (resp. the determinant) of the matrix (gij). the laplacian ∆ on s is, in turn, given by (2.3) ∆ = −1√ |d| ∑ ij ∂ ∂ri (√ |d|gij ∂ ∂rj ) . if r = r(x,y) = (r1 = r1(x,y),r2 = r2(x,y),r3 = r3(x,y)) is a function of class c 2 then we set ∆r = (∆r1, ∆r2, ∆r3). 3. surfaces as graphs of finite type in h2 ×r let s be a graph of a smooth function f : ω ⊂ r2 → r. we consider the following parametrization of s r(x,y) = (x,y, f(x, y)), (x,y) ∈ ω. int. j. anal. appl. 18 (5) (2020) 842 theorem 3.1. a beltrami formula in h2 ×r is given by the following: ∆r = 2h, where ∆ is the laplacian of the surface and h is the mean curvature vector field of s. proof. a basis of the tangent space tps associated to this parametrization is given by rx = ∂ ∂x + fx ∂ ∂z = 1 y e1 + fxe3, ry = ∂ ∂y + fy ∂ ∂z = 1 y e2 + fye3, the coefficients of the first fundamental form of s are given by e = g(rx, rx) = 1 y2 + f2x, f = g(rx, ry) = fxfy, g = g(ry, ry) = 1 y2 + f2y . the unit normal vector field n on s is given by n = 1 w ( − 1 y fxe1 − 1 y fye2 + 1 y2 e3 ) , where w = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . to compute the second fundamental form of s, we have to calculate the following rxx = ∇rxrx = 1 y2 e2 + fxxe3, rxy = ∇rxry = ∇ryrx = − 1 y2 e1 + fxye3,(3.1) ryy = ∇ryry = − 1 y2 e2 + fyye3. so, the coefficients of the second fundamental form of s are given by l = g(∇rxrx, n) = 1 wy2 ( fxx − 1 y fy ) , m = g(∇rxry, n) = 1 wy2 ( fxy + 1 y fx ) , n = g(∇ryry, n) = 1 wy2 ( fyy + 1 y fy ) , where w = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . thus, the mean curvature h of s is given by h = en − 2fm + gl 2w2 . h = 1 2w3y2 [ 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ] . int. j. anal. appl. 18 (5) (2020) 843 by (2.3), the laplacian operator ∆ of r can be expressed as (3.2) ∆ = − 1 w4 [ w2 ( g ∂2 ∂x2 − 2f ∂2 ∂x∂y + e ∂2 ∂y2 ) + ∆1 ∂ ∂x + ∆2 ∂ ∂y ] , where ∆1 = 2 y2 fyf 2 xfxy − 1 y4 fxfxx − 1 y2 fxf 2 y fxx − 1 y4 fxfyy − 1 y2 f3xfyy − 2 y5 fxfy − 1 y3 f3xfy − 1 y3 fxf 3 y , and ∆2 = 2 y2 fxf 2 y fxy − 1 y4 fyfyy − 1 y2 f2xfyfyy − 1 y4 fyfxx − 1 y2 f3y fxx − 1 y5 f2y + 1 y5 f2x + 1 y3 f4x + 1 y3 f2xf 2 y . by a straightforward computation, the laplacian operator ∆ of r with the help of (3.1) and (3.2) turns out to be ∆r = − 1 w4   ( 2 y3 f2xfyfxy − 1 y5 fxfxx − 1 y3 fxf 2 y fxx − 1 y5 fxfyy − 1 y3 f3xfyy + 1 y4 f3xfy + 1 y4 fxf 3 y ) e1 + ( 2 y3 fxf 2 y fxy − 1 y5 fyfyy − 1 y3 f2xfyfyy − 1 y5 fyfxx − 1 y3 f3y fxx + 1 y4 f2xf 2 y + 1 y4 f4y ) e2 + ( − 2 y4 fxfyfxy − 1 y5 f2xfy − 1 y5 f3y + 1 y6 fxx + 1 y4 f2y fxx + 1 y6 fyy + 1 y4 f2xfyy ) e3   , ∆r =   ( −fx wy ) 1 w3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e1 + ( −fy wy ) 1 w3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e2 + ( 1 wy2 ) 1 w3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ) e3   , ∆r = 1 w3y2 ( 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy )   ( −fx wy ) e1 + ( −fy wy ) e2 + ( 1 wy2 ) e3   , thus we get ∆r = 2hn,(3.3) = 2h, int. j. anal. appl. 18 (5) (2020) 844 where h is the mean curvature vector field of s. s is a minimal surfaces in h2 ×r if and only if its coordinate functions are harmonic . � 4. surfaces as graphs in h2 ×r satisfying 4xi = λixi let s be an immersed surface in h2 × r given as the graph of function z = f(x,y). hence, the vector position is described by r(x,y) = (x,y,f(x,y)). we have rx = 1 y e1 + fxe3, ry = 1 y e2 + fye3, where rx = ∂r ∂x , ry = ∂r ∂x , and fx = ∂f ∂x , fy = ∂f ∂y . from an earlier results the mean curvature h of s and the unit normal vector field n on s are given by h = 1 2w3y2 [ 1 y2 (fxx + fyy) + (f 2 xfyy + f 2 y fxx) − 1 y (f2xfy + f 3 y ) − 2fxfyfxy ] , and (4.1) n = 1 w ( − 1 y fxe1 − 1 y fye2 + 1 y2 e3 ) , where w = √ 1 y4 + 1 y2 f2x + 1 y2 f2y . if the vector position on the tangent space tps is described by r = (x,y,f(x,y)) r(x,y) = x ∂ ∂x + y ∂ ∂y + f(x,y) ∂ ∂z , then (4.2) r(x,y) = x y e1 + e2 + f(x,y)e3. the equation (1.3) by means of (3.3), (4.1) and (4.2) gives rise to the following system of ordinary differential equations ( 2h w ) fx = −λ1x,(4.3) ( 2h w ) fy = −λ2y,(4.4) 2h w = λ3y 2f.(4.5) int. j. anal. appl. 18 (5) (2020) 845 therefore, the problem of classifying the surfaces s of (1.3) is reduced to the integration of this system of ordinary differential equations. next we study it according to the constants λ1, λ2 and λ3. case 1. let λ3 = 0. in this case the system (4.3), (4.4) and (4.5) is reduced equivalently to( 2h w ) fx = −λ1x,(4.6) ( 2h w ) fy = −λ2y,(4.7) 2h w = 0.(4.8) the equation (4.8) implies that the mean curvature h is identically zero. thus, the surface s is minimal; and we get also λ1 = λ2 = 0. case 2. let λ3 6= 0. in this case we study the general system (4.3), (4.4) and (4.5). 2-i): if λ1 = λ2 = 0, then h = 0. from (4.5) we obtain λ3 = 0, so we get a contradiction. 2-ii): if λ1 = 0 and λ2 6= 0., from (4.3) we obtain hfx = 0. 2-ii-a: if h = 0 (4.4), (4.5) implies that λ2 = λ3 = 0. so we get a contradiction. 2-ii-b: if fx = 0, then f(x,y) = ϕ(y), where ϕ is smooth function of y. the mean curvature h turns to (4.9) h = 1 2wy3 ( 1 y ϕ′′ −ϕ′3 ) , where ϕ′ = dϕ dy . using (4.4) and (4.5) we obtain ϕ′ = −λ2 λ3yϕ , which leads to, λ3ϕ ′ϕ = −λ2 y . after integrating with respect to y, we obtain int. j. anal. appl. 18 (5) (2020) 846 λ3 2 ϕ2(y) = −λ2 ln y + φ(x); y > 0, where φ is smooth function of x, and hence f(x,y) = ϕ(y) = ± √ λ2 λ3 ln 1 y2 + φ(x). using the condition fx = 0 we get φ(x) = a, a ∈ r. thus, f(x,y) = ϕ(y) = ± √ λ2 λ3 ln 1 y2 + c; c = 2 λ3 a, in this subcase, the surfaces s are given by r(x,y) = ( x,y,± √ λ2 λ3 ln 1 y2 + c ) ; λ2 6= 0, λ3 6= 0, c ∈ r. 2-iii): if λ1 6= 0 and λ2 = 0., from (4.4) we obtain hfy = 0. 2-iii-a: if h = 0, (4.3) and (4.5) implies that λ2 = λ3 = 0. so we get a contradiction. 2-iii-b: if fy = 0,then f(x,y) = ψ(x), where ψ is smooth function of x. the mean curvature h turns to (4.10) h = 1 2wy4 ψ′′, where ψ′ = dψ dx . using (4.3) and (4.5) we get ψ′ = −λ1x λ3y2ψ , so we can write (4.11) λ3y 2 + λ1 x ψψ′ = 0, a differentiation with respect to y gives λ3y = 0, this implies that λ3 = 0 and from (4.8) we get the mean curvature h is identically zero. from (4.6) and (4.7) we obtain λ1 = λ2 = 0, which leads to a contradiction. int. j. anal. appl. 18 (5) (2020) 847 2-iv): if λ1 6= 0 and λ2 6= 0 from (4.3), we have (4.12) 2h w = − λ1x ψ′ . substituting (4.12) into (4.5), we get − λ1x ψ′ = λ3y 2ψ, a differentiation with respect to x gives −λ1 ( ψ −xψ′′ ψ′2 ) = λ3y 2ψ′, this equation gives (4.13) λ1 ( ψ′ −xψ′′ ψ′3 ) + λ3y 2 = 0. a differentiation with respect to y gives λ3y = 0, this implies that λ3 = 0 and from (4.8) we get the mean curvature h is identically zero. from (4.6) and (4.7) we obtain λ1 = λ2 = 0, which leads to a contradiction. therefore, we have the following theorem, theorem 4.1. let s be a surface as graph of function parametrized by r(x,y) = (x,y,f(x,y)) in h2 × r then, s satisfies the equation ∆ri = λiri, λi ∈ r if and only if s is minimal surfaces or parametrized as s : r(x,y) = ( x,y,± √ λ2 λ3 ln 1 y2 + c ) ; λ2 6= 0, λ3 6= 0, c ∈ r. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. bekkar, b. senoussi, translation surfaces in the 3-dimensional space satisfying ∆iiiri = µiri, j. geom. 103 (2012), 367-374. [2] m. bekkar, h, zoubir, surfaces of revolution in the 3-dimensional lorentz minkowski space satisfying ∆jri = µiri, int. j. contemp. math. sci. 3 (2008), 1173-1185. [3] b-y. chen, total mean curvature and submanifolds of finite type, (2nd edition), world scientific publisher, singapore, 1984. int. j. anal. appl. 18 (5) (2020) 848 [4] f. dillen, l. verstraelen, g. zafindratafa, a generalization of the translation surfaces of scherk. differential geometry in honor of radu rosca: meeting on pure and applied differential geometry, leuven, belgium, 1989, ku leuven, departement wiskunde (1991), pp. 107–109. [5] d. hoffman, h. matisse, the computer-aided discovery of new embedded minimal surfaces, math. intell. 9 (1987) 8–21. [6] p. scott, the geometries of 3-manifolds, bull. london math. soc. 15 (1983), 401–487. [7] b. senoussi, m. bekkar, translation surfaces of finite type in h3 and sol3, anal. univ. orad. fasc. math. tom, xxvi (1) (2019), 17-29. [8] d.w. yoon, on translation surfaces with zero gaussian curvature in h2×r, int. j. pure appl. math. 99 (3) 2015, 289-297. [9] d.w. yoon, coordinate finite type invariante surfaces in sol spaces, bull. iran. math. soc. 43 (2017), 649-658. 1. introduction 2. preliminaries 3. surfaces as graphs of finite type in h2r 4. surfaces as graphs in h2r satisfying xi=ixi references international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 79-95 http://www.etamaths.com analysis of nonlinear fractional nabla difference equations jagan mohan jonnalagadda abstract. in this paper, we establish sufficient conditions on global existence and uniqueness of solutions of nonlinear fractional nabla difference systems and investigate the dependence of solutions on initial conditions and parameters. 1. introduction discrete fractional calculus deals with sums and differences of arbitrary orders. looking into the literature of fractional difference calculus, two approaches are found: one using the ∆ point of view (called the fractional delta difference approach) and another using the ∇ perspective (called the nabla fractional difference approach). the theory for fractional nabla difference calculus was initiated by gray and zhang [18], atici and eloe [9] and anastassiou [17], where basic approaches, definitions and properties of fractional sums and differences were reported. recently, a series of papers continuing research on fractional nabla difference equations has appeared [10, 11, 12, 14, 16, 19, 20, 21, 22, 23, 26]. but a very little progress has been made to develop fractional nabla difference systems [11, 24]. in the following example, we illustrate the advantage of fractional order nabla difference system over integer order nabla difference system. example 1. consider the following two systems. ∇0u(t) = βtβ−1, 0 < β < 1, t ∈ n1,(1.1) ∇α0∗u(t) = βt β−1, 0 < α < β < 1, t ∈ n1,(1.2) where ∇α0∗ is the caputo type fractional nabla difference operator. the solution of (1.1) is given by (1.3) u(t) = u(0) + tβ, t ∈ n0, clearly (1.3) tends to ∞ as t →∞ for 0 < β < 1 and thus it is unstable. but the solution of (1.2) is given by (1.4) u(t) = u(0) + γ(1 −β) γ(1 −β + α) tα−β, t ∈ n0. 2010 mathematics subject classification. 39a10, 39a99. key words and phrases. fractional order; nabla difference; fixed point; global existence; uniqueness; stability. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 79 80 jonnalagadda clearly (1.4) tends to 0 as t →∞ for 0 < α < β < 1 and therefore it is stable, which implies that the fractional order system may have additional attractive feature over the integer order system. on the other hand, several authors [13, 15, 31, 32] used fixed point theorems to discuss existence, uniqueness and stability properties of fractional differential systems. motivated by this fact, in this paper, we initiate the study on global existence and uniqueness of solutions of nonlinear fractional nabla difference systems. the present paper is organized as follows: section 2 contains preliminaries on nabla discrete fractional calculus and functional analysis. we consider a system of nonlinear fractional nabla difference equations and obtain sufficient conditions on global existence and uniqueness of solutions and the dependence of solutions on initial conditions and parameters in sections 3 and 4 respectively. 2. preliminaries we shall use the following notations, definitions and known results of discrete fractional calculus [8, 9, 24, 29] throughout this article. for any a, b ∈ r, na = {a,a + 1,a + 2, ...........}, na,b = {a,a + 1,a + 2, ...........,b} where a < b. definition 2.1. for any α, t ∈ r, the α rising function is defined by tα = γ(t + α) γ(t) , t ∈ r\{......,−2,−1, 0}, 0α = 0. we observe the following properties of rising factorial function. lemma 2.1. assume the following factorial functions are well defined. (1) tα(t + α)β = tα+β. (2) if t ≤ r then tα ≤ rα. (3) if α < t ≤ r then r−α ≤ t−α. definition 2.2. let u : na → r, α ∈ r+ and choose n ∈ n1 such that n − 1 < α < n. (1) (nabla difference) the first order backward difference or nabla difference of u is defined by ∇u(t) = u(t) −u(t− 1), t ∈ na+1, and the nth order nabla difference of u is defined recursively by ∇nu(t) = ∇(∇n−1u(t)), t ∈ na+n. in addition, we take ∇0 as the identity operator. (2) (fractional nabla sum) the αth order fractional nabla sum of u is given by (2.1) ∇−αa u(t) = 1 γ(α) t∑ s=a+1 (t−ρ(s))α−1u(s), t ∈ na where ρ(s) = s− 1. also, we define the trivial sum by ∇−0a u(t) = u(t) for t ∈ na. nonlinear fractional nabla difference equations 81 (3) (r l nabla fractional difference) the αth order riemann liouville type nabla fractional difference of u is given by (2.2) ∇αau(t) = ∇ n [ ∇−(n−α)a u(t) ] , t ∈ na+n. for α = 0, we set ∇0au(t) = u(t), t ∈ na. (4) (caputo fractional nabla difference) the αth order caputo type fractional nabla difference of u is given by (2.3) ∇αa∗u(t) = ∇ −(n−α) a [ ∇nu(t) ] , t ∈ na+n. for α = 0, we set ∇0a∗u(t) = u(t), t ∈ na. theorem 2.2. (power rule) let α > 0 and µ > −1. then, (1) ∇−αa (t−a)µ = γ(µ+1) γ(µ+α+1) (t−a)µ+α, t ∈ na. (2) ∇αa (t−a)µ = γ(µ+1) γ(µ−α+1) (t−a) µ−α, t ∈ na+n . let f : na × r → r, u : na → r and 0 < α < 1. consider a nonautonomous fractional nabla difference equation of riemann liouville type together with an initial condition of the form ∇αa−1u(t) = f(t,u(t)), t ∈ na+1,(2.4) ∇−(1−α)a−1 u(t) ∣∣∣ t=a = u(a) = u0.(2.5) then, from [30], u is a solution of the initial value problem (2.4) (2.5) if and only if it has the following representation (2.6) u(t) = (t−a + 1)α−1 γ(α) u0 + 1 γ(α) t∑ s=a+1 (t−ρ(s))α−1f(s,u(s)), t ∈ na. if we consider a nonautonomous fractional nabla difference equation of caputo type together with an initial condition of the form ∇αa∗u(t) = f(t,u(t)), t ∈ na+1,(2.7) u(a) = u0.(2.8) then, u is a solution of the initial value problem (2.7) (2.8) if and only if it has the following representation (2.9) u(t) = u0 + 1 γ(α) t∑ s=a+1 (t−ρ(s))α−1f(s,u(s)), t ∈ na. now we present some important definitions and theorems of functional analysis [3, 7] which will be useful in establishing main results. definition 2.3. rn is the space of all ordered n-tuples of real numbers. clearly, rn is a banach space with respect to the supremum norm. a closed ball with radius r centered at the origin of rn is defined by b∞0 (r) = {u = (u1,u2, ...,un) ∈ r n : ‖u‖∞ ≤ r}. 82 jonnalagadda definition 2.4. l∞ = l∞(r) is the space of all real sequences defined on the set of positive integers where any individual sequence is bounded with respect to the usual supremum norm. clearly l∞ is a banach space under the supremum norm. a closed ball with radius r centered on the null sequence of l∞ is defined by b∞0 (r) = {u = {u(t)} ∞ t=0 ∈ l ∞ : ‖u‖∞ ≤ r}. definition 2.5. a subset s of l∞ is uniformly cauchy (or equi cauchy), if for every � > 0, there exists k ∈ n1 such that |u(t1) −u(t2)| < � whenever t1, t2 ∈ nk+1, for any u = {u(t)}∞t=0 in s. theorem 2.3. (discrete arzela ascoli’s theorem) a bounded uniformly cauchy subset s of l∞ is relatively compact. theorem 2.4. (krasnoselskii’s fixed point theorem) let s be a nonempty, closed, convex and bounded subset of a banach space x, and let a : x → x and b : s → x be two operators such that (1) a is a contraction with constant l < 1, (2) b is continuous, bs resides in a compact subset of x, (3) [x = ax + by, y ∈ s] =⇒ x ∈ s. then the operator equation ax + bx = x has a solution in s. theorem 2.5. (generalized banach fixed point theorem) let s be a nonempty, closed subset of a banach space (x,‖.‖), and let a γn ≥ 0 for every n ∈ n0 and such∑∞ n=0 γn converges. moreover, let the mapping t : s → s satisfy the inequality ‖tnu−tnv‖≤ γn‖u−v‖ for every n ∈ n1 and any u,v ∈ s. then, t has a uniquely defined fixed point u∗. furthermore, for any u0 ∈ s, the sequence (tnu0)∞n=1 converges to this fixed point u∗. theorem 2.6. (schauder fixed point theorem) let s be a nonempty, closed and convex subset of a banach space x. let t : s → s be a continuous mapping such that ts is a relatively compact subset of x. then t has at least one fixed point in s. that is, there exists an x ∈ s such that tx = x. definition 2.6. let x be a banach space with respect to a norm ‖.‖. define the set s = s(x) = {u : u = {u(t)}∞t=0, u(t) ∈ x}. then, s is a linear space of sequences of elements of x under obvious definition of addition and scalar multiplication. now we employ the notation u = {u(t)}∞t=0, ‖u‖∞ = sup t∈n0 |u(t)|, and define the set s∞(x) = {u : u ∈ s(x) with ‖u‖∞ < ∞}. clearly s∞(x) is a banach space consisting of elements of s(x), with respect to the supremum norm. definition 2.7. from definitions 2.4 and 2.6, we observe that l∞ = l∞(r) = s∞(r). now we choose x = rn in definition 2.6 to define l∞ = l∞(rn) = s∞(rn) = {u : u = {u(t)}∞t=0, u(t) ∈ r n with ‖u‖∞ < ∞}. nonlinear fractional nabla difference equations 83 thus, l∞ denotes the banach space comprising sequences of vectors with respect to the supremum norm ‖.‖∞ defined by ‖u‖∞ = sup t∈n0 ‖u(t)‖. a closed ball with radius r centered on the null sequence in l∞ is defined by b∞0 (r) = {u = {u(t)} ∞ t=0 ∈ l ∞ : ‖u‖∞ ≤ r}. 3. existence & uniqueness in this section we prove existence and uniqueness theorems pertaining to the initial value problems associated with a system of fractional nabla difference equations of the form (3.1) ∇α−1u(t) = f (t, u(t)), ∇ −(1−α) −1 u(t) ∣∣∣ t=0 = u(0) = c, 0 < α < 1, t ∈ n1 and (3.2) ∇α0∗u(t) = f (t, u(t)), u(0) = c, 0 < α < 1, t ∈ n1, where ∇α−1 and ∇α0∗ are the riemann liouville and caputo type fractional difference operators, u(t) is an n-vector whose components are functions of the variable t, c is a constant n-vector and f (t, u(t)) is an n-vector whose components are functions of the variable t and the n-vector u(t). let u : n0 → l∞ and f : n0 × l∞ → l∞. analogous to (2.6), u = {u(t)}∞t=0 ∈ l∞ is any solution of the initial value problem (3.1) if and only if (3.3) u(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0. analogous to (2.9), u = {u(t)}∞t=0 ∈ l∞ is any solution of the initial value problem (3.2) if and only if (3.4) u(t) = c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0. define the operators tu(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0,(3.5) t ′u(t) = c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0,(3.6) au(t) = (t + 1)α−1 γ(α) c, t ∈ n0,(3.7) bu(t) = 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0.(3.8) it is evident from (3.3) (3.6) that u is a fixed point of t if and only if u is a solution of (3.1) and u is a fixed point of t ′ if and only if u is a solution of (3.2). first we use krasnoselskii’s fixed point theorem (theorem 2.4) to establish global existence of solutions of (3.1). clearly a is a contraction mapping with constant 0, implies condition (1) of theorem 2.4 holds. 84 jonnalagadda theorem 3.1. (global existence) if f is continuous with respect to the second variable and there exist constants β1 ∈ [α, 1) and l1 ≥ 0 such that (3.9) ‖f (t, u(t))‖≤ l1t−β1, t ∈ n1, then the nonautonomous initial value problem (3.1) has at least one bounded solution in l∞. proof. to prove condition (2) of theorem 2.4, we define a set s1 = {u : ‖u(t)‖≤‖c‖ + l1γ(1 −β1), t ∈ n1}. clearly s1 is a nonempty, closed, bounded and convex subset of l ∞. first, we show that b maps s1 into s1. using lemma 2.1, theorem 2.2 and (3.9), we have ‖bu(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1 ‖f (s, u(s))‖ ≤ l1 γ(α) t∑ s=1 (t−ρ(s))α−1s−β1 = l1∇−α0 t −β1 = l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) ≤ l1γ(1 −β1) γ(1 −β1 + α) (1)−(β1−α) = l1γ(1 −β1) ≤ ‖c‖ + l1γ(1 −β1), t ∈ n1, implies bs1 ⊂ s1. next, we show that b is continuous on s1. let � > 0 be given. then there exists m ∈ n1 such that, for t ∈ nm+1, l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) < � 2 . let {uk}, (k = 1, 2, .....) be a sequence in s1 such that uk → u in s1. then, we have ‖uk − u‖∞ → 0 as k → ∞. since f is continuous with respect to the second variable, we get ‖f (t, uk) − f (t, u)‖∞ → 0 as k →∞. for t ≤ m, ‖buk(t) −bu(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, uk(s)) − f (s, u(s))‖ ≤ [ 1 γ(α) t∑ s=1 (t−ρ(s))α−1 ][ sups∈{1,2,......,m}‖f (s, uk(s)) − f (s, u(s))‖ ] = tα γ(α + 1) ‖f (s, uk) − f (s, u)‖∞ → 0 as k →∞. nonlinear fractional nabla difference equations 85 for t ∈ nm+1, ‖buk(t) −bu(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1[‖f (s, uk(s))‖ + ‖f (s, u(s))‖] ≤ 2l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) < �. thus we have, ‖buk −bu‖∞ → 0 as k → ∞, implies b is continuous. now, we show that bs1 is relatively compact. let t1, t2 ∈ nm+1 such that t2 > t1. then, we have ‖bu(t1) −bu(t2)‖ ≤ 1 γ(α) t1∑ s=1 (t1 −ρ(s))α−1‖f (s, u(s))‖ + 1 γ(α) t2∑ s=1 (t2 −ρ(s))α−1‖f (s, u(s))‖ ≤ l1γ(1 −β1) γ(1 −β1 + α) t −(β1−α) 1 + l1γ(1 −β1) γ(1 −β1 + α) t −(β1−α) 2 < �. thus {bu : u ∈ s1} is a bounded and uniformly cauchy subset of l∞. hence, by theorem 2.3, bs1 is relatively compact. now we prove condition (3) of theorem 2.4. let us suppose, for a fixed v ∈ s1, u = au + bv. using lemma 2.1, theorem 2.2 and (3.9), we have ‖u(t)‖ ≤ ‖au(t)‖ + ‖bv(t)‖ ≤ (t + 1)α−1 γ(α) ‖c‖ + 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, v(s))‖ ≤ (1)α−1 γ(α) ‖c‖ + l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) ≤ ‖c‖ + l1γ(1 −β1) γ(1 −β1 + α) (1)−(β1−α) = ‖c‖ + l1γ(1 −β1), t ∈ n1. thus u ∈ s1. according to theorem 2.4, t has a fixed point in s1 which is a solution of (3.1). hence the proof. � theorem 3.2. (global existence) if f is continuous with respect to the second variable and there exist constants β2 ∈ [α, 1) and l2 ≥ 0 such that (3.10) ‖f (t, u(t))‖≤ l2t−β2 ‖u(t)‖ , t ∈ n1, then the nonautonomous initial value problem (3.1) has at least one bounded solution in l∞ provided that (3.11) l2γ(1 −β2) < 1. proof. define s2 = { u : ‖u(t)‖≤ ‖c‖ [1 −l2γ(1 −β2)] , t ∈ n1 } . 86 jonnalagadda clearly s2 is a nonempty, closed, bounded and convex subset of l ∞. first, we show that b maps s2 into s2. using lemma 2.1, theorem 2.2 and (3.10), we have ‖bu(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1 ‖f (s, u(s))‖ ≤ l2 γ(α) t∑ s=1 (t−ρ(s))α−1s−β2 ‖u(s)‖ ≤ l2‖c‖ [1 −l2γ(1 −β2)] 1 γ(α) t∑ s=1 (t−ρ(s))α−1s−β2 = l2‖c‖ [1 −l2γ(1 −β2)] ∇−α0 t −β2 = l2‖c‖ [1 −l2γ(1 −β2)] γ(1 −β2) γ(1 −β2 + α) t−(β2−α) ≤ l2‖c‖ [1 −l2γ(1 −β2)] γ(1 −β2) γ(1 −β2 + α) (1)−(β2−α) = l2‖c‖γ(1 −β2) [1 −l2γ(1 −β2)] = ‖c‖ [1 −l2γ(1 −β2)] −‖c‖ ≤ ‖c‖ [1 −l2γ(1 −β2)] , t ∈ n1, implies bs2 ⊂ s2. the remaining proof of condition (2) is similar to that of theorem 3.1 and we omit it. now we prove condition (3) of theorem 2.4. let us suppose, for a fixed v ∈ s2, u = au + bv. using lemma 2.1, theorem 2.2 and (3.10), we have ‖u(t)‖ ≤ ‖au(t)‖ + ‖bv(t)‖ ≤ (t + 1)α−1 γ(α) ‖c‖ + 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, v(s))‖ ≤ (1)α−1 γ(α) ‖c‖ + l2‖c‖γ(1 −β2) [1 −l2γ(1 −β2)] ≤ ‖c‖ + l2‖c‖γ(1 −β2) [1 −l2γ(1 −β2)] = ‖c‖ [1 −l2γ(1 −β2)] , t ∈ n1. thus u ∈ s2. according to theorem 2.4, t has a fixed point in s2 which is a solution of (3.1) (3.2). hence the proof. � now we apply schauder fixed point theorem (theorem 2.6) to establish global existence of solutions of (3.2). theorem 3.3. (global existence) if f satisfies the hypothesis of theorem 3.1, then the nonautonomous initial value problem (3.2) has at least one bounded solution in l∞. nonlinear fractional nabla difference equations 87 proof. define a set s3 = {u : u(0) = c, ‖u(t) − c‖≤ l1γ(1 −β1), t ∈ n1}. clearly s3 is a nonempty, closed, bounded and convex subset of l ∞. first, we show that t ′ maps s3 into s3. using lemma 2.1, theorem 2.2 and (3.9), we have ‖t ′u(t) − c‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1 ‖f (s, u(s))‖ ≤ l1 γ(α) t∑ s=1 (t−ρ(s))α−1s−β1 = l1∇−α0 t −β1 = l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) ≤ l1γ(1 −β1) γ(1 −β1 + α) (1)−(β1−α) = l1γ(1 −β1), t ∈ n1, and t ′u(0) = c, implies t ′s3 ⊂ s3. next, we show that t ′ is continuous on s3. let � > 0 be given. then there exists m ∈ n1 such that, for t ∈ nm+1, (3.12) l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) < � 2 . let {uk}, (k = 1, 2, .....) be a sequence in s3 such that uk → u in s3. then, we have ‖uk − u‖∞ → 0 as k → ∞. since f is continuous with respect to the second variable, we get ‖f (t, uk) − f (t, u)‖∞ → 0 as k →∞. for t ≤ m, ‖t ′uk(t) −t ′u(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, uk(s)) − f (s, u(s))‖ ≤ [ 1 γ(α) t∑ s=1 (t−ρ(s))α−1 ][ sups∈{1,2,......,m}‖f (s, uk(s)) − f (s, u(s))‖ ] = tα γ(α + 1) ‖f (s, uk) − f (s, u)‖∞ → 0 as k →∞. for t ∈ nm+1, ‖t ′uk(t) −t ′u(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1[‖f (s, uk(s))‖ + ‖f (s, u(s))‖] ≤ 2l1γ(1 −β1) γ(1 −β1 + α) t−(β1−α) < �. thus we have, ‖t ′uk −t ′u‖∞ → 0 as k → ∞, implies t ′ is continuous. now, we show that t ′s3 is relatively compact. let t1, t2 ∈ nm+1 such that t2 > t1. then, 88 jonnalagadda we have ‖t ′u(t1) −t ′u(t2)‖ ≤ 1 γ(α) t1∑ s=1 (t1 −ρ(s))α−1‖f (s, u(s))‖ + 1 γ(α) t2∑ s=1 (t2 −ρ(s))α−1‖f (s, u(s))‖ ≤ l1γ(1 −β1) γ(1 −β1 + α) t −(β1−α) 1 + l1γ(1 −β1) γ(1 −β1 + α) t −(β1−α) 2 < �. thus {t ′u : u ∈ s3} is a bounded and uniformly cauchy subset of l∞. hence, by theorem 2.3, t ′s3 is relatively compact. according to theorem 2.6, t ′ has a fixed point in s3 which is a solution of (3.2). hence the proof. � we use generalized banach fixed point theorem (theorem 2.5) to prove the uniqueness of solutions of (3.1) and (3.2). theorem 3.4. (global uniqueness) if f is continuous with respect to the second variable and there exist constants γ ∈ [α, 1) and m ≥ 0 such that (3.13) ‖f (t, u) − f (t, v)‖∞ ≤ mt −γ‖u − v‖∞, t ∈ n1, for any pair of elements u and v in l∞. then the initial value problems (3.1) and (3.2) have unique bounded solution in l∞ provided that (3.14) c = mγ(1 −γ) < 1. proof. let us define the iterates of operator t as follows: t 1 = t, tn = totn−1, n ∈ n1. it is sufficient to prove that tn is a contraction operator for sufficiently large n. actually, we have (3.15) ‖tnu −tnv‖∞ ≤ cn‖u − v‖∞ where the constant c depends only on m and γ. in fact, using lemma 2.1, theorem 2.2 and (3.13), we get ‖tu(t) −tv(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖ ≤ m γ(α) t∑ s=1 (t−ρ(s))α−1s−γ‖u − v‖∞ = m∇−α0 t −γ‖u − v‖∞ = mγ(1 −γ) γ(1 −γ + α) t−(γ−α)‖u − v‖∞ ≤ mγ(1 −γ) γ(1 −γ + α) (1)−(γ−α)‖u − v‖∞ = c‖u − v‖∞, implies (3.16) ‖tu −tv‖∞ ≤ c‖u − v‖∞. nonlinear fractional nabla difference equations 89 therefore (3.15) is true for n = 1. assuming (3.15) is valid for n, we obtain similarly ‖tn+1u(t) −tn+1v(t)‖ = ‖(totn)u(t) − (totn)v(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s,tnu(s)) − f (s,tnv(s))‖ ≤ m γ(α) t∑ s=1 (t−ρ(s))α−1s−γ‖tnu −tnv‖∞ ≤ mcn∇−α0 t −γ‖u − v‖∞ = mcnγ(1 −γ) γ(1 −γ + α) t−(γ−α)‖u − v‖∞ ≤ mcnγ(1 −γ) γ(1 −γ + α) (1)−(γ−α)‖u − v‖∞ = cn+1‖u − v‖∞. thus, by the principle of mathematical induction on n, the statement (3.15) is true for each n ∈ n1. since c < 1, the geometric series ∑∞ n=0 c n converges. hence t has a uniquely defined point u∗ in s1 (or s2). this completes the proof. similarly we can prove that t ′ has a uniquely defined point u∗ in s3. � 4. dependence of solutions on initial conditions and parameters the initial value problems (3.1) and (3.2) describes a model of a physical problem in which often some parameters such as lengths, masses, temperature, etc. are involved. the values of these parameters can be measured only up to a certain degree of accuracy. thus, in (3.1) and (3.2), the initial value c, the order of the difference operator α and the function f , may be subject to some errors either by necessity or for convenience. hence, it is important to know how the solution changes when these parameters are slightly altered. we shall discuss this question quantitatively in the following theorems. theorem 4.1. assume that f is continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α+�−1 u(t) = f (t, u(t)), ∇ −(1−α−�) −1 u(t) ∣∣∣ t=0 = u(0) = c, t ∈ n1,(4.1) ∇α−1v(t) = f (t, v(t)), ∇ −(1−α) −1 v(t) ∣∣∣ t=0 = v(0) = c, t ∈ n1,(4.2) respectively, where � > 0 and 0 < α < α + � < 1. then (4.3) ‖u − v‖∞ = o(�) provided that (3.14) holds. proof. we have u(t) = (t + 1)α+�−1 γ(α + �) c + 1 γ(α) t∑ s=1 (t−ρ(s))α+�−1f (s, u(s)), t ∈ n0, v(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, v(s)), t ∈ n0. 90 jonnalagadda consider ‖u(t) − v(t)‖ ≤ ∣∣∣(t + 1)α+�−1 γ(α + �) − (t + 1)α−1 γ(α) ∣∣∣‖c‖ + ∥∥∥ 1 γ(α + �) t∑ s=1 (t−ρ(s))α+�−1f (s, u(s)) − 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, v(s)) ∥∥∥ ≤ ∣∣∣ γ(α) γ(α + �) (t + α)� − 1 ∣∣∣(t + 1)α−1 γ(α) ‖c‖∞ + ∥∥∥ 1 γ(α + �) t∑ s=1 (t−ρ(s))α+�−1[f (s, u(s)) − f (s, v(s))] ∥∥∥ + ∥∥∥ 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, v(s)) [ 1 − γ(α) γ(α + �) (t−s + α)� ]∥∥∥ ≤ ∣∣∣ γ(α) γ(t + α) γ(� + t + α) γ(� + α) − 1 ∣∣∣(2)α−1 γ(α) ‖c‖∞ + 1 γ(α + �) t∑ s=1 (t−ρ(s))α+�−1‖f (s, u(s)) − f (s, v(s))‖ + 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, v(s))‖ ∣∣∣1 − γ(α) γ(t−s + α) γ(� + t−s + α) γ(� + α) ∣∣∣, t ∈ n1. (4.4) since lim �→0 1 � [ γ(α) γ(t + α) γ(� + t + α) γ(� + α) − 1 ] = c1 (a constant independent of �) and lim �→0 1 � [ 1 − γ(α) γ(t−s + α) γ(� + t−s + α) γ(� + α) ] = c2 (a constant independent of �), we have [ γ(α) γ(t + α) γ(� + t + α) γ(� + α) − 1 ] = o(�),(4.5) [ 1 − γ(α) γ(t−s + α) γ(� + t−s + α) γ(� + α) ] = o(�).(4.6) nonlinear fractional nabla difference equations 91 using (4.5) and (4.6) in (4.4), we get ‖u(t) − v(t)‖ ≤ o(�)α‖c‖∞ + m‖u − v‖∞ 1 γ(α + �) t∑ s=1 (t−ρ(s))α+�−1s−γ +o(�)‖f‖∞ 1 γ(α) t∑ s=1 (t−ρ(s))α−1s−γ = o(�)α‖c‖∞ + m‖u − v‖∞∇ −(α+�) 0 t −γ + o(�)‖f‖∞∇−α0 t −γ = o(�)α‖c‖∞ + m‖u − v‖∞ γ(1 −γ) γ(1 + α + �−γ) tα+�−γ + o(�)‖f‖∞ γ(1 −γ) γ(1 + α−γ) tα−γ ≤ o(�)α‖c‖∞ + m‖u − v‖∞ γ(1 −γ) γ(1 + α + �−γ) (1)α+�−γ + o(�)‖f‖∞ γ(1 −γ) γ(1 + α−γ) (1)α−γ = o(�)α‖c‖∞ + m‖u − v‖∞γ(1 −γ) + o(�)‖f‖∞γ(1 −γ), t ∈ n1. then, we have the relation ‖u − v‖∞ ≤ [α‖c‖∞ + ‖f‖∞γ(1 −γ)] [1 −mγ(1 −γ)] o(�) implies ‖u − v‖∞ = o(�). � corollary 1. assume that f is continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α+�0∗ u(t) = f (t, u(t)), u(0) = c, t ∈ n1,(4.7) ∇α0∗v(t) = f (t, v(t)), v(0) = c, t ∈ n1,(4.8) respectively, where � > 0 and 0 < α < α + � < 1. then (4.9) ‖u − v‖∞ = o(�) provided that (3.14) holds. theorem 4.2. assume that f is continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α−1u(t) = f (t, u(t)), ∇ −(1−α) −1 u(t) ∣∣∣ t=0 = u(0) = c, t ∈ n1,(4.10) ∇α−1v(t) = f (t, v(t)), ∇ −(1−α) −1 v(t) ∣∣∣ t=0 = v(0) = d, t ∈ n1,(4.11) respectively, where 0 < α < 1. then (4.12) ‖u − v‖∞ = o(‖c − d‖∞) provided that (3.14) holds. proof. we have u(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0, v(t) = (t + 1)α−1 γ(α) d + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, v(s)), t ∈ n0. 92 jonnalagadda consider ‖u(t) − v(t)‖ ≤ ‖c − d‖ (t + 1)α−1 γ(α) + 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖ ≤ ‖c − d‖∞ (2)α−1 γ(α) + m‖u − v‖∞ 1 γ(α) t∑ s=1 (t−ρ(s))α−1s−γ = α‖c − d‖∞ + m‖u − v‖∞∇−α0 t −γ = α‖c − d‖∞ + m‖u − v‖∞ γ(1 −γ) γ(1 + α−γ) tα−γ ≤ α‖c − d‖∞ + m‖u − v‖∞ γ(1 −γ) γ(1 + α−γ) (1)α−γ = α‖c − d‖∞ + m‖u − v‖∞γ(1 −γ), t ∈ n1. then, we have the relation ‖u − v‖∞ ≤ α‖c − d‖∞ [1 −mγ(1 −γ)] implies ‖u − v‖∞ = o(‖c − d‖∞). � corollary 2. assume that f is continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α0∗u(t) = f (t, u(t)), u(0) = c, t ∈ n1,(4.13) ∇α0∗v(t) = f (t, v(t)), v(0) = d, t ∈ n1,(4.14) respectively, where 0 < α < 1. then (4.15) ‖u − v‖∞ = o(‖c − d‖∞) provided that (3.14) holds. theorem 4.3. assume that f and g are continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α−1u(t) = f (t, u(t)), ∇ −(1−α) −1 u(t) ∣∣∣ t=0 = u(0) = c, t ∈ n1,(4.16) ∇α−1v(t) = g(t, v(t)), ∇ −(1−α) −1 v(t) ∣∣∣ t=0 = v(0) = c, t ∈ n1,(4.17) respectively, where 0 < α < 1. then (4.18) ‖u − v‖∞ = o(‖f − g‖∞) provided that (3.14) holds. proof. we have u(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1f (s, u(s)), t ∈ n0, v(t) = (t + 1)α−1 γ(α) c + 1 γ(α) t∑ s=1 (t−ρ(s))α−1g(s, v(s)), t ∈ n0. nonlinear fractional nabla difference equations 93 consider ‖u(t) − v(t)‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, u(s)) − g(s, v(s))‖ = 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s)) + f (s, v(s)) − g(s, v(s))‖ ≤ 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, u(s)) − f (s, v(s))‖ + 1 γ(α) t∑ s=1 (t−ρ(s))α−1‖f (s, v(s)) − g(s, v(s))‖ ≤ [m‖u − v‖∞ + ‖f − g‖∞] 1 γ(α) t∑ s=1 (t−ρ(s))α−1s−γ = [m‖u − v‖∞ + ‖f − g‖∞]∇−α0 t −γ = [m‖u − v‖∞ + ‖f − g‖∞] γ(1 −γ) γ(1 + α−γ) tα−γ ≤ [m‖u − v‖∞ + ‖f − g‖∞] γ(1 −γ) γ(1 + α−γ) (1)α−γ = [m‖u − v‖∞ + ‖f − g‖∞]γ(1 −γ), t ∈ n1. then, we have the relation ‖u − v‖∞ ≤ γ(1 −γ) [1 −mγ(1 −γ)] ‖f − g‖∞ implies ‖u − v‖∞ = o(‖f − g‖∞). � corollary 3. assume that f and g are continuous and satisfies (3.13) with respect to the second variable. suppose u and v are the solutions of the initial value problems ∇α0∗u(t) = f (t, u(t)), u(0) = c, t ∈ n1,(4.19) ∇α0∗v(t) = g(t, v(t)), v(0) = c, t ∈ n1,(4.20) respectively, where 0 < α < 1. then (4.21) ‖u − v‖∞ = o(‖f − g‖∞) provided that (3.14) holds. definition 4.1. a solution ũ ∈ l∞ is said to be stable, if given � > 0 and t0 ≥ 0, there exists δ = δ(�,t0) such that ‖u(t0) − ũ(t0)‖∞ < δ ⇒ ‖u − ũ‖∞ < � for all t ≥ t0. theorem 4.4. assume that f is continuous and satisfies (3.13) with respect to the second variable. then the solutions of (3.1) and (3.2) are stable provided that (3.14) holds. proof. the proof is a direct consequence of theorem 5.2 and corollary 2. � 94 jonnalagadda references [1] agarwal, r.p., difference equations and inequalities, marcel dekker, new york, 1992. [2] atsushi nagai, discrete mittag leffler function and its applications, publ. res. inst. math. sci., kyoto univ., 1302 (2003), 1 20. [3] balmohan vishnu limaye, functional analysis, second edition, new age international, india, 1996. [4] bohner, m. and peterson, a. advances in dynamic equations on time scales, birkhauser, boston, 2002. [5] bohner, m. and peterson, a. dynamic equations on time scales, birkhauser, boston, 2001. [6] elaydi, s., an introduction to difference equations, undergraduate texts in mathematics, 3rd edition, springer, new york, 2005. [7] erwin kreyszig, introductory functional analysis with applications, john wiley & sons, canada, 1978. [8] fahd jarad, billur kaymakcalan and kenan tas, a new transform method in nabla discrete fractional calculus, advances in difference equations 2012 (2012), article id 190. [9] ferhan m. atici and paul w. eloe, discrete fractional calculus with the nabla operator, electron. j. qual. theory differ. equat., special edition i 2009(2009), 1-12. [10] ferhan m. atici and paul w. eloe, gronwall’s inequality on discrete fractional calculus, computers and mathematics with applications, 64 (2012), 3193 3200. [11] ferhan m. atici and paul w. eloe, linear systems of nabla fractional difference equations, rocky mountain journal of mathematics, 41 (2011), number 2, 353 370. [12] fulai chen, fixed points and asymptotic stability of nonlinear fractional difference equations, electronic journal of qualitative theory of differential equations, 2011(2011), number 39, 1 18. [13] fulai chen, juan j. nieto and yong zhou, global attractivity for nonlinear fractional differential equations, nonlinear analysis: real world applications, 13 (2012), 287 298. [14] fulai chen, xiannan luo and yong zhou, existence results for nonlinear fractional difference equation, advances in difference equations, 2011 (2011), article id 713201, 12 pages. [15] fulai chen and yong zhou, attractivity of fractional functional differential equations, computers and mathematics with applications, 62 (2011), 1359 1369. [16] fulai chen and zhigang liu, asymptotic stability results for nonlinear fractional difference equations, journal of applied mathematics, 2012 (2012), article id 879657, 14 pages. [17] george a. anastassiou, nabla discrete fractional calculus and nabla inequalities, mathematical and computer modelling, 51 (2010), 562 571. [18] gray, h.l. and zhang, n.f., on a new definition of the fractional difference, mathematics of computaion, 50 (1988), number 182, 513 529. [19] hein, j., mc carthy, s., gaswick, n., mc kain, b. and spear, k., laplace transforms for the nabla difference operator, pan american mathematical journal, 21 (2011), number 3, 79 96. [20] jagan mohan, j. and deekshitulu, g.v.s.r., solutions of nabla fractional difference equations using n transforms, commun. math. stat., 2 (2014), 1 16. [21] jagan mohan, j. and deekshitulu, g.v.s.r. and shobanadevi, n., stability of nonlinear nabla fractional difference equations using fixed point theorems, italian journal of pure and applied mathematics, 32 (2014), 165 184. [22] jagan mohan jonnalagadda, solutions of perturbed linear nabla fractional difference equations, differential equations and dynamical systems, 22 (2014), number 3, 281 292. [23] jan cermák, tomáš kisela and ludek nechvátal, stability and asymptotic properties of a linear fractional difference equation, advances in difference equations 2012 (2012), article id 122. [24] malgorzata wyrwas, dorota mozyrska and ewa girejko, stability of discrete fractional order nonlinear systems with the nabla caputo difference, 6th workshop on fractional differentiation and its applications, part of 2013 ifac joint conference sssc, fda, tds grenoble, 167 171, france, february 4 6, 2013. [25] miller, k.s. and ross, b., fractional difference calculus, proceedings of the international symposium on univalent functions, fractional calculus and their applications, 139 152, nihon university, koriyama, japan, 1989. [26] nihan acar, ferhan m. atici, exponential functions of discrete fractional calculus. applicable analysis and discrete mathematics, 7 (2013), 343 353. nonlinear fractional nabla difference equations 95 [27] podlubny, i., fractional differential equations, academic press, san diego, 1999. [28] thabet abdeljawad, on riemann and caputo fractional differences, computers and mathematics with applications, 62 (2011), 1602 1611. [29] thabet abdeljawad, fahd jarad and dumitru baleanu, a semigroup like property for discrete mittag leffler functions, advances in difference equations 2012 (2012), article id 190. [30] thabet abdeljawad and ferhan m. atici, on the definitions of nabla fractional operators, abstract and applied analysis, 2012 (2012), article id 406757, 13 pages. [31] varsha daftardar gejji and babakhani, a., analysis of a system of fractional differential equations, j. math. anal. appl., volume 293 (2004), 511 522. [32] yong zhou, existence and uniqueness of solutions for a system of fractional differential equations, fractional calculus and applied analysis, 12 (2009), number 2, 195 204. department of mathematics, birla institute of technology and science pilani, hyderabad campus, hyderabad 500078, telangana, india international journal of analysis and applications volume 18, number 5 (2020), 699-717 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-699 received april 20th, 2020; accepted may21st, 2020; published june 17th, 2020. 2010 mathematics subject classification. 90b50. key words and phrases. motivation; direction; psychological; goal-setting; work environment; expectancy; intensity; persistence; effort; empowerment; self-actualization. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 699 a study on millennial generation in vietnam on the factors to motivate employees nhu-ty nguyen1,2,* luong-hoai-thuong pham1,2 1school ofbusiness, international university, quarter 6, linh trung ward, thu duc district, hcmc, vietnam 2vietnam national university hcmc, quarter 6, linh trung ward, thu duc district, hcmc, vietnam *corresponding author: nhutynguyen@hcmiu.edu.vn; nhutynguyen@gmail.com abstract. in this paper, we will overview the most of theories in the field of motivation and retest the practical contexts in reality to tackle the questions of the ways to motivate employees. the test of this paper is expected to find out the factors which could upgrade the motivation index of employees in work environment. the model of contexts that influence the work motivation in special case of vietnamese youngsters will perform and organize into 4 components: organizational and team culture/ climate, leadership/ social relations, organizational practice and policies and work role and job demand/ design in the third chapter. last but not least, the final sector would figure out the gap in this research in point out the direction for future research performances in the field of motivating employees. 1. introduction nowadays, motivation is playing an important role in the fact of becoming increasingly prominent as a strategy for attempting to improve simultaneously the productivity, and the quality of the work optimization of employees is in contemporary organizations ([3]; [10]; [14]). https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-699 int. j. anal. appl. 18 (5) (2020) 700 in this paper, we will overview the most of theories in the field of motivation and retest the practical contexts in reality to tackle the questions: which extrinsic components will affect perspective of millennials the most and which one does not have any relationship with total work motivation? “due to its geographic position, vietnam occupies a privileged place in asia, in the heart of a region that in the 1990s, was seen as a major pole of economic development between the european continent and pacific asia” ([2]). besides, the population of over 97 million citizens (in 2019), vietnam has an exuberant workforce. in addition, “the vietnamese are hard workers, energetic and disciplined. they also represent a large portion of consumers interested and enthusiastic about foreign products. this advantage attracted many investors to vietnam.” however, motivation in workplace is perhaps one of the most important challenges that a large amount of managers is facing, especially in developing countries ([15]; [16]; [18]) due to the fact of this problem, when it comes to this transform, the test of this paper is expected to find out the factors which could upgrade the motivation index of employees in work environment. 2. literature review 2.1 research model and hypothesizes: figure 1: research model int. j. anal. appl. 18 (5) (2020) 701 h1: the organizational and team culture/climate is positively related to work motivation. h2: leadership/social relation is positively related to work motivation. h3: organizational practice and policies is positively related to work motivation. h4: work role and job demand/ design is positively related to work motivation. maslow’s hierarchy theory of needs first of all, the maslow’s theory would be introduced to give us a general concept of needs, in the earliest and most widespread version of maslow's ([11]) hierarchy of basic needs, maslow ([11]) has pointed out five main sets of goals. besides, according to the opinion of mcleod ([13]), “we are motivated by the desire to achieve or maintain the various conditions upon which these basic satisfactions rest and by certain more intellectual desires”. the hierarchy of needs five-stage model includes figure 2:maslow’s hierarchy theory of needs the levels of needs have closed relationship and impact with each other, the position and order of them, perhaps, change based on the effect of national culture and norms ([30]; [31]; [32]; [33]; physiological food water shelter warmth organizational and team culture/ climate self-esteem achievement mastery recognition respect selfactualizati on belonging love friends family spouse lover safety security stability freedom from fear int. j. anal. appl. 18 (5) (2020) 702 [34]; [36]). moreover, according to saul mcleod, the prepotency of goal will monopolize consciousness and will tend of itself to organize the recruitment of the various capacities of the organism. the needs which are not considered essential would be put less attention or maybe, be eliminated or rejected. however, if one needed exigency can be satisfied or achieved, this demand will not be a motivator anymore, in this situation the “higher” or “at the top maslow’s pyramid” becomes emerge. respectively, to dominate the conscious life and to serve as the center of organization of behavior, since gratified needs are not active motivators ([13]). in conclusion, the principle of hierarchy often observes in fact of increasing proportions of nonsatisfaction as we go up the hierarchy; and the reversals in middle order of this are sometimes observed. furthermore, the research finds out that a person; perhaps, lose their higher demand/ motivators in some special conditions permanently ([37]; [38]; [39]; [40]; [41]; [42]). there are not only ordinarily multiple motivations for usual behavior, but in addition many determinants other than motives ([3]; [10]). mcgregor’s theory x and theory y in the landmark book called the human side of enterprise ([12]), douglas mcgregor thinks that the pathway of practical management has already changed and the roles of managers and leaders play an important position in organize and support the employees. theory x: the traditional viewpoint of direction and control is based on the hypothesis below: 1. individuals have tendency to avoid task or deny responsibility if they can. 2. due to the fact dislike of work, majority of them must be directed, controlled and coerced; the strategy of management usually apply punishment to force them pay attention and effort to achieve the company’s targets. 3. the average human prefers to be guided clearly, hopes to avoid job and duty. with them, the secure and stable position is more attractive than ambitiousness. style of leadership or management for theory x people usually apply the form of autocratic leadership to behave with their subordinates; giving details and clearly guide step by step with each work task, threat by using punishment and higher payment are some illustration for this type of management. however, all of those int. j. anal. appl. 18 (5) (2020) 703 controlling can result in lack of loyalty and resentment from employee of these leaders.mcgregor acknowledges that the `carrot and stick' approach can have a place, but will not work when the needs of people are predominantly social and egoistic. ultimately, the assumption that a manager’s objective is to persuade people to be docile, to do what they are told in exchange for reward or escape from punishment, is presented as flawed and in need of re-evaluation ([14]; [15]). theory y: the direction of each person and objectives of organization are based on the hypothesis below: 1. based on the atmosphere in working environment, conducting and finishing job, perhaps satisfy or punish the one who take that task. besides, the expenditure of physical and mental effort in work is as natural as play or rest. the average human being does not inherently dislike work. 2. human will practice direct and control by themselves to which they are take responsibilities, therefore, the outside oversee or punishment are not all to bring the exertion for goals of company. 3. the most impressiveness of rewards… that is the satisfaction of ego and achievement of highest maslow’s need in pyramid (self-actualization), they could be the directly persuade the efforts to get company’s goals. in the other words, the commitment to objectives is a function of the rewards associated with their achievement. 4. the fact of avoid taking responsibility, deny ambitious behaviors or focus on stability is usually come from experience, not inherent human characteristics. with theory y, average people being learn, under proper conditions, not only to accept but to seek responsibility. 5. the capacity to exercise a relatively high degree of imagination, ingenuity, and creativity in the solution of organizational problems is widely, not narrowly, distributed in the population. 6. under the conditions of modern industrial life, the intellectual potentialities of the average human being are only partially utilized. theory y assumptions can lead to more cooperative relationships between managers and workers. a theory y int. j. anal. appl. 18 (5) (2020) 704 management style seeks to establish a working environment in which the personal needs and objectives of individuals can relate to, and harmonies with, the objectives of the organization. based on the concept of the human side of enterprise, mcgregor consumed that theory y was not a panacea for all ills and he wished that each managers have could apply suitable tools or techniques for theory y while have accepted the limiting assumptions of theory x ([16]; [17]). herzberg’s two-factor theory the theoretical formulation of frederick herzberg was popular and applied widely respected theories for explaining motivation and job satisfaction ([4]). there are two main components of factors in the research of herzberg et al. [6] in the field of job satisfaction and performance in workplace. the first element named “motivators” or “satisfiers”, which is considered to lead to the satisfaction if being supplied enough typically. the satisfiers and satisfiers are usually belonging to internal factors: they are key component of content and are largely administered by the employee or the student. by the other hand, the second one called “dissatisfiers” or “hygiene factor”, this element could not bring the satisfaction if being give redundantly, however, it causes dissatisfaction when deficient. the hygiene or dissatisfier factors are typically external factors and are under the control of the managers, leaders or any other person rather than the employee or student ([19]; [20]; [21]; [22]; [23]). one of the most serious points of view in theory is that herzberg did not define satisfaction and dissatisfaction as being at opposite ends of the same continuum in his paper. “the opposite of satisfaction is not dissatisfaction, but no satisfaction. the opposite of dissatisfaction is not satisfaction, but no dissatisfaction. the extrinsic factors affect job satisfaction and if not adequately fulfilled can cause dissatisfaction, even if the motivating factors themselves are addressed satisfactorily” ([4]). goal-setting theory the goal-setting theory has already been developed by two authors locke and latham in the field of motivation. paper gave a hand to predict and forecast that if the targets include enough details and challenges, they will bring back the highest effectiveness, when they are used to int. j. anal. appl. 18 (5) (2020) 705 evaluate performance and linked to feedback on results, and create commitment and acceptance. there are some factors could influence or impact the motivation of objectives like ability and self-efficacy, besides, deadlines also improve the effectiveness of goals. a learning goal orientation leads to higher performance than a performance goal orientation, and group goal-setting is as important as individual goalsetting. according to locke and latham [9], goals impact pervasively on behavior and exertion in practicing organizations and management; in addition, most of recent companies have some specific form of goal setting in process. management by objectives (mbo), high-performance work practices (hpwps), management information systems (mis), benchmarking, stretch targets, as well as systems thinking and strategic planning are some illustration for programs which cover the development of goals. moreover, in lots of constructions such as vroom’s [35] vie theory, maslow’s [11] or herzberg’s [6] motivation theories, bandura’s [1] social cognitive theory, or operant-based behaviorism ([28]), goal setting was used for explanation for nearly theories of work motivation. managers widely accept goal setting as a means to improve and sustain performance ([5]).based on hundreds of studies, the major finding of goal setting is that individuals who are provided with specific, difficult but attainable goals perform better than those given easy, nonspecific, or no goals at all. at the same time, however, the individuals must have sufficient ability, accept the goals, and receive feedback related to performance ([7]). in the definition of theory, values and intentions (goals) would be the cognitive determinants of behavior. besides, they also simply defined a goal as a specific thing that the person is consciously trying to do. with locke and latham, goals guide the attention and action directly because one’s values create a desire to do things consistent with them; goals also affect behavior (job performance) through other mechanisms. also, locke and latham postulate that the form in which one experiences one’s value judgments is emotional. moreover, “challenging goals mobilize energy, lead to higher effort, and increase persistent effort. goals motivate people to develop strategies that will enable them to perform at the required goal levels”. furthermore, achieving the final goal could bring back the fulfillment as well as motivation or disappointment, and decline motivation whether the goal is not accomplished ([24]; [25]). int. j. anal. appl. 18 (5) (2020) 706 2.2 implications for practice • goals need to be specific • goals must be difficult but achievable • goals must be accepted • feedback must be provided on goal attainment • goals are more effective when they are used to evaluate performance • deadlines improve the effectiveness of goals • a learning goal orientation leads to higher performance than a performance goal orientation • group goal-setting is as important as individual goal-setting expectancy theory or vroom’s theory being published in the year of 1964, the expectancy theory of professor victor vroom was considered as one of the process theories of motivation. vroom [35] instrumentality theory represents the first attempt to use cognitively-oriented assumptions as the basis for a general theory of work motivation ([8]). motivation plays a role as the “force” to persuade an individual to perform a determined task, as figure out by the coloration of (a) the person's expectancy that his act will be followed by a particular outcome, and (b) the valence of that (first-level) outcome ([35]). based on the thinking of stephan robbins, the meaning of expectancy theory is that the strong point of a trend to conduct in a specific pathway is dependable to the strength of an expected willing if the action can be strictly done by a given result and on the attractiveness of that output to the person ([26]; [27]; [18]; [29]). there are four components for an employee that matter him/her in motivation. these factors are: individual effort (1), individual performance (2), organizational rewards/work outcomes (3) and personal goals (4). similarly, there are connections based on these components: 1. relationship between efforts and performance. this is known as expectancy (e). int. j. anal. appl. 18 (5) (2020) 707 2. relationship between performance and rewards/work outcomes. this relationship is called instrumentality (i) 3. relationship between rewards/work outcomes and personal goals. this is known as valence (v) throughout the reward, benefits and outcomes, the organization can fulfill the personal goals of each individual, which is the thing employees want to achieve and also the reason they keep work for company. as consequences, the connection or relationship among organizational rewards/ work outcomes and personal goals is essential and important... to what extent organizational rewards satisfy an employee’s personal goals and how attractive are those rewards to the employee.this relationship can also be expressed as the value the employee gives to the work outcomes. in addition, the personal performance of individual impacts on the organizational rewards/work outcomes. the level of belief that the employee’s behavior and enforcement would lead to achievement of company’s rewards and work outcomes needed put as much as attention. last but not least, the perception of the chances by the individual employee that personal effort on his/her will lead to high performance is again important. therefore, the expectancy theory says that: effort or motivation = e x i x v. 3. data analysis 3.1. sample demographic overall, there is a large amount of people (more than 320) that agree to do the questionnaire and most of them are belonging to the young generation. the special point in table 1 that is a separation into 2 groups in the age of responders, the first group was born from 1996 to 2000 (18-24 years old) and the second one was born from 19831995 (25-37 years old). this division could help us easy and convenient in interview the focus group or responders because the persons who were in first one are usually the students or the one having part-time job whereas the rest maybe official employee and intend to keep loyalty with their current organization. int. j. anal. appl. 18 (5) (2020) 708 table 1: respondents’ demographics variable items n = 321 scale gender male 180 (49.05) nominal female 182 (49.59) other 5 (1.36) age group from 18 to 24 110 (29.97) nominal from 25 to 37 257 (70.03) education secondary school 4 (1.09) high school 18 (4.9) college level 99 (26.98) graduate level 246 (67.03) industry it 22 nominal information 8 manufacturing 29 medical and bio 21 contents and culture 17 educational services 23 fashion and clothes 40 retailing 22 wholesale trade 8 agriculture/fisheries/forestry/ hunting 6 real estate / rental / leasing 14 travelling 18 transportation / warehousing 16 finance and insurance 24 professional, scientific, or technical services 11 management of companies and enterprises 6 health care and social assistance 8 arts, entertainment, and recreation 13 accommodation and food services 35 public administration 5 other 21 int. j. anal. appl. 18 (5) (2020) 709 in addition, the education was also divided into four specific levels. from that, we would observe its difference in the anova analysis through spss software. 3.2 reliability test and validity test based on the theory of nguyen [14] every quantitative research has to be measured with the accuracy or dependability or reliability of measurement. the consistency of index over time would be tested by reliability measures; a reliability coefficient demonstrates whether the test designer was correct in expecting a certain collection of items to yield interpretable statements about individual differences. table 2: cronbach's alpha of variables construct cronbach’s item-total cronbach's alpha if item deleted items alpha correlation deleted organizational and teamculture/ climate 0.868 none leadership/social relations 0.890 none organizational practice and policies 0.803 none work role and job demand/ design 0.968 none working motivation 0.902 none according to the given date, we can recognize easily that all the cronbach alpha of sub components and variables are good enough to be kept (they was  0.6), so the instrument was reliable and valid (table 2). table 3: kmo and bartlett's test kmo and bartlett's test kaiser-meyer-olkin measure of sampling adequacy. 0.925 bartlett's test of sphericity approx. chi-square 6989.987 df 465 sig. 0 int. j. anal. appl. 18 (5) (2020) 710 in the first time running reliability testing, the kmo of independent components was shown at 0.925(>0.600) as well as the bartlett’s test of sphericity is significant (sig. = 0.000<0.05). moreover, according to the table rotated component matrix, all the sub-variables contributed just in one column which are respective with their main component (table 3).consequently, all the variables are valid enough and we could perform the next analysis to check to meaning of data. after the test, there were no variables eliminated and still 31 ones remained, likes the beginning. all the remaining variables whose factor loadings were greater than 0.5 grouped into 4 factors as expected in proposed theoretical model. factor 1: o1, o2, o3, o4, o5, o6, o7, o8 factor 2: l1, l2, l3, l4, l5, l6 factor 3: p1, p2, p3, p4, p5, p6, p7 factor 4: w1, w2, w3, w4, w5, w6, w7 regression analysis multiple linear regression is a bivariate correlation’s expansion. the result of regression is an equation show the relationship between independent variables and dependent variable (table 4). the framework can be represented by the following equation: i= β0 + β1h+ β2r+ β3o + β4s+ e table 4: model summary model summaryb model r r square adjusted r square std. error of the estimate 1 .913a 0.833 0.831 0.155927 this multiple linear regression model, with four explanatory variables, now has an r squared value of 0.831 or 83.1 % of the variation in working motivation can be explained by this model. int. j. anal. appl. 18 (5) (2020) 711 table 5: anova anovaa model sum of squares df mean square f sig. 1 regression 43.794 4 10.949 450.311 .000b residual 8.801 362 0.024 total 52.595 366 table 6: coefficients coefficientsa model unstandardized coefficients standardized coefficients t sig. collinearity statistics b std. error beta tolerance 1 (constant) 2.57 0.141 18.245 0 o -0.015 0.018 -0.018 -0.803 0.423 0.953 l 0.053 0.013 0.088 3.968 0 0.933 p 0.06 0.022 0.064 2.807 0.005 0.88 w 0.386 0.01 0.904 40.164 0 0.913 table 7: vif coefficientsa model collinearity statistics vif 1 (constant) o 1.05 l 1.072 p 1.136 w 1.095 independent variable: m int. j. anal. appl. 18 (5) (2020) 712 based on the output of tables 5, 6 and 7, the research has already figured out the sig value of variables l, p, w and constant index that are lower than 0.05. this proved all of them are relevant with the model and have a deep effect on motivation. on the other hands, the significant level of o (sig=0.423) does not satisfy conditions to make the model right or have any closed intercommunion with dependent factor. thus, in this multiple linear regression test, scores of o would be deleted or rejected out of the equation. from the analysis above, a new detailing equation delineating the fluctuation of the dimensions which impact motivation in work of labors is: m = 0.088l + 0.064p + 0.904w + 0 in addition, the variance inflation factor (vif) measures the amount by which the variance of a parameter estimator is inflated due to predictor variables being correlated with each other, rather than being orthogonal ([16]). if value of vifs are higher than 10, regression violates multi-colinearity between independent factors; in reality, this index should lower than 2. nevertheless, the regression model has satisfied the rules with large tolerances and small vif scores of variables. from the spss source, it informs that three in four factors (l, p, and w) have positive and direct effect on dependent one with the standardized coefficinets beta are 0.088, 0.064 and 0.904 respectively. hypothesis h1: organizational climate (o) of millennials had not any direct relationship with the fact of motivation (m). the sig value of this item was higher than 0.05 (pvalue = 0.423), therefore, this factor would be reject out of the final equation. furthermore, the beta of o is just about -0.018, this mean that it can just explain 1.8% the dependent motivation negatively. as a result, organizational climate/culture did not have direct relation with the factor motivation in work of young generation. hypothesis h2: leadership/social relations (l) has considerable effects on their start-up intention. owning to the similarity in sig value of w (p = 0), leadership is accepted in this assumption. simultaneously, if leadership relations improves in 1 unit, the job would motivate also raises 0.088 as a result of standardized beta. with its index, l is the second most influential factor. int. j. anal. appl. 18 (5) (2020) 713 hypothesis h3: organizational policies has considerable effects on their motivaton (m). the p-value (sig=.005<0.05) is illustration of the acceptance of p variable in regression analysis. subsequently, with a standardized beta of this factor is 0.064, when the p increases by 1 unit, dependent value would increase by 6.4 percentages and it is the weakest component to influence m. hypothesis h4: work design/ demand (w) of millennials has considerable is the most concern on their willing to conduct the job. factor w is easily recognized having the strongest impact with the significant p-value and the highest beta; this means if the increasing of w is 1 unit, the m would increase by 0.904 unit. 4. discussions and conclusions this research focuses on summarize some theories which cover method in the field of motivation such as maslow’s hierarchy theory of needs ([11]), mcgregor’s theory x and theory y ([12]) herzberg’s two-factor theory ([6]), goal-setting theory ([9]) locke and latham provide a well-developed goal-setting theory of motivation. the theory emphasizes the important relationship between goals and performance; the motivational impact of goals may be affected by moderators such as ability and self-efficacy that is illustration for this explanation. practical implications the target for implication of this report is that it can give the audiences the overview of subjects may be improved the willing and ambitiousness of employees while they are in a specific organization. the income, salary, bonus, benefits, commission and so on in someone’s belief may directly have relationship with the inspiration of employees, and if those of benefits are given as high will lead to higher motivation and could force the workers put more enthusiasm in finishing the job. nevertheless, there are a large amount of other factors more essential to pay attention if managers or owners of companies to inspire their workforce. for example, with the circumstance of this report, the demand or design of the task is special significant (if the characteristic of the task is suitable with the demand of labour, 90 percent it is going to bring high effectiveness in finishing job). not only role of task but also int. j. anal. appl. 18 (5) (2020) 714 leadership style, and policies of organization are momentous to be care to, which if are appropriate with willing, norms, and beliefs of millennials could positive upgrade the motivation and indirectly increase productivity as well efficiency. as understand the features of both assigns and employees, introduce as well as imply valid rules, regulations, strategies and leadership for each kinds of person, the managers can develop and add more value for labors’ needs meaningfully. limitations and recommendations one of limitations of this paper is that the collection of questionnaire was conducted mainly in ho chi minh city due the fact of limited budget, financial capability and lack of time. as the results, the final conclusion of research hardly represents and expresses the thinking and comprehension of young generation in vietnam. besides, the research pay attention to quantitative method while perform the survey instead of using both qualitative and quantitative one. if it could have conducted in both processes, it would have figure out deeper the thought and idea or understand insight needs of workforce. one advice for future researches, that should considers apply both of method so that it finds all the aspects and problems such as using focus group, interview… if this difficult challenge might be solved, the core requirement can be satisfied. acknowledgments: this research is funded by international university vnuhcm under grant number t2019-06-ba. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. bandura, social foundations of thought and action: a social cognitive theory, prentice-hall, englewood cliffs, n.j, 1986. [2] v. bodolica, m. waxin, motivating vietnamese employees and managers in an american jointventure: what a challenge!, asian case res. j. 11 (2007), 59–77. int. j. anal. appl. 18 (5) (2020) 715 [3] w.chia-nan,n. n. ty,forecasting the manpower requirement in vietnamese tertiary institutions. asian j. empiric. res. 3(5) (2013), 563-575. [4] o.w. deshields, a. kara, e. kaynak, determinants of business student satisfaction and retention in higher education: applying herzberg’s two‐factor theory, int. j. educ. manage. 19 (2005), 128–139. [5] a.j. dubrin, essentials of management, 9th ed, south-western, australia ; mason, oh, 2010. [6] f. herzberg, one more time: how do you motivate employees?, harvard business press, boston, mass, 2008. [7] g.p. latham, goal setting:, organizational dynamics. 32 (2003), 309–318. [8] e.e. lawler, j.l. suttle, expectancy theory and job behavior, organ.behav. human perform. 9 (1973), 482–503. [9] e.a. locke, g.p. latham, a theory of goal setting & task performance, prentice hall, englewood cliffs, n.j, 1990. [10] l.h.t. pham, n. t.nguyen,t. t. tran, on the factors affecting start-up intention of millennials in vietnam. int. j. adv. appl. sci. 6(1) (2019), 1-8. [11] a. h. maslow, a theory of human motivation. psychological review, 50(4)(1943), 370. [12] d. mcgregor, theory x and theory y. organization theory, 358(1960), 374. [13] s.mcleod, maslow's hierarchy of needs. simply psychology, 2007. [14] n. t. nguyen, optimizing factors for accuracy of forecasting models in food processing industry: a context of cacao manufacturers in vietnam. ind. eng. manage. syst. 18(4)(2019), 808-824. [15] n. t.nguyen,performance evaluation in strategic alliances: a case of vietnamese construction industry. glob. j. flex. syst. manage. 21(1)(2020), 85-99. [16] n. t. nguyen, attitudes and repurchase intention of consumers towards functional foods in ho chi minh city, vietnam. int. j. anal. appl. 18(2)(2020b), 212-242. [17] p.nguyen, t. nguyen, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese steel industry. int. j. data envelop. anal. 7(2)(2019), 45-64. [18] n. t.nguyen, l. x. t. nguyen, applying dea model to measure the efficiency of hospitality sector: the case of vietnam. int. j. anal. appl. 17(6)(2019b), 994-1018. [19] n. t.nguyen, t. t. tran, mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dyn. nat. soc. 2015(2015), 858157. [20] n. t.nguyen, t. t. tran, facilitating an advanced product layout to prioritize hot lots in 450 mm wafer foundry in the semiconductor industry. int. j. adv. appl. sci. 3(6)(2016), 14-23. int. j. anal. appl. 18 (5) (2020) 716 [21] n. t.nguyen, t. t. tran, a novel integration of dea, gm (1, 1) and neural network in strategic alliance for the indian electricity organizations. j. grey syst. 29(2)(2017), 80-101. [22] n. t.nguyen, t. t. tran, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9)(2018), 73-81. [23] n. t.nguyen, t. t. tran, a study of the strategic alliance for vietnam domestic pharmaceutical industry: a dynamic integration of a hybrid dea and gm (1, 1) approach. j. grey syst. 30(4)(2018), 134-151. [24] n. t.nguyen, t. t. tran, optimizing mathematical parameters of grey system theory: an empirical forecasting case of vietnamese tourism. neural comput. appl. 31(2)(2019a), 1075-1089. [25] n. t.nguyen, t. t. tran, raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl. 31(11)(2019b), 79637974. [26] n. t.nguyen,t. t.tran,c. n.wang,n. t.nguyen, optimization of strategic alliances by integrating dea and grey model. j. grey syst. 27(1)(2015), 38-56. [27] n. t.nguyen, t. t. tran, c. n. wang, management styles and organisational effectiveness in vietnam: a comparison in terms of management practices between state-owned and foreign enterprises. res. world econ. 6(1)(2015), 85-98. [28] b.f. skinner, the shaping of a behaviorist: part two of an autobiography, 1st ed, knopf : distributed by random house, new york, 1979. [29] t.thanh-tuyen, n. t. nguyen, determinants affecting vietnamese laborers’ decision to work in enterprises in taiwan. j. stock forex trad. 5(2)(2016), 1000173. [30] t. t.tran, an investigation about factors that affecting satisfaction and efficiency in vietnamese tourism. int. j. adv. appl. sci. 5(12)(2018), 7-15. [31] t. t.tran, an empirical research on selecting the targeted suppliers and purchasing process of supermarket. int. j. adv. appl. sci. 4(4)(2017), 96-109. [32] t. t.tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci, 3(12)(2016), 5-20. [33] t. t.tran, factors affecting to the purchase and repurchase intention smart-phones of vietnamese staff, international journal of advanced and applied sciences, 5 (3) (2018), 107-119. [34] t. t. tran, forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci. 4(9)(2017), 86-95. [35] v. h. vroom, work and motivation (vol. 54). wiley, new york, 1964. int. j. anal. appl. 18 (5) (2020) 717 [36] c. n.wang, n. t. nguyen, t. t. tran, an empirical study of customer satisfaction towards bank payment card service quality in ho chi minh banking branches. int. j. econ. finance, 6(5)(2014), 170181. [37] c. n.wang, n. t. nguyen, t. t. tran, the study of staff satisfaction in consulting center system-a case study of job consulting centers in ho chi minh city, vietnam. asian econ. financ. rev. 4(4)(2014), 472-491. [38] c. n.wang, n. t. nguyen, t. t. tran, integrated dea models and grey system theory to evaluate past-to-future performance: a case of indian electricity industry.sci. world j. 2015(2015), 638710. [39] c. n.wang, n. t. nguyen, t. t. tran,b. b.huong, a study of the strategic alliance for ems industry: the application of a hybrid dea and gm (1, 1) approach.sci. world j. 2015(2015), 948793. [40] l. w.wang, t. t. tran, n. t. nguyen, analyzing factors to improve service quality of local specialties restaurants: a comparison with fast food restaurants in southern vietnam. asian econ. financ. rev. 4(11)(2014), 1592-1606. [41] l. w.wang, t. t. tran, n. t. nguyen, an empirical study of hybrid dea and grey system theory on analyzing performance: a case from indian mining industry. j. appl. math. 2015(2015), 395360. [42] l. w.wang, t. tran, n. t. nguyen, an analysis of manpower in vietnamese undergraduate educational system. int. j. econ. bus. finance, 1(11)(2013), 398-408. international journal of analysis and applications volume 19, number 3 (2021), 341-359 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-341 intuitionistic fuzzy normal subrings over normed rings nour abed alhaleem∗, abd ghafur ahmad department of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, bangi 43600, selangor, malaysia ∗corresponding author: noorb@ymail.com abstract. this paper presents the notion of intuitionistic fuzzy normed normal subrings. we investigate the concept of intuitionistic fuzzy normal subrings over normed rings and characterize relevant properties of such subrings. further, we define direct product of fuzzy normal subrings over normed rings and investigate some related fundamental properties. 1. introduction after an introduction of fuzzy sets by zadeh [1] several researchers investigated on the generalization of the concept of fuzzy set. in 1971, rosenfeld [2] initiated the studies of fuzzy group theory by introducing the concepts of fuzzy subgroupoid and fuzzy subgroup. later in 1981 [3], wu introduced the notion of fuzzy normal subgroups of an ordinary group and liu [4] defined a fuzzy invariant (normal) subgroup, liu also extended the notion of a subring of a ring and the product of complexes to the fuzzy setting in the same paper. in 1984 [5], mukharjee and bhattacharya introduced the concept of fuzzy cosets and studied their relation with normal fuzzy subgroups, proved that for a group g a fuzzy subgroup is fuzzy normal if and only if it is constant on the conjugate classes of g and showed that the level subgroups of a fuzzy normal subgroup are all normal. moreover, wu in [6] introduced the concept of a normal fuzzy subgroup of a fuzzy group and used this concept to formulate the quotient structure of a fuzzy group. in [7], mishref defined the received february 1st, 2021; accepted march 3rd, 2021; published april 1st, 2021. 2010 mathematics subject classification. 03f55, 03e72. key words and phrases. intuitionistic fuzzy normed subrings; intuitionistic fuzzy normed normal subrings; direct product of intuitionistic fuzzy normed normal subrings. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 341 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-341 int. j. anal. appl. 19 (3) (2021) 342 maximal normal fuzzy subgroup and gave some of its properties in analogy to the crisp case, also he defined the subnormal, normal and composition series of normal fuzzy subgroups and explained the interrelationship between them and the series in the crisp case. in [8], a new type of fuzzy normal subgroups and fuzzy cosets was presented. the notion of intuitionistic fuzzy set was introduced by atanassov [9] as a generalization of fuzzy sets. after that many researches applied this notion in various branches of mathematics especially in algebra and defined the notions of intuitionistic fuzzy subgroups, intuitionistic fuzzy subrings and intuitionistic fuzzy normal subgroups. hur et al in [10], studied and characterized some properties of intuitionistic fuzzy normal subgroups of a group. in [11], marashdeh and salleh studied intuitionistic fuzzy groups by generalizing the notion of the fuzzy normal subgroup to the intuitionistic fuzzy normal subgroup. later, some properties of intuitionistic fuzzy normal subrings were studied by veeramani in [12], he described the algebraic nature of intuitionistic fuzzy normal subrings of a ring under homomorphism and anti-homomorphism. in [13], sharma defined intuitionistic fuzzy magnified translation of intuitionistic fuzzy (normal) subring and ideal of a ring. the notion of intuitionistic fuzzy normed rings was introduced by abed alhaleem and ahmad in [14]. the concepts of intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals was presented as an extension of fuzzy normed rings which were studied by emniyet and sahin in [15], which presented the notions of fuzzy normed subrings and fuzzy normed ideals. a generalization of normed ring was studied by aren in [16]. later, naimark introduced normed rings in [17], which gave the first comprehensive treatment of banach algebras. gel’fand defined commutative normed rings in [18] and addressed them as complex banach spaces with introduction of the notion of commutative normed rings. in this paper, we introduce the notion of intuitionistic fuzzy normed normal subrings. we study the algebraic nature of intuitionistic fuzzy normed normal subrings and characterize relevant properties. we introduce and study the notions of direct product of intuitionistic fuzzy normal subrings over normed rings. further we define the relations between the intuitionistic characteristic function and intuitionistic fuzzy normed normal subrings. 2. preliminaries in this section, we outline the most significant definitions and results needed for the following sections. we review some basic ideas of intuitionistic fuzzy set, intuitionistic fuzzy normed subring, definitions of normed linear spaces, t-norm and s-norm. int. j. anal. appl. 19 (3) (2021) 343 definition 2.1. [19] the fuzzy set a in a universal x is a set of ordered pairs: a = {(v,µa(v)) : v ∈ x} where, µa(v) is the membership function of v in a which associates each element in x with a real number in the interval [0, 1]. definition 2.2. [20] an intuitionistic fuzzy set (briefly, ifs) a in a nonempty set x is an object having the form ifs a = {(v,µa(v),γa(v) : v ∈ x}, where the functions µa(v) : x → [0, 1] and γa(v) : x → [0, 1] denote the degree of membership and the degree of nonmembership, respectively, where 0 ≤ µa(v)+γa(v)) ≤ 1 for all v ∈ x. an intuitionistic fuzzy set a is written symbolically in the form a = (µa,γa). definition 2.3. [17] a ring a is said to be a normed ring (nr) if a possesses a norm ‖‖, that is, a non-negative real-valued function ‖‖ : a → r such that for any v,r ∈ a, (1) ‖v‖ = 0 ⇔ v = 0, (2) ‖v + r‖≤‖v‖ + ‖r‖ , (3) ‖v‖ = ‖−v‖, and (4) ‖vr‖≤‖v‖‖r‖. definition 2.4. [14] let a = {(v,µa(v),γa(v)) : v ∈ nr} and b = {(v,µb(v),γb(v)) : v ∈ nr} be two intuitionistic fuzzy normed rings over the normed ring nr. then a is an intuitionistic fuzzy normed subring of b if µa(v) ≤ µb(v) and γa(v) ≥ γb(v). proposition 2.5. [14] let a be an intuitionistic fuzzy normed ring and let 0 be the zero of the normed ring nr, then the following is true for every v ∈ nr: (i) µa(v) ≤ µa(0) , γa(0) ≤ γa(v), (ii) µa(v) = µa(−v) , γa(v) = γa(−v). lemma 2.6. [14] let 1nr be the multiplicative identity of nr then for all v ∈ nr: 1. µa(v) ≥ µa(1nr) 2. γa(v) ≤ γa(1nr). proposition 2.7. [21] let a be an intuitionistic fuzzy set in a ring r, we denote the (α,β)-cut set by aα,β = {v ∈ r : µa ≥ α and γa ≤ β} where α + β ≤ 1 and α,β ∈ [0, 1]. definition 2.8. [22] let ∗ : [0, 1] × [0, 1] → [0, 1] be a binary operation. then ∗ is a t-norm if ∗ satisfies the conditions of commutativity, associativity, monotonicity and neutral element 1. we shortly use t-norm and write v∗r instead of ∗(v,r). some examples of t-norm are v∗r= min {v,r} and v ∗r = v ·r. int. j. anal. appl. 19 (3) (2021) 344 definition 2.9. [23] let � : [0, 1] × [0, 1] → [0, 1] be a binary operation. then � is a s-norm if � satisfies the conditions of commutativity, associativity, monotonicity and neutral element 0. we shortly use s-norm and write v � r instead of �(v,r). some examples of s-norm are v � r= max {v,r} and v �r = v + r −v ×r. 3. some properties of intuitionistic fuzzy normed normal subrings throughout the rest of this paper, r is the set of real numbers and nr is a normed ring. we define the intuitionistic fuzzy normed normal subrings and some basic properties related to it. definition 3.1. [14] let ∗ be a continuous t-norm and � be a continuous s-norm. an intuitionistic fuzzy set a = {(v,µa(v),γa(v)) : v ∈ nr} is called an intuitionistic fuzzy normed subring (ifnsr) of the normed ring (nr, +, .) if it satisfies the following conditions for all v,r ∈ nr: (i) µa(v −r) ≥ µa(v) ∗µa(r), (ii) µa(vr) ≥ µa(v) ∗µa(r), (iii) γa(v −r) ≤ γa(v) �γa(r), (iv) γa(vr) ≤ γa(v) �γa(r). definition 3.2. let nr be a normed ring. an intuitionistic fuzzy subring a of nr is said to be an intuitionistic fuzzy normed normal subring (ifnnsr) of nr if it satisfies the following for all v,r ∈ r: (i) µa(vr) = µa(rv), (ii) γa(vr) = γa(rv) proposition 3.3. let (nr, +, .) be a ring. if a and b are two intuitionistic fuzzy normed normal subrings of nr, then their intersection (a∩b) is an intuitionistic fuzzy normed normal subring of nr. proof. let v,r ∈ nr. let a = {(v,µa(v),γa(v)) : v ∈ nr} and b = {(v,µb(v),γb(v)) : v ∈ nr} be intuitionistic fuzzy normed normal subrings. let c = a ∩ b such that c = {(v,µc(v),γc(v)) : v ∈ nr} where µc(v)=min {µa(v),µb(v)} and γc(v)= max{γa(v),γb(v)}. µc(v −r) = min{µa(v −r),µb(v −r)} = µa(v −r) ∗µb(v −r) ≥{µa(v) ∗µa(r)}∗{µb(v) ∗µb(r)} = µa(v) ∗{µa(r) ∗µb(v)}∗µb(r) = µa(v) ∗{µb(v) ∗µa(r)}∗µb(r) = {µa(v) ∗µb(v)}∗{µa(r) ∗µb(r)} = µc(v) ∗µc(r) int. j. anal. appl. 19 (3) (2021) 345 and µc(vr) = min{µa(vr),µb(vr)} = µa(vr) ∗µb(vr) ≥{µa(v) ∗µa(r)}∗{µb(v) ∗µb(r)} = µa(v) ∗{µa(r) ∗µb(v)}∗µb(r) = µa(v) ∗{µb(v) ∗µa(r)}∗µb(r) = {µa(v) ∗µb(v)}∗{µa(r) ∗µb(r)} = µc(v) ∗µc(r). similarly γc(v −r) ≤ γc(v) �γc(r) and γc(vr) ≤ γc(v) �γc(r). thus c is an intuitionistic fuzzy normed subring of nr. now, µc(vr) = µa(vr) ∗µb(vr) = µa(rv) ∗µb(rv) = µc(rv). therefore µc(vr) = µc(rv). also γc(vr) = γa(vr) �γb(vr) = γa(rv) �γb(rv) = γc(rv). therefore γc(vr) = γc(rv). then, the intersection of any two intuitionistic fuzzy normed normal subrings is an intuitionistic fuzzy normed normal subring of nr. � definition 3.4. let a be a non-empty subset of the normed ring nr, the intuitionistic characteristic function of a is defined as λa = (µλa,γλa ), where µλa (r) =   1 , if r ∈ a0 , if r /∈ a ,γλa (r) =   0 , if r ∈ a1 , if r /∈ a lemma 3.5. if a = (µa,γa) is a subring of nr then λa = (µλa,γλa ) is an intuitionistic fuzzy normed normal subring of nr. int. j. anal. appl. 19 (3) (2021) 346 proof. it shown in [14] that λa = (µλa,γλa ) is an intuitionistic fuzzy normed subring of nr when a is a subring. now, we need to show that λa = (µλa,γλa ) is an intuitionistic fuzzy normed normal subring, since vr and rv are in a, it follows that µλa (vr) = 1 = µλa (rv) and γλa (vr) = 0 = γλa (rv). consequently, µλa (vr) = µλa (rv) and γλa (vr) = γλa (rv). thus the intuitionistic characteristic function λa = (µλa,γλa ) of a is an intuitionistic fuzzy normed normal subring of nr. � lemma 3.6. if a and b are two subrings of the ring nr, then their intersection a∩b is a subring of nr if and only if the intuitionistic characteristic function λc = (µλc,γλc ) of c = a ∩ b is an intuitionistic fuzzy normed normal subring of nr. proof. let c = a ∩ b be a subring of nr and v,r ∈ nr. if v,r ∈ c, then by definition of intuitionistic characteristic function µλc (v) = 1 = µλc (r) and γλc (v) = 0 = γλc (r). since v−r, vr in a and b, it follows that v−r , vr in c. thus, µλc (r−v) = 1 = 1∗1 = µλc (r)∗µλc (v) and µλc (rv) = 1 = 1∗1 = µλc (r)∗µλc (v). thus µλc (r − v) ≥ µλc (r) ∗ µλc (v) and µλc (rv) ≥ µλc (r) ∗ µλc (v). now γλc (v − r) = 0 = 0 � 0 = γλc (v) � γλc (r) and γλc (vr) = 0 = 0 � 0 = γλc (v) � γλc (r). thus, γλc (v − r) ≤ γλc (v) � γλc (r) and γλc (vr) ≤ γλc (v) � γλc (r). as vr and rv ∈ c, so µλc (rv) = 1 = µλc (vr) and γλc (vr) = 0 = γλc (rv). accordingly, µλc (rv) = µλc (vr) and γλc (vr) = γλc (rv). similarly we have when v,r /∈ c: µλc (v −r) ≥ µλc (v) ∗µλc (r) and µλc (vr) ≥ µλc (v) ∗µλc (r), γλc (v −r) ≤ γλc (v) �γλc (r) and γλc (vr) ≤ γλc (v) �γλc (r), µλc (vr) = µλc (rv) and γλc (vr) = γλc (rv). hence the intuitionistic characteristic function λc = (µλc,γλc ) of c is an intuitionistic fuzzy normed normal subring of nr. conversely, assume that the intuitionistic characteristic function λc = (µλc,γλc ) of c is an intuitionistic fuzzy normal normed subring of nr. let v,r ∈ c, this imply that µλc (v) = 1 = µλc (r) and γλc (v) = 0 = γλc (r), then: µλc (v −r) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, µλc (vr) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, γλc (v −r) ≤ γλc (v) �γλc (r) = 0 � 0 = 0, γλc (vr) ≤ γλc (v) �γλc (r) = 0 � 0 = 0. this implies that µλc (v − r) = 1, µλc (vr) = 1 and γλc (v − r) = 0, γλc (vr) = 0. thus, v − r and vr ∈ c. hence c is a subring of nr. � proposition 3.7. if a is an intuitionistic fuzzy normed normal subring of a ring nr. then 4a = (µa,µca) is an intuitionistic fuzzy normed normal subring of a ring nr. int. j. anal. appl. 19 (3) (2021) 347 proof. let v,r ∈ nr µca(v −r) = 1 −µa(v −r) ≤ 1 − (µa(v) ∗µa(r)) ≤ 1 −min{µa(v),µa(r)} = max{1 −µa(v), 1 −µa(r)} = max{µca(v),µ c a(r)}. then, µca(v −r) ≤ µ c a(v) �µ c a(r). µca(vr) = 1 −µa(vr) ≤ 1 − (µa(v) ∗µa(r)) ≤ 1 −min{µa(v),µa(r)} = max{1 −µa(v), 1 −µa(r)} = max{µca(v),µ c a(r)}. then, µca(vr) ≤ µ c a(v) �µ c a(r). also, µca(vr) = 1 −µa(vr) = 1 −µa(rv) = µ c a(rv), then µ c a(vr) = µ c a(rv). therefore, 4a = (µa,µca) is an intuitionistic fuzzy normed normal subring of nr. � proposition 3.8. if a is an intuitionistic fuzzy normed normal subring of a ring nr. then ♦a = (γca,γa) is an intuitionistic fuzzy normed normal subring of a ring nr. proof. let v,r ∈ nr γca(v −r) = 1 −γa(v −r) ≥ 1 − (γa(v) �γa(r)) ≥ 1 −max{γa(v),γa(r)} = min{1 −γa(v), 1 −γa(r)} = min{γca(v),γ c a(r)}. then, γca(v −r) ≥ γ c a(v) ∗γ c a(r). γca(vr) = 1 −γa(vr) ≥ 1 − (γa(v) �γa(r)) ≥ 1 −max{γa(v),γa(r)} = min{1 −µa(v), 1 −γa(r)} = min{γca(v),γ c a(r)}. then, γca(vr) ≥ γ c a(v) ∗γ c a(r). also, γca(vr) = 1 −γa(vr) = 1 −γa(rv) = γ c a(rv), then γ c a(vr) = γ c a(rv). therefore, ♦a = (γca,γa) is an intuitionistic fuzzy normed normal ideal of nr. � int. j. anal. appl. 19 (3) (2021) 348 proposition 3.9. if a is an intuitionistic fuzzy normed normal subring of a ring nr. then a = (µa,γa) is an intuitionistic fuzzy normed normal subring of nr if the fuzzy subsets µa and γ c a are intuitionistic fuzzy normed normal subrings of nr. proof. clearly, µa is an intuitionistic fuzzy normed normal subring of nr, we need to show that γa is an intuitionistic fuzzy normed normal subring of nr. 1 −γa(v −r) = γca(v −r) ≥ γca(v) ∗γ c a(r) ≥ min{γca(v),γ c a(r)} = min{1 −γa(v), 1 −γa(r)} = 1 −max{γa(v),γa(r)}. then, γa(v −r) ≤ γa(v) �γa(r). 1 −γa(vr) = γca(vr) ≥ γca(v) ∗γ c a(r) ≥ min{γca(v),γ c a(r)} = min{1 −γa(v), 1 −γa(r)} = 1 −max{γa(v),γa(r)}. then, γa(vr) ≤ γa(v) �γa(r). also, 1 −γa(vr) = γca(vr) = γ c a(rv) = 1 −γa(rv). then, γa(vr) = γa(rv). hence, a = (µa,γa) is an intuitionistic fuzzy normed normal subring of nr. � proposition 3.10. if a is an intuitionistic fuzzy normed normal subring of a ring nr. then a = (µa,γa) is an intuitionistic fuzzy normed normal subring of nr if the fuzzy subsets µca and γa are intuitionistic fuzzy normed normal subrings of nr. proof. clearly, γa is an intuitionistic fuzzy normed normal subring of nr. we need to show that µa is an intuitionistic fuzzy normed normal subring of nr. 1 −µa(v −r) = µca(v −r) ≤ µca(v) �µ c a(r) ≤ max{µca(v),µ c a(r)} = max{1 −µa(v), 1 −µa(r)} = 1 −min{µa(v),γa(r)}. int. j. anal. appl. 19 (3) (2021) 349 then, µa(v −r) ≥ µa(v) ∗µa(r). 1 −µa(vr) = µca(vr) ≤ µca(v) �µ c a(r) ≤ max{µca(v),µ c a(r)} = max{1 −µa(v), 1 −µa(r)} = 1 −min{µa(v),µa(r)}. then, µa(vr) ≥ µa(v) ∗µa(r). also, 1 −µa(vr) = µca(vr) = µ c a(rv) = 1 −µa(rv). then, µa(vr) = µa(rv) hence, a = (µa,γa) is an intuitionistic fuzzy normed normal subring of nr. � 4. direct product of intuitionistic fuzzy normed normal subrings in this section, we define the direct product of intuitionistic fuzzy sets a1,a2 of normed rings r1,r2, respectively and examine some fundamental properties of direct product of intuitionistic fuzzy normed normal subrings. if nr1,nr2 are normed rings, then the direct product nr1 × nr2 of nr1 and nr2 is a normed ring with addition + defined as (v,r) + (z,d) = (v + z,r + d) and multiplication ◦ defined as (v,r) ◦ (z,d) = (vz,rd) for every (v,r), (z,d) in nr1 ×nr2. definition 4.1. an intuitionistic fuzzy set (ifs) a×b = (µa×b,γa×b) of nr1×nr2 is an intuitionistic fuzzy normed subring (ifnsr) of nr1×nr2 if for all v = (v1,v2) and r = (r1,r2) in nr1×nr2, satisfies: (i) µa×b(v −r) ≥ µa×b(v) ∗µa×b(r); (ii) µa×b(vr) ≥ µa×b(v) ∗µa×b(r); (iii) γa×b(v −r) ≤ γa×b(v) �γa×b(r); (iv) γa×b(vr) ≤ γa×b(v) �γa×b(r). definition 4.2. an intuitionistic fuzzy normed subring a×b = (µa×b,γa×b) of ring nr1 ×nr2 is an intuitionistic fuzzy normed normal subring of r1 ×r2 if for all v = (v1,v2) and r = (r1,r2) in r1 ×r2: µa×b(vr) = µa×b(rv) and γa×b(vr) = γa×b(rv). lemma 4.3. if a and b are intuitionistic fuzzy normed subrings of the rings nr1 and nr2, respectively, then a × b is an intuitionistic fuzzy normed subring of the ring nr1 × nr2 under the same operations defined in nr1 ×nr2. let a and b be two intuitionistic fuzzy normed subsets of nr1 and nr2, respectively. the direct product of a and b, is denoted by a×b, and defined as a×b = {((v,r),µa×b(v,r),γa×b(v,r)): for all v ∈ nr1 and r ∈ nr2} int. j. anal. appl. 19 (3) (2021) 350 where µa×b(v,r) = min{µa(v),µb(r)} and γa×b(v,r) = max{γa(v),γb(r)}. lemma 4.4. if a and b are intuitionistic fuzzy normed normal subrings of rings nr1 and nr2, respectively, then a×b is also an intuitionistic fuzzy normed normal subring nr1 ×nr2. proof. since the direct product of a and b is denoted by a × b = (µa×b,γa×b). let (v,r), (z,d) be in nr1 ×nr2, then: µa×b((v,r) − (z,d)) = µa×b(v −z,r −d) = min{µa(v −z),µb(r −d)} = µa(v −z) ∗µb(r −d) ≥{µa(v) ∗µa(z)}∗{µb(r) ∗µb(d)} = µa(v) ∗{µa(z) ∗µb(r)}∗µb(d) = µa(v) ∗{µb(r) ∗µa(z)}∗µb(d) = {µa(v) ∗µb(r)}∗{µa(z) ∗µb(d)} = µa×b(v,r) ∗µa×b(z,d) and µa×b((v,r) ◦ (z,d)) = µa×b(vz,rd) = min{µa(vz),µb(rd)} = µa(vz) ∗µb(rd) ≥{µa(v) ∗µa(z)}∗{µb(r) ∗µb(d)} = µa(v) ∗{µa(z) ∗µb(r)}∗µb(d) = µa(v) ∗{µb(r) ∗µa(z)}∗µb(d) = {µa(v) ∗µb(r)}∗{µa(z) ∗µb(d)} = µa×b(v,r) ∗µa×b(z,d). therefore, a×b is an intuitionistic fuzzy normed subring of nr1 ×nr2. now, µa×b((v,r) ◦ (z,d)) = µa×b(vz,rd) = min{µa(vz),µb(rd)} = min{µa(zv),µb(dr)} = µa×b(zv,dr) = µa×b((z,d) ◦ (v,r)). similarly, γa×b((v,r) − (z,d)) ≤ γa×b(v,r) �γa×b(z,d), γa×b((v,r) ◦ (z,d)) ≤ γa×b(v,r) �γa×b(z,d). and γa×b((v,r) ◦ (z,d)) = γa×b((z,d) ◦ (v,r)). hence, a×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. � int. j. anal. appl. 19 (3) (2021) 351 proposition 4.5. let a and b be an intuitionistic fuzzy subsets of the rings nr1 and nr2 with identities 1nr1 and 1nr2 , respectively. if a×b is an intuitionistic fuzzy normed subring of nr1×nr2, then at least one of the following must holds: (i) µa(v) ≤ µb(1nr2 ) and γa(v) ≥ γb(1nr2 ); for all v ∈ nr1, (ii) µb(r) ≤ µa(1nr1 ) and γb(r) ≥ γa(1nr1 ); for all r ∈ nr2. proof. let a×b be an intuitionistic fuzzy normed subring of nr1 ×nr2, and let the statements (i) and (ii) does not holds, we can find v ∈ nr1 and r ∈ nr2 such that µa(v) > µb(1nr2 ), γa(v) < γb(1nr2 ) and µb(r) > µa(1nr1 ), γb(r) < γa(1nr1 ). thus we have µa×b(vr) = min{µa(v),µb(r)} > min{µa(1nr1 ),µb(1nr2 )} = µa×b(1nr1, 1nr2 ) and γa×b(vr) = max{γa(v),γb(r)} < max{γa(1nr1 ),γb(1nr2 )} = γa×b(1nr1, 1nr2 ). which implies that a×b is not an intuitionistic fuzzy normed subring of nr1×nr2 which a contradiction. therefore, at least one of the statements must hold. � lemma 4.6. let a and b be an intuitionistic fuzzy subsets of the rings nr1 and nr2 with identities 1nr1 and 1nr2 , respectively. if a×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2, then the following are true: (i) if µa(v) ≤ µb(1nr2 ) and γa(v) ≥ γb(1nr2 ), then a is an intuitionistic fuzzy normed normal subring of nr1. (ii) if µb(v) ≤ µa(1nr1 ) and γb(v) ≥ γa(1nr1 ), then b is an intuitionistic fuzzy normed normal subring of nr2. proof. let a × b be an intuitionistic fuzzy normed normal subring of nr1 × nr2 with v,r ∈ nr1 and 1nr2 ∈ nr2. then (v, 1nr2 ) and (r, 1nr2 ) are in nr1 × nr2. obviously, a is an intuitionistic fuzzy normed subring of nr1, then int. j. anal. appl. 19 (3) (2021) 352 i. µa(v −r) = µa(v + (−r)) = min{µa(v + (−r)),µb(1nr2 + (−1nr2 ))} = µa×b((v + (−r)), (1nr2 + (−1nr2 )) = µa×b((v, 1nr2 ) + (−r,−1nr2 )) = µa×b((v, 1nr2 ) − (r, 1nr2 )) ≥ µa×b(v, 1nr2 ) ∗µa×b(r, 1nr2 ) = min{µa(v),µb(1nr2 )}∗min{µa(r),µb(1nr2 )} = µa(v) ∗µa(r). also, µa(vr) = min{µa(vr),µb(1nr2 1nr2 )} = µa×b(vr, 1nr2 1nr2 ) = µa×b((v, 1nr2 ) ◦ (r, 1nr2 )) ≥ µa×b(v, 1nr2 ) ∗µa×b(r, 1nr2 ) = min{µa(v),µb(1nr2 )}∗min{µa(r),µb(1nr2 )} = µa(v) ∗µa(r) and with, µa(vr) = min{µa(vr),µb(1nr2 1nr2 )} = µa×b((vr), (1nr2 1nr2 )) = µa×b((v, 1nr2 ) ◦ (r, 1nr2 )) = µa×b((r, 1nr2 ) ◦ (v, 1nr2 )) = µa×b((rv), (1nr2 1nr2 )) = min{µa(rv),µb(1nr2 1nr2 )} = µa(rv). similarly, we can prove that γa(v−r) ≤ γa(v) �γa(r), γa(vr) ≤ γa(v) �γa(r) and γa(vr) = γa(rv) for all v,r ∈ nr1. hence, a is an intuitionistic fuzzy normed normal subring of nr1. ii. the proof is similar to the above. � definition 4.7. let a×b be a non-empty subset of the ring nr1 ×nr2. the intuitionistic characteristic function of a×b is denoted by λa×b = (µλa×b,γλa×b ) and defined as: µλa×b (r) =   1 , if r ∈ a×b0 , if r /∈ a×b ,γλa×b (r) =   0 , if r ∈ a×b1 , if r /∈ a×b theorem 4.8. let a and b be two subrings of the rings nr1 and nr2, respectively. then a × b is a subring of nr1×nr2 if and only if the intuitionistic characteristic function λc = (µλc,γλc ) of c = a×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. int. j. anal. appl. 19 (3) (2021) 353 proof. let c = a × b be a subring of nr1 × nr2 and v,r ∈ nr1 × nr2. if v,r ∈ c = a × b, then by definition of intuitionistic characteristic function µλc (v) = 1 = µλc (r) and γλc (v) = 0 = γλc (r). since v − r and vr ∈ c and c is a subring. it follows that µλc (v − r) = 1 = 1 ∗ 1 = µλc (v) ∗ µλc (r) and µλc (vr) = 1 = 1∗1 = µλc (v)∗µλc (r). thus µλc (v−r) ≥ µλc (v)∗µλc (r) and µλc (vr) ≥ µλc (v)∗µλc (r). now γλc (v − r) = 0 = 0 � 0 = γλc (v) � γλc (r) and γλc (v − r) = 0 = 0 � 0 = γλc (v) � γλc (r). thus γλc (v −r) ≤ γλc (v) �γλc (r) and γλc (vr) ≤ γλc (v) �γλc (r). as vr and rv ∈ c, so µλc (vr) = 1 = µλc (rv) and γλc (vr) = 0 = γλc (rv). this implies that µλc (vr) = µλc (rv) and γλc (vr) = γλc (rv). similarly we have µλc (v −r) ≥ µλc (v) ∗µλc (r) and µλc (vr) ≥ µλc (v) ∗µλc (r), γλc (v −r) ≤ γλc (v) �γλc (r) and γλc (vr) ≤ γλc (v) �γλc (r), µλc (vr) = µλc (rv) and γλc (vr) = γλc (rv). when v,r /∈ c. hence the intuitionistic characteristic function λc = (µλc,γλc ) of c = a × b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. on the other hand, assume that the intuitionistic characteristic function λc = (µλc,γλc ) of c = a×b is an intuitionistic fuzzy normed normal subring of nr1 × nr2. now we have to show that c = a × b is a subring of nr. let v,r ∈ c, where v = (v ′ ,r ′ ) and r = (v ′′ ,r ′′ ), where v ′ ,v ′′ ∈ a and r ′ ,r ′′ ∈ b. by definition µλc (v) = 1 = µλc (r) and γλc (v) = 0 = γλc (r), µλc (v −r) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, µλc (vr) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, γλc (v −r) ≤ γλc (v) �γλc (r) = 0 � 0 = 0, γλc (vr) ≤ γλc (v) �γλc (r) = 0 � 0 = 0. this implies that µλc (v − r) = 1, µλc (vr) = 1 and γλc (v − r) = 0, γλc (vr) = 0. thus v − r and vr ∈ c. hence c = a×b is a subring of nr1 ×nr2. � lemma 4.9. if v = a×b and q = c ×d are two subrings of nr1 ×nr2 then their intersection v ∩q is also a subring of nr1 ×nr2. theorem 4.10. let v = a×b and q = c×d be two intuitionistic fuzzy normed subrings of nr1×nr2. then v ∩q is subring of nr1×nr2 if and only if the intuitionistic characteristic function λz = (µλz,γλz ) of z = v ∩q is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. proof. let z = v ∩ q be a subring of ring nr1 × nr2 and let v = (v1,v2), r = (r1,r2) ∈ nr1 × nr2. if v,r ∈ z = v ∩ q, then by properties of intuitionistic characteristic function µλz (v) = 1 = µλz (r) and γλz (v) = 0 = γλz (r). since v−r and vr ∈ z. then, µλz (v−r) = 1 = 1∗1 = µλz (v)∗µλz (r), µλz (vr) = 1 = int. j. anal. appl. 19 (3) (2021) 354 1∗1 = µλz (v)∗µλz (r) and γλz (v−r) = 0 = 0�0 = γλz (v)�γλz (v), γλz (vr) = 0 = 0�0 = γλz (v)�γλz (r). therefore, µλz (v −r) ≥ µλz (v) ∗µλz (r), µλz (vr) ≥ µλz (v) ∗µλz (r), γλz (v −r) ≤ γλz (v) �γλz (r), γλz (vr) ≤ γλz (v) �γλz (r). since, vr and rv ∈ z, then µλz (vr) = 1 = µλz (rv) and γλz (vr) = 0 = γλz (rv) so µλz (vr) = µλz (rv) and γλz (vr) = γλz (rv). we also have when v,r /∈ z: µλc (v −r) ≥ µλc (v) ∗µλc (r) and µλc (vr) ≥ µλc (v) ∗µλc (r), γλc (v −r) ≤ γλc (v) �γλc (r) and γλc (vr) ≤ γλc (v) �γλc (r), µλc (vr) = µλc (rv) and γλc (vr) = γλc (rv). hence the intuitionistic characteristic function λz = (µλz,γλz ) of z is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. conversely, assume that the intuitionistic characteristic function λz = (µλz,γλz ) is an intuitionistic fuzzy normed normal subring. let v,r ∈ z = v ∩q, then µλz (v) = 1 = µλz (r) and γλz (v) = 0 = γλz (r), hence: µλc (v −r) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, µλc (vr) ≥ µλc (v) ∗µλc (r) = 1 ∗ 1 = 1, γλc (v −r) ≤ γλc (v) �γλc (r) = 0 � 0 = 0, γλc (vr) ≤ γλc (v) �γλc (r) = 0 � 0 = 0. thus µλc (v − r) = 1 = µλc (vr) and γλc (v − r) = 0 = γλc (vr). this implies that v − r and vr ∈ z. hence z is a subring of ring nr1 ×nr2. � proposition 4.11. if the ifs a×b is an intuitionistic fuzzy normed normal subring of the ring nr1×nr2, then 4a×b = (µa×b,µca×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. proof. let a × b be an intuitionistic fuzzy normed normal subring of nr1 × nr2 and let (v,r), (z,d) ∈ nr1 ×nr2. then µca×b((v,r) − (z,d)) = 1 −µa×b((v,r) − (z,d)) ≤ 1 − (µa×b(v,r) ∗µa×b(z,d)) = 1 −min{µa×b(v,r),µa×b(z,d)} = max{1 −µa×b(v,r), 1 −µa×b(z,d)} = max{µca×b(v,r),µ c a×b(z,d)} = µca×b(v,r) �µ c a×b(z,d) int. j. anal. appl. 19 (3) (2021) 355 and µca×b((v,r) ◦ (z,d)) = 1 −µa×b((v,r) ◦ (z,d)) ≤ 1 − (µa×b(v,r) ∗µa×b(z,d)) = 1 −min{µa×b(v,r),µa×b(z,d)} = max{1 −µa×b(v,r), 1 −µa×b(z,d)} = max{µca×b(v,r),µ c a×b(z,d)} = µca×b(v,r) �µ c a×b(z,d). thus 4a×b = (µa×b,µca×b) is an intuitionistic fuzzy normed subring nr1 ×nr2. µca×b((v,r) ◦ (z,d)) = 1 −µa×b((v,r) ◦ (z,d)) = 1 −µa×b((z,d) ◦ (v,r)) = µca×b((z,d) ◦ (v,r)) hence, 4a×b = (µa×b,µca×b) is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. � proposition 4.12. if the ifs a×b is an intuitionistic fuzzy normed normal subring of the ring nr1×nr2, then ♦a×b = (γca×b,γa×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. proof. similar to the proof of proposition 4.11 � corollary 4.13. an ifs a×b is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2 if and only if 4a × b = (µa×b,µca×b) (resp.♦a × b = (γ c a×b,γa×b)) is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. theorem 4.14. an ifs a×b is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2 if and only if the fuzzy subsets µa×b and γ c a×b are intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. proof. let a×b = (µa×b,γa×b) be an intuitionistic fuzzy normed normal subring of the ring nr1×nr2. this implies that µa×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. we have to show that γca×b is also an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. let (v,r), (z,d) ∈ nr1 ×nr2. then γca×b((v,r) − (z,d)) = 1 −γa×b((v,r) − (z,d)) ≥ 1 − (γa×b(v,r) �γa×b(z,d)) = 1 −max{γa×b(v,r),γa×b(z,d)} = min{1 −γa×b(v,r), 1 −γa×b(z,d)} = min{γca×b(v,r),γ c a×b(z,d)} = γca×b(v,r) ∗γ c a×b(z,d) int. j. anal. appl. 19 (3) (2021) 356 and γca×b((v,r) ◦ (z,d)) = 1 −γa×b((v,r) − (z,d)) ≥ 1 − (γa×b(v,r) �γa×b(z,d)) = 1 −max{γa×b(v,r),γa×b(z,d)} = min{1 −γa×b(v,r), 1 −γa×b(z,d)} = min{γca×b(v,r),γ c a×b(z,d)} = γca×b(v,r) ∗γ c a×b(z,d) hence, γca×b is also an intuitionistic fuzzy normed subring of the ring nr1 ×nr2. γca×b((v,r) ◦ (z,d)) = 1 −γa×b((v,r) ◦ (z,d)) = 1 −γa×b((z,d) ◦ (v,r)) = γca×b((z,d) ◦ (v,r)). hence, γca×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. conversely, suppose that µa×b and γ c a×b are intuitionistic fuzzy normed normal subring of the ring nr1 × nr2. we have to show that a × b = (µa×b,γa×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. then 1 −γa×b((v,r) − (z,d)) = γca×b((v,r) − (z,d)) ≥ γca×b(z,d) ∗γ c a×b(v,r) = min{γca×b(z,d),γ c a×b(v,r)} = min{1 −γa×b(z,d), 1 −γa×b(v,r)} = 1 −max{γa×b(z,d),γa×b(v,r)} = 1 − (γa×b(z,d) �γa×b(v,r)) and 1 −γa×b((v,r) ◦ (z,d)) = γca×b((v,r) ◦ (z,d)) ≥ γca×b(z,d) ∗γ c a×b(v,r) = min{γca×b(z,d),γ c a×b(v,r)} = min{1 −γa×b(z,d), 1 −γa×b(v,r)} = 1 −max{γa×b(z,d),γa×b(v,r)} = 1 − (γa×b(z,d) �γa×b(v,r)). therefore, a×b = (µa×b,γa×b) is an intuitionistic fuzzy normed subring of the ring nr1 ×nr2. 1 −γa×b((v,r) ◦ (z,d)) = γca×b((v,r) ◦ (z,d)) = γca×b((z,d) ◦ (v,r)) = 1 −γa×b((z,d) ◦ (v,r)). int. j. anal. appl. 19 (3) (2021) 357 therefore, a × b = (µa×b,γa×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 × nr2. � theorem 4.15. an ifs a×b is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2 if and only if the fuzzy subsets µca×b and γa×b are intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. proof. let a×b = (µa×b,γa×b) be an intuitionistic fuzzy normed normal subring of the ring nr1×nr2. this implies that γa×b is an intuitionistic fuzzy normed normal subring of nr1 ×nr2. we have to show that µca×b is also an an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. the proof of the first part is similar to the first part of proposition 4.11. conversely, suppose that µc a×b and γa×b are intuitionistic fuzzy normed normal subring of the ring nr1 × nr2. we have to show that a×b = (µa×b,γa×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 ×nr2. then 1 −µa×b((v,r) − (z,d)) = µca×b((v,r) − (z,d)) ≤ µca×b(z,d) �µ c a×b(v,r) = max{µca×b(z,d),µ c a×b(v,r)} = max{1 −µa×b(z,d), 1 −µa×b(v,r)} = 1 −min{µa×b(z,d),µa×b(v,r)} = 1 − (µa×b(z,d) ∗µa×b(v,r)) and 1 −µa×b((v,r) ◦ (z,d)) = µca×b((v,r) ◦ (z,d)) ≤ µca×b(z,d) �µ c a×b(v,r) = max{µca×b(z,d),µ c a×b(v,r)} = max{1 −µa×b(z,d), 1 −µa×b(v,r)} = 1 −min{µa×b(z,d),µa×b(v,r)} = 1 − (µa×b(z,d) ∗µa×b(v,r)). therefore, a×b = (µa×b,γa×b) is an intuitionistic fuzzy normed subring of the ring nr1 ×nr2. 1 −µa×b((v,r) ◦ (z,d)) = µca×b((v,r) ◦ (z,d)) = µca×b((z,d) ◦ (v,r)) = 1 −µa×b((z,d) ◦ (v,r)). therefore, a × b = (µa×b,γa×b) is an intuitionistic fuzzy normed normal subring of the ring nr1 × nr2. � int. j. anal. appl. 19 (3) (2021) 358 5. conclusion the objective of this paper was to initiate the notion of intuitionistic fuzzy normed normal subrings and to establish some relevant properties. we extended the notion of intuitionistic fuzzy normed subrings to intuitionistic fuzzy normed normal subrings. also, we established the direct product of intuitionistic fuzzy normed normal subrings and examined some fundamental properties of direct product of intuitionistic fuzzy normed normal subrings. further research could be done is to study the intuitionistic anti fuzzy normed normal subrings. we hope that in future, this concept would be a useful contribution to the study of intuitionistic fuzzy normed rings by generalizing other fundamental properties. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. [2] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 512-517. [3] w. wu, normal fuzzy subgroups, fuzzy math. 1 (1981), 21-30. [4] w. liu, fuzzy invariant subgroups and fuzzy ideals, fuzzy sets syst. 8 (1982), 133–139. [5] n. mukherjee, fuzzy normal subgroups and fuzzy cosets, inform. sci. 34 (1984), 225–239. [6] w. wu, fuzzy congruences and normal fuzzy subgroups, math. appl. 1 (1988), 9-20. [7] m. mishref, normal fuzzy subgroups and fuzzy normal series of finite groups, fuzzy sets syst. 72 (1995), 379-383. [8] s. abdullah, m. aslam, t. a. khan, m. naeem, a new type of fuzzy normal subgroups and fuzzy cosets, j. intell. fuzzy syst. 25 (2013), 37-47. [9] k.t. atanassov, intuitionistic fuzzy sets, in intuitionistic fuzzy sets, springer, berlin, germany, 1999, 1-137. [10] k. hur, s.y. jang, h.w. kang, intuitionistic fuzzy normal subgroups and intuitionistic fuzzy cosets, honam math. j. 26 (2004), 559-587. [11] m.f. marashdeh, a.r. salleh, the intuitionistic fuzzy normal subgroup, int. j. fuzzy log. intell. syst. 10 (2010), 82-88. [12] v. veeramani, k. arjunan, n. palaniappan, some properties of intuitionistic fuzzy normal subrings, appl. math. sci. 4 (2010), 2119-2124. [13] p. sharma, on intuitionistic fuzzy magnified translation in rings, int. j. algebra. 1 (2011), 1451-1458. [14] n. abed alhaleem, a.g. ahmad, intuitionistic fuzzy normed subrings and intuitionistic fuzzy normed ideals, mathematics. 8 (2020), 1594. [15] a. emniyet, m. şahin, fuzzy normed rings, symmetry. 10 (2018), 515. [16] r. arens, a generalization of normed rings, pac. j. math. 2 (1952), 455-471. [17] m.a. najmark, normed rings, noordhoff, groningen, the netherlands, 1964. [18] i. gelfand, d. raikov, g. shilov, commutative normed rings, american mathematical society, usa, 2002. [19] a.u alkouri, a.r. salleh, complex atanassov’s intuitionistic fuzzy relation, abstr. appl. anal. 2013 (2013), 287382. [20] m.o. alsarahead, a.g. ahmad, complex intuitionistic fuzzy ideals, aip conf. proc. 1940 (2018), 020118. [21] p. sharma, g. kaur, on intuitionistic fuzzy prime submodules, notes ifs. 24 (2018), 97-112. int. j. anal. appl. 19 (3) (2021) 359 [22] a. al-masarwah, a.g. ahmad, structures on doubt neutrosophic ideals of bck/bci-algebras under (s, t)-norms, neutrosophic sets syst. 33 (2020), 17. [23] m. gupta, j. qi, theory of t-norms and fuzzy inference methods, fuzzy sets syst. 40 (1991), 431-450. 1. introduction 2. preliminaries 3. some properties of intuitionistic fuzzy normed normal subrings 4. direct product of intuitionistic fuzzy normed normal subrings 5. conclusion references international journal of analysis and applications volume 19, number 1 (2021), 1-19 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-1 on reciprocals leap indices of graphs ammar alsinai1,∗, anwar alwardi2, n.d. soner1 1department of studies in mathematics, university of mysore, manasagangotri,mysuru 570 006, india 2department of mathematics, university of aden, yemen ∗corresponding author: aliiammar1985@gmail.com abstract. in the field of chemical graph theory, topological indices are calculated based on the molecular graph of a chemical compound. topological indices are used in the development of quantitative structure activity/propoerty relations. to study the physico-chemical properties of molecules most commonly used are the zagreb indices. in this paper, we introduce reciprocals leap indices as a modified version of leap zagreb indices. the exact values of reciprocals leap indices of some well-known classes of graphs are calculated. lower and upper bounds on the reciprocals leap indices of graphs are established. the relationship between reciprocals leap indices and leap zagreb indices are presented. 1. introduction in last decade, graph theory has found a considerable use in the mathematical chemistry. in this area we can apply tools of graph theory to model the chemical phenomenon mathematically. this theory contributes a prominent in chemical science. a chemical structure of molecules can be represent by molecular graph, where vertices represent the atoms and edges represent the bonds between them. the graph theory based structure descriptors can be determined by considering graph vertices and edges. a simply arithmetic operators are carried out to get numerical indices. topological indices are used in the development of quantitative structure activity/property relations (qsar/qspr). a graph is a collection of points and received august 20th, 2020; accepted september 15th, 2020; published november 24th, 2020. 2010 mathematics subject classification. 05c07,05c05, 05c90. key words and phrases. second degree of vertex; inverse degree; modified zagreb indices; leap zagreb indices; reciprocals leap indices. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-1 int. j. anal. appl. 19 (1) (2021) 2 lines connecting a subset of them. the points and lines of a graph are also called vertices and edges of the graph, respectively. the vertex and edge sets of a graph g are denoted by v (g) and e(g), respectively. let |v (g)| = n and |e(g)| = m, if two vertices u and v of the graph g are adjacent, then the edge connecting them will be denoted by uv. if u,v ∈ v (g) then the distance dg(u,v) between u and v is defined as the length of a shortest path in g connecting them. the diameter of a connected graph g is the length of any longest geodesic, denoted by diam(g). in a graph g, the degree (first degree) of a vertex v, denoted d(v), is the number of first neighbors (the number of edges incident with v) and the second degree of v, denoted d2(v), is the number of second neighbors. the maximum and minimum degrees among the vertices of g, are denoted by ∆ = ∆(g) and δ = δ(g), respectively. the join graph g + h is the graph with vertex set v (g + h) = v (g) ∪ v (h) and edge set e(g + h) = e(g) ∪ e(h) ∪{uv : u ∈ v (g) andv ∈ v (h)}. a wheel w1,n and a friendship graphs are defined as w1,n = k1 + cn and fn = k1 + n−1 2 k2, respactively. a graph g is called f-free graph if no induced subgraph of g is isomorphic to f. in this paper, we only conceder with a simple connected graphs. any undefined term or notation in this paper can be found in ( [2] [7]). one of the oldest and most commonly used to study the physico-chemical properties of molecules topological index are zagreb indices introduced by gutman and trinajstic on based degree of vertices of g. the first and second zagreb indices of a graph g are defined as: m1(g) = ∑ v∈v (g) d(v)2 m2(g) = ∑ uv∈e(g) d(u)d(v). the quantity m1(g) was first time considered in 1972 [4], whereas m2(g) in 1975 [5]. for more information on zagreb and beyond topological indices, readers are referred to the survey [6], and the references therein. in 2017, naji et al. [9] have introduced a new distance-degree-based topological indices conceived depending on the second degrees of vertices, and are so-called leap zagreb indices of a graph g and are defined as: lm1(g) = ∑ v∈v (g) d2(v) 2 lm2(g) = ∑ uv∈e(g) d2(u)d2(v) lm3(g) = ∑ v∈v (g) d(v)d2(v). the leap zagreb indices have several chemical applications. surprisingly, the first leap zagreb index has very good correlation with physical properties of chemical compound, like bolling point, entropy, dhvap, int. j. anal. appl. 19 (1) (2021) 3 hvap and eccentric factor [1]. consequently, the new class of graphs, that so called leap graphs was defined and studied in [10], and was defined as, a graph g is said to be a leap graph, if and only if for every vertex v ∈ v (g), d(v) = d2(v). the inverse degree index of a graph the first was introduced in 2005 [11], and was defined by id(g) = ∑ v∈v (g) 1 d(v) . the inverse degree has attracted attention through a conjecture generating computer graffiti [3]. the modified first zagreb index mm1(g) was first introduced in [8], and defined as mm1(g) = ∑ v∈v (g) 1 d(v)2 . motivated by the inverse degree index, we herewith define the inverse second degree index of a graph, as following id2(g) = ∑ v∈v (g) 1 d2(v) + 1 . note that,we added one to the second degree of a vertex, because there are infinity graph with some vertex whose d(v) = n− 1 and so d2(v) = 0. thus, the id2(g) defined here is well-defined for every graph. lemma 1.1. let g be the connected graph with δ ≥ 1. then n n−δ ≤ id2(g) ≤ n. motivated by the modified zagreb and leap zagreb indices of graphs and the huge applications of them, we in this work define the first, second and third reciprocals leap indices of a graph as a modified version of leap zagreb indices. the exact values of some well-known graphs are computed. some upper and lower bounds on reciprocals leap indices of a graph are established. finally, we investigate and present the relationship between reciprocals leap indices and leap zagreb indices of graphs. we need the following fundamental results to prove our main results. lemma 1.2. (a): let g be a connected graph with n vertices and m edges. then for every vertex v ∈ v (g), d2(v) ≤ ( ∑ u∈n(v) d(u)) −d(v). equality holds if and only if g is a (c3,c4) free. int. j. anal. appl. 19 (1) (2021) 4 lemma 1.3. (b): let g be a connected graph with n vertex. then for every vertex v ∈ v (g), d2(v) ≤ n− 1 −d(v). equality holds if and only if g having diameter of most two. lemma 1.4. (c): let g be k-regular (c3,c4)-free graph. then for every vertex v ∈ v (g) d2(v) = k(k − 1). 2. reciprocals leap indices of graphs in this section, we present the definitions of first, second and third reciprocals leap indices of a graph and explore how we can calculation them for a connected simple graph. definition 2.1. for a connected graph g, the first, second and third reciprocals leap indices are defined as: rl1 = rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 rl2 = rl2(g) = ∑ uv∈e(g) 1 (d2(u) + 1)(d2(v) + 1) rl3 = rl3(g) = ∑ v∈v (g) 1 d(v)(d2(v) + 1) . to illustrate these invariants, we compute them for a graph g shown in figure 1. x xx x v1 v4 v3 v2 figure 1. it is easy to show that d2(v1) = 2, d2(v2) = 0, d2(v3) = d2(v4) = 1. hence, rl1(g) = 1 (2 + 1)2 + 1 (0 + 1)2 + 1 (1 + 1)2 + 1 (1 + 1)2 = 1 4 + 1 1 + 1 4 + 1 4 = 29 18 int. j. anal. appl. 19 (1) (2021) 5 rl2(g) = 1 (2 + 1)(0 + 1) + 1 (0 + 1)(1 + 1) + 1 (0 + 1)(1 + 1) + 1 (1 + 1)(1 + 1) = 1 3 + 1 2 + 1 2 + 1 4 = 19 12 , rl3(g) = 1 1(2 + 1) + 1 3(0 + 1) + 1 2(1 + 1) + 1 2(1 + 1) = 1 3 + 1 3 + 1 4 + 1 4 = 7 6 . 3. reciprocals leap indices for some families of graphs in this section, we establish the formulaes of the exact values of reciprocals leap indices for some wellknown graph classes. proposition 3.1. for a positive integer n ≥ 2, (1) for the complete graph kn, • rl1(kn) = n, • rl2(kn) = n(n−1) 2 , • rl3(kn) = n(n−1) . (2) for the path pn, • rl1(pn) =   2, if n = 2 ; 3 2 , if n = 3 ; n+5 9 , otherwise. • rl2(pn) =   1, if n = 2, 3 ; 3 4 , if n = 4 ; 2n+5 18 , otherwise. • rl3(pn) =   1, if n = 2 ; 3 2 , if n = 3 ; n+5 6 , otherwise. (3) for the cycle cn, • rl1(cn) = rl2(cn) =   3, if n = 3 ; 1, if n = 4 ; n 9 , otherwise. • rl3(cn) =   3 2 , if n = 3 ; 1, if n = 4 ; n 6 , otherwise. int. j. anal. appl. 19 (1) (2021) 6 (4) for the complete bipartite graph kr,s, 1 ≤ r ≤ s ≤ n− 1, • rl1(kr,s) = rl3(kr,s) = r+srs . • rl2(kr,s) = 1. (5) for the star graph k1,n−1, • rl1(k1,n−1) = rl3(k1,n−1) = nn−1 • rl2(k1,n−1) = 1. (6) for the wheel graph, w1,n,n ≥ 3, • rl1(w1,n) =   4, if n = 3 ;n2−3n+4 (n−2)2 , otherwise. • rl2(w1,n) =   6, if n = 3 ;n(n−1) (n−2)2 , otherwise. • rl3(w1,n) =   4 3 , if n = 3 ; n2+3n−6 3n(n−2) , otherwise. (7) for the friendship graph fn, n ≥ 3, • rl1(fn) = 1 + n−1(n−2)2 , • rl2(fn) = (n−1)(2n−3) 2(n−2)2 , • rl3(fn) = (n2−3) 2(n−1)(n−2) . proposition 3.2. let g be a connected k-regular c3,c4-free graph. then (1) rl1(g) = n (k2−k+1)2 , (2) rl2(g) = nk (k2−k+1)2 , (3) rl3(g) = n k(k2−k+1) . proof. the proof is immediately consequence of lemma 1.4 and the definitions of reciprocals leap indices of graphs. � 4. bounds on reciprocals leap indices of graphs in this section, we present some upper and lower bounds on reciprocals leap indices of a graph, in term of number of vertices, number of edges, minimum (maximum) degree, inverse degree and second inverse degree indices of a graph. theorem 4.1. let g be a connected graph with n ≥ 2 vertices. then, rl1(g) ≤ n. equality holds if and only if g is a complete graph. int. j. anal. appl. 19 (1) (2021) 7 proof. let g be a connected graph with n ≥ 2 vertices. since for every v ∈ v (g), d2(v) ≥ 0, which led to d2(v) + 1 ≥ 1. thus, 1(d2(v)+1)2 ≤ 1, for every v ∈ v (g) and so, rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 ≤ ∑ v∈v (g) 1 = n. to prove the equality, we assume, on the contrary, that g 6= kn, for n ≥ 2. then there are at least two vertices u,v ∈ v (g) such that d2(u) ≥ 1 and d2(v) ≥ 1. thus, rl1(g) = 1 (d2(u) + 1)2 + 1 (d2(v) + 1)2 + ∑ w∈v (g)−{u,v} 1 (d2(w) + 1)2 ≤ 1 (2)2 + 1 (2)2 + ∑ w∈v (g)−{u,v} 1 (0 + 1)2 = 1 4 + 1 4 + (n− 2) = 4(n− 2) + 2 4 = 4n− 6 4 = n− 3 2 < n. which is a contradiction. conversely, if g is a complete graph, then it is immediate the result follow from proposition 3.1. � corollary 4.1. for any connected graph g, with n vertices, rl1(g) = n if and only if g is a complete graph. theorem 4.2. let g be a connected graph with m ≥ 1 edegs. then, rl2(g) ≤ m. equality holds if and only if g is a complete graph. proof. the proof is similar to the proof of theorem 4.1, then we omit it. � corollary 4.2. for any connected graph g, with m edges, rl2(g) = m, if and only if g is a complete graph. theorem 4.3. let g be a connected graph with n ≥ 2 vertices. then, rl3(g) ≤ n. equality holds if and only if g = k2. int. j. anal. appl. 19 (1) (2021) 8 proof. let g be a connected graph with n ≥ 2 vertices. since d(v) ≥ 1 and d2(v) ≥ 0, for every v ∈ v (g). so d(v)(d2(v) + 1) ≥ 1, for every v ∈ v (g). hence, rl3(g) = ∑ v∈v (g) 1 d(v)(d2(v) + 1) ≤ ∑ v∈v (g) 1 1 = n. suppose the equality rl3(g) = n holds. then d(v)(d2(v) + 1) = 1, for every vertex v ∈ v (g). since d(v) and d2(v) are a positive integers numbers. then d(v)(d2(v) + 1) = 1, if and only if d(v) = 1 and d2(v) = 0. since for any graph g and any vertex v ∈ v (g), d2(v) = 0 if and only if d(v) = n− 1. then, d2(v) = 0 and d(v) = 1, if and only if n = 2 and g is a complete graph. therefore, rl3(g) = n, if and only if g = k2. � corollary 4.3. for any connected graph g, with n vertices, rl3(g) = n if and only if n = 2. theorem 4.4. let g be a connected graph with n vertices. then, n ∆ ≤ rl3(g) ≤ n δ . the lower bound attains on k1,n−1 and kn, whereas the upper bound attains on kn. proof. let g be a connected graph with n vertices. since for every vertex v ∈ v (g), δ ≤ d(v) ≤ ∆ and d2(v) ≥ 0. then, δ ≤ d(v)(d2(v) + 1) ≤ ∆. so rl3(g) = ∑ v∈v (g) 1 d(v)(d2(v) + 1) ≥ ∑ v∈v (g) 1 ∆(1) = n ∆ . = ∑ v∈v (g) 1 d(v)(d2(v) + 1) ≤ ∑ v∈v (g) 1 δ(1) = n δ . � theorem 4.5. for any connected graph g with n vertices, rl1(g) ≥ n (n− δ)2 . equality holds if and only if g is a regular graph with diameter at most two. int. j. anal. appl. 19 (1) (2021) 9 proof. since for any connected graph d(v) ≥ δ, for every v ∈ v (g) and by using lemma 1.3, d2(v) ≤ n− 1 −d(v), for every v ∈ v (g). then, d2(v) + 1 ≤ n−d(v) ≤ n−δ, for every v ∈ v (g). hence, rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 ≥ ∑ v∈v (g) 1 (n− δ)2 = n (n− δ)2 . suppose the equality rl1(g) = n (n−δ)2 is holding. then by lemma 1.3, equality d2(v) = n− 1 −d(v) holds for every vertex v ∈ v (g), if and only if diam(g) ≤ 2. thus, d2(v) + 1 = n − d(v) = n − δ, for every v ∈ v (g) if and only if diam(g) ≤ 2 and g is δ-regular graph. therefore, equality rl1(g) = n(n−δ)2 hold if and only if g is a regular graph with diam(g) ≤ 2. � the following results immediately follow from the facts that, δ(g) ≥ 1 for any connected graph g and for every vertex v ∈ v (g) and if g is a k-regular graph, then δ(g) = k for every vertex v ∈ v (g). corollary 4.4. for any connected graph g, rl1(g) ≥ n (n− 1)2 . equality holds if and only if g = k2. corollary 4.5. for any k-regular graph, rl1(g) ≥ n (n−k)2 . equality holds if and only if diam(g) ≤ 2. theorem 4.6. for any connected graph g, rl2(g) ≥ m (n− δ)2 . equality holds if and only if g is a regular graph with diameter at most two. proof. the proof is similar to the proof of theorem 4.5. � corollary 4.6. for any connecte k-regular graph, rl2(g) ≥ nk (n−k)2 . equality holds if and only if g having diameter at most two. int. j. anal. appl. 19 (1) (2021) 10 theorem 4.7. for any connected graph g, rl3(g) ≥ n ∆(n−δ) . equality holds if and only if g is regular graph with diameter at most two. proof. let g be a connected graph with minimum and maximum degrees 1 ≤ δ ≤ ∆. then by lemma 1.3, d2(v) + 1 ≤ n−d(v). since δ ≤ d(v) ≤ ∆, then d(v)(d2(v) + 1) ≤ ∆(n− δ). hence, rl3(g) ≥ ∑ v∈v (g) 1 ∆(n− δ) = n ∆(n− δ) . suppose the equality rl3(g) = n ∆(n−δ) holds. then by lemma 1.3, d2(v) + 1 = n−d(v), for every v ∈ v (g), if and only if diam(g) ≤ 2 and d(v)(d2(v) + 1) = ∆(n− δ), if and only if diam(g) ≤ 2 and d(v) = δ = ∆. this complete the proof. � since, for every vertex v in a graph g, δ ≤ ∆ which implies that δ2 ≤ δ∆ and ∆n−δ2 ≥ ∆n−δ∆. then the following results strieghtforward. proposition 4.1. for any connected graph g, rl3(g) ≥ n ∆n− δ2 . equality holds if and only if g is regular graph with diameter at most two. corollary 4.7. for any k−regular graph, rl3(g) ≥ n k(n−k) . equality holds if and only if diam(g) ≤ 2. theorem 4.8. for any connected graph g, rl1(g) ≤ id2(g). equality holds if and only if g is a complete. proof. since d2(v) is a positive integer number for every v ∈ v (g). then (d2(v) + 1) 2 ≥ d2(v) + 1 and hence 1(d2(v)+1)2 ≤ 1 d2(v)+1 . therefore, rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 ≤ ∑ v∈v (g) 1 d2(v) + 1 = id2(g). int. j. anal. appl. 19 (1) (2021) 11 suppose the equality rl1(g) = id2(g) holds. then (d2(v) + 1) 2 = d2(v) + 1, if and only if d2(v) + 1 = 1, if and only if d2(v) = 0, for every v ∈ v (g), if and only if g = kn. � theorem 4.9. for any connected graph g, rl3(g) ≤ id(g). equality holds if and only if g = kn. proof. let g be a connected graph. since d2(v) + 1 ≥ 1, for every v ∈ v (g). then 1d(v)(d2(v)+1) ≤ 1 d(v) and hence, rl3(g) = ∑ v∈v (g) 1 d(v)(d2(v) + 1) ≤ ∑ v∈v (g) 1 d(v) = id(g). if the equality rl3(g) = id(g) holds, then d(v)(d2(v) + 1) = d(v), for every v ∈ v (g), that implies that d2(v) = 0 for every v ∈ v (g) and hence g = kn. conversely, it is easy to check that if g = kn, then rl3(g) = id(g). � theorem 4.10. let g be a connected graph of order n and with maximum degree ∆ ≥ 1, rl3(g) ≥ id2(g) ∆ . the equality holds if and only if g is regular. proof. let g be a connected graph with n vertices and ∆ ≥ 1. the chebyshers sum inequality stat that if a1 ≥ a2 ≥ ... ≥ an and b1 ≥ b2 ≥ ... ≥ bn, then n n∑ i=1 aibi ≥ n∑ i=1 ai n∑ i=1 bi. hence, by put ai = 1 d(vi) and bi = 1 d2(vi)+1 , for every i = 1, 2, ..,n. we get nrl3(g) ≥ ( n∑ i=1 1 d(vi) )( n∑ i=1 1 (d2(vi) + 1) ) ≥ ( n∑ i=1 1 ∆ )(id2(g)) = n ∆ id2(g). therefore, rl3(g) ≥ 1∆id2(g). suppose the equality rl3(g) = 1 ∆ id2(g) holds. then d(v) = ∆ for every v ∈ v (g), if and only if g is a ∆-regular graph. conversely, let g be a k-regular graph. then rl3(g) = ∑ v∈v (g) 1 k(d2(v) + 1) = 1 k ∑ v∈v (g) 1 d2(v) + 1 = id2(g) k . � int. j. anal. appl. 19 (1) (2021) 12 theorem 4.11. let g be a connected graph of order n. then (1) rl1(g) ≤ n[1 − 2 ln(n− δ)]. (2) rl2(g) ≤ m[1 − 2 ln(n− δ)]. (3) rl3(g) ≤ n[1 − ln(∆n− δ2)]. the equality holds in (1),(2) and (3) if and only if g is a complete. proof. let g be a connected graph with n vertices, m edges and minimum (maximum) degrees δ ≥ 1 ( ∆ ≤ n− 1). we prove an inequality for rl1(g), and the proof of the inequalities (2) and (3) are similar. assume the function f(x) = x − ln x − 1. easy calculating gives us that f(x) ≥ 0, for every positive real number. thus, for every v ∈ v (g), we get 1 (d2(v) + 1)2 − ln( 1 (d2(v) + 1)2 ) − 1 ≥ 0. hence, 1 (d2(v) + 1)2 ≥ 1 + ln( 1 (d2(v) + 1)2 ). by taking the summation on two sides of the inequality over the vertex set of the graph, we get rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 ≥ ∑ v∈v (g) 1 + ∑ v∈v (g) ln( 1 (d2(v) + 1)2 ) = n + ln   n∏ j=1 ( 1 (d2(v) + 1)2 )   . by using the fact, for every v ∈ v (g), d2(v) + 1 ≤ n−δ, we get, rl1(g) ≥ n + ln   n∏ j=1 ( 1 (n−δ)2 )   = n + ln [ 1 (n−δ)2n ] = n− 2n ln(n− δ) = n[1 − 2 ln(n− δ)]. the equality holds if and only if f( 1 d2(v)+1 ) = 0, if and only if 1 d2(v)+1 = 1, if and only if d2(v) = 0, for every v ∈ v (g), if and only if g = kn, n ≥ 2. � theorem 4.12. let g be a leap graph of order n and minimum and maximum degrees δ ≥ 1 and ∆ ≤ n−1. then (1) n (∆+1)2 ≤ rl1(g) ≤ n(δ+1)2 . (2) m (∆+1)2 ≤ rl2(g) ≤ m(δ+1)2 . int. j. anal. appl. 19 (1) (2021) 13 (3) n ∆(∆+1) ≤ rl3(g) ≤ nδ(δ+1). equalities holds in (1), (2) and (3) if and only if g is a regular. proof. let g be a connected leap graph with n ≥ 4 vertex, minimum degree δ ≥ 1 and maximum degree ∆ ≤ n− 1. we prove part (1)only, and the proofs of (2) and (3) are similar. since g is a leap graph and δ ≤ d(v) ≤ ∆, for every v ∈ v (g).then δ ≤ d2(v) ≤ ∆, for every v ∈ v (g). hence, 1 (∆ + 1)2 ≤ 1 (d2(v) + 1)2 ≤ 1 (δ + 1)2 , by taken the summation over vertex set of g, we get ∑ v∈v (g) 1 (∆ + 1)2 ≤ rl1(g) ≤ ∑ v∈v (g) 1 (δ + 1)2 . so, n (∆ + 1)2 ≤ rl1(g) ≤ n (δ + 1)2 . suppose the equality holds in the lower bound. then d(v) + 1 = ∆ + 1, for every v ∈ v (g). that mean g is a ∆-regular graph. similarly, for the upper bound, which let g is δ-regular graph. conversely, let g be a k-regular leap graph. then ∆ = δ = k, and hence rl1(g) = ∑ v∈v (g) 1 (d(v) + 1)2 = ∑ v∈v (g) 1 (k + 1)2 = n (k + 1)2 . � 5. relationship between reciprocals leap indices and leap zagreb indices of graphs in this section, we investigate the relationship between reciprocals leap indices and leap zagreb indices of graphs. also, the relation between first and third reciprocals leap indices of a graph are presented. theorem 5.1. for any connected graph g with n vertices and m edges. (1) rl1(g) + lm1(g) ≥ n + 4m− 2m1(g). (2) rl2(g) + lm2(g) ≥ m−lm3(g). (3) rl3(g) + lm3(g) ≥ 2(n−m). equalities hold in (1) and (3) if and only if g = k2, whereas in (2) if and only if g is complete. proof. let g be a connected graph of order n and size m. assume the function f(x) = x + 1 x − 2. easy calculation gives f(x) ≥ 0, for every positive real number x. so, if we put x = 1 (d2(v)+1)2 , then for every v ∈ v (g), 1 (d2(v) + 1)2 + (d2(v) + 1) 2 − 2 ≥ 0 int. j. anal. appl. 19 (1) (2021) 14 and hence 1 (d2(v)+1)2 ≥ 2 − (d2(v) + 1)2. since, by lemma 1.2,∑ v∈v (g) d2(v) ≤ m1(g) − 2m. then rl1(g) = ∑ v∈v (g) 1 (d2(v) + 1)2 ≥ 2n− ∑ v∈v (g) ( d22(v) + 2d2(v) + 1 ) = 2n− (lm1(g) + 2m1(g) − 4m−n) . therefore, rl1(g) + lm1(g) ≥ n + 4m− 2m1(g). since the equality f(x) = 0, if and only if x = 1. then the equality holds in (1), if and only if 1 (d2(v)+1)2 = 1, if and only if d2(v) = 0, for every v ∈ v (g), if and only if g + kn, and by lemma 1.2, g is c3,c4-free graph. thus g = k2. to prove the inequality (2), put x = 1 (d2(v)+1)(d2(v)+1) . so for every uv ∈ e(g), we get 1 (d2(v)+1)(d2(v)+1) ≥ 2 − (d2(u) + 1)(d2(v) + 1). by taken the summation over edge set of graph g, we get rl2(g) ≥ 2m− ∑ uv∈e(g) (d2(u) + 1)(d2(v) + 1)) = 2m− ∑ uv∈e(g) [d2(u)d2(v) + d2(u) + d2(v) + 1] = 2m− [lm2(g) + lm3(g) + m] = m−lm2(g) −lm3(g). therefore, rl2(g) + lm2(g) ≥ m−lm3(g). the proof of an equality in (2) is very similar to the proof of part (1). now, by put x = 1 d(v)(d2(v)+1) , we get for every v ∈ v (g) 1 d(v)(d2(v) + 1) ≥ 2 −d(v)(d2(v) + 1) and hence, by taken the summation over vertex set of g, rl3(g) ≥ 2n− ∑ v∈v (g) (d(v)d2(v) + d(v)) = 2n−lm3(g) − 2m. therefore, rl3(g) + lm3(g) ≥ 2(n−m). the proof of equality in (3) is similar to the proof of part (1). � theorem 5.2. let g be a connected graph of order n and size m. then (1) 1 rl1(g) ≤ lm1(g)+n(2n−1)−4m n2 . int. j. anal. appl. 19 (1) (2021) 15 (2) 1 rl2(g) ≤ lm2(g)+lm3(g)+m m2 . (3) 1 rl3(g) ≤ lm3(g)+2m n2 . the equality holds in (1) and (2), if and only if g = kn, whereas in (3) if and only if g = k2 . proof. let g be a connected graph of order n and size m. cauchy-schwartz state, for every real numbers ai and bi, i = 1, 2, ...,n, that ( n∑ i=1 aibi) 2 ≤ ( n∑ i=1 a2i )( n∑ i=1 b2i ) if we put ai = 1 d2(v)+1 and bi = d2(v) + 1, for i = 1, 2, ...,n, and for every vertex v ∈ v (g), then n2 = ( n∑ i=1 1)2 =   ∑ v∈v (g) 1 (d2(v) + 1) (d2(v) + 1)  2 ≤ ( ∑ v∈v (g) 1 (d2(v) + 1)2 )( ∑ v∈v (g) (d2(v) + 1) 2) ≤ rl1(g) ∑ v∈v (g) (d22(v) + 2d2(v) + 1)) = rl1(g) [lm1(g) + 2(n(n− 1)) − 4m + n] = rl1(g) [lm1(g) + n(2n− 1) − 4m] . therefore, 1 rl1(g) ≤ lm1(g)+n(2n−1)−4m n2 . since the equality hold in cauchy-schwartiz, if and only if ai = bi, for every i = 1, 2, ...,n. then the equality holds in (1), if and only if 1 d2(v)+1 = d2(v) + 1, if and only if d2(v) = 0, for every v ∈ v (g), if and only if g is a complete graph. now, to prove an inequality (2), we put ai = 1√ (d2(u)+1)(d2(v)+1) and bi = √ (d2(u) + 1)(d2(v) + 1) for every edge uv ∈ e(g), we get m2 = ( ∑ uv∈e 1)2 =   ∑ uv∈e(g) 1√ (d2(u) + 1)(d2(v) + 1) · √ (d2(u) + 1)(d2(v) + 1)  2 ≤   ∑ uv∈e(g) 1 (d2(u) + 1)(d2(v) + 1)     ∑ uv∈e(g) (d2(u) + 1)(d2(v) + 1)   = rl2(g)   ∑ uv∈e(g) (d2(u)d2(v) + (d2(u) + d2(v)) + 1)   = rl2(g) [lm2(g) + lm3(g) + m] therefore, 1 rl2(g) ≤ lm2(g)+lm3(g)+m m2 . int. j. anal. appl. 19 (1) (2021) 16 finally, by putting ai = 1√ (d(v))(d2(v)+1) and bi = √ (d(v))(d2(v) + 1) in cauchy-schwartz inequality. then for every v ∈ v (g), n2 = ( ∑ v∈v (g) 1)2 =   ∑ v∈v (g) 1√ d(v)(d2(v) + 1) · √ d(v)(d2(v) + 1)  2 ≤   ∑ v∈v (g) 1 d(v)(d2(v) + 1)     ∑ v∈v (g) d(v)(d2(v) + 1)   = rl3(g)   ∑ v∈v (g) (d(v)d2(v) + d(v))   = rl3(g) [lm3(g) + 2m] . therefore, 1 rl3(g) ≤ lm3(g)+2m n2 . � theorem 5.3. let g be a connected graph of order n. then rl3(g) ≤ 1 δ √ nrl1(g). equality holds if and only if g is an n 2 -regular graph. proof. let g be a connected graph of order n. put ai = 1 d(v) and bi = 1 d2(v)+1 in cauchy schwartz inequality. then (rl3(g)) 2 = ( ∑ v∈v (g) 1 d(v)(d2(v) + 1) )2 ≤   ∑ v∈v (g) 1 d(v)     ∑ v∈v (g) 1 (d2(v) + 1)2   ≤   ∑ v∈v (g) 1 δ2   (rl1(g)) = n δ2 rl1(g). therefore rl3(g) ≤ 1δ √ nrl1(g). the equality rl3(g) = 1 δ √ nrl1(g) holds if and only if 1 d(v) = 1 d2(v)+1 , if and only if d(v) = d2(v) + 1, if and only if d2(v) = d(v) − 1, if and only if d2(v) = δ − 1, for every vertex v ∈ v (g). so g is a δ-regular graph with d2(v) = δ − 1, for every v ∈ v (g). hence by this conclusion and apply lemma 1.3, we get , δ = n 2 . therefore the equality rl3(g) = 1 δ √ nrl1(g) holds if and only if g is an n 2 -regular graph. � corollary 5.1. let g be a connected graph. then rl3(g) ≤ (mm1(g))(rl1(g)). int. j. anal. appl. 19 (1) (2021) 17 corollary 5.2. let g be a connected graph. then rl3(g) ≤ n(mm1(g)). theorem 5.4. let g be a connected graph of order n. then (1) rl1(g) ≤ (n−δ+1)id2(g)−n n−δ . (2) rl2(g) ≤ m− lm2(g)+lm3(g) n−δ . (3) rl3(g) ≤ nδ + nδ−lm3(g)−2m δ∆(n−δ) . equality holds in (1), (2) and (3) if and only if g is regular graph with diam(g) ≤ 2. proof. let g be a connected graph of order n, size m and δ ≥ 1, ∆ ≤ n− 1. diaz-metcalf inequality stat that if ai 6= 0 and bi for i = 0, 1, 2, ...,n satisfy t ≤ biai ≤ t , then n∑ i=1 b2i + tt n∑ i=1 a2i ≤ (t + t) n∑ i=1 aibi. equality holds if and only if bi = tai or bi = tai, for i = 1, 2, ...,n. 1) by diaz-matcalf inequality, we prove inequality (1). put bi = 1 and ai = 1 d2(vi)+1 for i = 1, 2, ...,n. since d2(v) + 1 ≤ n− δ, for every v ∈ v (g). then 1 ≤ biai ≤ n−δ and hence t = 1 and t = n− δ. thus, n∑ i=1 1 + (n− δ) n∑ i=1 1 (d2(vi) + 1)2 ≤ (n−δ + 1) n∑ i=1 1 d2(vi) + 1 . so, n + (n− δ)rl1(g) ≤ (n− δ + 1)id2(g). therefore, rl1(g) ≤ (n−δ+1)id2(g)−n (n−δ) . suppose the equality holds in (1). then by diaz-metcalf inequality 1 = 1 d2(v)+1 , which implies that d2(v) + 1 = 1, so d2(v) = 0 for every v ∈ v (g) or 1 = n−δ d1(v)+1 which implies that d2(v) + 1 = n−δ, so d2(v) = n−1−δ, which implies, by lemma 1.2, that d(v) = δ for every v ∈ v (g) and diam(g) ≤ 2. conversely, let g be a k-regular graph with diam(g) ≤ 2. then by lemma 1.2, d2(v) + 1 = n−k, for every v ∈ v (g) and hence, rl1(g) = ∑n i=1 1 (n−k)2 = n (n−k)2 , and id2(g) = ∑n i=1 1 (n−k) = n (n−k) . on the other hand, (n− δ + 1)id2(g) −n (n− δ) = (n−k + 1)( n n−k ) −n n−k = n(n−k −n) −n(n−k) (n−k)2 = n (n−k)2 = rl1(g). therefore, the equality holds. 2) to prove inequality (2), put in diaz-matcalf inequality bi = √ (d1(u) + 1)(d2(v) + 1) and ai = int. j. anal. appl. 19 (1) (2021) 18 1√ (d2(u)+1)(d2(v)+1 for every uv ∈ e(g). hence, 1 ≤ bi ai ≤ (n− δ)2 for every i = 1, 2, ...,n. thus, t = 1 and t = n−δ, and hence, ∑ uv∈e(g) (d2(u) + 1)(d2(v) + 1) + (n− δ)2 ∑ uv∈e(g) 1 (d1(u) + 1)(d2(v) + 1) ≤ [(n− δ)2] ∑ uv∈e(g) 1 ∑ uv∈e(g) [d2(u)d2(v) + d2(u) + d2(v) + 1] + (n− δ)2rl2(g) ≤ m(n− δ)2 + 1 lm2(g) + lm3(g) + m + (n− δ)2rl2(g) ≤ m(n− δ)2 + m. therefore, rl2(g) ≤ m(n−δ)2 + m−lm2(g) −lm3(g) −m (n− δ)2 = m− lm2(g) + lm3(g) (n− δ)2 . the proof of the equality is similar to the proof of inequality (1), so we left it. 3) to prove the inequality (3), we put bi = √ (d(v)(d2(v) + 1) and ai = 1√ (d(v))(d2(v)+1) for every v ∈ v (g).thus, δ ≤ bi ai ≤ ∆(n− δ), for every i = 1, 2, ...,n. hence, t = δ and t = ∆(n− δ). so, ∑ v∈v (g) d(v)(d2(v) + 1) + δ∆(n− δ) ∑ v∈v (g) 1 d(v)(d2(v) + 1) ≤ (∆(n− δ) + δ) ∑ v∈v (g) 1 lm3(g) + 2m + δ∆(n− δ)rl3(g) ≤ δn(n− δ) + δn. thus, rl3(g) ≤ ∆n(n− δ) + δn−lm3(g) − 2m δ∆(n− δ) = n δ + δn−lm3(g) − 2m δ∆(n− δ) . the proof of the equality is similar to the proof of inequality (1). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. basavanagoud, p. jakkannavar, computing first leap zagreb index of some nano structures, int. j. math. appl. 6(2-b) (2018), 141-150. [2] j.a. bondy, u.s.r. murty, graph theory with applications, north holland, new york, 1976. [3] s. fajtlowicz, on conjectures of grafitti ii, congr. numer. 60 (1987), 189-197. [4] i. gutman, n. trinajstic, graph theory and molecular orbitals. total φ-electron energy of alternant hydrocarbons, chem. phys. lett. 17 (1972), 535–538. [5] i. gutman, graph theory and molecular orbitals. xii. acyclic polyenes, j. chem. phys. 62 (1975), 3399. [6] i. gutman, e. milovanović, i. milovanović, beyond the zagreb indices, akce int. j. graphs comb. 17 (2020), 74–85. [7] f. harary, graph theory, addison wesley, reading mass, 1969. int. j. anal. appl. 19 (1) (2021) 19 [8] a. milicevic, s. nikolic, on variable zagreb indices, croat. chem. acta. 77 (2004), 97-101. [9] a. m. naji, n. d. soner, and i. gutman, on leap zagreb indices of graphs, commun. comb. optim. 2 (2017), 99-117. [10] a. m. naji, b. davvaz, s. s. mahde and n. d. soner, a study on some properties of leap graphs, commun. comb. optim. 5 (2020), 9-17. [11] z. zhang, j. zhang, x. lu, the relation of matching with inverse degree of a graph, discrete math. 301 (2005), 243-246. 1. introduction 2. reciprocals leap indices of graphs 3. reciprocals leap indices for some families of graphs 4. bounds on reciprocals leap indices of graphs 5. relationship between reciprocals leap indices and leap zagreb indices of graphs references int. j. anal. appl. (2022), 20:12 some results of conditionally sequential absorbing and pseudo reciprocally continuous mappings in probabilistic 2-metric space k. satyanna1,∗, v. srinivas2 1department of mathematics, m.a.l.d. government degree college, gadwal, palamoor univesity, mahaboobnagar, telangana state, 509125 india 2department of mathematics, university college of science, osmania univesity, hyderabad, telangana state 500004, india ∗corresponding author: satgjls@gmail.com abstract. the objective of this paper is to generate two results in probabilistic 2-metric space by using the concepts of conditionally sequential absorbing mappings and pseudo reciprocally continuous mappings. these results stand as generalizations of the theorem proved by v. k. gupta, arihant jain and rajesh kumar. further these two outcomes are justified by supporting examples. 1. introduction the metric space notion was introduced by fréchet [4]. afterwords many generalizations came into existence one such prominent one was banach contraction principle. gähler [5] used the notion of 2metric space as generalization of metric space. golet [6] presented the concept of probabilistic 2-metric space as generalization of 2metric space and gave some fundamental concepts like convergence, continuity. dwelling of fixed point results has got importance for researchers due to the newly emerging platforms like 2-metric space, fuzzy space, menger space and 2-menger space etc. in this aspect many fixed point theorems came into the light by using the concepts like compatibility, continuity and contraction. the notion of compatibility was coined in metric space by jungck and b. e. rhodes [9]. the weaker form of compatibility as weakly compatible mappings in 2-menger space used v. k. gupta et al. [7] and obtained some results. further abbas and b. e. rhodes [1] by using the concept of received: nov. 8, 2021. 2010 mathematics subject classification. 47h10. key words and phrases. self-mappings; pseudo reciprocally continuous mappings; conditionally sequentially absorbing maps; probabilistic 2-metric space. https://doi.org/10.28924/2291-8639-20-2022-12 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-12 2 int. j. anal. appl. (2022), 20:12 occasionally weakly compatible mappings proved some fixed point theorems. in the recent past some of fixed theorems have been evolved without using the condition of continuity [14], [3], [13]. out of these results weaker form continuity known as reciprocally continuity came into existence due to pant er al. [11] and became instrumental for deriving some fruitful results in metric and fuzzy spaces under some weaker conditions. u. mishra et al. [10] tried to find out coincidence points without using the concepts of compatibility and continuity conditions to generate some fixed point theorems in menger space. some more results witnessed by [2], [8] in menger space. in this process pseudo reciprocally continuous and conditionally sequential absorbing mappings were emerged in metric space [12] and resulted in the formation of some fixed point results. in this article we extend these notions to 2menger space and generate two results in 2menger space. for this we present some definitions and preliminaries. 2. preliminaries definition 2.1. f : r→ r+ is distribution function [13] if it is (i) non-decreasing, (ii) continuous from left, (iii) inf {f(t) : t ∈ r} = 0, (iv) sup {f(t) : t ∈ r} = 1. the letter l is used to refer to a collection of all distribution functions. definition 2.2. a probabilistic 2-metric space (2-pm space) [7] is a pair (x,f) with f : x×x×x→ l here l stands as the set of all distribution functions and the f value at (e,f ,g)∈ x×x×x is written as fe,f,g and obeys the following properties: (a) fe,f,g(0)=0 (b) ∃g ∈ x such that fe,f,g(t�) < 1,∀e,f ∈ x,e 6= f ,for some t� > 0 (c) fe,f,g(t�)=1 ∀t� > 0 if e = f = g or e = f or f = g or e = g (d) fe,f,g(t�)= ff ,g,e(t�)= fg,f,e,(t�) (e) fe,f,g(tx)= ff ,g,e(ty)= fg,f,e,(tz)=1 =⇒ fe,f,g(tx + ty + tz)=1. definition 2.3. the mapping t� : [0,1]3 → [0,1] is a t− norm [7] it has the following properties: (i) t�(0,0,0)=0 (ii) t�(υ,1,1)= υ (iii) t�(a0,b0,c0)= t�(b0,c0,a0)= t�(c0,a0,b0) (iv) t�(d,e,f )≥ t�(d1,e1, f1) for d ≥ d1,e ≥ e1, f ≥ f1 (v) t�(t�(a0,b0,c0)), r,s)= t�(a0,t�(b0,c0, r),s)= t�(a0,b0,t�(c0, r,s)). definition 2.4. a 2-menger space [7] is a triplet (x,f,t�) where (x,f) is a 2-pm space and t� is a t-norm having triangle inequality: fu,v,w(tx + ty + tz)≥ t(fu,v,p(tx),fu,p,w(ty),fp,v,w(tz)) ∀w,p,v,u ∈ x and tx,ty,tz ≥ 0. int. j. anal. appl. (2022), 20:12 3 definition 2.5. [7] a sequence (pn) in 2-menger space (x,f,t�) (i) converges to β if for each � > 0,t� > 0,∃n(�)∈ n =⇒ fpn,β,a(�) > 1− t�, ∀a ∈ x and n ≥ n(�); (ii) cauchy if for each � > 0,t� > 0,∃n(�)∈ n =⇒ fpn,pm,a(�) > 1− t�, ∀a ∈ x and n,m ≥ n(�); (iii) if each cauchy sequence converges in x then it is mentioned as complete 2-menger space. definition 2.6. self-mappings p, s in 2-menger space (x,f,t�) are known as (a) compatible [7] if fpsxn,spxn,a(β) → 1,∀a ∈ x and β > 0 whenever a sequence (xn) ∈ x such that pxn,sxn → θ where θ is some element of x as n →∞. (b) non compatible [14] if limn→∞fpsxn,spxn,a(β) not exists or limn→∞fpsxn,spxn,a(β) 6= 1∀a ∈ x and β > 0 whenever a sequence xn ∈ x such that pxn,sxn → θ where θ is an element of x as n →∞. (c) weakly compatible [7] if commute at their coincidence points. (d) occasionally weakly compatible (owc) [1] if there is a coincidence point at which the mappings are commuting. example 2.1. weakly compatible mappings always owc but the converse may not be true. define ∀t� ∈ [0,1] fυ,β,γ(t1)=   t� t�+d(υ,β) if t� > 0 0, if t� =0 (2.1) ∀υ,β and fixed γ =0,t� > 0. by considering x= [−2,2] and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. the mappings e,h : x→ x are defined as e(x)=4−x ∀x ∈ [−2,2] (2.2) h(x)=4−x 2 ∀x ∈ [−2,2]. (2.3) from (2.2) and (2.3) the mappings e,h have coincidence points 0, 1. at x =0,e(0)= h(0)=1,eh(0)= e(1)=4−1 and he(0)= h(1)=4−1. this shows that e(0)= h(0) =⇒ eh(0)= he(0). at x =1,e(1)= h(1)=4−1,eh(0)= e(4−1)=4−4 −1 and he(1)= h(4−1)=4−(4 −1)2. this gives e(1)= h(1) but eh(1) 6= he(1). as a result the mappings e, h are owc but not weakly compatible. definition 2.7. self-mappings p,s in 2-menger space (x,f,t�) are known as (a) conditionally sequential absorbing(csa) [12] if, whenever sequence 4 int. j. anal. appl. (2022), 20:12 {< xn >: limn→∞pxn = limn→∞sxn} 6= φ =⇒ there exists another sequence < yn > satisfying limn→∞pyn = limn→∞syn = t(say) such that limn→∞fpyn,psyn,a(β)=1 and limn→∞fsyn,spyn,a(β)=1 ∀a ∈ x and β > 0. (b) pseudo reciprocally continuous (prc) [12] (w.r.t to conditionally sequential absorbing) if, whenever sequence {< xn >: limn→∞pxn = limn→∞sxn} 6= φ =⇒ there exist another sequence < yn > satisfying limn→∞pyn = limn→∞syn = t(say) then limn→∞fpyn,psyn,a(β)=1 and limn→∞fsyn,spyn,a(β)=1 such that limn→∞fpsyn,pt,a(β)=1 and limn→∞fspyn,st,a(β)=1 ∀a ∈ x and for some β > 0. example 2.2. by considering x = [−3,5] and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. the mappings e,h : x→ x are defined as e(x)=  −3 if x ∈ [−3,0) 2−x if x ∈ [0,5] (2.4) and h(x)=  x if x ∈ [−3,0) 2−2x if x ∈ [0,5]. (2.5) from (2.4), (2.5) -3 and 0 are intersecting points for the mappings e, h. at x =0,e(0)= h(0)=1, eh(0)= e(1)= 1 2 and he(0)= h(1)= 1 4 =⇒ eh(0) 6= he(0). consequently the mapping h, e are not commuting at that coincidence point x =0. hence these are not weakly compatible. consider a sequence < xn >= √ 3 n ∀ n ≥ 1 then exn = e( √ 3 n )=2−( √ 3 n ) → 1 (2.6) and hxn = h( √ 3 n )=2−2( √ 3 n ) → 1 (2.7) as n →∞. from (2.6), (2.7) =⇒ lim n→∞ hxn = lim n→∞ exn. for a sequence < yn >=(−3+ 3n ∀ n ≥ 1. then eyn = e(−3+ 3 n =−3→−3, (2.8) hyn = h(−3+ 3 n )= (−3+ 3 n )→−3. (2.9) int. j. anal. appl. (2022), 20:12 5 as n →∞ and ehyn = e(−3+ 3 n )=−3→−3, (2.10) heyn = h(−3)=−3→−3 (2.11) as n →∞. from (2.8), (2.9), (2.10) and (2.11) lim n→∞ feyn,ehyn,a(β)=1 and lim n→∞ fhyn,heyn,a(β)=1. (2.12) further lim n→∞ fehyn,e(−3),a(β)=1 and limn→∞ fheyn,h(−3),a(β)=1. (2.13) from (2.12), (2.13) the mappings e, h are csa and psc (w.r.t csa) but not weakly compatible. consequently conditionally sequential absorbing and pseudo reciprocally continuous maps(w.r.t csa) are weaker than weakly compatible mappings. we discuss the following examples to find the relation between conditionally sequential absorbing and noncompatible mappings. example 2.3. by considering x = (0,3] and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. e,h : x→ x are defined as e(x)=  1−5x if 0 < x ≤ 1 10 x2, if 1 10 < x ≤ 3 (2.14) and h(x)=  5x if 0 < x ≤ 1 10 3, if 1 10 < x ≤ 3 (2.15) by considering sequence < xn >= 1 10 ∀n ≥ 1. then from (2.14), (2.15) exn = e( 1 10 )=1−5( 1 10 )= 1 2 → 1 2 , (2.16) hxn = h( 1 10 )=5( 1 10 )→ 1 2 (2.17) as n →∞. from (2.15), (2.16) =⇒ lim n→∞ hxn = lim n→∞ exn. (2.18) 6 int. j. anal. appl. (2022), 20:12 then there is a sequence < yn >= √ 3 ∀ n ≥ 1. then eyn = e( √ 3)=3→ 3, (2.19) hyn = h( √ 3)=3 (2.20) as n →∞. from (2.19), (2.20) =⇒ lim n→∞ hyn = lim n→∞ eyn. (2.21) ehyn = e(3)=9→ 9, (2.22) heyn = h(3)=3→ 3 (2.23) as n →∞. from (2.22), (2.23) lim n→∞ fehyn,heyn,a(t�) 6=1 (2.24) ∀a ∈ x and t� > 0. hence from (2.21), (2.24) the mappings e, h are non compatible. moreover from (2.19)(2.22) lim n→∞ feyn,ehyn,a(t�) 6=1 (2.25) and from (2.20),(2.23) lim n→∞ fhyn,heyn,a(t�) 6=1. (2.26) from (2.24), (2.25) and (2.26) demonstrate that the mappings e, h are non compatible but not conditionally sequential absorbing. example 2.4. by considering x = (0,13] and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. e,h : x→ x are defined as e(x)=  5x if 0≤ x < 2 7, if 2≤ x ≤ 8 (2.27) and h(x)=  6x if 0≤ x < 2 x, if 2≤ x ≤ 8 (2.28) by considering sequence < xn >= e 3n ∀ n ≥ 1. then from (2.27),(2.28) exn = e( e 3n )=5( e 3n )→ 0, (2.29) int. j. anal. appl. (2022), 20:12 7 hxn = h( e 3n )=6( e 3n )→ 0 (2.30) as n →∞. from (2.29), (2.30) =⇒ lim n→∞ hxn = lim n→∞ exn. there exists another sequence < yn >=7 ∀n ≥ 1. then from (2.27), (2.28) eyn = e(7)=7→ 7, (2.31) hyn = h(7)=7→ 7 (2.32) as n →∞. but ehyn = e(7)=7→ 7, (2.33) heyn = h(7)=7→ 7 (2.34) as n →∞. from (2.33), (2.34) lim n→∞ fehyn,heyn,a(t�)=1 (2.35) ∀a ∈ x and t� > 0. hence from (2.33) the mappings e, h are compatible. moreover from (2.31), (2.33) lim n→∞ feyn,ehyn,a(t�)=1 (2.36) and from (2.32), (2.34) lim n→∞ fhyn,heyn,a(t�)=1 (2.37) ∀a ∈ x and t� > 0. resulting that from (2.35), (2.36) and (2.37) the mappings h, e are compatible as well as conditionally sequential absorbing. the following theorem was proved in [7]. theorem 2.1. let a,b,s and t be self -mappings on a complete probabilistic 2-metric space (x,f,t�) satisfying : (i) a(x)⊆ t(x),b(x)⊆ s(x) (ii) one of a(x),b(x),t(x) or s(x) is complete (iii) pairs (a, s) and (b, t) are weakly compatible (iv) fax,by,γ(t�)≥ rfsx,ty,γ(t�) for all x, y in x and t� > 0 8 int. j. anal. appl. (2022), 20:12 where r : [0,1]→ [0,1] is some continuous function such that r(t�) > t� for each o < t� < 1. then the mappings a, b, s and t have unique common fixed point in x. now we generalize above theorem in next section. 3. main results theorem 3.1. let a, b, s and t be mappings on a complete probabilistic 2-metric space (x,f,t�) to itself satisfying a(x)⊆ t(x),b(x)⊆ s(x) (3.1) the pair of mappings (a, s) pseudo reciprocally continuous (w,r.t. conditionally sequential absorbing) and conditionally sequential absorbing and (b, t) is occasionally weakly compatible fax,by,γ(t�)≥ r(fsx,ty,γ(t�)) (3.2) whenever x, y in x and t� > 0 for some continuous self-map on [0, 1] such that r(t�) > t� for each o < t� < 1. then a, b, s and t have unique common fixed point in x. proof. by using (3.1) the sequence < yn > derived as < y2n >= ax2n = tx2n+1 (3.3) < y2n+1> = bx2n+1 = sx2n+2. (3.4) now our claim is to show < yn > is a cauchy sequence. by taking the values x = x2n,y = x2n+1 in (3.2) we get fax2n,bx2n+1,γ(t�)≥ r(fsx2n,tx2n+1,γ(t�)) fy2n,y2n+1,γ(t�)≥ r(fy2n−1,y2n,γ(t�)) > fy2n−1,y2n,γ(t�). in general we have fyn+1,yn,γ(t�) > fyn,yn−1,γ(t�) for all n ≥ 1. then we have {fyn+1,yn,γ(t�), ∀ n ≥ 1} is an increasing sequence of positive real numbers bounded above by 1 therefore it must be converge to a limit say l ≤ 1. int. j. anal. appl. (2022), 20:12 9 if l < 1 then fyn+1,yn,γ(t�)= l > r(1) > 1 which is a conflict. hence l =1. therefore for all n and p fyn+p,yn,γ(t�)=1. thus the cauchyness of the sequence (yn) in complete space x so it has limit z ∈ x, results every sub sequence has the same limit z. that is from (3.2) and (3.4) ax2n,sx2n → z (3.5) tx2n+1,bx2n+1 → z as n →∞. use the notion l{a,s}= {< xn >: limn→∞axn = limn→∞sxn}. since the pair (a, s) is conditionally sequential absorbing from (3.5) l{a,s} 6= φ =⇒ ∃ < yn > such that lim n→∞ ayn = lim n→∞ syn = θ(say) (3.6) =⇒ lim n→∞ fayn,asyn,γ(t�)=1 and fsyn,sayn,γ(t�)=1 (3.7) ∀γ ∈ x,t� > 0. also the pair (a, s) satisfies pseudo reciprocally continuous means whenever lim n→∞ ayn = lim n→∞ syn = θ(say) =⇒ lim n→∞ fayn,asyn,γ(t�)=1 and lim n→∞ fsyn,sayn,γ(t�)=1 such that lim n→∞ asyn = a(θ) and lim n→∞ sayn = s(θ). (3.8) using (3.6) and (3.8) in (3.7) we get aθ = sθ = θ. but aθ is element in a(x) by (3.1) there exists η such that θ = sθ = aθ = tη. (3.9) claim bη = tη. by putting x = θ,y = η in (3.2) faθ,bη,γ(t�)≥ r(fsθ,tη,γ(t)) 10 int. j. anal. appl. (2022), 20:12 from (3.9) faθ,bη,γ(t�)≥ r(fsθ,sθ,γ(t�))= r(1)=1. (3.10) =⇒ aθ = bη = tη. the pair (b, t) is occasionally weakly compatible gives btη = tbη =⇒ bθ = tθ from (3.9). claim θ = bθ. by taking x = y = θ in (3.2) faθ,bθ,γ(t�)≥ r(fsθ,tθ,γ(t�)) using (3.9) and bθ = tθ fθ,bθ,γ(t�)≥ r(fθ,bθ,γ(t�)) > fθ,bθ,γ(t�) fθ,bθ,γ(t�) > fθ,bθ,γ(t�) which is absurd. hence θ = bθ. resulting θ = bθ = tθ = aθ = sθ. (3.11) therefore θ is the required common fixed point. uniqueness: suppose θ1 is another fixed common fixed point for the mappings a, s, b and t. claim θ = θ1. suppose if θ 6= θ1, then by taking x = θ,y = θ1 in (3.2) faθ,bθ1,γ(t�)≥ r(fsθ,tθ1,γ(t�)). this gives fθ,θ1,γ(t�)≥ r(fθ,θ1,γ(t�)) > fθ,θ1,γ(t�) implies fθ,θ1,γ(t�) > fθ,θ1,γ(t�) which is absurd. hence θ = θ1. as a result four self mappings have unique common fixd point in x. � this result can be justified by the following example. int. j. anal. appl. (2022), 20:12 11 example 3.1. by considering x = [−2,3] and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. the mappings a,b,s,t : x→ x are defined as a(x)= b(x)=  −2 if x ∈ [−2,0) e−x 2 , if x ∈ [0,3] and (3.12) s(x)= t(x)=   x3 4 if x ∈ [−2,0) e−3x, if x ∈ [0,3]. (3.13) from (3.12) and (3.13) a(x)= {−2}∪ [e−9,1],s(x)= [−2,0)∪ [e−9,1] implies a(x)⊆ t(x),b(x)⊆ s(x). clearly -2 and 0 are coincidence points for the mappings a, s. at x =−2,s(−2)= a(−2) as(−2)= a(−2)=−2, (3.14) sa(−2)= ss(−2)=−2. (3.15) from (3.14) and (3.15) as(−2)= sa(−2). at x =0,a(0)= s(0)=1 and as(0)= a(1)= e−1, (3.16) sa(0)= s(1)= e−3. (3.17) from (3.16) and (3.17) as(0) 6= sa(0). (3.18) from (3.18) the pair (a, s) is not weakly compatible but owc. considering a sequence < xn >= √ 2 n ∀ n ≥ 1 then axn = a( √ 2 n )= e−( √ 2 n )2 → 1, (3.19) sxn = s( √ 2 n )= e−3( √ 2 n ) → 1 (3.20) as n →∞. from (3.19) and (3.20) lim n→∞ axn = lim n→∞ sxn. (3.21) 12 int. j. anal. appl. (2022), 20:12 from (3.21) l{a,s} 6= φ =⇒ ∃ < yn > such that (yn)=−2+ √ 3 n ∀n ≥ 1 such that ayn = a(−2+ √ 3 n )=−2→−2 (3.22) and syn = s(−2+ √ 3 n )= (−2+ √ 3 n )3 4 →−2 (3.23) as n →∞ and asyn = a( (−2+ √ 3 n )3 4 )=−2→−2, (3.24) sayn = s(−2)= (−2)3 4 =−2→−2 (3.25) as n →∞. from (3.22) , (3.23) . (3.24), (3.25) lim n→∞ asyn = lim n→∞ ayn and lim n→∞ sayn = lim n→∞ syn (3.26) lim n→∞ asyn = a(−2) and lim n→∞ sayn = s(−2). (3.27) from (3.26) and (3.27) the joint pairs (a, s), (b, t) are non-compatible pseudo reciprocally continuous (w,r.t. conditionally sequentially absorbing) and conditionally sequential absorbing having unique common fixed point at x = −2. further these joint pairs (a, s), (b, t) are not weakly compatible and satisfy all the conditions of theorem(3.1). now we present another generalization of theorem(2.1) on an incomplete 2-menger space. theorem 3.2. let a, b, s and t be mappings on a 2-menger space (x,f,t�) to itself satisfying (a) the pairs (a, s) and (b, t) non-compatible pseudo reciprocally continuous (w,r.t. csa ) and conditionally sequential absorbing (b) fax,by,γ(t�)≥ r(fsx,ty,γ(t�)) whenever x, y in x and t� > 0 for some continuous self-map on [0, 1] such that r(t�) > t� for each o < t� < 1. then a, b, s and t have unique common fixed point in x. moreover all these mappings are discontinuous at their fixed point. int. j. anal. appl. (2022), 20:12 13 proof. since the pairs ( a, s) are non-compatible implies some sequence < xn > with lim n→∞ axn = lim n→∞ sxn = θ(say) (3.28) for some θ ∈ x =⇒ lim n→∞ fasyn,asyn,γ(β) not exist or lim n→∞ fsyn,sayn,γ(β) 6=1. since the pair (a, s) is conditionally sequential absorbing from (3.28) l{a,s} 6= φ =⇒ there exists sequence < yn > such that lim n→∞ ayn = lim n→∞ syn = θ(say) =⇒ lim n→∞ fayn,asyn,γ(t�)=1 and lim n→∞ fsyn,sayn,γ(t�)=1 ∀γ ∈ x,t� > 0. also the pair (a, s) satisfy pseudo reciprocally continuous means whenever lim n→∞ ayn = lim n→∞ syn = θ(say) (3.29) =⇒ lim n→∞ fayn,asyn,γ(t�)=1 and lim n→∞ fsyn,sayn,γ(t�)=1 (3.30) such that lim n→∞ asyn = a(θ) and lim n→∞ sayn = s(θ). (3.31) using from (3.29) , (3.31) in (3.30) we get aθ = sθ = θ. (3.32) since the pair ( b, t) is non-compatible implies some sequence (xn) with lim n→∞ bxn = lim n→∞ txn = η(say) (3.33) for some η ∈ x =⇒ lim n→∞ fbtxn,tbxn,γ(t�) does not exist or lim n→∞ fbtxn,tbxn,γ(t�) 6=1. since the pair (b, t) is conditionally sequential absorbing from (3.33) l{b,t} 6= φ =⇒ there exists sequence < yn > such way that lim n→∞ byn = lim n→∞ tyn = w(say) =⇒ lim n→∞ fbyn,btyn,γ(t�)=1 and lim n→∞ ftyn,tbyn,γ(t�)=1 14 int. j. anal. appl. (2022), 20:12 ∀γ ∈ x,β > 0. also the pair (b, t) is pseudo reciprocally continuous implies whenever lim n→∞ byn = lim n→∞ styn = w(say) (3.34) =⇒ lim n→∞ fbyn,btyn,γ(t�)=1 and lim n→∞ ftyn,tbyn,γ(t�)=1 (3.35) such that lim n→∞ btyn = bw and lim n→∞ tbyn = tw. (3.36) using (3.34) and (3.26) in (3.25) we get bw = tw = w. (3.37) claim w = θ. on contrary if w 6= θ put x = θ and y = w in (3.2) faθ,bw,γ(t�)≥ r(fsθ,tw,γ(t�)) =⇒ fθ,w,γ(t�)≥ r(fθ,w,γ(t)) > fθ,w,γ(t�). from (3.32) and (3.37) =⇒ fθ,w,γ(t�) > fθ,w,γ(t�) which is contradiction hence θ = w. uniqueness follows as in theorem(3.1). suppose a is continuous at w from (3.29) then lim n→∞ syn = θ =⇒ lim n→∞ asyn = aθ(say). from (3.31) lim n→∞ sayn = sθ but aθ = sθ = θ =⇒ lim n→∞ asyn = lim n→∞ sayn (3.38) (3.38) demonstrates that (a, s) is compatible pair, despite the fact that it is non-compatible. therefore a should be discontinuous at w. similarly the other mappings are also discontinuous at w. � to justify our theorem, we now present a supporting example. int. j. anal. appl. (2022), 20:12 15 example 3.2. by considering x=(−2,22) and d is usual distance on x then by (2.1) (x,f,t�) forms 2-menger space. the mappings a,b,s,t : x→ x are defined as a(x)= b(x)=   x2 2 if x ∈ (−2,2] 3, if x ∈ (2,22) and (3.39) s(x)= t(x)=  2 if x ∈ (−2,2] logx, if x ∈ (2,22) (3.40) from (3.39),(3.40) e3 and 2 are intersecting points for the mappings a, s. atx =2,a(2)= s(2)=2 and as(2)= a(2)=2= s(2)= sa(2). at x = e3, s(e3)= a(e3)=3, as(e3)= a(3)=3. (3.41) sa(e3)= s(3)= log3. (3.42) (3.41) and (3.42) as(e2) 6= sa(e2). (3.43) (3.43) shows that this pair (a, s) is not weakly compatible. considering a sequence < xn >= e3 + 3 n ∀n ≥ 1. then axn = a(e 2 + 3 n )=3→ 3, sxn = s(e 3 + 3 n )= log(e3 + 3 n )→ 3 as n →∞ and asxn = a(log(e 3 + 3 n ))=3→ 3, (3.44) saxn = s(3)= log3→ log3 (3.45) as n →∞. (3.44),(3.45) demonstrate that the pair (a, s) is non-compatible implies there exists another sequence < yn >=2− √ 2 n ∀n ≥ 1 such that ayn = a(2− √ 2 n )= (2− √ 2 n )2 2 → 2, (3.46) 16 int. j. anal. appl. (2022), 20:12 syn = s(2− √ 2 n )=2→ 2 (3.47) as n →∞ and asyn = a((2)=2→ 2, (3.48) sayn = s(( (2− √ 2 n )2 2 )=2→ 2 (3.49) as n →∞. from (3.46),(3.48) lim n→∞ asyn = lim n→∞ ayn. (3.50) from (3.47),(3.49) lim n→∞ sayn = lim n→∞ syn. (3.51) further lim n→∞ asyn = a(2), (3.52) lim n→∞ sayn = s(2). (3.53) from (3.50),(3.51),(3.52) and (3.53) the joint pairs (a, s), (b, t) are non-compatible pseudo reciprocally continuous (w,r.t. conditionally sequential absorbing) and conditionally sequential absorbing, having unique common fixed point at x = 2. further the maps a, s, b and t have discontinuity at x = 2. moreover a(x),s(x),b(x)andt(x) are not closed sub spaces and also the pairs of (a, s), (b, t) are not weakly compatible and satisfy all the conditions of theorem (3.2). 4. conclusion in this paper we improve theorem (2.1) in two ways: (i) in theorem (3.1) the concepts of pseudo reciprocally continuous and conditionally sequential absorbing mapping are being used in place of weakly compatible mappings in the first pair and owc mappings in place of weakly compatible mappings in the second pair. (ii) in theorem (3.2) the concepts of non-compatible pseudo reciprocally continuous and conditionally sequential absorbing mappings are being used in place of weakly compatible mappings and further the completeness of x is being removed. moreover all the mappings are discontinuous at their fixed point. further these two results are justified with appropriate examples. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2022), 20:12 17 references [1] a. mujahid, b.e. rhodes, common fixed point theorems for occasionally weakly compatible mappings satisfying a generalized contractive condition, math. commun. 13 (2008), 295-301. [2] h.m. abu-donia, h.a. atia, o.m.a. khater, fixed point theorem by using ψ–contraction and (ψ,φ)–contraction in probabilistic 2–metric spaces, alexandria eng. j. 59 (2020), 1239–1242. https://doi.org/10.1016/j.aej. 2020.02.009. [3] h. bouhadjera, c. godet-thobie, common fixed point theorems for pairs of subcompatible maps, arxiv:0906.3159 [math]. (2011). http://arxiv.org/abs/0906.3159. [4] fréchet, m. maurice, sur quelques points du calcul fonctionnel, rend. circ. mat. palermo 22 (1906), 1-72. [5] s. gähler, 2-metrische räume und ihre topologische struktur, math. nachr. 26 (1963), 115–148. https://doi. org/10.1002/mana.19630260109. [6] i. golet, fixed point theorems for multivalued mapping in probabilistic 2-metric spaces, an. st. univ. ovidius constanta 3 (1995), 44-51. [7] v.k. gupta, a. jain, r. kumar, common fixed point theorem in probabilistic 2-metric space by weak compatibility, int. j. theor. appl. sci. 11 (2019), 09-12. [8] s. jafari, m. shams, fixed point theorems for ψ-contraction mappings in probabilistic generalized menger space, indian j. pure appl. math. 51 (2020), 519–532. https://doi.org/10.1007/s13226-020-0414-8. [9] g. jungck, b. e. rhodes, some fixed point theorems for compatible maps, int. j math. math. sci. 16 (1993), 417-428. [10] u. mishra, a.s. ranadive, d. gopal, fixed point theorems via absorbing maps, thai j. math. 6 (2012), 49-60. [11] r.p. pant, s. padaliya, reciprocal continuity and fixed point, jñānābha 29 (1999), 137-143. [12] d.k. patel, p. kumam, d. gopal, some discussion on the existence of common fixed points for a pair of maps, fixed point theory appl. 2013 (2013), 187. https://doi.org/10.1186/1687-1812-2013-187. [13] k. satyanna, v. srinivas, fixed point theorem using semi compatible and sub sequentially continuous mappings in menger space, j. math. comput. sci. 10 (2020), 2503-2515. https://doi.org/10.28919/jmcs/4953. [14] v. srinivas, k. satyanna, some results by using clr’s-property in probabilistic 2-metric space, int. j. anal. appl. 19 (2021), 904-914. https://doi.org/10.28924/2291-8639-19-2021-904. https://doi.org/10.1016/j.aej.2020.02.009 https://doi.org/10.1016/j.aej.2020.02.009 http://arxiv.org/abs/0906.3159 https://doi.org/10.1002/mana.19630260109 https://doi.org/10.1002/mana.19630260109 https://doi.org/10.1007/s13226-020-0414-8 https://doi.org/10.1186/1687-1812-2013-187 https://doi.org/10.28919/jmcs/4953 https://doi.org/10.28924/2291-8639-19-2021-904 1. introduction 2. preliminaries 3. main results 4. conclusion references international journal of analysis and applications volume 18, number 4 (2020), 594-613 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-594 received january 24th, 2020; accepted april 7th, 2020; published may 13th, 2020. 2010 mathematics subject classification. 90b50. key words and phrases. equity; airline; brand image; loyalty. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 594 analyzing the case of vietjet air to assess the vietnamese customers’ airline brand equity: an empirical research study nhu-ty nguyen1,2,* 1school of business, international university, vietnam 2vietnam national university hcmc; quarter 6, linh trung ward, thu duc district, hcmc, vietnam *corresponding author: nhutynguyen@gmail.com; abstract. from the recent marketing incidents of a typical airline company, this research study would like to analyze the sources and consequences of airline's consumer brand equity by using vietjet air vietnam as an example. to this end, we developed an empirical study based on the chen & tseng [5] brand equity airline model in taiwan, using structural equation modeling (sem) to investigate the interrelationship between the dimensions of the components of brand equity in the airline industry and how they directly affect brand equity. with 307 valid respondents, the questionnaire was designed to include 3 age groups: 18-25, 26-35, 36-50 and older than 50 years. the participants from some universities and some firms in ho chi minh city involved in conducting the survey. the findings suggest that brand equity factors in their interrelationships have a positive effect. in addition, brand image is a direct factor that has the most influence on airline brand equity, following perceived dimensions of quality and brand loyalty. this research also has relevant implications for vietjet marketing managers, who should reinforce their brand equity in the future to attract more customers. 1. introduction vietjet aviation joint stock company (vietjet air or vietjet) known as an airline with cheap faresin vietnam, it was established as the first privately owned airline. they started offering domestic service in vietnam since december 2011 such as civil domestic flights. in https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-594 int. j. anal. appl. 18 (4) (2020) 595 which, sovico holdings, hdbank, other organizational investors and individual stakeholders are the owners of vietjet air. by the end of 2018, vietjet operated 105 routes including 39 domestic and 66 international routes. vietjet was using ticket discount strategy to attract customers andexpand their marketing system. first, they sold tickets with 0 vnd or low-priced tickets for empty seats, those tickets weresold before a few weeks to improve the flight occupancy rate. this strategy will create word-of-mouth marketing effects through advertising and customer experiences. besides, low-priced tickets were also attracting potential customers who had the first time to experience airplane and customers who was using fscs (low-priced tickets of vietjet may be cheaper than fscs’ ticket price up to 60%). in short, vietjet is the low-cost airline; their strategies focus on customers from middle-income class or first-time experience flying. they are usually found young and dynamic, and especially expert in utilizing technological equipment such as smartphones, laptops, etc. vietjet’s report presentation on september 2018 shown that customers from 20 to 34 years oldaccounted for 25.9%, they are main consumers of vietjet’s tickets. after the achievement of the 2018 asian u23 final in changzhou city, china, vietjet air organized a bikini performance on vj7268 flight to welcome coach park hang-seo and vietnam u23 team to return to the country on january 28th2018. however, this activity received a fierce protest from fans all over the country because they thought that bikini models were too offensive. most people think that sexiness and glamour need to be put in the right context with the right level. especially when using a sexy idea, which is considered a double-edged sword, must be applied very carefully. with images contrary to the acceptance of vietnamese people from the use of bikini models in anoffensive way, vietjet's brand equity have been negatively affected, especially is brand image. consumers think they will be loyal to the airline based on strong brand image ([39]; [25]; [27]; [41]). in vietjet’s prospectus, they show that they have many unique risks, one of them is risk of accidents and incidents. like other airlines, this kind of risk can make vietjet bear some potential losses such as property damage, unexploited aircraft, and affection of the company’s image. in addtion, any such event will significantly increase the costs associated with passenger int. j. anal. appl. 18 (4) (2020) 596 compensation, repair and replacement of damaged. moreover, any aircraft accident, even if the passenger is fully insured, will greatly affect the positive identification of brand equity of the customers. accordingly, this research is to investigate an vietjet air’s customer-based brand equity including four dimensions: brand awareness, brand loyalty, brand image, and perceived quality. furthermore, this research also tries to discover relationships among those dimensions and their direct effects on overall brand equity. to investigate this study, some goals need to be achieved as follow: identify the brand equity dimensions of vietjet from the customer’s perspective evaluate the inter-relationships between those dimensions of brand equity significance of dimensions have direct effects on overall brand equity suggest recommendations to improve vietjet brand equity in order to reach those research objectives, research questions should be asked: what is the view of brand equity from customer’s perspective? what is the influence among brand awareness, brand loyalty, brand image, and perceived quality? what is the most important dimension affecting directly on overall brand equity? are there any aspects that vietjet air needs to be improved brand equity? 2. literature review 2.1. airline classification leick and wensveen [13] showed four basic types of airline today. the airlines dependence on what kind of strategy they are aiming for, they will choose for themselves the most suitable type of aviation. these include: full – service carriers (fscs) low – cost carriers (lccs) regional carriers (rcs) charter carriers (ccs) in vietnam’s aviation domestic market, there are two popular airlines – vietjet air and vietnam airlines – they account for almost the entire domestic market share. each airline presents for two kind of different business model, vietjet is the lccs model and vietnam int. j. anal. appl. 18 (4) (2020) 597 airlines is the fscs model. thus, the author just concentrates on giving clearly about these two type models, especially vietjet with low-cost carriers. full – service carriers: in spite of the fact that airline models are getting more and more modern, the full-service carrier with traditional network still delivers regular flight-service to many destinations and offers a number of ancillary services such as additional drinks, entertainment on flights, lounges at airport, and designated seats gillen [8]; huschelrath and muller [9]. in actuality, the full-service carrier sets the goal to become the air transport contributor with one-way for the communities that they are serving like supplying vacation by airplane to any destination for travelers (usually having cooperated with international airlines). low – cost carriers: focusing on no-frills and point to point service is a recently replacement model for fscs by low-cost carrier model. like the full-service carrier model, not all low-cost carriers are the same. nevertheless, the most successful of this model has been using as a popular cost reduction strategy. some of the cost factors from production functions are removed by this strategy and it also lessens the level of the remaining costs kenneth [2; 10]. similarly, low-cost airlines could offer more limited services and from that, they can charge separately for the extra services. 2.2 the dimensions of brand equity & research model according to chen & tseng [5] model in customer-based airline brand equity, this section provides a description of this model made up of four dimensions: brand awareness, brand image, perceived quality, and brand loyalty. a) brand awareness brand awareness relates to brand recognition and recall ability of consumers to the brand ([6]; [20]; [42]). brand recognition requires consumers to distinguish brands correctly when looking or hearing about them. the ability to recall brands requires consumers to recall the brand from their memory ([19]; [23]). in fact, the level of brand awareness is certain associations in consumers' minds such as brand names, logos, or images that are visible to the eyes ([21]; [22]) many studies suggest that the perceived quality will increase with the increase of brand awareness ([28]; [38]; [44]; [49]). nguyen [16] proposed that the quality of consumers' assessment is higher when brand awareness is in a high-level. the following hypothesis that based on those is proposed in this study: int. j. anal. appl. 18 (4) (2020) 598 h1: brand awareness has a positive effect on perceived quality h2: brand awareness has a positive effect on brand image b) perceived quality zeithaml (1998) expounded the perceived quality is the subjective perception of consumers about outstanding advantages, not the quality of the product.for the service industry, the perceived quality is consumers’ impression to the organization's performance [31]. motameni & shahrokhi [15] and yoo et al. [48] have noticed that brand equity and perceived quality have a positive relationship with each other. in brand equity measurement, aaker [1] started that perceived quality is one of the core components. the perceived quality is considered as the main premise of brand loyalty [3]. influence on brand image by perceived quality was presented by nguyen & nguyen [26]; nguyen & tran [24] and chen & tseng [5]. the following hypothesis that based on those is proposed in this study: h3: perceived quality has a positive effect on brand image h4: perceived quality has a positive effect on brand loyalty h5: perceived quality has a positive effect on brand equity c) brand image according to keller [11] brand images were defined as consumer perceptions reflected by brand associations in their minds. some associations are clearly show the abstract conceptual nature of brand image such as a client's association with business image through prestige, trust, morality or corporate social responsibility is invisible evaluations of brand image [47]. the brand image of airlines is an important factor to assess the overall business as well as the services of the business [30]. in addition, it affects consumer choice of services ([45]; [17]; [31]).these demonstrates are better methods to improve the brand image of airline, the more likely customers will go on with using its services in the future and may endow with attraction for others to choose this airline. when building brand equity, brand image often appears as a sufficient step instead of brand awareness ([14]; [35]; [34]; [36]; [37]; [33]). moreover, pham et al. [14] proved that brand equity is controlled by brand image. brand image has been detected to have a supportive impact on brand loyalty ([34]; [37]). the following hypothesis that based on those is proposed in this study: int. j. anal. appl. 18 (4) (2020) 599 h6: brand image has a positive effect on brand loyalty h7: brand image has a positive effect on brand equity d)brand loyalty bloemer and kasper [4] defined brand loyalty as “a situation which reflects how likely a customer will be to switch to another brand, especially when that brand makes a change, either in price or in product features”. brand loyalty refers to the tendency of consumers' loyalty to the brand and it is expressed by the intention of choosing to buy the product brand primarily. this proves that brand loyalty results from consumer satisfaction in their perception of a highquality product ([48]). consumer’s satisfaction with an airline service will make them continue to choose this airline for the next time and most likely recommend it to others [31]. unlike non-loyal customers, loyal customers are less likely to switch brands that they trust; they also repurchase and stick more on their favorite brands ([32]). therefore, brand equity will be affected directly and positively by brand loyalty ([18]).the following hypothesis that based on those is proposed in this study: h8: brand loyalty has a positive effect on brand equity 2.3 research model figure 1: research model of branding process. int. j. anal. appl. 18 (4) (2020) 600 3. research methodology this study will use questionnaires from the survey as a tool. the quantitative method will be an approach to research and information gathering. survey research was selected because it provided a fast, effective and accurate means to evaluate population information. a) sampling target population: people who have experienced vietjet air’s flight. sampling method: by two main methods: online survey (email, facebook, etc.) and offline survey sampling pool and criteria: being at least 18 years old and having experience of vietjet’s flight; studying and working at ho chi minh city and sample size: the sample size of this research is supposed to be around 300 respondents. b) questionnaire structure mr. nguyen tran tuan has conducted a research named “researching brand value components of vietjet airline: approached from the perspective of vietnamese consumers” in 2017. the author decided to use nguyen’s measurement scale because of a similarity among the elements in the research models of both.there are three main parts of questionnaire: part 1: general question that is concerned about the experience of using vietjet’ flight part 2: further detailed of survey. a likert-scale of 5 points is applied to evaluate the question for both dependent and independent variables. “the likert scale, developed by rensis likert, is the most frequently used variation of the summated rating scale”. the likert range in this study would be: 1 = strongly disagree 2 = disagree 3 = neutral 4 = agree 5 = strongly agree part 3: respondents’ demographics about socio-demographic and economic variables int. j. anal. appl. 18 (4) (2020) 601 4. research findings 4.1 response rate the number 355 surveys were collected online (292 respondents) as well as offline (63 respondents). there are 17.7% of offline and 82.3% of online survey were distributed through email, facebook, zalo. each questionnaire was answered by an experienced vietjet’s flight service customers. finally, due to some uncommitted and missing responses, only 307 surveys are qualified and valid among 355 returned surveys. consequently, the response rate is around 86%. based on these valid responses, this research will also be carried out. figure 2: response rate 4.2 descriptive statistics descriptive statistics are used to summarize and describe the basic features of the data this study. it creates the general observation about data collected. in this part, the sample size of 307 respondents is taken to make the overall descriptive observation about the impact of brand awareness, brand association, perceived quality and brand loyalty on consumer-based brand equity. the table of descriptive statistics includes maximum, minimum, mean and standard deviation of 27 items. 86% 14% valid invalid int. j. anal. appl. 18 (4) (2020) 602 table 1: descriptive statistics factor maximum mean minimum mean brand awareness (aw) aw3 (3.93) aw1 (3.77) perceived quality (pq) pq9 (3.57) pq5 (3.13) brand image (bi) bi5 (3.63) bi4 (3.42) brand loyalty (lo) lo4 (3.57) lo2, lo3, lo5 (3.44) brand equity (be) be3 (3.50) be2 (3.46) 4.3 reliability test according to george and mallery [7], the cronbach’s alpha indexes are represented as the following rule: excellent: alpha > 0.9 good: alpha >0.8 acceptable: alpha > 0.7 questionable: alpha > 0.6 poor: alpha > 0.5 unacceptable: alpha < 0.5 in addition, if the "cronbach's alpha if item deleted" is larger than the cronbach's alpha overall and the "corrected item-total correlation" is less than 0.3, the item will be removed from the list of variables. in this part, only two components – perceived quality and brand loyalty – have the elimination items due to the “corrected item-total correlation” is less than 0.3. they include pq8 (0.289), pq9 (0.136), and lo5 (0.108). others are meet with the conditional requirements above. after eliminating those unsatisfied items, the reliability statistics are showed in the table below. table 2: reliability statistics factor cronbach's alpha n of items brand awareness 0.864 4 perceived quality 0.869 7 brand image 0.778 5 brand loyalty 0.811 4 brand equity 0.854 4 int. j. anal. appl. 18 (4) (2020) 603 4.4 exploratory factor analysis (efa) the final result of the exploratory factor analysis process is presented as below after removing the unsatisfactory items. table 3: kmo and barlett’s test of final round efa kaiser-meyer-olkin measure of sampling adequacy. .800 bartlett's test of sphericity approx. chi-square 2306.571 df 171 sig. .000 table 4: total variance explained of final round – efa factor initial eigenvalues extraction sums of squared loadings total cumulative % total cumulative % 1 4.472 23.537 4.038 21.253 2 2.606 37.252 2.207 32.871 3 2.287 49.287 1.861 42.666 4 1.871 59.133 1.409 50.079 5 1.658 67.858 1.204 56.414 int. j. anal. appl. 18 (4) (2020) 604 table 5: pattern matrix of final round – efa factor 1 2 3 4 5 pq2 767 pq6 745 pq4 713 pq5 701 pq1 678 aw4 795 aw3 784 aw1 782 aw2 774 be3 841 be4 763 be1 744 be2 734 bi1 770 bi3 756 bi2 682 lo2 746 lo3 725 lo1 679 extraction method: principal axis factoring. rotation method: promax with kaiser normalization. a. rotation converged in 5 iterations. according to the results, the value of 0.8 kmo (> 0.5) and the bartlett's significant test (sig.=.000 < 0.05) fulfill the condition required. moreover, 61.2% of “cumulative % of extraction sums of squared loadings” shows that these final factors account for 61.2% of the variance in data. variable loads ranging from 0.678 to 0.841 (greater than 0.5) are satisfactory. the criterion for the efa analysis is thus guaranteed by these 19 observed variables. int. j. anal. appl. 18 (4) (2020) 605 4.5 confirmatory factor analysis figure 3: standardized measurement modeling the cfa results show that cmin / df= 1.227 (< 3), gfi=0.947 (> 0.9), tli=0.982 (> 0.9), cfi=0.985 (> 0.9), agfi= 0.928 (> 0.8), rmsea=0.027 (< 0.08) and pclose=0.999 (> 0.05). consequently, this result, which is considered to be satisfactory, shows that the measurement model is fit with survey data. moreover, standardized regression weights and regression weights show that all weights are higher than 0.5 and have statistically significant due to 0.000 p-value (<0.05). the measurement model therefore meets the standard of convergent validity based on the acceptable result of standardized regression weights. in addition, for each factor, the composite reliability (cr) and average variance extracted (ave) are calculated to assess the precise fit of the measurement model. microsoft excel 2019 programming has been utilized in this progression to physically compute cr and ave. int. j. anal. appl. 18 (4) (2020) 606 table 6: composite reliability (cr) and average variance extracted (ave) factor cr ave be 0.855 0.596 aw 0.864 0.614 pq 0.844 0.52 bi 0.781 0.543 lo 0.759 0.512 the measurement model is significant if the composite reliability (cr) of all factors is higher than 0.7 and the average extracted variance (ave) is higher than 0.5. according to table 6, all composite reliability and average extracted variance indexes meet the conditions. therefore; reliability and convergent validity are sufficient for all factors in the measurement model. figure 3 shows the final outcome of the cfa. 4.6 structural equation modeling (sem) structural equation modeling (sem) was used to test in a research model the hypothesized causal relationships. the criteria for examining whether sem fits data are similar to cfa's measuring model criteria: chi-square / df < 3, cfi > 0.9, gfi > 0.9, tli > 0.9, agfi > 0.8, pclose > 0.05 and rmsea < 0.08. the result is presented as below after conducting sem: figure 4: structural equation modeling (sem) int. j. anal. appl. 18 (4) (2020) 607 from figure 4, all measurement values are met with the chi-square / df= 1.225 (< 2), gfi=0.946 ((> 0.9), tli=0.982 (> 0.9),agfi=0.929 (> 0.8), cfi=0.985 (> 0.9) and rmsea=0.027 (< 0.08) criteria. this result, reaches the highest level of satisfaction with the standard model fit criteria. table 7: conclusion of research hypothesis hypothesis estimate p-value hypothesis test h1 brand awareness has a positive effect on perceived quality 0.170 .012 accepted h2 brand awareness has a positive effect on brand image 0.192 .006 accepted h3 perceived quality has a positive effect on brand image 0.235 *** accepted h4 brand image has a positive effect on brand loyalty 0.163 .034 accepted h5 perceived quality has a positive effect on brand loyalty 0.200 .008 accepted h6 brand image has a positive effect on brand equity 0.185 .010 accepted h7 perceived quality has a positive effect on brand equity 0.171 .015 accepted h8 brand loyalty has a positive effect on brand equity 0.147 .042 accepted 5. discussion, implication, and recommendations 5.1 discussion based on the research results, there exists a positive relationship among the components of brand equity including brand awareness, perceived quality, brand image, and brand loyalty, which corresponding to h1, h2, h3, h4, h5 hypotheses. thus, for factors that are closely related to each other, when we improve the independence factor, then at the same time we will improve the dependency factor. for example, when we propose good solutions to int. j. anal. appl. 18 (4) (2020) 608 improve brand awareness, then vietjet's brand image and perceived quality can also be improved in addition to the individual solutions of each one. specifically: for brand awareness, it is significant because it helps customers make decision in their choices of certain airlines. in other words, if a certain airline is not known to customers when they are searching for a certain airline, then choosing this airline is very unlikely for them chen & tseng (2010). this is consistent with konecnik and gartner [12], by which brand equity could not be created without brand awareness; even it does not have a direct effect on brand equity. vietjet airlines is therefore advised to carefully examine their brand communication strategy in order to help maintain customer recognition of the brand name of an airline compared to its competitors. for perceived quality, this result conforms to the research model of park, robertson and wu [31]; park [20]; yoo & donthu [48]; nguyen & tran [20]; chen and tseng [5]. this means consumers who appreciate the high-quality service provided by vietjet air will love vietjet air rather than other airlines and intend to choose it.so, marketing managers should focus their efforts primarily on perceived quality that will positively contribute to customers’ brand loyalty. according to wang, et al. [46], passenger recognition of airline companies’ quality of service depend soninternal flight cabin decoration and cleanliness, comfortable seats, ticket prices, schedules, on-time flights, and flight safety. thus, the ways to improve perceived quality are: vietjet airline needs to provide accurate operating procedures and professionalism throughout the entire service process to enable passengers to enjoy comfortable and convenient flight service quality. for brand image, by the similarity between the results of this study and chen & tseng [5], it shows that consumers are impressed and relate easily from their minds to the brand image of vietjet air, thereby increasing their popularity and affecting the choice of vietjet air instead of the other airlines, thus increasing vietjet air's brand equity. nguyen [18] recommended that brand image is very important, so we need to raise awareness of advertising, brand awareness and brand association through marketing activities. for brand loyalty, the findings follow the research models of chen & tseng [5]. this means consumers will have a tendency in the future to choose vietjet air more if they are loyal to vietjet air service. there are some ways to build brand loyalty of customers: making int. j. anal. appl. 18 (4) (2020) 609 impressive to customers through in-flight reward programs for food and beverage, or in holiday seasons, vietjet could offer rewards and flash deals to catch customers’ attention. as a result, they keep customers loyal. based on the results of the study, we can conclude that the incident with vietjet flights in 2018 will not affect the continued use of this airline's flight service if vietjet can guarantee the satisfaction of brand equity elements including brand awareness, brand image, perceived quality, and brand loyalty. besides, the solutions for each component will be discussed further. 5.2 recommendation for limitation regardless of the huge commitment made by the present investigation in the brand equity area, trying to duplicate this examination for development, a few components of the present investigation ought to be considered. first, the specific area where the research was conducted should be pointed out as the main limitation of this research. the findings rely only on data collected with 355vietnamese respondents in ho chi minh city so that the conclusions can not be generalized broadly for this reason. in fact, vietjet air not only has domestic flights but also international flights. thus, the generality will be higher if the survey is on foreign consumers.in addition, most respondents are between the ages of 18 25 and are currently university students. so as to sum up the goal results, future research ought to differentiate the objective populace with various statistic foundations. besides, this investigation centers exclusively around quantitative technique. future research would in this manner be prescribed to join both quantitative and subjective strategies so as to investigate all parts of research. next, the research focused mainly on the three-dimensional direct effects on brand equity. from the customer's point of view, there are some factors that affect brand equity indirectly. the nation of birthplace, for instance, demonstrates the positive and huge effect through its measurements on consumer-based brand equity ([20]; [26] [43]) are different variables influencing brand equity. in this manner, so as to improve the proposed brand value model, the following exploration would address these fascinating elements of brand equity. finally, the research topic focuses only on researching brand equity components approaching customer-based brand equity of vietjet airline only. that’s why the next research int. j. anal. appl. 18 (4) (2020) 610 direction is to expand to bigger section, vietnam aviation industry. from that, people can observe correctly the overall needs and demands of consumers in flight service. acknowledgements: the author would like to thank mr. le nhat duy from school of business, international university – vietnam national university, hcmc for his editorial assistance. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.a. aaker, measuring brand equity across products and markets, california manage. rev. 38 (1996) 102–120. [2] kenneth button, low-cost airlines:, transport. j. 51 (2012) 197. [3] g. biedenbach, a. marell, the impact of customer experience on brand equity in a business-tobusiness services setting, j brand manage. 17 (2010) 446–458. [4] j.m.m. bloemer, h.d.p. kasper, the complex relationship between consumer satisfaction and brand loyalty, j. econ. psychol. 16 (1995) 311–329. [5] c.f. chen, w.s. tseng, exploring customer-based airline brand equity: evidence from taiwan, transport. j. 49(2010), 24-34. [6] w. chia-nan, n. n. ty, forecasting the manpower requirement in vietnamese tertiary institutions. asian j. empiric. res. 3(5)(2013), 563-575. [7] d. george, p. mallery, spss for windows step by step: answers to selected exercises. a simple guide and reference, 1461-1470, 2003. [8] d. gillen, airline business models and networks: regulation, competition and evolution in aviation markets, rev. network econ. 5(2006), 366-385. [9] k. huschelrath, k. muller, low cost carriers and the evolution of the domestic u.s. airline industry, competit. regulat. network ind. 13 (2012), 133-159 [10] k. button, low-cost airline: a failed business model?, transport. j. 51(2012), 197-219. [11] k.l. keller, branding perspectives on social marketing, in na advances in consumer research volume 25, eds. joseph w. alba & j. wesley hutchinson, provo, ut: association for consumer research, pages: 299-302, 1998. int. j. anal. appl. 18 (4) (2020) 611 [12] m. konecnik, w.c. gartner, customer-based brand equity for a destination, ann. tour. res. 34(2007),400–421. [13] r. leick, j. wensveen, the airline business. the business of transportation, 65e99, 2014. [14] l.h.t. pham, n. t. nguyen, t. t. tran, on the factors affecting start-up intention of millennials in vietnam. int. j. adv. appl. sci. 6(1)(2019), 1-8. [15] r. motameni, m. shahrokhi, brand equity valuation: a global perspective, j. product brand manage. 7 (1998), 275-290. [16] n.t. nguyen, optimizing factors for accuracy of forecasting models in food processing industry: a context of cacao manufacturers in vietnam. ind. eng. manage. syst. 18(4)(2019), 808-824. [17] n.t. nguyen, performance evaluation in strategic alliances: a case of vietnamese construction industry. glob. j. flex. syst. manage. 21(1)(2020), 85-99. [18] n. t. nguyen, attitudes and repurchase intention of consumers towards functional foods in ho chi minh city, vietnam. int. j. anal. appl. 18(2) (2020), 212-242. [19] n. t. nguyen, l. x. t. nguyen, applying dea model to measure the efficiency of hospitality sector: the case of vietnam. int. j. anal. appl. 17(6)(2019), 994-1018. [20] n. t. nguyen, t. t. tran, mathematical development and evaluation of forecasting models for accuracy of inflation in developing countries: a case of vietnam. discrete dyn. nat. soc. 2015 (2015), art. id 858157. [21] n. t. nguyen, t. t. tran, facilitating an advanced product layout to prioritize hot lots in 450 mm wafer foundry in the semiconductor industry. int. j. adv. appl. sci. 3(6) (2016), 14-23. [22] n. t. nguyen, t. t. tran, a novel integration of dea, gm (1, 1) and neural network in strategic alliance for the indian electricity organizations. j. grey syst. 29(2)(2017), 80-101. [23] n. t. nguyen, t. t. tran, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese fertilizing industry. int. j. adv. appl. sci. 5(9)(2018), 73-81. [24] n. t. nguyen, t. t. tran, a study of the strategic alliance for vietnam domestic pharmaceutical industry: a dynamic integration of a hybrid dea and gm (1, 1) approach. j. grey syst. 30(4)(2018), 134-151. [25] n. t. nguyen, t. t. tran, optimizing mathematical parameters of grey system theory: an empirical forecasting case of vietnamese tourism. neural comput. appl. 31(2)(2019), 1075-1089. [26] n. t. nguyen, t. t. tran, raising opportunities in strategic alliance by evaluating efficiency of logistics companies in vietnam: a case of cat lai port. neural comput. appl. 31(11)(2019), 7963-7974. https://www.emerald.com/insight/search?q=reza%20motameni https://www.emerald.com/insight/search?q=manuchehr%20shahrokhi int. j. anal. appl. 18 (4) (2020) 612 [27] n. t. nguyen, t. t. tran, c. n. wang, management styles and organisational effectiveness in vietnam: a comparison in terms of management practices between state-owned and foreign enterprises. res. world econ. 6(1)(2015), 85-98. [28] n. t. nguyen, t. t. tran, c. n. wang, n. t. nguyen, optimization of strategic alliances by integrating dea and grey model. j. grey syst. 27(1)(2015), 38-56. [29] p. nguyen, n.t. nguyen, a two-stage study of grey system theory and dea in strategic alliance: an application in vietnamese steel industry. int. j. data envelop. anal. 7(2)(2019), 45-64. [30] j. park, passenger perceptions of service quality: korean and australian case studies, j. air transport manage. 13(2007),238 – 242. [31] j. park, r. robertson, c. wu, the effect of airline service quality on passengers’ behavioural intentions: a korean case study, j. air transport manage.10(2004),435 – 439. [32] t. thanh-tuyen, n. t. nguyen, determinants affecting vietnamese laborers’ decision to work in enterprises in taiwan. j. stock forex trad, 5(2)(173) (2016), 1000173. [33] t. t. tran, an investigation about factors that affecting satisfaction and efficiency in vietnamese tourism. int. j. adv. appl. sci. 5(12)(2018), 7-15. [34] t. t. tran, an empirical research on selecting the targeted suppliers and purchasing process of supermarket. int. j. adv. appl. sci. 4(4)(2017), 96-109. [35] t. t. tran, evaluating and forecasting performance using past data of an industry: an analysis of electronic manufacturing services industry. int. j. adv. appl. sci, 3(12)(2016), 5-20. [36] t. t. tran, factors affecting the purchase and repurchase intention smart-phones of vietnamese staff. int. j. adv. appl. sci. 5(3)(2018), 107-119. [37] t. t. tran, forecasting strategies and analyzing the numbers of incoming students: case in taiwanese vocational schools. int. j. adv. appl. sci. 4(9)(2017), 86-95. [38] m. wall, j. liefeld, l. a. heslop, impact of country of original cues on consumer judgment in multicue situation: a covariance analysis, j. acad. market. sci. 19(2)(1991), 105-113. [39] c. n. wang, n. t. nguyen, t. t. tran, an empirical study of customer satisfaction towards bank payment card service quality in ho chi minh banking branches. int. j. econ. finance, 6(5)(2014), 170181. [40] c. n. wang, n. t. nguyen, t. t. tran, the study of staff satisfaction in consulting center system-a case study of job consulting centers in ho chi minh city, vietnam. asian econ. financ. rev. 4(4)(2014), 472-491. int. j. anal. appl. 18 (4) (2020) 613 [41] c. n. wang, n. t. nguyen, t. t. tran, integrated dea models and grey system theory to evaluate past-to-future performance: a case of indian electricity industry. sci. world j. 2015(2015), 638710. [42] c. n. wang, n. t. nguyen, t. t. tran, b. b. huong, a study of the strategic alliance for ems industry: the application of a hybrid dea and gm (1, 1) approach. sci. world j. 2015(2015), 948793. [43] l. w. wang, t. t. tran, n. t. nguyen, analyzing factors to improve service quality of local specialties restaurants: a comparison with fast food restaurants in southern vietnam. asian econ. financ. rev. 4(11)(2014), 1592. [44] l. w. wang, t. t. tran, n. t. nguyen, an empirical study of hybrid dea and grey system theory on analyzing performance: a case from indian mining industry. j. appl. math. 2015 (2015), 395360. [45] l. w. wang, t. tran, n. t. nguyen, an analysis of manpower in vietnamese undergraduate educational system. int. j. econ. bus. finance, 1(1) (2013), 398-408. [46] r. wang, h. shu-li, y. h. lin, m.-l. tseng, evaluation of customer perceptions on airline service quality in uncertainty, procedia – soc. behavi. sci. 25(2011), 419–437. [47] f.e. webster, k.l. keller, a roadmap for branding in industrial markets, brand manage. 11(5)(2004), 388–402. [48] b. yoo, n. donthu, developing and validating a multidimensional customer-based brand equity scale, j. bus. res. 52(2001), 1-14. [49] v.a. zeithaml, consumer perceptions of price, quality, and value: a means-end model and synthesis of evidence, journal of marketing, 52(3) (1988), 2-22. international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 23-27 http://www.etamaths.com common fixed point theorems for g–contraction in c∗–algebra–valued metric spaces akbar zada1,∗, shahid saifullah1 and zhenhua ma2,3 abstract. in this paper we prove the common fixed point theorems for two mappings in complete c∗–valued metric space endowed with the graph g = (v,e), which satisfies g-contractive condition. also, we provide an example in support of our main result. 1. introduction and preliminaries the banach contraction principle [5] plays an important role in solving non linear problems. the banach contraction principle says that: if (x,d) be a complete metric space and f is a self mapping on x with the condition that there exists λ ∈ (0, 1) such that d(fx,fy) ≤ λd(x,y) for all x,y ∈ x, then f has a unique fixed point in x. since then a lot of publications are devoted to the study and solutions of many practical and theoretical problems by using this condition. due to a numerous applications of the fixed point theory, from the last few decades this theory is a central topic of research. in this theory one of the approach is the common fixed point theorems. the concept of the common fixed point theorems was investigated by jungck [1]. many authors studied the fixed and common fixed point theorems for different spaces, like in cone metric spaces [8], non-commutative banach spaces [22], fuzzy metric spaces [14] and uniform metric spaces [21]. for more information about this topic see ([1, 6, 7, 9, 17, 18, 23]). on the other hand the concept of c∗–algebra is well developed. here we recall some basic definitions, notations and results of c∗–algebra that may be found in [13]. a ∗-algebra a is a complex algebra with linear involution ∗ such that x∗∗ = x and (xy)∗ = y∗x∗, for any x,y ∈ a. if ∗-algebra together with complete sub multiplicative norm satisfying ||x∗|| = ||x|| for all x ∈ a, then ∗-algebra is said to be a banach ∗-algebra. a c∗–algebra is a banach ∗-algebra such that ||x∗x|| = ||x||2 for all x ∈ a. an element of a is called positive element, if a+ = {x∗ = x|x ∈ a} and σ(x) ⊂ r+, where σ(x) is the spectrum of an element x ∈ a, i.e., σ(x) = {λ ∈ c : λi −x is not invertible}. there is a natural partial ordering on a+ given by x � y if and only if x − y ∈ a+. in [12] z. ma et al., introduced the notion of c∗-algebra valued metric space and proved fixed point theorems for c∗-algebra valued contractive mapping. many researchers tried to obtain some fixed point theorems of banach type contraction endowed with the graph g, we recommend [2, 3, 4, 15, 16, 20]. recently, t. kamran et al., in [19] extended the results of ma et al., which was given in[12], by using c∗–valued metric spaces and g-contraction principles. now we give some definitions of graph theory which is found in any text on graph theory, for example [11]. following jachymski [10], let ∆ denote the diagonal of the x ×x in a metric space (x,d), and consider a directed graph g = (v (g),e(g)) = (v,e) the set in which v of its vertices and e of its edges, and ∆ ⊆ e. assume that g has no parallel edges. we may treat g as a weighted graph by assigning to each edge the distance between its vertices. in this paper we will continue to study common fixed points in the c∗–valued metric space endowed with the graph g under g–contractive condition. 2010 mathematics subject classification. 47h10, 47a56. key words and phrases. metric space; c∗–algebra valued metric spaces; g-contraction; common fixed point. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 23 24 zada, saifullah and ma definition 1.1. let x be a nonempty set, and the mapping d : x ×x →a endowed with the graph g = (v,e), if it satisfies the following conditions: (1) d(x,y) ≥ 0 for all x,y ∈ x and d(x,y) = 0 ⇔ x = y; (2) d(x,y) = d(y,x) for all x,y ∈ x; (3) d(x,y) ≤ d(x,z) + d(z,y) for all x,y,z ∈ x. then d is called a c∗–valued metric on x, and (x,d,a) is called c∗–valued metric space. definition 1.2. suppose that (x,d,a) is a c∗–valued metric space. let x ∈ (x,d,a) and {xn} be a sequence in x. the sequence {xn} is said to be convergent, if for any � > 0 there exists a positive integer n such that ||d(xn,x)|| ≤ � for all n ≥ n. the sequence {xn} is said to be cauchy, if for any � > 0 there exists a positive integer n such that ||d(xn,xm)|| ≤ � for all n,m ≥ n. if every cauchy sequence is convergent in (x,d,a), then (x,d,a) is said to be complete c∗–valued metric space. example 1.3. let x = r and a = m2(r). define d : x ×x →a such that d(x,y) = ( |x−y| 0 0 α|x−y| ) for all x,y ∈ r and α ≥ 0. it is essay to verify that d is a c∗–algebra valued metric space and (x,d,m2(r)) is a complete c∗– algebra valued metric space. definition 1.4. let (x,d,a) be a c∗–valued metric space. a mapping f : x → x is said to be a c∗–algebra–valued contraction mapping on x if there exists an a ∈a with ||a|| < 1 such that (1.1) d(fx,fy) ≤ a∗d(x,y)a, for all x, y ∈ x. theorem 1.5. [12] let (x,d,a) be a complete c∗–algebra-valued metric space and f satisfies (1.1), then f has a unique fixed point in x. property 1.6. [12] (1) for any {xn} ∈ x such that xn converges to x with (xn+1,xn) ∈ e for all n ≥ 1 there exists a subsequence {xnk} of {xn} such that (x,xnk ) ∈ e. (2) for any {fnx} ∈ x such that fnx converges to x ∈ x with (fn+1x,fnx) ∈ e there exists a subsequence {fnkx} and n0 ∈ n such that (x,fnkx) ∈ e for all k ≥ n0. 2. main result in this section, we prove common fixed point theorems for two mappings satisfying g–contractive condition in a complete c∗–valued metric space endowed with the graph g = (v,e). definition 2.1. let (x,d,a) be a c∗–valued metric space endowed with the graph g = (v,e). the mappings f,g : x → x are said to be c∗–valued g–contractive on x, if there exists an a ∈ a with ||a|| < 1 such that (2.1) d(fx,gy) ≤ a∗d(x,y)a, for all (x,y) ∈ e. theorem 2.2. let (x,d,a) is a complete c∗–valued metric space endowed with the graph g = (v,e). suppose that the mappings f,g : x → x are c∗–valued g–contractive mappings on x satisfying the property 1.6 (2) and the following conditions (1) if (x,y) ∈ e then (fx,gy) ∈ e, (2) there exists z0 ∈ x such that (z0,fz0), (z0,gz0) ∈ e. then f and g has a unique common fixed point in x. g–contraction in c∗–algebra–valued metric spaces 25 proof. let z1 ∈ x, and construct sequence {zn}∈ x, such that z2n+1 = fz2n, z2n+2 = gz2n+1, and (z2n−1,z2n) ∈ e for all n ∈ n. we have d(z2n+1,z2n+2) = d(gz2n+1,fz2n) ≤ a∗d(z2n+1,z2n)a ≤ (a∗)2d(z2n,z2n−1)(a)2 . . . ≤ (a∗)2n+1d(z1,z0)(a)2n+1. similarly, d(z2n+1,z2n) = d(fz2n,gz2n−1) ≤ a∗d(z2n,z2n−1)a . . . ≤ (a∗)2nd(z1,z0)(a)2n = (a∗)2nq(a)2n. let us denote d(z1,z0) by q ∈a. then for any n ∈ n d(zn+1,zn) = (a ∗)nd(z1,z0)(a) n = (a∗)nq(a)n, then for any q ∈ n and applying the triangular inequality (3) for the c∗–valued metric spaces, d(zn+q,zn) = d(zn+q,zn+q−1) + d(zn+q−1,zn+q−2) + · · · + d(zn+1,zn) ≤ n+q−1∑ j=n (a∗)jd(z1,z0)(a) j = n+q−1∑ j=n (a∗)jq(a)j = n+q−1∑ j=n (a∗)jq 1 2 q 1 2 (a)j = n+q−1∑ j=n (q 1 2 aj)∗(q 1 2 aj) = n+q−1∑ j=n |q 1 2 aj|2 ≤ n+q−1∑ j=n || |q 1 2 aj|2||.i = ||q 1 2 ||2 n+q−1∑ j=n ||a2j||. since ||a|| < 1, thus d(zn+q,zn) → 0 as n →∞. thus we conclude that the sequence {zn} is a cauchy sequence, with respect to a. using the completeness of x, there exists an element z0 ∈ x = v, such that zn → z0 as n →∞. 26 zada, saifullah and ma on the other hand, using the triangular inequality, we get d(z0,fz0) = d(z0,z2n+1) + d(z2n+1,fz0) = d(z0,z2n+1) + d(gz2n,fz0) ≤ d(z0,z2n+1) + a∗d(z2n,z0)a. thus if n → ∞, then d(z0,fz0) → 0 i.e. fz0 = z0. similarly we can prove that gz0 = z0. now we will show the uniqueness of common fixed points in x. for this we assume that there is another point z∗ ∈ x = v, such that(z0,z∗) ∈ e. consider d(z0,z ∗) = d(fz0,gz0) ≤ a∗d(z0,z∗)a. since ||a|| < 1, then the above inequality yields that 0 ≤ ||d(z0,z∗)|| ≤ ||a||2||d(z0,z∗)|| < ||d(z0,z∗)||. which is a contradiction. thus, ||d(z0,z∗)|| = 0 which implies that d(z0,z∗) = 0 i.e. z0 = z∗. thus the proof is complete. corollary 2.3. suppose that (x,d,a) is a c∗–valued metric space endowed with the graph g, and suppose that the mappings f, g : x → x are g–contractive, satisfying ||d(fx,gy)|| ≤ ||a||||d(x,y)||, for all (x,y) ∈ e, where a ∈a with ||a|| < 1. then f and g have a unique common fixed point in x. corollary 2.4. let (x,d,a) is a c∗–valued metric space endowed with the graph g, and suppose that the mapping f : x → x is g–contractive, satisfying ||d(fmx,fny)|| ≤ a∗d(x,y)a, for all (x,y) ∈ e, where a ∈a with ||a|| < 1 and m, n are positive integers. then f has a unique fixed point in x. remark 2.5. in theorem 2.2, if g = f, then we have (2.2) d(fx,fy) ≤ a∗d(x,y)a, for all (x,y) ∈ e. in this case we have the following corollary, which can also be found in [12]. corollary 2.6. let (x,d,a) be a complete c∗–valued metric space, and consider the mapping f : x → x such that it satisfies (2.2), then f has a unique fixed point in x. example 2.7. consider, a = m2×2(r), of all 2 × 2 matrices with the usual operation of addition, scalar multiplication, and matrix multiplication. thus a becomes c∗–algebra. let us define d : r×r → a by d(x,y) = ( |x−y| 0 0 |x−y| ) . it is essay to check that d satisfies all the conditions of definition 1.1. therefore (r,a,d) is c∗–valued metric space. define f,g : r → r by f(x) = x2 4 and g(x) = x2 3 , and consider the graph g = (v,e), where v = r and e = {( 1 4m , 1 32m+1 ) ; m = 1, 2, . . . } ∪ {( 1 4m , 0 ) ; m = 1, 2, . . . } ∪{(x,x); x ∈ r}. note that, for each m ∈ n, ( f( 1 4m ),g( 1 32m+1 ) ) = ( 1 42n+1 , 1 34n+3 ) ∈ e, and ( f( 1 4m ),g(0) ) = ( 1 42m+1 , 0 ) ∈ e. g–contraction in c∗–algebra–valued metric spaces 27 also, (fx,gx) = ( x 2 4 , x 2 3 ), for each x ∈ r, which is again in e. moreover, by taking a = ( 1√ 2 0 0 1√ 2 ) , we have ||a|| < 1, so all the conditions of theorem 2.2 are satisfied and thus the common fixed point of f and g is 0. references [1] m. abbas, g. jungck, common fixed points results for noncommuting mapping without continuity in cone metric space, j. math. anal. appl. 341 (2008), 416–420. [2] m. abbas, t. nazir, h. aydi, fixed points of generalized graphic contraction mappings in partial metric spaces endowed with a graph. j. adv. math. stud. 6 (2013), 130–139. [3] s. aleomraninejad, s. rezapour, n. shahzad, some fixed point results on a metric space with a graph. topol. appl. 159 (2012), 659–663. [4] m. ali, t. kamran, l. khan, a new type of multivalued contraction in partial hausdorff metric spaces endowed with a graph. j. inequal. appl. 2015 (2015), article id 205. [5] s. banach, sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. fundam. math. 3 (1922), 133–181. [6] b. choudhury, n. metiya, the point of coincidence and common fixed point for a pair of mappings in cone metric spaces, comput. math. appl. 60 (2010), 1686–1695. [7] l. cirić, b. samet, h. aydi, c. vetro, common fixed points of generalized contractions on partial metric spaces and an application, appl. math. comput. 218 (2011), 2398–2406. [8] l. haung, x. zhang, cone metric space and fixed point theorems of contractive mappings, j. math. anal., apal, vol. 332 2007, 1468–1476. [9] s. janković, z. golubović, s. radenović, common fixed point theorems for weakly compatible pairs on cone metric spaces, fixed point theory appl. 2009 (2009), article id 643840. [10] j. jachymski, the contraction principle for the mappings on a metric space with a graph. proc. am. math. soc. 136 (2008), 1359–1373. [11] r. johnsonbaugh, discrete mathematics, prentice-hall, englewood cliffs (1997). [12] z. ma, l. jiang, h. sun, c∗–algebra-valued metric spaces and related fixed point theorems. fixed point theory appl. 2014 (2014), article id 206. [13] gj. murphy, c∗–algebras and operator theory. academic press, london (1990). [14] m. a. osman, fuzzy metric space and fixed fuzzy set theorem, bull. malaysian math. soc. 6 (1983), 1–4. [15] m. samreen, t. kamran, fixed point theorems for weakly contractive mappings on a metric space endowed with a graph. filomat 28 (2014), 441–450. [16] m. samreen, t. kamran, n. shahzad, some fixed point theorems in b-metric space endowed with graph. abstr. appl. anal. 2013 (2013), article id 967132. [17] w. shatanawi, m. postolache, common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. fixed point theory appl. 2013 (2013), article id 271. [18] w. shatanawi, m. postolache, common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. fixed point theory appl. 2013 (2013), article id 60. [19] d. shehwar, t. kamran, c∗–valued g-contractions and fixed points, j. inequal. appl. 2015 (2015), article id 304. [20] t. sistani, m. kazemipour, fixed points for α-ψ-contractions on metric spaces with a graph. j. adv. math. stud. 7 (2014), 65–79. [21] e. tarafdar, an approach to fixed-point theorems on uniform spaces, trans. amer. math. soc. 191 (1974), 209–225. [22] q. xin, l. jiang, common fixed point theorems for generalized k-ordered contractions and b-contractions on noncommutative banach spaces, fixed point theory appl. 2015 (2015), article id 77. [23] a. zada, r. shah, t. li, integral type contraction and coupled coincidence fixed point theorems for two pairs in g–metric spaces, hacet. j. math. stat., in press. 1department of mathematics, university of peshawar, peshawar, pakistan 2school of mathematics and statistics, beijing institute of technology, beijing, 100081, china 3department of mathematics and physics, hebei institute of architecture and civil engineering, zhangjiakou, 075024, china ∗corresponding author: zadababo@yahoo.com international journal of analysis and applications volume 18, number 5 (2020), 890-899 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-890 applicable solution for a class of ordinary differential equations with singularity n. ameer ahamad∗ department of mathematics, faculty of science, university of tabuk, p.o. box 741, tabuk 71491, saudi arabia ∗corresponding author: n.ameer1234@gmail.com abstract. boundary value problems arise in many real applications such as nanofluids and other areas of applied sciences. the temperature/nanoparticles concentration are usually expressed as singular 2ndorder odes. so, it is a challenge to obtain the exact solution of these problems due to the difficulty of the singularity encountered in the governing equations. by means of a suitable transformation, a direct approach is introduced to solve a general class of 2nd-order odes. the efficiency of the obtained results is validated through selected problems in the literature. it is found that several existing solutions can be deduced as special cases of our generalized one. moreover, the present results may be invested for similar future problems in fluid mechanics, especially nanofluids. 1. introduction the field of nanofluid is of great importance in industry, engineering, and physics. the distributions of temperature and nanoparticles concentration of such fluids are originally governed by pdes which can be transformed into odes [1-9]. such odes are, basically, subjected to boundary conditions (bcs) given at infinity. in refs. [10-16], the authors implemented several numerical/analytical methods to solve such types of problems. however, the approaches [10-16] need a massive computational work to obtain accurate solution because of the difficulty of applying the bc at infinity. in addition, it was shown in the literature [17-19] that received july 5th, 2020; accepted july 27th, 2020; published august 20th, 2020. 2010 mathematics subject classification. 34b10. key words and phrases. ordinary differential equation; hypergeometric series; boundary value problem; exact solution; nanofluid. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 890 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-890 int. j. anal. appl. 18 (5) (2020) 891 some of the approximate methods have drawbacks. therefore, solving bvps with singularities is a challenge. usually, the bcs at infinity are transformed to new finite ones by applying certain substitutions. accordingly, the coefficients of the odes become polynomials. hence, the temperature/nanoparticles concentration of nanofluids are usually special cases of the following class: (1.1) z′′(t) + ( p t + q ) z′(t) + ( l t2 + r t ) z(t) = α tn−1, n > −1, α ∈ r, under the bcs: z(0) = 0, z(δ) = 1, δ ∈ r−{0},(1.2) where p , q, l, and r are physical parameters of the nanofluids [1-8]. the constant n takes a particular value according to the final form of the temperature equation, while δ 6= 0 depends on the final bc. the main objective of this paper is to introduce a direct analysis to exactly solving eqs. (1.1-1.2). then, the present generalized results will be invested to construct several exact solutions for some published nanofluids problems as special cases. 2. analysis firstly, we rewrite eq. (1.1) as (2.1) t2z′′(t) + ( pt + qt2 ) z′(t) + (l + rt) z(t) = α tn+1. suppose that (2.2) z(t) = tγψ(t). accordingly, (2.3) z′(t) = tγ−1 (tψ′(t) + γψ(t)) , and (2.4) z′′(t) = tγ−2 ( t2ψ′′(t) + 2γtψ′(t) + γ (γ − 1) ψ(t) ) , respectively. substituting eqs. (2.2-2.4) into eq. (2.1), we have (2.5) t2ψ′′(t) + ( (2γ + p) t + qt2 ) ψ′(t) + (( γ2 −γ + γp + l ) + (γq + r) t ) ψ(t) = α tn−γ+1. setting (2.6) γ2 −γ + γp + l = 0, int. j. anal. appl. 18 (5) (2020) 892 and solving for γ, we obtain (2.7) γ = 1 −p ± √ (1 −p)2 − 4l 2 . accordingly, eq. (2.5) becomes (2.8) tψ′′(t) + ((2γ + p) + qt) ψ′(t) + (γq + r) ψ(t) = α tn−γ. assuming that (2.9) p1 = 2γ + p, r1 = γq + r, then eq. (2.8) takes the form: (2.10) tψ′′(t) + (p1 + qt) ψ ′(t) + r1ψ(t) = α t n−γ. 3. exact solution and convergence analysis 3.1. exact solution. suppose that ψc(t) and ψp(t) are complementary and particular solution of eq. (2.10), respectively, then (3.1) ψ(t) = ψc(t) + ψp(t), following [20], we have (3.2) ψc(t) = c tµ1+µ2−1 γ(µ1 + µ2) 1f1[µ1,µ1 + µ2,−q t], where 1f1 is kummer’s function and c is a constant. µ1 and µ2 are defined as (3.3) µ1 = 1 −p1 + r1 q , µ2 = 1 − r1 q . from (2.9) and (3.3), we have (3.4) µ1 = 1 −γ −p + r q , µ2 = 1 −γ − r q . ψp(t) can be obtained as (see [20]) (3.5) ψp(t) = α tn−γ+1 (n−γ + 1) (n−γ + p1) , such that (3.6) r1 = −(n−γ + 1)q, (n−γ + 1) (n−γ + p1) 6= 0. from (2.9) and (3.6), we have (3.7) r = −(n + 1)q, (n−γ + 1) (n + γ + p) 6= 0, int. j. anal. appl. 18 (5) (2020) 893 and hence eq. (2.8) reduces to (3.8) tψ′′(t) + ((2γ + p) + qt) ψ′(t) − (n−γ + 1) qψ(t) = α tn−γ. inserting p1 given by (2.9) into (3.5), we obtain (3.9) ψp(t) = α tn−γ+1 (n−γ + 1) (n + γ + p) . therefore, the general solution of eq. (3.8) is given from (3.1) by (3.10) ψ(t) = c tµ1+µ2−1 γ(µ1 + µ2) 1f1[µ1,µ1 + µ2,−q t] + α tn−γ+1 (n−γ + 1) (n + γ + p) , where µ1 and µ2 are defined by (3.3). hence, the general solution of the original equation (2.1) such that r = −(n + 1)q is obtained as (3.11) z(t) = c tγ+µ1+µ2−1 γ(µ1 + µ2) 1f1[µ1,µ1 + µ2,−q t] + α tn+1 (n−γ + 1) (n + γ + p) , for the ode: (3.12) t2z′′(t) + ( pt + qt2 ) z′(t) + (l− (n + 1)qt) z(t) = α tn+1. it is noted from (3.11) that the first bc z(0) = 0 is automatically satisfied when (3.13) γ + µ1 + µ2 > 1, n > −1, (n−γ + 1) (n + γ + p) 6= 0. applying the second bc in (1.2) on eq. (3.11), yields (3.14) c = δ1−γ−µ1−µ2 γ(µ1 + µ2) 1f1[µ1,µ1 + µ2,−qδ] ( 1 − α δn+1 (n−γ + 1) (n + γ + p) ) . in such case, the solution given by eq. (3.11) becomes z(t) = (t/δ) γ+µ1+µ2−1 1f1[µ1,µ1 + µ2,−qt] 1f1[µ1,µ1 + µ2,−qδ] ( 1 − α δn+1 (n−γ + 1) (n + γ + p) ) + α tn+1 (n−γ + 1) (n + γ + p) .(3.15) therefore z(t) = (t/δ) 1−γ−p 1f1[−n−γ −p, 2 − 2γ −p,−qt] 1f1[−n−γ −p, 2 − 2γ −p,−qδ] ×(3.16) ( 1 − α δn+1 (n−γ + 1) (n + γ + p) ) + α tn+1 (n−γ + 1) (n + γ + p) , provided that (3.17) 1 −γ −p > 0, n > −1, (n−γ + 1) (n + γ + p) 6= 0. it can be verified by direct substitution that the solution (3.16) satisfies eq. (3.12) and the bcs (1.2). in a sequent section, it is declared that eq. (3.16) agrees with several existing results at prescribed values of the coefficients p , q, l, r, and the parameter δ. int. j. anal. appl. 18 (5) (2020) 894 3.2. convergence analysis. in order to prove the convergence of the solution given by eqs. (3.16-3.17), we begin with the definition of kummer’s function 1f1(a,b,t): (3.18) 1f1(a,b,t) = ∞∑ i=0 (a)i (b)i ti i! , where (a)i is pochhammer symbol defined as (3.19) (a)i = a(a + 1)(a + 2) . . . (a + i− 1), i = 1, 2, 3, . . . , (a)0 = 1. the series (3.18) is not defined for b = 0,−1,−2,−3, . . . , and if a is a negative integer, the series truncates. theorem 1: let b is neither a negative integer nor zero, then 1f1(a,b,−qt) converges for all (finite) t and finite q. proofs: from the definition (3.18), 1f1(a,b,−qt) is given as (3.20) 1f1(a,b,−qt) = ∞∑ i=0 (a)i (b)i (−qt)i i! . since b is neither a negative integer nor zero, then the series (3.20) is defined and its general term vi(x) is given by (3.21) vi(x) = (a)i (b)i (−qt)i i! . implementing the ratio test, we have lim i→∞ ∣∣∣∣vi+1(x)vi(x) ∣∣∣∣ = limi→∞ ∣∣∣∣∣(a)i+1(b)i+1 (−qt) i+1 (i + 1)! × (b)i (a)i i! (−qt)i ∣∣∣∣∣ , = lim i→∞ ∣∣∣∣∣(a)i(a + i)(b)i(b + i) (−q) i+1 (t) i+1 (i + 1)(i!) × (b)i (a)i i! (−q)i (t)i ∣∣∣∣∣ , = lim i→∞ ∣∣∣∣ (a + i)(b + i)(i + 1) ∣∣∣∣×|−q| |t| .(3.22) for finite t and finite q, we have from (3.22) that (3.23) lim i→∞ ∣∣∣∣vi+1(x)vi(x) ∣∣∣∣ = 0, and hence, 1f1(a,b,−qt) is convergent. lemma 1: for finite δ and q, the solution given by eqs. (3.16-3.17) converges if (2 − 2γ −p) is neither a negative integer nor zero. proofs: let a = −n−γ −p and b = 2 − 2γ −p, then the solution in eqs. (3.16-3.17) takes the form: (3.24) z(t) = (t/δ) 1−γ−p 1f1[a,b,−qt] 1f1[a,b,−qδ] ( 1 + α δn+1 (n−γ + 1) a ) − α tn+1 (n−γ + 1) a , such that (3.25) 1 −γ −p > 0, n > −1, (n−γ + 1) a 6= 0. int. j. anal. appl. 18 (5) (2020) 895 since b = 2 − 2γ − p is neither a negative integer nor zero, then z(t) in (36) is defined. also, since q is finite and t is finite in the domain of the problem, t ∈ [0,δ], then 1f1(a,b,−qt) is convergent by theorem 1. also, 1f1(a,b,−qδ) is convergent because δ is finite and therefore the solution given by eqs. (3.16-3.17) or its equivalent form (3.24-3.25) converges. 4. applications 4.1. at l = 0, α 6= 0, n > −1. at l = 0, γ in (2.7) reduces to (4.1) γ = 1 −p ±|1 −p | 2 . for p < 1, we have γ = 1 − p or γ = 0. however, γ = 1 − p doesn’t satisfy the first condition in (3.25) which is 1 −γ −p > 0. also, for p > 1, we have γ = 0 or γ = 1 −p . hence, γ = 0 is the only acceptable value when the special case l = 0 is considered. at such case (l = 0), eq. (3.12) reduces to (4.2) tz′′(t) + (p + qt) z′(t) − ((n + 1)q) z(t) = α tn, and its solution comes by setting γ = 0 in (3.16) which is z(t) = (t/δ) 1−p 1f1[−n−p, 2 −p,−qt] 1f1[−n−p, 2 −p,−qδ] ( 1 − α δn+1 (n + 1) (n + p) ) + α tn+1 (n + 1) (n + p) .(4.3) the result obtained by eq. (4.3) agrees with the published solution in literature (ref. [20], equation 31). consequently, the solution in literature is a special case of our analysis when l = 0. in addition, the solution (40) converges if (2−p) is neither a negative integer nor zero, i.e., p 6= 2, 3, 4, . . . , or p 6= (j + 1) ∀ j ∈ z+. 4.2. at l = 0, α 6= 0, n = 1, δ = 1. the temperature equation for the marangoni boundary layer was obtained by khaled [21] as (4.4) tz′′(t) + (l1 −m t) z′(t) + 2mz(t) = −λt, subject to (4.5) z(0) = 0, z(1) = 1. at n = 1, eq. (4.2), in the previous section, becomes (4.6) tz′′(t) + (p + qt) z′(t) − 2qz(t) = αt. comparing (4.4) with (4.2) and (4.5) with (1.2), we find that (4.7) p = l1, q = −m, α = −λ, δ = 1. int. j. anal. appl. 18 (5) (2020) 896 substituting these values of parameters into (4.3), we have (4.8) y(t) = ( 1 + λ 2(l1 + 1) ) t1−l1 1f1[−1 − l1, 2 − l1,m t] 1f1[−1 − l1, 2 − l1,m] − λ t2 2(l1 + 1) , which is the same obtained result in ref. [21]. as shown in the previous subsection, the solution (4.8) converges if l1 6= (j + 1) ∀ j ∈ z+. 4.3. at l = 0, α = 0, δ = sc β2 . qasim [22] obtained the mass transfer equation: (4.9) t z′′(t) + ( 1 − sc β2 + t ) z′(t) −m z(t) = 0, for a jeffrey fluid with heat source/sink, where (4.10) z(0) = 0, z ( sc β2 ) = 1. sc > 0 is the schmidt parameter and β is a positive parameter. it then follows that (4.11) p = 1 − sc β2 , q = 1, n = m− 1, α = 0, δ = sc β2 . inserting these values into (4.3), we obtain (4.12) z(t) = ( β2 sc t ) sc β2 1f1[ sc β2 −m, sc β2 + 1,−t] 1f1[ sc β2 −m, sc β2 + 1,−sc β2 ] . the solution (4.12) can be verified by substitution. it should be mentioned that the solution obtained by qasim [22] as (4.13) z(t) = ( β2 sc t ) sc β2 1f1[ sc β2 −m, 2 sc β2 + 1,−t] 1f1[ sc β2 −m, 2 sc β2 + 1,−sc β2 ] , does not satisfy eq. (4.9). since β > 0 and sc > 0, then the magnitude ( sc β2 + 1 ) is never a zero or a negative integer. hence, the solution (4.12) converges for all positive values of β and sc. 4.4. at l 6= 0, α = 0, δ = pr β2 . the following heat transfer equation (4.14) t2z′′(t) + (( 1 − pr β2 ) t + t2 ) z′(t) − ( mt− γ1 β2 ) z(t) = 0, was also obtained by qasim [22], where (4.15) z(0) = 0, z ( pr β2 ) = 1. pr > 0 is the prandtl number, β > 0, and γ1 is the heat generation/absorption parameter. in this case, we have (4.16) p = 1 − pr β2 , q = 1, n = m− 1, α = 0, l = γ1 β2 , δ = pr β2 . int. j. anal. appl. 18 (5) (2020) 897 at l = γ1 β2 , γ is given by (4.17) γ = 1 2  pr β2 − √ pr2 β4 − 4γ1 β2   , which be written as (4.18) γ = 1 2  pr β2 − √ pr2 β4 − 4γ1 β2   = k1 −k2, where (4.19) k1 = pr 2β2 , k2 = √ pr2 − 4γ1β2 2β2 . hence, the solution of the present model is (4.20) z(t) = ( β2 pr t )k1+k2 1f1[k1 + k2 −m, 2k1 + 1,−t] 1f1[k1 + k2 −m, 2k1 + 1,−prβ2 ] , which is the same exact solution obtained by qasim [22]. since β and pr are always positives, then (2k1+1) =( pr β2 + 1 ) is never a zero or a negative integer. hence, the solution (4.20) converges for all positive values of β and pr. 4.5. at l 6= 0, α = 0, δ = -sc∗. kameswaran et. al [2] obtained following equation for the mass transfer of nanofluids: (4.21) t2z′′(t) + ( (1 − sc∗) t− t2 ) z′(t) + (2t−γ2sc∗) z(t) = 0, subject to (4.22) z(0) = 0, z (-sc∗) = 1, where γ2 is the parameter of scaled chemical reaction and sc ∗ is the modified schmidt number. thus (4.23) p = 1 − sc∗, q = −1, n = 1, α = 0, l = −γ2sc∗, δ = -sc∗. at l = −γ2sc∗, we obtain γ as (4.24) γ = 1 2 ( sc∗ − √ (sc∗) 2 + 4γ2sc ∗ ) , or (4.25) γ = c1 −d1 2 , c1 = sc ∗, d1 = (sc ∗) 2 + 4γ2sc ∗. the solution of the present model is in the form: (4.26) z(t) = ( − t sc∗ )c1+d1 2 1f1[ c1+d1 2 − 2,d1 + 1, t] 1f1[ c1+d1 2 − 2,d1 + 1,−sc∗] , int. j. anal. appl. 18 (5) (2020) 898 which agrees with kameswaran et. al [2]. according to the physical values taken by the authors [2], the magnitude d1 + 1 = (sc ∗) 2 + 4γ2sc ∗ + 1 is always positive and this admits the convergence of the solution (4.26). 5. conclusion in this paper, a general solution was obtained for a class of singular bvps arise in the field of nanofluids. the solution was derived in terms of the hypergeometric series. the studied class reduced to several published physical models at particular choices of the involved parameters. the obtained solutions were compared with the corresponding results of several models in the literature. it was found that the results in the literature were recovered as special cases of the current ones. furthermore, this work can be extended in the near future to deal with the recently published physical models [23-25]. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.a.a. hamad, analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, int. commun. heat mass transfer 38 (2011), 487-492. [2] p.k. kameswaran, m. narayana, p. sibanda and p.v.s.n. murthy, hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects, int. j. heat mass transfer, 55 (2012), 7587-7595. [3] e.h. aly and a. ebaid, exact analytical solution for suction and injection flow with thermal enhancement of five nanofluids over an isothermal stretching sheet with effect of the slip model: a comparative study, abstr. appl. anal. 2013 (2013), article id 721578. [4] e.h. aly and a. ebaid, new exact solutions for boundary-layer flow of a nanofluid past a stretching sheet, j. comput. theor. nanosci. 10 (11) (2013), 2591-2594. [5] w.a. khan, z.h. khan and m. rahi, fluid flow and heat transfer of carbon nanotubes along a flat plate with navier slip boundary. appl. nanosci. 4 (2014), 633-641. [6] a. ebaid, f. al mutairi and s.m. khaled, effect of velocity slip boundary condition on the flow and heat transfer of cu-water and tio2-water nanofluids in the presence of a magnetic field, adv. math. phys. 2014 (2014), article id 538950. [7] a. ebaid and m. al sharif, application of laplace transform for the exact effect of a magnetic field on heat transfer of carbon-nanotubes suspended nanofluids, z. naturforsch. a. 70 (6) (2015), 471-475. [8] e.r. el-zahar, a.m. rashad and a.m. gelany, studying high suction effect on boundary-layer flow of a nanofluid on permeable surface via singular perturbation technique, j. comput. theor. nanosci. 12 (11) (2015), 4828-4836. [9] e.h. aly and a. ebaid, exact analysis for the effect of heat transfer on mhd and radiation marangoni boundary layer nanofluid flow past a surface embedded in a porous medium, j. molecular liquids, 215 (2016), 625-639. [10] j.p. boyd, pade-approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, comput. phys. 11 (3) (1997), 299-303. int. j. anal. appl. 18 (5) (2020) 899 [11] a.m. wazwaz, the modified decomposition method and pade approximants for solving the thomas-fermi equation, appl. math. comput. 105 (1) (1999), 11-19. [12] a.m. wazwaz, the modified decomposition method and pade approximants for a boundary layer equation in unbounded domain, appl. math. comput. 177 (2) (2006), 737-744. [13] e.r. el-zahar and y.s. hamed, an algorithm for solving second order nonlinear singular perturbation boundary value problems, j. mod. methods numer. math. 2 (1-2) (2011), 21-31. [14] e.r. el-zahar, approximate analytical solutions for singularly perturbed boundary value problems by multi-step differential transform method, j. appl. sci. 12 (19) (2012), 2026-2034. [15] e.r. el-zahar, applications of adaptive multi step differential transform method to singular perturbation problems arising in science and engineering, appl. math. inform. sci. 9 (1) (2015), 223-232. [16] e.r. el-zahar and s.m.m. el-kabeir, approximate analytical solution of nonlinear third-order singularly perturbed bvps using homotopy analysis-pade method, j. comput. theor. nanosci. 13 (2016), 8917-8927. [17] a. ebaid and e.h. aly, exact analytical solution of the peristaltic nanofluids flow in an asymmetric channel with flexible walls: application to cancer treatment, comput. math. methods med. 2013 (2013), article id 825376. [18] a. ebaid, remarks on the homotopy perturbation method for the peristaltic flow of jeffrey fluid with nano-particles in an asymmetric channel, comput. math. appl. 68 (3) (2014), 77-85. [19] a. ebaid and s.m. khaled, an exact solution for a boundary value problem with application in fluid mechanics and comparison with the regular perturbation solution, abstr. appl. anal. 2014 (2014), article id 172590. [20] a. ebaid, a.m. wazwaz, e. alali and b. masaedeh, hypergeometric series solution to a class of second-order boundary value problems via laplace transform with applications to nanouids, commun. theor. phys. 67 (2017), 231. [21] s.m. khaled, the exact effects of radiation and joule heating on magnetohydrodynamic marangoni convection over a flat surface, therm. sci. 22 (2018), 63-72. [22] m. qasim, heat and mass transfer in a jeffrey fluid over a stretching sheet with heat source/sink, alexandria eng. j. 52 (2013), 571-575. [23] a. ebaid, e. alali, h.s. ali, the exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, j. assoc. arab univ. basic appl. sci. 24 (2017), 156-159. [24] h.s. ali, elham alali, a. ebaid and f.m. alharbi, analytic solution of a class of singular second-order boundary value problems with applications, mathematics, 7 (2019), 172. [25] q.m. al-mdallal, n. indumathi, b. ganga and a.k. abdul hakeem, marangoni radiative effects of hybrid-nanofluids flow past a permeable surface with inclined magnetic field, case stud. therm. eng. 17 (2020), 100571. 1. introduction 2. analysis 3. exact solution and convergence analysis 3.1. exact solution 3.2. convergence analysis 4. applications 4.1. at l=0, =0, n>-1 4.2. at l=0, =0, n=1, =1 4.3. at l=0, =0, =sc2 4.4. at l=0, =0, =pr2 4.5. at l=0, =0, =-sc* 5. conclusion references int. j. anal. appl. (2023), 21:60 fuzzy soft boolean rings gadde sambasiva rao1, d. ramesh1, aiyared iampan2,∗, b. satyanarayana3 1department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, vaddeswaram, andhra pradesh-522302, india 2fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 3department of mathematics, acharya nagarjuna university, nagarjuna nagar, andhra pradesh-522510, india ∗corresponding author: aiyared.ia@up.ac.th abstract. this article introduces the idea of (fuzzy soft boolean rings) fsbrs and investigates their algebraic properties. the concepts of fuzzy soft ideals (fsis) of fsbrs, and idealistic fuzzy soft boolean rings (ifsbrs) are then defined and discussed. 1. introduction in classical mathematics, exact solutions to mathematical models are required. if the model is so complex that an exact solution cannot be determined, we can get a rough estimate. in 1999, russian researcher molodtsov [8] pioneered the idea behind soft set (ss) theory and began developing the foundations of the corresponding theory as a novel approach to modelling uncertainty. the ss is an approximate representation of an object. there are numerous potential applications for ss theory. ss theory and its applications are currently advancing at a rapid pace. maji et al. [7] proposed new ss definitions. pei and miao [9] looked into how sss and information systems interact. by fusing ss and fuzzy set (fs) designs, maji et al. [6] developed the concept of fuzzy soft sets (fsss) in 2001. to continue the investigation, ahmad and kharal [2] obtainable some more properties of fsss. there has been a surge of interest in the algebraic structure of sss in recent years. soft groups were defined received: apr. 10, 2023. 2020 mathematics subject classification. 03e72, 03g05, 28a60, 06d72. key words and phrases. boolean ring; fuzzy soft set; fuzzy soft boolean ring; fuzzy soft sub boolean ring; fuzzy ideal; fuzzy soft ideal; idealistic fuzzy soft boolean ring. https://doi.org/10.28924/2291-8639-21-2023-60 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-60 2 int. j. anal. appl. (2023), 21:60 by aktaş and çag̃man [3], and some properties were derived from them. additionally, they contrasted sss with rough and fss, two concepts that are related. the fundamental ideas of soft rings were first introduced by acar et al. [1] in 2010, which are a generalized family of subrings. liu [5] proposed the fuzzy ring concept in 1982. following that, dixit et al. [4] investigated the fuzzy ring and discovered some theoretical analogues. the algebraic characteristics of fsss in boolean ring (br) theory are examined in this paper. 2. preliminaries to begin, we will present maji et al. [6] and ahmad and kharal [2]’s fundamental definitions and notations. definition 2.1. let ê denote a set of parameters, z denote an initial universe, and î represents the closed unit interval. q(z) represents z’s power set. then the pair (m,ê) over z is a soft set, where m : ê → q(z) is a set valued function. definition 2.2. let ê denote a set of parameters, z denote an initial universe, and î represents the closed unit interval, i.e., î = [0,1]. q(z) represents z’s power set, where m : c → iz is set-valued function and iz represent the collection of all fuzzy sets on z. definition 2.3. consider (m,c) and (n,d) to be fsss. then (m,c) is a fss of (n,d) and we can write (m,c) ⊆ (n,d) if (i) c ⊆ d (ii) for each α ∈ c,mα ≤ nα implying that, mα is fuzzy subset of nα. definition 2.4. let us assume (m,c) and (n,d) be two fsss, with c∩d 6= ∅. then the fss (o,e) is formed by the intersection of (m,c) and (n,d), where e = c ∩d and oα = mα ∧nα,∀α ∈ e. we can write (m,c)e (n,d)= (o,e). definition 2.5. let us assume (m,c) and (n,d) are two fsss. the fss (o,e) is formed by the union of (m,c) and (n,d), where e = c ∪d and (∀α ∈ e)  oα =   mα if α ∈ c −d nα if α ∈ d−c mα,∨nα if α ∈ c ∩d   . (2.1) then we write (m,c)d (n,d)= (o,e). definition 2.6. let (mj,cj)j∈j be a family of fsss. the union of these fsss is a fss (o,e), where e =∪j∈jcj and o(α)=∨j∈jmj(α),∀α ∈ e. then we can write dj∈j(mj,cj)= (o,e). definition 2.7. let (mj,cj)j∈j be a family of fsss, with ∩j∈jcj 6= ∅. a fss is formed by the intersection of these fsss (o,e), where e =∩j∈jcj and o(α)=∧j∈jmj(α),∀α ∈ e. then we can write ej∈j(mj,cj)= (o,e). int. j. anal. appl. (2023), 21:60 3 definition 2.8. let (m,c) and (n,d) be two fsss, subsequently, (m,c) and (n,d) are represented by (m,c)∧̂(n,d) and it’s indicated by (o,c × d), where o(α,β) = oα,β = mα ∧ nβ for every (α,β)∈ c ×d. definition 2.9. let (m,c) and (n,d) be two fsss. then (m,c) or (n,d) are represented by (m,c)∨̂(n,d) and it’s indicated by (o,c × d), where o(α,β) = oα,β = mα ∨ nβ for every (α,β)∈ c ×d. definition 2.10. consider (m,c) to be a fss. the set supp(m,c)= {α ∈ c : m(α)= mα 6= ∅} is known support of the fss (m,c). if the support of a fss is greater than the empty set, it is said to be non-null. 3. fuzzy soft boolean rings the concept of soft rings was proposed by acar et al. [1]. in this section contains, we define fsbrs and discuss some of their fundamental properties. r denotes a br from now on, and all fsss are preferred over r. definition 3.1. let us assume (m,c) is a non-null ss. then (m,c) is referred to as a soft boolean ring (sbr) over r if for each α ∈ c,m(α) is a sub-br of r. definition 3.2. let us assume (m,c) is a non-null fss. then (m,c) is referred to as a fsbr over r if for each γ ∈ c, m(γ) = mγ is a f-sub-br of r, i.e., mγ(α − β) ≥ min(mγ(α),mγ(β)) and mγ(α ·β)≥min(mγ(α),mγ(β)),∀α,β ∈ r. example 3.1. let r = {0, j, t, r} be a nonempty set with two binary operations + and · defined as follows: + 0 j t r 0 0 j t r j j 0 r t t t r 0 j r r t j 0 · 0 j t r 0 0 0 0 0 j 0 j r t t 0 r t j r 0 t j r 4 int. j. anal. appl. (2023), 21:60 let a = {e11,e 1 2,e 1 3} be the set of parameters and now define a fss (m,c) on a br r by m(e11)= {(0,0.9),(j,0.8),(t,0.6),(r,0.4)}, m(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)}, m(e13)= {(0,0.9),(j,0.6),(t,0.5),(r,0.4)}. here (m,c) is a fss over r, which is also a f-sub-br of r, for all α ∈ c. hence (m,c) is a fsbr over r. example 3.2. because each ss can be thought of as a fss, and each characteristic function of a br is a f-sub-br of r, we can think of an sbr as a fsbr. theorem 3.1. let (m,c) and (n,d) are two fsbrs over r. if (m,c)∧̂(n,d) is non-null, then it’s a fsbr over r. proof. let (m,c)∧̂(n,d) = (o,c × d), where o(α,β) = mα ∧ nβ,∀(α,β) ∈ c × d. since (o,c ×d), is non-null, then there exists the pair (α,β)∈ c ×d such that oα,β = mα ∧nβ 6= or. we already know that mα,∀α ∈ c and nβ,∀β ∈ d are f-sub-br of r. since then the intersection of two f-sub-brs of r is also a f-sub-br of r, then o(α,β) = oα,β is a f-sub-br of r. hence (o,c ×d)= (m,c)∧̂(n,d) is a fsbr over r. � theorem 3.2. let (m,c) and (n,d) are two fsbrs over r. if (m,c)e(n,d) is non-null, then it’s a fsbr over r. proof. let (m,c)e (n,d) = (o,e), where e = c ∩d and oα = mα ∧nα,∀α ∈ e. since (o,e) is non-null, then there exists α ∈ e such that oα(β) 6=0 for some β ∈ r. we know mα ∧nα is a fsub-br of r, because oα 6=0r and mα,nα are f-sub-br of r. therefore, (o,e)= (m,c)e(n,d) is a fsbr over r. � theorem 3.3. let (mj,cj)j∈j be a family of fsbrs over r. there are also the following: (i) if ∧̂j∈j(mj,cj) is non-null, then it’s a fsbr over r. (ii) if ej∈j(mj,cj) is non-null, it’s a fsbr over r. proof. (i) let ∧̂j∈j(mj,cj) = (o,e), where e = ej∈jcj and oα = ∧j∈jmj(ej),∀α = (αj)j∈j ∈ e. suppose that the fss (o,e) is non-null. if α =(αj)j∈j ∈supp(o,e), then oα =∧j∈jmj(αj) 6=0r. since (mj,cj) is a fsbr over r, ∀j ∈ j, mj(αj) is a f-sub-br of r. as a result oα is a f-sub-br of r for all α ∈supp(o,e). consequently, ∧̂j∈j(mj,cj)= (o,e) is a fsbr over r. (ii) let ej∈j(mj,cj) = (o,e), where e = ∩j∈jcj and oα = ∧j∈jmj(αj),∀α ∈ e. suppose that the fss (o,e) is non-null. if α ∈ supp(o,e), then oα = ∧j∈jmj(αj) 6= 0r. since (mj,cj) is a fsbr over r, then mj(αj) is a f-sub-br of r for all j ∈ j. as a result oα is a f-sub-br of r for all α ∈supp(o,e). therefore ej∈j(mj,cj)= (o,e) is a fsbr over r. � int. j. anal. appl. (2023), 21:60 5 definition 3.3. let (m,c) and (n,d) be two fsbrs over r. then (n,d) is referred to as a fssbr of (m,c) if the circumstances listed below are true: (i) d ⊆ c, (ii) nα is a f-sub-br of mα,∀α ∈supp(n,e). theorem 3.4. let (m,c) and (n,d) be two fsbrs over r. if (m,c)e(n,d) is non-null, then it’s a fssbr of (m,c) and (n,d). proof. (m,c)e(n,d)= (o,e), where e = c∩d and oα = mα∧nα,∀α ∈ e. since e = c∩d ⊆ c and oα = mα ∧nα, is a f-sub-br of mα, then (o,e) is a fssbr of (m,c). similarly, we obtain that (o,e) is a fssbr of (n,d). � 4. fuzzy soft ideals of fuzzy soft boolean rings definition 4.1. assume (m,c) is a fsbr over r. a fss (n,d) is a fsi of (m,c), as indicated by (n,d)/̂(m,c). if it meets the following criteria: (i) d ⊆ c, (ii) nγ is a fuzzy ideal (fi) of a fuzzy br mγ for all γ ∈ supp(n,d), i.e., nγ is a fi, for each γ ∈supp(n,d), (a) nγ(α−β)≥ nγ(α)∧nγ(β), (b) nγ(αβ)≥ nγ(α)∧nγ(β), (c) nγ(α)≤ mγ(α),∀α,β ∈ r. example 4.1. take a look at the br (r,+, ·) established in example 3.1. m(e11)= {(0,0.9),(j,0.8),(t,0.6),(r,0.4)}, m(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)}, m(e13)= {(0,0.9),(j,0.6),(t,0.5),(r,0.4)}. here (m,c) is a fss over r, which is also a f-sub-br of r, ∀α ∈ c. hence (m,c) is a fsbr over r. let d = {e12} and n : d → q(r) be a function with a set of values defined by n(e12)= {(0,0.4),(1,0.3),(2,0.2),(3,0.2)}. obviously (n,d) is a fss of r. we also see that d ⊆ c and n(γ) is a fi of m(γ),∀γ ∈ i. as a result, (n,d) is a fsi of (m,c). theorem 4.1. let (n1,d1) and (n2,d2) be fsis of a fsbr (m,c). then (n1,d1)e (n2,d2) is a fsi of (m,c) if it is non-null. proof. let (n1,d1)/̂(m,c),(n2,d2)/̂(m,c). by the definition 2.4, we write (n1,d1)e((n2,d2)= (n,d), where d = d1∩d2 and n(γ)= n1(γ)∧n2(γ),∀γ ∈ d. since d1 ⊆ c and d2 ⊆ c, we have 6 int. j. anal. appl. (2023), 21:60 d1∩d2 = d ⊆ c. suppose that (n,d) is non-null. if γ ∈supp(n,d), then n(γ)= n1(γ)∧n2(γ) 6= 0r. since (n1,d1)/̂(m,c) and (n2,d2)/̂(m,c), n1(γ) and n2(γ) are both fis of m(γ), we conclude m(γ). as a result, n(γ) is a fi of m(γ),∀γ ∈supp(n,d). therefore, (n1,d1)e(n2,d2)= (n,d) is a fsi of (m,c). � theorem 4.2. let (n1,d1) and (n2,d2) be fsis of a fsbr (m,c). if d1 and d2 are disjoint, then (n1,d1)d (n2,d2) is a fsi of (m,c). proof. let (n1,d1)/̂(m,c),(n2,d2)/̂(m,c). by the definition 2.5, we write (n1,d1)d(n2,d2)= (n,d), where d = d1 ∪d2 and ∀γ ∈ d, (∀α ∈ e)  nγ =   n1(γ) if α ∈ d1 −d2 n2(γ) if α ∈ d2 −d1 n1(γ)∨n2(γ) if α ∈ d1 ∩d2   . (4.1) obviously, we have d ⊆ c. since d1 and d2 are disjoint, γ ∈ d1 − d2 or γ ∈ d2 − d1,∀γ ∈ supp(n,d). let γ ∈ d1−d2. since (n1,d1)/̂ is a fi of m(γ). thus, ∀γ ∈supp(n,d), (n1,d1)⊆ (m,c). consequently, (n,d) is a fsi of (m,c). � 5. idealistic fuzzy soft boolean rings definition 5.1. let (m,c) be a non-null fss. then (m,c) is referred to as an ifsbr over r, if mγ is a fi of r, ∀γ ∈ supp(m,c). in other words, for each γ ∈ supp(m,c),mγ is a fi of r defined in [4], i.e., mγ(α−β)≥ mγ(α)∧mγ(β) and mγ(α ·β)≥ mγ(α)∨mγ(β),∀α,β ∈ r. example 5.1. let r = {0, j, t, r} be a set with two binary operations + and · as shown: + 0 j t r 0 0 j t r j j 0 r t t t r 0 j r r t j 0 · 0 j t r 0 0 0 0 0 j 0 j r t t 0 r t j r 0 t j r then (r,+, ·) is a br. let a = {e11,e 1 2} represent the set of parameters. m(e11)= {(0,0.9),(j,0.7),(t,0.6),(r,0.4)}, m(e12)= {(0,0.8),(j,0.5),(t,0.3),(r,0.1)}. int. j. anal. appl. (2023), 21:60 7 here (m,c) is a fss over r. also, we can also see that m(γ) is a fi of r, ∀γ ∈ c. as a result, (m,c) is an ifsbr over r. theorem 5.1. assume (m,c) and (n,d) are two ifsbrs over r. then (m,c)e(n,d) is an ifsbr over r if it is non-null. proof. let (m,c)e (n,d) = (o,e), where o = c ∩d and oγ = mγ ∧nγ,∀γ ∈ e. suppose that (o,e) is non-null. if γ ∈ supp(o,e), then oγ = mγ ∧nγ 6= 0r. as a result, mγ and nγ are both fis of r. as a result, oγ is a fi of r, ∀γ ∈ supp(o,e). hence, (o,e) = (m,c)e (n,d) is an ifsbr over r. � theorem 5.2. assume (m,c) and (n,d) are two ifsbrs over r. if c and d are disjoint, then (m,c)d (n,d) is an ifsbr over r. proof. let (m,c)d (n,d)= (o,e), where e = c ∪d and (∀γ ∈ e)  oγ =   mγ if γ ∈ c −d nγ if ∈ d−c mγ ∨nγ if γ ∈ c ∩d   . (5.1) let us suppose that c ∩d = ∅. then either γ ∈ c −d or nγ ∈ d−c,∀γ ∈supp(o,e). if γ ∈ c −d,oγ = mγ is a fi of r. because (m,c) is an ifsbr over r. if γ ∈ d−c,oγ = nγ is a fi of r. because (n,d) is an ifsbr over r. thus, for all γ ∈ supp(o,d),oγ is a fi of r. consequently, (o,e)= (m,c)d (n,d) is an ifsbr over r. � theorem 5.2 is false generally if and only if c and d are not disjoint. consequently, the theorem is not generally true. because a ring’s fi may not be the union of two different fis of a ring r. theorem 5.3. assume (m,c) and (n,d) are two ifsbrs over r. then (m,c)∧̂(n,d) is an ifsbr over r if it is non-null. proof. let (m,c)∧̂(n,d) = (o,c × d), where o(α,β) = oα,β = mα ∧ nβ,∀(α,β) ∈ (c × d). assume (o,c×d) is non-null. if (α,β)∈supp(o,c×d), then oα,β = mα∧nβ 6= ∅. since (m,c) and (n,d) are ifsbrs over r, we can conclude that mα and nβ are both fis of r. as a result, oα,β is a fi of r, ∀(α,β)∈supp(o,c×d). thus, (o,c×d)= (m,c)∧̂(n,d) is an ifsbr over r. � 6. conclusion the concept of fsbrs is introduced and its individual properties are studied in this paper. the concepts of fsis of a fsbr and an ifsbr are also introduced. this research could be expanded to investigate the properties of fsss in other algebraic structures. 8 int. j. anal. appl. (2023), 21:60 acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] u. acar, f. koyuncu, b. tanay, soft sets and soft rings, computers math. appl. 59 (2010), 3458-3463. https: //doi.org/10.1016/j.camwa.2010.03.034. [2] b. ahmad, a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [3] h. aktaş, n. çag̃man, soft sets and soft groups, inform. sci. 177 (2007), 2726-2735. https://doi.org/10. 1016/j.ins.2006.12.008. [4] v.n. dixit, r. kumar, n. ajmal, on fuzzy rings, fuzzy sets syst. 49 (1992), 205-213. https://doi.org/10. 1016/0165-0114(92)90325-x. [5] w. liu, fuzzy invariant subgroups and fuzzy ideals, fuzzy sets syst. 8 (1982), 133-139. https://doi.org/10. 1016/0165-0114(82)90003-3. [6] p.k. maji, r. biswas, a.r. roy, fuzzy soft sets, j. fuzzy math. 9 (2001), 589-602. [7] p.k. maji, r. biswas, a.r. roy, soft set theory, computers math. appl. 45 (2003), 555-562. https://doi.org/ 10.1016/s0898-1221(03)00016-6. [8] d. molodtsov, soft set theory–first results, computers math. appl. 37 (1999), 19-31. https://doi.org/10. 1016/s0898-1221(99)00056-5. [9] d. pei, d. miao, from soft sets to information systems, in: 2005 ieee international conference on granular computing, ieee, beijing, china, 2005: pp. 617-621. https://doi.org/10.1109/grc.2005.1547365. https://doi.org/10.1016/j.camwa.2010.03.034 https://doi.org/10.1016/j.camwa.2010.03.034 https://doi.org/10.1155/2009/586507 https://doi.org/10.1155/2009/586507 https://doi.org/10.1016/j.ins.2006.12.008 https://doi.org/10.1016/j.ins.2006.12.008 https://doi.org/10.1016/0165-0114(92)90325-x https://doi.org/10.1016/0165-0114(92)90325-x https://doi.org/10.1016/0165-0114(82)90003-3 https://doi.org/10.1016/0165-0114(82)90003-3 https://doi.org/10.1016/s0898-1221(03)00016-6 https://doi.org/10.1016/s0898-1221(03)00016-6 https://doi.org/10.1016/s0898-1221(99)00056-5 https://doi.org/10.1016/s0898-1221(99)00056-5 https://doi.org/10.1109/grc.2005.1547365 1. introduction 2. preliminaries 3. fuzzy soft boolean rings 4. fuzzy soft ideals of fuzzy soft boolean rings 5. idealistic fuzzy soft boolean rings 6. conclusion references international journal of analysis and applications volume 19, number 5 (2021), 660-673 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-660 explicit and implicit crandall’s scheme for the heat equation with nonlocal nonlinear conditions bensaid souad1, dehilis sofiane2,∗, bouziani abdelfatah2 1institute of veterinary sciences, university of constantine i, constantine, 25000, algeria 2department of mathematics and informatics. larbi ben m’hidi university, oum el bouaghi, 04001, algeria ∗corresponding author: dehilissofiane@yahoo.fr abstract. in this paper, explicit and implicit crandall’s formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. the integrals in the boundary equations are approximated by the composite simpson quadrature rule. here nonlinear terms are approximated by richtmyer’s linearization method. finally, some numerical examples are given to show the effectiveness of the proposed method. 1. introduction this paper is concerned with the numerical solution of the heat equation (1.1) ∂u ∂t − ∂2u ∂x2 = f (x,t) , 0 < x < 1, 0 < t ≤ t, with the initial condition (1.2) u (x, 0) = ϕ (x) , 0 < x < 1, and the nonlinear nonlocal boundary conditions (1.3) u (0, t) = ∫ 1 0 p (x,t) uγ (x,t) dx + e (t) , 0 < t ≤ t, received may 4th, 2020; accepted may 21st, 2020; published august 2nd, 2021. 2010 mathematics subject classification. 35k58, 65l12. key words and phrases. nonlocal nonlinear conditions; explicit crandall’s scheme (ecs); implicit crandall’s scheme (ics). ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 660 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-660 int. j. anal. appl. 19 (5) (2021) 661 (1.4) u (1, t) = ∫ 1 0 q (x,t) uγ (x,t) dx + g (t) , 0 < t ≤ t, where f, ϕ, p, q, g and e are known functions, u (x,t) is unknown function to be determined. this kind of nonlocal boundary-value problem ( with γ = 1) occur in many fields of science and engineering, especially in thermoelasticity [7, 8, 10] thermodynamics [9], heat conduction [5, 6, 17, 28]. a lot of effort has been devoted in the past few years to the study of parabolic initial-boundary value problems which involve nonlocal boundary conditions of the type : u (0, t) = ∫ 1 0 p (x,t) u (x,t) dx + e (t) , 0 < t ≤ t, u (1, t) = ∫ 1 0 q (x,t) u (x,t) dx + g (t) , 0 < t ≤ t, the numerical solution has been considered in several papers [1, 2, 4, 11–13, 15, 16, 18, 19, 21–24, 26–28] . much less effort is given to the problem with nonlocal nonlinear type boundary conditions (3) and (4). recently, [3] proposed the implicit difference scheme for the solution of the heat equation with nonlinear nonlocal boundary condition. therefore this work is aimed at producing a very efficient techniques for solving the heat equation with nonlinear nonlocal boundary condition. the paper is organized as follows. the explicit and implicit crandall’s methods are employed for solving the one-dimensional heat equation with nonlinear nonlocal boundary value problem (1.1)-(1.4) in section 2 and section 3, respectively. to support our findings, we present results of numerical experiments in section 4. the conclusion is given in section 5. in [29], the authors consider to solve one-dimensional diffusion equation with nonlinear nonlocal boundary conditions (1.1)-(1.4), by using the forward time centered space (ftcs-nnc), dufort–frankel scheme (dfsnnc), backward time centered space (btcs-nnc), crank-nicholson method (cnm-nnc). crandall’s formulas in order to describe our method, we introduce the following notation. first, we take a positive integers n and m. we divide the intervals [0, 1] and [0,t] into m and n subintervals of equal lengths h = 1/m and k = t/n, respectively. by uni , we denote the approximation to u at the i th grid-point and nth time step. the grid point (xi, tn) are given by xi = ih, i = 0, 1, 2, . . . , m, tn = nk, n = 0, 1, 2, . . . , n. the notations uni , f n i , p n i , q n i , e n and gn are used for the finite difference approximations of u(xi, tn), f(xi, tn), p(xi, tn), q(xi, tn), e(tn) and g(tn), respectively. if we consider the modified equivalent equation of the crandall’s formula, the scheme shown in [20]. this scheme for the equation (1.1) can be written as: int. j. anal. appl. 19 (5) (2021) 662 (1.5) un+1i −u n i k − [( 1 2 + 1 12r ) uni+1 − 2u n i + u n i−1 h2 + ( 1 2 − 1 12r ) un+1i+1 − 2u n+1 i + u n+1 i−1 h2 ] = c00f n+1 i−1 + c01f n+1 i + c00f n+1 i+1 + c10f n i−1 + c11f n i + c10f n i+1, for i = 1, 2, ...,m −1, n = 0, 1, ...,n, and r = k/h2, where c00, c01, c10, c11 are coefficient to be determined . by using taylor series in (1.5), we obtain the local truncation error : −(2c00 + c01 + 2c10 + c11 − 1) f + 1 12 [ −6 ( 2c00 + 2c10 − 1 12 ) fxx − 6 (4c00 + 2c01 − 1) rft] h2 + o(h4). in order to achieve the fourth order, it is necessary that coefficients c00, c01, c10, c11 would satisfy conditions 2c00 + c01 + 2c10 + c11 − 1 = 0, 2c00 + 2c10 − 1 6 = 0, 4c00 + 2c01 − 1 = 0. we have chosen c00 = 1 12 , c10 = 0, c01 = 1 3 and c11 = 1 2 in our computation. after some rearrangement, the equation (1.5) becomes : (1.6) (1 − 6r)un+1i−1 + (10 + 12r)u n+1 i + (1 − 6r)u n+1 i+1 = (1 + 6r)uni−1 + (10 − 12r)u n i + (1 + 6r)u n i+1 +12k ( 1 12 fn+1i−1 + 1 3 fn+1i + 1 12 fn+1i+1 + 1 2 fni ) , for i = 1, 2, ...,m − 1, n = 0, 1, ...,n. 2. the explicit crandall’s scheme (ecs) if r = 1 6 , the crandall’s scheme (1.6) becomes explicite : (2.1) 12un+1i = 2u n i−1 + 8u n i + 2u n i+1 +12k ( 1 12 fn+1i−1 + 1 3 fn+1i + 1 12 fn+1i+1 + 1 2 fni ) , this scheme can be written as: (2.2) un+1i = 1 6 uni−1 + 2 3 uni + 1 6 uni+1 +k ( 1 12 fn+1i−1 + 1 3 fn+1i + 1 12 fn+1i+1 + 1 2 fni ) int. j. anal. appl. 19 (5) (2021) 663 for i = 1, 2, ...,m − 1, n = 0, 1, ...,n. we still have to determinates two unknowns u0 and um+1, for this we approximate integrals in (1.3) and (1.4) numerically by the composite simpson quadrature formula (we have chosen this approximation since it is of the same, fourth-order of accuracy in space as the methods used for the interior part of the problem) which requires m to be even. letting m = 2m, yields (2.3) un+10 = u ( 0, tn+1 ) = ∫ 1 0 p ( x,tn+1 ) u ( x,tn+1 ) dx = h 3 ( pn+10 ( un+10 )γ + 4 ∑m/2 i=1 p n+1 2i−1 ( un+12i−1 )γ +2 ∑m/2−1 i=1 p n+1 2i ( un+12i )γ + pn+1m ( un+1m )γ) + en+1 + o(h4), (2.4) un+1m = u ( 1, tn+1 ) = ∫ 1 0 q ( x,tn+1 ) u ( x,tn+1 ) dx = h 3 ( qn+10 ( un+10 )γ + 4 ∑m/2 i=1 q n+1 2i−1 ( un+12i−1 )γ +2 ∑m/2−1 i=1 q n+1 2i ( un+12i )γ + qn+1m ( un+1m )γ) + gn+1 + o(h4). thus, we can write (2.5) 3un+10 −hp n+1 0 (u n+1 0 ) γ −hpn+1m (u n+1 m ) γ = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ + 3en+1, (2.6) −hqn+10 (u n+1 0 ) γ + 3un+1m −hq n+1 m (u n+1 m ) γ = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ + 3gn+1. by applying the taylor’s expansion ( un+1i )γ = (uni ) γ + k ((uni )t) γ + .... = (uni ) γ + kγ (uni ) γ−1 ( un+1i −u n i k ) + ... = (uni ) γ + γ (uni ) γ−1 ( un+1i −u n i ) + ... hence to terms of order k, (2.7) ( un+1i )γ ≈ γ (uni ) γ−1 ( un+1i ) + (1 −γ) (uni ) γ , a result which replace the non-linear unknown ( un+1i )γ by approximation linear in un+1i (the richtmyer’s linearization method [25]). substituting (2.7) for i = 0 and i = m in (2.4) and (2.5), we have (2.8) ( 3 −hγpn+10 (u n 0 ) γ−1 ) un+10 −hp n+1 m γ(u n m ) γ−1un+1m = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ +h(1 −γ)pn+10 (u n 0 ) γ + h(1 −γ)pn+1m (u n m ) γ + 3en+1, int. j. anal. appl. 19 (5) (2021) 664 (2.9) −hqn+10 γ(u n 0 ) γ−1un+10 + (3 −hγq n+1 m (u n m ) γ−1)un+1m = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ +h(1 −γ)qn+10 (u n 0 ) γ + h(1 −γ)qn+1m (u n m ) γ + 3gn+1, hence we have: (2.10) un+10 = z1(3 −hγqn+1m (u n m ) γ−1) + z2hp n+1 m γ(u n m ) γ−1) y , (2.11) un+1m = z2(3 −hγpn+10 (u n 0 ) γ−1) + z1hq n+1 0 γ(u n 0 ) γ−1) y , where (2.12) z1 = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ +h(1 −γ)pn+10 (u n 0 ) γ + h(1 −γ)pn+1m (u n m ) γ + 3en+1, (2.13) z2 = 4 ∑m/2 i=1 p n+1 2i−1(u n+1 2i−1t) γ + 2 ∑m/2−1 i=1 p n+1 2i (u n+1 2i ) γ +h(1 −γ)qn+10 (u n 0 ) γ + h(1 −γ)qn+1m (u n m ) γ + 3gn+1, and (2.14) y = (3 −hγqn+1m (u n m ) γ−1)(3 −hγpn+10 (u n 0 ) γ−1) −h2γ2qn+10 (u n 0 ) γ−1pn+1m (u n m ) γ−1 6= 0 y 6= 0 for sufficiently small h. 3. the implicit crandall’s scheme (ics) if r 6= 1 6 , the crandall’s scheme (1.6) is implicite : (3.1) (1 − 6r)un+1i−1 + (10 + 12r)u n+1 i + (1 − 6r)u n+1 i+1 = (1 + 6r)uni−1 + (10 − 12r)u n i + (1 + 6r)u n i+1 +12k ( 1 12 fn+1i−1 + 1 3 fn+1i + 1 12 fn+1i+1 + 1 2 fni ) , for i = 1, 2, ...,m − 1, n = 0, 1, ...,n. equation (3.1) presents m − 1 linear equations in m + 1 unknowns u0,u1, · · · ,um. in order to solve the linear system, we need two more equations, that we can obtain approximating the integrals in (1.3) and int. j. anal. appl. 19 (5) (2021) 665 (1.4) numerically by the fourth-order simpson’s composite formula (which requires m to be even). letting m = 2m, yields (3.2) un+10 = u ( 0, tn+1 ) = ∫ 1 0 p ( x,tn+1 ) u ( x,tn+1 ) dx = h 3 ( pn+10 ( un+10 )γ + 4 ∑m/2 i=1 p n+1 2i−1 ( un+12i−1 )γ +2 ∑m/2−1 i=1 p n+1 2i ( un+12i )γ + pn+1m ( un+1m )γ) + en+1 + o(h4), (3.3) un+1m = u ( 1, tn+1 ) = ∫ 1 0 q ( x,tn+1 ) u ( x,tn+1 ) dx = h 3 ( qn+10 ( un+10 )γ + 4 ∑m/2 i=1 q n+1 2i−1 ( un+12i−1 )γ +2 ∑m/2−1 i=1 q n+1 2i ( un+12i )γ + qn+1m ( un+1m )γ) + gn+1 + o(h4). by applying the richtmyer’s linearization method (2.7) in (3.2) and (3.3), we get (3.4) un+10 = h 3 ( pn+10 ( γ (un0 ) γ−1 ( un+10 ) + (1 −γ) (un0 ) γ ) +4 ∑m/2 i=1 p n+1 2i−1 ( γ ( un2i−1 )γ−1 ( un+12i−1 ) + (1 −γ) ( un2i−1 )γ) +2 ∑m/2−1 i=1 p n+1 2i ( γ (un2i) γ−1 ( un+12i ) + (1 −γ) (un2i) γ )) +h 3 pn+1m ( γ (unm ) γ−1 ( un+1m ) + (1 −γ) (unm ) γ ) + en+1, (3.5) un+1m = h 3 ( qn+10 ( γ (un0 ) γ−1 ( un+10 ) + (1 −γ) (un0 ) γ ) +4 ∑m/2 i=1 q n+1 2i−1 ( γ ( un2i−1 )γ−1 ( un+12i−1 ) + (1 −γ) ( un2i−1 )γ) +2 ∑m/2−1 i=1 q n+1 2i ( γ (un2i) γ−1 ( un+12i ) + (1 −γ) (un2i) γ )) +h 3 qn+1m ( γ (unm ) γ−1 ( un+1m ) + (1 −γ) (unm ) γ ) + gn+1. thus, we can write (3.4) and (3.5) as follows (3.6) ( γhpn+10 (u n 0 ) γ−1 − 3 ) un+10 + 4γh ∑m/2 i=1 p n+1 2i−1 ( un2i−1 )γ−1 ( un+12i−1 ) +2γh ∑m/2−1 i=1 p n+1 2i (u n 2i) γ−1 ( un+12i ) + γhpn+1m (u n m ) γ−1 ( un+1m ) = (γ − 1) hpn+10 (u n 0 ) γ + 4 (γ − 1) h ∑m/2 i=1 p n+1 2i−1 ( un2i−1 )γ +2 (γ − 1) h ∑m/2−1 i=1 p n+1 2i (u n 2i) γ + (γ − 1) hpn+1m (u n m ) γ − 3en+1, (3.7) γhqn+10 (u n 0 ) γ−1 un+10 + 4γh ∑m/2 i=1 q n+1 2i−1 ( un2i−1 )γ−1 ( un+12i−1 ) +2γh ∑m/2−1 i=1 q n+1 2i (u n 2i) γ−1 ( un+12i ) + ( γhqn+1m (u n m ) γ−1 − 3 )( un+1m ) = (γ − 1) hqn+10 (u n 0 ) γ + 4 (γ − 1) h ∑m/2 i=1 q n+1 2i−1 ( un2i−1 )γ +2 (γ − 1) h ∑m/2−1 i=1 q n+1 2i (u n 2i) γ + (γ − 1) hqn+1m (u n m ) γ − 3gn+1, then, we have (3.8) an0u n+1 0 + a n 1u n+1 1 + a n 2u n+1 2 + ... + a n m−1u n+1 m−1 + a n mu n+1 m = l n m, int. j. anal. appl. 19 (5) (2021) 666 where (3.9)   an0 = γhp n+1 0 (u n 0 ) γ−1 − 3 anm = γhp n+1 m (u n m ) γ−1 an2i+1 = 4γhp n+1 2i+1 ( un2i+1 )γ−1 i = 0, 1, 2, · · · , m 2 − 1, an2i = 2γhp n+1 2i (u n 2i) γ−1 i = 1, 2, · · · , m 2 − 1, and (3.10) lnm = (γ − 1) hp n+1 0 (u n 0 ) γ + 4 (γ − 1) h ∑m/2 i=1 p n+1 2i−1 ( un2i−1 )γ +2 (γ − 1) h ∑m/2−1 i=1 p n+1 2i (u n 2i) γ + (γ − 1) hpn+1m (u n m ) γ , and also (3.11) bn0u n+1 0 + b n 1u n+1 1 + b n 2u n+1 2 + ... + b n m−1u n+1 m−1 + b n mu n+1 m = k n m, where (3.12)   bn0 = γhq n+1 0 (u n 0 ) γ−1 bnm = γhq n+1 m (u n m ) γ−1 − 3 bn2i+1 = 4γhq n+1 2i+1 ( un2i+1 )γ−1 i = 0, 1, 2, · · · , m 2 − 1, bn2i = 2γhq n+1 2i (u n 2i) γ−1 i = 1, 2, · · · , m 2 − 1, and (3.13) knm = (γ − 1) hq n+1 0 (u n 0 ) γ + 4 (γ − 1) h ∑m/2 i=1 q n+1 2i−1 ( un2i−1 )γ +2 (γ − 1) h ∑m/2−1 i=1 q n+1 2i (u n 2i) γ + (γ − 1) hqn+1m (u n m ) γ , combining (3.8), (3.11), with (3.1) yields an (m + 1) × (m + 1) linear system of equations. we write the system in the matrix form (3.14) an+1un+1 = bn+1, which an+1 =   an0 a n 1 a n 2 a n 3 a n m−2 a n m−1 a n m 1 − 6r 10 + 12r 1 − 6r 0 · · · 0 0 1 − 6r 10 + 12r 1 − 6r . . . ... ... . . . . . . . . . 0 0 . . . . . . 1 − 6r 10 + 12r 1 − 6r bn0 b n 1 b n 2 b n m−2 b n m−1 b n m   , int. j. anal. appl. 19 (5) (2021) 667 un+1 =   un+10 un+11 un+12 ... un+1m−1 un+1m   , b n+1 =   (γ − 1)h ( pn+10 (u n 0 ) γ + 4 ∑m/2 i=1 pn+12i−1 (u n 2i−1) γ + 2 ∑m/2−1 i=1 pn+12i (u n 2i) γ + pn+1m (u n m) γ ) − 3en+1 (1 + 6r)un0 + (10 − 12r)un1 + (1 + 6r)un2 + 12k( 112f n+1 0 + 1 3 fn+11 + 1 12 fn+12 + 1 2 fn1 ) ... (1 + 6r)unm−2 + (10 − 12r)u n m−1 + (1 + 6r)u n m + 12k( 1 12 fn+1m−2 + 1 3 fn+1m−1 + 1 12 fn+1m + 1 2 fnm−1) (γ − 1)h ( qn+10 (u n 0 ) γ + 4 ∑m/2 i=1 qn+12i−1 (u n 2i−1) γ + 2 ∑m/2−1 i=1 qn+12i (u n 2i) γ + qn+1m (u n m) γ ) − 3gn+1   where an0 , a n 1 ,a n 2 , ...,a n m−1,a n m and b n 0 ,b n 1 ,b n 2 , ...,b n m−1,b n m are the coefficients in (3.9) and (3.12), respectively. this procedure is unconditionally von neumann stable [14] for all r > 0. theorem 3.1. the icm scheme has a unique solution for sufficiently small h . proof. it is easy to see that |10 + 12r| > |2 + 12r|. the matrix (3.14) is diagonally dominant, if |an0 | > m∑ i=1 |ani | and |b n m| > m−1∑ i=0 |bni | i.e (3.15) h m∑ i=0 γωi|pn+1i (u n i ) γ−1 | < 1, and h m∑ i=0 γωi|qn+1i (u n i ) γ−1 | < 1 where ω0 = ωm = 1 3 , ω2i = 2 3 , i = 1, ...,m−1 and ω2i+1 = 43, i = 0, ...,m−1. as (3.15) is true for sufficiently small h, the existence and uniqueness of the solution of ics scheme are proved. � 4. numerical experiments to test the above algorithms described in section 2-3 , we use two examples with known analytical solutions as follows: example 4.1. we consider the following problem (test given in paper [3]) (4.1) ∂u ∂t − ∂2u ∂x2 = −2 ( x2 + t + 1 ) (t + 1) 3 , 0 < x < 1, 0 < t ≤ t, subject to the initial condition (4.2) u (x, 0) = x2, 0 ≤ x < 1, int. j. anal. appl. 19 (5) (2021) 668 and the nonlinear nonlocal boundary conditions (4.3) u (0, t) = ∫ 1 0 xu2 (x,t) dx− 1 6 (t + 1) 4 , 0 < t ≤ t, (4.4) u (1, t) = ∫ 1 0 xu2 (x,t) dx + 6x2 + 12t + 5 6 (t + 1) 4 , 0 < t ≤ t, the functions f,ϕ, p, q, e and g are chosen so that the function (4.5) u (x,t) = ( x t + 1 )2 . is the exact solution solution of the problem(1.1)-(1.4). in table 1 and table 2 we present results with h = 0.05, 0.005 using the crandall’s scheme discussed in section 2-3 together with the results from [3] for x = 0.1 and t = 0.01, 0.02, 0.03, ..., 0.1. table 3 gives the maximum errors of the numerical solutions experimental order of convergence. the maximum error is defined as follows er(h,k) = ‖u−uhk‖∞ = max 0≤k≤n { max 0≤i≤m |u(xi, tk) −uki |}, and the experiment order convergence is calculated using the formula : order = ln (er (hi−1,ki−1) /er (hi,ki)) ln (hi−1/hi) . ti exact ecs ics from [3] 0.01 0.0098029 0.00980407 0.0098037 0.0093 0.02 0.0096116 0.00961317 0.0096126 0.0091 0.03 0.0094259 0.00942762 0.0094270 0.0090 ... ... ... ... ... 0.1 0.0082644 0.00826636 0.0082657 0.0079 table 1. some numerical results at x = 0.1 with h = 0.05 for example 1 figs 1-2 illustrates the exact solution and an approximate solution of example 1 by ecs and ics, respectively. we plot the errors eni = u(xi, tn)−u n i , i = 0, 1, 2, ...,m, n = 0, 1, ...,n, for the schemes ecs and ics in figure 3. int. j. anal. appl. 19 (5) (2021) 669 ti exact ecs ics from [3] 0.01 0.0098029 0.0098029 0.0098029 0.0098 0.02 0.0096116 0.0096116 0.0096116 0.0096 0.03 0.0094259 0.0094259 0.0094259 0.0094 ... ... ... ... ... 0.1 0.0082644 0.0082644 0.0082644 0.0083 table 2. some numerical results at x = 0.1 with h = 0.005 for example 1 m ecs order ics order 12 1.700987 · 10−6 1.356601 · 10−6 24 1.064666 · 10−7 3.9978 8.483043 · 10−8 3.9992 48 6.658956 · 10−9 3.9989 5.302550 · 10−9 3.9998 96 4.163736 · 10−10 3.9993 3.314515 · 10−10 3.9998 table 3. the maximum errors and experiment order of convergence for example 1 . (a) (b) figure 1. (a) exact and (b) approximate solution by ics for example 1. example 4.2. the second test example to be solved is (4.6) ∂u ∂t − ∂2u ∂x2 = ( 1 + π2 ) exp (t) cos (πx) , 0 < x < 1, 0 < t ≤ t, with the initial condition (4.7) u (x, 0) = cos (πx) , 0 < x < 1, int. j. anal. appl. 19 (5) (2021) 670 (a) (b) figure 2. (a) exact and (b) approximate solution by ecs for example 1. and the nonlinear nonlocal boundary conditions (4.8) u (0, t) = ∫ 1 0 sin (πx) u3 (x,t) dx + exp (t) , 0 < t ≤ t, (4.9) u (1, t) = ∫ 1 0 sin (πx) u3 (x,t) dx− exp (t) , 0 < t ≤ t. the analytic solution is (4.10) u (x,t) = cos (πx) exp (t) . in table 4 and table 5 we present results with h = 0.05, 0.005 and r = 0.4 using the finite difference formulate discussed in section 2-3 for x = 0.1 and t = 0.01, 0.02, 0.03, ..., 0.1. figures 3-4 illustrates the exact solution and an approximate solution of example 2 by ecs and ics, respectively. ti exact ecs ics 0.01 0.96061479 0.96061492 0.96061503 0.02 0.97026913 0.97026933 0.97026948 0.03 0.98002050 0.98002074 0.98002092 ... ... ... ... 0.1 1.05108000 1.05108034 1.05108060 table 4. some numerical results at x = 0.1 for h = 0.05 for example 2 int. j. anal. appl. 19 (5) (2021) 671 ti exact ecs ics 0.01 0.96061479 0.96061479 0.96061479 0.02 0.97026913 0.97026913 0.97026913 0.03 0.98002050 0.98002050 0.98002050 ... ... ... ... 0.1 1.05108000 1.05108000 1.05108000 table 5. some numerical results at x = 0.1 for h = 0.005 for example 2 (a) (b) figure 3. (a) exact and (b) approximate solution by ics for example 2. (a) (b) figure 4. (a) exact and (b) approximate solution by ecs for example 2. int. j. anal. appl. 19 (5) (2021) 672 5. conclusion in this paper new techniques were applied to the one-dimensional heat equation with nonlinear nonlocal boundary conditions. the numerical results obtained by using the methods described in this article give acceptable results and suggests convergence to the exact solution when h goes to zero. the ecs method is explicit and require less computational time than the implicit crandall’s scheme, but the disadvantage of this discretization is the restriction in choosing the value of r ( r = 1 6 ) . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] w.t. ang, a method of solution for the one-dimensional heat equation subject to nonlocal conditions, southeast asian bull. math. 26 (2003), 185–191. [2] s. bensaid, a. bouziani, m. zereg, backward euler method for the diffusion equation with integral boundary specifications, j. pure appl. math.: adv. appl. 2 (2009), 169-185. [3] a. borhanifar, m.m. kabir, a.h. pour a numerical method for solution of the heat equation with nonlocal nonlinear condition, world appl. sci. j. 13 (2011), 2405-2409. [4] a. borhanifar, s. shahmorad, e. feizi, a matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions, numer. algorithms, 74 (2017), 1203–1221. [5] a. bouziani, mixed problem with boundary integral conditions for a certain parabolic equation, j. appl. math. stoch. anal. 9 (1996), 323–330. [6] j.r. cannon, the solution of the heat equation subject to the specification of energy, quart. appl. math. 21 (1963), 155-160. [7] w.a. day, extension of a property of the heat equation to linear thermoelasticity and other theories, quart. appl. math. 40 (1982), 319-330. [8] w.a. day, a decreasing property of solutions of parabolic equations with applications to thermoelasticity, quart. appl. math. 41 (1983), 468-475. [9] w.a. day, parabolic equations and thermodynamics, quart. appl. math. 50 (1992), 523-533. [10] g. dagan, the significance of heterogeneity of evolving scales to transport in porous formations, water resour. res. 30 (1994), 3327–3336. [11] s. delihis, a. bouziani, t.e. oussaeif, study of solution for a parabolic integrodifferential equation with the second kind integral condition, int. j. anal. appl. 16 (2018), 569-593. [12] m. dehghan, numerical solution of a parabolic equation with non-local boundary specifications, appl. math. comput. 145 (2003), 185-194. [13] m. dehghan, application of the adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications, appl. math. comput. 157 (2004), 549-560. [14] m. dehghan, efficient techniques for the second-odrer parabolic equation subject to nonlocal specifications, appl. numer. math. 52 (2005), 39-62. [15] g. ekolin, finite difference methods for a nonlocal boundary value problem for the heat equation, bit numer. math. 31 (1991), 245-261. int. j. anal. appl. 19 (5) (2021) 673 [16] m. javidi, the mol solution for the one-dimentional heat equation subject to nonlocal conditions, int. math. forum, 12 (2006), 597-602. [17] b. kawohl, remark on a paper by w.a. day on a maximum principle under nonlocal boundary conditions, quart. appl. math. 44 (1987), 751-752. [18] j. mart́ın-vaquero, j. vigo-aguiar, a note on efficient techniques for the second-order parabolic equation subject to non-local conditions, appl. numer. math. 59 (2009), 1258–1264. [19] j. mart́ın-vaquero, a. queiruga-dios, a.h. encinas, numerical algorithms for diffusion–reaction problems with nonclassical conditions, appl. math. comput. 218 (2012), 5487–5495. [20] j. mart́ın-vaquero, s. sajavičius, the two-level finite difference schemes for the heat equation with nonlocal initial condition, appl. math. comput. 342 (2019), 166–177. [21] k.w. morton, d.f. mayers, numerical solution of partial differential equations, cambridge university press, cambridge, 1994. [22] c.v. pao, numerical solutions of reaction–diffusion equations with nonlocal boundary conditions, j. comput. appl. math. 136 (2001), 227–243. [23] l.s. pulkina, a.b. beylin, nonlocal approach to problems on longitudinal vibration in a short bar, electron. j. differ. equ. 2019 (2019), 29. [24] m. slodička, and s.dehilis, a numerical approach for a semilinear parabolic equation with a nonlocal boundary condition, j. comput. appl. math. 231 (2009), 715-724. [25] g.d. smith, numerical solution of partial differential equations: finite difference methods, clarendon press, oxford, 1985 [26] s. wang, the numerical method for the conduction subject to moving boundary energy specification, numer. heat transfer, 130 (1990), 35-38. [27] s. wang, y. lin, a finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation, inverse probl. 5 (1989), 631–640. [28] z.-z. sun, a high-order difference scheme for a nonlocal boundary-value problem for the heat equation, comput. methods appl. math. 1 (2001), 398–414. [29] a. bouziani, s. bensaid, s. dehilis, a second order accurate difference scheme for the diffusion equation with nonlocal nonlinear boundary conditions, j. phys. math. 11 (2020), art. id 2, 1-7. 1. introduction crandall's formulas 2. the explicit crandall's scheme (ecs) 3. the implicit crandall's scheme (ics) 4. numerical experiments 5. conclusion references int. j. anal. appl. (2023), 21:58 analysis of the logistic growth model with taylor matrix and collocation method elçin çelik1,∗, deniz uçar2 1department of mathematics, faculty of science, izmir institute of technology, 35430, izmir, turkey 2department of mathematics, uşak university 1 eylul campus, 64200, uşak, turkey ∗corresponding author: elcincelik@iyte.edu.tr abstract. early analysis of infectious diseases is very important in the spread of the disease. the main aim of this study is to make important predictions and inferences for covid 19, which is the current epidemic disease, with mathematical modeling and numerical solution methods. so we deal with the logistic growth model. we obtain carrying capacity and growth rate with turkey epidemic data. the obtained growth rate and carrying capacity is used in the taylor collocation method. with this method, we estimate and making predictions close to reality with maple. we also show the estimates made with the help of graphics and tables. 1. introduction people have made many studies based on science to explain the natural phenomena they encounter. this situation paved the way for the birth of many sciences such as mathematics, physics, engineering, astronomy, chemistry and biology. these sciences converged at one point over time, allowing the emergence of many fields in interaction with each other, generally under the title of applied mathematics. in these areas, mathematical models are used to represent the behavior of a system mathematically. mathematical models are very important in that they provide information about the problem and allow predictions to be made. most mathematical models are built with differential equations. roughly, differential equations are equations that contain derivatives of an unknown function, which we call y(x) and which we want to determine from the equation. differential equations are of great importance in engineering, because many physical laws and relations appear mathematically in the form of differential equations. in addition, whenever a physical law involves a rate of change received: may 7, 2023. 2020 mathematics subject classification. 92d25, 92d30, 65h05, 65k05. key words and phrases. logistic growth model; covid 19; taylor polynomials and series; collocation points. https://doi.org/10.28924/2291-8639-21-2023-58 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-58 2 int. j. anal. appl. (2023), 21:58 of a function, such as velocity, acceleratin, etc., it will be lead to a differential equation. for this reason differential equations occur frequently in physics and engineering. analytical solutions of these differential equations can be made using various techniques, or many approximate solution methods are presented in cases where analytical solutions cannot be obtained or are difficult to obtain. in this study, the differential equation used in engineering, such as the speed of a falling object, the current in a series-connected electrical circuit and the cooling of the objects, radioactivity, radiocarbondating, newton laws of cooling, etc., has been used in the modeling of epidemic disease [3]. mathematical methods used in epidemiology get important contributions to the spread, occurrence, analysis and control of epidemics [5]. mathematical modeling is appropriate in epidemic diseases and models are generally intented to explain the course of the disease [13]. mathematical epidemiology is distinct from other epidemiological sciences. because experiments and observations are not possible for the validity of most models in this field [8]. verhulst has developed a linear mathematical model to the population problem, assuming the population is a continuous function of time. this mathematical model developed by verhulst is called logistic differential equation. the logistic growth model is written in the form: y ′ (t) = ry (t) ( 1− y (t) k ) (1.1) y (0) = λ where r > 0 and k > 0 are respectively growth rate and carrying capacity. according to the logistic growth model, growth increases rapidly at the beginning. however growth decreases as it approaches the carrying capacity. the logistic growth model can also be used in the growth of epidemics. as the carrying capacity in the growth of infectious deseases can be taken as the total number of people in the world, it can be defined and modeled as a population growth. therefore, the logistic growth model can be used to observe the largest outbreak epidemic disease of the present time, covid 19, and to predict its spread. covid 19, which emerged in wuhan, china in december 2019 and continues to spread by affecting the whole world in a very short time, is a virus that shows symptoms of respiratory infection. in order to observe the covid 19 epidemic and make prediction about the epidemic, some scientists have applied the epidemic data to the logistic growth model. for more details, we refer the reader to [4,6,7,9,10], the aim of study is to apply the taylor matrix and collocation method [11, 12], which has been used for differential equations in the literature, to the logistic growth model with covid 19 datas of turkey. we develop a method to determine the growth rate carrying capacity and the series truncate limit to give the best approach. 2. method in this study, we deal with the logistic growth model for modelling the total number of cases of covid 19. we use the taylor matrix and collocation method with maple to obtain the numerical solutions of this model. error analysis is performed to show the sensitivity and accuracy of the method presented. int. j. anal. appl. (2023), 21:58 3 2.1. carrying capacity in logistic differential equations. y ′ (t) = ry (t) ( 1− 1 k y (t) ) (2.1) y (0) = y0 the solution of the logistic initial value problem is y (t)= ky0 y0 +(k −y0)e−rt (2.2) if we choose y0 = k in (2.2), we obtain y = k. this is called equilibrium population. one can see easily that y ′ > 0 for 0 < y0 < k, we have y (t) < k. on the other hand if y0 > k, we have y ′ < 0 and y (t) > k. in all cases, the positive or negative numbers in the denominator will be less than y0 in absolute value and will approach 0 when t →∞. hence we obtain lim t→∞ y (t)= k . it can be concluded that a population that satisfies the logistic differential equation given the initial condition does not increase indefinitely. that is, when t →∞ converges to the finite limit population. that is called the carrying capacity [1]. 2.2. method of solution for logistic model. the logistic differential equation is defined as y ′ (t)= ry (t) ( 1− 1 k y (t) ) (2.3) where r > 0 k > 0, y (t)= n∑ n=0 ynt n, yn = yn n! , 0≤ t ≤ m (2.4) with the initial condition nth order. our aim in this section is to obtain the numerical solution in terms of truncated taylor series forms. we consider the approximate solution y (t) and its derivative defined by truncated taylor series (2.4). if we write (2.4) in the matrix form, we obtain y (t)= s (t)y (2.5) where s (t)= [ 1 t t2 · · · tn ] 1x(n+1) y =   y0 y1 ... yn   (n+1)x1 . if we write (2.4) derivative in the matrix form, we have y ′ (t)= s (t)by (2.6) 4 int. j. anal. appl. (2023), 21:58 where b =   0 1 0 · · · 0 0 0 2 · · · 0 ... ... ... ... ... 0 0 0 · · · n 0 0 0 · · · 0   (n+1)x(n+1) . similarly, we can write y2 (t)= s (t)s∗ (t)y ∗ (2.7) where s∗ (t)=   s (t) 0 · · · 0 0 s (t) · · · 0 ... ... ... ... 0 0 · · · s (t)   (n+1)x(n+1) 2 y ∗ =   y0y y1y ... yny   (n+1) 2 x1 . by substituting (2.5), (2.6) and (2.7) into eq (2.3), we obtain the matrix equation as: −rs (t)y +s (t)by + r k s (t)s∗ (t)y ∗ =0 . (2.8) thus, the matrix representation of the logistic differential equation can be written as m (t)y +d(t)y ∗ = g (t) (2.9) where m (t) = −rs (t)+s (t)b d(t) = r k s (t)s∗ (t) g (t) = 0 . if the collocation points which are defined as ti = m n i, i =0,1, ...,n substitude into eq (2.9), we obtain m (ti)y +d(ti)y ∗ = g (ti) , i =0,1, ...,n (2.10) or the matrix equation my +dỹ = g (2.11) int. j. anal. appl. (2023), 21:58 5 where m   m (t0) 0 · · · 0 0 m (t1) · · · 0 ... ... ... ... 0 0 · · · m (tn)   (n+1)x(n+1) 2 y =   y y ... y   (n+1) 2 x1 d =   d(t0) 0 · · · 0 0 d(t1) · · · 0 ... ... ... ... 0 0 · · · d(tn)   (n+1)x(n+1) 3 ỹ =   y ∗ y ∗ ... y ∗   (n+1) 3 x1 . the matrix form of the condition is written as s (0)y = λ (2.12) and we replaces row matrix (2.12) by any row of the matrix (2.11) and we get the new system depending on conditions. thus, the logistic equation is transformed into a nonlinear equation under initial conditions. then, if we calculate the unknown coefficients and replaced in truncated taylor series in nth order, a taylor polynomial solution [2]. 3. numerical application in this section, we apply the covid 19 data to the logistic growth model, which are received from ministry of health (turkey). we obtain necessary results by analyzing the taylor collocation method using maple program. 3.1. turkey and covid 19. 6 int. j. anal. appl. (2023), 21:58 3.1.1. the numbers of death. if we solve the logistic differential equation (2.2), we obtain 37= k 1+(k −1)e−6r (3.1) for t =6 and 356= k 1+(k −1)e−16r (3.2) for t = 16. we calculate the growth rate and carrying capacity for the numbers of death as 0.6193081195 and 362.3979687 using eqs (3.1) and (3.2) with maple program, respectively. we plot the figure 1 using these values with maple program. figure 1 gives for 480 days comparison of the approximate solution with the numbers of actual death in the following. we can see the rate of the numbers of death gradually increases. actual predicted 0 10000 20000 30000 40000 50000 number of people 100 200 300 400 day figure 1. time evolution of the numbers of death predicted by the logistic growth model with taylor collocation method and the numbers of actual daeath, which number of healthy people living in turkey from the date of the first case march 11, 2020 through july 3, 2021, obtained by eq. (2.1). the parameters are r=0.6193081195 and k=362.3979687. 3.1.2. the numbers of case. if we solve the logistic differential equation (2.2), we obtain 98= k 1+(k −1)e−6r (3.3) for t =6 and 460916= k 1+(k −1)e−258r (3.4) for t = 258. we calculate the growth rate and carrying capacity for the numbers of case as −3.548090934 and 1.979327005 using eqs (3.3) and (3.4) with maple program, respectively. we plot the figure 2 using these values with maple program. figure 2 gives for 480 days comparison of int. j. anal. appl. (2023), 21:58 7 the approximate solution with the numbers of actual case in the following. we can see the rate of the numbers of case gradually increases. actual predicted 0 1e+06 2e+06 3e+06 4e+06 5e+06 number of people 100 200 300 400 day figure 2. time evolution of the numbers of case predicted by the logistic growth model with taylor collocation method and the numbers of actual case, which number of healthy people living in turkey from the date of the first case march 11, 2020 through july 3, 2021, obtained by eq. (2.1). the parameters are r=−3.548090934 and k=1.979327005. 3.1.3. the numbers of total healing. if we solve the logistic differential equation (2.2), we obtain 10453= 42k 42+(k −42)e−22r (3.5) for t =22 and 11976= 42k 42+(k −42)e−23r (3.6) for t = 23. we calculate the growth rate and carrying capacity for the numbers of total healing as 0.4698047529 and 664420.9304 using eqs (3.5) and (3.6) with maple program, respectively. we plot the figure 3 using these values with maple program. figure 3 gives for 480 days comparison of the approximate solution with the numbers of actual total healing in the following. we can see the rate of the numbers of total healing gradually increases. 8 int. j. anal. appl. (2023), 21:58 actual predicted 0 10000 20000 30000 40000 50000 number of people 100 200 300 400 day figure 3. time evolution of the numbers of total healing predicted by the logistic growth model with taylor collocation method and the numbers of actual total healing, which number of healthy people living in turkey from the date of the first case march 11, 2020 through july 3, 2021, obtained by eq. (2.1). the parameters are r=0.4698047529 and k=664420.9304. 4. conclusion in this study, the numbers of death, the numbers of total healing and the numbers of case, associated with covid 19 are handled in turkey. applying datas to the logistic growth model, we have tried to estimates that can be used to quantify the temporal evolution of covid 19 in the country using taylor matrix and collocation method. the numbers of death, the numbers of case and the numbers of total healing are solved for n = 3 in figure 1 figure 3, respectively. it is possible to make predictions for the occurences, progression and analysis of covid 19 with this method. also, this process is very useful as the results can be obtained easily with a computer program. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] c.h. edwards, d.e. penny, d.t. calvis, differential equations and boundary value problems computing and modeling, 5th edition, pearson, london, 2015. [2] e. gökmen and e. çelik, a numerical method for solving continuous population models for single and interacting species, sakarya univ. j. sci. 23 (2019), 403-412. https://doi.org/10.16984/saufenbilder.410641. [3] e. kreyszig, advanced engineering mathematics, 9th edition, john wiley & sons, new york, 2006. [4] e. pelinovsky, a. kurkin, o. kurkina, et al. logistic equation and covid-19, chaos solitons fractals. 140 (2020), 110241. https://doi.org/10.1016/j.chaos.2020.110241. [5] j.m. last, a dictionary of epidemiology, 2nd edition, oxford university press, oxford, 1988. https://doi.org/10.16984/saufenbilder.410641 https://doi.org/10.1016/j.chaos.2020.110241 int. j. anal. appl. (2023), 21:58 9 [6] k. roosa, y. lee, r. luo, et al. real-time forecasts of the covid-19 epidemic in china from february 5th to february 24th, 2020, infect. dis. model. 5 (2020), 256-263. https://doi.org/10.1016/j.idm.2020.02.002. [7] k. wu, d. darcet, q. wang, et al. generalized logistic growth modeling of the covid-19 outbreak: comparing the dynamics in the 29 provinces in china and in the rest of the world, nonlinear dyn. 101 (2020), 1561-1581. https://doi.org/10.1007/s11071-020-05862-6. [8] l.j. allen, f. brauer, p. van den driessche, et al. mathematical epidemiology, vol. 1945, springer, berlin, 2008. [9] m. batista, estimation of the final size of the coronavirus epidemic by the logistic model, preprint, 2020. https://doi.org/10.1101/2020.02.16.20023606. [10] m. jain, p.k. bhati, p. kataria, et al. modelling logistic growth model for covid-19 pandemic in india, in: 2020 5th international conference on communication and electronics systems (icces), ieee, coimbatore, india, 2020: pp. 784–789. https://doi.org/10.1109/icces48766.2020.9138049. [11] m. sezer, a method for the approximate solution of the second-order linear differential equations in terms of taylor polynomials, int. j. math. educ. sci. technol. 27 (1996), 821-834. https://doi.org/10.1080/ 0020739960270606. [12] m. sezer, a. karamete, m. gülsu, taylor polynomial solutions of systems of linear differential equations with variable coefficients, int. j. computer math. 82 (2005), 755-764. https://doi.org/10.1080/ 00207160512331323336. [13] n.m. ferguson, mathematical prediction in infection, medicine. 37 (2009), 507-509. https://doi.org/10.1016/ j.mpmed.2009.07.004. https://doi.org/10.1016/j.idm.2020.02.002 https://doi.org/10.1007/s11071-020-05862-6 https://doi.org/10.1101/2020.02.16.20023606 https://doi.org/10.1109/icces48766.2020.9138049 https://doi.org/10.1080/0020739960270606 https://doi.org/10.1080/0020739960270606 https://doi.org/10.1080/00207160512331323336 https://doi.org/10.1080/00207160512331323336 https://doi.org/10.1016/j.mpmed.2009.07.004 https://doi.org/10.1016/j.mpmed.2009.07.004 1. introduction 2. method 2.1. carrying capacity in logistic differential equations 2.2. method of solution for logistic model 3. numerical application 3.1. turkey and covid 19 4. conclusion references international journal of analysis and applications volume 19, number 3 (2021), 455-464 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-455 on rough fuzzy prime ideals in left almost semigroups ahmed elmoasry1,2,∗ 1department of mathematics, college of science al-zulfi, majmaah university, al-zulfi, saudi arabia 2mathematics department faculty of science, aswan university, aswan, egypt ∗corresponding author: a.elmoasry@mu.edu.sa abstract. in this paper we shall introduce the notion of rough prime ideals and rough fuzzy prime ideals in la-semigroups. we proved that the lower and the upper approximation of a prime ideal is a prime ideal and we also proved that a fuzzy subset f of an la-semigroup s is a fuzzy prime ideal of s iff fλ 6= ∅ (fsλ 6= ∅) is a prime ideal of s for every λ ∈ [0, 1]. 1. introduction the notion of a rough set was originally proposed by z. pawlak [26] as a formal tool for modeling and processing incomplete information in information systems. the theory of rough set is an extension of set theory. the equivalence classes are the building blocks for the construction of the lower and upper approximations. the lower approximation of a given set is the union of all equivalence classes which are subsets of the set, and the upper approximation is the union of all equivalence classes which have a nonempty intersection with the set. some authors have studied the algebraic properties of rough sets. biswas and nanda [3], introduced the notion of rough subgroups. kuroki, in [14], introduced the notion of a rough ideal in a semigroup. also, kuroki and mordeson in [13] studied the structure of rough sets and rough groups. y. b. jun applied the rough set theory to bck-algebras [9]. received december 9th, 2020; accepted january 7th, 2021; published april 28th, 2021. 2010 mathematics subject classification. 20m10, 20m12. key words and phrases. la-semigroups; rough prime ideals; rough fuzzy prime ideals. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 455 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-455 int. j. anal. appl. 19 (3) (2021) 456 a fuzzy subset f of a set s is a function from s to a closed interval [0, 1], this concept of a fuzzy set was introduced by zadeh [31], in 1965. rosenfeld [27], was the first who studied fuzzy sets in the structure of groups. kuroki [16], has studied the fuzzy ideals and fuzzy bi-ideals in semigroups. m. banerjee [2], give the concept of roughness of a fuzzy set. the fuzzy theory provides the underlying structure for the generalization of many fields including logic, differential equations and groups. the fuzzy theory on algebraic structures have been widely explored. this paper concerns the relationship between rough fuzzy sets and left almost semigroups. the left almost semigroup abbreviated as an la-semigroup, was first introduced by m. a. kazim and m. naseerudin [10]. they generalized some useful results of semigroup theory. they introduced braces on the left of the ternary commutative law abc = cba, to get a new pseudo associative law, that is (ab)c = (cb)a, and named it as left invertive law. later, q. mushtaq (in [19], [20], [23] and [25]) and others investigated the structure further and added many useful results to the theory of la-semigroups. an la-semigroup is the midway structure between a commutative semigroup and a groupoid. despite the fact, the structure is non-associative and noncommutative. it nevertheless possesses many interesting properties which we usually find in commutative and associative algebraic structures. q. mushtaq and s. m. yusuf produced useful results [20], on locally associative la-semigroups in 1979. in this structure they defined powers of an element and congruences using these powers. they constructed quotient la-semigroups using these congruences. relations between la-semigroups, quasi-groups, commutative monoids and commutative groups were established in [18], [21] and [22]. it is a useful nonassociative structure with wide applications in theory of flocks. in this paper, we have studied ρ-upper and ρ-lower rough prime ideals and also studied ρ-upper and ρ-lower rough fuzzy prime ideals in la-semigroups. 2. preliminaries let s be an la-semigroup. a subset a of an la-semigroup s is called an la-subsemigroup of s if aa ⊆ a. a subset a of an la-semigroup s is called a left [right] ideal of s if sa ⊆ a [as ⊆ a], and a is called a two sided ideal of s if it is both a left and a right ideal of s. let s denote an la-semigroup unless otherwise specified. let ρ be a congruence relation on s, that is, ρ is an equivalence relation on s such that (a,b) ∈ ρ implies (ax,bx) ∈ ρ and (xa,xb) ∈ ρ for all x ∈ s. if ρ is a congruence relation on s, then for every x ∈ s, [x]ρ stands for the congruence class of x with respect to ρ. a congruence ρ on s is called complete if [a]ρ[b]ρ = [ab]ρ for all a,b ∈ s. int. j. anal. appl. 19 (3) (2021) 457 definition 2.1. [1] let a be a nonempty subset of an la-semigroup s and ρ be a congruence relation on s. then the sets apr ρ (a) = { x ∈ s : [x]ρ ⊆ a } and aprρ(a) = { x ∈ s : [x]ρ ∩a 6= ∅ } are called ρ-lower and ρ-upper approximations of a respectively. for a nonempty subset a of s, aprρ(a) = (aprρ(a),aprρ(a)) is called a rough set with respect to ρ if apr ρ (a) 6= aprρ(a). a subset a of an la-semigroup s is called a ρ-upper [ρ-lower] rough ideal of s if aprρ(a) [aprρ(a)] is an ideal of s. theorem 2.1. [1] let ρ be a congruence relation on an la-semigroup s. if a is a left [right, two-sided] ideal of s. then (1) aprρ(a) is a left [right, two-sided] ideal of s. (2) if ρ is complete, then apr ρ (a) is, if it is nonempty, a left [right, two-sided] ideal of s. 3. rough prime ideals in la-semigroups an ideal a of an la-semigroup s is said to be a prime ideal of s, if for x,y ∈ s, xy ∈ a implies x ∈ a or y ∈ a. let ρ be a congruence relation on an la-semigroup s. then a subset a of s is called a ρ-upper rough prime ideal of s if aprρ(a) is a prime ideal of s. a ρ-lower rough prime ideal of s is defined analogously. a is called a rough prime ideal of s if a is a ρ-upper and a ρ-lower rough prime ideal of s. theorem 3.1. let ρ be a complete congruence relation on an la-semigroup s. if a is a prime ideal of s. then a is a ρ-upper rough prime ideal of s. proof. since a is a prime ideal of s, then by theorem 2.1(1), aprρ(a) is an ideal of s. then for xy ∈ aprρ(a) for some x,y ∈ s. then we have [xy]ρ ∩a = [x]ρ[y]ρ ∩a 6= φ. thus there exist a ∈ [x]ρ and b ∈ [y]ρ such that ab ∈ a. since a is a prime ideal, we have a ∈ a or b ∈ a. thus a ∈ [x]ρ ∩a or b ∈ [y]ρ ∩a. this implies [x]ρ ∩a 6= φ or [y]ρ ∩a 6= φ, and so x ∈ aprρ(a) or y ∈ aprρ(a). therefore aprρ(a) is a prime ideal of s. � int. j. anal. appl. 19 (3) (2021) 458 the following example shows that the upper approximation of a prime ideal is not a prime ideal in general on the same conditions of theorem 2.1(1). example 3.1. let s = {0, 1, 2, 3}, the binary operation ”·” on s be defined as follows: · 0 1 2 3 0 2 2 2 3 1 0 2 2 3 2 2 2 2 3 3 3 3 3 3 clearly, 0 = 1·(1·0) 6= (1·1)·0 = 2. this shows that s is an la-semigroup. now let ρ be a congruence relation on s such that ρ-congruence classes are the subsets {0}, {1}, {2, 3}. then for a = {3}⊆ s, aprρ(a) = {2, 3}. it is clear that a is a prime ideal of s. the set aprρ(a) is not a prime ideal for 0 · 1 = 2 ∈ aprρ(a) but 0 /∈ aprρ(a) and 1 /∈ aprρ(a). theorem 3.2. let ρ be a complete congruence relation on an la-semigroup s and a is a prime ideal of s. then apr ρ (a) is, if it is nonempty, a prime ideal of s. proof. since a is a prime ideal of s, then by theorem 2.1(2), we know that apr ρ (a) is an ideal of s. let xy ∈ apr ρ (a) for some x,y ∈ s, then [x]ρ[y]ρ = [xy]ρ ⊆ a. suppose x /∈ apr ρ (a) and y /∈ apr ρ (a). this implies [x]ρ * a and [y]ρ * a, then there exist a ∈ [x]ρ and b ∈ [y]ρ such that a,b /∈ a. thus ab ∈ [x]ρ[y]ρ = [xy]ρ ⊆ a. since a is a prime ideal, we have a ∈ a or b ∈ a. it contradicts the supposition. this means that apr ρ (a) is, if it is nonempty, a prime ideal of s. � we call a a rough prime ideal of s if it is both a ρ-upper and a ρ-lower rough prime ideal of s. from the above, we know that a prime ideal is a rough prime ideal with respect to a complete congruence relation on an la-semigroup. the following example shows that the converse of theorems 3.1 and 3.2, does not hold in general. int. j. anal. appl. 19 (3) (2021) 459 example 3.2. let s = {0, 1, 2, 3, 4}, the binary operation ”·” on s be defined as follows: · 0 1 2 3 4 0 0 0 0 0 4 1 0 0 3 0 4 2 0 1 2 3 4 3 0 0 1 0 4 4 4 4 4 4 4 clearly, 3 = 1·(2·2) 6= (1·2)·2 = 1. this shows that s is an la-semigroup. now let ρ be a complete congruence relation on s such that ρ-congruence classes are the subsets {0, 1, 2, 3}, {4}. then for a = {0, 4} ⊆ s, aprρ(a) = {0, 1, 2, 3, 4}, and aprρ(a) = {4}. it is clear that aprρ(a) and aprρ(a) are prime ideals of s. the ideal a is not a prime ideal for 1 · 3 = 0 ∈ a but 1 /∈ a and 3 /∈ a. 4. rough prime ideals in the quotient la-semigroups let ρ be a congruence relation on an la-semigroup s and a be a subset of s. the ρ-upper and the ρ-lower approximations can be presented in an equivalent form as shown below aprρ(a) = {[x]ρ ∈ s/ρ : [x]ρ ∩a 6= ∅} and apr ρ (a) = {[x]ρ ∈ s/ρ : [x]ρ ⊆ a}. now we discuss these sets as subsets of a quotient la-semigroup s/ρ of an la-semigroup s. theorem 4.1. [1] let ρ be a congruence relation on an la-semigroup s. if a is a left [right, two-sided] ideal of s. then (1) aprρ(a) is a left [right, two-sided] ideal of s/ρ. (2) apr ρ (a) is, if it is nonempty, a left [right, two-sided] ideal of s/ρ. theorem 4.2. let ρ be a complete congruence relation on an la-semigroup s. if a is a ρ-upper rough prime ideal of s, then aprρ(a) is a prime ideal of s/ρ. proof. since a is a ρ-upper rough prime ideal of s, then by theorem 4.1(1), we know that aprρ(a) is an ideal of s/ρ. suppose [x]ρ[y]ρ ∈ aprρ(a) for some [x]ρ, [y]ρ ∈ s/ρ such that [xy]ρ ∈ aprρ(a) for some [x]ρ, [y]ρ ∈ s/ρ then [xy]ρ ∩a 6= φ. thus xy ∈ aprρ(a). since a is a ρ-upper rough prime ideal of s, that is aprρ(a) is a prime ideal, thus we have x ∈ aprρ(a) or y ∈ aprρ(a) int. j. anal. appl. 19 (3) (2021) 460 so [x]ρ ∩a 6= φ or [y]ρ ∩a 6= φ. hence [x]ρ ∈ aprρ(a) or [y]ρ ∈ aprρ(a). therefore aprρ(a) is a prime ideal of s/ρ. this completes the proof. � theorem 4.3. let ρ be a complete congruence relation on an la-semigroup s. if a is a ρ-lower rough prime ideal of s, then apr ρ (a) is a prime ideal of s/ρ. proof. since a is a ρ-lower rough prime ideal of s, then by theorem 4.1(2), we know that apr ρ (a) is an ideal of s/ρ. suppose [x]ρ[y]ρ ∈ apr ρ (a) for some [x]ρ, [y]ρ ∈ s/ρ such that [xy]ρ ∈ apr ρ (a) for some [x]ρ, [y]ρ ∈ s/ρ then [xy]ρ ⊆ a. thus xy ∈ aprρ(a). since a is a ρ-lower rough prime ideal of s, that is aprρ(a) is a prime ideal, we have x ∈ apr ρ (a) or y ∈ apr ρ (a) so [x]ρ ⊆ a or [y]ρ ⊆ a. hence [x]ρ ∈ apr ρ (a) or [y]ρ ∈ apr ρ (a). therefore apr ρ (a) is a prime ideal of s/ρ. this completes the proof. � 5. rough fuzzy prime ideals in la-semigroups a function f from s to the unit interval [0, 1] is called a fuzzy subset of s. a fuzzy subset f of an la-semigroup s is called a fuzzy subsemigroup of s if f(xy) ≥ f(x) ∧f(y) for all x,y ∈ s. a fuzzy subset f of an la-semigroup s is called a fuzzy ideal of s if f(xy) ≥ f(x) ∨f(y) for any x,y ∈ s. let f be a fuzzy subset of s and λ ∈ [0, 1]. then the sets fλ = {x ∈ s : f(x) ≥ λ} and fsλ = {x ∈ s : f(x) > λ} are called, respectively, λ-levelset and λ-strong levelset of the fuzzy set f. theorem 5.1. let f be a fuzzy subset of an la-semigroup s. then (1) f is a fuzzy ideal of s iff fλ 6= ∅ is an ideal of s for every λ ∈ [0, 1]. (2) f is a fuzzy ideal of s iff fsλ 6= ∅ is an ideal of s for every λ ∈ [0, 1]. int. j. anal. appl. 19 (3) (2021) 461 proof. (1) assume f is a fuzzy ideal of s. then f(xy) ≥ f(x) ∨f(y) for any x,y ∈ s. assume fλ 6= ∅. let x ∈ fλ, y ∈ s. thus f(x) ≥ λ. since f is a fuzzy ideal of s, f(xy) ≥ f(x) ∨ f(y) ≥ f(x) ≥ λ. therefore xy ∈ fλ. similarly, yx ∈ fλ. hence fλ is an ideal of s. conversely, assume for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is an ideal of s. let x,y ∈ s. case 1 : f(x) ≥ f(y). let λ = f(x). then x ∈ fλ. by assumption, we have fλ is an ideal of s. so xy ∈ fλ. then f(xy) ≥ λ = f(x) = f(x) ∨f(y). case 2 : f(x) < f(y). let λ = f(y). then y ∈ fλ. by assumption, we have fλ is an ideal of s. so xy ∈ fλ. then f(xy) ≥ λ = f(y) = f(x) ∨f(y). therefore f is a fuzzy ideal of s. (2) assume f is a fuzzy ideal of s. then f(xy) ≥ f(x) ∨ f(y) for any x,y ∈ s. assume fsλ 6= ∅. let x ∈ fsλ , y ∈ s. thus f(x) > λ. since f is a fuzzy ideal of s, f(xy) ≥ f(x) ∨ f(y) ≥ f(x) > λ. therefore xy ∈ fsλ . similarly, yx ∈ f s λ . hence f s λ is an ideal of s. conversely, assume for all λ ∈ [0, 1], if fsλ 6= ∅, then f s λ is an ideal of s. let x,y ∈ s. case 1 : f(x) ≥ f(y). thus x ∈ fsλ for all λ < f(x). by assumption, we have f s λ is an ideal of s for all λ < f(x). so xy ∈ fsλ for all λ < f(x). then f(xy) > λ for all λ < f(x). then f(xy) ≥ f(x) = f(x)∨f(y). case 2 : f(x) < f(y). thus y ∈ fsλ for all λ < f(y). by assumption, we have f s λ is an ideal of s for all λ < f(y). so xy ∈ fsλ for all λ < f(y). then f(xy) > λ for all λ < f(y). then f(xy) ≥ f(y) = f(x) ∨f(y). therefore f is a fuzzy ideal of s. � let f be a fuzzy subset of s. let aprρ(f)(x) and aprρ(f)(x) be fuzzy subsets of s defined by aprρ(f)(x) = ∨ a∈[x]ρ f(a) and apr ρ (f)(x) = ∧ a∈[x]ρ f(a) are called, respectively, the ρ-upper and ρ-lower approximations of the fuzzy set f. aprρ(f) = (aprρ(f),aprρ(f)) is called a rough fuzzy set with respect to ρ if aprρ(f) 6= aprρ(f). theorem 5.2. let ρ be a complete congruence relation on an la-semigroup s. let f be a fuzzy subset of s. if f is a fuzzy ideal of s. then (1) aprρ(f) is a fuzzy ideal of s. (2) apr ρ (f) is, if it is nonempty, a fuzzy ideal of s. proof. (1) assume f is a fuzzy ideal of s. let x,y ∈ s. then f(xy) ≥ f(x) ∨f(y). we have aprρ(f)(xy) = ∨ s∈[xy]ρ f(s) = ∨ s∈[x]ρ[y]ρ f(s) = ∨ p∈[x]ρ, q∈[y]ρ f(pq) ≥   ∨ p∈[x]ρ f(p)  ∨   ∨ q∈[y]ρ f(q)   = aprρ(f)(x) ∨aprρ(f)(y). int. j. anal. appl. 19 (3) (2021) 462 then aprρ(f)(xy) ≥ aprρ(f)(x) ∨aprρ(f)(y). therefore we obtain that aprρ(f) is a fuzzy ideal of s. (2) assume f is a fuzzy ideal of s. let x,y ∈ s. then f(xy) ≥ f(x) ∨f(y). we have apr ρ (f)(xy) = ∧ s∈[xy]ρ f(s) = ∧ s∈[x]ρ[y]ρ f(s) = ∧ p∈[x]ρ, q∈[y]ρ f(pq) ≥   ∧ p∈[x]ρ f(p)  ∨   ∧ q∈[y]ρ f(q)   = apr ρ (f)(x) ∨apr ρ (f)(y). then apr ρ (f)(xy) ≥ apr ρ (f)(x) ∨ apr ρ (f)(y). therefore we obtain that apr ρ (f) is, if it is nonempty, a fuzzy ideal of s. this completes the proof. � a fuzzy ideal f of an la-semigroup s is called a fuzzy prime ideal of s if f(xy) = f(x) or f(xy) = f(y) for all x,y ∈ s. theorem 5.3. let f be a fuzzy subset of an la-semigroup s. then f is a fuzzy prime ideal of s iff fλ 6= ∅ is a prime ideal of s for every λ ∈ [0, 1]. proof. assume f is a fuzzy ideal of s. then f is a fuzzy ideal of s. assume fλ 6= ∅. by theorem 5.1, fλ is a ideal of s. let x,y ∈ s such that xy ∈ fλ. since f is a fuzzy prime ideal of s, f(xy) = f(x) or f(xy) = f(y). this implies x ∈ fλ or y ∈ fλ. therefore fλ is a prime ideal of s. conversely, assume for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is a prime ideal of s. let x,y ∈ s. by theorem 5.1, f is a fuzzy ideal of s. this implies f(xy) ≥ f(x) and f(xy) ≥ f(y). let λ = f(xy). thus xy ∈ fλ. since fλ is a prime ideal of s, x ∈ fλ or y ∈ fλ. this implies that f(x) ≥ λ = f(xy) or f(y) ≥ λ = f(xy). hence f(xy) = f(x) or f(xy) = f(y). hence f is a fuzzy prime ideal of s. � theorem 5.4. let f be a fuzzy subset of an la-semigroup s. then f is a fuzzy prime ideal of s iff fsλ 6= ∅ is a prime ideal of s for every λ ∈ [0, 1]. proof. assume f is a fuzzy prime ideal of s. then f is a fuzzy ideal of s. assume fsλ 6= ∅. by theorem 5.1, fsλ is a ideal of s. let x,y ∈ s such that xy ∈ f s λ. then f(xy) > λ. since f is a fuzzy prime ideal of s, f(xy) = f(x) or f(xy) = f(y). this implies that f(x) > λ or f(y) > λ. hence x ∈ fsλ or y ∈ f s λ. therefore fsλ is a prime ideal of s. conversely, assume for all λ ∈ [0, 1], if fsλ 6= ∅, then f s λ is a prime ideal of s. let x,y ∈ s. by theorem 5.1, f is a fuzzy ideal of s. this implies f(xy) ≥ f(x) and f(xy) ≥ f(y). we have xy ∈ fsλ for all λ < f(xy). since fsλ is a prime ideal of s for all λ < f(xy), x ∈ f s λ or y ∈ f s λ for all λ < f(xy). this implies that f(x) > λ or f(y) > λ for all λ < f(xy). then f(x) ≥ f(xy) or f(y) ≥ f(xy). hence f(xy) = f(x) or f(xy) = f(y). hence f is a fuzzy prime ideal of s. � int. j. anal. appl. 19 (3) (2021) 463 let ρ be a congruence relation on an la-semigroup s. a fuzzy subset f of s is called a ρ-upper [a ρ-lower] rough fuzzy prime ideal of s if aprρ(f) [aprρ(f)] is a fuzzy prime ideal of s. we call f a rough fuzzy prime ideal of s if it is both a ρ-upper and a ρ-lower rough fuzzy prime ideal of s. lemma 5.1. let ρ be a congruence relation on an la-semigroup s. if f is a fuzzy subset of s and λ ∈ [0, 1]. then (i) (apr ρ (f))λ = aprρ(fλ) and (ii) (aprρ(f)) s λ = aprρ(f s λ) proof. (i) let x ∈ (apr ρ (f))λ ⇐⇒ aprρ(f)(x) ≥ λ ⇐⇒ ∧ y∈[x]ρ f(y) ≥ λ ⇐⇒ for all y ∈ [x]ρ, f(y) ≥ λ ⇐⇒ [x]ρ ⊆ fλ ⇐⇒ x ∈ aprρ(fλ). (ii) let x ∈ (aprρ(f))sλ ⇐⇒ aprρ(f)(x) > λ ⇐⇒ ∨ y∈[x]ρ f(y) > λ ⇐⇒ there exist y ∈ [x]ρ, f(y) > λ ⇐⇒ [x]ρ ∩fsλ 6= ∅⇐⇒ x ∈ aprρ(f s λ). � theorem 5.5. let f be a fuzzy prime ideal of an la-semigroup s and ρ be a complete congruence relation on s. then f is a rough fuzzy prime ideal of s. proof. let f be a fuzzy prime ideal of an la-semigroup s and ρ a complete congruence on s. by theorem 5.3, for all λ ∈ [0, 1], if fλ 6= ∅, then fλ is a prime ideal of s. by theorem 3.2, for all λ ∈ [0, 1], if apr ρ (fλ) 6= ∅, then aprρ(fλ) is a prime ideal of s. from this and lemma 5.1(i), for all λ ∈ [0, 1], if (apr ρ (f))λ 6= ∅, (aprρ(f))λ is a prime ideal of s. by theorem 5.3, aprρ(f) is a fuzzy prime ideal of s. hence f is a ρ-lower rough fuzzy prime ideal of s. similarly, f is a ρ-upper rough fuzzy prime ideal of s. therefore f is a rough fuzzy prime ideal of s. � theorem 5.6. let ρ be a congruence relation on an la-semigroup s. then f is a ρ-lower rough fuzzy prime ideal if and only if for all λ ∈ [0, 1], if apr ρ (fλ) 6= ∅, then fλ is a ρ-lower rough prime ideal of s. proof. by theorem 5.3 and lemma 5.1(i), we can obtain the conclusion easily. � theorem 5.7. let ρ be a congruence relation on an la-semigroup s. then f is a ρ-upper rough fuzzy prime ideal if and only if for all λ ∈ [0, 1], if fsλ 6= ∅, then f s λ is a ρ-upper rough prime ideal of s. proof. by theorem 5.4 and lemma 5.1(ii), we can obtain the conclusion easily. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 19 (3) (2021) 464 references [1] m. aslam, m. shabir, n. yaqoob, roughness in left almost semigroups, j. adv. res. pure math. 3 (3) (2011), 70-88. [2] m. banerjee, roughness of a fuzzy set, inform. sci, 93 (1996), 235–246. [3] r. biswas, s. nanda, rough groups and rough subgroups, bull. polish acad. sci. math. 42 (1994), 251-254. [4] f. bouaziz, n. yaqoob, rough hyperfilters in po-la-semihypergroups, discrete dyn. nat. soc. 2019 (2019), 8326124. [5] d. dubois, h. prade, rough fuzzy sets and fuzzy rough sets, int. j. general syst. 17 (2–3) (1990), 191–209. [6] a. elmoasry, n. yaqoob, m-polar fuzzy hyperideals in la-semihypergroups, int. j. anal. appl. 17(3) (2019), 329-341. [7] m. gulistan, a. elmoasry, n. yaqoob, n -version of the neutrosophic cubic set: application in the negative influences of internet, j. supercomput. (2021). https://doi.org/10.1007/s11227-020-03615-1. [8] m. gulistan, n. yaqoob, a. elmoasry, j. alebraheem, complex bipolar fuzzy sets: an application in a transport’s company, j. intell. fuzzy syst. 40(3) (2021), 3981-3997. [9] y. b. jun, roughness of ideals in bck-algebras, sci. math. jpn. 57 (1) (2003), 165–169. [10] m. a. kazim, m. naseerudin, on almost-semigroups, alig. bull. math. 2 (1972), 1-7. [11] m. m. khalaf, a. elmoasry, nanogeneralized-closed sets and slightly nanoseparation axioms, glob. j. pure appl. math. 11(2) (2015), 123-130. [12] n. kuroki, p.p. wang, the lower and upper approximations in a fuzzy group, inform. sci. 90 (1996), 203–220. [13] n. kuroki, j. n. mordeson, structure of rough sets and rough groups, j. fuzzy math. 5 (1) (1997), 183–191. [14] n. kuroki, rough ideals in semigroups, inform. sci. 100 (1997), 139–163. [15] n. kuroki, fuzzy bi-ideals in semigroups. comment. math. univ. st. pauli, 27 (1979), 17-21. [16] n. kuroki, on fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy sets syst. 5 (1981), 203-215. [17] j. ma, fuzzy algebra, xueyuan press, beijing, 1989. [18] q. mushtaq, left almost semigroups, m. phil. dissertation, quaid-i-azam university, islamabad, 1978. [19] q. mushtaq, s. m. yusuf, on la-semigroups, alig. bull. math. 8 (1978), 65-70. [20] q. mushtaq, s. m. yusuf, on locally associative la-semigroups, j. nat. sci. math. 19 (1979), 57-62. [21] q. mushtaq, abelian groups defined by la-semigroups, stud. sci. math. hunger, 18 (1983), 427-428. [22] q. mushtaq, m. s. kamran, on la-semigroups with weak associative law, sci. sci. khyb. 1(11) (1989), 69–71. [23] q. mushtaq, m. iqbal, partial ordering and congruences on la-semigroups, indian j. pure appl. math. 22(4) (1991), 331-336. [24] q. mushtaq, m. khan, ideals in ag-band and ag∗-groupoid, quasigroups related syst. 14 (2006), 207-215. [25] q. mushtaq, m. khan, m-systems in la-semigroups, southeast asian bull. math. 33 (2009), 321–327. [26] z. pawlak, rough sets, int. j. comput. inform. sci. 11 (1982), 341–356. [27] a. rosenfeld, fuzzy groups. j. math. anal. appl. 35 (1971), 512-517. [28] s. tariq, a. elmoasry, s. i. batool, m. khan, quantum harmonic oscillator and schrodinger paradox based nonlinear confusion component, int. j. theor. phys. 59(11) (2020), 3558-3573. [29] q. m. xiao, z. l. zhang, rough prime ideals and rough fuzzy prime ideals in semigroups, inform. sci. 176 (2006), 725-733. [30] n. yaqoob, approximations in left almost polygroups, j. intell. fuzzy syst. 36(1) (2019), 517-526. [31] l. a. zadeh, fuzzy sets. inform. control, 8 (1965), 338-353. https://doi.org/10.1007/s11227-020-03615-1 1. introduction 2. preliminaries 3. rough prime ideals in la-semigroups 4. rough prime ideals in the quotient la-semigroups 5. rough fuzzy prime ideals in la-semigroups references int. j. anal. appl. (2023), 21:5 computational approach for a singularly perturbed differential equations with mixed shifts using a non-polynomial spline kumar ragula1, g. bsl soujanya2,∗, d. swarnakar3 1 department mathematics, rajiv gandhi university of knowledge technologies, basar, india 2 department mathematics, university p.g college for women,kakatiya university, warangal, india 3department mathematics, vnr vignana jyothi institute of engineering and technology, hyderabad, india ∗corresponding author: gbslsoujanya@gmail.com abstract. in this paper, a second order singularly perturbed differential difference equation with both the negative and positive shifts is considered. a fitted non-polynomial spline approach is applied to solve the problem. taylor series expansion process is being used to produce an approximated form of the considered problem, and then a fitted non-polynomial spline approach is devised in the form of a three-term recurrence relation. the convergence of the method is examined, and a quadratic rate of convergence is achieved. the maximum absolute error with quadratic rate of convergence of the solution is recorded. layer profile is examined using the graphs. 1. introduction in applied science and engineering, there are many physical and biological problems with solutions that include boundary and interior layers for specific parameter values and these are known as singular perturbation problems(spps). a spp is one in which no asymptotic expansion is always true over the interval as the perturbation parameter approaches zero. due to the boundary or interior layer structure of the solutions,the numerical and analytical treatments of such spps are very difficult. there is a narrow transitional layer in which the solution changes most quickly, whereas the solution responds uniformly and slowly away from the layer. when the perturbation parameter approaches received: dec. 8, 2022. 2020 mathematics subject classification. 65l11, 65l12. key words and phrases. singularly perturbed differential-difference equation; boundary layer; non-polynomial spline; mixed shifts. https://doi.org/10.28924/2291-8639-21-2023-5 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-5 2 int. j. anal. appl. (2023), 21:5 zero, the regularity of the solution decreases. a singularly perturbed differential-difference equation (spdde) is formed by multiplying the highest order derivative term of a differential equation with a small positive perturbation parameter ε and involves a delay and advance terms. for the solutions such spddes in the interval of boundary conditions is a challenging in the modeling of a variety of physical and biological problems like, the initial exit time problem in neuronal variability activation models [25], oscillations of the human pupil light reflex with delayed and mixed responses [15], red cell system evaluation [27] and bifurcated gap in a hybrid optical system [3] etc. for a detailed information on the spddes is given in [2], [4], [5], [6], [17], [18]. a few numerical methods are proposed to obtain valid and realistic approaches for the solution for second order spddes with mixed shifts (negative and positive) with boundary constraints. in the articles [11] [14] the authors presented asymptotic analysis of spddes. in [13], [14] taylor series is used to analyze a term containing small shift. the same authors investigated the impact of small shifts on the oscillatory solution of the problem in [11]. in [1] the authors proposed an a nonstandard exponentially fitted finite difference method(fdm) to solve spddes with boundary layer on each sides of the interval. to solve spdde with delay, an impactful haar wavelet collocation methodology is developed in [20]. in the articles [21], [22] the authors suggested an exponentially fitted fdm to solve spddes with delay and advanced terms and turning point problems. the authors advised a fourth order fdm with a fitting factor in [10], [16] to solve the considered problem. in [8], [9] the authors developed some numerical techniques to solve spddes with mixed shifts. an asymptotic expansion estimation of the solution is designed to solve spddes in [23]. the author suggested successive complementary expansion method (scem) for solving this model in [19]. a mixed fdm is proposed to solve spddes with both the shifts delay and advanced in [24]. the approach given in this study gives a novel difference method to the spdde with mixed shifts. in section 2, the problem description is given and the non-polynomial spline is discussed. in section 3, a three term difference scheme is devised using the non-polynomial spline. convergence analysis of this computational approach is discussed in section 4. in section 5, the computational solutions for the test problems to illustrate the technique, along with comparisons to alternative approach and conclusions are presented. 2. definition of the problem and non-polynomial spline consider the following spdde with delay and advance terms εz′′ (θ)+p(θ)z′ (θ) + q (θ)z (θ−δ)+ r (θ)z (θ)+ s (θ)z (θ+η) = f (θ) , (2.1) for θ ∈ (0,1) with the boundary constraints z (θ)= ϕ (θ) , −δ ≤ θ ≤ 0; z (θ)= ψ(θ) ,1≤ θ ≤ 1+η. (2.2) int. j. anal. appl. (2023), 21:5 3 where 0 < ε � 1 and p(θ), q (θ), r (θ), f (θ), ϕ(θ) and ψ(θ) are sufficiently smooth functions on (0,1) and 0 < δ, η = o(ε), δ, η are delay and advance shifts respectively. the solution of equation (2.1) with (2.2) represents the layer behavior at each end of the interval if p(θ)−δq (θ)+ηs (θ) > 0 and p(θ)−δq (θ)+ηs (θ) < 0 (left end and right end). depending on the sign of q (θ)+ r (θ)+ s (θ), existence of boundary layers in two cases reported in [21]. the taylor’s series expansions of z (θ−δ), z (θ+η) in the neighborhood of the point θ, we have z (θ−δ) ≈ z (θ)−δz′ (θ)+o ( δ2 ) and z (θ+η)≈ z (θ)−ηz′ (θ)+o ( η2 ) (2.3) using (2.3) in (2.1), we get εz′′ (θ)+a(θ)z′ (θ)+b(θ)z (θ)= f (θ) , 0 < θ < 1 (2.4) with the boundary constraints z (0)= ϕ(0)= ϕ0, z (1)= ψ(1)= ψ1 (2.5) consider the domain [0,1] and it is split into n equal length of sub-domains with constant step size h. let 0 = θ0 < θ1 < ... < θn = 1 be the n grid points. then we have, θi = ih and in each sub-domain [θi,θi+1] , i =1,2, . . . ,n −1 the non-polynomial spline is of the form gi (θ)= aisinτ (θ−θi)+bi ( e−τ(θ−θi) +eτ(θ−θi) ) +ci (θ−θi)+di (2.6) where ai, bi, ci and di are constant coefficients, and τ 6= 0 arbitrary parameter. to derive the coefficients ai, bi, ci and di (2.6) in terms of zi, zi+1, qi and qi+1, we define li (θi) = zi, li (θi+1) = zi+1 (2.7) l′′i (θi) = qi, l ′′ i (θi+1) = qi+1 (2.8) by using the conditions (2.7) and (2.8), we calculate the coefficients in (2.6) as .   ai = qih 2(eω+e−ω) 2ω2sinω − qi+1h 2 2ω2sinω bi = qih 2 2ω2 ci = zi+1− zi h + qih 2(1 − eω−e−ω) hω2 + qi+1h 2 hω2 di = yi − qih 2 ω2 (2.9) where ω = τh. using the first derivative of continuity, li−1 (m) (θi)= li (m) (θi) ,m =1, we obtain the relation (z i−1 −2zi +zi+1)= h 2(αqi−1 +βqi +γqi+1), i =1,2, . . . ,n −1 (2.10) where, α = ω ( e−ω +eω ) cosω +ω ( − e−ω +eω ) sinω +2 ( 1−eω −e−ω ) sinω 2ω2sinω , 4 int. j. anal. appl. (2023), 21:5 β = −2ωcosω +2sinω−ω ( eω +e−ω ) −2 ( 1−eω −e−ω ) sinω 2ω2sinω , γ = ω − sinω ω2sinω if h → 0, then ω = hτ → 0, we have (α,β,γ)→ ( 1 6 , 4 6 , 1 6 ) , (2.10) reduces to cubic spline [7]. 3. numerical algorithm at the grid points θi, (2.4) can be written as εz′′i =−a(θi)z ′ i −b(θi)zi + f (θi) using l′′i (θi) = qi = z ′′ i in above equation, we get εqj =−aj (θi)z′i −bj (θi)zi + fj (θi) for j = i, i ±1 (3.1) substitute (3.1) in (2.10) and then using z′j for j = i, i ±1 i.e z′i ≈ 1 2h (zi+1 −zi−1) , z′i+1 ≈ 1 2h (3zi+1 − 4zi +zi−1) and z′i−1 ≈ 1 2h (−zi+1 +4zi −3zi−1) ε h2 (z i+1 −2zi +zi−1) =−αai−1 (−zi+1 +4zi −3zi−1) 2h −βai (−zi−1 +zi+1) 2h −γai+1 (zi−1 −4zi +3zi+1) 2h −αbi−1zi−1 −βbizi −γbi+1zi+1+( αf i−1 +βfi +γf i+1 ) (3.2) we incorporate a fitting parameter σi (ρ) in the present approach to increase the solution’s accuracy and manage the layer behaviour. then, we have εσi (ρ) h2 (z i+1 −2zi +zi−1) =−αai−1 (−zi+1 +4zi −3zi−1) 2h − βai (zi+1 −zi−1) 2h −γai+1 (zi−1 −4zi +3z i+1) 2h −αbi−1zi−1 −βbizi− γbi+1zi+1 + ( αf i−1 +βfi +γf i+1 ) (3.3) on simplification, we obtain the following tridiagonal system eizi−1 +fizi +gizi+1 = hi, i = 1,2, . . . , n −1 (3.4) where ei = εσi − 3αhai−1 2 − βhai 2 + γhai+1 2 + h2αbi−1 , fi = −2σiε + 2αhai−1 − 2γhai+1 + h2βbi , int. j. anal. appl. (2023), 21:5 5 gi = εσi − αhai−1 2 + βhai 2 + 3γhai+1 2 + h2γbi+1 , hi = h 2 ( α f i−1 +βfi +γf i+1 ) the system (3.4) can be solved using thomas algorithm with the constraints z (0)= ϕ0, z (1)= ψ1. to calculate the fitting parameter, we adopt the process given in [18]. the following is an approximation for the solution of the homogeneous problem of (2.1) z (θ)= z0 (θ)+ a(0) a(θ) (α−z0 (0))e − ∫ θ 0 ( a(θ) ε −b(θ) a(θ) ) dθ +o (ε) (3.5) where z0 (θ) is the solution of a(θ)z0′ (θ)+b(θ)z0 (θ)= f (θ) , z0 (1)= ψ1 by using the expansion for a(θ) and b(θ) about the point zero, then (3.5) becomes z (θ)= z0 (θ)+(ϕ0 −z0 (0))e − ( a(θ) ε ) θ +o (ε) from (3.5), we have lim h→0 z (ih)= z0 (0)+(ϕ0 −z0 (0))e−a(θ)iρ, using these limit values in (3.3), we obtain the following fitting factor σi (ρ)= ( α+ β 2 ) aiρ coth (aiρ 2 ) ,where ρ = h ε . 4. convergence analysis the local error estimate for the numerical scheme of (3.4) is ti (h)= h 2 [1− (α+β +γ)]εz′′i +h 3 (γ −α) [ b′izi +a ′ iz ′ i +biz ′ i − f ′ i ] +h4 (γ +α) 2 [ b′′i zi +a ′′ i z ′ i +2b ′ iz ′ i +2a ′ iz ′′ i +biz ′′ i − f ′′ i ] + (α−γ)o ( h5 ) +o ( h6 ) hence, with (α,β,γ)= ( 1 6 , 4 6 , 1 6 ) , truncation error is of fourth order. with the help of (2.2), the matrix form of (3.4) is( r̃+ g̃ ) z +m̃ +t (h)= o (4.1) where r̃ =   −2εσ εσ 0 0 ... 0 εσ −2εσ εσ .. ... 0 0 .. .. .. ... 0 .. .. .. .. .. .. .. .. .. .. .. .. 0 .. .. .. εσ −2εσ   , 6 int. j. anal. appl. (2023), 21:5 g̃ = [xi,vi,wi] =   v1 w1 0 0 ........ 0 x2 v2 w2 0 ........ 0 0 x3 v3 w3 ........ 0 .. .. .. .. .. .. .. .. .. .. .. .. 0 .. .. 0 xn−1 vn−1   xi =− 3αhai−1 2 − βhai 2 + γhai+1 2 + h2αbi−1 , vi =2αhai−1 − 2γhai+1 + h2βbi , wi =− αhai−1 2 + βhai 2 + 3γhai+1 2 + γh2bi+1 , for all i =1,2, . . . ,n −1 m̃ = [m1 +(εσ +x1)φ(0) ,m2,m3 , . . . ,mn−2,mn−1 +(εσ +wn−1)γ] t where, mi = h 2 ( γf i+1 +βfi +αf i−1 ) , for i =1,2, . . . ,n −1, t (h)= o ( h4 ) , z = [z1,z2, ...,zn−1] t , t (h)= [t1,t2, ...,tn−1] t and o = [0,0, ...,0] t are corresponding vectors of (4.1). let z = [z1,z2, ...,zn−1] t ∼= z which satisfies the equation( r̃+ g̃ ) z +m̃ = 0 (4.2) let ei = zi−zi, i =1,2 . . . ,n−1 denote the discretized error so that ẽ = [e1,e2, ...,en−1] t = z−z. from (4.1) and (4.2), we obtain the error equation( r̃+ g̃ ) ẽ = t(h) (4.3) let |ai | ≤ k1, | bi| ≤ k2 so that, if q̃i,j is the (i, j) th element of matrix g̃, then ∣∣q̃i, i+1∣∣ = |wi| ≤ ε+h(α+β +3γ)k1 +h2αk2, i =1,2, . . . , n −2 (4.4a) ∣∣q̃i, i−1∣∣ = |ui| ≤ ε+h(3α+β +γ)k1 +h2αk2, i =2,3, . . . ,n −1 (4.4b) as a result, for relatively small h (h → 0), we see that∣∣q̃i,i+1∣∣ < ε, ∀ i = 1,2, . . . , n −1, ∣∣q̃i,i−1∣∣ < ε,∀ i = 2,3, . . . ,n −1 (4.4c) hence, ( r̃+ g̃ ) is irreducible [26]. let the sum of elements of ith row of ( r̃+ g̃ ) be si, then we have si =−εσ + 3αhai−1 2 + βhai 2 − γhai+1 2 +h2 (γbi+1 +βbi) , for i =1 si = h2 ( αbi−1 +βbi +γbi+1 ) , for i = 2,3, . . . , n −2 si =−εσ + αhai−1 2 − βhai 2 − 3γhai+1 2 +h2 (αbi−1 +βbi) , for i = n −1 int. j. anal. appl. (2023), 21:5 7 let k1∗ = min 1 ≤i≤ n−1 | ai|, k1∗ = max 1 ≤i ≤n |ai| ,k2∗ = min 1 ≤i ≤n−1 | bi|, k2 ∗ = max 1 ≤i ≤n | bi| then 0≤ k1∗ ≤ k1 ≤ k1∗, 0≤ k2∗ ≤ k2 ≤ k2∗ for relatively small h, ( r̃+ g̃ ) is monotone [26]. hence ( r̃+ g̃ )−1 exists and ( r̃+ g̃ )−1 ≥ 0. thus from (4.3), we have ||ẽ|| ≤ ||r̃+ g̃||−1||t || (4.5) for relatively small h, we have let ( r̃+ g̃ )−1 i,k be the (i,k)th element of ( r̃+ g̃ )−1 and define ||r̃+ g̃||−1 = max 1≤ i≤ n−1 n−1∑ k=1 ( r̃+ g̃ )−1 i,k , and ||t(h)||= max 1≤ i≤ n−1 |ti|. since (r̃+ g̃)−1 i,k ≥ 0 and ∑n−1 k=1 (r̃+ g̃) −1 i,k .sk =1, for i =1(1)n −1( r̃+ g̃ )−1 i, k ≤ 1 si < 1 h2k2 , i =1. (4.6a) ( r̃+ g̃ )−1 i, k ≤ 1 si < 1 h2k2 , i = n −1 (4.6b) further more, n−1∑ k=1 ( r̃+ g̃ )−1 i,k ≤ 1 min 2≤i≤n−2 si < 1 h2k2 , for i =2,3, . . . , n −2 (4.6c) with the help of (4.6a), (4.6b), (4.6c) and using (4.5), we have ||ẽ|| ≤ o (h2) this illustrates the second-order convergence for the scheme (3.4) with (α,β,γ)= ( 1 6 , 4 6 , 1 6 ) 5. numerical examples to show the validity and robustness of the suggested technique, we reported the computational results of the four example problems in terms of maximum absolute errors (maes) with calculated rates of convergence (roc) in the tables. since the exact solutions are not known for considered examples, the maes are estimated using the double mesh approach using the formula en = max 0≤i≤n ∣∣zin −z2i2n∣∣ where zin and z2i2n are the computational solutions of the example problem for n and 2n grid points respectively. further, the rate of convergence is determine by the formula rn = log2 ∣∣∣∣ ene2n ∣∣∣∣ . 8 int. j. anal. appl. (2023), 21:5 example 5.1. εz′′ (θ)+z′ (θ)−z (θ−δ)+z (θ)−z (θ+η)=−1, subject to boundary constraints z (θ)=   1, − δ ≤ θ ≤ 0 1, 1≤ θ ≤ 1+η. example 5.2. εz′′ (θ)+2.5z′ (θ)−2exp(θ)z (θ−δ)−z (θ)−θz (θ+η)=1, subject to boundary constraints z (θ)=   1, − δ ≤ θ ≤ 0 1, 1≤ θ ≤ 1+η. example 5.3. εz′′ (θ)− ( 1+exp ( −θ2 )) z ′ (θ)−θz (θ−δ)−θ2z (θ)− (1.5−exp(−θ))z (θ+η)=1 , with boundary constraints z (θ)=   1, − δ ≤ θ ≤ 0 1, 1≤ θ ≤ 1+η. example 5.4. εz′′ (θ)− ( 1+exp ( θ2 )) z ′ (θ)−θz (θ−δ)+θ2z (θ)− (1−exp(−θ))z (θ+η)=1, with boundary constraints z (θ)=   1, − δ ≤ θ ≤ 0−1, 1≤ θ ≤ 1+η. 6. conclusion a novel finite difference algorithm is suggested for solving spdde of second order with mixed shifts using a non-polynomial cubic spline with fitting factor. to represent the validity and efficiency of the method, we solved four test problems for different values n and with δ = 0.5ε = η and recorded computational results in the form of maes and rocs. using matlab, the maes in the solutions of the examples 5.1, 5.2 and 5.3 are listed in comparison to the method given in [21] in tables 1, 2, 3 and 4. tables 5 and 6 compare the maes in example 5.4 solution to the method described in [16]. the mixed shifts have no significant impact on the layer behaviour of the problems with boundary layers at the left-side and right-side of the points in the given interval shown in figures. 1, 2, 3, 4, 5, 6, 7 and 8. based on the results, we observe that the thickness of the layer increases as the size of the delay parameter increases and decreases as the size of the advance parameter increases. the proposed method is simple and can be easily implemented on a computer. int. j. anal. appl. (2023), 21:5 9 table 1. maes of example 5.1 with various values of ε ε ↓ n → 23 24 25 26 27 28 present method η = δ =0.5ε 10−1 3.769e-03 9.401e-04 2.209e-04 5.439e-05 1.359e-05 3.396e-06 2.0034 2.0889 2.0223 2.0002 1.7786 10−2 8.914e-03 4.578e-03 2.006e-03 7.512e-04 1.893e-04 4.425e-05 0.9611 1.1905 1.4168 1.9885 2.0968 10−3 8.968e-03 5.034e-03 2.683e-03 1.386e-03 6.778e-04 2.835e-04 0.8330 0.9076 0.9528 1.0324 1.2573 10−4 8.970e-03 5.036e-03 2.684e-03 1.388e-03 7.060e-04 3.561e-04 0.8334 0.9075 0.9517 0.9752 0.9874 10−5 8.970e-03 5.036e-03 2.684e-03 1.388e-03 7.060e-04 3.561e-04 0.8334 0.9075 0.9516 0.9752 0.9874 10−6 8.970e-03 5.036e-03 2.684e-03 1.388e-03 7.061e-04 3.561e-04 0.8334 0.9075 0.9516 0.9752 0.9874 results in [21] η = δ =0.5ε 10−1 3.658e-03 9.595e-04 2.409e-04 6.759e-05 1.776e-05 1.232e-05 10−2 1.695e-02 7.297e-03 2.486e-03 6.964e-04 1.776e-04 2.616e-05 10−3 2.020e-02 1.047e-02 5.210e-03 2.461e-03 1.057e-03 3.771e-04 10−4 2.052e-02 1.079e-02 5.520e-03 2.769e-03 1.363e-03 6.539e-04 10−5 2.061e-02 1.088e-02 5.608e-03 2.858e-03 1.453e-03 7.417e-04 10−6 1.951e-02 9.783e-03 4.513e-03 1.762e-03 3.577e-04 3.729e-04 10 int. j. anal. appl. (2023), 21:5 table 2. maes in example 5.2 ε ↓ n → 101 102 103 104 present method η =0.5ε δ =0.7ε 10−1 1.243e-02 1.581e-04 1.585e-06 1.585e-08 10−2 2.466e-02 1.819e-03 2.077e-05 2.080e-07 10−3 2.488e-02 3.337e-03 1.914e-04 2.157e-06 10−4 2.490e-02 3.341e-03 3.458e-04 1.924e-05 results in [21] η =0.5ε δ =0.7ε 10−1 1.533e-02 1.917e-04 1.921e-06 1.917e-08 10−2 2.817e-02 1.865e-03 2.024e-05 2.026e-07 10−3 2.853e-02 3.389e-03 1.919e-04 2.162e-06 10−4 2.857e-02 3.395e-03 3.463e-04 1.925e-05 int. j. anal. appl. (2023), 21:5 11 table 3. maes and rocs in example 5.2 ε ↓ n → 25 26 27 28 29 210 present method η = δ =0.5ε 2−3 1.150e-03 2.919e-04 7.324e-05 1.832e-05 4.583e-06 1.145e-06 1.9789 1.9948 1.9987 1.9996 1.9999 2−4 2.582e-03 6.718e-04 1.697e-04 4.253e-05 1.064e-05 2.660e-06 1.9423 1.9851 1.9961 1.9990 1.9997 2−5 5.108e-03 1.437e-03 3.712e-04 9.358e-05 2.344e-05 5.864e-06 1.8290 1.9533 1.9880 1.9969 1.9992 2−6 8.048e-03 2.759e-03 7.653e-04 1.968e-04 4.955e-05 1.241e-05 1.5444 1.8501 1.9592 1.9896 1.9973 2−7 9.448e-03 4.293e-03 1.438e-03 3.959e-04 1.016e-04 2.556e-05 1.1381 1.5751 1.8615 1.9623 1.9904 2−8 9.619e-03 5.019e-03 2.221e-03 7.353e-04 2.015e-04 5.165e-05 0.9384 1.1762 1.5946 1.8674 1.9640 2−9 9.641e-03 5.099e-03 2.590e-03 1.130e-03 3.718e-04 1.017e-04 0.9191 0.9770 1.1965 1.6038 1.8703 results in [21] η = δ =0.5ε 2−3 1.378e-03 3.486e-04 8.742e-05 2.187e-05 5.469e-06 1.367e-06 2−4 2.880e-03 7.458e-04 1.881e-04 4.714e-05 1.179e-05 2.948e-06 2−5 5.477e-03 1.526e-03 3.930e-04 9.902e-05 2.480e-05 6.204e-06 2−6 8.487e-03 2.862e-03 7.898e-04 2.028e-04 5.105e-05 1.278e-05 2−7 9.922e-03 4.413e-03 1.466e-03 4.024e-04 1.031e-04 2.596e-05 2−8 1.009e-02 5.148e-03 2.252e-03 7.424e-04 2.032e-04 5.206e-05 2−9 1.011e-02 5.228e-03 2.624e-03 1.138e-03 3.736e-04 1.021e-04 12 int. j. anal. appl. (2023), 21:5 table 4. maes and rocs in example 5.3 ε ↓ n → 25 26 27 28 29 210 present method η = δ =0.5ε 2−3 5.102e-04 1.261e-04 3.161e-05 7.897e-06 1.973e-06 4.934e-07 2.0165 1.9960 2.0010 2.0002 2.0000 2−4 1.174e-03 2.837e-04 7.153e-05 1.784e-05 4.457e-06 1.114e-06 2.0489 1.9878 2.0034 2.0008 2.0002 2−5 2.676e-03 6.223e-04 1.550e-04 3.864e-05 9.656e-06 2.413e-06 2.1045 2.0049 2.0042 2.0007 2.0002 2−6 4.201e-03 1.413e-03 3.222e-04 8.159e-05 2.024e-05 5.067e-06 1.5715 2.1330 1.9815 2.0111 1.9978 2−7 4.818e-03 2.205e-03 7.283e-04 1.642e-04 4.195e-05 1.041e-05 1.1270 1.5986 2.1483 1.9692 2.0107 2−8 4.980e-03 2.520e-03 1.132e-03 3.700e-04 8.300e-05 2.129e-05 0.9823 1.1550 1.6131 2.1563 1.9629 2−9 4.997e-03 2.604e-03 1.290e-03 5.736e-04 1.865e-04 4.172e-05 0.9401 1.0127 1.1700 1.6206 2.1604 results in [21] η = δ =0.5ε 2−3 8.434e-04 2.112e-04 5.284e-05 1.321e-05 3.303e-06 8.260e-07 2−4 4.172e-03 1.047e-03 2.640e-04 6.602e-05 1.650e-05 4.127e-06 2−5 1.858e-02 4.743e-03 1.190e-03 2.980e-04 7.452e-05 1.864e-05 2−6 6.074e-02 1.988e-02 5.080e-03 1.275e-03 3.192e-04 7.981e-05 2−7 1.111e-01 6.451e-02 2.061e-02 5.270e-03 1.323e-03 3.311e-04 2−8 1.297e-01 1.176e-01 6.658e-02 2.101e-02 5.372e-03 1.349e-03 2−9 1.310e-01 1.372e-01 1.212e-01 6.766e-02 2.122e-02 5.425e-03 int. j. anal. appl. (2023), 21:5 13 table 5. maes and rocs in example 5.4 ε ↓ n → 25 26 27 28 29 present method η = δ =0.5ε 2−3 4.896e-03 1.129e-03 2.774e-04 6.934e-05 1.731e-05 2.1154 2.0257 2.0005 2.0018 2−4 1.114e-02 2.544e-03 5.859e-04 1.434e-04 3.587e-05 2.1312 2.1187 2.0302 1.9992 2−5 1.898e-02 5.677e-03 1.295e-03 2.979e-04 7.289e-05 1.7412 2.1315 2.1208 2.0309 2−6 2.371e-02 9.579e-03 2.864e-03 6.537e-04 1.501e-04 1.3076 1.7414 2.1317 2.1221 2−7 2.448e-02 1.188e-02 4.811e-03 1.438e-03 3.282e-04 1.0425 1.3047 1.7415 2.1319 2−8 2.449e-02 1.224e-02 5.950e-03 2.411e-03 7.210e-04 1.0000 1.0414 1.3032 1.7416 results in [16] η = δ =0.5ε 2−3 8.354e-03 2.013e-03 4.986e-04 1.249e-04 3.121e-05 2−4 1.719e-02 4.378e-03 1.041e-03 2.571e-04 6.429e-05 2−5 2.517e-02 8.889e-03 2.238e-03 5.290e-04 1.303e-04 2−6 3.154e-02 1.294e-02 4.516e-03 1.131e-03 2.664e-04 2−7 4.478e-02 1.622e-02 6.559e-03 2.276e-03 5.686e-04 2−8 7.878e-02 2.317e-02 8.224e-03 3.301e-03 1.142e-03 14 int. j. anal. appl. (2023), 21:5 table 6. mae in example 5.4 with ε =0.1 n → 101 102 103 104 present method δ ↓ η =0.5ε 0.00 8.123e-02 6.690e-03 3.573e-04 2.225e-05 0.05 8.066e-02 6.582e-03 3.529e-04 2.194e-05 0.09 8.019e-02 6.495e-03 3.494e-04 2.171e-05 η ↓ δ =0.5ε 0.00 8.051e-02 6.527e-03 3.508e-04 2.180e-05 0.05 8.066e-02 6.582e-03 3.529e-04 2.194e-05 0.09 8.077e-02 6.626e-03 3.546e-04 2.206e-05 results in [16] δ ↓ η =0.5ε 0.00 9.109e-02 1.112e-02 6.382e-04 4.004e-05 0.05 9.047e-02 1.095e-02 6.306e-04 3.950e-05 0.09 8.996e-02 1.082e-02 6.244e-04 3.906e-05 η ↓ δ =0.5ε 0.00 9.604e-02 1.116e-02 6.458e-04 3.924e-05 0.05 9.621e-02 1.124e-02 6.494e-04 3.950e-05 0.09 9.634e-02 1.131e-02 6.522e-04 3.970e-05 int. j. anal. appl. (2023), 21:5 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 1. layer profile in example 5.1 with δ , n =25, ε =10−1 and η =0.5ε 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 2. layer profile of example 5.1 with η , n =25, ε =10−1 and δ =0.5ε 16 int. j. anal. appl. (2023), 21:5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 3. layer profile of example 5.2 with δ , n =25, ε =10−1 and η =0.5ε 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 4. layer profile of example 5.2 with η , n =25, ε =10−1 and δ =0.5ε int. j. anal. appl. (2023), 21:5 17 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 5. layer profile of example 5.3 with δ , n =25, ε =10−1 and η =0.5ε 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 6. layer profile of example 5.3 with η , n =25, ε =10−1 and δ =0.5ε 18 int. j. anal. appl. (2023), 21:5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 7. layer profile of example 5.4 with δ , n =25, ε =10−1 and η =0.5ε 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 n um er ic al s ol ut io n of z ( ) =0.00 =0.05 =0.09 figure 8. layer profile of example 5.4 with η , n =25, ε =10−1 and δ =0.5ε int. j. anal. appl. (2023), 21:5 19 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. adilaxmi, d. bhargavi, y. reddy, an initial value technique using exponentially fitted nonstandard finite difference method for singularly perturbed differential-difference equations, appl. math. 14 (2019), 245–269. [2] r.e. bellman, k.l. cooke, differential-difference equations, academic press, new york, 1963. [3] m.w. derstine, h.m. gibbs, f.a. hopf, d.l. kaplan, bifurcation gap in a hybrid optically bistable system, phys. rev. a. 26 (1982), 3720–3722. https://doi.org/10.1103/physreva.26.3720. [4] e.p. doolan, j.j.h miller, w.h. schilders, uniform numerical methods for problems with initial and boundary layers, boole press, dublin, 1980. [5] r.d. driver, ordinary and delay differential equations, springer, new york, 1977. [6] m. holmes, introduction to perturbation methods, springer, berlin, 1995. [7] m.k. kadalbajoo, r.k. bawa, variable-mesh difference scheme for singularly-perturbed boundary-value problems using splines, j. optim. theory appl. 90 (1996), 405–416. https://doi.org/10.1007/bf02190005. [8] m.k. kadalbajoo, k.k. sharma, numerical analysis of boundary-value problems for singularly-perturbed differential-difference equations with small shifts of mixed type, j. optim. theory appl. 115 (2002), 145–163. https://doi.org/10.1023/a:1019681130824. [9] m.k. kadalbajoo, k.k. sharma, numerical treatment of a mathematical model arising from a model of neuronal variability, j. math. anal. appl. 307 (2005), 606–627. https://doi.org/10.1016/j.jmaa.2005.02.014. [10] d. kumara swamy, k. phaneendra, y.n. reddy, accurate numerical method for singularly perturbed differentialdifference equations with mixed shifts, khayyam j. math. 4 (2018), 110–122. https://doi.org/10.22034/kjm. 2018.57949. [11] c.g. lange, r.m. miura, singular perturbation analysis of boundary value problems for differential-difference equations, siam j. appl. math. 42 (1982), 502–531. https://doi.org/10.1137/0142036. [12] c.g. lange, r.m. miura, singular perturbation analysis of boundary value problems for differential-difference equations iii. turning point problems, siam j. appl. math. 45 (1985), 708–734. https://doi.org/10.1137/ 0145042. [13] c.g. lange, r.m. miura, singular perturbation analysis of boundary value problems for differential-difference equations. v. small shifts with layer behavior, siam j. appl. math. 54 (1994), 249–272. https://doi.org/10. 1137/s0036139992228120. [14] c.g. lange, r.m. miura, singular perturbation analysis of boundary-value problems for differential-difference equations. vi. small shifts with rapid oscillations, siam j. appl. math. 54 (1994), 273–283. https://doi.org/ 10.1137/s0036139992228119. [15] a. longtin, j.g. milton, complex oscillations in the human pupil light reflex with "mixed" and delayed feedback, math. biosci. 90 (1988), 183–199. https://doi.org/10.1016/0025-5564(88)90064-8. [16] m. ayalew, g.g. kiltu, g.f. duressa, fitted numerical scheme for second-order singularly perturbed differentialdifference equations with mixed shifts, abstr. appl. anal. 2021 (2021), 4573847. https://doi.org/10.1155/ 2021/4573847. [17] a.h. nayfeh, perturbation methods, wiley, new york, 1979. [18] r.e. o’malley jr, introduction to singular perturbations, academic press, new york, 1974. [19] s. priyadarshana, s.r. sahu, j. mohapatra, asymptotic and numerical methods for solving singularly perturbed differential difference equations with mixed shifts, iran. j. numer. anal. optim. 12 (2022), 55–72. https: //doi.org/10.22067/ijnao.2021.70731.1038. https://doi.org/10.1103/physreva.26.3720 https://doi.org/10.1007/bf02190005 https://doi.org/10.1023/a:1019681130824 https://doi.org/10.1016/j.jmaa.2005.02.014 https://doi.org/10.22034/kjm.2018.57949 https://doi.org/10.22034/kjm.2018.57949 https://doi.org/10.1137/0142036 https://doi.org/10.1137/0145042 https://doi.org/10.1137/0145042 https://doi.org/10.1137/s0036139992228120 https://doi.org/10.1137/s0036139992228120 https://doi.org/10.1137/s0036139992228119 https://doi.org/10.1137/s0036139992228119 https://doi.org/10.1016/0025-5564(88)90064-8 https://doi.org/10.1155/2021/4573847 https://doi.org/10.1155/2021/4573847 https://doi.org/10.22067/ijnao.2021.70731.1038 https://doi.org/10.22067/ijnao.2021.70731.1038 20 int. j. anal. appl. (2023), 21:5 [20] a. raza, a. khan, treatment of singularly perturbed differential equations with delay and shift using haar wavelet collocation method, tamkang j. math. 53 (2021), 303–322. https://doi.org/10.5556/j.tkjm.53.2022.3250. [21] r. ranjan, h.s. prasad, a novel approach for the numerical approximation to the solution of singularly perturbed differential-difference equations with small shifts, j. appl. math. comput. 65 (2020), 403–427. https://doi. org/10.1007/s12190-020-01397-6. [22] r. ranjan, h.s. prasad, a novel exponentially fitted finite difference method for a class of 2nd order singularly perturbed boundary value problems with a simple turning point exhibiting twin boundary layers, j. ambient intell. human comput. 13 (2022), 4207–4221. https://doi.org/10.1007/s12652-022-03902-0. [23] l.s. senthilkumar, r. mahendran, v. subburayan, a second order convergent initial value method for singularly perturbed system of differential-difference equations of convection diffusion type, j. math. computer sci. 25 (2021), 73–83. https://doi.org/10.22436/jmcs.025.01.06. [24] l. sirisha, k. phaneendra, y.n. reddy, mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, ain shams eng. j. 9 (2018), 647–654. https://doi.org/10.1016/j.asej.2016.03.009. [25] r.b. stein, some models of neuronal variability, biophys. j. 7 (1967), 37–68. https://doi.org/10.1016/ s0006-3495(67)86574-3. [26] r.s. varga, matrix iterative analysis, prentice-hall, englewood cliffs, 1962. [27] m. wazewska, a. lasota, mathematical models of the red cell system, mat. stosowana, 6 (1976), 25–40. https://doi.org/10.5556/j.tkjm.53.2022.3250 https://doi.org/10.1007/s12190-020-01397-6 https://doi.org/10.1007/s12190-020-01397-6 https://doi.org/10.1007/s12652-022-03902-0 https://doi.org/10.22436/jmcs.025.01.06 https://doi.org/10.1016/j.asej.2016.03.009 https://doi.org/10.1016/s0006-3495(67)86574-3 https://doi.org/10.1016/s0006-3495(67)86574-3 1. introduction 2. definition of the problem and non-polynomial spline 3. numerical algorithm 4. convergence analysis 5. numerical examples 6. conclusion references int. j. anal. appl. (2023), 21:69 a note on skew generalized power serieswise reversible property eltiyeb ali1,2,∗ 1department of mathematics, college of science and arts, najran university, ksa 2department of mathematics, faculty of education, university of khartoum, sudan ∗corresponding author: eltiyeb76@gmail.com abstract. the aim of this paper is to introduce and study (s,ω)-nil-reversible rings wherein we call a ring r is (s,ω)-nil-reversible if the left and right annihilators of every nilpotent element of r are equal. the researcher obtains various necessary or sufficient conditions for (s,ω)-nil-reversible rings are abelian, 2-primal, (s,ω)-nil-semicommutative and (s,ω)-nil-armendariz. also, he proved that, if r is completely (s,ω)-compatible (s,ω)-nil-reversible and j an ideal consisting of nilpotent elements of bounded index ≤ n in r, then r/j is (s,ω̄)-nil-reversible. moreover, other standard rings-theoretic properties are given. 1. introduction throughout this paper, all rings are associated with identity unless otherwise stated. we write p (r), nil(r), matn(r), tn(r, ) sn(r),r[x], end(r) and aut(r), respectively for the prime radical, the set of all nilpotent elements of r, full square matrices, upper square triangular matrices for a positive integer n with entries in r, the subring consisting of all upper square triangular matrices, the polynomial ring, the monoid of ring endomorphisms of r and the group of ring automorphisms of r. the purpose of this article is to examine (s,ω)-nil-reversible rings, where (s,≤) is a strictly ordered monoid and ω : s → end(r) is a monoid homomorphism. a ring r is considered (s,ω)-nil-reversible if the left and right annihilators of every nilpotent element in r are equal. the author provides various necessary or sufficient conditions for (s,ω)-nil-reversible rings to be abelian, 2-primal, (s,ω)-nilsemicommutative and (s,ω)-nil-armendariz. i have an example to illustrate, a (s,ω)-nil-reversible ring may not necessarily be (s,ω)-semicommutative or (s,ω)-reversible. additionally, it is shown that received: may 20, 2023. 2020 mathematics subject classification. 16s99, 16d80, 13c99. key words and phrases. armendariz ring; (s,ω)-reversible; ordered monoid (s,≤); semicommutative ring. https://doi.org/10.28924/2291-8639-21-2023-69 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-69 2 int. j. anal. appl. (2023), 21:69 if r is completely (s,ω)-compatible and j is an ideal consisting of nilpotent elements with bounded index ≤ n, then r/j is also (s,ω)-nil-reversible. furthermore, it is proven that a multiplicatively closed subset of a ring consisting of central non-zero divisors is (s,ω)-nil-reversible if and only if the entire ring itself is (s,ω)-nil-reversible. the article also covers other standard properties in ring theory. a ring r is said to be reversible if xy = 0, then yx = 0, where x,y ∈ r see cohn [1]. the article [2] defines a semicommutative ring r as one where xy = 0 implies xry = 0 for all x,y ∈ r. rings with no nonzero nilpotent elements are called reduced rings and are symmetric, reversible, and semicommutative according to [3, p. 361] and [3, proposition 1.3]. however, polynomial rings over reversible rings need not be reversible as shown in [4, example 2.1]. in [5], strongly reversible rings are introduced as reversible rings over which polynomial rings are also reversible. a ring r is strongly reversible if f (x)h(x) = 0 implies h(x)f (x) = 0 for all polynomials f (x),h(x) ∈ r[x]. reversible armendariz rings satisfy this property, but reduced rings may not be strongly reversible in general. a ring is called a 2-primal ring if its nilradical coincides with its prime radical, and an ni-ring if its upper nilradical coincides with its set of nilpotent elements. a ring is an ni-ring if and only if its set of nilpotent elements forms an ideal, while 2-primal rings are ni-rings. armendariz ring defined by the reference [2]. if the products two polynomials f (x)g(x) = 0, then aibj = 0, for all i, j. in our discussion, we use the following terminology: given non-empty subsets a and d of a monoid s, an element u0 ∈ ad = {st : s ∈ a,t ∈ d} is considered a single product element (abbreviated as s.p. element) in ad if it can be expressed singly in the form u = st. the following definition will be useful in the next section. definition 1.1. the article [6] defines an ordered monoid (s,≤) as an artinian narrow unique product monoid (or a.n.u.p. monoid) if, for any two artinian and narrow subsets a and d of s, there exists a unique product element in the set ad that is upper principal. a minimal artinian narrow unique product monoid (or m.a.n.u.p. monoid) is defined as an ordered monoid (s,≤) where, for any two artinian and narrow subsets a and d of s, there exist minimum elements a ∈ min(a) and b ∈ min(d) such that their product ab is an upper principal element of the set ad. a monoid is said to be totally orderable if it can be ordered with a total order ≤, while a quasitotally ordered monoid is one where the order ≤ can be refined to a strictly total order �. to start, we revisit the creation of the generalized power series ring, which was initially presented in [7]. let (s,≤) be an ordered set. if every strictly decreasing sequence of elements in s is finite, then (s,≤) is said to be artinian. assume that s is a commutative monoid with the operation denoted additively and the neutral element denoted by 0. for a ring r, (s,≤) be a strictly ordered monoid, and ω : s → end(r) be a monoid homomorphism. denote the image of s under ω as ωs = ω(s) for any s ∈ s. int. j. anal. appl. (2023), 21:69 3 let f be the set of all functions f : s → r such that the support supp(f ) = {s ∈ s : f (s) 6= 0} is both artinian and narrow. for any s ∈ s and f ,g ∈ f, the set xs(f ,g) consisting of all pairs (u,v) ∈ supp(f )×supp(g) such that s = uv is finite. therefore, we can define the product f g : s → r of f and g as follows: if (u,v) /∈ xs(f ,g) for all (u,v) ∈ supp(f ) × supp(g), then (f g)(s) = 0, otherwise, (f g)(s) = ∑ (u,v)∈xs(f ,g) f (u)ωu(g(v)) is conventionally considered to be 0. using the previously defined pointwise addition and multiplication, the set f becomes a ring known as the ring of skew generalized power series with coefficients in r and exponents in s, denoted by [[rs,≤,ω]] (or simply r[[s,ω]] if the order ≤ is unambiguous), as described in [8]. a subset p ⊆ r is considered to be s-invariant if it is ωt-invariant for every t ∈ s, meaning that ωt(p ) ⊆ p . for each element d ∈ r and each element t ∈ s, we have the elements cd and et in [[rs,≤,ω]] defined by cd(λ) =   d, λ = 1, 0, λ ∈ s\{1}, et(λ) =   1, λ = t, 0, λ ∈ s\{t}. the mapping d 7→ cd is a ring embedding of r into the ring [[rs,≤,ω]], while the mapping t 7→ et is a monoid embedding of s into the multiplicative monoid of that same ring. moreover, we have the relationship that etcd = cωt(d)et. 2. (s,ω)-nil-reversible rings in this section, we introduce the concept of (s,ω)-nil-reversible rings, which is a generalization of both (s,ω)-reversible rings and generalized power series reversible rings. we then utilize this concept to investigate the relationships between (s,ω)-nil-reversible rings and certain classes of rings. definition 2.1. for a ring r, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. r is to be (s,ω)-nil-reversible, if f g ∈ [[nil(r)s,≤,ω]], then gf ∈ [[nil(r)s,≤,ω]], for all f ,g ∈ [[rs,≤,ω]]. remark 2.2. by definition, it is clear that, skew generalized power series nil-reversible rings are closed under subrings. definition 2.3. in [6] a ring r is said to be s-compatible (or (s,ω)-compatible) if for every element d in the strictly ordered monoid s, the corresponding endomorphism ωd of r is compatible. similarly, a ring r is said to be s-rigid (or (s,ω)-rigid) if for every element d in s, the corresponding endomorphism ωd of r is rigid. here, ω : s → end(r) is a monoid homomorphism that maps elements of the monoid s to endomorphisms of the ring r. lemma 2.4. [6] for a ring r, (s,≤) a strictly ordered monoid and ω : s → end(r) a monoid homomorphism. a ring r is reduced ⇔ [[rs,≤,ω]] is reduced. 4 int. j. anal. appl. (2023), 21:69 lemma 2.5. [9] for ω : s →end(r) a monoid homomorphism. any elements r,t ∈ r and d ∈ s. the following results are correct: (1) rt ∈ nil(r) ⇔ rωd(t) ∈ nil(r). (2) rt ∈ nil(r) ⇔ ωd(r)t ∈ nil(r). we can provide an example of nil-reversible rings of skew generalized power series that do not fall under the categories of either skew generalized power series reversible or skew generalized power series semicommutative. it is important to note that skew generalized power series reversible rings are both skew generalized power series semicommutative and skew generalized power series nil-reversible by definition. this leads us to speculate that skew generalized power series nil-reversible rings may also be skew generalized power series semicommutative. however, the following examples disprove this possibility. to support this claim, we require the following propositions. proposition 2.6. for a ring r, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. suppose r is an (s,ω)-compatible with nil(r) an ideal, then r is (s,ω)-nil-reversible. proof. let f ,g ∈ [[rs,≤,ω]], satisfying f g is nilpotent. there exists a positive integer ` such that (f g)` = 0, so (f (r)ωr (g(t)))` = 0 for each r,t ∈ s. then by compatibility f (r)g(t) ∈ nil(r). hence g(t)f (r) is nilpotent. thus, gf is nilpotent. � proposition 2.7. for a ring r, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. suppose r is an (s,ω)-compatible. a ring r is (s,ω)-nil-reversible ring if and only if for any n, the n-by-n upper triangular matrix ring tn(r) is (s,ω)-nil-reversible. proof. assume that f ,g ∈ [[tn(r)s,≤,ω]], satisfying f g ∈ [[nil(tn(r))s,≤,ω]]. so by [10], nil(tn(r)) =   nil(r) r r · · · r 0 nil(r) r · · · r 0 0 nil(r) · · · r ... ... ... ... ... 0 0 0 · · · nil(r)   . if r is a ring with no nonzero nilpotent elements, then the nilradical of r is trivial, i.e., nil(r) = 0. therefore, the nilradical of the n-th triangular matrix ring over r, denoted by tn(r), is also trivial. hence, nil(tn(r)) forms an ideal in tn(r). by proposition 2.6, tn(r) is (s,ω)-nil-reversible. the if part follows remark 2.2. � example 2.8. for a ring r, (s,ω)-compatible, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. let r be (s,ω)-nil-reversible ring. then t =     a11 a12 a13 0 a22 a23 0 0 a33   | aij ∈ r   . int. j. anal. appl. (2023), 21:69 5 is (s,ω)-nil-reversible ring by proposition 2.7. note that f g = 0, where f = ce23 + ce13es and g = ce12 + ce22es, but we have gf 6= 0. so t is not (s,ω)-reversible. in fact, t is not (s,ω)semiccomutative by [11, example 2.5] (with n = 3). also, let s be an (s,ω)-nil-reversible ring. then the ring rn =     a a12 a13 · · · a1n 0 a a23 · · · a2n 0 0 a · · · a3n ... ... ... ... ... 0 0 0 · · · a   | a,aij ∈ s; n ≥ 3   . is not (s,ω)-reversible by [11, example 2.5]. but rn is (s,ω)-nil-reversible by proposition 2.7 since any subring of (s,ω)-nil-reversible ring is (s,ω)-nil-reversible. it is obvious that r4 is not (s,ω)semicommutative and it can be proved similarly that rn is not (s,ω)-semicommutative for n ≥ 5. proposition 2.9. for a ring r, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. assume that r is (s,ω)-nil-reversible and (s,ω)-compatible. suppose g1,g2, . . . ,gn ∈ [[rs,≤,ω]] satisfying g1g2 · · ·gn ∈ [[nil(r)s,≤,ω]], then g1(v1)g2(v2) · · ·gn(vn) ∈ nil(r) for all v1,v2, . . . ,vn ∈ s. proof. it is clear by the definition. � corollary 2.10. for a ring r, (s,≤) a strictly ordered monoid and w : s → end(r) a monoid homomorphism and r to be s-compatible. the following conditions are equal: (1) if g1,g2, . . . ,gn ∈ [[rs,≤,ω]] satisfy g1g2 · · ·gn ∈ [[nil(r)s,≤,ω]], then g1(v1)g2(v2) · · ·gn(vn) ∈ nil(r), for any v1,v2, . . . ,vn ∈ s. (2) r is ni ring. proposition 2.11. for a ring r, (s,≤) a strictly ordered monoid and w : s → end(r) a monoid homomorphism. assume r is an (s,ω)-compatible. if r is (s,ω)-nil-reversible, then nil(r[[s,ω]]) ⊆ nil(r)[[s,ω]]. proof. let g ∈ nil(r[[s,ω]]), suppose g` = 0 where ` ∈z+. then by proposition 2.9, g(v) ∈ nil(r) for each v ∈ s. thus nil(r[[s,ω]]) ⊆ nil(r)[[s,ω]]. � proposition 2.12. let r be a ring, (s,≤) a strictly ordered monoid and w : s → end(r) a monoid homomorphism. suppose that r to be s-compatible. if r is (s,ω)-nil-reversible, then (1) r is abelian. () r is 2-primal.. proof. let r be a (s,ω)-nil-reversible ring. (1) let e be an idempotent element of r. for any g(v) ∈ r,v ∈ s,eg(v) − eg(v)e ∈ nil(r). note 6 int. j. anal. appl. (2023), 21:69 that (eg(v) − eg(v)e)e = 0. by hypothesis, e(eg(v) − eg(v)e) = 0, so eg(v) = eg(v)e. again, g(v)e − eg(v)e ∈ nil(r) and e(g(v)e − eg(v)e) = 0. so by (s,ω)-nil-reversibility of r, we have (g(v)e −eg(v)e)e = 0, that is, g(v)e = eg(v)e. hence, eg(v) = g(v)e. (2) note that p (r) ⊆ nil(r). suppose g(v) ∈ nil(r). then there is a positive integer m ≥ 2 such that (g(v))m = 0. thus, r(g(v))m−1g(v) = 0. this implies that g(v)r(g(v))m−1 = 0 as r is (s,ω)-nil-reversible. this yields (rg(v))m = 0, so g(v) ∈ p (r). � according to [9], a ring r is called (s,ω)-nil-armendariz, if whenever f ,g ∈ r[[s,ω]] satisfying f g ∈ nil(r[[s,ω]]), then f (r)ωr (g(t)) ∈ nil(r) for all r,t ∈ s. proposition 2.13. for a ring r, s-compatible, (s,≤) totally ordered monoid and w : s → end(r) a homomorphism of monoid. then, every (s,ω)-nil-reversible rings are (s,ω)-nil-armendariz. proof. suppose 0 6= f ,g ∈ r[[s,ω]] satisfying f g ∈ nil(r[[s,ω]]). transfinite induction will be applied to the set that is strictly and totally ordered (s,≤) showing f (r)g(t) ∈ nil(r) for any r ∈ supp(f ) and t ∈ supp(g). in the ≤′ order, let s and d be the smallest elements in supp(f ) and supp(g), respectively. if r ∈ supp(f ) and t ∈ supp(g) satisfying r + t = s + d, then s ≤′ r and d ≤′ t. if s <′ r then s + d <′ r + t = s + d, a contradiction. thus r = s. similarly, t = d. hence 0 = (f g)(s + d) = ∑ (r,t)∈xs+d(f ,g) f (r)ωr (g(t)) = f (s)ωs(g(d)). now let w ∈ s with r + t <′ w,f (r)g(t) = 0. we need to show f (r)ωr (g(t)) ∈ nil(r) for al r ∈ supp(f ) and t ∈ supp(g) with r + t = w. writing xw (f ,g) = {(r,t) | r + t = w as {(ri,ti ) | i = 1, 2, . . . ,n} such that r1 <′ r2 <′ · · · <′ rn. since s is cancellative, r1 = r2 and r1 + t1 = r2 + t2 = w imply t1 = t2. since ≤′ is a strict order, r1 <′ r2 and r1 + t1 = r2 + t2 = w imply t2 <′ t1. thus we have tn <′ · · · <′ t2 <′ t1. now, 0 = (f g)(w) = ∑ (r,t)∈xw(f ,g) f (r)ωr (g(t)) = n∑ i=1 f (ri )ωri (g(ti )). (2.1) for each i ≥ 2, r1 + ti <′ ri + ti = w. therefore, using the induction hypothesis, we can conclude that f (r1)g(ti ) belongs to the nilradical of r. since r is a 2-primal ring (as shown in proposition 2.12), this implies that f (r1)g(ti ) also belongs to the nilradical of r. thus, by multiplying equation (2.1) on the right by f (r1)g(t1), we get:( n∑ i=1 f (ri )g(ti ) ) f (r1)g(t1) = f (r1)g(t1)f (r1)g(t1) = 0. then (f (r1)g(t1))2 = 0 and so f (r1)g(t1) ∈ nil(r). now (2.1) becomes n∑ i=2 f (ri )g(ti ) = 0. (2.2) by performing a right-hand side multiplication of (2.2) with f (r2)g(t2), we get f (r2)g(t2) = 0. by following the same method as described above, we can continue this process and establish proof int. j. anal. appl. (2023), 21:69 7 f (ri )g(ti ) = 0 for all i = 1, 2, . . . ,n. thus f (r)g(t) ∈ nil(r) with r + t = w. hence, utilizing transfinite induction, it follows that f (r)ωr (g(t)) belongs to the set of nilpotent elements in r for any r ∈ supp(f ) and t ∈ supp(g). � lemma 2.14. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s. we now examine the conditions for r. (1) r is (s,ω)-nil-reversible. (2) if ab is a nilpotent set, then so is ba for each subsets a,b in r. (3) if kz is nilpotent, then zk is nilpotent for right ideals (or left) k,z in r. then (1) ⇒ (2) ⇒ (3). proof. the proof is analog with the proof of [11, lemma 3.5] � lemma 2.15. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s. then every (s,ω)-nil-reversible rings are (s,ω)-nil-semicommutative. proof. suppose f ,g ∈ r[[s,ω]] satisfying f g ∈ nil(r[[s,ω]]). then gf ∈ nil(r[[s,ω]]) and g(t)ωt(h(w)ωw (f (r))) ∈ nil(r) for any r,t,w ∈ s and h(w) ∈ r, so f (t)h(w)g(r) ∈ nil(r) by compatibility. thus, f hg ∈ nil(r[[s,ω]]) by [4, lemma 1.1]. therefore, r is an (s,ω)-nilsemicommutative. � proposition 2.16. consider r is an ni ring and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s. if (s,ω)-nil-reversible with nil(r) is an ideal of r, then nil(r)[[s,ω]] = nil(r[[s,ω]]). proof. suppose d ∈ nil(r), by lemma 2.14, rdr is a nilpotent in r. since r is compatible with s, for any λ ∈ s, rωλ(d)r is a nilpotent ideal of r and so ωλ(d) ∈ nil(r). thus nil(r) is an invariant with s and so nil(r)[[s,ω]] is an ideal of r[[s,ω]]. by proposition 2.11, nil(r[[s,ω]]) ⊆ nil(r)[[s,ω]]. therefore, it is enough to demonstrate that nil(r)[[s,ω]] ⊆ nil([[rs,≤,ω]]). suppose f ∈ nil(r)[[s,ω]] then for any r ∈ s, f (r) ∈ nil(r). by proposition 2.11, there is a positive integer ` such that r ∈ s, (rf (r)r)` = 0. since r is compatible with s, then for any g,h ∈ r[[s,ω]], (gf h)` = 0. i have know, if g ∈ nil(r)[[s,ω]], then g(t) ∈ nil(r). so g ∈ nil(r[[s,ω]]). thus, nil(r)[[s,ω]] ⊆ nil(r[[s,ω]]). therefore, nil(r)[[s,ω]] = nil(r[[s,ω]]). � corollary 2.17. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s and (s,ω)-reversible. then g is a nil element of [[rs,≤,ω]] ⇔ f (r) ∈ nil(r) for all r ∈ s. 8 int. j. anal. appl. (2023), 21:69 proposition 2.18. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s. if a subdirect product of (s,ω)-nilreversible rings is finite, then it is also an (s,ω)-nil-reversible ring. proof. suppose we have ideals jk of r and r/jk is (s,ω̄)-nil-reversible for k = 1, . . . , l such that⋂l k=1jk = 0. assume that f ,g ∈ r[[s,ω]] satisfying f g ∈ nil(r[[s,ω]]). then, we have f g ∈ nil(r/jk[[s,ω]]). since r/jk is (s,ω̄)-nil-reversible, we have (f (r)g(t))dr,t,k ∈ jk for all r,t ∈ s and k = 1, . . . , l, where dr,t,k is the maximum value of dr,t over all ideals. thus, (f (r)g(t))dr,t ∈ ⋂l k=1jk = 0, which implies that f (r)g(t) ∈ nil(r) for all r,t ∈ s. therefore, we have g(t)f (r) ∈ nil(r) as well, and so gf ∈ nil(r[[s,ω]]) as desired. � proposition 2.19. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s and e2 = e ∈ r. if r is (s,ω)-nil-reversible, then so is ere. proof. suppose cef ce,cegce ∈ (ere)[[s,ω]] satisfying (cef ce)(cegce) ∈ nil(ere)[[s,ω]]. let e be an idempotent of r. ce is clearly an idempotent element of (ere)[[s,ω]], ceg = gce for each g ∈ r[[s,ω]]. then (cef )(ceg) ∈ nil(er)[[s,ω]]. since r is (s,ω)-nil-reversible, the elements f g ∈ nil(r)[[s,ω]], and so gf ∈ nil(r)[[s,ω]]. then there exists ` ∈ n such that ((cef ce)(cegce))` = 0. therefore (cegce)(cef ce) ∈ nil(ere)[[s,ω]]. � corollary 2.20. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). if and only if r is (s,ω)-nil-reversible, then both er and (1 − e)r are also (s,ω)-nil-reversible for a central idempotent e of the ring r. proof. assume that er and (1 −e)r are (s,ω)-nil-reversible. since the nil skew generalized power series reversibility property finite direct products preserve the closure property of the set, r ∼= er × (1 −e)r is (s,ω)-nil-reversible. the converse is true by proposition 2.19. � in [12], a homomorphic image of a nil-reversible ring may not be nil-reversible, so as (s,ω)-nilreversible by the next example. example 2.21. let (s,≤) a strictly ordered monoid and ω : s → end(r) a monoid homomorphism. assume that r = d[[s,≤]], where d is a division ring and i =< xy >, where xy 6= yx. as r is a domain, r is (s,ω)-nil-reversible. clearly yx ∈ nil(r/i)[[s,ω]] and x(yx) = xyx = 0. but, (yx)x = yx2 6= 0. this implies r/i is not (s,ω)-nil-reversible. definition 2.22. [9] consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). to express the concept of a ring being completely compatible with a set s, we define it as follows: a ring r is said to be completely s-compatible if every ideal j of r yields an s-compatible quotient ring r/j. in order to refer to the homomorphism ω, we may alternatively refer to r as completely compatible with s.. int. j. anal. appl. (2023), 21:69 9 it is evident that any ring that is completely (s,ω)-compatible also qualifies as (s,ω)-compatible. another way to express the complete (s,ω)-compatibility of a ring r is by stating that for any subset j of r and elements r and t in r, the condition rt ∈ j is equivalent to rω(t) ∈ j. this description will be frequently referenced in our discussions. theorem 2.23. let r be a ring, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. if r is completely s-compatible (s,ω)-nil-reversible and j an ideal consisting of nilpotent elements of bounded index ≤ n in r, then r/j is (s,ω)-nil-reversible. proof. suppose f̄ , ḡ ∈ (r/j)[[s,ω]] satisfying f̄ ḡ ∈ nil(r/j)[[s,ω]]. assuming that the order (s,≤) can be improved to a strict total order ≤ on s, we will utilize transfinite induction on the strictly totally ordered set (s,≤) to demonstrate that ḡf̄ ∈ nil(r/j)[[s,ω]]. to begin with, demonstrate through transfinite induction that g(t)f (s) ∈ nil(r) for every s ∈ supp(f ) and t ∈ supp(g). given that supp(f ) and supp(g) are non-empty subsets of s, there exist finite and non-empty sets of minimal elements in supp(f ) and supp(g), respectively. let s0 and t0 be the minimum elements in the ≤ order of these sets. by the same of the proof of [9, theorem 2.25], we need to show f (s0)ωs0(g(t0)) = 0. therefore, by transfinite induction, we can proof that f (s)g(t) = 0. since f̄ ḡ ∈ nil(r/j)[[s,ω]]. then, there is a positive integer ` ∈n such that (f̄ ḡ)` = 0̄. so (f (s)g(t))` ∈ j, for any s,t ∈ s. since j ⊆ nil(r), (f (s)g(t))` = 0. hence f (s)g(t) ∈ nil(r) by compatibility, so g(t)f (s) ∈ nil(r), by r is (s,ω)-nil-reversible, gf ∈ nil(r)[[s,ω]]. thus ḡf̄ ∈ nil(r/j)[[s,ω]]. therefore r/j is (s,ω)-nilreversible. � in the next, we utilize the prime radical of a ring to provide descriptions of skew generalized power series that exhibit nil-reversibility. corollary 2.24. let r be a ring, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. if r is completely (s,ω)-compatible (s,ω)-nil-reversible, then r/p (r) is (s,ω)nil-reversible. proof. the proof can be derived from theorem 2.23 due to the fact that all elements in p (r) are nilpotent. � proposition 2.25. let r be a ring, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. let j be a reduced ideal of a ring r such that r/j is (s,ω)-nil-reversible. then r is (s,ω)-nil-reversible. proof. suppose f ,g ∈ [[rs,≤,ω]] satisfying f g ∈ nil(r)[[s,ω]]. then f̄ ḡ ∈ nil(r/j)[[s,ω]] and so ḡf̄ ∈ nil(r/j)[[s,ω]] since r/j is (s,ω)-nil-reversible. there is a positive integer ` ∈ n and (f̄ ḡ)` = 0̄. therefore (f (s)g(t))` ∈ j for any s,t ∈ s. since j is reduced, we have f (s)g(t) = 0 yields g(t)f (s) = 0. thus, gf ∈ nil(r)[[s,ω]]. therefore, r is (s,ω)-nil-reversible. � 10 int. j. anal. appl. (2023), 21:69 3. weak annihilator of reversible property of skew generalized power series rings the concept of weak annihilators and its properties were introduced by ouyang in [13], with a focus on subsets x of a ring r put nrr(x) = {a ∈ r|xa ∈ nil(r)} and nlr(x) = {b ∈ r|bx ∈ nil(r)}. it can be easily calculated that nrr(x) = nlr(x). the set nrr(x) is called the weak annihilator of x. if r is a ni-ring, it is evident that nrr(x) forms an ideal of r. additionally, if r is reduced, then we have rr(x) = nrr(x) = lr(x) = nlr(x), and more information and findings on weak annihilators can be found in [14]. our investigation now focuses on the correlation between weak annihilators in a ring r and those in a skew generalized power series ring [[rs,≤,ω]]. let r be a ring and γ = c(f ) be the content of f , defined as c(f ) = {f (u)|u ∈ supp(f )}⊆ r. as r ' cr, we can equate the content of f with cc(f ) = {cf (ui)|ui ∈ supp(f )}⊆ [[r s,≤,ω]]. then we have two maps φ : nrannr(id(r)) → nrann[[rs,≤,ω]](id([[rs,≤,ω]])) and ψ : nlannr(id(r)) → nlann[[rs,≤,ω]](id([[rs,≤,ω]])) defined by φ(i) = i[[rs,≤,ω]] and ψ(j) = [[rs,≤,ω]]j for each i ∈ nrannr(id(r)) = {nrr(u)|u is an ideal of r} and j ∈ nlannr(id(r)) = {nlr(u)|u is an ideal of r}, respectively. it is evident that φ is a one-to-one function. the subsequent theorem demonstrates that φ and ψ are both bijective mappings if and only if r is (s,ω)-nil-reversible. theorem 3.1. let r be a ring, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. if r is reduced and nil(r) is a nilpotent ideal of r, then the following are equivalent: (1) r is (s,ω)-nil-reversible ring. (2) the function φ : nrannr(id(r)) → nrann[[rs,≤,ω]](id([[rs,≤,ω]])) is bijective, where φ(i) = i[[rs,≤,ω]] for each i ∈ nrannr(id(r)). (3) the function ψ : nlannr(id(r)) → nlann[[rs,≤,ω]](id([[rs,≤,ω]])) is bijective, where ψ(j) = [[rs,≤,ω]]j for every j ∈ nlannr(id(r)). proof. (1)⇒(2) suppose y ⊆ [[rs,≤,ω]] and γ = ∪f∈y c(f ). by proposition 2.9 it is enough to prove nr[[rs,≤,ω]](f ) = nrrc(f )r[[s,ω]] for every f ∈ y. we know that, if g ∈ nrr[[s,ω]](f ). then f g ∈ nil(r)[[s,ω]]. according to the premise f (di )ωdi (g(tj)) ∈ nil(r) for each di ∈ supp(f ) for all tj ∈ supp(g). for element di ∈ supp(f ) for every tj ∈ supp(g), 0 = f (di )ωdi (g(tj)) = (cf (di)g)(tj) and it follows that g ∈ nrr ∪di∈supp(f ) cf (di)r[[s,ω]] = nrrc(f )r[[s,ω]]. so nrr[[s,ω]](f ) ⊆ nrrc(f )r[[s,ω]]. conversely, suppose g ∈ nrrc(f )r[[s,ω]], so cf (di)g ∈ nil(r)[[s,ω]] for all di ∈ supp(f ). thus, (cf (di)g)(tj) = f (di )ωdi (g(tj)) ∈ nil(r) for all di ∈ supp(f ) and tj ∈ supp(g). therefore, (f g)(s) = ∑ (di,tj)∈xs(f ,g) f (di )ωdi (g(tj)) = 0 int. j. anal. appl. (2023), 21:69 11 it is evident that g ∈ nrr[[s,ω]](f ). hence nrrc(f )r[[s,ω]] ⊆ nrr[[s,ω]](f ) therefore nrrc(f )r[[s,ω]] = nrr[[s,ω]](f ). so nrr[[s,ω]](y ) = ∩f∈y nrr[[s,ω]](f ) = ∩f∈y c(f )r[[s,ω]] = nrr(γ)r[[s,ω]]. (2)⇒(1) assume the elements f ,g ∈ r[[s,ω]] satisfying f g ∈ nil(r)[[s,ω]]. then g ∈ nrr[[s,ω]](f ) according to the premise nrr[[s,ω]](f ) = γr[[s,ω]] for any right ideal γ of r. inversely, 0 = f cg(tj) and for every di ∈ supp(f ), (f cg(tj))(di ) = f (di )g(tj) ∈ nil(r) for every di ∈ supp(f ) and tj ∈ supp(g). thus by reduced ring, g(tj)f (di ) ∈ nil(r), then gf ∈ nil(r)[[s,ω]]. thus, r is (s,ω)-nil-reversible. the demonstration of the equivalence between (1)⇔(3) follows a similar approach to that used for proving the equivalence between (1)⇔(2). � according to [6], a ring r is defined (s,ω)-armendariz if for tow polynomial f ,g ∈ r[[s,ω]] satisfying f g = 0, then f (r)ωr (g(t)) = 0 for all r,t ∈ s. now we given a strong condition under which r[[s,ω]] is nil-reversible. theorem 3.2. let r be a ring, (s,≤) a strictly ordered monoid and ω : s →end(r) a monoid homomorphism. if r is (s,ω)-compatible. assume that r is (s,ω)-armendariz ring, then r is (s,ω)-nil-reversible if and only if r[[s,ω]] is nil-reversible. proof. assume that r is (s,ω)-nil-reversible. let f ,g ∈ r[[s,ω]] be such that f g ∈ nil(r)[[s,ω]]. by proposition 2.16, nil(r)[[s,ω]] = nil(r[[s,ω]]). so f (ri )g(tj) ∈ nil(r) for every r,t ∈ s,∀ i, j. by condition that r is (s,ω)-armendariz, f (ri )ωri (g(tj)) = 0, for all i, j. by compatibility nil-reversibility, g(tj)f (ri ) ∈ nil(r) for all i, j. so, gf ∈ nil(r)[[s,ω]]. thus, r[[s,ω]] is nil-reversible. the converse is clear. � theorem 3.3. consider a ring r and a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). suppose that r is compatible with s. let ∆ denotes a multiplicatively closed subset of r consisting of central non-zero divisors. then r is (s,ω)-nil-reversible if and only if ∆−1r is (s,ω)-nil-reversible. proof. suppose r is (s,ω)-nil-reversible and pi,dj,u,v ∈ r. let u−1cpi,v −1cdj ∈ ∆ −1r[[s,ω]] for all i, j satisfying that u−1cpiv −1cdj ∈ nil(∆ −1r[[s,ω]]). then (u−1cpiv −1cdj ) ` = 0 for some positive integer `. this implies (cpicdj ) ` = 0, so pidj ∈ nil(r) by using lemma 2.5 freely. for any u−1cpi,v −1cdj ∈ ∆ −1r[[s,ω]] having the property that (u−1cpi )(v −1cdj ) = 0, we have (uv)−1cpicdj = 0,cpicdj = 0 for every i, j. by condition that, r is (s,ω)-nil-reversible, djpi ∈ nil(r), so (v−1u−1)cdjcpi = 0 which further yields (v −1cdj )(u −1cpi ) ∈ nil(∆ −1r[[s,≤]]). hence ∆−1r is (s,ω)-nil-reversible. the converse part is clear. � a mccoy ring is a generalization of a reversible ring, defined as a ring where the equation f (x)g(x) = 0 implies the existence of a non-zero element d such that f (x)d = 0. left mccoy rings are defined 12 int. j. anal. appl. (2023), 21:69 similarly. mccoy rings are both left and right mccoy rings. it is known that every reversible ring is mccoy. however, it cannot be assumed that if a ring r is (s,ω)-nil-reversible, then it is also (s,ω)-mccoy. an example exists that disproves this assumption. example 3.4. assume that r is a reduced ring, a strictly ordered monoid (s,≤) with a monoid homomorphism w : s → end(r). let tn(r)) =     a11 a12 a13 · · · a1n 0 a22 a23 · · · a2n 0 0 a33 · · · a3n ... ... ... ... ... 0 0 0 · · · ann   | aij ∈ r   . then tn(r) is not (s,ω)-mccoy by the similar as argument of [16, example 2.6], but tn(r) to be (s,ω)-nil-reversible by proposition 2.7. as per lambek [17], a ring r is considered symmetric if for any x,y,z ∈ r, the condition xyz = 0 implies xzy = 0. it can be easily observed that commutative rings are symmetric and symmetric rings are reversible rings. theorem 3.5. let r be a ring, (s,ω)-compatible and reversible right noetherian ring, (s,≤) a strictly ordered monoid with nil(r) is a nilpotent ideal of r and ω : s →end(r) a monoid homomorphism. the ring r to be (s,ω)-nil-symmetric if and only if so is [[rs,≤,ω]]. proof. assume a ring r is (s,ω)-nil-symmetric such that f ,g,h ∈ [[rs,≤,ω]] satisfying f gh ∈ nil([[rs,≤,ω]]). hence by proposition 2.9, f (r)g(d)h(t) ∈ nil(r) for any r,d,t ∈ s. by assumption r is nil-symmetric, then f (r)h(t)g(d) ∈ nil(r). for all s ∈ s. thus (f hg)(s) = ∑ (r,t,d)∈xs(f ,h,g) f (r)ωr (h(t)ωt(g(d))). so, the reversibility of r, f hg ∈ nil([[rs,≤,ω]]), it follows that [[rs,≤,ω]] is nil-symmetric. on the other hand, if [[rs,≤,ω]] is nil-symmetric, then r is (s,ω)-nil-symmetric because subrings of (s,ω)-nil-symmetric rings is to be (s,ω)-nil-symmetric. � conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] p.m. cohn, reversible rings, bull. london math. soc. 31 (1999), 641-648. https://doi.org/10.1112/ s0024609399006116. [2] m.b. rege, s. chhawchharia, armendariz rings, proc. japan acad. ser. a math. sci. 73 (1997), 14-17. https: //doi.org/10.3792/pjaa.73.14. https://doi.org/10.1112/s0024609399006116 https://doi.org/10.1112/s0024609399006116 https://doi.org/10.3792/pjaa.73.14 https://doi.org/10.3792/pjaa.73.14 int. j. anal. appl. (2023), 21:69 13 [3] j. lambek, on the representation of modules by sheaves of factor modules, can. math. bull. 14 (1971), 359-368. https://doi.org/10.4153/cmb-1971-065-1. [4] n.k. kim, y. lee, extensions of reversible rings, j. pure appl. algebra. 185 (2003), 207-223. https://doi.org/ 10.1016/s0022-4049(03)00109-9. [5] g. yang, z.k. liu, on strongly reversible rings, taiwan. j. math. 12 (2008), 129-136. https://www.jstor.org/ stable/43833897. [6] g. marks, r. mazurek, m. ziembowski, a unified approach to various generalizations of armendariz rings, bull. aust. math. soc. 81 (2010), 361-397. https://doi.org/10.1017/s0004972709001178. [7] p. ribenboim, noetherian rings of generalized power series, j. pure appl. algebra. 79 (1992), 293-312. https: //doi.org/10.1016/0022-4049(92)90056-l. [8] r. mazurek, m. ziembowski, on von neumann regular rings of skew generalized power series, commun. algebra. 36 (2008), 1855-1868. https://doi.org/10.1080/00927870801941150. [9] e. ali, a. elshokry, some results on a generalization of armendariz rings, asia pac. j. math. 6 (2019), 1. https://doi.org/10.28924/apjm/6-1. [10] r. antoine, nilpotent elements and armendariz rings, j. algebra. 319 (2008), 3128-3140. https://doi.org/10. 1016/j.jalgebra.2008.01.019. [11] e. ali, a. elshokry, extended of generalized power series reversible rings, italian j. pure appl. math. in press. [12] s. subba, t. subedi, nil-reversible rings, (2021). https://doi.org/10.48550/arxiv.2102.11512. [13] l. ouyang, extensions of nilpotent p.p. rings, bull. iran. math. soc. 36 (2010), 169-184. [14] l. ouyang, special weak properties of generalized power series rings, j. korean math. soc. 49 (2012), 687-701. https://doi.org/10.4134/jkms.2012.49.4.687. [15] p.p. nielsen, semi-commutativity and the mccoy condition, j. algebra. 298 (2006), 134-141. https://doi.org/ 10.1016/j.jalgebra.2005.10.008. [16] s. yang, x. song, extensions of mccoy rings relative to a monoid, j. math. res. exposition. 28 (2008), 659-665. https://doi.org/10.3770/j.issn:1000-341x.2008.03.028. [17] j. lambek, on the representation of modules by sheaves of factor modules, can. math. bull. 14 (1971), 359-368. https://doi.org/10.4153/cmb-1971-065-1. https://doi.org/10.4153/cmb-1971-065-1 https://doi.org/10.1016/s0022-4049(03)00109-9 https://doi.org/10.1016/s0022-4049(03)00109-9 https://www.jstor.org/stable/43833897 https://www.jstor.org/stable/43833897 https://doi.org/10.1017/s0004972709001178 https://doi.org/10.1016/0022-4049(92)90056-l https://doi.org/10.1016/0022-4049(92)90056-l https://doi.org/10.1080/00927870801941150 https://doi.org/10.28924/apjm/6-1 https://doi.org/10.1016/j.jalgebra.2008.01.019 https://doi.org/10.1016/j.jalgebra.2008.01.019 https://doi.org/10.48550/arxiv.2102.11512 https://doi.org/10.4134/jkms.2012.49.4.687 https://doi.org/10.1016/j.jalgebra.2005.10.008 https://doi.org/10.1016/j.jalgebra.2005.10.008 https://doi.org/10.3770/j.issn:1000-341x.2008.03.028 https://doi.org/10.4153/cmb-1971-065-1 1. introduction 2. (s, )-nil-reversible rings 3. weak annihilator of reversible property of skew generalized power series rings references international journal of analysis and applications issn 2291-8639 volume 3, number 1 (2013), 47-52 http://www.etamaths.com existence of heteroclinic solutions to fourth order φ−laplacian dynamical equations k. r. prasad1, p. murali1 and n.v.v.s. suryanarayana2,∗ abstract. in this paper, we derive sufficient conditions for the existence of heteroclinic solutions to fourth order φ−laplacian dynamical equation,[ φ ( y∆ 2 (t) )]∆2 = f(y(t)), t ∈ t, on infinite time scales by using variational approach as minimizers of an action functional on special functional space. and also, as an application we demonstrate our result with an example. 1. introduction the study of heteroclinic solutions for p and φ-laplacian operators on infinite time scales have a certain impulse in recent years, which are motivated by applications in various biological, physical, mechanical and chemical models, such as phase transition, physical processes in which the variable transits from an unstable equilibrium to a stable one, or front propagation in reaction diffusion equation. these solutions provide an important information on the dynamics of the system. due to the importance in both theory and applications, the study of heteroclinic solutions gained momentum on real intervals, we list a few; avrameseu and vladimirecu [1], cabada and cid [3, 4], cabada and tersion [5], and marcelli and papalini [7, 8]. the history of p and φ-laplacian operators for boundary value problems also enjoys a good history, first for differential equations, then finite difference equations, and recently, unifying results for dynamic equations. these operators have been widely studied by many researchers. in this theory, the most investigated operator is the classical p-laplacian, generally φp(y) := y|y|p−2 with p > 1, which, in recent years, has been generalized to other types of differential operators that preserve the monotonicity of the p-laplacian, but are not homogeneous. these more general operators, which are usually referred to as φ-laplacian, are involved in the modeling of non-newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity and theory of capillary surfaces. the related nonlinear differential equation has the form [φ(y′)]′ = f(t,y,y′). in this paper, we are dealing with some special class of time scales namely infinite and semi infinite time scales because continuous time orbits and discrete time orbits 2010 mathematics subject classification. 34a40, 34b15, 34l30. key words and phrases. heteroclinic solution, φ-laplacian, variational approach, infinite time scales. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 47 48 prasad, murali and suryanarayana are topologically different. a time scale is an arbitrary nonempty closed subset of the real numbers and we denote the time scale by the symbol t. a time scale t is said to be infinite time scale, we mean, if it has no infimum and no supremum. (i.e., inf t = −∞ and sup t = +∞.) a time scale t is said to be semi infinite time scale, we mean, if it has either infimum but no supremum or supremum but no infimum. (i.e., inf t = a and sup t = +∞ or inf t = −∞ and sup t = b)and we denote as t+a = [a, +∞) and t − b = (−∞,b] respectively. for example if we consider time scale t = {−(a)n} n∈n ∪{0}∪{b n} n∈n, (a,b > 1) then it is an infinite time scale and if we remove either negative terms or positive terms then it is an example of semi infinite time scale. by an interval [a,b]t means the intersection of the real interval with a given time scale. i.e [a,b]t = [a,b] ∩t. most of the definitions and results on time scales are from the text book by bohner and peterson [2] and lakshmikantham, sivasundaram and kaymakcalan [6]. heteroclinic solutions to time scale dynamical systems are not available in the literature which will unify both the continuous and discrete dynamical systems. now, we consider fourth order φ−laplacian dynamical equation on infinite time scales, (1) [ φ ( y∆ 2 (t) )]∆2 = f(y(t)), t ∈ t, where f : t → r is continuous. by using variational approach, we establish sufficient conditions for the existence of heteroclinic solution to the dynamical equation (1) under certain assumptions. a heteroclinic solution of equation (1) connecting −1 to +1 in the phase-space, is a function y∆ 2 ∈ c2rd(t) such that y∆ 2 ∈ (−a,a), φ ◦ y∆ 2 ∈ c2rd(t) and y satisfies the dynamical equation (1) and with the property lim t→±∞ (y(t),y∆(t),y∆ 2 (t),y∆ 3 (t))) = (±1, 0, 0, 0). through out the paper we assume the following: (a1) f(y) = 0 if and only if y = ±1. (a2) there exists a primitive f of f such that f(−1) = f(+1) and f(y) ≥ 0 for all y ∈ r and lim |y|→+∞ inf f(y) > 0. (a3) φ : (−a,a) → r is a positive increasing homeomorphism with φ(0) = 0 and 0 < a < +∞. (a4) φ and f satisfy symmetric conditions φ(y) = φ(−y) and f(y) = f(−y). (a5) the energy is conserved. (i.e., φ̃(y 2 ) + f(y) = k, k ∈ r+) (a6) if {yn(t)} is a sequence of solutions of (1) for which for each n ∈ t + 0 an arbitrary compact interval [0,σ2(n)]t and there exists an m > 0 such that yn(t) ≤ m for all t ∈ [0,σ2(n)]t and for all t ∈ n, then there exists a subsequence {ynj (t)}, such that {y∆ i nj } converges uniformly on [0,σ2(n)]t, i = 0, 1. existence of heteroclinic solutions 49 the rest of the paper is organized as follows, in section 2, by using variational approach, we establish sufficient conditions for the existence of heteroclinic solution to the differential equation (1). as an application, we give an example to demonstrate our result. 2. existence of heteroclinic solutions in this section, by using variational approach, we establish sufficient conditions for the existence of heteroclinic solution to the differential equation (1). the equation (1) is the euler-lagrange equation corresponding to the action functional (2) f(y) = ∫ t (∫ φ(y 2 )∆y 2 + f(y) ) ∆t, where y 2 is the second delta derivative of y with respect to t and f(y) is the primitive of f(y). the action functional is defined in functional space (3) e = { y : t → r| y(0) = 0, y + 1 ∈ h2(t−), y − 1 ∈ h2(t+) } . theorem 2.1. the functional f : e → r defined by (2) on (3) is of class c1 and any critical point in c∞ is a heteroclinic solution of (1) connecting −1 to +1. proof. for any function η ∈ c2c (t), for all τ ∈ r, y+τη ∈f and f(y+τη) is delta differentiable as a real function of the parameter τ. since y minimizes f in the space e, the function f(y + τη) archives a minimum at τ = 0. let φ̃(y 2 ) = ∫ φ(y 2 )∆y 2 , f∆τ (y + τη)|τ=0 = [∫ t ( φ̃(y 2 ) + f(y) ) ∆t ]∆τ at τ = 0 f∆τη = ∫ t ( φ̃∆τ (y 2 )η∆ 2 τ + f(y)η ) ∆t. now let y be a critical point of f. we have f∆τ (y)η = 0 for every η ∈ h2(t) satisfies η(0) = 0. starting with η ∈ c2c (t) and du bois-reymond lemma, we have φ ◦ y∆ 2 ∈ c2(t) and y satisfies (1). hence y − 1 ∈ h4(t+). similarly we have y + 1 ∈ h4(t−). from the l2−integrability of the delta derivative implies that lim t→±∞ y(t) = ±1 and lim t→±∞ y∆ n (t) = 0 for n = 1, 2, 3, so that y is a heteroclinic solution of (1) connecting −1 to +1 and a straightforward argument given as y is of class c∞. � now, we prove main theorem which confirms the efficiency of a minimization approach. theorem 2.2. the functional f : e → r defined by (2) on (3) has a minimizer which is a heteroclinic solution of (1) connecting −1 to +1. furthermore, any minimizer is odd and positive in (0, +∞). proof. for convenience, we introduce the following spaces e+ = { y : t+ → r : y(0) = 0, y − 1 ∈ h2(t+) } e− = { y : t− → r : y(0) = 0, y + 1 ∈ h2(t−) } 50 prasad, murali and suryanarayana and consider the action functionals f± : e± → r by f±(y) = ∫ t± l(y,y∆,y∆ 2 )∆t, where l(y,y∆,y∆ 2 ) is the lagrangian given by l(y,y∆,y∆ 2 ) = φ̃(y 2 ) + f(y). let us denote the values c = inf e f and c± = inf e± f±. since f is symmetric, we have, for all y+ ∈e+, f+(y+) = f−(y−), where y− ∈e− is defined by y−(t) = −y+(−t). therefore, we have c+ = c− = c 2 . first we prove that the variational problem inf{f+(y) : y ∈ e+} has a positive solution. let (vn)n ⊂ e+ be a minimizing sequence for f+, i.e., vn ∈ e+ for all n ∈ n and f+(vn) → c+. for each n ≥ 0, we define tn = sup{t ≥ 0 : vn(t) = 0}. since limt→+∞vn(t) = 1, tn < +∞ for all n ≥ 0. we now consider the positive sequence (v+n )n ⊂e+, where v+n (t) = vn(t + tn) for t ≥ 0. we observe that∫ tn 0 l(vn,v ∆ n ,v ∆2 n )dt ≥ 0 so that f+(v+n ) ≤f+(vn) which implies that (v+n )n is also a minimizing sequence for f+. as the sequence f+(v+n ) is uniformly bounded, we deduce a uniform estimate for ‖ v+n − 1 ‖ h2(t+) . from the positivity of v + n that∫ t+ f(v+n )∆t ≤f +(v+n ) and there exists v+ ∈ h2(t+) + 1 such that v+n − 1 h 2(t+) −−−−−→ v+ − 1 and v+n c 1 loc(t + ) −−−−−−→ v+. as the first two terms in f+ are the sequence of seminorms and fatou’s lemma is applicable to the last one, we have f+(v+) ≤ lim n→+∞ inf f(v+n ) = c +. the converges being uniform an compact interval, we conclude that v+(0) = 0 so that v+ ∈e+ and f+(v+) = c+. observe that v+ is positive on (0, +∞) otherwise we could proceed as above to construct a positive function having smaller action. existence of heteroclinic solutions 51 secondly, we show that, if v ∈ e+ is such that (f+)∆τ (v) = 0, then v∆ 2 (0) = 0, v∗ ∈e defined by (4) v∗(t) = { v(t), if t ≥ 0 −v(−t) if t < 0 is a minimizer of f in e and v∗ is a heteroclinic solution of (1). we compute (5) (f+)∆τ (v)(η) = ∫ t+ ( φ̃∆τ (v 2 )η∆ 2 τ + f(v)η ) ∆t for all η ∈ h2(t+) ∩h10 (t + ). from theorem 2.1, we deduce that v is a solution of equation (1), φ ◦ v∆ 2 ∈ c2(t+) and (6) lim t→+∞ v(t) = +1 and lim t→+∞ v∆ n (t) = 0 for n = 1, 2, 3, integrating (5), we obtain for all n ∈ h2(t+) ∩h10 (t + ), as∫ t+ (∫ [φ(y∆ 2 )]∆ 2 τ + f(y) ) η(t)∆t = 0. take (6) into account, we now deduce that v∆ 2 (0)η∆(0) = 0 for all η ∈ h2(r+) ∩ h10 (r + ) which implies that v∆ 2 (0) = 0. the function v∗ : t → r defined by (4) is of class c4, and solution of (1) and also as f(v∗) = 2f+(v) = 2c+ = c. we conclude that v∗ is a minimization of f in e. finally, we prove that if y ∈e minimizes f, then y is odd. let us define y± = y|r±. as f(y) = c, we obviously have f+(y+) = f−(y−) = c 2 otherwise the odd extension of y+ or y− would have a lower action than c. define v+ ∈ e+ by v+(t) = −y−(−t). then v+ satisfies f+(v+) = c+ and therefore minimizer f+ in e+. it follows that both y+ and v+ are minimizers of f+ in e+. from claim 2, we have that y∆ 2 (0) = (v+)∆ 2 (0) = 0 and as (v+)∆(0) = (−y−)∆(0) = (y+)∆(0) and (v+)∆ 3 (0) = (−y−)∆ 3 (0) = (y+)∆ 3 (0), the functions v+ and y+ are the solutions of the cauchy problem,[ φ ( y∆ 2 (t) )]∆2 = f(y(t)), t ∈ t, y(0) = 0, y∆(0) = (y+)∆(0); y∆(0) = 0,y∆ 3 (0) = (y+)∆ 3 (0). by uniqueness, this implies y+(t) = v+(t) for t ∈ r+, that is y+(t) = −y−(−t) for all t ∈ t+. � 52 prasad, murali and suryanarayana example in this section, as an application, we give an example to demonstrate our result. we consider the following fourth order φ-laplacian differential equation on infinite time scale t = { {−2n} n∈n∪{0} ∪ [−1, 1] ∪{2 n} n∈n∪{0} } (7) [ φ(y∆ 2 ) ]∆2 = f(y), t ∈ t, where φ(y 2 ) = y2 2√ 1+y2 2 for y 2 ∈ t and f(y) = y2 − 1. then φ and f satisfy the conditions (a1)-(a6) . therefore, it follows from theorem 2.2 that the fourth order φ-laplacian differential (7) has a heteroclinic solution. acknowledgement. one of the authors (dr. p. murali) is thankful to c.s.i.r. of india for awarding him an ra. references [1] c. avrameseu and c. vladimirecu, existence of solution to second order differential equations having finite limits at ±∞, elec. j. diff. equa., vol.18(2004), pp. 1-12. [2] m.bohner and a.c.peterson, dynamic equations on time scales, an introduction with applications, birkhauser, boston,ma,(2001). [3] a. cabada and j. a. cid, solvability of some φ-laplacian singular difference equation defined on integers, the arab. j. scie. engi., vol 34(2009), no.id, pp.75-81. [4] a. cabada and j. a. cid, heteroclinic solution for non-autonomous boundary value problem with singular φ-laplacian operator, discr. cont. dyna. syst., vol 18(2009), pp. 118-112. [5] a. cabada and s. tersion, existence of heteroclinic solutions for discrete p-laplacian problems with a parameter, nonlinear; real world appl., doi:10.1016/j.nonwrg.2011.02.022. [6] v. lakshmikantham, s. sivasundaram and b. kaymakcalan, dynanic systems on measure chains, mathematics and its applications, 370, kluwer, dordrecht, 1996. [7] c. marcelli and f. papalini, heteroclinic solution for second order non-autonoms boundary value problem on real line, diff. equa. dyna. syst., vol11(2003), pp.333-353. [8] c. marcelli and f. papalini, heteroclinic connection for fully nonlinear non-autonums second order differential equations, j. diff. equa., 241(2007), pp.160-183. 1department of applied mathematics, andhra university, visakhapatnam, 530003, india 2department of mathematics, vitam college of engineering, visakhapatnam, 531173, india ∗corresponding author international journal of analysis and applications volume 19, number 3 (2021), 477-493 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-477 received february 28th, 2021; accepted april 7th, 2021; published may 4th, 2021. 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. coefficient bounds; univalent functions; starlike functions; toeplitz determinants ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 477 bounds on toeplitz determinant for starlike functions with respect to conjugate points daud mohamad, nur hazwani aqilah abdul wahid* faculty of computer and mathematical sciences, universiti teknologi mara, 40450 shah alam, selangor, malaysia *corresponding author: hazwaniaqilah@tmsk.uitm.edu.my abstract. this paper is concerned with the estimate of the upper bounds of the toeplitz determinants  2 3t and  3 3t for functions belonging to the subclass of starlike functions with respect to conjugate points. the results presented would extend the results for some existing subclasses in the literature. 1. introduction let  be the class of functions  f z which are analytic in an open unit disk  : 1e z z  and having the power series expansion (1.1)   2 n n n f z z a z     in .e let s be the class of functions  f z  and univalent in .e let p be the class of functions  p z of the form https://doi.org/10.28924/2291-8639-19-2021-477 int. j. anal. appl. 19 (3) (2021) 478 (1.2)   1 1 nn n p z p z     that is analytic in e and satisfying the condition  re 0, .p z z e  functions in p are called carathéodory functions. it is well known that if   ,p z p then a schwarz function  z exists with  0 0,    1,z  z e such that [1]       1 . 1 z p z z      for two functions  f z and  g z analytic in ,e we say that the function  f z is subordinate to  g z and we write it as    f z g z if there exists a schwarz function  z which is analytic in e with  0 0,    1,z  such that     .f z g z further, if  g z is univalent in ,e then        0 0f z g z f g  and    f e g e (see miller and mocanu [2, 3] for details). let *s denote the class of starlike functions in .s it is known that   *f z s if and only if (1.3)     re 0, . zf z z e f z          el-ashwah and thomas [4] defined the following class: (1.4)         * 2: re 0, .c zf z s f z z e f z f z                    functions in the class * cs are called starlike functions with respect to conjugate points. halim [5] defined the following class: (1.5)           * 2: re , 0 <1, .c zf z s f z z e f z f z                         in terms of subordination, dahhar and janteng [6] generalized the class *cs and it is denoted by  * , .cs a b this class is defined as follows: int. j. anal. appl. 19 (3) (2021) 479 (1.6)           * 2 1, : , 1 1, . 1 c zf z az s a b f z b a z e bzf z f z                  wahid et al. [7] introduced the subclass of tilted starlike functions with respect to conjugate points of order ,  * , , ,cs a b  and it is given by (1.7)         * 1 1, , , : sin , , 1 i c zf z az s a b f z e i z e g z t bz                          where       , 2 f z f z g z   cos 0,t     0 <1, 2    and 1 1.b a    in particular,  * *0 ,c cs s  * *1, 1c cs s  and   * *0,0,1, 1 .c cs s  toeplitz matrices are one of the well-studied classes of structured matrices. the concept of toeplitz matrices led to the development of the studies related to toeplitz determinants, toeplitz kernel, toeplitz operators, and q-deformed toeplitz matrices [8]. in a recent investigation, the toeplitz determinant has been studied by [9-18], and they succeeded in estimating the coefficient bounds for toeplitz determinant   ,qt n , 1n q  for the first few values of n and q over some subclasses of . the toeplitz determinant  ,qt n , 1n q  of functions  f z of the form (1.1), is defined by thomas and halim [9]   1 1 1 2 1 1 2 , 1. n n n q n n n q q n q n q n a a a a a a t n a a a a                   however, apart from these works, there was no study of finding estimates for  2 3t and  3 3t for the subclasses introduced by el-ashwah and thomas [4], halim [5], dahhar and janteng [6], and wahid et al. [7]. in fact, as far as we are concerned, no bound for  3 3t was obtained for the class of univalent functions and its subclasses in the existing literature. therefore, in this paper, we obtain the upper bounds for the toeplitz determinant for  * , , ,cs a b  as defined in (1.7) for the case of 3,n 2q  and 3,n 3q  namely int. j. anal. appl. 19 (3) (2021) 480 (1.8)   3 42 4 3 3 a a t a a  and (1.9)   3 4 5 3 4 3 4 5 4 3 3 . a a a t a a a a a a  we also give some results for the subclasses introduced by el-ashwah and thomas [4], halim [5], and dahhar and janteng [6]. we shall state the following lemmas to prove our main results. 2. preliminary results lemma 2.1. [19] for a function  p z p of the form (1.2), the sharp inequality 2np  holds for each 1.n  equality holds for the function   1 . 1 z p z z    lemma 2.2. [20] let  p z p of the form (1.2) and . then  2max 1, 2 1 , 1 1.n k n kp p p k n       if 2 1 1,   then the inequality is sharp for the function   1 1 z p z z    or its rotations. if 2 1 1,   then the inequality is sharp for the function   1 1 n n z p z z    or its rotations. 3. main results theorem 3.1. if the function  f z given by (1.1) belongs to the class  * , , , ,cs a b  then                    2 4 2 3 3 2 2 4 2 2 2 2 2 2 3 2 3 3 2 2 3 832 64 1 3 3 11 12 2 4 2304 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 t t                                                                   where ,ite     ,t a b t  cost    and 1 .b  int. j. anal. appl. 19 (3) (2021) 481 proof. from (1.7), since    * , , , ,cf z s a b  according to subordination relationship, so there exists a schwarz function  z such that (3.1)         11 sin , 1 i zf z a ze i g z t b z                  where       , 2 f z f z g z   cos .t    define a function       1 1 1 . 1 n n n z h z k z z           we have  h z p and (3.2)       1 . 1 h z z h z     using (3.2), from (3.1), we have (3.3)              1 1 1 1 i i i e b t h z e b tzf z e g z b h z b                   where   .t a b t  using the series expansion in (3.3), we get (3.4)                 2 3 4 2 3 4 2 3 4 2 3 2 3 4 1 2 3 2 3 4 2 3 4 2 3 4 2 3 2 3 4 1 2 3 1 2 3 4 1 2 3 4 1 1 1 1 . i i i i e b z a z a z a z e b z a z a z a z k z k z k z e b t z a z a z a z e b t z a z a z a z k z k z k z                                                equating the coefficients of 3z and 4z respectively in the expansion of (3.4) and for simplicity, we take ite   and 1 ,b  give us (3.5) 2 2 2 2 1 1 3 2 8 k k k a      int. j. anal. appl. 19 (3) (2021) 482 and (3.6) 2 3 3 3 2 3 2 3 1 2 1 2 1 1 1 4 8 6 8 3 2 . 48 k k k k k k k k a               squaring (3.5) and (3.6), respectively, we get (3.7) 2 2 4 2 2 2 2 4 3 2 3 4 4 2 2 1 1 2 1 1 2 1 3 4 4 2 4 64 k k k k k k k k a               and 2 2 2 6 4 6 3 6 2 2 6 4 6 2 3 6 3 2 6 4 4 3 1 1 1 1 1 1 1 3 2 3 2 3 3 2 3 2 1 2 3 1 2 3 1 3 1 3 1 3 1 3 1 3 2 2 2 2 2 2 2 2 4 3 4 1 2 1 2 1 2 1 2 1 2 64 3 2 3 11 12 4 2304 96 128 16 24 16 24 16 36 96 64 12 34 a k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k                                                  2 4 2 3 41 2 1 2 4 3 4 2 2 4 3 1 2 1 2 1 2 36 16 6 36 16 . k k k k k k k k k k                  (3.8) from the equations (1.8), (3.7), and (3.8), yield    2 2 2 4 3 2 2 6 4 6 3 6 2 2 6 4 6 2 3 6 3 2 6 4 3 1 1 1 1 1 1 1 3 2 3 2 3 3 2 3 2 1 2 3 1 2 3 1 3 1 3 1 3 1 3 1 3 2 2 2 2 2 2 2 2 4 3 1 2 1 2 1 2 1 2 3 64 3 2 3 11 12 4 2304 96 128 16 24 16 24 16 36 96 64 12 t a a k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k                                                          4 2 4 2 3 4 1 2 1 2 1 2 4 3 4 2 2 4 3 1 2 1 2 1 2 2 2 4 2 4 2 4 2 2 1 1 2 1 1 2 1 2 2 2 6 4 2 3 2 3 2 4 3 2 1 4 2 1 34 36 16 6 36 16 144 36 144 72 144 36 64 144 1 3 3 11 2 12 4 2304 36 72 36 k k k k k k k k k k k k k k k k k k k k k k k k                                                                               2 2 1 2 1 2 3 2 2 2 2 3 2 2 2 1 2 1 3 4 3 2 2 2 3 3 1 2 144 144 96 128 36 96 64 16 16 24 24 16 12 6 34 36 36 16 16 . k k k k k k k k k k k                                                         further, by suitably arranging the terms yield int. j. anal. appl. 19 (3) (2021) 483             2 2 2 6 4 2 3 2 3 2 4 2 3 2 1 2 2 2 2 1 2 1 2 2 1 2 3 1 2 3 2 2 2 1 3 64 144 1 3 3 11 2 12 4 2304 36 36 72 144 144 144 144 36 96 64 96 128 96 128 16 16 24 16 t k k k k k k k k k k k k                                                                                                    2 3 2 2 3 3 1 2 2 2 2 2 2 2 2 2 2 6 4 2 3 2 3 2 4 3 2 1 2 2 1 2 1 24 12 6 36 34 36 16 16 16 16 24 16 24 16 16 24 16 24 64 144 1 3 3 11 2 12 4 2304 144 144 k k k k k k k k k                                                                                              1 2 3 1 2 3 2 2 2 1 3 1 2 96 128 16 16 24 16 24 k k k k k k k k k                      (3.9) where 2 236 36 72 , 144 144             2 236 96 64 96 128             and 3 3 2 2 2 2 3 3 2 2 2 2 12 6 36 34 36 16 16 . 16 16 24 16 24                                  consequently, by the triangle inequality, from (3.9), we get              2 2 2 6 4 2 3 3 2 2 4 2 3 2 1 2 2 1 2 1 1 2 3 1 2 3 2 2 2 1 3 1 2 3 64 144 1 3 3 11 12 2 4 2304 144 144 96 128 16 16 24 16 24 . t t k k k k k k k k k k k k k k k                                                (3.10) by lemma 2.2, int. j. anal. appl. 19 (3) (2021) 484 (3.11)  22 1 2 2 2max 1, 2 1 72 144 72 144 144 2max 1, , 144 144 k k                         (3.12)  3 1 2 2 2 2max 1, 2 1 128 128 72 192 96 2max 1, 96 128 k k k                         and (3.13)         3 1 2 2 3 2 3 3 2 2 2 2 2 2 2 2 2 2 2max 1, 2 1 16 32 24 88 32 24 12 2 max 1, 16 16 24 16 24 16 92 72 . 16 16 24 16 24 k k k                                                          by making use of lemma 2.1 together with (3.11)-(3.13), we find that (3.14)  2 22 2 1 2 1 72 144 72 144 144 144 144 8 144 144 , 144 144 k k k                          (3.15)  2 2 1 2 3 1 2 128 128 72 192 96 96 128 8 96 128 96 128 k k k k k                       and                       3 2 2 2 1 3 1 2 2 3 2 3 2 2 2 2 2 2 3 2 2 2 2 2 16 16 24 16 24 16 32 24 88 32 16 16 16 24 16 24 16 16 24 16 24 24 12 16 92 72 . 16 16 24 16 24 k k k k                                                                  (3.16) again by applying lemma 2.1 along with (3.14)-(3.16), from (3.10), we obtain int. j. anal. appl. 19 (3) (2021) 485                    2 4 2 3 3 2 2 4 2 2 2 2 2 2 3 2 3 3 2 2 3 832 64 1 3 3 11 12 2 4 2304 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 . t t                                                                   the result is sharp for the function given by     1 1 sin . 1 i zf z ze i g z t z                this completes the proof of theorem 3.1. remark 3.1. for 0,  0 ,  1a  and 1,b   theorem 3.1 yields  2 3 25.t  this inequality coincides with the result obtained by ali et al. [14] for *.s theorem 3.2. if the function  f z given by (1.1) belongs to the class  * , , , ,cs a b  then                   3 2 2 2 3 3 2 2 2 4 2 3 2 3 2 3 4 4 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 t t t                                                                                         2 2 3 2 2 2 3 3 2 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16                                where ,ite     ,t a b t  cost    and 1 .b  proof. upon simplification of (1.9), the determinant  3 3t can be written as     2 23 3 5 3 4 3 53 2t a a a a a a    and by using the triangle inequality, we get   2 23 3 5 3 4 3 53 2 .t a a a a a a    now, equating the coefficient of 5z in the expansion of (3.4) and for simplicity, we take ite   and 1 ,b  give us (3.17) 2 2 2 2 4 4 4 3 4 1 3 1 3 2 2 1 1 5 4 2 2 4 3 2 3 2 2 2 2 1 1 1 2 1 2 1 2 48 32 48 12 24 6 384 11 6 12 44 36 . 384 k k k k k k k k k a k k k k k k k k                                int. j. anal. appl. 19 (3) (2021) 486 from the equations (3.5) and (3.17), we obtain   2 2 2 2 2 2 2 4 4 3 5 2 1 1 4 1 3 1 3 2 2 1 4 3 4 2 2 4 3 2 3 2 2 2 2 1 1 1 1 2 1 2 1 2 4 3 2 2 3 2 2 2 1 1 2 1 96 48 48 48 32 48 12 24 384 6 11 6 12 44 36 1 6 11 6 36 44 12 384 a a k k k k k k k k k k k k k k k k k k k k k k k                                                                             2 21 2 1 3 2 448 48 24 12 48 32 96 48 .k k k k k k                  (3.18) further, by suitably arranging the terms, we get             2 2 2 2 3 2 2 3 3 5 1 2 1 2 2 1 2 4 1 3 2 3 2 2 3 2 2 2 2 1 2 1 2 2 4 1 3 1 36 44 12 6 11 6 384 48 48 24 12 8 6 6 4 96 6 11 6 36 44 12 384 36 44 12 6 4 48 6 a a k k k k k k k k k k k k k k k                                                                                                2 2 1 2 2 2 2 2 2 1 2 1 4 1 3 2 2 1 2 2 48 48 24 12 96 36 44 12 48 384 48 48 24 12 96 k k k k k k k k k k k k                                             (3.19) where 3 2 2 3 2 2 6 11 6 36 44 12                    and 6 4 . 6      consequently, by the triangle inequality, from (3.19), we get (3.20)   2 2 2 2 3 5 1 2 1 4 1 3 2 2 2 1 2 36 44 12 48 384 96 48 48 24 12 . t a a k k k k k k k k k                         by making use of lemma 2.1 and lemma 2.2, we find that int. j. anal. appl. 19 (3) (2021) 487 (3.21)     2 2 2 2 2 2 1 2 1 3 2 2 2 3 2 2 36 44 12 8 36 44 12 12 36 22 44 12 12 2 36 44 12 k k k                                         and (3.22) 4 1 3 12 6 8 48 96 . 6 k k k        again by applying lemma 2.1 along with (3.21) and (3.22), from (3.20) yields (3.23)      3 2 2 2 3 3 5 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 . t a a                                  in view of (3.5), (3.7), (3.8), and (3.17), we have       2 2 3 4 3 5 2 2 4 2 2 2 2 1 1 2 2 2 6 4 3 2 2 4 2 3 3 2 4 3 1 3 2 2 2 2 1 2 3 1 3 2 2 1 2 2 576 144 144 288 576 576 9216 512 8 24 16 24 88 96 32 9216 768 1024 128 192 128 192 128 a a a a k k k k k k k k k k k k k                                                                           2 2 4 3 2 2 3 1 2 2 3 2 2 3 3 2 4 2 1 2 3 6 4 3 2 2 3 4 3 2 2 1 1 3 2 2 1 4 1 288 768 512 96 272 288 128 48 288 128 288 72 144 192 288 9216 3 21 51 51 18 96 240 144 144 144 k k k k k k k k k k k k k k                                                                                      2 2 2 2 4 3 2 2 3 1 2 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 4 2 2 1 2 1 2 4 108 372 288 42 204 306 144 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 144 144 288 288 k k k k k k k k k k k k k k                                                                                     3 2 2 2 2 2 1 2 3 1 2 3 2 2 2 1 3 4 3 2 2 2 3 3 1 2 72 144 576 736 180 396 224 32 192 48 128 16 54 48 68 288 18 16 128 . k k k k k k k k k                                                     int. j. anal. appl. 19 (3) (2021) 488 by suitably arranging the terms, we get             2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 2 2 1 2 1 2 4 2 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 144 144 288 72 144 576 576 288 576 576 288 a a a a k k k k k k k k k k k                                                                                          2 2 1 2 3 1 2 3 2 2 2 1 3 2 2 2 3 3 3 1 2 2 2 2 180 396 224 576 736 576 736 32 192 48 128 16 54 48 68 288 18 16 128 32 192 48 128 16 k k k k k k k k k                                                                          and further yields                    2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 1 2 1 2 4 2 1 2 3 1 2 3 2 2 2 1 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 288 576 736 32 192 48 128 16 a a a a k k k k k k k k k k k k k k k k k                                                                        3 1 2k k k    (3.24) where 2 2144 144 288 , 576 576             72 144 , 288       2 2180 396 224 576 736             int. j. anal. appl. 19 (3) (2021) 489 and           3 2 2 2 3 3 2 2 2 . 54 48 68 288 18 16 128 32 192 48 128 16                                consequently, by the triangle inequality, from (3.24), we obtain               2 2 3 4 3 5 2 2 2 2 2 3 1 4 6 4 2 3 2 3 2 4 3 4 1 2 2 2 1 2 1 2 4 2 1 2 3 1 2 3 2 2 2 1 3 1 2 2 576 512 144 144 9216 24 5 3 88 35 96 32 51 18 576 576 288 576 736 32 192 48 128 16 . a a a a t k k k k k k k k k k k k k k k k k k k k                                                             (3.25) by lemma 2.2, (3.26)     2 2 1 2 2 2max 1, 2 1 288 576 288 576 576 2max 1, , 576 576 k k                          (3.27)  24 2 2max 1, 2 1 144 288 288 2max 1, , 288 k k                  (3.28)     3 1 2 2 2 2max 1, 2 1 360 792 576 448 736 2max 1, 576 736 k k k                          and             3 1 2 3 2 2 2 3 3 2 2 2 2 2max 1, 2 1 108 96 328 576 32 92 32 48 256 16 2max 1, . 32 192 48 128 16 k k k                                             (3.29) hence, applying lemma 2.1 together with (3.26)-(3.29), we find that int. j. anal. appl. 19 (3) (2021) 490 (3.30)  2 22 2 1 2 1 288 576 288 576 576 576 576 8 576 576 , 576 576 k k k                          (3.31) 2 2 4 2 1152 144 288 288 288 , 288 k k k         (3.32)  2 2 1 2 3 1 2 360 792 576 448 736 576 736 8 576 736 576 736 k k k k k                          and                   3 2 2 2 1 3 1 2 2 2 2 3 2 2 2 3 3 2 2 2 2 32 192 48 128 16 16 32 192 48 128 16 108 96 328 576 32 92 32 48 256 16 . 32 192 48 128 16 k k k k                                                             (3.33) again by applying lemma 2.1 along with (3.30)-(3.33), from (3.25) yields                  2 2 3 4 3 5 2 2 2 4 2 3 2 3 2 3 4 4 2 2 3 2 2 2 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 8 360 792 576 448 736 16 108 96 32 328 576 48 92 a a a a t                                                                           2 3 3 232 256 16 .      (3.34) finally, from (3.23) and (3.34), we obtain                   3 2 2 2 3 3 2 2 2 4 2 3 2 3 2 3 4 4 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 8 288 576 288 576 576 9216 64 24 5 3 88 35 96 51 32 18 4 144 288 288 t t t                                                                                         2 2 3 2 2 2 3 3 2 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16 .                                int. j. anal. appl. 19 (3) (2021) 491 this completes the proof of theorem 3.2. by putting the specific values for the parameters , , a and b in theorem 3.1 and theorem 3.2, we obtain the coefficient bounds for the toeplitz determinants for the subclasses introduced by el-ashwah and thomas [4], halim [5], and dahhar and janteng [6], respectively as follows. corollary 3.1. for  * 0,0,1, 1 ,cf s  we obtain  2 3 25t  and  3 3 240.t  corollary 3.2. for  * 0, ,1, 1 ,cf s   we obtain                    2 4 2 2 2 3 2 4 1 3 832 64 16 1 8 288 1 288 1 2304 8 288 1 192 1 16 192 1 64 1 t                           and                                      2 3 3 2 2 4 2 3 2 2 1 3 8 48 1 16 1 16 6 16 1 4 96 1 4 24 1 192 384 4 1 2304 2048 8 288 1 8 1152 1 1152 1 9216 64 80 1 4 288 1 288 8 1440 1 1152 1 16 864 1 128 1 . t                                                         corollary 3.3. for  * 0,0, , ,cf s a b we obtain                                    2 4 32 2 23 2 4 22 22 2 3 2 3 3 22 3 832 64 1 3 3 11 2304 12 2 4 8 72 144 72 144 144 8 128 128 72 192 96 16 16 32 24 88 32 24 12 16 92 72 a b t a b a b a b a b a b a b a b a b a b a b a b                                                                    int. j. anal. appl. 19 (3) (2021) 492 and                                      2 33 2 2 3 2 22 4 3 22 2 3 3 3 8 12 36 22 44 12 12 2 384 16 12 6 8 4 48 48 4 24 12 192 2304 2048 8 144 144 9216 8 288 576 288 576 576 64 24 5 3 88 35 96 51 a b t a b a b a b a b a b a b a b a b a b a b a b a b a b                                                                                          4 4 2 32 22 2 3 3 2 32 18 4 144 288 288 8 360 792 576 448 736 16 108 96 32 328 576 48 92 32 256 16 . a b a b a b a b a b a b a b                                            it is observed that the result of  2 3t for *s and * cs are shown to be equivalent. 4. conclusion in this paper, we have obtained the coefficient bounds for  2 3t and  3 3t for the subclass of tilted starlike functions with respect to conjugate points of order ,  * , , , .cs a b  the results obtained can be reduced to the results for some existing subclasses in the literature by considering specific values for the parameters , , a and .b acknowledgements: the authors wish to thank the anonymous referees for their careful reading. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] i. graham, geometric function theory in one and higher dimensions, crc press, new york, 2003. [2] s. s. miller, p. t. mocanu, second order differential inequalities in the complex plane, j. math. anal. appl. 65(2) (1978), 289-305. [3] s. s. miller, p. t. mocanu, differential subordinations and univalent functions, michigan math. j. 28(2) (1981), 157-172. int. j. anal. appl. 19 (3) (2021) 493 [4] r. m. el-ashwah, d. k. thomas, some subclasses of close-to-convex functions, j. ramanujan math. soc. 2(1) (1987), 85-100. [5] s. halim, functions starlike with respect to other points, int. j. math. math. sci. 14(3) (1991), 451-456. [6] s. a. f. m. dahhar, a. janteng, a subclass of starlike functions with respect to conjugate points, int. math. forum, 4(28) (2009), 1373-1377. [7] n. h. a. a. wahid, d. mohamad, s. cik soh, on a subclass of tilted starlike functions with respect to conjugate points, menemui mat. (discover. math.) 37(1) (2015), 1-6. [8] k. ye, l. h. lim, every matrix is a product of toeplitz matrices, found. comput. math. 16(3) (2016), 577-598. [9] d. k. thomas and s. a. halim, toeplitz matrices whose elements are the coefficients of starlike and close-to-convex functions, bull. malaysian math. sci. soc. 40(4) (2016), 1781-1790. [10] v. radhika, s. sivasubramanian, g. murugusundaramoorthy, j. m. jahangiri, toeplitz matrices whose elements are the coefficients of functions with bounded boundary rotation, j. complex anal. 2016 (2016), art. id 4960704. [11] s. sivasubramanian, m. govindaraj, g. murugusundaramoorthy, toeplitz matrices whose elements are the coefficients of analytic functions belonging to certain conic domains, int. j. pure appl. math. 109(10) (2016), 39-49. [12] c. ramachandran, d. kavitha, toeplitz determinant for some subclasses of analytic functions, glob. j. pure appl. math. 13(2) (2017), 785-793. [13] n. magesh, ş. altınkaya, s. yalçın, construction of toeplitz matrices whose elements are the coefficients of univalent functions associated with q-derivative operator, arxiv:1708.03600 [math]. (2017). [14] m. f. ali, d. k. thomas, a. vasudevarao, toeplitz determinants whose elements are the coefficients of analytic and univalent functions, bull. aust. math. soc. 97(2) (2018), 253-264. [15] v. radhika, j. m. jahangiri, s. sivasubramanian, g. murugusundaramoorthy, toeplitz matrices whose elements are coefficients of bazilevič functions, open math. 16(1) (2018), 1161-1169. [16] h. m. srivastava, q. z. ahmad, n. khan, b. khan, hankel and toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain, mathematics, 7(2) (2019), 181. [17] h. y. zhang, r. srivastava, h. tang, third-order hankel and toeplitz determinants for starlike functions connected with the sine function, mathematics, 7(5) (2019), 404. [18] s. n. al-khafaji, a. al-fayadh, a. h. hussain, s. a. abbas, toeplitz determinant whose its entries are the coefficients for class of non-bazilevic functions, j. phys.: conf. ser. 1660 (2020), 012091. [19] p. l. duren, univalent functions vol. 259, springer, new york-berlin–heidelberg–tokyo, 1983. [20] i. efraimidis, a generalization of livingston's coefficient inequalities for functions with positive real part, j. math. anal. appl. 435(1) (2016), 369-379. international journal of analysis and applications volume 19, number 1 (2021), 20-28 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-20 solution of ambartsumian delay differential equation in the q-calculus abdulaziz m. alanazi, abdelhalim ebaid∗ department of mathematics, faculty of sciences, university of tabuk, p.o. box 741, tabuk 71491, saudi arabia ∗corresponding author: aebaid@ut.edu.sa abstract. the ambartsumian equation in view of the q-calculus is investigated in this paper. this equation is of practical interest in the theory of surface brightness in the milky way. two approaches are applied to obtain the closed form solution. the first approach implements a direct series assumption while the second approach is based on the adomian decomposition method. the two approaches lead to a unique power series of arbitrary powers. furthermore, the convergence of the obtained series is theoretically proven. in addition, we showed that the present solution reduces to the results in the relevant literature when the quantum calculus parameter tends to 1. 1. introduction in regular calculus, we usually use the limits to calculate the derivatives of any given real functions. while the quantum calculus (q-calculus) provides the derivatives without implementing limits. euler obtained the basic formulae in q-calculus in the eighteenth century. however, the notion of the definite q-derivative and q-integral were introduced by jackson [1] to the first time. in the present time, there is a great interest in the applications of the q-calculus in various fields such as mathematics, number theory, and combinatorics [2]. besides, ernst [3, 4] pointed out that the majority of scientists who use q-calculus are physicists. in addition, baxter [5] introduced the exact solutions of several models in statistical mechanics. also, bettaibi received october 8th, 2020; accepted november 3rd, 2020; published november 24th, 2020. 2010 mathematics subject classification. 34a08. key words and phrases. q-series; q-calculus; q-differential equation; ambartsumian equation. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 20 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-20 int. j. anal. appl. 19 (1) (2021) 21 and mezlini [6] obtained the solutions of some q-heat/q-wave equations. in the literature [7-12], several interesting results for the q-calculus have been discussed by several authors. this paper considers a q-calculus model of the ambartsumian delay equation (ade) in the form: (1.1) dqy dqt = −y(t) + 1 ρ y ( t ρ ) , q ∈ (0, 1], such that (1.2) y(0) = λ, where ρ > 1 and λ is a constant. the system (1-2) is a generalized form of the standard ade which describes the surface brightness in the milky way [13]. when q → 1, the system (1-2) was investigated by the authors [14-15]. however, the fractional model was solved in [16] using the homotopy transform analysis method by means of the caputo’s definition. here, we consider the q-derivative to deal with the present model. in the literature, there are several analytical methods to deal with the system (1-2) such as the adomian decomposition method (adm) [17-18], the homotopy perturbation method (hpm) [19-20], and the homotopy analysis method (ham) [16]. in this paper, two different approaches are suggested to analytically solve the system (1-2). the first approach is a regular power series approach while the second is based on the adm. such approaches are preferred here, especially, in proving the convergence of the resulting series solution. the paper is organized as follows. the main aspects of the q-calculus are presented in section 2. in addition, a basic lemma for the integrals arise from the adm is to be proved in section 2. sections 3 discusses the application of the q-calculus to solving the current model. section 4 is devoted to the application of the adm. in addition it will be shown that the present solution reduces to that one in the literature as q → 1. section 5 includes an analysis of convergence. finally, section 6 outlines the conclusions. 2. preliminaries let q ∈ r and n ∈ n, then [n]q is defined as (first chapter in [21]) (2.1) [n]q = 1 −qn 1 −q , and as q → 1, we have (2.2) lim q→1 [n]q = n. the q-factorial [n]q! of a positive integer n is given by (2.3) [n]q! = [1]q × [2]q × [3]q ×···× [n]q , int. j. anal. appl. 19 (1) (2021) 22 and as q → 1, we have lim q→1 [n]q! = [1]1 × [2]1 × [3]1 ×···× lim q→1 [n]q , = 1 × 2 × 3 ×···×n = n!(2.4) the definition of q-differential is dqf(t) = f(t) −f(qt) and the q-derivative of a function f(t) is defined by [21] (2.5) dqf(t) := dqf(t) dqt = f(t) −f(qt) (1 −q)t , t 6= 0, such that (2.6) lim q→1 dqf(t) = f ′(t), if f is differentiable at t, and we have at t = 0 that (2.7) dqf(0) = lim t→0 dqf(t). according to (2.5) we have (2.8) dq t n = [n]q t n−1. the definite jackson q-integral is defined by (2.9) ∫ t 0 f(τ) dqτ = (1 −q)t ∞∑ j=0 qjf(qjt), and hence, (2.10) ∫ t 0 dqf(τ) dqτ = f(t) −f(0). in order to apply the adm on the system (1.1-1.2), we need to introduce and prove the following lemma. 2.1. lemma 1. for q ∈ (0, 1], we have (2.11) ∫ t 0 τn dqτ = tn+1 [n + 1]q . proof: int. j. anal. appl. 19 (1) (2021) 23 from the definite jackson q-integral given by eq. (2.9), we obtain ∫ t 0 τn dqτ = (1 −q)t ∞∑ j=0 qj(qjt)n, = (1 −q)tn+1 ∞∑ j=0 ( qn+1 )j , = (1 −q)tn+1 ( 1 1 −qn+1 ) , = tn+1 [n + 1]q .(2.12) 3. direct series solution in order to solve eq. (1.1), we assume the solution in the series form: (3.1) y(t) = ∞∑ n=0 ant n, and therefore dqy dqt = ∞∑ n=0 [n]qant n−1, = ∞∑ n=1 [n]qant n−1, where [0]q = 0, = ∞∑ n=0 [n + 1]qan+1t n.(3.2) substituting (3.1) and (3.2) into (1.1), yields ∞∑ n=0 [n + 1]qan+1t n = − ∞∑ n=0 ant n + 1 ρ ∞∑ n=0 an ( t ρ )n , = − ∞∑ n=0 ant n + ∞∑ n=0 ( 1 ρn+1 ) ant n, = ∞∑ n=0 ( 1 ρn+1 − 1 ) ant n,(3.3) or (3.4) ∞∑ n=0 [ [n + 1]qan+1 − ( 1 ρn+1 − 1 ) an ] tn = 0, which requires that (3.5) [n + 1]qan+1 − ( 1 ρn+1 − 1 ) an = 0. int. j. anal. appl. 19 (1) (2021) 24 therefore (3.6) an+1 = ( ρ−(n+1) − 1 [n + 1]q ) an, n ≥ 0 from (3.6), we have a1 = ( ρ−1 − 1 [1]q ) a0, a2 = ( ρ−2 − 1 [2]q ) a1 = ( (ρ−1 − 1)(ρ−2 − 1) [1]q × [2]q ) a0, a3 = ( ρ−3 − 1 [3]q ) a2 = ( (ρ−1 − 1)(ρ−2 − 1)(ρ−3 − 1) [1]q × [2]q × [3]q ) a0, . . an = ( (ρ−1 − 1)(ρ−2 − 1)(ρ−3 − 1) . . . (ρ−n − 1) [1]q × [2]q × [3]q ×···× [n]q ) a0.(3.7) this n-term coefficient can expressed in terms of the q-factorial [n]q! as (3.8) an = a0 [n]q! n∏ i=1 ( ρ−i − 1 ) , n ≥ 1. thus y(t) = a0 + ∞∑ n=1 ant n, = a0 + a0 ∞∑ n=1 tn [n]q! n∏ i=1 ( ρ−i − 1 ) , = a0 [ 1 + ∞∑ n=1 tn [n]q! n∏ i=1 ( ρ−i − 1 )] .(3.9) applying the initial condition (1.2) on (3.9), yields a0 = λ. hence, the closed-form solution of the system (1.1-1.2) is finally given by (3.10) y(t) = λ [ 1 + ∞∑ n=1 tn [n]q! n∏ i=1 ( ρ−i − 1 )] , which is the solution of ade in the q-calculus. moreover, the solution (3.10) as q → 1 reduces to y(t) = λ [ 1 + lim q→1 ∞∑ n=1 tn [n]q! n∏ i=1 ( ρ−i − 1 )] , = λ [ 1 + ∞∑ n=1 tn (limq→1 [n]q!) n∏ i=1 ( ρ−i − 1 )] , = λ [ 1 + ∞∑ n=1 tn n! n∏ i=1 ( ρ−i − 1 )] ,(3.11) which is the same closed form solution obtained by the authors [14] for the standard model of ambartsumian equation. int. j. anal. appl. 19 (1) (2021) 25 4. application of the adm integrating eq. (1.1) and based on eq. (2.10), we have (4.1) y(t) = λ + ∫ t 0 ( 1 ρ y ( τ ρ ) −y(τ) ) dqτ. following the adm [17-18], we assume that (4.2) y(t) = ∞∑ k=0 yk(t), and hence, y0(t) = λ, yk+1(t) = ∫ t 0 ( 1 ρ yk ( τ ρ ) −yk(τ) ) dqτ, k ≥ 1.(4.3) at k = 1, eq. (4.3) gives y1(t) = ∫ t 0 ( 1 ρ y0 ( τ ρ ) −y0(τ) ) dqτ, = λ ( ρ−1 − 1 )∫ t 0 dqτ, = λ ( ρ−1 − 1 ) t [1]q ,(4.4) where lemma 1 is implemented to calculate the involved integral. similarly, at k = 2, we have y2(t) = ∫ t 0 ( 1 ρ y1 ( τ ρ ) −y1(τ) ) dqτ, = λ ( ρ−1 − 1 )( ρ−2 − 1 )∫ t 0 τ [1]q dqτ, = λ ( ρ−1 − 1 )( ρ−2 − 1 ) t2 [1]q[2]q .(4.5) proceeding as above, we obtain (4.6) yk(t) = λ ( ρ−1 − 1 )( ρ−2 − 1 )( ρ−3 − 1 ) . . . ( ρ−k − 1 ) tk [1]q[2]q[3]q . . . [k]q . from eq. (2.3), we can rewrite eq. (4.6) as (4.7) yk(t) = λ k∏ j=1 ( ρ−j − 1 )( tk [k]q! ) , k ≥ 1. int. j. anal. appl. 19 (1) (2021) 26 thus y(t) = y0(t) + ∞∑ k=1 yk(t), = λ + λ ∞∑ k=1 tk [k]q! k∏ j=1 ( ρ−j − 1 ) , = λ  1 + ∞∑ k=1 tk [k]q! k∏ j=1 ( ρ−j − 1 ) ,(4.8) which is the same closed form series solution that was obtained in the previous section. the series (3.11) which is equivalent to (4.8) will be proved for convergence in the next section. 5. analysis of convergence in order to proving the convergence of (3.11), we assume that (5.1) an = 1 [n]q! n∏ i=1 ( ρ−i − 1 ) , n ≥ 1. accordingly, we obtain the theorem below. theorem 1: the radius of convergence of the series (3.11) is ( 1 1−q ) ∀ q ∈ (0, 1]. proof: assume that µ is the radius of convergence. therefore, we have from (5.1) and the ratio test that 1 µ = lim n→∞ ∣∣∣∣an+1an ∣∣∣∣ , = lim n→∞ ∣∣∣∣∣ ∏n+1 i=1 ( ρ−i − 1 ) [n + 1]q! × [n]q!∏n i=1 (ρ −i − 1) ∣∣∣∣∣ .(5.2) from (2.3), we observe that (5.3) [n + 1]q! = [1]q × [2]q × [3]q ×···× [n]q × [n + 1]q = [n]q! [n + 1]q. inserting (5.3) into (5.2), we obtain (5.4) 1 µ = lim n→∞ ∣∣∣∣∣ ( ρ−(n+1) − 1 )∏n i=1 ( ρ−i − 1 ) [n]q! [n + 1]q × [n]q!∏n i=1 (ρ −i − 1) ∣∣∣∣∣ . int. j. anal. appl. 19 (1) (2021) 27 simplifying (5.4), it then follows 1 µ = lim n→∞ ∣∣∣∣ρ−(n+1) − 1[n + 1]q ∣∣∣∣ , = ∣∣∣ lim n→∞ ( ρ−(n+1) − 1 ) / lim n→∞ ([n + 1]q) ∣∣∣ , = ∣∣∣∣−1/ ( 1 1 −q )∣∣∣∣ , where ρ > 1, = |1 −q| , = 1 −q ∀ q ∈ (0, 1],(5.5) which completes the proof, where the following property: (5.6) lim n→∞ [n + 1]q = lim n→∞ ( 1 −qn+1 1 −q ) = 1 1 −q ∀ q ∈ (0, 1], was implemented in deducing (5.5). it is noticed from (5.6) that as q → 1 then 1 µ → 0. hence, the series (3.11) has an infinite radius of convergence at such special case which is in full agreement with the results obtained by [14] for the standard ade. 6. conclusion in this paper, the quantum calculus was applied to generalize the ambartsumian equation. the resulting q-differential equation was analytically solved via the power series approach and the adm. the convergence of the obtained power series was theoretically proven. the implemented approaches led to the same closed form series solution. in addition, we showed that the present solution reduces to the results in the literature when the quantum calculus parameter tends to 1. finally, the present work can be further extended to explore several physical models in view of the q-calculus. availability of data and material: not applicable. authors’ contributions: all authors read and approved the final manuscript and all authors have agreed to the authorship and the order of authorship for this manuscript. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] f.h. jackson, on a q-definite integrals, quart. j. pure appl. math. 41 (1910), 193-203. [2] g.e. andrews, q-series: their development and applications in analysis, number theory, combinatorics, physics, and computer algebra, cbms series, vol. 66, amer. math. soc., providence, 1986. int. j. anal. appl. 19 (1) (2021) 28 [3] t. ernst, the history of q-calculus and a new method (licentiatethesis), u.u.d.m. report 2000: 16; http://www.math.uu.se/thomas/lics.pdf. [4] t. ernst, a method for q-calculus, j. nonlinear math. phys. 10(4) (2003), 487-525. [5] r. baxter, exact solved models in statistical mechanics, academic press, new york, 1982. [6] n. bettaibi, k. mezlini, on the use of the q-mellin transform to solve some q-heat and q-wave equations, int. j. math. arch. 3(2) (2012), 446-455. [7] y. li, z.-m. sheng, a deformation of quantum mechanics, j. phys. a: math. gen. 25 (1992), 6779?6788. [8] m.h. annaby, z.s. mansour, q-fractional calculus and equations, springer, heidelberg, new york, 2012. [9] i. koca, e. demirci, on local asymptotic stability of q-fractional nonlinear dynamical systems, appl. appl. math. 11 (2016), 174-183. [10] n.a. rangaig, c.t. pada, v.c. convicto, on the existence of the solution for q-caputo fractional boundary value problem, appl. math. phys. 5 (2017), 99-102. [11] y. tang, t. zhang, a remark on the q-fractional order differential equations, appl. math. comput. 350 (2019), 198-208. [12] l. chanchlani, s. alha, j. gupta, generalization of taylor’s formula and differential transform method for composite fractional q-derivative, ramanujan j. 48 (2019), 21-32. [13] v.a. ambartsumian, on the fluctuation of the brightness of the milky way, doklady akad nauk ussr, 44 (1994), 223-226. [14] j. patade, s. bhalekar, on analytical solution of ambartsumian equation, natl. acad. sci. lett. 40 (2017), 291-293. [15] t. kato, j.b. mcleod, the functional-differential equation y′(x) = ay(λx) + by(x), bull. amer. math. soc. 77 (1971), 891-935. [16] d. kumar, j. singh, d. baleanu, s. rathore, analysis of a fractional model of the ambartsumian equation, eur. phys. j. plus, 133 (2018), 259. [17] a. alshaery, a. ebaid, accurate analytical periodic solution of the elliptical kepler equation using the adomian decomposition method, acta astron. 140 (2017), 27-33. [18] h.o. bakodah and a. ebaid, the adomian decomposition method for the slip flow and heat transfer of nanofluids over a stretching/shrinking sheet, rom. rep. phys. 70 (2018), 115. [19] a. patra, s. saha ray, homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation, int. j. heat mass transfer. 76 (2014), 162-170. [20] z. ayati, j. biazar, on the convergence of homotopy perturbation method, j. egypt. math. soc. 23 (2015), 424-428. [21] v.g. kac, p. cheung, quantum calculus, springer-verlag, new york, 2002. 1. introduction 2. preliminaries 2.1. lemma 1 3. direct series solution 4. application of the adm 5. analysis of convergence 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 104-109 http://www.etamaths.com some remarks concerning the jacobi-dunkl transform in the space lp(r,aα,β(t)dt) r. daher, s. el ouadih∗ and a. belkhadir abstract. in this paper, using a generalized jacobi-dunkl translation operator, we obtain a generalization of titchmarsh’s theorem for the dunkl transform for functions satisfying the (φ,p)-lipschitz jacobi-dunkl condition in the space lp(r,aα,β(t)dt),α ≥ β ≥ −12 ,α 6= −1 2 . 1. introduction and preliminaries titchmarsh’s [8,theorem 85] characterized the set of functions in l2(r) satisfying the cauchy-lipschitz condition by means of an asymptotic estimate growth of the norm of their fourier transform, namely we have theorem 1.1. [8] let α ∈ (0, 1) and assume that f ∈ l2(r). then the following are equivalents (a) ‖f(t + h) −f(t)‖ = 0(hα), as h → 0 (b) ∫ |λ|≥r |f̂(λ)|2dλ = o(r−2α) as r →∞, where f̂ stand for the fourier transform of f. in this paper, we prove a generalization of theorem 1.1 for the jacobi-dunkl transform for functions satisfying the (φ,p)-lipschitz jacobi-dunkl condition in the space lp(r,aα,β(t)dt), 1 < p ≤ 2. for this purpose, we use the generalized jacobi-dunkl translation operator. in this section, we recapitulate from [1,2,3,5,6] some results related to the harmonic analysis associated with jacobi-dunkl operator λα,β. the jacobi-dunkl function with parameters (α,β),α ≥ β ≥ −1 2 ,α 6= −1 2 , defined by the formula: ∀x ∈ r,ψα,βλ (x) =   ϕα,βµ (x) − i λ d dx ϕα,βµ (x) if λ ∈ c\{0} 1 if λ = 0 with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕα,βµ is the jacobi function given by: ϕα,βµ (x) = f ( ρ + iµ 2 , ρ− iµ 2 ,α + 1,−(sinh(x))2 ) , 2010 mathematics subject classification. 65r10. key words and phrases. jacobi-dunkl operator, jacobi-dunkl transform, generalized jacobidunkl translation. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 104 gelfand triple isomorphisms 105 f is the gausse hypergeometric function (see [1,7]). ψ α,β λ is the unique c ∞-solution on r of the differentiel-difference equation  λα,βu = iλu ,λ ∈ c u(0) = 1 where λα,β is the jacobi-dunkl operator given by: λα,βu(x) = du(x) dx + [(2α + 1) coth x + (2β + 1) tanh x] × u(x) −u(−x) 2 ]. the operator λα,β is a particular case of the operator d given by du(x) = du(x) dx + a′(x) a(x) × ( u(x) −u(−x) 2 ) , where a(x) = |x|2α+1b(x), and b a function of class c∞ on r, even and positive. the operator λα,β corresponds to the function a(x) = aα,β(x) = 2 ρ(sinh |x|)2α+1(cosh |x|)2β+1. using the relation d dx ϕα,βµ (x) = − µ2 + ρ2 4(α + 1) sinh(2x)ϕα+1,β+1µ (x), the function ψ α,β λ can be written in the form above (see [2]) ∀x ∈ r,ψα,βλ (x) = ϕ α,β µ (x) + i λ 4(α + 1) sinh(2x)ϕα+1,β+1µ (x). denote l p α,β(r) = l p α,β(r,aα,β(t)dt), 1 < p ≤ 2, the space of measurable functions f on r such that ‖f‖p,α,β = (∫ r |f(t)|paα,β(t)dt )1/p < +∞. using the eigenfunctions ψ α,β λ of the operator λα,β called the jacobi-dunkl kernels, we define the jacobi-dunkl transform by fα,βf(λ) = ∫ r f(t)ψ α,β λ (t)aα,β(t)dt, λ ∈ r, and the inversion formula by f(t) = ∫ r fα,βf(λ)ψ α,β −λ (t)dσ(λ), where dσ(λ) = |λ| 8π √ λ2 −ρ2|cα,β( √ λ2 −ρ2)| ir\]−ρ,ρ[(λ)dλ. here, cα,β(µ) = 2ρ−iµγ(α + 1)γ(iµ) γ( 1 2 (ρ + iµ))γ( 1 2 (α−β + 1 + iµ)) , µ ∈ c\(in) and ir\]−ρ,ρ[ is the characteristic function of r\] −ρ,ρ[. the jacobi-dunkl transform is a unitary isomorphism from l2α,β(r) onto l 2(r,dσ(λ)), i.e. ‖f‖2,α,β = ‖fα,β(f)‖l2(r,dσ(λ)).(1) 106 daher, ouadih and belkhadir plancherel’s theorem (1) and the marcinkiewics interpolation theorem (see [8]) we get for f ∈ lpα,β(r) with 1 < p ≤ 2 and q such that 1 p + 1 q = 1, ‖fα,β(f)‖lq(r,dσ(λ)) ≤ k‖f‖p,α,β,(2) where k is a positive constant (see [6]). the operator of jacobi-dunkl translation is defined by txf(y) = ∫ r f(z)dνα,βx,y (z), ∀x,y ∈ r where να,βx,y (z),x,y ∈ r are the signed measures given by dνα,βx,y (z) =   kα,β(x,y,z)aα,β(z)dz if x,y ∈ r∗ δx if y = 0 δy if x = 0 here, δx is the dirac measure at x. and, kα,β(x,y,z) = mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αiix,y × ∫ π 0 ρθ(x,y,z) × (gθ(x,y,z)) α−β−1 + sin 2β θdθ ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x|− |y||, |x| + |y|] ρθ(x,y,z) = 1 −σθx,y,z + σ θ z,x,y + σ θ z,y,x ∀z ∈ r,θ ∈ [0,π],σθx,y,z =   cosh(x)+cosh(y)−cosh(z) cos(θ) sinh(x) sinh(y) ,if xy 6= 0 0 ,if xy = 0 gθ(x,y,z) = 1 − cosh2(x) − cosh2(y) − cosh2(z) + 2 cosh(x) cosh(y) cosh(z) cos θ t+ =   t ,if t > 0 0 ,if t ≤ 0 and, mα,β =   2−2ργ(α+1)√ πγ(α−β)γ(β+ 1 2 ) ,if α > β 0 ,if α = β in [2], we have fα,β(thf)(λ) = ψ α,β λ (h)fα,β(f)(λ).(3) for α ≥ −1 2 , we introduce the bessel normalized function of the first kind defined by jα(z) = γ(α + 1) ∞∑ n=0 (−1)n(z 2 )2n n!γ(n + α + 1) , z ∈ c. moreover, we see that lim z→0 jα(z) − 1 z2 6= 0, by consequence, there exists c1 > 0 and η > 0 satisfying |z| ≤ η ⇒|jα(z) − 1| ≥ c1|z|2.(4) gelfand triple isomorphisms 107 lemma 1.1. let α ≥ β ≥ −1 2 ,α 6= −1 2 . then for |ν| ≤ ρ, there exists a positive constant c2 such that |1 −ϕα,βµ+iν(t)| ≥ c2|1 − jα(µt)|. proof. (see[4],lemma 9). 2. main result in this section we give the main result of this paper. we need first to define (φ,p)-lipschitz jacobi-dunkl class. denote nh by nh = th + t−h − 2i where i is the unit operator in the space l p α,β(r). definition 2.1. a function f ∈ lpα,β(r) is said to be in (φ,p)-lipschitz jacobidunkl class, denoted by lip(φ,p,α,β), if ‖nhf‖p,α,β = o(φ(h)), as h → 0, where φ(t) is a continuous increasing function on [0,∞), φ(0) = 0 and φ(ts) = φ(t)φ(s) for all t,s ∈ [0,∞). lemma 2.2. for f ∈ lpα,β(r), then(∫ r 2q|ϕα,βµ (h) − 1| q|fα,βf(λ)|qdσ(λ) )1 q ≤ k‖nhf‖p,α,β where 1 p + 1 q = 1. proof. we us formula (3), we conclude that fα,β(nhf)(λ) = (ψ α,β λ (h) + ψ α,β λ (−h) − 2)fα,β(f)(λ), since ψ α,β λ (h) = ϕ α,β µ (h) + i λ 4(α + 1) sinh(2h)ϕα+1,β+1µ (h), ψ α,β λ (−h) = ϕ α,β µ (−h) − i λ 4(α + 1) sinh(2h)ϕα+1,β+1µ (−h), and ϕα,βµ is even (see [2]), then fα,β(nhf)(λ) = 2(ϕα,βµ (h) − 1)fα,β(f)(λ). by formula (2), we have the result. theorem 2.3. let f(x) belong to lip(φ,p,α,β). then∫ |λ|≥r |fα,β(f)(λ)|qdσ(λ) = o(φ(r−q)), as r →∞, where 1 p + 1 q = 1. proof. assume that f ∈ lip(φ,p,α,β), then we have ‖nhf‖p,α,β = o(φ(h)), as h → 0. 108 daher, ouadih and belkhadir from lemma 2.2, we have∫ r |ϕα,βµ (h) − 1| q|fα,βf(λ)|qdσ(λ) ≤ kq 2q ‖nhf‖ q p,α,β by (4) and lemma 1.1, we get∫ η 2h ≤|λ|≤η h |1−ϕα,βµ (h)| q|fα,β(f)(λ)|qdσ(λ) ≥ c q 1c q 2 ∫ η 2h ≤|λ|≤η h |µh|2q|fα,β(f)(λ)|qdσ(λ). from η 2h ≤ |λ| ≤ η h we have( η 2h )2 − ρ2 ≤ µ2 ≤ (η h )2 −ρ2 ⇒ µ2h2 ≥ η2 4 −ρ2h2. take h ≤ η 3ρ , then we have µ2h2 ≥ c3 = c3(η). so,∫ η 2h ≤|λ|≤η h |1−ϕα,βµ (h)| q|fα,β(f)(λ)|qdσ(λ) ≥ c q 1c q 2c q 3 ∫ η 2h ≤|λ|≤η h |fα,β(f)(λ)|qdσ(λ). there exists then a positives constants c and k1 such that∫ η 2h ≤|λ|≤η h |fα,β(f)(λ)|qdσ(λ) ≤ c ∫ r |1 −ϕα,βµ (h)| q|fα,β(f)(λ)|qdσ(λ) ≤ k1φq(h) = k1φ(hq). for all 0 < h < η 3ρ . then we have∫ r≤|λ|≤2r |fα,β(f)(λ)|qdσ(λ) ≤ k2φ(r−q), r →∞, where k2 = k1φ(η q2−q). furthermore, we obtain∫ |λ|≥r |fα,β(f)(λ)|qdσ(λ) = (∫ r≤|λ|≤2r + ∫ 2r≤|λ|≤4r + ∫ 4r≤|λ|≤8r + · ·· ) |fα,β(f)(λ)|qdσ(λ) ≤ k2φ(r−q) + k2φ((2r)−q) + k2φ((4r)−q) + · · · ≤ k2φ(r−q) + k2φ(2−q)φ(r−q) + k2φ((2−q)2)φ(r−q) + · · · ≤ k2φ(r−q)(1 + φ(2−q) + φ((2−q)2) + · · ·). we have φ(2−q) < 1, then∫ |λ|≥r |fα,β(f)(λ)|qdσ(λ) ≤ k3φ(r−q), where k3 = k2(1 −φ(2−q))−1. finally, we get∫ |λ|≥r |fα,β(f)(λ)|qdσ(λ) = o(φ(r−q)), as r →∞. thus, the proof is finished. gelfand triple isomorphisms 109 references [1] ben mohamed. h and mejjaoli. h, distributional jacobi-dunkl transform and applications, afr.diaspora j.math 1(2004), 24-46. [2] ben mohamed. h, the jacobi-dunkl transform on r and the convolution product on new space of distributions, ramanujan j.21(2010), 145-175.. [3] ben salem. n and ahmed salem. a , convolution structure associated with the jacobi-dunkl operator on r, ramanuy j.12(3) (2006), 359-378. [4] bray. w. o and pinsky. m. a, growth properties of fourier transforms via module of continuity , journal of functional analysis.255(288), 2256-2285. [5] chouchane. f, mili. m and trimche. k, positivity of the intertwining operator and harmonic analysis associated with the jacobi-dunkl operator on r, j.anal. appl.1(4)(2003), 387-412. [6] johansen. t. r, remarks on the inverse cherednik-opdam transform on the real line, arxiv: 1502.01293v1 (2015). [7] koornwinder. t. h , a new proof of a paley-wiener type theorems for the jacobi transform , ark.math.13(1975), 145-159. [8] titchmarsh. e. c, introduction to the theory of fourier integrals , claredon, oxford. (1948), komkniga, moscow, (2005). departement of mathematics, faculty of sciences äın chock, university hassan ii, casablanca, morocco ∗corresponding author int. j. anal. appl. (2022), 20:15 generalized ulam-hyers stability results of a quadratic functional equation in felbin’s type fuzzy normed linear spaces john m. rassias1, s. karthikeyan2,∗, g. ganapathy3, m. suresh3 and t.r.k. kumar2 1pedagogical department e.e. , section of mathematics and informatics, national and capodistrian university of athens, 4, agamemnonos str., aghia paraskevi, athens 15342, greece 2department of mathematics, r.m.k. engineering college, kavaraipettai 601 206, tamil nadu, india 3department of mathematics, r.m.d. engineering college, kavaraipettai 601 206, tamil nadu, india ∗corresponding author: karthik.sma204@yahoo.com abstract. this paper presents the generalized ulam-hyers stability of the following quadratic functional equation f ( x +y 2 −z ) + f ( y +z 2 −x ) + f ( z +x 2 −y ) = 3 4 (f (z −x)+ f (z −y)+ f (x −y)) in felbin’s type fuzzy normed linear spaces (f-nls) using direct and fixed point methods. 1. introduction in 1940, s.m. ulam [37] posed the stability problem for approximate homomorphisms. in 1941, d.h. hyers [12] provided a partial solution to ulam’s problem for mappings between banach spaces. in 1950, t. aoki [2] generalized hyers’ theorem for additive mappings. in 1978, th.m.rassias [28] proved a further generalzation of hyers’ theorem by introducing the concept of the unbounded cauchy difference for the sum of powers of two p−norms. during the last three decades the stability theorem of th.m. rassias [28] provided a lot of influence for the development of stability theory of a large variety of functional equations. this new concept is known today with the term hyers-ulam-rassias received: dec. 9, 2021. 2010 mathematics subject classification. 39b52, 39b72, 39b05. key words and phrases. quadratic functional equation; generalized ulam-hyers stability; felbin’s fuzzy normed linear space. https://doi.org/10.28924/2291-8639-20-2022-15 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-15 2 int. j. anal. appl. (2022), 20:15 stability for functional equations. following the innovative approach of th.m. rassias similar theorems were formulated and proved by a number of mathematicians. for example four years later in 1982, j.m. rassias [27] proved a similar theorem for the case “the unbounded cauchy difference" is the “product of two p−norms". in 1994, the above stability results were further extended by p. gavruta [9] who considered a more general control function in the real variables x,y for the unbounded cauchy difference in the spirit of th.m. rassias’ stability approach. in 2008, a special case of gavruta’s theorem for the unbounded cauchy difference was obtained in [31] by considereing the summation of both the sum and product of two p−norms. the functional equation f (x + y) + f (x −y) = 2f (x) + 2f (y) (1.1) is said to be quadratic functional equation because the quadratic function f (x) = ax2 is a solution of the functional equation (1.1). this paper presents the generalized ulam-hyers stability of the following quadratic functional equation f ( x + y 2 −z ) + f ( y + z 2 −x ) + f ( z + x 2 −y ) = 3 4 (f (z −x) + f (z −y) + f (x −y)) (1.2) in felbin’s fuzzy normed linear spaces (f-nls) using direct and fixed point methods 2. some preliminaries on fuzzy real number this section, some preliminaries in the theory of fuzzy real numbers are given. furthermore, we give some definition which help to investigate the stability in felbin’s type normed linear spaces. in [10] grantner takes the fuzzy real number as a decreasing mapping from the real line to the unit interval or lattice in general. lowen [23] applies the fuzzy real numbers as non-decreasing, left continuous mapping from the real line to the unit interval so that its supremum over r is 1. also fuzzy arithmetic operations on l−fuzzy real line were studied by rodabaugh [34], where he showed that the binary addition is the only extension of addition to r((l)). hoehle [13] especially emphasized the role of fuzzy real numbers as modeling a fuzzy threshold softening the notion of dedekind cut. in this paper a fuzzy real number is taken as a fuzzy normal and convex mapping from the real line to the unit interval. the concept of the fuzzy metric space has been studied by kaleva [19,20] by using fuzzy number as a fuzzy set on the real axis. kaleva also has recently showed that a fuzzy metric space can be embedded in a complete fuzzy metric space [21]. in [8], felbin introduced the concept of fuzzy normed linear space (f-nls); xiao and zhu [38] studied its linear topological structures and some basic properties of a fuzzy normed linear space. it int. j. anal. appl. (2022), 20:15 3 is known that theories of classical normed space and menger probabilistic normed spaces are special cases of fuzzy normed linear spaces. let η be a fuzzy subset on r, i.e., a mapping η : r → [0, 1] associating with each real number t its grade of membership ηt. definition 2.1. [8] a fuzzy subset η on r is called a fuzzy real number, whose α−level set is denoted by [η]α i.e., [η]α = {t : η(t) ≥ α}, if it satisfies two axioms: (n1) there exists t0 ∈r such that η(t0) = 1. (n2) for each α ∈ (0, 1], [η]α = [η−α ,η+α ] where −∞ < η−α ≤ η+α < +∞. the set of all fuzzy real number is denoted by f (r). if η ∈ f (r) and η(t) = 0 whenever t < 0, then η is called a non-negative fuzzy real number and f∗(r) denotes the set of all non-negative fuzzy real numbers. the number 0̄ stands for the fuzzy real number as: 0̄ = { t, t = 0, 0, t 6= 0. clearly, 0̄ ∈ f∗(r). also the set of all real numbers can be embedded in f (r) because if r ∈ (−∞,∞), then r̄ ∈ f (r) satisfies r̄(t) = 0̄(t − r). definition 2.2. [8] fuzzy arithmetic operations ⊕, ,⊗,� on f (r) ×f (r) can be defined as: (1) (η ⊕δ)(t) = sup s∈r {η(s) ∧δ(t − s)},t ∈r, (2) (η δ)(t) = sup s∈r {η(s) ∧δ(s − t)},t ∈r, (3) (η ⊗δ)(t) = sup s∈r {η(s) ∧δ(t/s)},t ∈r, (4) (η �δ)(t) = sup s∈r {η(st) ∧δ(s)},t ∈r. the additive and multiplicative identities in f (r) are 0̄ and 1̄, respectively. let η be defined as 0̄ η. it is clear that η δ = η ⊕ ( δ). definition 2.3. [8] for k, 0 ∈r, fuzzy scalar multiplication k �η is defined as (k �η)(t) = η(t/k) and 0 �η is defined to be 0. lemma 2.1. let η,δ be fuzzy real numbers. then ∀ t ∈r, η(t) = δ(t) ⇔ ∀ α ∈ (0, 1], [η]α = [δ]α. lemma 2.2. let η,δ ∈ f (r) and [η]α = [η−α ,η+α ], [δ]α = [δ−α ,δ+α ]. then (i) [η ⊕δ]α = [η−α + δ−α ,η+α + δ+α ], 4 int. j. anal. appl. (2022), 20:15 (ii) [η δ]α = [η−α −δ−α ,η+α −δ+α ], (iii) [η ⊗δ]α = [η−αδ−α ,η+αδ+α ],η,δ ∈ f∗(r), (iv) [1̄ �δ]α = [1/δ+α , 1/δ−α ],δ−α > 0. definition 2.4. [8] define a partial ordering ≺ in f (r) by η≺δ if and only if η−α ≤ δ−α and η+α ≤ δ+α for all α ∈ (0, 1]. the strict inequality in f (r) is defined by η ≺ δ if and only if η−α < δ−α and η+α < δ+α for all α ∈ (0, 1]. definition 2.5. [38] let x be a real linear space, l and r (respectively, left norm and right norm) be symmetric and non-decreasing mappings from [0, 1]×[0, 1] → [0, 1] satisfying l(0, 0) = 0,r(1, 1) = 1. then || · || is called a fuzzy norm and (x, || · ||,l,r) is a fuzzy normed linear space (abbreviated to f-nls) if the mapping ||·|| : x → f∗(r) satisfies the following axioms, where [||x||]α = [||x||−α, ||x||+α ] for x ∈ x and α ∈ (0, 1]: (a1) ||x|| = 0 if and only if x = 0, (a2) ||rx|| = |r|� ||x|| for all x ∈ x and r ∈ (−∞,∞), (a3) for all x,y ∈ x, (a3l) if s ≤ ||x||−1 ,t ≤ ||y|| − 1 and s + t ≤ ||x + y|| − 1 , then ||x + y||(s + t) ≥ l(||x||(s), ||y||(t)), (a3r) if s ≥ ||x||−1 ,t ≥ ||y|| − 1 and s + t ≥ ||x + y|| − 1 , then ||x + y||(s + t) ≤ l(||x||(s), ||y||(t)), lemma 2.3. [39] let (x, || · ||,l,r) be an f-nls, and suppose that (r1) r(a,b) ≤ max(a,b), (r2) ∀ α ∈ (0, 1], ∃β ∈ (0,α] such that r(β,y) ≤ α for all y ∈ (0,α), (r3) lim a→0+ r(a,a) = 0. then (r1) ⇒ (r2) ⇒ (r3) but not conversely. lemma 2.4. [39] let (x, || · ||,l,r) be an f-nls. then we have the following: (a) if r(a,b) ≤ max(a,b), then ∀ α ∈ (0, 1], ||x + y||+α ≤ ||x||+α + ||y||+α for all x,y ∈ x. (b) if (r2) then for each α ∈ (0, 1] there is β ∈ (0,α] such that ||x + y||+α ≤ ||x|| + β + ||y||+α for all x,y ∈ x. (c) if lim a→0+ r(a,a) = 0, then for each α ∈ (0, 1] there is β ∈ (0,α] such that ||x + y||+α ≤ ||x||+ β + ||y||+ β for all x,y ∈ x. lemma 2.5. [39] let (x, || · ||,l,r) be an f-nls, and suppose that (l1) l(a,b) ≥ min(a,b), (l2) ∀ α ∈ (0, 1], ∃β ∈ [α, 1] such that l(β,γ) ≥ α for all γ ∈ [α, 1), (l3) lim a→1− l(a,a) = 1. then (l1) ⇒ (l2) ⇒ (l3). lemma 2.6. [39] let (x, || · ||,l,r) be an f-nls. then we have the following: int. j. anal. appl. (2022), 20:15 5 (a) if l(a,b) ≥ min(a,b), then ∀ α ∈ (0, 1], ||x + y||−α ≤ ||x||−α + ||y||−α for all x,y ∈ x. (b) if (l2) then for each α ∈ (0, 1] there is β ∈ [α, 1] such that ||x + y||−α ≤ ||x|| − β + ||y||−α for all x,y ∈ x. (c) if lim a→1− r(a,a) = 1, then for each α ∈ (0, 1] there is β ∈ [α, 1] such that ||x + y||−α ≤ ||x||− β + ||y||− β for all x,y ∈ x. lemma 2.7. [39] let (x, || · ||,l,r) be an f-nls. (a) if r(a,b) ≥ max(a,b) and ∀α ∈ (0, 1], ||x + y||+α ≤ ||x||+α + ||y||+α for all x,y ∈ x then (a3r). (b) if l(a,b) ≤ min(a,b) and ∀α ∈ (0, 1] ||x + y||−α ≤ ||x||−α + ||y||−α for all x,y ∈ x then (a3l). theorem 2.1. [35] let (x, || · ||,l,r) be an f-nls and lim a→0+ r(a,a) = 0. then (x, || · ||,l,r) is a hausdorff topological vector space, whose neighborhood base of origin is {n(�,α) : � > 0,α ∈ (0, 1]}, where n(�,α) = {x : ||x||+α ≤ �}. definition 2.6. let (x, || · ||,l,r) be an f-nls. a sequence {xn}∞n=1 ⊆ x converges to x ∈ x, if lim n→∞ ‖xn −x‖+α, for every α ∈ (0, 1] denoted by lim n→∞ xn = x. definition 2.7. let (x, ||·||,l,r) be an f-nls. a sequence {xn}∞n=1 ⊆ x is called a cauchy sequence if lim m,n→∞ ‖xm −xn‖+α = 0 for every α ∈ (0, 1]. definition 2.8. let (x, || · ||,l,r) be an f-nls. a subset a ⊆ x is said to be complete if every cauchy sequence in a, converges in a. the fuzzy normed space (x, || · ||,l,r) is said to be a fuzzy banach space (f-bs) if it is complete. 3. felbin’s stability results: direct method the generalized ulam-hyers stability of the quadratic functional equation is discussed in this section. through out this section, let u be a normed space and v be a banach space respectively. define a mapping f : u3 → v by f (x,y,z) =f ( x + y 2 −z ) + f ( y + z 2 −x ) + f ( z + x 2 −y ) − 3 4 (f (z −x) + f (z −y) + f (x −y)) for all x,y,z ∈ u. theorem 3.1. let j ∈{−1, 1}. let ξ : u3 → f∗(r) be a function such that ∞∑ n=0 ξ ( 2njx, 2njy, 2njz )+ α 22nj converges and lim n→∞ ξ ( 2njx, 2njy, 2njz )+ α 22nj < ∞ (3.1) for all x,y,z ∈ u and let f : u3 → v be an even function satisfying the inequality ‖f (x,y,z)‖+α ≺ ξ (x,y,z) + α (3.2) 6 int. j. anal. appl. (2022), 20:15 for all x,y,z ∈ u. then there exists a unique quadratic function q : u3 → v such that ‖f (x) −q(x)‖+α ≺ 1 22 � ∞∑ i= 1−j 2 ξ ( 2ijx, 2ijx,−2ijx )+ α 22ij (3.3) for all x ∈ u. the mapping q(x) is defined by q(x) = lim n→∞ f (2njx) 22nj (3.4) for all x ∈ u. proof. assume j = 1. replacing (x,y,z) by (x,x,−x) and dividing by 22 in (1.2), we get∥∥∥∥f (2x)22 − f (x) ∥∥∥∥+ α ≺ 1 22 �ξ (x,x,−x)+α (3.5) for all x ∈ u. replacing x by 2x in (3.5) and divided by 22, we get∥∥∥∥f (22x)24 − f (2x)22 ∥∥∥∥+ α ≺ 1 24 �ξ (2x, 2x,−2x)+α (3.6) for all x ∈ u. combining (3.5) and (3.6), we obtain∥∥∥∥f (22x)24 − f (x) ∥∥∥∥+ α ≺ 1 22 � ( ξ (x,x,−x)+α + 1 22 �ξ (2x, 2x,−2x)+α ) (3.7) for all x ∈ u. using induction on a positive integer n, we obtain that∥∥∥∥f (2nx)22n − f (x) ∥∥∥∥+ α ≺ 1 22 � n−1∑ i=0 ξ ( 2ix, 2ix,−2ix )+ α 22i (3.8) ≺ 1 22 � ∞∑ i=0 ξ ( 2ix, 2ix,−2ix )+ α 22i for all x ∈ u. in order to prove the convergence of the sequence { f (2nx) 22n } , replace x by 2mx and divided by 22m in (3.8), for any m,n > 0 , we arrive∥∥∥∥f (2n2mx)22n+2m − f (2 mx) 22m ∥∥∥∥+ α = 1 22m � ∥∥∥∥f (2n2mx)22nk − f (2mx) ∥∥∥∥+ α ≺ 1 22 � n−1∑ i=0 ξ ( 2i+mx, 2i+mx,−2i+mx )+ α 22(i+m) ≺ 1 22 � ∞∑ i=0 ξ ( 2i+mx, 2i+mx,−2i+mx )+ α 22(i+m) (3.9) for all x ∈ u. since the right hand side of the inequality (3.9) tends to 0 as m → ∞, the sequence{ f (2nx) 22n } is cauchy sequence. since y is complete, there exists a mapping q : u → y such that q(x) = lim n→∞ f (2nx) 22n , ∀ x ∈ u. int. j. anal. appl. (2022), 20:15 7 letting n → ∞ in (3.8), we see that (3.3) holds for all x ∈ u. now we need to prove q satisfies (1.2), replacing (x,y,z) by (2nx, 2ny, 2nz) and divided by 22n in (3.2), we arrive 1 22n �‖df (2nx, 2ny, 2nz)‖+α ≺ 1 22n �ξ (2nx, 2ny, 2nz)+α for all x,y,z ∈ u. hence we arrive ‖q (2nx, 2ny, 2nz)‖+α = 0. hence q satisfies (1.2) for all x,y,z ∈ u. in order to prove q is unique, let q′(x) be another quadratic mapping satisfying (3.3) and (1.2). then∥∥q(x) −q′(x)∥∥+ α = 1 22n � ∥∥q(2nx) −q′(2nx)∥∥+ α ≺ 1 22n � { ‖q(2nx) − f (2nx)‖+α + ∥∥f (2nx) −q′(2nx)∥∥} ≺ 2 22 � ∞∑ i=0 ξ(2n+ix, 2n+ix,−2n+ix)+α 22(n+i) → 0 as n →∞ for all x ∈ u. hence q is unique. for j = −1, we can prove the similar stability result. hence it completes the proof. � the following corollary is an immediate consequence of theorem 3.1 concerning the ulam-hyers, hyers-ulam-rassias, and rassias stabilities of (1.2). corollary 3.1. let λ and s be nonnegative real numbers. if an even function f : u → y satisfies the inequality ‖f (x,y,z)‖+α ≺   λ, λ⊗ (||x||s ⊕||y||s ⊕||z||s) , s 6= 2; λ⊗ (||x||s ⊗||y||s ⊗||z||s) , s 6= 2 3 ; λ⊗ { (||x||s ⊗||y||s ⊗||z||s) ⊕||x||3s ⊕||y||3s ⊕||z||3s } , s 6= 2 3 ; (3.10) for all x,y,z ∈ u, then there exists a unique quadratic function q : u → y such that ‖f (x) −q(x)‖+α ≺   λ |3| , 3 �λ⊗ (||x||s)+α |22 − 2s| , λ⊗ ( ||x||3s )+ α |22 − 23s| , 4 �λ⊗ ( ||x||3s )+ α |22 − 23s| (3.11) for all x ∈ u. 8 int. j. anal. appl. (2022), 20:15 4. felbin’s stability results: fixed point method this section deals with the generalized ulam hyers stability of the functional equation (1.2) in banach space using fixed point method. now we will recall the fundamental results in fixed point theory. theorem 4.1. (banach’s contraction principle) let (x,d) be a complete metric space and consider a mapping t : x → x which is strictly contractive mapping, that is (a1) d(tx,ty) ≤ ld(x,y) for some (lipschitz constant) l < 1. then, (i) the mapping t has one and only fixed point x∗ = t (x∗); (ii)the fixed point for each given element x∗ is globally attractive, that is (a2) limn→∞tnx = x∗, for any starting point x ∈ x; (iii) one has the following estimation inequalities: (a3) d(tnx,x∗) ≤ 1 1−l d(t nx,tn+1x),∀ n ≥ 0,∀ x ∈ x; (a4) d(x,x∗) ≤ 1 1−l d(x,x ∗),∀ x ∈ x. theorem 4.2. [24](the alternative of fixed point) suppose that for a complete generalized metric space (x,d) and a strictly contractive mapping t : x → x with lipschitz constant l. then, for each given element x ∈ x, either (b1) d(tnx,tn+1x) = ∞ ∀ n ≥ 0, or (b2) there exists a natural number n0 such that: (i) d(tnx,tn+1x) < ∞ for all n ≥ n0 ; (ii)the sequence (tnx) is convergent to a fixed point y∗ of t (iii) y∗ is the unique fixed point of t in the set y = {y ∈ x : d(tn0x,y) < ∞}; (iv) d(y∗,y) ≤ 1 1−l d(y,ty) for all y ∈ y. theorem 4.3. let f : u3 → v be a mapping for which there exist a function ϕ : u3 → [0,∞) with the condition lim n→∞ 1 µ2n i ϕ (µni x,µ n i y,µ n i z) + α = 0 (4.1) where µi = 2 if i = 0 and µi = 1 2 if i = 1 such that the functional inequality with ‖f (x,y,z)‖+α ≺ ϕ(x,y,z) + α (4.2) for all x,y,z ∈ u. if there exists l = l(i) < 1 such that the function x → γ(x)+α = ϕ ( x 2 , x 2 , −x 2 )+ α , (4.3) int. j. anal. appl. (2022), 20:15 9 has the property γ(x)+α ≺ l 1 µ2 i �γ (µix) + α (4.4) for all x ∈ u, then there exists a unique quadratic mapping q : u → y satisfying the functional equation (1.2) and ‖ f (x) −q(x) ‖+α ≺ l1−i 1 −l γ(x)+α (4.5) for all x ∈ u. proof. consider the set ω = {p/p : u → y, p(0) = 0} and introduce the generalized metric on ω, d(p,q) = dγ(p,q) = inf{k ∈ (0,∞) :‖ p(x) −q(x) ‖+α ≺ kγ(x) + α,x ∈ u}. it is easy to see that (ω,d) is complete. define t : ω → ω by tp(x) = 1 µ2 i �p(µix)+α, for all x ∈ u. now p,q ∈ ω, we have d(p,q) ≤ k ⇒‖ p(x) −q(x) ‖+α ≺ kγ(x) + α,x ∈ u. ⇒ ∥∥∥∥ 1µ2 i p(µix) − 1 µ2 i q(µix) ∥∥∥∥+ α ≺ 1 µ2 i �kγ(µix)+α,x ∈ u, ⇒ ∥∥∥∥ 1µ2 i p(µix) − 1 µ2 i q(µix) ∥∥∥∥+ α ≺ lkγ(x)+α,x ∈ u, ⇒‖ tp(x) −tq(x) ‖+α ≺ lkγ(x) + α,x ∈ u, ⇒ dγ(p,q) ≺ lk. this implies d(tp,tq) ≺ ld(p,q), for all p,q ∈ ω . i.e., t is a strictly contractive mapping on ω with lipschitz constant l. replacing (x,y,z) by (x,x,−x) in (1.2), we get,∥∥22f (x) − f (2x)∥∥+ α ≺ ϕ (x,x,−x)+α (4.6) hence from the above inequality, we have∥∥∥∥f (x) − f (2x)22 ∥∥∥∥+ α ≺ 1 22 �ϕ (x,x,−x)+α (4.7) for all x ∈ u. using (4.3) and (4.4) for the case i = 0, it reduces to∥∥∥∥f (x) − f (2x)22 ∥∥∥∥+ α ≺ 1 22 �γ(x)+α 10 int. j. anal. appl. (2022), 20:15 for all x ∈ u. i.e., dϕ(f ,tf ) ≺ l ⇒ d(f ,tf ) ≺ l ≺ l1 < ∞. again replacing x = x 2 in (4.6), we get,∥∥∥22f (x 2 ) − f (x) ∥∥∥+ α ≺ ϕ ( x 2 , x 2 , −x 2 )+ α (4.8) for all x ∈ u. using (4.3) and (4.4) for the case i = 1 it reduces to∥∥∥f (x) − 22f (x 2 )∥∥∥+ α ≺ γ(x)+α for all x ∈ u, i.e., dϕ(f ,tf ) ≤ 1 ⇒ d(f ,tf ) ≺ 1 ≺ l0 < ∞. in both cases, we arrive d(f ,tf )≺l1−i. therefore (a1) holds. by (a2), it follows that there exists a fixed point q of t in ω such that q(x) = lim n→∞ 1 µ2n i (f (µni x)) (4.9) for all x ∈ u. to prove q : u → y is quadratic. replacing (x,y,z) by( µni x,µ n i y,µ n i z ) in (4.2) and dividing by µ2ni , it follows from (4.1) that ‖q(x,y,z)‖+α = lim n→∞ ∥∥d f (µni x,µni y,µni z)∥∥+α µ2n i ≺ lim n→∞ ϕ(µni x,µ n i y,µ n i z) + α µ2n i = 0 for all x,y,z ∈ u. that is, q satisfies the functional equation (1.2). by (a3), q is the unique fixed point of t in the set ∆ = {q ∈ ω : d(f ,q) < ∞},q is the unique function such that ‖f (x) −q(x)‖+α ≺ kγ(x) + α for all x ∈ u and k > 0. finally by (a4), we obtain d(f ,q) ≺ 1 1 −l d(f ,tf ) this implies d(f ,q) ≺ l1−i 1 −l which yields ‖ f (x) −q(x) ‖+α ≺ l1−i 1 −l γ(x)+α. this completes the proof of the theorem. � int. j. anal. appl. (2022), 20:15 11 the following corollary is an immediate consequence of theorem 4.3 concerning the stability of (1.2). corollary 4.1. let f : u → y be a mapping and there exit real numbers λ and s such that ‖f (x,y,z)‖+α ≺   λ, λ⊗ (||x||s ⊕||y||s ⊕||z||s) , s 6= 2; λ⊗ (||x||s ⊗||y||s ⊗||z||s) , s 6= 2 3 ; λ⊗ { (||x||s ⊗||y||s ⊗||z||s) ⊕||x||3s ⊕||y||3s ⊕||z||3s } , s 6= 2 3 ; (4.10) for all x,y,z ∈ u, then there exists a unique quadratic function q : u → u such that ‖f (x) −q(x)‖+α ≺   λ |3| , 2 �λ+α ⊗ (||x||s) + α |22 − 2s| , λ+α ⊗ ( ||x||3s )+ α |22 − 23s| , 4 �λ+α ⊗ ( ||x||3s )+ α |22 − 23s| . (4.11) for all x ∈ u. proof. set ϕ(x,y,z) =   λ, λ⊗ (||x||s ⊕||y||s ⊕||z||s) , λ⊗ (||x||s ⊗||y||s ⊗||z||s) , λ⊗ { ||x||s ⊗||y||s ⊗||z||s ⊕||x||3s ⊕||y||3s ⊕||z||3s } for all x,y,z ∈ u. now, ϕ(µni x,µ n i y,µ n i z) µ2ni =   λ µ2ni , λ µ2ni ⊗ (||µni x|| s ⊕||µni y|| s ⊕||µni z|| s), λ µ2ni ⊗ (||µni x|| s ⊗||µni y|| s ⊗||µni z|| s) λ µ2ni ⊗{(||µni x|| s ⊗||µni y|| s ⊗||µni z|| s) ⊕||µni x|| 3s ⊕||µni y|| 3s ⊕||µni z|| 3s } =   λ⊗µ−2ni , λ⊗µ(s−2)ni ⊗ (||x|| s ⊕||y||s ⊕||z||s), λ⊗µ(3s−2)ni ⊗ (||x|| s ⊗||y||s ⊗||z||s) λ⊗µ(3s−2)ni ⊗{(||x|| s ⊗||y||s ⊗||z||s) ⊕||x||3s ⊕||y||3s ⊕||z||3s } =   → 0 as n →∞, → 0 as n →∞, → 0 as n →∞, → 0 as n →∞. 12 int. j. anal. appl. (2022), 20:15 thus, (4.1) holds. but we have γ(x) = ϕ ( x 2 , x 2 , −x 2 ) has the property γ(x) ≺ l · 1 µ2 i γ (µix) for all x ∈ u. hence γ(x) = ϕ ( x 2 , x 2 , −x 2 ) =   λ, λ⊗ 3 2s ⊗||x||s, λ⊗ 1 23s ⊗||x||3s, λ⊗ 4 23s ⊗||x||3s. now, 1 µ2 i γ(µix) =   λ µ2 i , λ µ2 i � 3 2s ⊗ (||µix||s), λ µ2 i � 1 23s ⊗ (||µix||ns), λ µ2 i � 4 23s ⊗ ( ||µix||3s ) =   µ−2 i γ(x)+α, µs−2 i γ(x)+α, µ3s−2 i γ(x)+α, µ3s−2 i γ(x)+α. now from (4.5), case:1 l = 2−2 for s = 0 if i = 0, ‖f (x) −q(x)‖+α ≺ ( 2−2 )1−0 1 − 2(−2) γ(x)+α ≺ λ + α � ( 2−2 1 − 2−2 ) ≺ λ+α 3 . case:2 l = 22 for s = 0 if i = 1, ‖f (x) −q(x)‖+α ≺ ( 22 )1−1 1 − 22 γ(x)+α ≺ λ + α � ( 1 1 − 22 ) ≺ λ+α −3 . case:3 l = 2s−2 for s > 2 if i = 0, ‖f (x) −q(x)‖+α ≺ ( 2s−2 )1−0 1 − 2s−2 γ(x)+α ≺ 3 �λ+α 22 − 2s ⊗ (||x||s)+α . case:4 l = 22−s for s < 2 if i = 1, ‖f (x) −q(x)‖+α ≺ ( 22−s )1−1 1 − 22−s γ(x)+α ≺ 3 �λ+α 2s − 22 ⊗ (||x||s)+α . case:5 l = 23s−2 for s > 2 3 if i = 0, ‖f (x) −q(x)‖ ≺ (( 2(2s−2) )1−0 1 − 2(2s−2) ) γ(x)+α ≺ λ+α 22 − 23s ⊗ ( ||x||3s )+ α . case:6 l = 22−3s for s < 2 3 if i = 1, ‖f (x) −q(x)‖+α ≺ (( 2(2−3s) )1−1 1 − 2(2−3s) ) γ(x)+α ≺ λ+α 23s − 22 ⊗ ( ||x||3s )+ α . hence it completes the proof. � int. j. anal. appl. (2022), 20:15 13 5. conclusion this article has proved the stability results of the quadratic functional equation in felbin’s fuzzy normed linear spaces (f-nls) using both the direct and fixed point methods. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. aczel, j. dhombres, functional equations in several variables, cambridge university press, cambridge, 1989. [2] t. aoki, on the stability of the linear transformation in banach spaces, j. math. soc. japan. 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064. [3] m. arunkumar, s. karthikeyan, solution and stability of n-dimensional additive functional equation, int. j. appl. math. 25 (2012), 163-174. [4] i.s. chang, e.h. lee, h.m. kim, on the hyers-ulam-rassias stability of a quadratic functional equations, math. ineq. appl. 6 (2003), 87-95. [5] s. czerwik, functional equations and inequalities in several variables, world scientific, river edge, nj, 2002. [6] s. czerwik, on the stability of the quadratic mapping in normed spaces, abh. math. semin. univ. hambg. 62 (1992), 59–64. https://doi.org/10.1007/bf02941618. [7] d.k. singha, s. grover, on the stability of a sum form functional equation related to entropies of type (α,β), j. nonlinear sci. appl. 14 (2021), 168-180. https://doi.org/10.22436/jnsa.014.03.06. [8] c. felbin, finite dimensional fuzzy normed linear space, fuzzy sets syst. 48 (1992), 239–248. https://doi. org/10.1016/0165-0114(92)90338-5. [9] p. gavruta, a generalization of the hyers-ulam-rassias stability of approximately additive mappings, j. math. anal. appl. 184 (1994), 431–436. https://doi.org/10.1006/jmaa.1994.1211. [10] t.e. gantner, r.c. steinlage, r.h. warren, compactness in fuzzy topological spaces, j. math. anal. appl. 62 (1978), 547–562. https://doi.org/10.1016/0022-247x(78)90148-8. [11] e.s. el-hady, on hyperstability of cauchy functional equation in (2, γ)-banach spaces, j. math. computer sci. 23 (2021), 354-363. https://doi.org/10.22436/jmcs.023.04.08. [12] d.h. hyers, on the stability of the linear functional equation, proc. nat. acad. sci. 27 (1941), 222–224. https: //doi.org/10.1073/pnas.27.4.222. [13] u. höhle, fuzzy real numbers as dedekind cuts with respect to a multiple-valued logic, fuzzy sets syst. 24 (1987), 263–278. https://doi.org/10.1016/0165-0114(87)90027-3. [14] d.h. hyers, g. isac, t.m. rassias, stability of functional equations in several variables, birkhauser, basel, 1998. [15] k.-w. jun, h.-m. kim, on the stability of an n-dimensional quadratic and additive functional equation, math. inequal. appl. 9 (2006), 153–165. https://doi.org/10.7153/mia-09-16. [16] s.m. jung, hyers-ulam-rassias stability of functional equations in mathematical analysis, hadronic press, palm harbor, 2001. [17] pl. kannappan, functional equations and inequalities with applications, springer, new york, 2009. [18] s. karthikeyan, c. park, p. palani, t.r.k. kumar, stability of an additive-quartic functional equation in modular spaces, j. math. computer sci. 26 (2021), 22–40. https://doi.org/10.22436/jmcs.026.01.04. [19] o. kaleva, s. seikkala, on fuzzy metric spaces, fuzzy sets syst. 12 (1984), 215–229. https://doi.org/10. 1016/0165-0114(84)90069-1. https://doi.org/10.2969/jmsj/00210064 https://doi.org/10.1007/bf02941618 https://doi.org/10.22436/jnsa.014.03.06 https://doi.org/10.1016/0165-0114(92)90338-5 https://doi.org/10.1016/0165-0114(92)90338-5 https://doi.org/10.1006/jmaa.1994.1211 https://doi.org/10.1016/0022-247x(78)90148-8 https://doi.org/10.22436/jmcs.023.04.08 https://doi.org/10.1073/pnas.27.4.222 https://doi.org/10.1073/pnas.27.4.222 https://doi.org/10.1016/0165-0114(87)90027-3 https://doi.org/10.7153/mia-09-16 https://doi.org/10.22436/jmcs.026.01.04 https://doi.org/10.1016/0165-0114(84)90069-1 https://doi.org/10.1016/0165-0114(84)90069-1 14 int. j. anal. appl. (2022), 20:15 [20] o. kaleva, the completion of fuzzy metric spaces, j. math. anal. appl. 109 (1985), 194-198. [21] o. kaleva, a comment on the completion of fuzzy metric spaces, fuzzy sets syst. 159 (2008), 2190–2192. https://doi.org/10.1016/j.fss.2008.03.011. [22] h.-m. kim, on the stability problem for a mixed type of quartic and quadratic functional equation, j. math. anal. appl. 324 (2006), 358–372. https://doi.org/10.1016/j.jmaa.2005.11.053. [23] r. lowen, fuzzy set theory, (ch. 5: fuzzy real numbers), kluwer, dordrecht, 1996. [24] j.b. diaz, b. margolis, a fixed point theorem of the alternative, for contractions on a generalized complete metric space, bull. amer. math. soc. 74 (1968), 305-309. [25] f. moradlou, s. rezaee, i. sadeqi, stability of cauchy functional equation in felbin’s type spaces: a fixed point approach, submitted. [26] a.k. mirmostafaee, m.s. moslehian, fuzzy versions of hyers–ulam–rassias theorem, fuzzy sets syst. 159 (2008), 720–729. https://doi.org/10.1016/j.fss.2007.09.016. [27] j.m. rassias, on approximation of approximately linear mappings by linear mappings, j. funct. anal. 46 (1982), 126–130. https://doi.org/10.1016/0022-1236(82)90048-9. [28] t.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72 (1978), 297–300. https://doi.org/10.1090/s0002-9939-1978-0507327-1. [29] t.m. rassias, on the stability of functional equations and a problem of ulam, acta appl. math. 62 (2000), 23–130. https://doi.org/10.1023/a:1006499223572. [30] t.m. rassias, functional equations, inequalities and applications, kluwer acedamic publishers, dordrecht, 2003. [31] k. ravi, m. arunkumar, j.m. rassias, ulam stability for the orthogonally general euler-lagrange type functional equation, int. j. math. stat. 3 (2008), 36-46. [32] j.m. rassias, r. saadati, g. sadeghi, j. vahidi, on nonlinear stability in various random normed spaces, j. inequal. appl. 2011 (2011), 62. https://doi.org/10.1186/1029-242x-2011-62. [33] j.m. rassias, n. pasupathi, r. saadati, m. de la sen, approximation of mixed euler-lagrange σ-cubic-quartic functional equation in felbin’s type f-nls, j. funct. spaces. 2021 (2021), 8068673. https://doi.org/10.1155/ 2021/8068673. [34] s.e. rodabaugh, fuzzy addition in the l-fuzzy real line, fuzzy sets syst. 8 (1982), 39–52. https://doi.org/ 10.1016/0165-0114(82)90028-8. [35] i. sadeqi, f. moradlou, m. salehi, on approximate cauchy equation in felbin’s type fuzzy normed linear spaces, iran. j. fuzzy syst. 10 (2013), 51-63. https://doi.org/10.22111/ijfs.2013.862. [36] i. sadeqi, m. salehi, fuzzy compact operators and topological degree theory, fuzzy sets syst. 160 (2009), 1277–1285. https://doi.org/10.1016/j.fss.2008.08.014. [37] s.m. ulam, problems in modern mathematics, science editions, wiley, new york, 1964. [38] j. xiao, x. zhu, on linearly topological structure and property of fuzzy normed linear space, fuzzy sets syst. 125 (2002), 153–161. https://doi.org/10.1016/s0165-0114(00)00136-6. [39] j. xiao, x. zhu, topological degree theory and fixed point theorems in fuzzy normed space, fuzzy sets syst. 147 (2004), 437–452. https://doi.org/10.1016/j.fss.2004.01.003. https://doi.org/10.1016/j.fss.2008.03.011 https://doi.org/10.1016/j.jmaa.2005.11.053 https://doi.org/10.1016/j.fss.2007.09.016 https://doi.org/10.1016/0022-1236(82)90048-9 https://doi.org/10.1090/s0002-9939-1978-0507327-1 https://doi.org/10.1023/a:1006499223572 https://doi.org/10.1186/1029-242x-2011-62 https://doi.org/10.1155/2021/8068673 https://doi.org/10.1155/2021/8068673 https://doi.org/10.1016/0165-0114(82)90028-8 https://doi.org/10.1016/0165-0114(82)90028-8 https://doi.org/10.22111/ijfs.2013.862 https://doi.org/10.1016/j.fss.2008.08.014 https://doi.org/10.1016/s0165-0114(00)00136-6 https://doi.org/10.1016/j.fss.2004.01.003 1. introduction 2. some preliminaries on fuzzy real number 3. felbin's stability results: direct method 4. felbin's stability results: fixed point method 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 21-25 http://www.etamaths.com on the growth of iterated entire functions dibyendu banerjee∗ and nilkanta mondal abstract. we consider iteration of two entire functions of (p, q)-order and study some growth properties of iterated entire functions to generalise some earlier results. 1. introduction for any two transcendental entire functions f(z) and g(z), lim r→∞ m(r,f◦g) m(r,f) = ∞ and clunie [2] proved that the same is true for the ratio t(r,f◦g) t(r,f) . in [7] singh proved some results dealing with the ratios of log t(r,f◦g) and t(r,f) under some restrictions on the orders of f and g. in this paper, we generalise the results of singh [7] for iterated entire functions of (p,q)-orders. following sato [6], we write log[0] x = x, exp[0] x = x and for positive integer m, log[m] x = log(log[m−1] x), exp[m] x = exp(exp[m−1] x). let f(z) = ∞∑ n=0 anz n be an entire function. then the (p,q)-order and lower (p,q)-order of f(z) are denoted by ρ(p,q)(f) and λ(p,q)(f) respectively and defined by [1] ρ(p,q)(f) = lim r→∞ sup log[p] t(r,f) log[q] r and λ(p,q)(f) = lim r→∞ inf log[p] t(r,f) log[q] r , p ≥ q ≥ 1. according to lahiri and banerjee [4] iff(z) and g(z) be entire functions then the iteration of f with respect to g is defined as follows: f1(z) = f(z) f2(z) = f(g(z)) = f(g1(z)) f3(z) = f(g(f(z))) = f(g2(z)) . . . . . . . . . . . . . . . . . . . fn(z) = f(g(f(g(....(f(z) or g(z) according as n is odd or even)))) and so are gn(z). clearly all fn(z) and gn(z) are entire functions. the main purpose of this paper is to study growth properties of iterated entire functions to that of the generating functions under some restriction on (p,q)-orders and lower (p,q)-orders of f and g. throughout we assume f, g etc., are non-constant entire functions having finite (p,q)-orders. 2010 mathematics subject classification. 30d35. key words and phrases. entire functions, iteration, growth. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 21 22 dibyendu banerjee and nilkanta mondal 2. lemmas following two lemmas will be needed throughout the proof of our theorems. lemma 1[5]. let f(z) and g(z) be entire functions. if m(r,g) > 2+� � |g(0)| for any � > 0, then t(r,f ◦g) ≤ (1 + �)t(m(r,g),f). in particular, if g(0) = 0 then t(r,f ◦g) ≤ t(m(r,g),f)for all r > 0. lemma 2[3]. if f(z) be regular in |z| ≤ r, then for 0 ≤ r < r t(r,f) ≤ log+ m(r,f) ≤ r+r r−rt(r,f). in particular if f be entire, then t(r,f) ≤ log+ m(r,f) ≤ 3t(2r,f). 3. main results first we shall show that if we put some restriction on (p,q)-orders of f and g then the limit superior of the ratio is bounded above by a finite quantity. the following two theorems admit the results. theorem 1. let f(z) and g(z) be two entire functions with f(0) = g(0) = 0 and ρ(p,q)(g) < λ(p,q)(f). then for even n lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t(2n−2r,f) ≤ ρ(p,q)(f). proof. we have by lemma 1 and lemma 2 log[p] t(r,fn) ≤ log[p] t(m(r,gn−1),f) < (ρ(p,q)(f) + �) log [q] m(r,gn−1), for all large values of r and � > 0 ≤ (ρ(p,q)(f) + �) log [q−1]{3t(2r,gn−1)} = (ρ(p,q)(f) + �) log [q−1] t(2r,gn−1) + o(1). so, log[p+(p+1−q)] t(r,fn) < log [p] t(2r,gn−1) + o(1) < (ρ(p,q)(g) + �) log [q−1] t(22r,fn−2) + o(1). proceeding similarly after (n− 2) steps we get log[p+(n−2)(p+1−q)] t(r,fn) < log [p] t(2n−2r,f(g)) + o(1) ≤ log[p] t(m(2n−2r,g),f) + o(1) < (ρ(p,q)(f) + �) log [q] m(2n−2r,g) + o(1) < (ρ(p,q)(f)+�){exp[p−q](log [q−1](2n−2r))ρ(p,q)(g)+�}+o(1) (3.1) for all large values of r < (ρ(p,q)(f)+�){exp[p−q](log [q−1](2n−2r))λ(p,q)(f)−�}+o(1) by choosing � > 0 so small that ρ(p,q)(g) + � < λ(p,q)(f) − �. on the other hand, t(r,f) > exp[p−1](log[q−1] r)λ(p,q)(f)−� , for all r ≥ r0 or, log[q−1] t(r,f) > exp[p−q](log[q−1] r)λ(p,q)(f)−� , for all r ≥ r0. therefore, from above log[p+(n−2)(p+1−q)] t(r,fn) [log[q−1] t(2n−2r,f) < (ρ(p,q)(f)+�){exp [p−q](log[q−1](2n−2r)) λ(p,q)(f)−�}+o(1) exp[p−q](log[q−1](2n−2r)) λ(p,q)(f)−� , for all r ≥ r0. hence, on the growth of iterated entire functions 23 lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t(2n−2r,f) ≤ ρ(p,q)(f) + �. the theorem now follows since � (> 0) is arbitrary. note 1. from the hypothesis it is clear that f must be transcendental. theorem 2. let f and g be two entire functions with f(0) = g(0) = 0 and ρ(p,q)(f) < λ(p,q)(g). then for odd n lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t(2n−2r,g) ≤ ρ(p,q)(g). the proof of the theorem is on the same line as that of theorem 1. if ρ(p,q)(g) > ρ(p,q)(f) holds in theorem 1 we shall show that the limit superior will tend to infinity. now we prove the following two theorems. theorem 3. let f(z) and g(z) be two entire functions of positive lower (p,q)orders with ρ(p,q)(g) > ρ(p,q)(f). then for even n lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t( r 4n−1 ,f) = ∞. proof. we have, t(r,fn) = t(r,f(gn−1)) ≥ 1 3 log m( 1 8 m( r 4 ,gn−1) + o(1),f) { see [7], page 100 } ≥ 1 3 log m( 1 9 m( r 4 ,gn−1),f) ≥ 1 3 t( 1 9 m( r 4 ,gn−1),f) > 1 3 exp[p−1]{log[q−1] 1 9 m( r 4 ,gn−1)}λ(p,q)(f)−�, for all r ≥ r0 = 1 3 exp[p−1]{log[q−1] m( r 4 ,gn−1)}λ(p,q)(f)−� + o(1), for all r ≥ r0. therefore, log[p] t(r,fn) > log{log[q−1] m( r4,gn−1)} λ(p,q)(f)−� + o(1), = (λ(p,q)(f) − �) log [q] m( r 4 ,gn−1) + o(1). (3.2) so, we have for all r ≥ r0 log[p+(p+1−q)] t(r,fn) > log [p][log m( r 4 ,gn−1)] + o(1) ≥ log[p] t( r 4 ,gn−1) + o(1) > (λ(p,q)(g) − �) log [q] m( r 42 ,fn−2) + o(1), using (3.2) or, log[p+2(p+1−q)] t(r,fn) > log [p] t( r 42 ,fn−2) + o(1) > (λ(p,q)(f) − �) log [q] m( r 43 ,gn−3) + o(1), using (3.2). proceeding similarly after some steps we get log[p+(n−2)(p+1−q)] t(r,fn) > (λ(p,q)(f) − �) log [q] m( r 4n−1 ,g) + o(1) > (λ(p,q)(f) − �) exp[p−q](log [q−1]( r 4n−1 ))ρ(p,q)(g)−� + o(1) (3.3) for a sequence of values of r →∞. on the other hand for all r ≥ r0 we have, t(r,f) < exp[p−1](log[q−1] r)ρ(p,q)(f)+� or, log[q−1] t(r,f) < exp[p−q](log[q−1] r)ρ(p,q)(f)+�. (3.4) so, from (3.3) and (3.4) we have for a sequence of values of r →∞, log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t( r 4n−1 ,f) > (λ(p,q)(f)−�) exp [p−q](log[q−1]( r 4n−1 )) ρ(p,q)(g)−� exp[p−q](log[q−1] r 4n−1 ) ρ(p,q)(f)+� + o(1) and so, 24 dibyendu banerjee and nilkanta mondal lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t( r 4n−1 ,f) = ∞ since we can choose � (> 0) such that ρ(p,q)(g) − � > ρ(p,q)(f) + �. this proves the theorem. an immediate consequence of theorem 3 for odd n is the following theorem. theorem 4. let f(z) and g(z) be two entire functions of positive lower (p,q)orders with ρ(p,q)(g) < ρ(p,q)(f). then for odd n lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[q−1] t( r 4n−1 ,g) = ∞. next if we consider the ratios log[p+(n−1)(p+1−q)] t(r,fn) log[p] t(2n−2r,g) or log[p+(n−2)(p+1−q)] t(r,fn) log[p] t( r 4n−1 ,g) we have obtained the following four theorems. theorem 5. let f(z) and g(z) be two transcendental entire functions with f(0) = g(0) = 0 and let λ(p,q)(g) > 0. then for even n lim sup r→∞ log[p+(n−1)(p+1−q)] t(r,fn) log[p] t(2n−2r,g) ≤ ρ(p,q)(g) λ(p,q)(g) . proof. we get from (3.1), for all large values of r log[p+(n−2)(p+1−q)] t(r,fn) < (ρ(p,q)(f)+�){exp[p−q](log [q−1](2n−2r))ρ(p,q)(g)+�}+ o(1) or, log[p+(n−1)(p+1−q)] t(r,fn) < (ρ(p,q)(g) + �) log [q](2n−2r) + o(1). on the other hand, log[p] t(r,g) > (λ (p,q) (g) − �) log[q] r, for all r ≥ r0. thus for all r ≥ r0 log[p+(n−1)(p+1−q)] t(r,fn) log[p] t(2n−2r,g) < (ρ(p,q)(g)+�) log [q](2n−2r)+o(1) (λ (p,q) (g)−�) log[q](2n−2r) . therefore, lim sup r→∞ log[p+(n−1)(p+1−q)] t(r,fn) log[p] t(2n−2r,g) ≤ ρ(p,q)(g) λ(p,q)(g) . hence the theorem is proved. theorem 6. let f(z) and g(z) be two transcendental entire functions with f(0) = g(0) = 0 and let λ(p,q)(f) > 0. then for odd n lim sup r→∞ log[p+(n−1)(p+1−q)] t(r,fn) log[p] t(2n−2r,f) ≤ ρ(p,q)(f) λ(p,q)(f) . the proof is omitted. theorem 7. let f(z) and g(z) be two transcendental entire functions of positive lower (p,q)-orders with ρ(p,q)(g) > 0. then for even n lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[p] t( r 4n−1 ,g) = ∞. proof. from (3.3), we have for a sequence of values of r →∞ log[p+(n−2)(p+1−q)] t(r,fn) > (λ(p,q)(f)−�) exp[p−q](log [q−1]( r 4n−1 ))ρ(p,q)(g)−�+ o(1). also, log[p] t(r,g) < (ρ(p,q)(g) + �) log [q] r, for all r ≥ r0. thus log[p+(n−2)(p+1−q)] t(r,fn) log[p] t( r 4n−1 ,g) ≥ (λ(p,q)(f)−�) (ρ(p,q)(g)+�) exp[p−q](log[q−1]( r 4n−1 )) ρ(p,q)(g)−� log[q] r 4n−1 which tends to infinity as r →∞, through this sequence since ρ(p,q)(g) > 0. theorem 8. let f(z) and g(z) be two transcendental entire functions of positive lower (p,q)-orders with ρ(p,q)(f) > 0. then for odd n on the growth of iterated entire functions 25 lim sup r→∞ log[p+(n−2)(p+1−q)] t(r,fn) log[p] t( r 4n−1 ,f) = ∞. the proof is omitted, since it follows easily as in theorem 7. note 2. if we put n = 2, p = q = 1 in the theorem 1 and theorem 5 we get the results of a.p. singh [7]. references [1] walter bergweiler, gerhard jank and lutz volkmann, results in mathematics 7 (1984), 35-53. [2] j. clunie, the composition of entire and meromorphic functions, mathematical essays dedicated to a. j. macintyre, ohio university press, 1970, 75-92. [3] w. k. hayman, meromorphic functions, oxford university press, 1964. [4] b. k. lahiri and d. banerjee, relative fix points of entire functions, j. indian acad. math., 19 (1), (1997), 87-97. [5] k. ninno and n. suita, growth of a composite function of entire functions, kodai math. j., 3 (1980), 374-379. [6] d. sato, on the rate of growth of entire functions of fast growth, bull. amer. math. soc., 69 (1963), 411-414. [7] a. p. singh, growth of composite entire functions, kodai math. j., 8 (1985), 99-102. department of mathematics, visva-bharati, santiniketan-731235, west bengal, india ∗corresponding author int. j. anal. appl. (2023), 21:63 received: oct. 31, 2020. 2020 mathematics subject classification. 90c26, 90c29, 90c48. key words and phrases. generalized subconvexlike maps; presubconvexlike maps; set-valued optimization; optimality conditions. https://doi.org/10.28924/2291-8639-21-2023-63 © 2023 the author(s) issn: 2291-8639 1 optimality conditions for set-valued optimization problems renying zeng* mathematics department, saskatchewan polytechnic, canada *correspondence: renying.zeng@saskpolytech.ca abstract. in this paper, we first prove that the generalized subconvexlikeness introduced by yang, yang and chen [1] and the presubconvelikeness introduced by zeng [2] are equivalent. we discuss set-valued nonconvex optimization problems and obtain some optimality conditions. 1. introduction set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objectives and/or the constraints are set-valued maps. corley [3] pointed out that the dual problem of a multiobjective optimization involves the optimization of a setvalued map, while klein and thompson [4] gave some examples in economics where it is necessary to use set-valued maps instead of single-valued maps. there are many recent developments about set-values optimization problems, e.g., [5-9]. convex and generalized convex optimization is a rich branch of mathematics. many interesting and useful definitions of generalized convexities were introduced. borwein [10] proposed the definition of cone convexity, fan [11] introduced the definition of convexlikeness. yang, yang and chen [1] defined the generalized subconvexlike functions, while zeng [2] introduced a presubconvexlikeness. in this paper, we first prove that the generalized subconvexlikeness introduced by yang, yang, and chen [1] and the presubconvexlikeness introduced by zeng [2] are equivalent, in locally convex https://doi.org/10.28924/2291-8639-21-2023-63 2 int. j. anal. appl. (2023), 21:63 topological spaces. and then, we deal with set-valued optimization problems and obtain some optimality conditions. a subset y + of a real linear topological space y is a cone if y y +  for all y y +  and 0  . we denote by 0 y the zero element in the linear topological space y and simply by 0 if there is no confusion. a convex cone is one for which 1 1 2 2 y y y  + +  for all 1 2 ,y y y +  and 1 2 , 0   . a pointed cone is one for which ( ) {0}y y + + − = . let y be a real linear topological space with pointed convex cone y + . we denote the partial order induced by y + as follows: 1 2 y y iff 1 2 y y y + −  , 1 2 y y iff 1 2 inty y y + −  , where inty + denotes the topological interior of a set y + . let x, zi, wj be real linear topological spaces and y be an ordered linear topological space with the partial order induced by a pointed convex cone y + . we recall some notions of generalized convexity of set-valued maps. first we recall the notion of cone-convexity of a set-valued map introduced by borwein [10]. definition 1.1 (convexity) let x, y be real linear topological spaces, d x a nonempty convex set and y + a convex cone in y. a set-valued map f : x y→ is said to be y + -convex on d if and only if 1 2 ,x x d  , [0,1]  , there holds 1 2 1 2 ( ) (1 ) ( ) ( (1 ) ) .f x f x f x x y    + + −  + − + the following notion of generalized convexity is a set-valued map version of ky fan convexity [11] (ky fan’s definition was for vector-valued optimization problems). definition 1.2 (convexlike) let x, y be real linear topological spaces, d x a nonempty set and y + be a convex cone in y. a set-valued map f : x y→ is said to be y + -convexlike on d if and only if 1 2 ,x x d  , [0,1]  , 3 x d  such that 1 2 3 ( ) (1 ) ( ) ( ) .f x f x f x y  + + −  + the following concept of generalized subconvexlikeness was introduced by yang, yang and chen [1] ([1] introduced subconvexlikeness for vector-valued optimization). 3 int. j. anal. appl. (2023), 21:63 definition 1.3 (generalized subconvexlike) let y be a linear topological space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is said to be generalized y + -subconvexlike on d if intu y +   such that 1 2 ,x x d  , 0  , [0,1]  , 3 x d  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y    + + + −  + . the following lemma 1.1 is from chen and rong [12, proposition 3.1]. lemma 1.1 a function f : yd → is generalized y + -subconvexlike on d if  + yu int , 1 2 ,x x d  ,  ]1,0[ ,  dx 3 ,  0 such that 1 2 3 ( ) (1 ) ( ) ( ) .u f x f x f x y   + + + −  + a bounded function in a real linear topological space can be fined as following definition 1.4 (e.g,, see yosida [13]). definition 1.4 (bounded set-valued map) a subset m of a real linear topological space y is said to be a bounded subset if for any given neighbourhood u of 0,  positive scalar  such that 1 m u −  , where 1 1 { ; ; }m y y y v v m  − − =  =  . a set-valued map f : d y→ is said to bounded map if f (y) is a bounded subset of y. the following definition 1.5 was introduced by zeng [2] for single-valued functions. definition 1.5 (presubconvexlike) let y be a linear topological space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is said to be y + -presubconvexlike on d if 1 2 ,x x d  , [0,1]  , 0  , 3 x d  , 0  ,  bounded set-valued map u: d y→ such that 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y    + + + −  + . it is obvious that y + -convex  y+ -convexlike  generalized y+ -subconvexlike  y+ presubconvexlike. it is important to note that the concept of convexlike or any weaker concepts are only nontrivial if y is not the one-dimensional euclidean space since any real-valued function is r+-convexlike. 4 int. j. anal. appl. (2023), 21:63 2. the equivalence of generalized subconvexlikeness and presubconvexlikeness in this section, we are going to prove that definition 1.4 (generalized subconvexlikeness) and definition 1.5 (presubconvexlikeness) are equivalent. definition 2.1 (1) a subset m of y is said to be convex, if 1 2 ,y y m and 0 1  implies 1 2 (1 )y y m + −  ; (2) m is said to be balanced if y m and | | 1  implies y m  ; (3) m is said to be absorbing if for any given neighbourhood u of 0, there exists a positive scalar  such that 1 m u −  , where 1 1 { ; ; }m y y y v v m  − − =  =  . definition 2.2 a real linear topological space y is called a locally convex, linear topological space (we call it a locally convex topological space, in the sequel) if any neighborhood of 0 y contains a convex, balanced, and absorbing open set. from [13, pp.26 theorem, pp.33 definition 1] one has lemma 2.1. lemma 2.1 banach spaces are locally convex topological spaces, so are finite dimensional euclidean spaces. proposition 2.1 let y be a locally convex topological space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is generalized y + -subconvexlike on d if and only if 0 ( ( ) int ) t tf d y  + + is convex. proof. the necessity. see [1, theorem 2.1]. the sufficiency. assume that 0 ( ( ) int ) t tf d y  + + is convex, aim to show that f : d y→ is generalized y + -subconvexlike on d. from lemma 1.1, we are going to show that,  + yu int , 1 2 ,x x d  ,  ]1,0[ ,  dx 3 ,  0 such that 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y   + + + −  + . int ,y y +   0,t  since int y + is a cone, one has int .ty y +  1 1 2 2 ( ), ( ),y f x y f x   ,r  one has 1 2 0 ( ) , ( ) ( ( ) int ). t f x ty f x ty tf d y + + +  + from the convexity of 0 ( ( ) int ) t tf d y  + + , 3 3 , intx d y y +     , 0  such that 5 int. j. anal. appl. (2023), 21:63 1 2 1 2 3 3 0 ( ( ) ) (1 )( ( ) ) ( ) (1 ) ( ) ( ) ( ( ) int ). t f x ty f x ty f x f x ty f x y tf d y      + + + − +  + − +  +  + for the given intu y +  , from definition 2.2, neighbourhood u of 0 such that u is convex, balanced, and absorbing, and intu u y + +  , where u u+ is a neighbourhood of u. therefore, we may take 0t  small enough, such that ty u−  . then, intty u u u y + − +  +  . this and the convexity of int y + imply that 3 int .y ty u y + − +  and so 1 2 3 3 3 3 ( ) (1 ) ( ) ( ) ( ) int ( ) .u f x f x f x y ty f x y f x y     + + + + −  + −  +  + proposition 2.2 let y be a locally convex topological space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is y + -presubconvexlike on d if and only if 0 ( ( ) int ) t tf d y  + + is convex. proof. the necessity. suppose that f is y + -presubconvexlike on, aim to show that 0 ( ( ) int ) t tf d y  + + is convex. 1 2 1 1 1 2 2 2 0 , ( ( ) int ) t v t y y v t y y tf d y + +  +  = + = +  + , 1 2 ,x x d  such that 1 1 ( ),y f x 2 2 ( )y f x . let 0 1 2 (1 ) ,y y y  + + + = + − then 0 int .y y + +  therefore, neighbourhood u of 0 such that 0 y u + + is a neighbourhood of 0 y + and 0 inty u y + + +  . by definition 2.2, without loss of generality, we may assume that u is convex, balanced, and absorbing. from the assumption of y + -presubconvexlikeness, 0  , 3 x d  ,  bounded function u, and  > 0 such that 1 2 1 2 3 1 2 1 2 (1 ) ( ) ( ) ( ) , (1 ) (1 ) t t f x f x f x u y t t t t         + − +  − + + − + − 6 int. j. anal. appl. (2023), 21:63 therefore, 3 3 ( )y f x  such that 1 2 1 2 1 1 2 2 01 2 1 2 1 2 1 2 1 2 01 2 1 2 1 2 1 2 1 2 0 1 2 3 1 (1 ) (1 ) (1 ) (1 ) ( (1 ) )[ ] (1 ) (1 ) (1 ) ( (1 ) )[ ( ) ( )] (1 ) (1 ) ( (1 ) )[ ( ) ] ( ( v v t y t y y y t t t t y y y t t t t t t t t f x f x y t t t t t t f x u y y t                            + + + + + + + − = + − + + − − = + − + + + − + − −  + − + + + − + −  + − − + + = + 0 2 3 1 2 1 ) ) ( ) ( (1 ) )( ) .t f x t t y u y     + + − + + − − + since u is convex, balanced, and absorbing, by definition 2.2, we may take 0  small enough such that 1 2 ( (1 ) )t t u u  − + −  . therefore 0 0 1 2 ( (1 ) ) intt t u y y u y   + + + − + − +  +  . and then 0 1 2 1 2 ( (1 ) ) ( (1 ) ) int int .t t y t t u y y y y     + + + + + + − − + − +  +  therefore 1 2 0 1 2 3 1 2 0 (1 ) ( (1 ) ) ( ) ( (1 ) )( ) ( ( ) int ). t v v t t f x t t y u y tf d y         + +  + + −  + − + + − − +  + hence 0 ( ( ) int ) t tf d y  + + is a convex set. the sufficiency. assume that 0 ( ( ) int ) t tf d y  + + is convex. from lemma 1.1 and proposition 2.1,  intu y +  such that for all 1 2 ,x x d  ,  ]1,0[ , 0  , 3 x d  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y    + + + −  + . the given intu y +  can be consider as a bounded function. by propositions 1 and 2 one has theorem 2.1. 7 int. j. anal. appl. (2023), 21:63 theorem 2.1 let y be a locally convex topological space and d x be a nonempty set, and y + a convex cone in y. a set-valued map f : d y→ is generalized y + -subconvexlike on d if and only if f is y + -presubconvexlike on d. 3. optimal conditions we consider the following optimization problem with set-valued maps: (vp) y+ -min f (x) s.t. ( ) ( ) 0 i i g x z + −  , 1, 2, ,i m=  0 ( ), 1, 2, j h x j = , n x d where f : x y→ , : i i g x z→ , jj wxh →: are set-valued maps, zi+ is a closed convex cone in zi and d is a nonempty subset of x. for a set-valued map f : x y→ , we denote by ( ) ( ) x d f d f x  = . we now explain the kind of optimality we consider here. let f be the feasible set of (vp), i.e. : { : ( ) ( ) , 1, 2, , ;0 ( ), 1, 2, , }. i i j f x d g x z i m h x j n + =  −  =   =  we are looking for a weakly efficient solution of (vp) defined as follows. definition 3.1 (weakly efficient solution) a point x f is said to be a weakly efficient solution of (vp) with a weakly efficient value ( )y f x if for every x f , there exists no ( )y f x satisfying y y . consider the set-valued optimization problem (vp). from now on we assume that y+, zi+ are pointed convex cones with nonempty interior of int y + , int i z + , respectively. the following three assumptions will be used in this paper. (a1) generalized convexity assumption. there exist 0 intu y +  , int i i u z +  such that for all 1 2 ,x x d , 0  , [0,1]  ,, there exist 3 x d , 0( 1, 2, , ), 0( 1, 2, , ) i j i m t j n  =   =  such that 0 1 2 0 3 1 2 3 1 2 3 ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) i i i i i i j j j j u f x f x f x y u g x g x g x z h x h x t h x           + + + + −  + + + −  + + −  (a2) interior point assumption. 8 int. j. anal. appl. (2023), 21:63 int ( ) j h d  , (j = 1, 2, …, n). (a3) finite dimension assumption. wj (j = 1, 2, …, n) are finite dimensional spaces. similar to the proof of propositions 2.1 or 2.2, one has proposition 3.1. proposition 3.1 assumption (a1) is satisfied if and only if the following set is convex: 1 1 0 : {( , , ) : , , 0, . . ( ) int , ( ) int , ( )}. m n i i j j i j i i i i j j j b y z w y z w x d t s t y f x y z g x z w t h x    = = + + =        +  +  proposition 3.2 (alternative theorem) assume that the assumption (a1) and either (a2) or (a3) are satisfied. consider the following generalized inequality-equality systems: [system 1] , . . ( ) ( int ) , ( ) ( ) , 0 ( ). i i j x d s t f x y g x z h x + +   −  −   [system 2] * * * 1 1 1 1 ( , , ) ( ) \ {0}, . . ( ( )) ( ( )) ( ( )) 0. m n i i j j m n i i j ji j y z w s t x d f x g x h x       + = + = = =       + +   then if system 1 has no solution x, then system 2 has a solution ( , , )   . if system 2 has a solution ( , , )   with 0  , then system 1 has no solution. proof. suppose that system 1 has no solution, then 0 b . since (a1) holds, the set b is convex. by assumption, 1 m i i z =  is infinite dimensional and (a2) holds (which is equivalent to saying that int b  ) or 1 n j j w =  is finite dimensional. therefore by the separation theorem, nonzero vector * * * 1 1 ( , , ) m n i i j j y z w   = =    such that 0 0 0 1 1 ( ) ( ) ( ) 0 m n i i i i j j ji j y y z z t w     = = + + + +   for all 0 0 , ( ), ( ), ( ), int , int , 0, 0. i i j j i i i j x d y f x z g x w h x y y z z t + +         since inty+, intzi+ are convex cones, we have 0 0 0 0 1 1 ( ) ( ) ( ) 0 m n i i i i i j j ji j y s y z s z t w     = = + + + +   for all 0 0 , ( ), ( ), ( ), int , int , 0, 0, 0( 0,1, 2, ). i i j j i i i j i x d y f x z g x w h x y y z z t s i m + +          =  9 int. j. anal. appl. (2023), 21:63 taking ),,2,1,0(0,0,0 mist iji =→→→ , we obtain 0 0 ( ) 0, inty y y +    , and consequently 0 0 ( ) 0, inty y y cly cl y + + +     = , where cly + is the topological closure of the set y + . similarly, we have ( ) 0, i i i i z z z +    , and hence * y +  , * i i z +  . let 1( 1, 2, , ), 1( 1, 2, , ) i j i m t j n = =  = =  and take 0( 0,1, 2, ) i s i m→ =  , we have 1 1 ( ) ( ) ( ) 0 m n i i j ji j y z w   = = + +   for , ( ), ( ), ( ) i i j j x d y f x z g x w h x    . hence, system 2 has a solution ( , , )   . conversely, suppose that system 2 has a solution ( , , )   with 0  . if system 1 has a solution x d  , there would exist ( ), ( ), ( ) i i j j y f x z g x w h x   such that int , , 0 i i j y y z z w + + − − = . thus, ( ) 0, ( ) 0, 0 i i j j y z w    = , i.e., 0)()()( 11 ++  == n j jj m i ii wzy  . which is a contradiction and hence system 1 does not have a solution. theorem 3.1 [fritz john type necessary optimality condition] assume that the generalized convexity assumption (a1) is satisfied and either (a2) or (a3) holds. if x f is a weakly efficient solution of (vp) with ( )y f x , nonzero vector * * * 1 1 ( , , ) m n i i j j y z w   = =    such that 1 1 1 ( ) min[ ( ( )) ( ( )) ( ( ))] min ( ( )) 0, m n i i j ji jx d m i ii y f x g x h x g x      = = = = + + =    where ( )1 1 min ( ( )) : min ( ) i i m m i i z g x i ii i g x z  = = =  . proof. since x f is a weakly efficient solution of (vp) with ( )y f x , by definition the following system 10 int. j. anal. appl. (2023), 21:63 , ( ( ) ) ( int ) , ( ) ( ) , 0 ( ) i i j x d f x y y g x z h x + +  − −   −    has no solution. by proposition 2.2, there exists a nonzero vector * * * 1 1 ( , , ) m n i i j j y z w   = =    such that x d  there holds 1 1 ( ( ) ) ( ( )) ( ( )) 0 m n i i j ji j f x y g x h x   = = − + +   . since x f , there exists ( ) i i z g x such that i i z z + − . for such i z , it follows * i i z +  that ( ) 0 i i z  . on the other hand, taking x x= we get 1 1 ( ( ) ) ( ) ( ( )) 0 m n i i j ji j f x y z h x   = = − + +   , and noticing that ( )y f x and 0 ( ) j h x we obtain 1 ( ) 0 m i ii z =  , and hence ( ) 0 i i z = . since 1 1 ( ) ( ) (0) ( ) m n i i ji j y z y    = = + + =  , taking x x= again we get 1 1 ( ( ) ) ( ( )) ( ( )) 0 m n i i j ji j f x y g x h x   = = − + +   . noticing that ( )y f x and 0 ( ) j h x , we obtain 1 ( ( )) 0 m i ii g x =  . we have shown previously that there exists ( ) i i z g x such that ( ) 0 i i z = . therefore 1 min ( ( )) 0 m i ii g x = = . theorem 3.2 (sufficient optimality condition) let x f and ( )y f x . if there exists a * * * 1 1 ( , , ) m n i i j j y z w   = =    with 0  such that 1 1 ( ) min[ ( ( )) ( ( )) ( ( ))] m n i i j ji jx d y f x g x h x    = =  + +  , then x is a weakly efficient solution of (vp) with ( )y f x . proof. by contradiction, we assume that x f is not a weakly efficient solution of (vp) with ( )y f x . then by definition,  0 x f and  0 0 ( )y f x such that 0 inty y y + −  , which implies 11 int. j. anal. appl. (2023), 21:63 that 0 ( ) 0y y −  . since 0 x f , 0 0 ( ) j h x and  0 0 ( ) i i z g x such that 0 i i z z + − , and hence 0 ( ) 0 i i z  . consequently, 0 0 1 1 ( ) ( ) (0) 0 m n i i ji j y y z   = = − + +   . hence x is a weakly efficient solution of (vp) with ( )y f x . from theorem 3.2 and 3.3 one has theorem 3.3. theorem 3.3 (strong duality) suppose all assumptions in theorem 3.1 hold and there is no nonzero vector ( , ) m n r r  +   satisfying the system: 1 1 min[ ( ) ( )] 0 ( ) 0. m n i i j ji jx d i i g x h x g x    = = + = =   let x be a solution of problem (p). then the strong duality holds. that is, 1 1( ) 0, ( ) 0, 0 ( ) min ( ) max min[ ( ) ( ) ( )]. m n i i j ji jg x h x x d x d f x f x f x g x h x    = = =   = = + +  4. applications to single-valued optimization problems consider the optimization problem: (p) min f (x) s.t. ( ) 0 i g x  (i = 1, 2, …, m) ( ) 0, ( 1, 2, j h x j= = ,n) x d where f, gi, hj: x r→ are functions and d is a nonempty subset of x. applying theorem 3.1 to the above single-valued optimization problem we have the following fritz john type necessary optimality condition. theorem 4.1 let x be an optimal solution of (p). suppose the following generalized convexity assumption holds:  0,( 0,1, 2, , )iu i m =  such that 1 2,x x d  , 0  , [0,1]  ,  3 x d ,  0, ( 1, 2, , ) i i m  =  ,  ,0jt ( 1, 2, , )j n=  there holds 0 1 2 0 3 1 2 3 1 2 3 ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) ( ) (1 ) ( ) ( ) i i i i i j j j j u f x f x f x r u g x g x g x r h x h x t h x           + + + + −  + + + −  + + − = ( 1, 2, ,i m=  ; 1, 2, , )j n=  . 12 int. j. anal. appl. (2023), 21:63 then, nonzero vector ( , , ) m n r r r   + +    such that 1 1 1 ( ) min[ ( ) ( ) ( )] min ( ) 0. m n i i j ji jx d m i ii f x f x g x h x g x      = = = = + + =    we now study some cases where the generalized convexity holds and consequently the fritz john condition in the above theorem holds. theorem 4.2 let x be an optimal solution of (p). suppose one of the following set of assumptions hold. (i) all functions gi are nonnegative on the set d and n = 0 (i.e. there is no equality constraints). (ii) all functions f, gi are nonnegative on the set d and n = 1. then, non-zero vector ( , , ) m n r r r   + +    such that 1 1 1 ( ) min[ ( ) ( ) ( )] min ( ) 0. m n i i j ji jx d m i ii f x f x g x h x g x      = = = = + + =    proof. from theorem 4.1, it suffices to prove that the generalized convexity assumption holds. first assume that assumption (i) holds. let 1 2 ,x x d , [0,1]  . case 1: 1 2 ( ) ( )f x f x .then 1 2 1 2 1 ( ) (1 ) ( ) ( ) (1 )( ( ) ( )) 0.f x f x f x f x f x  + − − = − −  let 3 1 x x= . then 1 2 3 ( ) (1 ) ( ) ( ) .f x f x f x r  + + −  + (1) since g is nonnegative on set d, for small enough (0, ]  one has 1 2 1 1 2 ( ) (1 ) ( ) ( ) ( ) ( ) (1 ) ( ) 0.g x g x g x g x g x     + − − = − + −  that is, 1 2 3 ( ) (1 ) ( ) ( ) m g x g x g x r   + + −  + . (2) case 2: 1 2 ( ) ( )f x f x . in this case by choosing 3 2 x x= similarly as in case 1 we can prove (1) and (2). hence the generalized convexity assumption holds. now assume that assumption (ii) holds. let 1 2 ,x x d and [0,1]  . if h( 2 x ) = 0 then 1 2 1 ( ) (1 ) ( ) ( )h x h x h x  + − = . 13 int. j. anal. appl. (2023), 21:63 let 3 1 x x= . then since f, gi are nonnegative, similarly as in (i) one can find a small enough 0 0  and 1 0  such that 1 2 0 3 ( ) (1 ) ( ) ( ) .f x f x f x r   + + −  + (3) 1 2 1 3 ( ) (1 ) ( ) ( ) . m g x g x g x r   + + −  + (4) otherwise if h( 2 x )  0, then one can find 2 0  such that 1 2 2 2 ( ) (1 ) ( ) ( )h x h x h x  + − = . let 3 2 x x= . then since f, gi are nonnegative, similarly as in (i) one can find a small enough 0 0  and 1 0  such that (3) and (4) hold. hence the generalized convexity assumption holds. theorem 4.3 (kuhn-tucker type necessary optimality condition) let x be an optimal solution of (p). suppose all assumptions in theorem 4.2 hold and there is no nonzero vector ( , ) m n r r  +   satisfying the system: 1 1( ) min [ ( ) ( )] 0 ( ) 0. m n i i j ji jx d u x i i g x h x g x    = = + = =   where )(xu is a neighbourhood of x , then,  nm rr  +),(  such that 1 1( ) ( ) min [ ( ) ( ) ( )] ( ) 0. m n i i j ji jx d u x i i f x f x g x h x g x    = = = + + =   5. conclusion remark yang, yang and chen [1] defined the following generalized subconvexlike functions. ([1] introduced subconvexlikeness for vector-valued optimization). let y be a topological vector space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is said to be generalized y + -subconvexlike on d if intu y +   , such that 1 2 ,x x d  , 0  , [0,1]  , 3 x d  , 0  there holds 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y    + + + −  + . (5) and zeng [2] introduced the presubconvexlikeness as follows. 14 int. j. anal. appl. (2023), 21:63 let y be a topological vector space and d x be a nonempty set and y + be a convex cone in y. a set-valued map f : d y→ is said to be y + -presubconvexlike on d if bounded set-valued map u: d y→ such that 1 2 ,x x d  , [0,1]  , 3 x d  , 0  , 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x y    + + + −  + . (6) the inclusions (5) and (6) may be written as 1 2 3 ( ) (1 ) ( ) ( )u f x f x f x   + + − , by the partial order induced by the convex cone y + . in this paper, we proved that the above two generalized convexities are equivalent. and then, we worked with nonconvex set-valued optimization problems and attained some optimality conditions. our fritz john type necessary optimality condition (theorem 3.1) and kuhn-tucker type necessary optimality condition (theorem 4.3) extend the classic results in clarke [14]. our proposition 3.1 are modifications of the alternative theorems in [15, 16]. our theorem 3.2 (sufficient optimality condition) extends theorem 23 in [9]. our strong duality theorem (theorem 3.3) extends theorem 7 in li and chen [8]. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] x.m. yang, x.q. yang, g.y. chen, theorems of the alternative and optimization with set-valued maps, j. optim. theory appl. 107 (2000), 627–640. https://doi.org/10.1023/a:1026407517917. [2] r. zeng, a general gordan alternative theorem with weakened convexity and its application, optimization. 51 (2002), 709–717. https://doi.org/10.1080/0233193021000031615. [3] h.w. corley, existence and lagrangian duality for maximizations of set-valued functions, j. optim. theory appl. 54 (1987) 489–501. https://doi.org/10.1007/bf00940198. [4] e. klein, a.c. thompson, theory of correspondences, wiley, new york, 1984. [5] z. li, a theorem of the alternative and its application to the optimization of set-valued maps, j. optim. theory appl. 100 (1999), 365–375. https://doi.org/10.1023/a:1021786303883. [6] y.w. huang, a farkas-minkowski type alternative theorem and its applications to set-valued equilibrium problems, j. nonlinear convex anal. 3 (2002), 17-24. [7] o. ferrero, theorems of the alternative for set-valued functions in infinite-dimensional spaces, optimization. 20 (1989), 167–175. https://doi.org/10.1080/02331938908843428. https://doi.org/10.1023/a:1026407517917 https://doi.org/10.1080/0233193021000031615 https://doi.org/10.1007/bf00940198 https://doi.org/10.1023/a:1021786303883 https://doi.org/10.1080/02331938908843428 15 int. j. anal. appl. (2023), 21:63 [8] z.f. li, g.y. chen, lagrangian multipliers, saddle points, and duality in vector optimization of setvalued maps, j. math. anal. appl. 215 (1997), 297–316. https://doi.org/10.1006/jmaa.1997.5568. [9] a. oussarhan, i. daidai, necessary and sufficient conditions for set-valued maps with set optimization, abstr. appl. anal. 2018 (2018), 5962049. https://doi.org/10.1155/2018/5962049. [10] j. borwein, multivalued convexity and optimization: a unified approach to inequality and equality constraints, math. program. 13 (1977), 183–199. https://doi.org/10.1007/bf01584336. [11] k. fan, minimax theorems, proc. natl. acad. sci. u.s.a. 39 (1953), 42–47. https://doi.org/10.1073/pnas.39.1.42. [12] g.y. chen, w.d. rong, characterizations of the benson proper efficiency for nonconvex vector optimization, j. optim. theory appl. 98 (1998), 365–384. https://doi.org/10.1023/a:1022689517921. [13] k. yosida, functional analysis, 6th edition, springer, 1980. [14] f.h. clarke, optimization and nonsmooth analysis, wiley-interscience, new york, 1983. [15] t.d. chuong, robust alternative theorem for linear inequalities with applications to robust multiobjective optimization, oper. res. lett. 45 (2017), 575–580. https://doi.org/10.1016/j.orl.2017.09.002. [16] m. ruiz galán, a theorem of the alternative with an arbitrary number of inequalities and quadratic programming, j. glob. optim. 69 (2017), 427–442. https://doi.org/10.1007/s10898-017-0525-x. https://doi.org/10.1006/jmaa.1997.5568 https://doi.org/10.1155/2018/5962049 https://doi.org/10.1007/bf01584336 https://doi.org/10.1073/pnas.39.1.42 https://doi.org/10.1023/a:1022689517921 https://doi.org/10.1016/j.orl.2017.09.002 https://doi.org/10.1007/s10898-017-0525-x international journal of analysis and applications volume 18, number 3 (2020), 409-420 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-409 on the solutions of falkner-skan equation labbaoui fatma1,2,∗, and aiboudi mohammed1,2, 1département de mathématique, faculté des sciences exactes et appliquées, université oran 1 ahmed ben bella, oran, algérie 2laboratoire de recherche d’analyse mathématique et application l.a.m.a, oran, algerie ∗corresponding author: fatimazahra107@yahoo.fr abstract. we consider the differential equation f ′′′ +ff ′′ +β ( f′2 − 1 ) = 0, with β > 0. in order to prove the existence of solutions satisfying the boundary conditions f (0) = a ≥ 0, f ′ (0) = b ≥ 0 and f ′ (+∞) = −1 or 1 for 0 < β ≤ 1 2 . we use shooting technique and consider the initial conditions f (0) = a, f ′ (0) = b and f ′′ (0) = c. we prove that there exists an infinitely many solutions such that f ′ (+∞) = 1. 1. introduction in 1931 the falkner-skan equation is introduced for studying the boundary layer flow past a semi infinite wedge, it is defined by f ′′′ + ff ′′ + β ( f ′2 − 1 ) = 0 (1.1) the solution of this equation have been studied by numerous authors as, for example, d. r. hartree (1937), h. weyl (1942), w. a. coppel (1960), p. hartman (1964) and g.c. yang (2003,2004). the more general equation of (1.1) is f ′′′ + ff ′′ + g ( f ′ ) = 0 (1.2) where g : r −→ r is some function. the solutions obtained are called similarity solutions. received february 12th, 2020; accepted march 9th, 2020; published may 1st, 2020. 2010 mathematics subject classification. 76d10, 34b15. key words and phrases. boundary layer; shooting technique; convex solution; concave solution; convex-concave solution. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 409 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-409 int. j. anal. appl. 18 (3) (2020) 410 the most famous example is perhaps the blasius equation (1908), which corresponds to g (x) = 0 and arises in the study of laminar boundary layer on a flat plate. more recently, the equation (1.2) with g (x) = βx2 and g (x) = βx (x− 1) has been considered. these cases occur, for example, in the study of free convection and of mixed convection boundary layer flows over a vertical surface embedded in a porous medium. most of the time, associated with the equation (1.1) is the boundary value problem:  f ′′′ + ff ′′ + β ( f ′2 − 1 ) = 0 f (0) = a f ′ (0) = b f ′ (t) −→ λ as t −→ +∞ (pβ;a,b,λ) to solve the boundary value problem (pβ; a, b, λ) we will use the shooting technique. to this end, let fc denote the solution of the initial value problem (qβ; a, b, c) consisting in the equation (1.1) together with the initial conditions fc (0) = a, f ′ c (0) = b and f ′′ c (0) = c, and let [0, tc[ be the right maximal interval of existence of fc. to obtain a solution of (pβ; a, b, λ) amounts to find a value of c such that tc = +∞ and f ′ c (t) −→ λ as t −→ +∞. we must assume that β ( λ2 − 1 ) = 0 to have solutions, in our case of falkner-skan equation, the only relevant conditions are f ′ c (t) −→−1 or f ′ c (t) −→ 1 as t −→ +∞. in the following, we will study the existence of concave, convex, concave-convex and convex-concave solutions to the boundary value problem (pβ; a, b, −1) and (pβ; a, b, 1) for 0 < β ≤ 12 , a ≥ 0 and b ≥ 0. 2. preliminary results let f be a solution of the equation (1.1) on some interval i, we consider the function hf : i −→ r defined by hf = f ′′ + f ( f ′ − 1 ) . (2.1) this function is obtained by integrating the equation (1.1), in fact, if f is a solution of (1.1) then h ′ f = ( f ′ − 1 )( (1 −β) f ′ −β ) . we give lemmas that will be useful later. lemma 2.1. let f be a solution of (1.1) on some maximal interval i. if there exists t0 ∈ i such that f ′ (t0) ∈{−1, 1} and f ′′ (t0) = 0, then i = r and f ′′ (t) = 0 for all t ∈ r. proof. see [4], proposition 3.1 item 3. � lemma 2.2. let β > 0 and f be a solution of equation (1.1) on some interval i, such that f ′ is not constant. int. j. anal. appl. 18 (3) (2020) 411 (1) if there exists s < r ∈ i such that f ′′ (s) ≤ 0 and ( f ′2 − 1 ) > 0 on ]s,r[, then f ′′ (t) < 0 for all t ∈ ]s, r] . (2) if there exists s < r ∈ i such that f ′′ (s) ≥ 0 and ( f ′2 − 1 ) < 0 on ]s, r[, then f ′′ (t) > 0 for all t ∈ ]s, r] . (3) if there exists s < r ∈ i such that f ′′ < 0 on ]s, r[ and f ′′ (r) = 0, then ( f ′2 (r) − 1 ) < 0. (4) if there exists s < r ∈ i such that f ′′ > 0 on ]s, r[ and f ′′ (r) = 0, then ( f ′2 (r) − 1 ) > 0. proof. let f denote any primitive function of f. from (1.1) we deduce the relation ( f ′′ exp f )′ = −β ( f ′2 − 1 ) exp f. all the assertions 1-4 follow easily from this relation and from previous lemma. let us verify the first and the third of these assertions. for the first one, since ψ = f ′′ exp f is decreasing on [s, r], we obtain t ≥ s =⇒ ψ (t) ≤ ψ (s) =⇒ f ′′ (t) exp f (t) ≤ f ′′ (s) exp f (s) =⇒ f ′′ (t) ≤ f ′′ (s) exp (f (s) −f (t)) =⇒ f ′′ (t) ≤ 0, ∀t ∈ ]s, r] . for the third one, since ψ < 0 on ]s, r[ and ψ (r) = 0, ψ ′ (t) ≥ 0 on [s, r], then ψ ′ (r) ≥ 0. ψ ′ (r) = −β ( f ′2 (r) − 1 ) exp f ≥ 0. this and lemma 2.1 imply that ( f ′2 (r) − 1 ) < 0. � lemma 2.3. let f be a solution of (1.1) on some maximal interval ]t−, t+[ . if t+ is finite, then f ′ and f ′′ are unbounded in any neighborhood of t+. proof. see [4], proposition 3.1 item 6. � lemma 2.4. let β 6= 0. if f is a solution of (1.1)on some interval ]τ, + ∞[ such that f ′ (t) −→ λ as t −→ +∞, then λ ∈{−1, 1}. moreover, if f is of constant sign at infinity, then f ′′ (t) −→ 0 as t −→ +∞. proof. see [4], proposition 3.1 item 4 and 5. � lemma 2.5. let β > 0 and f be a solution of equation (1.1) on some right maximal interval i = [τ, + ∞[. if f ≥ 0 and f ′ ≥ 0 on i, then t+ = +∞ and f ′ is bounded on i. int. j. anal. appl. 18 (3) (2020) 412 proof. let l = lf be the function defined on i by l (t) = 3f′′ 2 (t) + βf′ (t) ( 2f′ 2 (t) − 6 ) . (2.2) easily, using (1.1), we obtain that l′ (t) = −6f (t) f′′ 2 (t) ∀t ∈ i, and, since f ≥ 0 on i this implies that l is decreasing. hence ∀t ∈ i = [τ, t+[ : t > τ =⇒ l (t) ≤ l (τ) βf′ (t) ( 2f′ 2 (t) − 6 ) ≤ 3f′′ 2 (t) + βf′ (t) ( 2f′ 2 (t) − 6 ) ≤ l (τ) , ∀t ∈ i. it follows that f ′ is bounded on i and thanks to lemma (2.3) that t+ = +∞. � lemma 2.6. let β > 0 and f be a solution of equation (1.1) on some right maximal interval i = [τ, t+[ . if f (τ) ≥ 0, f ′ (τ) ≥ 1 and f ′′ (τ) > 0, then there exists t0 ∈ ]τ, t+[ such that f ′′ > 0 on [τ, t0[ and f ′′ (t0) = 0. proof. assume for contradiction that f ′′ > 0 on i. then f (t) ≥ 0, f ′ (t) ≥ 1 for all t ∈ i. then, we have f ′′′ = −ff ′′ −β ( f ′2 − 1 ) ≤ 0. (2.3) it follows that 0 < f ′′ (t) ≤ c for all t ∈ i and hence, by lemma (2.3), we have t+ = +∞. next, let s > τ and � = β ( f ′ (s) 2 − 1 ) . one has � > 0 and, comming back to (2.3), we obtain f ′′′ ≤−� on [s, + ∞[ . after integrating, we get ∀t ≥ s, f ′′ (t) −f ′′ (s) ≤−� (t−s) , and a contradiction with the fact that f ′′ (t) > 0. consequently, there exists t0 ∈ ]τ, t+[such that f ′′ > 0 on [τ, t0[and f ′′ (t0) = 0. � lemma 2.7. let β ∈ ] 0, 1 2 [ and f be a solution of equation (1.1) on some right maximal interval i = ]t−, t+[ . if there exists t0 ∈ i such that β1−β < f ′ (t0) < 1 and 0 ≤ f ′′ (t0) ≤ f (t0) ( 1 −f ′ (t0) ) , then t+ = +∞ and f ′ (t) −→ 1 as t −→ +∞. moreover f ′′ > 0 on [t0, + ∞[ . int. j. anal. appl. 18 (3) (2020) 413 proof. let δ = sup x (t0), where x (t0) = { t ∈ ]t0, t+[ : f ′ (t0) < f ′ < 1 and f ′′ > 0 on ]t0, t[ } . the set x (t0) is not empty. this is clear if f ′′ (t0) > 0, and if f ′′ (t0) = 0 it follows from the fact that f ′′′ (t0) = −β ( f ′2 (t0) − 1 ) > 0. we claim that δ = t+, assume for contradiction that δ < t+. from lemma (2.2), item 2, we get that f ′′ (δ) > 0, which implies, by definition of δ, that f ′ (δ) = 1. therefore, since the function hf defined by (2.1) is nonincreasing on [t0, δ] , we obtain δ ≥ t0 =⇒ hf (δ) ≤ hf (t0) =⇒ f ′′ (δ) ≤ f ′′ (t0) −f (t0) ( 1 −f ′ (t0) ) =⇒ f ′′ (δ) < 0, a contradiction. thus, we have δ = t+. from lemma 2.3, it follows that t+ = +∞. since f ′′ > 0 on [t0, + ∞[ , by virtue of lemma 2.4, we get that f ′ (t) −→ 1 as t −→ +∞. � lemma 2.8. let β ∈ ] 0, 1 2 ] and f be a solution of equation (1.1) on some right maximal interval i = ]t−, t+[ . if there exists t0 ∈ i such that f ′ (t0) > 1 and f (t0) ( 1 −f ′ (t0) ) ≤ f ′′ (t0) ≤ 0. then t+ = +∞ and f ′ (t) −→ 1 as t −→ +∞. moreover f ′′ < 0 on [t0, + ∞[ . proof. if we set ω = sup y (t0), where y (t0) = { t ∈ ]t0, t+[ : 1 < f ′ < f ′ (t0) and f ′′ < 0 on ]t0, t[ } . the conclusion will follow by proceeding in the same way as the previous proof. � 3. description of our approach when b ≥ 1 let β > 0, a ≥ 0 and b ≥ 1. as said in the introduction, the method we will use to obtain solutions of the boundary value problems (pβ; a, b, −1) and (pβ; a, b, 1) is the shooting technique. specifically, for c ∈ r, let us denote by fc the solution of equation (1.1) satisfying the initial conditions fc (0) = a, f ′ c (0) = b and f ′′ c (0) = c, (3.1) and let [0, tc[ be the right maximal interval of existence of fc. hence, finding a solution of one of the problems (pβ; a, b, −1) and (pβ; a, b, 1) amounts to finding a value of c such that t+ = +∞ and f ′ c (t) −→−1 or 1 as t −→ +∞. int. j. anal. appl. 18 (3) (2020) 414 to this end, let us partition r into the four sets c0, c1, c2 and c3 defined as follows. let c0 = ]0, + ∞[ and, according to the notations used in [4], let us set c1 = { c ≤ 0; 1 ≤ f ′ c ≤ b and f ′′ c ≤ 0 on [0, tc[ } , c2 =   c ≤ 0; ∃ tc ∈ [0, tc[ , ∃ �c > 0 such that f ′ c > 1 on ]0, tc[ , f ′ c < 1 on ]tc, tc + �c[ and f ′′ c ≤ 0 on [0, tc + �c[   , c3 =   c ≤ 0; ∃ rc ∈ [0, tc[ , ∃ ηc > 0 such that f ′′ c < 0 on ]0, rc[ f ′′ c > 0 on ]rc, rc + ηc[ and f ′ c > 1 on ]0, rc + ηc[   . this is obvious that c0, c1, c2 and c3 are disjoint sets and that their union is the whole line of real numbers. thanks to lemma 2.3 and 2.4 if c ∈ c1 then t+ = +∞ and f ′ c (t) −→ 1 as t −→ +∞. in fact, c1 is the set of values of c for which fc is a concave solution of (pβ; a, b, 1). since β > 0, the study done in [4] (specially in section 5.2) says, on the one hand, that c3 = ∅ (which can easily be deduced from lemma 2, item 1) and, on the other hand, that either c1 = ∅ and c2 = ]−∞, 0] , or there exist c∗ ≤ 0 such that c1 = [c ∗, 0] and c2 = ]−∞, c∗[. in addition, from the lemma 5.16 in [4], if β ∈ ] 0, 1 2 ] then we are in the second case and c∗ ≤−a (b− 1). in order to complete the study, let us divide the set c2 into the following two subsets c2,1 = { c ∈ c2 : f ′ c > −1 on [0, tc[ } , c2,2 = { c ∈ c2 : ∃ sc ∈ ]0, tc[ such that f ′ c > −1 on [0, sc[ and f ′ c (sc) = −1 } . and let us give properties of each of them that hold for all β ∈ ] 0, 1 2 ] . lemma 3.1. if c ∈ r such that f ′ c > 0 on [0, tc[, then tc = +∞ and f ′ c is bounded. moreover, if c ≤ 0, then f ′ c ≤ max { b, √ 3 } on [0, + ∞[. proof. let c ∈ r is such that f ′ c > 0 on [0, tc[, then fc ≥ a ≥ 0 on [0, tc[ and thanks to lemma 2.5, it follows that tc = +∞ and f ′ c is bounded. it remains to show that f ′ c ≤ max { b, √ 3 } in the case where c ≤ 0. as in [4], let us define the function lc on [0, + ∞[ by lc (t) = 3f ′′2 c (t) + βf ′ c (t) ( 2f′ 2 c (t) − 6 ) , and since fc ≥ 0, it implies that lc is decreasing. int. j. anal. appl. 18 (3) (2020) 415 if f ′′ c ≤ 0 on ]0, + ∞[, then f ′ c ≤ b. otherwise, there exists t0 such that f ′′ c < 0 on ]0, t0[ and f ′′ c (t0) = 0. by lemma 2.2 item 3, it follows that f ′ c < 1, and thus lc (t0) < 0. then, lc < 0 on ]t0, + ∞[ which implies that f ′ c ≤ √ 3 on ]t0, + ∞[. since f ′ c ≤ b on ]0, t0[, the proof is complete. � proposition 3.1. let c∗ = inf (c1 ∪c2,1). then c∗ is finite. proof. let c ∈ c1 ∪c2,1. by definition of c1 and c2,1, and thanks to lemma 2.3, we have tc = +∞ and 0 < f ′ c < d on [0, + ∞[ where d = max { b, √ 3 } . since ( f ′′ c + fcf ′ c )′ = f ′′′ c + fcf ′′ c + f ′2 c = −β ( f ′2 c − 1 ) + f ′2 c = −βf ′2 c + β + f ′2 c ≤ β + d 2. by integrating, we then have ∀t ≥ 0, f ′′ c (t) + fc (t) f ′ c (t) ≤ c + ab + ( β + d2 ) t. integrating once again, for all t ≥ 0, we get 0 < f ′ c (t) ≤ f ′ c (t) + 1 2 f2c (t) ≤ b + 1 2 a2 + (c + ab) + 1 2 ( β + d2 ) t2. which implies that c ≥−ab− √ (2b + a2) (β + d2). � remark 3.1. as we have seen above, if c1 6= ∅, then c1 = [c∗, 0] and thus c2,1 ⊂ [c∗,c∗[. 4. the case β ∈ ] 0, 1 2 ] and b > 1 in this section we assume that β ∈ ] 0, 1 2 ] , a > 0 and b > 1. proposition 4.1. if c > 0, then tc = +∞ and f′c (t) → 1 as t → +∞. proof. from lemma 2.6, there exists t0 ∈ ]0,tc[ such that f′′c > 0 on [0, t0[ and f ′′ c (t0) = 0. since fc (t0) > 0 and f ′ c (t0) > b > 1. thus fc (t0) ( 1 −f ′ c (t0) ) 6 f ′′ c (t0) = 0, the conclusion follows from lemma 2.8. � remark 4.1. thanks to the previous proposition, we see that fc is a convex-concave solution of (pβ; a, b, 1) for all c > 0. proposition 4.2. there exists c∗≤−a (b− 1) such that c1 = [c∗, 0]. int. j. anal. appl. 18 (3) (2020) 416 proof. if b = 1 then c1 = {0}. if b > 1, as we already said in the previous section, this result is proven in [4] (see corollary 5.13 and lemma 5.16), let us recall briefly the main arguments which where used to get it. on the one hand, from lemma 2.8 with t0 = 0 (or lemma 5.16 of [4]), it follows that [−a (b− 1) , 0] ⊂ c1. on the other hand, lemma 5.12 of [4] implies that c2 is an interval of the type ]−∞,c∗[. this complete the proof since c1 = ]−∞, 0] �c2. � remark 4.2. from the previous proposition, we have that 0 /∈ c2.2. proposition 4.3. if c ∈ c2.1, then tc = +∞ and f′c has a finite limit at infinity, equal either to −1 or to 1. proof. let c ∈ c2.1. by proposition 4.2, we have c < 0. assume first that f ′′ c < 0 on ]0, +∞[. then f ′ c is decreasing, and thus f ′ c has a finite limit λ at infinity. moreover, by definition of the set c2.1 we get ∃ tc ∈ [0, +∞[ such that f ′ c (tc) = 1 and by lemma 2.4, we finally get that λ = −1. assume now that f ′′ c vanishes on ]0, +∞[, let t0 be the first point where f ′′ c vanishes. thanks to lemma 2.2 item 3, we have 0 < f ′ c (t0) < 1, and the conclusion follows from lemma 2.7 (λ = 1). � remark 4.3. if c ∈ c2.1 then either fc is a concave solution of (pβ; a, b, −1) or fc is a concave-convex solution of (pβ; a, b, 1). proposition 4.4. let c be a point of the boundary of c2.2. then c ∈ c2.1 and f ′ c (t) −→−1 as t −→ +∞. proof. see [3] in the case of the mixed convection equation. � proposition 4.5. there exists at most one c such that f ′ c (t) −→−1 as t −→ +∞. proof. from proposition 4.2 and 4.3 , we see that if c is such that f ′ c (t) −→ −1 as t −→ +∞, then c < 0 and f ′′ c < 0. by the change of variable, as done in [4], section 4, we can define a function v : ] 0, b2 ] → r such that ∀ t ≥ 0, v ( f′ 2 c (t) ) = fc (t) . (4.1) by setting y = f′ 2 c (t), we get fc (t) = v (y) , f ′ c (t) = √ y, f′′c (t) = 1 2v′ (y) and f′′′c (t) = − v′′ (y) √ y 2v′ 3 (y) , int. j. anal. appl. 18 (3) (2020) 417 and using (1.1) we obtain ∀ y ∈ ] 0, b2 ] , v′′ (y) = v (y) v′ 2 (y) √ y + 2β (y − 1) √ y v′ 3 (y) 3 . (4.2) from (3.1), we deduce that v ( b2 ) = a and v ′ ( b2 ) = 1 2c . moreover, since fc is bounded, it is so for v. assume that there exists c1 > c2 such that f ′ c1 (t) → 0 and f ′ c2 (t) → 0 as t → +∞, and denote by v1 and v2 the functions associated to fc1 and fc2 by (4.1). if we set w = v1 −v2 then w ( b2 ) = 0 and w ′ ( b2 ) < 0. we claim that w ′ < 0 on ] 0, b2 ] . for contradiction, assume there exists x ∈ ] 0, b2 [ such that w ′ < 0 on ]0, x[ and w ′ (x) = 0. hence we have w ′′ (x) ≤ 0 and w (x) > 0. but thanks to (4.1), we have w′′ (x) = w (x) √ x v′ 2 1 (x) , and a contradiction. now, let us set vi = 1 v′i for i = 1, 2 and w = v1 −v2. then w ( b2 ) = 2 (c1 − c2) > 0 and w (y) → 0 as y → 0. in the other hand, thanks to (4.2); we have ∀ y ∈ ] 0, b2 ] , w ′ (y) = − w (y) √ y − 2β y − 1 √ y w′ (y) . therefore, we have w ( b2 ) = ∫ b2 0 w ′ (y) dy = − ∫ b2 0 ( w (y) √ y + 2β y − 1 √ y w ′ (y) ) dy = −2 [ √ yw (y)] b2 0 + 2 ∫ b2 0 ( (1 −β) √ y + 2β √ y ) w ′ (y) dy = 2 ∫ b2 0 ( (1 −β) √ y + 2β √ y ) w ′ (y) dy, (4.3) the last equality following from the fact that w (y) tends to finite limit as y → 0. since w ′ < 0, we finally obtain w ( b2 ) < 0 and a contradiction. � remark 4.4. the change of variable (4.1) is particularly efficient to obtain some uniqueness results. in [4], it is used for the general equation (1.2) (cf. section 4, lemma 5.4 and lemma 5.17). the case we examined in the previous proposition is part of lemma 5.17 of [4] with λ = −1. corollary 4.1. one has c2.2 = ]−∞, c∗[ and c2.1 = [c∗, c∗[. proof. from remark 4.2, proposition 4.4 and 4.5, we see that c2.2 is open, contains ]−∞, c∗[ . therefore, since c∗ = inf (c1 ∪c2,1), we necessarily have c2.2 = ]−∞, c∗[ and c2.1 = [c∗, c∗[. � int. j. anal. appl. 18 (3) (2020) 418 to finish this section, let us express the results of proposition 4.1, proposition 4.2 and corollary 4.1 in terms of the boundary problems (pβ; a, b, −1) and (pβ; a, b, 1). theorem 4.1. let β ∈ ] 0, 1 2 ] , a ≥ 0 and b ≥ 1. there exists c∗ < 0 such that: (1) fc is not defined on the whole interval [0, + ∞[ if c < c∗; (2) fc∗ is a concave solution of (pβ; a, b, −1); (3) fc is a solution of (pβ; a, b, 1) for all c ∈ ]c∗, + ∞[ moreover, there exists c∗ ∈ ]c∗, −a (b− 1)] ; (1) fc is a convex-concave solution of (pβ; a, b, 1) for all c ∈ ]0, + ∞[ ; (2) fc is a concave solution of (pβ; a, b, 1) for all c ∈ [c∗, 0] ; (3) fc is a concave-convex solution of (pβ; a, b, 1) for all c ∈ ]c∗, c∗[ ; 5. the case β ∈ ] 0, 1 2 ] and −1 < b < 1 let β ∈ ] 0, 1 2 ] , a ≥ 0 and −1 < b < 1. in this situation, it is easy to see that r can be partitioned into the four sets c ′ 0,1, c ′ 0,2, c ′ 1 and c ′ 2 where c ′ 0,1 = { c < 0 : f ′ c > −1 on [0, tc[ } , c ′ 0,2 = { c < 0 : ∃ sc ∈ ]0, tc[ such that f ′ c > −1 on [0, sc[ and f ′ c (sc) = −1 } , c ′ 1 = { c ≥ 0; b ≤ f ′ c ≤ 1 and f ′′ c ≥ 0 on [0, tc[ } , c ′ 2 =   c ≥ 0; ∃ tc ∈ [0, tc[ , ∃ �c > 0 such that f ′ c < 1 on ]0, tc[ , f ′ c > 1 on ]tc, tc + �c[ and f ′′ c > 0 on ]0, tc + �c[   . the arguments used in the previous section, can be applied here. first, since g (x) = β ( x2 − 1 ) < 0 for x ∈ ]−1, b] where b ∈ ]−1, 0[, the function g is nonincreasing on ]−1, b], it follows from theorem 5.5 of [4] that there exists a unique c∗ such that fc∗ is a concave solution of (pβ; a, b, −1). moreover, we have c∗ < 0. as in the previous section, this implies that c ′ 0,2 = ]−∞, c∗[. hence c ′ 0,1 = [c∗, 0[, and if c ∈ ]c∗, 0[, then f ′′ c vanishes at a first point where f ′ c < 1. next, in the same way as in the proof of proposition 3.1, we can proof that c∗ = sup c ′ 1 is finite, and hence that c ′ 1 = [0, c ∗] and c ′ 2 = ]c ∗, + ∞[. moreover, from lemma 2.7, we have c∗ ≥ a (1 − b). on the other hand, it follows from lemma 2.6 that, if c ∈ c ′ 2, then f ′′ c vanishes at a first point where f ′ c > 1. all this, combined with an appropriate use of lemmas 2.7 and 2.8 allows to state the following theorem. int. j. anal. appl. 18 (3) (2020) 419 theorem 5.1. let β ∈ ] 0, 1 2 ] and −1 < b < 1. there exist c∗ < 0 and c∗ ≥ a (1 − b) such that: (1) fc is not defined on the whole interval [0, + ∞[ if c < c∗; (2) fc∗ is a concave solution of (pβ; a, b, −1) if b ∈ ]−1, 0[ ; (3) fc is a concave-convex solution of (pβ; a, b, 1) for all c ∈ ]c∗, 0[ ; (4) fc is a convex solution of (pβ; a, b, 1) for all c ∈ [0, c∗] ; (5) fc is a convex-concave solution of (pβ; a, b, 1) for all c ∈ ]c∗, + ∞[ . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. aiboudi and b. brighi, on the solutions of a boundary value problem arising in free convection with prescribed heat ux, arch. math. 93(2), 2009, 165-174. [2] m. aiboudi, k. boudjema djeffal, and b. brighi, on the convex and convex-concave solutions of opposing mixed convection boundary layer flow in a porous medium, abstr. appl. anal. 2018 (2018), article id 4340204, 5 pages. [3] m. aiboudi, i. bensari and b. brighi, similarity solutions of mixed convection boundary-layer flows in a porous medium, differ. equ. appl. 9(1)(2017), 69-85. [4] b. brighi, the equation f ′′′ +ff ′′ +g ( f ′ ) = 0 and the associated boundary value problems, results math. 61(3-4)(2012), 355-391. [5] b. brighi, a. fruchard and t. sari, on the blasius problem, adv. differ. equ, 13(5-6), 2008, 509-600. [6] b. brighi and j.d. hoernel, on similarity solutions for boundary layer ows with prescribed heat flux, math. methods appl. sci. 28(4)(2005), 479-503. [7] b. brighi and j.d. hoernel, on the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, appl. math. lett. 19(2006), 69-74. [8] b. brighi and j-d. hoernel, on a general similarity boundary layer equation, acta math. univ. comenian. 77(1)(2008), 9-22. [9] d. r. hartree, on an equation occurring in falkner and skan’s approximate treatment of the equations of the boundary layer, math. proc. cambridge phil. soc. 33(2)(1937), 223-239. [10] g.c. yang, a note on f ′′′ + ff ′′ + β ( f ′2 − 1 ) = 0 with λ ∈ ( −1 2 , 0 ) arising in boundary layer theory, appl. math. lett. 17(2004), 1261-1265. [11] g.c. yang and j. li, a new result on f ′′′ + ff ′′ + β ( f ′2 − 1 ) = 0 arising in boundary layer theory, nonlinear funct. anal. appl. 10(2005), 117-122. [12] g.c. yang, l.l. shi and k.q. lan, properties of positive solutions of the falkner–skan equation arising in boundary layer theory, in: integral methods in science and engineering, birkhäuser, boston, boston, ma, 2008, 277–283. [13] g.c. yang, existence of solutions of laminar boundary layer equations with decelerating external flows, nonlinear anal., theory methods appl. 72(2010), 2063-2075. [14] g.c.yang, an extension result of the opposing mixed convection problem arising in boundary layer theory, appl. math. lett. 38(1)(2014), 180-185. [15] h. weyl, on the differential equations of the simplest boundary-layer problems, ann. math. 43(2)(1942), 381-407. int. j. anal. appl. 18 (3) (2020) 420 [16] w. a. coppel, on a differential equation of boundary-layer theory, phil. trans. r. soc. lond. ser. a, math. phys. eng. sci. 253(1023)(1960), 101-136. [17] p. hartman, ordinary differential equations, johnwiley & sons, new york, ny, usa, 1964. 1. introduction 2. preliminary results 3. description of our approach when b1 4. the case ] 0,12] and b1 5. the case ] 0, 12] and -1<b<1 references int. j. anal. appl. (2022), 20:4 exact and sinusoidal periodic solutions of lienard equation without restoring force jean akande, kolawolé kêgnidé damien adjaï, ayéna vignon régis yehossou, marc delphin monsia∗ department of physics, university of abomey-calavi, abomey-calavi, 01.bp.526, cotonou, benin ∗corresponding author: monsiadelphin@yahoo.fr abstract. in this paper we study a lienard equation without restoring force. although this equation does not satisfy the classical existence theorems, we show, for the first time, that such an equation can exhibit harmonic periodic solutions. as such the usual existence theorems are not entirely adequate and satisfactory to predict the existence of periodic solutions. 1. introduction the lienard equation ẍ +ϑ(x)ẋ +h(x)=0 (1.1) where overdot denotes derivative with respect to time, ϑ(x) and h(x) are arbitrary functions of x, is a generalization of the conservative equation ẍ +h(x)=0 (1.2) where h(x) is a restoring force function, to take into account energy dissipation in real world dynamical systems. as such, the use of equation (1.2) constitutes a first approximation to model oscillations in dynamical systems so that the determination of periodic solutions of equation (1.1) has become a fundamental problem in mathematics and physics. in this way, a rich and various study has been carried out on the existence of periodic solutions of equation (1.1). one can find a vast literature on received: nov. 11, 2021. 2010 mathematics subject classification. 34a34, 34c25, 34a05, 34a12. key words and phrases. lienard equation; exact periodic solution; harmonic and isochronous oscillations; restoring force; existence theorems. https://doi.org/10.28924/2291-8639-20-2022-4 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-4 2 int. j. anal. appl. (2022), 20:4 the theorems of existence and uniqueness ( [1], [2], [3], [4], [5], [6]) of periodic solutions of equation (1.1). however, if the limit cycle of the van der pol equation ( [1], [2], [3], [4]) ẍ +β(x2 −1)ẋ +x =0 (1.3) of type (1.1), where β is an arbitrary constant, is well studied, one can then notice that exact and explicit periodic solutions of equation (1.1) for specific expressions of ϑ(x) and h(x) are less obtained in the literature. a very limited number of authors have been able to find explicitly exact periodic solutions to equation (1.1) since its statement nearly a century ago, to model electrical circuits. in [7] chandrasekar and his group investigated the case ẍ +αxẋ +γx3 +λx =0 (1.4) to extract for the first time, exact and explicit harmonic and isochronous periodic solutions, so that the authors qualified this equation of an unusual lienard type nonlinear oscillator. thus a lot of publications can be found on this equation from various mathematical points of view. in [8] the authors investigated a more general form of equation (1.4). in [9] the lie point symmetries of the general class (1.1) are discussed. in a recent paper [10] doutètien and coworkers studied analytically equation (1.4). their study shows without ambiguity that the type of equation (1.4) investigated by chandrasekar et al. [7] can exhibit unbounded periodic solutions. in [11] a lienard nonlinear oscillator of type (1.1) is presented to represent periodic and isochronous oscillations in dynamical systems. the authors in [12] succeeded in exhibiting harmonic and isochronous periodic solution of a lienard equation of type (1.1) using the first integral approach. in [13] the authors have successfully calculated a general sinusoidal function with isochronicity property for the exact and explicit general solution of an equation of type (1.1). the above shows the importance to investigate the lienard type equation (1.1), as periodic solutions constitute an interesting and special class of solutions of differential equations. the theory of existence conditions of periodic solutions of equations of type (1.1) states ( [1], [2], [3], [4], [5], [6]) often the requirement xh(x) � 0, for x 6= 0, that is to say equation (1.1) with h(x) = 0, does not satisfy these conditions. in other words, h(x) must be odd, so that the theorems of existence for periodic solutions of equation (1.1) eliminate the case where the restoring force h(x)=0, for all x. it seems that the question of existence of periodic solutions of such equations of the form ẍ +ϑ(x)ẋ =0 (1.5) has not been formulated and solved in any study published to date explicitly. nevertheless, it is reasonable to investigate such a question due to the results obtained previously in ( [14], [15]). indeed, the authors in ( [14], [15]) succeeded in proving explicitly the existence of harmonic and isochronous periodic solutions of equations of type int. j. anal. appl. (2022), 20:4 3 ẍ +σ(x, ẋ)ẋ =0 (1.6) where σ(x, ẋ) denotes the damping, as a function of x and ẋ, and the restoring force is null, whereas equation (1.6), as such, can be viewed as a more general form of equation (1.5). it was for the first time such a result is reached for the study of a lienard type equation without restoring force in mathematics and physics. in this regard, the objective in the present paper is to show explicitly the existence of sinusoidal and isochronous periodic solutions of equation (1.5) under appropriate choice of function ϑ(x). for this purpose, the required theory is stated (section 2) so that we can present the equation of interest and solve it explicitly (section 3).the phase plane and existence theorem analysis is carried out (section 4) and a conclusion is given finally for the work. 2. theory let us consider the lienard type equation ( [14], [15], [16]) ẍ + g′(x) g(x) ẋ2 +a`x`−1 f (x) g(x) ẋ +ax` f ′(x) g(x) ẋ =0 (2.1) where prime means differentiation with respect to x, a and ` are arbitrary parameters, and f (x) and g(x) 6=0 are arbitrary functions of x. equation (2.1) can be written ẍ + g′(x) g(x) ẋ2 + ax` g(x) (` f (x) x + f ′(x))ẋ =0 (2.2) when x 6=0 . for ` =0, equation (2.2) becomes ẍ + g′(x) g(x) ẋ2 +a f ′(x) g(x) ẋ =0 (2.3) under the requirement g(x)=1, equation (2.3) reduces to ẍ +af ′(x)ẋ =0 (2.4) equation (2.4) is equivalent to the system ẋ = y, ẏ =−af ′(x)y (2.5) which can lead to dy dx =−af ′(x) so that one can get dy =−af ′(x)dx (2.6) that is, the integral curves 4 int. j. anal. appl. (2022), 20:4 y =−af (x)+c (2.7) where c is a constant of integration. the problem to solve becomes now the choice of function f (x) that reduces equation (2.7) to that of an ellipse centered at the origin if we expect to have harmonic periodic solution. 3. equation of interest let f (x) = √ µ2 −x2, where µ is an arbitrary constant. then, equation (2.4) turns into the equation of interest ẍ − ax√ µ2 −x2 ẋ =0 (3.1) where h(x) = 0. using equation (2.7), the solution of equation (3.1) is given by the quadrature, as expected ( [14], [15], [16]), defined in the form −a(t +k)= ∫ dx f (x) (3.2) where k is an arbitrary constant and c = 0. from the expression of f (x), equation (3.2) takes the form −a(t +k)= ∫ dx√ µ2 −x2 (3.3) so that, after integration, one can obtain sin−1( x µ )=−a(t +k) (3.4) in this context, by inverting, the exact harmonic and isochronous periodic solution of equation (3.1) is secured as x(t)= µsin(−a(t +k)) (3.5) where µ � 0 and a ≺ 0. solution (3.5) shows that equation (3.1) and the linear harmonic oscillator ẍ +(−a)2x =0 (3.6) with amplitude of oscillations µ � 0, and angular frequency ω = −a , have identical solutions. from physical point of view, the hamiltonian of oscillator (3.1) can, using equation (2.7), reads h = 1 2 y2 + 1 2 a2x2 (3.7) as h is a constant, the damped equation (3.1) is then, a conservative nonlinear system. int. j. anal. appl. (2022), 20:4 5 4. phase plane and existence theorem analysis according to equation (2.5), equation (3.1) is equivalent to the dynamical system ẋ = y, ẏ = axy√ µ2 −x2 (4.1) the equilibrium points in the (x,y) phase plane are given by y = 0, and axy√ µ2−x2 = 0. this means that if y = 0, then x 6= 0, and inversely, if x = 0, then y 6= 0. this is in opposition to results (3.5) and (3.7) showing that the origin is a center, that is a single equilibrium point. according, for example, to theorem 11.3 of the book [ [2], p.390] equation (3.1) has a center at the origin when h(x) � 0, for x � 0, and is odd. since h(x) = 0, then equation (3.1) cannot have a center at the origin. as previously, this prediction of the preceding existence theorem is in contradiction with the analytical results (3.5) and (3.7). as seen, the classical existence theorems exclude some cases of lienard nonlinear differential equations while the exact and explicit results show the existence of harmonic and isochronous periodic solutions. now, we can address a conclusion for the work. 5. conclusion in this paper a lienard equation without restoring force is studied. although the equation does not satisfy the usual existence theorems, we have successfully shown that it can exhibit sinusoidal and isochronous periodic solution as the linear harmonic oscillator. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. sandqvist, k. andersen, existence and uniqueness results for lienards equation having a dead band, ieee trans. circuits syst. 27 (1980), 1251–1254. https://doi.org/10.1109/tcs.1980.1084775. [2] d.w. jordan, p. smith, nonlinear ordinary differential equations: an introduction for scientists and engineers, 4th ed, oxford university press, oxford, 2007. [3] m. cioni, g. villari, an extension of dragilev’s theorem for the existence of periodic solutions of the liénard equation, nonlinear anal.: theory methods appl. 127 (2015), 55–70. https://doi.org/10.1016/j.na.2015.06.026. [4] g. villari, f. zanolin, on the qualitative behavior of a class of generalized liénard planar systems, j. dyn. differ. equ. (2021). https://doi.org/10.1007/s10884-021-09984-2. [5] s. strogatz, nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, repr., westview press, cambridge, mass, 2007. [6] m. sabatini, on the period function of liénard systems, j. differ. equ. 152 (1999), 467–487. https://doi.org/ 10.1006/jdeq.1998.3520. [7] v.k. chandrasekar, m. senthilvelan, m. lakshmanan, unusual liénard-type nonlinear oscillator, phys. rev. e. 72 (2005), 066203. https://doi.org/10.1103/physreve.72.066203. [8] r. iacono, f. russo, class of solvable nonlinear oscillators with isochronous orbits, phys. rev. e. 83 (2011), 027601. https://doi.org/10.1103/physreve.83.027601. https://doi.org/10.1109/tcs.1980.1084775 https://doi.org/10.1016/j.na.2015.06.026 https://doi.org/10.1007/s10884-021-09984-2 https://doi.org/10.1006/jdeq.1998.3520 https://doi.org/10.1006/jdeq.1998.3520 https://doi.org/10.1103/physreve.72.066203 https://doi.org/10.1103/physreve.83.027601 6 int. j. anal. appl. (2022), 20:4 [9] s. n. pandey, p. s. bindu, m. senthilvelan, m. lakshmanan, a group theoretical identification of integrable cases of the liénard-type equation ẍ+f (x)ẋ+g(x)=0. i. equations having nonmaximal number of lie point symmetries, j. math. phys. 50 (2009), 082702. https://doi.org/10.1063/1.3187783. [10] e.a. doutetien, a.b. yessoufou, y.j.f. kpomahou, m.d. monsia, unbounded periodic solution of a modified emden type equation. (2021). https://vixra.org/abs/2101.0002. [11] y.j.f. kpomahou, m. nonti, a.b. yessoufou, m.d. monsia, on isochronous periodic solution of a generalized emden type equation. (2021). https://rxiv.org/abs/2101.0024. [12] m. nonti, k.k.d. adjaï, a.b. yessoufou, m.d. monsia, on sinusoidal and isochronous periodic solution of dissipative lienard type equations. (2021). https://vixra.org/abs/2101.0142. [13] k.k.d. adjaï, a.b. yessoufou, j. akande, m.d. monsia, sinusoidal and isochronous oscillations of dissipative lienard type equations. (2021). https://vixra.org/abs/2101.0161. [14] j. akande, a.v.r. yehossou, k.k.d. adjaï, m.d. monsia, on identical harmonic and isochronous periodic oscillations of lienard type equations. (2021). https://vixra.org/abs/2102.0082. [15] k.k.d. adjaï, m. nonti, j. akande, m.d. monsia, on sinusoidal and isochronous periodic solutions of lienard type equations with only damping. (2021). https://vixra.org/abs/2102.0132. [16] j. akande, a.v.r yehossou, k.k.d. adjaï, m.d. monsia, sinusoidal and isochronous periodic oscillations of lienard type equations without restoring force. (2021). https://vixra.org/abs/2102.0158. https://doi.org/10.1063/1.3187783 https://vixra.org/abs/2101.0002 https://rxiv.org/abs/2101.0024 https://vixra.org/abs/2101.0142 https://vixra.org/abs/2101.0161 https://vixra.org/abs/2102.0082 https://vixra.org/abs/2102.0132 https://vixra.org/abs/2102.0158 1. introduction 2. theory 3. equation of interest 4. phase plane and existence theorem analysis 5. conclusion references international journal of analysis and applications volume 18, number 6 (2020), 1056-1065 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1056 on meromorphic functions defined by a new class of liu-srivastava integral operator syed ghoos ali shah1, saima noor2, maslina darus3,†,∗, wasim ul haq4, saqib hussain1 1department of mathematics, comsats university islamabad, abbottabad campus 22060, pakistan 2department of basic sciences, preparatory year deanship, king faisal university, hofuf 31982 al ahsa, saudia arabia 3department of mathematical sciences, universiti kebangsaan malaysia, bangi 43600, selangor, malaysia 4department of mathematics, abbottabad university of science and technology, abbottabad 22010, pakistan ∗corresponding author: maslina@ukm.edu.my abstract. in this work, we introduce and explore certain new subclasses of meromorphic functions. we aim to study some important properties such as coefficient estimates, growth rate and partial sums for these newly defined subclasses. it is important to mentioned that our results are generalization of number of existing results. 1. introduction let ∑ p denote the class of p-valent meromorphic function of the form: (1.1) λ (ω) = 1 ωp + ∞∑ t=p atω t, received september 5th, 2020; accepted september 30th, 2020; published october 28th, 2020. 2010 mathematics subject classification. primary 30c45, secondary 30c50. key words and phrases. integral operator; meromorphic function; starlike function. †this author is supported by the grant number gup-2019-032. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1056 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1056 int. j. anal. appl. 18 (6) (2020) 1057 which are analytic in the punctured open unit disc u∗ = {ω : ω ∈ c and 0 < {ω} < 1} = u −{0}, where u = u∗ ∪{0}. here we are listing some important subclasses of meromorphic functions which will be used in our subsequal work. in 1936, roberston [23] introduced the classes of meromorphic starlike and meromorphic convex functions of order α. by ∑ms (α) we mean the subclass of ∑ 1 consisting of all meromorphic starlike functions of order α. analytically (1.2) λ (ω) ∈ ms∑ (α) ⇔< ( ωλ ′ (ω) λ (ω) ) < −α, (0 ≤ α < 1; ω ∈ u∗). a closely related class of meromorphic convex functions of order α is denoted by ∑mc (α) and defined as: (1.3) λ (ω) ∈ mc∑ (α) ⇔−ωλ ′ (ω) ∈ ms∑ (α). in 1952, kaplan [16] introduced and studied an important class of analytic functions in the open unit disc u known as close-to-convex functions. a function λ belongs to ∑ 1 is in class ∑mk (α,β), of meromorphic close-to-convex functions of order α and type β, if there exist δ (ω) ∈ ∑ms (β) and (1.4) < ( ωλ ′ (ω) δ (ω) ) < −α. many differential and integral operators can be written in terms of convolution of certain holomorphic functions. let δ (ω) ∈ ∑ p and having series representation of the form (1.5) δ (ω) = 1 ωp + ∞∑ t=0 btω t, then convolution (hadamard product) is denoted by λ∗ δ and defined as: (1.6) (λ∗ δ) (ω) = 1 ωp + ∞∑ t=0 atbtω t = (δ ∗λ) (ω) , where λ (ω) as given by (1.1). following the current work of liu and srivastava [18] (see also [1][6]), now we defined the integral operator given below (1.7) mmp (a,b)λ (ω) = 1 ωp + ∞∑ t=p [ a a + b(p + t) ]m atω t (ab > 0; p ∈ n) . the above integral operator converts into the following operator when p = 1 (1.8) mm1 (a,b)λ (ω) = 1 ω + ∞∑ t=p [ a a + b(1 + t) ]m atω t (a > 0,b ≥ 0,m ∈ n) . it can be easily verified from (1.8) (1.9) λ (ω) (mm1 (a,b)λ (ω)) ′ = amm1 (a,b)λ (ω) − (a + b)m m+1 1 (a,b)λ (ω) (b > 0). for more details see [7–9, 12, 15, 20, 21, 24]. int. j. anal. appl. 18 (6) (2020) 1058 definition 1.1. a function λ (ω) is subordinate to δ (ω) in u and written as: λ (ω) ≺ δ (ω) , if there exists a schwarz function k(ω), which is holomorphic in u∗ with |k(ω)| < 1, such that λ (ω) = δ(k (ω)). furthermore, if the function δ (ω) is univalent in u∗, then we have the following equivalence (see [22]): (1.10) λ (ω) ≺ δ (ω) and λ (u) ⊂ δ (u) . further, λ (ω) is quasi-subordinate to δ (ω) in u∗ and written as: λ (ω) ≺q δ (ω) ( ω ∈ u∗) , if there exist two analytic functions ϕ (ω) and k (ω) in u∗ such that λ(ω) ϕ(ω) is analytic in u∗ and |ϕ (ω)| ≤ 1 and k (ω) ≤ |ω| < 1 ω ∈ u∗, satisfying (1.11) λ (ω) = ϕ (ω) δ (k (ω)) ω ∈ u∗. definition 1.2. for −1 ≤ s < t ≤ 1 the function λ ∈ ∑ p is in the class n m p (a,b; d,s,t) if it satisfies the inequality 1 − 1 d  ω(mmp (a,b)λ (ω))′ mmp (a,b)λ (ω) + 1   ≺ 1 + s (ω) 1 + t (ω) , or, equivalently to: (1.12) ∣∣∣∣∣∣∣ ω(mmp (a,b)λ(ω)) ′ mmp (a,b)λ(ω) + 1 t ω(mmp (a,b)λ(ω)) ′ mmp (a,b)λ(ω) + [|d|(s −t) + t ] ∣∣∣∣∣∣∣ < 1. let ∑∗ p denote the subclass of functions ∑ p consisting of functions of the form: (1.13) λ (ω) = 1 ωp + ∞∑ t=p |at|ωt ( p ∈ n = {1, 2, ...}) . now, we define the class n∗mp (a,b; d,s,t) = n m p (a,b; d,s,t) ∩ ∑∗ p . for recent work on meromorphic functions we refer [10, 11, 13, 14, 17, 19]. motivated, from the above cited work we obtained the following results. 2. main results in this section, in present the work to acquire sufficient conditions in which (1.13) gives the function λ (ω) within the class n∗mp (a,b; d,s,t), as well as demonstrates that this condition is required for function which belong to this class. in our first theorem, we begin with the necessary and sufficient condition for function λ in n∗mp (a,b; d,s,t). we also prove some other related theorems. int. j. anal. appl. 18 (6) (2020) 1059 theorem 2.1. let the function λ (ω) is of the form (1.1). then λ (ω) ∈ n∗mp (a,b; d,s,t) if and only if (2.1) ∞∑ t=p [ a a + b(p + t) ]m |at| [1 + t−{|d|(s −t) + (1 + t) t}] ≤ (1 −p) (t − 1) + |d|(s −t). proof. assuming that (2.1) holds true, we obtain∣∣∣∣∣∣∣ ω(mmp (a,b)λ(ω)) ′ mmp (a,b)λ(ω) + 1 t ω(mmp (a,b)λ(ω)) ′ mmp (a,b)λ(ω) + [|d|(s −t) + t ] ∣∣∣∣∣∣∣ = ∣∣∣∣∣∣ ω ( mmp (a,b)λ (ω) )′ + mmp (a,b)λ (ω) tω ( mmp (a,b)λ (ω) )′ + [|d|(s −t) + t ] mmp (a,b)λ (ω) ∣∣∣∣∣∣ = ∣∣∣∣∣∣∣∣ (1 −p) 1 ωp + ∞∑ t=p [ a a+b(p+t) ]m atω t [(1 −p) t + |d|(s −t)] ωp + ∑∞ t=1 ((t + 1) t + |d|(s −t)) |at|ωt ∣∣∣∣∣∣∣∣ < 1. then, by maximum modulus theorem, we have λ (ω) ∈ n∗mp (a,b; d,s,t). conversely, assume that λ (ω) is in the class n∗mp (a,b; d,s,t) with λ (ω) of the form (1.13), then we find from (1.12) that ∣∣∣∣∣∣ ω ( mmp (a,b)λ (ω) )′ + mmp (a,b)λ (ω) tω ( mmp (a,b)λ (ω) )′ + [|d|(s −t) + t ] mmp (a,b)λ (ω) ∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣ (1 −p) 1 ωp + ∞∑ t=p [ a a+b(p+t) ]m atω t [(1 −p) t + |d|(s −t)] ωp + ∑∞ t=1 ((t + 1) t + |d|(s −t)) |at|ωt ∣∣∣∣∣∣∣∣ < 1, since the above inequality is genuine for all ω ∈ u, let the value of ω on the real axis. letting ω −→ 1− through real values, we get ∞∑ t=p [ a a + b(p + t) ]m |at| [1 + t−{|d|(s −t) + (1 + t) t}] ≤ (1 −p) (t − 1) + |d|(s −t). which complete the proof. � corollary 2.1. if the function λ (ω) is of the form (1.1) is in the class n∗mp (a,b; d,s,t) then |at| ≤ (1 −p) (t − 1) + |d|(s −t) ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] , (t ≥ 1). the result is sharp for the function (2.2) λ (ω) = 1 ωp +   (1 −p) (t − 1) + |d|(s −t)∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}]  ωt. growth and distortion bounds for functions belonging to the class n∗mp (a,b; d,s,t) will be given in the following result: int. j. anal. appl. 18 (6) (2020) 1060 theorem 2.2. if a function λ (ω) given by (1.1) is in the class n∗mp (a,b; d,s,t) then for |ω| = r, we have: 1 rp −   (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}]  r ≤ |λ (ω)| ≤ 1 rp +   (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}]  r,(2.3) and −p |r|p+1 −   (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}]   ≤ ∣∣∣λ′ (ω)∣∣∣ ≤ −p |r|p+1 +   (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}]  (2.4) proof. in view of theorem 2.2, we have[ a a + 2bp ]m [2 −{|d|(s −t) + 2t}] ∞∑ t=p |at| ≤ ∞∑ t=p [ a a + b(p + t) ]m |at| [1 + t−{|d|(s −t) + (1 + t) t}] ≤ (1 −p) (t − 1) + |d|(s −t), which yield ∞∑ t=p |at| ≤ (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}] (t ∈ n). therefore, (2.5) |λ (ω)| ≤ 1 |ω|p + |ω| ∞∑ t=p |at| ≤ 1 |ω|p + |ω| (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}] , and (2.6) |λ (ω)| ≥ 1 |ω|p −|ω| ∞∑ t=p |at| ≤ 1 |ω|p −|ω| (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}] . now, by differentiating(1.13), we have (2.7) ∣∣∣λ′ (ω)∣∣∣ ≤ −p |ω|p+1 + ∞∑ t=p |at| ≤ −p |ω|p+1 + (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}] , and (2.8) ∣∣∣λ′ (ω)∣∣∣ ≥ −p |ω|p+1 − ∞∑ t=p |at| ≥ −p |ω|p+1 − (1 −p) (t − 1) + |d|(s −t)[ a a+2bp ]m [2 −{|d|(s −t) + 2t}] . int. j. anal. appl. 18 (6) (2020) 1061 we have thus completed the proof. � theorem 2.3. let the function λ (ω) given by (1.13) is in the class n∗mp (a,b; d,s,t). then we have (i) λ is meromorphically starlike of order q in the disc |ω| < r3, that is < ( − ωλ ′ (ω) λ (ω) ) > q (|ω| < r3, 0 ≤ q < 1), where (2.9) r3 = inf t≥1  − ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 −p) (t − 1) + |d|(s −t)   1 t+p . (ii) λ is meromorphically convex of order q in the disc |ω| < r4, that is < { − ( 1 + ωλ ′′ (ω) λ ′ (ω) )} > q (|ω| < r4, 0 ≤ q < 1), where (2.10) r4 = inf t≥1   ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] p (1 −q) (1 −p) (t − 1) + |d|(s −t) [t (1 + q)]   1 t+p . proof. (i) in order to the inequality (2.9), we set∣∣∣∣∣∣ ωλ ′ (ω) λ(ω) + 1 ωλ ′ (ω) λ(ω) − 1 + 2q ∣∣∣∣∣∣ ≤ (1 −p) + ∑∞ t=p(t + 1) |at| |ω| t+p (2q −p− 1) + ∑∞ t=1(2q − 1 + t) |at| |ω| t+p . then we have ∣∣∣∣∣∣ ωλ ′ (ω) λ(ω) + 1 ωλ ′ (ω) λ(ω) − 1 + 2q ∣∣∣∣∣∣ ≤ 1 (0 ≤ q < 1), if (2.11) ∞∑ t=1 |at| |ω| t+p ≤−1. thus, by theorem 2.1, the inequality (2.11) will be true if |ω|t+p ≤− ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 −p) (t − 1) + |d|(s −t) , then |ω| =  − ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 −p) (t − 1) + |d|(s −t)   1 t+p . the last inequality leads us immediately to the disc |ω| < r3, where r3 is given by (2.9). int. j. anal. appl. 18 (6) (2020) 1062 (ii) in order to prove the second affirmation of theorem 2.3, we find from (1.1) that: ∣∣∣∣∣∣∣ ωλ ′′ (ω) λ ′ (ω) + 2 ωλ ′′ (ω) λ ′ (ω) + 2q ∣∣∣∣∣∣∣ ≤ p (p− 1) + ∑∞ t=p t(t + 1) |at| |ω| t+p p (p + 1 − 2q) |ω|p−1 + ∑∞ t=p t(t− 1 − 2q) |at| |ω| t+p . thus we have desired inequality: ∣∣∣∣∣∣∣ ωλ ′′ (ω) λ ′ (ω) + 2 ωλ ′′ (ω) λ ′ (ω) + 2q ∣∣∣∣∣∣∣ ≤ 1 (0 ≤ q < 1), if (2.12) ∞∑ t=1 ( t (1 + q) p (1 −q) ) |at| |ω| t+1 ≤ 1. thus, by theorem 2.1, the inequality (2.12) will be true if ( t (1 + q) p (1 −q) ) |ω|t+p ≤ ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 −p) (t − 1) + |d|(s −t) , then |ω| =   ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] p (1 −q) (1 −p) (t − 1) + |d|(s −t) [t (1 + q)]   1 t+p . the last inequality readily yields the disc |ω| < r4, where r4 is given by (2.10), which complete the proof. � theorem 2.4. the class n∗mp (a,b; d,s,t), is closed under convex linear combinations. proof. let the function λi (ω) = 1 ωp + ∞∑ t=p |at,i|ωt (i = 1, 2) , are in n∗mp (a,b; d,s,t), it suffices to show that the function h defined by h (ω) = (1 − c)λ1 (ω) + cλ2 (ω) (0 ≤ c ≤ 1) , is in the class n∗mp (a,b; d,s,t). since h (ω) = 1 ωp + ∞∑ t=p [(1 − c) |at,1| + c |at,2|] ωt (0 ≤ c ≤ 1) . int. j. anal. appl. 18 (6) (2020) 1063 in view of theorem 2.1, we have ∞∑ t=p [ a a + b(p + t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] [(1 − c) |at,1| + c |at,2|] = ∞∑ t=p [ a a + b(p + t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 − c) |at,1| + ∞∑ t=p [ a a + b(p + t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] c |at,2| ≤ (1 − c) [(1 −p) (t − 1) + |d|(s −t)] + c [(1 −p) (t − 1) + |d|(s −t)] = [(1 −p) (t − 1) + |d|(s −t)] , which show that h (ω) ∈ n∗mp (a,b; d,s,t), which is required. � theorem 2.5. let λ0 (ω) = 1 ωp and λt (ω) = 1 ω +   (1 −p) (t − 1) + |d|(s −t)∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}]  ωt t ≥ 1, then λ ∈ n∗mp (a,b; d,s,t). if and only if it can be expressed in the form (2.13) λ (ω) = ∞∑ t=p vtλt (ω) , where vt ≥ 0, and ∞∑ t=p vt = 1. proof. let the function λ (ω) be expressed in the form given by (2.13), then λ (ω) = 1 ω +  vt (1 −p) (t − 1) + |d|(s −t)∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}]  ωt, and for this function, we have ∞∑ t=p [ a a + b(p + t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] ×vt (1 −p) (t − 1) + |d|(s −t) ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] ωt = ∞∑ t=p vt (1 −p) (t − 1) + |d|(s −t) = [1 −v0] (1 −p) (t − 1) + |d|(s −t) ≤ (1 −p) (t − 1) + |d|(s −t), int. j. anal. appl. 18 (6) (2020) 1064 the condition (2.1) is satisfied. thus, λ ∈ n∗mp (a,b; d,s,t). conversely, we suppose that λ ∈ n∗mp (a,b; d,s,t). since |at| ≤ (1 −p) (t − 1) + |d|(s −t) ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] , (t ≥ 1). we set vt = ∞∑ t=p [ a a+b(p+t) ]m [1 + t−{|d|(s −t) + (1 + t) t}] (1 −p) (t − 1) + |d|(s −t) |at| (t ≥ 1), and v0 = 1 − ∞∑ vt t=p , so it follows that λ (ω) = ∞∑ t=p vtλt (ω) . this completes the assertion of theorem 2.5. � 3. conclusion in our current investigation, we have presented and studied thoroughly some new subclasses of p−valent functions related with meromorphic convex and meromorphic starlike functions, in connection with the integral operator given by (1.7). we have obtained sufficient and necessary conditions in relation to these classes, including growth and distortion theorem along with a radius problem. the technique and ideas of this paper may stimulate further research in the theory of multivalent meromorphic functions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. k. aouf and t. m. seoudy, some properties of a certain subclass of multivalent analytic functions involving the liu–owa operator. comput. math. appl. 60 (2010), 1525-1535. [2] m. k. aouf and t. m. seoudy, some preserving subordination and super ordination of analytic functions involving the liu–owa integral operator, comput. math. appl. 62 (2011), 3575-3580. [3] m. k. aouf and t. m. seoudy, some preserving subordination and superordination of the liu–owa integral operator, complex anal. oper. theory. 7 (2013), 275-283. [4] k. r. alhindi and m. darus, a new class of meromorphic functions involving the polylogarithm function, j. complex anal. 2014 (2014), 864805. [5] r. bharati, r. parvatham and a. swaminathan, on subclasses of uniformly convex functions and corresponding class of starlike functions, tamkang j. math. 28 (1997), 17-32. [6] n. e. cho and s. owa, sufficient conditions for meromorphic starlikeness and close to-convexity of order α, int. j. math. sci. 26 (2001), 317-319. int. j. anal. appl. 18 (6) (2020) 1065 [7] m. darus, s. hussain, m. raza and j. sokol, on a subclass of starlike functions, res. math. 73 (2018), 22. [8] h. aldweby and m. darus, on harmonic meromorphic functions associated with basic hypergeometric functions, sci. world j. 2013 (2013), 164287. [9] r. m. el-ashwah, m. k. aouf, a. a. hassan and a. h. hassan, certain new classes of analytic functions with varying arguments, j. complex anal. 2013 (2013), art. id 958210. [10] e. a. elrifai, h. e. darwish and a. r. ahmed, on certain subclasses of meromorphic functions associated with certain differential operators. appl. math. lett. 25 (2012), 952-958. [11] s. elhaddad and m. darus, on meromorphic functions defined by a new operator containing the mittag-leffler fuction, symmetry. 11 (2019), art. id 210. [12] b. a. frasin, m. darus, on certain meromorphic functions with positive coefficients, southeast asian bull. math. 28 (2004), 615-623. [13] f. ghanim, m. darus, on class of hypergeometric meromorphic functions with fixed second positive coefficients, gen. math.17 (2009), 13–28. [14] h. m. hossen, h. m. srivastava and m. k. aouf, a unified presentation of some classes of meromorphically multivalent functions. comput. math. appl. 38 (1999), 63-70. [15] w. janowski, some extrenal problems for certain families of analytic functions, ann. pol. math. 28 (1973), 297–326. [16] w. kaplan, close-to-convex schlicht functions, michigan math. j. 1 (1952), 169-185. [17] a. y. lashin, on certain subclasses of meromorphic functions associated with certain integral operators. comput. math. appl. 59 (2010), 524-531. [18] j. l. liu and h. m. srivastava, classes of meromorphically multivalent functions associated with the generalized hyper geometric function. math. comput. model. 39 (2004), 21-34. [19] j. l. liu and s. owa, properties of certain integral operators, int. j.math. math. sci. 3 (2004), 69–75. [20] j. e. miller, convex meromrphic mapping and related functions, proc. amer. math. soc. 25 (1970), 220-228. [21] a. mannino, some inequalities concerning starlike and convex functions, gen. math. 12 (2004), 5-12. [22] s. s. miller and p. t. mocanu, differential subordinations theory and applications, in monographs and textbooks in pure and applied mathematics, 225, marcel dekker, new york, 2000. [23] m. s. roberston, on the theory of univalent functions. annal. math. 37 (1936), 374-408. [24] a. rasheed, s. hussain, s. g. a. shah, m. darus and s. lodhi, majorization problem for two subclasses of meromorphic functions associated with a convolution operator, aims math. 5(2020), 5157-5170. 1. introduction 2. main results 3. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 136-143 http://www.etamaths.com existence of positive periodic solutions for a third-order delay differential equation farid nouioua1,2, abdelouaheb ardjouni2,3,∗, abdelkerim merzougui1 and ahcene djoudi3 abstract. in this paper, the following third-order nonlinear delay differential equation with periodic coefficients x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = f (t,x (t) ,x(t − τ(t))) + c(t)x′(t − τ(t)), is considered. by employing green’s function and krasnoselskii’s fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order delay differential equation. 1. introduction delay differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering, see the monograph [8, 19] and the papers [1][18], [20][22], [24][27] and the references therein. the second order nonlinear delay differential equation with periodic coefficients x′′ (t) + p (t) x′ (t) + q (t) x (t) = r (t) x′ (t− τ (t)) + f (t,x (t) ,x (t− τ (t))) , has been investigated in [25]. by using krasnoselskii’s fixed point theorem and the contraction mapping principle, wang, lian and ge obtained existence and uniqueness of periodic solutions. in [22], ren, siegmund and chen discussed the existence of positive periodic solutions for the third-order differential equation x′′′ (t) + p (t) x′′ (t) + q (t) x′ (t) + c (t) x (t) = g (t,x (t)) . by employing the fixed point index, the authors obtained existence results for positive periodic solutions. inspired and motivated by the works mentioned above and the papers [1][18], [20][22], [24][27] and the references therein, we concentrate on the existence of positive periodic solutions for the third-order nonlinear delay differential equation x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = f (t,x (t) ,x(t− τ(t))) + c(t)x′(t− τ(t)). (1.1) where p, q, r are continuous real-valued functions. the function c : r −→ r+ is continuously differentiable, τ : r −→ r+ is twice continuously differentiable and f : r×r×r −→ r is continuous in their respective arguments. to show the existence of positive periodic solutions, we transform (1.1) into an integral equation and then use krasnoselskii’s fixed point theorem. the obtained integral equation splits in the sum of two mappings, one is a contraction and the other is compact. in this paper, we give the assumptions as follows that will be used in the main results. (h1) there exist differentiable positive t-periodic functions a1 and a2 and a positive real constant ρ such that   a1(t) + ρ = p(t), a′1 (t) + a2 (t) + ρa1(t) = q (t) , a′2 (t) + ρa2(t) = r (t) . received 14th august, 2016; accepted 6th october, 2016; published 1st march, 2017. 2010 mathematics subject classification. 34k13, 34a34, 34k30, 34l30. key words and phrases. fixed point; positive periodic solutions; third-order delay differential equations. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 136 a third-order delay differential equation 137 (h2) p,q,r,c ∈ c (r,r+) are t-periodic functions with τ (t) ≥ τ∗ > 0, τ′ (t) 6= 1 for all t ∈ [0,t] and ∫ t 0 p(s)ds > ρ, ∫ t 0 q(s)ds > 0. (h3) the function f(t,x,y) is continuous t-periodic in t and continuous in x and y. the organization of this paper is as follows. in section 2, we introduce some notations and lemmas, and state some preliminary results needed in later section, then we give the green’s function of (1.1), which plays an important role in this paper. in section 3, we present our main results on existence of positive periodic solutions of (1.1). lastly in this section, we state krasnoselskii’s fixed point theorem which enables us to prove the existence of positive periodic solutions to (1.1). for its proof we refer the reader to [23]. theorem 1.1 (krasnoselskii). let m be a closed convex nonempty subset of a banach space (b,‖.‖). suppose that h1 and h2 map m into b such that (i) x,y ∈ m, implies h1x + h2y ∈ m, (ii) h1 is compact and continuous, (iii) h2 is a contraction mapping. then there exists z ∈ m with z = h1z + h2z. 2. green’s function of third-order differential equation for t > 0, let pt be the set of all continuous scalar functions x, periodic in t of period t . then (pt ,‖.‖) is a banach space with the supremum norm ‖x‖ = sup t∈r |x(t)| = sup t∈[0,t ] |x(t)| . we consider x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = h (t) , (2.1) where h is a continuous t-periodic function. obviously, by the condition (h1), (2.1) is transformed into { y′(t) + ρy(t) = h(t), x′′(t) + a1(t)x ′(t) + a2(t)x(t) = y(t). lemma 2.1 ( [3]). if y,h ∈ pt , then y is a solution of equation y′(t) + ρy(t) = h(t), if only if y(t) = ∫ t+t t g1(t,s)h(s)ds, (2.2) where g1(t,s) = exp (ρ (s− t)) exp (ρt) − 1 . (2.3) corollary 2.1. green function g1 satisfies the following properties g1(t + t,s + t) = g1(t,s), g1(t,t + t) = g1(t,t) exp (ρt) , g1 (t + t,s) = g1(t,s) exp (−ρt) , g1(t,s + t) = g1(t,s) exp (ρt) , ∂ ∂t g1(t,s) = −ρg1(t,s), ∂ ∂s g1(t,s) = ρg1(t,s), and m1 ≤ g1(t,s) ≤ m1, where m1 = 1 exp (ρt) − 1 , m1 = exp (ρt) exp (ρt) − 1 . 138 nouioua, ardjouni, merzougui and djoudi lemma 2.2 ( [21]). suppose that (h1) and (h2) hold and r1 [ exp (∫ t 0 a1(v)dv ) − 1 ] q1t ≥ 1, (2.4) where r1 = max t∈[0,t ] ∣∣∣∣∣∣ ∫ t+t t exp (∫ t 0 a1(v)dv ) exp (∫ t 0 a1(v)dv ) − 1 a2 (s) ds ∣∣∣∣∣∣ , q1 = ( 1 + exp (∫ t 0 a1(v)dv ))2 r21. then there are continuous t -periodic functions a and b such that b(t) > 0, ∫ t 0 a(v)dv > 0, and a(t) + b(t) = a1(t), b ′(t) + a(t)b(t) = a2(t), for t ∈ r. lemma 2.3 ( [25]). suppose the conditions of lemma 2.2 hold and y ∈ pt . then the equation x′′(t) + a1(t)x ′(t) + a2(t)x(t) = y(t), has a t periodic solution. moreover, the periodic solution can be expressed by x(t) = ∫ t+t t g2(t,s)y(s)ds, (2.5) where g2(t,s) = ∫ s t exp [∫ v t b(u)du + ∫ s v a(u)du ] dv + ∫ t+t s exp [∫ v t b(u)du + ∫ s+t v a(u)du ] dv[ exp (∫ t 0 a(v)dv ) − 1 ][ exp (∫ t 0 b(v)dv ) − 1 ] . (2.6) corollary 2.2. green’s function g2 satisfies the following proprieties g2(t + t,s + t) = g2(t,s), g2(t,t + t) = g2(t,t), g2(t + t,s) = exp ( − ∫ t 0 b(v)dv )[ g2 (t,s) + ∫ t+t t e (t,u) f (u,s) du ] , ∂ ∂t g2(t,s) = −b(t)g2(t,s) + f (t,s) , ∂ ∂s g2(t,s) = a(t)g2(t,s) −e (t,s) , where e (t,s) = exp (∫ s t b(v)dv ) exp (∫ t 0 b(v)dv ) − 1 , f (t,s) = exp (∫ s t a (v) dv ) exp (∫ t 0 a (v) dv ) − 1 . lemma 2.4 ( [21]). let a = ∫ t 0 a1(v)dv and b = t 2 exp ( 1 t ∫ t 0 ln (a2(v)) dv ) . if a2 ≥ 4b, (2.7) then min {∫ t 0 a(v)dv, ∫ t 0 b(v)dv } ≥ 1 2 ( a− √ a2 − 4b ) = l, max {∫ t 0 a(v)dv, ∫ t 0 b(v)dv } ≤ 1 2 ( a + √ a2 − 4b ) = l. a third-order delay differential equation 139 corollary 2.3. functions g2, e and f satisfy m2 ≤ g2(t,s) ≤ m2, e (t,s) ≤ el el − 1 , f (t,s) ≤ el, where m2 = t (exp (l) − 1)2 , m2 = t exp (∫ t 0 a1 (v) dv ) (exp (l) − 1)2 . lemma 2.5 ( [11]). suppose the conditions of lemma 2.2 hold and h ∈ pt . then the equation x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = h (t) , has a t -periodic solution. moreover, the periodic solution can be expressed by x(t) = ∫ t+t t g(t,s)h(s)ds, (2.8) where g (t,s) = ∫ t+t t g2 (t,σ) g1 (σ,s) dσ. (2.9) corollary 2.4. green’s function g satisfies the following properties g(t + t,s + t) = g(t,s), g (t,t + t) = g (t,t) exp (ρt) , ∂ ∂t g(t,s) = (exp (−ρt) − 1) g1 (t,t) g2 (t,s) − b (t) g (t,s) + ∫ t+t t f(t,σ)g1 (σ,s) dσ, ∂ ∂s g(t,s) = ρg (t,s) , and m ≤ g(t,s) ≤ m, where m = t2 (exp (l) − 1)2 (exp (ρt) − 1) , m = t2 exp ( ρt + ∫ t 0 a (v) dv ) (exp (l) − 1)2 (exp (ρt) − 1) . 3. main results in this section we will study the existence of positive periodic solutions of (1.1). lemma 3.1. suppose (h1)-(h3) and (2.4) hold. the function x ∈ pt is a solution of (1.1) if and only if x (t) = z (t) (exp (ρt) − 1) g (t,t) x (t− τ (t)) + ∫ t+t t g (t,s){f (s,x (s) ,x (s− τ (s))) −r (s) x (s− τ (s))}ds, (3.1) where r (s) = (c′ (s) + c (s) ρ) (1 − τ′ (s)) + c (s) τ′′ (s) (1 − τ′ (s))2 , (3.2) z (t) = c (t) 1 − τ′ (t) . (3.3) proof. let x ∈ pt be a solution of (1.1). from lemma 2.5, we have x (t) = ∫ t+t t g (t,s) [f (s,x (s) ,x (s− τ (s))) + c (s) x′ (s− τ (s))] ds = ∫ t+t t g (t,s) f (s,x (s) ,x (s− τ (s))) ds + ∫ t+t t g (t,s) c (s) x′ (s− τ (s)) ds. (3.4) 140 nouioua, ardjouni, merzougui and djoudi performing an integration by parts, we get∫ t+t t g (t,s) c (s) x′ (s− τ (s)) ds = ∫ t+t t c (s) (1 − τ′ (s)) x′ (s− τ (s)) 1 − τ′ (s) g (t,s) ds = ∫ t+t t c (s) 1 − τ′ (s) g (t,s) dx (s− τ (s)) = c (s) 1 − τ′ (s) g (t,s) x (s− τ (s)) ∣∣∣∣t+t t − ∫ t+t t ∂ ∂s [ c (s) 1 − τ′ (s) g (t,s) ] x (s− τ (s)) ds = z (t) (exp (ρt) − 1) x (t− τ (t)) g (t,t) − ∫ t+t t r (s) g (t,s) x (s− τ (s)) ds, (3.5) where r and z are given by (3.2) and (3.3), respectively. we obtain (3.1) by substituting (3.5) in (3.4). since each step is reversible, the converse follows easily. this completes the proof. � define the mapping h : pt → pt by (hϕ) (t) = ∫ t+t t g (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −r (s) ϕ (s− τ (s))}ds + z (t) (exp (ρt) − 1) g (t,t) ϕ (t− τ (t)) . (3.6) note that to apply krasnoselskii’s fixed point theorem we need to construct two mappings, one is a contraction and the other is compact. therefore, we express (3.6) as (hϕ) (t) = (h1ϕ) (t) + (h2ϕ) (t) . where h1,h2 : pt → pt are given by (h1ϕ) (t) = ∫ t+t t g (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −r (s) ϕ (s− τ (s))}ds, (3.7) and (h2ϕ) (t) = z (t) (exp (ρt) − 1) g (t,t) ϕ (t− τ (t)) . (3.8) to simplify notations, we introduce the constants α = max t∈[0,t ] |z (t)| , β = max t∈[0,t ] {b(t)} , δ = exp (l) exp (l) − 1 , γ = exp (ρt) − 1. (3.9) in this section we obtain the existence of a positive periodic solution of (1.1) by considering the two cases; c (t) ≥ 0 and c (t) ≤ 0 for all t ∈ r. for a non-negative constant k and a positive constant l we define the set d ={ϕ ∈ pt : k ≤ ϕ ≤ l} , which is a closed convex and bounded subset of the banach space pt . in case c (t) ≥ 0, we assume that there exist a positive constant η such that η ≤ z (t) , for all t ∈ [0,t] , (3.10) αmγ < 1, (3.11) and for all s ∈ [0,t] , x,y ∈ d k (1 −ηmγ) mt ≤ f (s,x,y) −r (s) y ≤ l (1 −αmγ) mt . (3.12) lemma 3.2. suppose (h1)-(h3), (2.4), (2.7) and (3.10)-(3.12) hold. then h1 : d → pt is compact. a third-order delay differential equation 141 proof. let h1 be defined by (3.7). obviously, h1ϕ is continuous and it is easy to show that (h1ϕ) (t + t) = (h1ϕ) (t). for t ∈ [0,t] and for ϕ ∈ d, we have |(h1ϕ) (t)| = ∣∣∣∣∣ ∫ t+t t g (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −r (s) ϕ (s− τ (s))}ds ∣∣∣∣∣ ≤ mt l (1 −αmγ) mt = l (1 −αmγ) . thus from the estimation of |(h1ϕ) (t)| we have ‖h1ϕ‖≤ l (1 −αmγ) . this shows that h1 (d) is uniformly bounded. to show that h1 (d) is equicontinuous, let ϕn ∈ d, where n is a positive integer. next we calculate d dt (h1ϕn) (t) and show that it is uniformly bounded. by using (h1), (h2) and (h3) we obtain by taking the derivative in (3.7) that d dt (h1ϕn) (t) = ∫ t+t t [ (exp (−ρt) − 1) g1 (t,t) g2 (t,s) − b (t) g (t,s) + ∫ t+t t f(t,σ)g1 (σ,s) dσ ] × [f (s,ϕn (s) ,ϕn (s− τ (s))) −r (s) ϕn (s− τ (s))] ds. consequently, by invoking (3.9) and (3.12), we obtain∣∣∣∣ ddt (h1ϕn) (t) ∣∣∣∣ ≤ [(1 − exp (−ρt)) m1m2 + mβ + m1δt] l (1 −αmγ)m ≤ d, for some positive constant d. hence the sequence (h1ϕn) is equicontinuous. the ascoli-arzela theorem implies that a subsequence (h1ϕnk ) of (h1ϕn) converges uniformly to a continuous t-periodic function. thus h1 is continuous and h1 (d) is contained in a compact subset of d. � lemma 3.3. suppose that (3.11) holds. if h2 is given by (3.8), then h2 : d → pt is a contraction. proof. let h2 be defined by (3.8). it is easy to show that (h2ϕ) (t + t) = (h2ϕ) (t). let ϕ,ψ ∈ d, we have ‖h2ϕ−h2ψ‖ = sup t∈[0,t ] |(h2ϕ) (t) − (h2ψ) (t)| ≤ αγm ‖ϕ−ψ‖ . hence h2 : d → pt is a contraction by (3.11). � theorem 3.1. suppose that conditions (h1)-(h3), (2.4), (2.7) and (3.10)-(3.12) hold. then equation (1.1) has a positive t -periodic solution x in the subset d. proof. by lemma 3.2, the operator h1 : d → pt is compact and continuous. also, from lemma 3.3, the operator h2 : d → pt is a contraction. moreover, if ϕ,ψ ∈ d, we see that (h2ψ) (t) + (h1ϕ) (t) = γz (t) g (t,t) ϕ (t− τ (t)) + ∫ t+t t g (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −r (s) ϕ (s− τ (s))}ds ≤ γαml + l (1 −αmγ) = l. on the other hand (h2ψ) (t) + (h1ϕ) (t) = γz (t) g (t,t) ϕ (t− τ (t)) + ∫ t+t t g (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −r (s) ϕ (s− τ (s))}ds ≥ γαmk + k (1 −αmγ) = k. this shows that h2ψ +h1ϕ ∈ d. clearly, all the hypotheses of theorem 1.1, are satisfied. thus there exists a fixed point x ∈ d such that x = h1ψ + h2ϕ. by lemma 3.1 this fixed point is a solution of (1.1) and the proof is complete. � 142 nouioua, ardjouni, merzougui and djoudi in the case c (t) ≤ 0, we substitute conditions (3.10)-(3.12) with the following conditions respectively. we assume that there exist a negative constant z1 and a non-positive constant z2 such that z1 ≤ z (t) ≤ z2, for all t ∈ [0,t] , (3.13) −z1mγ < 1, (3.14) and for all s ∈ [0,t] , x,y ∈ d k−z1mγl mt ≤ f (s,x,y) −r (s) y ≤ l−z2mγk mt . (3.15) theorem 3.2. suppose that conditions (h1)-(h3), (2.4), (2.7) and (3.13)-(3.15) hold. then equation (1.1) has a positive t -periodic solution x in the subset d. the proof follows along the lines of theorem 3.1, and hence we omit it. references [1] a. ardjouni and a. djoudi, existence of periodic solutions for a second-order nonlinear neutral differential equation with variable delay, palestine journal of mathematics, 3(2) (2014), 191–197. [2] a. ardjouni, a. djoudi and a. rezaiguia, existence of positive periodic solutions for two types of third-order nonlinear neutral differential equations with variable delay, applied mathematics e-notes, 14 (2014), 86–96. [3] a. ardjouni and a. djoudi, existence of positive periodic solutions for a nonlinear neutral differential equations with variable delay, applied mathematics e-notes, 12 (2012), 94–101. [4] a. ardjouni and a. djoudi, existence of periodic solutions for a second order nonlinear neutral differential equation with functional delay, electronic journal of qualitative theory of differential equations, 2012 (2012), art. id 31, 1–9. [5] a. ardjouni and a. djoudi, periodic solutions for a second-order nonlinear neutral differential equation with variable delay, electron. j. differential equations, 2011 (2011), art. id 128, 1–7. [6] a. ardjouni and a. djoudi, periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale, rend. sem. mat. univ. politec. torino, 68 (4) (2010), 349-359. [7] t. a. burton, liapunov functionals, fixed points and stability by krasnoselskii’s theorem, nonlinear stud. 9 (2) (2002), 181-190. [8] t. a. burton, stability by fixed point theory for functional differential equations, dover publications, new york, 2006. [9] f. d. chen, positive periodic solutions of neutral lotka-volterra system with feedback control, appl. math. comput. 162 (3) (2005), 1279-1302. [10] f. d. chen and j. l. shi, periodicity in a nonlinear predator-prey system with state dependent delays, acta math. appl. sin. engl. ser. 21 (1) (2005), 49-60. [11] z. cheng and j. ren, existence of positive periodic solution for variable-coefficient third-order differential equation with singularity, math. meth. appl. sci. 37 (2014), 2281–2289. [12] z. cheng and y. xin, multiplicity results for variable-coefficient singular third-order differential equation with a parameter, abstract and applied analysis, 2014 (2014), article id 527162, 1–10. [13] s. cheng and g. zhang, existence of positive periodic solutions for non-autonomous functional differential equations, electron. j. differential equations, 2001 (2001), art. id 59, 1–8. [14] h. deham and a. djoudi, periodic solutions for nonlinear differential equation with functional delay, georgian mathematical journal, 15 (4) (2008), 635-642. [15] h. deham and a. djoudi, existence of periodic solutions for neutral nonlinear differential equations withvariable delay, electronic journal of differential equations, 2010 (2010), art. id 127, 1–8. [16] y. m. dib, m. r. maroun and y. n. rafoul, periodicity and stability in neutral nonlinear differential equations with functional delay, electronic journal of differential equations, 2005 (2005), art. id 142, 1-11. [17] m. fan and k. wang, p. j. y. wong and r. p. agarwal, periodicity and stability in periodic n-species lotkavolterra competition system with feedback controls and deviating arguments, acta math. sin. engl. ser. 19 (4) (2003), 801-822. [18] h. i. freedman, j. wu, periodic solutions of single-species models with periodic delay, siam j. math. anal. 23 (1992), 689–701. [19] y. kuang, delay differential equations with application in population dynamics, academic press, new york, 1993. [20] w. g. li and z. h. shen, an constructive proof of the existence theorem for periodic solutions of duffng equations, chinese sci. bull. 42 (1997), 1591–1595. [21] y. liu, w. ge, positive periodic solutions of nonlinear duffing equations with delay and variable coefficients, tamsui oxf. j. math. sci. 20 (2004), 235–255. [22] j. ren, s. siegmund and y. chen, positive periodic solutions for third-order nonlinear differential equations, electron. j. differential equations, 2011 (2011), art. id 66, 1–19. [23] d. r. smart, fixed points theorems, cambridge university press, cambridge, 1980. a third-order delay differential equation 143 [24] q. wang, positive periodic solutions of neutral delay equations (in chinese), acta math. sinica (n.s.) 6(1996), 789-795. [25] y. wang, h. lian and w. ge, periodic solutions for a second order nonlinear functional differential equation, applied mathematics letters, 20 (2007), 110-115. [26] w. zeng, almost periodic solutions for nonlinear duffing equations, acta math. sinica (n.s.) 13 (1997), 373-380. [27] g. zhang, s. cheng, positive periodic solutions of non autonomous functional differential equations depending on a parameter, abstr. appl. anal. 7 (2002) 279–286. 1faculty mathematics and informatics, department of mathematics, univ m’sila, p.o. box 166, 28000 m’sila, algeria 2faculty of sciences and technology, department of mathematics and informatics, univ souk ahras, p.o. box 1553, souk ahras, 41000, algeria 3applied mathematics lab, faculty of sciences, department of mathematics, univ annaba, p.o. box 12, annaba 23000, algeria ∗corresponding author: abd ardjouni@yahoo.fr 1. introduction 2. green's function of third-order differential equation 3. main results references international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 19-28 http://www.etamaths.com harmonic analysis associated with the generalized weinstein operator ahmed abouelaz, azz-edine achak , radouan daher and el mehdi loualid∗ abstract. in this paper we consider a generalized weinstein operator ∆d,α,n on rd−1×]0,∞[, which generalizes the weinstein operator ∆d,α, we define the generalized weinstein intertwining operator rα,n which turn out to be transmutation operator between ∆d,α,n and the laplacian operator ∆d. we build the dual of the generalized weinstein intertwining operator trα,n, another hand we prove the formula related rα,n and trα,n . we exploit these transmutation operators to develop a new harmonic analysis corresponding to ∆d,α,n. 1. introduction in this paper we consider a generalized weinstein operator ∆d,α,n on rd−1×]0,∞[, defined by (1) ∆d,α,n = d∑ i=1 ∂2 ∂x2i + 2α + 1 xd ∂ ∂xd − 4n(α + n) x2d , α > − 1 2 where n = 0, 1, ... . for n = 0, we regain the weinstein operator (2) ∆d,α = d∑ i=1 ∂2 ∂x2i + 2α + 1 xd ∂ ∂xd , α > − 1 2 through this paper, we provide a new harmonic analysis on rd−1×]0,∞[ corresponding to the generalized weinstein operator ∆d,α,n. the outline of the content of this paper is as follows. section 2 is dedicated to some properties and results concerning the weinstein transform. in section 3, we construct a pair of transmutation operators rα,n and trα,n, afterwards we exploit these transmutation operators to build a new harmonic analysis on rd−1×]0,∞[ corresponding to operator ∆d,α,n. 2. preliminaries throughout this paper, we denote by • rd+ = rd−1×]0,∞[. • x = (x1, ...,xd) = (x ′ ,xd) ∈ rd−1×]0,∞[. • λ = (λ1, ...,λd) = (λ ′ ,λd) ∈ cd. • e(rd) (resp. d(rd)) the space of c∞ functions on rd, even with respect to the last variable (resp. with compact support). 2010 mathematics subject classification. 42a38, 44a35, 34b30. key words and phrases. generalized weinstein operator; generalized weinstein transform; generalized convolution; generalized translation operators; harmonic analysis. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 19 • s(rd) the schwartz space of rapidly decreasing functions on rd which are even with respect to the last variable. in this section, we recapitulate some facts about harmonic analysis related to the weinstein operator ∆d,α. we cite here, as briefly as possible, some properties. for more details we refer to [2, 3, 4]. the weinstein operator ∆d,α defined on rd+ by (3) ∆d,α = d∑ i=1 ∂2 ∂x2i + 2α + 1 xd ∂ ∂xd , α > − 1 2 then ∆d,α = ∆d + bα where ∆d is the laplacian operator in rd−1 and bα the bessel operator with respect to the variable xd defined by (4) bα = ∂2 ∂x2d + 2α + 1 xd ∂ ∂xd , α > − 1 2 . the weinstein kernel is given by (5) ψλ,α(x) = e−i<x ′,λ′>jα(xdλd), for all (x,λ) ∈ rd ×cd. here x′ = (x1, ....,xd−1),λ′ = (λ1, ....,λd−1) and jα is the normalized bessel function of index α defined by (6) jα(z) = γ(α + 1) ∞∑ n=0 (−1)n(z 2 )2n n! γ(n + α + 1) (z ∈ c). proposition 1. ψλ,α satisfies the differential equation ∆d,αψλ,α = −‖λ‖2ψλ,α. definition 1. the weinstein intertwining operator is the operator rα defined on c(rd) by (7) rαf(x) = aαx−2αd ∫ xd 0 (x2d − t 2)α− 1 2 f(x′, t)dt, xd > 0 where (8) aα = 2γ(α + 1) √ πγ(α + 1 2 ) . proposition 2. rα is a topological isomorphism from e(rd) onto itself satisfying the following transmutation relation (9) ∆d,α(rαf) = rα(∆df), for all f ∈ e(rd), where ∆d is the laplacian on rd. proposition 3. ∆d,α is self-adjoint, i.e∫ rd + ∆d,αf(x)g(x)dµα(x) = ∫ rd + f(x)∆d,αg(x)dµα(x) for all f ∈ e(rd) and g ∈ d(rd). definition 2. the dual of the weinstein intertwining operator rα is the operator trα defined on d(rd) by (10) trα(f)(y) = aα ∫ ∞ yd (s2 −y2d) α−1 2 f(y′,s)sds. 20 proposition 4. trα is a topological isomorphism from s(rd) onto itself satisfying the following transmutation relation (11) trα(∆d,αf) = ∆d(trαf), for all f ∈ s(rd), where ∆d is the laplacian on rd. it satisfies for f ∈ d(rd) and g ∈ e(rd) the following relation (12) ∫ rd + trα(f)(y)g(y)dy = ∫ rd + f(y)rα(g)(y)dµα(y). definition 3. the weinstein transform fw,α is defined on l1α(rd+) by (13) fw,α(f)(λ) = ∫ rd + f(x)ψλ,α(x)dµα(x), for all λ ∈ rd. proposition 5. (i) for all f ∈ l1(rd+), the function fw,α(f) is continuous on rd and we have (14) ‖fw,α(f)‖α,∞ ≤‖f‖α,1. (ii) for all f ∈ s(rd) we have (15) fw,α(f)(y) = f0 ◦t rα(f)(y), ∀y ∈ rd+, where f0 is the transformation defined by, for all y ∈ rd+ (16) f0(f)(y) = ∫ rd + f(x)e−i<y ′,x′> cos(xdyd)dx, ∀f ∈ d(rd). (iii) for all f ∈ s(rd) and m ∈ n, we have (17) fw,α(∆d,αf)(λ) = −‖λ‖2fw,α(f)(λ). . theorem 1. (i) plancherel formula: for all f ∈ s(rd) we have (18) ∫ rd + |f(x)|2dµα(x) = c(α) ∫ rd + |fw,α(f)(λ)|2dµα(λ) where (19) c(α) = 1 (2π)d−122α(γ(α + 1))2 . (ii) for all f ∈ l1α(rd+), if fw,α(f) ∈ l1α(rd+), then (20) f(y) = c(α) ∫ rd + fw,α(f)(x)ψλ,α(−x)dµα(x) where c(α) is given by (19). definition 4. the translation operators τxα, x ∈ rd+, associated with the operator ∆d,α are defined by (21) τxαf(y) = γ(α + 1) √ πγ(α + 1 2 ) ∫ π 0 f(x′ + y′, √ x2d + y 2 d + 2xdyd cos θ)(sin θ) 2αdθ where f ∈ c(rd+). proposition 6. for all f ∈ lpα,n(rd), p ∈ [1,∞], and for all x ∈ rd+ τxα,n(φλ,α,n(y)) = φλ,α,n(x)φλ,α,n(y). 21 proposition 7. the translation operator τxα, x ∈ rd+ satisfies the following properties: (i) ∀x ∈ rd+, we have (22) ∆d,α ◦ τxα = τ x α ◦ ∆d,α. (ii) for all f in e(rd) and g in s(rd) we have (23) ∫ rd + τxαf(y)g(y)dµα(y) = ∫ rd + f(y)τxαg(y)dµα(y). (iii) for all f in lpα(r d +), p ∈ [1,∞], and x ∈ rd+ we have (24) ‖τxα‖p,α ≤‖f‖p,α. (iv) for f ∈ s(rd) and y ∈ rd+ we have (25) fw,α (τyαf) (x) = ψy,α(x)fw,α(f)(x). definition 5. the generalized convolution product f ∗w,α g of functions f,g ∈ l1α(rd+) is defined by (26) f ∗w,α g(x) = ∫ rd + τxαf(−y ′,y)g(y)dµα(y). proposition 8. for all f,g ∈ l1α(rd+), f ∗w,α g belongs to l1α(rd+) and (27) fw,α (f ∗w,α g) = fw,α(f)fw,α(g). 3. harmonic analysis associated with the generalized weinstein operator transmutation operators. • mn the map defined by mnf(x ′ ,xd) = x 2n d f(x ′ ,xd). • lpα,n(rd+) the class of measurable functions f on rd+ for which ‖f‖α,n,p = ‖m−1n f‖α+2n,p < ∞. • en(rd) (resp. dn(rd) and sn(rd)) stand for the subspace of e(rd) (resp. d(rd) and s(rd)) consisting of functions f such that f(x ′ , 0) = ( dkf dxkd ) (x ′ , 0) = 0, ∀k ∈{1, ...2n− 1}. lemma 1. (i) the map (28) mn(f)(x) = x2nd f(x) is an isomorphism – from e(rd) onto en(rd). – from s(rd) onto sn(rd). (ii) for all f ∈ e(rd) we have (29) bα,n ◦mn(f) = mn ◦bα+2n(f), where bα,n is the generalized bessel operator given by (4). (iii) for all f ∈ e(rd) (30) ∆d,α,n ◦mn(f)(x) = mn ◦ ∆d,α+2n where ∆d,α+2n is the weinstein operator of order α + 2n given by (3). 22 (iv) ∆d,α,n is self-adjoint, i.e (31) ∫ rd + ∆d,α,nf(x)g(x)dµα(x) = ∫ rd + f(x)∆d,α,ng(x)dµα(x) for all f ∈ e(rd) and g ∈ dn(rd). proof. assertion (i) and (ii) (see [1]). for assertion (iii) using (1) and (29) we obtain ∆d,α,n ◦mn(f)(x ′ ,xd) = (∆d + bα,n) ◦mn(f)(x ′ ,xd), = ∆d(mnf)(x ′ ,xd) + bα,n(mnf)(x ′ ,xd), = mn(∆df)(x ′ ,xd) + mn(bα+2nf)(x ′ ,xd), = mn ◦ ∆d,α+2nf(x ′ ,xd). which give (iii). if f ∈ e(rd) and g ∈ dn(rd), then by proposition 3 we get∫ rd + ∆d,α,nf(x)g(x)dµα(x) = ∫ rd + ( ∆d,αf(x) − 4n(α + n) x2d f(x) ) g(x)dµα(x), = ∫ rd + ∆d,αf(x)g(x)dµα(x) − ∫ rd + 4n(α + n) x2d f(x)g(x)dµα(x), = ∫ rd + f(x)∆d,αg(x)dµα(x) − ∫ rd + 4n(α + n) x2d f(x)g(x)dµα(x), = ∫ rd + f(x) ( ∆d,αg(x) − 4n(α + n) x2d g(x) ) dµα(x), = ∫ rd + f(x)∆d,α,ng(x)dµα(x). definition 6. the generalized weirstein intertwining operator is the operator rα,n defined on e(rd+1) by (32) rα,nf(x) = aα+2nx −2(α+n) d ∫ xd 0 (x2d − t 2)α+2n− 1 2 f(x′, t)dt, xd > 0 where aα+2n is given by 8. remark 1. by (7) and (32) we have (33) rα,n = mn ◦rα+2n. proposition 9. rα,n is a topological isomorphism from e(r) onto en(r) satisfying the following transmutation relation (34) ∆d,α,n(rα,nf) = rα,n(∆df), forallf ∈ e(rd+1) where ∆d is the laplacian on rd. proof. using (9), (30) and (33) we obtain ∆d,α,n(rα,nf) = ∆d,α,n (mn ◦rα+2n) (f), = mn ◦ ∆d,α+2n(rα+2nf) = mn (rα+2n ◦ ∆d) (f) = rα,n(∆df). 23 definition 7. the dual of the generalized weinstein intertwining operator rα,n is the operator trα,n defined on dn(rd) by (35) trα,n(f)(y) = aα+2n ∫ ∞ yd (s2 −y2d) α+2n−1 2 f(y′,s)s1−2nds. remark 2. from (10) and (35) we have (36) trα,n =t rα+2n ◦m−1n . proposition 10. trα,n is a topological isomorphism from sn(rd+1) onto s(rd+1) satisfying the following transmutation relation (37) trα,n(∆d,α,nf) = ∆d(trα,nf), for all f ∈ sn(rd+1) where ∆d is the laplacian on rd. proof. an easily combination of (11), (30) and (37) shows that trα,n(∆d,α,nf) = trα+2n ◦m−1n ( mn ◦ ∆d,α+2n ◦m−1n ) (f), = trα+2n ( ∆d,α+2n ◦m−1n ) (f), = ∆d ( rα+2n ◦m−1n ) (f), = ∆d( trα,nf). proposition 11. for all f ∈ dn(rd) and g ∈ e(rd) (38) ∫ rd + trα,n(f)(y)g(y)dy = ∫ rd + f(y)rα,n(g)(y)dµα(y). proof. using (12), (33) and (37)∫ rd + trα,n(f)(x)g(x)dx = ∫ rd + trα+2n ◦m−1n f(x)g(x)dx = ∫ rd + m−1n f(x) trα+2n(g)(x)dµα+2n(x) = ∫ rd + f(x)mn(rα+2n(g))(x)dµα(x) = ∫ rd + f(y)rα,n(g)(y)dµα(y). generalized weinstein transform. throughout this section assume α > −1 2 and n a non-negative integer. for all λ = (λ1, ....,λd) ∈ cd and x = (x1, ....,xd) ∈ rd, put (39) φλ,α,n(x) = x2nd ψλ,α+2n(x) where ψλ,α+2n(x) is the weinstein kernel of index α + 2n is given by (5). proposition 12. φλ,α,n satisfies the differential equation (40) ∆d,α,nφλ,α,n = −‖λ‖2φλ,α,n. 24 proof. from proposition 1 and (39) we obtain ∆d,α,nφλ,α,n = mn ◦ ∆d,α+2nm−1n φλ,α,n, = mn ◦ ∆d,α+2nψλ,α+2n, = −‖λ‖2mnψλ,α+2n, = −‖λ‖2φλ,α,n. definition 8. the generalized weinstein transform is defined on l1α,n(r d +) by, for all λ ∈ rd (41) fw,α,n(f)(λ) = ∫ rd + f(x)φλ,α,n(x)dµα(x). remark 3. by (5), (13) and (41), we have (42) fw,α,n = fw,α+2n ◦m−1n . theorem 2. (i) inverse formula: let f ∈ l1α,n(rd+), if fw,α,n ∈ l1α(rd+) then (43) f(x) = c(α + 2n) ∫ rd + fw,α,nf(λ)φλ,α,n(x)dµα+2n(λ). (ii) plancherel formula: (44) ∫ rd + |f(x)|2dµα(x) = c(α + 2n) ∫ rd + |fw,α,nf(λ)|2dµα+2n(λ) where c(α + 2n) is given by (19). proof. by (20), (39) and (42) we obtain c(α + 2n) ∫ rd + fw,α,nf(λ)φλ,α,n(x)dµα+2n(λ) = c(α + 2n) ∫ rd + fw,α,nf(λ)x2nd ψλ,α+2n(x)dµα+2n(λ), = x2nd c(α + 2n) ∫ rd + fw,α+2n ( m−1n f ) (λ)ψλ,α+2n(x)dµα+2n(λ), = x2nd m −1 n f(x), = f(x). which proves (i). for (ii) an easily combination of (18), (39) and (42) shows that∫ rd + |f(x)|2dµα(x) = ∫ rd + |m−1n f(x)| 2dµα+2n(x), = c(α + 2n) ∫ rd + ∣∣fw,α+2n (m−1n f(λ))∣∣2 dµα+2n(λ), = c(α + 2n) ∫ rd + |fw,α,nf(λ)|2dµα+2n(λ). proposition 13. (i) for all f ∈ l1α,n(rd+), we have ‖fw,α,n(f)‖α,∞ ≤‖f‖α,n,1. 25 (ii) for all f ∈ sn(rd) we have fw,α,n(f)(y) = f0 ◦t rα,n(f)(y), ∀y ∈ rd+, where f0 is the transformation defined by (). (iii) for all f ∈ sn(rd) and m ∈ n, we have fw,α,n(∆d,αf)(λ) = −‖λ‖2fw,α,n(f)(λ). proof. from (14) and (42) we have ‖fw,α,n(f)‖α,n,∞ = ‖fw,α+2n ◦m−1n (f)‖α,n,∞ ≤ ‖m−1n f‖α+2n,1 ≤ ‖f‖α,n,1 which proves assertion (i). by (15), (36) and (42) we obtain fw,α,n(f) = fw,α+2n ◦m−1n (f) = f0 ◦ trα+2n ◦m−1n (f) = f0 ◦ trα,n(f), which proves assertion (ii). due to (16), (33) and (42) we have fw,α,n(∆d,α,nf)(λ) = fd,α+2n ◦m−1n (∆d,α,nf)(λ) = fw,α+2n ◦m−1n (∆d,α,nf)(λ) = fw,α+2n(∆d,α+2nm−1n f)(λ) = −‖λ‖2fw,α+2n ◦m−1n (f)(λ) = −‖λ‖2fw,α,n(f)(λ). generalized convolution product. definition 9. the generalized translation operators τxα,n, x ∈ rd associated with ∆d,α,n are defined on rd+ by (45) τxα,nf = x 2n d mnτ x α+2nm −1 n f where τxα+2n are the weinstein translation operators of order α + 2n given by (21). definition 10. the generalized convolution product of two functions f ∈ e(rd) and g ∈ d(rd) is defined by: (46) f ∗w,α,n g(x) = ∫ rd + τxα,nf(y)g(y)dµα(y); ∀x ∈ r d +. proposition 14. let f and g in dn(rd), we have (47) f ∗w,α,n g = mn ( m−1n f ∗w,α+2n m −1 n g ) . 26 proof. using (23) and (45) we get f ∗w,α,n g(x) = ∫ rd + τxα,nf(y)g(y)dµα(y) = ∫ rd + x2nd mnτ x α+2nm −1 n f(y)g(y)dµα(y) = x2nd ∫ rd + τxα+2nm −1 n f(y)m −1 n g(y)dµα+2n(y) = mn ( m−1n f ∗w,α+2n m −1 n g ) (x). proposition 15. (i) for all f ∈ lpα,n(rd), p ∈ [1,∞], and for all x ∈ rd+ (48) ‖τxα,nf‖p,α,n ≤ x 2n d ‖f‖p,α,n. (ii) (49) τxα,n(φλ,α,n(y)) = φλ,α,n(x)φλ,α,n(y). proof. from (24) and (45) we have ‖τxα,nf‖p,α,n = x 2 d‖mnτ x α+2nm −1 n f‖p,α,n = x2d‖τ x α+2nm −1 n f‖p,α+2n ≤ x2d‖τ x α+2nm −1 n f‖p,α+2n ≤ x2d‖m −1 n f‖p,α+2n = x2nd ‖f‖p,α,n. which give (i). from (39), (45) and proposition 6 we get τxα,nφλ,α,n(y) = x 2n d mn ◦ τ x α+2n ◦m −1 n φλ,α,n(y) = x2nd mn ◦ τ x α+2nψλ,α,n(y) = x2nd y 2n d τ x α+2nψλ,α,n(y) = x2nd y 2n d ψλ,α,n(x)ψλ,α,n(y) = φλ,α,n(x)φλ,α,n(y). which prove (ii). theorem 3. (i) for f ∈ s(rd) and y ∈ rd+ (50) fw,α,n ( τxα,nf ) (λ) = φλ,α,n(x)fw,α,n(f(λ)), λ ∈ rd+. (ii) for all f ∈ e(rd) and g ∈ s(rd) (51) ∫ rd + τxα,nf(y)g(y)dµα(y) = ∫ rd + f(y)τxα,ng(y)dµα(y). (iii) for all f,g ∈ l1α(rd+), f ∗w,α,n g ∈ l1α(rd+), and (52) fw,α,n (f ∗w,α,n g) = fw,α,n(f)fw,α,n(g). 27 proof. an easily combination of (25), (39), (42) and (45) shows that fw,α,n ( τxα,nf ) (λ) = x2nd fw,α+2n ( τxα+2nm −1 n f ) (λ), = x2nd ψλ,α+2n(x)fw,α+2nm −1 n (f)(λ), = φλ,α,n(x)fw,α,n(f(λ)). which prove (i). for assertion (ii) using (23) and (45) we obtain∫ rd + τxα,nf(y)g(y)dµα(y) = x 2 d ∫ rd + τxα+2n ( m−1n f(y) )( m−1n g(y) ) dµα+2n(y), = x2d ∫ rd + ( m−1n f(y) ) τxα+2n ( m−1n g(y) ) dµα+2n(y), = ∫ rd + f(y)τxα,ng(y)dµα(y). which prove (ii). for the last assertion using (47) we get f ∗w,α,n g = mn [ (m−1n f) ∗w,α+2n (m −1 n g) ] using (27) and (42) we get fw,α,n(f ∗w,α,n g) = fw,α,n ◦mn [ (m−1n f) ∗w,α+2n (m −1 n g) ] = fw,α+2n ◦m−1n ◦mn [ (m−1n f) ∗w,α+2n (m −1 n g) ] = fw,α+2n [ (m−1n f) ∗w,α+2n (m −1 n g) ] = fw,α+2n(m−1n f)fw,α+2n(m −1 n g) = fw,α,n(f)fw,α,n(g). references [1] r. f. al subaie and m. a. mourou, transmutation operators associated with a bessel type operator on the half line and certain of their applications, tamsui oxford journal of information and mathematical sciences, 29(2013), 329-349. [2] hassen ben mohamed, nèji bettaibi, sidi hamidou jah, sobolev type spaces asociated with the weinstein operator, int. journal of math. analysis, 5(2011), , 1353-1373. [3] hatem mejjaoli, ahsaa, makren salhi, uncertainty principles for the weinstein transform, czechoslovak mathematical journal, 61(2011), 941-974. [4] youssef othmani and khalifa trimèche, real paley-wiener theorems associated with the weinstein operator, mediterr. j. math. 3(2006), 105-118. department of mathematics, faculty of sciences aïn chock, university of hassan ii, casablanca 20100, morocco ∗corresponding author 28 int. j. anal. appl. (2023), 21:23 received: dec. 25, 2022. 2020 mathematics subject classification. 91b70. key words and phrases. e-commerce; awareness; perceived value; online shopping. https://doi.org/10.28924/2291-8639-21-2023-23 © 2023 the author(s) issn: 2291-8639 1 an empirical research on customers’ awareness of e-commerce in the context of vietnamese developing economies nhu-ty nguyen1,2, thanh-tuyen tran3,*, anh-quan huynh1,2 1school of business, international university, vietnam 2vietnam national university hcmc, quarter 6, linh trung ward, thu duc district, hcmc, vietnam 3nguyen tat thanh university, hcmc, vietnam *corresponding author: tttuyen@ntt.edu.vn abstract. many research studies have been conducted in the field of e-commerce, which can contribute for academic background and development of e-commerce around the world. in vietnam, the awareness of customers toward ecommerce is limited for the emerging of this industry in vietnam recently. this study tries to figure out research model explains the factors and to study the awareness of customers that influence the behaviors and acceptance of users buying online in the developing economy. according to the research, buying online is quite risky. therefore, product safety would be the purchasing decision's priority. online sales providers need to provide a safe way of ensuring consumers ' purchasing process. through hard qualification process, they need to shortlist the good quality product. the sale would grow dramatically as consumers gained confidence. not only should they care about the short-term profit in order to get the products of low quality. i. introduction although there are many theories in the world today, the research model explains the factors that affect online buyers ' behavior and acceptance. however, it is difficult to apply the environment in vietnam due to the differences in the economic model, culture and society, while the country's research models are quite limited ([19]). therefore, it is urgent to study models in the world that have been based on domestic research in the past to build an appropriate model for vietnam's https://doi.org/10.28924/2291-8639-21-2023-23 2 int. j. anal. appl. (2023), 21:23 current situation. consumers in vietnam are still used to product trials or direct purchases because of low-quality psychology compared to online purchasing lustration images ([22]). an effective marketing strategy should be based on a rigid basis for understanding market segmentation expectations, cognition, decision-making, trends and behaviour. the internet has become the destination for business and mass media in recent years, leading to a dramatic change in the way people purchasing. in addition to the rapid development and national coverage of internet service providers, customers are now truly free from the constraints of brick-and-mortar stores and adapt quickly to the new era of online shopping ([21]). although there are now many theories in the world, the research model explains the factors that influence the behavior and acceptance of users buying online. but the application to the environment in vietnam is difficult due to differences in economics, culture and society, while the research models in the country are quite small ([20]). ii. liturature review 2.1. the concept of e-commerce e-commerce means the distribution of goods, services, information or payments via computer networks or other electronic means. e commerce is an application for technology that automates business transactions. e commerce is a tool that helps businesses; customers reduce service costs, improve the quality of the product, and speed up service. in short, e commerce is a business model that is enabled by information technology. there are two major e-commerce models: b2b (business-to-business) and b2c (business-toconsumer). b2b is an e-commerce model in which participants are businesses or organizations. currently, the majority of e-commerce is implemented under this model. b2c is the e-commerce model in which the business sells to the consumer directly. in addition to the two main models, the internet also forms many new models: c2c (consumer-to-consumer): personal sales direct to individuals such as car applications, consumer goods, real estate and software. c2b (consumer-to-business): individuals can find businesses to sell (goods, software) to businesses. g2c (government-to-citizens): state-owned organizations purchase and sell goods, services and information to businesses and citizens. 3 int. j. anal. appl. (2023), 21:23 online banking: access to banking services on individuals or businesses from commercial services online or via the internet. this thesis shall focus on business to customer (b2c) e-commerce. suggestion research model figure 1: proposed research model describe the elements in the proposed research model variable definition reference perceived usefulness the online shopping will bring convenience to consumers, they are no longer limited time and place when shopping. [18] peu interoperability between webbased and consumer web sites, easily handled when performing searches and transactions. [23] moon ji won et al., 2001 [24] venkatesh et al., 2003 4 int. j. anal. appl. (2023), 21:23 perceived enjoyment sense of the user when using the online shopping service. [23] moon ji won et al., 2001 perceived risk risks arising during the transaction such as passwords, online fraud, loss of credit card. [25] joongho ahn et al., 2001 reference group the introduction of the use of friends, relatives and coworker. [24] venkatesh et al., 2003 perceived price the price of the product on the web compared to the price at the store is a factor of concern consumers over the network. [18] 2.2. developmental hypothesis perceived usefulness: [18] also stated that consumers find online shopping timesaving, effortless and convenient. as a result, the study posits the following hypothesis: h1: perceived usefulness of e-buying affect purchase intention of customers positively (+) perceived ease of use (peu): peu is the degree of effortlessness a person believes a particular system could provide (davis 1985). in this study, the peu demonstrates that users are easy to get acquainted with, use electronic purchases online and become easy to become a proficient user of the service. therefore, it can be hypothesized that: h2: user’s peu of e-commerce will positively (+) influence their intention to purchase. perceived enjoyment: is defined as the enjoyment degree of using technology perceived by a person apart from any probable anticipation caused by performance consequences [26] . in a research by moon and kim (2001) [23], pleasure perception expresses three components: concentration, curiosity, and enjoyment. they also discovered that pleasure was the premise of the inner motivation of using the world-wide-web, and confirmed that the intrinsic motivation was strongly correlated with the decision to use the internet-based system. thus, the expectation that perceptions of pleasure affect the acceptance of e-buying services online. from that, it can be hypothesized that: 5 int. j. anal. appl. (2023), 21:23 h3: perceived enjoyment positively affects (+) intention to purchase. perceived risk: in risk-focused e-commerce adoption model, the perception of risk associated with the product or service reflects the consumer's anxiety about the use of the online product. the risks associated with using the online purchase service include personal information loss, account loss, credit card information, the actual product does not conform to the advertisement. h4: perceived risk negative effects (-) intention of consumer reference group: the concept of social influence is defined as the extent to which users perceive that other important people believe they should use new systems, information technology products. following to unified technology acceptance and use technology model of venkatesh et al. 2003 [24], social influence has a positive influence on intention. in this study, social influences are manifested by the perception that people around them, such as family, friends, co-workers or other institutions, influence the intention to use the service online purchases of their electronics. therefore, it can be hypothesized: h5: reference group relate positively (+) to users ‘intention to purchase. perceived price: the price is what consumers pay to get the desired product or service. perceived price is the consumer's perception of what you're going to sell at a cost. consumers will feel the price on two sides: the cost of money must be spent and the opportunity cost of abandoning the use of that money to buy other products or services. according to the "factors affecting online consumers" model, [18] mentioned that consumers believe that online purchases will save money and and be comparable in price. h6: perceived price have a positive (+) effect on the consumer's willingness to buy e-electronic products. iii. methodology methodology research perceived price scale perceived price refers to the extent to which an individual believes that using electronic purchase ecommerce will help them save money and be comparable in price to shopping. according to the model "factors affecting online consumers," [18] and "a study on the online purchasing behavior of women," eliasson malin (2009) [27] used the observation variables to measure the concept of "perceived price." after a qualitative study, the preliminary scale consists of 3 observation variables added: "e-commerce promotions help me save money." 6 int. j. anal. appl. (2023), 21:23 table 3. 1 scale of perceived price code research question price 01 the price of electronic goods on e-commerce cheaper than the price at the store price 02 using e-commerce services makes it easy for me to compare prices price 03 using e-commerce services helps me save on transport expenses to view the goods price 04 the promotions on e-commerce help me save money perceived usefulness scale perceived usefulness refers to the level of an individual who believes that using e-commerce for electronic purchase will help them gain benefits in work and life. following “factors affecting online consumer behaviour” model of [18] use the observation variables to measure the concept of "perceived usefullness ". the "perceived usefulness" scale initially had four observational variables. this scale has nothing to do with the original. table 3. 2 scale of perceived usefulness code research question conve_05 e-commerce service useful, saving time conve_06 e-commerce shopping service that helped me find information about the product quickly conve_07 using e-commerce service helps me buy products anywhere conve_08 using e-commerce service helps me buy products anytime perceived ease of use scale peu is defined as the level of ease related to the usage of system, it products. the "peu" preliminary scale has 4 observation variables. through a qualitative study to eliminate the variable " learning how easy it is to use e-commerce" and is replaced by "the features on e-commerce are extremely clear and easy to understand." 7 int. j. anal. appl. (2023), 21:23 table 3. 3 scale of peu code research question peasy_09 account registration procedures, e-commerce purchase and payment process is quite simple peasy_10 easy to find the product you want to use e-commerce peasy_11 the features on e-commerce web are extremely clear and easy to understand peasy_12 online shopping is easily compare one product to the others and make decision from that reference group scale social influences reflect the influence and impact of the surrounding people in encouraging and supporting users to use electronic purchase services online. the initial scale consists of four observation variables. this scale has nothing to do with the original. table 3. 4 scale of reference group code research question soinf_13 every member of the family uses e-commerce, so i use it soinf_14 friends, colleagues, customers recommend me to use the ecommerce service soinf_15 the media advertise e-commerce, so i join and try it out soinf_16 i would recommend that my friends shop online as well perceived enjoyment according to research by moon and kim (2001) [23], pleasure perception expresses three components: concentration, curiosity and enjoyment. the initial "perceived enjoyment" scale was initially proposed to include four variables. through qualitative research, rejecting the statement "using the internet every day is my hobby" as it does not focus on the interest of the e-commerce user. replaced by ” promotion on e-commerce attracts me a lot”. referring to the research by moon and kim (2001) [23], the preliminary scale and the observed variables for the "perceived enjoyment" component are as follows: 8 int. j. anal. appl. (2023), 21:23 table 3. 5 scale of perceived enjoyment code research question enjoy_17 i have accounts on multiple shopping sites enjoy_18 i spend over 2 hours daily use online shopping websites enjoy_19 whenever i find a good item on e-commerce i feel very excited enjoy_20 promotion on e-commerce attracts me a lot perceived risk in risk-focused e-commerce adoption model, the perception of risk associated with the product or service reflects the consumer's anxiety about the use of the online product. use the following observation variables to measure “perceived risk” when using e-commerce for electronic purchase. qualitative research helps to make words easier to understand, so there is no change in the variables. table 3. 6 scale of perceived risk code research question prisk_21 the quality of goods received from online purchases is lower than the advertised information prisk_22 i'm so worry that my personal information would be leaked out to the third party prisk_23 online purchases may not receive the item prisk_24 my credit card information may not be secure intention to purchase intention to use refers to the intention of the user will continue to use or will use e-commerce services. thus, one additional observation variable "i intend to use”. finally, the intended scale is as follows: table 3. 7 scale of intention to purchase code research question inten_25 i intend to use (or continue to use) e-commerce in the future inten_26 i will definitely use e-commerce inten_27 i will learn to use e-commerce in the future inten_28 i will introduce e-commerce for many people to use 9 int. j. anal. appl. (2023), 21:23 summary of qualitative research results qualitative research helped to calibrate the scale for the following research models: • edit words in scale to make them easier to understand • add 5 observation variables, remove 2 observation variables. • finally, the "research of factors affects purchasing decisions on e-commerce" model uses six conceptual components that influence the intention to use and a total of 28 observed variables in this model. quantitative research quantitative research was conducted through questionnaires. the results are used to evaluate the reliability and validity of the scale, test the scale, verify the fit of the model. data collection data collection was conducted using open questionnaire interviews. for research person who has a stable job and over 22 years old. the questionnaire survey was conducted as follows: design the questionnaire online and send the link to the survey respondent’s online, information recorded in the database. place of research: ho chi minh city research time: may. 2019 the variables used for this concept will be measured by the 5 points likert scale: • strongly disagree • disagree • neutral • agree • strongly agree likert type scales or frequencies use fixed-choice feedback formats and are designed to measure attitudes or opinions in the survey. (bowling, 1997; burns, & grove, 1997) data analysis the sequence of data analysis is as follows: step 1 prepare information: retrieve the answer sheet, filter the information, encrypt the necessary information in the answer table, input and analyze the data using spss 2.0. step 2 statistical: conduct statistics describing the data collected. step 3 estimate the reliability: cronbach alpha analysis was performed. 10 int. j. anal. appl. (2023), 21:23 step 4 efa analysic: scale analysis by efa analysis step 5 multivariate regression analysis: performed multivariate regression analysis and validated the hypotheses of the model with a significance level of 5%. analyze data based on demographic variables to analyze the differences between the following groups: male and female; high income and low income; young and old. iv. data analysis data description analysis questionnaires have been given to 550 people for the survey. following the elimination of invalid sa mples with missing information or from participants outside the target age range, 467 remains are up to quantitative analysis. table 4. 1 result of respondents. total samples valid samples invalid samples online survey 330 272 58 offline survey 220 195 25 total 550 467 83 4.1. gender table 4. 2 sample analysis by gender gender quantity percent male 210 44.97% female 257 55.03% total 467 the number of women using online electronics purchase is higher than that of men, according to the figures in the sample. specifically: 44.97% female and 55.03% male. 4.2. age 11 int. j. anal. appl. (2023), 21:23 table 4. 3 sample analysis by age age group quantity percent 22-24 27 5.78% 25-27 208 44.54% 28-30 85 18.20% 31-40 126 26.98% over 40 21 4.50% total 467 the most distributed group in the age group is 25-27 (about 44.54%), the age from 31 to 40 accounted for 26.98%, the age group 28-30 accounted for 19.20 % and lowest among the age group of 22-24 and over 40 (following by 5.78% and 4.50% respectively). 4.3. e-commerce access time table 4. 4 average time per visit e-commerce access time quantity percent never use 46 9.85% under 10 minutes 58 12.42% 10-30 minutes 287 61.46% over 30 minutes 76 16.27% total 467 although they know about e-commerce, only 9.85% of respondents (46 respondents) have never purchased online. most observers use online shopping sites over an average of 10-30 minutes at 61.46% and over 30 minutes at 16.27%. 4.4. internet habits table 4. 5 figures internet habits internet habit quantity percent under 3 years 1 0.21% 3-5 years 86 18.42% 5-7 years 182 38.97% over 7 years 198 42.40% total 467 12 int. j. anal. appl. (2023), 21:23 among 467 observed targets, internet users over 7 years is made up for almost half of number (42.40% of the targets) and only 0.21% that use internet less than 3 years. 4.5. reliability test 4.5.1. evaluation criteria cronbach's alpha analysis is a statistical test of the degree of correlation between the items in the scale. this is a necessary reflection scale analysis that is used to exclude inappropriate variables before analyzing the efa. acceptable range for exploration purposes when the alpha value of cronbach is 0.6. variable coefficient — sum is the variable correlation coefficient with the average of other variables in the same high group. coefficient of correlation-the sum must be greater than 0.3. variables with variable correlation-rubbish variable less than 0.3 is considered and removed from the scale. 4.5.2. cronbach’s alpha analysis results table 4. 6 cronbach's alpha analysis results factors item average intertem covariance standard deviation corrected item total correclation crombach's alpha if item deleted p e rc e iv e d p ri c e price_01 3.52 0.920 0.681 0.731 price_02 3.94 0.892 0.542 0.799 price_03 3.57 0.832 0.662 0.743 price_04 3.49 0.868 0.620 0.761 crombach's alpha: 0.808 p e rc e iv e d u se fu ln e ss conve_05 3.26 1.127 0.531 0.672 conve_06 3.45 1.035 0.718 0.654 conve_07 3.91 .843 0.478 0.778 conve_08 3.56 1.027 0.638 0.699 crombach's alpha : 0.781 p e u peasy_09 3.14 1.025 0.594 0.715 peasy_10 2.82 1.098 0.587 0.716 13 int. j. anal. appl. (2023), 21:23 peasy_11 3.21 1.244 0.531 0.748 peasy_12 3.15 1.185 0.610 0.703 crombach's alpha: .744 r e fe re n c e g ro u p soinf_13 3.16 1.224 0.720 0.745 soinf_14 3.10 1.170 0.714 0.748 soinf_15 3.47 1.081 0.594 0.803 soinf_16 3.33 1.100 0.574 0.812 crombach's alpha: 0.825 p e rc e iv e d e n jo y m e n t enjoy_17 3.79 .990 0.753 0.823 enjoy_18 3.71 .926 0.743 0.821 enjoy_19 3.71 1.030 0.793 0.797 enjoy_20 3.54 1.090 0.614 0.875 crombach's alpha: 0.867 p e rc e iv e d r is k prisk_21 2.81 1.186 0.706 0.738 prisk_22 2.67 1.357 0.755 0.706 prisk_23 2.70 1.346 0.721 0.724 prisk_24 2.96 1.283 0.390 0.874 crombach's alpha: 0.815 in te n ti o n t o u se inten_25 3.46 0.818 0.606 0.779 inten_26 3.39 0.819 0.696 0.737 inten_27 3.55 0.869 0.677 0.745 inten_28 3.51 0.803 0.558 0.800 crombach's alpha: 0.814 14 int. j. anal. appl. (2023), 21:23 comment: the concept of components has a coefficient of cronbach alpha higher than 0.6. the lowest is the "peu" component with a coefficient of 0.774 for the cronbach alpha and the highest is "perceived enjoyment" (0.867). this shows that in the same concept the variables are closely related to one another. 4.5.3. exploratory factor analysis (efa) the conceptual scale in the satisfactory model in the reliability assessment will be used in the efa analysis. 4.5.3.1. standard analysis this study uses the principal component extraction method with varimax rotation and stopping when extracting elements with eligen values higher than or equal to 1 for 24 variables of measurement observation. 4.5.3.2. first efa analysis at the first efa analysis, remove the soinf_16 and prisk_24 variables because of factor coefficients of <0.5. conducted the second efa analysis with the remaining 26 variables. hypothesis h0: the observed variables have no correlation in the whole. barlett test: sig = 0.000 <5%. rejection of h0, the observed variables in efa analysis are correlated in overall. • kmo = 0.883> 0.5: factor analysis is required for analytical data. • there are six factors extracted from the efa analysis with: eigenvalues of all factors are> 1: qualified observed variables have load coefficients > 0.5: qualified total variance value = 69.364% (> 50%): efa factor analysis was satisfactory. it can be said that these 6 factors explained 69.364% of the variance of the data. table 4. 7 the second factor loadings efa rotated component matrix a component code 1 2 3 4 5 6 7 enjoy_19 .847 enjoy_18 .816 15 int. j. anal. appl. (2023), 21:23 enjoy_17 .810 enjoy_16 .604 price_01 .842 price_03 .794 price_04 .754 price_02 .736 inten_26 .774 inten_25 .750 inten_27 .730 inten_28 .658 conve_06 .811 conve_07 .738 conve_08 .664 conve_05 .596 peasy_12 .778 peasy_11 .722 peasy_10 .680 peasy_09 .658 prisk_22 .944 prisk_23 .934 prisk_21 .750 soinf_13 .794 soinf_14 .758 soinf_15 .653 extraction method: principal component analysis. rotation method: varimax with kaiser normalization. 16 int. j. anal. appl. (2023), 21:23 the results of the second efa analysis showed that seven factors were extracted from these factors, corresponding to six original independence concepts: perceived enjoyment, perceived price, peu, reference group, perceived risk, perceived usefulness and 1 dependence concept is purchasing intent. factor analysis results show that these factors ' observed variables have a good factor load factor (0.596 and above) and the alpha coefficients of cronbach are higher than 0.7. so the model will still consist of 6 conceptual component elements as the proposed model after calibration. 4.6. research model after measuring scale there is no change in the composition of the intended use of e-commerce from the results of the analysis. the research model will remain the same as the original proposal: 6 independent variables are the variables that influence the intention to use e-commerce and the intention to buy is one dependent variable. table 4. 8 abstract hypothesis in the research model hypothesis content h1 perceived price have a positive (+) effect to users’intention to use e-commerce h2 perceived usefulness of e-buying relate positively (+) to users’intention to use e-commerce h3 user’s peou of e-commerce will positively (+) to users’intention to use e-commerce h4 perceived enjoyment positively affects (+) to users’intention to use e-commerce h5 reference group relate positively (+) to users’intention to use e-commerce h6 perceived risk negative affects (-) to users’intention to use e-commerce 17 int. j. anal. appl. (2023), 21:23 4.7. regression analysis and hypothesis 4.7.1. correlation analysis between inten and independent variables such as perceived price (price), perceived usefullness (conve), peu (peasy), perceived enjoyment (enjoy), reference group (soinf), perceived risk (prisk), correlational analysis is carried out. at the same time, it is also analyzed the correlation between the independent variables to find the strong correlation between the independent variables. because such correlations can have a major impact on the result of regression analysis as they result in multi-co linearity. pearson correlation analysis results as shown below: table 4. 9 pearson correlation analysis correlations inten price conve peasy soinf enjoy prisk pearson correlation 1.000 .237** .393** .435** .454** 389** -.276** sig. (2-tailed) .000 .000 .000 .000 .000 .000 n 467.000 467 467 467 467 467 467 pearson correlation .237** 1.000 .052 .026 .104* -.139** -.025 sig. (2-tailed) .000 .262 .574 .025 .003 .596 n 467 467.000 467 467 467 467 467 pearson correlation .393** .052 1.000 .357** .584** .480** -.086 sig. (2-tailed) .000 .262 .000 .000 .000 .063 n 467 467 467.000 467 467 467 467 pearson correlation .435** .026 .357** 1.000 .436** .412** -.156** sig. (2-tailed) .000 .574 .000 .000 .000 .001 n 467 467 467 467.000 467 467 467 pearson correlation .454** .104* .484 .436** 1000 .468** -.108* sig. (2-tailed) .000 .025 .000 .000 .000 .020 n 467 467 467 467 467.000 467 467 pearson correlation .389** -.139** .480** .412** .486** 1.000 -165** sig. (2-tailed) .000 .003 .000 .000 .000 .000 n 467 467 467 467 467 467.000 467 pearson correlation -.276** -.025 -086 -.156** -.108* -.165** 1.000 sig. (2-tailed) .000 .596 .063 .001 .020 .000 n 467 467 467 467 467 467 467.000 prisk inten price conve peasy soinf enjoy comment: independent variables have a strong linear correlation with dependent variables and statistically significant correlation coefficients (p<0.01). 18 int. j. anal. appl. (2023), 21:23 4.7.2. regression analysis multivariate regression results as shown in table 4.10 table 4. 10 the coefficients of the independent variables in the multivariate regression model sumary b model r r square adjusted r square std.error of the estimate durbin watson 1 620 .384 .376 .52246 2.076 a. predictors: (constant), prisk, price, conve, peasy, enjoy, soinf b. dependent variable: inten anova b model sum of squares df mean square f sig. 1 regression 78.422 6 13.070 47.883 .000 a residual 125.563 460 .273 total 203.985 466 a. predictors: (constant), prisk, price, conve, peasy, enjoy, soinf b. dependent variable: inten coefficients a unstandardized coefficients standardized coefficients collinearity statistics model b std.error beta t sig. tolerance vif 1 (constant) 1.289 .205 6.327 .000 price .214 .036 .225 5.991 .000 .939 1.065 conve .087 .040 .104 2.199 0.28 .599 1.067 peasy .163 .032 .217 5.121 .000 .744 1.345 soinf .118 .033 .177 3.634 .000 .567 1.764 enjoy .131 .036 .169 3.677 .000 .636 1.572 prisk -.103 .021 -.181 4.843 .000 .962 1.040 a. dependent variable: inten comment: relevance of the model: thus, the modified r2 model is 0.376, which means that 37.6% of the variance of the intended use (inten) is explained by the variation of the 19 int. j. anal. appl. (2023), 21:23 components: perceived price (price), perceived usefullness (conve), peu (peasy), perceived enjoyment (enjoy), reference group (soinf), perceived risk (prisk). test the hypothesis of model fit: hypothesis h0: β1 = β2 = β3 = β4 = β5 = 0 (all partial regression coefficients = 0) sig(β1), sig(β2), sig(β3), sig(β4), sig(β5), sig(β6) < significance (5%), the independent variables are price, peasy, soinf, enjoy, prisk with significant statistical significance at 5% significance. multi-collinearity testing: vif values <10: multiplicity phenomena of independent variables do not affect the interpretation of the model. remainder: from normal normalized frequency histogram (appendix ii. 2.4) with mean value = 1.5 * 10-16≅ 0; standard deviation = 0.994 ≅1: distribute the remainder of the form near the standard, satisfying the hypothetical assumption of the normal distribution of the residual. the durbin-watson coefficient of 2.076 shows that the errors in the model are independent of each other. 4.7.3. hypothesis verification perceived price hypothesis h1: perceived price have a positive (+) effect to users’intention to use e-commerce the reference standard deviation β1= 0.226, sig (β1)= 0.000 <5%: support the hypotheis h1 comment: survey results show that "perceived price" has a positive (+) effect on the intention of users to use e-commerce. the more about the price, the more interested users intend to use the ecommerce service. perceived usefulness hypothesis h2: perceived usefulness of e-buying relate positively (+) to users’s intention to use ecommerce. the refercence standard deviation β2 = 0.104, sig (β2) = 0.000 < 5% : support the hypothesis h2. comment: thus, e-buying perceived usefulness is positively (+) related to the intention of users to use e-commerce. the more convenience sellers bring to buyers, the more online shopping services they would use. peu hypothesis h3: user’s peou of e-commerce will positively (+) to users’intention to use ecommerce. the reference standard deviation β3 = 0.217, sig (β3) = 0.000 <5%: support the hypothesis h3. 20 int. j. anal. appl. (2023), 21:23 comment: so that user’s peou of e-commerce will positively (+) to users’s intention to use ecommerce that means that when the user realizes that the functionality and operation of ecommerce is easy to use, the intention to use the service for consumers will increase. perceived enjoyment hypothesis h4: perceived enjoyment positively affects (+) to users’intention to use e-commerce. the reference standard deviation β4 = 0.169, sig (β4) = 0.000 < 5% : support the hypothesis h4. comment: as the hypothesis mention, perceived enjoyment positively affects (+) to users’intention to use e-commerce. perceived enjoyment when buying electronic on e-commerce as an internal motive increases the individual's intention. when users find that content and activities on ecommerce websites are interesting, their intentions will increase. reference group hypothesis h5: reference group relate positively (+) to users’intention to use e-commerce. the reference standard deviation β5 = 0.117, sig (β5) = 0.000 < 5%: support the hypothesis h5. comment: positively (+) refers to the intention of users to use e-commerce. in other words, in the survey, the impact of the people around them affected consumers. the more influential people (family, family, colleagues, etc.) support and encourage, the greater the intention of consumers to use the e-commerce service. perceived risk hypothesis h6: perceived risk negative effects (-) to users’ intention to use e-commerce. the standard deviation β6 = -0.181, sig (β6) = 0.000 < 5%: support the hypothesis h6. comment: as we can see perceived risk negative effects (-) to users ‘intention to use ecommerce, consumers are more aware of the risk, the less they intend to use. today, cyber security is a top issue for online shopping sites in vietnam, which has had a negative impact on consumers' purchasing decisions. table 4. 11 summary of hypothesis verify results hypothesis content result h1 perceived price have a positive (+) effect to users’intention to use e-commerce support h1 h2 perceived usefulness of e-buying relate positively (+) to users’intention to use e-commerce support h2 21 int. j. anal. appl. (2023), 21:23 h3 user’s peou of e-commerce will positively (+) to users’intention to use e-commerce support h3 h4 perceived enjoyment positively affects (+) to users’intention to use e-commerce support h4 h5 reference group relate positively (+) to users’intention to use e-commerce support h5 h6 perceived risk negative affects (-) to users’intention to use e-commerce support h6 4.8. discriminant analysis in demographic variables income discrimination income differences hypothesis: • hypothesis h3,0: in terms of income, there is no difference in intention to use • hypothesis h3,1: in terms of income, there is no difference in perceived price • hypothesis h3,2: in terms of income, there is no difference in perceived usefullness • hypothesis h3,3: in terms of income, there is no difference in peu • hypothesis h3,4: in terms of income, there is no difference in social influence • hypothesis h3,5: in terms of income, there is no difference in perceived enjoyment • hypothesis h3,6: in terms of income, there is no difference in perceived risk homogeneity tests of the price and enjoy components 0.0%, 0.9% respectively, were higher than 5%, indicating the variance of income was equal to the anova analysis. homogeneity test of components inten = 78.9%, conve = 5.5%, peasy = 39.0%, soinf = 48.4%, prisk = 56.6% <5% variance of income is not equal, does not satisfy the anova analysis conditions. among factors that satisfy the condition of anova can be seen: sig (inten) = 1.9%, sig (soinf) = 4.1%< 5% rejection of h3,0 & h3,4 : there are difference in income group in the intention to use e-commerce and social influence. in particular, the intention to use low-income group e-commerce tends to be lower than that of high-income groups, while low-income groups tend to be more vulnerable than high-income groups to social impact. this may be due to the higher income group being more exposed to using e22 int. j. anal. appl. (2023), 21:23 commerce services, so their intentions to buy e-commer are higher. moreover, the level of impact from their social impact is lower than that of low-income groups due to more exposure to information sources. sig (conve) = 31.2%, sig(peasy) = 19.5%, sig(prisk) = 23.8% > 5% there is no basis for h3,2, h3,3, h3,6 rejection: meaning there is no evidence of income group differences with peu, perceived usefulness, and perceived risk. this can be explained by the fact that in recent years only e-commerce service has grown, making online payments one of the key strengths of e-commerce has not yet become widely known. therefore, there is no difference between income groups in ease of use, usefulness as well as risk. 4.9. conclusion section 4 presented information on survey samples, cronbach’s alpha and efa analysis, multivariate regression analysis, and control variables. the information from the observation sample showed that the sample was young, ranging from 25 to 27 years old. most of them have knowledge of using the internet, have knowledge of using ecommerce. the cronbach alpha reliability and efa analysis of soinf_16 and prisk_24 variables. v. conclusion the study suggests that e-commerce providers can improve and enhance customer service, depending on the extent to which each factor influences the intention to use electronic purchases. 5.1.1. perceived price service providers therefore need to pay attention to price-related factors in order to attract consumers in order to improve the intention to buy electronic goods through the user network. 5.1.2. perceived usefulness service providers need to improve the purchasing process, simple, convenient, complete installation information and guarantee to improve the intention to use consumer purchasing e-commerce services. there will also be a significant reduction in online purchases if you do not pay online. moreover, wider advertising is needed to enable consumers to see the convenience of buying electronics online. 5.1.3. peu this demonstrates that consumers are very keen on ease of use. however, there was not a high level of consumer feeling in the survey (from 2.82 to 3.21). in order to raise awareness of e23 int. j. anal. appl. (2023), 21:23 procurement providers ' ease of use, it is important to provide complete user information, displaying instructions in the process at prominent locations; access website. 5.1.4. perceived enjoyment the survey results agree with the perceived pleasure statements that the user's approval level for this factor is quite high (3.54 to 3.79 on average). this demonstrates that e-commerce is now a new trend that attracts attention to all ages. therefore, when implementing advertising programs to promote their services, e-commerce service providers need to pay attention to the aspect of exploring and discerning. 5.1.5. reference group as friends and relatives, colleagues, partners, media, the audience can influence consumers. the survey results agree with the social impact statements that the user's level of consent to family and family impact is low (from 3.10 to 3.16), while the organization's level of impact is higher (3.47), the data does not support media impact information. e-commerce providers should therefore focus on marketing programs for teams, organizations and referral discounts. consumers will introduce and invite their friends and colleagues to participate in promotional activities. 5.1.6. perceived risk the results show that users do not agree with perceived enjoyment (average of observation variables from 2.57 to 2.94) the impact of risk awareness tends to be greater for women, so online e-commerce providers need to have policies tailored to women in order for them to feel secure when purchasing electrical goods. death through the network. 5.2. contribution of research research with theoretical and practical contribution in online trading in vietnam 5.2.1. contribution in theory based on the utaut model combination (venkatesh et al., 2003) [24] model "factors affecting online consumers", e-cam (joongho ahn et al., 2001) [25] and extended tam for www (moon ji won & kim young gul, et al., 2001) [23], this research provided a more comprehensive overview of research and survey compared to a single model. on the other hand, it is designed for developed countries to measure this research. however, through practical data in ho chi minh city and surrounding provinces, this measurement is modified and evaluated to match the vietnamese 24 int. j. anal. appl. (2023), 21:23 environment. the data will play a role in measurement theory that will help academic and applied researchers gain a better understanding of the vietnamese market. 5.2.2. contribution in application this research has opened up a path for service providers to carry on similar research with other products such as: magazines, movies ... and eventually, service providers in both developing countries and vietnam will be able to improve e-commerce service. 5.3. proposal for e-commerce business 5.3.1. increase spending benefit for consumers. the business needs to concentrate on building up its customer services: the third party would organize fastest delivery activities, cod activities across the country. 5.3.2. minimize the risk for the purchasers when they use online sale services according to the research, buying online is quite risky. therefore, product safety would be the purchasing decision's priority. online sales providers need to provide a safe way of ensuring consumers ' purchasing process. through hard qualification process, they need to shortlist the good quality product. the sale would grow dramatically as consumers gained confidence. not only should they care about the short-term profit in order to get the products of low quality. that is the easiest way to kill their business. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] i. ajzen, the theory of planned behavior, organ. behav. human dec. processes. 50 (1991), 179–211. https://doi.org/10.1016/0749-5978(91)90020-t. [2] i. ajzen, explaining intentions and behavior, attitudes, personality and behavior, open university press, poland, second edition, (2005), pp.117-141. [3] a. haslinger, s. hodzic, c. opazo, consumer behaviour in online shopping, thesis, kristiantad university, (2007). [4] y.h. chen, i.c. hsu, c.c. lin, website attributes that increase consumer purchase intention: a conjoint analysis, j. bus. res. 63 (2010), 1007–1014. https://doi.org/10.1016/j.jbusres.2009.01.023. [5] b.j. corbitt, t. thanasankit, h. yi, trust and e-commerce: a study of consumer perceptions, electron. commerce res. appl. 2 (2003), 203–215. https://doi.org/10.1016/s1567-4223(03)00024-3. https://doi.org/10.1016/0749-5978(91)90020-t https://doi.org/10.1016/j.jbusres.2009.01.023 https://doi.org/10.1016/s1567-4223(03)00024-3 25 int. j. anal. appl. (2023), 21:23 [6] l.j. cronbach, test “reliability”: its meaning and determination, psychometrika. 12 (1947), 1–16. https://doi.org/10.1007/bf02289289. [7] f.d. davis, perceived usefulness, perceived ease of use, and user acceptance of information technology, mis quart. 13 (1989), 319-340. https://doi.org/10.2307/249008. [8] j.f. engel, r.d. blackwell, p.w. miniard, consumer behavior, 5th edition, dryden, (1986). [9] g. häubl, v. trifts, consumer decision making in online shopping environments: the effects of interactive decision aids, market. sci. 19 (2000), 4–21. https://doi.org/10.1287/mksc.19.1.4.15178. [10] p. kotler, k. keller, marketing management, 14th edition, pearson education, london, (2011). [11] r. likert, a technique for the measurement of attitudes, arch. psychol. 22 (1932), 1–55. [12] m.s. chowdhury, n. ahmad, factors affecting consumer participation in online shopping in malaysia: the case of university students, eur. j. bus. econ. 5 (2012), 49-53. https://doi.org/10.12955/ejbe.v5i0.171. [13] p. kotler, g. armstrong, principles of marketing, 5th ed. pearson education limited, essex, (2008), pp. 262-267. [14] m.o. richard, modeling the impact of internet atmospherics on surfer behavior, j. bus. res. 58 (2005), 1632–1642. https://doi.org/10.1016/j.jbusres.2004.07.009. [15] m. sam, h. tahir, website quality and consumer online purchase intention of air ticket, int. j. basic appl. sci. 9 (2009), 20-25. https://ssrn.com/abstract=2255286. [16] m. solomon, consumer behavior, (3rd edition), prentice hall, new jersey, (1995). [17] b. vietnam, background of vietnam's e-commerce market 2017, (2018). https://www.brandsvietnam.com/congdong/topic/8696-boi-canh-thi-truong-thuong-mai-dien-tu-vietnam-2017%20%5baccessed%207%20oct.%202018%5d. [18] g. haßlinger, j. mende, r. geib, t. beckhaus, f. hartleb, measurement and characteristics of aggregated traffic in broadband access networks, in: l. mason, t. drwiega, j. yan (eds.), managing traffic performance in converged networks, springer berlin heidelberg, berlin, heidelberg, 2007: pp. 998–1010. https://doi.org/10.1007/978-3-540-72990-7_86. [19] r. muradian, e. corbera, u. pascual, n. kosoy, p.h. may, reconciling theory and practice: an alternative conceptual framework for understanding payments for environmental services, ecol. econ. 69 (2010), 1202–1208. https://doi.org/10.1016/j.ecolecon.2009.11.006. [20] c.n. wang, n.t. nguyen, t.t. tuyen, the study of staff satisfaction in consulting center system-a case study of job consulting centers in ho chi minh city, vietnam, asian econ. financial rev. 4 (2014), 472–491. [21] w. ozuem, g. lancaster, technology-induced customer services in the developing countries, in: service science research, strategy and innovation: dynamic knowledge management methods (pp. 185-201), igi global, (2012). https://doi.org/10.1007/bf02289289 https://doi.org/10.2307/249008 https://doi.org/10.1287/mksc.19.1.4.15178 https://doi.org/10.12955/ejbe.v5i0.171 https://doi.org/10.1016/j.jbusres.2004.07.009 https://ssrn.com/abstract=2255286 https://www.brandsvietnam.com/congdong/topic/8696-boi-canh-thi-truong-thuong-mai-dien-tu-viet-nam-2017%20%5baccessed%207%20oct.%202018%5d https://www.brandsvietnam.com/congdong/topic/8696-boi-canh-thi-truong-thuong-mai-dien-tu-viet-nam-2017%20%5baccessed%207%20oct.%202018%5d https://doi.org/10.1007/978-3-540-72990-7_86 https://doi.org/10.1016/j.ecolecon.2009.11.006 26 int. j. anal. appl. (2023), 21:23 [22] l.w. wang, t.t. tran, n.t. nguyen, analyzing factors to improve service quality of local specialties restaurants: a comparison with fast food restaurants in southern vietnam, asian econ. financial rev. 4 (2014), 1592–1606. [23] j.w. moon, y.g. kim, extending the tam for a world-wide-web context, inform. manage. 38 (2001), 217–230. https://doi.org/10.1016/s0378-7206(00)00061-6. [24] v. venkatesh, m.g. morris, g.b. davis, f.d. davis, user acceptance of information technology: toward a unified view, mis quart. 27 (2003), 425-478. https://doi.org/10.2307/30036540. [25] j.h. ahn, t.h. yoon, j.w. chung, analysis of prognosis in patients with sudden sensorineural hearing loss and dizziness, korean j. otorhinolaryngol.-head neck surg. 44 (2001), 1032-1037. [26] j.h. davis, some compelling intuitions about group consensus decisions, theoretical and empirical research, and interpersonal aggregation phenomena: selected examples 1950–1990, organ. behav. human decision processes. 52 (1992), 3–38. https://doi.org/10.1016/0749-5978(92)90044-8. [27] m. eliasson, j. holkko lafourcade, s. smajovic, e-commerce: a study on women's online purchasing behavior, jönköping university, 2009. https://doi.org/10.1016/s0378-7206(00)00061-6 https://doi.org/10.2307/30036540 https://doi.org/10.1016/0749-5978(92)90044-8 int. j. anal. appl. (2022), 20:66 mappings and finite product of pairwise expandable spaces jamal oudetallah1, iqbal m. batiha2,3,∗ 1department of mathematics, faculty of science and information technology, irbid national university, p.o. box 2600, irbid, p.c. 21110, jordan 2department of mathematics, faculty of science and information technology, al-zaytoonah university of jordan, p.o. box 130 amman 11733, jordan 3nonlinear dynamics research center (ndrc), ajman university, ajman 346, uae ∗corresponding author: i.batiha@zuj.edu.jo abstract. in this work, we intend to study certain mappings in bitopological spaces such as pairwise perfect and pairwise countably perfect mappings that possess the property (e). some properties of such mappings are provided which have helped us to obtain some finite product theorems concerning with pairwise expandable and almost pairwise expandable spaces. two illustrative examples are given to demonstrate the effectiveness of some proposed results. 1. introduction a bitopological space (x,τ1,τ2) is a non-empty set x with two arbitrary topologies τ1,τ2. the idea of the bitopological space was induced by topologies generalized by the following two sets: bρ� = {y ∈ x | ρ(x,y) ≤ �} and bδ� = {y ∈ x | δ(x,y) ≤ �}, where ρ and δ are quasi-metric spaces of x with ρ(x,y) = δ(y,x). from 1963, when kellay introduced the concept of bitopological space, several topological properties, which are already included in a single topology, are generalized into bitopological spaces [1]. some of these properties are compactness, received: nov. 3, 2022. 2010 mathematics subject classification. 54a05. key words and phrases. pairwise expandable; pairwise paracompact; pairwise subparacompact; pairwise metacompact; bitopological space. https://doi.org/10.28924/2291-8639-20-2022-66 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-66 2 int. j. anal. appl. (2022), 20:66 paracompactness, separation axioms, connectedness, some special types of functions and many others [1–4]. many authors studied and investigated these bitopological spaces after kelly, like fletcher et al. [5], birsan [6], reilly [7], datta [8], hdeib and fora [9], bose et al. [10], killiman and salleh [11], abushaheen et al. [12], and qoqazeh et al. [13]. for further exploration in this field, this work aims to present some properties of the pairwise perfect and the pairwise countably perfect mappings for the purpose of using them to obtain some novel finite product theorems concerning with pairwise expandable and almost pairwise expandable spaces. to go forward in this paper and for more simplification, we state some notations and notions which will be used later on. in particular, the closure and interior of the set a will be denoted respectively by cl(a) and int(a) with noting that (x,τ) is a topological space. besides, if a is a subset of a bitopological space x = (x,τ1,τ2), then the relative topology, which is a subspace topology of x, on the set a inherited by τ will be denoted by τa. the cardinality of the set ∆ will be denoted by |∆|. the sets r, q, n, and z will denote the sets of real numbers, rational numbers, natural numbers and integer numbers, respectively. on the other hand, ω0 and ω1 will denote respectively the first two uncountable ordinals, and m will denote generally for an infinite cardinal. finally, the terms τu, τdis, τcof and τcoc will denote the usual, discrete, cofinite and the co-countable topologies, respectively. 2. basic definitions in the following content, we state certain definitions and preliminaries associated with the bitopological space for completeness. definition 2.1. [14] a collection subset f̃ = {fα : α ∈ ∆} of a bitopological space (x,τ1,τ2) is said to be pairwise locally finite if for each x ∈ x there exist an τ1–open set u containing x such that u intersects only finitely many members of f̃, or there exist τ2– open set v containing x such that v intersects only finitely many members of f̃. definition 2.2. [15] a p–open cover ṽ of a bitopological space (x,τ1,τ2) is called parallel refinement of a p–open cover ũ of x if each τi–open set of ṽ is contained in some τi–open set of ũ, where i = 1, 2. definition 2.3. [8] a bitopological space x is called p–m–paracompact, if every p–open cover ũ of x, so that ∣∣ũ∣∣ ≤ m, has a pairwise locally finite open parallel refinement. if m = ω0, then the space x is called p–countably paracompact. if the space x is p–m–paracompact for every m, then x is called p–paracompact. definition 2.4. [14] let m be an infinite cardinal, then the bitopological space (x,τ1,τ2) is called τι–m–expandable space with respect to τj if for every τι–locally finite f̃ = {fα : α ∈ ∆} with |4|≤ m, there exist τj–locally finite collection g̃ = {gα : α ∈ ∆} of open subsets of x such that fα ⊂ gα for all α ∈ ∆ and for i 6= j, where i, j = 1, 2. int. j. anal. appl. (2022), 20:66 3 definition 2.5. [14] a bitopological space (x,τ1,τ2) is called τi–expandable with respect to τj, if it is an τi–m–expandable for every cardinal m and i 6= j where i, j = 1, 2. definition 2.6. [14] a bitopological space (x,τ 1,τ2) is called a pairwise expandable (or simply p– expandable), if it is p–t2–space and it is τ1–expandable with respect to τ2 and τ2– expandable with respect to τ1. definition 2.7. let x = (x,τ1,τ2), y = (y,σ1,σ2) be two bitopological spaces and f : x −→ y be a given map. then f is called pairwise closed (p–closed) map if it maps a τi–closed subset of x onto a σi–closed subset of y , for each i = 1, 2. definition 2.8. let f : x −→ y be a p–closed, p–continuous map from a bitopological space x = (x,τ1,τ2) onto a bitopological space y = (y,σ1,σ2) such that f−1(y) is compact for each y ∈ y . then the map f is called pairwise perfect (p–perfect) map. definition 2.9. according to definition 2.2 above, if the preimage f−1(y ) is p–countably compact for each y ∈ y then f is called a pairwise countably perfect (p–countably perfect) map. definition 2.10. the topologies τi and τj on a nonempty set x are said to have the property (e) if a ∈ λc(x,τi ) and b ∈ λc(x,τj) imply a∩b ∈ λc(x,τi ). 3. main theoretical results in this part, we intend to state and derive some novel significant theoretical results in light of the aforesaid definitions. at the beginning, one can observe that it is easy to prove the following two lemmas: lemma 3.1. let f : (x,τ1,τ2) −→ (y,σ1,σ2) be a p–countably perfect map. then we have: • if f̃ = {fα : α ∈ ∆} be a p–locally finite collection of subsets of y , then f−1(f̃ ) = {f−1(fα) : α ∈ ∆} is a p–locally finite collection of x. • if f̃ = {fα : α ∈ ∆} be a p–locally finite collection of subsets of x, then f (f̃ ) = {f (fα) : α ∈ ∆} is also p–locally finite collection of subsets of y . lemma 3.2. let f : (x,τ1,τ2) −→ (y,σ1,σ2) be a p–closed map, if g̃ = {gα : α ∈ ∆} be a p–locally finite of subsets of x and ṽ = {vα : α ∈ ∆} be a σi–open in y for i = 1, 2 such that vα ⊆ f (gα) for α ∈ ∆, then the family {f (gα) : α ∈ ∆} is p–locally finite collection of subsets of y . next, based on the two lemmas provided above, we state and prove the following important theorem that concerns with the p–m-expandable space. theorem 3.1. let f be a p–countably perfect map from a bitopological space x = (x,τ1,τ2) onto a bitopological space y = (y,σ1,σ2). then x is p–m–expandable if and only if y is p–m–expandable. 4 int. j. anal. appl. (2022), 20:66 proof. ⇒) suppose x = (x,τ1,τ2) is p–m–expandable, and f̃ = {fα : α ∈ ∆} is a p–locally finite collection of subsets of y with |∆| ≤ m. then, by lemma 3.2, the following mapping: f−1(f̃ ) = {f−1(fα) : α ∈ ∆} will be a p–locally finite collection of subsets of x with |∆| ≤ m. now, since x is p–m–expandable, then there exists a p–locally finite collection g̃ = {gα : α ∈ ∆} of open subsets of x such that f−1(fα) ⊆ gα, for each α ∈ ∆. set vα = y − f (x −gα), α ∈ ∆. consequently, we have the following claim: claim fα ⊆ gα. to prove this claim, we have clearly: x −gα ⊆ x − f−1(fα), and hence f (x −gα) ⊆ y −fα. again, we have: fα ⊆ y − f (x −gα). this consequently implies that fα ⊆ vα, for each α ∈ ∆. now, it remains to show that ṽ = {vα : α ∈ ∆} is a p–locally finite collection of open subsets of y . for this purpose, we should note that vα is σi–open in y , for i = 1, 2. now, since f is p–closed map, vα ⊆ f (gα), and by lemma 3.2, the family {f (gα) : α ∈ ∆} will be a p–locally finite collection of subsets of y . therefore, we conclude that ṽ is locally finite collection of subsets of y , which implies that y is indeed a p–m–expandable. ⇐) in order to show the converse inclusion, we assume that y is p–m–expandable and f̃ = {fα : α ∈ ∆} is a p–locally finite collection of subsets of x such that |∆| ≤ m. by lemma 3.1, we can deduce that: f (f̃ ) = {f (fα) : α ∈ ∆} represents a p–locally finite collection of subsets of y . hence, there exists p–locally finite collection of subsets of σi–open subsets of y , for i = 1, 2, say g̃ = {gα : α ∈ ∆}, such that fα ⊆ gα for each α ∈ ∆. then: fα ⊆ f−1 (f (fα)) ⊆ f−1(gα). since f is p–continuous, we get f−1(gα) is τi–open in x for i = 1, 2. now, by lemma 3.1, the following collection f−1(g̃) = {f−1(gα) : α ∈ ∆} will be a τi–open locally finite collection for i = 1, 2. hence, x is p–m–expandable. � in the same context, we should notice that the p–closed continuous image of the p–expandable space need not be p–expandable. however, the following example aims to illustrate this assertion. int. j. anal. appl. (2022), 20:66 5 example 3.1. let w = (w,τ,τ) where w −[0,ω1]×[0,ω1) and τ be an order topology define on w and x = w ×n where n = (n,τdis,τdis) and τdis is a discrete topology. clearly, w is p–countably compact and hence x is p–expandable. now, let f1 : (x,τ ×τdis) −→ (y,σ1) be a p–closed onto continuous map. we have illustrated that y is not ωo–expandable. hence, f2 : (x,τ ×τdis,τ ×τdis) −→ (y,σ1,σ2) is a p–closed onto continuous map and then y is not p–ωo–expandable. on the other hand, in order to turn into the other theoretical results, we will recall bellow the following definition that relates with the p–expandable space. definition 3.1. let x = (x,τ1,τ2) be a bitopological space. a τ1–subset f is called p–expandable relative to x if and only if every p–open, p–aσ–cover of f in x has a p–open, p–locally finite refinement in x. thus, we are now ready to introduce the following important results that deal with the p–expandable space. lemma 3.3. let m be a τi–closed expandable subset of a bitopological space x = (x,τ1,τ2). if f is τi–closed in int(m), then f is p–expandable relative to x. proof. let ũ be a p–open, p–aσ–cover of f in x. then: b̃ = {u ∩m : u ∈ ũ}∪ (m −f ) will be an p–open, p–aσ–cover of m. since m is τi–expandable, then there exists a p–open, p– locally finite refinement of b̃, say ṽ , such that for each v ∈ ṽ , there exists b ∈ b̃ such that v ⊆ b. now, let w̃ = {v ∩ int(m) : v ∈ ṽ}, then w̃ will be a p–open locally finite refinement of ũ in x. therefore, f is indeed a p–expandable relative to x. � theorem 3.2. let i = 1, 2 and f : (x,τ1,τ2) −→ (y,σ1,σ2) be a p–continuous map from a bitopological space x onto a bitopological space y . then f is p–closed if and only if for each y ∈ y and each τi–open set u in x such that f−1(y) ⊆ u, there exists a σi–open set oy in y containing y and f−1(oy ) ⊆ u. proof. ⇒) let i = 1, 2 and suppose f is p–closed. let y ∈ y and let u be any τi–open set such that f−1(y) ⊆ u. if one lets a = x −u, then f (a) = f (x −u) will be σi–closed in y . moreover, letting oy = y − f (x −u) 6 int. j. anal. appl. (2022), 20:66 yields oy to be σi–open in y and y ∈ oy, which consequently implies that f−1(oy ) ⊆ u. hence, if we let t ∈ f−1(oy ), then f (t) ∈ oy and f (t) /∈ f (x −u), which implies t /∈ x −u, and therefore t ∈ u. ⇐) conversely, let a be any τi–closed subset of x and y ∈ y − f (a). since f is onto, then there exists x ∈ x such that y = f (x). now, we have the following claim: claim f−1(y) ∩a = φ. to prove this claim, suppose that f−1(y) ∩a 6= φ. then, there exists z ∈ f−1(y) and z ∈ a, which implies that f (z) = y. but f (z) = y − f (a), which gives that f (z) /∈ f (a). hence, z /∈ a. this yields a contradiction. thus, f−1(y)∩a = φ or f−1(y) ⊆ x−a. consequently, due to x−a is τi–open, then by assumption there exists a σi–open set oy in y such that f−1(oy ) ⊆ x −a. now, we have: y ∈ oy ⊆ f (x −a) = y − f (a). this implies that y − f (a) is a σi–open subset of y , which means that f (a) is σi–closed. hence, we have f is p–closed. � corollary 3.1. let x be a p–compact space and y = (y,σ1,σ2) be any bitopological space. then, the projection map π : x ×y −→ y is p–closed map. proof. the proof follows directly from theorem 3.2. � theorem 3.3. let x = (x,τ1,τ2) be a p–expandable space and y = (y,σ1,σ2) be a p–compact space. then, x ×y is p–expandable. proof. the projection map π : x ×y −→ x is a p–closed map by corollary 3.1 and because π−1(x) = {x}×y ∼= y is p–compact, for each x ∈ x. therefore, y is p–countably compact, which implies that π is a p–countably perfect map. hence, by theorem 3.2, we conclude that x ×y is p–expandable. � it is worth mentioning that if x and y are two p–expandable bitopological spaces, then x×y need not be in general p–expandable. however, this assertion can be illustrated by the following example. example 3.2. suppose x = (x,τs,τs) so that τs represents the sorgenfrey topology. due to x is p–lindelof and p–t3–space, then x is p–paracompact and thus p− expandable. but x ×x is not p–expandable since the collection f̃ = {(x,−x) : x ∈ x} is p–locally finite. however, one can show, by a category argument, that there is no p–locally finite collection of open sets {gx : x ∈ x} such that (x,−x) ∈ gx, for each x ∈ x. theorem 3.4. let x = (x,τs,τs) be a p–regular, p–subparacompact and p–expandable bitopological space and y = (y,σ1,σ2) be a p–expandable bitopological space. then x ×y is p–expandable int. j. anal. appl. (2022), 20:66 7 if and only if for each x ∈ x there is a τi–open set u(x) of x such that cl(u(x) is p–expandable, for each i = 1, 2. proof. ⇒) suppose x ×y is p–expandable. let x ∈ x and take x to be the τi–open set containing x. this implies that cl(x) ×y = x ×y is p–expandable. ⇐) conversely, suppose that for each x ∈ x there exist a τi–open set of x, u(x), such that cl(u(x))× y is p–expandable. it suffices to show that the following projection map: π : x ×y −→ x has property (e). let ũ be a p–open, p–aσ–cover of x×y . now, by p–regularity, and due to there is a τi–open set of x, u(x), such that x ∈ u(x), for each x ∈ x and for i = 1, 2, then there is a τi–open set v (x) of x such that: x ∈ v (x) ⊆ cl(v (x)) ⊆ u(x) ⊆ cl(u(x)). then v (x) ×y ⊆ cl(v (x)) ×y ⊆ cl(u(x)) ×y. by lemma 3.3, cl(v (x)) ×y is p–expandable relative to x×y , which consequently implies that ũ has a p–open, p–locally finite refinement in x ×y , say ã(x) such that: cl(v (x)) ×y ⊆∪{ã(x) : x ∈ x}. now, we have: ã = ∪{ã(x) : x ∈ x} is p–open, p–refinement of ũ. further, ṽ = {v (x) : x ∈ x} is a p–open cover of x such that: π−1(v (x)) = v (x) ×y ⊆ ∪ x∈x ã(x), where ã(x) is a p–locally finite subfamily of ã. since x is p–subparacompact, then ṽ is a p–open, p−aσ–cover of x. therefore, we have for each p–open, p–aσ–cover ũ of x×y , there is a p–open, p–aσ–cover ṽ of x and p–open refinement ã of ũ such that π−1(v (x)) is contained in the union of p–locally finite subfamily of ã, for each x ∈ x and v (x) ∈ ṽ . hence, π has property (e), and x×y is p–expandable space. � corollary 3.2. let x = (x,τs,τs) be a p–paracompact, p–locally compact bitopological space and y = (y,σ1,σ2) be a p–expandable bitopological space. then x ×y is p–expandable. proof. the proof follows immediately from theorem 3.3. � 8 int. j. anal. appl. (2022), 20:66 4. conclusions this paper has studied and explored some certain mappings in bitopological spaces like the pairwise perfect and pairwise countably perfect mappings. in order to gain some finite product theorems concerning with pairwise expandable and almost pairwise expandable spaces, some significant properties of such mappings have been stated and derived well. some illustrative examples have been provided for completeness. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.c. kelly, bitopological spaces, proc. london math. soc. s3-13 (1963), 71–89. https://doi.org/10.1112/plms/ s3-13.1.71. [2] j. oudetallah, i.m. batiha, on almost expandability in bitopological spaces, int. j. open probl. computer sci. math. 14 (2021), 43-48. [3] j. oudetallah, m.m. rousan, i.m. batiha, on d-metacompactness in topological spaces, j. appl. math. inform. 39 (2021), 919–926. https://doi.org/10.14317/jami.2021.919. [4] a.a. hnaif, a.a. tamimi, a.m. abdalla, i. jebril, a fault-handling method for the hamiltonian cycle in the hypercube topology, computers mater. continua. 68 (2021), 505-519. https://doi.org/10.32604/cmc.2021. 016123. [5] p. fletcher, h.b. hoyle, iii, c.w. patty, the comparison of topologies, duke math. j. 36 (1969), 325-331. https://doi.org/10.1215/s0012-7094-69-03641-2. [6] t. birsan, compacite dans les espaces bitopologiques, an. st. univ. iasi, s.i.a. mat. 15 (1969), 317-328. [7] i.l. reilly, bitopological local compactness, indagationes math. (proc.) 75 (1972), 407–411. https://doi.org/ 10.1016/1385-7258(72)90037-6. [8] m.c. datta, projective bitopological spaces, j. aust. math. soc. 13 (1972), 327–334. https://doi.org/10. 1017/s1446788700013744. [9] h. hdeib, a. fora, on pairwise paracompact spaces, dirasat, ix(2) (1982), 21-29. [10] a. mukherjee, a.r. choudhury, m.k. bose, on bitopological spaces, j. mech. continua math. sci. 3 (2008), 00003. [11] a. killiman, z. salleh, product properties for pairwise lindelöf spaces, bull. malays. math. sci. soc. 34 (2011), 231-246. [12] f. a. abushaheen and h. hdeib, on [a,b]-compactness in bitopological spaces, int. j. pure appl. math. 110 (2016), 519-535. https://doi.org/10.12732/ijpam.v110i3.11. [13] h. qoqazeh, h. hdeib, e.a. osba, on metacompactness in bitopological spaces, int. j. pure appl. math. 119 (2018), 191-206. [14] j. oudetallah, on feebly pairwise expandable space, j. math. comput. sci. 11 (2021), 6216-6225. https: //doi.org/10.28919/jmcs/6294. [15] r. engleking, general topology, heldermann verlag, berlin, 1989. https://doi.org/10.1112/plms/s3-13.1.71 https://doi.org/10.1112/plms/s3-13.1.71 https://doi.org/10.14317/jami.2021.919 https://doi.org/10.32604/cmc.2021.016123 https://doi.org/10.32604/cmc.2021.016123 https://doi.org/10.1215/s0012-7094-69-03641-2 https://doi.org/10.1016/1385-7258(72)90037-6 https://doi.org/10.1016/1385-7258(72)90037-6 https://doi.org/10.1017/s1446788700013744 https://doi.org/10.1017/s1446788700013744 https://doi.org/10.12732/ijpam.v110i3.11 https://doi.org/10.28919/jmcs/6294 https://doi.org/10.28919/jmcs/6294 1. introduction 2. basic definitions 3. main theoretical results 4. conclusions references international journal of analysis and applications volume 18, number 6 (2020), 1108-1122 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1108 monotone chromatic number of graphs anwar saleh1,∗, najat muthana1, wafa al-shammakh1, hanaa alashwali2 1department of mathematics, faculty of science, university of jeddah, jeddah, saudi arabia 2department of mathematics, king abdulaziz university, jeddah, saudi arabia ∗corresponding author: asaleh1@uj.edu.sa abstract. for a graph g = (v, e), a vertex coloring (or, simply, a coloring) of g is a function c : v (g) → {1, 2, ..., k} (using the non-negative integers {1, 2, ..., k} as colors). in this research work, we introduce a new type of graph coloring called monotone coloring, along with this new coloring, we define the monotone chromatic number of a graph and establish some related new graphs. basic properties and exact values of the monotone chromatic number of some graph families, like standard graphs, kragujevac trees and firefly graph are obtained. also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. exact values of the monotone chromatic number for some special case of cartesian product of graphs are found. finally, upper and lower bounds for monotone chromatic number of the cartesian product for non trivial connected graphs are presented. 1. introduction throughout this research work, by graph we mean finite graph without loops and parallel edges. any notations or terminology not specifically defined here, we refer the book [6]. more details about coloring in graph and its related are reported in ( [3, 8]). two interesting types of coloring are introduced and studied in ( [1, 2, 5]). as usual, pn,cn,kn and wn are the n−vertex path, cycle, complete, and wheel graph, respectively, kr,s is the complete bipartite graph on r+s vertices and sr is the star graph with r+ 1 vertices. received september 21st, 2020; accepted october 19th, 2020; published november 17th, 2020. 2010 mathematics subject classification. 05c76, 05c22, 05c15. key words and phrases. monotone coloring; monotone chromatic number; monotone clique set; montone bipartite graph; complete monotone graph. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1108 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1108 int. j. anal. appl. 18 (6) (2020) 1109 graph coloring is one of essential concepts in the theory of graphs. it has preoccupied a large number of people as a distraction puzzle during the 19th century and later in the framework of scientific research, since this conception exhibits a significant interest from a theoretical and practical point of view. many applications are modeled and investigated with the use of graph coloring. for a graph g = (v, e), a vertex coloring (or, simply, a coloring) of g is a function c : v (g) →{1, 2, ...,k} (using the non-negative integers {1, 2, ...,k} as colors). the huge applications of coloring motivated us to introduce a new type of graph coloring called monotone coloring of the graph, we define the monotone chromatic number of a graph, complete monotone graph and monotone clique set of a graph. some basic properties and relations with the other graph parameters and exact values of the monotone chromatic number of some graph families, like standard graphs, firefly graph and kragujevac trees obtained. also, we get a characterization for bipartite graphs by defining the monotone bipartite graph. exact values of the monotone chromatic number for some special case of cartesian product of graphs are found. lastly upper and lower bounds for monotone chromatic number of the cartesian product for non trivial connected graphs are presented. 2. monotone chromatic number of graphs in this section, we define the monotone chromatic coloring and monotone chromatic number of a graph and give several preliminary results and straightforward facts regarding the monotone coloring of graphs. also we found the monotone chromatic number for some families of graphs. definition 2.1. let g = (v,e) be a graph. a path p = [v1,v2, ...,vk+1] in g is a monotone path if either deg(vi) ≤ deg(vi+1) or deg(vi) ≥ deg(vi+1) for all i = 1, 2, ...,k. any two vertices u and v in g are called monotone adjacent if there exists a monotone path connected them. definition 2.2. a monotone k− coloring of the graph g is coloring the vertices of g with k colors such that no two monotone adjacent vertices share the same color. the smallest integer k such that g has a monotone k− kcoloring is called the monotone chromatic number of g and denoted by χmo(g). a graph g is said to be monotone k− colorable if it has monotone k− coloring. monotone coloring as function we can define as: definition 2.3. the monotone coloring function is a function f : v (g) → 1, 2, 3, ...,k ⊆ n which satisfy that for any two monotone adjacent vertices v and u, f(v) 6= f(u). proposition 2.1. int. j. anal. appl. 18 (6) (2020) 1110 (1) for any path pn with n ≥ 2 vertices, χmo(g) =   2, if n = 2;n− 1, if n ≥ 3. (2) for any connected regular graph g with n ≥ 3 vertices, χmo(g) = n. (3) for any complete bipartite graph kr,s, where 1 < r ≤ s, we have χmokr,s = 2. (4) for any wheel graph g ∼= wr with r + 1 vertices, χmo(g) = r + 1. (5) for any book graph g ∼= bm, we have χmo(g) = 4. (6) for any helm graph g ∼= hn with 2n + 1 vertices, χmo(g) = n + 2. (7) for any gear graph g with 2n + 1 vertices, we have χmo(g) =   4, if n = 3;3, if n ≥ 4. from the definition of the proper coloring and monotone coloring, it is obviously, that any monotone coloring is proper coloring but the converse is not true. as any two adjacent vertices in any graph are also monotone adjacent, then we have the following result. proposition 2.2. for any graph g, χ(g) ≤ χmo(g). the equality hold if and only if any two monotone adjacent vertices are adjacent. proposition 2.3. (1) for any monotone χmo(g)− coloring of any graph g, all the very weak vertices has the same color. (2) for any nontrivial connected graph g with n vertices, 2 ≤ χmo(g) ≤ n. (3) for any graph g ∼= ∪ri=1gi, we have, χmo(g) = max{χmo(gi) : i = 1, 2, ...r}. remark 2.1. let g and h be any two graphs such that h is subgraph of g. then the monotone chromatic number of g and h are not comparable. that means all the possibilities allowed. according to the monotone adjacency between vertices, we define the monotone bipartite graph, and complete monotone graph. definition 2.4. a bipartite graph g is called monotone bipartite graph if and only if any monotone path in g is of length at most one. the definition of monotone bipartite graph characterize the trees into two families monotone trees and non-monotone trees the double star graph denoted by b(r,s) with r+s+ 2 vertices, is a tree that containing exactly two non-pendent vertices. int. j. anal. appl. 18 (6) (2020) 1111 proposition 2.4. for any double star graph g ∼== b(r,s) with r + s + 2 vertices, such that r < s, χmo(g) = 3. theorem 2.1. let g be any graph. then g is monotone bipartite graph if and only if χmo(g) = 2. proof. let g be a graph such that χmo(g) = 2 and suppose contrary that there is monotone path uvw of length 2, we need at least three colors for monotone coloring of g that means χmo(g) ≥ 3 which is a contradiction. therefore any monotone is of length at most one. hence g is monotone bipartite graph. to prove the other direction, suppose, that g is monotone bipartite graph. then the result coming from the proposition 2.2. � corollary 2.1. for any monotone tree tn with n vertices, χmo(g) = 2. proposition 2.5. for any connected graph g with n ≥ 3 vertices , χmo(g) = 2 if and only if for any vertex v ∈ v (g), either v is very strong or very weak. proof. suppose that χmo(g) = 2, and assume in contrary that, there is a vertex v ∈ v (g) which is neither very strong nor very weak. then we have two cases: case 1 : if deg(v) = 1, then v is very weak which contradict our assumption. case 2: if deg(v) 6= 1, then v must belongs to some path say uvw such that deg(u) < deg(v) < deg(w) or deg(u) = deg(v) = deg(w) or one of the vertices u, or w has the same degree as v. for all of these cases, it needs at least 3 colors to monotone coloring which contradicts that χmo(g) = 2. now, suppose that for any vertex v ∈ v (g), either v is very strong and one very weak. so, any monotone path will contains only two vertices; one of them is very weak and the other is very strong. hence χmo(g) = 2. � corollary 2.2. a connected graph g = (v ; e) is monotone bipartite graph if and only if any vertex v ∈ v (g) is either very strong or very weak. proposition 2.6. for any connected graph g with n ≥ 2 vertices , χmo(g) = n if and only g is complete monotone graph. proof. let g be a monotone complete graph with n vertices and suppose that g in contrary not satisfy the condition g contains at most either one very weak or one very weak vertex. suppose the vertex set v (g) = {v1,v2, ...,vn}, such that v1 is very weak vertex, then if there is another very weak vertex say vi, then there is no monotone vertex between v1 and vi which is a contradiction with the definition of the monotone complete graph, similarly if there is very strong vertex say vj ,then there is no monotone path int. j. anal. appl. 18 (6) (2020) 1112 between v1 and at least one of the neighborhood of the vertex vj . the proof in the same way if we suppose v1 is very strong. hence g contains at most either one very weak or one very weak vertex. to prove the other direction. suppose g contains at most either one very weak or one very strong vertex. then clearly there exist a monotone path containing all the vertices that means between any two vertices there is a monotone. hence g is monotone complete graph. � definition 2.5. a subset x ⊆ v (g) of size k is called a monotone k− clique of g if between any two vertices u and v in the set x there is a monotone path. the monotone clique number of g denote by ωmo(g) is the largest positive integer k such that g contains a monotone k− clique. the monotone clique set with size ωmo(g) is called maximum monotone clique. definition 2.6. ( [4]) let p3 be the 3− vertex tree, rooted at one of its terminal vertices. for k = 2, 3, ·,k; construct the rooted tree bk by identifying the roots of k copies of p3 . the vertex obtained by identifying the roots of p3−trees is the root of bk. definition 2.7. ( [4]) let d ≥ 2 be an integer.let β1,β21, · · · ,βd, be rooted trees,specified in definition 2.6, i.e., β1,β2, · · · ,βd ∈ {b2.b3, ...}. a kragujevac tree t is a tree possessing a vertex of degree d, adjacent to the roots of β1,β1, · · · ,βd. this vertex is said to be the central vertex of t , whereas d is the degree of t. theorem 2.2. let g ∼= kgd,k, where kgd,k is the kragujevac tree of degree d ≥ 2 and with branches bki ,i = 1, 2, ..., where every branch bki contains ki pendant vertices and if t is the number of branches bki where d = ki . then χmo(g) =   3, if the central vertex is very weak vertex; 4, if the central vertex is very strong vertex; 5, if the central vertex is very typical vertex; d + 3, if the central vertex is regular vertex; t + 4, if the central vertex is weak or strong vertex; t + 5, if the central vertex is typical vertex. proof. let g be the kragujevac tree kgd,k of degree d ≥ 2 with enteral vertex v and with branches bki ,i = 1, 2, · · · , where every branch bki contains ki pendant vertices and if t is the number of branches bkj where d = kj . we have 7 possibilities for the type of the central vertex v of the kragujevac tree: case 1. the central vertex v is very strong, in this case, we can define monotone coloring function by partition the vertex set of the tree into 4 classes, first class, the pendant vertices, second class the support vertices and third class the roots vertices of the branches and fourth class the central vertex and give color for each class, that means χmo(g) ≤ 4 and it is not difficult to see that, any set contains the the pendent int. j. anal. appl. 18 (6) (2020) 1113 vertex with its support vertex and the root vertex in any branch along with the central vertex v is clique set with 4 vertices. that means in this case χmo(g) = 4. case 2. the central vertex v is very weak, then we can define monotone coloring function cmo by partition the vertex set into 4 subsets, s1 is the pendant vertices, s2 the support vertices and s3 be the roots vertices of the branches and s4 be the central vertex v and giving color for each subset as following: cmo(x) =   i, if x ∈ si and x not the central vertex;1 or 2, if x is the central vertex. therefore, χmo(g) ≤ 3 and since the set of vertices in any path between the pendant vertex of any branch and its root vertex is monotone clique set. hence in this case χmo(g) = 3. case 3. the central vertex v is regular vertex, then in this case, let us partition the vertex set into three subsets; the set of pendant vertices s1, the support vertices s2 and the set of roots vertices with the central vertex s4, we define a monotone coloring function which assign d + 1 different colors to the vertices of s3 and one color to all the vertices of s2 and another different color to the vertices of s1, so, we need to d + 3 colors to this monotone coloring function, therefore χmo(g) ≤ d + 3. obviously, if we take any pendant vertex along with its support vertex in any branches along with the set s3 will make monotone clique set of g of size d + 3. hence χmo(g) = d + 3. case 4. the central vertex v is weak vertex, in this case, let us partition the vertex set into the following subsets of vertices, s1, the set of pendant vertices, s2 the set of support vertices, s3 the central vertex and the set of root vertices of degree equal to d and s4 the set of root vertices with degree greater than d. now, |s3| = t + 1, we define monotone coloring function by assigning one color say 1 to the vertices of s1 and another color say 2 to the vertices of s2 and assign t + 1 different colors to the t + 1 vertices in s3 and assign one other color to the vertices in s4. so we need to t + 4 colors in this monotone coloring function and therefore, χmo(g) ≤ t + 4. also, it is obviously to see that the vertices of s3 with one pendent vertex with its support vertex and one root vertex from s4 will make clique set in g. hence χmo(g) = t + 4. case 5. the central vertex v is strong vertex, in this case let us partition the vertex set into the following subsets of vertices; s1 the set of pendant vertices, s2 the set of support vertices s3 the central vertex with the set of root vertices of degree equal to d and s4 the set of root vertices with degree less than d. now, if the number of root vertices with degree d is t, we define monotone coloring function by assigning one color say 1 to the vertices of s1 and another color say 2 to the vertices of s2 and assign t + 1 different colors to the t + 1 vertices in s3 and assign one other color to the vertices in s4. so we need to t + 4 colors in this monotone coloring function and therefore, χmo(g) ≤ t + 4. also, obviously the set of vertices in s3 with the pendant vertex and its support vertex from any branch bki with ki = d and the root vertex of any other branch bki with ki ≤ d make a clique set in g with size t + 4. hence χmo(g) = t + 4. int. j. anal. appl. 18 (6) (2020) 1114 case 6. the central vertex v is very typical vertex. suppose there are s root vertex with degree less than d. let s1 be the set of pendant vertices, s2 the set of support vertices s3 the set of roots vertices with degree less than d and let s4 be the root vertices of degree greater than d and s5 contains the central vertex. we can define monotone coloring function by assigning for any vertex x ∈ si the color i; cmo(x) = i. that means χmo(g) ≤ 5. also the set which contains one vertex along with its support vertex and the root vertex in the same branch bki where ki ≤ d and the central vertex and one root vertex from any branch bki where ki ≥ d is clique set of size 5. hence in this case χmo(g) = 5. case 7. the central vertex v is typical vertex. suppose there are t root vertex with degree equal to d. let s1 be the set of pendant vertices, s2 the set of support vertices s3 the set of roots vertices with degree equal to d and let s4 be the root vertices of degree greater than d and s5 be the roots vertices of degree less than d, s6 contains the central vertex. we construct monotone coloring function by assigning one color to the vertices in s1 and another different color to the vertices in s2 and t + 1 different colors for the vertices in s3 and s6 also one other different color to the vertices in s4 and one different color to the vertices in s5. therefore χmo(g) ≤ t + 5. now, we have clique set of size t+ 5 which contains one pendant vertex, support vertex and root vertex from any of the branch bki , where ki < d and the t vertices of s3 along with the central vertex, one root vertex from any branch bki , where ki > d. hence χmo(g) = t + 5. � theorem 2.3. for any nontrivial connected graph g, with monotone chromatic number χmo(g), there exist at least one maximum monotone clique set with size ωmo(g) = χmo(g). proof. let g be any nontrivial connected graph with monotone chromatic number χmo(g) = s. suppose to the contrary, that we have maximum clique set a with size |a| = ωmo(g) = t, where either t < s or t > s. case 1. if t < s. then there exists at least one coloring class say b and at least one monotone path between every two elements one from a and one from b which is contradict that a is the monotone clique with maximum size. case 2. if t > s, then at least we need to t + 1 colors for monotone coloring which is contradict that χmo(g) = s. hence, ωmo(g) = χmo(g). � theorem 2.4. let g ∼= km1,m2···mk , where m1 ≤ m2 ≤ ··· ≤ mk be any complete k− partite graph and there are ti partite sets of the same number of vertices λi, where i = 1, 2, · · · ,s for some positive integer i. then χmo(g) = k + s∑ i=1 ti(λi − 1). proof. let g ∼= km1,m2···mk , where m1 ≤ m2 ≤ ···≤ mk be any complete k− partite graph and there are ti partite sets of the same number of vertices λi, where i = 1, 2, · · · ,s for some positive integer i ,by reordering int. j. anal. appl. 18 (6) (2020) 1115 the partite sets which they have different number of elements as v1,v2, · · ·vk−∑s i=1 ti, we can define a monotone coloring function as the following. f : v (g) → [k + s∑ i=1 ti(λi − 1)], by assigning the color i for each vertex in vi and assigning for each vertex in the equal parite sets to different colors that means assign ∑s i=1 tiλi different colors to the vertices in v − ∑s i=1 ti⋃ i=1 vi. clearly, the function f which defined above is monotone coloring function on g. therefore, χmo(g) ≤ k + s∑ i=1 ti(λi − 1). (2.1) it is not difficult to check that the set which contains the vertices in∑s i=1 ti⋃ i=1 vi, and only one vertex from each vi will make clique set with k + ∑s i=1 ti(λi − 1). thus, χmo(g) ≥ k + s∑ i=1 ti(λi − 1). (2.2) hence by inequalities 2.1 and 2.2, we get, χmo(g) = k + s∑ i=1 ti(λi − 1). � theorem 2.5. let g be any connected graph with n ≥ 3 vertices and d is the set of distinct degrees of the vertices. then g is monotone complete graph if and only if for any set of vertices with the same degree in g is clique set in g. proof. let g be monotone complete graph with n ≥ 3 vertices, since g is monotone complete, then ωmo(g) = n. therefore v (g) is a monotone clique set contains all the vertices of g. if g is regular, then there is only one clique set containing all the vertices. suppose the graph is not regular that means the set d is of size greater than or equal to 2. let d = {k1,k2, . . . ,kt} for some integer t ≥ 2, also let c1,c2, . . . ,ct be the sets of vertices of degrees k1,k2, . . . ,kt respectively. suppose in contrary, that there exists one set cj, 1 ≤ j ≤ t such that cj not a monotone clique set, then there exist at least two vertices u and v which they are not monotone adjacent in g which is a contradict that v (g) is clique set. similarly, let c1,c2, . . . ,ct are monotone clique sets in g. suppose g is not monotone complete, then there int. j. anal. appl. 18 (6) (2020) 1116 exist at least two vertices u and v not monotone adjacent if they are with the same degree, then they will belong to same set cj, 1 ≤ j ≤ t which is a contradiction, similarly we will get a contradiction even if they have different degrees. � we recall, that in [7], a firefly graph fs,t,l where s ≥ 0, t ≥ 0, n− 2s− 2t− 1 ≥ 0 is a graph of order n that consists of s triangles, t pendent paths of length 2 and l pendant edges sharing a common vertex (see figure 1). proposition 2.7. for any firefly graph g ∼= fs,t,l, χmo(g) = 3. figure 1. firefly graph proof. let g ∼= fs,t,l where s ≥ 0, t ≥ 0, n− 2s− 2t− 1 ≥ 0 be a firefly graph and by labeling the vertices as in figure 1. by partition the vertex set into the following sunsets of vertices, s1 = {w1,w2, . . . ,ws}, s2 = {w′1,w′2, . . . ,w′s}, s3 = {v1,v2, . . . ,vl}, s4 = {u1,u2, . . . ,ut}, s5 = {u′1,u′2, . . . ,u′t} and s6 is the central vertex. by defining the monotone coloring function f : v (g) −→{1, 2, 3} as following, f(x) =   1, if x ∈ s1 ∪s3 ∪s4; 2, if x ∈ s2 ∪s5; 3, x is the central vertex. therefore, χmo(g) ≤ 3. the set of vertices in any triangle in the firefly graph generate a monotone clique set, that means χmo(g) ≥ 3. hence, χmo(g) = 3. � int. j. anal. appl. 18 (6) (2020) 1117 the corona product g1 ◦ g2 of two graphs g1 and g2, where v (g1),v (g2) are the set of vertices of g1,g2 respectively, is the graph obtained by taking |v (g1)| copies of g2 and joining each vertex of the i-th copy with the corresponding vertex u ∈ v (g1) [6]. proposition 2.8. for any positive integer a ≥ 3, there exists a graph g such that χ(g) = a and χmo(g) = a + 1. proof. let g be a graph which construct by the corona product between the complete graphs ka and k1. that means g ∼= ka◦k1 where a ≥ 3. then it is not difficult to see that χ(g) = a and χmo(g) = a+ 1. � definition 2.8. a subset s ⊆ v is called monotone independent set, if the set s does not contains any monotone adjacent vertices. the maximum cardinality of the monotone independent set is called monotone independence number of the graph and denoted by βmo(g). for example; for any path pn with n ≥ 3 vertices, βmo(pn) = 2. the monotone independence number of any complete monotone graph is one. theorem 2.6. let g be a graph with n vertices and with monotone independence number βmo(g). then χmo(g) ≥ n βmo(g) . further, the equality holds if g is complete monotone graph. proof. let χmo(g) = k and let cmo : v (g) −→ 1, 2, . . . ,k be monotone k-coloring function. let βmo(g) be the monotone independence number of g. we define vj = {v | cmo(v) = j} for j = 1, 2, . . . ,k. obviously, vj is monotone independent set and clearly |vj| ≤ βmo(g) and k∑ j=1 |vj| ≤ kβmo(g) = χmo(g)βmo(g). (2.3) also since k∑ j=1 |vj| = n. (2.4) thus, from inequalities 2.3 and 3.1 n ≤ χmo(g)βmo(g) ⇒ χmo(g) ≥ n βmo(g) . also, if g is complete monotone graph, then βmo(g) = 1. therefore, χmo(g) = n βmo(g) = n. � int. j. anal. appl. 18 (6) (2020) 1118 3. monotone chromatic number for some cartesian product of graphs the cartesian product g1�g2 of two graphs g1 and g2, where v (g1),e(g1) and v (g2),e(g2) are the sets of vertices and edges of g1 and g2, respectively, has the vertex set v (g1) × v (g2) and two vertices (u,u′) and (v,v′) are connected by an edge if and only if either (u = v and u′v′ ∈ e(g2)) or (u′ = v′ and uv ∈ e(g1)) [6]. theorem 3.1. let g ∼= pr�ps where r,s ≥ 3. then χmo(g) = (r − 2)(s− 1) + 1. proof. let g ∼= pr�ps where r,s ≥ 3. by labeling the vertices of the graph g from the left to the right the first line l1 of the vertices by v1,1,v1,2, . . . ,v1,s−1,v1,s, the second line of the vertices v2,1,v2,2, . . . ,v2,s−1,v2,s and so on till line lr−1 by vr−1,1,vr−1,2, . . . ,vr−1,s−1,vr−1,s and finally, the vertices of the line lr by vr,1,vr,2, . . . ,vr,s−1,vr,s. let us make partition for the vertex set to the following subsets s1 = {v1,1,v1,s,vr,1,vr,s}. s2 = {vi,j : 1 < i < r, 1 ≤ j < s}. s3 = {v1,j,vr,j : 1 < j < s}. s4 = {vi,s : 1 < i < r}. we can define a monotone coloring function as following f(vi,j) =   1, if vi,j ∈ s1 ; (i− 2)s + j + 1, if vi,j ∈ s2; f(vj, 1), if vi,j ∈ s3 ; f(vj, 1), if vi,j ∈ s4. clearly f is a monotone-k-coloring function, where k = (r − 2)(s− 1) + 1. so χmo(g) ≤ (r − 2)(s− 1) + 1. (3.1) the set s2 ∪{v1,1} is monotone clique set with (r − 2)(s− 1) + 1 vertices. therefore, χmo(g) ≥ (r − 2)(s− 1) + 1. (3.2) hence by inequalities 3.1 and 3.2, we get, χmo(g) = (r − 2)(s− 1) + 1. � int. j. anal. appl. 18 (6) (2020) 1119 theorem 3.2. let g ∼= cr�ps, where r,s ≥ 3. then χmo(g) = r(s− 1). proof. let g ∼= cr�ps, where r,s ≥ 3. by labeling the vertices of the internal and external cycles of g as the vertices of the internal cycle by v1,v2, . . . ,vr and the vertices of external cycle by u1,u2, . . . ,ur see as figure 2. let us partition the set of vertices as following subsets: figure 2. cr�ps s1 = {vi, 1 ≤ i ≤ r}. s2 = {ui, 1 ≤ i ≤ r}. s3 = {wj,wj ∈ v (g) and w /∈ (s1 ∪s2)}. by defining a the coloring function, f : v (g) −→{1, 2, . . . ,r(s− 1)} such that: for any vertex x f(x) =   i, if x = vi or x = ui; j + r, if wj ∈ s3. obviously, f is a monotone r(s− 1) coloring function. so χmo(g) ≤ r(s− 1). (3.3) the set s1 ∪s3 is monotone clique set with r(s− 1) vertices. therefore, χmo(g) ≥ r(s− 1). (3.4) int. j. anal. appl. 18 (6) (2020) 1120 hence by inequalities 3.3 and 3.4, we get, χmo(g) = r(s− 1). � the stacked book is defined as the graph which construct by cartesian product between star sr of r + 1 vertices and path ps of s vertices and denoted by br,s. that means br,s ∼= sr�ps. theorem 3.3. for any stacked book graph g ∼= sr�ps, where r ≥ 3 and s ≥ 2 χmo(g) =   4, if s = 2;2s− 3, if s ≥ 3. proof. let g ∼= sr�ps. by labeling the vertices of the graph by using the horizontal lines of the vertices, let the center line l0 and the outside lines are l1,l2, . . . ,lr. by labeling the vertices of lines from the left to the right according to the line, the vertices on the center line means the vertices on the line l0 are labeling as v01,v02, . . . ,v0s, the vertices on the line l1 are labeling as v11,v12, . . . ,v1s, the vertices on the line l2 are labeling as v21,v22, . . . ,v2s and so on till the vertices on the line lr are labeling as vr1,vr2, . . . ,vrs (see figure 3). figure 3. stacked book graph let us partition the vertex set of the graph g to following subsets: s1 = {vi1,vis; 1 ≤ i ≤ r}, s2 = {v0j; 1 ≤ j < s}, s3 = {v1j; 2 < j < s}, s4 = {vi2; 1 ≤ i ≤ r}, s5 = {vij; 2 ≤ i ≤ r, and 2 < j < s} and s6 = {v0s}. define a coloring function f : v (g) −→{1, 2, . . . ,k} such that: int. j. anal. appl. 18 (6) (2020) 1121 f(vij) =   1, if vij ∈ s1 ; j + 1, if vij ∈ s2; s + j − 2, if vij ∈ s3; f(v01), if vij ∈ s4 or vij = v0s ; f(v1j), if vij ∈ s5. clearly, f is a monotone (2s− 3) coloring function. so χmo(g) ≤ (2s− 3). (3.5) the set s2∪s3∪{v11} is monotone clique set in g and need (2s−3) colors for monotone coloring. therefore, χmo(g) ≥ (2s− 3). (3.6) hence by inequalities 3.5 and 3.6, we get, χmo(g) = (2s− 3). � lemma 3.1. let g and h be any non trivial connected graphs and g�h is the cartesian product. then any two vertices (u,v) and (x,y) are monotone adjacent in g�h, if one of the following satisfy: (1) x = u in g and y is monotone adjacent with v in h. (2) x and u are monotone adjacent vertices in g and y = v in h. (3) x is monotone adjacent with u in g and y is monotone adjacent with v in h. theorem 3.4. for any two graphs g and h, χmo(g) + χmo(h) − 1 ≤ χmo(g�h) ≤ χmo(g)χmo(h). proof. let χmo(g) = k1 and χmo(h) = k2. by theorem 2.3, ωmo(g) = k1 and ωmo(h) = k2. then there exist two monotone cliques: c1 = {u1,u2, . . . ,uk1} and c2 = {v1,v2, . . . ,vk2} in g and h respectively, with k1 and k2 vertices. therefore, {(u1,v1), (u2,v1), (u3,v1), . . . , (uk1,v1), (uk1,v2), (uk1,v3), . . . , (uk1,vk3 )} is monotone clique in g�h with k1 + k2 − 1 vertices.therefore, k1 + k2 − 1 ≤ ωmo(g�h) = χmo(g�h). then, ωmo(g) + ωmo(h) − 1 ≤ χmo(g�h) (3.7) now, assume that χmo(g�h) = m, that means ωmo(g�h) = m, then there exists at least one monotone clique set with m vertices, say c = {w1,w2, . . . ,wm}. by definition of cartesian product any two adjacent int. j. anal. appl. 18 (6) (2020) 1122 vertices wi = (ai,bi),wi+1 = (ai+1,bi+1) either aiai+1 is connected in g with bi = bi+1 or bibi+1 is connected in h with ai = ai+1. any sequence of adjacent vertices on c which they are mutually different in first component will make correspondence monotone clique in g. similarly, any sequence of adjacent vertices which they are mutually different in second component will make correspondence monotone clique in h. let the maximum number of vertices for the correspondence monotone clique in g which correspondence vertices in g�h is s and similarly, the maximum number of vertices for the correspondence monotone clique in h which correspondence vertices in g�h is t. then the number of vertices on c = {w1,w2, . . . ,wm} will be at most m = st, since s ≤ ωmo(g) and t ≤ ωmo(h). then ωmo(g�h) ≤ ωmo(g)ωmo(h) (3.8) thus, by inequalities 3.7 and 3.8, χmo(g) + χmo(h) − 1 ≤ χmo(g�h) ≤ χmo(g)χmo(h). � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.m. cardoso, j.o. cerdeira, j.p. cruz, c. dominic, injective edge chromatic index of a graph, arxiv:1510.02626 [math]. (2015). [2] j.l. fouquet , j.l. jolivet, strong edge-colorings of graphs and applications to multi k -gons, ars combin. 16a (1983), 141-150. [3] f. guthrie. note on the colouring of maps. proc. r. soc. edinburgh. 10 (1880), 727–728. [4] i. gutman, kragujevac trees and their energy, sci. publ. state univ. novi pazar ser. a., appl. math. inform. mech. 6 (2) (2014), 71-79. [5] g. hahn, j. kratochv́il, j. s̆irán̆ and d. sotteau, on the injective chromatic number of graphs. discrete math. 256 (2002), 179–192. [6] f. harary, graph theory, addison-wesley, reading mass. (1969). [7] j.x. li, j.m. guo, w.c. shiu, on the second largest laplacian eigenvalues of graphs, linear algebra appl. 438 (2013), 2438–2446. [8] n. zufferey, p. amstutz, p. giaccari, graph colouring approaches for a satellite range scheduling problem, j. schedul. 11 (4) (2008), 263–277. 1. introduction 2. monotone chromatic number of graphs 3. monotone chromatic number for some cartesian product of graphs references international journal of analysis and applications issn 2291-8639 volume 3, number 2 (2013), 93-103 http://www.etamaths.com ifαgs continuous and ifαgs irresolute mappings m. jeyaraman1, a. yuvarani2,∗ and o. ravi3 abstract. the objective of this paper is to establish intuitionistic fuzzy αgeneralized semi continuous mappings and to study some of their properties. finally we introduce intuitionistic fuzzy α-generalized semi irresolute mappings and investigate their characterizations. 1. introduction as a generalization of fuzzy sets, the concepts of intuitionistic fuzzy sets was introduced by atanassov [1]. recently, coker and demirci [3] introduced the basic definitions and properties of intuitionistic fuzzy topological spaces using the notion of intuitionistic fuzzy sets. in 2004, m.rajamani and k.viswanathan [8] introduced α generalized semi continuous maps and α generalized semi irresolute maps in topological spaces. in this paper we introduce intuitionistic fuzzy α-generalized semi continuous mappings and intuitionistic fuzzy α-generalized semi irresolute mappings. also the interconnections between the intuitionistic fuzzy continuous mappings and the intuitionistic fuzzy irresolute mappings are investigated. some examples are given to illustrate the results. 2. preliminaries definition 2.1. [1] let x be a non empty fixed set. an intuitionistic fuzzy set(ifs in short) a in x is an object having the form a = {〈 x, µa(x), νa(x) 〉 | x ∈ x} where the function µa(x):x → [0,1] denotes the degree of membership(namely µa(x)) and the function νa(x):x → [0,1] denotes the degree of non-membership(namely νa(x)) of each element x ∈ x to the set a, respectively and 0 ≤ µa(x) + νa(x) ≤ 1 for each x ∈ x. ifs(x) denote the set of all intuitionistic fuzzy sets in x. definition 2.2. [1] let a and b be ifss of the form a={〈 x, µa(x), νa(x) 〉 | x ∈x} and b={〈 x, µb(x), νb(x) 〉 | x ∈ x}. then (1) a ⊆ b if and only if µa(x) ≤ µb(x) and νa(x) ≥ νb(x) for all x ∈ x, (2) a = b if and only if a ⊆ b and b ⊆ a, 2010 mathematics subject classification. 54a02, 54a40, 54a99, 03f55. key words and phrases. intuitionistic fuzzy topology, intuitionistic fuzzy α-generalized semi closed set, intuitionistic fuzzy α-generalized semi continuous mapping and intuitionistic fuzzy α-generalized semi irresolute mapping. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 93 94 jeyaraman, yuvarani and ravi (3) ac = {〈 x, νa(x), µa(x) 〉 | x ∈ x}, (4) a∩b = {〈 x, µa(x) ∧ µb(x), νa(x) ∨ νb(x) 〉 | x ∈ x}, (5) a∪b = {〈 x, µa(x) ∨ µb(x), νa(x) ∧ νb(x) 〉 | x ∈ x}. for the sake of simplicity, we shall use the notation a=〈 x, µa, νa 〉 instead of a = {〈 x, µa(x), νa(x) 〉 | x ∈ x}. the intuitionistic fuzzy sets 0∼ = {〈 x, 0, 1 〉 : x ∈ x} and 1∼ = {〈 x, 1, 0 〉 : x ∈ x} are the empty set and the whole set of x respectively. definition 2.3. [3] an intuitionistic fuzzy topology (ift in short) on x is a family τ of ifss in x satisfying the following axioms: (1) 0∼, 1∼ ∈ τ, (2) g1 ∩ g2 ∈ τ for any g1, g2 ∈ τ, (3) ∪gi ∈ τ for any family {gi | i ∈ j} ⊆ τ. in this case the pair (x, τ) is called an intuitionistic fuzzy topological space(ifts in short) and any ifs in τ is known as an intuitionistic fuzzy open set(ifos in short) in x. the complement ac of an ifos a in an ifts (x, τ) is called an intuitionistic fuzzy closed set(ifcs in short) in x. definition 2.4. [3] let (x, τ) be an ifts and a = 〈 x, µa, νa 〉 be an ifs in x. then the intuitionistic fuzzy interior and the intuitionistic fuzzy closure are defined as follows: (1) int(a) = ∪{g | g is an ifos in x and g ⊆ a}, (2) cl(a) = ∩{k | k is an ifcs in x and a ⊆ k}. note that for any ifs a in (x, τ), we have cl(ac) = (int(a))c and int(ac) = (cl(a))c. definition 2.5. an ifs a = 〈 x, µa, νa 〉 in an ifts (x, τ) is said to be an (1) intuitionistic fuzzy regular closed set(ifrcs in short) if a = cl(int(a)) [3], (2) intuitionistic fuzzy α-closed set(ifαcs in short) if cl(int(cl(a))) ⊆ a [5], (3) intuitionistic fuzzy semiclosed set(ifscs in short) if int(cl(a)) ⊆ a [3], (4) intuitionistic fuzzy preclosed set(ifpcs in short) if cl(int(a)) ⊆ a [3], (5) intuitionistic fuzzy semipreclosed set(ifspcs in short) if there exists an ifpcs b such that int(b) ⊆ a ⊆ b[14]. definition 2.6. an ifs a = 〈 x, µa, νa 〉 in an ifts (x, τ) is said to be an (1) intuitionistic fuzzy regular open set(ifros in short) if a = int(cl(a))[3], (2) intuitionistic fuzzy α-open set(ifαos in short) if a ⊆ int(cl(int(a)))[5], (3) intuitionistic fuzzy semiopen set(ifsos in short) if a ⊆ cl(int(a))[3], (4) intuitionistic fuzzy preopen set (ifpos in short) if a ⊆ int(cl(a))[3], (5) intuitionistic fuzzy semipreopne set (ifspos in short) if there exists an ifpos b such that b ⊆ a ⊆ cl(b)[14] the family of all ifos(respectively ifsos, ifαos, ifros) of an ifts (x, τ) is denoted by ifos(x)(respectively ifsos(x), ifαos(x), ifros(x)). definition 2.7. [14] let a be an ifs in (x, τ), then semi interior of a(sint(a) in short) and semi closure of a ( scl(a) in short) are defined as (1) sint(a) = ∪{k | k is an ifsos in x and k ⊆ a}, (2) scl(a) = ∩{k | k is an ifscs in x and a ⊆ k}. ifαgs continuous and ifαgs irresolute mappings 95 definition 2.8. [12] let a be an ifs in (x, τ), then semipre interior of a(spint(a) in short) and semipre closure of a ( spcl(a) in short) are defined as (1) spint(a) = ∪{g | g is an ifspos in x and g ⊆ a}, (2) spcl(a) = ∩{k | k is an ifspcs in x and a ⊆ k}. definition 2.9. [9] let a be an ifs of an ifts (x, τ). then (1) αcl(a) = ∩{k | k is an ifαcs in x and a ⊆ k}, (2) αint(a) = ∪{k | k is an ifαos in x and k ⊆ a}. definition 2.10. an ifs a of an ifts (x, τ) is an (1) intuitionistic fuzzy generalized closed set(ifgcs in short) if cl(a) ⊆ u whenever a ⊆ u and u is an ifos in x [13], (2) intuitionistic fuzzy generalized semiclosed set(ifgscs in short) if scl(a) ⊆ u whenever a ⊆ u and u is an ifos in x [11], (3) intuitionistic fuzzy generalized semipreclosed set(ifgspcs in short) if spcl(a) ⊆ u whenever a ⊆ u and u is an ifos in x [12], (4) intuitionistic fuzzy alpha generalized closed set(ifαgcs in short) if αcl(a) ⊆ u whenever a ⊆ u and u is an ifos in x [9], (5) intuitionitic fuzzy generalized alpha closed set (ifgαcs in short) if αcl(a) ⊆ u whenever a ⊆ u and u is an ifαos in x [7]. the complements of the above mentioned intuitionistic fuzzy closed sets are called their respective intuitionistic fuzzy open sets. definition 2.11. [15] an ifs a of an ifts (x, τ) is said to be an intuitionistic fuzzy alpha generalized semi closed set(ifαgscs in short) if αcl(a) ⊆ u whenever a ⊆ u and u is an ifsos in (x, τ). an ifs a is said to be an intuitionistic fuzzy α-generalized semi openset(ifαgsos in short) in x if ac is an ifαgscs in x. the family of all ifαgscss(respective ifαgsoss) of an ifts (x, τ) is denoted by ifαgscs(x)(respective ifαgsos(x)). remark 2.12. [15] every ifcs, ifrcs, ifαcs is an ifαgscs but their separate converses may not be true in general. every ifαgscs is ifgscs, ifgαcs, ifαgcs but their separate converses may not be true in general. definition 2.13. let f be a mapping from an ifts (x, τ) into an ifts (y, σ). then f is said to be an (1) intuitionistic fuzzy continuous (if continuous in short) if f−1(b) ∈ ifos(x) for every b ∈ σ[4], (2) intuitionistic fuzzy α-continuous (ifα continuous in short) if f−1(b) ∈ ifαos(x) for every b ∈ σ[6], (3) intuitionistic fuzzy pre continuous (ifp continuous in short) if f−1(b) ∈ ifpos(x) for every b ∈ σ[6]. every if continuous mapping is an ifα-continuous mapping but not conversely. definition 2.14. let f be a mapping from an ifts (x, τ) into an ifts (y, σ). then f is said to be an (1) intuitionistic fuzzy generalized continuous(ifg continuous in short) if f−1(b) is an ifgcs for every ifcs b of (y, σ)[13], (2) intuitionistic fuzzy generalized semi continuous(ifgs continuous in short) if f−1(b) is an ifgscs for every ifcs b of (y, σ)[11], 96 jeyaraman, yuvarani and ravi (3) intuitionistic fuzzy generalized semi pre continuous(ifgsp continuous in short) if f−1(b) is an ifgspcs for every ifcs b of (y, σ)[12], (4) intuitionistic fuzzy α-generalized continuous(ifαg continuous in short) if f−1(b) is an ifαgcs for every ifcs b of (y, σ)[10], (5) intuitionistic fuzzy generalized α continuous(ifgα continuous in short) if f−1(b) is an ifgαcs for every ifcs b of (y, σ)[7]. definition 2.15. let f be a mapping from an ifts (x, τ) into an ifts (y, σ). then f is said to be an (1) intuitionistic fuzzy irresolute (if irresolute in short) if f−1(b) ∈ ifcs(x) for every ifcs b in y[11], (2) intuitionistic fuzzy generalized irresolute(ifg irresolute in short) if f−1(b) is ifgcs in x for every ifgcs b in y[11]. 3. intuitionistic fuzzy α-generalized semi continuous mappings in this section we introduce intuitionistic fuzzy α-generalized semi continuous mapping and study some of its properties. definition 3.1. a mapping f:(x, τ) → (y, σ) is called an intuitionistic fuzzy αgeneralized semi continuous(ifαgs continuous in short) if f−1(b) is an ifαgscs in (x, τ) for every ifcs b of (y, σ). example 3.2. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.6, 0.7), (0.4, 0.3)〉 and t2 = 〈 y, (0.9, 0.8), (0.1, 0.2) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. then f is an ifαgs continuous mapping. theorem 3.3. every if continuous mapping is an ifαgs continuous mapping. proof. let f:(x, τ) → (y, σ) be an if continuous mapping. let a be an ifcs in y. since f is an if continuous mapping, f−1(a) is an ifcs in x. since every ifcs is an ifαgscs, f−1(a) is an ifαgscs in x. hence f is an ifαgs continuous mapping. example 3.4. ifαgs continuous mapping 9 if continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.4, 0.2), (0.6, 0.7)〉 and t2 = 〈 y, (0.3, 0.2), (0.7, 0.8) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.7, 0.8), (0.3, 0.2)〉 is ifcs in y, f−1(a) is an ifαgscs but not ifcs in x. therefore f is an ifαgs continuous mapping but not an if continuous mapping. theorem 3.5. every ifα continuous mapping is an ifαgs continuous mapping. proof. let f:(x, τ) → (y, σ) be an ifα continuous mapping. let a be an ifcs in y. then by hypothesis f−1(a) is an ifαcs in x. since every ifαcs is an ifαgscs, f−1(a) is an ifαgscs in x. hence f is an ifαgs continuous mapping. example 3.6. ifαgs continuous mapping 9 ifα continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.4, 0.5), (0.6, 0.5)〉 and t2 = 〈 y, (0.2, 0.4), (0.8, 0.6) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.8, 0.6), (0.2, 0.4)〉 is ifcs in y, f−1(a) is an ifαgscs ifαgs continuous and ifαgs irresolute mappings 97 but not ifαcs in x. therefore f is an ifαgs continuous mapping but not an ifα continuous mapping. remark 3.7. ifg continuous mappings and ifαgs continuous mappings are independent of each other. example 3.8. ifαgs continuous mapping 9 ifg continuous mapping. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.4, 0.7), (0.5, 0.3)〉 and t2 = 〈 y, (0.6, 0.8), (0.3, 0.2) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. then f is ifαgs continuous mapping but not ifg continuous mapping. since a = 〈y, (0.3, 0.2), (0.6, 0.8)〉 is ifcs in y, f−1(a) = 〈 x, (0.3, 0.2), (0.6, 0.8)〉 is not ifgcs in x. example 3.9. ifg continuous mapping 9 ifαgs continuous mapping. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.6, 0.8), (0.4, 0.2)〉 and t2 = 〈 y, (0.3, 0.1), (0.7, 0.9) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. then f is ifg continuous mapping but not an ifαgs continuous mapping. since a = 〈y, (0.7, 0.9), (0.3, 0.1)〉 is ifcs in y, f−1(a) = 〈 x, (0.7, 0.9), (0.3, 0.1)〉 is not ifαgscs in x. theorem 3.10. every ifαgs continuous mapping is an ifgs continuous mapping. proof. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping. let a be an ifcs in y. then by hypothesis f−1(a) ifαgscs in x. since every ifαgscs is an ifgscs, f−1(a) is an ifgscs in x. hence f is an ifgs continuous mapping. example 3.11. ifgs continuous mapping 9 ifαgs continuous mapping. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.7, 0.8), (0.3, 0.1)〉 and t2 = 〈 y, (0.2, 0), (0.8, 0.8) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.8, 0.8), (0.2, 0)〉 is ifcs in y, f−1(a) is ifgscs in x but not ifαgscs in x. therefore f is an ifgs continuous mapping but not an ifαgs continuous mapping. remark 3.12. ifp continuous mappings and ifαgs continuous mappings are independent of each other. example 3.13. ifp continuous mapping 9 ifαgs continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.4, 0.3), (0.6, 0.5)〉 and t2 = 〈 y, (0.7, 0.8), (0.2, 0.1) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.2, 0.1), (0.7, 0.8)〉 is ifcs in y, f−1(a) is ifpcs in x but not ifαgscs in x. therefore f is an ifp continuous mapping but not ifαgs continuous mapping. example 3.14. ifαgs continuous mapping 9 ifp continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.3, 0.4), (0.7, 0.6)〉 and t2 = 〈 y, (0.4, 0.5), (0.6, 0.5) 〉 and t3 = 〈 y, (0.7, 0.4), (0.3, 0.6) 〉. then τ = { 0∼, t1, t2, 1∼ } and σ = { 0∼, t3, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.3, 0.6), (0.7, 98 jeyaraman, yuvarani and ravi 0.4)〉 is ifαgscs but not ifpcs in y, f−1(a) is ifαgscs in x but not ifpcs in x. therefore f is an ifαgs continuous mapping but not ifp continuous mapping. theorem 3.15. every ifαgs continuous mapping is an ifgsp continuous mapping. proof. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping. let a be an ifcs in y. then by hypothesis f−1(a) is an ifαgscs in x. since every ifαgscs is an ifgspcs, f−1(a) is an ifgspcs in x. hence f is an ifgsp continuous mapping. example 3.16. ifgsp continuous mapping 9 ifαgs continuous mapping. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.3, 0.1), (0.6, 0.8)〉 and t2 = 〈 y, (0.7, 0.8), (0.2, 0.0) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.2, 0.0), (0.7, 0.8)〉 is ifcs in y, f−1(a) is an ifgspcs but not ifαgscs in x. therefore f is an ifgsp continuous mapping but not an ifαgs continuous mapping. theorem 3.17. every ifαgs continuous mapping is an ifαg continuous mapping. proof. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping. let a be an ifcs in y. since f is ifαgs continuous mapping, f−1(a) is an ifαgscs in x. since every ifαgscs is an ifαgcs, f−1(a) is an ifαgcs in x. hence f is an ifαg continuous mapping. example 3.18. ifαg continuous mapping 9 ifαgs continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.1, 0.3), (0.7, 0.6)〉 and t2 = 〈 y, (0.6, 0.5), (0.3, 0.4) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.3, 0.4), (0.6, 0.5)〉 is ifcs in y, f−1(a) is ifαgcs in x but not ifαgscs in x. therefore f is an ifαg continuous mapping but not an ifαgs continuous mapping. theorem 3.19. every ifαgs continuous mapping is an ifgα continuous mapping. proof. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping. let a be an ifcs in y. since f is ifαgs continuous mapping, f−1(a) is an ifαgscs in x. since every ifαgscs is an ifgαcs, f−1(a) is an ifgαcs in x. hence f is an ifgα continuous mapping. example 3.20. ifgα continuous mapping 9 ifαgs continuous mapping let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.4, 0.2), (0.6, 0.8)〉 and t2 = 〈 y, (0.5, 0.4), (0.5, 0.6) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. since the ifs a = 〈y, (0.5, 0.6), (0.5, 0.4)〉 is ifcs in y, f−1(a) is ifgαcs in x but not ifαgscs in x. therefore f is an ifgα continuous mapping but not an ifαgs continuous mapping. remark 3.21. we obtain the following diagram from the results we discussed above. ifαgs continuous and ifαgs irresolute mappings 99 if cts ifαgs cts ifgα cts ifαg cts ifgs cts @ @@i � � �� @ @ @i 7 r� ��� ifα cts w � ifgsp cts none of the reverse implications are not true. theorem 3.22. a mapping f: x → y is ifαgs continuous if and only if the inverse image of each ifos in y is an ifαgsos in x. proof. ⇒ part let a be an ifos in y. this implies ac is ifcs in y. since f is ifαgs continuous, f−1(ac) is ifαgscs in x. since f−1(ac) = (f−1(a))c, f−1(a) is an ifαgsos in x. ⇐ part let a be an ifcs in y. then ac is an ifos in y. by hypothesis f−1(ac) is ifαgsos in x. since f−1(ac) = (f−1(a))c, (f−1(a))c is an ifαgsos in x. therefore f−1(a) is an ifαgscs in x. hence f is ifαgs continuous. theorem 3.23. let f:(x, τ) → (y, σ) be a mapping and f−1(a) be an ifrcs in x for every ifcs a in y. then f is an ifαgs continuous mapping. proof. let a be an ifcs in y and f−1(a) be an ifrcs in x. since every ifrcs is an ifαgscs, f−1(a) is an ifαgscs in x. hence f is an ifαgs continuous mapping. definition 3.24. an ifts (x, τ) is said to be an (1) intuitionistic fuzzy αgat1/2(in short ifαgat1/2)space if every ifαgscs in x is an ifcs in x, (2) intuitionistic fuzzy αgbt1/2(in short ifαgbt1/2)space if every ifαgscs in x is an ifgcs in x, (3) intuitionistic fuzzy αgct1/2(in short ifαgct1/2)space if every ifαgscs in x is an ifgscs in x. theorem 3.25. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping, then f is an if continuous mapping if x is an ifαgat1/2 space. proof. let a be an ifcs in y. then f−1(a) is an ifαgscs in x, by hypothesis. since x is an ifαgat1/2, f −1(a) is an ifcs in x. hence f is an if continuous mapping. theorem 3.26. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping, then f is an ifg continuous mapping if x is an ifαgbt1/2 space. proof. let a be an ifcs in y. then f−1(a) is an ifαgscs in x, by hypothesis. since x is an ifαgbt1/2, f −1(a) is an ifgcs in x. hence f is an ifg continuous mapping. 100 jeyaraman, yuvarani and ravi theorem 3.27. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping, then f is an ifgs continuous mapping if x is an ifαgct1/2 space. proof. let a be an ifcs in y. then f−1(a) is an ifαgscs in x, by hypothesis. since x is an ifαgct1/2, f −1(a) is an ifgscs in x. hence f is an ifgs continuous mapping. theorem 3.28. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping and g:(y, σ) → (z, δ) be an if continuous, then g◦f : (x, τ) → (z, δ) is an ifαgs continuous. proof. let a be an ifcs in z. then g−1(a) is an ifcs in y, by hypothesis. since f is an ifαgs continuous mapping, f−1(g−1(a)) is an ifαgscs in x. hence g◦f is an ifαgs continuous mapping. theorem 3.29. let f:(x, τ) → (y, σ) be a mapping from an ifts x into an ifts y. then the following conditions are equivalent if x is an ifαgat1/2 space. (1) f is an ifαgs continuous mapping. (2) if b is an ifos in y then f−1(b) is an ifαgsos in x. (3) f−1(int(b)) ⊆ int(cl(int(f−1(b)))) for every ifs b in y. proof. (1) ⇒ (2): is obviously true. (2) ⇒ (3): let b be any ifs in y. then int(b) is an ifos in y. then f−1(int(b)) is an ifαgsos in x. since x is an ifαgat1/2 space, f −1(int(b)) is an ifos in x. therefore f−1(int(b)) = int(f−1(int(b))) ⊆ int(cl(int(f−1(b)))). (3) ⇒ (1) let b be an ifcs in y. then its complement bc is an ifos in y. by hypothesis f−1(int(bc)) ⊆ int(cl(int(f−1(int(bc))))). this implies that f−1(bc) ⊆ int(cl(int(f−1(int(bc))))). hence f−1(bc) is an ifαos in x. since every ifαos is an ifαgsos, f−1(bc) is an ifαgsos in x. therefore f−1(b) is an ifαgscs in x. hence f is an ifαgs continuous mapping. theorem 3.30. let f:(x, τ) → (y, σ) be a mapping. then the following conditions are equivalent if x is an ifαgat1/2 space. (1) f is an ifαgs continuous mapping. (2) f−1(a) is an ifαgscs in x for every ifcs a in y. (3) cl(int(cl(f−1(a)))) ⊆ f−1(cl(a)) for every ifs a in y. proof. (1) ⇒ (2): is obviously true. (2) ⇒ (3): let a be an ifs in y. then cl(a) is an ifcs in y. by hypothesis, f−1(cl(a)) is an ifαgscs in x. since x is an ifαgat1/2 space, f −1(cl(a)) is an ifcs in x. therefore cl(f−1(cl(a))) = f−1(cl(a)). now cl(int(cl(f−1(a)))) ⊆ cl(int(cl(f−1(cl(a))))) ⊆ f−1(cl(a)). (3) ⇒ (1): let a be an ifcs in y. by hypothesis cl(int(cl(f−1(a)))) ⊆ f−1(cl(a)) = f−1(a). this implies f−1(a) is an ifαcs in x and hence it is an ifαgscs in x.therefore f is an ifαgs continuous mapping. definition 3.31. let (x, τ) be an ifts. the alpha generalized semi closure (αgscl(a) in short) for any ifs a is defined as follows. αgscl(a) = ∩ {k | k is an ifαgscs in x and a ⊆ k }. if a is ifαgscs, then αgscl(a) = a. theorem 3.32. let f:(x, τ) → (y, σ) be an ifαgs continuous mapping. then the following conditions are hold. ifαgs continuous and ifαgs irresolute mappings 101 (1) f(αgscl(a)) ⊆ cl(f(a)), for every ifs a in x. (2) αgscl(f−1(b)) ⊆ f−1(cl(b)), for every ifs b in y. proof. (i) since cl(f(a)) is an ifcs in y and f is an ifαgs continuous mapping, f−1(cl(f(a))) is ifαgscs in x. that is αgscl(a) ⊆ f−1(cl(f(a))). therefore f(αgscl(a)) ⊆ cl(f(a)), for every ifs a in x. (ii) replacing a by f−1(b) in (i) we get f(αgscl(f−1(b))) ⊆ cl(f(f−1(b))) ⊆ cl(b). hence αgscl(f−1(b)) ⊆ f−1(cl(b)), for every ifs b in y. remark 3.33. the composition of two ifαgs continuous mappings need not be ifαgs continuous as can be seen from the following example: example 3.34. let x = {a, b}, y = {u, v} and z = {s, t}. let τ = { 0∼, t1, 1∼ }, σ = { 0∼, t2, 1∼ } and δ = { 0∼, t3, 1∼ } be ifts on x, y and z respectively where t1 = 〈 x, (0.4, 0.3), (0.6, 0.7)〉, t2 = 〈 y, (0.3, 0.8), (0.7, 0.2)〉 and t3 = 〈 z, (0.4, 0.9), (0.6, 0.1)〉. define f:(x, τ) → (y, σ) by f(a) = u and f(b) = v and g:(y, σ) → (z, δ) by g(u) = s and g(v) = t. then f and g are ifαgs continuous mappings. since a = 〈z, (0.6, 0.1), (0.4, 0.9)〉 is an ifcs in z, f−1(a) is not an ifαgscs in x. therefore the composition map g ◦ f: (x, τ) → (z, δ) is not an ifαgs continuous. 4. intuitionistic fuzzy α-generalized semi irresolute mappings in this section we introduce intuitionistic fuzzy α-generalized semi irresolute mappings and study some of its characterizations. definition 4.1. a mapping f:(x, τ) → (y, σ) is called an intuitionistic fuzzy alpha-generalized semi irresolute( ifαgs irresolute) mapping if f−1(a) is an ifαgscs in (x, τ) for every ifαgscs a of (y, σ). theorem 4.2. let f:(x, τ) → (y, σ) be an ifαgs irresolute, then f is an ifαgs continuous mapping. proof. let f be an ifαgs irresolute mapping. let a be any ifcs in y. since every ifcs is an ifαgscs, a is an ifαgscs in y. by hypothesis f−1(a) is an ifαgscs in x. hence f is an ifαgs continuous mapping. example 4.3. ifαgs continuous mapping 9 ifαgs irresolute mapping. let x = {a, b}, y = {u, v}, t1 = 〈 x, (0.3, 0.4), (0.6, 0.5)〉 and t2 = 〈 y, (0.7, 0.3), (0.2, 0.6) 〉. then τ = { 0∼, t1, 1∼ } and σ = { 0∼, t2, 1∼ } are ifts on x and y respectively. define a mapping f:(x, τ) → (y, σ) by f(a) = u and f(b) = v. then f is an ifαgs continuous. we have b = 〈y, (0.1, 0.5), (0.8, 0.4)〉 is an ifαgscs in y but f−1(b) is not an ifαgscs in x. therefore f is not an ifαgs irresolute mapping. theorem 4.4. let f:(x, τ) → (y, σ) be an ifαgs irresolute, then f is an if irresolute mapping if x is an ifαgat1/2 space. proof. let a be an ifcs in y. then a is an ifαgscs in y. therefore f−1(a) is an ifαgscs in x, by hypothesis. since x is an ifαgat1/2 space, f −1(a) is an ifcs in x. hence f is an if irresolute mapping. theorem 4.5. let f:(x, τ) → (y, σ) and g:(y, σ) → (z, δ) be ifαgs irresolute mappings, then g◦f: (x, τ) → (z, δ) is an ifαgs irresolute mapping. 102 jeyaraman, yuvarani and ravi proof. let a be an ifαgscs in z. then g−1(a) is an ifαgscs in y. since f is an ifαgs irresolute mapping. f−1((g−1(a))) is an ifαgscs in x. hence g◦f is an ifαgs irresolute mapping. theorem 4.6. let f:(x, τ) → (y, σ) be an ifαgs irresolute and g:(y, σ) → (z, δ) be ifαgs continuous mappings, then g◦f: (x, τ) → (z, δ) is an ifαgs continuous mapping. proof. let a be an ifcs in z. then g−1(a) is an ifαgscs in y. since f is an ifαgs irresolute, f−1((g−1(a)) is an ifαgscs in x. hence g◦f is an ifαgs continuous mapping. theorem 4.7. let f:(x, τ) → (y, σ) be an ifαgs irresolute, then f is an ifg irresolute mapping if x is an ifαgbt1/2 space. proof. let a be an ifαgscs in y. by hypothesis, f−1(a) is an ifαgscs in x. since x is an ifαgbt1/2 space, f −1(a) is an ifgcs in x. hence f is an ifg irresolute mapping. theorem 4.8. let f:(x, τ) → (y, σ) be a mapping from an ifts x into an ifts y. then the following conditions are equivalent if x and y are ifαgat1/2 spaces. (1) f is an ifαgs irresolute mapping. (2) f−1(b) is an ifαgsos in x for each ifαgsos b in y. (3) cl(f−1(b)) ⊆ f−1(cl(b)) for each ifs b of y. proof. (1) ⇒ (2): let b be any ifαgsos in y. then bc is an ifαgscs in y. since f is ifαgs irresolute, f−1(bc) is an ifαgscs in x. but f−1(bc) = (f−1(b))c. therefore f−1(b) is an ifαgsos in x. (2) ⇒ (3): let b be any ifs in y and b ⊆ cl(b). then f−1(b) ⊆ f−1(cl(b)). since cl(b) is an ifcs in y, cl(b) is an ifαgscs in y. therefore (cl(b))c is an ifαgsos in y. by hypothesis, f−1((cl(b))c) is an ifαgsos in x. since f−1((cl(b))c) = (f−1(cl(b)))c, f−1(cl(b)) is an ifαgscs in x. since x is ifαgat1/2 space, f −1(cl(b)) is an ifcs in x. hence cl(f−1(b)) ⊆ cl(f−1(cl(b))) = f−1(cl(b)). that is cl(f−1(b)) ⊆ f−1(cl(b)). (3) ⇒ (1): let b be any ifαgscs in y. since y is ifαgat1/2 space, b is an ifcs in y and cl(b) = b. hence f−1(b) = f−1(cl(b)) ⊇ cl(f−1(b)). but clearly f−1(b) ⊆ cl(f−1(b)). therefore cl(f−1(b)) = f−1(b). this implies f−1(b) is an ifcs and hence it is an ifαgscs in x. thus f is an ifαgs irresolute mapping. references [1] atanassov. k. t., intuitionistic fuzzy sets, fuzzy sets and systems, 20(1986), 87-96. [2] chang. c., fuzzy topological spaces, j. math.anal.appl., 24(1968), 182-190. [3] coker. d., an introduction to fuzzy topological spaces, fuzzy sets and systems, 88(1997), 81-89. [4] gurcay. h., coker. d.,and es. a. haydar., on fuzzy continuity in intuitionistic fuzzy topological spaces, jour. of fuzzy math., 5(1997), 365-378. [5] hur. k. and jun. y. b., on intuitionistic fuzzy alpha continuous mappings, honam math. jour., 25(2003), 131-139. [6] joung kon jeon, young bae jun, and jin han park, intuitionistic fuzzy alpha continuity and intuitionistic fuzzy pre continuity, international journal of mathematics and mathematical sciences, 19(2005), 3091-3101. [7] kalamani. d, sakthivel. k and gowri. c. s., generalized alpha closed sets in intuitionistic fuzzy topological spaces, applied mathematical sciences, 6(2012), 4691-4700. ifαgs continuous and ifαgs irresolute mappings 103 [8] rajamani. m and k. viswanathan, on αgs-continuous maps in topological spaces, acta ciencia indica, xxxi m (1)(2005), 293-303. [9] sakthivel. k., intuitionistic fuzzy alpha generalized closed sets and intuitionistic fuzzy alpha generalized open sets, the mathematical education, 4(2012), submitted. [10] sakthivel. k., intuitionistic fuzzy alpha generalized continuous mappings and intuitionistic alpha generalized irresolute mappings, applied mathematical sciences, 4(37)(2010), 18311834. [11] santhi. r and sakthivel. k., intuitionistic fuzzy generalized semi continuous mappings, advances in theoretical and applied mathematics, 5(2009), 73-82. [12] santhi. r and jayanthi. d., intuitionistic fuzzy generalized semi-pre continuous mappings., int.j.contemp.math.sciences, 5(30)(2010), 1455-1469. [13] thakur. s. s and rekha chaturvedi., generalized closed sets in intuitionistic fuzzy topology, the journal of fuzzy mathematics, 16(2008), 559-572. [14] young baejin and seok-zun song, intuitionistic fuzzy semi-pre open sets and intuitionistic semi-pre continuous mappings, jour. of appl. math and computing, 19(2005), 467-474. [15] jeyaraman. m, yuvarani. a and ravi. o., intuitionistic fuzzy α-generalized semi closed sets(submitted) [16] zadeh. l. a., fuzzy sets, information and control, 8(1965), 338-353. 1department of mathematics, h. h.the rajah’s college, pudukkottai, tamil nadu, india 2department of mathematics, raja college of engineering and technology, madurai, tamil nadu, india 3department of mathematics, p. m. thevar college, usilampatti, madurai district, tamil nadu, india ∗corresponding author int. j. anal. appl. (2023), 21:57 a pde approach to the problems of optimality of expectations mahir hasanov∗ department of mathematics, istanbul beykent university, türkiye ∗corresponding author: hasanov61@yahoo.com abstract. let (x,z) be a bivariate random vector. a predictor of x based on z is just a borel function g(z). the problem of "least squares prediction" of x given the observation z is to find the global minimum point of the functional e[(x−g(z))2] with respect to all random variables g(z), where g is a borel function. it is well known that the solution of this problem is the conditional expectation e(x|z). we also know that, if for a nonnegative smooth function f : r×r → r, arg ming(z)e[f(x,g(z))]= e[x|z], for all x and z, then f(x,y) is a bregmann loss function. it is also of interest, for a fixed ϕ to find f(x,y), satisfying, arg ming(z)e[f(x,g(z))]=ϕ(e[x|z]), for all x and z. in more general setting, a stronger problem is to find f(x,y) satisfying arg miny∈re[f(x,y)] = ϕ(e[x]), ∀x. we study this problem and develop a partial differential equation (pde) approach to solution of these problems. 1. introduction and preliminary facts best approximation problems in mathematics have long history of study. it is known that for every given x in a hilbert space h and every given closed subspace l of h there is a unique best approximation to x out of l (namely, y = px, where p is the orthogonal projection of h onto l) (see [8] and [11]). theorem 1.1 below, regarding the optimality of conditional expectations with respect to l2 loss function f (x,y) = (x −y)2 follows from this result. theorem 1.1. (see [1], [9], [13] ) let (x,z) be a bivariate random vector and lz = {g(z)|g(z) ∈ l2(ω), g is a borel function}. let e[x2] < ∞. then there exists a borel function g0 : r → r with e[(g0(z)2] < ∞, such that e[(x − g0(z))2] = inf{e[(x − g(z))2 ∣∣g(z) ∈ lz}. moreover, g0(z) = e[x|z]. received: apr. 14, 2023. 2020 mathematics subject classification. 60a05, 49k45 . key words and phrases. expectation; conditional expectation; random variables; bregman loss functions: partial differential equations. https://doi.org/10.28924/2291-8639-21-2023-57 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-57 2 int. j. anal. appl. (2023), 21:57 this theorem means that the distance function ||x−y ||22 attains its minimum value at y = ψ(z) = e[x|z]. thus, arg miny∈lz||x −y || 2 2 = e[x|z]. (1.1) we recall some basic notions and facts from probability theory in the form we use in this paper ( [1], [9], [13]). expectation. let (ω,f,p) be a probability space and x : ω → r be a random variable. by the definition, a random variable is measurable, i.e., x−1(σb) ⊂ f, where σb is the borel algebra, consisting of all borel sets in r. the expectation of a random variable x is defined by the following integral, which is lebesgue integral with respect to the probability measure. e[x] = ∫ ω x dp. particularly, for a simple random variable x(w) = ∑n i=1 aiχai (w), e[x] = n∑ i=1 aip (ai). (1.2) l2(ω) = {x| ∫ ω |x|2 dp < ∞}. the norm in l2(ω) is defined by ||x||2 = (∫ ω |x|2 dp )1 2 . conditional expectation. let (x,z) be a bivariate random vector. the conditional expectation of x given z is denoted by e[x|z], which is a random variable, defined by ψ(z)(w) = ψ(z(w)) = e[x|z = z(w)],∀w ∈ ω. the following problem is a natural generalization of the problem (1.1), which has very important applications (see [2] and references therein); find a loss function f (x,y) satisfying the following condition arg miny∈re[f (x,y)] = ϕ(e[x]), ∀x, (1.3) where ϕ is a borel function. in this paper our main concern will be the problem (1.3). such problems arise in different contexts of statistics and probability theory (see [4]). in the case of ϕ(x) = x; f (x,y) = c(x−y) and f (x,y) = (x−y)2 the optimality of conditional expectations have been studied by many authors (see [1], [9], [10], [13]). for ϕ(x) = x and arbitrary function f (x,y) the bregman loss functions play an important role ( [5], [6], [7]). particularly, it was proved in [2] (see theorem 1.2 below) that if for a nonnegative smooth function f : r×r → r, arg ming(z)e[f (x,g(z))] = e[x|z], for all x and z, then f (x,y) is a bregmann loss function. definition 1.1. let f : r → r be a strictly convex differentiable function. then the bregman loss function (blf) df : r×r→r is defined as df (x,y) = f (x) − f (y) − f ′(y)(x −y) int. j. anal. appl. (2023), 21:57 3 in general, bregman loss functions are defined by using strictly convex differentiable functions f : rn → r. in this paper, for convenience we consider the case n = 1. all results can easily be extended to the case n > 1. for more information on bregman loss functions see [3] and [12]. the following theorem contains the most general result, regarding problem 1.3 in the case of ϕ(x) = x. theorem 1.2. ( [2]) let df : r×r→r be a blf. then, arg miny∈lze[df (x,y )] = e[x|z]. moreover, if f : r×r → r, f ≥ 0, f (x,x) = 0, f and fx are continuous functions and for all x and z, arg miny∈lze[df (x,y )] = e[x|z] then f is a blf. the rest of this paper will be organized as follows. in section 2 we present a theorem about optimality of expectations. section 3 consists of two subsections. in subsection 3.1 we develop a partial differential equation approach for critical points of e[f (x,y)]. the main problem studied in this subsection is: when y = ϕ(e[x]) is a critical point of the function e[f (x,y)] for every x ∈ l1(ω)? we present a partial differential equation approach for solving this problem and give a necessary and sufficient condition. in subsection 3.2 we study extreme problems. our main goal is to find the class of all f such that y = ϕ(e[x]) is a unique extremum point for e[f (x,y)], for all x ∈ l1(ω). 2. on the optimality of expectations we start with a slightly stronger version of theorem 1.2. theorem 2.1. let f : r×r → r, f ≥ 0, f (x,x) = 0, fx and fy are continuous. suppose that there exists a function ϕ : r→r such that ϕ(e[x]) is a unique minimizer for e[f (x,y)] in r for all x ∈ l1(ω),i.e., arg miny∈re[f (x,y)] = ϕ(e[x]),∀x ∈ l1(ω), provided that f (x,y) ∈ l1(ω). then f (x,y) is a blf if and only if ϕ(x) = x. proof. let f (x,y) be a blf. then, f (x,y) = f (x) − f (y) − f ′(y)(x −y). we can write f (x,y) = f (x) − f (y) − f ′(y)(x −y) and f (x,e[x]) = f (x) − f (e[x]) − f ′(e[x])(x −e[x]). hence, f (x,y) −f (x,e[x]) = f (e[x]) − f (y) + f ′(e[x])(x −e[x]) − f ′(y)(x −y). 4 int. j. anal. appl. (2023), 21:57 obviously, e [ f ′(e[x])(x −e[x]) ] = 0 and e [ f ′(y)(x −y) ] = f ′(y)(e[x] −y). then, e [ f (x,y) −f (x,e[x]) ] = f (e[x]) − f (y) − f ′(y)(e[x] −y). consequently, e [ f (x,y) −f (x,e[x]) ] = df (e[x],y) ≥ 0. (2.1) since f (x,y) = df (x,y) is a blf, df (e[x],y) = 0 ⇔ y = e[x]. thus y = e[x] is a minimum point of e[f (x,y)]. by the condition ϕ(e[x]) is a unique minimizer. then, it follows immediately that ϕ(x) = x. now let ϕ(x) = x. and arg miny∈re[f (x,y)] = e[x],∀x ∈ l1(ω). then it follows from this condition that f is a blf. this case was proved in [2] (see theorem 3). � 3. a pde approach to optimality problems 3.1. critical points. in this section we develop a partial differential equation (pde) approach for critical points of e[f (x,y)]. more precisely, the main question is: when y = ϕ(e[x]) is a critical point of the function e[f (x,y)] for every x? we give a necessary and sufficient condition for this question. the following assumption will be needed throughout this section. f : r×r → r, f (x,x) = 0, and the function f has first and second derivatives. now we prove a critical point theorem. theorem 3.1. let ϕ : r→r be an invertible function. then, y = ϕ(e[x]) is a critical point of the function e[f (x,y)] for all x ∈ l1(ω), if and only if f (x,y) is a solution of the following pde fxy(ϕ −1(y) −x) + fy = 0. (3.1) proof. let y = ϕ(e[x]) be a critical point of the function e[f (x,y)] for all x ∈ l1(ω). consider a simple random variable x such that p (x = a) = p, p (x = b) = q and p + q = 1. by (1.2) e[f (x,y)] = pf (a,y) + qf (b,y). and ϕ(e[x]) = ϕ(pa + qb). then pfy(a,ϕ(pa + qb)) + pfy(b,ϕ(pa + qb)) = 0. int. j. anal. appl. (2023), 21:57 5 it means that fy(a,ϕ(pa + qb)) q = − fy(b,ϕ(pa + qb)) p ⇔ fy(a,ϕ(pa + qb)) q(b−a) = − fy(b,ϕ(pa + qb)) p(b−a) . (3.2) y = ϕ(e[x]) ⇒ y = ϕ(pa + qb) ⇒ pa + qb = ϕ−1(y). note that pa + qb−a = q(b−a) and pa + qb−b = −p(b−a). hence, ϕ−1(y) −a = q(b−a) and ϕ−1(y) −b = −p(b−a). it follows from equation (3.2) that fy(a,y) ϕ−1(y) −a = fy(b,y) ϕ−1(y) −b . therefore, the function fy (x,y) ϕ−1(y)−x does not depend on x. then ∂ ∂x [ fy(x,y) ϕ−1(y) −x) ] = 0 and fxy(ϕ −1(y) −x) + fy (ϕ−1(y) −x)2 = 0. consequently, fxy(ϕ −1(y) −x) + fy = 0. to finish the proof of this theorem, we need to show that the (3.1) implies y = ϕ(e[x]) is a critical point of the function e[f (x,y)] for all x ∈ l1(ω). thus, by (3.1) fxy(ϕ −1(y) −x) + fy = 0. multiplying, this equation by the integrating factor µ(x,y) = 1 (ϕ−1(y)−x)2 we get 1 ϕ−1(y) −x fxy + 1 (ϕ−1(y) −x)2 fy = 0. then, ( 1 ϕ−1(y) −x fy ) x = 0 and fy ϕ−1(y) −x = c(y) ⇒ fy = (ϕ−1(y) −x)c(y). setting y = ϕ(e[x]) we get fy(x,ϕ(e[x])) = ( e[x] −x ) c(ϕ(e[x])) and e [ fy(x,ϕ(e[x]) ] = ( e[x] −e[x] ) c(ϕ(e[x])) = 0. � 6 int. j. anal. appl. (2023), 21:57 we next give an application of this theorem. example 3.1. let us find a general solution of the following problem fxy(ϕ −1(y) −x) + fy = 0, f (x,x) = 0 in the case of ϕ(y) = y. solution. we can write the equation in the form fxy + 1 y −x fy = 0. multiplying, this equation by the integrating factor µ(x,y) = 1 y−x we get 1 y −x fxy + 1 (y −x)2 fy = 0. then, ( 1 y −x fy ) x = 0 and fy y −x = c(y). let c(y) = f ′′(y). by using integration by parts we obtain that∫ y x fy(x,t) dt = ∫ y x f ′′(t)(t −x))dt = [ f ′(t)(t −x) ]t=y t=x − ∫ y x f (t) dt. consequently, f (x,y) = f (x) − f (y) − f ′(y)(x −y). the following corollary immediately follows from this example and theorem 3.1. corollary 3.1. if f (x,x) = 0 and y = e[x] is a critical point of the function e[f (x,y)] for all x ∈ l1(ω), then f (x,y) can be written in the form f (x,y) = f (x) − f (y) − f ′(y)(x − y) for a differentiable function f . not. by imposing additional conditions: f (x,y) ≥ 0 and e[x] is the unique minimizer, it was proved in [2] that f is a blf. 3.2. extreme points. let ϕ : r→r be an invertible function. in this subsection the main problem is to find the class of all f such that y = ϕ(e[x]) is a unique extremum point for e[f (x,y)], for all x ∈ l1(ω). we first prove the following theorem. theorem 3.2. let arg miny∈re[f (x,y)] = ϕ(e[x]),∀x ∈ l1(ω). then f (x,y) = ( ϕ−1(y) −x ) f ′(y) − ( ϕ−1(x) −x ) f ′(x) − ∫ y x f ′(t) ( ϕ−1(t) )′ dt, (3.3) int. j. anal. appl. (2023), 21:57 7 where f is a differentiable function satisfying the following condition ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x), (3.4) proof. by theorem 3.1 fxy(ϕ −1(y) −x) + fy = 0, f (x,x) = 0. then, ( 1 ϕ−1(y) −x fy ) x = 0 and fy ϕ−1(y) −x = c(y). setting c(y) = f ′′(y) we can write fy = ( ϕ−1(y) −x ) f ′′(y). (3.5) using integration by parts in (3.5) we obtain that∫ y x fy(x,t) dt = ∫ y x (ϕ−1(t) −x)) df ′(t) = [ f ′(t)(ϕ−1(t) −x) ]t=y t=x − ∫ y x f ′(t) ( ϕ−1(t) )′ dt. consequently, f (x,y) = ( ϕ−1(y) −x ) f ′(y) − ( ϕ−1(x) −x ) f ′(x) − ∫ y x f ′(t) ( ϕ−1(t) )′ dt and (3.3) holds. now we use the condition arg miny∈re[f (x,y)] = ϕ(e[x]). this condition means that e [ f (x,y) −f (x,ϕ ( e[x] )] > 0, provided that y 6= ϕ ( e[x] ) . using (3.3) we obtain that e [ f (x,y) −f (x,ϕ ( e[x] )] = ( ϕ−1(y) −e[x] ) f ′(y) + ∫ ϕ(e[x]) y f ′(t) ( ϕ−1(t) )′ dt > 0. thus, ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x). � note. in case of ϕ(x) = x, ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x) ⇒ f (x) − f (y) − f ′(y)(x −y) > 0, x 6= y. and 8 int. j. anal. appl. (2023), 21:57 f (x,y) = ( ϕ−1(y) −x ) f ′(y) − ( ϕ−1(x) −x ) f ′(x) − ∫ y x f ′(t) ( ϕ−1(t) )′ dt ⇒ f (x,y) = f (x) − f (y) − f ′(y)(x −y). therefore, in case of ϕ(x) = x, the condition (3.4) means that f is a is strictly convex function and (3.3) means simply that f (x,y) is a bregman loss function. corollary 3.2. let arg maxy∈re[f (x,y)] = ϕ(e[x]),∀x ∈ l1(ω). then f (x,y) = ( ϕ−1(y) −x ) f ′(y) − ( ϕ−1(x) −x ) f ′(x) − ∫ y x f ′(t) ( ϕ−1(t) )′ dt and ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt < 0, ∀y 6= ϕ(x). finally, we discus the condition (3.4), which is a generalization of the strictly convexity condition. the main question is: are there functions satisfying the following inequality( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x). regarding this question, we prove the following theorem. theorem 3.3. if ϕ(x) is an increasing function and f ′′(x) > 0,∀x ∈r. then ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x). proof. let us define g(x,y) = ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt. then, gy(x,y) = ( ϕ−1(y) )′ f ′(y) + ( ϕ−1(y) −x ) f ′′(y) − ( ϕ−1(y) )′ f ′(y) ⇒ gy(x,y) = ( ϕ−1(y) −x ) f ′′(y). we have y > ϕ(x) ⇔ ϕ−1(y) −x > 0 ⇔ gy(x,y) > 0, y < ϕ(x) ⇔ ϕ−1(y) −x < 0 ⇔ gy(x,y) < 0 and gy(x,ϕ(x)) = 0. consequently, g(x,y) = ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt > 0, ∀y 6= ϕ(x). � int. j. anal. appl. (2023), 21:57 9 corollary 3.3. if ϕ(x) is a decreasing function and f ′′(x) > 0,∀x ∈r. then ( ϕ−1(y) −x ) f ′(y) + ∫ ϕ(x) y f ′(t) ( ϕ−1(t) )′ dt < 0, ∀y 6= ϕ(x). conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] k.b. athreya, s.n. lahiri, measure theory and probability theory, springer texts in statistics, springer, new york, 2006. [2] a. banerjee, x. guo, h. wang, on the optimality of conditional expectation as a bregman predictor, ieee trans. inform. theory. 51 (2005), 2664–2669. https://doi.org/10.1109/tit.2005.850145. [3] h.h. bauschke, m.s. macklem, j.b. sewell, x. wang, klee sets and chebyshev centers for the right bregman distance, j. approx. theory. 162 (2010), 1225–1244. https://doi.org/10.1016/j.jat.2010.01.001. [4] a. ben-tal, a. charnes, m. teboulle, entropic means, j. math. anal. appl. 139 (1989), 537–551. https://doi. org/10.1016/0022-247x(89)90128-5. [5] y. censor, s. zenios, parallel optimization: theory, algorithms, and applications, oxford university press, london, 1998. [6] i. csiszar, why least squares and maximum entropy? an axiomatic approach to inference for linear inverse problems, ann. stat. 19 (1991), 2032–2066. https://doi.org/10.1214/aos/1176348385. [7] i. csiszar, generalized projections for non-negative functions, in: proceedings of 1995 ieee international symposium on information theory, ieee, whistler, bc, canada, 1995: p. 6. https://doi.org/10.1109/isit.1995. 531108. [8] f. deutsch, best approximation in inner product spaces, springer-verlag, new york, 2021. [9] g. grimmett, d. stirzaker, probability and random processes, oxford university press, oxford, 2004. [10] s. karlin, h.m. taylor, a second course in stochastic processes, 2nd ed. academic press, san diego, 1991. [11] m. hasanov, the spectra of two-parameter quadratic operator pencils, math. computer model. 54 (2011), 742–755. https://doi.org/10.1016/j.mcm.2011.03.018. [12] d. reem, s. reich, a. de pierro, re-examination of bregman functions and new properties of their divergences, optimization. 68 (2018), 279–348. https://doi.org/10.1080/02331934.2018.1543295. [13] d. williams, probability with martingales, cambridge mathematical textbooks, cambridge university press, cambridge, 2001. https://doi.org/10.1109/tit.2005.850145 https://doi.org/10.1016/j.jat.2010.01.001 https://doi.org/10.1016/0022-247x(89)90128-5 https://doi.org/10.1016/0022-247x(89)90128-5 https://doi.org/10.1214/aos/1176348385 https://doi.org/10.1109/isit.1995.531108 https://doi.org/10.1109/isit.1995.531108 https://doi.org/10.1016/j.mcm.2011.03.018 https://doi.org/10.1080/02331934.2018.1543295 1. introduction and preliminary facts 2. on the optimality of expectations 3. a pde approach to optimality problems 3.1. critical points 3.2. extreme points references international journal of analysis and applications volume 16, number 4 (2018), 605-613 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-605 hermite-hadamard type inequalities for quasi-convex functions via katugampola fractional integrals erhan set∗, i̇lker mumcu department of mathematics, faculty of science and arts, ordu university, ordu, turkey ∗corresponding author: erhanset@yahoo.com abstract. the paper deals with quasi-convex functions, katugampola fractional integrals and hermitehadamard type integral inequalities. the main idea of this paper is to present new hermite-hadamard type inequalities for quasi-convex functions using katugampola fractional integrals, hölder inequality and the identities in the literature. 1. introduction a function f : i ⊆ r → r is said to be convex if the inequality f (λu + (1 −λ) v) ≤ λf (u) + (1 −λ) f (v) holds for all u,v ∈ i and λ ∈ [0, 1]. this definition has been used in the following inequality that is called hermite-hadamard inequality: let f : i ⊆ r → r be a convex function and a,b ∈ i with a < b, then f ( a + b 2 ) ≤ 1 b−a ∫ b a f (x) dx ≤ f (a) + f (b) 2 . (1.1) this inequality has attracted many mathematicians. especially, in the last three decades, numerous generalizations, variants, and extensions of this inequality have been presented (see, e.g., [1, 3, 13, 14, 20] and the references cited therein). received 2017-10-26; accepted 2018-01-03; published 2018-07-02. 2010 mathematics subject classification. 26a33, 26d10, 33b20. key words and phrases. hermite-hadamard inequality; riemann-liouville fractional integrals; katugampola fractional integrals. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 605 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-605 int. j. anal. appl. 16 (4) (2018) 606 the notion of quasi-convex functions generalizes the notion of convex functions. more precisely, a function f : [a,b] → r is said quasi-convex on [a,b] if f (λu + (1 −λ) v) ≤ max{f(x),f(y)}, for any x,y ∈ [a,b] and λ ∈ [0, 1]. clearly, any convex function is a quasi-convex function. furthermore, there exist quasi-convex functions which are not convex (see [4]). let f ∈ l1[a,b] := l(a,b). the riemann-liouville integrals jαa+f and jαb−f of order α ∈ r + are defined, respectively, by jαa+f(x) = 1 γ(α) ∫ x a (x− t)α−1 f(t) dt (x > a) and jαb−f(x) = 1 γ(α) ∫ b x (t−x)α−1 f(t) dt (x < b) where γ is the familiar gamma function (see, e.g., [21, section 1.1]). it is noted that j1a+f(x) and j 1 b−f(x) become the usual riemann integrals. in the case of α = 1, the fractional integral reduces to classical integral. for further results related to hermite-hadamard type inequalities involving fractional integrals on can see [7, 11–19]. the beta function b(α,β) is defined by (see, e.g., [21, section 1.1] [10, p18]) b(α, β) =   ∫ 1 0 tα−1(1 − t)β−1 dt (<(α) > 0; <(β) > 0) γ(α) γ(β) γ(α + β) ( α, β ∈ c\z−0 ) . (1.2) the hypergeometric function [6]: 2f1(a,b; c; z) = 1 β(b,c− b) ∫ 1 0 tb−1(1 − t)c−b−1(1 −zt)−adt, c > b > 0, |z| < 1. a hypergeometric function can be written using euler’s hypergeometric transformations (t → 1 − t) in equivalent form: 2f1(a,b; c; z) = (1 −z)−a2f1 ( a,c− b; c; z z − 1 ) (1.3) lemma 1.1. [9] for 0 < α ≤ 1 and 0 ≤ a < b, we have |aα − bα| ≤ (b−a)α. in [11], sarıkaya et. al. proved a new version of hermite-hadamard’s inequalities in riemann-liouville fractional integral form as follows: int. j. anal. appl. 16 (4) (2018) 607 theorem 1.1. let f : [a,b] → r be a positive function with 0 ≤ a < b and f ∈ l1[a,b]. if f is a convex function on [a,b], then the following inequalities for fractional integrals holds: f ( a + b 2 ) ≤ γ(α + 1) 2(b−a)α [jαa+f(b) + j α b−f(a)] ≤ f(a) + f(b) 2 (1.4) with α > 0. in [8], özdemir et. al. gave following results for quasi-convex functions via riemann-liouville fractional integrals. theorem 1.2. let f : [a,b] → r, be a positive function with 0 ≤ a < b and f ∈ l1[a,b]. if f is a quasi-convex function [a,b], then the following inequality for fractional integrals holds: γ(α + 1) 2(b−a)α [jαa+f(b) + j α b−f(a)] ≤ max{f(a),f(b)} (1.5) with α > 0. theorem 1.3. let f : [a,b] → r, be a differentiable mapping on (a,b) with a < b. if |f′| is quasi-convex on [a,b] with α > 0, then the following inequality holds:∣∣∣∣f(a) + f(b)2 − γ(α + 1)2(b−a)α [jαa+f(b) + jαb−f(a)] ∣∣∣∣ ≤ b−a α + 1 ( 1 − 1 2α ) max{f′(a),f′(b)}. (1.6) theorem 1.4. let f : [a,b] → r, be a differentiable mapping on (a,b) such that f′ ∈ l1[a,b]. if |f′|q is quasi-convex on [a,b], and p > 1, then the following inequality for fractional integrals holds:∣∣∣∣f(a) + f(b)2 − γ(α + 1)2(b−a)α [jαa+f(b) + jαb−f(a)] ∣∣∣∣ ≤ b−a 2(αp + 1)1/p ( 1 − 1 2α ) (max{|f′(a)|q, |f′(b)|q})1/q . (1.7) where 1 p + 1 q = 1 and α ∈ [0, 1]. theorem 1.5. let f : [a,b] → r, be a differentiable mapping on (a,b) such that f′ ∈ l1[a,b]. if |f′|q is quasi-convex on [a,b], and q ≥ 1, then the following inequality for fractional integrals holds:∣∣∣∣f(a) + f(b)2 − γ(α + 1)2(b−a)α [jαa+f(b) + jαb−f(a)] ∣∣∣∣ ≤ b−a α + 1 ( 1 − 1 2α ) (max{|f′(a)|q, |f′(b)|q})1/q . (1.8) with α ∈ [0, 1]. katugampola gave a new fractional integral that generalizes the riemann-liouville and the hadamard fractional integrals into a single form. int. j. anal. appl. 16 (4) (2018) 608 definition 1.1. [5] let [a,b] ⊂ r be a finite interval. then, the leftand right-side katugampola fractional integrals of order (α > 0) of f ∈ xpc (a,b) are defined: ρiαa+f(x) = ρ1−α γ(α) ∫ x a tρ−1 (xρ − tρ)1−α f(t)dt and ρiαb−f(x) = ρ1−α γ(α) ∫ b x tρ−1 (tρ −xρ)1−α f(t)dt with a < x < b and ρ > 0, if the integral exist. theorem 1.6. [5] let α > 0 and ρ > 0. then for x > a, 1. limρ→1 ρiαa+f(x) = jαa+f(x), 2. limρ→0+ ρiαa+f(x) = hαa+f(x). similar results also hold for right-sided operators. in [2], chen and katugampola proved the following lemma: lemma 1.2. let f : [aρ,bρ] → r be a differentiable mapping on (aρ,bρ) with 0 ≤ a < b. then the following equality holds if the fractional integrals exist: f(aρ) + f(bρ) 2 − αραγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ] = bρ −aρ 2 ∫ 1 0 [(1 − tρ)α − tρα]tρ−1f′(tρaρ + (1 − tρ)bρ)dt (1.9) where g(x) = xρ. the main purpose of this paper is to establish hermite-hadamard’s inequalities for quasi-convex functions via katugampola fractional integral. we also obtain hermite-hadamard type inequalities of these classes functions. 2. main results theorem 2.1. let α > 0 and ρ > 0. let f : [aρ,bρ] → r be a positive function with 0 ≤ a < b and f ∈ xpc (aρ,bρ). if f is a quasi-convex function on [aρ,bρ], then the following inequalities holds: ραγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ] ≤ max{f(aρ),f(bρ)} (2.1) where g(x) = xρ. proof. since f is quasi-convex function on [aρ,bρ], we get f(tρaρ + (1 − tρ)bρ) ≤ max{f(aρ),f(bρ)} int. j. anal. appl. 16 (4) (2018) 609 and f((1 − tρ)aρ + tρbρ) ≤ max{f(aρ),f(bρ)}. by adding these inequalities we have 1 2 [f(tρaρ + (1 − tρ)bρ) + f((1 − tρ)aρ + tρbρ)] ≤ max{f(aρ),f(bρ)}. (2.2) multiplying both sides of (2.2) by tαρ−1 and integrating the resulting inequality with respect to t over [aρ,bρ], we obtain ∫ 1 0 tαρ−1f(tρaρ + (1 − tρ)bρ)dt + ∫ 1 0 tαρ−1f((1 − tρ)aρ + tρbρ)dt = ∫ b a ( bρ −xρ bρ −aρ )α−1 f(xρ) xρ−1 bρ −aρ dx + ∫ b a ( xρ −aρ bρ −aρ )α−1 f(xρ) xρ−1 bρ −aρ dx = 1 (bρ −aρ)α ∫ b a xρ−1 (bρ −xρ)1−α f(xρ)dx + 1 (bρ −aρ)α ∫ b a xρ−1 (xρ −aρ)1−α f(xρ)dx = γ(α) ρ1−α(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ] ≤ 2 ρα max{f(aρ),f(bρ)} so we get desired result. the proof is completed. � remark 2.1. in theorem 2.1, taking limit ρ → 1 we obtain inequality of (1.5). theorem 2.2. let α > 0 and ρ > 0. let f : [aρ,bρ] → r be a differentiable mapping on [aρ,bρ] with 0 ≤ a < b. if |f′| is a quasi-convex function on [aρ,bρ], then the following inequalities holds: ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ = bρ −aρ ρ(α + 1) ( 1 − 1 2ρ(α+1) ) max{|f′(aρ)| |f′(bρ)|} (2.3) where g(x) = xρ. int. j. anal. appl. 16 (4) (2018) 610 proof. using lemma 1.2 and quasi-convex of |f′| with modulus, we get ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ ≤ bρ −aρ 2 ∫ 1 0 |(1 − tρ)α − tρα|tρ−1|f′(tρaρ + (1 − tρ)bρ)|dt ≤ bρ −aρ 2 ∫ 1 0 |(1 − tρ)α − tρα|tρ−1max{|f′(aρ)| |f′(bρ)|}dt = bρ −aρ 2 max{|f′(aρ)| |f′(bρ)|} × {∫ 1/21/ρ 0 [(1 − tρ)α − tρα]tρ−1dt + ∫ 1 1/21/ρ [tρα + (1 − tρ)α]tρ−1dt } = bρ −aρ ρ(α + 1) ( 1 − 1 2α ) max{|f′(aρ)| |f′(bρ)|} where ∫ 1/21/ρ 0 [(1 − tρ)α − tρα]tρ−1dt + ∫ 1 1/21/ρ [tρα + (1 − tρ)α]tρ−1dt = 1 ρ {∫ 1/2 0 [(1 −u)α −uα] du + ∫ 1 1/2 [uα − (1 −u)α] du } = 2 ρ(α + 1) ( 1 − 1 2α ) . (2.4) the proof is completed. � remark 2.2. in theorem 2.2, taking limit ρ → 1 we obtain inequality of (1.6). theorem 2.3. let α > 0 and ρ > 0. let f : [aρ,bρ] → r be a differentiable mapping on [aρ,bρ] with 0 ≤ a < b. if |f′|q is a quasi-convex function on [aρ,bρ] and s > 1, then the following inequalities holds: ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ = bρ −aρ 2 (max{|f′(aρ)|q, |f′(bρ)|q})1/q (k1 + k2)1/s where k1 = 1 ρ2s+ 1−s ρ b(s + 1 −s ρ ,αs + 1), k2 = αs + 1 2ρ 2f1 ( 1 −s + s− 1 ρ , 1; αs + 2; 1 2 ) , 1 s + 1 q = 1 and g(x) = xρ. int. j. anal. appl. 16 (4) (2018) 611 proof. from lemma 1.1, lemma 1.2, hölder inequality and quasi-convex of |f′| with proporties of modulus, we have ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ ≤ bρ −aρ 2 ∫ 1 0 |(1 − tρ)α − tρα|tρ−1|f′(tρaρ + (1 − tρ)bρ)|dt ≤ bρ −aρ 2 (∫ 1 0 |(1 − tρ)α − tρα|sts(ρ−1)dt )1/s (∫ 1 0 |f′(tρaρ + (1 − tρ)bρ)|qdt )1/q ≤ bρ −aρ 2 (∫ 1 0 |1 − 2tρ|αsts(ρ−1)dt )1/s (max{|f′(aρ)|q, |f′(bρ)|q})1/q = bρ −aρ 2 (max{|f′(aρ)|q, |f′(bρ)|q})1/q × {∫ 1/21/ρ 0 (1 − 2tρ)αsts(ρ−1)dt + ∫ 1 1/21/ρ (2tρ − 1)αsts(ρ−1)dt }1/s = bρ −aρ 2 (max{|f′(aρ)|q, |f′(bρ)|q})1/q (k1 + k2)1/s (2.5) where k1 = ∫ 1/21/ρ 0 (1 − 2tρ)αsts(ρ−1)dt = 1 ρ2s+ 1−s ρ ∫ 1 0 us−1+ 1−s ρ (1 −u)αsdu = 1 ρ2s+ 1−s ρ b ( s + 1 −s ρ ,αs + 1 ) (2.6) k2 = ∫ 1 1/21/ρ (2tρ − 1)αsts(ρ−1)dt = 1 2s+ 1−s ρ ρ ∫ 1 0 uαs(1 + u)s−1+ 1−s ρ du = αs + 1 2ρ 2f1 ( 1 −s + s− 1 ρ , 1; αs + 2; 1 2 ) (2.7) so, if we use (2.6), (2.7) in (2.5), we obtain desired result. � remark 2.3. in theorem 2.3, taking limit ρ → 1 we obtain inequality of (1.7). theorem 2.4. let α > 0 and ρ > 0. let f : [aρ,bρ] → r be a differentiable mapping on [aρ,bρ] with 0 ≤ a < b. if |f′|q is a quasi-convex function on [aρ,bρ] and q ≥ 1, then the following inequalities holds: ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ ≤ bρ −aρ ρ(α + 1) ( 1 − 1 2α ) (max{|f′(aρ)|q, |f′(bρ)|q})1/q where g(x) = xρ. int. j. anal. appl. 16 (4) (2018) 612 proof. from lemma 1.2, quasi-convex of |f′| and using power-mean inequality with proporties of modulus, we have ∣∣∣∣f(aρ) + f(bρ)2 − αρ αγ(α + 1) 2(bρ −aρ)α [ ρiαa+(f ◦g)(b) + ρiαb−(f ◦g)(a) ]∣∣∣∣ ≤ bρ −aρ 2 ∫ 1 0 |(1 − tρ)α − tρα|tρ−1|f′(tρaρ + (1 − tρ)bρ)|dt ≤ bρ −aρ 2 (∫ 1 0 |(1 − tρ)α − tρα|tρ−1dt )1−1/q × (∫ 1 0 |(1 − tρ)α − tρα|tρ−1|f′(tρaρ + (1 − tρ)bρ)|qdt )1/q ≤ bρ −aρ 2 (∫ 1 0 |(1 − tρ)α − tρα|tρ−1dt )1−1/q ×(max{|f′(aρ)|q, |f′(bρ)|q})1/q (∫ 1 0 |(1 − tρ)α − tρα|tρ−1dt )1/q = bρ −aρ 2 (∫ 1 0 |(1 − tρ)α − tρα|tρ−1dt ) (max{|f′(aρ)|q, |f′(bρ)|q})1/q using (2.4) we get desired result. � remark 2.4. in theorem 2.4, taking limit ρ → 1 we obtain inequality of (1.8). 3. acknowledgement this research is supported by ordu university scientific research projects coordination unit (bap). project number: ykd-17224 references [1] m.u. awan, m.a. noor, m.v. mihai and k.i. noor, fractional hermite-hadamard inequalities for differentiablesgodunova-levin functions, filomat, 30(12) (2016), 3235c-3241. [2] h. chen, u.n. katugampola, hermite-hadamard and hermite-hadamard-fejer type inequalities for generalized fractional integrals, j. math. anal. appl., 446, 1274-1291. [3] s.s. dragomir and c.e.m. pearce, selected topics on hermite-hadamard inequalities and applications, rgmia monographs, victoria university, 2000. [4] d.a. ion, some estimates on the hermite-hadamard inequality through quasi-convex functions, ann. univ. craiova, math. comp. sci. ser., 34 (2007), 82-87. [5] u.n. katugampola, new approach to generalized fractional derivatives, bull. math. anal. appl., 6(4), (2014), 1-15. [6] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, amsterdam: elsevier; (2006). [7] m. a. noor, k. i. noor, and m. u. awan, generalized convexity and integral inequalities, appl. math. inform. sci., 9(1) (2015),233-243. [8] m. e. özdemir, ç. yıldız, the hadamards inequality for quasi-convex functions via fractional integrals, annals of the university of craiova, ann. univ. craiova, math. comp. sci. series, 40(2) (2013), 167-173. int. j. anal. appl. 16 (4) (2018) 613 [9] a.p. prudnikov, y.a. brychkov, o.i. marichev, integral and series. in: elementary functions, vol. 1. nauka, moscow; (1981). [10] e.d. rainville, special functions, the mcmillan company, new york, 1960. [11] m. z. sarıkaya, e. set, h. yaldız, n. başak, hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities, math. comput. model., 57 (2013), 2403-2407. [12] e. set, m.z. sarikaya, m.e. özdemir, h. yildirim, the hermite-hadamards inequality for some convex functions via fractional integrals and related results, j. appl. math. stat. inf., 10(2) (2014), 69-83. [13] e. set, i̇. i̇şcan, f. zehir, on some new inequalities of hermite-hadamard type involving harmonically convex functions via fractional integrals, konuralp j. math., 3(1) (2015), 42-55. [14] e. set, a.o. akdemir, i̇. mumcu, hermite-hadamard’s inequality and its extensions for conformable fractional integrals of any order α > 0, researchgate, https://www.researchgate.net/publication/303382221. [15] e. set, b. çelik, fractional hermite-hadamard type inequalities for quasi-convex functions, ordu univ. j. sci. tech., 6(1) (2016), 137-149. [16] e. set, a.o. akdemir, b. çelik, on generalization of fejer type inequalities via fractional integral operator, researchgate, https://www.researchgate.net/publication/311452467. [17] e. set, a.o. akdemir, i̇. mumcu, ostrowski type inequalities involving special functions via conformable fractional integrals, j. adv. math. stud., 10(3) (2017), 386-395. [18] e. set, m.a. noor, m.u. awan, a. gözpinar, generalized hermite-hadamard type inequalities involving fractional integral operators, j. ineq. appl., 2017(169) (2017), 1-10. [19] e. set,j. choi, a. gözpinar, hermite-hadamard type inequalities for the generalized k-fractional integral operators, j. ineq. appl., 2017 (2017), art. id 206. [20] e. set, m.e. özdemir, m.z. sarıkaya, inequalities of hermite-hadamard type for functions whose derivatives absolute values are m-convex, aip conf. proc., 1309(1) (2010), 861-863. [21] h.m. srivastava and j. choi, zeta and q-zeta functions and associated series and integrals, elsevier science publishers, amsterdam, london and new york, 2012. 1. introduction 2. main results 3. acknowledgement references int. j. anal. appl. (2022), 20:38 exact solutions and stability of fourth order systems of difference equations using padovan numbers marwa m. alzubaidi∗ department of mathematics, college of duba, university of tabuk, saudi arabia ∗corresponding author: mmialzubaidi@hotmail.com abstract. difference equations are widely utilized to describe some phenomena arising in nonlinear sciences. in particular, systems of difference equations play an important role in investigating most nonlinear applications. future behaviors of such phenomena can be sometimes known and understood by using exact solutions of systems of difference equations. therefore, this article investigates the exact solutions of fourth order systems of difference equations. we use successive iterations and padovan numbers to obtain the exact solutions in the form of rational functions. the stability of the considered systems are analyzed using jacobian matrix. real equilibrium points are found saddle. under some selected parameters, we plot some 2d figures to show the behavior of the obtained solutions. the used methods can be successfully applied for high order systems of difference equations. 1. introduction differential equations have been widely used to model various economical, physical, biological and artificial phenomena. these equations describe populations or objects in which time is continuous. in contrast, difference equations have been extensively used to describe populations or objects that evolve discrete time. the study of the theory of difference equations has strongly become an active topic for some researchers. the main reason behind that is that difference equations are used to model and describe most natural and non-natural phenomena such as those occurred in physics, biology, chemistry, economy, engineering, etc. difference equations appear as discrete mathematical models of these phenomena. recursive equations model wide spectrum of biomedical phenomena such as cell proliferation, cancer growth and genetics [1]. discrete dynamical systems are also used to describe received: jul. 3, 2022. 2010 mathematics subject classification. 39a30. key words and phrases. equilibrium; stability; system of difference equations; padovan numbers; plastic number. https://doi.org/10.28924/2291-8639-20-2022-38 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-38 2 int. j. anal. appl. (2022), 20:38 various applications in different sciences. moreover, difference equations play a key role in investigating the solutions of various differential equations. the solutions and the stability of many systems of difference equations have been recently discussed. for instance, tollu et al. [2] used fibonacci numbers to investigate the solutions of the difference equations xn+1 = 1 1 + xn , yn+1 = 1 −1 + yn . yazlik et al. [3] obtained the solutions of the following systems xn+1 = xn−1 ± 1 ynxn−1 , yn+1 = yn−1 ± 1 xnyn−1 . (1.1) the author in [4] investigated the solutions of the the system xn+1 = xn−1yn−3 yn−1 (−1 −xn−1yn−3) , yn+1 = yn−1xn−3 xn−1 (±1 ±yn−1xn−3) , n = 0, 1, ..., moreover, almatrafi and alzubaidi [5] investigated the stability, periodicity and the solutions of the difference equation xn+1 = c1xn−3 + c2xn−3 c3xn−3 −c4xn−7 . cinar [6] obtained the positive solutions of the following system xn+1 = 1 yn , yn+1 = yn yn−1xn−1 . in [7], elsayed extracted the solutions of the following system of difference equations xn+1 = xn−1 ±1 + xn−1yn , yn+1 = yn−1 ∓1 + yn−1xn . alayachi et al. [8] discussed the solutions of nonlinear rational systems of difference equations of third order given by xn+1 = ynzn−1 yn ±xn−2 , yn+1 = znxn−1 zn ±yn−2 , zn+1 = xnyn−1 xn ±zn−2 . alayachi et al. [9] obtained the solutions of the following systems xn+1 = xn−3yn−4 yn(1 + xn−1yn−2xn−3yn−4) , yn+1 = yn−3xn−4 xn(±1 ±yn−1xn−2yn−3xn−4) . more information about systems of difference equations and their stability can be obtained in [10–16]. the motivation of this paper arises from the investigation of system (1.1). hence, the main purpose of this paper is to investigate the solutions of the following system using padovan numbers: φn+1 = φn−3 + 1 ψn−1φn−3 , ψn+1 = ψn−3 + 1 φn−1ψn−3 , (1.2) φn+1 = φn−3 − 1 ψn−1φn−3 , ψn+1 = ψn−3 − 1 φn−1ψn−3 , (1.3) where the initial conditions are real numbers. we also present the future pattern of the considered systems under some selected values for the initial conditions. the padovan sequence {ρn}n∈n is defined by ρn+1 = ρn−1 + ρn−2, where ρ−2 = 0, ρ−1 = 0, ρ0 = 1. int. j. anal. appl. (2022), 20:38 3 this paper is organized as follows. section 4 presents the solutions of system (1.2) while section 3 gives the solutions of system (1.3). section (4) analyzes the stability of the equilibrium points. the numerical solutions of the considered systems are shown in section 5. finally, we conclude this article in section 6. 2. solutions of system (1.2) this section is devoted to give explicit forms for the solutions of system (1.2) using padovan numbers. theorem 2.1. let {φn, ψn}∞n=−3 be the solutions of system (1.2) and assume that φ−3 = α, φ−2 = β, φ−1 = γ, φ0 = δ, ψ−3 = �, ψ−2 = ζ, ψ−1 = η, and ψ0 = κ. then, for n = 0, 1, ..., we have φ4n−3 = α (ρ2n−1η + ρ2n) + ρ2n−2 α (ρ2n−2η + ρ2n−1) + ρ2n−3 , φ4n−2 = β(ρ2n−1κ + ρ2n) + ρ2n−2 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 , φ4n−1 = �(ρ2nγ + ρ2n+1) + ρ2n−1 �(ρ2n−1γ + ρ2n) + ρ2n−2 , φ4n = ζ (ρ2nδ + ρ2n+1) + ρ2n−1 ζ (ρ2n−1δ + ρ2n) + ρ2n−2 . and ψ4n−3 = � (ρ2n−1γ + ρ2n) + ρ2n−2 � (ρ2n−2γ + ρ2n−1) + ρ2n−3 , ψ4n−2 = ζ (ρ2n−1δ + ρ2n) + ρ2n−2 ζ (ρ2n−2δ + ρ2n−1) + ρ2n−3 , ψ4n−1 = α(ρ2nη + ρ2n+1) + ρ2n−1 α(ρ2n−1η + ρ2n) + ρ2n−2 , ψ4n = β(ρ2nκ + ρ2n+1) + ρ2n−1 β(ρ2n−1κ + ρ2n) + ρ2n−2 , where ρn+1 = ρn−1 + ρn−2 ∀n ∈n is the padovan sequence. proof. the forms of the solutions are true for n = 0. we assume that n > 0 and that our assumption holds for n− 1. that is, φ4n−7 = α (ρ2n−3η + ρ2n−2) + ρ2n−4 α (ρ2n−4η + ρ2n−3) + ρ2n−5 , φ4n−6 = β(ρ2n−3κ + ρ2n−2) + ρ2n−4 β(ρ2n−4κ + ρ2n−3) + ρ2n−5 , φ4n−5 = �(ρ2n−2γ + ρ2n−1) + ρ2n−3 �(ρ2n−3γ + ρ2n−2) + ρ2n−4 , 4 int. j. anal. appl. (2022), 20:38 φ4n−4 = ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 ζ(ρ2n−3δ + ρ2n−2) + ρ2n−4 , and ψ4n−7 = � (ρ2n−3γ + ρ2n−2) + ρ2n−4 � (ρ2n−4γ + ρ2n−3) + ρ2n−5 , ψ4n−6 = ζ (ρ2n−3δ + ρ2n−2) + ρ2n−4 ζ (ρ2n−4δ + ρ2n−3) + ρ2n−5 , ψ4n−5 = α(ρ2n−2η + ρ2n−1) + ρ2n−3 α(ρ2n−3η + ρ2n−2) + ρ2n−4 , ψ4n−4 = β(ρ2n−2κ + ρ2n−1) + ρ2n−3 β(ρ2n−3κ + ρ2n−2) + ρ2n−4 . next, it can be easily observed from system (1.2) that φ4n−3 = φ4n−7 + 1 ψ4n−5φ4n−7 = α (ρ2n−3η+ρ2n−2)+ρ2n−4 α (ρ2n−4η+ρ2n−3)+ρ2n−5 + 1( α(ρ2n−2η+ρ2n−1)+ρ2n−3 α(ρ2n−3η+ρ2n−2)+ρ2n−4 )( α (ρ2n−3η+ρ2n−2)+ρ2n−4 α (ρ2n−4η+ρ2n−3)+ρ2n−5 ) = α (ρ2n−3η+ρ2n−2)+ρ2n−4+α (ρ2n−4η+ρ2n−3)+ρ2n−5 α (ρ2n−4η+ρ2n−3)+ρ2n−5 α(ρ2n−2η+ρ2n−1)+ρ2n−3 α (ρ2n−4η+ρ2n−3)+ρ2n−5 = α (ρ2n−3η + ρ2n−2) + ρ2n−4 + α (ρ2n−4η + ρ2n−3) + ρ2n−5 α(ρ2n−2η + ρ2n−1) + ρ2n−3 = α ρ2n−3η + αρ2n−2 + ρ2n−4 + α ρ2n−4η + αρ2n−3 + ρ2n−5 α(ρ2n−2η + ρ2n−1) + ρ2n−3 = α η (ρ2n−3 + ρ2n−4) + α (ρ2n−2 + ρ2n−3) + (ρ2n−4 + ρ2n−5) α(ρ2n−2η + ρ2n−1) + ρ2n−3 . since ρn+1 = ρn−1 + ρn−2, we have φ4n−3 = α (ρ2n−1η + ρ2n) + ρ2n−2 α(ρ2n−2η + ρ2n−1) + ρ2n−3 . ψ4n−3 = ψ4n−7 + 1 φ4n−5ψ4n−7 = � (ρ2n−3γ+ρ2n−2)+ρ2n−4 � (ρ2n−4γ+ρ2n−3)+ρ2n−5 + 1( �(ρ2n−2γ+ρ2n−1)+ρ2n−3 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 )( � (ρ2n−3γ+ρ2n−2)+ρ2n−4 � (ρ2n−4γ+ρ2n−3)+ρ2n−5 ) = � (ρ2n−3γ+ρ2n−2)+ρ2n−4+� (ρ2n−4γ+ρ2n−3)+ρ2n−5 � (ρ2n−4γ+ρ2n−3)+ρ2n−5 �(ρ2n−2γ+ρ2n−1)+ρ2n−3 � (ρ2n−4γ+ρ2n−3)+ρ2n−5 = � (ρ2n−3γ + ρ2n−2) + ρ2n−4 + � (ρ2n−4γ + ρ2n−3) + ρ2n−5 �(ρ2n−2γ + ρ2n−1) + ρ2n−3 = �ρ2n−3γ + �ρ2n−2 + ρ2n−4 + �ρ2n−4γ + �ρ2n−3 + ρ2n−5 �(ρ2n−2γ + ρ2n−1) + ρ2n−3 int. j. anal. appl. (2022), 20:38 5 = �γ (ρ2n−3 + ρ2n−4) + � (ρ2n−2 + ρ2n−3) + ρ2n−4 + ρ2n−5 �(ρ2n−2γ + ρ2n−1) + ρ2n−3 = �γρ2n−1 + �ρ2n + ρ2n−2 �(ρ2n−2γ + ρ2n−1) + ρ2n−3 = �(ρ2n−1γ + ρ2n) + ρ2n−2 �(ρ2n−2γ + ρ2n−1) + ρ2n−3 . also, system (1.2) gives φ4n−2 = φ4n−6 + 1 ψ4n−4φ4n−6 = β(ρ2n−3κ+ρ2n−2)+ρ2n−4 β(ρ2n−4κ+ρ2n−3)+ρ2n−5 + 1( β(ρ2n−2κ+ρ2n−1)+ρ2n−3 β(ρ2n−3κ+ρ2n−2)+ρ2n−4 )( β(ρ2n−3κ+ρ2n−2)+ρ2n−4 β(ρ2n−4κ+ρ2n−3)+ρ2n−5 ) = β(ρ2n−3κ+ρ2n−2)+ρ2n−4+β(ρ2n−4κ+ρ2n−3)+ρ2n−5 β(ρ2n−4κ+ρ2n−3)+ρ2n−5 β(ρ2n−2κ+ρ2n−1)+ρ2n−3 β(ρ2n−4κ+ρ2n−3)+ρ2n−5 = β(ρ2n−3κ + ρ2n−2) + ρ2n−4 + β(ρ2n−4κ + ρ2n−3) + ρ2n−5 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 = βρ2n−3κ + βρ2n−2 + ρ2n−4 + βρ2n−4κ + βρ2n−3 + ρ2n−5 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 = βκ (ρ2n−3 + ρ2n−4) + β (ρ2n−2 + ρ2n−3) + ρ2n−4 + ρ2n−5 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 = βκρ2n−1 + βρ2n + ρ2n−2 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 = β (ρ2n−1κ + ρ2n) + ρ2n−2 β(ρ2n−2κ + ρ2n−1) + ρ2n−3 . ψ4n−2 = ψ4n−6 + 1 φ4n−4ψ4n−6 = ζ (ρ2n−3δ+ρ2n−2)+ρ2n−4 ζ (ρ2n−4δ+ρ2n−3)+ρ2n−5 + 1( ζ(ρ2n−2δ+ρ2n−1)+ρ2n−3 ζ(ρ2n−3δ+ρ2n−2)+ρ2n−4 )( ζ (ρ2n−3δ+ρ2n−2)+ρ2n−4 ζ (ρ2n−4δ+ρ2n−3)+ρ2n−5 ) = ζ (ρ2n−3δ+ρ2n−2)+ρ2n−4+ζ (ρ2n−4δ+ρ2n−3)+ρ2n−5 ζ (ρ2n−4δ+ρ2n−3)+ρ2n−5 ζ(ρ2n−2δ+ρ2n−1)+ρ2n−3 ζ (ρ2n−4δ+ρ2n−3)+ρ2n−5 = ζ (ρ2n−3δ + ρ2n−2) + ρ2n−4 + ζ (ρ2n−4δ + ρ2n−3) + ρ2n−5 ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 = ζ ρ2n−3δ + ζ ρ2n−2 + ρ2n−4 + ζ ρ2n−4δ + ζ ρ2n−3 + ρ2n−5 ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 = ζδ(ρ2n−3 + ρ2n−4) + ζ (ρ2n−2 + ρ2n−3) + ρ2n−4 + ρ2n−5 ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 = ζδρ2n−1 + ζρ2n + ρ2n−2 ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 = ζ (ρ2n−1δ + ρ2n) + ρ2n−2 ζ(ρ2n−2δ + ρ2n−1) + ρ2n−3 . furthermore, from system (1.2), we have 6 int. j. anal. appl. (2022), 20:38 φ4n−1 = φ4n−5 + 1 ψ4n−3φ4n−5 = �(ρ2n−2γ+ρ2n−1)+ρ2n−3 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 + 1( � (ρ2n−1γ+ρ2n)+ρ2n−2 � (ρ2n−2γ+ρ2n−1)+ρ2n−3 )( �(ρ2n−2γ+ρ2n−1)+ρ2n−3 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 ) = �(ρ2n−2γ+ρ2n−1)+ρ2n−3+�(ρ2n−3γ+ρ2n−2)+ρ2n−4 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 � (ρ2n−1γ+ρ2n)+ρ2n−2 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 = �(ρ2n−2γ + ρ2n−1) + ρ2n−3 + �(ρ2n−3γ + ρ2n−2) + ρ2n−4 � (ρ2n−1γ + ρ2n) + ρ2n−2 = �(ρ2nγ + ρ2n+1) + ρ2n−1 � (ρ2n−1γ + ρ2n) + ρ2n−2 . ψ4n−1 = ψ4n−5 + 1 φ4n−3ψ4n−5 = α(ρ2n−2η+ρ2n−1)+ρ2n−3 α(ρ2n−3η+ρ2n−2)+ρ2n−4 + 1( α (ρ2n−1η+ρ2n)+ρ2n−2 α (ρ2n−2η+ρ2n−1)+ρ2n−3 )( α(ρ2n−2η+ρ2n−1)+ρ2n−3 α(ρ2n−3η+ρ2n−2)+ρ2n−4 ) = α(ρ2n−2η+ρ2n−1)+ρ2n−3+α(ρ2n−3η+ρ2n−2)+ρ2n−4 α(ρ2n−3η+ρ2n−2)+ρ2n−4 α (ρ2n−1η+ρ2n)+ρ2n−2 α(ρ2n−3η+ρ2n−2)+ρ2n−4 = α(ρ2nη + ρ2n+1) + ρ2n−1 α (ρ2n−1η + ρ2n) + ρ2n−2 . moreover, from system (1.2), we have φ4n = 4n−4 + 1 4n−2φ4n−4 = ζ(ρ2n−2δ+ρ2n−1)+ρ2n−3 ζ(ρ2n−3δ+ρ2n−2)+ρ2n−4 + 1( ζ (ρ2n−1δ+ρ2n)+ρ2n−2 ζ (ρ2n−2δ+ρ2n−1)+ρ2n−3 )( ζ(ρ2n−2δ+ρ2n−1)+ρ2n−3 ζ(ρ2n−3δ+ρ2n−2)+ρ2n−4 ) = ζ(ρ2n−2δ+ρ2n−1)+ρ2n−3+ζ(ρ2n−3δ+ρ2n−2)+ρ2n−4 �(ρ2n−3γ+ρ2n−2)+ρ2n−4 ζ (ρ2n−1δ+ρ2n)+ρ2n−2 ζ(ρ2n−3δ+ρ2n−2)+ρ2n−4 = ζ(ρ2nδ + ρ2n+1) + ρ2n−1 ζ (ρ2n−1δ + ρ2n) + ρ2n−2 . ψ4n = ψ4n−4 + 1 φ4n−2ψ4n−4 = β(ρ2n−2κ+ρ2n−1)+ρ2n−3 β(ρ2n−3κ+ρ2n−2)+ρ2n−4 + 1( β(ρ2n−1κ+ρ2n)+ρ2n−2 β(ρ2n−2κ+ρ2n−1)+ρ2n−3 )( β(ρ2n−2κ+ρ2n−1)+ρ2n−3 β(ρ2n−3κ+ρ2n−2)+ρ2n−4 ) int. j. anal. appl. (2022), 20:38 7 = β(ρ2n−2κ+ρ2n−1)+ρ2n−3+β(ρ2n−3κ+ρ2n−2)+ρ2n−4 α(ρ2n−3η+ρ2n−2)+ρ2n−4 β(ρ2n−1κ+ρ2n)+ρ2n−2 β(ρ2n−3κ+ρ2n−2)+ρ2n−4 = β(ρ2nκ + ρ2n+1) + ρ2n−1 β(ρ2n−1κ + ρ2n) + ρ2n−2 . the proof has been completed. 3. solutions of system (1.3) in this section, we show analytic solutions for system (1.3) using padovan numbers. theorem 3.1. let {φn, ψn}∞n=−3 be the solutions of system (1.3) and assume that φ−3 = α, φ−2 = β, φ−1 = γ, φ0 = δ, ψ−3 = �, ψ−2 = ζ, ψ−1 = η, and ψ0 = κ. then, for n = 0, 1, ..., we have φ4n−3 = α (ρ2n −ρ2n−1η) −ρ2n−2 α (ρ2n−2η −ρ2n−1) + ρ2n−3 , φ4n−2 = β(ρ2n −ρ2n−1κ) −ρ2n−2 β(ρ2n−2κ−ρ2n−1) + ρ2n−3 , φ4n−1 = �(ρ2n+1 −ρ2nγ) −ρ2n−1 �(ρ2n−1γ −ρ2n) + ρ2n−2 , φ4n = ζ (ρ2n+1 −ρ2nδ) −ρ2n−1 ζ (ρ2n−1δ −ρ2n) + ρ2n−2 . and ψ4n−3 = � (ρ2n −ρ2n−1γ) −ρ2n−2 � (ρ2n−2γ −ρ2n−1) + ρ2n−3 , ψ4n−2 = ζ (ρ2n −ρ2n−1δ) −ρ2n−2 ζ (ρ2n−2δ −ρ2n−1) + ρ2n−3 , ψ4n−1 = α(ρ2n+1 −ρ2nη) −ρ2n−1 α(ρ2n−1η −ρ2n) + ρ2n−2 , ψ4n = β(ρ2n+1 −ρ2nκ) −ρ2n−1 β(ρ2n−1κ−ρ2n) + ρ2n−2 , where ρn+1 = ρn−1 + ρn−2 ∀n ∈n is the padovan sequence. proof. the above solutions are true for n = 0. we suppose that n > 0 and that our assumption holds for n− 1. that is, φ4n−7 = α (ρ2n−2 −ρ2n−3η) −ρ2n−4 α (ρ2n−4η −ρ2n−3) + ρ2n−5 , φ4n−6 = β(ρ2n−2 −ρ2n−3κ) −ρ2n−4 β(ρ2n−4κ−ρ2n−3) + ρ2n−5 , φ4n−5 = �(ρ2n−1 −ρ2n−2γ) −ρ2n−3 �(ρ2n−3γ −ρ2n−2) + ρ2n−4 , φ4n−4 = ζ(ρ2n−1 −ρ2n−2δ) −ρ2n−3 ζ(ρ2n−3δ −ρ2n−2) + ρ2n−4 . 8 int. j. anal. appl. (2022), 20:38 and ψ4n−7 = � (ρ2n−2 −ρ2n−3γ) −ρ2n−4 � (ρ2n−4γ −ρ2n−3) + ρ2n−5 , ψ4n−6 = ζ (ρ2n−2 −ρ2n−3δ) −ρ2n−4 ζ (ρ2n−4δ −ρ2n−3) + ρ2n−5 , ψ4n−5 = α(ρ2n−1 −ρ2n−2η) −ρ2n−3 α(ρ2n−3η −ρ2n−2) + ρ2n−4 , ψ4n−4 = β(ρ2n−1 −ρ2n−2κ) −ρ2n−3 β(ρ2n−3κ−ρ2n−2) + ρ2n−4 . system (1.3) leads to φ4n−3 = φ4n−7 − 1 ψ4n−5φ4n−7 = α (ρ2n−2−ρ2n−3η)−ρ2n−4 α (ρ2n−4η−ρ2n−3)+ρ2n−5 − 1( α(ρ2n−1−ρ2n−2η)−ρ2n−3 α(ρ2n−3η−ρ2n−2)+ρ2n−4 )( α(ρ2n−2−ρ2n−3η)−ρ2n−4 α (ρ2n−4η−ρ2n−3)+ρ2n−5 ) = α (ρ2n−2−ρ2n−3η)−ρ2n−4−(α (ρ2n−4η−ρ2n−3)+ρ2n−5) α (ρ2n−4η−ρ2n−3)+ρ2n−5 − α(ρ2n−1−ρ2n−2η)−ρ2n−3 α (ρ2n−4η−ρ2n−3)+ρ2n−5 = α (ρ2n−2 −ρ2n−3η) −ρ2n−4 − (α (ρ2n−4η −ρ2n−3) + ρ2n−5) −α(ρ2n−1 −ρ2n−2η) −ρ2n−3 = α (ρ2n−2 + ρ2n−3) −αη (ρ2n−3 + ρ2n−4) − (ρ2n−4 + ρ2n−5) α(ρ2n−2η −ρ2n−1) + ρ2n−3 = αρ2n −αηρ2n−1 −ρ2n−2 α(ρ2n−2η −ρ2n−1) + ρ2n−3 = α (ρ2n −ρ2n−1η) −ρ2n−2 α(ρ2n−2η −ρ2n−1) + ρ2n−3 . ψ4n−3 = ψ4n−7 − 1 φ4n−5ψ4n−7 = � (ρ2n−2−ρ2n−3γ)−ρ2n−4 � (ρ2n−4γ−ρ2n−3)+ρ2n−5 − 1( �(ρ2n−1−ρ2n−2γ)−ρ2n−3 �(ρ2n−3γ−ρ2n−2)+ρ2n−4 )( � (ρ2n−2−ρ2n−3γ)−ρ2n−4 � (ρ2n−4γ−ρ2n−3)+ρ2n−5 ) = � (ρ2n−2−ρ2n−3γ)−ρ2n−4−(� (ρ2n−4γ−ρ2n−3)+ρ2n−5) � (ρ2n−4γ−ρ2n−3)+ρ2n−5 − �(ρ2n−1−ρ2n−2γ)−ρ2n−3 � (ρ2n−4γ−ρ2n−3)+ρ2n−5 = � (ρ2n−2 −ρ2n−3γ) −ρ2n−4 − (� (ρ2n−4γ −ρ2n−3) + ρ2n−5) −�(ρ2n−1 −ρ2n−2γ) −ρ2n−3 = � (ρ2n−2 + ρ2n−3) − �γ (ρ2n−3 + ρ2n−4) − (ρ2n−4 + ρ2n−5) �(ρ2n−2γ −ρ2n−1) + ρ2n−3 = �ρ2n − �γρ2n−1 −ρ2n−2 �(ρ2n−2γ −ρ2n−1) + ρ2n−3 = � (ρ2n −ρ2n−1γ) −ρ2n−2 �(ρ2n−2γ −ρ2n−1) + ρ2n−3 . furthermore, we can obtain from system (1.3) that int. j. anal. appl. (2022), 20:38 9 φ4n−2 = φ4n−6 − 1 ψ4n−4φ4n−6 = β(ρ2n−2−ρ2n−3κ)−ρ2n−4 β(ρ2n−4κ−ρ2n−3)+ρ2n−5 − 1( β(ρ2n−1−ρ2n−2κ)−ρ2n−3 β(ρ2n−3κ−ρ2n−2)+ρ2n−4 )( β(ρ2n−2−ρ2n−3κ)−ρ2n−4 β(ρ2n−4κ−ρ2n−3)+ρ2n−5 ) = β(ρ2n−2−ρ2n−3κ)−ρ2n−4−(β(ρ2n−4κ−ρ2n−3)+ρ2n−5) β(ρ2n−4κ−ρ2n−3)+ρ2n−5 −β(ρ2n−1−ρ2n−2κ)−ρ2n−3 β(ρ2n−4κ−ρ2n−3)+ρ2n−5 = β(ρ2n−2 −ρ2n−3κ) −ρ2n−4 − (β(ρ2n−4κ−ρ2n−3) + ρ2n−5) −β(ρ2n−1 −ρ2n−2κ) −ρ2n−3 = β(ρ2n −ρ2n−1κ) −ρ2n−2 β(ρ2n−2κ−ρ2n−1) + ρ2n−3 . ψ4n−2 = ψ4n−6 − 1 φ4n−4ψ4n−6 = ζ (ρ2n−2−ρ2n−3δ)−ρ2n−4 ζ (ρ2n−4δ−ρ2n−3)+ρ2n−5 − 1( ζ(ρ2n−1−ρ2n−2δ)−ρ2n−3 ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4 )( ζ (ρ2n−2−ρ2n−3δ)−ρ2n−4 ζ (ρ2n−4δ−ρ2n−3)+ρ2n−5 ) = ζ (ρ2n−2−ρ2n−3δ)−ρ2n−4−(ζ (ρ2n−4δ−ρ2n−3)+ρ2n−5) ζ (ρ2n−4δ−ρ2n−3)+ρ2n−5 − ζ(ρ2n−1−ρ2n−2δ)−ρ2n−3 ζ (ρ2n−4δ−ρ2n−3)+ρ2n−5 = ζ (ρ2n−2 −ρ2n−3δ) −ρ2n−4 − (ζ (ρ2n−4δ −ρ2n−3) + ρ2n−5) −ζ(ρ2n−1 −ρ2n−2δ) −ρ2n−3 = ζ (ρ2n −ρ2n−1δ) −ρ2n−2 ζ(ρ2n−2δ −ρ2n−1) + ρ2n−3 . in addition, one can obtain from system (1.3) that φ4n−1 = φ4n−5 − 1 ψ4n−3φ4n−5 = �(ρ2n−1−ρ2n−2γ)−ρ2n−3 �(ρ2n−3γ−ρ2n−2)+ρ2n−4 − 1( � (ρ2n−ρ2n−1γ)−ρ2n−2 � (ρ2n−2γ−ρ2n−1)+ρ2n−3 )( �(ρ2n−1−ρ2n−2γ)−ρ2n−3 �(ρ2n−3γ−ρ2n−2)+ρ2n−4 ) = �(ρ2n−1−ρ2n−2γ)−ρ2n−3−(�(ρ2n−3γ−ρ2n−2)+ρ2n−4) �(ρ2n−3γ−ρ2n−2)+ρ2n−4 − � (ρ2n−ρ2n−1γ)−ρ2n−2 �(ρ2n−3γ−ρ2n−2)+ρ2n−4 = �(ρ2n−1 −ρ2n−2γ) −ρ2n−3 − (�(ρ2n−3γ −ρ2n−2) + ρ2n−4) −� (ρ2n −ρ2n−1γ) −ρ2n−2 = �(ρ2n+1 −ρ2nγ) −ρ2n−1 � (ρ2n−1γ −ρ2n) + ρ2n−2 . 10 int. j. anal. appl. (2022), 20:38 ψ4n−1 = ψ4n−5 − 1 φ4n−3ψ4n−5 = α(ρ2n−1−ρ2n−2η)−ρ2n−3 α(ρ2n−3η−ρ2n−2)+ρ2n−4 − 1( α (ρ2n−ρ2n−1η)−ρ2n−2 α (ρ2n−2η−ρ2n−1)+ρ2n−3 )( α(ρ2n−1−ρ2n−2η)−ρ2n−3 α(ρ2n−3η−ρ2n−2)+ρ2n−4 ) = α(ρ2n−1−ρ2n−2η)−ρ2n−3−(α(ρ2n−3η−ρ2n−2)+ρ2n−4) α(ρ2n−3η−ρ2n−2)+ρ2n−4 − α (ρ2n−ρ2n−1η)−ρ2n−2 α(ρ2n−3η−ρ2n−2)+ρ2n−4 = α(ρ2n−1 −ρ2n−2η) −ρ2n−3 − (α(ρ2n−3η −ρ2n−2) + ρ2n−4) −α (ρ2n −ρ2n−1η) −ρ2n−2 = α(ρ2n+1 −ρ2nη) −ρ2n−1 α (ρ2n−1η −ρ2n) + ρ2n−2 . finally, system (1.3) leads to φ4n = φ4n−4 − 1 ψ4n−2φ4n−4 = ζ(ρ2n−1−ρ2n−2δ)−ρ2n−3 ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4 − 1( ζ (ρ2n−ρ2n−1δ)−ρ2n−2 ζ (ρ2n−2δ−ρ2n−1)+ρ2n−3 )( ζ(ρ2n−1−ρ2n−2δ)−ρ2n−3 ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4 ) = ζ(ρ2n−1−ρ2n−2δ)−ρ2n−3−(ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4) ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4 − ζ (ρ2n−ρ2n−1δ)−ρ2n−2 ζ(ρ2n−3δ−ρ2n−2)+ρ2n−4 = ζ(ρ2n−1 −ρ2n−2δ) −ρ2n−3 − (ζ(ρ2n−3δ −ρ2n−2) + ρ2n−4) −ζ (ρ2n −ρ2n−1δ) −ρ2n−2 = ζ(ρ2n+1 −ρ2nδ) −ρ2n−1 ζ (ρ2n−1δρ2n) + ρ2n−2 . ψ4n = ψ4n−4 − 1 φ4n−2ψ4n−4 = β(ρ2n−1−ρ2n−2κ)−ρ2n−3 β(ρ2n−3κ−ρ2n−2)+ρ2n−4 − 1( β(ρ2n−ρ2n−1κ)−ρ2n−2 β(ρ2n−2κ−ρ2n−1)+ρ2n−3 )( β(ρ2n−1−ρ2n−2κ)−ρ2n−3 β(ρ2n−3κ−ρ2n−2)+ρ2n−4 ) = β(ρ2n−1−ρ2n−2κ)−ρ2n−3−(β(ρ2n−3κ−ρ2n−2)+ρ2n−4) β(ρ2n−3κ−ρ2n−2)+ρ2n−4 − β(ρ2n−ρ2n−1κ)−ρ2n−2 β(ρ2n−3κ−ρ2n−2)+ρ2n−4 = β(ρ2n−1 −ρ2n−2κ) −ρ2n−3 − (β(ρ2n−3κ−ρ2n−2) + ρ2n−4) −β(ρ2n −ρ2n−1κ) −ρ2n−2 = β(ρ2n+1 −ρ2nκ) −ρ2n−1 β(ρ2n−1κ−ρ2n) + ρ2n−2 . hence, the proof is completed. int. j. anal. appl. (2022), 20:38 11 4. stability of the equilibrium points this section discusses the stability of the equilibrium points of the considered systems. theorem 4.1. system (1.2) has a unique real equilibrium point (q1,q1) which is a saddle point. proof. the equilibrium point of system (1.2) is given by ψ∗ = ψ∗ + 1 φ∗ψ∗ , φ∗ = φ∗ + 1 φ∗ψ∗ . (4.1) subtract the second equation of system (4.1) from the first equation to have φ∗ 2 ψ∗ − φ∗ − 1 − φ∗ ψ∗ 2 + ψ∗ + 1 = 0, or, (φ∗ ψ∗ − 1) (φ∗ − ψ∗) = 0. hence, φ∗ ψ∗ − 1 = 0 , φ∗ ψ∗ = 1 , (4.2) φ∗ − ψ∗ = 0 , ψ∗ = φ∗ . (4.3) equation. (4.2) dose not satisfy system (4.1). therefore, the only real equilibrium point is obtained from substituting equation. (4.3) into any equation of system (4.1). this gives φ∗ 3 − φ∗ − 1 = 0. (4.4) solving equation. (4.4) gives q1 = α2 + 12 6α , q2,3 = − α2 + 12 6α ± √ 3 2 ( α 6 − 2 3α ) i, where, α = 3 √ 108 + 12 √ 69. hence, the only real equilibrium point is (q1,q1). now, we find the jacobian matrix. let f (u,v) = (f (u,v),g(u,v)), where f (u,v) = u + 1 uv , g(u,v) = v + 1 uv . then, jf =   −1 u2v −u+1 uv2 −v+1 u2v −1 uv2   . 12 int. j. anal. appl. (2022), 20:38 evaluating the jacobian matrix about (q1,q1) gives, jf (q1,q1) =   −1 q31 −q1+1 q31 −q1+1 q31 −1 q31   . thus, the characteristic equation of this matrix is given by ( λ + 1 q31 )( λ + 1 q31 ) − (q1 + 1) 2 q61 = 0. or, ( λ + 1 q31 )2 − (q1 + 1) 2 q61 = 0. note that (q1+1) 2 q61 = 1. hence, ( λ + 1 q31 )2 − 1 = 0. then, λ + 1 q31 = ±1. for λ + 1 q31 = 1, we have λ1 = |1− 1q31 | < 1. for λ + 1 q31 = −1, we have λ2 = |−1− 1q31 | = 1 + 1 q31 > 1. since λ1 < 1 and λ2 > 1, the point (q1,q1) is a saddle point. theorem 4.2. system (1.3) has a unique real equilibrium point (−q1,−q1) which is a saddle point. proof. the proof is similar to the proof of theorem 4.1. 5. behavior of the solutions this section presents the future pattern of the considered systems under specific initial conditions. we selected some random values for the initial conditions to illustrate the long behavior solutions. example 1. figure 1 (left) presents the dynamical behavior of the solutions of system (1.2) under the selected values φ−3 = 7, φ−2 = 3, φ−1 = 4, φ0 = 6, ψ−3 = 5, ψ−2 = 3, ψ−1 = 5, and ψ0 = 6. example 2. the exact solutions of system (1.2) are also plotted in figure 1 (right) under the initial conditions φ−3 = 1, φ−2 = 3, φ−1 = 1, φ0 = 2, ψ−3 = 1, ψ−2 = 4, ψ−1 = 2, and ψ0 = 4. int. j. anal. appl. (2022), 20:38 13 0 10 20 30 40 50 n 0 1 2 3 4 5 6 7 plot of the first system x(n) y(n) 0 10 20 30 40 50 n 0 0.5 1 1.5 2 2.5 3 3.5 4 plot of the first system x(n) y(n) figure 1. the dynamical behavior of the solutions of system (1.2). example 3. in figure 2 (left), we plot the dynamical behavior of the solutions of system (1.3) under the selected values φ−3 = 7, φ−2 = 2.5, φ−1 = 6, φ0 = 4.4, ψ−3 = 3.8, ψ−2 = 2, ψ−1 = 5, and ψ0 = 3.3. example 4. this example illustrates the behavior of the exact solutions of system (1.3) when we use the following initial values: φ−3 = 7, φ−2 = 3, φ−1 = 4, φ0 = 6, ψ−3 = 5, ψ−2 = 3, ψ−1 = 5, and ψ0 = 6. see figure 2 (right). 0 5 10 15 20 25 30 35 40 45 50 n -10 -8 -6 -4 -2 0 2 4 6 8 plot of the second system x(n) y(n) 0 5 10 15 20 25 30 35 40 45 50 n -4 -2 0 2 4 6 8 plot of the second system x(n) y(n) figure 2. the dynamical behavior of the solutions of system (1.3). 6. conclusion in brief, we have discussed the solutions of the considered systems using padovan numbers. successive iterations have been successfully used in extracting the exact solutions. theorem 2.1 presents the solutions of system 1.2 while theorem 3.1 gives the solutions of system 1.3. the obtained solutions 14 int. j. anal. appl. (2022), 20:38 are presented in the form of rational relations. jacobian matrix has been successfully used to examine the stability of the real equilibrium points. the equilibrium points are saddle. in section 5, we depict the behavior of the solutions for some random initial conditions. the constructed exact solutions are stable. the used approaches can be utilized to deal with high order systems of difference equations. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] j.d. murray, mathematical biology: i. an introduction, 3rd ed., springer-verlag, new york, 2001. https://doi.org/10.1007/b98868. [2] d.t. tollu, y. yazlik, n. taskara, on the solutions of two special types of riccati difference equation via fibonacci numbers, adv. differ. equ. 2013 (2013), 174. https://doi.org/10. 1186/1687-1847-2013-174. [3] y. yazlik, d.t. tollu, n. taskara, on the solutions of difference equation systems with padovan numbers, appl. math. 04 (2013), 15–20. https://doi.org/10.4236/am.2013.412a002. [4] m.b. almatrafi, solutions structures for some systems of fractional difference equations, open j. math. anal. 3 (2019), 52–61. https://doi.org/10.30538/psrp-oma2019.0032. [5] m.b. almatrafi, m.m. alzubaidi, analysis of the qualitative behaviour of an eighth-order fractional difference equation, open j. discret. appl. math. 2 (2019), 41–47. https://doi.org/ 10.30538/psrp-odam2019.0010. [6] c. çinar, on the positive solutions of the difference equation system xn+1 = 1 yn , yn+1 = yn yn−1xn−1 , appl. math. comput. 158 (2004), 303–305. https://doi.org/10.1016/j.amc.2003. 08.073. [7] e.m. elsayed, solutions of rational difference systems of order two, math. computer model. 55 (2012), 378–384. https://doi.org/10.1016/j.mcm.2011.08.012. [8] h.s. alayachi, a.q. khan, m.s.m. noorani, on the solutions of three-dimensional rational difference equation systems, j. math. 2021 (2021), 2480294. https://doi.org/10.1155/ 2021/2480294. [9] h.s. alayachi, a.q. khan, m.s.m. noorani, a. khaliq, displaying the structure of the solutions for some fifth-order systems of recursive equations, math. probl. eng. 2021 (2021), 6682009. https://doi.org/10.1155/2021/6682009. [10] h.s. alayachi, m.s.m. noorani, a.q. khan, m.b. almatrafi, analytic solutions and stability of sixth order difference equations, math. probl. eng. 2020 (2020), 1230979. https://doi.org/ 10.1155/2020/1230979. [11] s. elaydi, an introduction to difference equations, springer-verlag, new york, 2005. https: //doi.org/10.1007/0-387-27602-5. https://doi.org/10.1007/b98868 https://doi.org/10.1186/1687-1847-2013-174 https://doi.org/10.1186/1687-1847-2013-174 https://doi.org/10.4236/am.2013.412a002 https://doi.org/10.30538/psrp-oma2019.0032 https://doi.org/10.30538/psrp-odam2019.0010 https://doi.org/10.30538/psrp-odam2019.0010 https://doi.org/10.1016/j.amc.2003.08.073 https://doi.org/10.1016/j.amc.2003.08.073 https://doi.org/10.1016/j.mcm.2011.08.012 https://doi.org/10.1155/2021/2480294 https://doi.org/10.1155/2021/2480294 https://doi.org/10.1155/2021/6682009 https://doi.org/10.1155/2020/1230979 https://doi.org/10.1155/2020/1230979 https://doi.org/10.1007/0-387-27602-5 https://doi.org/10.1007/0-387-27602-5 int. j. anal. appl. (2022), 20:38 15 [12] m.b. almatrafi, e.m. elsayed, f. alzahrani, qualitative behavior of two rational difference equations, fund. j. math. appl. 1 (2018), 194–204. https://doi.org/10.33401/fujma. 454999. [13] m.b. almatrafi, e.m. elsayed, solutions and formulae for some systems of difference equations, mathlab j. 1 (2018), 356-369. [14] m.b. almatrafi, e.m. elsayed, f. alzahrani, qualitative behavior of a quadratic second-order rational difference equation, int. j. adv. math. 2019 (2019), 1-14. [15] m.b. almatrafi, exact solutions and stability of sixth order difference equations, electron. j. math. anal. appl. 10 (2022), 209-225. [16] m.b. almatrafi, abundant traveling wave and numerical solutions for novikov-veselov system with their stability and accuracy, appl. anal. (2022). https://doi.org/10.1080/00036811. 2022.2027381. https://doi.org/10.33401/fujma.454999 https://doi.org/10.33401/fujma.454999 https://doi.org/10.1080/00036811.2022.2027381 https://doi.org/10.1080/00036811.2022.2027381 1. introduction 2. solutions of system (1.2) 3. solutions of system (1.3) 4. stability of the equilibrium points 5. behavior of the solutions 6. conclusion references international journal of analysis and applications volume 16, number 4 (2018), 518-527 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-518 a new type of connected sets via bioperations hariwan z. ibrahim∗ department of mathematics, faculty of education, university of zakho, kurdistan-region, iraq ∗corresponding author: hariwan math@yahoo.com abstract. the purpose of this paper is to introduce the notion of α (γ,γ ′ ) -separated sets and study their properties in topological spaces, then we introduce the notions of α (γ,γ ′ ) -connected and α (γ,γ ′ ) -disconnected sets. we discuss the characterizations and properties of α (γ,γ ′ ) -connected sets and then properties under (α (γ,γ ′ ) , α (β,β ′ ) )-continuous functions. the α (γ,γ ′ ) -components in a space x is also introduced. 1. introduction njastad [5] introduced α-open sets in a topological space and studied some of their properties. ibrahim [1] introduced and discussed an operation of a topology αo(x) into the power set p(x) of a space x and also in [2] he introduced the notion of αo(x,τ)(γ,γ′ ), which is the collection of all α(γ,γ′ )-open sets in a topological space (x,τ). in addition, ibrahim [3] introduced the concept of (α(γ,γ′ ), α(β,β′ ))-closed and (α(γ,γ′ ), α(β,β′ ))-continuous functions and investigated some of their basic properties. mishra [4] introduced α-τ-disconnectedness and α-τ-connectedness in topological spaces. in this paper, the author introduce and study the characterizations and properties of α(γ,γ′ )-connected and α(γ,γ′ )-disconnected spaces and then properties under (α(γ,γ′ ), α(β,β′ ))-continuous functions. received 2018-01-18; accepted 2018-03-19; published 2018-07-02. 2010 mathematics subject classification. primary 22a05, 22a10, secondary 54c05. key words and phrases. α-open; bioperations; α (γ,γ ′ ) -connected set; α (γ,γ ′ ) -disconnected set. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 518 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-518 int. j. anal. appl. 16 (4) (2018) 519 2. preliminaries throughout the present paper, (x,τ) and (y,σ) (or simply x and y ) denotes a topological spaces on which no separation axioms is assumed unless explicitly stated. for a subset a of a space x, cl(a) and int(a) represent the closure of a and the interior of a, respectively. definition 2.1. [5] a subset a of a topological space (x,τ) is said to be α-open if a ⊆ int(cl(int(a))). the complement of an α-open set is said to be α-closed. the family of all α-open (resp., α-closed) sets in a topological space (x,τ) is denoted by αo(x,τ) (resp., αc(x,τ)). the intersection of all α-closed sets containing a is called the α-closure of a and is denoted by αcl(a). definition 2.2. [4] the subsets a and b of a topological space (x,τ) are called α-τ-separated sets if (αcl(a) ∩b) ∪ (a∩αcl(b)) = φ. definition 2.3. [1] an operation γ : αo(x,τ) → p(x) is a mapping satisfying the condition, v ⊆ v γ for each v ∈ αo(x,τ). we call the mapping γ an operation on αo(x,τ). the operation id : αo(x,τ) → p(x) is defined by v id = v for any set v ∈ αo(x,τ). this operation is called the identity operation on αo(x,τ). definition 2.4. [2] a nonempty subset a of (x,τ) is said to be α(γ,γ′ )-open if for each x ∈ a, there exist α-open sets u and v of x containing x such that uγ ∪v γ ′ ⊆ a. a subset f of (x,τ) is said to be α(γ,γ′ )-closed if its complement x \f is α(γ,γ′ )-open. the set of all α(γ,γ′ )-open sets of (x,τ) is denoted by αo(x,τ)(γ,γ′ ). definition 2.5. [2] let a be a subset of a topological space (x,τ). (1) the union of all α(γ,γ′ )-open sets contained in a is called the α(γ,γ′ )-interior of a and is denoted by α(γ,γ′ )-int(a). (2) the intersection of all α(γ,γ′ )-closed sets containing a is called the α(γ,γ′ )-closure of a and denoted by α(γ,γ′ )-cl(a). proposition 2.1. [2] let a and b be subsets of (x,τ). then the following hold: (1) a ⊆ α(γ,γ′ )-cl(a). (2) if a ⊆ b, then α(γ,γ′ )-cl(a) ⊆ α(γ,γ′ )-cl(b). (3) a is α(γ,γ′ )-closed if and only if α(γ,γ′ )-cl(a) = a. (4) α(γ,γ′ )-cl(a) is α(γ,γ′ )-closed. proposition 2.2. [2] for a point x ∈ x, x ∈ α(γ,γ′ )-cl(a) if and only if v ∩a 6= φ for every α(γ,γ′ )-open set v containing x. definition 2.6. [3] a function f : (x,τ) → (y,σ) is said to be (α(γ,γ′ ), α(β,β′ ))-closed if for α(γ,γ′ )-closed set a of x, f(a) is α(β,β′ )-closed in y . int. j. anal. appl. 16 (4) (2018) 520 proposition 2.3. [3] let f : (x,τ) → (y,σ) be a function. then, f is (α(γ,γ′ ), α(β,β′ ))-closed if and only if α(β,β′ )-cl(f(a)) ⊆ f(α(γ,γ′ )-cl(a)) for every subset a of x. theorem 2.1. [3] suppose that f : (x,τ) → (y,σ) is (α(γ,γ′ ), α(β,β′ ))-continuous. then, (1) f−1(v ) is α(γ,γ′ )-open for every α(β,β′ )-open set v of (y,σ). (2) for each point x ∈ x and each α(β,β′ )-open w of (y,σ) containing f(x), there exist α(γ,γ′ )-open u of (x,τ) containing x such that f(u) ⊆ w . 3. α(γ,γ′ )-connected and α(γ,γ′ )-disconnected sets throughout this section, let γ,γ ′ : αo(x,τ) → p(x) be operations on αo(x,τ) and β,β ′ : αo(y,σ) → p(y ) be operations on αo(y,σ). definition 3.1. two subsets a and b of a topological space (x,τ) are called α(γ,γ′ )-separated if (α(γ,γ′ )cl(a) ∩b) ∪ (a∩α(γ,γ′ )-cl(b)) = φ. remark 3.1. each two α(γ,γ′ )-separated sets are always disjoint, since a ∩ b ⊆ a ∩ α(γ,γ′ )-cl(b) = φ. the converse may not be true in general, as it is shown in the following example. example 3.1. let x = {1, 2, 3} and τ = {φ,x,{2}}. for each a ∈ αo(x), we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if 3 ∈ ax if 3 /∈ a. since αo(x,τ)(γ,γ′ ) = {φ,x,{2, 3}}, then {2} and {3} are disjoint subsets of x, but not α(γ,γ′ )-separated. from the fact that αcl(a) ⊆ α(γ,γ′ )-cl(a), for every subset a of x. then every α(γ,γ′ )-separated set is α-τ-separated. but the converse may not be true as shown in the following example. example 3.2. let x = {1, 2, 3, 4} and τ = {φ,x,{1},{2},{1, 2}}. for each a ∈ αo(x), we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if 4 ∈ ax if 4 /∈ a. since αo(x,τ)(γ,γ′ ) = {φ,x,{1, 2, 4}}, then the subsets {3} and {4} are α-τ-separated, but not α(γ,γ′ )separated. theorem 3.1. if a and b are any two nonempty subsets in a space x, then the following statements are true: (1) if a and b are α(γ,γ′ )-separated, a1 ⊆ a and b1 ⊆ b, then a1 and b1 are also α(γ,γ′ )-separated. int. j. anal. appl. 16 (4) (2018) 521 (2) if a ∩ b = φ such that each of a and b are both α(γ,γ′ )-closed (α(γ,γ′ )-open), then a and b are α(γ,γ′ )-separated. (3) if each of a and b is α(γ,γ′ )-closed (α(γ,γ′ )-open) and if h = a ∩ (x \ b) and g = b ∩ (x \ a), then h and g are α(γ,γ′ )-separated. proof. (1) since a1 ⊆ a, then α(γ,γ′ )-cl(a1) ⊆ α(γ,γ′ )-cl(a). then, b ∩ α(γ,γ′ )-cl(a) = φ implies b1 ∩α(γ,γ′ )-cl(a) = φ and b1 ∩α(γ,γ′ )-cl(a1) = φ. similarly a1 ∩α(γ,γ′ )-cl(b1) = φ. hence, a1 and b1 are α(γ,γ′ )-separated. (2) since a = α(γ,γ′ )-cl(a), b = α(γ,γ′ )-cl(b) and a ∩ b = φ, then α(γ,γ′ )-cl(a) ∩ b = φ and α(γ,γ′ )-cl(b) ∩ a = φ. hence, a and b are α(γ,γ′ )-separated. if a and b are α(γ,γ′ )-open, then their complements are α(γ,γ′ )-closed. hence, α(γ,γ′ )-cl(a) ⊆ x \ b and α(γ,γ′ )-cl(b) ⊆ x \ a. therefore, a and b are α(γ,γ′ )-separated. (3) if a and b are α(γ,γ′ )-open, then x \ a and x \ b are α(γ,γ′ )-closed. since h ⊆ x \ b, α(γ,γ′ )cl(h) ⊆ α(γ,γ′ )-cl(x \ b) = x \ b and so α(γ,γ′ )-cl(h) ∩ b = φ. thus g ∩ α(γ,γ′ )-cl(h) = φ. similarly, h∩α(γ,γ′ )-cl(g) = φ. hence h and g are α(γ,γ′ )-separated. if a and b are α(γ,γ′ )-closed, then α(γ,γ′ )-cl(h) ⊆ a and α(γ,γ′ )-cl(g) ⊆ b. thus, h and g are α(γ,γ′ )-separated. � theorem 3.2. the sets a and b of a space x are α(γ,γ′ )-separated if and only if there exist u and v in αo(x,τ)(γ,γ′ ) such that a ⊆ u, b ⊆ v and a∩v = φ and b ∩u = φ. proof. let a and b be α(γ,γ′ )-separated sets. set v = x \ α(γ,γ′ )-cl(a) and u = x \ α(γ,γ′ )-cl(b). then u,v ∈ αo(x,τ)(γ,γ′ ) such that a ⊆ u, b ⊆ v and a∩v = φ, b ∩u = φ. on the other hand, let u,v ∈ αo(x,τ)(γ,γ′ ) such that a ⊆ u, b ⊆ v and a∩v = φ, b∩u = φ. since x\v and x\u are α(γ,γ′ )closed, then α(γ,γ′ )-cl(a) ⊆ x\v ⊆ x\b and α(γ,γ′ )-cl(b) ⊆ x\u ⊆ x\a. thus α(γ,γ′ )-cl(a)∩b = φ and α(γ,γ′ )-cl(b) ∩a = φ. � theorem 3.3. in any topological space (x,τ), the following statements are equivalent: (1) φ and x are the only α(γ,γ′ )-open and α(γ,γ′ )-closed sets in x. (2) x is not the union of two disjoint nonempty α(γ,γ′ )-open sets. (3) x is not the union of two disjoint nonempty α(γ,γ′ )-closed sets. (4) x is not the union of two nonempty α(γ,γ′ )-separated sets. proof. (1) ⇒ (2): suppose (2) is false and that x = a∪b, where a,b are disjoint nonempty α(γ,γ′ )-open sets. since x \a = b is α(γ,γ′ )-open and nonempty, we have that a is a nonempty proper α(γ,γ′ )-open and α(γ,γ′ )-closed set in x, which shows that (1) is false. (2) ⇔ (3): this is clear. int. j. anal. appl. 16 (4) (2018) 522 (3) ⇒ (4): if (4) is false, then x = a∪b, where a,b are nonempty and α(γ,γ′ )-separated. since α(γ,γ′ )cl(b)∩a = φ, we conclude that α(γ,γ′ )-cl(b) ⊆ b, so b is α(γ,γ′ )-closed. similarly, a must be α(γ,γ′ )-closed. therefore, (3) is false. (4) ⇒ (1): suppose (1) is false and that a is a nonempty proper α(γ,γ′ )-open and α(γ,γ′ )-closed subset of x. then, b = x \ a is nonempty, α(γ,γ′ )-open and α(γ,γ′ )-closed, so a and b are α(γ,γ′ )-separated and x = a∪b, so (4) is false. � definition 3.2. a subset c of a space x is said to be α(γ,γ′ )-disconnected if there are nonempty α(γ,γ′ )separated subsets a and b of x such that c = a ∪ b, otherwise c is called α(γ,γ′ )-connected. if c is α(γ,γ′ )-disconnected, such a pair of sets a,b will be called an α(γ,γ′ )-disconnection of c. example 3.3. let x = {1, 2, 3} and τ = {φ,x,{1},{2},{1, 2},{2, 3}}. for each a ∈ αo(x), we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if 3 ∈ acl(a) if 3 /∈ a. then, x is α(γ,γ′ )-disconnected because there exist a pair {1},{2, 3} subsets of x such that {1}∪{2, 3} = x, and (α(γ,γ′ )-cl({1}) ∩{2, 3}) ∪ ({1}∩α(γ,γ′ )-cl({2, 3})) = ({1}∩{2, 3}) ∪ ({1}∩{2, 3}) = φ. example 3.4. let x = {1, 2, 3} and τ = {φ,x,{1},{3},{1, 3}}. for each a ∈ αo(x), we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if 2 ∈ ax if 2 /∈ a. then, x is α(γ,γ′ )-connected, since there does not exist a pair a,b of nonempty α(γ,γ′ )-separated subsets of x such that x = a∪b. remark 3.2. every indiscrete space is α(γ,γ′ )-connected. remark 3.3. every discrete space contains more than one element is α(id,id′ )-disconnected. remark 3.4. a space x is α(γ,γ′ )-connected if any (therefore all) of the conditions (1) − (4) in theorem 3.3 hold. remark 3.5. according to the definition 3.2 and remark 3.4, a space x is α(γ,γ′ )-disconnected if we can write x = a∪b, where the following (equivalent) statements are true: (1) a and b are disjoint, nonempty and α(γ,γ′ )-open. (2) a and b are disjoint, nonempty and α(γ,γ′ )-closed. (3) a and b are nonempty and α(γ,γ′ )-separated. int. j. anal. appl. 16 (4) (2018) 523 theorem 3.4. a space x is α(γ,γ′ )-disconnected if and only if there exists a nonempty proper subset a of x which is both α(γ,γ′ )-open and α(γ,γ′ )-closed in x. proof. follows from remark 3.5. � definition 3.3. let a be a subset of a space x, then the α(γ,γ′ )-boundary of a is defined as α(γ,γ′ )cl(a) \α(γ,γ′ )-int(a) and is denoted by α(γ,γ′ )-bd(a). proposition 3.1. let a be any subset of a topological space (x,τ). then, the following statements are hold: (1) α(γ,γ′ )-cl(a) = α(γ,γ′ )-int(a) ∪α(γ,γ′ )-bd(a). (2) α(γ,γ′ )-bd(a) = α(γ,γ′ )-cl(a) ∩α(γ,γ′ )-cl(x \a). proof. obvious. � theorem 3.5. a space x is α(γ,γ′ )-connected if and only if every nonempty proper subset of x has a nonempty α(γ,γ′ )-boundary. proof. suppose that a nonempty proper subset a of an α(γ,γ′ )-connected space x has empty α(γ,γ′ )boundary. since α(γ,γ′ )-cl(a) = α(γ,γ′ )-int(a) ∪α(γ,γ′ )-bd(a). thus, a is both α(γ,γ′ )-closed and α(γ,γ′ )open. by theorem 3.4, x is α(γ,γ′ )-disconnected. this contradiction, hence proves that a has a nonempty α(γ,γ′ )-boundary. conversely, suppose x is α(γ,γ′ )-disconnected. then by theorem 3.4, x has a nonempty proper subset a which is both α(γ,γ′ )-closed and α(γ,γ′ )-open. then, α(γ,γ′ )-cl(a) = a, α(γ,γ′ )-cl(x \ a) = x \ a and α(γ,γ′ )-cl(a) ∩α(γ,γ′ )-cl(x \a) = φ. so a has empty α(γ,γ′ )-boundary, this is a contradiction. hence, x is α(γ,γ′ )-connected. � lemma 3.1. suppose m,n are α(γ,γ′ )-separated subsets of x. if c ⊆ m ∪n and c is α(γ,γ′ )-connected, then c ⊆ m or c ⊆ n. proof. since c∩m ⊆ m and c∩n ⊆ n, then c∩m and c∩n are α(γ,γ′ )-separated and c = c∩(m∪n) = (c∩m)∪(c∩n). but c is α(γ,γ′ )-connected so (c∩m) and (c∩n) can not form an α(γ,γ′ )-disconnection of c. therefore, either c ∩m = φ, so c ⊆ n or c ∩n = φ, so c ⊆ m. � theorem 3.6. suppose c and ci (i ∈ i) are α(γ,γ′ )-connected subsets of x and that for each i, ci and c are not α(γ,γ′ )-separated. then, s = c ∪ci is α(γ,γ′ )-connected. proof. suppose that s = m ∪n, where m and n are α(γ,γ′ )-separated. by lemma 3.1, either c ⊆ m or c ⊆ n. without loss of generality, assume c ⊆ m. by the same reasoning we conclude that for each i, int. j. anal. appl. 16 (4) (2018) 524 either ci ⊆ m or ci ⊆ n. but if some ci ⊆ n, then c and ci would be α(γ,γ′ )-separated. hence every ci ⊆ m. therefore, n = φ and the pair m,n is not an α(γ,γ′ )-disconnection of s. � corollary 3.1. suppose that for each i ∈ i, ci is an α(γ,γ′ )-connected subset of x and that for all i 6= j, ci ∩cj 6= φ. then, ∪{ci : i ∈ i} is α(γ,γ′ )-connected. proof. if i = φ, then ∪{ci : i ∈ i} = φ is α(γ,γ′ )-connected. if i 6= φ, pick i0 ∈ i and let ci0 be the central set c in theorem 3.6. for all i ∈ i, ci ∩ci0 6= φ, so ci and ci0 are not α(γ,γ′ )-separated. by theorem 3.6, ∪{ci : i ∈ i} is α(γ,γ′ )-connected. � corollary 3.2. suppose that for all x,y ∈ x, there exists an α(γ,γ′ )-connected set cxy ⊆ x with x,y ∈ cxy. then, x is α(γ,γ′ )-connected. proof. certainly x = φ is α(γ,γ′ )-connected. if x 6= φ, choose a ∈ x. by hypothesis there is, for each y ∈ x, an α(γ,γ′ )-connected set cay containing both a and y. by corollary 3.1, x = ∪{cay : y ∈ x} is α(γ,γ′ )-connected. � corollary 3.3. suppose c is an α(γ,γ′ )-connected subset of x and a ⊆ x. if c ⊆ a ⊆ α(γ,γ′ )-cl(c), then a is α(γ,γ′ )-connected. proof. for each a ∈ a, {a} and c are not α(γ,γ′ )-separated. by theorem 3.6, a = c ∪ ⋃ {{a} : a ∈ a} is α(γ,γ′ )-connected. � remark 3.6. in particular, the α(γ,γ′ )-closure of an α(γ,γ′ )-connected set is α(γ,γ′ )-connected. theorem 3.7. let f : (x,τ) → (y,σ) be a function. consider the following statements. (1) f is (α(γ,γ′ ), α(β,β′ ))-continuous. (2) f−1(v ) ⊆ α(γ,γ′ )-int(f −1(v )) for every α(β,β′ )-open set v of y . (3) f(α(γ,γ′ )-cl(a)) ⊆ α(β,β′ )-cl(f(a)) for every subset a of x. (4) α(γ,γ′ )-cl(f −1(b)) ⊆ f−1(α(β,β′ )-cl(b)) for every subset b of y . then, the following implications are true: (1) ⇒ (2) ⇒ (3) ⇒ (4). proof. (1) ⇒ (2). let v be any α(β,β′ )-open set of y and x ∈ f −1(v ). then, f(x) ∈ v . since f is (α(γ,γ′ ), α(β,β′ ))-continuous, there exists an α(γ,γ′ )-open set u of x containing x such that f(u) ⊆ v and hence u ⊆ f−1(v ), this implies that x ∈ α(γ,γ′ )-int(f −1(v )). thus, it follows that f−1(v ) ⊆ α(γ,γ′ )int(f−1(v )). (2) ⇒ (3). let a be any subset of x and f(x) /∈ α(β,β′ )-cl(f(a)). then, by proposition 2.2, there exists an α(β,β′ )-open set v of y containing f(x) such that v ∩ f(a) = φ and hence f −1(v ) ∩ a = φ. also f(x) ∈ v implies x ∈ f−1(v ). then by (2) we obtain that x ∈ α(γ,γ′ )-int(f −1(v )). hence, there exists an int. j. anal. appl. 16 (4) (2018) 525 α(γ,γ′ )-open set u of x containing x such that u ⊆ f −1(v ). then u ∩ a = φ and so x /∈ α(γ,γ′ )-cl(a). this implies f(x) /∈ f(α(γ,γ′ )-cl(a)). thus, f(α(γ,γ′ )-cl(a)) ⊆ α(β,β′ )-cl(f(a)). (3) ⇒ (4). let b be any subset of y . since f(f−1(b)) ⊆ b, so, we have α(β,β′ )-cl(f(f −1(b))) ⊆ α(β,β′ )cl(b). also, f−1(b) ⊆ x, then by (3), we have f(α(γ,γ′ )-cl(f −1(b))) ⊆ α(β,β′ )-cl(f(f −1(b))) ⊆ α(β,β′ )cl(b). thus, α(γ,γ′ )-cl(f −1(b)) ⊆ f−1(α(β,β′ )-cl(b)). � corollary 3.4. let f : x → y be an (α(γ,γ′ ), α(β,β′ ))-continuous and injective function. if k is α(γ,γ′ )connected in x, then f(k) is α(β,β′ )-connected in y. proof. suppose that f(k) is α(β,β′ )-disconnected in y . there exist two α(β,β′ )-separated sets p and q of y such that f(k) = p∪q. set a = k∩f−1(p) and b = k∩f−1(q). since f(k)∩p 6= φ, then k∩f−1(p) 6= φ and so a 6= φ. similarly b 6= φ. now, a∪b = (k ∩f−1(p)) ∪ (k ∩f−1(q)) = k ∩ (f−1(p) ∪f−1(q)) = k ∩ f−1(p ∪ q) = k ∩ f−1(f(k)) = k. since f is (α(γ,γ′ ), α(β,β′ ))-continuous, then by theorem 3.7 , α(γ,γ′ )-cl(f −1(q)) ⊆ f−1(α(β,β′ )-cl(q)) and b ⊆ f −1(q), then α(γ,γ′ )-cl(b) ⊆ f −1(α(β,β′ )-cl(q)). since p ∩α(β,β′ )-cl(q) = φ, then a∩α(γ,γ′ )-cl(b) ⊆ a∩f −1(α(β,β′ )-cl(q)) ⊆ f −1(p)∩f−1(α(β,β′ )-cl(q)) = φ and then a ∩ α(γ,γ′ )-cl(b) = φ. similarly, b ∩ α(γ,γ′ )-cl(a) = φ. thus, a and b are α(γ,γ′ )-separated. therefore, k is α(γ,γ′ )-disconnected, this is contradiction. hence, f(k) is α(β,β′ )-connected. � theorem 3.8. if f : (x,τ) → (y,σ) is an onto (α(γ,γ′ ), α(β,β′ ))-continuous function and x is α(γ,γ′ )connected, then y is α(β,β′ )-connected. proof. suppose that y is α(β,β′ )-disconnected and a,b is an α(β,β′ )-disconnection of y . by remark 3.5, a and b are both α(β,β′ )-open sets. since f is (α(γ,γ′ ), α(β,β′ ))-continuous, so by theorem 2.1, f −1(a) and f−1(b) are both nonempty α(γ,γ′ )-open sets in x. now, f −1(a) ∪f−1(b) = f−1(a∪b) = f−1(y ) = x, and f−1(a) ∩ f−1(b) = f−1(a ∩ b) = f−1(φ) = φ. then by remark 3.5, f−1(a),f−1(b) is a pair of α(γ,γ′ )-disconnection of x. this contradiction shows that y is α(β,β′ )-connected. � corollary 3.5. for a bijective (α(γ,γ′ ), α(β,β′ ))-closed function f : x → y , if c is α(β,β′ )-connected in y , then f−1(c) is α(γ,γ′ )-connected in x. proof. suppose that f−1(c) is α(γ,γ′ )-disconnected in x. there exist two α(γ,γ′ )-separated sets m and n of x such that f−1(c) = m ∪ n. set k = c ∩ f(m) and l = c ∩ f(n). since c = f(m) ∪ f(n), then c ∩ f(m) 6= φ and so k 6= φ. similarly l 6= φ. now, k ∪ l = (c ∩ f(m)) ∪ (c ∩ f(n)) = c ∩ (f(m) ∪ f(n)) = c ∩ f(m ∪ n) = c ∩ f(f−1(c)) = c. since f is (α(γ,γ′ ), α(β,β′ ))-closed, then by proposition 2.3, α(β,β′ )-cl(f(n)) ⊆ f(α(γ,γ′ )-cl(n)) and l ⊆ f(n), then α(β,β′ )-cl(l) ⊆ f(α(γ,γ′ )-cl(n)). since m∩α(γ,γ′ )-cl(n) = φ, then k∩α(β,β′ )-cl(l) ⊆ k∩f(α(γ,γ′ )-cl(n)) ⊆ f(m)∩f(α(γ,γ′ )-cl(n)) = φ and then k ∩ α(β,β′ )-cl(l) = φ. similarly, l ∩ α(β,β′ )-cl(k) = φ. thus, k and l are α(β,β′ )-separated. therefore, c is α(β,β′ )-disconnected, this is contradiction. hence, f −1(c) is α(γ,γ′ )-connected. � int. j. anal. appl. 16 (4) (2018) 526 definition 3.4. a set c is called a maximal α(γ,γ′ )-connected set if it is α(γ,γ′ )-connected and if c ⊆ d ⊆ x where d is α(γ,γ′ )-connected, then c = d. a maximal α(γ,γ′ )-connected subset c of a space x is called an α(γ,γ′ )-component of x. if x is itself α(γ,γ′ )-connected, then x is the only α(γ,γ′ )-component of x. theorem 3.9. for each x ∈ x, there is exactly one α(γ,γ′ )-component of x containing x. proof. for any x ∈ x, let cx = ⋃ {a : x ∈ a ⊆ x and a is α(γ,γ′ )-connected}. then, {x}∈ cx, since cx is a union of α(γ,γ′ )-connected sets each containing x, cx is α(γ,γ′ )-connected by corollary 3.1. if cx ⊆ d and d is α(γ,γ′ )-connected, then d was one of the sets a in the collection whose union defines cx, so d ⊆ cx and therefore cx = d. therefore, cx is an α(γ,γ′ )-component of x that contains x. � corollary 3.6. a space x is the union of its α(γ,γ′ )-components. proof. follows from theorem 3.9. � corollary 3.7. two α(γ,γ′ )-components are either disjoint or coincide. proof. let cx and cy be α(γ,γ′ )-components and cx 6= cy. if p ∈ cx ∩cy, then by corollary 3.1, cx ∪cy would be an α(γ,γ′ )-connected set strictly larger than cx. therefore, cx ∩cy = φ. � theorem 3.10. each α(γ,γ′ )-connected subset of x is contained in exactly one α(γ,γ′ )-component of x. proof. let a be an α(γ,γ′ )-connected subset of x which is not in exactly one α(γ,γ′ )-component of x. suppose that c1 and c2 are α(γ,γ′ )-components of x such that a ⊆ c1 and a ⊆ c2. since c1 ∩ c2 6= φ and by corollary 3.1, c1∪c2 is another α(γ,γ′ )-connected set which contains c1 as well as c2, a contradiction to the fact that c1 and c2 are α(γ,γ′ )-components. this proves that a is contained in exactly one α(γ,γ′ )-component of x. � theorem 3.11. a nonempty α(γ,γ′ )-connected subset of x which is both α(γ,γ′ )-open and α(γ,γ′ )-closed is α(γ,γ′ )-component. proof. suppose that a is α(γ,γ′ )-connected subset of x which is both α(γ,γ′ )-open and α(γ,γ′ )-closed. by theorem 3.10, a is contained in exactly one α(γ,γ′ )-component c of x. if a is a proper subset of c, then c = (c∩a)∪(c∩(x\a)) and (c∩a), (c∩(x\a)) is an α(γ,γ′ )-disconnection of c, which is a contradiction. thus, a = c. � theorem 3.12. every α(γ,γ′ )-component of x is α(γ,γ′ )-closed. proof. suppose that c is an α(γ,γ′ )-component of x. then, by remark 3.6, α(γ,γ′ )-cl(c) is α(γ,γ′ )-connected containing α(γ,γ′ )-component c of x. this implies that c = α(γ,γ′ )-cl(c) and hence c is α(γ,γ′ )-closed. � int. j. anal. appl. 16 (4) (2018) 527 references [1] h. z. ibrahim, on a class of α (γ,γ ′ ) -open sets in a topological space, acta sci., technol., 35 (3) (2013), 539-545. [2] h. z. ibrahim, on α (γ,γ ′ ) -open sets in topological spaces, (submitted). [3] h. z. ibrahim, on (α (γ,γ ′ ) , α (β,β ′ ) )-functions, (submitted). [4] s. mishra, on α-τ-disconnectedness and α-τ-connectedness in topological spaces, acta sci., technol., 37 (3) (2015), 395399. [5] o. njastad, on some classes of nearly open sets, pac. j. math. 15 (1965), 961-970. 1. introduction 2. preliminaries 3. (, ')-connected and (, ')-disconnected sets references international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 54-67 http://www.etamaths.com existence of positive solutions for a coupled system of (p,q)-laplacian fractional higher order boundary value problems k. r. prasad1, b. m. b. krushna2,∗ and l. t. wesen1,3 abstract. in this paper, we establish the existence of at least three positive solutions for a system of (p, q)-laplacian fractional order two-point boundary value problems by applying five functionals fixed point theorem under suitable conditions on a cone in a banach space. 1. introduction in the universe, many real world problems can be formulated as mathematical models to analyze the situations and to predict future. most of these models involve the rate of change of the dependent variable which leads to formation of the differential equations. one goal of differential equations is to understand the phenomena of nature by developing mathematical models. fractional calculus is an extension of classical calculus and deals with the generalization of integration and differentiation to an arbitrary real order. a class of differential equations governed by nonlinear differential operators appears frequently and generated by great deal of interest in studying special types of problems. in this theory, the most applicable operator is the classical p-laplacian operator. these types of problems arise in mathematical modeling of viscoelastic flows, turbulent filtration in porous media, biophysics, plasma physics and chemical reaction design. for a detailed description on applications of p-laplacian operator, we refer [10]. the positivity of boundary value problems associated with ordinary differential equations were studied by many authors [14, 1, 2] and extended to p-laplacian boundary value problems [4, 22, 8]. later these results are further extended to fractional order boundary value problems [6, 5, 9, 21, 12, 19] by utilizing various fixed point theorems on cones. recently researchers are concentrating on the theory of fractional order boundary value problems associated with p-laplacian operator. yang and yan [22] studied the existence of positive solutions for third order sturm–liouville boundary value problems with p-laplacian operator by applying the fixed point index method. chai [7] obtained the existence and multiplicity of positive solutions for a class of p-laplacian fractional order boundary value problems by means of the fixed point theorem. prasad and krushna [18, 20] derived sufficient conditions for the existence of positive solutions to p-laplacian fractional order boundary value problems. 2010 mathematics subject classification. 34a08, 34b18, 35j05. key words and phrases. fractional order derivative; (p, q)-laplacian; boundary value problem; green’s function, positive solution. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 54 existence of positive solutions for (p,q)-laplacian fbvps 55 motivated by the papers mentioned above, in this paper, we are concerned with establishing the existence of positive solutions for a coupled system of (p,q)-laplacian fractional order differential equations (1.1) d β1 0+ ( φp ( dα1 0+ x(t) )) = f1 ( t,x(t),y(t) ) , t ∈ (0, 1), (1.2) d β2 0+ ( φq ( dα2 0+ y(t) )) = f2 ( t,x(t),y(t) ) , t ∈ (0, 1), satisfying the boundary conditions (1.3) x(j)(0) = 0, j = 0, 1, · · ·,n− 2, x(n−2)(1) = 0, φp ( dα1 0+ x(0) ) = 0, φp ( dα1 0+ x(1) ) = 0,   (1.4) y(j)(0) = 0, j = 0, 1, · · ·,n− 2, y(n−2)(1) = 0, φq ( dα2 0+ y(0) ) = 0, φq ( dα2 0+ y(1) ) = 0,   where αi ∈ (n−1,n],n ≥ 3, βi ∈ (1, 2], φp(s) = |s|p−2s, φq(s) = |s|q−2s, p,q > 1, φ−1p = φq, φ−1q = φp, 1 p + 1 q = 1, fi : [0, 1] × r 2 → r+ are continuous and dαi 0+ ,d βi 0+ , for i = 1, 2 are the standard riemann–liouville fractional order derivatives. by a positive solution of the coupled system of fractional order boundary value problem (1.1)-(1.4), we mean ( x(t),y(t) ) ∈ ( cα1+β1 [0, 1] × cα2+β2 [0, 1] ) satisfying the boundary value problem (1.1)-(1.4) with x(t) ≥ 0, y(t) ≥ 0, for all t ∈ [0, 1] and (x,y) 6= (0, 0). the rest of the paper is organized as follows. in section 2, the solution of the boundary value problems (1.1), (1.3) and (1.2), (1.4) are expressed in terms of green functions and the bounds for these green functions are estimated. in section 3, the existence of at least three positive solutions for a coupled system of (p,q)-laplacian fractional order boundary value problem (1.1)-(1.4) are established, by using five functionals fixed point theorem. in section 4, as an application, the results are demonstrated with an example. 2. green functions and bounds in this section, the solution of the boundary value problems (1.1), (1.3) and (1.2), (1.4) are expressed in terms of the equivalent integral equations involving green functions and the bounds for the green functions are estimated, which are essential to establish the main results. lemma 2.1. let h1(t) ∈ c[0, 1]. then the fractional order differential equation, (2.1) dα1 0+ x(t) + h1(t) = 0, t ∈ (0, 1), satisfying (2.2) x(j)(0) = 0, j = 0, 1, · · ·,n− 2, x(n−2)(1) = 0, has a unique solution, x(t) = ∫ 1 0 g1(t,s)h1(s)ds, where g1(t,s) is the green’s function for the problem (2.1)-(2.2) and is given by (2.3) g1(t,s) = { g11(t,s), 0 ≤ t ≤ s ≤ 1, g12(t,s), 0 ≤ s ≤ t ≤ 1, 56 prasad, krushna and wesen g11(t,s) = tα1−1(1 −s)α1−n+1 γ(α1) , g12(t,s) = tα1−1(1 −s)α1−n+1 − (t−s)α1−1 γ(α1) . for details refer to [19]. lemma 2.2. let h2(t) ∈ c[0, 1]. then the fractional order differential equation, (2.4) d β1 0+ ( φp ( dα1 0+ x(t) )) = h2(t), t ∈ (0, 1), satisfying (2.5) φp ( dα1 0+ x(0) ) = 0, φp ( dα1 0+ x(1) ) = 0, has a unique solution, (2.6) x(t) = ∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)h2(τ)dτ ) ds, where (2.7) h1(t,s) =   [ t(1 −s) ]β1−1 γ(β1) , 0 ≤ t ≤ s ≤ 1,[ t(1 −s) ]β1−1 − (t−s)β1−1 γ(β1) , 0 ≤ s ≤ t ≤ 1. proof. an equivalent integral equation for (2.4) is given by φp ( dα1 0+ x(t) ) = 1 γ(β1) ∫ t 0 (t− τ)β1−1h2(τ)dτ + k1tβ1−1 + k2tβ1−2. from (2.5), one gets that k2 = 0 and k1 = −1 γ(β1) ∫ 1 0 (1 − τ)β1−1h2(τ)dτ. then, φp ( dα1 0+ x(t) ) = 1 γ(β1) ∫ t 0 (t− τ)β1−1h2(τ)dτ − tβ1−1 γ(β1) ∫ 1 0 (1 − τ)β1−1h2(τ)dτ = − ∫ 1 0 h1(t,τ)h2(τ)dτ. therefore, dα1 0+ x(t) + φq (∫ 1 0 h1(t,τ)h2(τ)dτ ) = 0. hence x(t) in (2.6) is the solution to the fractional order boundary value problem (2.4), (1.3). � lemma 2.3. the green’s function g1(t,s) given in (2.3) is nonnegative, for all (t,s) ∈ [0, 1] × [0, 1]. proof. consider the green’s function g1(t,s) given by (2.3). let 0 ≤ t ≤ s ≤ 1. then, we have g11(t,s) = 1 γ(α1) [ tα1−1(1 −s)α1−n+1 ] ≥ 0. existence of positive solutions for (p,q)-laplacian fbvps 57 let 0 ≤ s ≤ t ≤ 1. then, we have g12(t,s) = 1 γ(α1) [ tα1−1(1 −s)α1−n+1 − (t−s)α1−1 ] ≥ 1 γ(α1) [ tα1−1(1 −s)α1−n+1 − (t− ts)α1−1 ] = tα1−1 γ(α1) [( 1 + (n− 2)s + 1 2 (n2 − 3n + 2)s2 + · · · ) − 1 ] (1 −s)α1−1 ≥0. � lemma 2.4. for t ∈ i = [ 1 4 , 3 4 ] , the green’s function g1(t,s) given in (2.3) satisfies the following inequalities (p1) g1(t,s) ≤ g1(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (p2) g1(t,s) ≥ (1 4 )α1−1 g1(1,s), for all (t,s) ∈ i × [0, 1]. proof. consider the green’s function g1(t,s) given by (2.3). let 0 ≤ t ≤ s ≤ 1. then, we have ∂g11(t,s) ∂t = 1 γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 ] ≥ 0. therefore, g11(t,s) is increasing in t, which implies g11(t,s) ≤ g11(1,s). let 0 ≤ s ≤ t ≤ 1. then, we have ∂g12(t,s) ∂t = 1 γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 − (α1 − 1)(t−s)α1−2 ] ≥ 1 γ(α1) [ (α1 − 1)tα1−2(1 −s)α1−n+1 − (α1 − 1)(t− ts)α1−2 ] ≥ tα1−2 γ(α1 − 1) [ 1 − ( 1 − (n− 3)s + 1 2 (n2 − 7n + 12)s2 + · · · )] (1 −s)α1−n+1 ≥ 0. therefore, g12(t,s) is increasing in t, which implies g12(t,s) ≤ g12(1,s). let 0 ≤ t ≤ s ≤ 1 and t ∈ i. then g11(t,s) = 1 γ(α1) [ tα1−1(1 −s)α1−n+1 ] ≥tα1−1 1 γ(α1) [ (1 −s)α1−n+1 − (1 −s)α1−1 ] =tα1−1g11(1,s) ≥ (1 4 )α1−1 g11(1,s). 58 prasad, krushna and wesen let 0 ≤ s ≤ t ≤ 1 and t ∈ i. then g12(t,s) = 1 γ(α1) [ tα1−1(1 −s)α1−n+1 − (t−s)α1−1 ] ≥ 1 γ(α1) [ tα1−1(1 −s)α1−n+1 − (t− ts)α1−1 ] =tα1−1g12(1,s) ≥ (1 4 )α1−1 g12(1,s). � lemma 2.5. for t,s ∈ [0, 1], the green’s function h1(t,s) given in (2.7) satisfies the following inequalities (q1) h1(t,s) ≥ 0, (q2) h1(t,s) ≤ h1(s,s). for details refer to [20]. lemma 2.6. let ξ1 ∈ ( 14, 3 4 ). then the green’s function h1(t,s) holds the inequality, (2.8) min t∈i h1(t,s) ≥ ϑ∗1(s)h1(s,s), for 0 < s < 1, where ϑ∗1(s) =   [ 3 4 (1 −s)]β1−1 − ( 3 4 −s)β1−1 [s(1 −s)]β1−1 , s ∈ (0,ξ1], 1 (4s)β1−1 , s ∈ [ξ1, 1). for details refer to [20]. lemma 2.7. let g1(t) ∈ c[0, 1], then the fractional order differential equation, (2.9) dα2 0+ y(t) + g1(t) = 0, t ∈ (0, 1), satisfying (2.10) y(j)(0) = 0, j = 0, 1, · · ·,n− 2, y(n−2)(1) = 0, has a unique solution, y(t) = ∫ 1 0 g2(t,s)g1(s)ds, where g2(t,s) is the green’s function for the problem (2.9)-(2.10) and is given by (2.11) g2(t,s) = { g21(t,s), 0 ≤ t ≤ s ≤ 1, g22(t,s), 0 ≤ s ≤ t ≤ 1, g21(t,s) = tα2−1(1 −s)α2−n+1 γ(α2) , g22(t,s) = tα2−1(1 −s)α2−n+1 − (t−s)α2−1 γ(α2) . for details refer to [19]. existence of positive solutions for (p,q)-laplacian fbvps 59 lemma 2.8. let g2(t) ∈ c[0, 1]. then the fractional order differential equation, (2.12) d β2 0+ ( φq ( dα2 0+ y(t) )) = g2(t), t ∈ (0, 1), satisfying (2.13) φq ( dα2 0+ y(0) ) = 0, φq ( dα2 0+ y(1) ) = 0, has a unique solution, y(t) = ∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)g2(τ)dτ ) ds, where (2.14) h2(t,s) =   [ t(1 −s) ]β2−1 γ(β2) , 0 ≤ t ≤ s ≤ 1,[ t(1 −s) ]β2−1 − (t−s)β2−1 γ(β2) , 0 ≤ s ≤ t ≤ 1. proof. proof is similar to lemma 2.2. � lemma 2.9. the green’s function g2(t,s) given in (2.11) is nonnegative, for all (t,s) ∈ [0, 1] × [0, 1]. proof. proof is similar to lemma 2.3. � lemma 2.10. for i = [ 1 4 , 3 4 ] , the green’s function g2(t,s) given in (2.11) satisfies the following inequalities (c1) g2(t,s) ≤ g2(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (c2) g2(t,s) ≥ (1 4 )α2−1 g2(1,s), for all (t,s) ∈ i × [0, 1]. proof. proof is similar to lemma 2.4. � lemma 2.11. for t,s ∈ [0, 1], the green’s function h2(t,s) given in (2.14) satisfies the following inequalities (d1) h2(t,s) ≥ 0, (d2) h2(t,s) ≤ h2(s,s). for details refer to [20]. lemma 2.12. let ξ2 ∈ ( 14, 3 4 ). then the green’s function h2(t,s) holds the inequality, (2.15) min t∈i h2(t,s) ≥ ϑ∗2(s)h2(s,s), for 0 < s < 1, where (2.16) ϑ∗2(s) =   [ 3 4 (1 −s)]β2−1 − ( 3 4 −s)β2−1 [s(1 −s)]β2−1 , s ∈ (0,ξ2], 1 (4s)β2−1 , s ∈ [ξ2, 1). for details refer to [20]. 60 prasad, krushna and wesen 3. existence of positive solutions in this section, we establish sufficient conditions for the existence of at least three positive solutions for a system of (p,q)-laplacian fractional order boundary value problem (1.1)-(1.4), by using five functionals fixed point theorem. let γ,β,θ be nonnegative continuous convex functionals on p and α,ψ be nonnegative continuous concave functionals on p , then for nonnegative numbers h′,a′,b′,d′ and c′, convex sets are defined. p(γ,c′) = { y ∈ p : γ(y) < c′ } , p(γ,α,a′,c′) = { y ∈ p : a′ ≤ α(y); γ(y) ≤ c′ } , q(γ,β,d′,c′) = { y ∈ p : β(y) ≤ d′; γ(y) ≤ c′ } , p(γ,θ,α,a′,b′,c′) = { y ∈ p : a′ ≤ α(y); θ(y) ≤ b′; γ(y) ≤ c′ } , q(γ,β,ψ,h′,d′,c′) = { y ∈ p : h′ ≤ ψ(y); β(y) ≤ d′; γ(y) ≤ c′ } . in establishing the positive solutions for a coupled system of (p,q)-laplacian fractional order boundary value problem (1.1)-(1.4), the following so called five functionals fixed point theorem is fundamental. theorem 3.1. [3] let p be a cone in the real banach space b. suppose α and ψ are nonnegative continuous concave functionals on p and γ,β,θ are nonnegative continuous convex functionals on p, such that for some positive numbers c′ and e′, α(y) ≤ β(y) and ‖ y ‖≤ e′γ(y), for all y ∈ p(γ,c′). suppose further that t : p(γ,c′) → p(γ,c′) is completely continuous and there exist constants h′,d′,a′ and b′ ≥ 0 with 0 < d′ < a′ such that each of the following is satisfied. (b1) { y ∈ p(γ,θ,α,a′,b′,c′) : α(y) > a′ } 6= ∅ and α(ty) > a′ for y ∈ p(γ,θ,α,a′,b′,c′), (b2) { y ∈ q(γ,β,ψ,h′,d′,c′) : β(y) > d′ } 6= ∅ and β(ty) > d′ for y ∈ q(γ,β,ψ,h′,d′,c′), (b3) α(ty) > a′ provided y ∈ p(γ,α,a′,c′) with θ(ty) > b′, (b4) β(ty) < d′ provided y ∈ q(γ,β,ψ,h′,d′,c′) with ψ(ty) < h′. then t has at least three fixed points y1,y2,y3 ∈ p(γ,c′) such that β(y1) < d′,a′ < α(y2) and d′ < β(y3) with α(y3) < a ′. consider the banach space b = e ×e, where e = { x : x ∈ c[0, 1] } equipped with the norm ‖(x,y)‖ = ‖x‖0 + ‖y‖0, for (x,y) ∈ b and the norm is defined as ‖x‖0 = max t∈[0,1] |x(t)|. define a cone p ⊂ b by p = { (x,y) ∈ b : x(t) ≥ 0, y(t) ≥ 0, t ∈ [0, 1] and min t∈i [ x(t) + y(t) ] ≥ η‖(x,y)‖ } , where (3.1) η = min {(1 4 )α1−1 , (1 4 )α2−1} . existence of positive solutions for (p,q)-laplacian fbvps 61 define the nonnegative continuous concave functionals α,ψ and the nonnegative continuous convex functionals β,θ,γ on p by α(x,y) = min t∈i { |x| + |y| } ,ψ(x,y) = min t∈i1 { |x| + |y| } , γ(x,y) = max t∈[0,1] { |x| + |y| } ,β(x,y) = max t∈i1 { |x| + |y| } ,θ(x,y) = max t∈i { |x| + |y| } , where i1 = [ 1 3 , 2 3 ] . for any (x,y) ∈ p , (3.2) α(x,y) = min t∈i { |x| + |y| } ≤ max t∈i1 { |x| + |y| } = β(x,y), (3.3) ‖(x,y)‖≤ 1 η min t∈i { |x| + |y| } ≤ 1 η max t∈[0,1] { |x| + |y| } = 1 η γ(x,y). let (3.4) ϑ∗(s) = min { ϑ∗1(s),ϑ ∗ 2(s) } . define l = min { 1∫ 1 0 g1(1,s)φq (∫ 1 0 h1(τ,τ)dτ ) ds , 1∫ 1 0 g2(1,s)φp (∫ 1 0 h2(τ,τ)dτ ) ds } , and m = max { 1∫ s∈i ηg1(1,s)φq (∫ τ∈i ϑ ∗(τ)h1(τ,τ)dτ ) ds , 1∫ s∈i ηg2(1,s)φp (∫ τ∈i ϑ ∗(τ)h2(τ,τ)dτ ) ds } . theorem 3.2. suppose there exist 0 < a′ < b′ < b′ η < c′ such that f1,f2 satisfies the following conditions: (a1) { f1 ( t,x(t),y(t) ) < φp (a′l 2 ) and f2 ( t,x(t),y(t) ) < φq (a′l 2 ) , t ∈ [0, 1] and x,y ∈ [ ηa′,a′ ] , (a2)   f1 ( t,x(t),y(t) ) > φp (b′m 2 ) and f2 ( t,x(t),y(t) ) > φq (b′m 2 ) , t ∈ i and x,y ∈ [ b′, b′ η ] , (a3)   f1 ( t,x(t),y(t) ) < φp (c′l 2 ) and f2 ( t,x(t),y(t) ) < φq (c′l 2 ) , t ∈ [0, 1] and x,y ∈ [ 0,c′ ] . then the (p,q)-laplacian fractional order boundary value problem (1.1)-(1.4) has at least three positive solutions, (x1,x2), (y1,y2) and (z1,z2) such that β(x1,x2) < a ′, b′ < α(y1,y2) and a ′ < β(z1,z2) with α(z1,z2) < b ′. 62 prasad, krushna and wesen proof. let t1,t2 : p → e and t : p → b be the operators defined by  t1(x,y)(t) = ∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds, t2(x,y)(t) = ∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds, and t(x,y)(t) = ( t1(x,y)(t),t2(x,y)(t) ) , for (x,y) ∈ b. it is obvious that a fixed point of t is the solution of the fractional order boundary value problem (1.1)-(1.4). three fixed points of t are sought. first, it is shown that t : p → p . let (x,y) ∈ p . clearly, t1 ( x,y ) (t) ≥ 0 and t2 ( x,y ) (t) ≥ 0, for t ∈ [0, 1]. also for (x,y) ∈ p,  ‖t1(x,y)‖0 ≤ ∫ 1 0 g1(1,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds, ‖t2(x,y)‖0 ≤ ∫ 1 0 g2(1,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds, and min t∈i t1(x,y)(t) = min t∈i ∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds ≥ η ∫ 1 0 g1(1,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds ≥ η‖t1(x,y)‖0. similarly, min t∈i t2 ( x,y ) (t) ≥ η‖t2 ( x,y ) ‖0. therefore, min t∈i { t1(x,y)(t) + t2(x,y)(t) } ≥ η‖t1(x,y)‖0 + η‖t2(x,y)‖0 = η ( ‖t1(x,y)‖0 + ‖t2(x,y)‖0 ) = η ∥∥∥(t1(x,y),t2(x,y))∥∥∥ = η‖t(x,y)‖. hence t(x,y) ∈ p and so t : p → p. moreover the operator t is completely continuous. from (3.2) and (3.3), for each (x,y) ∈ p, α(x,y) ≤ β(x,y) and ‖(x,y)‖ ≤ 1 η γ(x,y). it is shown that t : p(γ,c′) → p(γ,c′). let (x,y) ∈ p(γ,c′). then 0 ≤ |x| + |y| ≤ c′. the condition (a3) is used to obtain γ ( t(x,y)(t) ) = max t∈[0,1] [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ ∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)φp (c′l 2 ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)φq (c′l 2 ) dτ ) ds existence of positive solutions for (p,q)-laplacian fbvps 63 < c′l 2 ∫ 1 0 g1(1,s)φq (∫ 1 0 h1(τ,τ)dτ ) ds+ c′l 2 ∫ 1 0 g2(1,s)φp (∫ 1 0 h2(τ,τ)dτ ) ds < c′ 2 + c′ 2 = c′. therefore t : p(γ,c′) → p(γ,c′). now the conditions (b1) and (b2) of theorem 3.1 are to be verified. it is obvious that b′ ( η + 1 ) 2η ∈ { (x,y) ∈ p ( γ,θ,α,b′, b′ η ,c′ ) : α(x,y) > b′ } 6= ∅ and ηa′ + a′ 2 ∈ { (x,y) ∈ q ( γ,β,ψ,ηa′,a′,c′ ) : β(x,y) < a′ } 6= ∅. next, let (x,y) ∈ p ( γ,θ,α,b′, b′ η ,c′ ) or (x,y) ∈ q ( γ,β,ψ,ηa′,a′,c′ ) . then, b′ ≤ { |x(t)| + |y(t)| } ≤ b′ η and ηa′ ≤ { |x(t)| + |y(t)| } ≤ a′. now the condition (a2) is applied to get α ( t(x,y)(t) ) = min t∈i [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ))dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η [∫ 1 0 g1(1,s)φq (∫ 1 0 ϑ∗(τ)h1(τ,τ)φp (b′m 2 ) dτ ) ds+∫ 1 0 g2(1,s)φp (∫ 1 0 ϑ∗(τ)h2(τ,τ)φq (b′m 2 ) dτ ) ds ] > b′m 2 ∫ s∈i ηg1(1,s)φq (∫ τ∈i ϑ∗(τ)h1(τ,τ)dτ ) ds+ b′m 2 ∫ s∈i ηg2(1,s)φp (∫ τ∈i ϑ∗(τ)h2(τ,τ)dτ ) ds ≥ b′ 2 + b′ 2 = b′. clearly the condition (a1) leads to β ( t(x,y)(t) ) = max t∈i1 [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ ∫ 1 0 g1(1,s)φq (∫ 1 0 h1(s,τ)φp (a′l 2 ) dτ ) ds+∫ 1 0 g2(1,s)φp (∫ 1 0 h2(s,τ)φq (a′l 2 ) dτ ) ds 64 prasad, krushna and wesen < a′l 2 ∫ 1 0 g1(1,s)φq (∫ 1 0 h1(τ,τ)dτ ) ds+ a′l 2 ∫ 1 0 g2(1,s)φp (∫ 1 0 h2(τ,τ)dτ ) ds ≤ a′ 2 + a′ 2 = a′. to see that (b3) is satisfied, let (x,y) ∈ p ( γ,α,b′,c′ ) with θ ( t(x,y)(t) ) > b′ η . then α ( t(x,y)(t) ) = min t∈i [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η [∫ 1 0 g1(1,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(1,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η max t∈[0,1] [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≥ η max t∈i [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φq (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] = ηθ ( t(x,y)(t) ) > b′. finally it is shown that (b4) holds. let (x,y) ∈ q ( γ,β,a′,c′ ) with ψ ( t(x,y) ) < ηa′. then we have β ( t(x,y)(t) ) = max t∈i1 [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] existence of positive solutions for (p,q)-laplacian fbvps 65 ≤ max t∈[0,1] [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ 1 η min t∈i [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] ≤ 1 η min t∈i1 [∫ 1 0 g1(t,s)φq (∫ 1 0 h1(s,τ)f1 ( τ,x(τ),y(τ) ) dτ ) ds+∫ 1 0 g2(t,s)φp (∫ 1 0 h2(s,τ)f2 ( τ,x(τ),y(τ) ) dτ ) ds ] = 1 η ψ ( t(x,y)(t) ) < a′. it is been proved that all the conditions of theorem 3.1 are satisfied. therefore the system of (p,q)-laplacian fractional order boundary value problem (1.1)-(1.4) has at least three positive solutions (x1,x2), (y1,y2) and (z1,z2) such that β(x1,x2) < a ′,b′ < α(y1,y2) and a′ < β(z1,z2) with α(z1,z2) < b ′. this completes the proof. � 4. example in this section, as an application, the results are demonstrated with an example. consider the system of (p,q)-laplacian fractional order differential equations (4.1) d1.80+ ( φp ( d3.80+ x(t) )) = f1(t,x,y), t ∈ (0, 1), (4.2) d1.70+ ( φq ( d3.90+ y(t) )) = f2(t,x,y), t ∈ (0, 1), satisfying the boundary conditions (4.3) x(0) = x′(0) = x′′(0) = 0 and x′′(1) = 0, φp ( d3.8 0+ x(0) ) = φp ( d3.8 0+ x(1) ) = 0, } (4.4) y(0) = y′(0) = y′′(0) = 0 and y′′(1) = 0, φq ( d3.9 0+ y(0) ) = φq ( d3.9 0+ y(1) ) = 0, } where f1(t,x,y) =   e2t 57 + cos(x + y) 9 + 11(x + y)3 9 , 0 ≤ x + y ≤ 4, cos(x + y) 9 + e2t 57 + 5632 9 , x + y > 4, f2(t,x,y) =   sin(x + y) 9 + 11(x + y)3 9 + e2t 56 , 0 ≤ x + y ≤ 4, e2t 56 + sin(x + y) 9 + 5632 9 , x + y > 4. 66 prasad, krushna and wesen clearly fi, for i = 1, 2 are continuous and increasing on [0,∞). let p = 2. by direct calculations, one can determine η = 0.0205, l = 27.9632 and m = 314.5214. choosing a′ = 0.5,b′ = 4 and c′ = 200, then 0 < a′ < b′ < b′ η < c′ and f1,f2 satisfies (a) { f1 ( t,x,y ) < 0.5091 = φp (a′l 2 ) and f2 ( t,x,y ) < 0.5091 = φq (a′l 2 ) , t ∈ [0, 1] and x,y ∈ [ ηa′,a′ ] = [0.0103, 0.05], (b)   f1 ( t,x,y ) > 1356 = φp (b′m 2 ) and f2 ( t,x,y ) > 1356 = φq (b′m 2 ) , t ∈ i = [0.25, 0.75] and x,y ∈ [ b′, b′ η ] = [4, 195.12], (c)   f1 ( t,x,y ) < 2796.32 = φp (c′l 2 ) and f2 ( t,x,y ) < 2796.32 = φq (c′l 2 ) , t ∈ [0, 1] and x,y ∈ [ 0,c′ ] = [0, 200]. then all the conditions of theorem 3.2 are satisfied. thus by theorem 3.2, the (p,q)laplacian fractional order boundary value problem (4.1)-(4.4) has at least three positive solutions. references [1] r. p. agarwal, d. o’regan and p. j. y. wong, positive solutions of differential, difference and integral equations, kluwer academic publishers, dordrecht, the netherlands, 1999. [2] d. r. anderson and j. m. davis, multiple positive solutions and eigenvalues for third order right focal boundary value problems, j. math. anal. appl., 267(2002), 135–157. [3] r. i. avery, a generalization of the leggett-williams fixed point theorem, math. sci. res. hot-line, 3(1999), 9–14. [4] r. i. avery and j. henderson, existence of three positive pseudo-symmetric solutions for a onedimensional p-laplacian, j. math. anal. appl., 277(2003), 395–404. [5] c. bai, existence of positive solutions for boundary value problems of fractional functional differential equations, elec. j. qual. theory diff. equ., 30(2010), 1–14. [6] z. bai and h. lü, positive solutions for boundary value problem of nonlinear fractional differential equation, j. math. anal. appl., 311(2005), 495-505. [7] g. chai, positive solutions for boundary value problem of fractional differential equation with plaplacian operator, bound. value probl., 2012(2012), 1–18. [8] t. chen and w. liu, an anti-periodic boundary value problem for the fractional differential equation with a p-laplacian operator, appl. math. lett., 25(2012), 1671–1675. [9] r. dehghani and k. ghanbari, triple positive solutions for boundary value problem of a nonlinear fractional differential equation, bulletin of the iranian mathematical society, 33(2007), 1–14. [10] l. diening, p. lindqvist and b. kawohl, mini-workshop: the p-laplacian operator and applications, oberwolfach reports, 10(2013) 433–482. [11] l. h. erbe and h. wang, on the existence of positive solutions of ordinary differential equations, proc. amer. math. soc., 120(1994), 743–748. [12] c. goodrich, existence of a positive solution to systems of differential equations of fractional order, comput. math. appl., 62(2011), 1251–1268. [13] d. guo, v. lakshmikantham, nonlinear problems in abstract cones, acadamic press, san diego, 1988. [14] j. henderson and s. k. ntouyas, positive solutions for systems of nonlinear boundary value problems, nonlinear stud., 15(2008), 51-60. [15] a. a. kilbas, h. m. srivasthava and j. j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies, vol. 204, elsevier science, amsterdam, 2006. [16] l. kong and j. wang, multiple positive solutions for the one-dimensional p-laplacian, nonlinear anal., 42(2000), 1327–1333. [17] i. podulbny, fractional diffrential equations, academic press, san diego, 1999. existence of positive solutions for (p,q)-laplacian fbvps 67 [18] k. r. prasad and b. m. b. krushna, multiple positive solutions for a coupled system of p-laplacian fractional order two-point boundary value problems, int. j. differ. equ., 2014(2014), article id 485647, 1–10. [19] k. r. prasad and b. m. b. krushna, multiple positive solutions for the system of (n, p)-type fractional order boundary value problems, bull. int. math. virtual inst., 5(2015), 1–12. [20] k. r. prasad and b. m. b. krushna, solvability of p-laplacian fractional higher order two-point boundary value problems, commun. appl. anal., 19(2015), 659–678. [21] x. su, boundary value problem for a coupled system of nonlinear fractional differential equations, appl. math. lett., 22(2009), 64–69. [22] c. yang, j. yan, positive solutions for third order sturm–liouville boundary value problems with p-laplacian, comput. math. appl., 59(2010), 2059–2066. 1department of applied mathematics, andhra university, visakhapatnam, 530 003, india 2department of mathematics, mvgr college of engineering, vizianagaram, 535 005, india 3department of mathematics, jimma university, jimma, oromia, 378, ethiopia ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 33-44 http://www.etamaths.com existence and convergence of best proximity points for semi cyclic contraction pairs balwant singh thakur∗, ajay sharma abstract. in this article, we introduce the notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass. existence and convergence results of best proximity points for semi cyclic ϕcontraction pair of mappings are obtained. 1. introduction as it is well known, fixed point theory is an indispensable tool for solving various equations involving self-mappings, defined on subsets of a metric space or a normed linear space. nevertheless, when the mapping t (say) is a non-self one, then it is possible that the equation tx = x has no solution and, in this case, we have to focus the study on the problem of finding an element x which is in the closest proximity to tx in some sense; in such circumstances, it may be speculated to determine an element x for which the ”distance error” d(x,tx) is minimum. let a and b be nonempty subsets of a metric space (x,d) and t a mapping from a to b. since d(x,tx) is greater than or equals to the distance between a and b for all x in a, a best proximity theorem offers sufficient conditions for the existence of an element x, called a best proximity point of the mapping t, satisfying the condition that d(x,tx) = dist(a,b). also, it is interesting to see that best proximity point theorems emerge as a natural generalization of fixed point theorems, because a best proximity point reduces to a fixed point if the mapping under consideration turns out to be a self-mapping. now, let us consider s and t be given non-self mappings from a to b, where a and b are nonempty subsets of a metric space. as s and t are non-self mappings, the equations sx = x and tx = x do not necessarily have a common solution, called a common fixed point of the mappings s and t. therefore, in such cases of non-existence of a common fixed points, it is attempted to find a point x that is closest to both sx and tx in some sense. common best proximity theorems, explore the existence of such optimal solutions, known as best proximity point. in view of the fact that, for any element x in a, the distance between x and sx, and the distance between x and tx are at least the distance between the sets a and b, a common best proximity theorem states that, under certain conditions, there exists a point x satisfying d(x,sx) = d(x,tx) = dist(a,b). 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. best proximity point, fixed point, semi cyclic ϕ-contraction, metric space, banach space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 33 34 balwant singh thakur∗, ajay sharma in their elegant paper [1], kirk et al. introduced the notion of cyclical contractive mapping, and proved fixed point results for this class of mappings, while in [2], eldred and veeramani studied existence and convergence results of best proximity points in a more general case. in [3], thagafi and shahzad introduced a new class of mappings known as cyclic ϕ-contraction, and proved their convergence and existence results for best proximity points. recently, gabeleh and abkar [4], introduced results on best proximity points for semi-cylic contractive pairs in banach spaces. for other recent results on this topic, see chandok and postolache [5], shatanawi and postolache [6]. this paper aims to develop further these results. in this respect, we introduce the new notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass. existence and convergence results of best proximity points for semi cyclic ϕ-contraction pair of mappings, in the framework of a uniformly convex banach space [7], are obtained. our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. in particular, our results reduce to those of gabeleh and abkar [4]. 2. preliminaries let a and b be nonempty subsets of a metric space (x,d) and let t : a∪b → a ∪ b such that t(a) ⊆ b and t(b) ⊆ a. the mapping t is said to be cyclic contraction [1] if for some k ∈ (0, 1) we have d(tx,ty) ≤ k d(x,y), x ∈ a, y ∈ b. kirk et al. [1] proved that if a∩b 6= ø then the cyclic mapping t has a unique fixed point in a∩b. but what happens when a∩b is not necessarily nonempty. in this situation, a mapping t is said to be cyclic contraction [2] if for some k ∈ (0, 1) we have d(tx,ty) ≤ k d(x,y) + (1 −k)dist(a,b), for all x ∈ a and y ∈ b. eldred and veeramani [2] proved existence, uniqueness and convergence for best proximity points of cyclic contraction mapping t . thagafi and shahzad [3] introduced the following new class of mapping known as cyclic ϕcontraction: the mapping t is said to be cyclic ϕ-contraction if ϕ: [0, +∞) → [0, +∞) is strictly increasing mapping and d(tx,ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(dist(a,b)), for all x ∈ a and y ∈ b. we can see from the following example [3] that a cyclic ϕ-contraction mapping need not be a cyclic contraction. example 2.1. let x = r with usual metric. for a = b = [0, 1], define t : a∪b → a ∪ b by the formula tx = x 1+x . if ϕ(t) = t 2 1+t for t ≥ 0, then t is cyclic ϕcontraction mapping which is not a cyclic contraction. recently, gabeleh and akbar [4] introduced the concept of a semi-cyclic contraction pair: definition 2.1. let s,t be two self mappings on a∪b. the pair (s,t) is called a semi-cyclic contraction if the following conditions hold: existence and convergence of best proximity points 35 (i) s(a) ⊆ b, t(b) ⊆ a ; (ii) ∃α ∈ (0, 1), such that d(sx,ty) ≤ αd(x,y) + (1−α)dist(a,b) , x ∈ a, y ∈ b. clearly when s = t, a semi-cyclic contraction pair reduces to a cyclic contraction mapping, already studied by eldred and veeramani [2]. but there exist a semi-cyclic contraction pair which is not cyclic, the following example [4] illustrates it. example 2.2. let x = r2 and for all (x,y) ∈ r2 define ‖(x,y)‖ = max{|x|, |y|}. let a = { (x,y) ∈ r2 : 1 2 ≤ x ≤ 1, y = 0 } , b = { (x,y) ∈ r2 : x = 0 , 1 ≤ y ≤ 2 } . then a and b are closed and dist(a,b) = 1. define s,t : a∪b → a∪b by s(x,y) = { (0, 1), (x,y) ∈ a (x,y), (x,y) ∈ b, t(x,y) = { (x,y), (x,y) ∈ a (y/2, 0), (x,y) ∈ b. here s(a) ⊆ b and t(b) ⊆ a but s(b) * a and t(a) * b, hence neither s nor t is cyclic. on the other hand, if b = (0,y) ∈ b and a = (x, 0) ∈ a, then ‖tb−sa‖ = ‖t(0,y) −s(x, 0)‖ = ‖(y/2,−1)‖ = 1. similarly ‖a− b‖ = ‖(x,−y)‖ = max{x, |−y|} = y. therefore ‖tb−sa‖ = 1 ≤ 1 2 |y| + 1 2 < 1 2 ‖b−a‖ + 1 2 dist(a,b). hence (s,t) is a semi-cyclic contraction pair. we now introduce the following new class of semi-cyclic contraction pair. definition 2.2. let a and b be nonempty subsets of a metric space (x,d) and let s,t : a∪b → a∪b such that s(a) ⊆ b and t(b) ⊆ a. then (s,t) is said to be a semi-cyclic ϕ-contraction if ϕ: [0, +∞) → [0, +∞) is strictly increasing mapping and (1) d(sx,ty) ≤ d(x,y) −ϕ (d(x,y)) + ϕ (dist(a,b)) , for all x ∈ a and y ∈ b. a semi-cyclic contraction pair is semi-cyclic ϕ-contraction pair with ϕ(t) = (1− α)t for t ≥ 0 and 0 < α < 1. we now give an example to illustrate that a semi-cyclic ϕ-contraction pair need not necessary a semi-cyclic contraction pair example 2.3. let x = r with the usual metric. let a = [1, 2], b = [−2,−1], then dist(a,b) = 2. define t,s : a∪b → a∪b by s(x) =   −1 −x 2 , x ∈ a −1 + x 2 , x ∈ b, t(x) =   1 + x 2 , x ∈ a 1 −x 2 , x ∈ b. clearly s(a) ⊆ b and t(b) ⊆ a. take a ∈ a, b ∈ b and ϕ(t) = t 2 1+8t for t ≥ 0, then (s,t) is a semi-cyclic ϕ-contraction pair. on the other hand, if a = 2 ∈ a, b = −2 ∈ b and α ∈ ( 0, 1 2 ) , then d(tb,sa) = 3 > α · 4 + (1 −α) · 2 = α dist(a,b) + (1 −α)dist(a,b). hence (s,t) is not a semi-cyclic contraction pair. 36 balwant singh thakur∗, ajay sharma 3. main results consider x0 ∈ a, then sx0 ∈ b, so there exists y0 ∈ b such that y0 = sx0. now ty0 ∈ a, so there exists x1 ∈ a such that x1 = ty0. inductively, we define sequences {xn} and {yn} in a and b, respectively by (2) xn+1 = tyn, yn = sxn. for all x ∈ a and y ∈ b, we have dist(a,b) ≤ d(x,y). since ϕ is a strictly increasing function, we deduce that ϕ(dist(a,b)) ≤ ϕ(d(x,y)). also (s,t) is semi cyclic ϕ-contraction pair, hence d(sx,ty) ≤ d(x,y) −ϕ (d(x,y)) + ϕ (dist(a,b)) ≤ d(x,y). by (2), we have d(xn,sxn) = d(tyn−1,sxn) ≤ d(yn−1,xn) = d(xn−1,sxn−1), and d(xn+1,yn) = d(tyn,sxn) ≤ d(yn,xn) = d(yn,tyn−1). also, d(yn+1,tyn) = d(sxn+1,tyn) ≤ d(xn+1,yn) = d(tyn,sxn) ≤ d(yn,xn) = d(yn,tyn−1) . we summarize these results in: lemma 3.1. let (x,d) be a metric space and let a,b be nonempty subsets of x. let s,t : a ∪ b → a ∪ b such that the pair (s,t) is semi-cyclic ϕ−contraction. for x0 ∈ a ∪ b the sequences {xn} and {yn} are generated by (2). then for all x ∈ a, y ∈ b, and n ≥ 1, we have (i) −ϕ (d(x,y)) + ϕ (dist(a,b)) ≤ 0, (ii) d(sx,ty) ≤ d(x,y), (iii) d (xn,sxn) ≤ d (xn−1,sxn−1), (iv) d (xn+1,yn) ≤ d (yn,tyn−1), (v) d (yn+1,tyn) ≤ d (yn,tyn−1) . we now state and prove the following result which will be needed in what follows. theorem 3.1. let (x,d) be a metric space and let a,b be nonempty subsets of x. let s,t : a∪b → a∪b such that the pair (s,t) is semi-cyclic ϕ-contraction. for x0 ∈ a∪b the sequences {xn} and {yn} are generated by (2). then d(xn,sxn) → dist(a,b) and d(yn,tyn−1) → dist(a,b). proof. let dn = d(xn,sxn). it follows from lemma 3.1(iii), that {dn} is decreasing and bounded, so limn→∞dn = t0, for some t0 ≥ dist(a,b). if t0 = dist(a,b) there is nothing to prove, so assume t0 > dist(a,b). since d(sx,ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(dist(a,b)), for all x ∈ a, y ∈ b, existence and convergence of best proximity points 37 we have dn+1 = d (xn+1,sxn+1) = d (tyn,sxn+1) ≤ d (yn,xn+1) = d (sxn,tyn) ≤ d (xn,yn) −ϕ (d (xn,yn)) + ϕ (dist(a,b)) = d (xn,sxn) −ϕ (d (xn,sxn)) + ϕ (dist(a,b)) = dn −ϕ (dn) + ϕ (dist(a,b)) . hence, ϕ (dist(a,b)) ≤ ϕ(dn) = dn −dn+1 + ϕ (dist(a,b)) . thus ϕ (dist(a,b)) ≤ lim n→∞ ϕ (dn) ≤ ϕ (dist(a,b)) , which shows that (3) lim n→∞ ϕ (dn) = ϕ(t0) = ϕ (dist(a,b)) . on the other hand, since ϕ is strictly increasing and dn ≥ t0 > dist(a,b), we have lim n→∞ ϕ (dn) ≥ ϕ(t0) > ϕ (dist(a,b)) , which contradicts to (3). consequently, t0 = dist(a,b). similarly, using lemma 3.1, it can be shown that d(yn,tyn−1) → dist(a,b). � remark 3.1. proposition 3.1 of [4] is a special case of theorem 3.1. theorem 3.2. let (x,d) be a metric space and let a,b be nonempty subsets of x. let s,t : a∪b → a∪b such that the pair (s,t) is semi-cyclic ϕ-contraction. for x0 ∈ a ∪ b the sequences {xn} and {yn} are generated by (2). if both {xn} and {yn} have a convergent subsequence in a and b respectively, then there exists x ∈ a and y ∈ b such that d(x,sx) = dist(a,b) = d(y,ty). proof. let {ynk} be a subsequence of {yn} such that ynk → y. since dist(a,b) ≤ d (tynk,y) ≤ d (ynk,y) + d (ynk,tynk ) , letting k →∞, by theorem 3.1, we have d (y,tynk ) → dist(a,b). now, for each k ≥ 1 dist(a,b) ≤ d (ty,ynk ) = d (ty,sxnk ) ≤ d (y,xnk ) ≤ d (y,ynk ) + d (ynk,xnk ) = d (y,ynk ) + d (sxnk,xnk ) , 38 balwant singh thakur∗, ajay sharma i.e., dist(a,b) ≤ d (y,ynk ) + d (sxnk,xnk ) , letting k →∞ we conclude that d(ty,y) = dist(a,b). similarly, it can be proved that d(x,sx) = dist(a,b). � remark 3.2. proposition 3.2 of [4] is a special case of theorem 3.2. theorem 3.3. let (x,d) be a metric space and let a,b be nonempty subsets of x. let s,t : a∪b → a∪b such that the pair (s,t) is semi-cyclic ϕ-contraction. then the sequences {xn} and {yn} generated by (2) are bounded. proof. by theorem 3.1 we have that d (xn,sxn) → dist(a,b) as n → ∞, so it is enough to prove that the sequence {sxn} is bounded. if not, then for each m > 0 there exists n ∈ n such that d(x1,sxn ) > m and d (x1,sxn−1) ≤ m. we obtain m < d (x1,sxn ) = d (ty0,sxn ) , furthermore, we have m < d (x1,sxn ) = d (ty0,sxn ) ≤ d (y0,xn ) = d (sx0,tyn−1) ≤ d (x0,yn−1) −ϕ (d (x0,yn−1)) + ϕ (dist (a,b)) ≤ d (x0,x1) + d (x1,sxn−1) −ϕ (d (x0,sxn−1)) + ϕ (dist (a,b)) , i.e. m < d (x0,x1) + m −ϕ (d (x0,sxn−1)) + ϕ (dist (a,b)) , so, ϕ (d (x0,sxn−1)) < d (x0,x1) + ϕ (dist (a,b)) . since, ϕ is unbounded function, we can choose m such that ϕ(m) > d(x0,x1) + ϕ (dist(a,b)) . now, m < d (x1,sxn ) ≤ d (y0,xn ) = d (sx0,tyn−1) ≤ d (x0,yn−1) = d (x0,sxn−1) . we deduce that ϕ(m) < ϕ (d (x1,sxn )) ≤ ϕ (d (x0,sxn−1)) < d (x0,x1) + ϕ (dist (a,b)) , a contradiction. similarly, we can prove boundedness of {yn} in b. � remark 3.3. proposition 3.3 of [4] is a special case of theorem 3.3. in 1936, clarkson [7] introduced the notion of uniform convexity of norm in a banach space. existence and convergence of best proximity points 39 definition 3.1. a banach space x is said to be uniformly convex if and only if given ε > 0, there exists δ(ε) > 0 such that (4) ‖x‖≤ 1 ‖y‖≤ 1 ‖x−y‖≥ ε   ⇒ ∥∥∥∥x + y2 ∥∥∥∥ ≤ 1 − δ(ε), where δ : [0, 2] → [0, 1] given by δ(ε) = inf { 1 − ∥∥∥∥x + y2 ∥∥∥∥ : ‖x‖≤ 1,‖y‖≤ 1,‖x−y‖≥ ε } . the function δ is known as modulus of convexity of a banach space x. the implication (4) has following more general form. for x,y,p ∈ x, r > 0 and r ∈ [0, 2r] ‖x−p‖≤ r ‖y −p‖≤ r ‖x−y‖≥ r   ⇒ ∥∥∥∥p− x + y2 ∥∥∥∥ ≤ (1 −δ( rr )) r. now, define a sequence {zn} in a∪b in the following manner: (5) zn = { tyk , n = 2k sxk , n = 2k − 1. lemma 3.2. let a and b be nonempty convex subsets of a uniformly convex banach space x and let s,t : a∪b → a∪b are semi-cyclic ϕ-contraction mappings such that t(a) ⊆ b and s(b) ⊆ a. for x0 ∈ a∪b, the sequences {xn} and {yn} generated by (2). the sequences {zn} is generated by (5), then ‖z2n+2 −z2n‖→ 0 and ‖z2n+3 −z2n+1‖→ 0 as n →∞. proof. to show ‖z2n+2 −z2n‖ → 0 as n → ∞, assume the contrary. then there exists ε0 > 0 such that for each k ≥ 1, there exists nk ≥ k so that (6) ‖z2nk+2 −z2nk‖≥ ε0 . choose ε > 0 so that ( 1 −δ ( ε0 dist(a,b)+ε )) (dist(a,b) + ε) < dist(a,b). by theorem 3.1, we have ‖z2nk+2 −z2nk+1‖ = ‖tynk+1 −sxnk+1‖ ≤‖ynk+1 −xnk+1‖ = ‖sxnk+1 −xnk+1‖→ dist(a,b), hence, there exists n1 such that (7) ‖z2nk+2 −z2nk+1‖≤ dist(a,b) + ε, ∀nk ≥ n1 . also, ‖z2nk −z2nk+1‖ = ‖tynk −sxnk+1‖ ≤‖ynk −xnk+1‖ = ‖ynk −tynk‖ = ‖sxnk −tynk‖ ≤‖xnk −ynk‖ = ‖tynk−1 −ynk‖→ dist(a,b), 40 balwant singh thakur∗, ajay sharma so, there exists n2 such that (8) ‖z2nk −z2nk+1‖≤ dist(a,b) + ε, ∀nk ≥ n2 . let n = max{n1,n2}. it follows from the uniform convexity of x and (6)-(8) that∥∥∥∥z2nk+2 + z2nk2 −z2nk+1 ∥∥∥∥ ≤ ( 1 − δ ( ε0 dist(a,b) + ε )) (dist(a,b) + ε) , ∀nk ≥ n. since a is convex z2nk+2+z2nk 2 ∈ a, the choice of ε and the fact that δ is strictly increasing imply that∥∥∥∥z2nk+2 + z2nk2 −z2nk+1 ∥∥∥∥ < dist(a,b) , ∀nk ≥ n, a contradiction. by a similar argument, we can show that ‖z2n+3 −z2n+1‖→ 0, as n →∞. � theorem 3.4. let a and b be nonempty convex subsets of a uniformly convex banach space x and let s,t : a ∪ b → a ∪ b are semi-cyclic ϕ−contraction mappings such that t(a) ⊆ b and s(b) ⊆ a. for x0 ∈ a the sequences {zn} is generated by (5). then, for each ε > 0, there exists a positive integer n0 such that for all m ≥ n ≥ n0, ‖z2m −z2n+1‖ < dist(a,b) + ε. proof. suppose the contrary. then there exists ε0 > 0 such that for each k ≥ 1, there is mk > nk ≥ k satisfying (9) ‖z2mk −z2nk+1‖≥ dist(a,b) + ε0 and (10) ∥∥z2(mk−1) −z2nk+1∥∥ < dist(a,b) + ε0. it follows from (9) and (10), that dist(a,b) + ε0 ≤‖z2mk −z2nk+1‖ ≤ ∥∥z2mk −z2(mk−1)∥∥ + ∥∥z2(mk−1) −z2nk+1∥∥ < ∥∥z2mk −z2(mk−1)∥∥ + dist(a,b) + ε0, i.e. dist(a,b) + ε0 ≤‖z2mk −z2nk+1‖ < ∥∥z2mk −z2(mk−1)∥∥ + dist(a,b) + ε0, letting k →∞, lemma 3.2 implies that ∥∥z2mk −z2(mk−1)∥∥ → 0, hence (11) lim k→∞ ‖z2mk −z2nk+1‖ = dist(a,b) + ε0. existence and convergence of best proximity points 41 since (s,t) is semi cyclic ϕ−contraction pair, by lemma 3.1(i),(ii) we obtain ‖z2mk −z2nk+1‖≤‖z2mk −z2mk+2‖ + ‖z2mk+2 −z2nk+3‖ + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖tymk+1 −sxnk+2‖ + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + ‖ymk+1 −xnk+2‖ + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖sxmk+1 −tynk+1‖ + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + ‖xmk+1 −ynk+1‖−ϕ (‖xmk+1 −ynk+1‖) + ϕ (dist(a,b)) + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖tymk −sxnk+1‖−ϕ (‖tymk −sxnk+1‖) + ϕ (dist(a,b)) + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖z2mk −z2nk+1‖−ϕ (‖z2mk −z2nk+1‖) + ϕ (dist(a,b)) + ‖z2nk+3 −z2nk+1‖ . letting k →∞, using lemma 3.2 and (11), we get dist(a,b) + ε0 ≤ dist(a,b) + ε0 − lim k→∞ ϕ (‖z2mk −z2nk+1‖) + ϕ (dist(a,b)) ≤ dist(a,b) + ε0. hence, we obtain (12) lim k→∞ ϕ (‖z2mk −z2nk+1‖) = ϕ (dist(a,b)) . since ϕ is strictly increasing, from (9) and (12), it follows that ϕ (dist(a,b) + ε0) ≤ lim k→∞ ϕ (‖z2mk −z2nk+1‖) = ϕ (dist(a,b)) < ϕ (dist(a,b) + ε0) , a contradiction. � theorem 3.5. let a and b be nonempty closed convex subsets of a uniformly convex banach space x and let s,t : a∪b → a∪b are semi-cyclic ϕ-contraction mappings such that t(a) ⊆ b and s(b) ⊆ a. for x0 ∈ a the sequences {zn} is generated by (5). if dist(a,b) = 0 then (s,t) have a unique common fixed point in a∩b. proof. let ε > 0 be given. by theorem 3.1, we have ‖z2n −z2n+1‖ = ‖tyn −sxn+1‖ ≤‖yn −xn+1‖ ≤‖xn −yn‖ = ‖xn −sxn‖ → dist(a,b) = 0. hence, for given ε > 0, there exists a positive integer n1 such that ‖z2n −z2n+1‖ < ε, for all n ≥ n1. by theorem 3.4, there exists a positive integer n2 such that ‖z2m −z2n+1‖ < ε 42 balwant singh thakur∗, ajay sharma for all m > n ≥ n2. let n = max{n1,n2}. then, for all m > n ≥ n2, we have ‖z2m −z2n‖≤‖z2m −z2n+1‖ + ‖z2n+1 −z2n‖ < 2ε. thus {z2n} is a cauchy sequence in a, since a is closed subset of a complete space x then there exists z ∈ a such that z2n → z as n →∞. since {z2n−1}⊆ b, and b is closed, it follows that z ∈ b, and finally z ∈ a∩b. it follows from theorem 3.2 that ‖z −tz‖ = dist(a,b) = 0 = ‖z −sz‖ . so, z is a common fixed point of s and t and hence z ∈ f(t ∩s) ⊆ a∩b. we claim that the fixed point z is unique. in fact, if tw = w = sw for some w ∈ a∩b, z 6= w, then ‖z −w‖ = ‖tz −sw‖≤‖z −w‖−ϕ (‖z −w‖) + ϕ(0), it follows that ϕ(0) < ϕ (‖z −w‖) ≤ ϕ(0), a contradiction. let rn = ‖zn −z‖ for each n ≥ 0. as the sequence {rn} is bounded and decreasing and r2n → 0 as n → ∞, we conclude that rn → 0 as n → ∞, thus zn → z as n →∞. � theorem 3.6. let a and b be nonempty closed convex subsets of a uniformly convex banach space x and let s,t : a∪b → a∪b are semi-cyclic ϕ-contraction mappings such that t(a) ⊆ b and s(b) ⊆ a. for x0 ∈ a the sequences {zn} is generated by (5). then {z2n} and {z2n+1} are cauchy sequences. proof. if dist(a,b) = 0 then the result follows from theorem 3.5. so assume that dist(a,b) > 0. to show that {z2n} is a cauchy sequence in a we assume the contrary. then there exists ε0 > 0 such that for each k ≥ 1, there exists mk > nk ≥ k so that (13) ‖z2mk −z2nk‖≥ ε0. choose ε > 0 so that ( 1 −δ ( ε0 dist(a,b)+ε )) (dist(a,b) + ε) < dist(a,b). by theorem 3.1, we have ‖z2nk −z2nk+1‖ = ‖tynk −sxnk+1‖ ≤‖ynk −xnk+1‖ = ‖sxnk −tynk‖ ≤‖xnk −ynk‖ = ‖xnk −sxnk‖→ dist(a,b), hence, there exists a positive integer n1 such that (14) ‖z2nk −z2nk+1‖≤ dist(a,b) + ε, ∀nk ≥ n1. also, by theorem 3.4, there exists a positive integer n2 such that (15) ‖z2mk −z2nk+1‖≤ dist(a,b) + ε, ∀mk > nk ≥ n2 . existence and convergence of best proximity points 43 let n = max{n1,n2}. it follows from the uniform convexity of x and (13)-(15) that∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ ≤ ( 1 −δ ( ε0 dist(a,b) + ε )) (dist(a,b) + ε) , ∀mk > nk ≥ n. since a is convex z2mk +z2nk 2 ∈ a, the choice of ε and the fact that δ is strictly increasing imply that∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ < dist(a,b), ∀mk > nk ≥ n, a contradiction. thus {z2n} is a cauchy sequence in a. by a similar argument we can show that {z2n+1} is a cauchy sequence in b. � theorem 3.7. let a and b be nonempty closed convex subsets of a uniformly convex banach space x and let s,t : a∪b → a∪b are semi-cyclic ϕ-contraction mappings such that t(a) ⊆ b and s(b) ⊆ a. for x0 ∈ a, the sequences {zn} is generated by (5). then there exists unique x in a and y in b such that z2n → y, z2n+1 → x and ‖x−sx‖ = dist(a,b) = ‖y −ty‖ . proof. since {z2n} is a cauchy sequence, we can find a y ∈ b such that {z2n} converges to y. it follows from theorem 3.2 that ‖y −ty‖ = dist(a,b). similarly we can show that the sequence {z2n+1} is convergent to some x ∈ a and ‖x−sx‖ = dist(a,b). to prove uniqueness, assume that there is another w ∈ a such that ‖w −sw‖ = dist(a,b). since dist(a,b) ≤‖tsx−sx‖≤‖sx−x‖−ϕ (‖sx−x‖) + ϕ (dist(a,b)) = dist(a,b) −ϕ (dist(a,b)) + ϕ (dist(a,b)) = dist(a,b), it follows that ‖tsx−sx‖ = ‖x−sx‖, this in turn gives tsx = x. similarly, we can see that tsw = w. now if w 6= x, then ‖x−sw‖ > dist(a,b), from which we get ‖sx−w‖ = ‖sx−tsw‖≤‖x−sw‖−ϕ (‖x−sw‖) + ϕ (dist(a,b)) < ‖x−sw‖−ϕ (dist(a,b)) + ϕ (dist(a,b)) = ‖x−sw‖ = ‖tsx−sw‖ ≤‖sx−w‖−ϕ(‖sx−w‖) + ϕ (dist(a,b)) ≤‖sx−w‖ , which is a contraction. thus x = w. similarly, we can see the uniqueness of y ∈ b. this completes the proof. � 4. conclusion in this paper, the new notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass, is introduced. existence and convergence results of best proximity points for semi cyclic ϕ-contraction pair 44 balwant singh thakur∗, ajay sharma of mappings are obtained. our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. in particular, our results reduce to those of gabeleh and abkar [4]. references [1] kirk, w.a., srinivasan, p.s., veeramani, p.: fixed points for mappings satisfying cyclical contractive conditions. fixed point theory 4, 79-89 (2003) [2] eldred, a.a., veeramani, p.: existence and convergence of best proximinity points. j. math. anal. appl. 323, 1001-1006 (2006) [3] al-thagafi, m.a., shahzad, n.: convergence and existence results for best proximity points. nonlinear anal. 70, 3665–3671 (2009) [4] gabeleh, m., abkar, a.: best proximity points for semi-cylic contractive pairs in banach spaces, int. math. forum 6(44), 2179–2186 (2011) [5] chandok, s., postolache, m.: fixed point theorem for weakly chatterjea-type cyclic contractions. fixed point theory appl. id: 2013:28, 9 pp. (2013) [6] shatanawi, w., postolache, m.: common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. fixed point theory appl. id: 2013:60, 13 pp. (2013) [7] clarkson, j.a.: uniform convex spaces, trans. amer. math. soc. 40, 396–414 (1936) school of studies in mathematics, pt. ravishankar shukla university, raipur, 492010, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 132-135 http://www.etamaths.com bounding the difference and ratio between the weighted arithmetic and geometric means feng qi1,2,3,∗ abstract. in the paper, making use of two integral representations for the difference and ratio of the weighted arithmetic and geometric means and employing the weighted arithmetic-geometric-harmonic mean inequality, the author bounds the difference and ratio between the weighted arithmetic and geometric means in the form of double inequalities. 1. main results in [3, theorem 2.3, eq. (2.16)], the difference a(a,b; λ)−g(a,b; λ) between the weighted arithmetic mean a(a,b; λ) = λa + (1 −λ)b and the weighted geometric mean g(a,b; λ) = aλb1−λ was expressed as an integral representation a(a,b; λ) −g(a,b; λ) = sin(λπ) π ∫ b a g(t−a,b− t; λ) t d t (1.1) for b > a > 0 and λ ∈ (0, 1). in [4, remark 4.1], the ratio a(a,b;λ) g(a,b;λ) between the weighted arithmetic mean a(a,b; λ) and the weighted geometric mean g(a,b; λ) was expressed as an integral representation a(a,b; λ) g(a,b; λ) = 1 + sin(λπ) π ∫ b a g(t−a,b− t; 1 −λ) t2 d t (1.2) for b > a > 0 and λ ∈ (0, 1). in this paper, making use of the integral representations (1.1) and (1.2) and employing the weighted arithmetic-geometric-harmonic mean inequality a(a,b; λ) > g(a,b; λ) > h(a,b; λ) (1.3) for b > a > 0 and λ ∈ (0, 1), where h(a,b; λ) = 1λ a + 1−λ b , for b > a > 0 and λ ∈ (0, 1) is called the weighted harmonic mean, we will bound the difference a(a,b; λ) − g(a,b; λ) and the ratio a(a,b;λ) g(a,b;λ) of the weighted arithmetic mean a(a,b; λ) and the geometric mean g(a,b; λ) in the form of double inequalities. our main results can be stated as the following theorems. theorem 1.1. for b > a > 0 and λ ∈ (0, 1), the difference between the weighted arithmetic and geometric means can be bounded by sin(λπ) π ( (2λ− 1)(b−a) + [(1 −λ)b−λa] ln b a ) > [λa + (1 −λ)b] −aλb1−λ >   sin(λπ) π [ λ(1 −λ)(b−a)2 (2λ− 1)2[λb− (1 −λ)a] ln ( 1 λ − 1 ) − ab(ln b− ln a) λb− (1 −λ)a + b−a 2λ− 1 ] , λ 6= 1 2 ; 1 π b2 − 2ab(ln b− ln a) −a2 b−a , λ = 1 2 . (1.4) 2010 mathematics subject classification. primary 26e60; secondary 26d07, 47a64. key words and phrases. difference; ratio; weighted arithmetic mean; weighted geometric mean; weighted harmonic mean; integral representation; double inequality; weighted arithmetic-geometric-harmonic mean inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 132 bounding difference and ratio of arithmetic and geometric means 133 theorem 1.2. for b > a > 0 and λ ∈ (0, 1), the ratio between the weighted arithmetic and geometric means can be bounded by 1 + sin(λπ) π [ λ b2 −a2 ab + (1 − 2λ) ln b a + a b − 1 ] > λa + (1 −λ)b aλb1−λ >   1 + sin(λπ) π { (1 −λ)b2 −λa2 [λa− (1 −λ)b]2 ln b a + (b−a)[λa− (1 −λ)b] [λa− (1 −λ)b]2 + (1 −λ)λ(b−a)2 (2λ− 1)[λa− (1 −λ)b]2 ln ( 1 λ − 1 )} , λ 6= 1 2 ; 1 + 2 π [ (a + b) b−a ln b a − 2 ] , λ = 1 2 . (1.5) 2. proofs of theorems 1.1 and 1.2 now we start out to prove our main results. proof of theorem 1.1. by virtue of the inequality in the left-hand side of (1.3), we obtain∫ b a (t−a)λ(b− t)1−λ t d t < ∫ b a λ(t−a) + (1 −λ)(b− t) t d t = [(1 −λ)b−λa](ln b− ln a) + (2λ− 1)(b−a). substituting this into (1.1) yields [λa + (1 −λ)b] −aλb1−λ < sin(λπ) π { [(1 −λ)b−λa] ln b a + (2λ− 1)(b−a) } . by virtue of the inequality in the right-hand side of (1.3), we obtain ∫ b a (t−a)λ(b− t)1−λ t d t > ∫ b a 1 t 1 λ t−a + 1−λ b−t d t = λ(1 −λ)(b−a)2 (2λ− 1)2(λb− (1 −λ)a) ln ( 1 λ − 1 ) − ab(ln b− ln a) λb− (1 −λ)a + b−a 2λ− 1 . substituting this into (1.1) yields [λa + (1 −λ)b] −aλb1−λ > sin(λπ) π { λ(1 −λ)(b−a)2 (2λ− 1)2[λb− (1 −λ)a] ln ( 1 λ − 1 ) − ab(ln b− ln a) λb− (1 −λ)a + b−a 2λ− 1 } → 1 π b2 − 2ab(ln b− ln a) −a2 b−a as λ → 1 2 . the double inequality (1.4) is thus proved. the proof of theorem 1.1 is complete. � proof of theorem 1.2. by virtue of the inequality in the left-hand side of (1.3), we obtain∫ b a (t−a)1−λ(b− t)λ t2 d t < ∫ b a (1 −λ)(t−a) + λ(b− t) t2 d t = λ b2 −a2 ab + (1 − 2λ) ln b a + a b − 1. substituting this into (1.2) yields λa + (1 −λ)b aλb1−λ − 1 < sin(λπ) π [ λ b2 −a2 ab + (1 − 2λ) ln b a + a b − 1 ] . 134 f. qi by virtue of the inequality in the right-hand side of (1.3), we obtain∫ b a (t−a)1−λ(b− t)λ t2 d t > ∫ b a 1 t2 1 1−λ t−a + λ b−t d t = (1 − 2λ) [ λa2 − (1 −λ)b2 ] ln b a + (b−a) { (2λ− 1)[λa− (1 −λ)b] + (1 −λ)λ(b−a) ln ( 1 λ − 1 )} (2λ− 1)[λa− (1 −λ)b]2 → 2(a + b) b−a ln b a − 4 as λ → 1 2 . substituting this into (1.2) yields λa + (1 −λ)b aλb1−λ − 1 > sin(λπ) π { (1 −λ)b2 −λa2 [λa− (1 −λ)b]2 ln b a + (b−a)[λa− (1 −λ)b] [λa− (1 −λ)b]2 + (1 −λ)λ(b−a)2 (2λ− 1)[λa− (1 −λ)b]2 ln ( 1 λ − 1 )} → 2 π [ (a + b) b−a ln b a − 2 ] as λ → 1 2 . the double inequality (1.5) is thus proved. the proof of theorem 1.2 is complete. � 3. remarks finally we list several remarks on our main results. remark 3.1. when λ = 1 2 and b > a > 0, the double inequality (1.4) can be written as 1 π ( b−a 2 ln b a ) > a + b 2 − √ ab > 2 π ( a + b 2 −ab ln b− ln a b−a ) > 0. when λ = 1 2 and b > a > 0, the double inequality (1.5) can be written as b−a π ( a + b 2ab − 1 b ) > a + b 2 √ ab − 1 > 2 π [ (a + b) b−a ln b a − 2 ] > 0. remark 3.2. from the integral representations (1.1) and (1.2), we can easily see that all inequalities for bounding the (weighted) geometric mean can be used to construct inequalities for bounding the difference and ratio between the (weighted) arithmetic and geometric means. remark 3.3. let 0 < ak < ak+1 for 1 ≤ k ≤ n − 1, wk > 0 for 1 ≤ k ≤ n, and z ∈ c \ [−an,−a1]. theorem 3.1 in [7] states that the principal branch of the weighted geometric mean ∏n k=1(z + ak) wk has the integral representation n∏ k=1 (z + ak) wk = n∑ k=1 wkak + z − 1 π n−1∑ `=1 sin [(∑̀ j=1 wj ) π ]∫ a`+1 a` n∏ k=1 |ak − t|wk d t t + z . (3.1) by the same arguments as in proofs of theorems 1.1 and 1.2, we can derive from (3.1) lower and upper bounds for the difference ∑n k=1 wkak − ∏n k=1 a wk k between the weighted arithmetic mean ∑n k=1 wkak and the geometric mean ∏n k=1 a wk k . remark 3.4. in [4], it was obtained that, for ak < ak+1, z ∈ c \ [−an,−a1], and wk > 0 with∑n k=1 wk = 1, the principal branch of the reciprocal of the weight geometric mean ∏n k=1(z + ak) wk can be represented by 1∏n k=1(z + ak) wk = 1 π n−1∑ `=1 sin ( π ∑̀ k=1 wk )∫ a`+1 a` 1∏n k=1 |t−ak|wk 1 t + z d t. (3.2) the integral representation (3.2) generalizes corresponding results in [2, 3] and [5, lemma 2.4]. remark 3.5. this paper is a companion of the articles [1–4, 6–10]. bounding difference and ratio of arithmetic and geometric means 135 references [1] b.-n. guo and f. qi, on the degree of the weighted geometric mean as a complete bernstein function, afr. mat. 26 (7) (2015), 1253–1262. [2] f. qi and b.-n. guo, the reciprocal of the geometric mean of many positive numbers is a stieltjes transform, j. comput. appl. math. 311 (2017), 165–170. [3] f. qi and b.-n. guo, the reciprocal of the weighted geometric mean is a stieltjes function, bol. soc. mat. mex. (2016). doi:10.1007/s40590-016-0151-5. [4] f. qi and b.-n. guo, the reciprocal of the weighted geometric mean of many positive numbers is a stieltjes function, researchgate working paper (2016), doi:10.13140/rg.2.2.23822.36163. [5] f. qi, b.-n. guo, v. čerňanová, and x.-t. shi, explicit expressions, cauchy products, integral representations, convexity, and inequalities of central delannoy numbers, researchgate working paper (2016), doi:10.13140/rg.2.1.4889.6886. [6] f. qi, x.-j. zhang, and w.-h. li, an elementary proof of the weighted geometric mean being a bernstein function, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 77 (1) (2015), 35–38. [7] f. qi, x.-j. zhang, and w.-h. li, an integral representation for the weighted geometric mean and its applications, acta math. sin. (engl. ser.) 30 (1) (2014), 61–68. [8] f. qi, x.-j. zhang, and w.-h. li, lévy-khintchine representation of the geometric mean of many positive numbers and applications, math. inequal. appl. 17 (2) (2014), 719–729. [9] f. qi, x.-j. zhang, and w.-h. li, lévy-khintchine representations of the weighted geometric mean and the logarithmic mean, mediterr. j. math. 11 (2) (2014), 315–327. [10] f. qi, x.-j. zhang, and w.-h. li, the harmonic and geometric means are bernstein functions, bol. soc. mat. mex. (2016), doi:10.1007/s40590-016-0085-y. 1institute of mathematics, henan polytechnic university, jiaozuo city, henan province, 454010, china 2college of mathematics, inner mongolia university for nationalities, tongliao city, inner mongolia autonomous region, 028043, china 3department of mathematics, college of science, tianjin polytechnic university, tianjin city, 300387, china ∗corresponding author: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com 1. main results 2. proofs of theorems 1.1 and 1.2 3. remarks references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 123-129 http://www.etamaths.com (δ,γ)-jacobi-dunkl lipschitz functions in the space l2(r,aα,β(x)dx) r. daher, s. el ouadih∗ abstract. using a generalized jacobi-dunkl translation, we obtain an analog of theorem 5.2 in younis paper [7] for the jacobi-dunkl transform for functions satisfying the (δ,γ)-jacobi-dunkl lipschitz condition in the space l2(r,aα,β(x)dx),α ≥ β ≥ −12 ,α 6= −1 2 . 1. introduction and preliminaries younis ([7], theorem 5.2) characterized the set of functions in l2(r) satisfying the dini-lipschitz condition by means of an asymptotic estimate growth of the norm of their fourier transforms, namely we have the following statement. theorem 1.1. [7] let f ∈ l2(r). then the following are equivalents: (a) ‖f(x + h) −f(x)‖ = o ( hη (log 1 h )γ ) , as h → 0, 0 < η < 1,γ > 0 (b) ∫ |λ|≥r |f̂(λ)|2dλ = o ( r−2η (log r)2γ ) , as r →∞, where f̂ stand for the fourier transform of f. in this paper, we obtain an analog of theorem 1.1 for the jacobi-dunkl transform on the real line. for this purpose, we use a generalized jacobi-dunkl translation operator. in this section , we recapitulate from [1,2,3,5] some results related to the harmonic analysis associated with jacobi-dunkl operator λα,β. the jacobi-dunkl function with parameters (α,β),α ≥ β ≥ −1 2 ,α 6= −1 2 , defined by the formula ∀x ∈ r,ψα,βλ (x) =   ϕα,βµ (x) − i λ d dx ϕα,βµ (x) if λ ∈ c\{0} 1 if λ = 0, with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕα,βµ is the jacobi function given by ϕα,βµ (x) = f ( ρ + iµ 2 , ρ− iµ 2 ,α + 1,−(sinh(x))2 ) , 2010 mathematics subject classification. 65r10. key words and phrases. jacobi-dunkl operator, jacobi-dunkl transform, generalized jacobidunkl translation. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 123 124 daher and ouadih f is the gausse hypergeometric function (see [1,6]). ψ α,β λ is the unique c ∞-solution on r of the differential-difference equation  λα,βu = iλu ,λ ∈ c u(0) = 1, where λα,β is the jacobi-dunkl operator given by λα,βu(x) = du(x) dx + [(2α + 1) coth x + (2β + 1) tanh x] × u(x) −u(−x) 2 ]. the operator λα,β is a particular case of the operator d given by du(x) = du(x) dx + a′(x) a(x) × ( u(x) −u(−x) 2 ) , where a(x) = |x|2α+1b(x) and b a function of class c∞ on r, even and positive. the operator λα,β corresponds to the function a(x) = aα,β(x) = 2 ρ(sinh |x|)2α+1(cosh |x|)2β+1. using the relation d dx ϕα,βµ (x) = − µ2 + ρ2 4(α + 1) sinh(2x)ϕα+1,β+1µ (x), the function ψ α,β λ can be written in the form above (see [2]) ψ α,β λ (x) = ϕ α,β µ (x) + i λ 4(α + 1) sinh(2x)ϕα+1,β+1µ (x), x ∈ r. denote l2α,β(r) = l 2 α,β(r,aα,β(x)dx) the space of measurable functions f on r such that ‖f‖l2 α,β (r) = (∫ r |f(x)|2aα,β(x)dx )1/2 < +∞. using the eigenfunctions ψ α,β λ of the operator λα,β called the jacobi-dunkl kernels , we define the jacobi-dunkl transform of a function f ∈ l2α,β(r) by fα,βf(λ) = ∫ r f(x)ψ α,β λ (x)aα,β(x)dx, λ ∈ r, and the inversion formula f(x) = ∫ r fα,βf(λ)ψ α,β −λ (x)dσ(λ), where dσ(λ) = |λ| 8π √ λ2 −ρ2|cα,β( √ λ2 −ρ2)| ir\]−ρ,ρ[(λ)dλ. here, cα,β(µ) = 2ρ−iµγ(α + 1)γ(iµ) γ( 1 2 (ρ + iµ))γ( 1 2 (α−β + 1 + iµ)) , µ ∈ c\(in) and ir\]−ρ,ρ[ is the characteristic function of r\] −ρ,ρ[. the jacobi-dunkl transform is a unitary isomorphism from l2α,β(r) onto l 2(r,dσ(λ)), i.e. ‖f‖ := ‖f‖l2 α,β (r) = ‖fα,β(f)‖l2(r,dσ(λ)).(1) strong metrizability for closed operators 125 the operator of jacobi-dunkl translation is defined by txf(y) = ∫ r f(z)dνα,βx,y (z), ∀x,y ∈ r where να,βx,y (z),x,y ∈ r are the signed measures given by dνα,βx,y (z) =   kα,β(x,y,z)aα,β(z)dz if x,y ∈ r∗ δx if y = 0 δy if x = 0 here, δx is the dirac measure at x. and, kα,β(x,y,z) = mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αiix,y × ∫ π 0 ρθ(x,y,z) × (gθ(x,y,z)) α−β−1 + sin 2β θdθ ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x|− |y||, |x| + |y|] ρθ(x,y,z) = 1 −σθx,y,z + σ θ z,x,y + σ θ z,y,x ∀z ∈ r,θ ∈ [0,π],σθx,y,z =   cosh(x)+cosh(y)−cosh(z) cos(θ) sinh(x) sinh(y) ,if xy 6= 0 0 ,if xy = 0 gθ(x,y,z) = 1 − cosh2(x) − cosh2(y) − cosh2(z) + 2 cosh(x) cosh(y) cosh(z) cos θ t+ =   t ,if t > 0 0 ,if t ≤ 0 and, mα,β =   2−2ργ(α+1)√ πγ(α−β)γ(β+ 1 2 ) ,if α > β 0 ,if α = β in [2], we have fα,β(thf)(λ) = ψ α,β λ (h)fα,β(f)(λ); λ,h ∈ r.(2) for α ≥ −1 2 , we introduce the bessel normalized function of the first kind defined by: jα(x) = γ(α + 1) ∞∑ n=0 (−1)n(x 2 )2n n!γ(n + α + 1) . moreover, we see that lim x→0 jα(x) − 1 x2 6= 0, by consequence , there exists c1 > 0 and ε > 0 satisfying |x| ≤ ε ⇒|jα(x) − 1| ≥ c1|x|2(3) lemma 1.2. (see[8],lemma 3.1,lemma 3.2) the following inequalities are valid for jacobi functions ϕα,βµ (x) (c) |ϕα,βµ (x)| ≤ 1, (d) |1 −ϕα,βµ (x)| ≤ x2(µ2 + ρ2). 126 daher and ouadih lemma 1.3. (see[4],lemma 9) let α ≥ β ≥ −1 2 ,α 6= −1 2 . then for |ν| ≤ ρ, there exists a positive constant c2 such that |1 −ϕα,βµ+iν(x)| ≥ c2|1 − jα(µx)|. 2. main results in this section we give the main results of this paper. we need first to define (η,γ)-jacobi-dunkl lipschitz class. definition 2.1. let 0 < η < 1 and γ > 0. a function f ∈ l2α,β(r) is said to be in the (η,γ)-jacobi-dunkl lipschitz class, denoted by lip(η,γ, 2), if ‖thf(x) + t−hf(x) − 2f(x)‖ = o ( hη (log 1 h )γ ) , as h → 0. lemma 2.2. for f ∈ l2α,β(r), then ‖thf(x) + t−hf(x) − 2f(x)‖2 = 4 ∫ r |ϕα,βµ (h) − 1| 2|fα,βf(λ)|2dσ(λ). proof. we us formula (2), we conclude that fα,β(thf + t−hf − 2f)(λ) = (ψ α,β λ (h) + ψ α,β λ (−h) − 2)fα,β(f)(λ), since ψ α,β λ (h) = ϕ α,β µ (h) + i λ 4(α + 1) sinh(2h)ϕα+1,β+1µ (h), ψ α,β λ (−h) = ϕ α,β µ (−h) − i λ 4(α + 1) sinh(2h)ϕα+1,β+1µ (−h), and ϕα,βµ is even (see [2]), then fα,β(thf + t−hf − 2f)(λ) = 2(ϕα,βµ (h) − 1)fα,β(f)(λ). by parseval’s identity (formula (1)), we have the result. theorem 2.3. let 0 < η < 1 , γ > 0 and f ∈ l2α,β(r). then the following conditions are equivalents (i) f ∈ lip(η,γ, 2) (ii) ∫ |λ|≥r |fα,βf(λ)|2dσ(λ) = o ( r−2η (log r)2γ ) , as r →∞. proof. (i) ⇒ (ii). assume that f ∈ lip(η,γ, 2) . then we have ‖thf(x) + t−hf(x) − 2f(x)‖ = o ( hη (log 1 h )γ ) , as h → 0 from lemma 2.2, we have ‖thf(x) + t−hf(x) − 2f(x)‖2 = 4 ∫ r |ϕα,βµ (h) − 1| 2|fα,βf(λ)|2dσ(λ). by (3) and lemma 1.3, we get∫ ε 2h ≤|λ|≤ε h |1−ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≥ c21c 2 2 ∫ ε 2h ≤|λ|≤ε h |µh|4|fα,β(f)(λ)|2dσ(λ). strong metrizability for closed operators 127 from ε 2h ≤ |λ| ≤ ε h we have( ε 2h )2 − ρ2 ≤ µ2 ≤ (ε h )2 −ρ2 ⇒ µ2h2 ≥ ε2 4 −ρ2h2. take h ≤ ε 3ρ , then we have µ2h2 ≥ c3 = c3(ε). so,∫ ε 2h ≤|λ|≤ε h |1−ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ) ≥ c21c 2 2c 2 3 ∫ ε 2h ≤|λ|≤ε h |fα,β(f)(λ)|2dσ(λ). there exists then a positive constant c4 such that∫ ε 2h ≤|λ|≤ε h |fα,β(f)(λ)|2dσ(λ) ≤ c4 ∫ r |1 −ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ) = o ( h2η (log 1 h )2γ ) . we obtain ∫ r≤|λ|≤2r |fα,β(f)(λ)|2dσ(λ) ≤ c r−2η (log r)2γ , where c is a positive constant. now,∫ |λ|≥r |fα,β(f)(λ)|2dσ(λ) = ∞∑ i=0 ∫ 2ir≤|λ|≤2i+1r |fα,β(f)(λ)|2dσ(λ) ≤ c ( r−2η (log r)2γ + (2r)−2η (log 2r)2γ + (4r)−2η (log 4r)2γ + · · · ) ≤ c r−2η (log r)2γ ( 1 + 2−2η + (2−2η)2 + (2−2η)3 + · · · ) ≤ kη r−2η (log r)2γ , where kη = c(1 − 2−2η)−1 since 2−2η < 1. consequently∫ |λ|≥r |fα,βf(λ)|2dσ(λ) = o ( r−2η (log r)2γ ) , as r →∞. (ii) ⇒ (i). suppose now that∫ |λ|≥r |fα,βf(λ)|2dλ = o ( r−2η (log r)2γ ) , as r →∞, and write ∫ r |1 −ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ) = i1 + i2 where i1 = ∫ |λ|< 1 h |1 −ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ), i2 = ∫ |λ|≥1 h |1 −ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ). 128 daher and ouadih firstly, we use the formula |ϕα,βµ (h)| ≤ 1 and i2 ≤ 4 ∫ |λ|≥1 h |fα,β(f)(λ)|2dσ(λ) = o ( h2η (log 1 h )2γ ) .(4) set φ(λ) = ∫ ∞ λ |fα,β(f)(x)|2dσ(x). an integration by parts gives∫ x 0 λ2|fα,β(f)(λ)|2dσ(λ) = ∫ x 0 −λ2φ′(λ)dλ = −x2φ(x) + 2 ∫ x 0 λφ(λ)dλ ≤ 2 ∫ x 0 λ1−2δ(log λ)−2γdλ = o(x2−2δ(log x)−2γ). we use the formula (d) of lemma 1.2 i1 ≤ ∫ |λ|< 1 h |1 −ϕα,βµ (h)||fα,β(f)(λ)| 2dσ(λ) ≤ ∫ |λ|< 1 h (µ2 + ρ2)h2|fα,β(f)(λ)|2dσ(λ) ≤ h2 ∫ |λ|< 1 h λ2|fα,β(f)(λ)|2dσ(λ) = o ( h2h−2+2η ( log 1 h )−2γ) .. hence, i1 = o ( h2η (log 1 h )2γ ) .(5) finally, we conclude from (4) and (5) that∫ r |1 −ϕα,βµ (h)| 2|fα,β(f)(λ)|2dσ(λ) = o ( h2η (log 1 h )2γ ) . and this ends the proof. references [1] ben mohamed. h and mejjaoli. h, distributional jacobi-dunkl transform and applications, afr.diaspora j.math 1(2004), 24-46. [2] ben mohamed. h, the jacobi-dunkl transform on r and the convolution product on new space of distributions, ramanujan j.21(2010), 145-175.. [3] ben salem. n and ahmed salem. a , convolution structure associated with the jacobi-dunkl operator on r, ramanuy j.12(3) (2006), 359-378. [4] bray. w. o and pinsky. m. a, growth properties of fourier transforms via module of continuity , journal of functional analysis.255(288), 2256-2285. [5] chouchane. f, mili. m and trimche. k, positivity of the intertwining operator and harmonic analysis associated with the jacobi-dunkl operator on r, j.anal. appl.1(4)(2003), 387-412. strong metrizability for closed operators 129 [6] koornwinder. t. h, jacobi functions and analysis on noncompact semi-simple lie groups.in: askey.ra, koornwinder. t. h and schempp.w(eds) special functions: group theatrical aspects and applications.d.reidel, dordrecht (1984). [7] younis . m. s, fourier transforms of dini-lipschitz functions. int. j. math. math. sci. (1986), 9 (2), 301c312. doi:10.1155/s0161171286000376 [8] platonov. s, approximation of functions in l2-metric on noncompact rank 1 symmetric space . algebra analiz .11(1) (1999), 244-270. departement of mathematics, faculty of sciences äın chock, university hassan ii, casablanca, morocco ∗corresponding author international journal of analysis and applications volume 16, number 5 (2018), 643-653 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-643 wijsman rough lacunary statistical convergence on i cesàro triple sequences n. subramanian1 and a. esi2,∗ 1department of mathematics, sastra university, thanjavur-613 401, india 2department of mathematics, adiyaman university, 02040, adiyaman, turkey ∗corresponding author: aesi23@hotmail.com abstract. in this paper, we defined concept of wijsman i-cesàro summability for sequences of sets and investigate the relationships between the concepts of wijsman strongly i-cesàro summability and wijsman statistical i− cesàro summability by using the concept of a triple sequence spaces. 1. introduction the idea of statistical convergence was introduced by steinhaus and also independently by fast for real or complex sequences. statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence. let k be a subset of the set of positive integers n×n×n, and let us denote the set {(m,n,k) ∈ k : m ≤ u,n ≤ v,k ≤ w} by kuvw. then the natural density of k is given by δ (k) = limuvw→∞ |kuvw| uvw , where |kuvw| denotes the number of elements in kuvw. clearly, a finite subset has natural density zero, and we have δ (kc) = 1 − δ (k) where kc = n\k is the complement of k. if k1 ⊆ k2, then δ (k1) ≤ δ (k2) . throughout the paper, r denotes the real of three dimensional space with metric (x,d) . consider a triple sequence x = (xmnk) such that xmnk ∈ r,m,n,k ∈ n. received 2017-10-29; accepted 2018-01-13; published 2018-09-05. 2010 mathematics subject classification. 40f05, 40j05. key words and phrases. wijsman rough statistical convergence; natural density; triple sequences; i− cesàro. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 643 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-643 int. j. anal. appl. 16 (5) (2018) 644 a triple sequence x = (xmnk) is said to be statistically convergent to 0 ∈ r, written as st − lim x = 0, provided that the set { (m,n,k) ∈ n3 : |xmnk, 0| ≥ � } has natural density zero for any � > 0. in this case, 0 is called the statistical limit of the triple sequence x. if a triple sequence is statistically convergent, then for every � > 0, infinitely many terms of the sequence may remain outside the �− neighbourhood of the statistical limit, provided that the natural density of the set consisting of the indices of these terms is zero. this is an important property that distinguishes statistical convergence from ordinary convergence. because the natural density of a finite set is zero, we can say that every ordinary convergent sequence is statistically convergent. if a triple sequence x = (xmnk) satisfies some property p for all m,n,k except a set of natural density zero, then we say that the triple sequence x satisfies p for almost all (m,n,k) and we abbreviate this by a.a. (m,n,k). let ( xminjk` ) be a sub sequence of x = (xmnk). if the natural density of the set k = { (mi,nj,k`) ∈ n3 : (i,j,`) ∈ n3 } is different from zero, then ( xminjk` ) is called a non thin sub sequence of a triple sequence x. c ∈ r is called a statistical cluster point of a triple sequence x = (xmnk) provided that the natural density of the set { (m,n,k) ∈ n3 : |xmnk − c| < � } is different from zero for every � > 0. we denote the set of all statistical cluster points of the sequence x by γx. a triple sequence x = (xmnk) is said to be statistically analytic if there exists a positive number m such that δ ({ (m,n,k) ∈ n3 : |xmnk| 1/m+n+k ≥ m }) = 0 the theory of statistical convergence has been discussed in trigonometric series, summability theory, measure theory, turnpike theory, approximation theory, fuzzy set theory and so on. the idea of rough convergence was introduced by [8], who also introduced the concepts of rough limit points and roughness degree. the idea of rough convergence occurs very naturally in numerical analysis and has interesting applications [1] extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence [7] extended the notion of rough convergence using the concept of ideals which automatically extends the earlier notions of rough convergence and rough statistical convergence. let (x,ρ) be a metric space. for any non empty closed subsets a,amnk ⊂ x (m,n,k ∈ n) , we say that int. j. anal. appl. 16 (5) (2018) 645 the triple sequence (amnk) is wijsman statistical convergent to a is the triple sequence (d (x,amnk)) is statistically convergent to d (x,a) , i.e., for � > 0 and for each x ∈ x limrst 1 rst |{m ≤ r,n ≤ s,k ≤ t : |d (x,amnk) −d (x,a)| ≥ �}| = 0. in this case, we write st−limmnkamnk = a or amnk −→ a (ws) . the triple sequence (amnk) is bounded if supmnkd (x,amnk) < ∞ for each x ∈ x. in this paper, we introduce the notion of wijsman rough statistical convergence of triple sequences. defining the set of wijsman rough statistical limit points of a triple sequence, we obtain to wijsman statistical convergence criteria associated with this set. later, we prove that this set of wijsman statistical cluster points and the set of wijsman rough statistical limit points of a triple sequence. a triple sequence (real or complex) can be defined as a function x : n×n×n → r (c) , where n,r and c denote the set of natural numbers, real numbers and complex numbers respectively. the different types of notions of triple sequence was introduced and investigated at the initial by ( [9], [10]), ( [2], [3], [4]), [5], [11], [6], [12] and many others. throughout the paper let r be a nonnegative real number. 2. definitions and preliminaries definition 2.1. a triple sequence x = (xmnk) of real numbers is said to be statistically convergent to l ∈ r3 if for any � > 0 we have d (a (�)) = 0, where a (�) = { (m,n,k) ∈ n3 : |xmnk − l| ≥ � } . definition 2.2. a triple sequence x = (xmnk) is said to be statistically convergent to l ∈ r3, written as st− limx = l, provided that the set { (m,n,k) ∈ n3 : |xmnk − l| ≥ � } , has natural density zero for every � > 0. in this case, l is called the statistical limit of the sequence x. definition 2.3. let x = (xmnk)m,n,k∈n×n×n be a triple sequence in a metric space (x, |., .|) and r be a non-negative real number. a triple sequence x = (xmnk) is said to be r−convergent to l ∈ x, denoted by x →r l, if for any � > 0 there exists n� ∈ n×n×n such that for all m,n,k ≥ n� we have |xmnk − l| < r + � in this case l is called an r− limit of x. remark 2.1. we consider r− limit set x which is denoted by limrx and is defined by limrx = {l ∈ x : x →r l} . int. j. anal. appl. 16 (5) (2018) 646 definition 2.4. a triple sequence x = (xmnk) is said to be r− convergent if limrx 6= φ and r is called a rough convergence degree of x. if r = 0 then it is ordinary convergence of triple sequence. definition 2.5. let x = (xmnk) be a triple sequence in a metric space (x, |., .|) and r be a non-negative real number is said to be r− statistically convergent to l, denoted by x →r−st3 l, if for any � > 0 we have d (a (�)) = 0, where a (�) = {(m,n,k) ∈ n×n×n : |xmnk − l| ≥ r + �} . in this case l is called r− statistical limit of x. if r = 0 then it is ordinary statistical convergent of triple sequence. definition 2.6. a class i of subsets of a nonempty set x is said to be an ideal in x provided (i) φ ∈ i (ii) a,b ∈ i implies a ⋃ b ∈ i. (iii) a ∈ i,b ⊂ a implies b ∈ i. i is called a nontrivial ideal if x /∈ i. definition 2.7. a nonempty class f of subsets of a nonempty set x is said to be a filter in x. provided (i) φ ∈ f. (ii) a,b ∈ f implies a ⋂ b ∈ f. (iii) a ∈ f,a ⊂ b implies b ∈ f. definition 2.8. i is a non trivial ideal in x, x 6= φ, then the class f (i) = {m ⊂ x : m = x\a for some a ∈ i} is a filter on x, called the filter associated with i. definition 2.9. a non trivial ideal i in x is called admissible if {x}∈ i for each x ∈ x. case 2.1. if i is an admissible ideal, then usual convergence in x implies i convergence in x. remark 2.2. if i is an admissible ideal, then usual rough convergence implies rough i− convergence. definition 2.10. let x = (xmnk) be a triple sequence in a metric space (x, |., .|) and r be a non-negative real number is said to be rough ideal convergent or ri− convergent to l, denoted by x →ri l, if for any � > 0 we have {(m,n,k) ∈ n×n×n : |xmnk − l| ≥ r + �}∈ i. in this case l is called ri− limit of x and a triple sequence x = (xmnk) is called rough i− convergent to l with r as roughness of degree. if r = 0 then it is ordinary i− convergent. int. j. anal. appl. 16 (5) (2018) 647 case 2.2. generally, a triple sequence y = (ymnk) is not i− convergent in usual sense and |xmnk −ymnk| ≤ r for all (m,n,k) ∈ n×n×n or {(m,n,k) ∈ n×n×n : |xmnk −ymnk| ≥ r}∈ i. for some r > 0. then the triple sequence x = (xmnk) is ri− convergent. case 2.3. it is clear that ri− limit of x is not necessarily unique. definition 2.11. consider ri− limit set of x, which is denoted by i −limrx = { l ∈ x : x →ri l } , then the triple sequence x = (xmnk) is said to be ri− convergent if i −limrx 6= φ and r is called a rough i− convergence degree of x. definition 2.12. a triple sequence x = (xmnk) ∈ x is said to be i− analytic if there exists a positive real number m such that { (m,n,k) ∈ n×n×n : |xmnk| 1/m+n+k ≥ m } ∈ i. definition 2.13. a point l ∈ x is said to be an i− accumulation point of a triple sequence x = (xmnk) in a metric space (x,d) if and only if for each � > 0 the set{ (m,n,k) ∈ n3 : d (xmnk, l) = |xmnk − l| < � } /∈ i. we denote the set of all i− accumulation points of x by i (γx) . definition 2.14. a triple sequence x = (xmnk) is said to be wijsman r− convergent to a denoted by amnk →r a, provided that ∀� > 0 ∃(m�,n�,k�) ∈ n3 : m ≥ m�,n ≥ n�,k ≥ k� =⇒ limrst 1 rst |{m ≤ r,n ≤ s,k ≤ t : |d (x,amnk) −d (x,a)| < r + �}| = 0 the set limra = { l ∈ r3 : amnk →r a } is called the wijsman r− limit set of the triple sequences. definition 2.15. a triple sequence x = (xmnk) is said to be wijsman r− convergent if limra 6= φ. in this case, r is called the wijsman convergence degree of the triple sequence x = (xmnk). for r = 0, we get the ordinary convergence. definition 2.16. a triple sequence (xmnk) is said to be wijsman r− statistically convergent to a, denoted by amnk →rst a, provided that the set limrst 1 rst ∣∣{(m,n,k) ∈ n3 : |d (x,amnk) −d (x,a)| ≥ r + �}∣∣ = 0 has natural density zero for every � > 0, or equivalently, if the condition int. j. anal. appl. 16 (5) (2018) 648 st− lim sup |d (x,amnk) −d (x,a)| ≤ r is satisfied. in addition, we can write amnk →rst a if and only if the inequality limrst 1 rst |{m ≤ r,n ≤ s,k ≤ t : |d (x,amnk) −d (x,a)| < r + �}| = 0 holds for every � > 0 and almost all (m,n,k) . here r is called the wijsman roughness of degree. if we take r = 0, then we obtain the ordinary wijsman statistical convergence of triple sequence. definition 2.17. a triple sequence (xmnk) is said to be wijsman cesáro convergent to a, denoted by amnk →ces a, provided that the set limrst 1 rst ∑r m=1 ∑s n=1 ∑t k=1 d (x,amnk) = d (x,a) . definition 2.18. a triple sequence (xmnk) is said to be wijsman strongly cesáro convergent to a, denoted by amnk →stces a, provided that the set limrst 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| = 0. definition 2.19. a triple sequence (xmnk) is said to be wijsman p− cesáro convergent to a, denoted by amnk →stces a, if for each p positive real number and for each x ∈ x, limrst 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p = 0. definition 2.20. the triple sequence θi,`,j = {(mi,n`,kj)} is called triple lacunary if there exist three increasing sequences of integers such that m0 = 0,hi = mi −mi−1 →∞ as i →∞ and n0 = 0,h` = n` −n`−1 →∞ as ` →∞. k0 = 0,hj = kj −kj−1 →∞ as j →∞. let mi,`,j = min`kj,hi,`,j = hih`hj, and θi,`,j is determine by ii,`,j = {(m,n,k) : mi−1 < m < mi andn`−1 < n ≤ n` andkj−1 < k ≤ kj} ,qk = mkmk−1 ,q` = n` n`−1 ,qj = kj kj−1 . let θi,`,j be a lacunary sequence. a triple sequence (xmnk) is said to be wijsman strongly lacunary convergent to a, denoted by amnk →stlac a, if limrst 1 hrst ∑ (m,n,k)∈irst |d (x,amnk) −d (x,a)| = 0. in a similar fashion to the idea of classic wijsman rough convergence, the idea of wijsman rough statistical convergence of a triple sequence spaces can be interpreted as follows: assume that a triple sequence y = (ymnk) is wijsman statistically convergent and cannot be measured or calculated exactly; one has to do with an approximated (or wijsman statistically approximated) triple sequence x = (xmnk) satisfying |d (x−y,amnk) −d (x−y,a)| ≤ r for all m,n,k (or for almost all (m,n,k) , i.e., δ ( limrst 1 rst |{m ≤ r,n ≤ s,k ≤ t : |d (x−y,amnk) −d (x−y,a)| > r}| ) = 0. int. j. anal. appl. 16 (5) (2018) 649 then the triple sequence x is not statistically convergent any more, but as the inclusion limrst 1 rst {|d (y,amnk) −d (y,a)| ≥ �}⊇ limrst 1 rst {|d (x,amnk) −d (x,a)| ≥ r + �} (2.1) holds and we have δ ( limrst 1 rst ∣∣{(m,n,k) ∈ n3 : |ymnk − l| ≥ �}∣∣) = 0, i.e., we get δ ( limrst 1 rst |{m ≤ r,n ≤ s,k ≤ t : |d (x,amnk) −d (x,a)| ≥ r + �}| ) = 0, i.e., the triple sequence spaces x is wijsman r− statistically convergent in the sense of definition (2.21) in general, the wijsman rough statistical limit of a triple sequence may not unique for the wijsman roughness degree r > 0. so we have to consider the so called wijsman r− statistical limit set of a triple sequence x = (xmnk) , which is defined by st−limramnk = {l ∈ r : amnk →rst a} . the triple sequence x is said to be wijsman r− statistically convergent provided that st−limramnk 6= φ. it is clear that if st−limramnk 6= φ for a triple sequence x = (xmnk) of real numbers, then we have st−limramnk = [st− lim sup amnk −r,st− lim inf amnk + r] (2.2) we know that limr = φ for an unbounded triple sequence x = (xmnk) . but such a triple sequence might be wijsman rough statistically convergent. for instance, define d (x,amnk) =   (−1)mnk , if (m,n,k) 6= (i,j,`)2 (i,j,` ∈ n) , (mnk) , otherwise   . in r. because the set {1, 64, 739, · · ·} has natural density zero, we have st−limramnk =   φ, if r < 1, [1 −r,r − 1] , otherwise   and limramnk = φ for all r ≥ 0. as can be seen by the example above, the fact that st−limramnk 6= φ does not imply limramnk 6= φ. because a finite set of natural numbers has natural density zero, limramnk 6= φ implies st−limramnk 6= φ. therefore, we get limramnk ⊆ st−limramnk. this obvious fact means {r ≥ 0 : limramnk 6= φ} ⊆ {r ≥ 0 : st−limramnk 6= φ} in this language of sets and yields immediately inf {r ≥ 0 : limramnk 6= φ}≥ inf {r ≥ 0 : st−limramnk 6= φ} . moreover, it also yields directly diam (limramnk) ≤ diam (st−limramnk) . throughout the paper, we let (x; ρ) be a separable metric space, i ⊆ 2n 3 be an admissible ideal and a; amnk be any non-empty closed subsets of x. int. j. anal. appl. 16 (5) (2018) 650 definition 2.21. a triple sequence (xmnk) is said to be wijsman r − i convergent to a, if for every � > 0 and for each x ∈ x, a (x,�) = { (m,n,k) ∈ n3 : |d (x,amnk) −d (x,a)| ≥ r + � } ∈ i definition 2.22. a triple sequence (xmnk) is said to be wijsman r − i statistical convergent to a, if for every � > 0,δ > 0 and for each x ∈ x,{ (r,s,t) ∈ n3 : 1 rst |{(r,s,t) ≤ (m,n,k) : |d (x,amnk) −d (x,a)| ≥ r + �}|≥ δ } ∈ i. in this case, we write amnk →s(iw ) a. definition 2.23. let θ be a lacunary sequence. a triple sequence (xmnk) is said to be wijsman strongly r − i convergent to a, if for every � > 0 and for each x ∈ x,{ (r,s,t) ∈ n3 : 1 hrst ∑ (m,n,k)∈irst |d (x,amnk) −d (x,a)| ≥ r + � } ∈ i. in this case, we write amnk →nθ(iw ) a. definition 2.24. a triple sequence (xmnk) is said to be wijsman r−i cesáro convergent to a, if for every � > 0 and for each x ∈ x,{ (r,s,t) ∈ n3 : ∣∣∣ 1rst ∑rm=1 ∑sn=1 ∑tk=1 d (x,amnk) −d (x,a)∣∣∣ ≥ r + �} ∈ i. in this case, we write amnk →c(iw ) a. definition 2.25. a triple sequence (xmnk) is said to be wijsman strongly r−i cesáro convergent to a, if for every � > 0 and for each x ∈ x,{ (r,s,t) ∈ n3 : 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| ≥ r + � } ∈ i. in this case, we write amnk →c(iw ) a. definition 2.26. a triple sequence (xmnk) is said to be wijsman p strongly r−i cesáro convergent to a, if for each p positive real number, if for every � > 0 and for each x ∈ x,{ (r,s,t) ∈ n3 : 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p ≥ r + � } ∈ i. in this case, we write amnk →cp(iw ) a. 3. main results theorem 3.1. let the triple sequence (amnk) ∈ λ3. if (amnk) is wijsman r − i statistical convergent to a, then (amnk) is wijsman p strongly r − i cesáro convergent to a. proof: suppose that (amnk) is triple analytic and amnk →s(iw ) a. then, there is an m > 0 such that |d (x,amnk) −d (x,a)| 1/m+n+k ≤ m, int. j. anal. appl. 16 (5) (2018) 651 for each x ∈ x and for all m,n,k. given � > 0, we have 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p/m+n+k = 1 rst ∑r m=1 ∑s n=1 ∑t k=1,|d(x,amnk)−d(x,a)|≥r+� |d (x,amnk) −d (x,a)| p/m+n+k + 1 rst ∑r m=1 ∑s n=1 ∑t k=1,|d(x,amnk)−d(x,a)|<r+� |d (x,amnk) −d (x,a)| p/m+n+k ≤ 1 rst mp/m+n+k ∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,amnk) −d (x,a)|1/m+n+k ≥ r + �}∣∣∣ + 1 rst �p/m+n+k ∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,amnk) −d (x,a)|1/m+n+k < r + �}∣∣∣ ≤ m p/m+n+k rst ∣∣∣{(m,n,k) ≤ (r,s,t) : |d (x,amnk) −d (x,a)|1/m+n+k ≥ r + �}∣∣∣ + �p/m+n+k. then for any δ > 0{ (r,s,t) ∈ n3 : 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p/m+n+k ≥ δ } ⊆{ (r,s,t) ∈ n3 : 1 rst |{(m,n,k) ≤ (r,s,t) |d (x,amnk) −d (x,a)| ≥ r + �}|≥ δ p/m+n+k mp/m+n+k } ∈ i, for each x ∈ x. hence amnk →cp(iw ) a. theorem 3.2. let the triple sequence (amnk) is wijsman p strongly r − i cesáro convergent to a then (amnk) is wijsman r − i statistical convergent to a. proof: let amnk →cp(iw ) a and given � > 0. then, we have∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p ≥∑r m=1 ∑s n=1 ∑t k=1,|d(x,amnk)−d(x,a)|≥r+� |d (x,amnk) −d (x,a)| p �p |{(m,n,k) ≤ (r,s,t) : |d (x,amnk) −d (x,a)| ≥ r + �}| for each x ∈ x and so 1 �p(rst) ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p ≥ 1 rst |{(m,n,k) ≤ (r,s,t) : |d (x,amnk) −d (x,a)| ≥ r + �}| . hence for given δ > 0{ (r,s,t) ∈ n3 : 1 rst |{(m,n,k) ≤ (r,s,t) |d (x,amnk) −d (x,a)| ≥ r + �}|≥ δ } ⊆{ (r,s,t) ∈ n3 : 1 rst ∑r m=1 ∑s n=1 ∑t k=1 |d (x,amnk) −d (x,a)| p ≥ (r + �)p δ } ∈ i, for each x ∈ x. hence amnk →s(iw ) a. theorem 3.3. let θ be a triple lacunary sequence (amnk) . if liminfuvwquvw > 1 then, amnk →c1(iw ) a ⇒ amnk →nθ(iw ) a. proof: if liminfuvwquvw > 1, then there exists δ > 0 such that quvw ≥ 1 + δ for all u,v,w ≥ 1. since huvw = (munvkw) − (mu−1nv−1kw−1) , we have munvkw hu−1,v−1,w−1 ≤ 1+δ δ and mu−1nv−1kw−1 hu−1,v−1,w−1 ≤ 1 δ . let � > 0 and we define the set s = { (munvkw) ∈ n3 : 1munvkw ∑mu m=1 ∑nv n=1 ∑kw k=1 |d (x,amnk) −d (x,a)| < r + � } , for each x ∈ x and also s ∈ f (i) , which is a filter of the ideal i, we have 1 huvw ∑ mnk∈iuvw |d (x,amnk) −d (x,a)| = 1 huvw ∑mu m=1 ∑nv n=1 ∑kw k=1 |d (x,amnk) −d (x,a)|− 1 huvw ∑mu−1 m=1 ∑nv−1 n=1 ∑kw−1 k=1 |d (x,amnk) −d (x,a)| int. j. anal. appl. 16 (5) (2018) 652 = mrnvkw huvw 1 mrnvkw ∑mu m=1 ∑nv n=1 ∑kw k=1 |d (x,amnk) −d (x,a)|− mr−1nv−1kw−1 huvw 1 mr−1nv−1kw−1 ∑mu−1 m=1 ∑nv−1 n=1 ∑kw−1 k=1 |d (x,amnk) −d (x,a)| ≤ ( 1+δ δ )( r + � ′ ) − 1 δ ( r + � ′ ) for each x ∈ x and (munvkw) ∈ s. choose η = ( 1+δ δ )( r + � ′ ) + 1 δ ( r + � ′ ) . therefore, for each x ∈ x{ (u,v,w) ∈ n3 : 1 huvw ∑ mnk∈iuvw |d (x,amnk) −d (x,a)| < η } ∈ f (i) . theorem 3.4. let θ be a triple lacunary sequence (amnk) . if limsupuvwquvw < ∞ then, amnk →nθ(iw ) a ⇒ amnk →c1(iw ) a. proof: if limsupuvwquvw < ∞ then there exists m > 0 such that quvw < m, for all u,v,w ≥ 1. let amnk →nθ(iw ) a and we define the sets t and r such that t = { (u,v,w) ∈ n3 : 1 huvw ∑ mnk∈iuvw |d (x,amnk) −d (x,a)| < r + �1 } and r = { (a,b,c) ∈ n3 : 1 rst ∑a m=1 ∑b n=1 ∑c k=1 |d (x,amnk) −d (x,a)| < r + �2 } , for every �1,�2 > 0 for each x ∈ x. let aj = 1 hj ∑ mnk∈ij |d (x,amnk) −d (x,a)| < r + �1 for each x ∈ x and for all j ∈ t. it is obvious that t ∈ f (i) . choose (a,b,c) in any integer with (mu−1nv−1kw−1) < (a,b,c) < (munvkw) , where (u,v,w) ∈ t. then, we have 1 huvw ∑ mnk∈iuvw |d (x,amnk) −d (x,a)| ≤ 1 mu−1nv−1kw−1 ∑mu m=1 ∑nv n=1 ∑kw k=1 |d (x,amnk) −d (x,a)| = 1 mu−1nv−1kw−1  ∑(m,n,k)∈i111 |d (x,amnk) −d (x,a)| + ∑(m,n,k)∈i222 |d (x,amnk) −d (x,a)|+ · · · + ∑ (m,n,k)∈iuvw |d (x,amnk) −d (x,a)|   = m1n1k1 mu−1nv−1kw−1 ( 1 h111 ∑ (m,n,k)∈i111 |d (x,amnk) −d (x,a)| ) + (m2n2k2)−(m1n1k1) mu−1nv−1kw−1 ( 1 h222 ∑ (m,n,k)∈i222 |d (x,amnk) −d (x,a)| ) + · · · (munvkw)−(mu−1nv−1kw−1) mu−1nv−1kw−1 ( 1 huvw ∑ (m,n,k)∈iuvw |d (x,amnk) −d (x,a)| ) = m1n1k1 mu−1nv−1kw−1 a111 + (m2n2k2)−(m1n1k1) mu−1nv−1kw−1 a222 + · · · + (munvkw)−(mu−1nv−1kw−1) mu−1nv−1kw−1 auvw ≤ (supj∈taj) · m1n1k1mu−1nv−1kw−1 < r + �1 ·m for each x ∈ x. choose r + �2 = r+�1 m and the fact that⋃ {(a,b,c) : mu−1 < a < mu,nv−1 < b < nv,kw−1 < c < kw, (u,v,w) ∈ t}⊂ r, where t ∈ f (i) . it follows from our assumption on θ that the set r ∈ f (i) . competing interests: the authors declare that there is not any conflict of interests regarding the publication of this manuscript. references [1] s. aytar, rough statistical convergence, numer. funct. anal. optim. 29(3) (2008), 291-303. [2] a. esi , on some triple almost lacunary sequence spaces defined by orlicz functions, res. rev., discr. math. struct. 1(2) (2014), 16-25. int. j. anal. appl. 16 (5) (2018) 653 [3] a. esi and m. necdet catalbas,almost convergence of triple sequences, glob. j. math. anal. 2(1) (2014), 6-10. [4] a. esi and e. savas, on lacunary statistically convergent triple sequences in probabilistic normed space, appl. math. inf. sci. 9 (5)(2015), 2529-2534. [5] a. j. dutta,a. esi and b.c. tripathy,statistically convergent triple sequence spaces defined by orlicz function, j. math. anal. 4(2)(2013), 16-22. [6] s. debnath, b. sarma and b.c. das ,some generalized triple sequence spaces of real numbers , j. nonlinear anal. optim. 6(1) (2015), 71-79. [7] s.k. pal, d. chandra and s. dutta, rough ideal convergence, hacee. j. math. stat. 42(6)(2013), 633-640. [8] h.x. phu, rough convergence in normed linear spaces, numer. funct. anal. optim. 22(2001), 201-224. [9] a. sahiner, m. gurdal and f.k. duden, triple sequences and their statistical convergence, selcuk j. appl. math. 8 (2)(2007), 49-55. [10] a. sahiner and b.c. tripathy , some i related properties of triple sequences, selcuk j. appl. math. 9 (2)(2008), 9-18. [11] n. subramanian and a. esi, the generalized tripled difference of χ3 sequence spaces, glob. j. math. anal. 3 (2)(2015), 54-60. [12] m. aiyub, a. esi and n. subramanian, the triple entire difference ideal of fuzzy real numbers over fuzzy p− metric spaces defined by musielak orlicz function, j. intell. fuzzy syst. 33(2017), 1505-1512. 1. introduction 2. definitions and preliminaries 3. main results references int. j. anal. appl. (2023), 21:78 some aspects of rectifying curves on regular surfaces under different transformations sandeep sharma∗, kuljeet singh school of mathematics, shri mata vaishno devi university, katra-182320, jammu and kashmir, india; kulljeet83@gmail.com ∗corresponding author: sandeep.greater123@gmail.com abstract. an essential space curve in the study of differential geometry is the rectifying curve. in this paper, we studied the adequate requirement for a rectifying curve under the isometry of the surfaces. the normal components of the rectifying curves are also studied, and it is investigated that for rectifying curves, the christoffel symbols and the normal components along the surface normal are invariant under the isometric transformation. moreover, we also studied some properties for the first fundamental form of the surfaces. 1. introduction differential geometry is the area of geometry that employs calculus to study the characteristics of curves and surfaces of all kinds. it primarily focuses on the features of a small subset of geometric configurations of curves and surfaces. different types of curves are explored in differential geometry, but the regular curves are the most significant ones. the number of continuous derivatives is a characteristic that indicates how smooth a curve is. if a curve is differentiable and thus continuous everywhere, it is considered to be smooth. similar to this, if a curve can be differentiated and has no zero derivative, it is said to be regular. in differential geometry, the study of regular maps is a key area of research. for more information on the regular curve, we can refer the reader to see [3]. there are many ways to categorise motions, but we’ll concentrate on the ones that preserve particular geometrical characteristics. we categorise transformations generally into the following equivalence classes: conformal, isometric, homothetic, and non-conformal or general motion, depending on the received: may 8, 2023. 2020 mathematics subject classification. 53a04, 53a15, 53a35, 53b30. key words and phrases. geodesic curvature; first fundamental form; orthonormal frame; isometry of surfaces. https://doi.org/10.28924/2291-8639-21-2023-78 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-78 2 int. j. anal. appl. (2023), 21:78 varying nature of the mean curvature (m) and the gaussian curvature (g). in isometry, lengths and angles between curves on surfaces are both preserved. in terms of geometry, isometry preserves the gaussian curvature’s invariance while altering the mean curvature. the isometry between a helicoid and a catenoid, which suggests that they have the same g but different m, is one of the best known examples. the most significant transformation is a conformal transformation, which preserves angles in terms of magnitude and direction but not always in terms of length. conformal maps play a significant role in cartography. the stereographic projection, which maps a sphere onto a plane, is the most typical example of conformal transformation. in the year 1569, gerardus mercator initially used this conformal map characteristic to produce the legendary mercator’s globe map, the first conformal world map. we suggest that the reader watch an animated movie on conformal maps that was released by bobenko and gunn in 2018 along with springer videomath for additional details regarding the application of conformal maps [7]. angles and distances between any pair of intersecting curves are not preserved in the context of general motion. the use of motion, transformation, and maps is for same throughout the paper. the normal, rectifying, and osculating curves are the most often covered topics in differential geometry, therefore they are typically covered in every basic book on differential geometry of curves and surfaces. for more information on these topics, we can refer the reader to see [1, 3, 6]. in the euclidean 3-dimensional space r3, chen et al. [4, 10] studied the motion of rectifying curves and investigated some of the basic properties of such curves. shaikh et al. [1, 2, 6, 13] investigated the sufficient conditions for the invariance of the conformal image of osculating and normal curves on smooth immersed surfaces and found that there are various other properties of such curves that remain invariant under the isometry of surfaces. in the year 2003, chen [4] came across the following query regarding rectifying curves: what occurs when a space curve’s position vector is always within the range of its rectifying plane. it was found that, under the assumption that surfaces are isometric, the component of the position vector of a space curve along the surface normal remains constant. this paper’s primary objective is to expand on the work of lone et al. [5, 12, 13], they studied the geometric invariants of normal curves under conformal transformation in the euclidean space e3. in [5],the author explored the behaviour of the normal and tangential components of the normal curves under the same motion as well as the invariant characteristics of normal curves under conformal transformation. by motivating from the work of shaikh and ghosh in the recent papers [2,6], where they studied regarding the geometric invariants properties of rectifying curves on smooth surface under isometry of the surfaces. further in [1], they investigated the invariant properties of osculating curves under the isometry of surfaces. but a natural question arises: what happens with the geometric properties of rectifying curves with respect to conformal transformation in euclidean space r3. int. j. anal. appl. (2023), 21:78 3 in the present paper, we attempt to investigate the conformal image for rectifying curve on regular surfaces under the different transformation. moreover, we also study some geometric properties for the first fundamental form of the surfaces, as well as the derivation for the normal components of a rectifying curve under the conformal transformation. this article demonstrated that the christoffel symbols and normal component for the rectifying curves are also invariant under the isometry of the surfaces. the format of this paper is as follows: section 2 covers some fundamental definitions and information about dilation function, geodesic curvature, normal curvature, rectifying curves, and normal curves. section 3 deals with the understanding of rectifying curves on regular surfaces and their conformal image under various transformations. we also examine the major results in this section. 2. preliminaries this section includes some essential information on rectifying curves, including their first fundamental form, geodesic and normal curvature, and some basic definitions. let p and p̃ be two smooth and regular immersed surfaces in the euclidean space r3, and g : p → p̃ be a smooth map. a necessary and sufficient condition for g to be conformal is the first fundamental form quantities being proportional. in other words, the area element of p and p̃ are proportional to a differentiable function (factor), which is denoted by ζ(x,y) and is commonly known as dilation function. for more information on the dilation function, we can refer the reader to see [3,8,10]. a generalised class of certain motions is the conformal transformation, which is defined in the following way [8]: • when the dilation factor ζ(x,y) = c, where c is a constant with c 6= {0, 1}, then g is a homothetic transformation. • when the function ζ(x,y) = 1, then g becomes isometry. let δ : i ⊂ r → r3 be unit speed smooth parametrized curve with at least fourth order continuous derivative, by an arc length parameter (r). let the tangent, normal, and binormal of the curve δ is denoted by ~t, ~n and ~b respectively. at each point on the curve δ(r), the vectors ~t, ~n, and ~b are mutually perpandicular to each other and the triplet {~t,~n,~b} so forms an orthonormal frame. consider ~t′(r) 6= 0, the unit normal vector ~n along the tangents at a point on the curve δ, then we can write ~t′(r) = κ(r)~n(r), where ~t′(r) is the derivative of ~t with respect to arc length parameter ‘r’ and κ(r) is the curvature of δ(r). also the binormal vector field is denoted by ~b and is defined by ~b = ~t × ~n, and we can write ~b′(r) = τ(r)~n(r), where τ(r) is another curvature function known as torsion of the curve δ(r). 4 int. j. anal. appl. (2023), 21:78 in [3,11,15], serret-frenet equations are given as follows: ~t′(r) = κ(r)~n(r), ~n′(r) = −κ(r)~t(r) + τ(r)~b(r), ~b′(r) = −τ(r)~n(r), where the functions κ and τ are respectively called the curvature and torsion of the curve δ, satisfying the following conditions: ~t(r) = δ′(r),~n(r) = ~t′(r) κ(r) and ~b(r) = ~t(r) ×~n(r). from the arbitrary point δ(r) on the curve δ, we see that the plane spanned by {~t,~n} is called the osculating plane, and the plane spanned by {~t,~b} is called the rectifying plane. in the same way the plane spanned by {~n,~b} is called the normal plane. whenever we talk about the position vector of the curve, which defines the different kinds of curves [12,16,17] : • it is possible to define a curve as being a normal curve if its position vector is in the normal plane. • a curve is said to be a rectifying curve if its position vector is in the rectifying plane. • if a curve’s position vector is in the osculating plane, then the curve is said to be an osculating curve. firstly, try to investigate the properties of a rectifying curve on regular surfaces are invariant under conformal transformation. if the position vector of a curve is located in the rectifying plane, then the curve is said to be a rectifying curve [8,12,15], i.e., δ(r) = µ1(r)~t(r) + µ2(r)~b(r), (2.1) where µ1 and µ2 are two smooth functions. let σ : u → p be the coordinate chart map on the regular surface p and the smooth parametrized unit speed curve δ(r) : i → p, where i = (a , b) ⊂r and u⊂r2. as a result, the curve δ(r) is given by δ(r) = σ(x(r),y(r)). (2.2) by using chain rule to differentiate (2.2), with respect to r, we get δ′(r) = σxx ′ + σyy ′. (2.3) now, ~t(r) = δ′(r). then, from equation (2.3), we find that ~t(r) = σxx ′ + σyy ′. (2.4) int. j. anal. appl. (2023), 21:78 5 when we differentiate equation (2.4) again in terms of r, we obtained ~t′(r) = x′′σx + y ′′σy + x ′2σxx + 2x ′y ′σxy + y ′2σyy. if n is the normal to the surface p and κ(r) is the curvature of the curve δ(r), then the normal vector ~n(r) can be written as ~n(r) = 1 κ(r) (x′′σx + y ′′σy + x ′2σxx + 2x ′y ′σxy + y ′2σyy). (2.5) now the binormal vector ~b(r) can be written as ~b(r) = ~t(r) ×~n(r). by substituting the value of ~t(r) and ~n(r) from equation (2.4) and (2.5) we obtained ~b(r) = 1 κ(r) [(σxx ′ + σyy ′) × (x′′σx + y ′′σy + x′2σxx + 2x′y ′σxy + y ′2σyy)], = 1 κ(r) [(y ′′x′ −y ′x′′)n + x′3σx ×σxx + 2x′2y ′σx ×σxy + x′y ′2σx ×σyy +x′2y ′σy ×σxx + 2x′y ′2σy ×σxy + y ′3σy ×σyy]. (2.6) 2.1. first fundamental form. let σ = σ(x,y) represents the equation of a surface. let us define e = (σx ·σx), f = (σx ·σy), g = (σy ·σy). then the expression edx2 + 2fdxdy + gdy2 is called the first fundamental form, and e, f, and g are called the first fundamental form coefficient or the first fundamental form magnitude. note that in the above expression, dx and dy cannot vanish together. we denote notion of √ eg −f2 by h. a necessary and sufficient condition for the surfaces p and p̃ to be isometric is that the first fundamental form magnitude are invariant, i.e., ẽ = e, f̃ = f, g̃ = g. for more detail one can refer [5]. some of the main results concerning the first fundamental form are given as follows: theorem 2.1. let p and p̃ be two regular surfaces in the euclidean space r3 and e, f, g are the coefficients of the first fundamental form of the surfaces. then (i) the first fundamental form is the square of the metric. (ii) the first fundamental form is positive definite form. (iii) h = |σx ×σy|. proof. let x = x(r), y = y(r) be the curve on the surface σ = σ(x,y). let q(σ) and r(σ + dσ) be two neighbouring points on the curve corresponding to the parameter (x,y) and (x + dx,y + dy) respectively, such that arc (qr) = ds. then dσ = ∂σ ∂x dx + ∂σ ∂y dy = σxdx + σydy. 6 int. j. anal. appl. (2023), 21:78 since q and r are neighbouring points, therefore arc(qr) = chord (qr), i.e., ds2 = |dσ|2, = (dσ) · (dσ), = (σxdx + σydy) · (σxdx + σydy), = σx ·σxdx2 + 2σx ·σydxdy + σy ·σydy2, = edx2 + 2fdxdy + gdy2. this proves that the first fundamental form is the square of the metric. proof (ii): to prove the positive definiteness of the first fundamental form, it is sufficient to show that, for all real values of dx and dy, the expression for the first fundamental form is greater than 0. now, edx2 + 2fdxdy + gdy2 = 1 e [e2dx2 + 2efdxdy + egdy2], = 1 e [(edx + fdy)2 + (eg −f2)dy2], = 1 e [(edx + fdy)2 + h2dy2], ≥ 0 for all real values of dx and dy. if 1 e [(edx + fdy)2 + h2dy2] = 0 ⇒ edx + fdy = 0 and hdy = 0 ⇒ edx + fdy = 0 and dy = 0, as h 6= 0 ⇒ edx = 0, and dy = 0 ⇒ dx = 0, dy = 0, which are not possible because dx and dy cannot vanish together. thus, edx2 + 2fdxdy + gdy2 > 0, for all real values of dx and dy. it means that the first fundamental form for the surface is positive definite. proof (iii): we denote √ eg −f2 by h. therefore, we can write h2 = eg −f2, = (σx ·σx)(σy ·σy) − (σx ·σy)2, = σ2xσ 2 y −σ 2 xσ 2 ycos 2(θ), = σ2xσ 2 y(1 −cos 2(θ)), = σ2xσ 2 ysin 2(θ), = (σ2x ×σ 2 y) · (σ 2 x ×σ 2 y), = |σx ×σy|2. this proves (iii). � int. j. anal. appl. (2023), 21:78 7 definition 2.1. let p and p̃ be two regular surfaces in r3. then a diffeomorphism g : p→p̃ is an isometry if g maps the curve of same length from p to p̃. definition 2.2. [4] let p and p̃ be two regular surfaces in the euclidean space r3 and δ(r) be a curve having arc length parametrization lies on the surface p. then δ′(r) is perpandicular to the unit surface normal n, and also δ′(r) and δ′′(r) are perpandicular. thus, δ′′ can be represented as the linear combination of n and n×δ′, i.e., δ′′ = κnn + κgn×δ′, where the parameters κn and κg, which are commonly known as the normal and geodesic curvatures of the curve δ, and are given by κn = δ ′′·n, κg = δ ′′·(n×δ′). now from serret-frenet equations we have ~t′(r) = δ′′(r) = κ(r)~n(r). now, κn = δ ′′·n, = κ(r)~n(r)·n, = (x′′σx + y ′′σy + x ′2σxx + 2x ′y ′σxy + y ′2σyy)·n, = (x′′σx + y ′′σy + x ′2σxx + 2x ′y ′σxy + y ′2σyy)·(σx×σy). by solving the above expression and using the properties of vectors, we find that κn = x ′2x + 2x′y ′y + y ′2z, where x, y, and z are the magnitudes of second fundamental form [3,11,14]. this leads us to the conclusion that the curve δ(r) on the surface p, is asymptotic if and only if the normal curvature κn = 0 [9,13]. 3. conformal image of a rectifying curve consider two regular surfaces p and p̃ in the euclidean space r3 and δ(r) is a rectifying curve that is located on the surface p . then δ(r) can be written as: δ(r) = µ1(r)~t(r) + µ2(r)~b(r). now using equation (2.4) and (2.6) we get, δ(r) = µ1(r)(σxx ′ + σyy ′) + µ2(r) 1 κ(r) {(x′y ′′ −x′′y ′)n + x′3σx ×σxx + 2x′2y ′σx ×σxy +x′y ′2σx ×σyy + x′2y ′σy ×σxx + 2x′y ′2σy ×σxy + y ′3σy ×σyy}. (3.1) 8 int. j. anal. appl. (2023), 21:78 theorem 3.1. let g : p → p̃ be an isometry between two regular and smooth surfaces p and p̃ in the euclidean space r3 and δ(r) be a rectifying curve on the surface p . then the christoffel symbols for δ(r) are invariant under g. proof. since g : p → p̃ is an isometry and e = (σx ·σx), f = (σx ·σy), g = (σy ·σy). as e, f and g are the function of both x and y, then on differentiate with regard to x and y we find that ex = (σx ·σx)x = 2σxx ·σx ⇒ σxx ·σx = ex 2 (3.2) similarly, we can find σxx ·σy = fx − ey 2 , σxy ·σx = ey 2 , σxy ·σy = gx 2 , σyy ·σx = fy − gx 2 , σyy ·σy = gy 2 .   (3.3) let γnlm, where{l,m,n = 1, 2}, be the christoffel symbols of second kind. then we have, γ111 = 1 2h2 {gex + f [ey − 2fx]}, γ211 = 1 2h2 {e[2fx −ey] −fey}, γ212 = 1 2h2 {egx −fey} = γ221, γ222 = 1 2h2 {egy + f [gy − 2fy]}, γ122 = 1 2h2 {g[2fy −gx] −fgy}, γ121 = 1 2h2 {gey −fgx} = γ112,   (3.4) where h = √ eg −f2. now, for the conformal transformation these christoffel symbols taking the form γ̃111 = γ 1 11 + θ 1 11, γ̃ 2 11 = γ 2 11 + θ 2 11, γ̃ 1 12 = γ 1 12 + θ 1 12, γ̃212 = γ 2 12 + θ 2 12, γ̃ 1 22 = γ 1 22 + θ 1 22, γ̃ 2 22 = γ 2 22 + θ 2 22,   (3.5) where θ111 = egζx − 2f2ζx + feζy ζh2 θ211 = efζx −e2ζy ζh2 , θ112 = egζy −fgζx ζh2 , θ212 = egζx −feζy ζh2 , θ122 = gfζy −g2ζx ζh2 , θ222 = egζy − 2f2ζv + fgζu ζh2 .   (3.6) int. j. anal. appl. (2023), 21:78 9 now for the isometry between the surface we have ζ = 1, then on putting ζ = 1 in equation (3.6), we find that all θnlm = 0, for l,m,n = {1, 2}. on putting θ n l,m = 0 for l,m,n = {1, 2}, in equation (3.5) we get γ̃nlm = γ n lm. this proves that for an isometry, the christoffel symbols are invariant. � theorem 3.2. let g : p → p̃ be a conformal map between two regular and smooth surfaces p and p̃ in the euclidean space r3 and δ(r) be a rectifying curve on the surface p. then the normal component of the curve δ(r) along the surface normal n, satisfying the following conditions: δ̃(r) · ñ−ζ4δ(r) ·n = µ2(r) κ(r) ζ4h2(x′3θ211 −y ′3θ122 + 2x ′2y ′θ212 + x ′y ′2θ222 + x ′2y ′θ112 + 2x ′y ′2θ112). proof. given that p̃ is the conformal image of p under the map g, and δ(r) is a rectifying curve on p. let σ(x,y) and σ̃(x,y) be the surface patches of p and p̃ respectively, and σ̃(x,y) = g◦σ(x,y). we know that e = (σx ·σx), f = (σx ·σy), g = (σy ·σy). now for dilation function ζ we have ẽ = ζ2e, f̃ = ζ2f, g̃ = ζ2g. (3.7) on differentiating the above terms with respect to both x and y, we get ẽx = 2ζζxe + ζ 2ex, ẽy = 2ζζye + ζ 2ey, f̃x = 2ζζxf + ζ 2fx, f̃y = 2ζζyf + ζ 2fy, g̃x = 2ζζxg + ζ 2gx, g̃y = 2ζζyg + ζ 2gy.   (3.8) now for finding the normal component of the curve δ(r) along the surface normal n, we have δ(r) ·n = [µ1(r)(σxx′ + σyy ′) + µ2(r) 1 κ(r) {(x′y ′′ −x′′y ′)n + x′3σx ×σxx + 2x′2y ′σx ×σxy +x′y ′2σx ×σyy + x′2y ′σy ×σxx + 2x′y ′2σy ×σxy + y ′3σy ×σyy}] ·n, = µ1(r)(σxx ′ + σyy ′) ·n + [µ2(r) 1 κ(r) {(x′y ′′ −x′′y ′)n + x′3σx ×σxx + 2x′2y ′σx ×σxy +x′y ′2σx ×σyy + x′2y ′σy ×σxx + 2x′y ′2σy ×σxy + y ′3σy ×σyy}] ·n, = µ1(r)(σxx ′ + σyy ′) · (σx ×σy) + µ2(r) κ(r) {(x′y ′′ −x′′y ′)(eg −f2) + x′3(σx ×σxx) · (σx ×σy) +2x′2y ′(σx ×σxy)(σx ×σy) + x′y ′2(σx ×σyy) · (σx ×σy) + x′2y ′(σy ×σxx) · (σx ×σy) +y ′3(σy ×σyy) · (σx ×σy), = µ2(r) κ(r) {(x′y ′′ −x′′y ′)(eg −f2) + x′3{e(fx − ey 2 ) − fex 2 } + 2x′2y ′{ egx 2 − fey 2 } +x′y ′2{ egy 2 −f (fy − gx 2 )} + x′2y ′{ fgx 2 − gex 2 } + 2x′y ′2{ fgx 2 − gey 2 } +y ′3{ fgy 2 −g(fy − gx 2 )}}. (3.9) 10 int. j. anal. appl. (2023), 21:78 now on using equation (3.4) in equation (3.9), we obtained δ(r) ·n = µ2(r) κ(r) (eg −f2){(x′y ′′ −x′′y ′) + x′3γ211 + 2x ′2y ′γ212 + x ′y ′2γ222 + x ′2y ′γ112 +2x′y ′2γ112 −y ′3γ122}. (3.10) thus in view of (3.5), (3.7), and (3.10), the above equation can be written as δ̃(r) · ñ−ζ4δ(r) ·n = µ2(r) κ(r) ζ4h2(x′3θ211 + 2x ′2y ′θ212 + x ′y ′2θ222 + x ′2y ′θ112 + 2x ′y ′2θ112 −y ′3θ122). � corollary 3.1. let g : p → p̃ be an isometry between two regular and smooth surfaces p and p̃ in the euclidean space r3 and δ(r) be a rectifying curve on the surface p . then the normal component of the curve δ(r) along the surface normal n is invariant under the isometry g, i.e. δ̃(r) ·ñ = δ(r) ·n. proof. since for an isometry the dilation function ζ = 1, if we put ζ = 1 in the theorem (3.2), we obtained δ̃(r) · ñ = δ(r) ·n. � 4. conclusion in this article, we investigate some geometric properties for the first fundamental form of the surfaces. we also introduce the invariant properties for a class of curves, namely rectifying curves, and their geometric invariance under isometric transformations. we came up with a derivation for the rectifying curves’ normal components and discovered that both the christoffel symbols and the normal components are isometrically invariant. in future, one can study some other properties of the first and second fundamental forms of the surface. one can also make some new and intresting results about the conformal image of other classes of curves, namely osculating and normal curves under conformal transformation. moreover, one can also check for these curves, the normal and tangential components, normal and geodesic curvature, and the christoffel symbols of the first and second kinds are invariant under these transformations. ethical approval : there are no studies by any of the authors, either with human subjects or with animals, in the current article. data availability: since no new data was created or examined in this study, so data sharing is not applicable to this work. author contribution: all authors have made equal contributions while preparing this article. the author read and approved the final manuscript. funding: the authors received no specific funding to support this study. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2023), 21:78 11 references [1] a.a. shaikh, m.s. lone, p.r. ghosh, conformal image of an osculating curve on a smooth immersed surface, j. geom. phys. 151 (2020), 103625. https://doi.org/10.1016/j.geomphys.2020.103625. [2] a.a. shaikh, p.r. ghosh, rectifying curves on a smooth surface immersed in the euclidean space, indian j. pure appl. math. 50 (2019), 883-890. https://doi.org/10.1007/s13226-019-0361-4. [3] m.p. do carmo, differential geometry of curves & surfaces: revised & updated, second ed., dover publications, mineola, new york, (2016). [4] b.y. chen, when does the position vector of a space curve always lie in its rectifying plane?, amer. math. mon. 110 (2003), 147-152. https://doi.org/10.1080/00029890.2003.11919949. [5] m.s. lone, geometric invariants of normal curves under conformal transformation in e3, tamkang j. math. 53 (2022), 75-87. https://doi.org/10.5556/j.tkjm.53.2022.3611. [6] a.a. shaikh, p.r. ghosh, rectifying and osculating curves on a smooth surface, indian j. pure appl. math. 51 (2020), 67-75. https://doi.org/10.1007/s13226-020-0385-9. [7] a.i. bobenko, c. gunn, dvd-video pal, 15 minutes, in: springer videomath, springer, 2018. https://www. springer.com/us/book/9783319734736. [8] s. deshmukh, b.y. chen, s.h. alshammari, on rectifying curves in euclidean 3-space, turk. j. math. 42 (2018), 609-620. https://doi.org/10.3906/mat-1701-52. [9] m. he, d.b. goldgof, c. kambhamettu, variation of gaussian curvature under conformal mapping and its application, computers math. appl. 26 (1993), 63-74. https://doi.org/10.1016/0898-1221(93)90086-b. [10] b.y. chen, f. dillen, rectfying curve as centrode and extremal curve, bull. inst. math. acad. sinica, 33 (2005), 77-90. [11] k. ilarslan, e. nešović, timelike and null normal curves in minkowski space e31, indian j. pure appl. math. 35 (2004), 881-888. [12] a.a. shaikh, m.s. lone, p.r. ghosh, rectifying curves under conformal transformation, j. geom. phys. 163 (2021), 104117. https://doi.org/10.1016/j.geomphys.2021.104117. [13] a.a. shaikh, m.s. lone, p.r. ghosh, normal curves on a smooth immersed surface, indian j. pure appl. math. 51 (2020), 1343-1355. https://doi.org/10.1007/s13226-020-0469-6. [14] f. schwarz, transformation to canonical form, in: algorithmic lie theory for solving ordinary differential equations, 257-320, (2007). [15] a. yadav, b. pal, some characterizations of rectifying curves on a smooth surface in euclidean 3-space, arxiv:2104.02907 [math.dg], (2021). https://doi.org/10.48550/arxiv.2104.02907. [16] c. camci, l. kula, k. ilarslan, characterizations of the position vector of a surface curve in euclidean 3-space, an. s, tiint,. univ. "ovidius" constant,a, ser. mat. 19 (2011), 59-70. [17] a.a. shaikh, p.r. ghosh, curves on a smooth surface with position vectors lie in the tangent plane, indian j. pure appl. math. 51 (2020), 1097-1104. https://doi.org/10.1007/s13226-020-0452-2. https://doi.org/10.1016/j.geomphys.2020.103625 https://doi.org/10.1007/s13226-019-0361-4 https://doi.org/10.1080/00029890.2003.11919949 https://doi.org/10.5556/j.tkjm.53.2022.3611 https://doi.org/10.1007/s13226-020-0385-9 https://www.springer.com/us/book/9783319734736 https://www.springer.com/us/book/9783319734736 https://doi.org/10.3906/mat-1701-52 https://doi.org/10.1016/0898-1221(93)90086-b https://doi.org/10.1016/j.geomphys.2021.104117 https://doi.org/10.1007/s13226-020-0469-6 https://doi.org/10.48550/arxiv.2104.02907 https://doi.org/10.1007/s13226-020-0452-2 1. introduction 2. preliminaries 2.1. first fundamental form 3. conformal image of a rectifying curve 4. conclusion references international journal of analysis and applications volume 16, number 3 (2018), 353-367 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-353 characterization of nash equilibrium strategy for heptagonal fuzzy games f. madandar1, s. haghayeghi1 and s. m. vaezpour2,∗ 1department of mathematics, karaj branch, islamic azad university, karaj, iran 2department of mathematics and computer science, amirkabir university of technology, tehran, iran ∗corresponding author: vaez@aut.ac.ir abstract. in this paper, the nash equilibrium strategy of two-person zero-sum games with heptagonal fuzzy payoffs is considered and the existence of nash equilibrium strategy is studied. also, based on the fuzzy max order several models in heptagonal fuzzy environment is constructed and the existence of their equilibrium strategies is proposed. in the sequel, the existence of pareto nash equilibrium strategies and weak pareto nash equilibrium strategies is investigated for fuzzy matrix games. finally, the relation between pareto nash equilibrium strategy and parametric bi-matrix games is established. 1. introduction modern game theory was developed by the mathematician john von neumann in the mid-1940‘s and in 1944, he published the book of ”theory of games and economic behavior” joint with morgenstern [9]. the most important categories are as cooperative and non-cooperative games. in 1951, non-cooperative games was presented by john nash [8]. in this article we focus on a class of non-cooperative games namely two-person zero-sum matrix games. moreover, one of the most important concepts in game theory is the nash equilibrium. nash proves that if we approve mixed strategies, then every game with a finite number of players has at least one nash equilibrium. received 2017-12-13; accepted 2018-02-05; published 2018-05-02. 2010 mathematics subject classification. 91a05. key words and phrases. zero-sum matrix game; fuzzy payoffs; nash equilibrium; heptagonal fuzzy number. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 353 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-353 int. j. anal. appl. 16 (3) (2018) 354 in 1978, the idea of fuzziness was exhibited by zadeh [15], that is a type of subjective uncertainty. after then fuzzy numbers found many applications in various fields with incomplete information such as engineering, social and economics. in many science such as economics, business competition, auction and etc., the payoff for games is not realistic indeed the payoffs are fuzzy. many of mathematicians and reseachers study the fuzziness. zimmermann [16] in 1985, yazenin [13] in 1987 and sakawa [12] in 1993 applied the fuzzy theory to optimization problems. in 1989, campos [2] transformed the fuzzy games into fuzzy optimization problems. in 1999, liu [5] founded theory in the uncertain environments. in 2000, meada [7] constructed kind of concepts of minimax equilibrium strategies. in 2005, c. r. bector and s. chandra [1] provided fuzzy mathematical programming and fuzzy matrix games. in 2011, cunlin and zhang qiang [4] investigated two-person zero-sum games in the symmetric triangular fuzzy environment. they obtained nash equilibrium of two-person zero-sum games with fuzzy payoff. they also obtained pareto nash equilibrium strategy for fuzzy matrix game. in 2014, bapi dutta [3] extended their work in trapezoidal fuzzy environment and he introduced two special types of fuzzy games: constant and proportional fuzzy game. in [3, 4, 7] the uncertainty and imprecision in payoffs have been represented by either triangular or trapezoidal fuzzy numbers. the most frequently used fuzzy numbers in the different problems are triangular or trapezoidal fuzzy numbers. but, it is not possible to restrict the membership functions to take either triangular or trapezoidal form. therefore this paper focus on fuzzy payoffs of decision makers by heptagonal fuzzy numbers. in 2014, the arithmetic operations of heptagonal fuzzy numbers are defined by k. rathi and s. balamohan [10]. the heptagonal fuzzy number gives additional possibility to represent imperfect knowledge what leads to model many problems. heptagonal fuzzy numbers have different applications in optimization problems and decision making problems which need seven parameters. in this paper we define the k-heptagonal fuzzy numbers and generalize cunlin and qiang [4] and bapi dutta [3] nash equilibrium solution concepts. the paper is organized as follows: in section 2, the basic definitions and notations of fuzzy numbers are given. in section 3, we introduce the notation of two-person zero-sum matrix games with heptagonal fuzzy payoffs, the different types of equilibrium strategies and investigate their existence conditions for the fuzzy games. in section 4, parametric bi-matrix games are introduced and then the relation between parametric bi-matrix games and nash equilibrium strategies is studied. in section 5, we present some illustrative exampes. 2. preliminaries in this section, we suggest some basic definitions and concepts of fuzzy numbers, which were proposed by zadeh [14] in 1965. also, we introduce some notations of fuzzy sets, such as α-cut for heptagonal fuzzy numbers. int. j. anal. appl. 16 (3) (2018) 355 definition 2.1. [4] a fuzzy number ã is a fuzzy set on the real line r if its membership function µã(x) satisfies the following conditions. (i) µã(x) is a mapping from r to the closed interval [0, 1]; (ii) there exist a unique real number c, called the center of ã , such that ; (a) µã(c) = 1 ; (b) µã(x) nondecreasing on (−∞,c]; (c) µã(x) nonincreasing on [c, +∞). the α-cut or α-level of fuzzy number have an important role in parametric ordering of fuzzy numbers. the α-cut set of a fuzzy number ã, denoted by [ã]α. every α-cut is a closed interval [ã]α = [a l α,a u α ] ⊂ r, where alα = inf{x ∈ r|µã(x) ≥ α} and auα = sup{x ∈ r|µã(x) ≥ α} for any α ∈ [0, 1]. for more details see [4, 6]. definition 2.2. a fuzzy number ã = (a,c,b,h,l,r,m) is called a k-heptagonal if its membership function is defined as µã(x) =   k(x−a+h h ) ,a−h ≤ x ≤ a, k ,a ≤ x ≤ c− l, k + ( x−c+l l ) ,c− l ≤ x ≤ c, k + ( c+r−x r ) ,c ≤ x ≤ c + r, k ,c + r < x < b + m, 0 ,otherwise. where c is the center of ã and h,l,r,m are non-negative. in the rest of the paper, for simplicity, the k-heptagonal fuzzy number is denoted by k-hfn. let ã = (a1,c1,b1,h1, l1,r1,m1) and b̃ = (a2,c2,b2,h2, l2,r2,m2) are two k-hfn then • addition: ã + b̃ = (a1 + a2,c1 + c2,b1 + b2,h1 + h2, l1 + l2,r1 + r2,m1 + m2) • subtraction: ã− b̃ = (a1 −m2,c1 −r2,b1 − l2,h1 −h2, l1 − b2,r1 − c2,m1 −a2) • scalar multiplication : λ > 0,λ ∈ r; λã = (λa1,λc1,λb1,λh1,λl1,λr1,λm1) λ < 0,λ ∈ r; λã = (λm1,λr1,λl1,λh1,λb1,λc1,λa1). int. j. anal. appl. 16 (3) (2018) 356 by definition of α-cut we have the following lemma. lemma 2.1. let ã = (a1,c1,b1,h1, l1,r1,m1) be a k-hfn. then for α ∈ (0, 1], the α-cut of ã is, [ã]α =   [ a− (1 − α k )h,b + (1 − α k )m ] ; α ∈ (0,k] [c− ( 1−α 1−k ) l,c + ( 1−α 1−k ) r] ; α ∈ [k, 1]. definition 2.3. [4] let x = (ξ1,ξ2, ...,ξn) and y = (η1,η2, ...,ηn) be vectors in rn. then (i) x = y if and only if ξi ≥ ηi for all i = 1, 2, ...,n, (ii) x ≥ y if and only if x = y and x 6= y, (iii) x > y if and only if ξi > ηi for all i = 1, 2, ...,n. definition 2.4. [4] let ã and b̃ be two fuzzy numbers.then, (i) ã v b̃ if and only if (alα,a u α ) = (b l α,b u α ), for all α ∈ [0, 1], (ii) ã % b̃ if and only if (alα,a u α ) ≥ (blα,buα ), for all α ∈ [0, 1], (iii) ã � b̃ if and only if (alα,auα ) > (blα,buα ), for all α ∈ [0, 1]. the following theorem characterize the orders for heptagonal fuzzy numbers. theorem 2.1. let ã = (a1,c1,b1,h1, l1,r1,m1), b̃ = (a2,c2,b2,h2, l2,r2,m2) be two k-hfn. then (i) ã w b̃ if and only if max{h2 −h1, 0}≤ a2 −a1 , max{m1 −m2, 0}≤ b2 − b1 , max{l2 − l1, 0}≤ c2 − c1 and max{r1 −r2, 0}≤ c2 − c1, (ii) ã ≺ b̃ if and only if max{h2 −h1, 0} < a2 −a1 , max{m1 −m2, 0} < b2 − b1 , max{l2 − l1, 0} < c2 − c1 and max{r1 −r2, 0} < c2 − c1. proof. by using definition (2.4) ã w b̃ if and only if for all α ∈ [0, 1], (alα,a u α ) 5 (b l α,b u α ) or equivalently a l α 6 b l α and a u α 6 b u α . but by lemma (2.1) a l α 6 b l α if and only if a1 − (1 − αk )h1 6 a2 − (1 − α k )h2 for all α ∈ [0,k] and c1 − ( 1−α 1−k ) l1 6 c2 − ( 1−α 1−k ) l2 for all α ∈ [k, 1], which are equivalent to (1 − α k )(h2 −h1) 6 a2 −a1 for all α ∈ [0,k], and ( 1 −α 1 −k )(l2 − l1) 6 c2 − c1 for all α ∈ [k, 1]. which are equivalent to max{h2 − h1, 0} ≤ a2 − a1 and max{m1 − m2, 0} ≤ b2 − b1. similarly, by using lemma(2.1) it can be conclude auα 6 b u α if and only if max{l2−l1, 0}≤ c2−c1 and max{r1−r2, 0}≤ c2−c1 and the proof of part (i) is complete. part (ii) can be proved, similarly. � int. j. anal. appl. 16 (3) (2018) 357 3. two-person zero-sum matrix fuzzy games in this section, we shall consider two-person zero-sum games with fuzzy payoffs. let p = {1, 2, ...,p} and q = {1, 2, ...,q} be the sets of pure strategies of player i and player ii , respectively. let a = (aij)p×q be the payoff matrix whose entries aij denote the payoff that player i gains and player ii loses. in the zero-sum games −aij is the amount paid by player i to player ii i.e. the gain of one player is the loss of other player. the mixed strategies of players i and player ii are probability distributions on the set of pure strategies, represented by x = {(ξ1,ξ2, ...,ξp) ∈ rp|ξi ≥ 0, i = 1, 2, ...,p, p∑ i=1 ξi = 1}, y = {(η1,η2, ...,ηq) ∈ rq|ηj ≥ 0, i = 1, 2, ...,q, q∑ j=1 ηj = 1}. respectively. in this section, the payoffs of the pair (x,y) ∈ x × y are modeled by k-heptagonal fuzzy number ã = (a,c,b,h,l,r,m). let player i choose a mixed strategy x ∈ x and player ii choose mixed strategy y ∈ y . the k-heptagonal fuzzy number ãij = (aij,cij,bij,hij, lij,rij,mij) indicates the payoffs that player i receives and player ii loses, the fuzzy payoff matrix of the game is given by ã =   ã11 · · · ã1q ... . . . ... ãp1 · · · ãpq   . the fuzzy two-person zero-sum games is denoted by γ̃ ≡ (x,y,ã). the fuzzy payoffs of the players i and ii are xtãy = p∑ i=1 q∑ j=1 ξiãijηj =(xtay,xtcy,xtby,xthy,xtly,xtry,xtmy), which is a k-heptagonal fuzzy number, for more details see [11]. in the rest of this paper, we set ã = (ãij), a = (aij), c = (cij), b = (bij), h = (hij), r = (rij), m = (mij), where ã, a, b, h, l, r and m are p × q matrix. also ã is a fuzzy k-heptagonal payoff matrix. definition 3.1. [3] a pair (x∗,y∗) ∈ x ×y is called a nash equilibrium strategy for a game γ̃ if (i) xtãy∗ w x∗tãy∗, ∀x ∈ x, (ii) x∗tãy∗ w x∗tãy, ∀y ∈ y. int. j. anal. appl. 16 (3) (2018) 358 theorem 3.1. let γ̃ = (x,y,ã) be a two-person zero-sum game with fuzzy payoffs, the pair (x∗,y∗) is the expected nash equilibrium strategy of γ̃ if and only if : (i) xtay∗ 6 x∗tay∗ 6 x∗tay, (ii) xtby∗ 6 x∗tby∗ 6 x∗tby, (iii) xtcy∗ 6 x∗tcy∗ 6 x∗tcy, (iv) xt (a−h)y∗ 6 x∗t (a−h)y∗ 6 x∗t (a−h)y, (v) xt (b −l)y∗ 6 x∗t (b −l)y∗ 6 x∗t (b −l)y, (vi) xt (b + r)y∗ 6 x∗t (b + r)y∗ 6 x∗t (b + r)y, (vii) xt (c + m)y∗ 6 x∗t (c + m)y∗ 6 x∗t (c + m)y. proof. let γ̃ be a two-person zero-sum game with the fuzzy k-heptagonsl payoff matrix ã = (a,c,b,h,l,r,m). let (x∗,y∗) ∈ x×y be the nash equilibrium strategy of the game γ̃. therefore by definition (3.1) we have x∗tãy∗ w x∗tãy, ∀y ∈ y. since xtãy∗ = (xtay∗,xtcy∗,xtby∗,xthy∗,xtly∗,xtry∗,xtmy∗), and x∗tay∗ = (x∗tay∗,x∗tcy∗,x∗tby∗,x∗thy∗,x∗tly∗,x∗try∗,x∗tmy∗), so, by theorem (2.1), xtãy∗ w x∗tãy∗ if and only if max{x∗thy∗ −xthy∗, 0} 6 x∗tay∗ −xtay∗, max{xtmy∗ −x∗tmy∗, 0} 6 x∗tby∗ −xtby∗, max{x∗tly∗ −xtly∗, 0} 6 x∗tcy∗ −xtcy∗, max{xtry∗ −x∗try∗, 0} 6 x∗tcy∗ −xtcy∗. consequently xtãy∗ w x∗tãy∗ if and only if xt (a−h)y∗ 6x∗t (a−h)y∗, xtay∗ 6x∗tay∗, (3.1) xt (c −l)y∗ 6x∗t (c −l)y∗, xtcy∗ 6x∗tcy∗, (3.2) xt (c + r)y∗ 6x∗t (c + r)y∗, xtcy∗ 6x∗tcy∗. (3.3) also, since x∗tãy∗ = (x∗tay∗,x∗tcy∗,x∗tby∗,x∗thy∗,x∗tly∗,x∗try∗,x∗tmy∗), int. j. anal. appl. 16 (3) (2018) 359 and x∗tãy = (x∗tay,x∗tcy,x∗tby,x∗thy,x∗tly,x∗try,x∗tmy), similary by theorem (2.1), x∗tãy∗ w x∗tãy if and only if x∗t (a−h)y∗ 6x∗t (a−h)y, x∗tay∗ 6x∗tay, (3.4) x∗t (b + m)y∗ 6x∗t (b + m)y, x∗tby∗ 6x∗tby, (3.5) x∗t (c −l)y∗ 6x∗t (c −l)y, x∗tcy∗ 6x∗tcy, (3.6) x∗t (c + r)y∗ 6x∗t (c + r)y, x∗tcy∗ 6x∗tcy. (3.7) now, from (3.1) and (3.5) we get xtay∗ 6 x∗tay∗ 6 x∗tay, xt (a−h)y∗ 6 x∗t (a−h)y∗ 6 x∗t (a−h)y, from (3.2) and (3.6) we obtain xtby∗ 6x∗tby∗ 6 x∗tby, xt (b + m)y∗ 6x∗t (b + m)y∗ 6 x∗t (b + m)y, from (3.3) and (3.8) we have xtcy∗ 6x∗tcy∗ 6 x∗tcy, xt (c −l)y∗ 6x∗t (c −l)y∗ 6 x∗t (c −l)y. and from (3.4) and (3.8) we get xtcy∗ 6x∗tcy∗ 6 x∗tcy, xt (c + r)y∗ 6x∗t (c + r)y∗ 6 x∗t (c + r)y. hence, we have the required inequalities (i)-(vii). � in the rest of the paper, we purpose the following notations: al0 = a−h, c l 0 = c −l, c u 0 = c + r, b u 0 = b + m, where a,c,b,h,l,r,m are the p×q matrix. using these notations theorem(3.1) can be rewrite as follows. int. j. anal. appl. 16 (3) (2018) 360 corollary 3.1. let γ̃ be a two-person zero-sum game with fuzzy payoffs, the pair (x∗,y∗) is the nash equilibrium strategy of γ̃ if and only if the followings hold xt (a,c,b,al0 ,c l 0 ,c u 0 ,b u 0 )y ∗ 6 x∗t (a,c,b,al0 ,c l 0 ,c u 0 ,b u 0 )y ∗ 6 x∗t (a,c,b,al0 ,c l 0 ,c u 0 ,b u 0 )y. in the view of theorem 3.1, we understand that to solve the fuzzy game γ̃, it is enough to consider seven crisp two-person zero-sum games and attempt to determine a point (x∗,y∗) ∈ x × y which is simultaneously a saddle point of them. definition 3.2. a two-person zero-sum fuzzy game γ̃ = (x,y,ã) is called to be a proportional fuzzy game if and only if there exists γn ∈ (0, 1]; n = 1, ..., 4 such that hij = γ1aij, lij = γ2cij, rij = γ3cij and mij = γ4bij for all i = 1, 2, ...,p and for all j = 1, 2, ...,q. theorem 3.2. a pair of mixed strategies (x∗,y∗) ∈ x×y is a nash equilibrium strategy of the proportional fuzzy matrix game γ̃ = (x,y,ã) if and only if (x∗,y∗) ∈ x ×y is the nash equilibrium of crisp two-person zero-sum games γa = (x,y,a), γb = (x,y,b) and γc = (x,y,c). proof. let γ̃ = (x,y,ã) be a proportional fuzzy matrix game. therefore by definition (3.2) ã = (a,c,b,γ1a,γ2c,γ3c,γ4b) is the payoff matrix of the game. by theorem (3.1), (x ∗,y∗) ∈ x ×y is a nash equilibrium of γ̃ if and only if (i) xtay∗ 6 x∗tay∗ 6 x∗tay, (ii) xtby∗ 6 x∗tby∗ 6 x∗tby, (iii) xtcy∗ 6 x∗tcy∗ 6 x∗tcy, because the other inequalities came to these one. equivalently, (x∗,y∗) ∈ x × y is a nash equilibrium of crisp two-person zero-sum games γa = (x,y,a), γb = (x,y,b) and γc = (x,y,c). the proof is complete. � the following corollary is a direct result of theorem(3.2). corollary 3.2. let ã = (ãij) be a payoff matrix of proportional fuzzy game γ̃. suppose that bij = γ5aij,cij = γ6aij for all i,j with γ5,γ6 ≥ 1. then a pair of mixed strategies (x∗,y∗) ∈ x ×y is the nash equilibrium strategy for γ̃ if and only if (x∗,y∗) is a nash equilibrium of crisp two-person zero-sum game γa = (x,y,a). definition 3.3. let γ̃ be a two-person zero-sum fuzzy game. it is called constant fuzzy game if and only if there exist h,l,r,m > 0 such that hij = h,lij = l , rij = r and mij = m for all i = 1, 2, ...,p and j = 1, 2, ...,q. lemma 3.1. let γ̃ = (x,y,ã) be a constant fuzzy game. a pair of mixed strategies (x∗,y∗) ∈ x×y is the nash equilibrium strategy for γ̃ if and only if (x∗,y∗) is a nash equilibrium of crisp two-person zero-sum games γa, γb and γc. int. j. anal. appl. 16 (3) (2018) 361 proof. by definition(3.3) h,l,r and m are constant matrices which all the entries are h,l,r and m , respectively. hence xthy = h,xtly = l,xtry = r and xtmy = m for all x ∈ x,y ∈ y . by theorem(3.1) the result can be obtained, directly. � definition 3.4. [3] a pair of mixed strategies (x∗,y∗) ∈ x×y is called a pareto nash equilibrium strategy of the game γ̃ if (i) there does not exist any x ∈ x such that x∗tãy∗ xtãy∗, (ii) there does not exist any y ∈ y such that x∗tãy x∗tãy∗. theorem 3.3. let γ̃ ≡ (x,y,ã) be a fuzzy two-person zero-sum game. a pair (x∗,y∗) ∈ x ×y is the pareto nash equilibrium strategy for the game γ̃ if and only if (i) there exist no x ∈ x such that x∗tay∗ ≤ xtay∗, x∗tby∗ ≤ xtby∗, x∗tcy∗ ≤ xtcy∗ and (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗) ≤ (xtal0 y ∗,xtcl0 y ∗,xtcu0 y ∗,xtbu0 y ∗); (3.9) (ii) there exist no y ∈ y such that x∗tay ≤ xtay∗, x∗tby ≤ xtby∗, x∗tcy ≤ xtcy∗ and (x∗tal0 y,x ∗tcl0 y,x ∗tcu0 y,x ∗tbu0 y) ≤ (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗).(3.10) proof. by contradiction, let (x∗,y∗) ∈ x ×y be the pareto nash equilibrium strategy of γ̃. assume that there exist x1 ∈ x such that following relationships are established (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗) ≤ (xt1 a l 0 y ∗,xt1 c l 0 y ∗,xt1 c u 0 y ∗,xt1 b u 0 y ∗), and x∗tay∗ ≤ xt1 ay ∗, x∗tby∗ ≤ xt1 by ∗, x∗tcy∗ ≤ xt1 cy ∗. it implies that x∗tal0 y ∗ ≤ xt1 a l 0 y ∗, x∗tcl0 y ∗ ≤ xt1 c l 0 y ∗, x∗tcu0 y ∗ ≤ xt1 c u 0 y ∗, x∗tbu0 y ∗ ≤ xt1 b u 0 y ∗. but, by definition (2.3) the above inequalities do not occur simultaneously. therefore, we get int. j. anal. appl. 16 (3) (2018) 362 ( x∗t ( α k a+(1 − α k )(a−h))y∗,x∗t ( α k b + (1 − α k )(b + m))y∗ ) ≤( xt1 ( α k a + (1 − α k )(a−h))y∗,xt1 ( α k b + (1 − α k )(b + m))y∗ ) , for α ∈ [0,k] and obtain( x∗t ( α − k 1 − k )c+(1 − α − k 1 − k )(c − l))y∗,x∗t ( α − k 1 − k c + (1 − α − k 1 − k )(c + r))y∗ ) ≤ ( xt1 ( α − k 1 − k )c + (1 − α − k 1 − k )(c − l))y∗,xt1 ( α − k 1 − k c + (1 − α − k 1 − k )(c + r))y∗ ) , for α ∈ [k, 1]. by rearranging, it follows that ( x∗t (a− (1 − α k )h)y∗,x∗t (b + (1 − α k )m)y∗ ) ≤( xt1 (a− (1 − α k )h)y∗,xt1 (b + (1 − α k )m)y∗ ) , and ( x∗t (c − ( 1 −α 1 −k )l)y∗,x∗t (c + ( 1 −α 1 −k )r)y∗ ) ≤( xt1 (c − ( 1 −α 1 −k )l)y∗,xt1 (c + ( 1 −α 1 −k )r)y∗ ) . using definition (3.4) it implies that x∗tãy∗ � xt1 ãy∗. this is a contradiction. conversely, we assume that the pair of mixed strategy (x∗,y∗) ∈ x×y be satisfy (3.9) and (3.10). suppose that there exists a strategy x1 ∈ x such that x∗tãy∗ � xt1 ãy∗. by definition 2.4, we have for all α ∈ [0, 1] (x∗talαy ∗,x∗tauαy ∗) ≤ (xt1 a l αy ∗,xt1 a u αy ∗) which (a− (1 − α k )h) = alα, (b + (1 − α k )m) = auα for α ∈ [0,k], and (c − ( 1 −α 1 −k )l) = alα, (c + ( 1 −α 1 −k )r) = auα for α ∈ [k, 1]. set α = 0, then x∗t (al0 ,c l 0 ,c u 0 ,b u 0 )y ∗ ≤ xt1 (a l 0 ,c l 0 ,c u 0 ,b u 0 )y ∗, and x∗tay∗ ≤ xtay∗,x∗tby∗ ≤ xtby∗,x∗tcy∗ ≤ xtcy∗. this is contradict (i). similarly, we can show that there does not exist any y ∈ y such that x∗tãy � x∗tãy∗. then proof of the theorem is complete. � int. j. anal. appl. 16 (3) (2018) 363 definition 3.5. a pair of mixed strategies (x∗,y∗) ∈ x ×y is a weak pareto nash equilibrium strategy of the game γ̃ if (i) there does not exist any x ∈ x such that x∗tãy∗ ≺ xtãy∗, (ii) there does not exist any y ∈ y such that x∗tãy ≺ x∗tãy∗. following theorem is obtaine directly from definition (3.5) and theorem (3.3). theorem 3.4. let γ̃ ≡ (x,y,ã) be a fuzzy two-person zero-sum game. a pair (x∗,y∗) ∈ x ×y is the weak pareto nash equilibrium strategy for the game γ̃ if and only if (i) there exist no x ∈ x such that (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗) < (xtal0 y ∗,xtcl0 y ∗,xtcu0 y ∗,xtbu0 y ∗) and x∗tay∗ < xtay∗, x∗tby∗ < xtby∗, x∗tcy∗ < xtcy∗; (ii) there exist no y ∈ y such that (x∗tal0 y,x ∗tcl0 y,x ∗tcu0 y,x ∗tbu0 y) < (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗) and x∗tay < xtay∗, x∗tby < xtby∗, x∗tcy < xtcy∗. 4. parametric bi-matrix games in this section we characterize parametric bi-matrix games and investigate other types of nash equilibrium strategies for parametric bi-matrix games. let sp = {η1,η2, ...,ηp} and sq = {ξ1,ξ2, ...,ξq} be sets of pure strategies of player i and player ii , respectively. we set u = (uij)p×q to be payoffs matrices of player i and v = (vij)p×q to be payoffs matrices of player ii , respectively. suppose β,γ ∈ [0, 1] and let (1 −β)(aij + cij −hij − lij) + β(cij + rij + bij + mij) be the gain of player i and (1 −γ)(aij + cij −hij − lij) + γ(cij + rij + bij + mij) be the losses of player ii when player i emploing pure strategy i and player ii emploing pure strategy j. then the game γ = (x,y,u,v ) is called a bi-matrix game. the notation of parametric as follow; suppose β,γ ∈ [0, 1], then we set a(β) = (1 −β)(a + c −h −l) + β(c + r + b + m), (4.1) int. j. anal. appl. 16 (3) (2018) 364 and −a(γ) = − [(1 −γ)(a + c −h −l) + γ(c + r + b + m)] . (4.2) now, we consider the parametric bi-matrix game γ(β,γ) = (x,y,a(β),a(γ)). definition 4.1. [8] let γ(β,γ) be a parametric bi-matrix game. a pair of mixed strategies (x∗,y∗) ∈ x×y is a nash equilibrium strategy of γ if (i) xta(β)y∗ ≤ x∗ta(β)y∗ for all x ∈ x, (ii) x∗ta(γ)y∗ ≤ x∗ta(γ)y for all y ∈ y . theorem 4.1. let γ(β,γ) be a prametric bi-matrix game and the pair of mixed strategy (x∗,y∗) ∈ x×y be nash equilibrium strategy of γ. then (x∗,y∗) ∈ x×y is the pareto nash equilibrium strategy of the fuzzy two-person zero-sum game γ̃. proof. let (x∗,y∗) ∈ x × y be the nash equilibrium strategy of the parametric bi-matrix game γ(β,γ), which β,γ ∈ [0, 1]. by definition (4.1) we obtain (1 −β)xt (a + c −h −l)y∗ + βxt (c + r + b + m)y∗ ≤ (1 −β)x∗t (a + c −h −l)y∗ + βx∗t (c + r + b + m)y∗, and (1 −γ)x∗t (a + c −h −l)y∗ + γx∗t (c + r + b + m)y∗ ≤ (1 −γ)x∗t (a + c −h −l)y + γx∗t (c + r + b + m)y. in order to show that (x∗,y∗) ∈ x×y is pareto nash equilibrium strategy of γ̃, we have to prove that there exist x1 ∈ x such that x∗tãy∗ � xt1 ãy∗ holds. from definition (2.4), we get (x∗tal0 y ∗,x∗tcl0 y ∗,x∗tcu0 y ∗,x∗tbu0 y ∗) ≤ (xt1 a l 0 y ∗,xt1 c l 0 y ∗,xt1 c u 0 y ∗,xt1 b u 0 y ∗). moreover, by definition (2.3) x∗tal0 y ∗ = xt1 a l 0 y ∗, x∗tcl0 y ∗ = xt1 c l 0 y ∗, x∗tcu0 y ∗ = xt1 c u 0 y ∗, x∗tbu0 y ∗ = xt1 b u 0 y ∗, do not occur simultaneously. then we have (1 −β)x∗t (a + c −h −l)y∗ + βx∗t (c + r + b + m)y∗ < (1 −β)xt1 (a + c −h −l)y ∗ + βxt1 (c + r + b + m)y ∗. int. j. anal. appl. 16 (3) (2018) 365 this is a contradiction. the condition (ii) can de proved, similarly. � theorem 4.2. let the pair of mixed strategies (x∗,y∗) ∈ x ×y be nash equilibrium strategy of prametric bi-matrix game γ(β,γ) with β,γ ∈ [0, 1]. then (x∗,y∗) ∈ x×y is the weak pareto nash equilibrium strategy of fuzzy two-person zero-sum game γ̃. the following corollary is direct result of theorem (4.1) and theorem (4.2). corollary 4.1. a fuzzy two-person zero-sum game γ̃ satisfies the following properties: (i) there exsist at least one pareto nash equilibrium strategy of fuzzy game γ̃, (ii) there exsist at least one weak pareto nash equilibrium strategy of fuzzy game γ̃. 5. illustrative examples example 5.1. let γ̃ be a fuzzy two-person zero-sum game and ã be the fuzzy payoff matrix of γ̃ given as follows: ã =   (20, 40, 60, 2, 8, 12, 24) (70, 140, 210, 7, 28, 42, 84) (50, 100, 150, 5, 20, 30, 60) (10, 20, 30, 1, 4, 6, 12)   . find the nash equilibrium strategy for the game γ̃. obviously, γ̃ is a proportional fuzzy game. note that γ1 = 0.1, γ2 = 0.2, γ3 = 0.3 and γ4 = 0.4. let x∗t = ( p, 1 −p ) and y∗t = ( q, 1 −q ) be the mixed strategy of player i and ii , respectively. by theorem(3.2), the nash equilibrium strategy of game γ̃ can be obtined by solving a bi-matrix game whose payoff matrices are a =  20 70 50 10   ,c =   40 140 100 20   ,b =   60 210 150 30   . we have ( 1 0 )20 70 50 10     q 1 −q   ≤ (p 1 −p)  20 70 50 10     q 1 −q   , and ( p 1 −p )20 70 50 10     q 1 −q   ≤ (p 1 −p)  20 70 50 10    0 1   . it is easy to obtain that the nash equilibrium strategy of the crisp matrix game γa is (x ∗,y∗) =( ( 4 9 , 5 9 ), ( 2 3 , 1 3 ) ) and similarly the nash equilibrium strategy of the crisp matrix games γb and γc can be obtained. so expected value of the gasme γ̃ is ( 4 9 , 5 9 )ã( 2 3 , 1 3 )t = ( 990 27 , 1980 27 , 2870 27 , 99 27 , 396 27 , 594 27 , 1188 27 ). int. j. anal. appl. 16 (3) (2018) 366 example 5.2. let ã be the payoff matrix of the fuzzy two-person zero-sum game γ̃, given as follows: ã =   (50, 100, 150, 10, 15, 10, 40) (80, 160, 240, 10, 15, 10, 40) (100, 200, 300, 10, 15, 10, 40) (20, 40, 60, 10, 15, 10, 40)   . find the nash equilibrium strategy for the game γ̃. by definition(3.3) γ̃ is a proportional fuzzy game and h = 10, l = 15, r = 10 and m = 40. let x∗t = ( p, 1 −p ) and y∗t = ( q, 1 −q ) be the mixed strategy of player i and ii , respectively. by theorm(3.1), it is easy to show that the nash equilibrium strategy of γ̃ is (x∗,y∗) = ( ( 8 11 , 3 11 ), ( 6 11 , 5 11 ) ) and the expected value of γ̃ is given by ( 8 11 , 3 11 )ã( 6 11 , 5 11 )t = ( 7700 121 , 15400 121 , 2310 121 , 10, 15, 10, 40) example 5.3. consider the fuzzy two-person zero-sum game γ̃ with heptagonal fuzzy payoff matrix ã given by ã =  (90, 100, 120, 10, 5, 10, 15) (70, 80, 100, 15, 5, 10, 20) (60, 90, 100, 15, 10, 5, 10) (170, 180, 210, 20, 5, 20, 10)   . find the nash, pareto nash and weak pareto nash equilibrium strategy of the game γ̃. let x∗t = ( p, 1 −p ) and y∗t = ( q, 1 −q ) be the mixed strategy of player i and ii , respectively. since there is no (x,y) ∈ x ×y satisfying the conditions of theorem(3.1), so there is no nash equilibrium strategy for the game γ̃. by theorem(4.2) to finding the pareto nash equilibrium strategy, it is enough to find the nash equilibrium strategy of parametric bi-matrix game γ̃. so, we construct the bi-matrix game γ(β,γ) from fuzzy matrix game γ̃. using relations (4.1) and (4.2) we obtain a(β) =  175 + 70β 130 + 80β 125 + 80β 325 + 95β   , a(γ) =  175 + 70γ 130 + 80γ 125 + 80γ 325 + 95γ   , where β,γ ∈ [0, 1]. it is easy to see that (x∗,y∗) is the nash equilibrium strategy for the parametric bi-matrix game γ(β,γ) if it satisfies the following: (1, 0)a(β)y∗ ≤ x∗ta(β)y∗, (0, 1)a(β)y∗ ≤ x∗ta(β)y∗, x∗ta(γ)y∗ ≤ x∗ta(γ)(0, 1)t , x∗ta(γ)y∗ ≤ x∗ta(γ)(1, 0)t , which are equivalent to   (245 + 5β)(1 −p)q − (195 + 15β)(1 −p) ≤ 0, (245 + 5β)pq − (195 + 15β)p ≥ 0. int. j. anal. appl. 16 (3) (2018) 367   (245 + 5γ)(1 −q)p− (200 + 15γ)(1 −q) ≥ 0, (245 + 5γ)pq − (200 + 15γ)q ≤ 0. thus for β,γ ∈ [0, 1] , nash equilibrium strategy for the parametric game γ(β,γ) are as follows (x∗1,x ∗ 2) = ( 200 + 15γ 245 + 5γ , 45 − 10γ 245 + 5γ ) , (y∗1,y ∗ 2 ) = ( 195 + 15β 245 + 5β , 50 − 10β 245 + 5β ) . therfore by theorem(4.1) and (4.2) the pareto nash and weak pareto nash equilibrium strategy of the game γ̃ are as following{ (x∗,y∗)t = (( 200 + 15γ 245 + 5γ , 45 − 10γ 245 + 5γ ) , ( 195 + 15β 245 + 5β , 50 − 10β 245 + 5β ))∣∣∣β,γ ∈ [0, 1]}, { (x∗,y∗)t = (( 200 + 15γ 245 + 5γ , 45 − 10γ 245 + 5γ ) , ( 195 + 15β 245 + 5β , 50 − 10β 245 + 5β ))∣∣∣β,γ ∈ (0, 1)}, respectively. references [1] c. r. bector, s. chandra, fuzzy mathematical programming and fuzzy matrix games, springer, berlin, 2005. [2] l. campos, fuzzy linear programming model to solve fuzzy matrix game, fuzzy sets syst., 32(3)(1989), 275-289. [3] bapi dutta, s. k. gupta, on nash equilibrium strategy of two-person zero-sum games with trapezoidal fuzzy payoffs, fuzzy inf. eng., 6(3)(2014), 299-314. [4] li cunlin, zhang qiang, nash equilibrium strategy for fuzzy non-cooperative games, fuzzy sets syst., 176(1)(2011), 46-55. [5] b. liu, uncertain programming. new york:wiley, 1999. [6] b. liu , uncertainty theory. an introduction to its axiomatic foundations, studies in fuzziness and soft computing, 154, springer-verlag, berlin, 2004. [7] t. maeda ,on characterization of equilibrium strategy of bimatrix games with fuzzy payoffs, j. math. anal. appl., 251(2)(2000), 885-896. [8] j. f nash, non-cooperative games , ann. math, 54(2)(1951), 286-295. [9] j. von neumann, o. morgenstern, theory of games and economic behavior, princeton university press, princeton, new jersey, 1944. [10] k. rathi, s. balamohan, representation and ranking of fuzzy numbers with heptagonal membership function using value and ambiguity index, appl. math. sci., 87(8)(2014), 4309-4321. [11] j. ramik, j. r imanek, inequality relation between fuzzy numbers and its use in fuzzy optimization, fuzzy sets syst., 16(2)(1985), 123-138. [12] m. sakawa, fuzzy sets and interactive multiobjective optimization, plenum press, new york, 1993. [13] a. v. yazenin, fuzzy and stochastic programming, fuzzy sets syst., 22(1-2)(1987), 171-180. [14] l. a. zadeh, fuzzy sets, information and control, (8)(1965), 338-353. [15] l. a. zadeh, fuzzy set as a basis for a theory of possibility, fuzzy sets syst., 1(1978), 3-28. [16] h. j. zimmermann, application of fuzzy set theory to mathematical programming, inform. sci., 36(1-2)(1985), 29-58. [17] h. j. zimmermann, fuzzy set theory and its applications, second edition, kluwer academic publishers, boston, ma, 1992. 1. introduction 2. preliminaries 3. two-person zero-sum matrix fuzzy games 4. parametric bi-matrix games 5. illustrative examples references international journal of analysis and applications volume 18, number 6 (2020), 1015-1028 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1015 valuation of european style compound option written on european style currency and power options javed hussain∗ sukkur iba university, pakistan ∗corresponding author: javed.brohi@iba-suk.edu.pk abstract. the aim of the paper is paper is twofold. firstly, we will derive an explicit closed formula for pricing the compound call option contingent upon a currency call option. secondly, we will develop a pricing formula for the compound call option contingent upon the power call option. 1. introduction a party with the long position in a currency option has the right ( but not obligation) to trade (buy/sell) a predetermined amount of the foreign currency at a fixed predetermined price (strike pricier in domestic currency) on the fixed agreed date (maturity/expiry). this option can be found in both american and eu styles. certainly, the investor would prefer to enter into the long position currency call if he/she expects that the foreign currency is going to go high, on contrast, if the investor expects that the value of a foreign currency is going decline he/she will prefer to enter input option. there are numerous reasons that why currencies fluctuate, obviously with the normal fluctuations in currency can be explained by demand-supply principle, but a case of political uncertainty, financial crises, and natural disasters (such covid-19 pandemic) the currencies fluctuated gigantically. this makes the currency option a major tool for the management of risk associated with the evolution of currencies. in [15, 1979], introduced the currency option but they could not give an explicit formula for the premium of the currency option. in the black-scholes framework, received august 22nd, 2020; accepted september 14th, 2020; published october 7th, 2020. 2010 mathematics subject classification. 91g20. key words and phrases. financial derivatives; option pricing; compound option; currency option; power option. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1015 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1015 int. j. anal. appl. 18 (6) (2020) 1016 an explicit formula for pricing currency option was studied by grabbe in [16, 1983], where the underlying currency option is driven by geometric brownian motion. they also deduced that american style currency options are more expensive than that of their eu counterpart. amin and jarrow in [17, 1991], studied and priced the eu style currency option in the heath-jarrow morton (hjm) framework, under a strong assumption of constancy of volatility. to deal with the situations where the foreign currency fluctuates with high magnitude i.e. volatility several attempts have been made under the assumption that foreign currency follows a stochastic differential driven various version of sophisticated jump-diffusion processes, for instance, see hung [18, 2007], li, peng [19, 2013], ming [21, 2019] and david liu [20, 20]. power option is a type of polynomial option in which payoff depends on the basic resources (underlying assets) raised to a certain power. it is structured to allow the buyer (holder) to take a securitized view of a particular asset or its volatility. they have been created to modify the patterns of plain vanilla options. the payoff of a power option is the difference between the underlying asset price at maturity raised to some strictly positive power. for power call options, payoff is max ( si–x, 0 ) while for power put options, payoff is max ( x–si, 0 ) where i is some positive integer power i > 0. in [2, 1973] black, f. and scholes, m., suggested that power options can be beneficial for an investor in bringing more premium than an ordinary option which is often the reason of a sharp-sighted investor’s attraction. in [3, 2006], macovshi has studied power option and shown that polynomial option can be priced through expressing it as a combination of different power options with adapted strikes. in zhang [4, 2016], the pricing formula for the power option is driven for liu’s uncertain stock model under the assumption that the underlying stock price follows some probabilistic (stochastic) differential equation. in dastranj [5, 2017] option pricing under double heston model and double heston with three jumps are discussed, concluding that in comparison to all other options, power option produces more premium income under the double heston model. in saberi [6, 2018] power option pricing is derived using time-fractional pde for the dynamics of underlying asset price that follows a regime-switching model in which the underlying risky asset mainly depends on a continuous-time hidden markov chain process. they’ve further produced an exact solution for their proposed power option. in dastranj [7, 2020] a hypothetical option pricing based on fractional heston model is discussed for iran’s gold market. using fast fourier transformation, the pricing of the power option has been driven. the analysis brings forward that in most of the three months periods the power option prevented from creating arbitrage opportunity while related to 2018, two periods have been observed where power option created an arbitrage opportunity due to high volatilities. a compound option also called the split-fee option is an option over another option that enables a holder with the right to purchase or sell another option. one can have four possible types of compound options, int. j. anal. appl. 18 (6) (2020) 1017 namely’ call on call, call on put, put on put or put on call each with two dates of expiration and two strike prices. the usage of compound options mainly involves currency or fixed income markets. in geske [8, 1977] risky coupon bond problem is tackled using the compound option valuing technique. in geske [9, 1977] compound option is computed by defining a call option on a stock which is itself an option on the firm assets. this valuation result was extended by hull [10, 2003] to a compound option with underlying assets paying a known dividend return. elettra [11, 2003] considered the time-dependent volatility and interest rate for deriving the pricing model which is a generalized and more realistic version of geske [9, 1979]. sequential exchange options in carr [12] are priced using the compound options. american type compound options are priced for stochastic interest rates and stochastic volatility in chiarella [13, 2013]. in li [14, 2020] analytic pricing formula for compound options is driven in terms of fourier integral of the characteristic function. the obtained formula was used along with the fft algorithm to calculate the compound option’s price across the entire spectrum of the exercise price. section 1 is presenting the introduction based on the available literature. section 2 introduces foreign currency options, gives expressions for their payoff and later explains compound options written on currency options using feynman-kac formula. in section 3 european power option is explained followed by expressions for its payoff and then european compound options over power call option are derived in detail in detail. 2. compound option written on foreign currency option before moving towards the option pricing we will would like to mention a result that we will use frequently throughout the paper. theorem 2.1. (identity for bivariate standard normal distribution) [23] let x ∼ n (0, 1) and y ∼ n (0, 1) be jointly normally distributed with correlation coefficient ρ ∈ (−1, 1), then joint cumulative distribution function can be written as following (2.1) φ (α,β,ρxy) = ∫ α −∞ n(x)φ ( β −ρx√ 1 −ρ2 ) dx here φ denotes cumulative distribution of standard normal random, n(x) denotes the probability distribution function of standard normal random variable and φ denotes the bivariate cumulative distribution function. 2.1. foreign currency option. [10] a foreign currency option is like a european option give the right to option holder to buy or to sell one unit of foreign currency at time t for the price k at predetermined exchange rate k. the difference between foreign and european options is that in currency options risk-free rate is replaced by foreign interest rate. the payoff of the foreign currency option at time t for the strike price k is given by: ccur (t,k) = max [(st −k) , 0] . int. j. anal. appl. 18 (6) (2020) 1018 the derivation of the currency option is the same as the black-scholes methodology under the following assumption. the foreign currency option which is underlying assets follows the geometric brownian motion. the foreign and domestic interest rates are constant during the whole life of the contract. hence the pricing formula for currency call are, ccur(t,st,k) = ste −rf(t−t)φ(d2)−k2e−rd(t−t)n(d1). where, d1 = ln ( st k ) + ( rd −rf + σ 2 2 ) (t − t) σ √ t − t d2 = ln ( st k ) − ( rd −rf − σ 2 2 ) (t − t) σ √ t − t . where rf is the foreign interest rate and rd is the domestic interest rate, the value of call option will increase for currency if domestic interest rate increases and the value will down if the domestic interest rate will fall. 2.2. compound option written on currency option. theorem 2.2. let {wt : t ≥ 0} be a p -standard brownian motion on the probability space (ω,f,p) and let st, the price of unit of foreign currency, follow a gbm with the following sde dst = µstdt + σstdwt where µ is the drift parameter and σ is the volatility parameter. in addition, we let r be the risk-free interest rate. then the price of compound call option, with strike k1 and maturity t1, contingent upon a currency call option, with strike k2 and maturity t2, is given by, cccur (t,st,k1,k2,t1,t2) = φ(w1,w3; ρ)−k2e −rd(t2−t)φ(w2,w4; ρ)−k1e−rd(t1−t)φ(w1), where, ρ = √ t1 − t√ t2 − t , w1 = ln ( k1 st ) − ( rd − σ 2 2 ) (t1 − t) σ √ t1 − t , w2 = ln ( k1 st ) − ( rd + σ2 2 ) (t1 − t) σ √ t1 − t w3 = ln ( st k2 ) + ( rd − σ 2 2 ) (t2 − t) −rf (t2 −t1) σ √ t2 −t1 , w4 = ln ( st k2 ) + ( rd + σ2 2 ) (t2 − t) −rf (t2 −t1) σ √ t2 −t1 . in this pricing formula, φ represents distribution function of standard normal distribution and φ represents the bivariate cumulative distribution function of standard normal distribution and ρ is the correlation. int. j. anal. appl. 18 (6) (2020) 1019 proof. we will start by providing an explicit black-scholes price (cf. [16]) of the underlying currency option with maturity t2 and strike k2, with inception at t1, can be given as, ccur = st1e −rf(t2−t1)φ(d2)−k2e−rd(t2−t1)φ(d1). here d1 = ln ( st1 k ) + ( rd −rf + σ 2 2 ) (t2 −t1) σ √ t2 −t1 , d2 = ln ( st1 k ) − ( rd −rf − σ 2 2 ) (t2 −t1) σ √ t2 −t1 . consider the risk-neutral measure q, such that st follows dst = (rd −rf )stdt + σdw q t , where w q t = wt + ( µ−(rd−rf) σ ) t is a q -standard wiener process. indeed for t > t, log returns ln ( st st ) follows n [( (rd −rf ) − σ 2 2 ) (t − t),σ2(t − t) ] with conditional density function, n (st | st) = 1 stσ √ 2π(t − t) e −1 2   ln( stst )− ( (rd−rf )− σ2 2 ) (t−t) σ √ t−t  2 . next using the feynman-kac formula [22, theorem 4.33], the price of eu. compound option, contingent upon currency option, under risk-neutral probability measure q, at time t ≤ t1 ≤ t2, can be given as, cccur (t,st,k1,k2,t1,t2) = e−rd(t1−t)eq [ ccur (st1,t1; k2,t2) −k1) + | ft ] = e−rd(t1−t)eq [( st1e −rf(t2−t1)φ(d2)−k2e−rd(t2−t1)φ(d1) −k1 )+ | ft ] = e−rd(t1−t) ∫ ∞ 0 [( st1e −rf(t2−t1)φ(d2)−k2e−rd(t2−t1)φ(d1) ) −k1 ]+ ×n (st1 | st) dst1. let s∗ be price of unit of foreign currency such that, the compound option is at the money i.e. c (s∗,t1; k2,t2) = k1, and using this equation and any suitable method (say newton-raphson) an approximate value of s∗ can be computed. now, cccur (t,st,k1,k2,t1,t2) = e−rd(t1−t) ∫ ∞ s∗ (( st1e −rf(t2−t1)φ(d2) −k2e−rd(t2−t1)φ(d1) ) −k1 ) n (st1 | st) dst1, = e−rd(t1−t) ∫ ∞ s∗ (st1e −rf(t2−t1)φ(d2))n (st1 | st) dst1 −k2e−rd(t2−t) ∫ ∞ s∗ φ(d1)n (st1 | st) dst1 −k1e −rd(t1−t) ∫ ∞ s∗ n (st1 | st) dst1 =: i1 − i2 − i3. int. j. anal. appl. 18 (6) (2020) 1020 let us begin by computing the integral i1, i1 = e −rd(t1−t) ∫ ∞ s∗ (st1e −rf(t2−t1)φ(d2))n (st1 | st) dst1, = e−rd(t1−t) ∫ ∞ s∗ (st1e −rf(t2−t1)φ(d2))n (st1 | st) dst1. now the compound option will be in the money and hence will be exercised iff st1 > s∗ iff ste ( (rd−rf)(t1−t)+σ(wqt1−w q t ) ) > s∗ iff ste (( rd−rf−σ 2 2 ) (t1−t)+σ √ t1−tx ) > s∗ iff x > ln( s ∗ st )− ( rd−rf−σ 2 2 ) (t1−t) σ √ t1−t =: w1. hence we rewrite i1 as, i1 = e −rd(t1−t)ste ( rd−rf−σ 2 2 ) (t1−t)−rf(t2−t1) ×∫ ∞ w1 φ  ln ( st k2 ) + ( rd −rf + σ 2 2 ) (t2 −t1) + ( rd −rf − σ 2 2 ) (t1 − t) + xσ √ t1 − t σ √ t2 −t1   × e− (x2−2xσ √ t1−t) 2 √ 2π dx = ste −rf(t2−t)−σ 2 2 (t1−t)e σ2 2 (t1−t) ×∫ ∞ w1 φ  ln ( st k2 ) + ( rd −rf + σ 2 2 ) (t2 −t1) + ( rd −rf − σ 2 2 ) (t1 − t) + xσ √ t1 − t σ √ t2 −t1   × e− (x−σ √ t1−t) 2 2 √ 2π dx to simplify the above integral, let using substitution u = −(x−σ √ t1 − t), the integral simplifies to, i1 = ste −rf(t2−t) ∫ w2 −∞ φ  ln ( st k2 ) + ( rd −rf + σ 2 2 ) (t2 − t) −uσ √ t1 − t σ √ t2 −t1   e−u22√ 2π du where w2 = σ √ t1 − t − w1. if we set ρ = √ t1−t t2−t and w4 = ln ( st k2 ) + ( rd−rf+σ 2 2 ) (t2−t) σ √ t2−t along with using bivariate identity (2.1) the last integral can be written as, i1 = ste −rf(t2−t) ∫ ψ2 −∞ φ ( w4 −ρu√ 1 −ρ2 ) e− u2 2 √ 2π du = e−rf(t2−t)φ(w2,w4; ρ). now let us compute the second integral i2, i2 = k2e −rd(t2−t) ∫ ∞ s∗ φ(d1)n (st1 | st) dst1 by arguing same as previously, the compound option will be in the money and hence will be exercised iff st1 > s ∗ iff ste ( (rd−rf)(t1−t)+σ(wqt1−w q t ) ) > s∗ iff ste (( rd−rf−σ 2 2 ) (t1−t)+σ √ t1−tx ) > s∗ iff x > − ln( sts∗ )− ( rd−rf−σ 2 2 ) (t1−t) σ √ t1−t = −w1. hence we rewrite i2 as, i2 = k2e −rd(t1−t) ∫ ∞ −w1 φ  ln ( st k2 ) + ( rd −rf − σ 2 2 ) (t2 − t) + xσ √ t1 − t σ √ t2 −t1   e−x22√ 2π dx. int. j. anal. appl. 18 (6) (2020) 1021 using the substitution, u = −x, the above integral can be simplified to, i2 = −k2e−rd(t1−t) ∫ −∞ w1 φ  ln ( st k2 ) + ( rd −rf − σ 2 2 ) (t2 − t) −uσ √ t1 − t σ √ t2 −t1   e−u22√ 2π du. on setting, if we set ρ = √ t1−t t2−t and w3 = ln ( st k2 ) + ( rd−rf−σ 2 2 ) (t2−t) σ √ t2−t along with using bivariate identity (2.1) the last integral can be written as, i2 = k2e −rd(t1−t) ∫ w1 −∞ φ ( w3 −ρu√ 1 −ρ2 ) e− u2 2 √ 2π du, = k2e −rd(t1−t)φ (w1,w3; ρ) . finally, its fairly easy to compute the 3rd integral i3, we will make use of fact w q t1 − wqt ∼ n (0,t1 − t) we have i3 = k1e −rd(t1−t) ∫ ∞ s∗ n (st1 | st) dst1 = k1e −rd(t1−t)q (st1 > s ∗ | st) = k1e −rd(t1−t)q ( ste ( rd−rf−σ 2 2 ) (t1−t)+σ(wqt1−w q t ) > s∗ | st ) = k1e −rd(t1−t)q ( ste ( rd−rf−σ 2 2 ) (t1−t)+σ √ t1−tz > s∗ | st ) = k1e −rd(t1−t)q  z > ln ( s∗ st ) − ( rd −rf − σ 2 2 ) (t1 − t) σ √ t1 − t   = k1e −rd(t1−t)φ (w1) combine the integrals i1, i2 and i3 in (2.2) we get the pricing formula for european compound option written on foreign currency call option, cccur (t,st,k1,k2,t1,t2) = e −rf(t2−t)φ(w2,w4; ρ)−k2e−rd(t2−t)φ(w1,w3; ρ) −k1e−rd(t1−t)φ(w1). where w1 = ln ( s∗ st ) − ( rd −rf − σ 2 2 ) (t1 − t) σ √ t1 − t , w3 = ln ( st k2 ) + ( rd −rf + σ 2 2 ) (t2 − t) σ √ t2 − t w2 = ln ( st s∗ ) + ( rd −rf − σ 2 2 ) (t1 − t) σ √ t1 − t , w4 = ln ( st k2 ) + ( rd −rf − σ 2 2 ) (t2 − t) σ √ t2 − t . and ρ = √ t1 − t√ t2 − t . in this pricing formula, φ represents cumulative function of standard normal distribution and φ represents the bivariate cumulative normal distribution and ρ is the correlation. � int. j. anal. appl. 18 (6) (2020) 1022 3. compound option written on power option 3.1. power option. [10] if an option whose payoff price is related to exponent of r+ underlying assets is called european power option. this kind of option may have european power call option and european power put option whose payoff are the following, cpower = max [(s n t −k) , 0] , ppower = max [(k −snt ) , 0] . the valuation of power option can be calculated by using black-scholes strategy, then the pricing formula for european call and put option is, cpower (t,st,t,k) = s n t e [ (n−1) ( r+nσ 2 2 ) −nd ] (t−t) φ (d1) −ke−r(t−t)φ (d2) , where d1 = ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t − t) σ √ t − t , d2 = ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t − t) σ √ t − t , 3.2. compound option written on power option. theorem 3.1. let {wt : t ≥ 0} be a p -standard brownian motion on the probability space (ω,f,p) and let st, the price of unit of stock, follow a gbm with the following sde dst = (µ−d)stdt + σstdwt where µ is the drift parameter and σ is the volatility parameter. in addition, we let r be the risk-free interest rate. then the price of compound call option, with strike k1 and maturity t1, contingent upon a power call option, with strike k2 and maturity t2, is given by ccpow (t,st,k1,k2,t1,t2) = s n t e ( (n−1) ( r+σ 2 2 ) −nσ2 ) (t1−t)+(n−1) ( r+nσ 2 2 ) (t2−t1)−nd(t2−t) ×φ(ψ2,ψ4; ρ) −k2e−r(t1−t)φ (ψ1,ψ3; ρ) −k1e−r(t1−t)φ (ψ1) . int. j. anal. appl. 18 (6) (2020) 1023 where ψ1 = ln ( s∗ snt ) −n ( r −d − σ 2 2 ) (t1 − t) σ √ t1 − t , ψ2 = ln ( snt s∗ ) + n ( r −d + σ 2 2 ) (t1 − t) nσ √ t1 − t , ψ3 = ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 − t) nσ √ t2 − t , ψ4 = ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r + σ 2 2 ) (t1 − t) σ √ t2 − t , ρ = √ t1 − t√ t2 − t . in this pricing formula, φ represents distribution function of standard normal distribution and φ represents the bivariate cumulative distribution function of standard normal distribution and ρ is the correlation. proof. let us begin by explicitly giving the black-scholes price of the underlying power option with maturity t2 and strike k2, with inception at t1, can be given as: cpow (st1,k2,t2 −t1) = s n t1 e [ (n−1) ( r+nσ 2 2 ) −nd ] (t2−t1) φ (d1) −k2e−r(t2−t−1)φ (d2) . here d1 = ln ( st1 k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 −t1) σ √ t2 −t1 , d2 = ln ( st1 k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) σ √ t2 −t1 . before moving ahead towards the derivation of the price of compound option contigent upon a power, let us make sense of distribution and dynamics of the process (snt )t≥0 under a risk-neutral measure. from girsanov’s theorem (cf. [22]), under the risk-neutral measure q we can write dst = (r −d)stdt + σstdwqt . such that w q t = wt + ( µ−r σ ) t is a q -standard wiener process. by application of ito’s lemma, we may infer dsnt = ns n t [ (r −d)dt + σdwqt ] + n(n− 1)σ2 2 snt dt = nsnt [( (r −d) − σ2(n− 1) 2 ) dt + σdw q t ] . int. j. anal. appl. 18 (6) (2020) 1024 moreover, application of ito’s lemma on process (ln (snt ))t≥0, it follows that, d ln (snt ) = ( n(r −d) + n(n− 1)σ2 2 − n2σ2 2 ) dt + nσdw q t , = ( n(r −d) − nσ2 2 ) dt + nσdw q t , = n ( r −d − σ2 2 ) dt + nσdw q t . on integrating both sides we may infer, ∫ t t d ln (snu ) = ∫ t t n ( r −d − σ2 2 ) du + ∫ t t nσdwqu , ln ( snt snt ) = n ( r −d − σ2 2 ) (t − t) + nσ ( w q t −w q t ) i.e. snt = s n t e n ( r−d−σ 2 2 ) (t−t)+nσ(wqt −w q t ). as w q t −w q t = w q t−t ∼n(0,t − t). so it is easy to see that, ln ( snt snt ) ∼n [ n ( r −d − σ2 2 ) (t − t),n2σ2(t − t) ] . we aim to find pricing formula for the price ccpow (t,st,k1,k2,t1,t2) of compound option with maturity t1 and strike k1, contingent upon power option whose price mentioned above. using the feynman-kac formula [22, theorem 4.33] ccpow (t,st,k1,k2,t1,t2) = e−r(t1−t)eq [ cpow (st1,k2,t2 −t1) −k1) + | ft ] = e−r(t1−t)eq [( snt1e ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1) φ (d1) −k2e−r(t2−t1)φ (d2) −k1 )+ | ft ] = e−r(t1−t) ∫ ∞ 0 ( snt1e ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1) φ (d1) −k2e−r(t2−t1)φ (d2) −k1 )+ ×n (st1 | st) dst1 let s∗ be price of unit of foreign currency such that, the compound option is at the money i.e. c (s∗,t1; k2,t2) = k1, and using this equation and any suitable method (say newton-raphson) an approximate value of s∗ can be computed. now, int. j. anal. appl. 18 (6) (2020) 1025 ccpow (t,st,k1,k2,t1,t2) = e−r(t1−t) ∫ ∞ s∗ ( snt1e ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1) φ (d1) −k2e−r(t2−t1)φ (d2) −k1 ) ×n (st1 | st) dst1, = e −r(t1−t)+ ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1) ∫ ∞ s∗ snt1 φ(d2) n (st1 | st) dst1 −e−r(t2−t)k2 ∫ ∞ s∗ φ(d1) n (st1 | st) dst1 −e−r(t1−t)k1 ∫ ∞ s∗ n (st1 | st) dst1 =: j1 −j2 −j3. let start by computing j1 explicitly, j1 = e −r(t1−t)+ ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1) ∫ ∞ s∗ snt1 φ(d2) n (st1 | st) dst1, now the compound option will be in the money and hence will be exercised iff snt1 > s ∗ iff snt e n ( r−d−σ 2 2 ) (t1−t)+nσ ( w q t1 −wqt ) > s∗ iff snt e n ( r−d−σ 2 2 ) (t1−t)+nσ √ t1−tx > s∗ iff x > ln ( s∗ sn t ) −n ( r−d−σ 2 2 ) (t1−t) nσ √ t1−t =: ψ1. hence we rewrite j1 as, j1 = s n t e −r(t1−t)+ ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1)+n ( r−d−σ 2 2 ) (t1−t) ×∫ ∞ ψ1 φ  ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r − σ 2 2 ) (t1 − t) + nxσ √ t1 − t σ √ t2 −t1   × e− (x2−2xσ √ t1−t) 2 √ 2π dx, = snt e −r(t1−t)+ ( (n−1) ( r+nσ 2 2 ) −nd ) (t2−t1)+n ( r−d−σ 2 2 ) (t1−t)−σ 2 2 (t1−t) ×∫ ∞ ψ1 φ  ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r −d − σ 2 2 ) (t1 − t) + nxσ √ t1 − t σ √ t2 −t1   × e− (x−σ √ t1−t) 2 2 √ 2π dx. to simplify the above integral, let using substitution u = − ( x−σ √ t1 − t ) , the integral simplifies to, j1 = s n t e ( (n−1) ( r+σ 2 2 ) −nσ2 ) (t1−t)+(n−1) ( r+nσ 2 2 ) (t2−t1)−nd(t2−t)× ∫ ψ2 −∞ φ  ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r −d + σ 2 2 ) (t1 − t) −nσ √ t1 − tu σ √ t2 −t1   e−u22√ 2π du where ψ2 = σ √ t1 − t−ψ1. if we set ρ = √ t1 − t t2 − t and ψ4 = ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r + σ 2 2 ) (t1 − t) σ √ t2 − t int. j. anal. appl. 18 (6) (2020) 1026 along with using bivariate identity (2.1) the last integral can be written as, j1 = s n t e ( (n−1) ( r+σ 2 2 ) −nσ2 ) (t1−t)+(n−1) ( r+nσ 2 2 ) (t2−t1)−nd(t2−t) ∫ ψ2 −∞ φ ( ψ4 −ρu√ 1 −ρ2 ) e− u2 2 √ 2π du = snt e ( (n−1) ( r+σ 2 2 ) −nσ2 ) (t1−t)+(n−1) ( r+nσ 2 2 ) (t2−t1)−nd(t2−t) φ(ψ2,ψ4; ρ). now let us compute the second integral j2, j2 = −e−r(t2−t)k2 ∫ ∞ s∗ φ(d1) n (st1 | st) dst1. by arguing same as previously, the compound option will be in the money and hence will be exercised iff st1 > s ∗ iff snt e n ( r−d−σ 2 2 ) (t1−t)+nσ ( w q t1 −wqt ) > s∗ iff snt e n ( r−d−σ 2 2 ) (t1−t)+nσ √ t1−tx > s∗ iff x > − ln ( snt s∗ ) − ( r−d−σ 2 2 ) (t1−t) nσ √ t1−t = −ψ1. hence we rewrite j2 as, j2 = k2e −r(t1−t) ∫ ∞ −ψ1 φ  ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 − t) + xσ √ t1 − t σ √ t2 −t1   e−x22√ 2π dx. using the substitution, u = −x, the above integral can be simplified to, j2 = −k2e−r(t1−t) ∫ −∞ ψ1 φ  ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 − t) −uσ √ t1 − t σ √ t2 −t1   e−u22√ 2π du = k2e −r(t1−t) ∫ ψ1 −∞ φ  ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 − t) −uσ √ t1 − t σ √ t2 −t1   e−u22√ 2π du. on setting, if we set ρ = √ t1−t t2−t and ψ3 = ln ( st k 1 n ) +(r−d+(n−12 )σ 2)(t2−t) σ √ t2−t along with using bivariate identity (2.1) the last integral can be written as, j2 = k2e −r(t1−t) ∫ ψ1 −∞ φ ( ψ3 −ρu√ 1 −ρ2 ) e− u2 2 √ 2π du, = k2e −r(t1−t)φ (ψ1,ψ3; ρ) . int. j. anal. appl. 18 (6) (2020) 1027 finally, its fairly easy to compute the 3rd integral i3, we will make use of fact w q t1 −wqt ∼ n (0,t1 − t) we have j3 = k1e −r(t1−t) ∫ ∞ s∗ n (st1 | st) dst1 = k1e −r(t1−t)q (st1 > s ∗ | st) = k1e −r(t1−t)q ( snt e n ( r−d−σ 2 2 ) (t1−t)+nσ ( w q t1 −wqt ) > s∗ | st ) = k1e −r(t1−t)q ( snt e n ( r−d−σ 2 2 ) (t1−t)+nσ √ t1−tz > s∗ | st ) = k1e −r(t1−t)q  z > ln ( s∗ snt ) −n ( r −d − σ 2 2 ) (t1 − t) nσ √ t1 − t   = k1e −r(t1−t)φ (ψ1) . combine the integrals j1, j2 and j3 in (2.2) we get the pricing formula for european compound option written on foreign currency call option, ccpow (t,st,k1,k2,t1,t2) = s n t e ( (n−1) ( r+σ 2 2 ) −nσ2 ) (t1−t)+(n−1) ( r+nσ 2 2 ) (t2−t1)−nd(t2−t) ×φ(ψ2,ψ4; ρ) −k2e−r(t1−t)φ (ψ1,ψ3; ρ) −k1e−r(t1−t)φ (ψ1) . where ψ1 = ln ( s∗ snt ) −n ( r −d − σ 2 2 ) (t1 − t) σ √ t1 − t , ψ2 = ln ( snt s∗ ) + n ( r −d + σ 2 2 ) (t1 − t) nσ √ t1 − t , ψ3 = ln ( st k 1 n ) + ( r −d + ( n− 1 2 ) σ2 ) (t2 − t) nσ √ t2 − t , ψ4 = ln ( st k 1 n ) + ( r −d − σ 2 2 ) (t2 −t1) + n ( r + σ 2 2 ) (t1 − t) σ √ t2 − t , ρ = √ t1 − t√ t2 − t . in this pricing formula, φ represents cumulative function of standard normal distribution and φ represents the bivariate cumulative normal distribution and ρ is the correlation. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] z. brzezniak, t. zastawniak, basic stochastic processes: a course through exercises. springer science & business media. 2000. int. j. anal. appl. 18 (6) (2020) 1028 [2] f. black, m. scholes, the pricing of options and corporate liabilities. j. polit. econ. 81(3)(1973), 637-654. [3] s. macovschi, f. quittard-pinon, on the pricing of power and other polynomial options. j. deriv. 13(4)(2006), 61-71. [4] z. zhang, w. liu, y. sheng, valuation of power option for uncertain financial market. appl. math. comput. 286(2016), 257-264. [5] e. dastranj, r. latifi, a comparison of option pricing models, int. j. financ. eng. 4(02n03)(2017), 1750024. [6] e. saberi, s.r. hejazi, e. dastranj, a new method for option pricing via time fractional pde, asian-eur. j. math. 11 (5) (2018) 1850074. [7] e. dastranj, h.s. fard, a. abdolbaghi, s.r. hejazi, power option pricing under the unstable conditions (evidence of power option pricing under fractional heston model in the iran gold market). physica a. stat. mech. appl. 537(2020), 122690. [8] r. geske, the valuation of corporate liabilities as compound options. j. financ. quant. anal. 12(1977), 541-552. [9] r. geske, the valuation of compound options. j. financ. econ. 7(1)(1979), 63-81. [10] j.c. hull, options futures and other derivatives. pearson education india. 2003. [11] a. elettra, a. rossella, a generalization of the geske formula for compound options. math. soc. sci. 45(1)(2003), 75-82. [12] p. carr, the valuation of sequential exchange opportunities. j. finance, 43(5)(1988), 1235-1256. [13] c. chiarella, b. kang, the evaluation of american compound option prices under stochastic volatility and stochastic interest rates. j. comput. finance. 17(2013), 71-92. [14] c. li, h. liu, m. wang, w. li, the pricing of compound option under variance gamma process by fft, commun. stat., theory meth. (2020) 1–15. https://doi.org/10.1080/03610926.2020.1740268. [15] g. feiger, b. jacquillat, currency option bonds, puts and calls on spot exchange and the hedging of contingent foreign earnings, j. finance. 34(1979), 1129–1139. [16] j.o. grabbe, the pricing of call and put options on foreign exchange. j. int. money finance, 2(3)(1983), 239-253. [17] k.i. amin, r.a. jarrow, pricing foreign currency options under stochastic interest rates. j. int. money finance, 10(3)(1991), 310-329. [18] j.h. guo, m.w. hung, pricing american options on foreign currency with stochastic volatility, jumps, and stochastic interest rates. j. futures mark., fut. opt. other deriv. prod. 27(9)(2007), 867-891. [19] s. li, j. peng, b. zhang, the uncertain premium principle based on the distortion function. insurance, math. econ. 53(2)(2013), 317-324. [20] d. liu, markov modulated jump-diffusions for currency options when regime switching risk is priced. int. j. financ. eng. 6(04)(2019), 1950038. [21] m.c. chuang, c.h. wen, s.k. lin, valuation and empirical analysis of currency options. int. rev. econ. finance, 66(2020),71-91. [22] m. capi’nski, e. kopp, j. traple, stochastic calculus for finance, cambridge university press. 2012. [23] e. chin, s. ólafsson, d. nel, problems and solutions in mathematical finance: stochastic calculus, vol. 1, the wiley finance series. 2014. 1. introduction 2. compound option written on foreign currency option 2.1. foreign currency option 2.2. compound option written on currency option 3. compound option written on power option 3.1. power option 3.2. compound option written on power option references bibliography int. j. anal. appl. (2023), 21:77 on (fuzzy) weakly almost interior γ-hyperideals in ordered γ-semihypergroups warud nakkhasen1,∗, ronnason chinram2, aiyared iampan3 1department of mathematics, faculty of science, mahasarakham university, maha sarakham 44150, thailand 2division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 3department of mathematics, school of science, university of phayao, phayao 56000, thailand ∗corresponding author: warud.n@msu.ac.th abstract. in this paper, we concentrate on studying the generalization of almost interior γ-hyperideals in ordered γ-semihypergroups. the notion of weakly almost interior γ-hyperideals of ordered γsemihypergroups is introduced. this concept generalizes the notion of almost interior γ-hyperideals in ordered γ-semihypergroups. then, the characterization of ordered γ-semihypergroups having no proper weakly almost interior γ-hyperideals is provided. next, we introduce the concept of fuzzy weakly almost interior γ-hyperideals of ordered γ-semihypergroups. also, some properties of fuzzy weakly almost interior γ-hyperideals are considered. moreover, the concepts of weakly almost interior γ-hyperideals and fuzzy weakly almost interior γ-hyperideals of ordered γ-semihypergroups are characterized. the connections between strongly prime (resp., prime, semiprime) weakly almost interior γ-hyperideals and fuzzy strongly prime (resp., prime, semiprime) weakly almost interior γ-hyperideals in ordered γ-semihypergroups are presented. 1. introduction when it comes to studying in semigroups, ideal theory is essential. grošek and satko [5] extended the concept of ideals in semigroups to the concept of almost ideals in 1980, characterizing the semigroups that have proper almost ideals. afterwards, bogdanović [2] introduced the concept almost bi-ideals in semigroups, as a generalization of bi-ideals, by using the concepts of almost ideals and received: may 20, 2023. 2020 mathematics subject classification. 03e72, 20m12. key words and phrases. weakly almost interior γ-hyperideals; fuzzy weakly almost interior γ-hyperideals; ordered γ-semihypergroups. https://doi.org/10.28924/2291-8639-21-2023-77 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-77 2 int. j. anal. appl. (2023), 21:77 bi-ideals of semigroups. zadeh [22] introduced the concept of fuzzy subsets as a function from a nonempty set x to the unit interval [0, 1]. wattanatripop et al. [21] applied the concept of fuzzy subsets to define the notion of fuzzy almost bi-ideals of semigroups in 2018, they examined at some of the connections between almost bi-ideals and fuzzy almost bi-ideals in semigroups. the concepts of (resp., weakly) almost interior ideals and fuzzy (resp., weakly) almost interior ideals in semigroups were introduced and discussed by kaopusek et al. [8] and krailoet et al. [9], respectively. in 2022, chinram and nakkhasen [3] introduced the concept of almost bi-quasi-interior ideals of semigroups and considered some relationships between almost bi-quasi-interior ideals and their fuzzification in semigroups. the notion of γ-semigroups generalized from the classical semigroups, was first introduced by sen and saha [15]. then, simuen et al. [16] defined the concepts of almost quasi-γ-ideals and fuzzy almost quasi-γ-ideals of γ-semigroups. later, jantanan et al. [7] studied the concepts of almost interior γ-ideals and fuzzy almost interior γ-ideals in γ-semigroups. the notion of ordered semigroups is another generalization of the semigroups. in 2022, suebsung et al. [17] introduced the concepts of (resp., fuzzy) almost bi-ideals and (resp., fuzzy) almost quasi-ideals of ordered semigroups, and they have investigated the characterizations of these concepts. since 1934, the research of marty [10], who developed the notion of hyperstructures, has been studied by many mathematicians. the concept of almost hyperideals in semihypergroups, which is a generalization of hyperideals, was introduced and presented some properties by suebsung et al. [18]. then, they have defined the concept of almost quasi-hyperideals in semihypergroups and gave some interesting properties, see [19]. next, muangdoo et al. [11] introduced the notions of (resp., fuzzy) almost bi-hyperideals of semihypergroups and discussed some connections between almost bihyperideals and their fuzzification in semihypergroups. in 2022, nakkhasen et al. [12] surveyed some properties of fuzzy almost interior hyperideals in semihypergroups and considered some links between almost interior hyperideals and fuzzy almost interior hyperideals in semihypergroups. it is known that ordered γ-semihypergroups are a generalization of semihypergroups. recently, rao et al. [14] defined the concept of almost interior γ-hyperideals of ordered γ-semihypergroups and provided the relationships between interior γ-hyperideals and almost interior γ-hyperideals in ordered γ-semihypergroups. this article presents the notions of weakly almost interior γ-hyperideals in ordered γ-semihypergroups, which extend the idea of almost interior γ-hyperideals, and provides certain characteristics of these hyperideals. furthermore, we define the concept of fuzzy weakly almost interior γ-hyperideals of ordered γ-semihypergroups, and consider some connections between weakly almost interior γ-hyperideals and fuzzy weakly almost interior γ-hyperideals of ordered γ-semihypergroups. 2. preliminaries firstly, we recall some of the basis definitions and properties, which are necessary for this paper. int. j. anal. appl. (2023), 21:77 3 a hypergroupoid (h,◦) is a nonempty set h together with a mapping ◦ : h×h →p∗(h) called a hyperoperation, where p∗(h) denotes the set of all nonempty set of h (see [4,10]). we denote by a◦b the image of the pair (a,b) in h ×h. if x ∈ h and a,b ∈p∗(h), then we denote a◦b := ⋃ a∈a,b∈b a◦b,a◦x := a◦{x} and x ◦b := {x}◦b. definition 2.1. (see [6]) a hypergroupoid (s,◦) is called a semihypergroup if (x ◦y) ◦z = x ◦ (y ◦z) for all x,y,z ∈ s. in 2010, anvariyeh et al. [1] introduced the notion of γ-semihypergroups, which is a generalization of semihypergroups. definition 2.2. (see [1]) let s and γ be two nonempty sets. then, (s, γ) is called a γ-semihypergroup if for each γ ∈ γ is a hyperoperation on s, i.e., xγy ⊆ s for all x,y ∈ s, and for any α,β ∈ γ and x,y,z ∈ s, (xαy)βz = xα(yβz). let a and b be two nonempty subsets of a γ-semihypergroup (s, γ). we define aγb := ⋃ γ∈γ aγb = ⋃ γ∈γ {aγb | a ∈ a,b ∈ b}. particularly, if a = {a} and b = {b}, then we define aγb := {a}γ{b}. definition 2.3. (see [20]) let s and γ be two nonempty sets and ≤ be an order relation on s. an algebraic hyperstructure (s, γ,≤) is called an ordered γ-semihypergroup if the following conditions are satisfied: (i) (s, γ) is a γ-semihypergroup; (ii) (s,≤) is a partially ordered set; (iii) for every x,y,z ∈ s and γ ∈ γ, x ≤ y implies xγz ≤ yγz and zγx ≤ zγy. here, a ≤ b means that for each a ∈ a, there exists b ∈ b such that a ≤ b, for all nonempty subsets a and b of s. throughout this paper, we say an ordered γ-semihypergroup s instead of an ordered γsemihypergroup (s, γ,≤), unless otherwise mentioned. for any nonempty subset a of an ordered γ-semihypergroup s, we denote (a] := {t ∈ s | t ≤ a for some a ∈ a}. for a = {a}, we write (a] instead of ({a}]. lemma 2.1. [20] let a and b be nonempty subsets of an ordered γ-semihypergroup s. then, the following statements holds: (i) a ⊆ (a]; 4 int. j. anal. appl. (2023), 21:77 (ii) if a ⊆ b, then (a] ⊆ (b]; (iii) (a]γ(b] ⊆ (aγb] and ((a]γ(b]] = (aγb]; (iv) ((a]] = (a]. the notion of almost interior γ-hyperideals in ordered γ-semihypergroups, as a generalization of interior γ-hyperideals, has been introduced by rao et al. [14] in 2021 as follows. definition 2.4. [14] let s be an ordered γ-semihypergroup. a nonempty subset k of s is called an almost interior γ-hyperideal of s if (i) (xγkγy] ∩k 6= ∅ for every x,y ∈ s, (ii) (k] ⊆ k. now, we review the concept of fuzzy subsets, was defined by zadeh [22]. we say that µ is a fuzzy subset [22] of a nonempty set x if µ : x → [0, 1]. for any two fuzzy subsets µ and λ of a nonempty set x, we denote (i) µ ⊆ λ if and only if µ(x) ≤ λ(x) for all x ∈ x, (ii) (µ∩λ)(x) := min{f (x),g(x)} for all x ∈ x, (iii) (µ∪λ)(x) := max{f (x),g(x)} for all x ∈ x. for any fuzzy subset µ of a nonempty set x, the support of µ is defined by supp(µ) := {x ∈ x | µ(x) 6= 0}. the characteristic mapping ca of a, where a is a subset of a nonempty set x, is a fuzzy subset of x defined by for every x ∈ x, ca(x) :=  1 if x ∈ a, 0 otherwise. lemma 2.2. [11] let a and b be nonempty subsets of a nonempty set x and let µ and λ be fuzzy subsets of x. then, the following statements hold: (i) ca∩b = ca ∩cb; (ii) a ⊆ b if and only if ca ⊆ cb; (iii) supp(ca) = a; (iv) if µ ⊆ λ, then supp(µ) ⊆ supp(λ). for any element s of x and α ∈ (0, 1], a fuzzy point sα [13] of x is a fuzzy subset of x defined by for every x ∈ x, sα(x) :=  α if x = s, 0 otherwise. int. j. anal. appl. (2023), 21:77 5 let s be an ordered γ-semihypergroup. for each x ∈ s, we define hx := {(y,z) ∈ s×s | x ≤ yγz}. then, for any two fuzzy subsets µ and λ of s, the product µ◦λ [20] of µ and λ is defined by (µ◦λ)(x) =   sup (y,z)∈hx [min{µ(y),λ(z)}] if hx 6= ∅, 0 if hx = ∅, for all x ∈ s. let µ be a fuzzy subset of an ordered γ-semihypergroup s. then, we define (µ] : s → [0, 1] by (µ](x) = sup x≤y µ(y) for all x ∈ s (see [20]). the following results can be verified straightforward. lemma 2.3. let a and b be subsets of an ordered γ-semihypergroup s. then ca ◦cb = c(aγb]. proposition 2.1. let µ,λ and ν be fuzzy subsets of an ordered γ-semihypergroup s. then, the following conditions hold: (i) µ ⊆ (µ]; (ii) if µ ⊆ λ, then (µ] ⊆ (λ]; (iii) if µ ⊆ λ, then (µ◦ν] ⊆ (λ◦ν] and (ν ◦µ] ⊆ (ν ◦λ]. proposition 2.2. let µ be a fuzzy subset of an ordered γ-semihypergroup s. then, the following statements are equivalent: (i) if x ≤ y, then µ(x) ≥ µ(y) for all x,y ∈ s; (ii) (µ] = µ. 3. weakly almost interior γ-hyperideals in this section, we present and study the notion of weakly almost interior γ-hyperideals of ordered γ-semihypergroups as a generalization of almost interior γ-hyperideals. definition 3.1. let s be an ordered γ-semihypergroup. a nonempty subset i of s is called a weakly almost interior γ-hyperideal of s if it satisfies the following conditions: (i) (xγiγx] ∩ i 6= ∅ for all x ∈ s; (ii) (i] ⊆ i. the following proposition obtains direct from the definition of almost interior γ-hyperideals and weakly almost interior γ-hyperideals in ordered γ-semihypergroups. proposition 3.1. every almost interior γ-hyperideal of an ordered γ-semihypergroup s is also a weakly almost interior γ-hyperideal of s. the converse of proposition 3.1 is not true in general, as shown by the following example below. 6 int. j. anal. appl. (2023), 21:77 example 3.1. let s = {a,b,c,d,e,f} and γ = {γ} with the hyperoperation on s defined by γ a b c d e f a {a} {b} {c} {d} {e} {f} b {b} {c} {a} {f} {d} {e} c {c} {a} {b} {e} {f} {d} d {d} {f} {e} {a,b} {a,c} {b,c} e {e} {d} {f} {a,c} {b,c} {a,b} f {f} {e} {d} {b,c} {a,b} {a,c} then, (s, γ,≤) is an ordered γ-semihypergroup, where the order relation ≤ on s defined by ≤:= {(x,y) | x = y}. let i = {a,b}. hence, by routine calculation, we have that i is a weakly almost interior γ-hyperideal of s. but i is not an almost interior γ-hyperideal of s, because (dγiγa] ∩ i = ∅. theorem 3.1. let i be a weakly almost interior γ-hyperideal of ordered γ-semihypergroup s. if a is any subset of s containing i, then a is also a weakly almost interior γ-hyperideal of s. proof. assume that a is a subset of s such that i ⊆ a. let x ∈ s. then, (xγiγx] ∩ i 6= ∅. thus, ∅ 6= (xγiγx]∩i ⊆ (xγaγx]∩a. it follows that (xγaγx]∩a 6= ∅. hence, a is a weakly almost interior γ-hyperideal of s. � corollary 3.1. let s be an ordered γ-semihypergroup. if i1 and i2 are weakly almost interior γhyperideals of s, then i1 ∪ i2 is a weakly almost interior γ-hyperideal of s. example 3.2. let s = {a,b,c,d,e} and γ = {α} be the nonempty sets. define the hyperoperation as: α a b c d e a {d} {a,b,d} {a,b,d} {d} {a,b,d,e} b {a,b,d} {a,b,d} {a,b,d} {a,b,d} {a,b,d,e} c {a,b,d} {a,b,d} {a,b,d} {a,b,d} {a,b,d,e} d {d} {a,b,d} {a,b,d} {d} {a,b,d,e} e {a,b,d} {a,b,d} {a,b,d,e} {a,b,d} {a,b,d,e} next, we define an order relation ≤ on s as: ≤:={(a,a), (b,b), (c,c), (d,d), (e,e), (a,b), (a,c), (a,e), (b,c), (b,e), (d,b), (d,c), (d,e)}. then, (s, γ,≤) is an ordered γ-semihypergroup. let i1 = {a,b} and i2 = {d}. verifying that i1 and i2 are weakly almost interior γ-hyperideals of s is a routine process. however, i1 ∩ i2 is not a weakly almost interior γ-hyperideal of s. the intersection of any two weakly almost interior γ-hyperideals of an ordered γ-semihypergroup s does not necessarily have to be a weakly almost interior γ-hyperideal of s, as shown by example 3.2. int. j. anal. appl. (2023), 21:77 7 theorem 3.2. let s be an ordered γ-semihypergroup and |s| > 1. then, the following statements are equivalent: (i) s has no proper weakly almost interior γ-hyperideal; (ii) for every x ∈ s, there exists ax ∈ s such that (ax γ(s \{x})γax ] = {x}. proof. (i) ⇒ (ii) assume that (i) holds. for any x ∈ s, we have that s\{x} is not a weakly almost interior γ-hyperideal of s. so, there exists ax ∈ s such that (ax γ(s \{x})γax ] ∩ (s \{x}) = ∅. we obtain that (ax γ(s \{x}γax )] ⊆ s \ (s \{x}) = {x}. it turns out that (ax γ(s \{x})γax ] = {x}. (ii) ⇒ (i) assume that (ii) holds. let a be any a proper subset of s. then, a ⊆ s\{x} for some x ∈ s. by assumption, there exists ax ∈ s such that (ax γ(s \{x})γax ] = {x}. thus, (ax γaγax ] ∩a ⊆ (ax γ(s \{x})γax ] ∩ (s \{x}) = {x}∩ (s \{x}) = ∅. hence, a is not a weakly almost interior γ-hyperideal of s. this shows that s has no proper weakly almost interior γ-hyperideal of s. � 4. fuzzy weakly almost interior γ-hyperideals the concept of fuzzy weakly almost interior γ-hyperideals of ordered γ-semihypergroups and some of the relationships between them are discussed in this section. definition 4.1. let µ be a nonzero fuzzy subset of an ordered γ-semihypergroup s. then, µ is called a fuzzy weakly almost interior γ-hyperideal of s if for every fuzzy point sα of s, (sα◦µ◦sα] ∩µ 6= 0. from the definition 4.1, we obtain that the following remark holds. remark 4.1. let sα be any fuzzy point of an ordered γ-semihypergroup s. then, (sα◦µ◦sα]∩µ 6= 0 if and only if there exist x,a ∈ s such that x ≤ sγaγs and µ(x),µ(a) 6= 0. theorem 4.1. let µ be a fuzzy weakly almost interior γ-hyperideal of an ordered γ-semihypergroup s. if λ is a fuzzy subset of s such that µ ⊆ λ, then λ is also a fuzzy weakly almost interior γ-hyperideal of s. proof. assume that λ is a fuzzy subset of s such that µ ⊆ λ. let sα be a fuzzy point of s. then, (sα◦µ◦sα]∩µ 6= 0. since µ ⊆ λ, 0 6= (sα◦µ◦sα]∩µ ⊆ (sα◦λ◦sα]∩λ. also, (sα◦λ◦sα]∩λ 6= 0. hence, λ is a fuzzy weakly almost interior γ-hyperideal of s. � corollary 4.1. let µ and λ be fuzzy weakly almost interior γ-hyperideals of an ordered γsemihypergroup s. then, µ∪λ is a fuzzy weakly almost interior γ-hyperideal of s. 8 int. j. anal. appl. (2023), 21:77 example 4.1. consider the ordered γ-semihypergroup (s, γ,≤) in example 3.2, we define two fuzzy subsets µ and λ of s by for every x ∈ s, µ(x) =  0.8 if x ∈{a,b}, 0 otherwise and λ(x) =  0.5 if x = d, 0 otherwise. by routine computations, we find out that µ and λ are fuzzy weakly almost interior γ-hyperideals of s. however, µ∩λ is not a fuzzy weakly almost interior γ-hyperideal of s, because µ∩λ = 0. from example 4.1, we know that the intersection of two fuzzy weakly almost interior γ-hyperideals of an ordered γ-semihypergroup s need not be a fuzzy weakly almost interior γ-hyperideal of s. theorem 4.2. let i be a nonempty subset of an ordered γ-semihypergroup s. then, i is a weakly almost interior γ-hyperideal of s if and only if ci is a fuzzy weakly almost interior γ-hyperideal of s. proof. assume that i is a weakly almost interior γ-hyperideal of s. let sα be any fuzzy point of s. then, (sγiγs] ∩ i 6= ∅. thus, there exists a ∈ s such that a ∈ (sγiγs] and a ∈ i. so, ci(a) = 1 and a ≤ sγxγs for some x ∈ i. since x ∈ i, ci(x) = 1. it follows that (sα ◦ci ◦ sα](a) ≥ min{sα(s),ci(x),sα(s)} 6= 0. we obtain that [(sα◦ci ◦sα]∩ci](a) 6= 0. thus, ci is a fuzzy weakly almost interior γ-hyperideal of s. conversely, assume that ci is a fuzzy weakly almost interior γ-hyperideal of s. let s ∈ s. choose t = 1. then, (s1◦ci◦s1]∩ci 6= 0. so, there exist x,a ∈ s such that x ≤ sγaγs and ci(x),ci(a) 6= 0. this implies that x,a ∈ i. also, x ∈ (sγiγs]. thus, x ∈ (sγiγs] ∩ i, and then (sγiγs] ∩ i 6= ∅. therefore, i is a weakly almost interior γ-hyperideal of s. � theorem 4.3. let µ be a fuzzy subset of an ordered γ-semihypergroup s. then, µ is a fuzzy weakly almost interior γ-hyperideal of s if and only if supp(µ) is a weakly almost interior γ-hyperideal of s. proof. assume that µ is a fuzzy weakly almost interior γ-hyperideal of s. let s ∈ s. choose t = 1. so, (s1◦µ◦s1]∩µ 6= 0. so, there exist x,a ∈ s such that x ≤ sγaγs and µ(x),µ(a) 6= 0. also, x,a ∈ supp(µ). since x ≤ sγaγs, x ∈ (sγ(supp(µ))γs]. it turns out that x ∈ (sγ(supp(µ))γs] ∩supp(µ), that is, (sγ(supp(µ))γs] ∩ supp(µ) 6= ∅. hence, supp(µ) is a weakly almost interior γ-hyperideal of s. conversely, assume that supp(µ) is a weakly almost interior γ-hyperideal of s. let sα be any fuzzy point of s. then, (sγ(supp(µ))γs] ∩ supp(µ) 6= ∅. thus, there exists x ∈ s such that x ∈ (sγ(supp(µ))γs] and x ∈ supp(µ). so, x ≤ sγaγs for some a ∈ supp(µ). this means that µ(x),µ(a) 6= 0. we have that (sα ◦µ◦ sα] ∩µ 6= 0. therefore, µ is a fuzzy weakly almost interior γ-hyperideal of s. � int. j. anal. appl. (2023), 21:77 9 let s be an ordered γ-semihypergroup. a weakly almost interior γ-hyperideal i of s is called minimal if for any weakly almost interior γ-hyperideal a of s such that a ⊆ i implies that a = i. definition 4.2. let s be an ordered γ-semihypergroup. a fuzzy weakly almost interior γ-hyperideal µ of s is called minimal if for any fuzzy weakly almost interior γ-hyperideal λ of s such that λ ⊆ µ implies that supp(λ) = supp(µ). now, the relationship between minimal weakly almost interior γ-hyperideals and minimal fuzzy weakly almost interior γ-hyperideals in ordered γ-semihypergroups is then briefly examined. theorem 4.4. let s be an ordered γ-semihypergroup, and i be a nonempty subset of s. then, i is a minimal weakly almost almost interior γ-hyperideal of s if and only if ci is a minimal fuzzy weakly almost interior γ-hyperideal of s. proof. assume that i is a minimal weakly almost interior γ-hyperideal of s. by theorem 4.2, ci is a fuzzy weakly almost interior γ-hyperideal of s. let λ be any fuzzy weakly almost interior γhyperideal of s such that λ ⊆ ci. by lemma 2.2 and theorem 4.3, we have that supp(λ) is a weakly almost interior γ-hyperideal of s such that supp(λ) ⊆ supp(ci). since i is minimal, supp(λ) = i = supp(ci). hence, ci is a minimal fuzzy weakly almost interior γ-hyperideal of s. conversely, assume that ci is a minimal fuzzy weakly almost interior γ-hyperideal of s. thus, i is a weakly almost γ-hyperideal of s by theorem 4.2. now, let a be any weakly almost interior γ-hyperideal of s such that a ⊆ i. then, ca is a fuzzy weakly almost interior γ-hyperideal of s such that ca ⊆ ci. since ci is minimal and by lemma 2.2, we have that a = supp(ca) = supp(ci) = i. therefore, i is a minimal weakly almost interior γ-hyperideal of s. � the following corollary can be achieved by theorem 4.2 and theorem 4.3. corollary 4.2. let s be an ordered γ-semihypergroup. then, s has no proper weakly almost interior γ-hyperideal if and only if for every fuzzy weakly almost interior γ-hyperideal µ of s, supp(µ) = s. let s be an ordered γ-semihypergroup and p be a weakly almost interior γ-hyperideal of s. then: (i) p is said to be prime if for any weakly almost interior γ-hyperideals a and b of s such that (aγb] ⊆ p implies that a ⊆ p or b ⊆ p; (ii) p is said to be semiprime if for any weakly almost interior γ-hyperideal a of s such that (aγa] ⊆ p implies that a ⊆ p ; (iii) p is said to be strongly prime if for any weakly almost interior γ-hyperideals a and b of s such that (aγb] ∩ (bγa] ⊆ p implies that a ⊆ p or b ⊆ p. definition 4.3. let µ be a fuzzy weakly almost interior γ-hyperideal of an ordered γ-semihypergroup s. then, µ is said to be a fuzzy prime weakly almost interior γ-hyperideal of s if for any fuzzy weakly almost interior γ-hyperideals λ and ν of s such that λ◦ν ⊆ µ implies that λ ⊆ µ or ν ⊆ µ. 10 int. j. anal. appl. (2023), 21:77 definition 4.4. let µ be a fuzzy weakly almost interior γ-hyperideal of an ordered γ-semihypergroup s. then, µ is said to be a fuzzy semiprime weakly almost interior γ-hyperideal of s if for any fuzzy weakly almost interior γ-hyperideal λ of s such that λ◦λ ⊆ µ implies that λ ⊆ µ. definition 4.5. let µ be a fuzzy weakly almost interior γ-hyperideal of an ordered γ-semihypergroup s. then, µ is said to be a fuzzy strongly prime weakly almost interior γ-hyperideal of s if for any fuzzy weakly almost interior γ-hyperideals λ and ν of s such that (λ◦ν) ∩ (ν ◦λ) ⊆ µ implies that λ ⊆ µ or ν ⊆ µ. it is obvious that every fuzzy strongly prime weakly almost interior γ-hyperideal of an ordered γ-semihypergroup is a fuzzy prime weakly almost interior γ-hyperideal, and every fuzzy prime weakly almost interior γ-hyperideal of an ordered γ-semihypergroup is a fuzzy semiprime weakly almost interior γ-hyperideal. finally, we consider the connections between strongly prime (resp., prime, semiprime) weakly almost interior γ-hyperideals and their fuzzifications in ordered γ-semihypergroups. theorem 4.5. let s be an ordered γ-semihypergroup and p be a nonempty subset of s. then, p is a strongly prime weakly almost interior γ-hyperideal of s if and only if cp is a fuzzy strongly prime weakly almost interior γ-hyperideal of s. proof. assume that p is a strongly prime weakly almost interior γ-hyperideal of s. also, cp is a fuzzy weakly almost interior γ-hyperideal of s by theorem 4.2. let λ and ν be any two fuzzy weakly almost interior γ-hyperideals of s such that (λ ◦ ν) ∩ (ν ◦ λ) ⊆ cp . suppose that λ 6⊆ cp and ν 6⊆ cp . thus, there exist x,y ∈ s such that λ(x) 6= 0 and ν(y) 6= 0, but cp (x) = 0 and cp (y) = 0. so, x,y 6∈ p. by using theorem 4.3, we have that supp(λ) and supp(ν) are weakly almost interior γ-hyperideals of s such that x ∈ supp(λ) and y ∈ supp(ν). we obtain that, supp(λ) 6⊆ p and supp(ν) 6⊆ p. by assumption, ((supp(λ))γ(supp(ν))] ∩ ((supp(ν))γ(supp(λ))] 6⊆ p. then, there exists t ∈ ((supp(λ))γ(supp(ν))] ∩ ((supp(ν))γ(supp(λ))], but t 6∈ p. it follows that cp (t) = 0, and then [(λ◦ν)∩(ν◦λ)](t) = 0. since t ∈ ((supp(λ))γ(supp(ν))] and t ∈ ((supp(ν))γ(supp(λ))], we have that t ≤ a1γb1 and t ≤ b2γa2 for some a1,a2 ∈ supp(λ) and b1,b2 ∈ supp(ν). it turns out that (λ◦ν)(t) = sup t≤a1γb1 [min{λ(a1),ν(b1)}] 6= 0 and (ν ◦λ)(t) = sup t≤b2γa2 [min{ν(b2),λ(a2)}] 6= 0. this implies that [(λ◦ν) ∩ (ν ◦λ)](t) 6= 0, as a contradiction. so, λ ⊆ cp or ν ⊆ cp . this shows that cp is a fuzzy strongly prime weakly almost interior γ-hyperideal of s. conversely, assume that cp is a fuzzy strongly prime weakly almost interior γ-hyperideal of s. then, p is a weakly almost interior γ-hyperideal of s by theorem 4.2. let a and b be any two weakly almost interior γ-hyperideals of s such that (aγb] ∩ (bγa] ⊆ p. by using lemma 2.2 and int. j. anal. appl. (2023), 21:77 11 lemma 2.3, it follows that (ca ◦cb) ∩ (cb ◦ca) = c(aγb] ∩c(bγa] = c(aγb]∩(bγa] ⊆ cp . by the hypothesis, ca ⊆ cp or cb ⊆ cp . it follows that a ⊆ p or b ⊆ p. therefore, p is a strongly prime weakly almost interior γ-hyperideal of s. � theorem 4.6. let p be a nonempty subset of an ordered γ-semihypergroup s. then, p is a prime weakly almost interior γ-hyperideal of s if and only if cp is a fuzzy prime weakly almost interior γ-hyperideal of s. proof. assume that p is a prime weakly almost interior γ-hyperideal of s. by using theorem 4.2, we obtain that cp is a fuzzy weakly almost interior γ-hyperideal of s. let λ and ν be any two fuzzy weakly almost interior γ-hyperideals of s such that λ◦ν ⊆ cp . suppose that λ 6⊆ cp and ν 6⊆ cp . then, there exist x,y ∈ s such that λ(x) 6= 0 and ν(y) 6= 0, while cp (x) = 0 and cp (y) = 0. so, x ∈ supp(λ), y ∈ supp(ν) with x,y 6∈ p. by theorem 4.3, we have that supp(λ) and supp(ν) are weakly almost interior γ-hyperideals of s. this implies that supp(λ) 6⊆ p and supp(ν) 6⊆ p. by assumption, it follows that ((supp(λ)γ(supp(ν)))] 6⊆ p. also, there exists t ∈ ((supp(λ)γ(supp(ν)))] such that t 6∈ p . this means that cp (t) = 0. it turns out that (λ ◦ ν)(t) = 0, because λ ◦ ν ⊆ cp . since t ∈ ((supp(λ)γ(supp(ν)))], t ≤ aγb for some a ∈ supp(λ) and b ∈ supp(ν). thus, (λ◦ν)(t) = sup t≤aγb [min{λ(a),ν(b)}] 6= 0. this is a contradiction to the fact that (λ◦ν)(t) = 0. this shows that λ ⊆ cp or ν ⊆ cp . hence, cp is a fuzzy prime weakly almost interior γ-hyperideal of s. conversely, assume that cp is a fuzzy prime weakly almost interior γ-hyperideal of s. by theorem 4.2, p is a weakly almost γ-hyperideal of s. let a and b be any weakly almost interior γ-hyperideals of s such that (aγb] ⊆ p. by lemma 2.2 and lemma 2.3, it follows that ca ◦cb = c(aγb] ⊆ cp . by the given assumption, ca ⊆ cp or cb ⊆ cp . this implies that, a ⊆ p or b ⊆ p. therefore, p is a prime weakly almost interior γ-hyperideal of s. � theorem 4.7. let s be an ordered γ-semihypergroup and p be a nonempty subset of s. then, p is a semiprime weakly almost interior γ-hyperideal of s if and only if cp is a fuzzy semiprime weakly almost interior γ-hyperideal of s. proof. assume that p is a semiprime weakly almost interior γ-hyperideal of s. by theorem 4.2, we obtain that cp is a fuzzy weakly almost γ-hyperideal of s. let λ be any fuzzy weakly almost interior γ-hyperideal of s such that λ ◦ λ ⊆ cp . suppose that λ 6⊆ cp . so, there exists x ∈ s such that λ(x) 6= 0 and cp (x) = 0. also, x ∈ supp(λ) and x 6∈ p. by theorem 4.3, supp(λ) is a weakly almost interior γ-hyperideal of s where supp(λ) 6⊆ p. by assumption, ((supp(λ)γ(supp(λ)))] 6⊆ p. thus, there exists t ∈ s such that t ∈ ((supp(λ)γ(supp(λ)))], but t 6∈ p. this implies that cp (t) = 0. it 12 int. j. anal. appl. (2023), 21:77 follows that (λ◦λ)(t) = 0, because λ◦λ ⊆ cp . since t ∈ ((supp(λ)γ(supp(λ)))], t ≤ aγb for some a,b ∈ supp(λ). it turns out that (λ◦λ)(t) = sup t≤aγb [min{λ(a),λ(b)}] 6= 0, which is a contradiction. hence, λ ⊆ cp . therefore, cp is a fuzzy semiprime weakly almost γ-hyperideal of s. conversely, assume that cp is a fuzzy semiprime weakly almost γ-hyperideal of s. it follows that p is a weakly almost interior γ-hyperideal of s by theorem 4.2. let a be a weakly almost interior γ-hyperideal of s such that (aγa] ⊆ p. by using lemma 2.2 and lemma 2.3, we have that ca ◦ca = c(aγa] ⊆ cp . since cp is semiprime, ca ⊆ cp . it follows that a ⊆ p. this shows that p is a semiprime weakly interior γ-hyperideal of s. � 5. conclusions in 2021, rao et al. [14] introduced the concept of almost interior γ-hyperideals as a generalization of interior γ-hyperideals of ordered γ-semihypergroups. in this paper, we introduced the notion of weakly almost interior γ-hyperideals of ordered γ-semihypergroups which is a generalization of almost interior γ-hyperideals. next, we shown that the union of (fuzzy) weakly almost interior γ-hyperideals is also a (fuzzy) weakly almost interior γ-hyperideal, but the intersection of them need not to be a (fuzzy) weakly almost interior γ-hyperideal in ordered γ-semihypergroups. then, we characterized the ordered γ-semihypergroups having no proper weakly almost interior γ-hyperideal. finally, we discussed the connections between weakly almost interior γ-hyperideals and their fuzzification in ordered γsemihypergroups. in our future study, we plan to investigate other kinds of almost γ-hyperideals and their fuzzifications in ordered γ-semihypergroups or other algebraic structures. acknowledgements: this research project was financially supported by mahasarakham university. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s.m. anvariyeh, s. mirvakili, b. davvaz, on γ-hyperideals in γ-semihypergroups, carpathian j. math. 26 (2010), 11-23. https://www.jstor.org/stable/43999427. [2] s. bogdanović, semigroups in which some bi-ideal is a group, rev. res. fac. sci. univ. novi sad, 11 (1981), 261-266. [3] r. chinram, w. nakkhasen, almost bi-quasi-interior ideals and fuzzy almost bi-quasi-interior ideals of semigroups, j. math. computer sci. 26 (2021), 128-136. https://doi.org/10.22436/jmcs.026.02.03. [4] p. corsini, prolegomena of hypergroup theory, aviani editore publisher, tricesimo, italy, 1993. [5] o. grošek, l. satko, a new notion in the theory of semigroup, semigroup forum, 20 (1980), 233-240. https: //doi.org/10.1007/bf02572683. [6] d. heidari, b. davvaz, on ordered hyperstructures, upb sci. bull. ser. a: appl. math. phys. 73 (2011), 85-96. [7] w. jantanan, a. simuen, w. yonthanthum, r. chinram, almost interior gamma-ideals and fuzzy almost interior gamma-ideals in gamma-semigroups, math. stat. 9 (2021), 302-308. https://doi.org/10.13189/ms.2021. 090311. https://www.jstor.org/stable/43999427 https://doi.org/10.22436/jmcs.026.02.03 https://doi.org/10.1007/bf02572683 https://doi.org/10.1007/bf02572683 https://doi.org/10.13189/ms.2021.090311 https://doi.org/10.13189/ms.2021.090311 int. j. anal. appl. (2023), 21:77 13 [8] n. kaopusek, t. kaewnoi, r. chinram, on almost interior ideals and weakly almost interior ideals of semigroups, j. discrete math. sci. cryptography. 23 (2020), 773-778. https://doi.org/10.1080/09720529.2019.1696917. [9] w. krailoet, a. simuen, r. chinram, p. petchkaew, a note on fuzzy almost interior ideals in semigroups, int. j. math. computer sci. 16 (2021), 803-808. [10] f. marty, sur une generalization de la notion de group, in: 8th congres des mathematiciens scandinaves, stockholm, 45-49, 1934. [11] p. muangdoo, t. chuta, w. nakkhasen, almost bi-hyperideals and their fuzzification of semihypergroups, j. math. comput. sci. 11 (2021), 2755-2767. https://doi.org/10.28919/jmcs/5609. [12] w. nakkhasen, p. khathipphathi, s. panmuang, a note on fuzzy almost interior hyperideals of semihypergroups, int. j. math. computer sci. 17 (2022), 1419-1426. [13] p. pao-ming, l. ying-ming, fuzzy topology. i. neighborhood structure of a fuzzy point and moore-smith convergence, j. math. anal. appl. 76 (1980), 571-599. https://doi.org/10.1016/0022-247x(80)90048-7. [14] y. rao, s. kosari, z. shao, m. akhoundi, s. omidi, a study on a-i-γ-hyperideals and (m,n)-γ-hyperfilters in ordered γ-semihypergroups, discr. dyn. nat. soc. 2021 (2021), 6683910. https://doi.org/10.1155/2021/ 6683910. [15] m.k. sen, n.k. saha, on γ-semigroup-i, bull. calcutta math. soc. 78 (1986), 180-186. [16] a. simuen, k. wattanatripop, r. chinram, characterizing almost quasi-γ-ideals and fuzzy almost quasi-γ-ideals of γ-semigroups, commun. math. appl. 11 (2020), 233-240. [17] s. suebsung, r. chinram, w. yonthanthum, k. hila, a. iampan, on almost bi-ideals and almost quasi-ideals of ordered semigroups and their fuzzifications, icic express lett. 16 (2022), 127-135. [18] s. suebsung, t. kaewnoi, r. chinram, a note on almost hyperideals in semihypergroups, int. j. math. computer sci. 15 (2020), 127-133. [19] s. suebsung, w. yonthanthum, k. hila, r. chinram, on almost quasi-hyperideals in semihypergroups, j. discr. math. sci. cryptography. 24 (2021), 235-244. https://doi.org/10.1080/09720529.2020.1826167. [20] j. tang, b. davvaz, x. xie, a study on (fuzzy) quasi-γ-hyperideals in ordered γ-semihypergroups, j. intell. fuzzy syst. 32 (2017), 3821-3838. https://doi.org/10.3233/ifs-162117. [21] k. wattanatripop, r. chinram, t. changphas, fuzzy almost bi-ideals in semigroups, int. j. math. computer sci. 13 (2018), 51-58. [22] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.1080/09720529.2019.1696917 https://doi.org/10.28919/jmcs/5609 https://doi.org/10.1016/0022-247x(80)90048-7 https://doi.org/10.1155/2021/6683910 https://doi.org/10.1155/2021/6683910 https://doi.org/10.1080/09720529.2020.1826167 https://doi.org/10.3233/ifs-162117 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. weakly almost interior -hyperideals 4. fuzzy weakly almost interior -hyperideals 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 54-60 http://www.etamaths.com the role of complete parts in topological polygroups m. salehi shadkami, m.r. ahmadi zand and b. davvaz∗ abstract. a topological polygroup is a polygroup p together with a topology on p such that the polygroup’s binary hyperoperation and the polygroup’s inverse function are continuous with respect to the topology. in this paper, we present some facts about complete parts in polygroups and we use these facts to obtain some new results in topological polygroups. we define the concept of cp-resolvable topological polygroups. a non-empty subset x of a topological polygroup is called cpresolvable if there exist disjoint dense subsets a and b such that at least one of them is a complete part. then, we investigate a few properties of cp-resolvable topological polygroups. 1. introduction hypergroup that is based on the notion of hyperoperation was introduced by marty in [13] and studied extensively by many mathematicians. applications of hypergroups have mainly appeared in special subclasses. for example, polygroups that form an important subclass of hypergroups are studied by comer [3, 4]. polygroups have been applied in many area, such as geometry, lattices, combinatorics and color scheme. there exists a rich bibliography on polygroups [6]. this book contains the principal definitions endowed with examples and the basic results of the theory. aghabozorgi et al. [1] defined perfect and solvable polygroups. also see [11]. till now, only a few papers treated the notion of topological hyperstructures, for example see [2, 8, 10]. heidari et al. [9] defined the notion of topological polygroups. by considering the relative topology on subpolygroups they proved some properties of them. in particular, they proved the topological isomorphism theorems of topological polygroups. the purpose of this paper is as stated in the abstract. 2. basic definitions and results let h be a non-empty set and p∗(h) be the set of all non-empty subsets of h. then, the mapping ◦ : h × h → p∗(h) is called a hyperoperation and (h,◦) is called a hypergroupoid. if a and b are two non-empty subsets of h and x ∈ h, then we define a◦b = ⋃ a∈a b∈b a◦ b, a◦x = a◦{x} and x◦a = {x}◦a. if a◦ (b◦c) = (a◦b) ◦c for all a,b,c ∈ h (associativity axiom), then (h,◦) is called a semihypergroup and it is called a quasihypergroup if for every x ∈ h, we have x◦h = h = h◦x (reproduction axiom). the couple (h,◦) is called a hypergroup if it is a semihypergroup and a quasihypergroup. thus, a hypergroup is a generalization of a group. let (h,◦) be a semihypergroup and a be a non-empty subset of h. we say that a is a complete part of h if for any natural number n and for all a1,a2, . . . ,an ∈ h , the following implication holds: a∩ n∏ i=1 ai 6= ∅⇒ n∏ i=1 ai ⊆ a. the complete parts are introduced for the first time by koskas[12] and are studied by corsini [5] and many others. a special subclass of hypergroups is the class of polygroups. we recall the following 2010 mathematics subject classification. 20n20, 22a30. key words and phrases. hyperstructure; polygroup; topological polygroup; complete part; cp-resolvable polygroup; natural mapping. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 54 the role of complete parts in topological polygroups 55 definition from [3]. a polygroup is a system p =< p,◦,e,−1 >, where ◦ : p ×p → p∗(p), e ∈ p, −1 is a unitary operation on p and the following axioms hold for all x,y,z ∈ p : (1) (x◦y) ◦z = x◦ (y ◦z); (2) e◦x = x◦e = x; (3) x ∈ y ◦z implies y ∈ x◦z−1 and z ∈ y−1 ◦x. the following elementary facts about polygroups follow easily from the axioms: e ∈ x◦x−1 ∩x−1 ◦x , e−1 = e , (x−1)−1 = x, and (x ◦ y)−1 = y−1 ◦ x−1. a non-empty subset k of a polygroup p is a subpolygroup of p if and only if a,b ∈ k implies a ◦ b ⊆ k and a ∈ k implies a−1 ∈ k. the subpolygroup n of p is normal in p if and only if a−1 ◦n ◦a ⊆ n for every a ∈ p . proposition 2.1. let a and b be non-empty subsets of a polygroup p =< p,◦,e,−1 > such that a is a complete part and x ∈ p . then, (1) x−1 ◦x◦a = x◦x−1 ◦a = a; (2) a−1 is a complete part; (3) x◦a and a◦x are complete parts; (4) b ⊆ x−1 ◦a if and only if x◦b ⊆ a. remark 1. suppose that a is a complete part of a polygroup p , n,m ∈ n and (x1,x2, . . . ,xn) ∈ pn, (y1,y2, . . . ,ym) ∈ pm. by proposition 2.1, n∏ i=1 xi ◦a◦ m∏ j=1 yj and n∏ i=1 xi ◦a−1 ◦ m∏ j=1 yi are complete parts. let {as}s∈s be a family of complete parts of p. then, we observe that ⋃ s∈s as is a complete part and if i = ⋂ s∈s as 6= ∅, then i is a complete part. proposition 2.2. if a and b are two non-empty subsets of a polygroup p such that a is a complete part, then a◦b and b ◦a are complete parts. proof. suppose that n ∈ n and n∏ i=1 ai∩(a◦b) 6= ∅ where (a1,a2, . . . ,an) ∈ pn. then, there exists b ∈ b such that n∏ i=1 ai∩(a◦b) 6= ∅. by proposition 2.1, a◦b is a complete part and so n∏ i=1 ai ⊆ a◦b ⊆ a◦b, i.e., a◦b is a complete part. similarly, b ◦a is a complete part. � we observe that if a is a complete part and bi,cj are non-empty subsets of p for i = 1, 2, ...,n, j = 1, ...,m, where n,m ∈ n, then c1 ◦ ...◦cm ◦a◦b1 ◦b2...◦bn is a complete part. proposition 2.3. let p be a polygroup and bα be a complete part for every α ∈ i. then, a◦ ⋂ α∈i bα =⋂ α∈i (a◦bα) and ( ⋂ α∈i bα) ◦a = ⋂ α∈i (bα ◦a). proof. clearly, a◦ ⋂ α∈i bα ⊆ ⋂ α∈i a◦bα. conversely, suppose that t ∈ ⋂ α∈i a◦bα. so, for every α ∈ i, t ∈ a◦bα. now, if s ∈ i, then t ∈ a◦bs. hence, there exists bs ∈ bs such that t ∈ a◦ bs. thus, for every α ∈ i, t ∈ a◦bs∩a◦bα. so, for every α ∈ i , a◦bs ⊆ a◦bα. therefore, by proposition 2.1, for every α ∈ i , bs ∈ a−1 ◦a◦ bs ⊆ a−1 ◦a◦bα = bα, i.e., bs ∈ ⋂ α∈i bα. thus, t ∈ a◦ bs ⊆ a◦ ( ⋂ α∈i bα), and so a◦ ⋂ α∈i bα = ⋂ α∈i (a◦bα). similarly the equality ( ⋂ α∈i bα) ◦a = ⋂ α∈i (bα ◦a) holds. � remark 2. being complete part is necessary in proposition 2.3 as it is illustrated in the following example. example 1. consider the set of integer numbers z and define the hyperoperation ◦ on it as follows: for every m ∈ z, m◦ 0 = m and if m,n ∈ z\{0}, then m◦n = { e, m + n ∈ e ec, m + n ∈ ec where e = 2z. if a = {1, 2} and b = {1, 4}, then a and b are not complete parts and 1◦a = 1◦b = z. but the equality 1 ◦ (a∩b) = 2z holds. 56 shadkami, zand and davvaz the following result is a direct consequence of proposition 2.3. corollary 2.4. let a and c be non-empty subsets of a polygroup p . if bα is a complete part for every α ∈ i, then a◦ ( ⋂ α∈i bα) ◦c = ⋂ α∈i (a◦bα ◦c). 3. on topological polygroups and cp-resolvable topological polygroups by using complete parts in topological polygroups, some interesting results were obtained by a few authors, in the classical case, see [2, 8, 9, 10, 14]. in this section, we study subpolygroups and dense subsets of a topological polygroup. let (h,τ) be a topological space. then, the family b = {sv | v ∈ τ}, where sv = {u ∈ p∗(h) | u ⊆ v} is a base for a topology on p∗(h). this topology is denoted by τ∗ [10]. let (h,τ) be a topological space. we consider the product topology on h × h and the topology τ∗ on p∗(h). let h be a topological space and a ⊂ y ⊂ h, inty a denotes the interior of a in the subspace y , and the closure of a in the subspace y is denoted by cly a; the interior of a in h is denoted by a ◦, and the closure of a in h is denoted by a. definition 3.1. [9] let p =< p,◦,e,−1 > be a polygroup and (p,τ) be a topological space. then, the system p = (p,◦,e,−1 ,τ) is called a topological polygroup if the mappings µ : p × p → p∗(p) and ι : p → p defined by µ(x,y) = x◦y and ι(x) = x−1 are continuous. let u be an open subset of a topological polygroup p such that u is a complete part. then, a ◦ u and u ◦ a are open subsets of p for every a ∈ p [9]. let p be a topological polygroup such that every open subset of p is a complete part. let u be an open base at e. then, the families {x◦u | x ∈ p ,u ∈u} and {u ◦x | x ∈ p ,u ∈u} are open bases for p [9]. theorem 3.2. [9] let p = (p,◦,e,−1 ,τ) be a topological polygroup and u be a base at e. then, the following assertions hold. (1) for every u ∈u and x ∈ u there exists v ∈u such that x◦v ⊆ u. (2) for every u ∈u there exists v ∈u such that v ◦v ⊆ u. (3) for every u ∈u there exists v ∈u such that v −1 ⊆ u. let p = (p,◦,e,−1 ,τ) be a topological polygroup. we denote by νp (e) the set of all neighborhoods of e. theorem 3.3. [9] let p be a topological polygroup such that every open subset of p is a complete part. then, for every u ∈ νp (e) there exists v ∈ νp (e) such that v ⊆ u. let every open subset of a topological polygroup p be a complete part. if f is a compact subset of p , then for every a ∈ p , a◦f and f ◦a are compact [14]. theorem 3.4. [14] let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part, f be a compact subset of p and g be a closed subset of p . then, the sets f ◦g and g◦f are closed in p . lemma 3.5. [9] if h is a subpolygroup of a topological polygroup p = (p,◦,e,−1 ,τ) and every open subset of p is a complete part, then h is a subpolygroup of p . theorem 3.6. [9] let p be topological polygroup such that every open subset of p is a complete part. then, a subpolygroup k of p is open if and only if its interior is non-empty. theorem 3.7. [9] let p be a topological polygroup such that every open subset of p is a complete part. then, every open subpolygroup is closed. theorem 3.8. [14] let h be a non-empty subset of a topological polygroup p = (p,◦,e,−1 ,τ) and every open subset of p is a complete part. then, we have (1) h = ⋂ u∈ νp (e) u ◦h = ⋂ u∈νp (e) h ◦u = ⋂ u∈νp (e),v∈νp (e) u ◦h ◦v ; (2) if h is a normal subpolygroup, then h is a normal subpolygroup. the role of complete parts in topological polygroups 57 lemma 3.9. if p is a topological polygroup such that every open subset of p is a complete part, then a◦a = a◦a. proof. by theorem 3.8 and proposition 2.3, a◦a = ⋂ u∈νp (e) (a ◦ a) ◦ u = ⋂ u∈νp (e) a ◦ (a ◦ u) = a◦ ⋂ u∈νp (e) (a◦u) = a◦a. � remark 3. begin complete part is necessary in lemma 3.9 as it is illustrated in the following example. example 2. suppose that the multiplication table for a polygroup p =< p,◦, 1,−1 >, where p = {1, 2}, 1−1 = 1 and 2−1 = 2 is ◦ 1 2 1 {1} {2} 2 {2} {1, 2} if τ = {∅,{1},{1, 2}}, then p = (p,◦,−1 ,τ) is a topological polygroup. if a = {1}, then a = {1, 2} , 2 ◦a = {2} and 2 ◦a = {2}. but the equality 2 ◦a = {1, 2} holds. let x be a topological space and x ∈ x. recall that a family β of open subsets of x is called a base for x at the point x if all elements of β contain x and, for every neighborhood o of x, there exists u ∈ β such that x ∈ u ⊆ o, and that the character of x at the point x is denoted by χ(x,x) where χ(x,x) = min{ |β| | β is a base for x at the point x} and that if x has a countable base at each point x ∈ x, then x is called first-countable [7]. proposition 3.10. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and y is a dense subspace of p . then, χ(y,y ) = χ(y,p) for each y ∈ y . proof. suppose that β is a base for p at a point y ∈ y such that χ(y,p) = |β|. then, the family βy = {u ∩y | u ∈ β } is a base for y at y. thus, χ(y,y ) = min{ |β1| | β1 is a base for y at y}≤ |βy | ≤ |β| = χ(y,p). conversely, let βy be a base for y at a point y such that χ(y,y ) = |βy |. for every u ∈ βy , choose an open set vu in p such that u = vu ∩y . we claim that the family β = { vu| u ∈ βy} is a base for p at the point y. indeed, we have (1) for every vu ∈ β, y ∈ vu . (2) let o be an open subset of p containing y. then, by theorem 3.2, there exists w1 ∈ νp (e) such that y◦w1 ⊆ o. so, by theorem 3.3, there exists w ∈ νp (e) such that w ⊆ w1. hence, y ◦w ⊆ y ◦w1 ⊆ o. since w2 = y ◦w ∩y is an open subset of y and y ∈ w2, there exists u ∈ βy such that u ⊆ w2. since u = vu ∩y and y = p , it follows that vu = u ⊆ y ◦w = y ◦w ⊆ o by lemma 3.9. thus, y ∈ vu ⊆ o and so χ(y,p) = min{ |β1| | β1 is a base for p at y}≤ |β| ≤ |βy | = χ(y,y ). � lemma 3.11. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and let h be a subpolygroup of p . then, every open subset of h is a complete part. proof. suppose that u ∈p∗(h) is an open subest of h and for n ∈ n, n∏ i=1 ai ∩u 6= ∅, where ai ∈ h. thus, there exists an open subset v of p such that u = v ∩h. hence, n∏ i=1 ai∩v 6= ∅. so, n∏ i=1 ai ⊆ v . on the other hand n∏ i=1 ai ⊆ h and hence, n∏ i=1 ai ⊆ v ∩h = u. � lemma 3.12. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part. if y is a first-countable subpolygroup of p , then y is also first-countable. proof. suppose that k = y . then, by lemma 3.5, k is a subpolygroup of p and so by proposition 3.10, χ(e,k) = χ(e,y ) ≤ℵ0. � definition 3.13. let p = (p,◦,e,−1 ,τ) be a topological polygroup and u ∈ νp (e). a subset a of p is called u-disjoint, if b /∈ a◦u, for any distinct a,b ∈ a. 58 shadkami, zand and davvaz example 3. suppose that the multiplication table for a topological polygroup p = (p,τ,◦, 1,−1 ) where p = {e,a,b}, e−1 = e, a−1 = a and b−1 = b is ◦ e a b e {e} {a} {b} a {a} {e} {b} b {b} {b} {e,a} and τ = {∅,{e,a},{e,a,b}}. let a = {a,b} , u = {e,a}. then, a ◦ u = {a,e} , b ◦ u = {b} and b /∈ a◦u , a /∈ b◦u. thus, a is u-disjoint but a∩u 6= ∅. we recall that a family {as}s∈s of subsets of a topological space x is called discrete if every point x ∈ x has a neighborhood that intersects at most one set of the given family [7]. lemma 3.14. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and u,v ∈ νp (e) such that v 4 ⊆ u and v −1 = v . if a subset a of p is u-disjoint, then the family of open sets {a◦v | a ∈ a} is discrete in p . proof. suppose that x ∈ p , we claim that x◦v intersects at most one element of the family {a◦v | a ∈ a}. suppose to the contrary that, there exist distinct elements a,b ∈ a such that x◦v ∩a◦v 6= ∅ and x◦v ∩b◦v 6= ∅. let t ∈ x◦v ∩a◦v , hence there exists v1,v2 ∈ v such that t ∈ x◦v1 , t ∈ a◦v2. thus, x ∈ t◦v−11 , a ∈ t◦v −1 2 that is x −1◦a ⊆ v1◦t−1◦t◦v−12 ⊆ v ◦t −1◦t◦v −1 = v ◦v since v is a complete part. similarly, we can prove that b−1 ◦x ⊆ v ◦v . so, b−1 ◦a ⊆ b−1 ◦x◦x−1 ◦a ⊆ v 4 ⊆ u. then, a ∈ b◦ b−1 ◦a ⊆ b◦u which is a contradiction. � let (p,◦,e,−1 ,τ) be a topological polygroup and n be a normal subpolygroup of p . let π : p → p/n be the natural mapping defined by π(x) = n ◦ x. recall that (p/n,τ) is a topological space, where τ is the quotient topology induced by π, that is τ = {u ⊆ p/n | π−1(u) ⊆ p is open}. lemma 3.15. [9] let p = (p,◦,e,−1 ,τ) be a topological polygroup and n be a normal subpolygroup of p . let π : p → p/n be the natural mapping. then, (1) π−1({n ◦x | x ∈ x}) = n ◦x for every nonempty subset x of p and π−1(π(x)) = n ◦x; (2) {n ◦x | x ∈ x} = {n ◦y | y ∈ n ◦x} for every nonempty subset x of p ; (3) if every open subset of p is a complete part, then the natural mapping π is open. theorem 3.16. [9] let n be a normal subpolygroup of a topological polygroup p = (p,◦,e,−1 ,τ) and every open subset of p is a complete part. then, < p/n,�,n,−i ,τ > is a topological polygroup, where n ◦x�n ◦y = {n ◦z|z ∈ x◦y} and (n ◦x)−i = n ◦x−1. theorem 3.17. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and n is a normal subpolygroup of p . let π : p → p/n be the natural mapping. then, the family {π(u ◦x) | u ∈ νp (e)} is a local base of the space p/n at the point n ◦x ∈ p/n. proof. since u ∈ νp (e) is a complete part, it follows that u ◦ x is an open subset of p. then, for every y ∈ n, y ◦ (u ◦x) is open. so, π−1(π(u ◦x)) = n ◦ (u ◦x) = ⋃ y∈n y ◦ (u ◦x) is an open subset of p . thus, π(u ◦ x) is an open subset of p/n by lemma 3.15. now, suppose that w is an open neighborhood of n ◦x in p/n and o = π−1(w). clearly o ⊆ p is open. since n ◦x ⊆ w , it follows that x ∈ π−1(n ◦ x) ⊆ π−1(w) = o. so, there exists u ∈ νp (e) such that u ◦ x ⊆ o. therefore, n ◦x ∈ π(u ◦x) ⊆ π(o) = w and this completes the proof. � lemma 3.18. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part. let n be a normal subpolygroup of p and π : p → p/n be the natural mapping. if u ,v ∈ νp (e) and v ◦v −1 ⊆ u, then π(v ) ⊆ π(u). proof. suppose that π(x) ∈ π(v ), where x ∈ p . we prove that π(x) ∈ π(u). since v is a complete part, it follows that x◦v ⊆ p is open. by lemma 3.15, since π is open, it follows that π(x◦v ) is open in p/n. hence, π(x◦v ) ∩π(v ) 6= ∅. thus, there exists a y ∈ p such that π(y) ∈ π(x◦v ) ∩π(v ). therefore, there exist t ∈ x◦v and z ∈ v such that π(y) = π(t) = n ◦ t and π(y) = π(z) = n ◦ z. since t ∈ x◦v , there exists b ∈ v such that t ∈ x◦ b. so, x ∈ t◦ b−1 and x ∈ n ◦x ⊆ n ◦ t◦ b−1 = n ◦z◦b−1 ⊆ n ◦v ◦v −1 ⊆ n ◦u. then, π(x) ∈ π(n ◦u) = π(u) and this completes the proof. � the role of complete parts in topological polygroups 59 theorem 3.19. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part , n is a normal subpolygroup of p and w ∈ νp/n (n). then, there exists v ∈ νp/n (n) such that v ⊆ w . proof. suppose that π : p → p/n is the natural mapping. since π is continuous, it follows that u = π−1(w) ⊆ p is open and e ∈ u. then, u ∈ νp (e). thus, by theorem 3.2, there exists v1 ∈ νp (e) such that v1 ◦v −11 ⊆ u. hence, by lemma 3.18, π(v1) ⊆ π(u) = w . since π is open and v1 ∈ νp (e), v = π(v1) ∈ νp/n (n) and this completes the proof. � recall that a continuous mapping f : x → y is perfect if f is a closed mapping and all fibers f−1(y) are compact subsets of x. theorem 3.20. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and let n be a compact normal subpolygroup of p . then, the natural mapping π : p → p/n is perfect. proof. suppose that h ⊆ p is a closed subset. then, by theorem 3.4, n ◦ h is closed in p. since π−1(π(h)) = n ◦h, so π−1(π(h)) ⊆ p is closed. then, π(h) is closed in p/n. hence, π is closed. now, suppose that y ∈ p/n. then, there exists x ∈ p such that π(x) = y. since n is compact, it follows that π−1(y) = π−1(π(x)) = n ◦x is compact. therefore, π−1(y) is compact. � lemma 3.21. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and n is a normal subpolygroup of p . (1) if p is compact, then p/n is compact. (2) if n and p/n are compact, then p is compact. proof. (1) since π is continuous and p is compact, it follows that π(p) = p/n is compact. (2) since n is compact, by theorem 3.20, the natural mapping π : p → p/n is perfect. then, π−1(p/n) = p is compact [[7], theorem 3.7.2]. � definition 3.22. let x be a non-empty subset of a topological polygroup p = (p,◦,e,−1 ,τ). then, x is called cp-resolvable if there exist two disjoint dense subsets of x such that at least one of them is a complete part of p . lemma 3.23. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and let h be a cp-resolvable subpolygroup of p . if x ∈ p , then x◦h is cp-resolvable. proof. suppose that a and b are disjoint dense subspaces of h such that one of them is a complete part of p . then, by lemma 3.9, clx◦h(x ◦ a) = x◦a ∩ x ◦ h = x ◦ a ∩ x ◦ h = x ◦ h since h = clh(a) = a∩h. similarly, clx◦h(x◦b) = x◦h. without lose of generality we can assume that a is a complete part of p . thus, by proposition 2.1, x ◦ a is a complete part and we observe that x◦a∩x◦b = ∅. thus, x◦h is cp-resolvable. � we recall that if n is a normal subpolygroup of p , then the relation x,y ∈ p, xnpy if and only if x−1 ◦y ∩n 6= ∅ is an equivalence relation. the equivalence class of the element x ∈ p is denoted by np (x) [6]. lemma 3.24. [6] if n is a normal subpolygroup of p , then x◦n = np (x). theorem 3.25. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part and let {pα|α ∈ i} be a pairwise disjoint family of non-empty subsets of p such that every pα is cp-resolvable. if p = ⋃ α∈i pα, then p is cp-resolvable. proof. since each pα is cp-resolvable, there exist disjoint dense subsets aα and bα of pα such that aα is a complete part of p and so aα ∩pα ∩ pα = clpαaα = pα. if a = ⋃ α∈i aα and b = ⋃ α∈i bα, then a is a complete part by remark 1 and a and b are disjoint dense subsets of p . thus, p is cp-resolvable. � theorem 3.26. let p = (p,◦,e,−1 ,τ) be a topological polygroup such that every open subset of p is a complete part. then, the following conditions hold. 60 shadkami, zand and davvaz (1) if h is a proper dense subpolygroup of p and h is a complete part, then p is cp-resolvable. (2) if a normal subpolygruop n of p is cp-resolvable, then p is cp-resolvable. (3) if p contains a non-closed normal subpolygroup n which is a complete part, then p is cpresolvable. proof. (1) suppose that a = h and b = p \ h. since the subpolygroup a is not closed, then by theorem 3.7, a is not open. thus, int(a) = ∅. therefore, a = b = p . (2) by lemma 3.24, p is equal to the disjoint unions ( ⋃ x/∈n x◦n) ∪n. thus, p is cp-resolvable by lemma 3.23 and theorem 3.25. (3) by theorem 3.8, b = n is a normal subpolygroup of p , and so by (1); b is cp-resolvable. therefore, by (2), p is cp-resolvable. � references [1] h. aghabozorgi, b. davvaz, m. jafarpour, solvable polygroups and derived subpolygroups, comm. algebra 41 (2013), 3098–3107. [2] r. ameri, topological (transposition) hypergroups, ital. j. pure appl. math. 13 (2003), 171–176. [3] s.d. comer, polygroups derived from cogroups, j. algebra 89 (1984), 397–405. [4] s.d. comer, extension of polygroups by polygroups and their representations using colour schemes, lecture notes in meth. 1004, universal algebra and lattice theory, 1982, 91–103. [5] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. [6] b. davvaz, polygroup theory and related systems, world scientific publishing co. pte. ltd., hackensack, nj, 2013. [7] r. engelking, general topology. pwn-polish scientific publishers, warsaw, 1997. [8] d. heidari, b. davvaz, s.m.s. modarres, topological hypergroups in the sense of marty, comm. algebra 42 (2014), 4712–4721. [9] d. heidari, b. davvaz, s.m.s. modarres, topological polygroups, bull. malays. math. sci. soc., 39 (2016), 707–721. [10] s. hoskova-mayerova, topological hypergroupoids, comput. math. appl. 64 (2012), 2845–2849. [11] m. jafarpour, h. aghabozorgi, b. davvaz, on nilpotent and solvable polygroups, bull. iranian math. soc. 39 (2013), 487–499. [12] m. koskas, groupoides, demi-hypergroupes et hypergroupes, j. math. pures appl. 49(1970), 155–192. [13] f. marty, sur une généralization de la notion de groupe, 8iem, congress math. scandinaves, stockholm, 1934, 45–49. [14] m. salehi shadkami, m. r. ahmadi zand and b. davvaz, left big subsets of topological polygroups, filomat, in press. [15] y. sureau, contribution à la théorie des hypergroupes et hypergroupes opérant transitivement sur un ensemble, doctoral thesis, 1980. mathematics department, yazd university, yazd, iran ∗corresponding author: davvaz@yazd.ac.ir international journal of analysis and applications volume 18, number 5 (2020), 799-818 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-799 a new generalized exponential distribution: properties and applications jawad hussain ashraf1,∗, munir ahmad1, a. khalique1, zafar iqbal2 1school of social sciences, national college of business administration & economics lahore, pakistan 2department of statistics, government post graduate college gujranwala, pakistan ∗corresponding author: miyanjawad@gmail.com abstract. the exponential distribution is a popular statistical distribution to study the problems in lifetime and reliability theory. we proposed a new generalized exponential distribution, wherein exponentiated exponential and exponentiated generalized exponential distributions are sub-models of the proposed distribution. we study several important statistical and mathematical properties of the newly developed model and provide the simple expressions for the generating function, moments and mean deviations. parameters of the proposed distribution are estimated by the technique of maximum likelihood. for two real data sets from the field of biology and engineering, the proposed distribution is compared to some existing distributions. it is found that the proposed model is more suitable and useful to study lifetime data. thus, it gives us another alternative model for existing models. 1. introduction some authors discussed the gompertz-verhulst distribution whose distribution is defined by g(x) = (1 − ξe−ϑx)α, x > 1 ϑ ln ξ. (1.1) for ξ = 1, the exponentiated exponential (ee) or generalized exponential (ge) distribution by [5] is submodel of eq.(1.1). they developed the exponentiated-g family for any parent distribution by g(x) = h(x)δ, received may 26th, 2020; accepted june 23rd, 2020; published july 17th, 2020. 2010 mathematics subject classification. 60e05, 62n05, 62f10, 62p10, 62p25, 62p30. key words and phrases. exponential distribution; exponentiated type distributions; generalized exponential distribution; lifetime distributions; moments; maximum likelihood estimation. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 799 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-799 int. j. anal. appl. 18 (5) (2020) 800 where h(x) is an arbitrary cumulative distribution function (cdf) and δ is a positive real number. it is also called lehmann type i alternative. various authors developed different exponentiated distributions for various standard distributions and studied their properties. [25] provided a list of thirty exponentiated type distributions. the ee distribution is comprehensively studied in different papers by [6], [21], [7,8], and [22,23] is defined for (x > 0) by g(x) = (1 − e−ϑx)α. (1.2) for α = 1, the special case of eq.(1.2) is the exponential distribution. the ee distribution is effectively used for studying the skewed data. [9, 10] and [13] presented important works on comparison of the ee distribution with some will-known close distributions like the gamma, log-normal and weibull distributions. [11] further used the information to compare the ee and weibull distributions because both of them provide very close fits. [12] revisted some existing properties of the ee model. the beta exponential (be) distribution is developed by using the beta-g family by [18], where the ee distribution is a particular case of the be distribution. [1] generalized the ee distribution and developed the beta generalized exponential (bge) distribution, wherein the be and ee distributions are particular cases of the bge distribution. [17] reviewed the several important characteristics of the ee distribution and derived simple explicit expressions for some properties. [24] generalized the exponential distribution by the marshall-olkin family. [19] generalized the gamma and weibull distributions in similar way the ee distribution [16]. however, they used another type of exponentiated distribution g(x) = 1 − [1 −h(x)]β to generalize the gumbel and fréchet distributions. it is also called lehmann type ii alternative. [3] introduced the exponentiated generalized (eg) family as g(x) = [1 −{1 −h(x)}γ]δ, where γ > 0 and δ > 0 are additional shape parameters. this class is an extension of both exponentiated type distributions. in recent past, several authors used the eg family for generalizing the distributions. the rest of the paper is organized as follows: section 2 introduce the new generalized exponential distribution and discuss some important aspects of the proposed distribution. section 3 extensively studies the basic properties of the proposed distribution. section 4 deals with linear presentation of the distribution. in section 5, we derive the mathematical properties by employing the linear representation of the proposed distribution given in the previous section. section 6 studies the maximum likelihood estimation and provides the application of the proposed model to real data sets. finally, concluding remarks are presented in section 7. int. j. anal. appl. 18 (5) (2020) 801 2. new generalized exponential distribution by using the exponentiated-g and eg families, we proposed a three-parameter new generalized (ng) class of distributions as f(x) = h(x)α[1 − h̄(x)β]γ, (2.1) further, by using the exponential distribution with unity scale parameter (ϑ = 1) in eq.(2.1), we developed new generalized exponential (nge) distribution as f(x) = (1 − e−x)α(1 − e−βx)γ. (2.2) hereafter, we assume ψ(x) = (1 − e−x) and ψ(x; β) = (1 − e−βx) for simplicity. then, the corresponding probability density function (pdf) of the nge distribution is f(x) = αe−xψ(x)α−1ψ(x; β)γ + γβe−βxψ(x)αψ(x; β)γ−1. (2.3) henceforth, if x is a random variable that follows the pdf (2.3), it can be defined by x ∼ nge(α,β,γ). for simplicity, we can consider f(x) = f(x; α,β,γ). the nge distribution shares a interesting physical interpretation. if we consider α and β are positive integers and γ = 1, the cdf of the nge distribution is product of the distributions of maximum and minimum of a sample of size α and β, respectively, from the exponential distribution. in other words, when the components are identical and independently distributed as exponential distribution g(x), for γ = 1, the nge model is a product of weibull competing risk model [1 − ∏n i=1(1 − gi(x))] and multiplicative weibull model ∏n i=1 gi(x). these models are among the various extensions of the weibull distribution discussed by [14]. the nge distribution with three shape parameters is a flexible model which has some popular sub-models as particular cases are given in table 1. α β γ sub-model 1 1 0 exponential distribution α 1 0 exponentiated exponential distribution 0 β γ exponentiated generalized exponential distribution table 1. sub-model of the nge distribution int. j. anal. appl. 18 (5) (2020) 802 3. some basic properties 3.1. the mode. the silent features of a distribution can be investigated by the first two derivatives of the pdf. therefore, we differentiate (2.3) with respect to x and equate to zero, namely f ′ (x) = αe−xψ(x)α−2ψ(x; β)γ(αe−x − 1) + γβ2e−βxψ(x)αψ(x; β)γ−2(γe−βx − 1) + 2αβγe−(β+1)xψ(x)α−1ψ(x; β)γ−1 = 0. therefore, the critical values of f(x) can be obtained from the equation α(αe−x − 1) + 2αβγe−βxψ(x) ψ(x; β) = β2γe−(β−1)x(1 −γe−βx). (3.1) if the root of eq.(3.1) is x = x0, then the local maximum, local minimum and inflection points obtained when f ′′ (x0) < 0,f ′′ (x0) > 0 and f ′′ (x0) = 0 respectively. for γ = 0, the mode of x reduces to log α. for α = 0, the mode of the density of the nge model reduces to log γ β . for β = 1 the mode becomes log(θ), where θ = α + γ. for all these three special cases, the nge distribution becomes the ee distribution. for α = γ = 1, the mode of x becomes x = log [ (β + 1)2 1 + β2e−(β−1)x ] 1 β for β = 1, the last equation reduces to x = log(2). however, f ′ (∞) = 0 and when x → 0 the limit of f ′ (x) is lim x→0 f ′ (x) =   0 if α + γ > 2 2βγ if α + γ = 2 ∞ if α + γ < 2 . 3.2. density shapes. i. for β = 1 (a) α + γ ≤ 1 (b) γ = 0,α ≤ 1, (c) α = 0,γ ≤ 1 the nge density is strictly decreasing. ii. for β = 1 (a) α + γ > 1 (b) γ = 0,α > 1 (c) α = 0,γ > 1 the nge density is uni-modal skewed. iii. for α = β = γ = 1, the nge density is positively skewed with mode at x = 0.6931. iv. for α = β = 1,γ = 0, the nge density is reversed type j-shaped (exponential). v. for α = 0,β = γ = 1, the nge density is reversed type j-shaped (exponential). vi. for β = 1, the shape of the nge density remains similar by interchanging the values of α and γ. vii. for α + γ > 1 and β > 0, the nge density is unimodel and as the value of β increases the density becomes more skewed with longer tails. int. j. anal. appl. 18 (5) (2020) 803 viii. for α + γ ≤ 1 and β > 0, the nge density is reversed j-shaped with longer tails and when β →∞ the tails become shorter. ix. for α = 1,γ = 1 and β > 0, the nge density is unimodal and skewed and when β →∞ the skewness increases. x. for α > 1,γ > 1 and β > 0, the nge density is unimodal and positively skewed, and for β > 2, the mode of the nge density does not significantly change. xi. for β < 1,γ < 1 and 0 < α ≤ 1(α > 1), the nge density is reversed j-shaped with shorter height (unimodal positively skewed with increasing mode). xii. for β > 1,γ > 1 and 0 < α ≤ 1(α > 1), the nge density is unimodal positive skewed with sharp peak (unimodal positive skewed with shorter height), and when α → ∞, the mode of the density increases with heavy tails. xiii. for α < 1,β < 1 and 0 < γ ≤ 1(γ > 1), the nge density is reversed j-shaped (unimodal positive skewed with increasing mode). xiv. for α > 1,β > 1 and 0 < γ < ∞, an insignificant change is noted in the nge density. we displayed explanatory graphs of the densities of x for different specific parameter values in figure 1. the density of the x takes various forms for different specific parameter values of the nge distribution. it can be noted that the nge density is more flexible than its competitors. so, the proposed distribution can effectively be used for modeling the positive data. 3.3. hazard rate function (hrf ). an interesting feature of the proposed model that it can have the monotonically increasing, decreasing, bathtub and uni-modal (reversed bathtub) hrf for different parameter values. the hrf of the nge distribution is h(x) = αe−xψ(x)α−1ψ(x; β)γ + γβe−βxψ(x)αψ(x; β)γ−1 1 − [ψ(x)αψ(x; β)γ] . (3.2) for γ = 0 or β = 1, the hrf of x reduces to the hrf of the ee distribution and for α = 1 and γ = 0, the hrf becomes constant. in fig.2, we provide the visual representation of the hrfs of the x which illustrate important characteristics of the proposed model. it can be seen that the hrf is monotonically increasing, decreasing, uni-model, decreasing-increasing-decreasing, increasing-decreasing-increasing and constant. so, the proposed distribution is quite flexible to model the data which have different hrfs. 3.4. shapes of the hrf. i. for β = 1 (a) α + γ ≤ 1 (b) γ = 0,α ≤ 1 (c) α = 0,γ ≤ 1, the hrf of x is decreasing from ∞ to ϑ. ii. for β = 1 (a) α + γ > 1 (b) γ = 0,α > 1, (c) α = 0,γ > 1, the hrf of x is increasing from 0 to ϑ. iii. for β = 1 (a) α + γ = 1 (b) γ = 0,α = 1 (c) α = γ = 1, the hrf of x is constant ϑ = 1. int. j. anal. appl. 18 (5) (2020) 804 0 2 4 6 8 10 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 (a) x f( x ) α=7. 5, β=0. 3, γ=7. 5 α=7. 5, β=0. 5, γ=7. 5 α=1. 0, β=1. 0, γ=7. 5 α=7. 5, β=1. 5, γ=7. 5 α=3. 5, β=1. 0, γ=7. 5 α=7. 5, β=5. 5, γ=7. 5 0 2 4 6 8 10 0 .0 0 .4 0 .8 1 .2 (b) x f( x ) α=0. 5, β=0. 5, γ=0. 5 α=0. 5, β=1. 5, γ=0. 5 α=0. 5, β=0. 5, γ=0. 6 α=0. 5, β=1. 5, γ=0. 6 α=1. 5, β=0. 5, γ=1. 5 α=1. 5, β=1. 5, γ=1. 5 0 2 4 6 8 10 0 .0 0 .2 0 .4 0 .6 (c) x f( x ) α=0. 50, β=1. 0, γ=1. 0 α=1. 00, β=1. 0, γ=1. 0 α=5. 00, β=1. 0, γ=1. 0 α=10. 0, β=1. 0, γ=1. 0 α=30. 0, β=1. 0, γ=1. 0 α=50. 0, β=1. 0, γ=1. 0 0 2 4 6 8 10 0 .0 0 0 .0 2 0 .0 4 0 .0 6 (d) x f( x ) α=0. 05, β=0. 05, γ=1. 0 α=0. 05, β=0. 05, γ=1. 5 α=0. 50, β=0. 05, γ=1. 0 α=3. 50, β=0. 05, γ=1. 0 α=40. 0, β=0. 05, γ=2. 0 α=5. 50, β=0. 05, γ=5. 5 figure 1. plots of the density (2.3) for some specific parameter values. iv. for α + γ > 1 and 0 < β < 1 and 1 < β < ∞, the shape of the hrf of x is reversed bathtub. v. for α + γ ≤ 1 and 0 < β < 1 and 1 < β < ∞, the hrf of x is bathtub type. vi. for α → 0,β > 1.5,γ < 1, the hrf of x is decreasing-increasing-decreasing towards ϑ = 1. vii. for α → 0,β > 1.5,γ > 1, the hrf of x is uni-model. 3.5. asymptotics of the density and hrf. the asymptotics of (2.3) and (3.2) are (α + γ)βγxα+γ−1 as x → 0, further f(x) ∼ αe−x + βγe−βx and h(x) ∼ αe −x+βγe−βx α(α−2)e−x+2αβγe−(β+1)x+γβ2e−βx as x → ∞. it is noted that the tail behaviors of both equations are polynomial and also exponential. theorem 3.1. let x ∼ nge(α,β,γ). then the asymptotic behaviors of (2.3) and (3.2) are: lim x→0 f(x) =   0 if (α + γ) > 1 βγ if (α + γ) = 1 ∞ if (α + γ) < 1 , int. j. anal. appl. 18 (5) (2020) 805 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 (a) x h(x ) λ=1. 0, α=5. 5, β=3. 5, γ=5. 5 λ=1. 0, α=0. 7, β=1. 0, γ=0. 5 λ=1. 0, α=0. 7, β=2. 0, γ=0. 5 λ=1. 0, α=0. 4, β=1. 0, γ=0. 5 λ=1. 0, α=0. 4, β=2. 0, γ=0. 5 0 2 4 6 8 10 0 1 2 3 4 (b) x h(x ) λ=1. 0, α=0. 0005, β=2. 5, γ=1. 9 λ=1. 0, α=0. 0100, β=2. 5, γ=5. 0 λ=1. 5, α=0. 0100, β=2. 5, γ=0. 5 λ=1. 0, α=0. 1000, β=5. 5, γ=0. 8 λ=1. 0, α=0. 0005, β=2. 5, γ=0. 8 0 2 4 6 8 10 0.0 0 0.0 1 0.0 2 0.0 3 0.0 4 0.0 5 0.0 6 (c) x h(x ) λ=1. 0, α=0. 5, β=0. 05, γ=1. 0 λ=1. 0, α=0. 5, β=0. 05, γ=1. 5 λ=1. 0, α=0. 5, β=0. 05, γ=5. 5 λ=1. 0, α=5. 0, β=0. 10, γ=5. 5 λ=1. 0, α=3. 0, β=0. 10, γ=3. 0 figure 2. plots of the hrf of x for different selected values of the shape parameters when ϑ = 1. and lim x→∞ f(x) = lim x→∞ h(x) = 0. proof. (1 − e−x) = x when x → 0, and e−x → 0 as x → ∞. the limits of f(x) can then be obtained. similarly, when x → 0, the limiting behavior of h(x) is the same as the limit of f(x). as x → ∞ both numerator and denominator become 0 of the hrf of x. thus, the hrf is indeterminate. we evaluate the limit by using l’hôpital’s rule. so, it completes the proof. � 3.6. reversed hazard rate function (rhrf ). the rhrf of the x is r(x) = αe−x ψ(x) + γβe−βx ψ(x; β) . (3.3) equation (3.3) shows that the rhrf of x is a sum of the proportional rhrf of the ee and ege distributions. for γ = 0, eq.(3.3) becomes r(x) = αe−x ψ(x) . therefore, rhrf of x is proportionally equals to the rhrf of the exponential distribution. 4. linear representation we use the power series for any non-integer real θ as (1 −z)θ = ∞∑ k=0 (−1)kγ(θ) γ(θ −k)k! zi, ∀ |z| < 1. (4.1) int. j. anal. appl. 18 (5) (2020) 806 by using (4.1) in (2.2), we obtain f(x) = ∞∑ j=0 τjψ(x) α+j, (4.2) where the coefficients τj = τj(β,γ) are τj = (−1)jγ(γ + 1) j! ∞∑ k=0 (−1)kγ(kβ + 1) γ(kβ − j + 1)γ(γ −k + 1)k! we can rewrite (4.2) as f(x) = ∞∑ j=0 τjwα+j(x), (4.3) where wα+j(x) = ψ(x) α+j is the cdf of the ee distribution with power parameter is α + j. by differentiating (4.3), a simple linear representation of (2.3) is f(x) = ∞∑ j=0 τjwα+j(x), (4.4) where wα+j(x) = (α + j)ψ(x) α+j−1e−x is the pdf of the ee distribution with power parameter is α + j. equation(4.4) informs that the pdf of the nge distribution is linear mixture of well-known ee densities. hence, various important statistical properties of the nge distribution can be developed from simple properties of the ee distribution. 5. mathematical properties 5.1. generation function. we can use the moment generating function (mgf) to characterize the distributions. it is also used for generating the moments of a distribution. theorem 5.1. the mgf of x is defined by m(t) = ∞∑ j=0 τj γ(α + j + 1)γ(1 − t) γ(α− t + j + 1) . (5.1) proof. by using (4.4), we have m(t) = ∞∑ j=0 τj(α + j) ∫ ∞ 0 e(t−1)xψ(x)α+j−1d(x). setting e−x = u, m(t) can be written as m(t) = ∞∑ j=0 τj(α + j) ∫ 1 0 u−t(1 −u)α+j−1dx, which after simplification leads to (5.1). int. j. anal. appl. 18 (5) (2020) 807 m(t) = ∞∑ j=0 τj γ(α + j + 1)γ(1 − t) γ(α− t + j + 1) � if γ = 0 and ϑ = 1, the mgf of the nge distribution reduces to the mgf of the ee distribution given by [7]. 5.2. moments. moments play important role to know about the different aspects of a probability distribution such as the central tendency and variation of a distribution. furthermore, the moments can also be employed to examine the skewness and kurtosis of a probability distribution. we can write from (4.4), setting e−x = u, and after some algebra µr = (−1)r ∞∑ j=0 (α + j)τj ∂r ∂pr b(α + j,p + 1) ∣∣∣∣ p=0 . (5.2) alternatively, we obtain another expression µr = r! ∞∑ j=0 (α + j)τj ∞∑ p=0 (−1)p ( α+j−1 p ) (p + 1)r+1 . a comparison is also made of the mean-variance for different parameter values in figure 3. the numerical values of the first two moments, standard deviation, median, mode, skewness and kurtosis for various specific parameter values of x are given in table 2. for some different specific parameter values of the x, the skewness and kurtosis measures are illustrated in figure 4. it examined that these measures depend only on the shape parameters. int. j. anal. appl. 18 (5) (2020) 808 0 2 4 6 8 10 0 1 2 3 4 5 (a) β m e a n − v a ri a n c e α, γ=1. 5 α, γ=1. 5 0 2 4 6 8 10 0 1 2 3 4 5 (b) β m e a n − v a ri a n c e α, γ=0. 4 α, γ=0. 4 0 2 4 6 8 10 0 1 2 3 4 5 (c) α m e a n − v a ri a n c e β, γ=1. 5 β, γ=1. 5 0 2 4 6 8 10 0 1 2 3 4 5 (d) α m e a n − v a ri a n c e β, γ=0. 4 β, γ=0. 4 0 2 4 6 8 10 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 (e) γ m e a n − v a ri a n c e α, β=1. 5 α, β=1. 5 0 2 4 6 8 10 0 1 2 3 4 5 6 (f) γ m e a n − v a ri a n c e α, β=0. 4 α, β=0. 4 figure 3. plots for the mean and variation of x, where black line represent the mean of x and doted lines represent the variance of x when ϑ = 1. int. j. anal. appl. 18 (5) (2020) 809 (a) β=4.5 1 2 3 4 1 2 3 4 5 10 15 α γ sk1 (b) β=0.5 1 2 3 4 1 2 3 4 2 4 6 8 10 12 14 α γ sk2 (c) β=0.01 1 2 3 4 1 2 3 4 5 10 15 α γ sk3 (d) β=4.5 1 2 3 4 1 2 3 4 100 200 300 α γ kur1 (e) β=0.5 1 2 3 4 1 2 3 4 50 100 150 200 250 α γ kur2 (f) β=0.01 1 2 3 4 1 2 3 4 100 200 300 400 α γ kur3 figure 4. plots of skewness (a,b,c) and kurtosis (d,e,f) of the x when ϑ = 1. 5.3. quantile function. if x ∼ nge(α,β,γ), the quantile function of x is determined numerically from [ ψ(x) α ψ(x; β) γ ] = u. first, second and third quartile of the distribution can be calculated by setting u = 0.25, 0.50 and 0.75, respectively. 5.4. mean deviations. the mean deviations can be used to compute the variation of x . then, the mean deviations about the average value (mean) and median can be determined by φ1(x) = 2µ1f(µ) − 2m(µ1) and φ2(x) = µ1 − 2m(m), (5.3) respectively, here µ1 is the mean of x calculated from (5.2) and m is its median. the first incomplete moment is required for measuring (5.3). it is defined by m(z) = ∫ z −∞wf(w)dw. now by using (4.4) and (4.1) m(z) reduces to m(z) = ∞∑ j=0 (α + j)τj ∞∑ p=0 (−1)p ( α + j − 1 p )∫ z 0 xe−(p+1)xdx, int. j. anal. appl. 18 (5) (2020) 810 table 2. the values of first two moments, standard deviation, median, mode, skewness and kurtosis of x. (α,β,γ) e(x) e(x2) sd median mode sk kur (1,0.3,1) 3.6 23 3.2 2.59 1.14 2.1 9.7 (1,0.5,1) 2.3 9.1 1.9 1.82 0.95 2.0 9.3 (1,1,1) 1.5 3.5 1.1 1.23 0.69 1.6 7.1 (1,1.5,1) 1.3 2.6 0.98 1.02 0.6 1.8 8.2 (1,2,1) 1.2 2.3 0.96 0.91 0.47 2.0 9.3 (1,5,1) 1.0 2.0 0.98 0.72 0.26 2.1 9.5 (0.5,0.5,0.5) 1.5 5.1 1.7 1.00 1.6 × 10−10 2.4 12 (1,0.5,0.5) 1.8 5.8 1.6 1.29 0.49 2.3 12 (2,0.5,0.5) 2.1 7.0 1.6 1.68 1.03 2.2 11 (5,0.5,0.5) 2.7 9.6 1.5 2.34 1.82 2.0 10 (2,2,2) 1.6 3.8 1.1 1.40 0.98 1.7 7.7 (3,2,2) 1.9 5.0 1.1 1.68 1.24 1.5 6.9 (5,2,2) 2.3 6.8 1.2 2.09 1.66 1.4 6.3 (1.5,1.5,0.5) 1.4 3.0 1.1 1.23 0.62 1.7 7.6 (1.5,1.5,1.5) 1.6 3.5 1.0 1.32 0.89 1.6 7.5 (1.5,1.5,5.5) 2.0 4.8 0.98 1.76 1.42 1.5 7.2 (0.5,1.5,0.4) 0.8 1.4 0.86 0.52 2.3 × 10−9 2.3 11 (0.5,1.5,1.5) 1.1 2.1 0.88 0.92 0.51 1.8 8.7 (0.5,1.5,5.5) 1.7 3.7 0.89 1.54 1.24 1.5 7.0 (1.5,0.5,0.5) 1.9 4.3 1.4 1.21 0.57 2.3 13 (1.5,0.5,1.5) 2.9 12 2.0 2.37 1.53 1.8 8.2 (1.5,0.5,5.5) 4.8 29 2.4 4.32 3.46 1.4 6.1 (0.5,0.5,0.5) 1.5 5.1 1.7 1.00 1.6 × 10−10 2.4 12 (0.5,0.5,1) 2.2 8.6 2.0 1.63 0.62 2.0 9.1 (0.5,0.5,1.5) 2.7 12 2.1 2.14 1.14 1.8 7.9 by setting (p + 1)x = u and using γ(a,x) = ∫∞ 0 za−1e−zdz, we obtain m(z) = ∞∑ j=0 (α + j)τj ∞∑ p=0 (−1)p ( α+j−1 p ) γ(2, (p + 1)z) (p + 1)2 . (5.4) int. j. anal. appl. 18 (5) (2020) 811 the values of mean deviations of the nge distribution about the mean and median can be obtained by using eq.(5.4). we can also use (5.4) to construct the bonferroni and lorenz curves which have much importance in economics, renewal theory and reliability theory. for a given probability p, we can define these curves by b(p) = m(q)/(pµ1) and l(p) = m(q)/µ1, where q = q(p) is the quantile function of x at p and µ1 = e(x). 6. estimation and inference there are different methods of point estimation but the maximum likelihood is the famous technique to calculate the point estimates of the unknown quantities of a distribution. the maximum likelihood estimates (mles) have some nice and useful characteristics. moreover, we can use the mles to calculate the confidence intervals for the unknown parameters of a distribution. thus, we choose the technique of maximum likelihood to estimate the unknown quantities of the x from the given sample. let x1,x2, . . . ,xn be observed sample from (2.3) and ω = (α, β, γ)t be a vector of parameters of x. then, we will define the log-likelihood function by l(ω) = (α− 1) n∑ i=1 log(ψ(xi)) + γ n∑ i=1 log(ψ(xi; β)) − n∑ i=1 xi + n∑ i=1 log { αψ(xi; β) + βγe −(β−1)xiψ(xi) } − n∑ i=1 log(ψ(xi; β)). the elements of the score are ∂l(ω) ∂α = n∑ i=1 log(ψ(xi)) + n∑ i=1 ψ(xi; β) ν(xi) , ∂l(ω) ∂β = (γ − 1) n∑ i=1 xie −βxi ψ(xi; β) + n∑ i=1 αxie −βxi + γe−(β−1)xiψ(xi)ψ(xi; β) ν(xi) , ∂l(ω) ∂γ = n∑ i=1 log(ψ(xi; β)) + β n∑ i=1 e−(β−1)xiψ(xi) ν(xi) , where ν(xi) = αψ(xi; β) + βγe −(β−1)xiψ(xi). the mles ω̂ of ω can be calculated by solving the above equations simultaneously after equate these equations to zero. there is no analytical solution for these equations. therefore, we will solve the score equations numerically by using some nonlinear optimization algorithm in open source software environment for statistical programming r. to determine the test of hypothesis and find the interval estimate of the unknown parameters of x, we require the 3 × 3 observed information matrix int. j. anal. appl. 18 (5) (2020) 812 j(ω) =   jα,α jα,β jα,γ jα,β jβ,β jβ,γ jα,γ jβ,γ jγ,γ   under the relevant regularity conditions, the asymptotic distribution of √ n(ω̂ − ω) follows multivariate n3(0,j(ω̂) −1) which can be employed to establish the confidence intervals for the unknown parameters of the x. here, we consider j(ω̂) as the observed information matrix which is evaluated at ω = ω̂. 6.1. application. in the following important section, we apply the proposed model to real data sets to highlight the capability of the nge distribution. here, we compare the ege, ee and be distributions with the nge distribution. the study is conducted in statistical computing r developed and maintained by [20]. first, we provide the description of two real data sets. then, for each model, we calculate the mles of the unknown paramaters and their crossponding standard errors by using the r package adequacymodel [4]. the proposed distribution is compared with its competing models by using the well-known statistics in literature like the cramér-von mises (w*), anderson-darling (a*), bayesian information criterion (bic), akaike information criterion (aic) and kolmogrov-smirnov (k-s). generally, the fit is better if the values of these statistics are small. the first real data is repair times of 46 failures in (hours) of an airborne communications receiver is analysed by [2] is: 0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.8, 0.8, 1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 2.7, 3.0, 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5. the second data set have 128 observations which represent the remission times (month) of patients of bladder cancer and studied by [15] is: 0.08, 0.20, 0.40, 0.50, 0.51, 0.81, 0.90, 1.05, 1.19, 1.26, 1.35, 1.40, 1.46, 1.76, 2.02, 2.02, 2.07, 2.09, 2.23, 2.26, 2.46, 2.54, 2.62, 2.64, 2.69, 2.69, 2.75, 2.83, 2.87, 3.02, 3.25, 3.31, 3.36, 3.36, 3.48, 3.52, 3.57, 3.64, 3.70, 3.82, 3.88, 4.18, 4.23, 4.26, 4.33, 4.34, 4.40, 4.50, 4.51, 4.87, 4.98, 5.06, 5.09, 5.17, 5.32, 5.32, 5.34, 5.41, 5.41, 5.49, 5.62, 5.71, 5.85, 6.25, 6.54, 6.76, 6.93, 6.94, 6.97, 7.09, 7.26, 7.28, 7.32, 7.39, 7.59, 7.62, 7.63, 7.66, 7.87, 7.93, 8.26, 8.37, 8.53, 8.65, 8.66, 9.02, 9.22, 9.47, 9.74, 10.06, 10.34, 10.66, 10.75, 11.25, 11.64, 11.79, 11.98, 12.02, 12.03, 12.07, 12.63, 13.11, 13.29, 13.80, 14.24, 14.76, 14.77, 14.83, 15.96, 16.62, 17.12, 17.14, 17.36, 18.10, 19.13, 20.28, 21.73, 22.69, 23.63, 25.74, 25.82, 26.31, 32.15, 34.26, 36.66, 43.01, 46.12, 79.05. int. j. anal. appl. 18 (5) (2020) 813 table 3. descriptive statistics for the data sets. statistic first data set second data set n 46 128 median 1.75 6.40 mode 2.50 5.0 mean 3.61 9.37 variance 24.45 110.43 skewness 2.88 3.29 kurtosis 8.80 15.48 table 4. mles for the parameters of the nge, ege, be and ee distributions with their corresponding standard errors given in (parentheses) and statistics a* and w* for repair times data. model ϑ α β γ a* w* nge 0.1830 0.5212 18.0765 4.7498 02211 0.0241 (0.05325) (0.1643) (7.8974) (3.1616) ege 0.5189 0.5190 0.9583 1.0004 0.1442 (9.8776) (9.8776) (0.1897) be 0.0688 0.9344 3.7971 0.9953 0.1435 (0.3029) (0.1726) (16.5490) ee 0.2694 0.9583 1.0004 0.1442 (0.0544) (0.1897) table 5. other statistics to compare the models for the repair times data. model −ll aic bic k-s p-value (k-s) nge 98.8585 205.7169 213.0315 0.0635 0.9925 ege 104.9829 215.9658 221.4517 0.1520 0.2385 be 104.9368 215.8735 221.3594 0.1460 0.2804 ee 104.9829 213.9658 217.6231 0.1520 0.2385 int. j. anal. appl. 18 (5) (2020) 814 table 6. mles for the parameters of the nge, ege, be and ee distributions with their corresponding standard errors given in (parentheses) and statistics a* and w* for remission times data. model ϑ α β γ a* w* nge 0.1969 1.3169 0.2851 0.2003 0.1437 0.0210 (0.0582) (0.2761) (0.1148) (0.2310) ege 0.1370 0.8846 1.2179 0.6741 0.1122 (0.7036) (4.5442) (0.1488) be 0.6435 1.4480 0.1798 3.3864 0.5264 (0.6158) (0.3292) (0.1791) ee 0.1213 1.2219 0.6733 0.1120 (0.0139) (0.1494) table 7. other statistics to compare the models for the remission times data. model −ll aic bic k-s p-value (k-s) nge 409.7355 827.4711 838.8792 0.0381 0.9925 ege 413.0776 832.1552 840.7113 0.0725 0.5113 be 412.3440 830.6880 839.2441 0.0664 0.6244 ee 413.0779 830.1560 835.8600 0.0728 0.5053 int. j. anal. appl. 18 (5) (2020) 815 (a) data (repair time) d e n s it y 0 5 10 15 20 25 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 nge ege be ge 0 5 10 15 20 25 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 (b) data (repair time) c d f nge ege be ge (c) data (remission time) d e n s it y 0 20 40 60 80 0 .0 0 0 .0 4 0 .0 8 nge ege be ge 0 20 40 60 80 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 (d) data (remission time) c d f nge ege be ge figure 5. for repair times data: (a) plots of estimated density functions (b) estimated cdfs of the nge and their competing models ege, be and ee, for remission times data: (c) plots of estimated densities (d) estimated cdfs of the nge, ege, be and ee distributions. int. j. anal. appl. 18 (5) (2020) 816 0.0 0.2 0.4 0.6 0.8 1.0 0 .2 0 .4 0 .6 0 .8 1 .0 ge fitted ge distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .2 0 .4 0 .6 0 .8 1 .0 ege fitted ege distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 nge fitted nge distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .2 0 .4 0 .6 0 .8 1 .0 be fitted be distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n figure 6. fitted and empirical distribution functions for repair times data. 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 ge fitted ge distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 ege fitted ege distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 nge fitted nge distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n 0.0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 be fitted be distribution function e m p ir ic a l d is tr ib u ti o n f u n c ti o n figure 7. fitted and empirical distribution functions for remission times data. int. j. anal. appl. 18 (5) (2020) 817 7. final remarks we develop a new continuous probability distribution, named new generalized exponential (nge) distribution. we study its properties theoretically and numerically. we provide the linear presentation of the density function which is quite useful to drive the simple expressions of several statistical properties of the proposed distribution. moreover, the simple expressions are derived for some properties of the x. the parameters of the x are estimated by the technique of maximum likelihood. to compare the proposed model with other models, we apply these models to two sets of real data from different fields of science such as biology and engineering and fit is examined by using well-known statistics. we conclude that the nge distribution fit better than the ege, be and ee distributions. we are hopeful that the nge distribution is another very useful distribution to study the problems in several fields such as economics, biology, engineering and reliabiliy theory. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] w. barreto-souza, a.h.s. santos, g.m. cordeiro, the beta generalized exponential distribution, j. stat. comput. simul. 80 (2010), 159–172. [2] r.s. chhikara, j.l. folks, the inverse gaussian distribution as a lifetime model, technometrics. 19 (1977), 461–468. [3] g.m. cordeiro, e.m. ortega, d.c. da cunha, the exponentiated generalized class of distributions, j. data sci. 11 (2013), 1–27. [4] p. r. diniz marinho, m. bourguignon, and c. r. barros dias, adequacymodel: adequacy of probabilistic models and general purpose optimization, r package version 2.0.0. 2016. https://cran.r-project.org/package=adequacymodel [5] r.c. gupta, p.l. gupta, r.d. gupta, modeling failure time data by lehman alternatives, commun. stat. theory methods. 27 (1998), 887–904. [6] r.d. gupta, d. kundu, theory & methods: generalized exponential distributions, aust. n.z. j. stat. 41 (1999), 173–188. [7] r.d. gupta, d. kundu, exponentiated exponential family: an alternative to gamma and weibull distributions, biom. j. 43 (2001), 117–130. [8] r.d. gupta, debasis. kundu, generalized exponential distribution: different method of estimations, j. stat. comput. simul. 69 (2001), 315–337. [9] r.d. gupta, d. kundu, closeness of gamma and generalized exponential distribution, commun. stat. theory methods. 32 (2003), 705–721. [10] r.d. gupta, d. kundu, discriminating between weibull and generalized exponential distributions, comput. stat. data anal. 43 (2003), 179–196. [11] r.d. gupta, d. kundu, on the comparison of fisher information of the weibull and ge distributions, j. stat. plan. inference. 136 (2006), 3130–3144. [12] r.d. gupta, d. kundu, generalized exponential distribution: existing results and some recent developments, j. stat. plan. inference. 137 (2007), 3537–3547. int. j. anal. appl. 18 (5) (2020) 818 [13] d. kundu, r.d. gupta, a. manglick, discriminating between the log-normal and generalized exponential distributions, j. stat. plan. inference. 127 (2005), 213-227. [14] c.d. lai, d.n.p. murthy, m. xie, weibull distributions, wires comput. stat. 3 (2011), 282–287. [15] e.t. lee, j.w. wang, statistical methods for survival data analysis, 3rd ed, j. wiley, new york, 2003. [16] g.s. mudholkar, d.k. srivastava, exponentiated weibull family for analyzing bathtub failure-rate data, ieee trans. reliab. 42 (1993), 299–302. [17] s. nadarajah, the exponentiated exponential distribution: a survey, asta adv. stat. anal. 95 (2011), 219–251. [18] s. nadarajah, s. kotz, the beta exponential distribution, reliab. eng. syst. safety. 91 (2006), 689–697. [19] s. nadarajah, s. kotz, the exponentiated type distributions, acta appl. math. 92 (2006), 97–111. [20] r core team, r: a language and environment for statistical computing, r foundation for statistical computing, vienna, austria, 2019. [21] m.m. raqab, m. ahsanullah, estimation of the location and scale parameters of generalized exponential distribution based on order statistics, j. stat. comput. simul. 69 (2001), 109–123. [22] m.z. raqab, inferences for generalized exponential distribution based on record statistics, j. stat. plan. inference. 104 (2002), 339–350. [23] m.z. raqab, generalized exponential distribution: moments of order statistics, statistics. 38 (2004), 29–41. [24] m.m. ristić, d. kundu, marshall-olkin generalized exponential distribution, metron. 73 (2015), 317–333. [25] m.h. tahir, s. nadarajah, parameter induction in continuous univariate distributions: well-established g families, an. acad. bras. ciênc. 87 (2015), 539–568. 1. introduction 2. new generalized exponential distribution 3. some basic properties 3.1. the mode 3.2. density shapes 3.3. hazard rate function (hrf) 3.4. shapes of the hrf 3.5. asymptotics of the density and hrf 3.6. reversed hazard rate function (rhrf) 4. linear representation 5. mathematical properties 5.1. generation function 5.2. moments 5.3. quantile function 5.4. mean deviations 6. estimation and inference 6.1. application 7. final remarks references international journal of analysis and applications volume 19, number 3 (2021), 465-476 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-465 existence and location of a unique solution of caputo-liouville type langevin equation with finitely many nonlinearities and nonlocal boundary conditions bashir ahmad1,∗, ahmed alsaedi1, hanan al-johany1, sotiris k. ntouyas2 1nonlinear analysis and applied mathematics (naam)-research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia 2department of mathematics, university of ioannina, 451 10 ioannina, greece ∗corresponding author: bashirahmad−qau@yahoo.com abstract. in this paper, we discuss the existence of a unique solution of caputo-liouville type langevin equation involving two fractional orders and finitely many nonlinearities, equipped with nonlocal boundary conditions via banach contraction mapping principle. the location of the unique solution of the given problem is also presented. in addition, we discuss the existence of solutions for the problem at hand by means of krasnosel’skĭi’s fixed point theorem. examples are constructed for the illustration of the obtained results. the paper concludes with some interesting remarks. 1. introduction fractional order differential equations received overwhelming attention of many researchers as these equations extensively appear in the mathematical modeling of several scientific and technical phenomena. examples include physics, biology, chemistry, control theory, electrical circuits, wave propagation, blood flow phenomena, signal and image processing, etc. for further details, see [1][5]. received february 28th, 2021; accepted april 9th, 2021; published april 28th, 2021. 2010 mathematics subject classification. 26a33, 34a08, 34b15. key words and phrases. langevin equation; nonlinearities; nonlocal boundary condition; uniqueness; location. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 465 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-465 int. j. anal. appl. 19 (3) (2021) 466 langevin equation, formulated in terms of integer-order derivatives by langevin [6] in 1908, is a wellknown equation of mathematical physics, which is used to describe the evolution of physical phenomena, such as brownian motion in fluctuating environments. langevin equation is also known as a stochastic differential equation as it is related to the fast motion of microscopic variables of the dynamical systems. however, the failure of classical langevin equation to describe the complex systems led to its several generalizations, which successfully modeled the physical phenomena in disordered regions [7], anomalous diffusion processes in complex and viscoelastic environment [8, 9], etc. among these generalizations includes the one obtained by replacing the ordinary derivative by fractional order derivative in it; the resulting form is known as fractional langevin equation and can take care of the fractal and memory properties of the phenomena under investigation. applications of fractional langevin equation include motor control system [10], single-file diffusion [11], transformation of the fokkerplanck equation into the wiener process [12], association of kramers-fokker-planck equation with langevin equation [13], etc. in order to obtain a more flexible model for fractal processes, lim et al. [14] introduced a new form of langevin equation involving two different fractional orders. for some recent results on fractional langevin equation, for instance, see [15][24] and references therein. modern tools of functional analysis have played a key role in developing the theory (existence and uniqueness of solutions) for fractional order initial and boundary value problems, for example, see [25][30]. in this paper, motivated by the recent development on langevin equation, we study the following nonlocal boundary value problem involving langevin equation with finitely many nonlinearities: (1.1) cdα(cdβ + µ)y(t) = m∑ i=1 aifi(t,y(t)), 0 < α ≤ 1, 1 < β ≤ 2, (1.2) y(0) = 0, y(ξ1) = 0, y(1) = ω y(ξ2), 0 < ξ1 < ξ2 < 1, where cd denotes the caputo-liouville fractional derivative operator, fi : [0, 1] × r → r are continuous functions, and ω ∈ r. by using banach contraction mapping principle we prove the existence of a unique solution of boundary value problem (1.1)-(1.2) and moreover we study the location of the unique solution. an existence result is also obtained via krasnosel’skĭi’s fixed point theorem. the paper is organized as follows. in section 2 we recall some basic definitions and properties from fractional calculus and solve a linear variant of the boundary value problem (1.1)-(1.2). the main results are presented in section 3. examples illustrating the obtained results are also constructed. 2. preliminaries let us recall some basic definitions on fractional calculus. int. j. anal. appl. 19 (3) (2021) 467 definition 2.1. ( [2, 3]). the riemann–liouville fractional integral iαa y of order α > 0 for a function y ∈ l1[a,b],−∞ < a < b < +∞, existing almost everywhere on [a,b], is defined by iαa y (t) = 1 γ (α) t∫ a (t−s)α−1 y (s)ds, where γ denotes the euler gamma function. definition 2.2. [2, 3]. let y,y(m) ∈ l1[a,b]. then the riemann–liouville fractional derivative dαay of order α ∈ (m− 1,m],m ∈ n, existing almost everywhere on [a,b], is defined as dαay (t) = dm dtm im−αa y (t) = 1 γ (m−α) dm dtm t∫ a (t−s)m−1−α y (s)ds. the caputo fractional derivative cdαay of order α ∈ (m− 1,m],m ∈ n is defined as cdαay (t) = d α a [ y (t) −y (a) −y′ (a) (t−a) 1! − . . .−y(m−1) (a) (t−a)m−1 (m− 1)! ] . remark 2.1. [2]. if y ∈ acm[a,b], then the caputo fractional derivative cdαay of order α ∈ (m−1,m],m ∈ n, existing almost everywhere on [a,b], is defined as cdαay(t) = i m−α a y (m) (t) = 1 γ (m−α) t∫ a (t−s)m−1−α y(m) (s)ds. proposition 2.1. ( [2]) for κ > 0 and α > 0 with n− 1 < α ≤ n, and y ∈ l1[a,b], we have the following properties: (i) iαa i κ ay(t) = i κ ai α a y(t) = i α+κ a y(t); (ii) iαa (t−a) η = γ (η + 1) γ (α + η + 1) (t−a)α+η, η > −1; (iii) cdαa [i α a y (t)] = y(t); (iv) iαa [ cdαay (t)] = y (t) − n−1∑ p=0 y(p) (a) (t−a)p p! , y ∈ cn[a,b]. in the sequel, we write iσ and cdσ instead of iσ0 and cdσ0 respectively. to study the nonlinear problem (1.1)-(1.2), we first solve its linear variant in the following lemma. lemma 2.1. for a given ρ ∈ c([0, 1],r), the unique solution of the boundary value problem (2.1) cdα(cdβ + µ)y(t) = ρ(t) 0 < α ≤ 1, 1 < β ≤ 2, (2.2) y(0) = 0, y(ξ1) = 0, y(1) = ωy(ξ2), int. j. anal. appl. 19 (3) (2021) 468 is given by y(t) = 1 γ(β + α) ∫ t 0 (t−s)β+α−1ρ(s)ds− µ γ(β) ∫ t 0 (t−s)β−1y(s)ds + ρ1(t) [∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ρ(s)ds−µ ∫ ξ1 0 (ξ1 −s)β−1 γ(β) y(s)ds ] + ρ2(t) [∫ 1 0 (1 −s)β+α−1 γ(β + α) ρ(s)ds−µ ∫ 1 0 (1 −s)β−1 γ(β) y(s)ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ρ(s)ds + µω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) y(s)ds ] , (2.3) where ρ1(t) = 1 ∆γ(β + 1) ( t(1 −ωξβ2 ) − (1 −ωξ2)t β ) , ρ2(t) = 1 ∆γ(β + 1) ( tβξ1 − tξ β 1 ) ,(2.4) and it is assumed that (2.5) ∆ = [ξ β 1 (1 −ωξ2) − ξ1(1 −ωξ β 2 )] γ(β + 1) 6= 0. proof. applying the riemann-liouville integral operator iα to both sides of (2.1) and using proposition 2.1 (iv), we obtain (2.6) (cdβ + µ)y(t) + c0 = i αρ(t), where c0 ∈ r is an unknown constant. next, operating iβ on both sides of (2.6), we obtain y(t) = ∫ t 0 (t−s)β+α−1 γ(β + α) ρ(s)ds− µ γ(β) ∫ t 0 (t−s)β−1y(s)ds − c0t β γ(1 + β) − c1 − c2t. (2.7) using the conditions y(0) = 0 and y(ξ1) = 0 in (2.7) yields c1 = 0 and (2.8) c0ξ β 1 γ(β + 1) + c2ξ1 = a, where a = ∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ρ(s)ds−µ ∫ ξ1 0 (ξ1 −s)β−1 γ(β) y(s)ds. now using y(1) = ωy(ξ2) in (2.7), we obtain (2.9) (1 −ωξβ2 ) γ(β + 1) c0 + (1 −ωξ2)c2 = b, where b = ∫ 1 0 (1 −s)β+α−1 γ(β + α) ρ(s)ds−µ ∫ 1 0 (1 −s)β−1 γ(β) y(s)ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ρ(s)ds + µω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) y(s)ds. int. j. anal. appl. 19 (3) (2021) 469 solving (2.8) and (2.9) for c0 and c2, we find that (2.10) c0 = 1 ∆ (σ3a− ξ1b), c2 = 1 ∆ (σ1b −σ2a), where (2.11) ∆ = σ3σ1 − ξ1σ2, σ1 = ξ β 1 γ(β + 1) , σ2 = (1 −ωξβ2 ) γ(β + 1) , σ3 = (1 −ωξ2). inserting c1 = 0 and the values of c0 and c2 from (2.10) into (2.7), together with (2.11), leads to the solution (2.3). the converse follows by direct computation. this completes the proof. � definition 2.3. a function y ∈ c([0, 1] ,r) is a solution of the boundary value problem (1.1)-(1.2) if and only if it satisfies the integral equation: y(t) = ∫ t 0 (t−s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ t 0 (t−s)β−1 γ(β) y(s)ds + ρ1(t) [∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ ξ1 0 (ξ1 −s)β−1 γ(β) y(s)ds ] + ρ2(t) [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ 1 0 (1 −s)β−1 γ(β) y(s)ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds + µω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) y(s)ds ] . (2.12) 3. uniqueness and location results in the following theorem, we prove the existence of a unique solution for the problem (1.1)-(1.2) by applying banach contraction mapping principle. theorem 3.1. let fi(t,y) : [0, 1] × r → r, i = 1, . . . ,m be continuous functions satisfying the lipschitz condition: (a1) |fi(t,x) −fi(t,y)| ≤ li |x−y| , ∀t ∈ [0, 1] , x,y ∈ r, li > 0, i = 1, 2, . . . ,m. then, the boundary value problem (1.1)-(1.2) has a unique solution on [0, 1] if (3.1) ω1 < 1, where ω1 = ( ∑m i=1 |ai|li) γ(β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )] + |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2 ( 1 + |ω|ξβ2 )] , (3.2) and ρ̄1 = max t∈[0,1] |ρ1(t)|, ρ̄2 = max t∈[0,1] |ρ2(t)|. int. j. anal. appl. 19 (3) (2021) 470 proof. to transform the problem (1.1) and (1.2) into a fixed-point problem, we introduce an operator n : c([0, 1] ,r) → c([0, 1] ,r) as (ny)(t) = ∫ t 0 (t−s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ t 0 (t−s)β−1 γ(β) y(s)ds + ρ1(t) [∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ ξ1 0 (ξ1 −s)β−1 γ(β) y(s)ds ] + ρ2(t) [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds−µ ∫ 1 0 (1 −s)β−1 γ(β) y(s)ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds + µω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) y(s)ds ] , (3.3) where c([0, 1],r) is the banach space of all continuous functions from [0, 1] into r equipped with the norm ‖y‖ = supt∈[0,1] |y(t)| . observe that the fixed points of the operator n are solutions of the problem (1.1) and (1.2) by definition 2.3. further, it is an immediate consequence of the dominated convergence theorem that ny ∈ c([0, 1],r) for every y ∈ c([0, 1],r). the proof will be complete by means of banach contraction mapping principle once it is shown that that the operator n defined by (3.3) is a contraction. for x,y ∈ r and ∀t ∈ [0, 1], we have ‖(nx) − (ny)‖ = sup t∈[0,1] ∣∣∣∣∣ ∫ t 0 (t−s)β+α−1 γ(β + α) ( m∑ i=1 ai[fi(s,x(s)) −fi(s,y(s))] ) ds −µ ∫ t 0 (t−s)β−1 γ(β) [x(s) −y(s)]ds +ρ1(t) [∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 ai[fi(s,x(s)) −fi(s,y(s))] ) ds −µ ∫ ξ1 0 (ξ1 −s)β−1 γ(β) [x(s) −y(s)]ds ] +ρ2(t) [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 ai[fi(s,x(s)) −fi(s,y(s))] ) ds −µ ∫ 1 0 (1 −s)β−1 γ(β) [x(s) −y(s)]ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 ai[fi(s,x(s)) −fi(s,y(s))] ) ds +µω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) [x(s) −y(s)]ds ]∣∣∣∣∣ ≤ ( ∑m i=1 |ai|li) γ(β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )] ‖x−y‖ int. j. anal. appl. 19 (3) (2021) 471 + |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2 ( 1 + |ω|ξβ2 )] ‖x−y‖, which leads to ‖(nx) − (ny)‖≤ ω1 ‖x−y‖ . evidently, we deduce by (3.1) that n : c([0, 1] ,r) → c([0, 1] ,r) is a contraction. hence, by banach contraction mapping principle, the operator n has a unique fixed point, which corresponds to a unique solution of the boundary value problem (1.1)-(1.2) on [0, 1]. this completes the proof. � example 3.1. consider the following boundary value problem: cd 1 2 ( cd 3 2 + 1 5 ) y(t) = 3∑ i=1 aifi(t,y(t)), t ∈ [0, 1], y(0) = 0, y (1 3 ) = 0, y(1) = y (2 3 ) . (3.4) here α = 1/2,β = 3/2,µ = 1/5,ω = 1,ξ1 = 1/3,ξ2 = 2/3,m = 3,a1 = 1/2,a2 = 1,a3 = 3/4 and f1 (t,y) = 1 √ t2 + 100 |y| |y| + 1 + e−t, f2 (t,y) = 1 t2 + 20 tan−1 y + 2, f3 (t,y) = 1 15 ( e−t t2 + 1 ) sin y + t2 1 + t2 . it is easy to find that |fi (t,x) −fi (t,y)| ≤ li |x−y| , i = 1, 2, 3, with l1 = 1/10,l2 = 1/20,l3 = 1/15 and 3∑ i=1 aili = 3/20. furthermore, we have |∆| = |[ξ β 1 (1−ωξ2)−ξ1(1−ωξ β 2 )]| γ(β+1) ≈ 0.000499, ρ̄1 = maxt∈[0,1] |ρ1(t)| = ρ1(t)|t=0.830537 ≈ 1.437777, ρ̄2 = maxt∈[0,1] |ρ2(t)| = ρ2(t)|t=1 ≈ 1.605694 and ω1 = 0.826088 < 1. clearly all the assumptions of theorem 3.1 are satisfied. therefore the problem (3.4) has a unique solution on [0, 1]. in the following result, we present location of the unique solution of the boundary value problem (1.1)(1.2). theorem 3.2. let the hypotheses of theorem 3.1 hold. then the unique solution y of problem (1.1)-(1.2) satisfies (3.5) ‖y‖≤ ω2 1 − ω1 , where ω1 is given by (3.2) and (3.6) ω2 = ( m∑ i=1 mi ){ 1 γ(β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )]} , with mi = supt∈[0,1] |fi (t, 0)| . int. j. anal. appl. 19 (3) (2021) 472 proof. by theorem 3.1, the solution (2.12) of the boundary value problem (1.1)-(1.2) is unique. in view of the inequality: ∣∣∣ m∑ i=1 aifi(s,y(s)) ∣∣∣ ≤ m∑ i=1 |ai|(li‖y‖ + mi), where mi = supt∈[0,1] |fi (t, 0)| , it follows from (2.12) that ‖y‖≤ m∑ i=1 |ai|(li‖y‖ + mi) sup t∈[0,1] {∫ t 0 (t−s)β+α−1 γ(β + α) ds + |ρ1(t)| ∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ds + |ρ2(t)| [∫ 1 0 (1 −s)β+α−1 γ(β + α) ds + |ω| ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ds ]} + |µ|‖y‖ sup t∈[0,1] {∫ t 0 (t−s)β−1 γ(β) ds + |ρ1(t)| ∫ ξ1 0 (ξ1 −s)β−1 γ(β) ds + |ρ2(t)| [∫ 1 0 (1 −s)β−1 γ(β) ds + |ω| ∫ ξ2 0 (ξ2 −s)β−1 γ(β) ds ]} ≤‖y‖ { ( ∑m i=1 |ai|li) γ(β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )] + |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2 ( 1 + |ω|ξβ2 )]} + ( m∑ i=1 mi ){ 1 γ(β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )]} = ‖y‖ω1 + ω2. solving the above inequality for ‖y‖ yields ‖y‖≤ ω2 1 − ω1 . this completes the proof. � example 3.2. in relation to example 3.1, it is found that ‖y‖≤ δ, where δ ≈ 35.008543. 4. an existence result in this section, we establish an existence result for the boundary value problem (1.1)-(1.2) via krasnosel’skĭi’s fixed point theorem [31]. theorem 4.1. let fi : [0, 1] × r → r be continuous functions satisfying the condition (a2): |fi(t,y)| 6 h(t), ∀ i = 1, 2, . . . ,m, (t,y) ∈ [0, 1] × r, h ∈ c([0, 1],r+). then the boundary value problem (1.1)-(1.2) has at least one solution on [0, 1], provided that (4.1) q1 := |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2(1 + |ω|ξ β 2 ) ] < 1. int. j. anal. appl. 19 (3) (2021) 473 proof. consider br = {y ∈ c([0, 1],r) : ‖y‖≤ r}, with r > q2‖h‖ 1 −q1 , where (4.2) q2 := m∑ i=1 |ai| { 1 γ (β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )]} and ‖h‖ = sup t∈[0,1] |h(t)|. then we define the operators p and q on br to c([0, 1],r) as p(t) = ∫ t 0 (t−s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds +ρ1(t) ∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds +ρ2(t) [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds −ω ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds ] ,(4.3) q(t) = −µ ∫ t 0 (t−s)β−1 γ(β) y(s)ds−µρ1(t) ∫ ξ1 0 (ξ1 −s)β−1 γ(β) y(s)ds −µρ2(t) [∫ 1 0 (1 −s)β−1 γ(β) y(s)ds−ω ∫ ξ2 0 (ξ2 −s)β−1 γ(β) y(s)ds ] .(4.4) we show that px + qy ∈ br. for x,y ∈ br, we find that ‖(px) + (qy)‖ ≤ sup t∈[0,1] {∫ t 0 (t−s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds + |µ| ∫ t 0 (t−s)β−1 γ(β) |y(s)|ds +|ρ1(t)| [∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds +|µ| ∫ ξ1 0 (ξ1 −s)β−1 γ(β) |y(s)|ds ] + |ρ2(t)| [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds +|µ| ∫ 1 0 (1 −s)β−1 γ(β) |y(s)|ds + |ω| ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds +|µ||ω| ∫ ξ2 0 (ξ2 −s)β−1 γ(β) |y(s)|ds ]} ≤ m∑ i=1 |ai|‖h‖ { 1 γ (β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )]} + |µ|r γ (β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2 ( 1 + |ω|ξβ2 )] ≤ q2 ‖h‖ + q1r ≤ r int. j. anal. appl. 19 (3) (2021) 474 thus, px + qy ∈ br. next we show that q is a contraction mapping. for x,y ∈ c([0, 1],r) and each t ∈ [0, 1], we obtain ‖(qx) − (qy)‖ ≤ |µ| ∫ t 0 (t−s)β−1 γ(β) |x(s) −y(s)|ds + |µ||ρ1(t)| ∫ ξ1 0 (ξ1 −s)β−1 γ(β) (x(s) −y(s)|ds +|µ||ρ2(t)| [∫ 1 0 (1 −s)β−1 γ(β) |x(s) −y(s)|ds +|ω| ∫ ξ2 0 (ξ2 −s)β−1 γ(β) |x(s) −y(s)|ds ] ≤ |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2(1 + |ω|ξ β 2 ) ] ‖x−y‖, which is a contraction in view of the condition (4.1). we show that p is compact and continuous. the continuity of fi implies that the operator p is continuous. p is uniformly bounded on br as ‖px‖≤ q2 ‖h‖ , where q2 is given by (5.1). we shall prove now that p is equicontinuous. for t1, t2 ∈ [0, 1] with t1 > t2, we have |py(t1) −py(t2)| ≤ ∣∣∣∫ t1 0 (t1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) − ∫ t2 0 (t2 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) )∣∣∣ +|ρ1(t1) −ρ1(t2)| ∫ ξ1 0 (ξ1 −s)β+α−1 γ(β + α) ( m∑ i=1 aifi(s,y(s)) ) ds +|ρ2(t1) −ρ2(t2)| [∫ 1 0 (1 −s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds +|ω| ∫ ξ2 0 (ξ2 −s)β+α−1 γ(β + α) ( m∑ i=1 |ai||fi(s,y(s))| ) ds ] ≤ ∑m i=1 |ai|‖h‖ γ (β + α + 1) { 2 (t1 − t2) β+α + |tβ+α1 − t β+α 2 | + |ρ1(t1) −ρ1(t2)|ξ β+α 1 +|ρ2(t1) −ρ2(t2)|(1 + |ω|ξ β+α 2 ) } , which is independent of y and tends to zero as t1 −t2 → 0. so p is equicontinuous. hence, by arzelá-ascoli theorem, p is compact on br. thus all the assumptions of krasnosel’skĭi’s fixed point theorem [31] are verified and hence its conclusion implies that the boundary value problem (1.1)-(1.2) has at least one solution on [0, 1]. the proof is completed. � 5. concluding remarks we have discussed the solvability of caputo-liouville type langevin equation involving two fractional orders and finitely many nonlinearities subject to nonlocal boundary conditions by means of standard fixed int. j. anal. appl. 19 (3) (2021) 475 point theorems. it is imperative to notice that the right-hand side of (1.1) provides a leverage to consider a variety of nonlinearities, for instance, some of the terms in the given sum may be of the type fi(t,y) =∫ t 0 ki(t,s)y(s)ds or ∫ t 0 gi(s,y(s))ds or i pgi(t,y(t)) or some of the functions may be non-lipschitz type. here are two examples. (a): choosing the right-hand side of (1.1) as riemann-liouville type integral nonlinearities of the form:∑m i=1 aii qifi(t,y(t)), qi > 0 instead of ∑m i=1 aifi(t,y(t)), the results for the problem (1.1) and (1.2) obtained in the previous sections become the ones for the problem with riemann-liouville type integral nonlinearities by replacing ω1 and q2 with ω̂1 and q̂2 respectively, where ω̂1 = 1 γ(β + α + 1) m∑ i=1 (|ai|li) γ(qi + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )] + |µ| γ(β + 1) [ 1 + ρ̄1ξ β 1 + ρ̄2 ( 1 + |ω|ξβ2 )] , q̂2 := m∑ i=1 { |ai| γ(qi + 1)γ (β + α + 1) [ 1 + ρ̄1ξ β+α 1 + ρ̄2 ( 1 + |ω|ξβ+α2 )]} . (b): by a simple manipulation, we can obtain the results for the problem (1.1) and (1.2) with the right-hand side of (1.1) of the form: m∑ i=1 aifi(t,y(t)) + κ∑ j=1 aji qjgj(t,y(t)), qj > 0, where fi and gj are given continuous functions. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.l. magin, fractional calculus in bioengineering, begell house, chicago, 2006. [2] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies 204, elsevier science b.v, amsterdam, 2006. [3] k. diethelm, the analysis of fractional differential equations. an application-oriented exposition using differential operators of caputo type, lecture notes in mathematics, 2004, springer-verlag, berlin, 2010. [4] f. mainardi, fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, imperial college press, singapore, 2010. [5] h.a. fallahgoul, s.m. focardi, f.j. fabozzi, fractional calculus and fractional processes with applications to financial economics: theory and application, elsevier/academic press, london, 2017. [6] p. langevin, sur la théorie du mouvement brownien (in french), on the theory of brownian motion, cr acad. sci. paris, 146 (1908) 530-533. [7] r. klages, g. radons, i.m. sokolov, anomalous transport: foundations and applications, wiley-vch, weinheim, 2008. [8] r. kubo, the fluctuation-dissipation theorem, rep. prog. phys., 29 (1966) 255-84. [9] r. kubo, m. toda, n. hashitsume, statistical physics ii, second ed., springer-verlag, berlin, 1991. int. j. anal. appl. 19 (3) (2021) 476 [10] b.j. west, m. latka, fractional langevin model of gait variability, j. neuroeng. rehabil. 2 (2005), 24, [11] c.h. eab, s.c. lim, fractional generalized langevin equation approach to single-file diffusion, phys. a, 389 (2010), 25102521. [12] s.f. kwok, langevin equation with multiplicative white noise: transformation of diffusion processes into the wiener process in different prescriptions, ann. phys., 327 (2012), 1989-1997. [13] s. eule, r. friedrich, f. jenko, d. kleinhans, langevin approach to fractional diffusion equations including inertial effects, j. phys. chem. b, 111 (2007), 11474-11477. [14] s.c. lim, m. li, l.p. teo, langevin equation with two fractional orders, phys. lett. a, 372 (2008), 6309-6320. [15] b. ahmad, j.j. nieto, a. alsaedi, m. el-shahed, a study of nonlinear langevin equation involving two fractional orders in different intervals, nonlinear anal. real world appl. 13 (2012), 599-606. [16] o. baghani, on fractional langevin equation involving two fractional orders, commun. nonlinear sci. numer. simul. 42 (2017), 675-681. [17] b. li, s. sun, y. sun, existence of solutions for fractional langevin equation with infinite-point boundary conditions, j. appl. math. comput. 53 (2017), 683-692. [18] h. fazli, j.j. nieto, fractional langevin equation with anti-periodic boundary conditions, chaos solitons fractals, 114 (2018), 332-337. [19] z. zhou, y. qiao, solutions for a class of fractional langevin equations with integral and anti-periodic boundary conditions, bound. value probl. 2018 (2018), 152. [20] b. ahmad, a. alsaedi, s. salem, on a nonlocal integral boundary value problem of nonlinear langevin equation with different fractional orders, adv. difference equ. 2019 (2019), 57. [21] y. liu, r. agarwal, existence of solutions of bvps for impulsive fractional langevin equations involving caputo fractional derivatives, turk. j. math. 43 (2019), 2451-2472. [22] z. laadjal, b. ahmad, n. adjeroud, existence and uniqueness of solutions for multi-term fractional langevin equation with boundary conditions, dyn. contin. discrete impuls. syst. ser. a math. anal. 27 (2020), 339-350. [23] a. wongcharoen, b. ahmad, s.k. ntouyas, j. tariboon, three-point boundary value problems for the langevin equation with the hilfer fractional derivative, adv. math. phys. 2020 (2020), 9606428. [24] h. fazli, h. sun, j.j. nieto, new existence and stability results for fractional langevin equation with three-point boundary conditions, comput. appl. math. 40 (2021), 48. [25] f. jiao, y. zhou, existence of solution for a class of fractional boundary value problems via critical point theory, comput. math. appl. 62 (2011), 1181-1199. [26] s. sun, y. zhao, z. han, y. li, the existence of solutions for boundary-value problem of fractional hybrid differential equations, commun. nonlinear sci. numer. simul. 14 (2012), 4961-4967. [27] j. henderson, n. kosmatov, eigenvalue comparison for fractional boundary value problems with the caputo derivative, fract. calc. appl. anal. 17 (2014), 72-880. [28] j. henderson, r. luca, nonexistence of positive solutions for a system of coupled fractional boundary value problems, bound. value probl. 2015 (2015), 138. [29] b. ahmad, r. luca, existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions, fract. calc. appl. anal. 21 (2018), 423-441. [30] r.p. agarwal, r. luca, positive solutions for a semipositone singular riemann-liouville fractional differential problem, int. j. nonlinear sci. numer. simul. 20 (2019), 823-831. [31] m.a. krasnosel’skĭi, two remarks on the method of successive approximations, uspekhi mat. nauk, 10 (1955), 123-127. 1. introduction 2. preliminaries 3. uniqueness and location results 4. an existence result 5. concluding remarks references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 28-39 http://www.etamaths.com on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators vishnu narayan mishra1,2,∗ and shikha pandey1 abstract. in the present paper, we introduce the chlodowsky variant of (p,q) kantorovich-stancuschurer operators on the unbounded domain which is a generalization of (p,q) bernstein-stancukantorovich operators. we have also derived its korovkin type approximation properties and rate of convergence. 1. introduction and preliminaries approximation theory has an important role in mathematical research because of its great potential for applications. korovkin gave his famous approximation theorem in 1950, since then the study of the linear methods of approximation given by sequences of positive and linear operators became a deeprooted part of approximation theory. considering it, various operators as bernstein, szász, baskakov etc. and their generalizations are being studied. in recent years, many results about the generalization of positive linear operators have been obtained by several mathematicians ([6], [8]-[12], [21]). in last two decades, the applications of q-calculus has played an important role in the area of approximation theory, number theory and theoretical physics. in 1987, lupas, and in 1997, phillips introduced a sequence of bernstein polynomials based on q-integers and investigated its approximation properties. several researchers obtained various other generalizations of operators based on q-calculus(see [3], [17]-[19]). recently, mursaleen et al. applied (p,q)-calculus in approximation theory and introduced first (p,q)analogue of bernstein operators. they investigated uniform convergence of the operators and order of convergence, obtained voronovskaja theorem as well. also, (p,q)-analogue of bernstein operators, bernstein-stancu operators and bernstein-schurer operators, kontorovich bernstein schurer, and bleimann-butzer-hahn operators were introduced in ([13]-[16]), respectively. further, acar [1] have studied recently, (p,q)-generalization of szászmirakyan operators. in the present paper, we introduce the chlodowsky variant of (p,q) kantorovich-stancu-schurer operators on the unbounded domain. we begin by recalling certain notations of (p,q)-calculus. for 0 < q < p ≤ 1, the (p,q) integer [n]p,q is defined by [n]p,q := pn −qn p−q . (p,q) factorial is expressed as [n]p,q! = [n]p,q[n− 1]p,q[n− 2]p,q . . . 1. (p,q) binomial coefficient is expressed as[ n k ] p,q := [n]p,q! [k]p,q![n−k]p,q! . 2010 mathematics subject classification. 41a25, 41a36, 41a10, 41a30. key words and phrases. (p,q)-integers; (p,q) bernstein operators; chlodowsky polynomials; (p,q) bernstein-kantorovich operators; modulus of continuity; linear positive operator; korovkin type approximation theorem; rate of convergence. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 28 on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 29 (p,q) binomial expansion as (ax + by)np,q := n∑ k=0 [ n k ] p,q an−kbkxn−kyk. (x + y)np,q := (x + y)(px + qy)(p 2x + q2y) . . . (pn−1x + qn−1y). the definite integrals of the function f are given by∫ a 0 f(x)dp,qx = (q −p)a ∞∑ k=0 pk qk+1 f ( pk qk+1 a ) , ∣∣∣∣pq ∣∣∣∣ < 1, and ∫ a 0 f(x)dp,qx = (p−q)a ∞∑ k=0 qk pk+1 f ( qk pk+1 a ) , ∣∣∣∣pq ∣∣∣∣ > 1. further (p,q) analysis can be found in [2]. in 1932, chlodowsky [7] presented a generalization of bernstein polynomials on an unbounded set, known as bernstein-chlodowsky polynomials given by, (1.1) bn(f,x) = n∑ k=0 f( k n bn) ( n k )( x bn )k ( 1 − x bn )n−k , 0 ≤ x ≤ bn, where bn is an increasing sequence of positive terms with the properties bn → ∞ and bnn → 0 as n →∞. in 2015, vedi and özarslan [20] investigated chlodowsky-type q-bernstein-stancu-kantorovich operators, and wafi and rao investigated (p,q) form of kantorovich type bernstein-stancu-schurer operator. mursaleen and khan [15] defined the kantorovich type (p,q)-bernstein-schurer operators, given by tn,m(f; x,p,q) = n+m∑ k=0 [ n + m k ] p,q xk n+m−k−1∏ s=0 (ps −qsx) ∫ 1 0 f ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q ) dp,qt k = 0, 1, 2, . . . ,n = 1, 2, 3, . . . where lemma 1. (see [15]) for the operators t (α,β) n,m , we have tn,m(1; x,p,q) = 1, tn,m(t; x,p,q) = (px + 1 −x)n+mp,q [2]p,q[n + 1]p,q + (p + 2q − 1)[n + m]p,q [2]p,q[n + 1]p,q x, tn,m(t 2; x,p,q) = (p2x + 1 −x)n+mp,q [3]p,q[n + 1]2p,q + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q [n + 1]2p,q (px + 1 −x)n+m−1p,q x + { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q [n + 1]2p,q x2. 2. construction of the operators we construct the chlodowsky variant of (p,q) kontorovich-stancu-schurer operators as (2.1) k (α,β) n,m (f; x,p,q) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k ∫ 1 0 f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) dp,qt, where n ∈ n, m,α,β ∈ n0 with 0 ≤ α ≤ β, 0 ≤ x ≤ bn, 0 < q < p ≤ 1. obviously, k (α,β) n,m is a linear and positive operator. to begin with, we obtain the following important lemma. 30 mishra and pandey lemma 2. let k (α,β) n,m (f; x,p,q) be given by (2.1). the first few moments of the operators are (i) k(α,β)n,m (1; x,p,q) = 1, (ii) k(α,β)n,m (t; x,p,q) = ( 1 [n + 1]p,q + β )( αbn + (p x bn + 1 − x bn )n+mp,q [2]p,q bn + (p + 2q − 1)[n + m]p,q [2]p,q x ) , (iii) k(α,β)n,m (t 2; x,p,q) = ( 1 [n + 1]p,q + β )2[( α2 + 2α [2]p,q ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q ) b2n + ( 2α [2]p,q (p + 2q − 1) + { 1 + 2q [2]p,q + q2 − 1 [3]p,q }( p x bn + 1 − x bn )n+m−1 p,q ) [n + m]p,qbnx + { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,qx2 ] , (iv) k(α,β)n,m ((t−x); x,p,q) = [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x (v) k(α,β)n,m ((t−x) 2; x,p,q) = [ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + [ 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ] bnx + [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ] x2. proof. from operator 2.1, k (α,β) n,m (t u; x,p,q) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn )u dp,qt = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k [n + 1]up,qbun ([n + 1]p,q + β)u × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q )u dp,qt = [n + 1]up,qb u n ([n + 1]p,q + β)u n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) u∑ i=0 ( u i )( α [n + 1]p,q )u−i × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q )u dp,qt = [n + 1]up,qb u n ([n + 1]p,q + β)u u∑ i=0 ( u i )( α [n + 1]p,q )u−i n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) × ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt [n + 1]p,q )i dp,qt k (α,β) n,m (t u; x,p,q) = [n + 1]up,qb u n ([n + 1]p,q + β)u u∑ i=0 ( u i )( α [n + 1]p,q )u−i tn,m(t i; x bn ,p,q).(2.2) thus for u=0,1,2 we get k(α,β)n,m (1; x,p,q) = tn,m(1; x bn ,p,q), on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 31 k(α,β)n,m (t; x,p,q) = [n + 1]p,qbn [n + 1]p,q + β 1∑ i=0 ( 1 i )( α [n + 1]p,q )1−i tn,m(t i; x bn ,p,q) = ( [n + 1]p,q [n + 1]p,q + β ) bn { α [n + 1]p,q + tn,m(t; x bn ,p,q) } , k(α,β)n,m (t 2; x,p,q) = [n + 1]2p,qb 2 n ([n + 1]p,q + β)2 2∑ i=0 ( 2 i )( α [n + 1]p,q )2−i tn,m(t i; x bn ,p,q) = [n + 1]2p,qb 2 n ([n + 1]p,q + β)2 [( α [n + 1]p,q )2 +2 ( α [n + 1]p,q ) tn,m(t; x bn ,p,q) + tn,m(t 2; x bn ,p,q) ] . using lemma 1 and in view of the above relations we get the statements (i), (ii) and (iii). using linear property of operators, we have k(α,β)n,m ((t−x); x,p,q) = k (α,β) n,m (t; x,p,q) −xk (α,β) n,m (1; x,p,q) = [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x. hence, we get (iv). similar calculations give k(α,β)n,m ((t−x) 2; x,p,q) = k(α,β)n,m (t 2; x,p,q) − 2xk(α,β)n,m (t; x,p,q) + x 2k(α,β)n,m (1; x,p,q). then we obtain, k(α,β)n,m ((t−x) 2; x,p,q) = [ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + [ 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ] bnx + [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ] x2. this proves (v). � 3. korovkin-type approximation theorem assume cρ is the space of all continuous functions f such that |f(x)| ≤ mρ(x), −∞ < x < ∞. then cρ is a banach space with the norm ‖f‖ρ = sup −∞<x<∞ |f(x)| ρ(x) . the subsequent results are used for proving korovkin approximation theorem on unbounded sets. theorem 1. (see [5]) there exists a sequence of positive linear operators un, acting from cρ to cρ, satisfying the conditions (1) limn→∞‖un(1; x) − 1‖ρ = 0, (2) limn→∞‖un(φ; x) −φ‖ρ = 0, (3) limn→∞‖un(φ2; x) −φ2‖ρ = 0, 32 mishra and pandey where φ(x) is a continuous and increasing function on (−∞,∞) such that limx→±∞φ(x) = ±∞ and ρ(x) = 1 + φ2, and there exists a function f∗ ∈ cρ for which limn→∞‖unf∗ −f∗‖ρ > 0. theorem 2. (see [5]) conditions (1),(2),(3) of above theorem implies that lim n→∞ ‖unf −f‖ρ = 0 for any function f belonging to the subset c0ρ := {f ∈ cρ : lim|x|→∞ |f(x)| ρ(x) is finite} . consider the weight function ρ(x) = 1 + x2 and operators: uα,βn,m(f; x,p,q) = { kα,βn,m(f; x,p,q) if x ∈ [0,bn], f(x) if x ∈ [0,∞)/[0,bn]. thus for f ∈ c1+x2, we have ‖uα,βn,m(f; ·,p,q)‖≤ sup x∈[0,bn] |uα,βn,m(f; x,p,q)| 1 + x2 + sup bn<x<∞ |f(x)| 1 + x2 ≤‖f‖1+x2 [ sup x∈[0,∞) |uα,βn,m(1 + t2; x,p,q)| 1 + x2 + 1 ] . now, using lemma 2, we will obtain, ‖uα,βn,m(f; ·,p,q)‖1+x2 ≤ m‖f‖1+x2 if p := (p)n,q := (q)n with 0 < qn < pn ≤ 1, limn→∞pn = 1, limn→∞qn = 1, limn→∞pnn = limn→∞q n n = n,n < ∞ and limn→∞ bn [n] = 0. theorem 3. for all f ∈ c0 1+x2 , we have lim n→∞ ‖uα,βn,m(f; ·,pn,qn) −f(·)‖1+x2 = 0 provided that p := (p)n,q := (q)n with 0 < qn < pn ≤ 1, limn→∞pn = 1, limn→∞qn = 1, limn→∞pnn = limn→∞q n n = n,n < ∞ and limn→∞ bn [n] = 0. proof. using the results of theorem 1 and lemma 2(i),(ii) and (iii), we will achieve the following assessments, respectively: sup x∈[0,∞) |uα,βn,m(1; x,pn,qn) − 1| 1 + x2 = sup 0≤x≤bn |kα,βn,m(1; x,pn,qn) − 1| 1 + x2 = 0 sup x∈[0,∞) |uα,βn,m(t; x,pn,qn) − t| 1 + x2 = sup 0≤x≤bn |kα,βn,m(t; x,pn,qn) −x| 1 + x2 ≤ sup 0≤x≤bn [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x 1 + x2 ≤ [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ∣∣∣∣∣(p + 2q − 1)[n + m]p,q[2]p,q([n + 1]p,q + β) − 1 ∣∣∣∣∣→ 0 and sup x∈[0,∞) |uα,βn,m(t2; x,pn,qn) − t2| 1 + x2 = sup 0≤x≤bn |kα,βn,m(t2; x,pn,qn) −x2| 1 + x2 ≤ sup 0≤x≤bn 1 1 + x2 ([ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + [ 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 33 − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ] bnx + [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ] x2 ) ≤ ([ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] b2n + ∣∣∣∣∣2α(p + 2q − 1)[n + m]p,q[2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ∣∣∣∣∣bn + ∣∣∣∣∣ { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ∣∣∣∣∣ ) −→ 0 whenever n →∞, because we have limn→∞pn = limn→∞qn = 1 and bn[n]p,q = 0 as n →∞. � theorem 4. assuming c as a positive and real number independent of n and f as a continuous function which vanishes on [c,∞). let p := (pn),q := (qn) with 0 < qn < pn ≤ 1, limn→∞pn = limn→∞qn = 1, limn→∞p n n = limn→∞q n n = n < ∞ and limn→∞ b2n [n]p,q = 0. then we have lim n→∞ sup 0≤x≤bn ∣∣kα,βn,m(f; x,pn,qn) −f(x)∣∣ = 0. proof. from the hypothesis on f, it is bounded i.e. |f(x)| ≤ m (m > 0). for any � > 0, we have∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣ < � + 2mδ2 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 , where x ∈ [0,bn] and δ = δ(�) are independent of n. now since we know, k (α,β) n,m ((t−x)2; x,pn,qn) = n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps−qs x bn ) ( x bn )k ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn )2 dt. we can conclude by theorem 3, sup 0≤x≤bn ∣∣∣kα,βn,m(f; x,pn,qn) −f(x)∣∣∣ ≤ � + 2m δ2 b2n ([ α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ] + ∣∣∣∣∣2α(p + 2q − 1)[n + m]p,q[2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ∣∣∣∣∣ + ∣∣∣∣∣ { 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ∣∣∣∣∣ ) . since b2n [n]p,q = 0 as n →∞, we have the desired result. � 4. rate of convergence we will find the rate of convergence in terms of the lipschitz class lipm (γ), for 0 < γ ≤ 1. assume that cb[0,∞) denote the space of bounded continuous functions on [0,∞). a function f ∈ cb[0,∞) belongs to lipm (γ) if |f(t) −f(x)| ≤ m|t−x|γ, t,x ∈ [0,∞) is satisfied. 34 mishra and pandey theorem 5. let f ∈ lipm (γ), then kα,βn,m(f; x,p,q) ≤ m(µn,p,q(x)) γ/2, where µn,p,q(x) = k α,β n,m((t−x)2; x,p,q). proof. since f ∈ lipm (γ), and the operator kα,βn,m(f; x,p,q) is linear and monotone, |kα,βn,m(f; x,p,q) −f(x)| = ∣∣∣∣∣ n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ( f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ) dp,qt ∣∣∣∣∣ ≤ n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣dp,qt ≤ m n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × ∫ 1 0 ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣γ dp,qt . applying hölder’s inequality with the values p = 2 γ and q = 2 2−γ , we get following inequality,∫ 1 0 ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣γ dp,qt ≤ {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 {∫ 1 0 dp,qt }2−γ 2 = {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 using this, we get |kα,βn,m(f; x,p,q) −f(x)| ≤ m n+m∑ k=0 [ n + m k ] p,q n+m−k−1∏ s=0 (ps −qs x bn ) ( x bn )k × {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 = m n+m∑ k=0 {∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 wn,k(p,q; x), where wn,k(p,q; x) = [ n + m k ] p,q ∏n+m−k−1 s=0 (p s−qs x bn ) ( x bn )k . again using hölder’s inequality with p = 2 γ and q = 2 2−γ , we have |kα,βn,m(f; x,p,q) −f(x)| ≤ m { n+m∑ k=0 ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt wn,k(p,q; x) }γ 2 { n+m∑ k=0 wn,k(p,q; x) }2−γ 2 on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 35 = m { n+m∑ k=0 wn,k(p,q; x) ∫ 1 0 ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 dp,qt }γ 2 = m(µn,p,q(x)) γ/2, where (µn,p,q(x)) γ/2 = kα,βn,m((t−x)2; x,p,q). � in order to obtain rate of convergence in terms of modulus of continuity ω(f; δ), we assume that for any uniformly continuous f ∈ cb[0,∞) and x ≥ 0, modulus of continuity of f is given by (4.1) ω(f; δ) = max |t−x|≤δ t,x∈[0,∞) |f(t) −f(x)|. thus it implies for any δ > 0 (4.2) |f(x) −f(y)| ≤ ω(f; δ) ( |x−y| δ + 1 ) , is satisfied. theorem 6. if f ∈ cb[0,∞), we have |kα,βn,m(f; x,p,q) −f(x)| ≤ 2ω(f; √ µn,p,q(x)), where ω(f; ·) is modulus of continuity of f and λn,p,q(x) be the same as in theorem 5. proof. using triangular inequality, we get |kα,βn,m(f; x,p,q) −f(x)| = ∣∣∣∣∣ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ( f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) )∣∣∣∣∣ ≤ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ∣∣∣∣f ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn ) −f(x) ∣∣∣∣ , now using (4.2) and hölder’s inequality, we get |kα,βn,m(f; x,p,q) −f(x)| = n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn )  |(1−t)[k]p,q+[k+1]p,qt+α[n+1]p,q+β bn −x| δ + 1  ω(f; δ) ≤ ω(f; δ) n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) + ω(f; δ) δ n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) ∣∣∣∣(1 − t)[k]p,q + [k + 1]p,qt + α[n + 1]p,q + β bn −x ∣∣∣∣ = ω(f; δ) + ω(f; δ) δ { n+m∑ k=0 [ n + m k ] p,q ( x bn )k n+m−k−1∏ s=0 (ps −qs x bn ) × ( (1 − t)[k]p,q + [k + 1]p,qt + α [n + 1]p,q + β bn −x )2 }12 = ω(f; δ) + ω(f; δ) δ { kα,βn,m((t−x) 2; x,p,q) }1/2 . now choosing δ = µn,p,q(x) as in theorem 5, we have |kα,βn,m(f; x,p,q) −f(x)| ≤ 2ω(f; √ µn,p,q(x)). � 36 mishra and pandey now let us denote by c2b[0,∞) the space of all functions f ∈ cb[0,∞) such that f ′,f′′ ∈ cb[0,∞). let ‖f‖ denote the usual supremum norm of f . the classical peetre’s k-functional and the second modulus of smoothness of the function f ∈ cb[0,∞) are defined respectively as k(f,δ) := inf g∈c2 b [0,∞) [‖f −g‖ + δ‖g′′‖] and ω2(f,δ) = sup 0<h<δ, x,x+h∈i |f(x + 2h) − 2f(x + h) + f(x)|, where δ > 0. it is known that [see [4], p. 177] there exists a constant a > 0 such that (4.3) k(f,δ) ≤ aω2(f,δ). theorem 7. let x ∈ [0,bn] and f ∈ cb[0,∞). then, for fixed p ∈ n0, we have |kα,βn,m(f; x,p,q) −f(x)| ≤ cω2(f, √ αn,p,q(x)) + ω(f,βn,p,q(x)) for some positive constant c, where αn,p,q(x) = [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q + (p + 2q − 1)2 [2]2p,q } [n + m]2p,q ([n + 1]p,q + β)2 − 4 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 2 ] x2 + [{ 1 + 2q [2]p,q + q2 − 1 [3]p,q + 2 (p + 2q − 1) [2]2p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q +4 α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 − 4 (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) − 4 α ([n + 1]p,q + β) ] bnx + [ (p2 x bn + 1 − x bn )n+mp,q [3]p,q + (p x bn + 1 − x bn )2n+2mp,q [2]2p,q + 4 α [2]p,q (p x bn + 1 − x bn )n+mp,q + 2α 2 ] b2n ([n + 1]p,q + β)2 ,(4.4) and (4.5) βn,p,q(x) = ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ) . proof. consider an auxiliary operator k∗n,m(f; x,p,q) : cb[0,∞) → cb[0,∞) by (4.6) k∗n,m(f; x,p,q) := k α,β n,m(f; x,p,q) −f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) + f(x). then by lemma 2 we get k∗n,m(1; x,p,q) = 1, k∗n,m((t−x); x,p,q) = 0. (4.7) for given g ∈ cb[0,∞), it follows by the taylor formula that g(y) −g(x) = (y −x)g′(x) + ∫ y x (y −u)g′′(u) du. taking into account 4.6 and using 4.7, we get |k∗n,m(g; x,p,q) −g(x)| = |k ∗ n,m(g(y) −g(x); x,p,q)| = ∣∣∣∣g′(x)k∗n,m((y −x); x,p,q) + k∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣ = ∣∣∣∣k∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣ then by 4.6 |k∗n,m(g; x,p,q) −g(x)| = ∣∣∣∣∣k∗n,m (∫ y x (y −u)g′′(u) du; x,p,q ) − ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 37( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣ ≤ ∣∣∣∣∣k∗n,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣∣∣+ ∣∣∣∣∣ ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣. since, ∣∣∣kα,βn,m (∫ y x (y −u)g′′(u) du; x,p,q )∣∣∣≤‖g′′(x)‖ kα,βn,m((y −x)2; x,p,q) and ∣∣∣∣∣ ∫ ( [2]p,qα+(p xbn +1− xbn )n+mp,q [2]p,q([n+1]p,q+β) bn+ (p+2q−1)[n+m]p,q [2]p,q([n+1]p,q+β) x ) x( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x−u ) g′′(u) du ∣∣∣∣∣ ≤ ‖g′′‖ [ [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ]2 , we get |k∗n,m(g; x,p,q) −g(x)| ≤ ‖g ′′‖kα,βn,m((y −x) 2; x,p,q) + ‖g′′‖ [ [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ]2 . hence lemma 2 implies that |k∗n,m(g; x,p,q) −g(x)|(4.8) ≤‖g′′‖ [( α2 ([n + 1]p,q + β)2 + 2α [2]p,q([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + (p2 x bn + 1 − x bn )n+mp,q [3]p,q([n + 1]p,q + β)2 ) b2n + ( 2α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 + { 1 + 2q [2]p,q + q2 − 1 [3]p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m−1 p,q − 2α ([n + 1]p,q + β) − 2(p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) ) bnx + ({ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q } [n + m]p,q[n + m− 1]p,q ([n + 1]p,q + β)2 − 2 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 1 ) x2 + ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x )2] .(4.9) since k∗n,m(f; x,p,q) ≤ 3‖f‖, considering 4.4 and 4.5,for all f ∈ cb[0,∞) and g ∈ c2b[0,∞), we may write from 4.8 that |kα,βn,m(f; x,p,q) −f(x)| ≤ |k ∗ n,m(f −g; x,p,q) − (f −g)(x)| + |k ∗ n,m(g; x,p,q) −g(x)| + ∣∣∣∣∣f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) −f(x) ∣∣∣∣∣ ≤ 4‖f −g‖ + αn,p,q(x)‖g′‖ + ∣∣∣∣∣f ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) x ) −f(x) ∣∣∣∣∣ ≤ 4‖f −g‖ + αn,p,q(x)‖g′‖ + ω(f,βn,p,q(x)), 38 mishra and pandey which yields that |kα,βn,m(f; x,p,q) −f(x)| ≤ 4k(f,αn,p,q(x)) + ω(f,betan,p,q(x)) ≤ cω2(f, √ αn,p,q(x)) + ω(f,βn,p,q(x)), where αn,p,q(x) = [{ 1 + 2(q − 1) [2]p,q + (q − 1)2 [3]p,q + (p + 2q − 1)2 [2]2p,q } [n + m]2p,q ([n + 1]p,q + β)2 − 4 (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) + 2 ] x2 + [{ 1 + 2q [2]p,q + q2 − 1 [3]p,q + 2 (p + 2q − 1) [2]2p,q } [n + m]p,q ([n + 1]p,q + β)2 ( p x bn + 1 − x bn )n+m p,q + 4 α(p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β)2 − 4 (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) − 4 α ([n + 1]p,q + β) ] bnx + [ (p2 x bn + 1 − x bn )n+mp,q [3]p,q + (p x bn + 1 − x bn )2n+2mp,q [2]2p,q + 4 α [2]p,q (p x bn + 1 − x bn )n+mp,q + 2α 2 ] b2n ([n + 1]p,q + β)2 , and βn,p,q(x) = ( [2]p,qα + (p x bn + 1 − x bn )n+mp,q [2]p,q([n + 1]p,q + β) bn + ( (p + 2q − 1)[n + m]p,q [2]p,q([n + 1]p,q + β) − 1 ) x ) . hence we get the result. � references [1] t. acar, (p,q)-generalization of szász-mirakyan operators, math. meth. appl. sci., 2015 (2015), doi: 10.1002/mma.3721. [2] i. m. burban, a. u. klimyk, (p,q)-differentiation, (p,q)-integration, and (p,q)-hypergeometric functions related to quantum groups, integral transforms and special functions, 2 (1994), 15-36. [3] n. l. braha, h. m. srivastava and s. a. mohiuddine, a korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, appl. math. comput. 228 (2014), 162-169. [4] r.a. devore, g.g. lorentz, constructive approximation, springer, berlin,1993. [5] a. d. gadjiev, the convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of p.p. korovkin, dokl. akad. nauk sssr, 218 (5), 1001-1004. english translation in sov. math. dokl., 15 (1974), 1433-1436. [6] a.r. gairola, deepmala, l.n. mishra, rate of approximation by finite iterates of q-durrmeyer operators, proceedings of the national academy of sciences, india section a: physical sciences, 2016 (2016), doi: 10.1007/s40010016-0267-z. [7] e. ibikli, approximation by bernstein-chlodowsky polynomials, hacettepe journal of mathematics and statistics, 32 (2003), 1-5. [8] v.n. mishra, k. khatri, l.n. mishra, deepmala; inverse result in simultaneous approximation by baskakovdurrmeyer-stancu operators, journal of inequalities and applications, 2013 (2013), article id 586. [9] v.n. mishra, k. khatri, l.n. mishra; on simultaneous approximation for baskakov-durrmeyer-stancu type operators, journal of ultra scientist of physical sciences, 24 (2012), 567-577. [10] v.n. mishra, k. khatri, l.n. mishra; statistical approximation by kantorovich type discrete q−beta operators, advances in difference equations 2013 (2013), article id 345. [11] v.n. mishra, s. pandey, on (p,q) baskakov-durrmeyer-stancu operators, arxiv:1602.06719 [math.ca] [12] v.n. mishra, s. pandey, (p,q)-szász-mirakyan-baskakov-stancu type operators, arxiv:1602.06312 [math.ca]. [13] m. mursaleen, k.j. ansari, a. khan, on (p,q)-analogue of bernstein operators, appl. math. comput., 266 (2015), 874-882. [14] m. mursaleen, k.j. ansari, a. khan, some approximation results by (p,q)-analogue of bernstein-stancu operators, appl. math. comput., 264 (2015), 392-402. [15] m. mursaleen, f. khan, approximation by kantorovich type (p,q)-bernstein schurer operators, arxiv:1506.02492 [math.ca]. [16] m. mursaleen, md. nasiruzzaman, a. khan, k.j. ansari, some approximation results on bleimann-butzer-hahn operators defined by (p,q)-integers, arxiv:1505.00392, [math.ca]. [17] m. mursaleen, a. khan, h. m. srivastava and k. s. nisar, operators constructed by means of q-lagrange polynomials and a-statistical approximation, appl. math. comput. 219 (2013), 6911-6818. [18] h. m. srivastava, some generalizations and basic (or q-) extensions of the bernoulli, euler and genocchi polynomials, appl. math. inform. sci. 5 (2011), 390-444. [19] h. m. srivastava and j. choi, zeta and q-zeta functions and associated series and integrals, elsevier science publishers, amsterdam, london and new york, 2012. on chlodowsky variant of (p,q) kantorovich-stancu-schurer operators 39 [20] t. vedi and mehmet ali özarslan, chlodowsky-type q-bernstein-stancu-kantorovich operators, j. inequal. appl., 2015 (2015), article id 91. [21] a. wafi, n. rao, deepmala, approximation properties by generalized-baskakov-kantorovich-stancu type operators, appl. math. inf. sci. lett., 4 (2016), 1-8. 1department of applied mathematics & humanities, sardar vallabhbhai national institute of technology, ichchhanath mahadev dumas road, surat -395 007 (gujarat), india 2l. 1627 awadh puri colony beniganj, phase -iii, opp. industrial training institute, ayodhya main road, faizabad-224 001, (uttar pradesh), india ∗corresponding author: vishnunarayanmishra@gmail.com international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 29-38 http://www.etamaths.com reverse of the triangle inequality in hilbert c∗-modules nordine bounader, abdellatif chahbi∗ and samir kabbaj abstract. in this paper we prove the reverse of triangle inequality via selberg’s inequalities in the framework of hilbert c∗-modules. 1. introduction in 1966, diaz and matcalf [4] proved the following reverse triangle inequality in setting of hilbert spaces as follows . theorem 1.1. let x1, ...,xn be vectors in a hilbert space h. if e is a unit vector of h such that 0 ≤ r ≤ re〈xi,e〉||xi|| for some r ∈ r and each 1 ≤ i ≤ n, then r n∑ j=1 ‖xi‖≤ ∥∥∥∥∥ n∑ i=1 xi ∥∥∥∥∥ a number of mathematicians have represented several refinements of the reverse triangle inequality in hilbert spaces and normed spaces, see[1, 2, 5, 8, 9, 12, 13] recently, m. khosravi, h. mahyar and m.s. moslehian [12] obtained the following reverse of the triangle inequality in the framework of hilbert c∗-modules. theorem 1.2. let x be a hilbert a-module and e1, ...,em ∈ x be a family of vectors with 〈ei,ej〉 = 0 (1 ≤ i 6= j ≤ m) and ||ei|| = 1 (1 ≤ i ≤ m). if the vectors x1, . . . ,xn in x satisfy the conditions re〈ek,xj〉≥ ρk ‖xj‖ , im〈ek,xj〉≥ µk ‖xj‖ for j ∈{1, . . . ,n} , k ∈{1, . . . ,m} , where ρk,µk ∈ [0,∞) k ∈{1, . . . ,m} , then ( m∑ k=1 (ρ2k + µ 2 k)) 1 2 n∑ j=1 ‖xj‖≤ ∥∥∥∥∥∥ n∑ j=1 xj ∥∥∥∥∥∥ . in [3] we obtained an extension of selberg’s inequality in the framework of hilbert c∗-modules. the goal of this paper is to show the reverse of triangle inequality via a extension of selberg’s inequality in the framework of hilbert c∗-modules. our results are extensions of theorem 2.1 and corollary 2.3 obtained by dragomir in [5] and theorem 9 obtained by fujii and nakamoto see [9] in the setting of hilbert c∗-modules. 2010 mathematics subject classification. primary 46l08; secondary 26d15, 46l05. key words and phrases. triangle inequality; reverse inequality; hilbert c∗−module; c∗−algebra. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 29 30 bounader, chahbi and kabbaj 2. preliminaries in this section we briefly recall the definitions and examples of hilbert c∗modules. for information about hilbert c∗-module, we refer to [10, 13]. our reference for c∗-algebras as [15]. let a be a c∗-algebra (not necessarily unitary) and x be a complex linear space. definition 2.1. a pre-hilbert a-module is a right a-module x equipped with a sesquilinear map 〈., .〉 : x ×x →a satisfying (1) 〈x,x〉≥ 0;〈x,x〉 = 0 if and only if x = 0 for all x in x , (2) 〈x,αy + βz〉 = α〈x,y〉 + β(〈x,z)〉 for all x,y,z in x ,α,β in c, (3) 〈x,y〉 = 〈y,x〉∗ for all x,y in x , (4) 〈x,y.a〉 = 〈x,y〉a for all x,y in x , a in a. the map 〈., .〉 is called an a-valued inner product of x and for x ∈x , we define ||x|| = ||〈x,x〉|| 1 2 as a norm on x , where the latter is a norm in the c∗−algebra a. this norm makes x into a right normed module over a. a pre-hilbert module x is called a hilbert a-module if it is complete with respect to its norm. two typical examples of hilbert c∗-modules are as follows: (i) every hilbert space is a hilbert c∗-module. (ii) every c∗algebra a is a hilbert a -module via 〈a,b〉 = a∗b (a,b ∈a). notice that the inner product structure of a c∗-algebra is essentially more complicated than complex numbers. one may define an a -valued norm |.| by |x| = 〈x,x〉 1 2 . clearly, ‖x‖ = ‖|x|‖ for each x ∈x . it is known that |.| does not satisfy the triangle inequality in general (see [[13], p.4]). we also use the elementary c∗-algebra theory, we use the following property: if a ≤ b then a 1 2 ≤ b 1 2 ,where a, b are positive elements of c∗-algebra a, and the relation 1 2 (aa∗ + a∗a) = re(a)2 + im(a)2 where a is an arbitrary element of a 3. main result let x be a right hilbert a-module, which is an algebraic left a-module satisfying: 〈x,ay〉 = a〈x,y〉 for all x,y ∈x and a ∈a. for example if a is a unital c∗-algebra and i is a commutative right ideal of a, then i is a right hilbert module over a and 〈x,ay〉 = x∗(ay) = ax∗y (x,y ∈i,a ∈a). for a reverse of triangle inequality, we use the following lemma. lemma 3.1. let x be a right hilbert amodule which is an algebraic left a-module and y1, . . . ,ym be a non zero vectors in x. if x ∈x then (3.1) m∑ j=1 |〈yj,x〉| 2∑m k=1 ‖〈yj,yk〉‖ ≤ |x|2 and (3.2) m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ ≤ |x|2 reverse of the triangle inequality 31 proof. inequality (3.1) is proved in [[3], theorem 3.1 ]. next we prove the inequality (3.2). let αj ∈a,j = 1, . . . ,n. we know that 0 ≤ ∣∣∣∣∣∣x− m∑ j=1 αjyj ∣∣∣∣∣∣ 2 = 〈 x− m∑ j=1 αjyj,x− m∑ j=1 αjyj 〉 = 〈x,x〉− 〈 x, m∑ j=1 αjyj 〉 − 〈 m∑ j=1 αjyj,x 〉 + 〈 m∑ j=1 αjyj, m∑ j=1 αjyj 〉 = |x|2 − m∑ j=1 αj 〈x,yj〉− m∑ j=1 〈yj,x〉α∗j + m∑ j,k=1 αj 〈yk,yj〉α∗k = |x|2 − m∑ j=1 αj 〈x,yj〉− m∑ j=1 〈yj,x〉α∗j + 1 2 m∑ j,k=1 (αj 〈yk,yj〉α∗k + αk 〈yj,yk〉α ∗ j ). it follows from [[6], lemma 3.2] that αj 〈yj,yk〉α∗k + αk 〈yk,yj〉α ∗ j ≤ ∣∣α∗j∣∣2 ‖〈yj,yk〉‖ + |α∗k|2 ‖〈yj,yk〉‖ , so ∣∣∣∣∣∣x− m∑ j=1 αjyj ∣∣∣∣∣∣ 2 ≤ |x|2 − m∑ j=1 αj 〈x,yj〉− m∑ j=1 〈yj,x〉α∗j + 1 2 m∑ j,k=1 (|α∗j| 2| 〈yj,yk〉 | + |α∗k| 2 ‖〈yk,yj〉‖), take αj = 〈yj,x〉∑m k=1 ‖〈yj,yk〉‖ , a simple calculation shows that ∣∣∣∣∣∣x− m∑ j=1 αjyj ∣∣∣∣∣∣ 2 ≤ |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ + 1 2 m∑ j,k=1 |〈x,yj〉| 2 ‖〈yj,yk〉‖ ( ∑m k=1 ‖〈yj,yk〉‖)2 + 1 2 n∑ j,k=1 |〈x,yj〉| 2 ‖〈yj,yk〉‖ ( ∑m k=1 ‖〈yj,yk〉‖)2 . 32 bounader, chahbi and kabbaj since |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ + 1 2 m∑ j,k=1 |〈x,yj〉| 2 ‖〈yj,yk〉‖ ( ∑m k=1 ‖〈yj,yk〉‖)2 + 1 2 m∑ j,k=1 |〈x,yj〉| 2 ‖〈yj,yk〉‖ ( ∑m k=1 ‖〈yj,yk〉‖)2 = |x|2 − 2 m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ + n∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ = |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ . it follows that ∣∣∣∣∣∣x− m∑ j=1 αjyj ∣∣∣∣∣∣ 2 ≤ |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ , hence |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ ≥ 0. the proof is then completed. � corollary 3.2. let x be a right hilbert a− module which is an algebraic left a-module, then : | 〈y,x〉 |2 ≤‖y‖2 |x|2 and | 〈x,y〉 |2 ≤‖y‖2 |x|2 . corollary 3.3. let x be a right hilbert a− module which is an algebraic left amodule. if y1, . . . ,yn is a sequence of unit vectors in x such that 〈yj,yk〉 = 0 for 1 ≤ j 6= k ≤ n. then m∑ j=1 |〈yj,x〉| 2 ≤ |x|2 . and m∑ j=1 |〈x,yj〉| 2 ≤ |x|2 . theorem 3.4. let x be a right hilbert amodule which is an algebraic left amodule, x1, . . . ,xn and y1, . . . ,ym be a non zero vectors in x such that there exist the non-negative real numbers ρj,µj,j ∈{1, . . . ,m} with (3.3) re〈yj,xi〉≥ ρj ‖xi‖‖yj‖ , im〈yj,xi〉≥ µj ‖xi‖‖yj‖ for each i ∈{1, . . . ,n} , j ∈{1, . . . ,m} . then ( m∑ j=1 (ρ2j + µ 2 j)‖yj‖ 2∑m k=1 ‖〈yj,yk〉‖ ) 1 2 n∑ i=1 |xi| ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ . reverse of the triangle inequality 33 proof. by (3.3), we have ( n∑ i=1 re〈yj,xi〉)2 + ( n∑ i=1 im〈yj,xi〉)2 ≥ ρ2j|yj| 2( n∑ i=1 |xi|)2 + µ2kj ‖yj‖ 2 ( n∑ i=1 ‖xi‖)2 = (ρ2j + µ 2 j)|yj| 2( n∑ i=1 |xi|)2. by combining the above inequality and this equality 1 2 (|〈 ∑n i=1 xi,yj〉| 2 + |〈yj, ∑n i=1 xi〉| 2 ) = ( ∑n i=1 re〈yj,xi〉) 2 + ( ∑n i=1 im〈yj,xi〉) 2, we deduce 1 2 ∑m j=1 |〈yj, ∑n i=1 xi〉| 2∑ m k=1 ‖〈yj,yk〉‖ + 1 2 ∑m j=1 |〈∑ni=1 xi,yj〉|2∑ m k=1 ‖〈yj,yk〉‖ ≥ ( ∑m j=1 (ρ2j +µ 2 j )‖yj‖ 2∑ m k=1 ‖〈yj,yk〉‖ )( ∑n i=1 ‖xi‖) 2. apply lemma 3.1, we get 1 2 m∑ j=1 |〈yj, ∑n i=1 xi〉| 2∑m k=1 ‖〈yj,yk〉‖ + 1 2 m∑ j=1 |〈 ∑n i=1 xi,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ ≤ | n∑ i=1 xi|2. then ( m∑ j=1 (ρ2j + µ 2 j)‖yj‖ 2∑m k=1 ‖〈yj,yk〉‖ )( n∑ i=1 ||xi| |)2 ≤ ∥∥∥∥∥ n∑ i=1 xi ∥∥∥∥∥ 2 . and since |x| ≤ ‖x‖ and |x|2 ≤‖x‖2 for all x ∈x , then ( m∑ j=1 (ρ2j + µ 2 j)‖yj‖ 2∑m k=1 ‖〈yj,yk〉‖ )( n∑ i=1 |xi|)2 ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ 2 . the desired result follows by taking the square roots. � remark 3.5. if the first condition of (3.3) is the only one available, then ( m∑ j=1 ρ2j ‖yj‖ 2∑m k=1 ‖〈yj,yk〉‖ ) 1 2 n∑ i=1 |xi| ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ . corollary 3.6. let x be a right hilbert a−module which is an algebraic left amodule, x1, . . . ,xn and y1, . . . ,ym be a non zero vectors in x such that there exist the non-negative real numbers ρj,µj,j ∈{1, . . . ,m} with re〈yj,xi〉≥ ρj ‖xi‖‖yj‖ , im〈yj,xi〉≥ µj ‖xi‖‖yj‖ . then ( m∑ j=1 (ρ2j + µ 2 j)|yj| 2 max1≤j≤m |yj|2 + (m− 1) maxk 6=j | 〈yj,yk〉 | ) 1 2 n∑ i=1 |xi| ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ . proof. it is easy to show that m∑ k=1 ‖〈yj,yk〉‖≤ max 1≤j≤m ‖yj‖ 2 + (m− 1) max j 6=k ‖〈yj,yk〉‖ . we thus have that 1 max1≤j≤m‖yj‖ 2 + (m− 1) maxj 6=k ‖〈yj,yk〉‖ ≤ 1∑m k=1 ‖〈yj,yk〉‖ , 34 bounader, chahbi and kabbaj and (ρ2j + µ 2 j)‖yj‖ 2 max1≤j≤m‖yj‖ 2 + (m− 1) maxj 6=k ‖〈yj,yk〉‖ ≤ (ρ2j + µ 2 j)‖yj‖ 2∑m k=1 ‖〈yj,yk〉‖ . consequently ( ∑n j=1 (ρ2j +µ 2 j )‖yj‖ 2 max1≤j≤n‖yj‖2+(m−1) maxj 6=k‖〈yj,yk〉‖ ) 1 2 ∑n i=1 |xi| ≤ ( ∑n j=1 (ρ2j +µ 2 j )‖yj‖ 2∑ n k=1‖〈yj,yk〉‖ ) 1 2 ∑n i=1 |xi| . we apply the theorem 3.4 to get the result � the next corollary is the theorem 2.5 in [12]. corollary 3.7. let x be a hilbert a-module, x1, . . . ,xn be a family of vectors in x and ej be a unitary orthogonal vectors for j ∈ {1, . . . ,m} in x such that there exist the real numbers ρj,µj,j ∈{1, . . . ,m} with re〈ρjxi,ej〉≥ ρ2j ‖xi‖ , im〈µjxi,ej〉≥ µ 2 j ‖xi‖ for each i ∈{1, . . . ,n} , j ∈{1, . . . ,m} . then ( m∑ j=1 (ρ2j + µ 2 j)) 1 2 n∑ j=1 ‖xi‖≤ ∥∥∥∥∥ n∑ i=1 xi ∥∥∥∥∥ . corollary 3.8. let x be a right hilbert a−module which is an algebraic left amodule, x1, . . . ,xn and y1 . . . ,ym be a non zero vectors in x , such that there exist the non-negative real number in [0; 1] pj,qj,j ∈{1, . . . ,m} with (3.4) ||xi||2 − 2re〈yj,xi〉 + ||yj||2 ≤ p2j ≤ ||yj|| 2 and (3.5) ||xi||2 − 2 im〈yj,xi〉 + ||yj||2 ≤ p2j ≤ ||yj|| 2 for each i ∈{1, . . . ,n} , j ∈{1, . . . ,m} . then ( m∑ j=1 (2‖yj‖ 2 −p2j −qj 2)∑m k=1 ‖〈yj,yk〉‖ ) 1 2 n∑ i=1 |xi| ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ . proof. by the inequality (3.4), we get ||xi||2 + ||yj||2 −p2j ≤ 2re〈yj,xi〉 . since ||yj||2 −p2j ≥ 0, then 2||xi|| √ ||yj||2 −p2j ≤ ||xi|| 2 + ||yj||2 −p2j ≤ 2re〈yj,xi〉 . this implies that re〈yj,xi〉≥ √ ||yj||2 −p2j ||yj|| ||yj||||xi||. even from (3.5), we get im〈yj,xi〉≥ √ ||yj||2 −q2j ||yj|| ||yj||||xi||, and if we let ρj = √ ||yj||2−p2j ||yj and µj = √ ||yj||2−p2j ||yj in theorem 3.4, then by simple computation, we get the desired result. � reverse of the triangle inequality 35 the following lemma gives a refinement of selberg’s inequality in a right hilbert a− module which is an algebraic left a-module. lemma 3.9. let x be a right hilbert a− module which is an algebraic left amodule and y1, . . . ,ym be a non zero vectors in x . if x ∈x then (3.6) | 〈y,x〉 |2 + m∑ j=1 |〈yj,x〉| 2∑m k=1 ‖〈yj,yk〉‖ ||y||2 ≤ |x|2 ||y||2, (3.7) | 〈x,y〉 |2 + m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ ||y||2 ≤ |x|2 ||y||2, (3.8) | 〈x,y〉 |2 + m∑ j=1 |〈yj,x〉| 2∑m k=1 ‖〈yj,yk〉‖ ||y||2 ≤ |x|2 ||y||2, and (3.9) | 〈y,x〉 |2 + m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖ ||y||2 ≤ |x|2 ||y||2. proof. inequality ( 3.6) is proved in [ [3],theorem 3.3]. now we prove the inequality (3.7), let u = x− m∑ j=1 αjyj where αj = 〈yj,x〉∑n k=1 ‖〈yj,yk〉‖ . according to the proof of lemma 3.1, we have |u|2 = |x− m∑ j=1 αjyj|2 ≤ |x|2 − m∑ j=1 | 〈yj,x〉 |2∑m k=1 | 〈yj,yk〉 | . hence it follows that ‖y‖2  |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖   ≥‖y‖2 |u|2 . applying cauchy schwartz inequality, we get ‖y‖2 |u|2 ≥ |〈u,y〉|2 . and since 〈y,yj〉 = 0, so |〈u,y〉|2 = ∣∣∣∣∣∣ 〈 x− m∑ j=1 αjyj,y 〉∣∣∣∣∣∣ 2 = |〈x,y〉|2 . 36 bounader, chahbi and kabbaj it follows that ‖y‖2  |x|2 − m∑ j=1 |〈x,yj〉| 2∑m k=1 ‖〈yj,yk〉‖   ≥ |〈x,y〉|2 , which completes the proof of the inequality (3.7). similarly, we can get inequalities (3.8) and (3.9) . � lemma 3.10. let x be a hilbert a module, y1, · · · , ym,y be non zero vectors in x and x1, · · · , xn ∈x such that such that there exist the real numbers ρj,µj,j ∈ {1, · · · ,m} with (3.10) 0 ≤ ρj ‖xi‖||yj|| ≤ re〈yj,xi〉 , 0 ≤ µj ‖xi‖||yj|| ≤ im〈yj,xi〉 and 〈y,yj〉 = 0, for each i ∈{1, . . . ,n} , j ∈{1, . . . ,m} . then (3.11) |(y, n∑ i=1 xi)|2 + ( m∑ k=1 ρ2j + µ 2 j cj ||y2j )( n∑ i=1 |xi|)2||y||2 ≤ | n∑ i=1 xi|2||y||2, and (3.12) |( n∑ i=1 xi,y)|2 + ( m∑ j=1 ρ2j + µ 2 j cj ||yj||2)( n∑ i=1 |xi|)2||y||2 ≤ | n∑ i=1 xi|2||y||2, where cj = ∑m k=1 ‖〈yj,yk〉‖ . proof. let x = ∑n i=1 xi, from (3.10), we get ||y||2{|x|2 − ∑m j=1 ρ2j +µ 2 j cj ( ∑n i=1 ||xi||) 2} ≥ ||y||2{|x|2 − ∑m j=1 re〈x,yj〉2+im〈x,yj〉2 cj }, since ||y||2{|x|2− ∑m j=1 re〈x,yj〉2+im〈x,yj〉2 cj } = ‖y‖2 {|x|2−1 2 ∑m j=1 |〈x,yj〉|2 cj −1 2 ∑m j=1 〈yj,x〉2 cj }. then, from (3.6) and (3.9), we get ‖y‖2 {|x|2 − 1 2 m∑ j=1 | 〈x,yj〉 |2 cj − 1 2 m∑ j=1 〈yj,x〉 2 cj }≥ |〈y,x〉 |2, it follows that ||y||2{|x|2 − m∑ j=1 ρ2j + µ 2 j cj (‖x1‖ + · · · + ‖xn‖)2}≥ |〈y,x〉| 2 . by using (3.7) and (3.8) and by similar argument, we get (3.12). � theorem 3.11. let x be a hilbert a module, y1, · · · , ym be non zero vectors in x and x1, · · · , xn ∈ x such that such that there exist the real numbers a,b,ρj,µj,j ∈{1, · · · ,m} with (3.13) 0 ≤ ρj ‖xi‖||yj|| ≤ re〈yj,xi〉 , 0 ≤ µj ‖xi‖||yj|| ≤ im〈yj,xi〉 , (3.14) 0 ≤ a‖xi‖||y|| ≤ re〈y,xi〉 , 0 ≤ b‖xi‖||yj|| ≤ im〈y,xi〉 reverse of the triangle inequality 37 and 〈y,yj〉 = 0, for each i ∈{1, . . . ,n} , j ∈{1, . . . ,m} . then (a2 + b2 + m∑ j=1 ρ2j + µ 2 j cj )||y2j || 1 2 (|x1| + · · · + |xn|) ≤ | n∑ i=1 xi|. where cj = ∑m k=1 ‖〈yj,yk〉‖ . proof. from (3.11) and (3.12), we get 1 2 (| 〈 ∑n i=1 xi,y〉 | 2 + | 〈y, ∑n i=1 xi〉 | 2) +( ∑m j=1 ρ2j +µ 2 j cj ||y||2)( ∑n i=1 |xi|) 2||y||2 ≤ | ∑n i=1 xi| 2||y||2, since 1 2 (| 〈 ∑n i=1 xi,y〉 | 2 + | 〈y, ∑n i=1 xi〉 | 2) ≥ (re〈y,x〉)2 + (im〈y,x〉)2 applying (3.13) and (3.14) and taking the square root, the desired result follows. � remark 3.12. if in theorem 3.11 y1, . . . ,ym is a sequence orthonormal vectors, then (a2 + b2 + m∑ j=1 (ρ2j + µ 2 j)) 1 2 ( n∑ i=1 |xi|) ≤ ∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ . this inequality is an extension of diaz-metcalf [4] inequality in c∗-module. references [1] a.h. ansari and m.s.moslehian, refinement of reverse triangle inequality in inner product spaces.j. inequalities in pure and applied math., 6(2005). article id 64. [2] a.h. ansari and m.s.moslehian, more on reverse triangle inequality in inner product spaces. inter j. math. math sci, 18(2005), 2883-2893. [3] n.bounader and a.chahbi, selberg type inequalities in hilbert c*-modules. int. journal of math. analysis, 7 (2013), 385-391 [4] j.b. diaz and f.t. metcalf,a complementary triangle inequality in hilbert and banach spaces. proc. amer. math. soc., 17(1966), 88-97. [5] s. dragomir, reverse of the triangle inequality via selberg’s and boas-bellman’s inequalities.facta universitatis (nis), ser. math. inform. 21(2006), 29c39 [6] s. s. dragomir, m. khosravi and m. s. moslehian, bessel type inequality in hilbert c∗modules. linear and multilinear algebra, 8(2010), 967-975. [7] a. m. fink, d.s. mitrinovic and j. e.pecaric classical and new inequalities in analysis. kluwer academic, dordrecht, 1993. [8] m. fujii, selberg inequalty.(1991), 70-76. [9] m. fujii and r. nakamoto, simultaneous extensions of selberg inequality and heinz-katofuruta inequality. nihonkai math. j. 9(1998), 219-225. [10] i. kaplansky, modules over operateur algebras. amer. j. math, (1953), 839-858. [11] m.kato, k.s. saito and t. tamura, sharp triangle inequality and its reverse in banach spaces. math. inegal. appl. 10(2007), 451-460. [12] m.khosravi, h.mahyar, m.s. moslehyan,reverse triangle inequality hilbert c∗-modules. j. inequal. pure appl. math, 10(2009), article 110, 11 pages. [13] e. c. lance, hilbert c∗-modules. london mathematical society lecture note series, cambridge university press, cambridge, 210(1995). [14] c.-s. lin, heinz’s inequality and bernstein’s inequality. proceedings of the american mathematical society, 125(1997), 2319-2325. [15] j.g. murphy, c∗-algebras and operator theory. academic press, boston, 1990. [16] m. nakai and t. tada, the reverse triangle inequality in normed spaces. new zealand j.math., 25(1996), 181-193. 38 bounader, chahbi and kabbaj department of mathematics, faculty of sciences, university of ibn tofail, kenitra, morocco ∗corresponding author international journal of analysis and applications volume 18, number 6 (2020), 998-1014 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-998 fuzziness and roughness in hyperquantales raees khan1,∗, maslina darus2,∗, muhammad farooq3, asghar khan3 and nasir khan1 1department of mathematics, fata university, tsd darra nmd kohat, kp, pakistan 2school of mathematical sciences, faculty of science and technology, universiti kebangsaan malaysia, 43600 bangi, selangor, malaysia 3department of mathematics, abdul wali khan university, mardan, kp, pakistan ∗corresponding authors: raeeskhan@fu.edu.pk, maslina@ukm.edu.my abstract. theories of fuzzy set and rough set are powerful mathematical tools for modelling various types of uncertainty. in this paper, we introduce the notions of bi-hyperideal, fuzzy bi-hyperideals of hyperquantales and their related properties is given. furthermore we introduce the notion of generalized rough fuzzy bi-hyperideals. moreover, we will describe the set-valued homomorphism and strong set-valued homomorphism of hyperquantales and some related properties will be study. 1. introduction the theory of rough sets was introduced by pawlak [15, 16], to deal with uncertain knowledge in information systems. the rough set theory has been emerged as another major mathematical approach for managing uncertainty that arises from inexact, noisy or incomplete information. it has turned out to be fundamentally important in artificial intelligence and cognitive sciences, especially in fields such as machine learning, knowledge acquisition, decision analysis, expert systems, pattern recognition. with the development of rough set theory, possible connections between rough sets and various algebraic systems were considered by many authors. inspired by the construction of pawlak rough set algebras and the investigation in algebraic properties received july 7th, 2020; accepted august 24th, 2020; published october 7th, 2020. 2010 mathematics subject classification. 03e72. key words and phrases. rough set; fuzzy set; hyperquantale; fuzzy bi-hyperideal; generalized rough fuzzy bi-hyperideal. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 998 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-998 int. j. anal. appl. 18 (6) (2020) 999 of rough sets in [17,18]. as a combination of algebraic structures and partially ordered structures, the theory of quantales was initiated by mulvey [20] to study the spectrum of c-algebras and the foundations of quantum mechanics. wang and zhao [22, 23] proposed the concepts of ideals and prime ideals of quantales. yang and xu [24], considered quantales as universal sets and introduced the notions of rough (prime, semi-prime, primary) ideals and prime radicals of upper rough ideals of quantales. wang in [21] studied prime radical theorem in quantales. the concept of fuzzy sets was introduced by zadeh [19] in 1965. the theory of fuzzy sets has been developed fast and has many applications in many branches of sciences. luo and wang in [25], studied roughness and fuzziness in quantales. davvaz et al. in [6] applied atanassov’s intuitionistic fuzzy set theory to quantales. in [29, 30], saqib and shabir studied relationship between generalized rough sets and quantale by using fuzzy ideals of quantale. algebraic hyperstructures represent a natural extension of classical algebraic structures and they were originally proposed in 1934 by a french mathematician marty [1], at the 8th congress of scandinavian mathematicians. one of the main reason which attracts researches towards hyperstructures is its unique property that in hyperstructures composition of two elements is a set, while in classical algebraic structures the composition of two elements is an element. thus algebraic hyperstructures are natural extension of classical algebraic structures. since then, hyperstructures are widely investigated from the theoretical point of view and for their applications to many branches of pure and applied mathematics (see [2–5, 8]). since then, there appeared many components of hyperalgebras such as hypergroups in [9], hyperrings etc in [10, 11]. konstantinidou and mittas have introduced the concept of hyperlattices in [12], fuzzy ideal of hyperlattices have been introduced in [7]. the notion of hyperlattices is a generalization of the notion of lattices and there are some intimate connections between hyperlattices and lattices. in particular, rasouli and davvaz further studied the theory of hyperlattices and obtained some interesting results [13, 14], which enrich the theory of hyperlattices. in [28], estaji and bayati studied rough sets in terms of hyperlattices. in [26], khan et al. introduced the notions of hyperideals and fuzzy hyperideals of hyperquantales. in this paper, we introduce the notions of bi-hyperideal and fuzzy bi-hyperideals of hyperquantales and give several characterizations. in addition, we will introduce the notions of generalized rough fuzzy bi-hyperideal in hyperquantales and some new properties will be obtain. 2. preliminaries a map ∗ : s ×s → p∗(s) is called hyperoperation or join operation on the set s, where s is a non-empty set and p∗(s) = p(s)\{∅} denotes the set of all non-empty subsets of s. a hyperstructure is called the pair (s,∗) where ∗ is a hyperoperation on the set s. int. j. anal. appl. 18 (6) (2020) 1000 definition 2.1. (see [26]). a hyperquantale is a complete hyperlattice q with an associative binary operation ∗ satisfying x∗ (∨ i∈i yi ) = ∨ i∈i (x∗yi) , (∨ i∈i xi ) ∗ y = ∨ i∈i (xi ∗y) for all x,y,xi,yi ∈ q (i ∈ i) where i is an index set. a hyperquantale q is called commutative if x∗y = y ∗x for all x,y ∈ q. throughout this paper, we denote the least and greatest elements of a hyperquantale denoted by ⊥ and > respectively. definition 2.2. (see [26]). let q be a hyperquantale. a non-empty subset a of q is called a left (resp. right) hyperideal of q if it satisfies the following conditions: (1) x,y ∈ a implies x∨y ⊆ a. (2) (∀ x,y ∈ q) x ∈ a and y ≤ x imply y ∈ a. (3) ∀ x ∈ q and a ∈ a, we have x∗a ⊆ a (resp. a∗x ⊆ a). a non empty subset a of q is called a two sided hyperideal or simply a hyperideal of q if it is both a left hyperideal and right hyperideal of q. definition 2.3. let q be a hyperquantale. a non-empty subset b of q is called a bi-hyperideal of q if it satisfies the following conditions: (1) x,y ∈ b implies x∨y ⊆ b. (2) x,y ∈ b implies x∗y ⊆ b. (3) (∀ x,y ∈ q) x ∈ b and y ≤ x imply y ∈ b. (4) ∀ y ∈ q and x,z ∈ b, we have x∗y ∗z ⊆ b. example 2.1. let q = {⊥,e1,e2,e3,>} and define ∗ and ∨ by the following cayley tables: ∗ ⊥ e1 e2 e3 > ⊥ {⊥} {⊥} {⊥} {⊥} {⊥} e1 {⊥} {e1} {e1} {e1} {e1} e2 {⊥} {e1} {e2} {e1} {e2} e3 {⊥} {e1} {e1} {e3} {e3} > {⊥} {e1} {e2} {e3} {>} int. j. anal. appl. 18 (6) (2020) 1001 and ∨ ⊥ e1 e2 e3 > ⊥ {⊥} {e1} {e2} {e3} {>} e1 {e1} {⊥,e1} {⊥,e2} {⊥,e3} {⊥,>} e2 {e2} {⊥,e2} {⊥,e1,e2} {⊥,>} {⊥,e3,>} e3 {e3} {⊥,e3} {⊥,>} {⊥,e1,e3} {⊥,e2,>} > {>} {⊥,>} {⊥,e3,>} {⊥,e2,>} {⊥,e1,e2,e3,>} thus all bi-hyperideals of q are {⊥} , {⊥,e1} , {⊥,e1,e2} , {⊥,e1,e3} and q. for a,b ⊆ q, we have a∗b := ⋃ {a∗ b : a ∈ a, b ∈ b} and a∨b := ⋃ {a∨ b : a ∈ a, b ∈ b}. for a ⊆ q,we denote (a] := {a ∈ q : a ≤ b for some b ∈ a}. 3. fuzzy hyperideals of hyperquantale let q be a hyperquantale. a function f from a nonempty set x to the unit interval [0, 1] is called a fuzzy subset of q. let q be a hyperquantale and f be a fuzzy subset of q. then for every t ∈ [0, 1] the set u (f; t) = {x | x ∈ q, f (x) ≥ t} . for x ∈ q, we define ax = {(y,z) ∈ q×q | x ≤ y ∗z}. definition 3.1. (see [26]). let q be a hyperquantale and f,g are any two fuzzy subsets of q. we define the product f ◦g of f and g as follows: (f ◦g) (x) =   ∨ (y,z)∈ax {f (y) ∧ g (z)} , if ax 6= ∅ 0, if ax = ∅ . for two functions f and g then f ⊆ g if and only if f (x) ≤ g (x) . let q be a hyperquantale and ∅ 6= a ⊆ q. then the characteristic function χa of a is defined as: χa : q −→ [0, 1] ,−→ χa (x) =   1 if x ∈ a0 if x /∈ a definition 3.2. (see [26]). let q be a hyperquantale. a fuzzy subset f of q is called a fuzzy subhyperquantale of q if it satisfies the following conditions: (1) (∀x,y ∈ q) ∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) . (2) (∀x,y ∈ q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . int. j. anal. appl. 18 (6) (2020) 1002 definition 3.3. (see [26]). let q be a hyperquantale. a fuzzy subset f of q is called a fuzzy left (resp. right) hyperideal of q if it satisfies the following conditions: (1) (∀x,y ∈ q) ∧ α∈x∗y f (α) ≥ f (y) (resp. ∧ α∈x∗y f (α) ≥ f (x)). (2) (∀x,y ∈ q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . (3) (∀x,y ∈ q) x ≤ y then f (x) ≥ f (y) . 4. fuzzy bi-hyperideal of hyperquantale in this section, we introduce the notion of fuzzy bi-hyperideal of hyperquantale and investigate some related properties. definition 4.1. let q be a hyperquantale. a fuzzy subset f of q is called a fuzzy bi-hyperideal of q if it satisfies the following conditions: (1) (∀x,y ∈ q) ∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) . (2) (∀x,y ∈ q) ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . (3) (∀x,y,z ∈ q) ∧ γ∈(x∗y∗z) f (γ) ≥ f (x) ∧ f (z) . (4) (∀x,y ∈ q) x ≤ y then f (x) ≥ f (y) . example 4.1. let q = {⊥,e1,e2,>} and define ∗ and ∨ by the following cayley tables: ∗ ⊥ e1 e2 > ⊥ {⊥} {⊥} {⊥} {⊥} e1 {⊥} {e1} {⊥} {e1} e2 {⊥} {⊥} {e2} {e2} > {⊥} {e1} {e2} {>} and ∨ ⊥ e1 e2 > ⊥ {⊥} {e1} {e2} {>} e1 {e1} {⊥,e1} {>} {e2,>} e2 {e2} {>} {⊥,e2} {e1,>} > {>} {e2,>} {e1,>} q let us define a fuzzy subset f : q −→ [0, 1] as follows: f (x) =   1 if x = ⊥0.4 if x ∈{e1,e2,>} then it is easy to verify that f is a fuzzy bi-hyperideal of q. int. j. anal. appl. 18 (6) (2020) 1003 theorem 4.1. let b be a non empty subset of a hyperquantale q. then b is a bi-hyperideal of q if and only if χb is a fuzzy hyperideal of q. proof. suppose that b is a hyperideal of q. let x,y ∈ q. if x,y ∈ b then x∨y ⊆ b and x∗y ⊆ b. since x,y ∈ b, we have χb (x) = χb (y) = 1, for any α ∈ x∨y ⊆ b, we have ∧ α∈x∨y χb (α) = 1 = χb (x) ∧ χb (y) . also for any β ∈ x∗y ⊆ b, we have ∧ β∈x∗y χb (β) = 1 = χb (x) ∧ χb (y) . if x /∈ b or y /∈ b. then x∨y ⊆ b or x∨y * b and x∗y ⊆ b or x∗y * b. in all the cases we have χb (x) ∧ χb (y) = 0 ≤ ∧ α∈x∨y χb (α) and χb (x) ∧ χb (y) = 0 ≤ ∧ β∈x∗y χb (β) . let now x,y ∈ q, x ≤ y. then χb (x) ≥ χb(y). in fact, if y ∈ b, then χb(y) = 1. since q 3 x ≤ y ∈ b, by hypothesis we have x ∈ b, then χb (x) = 1. thus χb (x) ≥ χb (y) . if y /∈ b, then χb(y) = 0. since x ∈ q, we have χb (x) ≥ 0 = χb(y). let x,y and z be any elements of q. if x,z ∈ b, then χb (x) = χb (z) = 1 and since for every α ∈ x ∗ y ∗ z ⊆ b, we have χb (α) = 1 = χb (x) ∧ χb (z) . thus ∧ α∈x∗y∗z χb (α) = 1 = χb (x) ∧ χb (z) . if x /∈ b or z /∈ b, then χb (x) = 0 or χb (z) = 0, and so we have χb (α) ≥ 0 = χb (x) ∧ χb (z) . thus ∧ α∈x∗y∗z χb (α) ≥ χb (x) ∧ χb (z) . therefore χb is a fuzzy bi-hyperideal of q. conversely, assume that χb is a fuzzy bi-hyperideal of q. let x,y ∈ b. then ∧ α∈x∨y χb (α) = χb (x) ∧ χb (y) = 1, and thus α ∈ x∨y ⊆ b. since x,y ∈ b. then for any z ∈ x∗y, we have ∧ z∈x∗y χa (z) ≥ χa (x) ∧ χa (y) = 1. implies that ∧ z∈x∗y χa (z) = 1. thus x∗y ⊆ a. if x ≤ y and y ∈ b, then χb (x) ≥ χb (y) = 1, implies that x ∈ b. let x,z ∈ b and y ∈ s such that for any α ∈ x∗y ∗z, we have since∧ α∈x∗y∗z χb (α) ≥ χb (x) ∧ χb (z) = 1 ∧ 1 = 1. hence for each α ∈ x ∗ y ∗ z, we have χb (α) = 1, and so α ∈ b. thus x ∗ y ∗ z ⊆ b. thus b is a bi-hyperideal of q. � theorem 4.2. let q be a hyperquantale. a fuzzy subset f of q is a fuzzy bi-hyperideal of q if and only if for each t ∈ [0, 1], u(f; t) 6= ∅ is a bi-hyperideal of q. proof. assume that u(f; t) is a bi-hyperideal of q. let x,y ∈ q such that x ≤ y. if f (y) = 0 then f (x) ≥ f (y) . if f (y) = t then y ∈ u(f; t). since x ≤ y and u(f; t) is a bi-hyperideal of q, we have x ∈ u(f; t). then f (x) ≥ t = f (y) . since u(f; t) 6= ∅ is a bi-hyperideal of q. if ∧ α∈x∗y f (α) < f (x) ∧ f (y) for some x,y ∈ q, then there exists t0 ∈ [0, 1] such that ∧ α∈x∗y f (α) < t0 ≤ f (x) ∧ f (y) , which implies that x,y ∈ u(f; t) and x∗y * u(f; t). it contradicts the fact that u(f; t) is a bi-hyperideal of q. consequently,∧ α∈x∗y f (α) ≥ f (x) ∧ f (y) for all x,y ∈ q. next we show that ∧ b∈x∨y f (α) ≥ f (x) ∧ f (y) for all x,y ∈ q. if there exist x,y ∈ q and t0 ∈ [0, 1] such that ∧ β∈x∨y f (β) < t0 ≤ f (x) ∧ f (y). then x,y ∈ u(f; t) and int. j. anal. appl. 18 (6) (2020) 1004 β ∈ x∨y * u(f; t). it is again contradicts the fact that u(f; t) is a bi-hyperideal of q. thus ∧ β∈x∨y f (β) ≥ f (x) ∧ f (y) . now let x,y,z ∈ u(f; t). then x ∗ y ∗ z ⊆ u(f; t). since x,z ∈ u(f; t). then f (x) ≥ t and f (z) ≥ t. so for any α ∈ x∗y ∗z, we have f (α) ≥ t. thus f (x) ∧ f (z) = t ≤ ∧ α∈x∗y∗z f (α). therefore f is a fuzzy bi-hyperideal of q. conversely, suppose that f be a fuzzy bi-hyperideal of q. let x,y ∈ u(f; t). then f(x) ≥ t, f(y) ≥ t. since f is a fuzzy bi-hyperideal of q, so we have ∧ α∈x∗y f (α) ≥ f(x) ∧ f (y) = t. hence f(α) ≥ t for all α ∈ x ∗ y, this implies α ∈ u(f; t) that is x ∗ y ⊆ u(f; t). as f is a fuzzy bi-hyperideal of q. then∧ w∈x∨y f (w) ≥ f (x) ∧ f (y) ≥ t. hence f (w) ≥ t for any w ∈ x ∨ y implies that w ∈ u(f; t). thus x ∨ y ⊆ u(f; t). now let x,y,z ∈ u(f; t). then f(x) ≥ t, f(y) ≥ t and f(z) ≥ t. since f is a fuzzy bi-hyperideal of q, we have ∧ β∈x∗y∗z f (β) ≥ f (x) ∧ f (z) = t. so f (β) ≥ t. hence x ∗ y ∗ z ⊆ u(f; t). let x ∈ u(f; t) and y ∈ q with y ≤ x. then t ≤ f (x) ≤ fa (y) , we get y ∈ u(f; t). therefore u(f; t) is a bi-hyperideal of q. � theorem 4.3. let {fi | i ∈ i} be a family of fuzzy bi-hyperideals of q. then f = ⋂ i∈i fi is a fuzzy bihyperideal of q where (⋂ i∈i fi ) (x) = ∧ i∈i (fi (x)) . proof. let x,y ∈ q. then, since each fi (i ∈ i) is a fuzzy bi-hyperideal of q, so ∧ α∈x∨y fi (α) ≥ fi (x) ∧ fi (y) . thus for any α ∈ x∨y, fi (α) ≥ fi (x) ∧ fi (y) , and we have f (α) = (⋂ i∈i fi ) (α) = ∧ i∈i (fi (α)) ≥ ∧ i∈i ( fi (x) ∧ fi (y) ) = (∧ i∈i (fi (x)) )∧(∧ i∈i (fi (y)) ) = (⋂ i∈i fi ) (x) ∧(⋂ i∈i fi ) (y) = f (x) ∧ f (y) , which implies that ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) . let β ∈ x∗y and ∧ β∈x∗y fi (β) ≥ fi (x) ∧ fi (y) . thus for any β ∈ x∗y, fi (β) ≥ fi (x) ∧ fi (y) . then int. j. anal. appl. 18 (6) (2020) 1005 f (β) = (⋂ i∈i fi ) (β) = ∧ i∈i (fi (β)) ≥ ∧ i∈i ( fi (x) ∧ fi (y) ) = (⋂ i∈i fi ) (x) ∧(⋂ i∈i fi ) (y) = f (x) ∧ f (y) . thus ∧ β∈x∗y f (β) ≥ f (x) ∧ f (y). now let x,y,z ∈ q. then for any γ ∈ x∗y ∗z, we have f (γ) = (⋂ i∈i fi ) (γ) = ∧ i∈i (fi (γ)) ≥ ∧ i∈i ( fi (x) ∧ f (z) ) = (∧ i∈i fi (x) )∧(∧ i∈i fi (z) ) = (⋂ i∈i fi ) (x) ∧(⋂ i∈i fi ) (z) = f (x) ∧ f (z) . thus ∧ γ∈x∗y∗z f (γ) ≥ f (x) ∧ f (z) . furthermore, if x ≤ y, then f (x) ≥ f (y) . indeed: since every fi (i ∈ i) is a fuzzy bi-hyperideal of q, it can be obtained that fi (x) ≥ fi (y) for all i ∈ i. thus f (x) = (⋂ i∈i fi ) (x) = ∧ i∈i (fi (x)) ≥ ∧ i∈i (fi (y)) = (⋂ i∈i fi ) (y) = f (y) . thus f = ⋂ i∈i fi is a fuzzy bi-hyperideal of q. � int. j. anal. appl. 18 (6) (2020) 1006 5. homomorphism and generalized rough fuzzy bi-hyperideals of hyperquantales definition 5.1. (see [27]). let x and y be two nonempty universes. let f be a set-valued mapping given by f : x −→p (y ), where p (y ) is the power set of y . then the triple (x,y,f) is referred to as a generalized approximation space or generalized rough set. any set-valued function from x to p (y ) defines a binary relation from x to y by setting ρf = {(a,b) | b ∈ f (a)}. obviously, if ρ is an arbitrary relation from x to y , then a set-valued mapping fρ : x −→p (y ) can be defined by fρ (a) = {b ∈ y | (a,b) ∈ ρ} where a ∈ x. for any set a ⊆ y , the lower and upper approximations represented by f− (a) and f+ (a) respectively, are defined as f− (a) = {a ∈ x | f (a) ⊆ a} , f+ (a) = {a ∈ x | f (a) ∩a 6= ∅} . we call the pair (f− (a) ,f+ (a)) generalized rough set, and f−,f+ are termed as lower and upper generalized approximation operators, respectively. definition 5.2. let (q1,∗1) and (q2,∗2) be two hyperquantales. a set-valued mapping f : q1 −→p∗ (q2) , where p∗ (q2) represents the collection of all nonempty subsets of q2 is called a set-valued homomorphism if, for all ai,a,b ∈ q1 (i ∈ i) , (1) f (a) ∗2 f (b) ⊆ f (a∗1 b) . (2) ∨ i∈i f (ai) ⊆ f (∨ i∈i ai ) . a set-valued mapping f : q1 −→p∗ (q2) is called a strong set-valued homomorphism if we replace ⊆ by = in (1) and (2). definition 5.3. let (q1,∗1) and (q2,∗2) be two hyperquantales and let f be a set-valued homomorphism. let f be any fuzzy subset of q2. then for every x ∈ q1, we defines f− (f) (x) = ∧ y∈f(x) f (y) , f+ (f) (x) = ∨ y∈f(x) f (y) . here f− (f) is the generalized lower approximation and f+ (f) is the generalized upper approximation of the fuzzy subset of f. the pair (f− (f) ,f+ (f)) is called generalized rough fuzzy subset of q1, if f − (f) 6= f+ (f) . definition 5.4. let f be a set-valued homomorphism. a fuzzy subset f of the hyperquantale q2 is called a lower (resp. upper) generalized rough fuzzy bi-hyperideal of q2 if f − (f) (resp. f+ (f)) is a fuzzy bihyperideal of q1. a fuzzy subset f of q2, which is both an upper and a lower generalized rough fuzzy bi-hyperideal of q2, is called generalized rough fuzzy bi-hyperideal of q2. int. j. anal. appl. 18 (6) (2020) 1007 theorem 5.1. let f be a strong set-valued homomorphism and let f be a fuzzy bi-hyperideal of q2. then set f− (f) is a fuzzy bi-hyperideal of q1. proof. assume that f is a fuzzy bi-hyperideal of q2, then we have ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) imply that f (α) ≥ f (x) ∧ f (y) ∀x,y ∈ q2 and α ∈ x∨y. also f is a strong set-valued homomorphism, so f (x∨y) = f (x) ∨f (y) ∀x,y ∈ q1. therefore for any α ∈ x∨y f− (f) (α) = f− (f) (x∨y) = ∧ α∈f(x∨y) f (α) = ∧ α∈f(x)∨f(y) f (α) . since α ∈ f (x) ∨f (y), there exist a ∈ f (x) and b ∈ f (y) such that α ∈ a∨ b. hence f− (f) (x∨y) = ∧ a∨b∈f(x)∨f(y) f (a∨ b) ≥ ∧ a∈f(x),b∈f(y) ( f (a) ∧ f (b) ) =     ∧ a∈f(x) f (a)  ∧   ∧ b∈f(y) f (b)     = f− (f) (x) ∧ f− (f) (y) . hence ∧ α∈x∨y f− (f) (α) ≥ f− (f) (x) ∧ f− (f) (y) ∀x,y ∈ q1. again since f is a strong set-valued homomorphism, so we have f (x∗1 y) = f (x) ∗2 f (y) ∀x,y ∈ q1. thus for any β ∈ x∗1 y we have, f− (f) (x∗1 y) = ∧ β∈f(x∗1y) f (β) = ∧ β∈f(x)∗2f(y) f (β) . since β ∈ f (x) ∗2 f (y), there exist a ∈ f (x) and b ∈ f (y) such that β ∈ a∗2 b. hence f− (f) (β) = f− (f) (x∗1 y) = ∧ a∗2b∈f(x)∗2f(y) f (a∗2 b) ≥ ∧ a∈f(x),b∈f(y) ( f (a) ∧ f (b) ) =     ∧ a∈f(x) f (a)  ∧   ∧ b∈f(y) f (b)     = f− (f) (x) ∧ f− (f) (y) . hence ∧ β∈x∗1y f− (f) (β) ≥ f− (f) (x) ∧ f− (f) (y) ∀x,y ∈ q1. again since f is a fuzzy bi-hyperideal of q2, so for any γ ∈ x∗1 y ∗1 z, we have int. j. anal. appl. 18 (6) (2020) 1008 f− (f) (γ) = f− (f) (x∗1 y ∗1 z) = ∧ γ∈f(x∗1y∗1z) f (γ) = ∧ γ∈(f(x)∗2f(y)∗2f(z)) f (γ) . since γ ∈ f (x)∗2 f (y)∗2 f (z), there exist a ∈ f (x) and b ∈ f (y) and c ∈ f (z) such that γ ∈ a∗2 b∗2 c. hence f− (f) (γ) = f− (f) (x∗1 y ∗1 z) = ∧ a∗2b∗2c∈f(x)∗2f(y)∗2f(z) f (a∗2 b∗2 c) ≥ ∧ a∈f(x),c∈f(z) ( f (a) ∧ f (c) ) =     ∧ a∈f(x) f (a)  ∧   ∧ c∈f(z) f (c)     = f− (f) (x) ∧ f− (f) (z) . hence ∧ γ∈(x∗1y∗1z) f− (f) (γ) ≥ f− (f) (x) ∧ f− (f) (z) ∀x,y,z ∈ q1. � theorem 5.2. let f be a strong set-valued homomorphism and let f be a fuzzy bi-hyperideal of q2. then f+ (f) is a fuzzy bi-hyperideal of q1. proof. assume that f is a fuzzy bi-hyperideal of q2, then we have ∧ α∈x∨y f (α) ≥ f (x) ∧ f (y) imply that f (α) ≥ f (x) ∧ f (y) ∀x,y ∈ q2 and α ∈ x∨y. also f is a strong set-valued homomorphism, so f (x∨y) = f (x) ∨f (y) ∀x,y ∈ q1. therefore for any α ∈ x∨y f+ (f) (α) = f+ (f) (x∨y) = ∨ α∈f(x∨y) f (α) = ∨ α∈f(x)∨f(y) f (α) . since α ∈ f (x) ∨f (y), there exist a ∈ f (x) and b ∈ f (y) such that α ∈ a∨ b. hence f+ (f) (α) = f+ (f) (x∨y) = ∨ a∨b∈f(x)∨f(y) f (a∨ b) ≥ ∨ a∈f(x),b∈f(y) ( f (a) ∧ f (b) ) =     ∨ a∈f(x) f (a)  ∧   ∨ b∈f(y) f (b)     = f+ (f) (x) ∧ f+ (f) (y) . hence ∧ α∈x∨y f+ (f) (α) ≥ f+ (f) (x) ∧ f+ (f) (y) ∀x,y ∈ q1. again since f is a strong set-valued homomorphism, so we have f (x∗1 y) = f (x) ∗2 f (y) ∀x,y ∈ q1. thus for any β ∈ x∗1 y we have, int. j. anal. appl. 18 (6) (2020) 1009 f+ (f) (β) = f+ (f) (x∗1 y) = ∨ β∈f(x∗1y) f (β) = ∨ β∈f(x)∗2f(y) f (β) . since β ∈ f (x) ∗2 f (y), there exist a ∈ f (x) and b ∈ f (y) such that β ∈ a∗2 b. hence f+ (f) (β) = f+ (f) (x∗1 y) = ∨ a∗2b∈f(x)∗2f(y) f (a∗2 b) ≥ ∨ a∈f(x),b∈f(y) ( f (a) ∧ f (b) ) =     ∨ a∈f(x) f (a)  ∧   ∨ b∈f(y) f (b)     = f+ (f) (x) ∧ f+ (f) (y) . hence ∧ β∈x∗1y f+ (f) (β) ≥ f+ (f) (x) ∧ f+ (f) (y) ∀x,y ∈ q1. again since f is a fuzzy bi-hyperideal of q2, so for any γ ∈ x∗1 y ∗1 z, we have f+ (f) (γ) = f+ (f) (x∗1 y ∗1 z) = ∨ γ∈f(x∗1y∗1z) f (γ) = ∨ γ∈(f(x)∗2f(y)∗2f(z)) f (γ) . since γ ∈ f (x)∗2 f (y)∗2 f (z), there exist a ∈ f (x) and b ∈ f (y) and c ∈ f (z) such that γ ∈ a∗2 b∗2 c. hence f+ (f) (γ) = f+ (f) (x∗1 y ∗1 z) = ∨ a∗2b∗2c∈(f(x)∗2f(y)∗2f(z)) f (a∗2 b∗2 c) ≥ ∨ a∈f(x),c∈f(z) ( f (a) ∧ f (c) ) =     ∨ a∈f(x) f (a)  ∧   ∨ c∈f(z) f (c)     = f+ (f) (x) ∧ f+ (f) (z) . hence ∧ γ∈(x∗1y∗1z) f+ (f) (γ) ≥ f+ (f) (x) ∧ f+ (f) (z) ∀x,y,z ∈ q1. � proposition 5.1. let f be a strong set-valued homomorphism and let {fi}i∈i be a family of fuzzy bihyperideal of q2. then f − (∧ i∈i (fi) ) is a fuzzy bi-hyperideal of q1. int. j. anal. appl. 18 (6) (2020) 1010 proof. since every fi is a fuzzy bi-hyperideals for every i ∈ i, and for every x,y ∈ q1, f− (∧ i∈i (fi) ) (α) = f− (∧ i∈i (fi) ) (x∨y) = (∧ i∈i f− (fi) ) (x∨y) = ∧ i∈i f− (fi) (x∨y) ≥ ∧ i∈i ( f− (fi) (x) ∧ f− (fi) (y) ) = {(∧ i∈i f− (fi) ) (x) ∧(∧ i∈i f− (fi) ) (y) } = f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (y) . hence ∧ α∈x∨y f− (∧ i∈i fi ) (α) ≥ f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (y) ∀x,y ∈ q1. now, f− (∧ i∈i (fi) ) (β) = f− (∧ i∈i (fi) ) (x∗1 y) = (∧ i∈i f− (fi) ) (x∗1 y) = ∧ i∈i f− (fi) (x∗1 y) ≥ ∧ i∈i ( f− (fi) (x) ∧ f− (fi) (y) ) = {(∧ i∈i f− (fi) ) (x) ∧(∧ i∈i f− (fi) ) (y) } = f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (y) . hence ∧ β∈x∗1y f− (∧ i∈i fi ) (β) ≥ f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (y) ∀x,y ∈ q1. int. j. anal. appl. 18 (6) (2020) 1011 again since f is a strong set-valued homomorphism, and f is a fuzzy bi-hyperideal of q2, so for any γ ∈ x∗1 y ∗1 z, we have, f− (∧ i∈i (fi) ) (γ) = ( f− ∧ i∈i (fi) ) (x∗1 y ∗1 z) = ∧ i∈i f− (fi) (x∗1 y ∗1 z) ≥ ∧ i∈i ( f− (fi) (x) ∧ f− (fi) (z) ) = {(∧ i∈i f− (fi) ) (x) ∧(∧ i∈i f− (fi) ) (z) } = f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (z) . hence ∧ γ∈x∗1y∗1z f− (∧ i∈i fi ) (γ) ≥ f− (∧ i∈i fi ) (x) ∧ f− (∧ i∈i fi ) (z) ∀x,y,z ∈ q1. � for the following theorem we define the set fα where α ∈ [0, 1] as following fα = {x ∈ q | f (x) ≥ α} . theorem 5.3. let f be a strong set-valued homomorphism and f be a fuzzy bi-hyperideal of q2. then f− (f) (resp. f+ (f)) is a fuzzy bi-hyperideal of q1 if and only if for each α ∈ [0, 1] , f− (fα) (resp. f+ (fα)), where fα 6= ∅, is a bi-hyperideal of q1. proof. assume that f− (f) is a fuzzy bi-hyperideal of q1. we need to show that f − (fα) is a bi-hyperideal of q1. let x1,x2 ∈ f− (fα) . then f− (f) (x1) ≥ α and f− (f) (x2) ≥ α. but since f− (f) is a fuzzy bi-hyperideal, so ∧ z∈x1∨x2 f− (f) (z) ≥ f− (f) (x1) ∧ f− (f) (x2) ≥ α. implies that f− (f) (z) ≥ α. hence x1 ∨ x2 ⊆ f− (fα) . let y ∈ f− (fα) , x ∈ q1, and x ≤ y. then f− (f) (x) ≥ f− (f) (y) ≥ α. hence f− (f) (x) ≥ α. hence x ∈ f− (fα) . let y1,y2 ∈ f− (fα) , then f− (f) (y1) ≥ α and f− (f) (y2) ≥ α. since f− (f) is a fuzzy bi-hyperideal of q1, so we have ∧ z∈y1∗1y2 f− (f) (z) ≥ f− (f) (y1) ∧ f− (f) (y2) = α. hence f− (f) (z) ≥ α, for all z ∈ y1 ∗1 y2, this implies that z ∈ f− (fα) . hence y1 ∗1 y2 ⊆ f− (fα) . now let u,v,w ∈ f− (fα) . then f− (f) (u) ≥ α, f− (f) (v) ≥ α and f− (f) (w) ≥ α. again since f− (f) is a fuzzy bi-hyperideal of q1, so we have ∧ β∈u∗1v∗1w f− (f) (β) ≥ f− (f) (u) ∧ f− (f) (w) = α. hence f− (f) (β) ≥ α. thus u∗1 v ∗1 w ⊆ f− (fα) . therefore f− (fα) is a bi-hyperideal of q1. conversely, assume that f− (fα) is a bi-hyperideal of q1. we shall show that f − (f) is a fuzzy bihyperideal of q1. for any x,y ∈ q1, let α = f− (f) (x) ∧ f− (f) (y) ∈ range(f− (f)) . then f− (f) (x) ≥ α and f− (f) (y) ≥ α. so x,y ∈ f− (fα) . hence x∨y ⊆ f− (fα) . consider int. j. anal. appl. 18 (6) (2020) 1012 f− (f) (x∨y) = ∧ z∈f(x∨y) f (z) = ∧ z∈f(x)∨f(y) f (z) . since z ∈ f (x) ∨f (y), there exist a ∈ f (x) and b ∈ f (y) such that z ∈ a∨ b. hence f− (f) (x∨y) = ∧ a∨b∈f(x)∨f(y) f (a∨ b) ≥ ∧ a∈f(x),b∈f(y) ( f (a) ∧ f (b) ) =     ∧ a∈f(x) f (a)  ∧   ∧ b∈f(y) f (b)     = f− (f) (x) ∧ f− (f) (y) . hence ∧ z∈x∨y f− (f) (z) ≥ f− (f) (x) ∧ f− (f) (y) ∀x,y ∈ q1. now for x,y ∈ f− (fα), we have x∗1 y ⊆ f− (fα) . hence for β ∈ x∗1 y, we have f− (f) (β) ≥ α. since, x,y ∈ f− (fα) , so f− (f) (x) ≥ α and f− (f) (y) ≥ α. thus ∧ β∈x∗1y f− (f) (β) ≥ f− (f) (x) ∧ f− (f) (y) . let x,y ∈ q1 such that x ≤ y. if f− (f) (y) = 0 then f− (f) (x) ≥ f− (f) (y) . if f− (f) (y) = α then y ∈ f− (fα) . since x ≤ y and f− (fα) is a bi-hyperideal of q1, we have x ∈ f− (fα) . then f− (f) (x) ≥ α = f− (f) (y) . now let x,y,z ∈ f− (fα) . then x∗1y∗1z ⊆ f− (fα) . since x,z ∈ f− (fα) . then f− (f) (x) ≥ α and f− (f) (z) ≥ α. so for any γ ∈ x∗1 y∗1 z, we have f− (f) (γ) ≥ α. thus f− (f) (x) ∧ f− (f) (z) = α ≤∧ γ∈(x∗1y∗1z) f− (f) (γ). therefore f− (f) is a fuzzy bi-hyperideal of q1. � 6. conclusion in the present paper, we introduced the notion of bi-hyperideals of hyperquantales. furthermore we introduced the notions of fuzzy bi-hyperideals and generalized rough fuzzy bi-hyperideals of hyperquantales and their related properties is provided. finally we discussed the strong set-valued homomorphism and setvalued homomorphism of hyperquantales and generalized rough fuzzy bi-hyperideals and shown that how they are related. in our future study of hyperquantales, we will apply the above new idea to other algebraic structures for more applications. 7. acknowledgements the second author is supported by ukm grant: gp-2020-k006392. declaration: all authors agreed with the contents of the manuscript. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. 18 (6) (2020) 1013 references [1] f. marty, sur une generalisation de la notion de group, in proceedings of the 8th congress des mathematiciens scandinaves, stockholm, 1934, pp. 45–49. [2] j. tang, b. davvaz and y. f. luo, a study on fuzzy interior hyperideals in ordered semihypergroups, ital. j. pure appl. math. 36 (2016), 125-146. [3] j. tang, a. khan and y. f. luo, characterization of semisimple ordered semihypergroups in terms of fuzzy hyperideals, j. intell. fuzzy syst. 30 (2016), 1735-1753. [4] b. davvaz, a. khan and m. farooq, int-soft structures applied to ordered semihypergroups, le matematiche, lxxiii (2018), 235–259. [5] a. khan, m. farooq and h. u. khan, uni-soft hyperideals of ordered semihypergroups, j. intell. fuzzy syst. 35 (2018), 4557–4571. [6] b. davvaz, a. khan and m. khan, atanassov’s intuitionistic fuzzy set theory applied to quantales, novi sad j. math. 47 (2) (2017), 47-61. [7] b. b. n. koguep, on fuzzy ideals of hyperlattice, int. j. algebra, 2 (15) (2008), 739-750. [8] p. corsini, v. leoreanu, applications of hyperstructure theory, kluwer academic publishers, dordrecht, 2003. [9] j. zhan, b. davvaz, p. k. shun, probability n-ary bypergroups. inform. sci. 180 (2010), 1159-1166. [10] j. zhan, c. irina, γ-hypermodules: isomorphism theorems and regular relations. u. p. b. sci bull. ser. a. 73 (2011), 71-78. [11] j. zhan, b. davvaz, p. k. shun., a new view of fuzzy hyperrings. inform. sci. 178 (2008), 425-438. [12] m. konstantinidou and j. mittas, an introduction to the theory of hyperlattices, math. balkanica, 7 (1977), 187-193. [13] s. rasouli and b. davvaz, lattices derived from hyperlattices. commun. algebra, 38 (8) (2010), 2720-2737. [14] s. rasouli and b. davvaz, construction and spectral topology on hyperlattices, mediterr. j. math. 7 (2) (2010), 249-262. [15] z. pawlak, information systems–theoretical foundations, inform. syst. 6 (1981), 205–218. [16] z. pawlak, rough sets, int. j. comput. inform. sci. 11 (1982), 341–356. [17] j. järvinen, lattice theory for rough sets, transactions on rough sets vi, lncs 4374, springer-verlag, berlin heidelberg, 2007 (pp. 400–498). [18] g.l. liu and w. zhu, the algebraic structures of generalized rough set theory, inform. sci. 178 (2008), 4105–4113. [19] l. a. zadeh, fuzzy sets, inform. control, 8 (1965), 338-353. [20] c. j. mulvey, supplemento ai rendiconti del circolo matematico di palermo ii 12 (1986), 99–104. [21] k. y. wang, prime radical theorem in quantales, fuzzy syst. math. 25 (2) (2011), 60–64 (in chinese). [22] s. q. wang and b. zhao, ideals of quantales, j. shaanxi normal univ. (nat. sci. ed.) 31 (4) (2003), 7–10 (in chinese). [23] s. q. wang and b. zhao, prime ideal and weakly prime ideal of the quantale, fuzzy syst. math. 19 (1) (2005), 78–81 (in chinese). [24] l. y. yang, l.s. xu, roughness in quantales, inform. sci. 220 (2013), 568–579. [25] q. luo and g. wang, roughness and fuzziness in quantales, inform. sci. 271 (2014), 14–30. [26] m. farooq, t. mahmood, a. khan, m. izhar and b. davvaz, fuzzy hyperideals of hyperquantale, j. intell. fuzzy syst. 36 (2019), 5605–5615. [27] s. yamak, o. kazancı, and b. davvaz, generalized lower and upper approximations in a ring, inform. sci. 180 (9) (2010), 1759–1768. [28] a. k. estaji and f. bayati, on rough sets and hyperlattices, ratio math. 34 (2018), 15-33. int. j. anal. appl. 18 (6) (2020) 1014 [29] s. m. qurashi and m. shabir, generalized rough fuzzy ideals in quantales, discrete dyn. nat. soc. 2018 (2018), article id 1085201. [30] s. m. qurashi and m. shabir, generalized approximations of (∈, ∈ ∨q)-fuzzy ideals in quantales, comput. appl. math. 37 (2018), 6821–6837. 1. introduction 2. preliminaries 3. fuzzy hyperideals of hyperquantale 4. fuzzy bi-hyperideal of hyperquantale 5. homomorphism and generalized rough fuzzy bi-hyperideals of hyperquantales 6. conclusion 7. acknowledgements references international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 90-94 http://www.etamaths.com characterization of multiplicative metric completeness badshah e rome and muhammad sarwar∗ abstract. we established fixed point theorems in multiplicative metric spaces. the obtained results generalize banach contraction principle in multiplicative metric spaces and also characterize completeness of the underlying multiplicative metric space. 1. introduction and preliminaries in 1970 michael grossman and robert katz [11] established a new calculus called multiplicative calculus also termed as exponential calculus. florack and van assen [10] used the idea of multiplicative calculus in biomedical image analysis. bashirov et al.[3] demonstrated the efficiency of multiplicative calculus over the newtonian calculus. they elaborated that multiplicative calculus is more effective than newtonian calculus for modeling various problems from different fields . bashirov and bashirova [4] used the concept of multiplicative calculus for deriving function that shows dynamics of literary text. bashirov et al.[2] further demonstrated the usefulness of multiplicative calculus by proving the fundamental theorem of multiplicative calculus. by defining multiplicative distance they provided foundation for multiplicative metric spaces. özavsar and cevikel [13] presented the notion of multiplicative contraction mapping. besides some other results, they proved the well known banach contraction principle for such contraction in multiplicative metric spaces. hxiaoju et al.[12] established common fixed point theorems for weak commutative mappings in the setting of multiplicative metric space. abbas et al.[1] established common fixed point results of quasi-weak commutative mappings on a closed ball in the framework of multiplicative metric spaces. banach contraction principle has been a very advantageous and effectual means in nonlinear analysis. generalization of the banach contraction principle has been one of the most enquired branch of research. banach theorem has many generalizations; (see [5, 6, 7, 8, 17]). sarwar and rome [16] established several generalizations of banach contraction principle and proved cantor intersection theorem in the framework of multiplicative metric spaces. tomonari suzuki [18] proved a fixed point result which generalizes banach theorem and characterizes metric completeness. in the current article we prove fixed point results in the set up of multiplicative metric spaces. the derived results results generalized banach contraction principle in multiplicative metric spaces and characterize completeness of the underlying multiplicative metric space. for various definitions and elements of multiplicative calculus we refer the reader to [1, 2, 3, 9, 11, 12, 13, 14, 15]. definition 1.1. [2] let m be a nonempty set. a mapping d : m × m → [1,∞) is said to be multiplicative metric on m if the following condition are satisfied: (1) d(x,y) ≥ 1 for all x,y ∈ m; (2) d(x,y) = 1 if and only if x = y; (3) d(x,y) = d(y,x) for all x,y ∈ m; (4) d(x,z) ≤ d(x,y).d(y,z) for all x,y,z ∈ m. and the pair (m,d) is called multiplicative metric space. 2010 mathematics subject classification. primary 47h10, 54h25; secondary 55m20. key words and phrases. multiplicative metric space; multiplicative contraction mapping; multiplicative cauchy sequence; fixed point. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 90 characterization of multiplicative metric completeness 91 example 1.1. [2] the mapping d∗ : (0,∞) × (0,∞) → [1,∞) defined as d∗(x,y) = |x y |∗, where |a|∗ = { a if a ≥ 1 1 a if a < 1 . is a multiplicative metric. definition 1.2. [1, 12, 13] a sequence {xn} in a multiplicative metric space (x,d) is said to be multiplicative cauchy sequence if for all � > 1 there exits a positive integer n0 such that d(xn,xm) < � ∀n,m ≥ n0. definition 1.3. [1, 12, 13] a multiplicative metric space (x,d) is said to be complete if every multiplicative cauchy sequence in xconverges in x. definition 1.4. [1, 12, 13] let (x,d) be a multiplicative metric space. a mapping f : x → x is called multiplicative contraction if there exists a real constant λ ∈ [0, 1) such that d(f(x1),f(x2)) ≤ d(x1,x2) λ ∀x,y ∈ x. 2. main results this section studied two fixed point theorems in the setting of multiplicative metric spaces. the first result generalized the banach contraction principal while the second one characterizes multiplicative metric completeness. theorem 2.1. let (m,d) be a complete multiplicative metric space. let f be a mapping on m and ϕ : [0, 1) → (1/2, 1] a non increasing function defined as follows ϕ(γ) =   1 if 0 ≤ γ ≤ ( √ 5 − 1)/2, (1 −γ)γ−2 if ( √ 5 − 1)/2 < γ < 1/ √ 2, (1 + γ)−1 if 1/ √ 2 ≤ γ < 1. let there exists γ ∈ [0, 1) such that (1) d(x,fx)ϕ(γ) ≤ d(x,y) ⇒ d(fx,fy) ≤ d(x,y)γ ∀ x,y ∈ m. then f has a unique fixed point z. furthermore limnf nx = z for all x ∈ m. proof. as ϕ(γ) ≤ 1 therefor d(x,fx)ϕ(γ) ≤ d(x,fx) ∀ x ∈ m. condition (1) implies (2) d(fx,f2x) ≤ d(x,fx)γ ∀ x ∈ m. fix v ∈ m and define a sequence {vn} in m by vn = fnv. relation (2) implies that d(vn,vn+1) ≤ d(v,fv)γ n . therefor ∏∞ n=1 d(vn,vn+1) ≤ d(v,fv) γ 1−γ < ∞. it means {vn} is a cauchy sequence. as m is complete so {vn} converges to some point z ∈ m. we show that (3) d(fx,z) ≤ d(x,z)γ ∀ x ∈ m\{z}. for x ∈ m\{z} there will be some positive integer m such that d(vn,z) ≤ d(x,z)1/3 ∀ n ≥ m. we have d(vn,fvn) ϕ(γ) ≤ d(vn,fvn) = d(vn,vn+1) ≤ d(vn,z)d(z,vn+1) ≤ d(x,z)2/3 = d(x,z) d(x,z)1/3 ≤ d(x,z) d(vn,z) ≤ d(vn,x). using hypothesis of theorem, we get d(vn+1,fx) ≤ d(vn,x)γ for n ≥ m. letting n → ∞, we get d(z,fx) ≤ d(z,x)γ. hence (3) is proved. now let us suppose by the way of contradiction that fiz 6= z for all i ∈ n. then (3) gives (4) d(fi+1z,z) ≤ d(fz,z)γ i ∀ i ∈ n. now consider the following cases. • 0 ≤ γ ≤ ( √ 5 − 1)/2, • ( √ 5 − 1)/2 < γ < 1/ √ 2, • 1/ √ 2 ≤ γ < 1. when 0 ≤ γ ≤ ( √ 5 − 1)/2, then γ2 + γ − 1 ≤ 0, also 2γ2 ≤ 3 − √ 5 < 1. if d(f2z,z) < d(f2z,f3z), then d(z,fz) ≤ d(z,f2z)d(f2z,fz) < d(f2z,f3z)d(f2z,fz) ≤ d(z,fz)γ 2+γ ≤ d(z,fz). 92 rome and sarwar which is contradiction. therefor d(f2z,z) ≥ d(f2z,f3z) = d(f2z,f ◦f2z)ϕ(γ). using hypothesis of the theorem and (4),we have d(z,fz) ≤ d(z,f3z)d(f3z,fz) ≤ d(z,fz)γ 2 d(f2z,z)γ ≤ d(z,fz)γ 2 d(fz,z)γ 2 = d(fz,z)2γ 2 < d(fz,z). which is contraction. and when ( √ 5 − 1)/2 < γ < 1/ √ 2 then 2γ2 < 1. if we suppose d(f2z,z) < d(f2z,f3z)ϕ(γ), then using (2) we have d(z,fz) ≤ d(z,f2z)d(f2z,fz) < d(f2z,f3z)ϕ(γ)d(f2z,fz) ≤ d(z,fz)ϕ(γ)γ 2 d(z,fz)γ = d(z,fz)ϕ(γ)γ 2+γ = d(z,fz)(1−γ)γ −2γ2+γ = d(z,fz), giving a contradiction. hence d(f2z,z) ≥ d(f2z,f3z)ϕ(γ) = d(f2z,f ◦ f2z)ϕ(γ). and this, like the preceding case, produces the following contradiction. d(z,fz) ≤ d(z,fz)2γ 2 < d(z,fz). finally when 1/ √ 2 ≤ γ < 1. then for x,y ∈ m, either d(x,fx)ϕ(γ) ≤ d(x,y) or d(fx,f2x)ϕ(γ) ≤ d(fx,y). in case d(x,fx)ϕ(γ) > d(x,y) and d(fx,f2x)ϕ(γ) > d(fx,y), then using multiplicative triangular inequality and (2), we have d(x,fx) ≤ d(x,y)d(y,fx) < d(x,fx)ϕ(γ)d(fx,f2x)ϕ(γ) = (d(x,fx)d(fx,f2x))ϕ(γ) = d(x,fx) (1+γ)ϕ(γ) = d(x,fx) (1+γ)(1+γ)−1 = d(x,fx). which is again contradiction. now since d(v2n,v2n+1) ϕ(γ) ≤ d(v2n,z) or d(v2n+1,v2n+2) ϕ(γ) ≤ d(v2n+1,z) ∀n ∈ n. therefore using hypothesis of the theorem, either d(v2n+1,fz) ≤ d(v2n,z)γ ≤ d(v2n,z) or d(v2n+2,fz) ≤ d(v2n+1,z)γ ≤ d(v2n+1,z) ∀ n ∈ n. now {vn} converges to z, but the above inequalities indicate that there is a subsequence of {vn} which converges to fz. therefore fz = z. this contradicts the supposition. hence in all the above cases, there will be some i ∈ n such that fiz = z. as {fnz} is a cauchy sequence, therefore fz = z. in order to show uniqueness of the fixed point of f, let w ∈ m\{z} be another fixed point of f. then using (3), we have the contradiction, d(w,z) = d(fw,z) ≤ d(w,z)γ < d(w,z). hence z is the only fixed point of f in m. � theorem 2.2. let (m,d) be a multiplicative metric space and ϕ be a mapping as defined in theorem 2.1. for γ ∈ [0, 1) and β ∈ (0,ϕ(γ)], let sγ,β be the family of mappings f on m satisfying the following: (1) for x,y ∈ m, d(x,fx)β ≤ d(x,y) ⇒ d(fx,fy) ≤ d(x,y)γ. let tγ,β be the family of mappings f on m satisfying (1) and the following: (2) f(m) is countably infinite. (3) every subset of f(m) is closed. then the following are equivalent: (a) m is complete. (b) every mapping f ∈ sγ,β has a fixed point for all γ ∈ [0, 1). (c) there exist γ ∈ (0, 1) and β ∈ (0,ϕ(γ)] such that every mapping f ∈ tγ,β has a fixed point. proof. as β ≤ ϕ(γ), therefore using theorem 2.1, (a) implies (b). and as tγ,β ⊂ sγ,β, therefore (b) implies (c). next we prove that (c) implies (a). let (c) holds but m is not complete. it means there exists a cauchy sequence {vn} which doesn’t converge in m. define a mapping g : m → [1,∞) by g(x) = limnd(x,vn) for x ∈ m. with the properties of multiplicative metric, the following are obvious: (i) g(x)/g(y) ≤ d(x,y) ≤ g(x)g(y) for all x,y ∈ m, (ii) g(x) > 1 for all x ∈ m and (iii) limng(vn) = 1. define a mapping f on m as follows: as for each x ∈ m, g(x) > 1 and limng(vn) = 1, therefore there exists η ∈ n such that g(vη) ≤ g(x) γβ 3+γβ . for f(x) = vη, (5) obviously g(fx) ≤ g(x) γβ 3+γβ and fx ∈{vn : n ∈ n} for all x ∈ m. this implies that g(fx) < g(x) for all x ∈ m therefore fx 6= x for all x ∈ m. that is f has no fixed point. now since f(m) ⊂{vn : n ∈ n}, therefore condition (2) is satisfied. obviously every subset of characterization of multiplicative metric completeness 93 f(m) is closed, that is (3) holds. in order to prove (1), fix x,y ∈ m such that d(x,fx)β ≤ d(x,y). in case where g(y) > (g(x))2, (i) and (5) imply that d(fx,fy) ≤ g(fx)g(fy) ≤ (g(x)g(y)) γβ 3+γβ ≤ (g(x)g(y)) γ 3 < (g(x)g(y)) γ 3 ( g(y) (g(x))2 ) 2γ 3 ≤ (g(y) g(x) )γ ≤ d(x,y)γ. and when g(y) ≤ (g(x))2, then again using (i) and (5), we have d(x,y) ≥ d(x,fx)β ≥ ( g(x) g(fx) )β ≥ ( g(x) g(x) γβ 3+γβ )β = g(x) 3β 3+γβ . and therefore d(fx,fy) ≤ g(fx)g(fy) ≤ (g(x)g(y)) γβ 3+γβ ≤ (g(x)(g(x))2) γβ 3+γβ ≤ g(x) 3γβ 3+γβ = ( g(x) 3β 3+γβ )γ ≤ d(x,y)γ. therefore (1) is proved. hence f ∈ tγ,β. and by (c), f has a fixed point. which is contradiction. consequently m is complete. this completes the proof. � we conclude with the following example which supports theorem 2.1. example 2.1. let m = r+,set of positive real numbers. consider the multiplicative metric d : m ×m → [1,∞) defined by d(x,y) = e|x−y|. then (m,d) is complete multiplicative metric space. let ϕ be a mapping as defined in theorem 2.1. t : m → m be mapping defined by t(x) = 1 5+x , such that d(x,fx)ϕ(γ) = d(x,fx)ϕ( 1 2 ) = d(x,fx) = e|x− 1 5+x | ≤ e|x−y| = d(x,y) then d(fx,fy) = e|x−y|| 1 (5+x)(5+y) | ≤ e 1 2 |x−y| = d(x,y)γ ∀x,y ∈ m. obviously t has unique fixed point 0.1925824036 ∈ m. references [1] m. abbas, bashir ali and yusuf i. suleiman, common fixed points of locally contractive mappings in multiplicative metric spaces with application, international journal of mathematics and mathematical sciences, 2015 (2015), article id 218683. [2] a.e. bashirov, e.m. kurpnar and a. ozyapc, multiplicative calculus and its applications j. math. anal. appl., 337 (2008), 36-48. [3] a. e. bashirov, e. misirli, y. tandogdu, and a. ozyapici, on modeling with multiplicative differential equations, a journal of chinese universities, 26 ( 2011), 425–438. [4] a bashirov, g bashirova, dynamics of literary texts and diffusion, online journal of communication and media technologies, 1 (2011), 60–82. [5] j. caristi, fixed point theorems for mappings satisfying inwardness conditions, trans. amer. math. soc., 215 (1976), 241-251. [6] j. caristi and w. a. kirk, geometric xed point theory and inwardness conditions, lecture notes in math., 490 (1975), 74–83. [7] lj. b. ćirić, a generalization of banachs contraction principle, proc. amer. math. soc., 45 (1974), 267–273. [8] i. ekeland, on the variational principle, j. math. anal. appl., 47 (1974), 324-353. [9] j englehardt, j swartout and cloewenstine. a new theoretical discrete growth distribution with verification for microbial counts in water, risk analysis, 29 (2009), 841–856. [10] l. florack and h. v. assen, multiplicative calculus in biomedical image analysis, journal of mathematical imaging and vision, 42 (2012), 64–75. [11] m grossman, rkatz. non-newtonian calculus, pigeon cove, lee press, massachusats, 1972. [12] h. hxiaoju , m. song m and d. chen, common fixed points for weak commutative mappings on a multiplicative metric space, fixed point theory and applications 2014 (2014), article id 48. [13] m. őzavsar and a. c. cevikel, fixed point of multiplicative contraction mappings on multiplicative metric space, arxiv:1205.5131v1 [matn.gn] (2012). [14] m riza, a özyapc, e kurpnar, multiplicative finite difference methods, quarterly of applied mathematics, 67 (2009), 745–754. [15] d stanley. a multiplicative calculus, primus, 9(1999), 310-326. [16] m. sarwar and b. rome, extensions of the banach contraction principle in multiplicative metric spaces, pacific journal of optimization, in press. [17] t. suzuki, generalized distance and existence theorems in complete metric spaces, j. math. anal. appl., 253 (2001), 440-458. [18] t. suzuki, a generalized banach contraction principle that characterizes metric completeness, proceedings of the american mathematical society, 136 (2008), 1861–1869. 94 rome and sarwar department of mathematics, university of malakand, chakdara dir(l), pakistan ∗corresponding author: sarwarswati@gmail.com int. j. anal. appl. (2022), 20:70 testing the difference of means of populations with respect to intuitionistic fuzzy sets sasiwimon iwsakul1, khamika urawong1, thammarat panityakul1, ronnason chinram1, pattarawan singavananda2,∗ 1division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 2program in mathematics, faculty of science and technology, songkhla rajabhat university, songkhla 90000, thailand ∗corresponding author: pattarawan.pe@skru.ac.th abstract. in this study, the tests of statistical hypotheses with crisp data using small samples are extended to with the membership function and the non-membership function of the intuitionistic fuzzy set. the test procedure of statistical hypotheses for means of two normally distributed populations with respect to any intuitionistic fuzzy set is proposed. 1. introduction a concept of fuzzy set was introduced by zadeh [7] in 1965. a fuzzy set is a class of objects with a continuum of membership grades which are allocated to each object a grade of membership ranging unit close from zero to one. later, the original of fuzzy set theory were generalized to other fuzzy sets. one popular generalized fuzzy sets proposed by atanassov [1] starts by the specification of membership and non-membership functions. it is called an intuitionistic fuzzy set (ifs). hypothesis testing is one of the major aspects of data analysis. the observations of the sample are crisp and a statistical test leads to the binary conclusion. in contrast, the data occasionally cannot be collected precisely. the data sometimes collected in the form of statistical hypotheses testing under fuzzy environments. casals, gil and gil [2] approach to the problem of testing statistical hypotheses with fuzzy information. the statistical hypotheses testing for fuzzy data by proposing the notions of received: nov. 5, 2022. 2010 mathematics subject classification. 62f03, 03e72. key words and phrases. tests of statistical hypotheses; populations; intuitionistic fuzzy sets; means; variances. https://doi.org/10.28924/2291-8639-20-2022-70 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-70 2 int. j. anal. appl. (2022), 20:70 degrees of optimism and pessimism was investigated by wu [5]. moreover, wu [6] proposed some approaches to construct fuzzy confidence intervals for the unknown fuzzy parameter. kalpanapriya and pandian [3] proposed tests of hypothesis for means of populations using imprecise samples. the proposed test is extended to statistical hypotheses testing for fuzzy data. moreover, kalpanapriya and pandian [4] investigated the method of tests of statistical hypotheses with respect to a fuzzy set. in this paper, we propose statistical hypothesis tests using small sample (or samples) based on the ifs. we focus not only on testing of significance for the difference between two independent population means but also between two dependent population means with respect to an ifs. 2. basic concepts this section outlines some of the most important definitions relevant to the study. 2.1. intuitionistic fuzzy sets. definition 2.1. let s be a nonempty set. a fuzzy set (fs) a drawn from a set s is defined as a= {< x,µa(x) >| x ∈ s} where µa : s → [0,1] is the membership function of the fuzzy set a. definition 2.2. let s be a nonempty set. an intuitionistic fuzzy set (ifs) a in a set s is an object having the form a= {< x,µa(x),νa(x) >| x ∈ s} where the functions µa : s → [0,1] and νa : s → [0,1] defined respectively, the degree of membership and degree of non-membership of the element x to the set a, which is a subset of s, and for every element x ∈ s, 0≤ µa(x)+νa(x)≤ 1. for any fs a = {< x,µa(x) >| x ∈ s}, we can define νa(x) = 1− µa(x) and we have {< x,µa(x),νa(x) >| x ∈ s} is an ifs. then every fs can be considered as an ifs. 2.2. one-sample t-test. let x1,x2, ...,xn be a random small sample of size n (where n < 30) from a normally distributed population. now, the mean value, denoted by x and the sample standard deviation, denoted by s of the above small sample are given by x = n∑ i=1 xi n and s2 = n∑ i=1 (xi −x)2 n−1 . note that s2 is called a sample variance. in testing the null hypothesis that the population mean µ is equal to a specified value µo, one uses the statistic t = x −µ0 s/ √ n . we will use that the degrees of freedom used in this test is ν = n−1. int. j. anal. appl. (2022), 20:70 3 3. main results in this section, we propose the following two types of tests of statistical hypotheses: (3.1) test for the difference between the means of two independent populations using their small sample size with respect to an ifs. (3.2) test for the difference between the means of two dependent populations using their small sample size with respect to an ifs. definition 3.1. let {x1,x2, . . . ,xn} be a random sample of size n from a crisp set x with membership grades and non-membership grades µa(xi) and νa(xi), respectively of an ifs a. the average of membership grades and non-membership grades of an ifs a over the random sample or the sample mean of the membership grades and non-membership grades of the ifs a denoted by µa(x) and νa(x) respectively defined as follows: µa(x)= n∑ i=1 µa(xi) n and νa(x)= n∑ i=1 νa(xi) n . definition 3.2. let {x1,x2, . . . ,xn} be a random sample of size n from a crisp set x with membership grades and non-membership grades µa(xi) and νa(xi), respectively of an ifs a. the variance of membership grades and non-membership grades of an ifs a over the random sample or the sample variance of the membership grades and non-membership grades of the ifs a denoted by (sµ(x))2 and (s2ν(x)) 2 respectively defined as follows: (sµ(x)) 2 = n∑ i=1 (µa(xi)−µa(x)) 2 n−1 and (sν(x)) 2 = n∑ i=1 (νa(xi)−νa(x))2 n−1 . next, we will present the methods of two main problems. 3.1. testing of significance for the difference between two independent population means with respect to an ifs. let x and y be two crisp populations and a be an ifs defined on x and y . let {x1,x2, . . . ,xm} be a linguistic random sample of x with membership grades and non-membership grades µa(xi) and νa(xi), respectively where i =1,2, . . . ,m. let {y1,y2, . . . ,yn} be another linguistic random sample of y with membership grades and non-membership grades µa(yj) and νa(yj), respectively where j =1,2, . . . ,n. suppose that µa(xi), νa(xi), µa(yj) and νa(yj) are normally distributed. based on the samples, we test that the mean of the population x with respect to a , µ(a,x) and the mean of the population y with respect to a , µ(a,y ) are the same. 4 int. j. anal. appl. (2022), 20:70 now, we have the null hypothesis h0: µ(a,x)= µ(a,y ). if both populations have equal standard deviations with respect to a, we use the test statistic for testing the null hypothesis, tµ = µa(x)−µa(y ) sµ √ 1/n+1/m and tν = νa(x)−νa(y ) sν √ 1/n+1/m where sµ = √ (m−1)(sµ(x))2 +(n−1)(sµ(y ))2 m+n−2 and sν = √ (m−1)(sν(x))2 +(n−1)(sν(y ))2 m+n−2 . if both populations standard deviations with respect to a are not the same, we use the test statistic for testing the null hypothesis, tµ = µa(x)−µa(y )√ (sµ(x)) 2/n+(sµ(y )) 2/m and tν = νa(x)−νa(y )√ (sν(x)) 2/n+(sν(y )) 2/m . we let t = max{|tµ|, |tν|}. now, the degree of freedom used in this test is df = m + n − 2 and we let tα,d denote the critical value of t for df degrees of freedom at the level of significance α. now, for the level of significance α, the critical region of the alternative hypothesis, ha is given below: alternative hypothesis critical region µ(a,x) > µ(a,y )(upper-tailed test) t ≥ tα,df µ(a,x) < µ(a,y )(lower-tailed test) t ≤−tα,df µ(a,x) 6= µ(a,y )(two-tailed test) |t| ≥ tα/2,df if |t| ≤ |tα,df | (one-tailed test), we cannot reject the null hypothesis. there is insufficient evidence to conclude that the µ(a,x) is greater than µ(a,y )(the means of populations with respect to a) for (upper-tailed test) or µ(a,x) is less than µ(a,y ) (for lower-tailed test) at α level. otherwise, the null hypothesis µ(a,x)≤ µ(a,y ) (for upper-tailed test) or µ(a,x)≥ µ(a,y ) (for lowertailed test)) is rejected. if |t| ≤ tα/2,df (two-tailed test), there is not enough evidence to conclude that the difference between µ(a,x) and µ(a,y ) at α level is significant. therefore, the null hypothesis is not rejected. otherwise, the null hypothesis is rejected, that is, the population means of the membership and non-membership functions of a are different. int. j. anal. appl. (2022), 20:70 5 now, we get the following example. example 3.1. let x and y be two populations where x be the sets of all students in prince of songkla university and y be the sets of all students in songkhla rajabhat university. we let the ifs a, where µ and ν be the membership and non-membership functions of students’ satisfaction toward online learning, defined on x and y . it is assumed that µa(xi), νa(xi), µa(yj) and νa(yj) are normally distributed. now, we are going to test that the a in x is better than the a in y , that is, µ(a,x) > µ(a,y ). let s1 = {x1,x2,x3,x4,x5} be the sample of size five taken from students in prince of songkla university (the population x) and s2 = {y1,y2,y3,y4,y5,y6} be the sample of size six taken from students in songkhla rajabhat university (the population y ). then, the membership grades and the non-membership grades of the given two samples based on their information concerning the ifs a are given below. now, with the help of the numerical example given below, the procedure of the above said testing of hypothesis is explained. x1 x2 x3 x4 x5 µa(xi) 0.7 0.8 0.9 0.8 0.7 νa(xi) 0.2 0.2 0 0.1 0.3 and y1 y2 y3 y4 y5 y6 µa(yi) 0.8 0.8 0.7 0.8 0.7 0.6 νa(yi) 0.2 0.1 0.1 0.1 0.2 0.1 our hypotheses are h0 : µ(a,x)≤ µ(a,y ) and h1 : µ(a,x) > µ(a,y ). now, we have µa(x)=0.78,µa(y )=0.73,νa(x)=0.16 and νa(y )=0.13. then sµ(x)=0.08,sµ(y )=0.08,sν(x)=0.11,sν(x)=0.05. since σ2µ = σ 2 ν, sµ and sν are computed as follows: sµ = √ (m−1)(sµ(x))2 +(n−1)(sµ(y ))2 m+n−2 =0.08 and sν = √ (m−1)(sν(x))2 +(n−1)(sν(y ))2 m+n−2 =0.09. 6 int. j. anal. appl. (2022), 20:70 therefore, the tests of statistic are tµ = µa(x)−µa(y )√ (sµ(x)) 2/n+(sµ(y )) 2/m =0.93 and tν = νa(x)−νa(y )√ (sν(x)) 2/n+(sν(y )) 2/m =0.52. then t = max{|tµ|, |tν|} = max{|0.93|, |0.52|} = 0.93. the critical value of t0.05,9 is 1.83. since 0.93 < 1.83, we cannot reject the null hypothesis. there is insufficient evidence to conclude that the a in x is better than the a in y. 3.2. testing of significance for the difference between two dependent population means with respect to an ifs. let x and y be two crisp populations and a be an ifs defined on x and y . let {x1,x2, . . . ,xn} be a linguistic random sample of x with membership and non-membership grades measured at the first time point µa(xi) and νa(xi), respectively where i = 1,2, . . . ,m. let {y1,y2, . . . ,yn} be another linguistic random sample of y with membership and non-membership grades measured at the second time point µa(yi) and νa(yi), respectively where i = 1,2, . . . ,m. suppose that µa(xi), νa(xi), µa(yi) and νa(yi) are normally distributed. the mean difference in the population x and y with respect to a, µd(a). if our objective is to test whether the mean differences in the population x and y with respect to a , µd(a) are different. then the null hypothesis is h0: µd(a)= 0. the difference between the two observations on each pair can be calculated by d(µi) = µa(xi)− µa(yi) and d(νi) = νa(xi)− νa(yi). the sample means of the differences are denoted by d(µ) and d(ν), respectively. the sample means of the differences can be defined as follows: d(µ)= n∑ i=1 d(µi) n and d(ν)= n∑ i=1 d(νi) n . the sample variances are given by (sµ(d)) 2 = n∑ i=1 (d(µi)−d(µ))2 n−1 and (sν(d)) 2 = n∑ i=1 (d(νi)−d(ν))2 n−1 . the test statistics are calculated as: tµ = d(µ)−0 (sµ(d))√ n int. j. anal. appl. (2022), 20:70 7 and tν = d(ν)−0 (sν(d))√ n . then t = max{|tµ|, |tν|}. the degree of freedom is df = n − 1. hence, the critical value is tα,df where α is significance level of the test. the critical region of the alternative hypothesis, ha is given below: alternative hypothesis critical region µd(a) > 0(upper-tailed test) t ≥ tα,df µd(a) < 0(lower-tailed test) t ≤−tα,df µd(a) 6=0(two-tailed test) |t| ≥ tα/2,df if |t| ≤ |tα,df | (one-tailed test), we fail to reject the null hypothesis. there is insufficient evidence to conclude that µd(a) > 0 (the mean difference in the populations with respect to a is greater than 0) for (upper-tailed test) or µd(a) < 0 (the mean difference in the populations with respect to a is less than 0) for (lower-tailed test) at α level. otherwise, the null hypothesis µd(a) ≤ 0 (for upper-tailed test) or µd(a)≥ 0 (for lower-tailed test) is rejected. if |t| ≤ tα/2,df (two-tailed test), there is not enough evidence to conclude that there is a difference in the mean populations with respect to a at α level . therefore, the null hypothesis is failed to reject. otherwise, the null hypothesis is rejected, that is, the mean difference in the populations with respect to a is not equal to zero. example 3.2. suppose that we assign a homework to students in prince of songkla university. let x and y be two populations where x be the sets of all assignment scores evaluated by lecturer 1 and y be the sets of all assignment scores evaluated by lecturer 2. we let the ifs a, where µ and ν be the membership and non-membership functions of lecturers’ satisfaction toward assignment, defined on x and y . it is assumed that µa(xi), νa(xi), µa(yi) and νa(yi) are normally distributed. now, we perform the test, that is, the mean differences in the population x and y with respect to a , µd(a) are different. then the null hypothesis is h0: µd(a)=0. we randomly select a sample of five students. let s1 = {x1,x2,x3,x4,x5} be the size five taken from assignment scores evaluated by lecturer 1 (the population x) and s2 = {y1,y2,y3,y4,y5} be the sample of size five taken from assignment scores evaluated by lecturer 2 (the population y ). then, the membership grades and the non-membership grades of the given two samples based on their information concerning the ifs a are given below. 8 int. j. anal. appl. (2022), 20:70 student 1 2 3 4 5 x1 x2 x3 x4 x5 µa(xi) 0.7 0.8 0.6 0.8 0.9 νa(xi) 0.3 0.1 0.2 0.1 0 y1 y2 y3 y4 y5 µa(yi) 0.7 0.9 0.7 0.8 0.7 νa(yi) 0.2 0.1 0.1 0.1 0 it is assumed that d(µi) = µa(xi)− µa(yi) and d(νi) = νa(xi)− νa(yi). hence, d(µ) = 0, d(ν)=0.04, (sµ(d))=0.12 and (sν(d))=0.05. the test statistics are tµ = d(µ)−0 (sµ(d))√ n =0 and tν = d(ν)−0 (sν(d))√ n =1.633. therefore t = max{|tµ|, |tν|} = 1.633. the critical value of t0.025,4 is 2.77. since 1.63 < 2.77, we cannot reject the null hypothesis. there is insufficient evidence to conclude that the mean differences in the population x and y with respect to a , µd(a) are different. 4. conclusion this article proposes two types of tests of statistical hypotheses based on the membership and non-membership functions of intuitionistic fuzzy sets which are completely different from conventional statistical hypothesis testing. in the proposed tests of hypotheses, the differences of means of the populations are investigated with the help of intuitionistic fuzzy sets. the rules for making decisions regarding the hypotheses are provided. each proposed statistical hypothesis test is a characteristic or attribute based test on the population. the proposed statistical hypotheses tests can assist decision makers for selecting an appropriate decision with satisfaction. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2022), 20:70 9 references [1] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [2] m.r. casals, m.a. gil, p. gil, the fuzzy decision problem: an approach to the problem of testing statistical hypotheses with fuzzy information, eur. j. oper. res. 27 (1986), 371-382. https://doi.org/10.1016/ 0377-2217(86)90333-4. [3] d. kalpanapriya, p. pandian, statistical hypotheses testing with imprecise data, appl. math. sci. 6 (2012), 5285-5292. [4] d. kalpanapriya, p. pandian, tests of statistical hypotheses with respect to a fuzzy set, modern appl. sci. 8 (2014), 25-35. https://doi.org/10.5539/mas.v8n1p25. [5] h.c. wu, statistical hypotheses testing for fuzzy data, inform. sci. 175 (2005), 30-56. https://doi.org/10. 1016/j.ins.2003.12.009. [6] h.c. wu, statistical confidence intervals for fuzzy data, expert syst. appl. 36 (2009), 2670-2676. https://doi. org/10.1016/j.eswa.2008.01.022. [7] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/0377-2217(86)90333-4 https://doi.org/10.1016/0377-2217(86)90333-4 https://doi.org/10.5539/mas.v8n1p25 https://doi.org/10.1016/j.ins.2003.12.009 https://doi.org/10.1016/j.ins.2003.12.009 https://doi.org/10.1016/j.eswa.2008.01.022 https://doi.org/10.1016/j.eswa.2008.01.022 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. basic concepts 2.1. intuitionistic fuzzy sets 2.2. one-sample t-test 3. main results 3.1. testing of significance for the difference between two independent population means with respect to an ifs 3.2. testing of significance for the difference between two dependent population means with respect to an ifs 4. conclusion references int. j. anal. appl. (2023), 21:18 on certain fixed point theorems in sb-metric spaces with applications n. mangapathi1,3,∗, b. srinuvasa rao2, k.r.k. rao3, m.i. pasha1,3 1department of mathematics, b v raju institute of technology, narsapur, medak-502313, telangana, india 2department of mathematics, dr.b.r.ambedkar university, srikakulam, etcherla-532410, andhra pradesh, india 3department of mathematics, gitam school of science, gitam deemed to be university, hyderabad, rudraram-502329, telangana, india ∗corresponding author: nmp.maths@gmail.com abstract. in this paper, we introduce the notion of generalized (α,φ,ψ)geraghty contractive type mappings in the setup of sb-metric spaces and α-orbital admissible mappings with respect to φ. furthermore, the fixed-point theorems for such mappings in complete sb-metric spaces are proven without assuming the subadditivity of ψ. some examples are provided for supporting of our main results. also, we gave an application to integral equations as well as homotopy. 1. introduction the banach contraction principle [1] is one of the most significant findings in fixed point theory since it has applications in many areas of mathematics and mathematical sciences. by combining the ideas of s and b-metric spaces, sedghi et al. [2] created sb-metric spaces and established common fixed point outcomes in sbmetric spaces. in order to improve, numerous authors developed numerous findings in sb-metric spaces (see e.g. [3], [4], [5], [6], [7], [8]). one of the more intriguing findings is geraghty’s [9] generalisation of the banach contraction theorem. multiple researchers have since studied this type of research in different metric spaces (see e.g [10], [11], [12], [13], [14], [15], [16], [17], [18]). received: dec. 9, 2022. 2020 mathematics subject classification. 54h25, 47h10, 54e50. key words and phrases. (α,φ,ψ)-geraghty type contraction mappings; sb-metric spaces; α-orbital admissible mappings with respect to φ and completeness. https://doi.org/10.28924/2291-8639-21-2023-18 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-18 2 int. j. anal. appl. (2023), 21:18 triangular α-admissible mappings are a novel idea that karapinar et al. [15] introduced to study fixed points for such mappings in metric spaces. three new concepts triangular α-orbital admissible, α -orbital admissible and α-orbital attractive mappings were developed by popescu [17] in 2014. the idea of triangular α-orbital admissible mappings with respect to η was first suggested in 2016 by chuadchawna et al. [19]. the concept of generalised α−η−ψ-geraghty contractive type mappings and α-orbital attractive mappings with regard to is introduced by farajzadeh et al. in 2018 [20] in the framework of partial b-metric spaces, which will be effectively applied for establishing our key findings. in the context of sb-metric spaces and α-orbital admissible mappings with regard to φ, this paper aims to demonstrate unique fixed point theorems for generalised (α,φ,ψ)-geraghty contractive type mapping. additionally, we may provide relevant applications for homotopy, integral equations, and appropriate examples. we first review some fundamental findings. 2. preliminaries definition 2.1. ( [2]) let p be a non-empty set and b ≥ 1 be given real number. suppose that a mapping sb : p3 → [0,∞) be a function satisfying the following properties : (sb1) 0 < sb(σ,ς,τ) for all σ,ς,τ ∈ p with σ 6= ς 6= τ 6= σ, (sb2) sb(σ,ς,τ) = 0 ⇔ σ = ς = τ, (sb3) sb(σ,ς,τ) ≤ b(sb(σ,σ,a) + sb(ς,ς,a) + sb(τ,τ,a)) for all σ,ς,τ,a ∈ p. the function sb is then referred to as a sb-metric on p, and the pair (p,sb) is referred to as a sb-metric space. remark 2.1. ( [2]) it should be noted that the sb-metric space class is effectively larger than the s-metric space class. each s-metric space is, in fact, a sb-metric space with b = 1. example 2.1. ( [2]) let (p,s) be s-metric space and s∗(σ,ς,τ) = s(σ,ς,τ)p, where p > 1 is a real number.then obviously, s∗ is a sb-metric with b = 22(p−1) but (p,s∗) is not necessarily a s-metric space. definition 2.2. ( [2]) let (p,sb) be a sb-metric space. then, for σ ∈ p, λ > 0 we defined the open ball bsb (σ,λ) and closed ball bsb [σ,λ] with center σ and radius λ as follows respectively: bsb (σ,λ) = {ς ∈ p : sb(ς,ς,σ) < λ} and bsb [σ,λ] = {ς ∈ p : sb(ς,ς,σ) ≤ λ}. lemma 2.1. ( [2]) in the sb-metric space, we have sb(µ,µ,ν) ≤ 2bsb(µ,µ,ξ) + b2sb(ξ,ξ,ν). definition 2.3. ( [2]) let {σn} be a sequence in sb-metric space (p,sb) is said to be: (1) sb-cauchy sequence if, for each � > 0, there exists n0 ∈n such that sb(σn,σn,σm) < � for each m,n ≥ n0. int. j. anal. appl. (2023), 21:18 3 (2) sb-convergent to a point σ ∈ p if, for each � > 0, there exists a positive integer n0 such that sb(σn,σn,σ) < � or sb(σ,,σ,σn) < � for all n ≥ n0 and we denote by lim n→∞ σn = σ. (3) (p,sb) is sb-complete if every sb-cauchy sequence is sb-convergent in p. lemma 2.2. ( [2]) if (p,sb) be a sb-metric space with b ≥ 1 and suppose that {σn} is a sb-convergent to σ, then we have (i) 1 2b sb(ς,ς,σ) ≤ lim n→∞ inf sb(ς,ς,σn) ≤ lim n→∞ sup sb(ς,ς,σn) ≤ 2bsb(ς,ς,σ) (ii) 1 b2 sb(σ,σ,ς) ≤ lim n→∞ inf sb(σn,σn,ς) ≤ lim n→∞ sup sb(σn,σn,ς) ≤ b2sb(σ,σ,ς) for all ς ∈ p. in particular, if σ = ς, then we have lim n→∞ sb(σn,σn,ς) = 0. we should always consider the following factors in order to obtain our results. 3. main results we say f be the class of all functions β : [0,∞) → [0, 1) satisfying the following condition: lim n→∞ β(tn) = 1 implies lim n→∞ tn = 0 definition 3.1. let g : p → p be a self mapping defined on non-empty set p and α,φ : p × p × p → r+ be two functions. we say that g is an αadmissible mapping with respect to φ, if α(σ,σ,ς) ≥ φ(σ,σ,ς) implies α(gσ,gσ,gς) ≥ φ(gσ,gσ,gς) for all σ,ς ∈ p. we say that g is an αadmissible mapping if for all σ,ς ∈ p, α(σ,σ,ς) ≥ 1 implies α(gσ,gσ,gς) ≥ 1. definition 3.2. let p be a non-empty set. g : p → p be a self mapping and α : p×p×p →r+. we say that g is a triangular αadmissible mapping, if (a) g is an αadmissible mapping; (b) α(σ,σ,ς) ≥ 1 and α(ς,ς,τ) ≥ 1 implies α(σ,σ,τ) ≥ 1 for all σ,ς,τ ∈ p. definition 3.3. let p be a non-empty set. g : p → p be a self mapping and α,φ : p×p×p →r+ be two functions. we say that g is an αorbital admissible mapping with respect to φ, if α(σ,σ,gσ) ≥ φ(σ,σ,gσ) implies α(gσ,gσ,g2σ) ≥ φ(gσ,gσ,g2σ) for all σ ∈ p. definition 3.4. let g : p → p be a self mapping defined on nonempty set p and α,φ : p × p × p →r+. we say that g is a triangular α-orbital admissible mapping with respect to φ, if (a) g is an α-orbital admissible mapping with respect to φ; (b) α(σ,σ,ς) ≥ φ(σ,σ,ς) and α(ς,ς,gς) ≥ φ(ς,ς,gς) imply α(σ,σ,gς) ≥ φ(σ,σ,gς) for all σ,ς ∈ p. 4 int. j. anal. appl. (2023), 21:18 definition 3.5. let ω = {γ/γ : [0,∞) → [0,∞)} be a family of functions that satisfy the following properties; (i) γ is a continuously nondecreasing map; (ii) γ(t) = 0 if and only if t = 0; (iii) γ(t) is subadditive, γ(p + q) ≤ γ(p) + γ(q). definition 3.6. let (p,sb) be an sb-metric space, a mapping g : p → p is said to be a generalized (α,φ, γ)-geraghty contractive type mapping if there exist γ ∈ ω, α,φ : p × p × p → [0,∞) and β ∈ f such that α(σ,σ,ς) ≥ φ(σ,σ,ς) implies γ ( ( 1 + b3 2 )sb(gσ,gσ,gς) ) ≤ β ( γ(mg b (σ,σ,ς)) ) γ(mg b (σ,σ,ς)) (3.1) where mg b (σ,σ,ς) = max { sb (σ,σ,ς) ,sb (σ,σ,gσ) , sb (ς,ς,gς) , sb(σ,σ,gς)+sb(ς,ς,gσ) 4b3 } ∀ σ,ς ∈ p lemma 3.1. let g : p → p be a triangular α-orbital admissible mapping with respect to φ. assume that there exists σ1 ∈ p such that α(σ1,gσ1) ≥ φ(σ1,gσ1). define a sequence {σn} by σn+1 = gσn. then we have α(σn,σm) ≥ φ(σn,σm) for all m,n ∈ n with n < m. proof. since α(σ1,σ1,gσ1) ≥ φ(σ1,σ1,gσ1) and g is α-orbital admissible with respect to φ, we obtain that α(σ2,σ2,σ3) = α(gσ1,gσ1,g(gσ1)) ≥ φ(gσ1,gσ1,g(gσ1)) = φ(σ2,σ2,σ3). by continuing the process as above, we have α(σn,σn,σn+1) ≥ γ(σn,σn,σn+1) for all n ∈ n. suppose that α(σn,σn,σm) ≥ φ(σn,σn,σm) (3.2) and we will prove that α(σn,σn,σm+1) ≥ φ(σn,σn,σm+1), where m > n. since α(σm,σm,σm+1) ≥ φ(σm,σm,σm+1), we obtain that α(σm,σm,gσm) = α(σm,σm,σm+1) ≥ φ(σm,σm,σm+1) = φ(σm,σm,gσm). (3.3) by (3.2), (3.3) and triangular α-orbital admissibility of g, we have α(σn,σn,gσm) ≥ φ(σn,σn,gσm). this implies that α(σn,σn,σm+1) ≥ φ(σn,σn,σm+1). hence, α(σn,σm) ≥ φ(σn,σm) for all m,n ∈ n with n < m. � theorem 3.1. let (p,sb) be a complete sb-metric space with coefficient b ≥ 1. let g : p → p be an be a generalized (α,φ, γ)-geraghty contractive type mapping. assume the following conditions are hold: (i) g is a triangular α-orbital admissible mapping w.r.t φ; (ii) there exists σ1 ∈ p such that α(σ1,σ1,gσ1) ≥ φ(σ1,σ1,gσ1); int. j. anal. appl. (2023), 21:18 5 (iii) if {σn} is a sb-convergent sequence to ν in p and α(σn,σn,σn+1) ≥ φ(σn,σn,σn+1) for each n ∈ n then α(ν,ν,ν) ≥ φ(ν,ν,ν); (iv) g is continuous. then g has a fixed point. proof. let σ1 ∈ p such that α(σ1,σ1,gσ1) ≥ φ(σ1,σ1,gσ1). define the sequence {σn} in p by σn+1 = tσn for all n ∈ n. by lemma(2.1), we get that α(σn,σn,σn+1) ≥ φ(σn,σn,σn+1) for all n ∈ n. (3.4) if σn = σn+1 for some n ∈ n, then σn is a fixed point of g. assume that σn 6= σn+1 for all n ∈ n. the sequence {sb(σn,σn,σn+1)} is first shown to be non-increasing and to trend to 0 as n →∞. by using (3.4), for each n ∈ n, we have γ ( ( 1 + b3 2 )sb(σn+1,σn+1,σn+2) ) = γ ( ( 1 + b3 2 )sb(gσn,gσn,gσn+1) ) ≤ β ( γ(mg b (σn,σn,σn+1)) ) γ(mg b (σn,σn,σn+1)) < γ(mg b (σn,σn,σn+1)) (3.5) where, mg b (σn,σn,σn+1) = max { sb (σn,σn,σn+1) ,sb (σn,σn,gσn) , sb (σn+1,σn+1,gσn+1) , sb(σn,σn,gσn+1)+sb(σn+1,σn+1,gσn) 4b3 } = max { sb (σn,σn,σn+1) ,sb (σn,σn,σn+1) , sb (σn+1,σn+1,σn+2) , sb(σn,σn,σn+2)+sb(σn+1,σn+1,σn+1) 4b3 } = max { sb (σn,σn,σn+1) ,sb (σn,σn,σn+1) , sb (σn+1,σn+1,σn+2) , 2bsb(σn,σn,σn+1)+b 2sb(σn+1,σn+1,σn+2) 4b3 } = max { sb (σn,σn,σn+1) ,sb (σn+1,σn+1,σn+2) } . if max { sb (σn,σn,σn+1) ,sb (σn+1,σn+1,σn+2) } = sb (σn+1,σn+1,σn+2) then γ ( ( 1+b 3 2 )sb(σn+1,σn+1,σn+2) ) < γ(sb (σn+1,σn+1,σn+2)) implies that ( 1+b 3 2 )sb(σn+1,σn+1,σn+2) < sb (σn+1,σn+1,σn+2) which is contradiction. hance, max { sb (σn,σn,σn+1) ,sb (σn+1,σn+1,σn+2) } = sb (σn,σn,σn+1). it follows that 0 < sb(σn+1,σn+1,σn+2) ≤ sb(σn,σn,σn+1). hence the sequence {sb(σn,σn,σn+1)} is nonnegative non-increasing and bounded below. thus there exist some ξ ≥ 0 such that lim n→∞ sb (σn,σn,σn+1) = ξ. suppose that ξ > 0. 6 int. j. anal. appl. (2023), 21:18 by using (3.5), we have γ (sb(σn+1,σn+1,σn+2)) γ (sb(σn,σn,σn+1)) ≤ γ ( ( 1+b 3 2 )sb(σn+1,σn+1,σn+2) ) γ (sb(σn,σn,σn+1)) ≤ β ( γ(mg b (σn,σn,σn+1)) ) < 1, for all n ∈ n. on letting n →∞ in above inequality, we have lim n→∞ β ( γ(mg b (σn,σn,σn+1)) ) = 1. since β ∈ f, we have lim n→∞ γ(mg b (σn,σn,σn+1)) = 0 and so ξ = lim n→∞ sb (σn,σn,σn+1) = 0. we now demonstrate that the (p,sb) cauchy sequence is {σn}. on the other hand, we assume that {σn} is not cauchy. in this case, a monotonically rising sequence of the natural numbers {mk} and {nk} exists, where nk > mk. sb ( σmk,σmk,σnk ) ≥ � (3.6) and sb ( σmk,σmk,σnk−1 ) < �. (3.7) from (3.6) and (3.7), we have � ≤ sb ( σmk,σmk,σnk ) ≤ 2bsb ( σmk,σmk,σmk +1 ) + b2sb ( σmk +1,σmk +1,σnk ) . so that ( 1+b 3 2b2 )� ≤ ( 1+b 3 b )sb ( σmk,σmk,σmk +1 ) + ( 1+b 3 2 )sb ( σmk +1,σmk +1,σnk ) . we obtain that by applying γ on both sides and letting k →∞. γ(( 1 + b3 2b2 )�) ≤ lim k→∞ γ ( ( 1 + b3 2 )sb ( σmk +1,σmk +1,σnk )) ≤ lim k→∞ γ ( ( 1 + b3 2 )sb ( gσmk,gσmk,gσnk−1 )) . (3.8) by applying the triangular inequality, we get that sb ( σmk +1,σmk +1,σnk ) ≤ 2bsb ( σmk +1,σmk +1,σmk ) + b2sb ( σmk,σmk,σnk ) . ≤ 2bsb ( σmk +1,σmk +1,σmk ) + 2b3sb ( σmk,σmk,σnk−1 ) +b3sb ( σnk,σnk,σnk−1 ) ≤ 2bsb ( σmk +1,σmk +1,σmk ) + 2b3� + b3sb ( σnk,σnk,σnk−1 ) . in the above inequality, we get that by taking the limit as k →∞. lim k→∞ sb ( σmk +1,σmk +1,σnk ) ≤ 2b3�. (3.9) int. j. anal. appl. (2023), 21:18 7 we obtain (3.5) because g is a triangular α-orbital admissible mapping with respect to φ. and α(σmk,σmk,σnk−1) ≥ φ(σmk,σmk,σnk−1). by using (3.1), we have γ ( ( 1 + b3 2 )sb(σmk +1,σmk +1,σnk ) ) ≤ β ( γ(mgb (σmk,σmk,σnk−1)) ) γ(mgb (σmk,σmk,σnk−1)) (3.10) where, mg b (σmk,σmk,σnk−1) = max   sb ( σmk,σmk,σnk−1 ) ,sb ( σmk,σmk,gσmk ) , sb ( σnk−1,σnk−1,gσnk−1 ) , sb(σmk ,σmk ,gσnk−1)+sb(σnk−1,σnk−1,gσmk ) 4b3   ≤ max   sb ( σmk,σmk,σnk−1 ) ,sb ( σmk,σmk,σmk +1 ) ,sb ( σnk−1,σnk−1,σnk ) , 2bsb(σmk ,σmk ,σnk−1)+b 2sb(σnk−1,σnk−1,σnk )+sb(σnk−1,σnk−1,σmk +1) 4b3   ≤ max   sb ( σmk,σmk,σnk−1 ) ,sb ( σmk,σmk,σmk +1 ) ,sb ( σnk−1,σnk−1,σnk ) , 2bsb(σmk ,σmk ,σnk−1)+b 2sb(σnk−1,σnk−1,σnk ) 4b3 + 2bsb(σnk−1,σnk−1,σnk )+bsb(σmk +1,σmk +1,σnk ) 4b3   using (3.7) and (3.9) and treating the limit of the inequality above as k →∞, this results. lim k→∞ mg b (σmk,σmk,σnk−1) ≤ lim k→∞ max   sb ( σmk,σmk,σnk−1 ) ,sb ( σmk,σmk,σmk +1 ) ,sb ( σnk−1,σnk−1,σnk ) , 2bsb(σmk ,σmk ,σnk−1)+b 2sb(σnk−1,σnk−1,σnk ) 4b3 + 2bsb(σnk−1,σnk−1,σnk )+bsb(σmk +1,σmk +1,σnk ) 4b3   ≤ max { �, 0, 0,�( 1+b 3 2b2 ) } = �( 1 + b3 2b2 ) (3.11) by taking the limit in (3.10) as k →∞ and using (3.8) and (3.11), we have γ(( 1 + b3 2b2 )�) ≤ γ ( lim k→∞ ( 1 + b3 2 )sb ( σmk +1,σmk +1,σnk )) . ≤ β ( γ( lim k→∞ mg b (σmk,σmk,σnk−1)) ) γ( lim k→∞ mg b (σmk,σmk,σnk−1)) ≤ β ( γ( lim k→∞ mg b (σmk,σmk,σnk−1)) ) γ(�( 1 + b3 2b2 )) this implies that γ(( 1+b 3 2b2 )�) γ(( 1+b 3 2b2 )�) ≤ β ( γ ( lim k→∞ mg b (σmk,σmk,σnk−1) )) . since β ∈ f, we have lim n→∞ β ( γ( lim k→∞ mg b (σmk,σmk,σnk−1)) ) = 1. 8 int. j. anal. appl. (2023), 21:18 it follows that γ( lim k→∞ mg b (σmk,σmk,σnk−1)) = 0. by using (3.10) we obtain lim n→∞ sb(σmk +1,σmk +1,σnk ) = 0. which contradicts to(3.9).in the sb-metric space (p,sb), the sequence {σn} is a sb-cauchy sequence. the sequence {σn} → ν ∈ (p,sb) emerges from the completeness of (p,sb). we begin by presuming that g is continuous. therefore, we have ν = lim n→∞ σn+1 = lim n→∞ gσn = g lim n→∞ σn = gν. since {σn} is a sb-convergent sequence to ν in p and α(ν,ν,ν) ≥ φ(ν,ν,ν). then to prove ν = gν. suppose that ν 6= gν. from (3.1), we have γ (sb(ν,ν,gν)) ≤ γ ( ( 1 + b3 2 )sb(ν,ν,gν) ) = γ ( ( 1 + b3 2 ) lim n→∞ sb(σn+1,σn+1,gσn+1) ) = γ ( ( 1 + b3 2 ) lim n→∞ sb(gσn,gσn,gσn+1) ) ≤ β ( γ( lim n→∞ mg b (σn,σn,σn+1)) ) γ( lim n→∞ mg b (σn,σn,σn+1)) (3.12) where lim n→∞ mg b (σn,σn,σn+1) = lim n→∞ max   sb (σn,σn,σn+1) ,sb (σn,σn,gσn) , sb (σn+1,σn+1,gσn+1) , sb(σn,σn,gσn+1)+sb(σn+1,σn+1,gσn) 4b3   = max { sb (ν,ν,ν) ,sb (ν,ν,gν) , sb (ν,ν,gν) , sb(ν,ν,gν) 2b3 } = sb (ν,ν,gν) . by taking limit as n →∞ in (3.12), we have γ (sb(ν,ν,gν)) ≤ β ( γ( lim n→∞ mg b (σn,σn,σn+1)) ) γ( lim n→∞ mg b (σn,σn,σn+1)) ≤ β (γ(sb (ν,ν,gν))) γ(sb (ν,ν,gν)) we can deduce that γ(sb(ν,ν,gν)) γ(sb(ν,ν,gν)) ≤ β (γ(sb (ν,ν,gν))) we obtain that lim n→∞ β (γ(sb (ν,ν,gν))) = 1. therefore, sb (ν,ν,gν) = 0 implies gν = ν. and thus ν is a fixed point of g. assume further that ν and ν∗ are two fixed points of g such that ν 6= ν∗. consider γ (sb(ν,ν,ν ∗)) ≤ γ ( ( 1 + b3 2 )sb(gν,gν,gν∗) ) ≤ β ( γ(mg b (ν,ν,ν∗)) ) γ(mg b (ν,ν,ν∗)) (3.13) where mg b (ν,ν,ν∗) = max { sb (ν,ν,ν ∗) ,sb (ν,ν,gν) , sb (ν ∗,ν∗,gν∗) , sb(ν,ν,gν ∗)+sb(ν ∗,ν∗,gν) 4b3 } ≤ max { sb (ν,ν,ν ∗) ,sb (ν,ν,ν) , sb (ν ∗,ν∗,ν∗) , sb(ν,ν,ν ∗)+2bsb(ν ∗,ν∗,ν∗)+sb(ν,ν,ν ∗) 4b3 } = sb (ν,ν,ν ∗) int. j. anal. appl. (2023), 21:18 9 using by (3.13), we have γ (sb(ν,ν,ν∗)) ≤ β (γ(sb (ν,ν,ν∗))) γ(sb (ν,ν,ν∗)). we can deduce that γ(sb(ν,ν,ν ∗)) γ(sb(ν,ν,ν ∗)) ≤ β (γ(sb (ν,ν,ν∗))). we obtain that lim n→∞ β (γ(sb (ν,ν,ν ∗))) = 1. therefore, sb (ν,ν,ν∗) = 0 implies ν = ν∗. consequently, ν is a unique fixed point of g. � example 3.1. let sb : p3 → r+ be defined as sb(µ,ν,ξ) = (|ν + ξ − 2µ| + |ν − ξ|)2 where p = [0,∞). it is obvious that (p,sb) is a complete with b = 4. define g : p → p by g(µ) = µ43 and γ : [0,∞) → [0,∞) and β : [0,∞) → [0, 1) as γ(t) = t, β(t) =   e − 4 5 3969 t 1+ 4 5 3969 t , t ∈ (0,∞) 0, t = 0 also define α,φ : p×p×p →r+ α(µ,µ,ν) = { 4, (µ,µ,ν) ∈ [0, 1] 0, otherwise φ(µ,µ,ν) = { 1, (µ,µ,ν) ∈ [0, 1] 0, otherwise let α(µ,µ,gµ) ≥ φ(µ,µ,gµ). thus µ,gµ ∈ [0, 1] and so g2µ = g(gµ) ∈ [0, 1] which implies that α(gµ,gµ,g2µ) ≥ (gµ,gµ,g2µ) that is g is α-orbital admissible with respect to φ. now, let α(µ,µ,ν) ≥ φ(µ,µ,ν) and α(ν,ν,gν) ≥ φ(ν,ν,gν), we get that µ,ν,gν ∈ [0, 1] and so α(µ,µ,gν) ≥ φ(µ,µ,gν). therefore g is triangular α-orbital admissible with respect to φ. let {µn} be a sequence such that {µn} is sb-convergent to χ and α(µn,µn,µn+1) ≥ φ(µn,µn,µn+1) for all n ∈ n. then {µn} ∈ [0, 1] for any n ∈ n and so χ ∈ [0, 1] which we have, α(χ,χ,χ) ≥ φ(χ,χ,χ) and obviously the function g is continuous. following that, we show that g is a generalised (α,φ, γ)-geraghty contraction type mapping. let µ,ν ∈ p with α(µ,µ,ν) ≥ φ(µ,µ,ν). thus µ,ν ∈ [0, 1].we can assume without losing generality that 0 ≤ ν ≤ µ ≤ 1. therefore, sb(gµ,gµ,gν) = (|gµ + gν − 2gµ| + |gµ−gν|)2 = ( 2 ∣∣∣ µ 43 − ν 43 ∣∣∣)2 = 1 46 sb(µ,µ,ν) and mg b (µ,µ,ν) = max { (2|µ−ν|)2, 3969 45 µ2, 3969 45 ν2 ( (|43µ−ν|)2+(|µ−43ν|)2 46b3 } = 3969 45 µ2 since 65 2×46 ≤ 1 2e ≤ e −µ2 1+µ2 so that 65 2×46 3969 45 µ2 ≤ e −µ2 1+µ2 3969 45 µ2 γ ( ( 1+b 3 2 )sb(gµ,gµ,gν) ) = γ ( 65 2×46 sb(µ,µ,ν) ) = 65 2×46 sb(µ,µ,ν) ≤ 65 2×46 3969 45 µ2 ≤ e −µ2 1+µ2 3969 45 µ2 ≤ β ( γ( 3969 45 µ2) ) γ( 3969 45 µ2) ≤ β ( γ(mg b (µ,µ,ν)) ) γ(mg b (µ,µ,ν)) as a result, all of theorem (3.1)’s requirements are satisfied, and 0 is the only fixed point of g. 10 int. j. anal. appl. (2023), 21:18 4. application to integral equations as an application of theorem (3.1), we will look at the existence of a unique solution to an initial value problem in this section. theorem 4.1. consider the i. v. p. σ′(t) = g(t,σ(t)), t ∈ i = [0, 1], σ(0) = σ0 (4.1) where g : i ×r→r is a continuous function and σ0 ∈r. let γ : [0,∞) → [0,∞),β : [0,∞) → [0, 1) be a two functions defined as γ(t) = t and β(t) = 1 3 . also examine the following conditions, (i) if there exist a function α,φ : r3 →r such that there is an σ1 ∈ c(i), for all t ∈ i,we’ve α  σ1(t),σ1(t), t∫ 0 g(s,σ1(s))ds   ≥ φ  σ1(t),σ1(t), t∫ 0 g(s,σ1(s))ds   . (ii) ∀ t ∈ i, and ∀ x,y ∈ c(i), α (σ(t),σ(t),ς(t)) ≥ α (σ(t),σ(t),ς(t)) ⇒ α ( σ0 3b3 + t∫ 0 g(s,σ(s))ds, σ0 3b3 + t∫ 0 g(s,σ(s))ds, ς0 3b3 + t∫ 0 g(s,ς(s))ds ) ≥ φ ( σ0 3b3 + t∫ 0 g(s,σ(s))ds, σ0 3b3 + t∫ 0 g(s,σ(s))ds, ς0 3b3 + t∫ 0 g(s,ς(s))ds ) . (iii) for any point σ of a sequence {σn} of points in c(i) with α (σn,σn,σn+1) ≥ φ (σn,σn,σn+1), lim n→∞ inf α (σn,σn,σ) ≥ lim n→∞ inf φ (σn,σn,σ). then (4.1) has unique solution . proof. the integral equation of i. v. p. ( 4.1 ) is σ(t) = σ0 + 3( 1 + b3 2 ) t∫ 0 g(s,σ(s))ds. let p = c (i) and sb(σ,ς,τ) = (|ς + τ −2σ|+ |ς−τ|)2 for σ,ς,ς ∈ p. then (p,sb) is a complete, also define t : p → p by t (σ)(t) = 2σ0 3(1 + b3) + t∫ 0 g(s,σ(s))ds. (4.2) now γ ( ( 1 + b3 2 )sb(tσ(t),tσ(t),tς(t)) ) = ( 1 + b3 2 ){| tσ(t) + tς(t) − 2tσ(t) | + | tσ(t) −tς(t) |}2 int. j. anal. appl. (2023), 21:18 11 = 8(1+b3) 9(1+b3)2 ∣∣∣∣ σ0 + 3( 1+b32 ) t∫ 0 g(s,σ(s))ds − ς0 − 3( 1+b 3 2 ) t∫ 0 g(s,ς(s))ds ∣∣∣∣2 = 8 9(1+b3) | σ(t) − ς(t) |2 = 8 9(1+b3) sb(σ,σ,ς) ≤ 13 sb(σ,σ,ς) ≤ β (γ(sb(σ,σ,ς))) γ(sb(σ,σ,ς)) ≤ β ( γ(mtb (σ,σ,ς)) ) γ(mtb (σ,σ,ς)) thus we have γ ( ( 1+b 3 2 )sb(tσ(t),tσ(t),tς(t)) ) ≤ β ( γ(mtb (σ,σ,ς)) ) γ(mtb (σ,σ,ς))∀σ,ς ∈ p let us define α : p×p×p → [0,∞) and φ : p×p×p → [0,∞) by α(σ,σ,ς) = { 6, σ,ς ∈ [0, 1] 0, σ,ς ∈ (1,∞) , φ(σ,σ,ς) = { 2, σ,ς ∈ [0, 1] 1, σ,ς ∈ (1,∞) then obviously, t is triangular α-orbital admissible with respect to φ. let σ,ς ∈ p, if α(σ,σ,ς) = 6 and φ(σ,σ,ς) = 2, then α (σ(t),σ(t),ς(t)) ≥ φ (σ(t),σ(t),ς(t)). from (ii) we have α (tσ(t),tσ(t),tς(t)) ≥ φ (tσ(t),tσ(t),tς(t)) and so α(tσ,tσ,tς) = 6 and φ(tσ,tσ,tς) = 2. thus t is is triangular α-orbital admissible with respect to φ. from (i), there exist σ1,ς1 ∈ p such that α(σ1,σ1,tσ1) = 6 and φ(ς1,ς1,tς1) = 2. by (iii), we have that for any point σ of a sequence {σn} of points in c(i) with α (σn,σn,σn+1) = 6, lim n→∞ inf α (σn,σn,σ) = 6 and φ (σn,σn,σn+1) = 2, lim n→∞ inf φ (σn,σn,σ) = 2. therefore, for all σ,ς ∈ p and t ∈ i, we have α (σ(t),σ(t),ς(t)) ≥ φ (σ(t),σ(t),ς(t)) =⇒ γ ( ( 1+b 3 2 )sb(tσ(t),tσ(t),tς(t)) ) ≤ β ( γ(mtb (σ,σ,ς)) ) γ(mtb (σ,σ,ς)) theorem (3.1) states that t has a unique solution in p. � 5. application to homotopy the existence of a unique homotopy theory solution is investigated in this section. theorem 5.1. let (p,sb) be a complete sb-metric space, u and u be a open and closed subset of p such that u ⊆ u . suppose α,φ : p × p × p → [0,∞), hb : u × [0, 1] → p is a triangular α-orbital admissible operator with respect to φ and β ∈ f satisfying the following conditions: (τ0) σ 6= hb(σ,κ), for each σ ∈ ∂u and κ ∈ [0, 1] (here ∂u is boundary of u in p) (τ1) for all σ,ς ∈u and κ ∈ [0, 1], α(σ,σ,hb(σ,κ)) ≥ φ(σ,σ,hb(σ,κ)) implies γ ( ( 1 + b3 2 )sb(hb(σ,κ),hb(σ,κ),hb(ς,κ)) ) ≤ β (γ(sb(σ,σ,ς))) γ(sb(σ,σ,ς)) (τ2) ∃ mb ≥ 0 3 sb(hb(σ,κ),hb(σ,κ),hb(σ,ζ)) ≤ mb|κ−ζ| for every σ ∈uand κ,ζ ∈ [0, 1]. then hb(., 0) has a fixed point ⇐⇒ hb(., 1) has a fixed point. 12 int. j. anal. appl. (2023), 21:18 proof. let a = {κ ∈ [0, 1] : σ = hb(σ,κ) for some σ ∈ u}. we have that 0 ∈ a since hb(., 0) has a fixed point in u. as a result, the set a is not empty. by demonstrating that a is both open and closed in [0, 1], we shall establish that a = [0, 1]. as a result, in u, hb(, 1) has a fixed point.first, we demonstrate that a is a closed set in [0, 1]. let {κn}∞n=1 ⊆ a with κn → κ ∈ [0, 1] as n → ∞. we have to demonstrate that κ ∈ a. since κn ∈ a for n = 0, 1, 2, · · · . there exists σn ∈ u with σn+1 = hb(σn,κn). since hb is a triangular α-orbital admissible operator with respect to φ. α(σ0,σ0,hb(σ0,κ0)) ≥ φ(σ0,σ0,hb(σ0,κ0)). we can deduce from lemma (2.1) that α(σn,σn,σn+1) ≥ φ(σn,σn,σn+1) for all n ≥ 0 consider, sb(σn+1,σn+1,σn+2) = sb (hb(σn,κn),hb(σn,κn),hb(σn+1,κn+1)) ≤ 2bsb (hb(σn,κn),hb(σn,κn),hb(σn+1,κn)) +b2sb (hb(σn+1,κn),hb(σn+1,κn),hb(σn+1,κn+1)) ≤ 2bsb (hb(σn,κn),hb(σn,κn),hb(σn+1,κn)) + b2mb|κn −κn+1| letting n →∞, we get lim n→∞ sb(σn+1,σn+1,σn+2) ≤ lim n→∞ 2bsb (hb(σn,κn),hb(σn,κn),hb(σn+1,κn)) . we get γ since it is continuous and non-decreasing. lim n→∞ γ ( ( 1 + b3 4b )sb(σn+1,σn+1,σn+2) ) = lim n→∞ γ ( ( 1 + b3 2 )sb (hb(σn,κn),hb(σn,κn),hb(σn+1,κn)) ) ≤ lim n→∞ β (γ(sb(σn,σn,σn+1))) γ(sb(σn,σn,σn+1)) therefore, lim n→∞ γ ( ( 1+b 3 4b )sb(σn+1,σn+1,σn+2) ) lim n→∞ γ(sb(σn,σn,σn+1)) ≤ lim n→∞ β (γ(sb(σn,σn,ςn+1))) < 1. in above inequality, we have lim n→∞ β (γ(sb(σn,σn,σn+1))) = 1. since β ∈ f, we have lim n→∞ γ(sb(σn,σn,σn+1)) = 0 and so lim n→∞ sb (σn,σn,σn+1) = 0. it is now time to demonstrate {σn}, a sb-cauchy sequence in (p,sb). on the other hand, suppose {σn} is not a sb-cauchy sequence. there is a monotone increasing sequence with � > 0 and natural numbers with the property that {mk} and {nk} such that nk > mk. sb ( σmk,σmk,σnk ) ≥ � (5.1) and sb ( σmk,σmk,σnk−1 ) < �. (5.2) int. j. anal. appl. (2023), 21:18 13 from (5.1) and (5.2), we have � ≤ sb ( σmk,σmk,σnk ) ≤ 2bsb ( σmk,σmk,σmk +1 ) + b2sb ( σmk +1,σmk +1,σnk ) . so that ( 1+b 3 2b2 )� ≤ ( 1+b 3 b )sb ( σmk,σmk,σmk +1 ) + ( 1+b 3 2 )sb ( σmk +1,σmk +1,σnk ) . we get that by setting k →∞ and γ is applied on both sides, γ(( 1 + b3 2b2 )�) ≤ lim k→∞ γ ( ( 1 + b3 2 )sb ( σmk +1,σmk +1,σnk )) ≤ lim n→∞ β ( γ(sb(σmk,σmk,σnk−1)) ) γ(sb(σmk,σmk,σnk−1)) ≤ lim n→∞ β ( γ(sb(σmk,σmk,σnk−1)) ) γ(( 1 + b3 2b2 )�). that is 1 ≤ lim n→∞ β ( γ(sb(σmk,σmk,σnk−1)) ) ⇒ lim n→∞ β ( γ(sb(σmk,σmk,σnk−1)) ) = 1 . this leads to the result lim n→∞ sb(σmk,σmk,σnk−1) = 0. and hence, lim n→∞ sb(σmk +1,σmk +1,σnk ) = 0.it contradicts itself. in the sb-metric space (p,sb), the sequence {σn} is a sb-cauchy sequence.the sequence {σn} → ν ∈ (p,sb) emerges from the completeness of (p,sb). lim n→∞ σn+1 = ν = lim n→∞ σn. since {σn} is a sb-convergent sequence to ν in x and α(ν,ν,ν) ≥ φ(ν,ν,ν). then to prove ν = hb(ν,κ). now γ ( 1 2b sb(hb(ν,κ),hb(ν,κ),ν) ) ≤ lim n→∞ inf γ (sb(hb(ν,κ),hb(ν,κ),hb(σn,κ))) ≤ lim n→∞ inf γ ( ( 1 + b3 2 )sb(hb(ν,κ),hb(ν,κ),hb(σn,κ)) ) ≤ lim n→∞ β (γ(sb(ν,ν,σn))) γ(sb(ν,ν,σn)). so that γ ( 1 2b sb(hb(ν,κ),hb(ν,κ),ν) ) lim n→∞ γ(sb(ν,ν,σn)) ≤ lim n→∞ β (γ(sb(ν,ν,σn))) that is 1 ≤ lim n→∞ β (γ(sb(ν,ν,σn))) implies lim n→∞ β (γ(sb(ν,ν,σn))) = 1. as a result, we obtain lim n→∞ γ(sb(ν,ν,σn)) = 0 and hence sb(hb(ν,κ),hb(ν,κ),ν) = 0. thus, it follows ν = hb(ν,κ). thus κ ∈ a. clearly, [0, 1] closes a. let κ0 ∈ a.consequently, there is σ0 ∈ u such that σ0 = hb(σ0,κ0). due to the fact that uis open, r > 0 exists such that bsb (σ0, r) ⊆ u. choose κ ∈ (κ0 − �,κ0 + �) such that |κ−κ0| ≤ 1mn < �. then, for bp(σ0, r) = {σ ∈ p : sb(σ,σ,σ0) ≤ r + b2sb(σ0,σ0,σ0)}. now sb (hb(σ,κ),hb(σ,κ),σ0)) = sb (hb(σ,κ),hb(σ,κ),hb(σ0,κ0)) ≤ 2bsb (hb(σ,κ),hb(σ,κ),hb(σ,κ0)) +b2sb (hb(σ,κ0),hb(σ,κ0),hb(σ0,κ0)) ≤ 2bm|κ−κ0| + b2sb (hb(σ,κ0),hb(σ,κ0),hb(σ0,κ0)) 14 int. j. anal. appl. (2023), 21:18 upon letting n →∞ and applying γ to both sides, γ (sb (hb(σ,κ),hb(σ,κ),σ0))) ≤ γ ( b2sb (hb(σ,κ0),hb(σ,κ0),hb(σ0,κ0)) ) ≤ γ ( ( 1 + b3 2 )sb (hb(σ,κ0),hb(σ,κ0),hb(σ0,κ0)) ) ≤ β (γ(sb(σ,σ,σ0))) γ(sb(σ,σ,σ0)) ≤ γ(sb(σ,σ,σ0)). therefore, sb (hb(σ,κ),hb(σ,κ),σ0)) ≤ sb(σ,σ,σ0) ≤ r + b2sb(σ0,σ0,σ0). thus for each fixed κ ∈ (κ0 −�,κ0 + �), hb(.; κ) : bp(σ0, r) → bp(σ0, r). thus, theorem (5.1)’s criteria are met in full. consequently, it may be said that hb(.; κ) has a fixed point in u. but this must be in u. therefore, κ ∈ a for κ ∈ (κ0 −�,κ0 + �). hence (κ0 −�,κ0 + �) ⊆ a. in [0, 1], a is clearly open. the converse can be proven using a similar method. � conclusion: using generalised (α,φ, γ)-geraghty contractive type fixed point theorems in the setup of sb-metric spaces through α-orbital admissible mappings with respect to φ, we conclude several applications to homotopy theory and integral equations in this study. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. banach, sur les operations dans les ensembles abstraits et leur applications aux equations integrales, fund. math. 3 (1922), 133-181. https://eudml.org/doc/213289. [2] s. radenovic, s. sedghi, a. gholidahneh, t. dosenovic, j. esfahani, common fixed point of four maps in -metric spaces, j. linear topol. algebra, 5 (2016), 93-104. [3] n. souayaha, n. mlaikib, a fixed point theorem in sb-metric spaces, j. math. computer sci. 16 (2016), 131-139. https://doi.org/10.22436/jmcs.016.02.01. [4] m.v.r. kameswari, k. sujatha, d.m.k kiran, fixed point results in sb-metric spaces via (α,ψ,φ)generalized weakly contractive maps, glob. j. pure appl. math. 15 (2019), 411-428. [5] n. souayah, a fixed point in partial sb-metric spaces, an. univ. "ovidius" constanta ser. mat. 24 (2016), 351-362. https://doi.org/10.1515/auom-2016-0062. [6] y. rohen, t. dosenovic, s. radenovic, a note on the paper "a fixed point theorems in sb-metric spaces", filomat. 31 (2017), 3335-3346. https://doi.org/10.2298/fil1711335r. [7] b. srinuvasa, g.n. venkata ki, m. sarwar, n. konda redd, fixed point theorems in ordered sb-metric spaces by using (α,β)-admissible geraghty contraction and applications, j. appl. sci. 18 (2017), 9-18. https://doi.org/ 10.3923/jas.2018.9.18. [8] g. kishore, k. rao, d. panthi, b. srinuvasa rao, s. satyanaraya, some applications via fixed point results in partially ordered sb-metric spaces, fixed point theory appl. 2017 (2016), 10. https://doi.org/10.1186/ s13663-017-0603-2. [9] m.a. geraghty, on contractive mappings, proc. amer. math. soc. 40 (1973), 604-608. [10] a. amini-harandi, h. emami, a fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, nonlinear anal.: theory methods appl. 72 (2010), 2238-2242. https://doi.org/10.1016/j.na.2009.10.023. https://eudml.org/doc/213289 https://doi.org/10.22436/jmcs.016.02.01 https://doi.org/10.1515/auom-2016-0062 https://doi.org/10.2298/fil1711335r https://doi.org/10.3923/jas.2018.9.18 https://doi.org/10.3923/jas.2018.9.18 https://doi.org/10.1186/s13663-017-0603-2 https://doi.org/10.1186/s13663-017-0603-2 https://doi.org/10.1016/j.na.2009.10.023 int. j. anal. appl. (2023), 21:18 15 [11] m.m.m. jaradat, z. mustafa, a.h. ansari, p.s. kumari, d. dolicanin-djekic, h.m. jaradat, some fixed point results for fα−ω,ϕ-generalized cyclic contractions on metric-like space with applications to graphs and integral equations, j. math. anal. 8 (2017), 28-45. [12] j.r. roshan, v. parvaneh, s. radenovic, m. rajovic, some coincidence point results for generalized (α,β)-weakly contractions in ordered b-metric spaces, fixed point theory appl. 2015 (2015), 68. https://doi.org/10.1186/ s13663-015-0313-6. [13] m. dinarvand, some fixed point results for admissible geraghty contraction type mappings in fuzzy metric spaces, iran. j. fuzzy syst. 14 (2017), 161-177. [14] s.h. cho, j.s. bae, e. karapinar, fixed point theorems for α-geraghty contraction type maps in metric spaces, fixed point theory appl. 2013 (2013), 329. https://doi.org/10.1186/1687-1812-2013-329. [15] e. karapinar, α − ψ-geraghty contraction type mappings and some related fixed point results, filomat. 28 (2014), 37-48. https://www.jstor.org/stable/24896705. [16] a. mukheimer, α − ψ − ϕ-contractive mappings in ordered partial b-metric spaces, j. nonlinear sci. appl. 7 (2014), 168-179. [17] o. popescu, some new fixed point theorems for α-geraghty contraction type maps in metric spaces, fixed point theory appl. 2014 (2014), 190. https://doi.org/10.1186/1687-1812-2014-190. [18] p. salimi, a. latif, n. hussain, modified α−ψ-contractive mappings with applications, fixed point theory appl. 2013 (2013), 151. https://doi.org/10.1186/1687-1812-2013-151. [19] p. chuadchawna, a. kaewcharoen, s. plubtieng, fixed point theorems for generalized α − η − ψ-geraghty contraction type mappings in α − η-complete metric spaces, j. nonlinear sci. appl. 09 (2016), 471-485. https://doi.org/10.22436/jnsa.009.02.13. [20] a. farajzadeh, c. noytaptim, a. kaewcharoen, some fixed point theorems for generalized α−η −ψ-geraghty contractive type mappings in partial b-metric spaces, j. inform. math. sci. 10 (2018), 455–478. https://doi.org/10.1186/s13663-015-0313-6 https://doi.org/10.1186/s13663-015-0313-6 https://doi.org/10.1186/1687-1812-2013-329 https://www.jstor.org/stable/24896705 https://doi.org/10.1186/1687-1812-2014-190 https://doi.org/10.1186/1687-1812-2013-151 https://doi.org/10.22436/jnsa.009.02.13 1. introduction 2. preliminaries 3. main results 4. application to integral equations 5. application to homotopy references int. j. anal. appl. (2023), 21:65 large fractional linear type differential equations ma’mon abu hammad1,∗, iqbal jebril1, roshdi khalil2 1department of mathematics, al zaytoonah university of jordan, amman 11942, jordan 2department of mathematics, the university of jordan, amman, jordan ∗corresponding author: m.abuhammad@zuj.edu.jo abstract. this paper aims to handle some types of fractional differential equations with a fractionalorder values β > 1. in particular, we propose a novel analytical solution called an atomic solution for certain fractional linear type differential equations as well as for some other types of partial differential equations with fractional-order values exceeding one. some examples are provided to validate our findings. 1. introduction one of the most principal connections between pure and applied mathematics is differential equations in their three main types; ordinary, partial and fractional differential equations. a large number of applications that arise naturally in many fields of science and engineering might be described by fractional differential equations. their solutions and investigations provide a remarkable growth to several mathematical approaches, see e.g. [1–5]. in 2014, r. khalil introduced in [8] a newly scheme for dealing differential equations in their two types; ordinary and fractional ones. such novel scheme relies on the theory of tensor product of banach spaces, which can be employed for the aim of obtaining the so-called atomic solutions for ordinary/fractional differential equations. it is commonly known that most of fractional (ordinary/partial) differential equations, which were treated in literature, are handled with fractional-order values between 0 and 1. hence obtained many interesting results can be. by taking this notion into account. from received: apr. 19, 2023. 2020 mathematics subject classification. 26a33. key words and phrases. fractional calculus; fractional linear type equations atomic. https://doi.org/10.28924/2291-8639-21-2023-65 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-65 2 int. j. anal. appl. (2023), 21:65 this point of view, the object of this paper is to study α−fractional (ordinary/partial) differential equations for fractional-order value greater than one, i.e. β > 1. accordingly, many notable theoretical results are proved and then used to some given fractional differential equations. the rest of this paper is arrange as follows: section 2 recalls some basic facts and preliminaries related to the fractional calculus and atomic solution. section 3 provides the main results of this work. section 4 illustrates some applications of the derived theoretical results, followed by section 5 that summarizes the whole conclusions of this work. 2. preliminaries in this section, some needed basic facts and preliminaries in connection to the fractional calculus and the so-called atomic solution are recalled. in particular, the definition of the so-called α-conformable fractional derivative dα is introduced , [8]. so, for x ∈ e ⊆ (0,∞), we have: dαf (x)= lim �→0 f (x + �x1−α)− f (x) � . if the above limit exists, then dαf (x) is called the α-conformable fractional derivative of f at x. as a result, if f is α-differentiable on (0, r) for some r > 0 and lim �→0+ dαf (x) exists, then we define dαf (0)= lim x→0 dαf (x). for α ∈ (0,1] and f , g are α-differentiable at a point t, one can easily see that the conformable fractional derivative satisfies the following properties: (i) dα(bf +cg)= bdα(f )+cdα(g), for all b,c ∈r, (ii) dα(λ)=0, for all constant functions f (t)= λ, (iii) dα(f g)= f dα(g)+gdα(f ), (iv) dα(f g )= gdα(f )−f dα(g) g2 ,g(t) 6=0. in this regard, we list below the conformable fractional derivatives for certain functions: (i) dα(tp)= ptp−α, (ii) dα(sin(1 α tα))= cos(1 α tα), (iii) dα(cos(1 α tα))=−sin(1 α tα), (iv) dα(e 1 α tα)= e 1 α tα. on letting α = 1 in the above derivatives, we get their corresponding classical rules for ordinary derivatives. further, one should notice that a function could be α-conformable differentiable at a point but it is not differentiable at that point. for example, if one takes f (t) = 2 √ t, then d 1 2(f )(0) = 1, while for the classical fractional derivative, d1(f )(0) does not exist. for more overview about fractional calculus and its applications, we refer to the references [6–12]. many differential equations might be transformed to their corresponding fractional-order forms. these equations can have many applications in many branches of science and engineering, see [13–15]. int. j. anal. appl. (2023), 21:65 3 it is commonly known that the main technique used to solve many partial differential equations is using fourier series. from this point of view, fractional fourier series was introduced in [16]. such a concept proved to be very fruitful in solving fractional partial differential equations. but sometimes it is not possible to use separation of variables technique to deal with these equations. here comes the concept of atomic solution, which would help us to face this problem. this concept coupled with how we can use it in solving fractional (ordinary/partial) differential equations will be provided in the following content. definition 2.1. (atomic solution) let x and y be two banach spaces and x∗ be the dual of x. assume x ∈ x and y ∈ y . the operator t : x∗ → y defined by: t(x∗)= x∗(x)y, is a bounded one rank linear operator. then, we write x ⊗y for t . such operators are called atoms, which are among the main ingredients in the theory of tensor products. besides, the atoms are used in theory of best approximation in banach spaces. in what follow, we recall one of the most known results that we shall need in our investigation. this result was extensively utilized in many research papers, see [17–20]. lemma 2.1. if the sum of two atoms is an atom, then either the first components are dependent or the second ones are dependent. in other words, if x1⊗y1+x2⊗y2 = x3⊗y3 , then either x1 = x2 = x3. 3. main results for n < β < n+1, the β-fractional derivatives of the function f : [0,∞)→r was defined in [8] as dβf (x)= lim �→0 f [β]−1(x + �x[β]−α)− f [β]−1(x) � . this turned out to be equivalent to the following assertion: dβ(f (x))= x[β]−βf [β](x). (3.1) it is worth mentioning that if f is (n+1)-differentiable and n < β < n+1, then we have: dβ(f (x))= xn+1−αf (n+1)(x)= xn+1−αdn+1x f (x), for n < β < n +1. now, if one lets 1 < β < 2, then β = 1+ α, where 0 < α < 1. hence, dβf = d1+αf . similarly, if n < β < n+1, then β = n+α, where 0 < α < 1, and so, dβf = dn+αf , for 0 < α < 1. proposition 3.1. let n < β < n+1 and α = β −n. then we have: dβf (x) = dαdnf (x)= dαf (n)(x) = x1−αf (n+1)(x). 4 int. j. anal. appl. (2023), 21:65 proof. to prove this result, it should be noted that dβf (x) = dn+αf (x) = x(n+1)−βf (n+1)(x). but, from identity (3.1), one can write: dβf (x) = x(n+1)−(n+α)f (n+1)(x) = x1−αf (n+1)(x). � definition 3.1. an equation of the form: and n+αy +an−1d n−1+αy + · · ·+a1dαy +a0x1−αy =0 (3.2) is called an α-fractional linear equation of order n with 0 < α < 1. in the following content, we state and prove the main result of this section, which would be about the general solution of an α-fractional linear type equation. theorem 3.1. let 0 < α < 1 and and n+αy +an−1d n−1+αy + · · ·+a1dαy +a0x1−αy =0 (3.3) be an α-fractional linear type equation of order n. let r0, r1, r2, · · · , rn, rn+1 be the roots of the equation an+1r n+1 +anr n + · · ·+a1r +a0 =0. then by assuming no repetition of the roots, the general solution of (3.3) has the form: y = c0y0 + · · ·+cnyn +cn+1yn+1, where yk = erkx. proof. by proposition 3.1, equation (3.2) can be written in the form: and α(dny)+ · · ·+a1dαy +a0x1−αy =0. this is just of the form: anx 1−αy(n+1) + · · ·+a1x1−αy 8 +a0x1−αy =0. (3.4) thus (3.3) becomes: an+1y (n+1) +any (n) + · · ·+a1y 8 +a0y =0. actually, the above equation represents an ordinary linear differential equation. so, the associated characteristic equation is of the form: an+1r n+1 +anr n + · · ·+a1r +a0 =0. (3.5) int. j. anal. appl. (2023), 21:65 5 solving the above equation to get the roots r0, r1, · · · , rn, rn+1. consequently, if all the roots are distinct and real, then there are (n+1) solutions that have the forms: y0 = e r0x,y1 = e r1x, · · · ,yn+1 = ern+1x. therefore, the general solution of the α-fractional linear type equation (3.3) is: yg = c0y0 + · · ·+cnyn +cn+1yn+1. � remark 3.1. if a root is repeated in 3.5, then as is known in the theory of ordinary differential equations, some of the solutions are multiplied by some factors. 4. applications in this part, we intend to provide a solution of a fractional ordinary differential equation, and the atomic solution of a fractional partial differential equation. example 4.1. let us have the following fractional differential equation: y( 3 2 ) +2y( 1 2 ) −3 √ xy =0. (4.1) hence, we have, α = 1 2 and n =1. in other words, (3.4) becomes: d 1 2y ′ +2 √ xy ′ −3 √ xy =0. so, we have: √ xy ′′ +2 √ xy ′ −3 √ xy =0. this is equivalent to say: y ′′ +2y ′ −3y =0. (4.2) the characteristic equation of (4.1) is then of the form: r2 +2r −3=0, with the roots r1 =−3 and r2 =1. hence, we obtain: yg = c1e −3x +c2e x. (4.3) one can easily check that (4.3) is the general solution of (4.1). in the following content, we aim to find the atomic solution of a certain fractional partial differential equation of order α > 3 2 . 6 int. j. anal. appl. (2023), 21:65 example 4.2. for x > 0 and y > 0, consider the following fractional partial differential equation: d 3 2 x d 3 2 y u +d 1 2 x d 1 2 y u = √ xyu, (4.4) with the following initial conditions u(0,0)=1, ∂u ∂x (0,0)=1, ∂u ∂y (0,0)=1. (4.5) if one wants to solve the above problem with the use of our proposed approach, we firs assume: u(x,y)= p(x)q(y). now, conditions (4.5) yield the following condition on p and q: p ′(0)= p(0)=1 and q′(0)= q(0)=1. (4.6) this means: p 3 2(x)q 3 2(y)+p 1 2(x)q 1 2(y)= √ xyp(x)q(y). from the analysis of dn+αu, we get: √ xp ′′(x) √ yq′′(y)+ √ xp ′(x) √ yq′(y)= √ xyp(x)q(y). hence, we have: p ′′(x)q′′(y)+p ′(x)q′(y)= p(x)q(y). (4.7) with the use of lemma 2.1, we have two cases: case (a): p ′′(x)= p ′(x)= p(x). case (b): q′′(x)= q′(x)= q(x). now for case (a), we have the following three situations: (1) p ′′(x)= p ′(x). (2) p ′′(x)= p(x). (3) p ′(x)= p(x). observe that situation 1 can be solved to get p(x) = c1 + c2ex. but from conditions (4.6, we get p(x) = ex. in the same regard, solving both situations 2 and 3 gives also the same solution p(x)= ex. now substituting this result in (4.5), we get: exq′′(y)+exq′(y)= exq(y). immediately, we have: q′′(y)+q′(y)−q(y)=0. so, we have r2+ r −1=0 with the roots r1 = −1+ √ 5 2 and r2 = −1− √ 5 2 . now, from conditions (4.6), we get: q(y)= (1+ √ 5)er1y − √ 5er2y. int. j. anal. appl. (2023), 21:65 7 therefore, our proposed atomic solutions has the form: u(x,y)= p(x)q(y)= ex((1+ √ 5)er1y − √ 5er2y). since equation (4.6) is symmetric in p and q, then case (b) gives the following atomic solution: u(x,y)= ((1+ √ 5)er1x − √ 5er2x)ey. (4.8) however, in order to see how the atomic solution (4.8) of equation (4.4) looks like, we plot such solution in figure 4.2. figure 1. the atomic solution (4.8) of equation (4.4). 5. conclusions this paper has successfully introduced a new analytical method for handling the study of a class of α-fractional (ordinary/partial) differential equations with fractional-order value greater than one via atomic solutions method. the theory of tensor product of banach spaces coupled with some properties of atoms operators have been utilized for achieving such a notion. acknowledgment: this research was supported by al-zaytoonah university of jordan fund. project number 32 / 17 / 2022-2023. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] i.m. batiha, s. alshorm, a. ouannas, s. momani, o.y. ababneh, m. albdareen, modified three-point fractional formulas with richardson extrapolation, mathematics. 10 (2022), 3489. https://doi.org/10.3390/ math10193489. [2] i.m. batiha, s. alshorm, i.h. jebril, m.a. hammad, a brief review about fractional calculus, int. j. open probl. comput. math. 15 (2022), 39-56. https://doi.org/10.3390/math10193489 https://doi.org/10.3390/math10193489 8 int. j. anal. appl. (2023), 21:65 [3] i.m. batiha, a. obeidat, s. alshorm, a. alotaibi, h. alsubaie, s. momani, m. albdareen, f. zouidi, s.m. eldin, h. jahanshahi, a numerical confirmation of a fractional-order covid-19 model’s efficiency, symmetry. 14 (2022), 2583. https://doi.org/10.3390/sym14122583. [4] i.m. batiha, o.y. ababneh, a.a. al-nana, w.g. alshanti, s. alshorm, s. momani, a numerical implementation of fractional-order pid controllers for autonomous vehicles, axioms. 12 (2023), 306. https://doi.org/10.3390/ axioms12030306. [5] i.h. jebril, i.m. batiha, on the stability of commensurate fractional-order lorenz system, progr. fract. differ. appl. 8 (2022), 401-407. https://doi.org/10.18576/pfda/080305. [6] t. abdeljawad, on conformable fractional calculus, j. comput. appl. math. 279 (2015), 57-66. https://doi. org/10.1016/j.cam.2014.10.016. [7] m. adil khan, y.m. chu, a. kashuri, r. liko, g. ali, conformable fractional integrals versions of hermitehadamard inequalities and their generalizations, j. funct. spaces. 2018 (2018), 6928130. https://doi.org/10. 1155/2018/6928130. [8] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65-70. https://doi.org/10.1016/j.cam.2014.01.002. [9] e. mahmoudi, the beta generalized pareto distribution with application to lifetime data, math. computers simul. 81 (2011), 2414-2430. https://doi.org/10.1016/j.matcom.2011.03.006. [10] m.a. hammad, conformable fractional martingales and some convergence theorems, mathematics. 10 (2021), 6. https://doi.org/10.3390/math10010006. [11] m.a. hammad, m.a. horani, a. shmasenh, r. khalil, reduction of order of fractional differential equations, j. math. comput. sci. 8 (2018), 683-688. https://doi.org/10.28919/jmcs/3806. [12] a. dababneh, b. albarmawi, m.a. hammad, a. zraiqat, t. hamadneh, conformable fractional bernoulli differential equation with applications, in: 2019 ieee jordan international joint conference on electrical engineering and information technology (jeeit), ieee, amman, jordan, 2019: pp. 421-424. https://doi.org/10.1109/jeeit. 2019.8717456. [13] w.s. chung, fractional newton mechanics with conformable fractional derivative, j. comput. appl. math. 290 (2015), 150-158. https://doi.org/10.1016/j.cam.2015.04.049. [14] m. horani, m.a. hammad, r. khalil, variation of parameters for local fractional nonhomogenous lineardifferential equations, j. math. computer sci. 16 (2016), 147-153. [15] w. deeb, r. khalil, best approximation in l(x,y ), math. proc. camb. phil. soc. 104 (1988), 527-531. https: //doi.org/10.1017/s0305004100065713. [16] m.a. hammad, r. khalil, fractional fourier series with applications, amer. j. comput. appl. math. 4 (2014), 187-191. [17] m. al-horani, r. khalil, i. aldarawi, fractional cauchy euler differential equation, j. comput. anal. appl. 28 (2020), 226-233. [18] r. khalil, isometries of lp∗ a ⊗lp, tam. j. math. 16 (1985), 77-85. [19] h. al-zoubi, a. dabaneh, m. al-sabbagh, ruled surfaces of finite ii-type, wseas trans. math. 18 (2019), 1-5. [20] f. bekraoui, m. al-horani, r. khalil, atomic solution of fractional abstract cauchy problem of high order in banach spaces, eur. j. pure appl. math. 15 (2022), 106-125. https://doi.org/10.3390/sym14122583 https://doi.org/10.3390/axioms12030306 https://doi.org/10.3390/axioms12030306 https://doi.org/10.18576/pfda/080305 https://doi.org/10.1016/j.cam.2014.10.016 https://doi.org/10.1016/j.cam.2014.10.016 https://doi.org/10.1155/2018/6928130 https://doi.org/10.1155/2018/6928130 https://doi.org/10.1016/j.cam.2014.01.002 https://doi.org/10.1016/j.matcom.2011.03.006 https://doi.org/10.3390/math10010006 https://doi.org/10.28919/jmcs/3806 https://doi.org/10.1109/jeeit.2019.8717456 https://doi.org/10.1109/jeeit.2019.8717456 https://doi.org/10.1016/j.cam.2015.04.049 https://doi.org/10.1017/s0305004100065713 https://doi.org/10.1017/s0305004100065713 1. introduction 2. preliminaries 3. main results 4. applications 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 77-87 http://www.etamaths.com modified homotopy analysis method for nonlinear fractional partial differential equations d. ziane∗ and m. hamdi cherif abstract. in this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. this method is called the fractional homotopy analysis natural transform method (fhantm). the fhantm can easily be applied to many problems and is capable of reducing the size of computational work. the fractional derivative is described in the caputo sense. the results show that the fhantm is an appropriate method for solving nonlinear fractional partial differentia equation. 1. introduction the natural transform is a transform defined by an integral like all other transformations defined by integrals, such as the laplace transform as well, as the sumudu transform, where we find only used in solving of linear differential equations. this transform it was defined by z. h. khan and w. a. khan [1] in 2008 and it has been used by many researchers in the resolution of linear differential equations ( [2], [3], [4], [5]). but with the presence of some methods, such as the homotopy analysis method (ham) ( [6], [7], [8]) that used in the solution of linear and nonlinear differential equations. then, with the advent of the compositions of this method with the natural transform, lead to facilitating the resolution of nonlinear fractional partial differential equations. the objective of this study is to combine two powerful methods, the first method is ”homotopy analysis method”, the second is called ”the natural transform method”, the fractional derivative is described in the caputo sense, thus, we get the modified method ”fractional homotopy analysis natural transform method” (fhantm), and we apply this modified method to solve somme exemples of nonlinear fractional partial differential equations. the present paper has been organized as follows: in section 2 some basic definitions and properties of natural transform. in section 3 we will propose an analysis of the modified method. in section 4 we present three examples explaining how to apply the proposed method (fhantm). finally, the conclusion follows. 2. basic definitions in this section, we give some basic definitions and properties of fractional calculus, natral transform and natural transform of fractional derivatives which are used further in this paper. 2.1. fractional calculus. there are several definitions of a fractional derivative of order α > 0 (see [9], [10], [11]). the most commonly used definitions are the riemann–liouville and caputo. we give some basic definitions and properties of the fractional calculus theory which are used further in this paper. definition 2.1. let ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis r. the riemann–liouville fractional integral iα0+f of order α ∈ r (α > 0) is defined by received 14th january, 2017; accepted 3rd april, 2017; published 2nd may, 2017. 2010 mathematics subject classification. caputo fractional derivative; natural transform; homotopy analysis method; nonlinear fractional equations. key words and phrases. hermite-hadamard inequality; local fractional integral; fractal space; generalized convex function. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 77 78 d. ziane and m. hamdi cherif (iα0+f)(t) = 1 γ (α) ∫ t 0 f(τ)dτ (t− τ)1−α , t > 0, α > 0, (2.1) (i00+f)(t) = f(t). here γ(·) is the gamma function. theorem 2.1. let α > 0 and let n = [α]+1. if f(t) ∈ acn [a,b] , then the caputo fractional derivative (cdα0+f)(t) exist almost evrywhere on [a,b] . if α /∈ n, (cdα 0+ f)(t) is represented by (cdα0+f)(t) = 1 γ (n−α) ∫ t 0 f(n)(τ)dτ (t− τ)α−n+1 , (2.2) where d = d dx and n = [α] + 1. proof. (see [10]). � remark 2.1. in this paper, we consider the time-fractional derivative in the caputo’s sense. when α ∈ r+, the time-fractional derivative is defined as (cdαt u)(x,t) = ∂αu(x,t) ∂tα = { 1 γ(m−α) ∫ t 0 (t− τ)m−α−1 ∂ mu(x,τ) ∂τm , m− 1 < α < m, ∂mu(x,t) ∂tm , α = m (2.3) where m ∈ n∗. 2.2. definitions of the n-transform. we give some basic definitions and properties of the ntransform which are used further in this paper (see [1], [12], [13]). when the real function f(t) > 0 and f(t) = 0 for t < 0 is sectionwise continuous, exponential order and defined in the set a = { f(t) : ∃m, k1,k2 > 0, |f(t)| < me |t| kj , if t ∈ (−1)j × [0, ∞) } . definition 2.2. [12] the n-transform of the function f(t) > 0 and f(t) = 0 for t < 0 is defined by n+ [f(t)] = r(s,u) = ∫ ∞ 0 e−stf(ut)dt; s > o,u > o. (2.4) where s and u are the transform variables. the original function f(t) in (2.4) is called the inverse transform or inverse of r(s,u) and it is defined by n−1 {r(s,u)} = f(t) = 1 2πi ∫ c+i∞ c−i∞ e st u r(s,u)ds, (2.5) 2.2.1. n-transform of fractional derivatives. proposition 2.1. if r(s,u) is the n-transform of the function f(t), then the n-transform of fractional integral of order α is defined by n+ [ (iα0+f)(t) ] = sα uα r(s,u). (2.6) proposition 2.2. if r(s,u) is the n-transform of the function f(t), then the n-transform of fractional derivative of order α is defined as n+ [ (cdα0+f)(t) ] = sα uα r(s,u) − n−1∑ k=0 sα−(k+1) uα−k f(k)(0). (2.7) modified homotopy analysis method for fde 79 2.2.2. somme properties of the n-transform. here are some properties of the n-transform: 1. if n+ {f(t)} = r(s,u) then, n+ {f(at)} = 1 a r(s,u). 2. generalised n-transform for any value of n the generalised n-transform of function f(t) > 0 is defined by n+ {f(t)} = r(s,u) = ∞∑ n=0 n!anu n sn+1 . (2.8) 3. n-transform of derivative if f(n)(t) is the nth derivative of function f(t), then its ntransform is given by n+ { f(n)(t) } = rn(s,u) = sn un r(s,u) − n−1∑ k=0 sn−(k+1) un−k f(k)(0). (2.9) for n = 1 and n = 2, (2.9) gives the n-transform of first and second derivatives of f(t) n+ { f ′ (t) } = r1(s,u) = s u r(s,u) − 1 u f(0). (2.10) n+ { f ′′ (t) } = r2(s,u) = s2 u2 r(s,u) − s u2 f(0) − 1 u f ′ (0). (2.11) 4. n-transform of integral n+ {f(t)} = r(s,u) then n+ {∫ t 0 f(r)dr } = u s r(s,u). 5. the function f(t) in set a is multiplied with shift function tn then n+ {tnf(t)} = sn un dn dun unr(s,u). (2.12) and n+ { tα γ(α + 1) } = uα sα+1 , α > 0. 3. fractional homotopy analysis n-transform method (fhantm) to illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous timefractional partial differential equation cdαt u(x,t) + lu(x,t) + ru(x,t) = g(x,t), (3.1) where m = 1, 2, . . . , and the initial conditions ∂m−1u(x,t) ∂tm−1 ∣∣∣∣ t=0 = fm−1(x), m = 1, 2, . . . , (3.2) where cdαt u(x,t) is the caputo fractional derivative of the function u(x,t), l is the linear differential operator, r represents the general nonlinear differential operator, and g(x,t) is the source term. applying the n-transform (denoted in this paper by n+) on both sides of (3.1), we get n+ [cdαt u(x,t)] + n + [lu(x,t) + ru(x,t) −g(x,t)] = 0. (3.3) using the property of the n-transform, we have the following form n+ [u(x,t)] − uα sα n−1∑ k=0 sα−(k+1) uα−k u(k)(x, 0) + uα sα n+ [lu(x,t) + ru(x,t) −g(x,t)] = 0 (3.4) or 80 d. ziane and m. hamdi cherif n+ [u(x,t)] − n−1∑ k=0 uk sk+1 u(k)(x, 0) + uα sα n+ [lu(x,t) + ru(x,t) −g(x,t)] = 0. (3.5) define the nonlinear operator r[φ(x,t; p)] = n+ [φ(x,t; p)] − n−1∑ k=0 uk sk+1 φ(k)(x, 0,p) (3.6) + uα sα n+ [lφ(x,t; p) + rφ(x,t; p) −g(x,t; p)] by means of homotopy analysis method [6], we construct the so-called the zero-order deformation equation (1 −p)n+[φ(x,t; p) −u0(x,t)] = phh(x,t)r[φ(x,t; p)], (3.7) where p is an embedding parameter and p ∈ [0, 1], h(x; t) 6= 0 is an auxiliary function, h 6= 0 is an auxiliary parameter, n+ is an auxiliary linear n-transform operator. when p = 0 and p = 1, we have{ φ(x,t; 0) = u0(x,t), φ(x,t; 1) = u(x,t). (3.8) when p increases from 0 to 1, the φ(x,t,p) various from u0(x,t) to u(x,t). expanding φ(x,t; p) in taylor series with respect to p, we have φ(x,t; p) = u0(x,t) + +∞∑ m=1 um(x,t)p m, (3.9) where um(x,t) = 1 m! ∂mφ(x,t; p) ∂pm |p=0 . (3.10) when p = 1, the (3.9) becomes u(x,t) = u0(x,t) + +∞∑ m=1 um(x,t). (3.11) we define the vectors −→ u n = {u0(x,t),u1(x,t),u2(x,t), . . . ,un(x,t)}. (3.12) differentiating (3.7) m−times with respect to p, then setting p = 0 and finally dividing them by m!, we obtain the so-called mth-order deformation equation n+[um(x,t) −χmum−1(x,t)] = hph(x,t)<m( −→ u m−1(x,t)), (3.13) where <m( −→ u m−1(x,t)) = 1 (m− 1)! ∂m−1r[φ(x,t; p)] ∂pm−1 |p=0 , (3.14) and χm = { 0, m 6 1, 1, m > 1. applying the inverse n-transform on both sides of (3.13), we have um(x,t) = χmum−1(x,t) + n−1 [ hph(x,t)<m( −→ u m−1(x,t)) ] . (3.15) the mth deformation equation (3.15) is a linear which can be easily solved. so, the solution of (3.1) can be written into the following form modified homotopy analysis method for fde 81 u(x,t) = m∑ m=0 um(x,t), (3.16) when m →∞, we can obtain an accurate approximation solution of (3.1). 4. application of the fhantm in this section, we apply the homotopy analysis method (ham) coupled with the n-transform for solving somme exemples of nonlinear time-fractional partial differential equations. example 4.1. consider the following nonlinear time-fractional kdv equation cdαt u − 3(u 2)x + uxxx = 0, 0 < α 6 1, (4.1) with the initial condition u(x, 0) = 6x. (4.2) applying the n-transform on both sides of (4.1), we get n+ [u] − 1 s u(x, 0) + uα sα n+ [ −3(u2)x + uxxx ] = 0. (4.3) from (4.3) and the initial condition (4.2), we have n+ [u] − 1 s 6x− uα sα n+ [ 3(u2)x −uxxx ] = 0. (4.4) we take the nonlinear part as r[φ(x,t,p)] = n+ [φ] − 1 s 6x− uα sα n+ [ 3(φ2)x −φxxx ] . (4.5) we construct the so-called the zero-order deformation equation with assumption h(x; t) = 1, we have (1 −p)n+[φ(x,t; p) −u0(x,t)] = phr[φ(x,t; p)]. (4.6) when p = 0 and p = 1, we can obtain{ φ(x,t; 0) = u0(x,t), φ(x,t; 1) = u(x,t). therefore, we have the mth order deformation equation n+[um(x,t) −χmum−1(x,t)] = h<m( −→ u m−1(x,t)). (4.7) operating the inverse n-transform operator on both sides of (4.7), we get um(x,t) = χmum−1(x,t) + n−1[h<m( −→ u m−1(x,t))]. (4.8) from (4.8), we have u1(x,t) = hn−1[ <1( −→ u 0(x,t))], u2(x,t) = u1 + hn−1[ <2( −→ u 1(x,t))], (4.9) u3(x,t) = u2 + hn−1[ <3( −→ u 2(x,t))], ... where <1( −→ u 0(x,t)) = n+ [u0] − 1 s 6x− uα sα n+ [ 3(u20 )x − (u0)xxx ] , <2( −→ u 1(x,t)) = n+ [u1] − uα sα n+ [3(2u0u1)x − (u1)xxx] , (4.10) 82 d. ziane and m. hamdi cherif <3( −→ u 2(x,t)) = n+ [u2] − uα sα n+ [ 3(2u0u2 + u 2 1 )x − (u2)xxx ] , ... using the initial condition (4.2), the iteration formulas (4.9) and (4.10), we obtain u0(x,t) = 6x, u1(x,t) = (−h)(6x) 36 γ(α + 1) tα, u2(x,t) = (−h)(1 + h)(6x) 36 γ(α + 1) tα + 2h2(6x) (36)2 γ(2α + 1) t2α, (4.11) u3(x,t) = (−h)(1 + h)2(6x) 36 γ(α + 1) tα + 4(h2 + h3)(6x) (36)2 γ(2α + 1) t2α +(−h3)(6x)(36)3 [ 4 γ(2α + 1) + 1 γ2(α + 1) ] γ(2α + 1) γ(3α + 1) t3α, ... the other components of the fhantm can be determined in a similar way. finally, the approximate solution of (4.1) in a series form is u(x,t) = 6x + (−h)(3 + 3h + h2)(6x) 36 γ(α + 1) tα + (6h2 + 4h3)(6x) (36)2 γ(2α + 1) t2α +(−h3)(6x)(36)3 [ 4 γ(2α + 1) + 1 γ2(α + 1) ] γ(2α + 1) γ(3α + 1) t3α + · · · (4.12) when h = −1, the approximate solution of (4.1), is given by u(x,t) = 6x [ 1 + 36 γ(α + 1) tα + 2 (36)2 γ(2α + 1) t2α + (36)3 ( 4 γ(2α + 1) + 1 γ2(α + 1) ) γ(2α + 1) γ(3α + 1) t3α + · · · ] , (4.13) and when α = 1, we obtain u(x,t) = 6x [ 1 + 36t + (36t)2 + (36t)3 + · · · ] . (4.14) that gives u(x,t) = 6x 1 − 36t , |36t| < 1, (4.15) which is an exact solution to the nonlinear kdv equation as presented in [14]. example 4.2. consider the following nonlinear time-fractional partial differential equation cdαt u − 2 x2 t uux = 0, t > 0, 1 < α 6 2, (4.16) with the initial conditions u(x, 0) = 0, ut(x, 0) = x. (4.17) applying the n-transform on both sides of (4.16), we get n+ [u] − 1 s u(x, 0) − u s2 ut(x, 0) + uα sα n+ [ −2 x2 t uux ] = 0. (4.18) from (4.18) and the initial conditions (4.17), we have modified homotopy analysis method for fde 83 figure 1. shows the exact solution and approximate solutions of (4.1) for different values of α when x = 1. we note that the graph has changed its position according to the values of α, if the value of α is closer to 1, we see that the graph corresponding to this value takes the position closest to the graph of the exact solution. n+ [u] − u s2 x− uα sα n+ [ 2 x2 t uux ] = 0. (4.19) we take the nonlinear part as r[φ(x,t,p)] = n+ [φ] − u s2 x− uα sα n+ [ 2 x2 t φφx ] . (4.20) we construct the so-called the zero-order deformation equation with assumption h(x; t) = 1, we have (1 −p)n+[φ(x,t; p) −u0(x,t)] = phr[φ(x,t; p)]. (4.21) when p = 0 and p = 1, we can obtain{ φ(x,t; 0) = u0(x,t), φ(x,t; 1) = u(x,t). therefore, we have the mth order deformation equation n+[um(x,t) −χmum−1(x,t)] = h<m( −→ u m−1(x,t)). (4.22) operating the inverse n-transform operator on both sides of (4.22), we get um(x,t) = χmum−1(x,t) + n−1[h<m( −→ u m−1(x,t))]. (4.23) from the formula (4.23), we have u1(x,t) = hn−1[<1( −→ u 0(x,t))], u2(x,t) = u1 + hn−1[<2( −→ u 1(x,t))], (4.24) u3(x,t) = u2 + hn−1[ <3( −→ u 2(x,t))], ... where <1( −→ u 0(x,t)) = n+ [u0] −x u s2 − uα sα n+ [ 2 x2 t u0u0x ] , <2( −→ u 1(x,t)) = n+ [u1] − uα sα n+ [ 2 x2 t (u0u1x + u1u0x) ] , (4.25) 84 d. ziane and m. hamdi cherif <3( −→ u 2(x,t)) = n+ [u2] − uα sα n+ [ 2 x2 t (u0u2x + u1u1x + u2u0x) ] , ... and so on. consequently, while taking the initial conditions (4.17), and according to (4.24) and (4.25), the first few components of the homotopy analysis n-transform method of (4.16) are derived as follows u0(x,t) = xt, u1(x,t) = (−h) 2x3 γ(α + 2) tα+1, u2(x,t) = (−h)(1 + h) 2x3 γ(α + 2) tα+1 + h2 16x5 γ(2α + 2) t2α+1, (4.26) u3(x,t) = (−h)(1 + h)2 2x3 γ(α + 2) tα+1 + h2(1 + h) 32x5 γ(2α + 2) t2α+1 +(−h3) [ 32 × 6 γ(2α + 2) + 24 γ2(α + 2) ] γ(2α + 2) γ(3α + 2) x7t3α+1, ... the other components of the (fhantm) can be determined in a similar way. finally, the approximate solution of (4.16) in a series form is u(x,t) = xt + (−h) [ 2 + h + (1 + h)2 ] 2x3 γ(α + 2) tα+1 + h2(3 + 2h) 16x5 γ(2α + 2) t2α+1 (4.27) +(−h3) [ 32 × 6 γ(2α + 2) + 24 γ2(α + 2) ] γ(2α + 2) γ(3α + 2) x7t3α+1 + · · · when h = −1, the approximate solution of (4.16) is given by u(x,t) = xt + 2x3 γ(α + 2) tα+1 + 16x5 γ(2α + 2) t2α+1 (4.28) + [ 32 × 6 γ(2α + 2) + 24 γ2(α + 2) ] γ(2α + 2) γ(3α + 2) x7t3α+1 + · · · . when α = 2, the (4.28) becomes u(x,t) = xt + 1 3 (xt)3 + 2 15 (xt)5 + 17 315 (xt)7 + · · · . (4.29) on the other hand, we see that the development of the function u(x,t) = tan(xt) according to the taylor series in the vicinity of t = 0 is given by u(x,t) = xt + 1 3 (xt)3 + 2 15 (xt)5 + 17 315 (xt)7 + 0( t x )8. (4.30) therefore, we conclude that u(x,t) = tan(xt), (4.31) which is the exact solution of (4.16) in the case α = 2. modified homotopy analysis method for fde 85 figure 2. shows the exact solution and approximate solutions of (4.16) for different values of α when x = 1/2. we note that the graph has changed its position according to the values of α, if the value of α is closer to 2, we see that the graph corresponding to this value takes the position closest to the graph of the exact solution. example 4.3. finaly, we consider the nonlinear time-fractional partial differential equation cdαt u − 3 8 [ (uxx) 2 ] x = 3 2 t, 2 < α 6 3, (4.32) with the initial conditions u(x, 0) = − 1 2 x2, ut(x, 0) = 1 3 x3, utt(x, 0) = 0. (4.33) applying the n-transform on both sides of (4.32), we get n+ [u] − 1 s u(x, 0) − u s2 ut(x, 0) + uα sα n+ [ − 3 8 [ (uxx) 2 ] x ] = 3 2 uα+1 sα+2 . (4.34) from (4.34) and the initial conditions (4.33), we have n+ [u] + 1 s x2 2 − u s2 x3 3 − uα sα n+ [ 3 8 [ (uxx) 2 ] x ] = 3 2 uα+1 sα+2 . (4.35) we take the nonlinear part as r[φ(x,t,p)] = n+ [φ] + 1 s x2 2 − u s2 x3 3 − 3 2 uα+1 sα+2 − uα sα n+ [ 3 8 [ (φxx) 2 ] x ] . (4.36) we construct the so-called the zero-order deformation equation with assumption h(x; t) = 1, we have (1 −p)n+[φ(x,t; p) −u0(x,t)] = phr[φ(x,t; p)]. (4.37) when p = 0 and p = 1, we can obtain{ φ(x,t; 0) = u0(x,t), φ(x,t; 1) = u(x,t). therefore, we have the mth order deformation equation n+[um(x,t) −χmum−1(x,t)] = h<m( −→ u m−1(x,t)). (4.38) operating the inverse n-transform operator on both sides of (4.38), we get um(x,t) = χmum−1(x,t) + n−1[h<m( −→ u m−1(x,t))]. (4.39) from the formula (4.23), we have 86 d. ziane and m. hamdi cherif u1(x,t) = hn−1[<1( −→ u 0(x,t))], u2(x,t) = u1 + hn−1[<2( −→ u 1(x,t))], (4.40) u3(x,t) = u2 + hn−1[ <3( −→ u 2(x,t))], ... where <1( −→ u 0(x,t)) = n+ [u0] + 1 s x2 2 − u s2 x3 3 − 3 2 uα+1 sα+2 − uα sα n+ [ 3 8 [ (u0xx) 2 ] x ] , <2( −→ u 1(x,t)) = n+ [u1] − uα sα n+ [ 3 8 (u0u1xx + u1u0xx)x ] , (4.41) <3( −→ u 2(x,t)) = n+ [u2] − uα sα n+ [ 3 8 (u0u2xx + u1u1xx + u2u0xx)x ] , ... consequently, while taking (4.33), and according to (4.40) and (4.41), the first few components of the homotopy analysis n-transform method of (4.32), are derived as follows u0(x,t) = − 1 2 x2 + 1 3 x3t, u1(x,t) = (−h)6x tα+2 γ(α + 3) , u2(x,t) = (−h)(1 + h)6x tα+2 γ(α + 3) , (4.42) u3(x,t) = (−h)(1 + h)26x tα+2 γ(α + 3) , ... un(x,t) = (−h)(1 + h)n−16x tα+2 γ(α + 3) the other components of the (fntham) can be determined in a similar way. finally, the approximate solution of (4.32) in a series form is given by u(x,t) = − 1 2 x2 + 1 3 x3t + (−h)6x tα+2 γ(α + 3) + (−h)(1 + h)6x tα+2 γ(α + 3) (4.43) +(−h)(1 + h)26x tα+2 γ(α + 3) + · · · + (−h)(1 + h)n−16x tα+2 γ(α + 3) . when h = −1, we obtain the exact solution of (4.32) u(x,t) = − 1 2 x2 + 1 3 x3t + 6x tα+2 γ(α + 3) . (4.44) we note that from the term u2, all terms are equal to zero. when α = 3, the (4.44) becomes u(x,t) = − 1 2 x2 + 1 3 x3t + 1 20 xt5, (4.45) which is the exact solution of (4.32) in the case α = 3. modified homotopy analysis method for fde 87 figure 3. shows the exact solution and approximate solutions of (4.32) for different values of α when x = 1/2. we note that the graph has changed its position according to the values of α, if the value of α is closer to 3, we see that the graph corresponding to this value takes the position closest to the graph of the exact solution. 5. conclusion the coupling of homotopy analysis method (ham) and the natural transform method proved very effective to solve nonlinear fractional partial differential equations. the proposed algorithm provides the solution in a series form that converges rapidly to the exact solution if it exists. the advantage of this method is its ability to combine two powerful methods for obtaining exact or approximate solutions for nonlinear fractional partial differential equations. from the obtained results, it is clear that the fhantm yields very accurate approximate solutions using only a few iterates. references [1] z. h. khan and w. a. khan, n-transform properties and applications, nust j. eng. sci, 1 (2008), 127-133. [2] r. silambarasn and f. b. m. belgacem, applications of the natural transform to maxwell’s equations, prog. elect. res. symp. proc. suzhou. china, (2011), 899-902. [3] s. k. q. al-omari, on the application of natural transforms, int. j. pure appl. math. 85 (2013), 729-744. [4] m. junaid, application of natural transform to newtonian fluid problems, int. j. sci. technol. 5 (2016), 138-147. [5] d. loonker and p. k. banerji, solution of fractional ordinary differential equations by natural transform, int. j. math. eng. sci. 2 (2013), 1-7. [6] s.j. liao, the proposed homotopy analysis technique for the solution of nonlinear problems, ph.d. thesis, shanghai jiao tong university, 1992. [7] s. j. liao, beyond perturbation: introduction to homotopy analysis method, chapman and hall/crc press, boca raton, 2003. [8] s. j. liao, on the homotopy analysis method for nonlinear problems, appl. math. comput. 147 (2004), 499-513. [9] i. podlubny, fractional differential equations, academic press, new york, 1999. [10] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [11] k. diethelm, the analysis fractional differential equations, springer-verlag berlin heidelberg 2010. [12] f. b. m. belgacem and r. silambarasn, theory of natural transform, math. eng. sci. aerospace, 3 (2012), 105-135. [13] d. loonker and p.k. banerji, solution of fractional ordinary differential equations by natural transform, int. j. math. eng. sci. 2 (2013), 1-7. [14] a. wazwaz, the variational iteration method for rational solutions for kdv, k(2,2), burgers, and cubic boussinesq equations, j. comput. appl. math, 207 (2007), 18-23. laboratory of mathematics and its applications (lamap), university of oran1 ahmed ben bella, oran, 31000, algeria ∗corresponding author: djeloulz@yahoo.com 1. introduction 2. basic definitions 2.1. fractional calculus 2.2. definitions of the n-transform 3. fractional homotopy analysis n-transform method (fhantm) 4. application of the fhantm 5. conclusion references international journal of analysis and applications volume 16, number 5 (2018), 614-627 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-614 on dual curves of daw(k)−type and their evolutes h. s. abdel-aziz1, m. khalifa saad1,2,∗ and s. a. mohamed1 1department of mathematics, faculty of science, sohag university, 82524 sohag, egypt 2department of mathematics, faculty of science, islamic university of madinah, 170 madinah, ksa ∗corresponding author: m khalifag@yahoo.com abstract. in this paper, we study to express the theory of curves including a wide section of euclidean geometry in terms of dual vector calculus which has an important place in the three -dimensional dual space d3. in other words, we study daw (k)-type curves (1 ≤ k ≤ 3) by using bishop frame defined as alternatively of these curves and give some of their properties in d3. moreover, we define the notion of evolutes of dual spherical curves for ruled surfaces. finally, we give some examples to illustrate our findings. 1. introduction the analytical tools in the study of 3-dimensional kinematics and differential geometry of ruled surfaces are based on dual vector calculus. dual numbers were introduced in the 19th century by clifford as a tool for his geometrical investigations. in addition, their applications to rigid body kinematics were generalized by study in their principal of transference. the application of dual numbers to the lines of 3-space is carried out by the principle of transference which has been formulated by e. study [1]. it allows a complete generalization of the mathematical expression for the spherical point geometry to the spatial line geometry by means of dual-number extension, i.e., replacing all ordinary quantities by the corresponding dual-number quantities. in other words, in the euclidean 3-space e3, lines combined with one of their two directions can be represented received 2017-11-10; accepted 2018-01-17; published 2018-09-05. 2010 mathematics subject classification. 53a05, 53a17, 53a25. key words and phrases. curves of daw (k)−type; evolute; ruled surface; e. study map. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 614 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-614 int. j. anal. appl. 16 (5) (2018) 615 by unit dual vectors over the ring of dual numbers. the most important properties of real vector analysis are valid for the dual vectors and the oriented lines in e3 are in one-to-one correspondence with the points of a dual unit sphere. a dual point in dual space d3 corresponds to an oriented line in e3 and two different points in d3 represent two skew-lines in e3 in general. a differentiable curve on dual unit sphere in d3 represents a ruled surface in e3 [2]. ruled surfaces are those surfaces which are generated by moving a straight line continuously in the space [3]. in the light of this correspondence, dual spherical motion, expressed with the help of dual unit vectors, is closely analogous to real spherical motion, expressed with the help of real unit vectors. therefore, the properties of elementary real spherical motion can also be carried over by analogy into the motion of lines in e3. the notion of aw(k)− type submanifolds was defined by k. arslan and a. west [4]. after that, a lot of work related to curves of aw(k)− type has been done (see for example [5–7]). in [8], the authors studied daw(k)− type curves on the dual unit sphere using frenet frame. in this paper, we investigate the daw(k)−type curves and give the curvature conditions of these curves using a bishop frame which has many properties that make it ideal for mathematical research. it also has applications in the area of biology and computer graphics, for example it may be possible to compute information about the shape of sequences of dna using a curve defined by bishop frame. also, it may provide a new way to control virtual cameras in computer animations. 2. fundamental concepts in this section, we briefly give the mathematical formulations and basic concepts that are needed in our study. 2.1. bishop frame. let α : i ⊂ r → e3 be a unit speed curve in the euclidean 3-space e3 (i.e. its tangent vectors are normed over the whole parameter interval i). therefore, the frenet equations along α are defined as follows [9–11]:   t′ n′ b′   =   0 κ(s) 0 −κ(s) 0 τ(s) 0 −τ(s) 0     t n b   , (2.1) where {t(s), n(s), b(s)} is the moving frenet frame along α and the functions κ(s) and τ(s) are respectively, the curvature and torsion of α. the bishop frame of 1-type or parallel transport frame is an alternative approach to define a moving frame that is well defined even when the curve has vanishing second derivative. the tangent vector and any convenient arbitrary basis for the remainder of the frame are used. to define the frenet frame, curvature and torsion of a curve, this curve needs to be 3-times continuously differentiable non-degenerate. but the int. j. anal. appl. 16 (5) (2018) 616 curvature function may vanish at some points on the curve, i.e., second derivative of the curve may be zero at some parameter values. in this situation, we need an alternative frame in e3. we can parallel transport an orthonormal frame along α(s) simply by parallel transporting each component of the frame. the tangent vector and any convenient arbitrary basis for the remainder of the frame are used. this frame is denoted by {t(s), n1(s), n2(s)}, and its derivative is expressed using matrix coefficients k1 and k2.   t′ n′1 n′2   =   0 k1(s) k2(s) −k1(s) 0 0 −k2(s) 0 0     t n1 n2   . (2.2) thereby the relation between frenet and bishop frames are given as follows  t n b   =   1 0 0 0 cos θ sin θ 0 −sin θ cos θ     t n1 n2   , (2.3) where θ(s) = arctan( k2 k1 ); k1 6= 0, τ(s) = dθ(s) ds , κ(s) = √ k21 + k 2 2, k1 = κ cos θ and k2 = κ sin θ, (2.4) and k1, k2 are called the first and second bishop curvatures and effectively correspond to a cartesian coordinate system for the polar coordinates κ,θ. 2.2. dual space. here, we give the notions of dual space and dual spherical curves of ruled surfaces. for more detailed descriptions, see [ [12–16]]. let a and a∗ be two real numbers and ε 6= 0, ε2 = 0. a dual number â is an ordered pair of the form (a, a∗) for all a, a∗ ∈ r. let r×r be a set denoted as d, where d = â = a + εa∗ : a, a∗ ∈ r, (2.5) and the dual numbers form a ring over the real number field. two inner operations and an equality on d are defined as follows: (1) ⊕ : d × d → d for â = (a, a∗), b̂ = (b, b∗) defined as â⊕ b̂ = (a + b) + ε( a∗ + b∗), is called the addition in d. (2) � : d × d → d for â = (a, a∗), b̂ = (b, b∗) defined as â � b̂ = ab + ε( a∗b + ab∗), is called the multiplication in d. (3) for the equality of â and b̂ we have â = b̂ ⇔ a = b, and a∗ = b∗. int. j. anal. appl. 16 (5) (2018) 617 the dual number â = a + εa∗ divided by the dual number b̂ = b + εb∗ provided b 6= 0 can be defined as â b̂ = a + ε a∗ b + εb∗ = a b + ε a∗b− b∗a b2 . (2.6) obviously, the set d with the operations of addition, multiplication and equality on d = r × r is a ring with non trivial zero devisors. in a dual number â = (a,a∗) ∈ d, the real number a is called the real part of â and the real number a∗ is called the dual part of â. the dual number (1, 0) = 1 is called a real unit in d and the dual number (0, 1) is to be denoted with ε in short, and is called dual unit [9]. the set of d3 = d×d×d = {â : â = a + εa∗, a, a∗ ∈ r3}; (2.7) a = (a 1 ,a 2 ,a 3 ), a∗ = (a∗1,a ∗ 2,a ∗ 3) is a module over the ring d. for any â, b̂ ∈ d3, the inner product and the vector product are defined as follows: < â, b̂ >=< a, b > +ε(< a, b∗ > + < a∗, b >), â × b̂ = a × b + ε(a × b∗ + a∗ × b), (2.8) respectively. if a 6= 0, then the norm is defined by ‖â‖ = √ < â, â > = ‖a‖ + ε < a, a∗ > ‖a‖ . (2.9) a dual vector â with norm 1 is called a dual unit vector. let â = a + εa∗ ∈d3, the set ŝ2 = {â = a + εa∗ : ‖â‖ = (1, 0); a, a∗ ∈ r3}, (2.10) is called the dual unit sphere with center ô in d3. via this we have the following map (e. study’s map, c.f. figure 1): the set of all oriented lines in euclidean space e3 is in one-to-one correspondence with set of points of dual unit sphere in d3−space [13, 14, 16]. dual function of dual number presents a mapping of a dual number space on itself. properties of dual functions were thoroughly investigated by dimentberg [17]. he derived the general expression for dual analytic (differentiable) function as follows: f(â) = f(a + εa∗) = f(a) + εa∗ f′(a), where f′(a) is the derivative of f(a) and a, a∗ ∈ r. this definition allows us to write the dual forms of some well-known function as follows:  sin(â) = sin(a + εa∗) = sin(a) + εa∗ cos(a), cos(â) = cos(a + εa∗) = cos(a) −εa∗ sin(a), √ â = √ a + εa∗ = √ a + ε a ∗ 2 √ a , (a > 0). int. j. anal. appl. 16 (5) (2018) 618 the e. study’s map allows us to write a ruled surface by a dual vector function. so, ruled surfaces and dual curves are synonymous in this paper. consider the dual serret–frenet frame {t̂(ŝ), n̂(ŝ), b̂(ŝ)} associated with a dual curve α̂(ŝ), then the serret–frenet formulae read:   t̂ ′ n̂ ′ b̂ ′   =   0 k̂ 0 −k̂ 0 τ̂ 0 −τ̂ 0     t̂ n̂ b̂   , ( ′ = d dŝ ) , (2.11) where k̂ = k̂(ŝ) and τ̂ = τ̂(ŝ) are called the dual curvature function and the dual torsion function, respectively. 3. daw(k)−type curves in this section, we consider aw(k)−type curves in the dual space d3 and denote this type of curves by daw(k)−type. for this purpose, let {t̂, n̂1, n̂2} be a dual bishop frame of α̂(ŝ). then the bishop formulas of α̂ are given by   t̂′ n̂′1 n̂′2   =   0 k̂1 k̂2 −k̂1 0 0 −k̂2 0 0     t̂ n̂1 n̂2   , (3.1) where t̂ = t + εt∗, n̂1 = n1 + εn ∗ 1 , n̂2 = n2 + εn ∗ 2, k̂1 = k1 + εk ∗ 1 and k̂2 = k2 + εk ∗ 2. proposition 3.1. let α̂ be a unit speed dual curve with arc length parameter ŝ and {t̂, n̂1, n̂2} be its bishop frame. there within follows for its derivatives α̂8 = t̂, α̂88 = k̂1n̂1 + k̂2n̂2, α̂888 = − ( k̂21 + k̂ 2 2 ) t̂ + k̂81n̂1 + k̂ 8 2n̂2, α̂8888 = − (( k̂21 + k̂ 2 2 )8 + k̂1k̂ 8 1 + k̂2k̂ 8 2 ) t̂ + ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) n̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) n̂2. (3.2) int. j. anal. appl. 16 (5) (2018) 619 notation 3.1. from proposition 3.1, we can write v̂1 = k̂1n̂1 + k̂2n̂2, v̂2 = k̂ 8 1n̂1 + k̂ 8 2n̂2, v̂3 = ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) n̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) n̂2. (3.3) definition 3.1. [4] the unit speed dual curves of osculating order 3 are (i): of bishop daw(1)−type if and only if v̂3 = 0, (ii): of bishop daw(2)−type if and only if ∥∥∥v̂2∥∥∥2 v̂3 = 〈v̂3, v̂2〉 v̂2, (iii): of bishop daw(3)−type if and only if ∥∥∥v̂1∥∥∥2 v̂3 = 〈v̂3, v̂1〉 v̂1. (3.4) proposition 3.2. the unit speed dual curve is of bishop daw(1)−type if and only if bishop curvature equations k881 −k 3 1 −k1k 2 2 = 0, k 88 2 −k 3 2 −k 2 1k2 = 0, k∗881 − 3k 2 1k ∗ 1 −k 2 2k ∗ 1 − 2k1k2k ∗ 2 = 0, k ∗88 2 − 3k 2 2k ∗ 2 −k 2 1k ∗ 2 − 2k1k2k ∗ 1 = 0, (3.5) hold. proof. by the aid of definition 3.1 and notation 3.1, one can obtain ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) n̂1 + ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) n̂2 = 0. (3.6) since n̂1 and n̂2 are linearly independent, then k̂881 − k̂ 3 1 − k̂1k̂ 2 2 = 0, k̂882 − k̂ 3 2 − k̂ 2 1k̂2 = 0. (3.7) by separating eqs. (3.7) into real and dual parts we obtain the desired equations. � int. j. anal. appl. 16 (5) (2018) 620 proposition 3.3. the unit speed dual curve is of bishop daw(2)−type if and only if the bishop curvature equations ( k881 −k 3 1 −k1k 2 2 ) k82 = ( k882 −k 3 2 −k 2 1k2 ) k81, ( k881 −k 3 1 −k1k 2 2 ) k∗82 + ( k∗881 − 3k 2 1k ∗ 1 −k ∗ 1k 2 2 − 2k1k2k ∗ 1 ) k82 = ( k882 −k 3 2 −k 2 1k1 ) k∗81 + ( k∗882 − 3k 2 2k ∗ 2 − 2k1k ∗ 1k2 −k 2 1k ∗ 2 ) k81. (3.8) hold. proof. in the light of definition 3.1 and notation 3.1, it is easy to get ( k̂822 k̂ 88 1 − k̂ 3 1k̂ 82 2 − k̂1k̂ 4 2 ) n̂1 + ( k̂821 k̂ 88 2 − k̂ 3 2k̂ 82 1 − k̂ 2 1k̂2k̂ 82 1 ) n̂2 = ( k̂81k̂ 88 2 k̂ 8 2 − k̂ 3 2k̂ 8 1k̂ 8 2 − k̂ 2 1k̂2k̂ 8 1k̂ 8 2 ) n̂1 + ( k̂81k̂ 88 2 − k̂ 3 2k̂ 8 1 − k̂ 2 1k̂2k̂ 8 1 ) n̂2, then ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 ) k̂82 = ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) k̂81. (3.9) similarly, by separating eq. (3.9) into real and dual parts, we get eq. (3.8). � proposition 3.4. the unit speed dual curve is of bishop daw(3)−type if and only if the bishop curvature equations ( k881 −k 3 1 −k1k 2 2 ) k2 = ( k882 −k 3 2 −k 2 1k2 ) k1, ( k881 −k 3 1 −k1k 2 2 ) k∗2 + ( k∗881 − 3k 2 1k ∗ 1 −k ∗ 1k 2 2 − 2k1k2k ∗ 2 ) k2 = ( k882 −k 3 2 −k 2 1k1 ) k∗1 + ( k∗882 − 3k 2 2k ∗ 2 − 2k1k ∗ 1k2 −k 2 1k ∗ 2 ) k1. (3.10) hold. proof. using definition 3.1 and notation 3.1, we have ( k̂21k̂ 88 1 − k̂ 3 1k̂ 2 2 − k̂ 5 1 + k̂1k̂2k̂ 88 2 − k̂ 4 2k̂1 − k̂ 3 1k̂ 2 2 ) n̂1 + (( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 )( k̂21 + k̂ 2 1 )) n̂2 = ( k̂21k̂ 88 1 − k̂ 5 1 − k̂ 3 1k̂ 2 2 + k̂ 88 1 k̂ 2 2 − k̂ 3 1k̂ 2 2 − k̂1k̂ 4 2 ) n̂1 + ( k̂22 ( k̂882 − k̂ 3 2 − k̂ 2 1k̂2 ) + k̂1k̂2 ( k̂881 − k̂ 3 1 − k̂1k̂ 2 2 )) n̂2, it follows that { k̂881 − k̂ 3 1 − k̂1k̂ 2 2 } k̂2 = { k̂882 − k̂ 3 2 − k̂ 2 1k̂2 } k̂1. (3.11) separating eq. (3.11) into real and dual parts, we obtain eq. (3.10). � int. j. anal. appl. 16 (5) (2018) 621 4. evolutes of dual spherical curves for ruled surfaces in this section, we give the notions of dual spherical curves of ruled surfaces as well as evolutes of these curves. for more detailed descriptions (see [12, 15, 16, 18, 19]). a ruled surface is a surface swept out by a straight line l moving along a curve β = β(t). the various positions of the generating lines l are called the rulings of the surface. such a surface, thus, has a parametrization in the ruled form ψ (t,v) = β(t) + vα(t); v ∈ r. (4.1) here, β = β(t) is called the base curve, α = α(t) is the unit vector giving the direction of generating line, and t is the motion parameter. the base curve is not unique, because any curve of the form γ(t) = β(t) + η(t)α(t), (4.2) may be used as its base curve, η(t) is a smooth function. if there exists a common perpendicular to two neighboring rulings on ψ , then the foot of the common perpendicular on the main ruling is called a central point. the locus of the central points is called the striction curve. in eq. (4.2), if η(t) = − 〈β′(t),α′(t)〉 ‖α′(t)‖2 , (4.3) then γ(t) is called the striction curve on the ruled surface ψ, and it is unique. the base curve β(t) of the ruled surface is its striction curve if and only if 〈β′(t),α′(t)〉 = 0. while a differentiable curve on the dual unit sphere ŝ2 corresponds to a ruled surface in e3. a differentiable curve γ̂ on a dual unit sphere, depending on real parameter t, represents a differentiable family of straight lines in e3, which we call a ruled surface. the ruled surface ψ is written as the dual vector function γ̂ given by (according to the e. study’s dual-line coordinates) γ̂(t) = u(t) = α(t) + εγ(t) ∧α(t) = α(t) + εα∗(t), (4.4) where α∗ is the moment of α about the origin in e3, and ε is an indeterminate subject to the relation ε2 = 0. hence, ruled surfaces and dual curves are synonymous in this work. because 〈γ̂(t), γ̂(t)〉 = 1, thus, the ruled surface can be represented by the dual curve on the surface of a dual unit sphere ŝ2 (see fig. 1 and fig. 2). then γ̂(t) is called the dual spherical curve of ruled surface ψ. now, as in real spherical geometry, we define an orthonormal moving frame along this dual curve as follows [18]: u1 = u(t), u2(t) = u′1 ‖ u′1‖ , u3(t) = u1 ∧ u2. (4.5) from now on we consider the case without α(t) = constant vector and α∗(t) = 0. in the case α(t) = constant vector, the ruled surface ψ (t,v) is a cylinder and in the case α∗(t) = 0, the ruled surface ψ (t,v) is a cone. int. j. anal. appl. 16 (5) (2018) 622 figure 1. ruled surface mapped to the dual spherical curve. the dual unit vectors u1, u2 and u3 corresponds to three concurrent mutually orthogonal lines in e3. their point of intersection is the central point on the ruling u1, u3(t) is the limit position of the common perpendicular to u1(t), and is called the central tangent of the ruled surface u1 = u(t) at the central point. the line u2 = u2(t) is called the central normal of u1 = u(t) at the central point. moreover, the dual planes which correspond to the subspaces sp{u1, u2}, sp{u3, u2}, and sp{u1, u3}, respectively, are called the tangent plane, asymptotic plane and normal plane. by construction, the blaschke formula is  u′1 u′2 u′3   =   0 p 0 −p 0 q 0 −q 0     u1 u2 u3   ,′ = ddt, (4.6) where 〈u2, u2〉 = 1 = 〈u3, u3〉, 〈u1, u2〉 = 0, and p = p + εp∗ = ‖ u′1‖ , q = q + εq ∗ = det( u1, u ′ 1, u ′′ 1 ) ‖ u′1‖ 3 , (4.7) are called the blaschke’s invariants of the dual curve u1(t). one of the invariants of the dual curve u1 = u1(t) is σ : = q p , p 6= 0, (4.8) which is well-known as the dual geodesic curvature in ŝ2 [19, 20]. then, as in the case of real spherical curve, we may write for the dual curve u(t) the following formulas: k := κ + εκ∗ = √ 1 + σ2,t := τ + ετ∗ = ± σ ′ 1 + σ2 , (4.9) where k = k(t) is the dual curvature, and t = t (t) is the dual torsion of the dual curve u = u(t). due to [16], the evolute of the dual unit spherical curve γ̂ is the locus of all its centers of geodesic curvature. so, it can be defined through the following form eγ̂(ŝ) = 1 √ 1 + σ2 (σu1 + u3), (4.10) int. j. anal. appl. 16 (5) (2018) 623 where ŝ = t∫ t1 ‖(γ̂′(t))‖dt = s + εs∗ is the dual arc length of the curve γ̂(t) from t1 to t. under the previous notations about dual spherical curves and their evolutes we can summarize the following results: corollary 4.1. let γ̂ : i ⊂ d → ŝ2 be a dual regular unit spherical curve of a ruled surface, then γ̂ and its osculating dual circle have a four-point contact at γ̂(ŝ0) if and only if σ ′(ŝ) = 0 and σ′′(ŝ) 6= 0. corollary 4.2. [16] the evolute of the dual unit spherical curve γ̂ at ŝ0 is diffeomorphic to the ordinary cusp if σ′(ŝ0) = 0 and σ ′′(ŝ0) 6= 0. the ordinary cusp is ĉ = {(â1, â2) ∣∣â21 = â32)}. lemma 4.1. the dual spherical curve γ̂(ŝ) is a great circle if the dual geodesic curvature function σ(ŝ) of γ̂ is identically zero, and then the ruled surface is a right helicoid and the striction curve is a geodesic curve. 5. examples example 5.1. let α̂ be a dual curve in d3 defined by α̂ (ŝ) = ( sin ŝ, sin ŝ cos ŝ, cos2 ŝ ) = ( sin s, sin s cos s, cos2 s ) + εs∗ (cos s, cos 2s,−sin 2s) ; ŝ = s + εs∗. the corresponding ruled surface is given by rα(s,v) = α∧α∗ + vα = ( −cos2 s + v sin s, cos3 s + sin s sin 2s + v cos s sin s,−sin3 s + v cos2 s ) . after some calculations, we obtain t̂ = ( √ 2 cos ŝ √ 3 + cos 2ŝ , √ 2 cos 2ŝ √ 3 + cos 2ŝ ,− √ 2 sin 2ŝ √ 3 + cos 2ŝ ) , n̂ = ( − 2 sin ŝ √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ,− 12 sin 2ŝ + sin 4ŝ 2 √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ,− 4 ( cos4 ŝ + cos 2ŝ ) √ 3 + cos 2ŝ √ 13 + 3 cos 2ŝ ) , b̂ = ( − 2 √ 2 √ 13 + 3 cos 2ŝ , 2 √ 2 cos3 ŝ √ 13 + 3 cos 2ŝ , −3 sin ŝ− sin 3ŝ √ 26 + 6 cos 2ŝ ) . and κ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 , τ̂ = − 12 cos ŝ 13 + 3 cos 2ŝ , θ̂ = ∫ τ̂(ŝ)dŝ = 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ ) , k̂1 = κ̂ cos θ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 ( cos [ 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ )]) . int. j. anal. appl. 16 (5) (2018) 624 -1.0 -0.5 0.0 0.5 1.0x -1.0 -0.5 0.0 0.5 1.0 y -1.0 -0.5 0.0 0.5 1.0 z (a) -2 0 2 x -2 0 2 y -2 0 2 z (b) figure 2. the ruled surface corresponding to the dual unit spherical curve α̂. k̂2 = κ̂ sin θ̂ = 2 √ 13 + 3 cos 2ŝ (3 + cos 2ŝ)3/2 ( sin [ 1 2 √ 3 2 ln ( 8 √ 6 − 12 sin ŝ 2 √ 6 + 3 sin ŝ )]) . in the case that ŝ = 3π/2, we get k̂1 = −0.34, k̂2 = 2.21, k̂81 (s) = 0, k̂ 8 2 (s) = 0, k̂ 88 1 (s) = 3.46, k̂ 88 2 (s) = −4.90. according to proposition 3.3, the curve α̂ is of bishop daw(2)-type because( k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) ) k̂82 (s) = ( k̂882 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) ) k̂81 (s) = 0. also, according to propositions 3.2 and 3.4, α̂ is neither of bishop daw(1)-type nor bishop daw(3)-type because k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) 6= 0, k̂ 88 2 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) 6= 0, and { k̂881 (s) − k̂ 3 1 (s) − k̂1 (s) k̂ 2 2 (s) } k̂2 (s) 6= { k̂882 (s) − k̂ 3 2 (s) − k̂ 2 1 (s) k̂2 (s) } k̂1 (s) . example 5.2. let ψ(s,v) be a ruled surface of e3 defined by [16] ψ(s,v) = γ(s) + vδ(s); v ∈ r, where γ(s) = ( −2 1 + cos2 s , 2 cos3 s 1 + cos2 s , −sin s− 2 sin s cos2 s 1 + cos2 s ) and δ(s) = ( sin s, sin s cos s, cos2 s ) . int. j. anal. appl. 16 (5) (2018) 625 the ruled surface ψ is written as the dual vector function γ̂ given by γ̂(s) = û(s) = δ(s) + ε γ(s) ∧ δ (s) , which can be expressed as follows γ̂ (ŝ) = û(ŝ) = ( sin ŝ, sin ŝ cos ŝ, cos2 ŝ ) . now, we can write the orthonormal moving frame { û1, û2, û3 } along this dual curve as follows û1 = û(ŝ), û2(ŝ) = û′1∥∥∥ û′1∥∥∥, û3(ŝ) = û1 ∧ û2. û1 = { sin ŝ, cos ŝ sin ŝ, cos2 ŝ } , û2 = { √ 2 cos ŝ √ 3 + cos 2ŝ , √ 2 cos 2ŝ √ 3 + cos 2ŝ ,− √ 2 sin 2ŝ √ 3 + cos 2ŝ } , û3 = { − √ 2 cos2 ŝ √ 3 + cos 2ŝ ,− −5 cos ŝ + cos 3ŝ 2 √ 2 √ 3 + cos 2ŝ ,− √ 2 sin3 ŝ √ 3 + cos 2ŝ } , and p = ∥∥∥ û′1∥∥∥ = √ 3 + cos 2ŝ √ 2 , q = det( u1, u ′ 1, u ′′ 1 ) ‖ u′1‖ 3 = − √ 2(5 + cos 2ŝ) sin ŝ (3 + cos 2ŝ)3/2 , hence, the dual geodesic curvature σ̂ : = q p = − 2(5 + cos 2ŝ) sin ŝ (3 + cos 2ŝ)2 , the dual curvature k = k(ŝ) and the dual torsion t = t (ŝ) of γ̂ are calculated as follows k = √ 1 + σ̂2 = √ 1 + 4(5 + cos 2ŝ)2 sin2 ŝ (3 + cos 2ŝ)4 , t = ± σ̂ ′ 1 + σ̂2 = −106 cos ŝ + 9 cos 3ŝ + cos 5ŝ 2(3 + cos 2ŝ)3 ( 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 ). we obtain the evolute of the dual unit spherical curve of ruled surface as follows (see fig. 3) eγ̂ (ŝ) = ( â1 (ŝ) , â2 (ŝ) , â3 (ŝ) ) , where â1 (ŝ) = 1 − 8 (3+cos 2ŝ)2 − 2 3+cos 2ŝ + √ 2√ 3+cos 2ŝ − √ 3+cos 2ŝ√ 2√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 , int. j. anal. appl. 16 (5) (2018) 626 â2 (ŝ) = √ 2 cos3 ŝ√ 3+cos 2ŝ − 2 cos ŝ (5+cos 2ŝ) sin 2 ŝ (3+cos 2ŝ)2 + √ 2 sin ŝ sin 2ŝ√ 3+cos 2ŝ√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 , â3 (ŝ) = ( −1 + 4 (3+cos 2ŝ)2 − 2 √ 2√ 3+cos 2ŝ + √ 3+cos 2ŝ√ 2 ) sin ŝ√ 1 + 4(5+cos 2ŝ)2 sin2 ŝ (3+cos 2ŝ)4 . also, in the case that ŝ = 3π 2 , we get σ̂′e( 3π 2 ) = 0, σ̂′′e( 3π 2 ) = −8. then the evolute of the dual unit spherical curve γ̂ at ŝ = 3π 2 is diffeomorphic to the ordinary cusp and hence, the corollary 4.2 is satisfied. -1.0 -0.5 0.0 0.5 1.0x -1.0 -0.5 0.0 0.5 1.0 y -1.0 -0.5 0.0 0.5 1.0 z figure 3. the dual spherical curve γ̂ (the red color) of the ruled surface ψ and its evolute (the blue color). 6. conclusion in this work, we have studied dual curves in dual space d3 due to the notion of aw(k)-type curves which was defined by k. arslan and a. west [4] and denote it by daw(k) curves. besides, some conditions on curvatures of these curves to be of daw(k)-type using bishop frame were introduced. in addition, according to the e. study of the correspondence between the oriented lines in euclidean three space and the dual points of the dual unit sphere in dual three space, we have obtained evolutes of dual spherical curves for ruled surfaces. finally, the obtained results were confirmed by giving two examples. int. j. anal. appl. 16 (5) (2018) 627 references [1] e. study, geometry der dynamen. lepzig, 1901.clifford, w. k., preliminary sketch of biquaternions, proc. lond. soc., 4 (64) (1873), 381-395. [2] h. w. guggenheimer, differential geometry. mcgraw-hill book co., new york, 1963. [3] c.e. weatherburn, differential geometry of three dimensions. syndic of cambridge university press, 1981. [4] k. arslan, a. west, product submanifolds with pointwise 3-planar normal sections. glasgow math. j., 37(1) (1995), 73–81. [5] k. arslan, c. özgür, curves and surfaces of aw(k) type in geometry and topology of submanifolds. ix (valenciennes/lyon/leuven, 1997), world sci. publ., 1999, 21–26. [6] c. özgür, f. gezgin, on some curves of aw(k)-type. differ. geom. dyn. syst., 7 (2005), 74–80. [7] i̇lim kişi, günay öztürk, aw (k)-type curves according to the bishop frame. arxiv preprint arxiv:1305.3381, 2013. [8] g. öztürk, a.küçük, k. arslan, some characteristic properties of aw(k)-type curves on dual unit sphere. extracta mathematicae, 29(1-2) (2014), 167-175. [9] w. k. clifford, preliminary sketch of bi-quaternions. proc. london math. soc., s1-4(1) (1871), 381 – 395. [10] m. p. do carmo, differential geometry of curves and surfaces. prentice hall, englewood cliffs, n. j., 1976. [11] t. shifrin , differential geometry, a first course in curves and surfaces. (preliminary version), university of georgia, 2010. [12] s. suleyman, m. bilici and m. caliskan, some characterizations for the involute curves in dual space. math. combin. book ser., vol. 1, 2015, 113-125. [13] n. h. abdel-all, r. a. huesien and a. abdela ali, dual construction of developable ruled surface. j. amer. sci., 7(4) (2011), 789-793. [14] m. ōnder, h. uğurlu, dual darboux frame of a timelike ruled surface and darboux approach to mannheim offsets of timelike ruled surfaces. proc. nat. acad. sci., india sect. a: phys. sci., 83 (2013), 163–169. [15] r. baky, r. ghefari, on the one-parameter dual spherical motions. computer aided geometric design, 28 (2011), 23–37. [16] li. yanlin, pei. donghe, evolutes of dual spherical curves for ruled surfaces. math. meth. appl. sci., 39 (2016), 3005–3015. [17] f. m. dimentberg, the screw calculus and its applications in mechanics. (izdat. nauka, moscow, ussr) english translation: ad680993, clearinghouse for federal and scientific technical information, 1965. [18] m. t. aldossary, r. baky, on the bertrand offsets for ruled and developable surfaces. boll. unione mat. ital., 8 (2015), 53–64. [19] j.a. schaaf, curvature theory of line trajectories in spatial kinematics. doctoral dissertation, university of california, davis, ca, 1988. [20] g.r. veldkamp, on the use of dual numbers, vectors, and matrices in instantaneous spatial kinematics. mech. mach. theory, 11 (1976), 141–156. 1. introduction 2. fundamental concepts 2.1. bishop frame 2.2. dual space 3. daw(k)-type curves 4. evolutes of dual spherical curves for ruled surfaces 5. examples 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 209-217 http://www.etamaths.com ricci solitons in (ε,δ)-trans-sasakian manifolds c.s. bagewadi and gurupadavva ingalahalli∗ abstract. we study ricci solitons in (ε,δ)-trans-sasakian manifolds. it is shown that a symmetric parallel second order covariant tensor in a (ε,δ)-trans-sasakian manifold is a constant multiple of the metric tensor. using this it is shown that if lv g + 2s is parallel where v is a given vector field, then (g,v ) is ricci soliton. further, by virtue of this result, ricci solitons for 3-dimensional (ε,δ)-trans-sasakian manifolds are obtained. also an example of ricci solitons in 3-dimensional (ε,δ)-trans-sasakian manifold is provided in the region where trans-sasakian manifold is expanding (shrinking) the lorentzian trans-sasakian manifold is shrinking (expanding). 1. introduction in [12], gray-harvella classification of almost hermitian manifolds appears as a class w4 of hermitian manifolds which are closely related to locally conformal kähler manifolds. a class of almost contact metric manifolds known as trans-sasakian manifolds was introduced by j.a. oubina [24] in 1985. an almost contact metric structure on a manifold m is called a trans-sasakian structure [24] of the product manifold m ×r belongs to the class w4. the class c5 ⊕c6 [24] coincides with the class of trans-sasakian structure of (α,β). this class consists of both sasakian and kenmotsu structures. if α = 1, β = 0 then the class reduces to sasakian where as if α = 0, β = 1 then it reduces to kenmotsu. the above manifolds are studied by many authors like d.e. blair and j.a. oubina [6], j.c. marrero [20], k. kenmotsu [17], c.s. bagewadi and venkatesha [2], u.c. de and m.m. tripathi [9], a.a. shaikh et. al. [25] etc. the study of manifolds with indefinite metrics is of interest from the standpoint of physics and relatively. manifolds with indefinite metrics have been studied by several authors. the concept of (ε)-sasakain manifolds was initiated by bejancu and duggal [4] and further investigation was taken up by x. xufeng and c. xiaoli [31] and rakesh kumar et. al. [18]. the historical background of (ε)-sasakain manifolds can be traced back to the classification of sasakian manifolds with indefinite metrics which play crucial role in physics [10]. u.c. de and a. sarkar [8] studied the notion of (ε)kenmotsu manifolds with indefinite metric. s.s. shukla and d.d. sing [28] extended the study to (ε)trans-sasakian manifolds with indefinite metric which are natural generalization of both (ε)-sasakian and (ε)-kenmotsu manifolds. the authors h.g. nagaraja et. al. [22] studied (ε,δ)-trans-sasakian manifolds which are extensions of (ε)-trans-sasakian manifolds. ricci solitons move under the ricci flow introduced by hamilton [14] simply by diffeomorphisms of the initial metric that is they are stationary points of the ricci flow: ∂g ∂t = −2ric(g), (in this paper we use ric = s) in the space of metrics on m. definition 1.1. a ricci soliton (g,v,λ) on a riemannian manifold is defined by lv g + 2s + 2λg = 0, (1.1) where s is the ricci tensor, lv is the lie derivative along the vector field v on m and λ is a real. ricci soliton is said to be shrinking, steady or expanding according as λ < 0, λ = 0 and λ > 0. in 1923, l.p. eisenhart [11] proved that if a positive definite riemannian manifold (m,g) admits a second order parallel symmetric covariant tensor other than a constant multiple of the metric tensor, received 7th august, 2013; accepted 24th june, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 53c15, 53c20, 53c25, 53c44, 53d10. key words and phrases. ricci soliton; (ε,δ)-trans-sasakian manifold; einstein manifold. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 209 210 bagewadi and ingalahalli then it is reducible. in 1925, levy [19] obtained the necessary and sufficient conditions for the existence of such tensors. later, r. sharma [26] studied second order parallel tensor in real and complex space forms. in [27], r. sharma initiated the study of ricci solitons in contact geometry. thereafter ricci solitons in contact metric manifolds have been studied by various authors such as m.m. tripathi [29], constantin calin and mircea crasmareanu [7], amadendu ghosh and ramesh sharma [1], mircea crasmareanu [21], u.c. de et al. [30], h.g. nagaraja et al. [23], c.s. bagewadi et al. ( [3], [13]) and many others. motivated by the above results we studied ricci solitons in (ε,δ)-trans-sasakian manifolds. 2. preliminaries let m be an almost contact metric manifold of dimension n equipped with an almost contact metric structure (φ,ξ,η,g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a riemannian metric g satisfying φ2 = −i + η ⊗ ξ, η(ξ) = 1, η ◦φ = 0, φξ = 0. (2.1) almost contact metric manifold m is called (ε)-almost contact metric manifold if there exists a semiriemannian metric g, g(ξ,ξ) = ε, (2.2) g(φx,φy ) = g(x,y ) −εη(x)η(y ), η(x) = εg(x,ξ), (2.3) where ε is 1 or −1 according as ξ is space like or time like. in particular, if the metric g is positive definite, then an (ε)-almost contact metric manifold is the usual almost contact metric manifold [5]. an (ε)-almost contact metric manifold is called an (ε)-trans-sasakian manifold [28] if (∇xφ)y = α(g(x,y )ξ −εη(y )x) + β(g(φx,y )ξ −εη(y )φx), (2.4) holds for some smooth functions α and β on m. in (ε)-trans-sasakian manifolds ξ is never light like as per the definition of characteristic vector field ξ. according to the characteristic vector field ξ we have two classes of (ε)-trans-sasakian manifolds. when ε = −1 and index of g is odd, then m is a time-like trans-sasakian manifold and when ε = 1 and index of g is even, then m is a space-like trans-sasakian manifold. further, m is usual trans-sasakian manifold for ε = 1 and index of g is 0 and m is a lorentzian trans-sasakian manifold for ε = −1 and index of g is 1. an (ε)-almost contact metric manifold is said to be (ε,δ)-trans-sasakian manifold if it satisfies, (∇xφ)y = α(g(x,y )ξ −εη(y )x) + β(g(φx,y )ξ −δη(y )φx), (2.5) holds for some smooth functions α and β on m, where ε is 1 or −1 according as ξ is space like or time like and δ is alike ε. from (2.5), we have ∇xξ = −εαφx −βδφ2x, (2.6) (∇xη)y = βδ[εg(x,y ) −η(x)η(y )] −αg(φx,y ). (2.7) remark 2.1. from (2.5), we have the following remarks: (1) ε = δ, (ε,δ)-trans-sasakian manifold of type (α,β) reduces to (ε)-trans-sasakian manifold of type (α,β). (2) ε = δ = 1, (ε,δ)-trans-sasakian manifold of type (α,β) reduces to trans-sasakian manifold of type (α,β). (3) α 6= 0, β 6= 0, and ε = −1, δ = −1, (ε,δ)-trans-sasakian manifold of type (α,β) reduces to the form lorentzian trans-sasakian manifold of type (α,β). (4) α 6= 0, β 6= 0, and ε = 1, δ = −1, (ε,δ)-trans-sasakian manifold of type (α,β) reduces in the form α-sasakian lorentzian β-kenmostu manifold of type (α,β). (5) α 6= 0, β 6= 0, and ε = −1, δ = 1, (ε,δ)-trans-sasakian manifold of type (α,β) reduces in the form lorentzian α-sasakian β-kenmostu manifold of type (α,β). (6) α 6= 0, β = 0 and ε = 1 or ε = −1, the (ε,δ)-trans-sasakian manifold reduces to α-sasakian manifold or lorentzian α-sasakian manifold respectively. ricci solitons in (ε,δ)-trans-sasakian manifolds 211 (7) α = 0, β 6= 0 and δ = 1 or δ = −1, the (ε,δ)-trans-sasakian manifold reduces to β-kenmostu manifold or lorentzian β-kenmotsu manifold respectively. (8) if α and β are scalars and α = 1 and β = 0 or α = 0 and β = 1 then the (ε,δ)-trans-sasakian manifold reduces to (ε)-sasakian manifolds and (δ)-kenmostu manifolds. (a) again, if in (ε)-sasakian manifolds ε is 1 or −1 then the (ε)-sasakian manifolds reduces to sasakian manifolds or lorentzian sasakian manifolds. (b) further, if in (δ)-kenmostu manifolds δ is 1 or −1 then the (δ)-kenmostu manifolds reduces to kenmotsu manifold or lorentzian kenmotsu manifold. note: if (m,φ,ξ,η,g) is a lorentzian almost contact manifolds then it satisfies: φ2x = −x + η(x)ξ, η(ξ) = 1, (2.8) g(φx,φy ) = g(x,y ) + η(x)η(y ), η(x) = −g(x,ξ). (2.9) 2.1. example for trans-sasakian manifolds. the 3-dimensional manifold m = {(x,y,z) ∈ r3; z 6= 0} is a trans-sasakian manifold: if {e1,e2,e3} is a global frame field given by e1 = z ( ∂ ∂x + y ∂ ∂z ) , e2 = z ∂ ∂y , e3 = ∂ ∂z . (2.10) the (φ,ξ,η,g) is given by η = dz − ydx, ξ = e3 = ∂∂z , φe1 = e2, φe2 = −e1, φe3 = 0, g = 1 z2 [(1 − y2z2)dx ⊗ dx + dy ⊗ dy + z2dz ⊗ dz], g(e1,e2) = g(e2,e3) = g(e1,e3) = 0, g(e1,e1) = g(e2,e2) = g(e3,e3) = 1. the (φ,ξ,η,g) is trans-sasakian structure for ∇xξ = −αφx + β(x −η(x)ξ), with α = −1 2 z2 6= 0 and β = −1 z 6= 0. 2.2. example for (ε)-trans-sasakian manifolds. the 3-dimensional manifold m = {(x,y,z) ∈ r3; z 6= 0} is an (ε)-trans-sasakian manifold: if {e1,e2,e3} is a global frame field given by e1 = x z ∂ ∂x , e2 = y z ∂ ∂y , e3 = ε ∂ ∂z . (2.11) then (φ,ξ,η,g) is given by φe1 = e2, φe2 = −e1, φe3 = 0, η = εdz, ξ = e3 = ∂∂z , g = z2 x2 dx⊗dx + z2 y2 dy⊗dy + εdz ⊗dz, g(e1,e2) = g(e2,e3) = g(e1,e3) = 0, g(e1,e1) = g(e2,e2) = g(e3,e3) = ε. the (φ,ξ,η,g) is (ε)-trans-sasakian structure for ∇xξ = ε[−αφx + β(x −η(x)ξ)], with α = −1 and β = 1 z . in (ε,δ)-trans-sasakian manifold m, we have the following: r(x,y )ξ = ε[(y α)φx − (xα)φy ] + δ[(y β)φ2x − (xβ)φ2y ] + 2αβ(δ −ε)g(φx,y )ξ + 2εαβδ[η(y )φx −η(x)φy ] + (α2 −β2)[η(y )x −η(x)y ], (2.12) r(ξ,y )x = ε[(gradα)g(φx,y ) + (xα)φy ] + δ[(gradβ)g(φ2x,y ) − (xβ)φ2y ] + 2αβε(δ −ε)η(y )φx + 2εαβδ[εg(φx,y )ξ + η(x)φy ] + (α2 −β2)[εg(x,y )ξ −η(x)y ], (2.13) r(ξ,y )ξ = [(α2 −β2) − δ(ξβ)][−y + η(y )ξ] − [ε(ξα) + 2εαβδ](φy ), (2.14) s(x,ξ) = [(n− 1)(εα2 − δβ2) − (ξβ)]η(x) −ε((φx)α) − (n− 2)ε(xβ), (2.15) ε(ξα) + 2εαβδ = 0. (2.16) further, in 3-dimensional (ε,δ)-trans-sasakian manifold, we have φ(gradα) = gradβ. (2.17) 212 bagewadi and ingalahalli 3. parallel symmetric second order tensors and ricci solitons in (ε,δ)-trans-sasakian manifolds fix h a symmetric tensor field of (0, 2)-type which we suppose to be parallel with respect to ∇ that is ∇h = 0. applying the ricci identity [26] ∇2h(x,y ; z,w) −∇2h(x,y ; w,z) = 0, (3.1) we obtain the relation h(r(x,y )z,w) + h(z,r(x,y )w) = 0. (3.2) replacing z = w = ξ in (3.2) and by using (2.12) and by the symmetry of h, we have 2ε[(y α)h(φx,ξ) − (xα)h(φy,ξ)] + 2δ[(y β)h(φ2x,ξ) − (xβ)h(φ2y,ξ)] +2(α2 −β2)[η(y )h(x,ξ) −η(x)h(y,ξ)] + 4εαβδ[η(y )h(φx,ξ) −η(x)h(φy,ξ)] +4αβ(δ −ε)g(φx,y )h(ξ,ξ) = 0. (3.3) putting x = ξ in (3.3) and by virtue of (2.1), we obtain −2[ε(ξα) + 2εαβδ]h(φy,ξ) + 2[(α2 −β2) − δ(ξβ)][η(y )h(ξ,ξ) −h(y,ξ)] = 0. (3.4) by using (2.16) in (3.4), we have [(α2 −β2) −δ(ξβ)][η(y )h(ξ,ξ) −h(y,ξ)] = 0. (3.5) suppose (α2 −β2) −δ(ξβ) 6= 0, it results h(y,ξ) = η(y )h(ξ,ξ). (3.6) let us call a regular (ε,δ)-trans-sasakian manifolds with (α2 − β2) − δ(ξβ) 6= 0, where regularity means the non-vanishing of the ricci curvature with respect to the generator of (ε,δ)-trans-sasakian manifolds. differentiating (3.6) covariantly with respect to x, we have (∇xh)(y,ξ) + h(∇xy,ξ) + h(y,∇xξ) = [εg(∇xy,ξ) + εg(y,∇xξ)]h(ξ,ξ) + η(y )[(∇xh)(y,ξ) + 2h(∇xξ,ξ)]. (3.7) by using the parallel condition ∇h = 0, η(∇xξ) = 0 and by virtue of (3.6) in (3.7), we get h(y,∇xξ) = εg(y,∇xξ)h(ξ,ξ). by using (2.6) in the above equation, we obtain −εαh(y,φx) + βδh(y,x) = −αg(y,φx)h(ξ,ξ) + εβδg(y,x)h(ξ,ξ). (3.8) replacing x = φx in (3.8) and on simplification, we get h(x,y ) = εg(x,y )h(ξ,ξ), (3.9) which together with the standard fact that the parallelism of h implies that h(ξ,ξ) is a constant, via (3.6). now by considering the above conditions, we can state the following: theorem 3.1. a symmetric parallel second order covariant tensor in a regular (ε,δ)-trans-sasakian manifold is a constant multiple of the metric tensor. corollary 3.1. a locally ricci symmetric (∇s = 0) regular (ε,δ)-trans-sasakian manifold is an einstein manifold. remark 3.1. the following statements for indefinite (ε,δ)-trans-sasakian manifold are equivalent: (1) einstein, (2) locally ricci symmetric, (3) ricci semi-symmetric that is r ·s = 0. ricci solitons in (ε,δ)-trans-sasakian manifolds 213 the implication (1) −→ (2) −→ (3) is trivial. now we prove the implication (3) −→ (1) and r ·s = 0 means, (r(x,y ) ·s)(u,v ) = −s(r(x,y )u,v ) −s(u,r(x,y )v ). (3.10) considering r ·s = 0 and putting x = ξ in (3.10), we have s(r(ξ,y )u,v ) + s(u,r(ξ,y )v ) = 0. (3.11) by using (2.13) in (3.11), we obtain ε[g(φu,y )s(gradα,v ) + (uα)s(φy,v )] + δ[g(φ2u,y )s(gradβ,v ) − (uβ)s(φ2y,v )] +2αβε(δ −ε)η(y )s(φu,v ) + 2εαβδ[εg(φu,y )s(ξ,v ) + η(u)s(φy,v )] +(α2 −β2)[εg(y,u)s(ξ,v ) −η(u)s(y,v )] + ε[g(φv,y )s(u,gradα) + (v α)s(u,φy )] +δ[g(φ2v,y )s(u,gradβ) − (v β)s(u,φ2y )] + 2αβε(δ −ε)η(y )s(u,φv ) +2εαβδ[εg(φv,y )s(u,ξ) + η(v )s(u,φy )] + (α2 −β2)[εg(y,v )s(u,ξ) −η(v )s(u,y )] = 0. putting u = ξ in the above equation and by virtue of (2.1), (2.2), (2.15), (2.16) and on simplification, we get s(y,v ) = (n− 1)(εα2 − δβ2)g(y,v ). (3.12) in conclusion, we state the following: proposition 3.1. a ricci semi-symmetric regular indefinite (ε,δ)-trans-sasakian manifold is an einstein. corollary 3.2. suppose that on a regular (ε,δ)-trans-sasakian manifold the (0, 2)-type field lv g + 2s is parallel where v is a given vector field. then (g,v ) yield a ricci soliton. in particular, if the given regular (ε,δ)-trans-sasakian manifold is ricci semi-symmetric with lv g parallel, we have the same conclusion. proof. follows from theorem 3.1 and corollary 3.1. � hence we state the following result: corollary 3.3. a ricci soliton (g,ξ,λ) in an indefinite (ε,δ)-trans-sasakian manifold cannot be steady. proof. from linear algebra either the vector field v ∈ spanξ or v ⊥ ξ. however the second case seems to be complex to analyse in practice. for this reason we investigate for the case v = ξ. a simple computation of lξg + 2s gives (lξg)(x,y ) = 2βδ[g(x,y ) −εη(x)η(y )]. (3.13) from equation (1.1), we have h(x,y ) = −2λg(x,y ) and then putting x = y = ξ, we get h(ξ,ξ) = −2λε, (3.14) where h(x,y ) = (lξg)(x,y ) + 2s(x,y ) and then if we put x = y = ξ and again by using (3.13) and (2.15), we obtain h(ξ,ξ) = 2βδ[g(ξ,ξ) −εη(ξ)η(ξ)] + 2{ε[(n− 1)(εα2 − δβ2) − (ξβ)]η(ξ) − ε((φξ)α) − (n− 2)ε(ξβ)}. by using (2.1), (2.2) and (2.17) in the above equation, we get h(ξ,ξ) = 2(n− 1)ε(εα2 − δβ2). (3.15) equating (3.14) and (3.15), we have λ = −(n− 1)(εα2 − δβ2). (3.16) since from (3.16), we have λ 6= 0. hence the proof. � an indefinite (ε,δ)-trans-sasakian manifold contains the indefinite sasakain and kenmotsu manifolds based on this conditions and from corollary (3.3) we can state the following proposition: 214 bagewadi and ingalahalli proposition 3.2. (1) a ricci soliton in a trans-sasakian manifold is shrinking if α > β. (2) a ricci soliton in an α-sasakian lorentzian β-kenmotsu manifold is shrinking if α > β. (3) a ricci soliton in an lorentzian α-sasakian β-kenmotsu manifold is expanding if α > β. (4) a ricci soliton in an lorentzian trans-sasakian manifold is expanding if α > β. (5) a ricci soliton in an trans-sasakian manifold is expanding if α < β. (6) a ricci soliton in an lorentzian α-sasakian β-kenmotsu manifold is expanding if α < β. (7) a ricci soliton in a lorentzian trans-sasakian manifold is shrinking if α < β. (8) a ricci soliton in an α-sasakian lorentzian β-kenmotsu manifold is shrinking if α < β. proof. proofs of the above condition (1) to (4) follow from equation (3.16) and remarks (2.1) of (2), (4), (5), (3). again conditions (5) to (8) follow similarly from equation (3.16) and remarks (2.1) (2), (5), (3), (4). � 4. ricci solitons in 3-dimensional (ε,δ)-trans-sasakian manifold corollary 4.1. a ricci soliton (g,ξ,λ) where λ = −2(εα2 − δβ2) in an indefinite 3-dimensional (ε,δ)-trans-sasakian manifold with varying scalar curvature cannot be steady. proof. the riemannian curvature tensor of 3-dimensional (ε,δ)-trans-sasakian manifold is given by r(x,y )z = g(y,z)qx −g(x,z)qy + s(y,z)x −s(x,z)y − r 2 [g(y,z)x −g(x,z)y ]. (4.1) putting z = ξ in (4.1) and by using (2.12) and (2.15) for 3-dimensional (ε,δ)-trans-sasakian manifold, we get ε[(y α)φx − (xα)φy ] + δ[(y β)φ2x − (xβ)φ2y ] + 2αβ(δ −ε)g(φx,y )ξ +2εαβδ[η(y )φx −η(x)φy ] + (α2 −β2)[η(y )x −η(x)y ] = εη(y )qx −εη(x)qy +ε[(εα2 −δβ2) − (ξβ)](η(y )x −η(x)y ) −ε[((φy )α)x + (y β)x] + ε[((φx)α)y + (xβ)y ].(4.2) again, putting y = ξ in the above equation and by using (2.1) and (2.17), we obtain qx = [r 2 − 2(εα2 − δβ2) + ε(α2 −β2) ] x + [ 4(εα2 −δβ2) − r 2 −ε(α2 −β2) ] η(x)ξ. (4.3) from (4.3), we have s(x,y ) = [r 2 − 2(εα2 −δβ2) + ε(α2 −β2) ] g(x,y ) + [ 4(εα2 −δβ2) − r 2 −ε(α2 −β2) ] εη(x)η(y ). (4.4) equation (4.4) shows that a 3-dimensional (ε,δ)-trans-sasakian manifold is η-einstein. now we show that the scalar curvature r is not a constant that is r is varying. now h(x,y ) = (lξg)(x,y ) + 2s(x,y ). (4.5) by using (3.13) and (4.4) in (4.5), we have h(x,y ) = [r − 4(εα2 − δβ2) + 2ε(α2 −β2) + 2βδ]g(x,y ) + [8(εα2 − δβ2) − 2βδ − 2ε(α2 −β2) −r]εη(x)η(y ). (4.6) differentiating (4.6) covariantly with respect to z, we get (∇zh)(x,y ) = [∇zr − 4(2εα(zα) − 2δβ(zβ)) + 2ε(2α(zα) − 2β(zβ)) + 2δ(zβ)]g(x,y ) + [8(2εα(zα) − 2δβ(zβ)) − 2δ(zβ) − 2ε(2α(zα) − 2β(zβ)) −∇zr]εη(x)η(y ) + [8(εα2 − δβ2) − 2βδ − 2ε(α2 −β2) −r][g(x,∇zξ)η(y ) + g(y,∇zξ)η(x)]. (4.7) substituting z = ξ, x = y ∈ (spanξ)⊥ in (4.7) and by virtue of ∇h = 0 and (2.17), we have ∇ξr − 4εα(ξα) = 0. ricci solitons in (ε,δ)-trans-sasakian manifolds 215 by using (2.16) in the above equation, we obtain ∇ξr = −8εα2βδ. (4.8) thus, r is not a constant. now we have to check the nature of the soliton that is ricci soliton (g,ξ,λ) where λ = −2(εα2−δβ2) in 3-dimensional (ε,δ)-trans-sasakian manifold: from (1.1), we have h(x,y ) = −2λg(x,y ) and then putting x = y = ξ, we get h(ξ,ξ) = −2λε. (4.9) if x = y = ξ in (4.6), we obtain h(ξ,ξ) = 4ε(εα2 −δβ2). (4.10) equating (4.9) and (4.10), we have λ = −2(εα2 −δβ2). (4.11) since from (4.11), we have λ 6= 0. therefore ricci soliton (g,ξ,λ) of 3-dimensional (ε,δ)-trans-sasakian manifold cannot be steady. � now by corollary (4.1) we can state the following proposition: proposition 4.1. prove the following: (1) a ricci soliton in an 3-dimensional trans-sasakian manifold is shrinking if α > β. (2) a ricci soliton in an 3-dimensional α-sasakian lorentzian β-kenmotsu manifold is shrinking if α > β. (3) a ricci soliton in an 3-dimensional lorentzian α-sasakian β-kenmotsu manifold is expanding if α > β. (4) a ricci soliton in an 3-dimensional lorentzian trans-sasakian manifold is expanding if α > β. (5) a ricci soliton in an 3-dimensional trans-sasakian manifold is expanding if α < β. (6) a ricci soliton in an 3-dimensional lorentzian α-sasakian β-kenmotsu manifold is expanding if α < β. (7) a ricci soliton in a 3-dimensional lorentzian trans-sasakian manifold is shrinking if α < β. (8) a ricci soliton in an 3-dimensional α-sasakian lorentzian β-kenmotsu manifold is shrinking if α < β. proof. proofs of the above condition (1) to (4) follow from equation (4.11) and remarks (2.1) of (2), (4), (5), (3). again conditions (5) to (8) follow similarly from equation (4.11) and remarks (2.1) (2), (5), (3), (4). � 5. example of ricci solitons for 3-dimensional (ε,δ)-trans-sasakian manifold we consider the 3-dimensional manifold m = {(x,y,z) : (x,y,z) ∈ r3,z 6= 0}. let {e1,e2,e3} be linearly independent global frame field on m given by e1 = z ( ∂ ∂x + δy ∂ ∂z ) , e2 = δz ∂ ∂y , e3 = ∂ ∂z . (5.1) let g be the riemannian metric defined by g(e1,e2) = g(e2,e3) = g(e1,e3) = 0, g(e1,e1) = g(e2,e2) = g(e3,e3) = ε, where g is given by g = ε z2 [(1 −y2z2)dx⊗dx + dy ⊗dy + z2dz ⊗dz]. the (φ,ξ,η) is given by η = dz − δydx, ξ = e3 = ∂∂z , φe1 = e2, φe2 = −e1, φe3 = 0. clearly (φ,ξ,η,g) structure is an indefinite (ε,δ)-trans-sasakian structure and satisfy, (∇xφ)y = α(g(x,y )ξ −εη(y )x) + β(g(φx,y )ξ −δη(y )φx), (5.2) ∇xξ = −εαφx −βδφ2x, (5.3) where α = −z 2δ 2ε 6= 0 and β = − 1 zδ 6= 0. hence (φ,ξ,η,g) structure defines indefinite (ε,δ)-transsasakian structure. thus m equipped with indefinite (ε,δ)-trans-sasakian structure is a (ε,δ)-transsasakian manifold. 216 bagewadi and ingalahalli using the above α and β in (4.8), we have ∇ξr = −8εδα2β = 8εδ ( z4δ2 4ε2 )( 1 zδ ) = 2εz3, =⇒ r = z4ε 2 . (5.4) using the above α and β in (4.11), we have λ = 4δ −εz6 2z2 . (5.5) hence ricci soliton (g,ξ,λ) is given by (5.5) with varying scalar curvature (5.4). (1) if ε = δ = 1, in (5.5), then λ = 4−z 6 2z2 = (2−z3)(2+z3) 2z2 . (a) λ > 0 in {z : −21/3 < z < 21/3} : hence by remark (2.1), ricci soliton of the given 3-dimensional trans-sasakian manifold is expanding in the region {(x,y,z) ∈ r3 : −21/3 < z < 21/3}. (5.6) (b) also λ < 0 in {z : −21/3 > z > 21/3} : hence by remark (2.1), ricci soliton of the given 3-dimensional trans-sasakian manifold is shrinking in the region {(x,y,z) ∈ r3 : −21/3 > z > 21/3}. (5.7) hence the regions (5.6) and (5.7) are complementary to one another that is, m = {(x,y,z) ∈ r3 : −21/3 < z < 21/3}∪{(x,y,z) ∈ r3 : −21/3 > z > 21/3}. (2) if ε = δ = −1, in (5.5), then λ = z 6−4 2z2 = (z3−2)(z3+2) 2z2 . (a) λ > 0 in {z : −21/3 > z > 21/3} : hence by remark (2.1), ricci soliton in lorentzian trans-sasakian manifold is expanding in the region {(x,y,z) ∈ r3 : −21/3 > z > 21/3}. (5.8) (b) also λ < 0 in {z : −21/3 < z < 21/3} : hence by remark (2.1), ricci soliton in lorentzian trans-sasakian manifold is shrinking in the region {(x,y,z) ∈ r3 : −21/3 < z < 21/3}. (5.9) hence the regions (5.8) and (5.9) are complementary to one another that is, m = {(x,y,z) ∈ r3 : −21/3 > z > 21/3}∪{(x,y,z) ∈ r3 : −21/3 < z < 21/3}. thus from cases (1) and (2) one can conclude that in a region where the trans-sasakian manifold is shrinking the lorentzian trans-sasakian manifold is expanding and in a region where the trans-sasakian manifold is expanding the lorentzian trans-sasakian manifold is shrinking. hence in given example trans-sasakian and lorentzian trans-sasakian manifolds are complementary to each other. (3) if ε = −1,δ = 1, in (5.5), then λ = z 6+4 2z2 > 0. by remark (2.1), ricci soliton in lorentzian α-sasakian β-kenmotsu manifold is expanding. (4) if ε = 1,δ = −1, in (5.5), then λ = −(z 6+4) 2z2 < 0. by remark (2.1), ricci soliton in α-sasakian lorentzian β-kenmotsu manifold is shrinking. ricci solitons in (ε,δ)-trans-sasakian manifolds 217 6. conclusion we know [15, 16] that any compact steady or expanding ricci soliton is einstein. in our case we have shown that the ricci soliton in regular indefinite (ε,δ)-trans-sasakian manifold is einstein but it is not steady and it is a manifold of varying scalar curvature because (ε,δ)-trans-sasakian structure contains both sasakian as well as kenmotsu structures for ε = δ = 1 and lorentzian condition and the manifold is not compact. hence it is expanding or shrinking depending upon α and β which characterize the sasakian and kenmotsu structure and ε, δ which characterize lorentz structure or indefinite case. references [1] amalendu ghosh and r. sharma, k-contact metrics as ricci solitons, beitr. algebra geom. 53 (1) (2012), 25-30. [2] c.s. bagewadi and venkatesha, some curvature tensors on a trans-sasakian manifold, turk. j. math. 31 (2007), 111-121. [3] c.s. bagewadi and gurupadavva ingalahalli, ricci solitons in lorentzian α-sasakian manifolds, acta math. acad. paedagog. nyhzi. (n.s.) 28(1) (2012), 59-68. [4] a. bejancu and k.l. duggal, real hypersurfaces of indefinite kaehler manifolds, int. j. math and math sci., 16(3) (1993), 545-556. [5] d.e. blair, contact manifolds in riemannian geometry, lecture notes in mathematics, 509, springer-verlag, berlin-new-york, (1976). [6] d.e. blair and j. a. oubina, conformal and related changes of metric on the product of two almost contact metric manifolds, publ. mat. 34 (1990), 199-207. [7] constantin calin and mircea crasmareanu, from the eisenhart problem to ricci solitons in f-kenmotsu manifolds, bull. malays. math. sci. soc. 33(3) (2010), 361-368. [8] u.c. de and avijit sarkar, on ε-kenmotsu manifolds, hadronic j. 32 (2009), 231-242. [9] u.c. de and m.m. tripathi, ricci tensor in 3-dimensional trans-sasakian manifolds, kyungpook math. j., 43(2) (2003), 247-255. [10] k.l. duggal, space time manifolds and contact structures, int. j. math and math sci., 13(3) (1990), 545-553. [11] l.p. eisenhart, symmetric tensors of the second order whose first covariant derivatives are zero, trans. amer. math. soc., 25(2) (1923), 297-306. [12] a. gray and l.m. harvella, the sixteen classes of almost hermitian manifolds and their linear invariants, ann. mat. pura appl., 123(4) (1980), 35-58. [13] gurupadavva ingalahalli and c.s. bagewadi, ricci solitons in (ε)-trans-sasakain manifolds, j. tensor soc. 6 (1) (2012), 145-159. [14] r.s. hamilton, the ricci flow on surfaces, mathematics and general relativity, (santa cruz. ca, 1986), contemp. math. 71, amer. math. soc., (1988), 237-262. [15] r.s. hamilton, three manifolds with positive ricci curvature, j. differ. geom. 17 (1982), 255-306. [16] t. ivey, ricci solitons on compact three-manifolds, differ. geom. appl. 3 (1993), 301-307. [17] k. kenmotsu, a class of almost contact riemannian manifolds, tohoku math. j. 24(2) (1972), 93-103. [18] r. kumar, r. rani and r.k. nagaich, on sectional curvature of (ε)-sasakian manifolds, int. j. math. math. sci. 2007 (2007) article id 93562, doi:10.1155/2007/93562. [19] h. levy, symmetric tensors of the second order whose covariant derivatives vanish, ann. math. 27(2) (1925), 91-98. [20] j.c. marrero, the local structure of trans-sasakian manifolds, annali di mat. pura ed appl. 162 (1992), 77-86. [21] mircea crasmareanu, parallel tensors and ricci solitons in n(k)-quasi einstein manifolds, indian j. pure appl. math., 43(4) (2012), 359-369. [22] h.g. nagaraja, c.r. premalatha and g. somashekhara, on (ε,δ)-trans-sasakian strucutre, proc. est. acad. sci. 61 (1) (2012), 20-28. [23] h.g. nagaraja and c.r. premalatha, ricci solitons in kenmotsu manifolds, j. math. anal. 3 (2) (2012), 18-24. [24] j.a. oubina, new classes of almost contact metric structures, publ. math. debrecen 32 (1985), 187-193. [25] a.a. shaikh, k. k. baishya and eyasmin, on d-homothetic deformation of trans-sasakian structure, demonstr. math., 41 (1) (2008), 171-188. [26] r. sharma, second order parallel tensor in real and complex space forms, internat. j. math. math. sci., 12(4) (1989), 787-790. [27] r. sharma, certain results on k-contact and (k,µ)-contact manifolds, j. geom., 89(1-2) (2008), 138-147. [28] s.s. shukla and d.d. singh, on (ε)-trans-sasakian manifolds, int. j. math. anal. 49(4) (2010), 2401-2414. [29] m.m. tripathi, ricci solitons in contact metric manifolds, arxiv:0801.4222 [math.dg]. [30] m. turan, u.c. de and a. yildiz, ricci solitons and gradient ricci solitons on 3-dimensional trans-sasakian manifolds, filomat, 26(2) (2012), 363-370. [31] x. xufeng and c. xiaoli, two theorems on (ε)-sasakain manifolds, int. j. math. math.sci., 21(2) (1998), 249-254. department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india, ∗corresponding author: gurupadavva@gmail.com 1. introduction 2. preliminaries 2.1. example for trans-sasakian manifolds 2.2. example for ()-trans-sasakian manifolds 3. parallel symmetric second order tensors and ricci solitons in (,)-trans-sasakian manifolds 4. ricci solitons in 3-dimensional (,)-trans-sasakian manifold 5. example of ricci solitons for 3-dimensional (,)-trans-sasakian manifold 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 180-192 http://www.etamaths.com generalized beta-convex functions and integral inequalities bandar bin-mohsin1, muhammad uzair awan2, muhammad aslam noor1,3,∗ khalida inayat noor3, sabah iftikhar3, awais gul khan2 abstract. in this paper, we introduce the concept of generalized beta-convex functions. this new class of convex functions includes several new and previous known classes of convex functions as special cases. we derive some integral inequalities of hermite-hadamard type via generalized betaconvex functions. some special cases are also discussed. results proved in this paper can be viewed as significant new contributions in this dynamic field. 1. introduction and preliminaries convexity theory had played a pivotal role in the development of every branch of pure and applied sciences. closely related to this theory is inequality theory. in fact, it is known that every function is a convex function, if and if only, if satisfies an integral inequality. these type of integral inequalities are known as hermite-hadanard, simpson, trapeziodal and newton. the integral inequalities are used to find the lower and upper bounds of natural phenomena. due to their important applications in various branches of pure and applied science, the concept of convexity has been generalized and generalized using some interesting and novel techniques and ideas, see [1–4, 8–10, 13–15, 17–20, 23–25, 27, 30–32]. these developments played an crucial role to establish integral inequalities via various classes of convex functions and their variant forms. see [3–7, 11–20, 23–26, 28–30] and the references therein. motivated and inspired by the research going on in these fields, we introduced and consider a new class of convex functions, which is called generalized beta-convex functions. we show that this class of generalized beta-convex functions includes several other classes of convex functions. we also derive some new integral inequalities via beta-convex functions. several special cases are considered which cab be obtained from our main results. our results can be viewed as a significant refinement and improvement of the of the known results. techniques and ideas of this paper may stimulate further research. we now recall some known basic results and concepts, which are needed to obtain the main results. definition 1.1 ( [32]). an interval i is said to be a p-convex set if mp(x,y; t) = [tx p + (1 − t)yp] 1 p ∈ i for all x,y ∈ i,t ∈ [0, 1], where p = 2k + 1 or p = n m ,n = 2r + 1,m = 2t + 1 and k,r,t ∈ n. for p = 1, and p = −1, p-convex set reduces to convex set and harmonic convex set, respectively. definition 1.2 ( [32]). let i be a p-convex set. a function f : i → r is said to be p-convex function or belongs to the class pc(i), if f(mp(x,y; t)) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ i,t ∈ [0, 1]. it is very much obvious that for p = 1 definition 1.2 reduces to the definition for classical convex functions. note that for p = −1, we have the definition of harmonically convex functions. received 5th march, 2017; accepted 27th april, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 26d15, 26a51. key words and phrases. convex; beta; function; hermite-hadamard; inequalities. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 180 generalized beta-convex functions 181 definition 1.3 ( [10]). a function f : i ⊂ r\{0}→ r is said to be harmonically convex function, if f ( xy (1 − t)x + ty ) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ i,t ∈ [0, 1]. also note that for t = 1 2 in definition 1.2, we have jensen p-convex functions or mid p-convex functions. f(mp(x,y; 1/2)) ≤ f(x) + f(y) 2 , ∀x,y ∈ i,t ∈ [0, 1]. we now define the concept of generalized bet-convex functions, which is the main motivation of this paper. definition 1.4. let i be a p-convex set. a function f : i → r is said to be a generalized beta-convex function, if f(mp(x,y; t)) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ i,t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. for p = 1, we have beta-convex functions. f(tx + (1 − t)y) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ r, t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. for p = −1, we have harmonic beta-convex functions, which were introduced and studies by noor et. al [21, 22]. f ( xy (1 − t)x + ty ) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ r, t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. we now consider some results, which are useful in obtaining our results. lemma 1.1. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(mp(x,y; t))dt. proof. the proof follows from simple calculations. � lemma 1.2 ( [18]). let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b. if f′ ∈ l [a,b], then, we have rf (a,b; p) = f(a) + f(b) 2 − p bp −ap ∫ b a f(x) x1−p dx = bp −ap 2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt. 2. main results in this section, we derive our main results. theorem 2.1. let f : i = [a,b] ⊂ r → r be a generalized beta-convex function. if f ∈ l [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]b(θ1 + 1,θ2 + 1). proof. let f be a generalized beta-convex function. then f ([ ap + bp 2 ]1 p ) ≤ 1 4 [ f ( [tap + (1 − t)bp] 1 p ) + f ( [(1 − t)ap + tbp] 1 p )] . 182 mohsin, awan, noor, noor, iftikhar and khan integrating both sides of above inequality with respect to t on [0, 1], we have 2f  [ap + bp 2 ]1 p   ≤ p bp −ap b∫ a f(x) x1−p dx. (2.1) also f ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y). integrating both sides of above inequality with respect to t on [0, 1], we have p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]b(θ1 + 1,θ2 + 1). (2.2) on summation of inequalities (2.1) and (2.2) the proof is complete. � we now discuss a new special case of theorem 2.1. if θ1 = θ = θ2 in theorem 2.1, then we have following new result for brecker type of generalized tgs-convex functions. corollary 2.1. let f : i = [a,b] ⊂ r → r be brecker type of tgs-convex function. if f ∈ l [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]b(θ + 1,θ + 1). if θ1 = −θ = θ2 in theorem 2.1, then we have following new result for godunova-levin-dragomir type generalized tgs-convex functions. corollary 2.2. let f : i = [a,b] ⊂ r → r be godunova-levin-dragomir generalized tgs-convex function. if f ∈ l [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]b(1 −θ, 1 −θ). if p = −1 in theorem 2.1, then we have following new result for harmonic beta-convex functions. corollary 2.3. let f : i\{0}⊂ r → r be a harmonic beta-convex function. if f ∈ l [a,b], then, we have 2f ( 2ab a + b ) ≤ ab b−a b∫ a f(x) x2 dx ≤ [f(a) + f(b)]b(θ1 + 1,θ2 + 1). we now derive a lower bound for hermite-hadamard’s inequality via product of two generalized beta-convex functions. theorem 2.2. let f,g : i = [a,b] ⊂ r → r be two generalized beta-convex functions. if fg ∈ l [a,b], then 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −b(θ1 + θ2 + 1,θ1 + θ2 + 1)m(a,b) + b(2θ1 + 1, 2θ2 + 1)n(a,b) ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ b(2θ1 + 1, 2θ2 + 1)m(a,b) + b(θ1 + θ2 + 1,θ1 + θ2 + 1)n(a,b), generalized beta-convex functions 183 where m(a,b) = f(a)g(a) + f(b)g(b), (2.3) and n(a,b) = f(a)g(b) + f(b)g(a), (2.4) respectively. proof. since f and g are generalized beta-convex functions respectively, so f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) ≤ 1 4 [ f ( [tap + (1 − t)bp] 1 p ) + f ( [(1 − t)ap + tbp] 1 p )] × 1 4 [ g ( [tap + (1 − t)bp] 1 p ) + g ( [(1 − t)ap + tbp] 1 p )] = 1 16 [ f ( [tap + (1 − t)bp] 1 p ) g ( [tap + (1 − t)bp] 1 p ) +f ( [(1 − t)ap + tbp] 1 p ) g ( [(1 − t)ap + tbp] 1 p ) +f ( [tap + (1 − t)bp] 1 p ) g ( [(1 − t)ap + tbp] 1 p ) +f ( [(1 − t)ap + tbp] 1 p ) g ( [tap + (1 − t)bp] 1 p )] ≤ 1 16 [ f([tap + (1 − t)bp] 1 p ) g([tap + (1 − t)bp] 1 p ) +f([(1 − t)ap + tbp] 1 p ) g([(1 − t)ap + tbp] 1 p ) +[2tθ1+θ2 (1 − t)θ1+θ2 ][f(a)g(a) + f(b)g(b)] +[t2θ1 (1 − t)2θ2 + t2θ2 (1 − t)2θ1 ][f(a)g(b) + f(b)g(a)] ] . integrating above inequality with respect to t on [0, 1], we have f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) ≤ 1 8 [ p bp −ap 1∫ 0 f(x)g(x) x1−p dx + b(θ1 + θ2 + 1,θ1 + θ2 + 1)m(a,b) + b(2θ1 + 1, 2θ2 + 1)n(a,b) ] . (2.5) also since f and g are generalized beta-convex functions, then f ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2f(a) + (1 − t)θ1tθ2f(b), and g ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2g(a) + (1 − t)θ1tθ2g(b). multiplying both sides of above inequality and then integrating it with respect to t on [0, 1], we have 1∫ 0 f ( [tap + (1 − t)bp] 1 p ) g ( [tap + (1 − t)bp] 1 p ) dt ≤ f(a)g(a) 1∫ 0 tθ1 (1 − t)θ2tθ1 (1 − t)θ2 dt + f(b)g(b) 1∫ 0 tθ2+θ2 (1 − t)θ1+θ1 dt + [f(a)g(b) + f(b)g(a)] 1∫ 0 tθ1 (1 − t)θ2tθ2 (1 − t)θ1 dt. 184 mohsin, awan, noor, noor, iftikhar and khan this implies p bp −ap b∫ a f(x)g(x) x1−p dx ≤ b(2θ1, 2θ2 + 1)m(a,b) + b(θ1 + θ2 + 1,θ1 + θ2 + 1)n(a,b). (2.6) combining (2.5) and (2.6) completes the proof. � next we discuss a new special case of theorem 2.2. if θ1 = θ = θ2 in theorem 2.2, then we have following new result for brecker generalized tgs-convex functions. corollary 2.4. let f,g : i = [a,b] ⊂ r → r be two brecker type of tgs-convex functions. if fg ∈ l [a,b], then, we have 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −b(2θ + 1, 2θ + 1)[m(a,b) + n(a,b)] ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ b(2θ1 + 1, 2θ2 + 1)m(a,b) + b(θ1 + θ2 + 1,θ1 + θ2 + 1)n(a,b), where m(a,b) and n(a,b) are given by (2.3) and (2.4) respectively. if θ1 = −θ = θ2 in theorem 2.2, then we have following new result for godunova-levin-dragomir generalized tgs-convex functions. corollary 2.5. let f,g : i = [a,b] ⊂ r → r be two godunova-levin-dragomir generalized tgs-convex functions. if fg ∈ l [a,b], then 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −b(1 − 2θ, 1 − 2θ)[m(a,b) + n(a,b)] ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ b(2θ1 + 1, 2θ2 + 1)m(a,b) + b(θ1 + θ2 + 1,θ1 + θ2 + 1)n(a,b), where m(a,b) and n(a,b) are given by (2.3) and (2.4) respectively. if p = −1 in theorem 2.2, then we have following new result for harmonic beta-convex functions. corollary 2.6. let f,g : i \{0} ⊂ r → r be two harmonic beta-convex functions. if fg ∈ l [a,b], then, we have 8f ( 2ab a + b ) g ( 2ab a + b ) −b(θ1 + θ2 + 1,θ1 + θ2 + 1)m(a,b) + b(2θ1 + 1, 2θ2 + 1)n(a,b) ≤ ab b−a 1∫ 0 f(x)g(x) x2 dx ≤ b(2θ1 + 1, 2θ2 + 1)m(a,b) + b(θ1 + θ2 + 1,θ1 + θ2 + 1)n(a,b), where m(a,b) and n(a,b) are given in (2.3) and (2.4) respectively. generalized beta-convex functions 185 theorem 2.3. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if f is generalized beta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] , where k1(θ) := b(α + θ1 + 1,β + θ2 + 1), (2.7) and k2(θ) := b(α + θ2 + 1,β + θ1 + 1), (2.8) respectively. proof. using lemma 1.1 and the fact that f is generalized beta-convex function, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(mp(x,y; t))dt ≤ (bp −ap)α+β+1 1∫ 0 tα(1 − t)β[tθ1 (1 − t)θ2f(a) + (1 − t)θ1tθ2f(b)]dt = (bp −ap)α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] . this completes the proof. � if θ1 = θ = θ2 in theorem 2.3, then we have corollary 2.7. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if f is breckner generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1k(θ) [ f(a) + f(b) ] , where k(θ) := b(α + θ + 1,β + θ + 1). (2.9) if θ1 = −θ = θ2 in theorem 2.3, then we have corollary 2.8. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if f is godunova-levin-dragomir generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1h(θ) [ f(a) + f(b) ] , where h(θ) := b(α−θ + 1,β −θ + 1). (2.10) if p = −1 in theorem 2.3, then we have corollary 2.9. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if f is harmonic beta-convex function, then b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] , 186 mohsin, awan, noor, noor, iftikhar and khan where k1(θ) := b(α + θ1 + 1,β + θ2 + 1), (2.11) and k2(θ) := b(α + θ2 + 1,β + θ1 + 1), (2.12) respectively. theorem 2.4. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f| r r−1 is generalizedbeta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1b(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } b(θ1 + 1,θ2 + 1) ]r−1 r . proof. using lemma 1.1, holder’s inequality and the fact that |f| r r−1 is generalized beta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(mp(x,y; t))dt ≤ (bp −ap)α+β+1   1∫ 0 trα(1 − t)rβdt   1 r   1∫ 0 |f(mp(x,y; t))| r r−1 dt   r−1 r ≤ (bp −ap)α+β+1b(rα + 1,rβ + 1)   1∫ 0 { tθ1 (1 − t)θ2|f(a)| r r−1 + (1 − t)θ2tθ1|f(b)| r r−1 } dt   r−1 r ≤ (bp −ap)α+β+1b(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } b(θ1 + 1,θ2 + 1) ]r−1 r . this completes the proof. � if θ1 = θ = θ2 in theorem 2.4, then we have corollary 2.10. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f| r r−1 is breckner generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1b(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } b(θ + 1,θ + 1) ]r−1 r . if θ1 = −θ = θ2 in theorem 2.4, then we have corollary 2.11. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f| r r−1 is godunova-levin-dragomir type of tgs-convex function, then, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1b(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } b(1 −θ, 1 −θ) ]r−1 r . if p = −1 in theorem 2.4, then we have generalized beta-convex functions 187 corollary 2.12. let f : i \{0} ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f| r r−1 is harmonic beta-convex function, then, we have b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 b(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } b(θ1 + 1,θ2 + 1) ]r−1 r . theorem 2.5. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f|r is beta-convex function, then, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [b(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r , where k1(θ) and k2(θ) are given by (2.11) and (2.12) respectively. proof. using lemma 1.1, holder’s inequality and the fact that |f|r is beta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(mp(x,y; t))dt ≤ (bp −ap)α+β+1   1∫ 0 (1 − t)αtβdt   r−1 r   1∫ 0 tα(1 − t)β |f(mp(x,y; t))| r dt   1 r ≤ (bp −ap)α+β+1 [b(α + 1,β + 1)] r−1 r ×   1∫ 0 tα(1 − t)β [ tθ1 (1 − t)θ2|f(a)|r + (1 − t)θ1tθ2|f(b)|r ] dt   1 r = (bp −ap)α+β+1 [b(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r . this completes the proof. � if θ1 = θ = θ2 in theorem 2.5, then we have corollary 2.13. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f|r is breckner type of tgs-convex function, then we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [b(α + 1,β + 1)] r−1 r k 1 r (t) [|f(a)|r + |f(b)|r] 1 r , where k(θ) is given by (2.9). if θ1 = −θ = θ2 in theorem 2.5, then we have corollary 2.14. let f : i = [a,b] ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f|r is godunova-levin-dragomir generalized tgs-convex function, then 188 mohsin, awan, noor, noor, iftikhar and khan b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (b−a)α+β+1 [b(1 −α, 1 −β)] r−1 r h 1 r (t) [|f(a)|r + |f(b)|r] 1 r , where h(t) is given by (2.10). if p = −1 in theorem 2.5, then we have corollary 2.15. let f : i \{0} ⊂ r → r be a continuous function such that f ∈ l [a,b]. if |f|r is harmonic beta-convex function, then, we have b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 [b(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r , where k1(θ) and k2(θ) are given by (2.11) and (2.12) respectively. now using lemma 1.2 we derive some hermite-hadamard type inequalities. theorem 2.6. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′| is beta-convex function, then |rf (a,b; p)| ≤ bp −ap 2p [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) := b p−1b(θ1 + 1,θ2 + 1) 2f1 (1 p − 1,θ1 + 1; θ1 + θ2 + 2; 1 − ap bp ) − 2bp−1b(θ1 + 2,θ2 + 1) 2f1 (1 p − 1,θ1 + 2; θ1 + θ2 + 3; 1 − ap bp ) , (2.13) and h2(θ1,θ2) := b p−1b(θ2 + 1,θ1 + 1) 2f1 (1 p − 1,θ2 + 1; θ1 + θ2 + 2; 1 − ap bp ) − 2bp−1b(θ2 + 2,θ1 + 1) 2f1 (1 p − 1,θ2 + 2; θ1 + θ2 + 3; 1 − ap bp ) , (2.14) respectively. generalized beta-convex functions 189 proof. using lemma 1.2, property of the modulus and the fact that |f′| is beta-convex function, we have |rf (a,b; p)| = ∣∣∣∣bp −ap2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt ∣∣∣∣ ≤ bp −ap 2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t) [ tθ1 (1 − t)θ2|f′(a)| + (1 − t)θ1tθ2|f′(b)| ] dt = bp −ap 2p [∫ 1 0 tθ1 (1 − t)θ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(a)|dt + ∫ 1 0 (1 − t)θ1tθ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(b)|dt ] = bp −ap 2p [{ bp−1b(θ1 + 1,θ2 + 1) 2f1 (1 p − 1,θ1 + 1; θ1 + θ2 + 2; 1 − ap bp ) −2bp−1b(θ1 + 2,θ2 + 1) 2f1 (1 p − 1,θ1 + 2; θ1 + θ2 + 3; 1 − ap bp )} |f′(a)| + { bp−1b(θ2 + 1,θ1 + 1) 2f1 (1 p − 1,θ2 + 1; θ1 + θ2 + 2; 1 − ap bp ) −2bp−1b(θ2 + 2,θ1 + 1) 2f1 (1 p − 1,θ2 + 2; θ1 + θ2 + 3; 1 − ap bp )} |f′(b)| ] = bp −ap 2p [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] . this completes the proof. � we now discuss some special cases of theorem 2.6. if θ1 = θ = θ2 in theorem 2.6, then we have a following new result. corollary 2.16. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′| is breckner type of tgs-convex function, then |rf (a,b; p)| ≤ bp −ap 2p h(θ) [|f′(a)| + |f′(b)|] , where h(θ) := bp−1b(θ + 1,θ + 1) 2f1 (1 p − 1,θ + 1; 2θ + 2; 1 − ap bp ) − 2bp−1b(θ + 2,θ + 1) 2f1 (1 p − 1,θ + 2; 2θ + 3; 1 − ap bp ) . (2.15) if θ1 = −θ = θ2 in theorem 2.6, then we have following new result. corollary 2.17. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′| is godunova-levin-dragomir generalized tgs-convex function, then |rf (a,b; p)| ≤ bp −ap 2p l(θ) [|f′(a)| + |f′(b)|] , where l(θ) := bp−1b(1 −θ1, 1 −θ2) 2f1 (1 p − 1, 1 −θ; 2 − 2θ; 1 − ap bp ) − 2bp−1b(2 −θ1, 1 −θ2) 2f1 (1 p − 1, 2 −θ; 3 − 2θ; 1 − ap bp ) . (2.16) if p = 1 in theorem 2.6, then we have following new result. 190 mohsin, awan, noor, noor, iftikhar and khan corollary 2.18. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′| is beta-convex function, then∣∣∣∣∣∣f(a) + f(b)2 − 1b−a 1∫ 0 f(x)dx ∣∣∣∣∣∣ ≤ b−a2 [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. if p = −1 in theorem 2.6, then we have following new result. corollary 2.19. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′| is harmonic beta-convex function, then∣∣∣∣∣∣f(a) + f(b)2 − abb−a 1∫ 0 f(x) x2 dx ∣∣∣∣∣∣ ≤ ab(b−a)2 [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. theorem 2.7. let f : i = [a,b] ⊂ r → r be a differentiable function on i0 (the interior of i) with a < b and f′ ∈ l [a,b]. if |f′|r is beta-convex function, then |rf (a,b; p)| ≤ bp −ap 2p ( b1−p 2f1 (1 p − 1, 1; 2; 1 − ap bp ) − 4b1−p 2f1 (1 p − 1, 2; 3; 1 − ap bp ))1−1r × [h1(θ1,θ2)|f′(a)|rdt + h2(θ1,θ2)|f′(b)|rdt] 1 r , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. proof. using lemma 1.2, property of the modulus, power mean’s inequality and the fact that |f′|r is beta-convex function, we have |rf (a,b; p)| = ∣∣∣∣bp −ap2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt ∣∣∣∣ ≤ bp −ap 2p (∫ 1 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p dt )1−1 r ×   1∫ 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p [ tθ1 (1 − t)θ2|f′(a)|r + (1 − t)θ1tθ2|f′(b)|r ] dt   1 r = bp −ap 2p (∫ 1 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p dt )1−1 r × [∫ 1 0 tθ1 (1 − t)θ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(a)|rdt + ∫ 1 0 (1 − t)θ1tθ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(b)|rdt ]1 r = bp −ap 2p ( b1−p 2f1 (1 p − 1, 1; 2; 1 − ap bp ) − 4b1−p 2f1 (1 p − 1, 2; 3; 1 − ap bp ))1−1r × [h1(θ1,θ2)|f′(a)|rdt + h2(θ1,θ2)|f′(b)|rdt] 1 r . this completes the proof. � generalized beta-convex functions 191 acknowledgements authors are pleased to acknowledge the ”support of distinguished scientist fellowship program (dsfp), king saud university, riyadh, saudi arabia”. references [1] w. w. breckner, stetigkeitsaussagen fiir eine klasse verallgemeinerter convexer funktionen in topologischen linearen raumen. pupl. inst. math. 23 (1978), 13-20. [2] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrecht, holland, 2002. [3] g. cristescu, m. a. noor, m. u. awan, bounds of the second degree cumulative frontier gaps of functions with generalized convexity, carpathian j. math. 31 (2) (2015), 173-180. [4] s. s. dragomir, inequalities of jensen type for ϕ-convex functions, fasciculi mathematici, (2015). [5] s. s. dragomir, r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett., 11 (5) (1998), 91-95. [6] s. s. dragomir, c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university, australia, 2000. [7] s. s. dragomir, j. pecaric, l. e. persson, some inequalities of hadamard type. soochow j. math. 21 (1995), 335-341. [8] z.b. fang, r. shi, on the (p, h)-convex function and some integral inequalities, j. inequal. appl. 2014 (2014), article id 45. [9] e. k. godunova, v. i. levin, neravenstva dlja funkcii sirokogo klassa, soderzascego vypuklye, monotonnye i nekotorye drugie vidy funkii. vycislitel. mat. i. fiz. mezvuzov. sb. nauc. trudov, mgpi, moskva. (1985) 138-142, (in russian). [10] i. iscan, hermite-hadamard type inequalities for harmonically convex functions, hacettepe j. math. stat. 43 (6) (2014), 935-942. [11] s. k. khattri, three proofs of the inequality e < ( 1 + 1 n )n+0.5 , amer. math. monthly, 117 (3) (2010), 273-277. [12] w. liu, new integral inequalities involving beta function via p-convexity, miskolc math. notes, 15 (2) (2014), 585-591. [13] m. a. noor, some developments in general variational inequalites, appl. math. comput. 151 (2004), 199-277. [14] m. a. noor, k. i. noor, harmonic variational inequalities, appl. math. inform. sci. 10 (2016), 1811-1814. [15] m. a. noor, k. i. noor, some implict methods for solving harmonic variational inequalities, int. j. anal. appl. 12 (1) (2016), 10-14. [16] m. v. mihai, m. a. noor, k. i. noor, m. u. awan, some integral inequalities for harmonic h-convex functions involving hypergeometric functions. appl. math. comput. 252 (2015), 257-262. [17] m. a. noor, m. u. awan, k. i. noor, some new bounds of the quadrature formula of gauss-jacobi type via (p, q)-preinvex functions, appl. math. inform. sci. lett., 5 (2) (2017), 51-56. [18] m. a. noor, m. u. awan, m. v. mihai, k. i. noor, hermite-hadamard inequalities for differentiable p-convex functions using hypergeometric functions, publications de l’nstitut mathematique, 100 (114) (2016), 251-257. [19] m. a. noor, k. i. noor, m. u. awan, integral inequalities for coordinated harmonically convex functions. complex var. elliptic equat. 60 (6) (2015), 776-786. [20] m. a. noor, k. i. noor, m. u. awan, s. costache, some integral inequalities for harmonically h-convex functions. u. p. b. sci. bull., series a. 77 (1) (2015), 5-16. [21] m. a. noor, k. i. noor, s. iftikhar, harmonic beta-preinvex functions and inequalities, int. j. anal. appl. (2017) 13(2), 144-160. [22] m. a. noor, k. i. noor, s. iftikhar,integral inequalities for differentiable relative harmonic preinvex functions, twms j. pure appl. math. 7(1) (2016), 3-19. [23] m. a. noor, k. i. noor, s. iftikhar, hermite-hadamard inequalities for strongly harmonic convex functions, j. inequal. special funct. 7 (3) (2016), 99-113. [24] m. a. noor, k. i. noor, s. iftikhar, fractional integral inequalities for harmonic geometrically h-convex functions, adv. studies contemp. math. 26 (3) (2016), 447-456. [25] m. a. noor,k. i. noor, s. iftikhat, m. u. awan, strongly generalized harmonic convex functions and integral inequalities, j. math. anal. 7 (3) (2016), 66-71. [26] m. e. ozdemir, e. set, m. alomari, integral inequalities via several kinds of convexity, creat. math, inform., 20 (1) (2011), 62-73. [27] c. e. m. pearce, j. pecaric, inequalities for differentiable mappings with application to special means and quadrature formula, appl. math. lett., 13 (2000), 51-55. [28] m. z. sarikaya, a. saglam, h. yildrim, on some hadamard-type inequalities for h-convex functons, j. math. inequal. 2 (2008), 335-341. [29] e. set, new inequalities of ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, comput. math. appl., 63 (7) (2012), 1147-1154. [30] m. tunc, e. gov, u. sanal, on tgs-convex function and their inequalities, facta universitatis (nis) ser. math. inform. 30 (5) (2015), 679-691. [31] s. varošanec, on h-convexity, j. math. anal. appl. 326 (2007), 303-311. [32] k. s. zhang, j. p. wan, p-convex functions and their properties. pure appl. math. 23 (1) (2007), 130-133. 192 mohsin, awan, noor, noor, iftikhar and khan 1department of mathematics, king saud university, riyadh, saudi arabia 2department of mathematics, gc university, faisalabad, pakistan 3department of mathematics, comsats institute of information technology, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com 1. introduction and preliminaries 2. main results acknowledgements references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 138-145 doi: 10.28924/2291-8639-15-2017-138 on the (p,q)−stancu generalization of a genuine baskakovdurrmeyer type operators i̇smet yüksel∗, ülkü di̇nlemez kantar and bi̇rol altin abstract. in this paper, we introduce a stancu generalization of a genuine baskakovdurrmeyer type operators via (p, q)− integer. we investigate approximation properties of these operators. furthermore, we study on the linear positive operators in a weighted space of functions and obtain the rate of these convergence using weighted modulus of continuity. 1. introduction in the field of approximation theory, the quantum calculus has been studied for a long time. the generalization of (p,q)− calculus was introduced by sahai and yadav in [15]. recently, a series of papers giving (p,q)− generalizations a sequence of linear positive operators have been published in [3, 4, 9–13]. our aim is to give stancu type generalization, via (p,q)− integer, defined by agrawal and thamer as follows bn(f,x) = (n− 1) ∞∑ k=1 bn,k(x) ∞∫ 0 bn,k−1(t)f (t) dt + f (0) (1 + x) −n q , (1.1) where bn,k(x) = ( n + k − 1 k ) xk (1 + x)n+k . in [5]. we refer reader to [2] for unexplained terminologies and notations. 2. preliminaries and notations let’s give a table of some basic formulas, motivated from q−calculus, used in (p,q)−calculus as the following table1 (p,q)−calculus relation with q−calculus [n]p,q = pn−qn p−q [n]p,q = p n−1 [n]q/p [n]p,q ! = [1]p,q [2]p,q ... [n]p,q [n]p,q ! = p (n2) [n]q/p ! (a⊕ b)np,q= (a + b) (ap + bq)...(ap n−1 +bqn−1) (a⊕ b)np,q= p (n2) (a + b) n q/p dp,qf(x) = f(px) −f(qx) dqf(x) = f(x) −f(qx) table 1 recall that the beta function, introduced [14], in q− calculus is defined by bq(n,k) = k(a,n) ∞/a∫ 0 tk−1 (1 + t)n+kq dqt, (2.1) received 14th april, 2017; accepted 5th june, 2017; published 1st november, 2017. 2010 mathematics subject classification. 41a25, 41a36. key words and phrases. baskakov-durrmeyer operators; weighted approximation; rates of approximation, (p, q)−calculus. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 138 baskakovdurrmeyer type operators 139 where k(a,n) = an 1 + a ( 1 + 1 a )n q (1 + a) 1−s q , a > 0. (2.2) in the formula (2.2), k(a,n) = qn(n−1)/2 and k(a, 0) = 1 for n ∈ n. inspiring the formula (2.1), we introduce (p,q)-beta functions bp,q(n,k), as a generalization of bq(n,k), a > 0 and n,k ∈ n\{0}, defined by bp,q(k,n) = p (n2)q( k 2) ∞/a∫ 0 tk−1 (1 ⊕ t)n+kp,q dp,qt. (2.3) if p = 1 is replaced in (2.3), then the formula is reduced to (2.1). 3. genuine type stancu generalization via (p,q)−integer let’s start to give a stancu type (p,q)−generalization of these operators in (1.1). for 0 ≤ α,β and 0 < q < p ≤ 1, these operators are defined as follows; bα,βp,q,n(f,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q,x)p (n−1)2+kqk(k−1) × ∞/a∫ 0 bn,k−1(p,q,t)f ( [n]p,qt + α [n]p,q + β ) dp,qt   +f ( α [n]p,q + β ) p( n 2)(1 ⊕x)−np,q , (3.1) where bn,k(p,q,t) = [ n + k − 1 k ] p,q tk (1 ⊕ t)n+kp,q . if one replaces p = q = 1 and α = β = 0 in (3.1), then the operators bα,βp,q,n are reduced to the operators bnin (1.1). similar type operators studied in [1, 7, 16]. to obtain our main results, we need calculating the values of korovkin monomial functions. lemma 3.1. the following equalities are satisfied for em(t) = t m, m = 0, 1, 2 and n > 3 bα,βp,q,n(1,x) = 1, bα,βp,q,n(t,x) = [n]2 p,q pq([n]p,q+β)[n−2]p,q x + α [n] p,q +β , bα,βp,q,n(t 2,x) = [n]3 p,q [n+1] p,q p2q4([n]p,q+β) 2 [n−2] p,q [n−1] p,q x2 + ( pn−4[2] p,q [n]3 p,q q3([n]p,q+β) 2 [n−2]p,q[n−1]p,q + 2α[n]2 p,q pq([n]p,q+β) 2 [n−2]p,q ) x + α2( [n]pn,qn + β )2 . proof. by the definition (p,q)−beta functions in (2.3), we obtain the estimate, ∞/a∫ 0 bn,k−1(p,q; t)t mdp,qt = [ n + k − 2 k − 1 ] p,q ∞/a∫ 0 tk+m−1 (1 ⊕ t)n+k−1p,q dp,qt = [k + m− 1]p,q! [n−m− 2]p,q! [k − 1]p,q! [n− 1]p,q!p (n−m−12 )q( k+m 2 ) . (3.2) 140 yüksel, di̇nlemez kantar and altin if we apply the operators in (3.1) to the equality (3.2) for m = 0, we get bα,βp,q,n(1,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) × ∞/a∫ 0 bn,k−1(p,q; t)dp,qt   + p( n 2)(1 ⊕x)−np,q = [n− 1]p,q ∞∑ k=1 bn,k(p,q; x)) p(n−1) 2+kqk(k−1) [n− 1]p,q p (n−12 )q( k 2) +p( n 2)(1 ⊕x)−np,q = ∞∑ k=0 [n + k − 1]p,q! [k]p,q! [n− 1]p,q! p( n 2)q( k 2) (px) k (1 ⊕x)n+kp,q = ∞∑ k=0 p( n 2)q( k 2)bn,k(p,q; px) = 1. and the proof of (i) is finished. with the direct computation, we obtain (ii) as follows: bα,βp,q,n(t,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) × ∞/a∫ 0 bn,k−1(p,q; t) ( [n]p,qt + α [n]p,q + β ) dp,qt   + α [n]p,q + β p( n 2)(1 ⊕x)−np,q = [n]p,q [n]p,q + β ∞∑ k=1 bn,k(p,q; x) p(n−1) 2+kqk(k−1)[k] p,q [n−2] p,q p (n−22 )q( k+1 2 ) + α [n]p,q + β bα,βp,q,n(1,x) = [n]2p,qx pq([n]p,q+β)[n−2]p,q ∞∑ k=0 p( n+1 2 )q( k 2)bn+1,k(p,q; px) + α [n]p,q + β = [n]2 p,q pq([n]p,q+β)[n−2]p,q x + α [n]p,q + β . for (iii), bα,βp,q,n(t 2,x) = [n− 1]p,q ∞∑ k=1 { bn,k(p,q; x)p (n−1)2+kqk(k−1) = × ∞/a∫ 0 bn,k−1(p,q; t) ( [n]p,qt + α [n]p,q + β )2 dp,qt   + ( α [n]p,q + β )2 p( n 2)(1 ⊕x)−np,q . baskakovdurrmeyer type operators 141 using the equality [k]p,q = q k−s[s]p,q + p s[k −s]p,q, 0 ≤ s ≤ k, (3.3) we have bα,βp,q,n(t 2,x) = [n] 2 p,q( [n]p,q + β )2 ∞∑ k=2 { [k] p,q(p 2[k−1] p,q +qk−1[2] p,q)p n2+3n+2k−10 2 q k2−5k−2 2 [n−2]p,q[n−3]p,q × bn,k(p,q; x)} + 2α [n]p,q [n]p,q + β ∞∑ k=1 [k]p,q [n− 2]p,q p n2+n+2k−4 2 q k2−3k 2 bn,k(p,q; x) + α2( [n]p,q + β )2 bα,βp,q,n(1,x) then bα,βp,q,n(t 2,x) = [n]3p,q[n+1]p,qx 2 p2q4([n]p,q+β) 2 [n−2]p,q[n−3]p,q ∞∑ k=0 p( n+2 2 )q( k 2)bn+2,k(p,q; px) + ( pn−4[2]p,q[n] 2 p,q q3([n]p,q+β) 2 [n−2] p,q [n−1] p,q + 2α[n]2p,q pq([n]p,q+β) 2[n−2] p,q ) x × ∞∑ k=0 p( n+1 2 )q( k 2)bn+1,k(p,q; x) + α2( [n]p,q + β )2 bα,βp,q,n(1,x). and so we have completed the proof of (iii). � now we consider that b[0,∞) denotes the set of all bounded functions from [0,∞) to r, b[0,∞) is a normed space with the norm ‖f‖b = sup{|f(x)| : x ∈ [0,∞)} and cb[0,∞) denotes the subspace of all continuous functions in b[0,∞). we denote first modulus of continuity on finite interval [0,b], b > 0 ω[0,b](f,δ) = sup 0<h≤δ,x∈[0,b] |f(x + h) −f(x)| . (3.4) the peetre’s k−functional is defined by k2(f,δ) = inf { ‖f −g‖b + δ‖g ′′‖b : g ∈ w 2 ∞ } , δ > 0 where w2∞ = {g ∈ cb[0,∞) : g′,g′′ ∈ cb[0,∞)} . by [6, p. 177, theorem 2.4], there exists a positive constant c such that k2(f,δ) ≤ cω2(f, √ δ) where ω2(f, √ δ) = sup 0<h≤ √ δ sup x∈[0,∞) |f(x + 2h) − 2f(x + h) −f(x)| . (3.5) the weighted korovkintype theorems were proved by gadzhiev [8]. we give the gadzhiev’s results in weighted spaces. bρ[0,∞) denotes the set of all functions f, from [0,∞) to r, satisfying growth condition |f(x)| ≤ nfρ(x), where ρ(x) = 1 + x2 and nf is a constant depending only on f. bρ[0,∞) is a normed space with the norm ‖f‖ρ = sup { |f(x)| ρ(x) : x ∈ r } . cρ[0,∞) denotes the subspace of all continuous functions in bρ[0,∞) and c∗ρ[0,∞) denotes the subspace of all functions f ∈ cρ[0,∞) for which lim |x|→∞ |f(x)| ρ(x) exists finitely. 142 yüksel, di̇nlemez kantar and altin we define a genuine type generalization to the operators bα,βp,q,n given in (3.1) as follows b̃α,βpn,qn,n(f,x) := { bα,βpn,qn,n(f,σ(α,β,pn,qn,n; x)) , x > α [n]pn,qn +β f(x) , 0 ≤ x ≤ α [n]pn,qn +β where σ(α,β,pn,qn,n; x) = pnqn ( [n]pn,qn + β ) [n− 2]pn,qn [n] 2 pn,qn ( x− α [n]pn,qn + β ) . notice that these operators b̃α,βpn,qn,n are defined from the space c ∗ ρ[0,∞) into bρ[0,∞). to satisfy hypothesis of korovkin’s theorem, we assume that lim n→∞ pnn and lim n→∞ qnn are real numbers when lim n→∞ pn = 1 and lim n→∞ qn = 1 for 0 < qn < pn ≤ 1 and n > 3. on the other hand, since the operators b̃α,βpn,qn,n(f,x) are defined as f(x) on the interval [ 0, α [n] pn,qn +β ] , it is enough to examine the approximation properties of these operators at the interval ( α [n]pn,qn +β ,∞ ) . the following lemma can be obtained with the help of lemma 3.1. lemma 3.2. the operators b̃α,βpn,qn,n satisfy the following equalities for x > α [n]pn,qn +β and em(t) = t m, m = 0, 1, 2 b̃α,βpn,qn,n(1,x) = 1, b̃α,βpn,qn,n(t,x) = x, b̃α,βpn,qn,n(t 2,x) = [n+1] pn,qn [n−2] pn,qn q2n[n−1]pn,qn [n]pn,qn x2 + ( −2α[n+1]pn,qn [n−2]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn [n]pn,qn + pn−3n [2]pn,qn [n]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x− pn−3n [2]pn,qn [n] pn,qn α q2n([n]pn,qn +β) 2 [n−1]pn,qn . we need computing the second moment before giving our main results . lemma 3.3. we have the following inequality b̃α,βpn,qn,n((t−x) 2 ,x) ≤ ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1) for x > α [n]pn,qn +β . proof. by lemma 3.2, we write the second moment as b̃α,βpn,qn,n((t−x) 2 ,x) = ( [n+1] pn,qn [n−2] pn,qn q2n[n−1]pn,qn [n]pn,qn − 1 ) x2 + ( −2α[n+1]pn,qn [n−2]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn [n]pn,qn + pn−3n [2]pn,qn [n] pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x − pn−3n [2]pn,qn [n] pn,qn α q2n([n]pn,qn +β) 2 [n−1] pn,qn ≤ ∣∣∣ [n+1]pn,qn [n−2]pn,qnq2n[n−1]pn,qn [n]pn,qn − 1 ∣∣∣x2 + ( pn−3n [2]pn,qn [n] pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x. considering the following equalities [n + 1]pn,qn = p 3 n [n− 2]pn,qn + q n−3 n [3]pn,qn , [n]pn,qn = p 2 n [n− 2]pn,qn + q n−2 n [2]pn,qn , [n− 1]pn,qn = pn [n− 2]pn,qn + q n−1 n [1]pn,qn , baskakovdurrmeyer type operators 143 we get b̃α,βpn,qn,n((t−x) 2 ,x) ≤ (( p3n −p3nq2n ) q2n + [3]pn,qn [n]pn,qn ) x2 + ( pn−3n [2]pn,qn [n]pn,qn q2n([n]pn,qn +β)[n−1]pn,qn + 2α ([n]pn,qn +β) ) x ≤ ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). and the proof of the lemma 3.3 is now finished. � thus we are ready to give direct results. lemma 3.4. we have the inequality for every x ∈ [0,∞) and f′′ ∈ cb[0,∞)∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ δα,βpn,qn,n(x)‖f′′‖b , where δα,βpn,qn,n(x) := ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). proof. using taylor’s expansion f(t) = f(x) + (t−x)f′(x) + t∫ x (t−u)f′′(u)du and the lemma 3.2, we have the following equality b̃α,βpn,qn,n(f,x) −f(x) = b̃ α,β pn,qn,n   t∫ x (t−u)f′′(u)du; x   . on the other hand, combining the inequality∣∣∣∣∣∣ t∫ x (t−u)f′′(u)du ∣∣∣∣∣∣ ≤‖f′′‖b (t−x) 2 2 , and lemma 3.3, we get ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ = ∣∣∣∣∣∣b̃α,βpn,qn,n   t∫ x (t−u)f′′(u)du,x   ∣∣∣∣∣∣ ≤ ‖f′′‖b 2 b̃α,βpn,qn,n((t−x) 2,x) ≤ ‖f′′‖b 2 ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1), as desired. � theorem 3.1. we have the following inequality∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ 2cω2 ( f, √ δ α,β pn,qn,n(x) ) , where δα,βpn,qn,n(x) = ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1). proof. for any g ∈ w2∞, we obtain the inequality∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ ∣∣∣b̃α,βpn,qn,n(f −g,x) − (f −g) (x) + b̃α,βpn,qn,n(g,x) −g(x)∣∣∣ . 144 yüksel, di̇nlemez kantar and altin then, lemma 3.4, we have∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ ≤ 2‖f −g‖b + δα,βpn,qn,n(x)‖g′′‖b . taking infimum over g ∈ w2∞ on the right side of the above inequality and using the inequality (3.5), we reach the desired result. � theorem 3.2. for every f ∈ c∗ρ[0,∞), we have the following the limit lim n→∞ ∥∥∥b̃α,βpn,qn,n(f) −f∥∥∥ ρ = 0. proof. from lemma 3.2 , it is obvious that ∥∥∥b̃α,βpn,qn,n(e0) −e0∥∥∥ ρ = 0 and ∥∥∥b̃α,βpn,qn,n(e1) −e1∥∥∥ ρ = 0. we have ∥∥∥b̃α,βpn,qn,n(e2) −e2∥∥∥ ρ = b̃α,βpn,qn,n((t−x) 2 ,x) 1 + x2 ≤ sup x∈[0,∞)   ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1) 1 + x2   ≤ 2 ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) . then we get lim n→∞ ∥∥∥b̃α,βpn,qn,n(e2) −e2∥∥∥ ρ = 0. thus, from a. d. gadzhiev’s theorem in [8], we obtain the proof of theorem 3.2. � lemma 3.5. we assume that α [n] pn,qn +β < b. then we have the following inequality∥∥∥b̃α,βpn,qn,n(f; x) −f(x)∥∥∥ c[0,b] ≤ n { (1 + b)2δα,βpn,qn,n(b) + ω[0,b+1](f; √ δ α,β pn,qn,n(b)) } , where δα,βpn,qn,n(x) = ( p3n −p3nq2n q2n + 10(α+β+1) q2n([n]pn,qn +β) ) x(x + 1), for every f ∈ c∗ρ[0,∞). proof. let x ∈ [ α [n]pn,qn +β ,b ] and t > b + 1. since t−x > 1, we have |f(t) −f(x)| ≤ kf (2 + (t−x + x)2 + x2) ≤ 3kf (1 + b)2(t−x)2. (3.6) let x ∈ [ α [n]pn,qn +β ,b ] , t < b + 1 and δ > 0. then, we have |f(t) −f(x)| ≤ ( 1 + |t−x| δ ) ω[0,b+1](f,δ). (3.7) due to (3.6) and (3.7), we can write |f(t) −f(x)| ≤ 3kf (1 + b)2(t−x)2 + ( 1 + |t−x| δ ) ω[0,b+1](f,δ). after, using cauchyschwarz’ s inequality , we get∣∣∣b̃α,βpn,qn,n(f; x) −f(x)∣∣∣ ≤ 3kf (1 + b)2bα,βpn,qn,n ( (t−x)2,x ) +ω[0,b+1](f; δ) [ 1 + 1 δ ( b̃α,βpn,qn,n ( (t−x)2,x ))1/2] ≤ 3kf (1 + b)2δα,βpn,qn,n(x) + ω[0,b+1](f; δ) [ 1 + 1 δ ( δα,βpn,qn,n(x) )1/2] . baskakovdurrmeyer type operators 145 considering lemma 3.3 and choosing δ2 := δα,βpn,qn,n(b) and n = max{3kf, 2}. we reach the proof of lemma 3.5. � theorem 3.3. for every f ∈ c∗ρ[0,∞) and γ > 0 , we have the limit lim n→∞ sup x≥0 ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ = 0. proof. for γ > 0, f ∈ c∗ρ[0,∞) and b ≥ 1, using ( 3.4) the following inequality is satisfied sup x≥0 ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ ≤ sup 0≤x<b ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ + sup b≤x ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2+γ ≤ ∥∥∥b̃α,βpn,qn,n(f) −f∥∥∥ c[0,b] + sup b≤x ∣∣∣b̃α,βpn,qn,n(f,x) −f(x)∣∣∣ 1 + x2 ≤ ∥∥∥b̃α,βpn,qn,n(f) −f∥∥∥ c[0,b] + ∥∥∥b̃α,βpn,qn,n(f) −f∥∥∥ ρ . using lemma 3.5 and theorem 3.2 , we get the proof of theorem 3.3 . � references [1] a. aral and v. gupta, generalized q−baskakov operators, math. slovaca 61 (4) (2011), 619–634. [2] a. aral, v. gupta and agarwal, r. p. , applications of q-calculus in operator theory. springer, new york, 2013. [3] a. aral and v. gupta, (p, q)−type beta functions of second kind, adv. oper. theory 1 (1) (2016), 134-146. [4] t. acar, s. a. mohiuddine and m. mursaleen, approximation by (p, q)−baskakov-durrmeyer-stancu operators, complex anal. oper. theory doi 10.1007/s11785-016-0633-5. [5] p. n. agrawal and k. j., approximation of unbounded functions by a new sequence of linear positive operators, j. math. anal. appl. 225 (2) (1998), 660–672. [6] r. a. devore and g. g. lorentz, constructive approximation, springer, berlin, 1993. [7] ü. dinlemez, i̇. yüksel and b. altın, a note on the approximation by the q-hybrid summation integral type operators, taiwanese j. math. 18 (3) (2014), 781–792. [8] a. d. gadzhiev, theorems of the type of p. p. korovkin type theorems, math. zametki, 20(5) (1976), 781-786; english translation, math. notes, 20(5/6) (1976), 996-998. [9] v. gupta, (p, q)-szász–mirakyan–baskakov operators, complex anal. oper. theory, doi 10.1007/s11785-0150521-4. [10] m. mursaleen, k. j. ansari and a. khan, on (p, q)-analogue of bernstein operators, appl. math. comput., 266 (2015) 874-882. [11] m. mursaleen, k. j. ansari and a. khan, some approximation results by (p, q)-analogue of bernstein-stancu operators, appl. math. comput., 264 (2015) 392-402 [corrigendum: appl. math. comput, 269 (2015) 744–746]. [12] m. mursaleen, md. nasiruzzaman, a. khan and k. j. ansari, some approximation results on bleimann-butzer-hahn operators defined by (p, q)−integers, filomat 30 (3) (2016), 639–648. [13] t. acar, a. aral and s. a. mohiuddine, on kantorovich modification of (p, q)−baskakov operators. j. inequal. appl., 2016:98 (2016), 14 pp. [14] a. de sole and v. g. kac, on integral representations of q−gamma and q−beta functions. atti accad. naz. lincei cl. sci. fis. mat. natur. rend. lincei (9) mat. appl. 16 (1) (2005), 11–29. [15] v. sahai and s. yadav, representations of two parameter quantum algebras and p, q−special functions. j. math. anal. appl. 335 (1) (2007), 268–279. [16] d. k. verma, v. gupta and p. n. agrawal, some approximation properties of baskakov-durrmeyer-stancu operators. appl. math. comput. 218 (11) (2012), 6549–6556. gazi university, faculty of sciences, department of mathematics, teknikokullar, 06500, ankara, turkey ∗corresponding author: iyuksel@gazi.edu.tr 1. introduction 2. preliminaries and notations 3. genuine type stancu generalization via (p,q)-integer references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 45-57 http://www.etamaths.com convergence theorems for asymptotically quasi-nonexpansive type mappings in convex metric spaces g. s. saluja abstract. the aim of this paper to study a noor-type iteration process with errors for approximating common fixed point of a finite family of uniformly l-lipschitzian asymptotically quasi-nonexpansive type mappings in the framework of convex metric spaces. we give a necessary and sufficient condition for strong convergence of said iteration scheme involving a finite family of above said mappings and also establish a strong convergence theorem by using condition (a). the results presented in this paper extend, improve and unify some existing results in the previous work. 1. introduction and preliminaries throughout this paper, we assume that e is a metric space, f(ti) = {x ∈ e : tix = x} be the set of all fixed points of the mappings ti (i = 1, 2, . . . ,n), d(t) be the domain of t and n is the set of all positive integers. the set of common fixed points of ti (i = 1, 2, . . . ,n) denoted by f , that is, f = ∩ni=1f(ti). definition 1.1. (see [1]) let t : d(t) ⊂ e → e be a mapping. (1) the mapping t is said to be l-lipschitzian if there exists a constant l > 0 such that d(tx,ty) ≤ ld(x,y), ∀x, y ∈ d(t).(1.1) (2) the mapping t is said to be nonexpansive if d(tx,ty) ≤ d(x,y), ∀x, y ∈ d(t).(1.2) (3) the mapping t is said to be quasi-nonexpansive if f(t) 6= ∅ and d(tx,p) ≤ d(x,p), ∀x ∈ d(t), ∀p ∈ f(t).(1.3) 2010 mathematics subject classification. 47h05, 47h09, 47h10. key words and phrases. asymptotically quasi-nonexpansive type mapping, noor-type iteration process with errors, common fixed point, strong convergence, convex metric space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 45 46 saluja (4) the mapping t is said to be asymptotically nonexpansive if there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 such that d(tnx,tny) ≤ knd(x,y), ∀x, y ∈ d(t), ∀n ∈ n.(1.4) (5) the mapping t is said to be asymptotically quasi-nonexpansive if f(t) 6= ∅ and there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 such that d(tnx,p) ≤ knd(x,p), ∀x ∈ d(t), ∀p ∈ f(t), ∀n ∈ n.(1.5) (6) the mapping t is said to be asymptotically nonexpansive type, if lim sup n→∞ { sup x,y∈d(t) ( d(tnx,tny) −d(x,y) )} ≤ 0.(1.6) (7) the mapping t is said to be asymptotically quasi-nonexpansive type, if f(t) 6= ∅ and lim sup n→∞ { sup x∈d(t),p∈f(t) ( d(tnx,p) −d(x,p) )} ≤ 0.(1.7) remark 1.2. it is easy to see that if f(t) is nonempty, then nonexpansive mapping, quasi-nonexpansive mapping, asymptotically nonexpansive mapping, asymptotically quasi-nonexpansive mapping and asymptotically nonexpansive type mapping all are the special cases of asymptotically quasi-nonexpansive type mappings. in recent years, the problem concerning convergence of iterative sequences (and sequences with errors) for asymptotically nonexpansive mappings or asymptotically quasi-nonexpansive mappings converging to some fixed points in hilbert spaces or banach spaces have been considered by many authors. in 1973, petryshyn and williamson [13] obtained a necessary and sufficient condition for picard iterative sequences and mann iterative sequences to converge to a fixed point for quasi-nonexpansive mappings. in 1994, tan and xu [16] also proved some convergence theorems of ishikawa iterative sequences satisfies opial’s condition or has a frechet differential norm. in 1997, ghosh and debnath [5] extended the result of petryshyn and williamson [13] and gave a necessary and sufficient condition for ishikawa iterative sequences to converge to a fixed point of quasi-nonexpansive mappings. also in 2001 and 2002, liu [10, 11, 12] obtained some necessary and sufficient conditions for ishikawa iterative sequences or ishikawa iterative sequences with errors to converge to a fixed point for asymptotically quasinonexpansive mappings. in 2004, chang et al. [1] extended and improved the result of liu [12] in convex metric space. further in the same year, kim et al. [8] gave the necessary and sufficient conditions for asymptotically quasi-nonexpansive mappings in convex metric convergence theorems 47 spaces which generalized and improved some previous known results. very recently, tian and yang [18] gave some necessary and sufficient conditions for a new noor-type iterative sequences with errors to approximate a common fixed point for a finite family of uniformly quasi-lipschitzian mappings in convex metric spaces. the purpose of this paper is to give some necessary and sufficient conditions for noor-type iteration process with errors to approximate a common fixed point for a finite family of uniformly l-lipschitzian asymptotically quasi-nonexpansive type mappings in convex metric spaces. the results presented in this paper generalize, improve and unify some main results of [1]-[3], [5]-[8], [10]-[17], [19] and [21]. let t be a given self mapping of a nonempty convex subset c of an arbitrary normed space. the sequence {xn}∞n=0 defined by x0 ∈ c and xn+1 = αnxn + βntyn + γnun, n ≥ 0, yn = anxn + bntzn + cnvn, zn = dnxn + entxn + fnwn,(1.8) is called the noor-type iterative procedure with errors [2], where αn,βn,γn,an,bn,cn, dn,en and fn are appropriate sequences in [0, 1] with αn +βn +γn = an +bn +cn = dn + en + fn = 1, n ≥ 0 and {un}, {vn} and {wn} are bounded sequences in c. if dn = 1(en = fn = 0), n ≥ 0, then (1.8) reduces to the ishikawa iterative procedure with errors [20] defined as follows: x0 ∈ c and xn+1 = αnxn + βntyn + γnun, n ≥ 0, yn = anxn + bntxn + cnvn.(1.9) if an = 1(bn = cn = 0), then (1.9) reduces to the following mann type iterative procedure with errors [20]: x0 ∈ c and xn+1 = αnxn + βntyn + γnun, n ≥ 0.(1.10) for the sake of convenience, we first recall some definitions and notations. definition 1.3. (see [1]) let (e,d) be a metric space and i = [0, 1]. a mapping w : e3 × i3 → e is said to be a convex structure on e if it satisfies the following condition: d(u,w(x,y,z; α,β,γ)) ≤ αd(u,x) + βd(u,y) + γd(u,z), for any u,x,y,z ∈ e and for any α,β,γ ∈ i with α + β + γ = 1. if (e,d) is a metric space with a convex structure w, then (e,d) is called a convex metric space and denotes it by (e,d,w). let (e,d) be a convex metric space, a nonempty subset c of e is said to be convex if w(x,y,z,λ1,λ2,λ3) ∈ c, ∀(x,y,z,λ1,λ2,λ3) ∈ c3 × i3. 48 saluja remark 1.4. it is easy to prove that every linear normed space is a convex metric space with a convex structure w(x,y,z; α,β,γ) = αx + βy + γz, for all x,y,z ∈ e and α,β,γ ∈ i with α + β + γ = 1. but there exist some convex metric spaces which can not be embedded into any linear normed spaces (see, takahashi [15]). definition 1.5. let (e,d,w) be a convex metric space and ti : e → e be a finite family of asymptotically quasi-nonexpansive type mappings with i = 1, 2, . . . ,n. let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en} and {fn} be nine sequences in [0, 1] with αn + βn + γn = an + bn + cn = dn + en + fn = 1, n = 0, 1, 2, . . . .(1.11) for a given x0 ∈ e, define a sequence {xn} as follows: xn+1 = w(xn,t n n yn,un; αn,βn,γn), n ≥ 0, yn = w(f(xn),t n n zn,vn; an,bn,cn), zn = w(f(xn),t n n xn,wn; dn,en,fn),(1.12) where tnn = t n n(mod n) , f : e → e is a lipschitz continuous mapping with a lipschitz constant ξ > 0 and {un}, {vn}, {wn} are any given three sequences in e. then {xn} is called the noor-type iterative sequence with errors for a finite family of asymptotically quasi-nonexpansive type mappings {ti}ni=1. if f = i (the identity mapping on e) in (1.12), then the sequence {xn} defined by (1.12) can be written as follows: xn+1 = w(xn,t n n yn,un; αn,βn,γn), n ≥ 0, yn = w(xn,t n n zn,vn; an,bn,cn), zn = w(xn,t n n xn,wn; dn,en,fn),(1.13) if dn = 1(en = fn = 0) for all n ≥ 0 in (1.12), then zn = xn for all n ≥ 0 and the sequence {xn} defined by (1.12) can be written as follows: xn+1 = w(xn,t n n yn,un; αn,βn,γn), n ≥ 0, yn = w(f(xn),t n n xn,vn; an,bn,cn).(1.14) if f = i and dn = 1(en = fn = 0) for all n ≥ 0, then the sequence {xn} defined by (1.12) can be written as follows: xn+1 = w(xn,t n n yn,un; αn,βn,γn), n ≥ 0, yn = w(xn,t n n xn,vn; an,bn,cn),(1.15) which is the ishikawa type iterative sequence with errors considered in [17]. further, if f = i and dn = an = 1(en = fn = bn = cn = 0) for all n ≥ 0, then zn = yn = xn for all n ≥ 0 and (1.12) reduces to the following mann type iterative sequence with errors [17]: xn+1 = w(xn,t n n yn,un; αn,βn,γn), n ≥ 0.(1.16) convergence theorems 49 recall that a family {ti : i ∈ n = 1, 2, . . . ,n} of n asymptotically quasinonexpansive type self mappings of c with f = ∩ni=1f(ti) 6= ∅ is said to satisfy condition (a) ([4]) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0,∞) such that ‖x−tx‖ ≥ f(d(x,f)) for all x ∈ c holds for at least one t ∈{ti : i ∈n}. in the sequel, we shall need the following lemmas. lemma 1.6. (see [11]) let {pn}, {qn}, {rn} be three nonnegative sequences of real numbers satisfying the following conditions: pn+1 ≤ (1 + qn)pn + rn, n ≥ 0, ∞∑ n=0 qn < ∞, ∞∑ n=0 rn < ∞.(1.17) then (1) limn→∞pn exists. (2) in addition, if lim infn→∞pn = 0, then limn→∞pn = 0. lemma 1.7. let (e,d,w) be a complete convex metric space and c be a nonempty closed convex subset of e. let ti : c → c be a finite family of uniformly llipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,n such that f = ∩ni=1f(ti) 6= ∅ and f : c → c be a contractive mapping with a contractive constant ξ ∈ (0, 1). let {xn} be the iterative sequence with errors defined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in c. let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. then the following conclusions hold: (1) for all p ∈ f and n ≥ 0, d(xn+1,p) ≤ (1 + 3βn)d(xn,p) + 3kσn + mσn,(1.18) where σn = βn + γn for all n ≥ 0 and m = sup p∈f, n≥0 { d(un,p) + d(vn,p) + d(wn,p) + 2d(f(p),p) } , (2) there exists a constant m1 > 0 such that d(xn+m,p) ≤ m1d(xn,p) + 3km1 n+m−1∑ k=n σk + mm1 n+m−1∑ k=n σk, ∀p ∈ f,(1.19) for all n,m ≥ 0. 50 saluja proof. (1) let p ∈ f. it follows from (1.7) that lim sup n→∞ { sup x∈e, p∈f ( d(tnx,p) −d(x,p) )} ≤ 0. this implies that for any given k > 0, there exists a positive integer n0 such that for n ≥ n0 we have sup x∈e, p∈f ( d(tnx,p) −d(x,p) ) < k.(1.20) since {xn},{yn},{zn}⊂ e, we have d(tnn xn,p) −d(xn,p) < k, ∀p ∈ f, ∀n ≥ n0 d(tnn yn,p) −d(yn,p) < k, ∀p ∈ f, ∀n ≥ n0 d(tnn zn,p) −d(zn,p) < k, ∀p ∈ f, ∀n ≥ n0.(1.21) thus for each n ≥ 0 and for any p ∈ f, using (1.12), and (1.21), we have d(xn+1,p) = d(w(xn,t n n yn,un; αn,βn,γn),p) ≤ αnd(xn,p) + βnd(tnn yn,p) + γnd(un,p) ≤ αnd(xn,p) + βn[d(yn,p) + k] + γnd(un,p) ≤ αnd(xn,p) + βnd(yn,p) + βnk + γnd(un,p),(1.22) and d(yn,p) = d(w(f(xn),t n n zn,vn; an,bn,cn),p) ≤ and(f(xn),p) + bnd(tnn zn,p) + cnd(vn,p) ≤ and(f(xn),f(p)) + and(f(p),p) +bn[d(zn,p) + k] + cnd(vn,p) ≤ anξd(xn,p) + and(f(p),p) + bnd(zn,p) +bnk + cnd(vn,p),(1.23) and d(zn,p) = d(w(f(xn),t n n xn,wn; dn,en,fn),p) ≤ dnd(f(xn),p) + end(tnn xn,p) + fnd(wn,p) ≤ dnd(f(xn),f(p)) + dnd(f(p),p) +en[d(xn,p) + k] + fnd(wn,p) ≤ dnξd(xn,p) + dnd(f(p),p) + end(xn,p) +enk + fnd(wn,p) ≤ (dnξ + en)d(xn,p) + dnd(f(p),p) +enk + fnd(wn,p).(1.24) convergence theorems 51 substituting (1.23) into (1.22) and simplifying it, we have d(xn+1,p) ≤ αnd(xn,p) + βn [ anξd(xn,p) + and(f(p),p) +bnd(zn,p) + bnk + cnd(vn,p) ] + βnk + γnd(un,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + bnβnk +bnβnd(zn,p) + cnβnd(vn,p) + βnk + γnd(un,p) = (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + (1 + bn)βnk +bnβnd(zn,p) + cnβnd(vn,p) + γnd(un,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + 2βnk +bnβnd(zn,p) + cnβnd(vn,p) + γnd(un,p).(1.25) substituting (1.24) into (1.25) and simplifying it, we have d(xn+1,p) ≤ (αn + anβnξ)d(xn,p) + anβnd(f(p),p) + 2βnk +bnβn [ (dnξ + en)d(xn,p) + dnd(f(p),p) + enk +fnd(wn,p) ] + cnβnd(vn,p) + γnd(un,p) ≤ [ αn + anβnξ + bnβn(dnξ + en) ] d(xn,p) +βn(an + bndn)d(f(p),p) + βnk(2 + bnen) +bnβnfnd(wn,p) + cnβnd(vn,p) + γnd(un,p) ≤ [ αn + βn(anξ + bndnξ + bnen) ] d(xn,p) +2βnd(f(p),p) + 3βnk + βnd(wn,p) +βnd(vn,p) + γnd(un,p) ≤ (1 + 3βn)d(xn,p) + 2βnd(f(p),p) +2γnd(f(p),p) + 3βnk + 3γnk + βnd(wn,p) +γnd(wn,p) + βnd(vn,p) + γnd(vn,p) +βnd(un,p) + γnd(un,p) = (1 + 3βn)d(xn,p) + 3k(βn + γn) + 2(βn + γn)d(f(p),p) +(βn + γn) [ d(un,p) + d(vn,p) + d(wn,p) ] = (1 + 3βn)d(xn,p) + 3k(βn + γn) + (βn + γn) [ d(un,p) +d(vn,p) + d(wn,p) + 2d(f(p),p) ] = (1 + 3βn)d(xn,p) + 3kσn + mσn, ∀n ≥ 0, p ∈ f,(1.26) where m = sup p∈f n≥0 { d(un,p) + d(vn,p) + d(wn,p) + 2d(f(p),p) } , σn = βn + γn. this completes the proof of part (1). 52 saluja (2) since 1 + x ≤ ex for all x ≥ 0, it follows from (1.26) that, for n,m ≥ 0 and p ∈ f , we have d(xn+m,p) ≤ (1 + 3βn+m−1)d(xn+m−1,p) + 3kσn+m−1 + mσn+m−1 ≤ e3βn+m−1d(xn+m−1,p) + 3kσn+m−1 + mσn+m−1 ≤ e3βn+m−1 [ e3βn+m−2d(xn+m−2,p) + 3kσn+m−2 + mσn+m−2 ] +3kσn+m−1 + mσn+m−1 ≤ e3(βn+m−1+βn+m−2)d(xn+m−2,p) + 3k [ e3βn+m−1σn+m−2 +σn+m−1 ] + m [ e3βn+m−1σn+m−2 + σn+m−1 ] ≤ . . . ≤ . . . ≤ m1d(xn,p) + 3km1 n+m−1∑ k=n σk + mm1 n+m−1∑ k=n σk, = m1d(xn,p) + (3k + m)m1 n+m−1∑ k=n σk,(1.27) where m1 = e 3 ∑∞ k=0 βk. this completes the proof of part (2). � 2. main results theorem 2.1. let (e,d,w) be a complete convex metric space and c be a nonempty closed convex subset of e. let ti : c → c be a finite family of uniformly llipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,n such that f = ∩ni=1f(ti) 6= ∅ and f : c → c be a contractive mapping with a contractive constant ξ ∈ (0, 1). let {xn} be the iterative sequence with errors defined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in c. let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. then the sequence {xn} converges to a common fixed point p in f if and only if lim infn→∞d(xn,f) = 0, where d(x,f) = infp∈f d(x,p). proof. the necessity is obvious. now, we prove the sufficiency. in fact, from lemma 1.7, we have d(xn+1,f) ≤ (1 + 3βn)d(xn,f) + (3k + m)σn, ∀n ≥ 0,(2.1) where σn = βn + γn. by conditions (i) and (ii), we know that ∞∑ n=0 σn < ∞, ∞∑ n=0 βn < ∞.(2.2) convergence theorems 53 it follows from lemma 1.6 that limn→∞d(xn,f) exists. since lim infn→∞d(xn,f) = 0, we have lim n→∞ d(xn,f) = 0.(2.3) next, we prove that {xn} is a cauchy sequence in c. in fact, for any given ε > 0, there exists a positive integer n1 ≥ n0 (where n0 is the positive integer appeared in lemma 1.7) such that for any n ≥ n1, we have d(xn,f) < ε 8m1 , ∞∑ n=n1 σn < ε 12(k + m)m1 , ∀n ≥ 0.(2.4) from (2.4), there exists p1 ∈ f and positive integer n2 ≥ n1 such that d(xn2,p1) < ε 4m1 .(2.5) thus lemma 1.7(2) implies that, for any positive integer n,m with n ≥ n2, we have d(xn+m,xn) ≤ d(xn+m,p1) + d(xn,p1) ≤ m1d(xn2,p1) + 3(k + m)m1 n+m−1∑ k=n2 σk +m1d(xn2,p1) + 3(k + m)m1 n+m−1∑ k=n2 σk ≤ 2m1d(xn2,p1) + 6(k + m)m1 n+m−1∑ k=n2 σk < 2m1. ε 4m1 + 6(k + m)m1. ε 12(k + m)m1 < ε.(2.6) this shows that {xn} is a cauchy sequence in a nonempty closed convex subset c of a complete convex metric space e. without loss of generality, we can assume that limn→∞xn = q ∈ e. now we will prove that q ∈ f. since xn → q and d(xn,f) → 0 as n → ∞, for any given ε1 > 0, there exists a positive integer n2 ≥ n1 ≥ n0 such that for n ≥ n2, we have d(xn,q) < ε1, d(xn,f) < ε1.(2.7) again from (2.7), there exists q1 ∈ f and positive integer n3 ≥ n2 such that d(xn3,q1) < 2ε1.(2.8) moreover, it follows from (1.20) that for any n ≥ n3, we have d(tnq,q1) −d(q,q1) < k.(2.9) 54 saluja thus for any i = 1, 2, . . . ,n, from (2.7) (2.9) and for any n ≥ n3, we have d(tni q,q) ≤ d(t n i q,q1) + d(q1,q) ≤ d(q,q1) + k + d(q1,q) = k + 2d(q,q1) ≤ k + 2[d(q,xn3 ) + d(xn3,q1)] < k + 2(ε1 + 2ε1) = k + 6ε1 = ε ′,(2.10) where ε′ = k + 6ε1, since k > 0 and ε1 > 0, it follows that ε ′ > 0. by the arbitrariness of ε′ > 0, we know that tni q = q for all i = 1, 2, . . . ,n. again since for any n ≥ n3, we have d(tni q,tiq) ≤ d(t n i q,q1) + d(tiq,q1) ≤ d(q,q1) + k + d(tiq,q1) ≤ d(q,q1) + k + ld(q,q1) = (1 + l)d(q,q1) + k ≤ (1 + l)[d(q,xn3 ) + d(xn3,q1)] + k < (1 + l)[ε1 + 2ε1] + k = 3(1 + l)ε1 + k = ε ′′,(2.11) where ε′′ = 3(1 + l)ε1 + k, since k > 0 and ε1 > 0, it follows that ε ′′ > 0. by the arbitrariness of ε′′ > 0, we know that tni q = tiq for all i = 1, 2, . . . ,n. from the uniqueness of limit, we have q = tiq for all i = 1, 2, . . . ,n, that is, q ∈ f. this shows that q is a common fixed point of the mappings {ti}ni=1. this completes the proof. � taking f = i in theorem 2.1, then we have the following result. theorem 2.2. let (e,d,w) be a complete convex metric space and c be a nonempty closed convex subset of e. let ti : c → c be a finite family of uniformly llipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,n such that f = ∩ni=1f(ti) 6= ∅. let {xn} be the iterative sequence with errors defined by (1.13) and {un}, {vn}, {wn} be three bounded sequences in c. let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfying the conditions (i) and (ii) of theorem 2.1. then the sequence {xn} converges to a common fixed point p in f if and only if lim inf n→∞ d(xn,f) = 0,(2.12) where d(x,f) = infp∈f d(x,p). taking dn = 1(en = fn = 0) for all n ≥ 0 in theorem 2.1, then we have the following result. theorem 2.3. let (e,d,w) be a complete convex metric space and c be a nonempty closed convex subset of e. let ti : c → c be a finite family of uniformly llipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,n such that f = ∩ni=1f(ti) 6= ∅ and f : c → c be a contractive mapping with a contractive constant ξ ∈ (0, 1). let {xn} be the iterative sequence with errors defined convergence theorems 55 by (1.14) and {un}, {vn} be two bounded sequences in c. let {αn}, {βn}, {γn}, {an}, {bn}, {cn} be six sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = 1, for all n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. then the sequence {xn} converges to a common fixed point p in f if and only if lim inf n→∞ d(xn,f) = 0,(2.13) where d(x,f) = infp∈f d(x,p). as an application of theorem 2.1, we establish another strong convergence result as follows. theorem 2.4. let (e,d,w) be a complete convex metric space and c be a nonempty closed convex subset of e. let ti : c → c be a finite family of uniformly llipschitzian asymptotically quasi-nonexpansive type mappings for i = 1, 2, . . . ,n such that f = ∩ni=1f(ti) 6= ∅ and f : c → c be a contractive mapping with a contractive constant ξ ∈ (0, 1). let {xn} be the iterative sequence with errors defined by (1.12) and {un}, {vn}, {wn} be three bounded sequences in c. let {αn}, {βn}, {γn}, {an}, {bn}, {cn}, {dn}, {en}, {fn} be nine sequences in [0, 1] satisfying the following conditions: (i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0; (ii) ∑∞ n=0(βn + γn) < ∞. assume that limn→∞d(xn,tlxn) = 0 for l = 1, 2, . . . ,n. if {ti : i ∈n} satisfies condition (a), then the sequence {xn} converges strongly to a point in f . proof. as in the proof of theorem 2.1, we have that limn→∞d(xn,f) exists. again by hypothesis of the theorem limn→∞d(xn,tlxn) = 0 for l = 1, 2, . . . ,n. so condition (a) guarantees that limn→∞f(d(xn,f)) = 0. since f is a non-decreasing function and f(0) = 0, it follows that limn→∞d(xn,f) = 0. therefore, theorem 2.1 implies that {xn} converges strongly to a point in f. this completes the proof. � remark 2.5. theorems 2.1 2.3 generalize, improve and unify some corresponding result in [1]-[3], [5]-[8], [10]-[17], [19] and [21]. remark 2.6. our results also extend the corresponding results of [18] to the case of more general class of uniformly quasi-lipschitzian mappings considered in this paper. example 2.7. let e = [−π, π] and let t be defined by tx = xcosx for each x ∈ e. clearly f(t) = {0}. t is a quasi-nonexpansive mapping since if x ∈ e and z = 0, then d(tx,z) = d(tx, 0) = |x||cosx| ≤ |x| = |x−z| = d(x,z), 56 saluja and t is asymptotically quasi-nonexpansive mapping with constant sequence {kn} = {1}. hence by remark 1.1, t is asymptotically quasi-nonexpansive type mapping. but it is not a nonexpansive mapping and hence asymptotically nonexpansive mapping. in fact, if we take x = π 2 and y = π, then d(tx,ty) = ∣∣∣π 2 cos π 2 −π cosπ ∣∣∣ = π, whereas d(x,y) = ∣∣∣π 2 −π ∣∣∣ = π 2 . example 2.8. let e = r and let t be defined by t(x) = { x 2 cos 1 x , if x 6= 0, 0, if x = 0. if x 6= 0 and tx = x, then x = x 2 cos 1 x . thus 2 = cos 1 x . this is impossible. t is a quasi-nonexpansive mapping since if x ∈ e and z = 0, then d(tx,z) = d(tx, 0) = | x 2 ||cos 1 x | ≤ |x| 2 < |x| = |x−z| = d(x,z), and t is asymptotically quasi-nonexpansive mapping with constant sequence {kn} = {1}. hence by remark 1.1, t is asymptotically quasi-nonexpansive type mapping. but it is not a nonexpansive mapping and hence asymptotically nonexpansive mapping. in fact, if we take x = 2 3π and y = 1 π , then d(tx,ty) = ∣∣∣ 1 3π cos 3π 2 − 1 2π cosπ ∣∣∣ = 1 2π , whereas d(x,y) = ∣∣∣ 2 3π − 1 π ∣∣∣ = 1 3π . 3. conclusion if f(t) is nonempty, then asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mappings are asymptotically quasi-nonexpansive type mappings by remark 1.2, thus our results are good improvement and extension of some previous work from the existing literature (see, e.g., [1]-[3], [5]-[8], [10]-[21] and many others). references [1] s.s. chang, j.k. kim and d.s. jin, iterative sequences with errors for asymptotically quasinonexpansive type mappings in convex metric spaces, archives of inequality and applications 2(2004), 365-374. [2] y.j. cho, h. zhou and g. guo, weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings, comput. math. appl. 47(2004), 707-717. [3] h. fukhar-ud-din and s.h. khan, convergence of iterates with errors of asymptotically quasinonexpansive mappings and applications, j. math. anal. appl. 328(2)(2007), 821-829. [4] h. fukhar-ud-din, a.r. khan and m.a.a. khan, a new implicit algorithm of asymptotically quasi-nonexpansive maps in uniformly convex banach spaces, iaeng int. j. appl. maths. 42(3)(2008). [5] m.k. ghosh and l. debnath, convergence of ishikawa iterates of quasi-nonexpansive mappings, j. math. anal. appl. 207(1997), 96-103. [6] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1972), 171-174. convergence theorems 57 [7] j.u. jeong and s.h. kim, weak and strong convergence of the ishikawa iteration process with errors for two asymptotically nonexpansive mappings, appl. math. comput. 181(2) (2006), 1394-1401. [8] j.k. kim, k.h. kim and k.s. kim, three-step iterative sequences with errors for asymptotically quasi-nonexpansive mappings in convex metric spaces, nonlinear anal. convex anal. rims vol. 1365(2004), 156-165. [9] w.a. kirk, fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, israel j. math. 17(1974), 339-346. [10] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 259(2001), 1-7. [11] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings with error member, j. math. anal. appl. 259(2001), 18-24. [12] q.h. liu, iterative sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex banach spaces, j. math. anal. appl. 266(2002), 468-471. [13] w.v. petryshyn and t.e. williamson, strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, j. math. anal. appl. 43(1973), 459-497. [14] g.s. saluja, convergence of fixed point of asymptotically quasi-nonexpansive type mappings in convex metric spaces, j. nonlinear sci. appl. 1(3)(2008), 132-144. [15] w. takahashi, a convexity in metric space and nonexpansive mappings i, kodai math. sem. rep. 22(1970), 142-149. [16] k.k. tan and h.k. xu, fixed point iteration processes for asymptotically nonexpansive mappings, proc. amer. math. soc. 122(1994), 733-739. [17] y.-x. tian, convergence of an ishikawa type iterative scheme for asymptotically quasinonexpansive mappings, compt. math. appl. 49(11-12)(2005), 1905-1912. [18] y.-x. tian and chun-de yang, convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-lipschitzian mappings in convex metric spaces, fixed point theory and applications, vol. 2009, article id 891965, 12 pages, doi:10.1155/2009/891965. [19] b.l. xu and m.a. noor, fixed point iterations for asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 267(2002), no.2, 444-453. [20] y. xu, ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations, j. math. anal. appl. 224(1)(1998), 91-101. [21] h. zhou; j.i. kang; s.m. kang and y.j. cho, convergence theorems for uniformly quasilipschitzian mappings, int. j. math. math. sci. 15(2004), 763-775. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications volume 16, number 3 (2018), 427-436 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-427 some properties of generalized strongly harmonic convex functions muhammad aslam noor∗, khalida inayat noor, sabah iftikhar and farhat safdar department of mathematics, comsats institute of information technology, islamabad, pakistan. ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we introduce a new class of harmonic convex functions with respect to an arbitrary trifunction f(·, ·, ·) : k×k×[0, 1] → r, which is called generalized strongly harmonic convex functions. we study some basic properties of strongly harmonic convex functions. we also discuss the sufficient conditions of optimality for unconstrained and inequality constrained programming under the generalized harmonic convexity. several special cases are discussed as applications of our results. ideas and techniques of this paper may motivate further research in different fields. 1. introduction the concept of convexity and generalized convexity in the study of optimality to solve mathematical programming, have been extended using innovative ideas and techniques. for example, in earlier papers, bector and singh [3] introduced a class of b-vex functions. chao et al. [4] considered new generalized sub-b-convex functions and sub-b-convex sets. they proved the sufficient conditions of optimality for both unconstrained and inequality constrained sub-b-convex programming. anderson et. al. [1] and iscan [5] have investigated various properties of harmonic convex functions. noor and noor [9] have shown that the minimum of the differentiable harmonic convex functions on the harmonic convex set can be characterized by a class of variational inequalities, which is called harmonic variational inequality. to the best of our received 2018-01-19; accepted 2018-03-14; published 2018-05-02. 2010 mathematics subject classification. 26d15; 26d10; 90c23. key words and phrases. harmonic convex functions; f-harmonic convex functions; optimality conditions. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 427 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-427 int. j. anal. appl. 16 (3) (2018) 428 knowledge, this field is new one and has not developed as yet. a significant class of convex functions is that of strongly harmonic convex functions introduced by noor et. al. [11]. for recent applications, generalizations and other aspects of convex and harmonic convex functions, see [2–4,6,10,12–19] and references therein. inspired by the research works [2–4, 6], we introduce a new class of harmonic convex functions with respect to an arbitrary function f(., ., .), which is called strongly generalized harmonic convex function. one can show that the generalized strongly harmonic convex functions is quite general and unified one. several new and old classes of convex and harmonic convex functions can be obtained from these general harmonic convex functions. we consider the sufficient conditions of optimality for both unconstrained and inequality constrained programming. some properties of generalized strongly harmonic convex functions and generalized strongly harmonic convex sets are discussed. results obtained in this paper may be considered as significant improvement of the known results. 2. preliminaries in this section, we recall some basic concepts and results. we also introduce some new concepts and discuss some special cases. definition 2.1. [1]. a set k = [a,b] ⊂ rn \{0} is said to be a harmonic convex set, if xy tx + (1 − t)y ∈ k, ∀x,y ∈ k, t ∈ [0, 1]. we now consider the concept of generalized strongly harmonic convex functions with respect to an arbitrary trifunction f(·, ·, ·) : k ×k × [0, 1] → r. definition 2.2. let k is a nonempty harmonic convex set in rn \{0}. a function f : k → r is said to be generalized strongly harmonic convex function on k with respect to map f(·, ·, ·) : k ×k × [0, 1] → r, if and only if, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) + f(x,y,t), ∀x,y ∈ k,t ∈ [0, 1]. (2.1) (i). if f(x,y,t) = 0 in definition 2.2, then it reduces to harmonic convex function. definition 2.3. [5]. a function f : k → r, where k is a nonempty harmonic convex set rn \{0}, is said to be a harmonic convex function on k, if and only if, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ k, t ∈ [0, 1]. this shows that every harmonic convex function f is harmonic sub-f-convex function with respect to the map f(x,y,t) = 0, but the converse may not be true. see also [8–10, 17]. int. j. anal. appl. 16 (3) (2018) 429 (ii). if f(x,y,t) = t(1−t)f( xy x−y ) in definition 2.2, then it reduces to f-strongly harmonic convex function definition 2.4. [18]. a function f : k ⊆ r \{0}→ r is said to be f-strongly harmonic convex function, if there exists a non-negative function f : x \{0}→ [0,∞), such that f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) − t(1 − t)f ( xy x−y ) , ∀x,y ∈ k,t ∈ (0, 1). (2.2) if t = 1 2 , then (2.2) reduces to f ( 2xy x + y ) ≤ f(x) + f(y) 2 − 1 4 f ( xy x−y ) and the functionf is called f-strongly harmonic mid-convex(harmonic jensen-convex) function. (iii). if f(x,y,t) = −ct(1 − t) ‖ x−y xy ‖ in definition 2.2, then it reduces to strongly harmonic convex function with modulus c > 0. definition 2.5. [11]. a function f : k → r, where k is a nonempty harmonic convex set in rn \{0}, is said to be strongly harmonic convex function on k with modulus c > 0, if and only if, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) − ct(1 − t) ∥∥x−y xy ∥∥2, ∀x,y ∈ i,t ∈ (0, 1). theorem 2.1. [17]. let k be a nonempty harmonic convex set in rn \ {0} and let f : k → r be differentiable on k. then f is harmonic quasi convex, if and only if, f(x) ≤ f(y) ⇒〈f′(y), xy y −x 〉≤ 0 ∀ x,y ∈ k. definition 2.6. let k is a nonempty harmonic convex set in rn \{0}. a function f : k → r is said to be generalized strongly harmonic quasi convex function on k with respect to map f(·, ·, ·) : k×k× [0, 1] → r, if and only if, f ( xy tx + (1 − t)y ) ≤ max{f(x),f(y)} + f(x,y,t), ∀x,y ∈ k,t ∈ [0, 1]. (2.3) (i). if f(x,y,t) = 0 in definition 2.6, then it reduces to harmonic quasi convex function. definition 2.7. [20]. a function f : k → r, where k is a nonempty harmonic convex set rn\{0}, is said to be a harmonic quasi convex function on k, if and only if, f ( xy tx + (1 − t)y ) ≤ max{f(x),f(y)}, ∀x,y ∈ k. we now define a new class of generalized strongly harmonic log-convex functions. definition 2.8. let k is a nonempty harmonic convex set in rn \{0}. a function f : k → r is said to be generalized strongly harmonic log-convex function on k with respect to map f(·, ·, ·) : k ×k × [0, 1] → r, if and only if, f ( xy tx + (1 − t)y ) ≤ [f(x)]1−t[f(y)]t + f(x,y,t), ∀ x,y ∈ k, t ∈ [0, 1]. (2.4) int. j. anal. appl. 16 (3) (2018) 430 from (2.4), it follows that f ( xy tx + (1 − t)y ) ≤ [f(x)]1−t[f(y)]t + f(x,y,t) ≤ (1 − t)f(x) + tf(y) + f(x,y,t) ≤ max{f(x),f(y)} + f(x,y,t), ∀ x,y ∈ k, t ∈ [0, 1], this shows that every generalized strongly harmonic log-convex function is generalized strongly harmonic convex and every generalized strongly harmonic convex function is generalized strongly harmonic quasi convex, but the converse is not true. 3. main results in this section, we introduce the concept of generalized harmonic convex set and harmonic convex functions with respect to an arbitrary map f(·, ·, ·) and discuss its properties. theorem 3.1. if fi : k → r, (i = 1, 2, 3, ...,m) are generalized strongly harmonic convex functions with respect to map fi : k ×k × [0, 1] → r, respectively. then the function f = m∑ i=1 aifi, a1 ≥ 0, i = 1, 2, 3, ...,m, is generalized strongly harmonic convex with respect to f = ∑m i=1 aibi. proof. for all x,y ∈ k and t ∈ [0, 1], we have f ( xy tx + (1 − t)y ) = m∑ i=1 aifi ( xy tx + (1 − t)y ) ≤ m∑ i=1 ai[(1 − t)fi(x) + tfi(y) + fi(x,y,t)] = (1 − t) m∑ i=1 aifi(x) + t m∑ i=1 aifi(y) + m∑ i=1 aifi(x,y,t) = (1 − t)f(x) + tf(y) + f(x,y,t). � theorem 3.2. if the function fi : k → r are generalized strongly harmonic convex functions with respect to fi(x,y,t) respectively. then f = max{fi, i = 1, 2, 3, ...,m} is also generalized strongly harmonic convex with respect to f = max{fi}. int. j. anal. appl. 16 (3) (2018) 431 proof. consider, f ( xy tx + (1 − t)y ) = max { fi ( xy tx + (1 − t)y ) , i = 1, 2, 3, ...,m } ≤ (1 − t) max{fi(x)} + t max{fi(y)} + max{fi(x,y,t) = (1 − t)f(x) + tf(y) + f(x,y,t). so, f(x) is harmonic sub-f convex function with respect to f(x,y,t) = max{fi, i = 1, 2, 3, ...,m}. � theorem 3.3. if the function f : k → r is a generalized strongly harmonic convex function with respect to f(x,y,t) and g : r → r is a linear function, then f ◦g is generalized strongly harmonic convex with respect to f ′ = g ◦f. proof. since f is generalized strongly harmonic convex function with respect to f(x,y,t) and g is a an increasing function, it follows that (g ◦f) ( xy tx + (1 − t)y ) = g ( f ( xy tx + (1 − t)y )) ≤ g((1 − t)f(x) + tf(y) + f(x,y,t) = (1 − t)g(f(x)) + tg(f(y)) + g(f(x,y,t)) = (1 − t)(g ◦f) + t(g ◦f) + (g ◦f)(x,y,t). that is, g◦f is generalized strongly harmonic convex function with respect to f ′ = g◦f and this completes the proof. � the following theorem gives a necessary and sufficient characterization of a differentiable generalized strongly harmonic convex function with respect to a map f(·, ·, ·). theorem 3.4. let k be a harmonic convex set. if f : k → r is a differentiable generalized strongly harmonic convex function on the harmonic convex set k with respect to the map f(x,y,t), then (1) 〈f′(x), xy x−y〉≤ f(y) −f(x) + limt→0+ f (x,y,t) t , ∀x,y ∈ k. (2) 〈f′(x) −f′(y), xy x−y〉≤ limt→0+ f (x,y,t) t + limt→0+ f (y,x,t) t , ∀x,y ∈ k. proof. (1). let f be a generalized strongly harmonic convex function. then f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) + f(x,y,t), which can be written as f(y) −f(x) + f(x,y,t) t ≥ f ( xy y+t(x−y) ) −f(x) t . int. j. anal. appl. 16 (3) (2018) 432 since f is differentiable function, so taking the limit in the above inequality, as t → 0, we have f(y) −f(x) + lim t→0+ f(x,y,t) t ≥〈f′(x), xy x−y 〉, (3.1) which is (1). changing the role of x and y, in (3.1), we obtain f(x) −f(y) + lim t→0+ f(y,x,t) t ≥〈f′(y), xy y −x 〉. (3.2) adding (3.1) and (3.2), we have 〈f′(x) −f′(y), xy x−y 〉≤ lim t→0+ f(x,y,t) t + lim t→0+ f(y,x,t) t , which is the required (2). this completes the proof. � definition 3.1. [17]. a function f : k → r, where k is a nonempty harmonic convex set rn \{0} is said to be a harmonic quasi convex function, if, for each x,y ∈ k with f(x) ≤ f(y), we have 〈f′(y), xy y−x〉≤ 0; or equivalently, if 〈f′(y), xy y−x〉 > 0, then f(x) > f(y). definition 3.2. [17]. a function f : k → r, where k is a nonempty harmonic convex set rn \{0} is said to be a harmonic pseudo-convex function, if, for each x,y ∈ k with 〈f′(y), xy y−x〉 ≥ 0, we have f(x) ≥ f(y); or equivalently, if f(x) < f(y), then 〈f′(y), xy y−x〉 < 0. theorem 3.5. let k be a harmonic convex set and f : k → r be a differentiable generalized strongly harmonic convex function on the harmonic convex set k with respect to the map f(x,y,t). if limt→0 f (x,y,t) t ≤ |f(x) −f(y)|, ∀ x,y ∈ k, then f is harmonic quasi-convex. furthermore, if limt→0 f (x,y,t) t < |f(x) −f(y)|, ∀ x,y ∈ k, then f is harmonic pseudo-convex. proof. suppose that f(x) ≤ f(y), for any x,y ∈ k and t ∈ (0, 1). then from theorem 3.4, we have 〈f′(y), xy y −x 〉≤ f(x) −f(y) + lim t→0+ f(y,x,t) t . if limt→0 f (y,x,t) t ≤ |f(x) − f(y)|, then f(x) − f(y) + limt→0 f (x,y,t) t ≤ 0. so, 〈f′(y), xy y−x〉 ≤ 0. therefore, from definition 3.1, we have f is harmonic quasi-convex function. similarly, if f(x) < f(y), we also have 〈f′(x), xy x−y〉 < 0. so, from the definition 3.2, we have f is harmonic pseudo-convex function. � now, we are going to introduce a new concept of generalized harmonic convex set. definition 3.3. let ks ⊂ rn+1\{0} be a nonempty set. a set ks is said to be generalized harmonic convex with respect to f(x,y,t) : rn ×rn × [0, 1] → r, if( xy tx + (1 − t)y , (1 − t)α + tβ + f(x,y,t) ) ∈ ks, ∀(x,α), (y,β) ∈ ks, int. j. anal. appl. 16 (3) (2018) 433 for all x,y ∈ rn, and t ∈ [0, 1]. we now investigate some characterizations of generalized strongly harmonic convex function f : k → r in term of their epigraph e(f). definition 3.4. [2]. let k be a nonempty in rn and let f : k → r be a function. then epigraph of f, denoted by e(f) is defined by e(f) = {(x,α) : x ∈ k,α ∈ r,f(x) ≤ α}. theorem 3.6. a function f : k → r is generalized strongly harmonic convex with respect to f(x,y,t) : rn×rn×[0, 1] → r, if and only if, epigraph of f is generalized harmonic convex set with respect to f(·, ·, ·). proof. suppose that f is harmonic sub-f-convex function with respect to f(x,y,t). let (x,α) and (y,β) ∈ e(f). then x,y ∈ k, f(x) ≤ α and f(y) ≤ β, we have f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) + f(x,y,t) ≤ (1 − t)α + tβ + f(x,y,t). hence from definition, one has( xy tx + (1 − t)y , (1 − t)α + tβ + f(x,y,t) ) ∈ e(f) thus e(f) is generalized harmonic convex set with respect to f. conversely, assume that e(f) is generalized harmonic convex set with respect to f. let x,y ∈ k, then (x,f(x)) and (y,f(y)) belong to e(f). thus for t ∈ [0, 1],( xy tx + (1 − t)y , (1 − t)f(x) + tf(y) + f(x,y,t) ) ∈ e(f). this further follows that, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) + f(x,y,t) that is f is a generalized strongly harmonic convex function with respect to f(., ., .). � theorem 3.7. if ksi is a family of generalized harmonic convex set with respect to a map f(x,y,t), then ∩i∈iksi is a generalized harmonic convex set with respect to f(x,y,t). proof. let (x,α), (y,β) ∈ ∩i∈iksi , t ∈ [0, 1]. then for each i ∈ i, (x,α), (y,β) ∈ ii. since ksi is a generalized harmonic convex set with respect to f(., ., .), it follows that( xy tx + (1 − t)y , (1 − t)f(x) + tf(y) + f(x,y,t) ) ∈ ksi,∀i ∈ i. int. j. anal. appl. 16 (3) (2018) 434 thus ( xy tx + (1 − t)y , (1 − t)f(x) + tf(y) + f(x,y,t) ) ∈∩i∈iksi. hence, ∩i∈iksi is generalized harmonic convex set with respect to f(x,y,t). � we apply the above results to the nonlinear programming problem. first, we consider the unconstraint problem. theorem 3.8. let f : k → r be differentiable and generalized strongly harmonic convex function with respect to map f(·, ·, ·). consider the optimal problem min{f(x)|x ∈ k}. if x̄ ∈ k and the relation 〈f′(x), x̄x x̄−x 〉− lim t→0+ f(x̄,x,t) t ≥ 0, (3.3) holds for each x ∈ k, then x̄ is the optimal solution of f on k. proof. for any x ∈ k, from theorem 3.4, we have 〈f′(x), x̄x x̄−x 〉− lim t→0+ f(x̄,x,t) t ≤ f(x) −f(x̄). (3.4) from (3.3) and (3.4), we have f(x) −f(x̄) ≥ 0. so, x̄ is an optimal solution of f on k. � next, we apply the above results to the nonlinear programming problem with inequality constraints: min f(x) (pg) s.t gi(x) ≤ 0, i ∈ i = {1, 2, 3, ...,m} x ∈ rn denote the feasible set of (pg) by sg = {x ∈ rn|gi(x) ≤ 0, i ∈ i}. definition 3.5. let k be a nonempty harmonic convex set in rn. the function f : k → r is said to be generalized strongly harmonic pseudo convex on k with respect to f : k × k × [0, 1] → r, if for each x,y ∈ k and t ∈ (0, 1), from 〈f′(y), xy y−x〉 + limt→0 f (y,x,t) t ≥ 0 one can have f(x) ≥ f(y). theorem 3.9. (karush-kuhn-tucker sufficient conditions) the function f is differentiable generalized strongly harmonic pseudo convex with respect to f(., ., .) : k×k×[0, 1] → r, gi(x) (i ∈ i) are differentiable and generalized strongly harmonic convex with respect to f : k ×k × [0, 1] → r. assume that x̄ ∈ sg is a kkt point of (pg), that is, there exist multiplier λi ≥ 0 (i ∈ i) such that 〈f′(x̄), x̄x x̄−x 〉 + ∑ i∈i(x̄) λi〈g′(x̄), x̄x x̄−x 〉 = 0, λi〈g′(x̄), x̄x x̄−x 〉 = 0. (3.5) int. j. anal. appl. 16 (3) (2018) 435 if lim t→0 f(x,x̄,t) t ≥ ∑ i∈i λi lim t→0 f(x,x̄,t) t , ∀x ∈ sg. (3.6) then x̄ is an optimal solution of the problem (pg). proof. for any x ∈ sg, we have gi(x) ≤ 0, gi(x̄) = 0, i ∈ i(x̄) = {i ∈ i | gi(x̄) = 0}. therefore, from the generalized strongly harmonic convexity of gi(x) and theorem 3.4, we obtain 〈g′(x̄), x̄x x̄−x 〉− lim t→0+ f(x̄,x,t) t ≤ 0, for i ∈ i(x̄). (3.7) from (3.5), one has 〈f′(x̄), x̄x x̄−x 〉 = − ∑ i∈i(x̄) λi〈g′(x̄), x̄x x̄−x 〉. in view of (3.6) and from (3.7), we have 〈f′(x̄), x̄x x̄−x 〉 + lim t→0+ f(x̄,x,t) t ≥ − ∑ i∈i(x̄) λi〈g′(x̄), x̄x x̄−x 〉 + ∑ i∈i(x̄) λi lim t→0+ f(x̄,x,t) t = − ∑ i∈i(x̄) λi [ 〈g′(x̄), x̄x x̄−x 〉− lim t→0+ f(x̄,x,t) t ] ≥ 0. so, 〈f′(x̄), x̄x x̄−x 〉 + lim t→0+ f(x̄,x,t) t ≥ 0. from the generalized strongly harmonic pseudo convexity of f(x), we have f(x) ≥ f(x̄). therefore x̄ is an optimal solution of the problem (pg). � acknowledgements the authors would like to thank the rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] m. s. bazaraa, d. hanif, c. m. shetty, nonlinear programming theory and algorithms john wiley and sons, new york, 1993. [3] c.r. bector, c. singh, b-vex functions, j. optim. theory appl., 71 (1991), 439-453. [4] m. t. chao, j. b. jian and d. y. liang, sub-b-convex functions and sub-f-convex programming, oper. res. trans., 16(2012), 1-8. int. j. anal. appl. 16 (3) (2018) 436 [5] i. iscan, hermite-hadamard type inequalities for harmonically convex functions. hacet, j. math. stats., 43(6)(2014), 935-942. [6] j. liao and t. du, on some characterizations of sub-b-s-convex functions, filomat, 30(14)(2016), 3885-3895. [7] c. p. niculescu and l. e. persson, convex functions and their applications, springer-verlag, new york, (2006). [8] m. a. noor, some developments in general variational inequalities, appl. math. comput. 251(2004), 199-277. [9] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inf. sci., 10(5)(2016), 1811-1814. [10] m. a. noor and k. i. noor, some implicit methods frsolving harmonic variational inequalites, inter. j. anal. appl. 12(1)(2016), 10-14. [11] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for strongly harmonic convex functions, j. inequ. special func.,7(3)(2016), 99-113. [12] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions(survey), twms j. pure appl. math. 7(1)(2016), 3-19. [13] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic preinvex functions, saussurea 6(1)(2016), 34-53. [14] m. a. noor, k. i. noor and s. iftikhar, integral inequalities of hermite-hadamard type for harmonic (h,s)-convex functions, int. j. anal. appl., 11(1)(2016), 61-69. [15] m. a. noor, k. i. noor, s. iftikhar and f. safdar, integral inequalities for relative harmonic (s,η)-convex functions, appl. math. comput. sci. 1(1)(2016), 27-34. [16] m. a. noor, k. i. noor, s. iftikhar and c. ionescu, hermite-hadamard inequalities for co-ordinated harmonic convex functions, u.p.b. sci. bull., ser: a, 79(1)(2017), 25-34. [17] m. a. noor, k. i. noor and s. iftikhar, some characterizations of harmonic convex functions, int. j. anal. appl. 15(2)(2017), 179-187. [18] m. a. noor, b. bin-mohsin, k. i. noor and s. iftikhar, relative strongly harmonic convex functions and their characterizations, j. nonlinear sci. appl. in press. [19] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [20] t.-y. zhang, ai-p. ji, f. qi, integral inequalities of hermite-hadamard type for harmonically quasi-convex functions, proc. jangjeon. math. soc., 16(3)(2013), 399-407. 1. introduction 2. preliminaries 3. main results acknowledgements references int. j. anal. appl. (2023), 21:49 sufficient conditions for convergence of sequences of henstock-kurzweil integrable functions yassin alzubaid∗ department of mathematical sciences, umm al-qura university, makkah almukarramah, saudi arabia ∗corresponding author: yazubaidi@uqu.edu.sa abstract. the main aim of this paper is to present our approach of obtaining sufficient conditions for convergence of sequences of henstock-kurzweil integrable functions. our approach involves the use of the concept of multiplier functions, where we define a class φ of multipliers for the henstock-kurzweil integral. we consider a sequence (fn) of henstock-kurzweil integrable functions on a non-degenerate interval [a,b] and we assume that (fn) converges point wise to a function f . then we show that f is henstock-kurzweil integrable and its integral is equal to the limit of the sequence ( ∫ b a fn) if there exists φ ∈ φ such that the defined functionals of the type f (φ,fn) satisfy the imposed conditions. beside the fact that the results regarding the convergence under the integral sign are always of great importance, the method introduced here can be imitated and used to obtain other results on the related areas. 1. introduction the henstock-kurzweil integral was originally introduced in [5] and [6]. it is a generalization of the riemann integral. it is a very powerful technique of integration in a way that the space of all henstockkurzweil integrable functions strictly contains the spaces of all lebesgue and riemann integrable functions. however, the space of all henstockkurzweil integrable functions, unlike lebesgue space, lacks the property of completeness. for further reading about the henstock-kurzweil integral reader may consult [4], [9], and [11]. in this paper, we investigate the sufficient conditions for the convergence of sequences of henstockkurzweil integrable functions. let (fn) be a sequence of henstock-kurzweil integrable functions that converges point-wise to f on [a,b]. the properties and integrability of the function f was studied by many authors. one of the most well-known results for f was obtained by received: apr. 10, 2023. 2020 mathematics subject classification. 26a42. key words and phrases. henstock-kurzweil integral; equi-lipschitz; equi-absolutely continuous; multipliers for integral. https://doi.org/10.28924/2291-8639-21-2023-49 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-49 2 int. j. anal. appl. (2023), 21:49 charles swartz in [8]. he showed that if the sequence (fn) is uniformly henstockkurzweil integrable then f is as well henstockkurzweil integrable and ∫b a f = limn−→∞ ∫b a fn. in the following section, we obtain a similar result under a different set of conditions. for the convenience of the reader, we state some main definitions and results those will be needed later. definition 1.1. a sequence of functions (fn) is said to be equi-lipschitz on an interval i if there exists c > 0 such that |fn(x) −fn(y)| ≤ c|x −y| for all x,y ∈ i and for all n. definition 1.2. a sequence of functions (fn) where fn(x) = ∫ x a fn is said to be equi-absolutely continuous on an interval i if for all � > 0 there exists δ > 0 such that n∑ k=1 | ∫ xk xk−1 fn| < � (1.1) for all n whenever {[xk−1,xk] : k = 1, ...,n} is a collection of disjoint subintervals of i satisfying n∑ k=1 |xk −xk−1| < δ. (1.2) definition 1.3. a sequence of functions (fn) is said to be uniformly cauchy on an interval i if for all � > 0 there exists n� such that |fn(x) −fm(x)| < � for all x ∈ i whenever n,m ≥ n� . a connection between a sequence of lebesgue integrable functions and the sequence of their primitives (indefinite integrals) is given via the following theorem (see [4] and [7]). theorem 1.1. let fn be a sequence of lebesgue integrable functions that converges point-wise to a function f on [a,b]. if the sequence fn(x) = ∫ x a fn is equi-absolutely continuous on [a,b], then f is lebesgue integrable and lim n−→∞ ∫ b a fn = ∫ b a f . (1.3) another result that will be needed in the coming sections is the so-called hake’s theorem (see [2]), which can be stated as follows. theorem 1.2. let f : [a,b] −→r be a function, then f is henstockkurzweil integrable on [a,b] if and only if f is henstockkurzweil integrable on [a,c] for all c ∈ [a,b) and lim c→b− ∫ c a f dt exists. (1.4) in this case ∫ b a f = lim c→b− ∫ c a f . (1.5) int. j. anal. appl. (2023), 21:49 3 2. the main results definition 2.1. φ([a,b]) =: {φ ∈ c1([a,b]) such that φ is monotonic and φ(b) = 0 6= φ′(b) } . in the above definition c1([a,b]) denotes the class of all continuously differentiable functions on [a,b]. we note that φ([a,b]) ⊆ bv ([a,b]) since φ consists of monotonic functions. therefore, the function φf is henstock-kurzwel integrable for all φ ∈ φ([a,b]) and f ∈ hk([a,b]), (see [11]). lemma 2.1. let fn be a sequence of hk-integrable functions that converges point-wise to a function f on [a,b]. if there exists φ ∈ φ such that ∫ x a φfn is equi-lipschitz continuous on [a,b], then i. φf ∈ l1([a,b]), and ∫b a φfn −→ ∫b a φf . ii. f ∈ l1([a,r]) and ∫ r a fn −→ ∫ r a f for all r ∈ [a,b). proof. it is clear that φfn converges point-wise to φf and the sequence ∫ r a φfn is equi-absolutely continuous on [a,b] since it is equi-lipschitz on [a,b] where m∑ k=1 | ∫ xk xk−1 φfn| = m∑ k=1 | ∫ xk a φfn − ∫ xk−1 a φfn| ≤ c m∑ k=1 |xk −xk−1|. therefore, (i) follows immediately from theorem 1.1. now since φ is continuous on [a,b] it attains its extreme values on any closed subinterval of [a,b]. thus, mr = min{|φ(t)| : t ∈ [a,r]} exists for all r ∈ [a,b]. moreover, since φ is a monotonic function and φ(b) = 0, then |φ(x)| is a decreasing function. also, since φ′(b) 6= 0, then φ is a non-constant function in [r,b] for all r ∈ [a,b). thus, there is s ∈ [r,b] such that φ(s) 6= φ(b) = 0 and hence, mr > 0 for all r ∈ [a,b) since |φ(t)| ≥ |φ(r)| ≥ |φ(s)| > 0 for all t ∈ [a,r]. using this result, we get∫ r a |f | = ∫ r a | φf φ | ≤ 1 mr ∫ r a |φf | for all r ∈ [a,b), and (2.1) | ∫ r a fn − f | = | ∫ r a φ(fn − f ) φ | ≤ 1 mr | ∫ r a φ(fn − f )| for all r ∈ [a,b). (2.2) therefore, we can obtain (ii) directly form (i), (2.1) and (2.2). � remark 2.1. if φf is lebesgue integrable on [a,b] for some f ∈ hk and φ ∈ φ, then f can only attain a point of lebesgue singularity at b (f oscillates very rapidly as approaching b). in fact, we would concentrate on the local case when f is lebesgue integrable on any subinterval [c,d] ⊂ [a,b] with d 6= b (see [4]). for the special case φ = b−x, the reader my refer to the results in [1]. in the coming sections we set f (r) = ∫ r a f and fn(r) = ∫ r a fn. lemma 2.2. let fn be a sequence of hk-integrable functions that converges point-wise to a function f on [a,b]. if there exists φ ∈ φ such that the sequence 1 φ(r) ∫b r fndφ(t) is uniformly cauchy on [a,b), then there are c > 0 and s ∈ [a,b) such that | 1 φ(r) ∫ b r fndφ(t)| ≤ c for all r ∈ (s,b) and for all n. (2.3) 4 int. j. anal. appl. (2023), 21:49 proof. since 1 φ(r) ∫b r fndφ(t) is uniformly cauchy on [a,b), then there is n1 such that∣∣∣∣ 1φ(r) ∫ b r (fn −fm)dφ(t) ∣∣∣∣ < 1 for all n,m ≥ n1. (2.4) therefore, ∣∣∣∣ ∫ b r (fn −fm)dφ(t) ∣∣∣∣ < |φ(r)| for all n,m ≥ n1. (2.5) also, since φ′ and fn are continuous we have that the function ∫ r a fndφ(t) is lipschitz . thus, for all n there is kn > 0 such that ∣∣∣∣ ∫ b r fndφ(t) ∣∣∣∣ ≤ kn(b− r). (2.6) choosing k = max{k1,k2, ...,kn1}, we get∣∣∣∣ ∫ b r fndφ(t) ∣∣∣∣ ≤ k(b− r) for all n ∈{1, 2, ...,n1}. (2.7) for the case n > n1, we have ∣∣∣∣ ∫ b r fndφ(t) ∣∣∣∣ ≤ ∣∣∣∣ ∫ b r (fn −fn1dφ(t) ∣∣∣∣ + ∣∣∣∣ ∫ b r fn1dφ(t) ∣∣∣∣ ≤ |φ(r)| + k(b− r) for all n ≥ n1 combining the above results, we get∣∣∣∣ ∫ b r fndφ(t) ∣∣∣∣ ≤ |φ(r)| + k(b− r) for all n. (2.8) now, using the fact that φ(b) = 0 6= φ′(b), we get lim r→b− b− r −φ(r) = 1 φ′(b) . (2.9) therefore, there exists s ∈ (a,b) such that (b− r) |φ(r)| ≤ 2 |φ′(b)| for all r ∈ (s,b). (2.10) letting ms = min{|φ(t)| : t ∈ [a,s]} be as defined in lemma 2.2, we obtain (b− r) |φ(r)| ≤ 2 |φ′(b)| + |b−a| ms for all r ∈ [a,b), (2.11) and hence, (b− r) ≤ ( 2 |φ′(b)| + |b−a| ms ) |φ(r)| for all r ∈ [a,b) (2.12) choosing c = 1 + k ( 2 |φ′(b)| + |b−a| ms ) and and using it with (2.8 ) and (2.12), we get the desired result. � theorem 2.1. let fn be a sequence of hk-integrable functions that converges point-wise to a function f on [a,b]. if there exists φ ∈ φ such that: int. j. anal. appl. (2023), 21:49 5 • the sequence 1 φ(r) ∫b r fndφ(t) is uniformly cauchy on [a,b), and • the sequence ∫b r φfndt is equi-lipschitz on [a,b], then f ∈ hk([a,b]) and lim n→∞ ∫ b a fndt = ∫ b a f dt. (2.13) proof. similar to the argument above and since ∫b r φfndt is equi-lipschitz, there exists m0 > 0 such that ∣∣∣∣ 1φ(r) ∫ b r φfndt ∣∣∣∣ ≤ m0 |b− r||φ(r)| for all r ∈ (a,b) and all n. (2.14) therefore, choosing s as in lemma 2.2 and c2 = 2m0 φ′(b) , we get∣∣∣∣ 1φ(r) ∫ b r φfndt ∣∣∣∣ ≤ c2 for all r ∈ (s,b) and all n. (2.15) now integrating by parts and using the fact fn(a) = 0, we get φ(r)fn(r) = ∫ r a φfndt + ∫ r a fndφ(t). (2.16) also, since φ(b) = 0, we have ∫ b a φfndt + ∫ b a fndφ(t) = 0, (2.17) which implies φ(r)fn(r) = − ∫ b r φfndt − ∫ b r fndφ(t) (2.18) and hence, fn(r) = − 1 φ(r) ∫ b r φfndt − 1 φ(r) ∫ b r fndφ(t). (2.19) applying lemma 2.2 and (2.15), we get |fn(r)| ≤ c (2.20) for all n and all r ∈ (s,b). thus, passing the limit as n −→∞, we get∣∣∣∣ limn→∞ ∫ r a fndt ∣∣∣∣ ≤ c. (2.21) applying lemma 2.1, we get ∣∣∣∣ ∫ r a f dt ∣∣∣∣ ≤ c for all r ∈ (s,b). (2.22) therefore, lim r→b− |φ(r)f (r)| ≤ lim r→b− c|φ(r)| = 0. (2.23) using this result with the equation φ(r)f (r) = ∫ r a φf dt + ∫ r a fdφ(t), (2.24) we get 6 int. j. anal. appl. (2023), 21:49 0 = ∫ b a φf dt + ∫ b a fdφ(t) (2.25) thus, in asimilar way to that for (2.19) we get f (r) = − 1 φ(r) ∫ b r φf dt − 1 φ(r) ∫ b r fdφ(t). (2.26) now, we use (2.19), (2.26) and lemma 2.2 to get lim n→∞ 1 φ(r) ∫ b r fndφ(t) = lim n→∞ − 1 φ(r) ∫ b r φfndt −fn(r) = − 1 φ(r) ∫ b r φf dt −f (r) = 1 φ(r) ∫ b r fdφ(t) combining the above point-wise convergence with the given that ( 1 φ(r) ∫b r fndφ(t) ) is uniformly cauchy, we obtain that lim n→∞ 1 φ(r) ∫ b r fndφ(t) = 1 φ(r) ∫ b r fdφ(t) uniformly (2.27) now, since φ′fn is continuous then by the fundamental theorem of calculus ∫ r a fndφ(t) = ∫ r a φ′fndt is differentiable and d dr ( ∫ r a fndφ(t)) = φ ′fn by l’hopital rule, we have lim r→b 1 φ(r) ∫ b r fndφ(t) = − lim r→b φ′(r) ·fn(r) φ′(r) = − lim r→b fn(r) (2.28) similarly, lim r→b 1 φ(r) ∫ b r fdφ(t) = − lim r→b f (r) (2.29) on the other hand, passing the limit in (2.20) as r −→ b and using theorem 1.2, we get |fn(b)| < c. (2.30) therefore, by the completeness of r lim nk→∞ ∫ b a fnkdt = a for some a ∈r (2.31) also, in view of (2.27) we have (see [10] theorem 7.11) lim r→b lim nk→∞ 1 φ(r) ∫ b r fnkdφ(t) = lim nk→∞ lim r→b 1 φ(r) ∫ b r fnkdφ(t). (2.32) int. j. anal. appl. (2023), 21:49 7 now, using (2.29), (2.27), (2.32), (2.28), theorem 1.2 and (2.31) respectively, we get lim r→b ∫ r a f dt = − lim r→b 1 φ(r) ∫ b r fdφ(t) = − lim r→b lim nk→∞ 1 φ(r) ∫ b r fnkdφ(t) = − lim nk→∞ lim r→b 1 φ(r) ∫ b r fnkdφ(t) = − lim nk→∞ lim r→b ∫ r a fnk = lim nk→∞ ∫ b a fnk = a. thus, by theorem 1.2 f is henstock-kurzwel integrable on [a,b] and∫ b a f dt = lim r→b− ∫ r a f dt = a (2.33) reversing the steps above and using the the result that f is henstock-kurzwel integrable, we obtain the convergence for the whole sequences as follows lim n→∞ ∫ b a fndt = − lim n→∞ lim r→b 1 φ(r) ∫ b r fndφ(t) = − lim r→b lim n→∞ 1 φ(r) ∫ b r fndφ(t) = − lim r→b 1 φ(r) ∫ b r fdφ(t) = ∫ b a f dt this completes the proof. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. alzubaidi, a complete normed space of a class of guage integrable functions, j. math. 2022 (2022), 2022/2354758. https://doi.org/10.1155/2022/2354758. [2] r.g. bartle, return to the riemann integral, amer. math. mon. 103 (1996), 625–632. https://doi.org/10. 1080/00029890.1996.12004798. [3] j. depree, c. swartz, introduction to real analysis, wily, new york, 1987. [4] r.a. gordon, the integrals of lebesgue, denjoy, perron, and henstock, american mathematical society, providence, r.i, 1994. [5] r. henstock, definitions of riemann type of the variational integrals, proc. london math. soc. s3-11 (1961), 402–418. https://doi.org/10.1112/plms/s3-11.1.402. [6] j. kurzweil, generalized ordinary differential equations and continuous dependence on a parameter, czechoslovak math. j. 82 (1957), 418-446. [7] h.l. royden, p.m. fitzpatrick, real analysis, 4th edition, pearson education, 2010. [8] c. swartz, norm convergence and uniform integrability for the henstock-kurzweil integral, real anal. exchange, 24 (1998/99), 423-426. [9] c. swartz, introduction to gauge integrals, world scientific, singapore, 2001. https://doi.org/10.1155/2022/2354758 https://doi.org/10.1080/00029890.1996.12004798 https://doi.org/10.1080/00029890.1996.12004798 https://doi.org/10.1112/plms/s3-11.1.402 8 int. j. anal. appl. (2023), 21:49 [10] w. rudin, principles of mathematical analysis, 3rd edition, mcgraw-hill, new york, 1976. [11] t.y. lee, henstock-kurzweil integration on euclidean spaces, world scientific, singapore, 2011. 1. introduction 2. the main results references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 198-206 http://www.etamaths.com an approach to the concept of soft vietoris topology i̇zzetti̇n demi̇r∗ abstract. in the present paper, we study the vietoris topology in the context of soft set. firstly, we investigate some aspects of first countability in the soft vietoris topology. then, we obtain some properties about its second countability. 1. introduction in 1999, molodtsov [22] initiated the concept of a soft set theory as a new approach for coping with uncertainties and also presented the basic results of the new theory. in [22], molodtsov successfully applied the soft set theory in several directions, such as smoothness of functions, game theory, operations research, riemann integration, perron integration and theory of measurement. after presentation of the operations of soft sets [21], the properties and applications of this theory have been studied increasingly ([4], [23], [25]). aktaş and çağman [3] introduced the soft group and also compared soft sets to fuzzy set and rough set. shabir and naz [29] initiated the study of soft topological spaces. rong [28] presented the notions of soft first countable and soft second countable spaces and investigated some their fundamental properties. recently, many papers concerning the soft set theory have been published ([7], [11], [14], [16], [18], [24], [26]). hyperspace theory had its early beginnings in the 1900’s, with the work of hausdorff and vietoris. this theory plays a fundamental role in mathematics and applied sciences, such as convex analysis, optimization, economics and image processing. over the years, a lot of research has been performed on this subject ([5], [6], [9], [17], [20]). as hyperspace of a topological space (x,τ), it means c(x), the set of closed subsets of x, equipped with a topology τh such that the function i : (x,τ) → (c(x),τh) defined by i(x) = {x} is a homeomorphism onto its image. one of the most important and well-studied hyperspace topologies on c(x) is the vietoris topology. the vietoris topology is a basic construct due to its usefulness in different areas of mathematics and applications. therefore, this topology has attracted the attention of many mathematicians in the last few decades ([8], [12], [13], [15], [19], [27], [32]). extensions of hypertopologies to the soft sets have been studied by some authors. akdağ and erol [1] and shakir [30] defined independently a hyperspace of soft sets, called soft vietoris topological space. later, akdağ and erol [2] studied on some hyperspaces of soft sets such as co-quasi h-closed soft topological spaces and d-soft topological spaces. in this paper, firstly we present a brief synopsis of all necessary definitions and results that will be required. next, we continue studying the soft vietoris topology and obtain some results about its first and second countability. 2. preliminaries in this section, we recollect some basic notions regarding soft sets. throughout this work, let x be an initial universe, p(x) be the power set of x and e be a set of parameters for x, 2010 mathematics subject classification. 06d72, 54a40, 54b20. key words and phrases. soft set; soft vietoris topological space; soft first countable space; soft second countable space; soft separable space. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 198 an approach to the concept of soft vietoris topology 199 definition 2.1 ([22]). a soft set f on the universe x with the set e of parameters is defined by the set of ordered pairs f = {(e,f(e)) : e ∈ e,f(e) ∈ p(x)} where f is a mapping given by f : e → p(x). throughout this paper, the family of all soft sets over x is denoted by s(x,e) [7]. definition 2.2 ([4], [21], [25]). let f,g ∈ s(x,e). then, (i) the soft set f is called a null soft set, denoted by ∅̃, if f(e) = ∅ for every e ∈ e. (ii) the soft set f is called an absolute soft set, denoted by x̃, if f(e) = x for every e ∈ e. (iii) f is a soft subset of g if f(e) ⊆ g(e) for every e ∈ e. it is denoted by f v g. (iv) the complement of f is denoted by fc, where fc : e → p(x) is a mapping defined by fc(e) = x −f(e) for every e ∈ e. clearly, (fc)c = f. (v) the union of f and g is a soft set h defined by h(e) = f(e) ∪ g(e) for every e ∈ e. h is denoted by f tg. (vi) the intersection of f and g is a soft set h defined by h(e) = f(e)∩g(e) for every e ∈ e. h is denoted by f ug. (vii) the difference of f and g is a soft set h defined by h(e) = f(e) −g(e) for every e ∈ e. h is denoted by f ^ g. definition 2.3 ([10], [19], [24]). a soft set p over x is said to be a soft point if there exists an e ∈ e such that p(e) = {x} for some x ∈ x and p(e′) = ∅ for every e′ ∈ e\{e}. this soft point is denoted as xe. a soft point xe is said to belongs to a soft set f , denoted by xe ∈̃f , if x ∈ f(e). from now on, let sp(x) be the family of all soft points over x. definition 2.4 ([29]). let τ be a collection of soft sets over x, then τ is said to be a soft topology on x if (st1) ∅̃,x̃ belong to τ. (st2) the union of any number of soft sets in τ belongs to τ. (st3) the intersection of any two soft sets in τ belongs to τ (x,τ,e) is called a soft topological space. the members of τ are called soft open sets in x. a soft set f over x is called a soft closed in x if fc ∈ τ. definition 2.5 ([7], [24]). let (x,τ,e) be a soft topological space. a subcollection b of τ is called a soft base for τ if every member of τ can be expressed as the union of some members of b. definition 2.6. let (x,τ,e) be a soft topological space and f ∈ s(x,e). (i) the soft interior of f is the soft set fo = t{g : g is soft open set and g v f} [31]. (ii) the soft closure of f is the soft set f = u{g : g is soft closed set and f v g} [29]. definition 2.7 ([24]). a soft set f in a soft topological space (x,τ,e) is called a soft neighborhood of the soft point xe if there exists a soft open set g such that xe ∈̃ g v f . the soft neighborhood system of a soft point xe, denoted by n(xe), is the family of all its soft neighborhoods. definition 2.8 ([29]). let (x,τ,e) be a soft topological space and y be a non-empty subset of x. then, τy = {ỹ u f : f ∈ τ} is called the soft relative topology on y and (y,τy ,e) is called a soft subspace of (x,τ,e). 200 demi̇r here, ỹ is the soft set over x defined by ỹ (e) = y for all e ∈ e. definition 2.9 ([28]). let (x,τ,e) be a soft topological space and xe ∈̃x̃. a subcollection b of τ is called a soft local base at a soft point xe if for every soft open set f containing xe, there exists a g ∈b such that xe ∈̃g v f . definition 2.10 ([11]). let (x,τ,e) be a soft topological space, {xenn : n ∈ n} be a sequence of soft points in (x,τ,e) and xe ∈ sp(x). the sequence {xenn : n ∈ n} is said to converge to xe, and we write xenn → xe, if for every f ∈n(xe), there exists an n0 ∈ n such that xenn ∈̃ f for all n ≥ n0. definition 2.11 ([24]). let (x,τ,e) be a soft topological space. a soft point xe ∈ sp(x) is called a limiting soft point of a soft set f over x if every soft open set containing xe contains at least one soft point of f other than xe, i.e., if f u (g ^ xe) 6= ∅̃ for every g ∈ τ containing xe. the union of all limiting soft points of f is called the derived soft set of f and is denoted by f ′. definition 2.12 ([28]). let (x,τ,e) be a soft topological space. (i) if each soft point in x has a countable soft local base, then it is called a soft first countable space. (ii) if there exists a countable soft base for τ, then it is a soft second countable space. definition 2.13 ([28]). let (x,τ,e) be a soft topological space. if there exists a family {xenn : n ∈ n} of countable many soft points in x such that ⊔ n∈n x en n = x̃, then (x,τ,e) is called a soft separable space. definition 2.14. let (x,τ,e) be a soft topological space. then, (i) it is called a soft t1-space if every soft point in x is a soft closed set [18]. (ii) it is called a soft hausdorff space or a soft t2-space if for any two distinct soft points x e1 1 , xe22 ∈ sp(x) there exist soft open sets f and g such that x e1 1 ∈̃f,x e2 2 ∈̃g and f ug = ∅̃ [11]. (iii) it is called a soft regular space if for every xe ∈̃x̃ and every soft closed set f such that xe /̃∈f , there exist soft open sets f1 and f2 such that x e ∈̃f1, f v f2 and f1 uf2 = ∅̃ [16]. theorem 2.15 ([16]). let (x,τ,e) be a soft topological space. then, the following statements are equivalent: (1) (x,τ,e) is a soft regular space. (2) for any soft open set f in (x,τ,e) and xe ∈̃f , there exists a soft open set g containing xe such that xe ∈̃g v f . (3) for any soft closed set h in (x,τ,e) and xe /̃∈h, there exists a soft open set g containing xe such that guh = ∅̃. definition 2.16 ([7], [31]). let (x,τ,e) be a soft topological space and f ∈ s(x,e). (i) a family c = {fi : i ∈ i} of soft sets over x is called a cover of f if it satisfies f v ⊔ i∈i fi. it is called a soft open cover if each member of c is a soft open set. a subfamily of c is called a subcover of c if it is also a cover of f . (ii) (x,τ,e) is called a soft compact space if every soft open cover of x̃ has a finite subcover. theorem 2.17 ([26]). let (y,τy ,e) be a soft subspace of a soft topological space (x,τ,e). then, (y,τy ,e) is a soft compact space if and only if every cover of ỹ by soft open sets over x contains a finite subcover. theorem 2.18 ([26]). every soft compact subspace of a soft hausdorff space is soft closed. an approach to the concept of soft vietoris topology 201 definition 2.19 ([1]). let (x,τ,e) be a soft topological space and f ∈ τ. then, the families of soft sets f+ and f− are defined as follows: f+ = {g ∈ sc(x) : g v f} and f− = {g ∈ sc(x) : f ug 6= ∅̃}, where sc(x) is the family of non-null soft closed sets over x. proposition 2.20 ([1]). let (x,τ,e) be a soft topological space. for non-null soft sets f and g, the following statements are true: (i) f+ ∩g+ = (f ug)+. (ii) f+ ∪g+ ⊆ (f tg)+. (iii) (f ug)− ⊆ f− ∩g−. (iv) f− ∪g− = (f tg)−. (v) f v g if and only if f+ ⊆ g+. (vi) f v g if and only if f− ⊆ g−. proposition 2.21 ([1]). let (x,τ,e) be a soft topological space. then, the families δ+sv = {f + : f ∈ τ} and δ−sv = {f − : f ∈ τ} are subbases for the topological spaces τ+sv and τ − sv on sc(x), respectively. definition 2.22 ([1]). the topological spaces τ+sv and τ − sv on sc(x) which mentioned in above proposition are called a soft upper vietoris topological space and a soft lower vietoris topological space, respectively. definition 2.23 ([1]). a topological space on sc(x) with δsv = δ + sv ∪δ − sv as subbase is called a soft vietoris topological space, denoted by τsv . 3. first and second countability of the soft vietoris topology the aim of this section is to present some properties related to countability of soft vietoris topology. firstly, we focus on its first countability. theorem 3.1. let (x,τ,e) be a soft t1-space. then, the following statements are equivalent: (1) (sc(x),τsv ) is a first countable space. (2) (sc(x),τ+sv ) and (sc(x),τ − sv ) are first countable spaces. proof. (2) ⇒ (1) is obvious from definition 2.22 and 2.23. to prove (1) ⇒ (2), let (sc(x),τsv ) be a first countable space and take h ∈ sc(x). let l = { ` = ⋂ i∈i f+i ∩ ⋂ j∈j g−j : i,j is finite } be a countable local base at h for τsv . then, the family l+ = {⋂ i∈i f+i : ⋂ i∈i f+i occurs in some ` ∈l } ∪{sc(x)} forms a countable local base at h for τ+sv . indeed, if there is no ⋂ i∈i f + i ∈ ` with h ∈ ⋂ i∈i f + i , then sc(x) is the only open set in τ+sv containing h. if h ∈ f + and f+ ∈ τ+sv , where f 6= x̃, then there exists an ` ∈l such that h ∈ ` ⊆ f+. therefore, ` must be of the form ⋂ i∈i f + i ∩ ⋂ j∈j g − j where i is nonempty. suppose that i = ∅. then, since h ∈ ⋂ j∈j g − j , we obtain (h tf c) ∈ ⋂ j∈j g − j . also, we know that (h tfc) /∈ f+. but this contradicts the fact that ⋂ j∈j g − j ⊆ f +. next, observe that we have fc v ⊔ i∈i f c i , since otherwise, there would be an x e ∈̃x̃ such that xe ∈̃fc, xe /̃∈ ⊔ i∈i f c i and (h txe) ∈ `−f+. thus, by proposition 2.20 (i) and (v), we obtain ⋂ i∈i f + i ⊆ f +. now, we shall show that there exists a countable local base at h for τ−sv . let h ∈ g − and 202 demi̇r g− ∈ τ−sv , where g 6= x̃. since g − ∈ τsv , there exists an ` ∈l such that h ∈ ` ⊆ g−. without loss of generality we can suppose that in the expression of every element from l the family j is nonempty( for example, ⋂ i∈i f + i = ⋂ i∈i f + i ∩ x̃ − ) . so, we have ` = ⋂ i∈i f + i ∩ ⋂ j∈j g − j . for each j ∈ j, let us take a soft set kj = gj ^ ⊔ i∈i f c i and define k` = ⋂ j∈j k − j . it easy to see that h ∈ k`. then, l− = { k` : ` ∈l } forms a countable local base at h for τ−sv . indeed, since there exists a j ∈ j with kj v g, we obtain k` ⊆ g−. suppose that for each j ∈ j there exists an x ej j ∈̃kj ^ g. therefore, we get ⊔ j∈j x ej j ∈ `−g −, which yields a contradiction. theorem 3.2. let (x,τ,e) be a soft t1-space. then, (sc(x),τ−sv ) is a first countable space if and only if (x,τ,e) is a soft first countable space and each soft closed set over x is a soft separable. proof. let (sc(x),τ−sv ) be a first countable space. from the fact that each soft point in x is a soft closed set it follows that (x,τ,e) is a soft first countable space. let h ∈ sc(x) and {fn : n ∈ n} be a countable family of nonempty soft open sets which determines a countable local base at h for τ−sv . because h u fn 6= ∅̃ for each n ∈ n, we may choose a soft point xenn ∈̃h u fn. now, we shall show that ⊔ n∈n x en n = h. it is easy to see that ⊔ n∈n x en n v h. let xe ∈̃h. for each g ∈ τ containing xe, we have guh 6= ∅̃. then, there exist n1, ...,nk ∈ n such that f−n1 ∩ ...∩f − nk ⊆ g−. therefore, we obtain fnj v g for some j ∈{1, ...,k}. thus, since x ej j ∈̃fnj v g, we get gu (⊔ n∈n x en n ) 6= ∅̃ and so that xe ∈̃ ⊔ n∈n x en n . on the other hand, let h ∈ sc(x). then, by hypothesis, there exists a family {xenn : n ∈ n} of countable many soft points in h such that ⊔ n∈n x en n = h. now, let b(xenn ) be a countable soft local base at xenn for each n ∈ n. thus, one can readily verify that b(h) = {⋂ j∈j g − j : gj ∈ b(xejj ),j isfinite } is a countable local base at h for τ−sv . for each h ∈ sc(x), we define h∗ = {f ∈ sc(x) : f u h = ∅̃}. then, we get the following theorem. theorem 3.3. let (x,τ,e) be a soft t1-space. then, (sc(x),τ+sv ) is a first countable space if and only if for each h ∈ sc(x), there exists a countable family lh ⊆ h∗ such that for each f ∈ h∗ there exist f1,f2, ...,fn ∈lh with f v f1 tf2 t ...tfn. proof. the sufficiency is clear. to prove necessity, let (sc(x),τ+sv ) be a first countable space and let h ∈ sc(x). then, the family h = {⋂ i∈i f+i : f c i ∈ h ∗,i isfinite } ∪{sc(x)} is a countable local base at h for τ+sv . now, put lh = { g ∈ h∗ : goccursinsomeelementof h } . it is easy to see that lh is a countable family of h∗. let f ∈ h∗. then, we obtain h ∈ (fc)+ and (fc)+ ∈ τ+sv . therefore, there exists a member ⋂n i=1 f + i of h such that h ∈ ⋂n i=1 f + i ⊆ (f c)+. by proposition 2.20 (i) and (v), we have f v fc1 t ...tfcn. thus, it follows from fci ∈ lh for each i ∈{1, ...,n} that the family lh has the required properties. definition 3.4. let (x,τ,e) be a soft topological space, f ∈ s(x,e) and let f = { h : h v f } be a family of non-null soft closed sets. then, f is called a hemi-sc(x) if there exists a countable cofinal subfamily of f with respect to inclusion relation. an approach to the concept of soft vietoris topology 203 example 3.5. let x = {x,y,z}, e = {e1,e2} and let f be a soft set over x, where f = {(e1,{x,z}), (e2,{y})}. consider a discrete soft topology τ on x and a family f = { xe1,ze1,ye2,{(e1,{x}), (e2,{y})},{(e1,{z}), (e2,{y})},{(e1,{x,z})},f } of non-null soft closed sets contained in f . then, a subfamily g = {f}⊂f is cofinal in f since there exists an f ∈g such that h v f for every h ∈f. thus, f is a hemi-sc(x). definition 3.6. the character of a soft set f ∈ s(x,e) in a soft topological space (x,τ,e) is defined as the smallest cardinal number of the form |b(f)|, where b(f) is a soft local base at f for τ. this cardinal number is denoted by χ(f). example 3.7. let x = {x,y,z}, e = {e1,e2} and let f be a soft set over x, where f = {(e1,{z}), (e2,{x,y})}. let us define a soft topology on x as the following: τ = {g : xe1 ∈̃g}∪{∅̃}. then, b(f) = {h} is a soft local base at f for τ, where h = {(e1,{x,z}), (e2,{x,y})}. thus, the character of f is χ(f) = 1. using the above definitions and theorems, we can easily prove the following corollaries. corollary 3.8. let (x,τ,e) be a soft t1-space. then, the following statements are satisfied: (1) (sc(x),τ+sv ) is a first countable space if and only if each soft open set f , where f 6= x̃, is a hemi-sc(x). (2) (sc(x),τ+sv ) is a first countable space if and only if χ(f) ≤ |n| for each f ∈ sc(x), where |n| denotes the cardinal number of n. corollary 3.9. let (x,τ,e) be a soft t1-space. then, (sc(x),τsv ) is a first countable space if and only if the following three conditions hold: (i) (x,τ,e) is a soft first countable space. (ii) each soft closed set over x is a separable. (iii) each soft open set over x is a hemi-sc(x). lemma 3.10. let (x,τ,e) be a soft topological space and let {xenn : n ∈ n} be a sequence of soft points in x. if xenn → xe ∈ sp(x), then f = ⊔ n∈n x en n txe is a soft compact set. proof. let c = {fi : i ∈ i} be a cover of f by soft open sets over x. then, there exists an i0 ∈ i such that xe ∈̃fi0 . since xenn → xe, there exists an n0 ∈ n such that xenn ∈̃fi0 for all n ≥ n0. now, let us take an fn ∈c satisfying xenn ∈̃fn for all n < n0. therefore, we have xenn ∈̃ ⊔n0−1 i=1 fi for all n < n0 and so that f v fi0 t ⊔n0−1 i=1 fi. hence, the family {fi0}∪{fi : i = 1, ...,n0 − 1} is a finite subcover of c. thus, from theorem 2.17 it follows that f is a compact set. theorem 3.11. let (x,τ,e) be a soft hausdorff space. if (sc(x),τ+sv ) is a first countable space, then (x,τ,e) is a soft regular space. proof. let f be a soft open set and xe ∈̃f. by theorem 2.15, we shall show that there exists a soft open set g such that xe ∈̃g v g v f. without loss of generality we can suppose that f 6= x̃. the first countability of (sc(x),τ+sv ) implies that f is a hemi-sc(x). let {f1,f2, ...,fn, ...} be a countable cofinal subfamily in the family {h ∈ sc(x) : h v f}. also, one can easily verify that (x,τ,e) is a soft first countable space. let us denote by {un : n ∈ n} a countable soft local base at xe with un v f for each n ∈ n. now, we claim that there exists an n ∈ n such that xe ∈̃fon. indeed, suppose that xe /̃∈fon for each n ∈ n. then, there exists a soft point xenn ∈̃un ^ fn for each n ∈ n. therefore, we see that the sequence {xenn : n ∈ n} converges to xe. from lemma 3.10 it follows that l = ⊔ n∈n x en n t xe is a soft compact set. also, we have l v f and by theorem 2.18, we obtain 204 demi̇r l ∈ sc(x). hence, using the cofinality condition, we get l v fn for some n ∈ n, which yields a contradiction. thus, there exists an n ∈ n such that xe ∈̃fon v fon v f. definition 3.12. let (x,τ,e) be a soft topological space and {xeii : i ∈ i} a family of infinitely soft points in x. (x,τ,e) is called a countably soft compact space if the soft set ⊔ i∈i x ei i has a limiting soft point. theorem 3.13. let (x,τ,e) be a soft hausdorff space. if (sc(x),τ+sv ) is a first countable space, then the derived soft set x̃′ of x̃ is a countably soft compact. proof. suppose that x̃′ is not countably soft compact. then, there exists a family {xenn : n ∈ n} of countable many soft points in x̃′ such that f = ⊔ n∈n x en n does not have a limiting soft point. by theorem 3.11, (x,τ,e) is a soft regular space and therefore there exists a pairwise disjoint countable family of soft open sets {un : n ∈ n} such that xenn ∈̃un for every n ∈ n. also, from corollary 3.8(2) it follows that there exists a countable soft local base {gn : n ∈ n} of soft open sets at f. now, for every n ∈ n, we can choose a yknn ∈ sp(x) with yknn ∈̃(un ugn) ^ f because xenn is a limiting soft point of x̃. let g = ⊔ n∈n y kn n . then, we have f ug = ∅̃. indeed, suppose that there exists a soft point x en0 n0 such that x en0 n0 ∈̃f and x en0 n0 ∈̃g. take a soft set h = un0 ^ y kn0 n0 . since the family {un : n ∈ n} is pairwise disjoint, h is a soft open neighborhood of x en0 n0 such that h ug = ∅̃. this is a contradiction since x en0 n0 ∈̃g. hence, since f ug = ∅̃, there exists a gn such that f v gn v g c . but, this is a contradiction to the fact that yknn ∈̃g. thus, x̃′ is a countably soft compact. we now consider the second countability of the soft vietoris topology. theorem 3.14. let (x,τ,e) be a soft topological space. then, the following statements are equivalent: (1) (sc(x),τ−sv ) is a second countable space. (2) (x,τ,e) is a soft second countable space. proof. it is clear from proposition 2.20 (iii)-(iv) and (vi). theorem 3.15. let (x,τ,e) be a soft t1-space. then, the following statements are equivalent: (1) (sc(x),τsv ) is a second countable space. (2) (sc(x),τ+sv ) and (sc(x),τ − sv ) are second countable spaces. proof. (2) ⇒ (1) follows immediately from definition 2.22 and 2.23. to prove (1) ⇒ (2), let (sc(x),τsv ) be a second countable space. from the fact that every soft point in x is a soft closed set it follows that (x,τ,e) is a soft second countable space. hence, by theorem 3.14, (sc(x),τ−sv ) is a second countable space. let l = { ` = ⋂ i∈i f+i ∩ ⋂ j∈j g−j : i,j is finite } be a countable base for τsv . then, the family l+ = {⋂ i∈i f+i : ⋂ i∈i f+i occurs in some ` ∈l } ∪{sc(x)} forms a countable base for τ+sv . indeed, if there is no ⋂ i∈i f + i ∈ ` with h ∈ ⋂ i∈i f + i , then sc(x) is the only open set in τ+sv containing h. let h ∈ f + and f+ ∈ τ+sv , where f 6= x̃. since f + ∈ τsv , there exists an ` ∈ l such that h ∈ ` ⊆ f+. therefore, ` must be of the form ⋂ i∈i f + i ∩ ⋂ j∈j g − j where i is nonempty (see the proof of theorem 3.1). thus, as is shown in the proof of theorem 3.1, an approach to the concept of soft vietoris topology 205 we get ⋂ i∈i f + i ⊆ f +, which completes the proof. theorem 3.16. let (x,τ,e) be a soft t1-space. then, (sc(x),τ+sv ) is a second countable space if and only if there exists a countable family ∆ ⊆ sc(x) such that for each f ∈ sc(x) and for each g ∈ τ satisfying f v g there exist f1,f2, ...,fn ∈ ∆ with f v f1 tf2 t ...tfn v g. proof. let l be a countable base for τ+sv . we know that every element in l can be written as⋂{ h+j : j ∈ j } , where j is a finite set. take ∆ = { f ∈ sc(x) : f occurs in the presentation of some element from l } . one can readily verify that ∆ is a countable family of sc(x). let f ∈ sc(x) and g ∈ τ such that f v g. if f = x̃, then we are done. so, suppose that f 6= x̃ and also g 6= x̃. now, let us take a soft set k = gc. then, we have k ∈ sc(x) and k ∈ (fc)+. therefore, there exists a member ⋂ j∈j h + j of l such that k ∈ ⋂ j∈j h + j ⊆ (f c)+. hence, from proposition 2.20 (i) and (v) it follows that f v ⊔ j∈j hcj v k c = g. thus, since hcj ∈ ∆ for each j ∈ j, the family ∆ ⊆ sc(x) has the required properties. for the converse, the family of sets of the form ⋂{ (fcj ) + : j ∈ j } , where fj ∈ ∆ and j is a finite set, together with sc(x) is a countable base for τ+sv . references [1] m. akdağ and f. erol, on hyperspaces of soft sets, j. new theory 7 (2015), 86-97. [2] m. akdağ and f. erol, remarks on hyperspaces of soft sets, j. adv. stud. topol. 7 (2016), 1-11. [3] h. aktaş and n. çağman, soft sets and soft groups, inform. sci. 177 (2007), 2726-2735. [4] m.i. ali, f. feng, x. liu, w.k. min and m. shabir, on some new operations in soft set theory, comput. math. appl. 57 (2009), 1547-1553. [5] m. alimohammady, s. jafari, s.p. moshokoa and m.k. kalleji, a note on properties of hypermetric spaces, j. hyperstructures 3(2) (2014), 89-100. [6] d. andrijevic, m. jelic and m. mrsevic, some properties of hyperspaces of cech closure spaces, filomat 24 (2010), 53-61. [7] a. aygünoğlu and h. aygün, some notes on soft topological spaces, neural comput. appl. 21 (2012), 113–119. [8] a. bouziad, l. hola and l. zsilinszky, on hereditary baireness of the vietoris topology, topology appl. 115 (2001), 247-258. [9] j. chvalina, s. h. mayerova and a. d. nezhad, general actions of hyperstructures and some applications, an. st. univ. ovidius constanta 21 (2013), 59-82. [10] s. das and s.k. samanta, soft metric, ann. fuzzy math. inform. 6(1) (2013), 77–94. [11] i̇. demir and o.b. özbakır, soft hausdorff spaces and their some properties, ann. fuzzy math. inform. 8(5) (2014), 769-783. [12] g. di maio and l. hola, on hit-and-miss hyperspace topologies, rend. accad. sci. fis. mat. napoli 62 (1995), 103-124. [13] a. gavrilut and m. agop, approximation theorems for set multifunctions in vietoris topology. physical implications of regularity, iran. j. fuzzy syst. 12(1) (2013), 27-42. [14] a.ç. güler, e. d. yıldırım and o. b. özbakır, a fixed point theorem on soft g-metric spaces, j. nonlinear sci. appl. 9 (2016), 885-894. [15] l. hola and s. levi, decomposition properties of hyperspace topologies, set-valued anal. 5 (1997), 309-321. [16] s. hussain and b. ahmad, soft separation axioms in soft topological spaces, hacet. j. math. stat. 44 (3) (2015), 559-568. [17] w. kai-ting, a new fixed point theorem in noncompact hyperconvex metric spaces and its application to saddle point problems, j. math. res. expos. 28 (2008), 161-168. [18] a. kandil, o.a. tantawy, s.a. el-sheikh and a. zakaria, new structures of proximity spaces, inf. sci. lett. 3(3) (2014), 85-89. [19] f. lin, the vietoris topology on rectifiable spaces, semigroup forum 88 (2014), 273-278. [20] x. ma, b. huo and g. liao, chaos in hyperspace system, chaos solitons fractals 40 (2009), 653-660. [21] p.k. maji, r. biswas and a.r. roy, soft set theory, comput. math. appl. 45 (2003), 555-562. [22] d. molodtsov, soft set theory-first results, comput. math. appl. 37 (1999), 19–31. 206 demi̇r [23] d. molodtsov, v.y. leonov and d.v. kovkov, soft sets technique and its application, nechetkie sist. myagkie vychisl. 1 (2006), 8-39. [24] s. nazmul and s.k. samanta, neighbourhood properties of soft topological spaces, ann. fuzzy math. inform. 6(1) (2013), 1–15. [25] d. pei and d. miao, from soft sets to information systems, in: hu x, liu q, skowron a, lin ty, yager rr, zhang b, editors. proceedings of granular computing-ieee 2 (2005), 617-621. [26] e. peyghan, b. samadi and a. tayebi, some results related to soft topological spaces, facta universitatis ser. math. inform. 29(4) (2014), 325-336. [27] j. rodriguez-lopez and s. romaguera, the relationship between the vietoris topology and the hausdorff quasiuniformity, topology appl. 124 (2002), 451-464. [28] w. rong, the countabilities of soft topological spaces, int. j. comput. math. sci. 6 (2012), 159–162. [29] m. shabir and m. naz, on soft topological spaces, comput. math. appl. 61 (2011), 1786–1799. [30] q. r. shakir, on vietoris soft topology i, j. sci. res. 8(1) (2016), 13-19. [31] i. zorlutuna, m. akdag, w.k. min and s. atmaca, remarks on soft topological spaces, ann. fuzzy math. inform. 3(2) (2012), 171–185. [32] l. zsilinszky, baire spaces and hyperspace topologies, proc. amer. math. soc. 124 (1996), 2575-2584. department of mathematics, duzce university, 81620, duzce-turkey ∗corresponding author: izzettindemir@duzce.edu.tr international journal of analysis and applications volume 16, number 3 (2018), 374-399 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-374 majorization inequalities via peano’s representation of hermite’s polynomial n. latif1,∗, n. siddique2 and j. pečarić3,4 1,∗department of general studies, jubail industrial college, jubail industrial city 31961, kingdom of saudi arabia 2department of mathematics, govt. college university, faisalabad 38000, pakistan 3faculty of textile technology zagreb, university of zagreb, prilaz baruna filipovića 28a, 10000 zagreb, croatia 4rudn university, 6 miklukho-maklay st, moscow, 117198, russia ∗corresponding author: naveed707@gmail.com abstract. the peano’s representation of hermite polynomial and new green functions are used to construct the identities related to the generalization of majorization type inequalities in discrete as well as continuous case. čebyšev functional is used to find the bounds for new generalized identities and to develop the grüss and ostrowski type inequalities. further more, we present exponential convexity together with cauchy means for linear functionals associated with the obtained inequalities and give some applications. received 2018-01-05; accepted 2018-02-20; published 2018-05-02. 2010 mathematics subject classification. 26d07, 26d15, 26d20, 26d99. key words and phrases. classical majorization theorem; fuchs’s thorem; peano’s representation of hermite’s polynomial; green function for ’two point right focal’ problem; čebyšev functional; grüss type upper bounds; ostrowski-type bounds; n-exponentially convex function; mean value theorems; stolarsky type means. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 374 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-374 int. j. anal. appl. 16 (3) (2018) 375 1. introduction and preliminaries newton and lagrange gave the classical methods for constructing hermite interpolating polynomial. lagrange gave the method for such function f(t) is defined at the distinct increasing points a1,a2, ...,an but newton gave the method for such function f(t) is defined at the distinct (but not necessarily increasing) points a1,a2, ...,an (see [3, 7]). we start with a brief overview of divided differences and n-convex functions and give some basic results from the majorization theory. an nth order divided difference of a function φ : [α,β] → r at distinct points x0,x1, ...,xn ∈ [α,β] may be defined recursively by [xi; φ] = φ(xi), i = 0, ...,n, [x0, ...,xn; φ] = [x1, ...,xn; φ] − [x0, ...,xn−1; φ] xn −x0 . the value [x0, ...,xn; φ] is independent of the order of the points x0, ...,xn. a function φ is n-convex on [α,β] if [x0,x1, ...,xn; φ] ≥ 0 holds for all choices of (n + 1) distinct points xi ∈ [α,β], i = 0, ...,n. remark 1.1. from this definition it follows that 1-convex function is increasing function and 2-convex function is just convex function. if φ(n) exists, then φ is n-convex iff φ(n) ≥ 0. also, if φ is n-convex for n ≥ 2, then φ(k) exists and φ is (n−k)-convex for 1 ≤ k ≤ n− 2. for more informations see [13]. on the basis of various applications of the divided differences, several representations have been obtained like error representation, cauchy’s representation, newton’s representation and peano’s representation. in this paper, we give the generalized results with the connection of peano’s representation of hermite’s interpolating polynomial and newly defined green functions. majorization makes precise the vague notion that the components of a vector y are ”less spread out” or ”more nearly equal” than the components of a vector x. a complete and superb reference on the subject is the 2011 book by marshall et al. [12]. for fixed m ≥ 2 let x = (x1, ...,xm) , y = (y1, ...,ym) denote two real m-tuples. let x[1] ≥ x[2] ≥ ... ≥ x[m], y[1] ≥ y[2] ≥ ... ≥ y[m], x(1) ≤ x(2) ≤ ... ≤ x(m), y(1) ≤ y(2) ≤ ... ≤ y(m) be their ordered components. int. j. anal. appl. 16 (3) (2018) 376 definition 1.1. [13, p. 319] x is said to majorize y (or y is said to be majorized by x), in symbol, x � y, if l∑ i=1 y[i] ≤ l∑ i=1 x[i] (1.1) holds for l = 1, 2, ...,m− 1 and m∑ i=1 xi = m∑ i=1 yi. note that (1.1) is equivalent to m∑ i=m−l+1 y(i) ≤ m∑ i=m−l+1 x(i) holds for l = 1, 2, ...,m− 1. the following theorem is well-known as the majorization theorem given by marshall et al. [12, p. 14] (see also [13, p. 320]): theorem 1.1. let x = (x1, ...,xm) ,y = (y1, ...,ym) be two m-tuples such that xi, yi ∈ [α,β] (i = 1, ...,m). then m∑ i=1 f (yi) ≤ m∑ i=1 f (xi) (1.2) holds for every continuous convex function f : [α,β] → r if and only if x � y holds. the following theorem can be regarded as a weighted version of theorem 1.1 and is proved by fuchs in [8] ( [12, p. 580], [13, p. 323]): theorem 1.2. let x = (x1, ...,xm) ,y = (y1, ...,ym) be two decreasing real m-tuples with xi, yi ∈ [α,β] (i = 1, ...,m) and w = (w1,w2, ...,wm) be a real m-tuple such that l∑ i=1 wi yi ≤ l∑ i=1 wi xi for l = 1, ...,m− 1, (1.3) and m∑ i=1 wi yi = m∑ i=1 wi xi. (1.4) then for every continuous convex function f : [α,β] → r, we have m∑ i=1 wi f (yi) ≤ m∑ i=1 wi f (xi) . (1.5) the following integral version of theorem 1.2 is a simple consequence of theorem 12.14 in [15] (see also [13, p.328]): int. j. anal. appl. 16 (3) (2018) 377 theorem 1.3. let x,y : [a,b] → [α,β] be decreasing and w : [a,b] → r be continuous functions. if∫ ν a w(t) y(t) dt ≤ ∫ ν a w(t) x(t) dt for every ν ∈ [a,b], (1.6) and ∫ b a w(t) y(t) dt = ∫ b a w(t) x(t) dt (1.7) hold, then for every continuous convex function f : [α,β] → r, we have∫ b a w(t) f (y(t)) dt ≤ ∫ b a w(t) f (x(t)) dt. (1.8) let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points. for f ∈ cn[α,β] a unique polynomial ρh(s) of degree (n− 1) exists satisfying any of the following conditions: hermite conditions: ρ (i) h (aj) = f (i)(aj); 0 ≤ i ≤ kj, 1 ≤ j ≤ r, r∑ j=1 kj + r = n. (h) it is of great interest to note that hermite conditions include the following particular cases: type (m,n−m) conditions: (r = 2, 1 ≤ m ≤ n− 1, k1 = m− 1, k2 = n−m− 1) ρ (i) (m,n) (α) = f(i)(α), 0 ≤ i ≤ m− 1, ρ (i) (m,n) (β) = f(i)(β), 0 ≤ i ≤ n−m− 1, two-point taylor conditions: (n = 2m, r = 2, k1 = k2 = m− 1) ρ (i) 2t (α) = f (i)(α), ρ (i) 2t (β) = f (i)(β), 0 ≤ i ≤ m− 1. we have the following result from [3]. theorem 1.4. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]). then we have f(t) = ρh(t) + rh,n(f,t) (1.9) where ρh(t) is the hermite interpolating polynomial, i.e. ρh(t) = r∑ j=1 kj∑ i=0 hij(t)f (i)(aj); the hij are fundamental polynomials of the hermite basis defined by hij(t) = 1 i! ω(t) (t−aj) kj+1−i kj−i∑ k=0 1 k! dk dtk ( (t−aj) kj+1 ω(t) )∣∣∣∣∣ t=aj (t−aj) k , (1.10) int. j. anal. appl. 16 (3) (2018) 378 ω(t) = r∏ j=1 (t−aj) kj+1, (1.11) and the remainder is given by rh,n(f,t) = ∫ β α gh,n(t,s)f (n)(s)ds where gh,n(t,s) is defined by gh,n(t,s) =   l∑ j=1 kj∑ i=0 (aj−s)n−i−1 (n−i−1)! hij(t); s ≤ t, − r∑ j=l+1 kj∑ i=0 (aj−s)n−i−1 (n−i−1)! hij(t); s ≥ t, (1.12) for all al ≤ s ≤ al+1; l = 0, . . . ,r with a0 = α and ar+1 = β. remark 1.2. in particular cases, for type (m,n−m) conditions, from theorem 1.4 we have f(t) = ρ(m,n)(t) + r(m,n)(f,t) (1.13) where ρ(m,n)(t) is (m,n−m) interpolating polynomial, i.e ρ(m,n)(t) = m−1∑ i=0 τi(t)f i(α) + n−m−1∑ i=0 ηi(t)f i(β), with τi(t) = 1 i! (t−α)i ( t−β α−β )n−m m−1−i∑ k=0 ( n−m + k − 1 k )( t−α β −α )k (1.14) and ηi(t) = 1 i! (t−β)i ( t−α β −α )m n−m−1−i∑ k=0 ( m + k − 1 k )( t−β α−β )k , (1.15) and also the remainder r(m,n)(f,t) is given by r(m,n)(f,t) = ∫ β α g(m,n)(t,s)f (n)(s)ds with g(m,n)(t,s) =   m−1∑ j=0 [m−1−j∑ p=0 ( n−m+p−1 p )( t−α β−α )p] × (t−α)j(α−s)n−j−1 j!(n−j−1)! ( β−t β−α )n−m , α ≤ s ≤ t ≤ β, − n−m−1∑ i=0 [n−m−i−1∑ q=0 ( m+q−1 q )( β−t β−α )q] × (t−β)i(β−s)n−i−1 i!(n−i−1)! ( t−α β−α )m , α ≤ t ≤ s ≤ β. (1.16) for type two-point taylor conditions, from theorem 1.4 we have f(t) = ρ2t (t) + r2t (f,t) (1.17) int. j. anal. appl. 16 (3) (2018) 379 where ρ2t (t)is the two-point taylor interpolating polynomial i.e, ρ2t (t) = m−1∑ i=0 m−1−i∑ k=0 ( m + k − 1 k )[(t−α)i i! ( t−β α−β )m( t−α β −α )k f(i)(α) + (t−β)i i! ( t−α β −α )m( t−β α−β )k f(i)(β) ] (1.18) and the remainder r2t (f,t) is given by r2t (f,t) = ∫ β α g2t (t,s)f (n)(s)ds with g2t (t,s) =   (−1)m (2m−1)!p m(t,s) m−1∑ j=0 ( m−1+j j ) (t−s)m−1−jqj(t,s), s ≤ t; (−1)m (2m−1)!q m(t,s) m−1∑ j=0 ( m−1+j j ) (s− t)m−1−jpj(t,s), s ≥ t; (1.19) where p(t,s) = (s−α)(β−t) β−α , q(t,s) = p(s,t),∀t,s ∈ [α,β]. the following lemma describes the positivity of green’s function (1.12) see (beesack [4] and [levin [16]). lemma 1.1. the green’s function gh,n(t,s) has the following properties: (i) gh,n(t,s) w(t) > 0,a1 ≤ t ≤ ar,a1 ≤ s ≤ ar; (ii) gh,n(t,s) ≤ 1(n−1)!(β−α)|w(t)|; (iii) ∫β α gh,n(t,s)ds = w(t) n! . we arrange the paper in this manner, in section 2, we use peano’s representation of hermite interpolating polynomial and newly defined green functions to establish identities for majorization inequalities. we present generalized majorization inequalities and in particular we discuss the results for (m,n − m) interpolating polynomial and two-point taylor interpolating polynomial. in section 3, we give bounds for the identities related to the generalizations of majorization inequalities by using čebyšev functionals. we also give grüss type inequalities and ostrowski-type inequalities for these functionals. in section 4, we present lagrange and cauchy type mean value theorems related to the defined functionals and also give n-exponential convexity which leads to exponential convexity and then log-convexity. at the end, in section 5, we give some related analytical inequalities to our generalized results of upper bounds and also construct examples of exponentially convex functions. int. j. anal. appl. 16 (3) (2018) 380 2. main results via peano’s representation and new green functions as mentioned in [11], the complete reference about abel-gontscharoff polynomial and theorem for ’twopoint right focal’ problem is given in [3]: remark 2.1. as a special choice the abel-gontscharoff polynomial for ’two-point right focal’ interpolating polynomial for n = 2 can be given as: f(z) = f(α) + (z −α) f′(β) + ∫ β α gω,2(z,w)f ′′(w)dw, (2.1) where gω,2(z,w) is the green’s function for ’two-point right focal problem’ given as g1(z,w) = gω,2(z,w) =   (α−w) , α ≤ w ≤ z,(α−z) , z ≤ w ≤ β. (2.2) mehmood et al. (2017) [11] introduced some new types of green functions by keeping in view abelgontscharoff green’s function for ’two-point right focal problem’ that are: g2(z,w) =   (z −β) , α ≤ w ≤ z,(w −β) , z ≤ w ≤ β. (2.3) g3(z,w) =   (z −α) , α ≤ w ≤ z,(w −α) , z ≤ w ≤ β. (2.4) g4(z,w) =   (β −w) , α ≤ w ≤ z,(β −z) , z ≤ w ≤ β. (2.5) mehmood et al. (2017) gave the following lemma, using this we obtain the new generalizations of majorization inequality. lemma 2.1. let f : [α,β] → r be a twice differentiable function and gc, (c = 1, 2, 3, 4) be the new green functions defined above, then along with (2.1) the following identities holds: f(z) = f(β) + (z −β)f′(α) + ∫ β α g2(z,w)f ′′(w)dw, (2.6) f(z) = f(β) − (β −α)f′(α) + (z −α)f′(α) + ∫ β α g3(z,w)f ′′(w)dw, (2.7) f(z) = f(α) − (β −α)f′(α) − (β −z)f′(β) + ∫ β α g4(z,w)f ′′(w)dw. (2.8) equivalent statements between classical weighted majorization inequality and the inequality constructed by newly green functions are given as: int. j. anal. appl. 16 (3) (2018) 381 theorem 2.1. let x = (x1, ...,xm), y = (y1, ...,ym) ∈ im be two decreasing m-tuples and also w = (w1, ...,wm) be a real m-tuple such that satisfying (1.4) and gc (c = 1, 2, 3, 4) is defined as in (2.2)-(2.5) respectively. then the following statements are equivalent: (i) for every continuous convex function f : [α,β] → r, then m∑ l=1 wl f (yl) ≤ m∑ l=1 wl f (xl) . (2.9) (ii) for s ∈ [α,β], the following inequality holds m∑ l=1 wl gc (yl,s) ≤ m∑ l=1 wl gc (xl,s) , c = 1, 2, 3, 4. (2.10) moreover, the statements (i) and (ii) are also equivalent if we change the sign of inequality in both inequalities, in (2.9) and (2.10). proof. ”(i) ⇒ (ii)” suppose the statement (i) satisfies. fix c = 1, 2, 3, 4, the functions gc(.,s) (s ∈ [α,β]) are continuous and also convex, implies that these functions hold inequality (2.9) for each fix p, i.e., (2.10) holds. ”(ii) ⇒ (i)” since f : [α,β] → r be a convex function, f ∈ c2 ([α,β]) and (ii) holds. then the representation of the function f in the form (2.1), (2.6), (2.7) and (2.8) for the functions gc, c = 1, 2, 3, 4 implies that for all s ∈ [α,β], m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) (2.11) = ∫β α ( ∑m l=1 wlgc (xl,s) − ∑m l=1 wlgc (yl,s)) f ′′(s)ds, c = 1, 2, 3, 4. (2.12) since f is a convex function, then f ′′ (x) ≥ 0 for all x ∈ [α,β]. so, if for every s ∈ [α,β] the inequality (2.10) holds for each c = 1, 2, 3, 4, then it follows that for every convex function f : [α,β] → r, with f ∈ c2[α,β], inequality (2.9) holds. at the end, note that it is not necessary to demand the existence of the second derivative of the function f ( [12], p.172). the differentiability condition can be directly eliminated by using the fact that it is possible to approximate uniformly a continuous convex functions by convex polynomials. � we give some identities related to the generalizations of majorization inequality by using peano’s representation of hermite’s polynomial and new green functions: theorem 2.2. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]) and w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xl, yl int. j. anal. appl. 16 (3) (2018) 382 ∈ [α,β],wl ∈ r (l = 1, ...,m). also let hij,gh,n and gc(c = 1, 2, 3, 4) be as defined in (1.10), (1.12) and (2.2)-(2.5) respectively. then we have the following identities for c = 1, 2, 3, 4, m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) = ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) + ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt + ∫ β α f(n)(s) [∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] gh,n−2(t,s)dt ] ds, (2.13) where the peano’s kernel (green’s function) is defined as gh,n−2(t,s) =   l∑ j=1 kj∑ i=0 (aj−s)n−i−3 (n−i−3)! hij(t); s ≤ t, − r∑ j=l+1 kj∑ i=0 (aj−s)n−i−3 (n−i−3)! hij(t); s ≥ t, (2.14) for all al ≤ s ≤ al+1; l = 0, . . . ,r with a0 = α and ar+1 = β. proof. fix c = 1, 2, 3, 4, evaluating the identities one by one (2.1), (2.6), (2.7) and (2.8) into majorization difference, we get m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) = ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) + ∫ β α ( m∑ l=1 wlgc (xl, t) − m∑ l=1 wlgc (yl, t) ) f′′(t)dt. (2.15) by the peano’s representation of hermite’s interpolatinhg polynomial theorem 1.4, f′′(t) can be expressed as f′′(t) = r∑ j=1 kj∑ i=0 hij(t)f (i+2)(aj) + ∫ β α gh,n−2(t,s)f (n)(s)ds. (2.16) using (2.16) in (2.15) we get m∑ l=1 wl φ (xl) − m∑ l=1 wl φ (yl) = ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) + ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt + ∫ β α ( m∑ l=1 wl (gc(xl, t) −gc(yl, t)) )(∫ β α gh,n−2(t,s)f (n)(s)ds ) dt. after applying fubini’s theorm we get (2.13). � integral version of the above theorem can be stated as: int. j. anal. appl. 16 (3) (2018) 383 theorem 2.3. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, f ∈ cn([α,β]) and x,y : [a,b] → [α,β], w : [a,b] → r be continuous functions. also let hij,gh,n−2 and gc(c = 1, 2, 3, 4) be as defined in (1.10), (2.14) and (2.2)-(2.5) respectively. then we have the following identities for c = 1, 2, 3, 4,∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ = (∫ b a w(τ)x(τ)dτ − ∫ b a w(τ)y(τ)dτ ) f ′ (α) + ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt + ∫ β α f(n)(s) (∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] gh,n−2(t,s)dt ) ds. (2.17) theorem 2.4. let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,m) and hij, gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively. let f : [α,β] → r be n−convex and m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ≥ 0, t ∈ [α,β]. (2.18) consider the inequalities for c = 1, 2, 3, 4, m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) ≥ ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) + ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt. (2.19) (i) if kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.19) hold. (ii) if kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in (2.19) hold. proof. (i) since the function f is n−convex, therefore without loss of generality we can assume that φ is n−times differentiable and f(n) ≥ 0 see [13, p. 16 and p. 293]. also the given condition is that kj is odd for each j = 1, 2, ..,r implies that ω(t) = r∏ j=1 (t−aj) kj+1 ≥ 0. by using the first part of lemma 1.1 we have that the peano’s kernel gh,n−2(t,s) ≥ 0. hence, we can apply theorem 2.2 to obtain (2.19). (ii) if kr is even then (t−ar)kr+1 ≤ 0 for any t ∈ [α,β]. also clearly (t−a1)k1+1 ≥ 0 for any t ∈ [α,β] and ∏r−1 j=2 (t−aj) kj+1 ≥ 0 for t ∈ [α,β] if kj is odd for each j = 2, ..,r − 1, therefore combining all these we have ω(t) = ∏r j=1(t−aj) kj+1 ≤ 0 for any t ∈ [α,β] and by using the first part of lemma 1.1 we have gh,n−2(t,s) ≤ 0. hence, we can apply theorem 2.2 to obtain reverse inequality in (2.19). int. j. anal. appl. 16 (3) (2018) 384 � integral version of the above theorem can be stated as: theorem 2.5. let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be given points and x,y : [a,b] → [α,β], w : [a,b] → r be continuous functions and hij and gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively. let f : [α,β] → r be n−convex and∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ≥ 0, t ∈ [α,β]. (2.20) consider the inequalities for c = 1, 2, 3, 4,∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ ≥ (∫ b a w(τ)x(τ)dτ − ∫ b a w(τ)y(τ)dτ ) f ′ (α) + ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt. (2.21) (i) if kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.21) hold. (ii) if kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in (2.21) hold. by using type (m,n−m) conditions we can give the following result. corollary 2.1. let [α,β] be an interval and w = (w1, ...,wp), x = (x1, ...,xp) and y = (y1, ...,yp) be p-tuples such that xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,p). let gc(c = 1, 2, 3, 4) be the green functions as defined in (2.2)-(2.5) respectively and also τi,ηi be as defined in (1.14) and (1.15) respectively. let f : [α,β] → r be n−convex and the inequality (2.18) holds for p-tuples. consider the inequalities for c = 1, 2, 3, 4, p∑ l=1 wl φ (xl) − p∑ l=1 wl φ (yl) ≥ ( p∑ l=1 wlxl − p∑ l=1 wlyl ) f ′ (α) + ∫ β α [ p∑ l=1 wl (gc(xl, t) −gc(yl, t)) ]( m−1∑ i=0 τi(t)f (i+2)(α) + n−m−1∑ i=0 ηi(t)f (i+2)(β) ) dt. (2.22) (i) if n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.22) hold. (ii) if n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.22) hold. by using two-point taylor conditions we can give the following result. corollary 2.2. let [α,β] be an interval, w = (w1, ...,wp), x = (x1, ...,xp) and y = (y1, ...,yp) be p-tuples such that xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,p) and gc(c = 1, 2, 3, 4) be the green function as defined in (2.2)(2.5) respectively. let f : [α,β] → r be n−convex and the inequality (2.18) holds for p-tuples. consider the int. j. anal. appl. 16 (3) (2018) 385 inequalities for c = 1, 2, 3, 4, p∑ l=1 wl f (xl) − p∑ l=1 wl f (yl) ≥ ( p∑ l=1 wlxl − p∑ l=1 wlyl ) f ′ (α) + ∫ β α [ p∑ l=1 wl (gc(xl, t) −gc(yl, t)) ][ m−1∑ i=0 m−1−i∑ k=0 ( m + k − 1 k )[(t−α)i i! ( t−β α−β )m( t−α β −α )k f(i+2)(α) + (t−β)i i! ( t−α β −α )m( t−β α−β )k f(i+2)(β) ]] dt. (2.23) (i) if m is even, then the inequalities for c = 1, 2, 3, 4, in (2.23) hold. (ii) if m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.23) hold. remark 2.2. similarly we can give integral version of corollaries 2.1,2.2. the following generalization of classical majorization theorem (also known as karamata’s inequality) is valid. theorem 2.6. let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that y ≺ x with xl, yl ∈ [α,β] (l = 1, ...,m). let hij and gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively and also f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4, m∑ l=1 f (xl) − m∑ l=1 f (yl) ≥ ∫ β α [ m∑ l=1 (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt. (2.24) (i) if kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.24) hold. (ii) if kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in (2.24) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.24) hold and the function f(.) = r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(.) is non negative ( non positive), then the right hand side of (2.24) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.2) will hold. proof. (i) since the function g is convex and y ≺ x therefore by theorem 1.1, the inequalities for c = 1, 2, 3, 4, in (2.18) hold for wl = 1. hence by theorem 2.4(i) the inequalities for c = 1, 2, 3, 4, in (2.24) hold. also if the function f is convex then by using f in (1.2) instead of f we get that the right hand side of (2.24) is non negative for each c = 1, 2, 3, 4. similarly we can prove part (ii). � in the following theorem we give generalization of fuch’s majorization theorem. int. j. anal. appl. 16 (3) (2018) 386 theorem 2.7. let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, x = (x1, ...,xm) and y = (y1, ...,ym) be decreasing m-tuples and w = (w1, ...,wm) be any m-tuple with xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,m) which satisfy (1.3) and (1.4). let hij and gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively and also f : [α,β] → r be n−convex, then m∑ l=1 wlf (xl) − m∑ l=1 wlf (yl) ≥ ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt. (2.25) (i) if kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.25) hold. (ii) if kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in (2.25) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.25) hold and the function f(.) = r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(.) is non negative (non positive), then the right hand side of (2.25) will be non negative (non positive) for c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.5) will hold. proof. similar to the proof of theorem 2.6. � in the following theorem we give generalized majorization integral inequality. theorem 2.8. let −∞ < α = a1 < a2 · · · < ar = β < ∞, (r ≥ 2) be the given points, and x,y : [a,b] → [α,β] be decreasing and w : [a,b] → r be continuous functions such that (1.6) and (1.7) hold. also let hij and gc(c = 1, 2, 3, 4) be as defined in (1.10) and (2.2)-(2.5) respectively and also f : [α,β] → r be n−convex and consider the inequalities for c = 1, 2, 3, 4,∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ ≥ ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt. (2.26) (i) if kj is odd for each j = 2, ..,r, then the inequalities for c = 1, 2, 3, 4, in (2.26) hold. (ii) if kj is odd for each j = 2, ..,r − 1 and kr is even, then the reverse inequalities for c = 1, 2, 3, 4, in (2.26) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.26) hold and the function f(.) = r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(.) is non negative (non positive), then the right hand side of (2.26) will be non negative (non positive) for c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.8) will hold. by using type (m,n−m) conditions we can give generalization of majorization inequality for majorized tuples: int. j. anal. appl. 16 (3) (2018) 387 corollary 2.3. let [α,β] be an interval, x = (x1, ...,xp) and y = (y1, ...,yp) be any p-tuple such that y ≺ x with xl,yl ∈ [α,β] (l = 1, ...,p). let τi and ηi be as defined in (1.14) and (1.15) respectively. let gc(c = 1, 2, 3, 4) be defined as in (2.2)-(2.5) respectively and also f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4, p∑ l=1 f (xl) − p∑ l=1 f (yl) ≥ ∫ β α [ p∑ l=1 (gc(xl, t) −gc(yl, t)) ]( m−1∑ i=0 τi(t)f (i+2)(α) + n−m−1∑ i=0 ηi(t)f (i+2)(β) ) dt. (2.27) (i) if n−m is even, then the inequalities for c = 1, 2, 3, 4 in (2.27) hold. (ii) if n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.27) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.27) hold and the function f(.) = ∑m−1 i=0 f (i+2)(α)τi(.) + ∑n−m−1 i=0 f (i+2)(β)ηi(.) is non negative (non positive), then the right hand side of (2.27) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.2) will hold. by using two-point taylor conditions we can give generalization of majorization inequality for majorized tuples: corollary 2.4. let [α,β] be an interval and x = (x1, ...,xp), y = (y1, ...,yp) be decreasing p-tuples such that y ≺ x with xl,yl ∈ [α,β] (l = 1, ...,p). let f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4, p∑ l=1 f (xl) − p∑ l=1 f (yl) ≥ ∫ β α [ p∑ l=1 (gc(xl, t) −gc(yl, t)) ] f(t)dt, (2.28) where f(t) = m−1∑ i=0 m−1−i∑ k=0 ( m + k − 1 k )[ (t−α)i i! ( t−β α−β )m( t−α β −α )k f(i+2)(α) + (t−β)i i! ( t−α β −α )m( t−β α−β )k f(i+2)(β) ] . (i) if m is even, then the inequalities for c = 1, 2, 3, 4, in (2.28) hold. (ii) if m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.28) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.28) hold and the function f(.) is non negative (non positive), then the right hand side of (2.28) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.2) will hold. by using type (m,n−m) conditions we can give the following weighted majorization inequality. int. j. anal. appl. 16 (3) (2018) 388 corollary 2.5. let [α,β] be an interval and x = (x1, ...,xp) and y = (y1, ...,yp) be decreasing p-tuples and w = (w1, ...,wp) be any p-tuple such that xl,yl ∈ [α,β],wl ∈ r (l = 1, ...,p) which satisfy (1.3) and (1.4). let τi and ηi be as defined in (1.14) and (1.15) respectively and let f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4, p∑ l=1 wl f (xl) − p∑ l=1 wl f (yl) ≥ ∫ β α [ p∑ l=1 wl (gc(xl, t) −gc(yl, t)) ]( m−1∑ i=0 τi(t)f (i+2)(α) + n−m−1∑ i=0 ηi(t)f (i+2)(β) ) dt. (2.29) (i) if n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.29) hold. (ii) if n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.29) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.29) hold and the function f(.) = ∑m−1 i=0 f (i+2)(α)τi(.) + ∑n−m−1 i=0 f (i+2)(β)ηi(.) is non negative (non positive), then the right hand side of (2.29) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.5) will hold. by using two-point taylor conditions we can give the following weighted majorization inequality. corollary 2.6. let [α,β] be an interval and x = (x1, ...,xp), y = (y1, ...,yp) be decreasing p-tuples such that xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,p) which satisfy (1.3) and (1.4) and let f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4, p∑ l=1 wl f (xl) − p∑ l=1 wl f (yl) ≥ ∫ β α [ p∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] f(t)dt, (2.30) where f(t) = m−1∑ i=0 m−1−i∑ k=0 ( m + k − 1 k )[ (t−α)i i! ( t−β α−β )m( t−α β −α )k f(i+2)(α) + (t−β)i i! ( t−α β −α )m( t−β α−β )k f(i+2)(β) ] . (i) if m is even, then the inequalities for c = 1, 2, 3, 4, in (2.30) hold. (ii) if m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.30) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.30) hold and the function f(.) is non negative (non positive), then the right hand side of (2.30) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.5) will hold. the integral version of the above corollaries can be stated as: int. j. anal. appl. 16 (3) (2018) 389 corollary 2.7. let [α,β] be an interval and x,y : [a,b] → [α,β] be decreasing and w : [a,b] → r be continuous function such that (1.6), (1.7) hold. let τi and ηi be as defined in (1.14) and (1.15) respectively and f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4,∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ ≥ ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ]( m−1∑ i=0 τi(t)f (i+2)(α) + n−m−1∑ i=0 ηi(t)f (i+2)(β) ) dt. (2.31) (i) if n−m is even, then the inequalities for c = 1, 2, 3, 4, in (2.31) hold. (ii) if n−m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.31) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.31) holds and the function f(.) = ∑m−1 i=0 f (i+2)(α)τi(.) + ∑n−m−1 i=0 f (i+2)(β)ηi(.) is non negative (non positive), then the right hand side of (2.31) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.8) will hold. corollary 2.8. let [α,β] be an interval and x,y : [a,b] → [α,β] be decreasing and w : [a,b] → r be continuous functions such that (1.6) and (1.7) hold. let f : [α,β] → r be n−convex. consider the inequalities for c = 1, 2, 3, 4,∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ ≥ ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] f(t)dt, (2.32) where f(t) = m−1∑ i=0 m−1−i∑ k=0 ( m + k − 1 k )[ (t−α)i i! ( t−β α−β )m( t−α β −α )k f(i+2)(α) + (t−β)i i! ( t−α β −α )m( t−β α−β )k f(i+2)(β) ] . (i) if m is even, then the inequalities for c = 1, 2, 3, 4, in (2.32) hold. (ii) if m is odd, then the reverse inequalities for c = 1, 2, 3, 4, in (2.32) hold. if the inequalities (reverse inequalities) for c = 1, 2, 3, 4, in (2.32) hold and the function f(.) is non negative (non positive), then the right hand side of (2.32) will be non negative (non positive) for each c = 1, 2, 3, 4, that is the inequality (reverse inequality) in (1.8) will hold. 3. upper bounds for obtained generalized identities for two lebesgue integrable functions f,h : [α,β] → r we consider the čebyšev functional λ(f,h) = 1 β −α ∫ β α f(t)h(t)dt− 1 β −α ∫ β α f(t)dt · 1 β −α ∫ β α h(t)dt. in [6] the authors proved the following theorems: int. j. anal. appl. 16 (3) (2018) 390 theorem 3.1. let f : [α,β] → r be a lebesgue integrable function and h : [α,β] → r be an absolutely continuous function with (·−α)(β −·)[h′]2 ∈ l[α,β]. then we have the inequality |λ(f,h)| ≤ 1 √ 2 [λ(f,f)] 1 2 1 √ β −α (∫ β α (x−α)(β −x)[h′(x)]2dx )1 2 . (3.1) the constant 1√ 2 in (3.1) is the best possible. theorem 3.2. assume that h : [α,β] → r is monotonic nondecreasing on [α,β] and f : [α,β] → r is absolutely continuous with f′ ∈ l∞[α,β]. then we have the inequality |λ(f,h)| ≤ 1 2(β −α) ‖f′‖∞ ∫ β α (x−α)(β −x)dh(x). (3.2) the constant 1 2 in (3.2) is the best possible. in this section, we give the upper bounds like grüss-type and ostrowski-type for our generalized results. for m-tuples w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) with xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,m) and the green functions gc(c = 1, 2, 3, 4) and gh,n−2 be as defined in (2.2)-(2.5) and (2.14) respectively, denote l(s) = ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] gh,n−2(t,s)dt, s ∈ [α,β], (3.3) for c = 1, 2, 3, 4, similarly for continuous functions x,y : [a,b] → [α,β], w : [a,b] → r and the green function gc(c = 1, 2, 3, 4) and gh,n−2 be as defined in (2.2)-(2.5) and (2.14) respectively, denote j(s) = ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] gh,n−2(t,s)dt, s ∈ [α,β], (3.4) for c = 1, 2, 3, 4. consider the čebyšev functionals λ(l,l), λ(j,j) are given by: λ(l,l) = 1 β −α ∫ β α l2(s)ds− ( 1 β −α ∫ β α l(s)ds )2 , (3.5) λ(j,j) = 1 β −α ∫ β α j2(s)ds− ( 1 β −α ∫ β α j(s)ds )2 . (3.6) theorem 3.3. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]) such that (· − α)(β − ·)[f(n+1)]2 ∈ l[α,β] and w = (w1, ...,wm), x = (x1, ...,xm) and y = (y1, ...,ym) be m-tuples such that xl, yl ∈ [α,β],wl ∈ r (l = 1, ...,m). also let hij be the fundamental int. j. anal. appl. 16 (3) (2018) 391 polynomials of the hermite basis and the functions gc(c = 1, 2, 3, 4) and l be defined by (2.2)-(2.5) and (3.3) respectively. then m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) = ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) + r∑ j=1 kj∑ i=0 φ(i+2)(aj) ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] hij(t)dt + f(n−1)(β) −f(n−1)(α) β −α ∫ β α l(s)ds + rem(f; α,β), (3.7) where the remainder rem(f; α,β) satisfies the estimation |rem(f; α,β)| ≤ √ β −α √ 2 [λ(l,l)] 1 2 ∣∣∣∣∣ ∫ β α (s−α)(β −s)[f(n+1)(s)]2ds ∣∣∣∣∣ 1 2 . (3.8) proof. comparing (3.7) and (2.13) we have rem(f; α,β) = (β −α) λ(l,f(n)). applying theorem 3.1 on the functions l and f(n) we obtain (3.8). � the integral version of the above theorem can be stated as: theorem 3.4. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]) such that (· − α)(β − ·)[f(n+1)]2 ∈ l[α,β] and x,y : [a,b] → [α,β], w : [a,b] → r be continuous functions. also let hij be the fundamental polynomials of the hermite basis and the functions gc(c = 1, 2, 3, 4) and j be defined by (2.2)-(2.5) and (3.4) respectively. then∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ = (∫ b a w(τ)x(τ)dτ − ∫ b a w(τ)y(τ)dτ ) f ′ (α) + ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt + f(n−1)(β) −f(n−1)(α) β −α ∫ β α j(s)ds + ˜rem(f; α,β), (3.9) where the remainder ˜rem(f; α,β) satisfies the estimation ∣∣∣ ˜rem(f; α,β)∣∣∣ ≤ √β −α√ 2 [λ(j,j)] 1 2 ∣∣∣∣∣ ∫ β α (s−α)(β −s)[f(n+1)(s)]2ds ∣∣∣∣∣ 1 2 . (3.10) using theorem 3.2 we obtain the following grüss type inequalities. theorem 3.5. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]) such that f(n) is monotonic non decreasing on [α,β] and let l be defined by (3.3). then the int. j. anal. appl. 16 (3) (2018) 392 representation (3.7) holds and the remainder rem(f; α,β) satisfies the bound |rem(f; α,β)| ≤ ‖l′‖∞ { f(n−1)(β) + f(n−1)(α) 2 − f(n−2)(β) −f(n−2)(α) β −α } . (3.11) proof. since rem(f; α,β) = (β −α) λ(l,f(n)), applying theorem 3.2 on the functions l and f(n) we get (3.11). � integral case of the above theorem can be given: theorem 3.6. let −∞ < α < β < ∞ and α ≤ a1 < a2 · · · < ar ≤ β, (r ≥ 2) be the given points, and f ∈ cn([α,β]) such that f(n) is monotonic non decreasing on [α,β] and let x,y : [a,b] → [α,β], w : [a,b] → r be continuous functions and also gc(c = 1, 2, 3, 4) and j be defined by (2.2)-(2.5) and(3.4) respectively. then we have the representation (3.9) and the remainder ˜rem(f; α,β) satisfies the bound ∣∣∣ ˜rem(f; α,β)∣∣∣ ≤‖j′‖∞{f(n−1)(β) + f(n−1)(α) 2 − f(n−2)(β) −f(n−2)(α) β −α } . (3.12) we present the ostrowski-type inequalities related to generalizations of majorization inequality. theorem 3.7. suppose that all assumptions of theorem 2.2 hold. assume (u,v) is a pair of conjugate exponents, that is 1 ≤ u,v ≤ ∞, 1/u + 1/v = 1. let ∣∣f(n)∣∣u : [α,β] → r be an r-integrable function for some n ∈ n. then we have: ∣∣∣∣∣ m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) − ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt ∣∣∣∣∣∣ ≤ ∥∥∥f(n)∥∥∥ u ‖l‖v , (3.13) where l is defined in (3.3). the constant on the right-hand side of (3.13) is sharp for 1 < u ≤∞ and the best possible for u = 1. proof. by using (3.3) we have l(t) = ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] gh,n−2(t,s)dt, for c = 1, 2, 3, 4. using the identity (2.13) and applying hölder’s inequality we obtain∣∣∣∣∣ m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) − ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt ∣∣∣∣∣∣ = ∣∣∣∣∣ ∫ β α l(t)f(n)(t)dt ∣∣∣∣∣ ≤ ∥∥∥f(n)∥∥∥ u (∫ β α |l(t)|v dt )1 v . int. j. anal. appl. 16 (3) (2018) 393 for the proof of the sharpness of the constant (∫β α |l(t)|v dt )1 v is analog to one in proof of theorem 11 in [1]. � integral version of the above theorem can be given as: theorem 3.8. suppose that all assumptions of theorem 2.3 hold. assume (u,v) is a pair of conjugate exponents, that is 1 ≤ u,v ≤ ∞, 1/u + 1/v = 1. let ∣∣f(n)∣∣u : [α,β] → r be an r-integrable function for some n ∈ n. then we have: ∣∣∣∣∣ ∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ − (∫ b a w(τ)(x(τ) −y(τ))dτ ) f ′ (α) − ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt ∣∣∣∣∣∣ ≤ ∥∥∥f(n)∥∥∥ u ‖j‖v , (3.14) where j is defined in (3.4). the constant on the right-hand side of (3.14) is sharp for 1 < u ≤∞ and the best possible for u = 1. 4. n−exponential convexity and exponential convexity we begin this section by giving some definitions and notions which are used frequently in the results. for more details see e.g. [5], [9] and [14]. definition 4.1. a function f : i → r is n-exponentially convex in the jensen sense on i if n∑ i,j=1 ξiξj f ( xi + xj 2 ) ≥ 0, hold for all choices ξ1, . . . ,ξn ∈ r and all choices x1, . . . ,xn ∈ i. a function f : i → r is n-exponentially convex if it is n-exponentially convex in the jensen sense and continuous on i. definition 4.2. a function f : i → r is exponentially convex in the jensen sense on i if it is n-exponentially convex in the jensen sense for all n ∈ n. a function f : i → r is exponentially convex if it is exponentially convex in the jensen sense and continuous. proposition 4.1. if f : i → r is an n-exponentially convex in the jensen sense, then the matrix [ f ( xi+xj 2 )]m i,j=1 is a positive semi-definite matrix for all m ∈ n,m ≤ n. particularly, det [ f ( xi + xj 2 )]m i,j=1 ≥ 0, for all m ∈ n, m = 1, 2, ...,n. int. j. anal. appl. 16 (3) (2018) 394 remark 4.1. it is known that f : i → r+ is a log-convex in the jensen sense if and only if α2f(x) + 2αβf ( x + y 2 ) + β2f(y) ≥ 0, holds for every α,β ∈ r and x,y ∈ i. it follows that a positive function is log-convex in the jensen sense if and only if it is 2-exponentially convex in the jensen sense. a positive function is log-convex if and only if it is 2-exponentially convex. motivated by inequalities (2.19) and (2.21), under the assumptions of theorems 2.4 and 2.5 we define the following linear functionals: h1(f) = m∑ l=1 wl f (xl) − m∑ l=1 wl f (yl) − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) f ′ (α) − ∫ β α [ m∑ l=1 wl (gc(xl, t) −gc(yl, t)) ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt, c = 1, 2, 3, 4, (4.1) and h2(f) = ∫ b a w(τ)f(x(τ))dτ − ∫ b a w(τ)f(y(τ))dτ − (∫ b a w(τ)(x(τ) −y(τ))dτ ) f ′ (α) − ∫ β α [∫ b a w(τ) (gc(x(τ), t) −gc(y(τ), t)) dτ ] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt, c = 1, 2, 3, 4. (4.2) remark 4.2. under the assumptions of theorems 2.4 and 2.5, it holds hi(f) ≥ 0, i = 1, 2, for all n−convex functions f. lagrange and cauchy type mean value theorems related to defined functionals are given in the following theorems. theorem 4.1. let f : [α,β] → r be such that f ∈ cn[α,β]. if the inequalities in (2.18) (i = 1) and (2.20) (i = 2) hold, then there exist ξi ∈ [α,β] such that hi(f) = f(n)(ξi)hi(ϕ), i = 1, 2, (4.3) where ϕ(x) = x n n! and hi, i = 1, 2 are defined by (4.1) and(4.2). proof. similar to the proof of theorem 4.1 in [10]. � theorem 4.2. let f,g : [α,β] → r be such that f,g ∈ cn[α,β]. if the inequalities in (2.18) (i = 1), (2.20) (i = 2), hold, then there exist ξi ∈ [α,β] such that hi(f) hi(g) = f(n)(ξi) g(n)(ξi) , i = 1, 2, (4.4) provided that the denominators are non-zero and hi, i = 1, 2, are defined by (4.1) and(4.2). int. j. anal. appl. 16 (3) (2018) 395 proof. similar to the proof of theorem 4.2 in [10]. � now we will produce n−exponentially and exponentially convex functions applying defined functionals. we use an idea from [14]. in the sequel j will be interval in r. theorem 4.3. let ω = {ft : t ∈ j}, where j is an interval in r, be a family of functions defined on an interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is n−exponentially convex in the jensen sense on j for every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. then for the linear functionals hi(ft) (i = 1, 2) as defined by (4.1) and (4.2), the following statements hold: (i) the function t → hi(ft) is n-exponentially convex in the jensen sense on j and the matrix [hi(ftj+tl 2 )]mj,l=1 is a positive semi-definite for all m ∈ n,m ≤ n, t1, .., tm ∈ j. particularly, det[hi(ftj+tl 2 )]mj,l=1 ≥ 0 for all m ∈ n, m = 1, 2, ...,n. (ii) if the function t → hi(ft) is continuous on j, then it is n-exponentially convex on j. proof. the proof is similar to the proof of theorem 23 in [2]. � the following corollary is an immediate consequence of the above theorem. corollary 4.1. let ω = {ft : t ∈ j}, where j is an interval in r, be a family of functions defined on an interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is exponentially convex in the jensen sense on j for every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. then for the linear functionals hi(ft) (i = 1, 2) as defined by (4.1) and (4.2), the following statements hold: (i) the function t → hi(ft) is exponentially convex in the jensen sense on j and the matrix [hi(ftj+tl 2 )]mj,l=1 is a positive semi-definite for all m ∈ n,m ≤ n, t1, .., tm ∈ j. particularly, det[hi(ftj+tl 2 )]mj,l=1 ≥ 0 for all m ∈ n, m = 1, 2, ...,n. (ii) if the function t → hi(ft) is continuous on j, then it is exponentially convex on j. corollary 4.2. let ω = {ft : t ∈ j}, where j is an interval in r, be a family of functions defined on an interval [α,β] such that the function t 7→ [x0, . . . ,xn; ft] is 2-exponentially convex in the jensen sense on j for every (n + 1) mutually different points x0, . . . ,xn ∈ [α,β]. let hi, i = 1, 2 be linear functionals defined by (4.1) and (4.2). then the following statements hold: (i) if the function t 7→ hi(ft) is continuous on j, then it is 2-exponentially convex function on j. if t 7→ hi(ft) is additionally strictly positive, then it is also log-convex on j. furthermore, the following inequality holds true: [hi(fs)]t−r ≤ [hi(fr)] t−s [hi(ft)] s−r , i = 1, 2, int. j. anal. appl. 16 (3) (2018) 396 for every choice r,s,t ∈ j, such that r < s < t. (ii) if the function t 7→ hi(ft) is strictly positive and differentiable on j, then for every p,q,u,v ∈ j, such that p ≤ u and q ≤ v, we have µp,q(hi, ω) ≤ µu,v(hi, ω), (4.5) where µp,q(hi, ω) =   ( hi(fp) hi(fq) ) 1 p−q , p 6= q, exp ( d dp hi(fp) hi(fp) ) , p = q, (4.6) for fp,fq ∈ ω. proof. the proof is similar to the proof of corollary 2 in [2]. � remark 4.3. note that the results from theorem 4.3, corollary 4.1 and corollary 4.2 still hold when two of the points x0, ...,xl ∈ [α,β] coincide, say x1 = x0, for a family of differentiable functions ft such that the function t 7→ [x0, ...,xl; ft] is an n-exponentially convex in the jensen sense (exponentially convex in the jensen sense, log-convex in the jensen sense), and furthermore, they still hold when all (l + 1) points coincide for a family of l differentiable functions with the same property. the proofs are obtained by suitable characterization of convexity. 5. applications in this section, we give some applications of our generalized results about the upper bounds as well as exponential convex functions. firstly, we consider some related analytical inequalities by using our generalized results of upper bounds. example 5.1. by using ostrowski-type inequality (3.13) for n = 4 as an upper bound of our generalized results, • let f(x) = ex, x ∈ r, then 0 ≤| m∑ l=1 wle xl − m∑ l=1 wle yl − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) eα −ghc |≤ (euβ −euα) 1 u u 1 u ‖ l ‖v, • let f(x) = xr, [0,∞) for r > 3, then 0 ≤| m∑ l=1 wlx r l − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) rαr−1 −ghc | ≤ r(r − 1)(r − 2)(r − 3) (u(r − 4) + 1) 1 u ( βu(r−4)+1 −αu(r−4)+1 ) 1 u ‖ l ‖v, int. j. anal. appl. 16 (3) (2018) 397 • let f(x) = x log x, x ∈ (0,∞), then 0 ≤| m∑ l=1 wlxl log xl − m∑ l=1 wlyl log yl − ( m∑ l=1 wlxl − m∑ l=1 wlyl ) (log α + 1) −ghc | ≤ 2 (1 − 3u) 1 u ( β1−3u −α1−3u ) 1 u ‖ l ‖v, • let f(x) = − log x, x ∈ (0,∞), then 0 ≤| m∑ l=1 wl log yl − m∑ l=1 wl log xl + ( m∑ l=1 wlxl − m∑ l=1 wlyl ) 1 α −ghc | ≤ 6 (1 − 4u) 1 u ( β1−4u −α1−4u ) 1 u ‖ l ‖v, where, ghc = ∫β α [ ∑m l=1 wl (gc(xl, t) −gc(yl, t))] r∑ j=1 kj∑ i=0 f(i+2)(aj)hij(t)dt, (c = 1, 2, 3, 4), and l(s) =∫β α [ ∑m l=1 wl (gc(xl, t) −gc(yl, t))] gh,2(t,s)dt. we can also give the particular cases of above results for u = 1 and v = ∞. now, we construct exponentially convex function by using family of convex functions defined on (0,∞): example 5.2. let e1 = {θv : (0,∞) → (0,∞) : v ∈ r} be a family of continuous convex functions defined by θv(x) =   xevx v2 , v 6= 0; x3 2 , v = 0. we have v 7→ ( θv(x) x )′′ (t ∈ r) is exponentially convex for every fixed x ∈ r. using analogous arguing as in the proof of theorem 4.3 we also have that v 7→ θv[z0, ...,zt] is exponentially convex (and so exponentially convex in the jensen sense). using corollary 4.1 we conclude that v 7→ hi(θv) is exponentially convex in the jensen sense. it is easy to verify that this mapping is continuous (although mapping v 7→ θv is not continuous for v = 0), so it is exponentially convex. for this family of functions, µv,q (e1,hi) from (4.6), becomes µp,q(hi,e1) =   ( hi(θp) hi(θq) ) 1 p−q , p 6= q, exp ( hi(id·θp) hi(θp) − n p ) , p = q 6= 0, exp ( 1 n+1 hi(id·φ0) hi(θ0) ) , p = q = 0, where id is the identity function. int. j. anal. appl. 16 (3) (2018) 398 now using (4.5), µp,q is monotone function in parameters p and q. we observe here that ( d2θp dx2 d2θq dx2 ) 1 p−q (lnx) = x so using theorem 4.2 it follows that mp,q(e1,hi) = lnµp,q(e1,hi), satisfies α ≤ mp,q(e1,hi) ≤ β, i = 1, 2. this shows that mp,q(e1,hi) is mean. because of the above inequality (4.5), this mean is also monotonic. remark 5.1. we can construct other examples for exponentially convex functions as example 2 for the families of continuous convex functions: • e2 = {µt : (0,∞) → r : t ∈ r} where, µt(x) =   xt+1 t(t−1), t 6= 0, 1; −x log x, t = 0; x2 log x, t = 1. • e3 = {χt : (0,∞) → (0,∞) : t ∈ (0,∞)} where, χt(x) =   xt−x log2 t , t 6= 1; x3 2 , t = 1. • e4 = {δt : (0,∞) → (0,∞) : t ∈ (0,∞)} where, δt(x) := xe−x √ t t . conflict of interests: the authors declare that there is no conflict of interests. acknowledgment: the publication was supported by the ministry of education and science of the russian federation (the int. j. anal. appl. 16 (3) (2018) 399 agreement number no. 02.a03.21.0008.) this publication is partially supported by royal commission for jubail and yanbu, kingdom of saudi arabia. references [1] r. p. agarwal, s. ivelić bradanović and j. pečarić, generalizations of sherman’s inequality by lidstone’s interpolating polynomial, j. inequal. appl., 2016 (2016), art. id 6. [2] m. adil khan, n. latif and j. pečarić, generalization of majorization theorem, j. math. inequal., 9(3) (2015), 847-872. [3] r. p. agarwal and p. j. y. wong, error inequalities in polynomial interpolation and their applications, kluwer academic publishers, dordrecht/ boston/ london, 1993. [4] p. r. beesack, on the greens function of an n-point boundary value problem, pacific j. math. 12 (1962), 801-812. kluwer academic publishers, dordrecht / boston / london, 1993. [5] s. n. bernstein, sur les fonctions absolument monotones, acta math. 52 (1929), 1–66. [6] p. cerone and s. s. dragomir, some new ostrowski-type bounds for the čebyšev functional and applications, j. math. inequal. 8(1) (2014), 159–170. [7] p. j. davis, interpolation and approximation, blaisedell publishing co., boston, 1961. [8] l. fuchs, a new proof of an inequality of hardy-littlewood-polya, mat. tidsskr, b (1947), 53-54. [9] j. jakšetić and j. pečarić, exponential convexity method, j. convex anal. 20(2013), no. 1, 181-197. [10] j. jakšetić, j. pečarić and a. perušić, steffensen inequality, higher order convexity and exponential convexity, rend. circ. mat. palermo 63 (1) (2014), 109–127. [11] n. mahmood, r. p. agarwal, s. i. butt and j. pečarić, new generalization of popoviciu type inequalities via new green functions and montgomery identity, j. inequal. appl., 2017 (2017), art. id 108. [12] a. w. marshall, i. olkin and barry c. arnold, inequalities: theory of majorization and its applications (second edition), springer series in statistics, new york 2011. [13] j. pečarić, f. proschan and y. l. tong, convex functions, partial orderings and statistical applications, academic press, new york, 1992. [14] j. pečarić and j. perić, improvements of the giaccardi and the petrović inequality and related results, an. univ. craiova ser. mat. inform., 39(1) (2012), 65–75. [15] j. pečarić, on some inequalities for functions with nondecreasing increments, j. math. anal. appl., 98 (1984), 188-197. [16] a. yu. levin, some problems bearing on the oscillation of solutions of linear differential equations, soviet math. dokl., 4(1963), 121-124. 1. introduction and preliminaries 2. main results via peano's representation and new green functions 3. upper bounds for obtained generalized identities 4. n-exponential convexity and exponential convexity 5. applications references int. j. anal. appl. (2023), 21:7 existence fixed point solutions for c-class functions in bipolar metric spaces with applications g. upender reddy1, c. ushabhavani1,2,∗, b. srinuvasa rao3 1department of mathematics, mahatma gandhi university, nalgonda, telangana, india 2department of humanities & basic science, sreechaitanya college of engineering, thimmapur, karimnagar-505001,telangana, india 3department of mathematics, dr.b.r.ambedkar university, srikakulam, etcherla-532410, andhra pradesh, india ∗corresponding author: n.ushabhavani@gmail.com abstract. in this study, the idea of c-class functions is introduced in the process of building a bi-polar metric space, along with often coupled fixed point theorems for these mappings in complete bi-polar metric spaces that associate altering distance function and ultra-altering distance function. furthermore, we provide applications to integral equations as well as homotopy and we give an interpretation that demonstrates the relevance of the results obtained. 1. introduction fixed point theory is a crucial topic of non-linear analysis. numerous types of equations that exist in natural, biological, social, engineering, and other branches of science and technology are studied in order to understand their underlying relevance. examining the situations in which single or multi-valued mappings have solutions is a common application of this technique. coupled fixed points was originally understood by guo and lakshmikantham [1] in 1987. bhaskar and lakshmikantham [2] developed a novel fixed point theorem for mixed monotone mapping in a metric space with partial ordering after using a weak contractivity condition. for further information received: dec. 28, 2022. 2020 mathematics subject classification. 54h25, 47h10, 54e50. key words and phrases. complete bipolar metric space; ω-compatible mappings; c-class functions; common coupled fixed point. https://doi.org/10.28924/2291-8639-21-2023-7 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-7 2 int. j. anal. appl. (2023), 21:7 on coupled fixed point outcomes, see the study results ( [3], [4], [5], [6], [7], [8], [9]) and relevant references. in 2014, ansari [10] proposed the idea of c-class functions and the proofs of unique fixed point theorems for specific contractive mappings. this marked the beginning of a significant amount of work in this area (see.( [11], [12], [13], [14], [15], [16], [17] ). in addition to providing variant-related (coupled) fixed point solutions for co-variant and contravariant contractive mappings, muttu and gurdal [18] recently developed the concept of bi-polar metric spaces. later, we proved some fixed point theorems in our earlier papers (see. [19], [20], [21], [22], [23], [24]). the purpose of this article is to propose a coupled common fixed point theorem for a covariant mappings of c-class functions in relation to bi-polar metric spaces. examples that are appropriate and relevant applications to integral equations along with homotopy are also provided. what follows is in our subsequent conversations, we compile a few suitable definitions. 2. preliminaries definition 2.1. ( [18]) the mapping d : s×t → [0,∞) is said to be a bipolar-metric on pair of non empty sets (s,t ).if (b1) d(µ,ν) = 0 implies that µ = ν; (b2) µ = ν implies that d(µ,ν) = 0; (b3) if (µ,ν) ∈ (s,t ), then d(µ,ν) = d(ν,µ); (b4) d(µ1,ν2) ≤ d(µ1,ν1) + d(µ2,ν1) + d(µ2,ν2), for all µ,µ1,µ2 ∈s and ν,ν1,ν2 ∈t , and the triple (s,t ,d) is called a bipolar-metric space. example 2.1. ( [18]) let d : s×t → [0, +∞) be defined as d(ψ,a) = ψ(a), for all (ψ,a) ∈ (s,t ) where s = {ψ/ψ : r→ [1, 3]} be the set of all functions and t = r. then the triple (s,t ,d) is a disjoint bipolar-metric space. definition 2.2. ( [18]) let ω : s1 ∪t1 →s2 ∪t2 be a function defined on two pairs of sets (s1,t1) and (s2,t2) is said to be (i) covariant if ω(s1) ⊆s2 and ω(t1) ⊆t2. this is denoted as ω : (s1,t1) ⇒ (s2,t2); (ii) contravariant if ω(s1) ⊆t2 and ω(t1) ⊆s2. it is denoted as ω : (s1,t1) � (s2,t2). particularly, if d1 is bipolar metrics on (s1,t1) and d2 is bipolar metrics on (s2,t2), we often write ω : (s1,t1,d1) ⇒ (s2,t2,d2) and ω : (s1,t1,d1) � (s2,t2,d2) respectively. int. j. anal. appl. (2023), 21:7 3 definition 2.3. ( [18]) in a bipolar metric space (s,t ,d) for any ξ ∈s ∪t is left point if ξ ∈s, is right point if ξ ∈t and is central point if ξ ∈s∩t . also, {αi} in s and {βi} in t are left and right sequence respectively. in a bipolar metric space, we call a sequence, a left or a right one. a sequence {ξi} is said to be convergent to ξ iff either {ξi} is a left sequence, ξ is a right point and lim i→∞ d(ξi,ξ) = 0, or {ξi} is a right sequence, ξ is a left point and lim i→∞ d(ξ,ξi ) = 0. the bisequence ({αi},{βi}) on (s,t ,d) is a sequence on s×t . in the case where {αi} and {βi} are both convergent, then ({αi},{βi}) is convergent. the bi-sequence ({αi},{βi}) is a cauchy bisequence if lim i,j→∞ d(αi,βj) = 0. note that every convergent cauchy bisequence is biconvergent. the bipolar metric space is complete, if each cauchy bisequence is convergent (and so it is biconvergent). definition 2.4. ( [22]) let (s,t ,d) be a bipolar metric space and a pair (℘,$) is called (a) coupled fixed point of covariant mapping ω : ( s2,t 2 ) ⇒ (s,t ) if ω (℘,$) = ℘, ω ($,℘) = $ for (℘,$) ∈s2 ∪t 2 ; (b) coupled coincident point of ω : ( s2,t 2 ) ⇒ (s,t ) and λ : (s,t ) ⇒ (s,t ) if f (℘,$) = λ℘, ω ($,℘) = λ$; (c) coupled common point of ω : ( s2,t 2 ) ⇒ (s,t ) and λ : (s,t ) ⇒ (s,t ) if ω (℘,$) = λ℘ = ℘, ω ($,℘) = λ$ = $; (d) the pair (ω, λ) is weakly compatible if λ(ω(℘,$)) = ω(λ℘, λ$) and λ(ω($,℘)) = ω(λ$, λ℘) whenever ω (℘,$) = λ℘, ω ($,℘) = λ$ definition 2.5. ( [10]) let c = {∆/∆ : [0, +∞)×[0.+∞) → r} be a family of continuous functions is called a c-class function if for all s∗,t∗ ∈ [0,∞), (a) ∆(s∗,t∗) ≤ s?; (b) ∆(s∗,t∗) = s∗ ⇒ s∗ = 0 or t∗ = 0. example 2.2. ( [10]) each of the functions ∆ : [0, +∞) × [0. + ∞) → r defined below are elements of c. (a) ∆(s∗,t∗) = s? − t?; (b) ∆(s∗,t∗) = ms∗ where m ∈ (0, 1). (c) ∆(s∗,t∗) = s ∗ (1+t?)r where r ∈ (0,∞). (d) ∆(s∗,t∗) = s?η(s?) where η : [0,∞) → [0,∞) is continuous function. (e) ∆(s∗,t∗) = s? −ϕ(s?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function ϕ : [0,∞) → [0,∞) such that ϕ(s?) = 0 ⇔ s? = 0. (f ) ∆(s∗,t∗) = sω(s?,t?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function ω : [0,∞)2 → [0,∞) such that ω(s?,t?) < 1. khan et al. [25] and a. h. ansari et al. [11] both addressed a new category of contractive fixed point outcomes. the idea of an altering distance function and ultra altering distance functions, which 4 int. j. anal. appl. (2023), 21:7 are control functions that vary the distance between two locations in a metric space, were introduced in their work. we say f = {ψ?/ψ? : [0,∞) → [0,∞)} and g = {φ?/φ? : [0,∞) → [0,∞)} be the class of all altering distance and ultra altering distance functions satisfying the following condition: (ψ0) ψ? is nondecreasing and continuous; (ψ1) ψ?(t) = 0 if and only if t = 0. (ψ2) ψ?(t) is subadditivity, ψ?(a + b) ≤ ψ?(a) + ψ?(b); (φ0) φ? is continuous; (φ1) φ?(t) > 0, t > 0 and φ?(0) ≥ 0. 3. main results in this section, two covariant mappings that meet new type contractive criteria in bipolar metric spaces are given some common coupled fixed point theorems via c-class functions. theorem 3.1. let (s,t ,d) be a complete bipolar metric space. suppose that γ : ( s2,t 2 ) ⇒ (s,t ) and λ : (s,t ) ⇒ (s,t ) be two covariant mappings satiesfies ψ? (d(γ(u,v), γ(p,q))) ≤ ∆ (ψ? (m(u,v,p,q)) ,φ? (m(u,v,p,q))) (3.1) where, m(u,v,p,q) = ` max { d (λu, λp) ,d (λv, λq) } for all u,v ∈ s and p,q ∈ t and ∆ ∈ c, ψ? ∈ f, φ? ∈ g with ` ∈ (0, 1) (ξ0) γ(s2 ∪t 2) ⊆ λ(s∪t ) and λ(s∪t ) is a complete subspace of s∪t , (ξ1) pair (γ, λ) is ω-compatible. then there is a unique common coupled fixed point of γ and λ in s∪t . proof. let x0,y0 ∈ s and p0,q0 ∈ t be arbitrary, and from (ξ0), we construct the bisequences ({ακ} ,{ζκ}), ({βκ} ,{ηκ}) in (s,t ) as γ (xκ,yκ) = λxκ+1 = ακ, γ (pκ,qκ) = λpκ+1 = ζκ γ (yκ,yκ) = λyκ+1 = βκ, γ (qκ,pκ) = λqκ+1 = ηκ where κ = 0, 1, 2, . . . . then from (3.1), we can get ψ? (d(ακ,ζκ+1)) = ψ? (d(γ (xκ,yκ) , γ (pκ+1,qκ+1))) ≤ ∆ (ψ? (m(xκ,yκ,pκ+1,qκ+1)) ,φ? (m(xκ,yκ,pκ+1,qκ+1))) (3.2) int. j. anal. appl. (2023), 21:7 5 where, m(xκ,yκ,pκ+1,qκ+1) = ` max { d (λxκ, λpκ+1) ,d (λyκ, λqκ+1) } = ` max { d (ακ−1,ζκ) ,d (βκ−1,ηκ) } from (3.2), deduce that ψ? (d(ακ,ζκ+1)) ≤ ∆ ( ψ? ( ` max { d (ακ−1,ζκ) , d (βκ−1,ηκ) }) ,φ? ( ` max { d (ακ−1,ζκ) , d (βκ−1,ηκ) })) ≤ ψ? ( ` max { d (ακ−1,ζκ) ,d (βκ−1,ηκ) }) by using (ψ0), we have d(ακ,ζκ+1) ≤ ` max { d (ακ−1,ζκ) ,d (βκ−1,ηκ) } (3.3) similarly, we can prove d (βκ,ηκ+1) ≤ ` max { d (ακ−1,ζκ) ,d (βκ−1,ηκ) } (3.4) combining (3.3) and (3.4), we have max { d (ακ,ζκ+1) ,d (βκ,ηκ+1) } ≤ ` max { d (ακ−1,ζκ) ,d (βκ−1,ηκ) } ≤ `2 max { d (ακ−2,ζκ−1) ,d (βκ−2,ηκ−1) } ... ≤ `κ max { d (α0,ζ1) ,d (β0,η1) } → 0 as κ →∞. (3.5) on the other hand, we have ψ? (d(ακ+1,ζκ)) = ψ? (d(γ (xκ+1,yκ+1) , γ (pκ,qκ))) ≤ ∆ (ψ? (m(xκ+1,yκ+1,pκ,qκ)) ,φ? (m(xκ+1,yκ+1,pκ,qκ))) ≤ ψ? ( ` max { d (ακ,ζκ−1) ,d (βκ,ηκ−1) }) by using (ψ0), we have d(ακ+1,ζκ) ≤ ` max { d (ακ,ζκ−1) ,d (βκ,ηκ−1) } (3.6) because of m(xκ+1,yκ+1,pκ,qκ) = ` max { d (λxκ+1, λpκ) ,d (λyκ+1, λqκ) } = ` max { d (ακ,ζκ−1) ,d (βκ,ηκ−1) } 6 int. j. anal. appl. (2023), 21:7 similarly, we can prove d (βκ+1,ηκ) ≤ ` max { d (ακ,ζκ−1) ,d (βκ,ηκ−1) } (3.7) combining (3.6) and (3.7), we have max { d (ακ+1,ζκ) ,d (βκ+1,ηκ) } ≤ ` max { d (ακ,ζκ−1) ,d (βκ,ηκ−1) } ≤ `2 max { d (ακ−1,ζκ−2) ,d (βκ−1,ηκ−2) } ... ≤ `κ max { d (α1,ζ0) ,d (β1,η0) } → 0 as κ →∞. (3.8) moreover, ψ? (d(ακ,ζκ)) = ψ? (d(γ (xκ,yκ) , γ (pκ,qκ))) ≤ ∆ (ψ? (m(xκ,yκ,pκ,qκ)) ,φ? (m(xκ,yκ,pκ,qκ))) ≤ ψ? ( ` max { d (ακ−1,ζκ−1) ,d (βκ−1,ηκ−1) }) by using (ψ0), we have d(ακ,ζκ) ≤ ` max { d (ακ−1,ζκ−1) ,d (βκ−1,ηκ−1) } (3.9) because of m(xκ,yκ,pκ,qκ) = ` max { d (λxκ, λpκ) ,d (λyκ, λqκ) } = ` max { d (ακ−1,ζκ−1) ,d (βκ−1,ηκ−1) } similarly, we can prove d (βκ,ηκ) ≤ ` max { d (ακ−1,ζκ−1) ,d (βκ−1,ηκ−1) } (3.10) combining (3.9) and (3.10), we have max { d (ακ,ζκ) ,d (βκ,ηκ) } ≤ ` max { d (ακ−1,ζκ−1) ,d (βκ−1,ηκ−1) } ≤ `2 max { d (ακ−2,ζκ−2) ,d (βκ−2,ηκ−2) } ... ≤ `κ max { d (α0,ζ0) ,d (β0,η0) } → 0 as κ →∞. (3.11) int. j. anal. appl. (2023), 21:7 7 for each κ,δ ∈n with κ < δ. then, from (3.5), (3.8), (3.11) and using property (b4), we have d (ακ,ζδ) + d (βκ,ηδ) ≤ (d (ακ,ζκ+1) + d (βκ,ηκ+1)) + (d (ακ+1,ζκ+1) + d (βκ+1,ηκ+1)) + · · · + (d (αδ−1,ζδ−1) + d (βδ−1,ηδ−1)) + (d (αδ−1,ζδ) + d (βδ−1,ηδ)) ≤ 2 ( `κ + `κ+1 + · · · + `δ−1 ) max { d (α0,ζ1) ,d (β0,η1) } +2 ( `κ+1 + `κ+2 + · · · + `δ−1 ) max { d (α0,ζ0) ,d (β0,η0) } ≤ 2`κ 1 − ` max { d (α0,ζ1) ,d (β0,η1) } + 2`κ+1 1 − ` max { d (α0,ζ1) ,d (β0,η1) } → 0 as κ →∞. similarly, we can prove that (d (αδ,ζκ) + d (βδ,ηκ)) → 0 as κ,δ →∞. then the bisequence (ακ,ζδ) and (βκ,ηδ) are cauchy bisequences in (s,t ). suppose λ(s∪t ) is complete subspace of (s,t ,d), then the sequences {ακ} ,{βκ} and {ζκ} ,{ηκ} ⊆ f (s ∪ t ) are convergence in complete bipolar metric spaces (λ(s), λ(t ),d). therefore, there exist a,b ∈ λ(s) and l,m ∈ λ(t ) such that lim κ→∞ ακ = l lim κ→∞ βκ = m lim κ→∞ ζκ = a lim κ→∞ ηκ = b. (3.12) since λ : s ∪t → s ∪t and a,b ∈ λ(s) and l,m ∈ λ(t ), there exist x,y ∈ s and p,q ∈ t such that λx = a, λy = b and λp = l, λq = m. hence lim κ→∞ ακ = l = λp lim κ→∞ βκ = m = λq lim κ→∞ ζκ = a = λx lim κ→∞ ηκ = b = λy. claim that γ(x,y) = l, γ(y,x) = m and γ(p,q) = a, γ(q,p) = b. by using (3.1), (b4), (ψ1) and (ψ2), we have ψ? (d(γ(x,y), l)) ≤ ψ? (d(γ(x,y),ζκ+1)) + ψ? (d(ακ+1,ζκ+1)) + ψ? (d(ακ+1, l)) ≤ ψ? (d(γ(x,y), γ(pκ+1,qκ+1))) + ψ? (d(ακ+1,ζκ+1)) + ψ? (d(ακ+1, l)) ≤ ∆ (ψ? (m(x,y,pκ+1,qκ+1)) ,φ? (m(x,y,pκ+1,qκ+1))) +ψ? (d(ακ+1,ζκ+1)) + ψ? (d(ακ+1, l)) ≤ ψ? ( ` max { d (λx,ζκ) ,d (λy,ηκ) }) +ψ? (d(ακ+1,ζκ+1)) + ψ? (d(ακ+1, l)) → 0 as κ →∞. it follows that ψ? (d(γ(x,y), l)) = 0 implies that d(γ(x,y), l) = 0, which deduce that γ(x,y) = l. similarly, we can prove that γ(y,x) = m and γ(p,q) = a, γ(q,p) = b. therefore, it follows that γ(x,y) = l = λp, γ(y,x) = m = λq and γ(p,q) = a = λx, γ(q,p) = b = λy. 8 int. j. anal. appl. (2023), 21:7 since {γ, λ} is ω-compatible pair, we have γ(l,m) = λl, γ(m,l) = λm and γ(a,b) = λa, γ(b,a) = λb. now we prove that λl = l, λm = m and λa = a, λb = b. now we have ψ? (d(λa,ζκ)) ≤ ψ? (d(γ(a,b), γ(pκ,qκ))) ≤ ∆ (ψ? (m(a,b,pκ,qκ)) ,φ? (m(a,b,pκ,qκ))) ≤ ψ? ( ` max { d (λa,ζκ−1) ,d (λb,ηκ−1) }) by using (ψ0), we have d(λa,ζκ) ≤ ` max { d (λa,ζκ−1) ,d (λb,ηκ−1) } letting κ →∞, we have d(λa,a) ≤ ` max { d (λa,a) ,d (λb,b) } and ψ? (d(λb,ηκ)) ≤ ψ? (d(γ(b,a), γ(qκ,pκ))) ≤ ∆ (ψ? (m(b,a,qκ,pκ)) ,φ? (m(b,a,qκ,pκ))) ≤ ψ? ( ` max { d (λb,ηκ−1) ,d (λa,ζκ−1) }) by using (ψ0), we have d(λb,ηκ) ≤ ` max { d (λb,ηκ−1,d (λa,ζκ−1)) } letting κ →∞, we have d(λb,b) ≤ ` max { d (λb,b) ,d (λa,a) } therefore, max { d (λa,a) ,d (λb,b) } ≤ ` max { d (λa,a) ,d (λb,b) } which implies that d (λa,a) = 0 and d (λb,b) = 0 and hence λa = a and λb = b. therefore, γ(a,b) = λa = a, γ(b,a) = λb = b. similarly, we can prove γ(l,m) = λl = l, γ(m,l) = λm = m. therefore, γ(p,q) = λx = a = λa = γ(a,b) γ(x,y) = λp = l = λl = γ(l,m) γ(q,p) = λy = b = λb = γ(b,a) γ(y,x) = λq = m = λm = γ(m,l) on the other hand, from (3.12), we get d (λp, λx) = d(l,a) = d ( lim κ→∞ ακ, lim κ→∞ ζκ ) = lim κ→∞ d(ακ,ζκ) = 0 and d (λq, λy) = d(m,b) = d ( lim κ→∞ βκ, lim κ→∞ ηκ ) = lim κ→∞ d(βκ,ηκ) = 0. int. j. anal. appl. (2023), 21:7 9 thus a = l,b = m. therefore, (a,b) ∈ s2 ∩t 2 is a common coupled fixed point of γ and λ. in the following we will show the uniqueness. assume that there is another coupled fixed point (a′,b′) of γ, λ. then from (3.1), we have ψ? ( d(a,a′) ) = ψ? ( d(γ(a,b), γ(a′,b′) ) ≤ ∆ ( ψ? ( m(a,b,a′,b′) ) ,φ? ( m(a,b,a′,b′) )) ≤ ψ? ( ` max { d (λa, λa′) ,d (λb, λb′) }) ≤ ψ? ( ` max { d (a,a′) ,d (b,b′) }) by the property of (ψ0), we have d(a,a′) ≤ ` max { d (a,a′) ,d (b,b′) } therefore, we have max { d (a,a′) ,d (b,b′) } ≤ ` max { d (a,a′) ,d (b,b′) } hence, we get a = a′,b = b′. therefore, (a,b) is a unique common coupled fixed point of γ and λ. finally we will show a = b. ψ? (d(a,b)) = ψ? (d(γ(a,b), γ(b,a)) ≤ ∆ (ψ? (m(a,b,b,a)) ,φ? (m(a,b,b,a))) ≤ ψ? ( ` max { d (λa, λb) ,d (λb, λa) }) ≤ ψ? ( ` max { d (a,b) ,d (b,a) }) by the property of (ψ0), we have d(a,b) ≤ ` max { d (a,b) ,d (b,a) } therefore, we have max { d (a,b) ,d (b,a) } ≤ ` max { d (a,b) ,d (b,a) } hence, we get a = b. which means that γ and λ have a unique common fixed point of the form (a,a). � corollary 3.1. let (s,t ,d) be a complete bipolar metric space. suppose that γ : ( s2,t 2 ) ⇒ (s,t ) be a covariant mapping satisfy ψ? (d(γ(u,v), γ(p,q))) ≤ ∆ ( ψ? ( ` max { d (u,p) , d (v,q) }) ,φ? ( ` max { d (u,p) , d (v,q) })) for all u,v ∈ s and p,q ∈ t and ∆ ∈ c, ψ? ∈ f, φ? ∈ g with ` ∈ (0, 1) then there is a unique coupled fixed point of γ in s∪t . 10 int. j. anal. appl. (2023), 21:7 corollary 3.2. let (s,t ,d) be a complete bipolar metric space. suppose that γ : (s×t ,t ×s) ⇒ (s,t ) be a covariant mapping satisfy ψ? (d(γ(u,p), γ(q,v))) ≤ ∆ ( ψ? ( ` max { d (u,q) , d (v,p) }) ,φ? ( ` max { d (u,q) , d (v,p) })) for all u,v ∈ s and p,q ∈ t and ∆ ∈ c, ψ? ∈ f, φ? ∈ g with ` ∈ (0, 1) then there is a unique coupled fixed point of γ in s∪t . example 3.1. let s = un(r) and t = ln(r) be the set of all n × n upper and lower triangular matrices over r. define d : s×t → [0,∞) as d(x,y ) = κ∑ i,j=1 |αij −βij| for all x = (αij)n×n ∈ un(r) and y = (βij)n×n ∈ ln(r). then obviously (s,t ,d) is a bipolarmetric space. and define γ : s2 ∪t 2 →s∪t as γ(a,b) = ( aij−bij 10 )n×n where (a = (aij)n×n,b = (bij)n×n) ∈ un(r)2 ∪ln(r)2 and define λ : s ∪t → s ∪t as `(a) = (aij 2 )n×n and let ∆ : [0, +∞) × [0. + ∞) → r as ∆(s∗,t∗) = s∗ − t∗, also define ψ? : [0,∞) → [0,∞), φ? : [0,∞) → [0,∞) as ψ?(t∗) = t∗ and φ?(t∗) = t ∗ 2 respectively. then obviously, γ(s2 ∪t 2) ⊆ λ(s∪t ) and the pairs (γ, λ) is ω-compatible. in fact, we have ψ? (d(γ(a,b), γ(x,y ))) = d(γ(a,b), γ(x,y )) = κ∑ i,j=1 | aij −bij 10 − xij −yij 10 | ≤ 1 4   κ∑ i,j=1 | aij 2 − xij 2 | + κ∑ i,j=1 | bij 2 − yij 2 |   ≤ 1 4 (d(λa, λx) + d(λb, λy )) ≤ 1 2 ( 1 2 max{d(λa, λx),d(λb, λy )} ) ≤ ∆ ( ψ? ( ` max { d(λa, λx), d(λb, λy ) }) ,φ? ( ` max { d(λa, λx), d(λb, λy ) })) thus all the conditions of the theorem (3.1) are satisfied and (on×n,on×n) is unique coupled fixed point. 3.1. application to the existence of solutions of integral equations. let s = c (l∞(e1)) ,t = c (l∞(e2)) be the set of essential bounded measurable continuous functions on e1 and e2 where e1,e2 are two lebesgue measurable sets with m(e1 ∪ e2) < ∞. define d : s×t → r+ as d(`,σ) = ||`−σ|| for all ` ∈s,σ ∈t . therefore, (s,t ,d) is a complete bipolar metric space. in this section, we apply our theorem (3.1) to establish the existence and uniqueness solution of int. j. anal. appl. (2023), 21:7 11 nonlinear integral equation defined by: x(t) = f (t) + κ ∫ e1∪e2 ω(t,`, (x,y))d`. (3.13) where x,y ∈ c (l∞(e1) ∪l∞(e2)), κ ∈ r and t,` ∈ e1 ∪e2, ω : e21 ∪e 2 2 ×l ∞(e1) 2 ∪l∞(e2)2 → r and f : e1 ∪e2 → r are given continuous functions theorem 3.2. assume that the following conditions are fulfilled (i) define, ∆ : [0, +∞) × [0. + ∞) → r as ∆(s∗,t∗) = θs∗ where θ ∈ (0, 1), let ψ? : [0,∞) → [0,∞) as ψ?(t∗) = t∗. let λ : s ∪ t → s ∪ t as λ(x) = x and γ : s2 ∪t 2 →s∪t by γ(x,y)(t) = f (t) + κ ∫ e1∪e2 ω(t,`, (x,y))d` (ii) there exists a continuous function χ : e21 ∪e 2 2 → r + such that for all x,y ∈ s,p,q ∈ t , κ ∈ r and t,` ∈ e1 ∪e2, we get that ||ω(t,`, (x,y)) − ω(t,`, (p,q))|| ≤ χ(t,`)m(x,y,p,q) where, m(x,y,p,q) = λ max{d(λx, λp),d(λy, λq)} where λ ∈ (0, 1) (iii) ||κ|| ∫ e1∪e2 χ(t,`)d` ≤ θ (iv) γ ( s2 ∪t 2 ) ⊆ λ(s∪t ), λ(s∪t ) is closed and the pair (γ, λ) is weakly compatible. then there exists unique solution in c (l∞(e1) ∪l∞(e2)) for the initial value problem 3.13. proof. the existence of a solution of (3.13) is equivalent to the existence of a common fixed point of γ and λ. obviously, γ ( s2 ∪t 2 ) ⊆ λ(s ∪t ), λ(s ∪t ) is closed and the pair (γ, λ) is weakly compatible. using the inequalities, (i), (ii) and (iii), we have ψ? (d(γ(x,y), γ(p,q))) = d(γ(x,y), γ(p,q)) = ||κ ∫ e1∪e2 (ω(t,`, (x,y)))d`−κ ∫ e1∪e2 (ω(t,`, (p,q)))d`|| ≤ ||κ|| ∫ e1∪e2 ||ω(t,`, (x,y)) − ω(t,`, (p,q))||d` ≤ ||κ|| ∫ e1∪e2 χ(t,`)m(x,y,p,q)d` ≤ ||κ||   ∫ e1∪e2 χ(t,`)d`  m(x,y,p,q) ≤ θm(x,y,p,q) ≤ ∆ ( ψ? ( λ max { d(λx, λp), d(λy, λq) }) ,φ? ( λ max { d(λx, λp), d(λy, λq) })) hence, all the conditions of theorem (3.1) hold, we conclude that γ and λ have a unique solution in s∪t to the integral equation (3.13). � 12 int. j. anal. appl. (2023), 21:7 3.2. application to the existence of solutions of homotopy. in this part, we examine the possibility that homotopy theory has a unique solution. theorem 3.3. let (s,t ,d) be complete bipolar metric space, (p,q) and (p,q) be an open and closed subset of (s,t ) such that (p,q) ⊆ (p,q). suppose h : ( p ×q ) ∪ ( q×p ) × [0, 1] →s∪t be an operator with following conditions are satisfying, `0) ℘ 6= h(℘,$,s), $ 6= h($,℘,s), for each ℘ ∈ ∂p,$ ∈ ∂q and s ∈ [0, 1] (here ∂p ∪ ∂q is boundary of p ∪q in s∪t ); `1) for all ℘,$ ∈p, ı,  ∈q, s ∈ [0, 1] and ψ? ∈ f,φ? ∈ g ∆ ∈ c and ` ∈ (0, 1) such that ψ? (d (h(℘,ı,s),h(,$,s))) ≤ ∆ ( ψ? ( ` max { d (℘,) , d ($,ı) }) ,φ? ( ` max { d (℘,) , d ($,ı) })) `2) ∃ m ≥ 0 3 d(h(℘,ı,s),h(,$,t)) � m|s − t| for every ℘,$ ∈p, ı,  ∈q and s,t ∈ [0, 1]. then h(., 0) has a coupled fixed point ⇐⇒ h(., 1) has a coupled fixed point. proof. let the set θ = { s ∈ [0, 1] : h(℘,ı,s) = ℘,h(ı,℘,s) = ı for some ℘ ∈p, ı ∈q } . υ = { t ∈ [0, 1] : h(,$,t) = ,h($,,t) = $ for some $ ∈p,  ∈q } . suppose that h(., 0) has a coupled fixed point in (p ×q) ∪ (q×p), we have that (0, 0) ∈ (θ × υ) ∩ (υ × θ). now we show that (θ × υ) ∩ (υ × θ) is both closed and open in [0, 1] and hence by the connectedness θ = υ = [0, 1]. as a result, h(., 1) has a coupled fixed point in (θ × υ) ∩ (υ × θ). first we show that (θ × υ) ∩ (υ × θ) closed in [0, 1]. to see this, let ( { a p }∞ p=1 , { x p }∞ p=1 ) ⊆ (θ, υ) and ( { y p }∞ p=1 , { b p }∞ p=1 ) ⊆ (υ, θ) with (ap,xp) → (α,β), (yp,bp) → (β,α) ∈ [0, 1] as p →∞. we must show that (α,β) ∈ (θ × υ) ∩ (υ × θ). since (ap,xp) ∈ (θ, υ), (yp,bp) ∈ (υ, θ) for p = 0, 1, 2, 3, · · · , there exists sequences ({℘p} ,{$p}) and ({ıp} ,{p}) with ℘p+1 = h(℘p,$p,ap), $p+1 = h($p,℘p,xp) and ıp+1 = h(ıp, p,yp), p+1 = h(p, ıp,bp) consider ψ? (d(℘p, p+1)) = ψ? (d (h(℘p−1,$p−1,ap−1),h(p, ıp,bp))) ≤ ∆ ( ψ? ( ` max { d (℘p−1, p) , d (ıp,$p−1) }) ,φ? ( ` max { d (℘p−1, p) , d (ıp,$p−1) })) ≤ ψ? ( ` max { d (℘p−1, p) , d (ıp,$p−1) }) by using (ψ0), we have int. j. anal. appl. (2023), 21:7 13 d(℘p, p+1) ≤ ` max { d (℘p−1, p) , d (ıp,$p−1) } similar lines we can prove that d(ıp+1,$p) ≤ ` max { d (℘p−1, p) , d (ıp,$p−1) } therefore, we get max { d (℘p, p+1) , d (ıp+1,$p) } ≤ ` max { d (℘p−1, p) , d (ıp,$p−1) } ≤ `2 max { d (℘p−2, p−1) , d (ıp−1,$p−2) } ... ≤ `p max { d (℘0, 1) , d (ı1,$0) } (3.14) similarly, we can prove max { d (℘p+1, p) , d (ıp,$p+1) } ≤ `p max { d (℘1, 0) , d (ı0,$1) } (3.15) and max { d (℘p, p) , d (ıp,$p) } ≤ `p max { d (℘0, 0) , d (ı0,$0) } (3.16) for each p,q ∈n with p < q. then, from (3.14), (3.15), (3.16) and using property (b4), we have d (℘p, q) + d (ıp,$q) ≤ (d (℘p, p+1) + d (ıp,$p+1)) + (d (℘p+1, p+1) + d (ıp+1,$p+1)) + · · · + (d (℘q−1, q−1) + d (ıq−1,$q−1)) + (d (℘q−1, q) + d (ıq−1,$q)) ≤ (m|ap−1 −bp| + m|xp −yp−1|) + · · · + (m|aq−2 −bq−1| + m|xq−1 −yq−2|) +2 ( `p+1 + `p+2 + · · · + `q−1 ) max { d (℘0, 0) , d (ı0,$0) } ≤ (m|ap−1 −bp| + m|xp −yp−1|) + · · · + (m|aq−2 −bq−1| + m|xq−1 −yq−2|) + 2`p+1 1 − ` max { d (℘0, 0) , d (ı0,$0) } → 0 as p,q →∞. 14 int. j. anal. appl. (2023), 21:7 it follows that lim p,q→∞ (d (℘p, q) + d (ıp,$q)) = 0. similarly, we can prove that lim p,q→∞ (d (℘q, p) + d (ıq,$p)) = 0. therefore, ({℘p} ,{$p}) and ({ıp} ,{p}) are cauchy bisequences in (p,q). by completeness, there exist (a,x) ∈p ×q and (y,b) ∈q×p with lim p→∞ ℘p+1 = x lim p→∞ ıp+1 = y lim p→∞ $p+1 = a lim p→∞ p+1 = b (3.17) we have d (h(b,y,α),x) ≤ d (h(b,y,α), p+1) + d(℘p+1, p+1) + d(℘p+1,x) ≤ d (h(b,y,α),h(p, ıp,bp)) + m|ap −bp| + d(℘p+1,x) letting p →∞ in the above inequality and ψ? is continuous and non-decreasing, we have ψ? (d (h(b,y,α),x)) ≤ ψ? (d (h(b,y,α),h(p, ıp,bp))) ≤ ∆ ( ψ? ( ` max { d (b, p) , d (ıp,y) }) ,φ? ( ` max { d (b, p) , d (ıp,y) })) ≤ ψ? ( ` max { d (b, p) , d (ıp,y) }) by using (ψ0) and letting as p → ∞, we get that d (h(b,y,α),x) = 0 implies that h(b,y,α) = x. similarly, we can prove that h(y,b,β) = a and h(x,a,α) = y, h(a,x,β) = b. on the other hand, from (3.17), we get d (a,y) = d ( lim p→∞ $p, lim p→∞ ıp ) = lim p→∞ d(ıp,$p) = 0 and d (b,x) = d ( lim p→∞ p, lim p→∞ ℘p ) = lim p→∞ d(℘p, p) = 0. therefore, a = y and b = x and hence (α,β) ∈ (θ × υ) ∩ (υ × θ). clearly (θ×υ)∩(υ×θ) is closed in [0, 1]. let (α0,β0) ∈ θ×υ, there exists bisequences (℘0,$0) and (ı0, 0) with ℘0 = h(℘0,$0,α0), $0 = h($0,℘0,β0) and ı0 = h(ı0, 0,β0), 0 = h(0, ı0,α0). since (p×q) ∪ (q×p) is open, then there exist δ > 0 such that bd(℘0,δ) ⊆ (p×q) ∪ (q×p), bd($0,δ) ⊆ (p×q)∪(q×p), bd(ı0,δ) ⊆ (p×q)∪(q×p) and bd(0,δ) ⊆ (p×q)∪(q×p). choose α ∈ (α0 − �,α0 + �), β ∈ (β0 − �,β0 + �) such that |α−α0| ≤ 1mp < � 2 , |β −β0| ≤ 1mp < � 2 and |α0 −β0| ≤ 1mp < � 2 . then for,  ∈ bp∪q(℘0,δ) = {, 0 ∈q/d(℘0, ) ≤ d(℘0, 0) + δ}, ı ∈ bp∪q(δ,$0) = {ı, ı0 ∈p/d(ı,$0) ≤ d(ı0,$0) + δ} ℘ ∈ bp∪q(δ, 0) = {℘,℘0 ∈p/d(℘,0) ≤ d(℘0, 0) + δ} $ ∈ bp∪q(ı0,δ) = {$,$0 ∈q/d(ı0,$) ≤ d(ı0,$0) + δ} int. j. anal. appl. (2023), 21:7 15 d (h(℘,$,α), 0)) = d (h(℘,$,α),h(0, ı0,α0)) ≤ d (h(℘,$,α),h(, ı,α0)) + d (h(℘0,$0,α),h(, ı,α0)) +d (h(℘0,$0,α),h(0, ı0,α0)) ≤ 2m|α−α0| + d (h(℘0,$0,α),h(, ı,α0)) ≤ 2 mp−1 + d (h(℘0,$0,α),h(, ı,α0)) letting p →∞ and using (ψ0), then we have ψ? (d (h(℘,$,α), 0))) ≤ ψ? (d (h(℘0,$0,α),h(, ı,α0))) ≤ ∆ ( ψ? ( ` max { d (℘0, ) , d (ı,$0) }) ,φ? ( ` max { d (℘0, ) , d (ı,$0) })) ≤ ψ? ( ` max { d (℘0, ) , d (ı,$0) }) using the property of ψ?, we get d (h(℘,$,α), 0)) ≤ ` max { d (℘0, ) , d (ı,$0) } similarly we can prove d (ı0,h($,℘,β))) ≤ ` max { d (℘0, ) , d (ı,$0) } therefore, max { d (h(℘,$,α), 0)) , d (ı0,h($,℘,β))) } ≤ ` max { d (℘0, ) , d (ı,$0) } ≤ ` max { d (℘0, 0) + δ, d (ı0,$0) + δ } thus, d (h(℘,$,α), 0)) ≤ d (℘0, 0) + δ and d (ı0,h($,℘,β))) ≤ d (ı0,$0) + δ. similarly, we can prove d (h(ı, ,β),$0)) ≤ d (ı0,$0) + δ and d (℘0,h(, ı,α))) ≤ d (℘0, 0) + δ. on the other hand, d(℘0,$0) = d (h(℘0,$0,α0),h($0,℘0,β0)) ≤ m|α0 −β0| < 1 mp−1 → 0 as p →∞. and d(ı0, 0) = d (h(ı0, 0,β0),h(0, ı0,α0)) ≤ m|α0 −β0| < 1 mp−1 → 0 as p →∞. 16 int. j. anal. appl. (2023), 21:7 so ℘0 = $0 and ı0 = 0 and hence α = β. thus for each fixed α ∈ (α0 − �,α0 + �), h(.,α) : bθ∪υ(℘0,δ) → bθ∪υ(℘0,δ) and h(.,α) : bθ∪υ(ı0,δ) → bθ∪υ(ı0,δ). thus, we conclude that h(.,α) has a coupled fixed point in (p×q)∩(q×p). but this must be in (p×q)∪(q×p). therefore, (α,α) ∈ (θ × υ) ∩ (υ × θ) for α ∈ (α0 −�,α0 + �).hence, (α0 −�,α0 + �) ⊆ (θ × υ) ∩ (υ × θ). clearly, (θ×υ)∩(υ×θ) is open in [0, 1]. for the reverse implication, we use the same strategy. � theorem 3.4. let (s,t ,d) be complete bipolar metric space, (p,q) and (p,q) be an open and closed subset of (s,t ) such that (p,q) ⊆ (p,q). suppose h : ( p2 ∪q2 ) × [0, 1] →s∪t be an operator with following conditions are satisfying, `0) ℘ 6= h(℘,$,s), $ 6= h($,℘,s), for each ℘,$ ∈ ∂p ∪ ∂q and s ∈ [0, 1] (here ∂p ∪ ∂q is boundary of p ∪q in s∪t ); `1) for all ℘,$ ∈p, ı,  ∈q, s ∈ [0, 1] and ψ? ∈ f,φ? ∈ g ∆ ∈ c and ` ∈ (0, 1) such that ψ? (d (h(℘,$,s),h(ı, ,s))) ≤ ∆ ( ψ? ( ` max { d (℘,ı) , d ($,) }) ,φ? ( ` max { d (℘,ı) , d ($,) })) `2) ∃ m ≥ 0 3 d(h(℘,$,s),h(ı, , t)) � m|s − t| for every ℘,$ ∈p, ı,  ∈q and s,t ∈ [0, 1]. then h(., 0) has a coupled fixed point ⇐⇒ h(., 1) has a coupled fixed point. conclusion we ensured the existence and uniqueness of a common coupled fixed point for two covariant mappings in the class of complete bipolar metric spaces with examples via c-class functions. two illustrated application has been provided. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal.: theory methods appl. 11 (1987), 623-632. https://doi.org/10.1016/0362-546x(87)90077-0. [2] t.g. bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal.: theory methods appl. 65 (2006), 1379-1393. https://doi.org/10.1016/j.na.2005.10.017. [3] m. abbas, m. ali khan, s. radenovic, common coupled fixed point theorems in cone metric spaces for wcompatible mappings, appl. math. comput. 217 (2010), 195-202. https://doi.org/10.1016/j.amc.2010.05.042. [4] a. aghajani, m. abbas, e. pourhadi kallehbasti, coupled fixed point theorems in partially ordered metric spaces and application, math. commun. 17 (2012), 497-509. https://hrcak.srce.hr/clanak/137302. [5] e. karapinar, coupled fixed point on cone metric spaces, gazi univ. j. sci. 24 (2011), 51-58. https: //dergipark.org.tr/en/pub/gujs/issue/7418/96917. [6] m. abbas, b. ali, y.i. suleiman, generalized coupled common fixed point results in partially ordered a-metric spaces, fixed point theory appl. 2015 (2015), 64. https://doi.org/10.1186/s13663-015-0309-2. [7] w. long, b.e. rhoades, m. rajovic, coupled coincidence points for two mappings in metric spaces and cone metric spaces, fixed point theory appl. 2012 (2012), 66. https://doi.org/10.1186/1687-1812-2012-66. https://doi.org/10.1016/0362-546x(87)90077-0 https://doi.org/10.1016/j.na.2005.10.017 https://hrcak.srce.hr/clanak/137302 https://dergipark.org.tr/en/pub/gujs/issue/7418/96917 https://dergipark.org.tr/en/pub/gujs/issue/7418/96917 https://doi.org/10.1186/s13663-015-0309-2 https://doi.org/10.1186/1687-1812-2012-66 int. j. anal. appl. (2023), 21:7 17 [8] m. jain, s. kumar, r. chugh, coupled fixed point theorems for weak compatible mappings in fuzzy metric spaces, ann. fuzzy math. inform. 5 (2013), 321-336. [9] m. kir, e. yolacan, h. kiziltunc, coupled fixed point theorems in complete metric spaces endowed with a directed graph and application, open math. 15 (2017), 734-744. https://doi.org/10.1515/math-2017-0062. [10] a.h. ansari, note on ϕ−ψ-contractive type mappings and related fixed point, in: the 2nd regional conference on mathematics and applications, payame noor university, 2014. [11] a.h. ansari, a. kaewcharoen, c-class functions and fixed point theorems for generalized ℵ−η −ψ −ϕ−fcontraction type mappings in ℵ−η-complete metric spaces, j. nonlinear sci. appl. 9 (2016), 4177-4190. [12] h. huang, g. deng, s. radenovic, fixed point theorems for c-class functions in b-metric spaces and applications, j. nonlinear sci. appl. 10 (2017), 5853-5868. [13] a.h. ansari, w. shatanawi, a. kurdi, g. maniu, best proximity points in complete metric spaces with (p)property via c-class functions, j. math. anal. 7 (2016), 54-67. [14] v. ozturk, a.h. ansari, common fixed point theorems for mappings satisfying (e.a)-property via c-class functions in b-metric spaces, appl. gen. topol. 18 (2017), 45-52. https://doi.org/10.4995/agt.2017.4573. [15] t. hamaizia, common fixed point theorems involving c-class functions in partial metric spaces, sohag j. math. 8 (2021), 23–28. https://doi.org/10.18576/sjm/080103. [16] g.s. saluja, common fixed point theorems on s-metric spaces via c-class functions, int. j. math. combin. 3 (2022), 21-37. [17] w. shatanawi, m. postolache, a. h. ansari, w. kassab, common fixed points of dominating and weak annihilators in ordered metric spaces via c-class functions, j. math. anal. 8 (2017), 54-68. [18] a. mutlu, u. gürdal, bipolar metric spaces and some fixed point theorems, j. nonlinear sci. appl. 9 (2016), 5362-5373. [19] a. mutlu, k. özkan, u. gürdal, coupled fixed point theorems on bipolar metric spaces, eur. j. pure appl. math. 10 (2017), 655-667. [20] g.n.v. kishore, r.p. agarwal, b. srinuvasa rao, r.v.n. srinivasa rao, caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications, fixed point theory appl. 2018 (2018), 21. https://doi.org/10.1186/s13663-018-0646-z. [21] g.n.v. kishore, b. srinuvasa rao, r.s. rao, mixed monotone property and tripled fixed point theorems in partially ordered bipolar metric spaces, italian j. pure appl. math. 42 (2019), 598-615. [22] g.n.v. kishore, b.s. rao, s. radenović, h. huang, caristi type cyclic contraction and coupled fixed point results in bipolar metric spaces, sahand commun. math. anal. 17 (2020), 1-22. https://doi.org/10.22130/ scma.2018.79219.369. [23] g.n.v. kishore, k.p.r. rao, h. isik, b. srinuvasa rao, a. sombabu, covarian mappings and coupled fixed point results in bipolar metric spaces, int. j. nonlinear anal. appl. 12 (2021),1-15. https://doi.org/10.22075/ijnaa. 2021.4650. [24] g.n.v. kishorea, h. işık, , h. aydic, b.s. rao, d.r. prasad, on new types of contraction mappings in bipolar metric spaces and applications, j. linear topol. algebra, 9 (2020), 253-266. [25] m.s. khan, m. swaleh, s. sessa, fixed point theorems by altering distances between the points, bull. austral. math. soc. 30 (1984), 1-9. https://doi.org/10.1017/s0004972700001659. https://doi.org/10.1515/math-2017-0062 https://doi.org/10.4995/agt.2017.4573 https://doi.org/10.18576/sjm/080103 https://doi.org/10.1186/s13663-018-0646-z https://doi.org/10.22130/scma.2018.79219.369 https://doi.org/10.22130/scma.2018.79219.369 https://doi.org/10.22075/ijnaa.2021.4650 https://doi.org/10.22075/ijnaa.2021.4650 https://doi.org/10.1017/s0004972700001659 1. introduction 2. preliminaries 3. main results 3.1. application to the existence of solutions of integral equations 3.2. application to the existence of solutions of homotopy references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 1-10 http://www.etamaths.com donoho-stark uncertainty principle associated with a singular second-order differential operator fethi soltani abstract. for a class of singular second-order differential operators ∆, we prove a continuous-time principles for l1 theory and l2 theory, respectively. another version of continuous-time principle using l1 ∩ l2 theory is given. 1. introduction the classical uncertainty principle says that if a function f(t) is essentially zero outside an interval of length δt and its fourier transform f̂(w) is essentially zero outside an interval of length δw, then δt.δw ≥ 1; a function and its fourier transform cannot both be highly concentrated. the uncertainty principle is widely known for its ”philosophical” applications: in quantum mechanics, of course, it shows that a particle’s position and momentum cannot be determined simultaneously [10]; in signal processing it establishes limits on the extent to which the ”instantaneous frequency” of a signal can be measured [9]. however, it also has technical applications, for example in the theory of partial differential equations [8]. here we consider the second-order differential operator defined on ]0,∞[ by ∆u = u′′ + a′ a u′ + ρ2u, where a is a nonnegative function satisfying certain conditions and ρ is a nonnegative real number. this operator plays an important role in analysis. for example, many special functions (orthogonal polynomials) are eigenfunctions of an operator of ∆ type. the radial part of the beltrami-laplacian in a symmetric space is also of ∆ type. many aspects of such operators have been studied; we mention, in chronological order, in 1979 chébli [2]; in 1981 trimèche [15]; in 1989 zeuner [18]; in 1994 xu [17]; in 1997 trimèche [16]; in 1998 nessibi et al. [13]. in particular, the first two of these references investigate standard constructions of harmonic analysis, such as translation operators, convolution product, and fourier transform, in connection with ∆. 2010 mathematics subject classification. 42b10; 42b30; 33c45. key words and phrases. generalized fourier transform; l1 uncertainty principle; l2 uncertainty principles. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 fethi soltani many uncertainty principles have already been proved for the sturm-liouville operarator ∆, namely by rösler and voit [14] who established an uncertainty principle for hankel transforms. bouattour and trimèche [1] proved a beurling’s theorem for the sturm-liouville transform. daher et al. [3, 4, 5, 6] give some related versions of the uncertainty principle for the sturm-liouville transform (titchmarsh’s theorem, hardy’s theorem and miyachi’s theorem). ma [11, 12] proved a heisenberg uncertainty principle for the sturm-liouville transform. building on the ideas of donoho and stark [7] we show a continuous-time principle for the l1 theory. the analogous of this uncertainty principle in the l2 theory is also given. we prove another versions of continuous-time principle for the l2 theory and for the l1 ∩l2 theory. this paper is organized as follows. in section 2 we recall some basic properties of the fourier transform f associated to ∆ (plancherel theorem, inversion formula,...). in section 3 we prove a continuous-time principle for l1 theory. the last section of this paper is devoted to show another versions of continuous-time principles using l2 theory and l1 ∩l2 theory. 2. the operator ∆ we consider the second-order differential operator ∆ defined on ]0,∞[ by ∆u = u′′ + a′ a u′ + ρ2u, where ρ is a nonnegative real number and a(x) = x2α+1b(x), α > −1/2, for b a positive, even, infinitely differentiable function on r such that b(0) = 1. moreover we assume that a and b satisfy the following conditions: (i) a is increasing and lim x→∞ a(x) = ∞. (ii) a′ a is decreasing and lim x→∞ a′(x) a(x) = 2ρ. (iii) there exists a constant δ > 0 such that a′(x) a(x) = 2ρ + d(x) exp(−δx) if ρ > 0, a′(x) a(x) = 2α + 1 x + d(x) exp(−δx) if ρ = 0, where d is an infinitely differentiable function on ]0,∞[, bounded and with bounded derivatives on all intervals [x0,∞[, for x0 > 0. this operator was studied in [2, 13, 15], and the following results have been established: (i) for all λ ∈ c, the equation{ ∆u = −λ2u u(0) = 1, u′(0) = 0 admits a unique solution, denoted by ϕλ, with the following properties: ϕλ satisfies the product formula ϕλ(x)ϕλ(y) = ∫ ∞ 0 ϕλ(z)w(x,y,z)a(z)dz for x,y ≥ 0; donoho-stark uncertainty principle 3 where w(x,y, .) is a measurable positive function on [0,∞[, with support in [|x − y|,x + y], satisfying ∫ ∞ 0 w(x,y,z)a(z)dz = 1, w(x,y,z) = w(y,x,z) for z ≥ 0, w(x,y,z) = w(x,z,y) for z > 0; for x ≥ 0, the function λ → ϕλ(x) is analytic on c; for λ ∈ c, the function x → ϕλ(x) is even and infinitely differentiable on r; for all λ,x ∈ r, |ϕλ(x)| ≤ 1; (2.1) for all λ,x > 0, ϕλ(x) = 1√ b(x) jα(λx) + 1√ a(x) θλ(x), where jα is defined by jα(0) = 1 and jα(s) = 2 αγ(α + 1)s−αjα(s) if s 6= 0 (with jα the bessel function of first kind), and the function θλ satisfies |θλ(x)| ≤ c1 λα+ 3 2 (∫ x 0 |q(s)|ds ) exp (c2 λ ∫ x 0 |q(s)|ds ) with c1,c2 positive constants and q the function defined on ]0,∞[ by q(x) = 1 4 −α2 x2 + 1 4 (a′(x) a(x) )2 + 1 2 (a′(x) a(x) )′ −ρ2. (ii) for nonzero λ ∈ c, the equation ∆u = −λ2u has a solution φλ satisfying φλ(x) = 1√ a(x) exp(iλx)v (x,λ), with limx→∞v (x,λ) = 1. consequently there exists a function (spectral function) λ 7→ c(λ), such that ϕλ = c(λ)φλ + c(−λ)φ−λ for nonzero λ ∈ c. moreover there exist positive constants k1,k2,k3 such that k1|λ|α+1/2 ≤ |c(λ)|−1 ≤ k2|λ|α+1/2 for all λ such that imλ ≤ 0 and |λ| ≥ k3. notation 2.1. we denote by µ the measure defined on [0,∞[ by dµ(x) := a(x)dx; and by lp(µ), 1 ≤ p ≤∞, the space of measurable functions f on [0,∞[, such that ‖f‖lp(µ) := (∫ ∞ 0 |f(x)|pdµ(x) )1/p < ∞, 1 ≤ p < ∞, ‖f‖l∞(µ) := ess sup x∈[0,∞[ |f(x)| < ∞; ν the measure defined on [0,∞[ by dν(λ) := dλ 2π|c(λ)|2 ; and by lp(ν), 1 ≤ p ≤∞, the space of measurable functions f on [0,∞[, such that ‖f‖lp(ν) < ∞. 4 fethi soltani the fourier transform associated with the operator ∆ is defined on l1(µ) by f(f)(λ) := ∫ ∞ 0 ϕλ(x)f(x)dµ(x) for λ ∈ r. some of the properties of the fourier transform f are collected bellow (see [2, 13, 15, 16, 17]). (a) l1 −l∞-boundedness. for all f ∈ l1(µ), f(f) ∈ l∞(ν) and ‖f(f)‖l∞(ν) ≤‖f‖l1(µ). (2.2) (b) inversion theorem. let f ∈ l1(µ), such that f(f) ∈ l1(ν). then f(x) = ∫ ∞ 0 ϕλ(x)f(f)(λ)dν(λ), a.e. x ∈ [0,∞[. (2.3) (c) plancherel theorem. the dunkl transform f extends uniquely to an isometric isomorphism of l2(µ) onto l2(ν). in particular, ‖f‖l2(µ) = ‖f(f)‖l2(ν). (2.4) let t be measurable set of [0,∞[. we introduce the time-limiting operator pt by ptf(t) := { f(t), t ∈ t 0, t ∈ [0,∞[\t. (2.5) this operator is bounded from lp(µ), 1 ≤ p ≤∞ into itself and ‖ptf‖lp(µ) ≤‖f‖lp(µ), f ∈ lp(µ). (2.6) let w be measurable set of [0,∞[. we introduce the partial sum operator sw by f(swf) = f(f)1w . (2.7) this operator is bounded from l2(µ) into itself and ‖swf‖l2(µ) ≤‖f‖l2(µ), f ∈ l2(µ). (2.8) theorem 2.2. if ν(w) < ∞ and f ∈ l1(µ) or f ∈ l2(µ), swf(x) = ∫ w ϕλ(x)f(f)(λ)dν(λ). (2.9) proof. if f ∈ l1(µ), then by (2.2), ‖f(f)1w‖l1(ν) = ∫ w |f(f)(w)|dν(w) ≤ ν(w)‖f‖l1(µ), and ‖f(f)1w‖l2(ν) = (∫ w |f(f)(w)|2dν(w) )1/2 ≤ √ ν(w)‖f‖l1(µ). thus fk(f)1w ∈ l1(ν) ∩l2(ν) and by (2.7), swf = f−1 ( f(f)1w ) . this combined with (2.3) gives the result. if f ∈ l2(µ), then by (2.4), ‖f(f)1w‖l1(ν) ≤ √ ν(w)‖f‖l2(µ), and ‖f(f)1w‖l2(ν) ≤‖f‖l2(µ). donoho-stark uncertainty principle 5 thus f(f)1w ∈ l1(ν) ∩l2(ν). this yields the desired result. � 3. an l1 uncertainty principle let t and w be measurable sets of [0,∞[. we say that a function f ∈ l1(µ) is ε-concentrated to t if there is a measurable function g(t) vanishing outside t such that ‖f −g‖l1(µ) ≤ ε‖f‖l1(µ). if f is εt -concentrated on t in l 1(µ)-norm (g being the vanishing function) then ‖f −ptf‖l1(µ) = ∫ [0,∞[\t |f(t)|dµ(t) ≤‖f −g‖l1(µ) ≤ εt‖f‖l1(µ) and therefore f is εt -concentrated to t in l 1(µ)-norm if and only if ‖f−ptf‖l1(µ) ≤ εt‖f‖l1(µ). let b1(w) denote the set of functions g ∈ l1(µ) that are bandlimited to w (i.e. g ∈ b1(w) implies swg = g). we say that f is ε-bandlimited to w in l1(µ)-norm if there is a g ∈ b1(w) with ‖f −g‖l1(µ) ≤ ε‖f‖l1(µ). the space b1(w) satisfies the following property. lemma 3.1. let t and w be measurable sets of [0,∞[. for g ∈ b1(w), ‖ptg‖l1(µ) ‖g‖l1(µ) ≤ µ(t)ν(w). proof. if µ(t) = ∞ or ν(w) = ∞, the inequality is clear. assume that µ(t) < ∞ and ν(w) < ∞. for g ∈ b1(w), from theorem 2.2, g(t) = ∫ w ϕw(t)f(g)(w)dν(w) and by (2.1) and (2.2), |g(t)| ≤ ν(w)‖g‖l1(µ). hence ‖ptg‖l1(µ) = ∫ t |g(t)|dµ(t) ≤ µ(t)ν(w)‖g‖l1(µ). therefore, for g ∈ b1(w), ‖ptg‖l1(µ) ‖g‖l1(µ) ≤ µ(t)ν(w), which yields the result. � it is useful to have uncertainty principle for the l1(µ)-norm. theorem 3.2. let t and w be measurable sets of [0,∞[ and f ∈ l1(µ). if f is εt -concentrated to t and εw -bandlimited to w in l 1(µ)-norm, then µ(t)ν(w) ≥ 1 −εt −εw 1 + εw . proof. let f ∈ l1(µ). the triangle inequality gives ‖ptf‖l1(µ) ≥‖f‖l1(µ) −‖f −ptf‖l1(µ). since f is εt -concentrated to t in l 1(µ)-norm, ‖ptf‖l1(µ) ≥ (1 −εt )‖f‖l1(µ). (3.1) 6 fethi soltani on the other hand, f is εw -bandlimited to w in l 1(µ)-norm, by definition there is a g in b1(w) with ‖f −g‖l1(µ) ≤ εw‖f‖l1(µ). for this g and by (2.6), we have ‖ptg‖l1(µ) ≥ ‖ptf‖l1(µ) −‖pt (f −g)‖l1(µ) ≥ ‖ptf‖l1(µ) −εw‖f‖l1(µ) and also ‖g‖l1(µ) ≤ (1 + εw )‖f‖l1(µ). so that ‖ptg‖l1(µ) ‖g‖l1(µ) ≥ ‖ptf‖l1(µ) −εw‖f‖l1(µ) (1 + εw )‖f‖l1(µ) . thus, by (3.1) we deduce ‖ptg‖l1(µ) ‖g‖l1(µ) ≥ 1 −εt −εw 1 + εw . this combined with lemma 3.1 proves theorem 3.2. � 4. an l2 uncertainty principles let t and w be measurable sets of [0,∞[. we say that a function f ∈ l2(µ) is ε-concentrated to t if there is a measurable function g(t) vanishing outside t such that ‖f −g‖l2(µ) ≤ ε‖f‖l2(µ). similarly, we say that f(f) is ε-concentrated to w if there is a function h(w) vanishing outside w with ‖f(f) −h‖l2(ν) ≤ ε‖f‖l2(µ). if f is εt -concentrated to t in l 2(µ)-norm (g being the vanishing function) then ‖f −ptf‖l2(µ) = (∫ [0,∞[\t |f(t)|2dµ(t) )1/2 ≤‖f −g‖l2(µ) ≤ εt‖f‖l2(µ) and therefore f is εt -concentrated to t in l 2(µ)-norm if and only if ‖f−ptf‖l2(µ) ≤ εt‖f‖l2(µ). from (2.7) it follows as for pt that f(f) is εw -concentrated to w in l2(ν)-norm if and only if ‖f(f) −f(swf)‖l2(ν) = ‖f −swf‖l2(µ) ≤ εw‖f‖l2(µ). let b2(w) denote the set of functions g ∈ l2(µ) that are bandlimited to w (i.e. g ∈ b2(w) implies swg = g). we say that f is ε-bandlimited to w in l2(µ)-norm if there is a g ∈ b2(w) with ‖f −g‖l2(µ) ≤ ε‖f‖l2(µ). the space b2(w) satisfies the following property. lemma 4.1. let t and w be measurable sets of [0,∞[. for g ∈ b2(w), ‖ptg‖l2(µ) ‖g‖l2(µ) ≤ √ µ(t)ν(w). proof. assume that µ(t) < ∞ and ν(w) < ∞. for g ∈ b2(w), from (2.9), g(t) = ∫ w ϕw(t)f(g)(w)dν(w) and by (2.1) and hölder’s inequality, |g(t)| ≤ √ ν(w)‖g‖l2(µ). donoho-stark uncertainty principle 7 hence ‖ptg‖l2(µ) = (∫ t |g(t)|2dµ(t) )1/2 ≤ √ µ(t)ν(w)‖g‖l2(µ). therefore, for g ∈ b2(w), ‖ptg‖l2(µ) ‖g‖l2(µ) ≤ √ µ(t)ν(w) , which yields the result. � it is useful to have uncertainty principle for the l2(µ)-norm. theorem 4.2. let t and w be measurable sets of [0,∞[ and f ∈ l2(µ). if f is εt -concentrated to t and εw -bandlimited to w in l 2(µ)-norm, then√ µ(t)ν(w) ≥ 1 −εt −εw 1 + εw . proof. let f ∈ l2(µ). the triangle inequality gives ‖ptf‖l2(µ) ≥‖f‖l2(µ) −‖f −ptf‖l2(µ). since f is εt -concentrated to t in l 2(µ)-norm, ‖ptf‖l2(µ) ≥ (1 −εt )‖f‖l2(µ). (4.1) on the other hand, f is εw -bandlimited to w in l 2(µ)-norm, by definition there is a g in b2(w) with ‖f −g‖l2(µ) ≤ εw‖f‖l2(µ). for this g and by (2.6), we have ‖ptg‖l2(µ) ≥ ‖ptf‖l2(µ) −‖pt (f −g)‖l2(µ) ≥ ‖ptf‖l2(µ) −εw‖f‖l2(µ) and also ‖g‖l2(µ) ≤ (1 + εw )‖f‖l2(µ). so that ‖ptg‖l2(µ) ‖g‖l2(µ) ≥ ‖ptf‖l2(µ) −εw‖f‖l2(µ) (1 + εw )‖f‖l2(µ) . thus, by (4.1) we deduce ‖ptg‖l2(µ) ‖g‖l2(µ) ≥ 1 −εt −εw 1 + εw . this combined with lemma 4.1 proves theorem 4.2. � lemma 4.3. let t and w be measurable sets of [0,∞[. for f ∈ l2(µ), ‖swptf‖l2(µ) ‖f‖l2(µ) ≤ √ µ(t)ν(w). proof. assume that µ(t) < ∞ and ν(w) < ∞. let f ∈ l2(µ). from (2.5) and (2.9), swptf(s) = ∫ w ϕw(s)f(ptf)(w)dν(w) = ∫ w ϕw(s) ∫ t ϕw(t)f(t)dµ(t)dν(w). since by (2.1),∫ w ∫ t ∣∣∣ϕw(s)ϕw(t)f(t)∣∣∣dµ(t)dν(w) ≤ ν(w)√µ(t)‖f‖l2(µ) < ∞ 8 fethi soltani by fubini’s theorem, swptf(s) = ∫ t f(t) ∫ w ϕw(s)ϕw(t)dν(w)dµ(t), so that swptf(s) = ∫ t q(s,t)f(t)dµ(t), (4.2) where q(s,t) = ∫ w ϕw(s)ϕw(t)dν(w), t ∈ t,s ∈ [0,∞[. for t ∈ t , let gt(s) = q(s,t) = ∫ w ϕw(s)ϕw(t)dν(w). then the inversion formula (2.3) shows that f(gt)(w) = 1wϕw(t). by plancherel’s formula (2.4) it then follows∫ ∞ 0 |q(s,t)|2dµ(s) = ∫ ∞ 0 |gt(s)|2dµ(s) = ∫ ∞ 0 |f(gt)(w)|2dν(w) ≤ ν(w). by applying hölder’s inequality to (4.2), |swptf(s)|2 ≤‖f‖2l2(µ) ∫ t |q(s,t)|2dµ(t). hence ‖swptf‖l2(µ) ≤‖f‖l2(µ) (∫ ∞ 0 ∫ t |q(s,t)|2dµ(t)dµ(s) )1/2 . by fubini-tonnelli’s theorem, ‖swptf‖l2(µ) ≤‖f‖l2(µ) (∫ t ∫ ∞ 0 |q(s,t)|2dµ(s)dµ(t) )1/2 ≤‖f‖l2(µ) √ µ(t)ν(w). thus, the proof is complete. � another uncertainty principle for l2(µ)-norm is obtained. theorem 4.4. let t and w be measurable sets of [0,∞[ and f ∈ l2(µ). if f is εt -concentrated to t in l 2(µ)-norm and f(f) is εw -concentrated to w in l2(ν)-norm, then √ µ(t)ν(w) ≥ 1 −εt −εw . proof. let f ∈ l2(µ). from (2.8) it follows ‖f −swptf‖l2(µ) ≤ ‖f −swf‖l2(µ) + ‖swf −swptf‖l2(µ) ≤ εw‖f‖l2(µ) + ‖f −ptf‖l2(µ) ≤ (εt + εw )‖f‖l2(µ). the triangle inequality gives ‖swptf‖l2(µ) ≥‖f‖l2(µ) −‖f −swptf‖l2(µ) ≥ (1 −εw −εt )‖f‖l2(µ). it then follows that ‖swptf‖l2(µ) ≥ (1−εw −εt )‖f‖l2(µ). the lemma 4.3 show that √ µ(t)ν(w)‖f‖l2(µ) ≥ (1 −εt −εw )‖f‖l2(µ), which gives the desired result. � donoho-stark uncertainty principle 9 an uncertainty principle for l1(µ) ∩l2(µ) theory is obtained. theorem 4.5. let t and w be measurable sets of [0,∞[ and f ∈ l1(µ) ∩l2(µ). if f is εt -concentrated to t in l 1(µ)-norm and f(f) is εw -concentrated to w in l2(ν)-norm, then √ µ(t)ν(w) ≥ (1 −εt )(1 −εw ). proof. assume that µ(t) < ∞ and ν(w) < ∞. let f ∈ l1(µ) ∩l2(µ). since f(f) is εw -concentrated to w in l2(ν)-norm, then ‖f‖l2(µ) ≤ εw‖f‖l2(µ) + (∫ w |f(f)(w)|2dν(w) )1/2 ≤ εw‖f‖l2(µ) + √ ν(w)‖f(f)‖l∞(ν). thus by (2.2), (1 −εw )‖f‖l2(µ) ≤ √ ν(w)‖f‖l1(µ). (4.3) on the other hand, since f is εt -concentrated to t in l 1(µ)-norm, ‖f‖l1(µ) ≤ εt‖f‖l1(µ) + ∫ t |f(t)|dµ(t) ≤ εt‖f‖l1(µ) + √ µ(t)‖f‖l2(µ). thus (1 −εt )‖f‖l1(µ) ≤ √ µ(t)‖f‖l2(µ). (4.4) combining (4.3) and (4.4) we obtain the result of this theorem. � references [1] l. bouattour and k. trimèche, beurling-hörmander’s theorem for the chébli-trimèche transform, glob. j. pure appl. math. 1(3) (2005) 342–357 [2] h. chébli, théorème de paley-wiener associ??un op érateur différentiel singulier sur (0, ∞), j. math. pures appl. 58(1) (1979) 1–19. [3] r. daher, an analog of titchmarsh’s theorem of jacobi transform, int. j. math. anal. 6(20) (2012) 975 –981. [4] r. daher and t. kawazoe, generalized of hardy’s theorem for jacobi transform, hiroshima j. math. 36(3) (2006) 331–337. [5] r. daher and t. kawazoe, an uncertainty principle on sturm-liouville hypergroups, proc. japan acad. 83 ser. a (2007) 167–169. [6] r. daher, t. kawazoe and h. mejjaoli, a generalization of miyachi’s theorem, j. math. soc. japan 61(2) (2009) 551–558. [7] d.l. donoho and p.b. stark, uncertainty principles and signal recovery, siam j. appl. math. 49(3) (1989) 906–931. [8] c.l. fefferman, the uncertainty principle, bull. amer. math. soc. 9 (1983) 129–206. [9] d. gabor, theory of communication, j. inst. elec. engrg. 93 (1946) 429–457. [10] w. heisenberg, the physical principles of the quantum theory, dover, newyork, 1949 (the university of chicago press, 1930). [11] r. ma, heisenberg inequalities for jacobi transforms, j. math. anal. appl. 332 (2007) 155– 163. [12] r. ma, heisenberg uncertainty principle on chébli-trimèche hypergroups, pacific j. math. 235(2) (2008) 289–296. [13] m.m. nessibi, l.t. rachdi and k. trimèche, the local central limit theorem on the product of the chébli-trimèche hypergroup and the euclidean hypergroup rn, j. math. sci. (calcutta) 9(2) (1998) 109–123. [14] m. rösler and m. voit, an uncertainty principle for hankel transforms, proc. amer. math. soc. 127(1) (1999) 183–194. [15] k. trimèche, transformation intégrale de weyl et théorème de paley-wiener associés ?un opérateur différentiel singulier sur (0, ∞), j. math. pures appl. 60(1) (1981) 51–98. 10 fethi soltani [16] k. trimèche, inversion of the lions transmutation operators using generalized wavelets, appl. comput. harmon. anal. 4(1) (1997) 97–112. [17] z. xu, harmonic analysis on chébli-trimèche hypergroups, ph.d. thesis, murdoch university, perth, western australia, 1994. [18] h. zeuner, the central limit theorem for chébli-trimèche hypergroups, j. theoret. probab. 2(1) (1989) 51–63. department of mathematics, faculty of science, jazan university, p.o.box 114, jazan, kingdom of saudi arabia author partially supported by dgrst project 04/ur/15-02 and cmcu program 10g 1503 international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 108-113 http://www.etamaths.com a note on absolute cesàro ϕ−|c, 1; δ; l|k summability factor smita sonker1, xh. z. krasniqi2,∗ and alka munjal1 abstract. a positive non-decreasing sequence has been used to establish a theorem on a minimal set of sufficient conditions for an infinite series to be absolute cesàro ϕ−|c, 1; δ; l|k summable. for some well-known applications, suitable conditions have been applied on the presented theorem for obtaining the sub-result of the presented theorem. 1. introduction let {sn} be a sequence of partial sums of the series ∞∑ n=0 an and n th mean of {sn} is given by tn s.t. tn = ∞∑ k=0 tnksk (1.1) where {tnk} is the sequence of the coefficients of the matrix. if sequence of the means {tn} satisfied the following conditions: lim n→∞ tn = s, (1.2) and ∞∑ n=1 | tn − tn−1| < ∞, (1.3) then the series ∞∑ n=0 an is said to be absolute summable. if τn represent the n th (c, 1) means of the sequence (nan), then series ∞∑ n=0 an is said to be summable |c, 1|k,k ≥ 1 [9], if ∞∑ n=1 1 n |τn|k < ∞. (1.4) if the sequence {τn} satisfied the condition: ∞∑ n=1 ϕk−1n nk |τn|k < ∞, (1.5) then the series ∞∑ n=0 an is said to be summable ϕ−|c, 1|k, k ≥ 1, and if the sequence {τn} satisfied the following condition: ∞∑ n=1 ϕk−1n nk−δk |τn|k < ∞, (1.6) then the series ∞∑ n=0 an is ϕ−|c, 1; δ|k, summable, where k ≥ 1, δ ≥ 0 and (ϕn) be a sequence of positive real numbers. received 27th may, 2017; accepted 1st august, 2017; published 1st september, 2017. 2010 mathematics subject classification. 40f05, 40d15, 40g05. key words and phrases. absolute summability; infinite series; ϕ−|c, 1; δ; l|k summability; sequence space. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 108 absolute cesàro ϕ−|c, 1; δ; l|k summability 109 for ϕ−|c, 1; δ; l|k summability, the infinite series ∞∑ n=0 an satisfied ∞∑ n=1 ϕ l(k−1) n nl(k−δk) |τn|k < ∞ (1.7) where k ≥ 1, δ ≥ 0 and l is a real number. note: if we take l = 1, then ϕ − |c, 1; δ; l|k reduces to ϕ − |c, 1,δ|k, if ϕ = n, then ϕ − |c, 1; δ|k summability reduces to |c, 1; δ|k summability and if δ = 0, then |c, 1; δ|k reduces to |c, 1|k. in 1972, mazhar [8] determined the minimal set of sufficient conditions for an infinite series to be absolute |c, 1|k summable. this result became an essence of many results found in previous years. in 1980, balci [10] defined absolute ϕ-summability factors and determined a very interesting result. bor gave a number of theorems on absolute summability by a generalization of mazhar [8] results. in 1993 [2] and 1994 [3], he used |n̄,pn|k summability for enhancement of the results of mazhar [8] and apply it on fourier series. özarslan [4] generalized the result of bor [1] by a more general absolute ϕ−|c,α|k summability and in [6], he used absolute matrix ϕ−|a,δ|k summability and improve some well-known results. concerning the ϕ−|n,pn|k summability factors, saxena [7] gave a general theorem for an infinite series. 2. known results absolute ϕ−|c, 1|k summability has been used by özarslan [5] to establish the following theorem. theorem 2.1. let ϕn be a sequence of positive real numbers, if λm = o(1), m →∞, (2.1) m∑ n=1 n log n|∆2λn| = o(1), (2.2) m∑ v=1 ϕk−1v vk |tv|k = o(log m), as m →∞, (2.3) m∑ n=v ϕk−1n nk+1 = o ( ϕk−1v vk ) . (2.4) then the infinite series ∑ anλn is ϕ−|c, 1|k summable for k ≥ 1. 3. main results generalized cesáro ϕ−|c, 1; δ; l|k summability and a positive non-decreasing sequence have been used to moderate the conditions of özarslan [5] results for an infinite series. theorem 3.1. let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying the following conditions: λm = o(1), m →∞, (3.1) m∑ n=1 n log n|∆2λn| = o(1), (3.2) m∑ v=1 ϕ l(k−1) v vl(k−δk) |tv|k = o(log m.µm) as m →∞, (3.3) m∑ n=v ϕ l(k−1) n n1+l(k−δk) = o ( ϕ l(k−1) v vl(k−δk) ) , (3.4) n log nµn∆ ( 1 µn ) = o(1). (3.5) then the infinite series ∑ anλn/µn is ϕ−|c, 1; δ; l|k summable for k ≥ 1, δ ≥ 0 and l is a real number. 110 sonker, krasniqi and munjal 4. proof of the theorem let tn be the n th (c, 1) mean of the sequence (nanλn/µn). the series is ϕ−|c, 1; δ; l|k summable, if ∞∑ n=1 ϕ l(k−1) n nl(k−δk) |tn|k < ∞. (4.1) applying able’s transformation, we have tn = 1 n + 1 n∑ v=1 vavλv µv = 1 n + 1 ( n−1∑ v=1 ( v∑ r=1 rar ) ∆ ( λv µv ) + ( λv µv ) n∑ v=1 vav ) = 1 n + 1 n−1∑ v=1 (v + 1)tv ( 1 µv ) ∆λv + 1 n + 1 n−1∑ v=1 (v + 1)tvλv+1∆ ( 1 µv ) + tnλn µn = tn,1 + tn,2 + tn,3. (4.2) using minkowski’s inequality, |tn|k = |tn,1 + tn,2 + tn,3|k < 3k ( |tn,1|k + |tn,2|k + |tn,3|k ) . (4.3) in order to complete the proof of the theorem, it is sufficient to show that ∞∑ n=1 ϕ l(k−1) n nl(k−δk) |tn,r|k < ∞, for r = 1, 2, 3. (4.4) by using hölder’s inequality and abel’s transformation, we have m∑ n=2 ϕ l(k−1) n nl(k−δk) |tn,1|k = m∑ n=2 ϕ l(k−1) n nl(k−δk) ∣∣∣∣∣ 1n + 1 n−1∑ v=1 (v + 1)tv ∆λv µv ∣∣∣∣∣ k = o(1) m∑ n=2 ϕ l(k−1) n nk+l(k−δk) ( n−1∑ v=1 v|tv| |∆λv| µv )k = o(1) m∑ n=2 ϕ l(k−1) n nk+l(k−δk) n−1∑ v=1 v |∆λv| µv |tv|k ( n−1∑ v=1 v |∆λv| µv )k−1 = o(1) m∑ n=2 ϕ l(k−1) n n1+l(k−δk) ( n−1∑ v=1 v |∆λv| µv |tv|k ) = o(1) m∑ v=1 v |∆λv| µv |tv|k ( m∑ n=v ϕ l(k−1) n n1+l(k−δk) ) = o(1) m∑ v=1 v |∆λv| µv |tv|k ϕ l(k−1) v vl(k−δk) absolute cesàro ϕ−|c, 1; δ; l|k summability 111 = o(1) m−1∑ v=1 ∣∣∣∣∣∆ ( v |∆λv| µv )∣∣∣∣∣ v∑ r=1 ϕ l(k−1) r rl(k−δk) |tr|k + m |∆λm| µm v∑ r=1 ϕ l(k−1) r rl(k−δk) |tr|k = o(1) m−1∑ v=1 |∆λv| µv log v.µv + m−1∑ v=1 (v + 1)∆ ( |∆λv| µv ) log v.µv + m |∆λm| µm log m.µm = o(1) m−1∑ v=1 |∆λv| log v + m−1∑ v=1 (v + 1) 1 µv |∆2λv| log v.µv + m−1∑ v=1 (v + 1)|∆λv+1|∆ ( 1 µv ) log v.µv + m|∆λm| log m = o(1). (4.5) m∑ n=2 ϕ l(k−1) n nl(k−δk) |tn,2|k = m∑ n=2 ϕ l(k−1) n nl(k−δk) ∣∣∣∣∣ 1n + 1 n−1∑ v=1 λv+1(v + 1)tv∆ ( 1 µv )∣∣∣∣∣ k = o(1) m∑ n=2 ϕ l(k−1) n nk+l(k−δk) ( n−1∑ v=1 vλv+1|tv|∆ ( 1 µv ))k = o(1) m∑ n=2 ϕ l(k−1) n nk+l(k−δk) n−1∑ v=1 vλv+1∆ ( 1 µv ) |tv|k ( n−1∑ v=1 vλv+1∆ ( 1 µv ))k−1 = o(1) m∑ n=2 ϕ l(k−1) n n1+l(k−δk) ( n−1∑ v=1 vλv+1∆ ( 1 µv ) |tv|k ) = o(1) m−1∑ v=1 ∣∣∣∣∣vλv+1∆ ( 1 µv )∣∣∣∣∣ v∑ r=1 ϕ l(k−1) r r1+l(k−δk) |tr|k + mλm+1∆ ( 1 µm ) m∑ r=1 ϕ l(k−1) r r1+l(k−δk) |tr|k = o(1) m−1∑ v=1 λv+1∆ ( 1 µv ) µv log v + m−1∑ v=1 (v + 1)∆ ( λv+1∆ ( 1 µv )) µv log v + mλm+1∆ ( 1 µm ) log m.µm = o(1) m−1∑ v=1 λv+1 log v + m−1∑ v=1 (v + 1)∆λv+1∆ ( 1 µv ) µv log v + m−1∑ v=1 (v + 1)λv+2∆ 2 ( 1 µv ) µv log v + mλm+1 log m = o(1) as m →∞. (4.6) m∑ n=1 ϕ l(k−1) n nl(k−δk) |tn,3|k = m∑ n=1 ϕ l(k−1) n nl(k−δk) ∣∣∣∣∣tnλnµn ∣∣∣∣∣ k = o(1) m∑ n=1 ϕ l(k−1) n nl(k−δk) |tn|k ∣∣∣∣∣ ∞∑ v=n ∆ ( λv µv )∣∣∣∣∣ 112 sonker, krasniqi and munjal = o(1) ∞∑ v=1 ∣∣∣∣∣∆ ( λv µv )∣∣∣∣∣ v∑ n=1 ϕ l(k−1) n nl(k−δk) |tn|k = o(1) ∞∑ v=1 1 µv ∆λv log v.µv + ∞∑ v=1 λv+1∆ ( 1 µv ) log v.µv = o(1). (4.7) collecting (4.2) (4.7), we have ∞∑ n=1 ϕk−1n nk−δk |tn|k < ∞. (4.8) hence proof of the theorem is completed. 5. corollaries corollary 5.1. let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying (3.1)-(3.2), (3.5) and following conditions: m∑ v=1 ϕk−1v vk−δk |tv|k = o(log m.µm) as m →∞, (5.1) m∑ n=v ϕk−1n n1+k−δk = o ( ϕk−1v vk−δk ) . (5.2) then the infinite series ∑ anλn/µn is ϕ−|c, 1; δ|k summable for k ≥ 1 and δ ≥ 0. proof. by using specific value l = 1 in theorem 3.1, we will get (5.1) and (5.2). we omit the details as the proof is similar to that of theorem 3.1 and we use (5.1) and (5.2) instead of (3.3) and (3.4). � corollary 5.2. let (ϕn) is a sequence of positive real numbers and (µn) is positive non-decreasing sequence satisfying (3.1)-(3.2), (3.5) and following conditions: m∑ v=1 ϕk−1v vk |tv|k = o(log m.µm) as m →∞, (5.3) m∑ n=v ϕk−1n n1+k = o ( ϕk−1v vk ) . (5.4) then the infinite series ∑ anλn/µn is ϕ−|c, 1; δ|k summable for k ≥ 1 and δ ≥ 0. proof. by using specific value l = 1 and δ = 0 in theorem 3.1, we will get (5.3) and (5.4). we omit the details as the proof is similar to that of theorem 3.1 and we use (5.3) and (5.4) instead of (3.3) and (3.4). � hence theorem 3.1 is a generalization of above corollaries. 6. conclusion the aim of this research article is to formulate the problem of generalization of absolute cesáro (ϕ −|c, 1; δ; l|k, k ≥ 1, δ ≥ 0 and l is a real number) summability factor of infinite series which is a motivation for the researchers, interested in theoretical studies of an infinite series. further, this study has a number of direct applications in rectification of signals in fir filter (finite impulse response filter) and iir filter (infinite impulse response filter). in a nut shell, the absolute summability methods have vast potential in dealing with the problems based on infinite series. acknowledgments. the authors express their sincere gratitude to the department of science and technology (india) for providing financial support to the second author under inspire scheme (innovation in science pursuit for inspired research scheme). absolute cesàro ϕ−|c, 1; δ; l|k summability 113 references [1] h. bor, absolute summability factors, atti sem. mat. fis. univ. modena., 39 (1991), 419-422. [2] h. bor, on absolute summability factors, proc. amer. math. soc., 118(1) (1993), 71-75. [3] h. bor, on the absolute riesz summability factors, rocky mountain j. math., 24(4) (1994), 1263-1271. [4] h. s. özarslan, a note on absolute summability factors, proc. indian acad. sci., 113 (2003), 165-169. [5] h. s. özarslan, on absolute cesáro summability factors of infinite series, com. math. anal., 3(1) (2007), 53-56. [6] h. s. özarslan and t. ari, absolute matrix summability methods, applied math. lett., 24 (2011), 2102-2106. [7] s. k. saxena, on nörlund summability factors of infinite series, int. j. math. analysis, 22(2) (2008), 1097 1102. [8] s. m. mazhar, on (c, 1) summability factors of infinite series, indian j. math., 14 (1972), 45-48. [9] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. sci., 7 (1957), 113-141. [10] m. balci, absolute ϕ-summability factors, comm. fac. sci. univ. ankara, ser. a1, 29 (1980), 63-80. 1department of mathematics, national institute of technology, kurukshetra-136119, haryana, india 2faculty of education, university of prishtina ”hasan prishtina”, avenue ”mother theresa” 5, 10000 prishtina, kosovo ∗corresponding author: xhevat.krasniqi@uni-pr.edu 1. introduction 2. known results 3. main results 4. proof of the theorem 5. corollaries 6. conclusion references international journal of analysis and applications volume 19, number 1 (2021), 29-46 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-19-2021-29 on modelling and pricing weather derivatives driven by nonlinear brownian motion javed hussain∗, pervez ali department of mathematics, sukkur iba university, sindh, pakistan ∗corresponding author: javed.brohi@iba-suk.edu.pk abstract. in this paper, our focus is to derive the estimates satisfied by the risk-neutral prices of a class of weather derivatives, contingent upon temperature which satisfies g-stochastic differential equation driven by nonlinear g-brownian motion. 1. introduction in this work, we aim to focus on the weather derivatives, hdd call, and cdd call, where the underlying temperature is driven by a version of nonlinear brownian motion, known as gbrownian motion. robust finance is one of the emerging areas of modern finance, where the focus is on developing the risk management models, where the underlying asset is driven by uncertain volatility. classically, in the most risk management model/ financial asset pricing theory, the volatility is either assumed to be constant, deterministic or in case it is taken stochastic, it is driven by linear noise such as the wiener process. in all of these cases, models suffer from reversal disadvantages such as mispricing of financial assets. one very interesting proposal was given by levy in [3, 1995], to take the volatility to be uncertain i.e. lie in a closed interval, this study was a good start but suffered from the problem of risk management tools such as options were not dynamically priced. the solution to this problem came from peng in [24, 2007], where he introduced the motion of the probability space with independent nonlinear expectation known as g-expectation. this received september 6th, 2020; accepted november 3rd, 2020; published november 24th, 2020. 2010 mathematics subject classification. 60j65, 60j70. key words and phrases. financial derivatives; weather contracts; nonlinear brownian motion. ©2021 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 29 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-29 int. j. anal. appl. 19 (1) (2021) 30 was the point of inception of g-stochastic calculus and g-measure-theoretic probability. in [24, 2007] and [25, 2008], peng introduced this nonlinear version of probability and stochastics, more precisely peng introduced g-conditional expectations,g-normal distributions which can be treated as the solution of fully nonlinear g-heat equation. peng also introduced the notion of g-brownian motion, g-stochastic integrals and their function spaces, g-martingale, g-sub & super martingales, g-ito process, g-stochastic differential equations i.e. sde driven by g-brownian motion and multi-dimensional g-ito formulas. zhao in [33, 2016] numerically simulated g-normal distribution and g-brownian motion. in [27, 2019], peng introduced the feynman kac formula which is pivotal in solving nonlinear probabilistic motions in g-framework. using all these tools in g-framework, xu in [31, 2010] and [32, 2011] studied the eu call option and girsanov theorem in g-framework. recently julian in [15, 2020] has developed the theory of interest rate derivative in g-framework. the weather puts impacts all kinds of business activities directly or indirectly, so naturally, it is significant to make good predictions about the weather. production, transport, and preservation of agriculture crops; energy production, distribution (cf. [1, 2002]) and consumption; chain of supermarkets, tourism, and leisure industries are directly influenced by the weather. indeed, the key factors involved in weather that must be taken care of, include a variation of temperature, wind, rainfall, humidity level, snowfall, etc. in particular, we will focus on modeling the dynamics of temperature. the weather derivatives are becoming more and more popular due to several reasons. for instance, in the energy market, the energy producers are facing the challenges, firstly, as the energy sector is getting less and less regulated, and secondly since there is a positive correlation between the weather and the demand for energy (cf. [29, 2017]), hence, the prices of the energy is not the hands of energy produces. indeed this creates competition in the energy market, among the producers, and therefore the energy companies are interested to hedge their risk by entering into financial contracts based in the weather. the weather derivative market is not very liquid, it looks like many companies have not yet define hedging policies or even figured out themselves to weather risk. if actors outside the energy sector take interest in the weather derivatives market there will be exponential growth potential. there are some barriers that must be removed if the market is to grow. for example, the quality and cost of weather data. companies which want to analyze their performance against historical weather data, they should often buy information from the national meteorological offices and that is quite expensive. and the main issue is whether the gained information about the weather is good to rely on or not. the weather derivatives are a relatively new way to hedge the risk associated with weather, they were first introduced by marginson in [20, 2000]. this gives rise to interest in carefully studying the seasonal weather events el niño event (cf. trenberth [30, 1997]), and make a good prediction about the weather so that seasonal risks can be hedged. after that, the weather derivative market expanded rapidly and contracts have been traded over-the-counter(’otc’) as individually negotiated contracts, the primary user int. j. anal. appl. 19 (1) (2021) 31 of weather derivatives were the energy sectors. now the first market namely, chicago mercantile exchange (’cme’) started an electronic marketplace for weather derivatives in september 1999. the founder of cme is aquila energy, koch energy trading, southern energy, enron, and castle bridge weather markets, all were active in the otc market for weather derivatives. for more on weather derivatives we refer to recent works [18, 2011] [22, 2013], [4, 2013], [23, 2013], [21, 2019] and [28, 2020]. we now give a brief organization of the paper. section 2 comprises a very brief introduction to g-stochastic calculus. section 3 gives the reader a detailed account of the weather derivatives. section 4 comprises of our key results or estimates satisfied by the risk-neutral prices of weather derivatives (hdd call and cdd call), where the underlying non-trade able asset temperature is driven by g-brownian motion. 2. preliminaries 2.1. sublinear expectation. linear expectation e satisfies the following relations for random variables x and y; (2.1) e[ax] = ae[x], ∀ a ∈ r, and e[x + y ] = e[x] + e[y ]. a sub-linear expectation êg satisfies the following weaker condition; (2.2) êg[ax] = aêg[x] for a > 0, êg[x + y ] ≤ êg[x] + êg[y ]. the sublinear expectation êg follows the monotonicity property same as a linear expectation: if x ≥ y then êg[x] ≥ êg[y ]. the sublinear expectation is very important in volatility uncertainty, through sublinear expectation the people are doing work in super-hedging, super-pricing (cf. [3, 1995] and [19, 1995]) and measures of risk in finance which caused great attention in finance since the fundamental work of [2, 1999]. in sublinear expectation(non-linear expectation) space, one can introduce the distributions, of random variables, like distribution of a single variable, joint distribution, marginal distributions, etc. one still can show the independence and identically distributed random variables but in the sub-linear expectation if x is independent to y it does not directly imply that y is independent to x. we can still prove some important theorems in sublinear expectation theory, the law of large numbers, and the central limit theorem, see [24, 2007]. one can see the g-expectation in [26, 2010] and g-martingales in [17, 2008], also the girsanov theorem under g-framework in [32, 2011] is now available. g – brownian motion has a very rich and interesting new structure that generalizes the classical structure without triviality. we can determine the related stochastic calculus, particularly the g – itô integrals (see [16, 1987]). a short introduction of sublinear expectation and some definitions are the part of this section. definition 2.1. (sublinear expectation [27, 2019]) let χ be the linear subspace of real valued and bounded functions on ω and a functional êg[.] on χ satisfying the following properties is called sublinear expectation. int. j. anal. appl. 19 (1) (2021) 32 1. monotonicity: x ≥ y =⇒ êg[x] ≥ êg[y ]. 2. constant preserving: êg[c] = c, ∀ c ∈ r. 3. sub-additivity: for every x,y ∈ χ êg[x + y ] ≤ êg[x] + êg[y ]. 4. positive homogeneity: êg[λx] = λêg[x], ∀ λ ≥ 0. from the above properties(2.1) and(2.1) one can also show the convexity êg[αx + (1 −α)y ] ≤ αêg[x] + (1 −α)êg[y ], ∀ α ∈ [0, 1] definition 2.2. [27, 2019] let êg,1 and êg,2 be sublinear expectations if, êg,1[x] − êg,1[y ] ≤ êg,2[x −y ], ∀ x,y ∈ χ. the êg,2 is dominated by êg,2. from (2.1) of definition 2.1 a nonlinear expectation is dominated by itself and the strongest nonlinear expectation on χ is êg,∞[x] := sup ω∈ω x(ω). every sub linear expectation is dominated by the strongest sublinear expectation. we will denote pf set of of all finitely additive probability measures on (ω,f). definition 2.3. (distribution in linear expectation) [27, 2019] let p ∈ pf and let x : ω 7→ r be a fmeasurable function such that |x(ω)| < ∞ for every ω. the distribution of random variable x on (ω,f,p) is defined as, (2.3) fx[ϕ] = ep [ϕ(x)] : ϕ ∈ l∞(r,b(r)) 7→ r. here equation (2.3) can be written as (2.4) fx[ϕ] = ∫ r ϕ(x)fx(dx). int. j. anal. appl. 19 (1) (2021) 33 definition 2.4. [27, 2019] the distributions under nonlinear expectations, is defined as, let the random variable x = (x1,x2, . . . ,x3) be a n-dimensional random vector on a nonlinear expectation space (ω1, h1,êg) (2.5) f̂x[ϕ] := êg[ϕ(x)] : ϕ ∈ cl.lip(rn) 7→ (−∞,∞). where cl.lip(rn) is the space of lipschitz continuous functions. the following triple (rn , cl.lip(rn) , f̂x[.]) is called sublinear expectation space. f̂x is the distribution of x. definition 2.5. (mean and variance -uncertainty) [27, 2019] in general, under sublinear expectation mean and variance are uncertain, in robust statistics êg[x] 6= −êg[−x] similarly êg[x2] 6= −êg[−x2], so now these are four different parameters µ̄ := êg[x], µ̂ := −êg[−x], σ̄2 := êg[x2], σ̂2 := −êg[−x2]. the purpose of this study is to discuss only variance uncertainty if someone is interested to study the mean uncertainty he/she may (see [5, 2017] and [6, 2002]). so it is assumed that êg[x] = −êg[−x]. proposition 2.1. [27, 2019] let x,y ∈ h and êg[y ] = −êg[−y ] or y has not mean uncertain. then êg[x + y ] = êg[x] + êg[y ]. definition 2.6. (independence in sublinear expectation) [27, 2019] notion of independence in sublinear expectation space (rn , cl.lip(rn) , êg) is same as linear expectation space. let a random vector y = (y1, . . . ,yn), yi ∈ h is said to be independent to random vector x = (x1, . . . ,xm), xi ∈ h under êg[�] if for every function ϕ ∈ cl.lip(rm ×rn), the independence can be expressed as, êg[ϕ(x,y )] = êg[êg[ϕ(x,y )]x=x]. 2.2. g-normal distribution. in this section, some definitions has been discussed related to normal distribution under sublinear expectation as defined in [24, 2007]. definition 2.7. g-normal distribution: [27, 2019] a random variable x ∈ h in sublinear expectation space ( ω, h ,êg ) with σ̄2 = êg[x2], σ̂2 = −êg[−x2] > 0 is called n(0; [σ̂2, σ̄2])-distributed, if for every y ∈ h independent to x , y ∼ x and ax + by ∼ √ a2 + b2x, ∀ a,b ≥ 0. from above definition this is clear that êg[x] = −êg[−x] = 0 hence random variable x has no mean uncertainty. the g-normal distribution n(0; [σ̂2, σ̄2]) is generated by the parabolic pde defined for [0,∞) ×r: (2.6) ∂tu−g(∂2xxu) = 0, int. j. anal. appl. 19 (1) (2021) 34 with cauchy condition u|t=0 = ϕ where g, (2.7) g(α) := 1 2 êg[x2α] = 1 2 (σ̄2α+ − σ̂2α−),α ∈ r. is called the generating function. where α+ := max 0,α and α− := max 0,−α, the equation (2.6) is called generating heat equation. the solution of equation (2.6) is defined as (2.8) u(t,x) := êg [ ϕ ( x + √ tx )] , (t,x) ∈ [0,∞) ×r. the g-heat equations can be also written in the form, (2.9) ∂tu− 1 2 ( σ̄2(∂2xxu) + − σ̂2(∂2xxu) − ) = 0. 2.3. g-brownian motion. [27, 2019] g-brownian motion with respect to the g-normal distribution in a sublinear expectation space is defined as, definition 2.8. (g-brownian motion) a process ( bgt (ω) ) t≥0 is called g-brownian motion, if for n ∈ n , t1 < t2 <,.. . ,< tn and b g t1 ,bgt2, . . . ,b g tn ∈ h it satisfies the following conditions, • bg0 (ω) = 0 • for every t,s ≥ 0, bgt+s −bgt is n(0; [ σ̂2s, σ̄2s ] )-distributed and independent to ( bgt1,b g t2 , . . . ,bgtn ) g is the same as it was defined in section 2.7. it will be used, for the sake of simplicity, without loss of generality, in this work, let σ̄ = 1 and σ̂ ≤ 1, by this assumption now (2.10) g(α) := 1 2 ( α+ − σ̂2α− ) , α ∈ r now bgt ∼n ( 0; [ σ̂2s,s ]) . the existence of g-brownian motion has been proven in [24, 2007]. definition 2.9. (g-expectation) [27, 2019] sublinear expectation is also called g-expectation. the canonical process ( bgt ) t≥0 in the sublinear expectation space ( ω, h ,êg ) is g-brownian process. there are some properties conditional g-expectations which can be helpful in our study, let for any x,y ∈ h 0 here h 0 is the used for l0ip(f), (1) x ≥ y =⇒ êg [x|ht] ≥ êg [y |ht] (2) êg [η|ht] = η, for everyt ∈ [0,∞) and η ∈ h 0t , (3) for every x,y ∈ χ, êg [ x|ht] − êg[y |ht ] ≤ êg[x −y |ht]. (4) êg [ηx] = η+êg [x|ht] + η−êg [−x|ht] , for every η ∈ h 0t . sublinear expectation theory also have the tower property: (2.11) êg [ êg [x|ht] |hs ] = êg [x|ht∧s] int. j. anal. appl. 19 (1) (2021) 35 and (2.12) êg [x + η|ht] = êg[x] + η, for every t ∈ [0,∞) and η ∈ h 0t the proposition 2.1 can be defined similarly in the conditional g-expectation. some moments of gbrownian motion’s increments are, êg [ bgt −bgs |hs ] = 0, for every s < t, nth moment of increments is (2.13) êg [ |bgt −b g s | n|hs ] = 1√ 2π(t−s) ∫ ∞ −∞ |x|n exp− x2 2(t−s) dy, but, (2.14) êg [ −|bgt −b g s | n|hs ] = −σ̂nêg [∣∣bgt −bgs ∣∣n|hs] . now these are the formulas for nth moment, so one can easily calculate the moment which is needed, just like the classical case. definition 2.10. (g-martingale) [27, 2019] an (mt)t≥0 process is said to be g-martingale if for every t ∈ [0,∞) ,mt ∈ h 0 and for every t ∈ [0, t], êg [mt|hs] = ms. similarly, g-submartingale and g-super-martingale are defined as, êg [mt|hs] ≥ ms and êg [mt|hs] ≤ ms respectively. example 2.1. processes ( bgt ) t≥0 and ( −bgt ) t≥0 are g-martingales and (( bgt )2) t≥0 is g-submartingale. in classical brownian motion the quadratic variation of brownian motion is a deterministic function but, in g-brownian motion the quadratic variation is itself a process. the definition of quadratic variation (cf. [27, 2019]) is, (2.15) 〈bg〉t = bgt 2 − 2 ∫ t 0 bgs db g s . one can easily verify that, êg[〈bg〉t −〈bg〉s|ht] = t−s,(2.16) êg[−(〈bg〉t −〈bg〉s)|ht] = −σ̂2(t−s).(2.17) following some lemmas are being written without proofs, for proofs (see [24, 2007]) int. j. anal. appl. 19 (1) (2021) 36 lemma 2.1. [27, 2019] a) for every s ≥ 0, (〈 bg 〉 s+t − 〈 bg 〉 s ) t≥0 is independent of fs. this is the quadratic variation process of the brownian motion bgt s = bgs+t−bgs , t ≥ 0, i.e., 〈 bg 〉 s+t − 〈 bg 〉 s = 〈 bg s〉 t , moreover, êg [〈 bg 〉2 t ] ≤ 10t2. b) for square integrable process (ηt)t≥0 in the sense ∫t 0 êg [ |ηt| 2 ] dt, then êg [∫ t 0 η(s)dbs ] = 0 ,(2.18) êg  (∫ t 0 η(s)dbs )2 ≤ ∫ t 0 êg [ (η(t))2 ] dt,(2.19) êg [∫ t s ηud〈bg〉u|fs ] ≤ ∫ t s |ηu|du.(2.20) the distribution of quadratic variation contains mean and variance uncertainty, see equation (2.16) and above lemmas. 2.4. ito formula for g-brownian motion. like classical brownian motion, itô’s formula and integral can be defined under g-brownian motion, that is, theorem 2.1. (ito’s formula for g-brownian motion) [27, 2019] let the g-itô process of x is of form, xt = xs + αtdt + ηtd〈bg〉s)t + βtdbgt . then the g-itô formula of φ(xt) is given as, φ (xt) = φ (xs) + ∫ t s αu∂xφ (xu) du + ∫ t s βu∂xφ (xu) db g u + ∫ t s ( ηu∂xφ (xu) + 1 2 β2u∂xxφ (xu) ) d 〈 bg 〉 u . (2.21) this can be proved by using taylor series and g-itô table, dt dbgt d 〈 bg 〉 t dt 0 0 0 dbgt 0 d 〈 bg 〉 t 0 d 〈 bg 〉 t 0 0 0 2.5. product formula for g-ito processes. for the product g-itô formula, the technique is same as, it was in case of standard brownian motion, let xt and yt are two g-itô processes than product g-itô formula is [7, 2014], (2.22) d(xtyt) = ytdxt + xtdyt + dxtdyt. int. j. anal. appl. 19 (1) (2021) 37 3. the weather derivatives market 3.1. the weather derivative contract. weather derivatives are structured as futures, swaps, and put/call options against different underlying weather indices some of them are cooling and heating degreedays (defined in next section), snowfall and rain. but here we will discuss only underlying index temperature(degree days indices). we are giving some definitions and terminology. from now we speak only about the temperature index. some definitions which we will use in modeling temperature definition 3.1. (temperature) given a specific weather station, let tmaxi and t min i are the maximal and minimal temperatures(celsius) of ith day. we define temperature of day i as (3.1) ti ≡ tmaxi + t min i 2 . definition 3.2. (degree-days) let ti denote the temperature on day i. we define the heating degree-days, hddi and the cooling degree-days, cddi, as (3.2) hddi ≡ max{18 −ti, 0}. and (3.3) cddi ≡ max{ti − 18, 0}. respectively. in the above definitions, it can be seen that the hdds and cdds for a specific day are just the number of degrees that the temperature is deviating from a fixed level called reference level. the names cooling and heating degree-days originate from the us energy sector because if the temperature is below 18◦c people tend to use more energy to heat their homes, whereas if the temperature is above 18◦c people start to cool their homes. temperature based weather derivatives is based on the accumulation of hdds or cdds during a certain period, like one calendar month or a winter/summer period. mostly the hdd season includes the winter months from november to march and cdd season is from may to september. april and october are often referred to as the ’shoulder months’. 3.2. the cme contract. the cme deals with futures based on the cme degree day index, the aggregate amount of a calendar month’s average hdds or cdds, as well as options for those futures. for more than 11 u.s. cities, the cme degree day index is actually listed. the futures of the hdd and cdd index are agreements to purchase or sell the hdd and cdd index value at a specific future date. one contract’s notional value is $100 times the degree day index, and the contracts int. j. anal. appl. 19 (1) (2021) 38 are quoted as hdd and cdd index points. the futures are cash-settled, meaning that there is index-based regular labeling with the gain or loss added to the customer’s account a cme hdd or cdd call option is a contract that offers the owner the right to buy a hdd / cdd futures contract at a specific price usually called the strike or exercise price, but not the commitment. analogously, the hdd / cdd put option grants the owner the right to sell one hdd / cdd futures contract, but not the obligation. at the cme, the future options are european style which means that they can only be exercised at the expiration date, which means they can only be exercised on the expiry date. 3.3. weather options. there are several different contracts traded on the otc market as mentioned above. the option is the common type of contract. calls and puts are two main types of options. • a call option is the right to buy a specific asset for an agreed amount at a fixed time in the future, as you must pay the premium at the outset of the deal by buying the right to purchase or not. • a put option is the right to sell an asset at a fixed time in the future for an agreed amount. let someone purchase the call option for some fixed strike level then if the number of hdds for the contract period is greater than the agreed strike level, the buyer will receive a payout. the size of the payout is determined by the strike and the tick size. the tick size is the amount of money that the holder of the call receives for each degree-day above the strike level for the period. often the option has a cap on the maximum payout unlike, for example, traditional options on stocks. a generic weather option can be formulated by specifying the following parameters: • the contract nature (call or put) • the contract tenure (e.g. december 2019) • the underlying index (hdd or cdd) • an official weather station for temperature data • the strike level • the tick size • the maximum payout (if the option is capped) the aim of this study is to find the formula for the payout of contracted option, let k is the strike level and α is the tick size and contract period is m days. then the number of cdds and hdds for m days period are, (3.4) hm = m∑ i=1 hddi and cm = m∑ i=1 cddi. then the formula for payout of uncapped hdd call can be written as (3.5) χ = α max{hm −k, 0}. int. j. anal. appl. 19 (1) (2021) 39 formula for payout of uncapped cdd call can be written as (3.6) χ = α max{cm −k, 0}. payouts of hdd and cdd puts can be defined similarly. 4. pricing weather derivatives through g-brownian 4.1. temperature model under g-framework. theorem 4.1. let the test model for the temperature under g-brownian motion and from the girsanov theorem under g-framwork (cf. [32, 2011]) one can find, the risk neutral measure q such that, (4.1) dtt = ( dtmt dt + a(tmt −tt) −η(t) −λσt ) dt + σtdb g t + η(t)d〈b g〉t. here bgt and 〈bg〉t are g-brownian motion and quadratic variation of g-brownian motion respectively, η(t) is any integrable deterministic function. then solution of (4.12), tt = e −a(t−s) ( ts + ∫ t s βue −a(s−u)du + ∫ t s σue −a(s−u)dbgu + ∫ t s ηue −a(s−u)d〈bg〉u ) . moreover, the conditional expectation and conditional variance of tt can be given as, êg,q [tt|fs] = e−a(t−s) ( ts + ∫ t s βue −a(s−u)du + ξη ) .(4.2) where ξη := êg (∫ t s ηue −a(s−u)d〈bg〉u|fs ) and v arg,q [tt|fs] ≤ µ̄t (µ̄t + 2µ̂t ) + e−2a(t−s)   t 2s + i(t)2β + +êg,q[i(〈bg〉)2η|fs] + 2tsi(t)β×∫ t s ( 2tsηue −a(s−u) + 2i(t)βηue −a(s−u) + σ2ue −2a(s−u) ) du   , where i(t)β := ∫ t s βue −a(s−u)du, i(bg)σ = ∫ t s σue −a(s−u)dbgu , i(〈bg〉)η := ∫ t s ηue −a(s−u)d〈bg〉u. proof. let us start by rewriting the (4.12) in g-ito form, dtt = ( tmt dt + a(tmt −tt) −ηt −λσt ) dt + σtdb g t + ηtd 〈 bg 〉 t , = ( tmt dt + atmt −att −ηt −λσt ) dt + σtdb g t + ηtd 〈 bg 〉 t , = (( tmt dt + atmt −ηt −λσt ) −att ) dt + σtdb g t + ηtd 〈 bg 〉 t . int. j. anal. appl. 19 (1) (2021) 40 by setting βt = tmt dt + atmt −ηt −λσt, dtt = [βt −att] dt + σtdbgt + ηtd〈b g〉t,(4.3) by solving equation (4.3) through g-itô formula equation (2.21) and equation (2.22), first solving the homogeneous part of equation (4.3), that is dtt = −attdt.(4.4) integrating this from s to t, ln tt − ln ts = −a(t−s) tt = tse −a(t−s) set φs,t = tt −ts, and solving equation (4.3), by using following transformation. (4.5) u(t,tt) = ttφ −1 s,t . applying the product g-itô formula (i.e. (2.22)) on equation (4.5) and using the g-itô table 2.4, one can get the following results, d(ttφ −1 s,t ) = dttφ −1 s,t + ttdφ −1 s,t + dttdφ −1 s,t d(ttφ −1 s,t ) = ([βt −att]dt + σtdb g t + ηtd〈b〉t)φ −1 s,t + attφ −1 s,t + 0 d(ttφ −1 s,t ) = (βtdt + σtdb g t + ηtd〈b〉t)φ −1 s,t .(4.6) integrating equation (4.6) from s to t and by simplification, the solution of equation (4.3) is, tt = e −a(t−s) ( ts + ∫ t s βue −a(s−u)du + ∫ t s σue −a(s−u)dbgu + ∫ t s ηue −a(s−u)d〈bg〉u ) .(4.7) the equation (4.2) represents the stochastic model of temperature in g-framework. the conditional expected value is, êg,q [tt|fs] = êg,q [ e−a(t−s) ( ts + ∫ t s βue −a(s−u)du + ∫ t s σue −a(s−u)dbgu + ∫ t s ηue −a(s−u)d〈bg〉u ) |fs ] = e−a(t−s) ( ts + ∫ t s βue −a(s−u)du ) + êg,q (∫ t s e−a(s−u)σudb g u ) + êg,q (∫ t s e−a(s−u)ηud 〈 bg 〉 u | fs ) (4.8) int. j. anal. appl. 19 (1) (2021) 41 set ξη := êg (∫ t s ηue −a(s−u)d〈bg〉u|fs ) and using properties of sublinear expectations from section 2, proposition 2.3 and lemma 2.1, êg,q[tt|fs] = e−a(t−s) ( ts + ∫ t s βue −a(s−u)du + ξη ) .(4.9) the conditional variance can be defined same as it was in linear expectation notion. let µ̄t = êg[tt|fs], such that µ̄t ≥ 0 also let µ̂t = êg[−tt|fs] now variance is v arg,q [tt|fs] = êg,q[(tt − µ̄t ) 2 |fs] = êg,q[t 2t + µ̄ 2 t − 2µ̄ttt|fs] = µ̄2t + ê g,q[t 2t + 2µ̄t [−tt]|fs] ≤ µ̄2t + ê g,q[t 2t |fs] + ê g,q[2µ̄t (−tt)|fs] = µ̄2t + ê g,q[t 2t |fs] + 2µ̄t ê g,q[−tt|fs] = µ̄2t + ê g,q[t 2t |fs] + 2µ̄t µ̂t v arg,q [tt|fs] ≤ µ̄t (µ̄t + 2µ̂t ) + êg,q[t 2t |fs].(4.10) let n = µ̄t (µ̄t + 2µ̂t ) + êg[t 2t |fs] and from the fact that v ar [tt|fs] ≥ 0 one can write it as σ̄2 = v ar [tt|fs] ∈ [0,n]. here êg,q[t 2t |fs] can be estimated by following approximation of it, but for the sake of simplicity there are some conventional notations have been defined as, let i(t)β := ∫ t s βue −a(s−u)du, i(bg)σ := ∫ t s σue −a(s−u)dbgu , i(〈bg〉)η := ∫ t s ηue −a(s−u)d〈bg〉u. now êg,q [ t 2t |fs ] = êg,q [( e−a(t−s) ( ts + ∫ t s βue −a(s−u)du + ∫ t s σue −a(s−u)dbgu + ∫ t s ηue −a(s−u)d〈bg〉u ))2 ∣∣fs ] = e−2a(t−s)êg,q [ t 2s + i(t) 2 β + i(b g)2σ + i(〈b g〉)2η + 2tsi(t)β + 2tsi(b g)σ +2tsi(〈bg〉)η + 2i(t)βi(bg)σ + 2i(t)βi(〈bg〉)η + 2i(bg)σi(〈bg〉)η ∣∣fs] , using properties of sublinear expectations from section 2, proposition 2.3 and lemma 2.1, êg,q [ t 2t |fs ] ≤ e−2a(t−s) ( t 2s + i(t) 2 β + 2tsi(t)β + 2tsê g,q[i(bg)σ|fs] + 2tsêg,q[i(〈bg〉)η|fs] + 2i(t)βêg,q[i(bg)σ|fs] + 2i(t)βêg,q[i(〈bg〉)η|fs] +êg,q[i(bg)2σ|fs] + ê g,qi(〈bg〉)2η|fs] + 2ê g,q[i(bg)σi(〈b〉)η|fs] ) , int. j. anal. appl. 19 (1) (2021) 42 using some properties from [24, 2007], and properties of sublinear expectations from section 2, proposition 2.3 and lemma 2.1, êg,q[t 2t |fs] ≤ e −2a(t−s)   t 2s + i(t) 2 β + 2tsi(t)β + 2ts ∫ t s ηue −a(s−u)du +2i(t)β ∫ t s |ηu|e−a(s−u)du + ∫ t s σ2ue −2a(s−u)du +êg,q[i(〈bg〉)2η|fs]   , ≤ e−2a(t−s)   t 2s + i(t) 2 β + 2tsi(t)β + 2ts ∫ t s ηue −a(s−u)du +2i(t)β ∫ t s ηue −a(s−u)du + ∫ t s σ2ue −2a(s−u)du+ êg,q[i(〈bg〉)2η|fs]   ≤ e−2a(t−s)   t 2s + i(t) 2 β + 2tsi(t)β + 2ts ∫ t s ηue −a(s−u)du +2i(t)β ∫ t s ηue −a(s−u)du + ∫ t s σ2ue −2a(s−u)du+ êg,q[i(〈bg〉)2η|fs]   ≤ e−2a(t−s)   t 2s + i(t) 2 β + 2tsi(t)β×∫ t s ( 2tsηue −a(s−u) + 2i(t)βηue −a(s−u) + σ2ue −2a(s−u) ) du +êg,q[i(〈bg〉)2η|fs]   . (4.11) substituting the value of v arg,q [tt|fs] ≤ µ̄t (µ̄t + 2µ̂t ) + e−2a(t−s) ×  t 2s + i(t) 2 β + 2tsi(t)β×∫ t s ( 2tsηue −a(s−u) + 2i(t)βηue −a(s−u) + σ2ue −2a(s−u) ) du +êg,q[i(〈bg〉)2η|fs]   . � 4.2. pricing weather derivative. theorem 4.2. suppose that dynamics of temperatures process (tt)t≥0 satisfies following the g-sde, dtt = ( dtmt dt + a(tmt −tt) −η(t) −λσt ) dt + σtdb g t + η(t)d〈b g〉t. then the risk-neutral (arbitrage free) price hct of the uncapped hdd call (described in section 3) satisfies the following estimate, hct ≤ 18mα−e−(tm−t)kα− m∑ i=1 α ( e−a(ti−s) ( tti + ∫ ti s βue −a(ti−u)du + ξη(ti) )) . (4.12) int. j. anal. appl. 19 (1) (2021) 43 and the risk-neutral (arbitrage free) price cct of the uncapped cdd call (described in section 3) satisfies the following estimate, cct ≤ e−(tm−t)kα + 18mα + m∑ i=1 ( αe−a(ti−s) ( tti + ∫ ti s βue −a(ti−u)duξη(ti) )) . (4.13) where ξη = êg,q (∫ t s ηue −a(s−u)d〈bg〉u|fs ) . proof. let us recall, from section 3, the payout of hdd call with tick size α is (4.14) χ = α(hm −k)+, where hm = ∑m i=1 max (18 −tti ), it is known that tt is g-normally distributed but the hm is not gnormally distributed just because there is a maximum in its definition, so just for the simplicity if someone is interested to find out the explicit formula for option, then it will be possible for winter months and with an assumption that 18−tti ≥ 0. now just to be precise, let for winter months hm = 18m− ∑m i=1 tti , which is g-normally distributed. êg,q [hm|ft] = êg,q [ 18m− m∑ i=1 tti|ft ] = êg,q[18m] + êg,q [ m∑ i=1 (−tti )|ft ] ≤ 18m + m∑ i=1 êg,q [−tti|ft] ,(4.15) set µ̄m = êg,q [hm|ft] , µ̄m = êg,q [−hm|ft] , σ̂2m = v ar [hm|ft] , σ̂ 2 m = −v ar [−hm|ft] . now from feynman-kac formula (cf. [27]) hct = êg ( αe−(tm−t)(hm −k)+|ft ) the case when the derivative is in the money i.e. temperature is above 18 for some days, then hct = αe −(tm−t)êg,q ((hm −k)|ft) = αe−(tm−t)êg,q (hm|ft) −e−(tm−t)k, ≤ 18mα−e−(tm−t)kα + m∑ i=1 αêg,q[−tti|ft] ≤ 18mα−e−(tm−t)kα− m∑ i=1 ( αe−a(ti−s) ( tti + ∫ ti s βue −a(ti−u)du + ξη(ti) )) . (4.16) now lets move towards, uncapped cdd call. recall the payoff of the uncapped ccd call, (4.17) χ = α(k −cm)+, int. j. anal. appl. 19 (1) (2021) 44 where cm = ∑m i=1 max (tti−18), it is known that tt is g-normally distributed but the hm is not g-normally distributed, so to find out the explicit formula for option, then it will be possible for summer months there can be days where the temperature goes above 18 i.e. tti − 18 ≥ 0. now just to be precise, let for winter months cm = ∑m i=1 tti − 18m, which is g-normally distributed. êg,q [cm|ft] = êg,q [ m∑ i=1 tti − 18m|ft ] = êg,q [ m∑ i=1 (tti )|ft ] − 18m ≤ m∑ i=1 êg,q[tti|ft] − 18m,(4.18) now from feynman-kac formula (cf. [27]) ct = êg,q ( αe−(tm−t)(k −cm)+|ft ) the case when the derivative is in the money i.e. temperature is above 18 for some days, then cct = αe −(tm−t)êg ((k −cm)|ft) , = e−(tm−t)kα−αe−(tm−t)êg,q (cm|ft) , ≤ e−(tm−t)kα + 18mα + m∑ i=1 αêg,q[tti|ft] ≤ e−(tm−t)kα + 18mα + m∑ i=1 ( αe−a(ti−s) ( tti + ∫ ti s βue −a(ti−u)du + ξη(ti) )) . (4.19) � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. alaton, b. djehiche, d. stillberger, on modelling and pricing weather derivatives, appl. math. finance. 9 (2002), 1–20. [2] p. artzner, f. delbaen, j.-m. eber, d. heath, coherent measures of risk, math. finance. 9 (1999), 203–228. [3] m. avellaneda, a. levy, a. parás, pricing and hedging derivative securities in markets with uncertain volatilities, appl. math. finance. 2 (1995), 73–88. [4] a. alexandridis, a.d. zapranis, weather derivatives: modeling and pricing weather-related risk. springer science & business media, new york, (2012). [5] t.g. bali, s.j. brown, y. tang, is economic uncertainty priced in the cross-section of stock returns?, j. financ. econ. 126 (2017), 471–489. [6] z. chen, l. epstein, ambiguity, risk, and asset returns in continuous time, econometrica. 70 (2002), 1403–1443. [7] p. glasserman, x. xu, robust risk measurement and model risk, quant. finance. 14 (2014), 29–58. [8] m.a. soomro, j. hussain, on study of generalized novikov equation by reduced differential transform method, indian j. sci. technol. 12 (2019), 1-6. int. j. anal. appl. 19 (1) (2021) 45 [9] j. hussain, b. khan, on cox-ross-rubinstein pricing formula for pricing compound option, int. j. anal. appl. 18 (1) (2020), 129-148. [10] j. hussain, m.s. khan, on the pricing of call-put parities of asian options by reduced differential transform algorithm, int. j. anal. appl. 18 (3) (2020), 513-530. [11] j. hussain, valuation of european style compound option written on european style currency and power options, int. j. anal. appl. 18 (6) (2020), 1015-1028. [12] j. hussain, on existence and invariance of sphere, of solutions of constrained evolution equation, int. j. math. comput. sci. 15 (2020), 325–345. [13] m.-u. rehman, j. alzabut, j.h. brohi, a. hyder, on spectral properties of doubly stochastic matrices, symmetry. 12 (2020), 369. [14] m.-u. rehman, j. alzabut, j. hussain brohi, computing µ-values for lti systems, aims math. 6 (2021), 304–313. [15] j. hölzermann, pricing interest rate derivatives under volatility uncertainty, arxiv:2003.04606 [q-fin]. (2020). [16] k. ito, differential equations determining a markoff process. kiyosi itô selected papers (dw stroock and srs varadhan, eds.), springer-verlag, pp. 42–75. 1987. [17] l. jiang, convexity, translation invariance and subadditivity for g-expectations and related risk measures, ann. appl. probab. 18 (2008), 245–258. [18] g. leobacher, p. ngare, on modelling and pricing rainfall derivatives with seasonality, appl. math. finance. 18 (2011), 71–91. [19] t.j. lyons, uncertain volatility and the risk-free synthesis of derivatives, appl. math. finance. 2 (1995), 117–133. [20] s. marginson m. considine. the enterprise university: power, governance and reinvention in australia cambridge university press cambridge. (2000). [21] j. möllmann, m. buchholz, o. musshoff, comparing the hedging effectiveness of weather derivatives based on remotely sensed vegetation health indices and meteorological indices. weather climate soc. 11 (2019), 33–48. [22] f. pérez-gonzález, h. yun, risk management and firm value: evidence from weather derivatives: risk management and firm value, j. finance. 68 (2013), 2143–2176. [23] m. ritter, o. mußhoff, m. odening, minimizing geographical basis risk of weather derivatives using a multi-site rainfall model, comput. econ. 44 (2014), 67–86. [24] s. peng, g-brownian motion and dynamic risk measure under volatility uncertainty, arxiv:0711.2834 [math]. (2007). [25] s. peng, multi-dimensional g-brownian motion and related stochastic calculus under g-expectation, stoch. proc. appl. 118 (2008), 2223-2253. [26] s. peng, backward stochastic differential equation, nonlinear expectation and their applications, proceedings of the international congress of mathematicians 2010, pp. 393-432, (2010). [27] s. peng, nonlinear expectations and stochastic calculus under uncertainty: with robust clt and g-brownian motion, springer berlin heidelberg, 2019. [28] a. salgueiro, m. t. rodon, approaching rainfall-based weather derivatives pricing and operational challenges, rev. deriv. res. 23 (2020), 163–190. [29] i. štulec, effectiveness of weather derivatives as a risk management tool in food retail: the case of croatia, int. j. financ. stud. 5 (2017), 2. [30] k. e. trenberth. the definition of el nino. amer. meteorol. soc. 78 (12) (1997), 2771-2778. [31] j. xu, m.p. xu, european call option price under g-framework. math. practice theory, 4 (2010), 41-45. [32] j. xu, h. shang, b. zhang, a girsanov type theorem under g-framework, stoch. anal. appl. 29 (2011), 386–406. int. j. anal. appl. 19 (1) (2021) 46 [33] j. yang, w. zhao, numerical simulations for g-brownian motion, front. math. china. 11 (2016), 1625–1643. 1. introduction 2. preliminaries 2.1. sublinear expectation 2.2. g-normal distribution 2.3. g-brownian motion 2.4. ito formula for g-brownian motion 2.5. product formula for g-ito processes 3. the weather derivatives market 3.1. the weather derivative contract 3.2. the cme contract 3.3. weather options 4. pricing weather derivatives through g-brownian 4.1. temperature model under g-framework. 4.2. pricing weather derivative. references international journal of analysis and applications volume 17, number 1 (2019), 105-121 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-17-2019-105 left and right generalized drazin invertibility of an upper triangular operator matrices with application to boundary value problems kouider miloud hocine1, bekkai messirdi1 and mohammed benharrat2,∗ 1laboratoire de mathématiques fondamentales et appliquées d’oran (lmfao), département de mathématiques, université d’oran 1 ahmed ben bella; 31000 oran, algérie 2laboratoire de mathématiques fondamentales et appliquées d’oran (lmfao), département de mathématiques et informatique, ecole national polytechnique d’oran-maurice audin (ex. enset d’oran); b.p. 1523 oran-el m’naouar-, oran, algérie ∗corresponding author: mohammed.benharrat@gmail.com abstract. when a ∈ b(h) and b ∈ b(k) are given, we denote by mc the operator on the hilbert space h ⊕ k of the form mc =   a c 0 b   . in this paper we investigate the quasi-nilpotent part and the analytical core for the upper triangular operator matrix mc in terms of those of a and b. we give some necessary and sufficient conditions for mc to be left or right generalized drazin invertible operator for some c ∈ b(k, h). as an application, we study the existence and uniqueness of the solution for abstract boundary value problems described by upper triangular operator matrices with right generalized drazin invertible component. received 2017-10-26; accepted 2018-01-04; published 2019-01-04. 2010 mathematics subject classification. 47a10, 47b38. key words and phrases. generalized drazin inverse; left generalized drazin inverse; right generalized drazin inverse; upper triangular operator matrices. c©2019 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 105 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-105 int. j. anal. appl. 17 (1) (2019) 106 1. introduction and preliminaries let b(h) be the banach algebra of all bounded linear operators acting on an infinite-dimensional complex hilbert space h. associated with an operator t there are two (not necessarily closed) linear subspaces of h invariant by t, played an important role in the development of the generalized drazin inverse for t ∈b(h), the quasi-nilpotent part h0(t) of t : h0(t) = { x ∈ h : lim n→∞ ‖tnx‖ 1 n = 0 } and the analytical core k(t) of t : k(t) = {x ∈ h : there exist a sequence (xn) in h and a constant δ > 0 such that tx1 = x,txn+1 = xn and ‖xn‖≤ δn‖x‖ for all n ∈ n}. see also [1]. it is well-known that if k(t) and h0(t) are both closed, h = h0(t) ⊕k(t), the restriction of t to h0(t) is a quasi-nilpotent operator, and the restriction of t to k(t) is invertible, provided that t is generalized drazin invertible, (c.f. [18]). recently, by the use of this two subspaces, in [24], the authors defined and studied a new class of operators called left and right generalized drazin invertible operators as a generalization of left and right drazin invertible operators. definition 1.1. an operator t ∈ b(h) is said to be right generalized drazin invertible if k(t) is closed and complemented with a subspace n in h such that t(n) ⊂ n ⊆ h0(t). definition 1.2. an operator t ∈b(h) is said to be left generalized drazin invertible if h0(t) is closed and complemented with a subspace m in h such that t(m) ⊂ m and t(m) is closed. we also proved that t ∈b(h) is a right generalized drazin invertible if and only if 0 is an isolated point on the surjective spectrum σsu(t) of t and by duality t ∈ b(h) is a left generalized drazin invertible if and only if 0 is an isolated point on the approximate spectrum σap(t). so we are mainly interested in the case where the point 0 belongs to the spectrum σ(t) of t or in its various distinguished parts. here, we are interested in the analogous question for an upper triangular operator of the form mc =   a c 0 b   . (1.1) defined on the separable hilbert space h ⊕ k. recall that the problem of the relationship between the spectrum, various distinguished parts of the spectrum of an upper triangular and drazin invertibility and its diagonal has been considered by a number of authors in the recent past, we can see [4, 5, 13, 21, 22, 28–30] and the references therein for recent reviews on this topic. a related, and seemingly more demanding, problem int. j. anal. appl. 17 (1) (2019) 107 is the following. let h be a hilbert space, t is a bounded linear operator on h, and m is a t-invariant closed subspace of h, then t takes the form t =   ∗ ∗ 0 ∗   : m ⊕m⊥ → m ⊕m⊥ which motivated the interest in 2 × 2 upper-triangular operator matrices. in this paper we use the results of [24] to give a necessary and sufficient conditions for mc to be left (resp. right) generalized drazin invertible which generalizes the notion of generalized drazin invertible operators to the matrix case. we characterize the quasi-nilpotent part and the analytical core of the operator mc in term of the pair (a,b) of bounded operators. we apply our results to study the existence and uniqueness of solutions of boundary value problems described by an upper triangular operator matrices (2 × 2) acting in hilbert spaces with a complex spectral parameter λ : (p)   (ul −λmc)w = f γw = φ , where ul is right generalized drazin invertible, γ is a boundary operator, f and φ are given. for t ∈b(h) write n(t), r(t), σ(t) and ρ(t) respectively, the null space, the range, the spectrum and the resolvent set of t . the nullity and the deficiency of t are defined respectively by α(t) = dimn(t) and β(t) = dimh/r(t). here i denotes the identity operator in h. by isoσ(t) and accσ(t) we define the set of all isolated and accumulation spectral points of t. if m is a subspace of h then tm denote the restriction of t in m. assume that m and n are two subspaces of h such that h = m ⊕n (that is h = m + n and m ∩n = 0). we say that t is completely reduced by the pair (m,n), denoted as (m,n) ∈ red(t), if t(m) ⊂ m, t(n) ⊂ n and t = tm ⊕tn . in such case we have n(t) = n(tm )⊕n(tn ), r(t) = r(tm )⊕r(tn ) and tn = tnm ⊕t n n for all n ∈ n. an operator is said to be bounded below if it is injective with closed range. recall that (see, e.g. [14]) the ascent a(t) of an operator t ∈b(h) is defined as the smallest nonnegative integer p such that n(tp) = n(tp+1). if no such an integer exists, we set a(t) = ∞. analogously, the smallest nonnegative integer q such that r(tq) = r(tq+1) is called the descent of t and denoted by d(t). we set d(t) = ∞ if for each q, r(tq+1) is a proper subspace of r(tq). it is well known that if the ascent and the descent of an operator are finite, then they are equal. furthermore, if a(t) = p < ∞ then h0(t) = n(t p) and if d(t) = q < ∞ then k(t) = r(tq). an operator t ∈b(h) is said to be drazin invertible, if there exists an operator s ∈b(h) such that st = ts sts = s and tst = t + u where u is a nilpotent operator. (1.2) int. j. anal. appl. 17 (1) (2019) 108 the concept of drazin invertible operators has been generalized by koliha ( [18]) by replacing the nilpotent operator u in (1.2) by a quasi-nilpotent operator one. in this case, s is called a generalized drazin inverse of t , denoted by td. examples of generalized drazin invertible operators are the operators of the following classes: • invertible operators, right invertible operators and left invertible operators. • left drazin invertible operators, ld(h) = {t ∈b(h) : a(t) is finite and r(ta(t)+1) is closed}. • right drazin invertible operators, rd(h) = {t ∈b(h) : d(t) is finite and r(td(t)) is closed}. • drazin invertible operators, ld(h) ∩rd(h). according to the definitions 1.1 and 1.2, we also have right (resp. left) invertible operator =⇒ right (resp. left) drazin invertible operator =⇒ right (resp. left) generelazed drazin invertible operator. the left drazin spectrum, the right drazin spectrum, the drazin spectrum, the generalized drazin spectrum, the left generalized drazin spectrum and the right generalized drazin spectrum of t are, respectively, defined by σld(t) := {λ ∈ c : t −λi /∈ ld(h)}, σrd(t) := {λ ∈ c : t −λi /∈ rd(h)}, σd(t) = {λ ∈ c : t −λi /∈ ld(h) ∩rd(h)}, σgd(t) = {λ ∈ c : t −λi is not generalized drazin invertible operator}, σlgd(t) := {λ ∈ c : t −λi is not left generalized drazin invertible}, and σrgd(t) := {λ ∈ c : t −λi is not right generalized drazin invertible}. int. j. anal. appl. 17 (1) (2019) 109 it is well known that these spectra are compact sets in the complex plane, and we have, σgd(t) = σlgd(t) ∪σrgd(t) ⊂ σd(t) = σld(t) ∪σrd(t), σlgd(t) ⊂ σld(t) ⊂ σap(t), and σrgd(t) ⊂ σrd(t) ⊂ σsu(t), where σap(t) := {λ ∈ c : t −λi is not bounded below } and σsu(t) := {λ ∈ c : t −λi is not surjective} are respectively the approximate point spectrum and the surjective spectrum of t. the basic existence results of generalized drazin inverses and its relation to the quasi-nilpotent part and the analytical core are summarized in the following theorems. theorem 1.1 ( [18]). assume that t ∈b(h). the following assertions are equivalent: (i) t is generalized drazin invertible, (ii) 0 is an isolated point in the spectrum σ(t) of t ; (iii) k(t) is closed and h = k(t) ⊕h0(t), (iv) h0(t) is closed and h = k(t) ⊕h0(t), (v) there is a bounded projection p on h such that r(p) = k(t) and n(p) = h0(t). (vi) t = t1 ⊕t2, with t1 = tk(t) is invertible operator and t2 = th0(t) is quasi-nilpotent operator. ⊕ denotes the algebraic direct sum and tm denote the restriction of t to a subspace m of h. theorem 1.2. assume that t ∈b(h). the following assertions are equivalent: (i) t is left generalized drazin invertible; (ii) 0 is an isolated point in σap(t); (iii) t = t1 ⊕t2, with t1 = tm is left invertible operator and t2 = th0(t) is quasi-nilpotent operator. proof. the equivalence [(i)]⇐⇒[(iii)] follows by [24, proposition 3.2] and the implication [(i)]=⇒[(ii)] follows from [24, theorem 3.8]. now, if 0 is an isolated point in σap(t), then by [23, theorem 4.4] 0 is a singularity of the generalized resolvent, equivalently, t admits a generalized kato decomposition (m,n), and since t has the svep at 0, it follows from [1, theorem 3.14] that tm is injective and h0(t) = n. this proved the implication [(ii)]=⇒[(i)] and hence [(i)]⇐⇒[(ii)]. � int. j. anal. appl. 17 (1) (2019) 110 we know that the properties to be right generalized drazin invertible or to be left generalized drazin invertible are dual each other, (see [24, proposition 3.9]), then we have, theorem 1.3. let t ∈b(h). the following assertions are equivalent: (i) t is right generalized drazin invertible; (ii) 0 is an isolated point in σsu(t); (iii) t = t1 ⊕t2, with t1 = tk(t) is right invertible operator and t2 = tn is quasi-nilpotent operator. the reduced minimum modulus γ(t) of t is defined by γ(t) =   inf{‖tx‖ : dis(x,n(t)) = 1} if t 6= 00 if t = 0 and dis(x,n(t)) = inf{‖x−z‖ such that z ∈ n(t)}. it is well known that γ(t) > 0 if and only if r(t) is closed. the paper is organized as follows. in section 2 we give a relationship between the quasi-nilpotent part and the analytical core of a pair (a,b) of operators and that of 2 × 2 block triangular matrices mc, we show that the quasi-nilpotent part of mc is a direct sum of the quasi-nilpotent part of a and b and that is the same for analytical core. in section 3, we study the left (resp. right) generalized drazin invertibility of mc using the isolated point in the approximate spectrum (resp. the surjective spectrum) of a and b. finally in section 4 we illustrate our approach by studying a boundary value problems described by an upper triangular operator matrices. 2. the quasi-nilpotent part and the analytical core of the operator mc in the following, we find the relationship between the quasi-nipotent part (resp. the analytical core) of the pair (a,b) of operators and that of mc defined in (1.1) and we give fundamental results concerning this operator. proposition 2.1. if r(c) ⊂ h0(a), then h0(mc) = h0(a) ⊕h0(b). proof. suppose that   x y   ∈ h0(a) ⊕ h0(b) and consider h × k the product hilbert space equipped with the norm ∥∥∥∥∥∥   x y   ∥∥∥∥∥∥ 2 = ‖x‖2 + ‖y‖2 . int. j. anal. appl. 17 (1) (2019) 111 we have ∥∥∥∥∥∥mnc   x y   ∥∥∥∥∥∥ 2 = ‖anx + sy‖2 + ‖bny‖2 , with s = an−1c + an−2cb + · · · + acbn−2 + cbn−1. then ∥∥∥∥∥∥mnc   x y   ∥∥∥∥∥∥ 2 n ≤‖anx‖ 2 n + ‖bny‖ 2 n + ‖sy‖ 2 n . therefore, ‖s‖ 2 n ≤ ∑ p+k=n−1 ∥∥apcbky∥∥ 2n . if p →∞ then n →∞ and since r(c) ⊂ h0(a), we obtain limn→∞‖s‖ 2 n = 0. it follows that limn→∞ ∥∥∥∥∥∥mnc   x y   ∥∥∥∥∥∥ 2 n = 0. hence   x y   ∈ h0(mc). let now   x y   ∈ h0(mc). it is clear that ∥∥∥∥∥∥mnc   x 0   ∥∥∥∥∥∥ 2 n = ‖anx‖ 2 n and ∥∥∥∥∥∥mnc   0 y   ∥∥∥∥∥∥ 2 n ≥‖bny‖ 2 n . then limn→∞‖anx‖ 2 n = 0 and limn→∞‖bny‖ 2 n = 0. thus x ∈ h0(a) and y ∈ h0(b). � proposition 2.2. if r(b) ⊂ n(c), then k(mc) = k(a) ⊕k(b). proof. let x ∈ k(a) and y ∈ k(b), by definition there exist two sequences (xn) in h, (yn) in k and a constants δ1 > 0, δ2 > 0 such that ax1 = x,axn+1 = xn and ‖xn‖ ≤ δn1‖x‖ and by1 = y,byn+1 = yn and ‖yn‖≤ δn2‖y‖ for all n ∈ n. we have mc   xn+1 yn+1   =   axn+1 + cyn+1 byn+1   =   xn + cbyn+2 yn   . since r(b) ⊂ n(c), it follows that mc   xn+1 yn+1   =   xn yn   and mc   x1 y1   =   x y   . (2.1) furthermore ∥∥∥∥∥∥   xn yn   ∥∥∥∥∥∥ ≤ δn ∥∥∥∥∥∥   x y   ∥∥∥∥∥∥ , where δ = max(δ1,δ2). hence   x y   ∈ k(mc). int. j. anal. appl. 17 (1) (2019) 112 conversely, suppose that   x y   ∈ k(mc). then there exist a sequence   xn yn   in h×k and a constant δ > 0 such that mc   x1 y1   =   x y  , mc   xn+1 yn+1   =   xn yn   and ∥∥∥∥∥∥   xn yn   ∥∥∥∥∥∥ ≤ δn ∥∥∥∥∥∥   x y   ∥∥∥∥∥∥ for all n ∈ n. we obtain from (2.1) that byn+1 = yn, axn+1 = xn, ax1 = x and by1 = y. consequently, ‖xn‖ ≤ δn‖x‖ and ‖yn‖ ≤ δn‖y‖, that is   x y   ∈ k(a) ⊕ k(b). this completes the proof. � as a consequence of propositions 2.1 and 2.2 we have the following result. corollary 2.1. if r(c) ⊂ h0(a) and r(b) ⊂ n(c). then mc is generalized drazin invertible if and only if both a and b are generalized drazin invertible. 3. left and right generalized drazin invertibility of mc hwang and lee , [13], give a necessary and sufficient condition for mc to be bounded below for some c ∈b(k,h) and they are characterized the intersection of the approximate point spectrum, the surjective spectrum and the spectrum of mc. the next theorem is an extension of [13, theorem 1], we will give some necessary and sufficient conditions for mc to be left generalized drazin invertible operator for some c ∈b(k,h). theorem 3.1. for a given pair (a,b) of bounded operators, the following statements are equivalent: (i) mc is left generalized drazin invertible for some c ∈b(k,h), (ii) a is left generalized drazin invertible and there exists a constant δ such that  α(b −λi) ≤ β(a−λi) if r(b −λi) is closed, or β(a−λi) = ∞ if r(b −λi) is not closed, with 0 < |λ| < δ. to prove this theorem we need the following lemma. lemma 3.1. let t1, t2 and t3 ∈b(h) be ginven such that t2 is invertible. if α(t1) < ∞ and r(t1t2t3) is closed, then r(t3) is also closed. proof. follows from [12, theorem 1]. � proof of theorem 3.1. we first claim that if a is left generalized drazin invertible and there exists a constant δ such that for every λ with 0 < |λ| < δ, r(b − λi) is closed, then α(b − λi) ≤ β(a − λi) is equivalent int. j. anal. appl. 17 (1) (2019) 113 to mc is left generalized drazin invertible for some c ∈b(k,h). indeed, since a is left generalized drazin invertible, then by theorem 1.2, a−λi is bounded below for 0 < |λ| < δ. assume that α(b−λi) ≤ β(a−λi). then there exists an isometry j : n(b−λi) → r(a−λi)⊥. define an operator c : k → h by c :=   j 0 0 0   :   n(b −λi) n(b −λi)⊥   →   r(a−λi)⊥ r(a−λi)   . let   x y   ∈ n(mc −λi). then (mc −λi)   x y   =   0 0   ; implies that (a−λi)x + cy = 0 and (b−λi)y = 0. since a−λi is injective and the fact that n(c−λi) ⊆ n(b −λi)⊥, we get x = 0 and y = 0. then mc −λi is injective. now we prove that r(mc −λi) is closed. let   x y   ∈ n(mc −λi)⊥. then ∥∥∥∥∥∥(mc −λi)   x y   ∥∥∥∥∥∥ 2 = ‖(a−λi)x + cy‖2 + ‖(b −λi)y‖2 = ‖(a−λi)x‖2 + ‖cy‖2 + ‖(b −λi)y‖2 . write y := y1 + y2, where y1 ∈ n(b − λi) and y2 ∈ n(b − λi)⊥. then ‖cy‖ = ‖y1‖, ‖(a−λi)x‖ ≥ γ(a−λi)‖x‖ and ‖(b −λi)y2‖≥ γ(b −λi)‖y2‖ , because r(b −λi) and r(a−λi) are closed. hence∥∥∥∥∥∥(mc −λi)   x y   ∥∥∥∥∥∥ 2 ≥ γ2(a−λi)‖x‖2 + ‖y1‖ 2 + γ2(b −λi)‖y2‖ 2 ≥ min(γ2(a−λi),γ2(b −λi), 1) ∥∥∥∥∥∥   x y   ∥∥∥∥∥∥ 2 . then γ(mc −λi) > 0 and (mc −λi) is bounded below for 0 < |λ| < δ. it follows from theorem 1.2 that mc is left generalized drazin invertible. conversely, it suffices to show that if (a−λi) is bounded below with α(b −λi) > β(a−λi) for λ ∈ c, then (mc −λi) is not bounded below. assume that α(b −λi) > β(a−λi), so β(a−λi) < ∞. we now consider the following two cases. case (1). if n(c) ∩ n(b − λi) 6= {0}. then for all non-zero vector z ∈ n(c) ∩ n(b − λi) we have (mc −λi)z = 0. we conclude that (mc −λi) is not bounded below. case (2). suppose that n(c) ∩ n(b − λi) = {0}. then dim(c(n(b − λi))) = α(b − λi) > β(a − λi). int. j. anal. appl. 17 (1) (2019) 114 thus, c(n(b −λi)) ∩r(a−λi) 6= {0}. we take a non-zero vector z ∈ c(n(b −λi)) ∩r(a−λi). then there exist some x ∈ h and y ∈ k such that (a − λi)x = cy = z and (b − λi)y = 0. direct calculation shows that (mc −λi)   −x y   = 0. it follows that (mc −λi) is not bounded below. we next claim that if a is left generalized drazin invertible and there exists a constant δ such that for every λ with 0 < |λ| < δ, r(b−λi) is not closed, then β(a−λi) = ∞ is equivalent to mc is left generalized drazin invertible for some c ∈b(k,h). since r(b −λi) is not closed and β(a−λi) = ∞, there exists an isomorphism j : k → r(b −λi). define an operator c : k → h in the following way: c := ( j 0 ) : k →   r(a−λi)⊥ r(a−λi)   . by a similar proof we check easily that n(mc − λi) = {0} and γ(mc − λi) > 0. that is (mc − λi) is bounded below for 0 < |λ| < δ and by theorem 1.2 mc is left generalized drazin invertible. for the converse, suppose in the contrary that β(a−λi) < ∞. then dimn     (a−λi)∗ 0 0 i     = dimn((a−λi)∗) = β(a−λi) < ∞. since r((mc−λi)∗) is closed and   i 0 c∗ i   is invertible, by lemma 3.1, we have that r     i 0 0 (b −λi)∗     is closed, that is r((b − λi)∗) is closed. this contradicts our assumption. therefore we must have β(a−λi) = ∞. � by duality, we have: theorem 3.2. for a given pair (a,b) of operators, the following statements are equivalent: (i) mc is right generalized drazin invertible for some c ∈b(k,h), (ii) b is right generalized drazin invertible and there exists a constant δ such that  β(a−λi) ≤ α(b −λi) if r(a−λi) is closed, or α(b −λi) = ∞ if r(a−λi) is not closed, with 0 < |λ| < δ. as a direct application of theorem 3.1, the following corollary can be derived to give a characterization of σlgd(mc) for all c ∈b(k,h). int. j. anal. appl. 17 (1) (2019) 115 corollary 3.1. ⋂ c∈b(k,h) σlgd(mc) = σlgd(a) ⋃ {λ ∈ c : r(b −λi) is closed and β(a−λi) < α(b −λi)}⋃ {λ ∈ c : r(b −λi) is not closed and β(a−λi) < ∞} . the following is the dual statement of corollary 3.1. corollary 3.2. ⋂ c∈b(k,h) σrgd(mc) = σrgd(b) ⋃ {λ ∈ c : r(a−λi) is closed and β(a−λi) > α(b −λi)}⋃ {λ ∈ c : r(a−λi) is not closed and α(b −λi) < ∞} . by combining corollaries 3.1 and 3.2 we obtain; corollary 3.3. ⋂ c∈b(k,h) σgd(mc) = σlgd(a) ⋃ σrgd(b) ⋃ {λ ∈ c : β(a−λi) 6= α(b −λi)} . this result gives a generalization of [4, theorem 2.1]. 4. application to a spectral boundary value matrix problem this section is devoted to the study of boundary value problems described by an upper triangular operator matrices (2 × 2) acting in hilbert spaces with a complex spectral parameter λ, (p)   (ul −λmc)w = f γw = φ , where f and φ are given and ul is the matrix operator defined on h ⊕k by ul =   t l 0 d   , with l : k → h a given linear operator. we first define the boundary value problem (p) by ordered pairs (ul,mc) of an upper triangular operator matrix mc where ul is a right generalized drazin invertible and we construct the adapted boundary operator γ of ul. we prove the existence of an unique solution of (p) and we give an explicit expression for this solution. before this down, we define the boundary operator for a right generalized drazin invertible operator. int. j. anal. appl. 17 (1) (2019) 116 if s be a right generalized drazin inverse of the operator a ∈b(h), then k(a) = (r(s) ∩k(a)) ⊕ (n(a) ∩k(a)). (4.1) now, let e another complex hilbert space, called boundary space. definition 4.1. an operator γ : h → e is said to be a boundary operator for a right generalized drazin invertible operator a corresponding to its right generalized drazin inverse s ∈b(h) if, (i) k(a) ⊂ n(γ); (ii) there exists an operator π : e → h such that γπ = ie and r(π) = n(a) ∩k(a). theorem 4.1. let a ∈b(h) be a right generalized drazin invertible operator with a right generalized drazin inverse s. an operator γ : h −→ e is a boundary operator for a corresponding to s if and only if there exists a unique operator π : e −→ h, as in the definition 4.1, such that πγx = x−sax, for all x ∈ k(a). (4.2) proof. let γ : h −→ e be a boundary operator for a corresponding to s, then there exists π : e −→ h satisfying the conditions of the definition 4.1. let z = x−sax, then x = z +sax, sinse sax ∈ r(s)∩k(a) and x ∈ k(a) we have z ∈ n(a) ∩ k(a), then x − sax ∈ r(π), thus there exists ϕ ∈ e such that x−sax = πϕ. since n(γ) ⊂ k(a) and γπ = ie, we have γ(x−sax) = γπϕ and ϕ = γx, which implies (4.2). the uniqueness of π is directly obtained. conversely, suppose that γ and π satisfies the identity (4.2). then aπγx = 0, for all x ∈ k(a), so aπ = 0 on e. moreover, πγπγx = πγx−saπγx = πγx. hence γπγπγx = γπγx, so γπϕ = ϕ for all ϕ ∈ e, by taking ϕ = γπγx. finally, we have γsax = γx− γπγx = 0. thus k(a) ⊂ n(γ) � remark 4.1. if γ is a boundary operator for a right generalized drazin invertible operator a corresponding to its right generalized drazin inverse s, then k(a) = (r(s) ∩k(a)) ⊕r(π). (4.3) proposition 4.1 ( [15]). let a,b ∈b(h). then (i−λab) is invertible if and only if (i−λba) is invertible for all λ 6= 0. in this case, we have (i −λba)−1 = i + λb(i −λab)−1a, (4.4) int. j. anal. appl. 17 (1) (2019) 117 and (i −λab)−1 = i + λa(i −λba)−1b. (4.5) corollary 4.1. let a, b ∈b(h). if λ−1 ∈ ρ(ab) then (i −λab)−1a = a(i −λba)−1. proposition 4.2. let ul =   t l 0 d   defined on h ⊕k. assume that s1 and s2 are right generalized drazin inverses of t and d respectively. γ1 and γ2 are boundary operators for t and d with the boundary spaces e and z; respectively. if n(d) ⊂ n(l) then the operator γ =   γ1 0 0 γ2   from h ⊕k into e⊕z is a boundary operator for ul. proof. we have that k(t) ⊂ n(γ1), k(d) ⊂ n(γ2) and there exist π1 : e → h and π2 : z −→ k such that γ1π1 = ie, r(π1) = n(t) ∩ k(t) and γ2π2 = iz, r(π2) = n(d) ∩ k(d). denote by π =   π1 0 0 π2   : e ⊕z −→ h ⊕k. since t and d are right generalized drzain invertibles, then so is ul, hence k(ul) = k(t) ⊕k(d), that is k(ul) ⊂ n(γ) and γπ = ie⊕z. the condition n(d) ⊂ n(l) implies that n(ul) ∩k(ul) = (n(t) ⊕n(d)) ∩ (k(t) ⊕k(d)) = n(t) ∩k(t) ⊕n(d) ∩k(d) = r(π). � consider the operator ul defined as above and let a ∈b(h) and b ∈b(k) be given bounded operators on separable hilbert spaces h and k, and mc defined on h ⊕k by (1.1). we define the following spectral boundary value matrix problem for unknown w ∈ k(t) ×k(d) by (p)   (ul −λmc)w = f γw = φ , where f ∈ k(t)×k(d), φ ∈ e×z and λ ∈ c is a spectral parameter. we denote rλ[s1a] = (ih−λs1a)−1 and rλ[s2a] = (ik −λs2a)−1, s1 and s2 are right generalized drazin inverses of t and d, respectively. our purpose is to establish the existence and uniqueness of solutions for the boundary value problem (p). in the theorem below, we give an explicit expression for the solution of the problem (p). int. j. anal. appl. 17 (1) (2019) 118 theorem 4.2. if λ−1 ∈ ρ(s1a) ∩ ρ(s2b), the boundary value problem (p) is uniquely solvable for any f ∈ k(t) ×k(d) and φ ∈ e ×z, the solution is given by w f,φ λ = g(sf + πφ), where s =   s1 0 0 s2   and g =   rλ[s1a] −s1rλ[s1a](l−λc)rλ[s2b] 0 rλ[s2b]  . proof. we have (ul − λmc)w f,φ λ = (ul − λmc)gsf + (ul − λmc)gπφ. firstly, we calculate (ul − λmc)gsf. (ul −λmc)gsf = (ul −λmc)   rλ[s1a] −s1rλ[s1a](l−λc)rλ[s2b] 0 rλ[s2b]     s1f1 s2f2   =   (t −λa) (l−λc) 0 (d −λb)     rλ[s1a]s1f1 −s1rλ[s1a](l−λc)rλ[s2b]s2f2 rλ[s2b]s2f2   =   (t −λa)s1rλ[as1]f1 (d −λb)s2rλ[bs2]f2   = f, and (ul −λmc)gπφ =   (t −λa)[rλ[s1a]π1ϕ1 −s1rλ[s1a](l−λc)rλ[s2b]π2ϕ2] +(l−λc)rλ[s2b]π2ϕ2 (d −λb)rλ[s2b]π2ϕ2   =   (t −λa)rλ[s1a]π1ϕ1 (d −λb)rλ[s2b]π2ϕ2   =   (t −λa)[ih + λs1rλ[as1]a]π1ϕ1 (d −λb)[ik + λs2rλ[bs2]b]π2ϕ2   =   (t −λa)π1ϕ1 + λaπ1ϕ1 (d −λb)π2ϕ2 + λbπ2ϕ2   . int. j. anal. appl. 17 (1) (2019) 119 then (ul −λmc)gπφ = 0 since r(π1) = n(t) ∩k(t) and r(π2) = n(d) ∩k(d). using the fact that k(t) ⊂ n(γ1) and k(d) ⊂ n(γ2), we obtain, γw f,φ λ = γg(sf + πφ) =   γ1 0 0 γ2     rλ[s1a]s1f1 −s1rλ[s1a](l−λc)rλ[s2b]s2f2 rλ[s2b]s2f2   +   γ1 0 0 γ2     rλ[s1a]π1ϕ1 −s1rλ[s1a](l−λc)rλ[s2b]π2ϕ2 rλ[s2b]π2ϕ2   =   γ1rλ[s1a]π1ϕ1 − γ1s1rλ[s1a](l−λc)rλ[s2b]π2ϕ2 γ2rλ[s2b]π2ϕ2   =   γ1[ih + λs1rλ[as1]a]π1ϕ1 γ2[ik + λs2rλ[bs2]b]π2ϕ2   =   γ1π1ϕ1 γ2π2ϕ2   = φ. the uniqueness of the solution of (p) follows from standard arguments. that is, if w1,w2 ∈ k(t) ×k(d) are two solutions of (p), assume that w0 = w1 − w2 =   u0 v0   =   s1f0 + π1ϕ0 s2g0 + π2ψ0   with some f0 ∈ k(t), g0 ∈ k(d), ϕ0 ∈ e and ψ0 ∈ z. thus,   (ul −λmc)w0 = 0 γw0 = 0 . since k(ul) ⊂ n(γ) and γπ = ie⊕z, the second identity gives   ϕ0 ψ0   =   0 0   . then u0 = s1f0 and v0 = s2g0. so, 0 = (ul −λmc)w0 =   (t −λa) (l−λc) 0 (d −λb)     s1f0 s2g0   =   (t −λa)s1f0 + (l−λc)s2g0 (d −λb)s2g0   . then, f0 = g0 = 0, since λ −1 ∈ ρ(s1a)∩ρ(s1b), f0 ∈ k(t) and g0 ∈ k(d). hence w1 = w2. this complete the proof. � int. j. anal. appl. 17 (1) (2019) 120 references [1] p. aiena, fredholm and local spectral theory, with applications to multipliers, kluwer academic publishers, 2004. [2] m. barraa and m. boumazgour, on the perturbations of spectra of upper triangular operator matrices, j. math. anal. appl. 347 (1) (2008), 315–322. [3] m. benharrat and b. messirdi, on the generalized kato spectrum, serdica math. j. 37 (4) (2011), 283–294. [4] m. boumezgour, drazin invertibility of upper triangular operator matrices, linear and multilinear algebra 61 (5) (2013) , 627–634. [5] x. h. cao, m. z. guo and b. meng, drazin spectrum and weyl’s theorem for operator matrices, j. math. res. exposition, 26 (3) (2006), 413–422. [6] s. v. djordjevic and b. p. duggal, drazin invertibility of the diagonal of an operator, linear and multilinear algebra, 60 (1) (2012), 65–71. [7] h. k. du and j. pan, perturbation of spectrum of 2×2 operator matrices, proc. amer. math. soc. 121 (3) (1994), 761–766. [8] s. grabiner, ascent, descent, and compact perturbations, proc. amer. math. soc. 71 (1) (1978), 79–80. [9] j. k. han, h. y. lee and w. y. lee, invertible completions of 2×2 upper triangular operator matrices, proc. amer. math. soc. 128 (1) (1999), 119–123. [10] g.l. han and v. chen, on the right (left) invertible completions for operator matrices, integr. equ. oper. theory 67 (1) (2010), 79–93. [11] g. l. han and a. chen, perturbations of the right and left spectra for operator matrices, j. oper. theory 67 (1) (2012), 207–214. [12] r. harte, on kato non-singularity, studia math. 117 (2) (1996), 107–114. [13] i. s. hwang and w. y. lee, the boundedness below of 2×2 upper triangular operator matrices, integr. equ. oper. theory 39 (3) (2001), 267-276. [14] m. a. kaashoek and d.c. lay, ascent, descent, and commuting perturbations, trans. amer. math. soc. 169 (1972), 35–47. [15] n. khaldi, m. benharrat and b. messirdi, on the spectral boundary value problems and boundary approximate controllability of linear systems. rend. circ. mat. palermo 63 (1) (2014), 141–153. [16] n. khaldi, m. benharrat and b. messirdi, a spectral analysis for solving boundary value matrix problems: existence, uniqueness and application to symplectic elasticity, j. adv. res. appl. math. 6 (4) (2014), 68–80. [17] kato, t., perturbation theory for linear operators, springer-verlag, new york, 1966. [18] j. j. koliha, a generalized drazin inverse, glasgow math. j. 38 (3) (1996), 367–381, [19] v. kordula and v. müller, the distance from the apostol spectrum, proc. amer. math. soc. 124 (10) (1996), 3055–3061. [20] j-p. labrousse, les opérateurs quasi-fredholm une généralisation des opérateurs semi-fredholm, rend. circ. mat. palermo 29 (1) (1980), 161–258. [21] w. y. lee, weyl spectra of operator matrices, proc. amer. math. soc. 129 (1) (2001), 131–138. [22] y. li, x. h. sun, and h. k. du, the intersection of left (right) spectra of 2×2 upper triangular operator matrices, linear algebra appl. 418 (1) (2006), 112–121. [23] m. mbekhta, opérateurs pseudo-fredholm i : résolvant généralisé, j. operator theory 24 (2) ( 1990), 255–276. [24] k. miloud hocine, m. benharrat and b. messirdi, left and right generalized drazin invertible operators, linear and multilinear algebra 63 (8) (2015), 1635–1648. [25] v. müller, on the regular spectrum, j. operator theory 31 (2) (1994) , 363–380. [26] v. müller, spectral theory of linear operators and spectral systems in banach algebra, birkhauser, 2007. int. j. anal. appl. 17 (1) (2019) 121 [27] v. rakočević, generalized spectrum and commuting compact perturbations, proc. edinburgh. math. soc.36 (2) (1993), 197–209. [28] s. f. zhang, h. j. zhong, and q. f. jiang, drazin spectrum of operator matrices on the banach space, linear algebra appl. 429 (8-9) (2008), 2067–2075. [29] y. n. zhang, h. j. zhong and l. q. lin, browder spectra and essential spectra of operator matrices, acta math. sinica 24 (6) ( 2008), 947–954. [30] s. f. zhang, h. j. zhong and l. q. lin, generalized drazin spectrum of operator matrices, appl. math. j. chinese univ. 29 (2) (2014), 162–170. 1. introduction and preliminaries 2. the quasi-nilpotent part and the analytical core of the operator mc 3. left and right generalized drazin invertibility of mc 4. application to a spectral boundary value matrix problem references international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 39-44 http://www.etamaths.com dunkl lipschitz functions for the generalized fourier-dunkl transform in the space l2α,n r. daher, s. el ouadih∗ and m. el hamma abstract. in this paper, using a generalized translation operator, we prove the estimates for the generalized fourier-dunkl transform in the space l2α,n on certain classes of functions. 1. introduction and preliminaries in [5], e. c. titchmarsh’s characterizes the set of functions in l2(r) satisfying the cauchy-lipschitz condition by means of an asymptotic estimate growth of the norm of their fourier transform, namely we have theorem 1.1 let δ ∈ (0, 1) and assume that f ∈ l2(r). then the following are equivalents (i) ‖f(t + h) −f(t)‖ = o(hδ), as h → 0, (ii) ∫ |λ|≥r |f̂(λ)|2dλ = o(r−2δ) as r →∞, where f̂ stands for the fourier transform of f. in this paper, we consider a first-order singular differential-difference operator λ on r which generalizes the dunkl operator λα, we prove an analog of theorem 1.1 in the generalized fourier-dunkl transform associated to λ in l2α,n . for this purpose, we use a generalized translation operator. we point out that similar results have been established in the context of non compact rank one riemannian symetric spaces [4]. in this section, we develop some results from harmonic analysis related to the differential-difference operator λ. further details can be found in [1] and [6]. in all what follows assume where α > −1/2 and n a non-negative integer. consider the first-order singular differential-difference operator on r λf(x) = f′(x) + ( α + 1 2 ) f(x) −f(−x) x − 2n f(−x) x . for n = 0, we regain the differential-difference operator λαf(x) = f ′(x) + ( α + 1 2 ) f(x) −f(−x) x , 2010 mathematics subject classification. 46l08. key words and phrases. differential-difference operator; generalized fourier-dunkl transform; generalized translation operator. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 39 40 daher, ouadih and hamma which is referred to as the dunkl operator of index α + 1/2 associated with the reflection group z2 on r. such operators have been introduced by dunkl (see [3], [7]) in connection with a generalization of the classical theory of spherical harmonics. let m be the map defined by mf(x) = x2nf(x), n = 0, 1, ... let lpα,n, 1 ≤ p < ∞, be the class of measurable functions f on r for which ‖f‖p,α,n = ‖m−1f‖p,α+2n < ∞, where ‖f‖p,α = (∫ r |f(x)|p|x|2α+1dx )1/p . if p = 2, then we have l2α,n = l 2(r, |x|2α+1). the one-dimensional dunkl kernel is defined by eα(z) = jα(iz) + z 2(α + 1) jα+1(iz),z ∈ c,(1) where jα(z) = γ(α + 1) ∞∑ m=0 (−1)m(z/2)2m m!γ(m + α + 1) ,z ∈ c,(2) is the normalized spherical bessel function of index α. it is well-known that the functions eα(λ.), λ ∈ c, are solutions of the differential-difference equation λαu = λu,u(0) = 1. lemma 1.2 for x ∈ r the following inequalities are fulfilled i) |jα(x)| ≤ 1, ii) |1 − jα(x)| ≤ |x|, iii) |1 − jα(x)| ≥ c with |x| ≥ 1, where c > 0 is a certain constant which depends only on α. proof. similarly as the proof of lemma 2.9 in [2]. for λ ∈ c, and x ∈ r, put ϕλ(x) = x 2neα+2n(iλx), where eα+2n is the dunkl kernel of index α + 2n given by (1). proposition 1.3 i) ϕλ satisfies the differential equation λϕλ = iλϕλ. ii) for all λ ∈ c, and x ∈ r |ϕλ(x)| ≤ |x|2ne|imλ||x|. the generalized fourier-dunkl transform we call the integral transform fλf(λ) = ∫ r f(x)ϕ−λ(x)|x|2α+1dx,λ ∈ r,f ∈ l1α,n. dunkl lipschitz functions 41 let f ∈ l1α,n such that fλ(f) ∈ l1α+2n = l1(r, |x|2α+4n+1dx). then the inverse generalized fourier-dunkl transform is given by the formula f(x) = ∫ r fλf(λ)ϕλ(x)dµα+2n(λ), where dµα+2n(λ) = aα+2n|λ|2α+4n+1dλ, aα = 1 22α+2(γ(α + 1))2 . proposition 1.4 i) for every f ∈ l2α,n, fλ(λf)(λ) = iλfλ(f)(λ). ii) for every f ∈ l1α,n ∩l2α,n we have the plancherel formula∫ r |f(x)|2|x|2α+1dx = ∫ r |fλf(λ)|2dµα+2n(λ). iii) the generalized fourier-dunkl transform fλ extends uniquely to an isometric isomorphism from l2α,n onto l 2(r,µα+2n). the generalized translation operators τx, x ∈ r, tied to λ are defined by τxf(y) = (xy)2n 2 ∫ 1 −1 f( √ x2 + y2 − 2xyt) (x2 + y2 − 2xyt)n ( 1 + x−y√ x2 + y2 − 2xyt ) a(t)dt + (xy)2n 2 ∫ 1 −1 f(− √ x2 + y2 − 2xyt) (x2 + y2 − 2xyt)n ( 1 − x−y√ x2 + y2 − 2xyt ) a(t)dt, where a(t) = γ(α + 2n + 1) √ πγ(α + 2n + 1/2) (1 + t)(1 − t2)α+2n−1/2. proposition 1.5 let x ∈ r and f ∈ l2α,n. then τxf ∈ l2α,n and ‖τxf‖2,α,n ≤ 2x2n‖f‖2,α,n. furthermore, fλ(τxf)(λ) = x2neα+2n(iλx)fλ(f)(λ).(3) 2. main results in this section we give the main result of this paper. we need first to define (ψ,δ,β)-generalized dunkl lipschitz class. definition 2.1. let δ > 1 and β > 0. a function f ∈ l2α,n is said to be in the (ψ,δ,β)-generalized dunkl lipschitz class, denoted by dlip(ψ,δ,β), if ‖τhf(x) + τ−hf(x) − 2h2nf(x)‖2,α,n = o(hδ+2nψ(hβ)) as h → 0, where (a) ψ is a continuous increasing function on [0,∞), 42 daher, ouadih and hamma (b) ψ(0) = 0 , ψ(ts) = ψ(t)ψ(s) for all t,s ∈ [0,∞), (c) and ∫ 1/h 0 s1−2δψ(s−2β)ds = o(h2δ−2ψ(h2β)), h → 0. theorem 2.2. let f ∈ l2α,n. then the following are equivalents (a) f ∈ dlip(ψ,δ,β) (b) ∫ |λ|≥r |fλf(λ)|2dµα+2n(λ) = o(r−2δψ(r−2β)), as r →∞. proof. (a) ⇒ (b). let f ∈ dlip(ψ,δ,β). then we have ‖τhf(x) + τ−hf(x) − 2h2nf(x)‖2,α,n = o(hδ+2nψ(hβ)) as h → 0. from formulas (1), (2) and (3), we have the generalized fourier-dunkl transform of τhf(x) + τ−hf(x) − 2h2nf(x) is 2h2n(jα+2n(λh) − 1)fλf(λ). by plancherel equality, we obtain ‖τhf(x)+τ−hf(x)−2h2nf(x)‖22,α,n = 4h 4n ∫ +∞ −∞ |jα+2n(λh)−1|2|fλf(λ)|2dµα+2n(λ). if |λ| ∈ [ 1 h , 2 h ], then |λh| ≥ 1 and (iii) of lemma 1.2 implies that 1 ≤ 1 c2 |jα+2n(λh) − 1|2. then∫ 1 h ≤|λ|≤2 h |fλf(λ)|2dµα+2n(λ) ≤ 1 c2 ∫ 1 h ≤|λ|≤2 h |jα+2n(λh) − 1|2|fλf(λ)|2dµα+2n(λ) ≤ 1 c2 ∫ +∞ −∞ |jα+2n(λh) − 1|2|fλf(λ)|2dµα+2n(λ) ≤ h−4n 4c2 ‖τhf(x) + τ−hf(x) − 2h2nf(x)‖22,α,n = o(h2δψ(h2β)). we obtain ∫ r≤|λ|≤2r |fλf(λ)|2dµα+2n(λ) ≤ cr−2δψ(r−2β), r →∞, where c is a positive constant. now,∫ |λ|≥r |fλf(λ)|2dµα+2n(λ) = ∞∑ i=0 ∫ 2ir≤|λ|≤2i+1r |fλf(λ)|2dµα+2n(λ) ≤ cr−2δψ(r−2β) ∞∑ i=0 (2−2δψ(2−2β))i ≤ ccδ,βr−2δψ(r−2β), where cδ,β = (1 − 2−2δψ(2−2β)))−1 since 2−2δψ(2−2β) < 1. consequently∫ |λ|≥r |fλf(λ)|2dµα+2n(λ) = o(r−2δψ(r−2β)), as r →∞. dunkl lipschitz functions 43 (b) ⇒ (a). suppose now that∫ |λ|≥r |fλf(λ)|2dµα+2n(λ) = o(r−2δψ(r−2β)), as r →∞. and write ‖τhf(x) + τ−hf(x) − 2h2nf(x)‖22,α,n = 4h 4n(i1 + i2), where i1 = ∫ |λ|< 1 h |jα+2n(λh) − 1|2|fλf(λ)|2dµα+2n(λ), and i2 = ∫ |λ|≥1 h |jα+2n(λh) − 1|2|fλf(λ)|2dµα+2n(λ). firstly, we use the formulas |jα+2n(λh)| ≤ 1 and i2 ≤ 4 ∫ |λ|≥1 h |fλf(λ)|2dµα+2n(λ) = o(h2δψ(h2β)), as h → 0. set φ(x) = ∫ +∞ x |fλf(λ)|2dµα+2n(λ). integrating by parts we obtain∫ x 0 λ2|fλf(λ)|2dµα+2n(λ) = ∫ x 0 −λ2φ′(λ)dλ = −x2φ(x) + 2 ∫ x 0 λφ(λ)dλ ≤ c1 ∫ x 0 λ1−2δψ(λ−2β)dλ = o(x2−2δψ(x−2β)), where c1 is a positive constant. we use the formula (ii) of lemma 1.2∫ +∞ −∞ |jα+2n(λh) − 1|2|fλf(λ)|2dµα+2n(λ) = o ( h2 ∫ |λ|< 1 h λ2|fλf(λ)|2dµα+2n(λ) ) + o(h2δψ(h2β)) = o(h2h−2+2δψ(h2β)) + o(h2δψ(h2β)) = o(h2δψ(h2β)), and this ends the proof.� references [1] s. a. al sadhan, r. f. al subaie and m. a. mourou, harmonic analysis associated with a first-order singular differential-difference operator on the real line. current advances in mathematics research, 1(2014), 23-34. [2] e. s. belkina and s. s. platonov, equivalence of k-functionnals and modulus of smoothness constructed by generalized dunkl translations, izv. vyssh. uchebn. zaved. mat., 8(2008), 3-15. [3] c. f. dunkl, differential-difference operators associated to reflection groups. transactions of the american mathematical society, 311(1989), 167-183. [4] s. s. platonov, the fourier transform of function satisfying the lipshitz condition on rank 1 symetric spaces , siberian math.j.46(2005), 1108-1118. [5] e. c. titchmarsh , introduction to the theory of fourier integrals . claredon , oxford, 1948, komkniga.moxow. 2005. 44 daher, ouadih and hamma [6] r. f. al subaie and m. a. mourou, inversion of two dunkl type intertwining operators on r using generalized wavelets. far east journal of applied mathematics, 88(2014), 91-120. [7] c. f. dunkl, hankel transforms associated to finite reflection groups. contemporary mathematics, 138(1992), 128-138. department of mathematics, faculty of sciences äın chock, university hassan ii, casablanca, morocco ∗corresponding author int. j. anal. appl. (2022), 20:47 evolutes of fronts in de sitter and hyperbolic spheres m. khalifa saad1,∗, h. s. abdel-aziz2, a. a. abdel-salam2 1department of mathematics, faculty of science, islamic university of madinah, ksa 2department of mathematics, faculty of science, sohag university, 82524 sohag, egypt ∗corresponding author: mohammed.khalifa@iu.edu.sa abstract. the evolute of a regular curve is a classical object from the viewpoint of differential geometry. we study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de sitter and hyperbolic spaces. also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. finally, some computational examples in support of our main results are given and plotted. 1. introduction in 1915, einstein formulated general relativity as a theory of space, time and gravitation in semieuclidean space. however, this subject has remained dormant for much of its history because its understanding requires advanced mathematics knowledge. since the end of the twentieth century, semi-euclidean geometry has been an active area of mathematical research, and it has been applied to a variety of subjects related to differential geometry and general relativity. it is well known that many important results in the theory of curves in r3 were initiated by g. monge and g. darboux pionnered the moving frame idea. thereafter, frenet defined his moving frame and special equations which are playing an important role in mechanics and kinematics as well as in differential geometry [1]. at the beginning of the twentieth century, a. einstein’s theory opened a door to use of new geometries. one of them, minkowski space-time, which is simultaneously the geometry of special relativity. it is worth mentioning that the importance of the theory of singularity as a developing area which is received: aug. 8, 2022. 2010 mathematics subject classification. 53a25, 53c50. key words and phrases. frenet frames; evolute curves; front curves; de sitter space; hyperbolic space. https://doi.org/10.28924/2291-8639-20-2022-47 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-47 2 int. j. anal. appl. (2022), 20:47 related to nonlinear sciences, it has been extensively applied in studying the classifications of singularities associated with some objects in euclidean and semi-euclidean spaces [2–4]. therefore, it has been a field of research for many researchers. in this paper, we focus on the evolutes of curves at singular points in de sitter and hyperbolic spheres. the evolute of a plane curve is defined to be the locus of the center of its osculating circles [5]. in particular, the evolute of a regular curve is a classical object from the viewpoint of differential geometry. the evolute of a regular curve without inflection points is given by, not only, the locus of all its centers of curvature but also the envelope of its normal lines. the properties of evolutes can be discussed by frenet-serret formulas, distance squared functions and the theories of lagrangian and legendre singularities. in general, there exist singular points along the evolute of a regular curve and the singular points corresponding to the vertices of a regular curve. there are at least four vertices for a simple closed curve. one can not define the evolutes of curves at singular points. however, we can define evolutes of fronts under some conditions. in [6,7], t. fukunaga and m. takahashi defined legendre curves in euclidean plane and studied evolutes of legendre curves. moreover, s. izumiya, d. he pei, t.sano and e. torii defined the evolute curve in hyperbolic 2-space and found its equation (see [7]). the paper can be organized as follows: section 3 presents a framed curve and gives its moving frame in de sitter sphere. moreover, we define a pair of smooth functions of this curve as a geodesic curvature for a regular curve. we define evolutes of fronts in de sitter sphere. the evolute of a front is a generalization of the notion of an evolute of a regular curve. therefore, we discuss some properties of evolutes without inflection points. by the representation, we give properties for an evolute of the front. for example, the evolute of a front is also a front (see theorem 3.1). in section 4, similar to the way that considered in the study of the framed curves, fronts and the evolutes in de sitter sphere, we do it in the hyperbolic sphere, see theorem (4.1). we shall assume throughout the whole paper that all manifolds and maps are c∞ unless the contrary is explicitly stated. 2. geometric meanings and basis concepts in this section, we present some of classical differential geometric properties of de sitter and hyperbolic spaces of plane curves. we adopt s21 and h 2 0 as models of de sitter and hyperbolic spheres in minkowski 3-space e31, respectively. since s 2 1 and h 2 0 are a riemannian manifolds, so the explicit differential geometry of the curves in these spheres is analogous to the differential geometry of the curves in the euclidean plane (for more details see [5,8]). let r3 = {(x1,x2,x3) | x1,x2,x3 ∈ r} be a 3-dimensional vector space, and x = (x1,x2,x3) and y = (y1,y2,y3) be two vectors in r3. the pseudo scalar product of x and y is defined by 〈x,y〉 = −x1y1+x2y2+x3y3. we call (r3,〈,〉) a 3-dimensional pseudo euclidean space, or minkowski 3-space. we write e31 instead of (r 3,〈,〉). we say that a vector x in e31 is spacelike, lightlike or timelike if int. j. anal. appl. (2022), 20:47 3 〈x,x〉 > 0, 〈x,x〉=0 or 〈x,x〉 < 0, respectively. now, we define two spheres in e31 as follows: q2� =  h 2 0 = {x ∈e 3 1 | −x 2 1 +x 2 2 +x 2 3 =−1}, if � =− s21 = {x ∈e 3 1 | −x 2 1 +x 2 2 +x 2 3 =1}, if � =+, and we take h20 =  h 2 + = {x ∈e31 | −x 2 1 +x 2 2 +x 2 3 =−1, x1 ≥ 1} h2− = {x ∈e31 | −x 2 1 +x 2 2 +x 2 3 =−1, x1 ≤−1}, where h20 =h 2 + ∪h2−. we call h20 a hyperbolic sphere and s 2 1 a de sitter sphere. let γ : i −→q2� ⊂e31; γ(t)= (x1(t),x2(t),x3(t)) be a smooth regular curve in q 2 � (i.e.,γ ′(t) 6=0) for any t ∈ i, where i is an open interval. it is easy to show that 〈γ′(t),γ′(t)〉 > 0. throughout the remainder in this paper, we denote the parameter s of γ as the arc-length parameter. let us denote t(s)= γ̇(s), and we call t(s) a unit tangent vector of γ at s. for any x = (x1,x2,x3),y = (y1,y2,y3) ∈ r31, the pseudo vector product of x and y is defined as follows: x ∧y = ∣∣∣∣∣∣∣∣ −e1 e2 e3 x1 x2 x3 y1 y2 y3 ∣∣∣∣∣∣∣∣ =(−(x2y3 −x3y2),x3y1 −x1y3,x1y2 −x2y1). here, we set a vector e(s) = γ(s)∧t(s). by definition, we can calculate that 〈e(s),e(s)〉 = 1 and 〈γ(s),γ(s)〉 = −1. also, we can show that t(s)∧e(s) = −γ(s) and γ(s)∧e(s) = −t(s). therefore, we have a pseudo-orthonormal frame {γ(s),t(s),e(s)} along γ(s). the de sitter frenetserret formula of plane curve are:  γ̇(s) ṫ(s) ė(s)   =   0 1 0 −1 0 κg 0 κg 0     γ(s) t(s) e(s)   , (2.1) where κg is the geodesic curvature of γ in s21, which is given by κg(s)= det(γ(s) t(s) ṫ(s)). (2.2) for the general parameter t, we get t(t) = γ′(t) ‖γ′(t)‖ and e(t) = γ(t) ∧ t(t). then, de sitter frenet-serret formula of γ(t) is expressed as:  γ′(t) t′(t) e′(t)   =   0 ‖γ′(t)‖ 0 −‖γ′(t)‖ 0 ‖γ′(t)‖κg 0 ‖γ′(t)‖κg 0     γ(t) t(t) e(t)   , (2.3) where κg(t)= det (γ(t) γ′(t) γ′′(t)) ‖γ′(t)‖3 . (2.4) 4 int. j. anal. appl. (2022), 20:47 also, the hyperbolic frenet-serret formula is given by:  γ̇(s) ṫ(s) ė(s)   =   0 1 0 1 0 κg 0 −κg 0     γ(s) t(s) e(s)   , (2.5) where κg is the geodesic curvature of the curve γ in h20, which is defined as κg(s)= det(γ(s) t(s) ṫ(s)). (2.6) we have the following hyperbolic frenet-serret formula of γ(t)  γ′(t) t′(t) e′(t)   =   0 ‖γ′(t)‖ 0 ‖γ′(t)‖ 0 ‖γ′(t)‖κg 0 −‖γ′(t)‖κg 0     γ(t) t(t) e(t)   , (2.7) definition 2.1. under the assumption κ2g 6= ±1, the evolute of a regular curve γ is defined as eγ : i −→q2�; eγ(t)= 1√ |κ2g(t)−1| (κg(t)γ(t)+ �e(t)); (2.8) eγ is called hyperbolic evolute or de sitter evolute of γ when � =1 or � =−1, respectively (see [5,8]). remark 2.1. eγ(t) is located in h20 with κ 2 g(t) > 1, and it is in s 2 1 with κ 2 g(t) < 1. 3. evolutes of fronts in de sitter sphere s21 if γ has a singular point, we can not construct a moving frame of γ in a traditional way. however, we could define a moving frame of a front curve. for the case of euclidean plane, there are some creative works (see [6,7,9] ). now, we give the following definitions. definition 3.1. we say that (γ,ν) : i −→ s21 × s 2 1 is a framed curve, if 〈γ(t),ν(t)〉 = 0 and 〈γ′(t),ν(t)〉=0 for all t ∈ i. moreover, if (γ,ν) is an immersion, namely, (γ′(t),ν′(t)) 6=(0,0), we call (γ,ν) a framed immersion curve. definition 3.2. we say that γ : i −→ s21 is a frontal curve if there exists a smooth mapping ν : i −→ s 2 1 such that (γ,ν) is a framed curve. we also say that γ : i −→ s21 is a front curve if there exists a smooth mapping ν : i −→ s21 such that (γ,ν) is a framed immersion curve. throughout this paper, we assume that the pair (γ,ν) is co-oriented and the singular points of γ are finite. let (γ,ν) : i −→ s21×s 2 1 be a framed curve. if γ is singular at t0, then we can’t define a frame in a traditional way. however, ν always exists even if t is a singular point of γ. we take µ = ν ∧γ and int. j. anal. appl. (2022), 20:47 5 call the pair {γ,ν,µ} a moving frame of γ and then, de sitter frenet-serret formula is given by:  γ′(t) ν′(t) µ′(t)   =   0 0 m(t) 0 0 n(t) −m(t) n(t) 0     γ(t) ν(t) µ(t)   , where n(t) = 〈ν′(t),µ(t)〉, ν(t) and µ(t) are both unit spacelike vectors. we declare that (γ,−ν) is also a framed curve. in this case, m(t) dose not change, but n(t) changes to −n(t). if (γ,ν) is a framed immersion, we have (m(t),n(t)) 6= (0,0) for each t ∈ i. the pair (m,n) is an important pair of functions of the framed curves as the geodesic curvature of a regular curve. we call the pair (m,n) a geodesic curvature of the framed curve. also, we have ν ∧µ = γ and γ ∧µ = ν (for more details see [8,10]). in what follows, some properties of the meant curves are introduced. proposition 3.1. if γ be a regular curve and (γ,ν) : i −→ s21 ×s 2 1 be its framed curve, then the relationship between their geodesic curvatures is expressed as: κg(t)=− n(t) |m(t)| . (3.1) proof. by direct calculations, we have γ′(t)= m(t)µ(t), γ′′(t)= m′(t)µ(t)+m(t)µ′(t) = m′(t)µ(t)+m(t)(−m(t)γ(t)+n(t)ν(t)), where κg(t)= det(γ(t),γ′(t),γ′′(t)) ‖γ′(t)‖3 = det ( γ(t),mµ(t),−m2γ(t)+mnν(t)+m′µ(t) ) |m3| , so, we get κg(t)=− n(t) |m(t)| . for a framed immersion (γ,ν), we say that t0 is an inflection point of the front γ if n(t0) = 0. and then the condition m(t0) 6=0 and n(t0)=0, is equivalent to the condition κg(t0)=0. � if γ is not a regular curve, then we can not define the evolute as in equation (2.8), since the geodesic curvature may be divergence at a singular point. however, we can give the definition of the evolute of a front and then focus on its properties. hence, the notion of a parallel curve of γ can be presented as follows: 6 int. j. anal. appl. (2022), 20:47 let (γ,ν) : i −→ s21 ×s 2 1 be a framed curve which has the geodesic curvature (m,n). so, we can define a parallel curve γυ : i −→ s21 of γ as follows: γυ(t)= 1√ |υ2 −1| (γ(t)+υν(t)) , (3.2) where υ ∈r and υ 6=±1. proposition 3.2. if γυ is a regular curve, then κgυ(t)= −n(t)−υm(t) |m(t)+υn(t)| . (3.3) proof. from eq.(3.2), we have γ′υ(t)= m+υn√ |υ2 −1| µ(t), γ′′υ(t)= −m2 −υmn√ |υ2 −1| γ(t)+ mn+υn2√ |υ2 −1| ν(t)+ m′ +υn′√ |υ2 −1| µ(t), then, we find γυ ∧γ′υ = 1 (υ2 −1) ∣∣∣∣∣∣∣∣ −γ(t) ν(t) µ(t) 1 υ 0 0 0 m+υn ∣∣∣∣∣∣∣∣ = −υ(m+υn)γ(t)− (m+υn)ν(t) (υ2 −1) . since κgυ(t)= det(γυ(t),γ ′ υ(t),γ ′′ υ(t)) ‖γ′υ(t)‖3 . we obtain |m(t)+υn(t)|κgυ(t)= (−n(t)−υm(t)). thus, this completes the proof. � proposition 3.3. for a framed immersion curve (γ,ν) : i −→ s21×s 2 1, the parallel curve γυ : i −→ s 2 1 is a front for each υ 6=±1. proof. we take νυ : i −→ s21 by νυ(t)= 1√ |υ2 −1| (υγ(t)+ν(t)) , since   γυ(t)= 1√ |υ2 −1| (γ(t)+υν(t)) , γ′υ(t)= 1√ |υ2 −1| (γ′(t)+υν′(t)) . int. j. anal. appl. (2022), 20:47 7 if γ′υ(t0)=0 at a point t0 ∈ i, then we have γ′(t0)+υν ′(t0)=0. also, if ν′(t0)=0, then γ′(t0)=0. it is contradicted with the fact that (γ,ν) is a framed immersion and hence (γυ,νυ) is a framed immersion. by ‖ν(t)‖=1, we have 〈ν(t),ν′(t)〉=0. then 〈γ′υ(t),νυ(t)〉= 1 (υ2 −1) 〈γ′ +υν′,υγ +ν〉=0, therefore, it leads to the curve γυ is a front. � proposition 3.4. let (γ,ν) be a framed curve. if γ is a regular curve and υ 6=1/κg, then a parallel curve γυ is also a regular curve and its evolute is given by eγυ(t)=−eγ(t). (3.4) proof. since γυ(t)= 1√ |υ2 −1| (γ(t)+υe(t)) , γ′υ(t)= ‖γ′‖√ |υ2 −1| (1+υκg)t(t), γ′′υ(t)= ‖γ′‖√ |υ2 −1| ( −‖γ′‖(1+υκg)γ(t)+υκ′gt(t)+‖γ ′‖κg(1+υκg)e(t) ) . by the assumption υ 6=1/κg, γυ is a regular curve. therefore, we get γυ ∧γ′υ = 1 (υ2 −1) ∣∣∣∣∣∣∣∣ −γ(t) t(t) e(t) 1 0 υ 0 ‖γ′‖(1+υκg) 0 ∣∣∣∣∣∣∣∣ = ‖γ′‖υ(1+υκg)γ(t)+‖γ′‖(1+υκg)e(t) (υ2 −1) , and hence 〈γυ ∧γ′υ,γ ′′ υ〉= ‖γ′‖3υ(1+υκg)2 +‖γ′‖3κg(1+υκg)2 |υ2 −1| 3 2 , we get κgυ(t)= κg +υ |1+υκg| , and tυ(t)= γ′υ ‖γ′υ‖ = 1+υκg |1+υκg| t(t). since eυ = γυ ∧tυ, we obtain eυ(t)= 1+υκg |1+υκg| 1√ |υ2 −1| (e(t)+υγ(t)) . 8 int. j. anal. appl. (2022), 20:47 thus, from (2.8) we find eγυ(t)= 1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−eυ(t)) = 1√√√√∣∣∣∣∣ ( υ +κg |1+υκg| )2 −1 ∣∣∣∣∣ ( υ +κg |1+υκg| (γ(t)+υe(t))√ |υ2 −1| − 1+υκg |1+υκg| (e(t)+υγ(t))√ |υ2 −1| ) = 1√√√√∣∣∣∣∣ ( υ +κg |1+υκg| )2 −1 ∣∣∣∣∣ 1 |1+υκg| √ |υ2 −1| ( (1−υ2)κgγ(t)− (1−υ2)e(t) ) = 1√∣∣(υ2 −1)−κ2g(υ2 −1)∣∣ (1−υ2)√ |υ2 −1| (κgγ(t)−e(t)) =− 1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−e(t)) =−eγ(t). thus, the proof is completed. � definition 3.3. let (γ,ν) : i −→ s21 × s 2 1 be a framed immersion curve. we define an evolute eγ : i −→ s21 of γ as follows: if t is a regular point, then eγ(t)= 1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−e(t)) . (3.5) if t0 is a singular point, for any t ∈ (t0 −δ,t0 +δ), we get eγυ(t)= −1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−eυ(t)) , (3.6) where δ is a sufficiently small positive real number and υ ∈r satisfies the condition υ 6=1/κg(t). now, we give another representation of the evolute by using the moving frame {γ(t),ν(t),µ(t)} and its geodesic curvature {m(t),n(t)}. theorem 3.1. under the condition |m(t)| 6= |n(t)|, the evolute of a front curve eγ(t) : i −→ s21 is represented by eγ(t)= 1√ |n2(t)−m2(t)| (−n(t)γ(t)+m(t)ν(t)) , (3.7) and eγ(t) is a front curve. int. j. anal. appl. (2022), 20:47 9 proof. (i) suppose that γ is a regular curve. since γ′(t)= m(t)µ(t), we have |m(t)| 6=0 and t(t)= m(t) |m(t)| µ(t), e(t)=− m(t) |m(t)| ν(t). from eqs.(3.1) and (3.5), we get eγ(t)= 1√∣∣κ2g(t)−1∣∣ (κg(t)γ(t)−e(t)) = 1√√√√∣∣∣∣∣ ( − n(t) |m(t)| )2 −1 ∣∣∣∣∣ ( − n(t) |m(t)| γ(t)+ m(t) |m(t)| ν(t) ) = 1√ |n2(t)−m2(t)| (−n(t)γ(t)+m(t)ν(t)) . (ii) suppose that t0 is a singular point of γ and consider γυ in de sitter sphere, also we know that γυ is a regular curve around the neighbourhood of t0 with υ 6=1/κg(t). from eq.(3.2), we get γ′υ(t)= m+υn√ |υ2 −1| µ(t), then, |m+υn| 6=0 and tυ(t)= m+υn |m+υn| µ(t), where eυ(t)= γυ(t)∧tυ(t), we have eυ(t)= m+υn |m+υn| 1√ |υ2 −1| (−υγ(t)−ν(t)) . therefore, from eq.(3.3), we find κgυ =− n+υm |m+υn| , and from eqs.(3.4) and (3.6), we get eγ(t)=−eγυ(t) = 1√∣∣κ2gυ(t)−1∣∣ (κgυ(t)γυ(t)−eυ(t)) = 1√√√√∣∣∣∣∣ ( − (n+υm) |m+υn| )2 −1 ∣∣∣∣∣ (( − (n+υm) |m+υn| ) (γ(t)+υν(t))√ |υ2 −1| + m+υn |m+υn| (υγ(t)+ν(t))√ |υ2 −1| ) = 1√ |(n+υm)2 − (m+υn)2| ( nγ(t)+υ2mν(t)−υ2nγ(t)−mν(t) )√ |υ2 −1| = 1√ |(υ2 −1)(n2 −m2)| (υ2 −1)(mν(t)−nγ(t))√ |υ2 −1| 10 int. j. anal. appl. (2022), 20:47 = 1√ |n2 −m2| (−nγ(t)+mν(t)) = eγ(t). if we take ν̃(t)= µ(t), then (eγ(t), ν̃(t)) is a framed immersion. in fact, we have e′γ(t)= mm′ −nn′ |n2 −m2| 3 2 (−nγ(t)+mν(t))+ 1√ |n2 −m2| ( −n′γ(t)+m′ν(t) ) = m′n−mn′ |n2 −m2| 3 2 (−mγ(t)+nν(t)) = d dt (m n ) n2 |n2 −m2| 3 2 (−mγ(t)+nν(t)) , we have 〈eγ(t), ν̃(t)〉= 〈e′γ(t), ν̃(t)〉=0. and (γ,ν) is a framed immersion satisfying (m(t),n(t)) 6= (0,0). since, ν̃(t) = µ(t), we get ν̃′(t) = −mγ(t)+ nν(t) 6= 0. it follows that eγ(t) is a front. hence, this completes the proof. � 4. evolutes of fronts in hyperbolic sphere h20 in the hyperbolic sphere h20, if β has a singular point, we can not construct a moving frame of β in a traditional way. however, we could define a moving frame of a front curve. so, we give the following definitions. definition 4.1. we say that (β,ν) : i −→ h20 × h 2 0 is a framed curve, if 〈β(t),ν(t)〉 = 0 and 〈β′(t),ν(t)〉=0 for all t ∈ i. moreover, if (β,ν) is an immersion, namely, (β′(t),ν′(t)) 6=(0,0), we call (β,ν) a framed immersion curve. definition 4.2. we say that β : i −→ h20 is a frontal curve if there exists a smooth mapping ν : i −→ h20 such that (β,ν) is a framed curve. we also say that β : i −→ h 2 0 is a front curve if there exists a smooth mapping ν : i −→h20 such that (β,ν) is a framed immersion curve. let (β,ν) : i −→ h20 ×h 2 0 be a framed curve. if β is singular at t0, we can’t define a frame in a traditional way. however, ν always exists even if t is a singular point of β. we take µ = ν ∧β. we call the pair {β,ν,µ} is a moving frame of β and the hyperbolic frenet-serret matrix is given by  β′(t) ν′(t) µ′(t)   =   0 0 m(t) 0 0 n(t) m(t) −n(t) 0     β(t) ν(t) µ(t)   , where n(t) = 〈ν′(t),µ(t)〉, ν(t) and µ(t) are both unit spacelike vectors. we declare that (β,−ν) is also a framed curve. in this case, n(t) dose not change, but m(t) changes to −m(t). if (β,ν) is a framed immersion, we have (m(t),n(t)) 6= (0,0) for each t ∈ i and call the pair (m,n) geodesic curvature of the framed curve (for more details see [8,10]). in what follows, some important properties of the meant curves are introduced. int. j. anal. appl. (2022), 20:47 11 proposition 4.1. let β be a regular curve and (β,ν) : i −→ h20 ×h 2 0 be its framed curve, then the relationship between their geodesic curvatures is given by: κg(t)= n(t) |m(t)| . (4.1) proof. direct calculations lead to β′(t)= m(t)µ(t), β′′(t)= m′(t)µ(t)+m(t)µ′(t) = m′(t)µ(t)+m(t)(m(t)β −n(t)ν(t)), where κg(t)= det(β(t),β′(t),β′′(t)) ‖β′(t)‖3 = det ( β(t),mµ(t),m2β(t)−mnν(t)+m′µ(t) ) |m3| , then, we get κg(t)= n(t) |m(t)| . for the framed immersion (β,ν), we say that t0 is an inflection point of the front β if n(t0)=0. and then the condition m(t0) 6=0 and n(t0)=0, is equivalent to the condition κg(t0)=0. � if β is not a regular curve, then we can not define the evolute as in eq. (2.8). since the geodesic curvature may be divergence at a singular point. then, we can give the definition of the evolute of a front and focus on its properties. the notion of a parallel curve of β can be presented as follows: let (β,ν) : i −→ h20 ×h 2 0 be a framed curve which has the geodesic curvature (m,n). then, we define a parallel curve βλ : i −→h20 of β as βλ(t)= 1√ |λ2 −1| (β(t)+λν(t)) , (4.2) where λ ∈r and λ 6=±1. proposition 4.2. if βλ is a regular curve, then κgλ(t)= n(t)+λm(t) |m(t)+λn(t)| . (4.3) proof. from eq.(4.2), we have β′λ(t)= m+λn√ |λ2 −1| µ(t), β′′λ(t)= m2 +λmn√ |λ2 −1| β(t)+ −mn−λn2√ |λ2 −1| ν(t)+ m′ +λn′√ |λ2 −1| µ(t), 12 int. j. anal. appl. (2022), 20:47 which implies that βλ ∧β′λ = 1 (λ2 −1) ∣∣∣∣∣∣∣∣ −β(t) ν(t) µ(t) 1 λ 0 0 0 m+λn ∣∣∣∣∣∣∣∣ = −λ(m+λn)β(t)− (m+λn)ν(t) (λ2 −1) . since κgλ(t)= det ( βλ(t),β ′ λ(t),β ′′ λ(t) ) ‖β′ λ (t)‖3 , then, we get |m(t)+λn(t)|κgλ(t)= (n(t)+λm(t)), which completes the proof. � proposition 4.3. for a framed immersion curve (β,ν) : i −→h20×h 2 0, the parallel curve βλ : i −→h 2 0 is a front for each λ 6=±1. proof. we take νλ : i −→h20 as follows: νλ(t)= 1√ |λ2 −1| (λβ(t)+ν(t)) , since   βλ(t)= 1√ |λ2 −1| (β(t)+λν(t)) , β′λ(t)= 1√ |λ2 −1| (β′(t)+λν′(t)) . if β′λ(t0)=0 at a point t0 ∈ i, then we have β′(t0)+λν ′(t0)=0. if ν′(t0)= 0, then β′(t0)= 0. it is contradicted with the fact that (β,ν) is a framed immersion and hence (βλ,νλ) is a framed immersion. by ‖ν(t)‖=−1, we have 〈ν(t),ν′(t)〉=0. then 〈β′λ(t),νλ(t)〉= 1 (λ2 −1) 〈β′ +λν′,λβ +ν〉=0, and from this, the curve βλ is a front. � proposition 4.4. let (β,ν) be a framed curve. if β is a regular curve and λ 6=1/κg, then a parallel curve βλ is also a regular curve and its evolute is given by eβλ(t)= eβ(t). (4.4) int. j. anal. appl. (2022), 20:47 13 proof. since βλ(t)= 1√ |λ2 −1| (β(t)+λe(t)) , β′λ(t)= ‖β′‖√ |λ2 −1| (1−λκg)t(t), β′′λ(t)= ‖β′‖√ |λ2 −1| ( ‖β′‖(1−λκg)β(t)−λκ′gt(t)+‖β ′‖κg(1−λκg)e(t) ) . by the assumption λ 6=1/κg, βλ is a regular curve. by direct calculations, we obtain βλ ∧β′λ = 1 (λ2 −1) ∣∣∣∣∣∣∣∣ −β(t) t(t) e(t) 1 0 λ 0 ‖β′‖(1−λκg) 0 ∣∣∣∣∣∣∣∣ = ‖β′‖λ(1−λκg)β(t)+‖β′‖(1−λκg)e(t) (λ2 −1) , and hence 〈βλ ∧β′λ,β ′′ λ〉= −‖β′‖3λ(1−λκg)2 +‖β′‖3κg(1−λκg)2 |λ2 −1| 3 2 , then, we get κgλ(t)= κg −λ |1−λκg| , and tλ(t)= β′λ ‖β′ λ ‖ = 1−λκg |1−λκg| t(t), where eλ = βλ ∧tλ, we have eλ(t)= 1−λκg |1−λκg| 1√ |λ2 −1| (e(t)+λβ(t)) . from (2.8) we get eβλ(t)= 1√∣∣∣κ2gλ(t)−1∣∣∣ ( κgλ(t)βλ(t)+eλ(t) ) = 1√√√√∣∣∣∣∣ ( κg −λ |1−λκg| )2 −1 ∣∣∣∣∣ ( κg −λ |1−λκg| (β(t)+λe(t)) √ λ2 −1 + 1−λκg |1−λκg| (e(t)+λβ(t))√ |λ2 −1| ) = 1√√√√∣∣∣∣∣ ( κg −λ |1−λκg| )2 −1 ∣∣∣∣∣ 1 |1−λκg| √ |λ2 −1| ( (1−λ2)κgβ(t)+(1−λ2)e(t) ) = 1√∣∣κ2g(1−λ2)− (1−λ2)∣∣ (1−λ2)√ |λ2 −1| (κgβ(t)+e(t)) 14 int. j. anal. appl. (2022), 20:47 = 1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+e(t)) = eβ(t). in the light of the above calculations, the proof is completed. � definition 4.3. let (β,ν) : i −→ h20 ×h 2 0 be a framed immersion curve. we define an evolute eβ : i −→h20 of β as follows: if t is a regular point, then we find eβ(t)= 1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+e(t)) . (4.5) if t0 is a singular point, for any t ∈ (t0 −δ,t0 +δ), we get eβλ(t)= 1√∣∣∣κ2gλ(t)−1∣∣∣ ( κgλ(t)βλ(t)+eλ(t) ) , (4.6) where δ is a sufficiently small positive real number and λ ∈r satisfies the condition λ 6=1/κg(t). now, we give another representation of the evolute by using the moving frame {β(t),ν(t),µ(t)} and its geodesic curvature {m(t),n(t)}. theorem 4.1. under the condition of |m(t)| 6= |n(t)|, the evolute of a front curve eβ(t) : i −→ h20 is represented by eβ(t)= 1√ |n2(t)−m2(t)| (n(t)β(t)−m(t)ν(t)) , (4.7) and eβ(t) is a front curve. proof. (i) suppose that β is a regular curve. since β′(t)= m(t)µ(t), we have |m(t)| 6=0 and t(t)= m(t) |m(t)| µ(t), e(t)=− m(t) |m(t)| ν(t). from eqs.(4.1) and (4.5), we get eβ(t)= 1√∣∣κ2g(t)−1∣∣ (κg(t)β(t)+e(t)) = 1√√√√∣∣∣∣∣ ( n(t) |m(t)| )2 −1 ∣∣∣∣∣ ( n(t) |m(t)| β(t)− m(t) |m(t)| ν(t) ) = 1√ |n2(t)−m2(t)| (n(t)β(t)−m(t)ν(t)) . int. j. anal. appl. (2022), 20:47 15 (ii) suppose that t0 is a singular point of β and consider βλ in a hyperbolic sphere, also we know that βλ is a regular curve around the neighbourhood of t0 with λ 6=1/κg(t). from eq.(4.2), so we get β′λ(t)= m+λn√ |λ2 −1| µ(t), then, |m+λn| 6=0 and tλ(t)= m+λn |m+λn| µ(t), where eλ(t)= βλ(t)∧tλ(t), we find eλ(t)= m+λn |m+λn| 1√ |λ2 −1| (−λβ(t)−ν(t)) . therefore, from eq.(4.3), we obtain κgλ = (n+λm) |m+λn| , and from eqs.(4.4) and (4.6) we have eβ(t)= eβλ(t) = 1√∣∣∣κ2gλ(t)−1∣∣∣ ( κgλ(t)βλ(t)+eλ(t) ) = 1√√√√∣∣∣∣∣ ( (n+λm) |m+λn| )2 −1 ∣∣∣∣∣ (( (n+λm) |m+λn| ) (β(t)+λν(t)) √ λ2 −1 + m+λn |m+λn| (−λβ(t)−ν(t))√ |λ2 −1| ) = 1√ |(n+λm)2 − (m+λn)2| ( nβ(t)+λ2mν(t)−λ2nβ(t)−mν(t) )√ |λ2 −1| = 1√ |(1−λ2)(n2 −m2)| (1−λ2)(nβ(t)−mν(t))√ |λ2 −1| = 1√ |n2 −m2| (nβ(t)−mν(t)) . if we take ν̃(t)= µ(t), then (eβ(t), ν̃(t)) is a framed immersion. in fact, we have e′β(t)= mm′ −nn′ (n2 −m2) 3 2 (nβ(t)−mν(t))+ 1 √ n2 −m2 ( n′β(t)−m′ν(t) ) = m′n−mn′ (n2 −m2) 3 2 (mβ(t)−nν(t)) = d dt (m n ) n2 (n2 −m2) 3 2 (mβ(t)−nν(t)) , therefore, we find 〈eβ(t), ν̃(t)〉 = 〈e′β(t), ν̃(t)〉 = 0. and (β,ν) is a framed immersion satisfying (m(t),n(t)) 6= (0,0). since, ν̃(t) = µ(t), we get ν̃′(t) = mβ(t)−nν(t) 6= 0. it follows that eβ(t) is a front and thus, the proof is completed. � 16 int. j. anal. appl. (2022), 20:47 5. computational examples example 5.1. let γ : i −→ s21 be a regular curve given by γ(t)= ( sinh(t3),cos(t2)cosh(t3),sin(t2)cosh(t3) ) , then, we get γ′(t)= ( 3t2cosh(t3),−2t sin(t2)cosh(t3)+3t2cos(t2)sinh(t3),2t cos(t2)cosh(t3)+3t2 sin(t2)sinh(t3) ) . since, t =0 is a singular point on γ. if we take ν =(ν1,ν2,ν3), where  ν1 = 1 p ( 2cosh2(t3) ) ν2 = 1 p ( 2sinh(t3)cos(t2)cosh(t3)−3t sin(t2) ) ν3 = 1 p ( 2sin(t2)sinh(t3)cosh(t3)+3t cos(t2) ) , (5.1) and p = √∣∣9t2 −4cosh2(t3)∣∣, then we have 〈γ(t),ν(t)〉 = 〈γ′(t),ν(t)〉 = 0 and 〈ν(t),ν(t)〉 = 1. hence, (γ,ν) is a framed curve. also, from the relation µ = ν ∧γ, we get (see fig.(1a)). µ(t)= 1 p ∣∣∣∣∣∣∣∣ −i j k 2cosh2(t3) 2sinh(t3)cos(t2)cosh(t3)−3t sin(t2) 2sin(t2)sinh(t3)cosh(t3)+3t cos(t2) sinh(t3) cos(t2)cosh(t3) sin(t2)cosh(t3) ∣∣∣∣∣∣∣∣ , and we get µ = 1 p ( 3t cosh(t3),3t cos(t2)sinh(t3)−2sin(t2)cosh(t3),3t sin(t2)sinh(t3)+2cos(t2)cosh(t3) ) . (5.2) from the fact that 〈µ(t),µ(t)〉=1, one can obtain m(t)= ‖γ′(t)‖= t √∣∣9t2 −4cosh2(t3)∣∣. (5.3) also, from eq.(5.1) we have ν′(t)= −(9t −4cosh(t3)sinh(t3))∣∣9t2 −4cosh2(t3)∣∣32 (2cosh2(t3),2sinh(t3)cos(t2)cosh(t3)−3t sin(t2), 2sin(t2)sinh(t3)cosh(t3)+3t cos(t2)) + 1√∣∣9t2 −4cosh2(t3)∣∣(12t2cosh(t3)sinh(t3),6t2cos(t2)(cosh2(t3)+sinh2(t3)) −4t sinh(t3)sin(t2)cosh(t3)−3sin(t2)−6t2cos(t2),6t2 sin(t2)(cosh2(t3)+sinh2(t3)) +4t sinh(t3)cos(t2)cosh(t3)−3cos(t2)−6t2 sin(t2)), int. j. anal. appl. (2022), 20:47 17 which leads to n(t)= 〈ν′(t),µ(t)〉 = 2(−18t3 sinh(t3)+4t sinh(t3)cosh(t3)+3cosh(t3)) 9t2 −4cosh2(t3) , and we have (m(0),n(0)) 6=(0,0), thus, γ is a frontal curve. on the other hand, the evolute curve of γ is given as eγ(t)= (eγ1,eγ2,eγ3) , where eγ1(t)= 1 p √ |4f2 − t2p4| ( 2f sinh(t3)−2tp2cosh2(t3) ) , eγ2(t)= 1 p √ |4f2 − t2p4| ( 2f cos(t2)cosh(t3)−2tp22sinh(t3)cos(t2)cosh(t3)−3t2p2 sin(t2) ) , eγ3(t)= 1 p √ |4f2 − t2p4| ( 2f sin(t2)cosh(t3)− tp2 sin(t2)sinh(t3)cosh(t3)−3t2p2cos(t2) ) , keep in mind that f =−18t3 sinh(t3)+4t sinh(t3)cosh(t3)+3cosh(t3). (a) (b) figure 1. (a) the curve γ(t) with singular point at t = 0. (b) the curve ν(t) = (ν1,ν2,ν3). example 5.2. consider the de sitter asteroid curve β : i −→ s21, β(t)= (β1,β2,β3) expressed as [8]  β1 = √ |cos6(t)+sin6(t)−1| β2 =cos 3(t) β3 =sin 3(t), (5.4) 18 int. j. anal. appl. (2022), 20:47 then, we get β′(t)=3sin(t)cos(t) ( sin4(t)−cos4(t)√ |cos6(t)+sin6(t)−1| ,−cos(t),sin(t) ) . it is obvious that β is singular at t =0, π/2, π and 3π/2. if we take ν =(ν1,ν2,ν3), where  ν1 = 1 q ( sin(t)cos(t) √ |cos6(t)+sin6(t)−1| ) ν2 = 1 q ( sin(t) ( cos4(t)−1 )) ν3 = 1 q ( cos(t) ( sin4(t)−1 )) , (5.5) with the knowledge that q= √∣∣1− sin2(t)cos2(t)∣∣, then by a straightforward calculations, we have 〈β(t),ν(t)〉= 〈β′(t),ν(t)〉=0 and 〈ν(t),ν(t)〉=1. hence, (β,ν) is a framed curve. from the equation µ = ν ∧β, we have (see fig.(2a) and fig.(2b)). µ = 1 q ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ −i j k sin(t)cos(t) (|cos6(t)+sin6(t)−1|)− 1 2 sin(t) ( cos4(t)−1 ) cos(t) ( sin4(t)−1 ) √ |cos6(t)+sin6(t)−1| cos3(t) sin3(t) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣ , which gives µ(t)= √ |cos6(t)+sin6(t)−1|√∣∣1− sin2(t)cos2(t)∣∣ ( sin4(t)−cos4(t)√ |cos6(t)+sin6(t)−1| ,−cos(t),sin(t) ) . (5.6) from the previous equation, we have 〈µ(t),µ(t)〉=1. then, we obtain m(t)= ‖β′(t)‖=3sin(t)cos(t) √ cos6(t)+sin6(t) |cos6(t)+sin6(t)−1| . (5.7) also, from eq.(5.1) we have ν′(t)= sin(t)cos(t)(cos2(t)− sin2(t))∣∣1− sin2(t)cos2(t)∣∣32 (sin(t)cos(t) √ |cos6(t)+sin6(t)−1|, sin(t)(cos4(t)−1),cos(t)(sin4(t)−1)) + 1√∣∣1− sin2(t)cos2(t)∣∣( √ |cos6(t)+sin6(t)−1|(cos2(t)− sin2(t) + 3sin2(t)cos2(t)(sin4(t)−cos4(t)) cos6(t)+sin6(t)−1 ),cos3(t)(cos2(t)−4sin2(t)) −cos(t),−sin3(t)(sin2(t)−4cos2(t))+sin(t)). int. j. anal. appl. (2022), 20:47 19 hence, we get n(t)= 〈ν′(t),µ(t)〉 = √ |cos6(t)+sin6(t)−1| 1− sin2(t)cos2(t) ( 3sin2(t)cos2(t) ( 1− (cos2(t)− sin2(t))2 cos6(t)+sin6(t)−1 ) +1 ) . (5.8) in the light of the above calculations, we have (m(0),n(0)) 6=(0,0), thus, β is a frontal curve. also, from eqs.(3.7), (5.4), (5.5), (5.7), and (5.8), we get the evolute curve of β as eβ(t)= ( eβ1,eβ2,eβ3 ) , where eβ1(t)= −qn+m√ |n2 −m2| (√ |cos6(t)+sin6(t)−1|(1+sin(t)cos(t)) ) , eβ2(t)= −qn+m√ |n2 −m2| ( cos3(t)(1+sin(t)cos(t))− sin(t) ) , eβ3(t)= −qn+m√ |n2 −m2| ( sin3(t)(1+sin(t)cos(t))−cos(t) ) . (a) (b) figure 2. (a) the de sitter astroid curve β. (b) the curve ν(t)= (ν1,ν2,ν3). 6. conclusion in 2-dimensional de sitter and hyperbolic spaces, some types of curves such as framed curves, framed immersion curves, frontal curves and front curves are studied. also, the evolutes and some of their properties of fronts at singular points under some conditions are investigated. finally, two computational examples in support of our main results are given and plotted. acknowledgment: the researchers wish to extend their sincere gratitude to the deanship of scientific research at the islamic university of madinah for the support provided to the post-publishing program 1. 20 int. j. anal. appl. (2022), 20:47 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] c.g. gibson, elementary geometry of differentiable curves, cambridge university press, cambridge, 2001. [2] v.i. arnold, s.m. gusein-zade, a.n. varchenko, singularities of differentiable maps, birkha äuser, boston, 1985. [3] v.i. arnold, singularities of caustics and wave fronts, kluwer academic publishers, dordrecht, 1990. [4] j.w. bruce, p.j. giblin, curves and singularities, a geometrical introduction to singularity theory, cambridge university press, cambridge, 1992. [5] s. izumiya, d.h. pei, t. sano, e. torii, evolutes of hyperbolic plane curves, acta math. sinica. 20 (2004), 543–550. https://doi.org/10.1007/s10114-004-0301-y. [6] t. fukunaga, m. takahashi, existence and uniqueness for legendre curves, j. geom. 104 (2013), 297–307. https://doi.org/10.1007/s00022-013-0162-6. [7] t. fukunaga, m. takahashi, evolutes of fronts in the euclidean plane, j. singul. 10 (2014), 92-107. https: //doi.org/10.5427/jsing.2014.10f. [8] h. yu, d. pei, x. cui, evolutes of fronts on euclidean 2-sphere, j. nonlinear sci. appl. 8 (2015), 678-686. [9] l. chen, m. takahashi, dualities and evolutes of fronts in hyperbolic and de sitter space, j. math. anal. appl. 437 (2016), 133-159. https://doi.org/10.1016/j.jmaa.2015.12.029. [10] x. cui, d. pei, h. yu, evolutes of null torus fronts, j. nonlinear sci. appl. 8 (2015), 866-876. https://doi.org/10.1007/s10114-004-0301-y https://doi.org/10.1007/s00022-013-0162-6 https://doi.org/10.5427/jsing.2014.10f https://doi.org/10.5427/jsing.2014.10f https://doi.org/10.1016/j.jmaa.2015.12.029 1. introduction 2. geometric meanings and basis concepts 3. evolutes of fronts in de sitter sphere s12 4. evolutes of fronts in hyperbolic sphere h02 5. computational examples 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 95-100 http://www.etamaths.com on quasi-power increasing sequences and their some applications hüseyin bor∗ abstract. in [6], we proved a main theorem dealing with | n̄,pn,θn |k summability factors using a new general class of power increasing sequences instead of a quasi-σ-power increasing sequence. in this paper, we prove that theorem under weaker conditions. this theorem also includes some new results. 1. introduction a positive sequence x = (xn) is said to be a quasi-f-power increasing sequence if there exists a constant k = k(x,f) ≥ 1 such that kfnxn ≥ fmxm for all n ≥ m ≥ 1, where f = (fn) = {nσ(log n)η, η ≥ 0, 0 < σ < 1} (see [13]). if we set η=0, then we get a quasi-σ-power increasing sequence (see [10]). we write bvo = bv ∩ co, where co = { x = (xk) ∈ ω : limk |xk| = 0 }, bv ={ x = (xk) ∈ ω : ∑ k |xk −xk+1| < ∞ } and ω being the space of all real-valued sequences. let∑ an be a given infinite series with the sequence of partial sums (sn). we denote by u α n the nth cesàro mean of order α, with α > −1, of the sequence (sn), that is (see [7]), uαn = 1 aαn n∑ v=0 aα−1n−vsv(1) where (2) aαn = (α + 1)(α + 2)....(α + n) n! = o(nα), aα−n = 0 for n > 0. a series ∑ an is said to be summable | c,α |k, k ≥ 1, if (see [8]) (3) ∞∑ n=1 nk−1 | uαn −u α n−1 | k< ∞. if we take α = 1, then we get the | c, 1 |k summability. let (pn) be a sequence of positive real numbers such that (4) pn = n∑ v=0 pv →∞ as n →∞, (p−i = p−i = 0, i ≥ 1). the sequence-to-sequence transformation (5) vn = 1 pn n∑ v=0 pvsv defines the sequence (vn) of the riesz mean or simply the (n̄,pn) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [9]). let (θn) be any sequence of positive constants. the series∑ an is said to be summable | n̄,pn,θn |k,k ≥ 1, if (see [12]) ∞∑ n=1 θk−1n | vn −vn−1 | k < ∞.(6) 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g99, 46a45. key words and phrases. sequence spaces; riesz mean; summability factors; increasing sequences; infinite series; hölder inequality; minkowski inequality. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 95 96 hüseyin bor if we take θn = pn pn , then | n̄,pn,θn |k summability reduces to | n̄,pn |k summability (see [1]). also, if we take θn = pn pn and pn = 1 for all values of n, then we get | c, 1 |k summability. furthermore, if we take θn = n, then | n̄,pn,θn |k summability reduces to | r,pn |k summability (see [2]). 2. known results. the following theorems are known: theorem a ([4]). let ( θnpn pn ) be a non-increasing sequence. let (λn) ∈ bvo and let (xn) be a quasiσ-power increasing sequence for some σ (0 < σ < 1) . suppose also that there exist sequences (βn) and (λn) such that (7) | ∆λn |≤ βn, (8) βn → 0 as n →∞, (9) ∞∑ n=1 n | ∆βn | xn < ∞, (10) | λn | xn = o(1). if n∑ v=1 θk−1v | sv |k vk = o(xn) as n →∞,(11) and (pn) is a sequence such that (12) pn = o(npn), (13) pn∆pn = o(pnpn+1), then the series ∑∞ n=1 an pnλn npn is summable | n̄,pn,θn |k, k ≥ 1. remark. we can take (λn) ∈bv instead of (λn) ∈bvo and it is sufficient to prove theorem a. theorem b ([6]). let ( θnpn pn ) be a non-increasing sequence. let (λn) ∈ bv and let (xn) be a quasi-f-power increasing sequence for some σ (0 < σ < 1) and η ≥ 0. if the conditions (7)-(13) are satisfied, then the series ∑∞ n=1 an pnλn npn is summable | n̄,pn,θn |k, k ≥ 1. it should be noted that if we take η=0, then we obtain theorem a. 3. the main result. the purpose of this paper is to prove theorem b under weaker conditions. now, we shall prove the following general theorem. theorem. let ( θnpn pn ) be a non-increasing sequence. let (xn) be a quasi-f-power increasing sequence for some σ (0 < σ < 1) and η ≥ 0. if the conditions (7)-(10), (12)-(13), and n∑ v=1 θk−1v | sv |k vkxv k−1 = o(xn) as n →∞(14) are satisfied, then the series ∑∞ n=1 an pnλn npn is summable | n̄,pn,θn |k, k ≥ 1. remark. it should be noted that condition (14) is reduced to the condition (11), when k=1. when k > 1, the condition (14) is weaker than the condition (11), but the converse is not true. as in [14] we can show that if (11) is satisfied, then we get that n∑ v=1 θk−1v | sv |k vkxv k−1 = o( 1 xk−11 ) n∑ v=1 θk−1v | sv |k vk = o(xn). if (14) is satisfied, then for k > 1 we obtain that n∑ v=1 θk−1v | sv |k vk = n∑ v=1 θk−1v x k−1 v | sv |k vkxv k−1 = o(x k−1 n ) n∑ v=1 θk−1v | sv |k vkxv k−1 = o(x k n) 6= o(xn). also, it should be noted that the condition ”(λn) ∈bv ” has been removed. we require the following lemmas for the proof of the theorem. quasi-power increasing sequences 97 lemma 1 ([5]). under the conditions on (xn), (βn) and (λn) as expressed in the statement of the theorem, we have the following; (15) nxnβn = o(1), (16) ∞∑ n=1 βnxn < ∞. lemma 2 ([11]). if the conditions (12) and (13) are satisfied, then we have that ∆ ( pn npn ) = o ( 1 n ) .(17) 4. proof of the theorem. let (tn) be the sequence of (n̄,pn) mean of the series ∑∞ n=1 anpnλn npn . then, by definition, we have tn = 1 pn n∑ v=1 pv v∑ r=1 arprλr rpr = 1 pn n∑ v=1 (pn −pv−1) avpvλv vpv . then, for n ≥ 1 we obtain that tn −tn−1 = pn pnpn−1 n∑ v=1 pv−1pvavλv vpv . using abel’s transformation, we get tn −tn−1 = pn pnpn−1 n−1∑ v=1 sv∆ ( pv−1pvλv vpv ) + λnsn n = snλn n + pn pnpn−1 n−1∑ v=1 sv pv+1pv∆λv (v + 1)pv+1 + pn pnpn−1 n−1∑ v=1 pvsvλv∆ ( pv vpv ) − pn pnpn−1 n−1∑ v=1 svpvλv 1 v = tn,1 + tn,2 + tn,3 + tn,4. to prove the theorem, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 θk−1n | tn,r | k< ∞, for r = 1, 2, 3, 4.(18) firstly, by using abel’s transformation, we have that m∑ n=1 θk−1n | tn,1 | k = m∑ n=1 θk−1n n −k | λn |k−1| λn || sn |k = o(1) m∑ n=1 | λn | ( 1 xn )k−1 θk−1n n −k | sn |k = o(1) m−1∑ n=1 ∆ | λn | n∑ v=1 θk−1v | sv |k xv k−1vk + o(1) | λm | m∑ n=1 θk−1n | sn |k xn k−1nk = o(1) m−1∑ n=1 | ∆λn | xn + o(1) | λm | xm = o(1) m−1∑ n=1 βnxn + o(1) | λm | xm = o(1) as m →∞ 98 hüseyin bor by virtue of the hypotheses of the theorem and lemma 1. now, using (12) and applying hölder’s inequality, we have that m+1∑ n=2 θk−1n | tn,2 | k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pkn−1 | n−1∑ v=1 pvsv∆λv |k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pkn−1 { n−1∑ v=1 pv pv | sv | pv | ∆λv | }k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pn−1 n−1∑ v=1 ( pv pv )k | sv |k pv (βv)k × ( 1 pn−1 n−1∑ v=1 pv )k−1 = o(1) m∑ v=1 ( pv pv )k | sv |k pv (βv)k m+1∑ n=v+1 ( θnpn pn )k−1 pn pnpn−1 = o(1) m∑ v=1 ( pv pv )k | sv |k pv (βv)k ( θvpv pv )k−1 m+1∑ n=v+1 pn pnpn−1 = o(1) m∑ v=1 ( pv pv )k | sv |k (βv)k ( pv pv ) θk−1v ( pv pv )k−1 = o(1) m∑ v=1 (vβv) k−1vβv 1 vk θk−1v | sv | k = o(1) m∑ v=1 ( 1 xv )k−1 vβv 1 vk θk−1v | sv | k = o(1) m−1∑ v=1 ∆(vβv) v∑ r=1 θk−1r | sr |k rkxr k−1 + o(1)mβm m∑ v=1 θk−1v | sv |k vkxv k−1 = o(1) m−1∑ v=1 | ∆(vβv) | xv + o(1)mβmxm = o(1) m−1∑ v=1 | (v + 1)∆βv −βv | xv + o(1)mβmxm = o(1) m−1∑ v=1 v | ∆βv | xv + o(1) m−1∑ v=1 βvxv + o(1)mβmxm = o(1) as m →∞, in view of the hypotheses of the theorem and lemma 1. again, as in tn,1, we have that m+1∑ n=2 θk−1n | tn,3 | k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pkn−1 { n−1∑ v=1 pv | sv || λv | 1 v }k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pn−1 n−1∑ v=1 ( pv pv )k v−kpv | sv |k| λv |k × { 1 pn−1 n−1∑ v=1 pv }k−1 quasi-power increasing sequences 99 = o(1) m∑ v=1 ( pv pv )k v−k | sv |k pv | λv |k m+1∑ n=v+1 ( θnpn pn )k−1 pn pnpn−1 = o(1) m∑ v=1 ( pv pv )k−1 v−kθk−1v ( pv pv )k−1 | λv |k−1| λv || sv |k = o(1) m∑ v=1 | λv | ( 1 xv )k−1 θk−1v v −k | sv |k = o(1) m∑ v=1 | λv | θk−1v | sv |k vkxv k−1 = o(1) as m →∞, in view of the hypotheses of the theorem, lemma 1 and lemma 2. finally, using hölder’s inequality, as in tn,1 we have that m+1∑ n=2 θk−1n | tn,4 | k = m+1∑ n=2 θk−1n ( pn pn )k 1 pkn−1 | n−1∑ v=1 sv pv v λv |k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pkn−1 | n−1∑ v=1 sv pv vpv pvλ |k = o(1) m+1∑ n=2 θk−1n ( pn pn )k 1 pn−1 n−1∑ v=1 | sv |k ( pv pv )k v−kpv | λv |k × ( 1 pn−1 n−1∑ v=1 pv )k−1 = o(1) m∑ v=1 ( pv pv )k v−k | sv |k pv | λv |k 1 pv ( θvpv pv )k−1 = o(1) m∑ v=1 ( pv pv )k−1 v−k ( pv pv )k−1 θk−1v | λv | k−1| λv || sv |k = o(1) m∑ v=1 | λv | θk−1v | sv |k vkxv k−1 = o(1) as m →∞. this completes the proof of the theorem. if we set η ≥ 0, then we obtain theorem b under weaker conditions. if we take pn = 1 for all values of n, then we have a new result for | c, 1,θn |k summability. furthermore, if we take θn = n, then we have another new result for | r,pn |k summability. finally, if we take pn = 1 for all values of n and θn = n, then we get a new result dealing with | c, 1 |k summability factors. references [1] h. bor, on two summability methods, math. proc. camb. philos. soc., 97 (1985), 147-149. [2] h. bor, on the relative strength of two absolute summability methods, proc. amer. math. soc., 113 (1991), 10091012. [3] h. bor, a general note on increasing sequences, jipam. j. inequal. pure appl. math., 8 (2007), article id 82. [4] h. bor, new application of power increasing sequences, math. aeterna, 2 (2012), 423-429. [5] h. bor, a new application of generalized power increasing sequences, filomat, 26 (2012), 631-635. [6] h. bor, a new application of generalized quasi-power increasing sequences, ukrainian math. j., 64 (2012), 731-738. [7] e. cesàro, sur la multiplication des séries, bull. sci. math., 14 (1890), 114-120. [8] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc., 7 (1957), 113-141. [9] g. h. hardy, divergent series, oxford univ. press., oxford, (1949). [10] l. leindler, a new application of quasi power increasing sequences, publ. math. debrecen, 58 (2001), 791-796. [11] k. n. mishra and r. s. l. srivastava, on | n̄,pn | summability factors of infinite series, indian j. pure appl. math., 15 (1984), 651-656. [12] w. t. sulaiman, on some summability factors of infinite series, proc. amer. math. soc., 115 (1992), 313-317. 100 hüseyin bor [13] w. t. sulaiman, extension on absolute summability factors of infinite series, j. math. anal. appl., 322 (2006), 1224-1230. [14] w. t. sulaiman, a note on |a|k summability factors of infinite series, appl. math. comput., 216 (2010), 2645-2648. p. o. box 121, tr-06502 bahçelievler, ankara, turkey ∗corresponding author: hbor33@gmail.com international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 134-139 http://www.etamaths.com an application of δ-quasi monotone sequence hi̇kmet seyhan özarslan∗ abstract. in this paper, a known theorem dealing with |a,pn|k summability method of infinite series has been generalized to |a,pn; δ|k summability method. also, some results have been obtained. 1. introduction a sequence (dn) is said to be δ-quasi-monotone, if dn → 0, dn > 0 ultimately and ∆dn ≥ −δn, where ∆dn=dn −dn+1 and δ = (δn) is a sequence of positive numbers (see [1]). let ∑ an be a given infinite series with partial sums (sn). let (pn) be a sequence of positive numbers such that pn = n∑ v=0 pv →∞ as n →∞, (p−i = p−i = 0, i ≥ 1) . (1.1) the sequence-to-sequence transformation zn = 1 pn n∑ v=0 pvsv (1.2) defines the sequence (zn) of the riesz mean or simply the ( n̄,pn ) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [5]). the series ∑ an is said to be summable ∣∣n̄,pn∣∣k, k ≥ 1, if (see [2]) ∞∑ n=1 ( pn pn )k−1 |∆zn−1|k < ∞, (1.3) where ∆zn−1 = − pn pnpn−1 n∑ v=1 pv−1av, n ≥ 1. let a = (anv) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. then a defines the sequence-to-sequence transformation, mapping the sequence s = (sn) to as = (an(s)), where an(s) = n∑ v=0 anvsv, n = 0, 1, ... (1.4) the series ∑ an is said to be summable |a,pn; δ|k, k ≥ 1 and δ ≥ 0, if (see [6]) ∞∑ n=1 ( pn pn )δk+k−1 |∆̄an(s)|k < ∞, (1.5) where ∆̄an(s) = an(s) −an−1(s). if we set δ = 0, then |a,pn; δ|k summability reduces to |a,pn|k summability (see [8]). if we take anv = pv pn and δ = 0, then |a,pn; δ|k summability reduces to |n̄,pn|k summability. 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g99. key words and phrases. summability factors; absolute matrix summability; quasi-monotone sequences; infinite series; hölder inequality; minkowski inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 134 an application of δ-quasi monotone sequence 135 in the special case δ = 0 and pn = 1 for all n, |a,pn; δ|k summability is the same as |a|k summability (see [9]). also, if we take anv = pv pn , then |a,pn; δ|k summability is the same as |n̄,pn; δ|k summability (see [4]). before stating the main theorem we must first introduce some further notations. given a normal matrix a = (anv), we associate two lower semimatrices ā = (ānv) and â = (ânv) as follows: ānv = n∑ i=v ani, n,v = 0, 1, ... (1.6) and â00 = ā00 = a00, ânv = ānv − ān−1,v, n = 1, 2, ... (1.7) it may be noted that ā and â are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. then, we have an (s) = n∑ v=0 anvsv = n∑ v=0 ānvav (1.8) and ∆̄an (s) = n∑ v=0 ânvav. (1.9) 2. known results in [3], bor has proved the following theorem dealing with |n̄,pn|k summability. theorem 2.1. let (xn) be a positive non-decreasing sequence, (λn) → 0 as n → ∞ and (pn) be a sequence of positive numbers such that pn = o(npn) as n →∞. (2.1) suppose that there exist a sequence of numbers (bn) which is δ-quasi monotone with∑ nxnδn < ∞, ∑ bnxn is convergent and |∆λn| ≤ |bn| for all n. if m∑ n=1 pn pn |tn|k = o(xm) as m →∞, (2.2) then the series ∑ anλn is summable |n̄,pn|k, k ≥ 1. later on, in [7], özarslan and şakar have proved the following theorem dealing with |a,pn|k summability factors of infinite series. theorem 2.2. let a = (anv) be a positive normal matrix such that ān0 = 1, n = 0, 1, ..., (2.3) an−1,v ≥ anv for n ≥ v + 1, (2.4) ann = o ( pn pn ) , (2.5) |ân,v+1| = o (v |∆vânv|) . (2.6) if (xn) is a positive non-decreasing sequence and the conditions of theorem 2.1 are satisfied, then the series ∑ anλn is summable |a,pn|k, k ≥ 1. 136 özarslan 3. main result the purpose of this paper is to generalize theorem 2.2 for |a,pn; δ|k summability. now, we shall prove the following more general theorem. theorem 3.1. let a = (anv) be a positive normal matrix such that m+1∑ n=v+1 ( pn pn )δk |∆vânv| = o {( pv pv )δk−1} as m →∞. (3.1) if all conditions of theorem 2.2 with condition (2.2) replaced by: m∑ n=1 ( pn pn )δk−1 |tn|k = o(xm) as m →∞, (3.2) are satisfied, then the series ∑ anλn is summable |a,pn; δ|k , k ≥ 1 and 0 ≤ δ < 1/k. we require the following lemmas for the proof of theorem 3.1. lemma 3.1. ( [3]). under the conditions of theorem 3.1, we have that |λn|xn = o (1) as n →∞. (3.3) lemma 3.2. ( [3]). let (xn) be a positive non-decreasing sequence. if (bn) is δ-quasi monotone with∑ nxnδn < ∞ and ∑ bnxn is convergent, then nbnxn = o (1) as n →∞, (3.4) ∞∑ n=1 nxn|∆bn| < ∞. (3.5) 4. proof of theorem 3.1 let (in) denotes a-transform of the series ∑ anλn. then, by (1.8) and (1.9), we have ∆̄in = n∑ v=0 ânvavλv = n∑ v=1 ânvλv v vav. applying abel’s transformation to this sum, we get that ∆̄in = n−1∑ v=1 ∆v ( ânvλv v ) v∑ r=1 rar + ânnλn n n∑ r=1 rar = n−1∑ v=1 v + 1 v ∆v (ânv) λvtv + n−1∑ v=1 v + 1 v ân,v+1∆λvtv + n−1∑ v=1 ân,v+1λv+1 tv v + n + 1 n annλntn = in,1 + in,2 + in,3 + in,4. to complete the proof of theorem 3.1, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 ( pn pn )δk+k−1 |in,r| k < ∞, for r = 1, 2, 3, 4. an application of δ-quasi monotone sequence 137 first, when k > 1, applying hölder’s inequality with indices k and k ′ , where 1 k + 1 k ′ = 1, we have that m+1∑ n=2 ( pn pn )δk+k−1 |in,1|k = o(1) m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 |∆v(ânv)||λv||tv| )k = o(1) m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 |∆v(ânv)||λv|k|tv|k ) × ( n−1∑ v=1 |∆v(ânv)| )k−1 . by (1.6) and (1.7), we have that ∆v(ânv) = ânv − ân,v+1 = ānv − ān−1,v − ān,v+1 + ān−1,v+1 = anv −an−1,v. thus using (1.6), (2.3) and (2.4) n−1∑ v=1 |∆v(ânv)| = n−1∑ v=1 (an−1,v −anv) ≤ ann. hence, we get m+1∑ n=2 ( pn pn )δk+k−1 |in,1|k = o(1) m+1∑ n=2 ( pn pn )δk (n−1∑ v=1 |∆v(ânv)||λv|k|tv|k ) = o(1) m∑ v=1 |λv|k−1|λv||tv|k m+1∑ n=v+1 ( pn pn )δk |∆v(ânv)| = o(1) m∑ v=1 ( pv pv )δk−1 |λv||tv|k = o(1) m−1∑ v=1 ∆|λv| v∑ r=1 ( pr pr )δk−1 |tr|k + o(1)|λm| m∑ v=1 ( pv pv )δk−1 |tv|k = o(1) m−1∑ v=1 |∆λv|xv + o(1)|λm|xm = o(1) m−1∑ v=1 bvxv + o(1)|λm|xm = o(1) as m →∞, by virtue of the hypotheses of theorem 3.1 and lemma 3.1. 138 özarslan again, by using hölder’s inequality, we have that m+1∑ n=2 ( pn pn )δk+k−1 |in,2| k = o(1) m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 |ân,v+1||∆λv||tv| )k = o(1) m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 v |∆v(ânv)| |bv||tv|k ) × ( n−1∑ v=1 v |∆v(ânv)| |bv| )k−1 . by using (3.4), we get m+1∑ n=2 ( pn pn )δk+k−1 |in,2| k = o(1) m+1∑ n=2 ( pn pn )δk (n−1∑ v=1 v |∆v(ânv)| |bv||tv|k ) = o(1) m∑ v=1 v|bv||tv|k m+1∑ n=v+1 ( pn pn )δk |∆v(ânv)| = o(1) m∑ v=1 ( pv pv )δk−1 v|bv||tv|k. now, applying abel’s transformation to this sum, we have that m+1∑ n=2 ( pn pn )δk+k−1 |in,2| k = o(1) m−1∑ v=1 |∆ (v|bv|)| v∑ r=1 ( pr pr )δk−1 |tr|k + o(1)m |bm| m∑ v=1 ( pv pv )δk−1 |tv|k = o(1) m−1∑ v=1 v|∆bv|xv + o(1) m−1∑ v=1 bvxv + o(1)mbmxm = o(1) as m →∞, by virtue of the hypotheses of theorem 3.1 and lemma 3.2. also, as in in,1, we have that m+1∑ n=2 ( pn pn )δk+k−1 |in,3| k ≤ m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 |ân,v+1||λv+1| |tv| v )k = o(1) m+1∑ n=2 ( pn pn )δk+k−1 (n−1∑ v=1 |∆v(ânv)| |λv+1|k|tv|k ) × ( n−1∑ v=1 |∆v(ânv)| )k−1 = o(1) m∑ v=1 |λv+1||tv|k m+1∑ n=v+1 ( pn pn )δk |∆v(ânv)| = o(1) m∑ v=1 ( pv pv )δk−1 |λv+1||tv|k = o(1) as m →∞, by using (2.5), (2.6), (3.1), (3.2) and (3.3). an application of δ-quasi monotone sequence 139 finally, as in in,1, we have that m∑ n=1 ( pn pn )δk+k−1 |in,4|k = o(1) m∑ n=1 ( pn pn )δk+k−1 |λn|k|tn|kaknn = o(1) m∑ n=1 ( pn pn )δk−1 |λn||λn|k−1|tn|k = o(1) m∑ n=1 ( pn pn )δk−1 |λn||tn|k = o(1) as m →∞, by using (2.5), (3.1), (3.2) and (3.3). this completes the proof of theorem 3.1. it should be noted that if we take δ = 0 in theorem 3.1, then we get theorem 2.2. in this case, condition (3.2) reduces to condition (2.2). also, if we take δ = 0 and anv = pv pn , then we get theorem 2.1. references [1] r. p. boas, quasi-positive sequences and trigonometric series, proc. london math. soc. 14a (1965), 38-46. [2] h. bor, on two summability methods, math. proc. cambridge philos. soc. 97 (1985), 147-149. [3] h. bor, on quasi-monotone sequences and their applications, bull. austral. math. soc. 43 (1991), 187-192. [4] h. bor, on local property of | n̄,pn; δ |k summability of factored fourier series, j. math. anal. appl. 179 (1993), 646–649. [5] g. h. hardy, divergent series, oxford university press, oxford, 1949. [6] h. s. özarslan and h. n. öğdük, generalizations of two theorems on absolute summability methods, aust. j. math. anal. appl. 1 (1) (2004), article 13, 7 pp. [7] h. s. özarslan and m. ö. şakar, a new application of absolute matrix summability, math. sci. appl. e-notes 3 (2015), 36-43. [8] w. t. sulaiman, inclusion theorems for absolute matrix summability methods of an infinite series. iv, indian j. pure appl. math. 34 (11) (2003), 1547-1557. [9] n. tanovic̆-miller, on strong summability, glas. mat. ser. iii 14 (34) (1979), 87-97. department of mathematics, erciyes university, 38039 kayseri, turkey ∗corresponding author: seyhan@erciyes.edu.tr; hseyhan38@gmail.com 1. introduction 2. known results 3. main result 4. proof of theorem 3.1 references int. j. anal. appl. (2022), 20:68 fuzzy subalgebras and ideals with thresholds of hilbert algebras aiyared iampan1,∗, p. jayaraman2, s. d. sudha2, n. rajesh3 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2department of mathematics, bharathiyar university, coimbatore 641046, tamilnadu, india 3department of mathematics, rajah serfoji government college, thanjavur 613005, tamilnadu, india ∗corresponding author: aiyared.ia@up.ac.th abstract. the concepts of fuzzy subalgebras and ideals with thresholds of hilbert algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of fuzzy sets as a foundation. we also talk about the connections between fuzzy subalgebras (ideals) with thresholds and their level subsets. the homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in hilbert algebras are also studied and some related properties are investigated. 1. introduction the concept of fuzzy sets was proposed by zadeh [25]. the theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. after the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. the integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1,3,6]. the idea of intuitionistic fuzzy sets suggested by atanassov [2] is one of the extensions of fuzzy sets with better applicability. applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multicriteria decisionmaking [12–14]. the concept of hilbert algebras was introduced in early 50-ties by henkin [15] for received: nov. 7, 2022. 2010 mathematics subject classification. 03g25, 03e72. key words and phrases. hilbert algebra; fuzzy subalgebra with thresholds; fuzzy ideal with thresholds. https://doi.org/10.28924/2291-8639-20-2022-68 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-68 2 int. j. anal. appl. (2022), 20:68 some investigations of implication in intuitionistic and other non-classical logics. in 60-ties, these algebras were studied especially by diego [8] from algebraic point of view. diego [8] proved that hilbert algebras form a variety which is locally finite. hilbert algebras were treated by busneag [4, 5] and jun [16] and some of their filters forming deductive systems were recognized. dudek [10] considered the fuzzification of subalgebras/ideals and deductive systems in hilbert algebras. the concept of fuzzy sets with thresholds is presented in several articles as seen in [9,17–21,23,24]. this study builds on the theory of fuzzy sets to introduce fuzzy subalgebras and ideals with thresholds of hilbert algebras, explain some of their characteristics, and show how they may be extended. we also talk about the connections between fuzzy subalgebras (ideals) with thresholds and their level subsets. additionally, the homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in hilbert algebras are investigated and some associated features are examined. 2. preliminaries let’s go through the idea of hilbert algebras as it was introduced by diego [8] in 1966 before we start. definition 2.1. [8] a hilbert algebra is a triplet with the formula x =(x, ·,1), where x is a nonempty set, · is a binary operation, and 1 is a fixed member of x that is true according to the axioms stated below: (1) (∀x,y ∈ x)(x · (y ·x)=1), (2) (∀x,y,z ∈ x)((x · (y ·z)) · ((x ·y) · (x ·z))=1), (3) (∀x,y ∈ x)(x ·y =1,y ·x =1⇒ x = y). in [10], the following conclusion was established. lemma 2.1. let x =(x, ·,1) be a hilbert algebra. then (1) (∀x ∈ x)(x ·x =1), (2) (∀x ∈ x)(1 ·x = x), (3) (∀x ∈ x)(x ·1=1), (4) (∀x,y,z ∈ x)(x · (y ·z)= y · (x ·z)). in a hilbert algebra x =(x, ·,1), the binary relation ≤ is defined by (∀x,y ∈ x)(x ≤ y ⇔ x ·y =1), which is a partial order on x with 1 as the largest element. definition 2.2. [26] a nonempty subset d of a hilbert algebra x = (x, ·,1) is called a subalgebra of x if x ·y ∈ d for all x,y ∈ d. int. j. anal. appl. (2022), 20:68 3 definition 2.3. [7] a nonempty subset d of a hilbert algebra x =(x, ·,1) is called an ideal of x if the following conditions hold: (1) 1∈ d, (2) (∀x,y ∈ x)(y ∈ d ⇒ x ·y ∈ d), (3) (∀x,y1,y2 ∈ x)(y1,y2 ∈ d ⇒ (y1 · (y2 ·x)) ·x ∈ d). a fuzzy set [25] in a nonempty set x is defined to be a function µ : x → [0,1], where [0,1] is the unit closed interval of real numbers. definition 2.4. [22] a fuzzy set µ in a hilbert algebra x =(x, ·,1) is said to be a fuzzy subalgebra of x if the following condition holds: (∀x,y ∈ x)(µ(x ·y)≥min{µ(x),µ(y)}). definition 2.5. [11] a fuzzy set µ in a hilbert algebra x =(x, ·,1) is said to be a fuzzy ideal of x if the following conditions hold: (1) (∀x ∈ x)(µ(1)≥ µ(x)), (2) (∀x,y ∈ x)(µ(x ·y)≥ µ(y)), (3) (∀x,y1,y2 ∈ x)(µ((y1 · (y2 ·x)) ·x)≥min{µ(y1),µ(y2)}). 3. fuzzy subalgebras and ideals with thresholds for ε,δ ∈ [0,1] with ε < δ, we introduce the concepts of fuzzy subalgebras and ideals with thresholds ε and δ of hilbert algebras and investigate some related properties. definition 3.1. a fuzzy set µ in a hilbert algebra x = (x, ·,1) is called a fuzzy subalgebra with thresholds ε and δ (fstδε) of x, where ε,δ ∈ [0,1] with ε < δ if the following condition holds: (∀x,y ∈ x)(max{µ(x ·y),ε}≥min{µ(x),µ(y),δ}). example 3.1. let x = {1,x,y,z,0} with the following cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 then x is a hilbert algebra. we define a fuzzy set µ : a → [0,1] as follows: µ(1)=0.9,µ(x)=0.6,µ(y)=0.8,µ(z)=0.4,µ(0)=0.2. then µ is a fst0.90.8 of x. 4 int. j. anal. appl. (2022), 20:68 proposition 3.1. if µ is a fstδε of a hilbert algebra x =(x, ·,1), then (∀x ∈ x)(max{µ(1),ε}≥min{µ(x),δ}). proof. for all x ∈ x, we have max{µ(1),ε}=max{µ(x ·x),ε}≥min{µ(x),µ(x),δ}=min{µ(x),δ}. � definition 3.2. a fuzzy set µ in a hilbert algebra x =(x, ·,1) is called a fuzzy ideal with thresholds ε and δ (fitδε) of x if the following conditions hold: (1) (∀x ∈ x)(max{µ(1),ε}≥min{µ(x),δ}), (2) (∀x,y ∈ x)(max{µ(x ·y),ε}≥min{µ(y),δ}), (3) (∀x,y1,y2 ∈ x)(max{µ((y1 · (y2 ·x)) ·x),ε}≥min{µ(y1),µ(y2),δ}). example 3.2. let x = {1,x,y,z,0} with the following cayley table: · 1 x y z 0 1 1 x y z 0 x 1 1 y z 0 y 1 x 1 z z z 1 1 y 1 y 0 1 1 1 1 1 then x is a hilbert algebra. we define a fuzzy set µ : a → [0,1] as follows: µ(1)=0.9,µ(x)=0.5,µ(y)=0.4,µ(z)=0.3,µ(0)=0.2. then µ is a fit0.90.8 of x. proposition 3.2. if µ is fitδε of a hilbert algebra x =(x, ·,1), then (∀x,y ∈ x)(max{µa((y ·x) ·x),ε)≥min{µ(1),µ(y),δ}). proof. let x,y ∈ x. then max{µ((y ·x) ·x),ε}=max{µ((1 · (y ·x)) ·x),ε}≥min{µ(1),µ(y),δ}. � lemma 3.1. if µ is a fitδε of a hilbert algebra x =(x, ·,1), then (∀x,y ∈ x)(x ≤ y ⇒min{µ(1),µ(x),δ}≤max{µ(y),ε}). (3.1) int. j. anal. appl. (2022), 20:68 5 proof. let x,y ∈ x be such that x ≤ y. then x ·y =1 and so max{µ(y),ε} = max{µ(1 ·y),ε} = max{µ(((x ·y) · (x ·y)) ·y),ε} ≥ min{µ(x ·y),µ(x),δ} = min{µ(1),µ(x),δ}. � theorem 3.1. every fitδε of a hilbert algebra x =(x, ·,1) is a fst δ ε. proof. assume that µ is a fitδε of x. let x,y ∈ x. then max{µ(x ·y),ε}≥min{µ(y),δ}≥min{µ(x),µ(y),δ}. hence, µ is a fstδε of x. � theorem 3.2. let ε,δ ∈ [0,1] with ε < δ. if a fuzzy set µ in a hilbert algebra x = (x, ·,1) is such that µ(x)≤ ε for all x ∈ x, then it is a fitδε (resp., fst δ ε) of x. proof. let x ∈ x. then max{µ(1),ε} = ε ≥ µ(x) ≥ min{µ(x),δ}. let x,y ∈ x. then max{µ(x · y),ε} = ε ≥ µ(y) ≥ min{µ(y),δ}. let x,y1,y2 ∈ x. then max{µ((y1 · (y2 · x)) · x),ε} = ε ≥ µ(y1)≥min{µ(y1),µ(y2),δ}. hence, µ is a fitδε of x. let x,y ∈ x. then max{µ(x · y),ε} = ε ≥ µ(x) ≥ min{µ(x),µ(y),δ}. hence, µ is a fstδε of x. � by theorem 3.2, we get the following theorem. theorem 3.3. let ε,δ ∈ [0,1] with ε < δ. if a fuzzy set µ in a hilbert algebra x = (x, ·,1) is such that µ(x)≥ δ for all x ∈ x, then it is a fitδε (resp., fst δ ε) of x. theorem 3.4. let µ be a fuzzy set in a hilbert algebra x = (x, ·,1) and ε,δ ∈ [0,1] with ε < δ. then µ is a fstδε of x if and only if for all t ∈ (ε,δ],u(µ,t) := {x ∈ x | µ(x)≥ t} is a subalgebra of x if it is nonempty. proof. assume that µ is a fstδε of x. let t ∈ (ε,δ] be such that u(µ,t) 6= ∅ and let x,y ∈ u(µ,t). then µ(x)≥ t, µ(y)≥ t, and δ ≥ t. thus t is a lower bound of {µ(x),µ(y),δ}. since µ is a fstδε of x, we have max{µ(x ·y),ε}≥min{µ(x),µ(y),δ}≥ t > ε. so, max{µ(x ·y),ε}= µ(x ·y). since max{µ(x ·y),ε}≥ t, we get µ(x ·y)≥ t. hence, x ·y ∈ u(µ,t). therefore, u(µ,t) is a subalgebra of x. conversely, assume that for all t ∈ (ε,δ], u(µ,t) is a subalgebra of x if it is nonempty. let x,y ∈ x. then µ(x),µ(y) ∈ [0,1]. choose t = min{µ(x),µ(y)}. then µ(x) ≥ t and µ(y) ≥ t. thus x,y ∈ u(µ,t) 6= ∅. by the assumption, we have u(µ,t) is a subalgebra of x. so, x ·y ∈ u(µ,t), that is, µ(x · y) ≥ t = min{µ(x),µ(y)}. thus max{µ(x · y),ε} ≥ µ(x · y) ≥ min{µ(x),µ(y)} ≥ min{µ(x),µ(y),δ}. hence, µ is a fstδε of x. � 6 int. j. anal. appl. (2022), 20:68 the following theorem can be proved similarly to theorem 3.4. theorem 3.5. let µ be a fuzzy set in a hilbert algebra x = (x, ·,1) and ε,δ ∈ [0,1] with ε < δ. then µ is a fitδε of x if and only if for all t ∈ (ε,δ],u(µ,t) is an ideal of x if it is nonempty. definition 3.3. let f be a function from a nonempty set x to a nonempty set y . if µ is a fuzzy set in x, then the fuzzy set β in y defined by β(y)=   sup t∈f−1(y) {µ(t)} if f−1(y) 6= ∅ 0 otherwise is said to be the image of µ under f . similarly, if α is a fuzzy set in y , then the fuzzy set f−1(α)= α◦f in x (i.e., the fuzzy set is defined by f−1(α)(x) = α(f (x)) for all x ∈ x) is called the pre-image of α under f . definition 3.4. a fuzzy set µ in a hilbert algebra x =(x, ·,1) is said to have the sup property if for any nonempty subset t of x, there exists t0 ∈ t such that µ(t0)= supt∈t µ(t). definition 3.5. let x and y be any two nonempty sets and let f : x → y be any function. a fuzzy set µ in x is said to be f -invariant if (∀x,y ∈ x)(f (x)= f (y)⇒ µ(x)= µ(y)). lemma 3.2. let (x, ·,1x) and (y,?,1y ) be hilbert algebras and let f : x → y be a surjective homomorphism. let µ be an f -invariant fuzzy set in x with sup property. for any x,y ∈ y , there exist x0 ∈ f−1(x) and y0 ∈ f−1(y) such that β(x)= µ(x0),β(y)= µ(y0), and β(x ? y)= µ(x0 ·y0). proof. let x,y ∈ y . since f is surjective, we have f−1(x), f−1(y), and f−1(x·y)are nonempty subsets of x. since µ has sup property, there exist elements x0 ∈ f−1(x),y0 ∈ f−1(y), and z0 ∈ f−1(x ? y) such that β(x)= sups∈f−1(x){µ(s)}= µ(x0), β(y)= sups∈f−1(y){µ(s)}= µ(y0), β(x ? y)= sups∈f−1(x?y){µ(s)}= µ(z0). since f (z0)= x ? y = f (x0)? f (y0)= f (x0 ·y0), and µ is f -invariant, it follows that β(x ? y)= µ(z0)= µ(x0 ·y0). � int. j. anal. appl. (2022), 20:68 7 theorem 3.6. let (x, ·,1x) and (y,?,1y ) be hilbert algebras and let f : x → y be a surjective homomorphism. then the following statements hold: (1) if µ is an f -invariant fstδε of x with sup property, then β is a fst δ ε of y , (2) if µ is an f -invariant fitδε of x with sup property, then β is a fit δ ε of y . proof. (1) assume that µ is an f -invariant fstδε of x with sup property. let a,b ∈ y . then by lemma 3.2, there exist a0 ∈ f−1(a) and b0 ∈ f−1(b) such that β(a) = µ(a0), β(b) = µ(b0), and β(a ? b) = µ(a0 · b0). thus max{β(a ? b),ε} = max{µ(a0 · b0),ε} ≥ min{µ(a0),µ(b0),δ} = min{β(a),β(b),δ}. hence, β is a fstδε of y . (2) assume that µ is an f -invariant fitδε of x with sup property. since f (1x) = 1y , we have f−1(1y ) 6= ∅. by lemma 3.2, there exists x1 ∈ f−1(1y ) such that µ(x1) = β(1y ). thus f (x1) = 1y = f (1x). since µ is f -invariant, we have µ(x1) = µ(1x). so, β(1y ) = µ(1x). let y ∈ y . since f is surjective, we have f−1(y) 6= ∅. by lemma 3.2, there exists x ∈ f−1(y) such that µ(x)= β(y). thus max{β(1y ),ε} = max{µ(1x),ε} ≥ min{µ(x),δ} = min{β(y),δ}. now, let a,b ∈ y . then by lemma 3.2, there exist a0 ∈ f−1(a) and b0 ∈ f−1(b) such that µ(a0) = β(a), µ(b0) = β(b), and µ(a0 ·b0)= β(a ? b). thus max{β(a ? b),ε}=max{µ(a0 ·b0),ε}≥min{µ(b0),δ}=min{β(b),δ}. next, let a,b,y ∈ y . by lemma 3.2, there exist a0 ∈ f−1(a), b0 ∈ f−1(b) and x ∈ f−1(y) such that µ(a0) = β(a), µ(b0) = β(b), µ(y) = β(x) and µ((a0 · (b0 · x)) · x) = β((a ? (b ? x)) ? x). thus max{β((a ? (b ? x)) ? x),ε} = max{µ((a0 · (b0 · x)) · x) · x),ε} ≥ min{µ(a0),µ(b0),δ} = min{β(a),β(b),δ}. hence, β is a fitδε of y . � theorem 3.7. let (x, ·,1x) and (y,?,1y ) be hilbert algebras and let f : x → y be a homomorphism. then the following statements hold: (1) if α is a fstδε of y , then f −1(α) is a fstδε of x, (2) if α is a fitδε of y , then f −1(α) is a fitδε of x. proof. (1) assume that α is a fstδε of y . let x,y ∈ x. then max{f−1(α)(x ·y),ε} = max{(α◦ f )(x ·y),ε} = max{(α(f (x ·y)),ε} = max{(α(f (x)? f (y)),ε} ≥ min{α(f (x)),α(f (y)),δ} = min{(α◦ f )(x),(α◦ f )(y),δ} = min{f−1(α)(x), f−1(α)(y),δ}. hence, f−1(α) is a fstδε of x. 8 int. j. anal. appl. (2022), 20:68 (2) assume that α is a fitδε of y . let x ∈ x. then max{f−1(α)(1x),ε} = max{(α◦ f )(1x),ε} = max{(α(f (1x)),ε} = max{α(1y ),ε} ≥ min{α(f (x)),δ} = min{(α◦ f )(x),δ} = min{f−1(α)(x),δ}. let x,y ∈ x. then max{f−1(α)(x ·y),ε} = max{(α◦ f )(x ·y),ε} = max{α(f (x ·y)),ε} ≥ min{α(f (y)),ε} = min{(α◦ f )(y),δ} = min{f−1(α)(y),δ}. let x,y1,y2 ∈ x. then max{f−1(α)((y1 · (y2 ·x)) ·x),ε} = max{(α◦ f )((y1 · (y2 ·x)) ·x),ε} = max{α(f ((y1 · (y2 ·x)) ·x)),ε} = max{α(f (y1)? (f (y2)? f (x))? f (x)),ε} ≥ min{α(f (y1)),α(f (y2)),δ} = min{(α◦ f )(y1),(α◦ f )(y2),δ} = min{f−1(α)(y1), f−1(α)(y2),δ}. hence, f−1(α) is a fitδε of x. � definition 3.6. let f be a function from a nonempty set x to a nonempty set y . if µ is a fuzzy set in x, then the fuzzy set η in y defined by η(y)=   inft∈f−1(y){µ(t)} if f −1(y) 6= ∅ 1 otherwise is said to be the image of µ under f . definition 3.7. a fuzzy set µ in a hilbert algebra x =(x, ·,1) is said to have the inf property if for any nonempty subset t of x, there exists t0 ∈ t such that µ(t0)= inft∈t µ(t). the following lemma can be proved similarly to lemma 3.2. lemma 3.3. let (x, ·,1x) and (y,?,1y ) be hilbert algebras and let f : x → y be a surjective homomorphism. let µ be an f -invariant fuzzy set in x with inf property. for any x,y ∈ y , there exist x0 ∈ f−1(x) and y0 ∈ f−1(y) such that η(x)= µ(x0),η(y)= µ(y0), and η(x ? y)= µ(x0 ·y0). int. j. anal. appl. (2022), 20:68 9 the following theorem can be proved similarly to theorem 3.6. theorem 3.8. let (x, ·,1x) and (y,?,1y ) be hilbert algebras and let f : x → y be a surjective homomorphism. then the following statements hold: (1) if µ is an f -invariant fstδε of x with inf property, then η is a fst δ ε of y , (2) if µ is an f -invariant fitδε of x with inf property, then η is a fit δ ε of y . 4. conclusion in the present paper, we have introduced the concepts of fuzzy subalgebras and ideals with thresholds of hilbert algebras. the relationship between fuzzy subalgebras (ideals) and their level subsets is described. moreover, the homomorphic images and inverse images of fuzzy subalgebras and ideals with thresholds in hilbert algebras are also studied and some related properties are investigated. acknowledgment this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-rim032). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. ahmad, a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [2] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87–96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [3] m. atef, m.i. ali, t.m. al-shami, fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, comput. appl. math. 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x. [4] d. busneag, a note on deductive systems of a hilbert algebra, kobe j. math. 2 (1985), 29-35. https://cir. nii.ac.jp/crid/1570291224860431872. [5] d. busneag, hilbert algebras of fractions and maximal hilbert algebras of quotients, kobe j. math. 5 (1988), 161-172. https://cir.nii.ac.jp/crid/1570572702603831808. [6] n. caǧman, s. enginoǧlu, f. citak, fuzzy soft set theory and its application, iran. j. fuzzy syst. 8 (2011), 137-147. [7] i. chajda, r. halas, congruences and ideals in hilbert algebras, kyungpook math. j. 39 (1999), 429-429. [8] a. diego, sur les algébres de hilbert, collect. logique math. ser. a (ed. hermann, paris), 21 (1966), 1-52. [9] n. dokkhamdang, a. kesorn, a. iampan, generalized fuzzy sets in up-algebras, ann. fuzzy math. inform. 16 (2018), 171-190. [10] w.a. dudek, on fuzzification in hilbert algebras, contrib. gen. algebra, 11 (1999), 77-83. [11] w.a. dudek, y.b. jun, on fuzzy ideals in hilbert algebra, novi sad j. math. 29 (1999), 193-207. [12] h. garg, k. kumar, an advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, soft comput. 22 (2018), 4959–4970. https: //doi.org/10.1007/s00500-018-3202-1. https://doi.org/10.1155/2009/586507 https://doi.org/10.1155/2009/586507 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1007/s40314-021-01501-x https://cir.nii.ac.jp/crid/1570291224860431872 https://cir.nii.ac.jp/crid/1570291224860431872 https://cir.nii.ac.jp/crid/1570572702603831808 https://doi.org/10.1007/s00500-018-3202-1 https://doi.org/10.1007/s00500-018-3202-1 10 int. j. anal. appl. (2022), 20:68 [13] h. garg, k. kumar, distance measures for connection number sets based on set pair analysis and its applications to decision-making process, appl. intell. 48 (2018), 3346–3359. https://doi.org/10.1007/s10489-018-1152-z. [14] h. garg, s. singh, a novel triangular interval type-2 intuitionistic fuzzy set and their aggregation operators, iran. j. fuzzy syst. 15 (2018), 69-93. https://doi.org/10.22111/ijfs.2018.4159. [15] l. henkin, an algebraic characterization of quantifiers, fund. math. 37 (1950), 63-74. https://eudml.org/doc/ 213228. [16] y.b. jun, deductive systems of hilbert algebras, math. japon. 43 (1996), 51-54. https://cir.nii.ac.jp/crid/ 1571417124616097792. [17] y.b. jun, fuzzy subalgebras with thresholds in bck/bci-algebras, commun. korean math. soc. 22 (2007), 173–181. https://doi.org/10.4134/ckms.2007.22.2.173. [18] y.b. jun, on (α,β)-fuzzy subalgebras of bck/bci-algebras, bull. korean math. soc. 42 (2005), 703-711. https://doi.org/10.4134/bkms.2005.42.4.703. [19] f.f. kareem, m.m. abed, generalizations of fuzzy k-ideals in a ku-algebra with semigroup, j. phys.: conf. ser. 1879 (2021), 022108. https://doi.org/10.1088/1742-6596/1879/2/022108. [20] a. khan, y.b. jun, t. mahmood, generalized fuzzy interior ideals in abel grassmann’s groupoids, int. j. math. math. sci. 2010 (2010), 838392. https://doi.org/10.1155/2010/838392. [21] a. khan, m. shabir, (α,β)-fuzzy interior ideals in ordered semigroups, lobachevskii j. math. 30 (2009), 30-39. https://doi.org/10.1134/s1995080209010053. [22] k.h. kim, on t-fuzzy ideals in hilbert algebras, sci. math. japon. 70 (2009), 7-15. [23] a.b. saeid, redefined fuzzy subalgebra (with thresholds) of bck/bci-algebras, iran. j. math. sci. inform. 4 (2009), 19-24. https://doi.org/10.7508/ijmsi.2009.02.002. [24] m. siripitukdet, a. ruanon, fuzzy interior ideals with thresholds (s,t] in ordered semigroups, thai j. math. 11 (2013), 371-382. [25] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [26] j. zhan, z. tan, intuitionistic fuzzy deductive systems in hibert algebra, southeast asian bull. math. 29 (2005), 813-826. https://doi.org/10.1007/s10489-018-1152-z https://doi.org/10.22111/ijfs.2018.4159 https://eudml.org/doc/213228 https://eudml.org/doc/213228 https://cir.nii.ac.jp/crid/1571417124616097792 https://cir.nii.ac.jp/crid/1571417124616097792 https://doi.org/10.4134/ckms.2007.22.2.173 https://doi.org/10.4134/bkms.2005.42.4.703 https://doi.org/10.1088/1742-6596/1879/2/022108 https://doi.org/10.1155/2010/838392 https://doi.org/10.1134/s1995080209010053 https://doi.org/10.7508/ijmsi.2009.02.002 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. fuzzy subalgebras and ideals with thresholds 4. conclusion acknowledgment references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 179-187 doi: 10.28924/2291-8639-15-2017-179 some characterizations of harmonic convex functions muhammad aslam noor∗, khalida inayat noor, sabah iftikhar abstract. in this paper, we show that the harmonic convex functions have some nice properties, which convex functions enjoy. we also discuss some basic properties of harmonic convex functions. the techniques and ideas of this paper may be a starting point for future research. 1. introduction convexity theory played an important and fundamental role in the developments of various branches of engineering, financial mathematics, economics and optimization. in recent years, the concept of convex functions and its variant forms have been extended and generalized using innovative techniques to study complicated problems. it is well known that convexity is closely related to inequality theory. the optimality conditions of differentiable convex functions are characterized by variational inequalities, the origin of which can be traced back to euler, lagrange and newton. on the other hand, convex functions are related to integral inequalities. which are called the harmite-hadamard type integral inequalities. for recent developments, see [1, 3, 9, 16] and reference therein. related to the arithmetic means, we have harmonic means. the harmonic means have applications in electrical circuit theory and other branches of sciences. for example, the total resistance of a set of parallel resistors is obtained by adding up the reciprocal of the individual resistance value and then considering the reciprocal of their total. also the harmonic means are used in developing parallel algorithms for solving various problems, see noor [5]. a significant class of convex functions, called harmonic convex was introduced by anderson et al. [1] and iscan [3], independently. noor and noor [6, 7] have shown that the optimality conditions of the differentiable harmonic convex functions on the harmonic convex set can be expressed by a class of variational inequalities, which is called the harmonic variational inequality. for recent developments and applications, see [5–7, 9–14]. to the best of our knowledge, this field is new one and has not been developed as yet. in this paper, we show that the harmonic convex functions have some nice properties [2], which convex functions enjoy. we have investigated several basic properties of harmonic convex functions. we obtained the necessary and sufficient characterization of a differentiable harmonic convex and harmonic quasi convex functions. it is worth mentioning that the harmonic variational inequalities is a new class of variational inequalities and is not studied much. the interested readers are encouraged to study these harmonic variational inequalities. it is high time to find the applications of these inequalities along with efficient numerical methods. 2. preliminaries first of all, we recall the following basic concepts. definition 2.1. [1]. a set k ⊂ rn \{0} is said to be a harmonic convex set, if xy tx + (1 − t)y ∈ k, ∀ x,y ∈ k, t ∈ [0, 1]. received 4th august, 2017; accepted 10th october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 26d15; 26d10; 90c23. key words and phrases. convex functions; general preinvex functions; differentiability; hermite-hadamard inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 179 180 noor et al. definition 2.2. [3]. a function f : k → r, where k is a nonempty harmonic convex set in rn\{0}. the function f is said to be a harmonic convex function on k, if and only if, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y), ∀ x,y ∈ k, t ∈ [0, 1]. the function f is called strictly harmonic convex on k, if the above inequality is true as a strict inequality for each distinct x and y ∈ k and for each t ∈ (0, 1). the function f : k → r is called harmonic concave (strictly harmonic concave) on k if −f is harmonic convex (strictly harmonic convex) on k. remark 2.1. geometric interpretation we now consider the geometric interpretation of harmonic convex function. let x and y be two distinct points in the domain of f, and consider the point xy tx+(1−t)y , with t ∈ (0, 1). note that (1−t)f(x)+tf(y) gives the weighted arithmetic mean of f(x) and f(y), while f( xy tx+(1−t)y ) gives the value of f at the point xy tx+(1−t)y . so, for a harmonic convex function f, the value of f at the points on the path xy tx+(1−t)y whose initial point is x and terminal point is y, is less than or equal to the chord joining the points (x,f(x)) and (y,f(y)). for a harmonic concave function, the chord is (on or) below the function itself. definition 2.3. [16]. a function f : k → r, where k is a nonempty harmonic convex set in rn\{0}. the function f is said to be a harmonic quasi convex function on k, if and only if, f ( xy tx + (1 − t)y ) ≤ max{f(x),f(y)}, ∀ x,y ∈ k, t ∈ [0, 1]. the function f is said to be harmonic quasi concave, if −f is harmonic quasi convex. a function f is harmonic quasi convex, if whenever f(y) ≥ f(x), f(y) is greater than or equal to all the values of f at the points of the path xy tx+(1−t)y . a function is said to be strictly harmonic quasi convex, if strict inequality holds for f(x) 6= f(y). definition 2.4. [9]. a function f : k → r, where k is a harmonic convex set in rn \ {0}. the function f is said to be a harmonic log-convex function on k, if and only if, f ( xy tx + (1 − t)y ) ≤ [f(x)]1−t[f(y)]t, ∀ x,y ∈ k, t ∈ [0, 1]. (2.1) from (2.1), it follows that f ( xy tx + (1 − t)y ) ≤ [f(x)]1−t[f(y)]t ≤ (1 − t)f(x) + tf(y) ≤ max{f(x),f(y)}, ∀ x,y ∈ k, t ∈ [0, 1], this shows that, every harmonic log-convex function is harmonic convex and every harmonic convex function is harmonic quasi convex, but the converse is not true. also from (2.1), we have log f( xy tx + (1 − t)y ≤ (1 − t) log f(x) + t log f(y), ∀x,y ∈ k, t ∈ [0, 1]. for the properties and inequalities of harmonic log-convex functions, see [9, 13, 14]. 3. main results in this section, we discuss some properties of harmonic convex function and harmonic quasi convex function. theorem 3.1. if ki is a family of harmonic convex set, then ∩i∈ik is a harmonic convex set. characterizations of harmonic convex functions 181 proof. let x,y ∈∩i∈iki, t ∈ [0, 1]. then, for each i ∈ i, x,y ∈ ki, since ki is a harmonic convex set, it follows that xy tx + (1 − t)y ∈ ki, ∀ i ∈ i. thus xy tx + (1 − t)y ∈∩i∈iki. hence, ∩i∈iki is harmonic convex set. � theorem 3.2. let k be a harmonic set. if fi : k → r, (i = 1, 2, 3, ...,m) are harmonic convex functions. then the function f = m∑ i=1 aifi, ai ≥ 0, i = 1, 2, 3, ...,m, is harmonic convex function. proof. let k be a harmonic set. then ∀x,y ∈ k and t ∈ [0, 1], we have f ( xy tx + (1 − t)y ) = m∑ i=1 aifi ( xy tx + (1 − t)y ) ≤ m∑ i=1 ai[(1 − t)fi(x) + tfi(y)] = (1 − t) m∑ i=1 aifi(x) + t m∑ i=1 aifi(y) = (1 − t)f(x) + tf(y). this shows that f is a harmonic convex function. � theorem 3.3. let k be a harmonic set. if the functions fi : k → r are harmonic convex functions, then f = max{fi , i = 1, 2, 3, ...,m} is also harmonic convex. proof. consider, f ( xy tx + (1 − t)y ) = max { fi ( xy tx + (1 − t)y ) , i = 1, 2, 3, ...,m } = fw ( xy tx + (1 − t)y ) ≤ (1 − t)fw(x) + tfw(y) = (1 − t) max{fi(x)} + t max{fi(y)} = (1 − t)f(x) + tf(y). this implies that f(x) is harmonic convex function. � theorem 3.4. let k be a harmonic set. if the function f : k → r is harmonic convex function and g : r → r is a linear function, then f ◦g is a harmonic convex function. proof. let f be a harmonic convex function and g be a linear function. then (g ◦f) ( xy tx + (1 − t)y ) = g ( f ( xy tx + (1 − t)y )) ≤ g((1 − t)f(x) + tf(y) = (1 − t)g(f(x)) + tg(f(y)) = (1 − t)(g ◦f)(x) + t(g ◦f)(y). this shows that g ◦f is a harmonic convex function. � 182 noor et al. theorem 3.5. let k be a harmonic set. if the function f : k → r is a harmonic quasi convex function and g : r → r is a nondecreasing function, then f ◦g is a harmonic quasi convex function. proof. let f be a harmonic quasi convex function and g be a nondecreasing function. then (g ◦f) ( xy tx + (1 − t)y ) ≤ g[max{f(x),f(y)}] = max{g ◦f(x),g ◦f(y)}, which implies that (g ◦f) is a harmonic quasi convex function. � theorem 3.6. let f : k → r be a harmonic convex function. if µ = infx∈k f(x), then the set e = {x ∈ k : f(x) = µ} is a harmonic convex set. if f is strictly harmonic convex function, then e is a singleton. proof. let x,y ∈ e. since f is harmonic convex function, so f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) = µ, which implies that xy tx+(1−t)y ∈ e and hence e is a harmonic convex set. for the second part, assume that f(x) = f(y) = µ. since k is a harmonic convex set, so for t ∈ (0, 1), xy tx+(1−t)y ∈ k. further, since f is strictly harmonic convex function, so f ( xy tx + (1 − t)y ) < (1 − t)f(x) + tf(y) = µ. this contradicts that µ = infx∈k f(x) and hence e is singleton. � lemma 3.1. let k be a nonempty harmonic convex set in rn\{0} and let f : k → r be a harmonic convex function. then the level set α = {x ∈ k : f(x) ≤ α, α ∈ r}, (3.1) is a harmonic convex se. proof. let x,y ∈ kα. then f(x) ≤ α, f(y) ≤ α. consider, f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) ≤ (1 − t)α + tα = α. hence xy tx+(1−t)y ∈ kα and therefore kα is a harmonic convex set. � remark 3.1. we would like to mention that the converse of above result is not true. definition 3.1. [2]. let k be a nonempty in rn and let f : k → r be a function. then epigraph of f, denoted by e(f), is defined by e(f) = {(x,α) : x ∈ k,α ∈ r,f(x) ≤ α}. theorem 3.7. let k be a nonempty harmonic convex set in rn \{0} and let f : k → r. then f is harmonic convex, if and only if, e(f) is a harmonic convex set. proof. assume that f is harmonic convex function and let (x,α), (y,β) ∈ e(f). then f(x) ≤ α and f(y) ≤ β. consider , for t ∈ [0, 1], f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) ≤ (1 − t)α + tβ. characterizations of harmonic convex functions 183 thus ( xy tx+(1−t)y , (1 − t)α + tβ ) ∈ e(f) and hence e(f) is a harmonic convex set. conversely, assume that e(f) is a harmonic convex set and let x,y ∈ k. then (x,f(x)), (y,f(y)) ∈ e(f) and by the harmonic convexity of e(f), we have( xy tx + (1 − t)y , (1 − t)f(x) + tf(y) ) ∈ e(f). thus, f ( xy tx+(1−t)y ) ≤ (1−t)f(x) +tf(y) for each t ∈ (0, 1), that is, f is harmonic convex function. � remark 3.2. if the function f : k → r is harmonic log-convex function on the harmonic convex set k, then e(f) is a harmonic convex set. theorem 3.8. let k be a harmonic convex set and let f : k → r be a harmonic convex function. then any local minimum of f is a global minimum. proof. let x ∈ k be a local minimum of the harmonic convex function f.. assume the contrary, that is, f(y) < f(x), for some y ∈ k. since f is harmonic convex function, we have f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) ∀ x,y ∈ k, t ∈ [0, 1]. thus f ( xy tx + (1 − t)y ) −f(x) ≤ t[f(y) −f(x)], from which it follows that for some small t > 0, f ( xy tx + (1 − t)y ) < f(x). contradicting the local minimum. hence every local minimum of f is a global minimum. � strictly harmonic quasi convex and strictly harmonic quasi concave functions are especially important in nonlinear programming because they ensure that a local minimum and a local maximum over a harmonic convex set is a global minimum and a global maximum, respectively. theorem 3.9. let k be a harmonic convex set and let f : k → r be a strictly harmonic quasi convex function. consider the problem of minimum of f(x) subject to x ∈ k. if x̄ is a local optimal solution, then x̄ is also a global optimal solution. proof. assume, on the contrary, that there exists an x̂ ∈ k with f(x̂) < f(x̄). by the harmonic convexity of k, x̄x̂ tx̄+(1−t)x̂ ∈ k for each t ∈ (0, 1). since x̄ is a local minimum by assumption, f(x̄) < f ( x̄x̂ tx̄ + (1 − t)x̂ ) ∀ t ∈ (0, 1). but because f is strictly harmonic quasi convex and f(x̂) < f(x̄), we have f ( x̄x̂ tx̄ + (1 − t)x̂ ) < f(x̄) ∀ t ∈ (0, 1). this contradicts the local optimality of x̄, and the proof is complete. � lemma 3.2. let k be a nonempty harmonic convex set in rn\{0}. then a function f : k → r is a harmonic quasi convex function, if and only if, the level set kα defined by (3.1) is a harmonic convex set. proof. suppose that f is harmonic quasi-convex function and let x,y ∈ kα. therefore x,y ∈ k and max{f(x),f(y)}≤ α. also x1 = xytx+(1−t)y ∈ k, since k is a harmonic convex set. using the harmonic quasi-convexity of f, we have f ( xy tx + (1 − t)y ) ≤ max{f(x),f(y)}≤ α. 184 noor et al. hence x1 ∈ kα and therefore kα is harmonic convex set. conversely, suppose that kα is harmonic convex set for each real number α. let x,y ∈ kα. furthermore, let t ∈ (0, 1) and x1 = xytx+(1−t)y . note that x,y ∈ kα for α = max{f(x),f(y)}. by assumption, kα is harmonic convex set, so that x1 ∈ kα. therefore f ( xy tx + (1 − t)y ) ≤ α = max{f(x),f(y)}. hence, f is harmonic quasi-convex and the proof is complete. � remark 3.3. the level set defined as kα = {x ∈ k : f(x) ≤ α, α ∈ r}, is sometimes referred to as a lower-level set, to differentiate it from the upper-level set defined as kα = {x ∈ k : f(x) ≥ α, α ∈ r}, which is harmonic convex set, if and only if, f is harmonic quasi concave. theorem 3.10. if the function f : k → r is harmonic convex function such that f(x) < f(y), for all x,y ∈ k, then f is strictly harmonic quasi convex function. proof. by the harmonic convexity of f, we have f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y) < f(x); since f(x) < f(y), which shows that the harmonic convex function is strictly harmonic quasi convex function. � definition 3.2. the function f : k → r is said to be harmonic pseudo convex function with respect to a strictly positive function b(·, ·) such that f(y) < f(x) ⇒ f ( xy tx + (1 − t)y ) < f(x) + t(t− 1)b(x,y), ∀x,y ∈ k,t ∈ (0, 1). theorem 3.11. if the function f : k → r is harmonic convex function such that f(y) < f(x), then f is harmonic pseudo convex function f with respect to a strictly positive function b(·, ·). proof. let f(y) < f(x) and let f be a harmonic convex function. then f ( xy tx + (1 − t)y ) ≤ f(x) + t(f(y) −f(x)) < f(x) + t(1 − t)(f(y) −f(x)) = f(x) + t(t− 1)(f(x) −f(y)) < f(x) + t(t− 1)b(y,x), where b(x,y) = f(x) −f(y) > 0, the result result. � remark 3.4. if the function f : k → r is harmonic log-convex function such that f(y) < f(x), then the harmonic log-convex function is harmonic pseudo convex under the same sense. theorem 3.12. let k be a nonempty harmonic convex set in rn \{0} and let f : k → r be strictly harmonic quasi convex function and lower semicontinuous. then f is harmonic quasi convex function. proof. let x,y ∈ k. if f(x) 6= f(y), then by the strict harmonic quasi convexity of f, we must have f ( xy tx + (1 − t)y ) < max{f(x),f(y)} ∀ t ∈ (0, 1). now, suppose that f(x) = f(y). to show that f is harmonic quasi convex, we need to show that f ( xy tx + (1 − t)y ) ≤ f(y) ∀ t ∈ (0, 1). by contradiction, suppose that f ( xy tx + (1 − t)y ) > f(y) for some t ∈ (0, 1). characterizations of harmonic convex functions 185 denote xy tx+(1−t)y by x̄. since f is lower semicontinuous, there exists a t ∈ (0, 1) such that f(x̄) > f ( x̄y tx̄ + (1 − t)y ) > f(y) = f(x). (3.2) hence by the strict harmonic quasi convexity of f and since f ( x̄y tx̄+(1−t)y ) > f(y), we have f(x̄) < f ( x̄y tx̄+(1−t)y ) , contradicting (3.2). this completes the proof. � the following theorem gives a necessary and sufficient characterization of a differentiable harmonic quasi convex function. theorem 3.13. let k be a nonempty harmonic convex set in rn \ {0} and let f : k → r be differentiable on k. then f is harmonic quasi convex, if and only if, f(x) ≤ f(y) ⇒〈f′(y), xy y −x 〉≤ 0, ∀ x,y ∈ k. proof. let f be harmonic quasi convex and let x,y ∈ k be such that f(x) ≤ f(y). using the taylor series, we have f ( xy x + t(y −x) ) = f(y) + t〈f′(y), xy y −x 〉 + t ‖ xy y −x ‖ α [ y; t ( xy y −x )] . where α [ y; t ( xy y−x )] → 0, as t → 0. by the harmonic quasi convexity of f, we have f ( xy (1−t)x+ty ) ≤ f(y) and hence the above equation implies that t〈f′(y), xy y −x 〉 + t ‖ xy y −x ‖ α [ y; t ( xy y −x )] ≤ 0. dividing by t and taking the limit in the above inequality as t → 0, we have 〈f′(y), xy y −x 〉≤ 0. conversely, suppose that x,y ∈ k and that f(x) ≤ f(y). we need to show that f( xy tx+(1−t)y ) ≤ f(y), for all x ∈ k and t ∈ (0, 1). we do this by showing that the set l = {x′ : x′ = xy tx+(1−t)y , t ∈ (0, 1), f(x ′) > f(y)} is empty. by contradiction, suppose that there exists an x′ ∈ l. therefore x′ = xy tx+(1−t)y for some t ∈ (0, 1) and f(x′) > f(y). since f is differentiable, it is continuous and there must exits a δ ∈ (0, 1), such that f ( x′y (1 −µ)x′ + µy ) > f(y) for each µ ∈ (δ, 1), and f(x′) > f ( x′y (1−δ)x′+δy ) . by this inequality and the mean value theorem, we must have 0 < f(x′) −f ( x′y (1 −δ)x′ + δy ) = (1 − t)〈f′(x̂), x′y y −x′ 〉, (3.3) where x̂ = x ′y (1−µ′)x′+µ′y for some µ ′ ∈ (δ, 1). from it is clear that f(x̂) > f(y). dividing (3.3) by (1 − t) > 0, it follows that 〈f′(x̂), x ′y y−x′〉 > 0, which in turn implies that 〈f′(x̂), xy y −x 〉 > 0. (3.4) but on the other hand f(x̂) > f(y) ≥ f(x), and x̂ is harmonic combination of x and y. by the assumption of the theorem 〈f′(x̂), x̂x x̂−x〉≤ 0, and thus we must have 0 ≥〈f′(x̂), xy y −x 〉. the above inequality is not compatible with (3.4). therefore, l is empty, and the proof is complete. � 186 noor et al. the following theorem was derived by noor and noor [6, 7] and it gives a necessary and sufficient characterization of a differentiable harmonic convex function. theorem 3.14. [6]. let k be a harmonic convex set. if f : k → r is a differentiable harmonic convex function on the harmonic convex set k, then (1) f(y) −f(x) ≥〈f′(x), xy x−y〉, ∀ x,y ∈ k (2) 〈f′(x) −f′(y), xy x−y〉≤ 0, ∀ x,y ∈ k, but the converse is not true. theorem 3.15. [6]. let f be a differentiable harmonic convex function on the harmonic convex set k. then x ∈ k is a minimum of f, if and only if, x ∈ k satisfies 〈f′(x), xy x−y 〉≥ 0, ∀y ∈ k. remark 3.5. the inequality of the type 〈f′(x), xy x−y〉≥ 0 is known as harmonic variational inequality, which was introduced by noor and noor [6, 7]. definition 3.3. a function f : k → r, where k is a nonempty harmonic set rn\{0}. the function f is said to be a harmonic pseudo-convex function, if for each x,y ∈ k with 〈f′(y), xy y−x〉≥ 0, we have f(x) ≥ f(y); or equivalently, if f(x) < f(y), then 〈f′(y), xy y−x〉 < 0. definition 3.4. a function f : k → r, where k is a nonempty harmonic set rn\{0}. the function f is said to be a harmonic quasi convex function, if for each x,y ∈ k with f(x) ≤ f(y), we have 〈f′(y), xy y−x〉≤ 0; or equivalently, if 〈f ′(y), xy y−x〉 > 0, then f(x) > f(y). theorem 3.16. let k be a harmonic convex set and f : k → r a differentiable harmonic convex function on the harmonic convex set k. if f(x) ≤ f(y), ∀ x,y ∈ k, then f is harmonic quasi-convex function. furthermore, if f(x) < f(y), ∀ x,y ∈ k, then f is harmonic pseudo-convex function. proof. let f be differentiable harmonic convex function on the harmonic convex set k. then from theorem 3.14, we have 〈f′(y), xy y −x 〉≤ f(x) −f(y). if f(x) ≤ f(y), then 〈f′(y), xy y−x〉 ≤ 0. therefore, from theorem 3.13, we have f is harmonic quasiconvex function. similarly, if f(x) < f(y), we also have 〈f′(y), xy y−x〉 < 0. so, from the definition 3.3, we have f is harmonic pseudo-convex function. � acknowledgements the authors would like to thank rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] m. s. bazaraa, d. hanif, c. m. shetty, et al. nonlinear programming theory and algorithms (second edition)[m]. the united states of america: john wiley and sons, 1993. [3] i. iscan, hermite-hadamard type inequalities for harmonically convex functions. hacet. j. math. stats., 43(6)(2014), 935-942. [4] c. p. niculescu and l. e. persson, convex functions and their applications, springer-verlag, new york, (2006). [5] m. a. noor, advanced convex analysis and optimization, lecture notes, ciit, (2014-2017). [6] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inf. sci., 10(5)(2016), 1811-1814. [7] m. a. noor and k. i. noor, some implicit methods for solving harmonic variational inequalities, inter. j. anal. app. 12(1)(2016), 10-14 [8] m. a. noor and k. i. noor, auxiliary principle techinique for variational inequalities, appl. math. inf. sci., 11(1)(2017), 165-169. [9] m. a. noor, k. i. noor and m. u. awan. some characterizations of harmonically log-convex functions. proc. jangjeon. math. soc., 17(1)(2014), 51-61. characterizations of harmonic convex functions 187 [10] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic nonconvex functions, magnt research report. 4(1)(2016), 24-40. [11] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions(survey), twms j. pure appl. math. 7(1)(2016),3-19. [12] m. a. noor, k. i. noor and s. iftikhar, integral inequalities of hermite-hadamard type for harmonic (h, s)-convex functions, int. j. anal. appl., 11(1)(2016), 61-69. [13] m. a. noor, k. i. noor, s. iftikhar and c. ionescu, some integral inequalities for product of harmonic log-convex functions, u.p.b.sci. bull., series a,78(4)(2016),11-19. [14] m. a. noor, k. i. noor, s. iftikhar and c. ionescu ,hermite-hadmard inequalities for co-ordinated harmonic comvex functions, u.p.b.sci. bull., series a,79(1)(2017),24-34. [15] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [16] t.-y. zhang, ai-p. ji, f. qi, integral inequalities of hermite-hadamard type for harmonically quasi-convex functions, proc. jangjeon. math. soc., 16(3)(2013), 399-407. 2department of mathematics, comsats institute of information technology, islamabad, pakistan. ∗corresponding author: noormaslam@gmail.com 1. introduction 2. preliminaries 3. main results acknowledgements references int. j. anal. appl. (2023), 21:72 the security assignment problem and its solution paul ryan a. longhas∗, alsafat m. abdul, edcon b. baccay department of mathematics and statistics, college of science, polytechnic university of the philippines, manila, 1008, philippines ∗corresponding author: pralonghas@pup.edu.ph abstract. in this paper we derived a new method for finding the optimal solution to security assignment problems using projection onto a convex set. this study will help communities find the optimal number of assigned and reserved personnels in designating security officers to an area. this study is applicable as well to cctv assignment problems. the main goal of this study is to give a new method that can be applied in solving security assignment problems. we used some of the known properties of the convex optimization in proving the properties of the optimal solution, such as the concept of proximity operator, projection onto the convex set, and primal and dual problem. in addition to that, we used some basic knowledge in graph theory to answer our real-life application of this study. the main results of this paper showed that we found instances when the optimal solution to a security problem exists and when the solutions exist, we can determine the answer to the problem explicitly. also, we proved that there always exists a pseudo-solution in security assignment problems, and if the solution exists, then the pseudo-solution will coincide with it. the most important aspect of this paper is the introduction of the application of convex optimization in the security problem. 1. introduction there are many optimizations problem in optimization theory such as assignment problems, linear optimization problems, and convex optimization problems that are applicable in a real-life situation. for instance, linear programming is applicable in assignment problems [1–3] diet problems [4–6], and business problems in minimization of cost [7]. received: may 9, 2023. 2020 mathematics subject classification. 46n10, 47n10, 46c99. key words and phrases. security problem; assignment problem; convex optimization; proximity operator; projection onto set. https://doi.org/10.28924/2291-8639-21-2023-72 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-72 2 int. j. anal. appl. (2023), 21:72 the assignment problem is a problem so that there are n workers and m jobs where m ≤ n. any worker can be assigned to perform any job, with some costs that may vary depending on the workersjob assignment. it is required to perform a job by assigning at most one worker to each job and at most one job to each worker, in such a way that the total cost of the assignment is minimized. the assignment problem is a special type of {0,1}-linear programming [8,9]. in this paper, we will propose a model and solution to security assignment problems, cctv assignment problems, and any related problem. in the security assignment problem, if we let g be the map of the city, e be the major road, v be the intersection of the major road, w(e) be the criminal rate of the certain major road e ∈ e and π(e) be the time needed to travel the road e, then the goal is to assign a percentage of security personnel τ(e) ∈ [a,b] ⊆ [0,1] in any major road e ∈ e such that the crime will minimize. furthermore, we want to maximize the reserved personnel that we will assigned to the strategic location of the city. the crime will minimize in the road e ∈ e if the crime rate w(e) and the percentage of assigned personnel τ(e) to that road has small distance. thus, the goal is to find τ : e → [a,b] such that the crime in g will minimize, that is, τ satisfy the following equation min γ:e→[a,b] ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2. (1.1) in addition, we want to have a reserved personnel that we will assign to the center of gπ =(v,e,π) so that the total percentage of reserved personnel is maximized, and it is at least 1− s but not more than 1− r where r,s ∈ [0,1] and r < s. the quantity r and s are the lower bound and upper bound for the percentage of assigned personnel, respectively. for instance, if we want to avoid the case that the total percentage of reserved is at least 20% and it is at most the total percentage of assigned, then we set r =0.5 and s =0.8. this assumption can be formulated as 0≤ r ≤ ∑ e∈e τ(e)≤ s ≤ 1 (1.2) and max γ:e→[a,b] ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e) (1.3) where ∑ e∈e τ(e) is the percentage of assigned personnel and 1− ∑ e∈e τ(e) is the percentage of reserved personnel. we will assigned the total reserved to the central point u ∈ v of the graph gπ, that is, the vertex u where the greatest distance dgπ(u,v) to other vertices v is minimal. if there are more than one central points in the graph gπ, then we will divide the number of reserved personnel to the number of central points. thus, we will have τ∗(v)= { 1− ∑ e∈e τ(e) |c(gπ)| if v ∈ c(gπ) 0 if v /∈ c(gπ). (1.4) int. j. anal. appl. (2023), 21:72 3 here, the set c(gπ) is the set of all central points of the graph gπ, that is, the set of all vertices u where the greatest distance dgπ(u,v) to other vertices v is minimal. let gw =(v,e,w) and gπ(v,e,π) be a weighted graphs with weight function w : e → [0,1] and π : e →r, respectively. set y (e, [a,b], [r,s])= { w : e → [a,b] | r ≤ ∑ e∈e w(e)≤ s } . summarizing the equation in (1.1) to (1.4), then we will obtain the problem: find τ : e → [a,b] satisying the four condition: . (1.5) 1. 0≤ r ≤ ∑ e∈e τ(e)≤ s ≤ 1 2. max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e) 3. min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2 4. τ∗(v)= { 1− ∑ e∈e τ(e) |c(gπ)| if v ∈ c(gπ) 0 if v /∈ c(gπ) where r,s,a,b ∈ [0,1] such that a < b and r < s. the following definition is taken from [14]. definition 1.1. let g = (v,e,w) be a weighted graph where v is the vertex set and e is the edge set of g. then, the center of the graph g, denoted by c(g), is the set of all vertices u ∈ v where the greatest distance d(u,v) to other vertices v is minimal. the main objective of this study is to address the problem in definition 1.2. definition 1.2. let gw = (v,e,w) and gπ = (v,e,π) be a weighted graph with weight function w : e → [0,1] and π : e →r, respectively. let r,s,a,b ∈ [0,1] such that r < s and a < b. set y (e, [a,b], [r,s])= {w : e → [a,b] | r ≤ ∑ e∈e w(e)≤ s}. the (r,s,a,b)-security assignment problem of g is a problem of finding (τ,τ∗) where τ : e → [a,b] and τ∗ : v → [0,1] such that 1. r ≤ ∑ e∈e τ(e)≤ s ≤ 1 2. max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e) 3. min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2 4. τ∗(v)= { 1− ∑ e∈e τ(e) |c(gπ)| if v ∈ c(gπ) 0 if v /∈ c(gπ) 4 int. j. anal. appl. (2023), 21:72 where c(gπ) is the center of the graph gπ. the solution to the security assignment problem is the ordered pair (τ,τ∗). the paper is organized as follows. in section 2, we proved a necessary and sufficient condition when the security assignment problem has a solution. furthermore, we give an explicit formula for the solution to the security assignment problem in terms of projection onto the convex set if it has a solution. in section 3, we record some important properties of the pseudo-solution or solution to the security assignment problem. finally, in section 4 we discussed the real-life application of security assignment problems such as security, cctv assignment problems, and military assignment problems. in this study, || · || and 〈·, ·〉 are the standard norm and standard inner product in rn, respectively. 2. existence of solution to the security assignment problem first, we recall some terminology in convex analysis. the function f : rn → (−∞,+∞] is convex if and only if f (αx +(1−α)y)≤ αf (x)+(1−α)f (y) for all x,y ∈rn and for all α ∈ [0,1]. we say that f is proper if and only if f 6≡+∞. in this paper, the function f :rn → (−∞,+∞] is always convex, continuous and proper function. the following definition taken from [10,11] is vital in this paper. definition 2.1. let f : rn → (−∞,+∞] be a convex, continuous and proper function and u ∈ rn. the proximity of u in f is the unique point proxf (u)∈rn such that proxf (u)= arg min x∈rn ( 1 2 ||x −u||2 + f (x) ) . (2.1) in particular, if c is closed and convex set and f = ιc where ιc(x)= 0 if x ∈ c and +∞, otherwise, then proxf (u)= pc(u) where x0 =pc(u) is a unique point in c such that inf x∈c ||x −u||= ||x0 −u||. it is well-known that if g = f + α 2 || · −u||2 + 〈·,v〉 + β where α ≥ 0 and β ∈ r, then proxg(x) =prox(α+1)−1f ( (α+1)−1(x +(αu − v)) ) (see proposition 24.8 of [10]). consequently, if α =0, f = ιc, then proxg(x)=pc(x −v) where c is closed and convex set. the following lemma solves a minimization problem that can be apply in the security assignment problem. lemma 2.1. let u1,u2, . . . ,un ∈r and let r,s,a,b ∈ [0,1] such that r < s and a < b. consider the problem: min x1,x2,...,xn∈r ( n∑ i=1 (ui −xi)2 + n∑ i=1 xi ) (2.2) int. j. anal. appl. (2023), 21:72 5 subject to 1. a ≤ xi ≤ b 2. r ≤ n∑ i=1 xi ≤ s if c = {(x1,x2, . . . ,xn) : a ≤ xi ≤ b and 0 ≤ r ≤ n∑ i=1 xi ≤ s ≤ 1} is nonempty, then the problem above has exactly one solution. furthermore, pc(u − 12v) = (x ∗ 1,x ∗ 2, . . . ,x ∗ n) is the solution to the problem in (2.2). proof. let x =(x1,x2, . . . ,xn),u ∈ (u1,u2, . . . ,un)and v =(1,1,1, . . . ,1). let x∗ =(x∗1,x ∗ 2, . . . ,x ∗ n) be the solution to the problem in (2.2). set c = { (x1,x2, . . . ,xn) : a ≤ xi ≤ b and 0≤ r ≤ n∑ i=1 xi ≤ s ≤ 1 } . (2.3) then, the problem in (2.2) can be written as min x∈rn ( 1 2 ||x −u||2 + 1 2 〈x,v〉+ ιc(x) ) . (2.4) since c = [a,b]n ∩{(x1,x2, . . . ,xn) : 0 ≤ r ≤ n∑ i=1 xi ≤ s ≤ 1} is the intersection of two closed and convex sets, then c is closed and convex. therefore, the definition 2.1 guaranty that the problem in (2.2) has a unique solution proxg(u)= arg min x∈rn ( 1 2 ||x −u||2 + 〈x, 1 2 v〉+ ιc(x) ) (2.5) where g(x) = 1 2 〈x,v〉+ ιc(x). furthermore, applying proposition 24.8 of [10] by setting α = 0 and f = ιc, then we have proxg(u)=proxιc ( u − 1 2 v ) =pc ( u − 1 2 v ) as desired. � the following theorem gives an explicit form of the solution to the security assignment problem in terms of projection onto a convex set if it is existing. theorem 2.1. the (r,s,a,b)-security assignment problem of g has at most one solution (τ,τ∗). furthermore, if it has a solution (τ,τ∗), then τ(ei) = 〈pd(u − 12v),ui〉 where ui = (0,0,0, . . . ,0,1,0, . . . ,0) is the vector where the ith coordinate is 1, and 0 elsewhere where i ∈{1,2, . . . , |e|} and d = { (x1,x2, ...,x|e|)∈r |e| : 0≤ xi ≤ 1 and r ≤ |e|∑ i=1 xi ≤ s } . (2.6) 6 int. j. anal. appl. (2023), 21:72 proof. if the (r,s,a,b)-security assignment problem of g has a solution (τ,τ∗) and (γ,γ∗), then( τ(e1),τ(e2), ...,τ(e|e|) ) and ( γ(e1),γ(e2), ...,γ(e|e|) ) are the solution to the problem min x1,x2,...,x|e|∈r ( |e|∑ i=1 (w(ei)−xi)2 + |e|∑ i=1 xi ) (2.7) subject to 1. a ≤ xi ≤ b 2. r ≤ ∑|e| i=1 xi ≤ s applying lemma 2.1, then ( τ(e1),τ(e2), ...,τ(e|e|) ) = ( γ(e1),γ(e2), ...,γ(e|e|) ) , and therefore, τ = γ and τ∗ = γ∗. furthermore, if the solution (τ,τ∗) to the security assignment problem exists, then lemma 2.1 implies that ( τ(e1),τ(e2), ...,τ(e|e|) ) = pd ( u−1 2 v ) where u = ( w(e1),w(e2), ...,w(e|e|) ) and v = (1,1,1, ...,1). thus, for all i = 1,2, ..., |e|, we obtain τ(ei) = 〈pd ( u − 1 2 v ) ,ui〉, as desired. � lemma 2.1 gives the following definition. definition 2.2. suppose d = { (x1,x2, ...,x|v |)∈r|e| : 0≤ xi ≤ 1 and 0≤ r ≤ ∑|e| i=1 xi ≤ s ≤ 1 } 6= φ. the pseudo-solution of the (r,s,a,b)-security assignment problem of g is the unique ordered pair (θ,θ∗) where θ : e → [a,b] and θ∗ : v → [0,1] such that 1. 0≤ r ≤ ∑ e∈e θ(e)≤ s ≤ 1. 2. min γ∈y (e,[a,b],[r,s]) (∑ e∈e (w(e)−γ(e))2 + ∑ e∈e γ(e) ) = ∑ e∈e (w(e)−θ(e))2 + ∑ e∈e θ(e). 3. θ∗(v)= { 1− ∑ e∈e θ(e) |c(gπ)| if v ∈ c(gπ) 0 if v /∈ c(gπ) the vector solution to the (r,s,a,b)-security assignment problem is the vector (θ(e1),θ(e2), . . . ,θ(e|e|)). remark 2.1. if the (r,s,a,b)-security assignment problem of g has a solution (τ,τ∗), then it is unique, and its solution is its pseudo-solution (θ,θ∗). corollary 2.1. let (θ,θ∗) be the pseudo-solution of (r,s,a,b)-security assignment problem of g. then, the (r,s,a,b)-security assignment problem of g has a solution if and only if 1. max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e θ(e). 2. min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−θ(e))2. example 2.1: given the crime rate and distance of the road ei where i =1,2,3, ...,8 in the graph g below (see figure 2.1), determine the percentage of the number of assigned personnel for each road so that the crime will minimize. furthermore, we want that the total percentage of assigned personnel is int. j. anal. appl. (2023), 21:72 7 at most 90% and at least 85%, while the percentage of assigned personnel for each road is at most 25% and at least 5%. 5 km 15% e1 2 km 10% e2 3 km 10% e3 3 km 10% e4 7 km 25% e5 6 km 7% e6 3 km 15% e7 2 km 8% e8 figure 2.1: the graph g solution: we want to solve the (0.85,0.9,0.05,0.25)− security assignment problem, that is, we want to find τ : e → [0.05,0.25] satisfying the four condition: 1. 0.85≤ ∑ e∈e τ(e)≤ 0.90≤ 1 2. max γ∈y (e,[0.05,0.25],[0.85,0.9]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e) 3. min γ∈y (e,[0.05,0.25],[0.85,0.9]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2 4. τ∗(v)= { 1− ∑ e∈e τ(e) |c(gπ)| if v ∈ c(gπ) 0 if v /∈ c(gπ) where π(ei) is the length of the road ei (in km). let xi = τ(ei). then, from lemma 2.1 and theorem 2.1, (x1,x2,x3...,x8) is the solution to the minimization problem: min x1,x2,...,x8∈r ( 8∑ i=1 (w(ei)−xi)2 + 8∑ i=1 xi ) subject to 1. 0.05≤ xi ≤ 0.25 2. 0.85≤ ∑8 i=1xi ≤ 0.9. applying lagrange multiplier using matlab, then we obtain the solution (x1,x2,x3,x4,x5,x6,x7,x8) ≈ (0.135, 0.08, 0.08, 0.08, 0.23, 0.05, 0.135, 0.06). thus, the total percentage of assigned security personnel is x1+x2+ ...+x8 =85% and the total percentage of reserved personnel is 15%. the following table discussed the optimal percentage of assigned security personnel for each road. 8 int. j. anal. appl. (2023), 21:72 table 2.1: percentage of assigned number of personnel per road road crime rate percentage of assigned number of personnel e1 15% 13.5% e2 10% 8% e3 10% 8% e4 10% 8% e5 25% 23% e6 7% 5% e7 15% 13.5% e8 8% 6% we assigned 15% of the total security personnel to the center of the graph gπ where the center of gπ is the intersection of the road e5,e6 and e7. 3. properties of the pseudo-solution in this section, we record some important properties of the pseudo-solution that can be help in solving the security assignment problem. in this section, u = (w(e1),w(e2), . . . ,w(e|e|)),v = (1,1,1, . . . ,1) and d = { (x1,x2, . . . ,x|e|)∈r |e| : 0≤ xi ≤ 1 and0≤ r ≤ |e|∑ i=1 xi ≤ s ≤ 1 } . (3.1) theorem 3.1. suppose d 6= φ. the pseudo-solution (θ,θ∗) to the (r,s,a,b)-security assignment problem is a solution to the minimization problem: find θ : e → [a,b] such that min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e)+0.5)2 = ∑ e∈e (w(e)−θ(e)+0.5)2 (3.2) subject to 0≤ r ≤ ∑ e∈e θ(e)≤ s ≤ 1. (3.3) thus, if (x1,x2, . . . ,x|e|) is the solution to the minimization problem, then we have min y1,y2,. . . ,y|e|∈r |e|∑ i=1 (w(ei)−yi +0.5)2 = |e|∑ i=1 (w(ei)−xi +0.5)2 (3.4) and 1. a ≤ xi ≤ b 2. 0≤ r ≤ ∑|e| i=1 xi ≤ s ≤ 1. proof. by lemma 2.1, x0 is a solution to the problem min x∈r|e| ( 1 2 ||x −u||2 + 〈x, 1 2 (1,1,1, ...,1)+ ιd(x) ) (3.5) if and only if pd(u − 12v)= x0 if and only if int. j. anal. appl. (2023), 21:72 9 inf x∈d 1 2 ∣∣∣∣ ∣∣∣∣x − (u − 12v) ∣∣∣∣ ∣∣∣∣2 = 12 ∣∣∣∣ ∣∣∣∣x0 − (u − 12v) ∣∣∣∣ ∣∣∣∣2. (3.6) thus, x0 =(θ(e1),θ(e2), . . . ,θ(e|e|)) if and only if min γ∈y (e,[a,b],[r,s]) ∑ e∈e ( w(e)−γ(e)+0.5 )2 = ∑ e∈e ( w(e)−θ(e)+0.5 )2 (3.7) subject to 0≤ r ≤ ∑ e∈e θ(e)≤ s ≤ 1. (3.8) as desired. � the projection theorem state that x0 = pc(x) if and only if x0 ∈ c and 〈p − y,p − x〉 ≤ 0 for all y ∈ c (see theorem 3.16 in [10]). thus, the following corollary directly holds. corollary 3.1. suppose d 6= φ. the point x0 ∈r|e| is the solution to min x∈r|e| { 1 2 ||x −u||2 + 〈x, 1 2 v〉+ ιd(x) } . (3.9) if and only if x0 ∈ d and for all y ∈ d, we have 〈x0 −y,x0 −u + 1 2 v〉≤ 0. (3.10) proof. the point x0 is a solution to min x∈r|e| { 1 2 ||x −u||2 + 〈x, 1 2 v〉+ ιd(x) } (3.11) if and only if pd ( u − 1 2 v ) = x0. thus, the projection theorem implies that pd ( u − 1 2 v ) = x0 if and only if x0 ∈ d and for all y ∈ d, we have 〈x0 −u + 1 2 v,x0 −y〉≤ 0 (3.12) as desired. � the following definition taken from [10] state the notion of fenchel conjugate and gateauxdifferentiable of a function which is important in the next result. definition 3.1. let f :rn → (−∞,+∞] be a convex, continuous and proper function. the fenchel conjugate f ∗ of f is a function defined by f ∗ = sup x∈rn (〈x, ·〉− f (x)). definition 3.2. let f :rn → (−∞,+∞] be a convex, continuous and proper function. the function f is said to be gateaux differentiable at x0 if and only if the limit below exists for all u ∈rn. duf (x0)= lim α→0+ 1 α (f (x0 +αu)+ f (x0)). the function d :rn →r : u 7→ duf (x0) is called the gateaux derivative of f . the gateaux gradient of f is the unique vector 5f (x) satisfying the equation duf (x)= 〈u,5f (x)〉. 10 int. j. anal. appl. (2023), 21:72 the next results give a characterization of the pseudo-solution in terms of the solution to the dual problem of the primal problem min x∈r|e| { 1 2 ||x −u||2 + 〈x, 1 2 v〉+ ιd(x) } . (3.13) theorem 3.2. suppose d 6= φ. let x0 = (τ(e1),τ(e2), . . . ,τ(e|e|)) be the vector solution to the (r,s,a,b)-security assignment problem, let u = (w(e1),w(e2), . . . ,w(e|e|)) and v = (1,1,1, . . . ,1). then, the problem min x∈r|e| { 1 2 ||y||2 −〈y,u〉+sup x∈d 〈x,y − 1 2 v〉 } (3.14) has a unique solution u −x0. in addition, proxh(u)= x0 where h(y)= sup x∈d 〈x,y − 1 2 v〉. proof. consider the primal problem min x∈r|e| { 1 2 ||x −u||2 + 〈x, 1 2 v〉+ ιd(x) } . (3.15) then, x0 is the solution to the primal problem above. let f (x)= 1 2 ||x−u||2 and g(x)= 〈x, 1 2 v〉+ιd(x). then, f ∗(a)= 1 2 ||a||2 + 〈a,u〉 and g∗(a)= sup x∈r|e| {〈x,a− 1 2 v〉− ιd(x)}= sup x∈d 〈x,a− 1 2 v〉. (3.16) thus, the dual problem of the primal problem in (3.15) is min x∈r|e| { 1 2 ||y||2 + 〈y,u〉+sup x∈d 〈x,y − 1 2 v〉 } . (3.17) since y is the solution to the dual problem, x0 is the solution to the primal problem and f ∗ is gateaux differentiable everywhere, then by proposition 19.4 of [10], x0 =5f ∗(−y), and thus, x0 =5f ∗(−y)= −y +u, as desired. � 4. some real-life application in this section, we present some real-life application of security assignment problem. a. security assignment problem for a road let g represent the whole map of a city, e be the set of all roads (or major roads) of the city, v be the set of the intersection of roads. let w(e) be the criminal rate (or population rate, number of houses, etc.) of the road e ∈ e and π(e) be the time needed to travel the road e (or length of the road, the volume of a vehicle in the road, etc.). then, the (r,s,a,b)-security assignment problem is a problem of finding the percentage of security personnel τ(e) that will assign in road e and finding the percentage of number of reserved security personnel τ∗(e) that will assign to the center of the city c(gπ) so that the total percentage of reserved security personnel is at most 1− r and at least 1−s where r,s ∈ [0,1] and r < s. we want to find τ : e → [a,b] where a,b ∈ [0,1] and a < b such int. j. anal. appl. (2023), 21:72 11 that if we assign 100τ(e) percent of security personnel to road e, for all e ∈ e, then the crime will minimize, that is, we want to solve the problem of finding τ : e → [a,b] such that min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2. (4.1) in addition to that, we want to maximize the total percentage of reserved security personnel, that is, we have max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e). (4.2) since the total percentage of reserved security personnel is at-most 1− r and at least 1− s where r,s ∈ [0,1] and r < s, then we have 0≤ r ≤ ∑ e∈e τ(e)≤ s ≤ 1. (4.3) summarizing equation (4.1) to (4.3), then we recover the problem in definition 1.2. note that the security assignment problem can be applied in assigning security personnel for the checkpoint or any related problem. if the (r,s,a,b)-security assignment problem has no solution, then the strategic solution is the pseudo-solution of the (r,s,a,b)-security assignment problem. b. cctv assignment problem for a road let g represent the whole map of a certain town, e be the set of all roads (or major roads) of the town, v be the set of the intersection of roads. let w(e) be the criminal rate (or population rate, number of houses, etc.) of the road e ∈ e and π(e) be the length the road e (or population, number of establishments, the volume of vehicle in the road, etc.). then, the (r,s,a,b)-security assignment problem for cctv installation is a problem of finding the percentage of number of cctv τ(e)∈ [a,b] that will install in road e and finding the percentage of number of cctv τ∗(e) that will install to the center of the city c(gπ) so that the total percentage of cctv that will assign to the center of the city c(g) is at-most 1− r and at least 1− s where r,s ∈ [0,1] and r < s. we want to find τ : e → [a,b] where a,b ∈ [0,1] and a < b such that if we install 100τ(e) percent of cctv to road e, for all e ∈ e, then this is the best strategic installation, that is, we want to solve the problem of finding τ : e → [a,b] such that min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−τ(e))2. (4.4) in addition to that, we want to maximize the total percentage of cctv that will assign to the center of the city c(gπ), that is, we have max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e τ(e). (4.5) 12 int. j. anal. appl. (2023), 21:72 since the total percentage of cctv that will assign to the center of the city c(g) is at most 1− r and at least 1− s where r,s ∈ [0,1] and r < s, then we have 0≤ r ≤ ∑ e∈e τ(e)≤ s ≤ 1. (4.6) summarizing the equation (4.4) to (4.6), then we obtain the problem in definition 1.2. if the (r,s,a,b)-security assignment problem has no solution, then the strategic solution is the pseudosolution of (r,s,a,b)-security assignment problem. c. security assignment problem for a planar map before we discuss the application of this study in planar graph, we recall first the definition of line graph l(g) of a graph g and planar graph. definition 4.1. let g be a graph. the line graph l(g) of g is a graph such that each vertex of l(g) represents an edge of g and two vertices of l(g) are adjacent if and only if their corresponding edges share a common endpoint in g. definition 4.2. let h be a map with region u1,u2, ...,un. the planar graph of h is the graph g where the vertex of g represent the region of h and two vertices in g are adjacent if and only if their corresponding region are adjacent. consider the map with regions u1,u2, . . . ,un and let g = (v,e) be a planar graph of the map where v = {u1,u2, . . . ,un}. let µ(ui) be the criminal rate (or population rate, number of houses, etc.) of the region ui. let l(g) = (l(v ),l(e)) be the line graph of g and set w : l(e) → [0,1] such that w(ui,j,ui,k)= µ(ui)( degui 2 ) where ( degui 2 ) = degui! (degui!−2)2! = degui(degui −1) 2 . (4.7) set π = τ. the (r,s,a,b)-security assignment problem is a problem of finding the percentage of number of security personnel τ(e) that will assign in the edge e and finding the percentage of number of reserved security personnel τ∗(e) that will assign to the center of the graph l(g) so that the total percentage of reserved security personnel is at most 1− r and at least 1− s where r,s ∈ [0,1] and r < s. we want to find τ : e → [a,b] such that if we assign 100τ(e) percent of security personnel to edge e, for all e ∈ l(e), then the crime will minimize, that is, we want to solve the problem of finding τ : e → [0,1] such that min γ∈y (l(e),[a,b],[r,s]) ∑ e∈l(e) (w(e)−γ(e))2 = ∑ e∈l(e) (w(e)−τ(e))2. (4.8) in addition to that, we want to maximize the total percentage of reserved security personnel, that is, we have int. j. anal. appl. (2023), 21:72 13 max γ∈y (l(e),[a,b],[r,s]) ( 1− ∑ e∈l(e) γ(e) ) =1− ∑ e∈l(e) τ(e). (4.9) since the total percentage of reserved security personnel is at most 1− r and at least 1− s where r,s ∈ [0,1] and r < s, then we have 0≤ r ≤ ∑ e∈l(e) τ(e)≤ s ≤ 1. (4.10) the percentage of assigned security personnel in region ui is∑ j ∈ 1,2, ..., |v | ui,jui,k ∈ l(e) τ(ui,jui,k). (4.11) if the (r,s,a,b)-security assignment problem has no solution, then the strategic solution is the pseudo-solution of (r,s,a,b)-security assignment problem. example 4.1: consider the map below, which shows the crime rate of each region. we need to determine the best strategic assignment of security personnel from region 1 to region 5 in order to minimize crime. however, we must ensure that we assign a maximum of 30% of security personnel for each e ∈ l(g). additionally, we want to ensure that the reserve security personnel make up at least 5%. 40% 15% 15% 10% 20% figure 4.1 the planar region constructing the planar graph g of the map, then we obtain 40% 15% 15% 10% 20% figure 4.2: the planar graph and planar region by constructing the line graph of the planar graph g, then we obtain l(g) 14 int. j. anal. appl. (2023), 21:72 u1 u2 u4 u3 u5 u1,2 u1,4 u1,5 u4,5 u2,4 u2,3 u3,4 figure 4.3: the line graph and planar graph set w : l(e)→ [0,1] such that w(ui,jui,k)= µ(ui)( degui 2 ) where µ(ui) is the crime rate of region ui and ( degui 2 ) = degui! (degui!−2)2! = degui(degui −1) 2 . (4.12) thus, we obtain w(u1,2u1,4)= w(u1,2u1,5)= w(u1,4u1,5)= 20% 3 = 1 15 w(u2,4u1,2)= w(u2,4u2,3)= w(u1,2u2,3)= 15% 3 = 1 15 w(u2,3u3,4)= 10% 1 = 1 10 w(u1,4u2,4)= w(u2,4u3,4)= w(u3,4u4,5)= w(u1,4u4,5) = w(u2,4u4,5)= w(u1,4u3,4)= 40% 6 = 1 15 w(u1,5u4,5)= 15% 1 = 3 20 1 15 1 20 1 20 1 15 1 15 1 15 1 15 1 15 3 20 1 15 1 20 1 15 1 15 1 10 u1,2 u1,4 u1,5 u4,5 u2,4 u2,3 u3,4 figure 4.4: weighted line graph the problem wants to solve the (r,s,a,b)-security assignment problem of l(g) where r = 0, s =0.95, a =0 and b =0.3. the (0,0.95,0,0.3)-security assignment problem of l(g) is a problem of finding τ : l(e)→ [0,0.3] such that 1. min γ:l(e)→[0,0.3] ∑ e∈l(e) ( w(e)−γ(e) )2 = ∑ e∈l(e) ( w(e)−τ(e) )2 . int. j. anal. appl. (2023), 21:72 15 2. max γ:l(e)→[0,0.3] ( 1− ∑ e∈l(e) γ(e) ) =1− ∑ e∈l(e) τ(e). 3. 0≤ r ≤ ∑ e∈l(e)τ(e)≤ s ≤ 1. from theorem 3.1, we can obtain the vector solution of the (r,s,a,b)-security assignment problem of l(g) by finding the solution (x∗1,x ∗ 2, . . . ,x ∗ 14) of the problem min x1,x2,. . . ,x14∈r ( 14∑ i=1 (xi −w(ei)−0.5 )2 (4.13) subject to 1. 0≤ xi ≤ 0.3. 2. 0≤ ∑|l(e)| i=1 xi ≤ 0.95. thus, we have min x1,x2,. . . ,x14∈r (( x1 − 17 30 )2 + ( x2 − 17 30 )2 + ( x3 − 17 30 )2 + ( x4 − 11 20 )2 + ( x5 − 11 20 )2 + ( x6 − 11 20 )2 + ( x7 − 3 5 )2 + ( x8 − 17 30 )2 + ( x9 − 17 30 )2 + ( x10 − 17 30 )2 + ( x11 − 17 30 )2 + ( x12 − 17 30 )2 + ( x13 − 17 30 )2 + ( x14 − 13 20 )2) subject to 1. 0≤ xi ≤ 0.3. 2. 0≤ ∑|l(e)| i=1 xi ≤ 0.95. applying lagrange multiplier, then we obtain the approximate solution (0.0631, 0.0631, 0.0631, 0.0464, 0.0464, 0.0464, 0.0964, 0.0631, 0.0631, 0.0631, 0.0631, 0.0631, 0.0631, 0.1464). thus, we have the following: percentage of assigned number of personnel per region table 5.1: region crime rate percentage of assigned number of personnel 1 20% 0.0631+0.0631+0.0631=0.1893 (18.93%) 2 15% 0.0464+0.0464,+0.0464=0.13.92 (13.92%) 3 10% 0.0964 (9.64%) 4 40% 0.0631+0.0631+0.0631+0.0631+0.0631+0.0631=0.3786(37.86%) 5 15% 0.1464 (14.64%) therefore, by considering the graph and weight of the region, the optimal percentage of total assigned security personnel is 95% and the percentage of reserved security personnel is 5%. 5. conclusions this study proved that the (r,s,a,b)-security assignment problem of g has at most one solution (τ,τ∗). furthermore, if it has a solution (τ,τ∗), then τ(ei) = 〈pd(u − 12v),ui〉 where 16 int. j. anal. appl. (2023), 21:72 ui = (0,0,0, . . . ,0,1,0, . . . ,0) is the vector where the ith coordinate is 1, and 0 elsewhere where i ∈{1,2, . . . , |e|} and d = { (x1,x2, ...,x|e|)∈r |e| : 0≤ xi ≤ 1 and0 ≤ r ≤ |e|∑ i=1 xi ≤ s ≤ 1 } in addition, we proved that if (θ,θ∗) is the pseudo-solution of (r,s,a,b)-security assignment problem of g, then the (r,s,a,b)-security assignment problem of g has a solution if and only if 1. max γ∈y (e,[a,b],[r,s]) ( 1− ∑ e∈e γ(e) ) =1− ∑ e∈e θ(e). 2. min γ∈y (e,[a,b],[r,s]) ∑ e∈e (w(e)−γ(e))2 = ∑ e∈e (w(e)−θ(e))2. the model is applicable in security assignment problem and cctv assignment problem. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] z. ghazali, m.a.a. majid, m. shazwani, optimal solution of transportation problem using linear programming: a case of a malaysian trading company, j. appl. sci. 12 (2012), 2430-2435. https://doi.org/10.3923/jas. 2012.2430.2435. [2] a.j. mehta, a.k. rifai, goal programming application to assignment problem in marketing, j. acad. market. sci. 7 (1979), 108-116. https://doi.org/10.1007/bf02721918. [3] j.a. breslaw, a linear programming solution to the faculty assignment problem, socio-econ. plan. sci. 10 (1976), 227-230. https://doi.org/10.1016/0038-0121(76)90008-2. [4] c. van dooren, a review of the use of linear programming to optimize diets, nutritiously, economically and environmentally, front. nutr. 5 (2018), 48. https://doi.org/10.3389/fnut.2018.00048. [5] j. foytik, very low-cost nutritious diet plans designed by linear programming, j. nutr. educ. 13 (1981), 63-66. https://doi.org/10.1016/s0022-3182(81)80098-2. [6] a. briend, n. darmon, determining limiting nutrients by linear programming: a new approach to predict insufficient intakes from complementary foods, pediatrics. 106 (2000), 1288-1289. https://doi.org/10.1542/ peds.106.s4.1288. [7] a. briend, e. ferguson, n. darmon, local food price analysis by linear programming: a new approach to assess the economic value of fortified food supplements, food nutr. bull. 22 (2001), 184-189. https://doi.org/10. 1177/156482650102200210. [8] d.h. stimson, r.p. thompson, linear programming, busing and educational administration, socio-econ. plan. sci. 8 (1974), 195-206. https://doi.org/10.1016/0038-0121(74)90043-3. [9] j. reeb, s. leavengood, using duality and sensitivity analysis to interpret linear programming solutions, oregon state university, 2000. http://hdl.handle.net/1957/20129. [10] h.h. bauschke, p.l. combettes, convex analysis and monotone operator theory in hilbert spaces, springer, cham, 2017. https://doi.org/10.1007/978-3-319-48311-5. [11] y.l. yu, the proximity operators, (2014). https://www.cs.cmu.edu/~suvrit/teach/yaoliang_proximity.pdf. [12] j. dattorro, convex optimization in euclidean distance geometry, (2005). https://ccrma.stanford.edu/ /~dattorro/0976401304.pdf. [13] s.p. boyd, l. vandenberghe, convex optimization, cambridge university press, cambridge, 2004. https://doi.org/10.3923/jas.2012.2430.2435 https://doi.org/10.3923/jas.2012.2430.2435 https://doi.org/10.1007/bf02721918 https://doi.org/10.1016/0038-0121(76)90008-2 https://doi.org/10.3389/fnut.2018.00048 https://doi.org/10.1016/s0022-3182(81)80098-2 https://doi.org/10.1542/peds.106.s4.1288 https://doi.org/10.1542/peds.106.s4.1288 https://doi.org/10.1177/156482650102200210 https://doi.org/10.1177/156482650102200210 https://doi.org/10.1016/0038-0121(74)90043-3 http://hdl.handle.net/1957/20129 https://doi.org/10.1007/978-3-319-48311-5 https://www.cs.cmu.edu/~suvrit/teach/yaoliang_proximity.pdf https://ccrma.stanford.edu//~dattorro/0976401304.pdf https://ccrma.stanford.edu//~dattorro/0976401304.pdf int. j. anal. appl. (2023), 21:72 17 [14] j.a. bondy, u.s.r. murty, graph theory with applications, elsevier, new york, (1976). [15] b. nag, business applications of operations research, business expert press, new york, 2014. 1. introduction 2. existence of solution to the security assignment problem 3. properties of the pseudo-solution 4. some real-life application 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 157-167 http://www.etamaths.com existence and approximate solutions for nonlinear hybrid fractional integrodifferential equations b.c. dhage1,∗, g.t. khurpe1, a.y. shete2 and j.n. salunke2 abstract. in this paper we prove existence and approximation of the solutions for initial value problems of nonlinear hybrid fractional differential equations with maxima and with a linear as well as quadratic perturbation of second type. the main results rely on dhage iteration method embodied in the recent hybrid fixed point theorem of dhage (2014) in a partially ordered normed linear space. the approximation of the solutions of the considered nonlinear fractional differential equations are obtained under weaker mixed partial continuity and lipschitz conditions. our hypotheses and the main results are also illustrated by a numerical example. 1. introduction in this paper we prove existence and approximations of the solutions for initial value problems of nonlinear hybrid fractional differential equations. consider the following initial value problem of fractional differential equations, (1.1)   cdα ( x(t) − iβh(t,x(t)) f(t,x(t)) ) = g ( t,x(t), ∫ t 0 k(s,x(s)) ds ) , t ∈ j := [0,t], x(0) = x0 ∈ r+, where cdα denotes the caputo fractional derivative of order α, 0 < α < 1, iβ is the riemann-liouville fractional integral of order β, and f : j × r → r \{0}, h,k : j × r → r and g : j × r × r → r are given continuous functions. by a solution of the problem (1.1) we mean a function x ∈ c1(j,r) if (i) the function t 7→ x(t) − iβh(t,x(t)) f(t,x(t)) is caputo differentiable, and (ii) x satisfies the relations in ((1.1) on j, where c1(j,r) is the space of continuously differentiable real-valued functions defined on j. fractional differential equations have aroused great interest, which is caused by both the intensive development of the theory of fractional calculus and the applications to rheology, physics, mechanics and chemistry engineering [16, 17]. for some recent development on the topic see [1] and the references cited therein. for some recent results on hybrid fractional differential equations we refer to [1], [2], [14], [18], [19] and the references cited therein. the origin of the problem (1.1) lies in the initial value problems of first order quadratic differential equations with ordinary derivative wherein only existence of the solutions is proved using classical hybrid fixed point theorem of dhage [3]. the problem (1.1) considered here is general in the sense that it includes the following three well-known classes of initial value problems of fractional differential equations. 2010 mathematics subject classification. 34a08, 47h07, 47h10. key words and phrases. fractional differential equation; fixed point theorem; dhage iteration method; existence and uniqueness theorems. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 157 158 dhage, khurpe, shete and salunke case i: let f(t,x) = 1 and h(t,x) = 0 for all t ∈ j and x ∈ r. then the problem (1.1) reduces to standard initial value problem of fractional differential equation (1.2)   cdαx(t) = g ( t,x(t), ∫ t 0 k(s,x(s)) ds ) , t ∈ j := [0,t], xx(0) = x0 ∈ r. case ii: if h(t,x) = 0 for all t ∈ j and x ∈ r in (1.1), we obtain the following quadratic fractional differential equation, (1.3)   cdα ( x(t) f(t,x(t)) ) = g ( t,x(t), ∫ t 0 k(s,x(s)) ds ) , t ∈ j := [0,t], x(0) = x0 ∈ r. case iii: if f(t,x) = 1 for all t ∈ j and x ∈ r in (1.1), we obtain the following interesting fractional differential equation, (1.4)   cdα [ x(t) − iβh(t,x(t)) ] = g ( t,x(t), ∫ t 0 k(s,x(s)) ds ) , t ∈ j := [0,t], x(0) = x0 ∈ r. therefore, the main result of this paper also includes the existence as well as approximation results for the solutions of above mentioned initial value problems of fractional differential equations as special cases. again our approach here in this paper is different than that employed in the related paper of dhage [3]. in the present paper we prove the existence and approximations of the solutions of problem (1.1) under weaker partial compactness and partial lipschitz type conditions via dhage iteration method [7]. very recently, dhage iteration method has been applied in [7, 8, 9, 11, 12, 13] to nonlinear ordinary differential equations for proving the existence and algorithms of the solutions. we recall the basic definitions of fractional calculus [16, 17] which are useful in what follows. definition 1.1. the riemann-liouville fractional integral of order q with the lower limit zero for a function f is defined as iqf(t) = 1 γ(q) ∫ t 0 f(s) (t−s)1−q ds, t > 0, q > 0, provided the right hand-side is point-wise defined on [0,∞), where γ(·) is the gamma function, which is defined by γ(q) = ∫ ∞ 0 tq−1e−tdt. definition 1.2. the riemann-liouville fractional derivative of order q > 0, n− 1 < q < n, n ∈ n, is defined as d q 0+f(t) = 1 γ(n−q) ( d dt )n ∫ t 0 (t−s)n−q−1f(s)ds, where the function f(t) has absolutely continuous derivative up to order (n− 1). definition 1.3. the caputo derivative of order q for a function f ∈ cn(j,r) can be written as cdqf(t) = dq ( f(t) − n−1∑ k=0 tk k! f(k)(0) ) , t > 0, n− 1 < q < n. remark 1.4. if f ∈ cn(j,r), then cdqf(t) = 1 γ(n−q) ∫ t 0 f(n)(s) (t−s)q+1−n ds = in−qf(n)(t), t > 0, n− 1 < q < n. approximate solutions of fractional differential equations 159 lemma 1.5. for q > 0, the general solution of the fractional differential equation cdqx(t) = 0 is given by x(t) = c0 + c1t + . . . + cn−1t n−1, where ci ∈ r, i = 1, 2, . . . ,n− 1 (n = [q] + 1). remark 1.6. in view of lemma 1.5, it follows that (1.5) iq cdqx(t) = x(t) + c0 + c1t + . . . + cn−1t n−1, for some ci ∈ r, i = 1, 2, . . . ,n− 1 (n = [q] + 1). the rest of the paper will be organized as follows. in section 2 we give some preliminaries and key fixed point theorems that will be used in subsequent part of the paper. in section 3 we discuss the main existence and approximation result for initial value problems of fractional differential equations (1.1). an illustrative example is also discussed. 2. auxiliary results unless otherwise mentioned, throughout this paper we let e denote a partially ordered real normed linear space with the order relation � and the norm ‖ · ‖ in which addition and scalar multiplication by positive real numbers are preserved by �. a few details on such partially ordered normed linear spaces appear in dhage [5] and the references therein. two elements x and y in e are said to be comparable if either the relation x � y or y � x holds. a non-empty subset c of e is called a chain or totally ordered if all elements of c are comparable. we say that e is regular if for any nondecreasing (resp. nonincreasing) sequence {xn} in e such that xn → x∗ as n →∞, we have that xn � x∗ (resp. xn � x∗) for all n ∈ n. conditions guaranteeing the regularity of e may be found in heikkilä and lakshmikantham [15] and the references therein. we need the following definitions (see dhage [5, 6, 6] and the references therein) in what follows. definition 2.1. a mapping b : e → e is called isotone or nondecreasing if it preserves the order relation �, that is, if x � y implies bx �by for all x,y ∈ e. definition 2.2. a mapping b : e → e is called partially continuous at a point a ∈ e if for � > 0 there exists a δ > 0 such that ‖bx−ba‖ < � whenever x is comparable to a and ‖x−a‖ < δ. b called a partially continuous on e if it is partially continuous at every point of it. it is clear that if b is a partially continuous on e, then it is continuous on every chain c contained in e. definition 2.3. a non-empty subset s of the partially ordered banach space e is called partially bounded if every chain c in s is bounded. a nondecreasing mapping b : e → e is called partially bounded if b(c) is bounded for every chain c in e. b is called uniformly partially bounded if all chains b(c) in e are bounded by a unique constant. b is called bounded if b(e) is a bounded subset of e. definition 2.4. a non-empty subset s of the partially ordered banach space e is called partially compact if every chain c in s is compact. a nondecreasing mapping b : e → e is called partially compact if b(c) is a relatively compact subset of e for all totally ordered sets or chains c in e. b is called uniformly partially compact if b(c) is a uniformly partially bounded and partially compact on e. b is called partially totally bounded if for any totally ordered and bounded subset c of e, b(c) is a relatively compact subset of e. if b is partially continuous and partially totally bounded, then it is called partially completely continuous on e. definition 2.5. the order relation � and the metric d on a non-empty set e are said to be compatible if {xn}n∈n is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk}n∈n of {xn}n∈n converges to x ∗ implies that the whole sequence {xn}n∈n converges to x∗. similarly, given a partially ordered normed linear space (e,�,‖·‖), the order relation � and the norm ‖·‖ are said to be compatible if � and the metric d defined through the norm ‖·‖ are compatible. a subset s of e is called janhavi if the order relation � and the metric d or the norm ‖ · ‖ are compatible in it. in particular, if s = e, then e is called a janhavi metric or janhavi banach space. 160 dhage, khurpe, shete and salunke clearly, the set r of real numbers with usual order relation ≤ and the norm defined by the absolute value function | · | has this property. similarly, the finite dimensional euclidean space rn with usual componentwise order relation and the standard norm possesses the compatibility property. definition 2.6. an upper semi-continuous and nondecreasing function ψ : r+ → r+ is called a dfunction provided ψ(0) = 0. let (e,�,‖ · ‖) be a partially ordered normed linear space. a mapping t : e → e is called partially nonlinear d-lipschitz if there exists a d-function ψ : r+ → r+ such that (2.1) ‖t x−t y‖≤ ψ(‖x−y‖) for all comparable elements x,y ∈ e. if ψ(r) = k r, k > 0, then t is called a partially lipschitz with a lipschitz constant k. furthermore, if ψ(r) < r, r > 0, t is called a partially nonlinear d-contraction on e. let (e,�,‖ ·‖) be a partially ordered normed linear algebra. denote e+ = { x ∈ e | x � θ, where θ is the zero element of e } and (2.2) k = {e+ ⊂ e | uv ∈ e+ for all u,v ∈ e+}. the elements of the set k are called the positive vectors in e. then following lemma is immediate. lemma 2.7 (dhage [3]). if u1,u2,v1,v2 ∈k are such that u1 � v1 and u2 � v2, then u1u2 � v1v2. definition 2.8. an operator b : e → e is said to be positive if the range r(b) of b is such that r (b) ⊆k. the dhage iteration method is embodied in the following hybrid fixed point theorem proved in dhage [6] which are useful tools in what follows. a few other such hybrid fixed point theorems appear in dhage [5, 6]. theorem 2.9 (dhage [7]). let ( e,�,‖ · ‖ ) be a regular partially ordered complete normed linear algebra such that every compact chain c in e is janhavi. let a,b : e → k and c : e → e be three nondecreasing operators such that (a) a and c are partially bounded and partially nonlinear d-lipschitz with d-functions ψa and ψc respectively. (b) b is partially continuous and uniformly partially compact, (c) 0 < mψa(r) + ψc(r) < r, r > 0, where m = sup{‖b(c)‖ : c is a chain in e}, and (d) there exists an element x0 ∈ e such that x0 �ax0bx0 + cx0 or x0 �ax0 bx0 + cx0. then the operator equation axbx+cx = x has a solution x∗ in e and the sequence {xn} of successive iterations defined by xn+1 = axnbxn + cxn, n = 0, 1, . . . converges monotonically to x∗. remark 2.10. the compatibility of the order relation � and the norm ‖ · ‖ in every compact chain of e is held if every partially compact subset s of e possesses the compatibility property with respect to � and ‖ · ‖. this simple fact is used to prove the desired characterization of the mild solution of the problem (1.1) on j. 3. main existence result the equivalent integral form of the problem (1.1) is considered in the function space c(j,r) of continuous real-valued functions defined on j. we define a norm ‖ · ‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) approximate solutions of fractional differential equations 161 for all t ∈ j. clearly, c(j,r) is a banach space with respect to above supremum norm and also partially ordered w.r.t. the above partially order relation ≤. it is known that the partially ordered banach space c(j,r) is regular and a lattice so that every pair of elements of e has a lower and an upper bound in it. it is known that the partially ordered banach space c(j,r) has some nice properties w.r.t. the above order relation in it. the following lemma follows by an application of arzellá-ascoli theorem. lemma 3.1. let ( c(j,r),≤,‖ · ‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2) respectively. then every partially compact subset s of c(j,r) is janhavi. proof. the proof of the lemma is given in dhage and dhage [11]. since the proof is well-known, we omit the details of proof. � we need the following definition in what follows. definition 3.2. a function u ∈ c1(j,r) is said to be a lower solution of the problem (1.1) if the function t 7→ u(t) − iβh(t,u(t)) f(t,u(t)) is continuously differentiable and satisfies cdα ( u(t) − iβh(t,u(t)) f(t,u(t)) ) ≤ g ( t,u(t), ∫ t 0 k(s,u(s)) ds ) , t ∈ j, u(0) ≤ x0.   (∗) similarly, an upper solution v ∈ c1(j,r) to the problem (1.1) is defined on j, by the above inequalities with reverse sign. we consider the following set of assumptions in what follows: (a0) the map x 7→ x f(t,x) is injective for each t ∈ j. (a1) there exists a constant mf > 0 such that 0 < f(t,x) ≤ mf for all t ∈ j and x ∈ r. (a2) there exists a d-function ϕ such that 0 ≤ f(t,x) −f(t,y) ≤ ϕ(x−y) for all t ∈ j and x,y ∈ r, x ≥ y. (b1) there exists a constant mg > 0 such that 0 < g(t,x,y) ≤ mg for all t ∈ j and x,y ∈ r. (b2) the function g(t,x,y) is monotone nondecreasing in x and y for each t ∈ j. (b3) the function k(t,x) is monotone nondecreasing in x for each t ∈ j. (c1) there exists a constant mh > 0 such that 0 ≤ h(t,x) ≤ mh for all t ∈ j and x ∈ r. (c2) there exists a d-function ω such that 0 ≤ h(t,x) −h(t,y) ≤ ω(x−y) for all t ∈ j and x,y ∈ r, x ≥ y. (d1) the problem (1.1) has a lower solution u ∈ c1(j,r). (d2) the problem (1.1) has an upper solution v ∈ c1(j,r). remark 3.3. notice that hypothesis (a0) holds in particular if the function x 7→ x f(t,x) is increasing in r for each t ∈ j. the following lemma is useful in what follows and may be found in kilbas et.al. [16] and podlubny [17]. lemma 3.4. for a given continuous function h : j → r, a function u ∈ c1(j,r) is a solution of the qfde (3.3) cdqx(t) = h(t), t ∈ j, 0 < q < 1, x(0) = α0, } 162 dhage, khurpe, shete and salunke if and only if it is a solution of the nonlinear integral equation, (3.4) x(t) = α0 + 1 γ(q) ∫ t 0 (t−s)q−1 h(s) ds, t ∈ j. as an application of lemma 3.4, we obtain lemma 3.5. assume that the hypothesis (a0) holds. if a function x ∈ c1(j,r) is a solution of the qfde (3.5) cdq [ x(t) − iβh(t,x(t)) f(t,x(t)) ] = g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) , t ∈ j, 0 < q < 1, x(t0) = α0,   then it satisfies the nonlinear integral equation, x(t) = [ f(t,x(t)) ][ α0 f(t0,α0) + 1 γ(q) ∫ t 0 (t−s)q−1 g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds ] + iβh(t,x(t)), t ∈ j. (3.6) proof. assume first that x ∈ c1(j,r) is a solution to the qfde (1.1) defined on j. by lemma 1.5, we have (3.7) x(t) − iβh(t,x(t)) f(t,x(t)) = ∫ t 0 (t−s)α−1 γ(α) g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds + c0, where c0 ∈ r. since x(0) = α0,f(0,α0) 6= 0, it follows c0 = α0 f(0,α0) . thus (3.6) holds. � definition 3.6. a function x ∈ c1(j,r) which satisfies the qfie (3.7) is called a mild solution of the qfde (1.1) defined on j. theorem 3.7. assume that the hypotheses (a0)-(a2), (b1)-(b2), (c1)-(c7) and (d1) hold. furthermore, if (3.8) [∣∣∣ α0 f(t0,α0) ∣∣∣ + mgtα γ(α + 1) ] ϕ(r) + tβ γ(β + 1) ω(r) < r, then the problem (1.1) has a mild solution x∗ defined on j and the sequence {xn}∞n=1 of successive approximations defined by xn+1(t) = ∫ t 0 (t−s)β−1 γ(β) h(s,xn(s))ds + f(t,xn(t)) [ α0 f(t0,α0) + ∫ t 0 (t−s)α−1 γ(α) g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds ](3.9) for all t ∈ j, where x1 = u, converges monotonically to x∗. proof. by lemma 3.5, the mild solution x of the problem (1.1) satisfies the nonlinear integral equation x(t) = ∫ t 0 (t−s)β−1 γ(β) h(s,x(s))ds + f(t,x(t)) [ α0 f(t0,α0) + ∫ t 0 (t−s)α−1 γ(α) g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds ](3.10) for all t ∈ j. set e = c(j,r). then, from lemma 3.1 it follows that every compact chain in e possesses the compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in e. define the operators a, b, and c on e by (3.11) ax(t) = f(t,x(t)), t ∈ j, approximate solutions of fractional differential equations 163 (3.12) bx(t) = α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds, t ∈ j, and (3.13) cx(t) = 1 γ(β) ∫ t 0 (t−s)β−1h(s,x(s)) ds, t ∈ j. from the continuity of the integrals, it follows that a,b and c define the maps a,b : e →k and c : e → e. then, the problem (1.1) is equivalent to the operator equation (3.14) ax(t)bx(t) + cx(t) = x(t), t ∈ j. we shall show that the operators a,b and c satisfy all the conditions of theorem 2.9. this is achieved in the series of following steps. step i: a,b and c are nondecreasing operators on e. let x,y ∈ e be such that x ≥ y. then by hypothesis (a2), we obtain ax(t) = f(t,x(t)) ≥ f(t,y(t)) = ay(t), for all t ∈ j. this shows that a is nondecreasing operator on e into e. similarly, we have by (a5), bx(t) = α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds ≥ α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds = by(t), for all t ∈ j. this shows that b is nondecreasing operator on e into itself. the proof that c is nondecreasing operator on e into itself is similar. step ii: a and c are partially bounded and partially d-contraction on e. let x ∈ e be arbitrary. then by (a1), |ax(t)| ≤ |f(t,x(t))| ≤ mf, for all t ∈ j. taking supremum over t, we obtain ‖ax‖ ≤ mf and so, a is bounded. this further implies that a is partially bounded on e. next, let x,y ∈ e be such that x ≥ y. then, |ax(t) −ay(t)| = |f(t,x(t)) −f(t,y(t))| ≤ ϕ(|x(t) −y(t)|) ≤ ϕ(‖x−y‖). then, ‖ax−ay‖≤ ϕ(‖x−y‖) for all x,y ∈ e with x ≥ y and hence a is a partially d-lipschitz on e with d-functions ϕ(r), which further implies that a is also a partially continuous on e. again, we have |cx(t)| ≤ ∫ t 0 (t−s)β−1 γ(β) |h(s,x(s))|ds ≤ mh ∫ t 0 (t−s)β−1 γ(β) ds ≤ mh t β γ(β + 1) ≤ mh t β γ(β + 1) , which means that c is bounded and consequently partially bounded on e. 164 dhage, khurpe, shete and salunke next, let x,y ∈ e be such that x ≥ y. then, |cx(t) −cy(t)| = ∫ t 0 (t−s)β−1 γ(β) |h(s,x(s)) −h(s,y(s))|ds ≤ tβ γ(β + 1) ω(‖x−y‖). hence c is a partially d-lipschitz on e with d-functions tβ γ(β + 1) ω(r), which further implies that c is a partially continuous on e. step iii: b is a partially continuous operator on e. let {xn} be a sequence of points of a chain c in e such that xn → x for all n ∈ n. then, by dominated convergence theorem, we have lim n→∞ bxn(t) = lim n→∞ [ α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1g ( s,xn(s), ∫ s 0 k(τ,xn(τ)) dτ ) ds ] = α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1 [ lim n→∞ g ( s,xn(s), ∫ s 0 k(τ,xn(τ)) dτ )] ds = α0 f(t0,α0) + 1 γ(α) ∫ t 0 (t−s)α−1g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds = bx(t), for all t ∈ j. this shows that {bxn} converges to bx pointwise on j. next, we will show that {bxn} is an equicontinuous sequence of functions in e. let t1, t2 ∈ j be arbitrary with t1 < t2. then |bxn(t2) −bxn(t1)| ≤ 1 γ(α) ∣∣∣∣ ∫ t2 0 |(t2 −s)α−1 − (t1 −s)α−1| ∣∣∣g(s,x(s),∫ s 0 k(τ,x(τ)) dτ )∣∣∣ds∣∣∣∣ + 1 γ(α) ∣∣∣∣ ∫ t2 t1 (t1 −s)α−1 ∣∣∣g(s,x(s),∫ s 0 k(τ,x(τ)) dτ )∣∣∣ds∣∣∣∣ ≤ mg γ(α + 1) (tα2 − t α 1 ). consequently, we obtain |bxn(t2) −bxn(t1)|→ 0 as t2 → t1 uniformly for all n ∈ n. this shows that the convergence bxn → bx is uniformly and hence b is a partially continuous on e. step iv: b is a partially compact operator on e. let c be an arbitrary chain in e. we show that b(c) is a uniformly bounded and equicontinuous set in e. first we show that b(c) is uniformly bounded. let x ∈ c be arbitrary. then, |bx(t)| ≤ ∣∣∣ α0 f(t0,α0) ∣∣∣ + 1 γ(α) ∫ t 0 (t−s)α−1 ∣∣∣∣g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ )∣∣∣∣ ds ≤ ∣∣∣ α0 f(t0,α0) + ∣∣∣ + mgtα γ(α + 1) = r, for all t ∈ j. taking the supremum over t, we obtain ‖bx‖ ≤ r for all x ∈ c. hence b(c) is a uniformly bounded subset of e. next, we will show that b(c) is an equicontinuous set in e. let approximate solutions of fractional differential equations 165 t1, t2 ∈ j be arbitrary with t1 < t2. then, |bx(t2) −bx(t1)| ≤ 1 γ(α) ∫ t2 0 |(t2 −s)α−1 − (t1 −s)α−1| ∣∣∣g(s,x(s),∫ s 0 k(τ,x(τ)) dτ )∣∣∣ds + 1 γ(α) ∣∣∣∣ ∫ t2 t1 (t1 −s)α−1 ∣∣∣g(s,x(s),∫ s 0 k(τ,x(τ)) dτ )∣∣∣ds∣∣∣∣ ≤ mg γ(α + 1) (tα2 − t α 1 ). thus, we have that |bx(t2) −bx(t1)|→ 0 as t2 → t1 uniformly for all x ∈ c. this shows that b(c) is an equicontinuous set in e. hence b(c) is compact subset of e and consequently b is a partially compact operator on e into itself. step v: d-functions ϕ and ω satisfy the inequality 0 < mψa(r) + ψc(r) < r, r > 0. we have mψa(r) + ψc(r) = [∣∣∣ α0 f(t0,α0) ∣∣∣ + mgtα γ(α + 1) ] ϕ(r) + tβ γ(β + 1) ω(r) < r, by assumption. step vi: u satisfies the operator inequality u ≤aubu + cu. since the hypothesis (a6) holds, u is a lower solution of (1.1) defined on j. then, (3.15) cdα ( u(t) − iβh(t,u(t)) f(t,u(t)) ) ≤ g(t,u(t)), satisfying, (3.16) u(0) ≤ α0, for all t ∈ j. taking the riemann-livoulle integration of fractional order α from 0 to t on both sides of the above inequality (3.15), we obtain u(t) ≤ ∫ t 0 (t−s)β−1 γ(β) h(s,u(s))ds + [ f(t,u(t)) ][ α0 f(0,α0) + ∫ t 0 (t−s)α−1 γ(α) g ( s,x(s), ∫ s 0 k(τ,x(τ)) dτ ) ds ] , (3.17) for all t ∈ j. this show that u is a solution of the operator inequality u ≤aubu + cu. thus, the operators a,b and c satisfy all the conditions of theorem 2.9 in view of remark 2.9 and we apply it to conclude that the operator equation axbx + cx = x has a solution defined on j. consequently the integral equation has a solution x∗ defined on j which is also a mild solution of the qfde (1.1) and the sequence {xn} of successive approximations defined by (3.9) converges monotonically to x∗. this completes the proof. � remark 3.8. the conclusion of theorem 3.7 alsi ramains true if we replace the hypothesis (d1) with (d2). the oroof under this new hypothesis is similar with obvious modifications. 166 dhage, khurpe, shete and salunke example 3.9. given a closed and bounded interval j = [0, 1] in r, consider the initial value problem of quadratic fractional nonlinear integro-differential equation, (3.18)   cd1/2 [ x(t) − i3/2(arctan x(t)) f(t,x(t)) ] = 2 + tanh x(t) + tanh (∫ t 0 k(s,x(s)) ds ) 16 , x(0) = 0, for all t ∈ j := [0, 1], where cd1/2 denotes the caputo fractional derivative of order 1/2, k : j×r → r and f : j ×r → r\{0} are two continuous functions defined by f(t,x) =   1, if x ≤ 0, 1 + x 1 + x , if x > 0. and k(t,x) = { 0 if x ≤ 0, log(1 + x) if x > 0 for t ∈ j and x ∈ r. if we take h(t,x) = arctan x, g(t,x,y) = 2 + tanh x + tanh y 16 and then it is easy to check that the conditions of theorem 3.7 are satisfied with the lower solution u defined by u(t) = − 4t3/2 3 √ π + t1/2 6 √ π , t ∈ j. therefore, the problem (3.17) has a mild solution defined on [0, 1]. references [1] m. ammi, e. el kinani, d. torres, existence and uniqueness of solutions to functional integro-differential fractional equations, electron. j. diff. equ., 2012 (2012), art. id 103. [2] m. el borai, m. abbas, on some integro-differential equations of fractional orders involving carathéodory nonlinearities, int. j. mod. math. 2 (2007), 41-52. [3] b.c. dhage, fixed point theorems in ordered banach algebras and applications, panamer. math. j. 9(4) (1999), 93–102. [4] b.c. dhage, basic results in the theory of hybrid differential equations with mixed perturbations of second type, funct. diff. equ. 19 (2012), 1-20. [5] b.c. dhage, quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, differ. equ. appl. 2 (2010), 465–486. [6] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ. appl. 5 (2013), 155-184. [7] b.c. dhage, partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45(4) (2014), 397-426. [8] b.c. dhage, nonlinear d-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, malaya j. mat. 3(1)(2015), 62-85. [9] b.c. dhage, operator theoretic techniques in the theory of nonlinear hybrid differential equations, nonlinear anal. forum 20 (2015), 15-31. [10] b.c. dhage, a new monotone iteration principle in the theory of nonlinear fractional differential equations, intern. j. anal. appl. 8(2) (2015), 130-143. [11] b.c. dhage, s.b. dhage, approximating solutions of nonlinear first order ordinary differential equations, glob. j. math. sci. 2 (2014), 25-35. [12] b.c. dhage, s.b. dhage, s.k. ntouyas, existence and approximate solutions for fractional differential equations with nonlocal conditions, journal of fractional calculus and applications 7(1) (2016), 24-35. [13] b.c. dhage, s.b. dhage, s.k. ntouyas, dhage iteration method for existence and approximate solutions of nonlinear quadratic fractional differential equations, journal of fractional calculus and applications, 7(2) (2016), 132-144. [14] b.c. dhage, s.k. ntouyas, existence results for boundary value problems for fractional hybrid differential inclucions, topol. methods nonlinear anal. 44 (2014), 229-238. [15] s. heikkilä and v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. [16] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [17] i. podlubny, fractional differential equations, academic press, san diego, 1999. approximate solutions of fractional differential equations 167 [18] s. sun, y. zhao, z. han, y. li, the existence of solutions for boundary value problem of fractional hybrid differential equations, commun. nonlinear sci. numer. simul., 17 (2012), 4961-4967. [19] y. zhao, s. sun, z. han, q. li, theory of fractional hybrid differential equations, comput. math. appl. 62 (2011), 1312-1324. 1kasubai, gurukul colony, ahmedpur-413 515, dist: latur, maharashtra, india 2school of mathematical sciences, swami ramanand teerth marathwada university, nanded, maharashtra, india ∗corresponding author: bcdhage@email.email international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 96-113 http://www.etamaths.com convergence theorem for finite family of total asymptotically nonexpansive mappings e.u. ofoedu and agatha chizoba nnubia∗ abstract. in this paper we introduce an explicit iteration process and prove strong convergence of the scheme in a real hilbert space h to the common fixed point of finite family of total asymptotically nonexpansive mappings which is nearest to the point u ∈ h. our results improve previously known ones obtained for the class of asymptotically nonexpansive mappings. as application, iterative method for: approximation of solution of variational inequality problem, finite family of continuous pseudocontractive mappings, approximation of solutions of classical equilibrium problems and approximation of solutions of convex minimization problems are proposed. our theorems unify and complement many recently announced results. 1. introduction let k be a nonempty subset of a real hilbert space h. a mapping t : k −→ k is called nonexpansive if and only if for all x,y ∈ k, we have that ‖tx−ty‖≤‖x−y‖.(1) the mapping t is called asymptotically nonexpansive mapping if and only if there exists a sequence {µn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 such that for all x.y ∈ k, (2) ‖tnx−tny‖≤ (1 + µn)‖x−y‖ ∀ n ∈ n. the class of asymptotically nonexpansive mappings was introduced by goebel and kirk [11] as a generalisation of nonexpansive mappings. as further generalisation of class of nonexpansive mappings, alber, chidume and zegeye [2] introduced the class of total asymptotically nonexpansive mappings, where a mapping t : k −→ k is called total asymptotically nonexpansive if and only if there exist two sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ k, (3) ‖tnx−tny‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1 observe that if φ(t) = 0 ∀ t ∈ [0, +∞), then equation (3) becomes (4) ‖tnx−tny‖≤‖x−y‖ + ηn n ≥ 1, so that if k is bounded and tn is continuous for some integer n ≥ 1, then the mapping t is of asymptotically nonexpansive type. the class of asymptotically 2010 mathematics subject classification. 47h10, 47j25. key words and phrases. hilbert space; total asymptotically nonexpansive; nearest point approximation; variational inequality; viscosity approximation method; strong convergence. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 96 total asymptotically nonexpansive mappings 97 nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense and the class of nearly asymptotically nonexpansive mappings. these classes of mappings had been studied extensively by several authors (see e.g.[11], [15], [31]). if φ(t) = t ∀ t ∈ [0, +∞), then equation (3) becomes (5) ‖tnx−tny‖≤ (1 + µn)‖x−y‖ + ηn n ≥ 1 in addition, if ηn = 0 for all n ∈ n, then we easily see that every asymptotically nonexpansive mapping is total asymptotically nonexpansive. if µn = 0 and ηn = 0 ∀ n ≥ 1 we obtain from equation (3) the class of mappings which includes the class of nonexpansive mappings. the class of total asymptotically nonexpansive mappings properly includes the class of asymptotically nonexpansive mappings (see example 2 of [20]). a point x0 ∈ k is called a fixed point of a mapping t : k −→ k if and only if tx0 = x0. we denote the set of fixed points of t by f(t), that is, f(t) = {x ∈ k : tx = x}. a point x∗ ∈ k is called a minimum norm fixed point of t if and only if x∗ ∈ f(t) and ‖x∗‖ = min{‖x‖ : x ∈ f(t)}. let d1 and d2 be nonempty closed convex subsets of real hilbert spaces h1 and h2, respectively. the split feasibility problem is formulated as finding a point x satisfying (6) x ∈ d1 such that ax ∈ d2, where a is bounded linear operator from h1 into h2. a split feasibility problem in finite dimensional hilbert spaces was first studied by censor and elfving [8] for modeling inverse problems which arise in medical image reconstruction, image restoration and radiation therapy treatment planning (see e.g [6], [7], [8]). it is clear that x ∈ d1 is a solution of the split feasibility problem (6) if and only if ax−pd2ax = 0, where pd2 is the metric projection from h2 onto d2. consider the minimization problem: (7) find x∗ ∈ d1 such that 1 2 ‖ax∗ −pd2ax‖ 2 = min x∈d1 1 2 ‖ax−pd2ax‖ 2, then x∗ is a solution of (6) if and only if x∗ solves the minimization problem (7) with the minimum equal to zero. suppose that problem (6) has solution and let ω denote the (closed convex) set of solutions of (6) (or equivalently, solution of (7), then ω is a singleton if and only if it is a set of solutions of the following variational inequality problem: (8) find x ∈ d1 such that 〈a∗(i −pd2 )ax,y −x〉≥ 0 ∀ y ∈ d2, where a∗ is the adjoint of the linear operator a . moreover, problem (8) can be rewritten as (9) find x ∈ d1 such that 〈x−ra∗(i −pd2 )ax−x,y −x〉≤ 0 ∀ y ∈ d2, where r > 0 is any positive scalar. using the nature of projection, (9) is equivalent to the fixed point equation (10) x = pd1 (x−ra ∗(i −pd2 )ax). thus, finding a solution of split feasibility problem (7) is equivalent to finding the minimum-norm fixed point of the mapping x −→ pd1 (x−ra∗(i −pd2 )ax). approximation of solutions of equations involving nonexpansive mappings and their 98 ofoedu and nnubia generalization by iterative methods has been of increasing research interest for numerous mathematicians in recent years. one of the first results of this nature was obtained by browder [5] for nonexpansive self mappings in hilbert spaces. suppose k is a closed convex nonempty subset of a real hilbert space h. browder [5] studied the path u ∈ k, xt = tu+ (1−t)txt, t ∈ (0, 1), where t : k −→ k is a nonexpansive mapping. in [5], browder proved that lim t→0 xt exists and lim t→0 xt ∈ f(t). the result was extended by reich [24] to uniformly smooth real banach spaces. reich [24] proved, in fact, that lim t→0 xt is a sunny nonexpansive retraction of k onto f(t). in [12], halpern studied the convergence of the explicit iteration method defined from x1 ∈ k by (11) xn+1 = αnu + (1 −αn)txn; n ≥ 1 in the frame work of real hilbert spaces. under appropriate conditions on the iterative parameter αn, it had been shown by halpern [12], lions [16], wittmann [26] and banschke [3] that the sequence {xn} generated by (11) converges strongly to a fixed point of t nearest to u, that is, pf(t)u, browder and halpern iterative methods had motivated different iterative methods for approximation of fixed points of asymptotically nonexpansive mappings. in this regard, lim and xu [15] introduced and studied the following implicit iterative method for asymptotically nonexpansive mapping t, (12) zn = αnu + (1 −αn)tnzn; n ≥ 1. they showed that the sequence {zn}n≥1 generated by (12) converges strongly to a fixed point of t in the frame work of uniformly smooth real banach spaces under suitable conditions on the iterative parameters. in [10], chidume, li and udomene proved the strong convergence of the explicit iterative method generated from x1,u ∈ k by (13) xn+1 = αnu + (1 −αn)tnxn; n ≥ 1, where lim n→∞ αn = 0, ∞∑ n=0 αn = +∞ and t is asymptotically nonexpansive. yao, zhou and lion [28], studied a modified mann iteration algorithm {xn} generated from x1,∈ h by νn = (1 − tn)xn, xn+1 = (1 −αn)xn + αntνn, n ≥ 1,(14) where {tn}n≥1,{αn}n≥1 are sequences in (0, 1) satisfying appropriate conditions. they proved the strong convergence of the modified algorithm to the fixed point of a nonexpansive mapping t : h −→ h when f(t) 6= ∅. osilike etal [23] modified the algorithm (14) with {xn} generated from x1,∈ k by νn = pk[(1 − tn)xn], xn+1 = (1 −αn)xn + αntnνn; n ≥ 1,(15) where {tn}n≥1,{αn}n≥1 are sequences in (0, 1) satisfying appropriate conditions. they proved the strong convergence of the modified algorithm to the fixed point of assymptotically nonexpansive mapping t : k −→ k when f(t) 6= ∅. recently, alber, espinola and lorenzo [2] obtained strong convergence of (13) for total asymptotically nonexpansive mappings 99 a total asymptotically nonexpansive self map t on k in the setting of smooth reflexive real banach space with weakly sequentially continuous duality mapping. in connection with the iterative approximation of minimum norm fixed point of the mapping t , yang, lion and yao [27] introduced an explicit iterative method generated from x1 ∈ k by (16) xn+1 = βntxn + (1 −βn)pk[(1 −αn)xn]; n ≥ 1, they proved under appropriate conditions on {αn}n≥1 and{βn}n≥1 that the sequence {xn}n≥1 converges strongly to the minimum norm fixed point of t in hilbert spaces. yang et al [27] proved that the explicit iterative method generated from x1 ∈ k defined by (17) xn+1 = pk[(1 −αn)txn]; n ≥ 1, converges strongly to the minimum norm fixed point of nonexpansive mapping t : k −→ k provided that {αn}n≥1 satisfies appropriate condition. recently, zegeye and shahzad [31] proved that the iterative method generated from arbitrary x1 ∈ k by yn = pk[(1 −αn)xn], xn+1 = βnxn + (1 −βn)tnyn; n ≥ 1,(18) converges strongly to minimum norm fixed point of asymptotically nonexpansive self map t on k. motivated by the results of these authors, it is our aim in this paper to prove strong convergence theorem to the common fixed point of finite family of total asymptotically nonexpansive mappings which is nearest to the point u ∈ h. our theorems generalize and unify the corresponding results of osilike etal [23], yao, zhou and lion [28], yang, lion and yao [27], zegeye and shahzad [31]. our method of proof is of independent interest. 2. preliminaries we shall make use of the following lemmas and propositions. lemma 2.1. let h be a real hilbert space. then for all x,y ∈ h the following inequality holds. ‖x + y‖2 ≤‖x‖2 + 2〈y,x + y〉 lemma 2.2. for any x,y,z in a real hilbert space h and a real number λ ∈ [0, 1], ‖λx + (1 −λ)y −z‖2 = λ‖x−z‖2 + (1 −λ)‖y −z‖2 −λ(1 −λ)‖x−y‖2. lemma 2.3. [25] let k be a closed convex nonempty subset of a real hilbert space h. let x ∈ h, then x0 = pkx if and only if 〈z −x0,x−x0〉≤ 0 ∀ z ∈ k let t : k −→ k be a mapping and i be the identity mapping of k, we say that (i −t) is demiclose at zero if and only if for any sequence {xn}n≥1 in k such that xn converges weakly to x and xn −txn → 0, as n →∞, we have that x = tx. 100 ofoedu and nnubia lemma 2.4. (see corollary 2.6 of [1]) let e be a reflexive banach space with weakly continuous normalized duality mapping. let k be a closed convex subset of e and let t be a uniformly continuous total asymptotically nonexpansive mapping from k into itself with bounded orbit, then (i −t) is demiclose at zero. lemma 2.5. [1] let {an} be a sequence of nonegative real numbers satisfying the following relation: an+1 ≤ (1 −αn)an + δn; n ≥ 1. suppose that for n ≥ 1, δn αn ≤ c1 and αn ≤ α (for some α,c1 > 0), then an ≤ max{a1, (1 +α)c1}. moreover, if ∞∑ n=0 αn = ∞ and δn, = o(αn), then lim n→∞ an = 0. lemma 2.6. (see [17]) let {γn} be sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence {γnj}of {γn} which satisfies γnj < γnj+1 ∀ j ∈ n. define the sequence {τ(n)}n≥n0 of integers as follows τ(n) = max{k ≤ n : γk < γk+1}, where n0 ∈ n and that the set {k ≤ n0 : γk < γk+1} is not empty, then the following hold (i) τ(n0) ≤ τ(n0 + 1) and τ(n) →∞ as n →∞ (ii) γτ(n) ≤ γτ(n)+1 and γn ≤ γτ(n)+1 ∀ n ∈ n. proposition 2.1. (see proposition 8 of [21]) let h be a real hilbert space, let k be a nonempty closed convex subset of h and let ti : k −→ k , where i ∈ i = {1, 2, ...,m}, be m uniformly continuous total asymptotically nonexpansive mappings from k into itself with sequences {µn,i}n≥1,{ηn,i}n≥1 ⊂ [0, +∞) such that lim n→∞ µn,i = 0 = lim n→∞ ηn,i and with function φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ m0t ∀ t > m1 for some constants m0,m1 > 0. let µn = max i∈i {µn,i} and ηn = max i∈i {ηn,i} and, φ(t) = max i∈i {φi(t)}∀ t ∈ [0,∞). suppose that f(t) =⋂m i=1 f(ti), then f(t) is closed and convex. proposition 2.2. [20] let k be a nonempty subset of a real normed space e and ti : k −→ k , where i ∈ i = {1, 2, ...,m}, be m total asymptotically nonexpansive mappings, then there exist sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ k, (19) ‖tni x−t n i y‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn; n ≥ 1,∀ i ∈ i. 3. main results let k be a nonempty closed and convex subset of a real hilbert space h. let ti : k −→ k, where i ∈ i = {1, 2, ...,m}, be m total asymptotically nonexpansive mappings and {αn}n≥1,{βn}n≥1 be sequences in (0, 1), we define the explicit total asymptotically nonexpansive mappings 101 iteration process {xn}n≥1 from x1 ∈ k,u ∈ h by y1 = pk[α1u + (1 −α1)x1], x2 = (1 −β1)x1 + β1t1y1, y2 = pk[α2u + (1 −α2)x2], x3 = (1 −β2)x2 + β2t2y2, ... ym−1 = pk[αm−1u + (1 −αm−1)xm−1], xm = (1 −βm−1)xm + βm−1tm−1ym−1, ym = pk[αmu + (1 −α)xm], xm+1 = (1 −βm)xm + βmt1mym, ym+1 = pk[αm+1u + (1 −αm+1)xm+1],(20) xm+2 = (1 −βm+1)xm+1 + βm+1t21 ym+1 ym+2 = pk[αm+2u + (1 −αm+2)xm+2], xm+3 = (1 −βm+2)xm+2 + βm+2t22 ym+2 ... y2m−1 = pk[α2m−1u + (1 −α2m−1)x2m−1], x2m = (1 −β2m−1)x + β2m−1t2m−1y2m−1 y2m = pk[α2mu + (1 −α2m)x2m], x2m+1 = (1 −β2m)x2m + β2mt2my2m y2m+1 = pk[α2m+1u + (1 −α2m+1)x2m+1], x2m+2 = (1 −β2m+1)x2m+1 + β2m+1t31 y2m+1 ... (21) since ∀z ∈ z (where z is the set of integers), there exists j(z) ∈ i such that z − j(z) is divisible by m (that is j(z) = z mod(m)), then there exists q(z) ∈ z with lim z→∞ q(z) = +∞ such that (22) z = ( q(z) − 1 ) m + j(z) 102 ofoedu and nnubia so we may write (20) in a more compact form as x1 ∈ k,u ∈ h,yn = pk[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βnt q(n) j(n) yn. (23) remark 3.1. since n−m ∈ z∀n ∈ n, we obtain from (22) for z = n−m that (24) n−m = ( q(n−m) − 1 ) m + j(n−m). also, substituting n ∈ n for z in (22) and subtracting m from both sides of the resulting equation gives (25) n−m = ( (q(n) − 1) − 1 ) m + j(n) comparing (24) and (25) we obtain (by unique representation theorem) that (26) q(n−m) = q(n) − 1 and j(n−m) = j(n) ∀ n ∈ n. theorem 3.1. let h be a real hilbert space, let k be a closed convex nonempty subset of h and let ti : k −→ k, where i ∈ i = {1, 2, ...,m}, be m uniformly continuous total asymptotically nonexpansive mapping from k into itself with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin and with function φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ m0t ∀ t > m1 for some constants m0,m1 > 0, let µn = max i∈i {µin} and ηn = max i∈i {ηin} and, φ(t) = max i∈i {φi(t)}∀ t ∈ [0,∞). suppose that f = ⋂m i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = lim n→∞ α−1n ηn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to pf (u). proof. let x∗ ∈ f, then from (23) and hypothesis on ti we have that ‖yn −x∗‖ = ‖pk[αnu + (1 −αn)xn] −pkx∗‖ ≤ ‖αnu + (1 −αn)xn −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αn‖u−x∗‖(27) and ‖xn+1 −x∗‖ = ‖(1 −βn)xn + βnt q(n) j(n) yn −x∗‖ ≤ (1 −βn)‖xn −x∗‖ + βn‖t q(n) j(n) yn −x∗‖ ≤ (1 −βn)‖xn −x∗‖ + βn [ ‖yn −x∗‖ + µq(n)φ(‖yn −x∗‖) + ηq(n) ] .(28) since φ is continuous, it follows that φ attains its maximum (say m) on the interval [0,m1], moreover, φ(t) ≤ m0t whenever t > m1. thus, (29) φ(t) ≤ m + m0t ∀ t ∈ [0, +∞). total asymptotically nonexpansive mappings 103 using (27) and (29) we obtain from (28) that ‖xn+1 −x∗‖ ≤ (1 −βn)‖xn −x∗‖ +βn [ ‖yn −x∗‖ + µq(n) ( m + m0‖yn −x∗‖ ) + ηq(n) ] ≤ (1 −βn)‖xn −x∗‖ + βn [ (1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ] ≤ (1 −βn)‖xn −x∗‖ + βn [ (1 + µq(n)m0)‖yn −x∗‖ ] +βnµq(n)m + βnηq(n) ≤ (1 −βn)‖xn −x∗‖ + βnµq(n)m + βnηq(n) +βn [ (1 + µq(n)m0)(1 −αn)]‖xn −x∗‖ + αn‖u−x∗‖ ] = [ 1 −αnβn + (1 −αn)βnµq(n)m0 ] ‖xn −x∗‖ +αnβn(1 + µq(n)m0)‖u−x∗‖ + βnµq(n)m + βnηq(n) = [ 1 −αnβn + (1 −αn)βnµq(n)m0 ] ‖xn −x∗‖ + δn,(30) where δn = αnβn(1+µq(n)m0)‖u−x∗‖+βnµq(n)m+βnηq(n). since lim n→∞ α−1n µq(n) = 0 = lim n→∞ α−1n ηq(n), we may assume without loss of generality that there exists k0 ∈ (0, 1) and m2 > 0 such that α−1n µq(n) < (1−k0) (1−αn)m0 and δn αnβn < m2. thus, we obtain from (30) that (31) ‖xn+1 −x∗‖≤‖xn −x∗‖−k0αnβn‖xn −x∗‖ + δn. so, lemma 2.5 gives ‖xn−x∗‖≤ max{‖x1−x∗‖, (1+k1)m2}. therefore, {xn}n≥1 is bounded and by (27) we obtain that {yn}n≥1 is bounded. moreover, using lemma 2.1, we obtain that ‖yn −x∗‖2 = ‖pk[αnu + (1 −αn)xn] −pkx∗‖2 ≤ ‖αnu + (1 −αn)xn −x∗‖2 ≤ (1 −αn)2‖xn −x∗‖2 +2αn(1 −αn)〈u−x∗,xn −x∗〉 + 2α2n‖u−x ∗‖2.(32) furthermore, using lemma 2.2, we obtain that ‖xn+1 −x∗‖2 = ‖(1 −βn)xn + βnt q(n) j(z) yn −x∗‖2 = (1 −βn)‖xn −x∗‖2 + βn‖t q(n) j(n) yn −x∗‖2 −βn(1 −βn)‖xn −t q(n) j(n) yn‖2 (33) but ‖tq(n) j(n) yn −x∗‖2 ≤ [ (1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ]2 = (1 + µq(n)m0) 2‖yn −x∗‖2 +(µq(n)m + ηq(n)) [ 2(1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ]2 (34) 104 ofoedu and nnubia so that putting (32) in (34), we have ‖tq(n) j(n) yn −x∗‖2 ≤ (1 + µq(n)m0)2 ( (1 −αn)2‖xn −x∗‖2 +2αn(1 −αn)〈u−x∗,xn −x∗〉 + 2α2n‖u−x ∗‖2 ) +(µq(n)m + ηq(n)) [ 2(1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ]2 (35) and βn‖t q(n) j(n) yn −x∗‖2 ≤ βn(1 + µq(n)m0)2(1 −αn)2‖xn −x∗‖2 +2αnβn(1 + µq(n)m0) 2(1 −αn)〈u−x∗,xn −x∗〉 +2α2nβn(1 + µq(n)m0) 2‖u−x∗‖2 + (µq(n)m + ηq(n))[ 2(1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ] .(36) now, substituting (36) in (33), we have ‖xn+1 −x∗‖2 = ‖(1 −βn)xn + βnt q(n) j(z) yn −x∗‖2 ≤ (1 −βn)‖xn −x∗‖2 + βn(1 + µq(n)m0)2(1 −αn)2‖xn −x∗‖2 +2αnβn(1 + µq(n)m0) 2(1 −αn)〈u−x∗,xn −x∗〉 +2α2nβn(1 + µq(n)m0) 2‖u−x∗‖2 −βn(1 −βn)‖xn −t q(n) j(n) yn‖2 +(µq(n)m + ηq(n)) [ 2(1 + µq(n)m0)‖yn −x∗‖ + µq(n)m + ηq(n) ] ≤ (1 −γn)‖xn −x∗‖2 + 2γn(1 −αn)〈u−x∗,xn −x∗〉 +θn −βn(1 −βn)‖xn −t q(n) j(n) yn‖2,(37) where γn = βnαn(1+µnm0) 2 and θn = 2α 2 nβn(1+µnm0) 2‖u2−x∗‖2+βnµq(n)m0(2+ µq(n)m0 sup n≥1 ‖xn−x∗‖2+βn(µq(n)m+ηq(n)) [ 2(1+µq(n)m0) sup n≥1 ‖yn−x∗‖+µq(n)m+ ηq(n) ] . two cases arise case 1: suppose{‖xn − x∗‖}n≥1 is nonincreasing for n ≥ n0, for some n0 ∈ n, this implies that ‖xn+1 − x∗‖ ≤ ‖xn − x∗‖ ∀ n ≥ n0. thus, lim n→∞ ‖xn−x∗‖ exist and lim n→∞ ( ‖xn+1−x∗‖−‖xn−x∗‖ ) = 0. moreover, using the fact that 0 < ξ0 < βn < ζ0 < 1, we obtain that (38) lim n→∞ ‖xn −t q(n) j(n) yn‖ = 0. next, we observe that ‖xn+1 −xn‖ = ‖(1 −βn)xn + βnt q(n) j(n) yn −xn‖ = βn‖t q(n) j(n) yn −xn‖.(39) thus, by (38) (40) lim n→∞ ‖xn+1 −xn‖ = 0. total asymptotically nonexpansive mappings 105 observe that by(40) (41) lim n→∞ ‖xn −xn−i‖ = 0 = lim n→∞ ‖xn −xn+i‖, i ∈ i. moreso, ‖yn −xn‖ = ‖pk[αnu + (1 −αn)xn] −pkxn‖ ≤ ‖αnu + (1 −αn)xn] −xn‖ = αn‖u−xn‖(42) so that by our hypothesis (43) lim n→∞ ‖yn −xn‖ = 0. furthermore, (44) ‖yn+1 −yn‖≤‖yn+1 −xn+1‖ + ‖xn+1 −xn‖ + ‖xn −yn‖ which by (40) and (43) gives (45) lim n→∞ ‖yn+1 −yn‖ = 0. observe that by (45) we have (46) lim n→∞ ‖yn+i −yn‖ = 0 = lim n→∞ ‖yn−i −yn‖ ∀ i ∈ i now, (47) ‖yn −t q(n) j(n) yn‖≤‖yn −xn‖ + ‖xn −t q(n) j(n) yn‖. using (38) and(43) in (47) gives (48) lim n→∞ ‖yn −t q(n) j(n) yn‖ = 0. by uniform continuity of ti; i ∈ i there exists a continuous increasing function πi : r −→ r with πi(0) = 0 such that (49) ‖tix−tiy‖≤ πi(‖x−y‖)∀x,y ∈ k. thus, defining π0 : r −→ r by π0(t) = max i∈i {πi(t)} ∀ t ∈ r, we have that π0 is a continuous increasing function with π0(0) = 0 and ‖yn −tj(n)yn‖ ≤ ‖yn −t q(n) j(n) yn‖ + ‖t q(n) j(n) yn −tj(n)yn‖ ≤ ‖yn −t q(n) j(n) yn‖ + π0(‖t q(n)−1 j(n) yn −yn‖).(50) consider the argument of π0 in (50), ‖tq(n)−1 j(n) yn −yn‖ ≤ ‖t q(n)−1 j(n) yn −t q(n)−1 j(n−m)yn−m‖ + ‖t q(n)−1 j(n−m)yn−m −yn−m‖ +‖yn−m −yn‖ .(51) by (26) we have that ‖tq(n)−1 j(n) yn −t q(n)−1 j(n−m)yn−m‖ ≤ ‖t q(n)−1 j(n−m)yn −t q(n)−1 j(n−m)yn−m‖ ≤ ‖yn−m −yn‖ + µq(n)−1 + φ(‖yn−m −yn‖) +ηq(n)−1 .(52) using (46) in (52) and by hypothesis we have that (53) lim n→∞ ‖tq(n)−1 j(n) yn −t q(n)−1 j(n−m)yn−m‖ = 0. 106 ofoedu and nnubia moreso, by (26) we have that (54) ‖tq(n)−1 j(n−m)yn−m −yn−m‖ = ‖t q(n−m) j(n−m) yn−m −yn−m‖. thus, (55) lim n→∞ ‖tq(n)−1 j(n−m)yn−m −yn−m‖ = 0. now, using (53) and (54) in (50) we obtain that lim n→∞ ‖tq(n)−1 j(n) yn −yn‖ = 0.(56) consequently, we obtain from (48) and (50) that lim n→∞ ‖yn −tj(n)yn‖ = 0.(57) furthermore, we obtain for i ∈ i that ‖yn −tj(n)+iyn‖ ≤ ‖yn −yn+i‖ + ‖yn+i −tj(n)+iyn+i‖ + ‖tj(n)+iyn+i −tj(n)+iyn‖ ≤ ‖yn −yn+i‖ + ‖yn+i −tj(n)+iyn+i‖ + π0(‖yn+i −yn‖).(58) so,using (46), (57) and (58) we have lim n→∞ ‖yn −tj(n)+iyn‖ = 0 ∀i ∈ i.(59) but ∀i ∈ i there exists ϑi ∈ i such that j(n) + ϑi = i mod(m) so that from (59), we have that lim n→∞ ‖yn −tiyn‖ = lim n→∞ ‖yn −tj(n)+iyn‖ = 0 ∀i ∈ i.(60) but (61) ‖xn −tixn‖ = ‖xn −yn‖ + ‖yn −tiyn‖ + ‖tiyn −tixn‖ ∀ n ∈ n. hence, using (43), uniform continuity of the mapping t and (60) we obtain from (61) that (62) lim n→∞ ‖xn −tixn‖ = 0, ∀i ∈ i. now, let {xnk}k≥1 be a subsequence of {xn}n≥1 such that (63) lim sup n→∞ 〈u−x∗,xn −x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉, then, there exist a subsequence{xnkj}j≥1 of {xnk}k≥1 that converges weakly to some z ∈ h. thus, (63) gives (64) lim sup n→∞ 〈u−x∗,xn−x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉 = lim j→∞ 〈u−x∗,xnkj −x ∗〉. furthermore, by (62) limj→∞‖xnkj − tixnkj‖ = 0 and by lemma 2.4, i − ti is demiclose at 0, we obtain that z ∈ f. so using(64) and the fact that x∗ = pku, we obtain from lemma 2.3 that lim sup n→∞ 〈u−x∗,xn −x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉 = lim j→∞ 〈u−x∗,xnkj −x ∗〉 = 〈u−x∗,z −x∗〉≤ 0. .(65) therefore, defining total asymptotically nonexpansive mappings 107 (66) νn = max{0,〈u−x∗,xn −x∗〉}, then it is easy to see that lim n→∞ νn = 0. moreover we obtain from (37)(using (66)) that ‖xn+1 −x∗‖2 ≤ ‖xn −x∗‖2 −γn‖xn −x∗‖2 + 2γn(1 −αn)〈u−x∗,xn −x∗〉 + θn ≤ (1 −γn)‖xn −x∗‖2 + 2γn(1 −αn)νn + θn = (1 −γn)‖xn −x∗‖2 + σn(67) where σn = 2γn(1 −αn)νn + θn. conditions on our iterative parameter easily give that σn = o(γn). hence, we obtain from (67) using lemma 2.5 that {xn}n≥0 converges strongly to x∗ = pku case 2: suppose there exists a subsequence {xni} of {xn} such that ‖xni −x∗‖≤ ‖xni+1 − x∗‖ ∀ i ∈ n, then by lemma 2.6 there exist a nondecreasing sequence {τ(n)}⊂ n such that (i) lim n→∞ τ(n) = ∞ (ii) ‖xτ(n)−x∗‖≤‖xτ(n)+1−x∗‖ ∀ n ∈ n. so, from (37), we have that γn‖xτ(n) −x∗‖2 ≤ ‖xτ(n) −x∗‖2 −‖xτ(n)+1 −x∗‖2 +2γτ(n)〈u−x∗,xτ(n) −x∗〉 + θτ(n) ∀ n ∈ n(68) thus, using the fact that γτ(n) > 0, we have that (69) ‖xτ(n) −x∗‖2 ≤ 2〈u−x∗,xτ(n) −x∗〉 + θτ(n) γτ(n) ∀ n ∈ n. observe that following the argument of case 1 we have that lim n→∞ ‖xτ(n+1)−xτ(n)‖ = lim n→∞ ‖yτ(n) −tiyτ(n)‖ = 0 ∀ i ∈ i and lim sup n→∞ 〈u−x∗,xn −x∗〉 ≤ 0. thus, setting ντ(n) = max{0,〈u−x∗,xτ(n) −x∗〉}, we obtain that ντ(n) → 0 as n →∞. furthermore, from conditions on our iterative parameters, we obtain that θτ(n) γτ(n) → 0. so we obtain from (69) that ‖xτ(n) − x∗‖2 ≤ 2ντ(n) + θτ(n) γτ(n) ∀ n ∈ n. thus, lim n→∞ ‖xτ(n) −x∗‖ = 0. also from lemma 2.6 we have that ‖xn −x∗‖≤‖xτ(n+1) − x∗‖2 ∀ n ∈ n. thus we obtain using sandwich theorem that lim n→∞ ‖xn − x∗‖ = 0. hence, xn converges strongly to x ∗ = pku. remark 3.2. observe that if ti, i ∈ i in theorem 3.1 were asymptotically nonexpansive mappings, the condition there exist m0 > 0 and m1 > 0 such that φ(t) ≤ m0t ∀ t > m1 is not needed. moreso, every asymptotically nonexpansive mapping ti : k −→ k is uniformly l-lipschitzian thus uniformly continuous. hence we have the folowing theorems as an easy corollaries of theorem 3.1 above: theorem 3.2. let k be a closed convex nonempty subset of a real hilbert space h and let ti : k −→ k, i ∈ i, be asymptotically nonexpansive mappings such that f = ⋂m i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: 108 ofoedu and nnubia ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to pf (u). theorem 3.3. let k be a closed convex nonempty subset of a real hilbert space h and let ti : k −→ k, i ∈ i, be finite family of nonexpansive mappings from k into itself. suppose that f = ⋂m i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to pfu. . corollary 3.1. suppose in our theorems the finite family is a singleton (that is if m = 1), our results hold. remark 3.3. if u = 0 in the recursion formulas of our theorems, we obtain what authors now call the minimum norm iteration process. we observe that all our theorems in this paper carry over trivially to the so called minimum norm iteration process. remark 3.4. if f : k −→ k is a contraction map and we replace u by f(xn) in the recursion formulas of our theorems, we obtain what some authors now call viscosity iteration process. we observe that all our theorems in this paper carry over trivially to the so-called viscosity process. 4. application to approximation of fixed points of continuous pseudocontractive mappings the most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. these mappings are intimately connected with the important class of nonlinear monotone operators. for the importance of monotone operators and their connections with evolution equations, the reader may consult [9], [19]. due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors. recently, h. zegeye [29] established the following lemmas: lemma 4.1. [29] let k be a nonempty closed convex subset of a real hilbert space h. let t : k −→ h be a continuous pseudocontractive mapping, then for all r > 0 and x ∈ h, there exists z ∈ k such that (70) 〈y −z,tz〉− 1 r 〈y −z, (1 + r)z −x〉≤ 0; ∀ y ∈ k lemma 4.2. [29] let k be a nonempty closed convex subset of a real hilbert space h. let t : k −→ k be a continuous pseudocontractive mapping, then for all r > 0 and x ∈ h, there exists z ∈ k, define a mapping fr : h −→ k by (71) fr(x) = {z ∈ k : 〈y −z,tz〉− 1 r 〈y −z, (1 + r)z −x〉≤ 0 ∀ y ∈ k} then the following hold: total asymptotically nonexpansive mappings 109 (1) fr is single-valued (2) fr is firmly nonexpansive type mapping i.e for all x,y,z ∈ h (72) ‖fr(x) −fr(y)‖2 ≤〈fr(x) −fr(y),x−y〉 (3) fix(fr) is closed and convex; and fix(fr) = fix(t); for all r > 0. remark 4.1. we observe that lemmas 4.1 and 4.2 hold in particular for r = 1. thus, if ti, i ∈ i = {1, 2, ...,m} is finite family of continuous pseudocontractive mapping and we define f1(i) : h −→ k by (73) f1(i)(x) = {z ∈ k : 〈y −z,tiz〉−〈y −z, 2z −x〉≤ 0 ∀ y ∈ k} then f1(i) satisfies the conditions of lemma 4.2 ∀ i ∈ i. hence, we easily see that f1(i) is nonexpansive and fix(f1(i)) = fix(ti)∀ i ∈ i. thus, we have the following theorem. theorem 4.1. let k be a closed convex nonempty subset of a real hilbert space h and let ti : k −→ k i ∈ i be finite family of continous pseudocontractive mappings from k into itself. suppose that f ′ = ⋂n i=1 f(ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by x1 ∈ k,u ∈ h,yn = pk[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βnf q(n) 1j(n) yn, (74) where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to pf′u. furthermore, if u = 0,{xn}n≥1 converges strongly to a minimum norm fixed point of the finite family. 5. application to approximation of solutions of classical equilibrium problems let k be a closed convex nonempty subset of a real hilbert space h. let f : kxk −→ r be a bifunction. the classical equilibrium problem (abbreviated ep) for f is to find u∗ ∈ k such that (75) f(u∗,y) ≥ 0 ∀ y ∈ k the set of solutions of classical equilibrium problem is denoted by ep(f), where ep(f) = {u ∈ k : f(u,y) ≥ 0 ∀ y ∈ k}. the classical equilibrium problem (ep) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, nash equilibria in noncooperative games. furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the classical equilibrium problem. in other words, the classical equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer 110 ofoedu and nnubia science, optimization theory, operations research, economics and countless other fields. for the past 20 years or so, many existence results have been published for various equilibrium problems (see e.g.[4], [14],[30]). in the sequel, we shall require that the bifunction f : kxk −→ r satisfies the following conditions: (a1) f(x,x) = 0 ∀ x ∈ k; (a2)f is monotone, in the sense that f(x,y)+f(y,x) ≤ 0 ∀ x,y ∈ k; (a3)lim sup t→0+ f(tz+(1−t)x,y) ≤ f(x,y) ∀ x,y,z ∈ k;. (a4) the function y 7→ f(x,y) is convex and lower semicontinuous for all x ∈ k lemma 5.1. [(compare with lemma 2.4 of [14])] let k be a nonempty closed convex subset of a real hilbert space h. letfi : kxk −→ r be finite family of bifunction satisfying conditions (a1) (a4) for each i ∈ i = {1, 2, ...,m} then for all r > 0 and x ∈ h, there exists u ∈ k such that (76) fi(u,y) + 1 r 〈y −u,u−x〉≥ 0 ∀ y ∈ k i ∈ i. moreover’ if for all x ∈ h we define gir : h −→ 2k by (77) gir(x) = {u ∈ k : fi(u,y) + 1 r 〈y −u,u−x〉≥ 0 ∀ y ∈ k.} then the following hold: (1) gir is single-valued for all r ≥ 0 i ∈ i (2) fix(gir) = ep(fi) for all r > 0 (3) ep(fi) is closed and convex remark 5.1. we observe that lemmas 5.1 holds in particular for r = 1. thus, if we define gi1 : h −→ 2k by (78) gi1(x) = {u ∈ k : fi(u,y) + 〈y −u,u−x〉≥ 0 ∀ y ∈ k.} then gi1 satisfies the conditions of lemma 5.1 ∀ i ∈ i. hence, we easily see that gi1 is nonexpansive and fix(gi1) = ep(fi) ∀ i ∈ i. thus, we have the following theorem: theorem 5.1. let k be a closed convex nonempty subset of a real hilbert space h and let letfi : kxk −→ r be finite family of bifunction satisfying conditions (a1) (a4) for each i ∈ i = {1, 2, ...,m}. suppose that f ′′ = ⋂m i=1 ep(fi) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by x1 ∈ k,u ∈ h,yn = pk[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βng q(n) 1j(n) yn n ≥ 0 (79) where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to pf′′u. . furthermore, if u = 0,{xn}n≥1 converges strongly to a minimum norm fixed point of the finite family. total asymptotically nonexpansive mappings 111 remark 5.2. several authors (see e.g.[14], [18] and references therein) have studied the following problem:let k be a closed convex nonempty subset of a real hilbert space h. let f : kxk −→ r be a bifunction and φ : k −→ r be a proper extended real valued function, where r denotes the real numbers. let θ : k −→ h be a nonlinear monotone mapping. the generalised mixed equilibrium problem (abbreviated gmep) for f, φandθ is to find u∗ ∈ k such that (80) f(u∗,y) + φ(y) − φ(u∗) + 〈θu∗,y −u∗〉≥ 0 ∀ y ∈ k observe that if we defineγ : kxk −→ r (81) γ(x,y) = f(x,y) + φ(y) − φ(x) + 〈θx,y −x〉 then it could be easily checked that γ is a bi-function and satisfies properties (a1)to(a4). thus, the so called generalized mixed equilibrium problem reduces to the classical equilibrium problem for the bifunction γ. thus, consideration of the so called generalized mixed equilibrium problem in place of the classical equilibrium problem studied in this section leads to no further generalization. 6. applications to convex optimization) let us look at the problem of minimizing a continuously frechet-differentiable convex functional with minimum norm in hilbert spaces. let k be a closed convex subset of a real hilbert space h, consider the minimization problem given by (82) min x∈k φ(x) where φ is a frechet-differentiable convex functional. let ω the solution set of (82) be nonempty. it is known that a point z ∈ k is a solution of (82) if and only if the following optimality condition holds: (83) z ∈ k, 〈∇φ(z),x−z〉≥ 0,x ∈ k, where ∇ is the gradient of φ at x ∈ k. it is also known that the optimality condition (83) is equivalent to the following fixed point problem: (84) z = tγ(z), wheretγ := pk(i −γ∇φ), for all γ > 0. so, we have the following corollary deduced from theorem 3.1 theorem 6.1. let h be a real hilbert space, let k be a closed convex nonempty subset of h. let ψ be a continuously frechet-differentiable convex functional on k such that tγ(i) := pk(i − γ(i)∇ψ) be finite family of uniformly continuous total asymptotically nonexpansive mapping from k into itself with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin and with function φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ m0t ∀ t > m1 for some constants m0,m1 > 0, let µn = max i∈i {µin} and ηn = max i∈i {ηin} and φ(t) = max i∈i {φi(t)}∀ t ∈ [0,∞). suppose that f = ⋂n i=1 f(tγ(i) ) 6= ∅ and {xn}n≥1 is a sequence generated iteratively by x1 ∈ k, yn = pk[(1 −αn)xn], xn+1 = (1 −βn)xn + βn[pk(i −γ(i)∇ψ)] q(n) j(n) yn; n ≥ 1(85) 112 ofoedu and nnubia where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to the minimum norm solution of the minimization problem (82). a prototype of φi : [0, +∞) −→ [0, +∞) in theorem 3.1 is φ(λ) = λs, where 0 < s ≤ 1. moreso, prototype of the sequences used in the same theorem 3.1 are: take αn = 1 n , µn = 1 n1+� , for � > 0,ηn = 1 n log n . remark 6.1. our results extends and unify most of the results that have been proved for the class of assymptotically nonexpansive mappings of which the results obtained in [10], [15],[27], [31] are examples. references [1] y. alber, r. espinola and p. lorenzo, strongly convergent approximations to fixed points of total asymptotically nonexpansive mappings, acta mathematica sinica, english series, vol. 24 no. 6 (2008) 1005-1022. [2] ya. alber, c. e. chidume and h. zegeye, approximating fixed points of total asymptotically nonexpansive mappings.fixed point theory and appl. 2006 (2006), article id 10673. [3] h. h. bauschke, the approximation of fixed points of compositions of nonexpansive mappings in hilbert spaces, j. math. anal. appl. 202 (1996) 150-159. [4] e. blum and w. oettli, from optimization and variational inequalities to equilibrum problems, the mathematics student 63 nos. 1-4 (1994), 123-145. [5] f. e. browder, convergence theorems for sequences of nonlinear operators in banach spaces, math. zeitschr. 100 (1967) 201-225. [6] c. byrne, iterative oblique projection onto convex subsets and split feasibility problems, inverse problems, 18(2002)441-453. [7] y. censor, t. bortfeld. b.martin and trofimov, a unified approach for inversion problem in intensity modulated radiation therapy,pys.med. biol., 51(2006) 2353-2365. [8] y. censor and elfving, a multiprojection algorithm using bregman projection in a product space, numer. algorithms, 8(1994) 221-239. [9] c.e. chidume; geometric properties of banach spaces and nonlinear iterations, springer verlag series: lecture notes in mathematics vol. 1965 (2009). [10] c. e. chidume, jinlu li and a. udomene; convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, proc. amer. math. soc., 133 (2005), 473480. [11] k. goebel and kirk, a fixed point theorem for asymptotically nonexpansive mappings proc. amer. math. soc. 35(1972),171-174. [12] b. halpern, fixed point of nonexpansive maps, bull.amer. math. soc., 73(1967), 975-961. [13] s. ishikawa; fixed point by a new iteration method, proc. amer. math. soc. 44(1974) 147-150. [14] p. katchang, t. jitpeera and p. kumam, strong convergence theorems for solving generalized mixed equilibrium problems and general system of variational inequalities by the hybrid method, nonlinear analysis: hybrid systems 4 (2010) 838c852. [15] t.c.lim and h.k. xu fixed point theorems for asymtotically nonexpansive mappings, nonlinear anal., 22(1994), 1345-1355. [16] p. l. lions, approximation de points fixes de contractions, computes rendus de l’academie des sciences, serie i-mathematique, 284 (1997) 1357-1359. [17] p.e mainge, strong convergence of projected subgradient method for non smooth and non strickly convex minimizaton, set valued anal. 16 (2008)899-912. [18] e. u. ofoedu, a general approximation scheme for solutions of various problems in fixed point theory, international journal of analysis, 2013 (2013), article id 762831. [19] e. u. ofoedu and h. zegeye, further investigation on iteration processes for pseudocontractive mappings with application, nonlinear anal. tma 75 (2012) 153-162. total asymptotically nonexpansive mappings 113 [20] ofoedu, e. u. and madu, l. o. iterative procedure for finite family of total asymptotically nonexpansive mappings,journal of the nigerian mathematical society, 33(2014) 93-112. [21] e.u. ofoedu and agatha c. nnubia, strong convergence theorem for minimum-norm fixed point of total asymptotically nonexpansive mapping. afrika matematika(2014) doi: 1007/513370-014-0240-4. [22] j.g ohara p. pillary and h.k. xu, iterative appraoches to convex feasibility problem in benach spaces, nonlinear anal., 64(20060, 2022-2042. [23] m.o osilike, e. e.chima. p.u. nwokoro and f. u. ogbuisi, strong convergence of a modified averaging iterative algorithm for asymptotically nonexpansive maps, journal of the nigerian mathematical society, 32(2013), 241-251. [24] s. reich. s.reich, strong convergence theorems for resolvents of accretive operators in benach spaces, j.math. anal. appl. (1966), 276-284. [25] w takahashi, nonlinear functional analysisfixed point theory and applications, yokohanna publisher inc. yokohanna (2000). [26] r. wittmann approximation of fixed point of nonexpansive mappings ,arc math,58(1999) 486-491 [27] x. yang, y. c liou and y. yao finding minimum norm fixed point of nonexpansive mappings and applications, mathematical problems in engineering, v. 2011, article id 106450. [28] y. yao, h.k xu and y.c liou, strong convergence of a modified krasnoselskimann iterative algorithm for non-expansive mapping, j. appl. comput. 29(2009) 383-389. [29] h. zegeye, an iterativee approximation method for a common fixed point of two pseudocontractive mappings, isrn math. anal. 14 (2011). article id 621901. [30] h. zegeye, e. u. ofoedu and n. shahzad, convergence theorems for equilibrum problem,variational inequality problem and countably infinite relatively quasi-nonexpansive mappings,applied mathematics and computation, 216 (2010) 3439-3449 [31] h. zegeye and n. shahzad, approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings, fixed point theory and appl. 2013 (2013) article id 1. department of mathematics, nnamdi azikiwe university, p. m. b. 5025, awka, anambra state, nigeria ∗corresponding author int. j. anal. appl. (2023), 21:68 hypersurfaces with a common geodesic curve in 4d euclidean space e4 sahar h. nazra∗ department of mathematical sciences, college of applied sciences, umm al-qura university, ksa ∗corresponding author: shnazra@uqu.edu.sa abstract. in this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4d euclidean space e4. using the serret–frenet frame of the given geodesic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be geodesic. we illustrate this method by presenting some examples. 1. introduction in differential geometry, geodesic curves representing in some sense the shortest distance (arc) amidst two points in a surface, or more in general in a riemannian manifold [7–9]. from this explicitness we can immediately see that the geodesic among two points on a sphere is a great circle. but there are two arcs of a great circle amid two of their points, and only one of them gives the short distance, with the exclusion of the two points are the end points of a diameter. this model indicates that there may exist more than one geodesic among two points. therefore, for example, the passage of a verticil orbiting about a star is the projection of a geodesic of the curved 4d space-time geometry about the star onto 3d space. nowadays, numerous research results have concentrated on surfaces family having a common geodesic curve in a diversity of applications, such as the tent manufacturing, designing industry of shoes, cutting and painting path. in general, the goal of mainly works on geodesics is to define a family of surfaces with a given geodesic curve and express it as a linear combination of the serret–frenet frame (see for example [1, 2, 4, 5, 11, 12, 14, 16]). however, there is little written works on differential geometry of parametric surface family in euclidean, and non-euclidean 4-spaces [3, 6, 10, 13, 15]. thus, the current study hopes to serve such a need. in this paper, we consider the parametric representation of hypersurface family passing a given received: dec. 12, 2022. 2010 mathematics subject classification. primary 53a05, secondary 22e15. key words and phrases. hypersurface; serret–frenet formulae; marching-scale functions. https://doi.org/10.28924/2291-8639-21-2023-68 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-68 2 int. j. anal. appl. (2023), 21:68 isogeodesic curve, that is, both a geodesic and a parameter curve in e4. then, we insert three types of the marching-scale functions, and give some examples for the purpose of clarity of our method. 2. preliminaries in this section we list some formulas and conclusions for space curves, and surfaces in euclidean 4-space e4 which can be found in [7-9, 17]: a curve is smooth if it admits a tangent vector at whole point of the curve. in the following argumentations, all curves are assumed to be regular. let α = α(s) be a unit speed curve in 4d euclidean space e4. we set up α ′ (s) 6= 0 for all s ∈ [0,l]; since this would give us a straight line. in this paper, α ′ (s) indicate to the derivatives of α(s) with respect to arc-length parameter s. for whole point of α(s), if the set {t(s), n(s), b1(s), b2(s)} is the serret–frenet frame along α(s), then:  t ′ (s) n ′ (s) b ′ 1(s) b ′ 2(s)   =   0 κ1 0 0 −κ1 0 κ2 0 0 κ2 0 κ3 0 −κ3 0 0     t(s) n(s) b1(s) b2(s)   , (2.1) where t, n, b1, and b2 are the tangent, the principal normal, the first binormal, and the second binormal vector fields; κi(s) (i = 1, 2, 3) are the ith curvature functions (κ1, κ2 > 0) of the curve α(s). for any three vectors x, y, z∈e4, the vectorial product is defined by x∧y∧z= ∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 e4 a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 ∣∣∣∣∣∣∣∣∣∣∣ , (2.2) where ei (i =1, 2, 3,4) are the standard base vectors of e4. theorem 2.1. let α: i 7→ e4 be a unit-speed curve. then the serret–frenet vectors of the curve are given by t(s)= α ′ (s), n(s)= α ′′ (s)∥∥∥α′′(s)∥∥∥, b2(s)=− α ′ (s)∧α ′′ (s)∧α ′′′ (s)∥∥∥α′(s)∧α′′(s)∧α′′′(s)∥∥∥, b1(s)=b2(s)∧ t(s)∧n(s). theorem 2.2. let α: i 7→e4 be a unit-speed curve. then the curvatures of the curve are given by: κ2(s)= < b1,α ′′′ > κ1 , and κ3(s)= < b2,α (4) > κ1κ2 . we indicate a surface m in e4 by m :p(s,t,r)= (x1 (s,t,r) ,x2 (s,t,r) ,x3 (s,t,r) ,)x4 (s,t,r) , (s,t,r)∈ d ⊆r3. (2.3) int. j. anal. appl. (2023), 21:68 3 if pj(s,t,r)= ∂p ∂j , the normal vector field of m is defined as follows [12] n(s,t,r)=ps ∧pt ∧pr, (2.4) which is orthogonal to each of the vectors ps, pt, and pr. similar to the euclidean 3-space e3, the following definition can be given: definition 2.1 let α: i 7→ e4 be a unit-speed curve. then the hyperplanes which correspond to the subspaces sp{t,b1,b2}, sp{t,n,b1}, sp{t,n,b2}, and sp{n,b1,b2}, respectively, are named the rectifying hyperplane, first osculating hyperplane, second osculating hyperplane, and normal hyperplane. the projection of a hypersurface into 3-space generally leads to a 3-dimensional volume. if we fix whole of the three variables, one at a time, we obtain three distinguished families of 2-spaces in 4-space. the projections of these 2-surfaces into 3-space are surfaces in 3-space. thus, they can be displayed by 3d rendering methods [12]. take x4 = 0 subspace and assuming r =constant for example, then the surface is parametrized as m :px4(s,t)= (x1 (s,t) ,x2 (s,t) ,x3 (s,t)) , (s,t)∈ d ⊆r 2. (2.5) 3. hypersurfaces with a common geodesic curve in this section, we consider a new approach for constructing a hypersurface family with a common geodesic curve α(s), 0 ≤ s ≤ l, in which the hypersurface tangent plane is coincident with the rectifying hyperplane sp{t,b1,b2}. then, the construction of the surface over α(s) is: m :p(s,t,r)= α(s)+u(s,t,r)t(s)+v(s,t,r)b1(s)+w(s,t,r)b2(s), (3.1) where u(s,t,r), v(s,t,r), and w(s,t,r) are all regular functions; 0≤ t ≤ t, 0≤ r ≤ h. these functions are named the marching-scale functions. from now on, we shall often not write the parameters s, t, and r explicitly in the functions u(s,t,r) v(s,t,r), and w(s,t,r). our aim is to find necessary and sufficient conditions for which the given α(s) is an iso-parametric and geodesic (geodesic for short) on the hypersurface p(s,t,r). the p′s tangent vectors are: ps =(1+us)t+(uκ1 −vκ2)n+(vs −w)b1 +(ws +vκ3)b2, pt = utt+vtb1 +wtb2, pr = urt+vrb1 +wrb2.   (3.2) the normal vector field is n(s,t,r) :=ps ∧pt ∧pr = η1t(s)+η2n(s)+η3b1(s)+η4b2(s), (3.3) 4 int. j. anal. appl. (2023), 21:68 where η1(s,t,r) = ∣∣∣∣∣∣∣∣ 0 vs ws 0 vt wt 0 vr wr ∣∣∣∣∣∣∣∣ =0, η2(s,t,r)= ∣∣∣∣∣∣∣∣ 1+us vs ws ut vt wt ur vr wr ∣∣∣∣∣∣∣∣ , η3(s,t,r) = ∣∣∣∣∣∣∣∣ 1+us 0 vs ut 0 vt ur 0 vr ∣∣∣∣∣∣∣∣ =0, η4(s,t,r)= ∣∣∣∣∣∣∣∣ 1+us 0 vs ut 0 vt ur 0 vr ∣∣∣∣∣∣∣∣ =0. since the α(((s))) is an iso-parametric curve on the hypersurface there exists t = t0 ∈ [0,t], and r = r0 ∈ [0,h] such that p(s,t0, r0)= α(((s))); that is, u(s,t0, r0)= v(s,t0, r0)= w(s,t0, r0)=0, us(s,t0, r0)= vs(s,t0, r0)= ws(s,t0, r0)=0. } (3.4) therefore, when t = t0, and r = r0—i.e., along the curve α(s)—the hypersurface normal is n(s,t0, r0)= (vt(s,t0, r0)wr(s,t0, r0)−wt(s,t0, r0)vr(s,t0, r0))n(s). (3.5) coincidence of the hypersurface normal n with the principal normal n(s) identifies the curve as a geodesic curve. then, we can state the following theorem: theorem 3.1. the given spatial curve α(s) is a geodesic curve on the hypersurface p(s,t,r) iff u(s,t0, r0)= v(s,t0, r0)= w(s,t0, r0)=0, us(s,t0, r0)= vs(s,t0, r0)= ws(s,t0, r0)=0, vt(s,t0, r0)wr(s,t0, r0)−wt(s,t0, r0)vr(s,t0, r0) 6=0,   (3.6) where 0≤ t ≤ t, 0≤ r ≤ h. evidently, eqs. (3.6) is further elegant and simple for applications (compare with [5], eqs. (9)). we call the set of hypersurfaces given by eqs. (3.1) and satisfying eqs. (3.6) a geodesic hypersurface family. for get better the conditions in theorem 3.1, the marching-scale functions u(s,t,r) v(s,t,r), and w(s,t,r) can be formed into three the following types: type (a). let u(s,t,r)= l(s)u(t,r), v(s,t,r)= m(s)v (t,r), w(s,t,r)= n(s)w(t,r), (3.7) where u(t,r), v (t,r), w(t,r)∈ c1, and l(s), m(s), n(s) are not identically zero. then, α(((s))) being a geodesic curve on the hypersurface p(s,t,r) iff int. j. anal. appl. (2023), 21:68 5 u(t0, r0)= v (t0, r0)= w(t0, r0)=0, (vtwr −wtvr)(t0, r0) 6=0, m(s) 6=0, and n(s) 6=0; 0≤ t0 ≤ t, 0≤ r ≤ h.   (3.8) type (b). let u(s,t,r)= l(s,t)u(r), v(s,t,r)= m(s,t)v (r), w(s,t,r)= n(s,t)w(r), (3.9) where u(t,r), v (t,r), w(t,r)∈ c1, and l(s), m(s), n(s) are not identically zero. then, α(((s))) being a geodesic curve on the hypersurface p(s,t,r) iff l(s,t0)u(r0)= m(s,t0)v (r0)= n(s,t0)w(r0)=0, v (r0)mt(s,t0)n(s,t0) dw(r0) dr −w(r0)nt(s,t0)m(s,t0) dv (r0) dr 6=0, 0≤ t0 ≤ t, 0≤ r ≤ h.   (3.10) type (c). let u(s,t,r)= l(s,r)u(t), v(s,t)= m(s,r)v (t), w(s,t)= n(s,r)w(t), (3.11) where u(t), v (t), w(t)∈ c1, and l(s,r), m(s,r), n(s,r) are not identically zero. hence, α(s) being a geodesic curve on the hypersurface p(s,t,r) iff l(s,r0)u(t0)= m(s,r0)v (t0)= n(s,r0)w(t0)=0, m(s,r0) dv (r0) dt nr(s,r0)w(t0)−n(s,r0)dwdt mr(s,t0)v (t0) 6=0, 0≤ t0 ≤ t, 0≤ r ≤ h.   (3.12) 3.1. example. now, we are interesting with an example to emphasize the method. example 3.1. let the curve α(((s))) be α(s)= ( 1 2 coss, 1 2 sins, 1 2 s, 1 √ 2 s ) , 0≤ s ≤ 2π. then, t(s)= (−1 2 sins, 1 2 coss, 1 2 , 1√ 2 ), n(s)= (−coss,−sins,0,0), b2(s)= ( 0,0, √ 6 3 ,− √ 3 3 ) b1(s)= (− √ 3 2 sins, √ 3 2 coss,− √ 3 6 ,− √ 6 6 ).   6 int. j. anal. appl. (2023), 21:68 thus, the hypersurface family with a common geodesic curve α(((s))) can be expressed as m :p(s,t,r)=   1 2 coss − 1 2 u(s,t,r)sins − √ 3 2 v(s,t,r)sins 1 2 sins + 1 2 u(s,t,r)coss + √ 3 2 v(s,t,r)coss 1 2 s + 1 2 u(s,t,r)− √ 3 6 v(s,t,r)+ √ 6 3 w(s,t,r) 1√ 2 s + 1√ 2 u(s,t,r)− 1√ 6 v(s,t,r)− 1√ 3 w(s,t,r)   , (3.13) where 0 ≤ s ≤ 2π, 0 ≤ t0 ≤ t , and 0 ≤ r ≤ h. a thorough treatment on p(s,t,r) will be given in the following: marching-scale functions of type (a). taking l(s)= m(s)= n(s)=1, and u(t,r)= (t − t0)(r − r0), v (t,r)= t − t0, w(t,r)= r − r0, with 0≤ r,t ≤ 1. then, we obtain u(s,t,r)= (t − t0)(r − r0), v(s,t)= t − t0, w(s,t)= r − r0, where 0≤ r,t ≤ 1, and with 0≤ s ≤ 2π. thereby, eq. (3.13) become: m :p(s,t,r)=   1 2 coss − 1 2 (t − t0)(r − r0)sins − √ 3 2 (t − t0)sins 1 2 sins + 1 2 (t − t0)(r − r0)coss + √ 3 2 (t − t0)coss 1 2 s + 1 2 (t − t0)(r − r0)− √ 3 6 (t − t0)+ √ 6 3 (r − r0) 1√ 2 s + 1√ 2 (t − t0)(r − r0)− 1√6(t − t0)− 1√ 3 (r − r0)   , where 0 ≤ r, t ≤ 1, 0 ≤ t0, r0 ≤ 1, and 0 ≤ s ≤ 2π. the position of the curve α(s) can be set on the hypersurface by changing the parameters t0 and r0. setting t0 = 1 and r0 = 0. then, the hypersurface p(s,t,r) becomes m :p(s,t,r)=   1 2 coss − 1 2 r(t −1)sins − √ 3 2 (t −1)sins 1 2 sins + 1 2 r(t −1)coss + √ 3 2 (t −1)coss 1 2 s + 1 2 r(t −1)− √ 3 6 (t −1)+ √ 6 3 r 1√ 2 s + 1√ 2 r(t −1)− 1√ 6 (t −1)− 1√ 3 r   depending on the 3d rendering methods, if we (parallel) project the hypersurface p(s,t,r) into the x4 =0 subspace and fixing r = 1 2 the hypersurface is m :px4(s,t, 1 2 )=   1 2 coss − 1 2 (t −1) ( 1 2 + √ 3 ) sins 1 2 sins + 1 2 (t −1) ( 1 2 + √ 3 ) coss 1 2 s + 1 2 (t −1) ( 1 2 + 1√ 3 ) + 1√ 6   where 0≤ t ≤ 1, and 0≤ s ≤ 2π, in 3-space drawn in figure 1-type (a). int. j. anal. appl. (2023), 21:68 7 figure 1. projection of a member of the hypersurface family and its geodesic. let m(s,t) = s + t +1, n(s,t)= (s +1)(t − t0), u(r) = 0, v (r)= r − r0, w(r)=1. then, u(s,t,r)=0, v(s,t)= (s + t +1)(r − r0) , w(s,t)= (s +1)(t − t0). thus, the eq. (3.13) become: m :p(s,t,r)=   1 2 coss − √ 3 2 (s + t +1)(r − r0)sins 1 2 sins + √ 3 2 (s + t +1)(r − r0)coss 1 2 s +− √ 3 6 (s + t +1)(r − r0)+ √ 6 3 (s +1)(t − t0) 1√ 2 s − 1√ 6 (s + t +1)(r − r0)− 1√3 (s +1)(t − t0)   . similarly, we may choose t0 =1/2 and r0 =0, so that m :p(s,t,r)=   1 2 coss − √ 3 2 r (s + t +1)sins 1 2 sins + √ 3 2 r (s + t +1)coss 1 2 s +− √ 3 6 r (s + t +1)+ √ 6 3 (s +1)(t − 1 2 ) 1√ 2 s − 1√ 6 r (s + t +1)− 1√ 3 (s +1)(t − 1 2 )   , hence, if we (parallel) project the hypersurface p(s,t,r) into the x3 =0 subspace, and taking t = 1 2 we get m :px3(s, 1 2 , r)= ( 1 2 coss, 1 2 sins, 1 √ 2 s − 1 √ 3 r (s +1) ) where 0≤ r ≤ 1, and 0≤ s ≤ 2π, in 3-space drawn in figure 2-type (b). 8 int. j. anal. appl. (2023), 21:68 figure 2. projection of a member of the hypersurface family and its geodesic. m(s,r) = (r − r0)sins, n(s,r)= sr2, u(t) = 0, v (t)=1, w(r)= t − t0. then, we obtain u(s,t,r)=0, v(s,t,r)= (r − r0)sins, w(s,r)= sr2 (t − t0) . the eq. (3.13) become: m :p(s,t,r)=   1 2 coss − √ 3 2 (r − r0)sins sins 1 2 sins + √ 3 2 (r − r0)sins coss 1 2 s − √ 3 2 (r − r0)sins + √ 6 3 (r − r0) 1√ 2 s − 1√ 6 (r − r0)sins − 1√3sr 2 (t − t0)   . similarly, we can choose t0 =1 and r0 =1, so that m :p(s,t,r)=   1 2 coss − √ 3 2 (r −1)sins sins 1 2 sins + √ 3 2 (r −1)sins coss 1 2 s − √ 3 2 (r −1)sins + √ 6 3 (r −1) 1√ 2 s − 1√ 6 (r −1)sins − 1√ 3 sr2 (t −1)   . similarly, if we (parallel) project the hypersurface p(s,t,r) into the x1 = 0 subspace, and setting r =1 we get m :px1(s,t,1)= ( 1 2 sins, 1 2 s, 1 √ 2 s + s √ 6 (t −1) ) , where 0≤ t ≤ 1, and 0≤ s ≤ 2π, in 3-space drawn in figure 3-type (c). int. j. anal. appl. (2023), 21:68 9 figure 3. projection of a member of the hypersurface family and its geodesic. 4. conclusion in this study, we have considered a mathematical framework, for constructing a surface family whose members all share a given geodesic curve as an isoparametric curve in e4. given a regular spatial curve, we answer question about the necessary and sufficient condition for the given curve to be a geodesic. lastly, as an application of our approach one example for each type of marching-scale functions is given. hopefully these results will lead to a wider usage of surfaces in geometric modeling, garment-manufacture industry, and the manufacturing of products. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] r.a. abdel-baky, n. alluhaibi, surfaces family with a common geodesic curve in euclidean 3-space e3, int. j. math. anal. 13 (2019), 433–447. https://doi.org/10.12988/ijma.2019.9846. [2] r.a. al-ghefari, r.a. abdel-baky, an approach for designing a developable surface with a common geodesic curve, int. j. contemp. math. sci. 8 (2013), 875–891. https://doi.org/10.12988/ijcms.2013.39101. [3] m. altin, a. kazan, h.b. karadag, hypersurface families with smarandache curves in galilean 4-space, commun. fac. sci. univ. ankara ser. a1. math. stat. 70 (2021), 744–761. https://doi.org/10.31801/cfsuasmas.794779. [4] g.s. atalay, f. guler, e. bayram, e. kasap, an approach for designing a surface pencil through a given geodesic curve, (2015). http://arxiv.org/abs/1406.0618. [5] e. bayram, e. kasap, parametric representation of a hypersurface family with a common spatial geodesic, (2014). http://arxiv.org/abs/1305.0411. [6] e. bayram, e. kasap, hypersurface family with a common isoasymptotic curve, geometry. 2014 (2014), 623408. https://doi.org/10.1155/2014/623408. [7] m.p. do carmo, differential geometry of curves and surfaces, prentice hall, englewood cliffs, 1976. [8] g. farin, curves and surfaces for computer aided geometric design, 2nd ed., academic press, new york, 1990. [9] j. hoschek, d. lasser, fundamentals of computer aided geometric design, a.k. peters, wellesley, ma, 1993. [10] j. zhou, visualization of four-dimensional space and its applications, ph.d. thesis, purdue university, 1991. [11] e. kasap, f.t. akyildiz, surfaces with common geodesic in minkowski 3-space, appl. math. comput. 177 (2006), 260–270. https://doi.org/10.1016/j.amc.2005.11.005. [12] e. kasap, family of surface with a common null geodesic, int. j. phys. sci. 4 (2009), 428-433. https://doi.org/10.12988/ijma.2019.9846 https://doi.org/10.12988/ijcms.2013.39101 https://doi.org/10.31801/cfsuasmas.794779 http://arxiv.org/abs/1406.0618 http://arxiv.org/abs/1305.0411 https://doi.org/10.1155/2014/623408 https://doi.org/10.1016/j.amc.2005.11.005 10 int. j. anal. appl. (2023), 21:68 [13] r. makki, hypersurfaces with a common asymptotic curve in the 4d galilean space g4, asian-eur. j. math. 15 (2022), 2250199. https://doi.org/10.1142/s1793557122501996. [14] g.j. wang, k. tang, c.l. tai, parametric representation of a surface pencil with a common spatial geodesic, computer-aided design. 36 (2004), 447–459. https://doi.org/10.1016/s0010-4485(03)00117-9. [15] d.w. yoon, z.k. yuzbas, an approach for curve in the 4d galilean space g4, j. korean soc. math. educ. ser. b: pure appl. math. 25 (2018), 229-241. [16] z.k. yuzbas, m. bektas, on the construction of a surface family with common geodesic in galilean space g3. open phys. 14 (2016), 360-363. https://doi.org/10.1515/phys-2016-0041. [17] z.k. yuzbas, d.w. yoon, on constructions of surfaces using a geodesic in lie group, j. geom. 110 (2019), 29. https://doi.org/10.1007/s00022-019-0487-x. https://doi.org/10.1142/s1793557122501996 https://doi.org/10.1016/s0010-4485(03)00117-9 https://doi.org/10.1515/phys-2016-0041 https://doi.org/10.1007/s00022-019-0487-x 1. introduction 2. preliminaries 3. hypersurfaces with a common geodesic curve 3.1. example 4. conclusion references international journal of analysis and applications volume 16, number 5 (2018), 628-642 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-628 generalized complex fuzzy and anti-fuzzy hv-subgroups m. al tahan1 and b. davvaz2,∗ 1department of mathematics, lebanese international university, lebanon 2department of mathematics, yazd university, yazd, iran ∗corresponding author: davvaz@yazd.ac.ir abstract. in this paper, we introduce the concept of generalized complex fuzzy subhypergroup (hvsubgroup) as well as the generalized concept of complex anti-fuzzy subhypergroup (hv-subgroup). we investigate their properties and their relations with the generalized traditional fuzzy (anti-fuzzy) subhypergroup (hv-subgroup). 1. introduction hyperstructure theory was born in 1934, when marty [8] gave the definition of hypergroup as a natural generalization of the concept of group based on the notion of hyperoperation at the eighth congress of scandinavian mathematicians. he analyzed their properties and applied them to groups, illustrated some applications and showed its utility in the study of groups, algebraic functions and relational fractions. recently, the hypergroups are studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics: geometry, topology, cryptography and code theory, graphs and hypergraphs, probability theory, binary relations, theory of fuzzy and rough sets, automata theory, economy, etc. (see [4, 6]). a hypergroup is an algebraic structure similar to a group, but the composition of two elements is a non-empty set. on the other hand, the fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. it was introduced in 1965 after the publication of l. a. zadeh received 2017-10-30; accepted 2018-01-11; published 2018-09-05. 2010 mathematics subject classification. 20n20, 20n25, 03e72 . key words and phrases. hypergroup; complex fuzzy set; generalized fuzzy subhypergroup. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 628 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-628 int. j. anal. appl. 16 (5) (2018) 629 (see [15]), who is considered as the pioneer of this theory, as an extension of the classical notion of set, when he proposed the idea of a multi-valued logic, which extends the traditional concept of a bivalent logic, which becomes a particular case of the new theory. the fuzzy set theory is based on the principle called by l. a. zadeh “the principle of incompatibility”, that is “the closer a phenomenon is studied, the more indistinct its definition becomes”. fuzzy sets are sets whose elements have degrees of membership. in classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. by contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. many researchers worked on fuzzy set theory, its applications and its extensions. an important extension of this theory was proposed by raymot et al. [10, 11]. they introduced the concept of complex fuzzy sets in which the codomain of membership function was the unit disc of the complex plane. then they defined different fuzzy complex operations and relations such as the complement of a complex fuzzy set, intersection and union of complex fuzzy sets. davvaz et al. (see [7]) introduced the concept of generalized traditional fuzzy hv-subgroups and the authors in [1] introduced the concept of complex fuzzy and anti-fuzzy hv-subgroups. our paper extends their results to complex fuzzy sets, and it is constructed as follows: after an introduction, in section 2 we present some definitions and results about hyperstructures and traditional fuzzy subhyperstructures. in section 3, we present the results of generalized fuzzy hv-subgroups and introduce the concept of generalized anti-fuzzy hv-subgroups. in section 4, we extend the definitions of generalized fuzzy and anti-fuzzy hvsubgroups and define generalized complex fuzzy and anti-fuzzy hv-subgroups. we investigate their properties and present different examples on them. 2. preliminaries in this section, we present some definitions and theorems related to hyperstructures and fuzzy subhyperstructures that are used throughout the paper. definition 2.1. let h be a non-empty set. then, a mapping ◦ : h × h → p∗(h) is called a binary hyperoperation on h, where p∗(h) is the family of all non-empty subsets of h. the couple (h,◦) is called a hypergroupoid. in the above definition, if a and b are two non-empty subsets of h and x ∈ h, then we define: a◦b = ⋃ a∈a b∈b a◦ b, x◦a = {x}◦a and a◦x = a◦{x}. int. j. anal. appl. 16 (5) (2018) 630 definition 2.2. a hypergroupoid (h,◦) is called a: • semihypergroup if for every x,y,z ∈ h, we have x◦ (y ◦z) = (x◦y) ◦z; • quasihypergroup if for every x ∈ h, x ◦ h = h = h ◦ x (this condition is called the reproduction axiom); • hypergroup if it is a semihypergroup and a quasihypergroup; • hv-group if it is a quasihypergroup and for every x,y,z ∈ h, we have x◦ (y ◦z) ∩ (x◦y) ◦z 6= ∅. definition 2.3. let (h,◦) be a hypergroup (or hv-group) and k ⊆ h. then (k,◦) is a subhypergroup (or hv-subgroup) of (h,◦) if for all a ∈ k, we have that a◦k = k ◦a = k. definition 2.4. [15] a fuzzy set, defined on a universe of discourse u is characterized by a membership function µa(x) that assigns any element a grade of membership in a. the fuzzy set may be represented by the set of ordered pairs a = {(x,µa(x)) : x ∈ u}, where µa(x) ∈ [0, 1]. definition 2.5. [7] let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x) ∈ [0, 1]. then a is a fuzzy subhypergroup (or hv-subgroup) of h if the following conditions hold: (1) inf{µa(z) : z ∈ x◦y}≥ min{µa(x),µa(y)} for all x,y ∈ h; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and min{µa(x),µa(a)}≤ µa(y); (3) for all x,a ∈ h, there exists z ∈ h such that x ∈ z ◦a and min{µa(x),µa(a)}≤ µa(z). definition 2.6. [7] let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an anti-fuzzy subhypergroup (or hv-subgroup) of h if the following conditions hold: (1) sup{µa(z) : z ∈ x◦y}≤ max{µa(x),µa(y)} for all x,y ∈ h; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(y) ≤ max{µa(x),µa(a)}; (3) for all x,a ∈ h, there exists z ∈ h such that x ∈ z ◦a and µa(z) ≤ max{µa(x),µa(a)}. theorem 2.1. [7] let (h,◦) be a hypergroup (or hv-group) and µ be a fuzzy subset of h. then µ is a fuzzy subhypergroup (or hv-subgroup) of h if and only if its complement µ c is an anti-fuzzy subhypergroup (or hv-subgroup) of h. here, µ c(x) = 1 −µ(x) for all x ∈ h. 3. generalized traditional fuzzy subhyperstructures davvaz et al. (see [7]) introduced the concept of generalized traditional fuzzy hv-subgroup. in this section, we present their results. and we introduce the concept generalized traditional anti-fuzzy hvsubgroup. int. j. anal. appl. 16 (5) (2018) 631 notation 3.1. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). we say that: (1) xt ∈ µa if µa(x) ≥ t; (2) xt ∈ qµa if µa(x) + t > 1; (3) xt ∈∨qµa if xt ∈ µa or xt ∈ qµa. otherwise, we say that xt ∈∨qµa. definition 3.1. [7] let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup) of h if for all t,s ∈]0, 1] and x,y ∈ h, the following conditions hold: (1) xt,ys ∈ µ implies zt∧s ∈∨qµ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ h such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ h such that x ∈ y ◦a. theorem 3.1. [7] let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈ ∨q) fuzzy subhypergroup (or hv-subgroup) of h if and only if for all x,y ∈ h, the following conditions hold: (1) µa(x) ∧µa(y) ∧ 0.5 ≤ µa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∧µa(a) ∧ 0.5 ≤ µa(y); (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∧µa(a) ∧ 0.5 ≤ µa(y). definition 3.2. let a = {(x,µa(x)) : x ∈ u} be a fuzzy set. then the set aπ = {(x, 2πµa(x)) : x ∈ u} is said to be a π-fuzzy set. proposition 3.1. let (h,◦) be a hypergroup (or hv-group). a π-fuzzy set aπ is an (∈,∈ ∨q) π-fuzzy subhypergroup (or hv-subgroup) of h if and only if a is an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup) of h. proof. the proof is straightforward. � notation 3.2. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). we say that: (1) xt ∈ µa if µa(x) ≤ t; (2) xt ∈ q̂µa if µa(x) + t < 1; (3) xt ∈∨q̂µa if xt ∈ µa or xt ∈ q̂µa. otherwise, we say that xt ∈∨q̂µa. definition 3.3. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h if for all t,s ∈]0, 1] and x,y ∈ h, the following conditions hold: int. j. anal. appl. 16 (5) (2018) 632 (1) xt,ys ∈ µ implies zt∨s ∈∨q̂µ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ h such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ h such that x ∈ y ◦a. theorem 3.2. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h if and only if for all x,y ∈ h, the following conditions hold: (1′) µa(x) ∨µa(y) ∨ 0.5 ≥∨µa(z) for all z ∈ x◦y, (2′) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∨µa(a) ∨ 0.5 ≥ µa(y), (3′) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∨µa(a) ∨ 0.5 ≥ µa(y). proof. (1 ⇒ 1′): suppose that x,y ∈ h. we consider the following cases: • case µ(x)∨µ(y) < 0.5. assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) > 0.5. it is clear that x0.5,y0.5 ∈ µ and that z0.5 is not in µ. having that µ(z) + 0.5 > 1 implies that z0.5 is not in q̂µ. we get that z0.5 ∈∨q̂µ. • case µ(x) ∨µ(y) ≥ 0.5. assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) > µ(x) ∨µ(y) ∨ 0.5 ≥ 0.5. choose a real number t such that 0.5 ≤ µ(x) ∨µ(y) < t < µ(z). it is clear that xt,yt ∈ µ and that zt is not in µ. having that µ(z) + t > 1 implies that zt is not in q̂µ. we get that zt ∈∨q̂µ. (2 ⇒ 2′): suppose that x,a ∈ h. we consider the following cases: • case µ(x) ∨µ(a) < 0.5. assume, to get contradiction, that for every y ∈ h such that x ∈ a◦y we have µ(y) > µ(a) ∨µ(x) ∨ 0.5 = 0.5. it is clear that x0.5,a0.5 ∈ µ and that y0.5 is not in µ. having that µ(y) + 0.5 > 1 implies that y0.5 is not in q̂µ. we get that y0.5 ∈∨q̂µ. • case µ(x)∨µ(a) ≥ 0.5. assume, to get contradiction, that for every y ∈ h such that x ∈ a◦y we have µ(y) > µ(a)∨µ(x)∨0.5 = µ(a)∨µ(x). choose a real number t such that 0.5 ≤ µ(x)∨µ(a) < t < µ(y). it is clear that xt,at ∈ µ and that yt is not in µ. having that µ(y) + t > 1 implies that yt is not in q̂µ. we get that yt ∈∨q̂µ. (3 ⇒ 3′): can be done in a similar manner to (2 ⇒ 2′). (1′ ⇒ 1): let xt,ys ∈ µ. then µ(x) ≤ t,µ(y) ≤ t. let z ∈ x◦y. we consider the following cases: • case t∨s < 0.5. we obtain that µ(z) ≤ µ(x)∨µ(y)∨0.5 ≤ t∨s∨0.5 ≤ 0.5. then µ(z) + t∨s < 1. then zt∨s ∈ q̂µ. • case t∨s ≥ 0.5. we obtain that µ(z) ≤ µ(x) ∨µ(y) ∨ 0.5 ≤ t∨s∨ 0.5 ≤ t∨s. then zt∨s ∈ µ. (2′ ⇒ 2): let xt,as ∈ µ. then µ(x) ≤ t,µ(y) ≤ t. let y ∈ h such that x ∈ a ◦ y. we consider the following cases: int. j. anal. appl. 16 (5) (2018) 633 • case t∨s < 0.5. we get that µ(y) ≤ µ(x)∨µ(a)∨0.5 ≤ t∨s∨0.5 ≤ 0.5. then µ(y) + t∨s ∈ µ < 1. • case t∨s ≥ 0.5. we get that µ(y) ≤ µ(x) ∨µ(a) ∨ 0.5 ≤ t∨s∨ 0.5 ≤ t∨s. then yt∨s ∈ µ. (3′ ⇒ 3): can be done in a similar manner to (2′ ⇒ 2). � proposition 3.2. let (h,◦) be a hypergroup (or hv-group). a π-fuzzy set aπ is an (∈,∈ ∨q̂) π-antifuzzy subhypergroup (or hv-subgroup) of h if and only if a is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h. proof. the proof is straightforward. � theorem 3.3. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈ ∨q) fuzzy subhypergroup (or hv-subgroup) of h if and only if its complement, ac, is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h. proof. let µa be an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup). then the conditions of theorem 3.1 are satisfied and we need to show that the conditions of theorem 3.2 are satisfied. to prove (1′), let x,y ∈ h. for all z ∈ x◦y, we have that µa(x) ∧µa(y) ∧ 0.5 ≤ µa(z). we get now that 1 − min{µa(x),µa(y), 0.5}≥ 1 −µa(z). the latter implies that max{µca(x),µ c a(y), 0.5}≥ µ c a(z). to prove (2′), let x,a ∈ h. then there exists y ∈ h such that x ∈ a◦y and µa(x)∧µa(a)∧0.5 ≤ µa(y). we get now that 1−min{µa(x),µa(a), 0.5}≥ 1−µa(y). the latter implies that max{µca(x),µ c a(a), 0.5}≥ µ c a(y). in a similar manner, we can prove the validity of condition (3′). thus, µca, is an (∈,∈ ∨q) anti-fuzzy subhypergroup (or hv-subgroup) of h. let µca be an (∈,∈∨q̂) anti-fuzzy subhypergroup (or hv-subgroup). then the conditions of theorem 3.2 are satisfied and we need to show that the conditions of theorem 3.1 are satisfied.. to prove (1), let x,y ∈ h. for all z ∈ x◦y, we have that µca(x)∨µ c a(y)∨0.5 ≥ µ c a(z). we get now that 1 −max{µ c a(x),µ c a(y), 0.5}≤ 1 −µca(z). the latter implies that min{µa(x),µa(y), 0.5}≤ µa(z). in order to prove (2), let x,a ∈ h. then there exists s y ∈ h such that x ∈ a ◦ y and µca(x) ∨ µca(a) ∨ 0.5 ≥ µ c a(y). we get now that 1 − max{µ c a(x),µ c a(a), 0.5} ≤ 1 − µ c a(y). the latter implies that min{µa(x),µa(a), 0.5}≤ µa(y). in a similar manner, we can prove the validity of condition (3). thus, µa, is an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup) of h. � 4. generalized complex fuzzy and anti-fuzzy subhyperstructures in this section, we use the concept of generalized fuzzy subhypergroups, discussed in section 3, to define complex fuzzy (anti-fuzzy) subhypergroups. and we investigate their properties. 4.1. generalized complex fuzzy hv-subgroups. int. j. anal. appl. 16 (5) (2018) 634 definition 4.1. [10] a complex fuzzy set, defined on a universe of discourse u is characterized by a membership function µa(x) that assigns any element a complex-valued grade of membership in a. the complex fuzzy set may be represented by the set of ordered pairs a = {(x,µa(x)) : x ∈ u}, where µa(x) = r(x)e iw(x), i = √ −1, r(x) ∈ [0, 1] and w(x) ∈ [0, 2π]. remark 4.1. by setting w(x) = 0 in the above definition, we return back to the traditional fuzzy set. definition 4.2. let a = {(x,µa(x)) : x ∈ h} be complex fuzzy subset of a non-void set h with membership function µa(x) = ra(x)e iwa(x). then a is said to be homogeneous if for all x,y ∈ h, we have ra(x) ≤ ra(y) if and only if wa(x) ≤ wa(y). notation 4.1. let a = {(x,µa(x)) : x ∈ h} and b = {(x,µb(x)) : x ∈ h} be complex fuzzy subsets of a non-void set h with membership functions µa(x) = ra(x)e iwa(x) and µb(x) = rb(x)e iwb (x) respectively. by µa(x) ≤ µb(x), we mean that ra(x) ≤ rb(x) and wa(x) ≤ wb(x). throughout this paper, all complex fuzzy sets are considered homogeneous. notation 4.2. let (h,◦) be a hypergroup (or hv-group) and a be a complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). we say, for all 0e0i < t = seiθ ≤ 1e2πi, that: (1) xt ∈ µa if ra(x) ≥ s and wa(x) ≥ θ; (2) xt ∈ qµa if ra(x) + s > 1 and wa(x) + θ > 2π; (3) xt ∈∨qµa if xt ∈ µa or xt ∈ qµa. otherwise, we say that xt ∈∨qµa. definition 4.3. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h if for all t,s ∈]0e0i, 1e2πi] and x,y ∈ h, the following conditions hold: (1) xt,ys ∈ µ implies zt∧s ∈∨qµ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ h such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ h such that x ∈ y ◦a. theorem 4.1. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h if and only if for all x,y ∈ h, the following conditions hold: (1′): µa(x) ∧µa(y) ∧ 0.5eiπ ≤ µa(z) for all z ∈ x◦y, (2′): for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∧µa(a) ∧ 0.5eiπ ≤ µa(y), int. j. anal. appl. 16 (5) (2018) 635 (3′): for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∧µa(a) ∧ 0.5eiπ ≤ µa(y). proof. (1 ⇒ 1′): suppose that x,y ∈ h. we consider the following cases: • case µ(x) ∧ µ(y) < 0.5eiπ. assume, to get contradiction, that there exists z ∈ x ◦ y such that µ(z) < µ(x)∧µ(y)∧0.5eiπ ≤ 0.5eiπ. choose a real number t such that µ(z) < t ≤ µ(x)∧µ(y) < 0.5eiπ. it is clear that xt,yt ∈ µ and that zt is not in µ. having that µ(z) + t < 1e2iπ implies that zt is not in qµ. we get that zt ∈∨qµ. • case µ(x)∧µ(y) ≥ 0.5eiπ. assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) < 0.5eiπ. it is clear that x0.5eiπ,y0.5eiπ ∈ µ and that z0.5eiπ is not in µ. having that µ(z)+0.5eiπ < 1e2iπ implies that z0.5eiπ is not in qµ. we get that z0.5eiπ ∈∨qµ. (2 ⇒ 2′): suppose that x,a ∈ h. we consider the following cases: • case µ(x) ∧µ(a) ≤ 0.5eiπ. assume, to get contradiction, that for every y ∈ h such that x ∈ a◦y we have µ(y) < µ(a) ∧ µ(x) ∧ 0.5eiπ = µ(a) ∧ µ(x). choose a real number t such that µ(y) < t < µ(x) ∧µ(a) < 0.5eiπ. it is clear that xt,at ∈ µ and that yt is not in µ. having that µ(y) + t < 1e2iπ implies that yt is not in qµ. we get that yt ∈∨qµ. • case µ(x)∧µ(a) > 0.5eiπ. assume, to get contradiction, that for every y ∈ h such that x ∈ a◦y we have µ(y) < µ(a) ∧µ(x) ∧ 0.5eiπ = 0.5eiπ. it is clear that x0.5eiπ,a0.5eiπ ∈ µ and that y0.5eiπ is not in µ. having that µ(y) + 0.5eiπ < 1e2iπ implies that y0.5eiπ is not in qµ. we get that y0.5eiπ ∈∨qµ. (3 ⇒ 3′): can be done in a similar manner to (2 ⇒ 2′). (1′ ⇒ 1): let xt,ys ∈ µ. then µ(x) ≤ t,µ(y) ≤ s. let z ∈ x◦y. we consider the following cases: • case t∧s ≤ 0.5eiπ. we get that µ(z) ≥ µ(x) ∧µ(y) ∧ 0.5eiπ ≥ t∧s∧ 0.5eiπ ≥ t∧s. then zt∧s ∈ µ. • case t ∧ s > 0.5eiπ. we get that µ(z) ≥ µ(x) ∧ µ(y) ∧ 0.5eiπ ≥ t ∧ s ∧ 0.5eiπ ≥ 0.5eiπ. then µ(z) + t∧s > 1e2iπ. thus, zt∧s ∈ q̂µ. (2′ ⇒ 2): let xt,as ∈ µ. then µ(x) ≤ t,µ(y) ≤ t. let y ∈ h such that x ∈ a ◦ y. we consider the following cases: • case t∧s ≤ 0.5eiπ. we get that µ(y) ≥ µ(x) ∧µ(a) ∧ 0.5eiπ ≥ t∧s∧ 0.5eiπ ≥ t∧s. then yt∧s ∈ µ. • case t ∧ s > 0.5eiπ. we get that µ(y) ≥ µ(x) ∧ µ(a) ∧ 0.5eiπ ≥ t ∧ s ∧ 0.5eiπ ≥ 0.5eiπ. then µ(y) + t∧s ∈ µ > 1e2iπ. (3′ ⇒ 3): can be done in a similar manner to (2′ ⇒ 2). � theorem 4.2. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h if and only if ra is an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup) of h and wa is an (∈,∈∨q) π-fuzzy subhypergroup (or hv-subgroup) of h. int. j. anal. appl. 16 (5) (2018) 636 proof. let a be an (∈,∈ ∨q) complex fuzzy subhypergroup (or hv-subgroup) of h. this is equivalent to having the conditions of theorem 4.1 satisfied and we can rewrite them as follows: (1) ra(x) ∧ra(y) ∧ 0.5 ≤ ra(z) and wa(x) ∧wa(y) ∧π ≤ wa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a ◦ y and ra(x) ∧ ra(a) ∧ 0.5 ≤ ra(y) and wa(x) ∧wa(a) ∧π ≤ wa(y); (3) for all x,a ∈ h, there exists y ∈ h such that ra(x) ∧ra(a) ∧ 0.5 ≤ ra(y) and wa(x) ∧wa(a) ∧π ≤ wa(y). the latter conditions are equivalent to having ra an (∈,∈∨q) fuzzy subhypergroup (or hv-subgroup) of h and wa an (∈,∈ ∨q) π-fuzzy subhypergroup (or hv-subgroup) of h as the conditions of theorem 2.6 are satisfied for both: ra and wa. � proposition 4.1. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). if a is a complex fuzzy subhypergroup (or hv-subgroup) of h then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h. proof. let a be a complex fuzzy subhypergroup (or hv-subgroup) of h. then the following conditions are satisfied for all x,y ∈ h: (1) µa(x) ∧µa(y) ≤ µa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∧µa(a) ≤ µa(y); (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∧µa(a) ≤ µa(y). we have that: (1) µa(x) ∧µa(y) ∧ 0.5eiπ ≤ µa(x) ∧µa(y) ≤ µa(z) for all z ∈ x◦y, (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x)∧µa(a)∧0.5eiπ ≤ µa(x)∧µa(a) ≤ µa(y), (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y◦a and µa(x)∧µa(a)∧0.5eiπ ≤ µa(x)∧µa(a) ≤ µa(y). therefore, a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h. � remark 4.2. the converse of proposition 4.1 is not always true. i.e., if a is an (∈,∈ ∨q) complex fuzzy subhypergroup (or hv-subgroup) of h then a may not be a complex fuzzy subhypergroup (or hv-subgroup) of h. we illustrate remark 4.2 by the following example. int. j. anal. appl. 16 (5) (2018) 637 example 4.1. let h = {0, 1, 2} and define the hv-group (h, +) by the following table: + 0 1 2 0 0 {1, 2} 2 1 {1, 2} 2 0 2 2 0 1 and define a complex fuzzy subset µ of h as: µ(0) = 0.8ei2π, µ(1) = 0.7ei 3π 2 and µ(2) = 0.6eiπ. then µ is an (∈,∈∨q) complex fuzzy hv-subgroup of h but it is not a complex fuzzy hv-subgroup of h. theorem 4.3. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h if and only if for all 0e0i < t ≤ 0.5eπi, µt = {x ∈ h : µa(x) ≥ t} 6= ∅ is a subhypergroup (or hv-subgroup) of h. proof. the proof is similar to that in [7]. � proposition 4.2. let (h,◦) be the biset hypergroup, i.e., x◦y = {x,y} for all x,y ∈ h and let µ be any homogeneous complex fuzzy subset of h. then µ is an (∈,∈∨q) complex fuzzy subhypergroup of h. proof. the proof follows from proposition 4.1 and having µ a complex fuzzy subhypergroup of h [1]. � proposition 4.3. let (h,◦) be the total hypergroup, i.e., x ◦ y = h for all x,y ∈ h and let µ be any homogeneous complex fuzzy subset of h. then µ is an (∈,∈ ∨q) complex fuzzy subhypergroup of h if and only if µ is a constant complex function or 0.5eiπ ≤ µ(x) ≤ 1e2iπ for all x ∈ h. proof. it is easy to see that if if µ is a constant complex function or 0.5eiπ ≤ µ(x) ≤ 1e2iπ for all x ∈ h then µ is an (∈,∈∨q) complex fuzzy subhypergroup of h. let µ be an (∈,∈∨q) complex fuzzy subhypergroup of h such that µ is not a constant complex function. suppose, to get contradiction, that there exists x ∈ h such that µ(x) = t < 0.5eiπ. then, by theorem 4.3, µt = {x ∈ h : µa(x) ≥ t} 6= ∅ is a subhypergroup (or hv-subgroup) of h. the latter implies that for all z ∈ h = x◦x µ(z) ≥ t. since µ is not a constant function, it follows that we can find y ∈ h such that µ(y) = t0 6= t = µ(x). we have two cases: t0 < t and t0 > t. we consider the case t0 < t and the other case can be done in a similar manner. having µt a subhypergroup (or hv-subgroup) of h implies that y ∈ h = x◦x ⊆ µt. � definition 4.4. let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or hv-group) h. then µ is called a complex fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h if for all x,y ∈ h, the following conditions are satisfied: int. j. anal. appl. 16 (5) (2018) 638 (1) µa(x) ∧µa(y) ∧β ≤ µa(z) ∨α for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∧µa(a) ∧β ≤ µa(y) ∨α; (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∧µa(a) ∧β ≤ µa(y) ∨α. remark 4.3. if α = 0e0i,β = 1ei2π, then we obtain the complex fuzzy subhypergroup (or hv-subgroup). and if α = 0e0i,β = 0.5eiπ, we have an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h. theorem 4.4. let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or hv-group) h. then µ is a complex fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h if and only if µt 6= ∅ is a subhypergroup (or hv-subgroup) of h for all t ∈]α,β]. proof. the proof is similar to that in [7]. � 4.2. generalized complex anti-fuzzy hv-subgroups. notation 4.3. let (h,◦) be a hypergroup (or hv-group) and a be a complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). we say, for all 0e0i ≤ t = seiθ < 1e2πi, that: (1) xt ∈ µa if ra(x) ≤ s and wa(x) ≤ θ; (2) xt ∈ q̂µa if ra(x) + s < 1 and wa(x) + θ < 2π; (3) xt ∈∨q̂µa if xt ∈ µa or xt ∈ qµa. otherwise, we say that xt ∈∨qµa. definition 4.5. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x). then a is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup (or hvsubgroup) of h if for all t,s ∈ [0e0i, 1e2πi[ and x,y ∈ h, the following conditions hold: (1) xt,ys ∈ µ implies zt∨s ∈∨q̂µ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ h such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ h such that x ∈ y ◦a. theorem 4.5. let (h,◦) be a hypergroup (or hv-group) and a be a fuzzy subset of h with membership function µa(x). then a is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h if and only if for all x,y ∈ h, the following conditions hold: (1) µa(x) ∨µa(y) ∨ 0.5eiπ ≥ µa(z) for all z ∈ x◦y}; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∨µa(a) ∨ 0.5eiπ ≥ µa(y); (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∨µa(a) ∨ 0.5eiπ ≥ µa(y). proof. the proof is similar to that of theorem 3.2. � theorem 4.6. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). then a is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup int. j. anal. appl. 16 (5) (2018) 639 (or hv-subgroup) of h if and only if ra is an (∈,∈∨q̂) fuzzy subhypergroup (or hv-subgroup) of h and wa is an (∈,∈∨q̂) π-anti-fuzzy subhypergroup (or hv-subgroup) of h. proof. let a be an (∈,∈ ∨q̂) complex fuzzy subhypergroup (or hv-subgroup) of h. this is equivalent to having the conditions of theorem 4.5 satisfied and we can rewrite them as follows: (1) ra(x) ∨ra(y) ∨ 0.5 ≥ ra(z) and wa(x) ∨wa(y) ∨π ≥ wa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a ◦ y and ra(x) ∨ ra(a) ∨ 0.5 ≥ ra(y) and wa(x) ∨wa(a) ∨π ≥ wa(y); (3) for all x,a ∈ h, there exists y ∈ h such that ra(x) ∨ra(a) ∨ 0.5 ≥ ra(y) and wa(x) ∨wa(a) ∨π ≥ wa(y). the latter conditions are equivalent to having ra is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or hv-subgroup) of h and wa is an (∈,∈∨q̂) π-anti-fuzzy subhypergroup (or hv-subgroup) of h as the conditions of theorem 3.2 are satisfied for both: ra and wa. � theorem 4.7. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h if and only if its complement, ac, is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup (or hv-subgroup) of h. proof. the proof results from theorems 3.3, 4.2 and 4.6. � proposition 4.4. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x) = ra(x)e iwa(x). if a is a complex anti-fuzzy subhypergroup (or hv-subgroup) of h then a is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or hv-subgroup) of h. proof. let a be a complex anti-fuzzy subhypergroup (or hv-subgroup) of h. then the following conditions are satisfied for all x,y ∈ h: (1) µa(x) ∨µa(y) ≥ µa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∨µa(a) ≥ µa(y); (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∨µa(a) ≥ µa(y). we have that: (1) µa(x) ∨µa(y) ∨ 0.5eiπ ≥ µa(x) ∨µa(y) ≥ µa(z) for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x)∨µa(a)∨0.5eiπ ≥ µa(x)∨µa(a) ≥ µa(y); (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y◦a and µa(x)∨µa(a)∨0.5eiπ ≥ µa(x)∨µa(a) ≥ µa(y). int. j. anal. appl. 16 (5) (2018) 640 therefore, a is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or hv-subgroup) of h. � remark 4.4. the converse of proposition 4.1 is not always true, i.e., if a is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or hv-subgroup) of h then a may not be a complex anti-fuzzy subhypergroup (or hv-subgroup) of h. we illustrate remark 4.4 by the following example. example 4.2. let h = {0, 1, 2} and define the hv-group (h, +) by the following table: + 0 1 2 0 0 {1, 2} 2 1 {1, 2} 2 0 2 2 0 1 and define a complex fuzzy subset µ of h as: µ(0) = 0.2ei0, µ(1) = 0.3ei π 2 and µ(2) = 0.4eiπ. then µ is an (∈,∈∨q̂) complex anti-fuzzy hv-subgroup of h but it is not a complex anti-fuzzy hv-subgroup of h. theorem 4.8. let (h,◦) be a hypergroup (or hv-group) and a be a (homogeneous) complex fuzzy subset of h with membership function µa(x). then a is an (∈,∈∨q̂) complex fuzzy subhypergroup (or hv-subgroup) of h if and only if for all 0.5eπi ≤ t < 1e2πi, µt = {x ∈ h : µa(x) ≤ t} 6= ∅ is a subhypergroup (or hv-subgroup) of h. proof. let µa be an (∈,∈ ∨q) complex fuzzy subhypergroup (or hv-subgroup) of h and let 0.5eπi ≤ t < 1e2πi. we need to show that a◦µt = µt◦a = µt for all a ∈ µt. we prove that a◦µt = µt and µt◦a = µt can be done in a similar manner. let x ∈ µt. then µa(x) ≤ t and µa(a) ≤ t. since µa is an (∈,∈∨q) complex fuzzy subhypergroup (or hv-subgroup) of h, it follows by theorem 4.5 that µa(z) ≤ µa(x) ∨µa(a) ∨ 0.5eπi = leqt∨0.5eπi ≤ t for all z ∈ x◦a. the latter implies that a◦µt ⊆ µt. to prove that µt ⊆ a◦µt, let x ∈ µt. then, by theorem 4.5, there exists y ∈ h such that x ∈ a◦y and µa(y) ≤ µa(x)∨µa(a)∨0.5eπi =≤ t∨0.5eπi ≤ t. thus, y ∈ µt and a◦µt ⊆ µt. conversely, let 0.5eπi ≤ t < 1e2πi and µt = {x ∈ h : µa(x) ≥ t} 6= ∅ be a subhypergroup (or hv-subgroup) of h. let t0 = µa(x) ∨ µa(y) ∨ 0.5eπi. then x,y ∈ µt0 . since µt0 is a subhypergroup (or hv-subgroup) of h, it follows that for all z ∈ x◦y, we have z ∈ µt0 . i.e., µa(x) ∨µa(y) ∨ 0.5eπi = t0 ≥ µa(z). now let x,a ∈ h and t1 = µa(x) ∨µa(x) ∨ 0.5eπi. then x,a ∈ µt1 . since µt1 is a subhypergroup (or hv-subgroup) of h, it follows that there exists y ∈ µt1 such that x ∈ a◦y. i.e., µa(x) ∨µa(a) ∨ 0.5eπi = t1 ≥ µa(y). � proposition 4.5. let (h,◦) be the biset hypergroup, i.e., x◦y = {x,y} for all x,y ∈ h and let µ be any homogeneous complex fuzzy subset of h. then µ is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup of h. int. j. anal. appl. 16 (5) (2018) 641 proof. the proof follows from proposition 4.4 and having µ a complex anti-fuzzy subhypergroup of h ( [1]). � proposition 4.6. let (h,◦) be the total hypergroup, i.e., x ◦ y = h for all x,y ∈ h and let µ be any homogeneous complex fuzzy subset of h. then µ is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup of h if and only if µ is a constant complex function or µ(x) ≤ 0.5eiπ for all x ∈ h. proof. let µ is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup of h. the statement is equivalent, by theorem 4.7, we get that µc is an (∈,∈ ∨q) complex fuzzy subhypergroup of h. the latter is equivalent, using proposition 4.3, to that µc is a constant complex function or µc(x) ≥ 0.5eiπ for all x ∈ h. � definition 4.6. let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or hv-group) h. then µ is called a complex anti-fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h if for all x,y ∈ h, the following conditions are satisfied: (1) µa(x) ∨µa(y) ∨α ≥ µa(z) ∧β for all z ∈ x◦y; (2) for all x,a ∈ h, there exists y ∈ h such that x ∈ a◦y and µa(x) ∨µa(a) ∨α ≥ µa(y) ∧β; (3) for all x,a ∈ h, there exists y ∈ h such that x ∈ y ◦a and µa(x) ∨µa(a) ∨α ≥ µa(y) ∧β. remark 4.5. if α = 0e0i,β = 1e2πi, we get the complex anti-fuzzy subhypergroup (or hv-subgroup). and if α = 0.5eiπ,β = 1ei2π, we get an (∈,∈∨q̂) complex fuzzy subhypergroup of h. theorem 4.9. let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or hv-group) h. then µ is a complex anti-fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h if and only if µt 6= ∅ is a subhypergroup (or hv-subgroup) of h for all t ∈ [α,β[. proof. let µ is a complex fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h. we need to show that a◦µt = µt ◦a = µt for all a ∈ µt. we prove that a◦µt = µt and µt ◦a = µt can be done in a similar manner. let x ∈ µt. then µ(x) ≤ t and µ(a) ≤ t. since µ is a complex anti-fuzzy subhypergroup (or hv-subgroup) with thresholds (α,β) of h, it follows that µ(z) ∧β ≤ µ(x) ∨µ(a) ∨α = t∨α ≤ t for all z ∈ x◦a. since β ≥ t, it follows that µ(z) ≤ t. the latter implies that a◦µt ⊆ µt. to prove that µt ⊆ a◦µt, let x ∈ µt. then, there exists y ∈ h such that x ∈ a◦y and µ(y) ∧β ≤ µ(x) ∨µ(a) ∨α = t∨α ≤ t. thus, y ∈ µt and a◦µt ⊆ µt. conversely, let α < t ≤ β and µt = {x ∈ h : µ(x) ≤ t} 6= ∅ be a subhypergroup (or hv-subgroup) of h. suppose that there exists z ∈ x◦y such that µ(z) ∧β > µ(x) ∨µ(y) ∨α = t. it is clear that x,y ∈ µt and z is not in µt which contradicts our hypothesis that µt is a subhypergroup (or hv-subgroup) of h. thus, condition 1. of definition 4.6 is satisfied. now assume that there exist a,x ∈ h such that for all y ∈ h, x ∈ a ◦ y, we have µ(y) ∧ β > µ(x) ∨ µ(a) ∨ α = t0. it is clear that x,a ∈ µt0 and y is not in µt0 int. j. anal. appl. 16 (5) (2018) 642 which contradicts our hypothesis that µt0 is a subhypergroup (or hv-subgroup) of h. thus, condition 2. of definition 4.6 is satisfied. we can prove condition 3. in a similar manner. � 5. conclusion this paper contributed to the study of fuzzy subhyperstructures by introducing the concepts of generalized complex fuzzy (anti-fuzzy) hv-subgroups and investigating their properties. references [1] m. al-tahan, b. davvaz, complex fuzzy hv-subgroups of an hv-group, submitted. [2] p. corsini, join spaces, power sets, fuzzy sets, proceedings of the fifth int. congress of algebraic hyperstructures and appl., 1993, iasi, romania, hadronic press, 1994. [3] p. corsini, a new connection between hypergroups and fuzzy sets, southeast asian bull. math. 27(2003), 221-229. [4] p. corsini and v. leoreanu, applications of hyperstructures theory, advances in mathematics, kluwer academic publisher, 2003. [5] b. davvaz, fuzzy hv-groups, fuzzy sets syst. 101 (1999), 191-195. [6] b. davvaz, polygroup theory and related systems, world scientific publishing co. pte. ltd., hackensack, nj, 2013. viii+200 pp. [7] b. davvaz and i. cristea, fuzzy algebraic hyperstructuresan introduction, studies in fuzziness and soft computing 321. cham: springer, 2015. [8] f. marty, sur une generalization de la notion de group, in 8th congress math. scandenaves, (1934), 45-49. [9] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 512?17. [10] d. ramot, r. milo, m. friedman and a. kandel, complex fuzzy logic, ieee trans. fuzzy syst. 10(2) (2002), 171-186. [11] d. ramot, m. friedman, g. langholz and a. kandel, complex fuzzy sets, ieee trans. fuzzy syst. 11(4) (2003), 450-461. [12] t. vougiouklis, hyperstructures and their representations, aviani editor. hadronic press, palm harbor, usa, 1994. [13] t. vougiouklis, a new class of hyperstructures, j. combin. inform. syst. sci. 20 (1995), 229?35. [14] t. vougiouklis, the fundamental relation in hyperrings. the general hyperfield, in: proc of the 4th int. congress on algebraic hyperstructures and appl. (a.h.a 1990). world sientific, xanthi, greece, pp 203-211. [15] l. a. zadeh, fuzzy sets, inf. control 8 (1965), 338-353. 1. introduction 2. preliminaries 3. generalized traditional fuzzy subhyperstructures 4. generalized complex fuzzy and anti-fuzzy subhyperstructures 4.1. generalized complex fuzzy hv-subgroups 4.2. generalized complex anti-fuzzy hv-subgroups 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 192-200 http://www.etamaths.com convergence of a modified multi-step iterative scheme for pnearly uniformly l-lipschitzian asymptotically pseudocontrative mappings v.o. olisama∗, a.a. mogbademu, j.o. olaleru abstract. in this paper, it is shown that the modified multi-step iteration converges strongly to the common fixed point of a finite family of nearly uniformly llipschitzian asymptotically pseudocontractive mappings. the main result is an improvement and extension of well known results in the literature. 1. preliminary. let e be a real banach space and let e∗ be its dual space. the normalized duality mappping j : e → 2e ∗ is defined by j(x) = {f ∈ x∗ : < x,f > = ‖x‖‖f‖}, where < .,. > denotes the generalized duality pairing and ‖x‖ = ‖f‖. we shall denote the single-valued normalized duality pairing by j. j satisfies the following properties: (1) j is an odd mapping, i.e j(−x) = −j(x). (2) j is positive homogeneous , i.e for any number λ > 0, j(λx) = λj(x). (3) j is bounded, i.e. for any subset a of e j(a) is a bounded subset of e∗. (4) if e is smooth (or e∗ is strictly convex), then j is singled-valued. consistent with goebel and kirk [4] we give the following definitions, let k be a nonempty closed convex subset of e and t : k → k be a map. definition 1.1 a mapping t is said to be asymptotically nonexpansive if for each x,y ∈ k < tnx−tny >≤ kn‖x−y‖, 2010 mathematics subject classification. 47h10. key words and phrases. modified multi-step iteration process, asymptotically pseudocontrative maps, uniformly llipschitzian maps, nearly uniformly l-lipschitzian maps, nearly asymptotically pseudocontrative maps, banach spaces. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 192 convergence of a modified multi-step iterative scheme 193 ∀n ≥ 0, where the sequence {kn}⊂ [1,∞) with limn→∞kn = 1. definition 1.2 the mapping is said to be uniformly llipschitzian if there exists a constant l ≥ 0 such that ‖tn −tny‖≤ l‖x−y‖, for any x,y ∈ k and ∀n ≥ 0. we give the definition of asymptotically pseudocontractive as in [14]. definition 1.3[14] the mapping t is said to be asymptotically pseudocontractive if there exists a sequence {kn}⊂ [1,∞) with limn→∞kn = 1 and for any x,y ∈ k there exist j(x−y) ∈ j(x−y) such that < tnx−tny,j(x−y) >≤ kn‖x−y‖2, ∀n ≥ 0. remark 1.4 every asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly llipschitzian. the converse is not true in general. clearly, every operator which is asymptotically pseudocontractive in general may not admit a fixed point. the existence of fixed point result for asymptotically pseudocontractive maps depend on the space, nature of subset and further properties of the operators (see, [1]). infact, asymptotically nonexpansive and asymptotically pseudocontractive were first introduced by goebel and kirk [4] and schu [15] respectively. since then many authors have studied several iterative process for asymptotically nonexpansive and asymptotically pseudocontractive in both hilbert and banach spaces (see, [7, 8, 12]). schu[15] proved the convergence of mann[6] iterative sequence to the fixed point of uniformly l-lipschitzian and asymptotically pseudocontractive mappings in the setting of hilbert space. in 2001, chang kextended the work of schu to the setting of real uniformly smooth banach spaces. also, ofoedu [11] extended theorem 1.2 of chang[1] to the setting of arbitrary real banach spaces and dropped the boundedness assumption. he proved the following theorem: theorem 1.5[11] let e be a real banach space, k be a nonempty closed convex subspace of e and t : k → k uniformly l-lipschitzian and asymptotically pseudocontractive mappings with a sequence {kn}n≥0 ⊂ [1,∞),kn → 1 and x∗ ∈ f(t). let the sequence {αn} be a sequence in [0,1] satisfying the following conditions: (a-1) ∑∞ n=0 αn = ∞; (a-2) ∑∞ n=0 α 2 n < ∞; (a-3) ∑∞ n=0 αn(kn − 1) < ∞. for any x0 ∈ k, let {xn}∞n≥0 be an iterative sequence defined by xn+1 = (1 −αn)xn + αntnxn, (1.1) 194 olisama, a.a. mogbademu and j.o. olaleru for all n ≥ 0. if there exists a strictly increasing function φ : [0,∞) → [0,∞), φ(0) = 0 such that < tnx−x∗,j(x−x∗) > ≤ kn‖x−x∗‖2 − φ(‖x−x∗‖) ∀ n ≥ 0. then (1) {xn}∞n≥0 is bounded; (2) {xn}∞n≥0 converges strongly to x ∗ ∈ f(t). further more, chang cho and kim [2] improved on the theorem above by extending the parameters and modifying the iterative procedure. infact, they proved the following theorem: theorem 1.6[2] let e be a real banach space, k be a nonempty closed and convex subspace of e and t : k → k uniformly l-lipschitzian and asymptotically pseudocontractive mappings with a sequence {kn}n≥0 ⊂ [1,∞), kn → 1 and x∗ ∈ f(t). let {an},{bn} and{cn} be a real sequences in [0,1] satisfying the following conditions: (a-1) an + bn + cn = 1; (a-2) ∑∞ n=0(bn + cn) = ∞; (a-3) ∑∞ n=0(bn + cn) 2 < ∞; (a-4) ∑∞ n=0(bn + cn)(kn − 1) < ∞; (a-5) ∑∞ n=0 cn < ∞. for any x0 ∈ k, let {xn}∞n≥0 be a sequence in k iteratively defined by xn+1 = anxn + bnt nxn + cnun, for all n ≥ 0. where {un} is a bounded sequence in k. suppose that there exists a strictly increasing continuous function φ : [0,∞) → [0,∞), φ(0) = 0 such that < tnx−x∗,j(x−x∗) > ≤ kn‖x−x∗‖2 − φ(‖x−x∗‖), ∀ n ≥ 0. then (1) {xn}∞n≥0 is bounded; (2) {xn}∞n≥0 converges strongly to x ∗ ∈ f(t). another extension of the fixed point theory is the iterative processes for approximating fixed points of mappings. several authors have studied and extended the mann[6],ishikawa[5], noor[10] and multi-step[13] iterative process to evaluate the fixed point of uniformly l-lipschitzian and asymptotically pseudocontractive mappings in hilbert and banach spaces.( see[7,8,16,17]). we remarked that in all these theorems above, for certain application the continuity assumption becomes a rather strong condition. in this direction, a natural question arises that whether there is any class of(not necessarily continuous) mapping more general than the class of asymptotically nonexpansive and asymptotically pseudocontractive (which has asymptotically nonexpansiveness)? motivated by this inspired question, sahu[14] introduced the classes of nearly contraction and nearly asymptotically nonexpansive mappings. he gave the following definition: definition 1.7 [14] let k be a nonempty subset of a banach space e andj sequence convergence of a modified multi-step iterative scheme 195 {an} in [0,∞) with an → 0. a mapping t : k → k is called nearly lipschitzian with respect to {an} if for each n ∈ n, there exists a constant kn ≥ 0 such that : < tnx−tny > ≤ kn(‖x−y‖ + an) for all x,y ∈ k. remark 1.8 it is important to note that lipschitzian mappings are always continuous but nearly lipschitzian mappings need not be continuous. the class of nearly asymptotically nonexpansive mappings contains the class of asymptotically nonexpasive mappings and is contained in the class of mappings of asymptotically nonexpansive type. hence, according to sahu[14] nearly asymptotically pseudocontractive mapping is a generalisation of asymptotically pseudocontractive mapping. example 1.9[16] let e = r and t : k → k be defined by: t(x) =   x 2 x ∈ [0, 1); 0 x = 1. then t is a discontinuous mapping which is not lipschitzian, but nearly 1 2 lipschitzian with sequence { 1 2n }. sahu[14] proved some theorems in an attempt to develop asymptotically fixed point theory for a more general class of demicontinuous nearly lipschirzian mappings in banach spaces. and later extended this theorem to uniformly convex banach spaces. he proved the following theorem: theorem 1.10[14, theorem 3.8] let c be a nonempty closed convex subset of a uniformly convex banach space x and t : c → a demicontinuos nearly lipschitzian mapping with sequences {(an,η(tn))} such that limn→∞η(tn) ≤ 1. then the following statements are equivalent: (a) t has a fixed point; (b) there exists a bounded sequence {tnx0} in c; (c) there exists a bounded sequence {yn} in c such that limm→∞(limn→∞‖yn −tmyn‖) = 0. very recently, thakur[16] stated and proved the following theorem: theorem 1.11[16] let e be a real banach space, k be a nonempty closed convex subset of e and ti : k → k,i = 1, 2 be two asymptotically generalised φhemicontractive nearly uniformly li lipschitzian mappings with sequence {an} and f(t1) ∩f(t2) 6= φ where f(ti) is the set of fixed point of ti in k. let {αn} and {βn} be two sequences in [0, 1] satisfying the following conditions: (i) ∑∞ n=1 αn = ∞; (ii) ∑∞ n=1 α 2 n < ∞; (iii) ∑∞ n=1 βn < ∞; (iv) ∑∞ n=0 αn(kn − 1) < ∞. let {xn} be a sequence in k generated from arbitrary x1 ∈ k by xn+1 = (1 −αn)xn + αt 1nyn, yn = (1 −βn)xn + βt 2nxn, n ∈ n. (1.2) then, {xn} converges strongly to x∗ ∈ f(t1) ∩f(t2). 196 olisama, a.a. mogbademu and j.o. olaleru the purpose of this sequel is to improve theorem 1.10 and theorem 1.11 by extending to a finite family of nearly puniformly llipschitzian asymptotically pseudocontractive mappings. 2. main results the following concepts and lemmas will be used. definition 2.1 [13]. let t1,t2, ...,ti : k → k be finite family of maps. for any given x1 ∈ k, the multi-step iteration {xn}∞(n=1) ⊂ k is defined by xn+1 = (1 − bn)xn + bnt 1nyn1, n ≥ 1 yn i = (1 − bni)xn + bnitinyni+1, i = 1, 2, ...,p− 2, yn p−1 = (1 − bnp−1)xn + bnp−1tpnxn, p ≥ 2. (2.1) lemma 2.2 [9]. let j : e → 2e∗ be the normalized duality mapping. then for any x,y ∈ e, we have ‖x + y‖2 ≤‖x‖2 + 2 < y,j(x + y) >,∀j(x + y) ∈ j(x,y) lemma 2.3 [3]. let {dn},{en} and {hn} be three positive real sequences and φ : [0,∞) → [0,∞) be a strictly increasing function with φ(x) = 0 ⇔ x = 0 satisfying the following inequality: d2n+1 ≤ dn 2 −enφ(dn+1) + hn,∀n ≥ 0 where en ∈ [0, 1], with ∑∞ n=0 en = +∞ and hn = o(en). then limn→∞dn = 0. theorem 2.4 let k be a nonempty closed convex subset of a real banach space e, and ti : k → k, (i = 1, 2, .....,p, p ≥ 2) be a finite family of nearly li uniformly lipschitzian mappings with sequence {kn}n≥1 ⊂ [1,∞), kn → 1 and ∑ n≥1(kn − 1) < ∞ such that ∩ p≥2 i=1 f(ti) 6= φ. let {bn}n≥1,{b i n}n≥1 and {bi+1n }n≥1 be the real sequences in [0,1] satisfying: (i)bn,b i n,b i+1 n → 0, as n →∞, (i = 1, 2, ...,p− 2). (ii) ∑ n≥1 bn = ∞. for any x1 ∈ k, define {xn}n≥0 by the iterative process (2.1). suppose there exists a strictly increasing function φ : [0,∞) → [0,∞), φ(0) = 0 such that < ti nxn −ρ,j(xn −ρ) > ≤ kn‖xn −ρ‖2 − φ(‖xn −ρ‖) (2.2) ∀x ∈ k, (i = 1, 2, ....,p, p ≥ 2). then {xn}n≥0 converges strongly to ρ ∈∩ p≥2 i=1 f(ti). proof: first we show that for any n ≥ 1,{xn} is a bounded sequence. since t1,t2, ....,tp, p ≥ 2 are nearly uniformly li lipschizian mappings, we have that, ∀x,y ∈ k ‖tni x−t n i y‖≤ li(‖x−y‖ + an), (i = 1, 2, ....,p, p ≥ 2). convergence of a modified multi-step iterative scheme 197 setting max{kn : n ≥ 1} = k and l = max{l1,l2, ....,lp}. there exists x1 ∈ k with x1 6= tx1 such that a0 = (k + l)‖x1 − ρ‖2 ∈ d(φ). if φ(a) → +∞ as a → +∞, then a0 ∈ d(φ). if sup{φ(a) : a ∈ [0,∞)} = a1 < +∞ with a1 < a0, then there exists a sequence {τn} ⊂ k such that τn → ρ as n → ∞ with τn 6= ρ, thus there exist a positive integer n0 such that (k + l)‖x1 −ρ‖2 < a12 for all n ≥ n0. redefining x1 = τn0, (k + l)‖x1 −ρ‖2 ∈ d(φ) and setting d(φ) = φ−1(a0), we obtain ‖x1 −ρ‖ ≤ d. also let a1 = {x ∈ k : ‖x1 −ρ‖ ≤ d},a2 = {x ∈ k : ‖x1 −ρ‖ ≤ 2d}. now, we show that xn ∈ a1 for any n ≥ 1. if n = 1, then x1 ∈ a1. assume that for some n, xn ∈ a1, we show that xn+1 ∈ a1. suppose xn+1 is not in a1, then ‖xn −ρ‖ > d. now we denote ν0 = min{ d d(1 + 2l) , φ(d) 10d2 , φ(d) 10(2dl + 3dl2) } (2.3) since bn,b i n,b i+1 n ,kn−1 → 0, as n →∞, (i = 1, 2, ...,p−2), without loss of generality, let 0 ≤ bn,bin,bi+1n ,kn−1 ≤ ν0 for any n ≥ 1. thus we have ‖ynp−1 −ρ‖ = ‖(1 − bnp−1)xn + bnp−1tpnxn −ρ‖ = ‖(xn −ρ) − bp−1n (xn −tnp xn)‖ ≤ ‖xn −ρ‖ + bp−1n (‖xn −ρ‖ + ‖tnp xn −ρ‖) ≤ ‖xn −ρ‖ + ν0(‖xn −ρ‖ + l(‖xn −ρ‖ + an)) ≤ d + ν0(d + ld + lan) < 2d. ‖ynp−2 −ρ‖ = ‖(1 − bnp−2)xn + bnp−2tp−1nyp−1n −ρ‖ ≤ ‖xn −ρ‖ + bp−2n ‖tnp−1yp−1n −xn‖ ≤ ‖xn −ρ‖ + ν0[l(‖yp−1n −ρ‖ + an) + ‖xn −ρ‖)] ≤ d + ν0(l(2d + an) + d) ≤ 2d. recursively, we have ‖yi−1n −ρ‖≤ 2d, thus, ‖yin −ρ‖≤ 2d for i = 1, 2, ...,p− 2. also, ‖xn+1 −yni‖ ≤ ‖xn+1 −xn‖ + ‖yn −xn‖ ≤ ‖xn+1 −xn‖ + bnp−2‖tnyp−1n −xn‖ ≤ d(1 + 2l) + d(1 + l) ≤ d(2 + 3l). but, ‖xn −tni xn‖ ≤ ‖xn −ρ‖ + ‖t n i xn −ρ‖ ≤ d + l(d + an) ≤ d(1 + l) and 198 olisama, a.a. mogbademu and j.o. olaleru ‖xn −tni y i n)‖ ≤ ‖xn −ρ‖ + ‖tni y i n −ρ‖ ≤ d + l(‖yin −ρ‖ + an) ≤ d + l(2d + an) ≤ d(1 + 2l). then, ‖tnxn+1 −tnyin‖ ≤ l(‖xn+1 −yin‖ + an) ≤ l[(‖yin −xn‖ + ‖xn+1 −xn‖ + an)] ≤ binl((‖xn −tni xn)‖ + bn‖xn −t n i y i n‖ + an) ≤ binl((d(1 + l) + bn(d(1 + 2l)) + an ≤ ν0(l(d(1 + l) + d(1 + 2l) + an)) = ν0(2dl + 3dl 2) ≤ φ(d) 10d . applying lemma(2.2) and the estimates above we obtain, ‖xn+1 −ρ‖2 = ‖(1 − bn)xn + bnt 1ny1n −ρ‖2 = ‖xn −ρ− bn(tn1 y1n −xn)‖2 ≤ ‖xn −ρ‖2 − 2bn < tn1 y1nxn,j(xn+1 −ρ) > = ‖xn −ρ‖2 − 2bn < tn1 xn+1 −ρ,j(xn+1 −ρ) > −2bn < xn+1,j(xn+1 −ρ) > +‖2bn < tn1 y1n −tn1 xn+1,j(xn+1 −ρ) > +2bn < xn+1 −xn,j(xn+1 −ρ) > ≤ ‖xn −ρ‖2 + 2bn(kn‖xn −ρ‖2 − φ(‖xn+1 −ρ‖)) −2bn‖xn+1 −ρ‖2 + 2bn‖tn1 y1n −tn1 xn+1‖‖xn+1 −ρ‖ +2bn‖xn+1 −xn‖‖xn+1 −ρ‖ = ‖xn −ρ‖2 + 2bn(kn − 1)‖xn+1 −ρ‖2 − 2bnφ‖xn+1 −ρ‖) +2bnl(‖y1n −xn+1‖ + an)‖xn+1 −ρ‖ + 2bn‖xn+1 −xn‖‖xn+1 −ρ‖ ≤ ‖xn −ρ‖2 + 2bn(kn − 1)‖xn −ρ‖2 − 2bnφ(d) +2bn( φ(d) 10d ) × 2d + 2bn(d(1 + 2l))) × 2d ≤ d2 + 8bn(kn − 1)d2 − 2bnφ(d) +d2 + 8bn(kn − 1)d2 − 85bnφ(d) + 4bnd(d + 2l) (2.4) since bn → 0 and (kn − 1) → 1 as n →∞,then (2.4) becomes ‖xn −ρ‖2 ≤ d2 which is a contradiction. hence, xn+1 ∈ a1. therefore, {xn} is bounded. next we prove that ‖xn−ρ‖→ 0 as n →∞. we have shown above that {‖xn−ρ‖} is a bounded sequence and so is {‖yin −ρ‖}. let r0 = supn≥{‖xn −ρ‖} + sup{‖yn −ρ‖}. but ‖xn+1 −yin‖ ≤ (‖yin −xn‖ + ‖xn+1 −xn‖) ≤ bin(‖xn −tinxn‖) + bn‖xn −tni y i n‖ ≤ bin(‖xn −ρ‖ + ‖tinxn −ρ‖) + bn(‖xn −ρ + ‖tni y i n‖) ≤ (bn + bin)m0 + (bn + bin)(1 + l)m0 (2.5) using lemma(2.2),equations (2.4) and (2.5) we have convergence of a modified multi-step iterative scheme 199 ‖xn+1 −ρ‖2 = ‖(1 − bn)xn + bnt 1ny1n −ρ‖2 = ‖xn −ρ− bn(tn1 y1n −xn)‖2 ≤ ‖xn −ρ‖2 − 2bn < tn1 y1n −ρ,j(xn+1 −ρ) > = ‖xn −ρ‖2 − 2bn < tn1 xn+1 −ρ,j(xn+1 −ρ) > −2bn < xn+1,j(xn+1 −ρ) > +‖2bn < tn1 y1n −tn1 xn+1,j(xn+1 −ρ) > +2bn < xn+1 −xn,j(xn+1 −ρ) > ≤ +2bn(kn)‖xn −ρ‖2 − φ(‖xn+1 −ρ‖)) −2bn‖xn+1 −ρ‖2 + 2bn‖tn1 y1n −tn1 xn+1‖‖xn+1 −ρ‖ +2bn‖xn+1 −xn‖‖xn+1 −ρ‖ = ‖xn −ρ‖2 + 2bn(kn − 1)‖xn+1 −ρ‖2 − 2bnφ(xn+1 −ρ‖) +2bnl(‖y1n −xn+1‖ + an)‖xn+1 −ρ‖ + 2bn‖xn+1 −xn‖‖xn+1 −ρ‖ ≤ ‖xn −ρ‖2 + 2bn(kn − 1)r20 − 2bnφ(‖xn+1 −ρ‖) +2bnl(bn + b i n)r0 + (bn + b i n)(1 + l)r0) + an)r0 + 4bnr 2 (2.6) ≤ ‖xn −ρ‖2 − bnφ(‖xn+1 −ρ‖) + bn where bn = 2bn(kn − 1)r20 + 2bnl(bn + bin)r0 +(bn + b i n)(1 + l)r0 + an)r0 + 4bnr 2. taking dn = ‖xn −ρ‖2,en = bn, and hn = bn. then (2.6) becomes d2n+1 ≤ d 2 n −enφ(dn+1) + hn,∀n ≥ n0. therefore, by lemma 2.3, we obtain lim →∞dn = 0. hence xn → ρ as n →∞.this completes the proof. also, using lemma 2.3 and the conditions of the parameters, we obtain: dn → 0 as n →∞. this ends the proof. we make the following remarks: (1) clearly,is it possible to drop the continuity condition in theorem 1.5 and extend to a finite family of nearly puniformly llipschitzian asymptotically pseudocontractive mappings? (2) we have dropped the continuity condition in theorem 1.6 and show that the modified multi-step converges to the common fixed point of t? corollary 2.5 the result in theorem 2.4 is also true for two and three nearly li uniformly lipschitzian asymptotically pseudocontractive mappings which extends the work of sahu[14]. corollary 2.6 let k be a nonempty closed convex subset of a real banach space e, ti : k → k, (i = 1, 2, .....,p, p ≥ 2) be p uniformly li lipschitzian mappings with sequence {kn}n≥0 ⊂ [1,∞),kn → 1 and ∑ n≥0(kn − 1) < ∞ such that ρ ∈∩p≥2i=1 f(ti) 6= φ. let {bn}n≥0,{b i n}n≥0 and {bi+1n }n≥0 be the real sequences in [0,1] satisfying: 200 olisama, a.a. mogbademu and j.o. olaleru (i)bn,b i n,b i+1 n → 0, as n →∞, (i = 1, 2, ...,p− 2). (ii) ∑ n≥0 bn = ∞. for any x0 ∈ k, define {xn}n≥0 by the iterative process (2.1). suppose there exists a strictly increasing function φ : [0,∞) → [0,∞), φ(0) = 0 such that < ti nxn −ρ,j(xn −ρ) > ≤ kn‖xn −ρ‖2 − φ(‖xn −ρ‖) (2.7) ∀x ∈ k, (i = 1, 2, ....,p, p ≥ 2). then {xn}n≥0 converges strongly to ρ ∈∩ p≥2 i=1 f(ti). references [1] s. s. chang, some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings , proc. amer. math. soc., 129(2001), 845-853. [2] s. s. chang, y. j. cho and j. k. kim, some results for uniformly llipschitzian mappings in banach spaces, applied mathematics letters, 22(2009), 121-125. [3] k. deimling, functional analysis, springer, berlin(1980). [4] k. goebel and w. a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc., 35(1972),171-174. [5] s. ishikawa, fixed point for a new iterative method, proc. amer. math. soc. 44(1974), 147-150. [6] w. r. mann, mean value method in iteration, proc. amer. math. soc. 4(1953), 506-510. [7] a. a. mogbademu, modified multi-step iterative methods for a family of p strongly pseudocontractive maps, bulletin of mathematical analysis and application, 4(2012), 83-90. [8] a .a. mogbademu, modified noor iterative procedure for three asymptoticallly pseudocontractive maps, international journal of open problems compt., mathematics, 6(2013), 95102. [9] c. moore and b. v. c. nnoli, iterative solution of nonlinear equations involving set-valued uniformly accretive operators, comput. math. anal.and appl., 42(2001), 131-140. [10] m. a. noor, three steps iterative algorithms for multi-valued quasi variational inclusions, jour. math. anal. appl., 225(2001), 589-604. [11] e. u. ofoedu, strong convergence theorem for uniformly l-lipschitzian and asymptotically nonexpansive mapping in banach space, jour.of math. anal. and appl., 321(2006), 722-728. [12] j. o. olaleru and a.a. mogbademu, modified noor iterative procedure for uniformly continuous mappings in banach space, boletin de la asociacion matematica venezolana, 18(2011), 127. [13] b. e. rhoades, s. m. soltuz, the equivalence between mann-ishikawa iterations and multistep iteration, nonlinear analysis theory, methods and applications, 58(2004), 219-228. [14] d. r. sahu, fixed point of demicontinuos nearly lipschitzian mappings in banach spaces, comment math.univ.carolin.46, 4(2005), 653-666. [15] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, jour.of math. and appl., 158(1999), 407-413. [16] b. s. tharkur, strong convergence for asymptotically generalised φ -hemicontractive mappings, romai j. 8(2012), 165-171. [17] zhique xue and guiwen lv, strong convergence theorems for uniformly llipschitzian asymptotically pseudocontractive mappings in banach spaces xue and lv jour. of ineq. and appl.. (2013) 1/79. department of mathematics, university of lagos. nigeria ∗corresponding author international journal of analysis and applications volume 16, number 4 (2018), 594-594 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-594 retraction notice to “differential subordinations for higher-order derivatives of multivalent analytic functions associated with dziok-srivastava operator” this article has been retracted for multiple submissions and duplicate publication (see e.g. [1]). on making each submission, authors are asked to confirm: the submission has not been previously published, and it is not under consideration for publication elsewhere. this article is a severe abuse of the scientific publishing system. references [1] a.k. wanas and a.h. majeed, differential subordinations for higher-order derivatives of multivalent analytic functions associated with dziok-srivastava operator, analele universitatii oradea fasc. matematica, tom xxv (2018), issue no. 1, 33–42. https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-594 int. j. anal. appl. (2023), 21:66 topological evaluation of four para-line graphs absolute pentacene graphs using topological indices mukhtar ahmad1, muhammad jafar hussain2, gulnaz atta3, sajid raza2, irfan waheed2, ather qayyum2,∗ 1department of mathematics, khawaja fareed university of engineering and information technology rahim yar khan, pakistan 2department of mathematics, institute of southern punjab multan, pakistan 3department of mathematics, university of education lahore dgk compuse, pakistan ∗corresponding author: atherqayyum@isp.edu.pk abstract. a real-number to molecular structure mapping is a topological index. it is a graph invariant method for describing physico-chemical properties of molecular structures specific substances. in that article, we examined pentacene’s chemical composition. the research on the subsequent indices is reflected in our paper, we conducted an analysis of several indices including general randic connectivity index, first general zagreb index, general sum-connectivity index, atomic bond connectivity index, geometric-arithmetic index, fifth class of geometric-arithmetic indices, hyper-zagreb index, first and second multiple zagreb indices for a four para-lines graphs of linear [n]-pentacene and multi-pentacene. 1. introduction and preliminaries all substances molecule possesses qualities, both chemical and physical, and certain may also exhibit physiologically active characteristics. several pharmaceutical companies are really hunting for novel antibacterial chemicals. for this reason, hundreds of compounds are examined, however costly examinations for biology. in order to circumvent such issue, additional methods for investigating potential antibiotics employ the relationship between structural features and biological activity or features of chemical and physical nature. topological indices, or molecular descriptors, provide insights into the physicochemical properties of molecules. they are valuable tools for understanding and explaining received: may 22, 2023. 2020 mathematics subject classification. 92c40. key words and phrases. topological indices; graphs of four para-lines; nanostructures; pentacene. https://doi.org/10.28924/2291-8639-21-2023-66 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-66 2 int. j. anal. appl. (2023), 21:66 the characteristics of chemical compounds. several graph invariants have been created in recent years and have been used in many academic fields such as structural chemistry, theoretical chemistry, environmental chemistry, toxicology, and pharmacology. because of the substantial industrial need, researchers are urged to study topological indices. more than 400 topological indexes have been opened a consequence of research. chemical compounds’ topological structures and chemical characteristics are tightly related, since each compound’s shape is critical to determining its functionality. topological indices are often used in multilinear regression modelling, chemical documentation, drug design, qsar/qspr modelling, and database selection. molecular descriptors are utilized to describe the physicochemical properties of molecular structures. these descriptors can be classified into three main types. degree-based indices [1–5], distance-based indices [6–11] and spectrum-based indices [12–15]. studies that have been documented in the literature (see [16–18]) use indicators that are based on both distances and degrees. due to pentacene’s important functions in both electrical devices and organic solar cells, a popular hydrocarbon semiconductor, it is necessary to optimise organic solar cells for less expensive energy sources [19]. the georgia institute of technology researchers have developed method to produce portable artificial solar cells. pentacene has been shown to be a very efficient means of converting sunlight into energy. in contrast to other materials, pentacene functions well as a semiconductor due to its crystalline properties. pentacene’s relevance motivated us to do topological study on it, and as a result, we have made several important discoveries that could be helpful for analysing pentacene’s physical and chemical characteristics. see [20,21] for further topological research on pentacene. consider an easy graph g consisting of a edge set e(g) and vertex set v (g), where loops and several edges present are excluded. the set x ∈ v (g),nx of neighbors in g is represented by nx, and the valence (degree) of x is equal to dx1 = |nx| and sx1 = ∑ y∈nx dx2. by inserting a vertex between every edge of the given graph, the edges are divided into two, resulting in the graph being subdivided. this operation, known as graph subdivision and denoted as s(g), leads to the formation of a line graph where adjacent edges in g become connected vertices in the new graph. the resulting graph, denoted as l(g), represents the line graph of the subdivision graph. in this article, the four para-line graph of g is represented by l(s(g)) (referred to as g?). conversely, g? can be constructed from g using the following procedure: 1. replace each vertex x1 ∈ v (g) with kx1, complete graph on dx1 vertices; 2. there is an edge connecting the vertex kx1 and the vertex kx2 in g ? if and only if there is an edge that coincides with x1 and x2 in g; 3. for each vertex x2 in kx1, in g ?, the valency (degree) of x2 is equal to the valency (degree) of x1 in g. structural chemistry commonly utilizes these diagrams. the research focus on four para-line graphs has diminished in recent times, but there is a shift happening. one appealing aspect of these graphs int. j. anal. appl. (2023), 21:66 3 is their straightforward construction process. the carbon skeleton, in which each atom acts as the vertex and each link between nearby atoms as the edge, may be used to generate any chemical compound. for example, butane is an organic compound with the formula (c4h10). butane is a saturated hydrocarbon containing 4 of carbon atoms, with an unbranched structure. butane is mainly used as a gasoline blend, alone or mixed with propane. it is also used as a feedstock for the production of ethylene and butadiene. butane, like propane, is obtained from natural gas or refineries, and the two gases usually occur together. butane is stored under pressure as a liquid. when the curler is turned on, butane is released and turns into a gas. figure 1(a) depicts the the molecular graph and its structure of butane. furthermore, figure 2(b) and (c) exhibit the four para-line graphs derived from the molecular plot of butane. now figure is; figure 1. (a) the molecular architecture of butane figure 2. (b) the molecular architecture of butane (c) four para-line graph of butane to accurately represent it the general randic connectivity index g is defined as [12]. rα(g) = ∑ x1x2∈e(g) (dx1dx2) α (1.1) the first universal zagreb index was presented by li and zhao [22]: mα(g) = ∑ x1∈v (g) (dx1) α (1.2) 4 int. j. anal. appl. (2023), 21:66 the the general sum connectivity index of the g chart was introduced in 2010 [23]: χα(g) = ∑ x1x2∈e(g) (dx1 + dx2) α (1.3) the index (abc) was proposed by estrada [24]. it is expressed as follows for a graph g: abc(g) = ∑ x1x2∈e(g) √ dx1 + dx2 − 2 dx1dx2 (1.4) the geometric-arithmetic index (ga) was introduced by vukicevic and furtula [25]. it is denoted as ga and is defined as follows for a graph g resently a. asghar et.al [31]: ga(g) = ∑ x1x2∈e(g) 2 √ (dx1dx2) (dx1 + dx2) (1.5) ghorbani et al. [26] described another index belonging to the 4th class of indices, denoted as (abc), which is defined as follows resently zaib hassan niazi et.al [32]: abc4(g) = ∑ x1x2∈e(g) √ ds1 + sx2 − 2 sx1sx2 (1.6) graovac et al. [27] introduced a fifth class of geometric-arithmetic indices denoted as ga5, which is defined as follows: ga5(g) = ∑ x1x2∈e(g) 2 √ (sx1sx2) (sx1 + sx2) (1.7) established the hyper-zagreb index in 2013 as follows resently mukhtar ahmad et.al [33]: hm(g) = ∑ x1x2∈e(g) (dx1 + dx2) 2 (1.8) in 2012, ghorbani and azimi introduced two new types of zagreb graph indices. the first is the first multiple zagreb index, denoted as pm1(g). the second multiple zagreb index is used, denoted as pm2(g). additionally, the first and second zagreb polynomials, m1(g,p) and m2(g,p), respectively, are characterised as: pm1(g) = πx1x2∈e(g)(dx1 + dx2) (1.9) pm2(g) = πx1x2∈e(g)(dx1 ×dx2) (1.10) m1(g,p) = ∑ x1x2∈e(g) p(dx1+dx2) (1.11) m2(g,p) = ∑ x1x2∈e(g) p(dx1×dx2) (1.12) int. j. anal. appl. (2023), 21:66 5 2. topological index of four para-line graphs for an index that schultz offered, ranjini created the independent relations. under the watchful eye of the schultz index, these researchers looked at the subdivision of a number of graphs, including helm, ladder, tadpole, and wheel [28]. they also looked at the ladder, tadpole, and wheel four para-line graph under the zagreb index [29]. in 2015, xu and su conducted an analysis of two indices specific to ladder, tadpole, and wheel graphs constructed using tare lines and named the total connectivity index of the sum and the co-index [30]. nadim et al. calculated the atomic bond connectivity index and fifth class of geometric arithmetic indices for four para-line tadpole, wheel, and ladder graphs. they also investigated several other indices, including randic general connectivity index, first zagreb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index, the first and second multiple zagreb index for a four paralinear graphs of linear [n]-pentacene and multiple pentacene., lattice plot in nanotorus tuc4c8[p,q] and 2d nanotube. in our study, we computed various indices, including randic general connectivity index, first zagreb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index. 2.1. molecular characteristics of the linear [n]-pentacene four para-line graph. figure 3 depicts the linear [n]-pentacene molecular graph, which is indicated by the symbol tn. tn consists of 28n− 2 edges and 22n vertices. theorem 2.1. consider a four para-line graph g? derived from the graph tn. mα(g ?) = (5n + 2)2α+2 + 3α+1(12n− 4). proof. in figure 3, the graph g? is displayed. there are 56n − 4 vertices in total in g?, this has 36n− 12 vertices of degree and 20n + 8 vertices of degree, where mα(g ?) = (5n + 2)2α+2 + 3α+1(12n− 4). theorem 2.2 consider a four para-line graph g? derived from the graph tn. 1. rα(g?) = (10n + 10)16α + (20n− 4)20α + (44n− 16)25α. 2. χα(g?) = (10n + 10)8α + (20n− 4)9α + (44n− 16)10α. 3. abc(g?) = (15 √ 2 + 88 3 )n + 3 √ 2 − 32 3 . 4. ga(g?) = (54 + 8 √ 6)n− 6 − 8 5 √ 6. proof. the total number of edges in g? is determined by the formula 74n− 10. the edges in g? can be divided into three sets, e1(g?), e2(g?), and e3(g?), which do not intersect with each other. the edge partition e1(g?) contains 10n + 10 edges x1,x2, where dx1 = dx2 = 4, edge the partition e2(g ?) contains 20n − 4 edges x1,x2, where dx1 = 4 and dx2 = 5, and the edge partition e3(g ?) consists of 44n−16 edges. this partition includes edges x1 and x2, where dx1 = dx2 = 5. by utilizing we get the required outcomes using formulae (1), (3), (4), and (5). theorem 2.3 consider a four para-line graph g? derived from the graph tn. 6 int. j. anal. appl. (2023), 21:66 figure 3. linear pentacene 1. abc 4 (g?) = ( √ 110 + 4 √ 2 + 2 √ 30 + 16 3 )n + 5 2 + 2 5 − 8 5 √ 2 − 2 3 √ 30 − 1 5 √ 110 − 32 9 2. ga5(g?) = (30 + 8013 √ 10 + 288 17 √ 2)n− 2 + 16 9 √ 5 − 16 13 √ 10 − 96 17 √ 2 proof. assuming that the set of edges depends on the sum of the degrees of the neighbors of the end vertices, we can partition edges that divide (g?) into seven distinct sets: e6(g?), e7(g?), ..., e12(g ?). thus, we have e(g?) = ⋃12 i=6ei (g ?). the edge assortment e6(g?) comprises 12 edges x1x2, where sx1 = sx2 = 6, the edge collection e7(g ?) holds 6 edges x1x2, where sx1 = 6 and sx2 = 7, the edge collection e8(g?) holds 11n − 5 edges x1x2, where sx1 = sx2 = 7, set of edges e9(g ?) contains 22n− 5 edges x1x2, where sx1 = 7 and sx2 = 10, edge the collection e10(g ?) contains 10n edges x1x2, where sx1 = sx2 = 10, the edge set e11(g ?) contains 26n − 9 edges x1x2, where sx1 = 10 and sx2 = 11 and the set of edges e12(g ?) is satisfied 13n− 9 edges x1x2, where sx1 = sx2 = 11. by utilizing we can get the required outcomes using formulae 6 and 7. theorem 2.4 consider a four para-line graph g? derived from the graph tn 1. hm(g) = 6480n− 1464. 2. pm1(g?) = 810n+10 × 920n−4 × 1044n−16. 3. pm2(g) = 1610n+10 × 2020n−4 × 2544n−16. proof. consider a four para-line graph g? of a linear pentacene. based on the angles of the final vertex, the collection of edges e(g?) might be categorised as three distinct groups. the first category, e1(g ?), consists of 10n + 10 edges x1x2, where dx1 = dx2 =4. the second category, e2(g ?), includes 20n−4 edges x1x2, where dx1 = 4 and dx2 =5. the third category, e3(g ?), comprises 44n−16 edges x1x2, where dx1 = dx2 = 5. let |e1(g)| = e4,4, |e2(g)| = e4,5, and |e3(g)| = e5,5. therefore, 1. hm(g) = ∑ x1x2∈e(g)(dx1 + dx2) 2 hm(g) = ∑ x1x2∈e1(g)[dx1 + dx2] 2 + ∑ x1x2∈e2(g)[dx1 + dx2] 2+ ∑ x1x2∈e3(g)[dx1 + dx2] 2 hm(g) = 64|e1(g)| + 81|e2(g)| + 100|e3(g)| hm(g) = 64(10n + 10) + 81(20n− 4) + 100(44n− 16) hm(g) = 460n + 460 + 1620n− 324 + 4400n− 1600 this implies that hm(g) = 6480n− 1464. 2. pm1(g) = πx1x2∈e1(g)(dx1 + dx2) × πx1x2∈e2(g)(dx1 + dx2) × πx1x2∈e3(g)(dx1 + dx2) pm1(g) = 8|e1(g)| × 9|e2(g)| × 10|e1(g)| pm1(g) = 810n+10 × 920n−4 × 1044n−16 int. j. anal. appl. (2023), 21:66 7 3. pm2(g) = πx1x2∈e1(g)(dx1 ×dx2) × πx1x2∈e2(g)(dx1 × (dx2) × πx1x2∈e3(g)(dx1 ×dx2) pm2(g) = 16|e1(g)| × 20|e2(g)| × 25|e1(g)| pm2(g) = 16|e1(g)| × 20|e2(g)| × 25|e1(g)| pm2(g) = 1610n+10 × 2020n−4 × 2544n−16 2.2. molecular descriptors of four paraline graphs for multiple pentacenes. the chemical diagram tm,n representing multiple pentacene is depicted in figure 4. this graph consists of 22mn vertices and 33mn− 2m− 5n edges. theorem 2.5 consider a four para-line graph g? derived from the graph tm,n. mα(g?) = (5n + 2)2α+2 + 3α+1(12n− 4). proof. figure 5 shows the graph g? in a visual format. it has 56n−4 worth of vertices in total, of which 20n + 8 and 36n− 12 have degrees of 3 and 4, respectively. using formula 2, we can calculate mα(g?). theorem 2.6 consider a four para-line graph g? derived from the graph tm,n. 1. rα(g?) = (10n + 6m + 4)16α+ (4m + 20n− 8)20α + (99mn− 20m− 55n + 4)25α. 2. χα(g?) = (10n + 6m + 4)8α + (4m + 20n− 8)9α + (99mn− 20m− 55n + 4)10α. 3. abc(g?) = (15 √ 2 110 3 )n + (5 √ 2 40 3 )m− 2 √ 2 + 66mn + 8 3 . 4. ga(g?) = (−45 + 8 √ 6)n+(8 5 √ 6 − 14)m + 99mn+ 8 16 5 √ 6. proof. the division graph s(tm,n) comprises a total of 198mn − 20m − 50 vertices and 99mn − 10m − 25n edges. there are 8m + 20nverticesof degree2and66mn-12m-30n vertices of degree 3, according to the vertex division. the edge set e(g?) of the four para-line graph g? consists of 99mn − 20m − 55n + 4 edges. based on the angles of the end vertices, these edges are divided into three groups, i.e, e(g?) = e1(g?) ∪e2(g?) ∪e3(g?). the edge separation e1(g?) consists of 10n+ 6m+ 4 edges x1x2. where dx1 = dx2 = 4. edge separation 4m+ 20n−8 with e2(g ?) edge x1x2, where dx1 = 4 and dx2 = 5. lastly, separating the edges e3(g ?) comprises 99mn − 20m − 55n + 4 edges x1x2, where dx1 = dx2 = 5. by applying the required outcome may be produced using formulae (1), (3), (4) and (5). figure 4. multiple pentacene theorem 2.7 consider a four para-line graph g? derived from the graph tm,n. 1. abc4(g?) = (44m + √ 14 + 4 √ 2 + √ 110 + 2 √ 30 − 116 3 )n + (1 2 √ 6 8 int. j. anal. appl. (2023), 21:66 +1 5 √ 110 + 2 5 √ 35 − 112 9 + 2 3 √ 30)m + 2 √ 6 − 8 5 √ 2 − 2 5 √ 110 − 4 3 √ 30 + 80 9 . 2. ga5(g?) = (8013 √ 10 + 99m + 288 17 √ 2 − 69)n + (−26 + 16 13 √ 10 + 16 9 √ 5 + 96 17 √ 2)m− 192 17 √ 10 − 32 13 √ 10 + 24 proof. seven distinct edge sets may be formed from the set of edges by taking into account the degree sum of end vertices’ neighbours. ei (g?), where i = 6, 7, ..., 12. thus, we have e(g?) =⋃12 i=6ei (g ?). the edge partition e6(g?) contains 2m + 8 edges x1x2, where sx1 = sx2 = 6. the edge partition e7(g?) consists of 4m edges x1x2, where sx1 = 6 and sx2 = 7. edge partition e8(g ?) contains 10n − 4 edges x1x2. where sx1 = sx2 = 7. edge partition e9(g ?) contains 20n + 4m − 8 edges x1x2. where sx1 = 8 and sx2 = 9. edge partition e10(g ?) consists of 10n edges x1x2. where sx1 = sx2 = 9. edge partition e11(g ?) contains 8m + 24n − 16 edge x1x2. where sx1 = 10 and sx2 = 11. finally, edge partition e12(g ?) contains 99mn − 28m − 87n + 20 edge x1x2. where sx1 = sx2 = 11. by utilizing formulas (6) and (7), we obtain the desired result. figure 5. four para-line graph multiple of pentacene by performing computations on the chemical structures of multiple-pentacene, we obtain the following indices: hm(g),pm1(g),pm2(g). theorem 2.8 consider a four para-line graph g? derived from the graph tm,n. 1. hm(g?) = 9900mn− 1292m− 3420n + 8 2. pm1(g?) = 810n+6m+4 × 94m+20n−8 × 1099mn−20m−55n+4. 3. pm2(g?) = 1610n+6m+4 × 204m+20n−8 × 2599mn−20m−55n+4 4. m1(g,p) = (10n + 6m + 4)p8 + (4m + 20n− 8)p9 + (99mn− 20m− 55n + 4)p10. 5. m2(g,p) = (10n + 6m + 4)p16 + (4m + 20n− 8)p20 + (99mn− 20m− 55n + 4)p25. proof. consider a graph g? with its edges broken down into three parts categories due to the degrees of the final vertex. the initial category, denoted as e1(g), consists of 10n + 6m + 4 edges x1x2, which both vertices x1 and x2 have a degree of 4. the second category, denoted as e2(g), contains 4m + 20n − 8 edges x1x2, which x1 has a degree of 4 and x2 has a degree of 5. the third category, denoted as e3(g), includes 99mn− 20m− 55n + 4 edges x1x2, where both vertices x1 and x2 have a degree of 5. we can observe that the cardinality of e1(g) is equal to e4,4, e2(g) is equal to e4,5, and e3(g) is equal to e5,5. 1. hm(g?) = ∑ x1x2∈e(g)(dx1 + dx2) 2 hm(g?) = ∑ x1x2∈e1(g)[dx1 + dx2] 2 + ∑ x1x2∈e2(g)[dx1 + dx2] 2 + ∑ x1x2∈e3(g)[dx1 + dx2] 2 int. j. anal. appl. (2023), 21:66 9 hm(g?) = 64|e1(g)| + 81|e2(g)| + 100|e3(g)| hm(g?) = 64(10n + 6m + 4) + 81(4m + 20n− 8) + 100(99mn− 20m− 55n + 4) hm(g?) = 460n + 384m + 256 + 324m + 1620n− 648 +9900mn− 2000m− 5500n + 400 this implies that hm(g?) = 9900mn− 1292m− 3420n + 8 since, 2. pm1(g?) = πx1x2∈e(g)(dx1 + dx2) pm1(g ?) = πx1x2∈e1(g)(dx1 + dx2) × πx1x2∈e2(g)(dx1 + dx2) × πx1x2∈e3(g)(dx1 + dx2) pm1(g ?) = 810n+6m+4 × 94m+20n−8 × 1099mn−20m−55n+4. now that 3. pm2(g?) = πx1x2∈e(g)(dx1 ×dx2) pm2(g ?) = πx1x2∈e1(g)(dx1 timesdx2) × πx1x2∈e2(g)(dx1 ×dx2) × πx1x2∈e3(g)(dx1 ×dx2) pm2(g ?) = 16|e1(g)| × 20|e1(g)| × 25|e1(g)| pm2(g ?) = 1610n+6m+4 × 204m+20n−8 × 2599mn−20m−55n+4. 4. m1(g,p) = ∑ x1x2∈e(g)p (dx1+dx2 m1(g,p) = ∑ x1x2∈e1(g)p (dx1+dx2) + ∑ x1x2∈e2(g)p (dx1+dx2) ∑ x1x2∈e1(g)p (dx1+dx2) m1(g,p) = ∑ x1x2∈e1(g)p 8 + ∑ x1x2∈e2(g)p 9 + ∑ x1x2∈e1(g)p 10 m1(g,p) = |e1(g)|p8 + |e2(g)|p9 + |e3(g)|p10 m1(g,p) = (10n + 6m + 4)p8 + (4m + 20n− 8)p9 + (99mn− 20m− 55n + 4)p10. 5. m2(g,p) = ∑ x1x2∈e(g)p (dx1+dx2 m2(g,p) = ∑ x1x2∈e1(g)p (dx1×dx2) + ∑ x1x2∈e2(g)p (dx1×dx2 ∑ x1x2∈e1(g)p (dx1×dx2) m2(g,p) = ∑ x1x2∈e1(g)p 16 + ∑ x1x2∈e2(g)p 20 + ∑ x1x2∈e1(g)p 20 m2(g,p) = |e1(g)|p16 + |e2(g)|p20 + |e3(g)|p25 m2(g,p) = (10n + 6m + 4)p16 + (4m + 20n− 8)p20 + (99mn− 20m− 55n + 4)p25. this makes the proof whole. 3. conclusion and future studies in our research article, we investigated indices randic general connectivity index, first zagreb general index, summation general connectivity index, atomic bond connectivity index, geometric arithmetic index, fifth class of geometric arithmetic indices, hyperzagreb index, the initial and secondly multiple [n]-pentacene zagreb indices for a four paraline graphs of these two types of pentacenes. these indices play a crucial role in chemical informatics, specifically in the analysis of organic compounds. the randic index (rα) is commonly used to explore the physicochemical properties of alkanes, such as boiling point, surface area, and enthalpy of formation. it provides valuable insights into the characteristics of organic molecules. the abc index is a useful tool for predicting the stability of hydrocarbons, encompassing both linear and branched alkanes. the stability of cycloalkanes can be assessed by the indicated index, which is associated with their strain energy stability. this provides significant insights 10 int. j. anal. appl. (2023), 21:66 into the overall stability of cycloalkanes. in terms of predicting physicochemical characteristics, chemical reactivity, and biological activities, the ga index demonstrates superior performance compared to the abc index. our investigation of pentacene was approached from a philosophical standpoint rather than relying solely on empirical observations. our theoretical understanding of pentacenes can substantially benefit in understanding their physical properties, chemical activity and biological activity. a variety from physical feature-related data may be correlated with the chemical structure of pentacenes according to this study’s major results, which may be useful for the power industry. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] x. li, y. shi, a survey on the randić index, match commun. math. comput. chem. 59 (2008), 127-156. [2] x. li, y. shi, l. wang, an updated survey on the randić index, in: i. gutman, b. furtula (eds.), recent results in the theory of randić index, university kragujevac: kragujevac, serbia, 2008, pp. 9–47. [3] z.s. mufti, s. zafar, z. zahid, m.f. nadeem, study of the paraline graphs of certain benzenoid structures using topological indices, magnt res. rep. 4 (2017), 110-116. [4] j. rada, r. cruz, vertex-degree-based topological indices over graphs, match commun. math. comput. chem. 72 (2014), 603-616 [5] a.m. hinz, d. parisse, the average eccentricity of sierpiński graphs, graphs comb. 28 (2011), 671-686. https: //doi.org/10.1007/s00373-011-1076-4. [6] j. devillers, a.t. balaban, topological indices and related descriptors in qsar and qspar, crc press, amsterdam, 2000. [7] m. azari, a. iranmanesh, harary index of some nano-structures, match commun. math. comput. chem. 71 (2014), 373-382. [8] l. feng, w. liu, g. yu, s. li, the hyper-wiener index of graphs with given bipartition, utilitas math. 96 (2014), 99-108. [9] a. ali, w. nazeer, m. munir, s. min kang, m-polynomials and topological indices of zigzag and rhombic benzenoid systems, open chem. 16 (2018), 73-78. https://doi.org/10.1515/chem-2018-0010. [10] m. knor, b. luzar, r. skrekovski, i. gutman, on wiener index of common neighborhood graphs, match commun. math. comput. chem. 72 (2014), 321-332. [11] k. xu, m. liu, k.c. das, i. gutman, b. furtula, a survey on graphs extremal with respect to distance-based topological indices, match commun. math. comput. chem. 71 (2014), 461-508. [12] h.p. schultz, topological organic chemistry. 1. graph theory and topological indices of alkanes, j. chem. inf. comput. sci. 29 (1989), 227–228. https://doi.org/10.1021/ci00063a012. [13] k. xu, k.ch. das, h. liu, some extremal results on the connective eccentricity index of graphs, j. math. anal. appl. 433 (2016), 803-817. https://doi.org/10.1016/j.jmaa.2015.08.027. [14] m.r.r. kanna, r. jagadeesh, topological indices of vitamin a, int. j. math. appl. 6 (2018), 271-279. [15] a.u.r. virk, w. nazeer, s.m. kang, on computational aspects of bismuth tri-iodide, preprints, 2018. https: //doi.org/10.20944/preprints201806.0209.v1. [16] m. dehmer, f. emmert-streib, m. grabner, a computational approach to construct a multivariate complete graph invariant, inf. sci. 260 (2014), 200-208. https://doi.org/10.1016/j.ins.2013.11.008. https://doi.org/10.1007/s00373-011-1076-4 https://doi.org/10.1007/s00373-011-1076-4 https://doi.org/10.1515/chem-2018-0010 https://doi.org/10.1021/ci00063a012 https://doi.org/10.1016/j.jmaa.2015.08.027 https://doi.org/10.20944/preprints201806.0209.v1 https://doi.org/10.20944/preprints201806.0209.v1 https://doi.org/10.1016/j.ins.2013.11.008 int. j. anal. appl. (2023), 21:66 11 [17] l. feng, w. liu, a. ilić, g. yu, degree distance of unicyclic graphs with given matching number, graphs comb. 29 (2012), 449-462. https://doi.org/10.1007/s00373-012-1143-5. [18] i. gutman, selected properties of the schultz molecular topological index, j. chem. inf. comput. sci. 34 (1994), 1087-1089. https://doi.org/10.1021/ci00021a009. [19] m.r. farahani, m.f. nadeem, s. zafar, z. zahid, m.n. husin, study of the topological indices of the line graphs of hpantacenic nanotubes, new front. chem. 26 (2017), 31-38. [20] n. soleimani, e. mohseni, n. maleki, n. imani, some topological indices of the family of nanostructures of polycyclic aromatic hydrocarbons (pahs), j. natn. sci. found. sri lanka. 46 (2018), 81-88. https://doi.org/ 10.4038/jnsfsr.v46i1.8267. [21] n. soleimani, m.j. nikmehr, h.a. tavallaee, theoretical study of nanostructures using topological indices, stud. u. babes-bol, che. 59 (2014), 139-148. [22] x. li, h. zhao, trees with the first smallest and largest generalized topological indices, match commun. math. comput. chem. 50 (2004), 57–62. [23] b. zhou, n. trinajstić, on general sum-connectivity index, j. math. chem. 47 (2009), 210-218. https://doi. org/10.1007/s10910-009-9542-4. [24] e. estrada, l. torres, l. rodriguez, i. gutman, an atom-bond connectivity index: modelling the enthalpy of formation of alkanes, indian j. chem. 37a (1998), 849-855. [25] d. vukičević, b. furtula, topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, j. math. chem. 46 (2009), 1369-1376. https://doi.org/10.1007/ s10910-009-9520-x. [26] m. ghorbani, m.a. hosseinzadeh, computing abc4 index of nanostar dendrimers, optoelectronics adv. mater. rapid commun. 4 (2010), 1419-1422. [27] a. graovac, m. ghorbani, m.a. hosseinzadeh, computing fifth geometric-arithmetic index for nanostar dendrimers, j. math. nanosci. 1 (2011), 33-42. https://doi.org/10.22061/jmns.2011.461. [28] p.s. ranjini, v. lokesha, m.a. rajan, on the schultz index of the subdivision graphs, adv. stud. contemp. math. 21 (2011), 279-290. [29] p.s. ranjini, v. lokesha, i.n. cangül, on the zagreb indices of the line graphs of the subdivision graphs, appl. math. comput. 218 (2011), 699-702. https://doi.org/10.1016/j.amc.2011.03.125. [30] g. su, l. xu, topological indices of the line graph of subdivision graphs and their schur-bounds, appl. math. comput. 253 (2015), 395-401. https://doi.org/10.1016/j.amc.2014.10.053. [31] a. asghar, a. qayyum, n. muhammad, different types of topological structures by graphs, eur. j. math. anal. 3 (2022), 3. https://doi.org/10.28924/ada/ma.3.3. [32] z.h. niazi, m.a.t. bhatti, m. aslam, y. qayyum, m. ibrahim, a. qayyum, d-lucky labeling of some special graphs, amer. j. math. anal. 10 (2022), 3-11. https://doi.org/10.12691/ajma-10-1-2. [33] m. ahmad, s. hussain, u. parveen, i. zahid, m. sultan, a. qayyum, on degree-based topological indices of petersen subdivision graph, eur. j. math. anal. 3 (2023), 20. https://doi.org/10.28924/ada/ma.3.20. https://doi.org/10.1007/s00373-012-1143-5 https://doi.org/10.1021/ci00021a009 https://doi.org/10.4038/jnsfsr.v46i1.8267 https://doi.org/10.4038/jnsfsr.v46i1.8267 https://doi.org/10.1007/s10910-009-9542-4 https://doi.org/10.1007/s10910-009-9542-4 https://doi.org/10.1007/s10910-009-9520-x https://doi.org/10.1007/s10910-009-9520-x https://doi.org/10.22061/jmns.2011.461 https://doi.org/10.1016/j.amc.2011.03.125 https://doi.org/10.1016/j.amc.2014.10.053 https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.12691/ajma-10-1-2 https://doi.org/10.28924/ada/ma.3.20 1. introduction and preliminaries 2. topological index of four para-line graphs 2.1. molecular characteristics of the linear [n]-pentacene four para-line graph 2.2. molecular descriptors of four paraline graphs for multiple pentacenes 3. conclusion and future studies references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 58-67 http://www.etamaths.com approximating derivatives by a class of positive linear operators bramha dutta pandey1,∗ and b. kunwar2 abstract. some direct theorems for the linear combinations of a new class of positive linear operators have been obtained for both, pointwise and uniform simultaneous approximations. a number of well known positive linear operators such as gamma operators of muller, post-widder and modified post-widder operators are special cases of this class of operators. 1. introduction during past few decades a number of sequences of positive linear operators ( henceforth written as operator) both, of the summation and those defined by integrals have been introduced and studied by a number of authors. some of wellknown operators of latter type are the gamma operators of müller [7], post-widder and modified post-widder operators [6], kunwar [4], sikkema and rathore [11]. now we define our linear operator ln [4] as (1) ln(f; x) = d(m,n,α)x mn+α−1 ∫∞ 0 u−mn−αe−n( x u )mf(u)du where d(m,n,α) = |m|nn+ α−1 m γn+ α−1 m ,m ∈ ir−{0},n > 0,α ∈ ir. the equation (1) defines a linear positive approximation methods, which contains as particular cases, a number of well known linear positive operators; e.g. post-widder and modified post-widder operators [6], and the gamma-operators of muller [7] . in the present paper we study the following problems: (i) is it possible to approximate the derivatives of f by the derivatives of ln(f)? (ii) can we use certain linear combinations of ln to obtain a better order of approximation? we introduce notations and definitions used in this paper. throughout the paper ir+ denotes the interval (0,∞),< a,b > open interval containing [a,b] ⊆ ir+,χδ,x(χcδ,x) the characteristic function of the interval (x − δ,x+δ) {ir+−(x−δ,x+δ)}. the spaces m(ir+),mb(ir+),loc(ir+),l1(ir+) respectively denote the sets of complex valued measurable, bounded and measurable, locally integrable and lebesgue integrable functions on ir+. let ω(> 1) be a continuous function defined on ir+.we call ω a bounding function if for each k ⊆ ir+ there exist positive numbers nk and mk such that lnk (ω; x) < mk, x ∈ k. 2000 mathematics subject classification. primary 41a35, 41a38; secondary 41a25, 41a60. key words and phrases. positive linear operators, simultaneous approximation, linear combinations. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 58 approximating derivatives by a class of positive linear operators 59 here ω(u) = u−a + ebu m + uc, where a,b,c > 0. for this bounding function dω = {f : f is locally integrable on ir+ and is such that lim supu→0 f(u) ω(u) and lim supu→∞ f(u) ω(u) exist} d (k) ω = {f : f ∈ dω and f is ktimes cotinuously differentiable on ir+ and f(i) ∈ dω, i = 1, 2, ...,k} cmb (ir +) = {f : f is m-times continuously differentiable and is such that f(k), k = 0, 1, 2, ...m are bounded on ir+}. 2. simultaneous approximation for continuous derivatives we consider the elementary case of simultaneous approximation by the operators ln wherein the derivatives of f are assumed to be continuous. we have termed this case elementary, for it is possible here to deduce the results on the simultaneous approximation: (lnf) (k) → f(k)(k ∈ in) from the corresponding results on the ordinary approximation: lnf → f. theorem 1. : if f ∈ d(k)ω ,then l (k) n (f; x) for x ∈< a,b > exists for all sufficiently large n and (2) limn→∞l (k) n (f; x) = f (k)(x), uniformly for x ∈ [a,b]. proof. we have ln(f; x) = d(m,n,α)x mn+α−1 ∫∞ 0 u−mn−αe−n( x u )mf(u)du a formal k-times differentiation within the integral sign and replacing α by α−k, let the new operator be denoted by l∗n and the corresponding d(m,n,α) be denoted by d∗(m,n,α).then (3) l (k) n (f; x) = d(m,n,α) d∗(m,n,α) l∗n(f (k)(x)) applying the known approximation lnf → f to (3), we find that l (k) n (f; x) = d(m,n,α) d∗(m,n,α) l∗n(f (k)(x)) → f(k)(x) as n →∞. this completes the proof of the theorem. � theorem 2. :if f ∈ d(k)ω . then at each x ∈ ir + where f(k+2) exists (4) l (k) n (f; x) −f(k)(x) = 12nm2 [(m + k − 2α + 2)kf (k)(x)+ +(m+2k−2α+3)xf(k+1)(x)+x2f(k+2)(x)]+o( 1 n ),n → ∞. further if f(k+2) exists and is continuous on < a,b >, then (4) holds uniformly in x ∈ [a,b]. proof. using voronovskaya formula [1], [6], [10],[11], [12] for l∗n and (3), the result follows. � in a similer manner one can prove the following results: theorem 3. : if f is such that f(k) exists and is continuous on ir+, then (5) ∣∣∣l(k)n (f; x) −f(k)(x)∣∣∣ ≤ ωf(k) (n−12 )[1+ + min{x2( 1 m2 + o(1)),x( 1 m2 + o(1)) 1 2 ] + o( 1 n ), (n →∞,x ∈ ir+). where ωf(k) is the modulus of continuity of f (k) [13] [2] [3]. 60 bramha dutta pandey1,∗ and b. kunwar2 theorem 4. : let f be such that f(k+1) exists on ir+.then for x ∈ ir+ (6) ∣∣∣l(k)n (f; x) −f(k)(x)∣∣∣ ≤ k|f(k)(x)| 2nm2 {|m + k − 2α + 2|} + x| f(k+1)(x)| 2nm2 {|m + k − 2α + 3|}+ +ωf(k+1) (n −1 2 )[xn− 1 2{ 1 m2(m−3) + o(1)} + x2 2n 1 2 { 1 m2(m−3) + o(1)}], (n →∞,x ∈ ir+). 3. pointwise simultaneous approximation in the present section we consider the “non-elementary” case of simultaneous approximation wherein assuming only that f(k)(x) exist at some point x, we solve the problem of pointwise approximation. before proving this result we establish: lemma 1. :let n > p ∈ in (set of natural numbers). then (7) ∂ p ∂xp {xα+mn−1u−mne−n( x u )m} = xmn+α−p−1up−mne−(n−p)( x u )m × ∑p k=0 ∑[ p−k 2 ] ν=0 ( m u )knν+k( x u )k(m−1)e−k( x u )m[1− ( x u )m]kgν,k,p(x,u) where [x] denotes the integral part of x ∈ ir+ and the function gν,k,p(x,u) are certain linear combinations of products of the powers of u−1,x−1 and ∂ k ∂xk {( x u )me−( x u )m},k = 0, 1, 2, ...,p and are independent of n. proof. we proceed by induction on p. we note that (8) ∂ ∂x {xmn+α−1u−mne−n( x u )m} = x(α−1)( x u )m(n−1)e−(n−1)( x u )m[ (mn+α−1) x ( x u )me−( x u )m−mn u ( x u )2m−1e−( x u )m] putting g0,0,1(x,u) = (α−1) x ( x u )me−( x u )m g0,1,1(x,u) = u −1 we observe that (8) is of the form (7). hence the result is true for p = 1. next, let us assume that the lemma holds for a certain p. then by the induction hypothesis, (9) ∂ p+1 ∂xp+1 {xα+mn−1u−mne−n( x u )m} = xα−1( x u )m(n−p−1)e−(n−p−1)( x u )m × ∑p+1 k=0 ∑[( p−k+1 2 )] ν=0 ( m u )knν+k{( x u )m−1e−( x u )m − ( x u )2m−1e−( x u )m}kgν,k,p+1(x,u) wherewith gν,k,p ≡ 0 for k > p or k < 0,ν < 0 or ν > [p−k2 ], we have put gν,k,p+1(x,u) = mn+α−1 x gν,k,p(x,u)( x u )me−( x u )m −mn u {m u ( x u )m−1e−( x u )m−m u ( x u )2m−1e−( x u )m}gν,k,p(x,u)+ + ∂ ∂x gν,k,p(x,u) + 1 u gν,k−1,p(x,u)+ +(k+1 u ){m(m−1) u2 ( x u )m−2e−( x u )m−(m u )2( x u )2(m−1)e−( x u )m− −m(2m−1) u2 ( x u )2(m−1)e−( x u )m+(m u )2( x u )3m−2e−( x u )m}gν−1,k+1,p(x,u). for k = 0, 1, 2, ...,p + 1 and ν = 0, 1, 2, ...., [p+1−k 2 ] it is clear that gν,k,p+1(x,u) satisfies the other required properties and hence the result is true for p + 1. hence it follows that (8) holds for all p = 1, 2, .... this completes the proof. � theorem 5. : let m ∈ in and f ∈ dω, then (10) lim n→∞ l (k) n (f; x) = f (k)(x). whenever x ∈ ir+ is such that f(k)(x) exists. moreover if f(k) exists and is continuous on < a,b >, (10) holds uniformly in x ∈ [a,b]. approximating derivatives by a class of positive linear operators 61 proof. if f(k)(x) exists at some x ∈ ir+, given an arbitrary � > 0 we can find a δ satisfying x > δ > 0 s.t. f(u) = ∑k p=0 f(p)(x) p! (u−x)p + hx(u)(u−x)k; |u−x| ≤ δ, where hx(u) is certain measurable function on [x − δ,x + δ] satisfying the inequality |hx(u)| ≤ �, |u−x| ≤ δ. hence (11) l (k) n (f; x) = ∑k p=0 f(p)(x) p! ∑p j=0 ( p j ) (−1)jl(k)n (up−j; x)+ +l (k) n (hx(u)(u−x)kχδ,x(u); x) + l (k) n (fχ c δ,x; x) = ∑ 1 + ∑ 2 + ∑ 3, (say). using the fact that ln maps polynomials to polynomials and the basic convergence theorem3, we obtain (12) ∑ 1 = f (k)(x)ln(u k; 1) → f(k)(x),n →∞. it follows from lemma1 that l (k) n (hx(u)(u−x)kχδ,x(u); x) = xmn+α−1d(m,n,α) ∑k p=0 ∑[ k−p 2 ] ν=0 n ν+p × ∫x+δ x−δ u −mn−αhx(u)(u− x)m{( x u )me−( x u )m}(n−k) ×[ ∂ ∂x {( x u )me−( x u )m}]kgν,p,k(x,u)du the δ above can be chosen so small that∣∣ ∂ ∂x {( x u )me−( x u )m} ∣∣ ≤ a |u−x| , |u−x| < δ, where a is some constant. since the functions gν,p,k(x,u) are bounded on [x− δ,x + δ], it is clear that there exists a constant m1 independent of n,� and δ s.t. for all n sufficiently large,∣∣∣ l(k)n (hx(u)(u−x)kχδ,x(u); x)∣∣∣ ≤ �m1 ∑kp=0 ∑[ k−p2 ]ν=0 nν+p−k+p2 by (3) where m2 is another constant not depending on n,� and δ. since ν ≤ [k−p 2 ],ν + p− p+k 2 − [k−p 2 ] − k−p 2 ≤ 0 there exists a constant m independent of n,� and δ s.t. (13) | ∑ 2| ≤ m for all sufficiently large n. to estimate ∑ 3, first of all we notice that there exist a positive integer p and a positive constant p such that∣∣[{(m u )e−( x u )m( x u )m−1}{1 − ( x u )m−1}]kgν,p,k(x,u) ∣∣ ≤ p(1 + u−m),u ∈ ir+ and 0 ≤ p ≤ k, 0 ≤ ν ≤ [k−p 2 ]. hence by lemma1, we have | ∑ 3| ≤ p ∑k p=0 ∑[ k−p 2 ] ν=0 n ν+pxmn+α−1d(m,n,α) × ∫∞ 0 u−mn−α(1+u−m)( x u )mn−ke−(n−k)( x u )mf(u)χcδ,x(u)du = p ∑k p=0 ∑[ k−p 2 ] ν=0 n ν+p d(m,n,α) d(m,n−k,α)ln−k(fχδ,x; x) + d(m,n,α) d∗∗(m,n−k,α)l ∗∗ n−k(fχ c δ,x; x) where l∗∗n corresponds to to the operator (1) with α replaced by α + m and d∗∗(m,n,α) refers to d(m,n,α) for l∗∗n . we observe that limn→∞ d(m,n,α) d(m,n−k,α) = limn→∞ d(m,n,α) d∗∗(m,n−k,α) also, by the definition of the operator ln, we have limn→∞n ν+pln−k(fχ c δ,x; x) = limn→∞n ν+pl∗∗n−k(fχ c δ,x; x) = 0 it follows that ∑ 3 → 0 as n → ∞. in view of this fact and (11) −−(13), it follows that there exists an n0 s.t.∣∣∣l(k)n (f; x) −f(k)(x)∣∣∣ < (2 + m)�,n > n0. since m does not depend on � we have (10). 62 bramha dutta pandey1,∗ and b. kunwar2 the uniformity part is easy to derive from the above proof by noting that, to begin with, δ can be chosen independent of x ∈ [a,b] so that |hx(u)| ≤ � for x ∈ [a,b] whenever |u−x| ≤ δ. then, it is clear that the various constants occuring in the above proof can be chosen independent of x ∈ [a,b]. this completes the proof of the theorem. � finally, we show that the asymptotic formula of theorem2 remains valid in the pointwise simultaneous approximation as well. we observe that the difference between theorem2 and the following one lies in the assumptions of f . we have theorem 6. : if f ∈ dω, then (14) l (k) n (f; x) −f(k)(x) = − 12nm2 [f (k)(x)k{(2α−k − 5)}+ +xf(k+1)(x){2(α−k−3) + (3−k)}+x2f(k+2)(x)]+ +o( 1 n ),n →∞. whenever x ∈ ir+ is s.t. f(k+2)(x) exists. also if f(k+2)(x) exists and is continuous on < a,b >, (14) holds uniformly in x ∈ [a,b]. proof. if f(k+2) exists, we have f (u) = ∑k+2 p=0 f(p)(x) p! (u−x)p + h(u,x), where h(u,x) ∈ dω and for any � > 0, there exist a δ > 0 s.t. |h(u,x)| ≤ � |u−x|k+2 for all sufficiently |u−x| ≤ δ. thus, (15) l (k) n (f; x) = l (k) n (q; x) + l (k) n (h(u,x); x), where q = ∑k+2 p=0 f(p)(x) p! (u − x)p is a polynomial in u.clearly q ∈ d(k)ω . also, q(p)(x) = f(p)(x), for p = k,k + 1,k + 2. hence, applying theorem2, we have (16) l (k) n (q; x) −f(k)(x) = − 12nm2 [k(2α−k −m− 2)f (k)(x)+ +(2α− 2k −m− 3)xf(k+1)(x) + x2f(k+2)(x)] + o( 1 n ), n →∞. to establish (14), it remains to show that (17) ∣∣∣l(k)n (h(u,x); x)∣∣∣ ≤ d(m,n,α)xα−1 ∑kp=0 ∑[ k−p2 ]ν=0 mnν+p ∫∞0 xmnu−mn−α−1e−n( xu )m × ∣∣( x u )m−1e−( x u )m{1 − ( x u )}m−1 ∣∣gν,p,k(x,u){h(u,x)χcδ,x(u) + � |u−x|k+2}du proceeding as in the proof of theorem5, we find that the term corresponding to � in the above is bounded by �m n for some m independent of � and n and χcδ,x− term contributes only a o( 1 n ) quantity (in fact o( 1 ns ) for an arbitrary s > 0). then in view of arbitraryness of � > 0, (17) follows. the uniformity part follows as a remark similar to that made for the proof of the uniformity part of theorem5. this completes the proof of the theorem. � in the rest of the paper, we study the second problem. 4. some direct theorems for linear combinations in this section we give some direct theorems for the the linear combinations of the operators ln. first, we give some definitions. the k th-moment µn,k(x),k ∈ in0 (set of non-negative integers) of the operators ln [5] is defined by approximating derivatives by a class of positive linear operators 63 (18) µn,k(x) = ln((u−x)k; x) = xkτn,k (say). clearly, τn,k is independent of x.now we first prove the lemma on the moments µn,k. lemma 2. : if k ∈ in0.then there exist constants γk,ν,ν ≥ [k+12 ] s.t. the following asymptotic expansion is valid: (19) τν,k = ∑∞ ν=[ k+1 2 ] γk,νn ν, n →∞. proof. let 1 3 < γ < 1 2 . then τn,k = ∫ 1+n−γ 1−n−γ s α−k−2(1 −s)k exp[n log{e−1 −m2 (s−1) 2 2! e−1+ ... + (s−1)2p 2p! ( d 2p dx2p {( x u )me−( x u )m}) x u =1 + o((s− 1)2p)}]ds, (p ≥ 2) = e−n ∫ 1+n−γ 1−n−γ s α−k−2(1 −s)k exp[−nm2 (s−1) 2 2 ] ×exp[{c3(s−1)3 +c4(s−1)4 +...+c2p(s−1)2p +o((s−1)2p)}]ds c′is being constants. = e−n ∫ 1+n−γ 1−n−γ s α−k−2(1−s)k exp[−nm2 (s−1) 2 2 ]{1+ ∑ 3≤3i≤j≤[2p+ i−1 γ ] bijn i(s− 1)j + o(n1−2pγ)}ds b′ijs depending on c ′ is. = e−n ∫ 1+n−γ 1−n−γ exp[−nm 2 (s−1)2 2 ][{ ∑2p−1 γ l=0 al(s− 1) k+l} ×{1 + ∑ 3≤3i≤j≤[2p+ i−1 γ ] bijn i(s−1)j}+ o(n1−(2p+k)γ)]ds = e−n ∫ 1+n−γ 1−n−γ exp[−nm 2 (s−1)2 2 ][ ∑ 3≤3i≤j≤[2p+ i−1 γ ] 0≤l≤[2p−1 γ ] dijln i(s − 1)j+k+l + o(n1−(2p+k)γ)]ds where d′ijls are certain constants depending on a ′ ls and b ′ ijs and vanish if j +k+l is odd. using substitutions we get = 2 1 2 e−n mn 1 2 ∫−n 0 t [ j+k+l+1 2 ]− 1 2 et [1 + γm2 ∑ (0≤3i≤j≤[2p+ i−1γ ]) d∗ijln i−[ j+k+l−1 2 ] + o(n1−(2p+k)γ+1−2γ)]dt where d∗ijl = dijl{ 2 m2 }[ j+k+l−1 2 ]. = 2 1 2 e−n mn 1 2 [ ∑ 0≤3i≤j≤[2p+ i−1 γ ] 0≤l≤[2p−1 γ ] d∗∗ijln i−[ j+k+l−1 2 ] + o(n2−(2p+2+k)γ)] where d∗∗ijl = d ∗ ijlγ(([ j+k+l−1 2 ])γ + 1 2 ) and we have made use of the fact that by enlarging the integral in the above from 0 to ∞, we are only adding the terms in n which decay exponentially and therefore can be absorbed in the o-term. next, we analyse the expression∫ (0,∞)−(1−n−γ,1+n−γ) s mn+α−k−2(1 −s)ke−ns m ds = e(n) (say). we have for any positive integer q, |e(n)| ≤ nγqd∗∗(m,n,α)l∗∗n (|u− 1| k+q ; 1), where d∗∗(m,n,α) and l∗∗n are the same as considered in the proof of theorem5. by making use of an estimate for the operators l∗∗n , we have |e(n)| ≤ anγq− k+q 2 d∗∗(m,n,α), 64 bramha dutta pandey1,∗ and b. kunwar2 where a is certain constant not depending upon n. again making use of the same estimate as above for d∗∗(m,n,α), we have en |e(n)| = o(nγq− k+q+1 2 ). thus, choosing q s.t. p ≥ 2(2p+2+k) 1−2γ , we have∫∞ 0 smn+α−k−2(1 −s)ke−ns m ds = 2 1 2 e−n mn 1 2 [ ∑ (0≤3i≤j≤[2p+ i−1γ ]) 0≤l≤[2p−1 γ ] d∗∗ijln i−[ j+k+l−1 2 ] + o(n2−(2p+k+2)γ)]. now, for all indices under consideration we have [j+k+l+1 2 ] − i = [j−2i+k+l+1 2 ] ≥ [k+1 2 ], and since p could be chosen arbitrarily large, there exist constants ck,ν,ν ≥ [k+12 ] s.t. we have the following asymptotic expansion∫∞ 0 smn+α−k−2(1 −s)ke−ns m ds = 2 1 2 e−n mn 1 2 ∑∞ ν=[ k+1 2 ] ck,ν nν noting that c0,0 = 1, it follows that there exist constants γk,ν,ν ≥ [k+12 ] s.t. (19) holds. this completes the proof of lemma2. � for any fixed set of positive constants αi, i = 0, 1, 2, ...,k following [9] the linear combination ln,k of the operators lαi,n, i = 0, 1, 2, ...k is defined by (20) ln,k(f; x) = 1 4 ∣∣∣∣∣∣∣∣∣∣ lα0n(f; x) α −1 0 α −2 0 ... ... α −k 0 lα1n(f; x) α −1 1 α −2 1 ... ... α −k 1 .... ... ... ... ... ... .... ... ... ... ... ... lαkn(f; x) α −1 k α −2 k ... ... α −k k ∣∣∣∣∣∣∣∣∣∣ where 4 is the determinant obtained by replacing the operator column by the entries ′1′. clearly (21) ln,k = ∑k j=0 c(j,k)lαjn, for constants c(j,k),j = 0, 1, 2, ...,k which satisfy ∑k j=1 c(j,k) = 1. ln,k is called a linear combination of order k. ln,0 denotes the operator ln itself. theorem 7. : if f ∈ dω. if at a point x ∈ ir+,f(2k+2)exists, then (22) |ln,k(f; x) −f(x)| = o(n−(k+1)), (23) |ln,k+1(f; x) −f(x)| = o(n−(k+1)), where k = 0, 1, 2, ... . also, if f(2k+2) exists and is continuous on < a,b >⊂ ir+, (22) −−(23) hold uniformly on [a,b]. proof. first we show that (24) ln(f; x) −f(x) = ∑2k+2 j=1 xjf(j)(x) j! τn,j + o(n −(k+1)), if x ∈ ir+ is such that f(2k+2) exists and f ∈ dω.to prove (24) with the assumption on f, we have f(u) −f(x) = ∑2k+2 j=1 f(j)(x) j! (u−x)j + rx(u); u → x, approximating derivatives by a class of positive linear operators 65 where rx(u) = o((u−x)2k+2),u → x. it is clear from the definition of τn,j that we only have to show that (25) ln(rx(u); x) = o(n −(k+1)). obviously, rx(u) ∈ dω. now, given an arbitrary � > 0, we can choose a δ > 0 s.t. |rx(u)| ≤ �(u−x)2k+2, |u−x| ≤ δ. hence, by using the basic properties of ln[1], we note that the result follows.in this case the uniformity part is obvious. now, using lemma2 and (24) we get (26) ln(f; x) −f(x) = ∑2k+2 j=1 xjf(j)(x) j! ∑k+1 ν=[ j+1 2 ] γj,ν nν + o(n−(k+1)), which, in the uniformity case holds uniformly in x ∈ [a,b].since the coefficients c(j,k) in (21) obviously satisfy the relation (27) ∑k j=0 c(j,k)α −p j = 0,p = 1, 2, 3, ...,k. in view of (26), (22) −−(23) are immediate and so is the uniformity part. this completes the proof of theorem7. � in the same spirit we have, theorem 8. :.let f ∈ dω. if 0 ≤ p ≤ 2k + 2 and f(p) exists and is continuous on < a,b >⊂ ir+, for each x ∈ [a,b] and sufficiently large n then (28) |ln,k(f; x) −f(x)| ≤ max[cn− p 2 ω(f(p); n− 1 2 ),c′n−(k+1)] where c = c(k) and c′ = c′(k,f) are constants and ω(f(p); δ) denotes the local modulus of continuity of f(p) on < a,b > . proof. :there exists a δ > 0 s.t. [a − δ,b + δ] ⊂< a,b > . it is clear that if u ∈< a,b >, there exists an η lying between x ∈ [a,b] and u s.t. (29) ∣∣∣f(u) −f(x) −∑pj=1 f(j)(x)j! (u−x)j∣∣∣ ≤ |u−x|pp! (1+|u−x|n12 )ω(f(p); n−12 ), using a well known result on modulus of continuity [13]. if the expression occuring within the modulus sign on l.h.s. of the above inequality is denoted by fx(u), by a well known property of ln, it follows that lαjn(fx(u)χ c δ,x(u); x) = o(n −(k+1)), uniformly in x ∈ [a,b]. by (29), we have (30) ∣∣∣lαjn(fx(u)χcδ,x(u); x)∣∣∣ ≤ bpp! (ap + ap−1)(αjn)−p2 ω(f(p); n−12 ) for all n sufficiently large and x ∈ [a,b]. here ap,ap−1 are constants depending on p. hence, for a constant cp independent of f such that for all x ∈ [a,b], (31) ∣∣∣ln,k(fx(u)χcδ,x(u); x)∣∣∣ ≤ cpn−p2 ω(f(p); n−12 ). applying the result (22) for the functions 1,u,u2,u3, ...,up, we find that there exists a constant c′′ depending on max{|f′(x)| , ..., ∣∣f(p)(x)∣∣ ; x ∈ [a,b]} and p such that for all x ∈ [a,b], (32) ∣∣∣ln,k(∑pj=1 f(j)(x)j! (u−x)j; x)∣∣∣ ≤ c′′n−(k+1) now, (28) is clear from (30) −−(32). this completes the proof of the theorem. � theorem 9. :let f ∈ dω. if at a point x ∈ ir+,f(2k+p+2) exists then (33) ∣∣∣l(p)n,k(f; x) −f(p)(x)∣∣∣ = o(n−(k+1)),and (34) ∣∣∣l(p)n,k+1(f; x) −f(p)(x)∣∣∣ = o(n−(k+1)), where k = 0, 1, 2, ... . also, if f(2k+p+2) exists and is continuous on < a,b >⊂ ir+, (33) −−− (34) hold uniformly in x ∈ [a,b]. 66 bramha dutta pandey1,∗ and b. kunwar2 proof. if f(2k+p+2) exists , we can find a neighbourhood (a′,b′) of x s.t. f(p) exists and is continuous on (a′,b′). let g(u) be an infinitely differentiable function with supp g ⊆ (a′,b′) s.t. g(u) = 1 for u ∈ [x − δ,x + δ] for some δ > 0. then an application of lemma1 shows that (35) l (p) n,k(f(u) −f(u)g(u); x) = o(n −(k+1)). in the uniformity case, we consider a g(u) with supp g ⊂< a,b > with g(u) = 1, for u ∈ [a − δ,b + δ] ⊆< a,b > and then (34) holds uniformly in x ∈ [a,b]. since f(u)g(u) ∈ c(p)b ir + we have (36) l (p) n (fg; x) = x −pln(u p{f(u)g(u)}(p); x). now, since up{f(u)g(u)}(p) is (2k+2)−times differentiable at x (and continuously differentiable on (a−δ,b + δ) in the uniformity case), applying theorem7 we have (37) ∣∣∣l(p)n,k(fg; x) −f(p)(x)∣∣∣ = o(n−(k+1)),and (38) ∣∣∣l(p)n,k+1(fg; x) −f(p)(x)∣∣∣ = o(n−(k+1)), where, in the uniformity case these holds in x ∈ [a,b]. thus, combining (35) − −− (38), we get (33) −−− (34). this completes the proof . � theorem 10. : let m ∈ in, and f ∈ dω. if 0≤ q ≤ 2k + 2 and f(p+q) exists and is continuous on < a,b >⊆ ir+ for each x ∈ [a,b],then for all sufficiently large n, (39) ∣∣∣l(p)n,k(f; x) −f(p)(x)∣∣∣ ≤ max{cpn−( k2 )ω(f(p+q); n−12 ),c′pn−(k+1)} where cp = cp(k),c ′ p = c ′ p(k,f) are constants and ω(f (p+q); δ) denotes the local modulus of continuity of f(p+q) on < a,b > . proof. : the proof of this theorem follows from lemma1 and theorem5−−9. � references [1] bramha dutta pandey and b. kunwar, on a class of positive linear operators(communicated). [2] r. a. devore and g. g. lorentz, constructive approximation. springer-verlag berlin heidelberg, new york, (1993). [3] p.p. korovkin, linear operators and approximation theory (1960), delhi. (translated from russian edition of 1959). [4] b. kunwar, approximation of analytic functions by a class of linear positive operators, j. approx. theory 44. 173-182 (1985). [5] g. g. lorentz, bernstein polynomials, toronto (1953). [6] c. p. may, saturation and inverse theorems for combination of a class of exponential type operators. canad. j. math. 28,(1976), 1224-50. [7] m. w.muller, die folge der gamma operatoren.thesis,technische hoschule,stuttgort.(1967). [8] paula anamaria piţul, evaluation of the approximation order by positive linear operators, ph. d. thesis, universitatea babeş bolyai(romania) 2007. [9] r. k. s. rathore, linear combinations of linear positive operators and generating relations in special functions, dissertation, iit delhi (india) (1973). [10] r. k. s. rathore, approximation of unbounded functions with linear positive operators, doctoral dissertation, technische hogeschool delft (1974). [11] p. c. sikkema and r. k. s. rathore,convolution with powers of bell shaped functions. report, dept. of math. technische hogeschool (1976). approximating derivatives by a class of positive linear operators 67 [12] p. c. sikkema, approximation formulae of voronovskayatype for certain convolution operators, j. approx. theory 26(1979), 26-45. [13] a. f. timan, theory of approximation of functions of a real variable, peargamon press (1963). 1department of applied sciences and humanities, institute of engineering and technology, lucknow-21 (india) 2department of applied sciences and humanities, institute of engineering and technology, lucknow-21 (india) ∗corresponding author the author is thankful to councel of scientific and industrial research, india for providing financial assistance for this research work under grant no.09/827(0004). international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 31-45 http://www.etamaths.com twin hypercube for intuitionistic fuzzy sets and their application in medicine b. davvaz1,∗, e. hassani sadrabadi1, juan j. nieto2 and angela torres3 abstract. in this paper, we present a description of intuitionistic fuzzy sets as vectors in twin hypercube. finally we verify an application of intuitionistic fuzzy sets in medicine using twin hypercube for intuitionistic fuzzy sets. 1. introduction the concept of intuitionistic fuzzy sets was introduced by atanassov [1]. the intuitionistic fuzzy sets have some applications in different sciences. in the papers [6, 9, 12, 13] has been developed the concept of applications of intuitionistic fuzzy sets in medical diagnosis using from relation between intuitionistic fuzzy sets and symptoms. also this paper is a generalization of the paper [11] to intuitionistic fuzzy sets. we present some applications of intuitionistic fuzzy sets in diagnosis. actually by employing twin hypercube for intuitionistic fuzzy sets we determine the kind of patient. this paper is organized as follows: in first section we recall basic facts about intuitionistic fuzzy sets. in second section we present twin hypercube. in third section we describe distance and entropy of intuitionistic fuzzy sets and fourth section we present intuitionistic fuzzy segments and intuitionistic fuzzy midpoints. in the final section we study an application of intuitionistic fuzzy sets in medicine. 2. basic facts about intuitionistic fuzzy sets after the introduction of fuzzy sets by zadeh [16], several researches consider some generalization of fuzzy sets. as an important generalization of the notion of fuzzy sets on a non-empty set x, atanassov introduced in [1, 3], the concept of intuitionistic fuzzy sets defined on a non-empty set x as an object having the form a = {〈x,µa(x),νa(x)〉 : x ∈ x}, (2.1) where the functions µa : x → [0, 1], νa : x → [0, 1], denote the degree of membership and the degree of non-membership of each element x ∈ x to the set a respectively, and 0 ≤ µa(x) + νa(x) ≤ 1 for all x ∈ x. if νa(x) = 1 −µa(x) , then a is a classical fuzzy set. such defined objects are studied by many authors and have many interesting applications not only in mathematics (see [5]). let a and b be two intuitionistic fuzzy sets in x. then, the following expressions are defined in [1, 3]. (1) a ⊆ b if and only if µa(x) ≤ µb(x) and νa(x) ≥ νb(x), (2) a = b if and only if a ⊆ b and b ⊆ a, (3) ac = {〈x,νa(x),µa(x)〉 : x ∈ x}, (4) a∩b = {〈x, min{µa(x),µb(x)}, max{νa(x),νb(x)}〉 : x ∈ x}, (5) a∪b = {〈x, max{µa(x),µb(x)}, min{νa(x),νb(x)}〉 : x ∈ x}, (6) �a = {〈x,µa(x), 1 −µa(x)〉 : x ∈ x}, (7) ♦a = {〈x, 1 −νa(x),νa(x)〉 : x ∈ x}. received 3rd may, 2017; accepted 26th june, 2017; published 1st september, 2017. 2010 mathematics subject classification. 92c50, 03e72. key words and phrases. intuitionistic fuzzy set; hamming distance; intuitionistic fuzzy midpoint; intuitionistic fuzzy segment; medicine. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 31 32 davvaz, hassani sadrabadi, nieto and torres we recall the following two examples from [7]. example 2.1. consider the universe x = {10, 100, 500, 1000, 1200}. an intutionistic fuzzy set “large” of x denoted by l and may be defined by l = {〈10, 0.01, 0.9〉, 〈100, 0.1, 0.88〉, 〈500, 0.4, 0.5〉, 〈1000, 0.8, 0.1〉, 〈1200, 1, 0〉}. one may define an intuitionistic fuzzy set “very large” (denoted by v l) as follows: µv l(x) = (µl(x)) 2 and νv l(x) = 1 − (1 −νl(x)) 2 , for all x ∈ x. thus, v l = {〈10, 0.0001, 0.99〉, 〈100, 0.01, 0.9856〉, 〈500, 0.16, 0.75〉, 〈1000, 0.64, 0.19〉, 〈1200, 1, 0〉}. example 2.2. consider the universe {a1,a2,a3,a4,a5,a6}. let a and b be two intuitionistic fuzzy sets of x given by a = {〈a1, 0.2, 0.6〉, 〈a2, 0.3, 0.7〉, 〈a3, 1, 0〉, 〈a4, 0.8, 0.1〉, 〈a5, 0.5, 0.4〉} and b = {〈a1, 0.4, 0.4〉, 〈a2, 0.5, 0.2〉, 〈a3, 0.6, 0.2, 〈a4, 0.1, 0.7〉, 〈a5, 0, 1〉}. then, ac = {〈a1, 0.6, 0.2〉, 〈a2, 0.7, 0.3〉, 〈a3, 0, 1〉, 〈a4, 0.1, 0.8〉, 〈a5, 0.4, 0.5〉}, a∩b = {〈a1, 0.2, 0.6〉, 〈a2, 0.3, 0.7〉, 〈a3, 0.6, 0.2〉, 〈a4, 0.1, 0.7〉, 〈a5, 0, 1〉}, a∪b = {〈a1, 0.4, 0.4〉, 〈a2, 0.5, 0.2〉, 〈a3, 1, 0〉, 〈a4, 0.8, 0.1〉, 〈a5, 0.5, 0.4〉}, �a = {〈a1, 0.2, 0.8〉, 〈a2, 0.3, 0.7〉, 〈a3, 1, 0〉, 〈a4, 0.8, 0.2〉, 〈a5, 0.5, 0.5〉}, ♦b = {〈a1, 0.6, 0.4〉, 〈a2, 0.8, 0.2〉, 〈a3, 0.8, 0.2〉, 〈a4, 0.3, 0.7〉, 〈a5, 0, 1〉}. for each intuitionistic fuzzy set in x, we call πa(x) = 1 −µa(x) −νa(x), the intuitionistic index of x in a. it is a hesitancy degree of x to a [2, 4]. 3. twin hypercube similar to kosko [10], we introduce a geometrical interpretation of intuitionistic fuzzy sets as vectors in twin hypercube. indeed, for a given set x = {x1, . . . ,xn}, an intuitionistic fuzzy set a is determined just by two mappings µa : x −→ i = [0, 1] and νa : x −→ i = [0, 1], and the values µa(x) and νa(x) giving the degree of membership and the degree of non-membership of each element x ∈ x to the intuitionistic fuzzy set a, respectively. thus, the set of all intutionistic fuzzy sets in x is determined precisely by two unit hypercubes. one of them is the unit hypercube of membership degree and another one is the unit hypercube of non-membership degree. any intuitionistic fuzzy set a determines a vector p ∈ in × in given by 〈(µa(x1), . . . ,µa(xn), (νa(x1), . . . ,νa(xn)〉. conversely, any vector q = 〈(a1, . . . ,an), (b1, . . . ,bn)〉 generates an intuitionistic fuzzy set a defined by µa(xi) = a1 and νa(xi) = b1, for i = 1, . . . ,n. cardinality of intuitionistic fuzzy sets [15]. let a be an intuitionistic fuzzy set in x. first, we consider the following two cardinalities of an intuitionistic fuzzy set. twin hypercube for intuitionistic fuzzy sets and their application in medicine 33 (1) the least cardinality of a is equal to the so-called sigma-count (cf. [17, 18]), and is called the min ∑ count (min-sigma-count): min ∑ count(a) = n∑ i=1 µa(xi). (2) the biggest cardinality of a, which is possible due to πa, is called the max ∑ count (maxsigma-count), and is equal to max ∑ count(a) = n∑ i=1 (µa(xi) + πa(xi)). clearly, for ac , we have min ∑ count(ac) = n∑ i=1 νa(xi), max ∑ count(ac) = n∑ i=1 (νa(xi) + πa(xi)). now, the cardinality of an intuitionistic fuzzy set a is defined as the interval card(a) = |a| = [ min ∑ count(a), max ∑ count(a) ] . therefore, we have |a| = [ n∑ i=1 µa(xi), n∑ i=1 (1 −νa(xi)) ] . (3.1) theorem 3.1. if a and b are two intuitionistic fuzzy sets in x, then |a∪b| = |a| + |b|− |a∩b|. (3.2) proof. according to eq. (3.1), we have |a∪b| = [ n∑ i=1 µa∪b(xi), n∑ i=1 (1 −νa∪b(xi)) ] = [ n∑ i=1 max{µa(xi),µb(xi)}, n∑ i=1 (1 − max{νa(xi),νb(xi)}) ] = [ n∑ i=1 max{µa(xi),µb(xi)}, n∑ i=1 min{1 −νa(xi), 1 −νb(xi)} ] , |a∩b| = [ n∑ i=1 µa∩b(xi), n∑ i=1 (1 −νa∩b(xi)) ] = [ n∑ i=1 min{µa(xi),µb(xi)}, n∑ i=1 (1 − min{νa(xi),νb(xi)}) ] = [ n∑ i=1 min{µa(xi),µb(xi)}, n∑ i=1 max{1 −νa(xi), 1 −νb(xi)} ] . therefore, we obtain |a∪b| + |a∩b| = [ n∑ i=1 ( µa(xi) + µb(xi) ) , ( (1 −νa(xi)) + (1 −νb(xi)) )] = [ n∑ i=1 µa(xi), n∑ i=1 (1 −νa(xi)) ] + [ n∑ i=1 µb(xi), n∑ i=1 (1 −νb(xi)) ] = |a| + |b|. � remark 3.1. when i = 1, for simplicity we use the symbol max count(a) insteas of max ∑ count(a). 34 davvaz, hassani sadrabadi, nieto and torres 4. distance and entropy of intuitionistic fuzzy sets the above discussion allows us to consider the following distance between intutionistic fuzzy sets. for two intutionistic fuzzy sets a and b, the hamming distance [14, 15] between a and b are given by d(a,b) = n∑ i=1 ( |µa(xi) −µb(xi)| + |νa(xi) −νb(xi)| + |πa(xi) −πb(xi)| ) . (4.1) each summand in (4) is between 0 and 1. for a,b,c intuitionistic fuzzy sets, we have: (1) 0 ≤ d(a,b) ≤ n, (2) d(a,b) = 0 if and only if a = b, (3) d(a,b) = d(b,a), (4) d(a,b) ≤ d(a,c) + d(c,b). a geometric interpretation of intutionistic fuzzy sets is presented in [15]. an intuitionistic fuzzy set is represented by a point in the triangle a′b′d′, where a′(1, 0, 0), b′(1, 0, 0) and d′(0, 0, 1). an intutionistic fuzzy set a in x is mapped into triangle a′b′d′ in that each element of x corresponds to an element of a′b′d′, as an example, a point x′ ∈ a′b′d′ corresponding to x ∈ x is marked the values of µa(x), νa(x) and πa(x). let e be a fuzzy set defined on fuzzy sets. then, e is an entropy measure if it satisfies the four de luca and termini axioms [8]. according to a geometric representation of an intuitionistic fuzzy set [15], we have e(a) = a b , (4.2) where a is the distance (a,anear) from a to the nearest point anear among a ′ and b′, and b is the distance (a,afar) from a to the fareast point afar among a ′ and b′. the eq. (4.2) describes the degree of fuzziness for a single point belonging to an intuitionistic fuzzy set. for n points belonging to an intuitionistic fuzzy set we have e(a) = 1 n n∑ i=1 e(ai). (4.3) theorem 4.1. [15] a generalized entropy measure of an intuitionistic fuzzy set a in x is e(a) = 1 n n∑ i=1 (max count(ai ∩aci ) max count(ai ∪aci ) ) = 1 n n∑ i=1 n∑ j=1 µai∩aci (xj) + πai∩a c i (xj) n∑ j=1 µai∪aci (xj) + πai∪a c i (xj) = 1 n n∑ i=1 n∑ j=1 min{µai (xj),νai (xj)} + 1 −µai∩aci −νai∩aci n∑ j=1 max{µai (xj),νai (xj)} + 1 −µai∪aci −νai∪aci = 1 n n∑ i=1 n∑ j=1 min{µai (xj),νai (xj)} + 1 − 2 min{µai (xj),νai (xj)} n∑ j=1 max{µai (xj),νai (xj)} + 1 − 2 max{µai (xj),νai (xj)} = 1 n n∑ i=1 n∑ j=1 −min{µai (xj),νai (xj)} + 1 n∑ j=1 −max{µai (xj),νai (xj)} + 1 . twin hypercube for intuitionistic fuzzy sets and their application in medicine 35 where ai denotes the single element intutionistic fuzzy set corresponding to the i th element of the universe x. in other words, ai is the i th component of a. for two intuitionistic fuzzy subsets a and b, it is possible to define a degree of subsethood s(a,b) = |a∩b| |a| . for the midpoint m, since µm (xj) = νm (xj) = 1 2 , we have e(m) = 1 n n∑ i=1 n∑ j=1 1 2 n∑ j=1 1 2 = 1 n n∑ i=1 n 2 n 2 = 1 n n∑ i=1 1 = n n = 1. since for any intutionistic fuzzy subset a 6= m, we have e(a) < 1, therefore the midpoint m is maximally intutionistic fuzzy set. now we present the following definition. the intuitionistic fuzzy subset c is between intuitionistic fuzzy subsets a and b, if d(a,c) + d(c,b) = d(a,b). the metric segment between a and b is defined as the following: segment(a,b) = {c : d(a,c) + d(c,b) = d(a,b)}, that is, n∑ i=1 |µa(xi) −µc (xi)| + |µc (xi) −µb(xi)| = n∑ i=1 |µa(xi) −µb(xi)|, n∑ i=1 |νa(xi) −νc (xi)| + |νc (xi) −νb(xi)| = n∑ i=1 |νa(xi) −νb(xi)|. another concept is the idea of equidistant points: equid(a,b) = {c : d(a,c) = d(c,b)}, that is, n∑ i=1 |µa(xi) −µc (xi)| = n∑ i=1 |µc (xi) −µb(xi)|, n∑ i=1 |νa(xi) −νc (xi)| = n∑ i=1 |νc (xi) −νb(xi)|. if c ∈ segment(a,b), and it is equidistant to a and b, then d(a,c) = d(c,b) = 1 2 d(a,b). therefore we introduce the set of midpoints between a and b: mid(a,b) = {c : d(a,c) = d(c,b) = 1 2 d(a,b)}, that is, n∑ i=1 |µa(xi) −µc (xi)| = n∑ i=1 |µc (xi) −µb(xi)| = 1 2 n∑ i=1 |µa(xi) −µb(xi)|, n∑ i=1 |νa(xi) −νc (xi)| = n∑ i=1 |νc (xi) −νb(xi)| = 1 2 n∑ i=1 |νa(xi) −νb(xi)|. obviously for any two intuitionistic fuzzy subsets a and b we have, mid(a,b) ⊆ equid(a,b). 36 davvaz, hassani sadrabadi, nieto and torres the canonical midpoint a and b is denoted by µc (xi) = 1 2 (µa(xi) + µb(xi)), νc (xi) = 1 2 (νa(xi) + νb(xi)), i = 1, 2, . . . ,n. this canonical midpoint is not the unique midpoint and there are more midpoints. actually we show c is a midpoint between a and b: n∑ i=1 |µa(xi) − 1 2 (µa(xi) + µb(xi))| = n∑ i=1 1 2 |µa(xi) −µb(xi)|, n∑ i=1 |νa(xi) − 1 2 (νa(xi) + νb(xi))| = n∑ i=1 1 2 |νa(xi) −νb(xi)|. therefore d(a,c) = 1 2 d(a,b) and similarly d(c,b) = 1 2 d(a,b), then d(a,c) = d(c,b) = 1 2 d(a,b). if a is a crisp subset of x, that is, a is a vertex of the hypercube, then µa : x → {0, 1}, νa : x →{0, 1}, therefore d(a,m) = n∑ i=1 |µa(xi) − 1 2 | + |νa(xi) − 1 2 | + |πa(xi) − (1 −µm (xi) −νm (xi))| = n∑ i=1 1 2 + 1 2 = n∑ i=1 1 = n, and d(a,ac) = n∑ i=1 |µa(xi) −µac (xi)| + |νa(xi) −νac (xi)| + |πa(xi) −πac (xi)| = n∑ i=1 |µa(xi) −νa(xi)| + |νa(xi) −µa(xi)| + |πa(xi) − (1 −νa(xi) −µa(xi))| = n∑ i=1 2|µa(xi) −νa(xi)| = n∑ i=1 2 = 2n. similarly d(ac,m) = n, and then d(a,m) = d(ac,m) = 1 2 d(a,ac). hence n∑ i=1 |µa(xi) − 1 2 | = 1 2 n∑ i=1 |µa(xi) −µac (xi)| = n 2 , n∑ i=1 |νa(xi) − 1 2 | = 1 2 n∑ i=1 |νa(xi) −νac (xi)| = n 2 . since d(a,c) = d(c,b) = 1 2 d(a,b), it follows that d(a,c)+d(c,b) = 1 2 d(a,b) and so mid(a,b) ⊂ segment(a,b). hence, c ∈ segment(a,b). in the bellow example we describe the definitions example 4.1. let x = {x1,x2}, µa = {0, 0},νa = {1, 1},µb = {1, 1},νb = {0, 0}. then, we have µa + µb 2 = ( 1 2 , 1 2 ), νa + νb 2 = ( 1 2 , 1 2 ) ∈ mid(a,b). if µc = (c1,c2),νc = (c3,c4) is another midpoint between a and b then: d(a,c) = c1 + c2 + (1 − c3) + (1 − c4), d(a,b) = n∑ i=1 |µa(xi) −µb(xi)| + |νa(xi) −νb(xi)| = n∑ i=1 (2 + 2) = 4, d(c,b) = 1 − c1 + 1 − c2 + c3 + c4 ⇒ c1 + c2 = 1,c3 + c4 = 1. therefore • d(a,c) = 1 2 d(a,b) ⇒ 1 − c1 + 1 − c2 = 1 ⇒ c1 + c2 = 1, twin hypercube for intuitionistic fuzzy sets and their application in medicine 37 • d(c,b) = 1 2 d(a,b) ⇒ c3 + c4 = 1, so mid(a,b) = {c : (c1,c2), (c3,c4)|c1 + c2 = 1,c3 + c4 = 1}, and we have an infinite set of midpoints. if c ∈ equid(a,b), then c1 + c2 = (1 − c1) + (1 − c2),c3 + c4 = (1 − c3) + (1 − c4). therefore c1 + c2 = 1,c3 + c4 = 1. then mid(a,b) = equid(a,b). as we can see these sets coincide. suppose that µc = ( 1 2 , 1 2 ), νc = ( 1 2 , 1 2 ). then, d(a,b) = 2,d(a,c) = 2,d(c,b) = 2. thus, c ∈ equid(a,b) but c /∈ segment(a,b), since d(a,c) + d(c,b) 6= d(a,b), therefore equid(a,b) * segment(a,b) figure 1. midpoints between µa = (0, 0), νa = (1, 1) and µb = (1, 1), νb = (0, 0) example 4.2. now consider µa = (0, 0), νa = (1, 1), µb = (1, 0), νb = (0, 1), then d(a,b) = 2, c ∈ mid(a,b) if and only if • c1 + c2 = 1 − c1 + c2 = 1 ⇒ c1 = 1 2 ,c2 = 1 2 , • 1 − c3 + 1 − c4 = c3 + 1 − c4 = 1 ⇒ c3 = 1 2 ,c4 = 1 2 , so mid(a,b) = { ( 1 2 , 1 2 ), ( 1 2 , 1 2 ) } . also, c ∈ equid(a,b) if and only if • c1 + c2 = 1 − c1 + c2 ⇒ c1 = 1 2 , • 1 − c3 + 1 − c4 = c3 + 1 − c4 ⇒ c3 = 1 2 , then equid(a,b) = { ( 1 2 ,c2), ( 1 2 ,c4) } , hence mid(a,b) ⊂ equid(a,b). figure 2. equidistant points between µa = (0, 0), νa = (1, 1) and µb = (1, 0), νb = (0, 1) 38 davvaz, hassani sadrabadi, nieto and torres 5. intuitionistic fuzzy segments and intuitionistic fuzzy midpoints in this section, first we verify the segment between two points in a square and determine the set of midpoints between a and b. let µa = (a1,a2),νa = (a3,a4),µb = (b1,b2),νb = (b3,b4) be intuitionistic fuzzy subsets of the set x = {x1,x2}. if µc = (c1,c2),νc = (c3,c4) ∈ segment(a,b), then: |a1 − c1| + |a2 − c2| + |b1 − c1| + |b2 − c2| = |a1 − b1| + |a2 − b2|, (5.1) |a3 − c3| + |a4 − c4| + |b3 − c3| + |b4 − c4| = |a3 − b3| + |a4 − b4|. (5.2) so, we obtain min{ai,bi}≤ ci ≤ max{ai,bi}, i = 1, 2, 3, 4. if min{ai,bi} > ci, then • |a1 − c1| + |b1 − c1| > |b1 −a1|, |a2 − c2| + |b2 − c2| > |b2 −a2|, so |a1 −c1| + |a2 −c2| + |b1 −c1| + |b2 −c2| > |b1 −a1| + |b2 −a2| that is contradiction with 5.1, also • |a3 − c3| + |b3 − c3| > |b3 −a3|, |a4 − c4| + |b4 − c4| > |b4 −a4|, therefore |a3 − c3| + |a4 − c4| + |b3 − c3| + |b4 − c4| > |b3 −a3| + |b4 −a4|, this relation is contradiction with 5.2. theorem 5.1. the points in the segment between a and b is as follows: segment(a,b) = {(c1,c2), (c3,c4)|min{ai,bi}≤ ci ≤ max{ai,bi}, i = 1, 2, 3, 4}. proof: suppose that a1 ≤ b1 and a3 ≥ b3 (if a1 > b1 and a3 ≤ b3 we interchange the roles of µa,µb and νa,νb). now, there are four possibilities, if µc = (c1,c2),νc = (c3,c4) then: (1) a1 ≤ b1, a2 ≤ b2, a3 ≥ b3, a4 ≤ b4 ⇒ a1 < c1 < b1, a2 < c2 < b2, b3 < c3 < a3, a4 < c4 < b4, (2) a1 ≤ b1, a2 > b2, a3 ≥ b3, a4 ≤ b4 ⇒ a1 < c1 < b1, b2 < c2 < a2, b3 < c3 < a3, a4 < c4 < b4, (3) a1 ≤ b1, a2 ≤ b2, a3 ≥ b3, a4 > b4 ⇒ a1 < c1 < b1, a2 < c2 < b2, b3 < c3 < a3, b4 < c4 < a4, (4) a1 ≤ b1, a2 > b2, a3 ≥ b3, a4 > b4 ⇒ a1 < c1 < b1, b2 < c2 < a2, b3 < c3 < a3, b4 < c4 < a4. suppose that min{ai,bi} ≤ ci ≤ max{ai,bi} and we show c ∈ segment(a,b) that is: d(a,c) + d(c,b) = d(a,b). proof: 1. if a1 ≤ b1, a2 ≤ b2, a3 ≥ b3, a4 ≤ b4, then d(a,c) = c1 −a1 + c2 −a2 + a3 − c3 + c4 −a4, d(c,b) = b1 − c1 + b2 − c2 + c3 − b3 + b4 − c4, d(a,c) = b1 −a1 + b2 −a2 + a3 − b3 + b4 −a4. therefore d(a,b) = d(a,c) + d(c,b). similarly in all of cases 2,3,4, d(a,b) = d(a,c) + d(c,b). theorem 5.2. the set of midpoints between a and b has the following possibilities: (1) if either a1 = b1, a3 = b3 or a2 = b2, a4 = b4, then there is a midpoint given by mid(a,b) = { µa + µb 2 , νa + νb 2 }. (2) if a1 < b1, a2 < b2, a3 > b3 and a4 < b4, then the set of midpoints is as follows. if µc = (c1,c2), νc = (c3,c4), then: • c1 + c2 = 1 2 (a1 + a2 + b1 + b2), • c3 + c4 = 1 2 (a3 + a4 + b3 + b4), (3) if a1 < b1, a2 > b2, a3 > b3, a4 > b4, then: • c1 − c2 = 1 2 (a1 −a2 + b1 − b2), • c3 − c4 = 1 2 (a3 −a4 + b3 − b4), (4) if a1 < b1, a2 < b2, a3 > b3, a4 > b4, then: • c1 + c2 = 1 2 (a1 + a2 + b1 + b2), twin hypercube for intuitionistic fuzzy sets and their application in medicine 39 figure 3. segment between a and b: 1. a1 ≤ b1, a2 ≤ b2, a3 ≥ b3, a4 ≤ b4, 2. a1 ≤ b1, a2 > b2, a3 ≥ b3, a4 ≤ b4, 3. a1 ≤ b1, a2 ≤ b2, a3 ≥ b3, a4 > b4, 4. a1 ≤ b1, a2 > b2, a3 ≥ b3, a4 > b4. • c3 − c4 = 1 2 (a3 −a4 + b3 − b4), (5) if a1 < b1, a2 > b2, a3 > b3, a4 < b4, then: • c1 − c2 = 1 2 (a1 −a2 + b1 − b2), • c3 + c4 = 1 2 (a3 + a4 + b3 + b4), proof: if c ∈ mid(a,b), then d(a,c) = d(c,b) = 1 2 d(a,b). so, we have: • d(a,c) = d(c,b), therefore: |a1 − c1| + |a2 − c2| + |a3 − c3| + |a4 − c4| = |b1 − c1| + |b2 − c2| + |b3 − c3| + |b4 − c4|, 40 davvaz, hassani sadrabadi, nieto and torres • d(a,c) = 1 2 d(a,b), then we have: |a1 − c1| + |a2 − c2| + |a3 − c3| + |a4 − c4| = 1 2 (|a1 − b1| + |a2 − b2| + |a3 − b3| + |a4 − b4|), • d(c,b) = 1 2 d(a,b), so we conclude: |b1 − c1| + |b2 − c2| + |b3 − c3| + |b4 − c4| = 1 2 (|a1 − b1| + |a2 − b2| + |a3 − b3| + |a4 − b4|), such that |a1 − c1| + |a2 − c2| = |b1 − c1| + |b2 − c2| = 1 2 (|a1 − b1| + |a2 − b2|), (5.3) |a3 − c3| + |a4 − c4| = |b3 − c3| + |b4 − c4| = 1 2 (|a3 − b3| + |a4 − b4|). (5.4) (1) suppose that a1 = b1 and a3 = b3. then we have c1 = a1 = b1 and c3 = a3 = b3. by using the above equalities we obtain • |a2 − c2| = |b2 − c2|, • |a4 − c4| = |b4 − c4| = 1 2 |a4 − b4|. then from the above equalities we conclude c2 = 1 2 (a2 + b2) and c4 = 1 2 (a4 + b4). similarly if a2 = b2 and a4 = b4 then we have c2 = a2 = b2 and c4 = a4 = b4. therefore, c1 = 1 2 (a1 + b1), c3 = 1 2 (a3 + b3). figure 4. unique midpoint between a and b. (2) now, suppose that a1 < b1, a2 < b2, a3 > b3, a4 < b4. then, by using 5.3 we find: • c1 −a1 + c2 −a2 = b1 − c1 + b2 − c2, • c1 −a1 + c2 −a2 = 1 2 (b1 −a1 + b2 −a2), • b1 − c1 + b2 − c2 = 1 2 (b1 −a1 + b2 −a2). so, c1 + c2 = 1 2 (a1 + a2 + b1 + b2), also by 5.3 we have: • c3 −a3 + c4 −a4 = b3 − c3 + b4 − c4, • c3 −a3 + c4 −a4 = 1 2 (a3 − b3 + b4 −a4), • b3 − c3 + b4 − c4 = 1 2 (a3 − b3 + b4 −a4). therefore, we obtain c3 + c4 = 1 2 (a3 + a4 + b3 + b4). (3) if a1 < b1, a2 > b2, a3 > b3, a4 > b4, then by using 5.3 we obtain: • c1 −a1 + a2 − c2 = b1 − c1 + c2 − b2, • c1 −a1 + a2 − c2 = 1 2 (b1 −a1 + a2 − b2), • b1 − c1 + c2 − b2 = 1 2 (b1 −a1 + a2 − b2). twin hypercube for intuitionistic fuzzy sets and their application in medicine 41 figure 5. midpoints between a and b: (a) b1 − a1 = b2 − a2, b3 − a3 = b4 − a4, (b) b1 −a1 < b2 −a2, a3 −b3 < b4 −a4, (c) b1 −a1 > b2 −a2, a3 −b3 > b4 −a4. thus, c1 − c2 = 1 2 (a1 −a2 + b1 − b2), also by using the relation 5.4 we have: • c3 −a3 + a4 − c4 = b3 − c3 + c4 − b4, • c3 −a3 + a4 − c4 = 1 2 (a3 − b3 + a4 − b4), • b3 − c3 + c4 − b4 = 1 2 (a3 − b3 + a4 − b4). so, we conclude c3 − c4 = 1 2 (a3 −a4 + b3 − b4). (4) let a1 < b1, a2 < b2, a3 ≥ b3, a4 > b4. according to the case 2, if a1 < b1, a2 < b2, we have c1 + c2 = 1 2 (a1 + a2 + b1 + b2). now, if a3 > b3, a4 > b4, we find c3 − c4 = 1 2 (a3 −a4 + b3 − b4). (5) let a1 < b1, a2 > b2, a3 > b3, a4 < b4. according to the case 3, if a1 < b1, a2 > b2, we obtain c1 − c2 = 1 2 (a1 −a2 + b1 − b2). also, if a3 > b3, a4 < b4, we have c3 + c4 = 1 2 (a3 + a4 + b3 + b4). now in this section we determine the segment between two points µa,νa and µb,νb and the set of midpoints between µa,νa and µb,νb in a hypercube i 3. 42 davvaz, hassani sadrabadi, nieto and torres as in the two-dimensional hypercube i2, we find the following result. theorem 5.3. the points in the segment between µa = (a1,a2,a3),νa = (a4,a5,a6) and µb = (b1,b2,b3),νb = (b4,b5,b6) are given by segment(a,b) = {c = (c1,c2,c3), (c4,c5,c6) : min{ai,bi}≤ ci ≤ max{ai,bi}, i = 1, 2, 3, 4, 5, 6}. example 5.1. suppose the first variable x1 depends on several blood tests, the second one x2 on cardiac tests, and the third variable x3 on some non-invasive vascular tests. for example, patient 1 is the point p1 = (0.9, 0.2, 0.9), (0.1, 0.8, 0.1) in twin three dimensional hypercube and patient 2 is p2 = (0.7, 0.8, 0.5), (0.3, 0.2, 0.5), patient 3 is p3 = (0.8, 0.2, 0.9), (0.2, 0.8, 0.1), then segment(p1,p2) = {(c1,c2,c3), (c4,c5,c6) : 0.7 ≤ c1 ≤ 0.9, 0.2 ≤ c2 ≤ 0.8, 0.5 ≤ c3 ≤ 0.9, 0.1 ≤ c4 ≤ 0.3, 0.2 ≤ c5 ≤ 0.8, 0.1 ≤ c6 ≤ 0.5}. this segment is represented in grey in figure 6. the point p3 = (0.8, 0.2, 0.9), (0.2, 0.8, 0.1) belongs to the segment between p1, p2. indeed d(p1,p2) = 2.4, d(p1,p2) = 0.2, d(p3,p2) = 2.2 and then d(p1,p2) = d(p1,p3) + d(p3,p2). figure 6. segment between p1 and p2 now, consider the intuitionistic fuzzy set µp1 = (0.7, 0.8, 0.5),νp1 = (0.3, 0.2, 0.5) and µp2 = (0.9, 0.2, 0.9), νp2 = (0.1, 0.8, 0.1). let p3 = (c1,c2,c3), (c4,c5,c6) be midpoint between µp1,νp1 and µp2,νp2 . then, d(p2,p3) = 1 2 d(p1,p2). therefore, • c1 − 0.7 + 0.8 − c2 + c3 − 0.5 = 0.6, • 0.3 − c4 + c5 − 0.2 + 0.5 − c6 = 0.6, also d(p1,p3) = 1 2 d(p1,p2), so • 0.9 − c1 + c2 − 0.2 + 0.9 − c3 = 0.6, • c4 − 0.1 + 0.8 − c5 + c6 − 0.1 = 0.6, therefore we obtain: c1 − c2 + c3 = 1,c4 − c5 + c6 = 0 such that c1,c2,c3,c4,c5,c6 have to satisfy 0.7 ≤ c1 ≤ 0.9, 0.2 ≤ c2 ≤ 0.8, 0.5 ≤ c3 ≤ 0.9, 0.1 ≤ c4 ≤ 0.3, 0.2 ≤ c5 ≤ 0.8, 0.1 ≤ c6 ≤ 0.5. twin hypercube for intuitionistic fuzzy sets and their application in medicine 43 figure 7. shows the set of midpoints between µa,νa and µb,νb. note that p3 = (0.8, 0.2, 0.9), (0.2, 0.8, 0.1) ∈ segment(a,b) but p3 /∈ mid(a,b) 6. an application of intuitionistic fuzzy sets in medicine we now state a complete description of the medical problem. as in the paper [11], we consider the following intuitionistic fuzzy variables: smoking and alcohol drinking. let x = {x1,x2}, for a nonsmoker we consider the degree of membership is 0 and the degree of nonmembership is 1, this point is the ideal situation for your health. also for example if you smoke six cigarettes per day we say that your degree of a smoker is 0.8, and we suppose the degree of nonmembership is 0.2. if the consumption of cigarettes is 10 or more, the degree of membership is 1 and the degree of nonmembership is 0. now for other intuitionistic fuzzy variables, if you drink no alcohol, the degree of membership is 0, and the the degree of nonmembership is 1. if you drink more than 75 cc of alcohol per day, the degree of alcoholism is 1 and the degree of non alcoholism is 0, for 25 cc per day, the degree of membership could be 0.4 and the degree of nonmembership is 0.6 and for 50 cc per day the degree of membership is 0.8 and the degree of nonmembership is 0.2. therefore the intuitionistic fuzzy set µa = (0, 0),νa = (1, 1) corresponds to a non smoker and teetotaler, also the intuitionistic fuzzy set µb = (1, 0),νb = (0, 1) showes a heavy smoker but a non teetotaler and the intuitionistic fuzzy set µc = (0.8, 1),νc = (0.2, 0) represents that person smokes about six cigarettes per day and 75 cc of alcohol per day. according to the above text, µa = (0, 0),νa = (1, 1) is the ideal situation for your health that is difficult to achieve. also for µb = (1, 1),νb = (0, 0) your physician has suggested you to reduce your consumption of cigarettes and alcohol by half, therefore you may achieve a midpoint between intuitionistic fuzzy set µa,νa and µb,νb. the intuitionistic fuzzy set µm = µa + µb 2 = (0.5, 0.5),νm = νa + νb 2 = (0.5, 0.5) is a moderate number that you consume four cigarettes per day and 30 cc alcohol per day. figure 8. midpoint between a non smoker and teetotaler µa = (0, 0),νa = (1, 1) and a heavy smoker and heavy drinker µb = (1, 1),νb = (0, 0). according to the previous sections, a midpoint between µa,νa and µb,νb is any intuitionistic fuzzy set (a1,a2), (a3,a4) with a1 + a2 = 1, a3 + a4 = 1, therefore the points µa = (0.2, 0.8),νa = (0.8, 0.2) 44 davvaz, hassani sadrabadi, nieto and torres and µb = (0.7, 0.3),νb = (0.3, 0.7) are also valid midpoints. any intuitionistic fuzzy set with ν = 1−µ is on the line a1 + a2 = 1, a3 + a4 = 1. if we calculate the distance of any intuitionistic fuzzy points that is on the line a1 + a2 = 1, a3 + a4 = 1 to point m = (0.5, 0.5), (0.5, 0.5), the point that has the least distance, has better condition. figure 9 example 6.1. consider the intuitionistic fuzzy sets a = (0.3, 0.7), (0.7, 0.3),b = (0.4, 0.6), (0.6, 0.4),c = (0.1, 0.9), (0.9, 0.1), by above description we find the best point: the midpoint is m = (0.5, 0.5), (0.5, 0.5), therefore d(a,m) = 0.2 + 0.2 + 0.2 + 0.2 = 0.8, d(b,m) = 0.4, d(c,m) = 1.2, that c has the most distance and b has the lest distance. therefore the best point is b and the worst point is c. any intuitionistic fuzzy set with ν 6= 1−µ is not on the line a1 + a2 = 1, a3 + a4 = 1. for foundation the best point, we calculate distance of points to ideal point, that is a = (0, 0), (1, 1). any point that it’s distance is the least from ideal point, this point is the best point. example 6.2. consider the intuitionistic fuzzy sets a = (0.3, 0.8), (0.5, 0.1), b = (0.3, 0.8), (0.7, 0.2) and c = (0.3, 0.8), (0.6, 0.1). for point a the degree of membership of smoking is 0.3 and the degree of nonmembership is 0.5 and the degree of membership of alcoholism is 0.8 and the degree of nonmembership is 0.1. we calculate distances a,b,c to ideal point, that is i = (0, 0), (1, 1), therefore d(a,i) = 2.5, d(b,i) = 2.2, d(c,i) = 2.4. as we can see the distance of b to ideal point is minimum and it is the best point, also the distance of a to point i is maximum, and it is the worst point. example 6.3. suppose that µp1 = (0.1, 0.6, 0.9), νp1 = (0.9, 0.4, 0.1), µp2 = (0.4, 0.4, 0.9), νp2 = (0.6, 0.6, 0.1), µp3 = (1, 0.4, 0.5), νp3 = (0, 0.6, 0.5). now, we find distances p1,p2,p3 to ideal point. by consideration µi = (0, 0, 0), νi = (1, 1, 1), then d(p1,i) = 2.8, d(p2,i) = 2, d(p3,i) = 2.4. therefore the best point is p2 and the worst point is p1. references [1] k. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. [2] k. atanassov, more on intuitionistic fuzzy sets, fuzzy sets syst. 33 (1989), 37-46. [3] k. atanassov, new operations defined over the intuitionistic fuzzy sets, fuzzy sets syst. 61 (1994), 137-142. [4] k. atanassov, operators over interval valued intuitionistic fuzzy sets, fuzzy sets syst. 64 (1994), 159-174. [5] k. atanassov, intuitionistic fuzzy sets, theory and applications, studies in fuzziness and soft computing, 35 (1999), physica-verlag, heidelberg. [6] b. davvaz and e. hassani sadrabadi, an application of intuitionistic fuzzy sets in medicine, int. j. biomath. 9 (3) (2016), art. id 1650037. [7] b. davvaz and s.k. majumder, atanassov’s intuitionistic fuzzy interior ideals of γ-semigroups, sci. bull., politeh. univ. buchar., ser. a 73(3) (2011), 45-60. [8] a. de luca and s. termini, a defnition of a non-probabilistic entropy in the setting of fuzzy sets theory, inform. control 20 (1972), 301-312. [9] s.k. de, r. biswas and a.r. roy, an application of intuitionistic fuzzy sets in medical diagnosis, fuzzy sets syst. 117 (2001), 209-213. [10] b. kosko, neural networks and fuzzy systems, prentice-hall, englewood cliffs, 1992. [11] j.j. nieto and a. torres, midpoints for fuzzy sets and their applications in medicine, artif. intell. med. 27 (2003), 81-101. [12] e. sanches, solutions in composite fuzzy relation equation. application to medical diagnosis brouwerian logic. in: m.m. gupta, g.n. saridis, b.r. gaines (eds.), fuzzy automata and decision process, elsevier, north-holland, 1977. twin hypercube for intuitionistic fuzzy sets and their application in medicine 45 [13] e. szmidt and j. kacprzyk, intuitionistic fuzzy sets in some medical applications. in: reusch b. (eds) computational intelligence. theory and applications. fuzzy days 2001. lecture notes in computer science, vol 2206. springer, berlin, heidelberg (2001). [14] e. szmidt and j. kacprzyk, on distances between intuitionistic fuzzy sets, fuzzy sets syst. 114 (2000), 505-518. [15] e. szmidt and j. kacprzyk, entropy for intuitionistic fuzzy sets, fuzzy sets syst. 118 (2001), 467-477. [16] l.a. zadeh, fuzzy sets, infor. control 8 (1965), 338-353. [17] l.a. zadeh, a computational approach to fuzzy quantifers in natural languages, comput. math. appl. 9 (1983), 149-184. [18] l.a. zadeh, the role of fuzzy logic in the management of uncertainty in expert systems, fuzzy sets syst. 11 (1983), 199-227. 1department of mathematics, yazd university, yazd, iran 2departamento de análisis matemático estadstica y optimizacion,, facultad de matemáticas, universidad de santiago de compostela, 15782 santiago de compostela, spain 3departamento de psiquiatria, radiologia y salud pública, facultad de medicina, universidad de santiago de compostela, 15782 santiago de compostela, spain ∗corresponding author: davvaz@yazd.ac.ir 1. introduction 2. basic facts about intuitionistic fuzzy sets 3. twin hypercube 4. distance and entropy of intuitionistic fuzzy sets 5. intuitionistic fuzzy segments and intuitionistic fuzzy midpoints 6. an application of intuitionistic fuzzy sets in medicine references international journal of analysis and applications issn 2291-8639 volume 8, number 2 (2015), 130-143 http://www.etamaths.com a new monotone iteration principle in the theory of nonlinear fractional differential equations bapurao c. dhage abstract. in this paper the author proves the algorithms for the existence as well as approximations of the solutions for the initial value problems of nonlinear fractional differential equations using the operator theoretic techniques in a partially ordered metric space. the main results rely on the dhage iteration principle embodied in the recent hybrid fixed point theorems of dhage (2014) in a partially ordered normed linear space and the existence and approximations of the solutions of the considered nonlinear fractional differential equations are obtained under weak mixed partial continuity and partial lipschitz conditions. our hypotheses and existence and approximation results are also well illustrated by some numerical examples. 1. introduction the dhage iteration principle or method (in short dip or dim) developed in [3, 4, 5, 6, 7] is relatively new to the literature on nonlinear analysis, particularly in the theory of nonlinear differential and integral equations. but it is is a powerful tool in the subject of nonlinear analysis because of its utility of applications to nonlinear equations for different qualitative aspects of the solutions. see dhage and dhage [8], dhage et al. [13], pathak and lopez [16] and the references therein. very recently, the above method has been applied in dhage [3, 4, 6, 7] and dhage and dhage [8, 9, 10, 11] to nonlinear ordinary differential and integral equations for proving the existence and algorithms of the solutions. similarly, dip has also some interesting applications to the theory of nonlinear fractional differential and integral equations and in the present paper we prove the existence as well as approximations of the solutions for the initial value problems of fractional differential equations. before stating the main differential problems of the paper, we recall the following basic definitions of fractional calculus [15, 17] which are useful in what follows. definition 1.1. let j = [t0, t0 + a] be a closed and bounded of the real line r for some t0,a ∈ r with t0 ≥ 0 and a > 0. if x ∈ acn(j,r), then the caputo derivative cdqx of x of fractional order q is defined as cdqx(t) = 1 γ(n−q) ∫ t t0 (t−s)n−q−1x(n)(s)ds, n− 1 < q < n,n = [q] + 1, where [q] denotes the integer part of the real number q, and γ is the euler’s gamma function. here acn(j,r) denote the space of real valued functions x(t) which have continuous derivatives up to order n− 1 on j such that the nth derivative x(n) ∈ ac(j,r). definition 1.2. if j∞ = [t0,∞) is a closed interval of the real line r for some t0 ∈ r with t0 ≥ 0, then for any x ∈ c(j,r), the riemann-liouville fractional integral of order q > 0 is defined as iqx(t) = 1 γ(q) ∫ t t0 x(s) (t−s)1−q ds, t ∈ j∞, 2010 mathematics subject classification. 34a12, 34h34, 47h07, 47h10. key words and phrases. fractional differential equation; dhage iteration principle; hybrid fixed point theorem; existence and uniqueness theorems. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 130 nonlinear fractional differential equations 131 provided the right hand side is defined pointwise on (t0,∞), where γ is the euler’s gamma function. given a closed and bounded interval j = [t0, t0 + a] of the real line r for some t0,a ∈ r with t0 ≥ 0 and a > 0, consider the initial value problem (in short ivp) of nonlinear fractional differential equation (fde), (1.1) cdqx(t) = f(t,x(t)), t ∈ j, x(t0) = α0,   where cdq is the caputo derivative of fractional order q, 0 < q < 1 and f : j × r → r is a continuous function. by a solution of the fde (1.1) we mean a function x ∈ c1(j,r) that satisfies equation (1.1), where c1(j,r) is the space of continuously differentiable real-valued functions defined on j. the nonlinear fde (1.1) is well-known and it has been discussed at length for the existence and the uniqueness of the solutions under compactness and lipschitz conditions which are considered to be very strong in the theory of nonlinear differential and integral equations. in the present paper we prove the existence and uniqueness of the solutions of fde (1.1) under weaker partially compactness and partially lipschitz type conditions via dhage iteration principle and also indicate some realizations. the rest of the paper will be organized as follows. in section 2 we give some preliminaries and key fixed point theorem that will be used in subsequent part of the paper. in section 3 we discuss the existence result for initial value problems and in section 4 we discuss the existence result for initial value problems of hybrid differential differential equations with linear perturbation of first type. 2. auxiliary results unless otherwise mentioned, throughout this paper that follows, let e denote a partially ordered real normed linear space with an order relation � and the norm ‖ · ‖ in which the addition and the scalar multiplication by positive real numbers are preserved by � . a few details of such partially ordered spaces appear in dhage [2] and the references therein. two elements x and y in e are said to be comparable if either the relation x � or y � x holds. a non-empty subset c of e is called a chain or totally ordered if all elements of c are comparable. it is known that e is regular if {xn} is a nondecreasing (resp. nonincreasing) sequence in e such that xn → x∗ as n → ∞, then xn � x∗ (resp. xn � x∗) for all n ∈ n. the conditions guaranteeing the regularity of e may be found in heikkilä and lakshmikantham [14] and the references therein. we need the following definitions in the sequel. definition 2.1. a mapping t : e → e is called isotone or monotone nondecreasing if it preserves the order relation �, that is, if x � y implies t x � t y for all x,y ∈ e. similarly, t is called monotone nonincreasing if x � y implies t x � t y for all x,y ∈ e. finally, t is called monotonic or simply monotone if it is either monotone nondecreasing or monotone nonincreasing on e. the following terminologies may be found in any book on nonlinear analysis and applications. definition 2.2. an operator t on a normed linear space e into itself is called compact if t (e) is a relatively compact subset of e. t is called totally bounded if for any bounded subset s of e, t (s) is a relatively compact subset of e. if t is continuous and totally bounded, then it is called a completely continuous on e. definition 2.3 (dhage [3]). a mapping t : e → e is called partially continuous at a point a ∈ e if for � > 0 there exists a δ > 0 such that ‖t x−t a‖ < � whenever x is comparable to a and ‖x−a‖ < δ. t is called a partially continuous on e if it is partially continuous at every point of it. it is clear that if t is a partially continuous on e, then it is continuous on every chain c contained in e. 132 dhage definition 2.4 (dhage [3, 4]). a non-empty subset s of the partially ordered banach space e is called partially bounded if every chain c in s is bounded. an operator t on a partially normed linear space e into itself is called partially bounded if t (c) is bounded for every chain c in e. t is called uniformly partially bounded if all chains t (c) in e are bounded by a unique constant. definition 2.5 (dhage [3, 4]). a non-empty subset s of the partially ordered banach space e is called partially compact if every chain c in s is compact. the operator t is called partially compact if t (c) is a relatively compact subset of e for all totally ordered sets or chains c in e. t is called uniformly partially compact if t is a uniformly partially bounded and partially compact operator on e. t is called partially totally bounded if for any totally ordered and bounded subset c of e, t (c) is a relatively compact subset of e. if t is partially continuous and partially totally bounded, then it is called a partially completely continuous operator on e. remark 2.6. note that every compact mapping on a partially normed linear space is partially compact and every partially compact mapping is partially totally bounded, however the reverse implications do not hold. again, every completely continuous mapping is partially completely continuous and every partially completely continuous mapping is partially continuous and partially totally bounded, but the converse may not be true. definition 2.7 (dhage [3]). the order relation � and the metric d on a non-empty set e are said to be compatible if {xn} is a monotone sequence, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk} of {xn} converges to x ∗ implies that the original sequence {xn} converges to x∗. similarly, given a partially ordered normed linear space (e,�,‖ · ‖), the order relation � and the norm ‖ · ‖ are said to be compatible if � and the metric d defined through the norm ‖ ·‖ are compatible. clearly, the set r of real numbers with usual order relation ≤ and the norm defined by the absolute value function has this property. similarly, every finite dimensional euclidean space rn possesses compatibility property with respect to the usual component-wise order relation ≤ and the standard norm ‖ · ‖ in rn. the dhage iteration principle or method developed in dhage [3, 4, 6] may be described as “the monotonic convergence of the sequence of successive approximations to the solution of a nonlinear equation beginning with a lower or an upper solution of the equation as its initial or first approximation ” which forms a useful tool in the subject of existence theory of nonlinear analysis. it is clear that dhage iteration method is different from usual picard’s successive iterations and embodied in some of the following applicable hybrid fixed point theorems of dhage [4] which are the key tools for our work contained in the present paper. a few other hybrid fixed point theorems containing the dhage iteration method along with their applications also appear in dhage [3, 4]. theorem 2.8 (dhage [4]). let ( e,�,‖ · ‖ ) be a regular partially ordered complete normed linear space such that the order relation � and the norm ‖ · ‖ in e are compatible in every compact chain c of e. let t : e → e be a partially continuous, nondecreasing and partially compact operator. if there exists an element x0 ∈ e such that x0 �t x0 or t x0 � x0, then the operator equation t x = x has a solution x∗ in e and the sequence {t nx0} of successive iterations converges monotonically to x∗. remark 2.9. the regularity of e in above theorem 2.8 may be replaced with a stronger continuity condition of the operator t on e which is a result proved in dhage [4]. the following hybrid fixed point theorems are employed for proving the existence and uniqueness of the solutions for the fde considered in the subsequent section of the paper. before stating these results, we consider the following definition in what follows. definition 2.10. an upper semi-continuous and nondecreasing function ψ : r+ → r+ is called a d-function provided ψ(0) = 0. an operator t : e → e is called partially nonlinear d-contraction if nonlinear fractional differential equations 133 there exists a d-function ψ such that (2.1) ‖t x−t y‖≤ ψ ( ‖x−y‖ ) for all comparable elements x,y ∈ e, where 0 < ψ(r) < r for r > 0. in particular if ψ(r) = k r, t is a partially linear contraction on e with a contraction constant k. theorem 2.11 (dhage [3]). let (e,�,‖·‖) be a partially ordered banach space and let t : e → e be a nondecreasing and partially nonlinear d-contraction. suppose that there exists an element x0 ∈ e such that x0 �t x0 or x0 �t x0. if t is continuous or e is regular, then t has a fixed point x∗ and the sequence {t nx0} of successive iterations converges monotonically to x∗. moreover, the fixed point x∗ is unique if every pair of elements in e has a lower and an upper bound. theorem 2.12 (dhage [4]). let ( e,�,‖ · ‖ ) be a regular partially ordered complete normed linear space such that the order relation � and the norm ‖·‖ in e are compatible in every compact chain c of e. let a,b : e → e be two nondecreasing operators such that (a) a is partially bounded and partially nonlinear d-contraction, (b) b is partially continuous and partially compact, and (c) there exists an element x0 ∈ e such that x0 �ax0 + bx0 or x0 �ax0 + bx0. then the operator equation ax + bx = x has a solution x∗ in e and the sequence {xn} of successive iterations defined by xn+1 = axn + bxn, n=0,1,. . . , converges monotonically to x∗. remark 2.13. the compatibility of the order relation � and the norm ‖ ·‖ in every compact chain of e holds if every partially compact subset of e possesses the compatibility property with respect to � and ‖ · ‖. this simple fact has been utilized in proving the main existence and approximations results for the considered the fde (3.1) on j. note that the dhage iteration method presented in the above hybrid fixed point theorems have been employed in dhage and dhage [9, 10, 11] for approximating the solutions of initial value problems of nonlinear first order ordinary differential equation under some natural hybrid conditions. in the following section we approximate the solutions of certain ivps of nonlinear integro-differential equations via successive approximations beginning with the loser or upper solution. 3. existence and uniqueness theorems the equivalent integral form of the fde (1.1) is considered in the function space c(j,r) of continuous real-valued functions defined on j. we define a norm ‖ · ‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) for all t ∈ j. clearly, c(j,r) is a banach space with respect to above supremum norm and also partially ordered w.r.t. the above partially order relation ≤. it is known that the partially ordered banach space c(j,r) is regular and a lattice so that every pair of elements of e has a lower and an upper bound in it. it is known that the partially ordered banach space c(j,r) has some nice properties w.r.t. the above order relation in it. the following crucial lemma concerning the compatibility of the order relation and the norm in c(j,r) follows by an application of arzellá-ascoli theorem. lemma 3.1. let ( c(j,r),≤,‖ · ‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2) respectively. then ‖ · ‖ and ≤ are compatible in every partially compact subset of c(j,r). 134 dhage proof. let s be a partially compact subset of c(j,r) and let {xn}n∈n be a monotone nondecreasing sequence of points in s. then we have x1(t) ≤ x2(t) ≤ ···≤ xn(t) ≤ ··· , (nd) for each t ∈ j. suppose that a subsequence {xnk}k∈n of {xn}n∈n is convergent and converges to a point x in s. then the subsequence {xnk (t)}k∈n of the monotone real sequence {xn(t)}n∈n is convergent. by monotone characterization, the whole sequence {xn(t)}n∈n is convergent and converges to a point x(t) in s for each t ∈ j. this shows that the sequence {xn}n∈n converges to x point-wise on j. to show the convergence is uniform, it is enough to show that the sequence {xn(t)}n∈n is equicontinuous. since s is partially compact, every chain or totally ordered set and consequently {xn}n∈n is an equicontinuous sequence by arzelá-ascoli theorem. hence {xn}n∈n is convergent and converges uniformly to x. as a result, ‖ ·‖ and ≤ are compatible in s. this completes the proof. � we need the following definition in what follows. definition 3.2. a function u ∈ c1(j,r) is said to be a lower solution of the fde (1.1) if it satisfies cdqu(t) ≤ f(t,u(t)), t ∈ j, u(t0) ≤ α0,   (∗) similarly, an upper solution v ∈ c1(j,r) to the fde (3.1) is defined on j, by the above inequalities with reverse sign. 3.1. existence theorem. we consider the following set of assumptions in what follows: (h1) there exists a constant mf > 0 such that |f(t,x)| ≤ mf for all t ∈ j and x ∈ r. (h2) the function f(t,x) is monotone nondecreasing in x for each t ∈ j. (h3) the fde (3.1) has a lower solution u ∈ c1(j,r). the following lemma is useful in what follows and may be found in kilbas et.al. [15] and podlubny [17]. lemma 3.3. for a given continuous function h : j → r, a function x ∈ c1(j,r) is a solution of the fde (3.3) cdqx(t) = h(t), t ∈ j, x(t0) = α0, } if and only if it is a solution of the nonlinear integral equation, (3.4) x(t) = α0 + 1 γq ∫ t t0 (t−s)q−1 h(s) ds, t ∈ j. theorem 3.4. assume that the hypotheses (h1) through (h3) hold. then the fde (3.1) has a solution x∗ defined on j and the sequence {xn}∞n=0 of successive approximations defined by (3.5) x0 = u, xn+1(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f((s,xn(s)) ds, for all t ∈ j, converges monotonically to x∗. proof. by lemma 3.3, the fde (1.1) is equivalent to the nonlinear integral equation (3.6) x(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds, t ∈ j. set e = c(j,r). then, from lemma 3.1 it follows that every compact chain in e possesses the compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in e. nonlinear fractional differential equations 135 define the operator t on e by (3.7) t x(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds, t ∈ j. from the continuity of the integral, it follows that t defines the map t : e → e. then, the fde (3.1) is equivalent to the operator equation (3.8) t x(t) = x(t), t ∈ j. we shall show that the operator t satisfies all the conditions of theorem 2.8. this is achieved in the series of following steps. step i: t is nondecreasing operator on e. let x,y ∈ e be such that x ≤ y. then by hypothesis (h2), we obtain t x(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds ≤ α0 + 1 γq ∫ t t0 (t−s)q−1f(s,y(s)) ds = t y(t), for all t ∈ j. this shows that t is nondecreasing operator on e into e. step ii: t is a partially continuous operator on e. let {xn} be a sequence of points of a chain c in e such that xn → x for all n ∈ n. then, by dominated convergence theorem, we have lim n→∞ t xn(t) = lim n→∞ [ α0 + 1 γq ∫ t t0 (t−s)q−1f(s,xn(s)) ds ] = α0 + 1 γq ∫ t t0 (t−s)q−1 [ lim n→∞ f(s,xn(s)) ] ds = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds = t x(t), for all t ∈ j. this shows that {t xn} converges to t x pointwise on j. next, we will show that {t xn} is an equicontinuous sequence of functions in e. let t1, t2 ∈ j be arbitrary with t1 < t2. then |t xn(t2) −t xn(t1)| ≤ ∣∣∣∣ 1γq ∫ t2 t0 (t2 −s)q−1f(s,xn(s)) ds− 1 γq ∫ t2 t0 (t1 −s)q−1f(s,xn(s)) ds ∣∣∣∣ + ∣∣∣∣ 1γq ∫ t2 t0 (t1 −s)q−1f(s,xn(s)) ds− 1 γq ∫ t1 t0 (t1 −s)q−1f(s,xn(s)) ds ∣∣∣∣ ≤ 1 γq ∣∣∣∣ ∫ t2 t0 |(t2 −s)q−1 − (t1 −s)q−1| |f(s,xn(s))|ds ∣∣∣∣ + 1 γq ∣∣∣∣ ∫ t2 t1 (t1 −s)q−1|f(s,xn(s))|ds ∣∣∣∣ 136 dhage ≤ mf γq ∣∣∣∣ ∫ t2 t0 |(t2 −s)q−1 − (t1 −s)q−1|ds ∣∣∣∣ + mfγq ∣∣∣∣ ∫ t2 t1 (t1 −s)q−1 ds ∣∣∣∣ ≤ mf γq ∣∣∣∣ ∫ t0+a t0 |(t2 −s)q−1 − (t1 −s)q−1|ds ∣∣∣∣ + mfγq |p(t1) −p(t2)| where, p(t) = mf γq ∫ t t0 (t0 + a−s)q−1 ds. since the functions t 7→ (t−s)q−1 and t 7→ p(t) are uniformly continuous on compact j = [t0, t0 +a], we have that |t xn(t2) −t xn(t1)|→ 0 as t2 → t1 uniformly for all n ∈ n. this shows that the convergence t xn → t x is uniformly and hence t is a partially continuous on e. step iii: t is a partially compact operator on e. let c be an arbitrary chain in e. we show that t (c) is a uniformly bounded and equicontinuous set in e. first we show that t (c) is uniformly bounded. let x ∈ c be arbitrary. then, |t x(t)| ≤ |α0| + 1 γq ∫ t t0 (t−s)q−1|f(s,x(s))|ds ≤ |α0| + 1 γq ∫ t t0 (t−s)q−1|f(s,x(s))|ds ≤ |α0| + aq mf γ(q + 1) = r, for all t ∈ j. taking the supremum over t, we obtain ‖t x‖ ≤ r for all x ∈ c. hence t (c) is a uniformly bounded subset of e. next, we will show that t (c) is an equicontinuous set in e. let t1, t2 ∈ j be arbitrary with t1 < t2. then |t x(t2) −t x(t1)| ≤ ∣∣∣∣ 1γq ∫ t2 t0 (t2 −s)q−1f(s,x(s)) ds− 1 γq ∫ t2 t0 (t1 −s)q−1f(s,x(s)) ds ∣∣∣∣ + ∣∣∣∣ 1γq ∫ t2 t0 (t1 −s)q−1f(s,x(s)) ds− 1 γq ∫ t1 t0 (t1 −s)q−1f(s,x(s)) ds ∣∣∣∣ ≤ 1 γq ∣∣∣∣ ∫ t2 t0 |(t2 −s)q−1 − (t1 −s)q−1| |f(s,x(s))|ds ∣∣∣∣ + 1 γq ∣∣∣∣ ∫ t2 t1 (t1 −s)q−1|f(s,x(s))|ds ∣∣∣∣ ≤ mf γq ∣∣∣∣ ∫ t2 t0 |(t2 −s)q−1 − (t1 −s)q−1|ds ∣∣∣∣ + mfγq ∣∣∣∣ ∫ t2 t1 (t1 −s)q−1 ds ∣∣∣∣ ≤ mf γq ∣∣∣∣ ∫ t0+a t0 |(t2 −s)q−1 − (t1 −s)q−1|ds ∣∣∣∣ + mfγq |p(t1) −p(t2)| . since the functions t 7→ (t−s)q−1 and t 7→ p(t) are uniformly continuous on compact j = [t0, t0 +a], we have that |t x(t2) −t x(t1)|→ 0 as t2 → t1 nonlinear fractional differential equations 137 uniformly for all x ∈ c. this shows that t (c) is an equicontinuous set in e. hence t (c) is compact subset of e and consequently t is a partially compact operator on e into itself. step iv: u satisfies the operator inequality u ≤t u. since the hypothesis (h3) holds, u is a lower solution of (3.1) defined on j. then, (3.9) cdqu(t) ≤ f(t,u(t)), satisfying, (3.10) u(t0) ≤ α0, for all t ∈ j. integrating (3.9) from t0 to t, we obtain (3.11) u(t) ≤ α0 + 1 γq ∫ t t0 (t−s)q−1f(s,u(s)) ds, for all t ∈ j. this show that u is a lower solution of the operator equation x = t x. thus t satisfies all the conditions of theorem 2.8 in view of remark 2.9 and we apply to conclude that the operator equation t x = x has a solution. consequently the fractional integral equation (3.6) and the fde (1.1) has a solution x∗ defined on j. furthermore, the sequence {xn} of successive approximations defined by (3.5) converges monotonically to x∗. this completes the proof. � remark 3.5. the conclusion of theorem 3.4 also remains true if we replace the hypothesis (h3) with the following one: (h′3) the fde (3.1) has an upper solution v ∈ c1(j,r). example 3.6. given a closed and bounded interval j = [0, 1], consider the fde, (3.12) cd1/2x(t) = π + tanh x(t), t ∈ j, x(0) = 1.   here, f(t,x) = π + tanh x. clearly, the function f is continuous on j ×r. the function f satisfies the hypothesis (h1) with mf = π + 2. moreover, the function f(t,x) = π + tanh x is nondecreasing in x for each t ∈ j and so the hypothesis (h2) is satisfied. finally, the fde (3.12) has a lower solution u defined by u(t) = 1 − 1 γ(1/2) ∫ t 0 (t−s)−1/2 ds = 1 + 2 t1/2 √ π defined on j. thus all the hypotheses of theorem 3.4 are satisfied. hence we conclude that the fde (3.12) has a solution x∗ defined on j and the sequence {xn}∞n=0 defined by x0 = u, xn+1(t) = 1 + 1 √ π ∫ t 0 (t−s)−1/2 tanh xn(s) ds, for all t ∈ j, converges monotonically to x∗. 3.2. uniqueness theorem. next, we prove the uniqueness theorem for the fde (3.1) under weak lipschitz condition. we need the following hypothesis in what follows. (h4) there exists a d-function φ such that 0 ≤ f(t,x) −f(t,y) ≤ φ(x−y) for all x,y ∈ r, x ≥ y. moreover, 0 < aq γ(q + 1) φ(r) < r for each r > 0. theorem 3.7. assume that hypotheses (h2) through (h4) hold. then the fde (3.1) has a unique solution x∗ defined on j and the sequence {xn} of successive approximations defined by (3.5) converges monotonically to x∗. 138 dhage proof. set e = c(j,r). clearly, e is a lattice w.r.t. the order relation ≤ and so the lower and the upper bound for every pair of elements in e exist. define the operator t by (3.7). then, the fde (1.1) is equivalent to the operator equation (3.8). we shall show that t satisfies all the conditions of theorem 2.11 in e. clearly, t is a nondecreasing operator on e into itself. we shall simply show that the operator t is a nonlinear d-contraction on e. let x,y ∈ e be any two elements such that x ≥ y. then, by hypothesis (h4), |t x(t) −t y(t)| ≤ ∣∣∣∣ 1γq ∫ t t0 (t−s)q−1f(s,x(s)) ds− 1 γq ∫ t t0 (t−s)q−1f(s,y(s)) ds ∣∣∣∣ ≤ 1 γq ∣∣∣∣ ∫ t t0 (t−s)q−1|f(s,x(s)) −f(s,y(s))|ds ∣∣∣∣ ≤ 1 γq ∣∣∣∣ ∫ t t0 (t−s)q−1φ(|x(s) −y(s)|) ds ∣∣∣∣ ≤ 1 γq ∣∣∣∣ ∫ t t0 (t−s)q−1 ds ∣∣∣∣φ(‖x−y‖) ≤ aq γ(q + 1) φ(‖x−y‖) = ψ(‖x−y‖)(3.13) for all t ∈ j, where ψ(r) = aq γ(q + 1) φ(r) < r, r > 0. taking the supremum over t, we obtain ‖t x−t y‖≤ ψ(‖x−y‖) for all x,y ∈ e, x ≥ y. as a result t is a partially nonlinear d-contraction on e. furthermore, it can be shown as in the proof of theorem 3.4 that the function u given in hypothesis (h3) satisfies the operator inequality u ≤t u on j. now a direct application of theorem 2.11 yields that the fde (1.1) has a unique solution x∗ and the sequence {xn}∞n=0 of successive approximations defined by (3.5) converges monotonically to x∗. � remark 3.8. the conclusion of theorem 3.7 also remains true if we replace the hypothesis (h3) with the following one: (h′3) the fde (3.1) has an upper solution v ∈ c1(j,r). example 3.9. given a closed and bounded interval j = [0, 1], consider the fde, (3.14) cd1/2x(t) = π + 1 2 arctan x(t), t ∈ j, x(0) = 1.   here, f(t,x) = π + 1 2 arctan x. clearly, the function f is continuous on j × r. the function f satisfies the hypothesis (h1) with mf = π + π 4 . moreover, the function f(t,x) is nondecreasing in x for each t ∈ j and so the hypothesis (h2) is satisfied. we show that f satisfies the hypothesis (h4) on j ×r. let x,y ∈ r be such that x ≥ y. then, we have 0 ≤ f(t,x) −f(t,y) ≤ 1 2 [ arctan x− arctan y ] ≤ φ(x−y) nonlinear fractional differential equations 139 for all t ∈ j, where ψ is d-function defined by φ(r) = 1 2 · r 1 + ξ2 and x > ξ > y. furthermore, aq γ(q + 1) φ(r) = 1 √ π · r 1 + ξ2 < r for each 0 < ξ < r. finally, the fde (3.1) has a lower solution u(t) = 1 − π 2 · 1 γ(1/2) · ∫ t 0 (t−s)−1/2 ds = 1 + √ π t1/2 defined on j. thus all the hypotheses of theorem 3.7 are satisfied and hence we conclude that the fde (3.14) has a unique solution x∗ defined on j and the sequence {xn}∞n=0 defined by x0 = u, xn+1(t) = 1 + 1 2 √ π ∫ t 0 (t−s)−1/2 arctan xn(s) ds, for all t ∈ j, converges monotonically to x∗. 4. linear perturbations of first type it is possible sometimes that the nonlinearity f involved in the fde (1.1) neither satisfies the hypothesis of theorem 3.4 nor satisfies the hypothesis of theorem 3.7, however the splitting functions f1 and f2 of f in the form f = f1 + f2 satisfy the conditions of theorems 3.4 and 3.7 respectively. in the terminology of dhage [2] it is called a hybrid fractional differential equation with a linear perturbation of the fde (1.1) of first type. then in this case it of interest to obtain the conclusion of theorem 3.4 with stated algorithm for the solutions which is a problem of this section. given the notations of previous section, we consider the following nonlinear hybrid fractional differential equation (in short hfde), (4.1) cdqx(t) = f(t,x(t)) + g(t,x(t)), t ∈ j, x(t0) = α0,   where f,g : j ×r → r are continuous functions. by a solution of the hfde (4.1) we mean a function x ∈ c1(j,r) that satisfies equation (4.1) on j. the hfde (4.1) is a hybrid fractional differential equation with a linear perturbation of first type. see dhage [1, 2] and the references therein. the hfde (4.1) is well-known in the literature and discussed at length for existence as well as other aspects of the solutions. in the present discussion, it is proved that the existence of the solutions may be proved under mixed partially lipschitz and partially compactness type conditions. we need the following definition in what follows. definition 4.1. a function u ∈ c1(j,r) is said to be a lower solution of the hfde (4.1) if it satisfies cdqu(t) ≤ f(t,u(t)) + g(t,u(t)), t ∈ j, u(t0) ≤ α0.   (∗∗) similarly, an upper solution v ∈ c1(j,r) to the hfde (4.1) is defined on j by the above inequalities with reverse sign. we consider the following set of assumptions in what follows: (h5) there exists a constant mg > 0 such that |g(t,x)| ≤ mg for all t ∈ j and x ∈ r. (h6) the mapping g(t,x) is monotone nondecreasing in x for each t ∈ j. (h7) the hfde (4.1) has a lower solution u ∈ c1(j,r). 140 dhage theorem 4.2. assume that the hypotheses (h1) and (h4) through (h7) hold. then the hfde (4.1) has a solution x∗ defined on j and the sequence {xn}∞n=0 of successive approximations defined by x0 = u, xn+1(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,xn(s)) ds + 1 γq ∫ t t0 (t−s)q−1g(s,xn(s)),(4.2) for all t ∈ j, converges monotonically to x∗. proof. set e = c(j,r). then, from lemma 3.1 it follows that every compact chain c in e possesses the compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in e. by lemma 3.3, the hfde (4.1) is equivalent to the nonlinear integral equation x(t) = α0 + 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds + 1 γq ∫ t t0 (t−s)q−1g(s,x(s)) ds, t ∈ j.(4.3) set e = c(j,r) and define the operators a and b on e by (4.4) ax(t) = 1 γq ∫ t t0 (t−s)q−1f(s,x(s)) ds, t ∈ j, and (4.5) bx(t) = α0 + 1 γq ∫ t t0 (t−s)q−1g(s,x(s)) ds, t ∈ j. from the continuity of the integrals, it follows that a and b define the operators a,b : e → e. now, the fde (4.1) is equivalent to the operator equation (4.6) ax(t) + bx(t) = x(t), t ∈ j. then following the arguments similar to those given in theorems 3.4 and 3.7, it can be shown that the operator a is partially bounded and nonlinear d-contraction and b is partially continuous and partially compact operator on e into itself. finally, by hypothesis (h7), we have an element u ∈ c1(j,r) such that u ≤au + bu. now by a direct application of theorem 2.12 we conclude that the operator equation ax + bx = x has a solution x∗. consequently the fde (4.1) has a solution x∗ and the sequence {xn}∞n=1 defined by (4.2) converges monotonically to x∗. this completes the proof. � the conclusion of theorems 4.2 also remains true if we replace the hypothesis (h7) with the following one: (h′7) the hfde (4.1) has an upper solution v ∈ c1(j,r). example 4.3. given a closed and bounded interval j = [0, 1], consider the hfde, (4.7) cd1/2x(t) = π + 1 2 arctan x(t) + tanh x(t), t ∈ j, x(0) = 1.   here, f(t,x) = π + 1 2 arctan x and g(t,x) = tanh y. then the function f satisfies the hypothesis (h1) with mf = π + π 4 . again, f satisfies (h4) with ψ(r) = 1 2 · r 1 + ξ2 , 0 < ξ < r. next, g satisfies nonlinear fractional differential equations 141 (h5) with mg = 1. similarly, g satisfies the hypothesis (h6). finally, the function u(t) = 1 − 1 γ(1/2) ∫ t 0 (t−s)−1/2 ds− 1 γ(1/2) · ∫ t 0 (t−s)−1/2 ds = 1 + 4 √ π t1/2, for all t ∈ j is a lower solution of the hfde (4.7) defined on j. as a result, we apply theorem 4.2 and conclude that the fde (4.7) has a solution x∗ on j and the sequence {xn}∞n=0 defined by x0 = u, xn+1(t) = 1 + 1 2 √ π ∫ t 0 (t−s)−1/2 arctan x(s) + 1 √ π ∫ t 0 (t−s)−1/2 tanh xn(s) ds for each t ∈ j, converges monotonically x∗. remark 4.4. in this paper we have proved only the existence and uniqueness results for the fde (1.1) and its linear perturbations of first type (4.1). however, other aspects of the solutions of these fdes such as the existence of minimal and maximal solutions and comparison theorems could also be proved using the same dhage iteration method with appropriate modifications. furthermore, if the fde (1.1) has a lower solution u and an upper solution v such that u ≤ v, then the corresponding solutions x∗ and x ∗ of the fde (1.1) satisfy x∗ ≤ x∗ and are the minimal and maximal solutions in the vector segment [u,v] of the banach space e = c(j,r). because the order relation ≤ defined by (3.1) is equivalent to the order relation defined by the order cone (4.8) k = { x ∈ c(j,r) | x(t) ≥ 0 for all t ∈ j } , which is a closed set in c(j,r). 5. dhage iteration and other principles we remark that the existence theorem for the fde (1.1) can also be proved via classical schauder fixed point principle under the compactness type arguments. the procedure involves the construction of a closed convex and bounded subset s of the banach space c(j,r) and a continuous and compact operator t which maps s into itself. this is somewhat a troublesome job and moreover, it does not yield any computational scheme for the solutions. in the present approach of the dhage iteration principle, we do not need any convexity argument and along with the existence we obtain an algorithm for the numerical solutions via construction of a sequence of successive approximations. again, the existence and uniqueness theorem for the fde (1.1) is generally proved via classical banach contraction mapping principle which involves a strong lipschitz condition on the nonlinearity f, but here in the present approach, the lipschitz condition is replaced by a weaker partial nonlinear lipschitz condition and obtained the same conclusion with some additional information about the characterizations of convergence of the sequence of successive approximations. similarly, the existence theorem for the hfde (4.1) may be proved via classical krasnoselskii type fixed point principle under the usual mixed lipschitz and compactness type conditions which again does not yield any numerical or approximate solution. see dhage [1] and the references therein. but in the case of dhage iteration principle for such nonlinear hybrid equations, we need the weaker partial lipschitz and partial compactness type conditions and we obtain an algorithm in terms of a sequence of successive approximations for the numerical solutions. thus in all cases, the dhage iteration principle has some advantages over the classical existing methods involving the basic schauder, banach and krasnoselskii fixed point principles. the limitation of the dhage iteration principle is that one needs an extra monotone feature of the nonlinearity in the unknown variable, but as a result we obtain an additional information about the monotonic characterization of the convergence of the sequence of successive approximations to the solutions of the problems in question. 142 dhage another approach of classic upper and lower solution method in the existence theory for the fdes (1.1), (4.1) and similar nonlinear problems is well-known in the literature in which one needs both the upper and lower solutions to exist in order to prove the existence of the solutions of the related differential equation. moreover, the lower solution u and an upper solution v of the related problem should satisfy the inequality u ≤ v. the work along this line may be found in heikkilä and lakshmikatham [14] and the references therein. the novelty of our present approach with respect to the existing ones lies in the fact that we do not assume both lower and upper solutions in our existence theorems even if they exist, but assume only one of them and prove the existence of solutions in a constructive way. furthermore, even if the lower and an upper solution exist, they may not satisfy the inequality mentioned above. therefore, in such a case the usual classic upper and lower solution method does not work. this is the main advantage of the present approach over the previous classic upper and lower solution method for the nonlinear differential and other different types of equations. 6. conclusion from the foregoing discussion it is clear that the dhage iteration principle forms an interesting powerful tool for discussing the existence results of certain nonlinear hybrid fractional differential equations. however, it has some limitations that unlike picard’s method, the new method does not give any idea about the rate of convergence of the sequence of successive approximations. but in a way we have been able to prove the numerical solutions of the considered nonlinear fractional differential equation (1.1) and its perturbation (4.1). finally, while concluding this paper we mention that the fractional differential equation considered here is of very simple nature for which we have illustrated the dhage iteration principle to obtain the algorithms for the solutions under weaker partially lipschitz and compactness conditions. however, an analogous study could also be made for other complex fractional differential equations using similar method with appropriate modifications. some of the results along this line will be reported elsewhere. references [1] b.c. dhage, a nonlinear alternative with applications to nonlinear perturbed differential equations, nonlinear studies 13 (4) (2006), 343-354. [2] b.c. dhage, quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, differ. equ. & appl. 2 (2010), 465–486. [3] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ. appl. 5 (2013), 155-184. [4] b.c. dhage, partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45(4) (2014), 397-426. [5] b.c. dhage, global attractivity results for comparable solutions of nonlinear hybrid fractional integral equations, differ. equ. appl. 6 (2014), 165186. [6] b.c. dhage, nonlinear d-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, malaya j. mat. 3(1)(2015), 62-85. [7] b.c. dhage, operator theoretic techniques in the theory of nonlinear hybrid differential equations, nonlinear anal. forum 20 (2015), 15-31. [8] b.c. dhage, s.b. dhage, global attractivity and stability results for comparable solutions of nonlinear fractional integral equations, nonlinear studies 21 (2014), 255-268. [9] b.c. dhage, s.b. dhage, approximating solutions of nonlinear first order ordinary differential equations, global j. math. sci. 3 (2014), 1-10. [10] b.c. dhage, s.b. dhage, approximating positive solutions of pbvps of nonlinear first order ordinary quadratic differential equations, appl. math. lett. 46 (2015), 133-142. [11] b.c. dhage, s.b. dhage, approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, indian j. math. 57(1) (2015), 103-119. [12] b.c. dhage, s.b. dhage, approximating solutions of nonlinear pbvp of second order differential equations via hybrid fixed point theory, electronic j. diff. equation 2015(2015), no.20, pp.1-10. [13] b.c. dhage, s.b. dhage, s.k. ntouyas, approximating solutions of nonlinear hybrid differential equations, appl. math. lett. 34 (2014), 76-80. nonlinear fractional differential equations 143 [14] s. heikkilä and v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. [15] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [16] h.k. pathak, r.r. lopez, existence and approximation of solutions to nonlinear hybrid ordinary differential equations, appl. math. lett. 39 (2015), 101-106. [17] i. podlubny, fractional differential equations, academic press, san diego, 1999. kasubai, gurukul colony, ahmedpur-413 515, dist: latur, maharashtra, india ∗corresponding author international journal of analysis and applications volume 16, number 2 (2018), 222-231 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-222 quasi-almost lacunary statistical convergence of sequences of sets esra gülle∗, uğur ulusu department of mathematics, faculty of science and literature, afyon kocatepe university, 03200, afyonkarahisar, turkey ∗corresponding author: egulle@aku.edu.tr abstract. in this study, we defined concepts of wijsman quasi-almost lacunary convergence, wijsman quasi-strongly almost lacunary convergence and wijsman quasi q-strongly almost lacunary convergence. also we give the concept of wijsman quasi-almost lacunary statistically convergence. then, we study relationships among these concepts. furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too. 1. introduction and backgrounds the concept of statistical convergence was first introduced by fast [10]. also this concept was studied by fridy [12], šalát [17] and many others. a sequence x = (xk) is statistically convergent to the number l if for every ε > 0, lim n→∞ 1 n ∣∣∣{k ≤ n : |xk −l| ≥ ε}∣∣∣ = 0 where the vertical bars indicate the number of elements in the enclosed set. freedman et al. [1] established the connection between the strongly cesàro summable sequences space |σ1| and the strongly lacunary summable sequences space nθ. received 2017-10-26; accepted 2017-12-20; published 2018-03-07. 2010 mathematics subject classification. 40a05, 40a35. key words and phrases. almost convergence; quasi-almost convergence; quasi-almost statistical convergence; lacunary sequence; sequences of sets; wijsman convergence. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 222 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-222 int. j. anal. appl. 16 (2) (2018) 223 by a lacunary sequence we mean an increasing integer sequence θ = {kr} such that k0 = 0 and hr = kr − kr−1 → ∞ as r → ∞. throughout this study the intervals determined by θ will be denoted by ir = (kr−1,kr] and ratio kr kr−1 will be abbreviated by qr. the concept of lacunary statistical convergence was introduced by fridy and orhan [13]. let θ = {kr} be a lacunary sequence. a sequence x = (xk) is lacunary statistically convergent to l if for every ε > 0, lim r→∞ 1 hr ∣∣∣{k ∈ ir : |xk −l| ≥ ε}∣∣∣ = 0. the idea of almost convergence was introduced by lorentz [9]. maddox [11] and (independently) freedman [1] gave the concept of strong almost convergence. similar concepts can be seen in [2]. let x be any non-empty set and n be the set of natural numbers. the function f : n → p(x) is defined by f(k) = ak ∈ p(x) for each k ∈ n, where p(x) is power set of x. the sequence {ak} = (a1,a2, . . .), which is the range’s elements of f, is said to be sequences of sets. let (x,ρ) be a metric space. for any point x ∈ x and any non-empty subset a of x, we define the distance from x to a by d(x,a) = inf a∈a ρ(x,a). throughout the paper we take (x,ρ) as a metric space and a,ak as any non-empty closed subsets of x. there are different convergence notions for sequence of sets. one of them handled in this paper is the concept of wijsman convergence (see, [6–8, 14–16]). a sequence {ak} is said to be wijsman convergent to a if for each x ∈ x lim k→∞ d(x,ak) = d(x,a) and denoted by ak w→ a or w − lim ak = a. a sequence {ak} is said to be bounded if for each x ∈ x sup k { d(x,ak) } < ∞. the set of all bounded sequences of sets is denoted by l∞. the concepts of wijsman statistical convergence was introduced by nuray and rhoades [6]. a sequence {ak} is wijsman statistically convergent to a if for each x ∈ x and every ε > 0 lim n→∞ 1 n ∣∣∣{k ≤ n : |d(x,ak) −d(x,a)| ≥ ε}∣∣∣ = 0 and it is denoted by st− limw ak = a. the concepts of wijsman lacunary summability ( (wnθ), [wn]θ, [wn] p θ ) and concept of wijsman lacunary statistical convergence ( (wsθ) ) were introduced by ulusu and nuray [20, 21]. int. j. anal. appl. 16 (2) (2018) 224 let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman lacunary summable to a if for each x ∈ x, lim r→∞ 1 hr ∑ k∈ir d(x,ak) = d(x,a) and it is denoted by ak (wnθ)−→ a. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman strongly lacunary summable to a if for each x ∈ x, lim r→∞ 1 hr ∑ k∈ir |d(x,ak) −d(x,a)| = 0 and it is denoted by ak [wn]θ−→ a. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman p-strongly lacunary summable to a if for each x ∈ x and 0 < p < ∞, lim r→∞ 1 hr ∑ k∈ir |d(x,ak) −d(x,a)| p = 0 and it is denoted by ak [wn] p θ−→ a. a sequence {ak} is wijsman lacunary statistically convergent to a if for every ε > 0 and each x ∈ x, lim r→∞ 1 hr ∣∣∣{k ∈ ir : |d(x,ak) −d(x,a)| ≥ ε}∣∣∣ = 0 and it is denoted by ak (wsθ)−→ a. also the concepts of wijsman almost lacunary convergence and wijsman almost lacunary statistical convergence were introduced by ulusu [18, 19], too. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman almost lacunary convergent to a if for each x ∈ x, lim r→∞ 1 hr ∑ k∈ir d(x,ak+i) = d(x,a) uniformly in i = 0, 1, 2, . . .. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman strongly almost lacunary convergent to a if for each x ∈ x, lim r→∞ 1 hr ∑ k∈ir |d(x,ak+i) −d(x,a)| = 0 uniformly in i. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman strongly p-almost lacunary convergent to a if for each x ∈ x and 0 < p < ∞, lim r→∞ 1 hr ∑ k∈ir |d(x,ak+i) −d(x,a)|p = 0 uniformly in i. int. j. anal. appl. 16 (2) (2018) 225 a sequence {ak} is wijsman almost lacunary statistically convergent to a if for every ε > 0 and each x ∈ x, lim r→∞ 1 hr ∣∣∣{k ∈ ir : |d(x,ak+i) −d(x,a)| ≥ ε}∣∣∣ = 0 uniformly in i. the idea of quasi-almost convergence in a normed space was introduced by hajduković [3]. then, nuray [5] studied concepts of quasi-invariant convergence and quasi-invariant statistical convergence in a normed space. the concepts of wijsman quasi-strongly almost convergence and wijsman quasi-almost statistically convergence were studied by gülle and ulusu [4]. a sequence {ak}∈ l∞ is wijsman quasi-strongly almost convergent to a if for each x ∈ x, 1 p np+p−1∑ k=np ∣∣dx(ak) −dx(a)∣∣ → 0 (as p →∞) uniformly in n and it is denoted by ak [wqf] −→ a. a sequence {ak} is wijsman quasi-almost statistically convergent to a if for each x ∈ x and every ε > 0 lim p→∞ 1 p ∣∣∣{k ≤ p : |dx(ak+np) −dx(a)| ≥ ε}∣∣∣ = 0, uniformly in n and it is denoted by ak wqs−→ a. the set of all wijsman quasi-almost statistically convergence sequences will be denoted by { wqs } . 2. main results in this section, we defined concepts of wijsman quasi-almost lacunary convergence, wijsman quasistrongly almost lacunary convergence and wijsman quasi q-strongly almost lacunary convergence. also we give the concept of wijsman quasi-almost lacunary statistically convergence. then, we study relationships among these concepts. furthermore, we investigate relationship between these concepts and some convergences types given earlier for sequences of sets, too. definition 2.1. let θ = {kr} be a lacunary sequence. a sequence {ak} ∈ l∞ is wijsman quasi-almost lacunary convergent to a if for each x ∈ x,∣∣∣∣∣ 1hr ∑ k∈ir dx(ak+nr) −dx(a) ∣∣∣∣∣ −→ 0 (as r →∞), (2.1) uniformly in n = 0, 1, 2, . . . where dx(ak+nr) = d(x,ak+nr) and dx(a) = d(x,a). in this case, we will write (wqf)θ − lim ak = a or ak (wqf)θ−→ a. int. j. anal. appl. 16 (2) (2018) 226 example 2.1. let we define a sequence {ak} as follows: ak :=   {x ∈ r : 2 ≤ x ≤ kr −kr−1} , if k ≥ 2 and k is square integer, {1} , otherwise. this sequence is not wijsman lacunary summable. but, since for each x ∈ x lim r→∞ ∣∣∣∣∣ 1hr ∑ k∈ir dx(ak+nr) −dx({1}) ∣∣∣∣∣ = 0 uniformly n, this sequence is wijsman quasi-almost lacunary convergent to the set a = {1}. theorem 2.1. if a sequence {ak}∈ l∞ is wijsman almost lacunary convergent to a, then {ak} is wijsman quasi-almost lacunary convergent to a. proof. suppose that the sequence {ak} is wijsman almost lacunary convergent to a. then, for each x ∈ x and every ε > 0 there exists an integer r0 > 0 such that for all r > r0∣∣∣∣∣ 1hr ∑ k∈ir dx(ak+i) −dx(a) ∣∣∣∣∣ < ε, uniformly in i. if i is taken as i = nr, then we have∣∣∣∣∣ 1hr ∑ k∈ir dx(ak+nr) −dx(a) ∣∣∣∣∣ < ε, uniformly in n. since ε > 0 is an arbitrary, the limit is taken for r →∞ we can write∣∣∣∣∣ 1hr ∑ k∈ir dx(ak+nr) −dx(a) ∣∣∣∣∣ −→ 0 uniformly in n. that is, the sequence {ak} is wijsman quasi-almost lacunary convergent to a. � theorem 2.2. if a sequence {ak} ∈ l∞ is wijsman quasi-almost lacunary convergent to a, then {ak} is wijsman lacunary summable to a. proof. assume that the sequence {ak} ∈ l∞ is wijsman quasi-almost lacunary convergent to a. then, equation (2.1) is true which for n = 0 implies for every ε > 0 and each x ∈ x,∣∣∣∣∣ 1hr ∑ k∈ir dx(ak) −dx(a) ∣∣∣∣∣ −→ 0 (as r →∞); so, {ak} is wijsman lacunary summable to a. � definition 2.2. let θ = {kr} be a lacunary sequence. a sequence {ak} is wijsman quasi-almost lacunary statistically convergent to a if for each x ∈ x and every ε > 0 lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ = 0, int. j. anal. appl. 16 (2) (2018) 227 uniformly in n. in this case, we will write (wqs)θ − lim ak = a or ak (wqs)θ−→ a. the set of all wijsman quasi-almost lacunary statistically convergence sequences will be denoted by{ wqsθ } : { wqsθ } = { {ak} : lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ = 0} . theorem 2.3. if a sequence {ak} is wijsman almost lacunary statistically convergent to a, then {ak} is wijsman quasi-almost lacunary statistically convergent to a. proof. suppose that the sequence {ak} is wijsman almost lacunary statistically convergent to a. then, for every ε,δ > 0 and for each x ∈ x there exists an integer r0 > 0 such that for all r > r0 1 hr ∣∣∣{k ∈ ir : |dx(ak+i) −dx(a)| ≥ ε}∣∣∣ < δ, uniformly in i. if i is taken as i = nr, then we have 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ < δ, uniformly in n. since δ > 0 is an arbitrary, we have lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ = 0, uniformly in n which means that ak (wqs)θ−→ a. � theorem 2.4. for any lacunary sequence θ = {kr}; if lim infr qr > 1, then { wqs } ⊂ { wqsθ } . proof. suppose that lim infr qr > 1. then for each r ≥ 1, there is a number δ ≥ 0 such that qr ≥ 1 + δ. since qr ≥ 1 + δ and hr = kr −kr−1, we have hr kr ≥ δ 1 + δ . assume that ak wqs−→ a. for each x ∈ x, we can write 1 kr ∣∣∣{k ≤ kr : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣ ≥ 1kr ∣∣∣{k ∈ ir : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣ = hr kr ( 1 hr ∣∣∣{k ∈ ir : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣) ≥ δ 1 + δ ( 1 hr ∣∣∣{k ∈ ir : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣) that is, 1 kr ∣∣∣{k ≤ kr : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣ ≥ δ1 + δ ( 1 hr ∣∣∣{k ∈ ir : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣) int. j. anal. appl. 16 (2) (2018) 228 uniformly in n. if the limit is taken for the above inequality; since ak wqs−→ a, we have 0 ≥ δ 1 + δ · lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nkr ) −dx(a)| ≥ ε}∣∣∣. by the definition of lacunary sequence, we can write r instead of kr. hence, for each x ∈ x we have lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ = 0, uniformly in n, that is ak (wqs)θ−→ a. � definition 2.3. let θ = {kr} be a lacunary sequence. a sequence {ak} ∈ l∞ is wijsman quasi-strongly almost lacunary convergent to a if for each x ∈ x, 1 hr ∑ k∈ir |dx(ak+nr) −dx(a)| −→ 0 (as r →∞); uniformly in n. in this case, we will write [wqf]θ − lim ak = a or ak [wqf]θ−→ a. theorem 2.5. for any lacunary sequence θ = {kr}; if lim infr qr > 1, then ak [wqf] −→ a ⇒ ak [wqf]θ−→ a. proof. let lim infr qr > 1. then for each r ≥ 1, there is a number δ ≥ 0 such that qr ≥ 1 + δ. since qr ≥ 1 + δ and hr = kr −kr−1, we have kr hr ≤ 1 + δ δ and kr−1 hr ≤ 1 δ · (2.2) assume that ak [wqf] −→ a. for each x ∈ x, we can write 1 hr ∑ k∈ir |dx (ak+nr) −dx (a)| = 1 hr kr∑ i=1 |dx (ai+nr) −dx (a)|− 1 hr kr−1∑ i=1 |dx (ai+nr) −dx (a)| = kr hr ( 1 kr kr∑ i=1 |dx (ai+nr) −dx (a)| ) − kr−1 hr ( 1 kr−1 kr−1∑ i=1 |dx (ai+nr) −dx (a)| ) . hence, for each x ∈ x we have lim r→∞ 1 hr ∑ k∈ir |dx (ak+nr) −dx (a)| = lim r→∞ kr hr ( 1 kr kr∑ i=1 |dx (ai+nr) −dx (a)| ) − lim r→∞ kr−1 hr ( 1 kr−1 kr−1∑ i=1 |dx (ai+nr) −dx (a)| ) uniformly in n. since ak [wqf] −→ a, for each x ∈ x we have 1 kr kr∑ i=1 |dx (ai+nr) −dx (a)|→ 0 and 1 kr−1 kr−1∑ i=1 |dx (ai+nr) −dx (a)|→ 0 (2.3) int. j. anal. appl. 16 (2) (2018) 229 uniformly in n. by using the inequalities (2.2) and the status (2.3), we handle lim r→∞ 1 hr ∑ k∈ir |dx (ak+nr) −dx (a)| = 0. the proof of theorem is completed. � definition 2.4. let θ = {kr} be a lacunary sequence. a sequence {ak}∈ l∞ is wijsman quasi q-strongly almost lacunary convergent to a if for each x ∈ x and 0 < q < ∞, 1 hr ∑ k∈ir |dx(ak+nr) −dx(a)|q −→ 0 (as r →∞); (2.4) uniformly in n. in this case, we will write [wqf] q θ − lim ak = a or ak [wqf] q θ−→ a. theorem 2.6. let 0 < q < ∞. then, we have following assertions: i. if a sequence {ak} is wijsman quasi q-strongly almost lacunary convergent to a, then the sequence {ak} is wijsman quasi-almost lacunary statistically convergent to a. ii. if a sequence {ak}∈ l∞ and wijsman quasi-almost lacunary statistically convergent to a, then the sequence {ak} is wijsman quasi q-strongly almost lacunary convergent to a. proof. (i) let ε > 0 be given. then, for each x ∈ x following inequality is proved∑ k∈ir |dx(ak+nr) −dx(a)|q ≥ εq ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣, (2.5) uniformly in n. since the sequence {ak} is wijsman quasi q-strongly almost lacunary convergent to a; if the both side of inequality (2.5) are multipled by 1 hr and after that the limit is taken for r →∞, then we have lim r→∞ 1 hr ∑ k∈ir |dx(ak+nr) −dx(a)|q ≥ εq lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ 0 ≥ εq lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣. hence, we handle lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak+nr) −dx(a)| ≥ ε}∣∣∣ = 0, uniformly in n. that is ak (wqs)θ−→ a. (ii) since {ak} is bounded, we can write sup k { dx(ak) } + dx(a) = m, (0 < m < ∞), for each x ∈ x. int. j. anal. appl. 16 (2) (2018) 230 if {ak} is wijsman quasi-almost lacunary statistically convergent to a, then for a given ε > 0 a number nε ∈ n can be chosen such that for all r > nε and each x ∈ x 1 hr ∣∣∣∣ { k ∈ ir : |dx(ak+nr) −dx(a)| ≥ (ε 2 )1/q}∣∣∣∣ < ε2mq uniformly in n. let take the set lp = { k ≤ p : |dx(ak+nr) −dx(a)| ≥ (ε 2 )1/q} . thus, for each x ∈ x we have 1 hr ∑ k∈ir ∣∣dx(ak+nr) −dx(a)∣∣q = 1 hr ( ∑ k∈ir k∈lp |dx(ak+nr) −dx(a)|q + ∑ k∈ir k/∈lp |dx(ak+nr) −dx(a)|q ) < 1 hr hr ε 2mq mq + 1 hr hr ε 2 = ε 2 + ε 2 = ε uniformly in n. so, the proof is completed. � theorem 2.7. if the sequence {ak} is wijsman quasi q-strongly almost lacunary convergence to a, then {ak} is wijsman q-strongly lacunary summable to a. proof. suppose that the sequence {ak}∈ l∞ is wijsman quasi q-strongly almost lacunary convergent to a. then, equation (2.4) is true which for n = 0 implies for every ε > 0 and each x ∈ x, 1 hr ∑ k∈ir |dx(ak) −dx(a)|q −→ 0 (as r →∞); so, {ak} is wijsman q-strongly lacunary summable to a. � theorem 2.8. if a sequence {ak} is wijsman quasi q-strongly almost lacunary convergence to a, then the sequence {ak} is wijsman lacunary statistically convergent to a. proof. assume that the sequence {ak} is wijsman quasi q-strongly almost lacunary convergence to a. then, by theorem 2.7, the sequence {ak} is wijsman q-strongly lacunary summable to a. for each x ∈ x and every ε > 0, we can write∑ k∈ir |dx(ak) −dx(a)|q ≥ εq ∣∣∣{k ∈ ir : |dx(ak) −dx(a)| ≥ ε}∣∣∣. (2.6) since the sequence {ak} is wijsman q-strongly lacunary summable to a; if the both side of inequality (2.6) are multipled by 1 hr and after that the limit is taken for r →∞, left side of the inequality (2.6) is equal to 0. hence, we handle lim r→∞ 1 hr ∣∣∣{k ∈ ir : |dx(ak) −dx(a)| ≥ ε}∣∣∣ = 0. int. j. anal. appl. 16 (2) (2018) 231 the proof of theorem is completed. � references [1] a. r. freedman, j. j. sember and m. raphael, some cesàro-type summability spaces, proc. london math. soc. 37 (3) (1978), 508–520. [2] d. hajduković, almost convergence of vector sequences, mat. vesnik 12 (27) (1975), 245–249. [3] d. hajduković, quasi-almost convergence in a normed space, univ. beograd. publ. elektrotehn. fak. ser. mat. 13 (2002), 36–41. [4] e. gülle and u. ulusu, quasi-almost convergence of sequences of sets, j. inequal. spec. funct. (in press). [5] f. nuray, quasi-invariant convergence in a normed space, annals of the university of craiova, mathematics and computer science series 41 (1) (2014), 1–5. [6] f. nuray and b.e. rhoades, statistical convergence of sequences of sets, fasc. math. 49 (2012), 87–99. [7] g. beer, on convergence of closed sets in a metric space and distance functions, bull. aust. math. soc. 31 (1985), 421–432. [8] g. beer, wijsman convergence: a survey, set-valued anal. 2 (1994), 77–94. [9] g. g. lorentz, a contribution to the theory of divergent sequences, acta math. 80 (1948), 167–190. [10] h. fast, sur la convergence statistique, colloq. math. 2 (1951), 241–244. [11] i. j. maddox, a new type of convergence, math. proc. cambridge philos. soc. 83 (1978), 61–64. [12] j. a. fridy, on statistical convergence, analysis 5 (1985), 301–313. [13] j. a. fridy and c. orhan, lacunary statistical convergence, pac. j. math. 160 (1) (1993), 43–53. [14] m. baronti and p. papini, convergence of sequences of sets, in: methods of functional analysis in approximation theory, isnm 76, birkhauser-verlag, basel 1986. [15] r. a. wijsman, convergence of sequences of convex sets, cones and functions, bull. amer. math. soc. 70 (1964), 186–188. [16] r. a. wijsman, convergence of sequences of convex sets, cones and functions ii, trans. amer. math. soc. 123 (1) (1966), 32–45. [17] t. šalát, on statistically convergent sequences of real numbers, math. slovaca 30 (1980), 139–150. [18] u. ulusu, on almost asymptotically lacunary statistical equivalence of sequences of sets, electr. j. math. anal. appl. 2 (2) (2014), 56–66. [19] u. ulusu, lacunary statistical convergence of sequences of sets, ph.d. thesis, afyon kocatepe university, institue of science and technology (2013). [20] u. ulusu and f. nuray, on strongly lacunary summability of sequences of sets, j. appl. math. bioinf. 3 (3) (2013), 75–88. [21] u. ulusu and f. nuray, lacunary statistical convergence of sequences of sets, progr. appl. math. 4 (2) (2012), 99–109. 1. introduction and backgrounds 2. main results references international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 206-215 http://www.etamaths.com dunkl generalization of q-parametric szász-mirakjan operators m. mursaleen∗, md. nasiruzzaman and a.a.h. al-abied abstract. in this paper, we construct q-parametric szász-mirakjan operators generated by the qdunkl generalization of the exponential function. we obtain korovkin’s type approximation theorem and compute convergence of these operators by using the modulus of continuity. furthermore, we obtain the rate of convergence of these operators for functions belonging to the lipschitz class. 1. introduction in 1912, bernstein [5] introduced the following sequence of operators bn : c[0, 1] → c[0, 1] defined by bn(f; x) = n∑ k=0 ( n k ) xk(1 −x)n−kf ( k n ) , (n ∈ n) x ∈ [0, 1], f ∈ c[0, 1]. (1.1) in 1950, szász [27] introduced the operators sn(f; x) = e −nx ∞∑ k=0 (nx)k k! f ( k n ) , x ≥ 0, f ∈ c[0,∞). (1.2) for the last two decades, the application of q-calculus emerged as a new area in the field of approximation theory. the first q-analogue of bernstein polynomials was introduced by lupaş [15] and later phillips [23] considered another q-analogue of the bernstein polynomials. later on, many authors introduced q-generalization of various operators and investigated several approximation properties. for instance, [1], [2], [3], [8], [9], [10], [12], [16]– [22], [24]. the q-integer [n]q, the q-factorial [n]q! and the q-binomial coefficient are defined by (see [13]) [n]q := { 1−qn 1−q , if q ∈ r + \{1} n, if q = 1, for n ∈ n and [0]q = 0, [n]q! := { [n]q[n− 1]q · · · [1]q, n ≥ 1, 1, n = 0,[ n k ] q := [n]q! [k]q![n−k]q! , respectively. the q-analogue of (1 + x)n is the polynomial (1 + x)nq := { (1 + x)(1 + qx) · · ·(1 + qn−1x) n = 1, 2, 3, · · · 1 n = 0. a q-analogue of the common pochhammer symbol also called a q-shifted factorial is defined as (x; q)0 = 1, (x; q)n = n−1∏ j=0 (1 −qjx), (x; q)∞ = ∞∏ j=0 (1 −qjx). received 1st september, 2016; accepted 8th november, 2016; published 1st march, 2017. 2010 mathematics subject classification. 41a25, 41a36, 33c45. key words and phrases. q-integers; szász operator; szász-mirakjan operator; dunkl analogue; modulus of continuity; lipschitz class. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 206 dunkl generalization of q-parametric szász-mirakjan operators 207 the gauss binomial formula is given by (x + a)nq = n∑ k=0 [ n k ] q qk(k−1)/2akxn−k. there are two q-analogue of the exponential function ez, defined as (see also [14]) for | z |< 1 1−q and | q |< 1, e(z) = ∞∑ k=0 zk k! = 1 1 − ((1 −q)z)∞q , (1.3) and for | q |< 1, e(z) = ∞∏ j=0 ( 1 + (1 −q)qjz )∞ q = ∞∑ k=0 q k(k−1) 2 zk k! = (1 + (1 −q)z)∞q , (1.4) where (1 −x)∞q = ∏∞ j=0(1 −q jx). the q−analogue of bernstein operators [23] is defined as follows: bn,q(f; x) = n∑ k=0 [ n k ] q xk n−k−1∏ s=0 (1 −qsx) f ( [k]q [m]q ) , x ∈ [0, 1],n ∈ n. (1.5) in [4] q-szász-mirakjan operators were defined as follows: sn,q(f; x) := e ( − [n]qx bn ) ∞∑ k=0 f ( [k]qbn [n]q ) [n]kqx k [k]q!bkn , (1.6) where 0 ≤ x < bn (1−q)[n]q , f ∈ c[0,∞) and {bn} is a sequence of positive numbers such that limn→∞ bn = ∞. sucu [26] defined a dunkl analogue of szász operators via a generalization of the exponential function given by [25] as follows: s∗n(f; x) := 1 eµ(nx) ∞∑ k=0 (nx)k γµ(k) f ( k + 2µθk n ) (n ∈ n), (1.7) where x ≥ 0,f ∈ c[0,∞),µ ≥ 0 and eµ(x) = ∞∑ n=0 xn γµ(n) . also here γµ(2k) = 22kk!γ ( k + µ + 1 2 ) γ ( µ + 1 2 ) , and γµ(2k + 1) = 22k+1k!γ ( k + µ + 3 2 ) γ ( µ + 1 2 ) . a recursion formula for γµ is given by γµ(k + 1) = (k + 1 + 2µθk+1)γµ(k), k = 0, 1, 2, · · · , where θk = { 0 if k ∈ 2n 1 if k ∈ 2n + 1 in [6], cheikh et al. studied the q-dunkl classical q-hermite type polynomials and presented the definitions of q-dunkl analogues of exponential functions, recursion relations and notaions for µ > −1 2 and 0 < q < 1, respectively. eµ,q(x) = ∞∑ n=0 xn γµ,q(n) , x ∈ r, (1.8) 208 mursaleen, nasiruzzaman and al-abied eµ,q(x) = ∞∑ n=0 q n(n−1) 2 xn γµ,q(n) , x ∈ r, (1.9) γµ,q(n + 1) = ( 1 −q2µθn+1+n+1 1 −q ) γµ,q(n), n ∈ n, (1.10) θn = { 0 if n ∈ 2n, 1 if n ∈ 2n + 1. an explicit formula for γµ,q(n) is given by γµ,q(n) = (q2µ+1,q2)[ n+1 2 ](q 2,q2)[ n 2 ] (1 −q)n γµ,q(n), n ∈ n. some of the special cases of γµ,q(n) are listed as: γµ,q(0) = 1, γµ,q(1) = 1 −q2µ+1 1 −q , γµ,q(2) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q ) , γµ,q(3) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q )( 1 −q2µ+3 1 −q ) , γµ,q(4) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q )( 1 −q2µ+3 1 −q )( 1 −q4 1 −q ) . in [11], içöz and çekim gave a dunkl generalization of szász operators via q-calculs as: dn,q(f; x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) f ( 1 −q2µθk+k 1 −qn ) (n ∈ n), (1.11) where µ > 1 2 , x ≥ 0, 0 < q < 1 and f ∈ c[0,∞). in this paper, we define a dunkl generalization of q-parametric szász-mirakjan operators: for any x ∈ [0,∞), f ∈ c[0,∞), 0 < q < 1, and µ > 1 2 , we define d∗n,q(f; x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 f ( 1 −q2µθk+k qk−2(1 −qn) ) (n ∈ n). (1.12) 2. main results lemma 2.1. let d∗n,q(. ; .) be the operators given by (1.12). then we have the following identities: (1) d∗n,q(e0; x) = 1 (2) d∗n,q(e1; x) = qx (3) qx2 + q 2(1+µ) [n]q [1 − 2µ]qx ≤ d∗n,q(e2; x) ≤ qx2 + q2(1+µ) [n]q [1 + 2µ]qx, where ej(t) = t j, j = 0, 1, 2, · · · . proof. (1) d∗n,q(1; x) = 1 eµ,q([n]qx) ∑∞ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 = 1 (from (1.9)). dunkl generalization of q-parametric szász-mirakjan operators 209 (2) d∗n,q(e1; x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) ) = q [n]q 1 eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q (k−1)(k−2) 2 = q [n]q 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−1) 2 = qx eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 = qx. (3) d∗n,q(e2; x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) )2 = 1 [n]2q 1 eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q k2−5k+8 2 ( 1 −q2µθk+k (1 −q) ) = q2 [n]2q 1 eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q (k−1)(k−4) 2 ( 1 −q2µθk+k (1 −q) ) , hence d∗n,q(e2; x) = q2 [n]2q 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−3) 2 ( 1 −q2µθk+1+k+1 (1 −q) ) . (2.1) a simple calculation yields [2µθk+1 + k + 1]q = [2µθk + k]q + q 2µθk+k[2µ(−1)k + 1]q. (2.2) replacing k by 2k, then (2.2) implies that [2µθ2k+1 + 2k + 1]q = ( 1 −q2µθ2k+2k 1 −q ) + q2µθ2k+2k[1 + 2µ]q, (2.3) and by replacing k by 2k + 1, we have [2µθ2k+2 + 2k + 2]q = ( 1 −q2µθ2k+1+2k+1 1 −q ) + q2µθ2k+1+2k+1[1 − 2µ]q. (2.4) now by separating (2.1), to even and odd terms and using (2.3) and (2.4) d∗n,q(e2; x) = q2 [n]2q 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−3) 2 ( 1 −q2µθk+k (1 −q) )∣∣∣∣ for k=2k,2k+1 + q2 [n]2q 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) 2k+1 γµ,q(2k) qk(2k−3)q2µθ2k+2k[1 + 2µ]q + q2 [n]2q 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) 2k+2 γµ,q(2k + 1) q(k−1)(2k+1)q2µθ2k+1+2k+1[1 − 2µ]q. since [1 − 2µ]q ≤ [1 + 2µ]q, (2.5) 210 mursaleen, nasiruzzaman and al-abied using the inequality (2.5), we have d∗n,q(e2; x) ≤ qx 2 + q2 [n]q x eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 (q[n]qx) 2k γµ,q(2k) qk(2k−3) + q2(µ+1) [n]q x eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 (q[n]qx) 2k+1 γµ,q(2k + 1) q(k−1)(2k+1) ≤ qx2 + q2(µ+1) [n]q x eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ≤ qx2 + q2(µ+1) [n]q [1 + 2µ]qx. similarly, we can show that d∗n,q(e2; x) ≥ qx 2 + q2(µ+1) [n]q [1 − 2µ]qx. � lemma 2.2. let the operators d∗n,q(. ; .) be given by (1.12). then we have the following identities: (1) d∗n,q(e1 − 1; x) = qx− 1 (2) d∗n,q(e1 −x; x) = (q − 1)x (3) (1 −q)x2 + q 2(1+µ) [n]q [1 − 2µ]qx ≤ d∗n,q((e1 −x)2; x) ≤ (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx. next, we obtain the korovkin’s type approximation properties for our operators defined by (1.12). in order to obtain the convergence results for the operators d∗n,q(., .), we write q = qn where qn ∈ (0, 1) such that, lim n qn → 1 (2.6) theorem 2.1. let q = qn satisfy (2.6), for 0 < qn < 1 and if d ∗ n,qn (. ; .) be the operators given by (1.12). then for any function f ∈ x[0,∞) ∩h, d∗n,qn(f; x) = f(x) uniformly on each compact subset of [0,∞). proof. the proof is based on the well known korovkin’s theorem regarding the convergence of a sequence of linear positive operators, so it is enough to prove the conditions d∗n,qn((ej; x) = x j, j = 0, 1, 2, {as n →∞} uniformly on [0, 1]. clearly from (2.6) and 1 [n]qn → 0, (n →∞), we have lim n→∞ d∗n,qn(e1; x) = x, limn→∞ d∗n,qn(e2; x) = x 2. which completes the proof. � let cb(r+) be the set of all bounded and continuous functions on r+ = [0,∞), which is linear normed space with ‖ f ‖cb = sup x≥0 | f(x) | . we write h := {f : x ∈ [0,∞), f(x) 1 + x2 is convergent as x →∞}. dunkl generalization of q-parametric szász-mirakjan operators 211 we recall the weighted spaces defined as follows: pρ(r+) = {f :| f(x) |≤ mfρ(x)} , qρ(r+) = { f : f ∈ pρ(r+) ∩c[0,∞) } , qkρ(r +) = { f : f ∈ qρ(r+) and lim x→∞ f(x) ρ(x) = k(k is a constant) } , where ρ(x) = 1 + x2 is a weight function and mf is a constant depending only on f. qρ(r+) is a normed space with the norm ‖ f ‖ρ= supx≥0 |f(x)| ρ(x) . theorem 2.2. let q = qn satisfy (2.6), for 0 < qn < 1 and if d ∗ n,qn (. ; .) be the operators given by (1.12). then for any function f ∈ qkρ(r+) we have lim n→∞ ‖ d∗n,qn(f; x) −f ‖ρ= 0. proof. from lemma 2.1, the first condition of (1) is fulfilled for τ = 0. now for τ = 1, 2 it is easy to see from (2), (3) of lemma 2.1 by using (2.6) that ‖ d∗n,qn (e1) τ ; x) −xτ ‖ρ= 0. this complete the proof. � 3. rate of convergence next, we calculate the rate of convergence of operators (1.12) by means of modulus of continuity and lipschitz type maximal functions. let f ∈ c[0,∞]. the modulus of continuity of f denoted by ω(f,δ) gives the maximum oscillation of f in any interval of length not exceeding δ > 0 and it is given by the relation ω(f,δ) = sup |y−x|≤δ | f(y) −f(x) |, x,y ∈ [0,∞). (3.1) it is known that limδ→0+ ω(f,δ) = 0 for f ∈ c[0,∞) and for any δ > 0 one has | f(y) −f(x) |≤ ( | y −x | δ + 1 ) ω(f,δ). (3.2) theorem 3.1. let q = qn satisfy (2.6) for x ≥ 0, 0 < qn < 1 and if d∗n,qn(. ; .) be the operators defined by (1.12). then for any function f ∈ c̃[0,∞), we have | d∗n,qn(f; x) −f(x) |≤ { 1 + √ (1 −qn)[n]qnx2 + q 2(1+µ) n [1 + 2µ]qnx } ω ( f; 1√ [n]qn ) , where c̃[0,∞) is the space of uniformly continuous functions on r+ and ω(f,δ) is the modulus of continuity of the function f ∈ c̃[0,∞) defined in (3.2). 212 mursaleen, nasiruzzaman and al-abied proof. we prove it by using the result (3.2),(3.3) and cauchy-schwarz inequality: | d∗n,q(f; x) −f(x) | ≤ 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣f ( 1 −q2µθk+k qk−2(1 −qn) ) −f(x) ∣∣∣∣ ≤ 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 { 1 + 1 δ ∣∣∣∣ ( 1 −q2µθk+k qk−2(1 −qn) ) −x ∣∣∣∣ } ω(f; δ) = { 1 + 1 δ ( 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣ 1 −q2µθk+kqk−2(1 −qn) −x ∣∣∣∣ )} ω(f; δ) ≤  1 + 1δ ( 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) −x )2)12 ( d∗n,q(e0; x) )1 2  ω(f; δ) = { 1 + 1 δ ( d∗n,q(e1 −x) 2; x )1 2 } ω(f; δ) ≤ { 1 + 1 δ √ (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx } ω(f; δ), if we choose δ = δn = √ 1 [n]q , then we get our result. � now we give the rate of convergence of the operators d∗n,q(f; x) defined in (1.12) in terms of the elements of the usual lipschitz class lipm (ν). let f ∈ c[0,∞), m > 0 and 0 < ν ≤ 1. we recall that f belongs to the class lipm (ν) if lipm (ν) = {f :| f(ζ1) −f(ζ2) |≤ m | ζ1 − ζ2 |ν (ζ1,ζ2 ∈ [0,∞))} (3.3) is satisfied. theorem 3.2. let d∗n,q(. ; .) be the operator defined in (1.12). then for each f ∈ lipm (ν), (m > 0, 0 < ν ≤ 1) satisfying (3.3), we have | d∗n,q(f; x) −f(x) |≤ m (λn(x)) ν 2 where λn(x) = d ∗ n,q ( (e1 −x)2; x ) . proof. we prove it by using the result (3.3) and hölder inequality: | d∗n,q(f; x) −f(x) |≤| d ∗ n,q(f(e1) −f(x); x) |≤ d ∗ n,q (| f(e1) −f(x) |; x) ≤| md ∗ n,q (| e1 −x | ν; x) . therefore | d∗n,q(f; x) −f(x) | ≤ m 1 eµ,q([n]qx) ∑∞ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣ν ≤ m 1 eµ,q([n]qx) ∑∞ k=0 ( ([n]qx) kq k(k−1) 2 γµ,q(k) )2−ν 2 ( ([n]qx) kq k(k−1) 2 γµ,q(k) )ν 2 ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣ν ≤ m ( 1 (eµ,q([n]qx)) ∑∞ k=0 ([n]qx) kq k(k−1) 2 γµ,q(k) )2−ν 2 ( 1 (eµ,q([n]qx)) ∑∞ k=0 ([n]qx) kq k(k−1) 2 γµ,q(k) ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣2 )ν 2 ≤ m ( d∗n,q(e1 −x)2; x )ν 2 . this complete the proof. � we denote cb[0,∞) for the space of all bounded and continuous functions on r+ = [0,∞), and c2b(r +) = {g ∈ cb(r+) : g′,g′′ ∈ cb(r+)}, (3.4) dunkl generalization of q-parametric szász-mirakjan operators 213 with the norm ‖ g ‖c2 b (r+)=‖ g ‖cb(r+) + ‖ g ′ ‖cb(r+) + ‖ g ′′ ‖cb(r+), (3.5) also ‖ g ‖cb(r+)= sup x∈r+ | g(x) | . (3.6) theorem 3.3. let d∗n,q(. ; .) be the operators defined by (1.12). then for any g ∈ c2b(r +), we have | d∗n,q(f; x) −f(x) |≤ ( (1 −q)x + λn(x) 2 ) ‖ g ‖c2 b (r+) where λn(x) is given as in theorem 3.2. proof. let g ∈ c2b(r +). then by using the generalized mean value theorem in the taylor series expansion we have g(e1) = g(x) + g ′(x)(e1 −x) + g′′(ψ) (e1 −x)2 2 , ψ ∈ (x,e1). by applying linearity property on d∗n,q, we have d∗n,q(g,x) −g(x) = g ′(x)d∗n,q ((e1 −x); x) + g′′(ψ) 2 d∗n,q ( (e1 −x)2; x ) , which implies that, | d∗n,q(g; x) −g(x) |≤ (1 −q)x ‖ g ′ ‖cb(r+) + ( (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx ) ‖ g′′ ‖cb(r+) 2 from (3.5) we have ‖ g′ ‖cb[0,∞)≤‖ g ‖c2b[0,∞). | d∗n,q(g; x) −g(x) |≤ (1 −q)x ‖ g ‖c2b(r+) + ( (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx ) ‖ g ‖c2 b (r+) 2 . from 3 of lemma 2.2, we get the required result. � the peetre’s k-functional is defined by k2(f,δ) = inf c2 b (r+) {( ‖ f −g ‖cb(r+) +δ ‖ g ′′ ‖c2 b (r+) ) : g ∈w2 } , (3.7) where w2 = { g ∈ cb(r+) : g′,g′′ ∈ cb(r+) } . (3.8) then there exits a positive constant c > 0 such that k2(f,δ) ≤ cω2(f,δ 1 2 ), δ > 0, where the second order modulus of continuity is given by ω2(f,δ 1 2 ) = sup 0<h<δ 1 2 sup x∈r+ | f(x + 2h) − 2f(x + h) + f(x) | . (3.9) theorem 3.4. let d∗n,q(. ; .) be the operators defined by (1.12) and cb[0,∞) be the space of all bounded and continuous functions on r+. then for x ∈ r+, f ∈ cb(r+), we have | d∗n,q(f; x)−f(x) |≤ 2m { ω2 ( f; √ 2x(1 −q) + λn(x) 4 ) + min ( 1, 2x(1 −q) + λn(x) 4 ) ‖ f ‖cb(r+) } , where m is a positive constant, λn(x) is given in theorem 3.2 and ω2(f; δ) is the second order modulus of continuity of the function f defined in (3.9). proof. we prove this by using the theorem (3.3) | d∗n,q(f; x) −f(x) | ≤ | d ∗ n,q(f −g; x) | + | d ∗ n,q(g; x) −g(x) | + | f(x) −g(x) | ≤ 2 ‖ f −g ‖cb(r+) +(1 −q)x ‖ g ‖cb(r+) + λn(x) 2 ‖ g ‖c2 b (r+) 214 mursaleen, nasiruzzaman and al-abied from (3.5) clearly we have ‖ g ‖cb[0,∞)≤‖ g ‖c2b[0,∞). therefore, | d∗n,q(f; x) −f(x) |≤ 2 ( ‖ f −g ‖cb(r+) + 2x(1 −q) + λn(x) 4 ‖ g ‖c2 b (r+) ) , where λn(x) is given in theorem 3.2. by taking infimum over all g ∈ c2b(r +) and by using (3.7), we get | d∗n,q(f; x) −f(x) |≤ 2k2 ( f; 2x(1 −q) + λn(x) 4 ) now for an absolute constant c > 0 in [7] we use the relation k2(f; δ) ≤ c{ω2(f; √ δ) + min(1,δ) ‖ f ‖}. this complete the proof. � acknowledgement. research of the first author was supported by the department of science and technology, new delhi, under grant no.sr/s4/ms:792/12. the second author acknowledges the financial support of university grants commission for awarding bsr fellowship. references [1] m. altınok, m. küçükaslan, a-statistical supremum-infimum and a-statistical convergence, azerbaijan journal of mathematics, 4 (2) (2014), 31-42. [2] a. aral, v. gupta, on q-analogue of stancu-beta operators, appl. math. letters, 25 (2012), 67–71. [3] a. aral, o. doǧru, bleimann butzer and hahn operators based on q-integers, j. ineq. appl., 2007 (2007), art. id 79410. [4] a. aral, v. gupta: the q-derivative and applications to q-szász mirakyan operators, calcolo 43 (3) (2006), 151–170. [5] s.n. bernstein, démonstration du théoréme de weierstrass fondée sur le calcul des probabilités, commun. soc. math. kharkow, 2 (13) (1912), 1–2. [6] b. cheikh, y. gaied, m. zaghouani, a q-dunkl-classical q-hermite type polynomials. georgian math. j., 21 (2) (2014), 125–137. [7] a. ciupa, a class of integral favard-szász type operators. stud. univ. babeş-bolyai, math., 40 (1) (1995), 39–47. [8] s. ersan, o. doğru, statistical approximation properties of q-bleimann, butzer and hahn operators, math. comput. modell., 49 (2009), 1595–1606. [9] a.d. gadjiev, simultaneous statistical approximation of analytic functions and their derivatives by k-positive linear operators, azerbaijan journal of mathematics, 1 (1) (2011), 57-66. [10] v. gupta, c. radu, statistical approximation properties of q-baskokov-kantorovich operators, cent. eur. j. math., 7 (4) (2009), 809–818. [11] g. i̇çöz, bayram çekim, dunkl generalization of szász operators via q-calculus, jour. ineq. appl., 2015 (2015), art. id 284. [12] n. ispir, approximation by modified complex szasz-mirakjan operators, azerbaijan journal of mathematics, 3 (2) (2013), 95-107. [13] v. kac, p. cheung, quantum calculus. springer-verlag new york, 2002. [14] v. kac, p. cheung, quantum calculu., universitext, springer-verlag, new york, 2002. [15] a. lupaş, a q-analogue of the bernstein operator, university of cluj-napoca, seminar on numerical and statistical calculus, no. 9, 1987. [16] n.i. mahmudov, v. gupta, on certain q-analogue of szász kantorovich operators, j. appl. math. comput., 37 (2011) 407–419. [17] m. mursaleen, asif khan, h.m. srivastava, k.s. nisar, operators constructed by means of q-lagrange polynomials and a-statistical approximation, appl. math. comput., 219 (2013), 6911–6918. [18] m. mursaleen, asif khan, generalized q-bernstein-schurer operators and some approximation theorems, jour. function spaces appl., 2013 (2013), article id 719834, 7 pages. [19] m. mursaleen, asif khan, statistical approximation properties of modified q-stancu-beta operators, bull. malaysian math. sci.soc.(2), 36 (3) (2013), 683–690. [20] m. mursaleen, faisal khan and asif khan, approximation properties for modified q-bernstein-kantorovich operators, numer. funct. anal. optim., 36(9) (2015), 1178–1197. [21] m. mursaleen, faisal khan and asif khan, approximation properties for king’s type modified q-bernsteinkantorovich operators, math. meth. appl. sci., 38 (2015), 5242–5252. [22] m. örkcü, o. doǧru, weighted statistical approximation by kantorovich type q-szász mirakjan operators, appl. math. comput., 217 (2011), 7913–7919. [23] g.m. phillips, bernstein polynomials based on the qintegers, the heritage of p.l. chebyshev, a festschrift in honor of the 70th-birthday of professor t. j. rivlin. ann. numer. math., 4 (1997), 511–518. dunkl generalization of q-parametric szász-mirakjan operators 215 [24] c. radu, statistical approximation properties of kantorovich operators based on q-integers, creat. math. inform., 17 (2) (2008), 75–84. [25] m. rosenblum, generalized hermite polynomials and the bose-like oscillator calculus, oper. theory, adv. appl., 73 (1994), 369-396. [26] s. sucu, dunkl analogue of szász operators, appl. math. comput., 244, (2014), 42–48. [27] o. szász, generalization of s. bernstein’s polynomials to the infinite interval, j. res. natl. bur. stand., 45 (1950), 239–245. department of mathematics, aligarh muslim university, aligarh–202002, india ∗corresponding author: mursaleenm@gmail.com 1. introduction 2. main results 3. rate of convergence references international journal of analysis and applications issn 2291-8639 volume 3, number 1 (2013), 1-13 http://www.etamaths.com exponential decay and numerical solution for a timoshenko system with delay term in the internal feedback c. a. raposo1,∗, j. a. d. chuquipoma1, j. a. j. avila1, m. l. santos2 abstract. in this work we study the asymptotic behavior as t → ∞ of the solution for the timoshenko system with delay term in the feedback. we use the semigroup theory for to prove the well-posedness of the system and for to establish the exponential stability. as far we know, there exist few results for problems with delay, where the asymptotic behavior is based on the gearhartherbst-pruss-huang theorem to dissipative system. see [4],[5],[6]. finally, we present numerical results of the solution of the system. 1. introduction in this paper we consider the following timoshenko system ρ1φtt(x,t) − k(φx + ψ)x(x,t) + µ1φt(x,t) + µ2φt(x,t − τ) = 0,(1) ρ2ψtt(x,t) − bψxx(x,t) + k(φx + ψ)(x,t) + µ3ψt(x,t) + µ4ψt(x,t − τ) = 0,(2) where φ is the transverse displacement of the beam, ψ is the rotation angle of the filament of the beam, (x,t) ∈ (0,l) × (0,∞), τ > 0 represents the time delay and ρ1,ρ2,b,k,µi, i = 1,2,3,4, are positive constants. this beam, of length l is subjected to the following boundary conditions φ(0, t) = φ(l,t) = ψ(0, t) = ψ(l,t) = 0, t > 0,(3) and initial conditions (φ0,φ1,ψ0,ψ1,f0,g0) belongs to a suitable functional space, defined for all x ∈ (0,l) by φ(x,0) = φ0(x), φt(x,0) = φ1(x), ψ(x,0) = ψ0(x), ψt(x,0) = ψ1(x),(4) and for (x,t) ∈ (0,l) × [0,τ], that implies past history with t − τ ≤ 0, by φt(x,t − τ) = f0(x,t − τ), ψt(x,t − τ) = g0(x,t − τ).(5) note that f0(x,0) = φ1(x) and g0(x,0) = ψ1(x). in the study of the asymptotic behavior, we use the result due to gearhart. see [4, 5, 6]. 2010 mathematics subject classification. 35b40, 93d15. key words and phrases. timoshenko system; weak damping; exponential decay; delay. c⃝2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 raposo, chuquipoma, santos theorem 1.1. let s(t) = ea t be a c0−semigroup of contractions on a hilbert space. then s(t) is exponentially stable if and only if ρ(a) ⊇ iβ,β ∈ r(6) and lim |β|→∞ ||(iβi − a)−1|| < ∞(7) hold. certainly, this approach is very different from other works in the literature, especially for problems with delay, where the exponential decay is made by the method of energy, see, for example [8, 9, 10], and references therein. the method of energy, in general, imposes a additional condition on the wave speeds, that is, kρ2 = bρ1, see [1, 2, 3]. here we do not use any additional condition for the coefficients of the system. our work improves the result obtained in [7] in the sense that delays has been introduced in the control ( damping terms ). the delays µ2φt(x,t − τ), µ4ψt(x,t − τ) makes the problem different from that considered in the literature. it is well known that small delays in the controls might turn such well-behaving system into a wild one. in recent years, the pdes with time delay effects have become an active area of research. the plan of this work is follows: in the next section, we introduce the energy space and prove that the full energy of the system decay. in the section 3, we introduce the semigroup representation for the system and prove that a the infinitesimal generator of the semigroup is dissipative, and more, that a generates a ea t, c0-semigroup of contractions, that implies, prove the existence and regularity of solution. finally in the section 4 by theorem of gearhart we prove that ea t is exponentially stably. 2. energy space for the sobolev spaces we use the standard notation as in [11]. let us proceed as [12]. we introduce the followings new dependents variables as in z(x,ρ,t) = φt(x,t − τρ), w(x,ρ,t) = ψt(x,t − τρ), ρ ∈ (0,1),(8) that satisfies for (x,ρ,t) ∈ (0,l) × (0,1) × (0,∞) τ zt(x,ρ,t) + zρ(x,ρ,t) = 0, τ wt(x,ρ,t) + wρ(x,ρ,t) = 0.(9) therefore, problem (1)-(2) is equivalent to ρ1φtt(x,t) − k(φx + ψ)x(x,t) + µ1φt(x,t) + µ2z(x,1, t) = 0, ρ2ψtt(x,t) − bψxx(x,t) + k(φx + ψ)(x,t) + µ3ψt(x,t) + µ4w(x,1, t) = 0,(10) τ zt(x,ρ,t) + zρ(x,ρ,t) = 0, τ wt(x,ρ,t) + wρ(x,ρ,t) = 0, exponential decay and numerical solution 3 the above system subjected to the following initial and boundary conditions φ(0, t) = φ(l,t) = ψ(0, t) = ψ(l,t) = 0, t > 0, z(x,0, t) = φt(x,t), w(x,0, t) = ψt(x,t), x ∈ (0,l), t > 0, φ(·,0) = φ0, φt(·,0) = φ1, x ∈ (0,l),(11) ψ(·,0) = ψ0, ψt(·,0) = ψ1, x ∈ (0,l), z(x,1,0) = f0(x,t − τ), w(x,1,0) = g0(x,t − τ), in (0,l) × (0,τ). now, the energy space h is defined as h = {h10 × l 2 × h10 × l 2 × l2(0,1; l2) × l2(0,1; l2)}. for µ1 > µ2, µ3 > µ4 satisfying τµ2 < ξ < τ(2µ1 − µ2), τµ4 < η < τ(2µ3 − µ4)(12) respectively, we define the full energy of the system in the energy space as e(t) = 1 2 ∫ l 0 (ρ1|φt|2 + ρ2|ψt|2 + k|φx + ψ|2 + b|ψx|2)dx + ξ 2 ∫ l 0 ∫ 1 0 z2(x,ρ,t)dρdx + η 2 ∫ l 0 ∫ 1 0 w2(x,ρ,t)dρdx. lemma 2.1. there exists a positive constant c such that for any regular solution (φ,ψ,z,w) of the problem (10)-(11) and for any t ≥ 0, we have d dt e(t) ≤ −c ∫ l 0 (|φt|2 + |ψt|2 + z2(x,1) + w2(x,1))dx.(13) proof. 2.1. we multiplying (1) by φt, (2) by ψt, and using integration by part to get 1 2 d dt ∫ l 0 (ρ1|φt|2 + ρ2|ψt|2 + k|φx + ψ|2 + b|ψx|2)dx = − µ1 ∫ l 0 |φt|2 dx − µ2 ∫ l 0 φt z(1, t)dx − µ3 ∫ l 0 |ψt|2 dx − µ4 ∫ l 0 ψt w(1, t)dx and using the energy e(t) of the system, we obatin d dt e(t) = −µ1 ∫ l 0 |φt|2 dx − µ2 ∫ l 0 φt z(1, t)dx − d dt { ξ 2 ∫ l 0 ∫ 1 0 z2(x,ρ,t)dρdx } −µ3 ∫ l 0 |ψt|2 dx − µ4 ∫ l 0 ψt w(1, t)dx − d dt { η 2 ∫ l 0 ∫ 1 0 w2(x,ρ,t)dρdx } 4 raposo, chuquipoma, santos using (14) and (15) d dt ξ 2 ∫ l 0 ∫ 1 0 z2(x,ρ,t)dρdx = − ξ τ ∫ l 0 ∫ 1 0 z(x,ρ,t)zρ(x,ρ,t)dρdx = − ξ 2τ ∫ l 0 ∫ 1 0 ∂ ∂ ρ z2(x,ρ,t)dρdx = ξ 2τ ∫ l 0 (z2(x,0) − z2(x,1))dx(14) d dt η 2 ∫ l 0 ∫ 1 0 w2(x,ρ,t)dρdx = − η τ ∫ l 0 ∫ 1 0 w(x,ρ,t)wρ(x,ρ,t)dρdx = − η 2τ ∫ l 0 ∫ 1 0 ∂ ∂ ρ w2(x,ρ,t)dρdx = η 2τ ∫ l 0 (w2(x,0) − w2(x,1))dx(15) we obtain d dt e(t) = −µ1 ∫ l 0 |φt|2 dx − µ2 ∫ l 0 φt z(1, t)dx(16) + ξ 2τ ∫ l 0 (z2(x,0) − z2(x,1))dx −µ3 ∫ l 0 |ψt|2 dx − µ4 ∫ l 0 ψt w(1, t)dx + η 2τ ∫ l 0 (w2(x,0) − w2(x,1))dx(17) now, using youngs’s inequality we can rewritten the last equation as d dt e(t) ≤ − ( µ1 − ξ 2τ − µ2 2 ) ∫ l 0 |φt|2 dx − ( ξ 2τ − µ2 2 ) ∫ l 0 z2(x,1)dx − ( µ3 − η 2τ − µ4 2 ) ∫ l 0 |ψt|2 dx − ( η 2τ − µ4 2 ) ∫ l 0 w2(x,1)dx from where our conclusion holds. 3. existence of solution let us introduce the semigroup representation. to this end, let u = (φ,φt,ψ,ψt,z,w) t and rewrite the problem (10)-(11) as ut = au u(0) = u0(18) exponential decay and numerical solution 5 where the operator a is defined for u = (φ,u = φt,ψ,v = ψt,z,w)t by au =   u k ρ1 (φx + ψ)x − µ1ρ1 u − µ2 ρ1 z(·,1) v b ρ2 ψxx − kρ2 (φx + ψ) − µ3 ρ2 u − µ4 ρ2 w(·,1) − 1 τ zρ − 1 τ wρ   , with domain d(a) = {(φ,φt,ψ,ψt,z,w)t ∈ h : u = z(·,0, ·), v = w(·,0, ·), in(0,1)}, where for x ∈ (0,l) we denote h = h(0,l), l = l(0,l) and h = {(h2 ∩ h10) × h 1 0 × (h 2 ∩ h10) × h 1 0 × l 2(0,1; h1) × l2(0,1; h1)}. for u = (φi,ui,ψi,vi,zi,wi)t ∈ h, i = 1,2 and ξ, η as in (12) we defined the following inner product in the energy space as ⟨u1,u2⟩ = ∫ l 0 [ρ1u 1u2 + ρ2v 1v2 + k(φ1 + ψ1)(φ2 + ψ2) + bψ1ψ2)dx + ξ ∫ l 0 ∫ 1 0 z1(x,ρ)z2(x,ρ)dρdx + η ∫ l 0 ∫ 1 0 w1(x,ρ)w2(x,ρ)dρdx. for to prove the existence of solution we begin with the proof that the operator a is dissipative. lemma 3.1. for u = (φ,u = φt,ψ,v = ψt,z,w) t ∈ d(a), we have ⟨au,u⟩ ≤ 0. proof. 3.1. ⟨au,u⟩ = −µ1 ∫ l 0 |φt|2 dx − µ2 ∫ l 0 φt z(1, t)dx − ξ τ ∫ l 0 ∫ 1 0 z(x,ρ,t)zρ(x,ρ,t)dρdx −µ3 ∫ l 0 |ψt|2 dx − µ4 ∫ l 0 ψt w(1, t)dx − η τ ∫ l 0 ∫ 1 0 w(x,ρ,t)wρ(x,ρ,t)dρdx. using (14) and (15) in the equation above we obtain ⟨au,u⟩ = −µ1 ∫ l 0 |φt|2 dx − µ2 ∫ l 0 φt z(1, t)dx + ξ 2τ ∫ l 0 (z2(x,0) − z2(x,1))dx −µ3 ∫ l 0 |ψt|2 dx − µ4 ∫ l 0 ψt w(1, t)dx + η 2τ ∫ l 0 (w2(x,0) − w2(x,1))dx. now using (16) and lemma 3.1 we concludes ⟨au,u⟩ = d dt e(t) ≤ −c ∫ l 0 (|u|2 + |v|2 + z2(x,1) + w2(x,1))dx.(19) in the next lemma, we will prove an important property of resolvent of the operator a. lemma 3.2. 0 ∈ ρ(a). 6 raposo, chuquipoma, santos proof. 3.2. for any f = (f1,f2,f3,f4,f5,f6) t ∈ h consider the equation au = f. this implies u = f1,(20) k(φx + ψ)x − µ1u − µ2z(·,1) = ρ1f2,(21) v = f3,(22) bψxx − k(φx + ψ) − µ3v − µ4w(·,1) = ρ2f4,(23) −zρ = τf5,(24) −wρ = τf6,(25) we plug u = f1 obtained from (20) into (21) to get k(φx + ψ)x = µ1u + µ2z(·,1) + ρ1f2 ∈ l2(0,l). by poincarè inequality we have k(φx + ψ) ∈ l2(0,l).(26) now we plug v = f3 obtained from (26) and (22) into (23) to get bψxx = k(φx + ψ) + µ3v + µ4w(·,1) + ρ2f4 ∈ l2(0,l).(27) by the standard theory in the linear elliptic equations , we have a unique ψ ∈ h2 ∩ h10 satisfying (27). then we plug ψ just obtained from solving (27) into (21) to get kφxx = −kψx + µ1u + µ2z(·,1) + ρ1f2 ∈ l2(0,l).(28) applying the standard theory in the linear elliptic equations again yields a unique solvability of φ ∈ h2 ∩ h10 for (28). from (24) we have using poincarè inequality, 1 cp ∫ l 0 ∫ 1 0 |z|2 dρdx ≤ ∫ l 0 ∫ 1 0 |zρ|2 dρdx ∈ l2(0,1 : l2(0,l)), then z ∈ l2(0,1 : h1(0,l)). the same idea we use for w. thus the unique solvability of au = f follows. it is clear from the theory of the linear elliptic equation, see chapter 1 of [13], that ||u||h ≤ c||f||h with c being a positive constant independent of u, and then 0 ∈ ρ(a). now we will to prove that a generates a c0−semigroup of contractions. lemma 3.3. the operator a generates a c0−semigroup of contractions on a hilbert space h. proof. 3.3. from lemma 3.1 we have that a is dissipative operator, and from lemma 3.2 follows that 0 ∈ ρ(a), them from theorem 1.2.4, page 3 of [13], we concludes that a generates a c0−semigroup of contractions on h. in this step, we prove that the problem (10)-(11) is well-posedness, and in this direction, we have the following result theorem 3.4. if µ2 ≤ µ1 and µ4 ≤ µ3, then there exists a unique solution u ∈ c([0,∞),h) of the (10)-(11). moreover if u0 ∈ d(a), then u ∈ c([0,∞),d(a))∩ c1([0,∞),h). exponential decay and numerical solution 7 proof. 3.5. from the classical semigroup theory, see for example [14], follows by lemma 3.3 that u(t) = ea t u0 is the unique solution of the problem (10)-(11) in the conditions of theorem. the proof is complete. 4. asymptotic behavior now we are in position to present our principal result theorem 4.1. the semigroup ea t is exponentially stably. proof. 4.2. we now use theorem 1.1 and we use a contradiction argument. we first prove (6). from lemma 3.2 we have that 0 ∈ ρ(a) and follows from this fact and the contraction mapping theorem that for any real number β with |β| < ||a||−1, the operator iβi − a = a(iβa−1 − i) is invertible. moreover, ||(iβi − a)−1|| is a continuous function of β in the interval (−||a||−1, ||a||−1). if sup{||(iβi − a)−1|| : |β| < ||a||−1} = m < ∞, then by the contraction mapping theorem, the operator iβi − a = (iβ0i − a)(i + i(β − β0)(iβ0i − a)−1) with |β0| < ||a||−1 is invertible for |β − β0| < 1/m. it turns out that by choosing |β0| as close to ||a||−1 as we can, we conclude that {β : |β| < ||a||−1 + 1/m} ⊂ ρ(a) and ||(iβi − a)−1|| is continuous function of β in the interval (−||a||−1 − 1/m , ||a||−1 + 1/m). from argument above, it follows that if (6) is not true, then there is w ∈ r with ||a||−1 ≤ |w| < ∞ such that {iβ ; |β| < |w|} ⊂ ρ(a) and sup{||(iβ − a)−1|| : |β| < |w|} = ∞. it turns out that there exists a sequence βn ∈ r with βn → w, |βn| < |w| and a sequence of complex vector functions un = (φn,un,ψn,vn,zn,wn)t satisfying un ∈ d(a) with ||un||h = 1 such that ||(iβn − a)un|| → 0, as n → ∞, and then iβnφn − un → 0 in h10(29) iβnρ1u n − k(φnx + ψ n)x + µ1u n + µ2z n(·,1) → 0 in l2 iβnψn − vn → 0 in h10(30) iβnρ2v n − bψnxx + k(φ n x + ψ n) + µ3v n + µ4w n(·,1) → 0 in l2 iβnτzn − znρ → 0 in l 2(0,1; l2) iβnτwn − wnρ → 0 in l 2(0,1; l2) making the inner product of (iβni − a)un with un in h, taking the real part, and using (19) we have∫ l 0 (|un|2 + |vn|2 + zn(x,1)2 + wn(x,1)2)dx → 0, 8 raposo, chuquipoma, santos from where follows that un → 0(31) vn → 0(32) zn → 0(33) wn → 0(34) using (31) into (29) we obtain φn → 0,(35) and using (32) into (30) we obtain φn → 0.(36) now using (31),(32),(33),(34),(35),(36) we concludes that ||un|| → 0 which is a contradiction with ||un|| = 1 and the proof of (6) is complete. finally we prove (7) by a contradiction argument again. suppose that (7) is not true. then there exists a sequence βn with |βn| → ∞ and a sequence of complex vector functions un ∈ d(a) with unit norm in h such that ||(iβni − a)un|| → 0, as n → ∞. again we have∫ l 0 (|un|2 + |vn|2 + zn(x,1)2 + wn(x,1)2)dx = −⟨aun,un⟩ → 0.(37) making the inner product of (iβni − a)un with un in h we obtain iβn||un||2 − ⟨aun,un⟩ → 0. from (37) we get βn||un||2 → 0.(38) as βn → ∞ and ||un|| is limited, we concludes that (38) is true only if ||un|| → 0 contradict ||un|| = 1. the proof of theorem is complete. 5. numerical solution we will solve numerically the system of timoshenko (1)-(5) in the one-dimension domain ω of the length l, using high-order schemes. we used the implicit compact finite difference method of fourth-order for discretization of spacial variable and the classic finite difference for discretization of temporal variable. 5.1. discretization. in order to get the discretization of the problem (1)-(5), we define the following sets: ωh = {xi : xi = ih, i = 0,1, ... ,i + 1; h = l/(i + 1)}, ⊤k = {tn : tn = nk, n = 0,1, ...,n; k = ch}, †k = {tn : tn = nk, n = −m,−m + 1, ...,0; 0 < m < n} where qkh = ωh × ⊤ k and dkh = ωh × † k are the computational mesh, and mesh of delay, respectively. the width of mesh dkh is τ = mk. in figure 1 we show a mesh model for the full-domain qkh ∪ d k h. the points (xi, tn) are called nodes of exponential decay and numerical solution 9 ( ,1)i ( , )i n (1, )n ( , )i n ( , 0)i ( 1, )n( 2, )i n+( 1, )i n+(0, )n ( , )i n ( , 1)i ( , )i m( , )i m k h q k h d figure 1. model mesh for the full-domain: qkh ∪ d k h. the mesh and, usually denote by (i,n). the classification of nodes is as follows: interiors (circles), boundaries (stars), initials (squares) and ghosts (diamonds). let χ = χ(x,t) be a function with second order partial derivatives. henceforth consider the following notation χni ≡ χ(xi, tn). we define the following approximation of the derivatives of χ, according to taylor, (χt) n i ≈ 1 k δ−t χ n i , (χt) n i ≈ 1 2k δ0t χ n i , (χt) (n−m) i ≈ 1 k δ−t χ (n−m) i (χx) n i ≈ 1 2h δ0xχ n i , (χtt) n i ≈ 1 k2 δ2t χ n i , (χxx) n i ≈ 1 h2 [ δ2x 1 + 1 12 δ2x ] χni(39) where the finite difference operators are given by δ−t χ n i := χ n i − χ n−1 i , δ 0 t χ n i := χ n+1 i − χ n−1 i , δ−t χ (n−m) i := χ (n−m) i − χ (n−m)−1 i , δ 0 xχ n i := χ n i+1 − χ n i−1, δ2t χ n i := χ n+1 i − 2χ n i + χ n−1 i , δ 2 xχ n i := χ n i+1 − 2χ n i + χ n i−1, (40) [ 1 + 1 12 δ2x ] χni := 1 12 χni+1 + 5 6 χni + 1 12 χni−1 the discrete formulation of equations (1)-(5) is obtained using (39), ρ1 [ 1 + 1 12 δ2x ] δ2t φ n i − α1δ 2 xφ n i − α2 [ 1 + 1 12 δ2x ] δ0xψ n i + α3 [ 1 + 1 12 δ2x ] δ0t φ n i + α4 [ 1 + 1 12 δ2x ] δ−t φ (n−m) i = 0 in (xi, tn) ∈ qkh(41) 10 raposo, chuquipoma, santos ρ2 [ 1 + 1 12 δ2x ] δ2t ψ n i − β1δ 2 xψ n i + β2 [ 1 + 1 12 δ2x ] δ0xφ n i + β0 [ 1 + 1 12 δ2x ] ψni + β3 [ 1 + 1 12 δ2x ] δ0t ψ n i + β4 [ 1 + 1 12 δ2x ] δ−t ψ (n−m) i = 0 in (xi, tn) ∈ qkh(42) (43) φ0i = (φ0)i, ψ 0 i = (φ0)i, (φt) 0 i = (φ1)i, (ψt) 0 i = (ψ1)i in xi ∈ ω̊h (44) φn0 = φ n i+1 = ψ n 0 = ψ n i+1 = 0 on tn ∈ ⊤ k (45) φ (n−m) i = (f0) (n−m) i , ψ (n−m) i = (g0) (n−m) i , in (xi, t(n−m)) ∈ d k h where, the parameters, are defined by α1 = kk 2/h2, α2 = kk 2/2h, α3 = µ1k/2, α4 = µ2k, β1 = bk 2/h2, β2 = α2, β0 = kk 2, β3 = µ3k/2, β4 = µ4k substituting (40) in (41)-(42), we have the following linear algebraic system: a1φ n+1 = b1φ n + c1ψ n + d1φ n−1 − e1δ−t φ (n−m) + υn1(46) a2ψ n+1 = b2ψ n + c2φ n + d2ψ n−1 − e2δ−t ψ (n−m) + υn2(47) where, φn+1 = (φn+11 ,φ n+1 2 , ...,φ n+1 i ) t and ψn+1 = (ψn+11 ,ψ n+1 2 , ...,ψ n+1 i ) t, n = 0,1, ...,n − 1, are unknown vectors, a1 = tridiag ( 1 12 (ρ1 + α3), 5 6 (ρ1 + α3), 1 12 (ρ1 + α3) ) , b1 = tridiag (1 6 (ρ1 + 6α1), 1 3 (5ρ1 − 6α1), 1 6 (ρ1 + 6α1) ) , c1 = pentadiag ( − 1 12 α2,− 5 6 α2,0, 5 6 α2, 1 12 α2 ) , d1 = tridiag ( 1 12 (−ρ1 + α3), 5 6 (−ρ1 + α3), 1 12 (−ρ1 + α3) ) , e1 = tridiag ( 1 12 α4, 5 6 α4, 1 12 α4 ) , a2 = tridiag ( 1 12 (ρ2 + β3), 5 6 (ρ2 + β3), 1 12 (ρ2 + β3) ) , b2 = tridiag ( 1 12 (2ρ2 + 12β1 − β0), 1 6 (10ρ2 − 12β1 − 5β0), 1 12 (2ρ2 + 12β1 − β0) ) , c2 = −c1, d2 = tridiag ( 1 12 (−ρ2 + β3), 5 6 (−ρ2 + β3), 1 12 (−ρ2 + β3) ) , e2 = tridiag ( 1 12 β4, 5 6 β4, 1 12 β4 ) , are matrices of order i×i. υn1 and υn2 are vectors of order i that load the boundary data. exponential decay and numerical solution 11 5.2. numerical test. in order to verify the asymptotic behavior of the solution of the timoshenko system, we consider the following data: l = 2π, ρ1 = ρ2 = k = b = 1. boundary condition: φ(0, t) = φ(2π,t) = ψ(0, t) = ψ(2π,t) = 0 initial condition: φ0(x) = 0, ψ0(x) = 0, φ1(x) = sin(x), ψ1(x) = cos(x) delay condition: f0(x,t − τ) = sin(x) cos(t − τ), g0(x,t − τ) = cos(x) cos(t − τ) numerical data: i = 18, c = 0.3, τ = 10% of the width of the mesh qkh, tol = 4 × 10−5 (tolerance). table 1 shows seven cases where timoshenko system may behave differently with the presence the terms of delay and damping. each of these cases are plotted in figure 2-8. note that the asymptotic behavior of the solution was calculated by taking the maximum value of the function φ, in x ∈ [0,2π], throughout time. in figure 2, it is observed that there is no asymptotic behavior of the solution, in contrast to figures 3-6, where the asymptotic behavior of the solution is increasingly more acute. figure 7 represents the case without delay, the presence of damping is very evident, obtaining the asymptotic behavior of the solution immediately. figure 8 represents the case without delay and damping, and as was expected, there is no convergence of the solution. in figure 9 we show the graph of function φ(x,t), where x ∈ [0,2π], t ∈ [−2.97,29.76], µ1 = µ3 = 1, µ2 = µ4 = 0.8 and we choose only 300 iterations along time. with respect to rotation angle ψ we observe that it exhibits the same behavior that the function φ. table 1. table for different cases. case damping delay iterations in time asymptotic behavior 1 µ1 = µ3 = 1 µ2 = µ4 = 1 3000 diverges 2 µ1 = µ3 = 1 µ2 = µ4 = 0.9 3000 converges 3 µ1 = µ3 = 1 µ2 = µ4 = 0.8 3000 converges 4 µ1 = µ3 = 1 µ2 = µ4 = 0.7 3000 converges 5 µ1 = µ3 = 1 µ2 = µ4 = 0.6 3000 converges ... 6 µ1 = µ3 = 1 µ2 = µ4 = 0 159 converges 7 µ1 = µ1 = 0 µ2 = µ4 = 0 3000 diverges 12 raposo, chuquipoma, santos t p h i 500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 1 figure 2. case 1. t p h i 500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 figure 3. case 2. t p h i 500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 figure 4. case 3. t p h i 500 1000 1500 2000 2500 3000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 figure 5. case 4. t p h i 500 1000 1500 2000 2500 3000 0.1 0.2 0.3 0.4 0.5 0.6 figure 6. case 5. t p h i 50 100 150 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 figure 7. case 6. exponential decay and numerical solution 13 t p h i 500 1000 1500 2000 2500 3000 0.2 0.4 0.6 0.8 1 figure 8. case 7. x 0 2 4 6 t 0 10 20 30 p h i -1 0 1 figure 9. graph of φ(x,t). 6. conclusion we have demonstrated the well-posedness and asymptotic behavior solution of the timoshenko system. thus, it also was obtained numerically the asymptotic behavior of the solution confirming the theory developed. references [1] a. soufyane, a. wehbe, exponential stability for the timoshenko beam by a locally distributed damping, electron. j. differ. equ. 29 (2003), 1-14. [2] j. e. m. rivera, r. racke, timoshenko systems with indefinite damping. j. math. anal. appl. 341 (2008), 1068-1083. [3] f. amar-khodja. a. benabdallah, j. e. m. rivera, r. racke, energy decay for timoshenko systems of memory type, j. differ. equ. 194 (2003), 82-115. [4] l. gearhart, spectral theory for the contractions semigroups on hilbert spaces. trans. of the american mathematical society . 236 (1978), 385-349. [5] f. huang, characteristic conditions for exponential stability of the linear dynamical systems in hilbert spaces. annals of differential equations. 1 (1985), 43-56. [6] j. prüss, on the spectrumm of c0-semigroups. trans. of the american mathematical society 284 (1984) 847-857. [7] c. a. raposo, j. ferreira, m. l. santos, n. n. castro. exponential stability for the timoshenko system with two weak dampings. applied mathematics letters. 18 (2010), 535-541. [8] b. said-houari, y. laskri. a stabilit result of a timoshenko system with a delay term in the internal feedback. applied mathematics and computation . 217 (2010), 2857-2869. [9] c. abdallah, p. dorato, j. benitez-read, r. byrne. delayed positive feedback can stabilize oscillatory system. acc, san francisco (1993) 3016-3107. [10] l. h suh, z. bien. use of time delay action in the controller designe. ieee trans. autom. control. 25 (1980) 600-603. [11] r. a. adams, sobolev spaces, academic press, new york, 1975. [12] s. nicaise, j. valein. stabilization of second order evolution eqaution with unbounded feedback with delay. esaim control. optim. calc. var. 16 (2010). [13] z. liu, s. zheng, semigroups associated with dissipative systems, chapman, new y & hallo/crc, new york, 1999. [14] a. pazy, semigroups of linear operators and applications to partial differential equations. springer, new york, 1993. 1federal university of são joão del-rei, 36.307-352, são joão del-rei mg, brazil 2federal university of pará, 36.307-352, belém pa, brazil ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 100-106 http://www.etamaths.com some common fixed point theorems in generalized vector metric spaces rajesh shrivastava1, rajendra kumar dubey2, pankaj tiwari1,∗ abstract. in this paper we give some theorems on point of coincidence and common fixed point for two self mappings satisfying some general contractive conditions in generalized vector spaces. our results generalize some well-known recent results in this direction. 1. introduction and preliminaries in 2003, mustafa and sims [6] introduced a more appropriate and robust notion of a generalized metric space as follows. definition 1.1. [6]. let x be a nonempty set, and let g : x ×x ×x → [0,∞) be a function satisfying the following axioms: (1) g(x,y,z) = 0 if and only if x = y = z; (2) g(x,x,y) > 0, for all x 6= y; (3) g(x,y,z) ≥ g(x,x,y), for all x,y,z ∈ x; (4) g(x,y,z) = g(x,z,y) = g(z,y,x) = · · · (symmetric in all three variables); (5) g(x,y,z) ≤ g(x,w,w) + g(w,y,z), for all x,y,z,w ∈ x. then the function g is called a generalized metric, or, more specifically a g-metric on x, and the pair (x,g) is called a g-metric space. a riesz space is an ordered vector space and a lattice. let e be a riesz space with the positive cone e+ = {x ∈ e : x ≥ 0}. if {an} is a decreasing sequence in e such that inf an = a, write an ↓ a. definition 1.2. the riesz space e is said to be achimedean if 1 n an ↓ 0 holds for every e+. definition 1.3. a sequence {bn} is said to be order convergent (or o-convergent) to b if there is a sequence {an} in e satisfying an ↓ 0 and |bn − b| ≤ an for all n, and written bn o−→ b or o− lim bn = b, where |a| = sup{a,−a} for any a ∈ e. definition 1.4. a sequence {bn} is said to be order-cauchy (or o-cauchy) if there exists a sequence {an} in e such that an ↓ 0 and |bn − bn+p| ≤ an holds for all n and p. definition 1.5. the riesz space e is said to be o − cauchy complete if every o−cauchy sequence in o− convergent. for notion and other facts regarding riesz spaces we refer to [1]. 2010 mathematics subject classification. 47h10. key words and phrases. reisz space, generalized vector space, coincidence point, fixed point. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 100 some common fixed point theorems 101 2. vector g-metric spaces in this section we introduce the following concepts and properties of vector gmetric spaces. definition 2.1. let x be a non-empty set and e be a riesz space. the function g : x ×x ×x → e is said to be vector g−metric if it is satisfying the following properties : (vgm-1) g(x,y,z) = 0 if and only if x = y = z, (vgm-2) g(x,y,z) ≤ g(x,w,w) + g(w,y,z), for all x,y,z,w ∈ x. also the triple (x,g,e) is said to be vector g−metric space. for arbitrary elements x,y,z,w ∈ x of a vector g−metric space, the following statements are satisfied (1) g(x,x,y) > 0, for all x 6= y; (2) g(x,y,z) ≥ g(x,x,y), for all x,y,z ∈ x; (3) g(x,y,z) = g(x,z,y) = g(z,y,x) = · · · (symmetric in all three variables). example 2.2. a riesz space e is a vector g−metric space with g : e×e×e → e defined by g(x,y,z) = |x − y| + |y − z| + |z − x|. this vector g−metric space is called to be absolute valued g−metric space on e. it is well known that r2 is a riesz space with coordinatwise ordering defined by (x1,y1) ≤ (x2,y2) ⇐⇒ x1 ≤ x2 and y1 ≤ y2 for (x1,y1), (x2,y2) ∈ r2. again r2 is a riesz space with lexicographical ordering defined by (x1,y1) ≤ (x2,y2) ⇐⇒ x1 < x2 or x1 = x2 and y1 ≤ y2. note that r2 is archimedean with coordinatwise ordering but not with lexicographical ordering. example 2.3. let g : r2 ×r2 ×r2 → r2 defined by g((x1,y1), (x2,y2), (x3,y3)) = (αx ∗,βy∗) where x∗ = |x1−x2|+|x2−x3|+|x3−x1| and y∗ = |y1−y2|+|y2−y3|+|y3−y1| also α,β are positive real numbers. then g is a vector g−metric space. let g : r×r×r → r2 defined by g(x,y,z) = (αw,βw) where w = |x−y|+ |y−z|+ |z−x| , α,β ≥ 0 and α + β > 0. then g is a vector g−metric space. definition 2.4. a sequence {xn} in a vector g−metric space (x,g,e) vectorial g−convergence to some x ∈ e, written xn g,e−−→ x, if there is a sequence {an} in e such that an ↓ 0 and satisfying, (1) g(xn,xn,x) ≤ an, (2) g(xn,x,x) ≤ an, 102 shrivatava, dubey and tiwari (3) g(xn,xm,x) ≤ an, for all n. definition 2.5. a sequence {xn} is called ge−cauchy sequence whenever there exists a sequence {an} in e such that an ↓ 0 and g(xn,xm,xm) ≤ an holds for all n and m. definition 2.6. a vector g-metric space is said to be complete if each ge − cauchy sequence in x is e − convergens to a limit in x. using the above definitions, we have the following properties. if xn g,e−−→ x, then (1) the limit x unique, (2) every subsequence of {xn} e−converges to x, (3) if also yn g,e−−→ y and zn g,e−−→ z, then g(xn,yn,zn) o−→ g(x,y,z). when e = r then the concepts of vectorial ge −convergence and convergence in g−metric space are the same also the concepts of ge −cauchy sequence and g−cauchy sequence are the same. remark 2.7. it e is a riesz space and a ≤ ka where a ∈ e+ k ∈ [0, 1), then a = 0. proof. the condition a ≤ ka means that −(1 −k)a = ka−a ∈ e+. since a ∈ e+ and 1 −k > 0, then also (1 −k)a ∈ e+. thus we have (1 −k)a = 0 and a = 0. � 3. main results theorem 3.1. let x be a vector g−metric space with e is archimedean. suppose the mappings s,t : x → x satisfying the following conditions, (i) for all x,y,z ∈ x and α,β,γ,δ ∈ [0, 1) such that 0 ≤ α + β + γ + δ < 1 g(tx,ty,tz) ≤ αg(sx,sy,sz) + βg(sx,tx,tx) + γg(sy,ty,ty) + δg(sz,tz,tz)(3.1) or g(tx,ty,tz) ≤ αg(sx,sy,sz) + βg(sx,sx,tx) + γg(sy,sy,ty) + δg(sz,sz,tz)(3.2) (ii) t(x) ⊆ s(x), (iii) t(x) or s(x) is complete subspace of x. then s and t have a unique point of coincidence in x. moreover, if s and t are weakly compatible, then they have a unique common fixed point in x. proof. let x0 be an arbitrary point in x, since t(x) ⊆ s(x) so we can choose a point x1 ∈ x such that sx1 = tx0. in general we can choose sxn+1 = txn = yn for all n. now, form 3.1 we have g(sxn,sxn+1,sxn+1) = g(txn−1,txn,txn) ≤ (α + β)g(sxn−1,sxn,sxn) + (γ + δ)g(sxn,xn+1,xn+1) g(sxn,sxn+1,sxn+1) ≤ α + β 1 − (γ + δ) g(sxn−1,sxn,sxn).(3.3) let q = α+β 1−(γ+δ) , then 0 ≤ q < 1 since 0 ≤ α + β + γ + δ < 1. so g(sxn,sxn+1,sxn+1) ≤ qg(sxn−1,sxn,sxn).(3.4) some common fixed point theorems 103 continuing in the same way, we have g(sxn,sxn+1,sxn+1) ≤ qng(sx0,sx1,sx1).(3.5) therefore, for all n,m ∈ n,n < m, we have by (vgm-2) g(yn,ym,ym) ≤ g(yn,yn+1,yn+1) + g(yn+1,yn+2,yn+2) + ... + g(ym−1,ym,yn) ≤ (qn + qn+1 + qn+2 + ... + qm−1)g(y0,y1,y1) ≤ qn 1 −q g(y0,y1,y1). now, since e is archimedean then {yn} is an ge−cauchy sequence in x. since the range of s contains the range of t and the range of at least one is ge−complete, so there is w ∈ x such that sxn g,e−−→ w. hence there exists a sequence {an} ∈ e such that an ↓ 0 and g(sxn,w,w) ≤ an. on the other hand, we can find u ∈ x such that sw = u. let us show that tw = u, we have g(tw,u,u) ≤ g(tw,txn,txn) + g(txn,u,u) ≤ αg(sw,sxn,sxn) + βg(sw,tw,tw) + (γ + δ)g(sxn,txn,txn) + an+1 ≤ (α + β + γ + δ + 1)an+1. since the infimum of the sequence on the right side of the above inequality are zero, then tw = u. therefore, w is a point of coincidence of t and s. if w1 is another point of coincidence then there is w1 ∈ x with w1 = tw1 = sw1. now from 3.1 it follows that g(w,w1,w1) = 0, that is w = w1. if s and t are weakly compatible, then it is obvious that w is unique common fixed point of t and s in x. if s and t satisfies condition 3.2, then the argument is similar to that above. however to show that the sequence {xn} is ge −cauchy sequence, we start with g(sxn,sxn,sxn+1) = g(txn−1,txn−1,txn) ≤ (α + β + γ)g(sxn−1,sxn−1,sxn) + δg(sxn,xn+1,xn+1) g(sxn,sxn+1,sxn+1) ≤ α + β + γ 1 − δ g(sxn−1,sxn,sxn).(3.6) let q = α+β+γ 1−δ , then 0 ≤ q < 1 since 0 ≤ α + β + γ + δ < 1. so g(sxn,sxn,sxn+1) ≤ qg(sxn−1,sxn−1,sxn).(3.7) continuing in the same way, we have g(sxn,sxn,sxn+1) ≤ qng(sx0,sx0,sx1).(3.8) then for all n,m ∈ n,n < m, we have by (vgm-2) we prove the remaining part of the proof. � corollary 3.2. let x be a vector g−metric space with e is archimedean. suppose the mappings s,t : x → x satisfying the following conditions, 104 shrivatava, dubey and tiwari (i) for all x,y,z ∈ x and α,β,γ,δ ∈ [0, 1) such that 0 ≤ α + β + γ + δ < 1 g(tmx,tmy,tmz) ≤ αg(smx,smy,smz) + βg(smx,tmx,tmx) +γg(smy,tmy,tmy) + δg(smz,tmz,tmz) or g(tmx,tmy,tmz) ≤ αg(smx,smy,smz) + βg(smx,smx,tmx) +γg(smy,smy,tmy) + δg(smz,smz,tmz) (ii) t(x) ⊆ s(x), (iii) t(x) or s(x) is complete subspace of x. then s and t have a unique point of coincidence in x. moreover, if s and t are weakly compatible, then they have a unique common fixed point in x, also tm and sm are ge − continuous at u. . proof. from theorem 3.1, we see that tm and sm have a unique common fixed point (say u), that is, tm(u) = u. but t(u) = t(tm(u)) = tm+1(u) = tm(t(u)), so t(u) is another fixed point for tm and by uniqueness tu = u.similarly we can show that su = u. � theorem 3.3. let x be a vector g−metric space with e is archimedean. suppose the mappings s,t : x → x satisfying the following conditions, (i) for all x,y,z ∈ x and α ∈ [0, 1) such that, g(tx,ty,tz) ≤ α{g(sx,sy,sz),g(sx,tx,tx),g(sy,ty,ty),g(sz,tz,tz)}(3.9) or g(tx,ty,tz) ≤ α{g(sx,sy,sz),g(sx,sx,tx),g(sy,sy,ty),g(sz,sz,tz)}(3.10) (ii) t(x) ⊆ s(x), (iii) t(x) or s(x) is complete subspace of x. then s and t have a unique point of coincidence in x. moreover, if s and t are weakly compatible, then they have a unique common fixed point in x, also t and s are ge − continuous at u. proof. let x0 be an arbitrary point in x, since t(x) ⊆ s(x) so we can choose a point x1 ∈ x such that sx1 = tx0. in general we can choose sxn+1 = txn = yn for all n. now, form 3.9 we have g(sxn,sxn+1,sxn+1) = g(txn−1,txn,txn) ≤ α{g(sxn−1,sxn,sxn),g(sxn,sxn+1,sxn+1)} g(sxn,sxn+1,sxn+1) ≤ αg(sxn−1,sxn,sxn) g(sxn,sxn+1,sxn+1) ≤ αg(sxn−1,sxn,sxn).(3.11) continuing in the same way, we have g(sxn,sxn+1,sxn+1) ≤ αng(sx0,sx1,sx1).(3.12) therefore, for all n,m ∈ n,n < m, we have by (vgm-2) some common fixed point theorems 105 g(yn,ym,ym) ≤ g(yn,yn+1,yn+1) + g(yn+1,yn+2,yn+2) + ... + g(ym−1,ym,yn) ≤ (αn + αn+1 + αn+2 + ... + αm−1)g(y0,y1,y1) ≤ αn 1 −α g(y0,y1,y1). now, since e is archimedean then {yn} is an ge−cauchy sequence in x. since the range of s contains the range of t and the range of at least one is ge−complete, so there is w ∈ x such that sxn g,e−−→ w. hence there exists a sequence {an} ∈ e such that an ↓ 0 and g(sxn,w,w) ≤ an. on the other hand, we can find u ∈ x such that sw = u. let us show that tw = u, we have g(tw,u,u) ≤ g(tw,txn,txn) + g(txn,u,u) ≤ α max{g(sw,sxn,sxn),g(sw,tw,tw), g(sxn,sxn+1,sxn+1),g(sxn,sxn+1,sxn+1)} + an+1 ≤ (α + 1)an since the infimum of the sequence on the right side of the above inequality are zero, then tw = u. therefore, w is a point of coincidence of t and s. if w1 is another point of coincidence then there is w1 ∈ x with w1 = tw1 = sw1. now from 3.9 it follows that g(w,w1,w1) = 0, that is w = w1. if s and t are weakly compatible, then it is obvious that w is unique common fixed point of t and s in x. if s and t satisfies condition 3.10, then the argument is similar to that above. however to show that the sequence {xn} is ge −cauchy sequence, we start with g(sxn,sxn,sxn+1) = g(txn−1,txn−1,txn) ≤ α max{g(sxn−1,sxn−1,sxn),g(sxn,xn+1,xn+1)} g(sxn,sxn+1,sxn+1) ≤ αg(sxn−1,sxn−1,sxn). continuing in the same way, we have g(sxn,sxn,sxn+1) ≤ αng(sx0,sx0,sx1). then for all n,m ∈ n,n < m, we have by (vgm-2) we prove the remaining part of the proof. � 4. acknowledgement the authors thank the referees for their careful reading of the manuscript and for their suggestions. references 1. c.d. aliprantis, k.c. border, ”infinite dimensional analysis”, springer-verlag, derlin, 1999. 2. c.d. aliprantis and r. tourky, cones and duality, in: graduate studies in mathematics, amer. math. soc. 84 (2007). 215–240. 106 shrivatava, dubey and tiwari 3. i. beg, m. abbas and t. nazir, generalized cone metric spaces, j. nonlinear sci. appl. 3 (2010), no. 1, 23–31. 4. l. g. huang and x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 322 (2007), 1468–1476. 5. a. meir and e. keeler, a theorem on contraction mappings, j. math. anal. appl. 28 (1969), 326–329. 6. z. mustafa and b. sims, a new approach to generalized metric spaces, j. nonlinear and convex anal. 7 (2006), no. 2, 289–297. 7. z. mustafa, w. shatanawi and m. bataineh, existence of fixed point result in g-metric spaces, , int. j. math. math. sci. 2009(2009), page 10, article id 283028. 8. z. mustafa and b. sims, fixed point theorems for contractive mappings in complete g-metric space, fixed point theory appl. 2009(2009), page 10, article id 917175. 9. z. mustafa, h. obiedat and f. awawdeh, some of fixed point theorem for mapping on complete g-metric spaces, fixed point theory appl., 2008(2008), article id 189870,page 12. 10. w. shatanawi: some fixed point theorems in ordered g-metric spaces and applications, abstr. appl. anal.,(2011), article id 126205, 11 p. 11. w. shatanawi, fixed point theory for contractive mappings satisfying maps in g-metric spaces, fixed point theory appl. 2010(2010), article id 181650, 9 pages. 12. m. abbas, t. nazir and p. vetro: common fixed point results for three maps in g-metric spaces, filomat, 25(2011), 1-17. 13. h. aydi: a fixed point result involving a generalized weakly contractive condition in g-metric spaces, bull. math. anal. appl., 3(2011), no. 4, 180-188. 1department of mathematics, govt. science & commerce college, benazir bhopalindia 2department of mathematics, govt. science p.g. college, reewa -india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 87-99 http://www.etamaths.com quadruple fixed point of multivalued nonlinear contraction mappings animesh gupta1∗, r.n. yadava2, s.s. rajput3 abstract. the notion of quadruple fixed point is introduced by karapinar e. [6]. samet and vetro [12] established some coupled fixed point theorems for multivalued non linear contraction mapping in partially ordered metric spaces. in this paper, we obtain existence of quadrupled fixed point of multivalued non linear contraction mappings in framework work of partially ordered metric spaces. also, we give an example. 1. introduction and preliminary let (x,d) be a metric space. we denote by cb(x) the collection of nonempty closed bounded subsets of x. for a,b ∈ cb(x) and x ∈ x, suppose that d(x,a) = inf a∈a d(x,a) h(a,b) = max{sup a∈a d(a,b), sup b∈b d(b,a)}. such mapping h is called a housdorff metric on cb(x) induced by d. definition 1. an element x ∈ x is said to be a fixed point of a multivalued mapping t : x → cb(x) iff x ∈ tx. in 1969, nadlar [8] extended the famous banach contraction principle from single valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction which state as follows, theorem 2. let (x,d) be a complete metric space and let t be a mapping from x into cb(x). assume that there exists c ∈ [0, 1) such that h(tx,ty) ≤ cd(x,y) for all x,y ∈ x. then t has a fixed point. the existence of fixed points for various multi valued contraction mappings has been studied by many authors under different conditions. in 1989, mizoguchi and takahashi [7] proved the following interesting fixed point theorem for a weak contraction. theorem 3. let (x,d) be a complete metric space and let t be a mapping form x into cb(x). assume that there exists c ∈ [0, 1) such that h(tx,ty) ≤ α(d(x,y))d(x,y) for all x,y ∈ x, where α is a function from [0,∞) into [0, 1), 2010 mathematics subject classification. 47h10, 54h25, 46j10, 46j15. key words and phrases. quadrupled coincidence point, quadrupled fixed point, mixed monotone, mixed gmonotone , nonlinear contraction. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 87 88 gupta,yadava and rajput satisfying the condition lim sups→t+ α(s) < 1 for all t ∈ [0,∞). then t has a fixed point. several authors studies the problem of existence of fixed point of multivalued mappings satisfying different contractive conditions (see e.g., [1, 2, 3, 4, 5, 7, 10, 11]). the theory of multivalued mapping has application in control theory, convex optimization, differential equations, and economics. existence of fixed points in ordered metric spaces has been initiated in 2004 by ran and reurings [10] further studied by nieto and rodriguez lopez [9]. samet and vetro [12] introduced the notion of fixed point of n order in case of single-valued mappings. in particular for n = 4 (quadruple case) i.e., let (x,�) be partially ordered set and (x,d) be a complete metric space. we consider the following partial order on the product space x4 = x ×x ×x ×x: (u,v,r,t) � (x,y,z,w) iff x � u, y � v, z � r, t � w,(1.1) where (u,v,r,t), (x,y,z,w) ∈ x4. regarding this partial order karapinar [6] give the following definitions, definition 4. let (x,�) be partially ordered set and f : x4 → x. we say that f has the mixed monotone property if f(x,y,z,w) is monotone non decreasing in x and z and it is monotone non increasing in y and w, that is, for any x,y,z,w ∈ x x1,x2 ∈ x, x1 � x2 =⇒ f(x1,y,z,w) � f(x2,y,z,w) y1,y2 ∈ x, y1 � y2 =⇒ f(x,y2,z,w) � f(x,y1,z,w) z1,z2 ∈ x, z1 � z2 =⇒ f(x,y,z1,w) � f(x,y,z2,w) w1,w2 ∈ x, w1 � w2 =⇒ f(x,y,z,w2) � f(x,y,z,w1).(1.2) definition 5. an element (x,y,z,w) ∈ x4 is called a quadruple fixed point of f : x4 → x if f(x,y,z,w) = x, f(y,z,w,x) = y, f(z,w,x,y) = z, f(w,x,y,z) = w.(1.3) for a metric space (x,d) the function ρ : x4 → [0,∞), given by ρ((x,y,z,w), (u,v,r,t)) = d(x,u) + d(y,v) + d(z,r) + d(w,t)(1.4) forms a metric space on x4, that is, (x4,ρ) is a metric induced by (x,d). 2. quadruple fixed point result for multivalued mappings first we introduced the following concepts. definition 6. an element (x,y,z,w) ∈ x4 is called a quadruple fixed point of f : x4 → cl(x) if x ∈ f(x,y,z,w), y ∈ f(y,z,w,x), z ∈ f(z,w,x,y), w ∈ f(w,x,y,z)(2.1) definition 7. a mapping f : x4 → r is called lower semi continuous if, for the sequences {xn},{yn},{zn},{wn} in x and (x,y,z,w) ∈ x4, one has quadruple fixed point 89 lim n→∞ ({xn},{yn},{zn},{wn}) = (x,y,z,w) =⇒ f(x,y,z,w) � lim n→∞ inf{{xn},{yn},{zn},{wn}}(2.2) let (x,d) be a metric space endowed with the partial order � and t : x → x. define the set ψ ⊂ x4 by, ψ = {(x,y,z,w) ∈ x4 : t(x) � t(y) � t(z) � t(w)}(2.3) definition 8. a mapping f : x4 → x is said to have a ψproperty if, (x,y,z,w) ∈ ψ =⇒ f(x,y,z,w) ×f(y,z,w,x) ×f(z,w,x,y) ×f(w,x,y,z) ⊂ ψ.(2.4) we give some examples to illustrate definition 8. example 9. let x = r be endowed with the usual order � and t : x → x. define f : x4 → cl(x) by, f(x,y,z,w) = {x}(2.5) obviously f has the ψproperty. example 10. let x = r+ be endowed with the usual order ≤ and t : x → x be defined by tx = exp(x). define f : x4 → cl(x) by, f(x,y,z,w) = {x + w} ∀x,y,z,w ∈ r+(2.6) we have ψ = {(x,y,z,w) ∈ x4,exp(x) � exp(y) � exp(z) � exp(w)}. moreover, f has the ψ− property. now, we prove the following theorem. theorem 11. let (x,d) be a complete metric space endowed with a partial order � and ψ 6= φ that is there exists (x0,y0,z0,w0) ∈ psi. suppose that f : x4 → cl(x) has a ψ− property such that f : x4 → [0,∞) given by for all x,y,z,w ∈ x, f(x,y,z,w) = d(x,f(x,y,z,w)) + d(y,f(y,z,w,y)) + d(z,f(z,w,x,y)) + d(w,f(w,x,y,z))(2.7) is lower semi continuous and there exists a function φ : [0,∞) → [m, 1), 0 < m < 1 satisfying lim r→s+ sup φ(r) < 1 for each s ∈ [0,∞)(2.8) if for any (x,y,z,w) ∈ ψ there exist u ∈ f(x,y,z,w),v ∈ f(y,z,w,x),r ∈ f(z,w,x,y), t ∈ f(w,x,y,z) with √ φ(f(x,y,z,w))[d(x,u) + d(y,v) + d(z,r) + d(w,t)] ≤ f(x,y,z,w)(2.9) such that f(u,v,r,t) ≤ φ(f(x,y,z,w))[d(x,u) + d(y,v) + d(z,r) + d(w,t)](2.10) then f has a quadruple fixed point. 90 gupta,yadava and rajput proof. by our assumption, φ(f(x,y,z,w)) < 1 for each (x,y,z,w) ∈ x4. hence , for any (x,y,z,w) ∈ x4, there exist u ∈ f(x,y,z,w),v ∈ f(y,z,w,x),r ∈ f(z,w,x,y), t ∈ f(w,x,y,z) satisfying √ φ(f(x,y,z,w))d(x,u) � d(x,f(x,y,z,w))√ φ(f(x,y,z,w))d(y,v) � d(y,f(y,z,w,x))√ φ(f(x,y,z,w))d(z,r) � d(z,f(z,w,x,y))√ φ(f(x,y,z,w))d(w,t) � d(w,f(w,x,y,z)).(2.11) let (x0,y0,z0,w0) be an arbitrary point in ψ. from (2.8) and (2.9) we can choose x1 ∈ f(x0,y0,z0,w0),y1 ∈ f(y0,z0,w0,x0),z1 ∈ f(z0,w0,x0,y0),w1 ∈ f(w0,x0,y0,z0) satisfying √ φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)] � f(x0,y0,z0,w0) (2.12) such that f(x1,y1,z1,w1) � φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)] (2.13) by 2.12 and 2.13, we obtain f(x1,y1,z1,w1) � φ(f(x0,y0,z0,w0))[d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)] � √ φ(f(x0,y0,z0,w0)) (φ(f(x0,y0,z0,w0)) [d(x0,x1) + d(y0,y1) + d(z0,z1) + d(w0,w1)]) f(x1,y1,z1,w1) � √ φ(f(x0,y0,z0,w0))f(x0,y0,z0,w0). since f has a ψ− property and (x0,y0,z0,w0) ∈ ψ, so we have f(x0,y0,z0,w0) ×f(y0,z0,w0,x0) ×f(z0,w0,x0,y0) ×f(w0,x0,y0,z0) ⊂ ψ (2.14) which implies that (x1,y1,z1,w1) ∈ ψ. again by 2.9 and 2.10 we can choose , x2 ∈ f(x1,y1,z1,w1),y2 ∈ f(y1,z1,w1,x1),z2 ∈ f(z1,w1,x1,y1),w2 ∈ f(w1,x1,y1,z1) satisfying √ φ(f(x1,y1,z1,w1))[d(x1,x2) + d(y1,y2) + d(z1,z2) + d(w1,w2)] � f(x1,y1,z1,w1) (2.15) such that quadruple fixed point 91 f(x1,y1,z1,w1) � φ(f(x1,y1,z1,w1))[d(x1,x2) + d(y1,y2) + d(z1,z2) + d(w1,w2)] (2.16) thus we have f(x1,y1,z1,w1) � √ φ(f(x1,y1,z1,w1))f(x1,y1,z1,w1) (2.17) which implies that (x2,y2,z2,w2) ∈ ψ. continuing this process, we can choose sequences {xn},{yn},{zn},{wn}, in x such that for each n ∈ n with (xn,yn,zn,wn) ∈ ψ. now xn+1 ∈ f(xn,yn,zn,wn), yn+1 ∈ f(yn,zn,wn,xn), zn+1 ∈ f(zn,wn,xn,yn), wn+1 ∈ f(wn,xn,yn,zn) satisfying √ φ(f(xn,yn,zn,wn))[d(xn,xn+1) + d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)] � f(xn,yn,zn,wn) (2.18) such that f(xn+1,yn+1,zn+1,wn+1) � φ(f(xn,yn,zn,wn)[d(xn,xn+1) +d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)]. (2.19) hence, we obtain f(xn+1,yn+1,zn+1,wn+1) � √ φ(f(xn,yn,zn,wn)f(xn,yn,zn,wn)(2.20) with (xn+1,yn+1,zn+1,wn+1) ∈ ψ.(2.21) we claim that f(xn,yn,zn,wn) → 0 as n →∞. if f(xn,yn,zn,wn) = 0 for some n ∈ n, then d(xn,f(xn,yn,zn,wn)) = 0 implies that xn ∈ f(xn,yn,zn,wn) = f(xn,yn,zn,wn). analogously, d(yn,f(yn,zn,wn,xn)) = 0 implies that yn ∈ f(yn,zn,wn,xn) = f(yn,zn,wn,xn) , d(zn,f(zn,wn,xn,yn)) = 0 implies that zn ∈ f(zn,wn,xn,yn) = f(zn,wn,xn,yn) , 92 gupta,yadava and rajput d(wn,f(wn,xn,yn,zn)) = 0 implies that wn ∈ f(wn,xn,yn,zn) = f(wn,xn,yn,zn). hence (xn,yn,zn,wn) become a quadruple fixed point of f for such n and the result follows. suppose that f(xn,yn,zn,wn) > 0 for all n ∈ n. using 2.20 and φ(t) < 1, we conclude that {f(xn,yn,zn,wn)} is decreasing sequence of positive real numbers. thus, there exists a δ ≥ 0 such that lim n→∞ f(xn,yn,zn,wn) = δ(2.22) we will show that δ = 0. assume on contrary that δ > 0. let n → ∞ in 2.20 and by assumption 2.8 we obtain δ � lim f(xn,yn,zn,wn)→δ+ sup √ φ(f(xn,yn,zn,wn))δ < δ,(2.23) a contradiction, hence lim n→∞ f(xn,yn,zn,wn) = 0 +(2.24) now, we prove that sequences {xn},{yn},{zn},{wn} in x are cauchy sequences in (x,d). assume that α = lim f(xn,yn,zn,wn) → 0+ sup √ φ(f(xn,yn,zn,wn)).(2.25) by 2.8 we conclude that α < 1. let k be a real number such that α < k < 1. thus there exists n0 ∈ n such that √ φ(f(xn,yn,zn,wn)) � k for each n ≥ n0.(2.26) using 2.20 we obtain f(xn+1,yn+1,zn+1,wn+1) � kf(xn,yn,zn,wn) for each n ≥ n0.(2.27) by mathematical induction, f(xn+1,yn+1,zn+1,wn+1) � kn+1−n0f(xn0,yn0,zn0,wn0 ) for each n ≥ n0. (2.28) since φ(t) ≥ m < 0 for all t ≥ 0 so 2.18 and 2.28 gives that [d(xn,xn+1) + d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1)] � kn−n0 √ m (xn0,yn0,zn0,wn0 ) (2.29) for each n ≥ n0, which yields that the sequences {xn},{yn},{zn},{wn} in x are cauchy sequences in (x,d). since x is complete then there exists (a,b,c,d) ∈ x4 such that quadruple fixed point 93 lim n→∞ xn = a, lim n→∞ yn = b, lim n→∞ zn = c, lim n→∞ wn = d.(2.30) finally we show that (a,b,c,d) ∈ x4 is quadruple fixed point of f . as f is lower semi continuous 2.24 implies that 0 � f(a,b,c,d) = d(a,f(a,b,c,d)) + d(b,f(b,c,d,a)) + d(c,f(c,d,a,b)) + d(d,f(d,a,b,c)) � lim n→∞ inf f(xn,yn,zn,wn) = δ.(2.31) hence, d(a,f(a,b,c,d)) = d(b,f(b,c,d,a)) = 0 d(c,f(c,d,a,b)) = d(d,f(d,a,b,c)) = 0 gives that (a,b,c,d) is a quadruple fixed point of f. � theorem 12. let (x,d) be a complete metric space endowed with a partial order � and ψ 6= φ that is there exists (x0,y0,z0,w0) ∈ psi. suppose that f : x4 → cl(x) has a ψ− property such that f : x4 → [0,∞) given by f(x,y,z,w) = d(x,f(x,y,z,w)) + d(y,f(y,z,w,y)) + d(z,f(z,w,x,y)) + d(w,f(w,x,y,z))(2.32) for all x,y,z,w ∈ x and f is lower semi continuous and there exists a function φ : [0,∞) → [m, 1), 0 < m < 1, satisfying lim r→s+ sup φ(r) < 1 for each s ∈ [0,∞)(2.33) if for any (x,y,z,w) ∈ ψ there exist u ∈ f(x,y,z,w),v ∈ f(y,z,w,x),r ∈ f(z,w,x,y), t ∈ f(w,x,y,z) with √ φ(∆)∆ � d(x,f(x,y,z,w)) + d(y,f(y,z,w,y)) + d(z,f(z,w,x,y)) + d(w,f(w,x,y,z))(2.34) such that d(u,f(u,v,r,t)) + d(v,f(v,r,t,u)) + d(r,f(r,t,u,v)) + d(t,f(t,u,v,r))) � φ(∆)∆ (2.35) where ∆ = ∆((x,y,z,w), (u,v,r,t)) = [d(x,u) + d(y,v) + d(z,r) + d(w,t)] then f has a quadruple fixed point. 94 gupta,yadava and rajput proof. by replacing φ(f(x,y,z,w)) with [d(x,u) + d(y,v) + d(z,r) + d(w,t)] in the proof of theorem 11 we obtain sequences {xn},{yn},{zn},{wn}, in x such that for each n ∈ n with, (xn,yn,zn,wn) ∈ ψ xn+1 ∈ f(xn,yn,zn,wn), yn+1 ∈ f(yn,zn,wn,xn) zn+1 ∈ f(zn,wn,xn,yn), wn+1 ∈ f(wn,xn,yn,zn)(2.36) such that √ φ(∆n)∆n � d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn))(2.37) d(xn+1,f(xn+1,yn+1,zn+1,wn+1)) + d(yn+1,f(yn+1,zn+1,wn+1,yn+1)) + d(zn+1,f(zn+1,wn+1,xn+1,yn+1)) + d(wn+1,f(wn+1,xn+1,yn+1,zn+1)) � √ φ(∆n)(d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn))). (2.38) where ∆n = ∆((xn,yn,zn,wn)(xn+1,yn+1,zn+1,wn+1))(2.39) = d(xn,xn+1) + d(yn,yn+1) + d(zn,zn+1) + d(wn,wn+1).(2.40) again following arguments similar to those given in proof of theorem 11 we deduce that d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)).(2.41) is a decreasing sequence of real numbers. thus, there exists a δ > 0 such that lim n→∞ (d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn))) = δ.(2.42) now we need to proof that {∆n} admits a subsequence converging to certain η+ for some η ≥ 0. since φ(t) ≤ m > 0, using 2.37 we obtain δn � 1 √ a (d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)))(2.43) from 2.42 and 2.43 it is clear that the sequence quadruple fixed point 95 (d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)))(2.44) is bounded. therefore, there is some θ ≥ 0 such that lim n→∞ inf ∆n = θ(2.45) from 2.36 we have xn+1 ∈ f(xn,yn,zn,wn),yn+1 ∈ f(yn,zn,wn,xn), zn+1 ∈ f(zn,wn,xn,yn),wn+1 ∈ f(wn,xn,yn,zn), ∆n � d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)).(2.46) by comparing 2.42 to 2.45 we get that θ ≥ δ. now, we shall show that θ = δ. if δ = 0, by 2.42 and 2.43 we get θ = lim infn→∞+ ∆n = 0 and consequently θ = δ = 0. suppose that δ > 0. assume on contrary that θ > δ. from 2.42 and 2.45 there is a positive integer n0 such that d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)) � δ + θ − δ 4 (2.47) δ − θ −δ 4 � ∆n(2.48) for all n ≥ n0. we combine 2.37, 2.47 and 2.48 to obtain√ φ((∆n) ( δ − θ −δ 4 ) � √ φ((∆n)∆n � d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)) � δ + θ −δ 4 (2.49) for all n ≥ n0. it follows that √ φ((∆n) � θ + 3δ 3θ + δ ∀n ≥ n0.(2.50) by 2.38 and 2.50 we have 96 gupta,yadava and rajput d(xn+1,f(xn+1,yn+1,zn+1,wn+1)) + d(yn+1,f(yn+1,zn+1,wn+1,yn+1)) + d(zn+1,f(zn+1,wn+1,xn+1,yn+1)) + d(wn+1,f(wn+1,xn+1,yn+1,zn+1)) ≤ hd(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn)). (2.51) where h = θ+3δ 3θ+δ . since θ > δ > 0, therefore h < 1, so proceeding by induction and combining the above inequalities, it follows that δ � d(xn0+k0,f(xn0+k0,yn0+k0,zn0+k0,wn0+k0 )) + d(yn0+k0,f(yn0+k0,zn0+k0,wn0+k0,yn0+k0 )) + d(zn0+k0,f(zn0+k0,wn0+k0,xn0+k0,yn0+k0 )) + d(wn0+k0,f(wn0+k0,xn0+k0,yn0+k0,zn0+k0 )) � hk0 [d(xn0,f(xn0,yn0,zn0,wn0 )) + d(yn0,f(yn0,zn0,wn0,yn0 )) + d(zn0,f(zn0,wn0,xn0,yn0 )) + d(wn0,f(wn0,xn0,yn0,zn0 ))]δ. (2.52) for a positive integer k0. then, we obtain a contradiction, so we must have θ = δ. now, we shall show that θ = 0. since θ = δ � d(xn,f(xn,yn,zn,wn)) + d(yn,f(yn,zn,wn,yn)) + d(zn,f(zn,wn,xn,yn)) + d(wn,f(wn,xn,yn,zn))∆n(2.53) then we rewrite 2.45 as (2.54) lim n→∞+ inf ∆n = θ +. hence, there exists a subsequence {∆nk} of {∆n} such that limk→∞+inf∆nk = θ+. by 2.33 we have lim ∆nk→∞ + sup √ φ(∆nk ) < 1.(2.55) from 2.38 we obtain quadruple fixed point 97 d(xnk+1,f(xnk+1,ynk+1,znk+1,wnk+1)) + d(ynk+1,f(ynk+1,znk+1,wnk+1,ynk+1)) +d(znk+1,f(znk+1,wnk+1,xnk+1,ynk+1)) + d(wnk+1,f(wnk+1,xnk+1,ynk+1,znk+1)) � √ φ(∆nk )[d(xnk,f(xnk,ynk,znk,wnk )) + d(ynk,f(ynk,znk,wnk,ynk )) + d(znk,f(znk,wnk,xnk,ynk )) + d(wnk,f(wnk,xnk,ynk,znk ))]. (2.56) taking the limit as k →∞ and using 2.42 we have δ = lim k→∞+ {sup[d(xnk+1,f(xnk+1,ynk+1,znk+1,wnk+1)) + d(ynk+1,f(ynk+1,znk+1,wnk+1,ynk+1)) + d(znk+1,f(znk+1,wnk+1,xnk+1,ynk+1)) + d(wnk+1,f(wnk+1,xnk+1,ynk+1,znk+1))]} � lim k→∞+ sup [√ φ(∆nk ) ] (2.57) lim k→∞+ {sup[d(xnk,f(xnk,ynk,znk,wnk )) + d(ynk,f(ynk,znk,wnk,ynk )) + d(znk,f(znk,wnk,xnk,ynk )) + d(wnk,f(wnk,xnk,ynk,znk ))] � ( lim k→∞+ sup √ φ(∆nk ) ) δ.(2.58) assume that δ > 0, then from 2.57 we get that 1 � lim k→∞+ sup √ φ(∆nk )(2.59) a contradiction with respect to 2.55 so δ = 0. now, from 2.38 and 2.42 we have α = lim ∆n→0 sup √ φ(∆n) < 1(2.60) the rest of the proof is similar to the proof of the theorem 11 so it is omitted. � we improve and corrected the example of samet and vetro [12]. 3. examples example 13. let x = [0, 2], and let d : x × x → [0,∞) be the usual metric. suppose that t(x) = m for all x ∈ [0, 2] where m is a constant in [0,2], and f : x4 → cl(x) is defined for all x,y,z,w ∈ x as follows f(x,y,z,w) = { x2 4 if x ∈ [ 0, 15 32 ) ∪ ( 15 32 , 2 ] {15 96 , 1 5 } if x = 15 32 98 gupta,yadava and rajput oviously, f has the ψproperty. set φ : [0,∞) → [0,∞) φ(s) = { 11 12 s if s ∈ [ 0, 2 3 ] 10 16 if s ∈ ( 2 3 ,∞ ) consider a function f(x,y,z,w) =   a if x,y,z,w ∈ [ 0, 15 32 ) ∪ ( 15 32 , 2 ] b if x,y,z ∈ [ 0, 15 32 ) ∪ ( 15 32 , 2 ] with w = 15 32 c if x,y ∈ [ 0, 15 32 ) ∪ ( 15 32 , 2 ] with z = w = 15 32 d if x ∈ [ 0, 15 32 ) ∪ ( 5 32 , 2 ] with y = z = w = 15 32 e if x = y = z = w = 15 32 where, a = x+y+z+w−1 4 (x2 +y2 +z2 +w2), b = x+y+z−1 4 (x2 +y2 +z2)+ 43 160 , c = x + y − 1 4 (x2 + y2) + 86 160 , d = x− 1 4 (x2) + 129 160 , e = 172 160 which is lower semicontinuous. thus for all x,y,z,w ∈ x with x,y,z,w 6= 5 32 , there exists u ∈ f(x,y,z,w) = x 2 4 , v ∈ f(y,z,w,x) = y 2 4 , r ∈ f(z,w,x,y) = z 2 4 , t ∈ f(w,x,y,z) = w 2 4 such that d(u,f(u,v,r,t)) + d(v,f(v,r,t,u)) + d(r,f(r,t,u,v)) + d(t,f(t,u,v,r)) = x2 4 − x4 16 + y2 4 − y4 16 + z2 4 − z4 16 + w2 4 − w4 16 = 1 4 [( x + x2 4 )( x− x2 4 ) + ( y + y2 4 )( y − y2 4 ) + ( z + z2 4 )( z − z2 4 ) + ( w + w2 4 )( w − w2 4 )] ≤ 1 4 [( x + x2 4 ) d(x,u) + ( y + y2 4 ) d(y,v) + ( z + z2 4 ) d(z,r) + ( w + w2 4 ) d(w,t) ] � 1 4 max{ ( x + x2 4 ) , ( y + y2 4 ) , ( z + z2 4 ) , ( w + w2 4 ) } d(x,u) + d(y,v) + d(z,r) + d(w,t) � 10 12 max{ ( x− x2 4 ) , ( y − y2 4 ) , ( z − z2 4 ) , ( w − w2 4 ) } d(x,u) + d(y,v) + d(z,r) + d(w,t) � φ(d(x,u) + d(y,v) + d(z,r) + d(w,t))[d(x,u) + d(y,v) + d(z,r) + d(w,t)] hence for all x,y,z,w ∈ x with x,y,z,w 6= 15 32 , the conditions 2.9 and 2.10 are satisfied. analogously, one can easy show that conditions 2.9 and 2.10 are satisfied for the cases x,y,z ∈ [ 0, 15 32 ) ∪ ( 15 32 , 2 ] with w = 15 32 and x ∈ [ 0, 15 32 ) ∪( 15 32 , 2 ] with y = z = w = 15 32 . for the last case, that is x = y = z = w = 15 36 , we assume that u = v = r = t = 15 96 , it follows that, [d(x,u) + d(y,v) + d(z,r) + d(w,t)] = 5 4 > 2 3 quadruple fixed point 99 as a consequence, we conclude that all the conditions of theorem 2.7 are satisfied and f admits a quadruple fixed point i.e. (0, 0, 0, 0). 4. acknowledgement the authors thank the referees for their careful reading of the manuscript and for their suggestions. references [1] beg, i, butt, ar: coupled fixed points of set valued mappings in partially ordered metric spaces. j nonlinear sci appl. 3, 179-185 (2010) [2] ciric, ljb: multi-valued nonlinear contraction mappings. nonlinear anal. 71, 2716-2723 (2009). doi:10.1016/j. na.2009.01.116 [3] ciric, ljb: fixed point theorems for multi-valued contractions in complete metric spaces. j math anal appl. 348, 499-507 (2008). doi:10.1016/j.jmaa.2008.07.062 [4] du, ws: coupled fixed point theorems for nonlinear contractions satisfied mizoguchitakahashis condition in quasiordered metric spaces. fixed point theory appl. 9 (2010). 2010, article id 876372 [5] hussain, n, shah, mh, kutbi, ma: coupled coincidence point theorems for nonlinear contractions in partially ordered quasi-metric spaces with a q-function. fixed point theory appl. 21 (2011). 2011, article id 703938 [6] karapiner e., quadruple fixed point theorems for weak φcontraction, isrn math. anal. (2011), id 989423, 15 pages, doi:10.5402/2011/989423. [7] mizoguchi, n, takahashi, w: fixed point theorems for multivalued mappings on complete metric spaces. j math anal appl. 141, 177-188 (1989). doi:10.1016/0022-247x(89)90214-x [8] nadler, sb: multivalued contraction mappings. pacific j math. 30, 475-488 (1969) [9] nieto, jj, rodriguez-lopez, r: contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order. 22, 223-239 (2005) [10] nieto, jj, rodriguez-lopez, r: existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. acta math sin (engl ser). 23, 22052212 (2007). doi:10.1007/s10114-005-0769-0 [11] nieto, jj, pouso, rl, rodriguez-lopez, r: fixed point theorems in ordered abstract spaces. proc am math soc. 135, 2505-2517 (2007). doi:10.1090/s0002-9939-07-08729-1 [12] samet, b, vetro, c: coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces. nonlinear anal. 74, 4260-4268 (2011). doi:10.1016/j.na.2011.04.007 1department of applied mathematics, vidhyapeeth institute of science & technology, near sos balgram, bhopal (m.p.) india 2patel group of institutions, ratibad, bhopal (m.p.), india 3department of mathematics, govt. p.g. collage, gadarwara dist. narsingpur (m.p.), india ∗corresponding author int. j. anal. appl. (2023), 21:51 received: apr. 24, 2023. 2020 mathematics subject classification. 05b30, 60g55, 62k99. key words and phrases. design of experiments; computer experiment designs; point process; continuum random cluster model; connected component process; markov point processes; nearest-neighbour markov point process; strauss process; markov chain monte carlo (mcmc). https://doi.org/10.28924/2291-8639-21-2023-51 © 2023 the author(s) issn: 2291-8639 1 new computer experiment designs using continuum random cluster point process hichem elmossaoui*, nadia oukid department of mathematics, faculty of science, university saad daheb, blida, algeria *correspondence: elmossaoui.hichem@yahoo.com abstract. in this paper, we propose a new approach for building computer experiment designs using the continuum random cluster point process, also referred to as the connected component markov point process. our method involves generating designs through the markov chain monte carlo method (mcmc) and the random walk metropolis hastings algorithm (rwmh algorithm), which can be easily scaled to meet various objectives. we have conducted a comprehensive study on the convergence of the markov chain and compared our approach with existing computer experiment designs. overall, our approach offers a novel and flexible solution for constructing computer experiment designs. 1. introduction the development of modelling techniques, boosted by increased computing power, has led to the development of simulators with challenging complexity. the main difficulties stem from the high computing cost of the simulator, and from the size of the problem to deal with. it may become impossible to consider the direct use of the simulator with some applications, even after significantly reducing the size of the system. hence, the alternative option would be to use one or more functions instead of the simulator. these functions are generally relatively simple, and can be obtained by means of approximation or interpolation from computer experiment designs. in order to achieve a more thorough exploration of the parameter space and obtain information throughout the entire experimental area, we propose a method for constructing computer https://doi.org/10.28924/2291-8639-21-2023-51 mailto:elmossaoui.hichem@yahoo.com 2 int. j. anal. appl. (2023), 21:51 experiment designs with points uniformly distributed in the unit hypercube. to accomplish this, we employ the continuum random cluster process (crcp) ([1], [2], [3], [4]) to simulate the 𝑛 computer experiments, which make up the computer experiment designs. although the process is not markovian under ripley-kelly [5], it is markovian with respect to a new neighborhood relationship that depends on the configuration 𝑥 (as defined in definition 2.1) and the relation ∼ 𝑥 (as defined in definition 2.3) for nearest neighbours. franco (2008) [6] introduced computer experiment designs based on strauss point processes that incorporate the concept of interaction between pairs of points. elmossaoui et al (2020) ([7], [8]) proposed computer experiment designs using marked strauss point processes that can achieve multiple objectives simultaneously. the first objective is related to the distribution of points, while the second objective concerns the specification of the marks of those points. in contrast to the aforementioned approaches, crcps are an alternative that allows for the creation of models with more or less regular spatial distributions without constraints on parameters. to generate the designs proposed in this work, we will use simulation techniques via mcmc methods and metropolis-hastings ([9], [10]) algorithm. there are several sub-types of metropolis-hastings algorithms, depending on the chosen transition density 𝑞. in this regard, we will develop the algorithm to generate a markov chain of the random walk type, where the density 𝑞 is centred on the current value of the chain and is symmetrical. the proposed value 𝑥𝑡+1 takes the form 𝑥𝑡+1 + , where ε is a random perturbation that follows the 𝑞 density distribution and is independent of 𝑥𝑡. this paper is organised as follows: section 2 is about some general definitions and notations. section 3 concerns the construction of new computer experiment designs based on the use of crcp by means of mcmc method and metropolis-hastings algorithm. section 4 is about the study of the convergence of the algorithm. finally, in section 5 we have compared our results with the existing ones. 2. preliminaries and general definitions let (ω, ℬ, p) be a probability space that models the random aspects of the experiments. let 𝜒 be a nonempty set equipped with a euclidean distance 𝑑, making it a complete and separable metric space. in most cases, 𝜒 will be equal to [0,1]𝑝 (a subset of ℝ𝑝), where 𝑝 is the number of continuous factors of interest (𝑝 ≥ 1). we will use μ to denote the lebesgue measure associated with this space, considered with its borel σ-algebra ℬ. 3 int. j. anal. appl. (2023), 21:51 definition 2.1. a completion 𝑥 of a point process 𝑋 on 𝜒 is defined as any locally finite collection of points from 𝜒, 𝑥 = ( 𝑥1, 𝑥2, … , 𝑥𝑛 ), with 𝑥𝑖 ∈ (𝜒, 𝑑) and 𝑖 ∈ ℕ. in other words, it is a part 𝑥 ⊂ 𝜒 , so that 𝑥 ∩ b is finite for every part b of ℬ that is borelean and limited. let 𝑁𝑙𝑓 denote the set of the locally finite point configurations, x, y … are configurations on 𝜒 , and 𝑥𝑖 , 𝑦𝑖 … points from these configurations. definition 2.2. a point process on 𝜒 is an application 𝑋 of a probability space (ω, ℬ, p) within the set of the locally finite point configurations 𝑁𝑙𝑓, so that for every limited borelean b, the number of points 𝑁(b) = 𝑁𝑋 (b) of points of 𝑋 falling on b is a finite random discrete variable. in this definition, 𝜒 can be replaced with a general complete metric space. however, it is important to note that the implementation of a point process is, at most, countable and without accumulation points. if 𝜒 is bounded, then 𝑁𝑋 (𝜒) is almost surely finite and the point process is said to be finite. we shall consider here only simple point processes that do not allow the repetition of points, in which case the realization 𝑥 of the point process coincides with a subset of 𝜒. definition 2.3. let ∼ be a binary relation that is symmetrical and reflexive on χ. two points 𝑥𝑖 and 𝑥𝑖′ are said to be neighbours if 𝑥𝑖 ∼ 𝑥𝑖′. the neighbourhood of 𝑦 ⊂ 𝑥 is given by: ∂(𝑦|𝑥) = {𝑥𝑖 ∈ 𝑥 𝑎𝑛𝑑 𝑥𝑖 ∉ 𝑦 ∶ ∃ 𝑥𝑖′ ∈ 𝑦 𝑠𝑜 𝑡ℎ𝑎𝑡 𝑥𝑖′ ∼ 𝑥𝑖 } for example, 𝑥𝑖 ∼ 𝑥𝑖′ if 𝑑(𝑥𝑖 , 𝑥𝑖′ ) ≤ 𝑟 is the r-neighbourhood relation on (𝜒, 𝑑), and 𝑟 > 0 being a fixed radius. definition 2.4. let 𝐵(𝑥) = {⋃ 𝐵(𝑥𝑖 , 𝑟 2 )𝑥𝑖∈𝑥 } ∩ 𝜒 be the reunion of balls centered at points 𝑥𝑖 of 𝑥, of radius 𝑟 2 , and limited at 𝜒. two points 𝑥𝑖 and 𝑥𝑖′ of 𝑥 are said to be connected for 𝑥 if 𝑥𝑖 and 𝑥𝑖′ are in the same connected component of 𝐵(𝑥). such relation shall be noted 𝑥𝑖 ~𝑥 𝑥𝑖′ . definition 2.5. let 𝑋 be a point process of density 𝜋 in relation to a poisson point process of law 𝜋𝜈 (∙) and intensity 𝜈(∙). the process 𝑋 is the nearest-neighbour markov point processes with regard to the relation ~𝑥 if, for every configuration 𝑥 ∈ 𝑁 𝑙𝑓, where 𝜋(𝑥) > 0, then we have what follows: (i) 𝜋(𝑦) > 0 for every 𝑦 ⊂ 𝑥 . (ii) ∀ 𝑥𝑖 ∈ 𝜒, then : 𝜆(𝑥𝑖 , 𝑥) = 𝑓((𝑥∪{𝑥𝑖 }) 𝑓(𝑥) depends only on 𝑥𝑖, ∂({𝑥𝑖 }|𝑥⋃{𝑥𝑖 }) ∩ 𝑥, and the two relations ~𝑥 and ~𝑥⋃{𝑥𝑖} . the quotient 𝜆(𝑥𝑖 , 𝑥) being the papangelou conditional density. 4 int. j. anal. appl. (2023), 21:51 3. computer experiment design using nearest-neighbour markov point process identically to the way we proceeded in ([7], [8]), we consider each experiment to be a point or particle defined on the interval [0,1]𝑝 , and each configuration 𝑥 as an experiment design. we thus simulate 𝑛 experiments to implement a continuum random cluster process. it is worth noting that the continuum random cluster process has interaction potentials. these interactions are defined by the neighbourhood properties as given by a markov chain on the nearest neighbours [3]. the interaction potential used is the connectedness interaction. these object processes are important in modelling repulsive phenomena. the probability density of the process is given as: 𝜋(𝑥) = 𝑘 𝛽𝑛(𝑥)𝛾ℎ(𝑥) where k is a positive normalization constant which makes 𝜋 a density, 𝛽 a positive scaling parameter, 𝑛(𝑥) denotes the number of points of the configuration 𝑥, 𝛾 is a repulsion parameter such as 0 < 𝛾 < 1, and ℎ(𝑥) = −𝑎(𝑥), where 𝑎(𝑥) refers to the area of the reunion of balls 𝐵(𝑥), or ℎ(𝑥) = −𝑐(𝑥), with 𝑐(𝑥) referring to the number of connected components of 𝐵(𝑥). through this article, we aim to study the connected component process with a probability density: 𝜋(𝑥) = 𝑘 𝛽𝑛(𝑥)𝛾−𝑐(𝑥) we note that the connected component process is not a markov process under ripley and kelly [5]. actually, two points of 𝜒 can be neighbours with the relation ∼ 𝑥 while being arbitrarily far from one another in 𝜒 with regard to the euclidean distance. connectedness through connected components shall link two points in case there exists a string of balls of radius 𝑟, centred at points 𝑥𝑖 of 𝑥, and joining with one another. the papangelou conditional intensity function of the process with connectedness interaction is given as follows [11]: 𝜆(𝑥𝑖 , 𝑥) = 𝛽𝛾 𝑐(𝑥)−𝑐(𝑥∪{𝑥𝑖}) nevertheless, 𝜆(𝑥𝑖 , 𝑥) can depend on a point 𝑥𝑖′ ∈ 𝑥 arbitrarily far from 𝑥𝑖 in 𝜒, and we can select a configuration 𝑦 so that, for every 𝑥 = 𝑦 ∪ {𝑥𝑖′ }, 𝑐(𝑦) = 2 and 𝑐(𝑥) = 𝑐(𝑥 ∪ {𝑥𝑖 }) = 1. hence, for every r > 0, the process is not markovian under ripley-kelly for the usual r-neighbourhood relation. however, if 𝑥𝑖′ ∈ 𝑥 is not connected to 𝑥𝑖 in 𝐵(𝑥 ∪ {𝑥𝑖 }), then 𝑥𝑖′ will not contribute to (3.1) (3.2) 5 int. j. anal. appl. (2023), 21:51 the difference 𝑐(𝑥) − 𝑐(𝑥 ∪ {𝑥𝑖 }), and 𝜆(𝑥𝑖 , 𝑥) will not depend on 𝑥𝑖′ , in this case the process is nearest-neighbour markov point process for the nearest neighbour ~𝑥 relation. 3.1. generating computer experiment designs via the connected component processes using mcmc method and rwmh algorithm mcmc method allows the simulation of a 𝜋 density using an ergodic markov chain {𝑋0, 𝑋1, … , 𝑋𝑁𝑀𝐶𝑀𝐶 }; the former being a stationary distribution. to establish such a chain, we use rwmh algorithm. the basic idea of the algorithm is to build a transition 𝑃𝑀𝐻 , which is 𝜋reversible. 𝑃𝑀𝐻 shall, then, be 𝜋-invariant. this shall be done in two steps: • change proposition transition: we start by proposing a change x → y according to a density 𝑞(𝑥,∙). • change acceptance probability: change is, then, accepted with the probability 𝑎(𝑥, 𝑦), a : ω × ω → [0,1]. rwmh algorithm includes an instrumental markov chain the transition density of which depends on the current state; more precisely it satisfies the equation: 𝑞(𝑥, 𝑦) = 𝑞(𝑦 − 𝑥). in addition, we can choose an instrumental density 𝑁(𝑥, 𝜎2). so, given a current state 𝑥, the algorithm generates a potential state 𝑦 stemming from a normal distribution centered at x and having 𝜎2 as a variance. if the new state is accepted the next potential state shall be generated according to a distribution 𝑁(𝑦, 𝜎2). else, 𝑥 shall be maintained as current state and another possible state y shall be proposed according to a distribution 𝑁(𝑥, 𝜎2). the algorithm, as defined, generates a markov chain the transition probabilities of which are given as follows [12]: 𝑃𝑀𝐻 (𝑥, 𝑦) = 𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦) + 1(𝑥 = 𝑦) [1 − ∫ 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝑃𝑀𝐻 is 𝜋-reversibility if and only if the balance equation below is satisfied: ∀𝑥, 𝑦 ∈ 𝛺 ∶ 𝜋(𝑥) × 𝑞(𝑥, 𝑦) × 𝑎(𝑥, 𝑦) = 𝜋(𝑦) × 𝑞(𝑦, 𝑥) × 𝑎(𝑦, 𝑥) for a couple (𝑥, 𝑦), let’s define metropolis-hastings acceptance probability as : 𝑎(𝑥, 𝑦) = 𝜋(𝑦) × 𝑞(𝑦, 𝑥) 𝜋(𝑥) × 𝑞(𝑥, 𝑦) the rwmh algorithm with the distribution 𝑁(𝑥, 𝜎2) has an additional property conferred by the symmetry of the instrumental density, mainly due to the fact that 𝑞(𝑥, 𝑦) = 𝑞(𝑦, 𝑥). this characteristic simplifies the calculation of a: 6 int. j. anal. appl. (2023), 21:51 𝑎(𝑥, 𝑦) = 𝜋(𝑦) 𝜋(𝑥) = 𝑘𝛽𝑛(𝑦)𝛾−𝑐(𝑦) 𝑘𝛽𝑛(𝑥)𝛾−𝑐(𝑥) = 𝛾𝑐(𝑥)−𝑐(𝑦) 3.2. algorithm for the construction of proposed experiment designs the computer experiment designs proposed in this work, referred to as connected component designs (ccd), are generated through the algorithm presented below, which is a variation of the random walk metropolis hastings algorithm. algorithm • initialisation – choose an initial experiment design 𝑥 = ( 𝑥1, 𝑥2, … , 𝑥𝑛 ) according to a given probability distribution. for instance the normal probability distribution. – take 𝑋0 = 𝑥. • for 𝑁 = 1, 2, . . . , 𝑁𝑀𝐶𝑀𝐶 for 𝑘 =, …, 𝑛 – randomly choose a spin s uniformly on {1, … , 𝑛} – simulate an experiment 𝑦𝑗 according to a proposed distribution q~𝑁(𝑥𝑠, 𝜎 2). thus, we take as a new configuration: 𝑦 = ( 𝑥1, 𝑥2, … 𝑥𝑠−1, 𝑦𝑗 , 𝑥𝑠+1 … , 𝑥𝑛 ) – calculate the acceptance probability: 𝑎(𝑥, 𝑦) = 𝑀𝑖𝑛 (1, 𝛾𝑐(𝑥)−𝑐(𝑦)) – take 𝑥 = { 𝑦 with a probability 𝑎 𝑥 with a probability 1 − 𝑎 end for k take 𝑋𝑁 = 𝑥. end for 𝑁 for n=1000, figure 1 shows the convergence towards a configuration that satisfies the connected component property starting from an initial configuration of 30 points, drawn from a normal distribution with mean 0 and variance 1. 7 int. j. anal. appl. (2023), 21:51 figure 1. left, an initial configuration of 30 points with c(x)=6. right, a final configuration with c(x)=30 ( γ =0,01, r=0,19 and 𝜎 = 0,05 ). 3.3. influence of parameters the figure below shows the influence of the parameter 𝑟 on the final distribution. the choice of the radius proves to be important. a radius that is too small generates a distribution without interaction, but with numerous deficiencies. however, a radius that is too large leads to a distribution with clusters. figure 2. left, a configuration of 30 points γ =0.01 and r =0.1. right, a configuration for γ=0.01 and r =0.3 8 int. j. anal. appl. (2023), 21:51 the same as with the interaction radius, it is important to adequately set the attraction parameter γ. the figure below shows that it is easier to generate a distribution that meets the criteria of filling the space with a strong attraction parameter. figure 3. left, a configuration of 30 points γ =0.01 and r =0.19. right, a configuration for γ =0.1 and r =0.19 4. study of convergence in this section, we shall prove the convergence of the sequence of computer experiment designs {𝑋0, 𝑋1, … , 𝑋𝑁𝑀𝐶𝑀𝐶 } generated with the construction algorithm previously introduced towards the invariant distribution π of a connected component process. this sequence is the realization of a markov chain with transition kernel: 𝑃(𝑥, 𝑦) = 𝑃𝑀𝐻 𝑛 (𝑥, 𝑦) moreover, it is important to know whether the distribution of the last generated design 𝑋𝑁𝑀𝐶𝑀𝐶 is close to the distribution 𝜋. to this end, let us state the main results that are of interest to us here: proposition 4.1. on a finite space, the transition kernel 𝑃 = 𝑃𝑀𝐻 𝑛 of the markov chain (𝑋𝑁 )𝑁 ≥ 0 obtained from the construction algorithm is irreducible and positive recurrent. the distribution 𝜋 is the unique stationary distribution of 𝑃; 𝑃 being aperiodic and a primitive kernel. proof of proposition 4.1. let us prove that the transition mechanism 𝑃𝑀𝐻 meets the three following conditions related to the distribution 𝜋 of the connected component process, defined in (3.2): 𝜋 − reversibility, 𝜋 −stationary and 𝜋 −irreductibility. 9 int. j. anal. appl. (2023), 21:51 the chain is said to be reversible with respect to the target distribution 𝜋(∙) if its transition kernel satisfies: ∀𝑥, 𝑦 ∈ 𝛺, b ∈ ℬ: ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑃𝑀𝐻 (𝑥, 𝑦)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑦)𝑃𝑀𝐻 (𝑦, 𝑥)𝑑𝑦 𝛺 let 𝑥, 𝑦 ∈ ω and b ∈ ℬ. then we have: ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑃𝑀𝐻 (𝑥, 𝑦)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦)𝑑𝑥 𝛺 + ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥) 𝛺 [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝛿𝑥 (𝑦)𝑑𝑥 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦)𝑑𝑥 𝛺 + ∫ 1𝐵(𝑥,𝑥)𝜋(𝑥) 𝛺 [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝑑𝑥 by construction 𝑎 and 𝑞 satisfy, 𝜋(𝑥)𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦) = 𝑘𝛽𝑛(𝑥)𝛾−𝑐(𝑥)𝑚𝑖𝑛 {1, 𝛾𝑐(𝑥)−𝑐(𝑦)}𝑞(𝑥, 𝑦) = 𝑘 𝑚𝑖𝑛 {𝛽𝑛(𝑥)𝛾−𝑐(𝑥), 𝛽𝑛(𝑥)𝛾−𝑐(𝑦)}𝑞(𝑥, 𝑦) and since 𝑛(𝑥) = 𝑛(𝑦) and 𝑞(𝑥, 𝑦) = 𝑞(𝑦, 𝑥). then we have: 𝜋(𝑥)𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦) = 𝑘 min{𝛽𝑛(𝑥)𝛾−𝑐(𝑥), 𝛽𝑛(𝑦)𝛾−𝑐(𝑦)} 𝑞(𝑦, 𝑥) =𝑘𝛽𝑛(𝑦)𝛾−𝑐(𝑦)𝑚𝑖𝑛 {𝛾𝑐(𝑦)−𝑐(𝑥), 1}𝑞(𝑦, 𝑥) = 𝜋(𝑦)𝑎(𝑦, 𝑥)𝑞(𝑦, 𝑥). finally, ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑃𝑀𝐻 (𝑥, 𝑦)𝑑𝑥𝛺 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑦)𝑎(𝑦, 𝑥)𝑞(𝑦, 𝑥)𝑑𝑥𝛺 + ∫ 1𝐵(𝑦,𝑦)𝜋(𝑦) 𝛺 [∫ 1 − 𝑎(𝑦, 𝑧)𝑞(𝑦, 𝑧)𝑑𝑧 𝛺 ] 𝑑𝑦 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑦)𝑃𝑀𝐻 (𝑦, 𝑥)𝑑𝑦. 𝛺 this is the condition of 𝜋 − reversibility of the transition mechanism 𝑃𝑀𝐻. a measure 𝜋(∙) is said to be stationary for the transition kernel 𝑃𝑀𝐻 if: ∀𝑥, 𝑦 ∈ 𝛺; b, a ∈ ℬ: ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑃𝑀𝐻 (𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑑𝑥 𝛺 let 𝑥 ∈ 𝛺, and b ∈ ℬ. then for every borelean a of ℬ we have: 10 int. j. anal. appl. (2023), 21:51 ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑃𝑀𝐻 (𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥) [∫ 𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦)𝑑𝑦 𝛺 ] 𝑑𝑥 𝛺 + ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥) 𝛺 [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝛿𝑥 (𝑦)𝑑 = ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥) [∫ 𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦)𝑑𝑦 𝛺 ] 𝑑𝑥 𝛺 + ∫ 1𝐵(𝑥,𝑥)𝜋(𝑥) 𝛺 [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝑑𝑥 = ∫ ∫ 1𝐵(𝑥,𝑦)𝜋(𝑥)𝑎(𝑥, 𝑦)𝑞(𝑥, 𝑦)𝑑𝑦𝑑𝑥 𝛺𝛺 + ∫ 1𝐵(𝑥,𝑥)𝜋(𝑥)𝑑𝑥 𝛺 − ∫ ∫ 𝜋(𝑥)𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧) 𝛺 𝑑𝑧𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,𝑥)𝜋(𝑥)𝑑𝑥𝛺 . consequently, the chain assumes 𝜋 as a stationary distribution. a measure 𝜋(∙) is said to be irreducible for the transition kernel 𝑃𝑀𝐻 of a markov chain if: ∀a ∈ ℬ so that 𝜋(a) > 0 ⇒ ∃𝑡 : 𝑃𝑀𝐻 𝑡 (𝑥, a) > 0 let a be a borelean of ℬ, and for t=1 we have: ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻(𝑥, a)𝑑𝑥 ω = ∫ 1𝐵(𝑥,a)𝑎(𝑥, a)𝑞(𝑥, a)𝑑𝑥 𝛺 + ∫ 1𝐵(𝑥,a) 𝛺 [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺 ] 𝛿𝑥 (𝐴)𝑑𝑥 = ∫ 1𝐵(𝑥,a)𝑎(𝑥, a)𝑞(𝑥, a)dx ω + ∫ 1b(𝑥,𝑥) ω [∫ 1 − 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 ω ] 𝑑𝑥 = ∫ 1𝐵(𝑥,𝐴)𝑎(𝑥, a)𝑞(𝑥, a)𝑑𝑥 ω + 1 − ∫ ∫ 𝑎(𝑥, 𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥. since 𝑎(𝑥, a) = 𝑚𝑖𝑛 (1; 𝛾𝑐(𝑥)−𝑐(a)) and 𝑎(𝑥, 𝑧) = 𝑚𝑖𝑛 (1; 𝛾𝑐(𝑥)−𝑐(𝑧)), then four possible cases can be distinguished: • if 𝑎(𝑥, a) = 1 and 𝑎(𝑥, 𝑧) = 1 then: ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻 (𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,a)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − ∫ ∫ 𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥 = ∫ 1𝐵(𝑥,a)𝑞(𝑥, a)𝑑𝑥 𝛺 > 0. • if 𝑎(𝑥, a) = 1 and 𝑎(𝑥, 𝑧) = 𝛾𝑐(𝑥)−𝑐(𝑧) then: ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻 (𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,a)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − ∫ ∫ 𝛾𝑐(𝑥)−𝑐(𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥 = ∫ 1𝐵(𝑥,a)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − 𝛾𝑐(𝑥)−𝑐(𝑧) > 0. • if 𝑎(𝑥, a) = 𝛾𝑐(𝑥)−𝑐(a) and 𝑎(𝑥, 𝑧) = 1 then: 11 int. j. anal. appl. (2023), 21:51 ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻(𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,a)𝛾 𝑐(𝑥)−𝑐(a)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − ∫ ∫ 𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥 = 𝛾𝑐(𝑥)−𝑐(a) ∫ 1𝐵(𝑥,a)𝑞(𝑥, a)𝑑𝑥 > 0. 𝛺 • if 𝑎(𝑥, a) = 𝛾𝑐(𝑥)−𝑐(𝐴) and 𝑎(𝑥, 𝑧) = 𝛾𝑐(𝑥)−𝑐(𝑧) then : ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻(𝑥, a)𝑑𝑥 𝛺 = ∫ 1𝐵(𝑥,a)𝛾 𝑐(𝑥)−𝑐(a)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − ∫ ∫ 𝛾𝑐(𝑥)−𝑐(𝑧)𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥 = 𝛾𝑐(𝑥)−𝑐(a) ∫ 1𝐵(𝑥,𝐴)𝑞(𝑥, a)𝑑𝑥 𝛺 + 1 − 𝛾𝑐(𝑥)−𝑐(𝑧) ∫ ∫ 𝑞(𝑥, 𝑧)𝑑𝑧 𝛺𝛺 𝑑𝑥 = 𝛾𝑐(𝑥)−𝑐(a) ∫ 1𝐵(𝑥,𝐴)𝑞(𝑥, a)𝑑𝑥𝛺 + 1 − 𝛾 𝑐(𝑥)−𝑐(𝑧) > 0. so ∫ 1𝐵(𝑥,a)𝑃𝑀𝐻 𝑡 (𝑥, a)𝑑𝑥 𝛺 > 0 ∀ 𝑡 ≥ 0, then 𝑃𝑀𝐻 is 𝜋 −irreducible. since 𝜋 is the invariant distribution of 𝑃𝑀𝐻, then it remains so with 𝑃. as a matter of fact, 𝜋𝑃𝑀𝐻 = 𝜋, and by recurrence on 𝑛 we get: 𝜋𝑃𝑀𝐻 = 𝜋𝑃𝑀𝐻 2 = 𝜋𝑃𝑀𝐻 3 = ⋯ = 𝜋𝑃𝑀𝐻 𝑛 = 𝜋 yet 𝑃 = 𝑃𝑀𝐻 𝑛 , then we get: 𝜋𝑃 = 𝜋. on the other hand, 𝜋 − reversibility of 𝑃𝑀𝐻 leads to 𝜋 − reversibility of 𝑃, i.e: 𝜋(𝑥)𝑃𝑀𝐻 (𝑥, 𝑦) = 𝜋(𝑦)𝑃𝑀𝐻 (𝑦, 𝑥) ⇒ 𝜋(𝑥)𝑃(𝑥, 𝑦) = 𝜋(𝑦)𝑃(𝑦, 𝑥) since 𝜋𝑃𝑀𝐻 = 𝜋𝑃𝑀𝐻 𝑛 = 𝜋, on then we shall get: 𝜋(𝑥)𝑃𝑀𝐻 (𝑥, 𝑦) = 𝜋(𝑥)𝑃𝑀𝐻 𝑛(𝑥, 𝑦) = 𝜋(𝑦)𝑃𝑀𝐻 (𝑦, 𝑥) = 𝜋(𝑦)𝑃𝑀𝐻 𝑛(𝑦, 𝑥) and since 𝑃𝑀𝐻 𝑛 = 𝑃, then we get: 𝜋(𝑥)𝑃(𝑥, 𝑦) = 𝜋(𝑦)𝑃(𝑦, 𝑥). by constructing 𝑃 = 𝑃𝑀𝐻 𝑛 , the 𝜋 −irreducibility of 𝑃𝑀𝐻 leads to 𝜋 −irreducibility of 𝑃. if 𝑃 is 𝜋 −irreducibile and has an invariant 𝜋 distribution, then 𝑃 is positive recurrent and π is the unique invariant distribution of 𝑃 [13] (see, proposition 1). on the other hand, the chain generated by the construction algorithm shall be aperiodic as well provided there exists at least a pair of configuration (𝑥, 𝑦) so that 𝑎(𝑥, 𝑦) < 1, and we shall ultimately get 𝑃(𝑥, 𝑥) > 0. we are quick to notice that the chain is aperiodic since the event 𝑋(𝑁+1) = 𝑋(𝑁) is likely at practically any moment. as a matter of fact, each state can, then, be visited at two subsequent iterations, thus 𝑃1(𝑥, 𝑥) > 0, and their period is then equal to 1. since the chain generated with the algorithm is irreducible and aperiodic, then its kernel with transition 𝑃 is primitive (a characterization of primitive markov kernel more common in probability theory is to say that are irreducible and aperiodic [14]). 12 int. j. anal. appl. (2023), 21:51 theorem 4.1. the markov chain (𝑋𝑘 )𝑘 ≥ 0 obtained from the construction algorithm is uniformly ergodic, and its kernel 𝑃 realizes the simulation of the connected component process with density 𝜋(𝑥) = 𝑘𝛽𝑛(𝑥)𝛾−𝑐(𝑥), i.e. 𝑣𝑃𝑚 converges to 𝜋 as 𝑚 approaches infinity; where 𝑣 is an initial distribution, and we have : ‖𝑣𝑃𝑚 − 𝜋‖ → 0, 𝑚 → ∞ proof of theorem 4.1 let 𝑣 be an initial distribution, for every integer 𝑚 and ∀ 𝑥 ∈ 𝑁𝑓 we have: ‖𝑣𝑃𝑚 (𝑥,∙) − 𝜋‖ = ‖𝑣𝑃𝑚 − 𝜋𝑃𝑚 ‖ on the other hand, ‖𝑣𝑃𝑚 − 𝜋𝑃𝑚 ‖ ≤ 2𝐶(𝑃𝑚 ) and 𝐶(𝑃𝑚 ) ≤ (𝐶(𝑃))𝑚 [15] (see lemma 4.2.2, p.71), this implies that: ‖𝑣𝑃𝑚 (𝑥,∙) − 𝜋‖ ≤ 2(𝐶(𝑃))𝑚 where, 𝐶(𝑃)is the dobrushin contraction coefficient of 𝑃 [16]. according to proposition 4.1, the kernel 𝑃 is primitive, then 𝐶(𝑃) < 1 [15] (see lemma 4.2.3, p.72), so when 𝑚 approaches infinity, ‖𝑣𝑃𝑚 − 𝜋‖ tends towards zero. hence, the chain is uniformly ergodic and converges towards the distribution defined in (3.2). 5. simulation study and discussion in this section, we will compare the quality of point distributions in the proposed designs using standard criteria to evaluate the degree of filling of the experimental space and the quality of the uniform distribution. • criterion of distance (mindist): it is about maximizing the minimal distance between two points of the design [17]. 𝑀𝑖𝑛𝑑𝑖𝑠𝑡 = min 𝑖 min 𝑗≠𝑖 𝑑(𝑥𝑖 , 𝑥𝑗 ) where, 𝑑(𝑥𝑖 , 𝑥𝑗 ) is the euclidean distance between the points 𝑥𝑖 and 𝑥𝑗. • the coverage criterion (cov): it helps to measure the gap between the points of the design and those of a regular grid. this criterion is invalid for a regular grid. our goal, here, is to minimize this gap to get closer to a regular grid, hence ensuring the filling of the space without, however, reaching it to comply with a uniform distribution, mainly in projection on factorial axes [18]. 13 int. j. anal. appl. (2023), 21:51 𝑐𝑜𝑣 = 1 𝛿̅ √ 1 𝑛 ∑(𝛿𝑖 − 𝛿̅) 2 𝑛 𝑖=1 where 𝛿𝑖 = min 𝑖≠𝑗 (𝑑(𝑥𝑖 , 𝑥𝑗 )) and �̅� = 1 𝑛 ∑ 𝛿𝑖 𝑛 𝑖=1 . for regular grid, 𝛿1 = 𝛿2 = ⋯ = 𝛿𝑛, then cov=0. • the mesh ratio (r): is the ratio between maximal and minimal distances separating the points of the experimental design. in the case of a regular grid, r=1. thus, when r is close to the value 1 the points are close to those of a regular grid. 𝑅 = max 𝑖:1…𝑛 𝛿𝑖 min 𝑖:1…𝑛 𝛿𝑖 • the discrepancy criterion (disc): discrepancy measures the difference between the empirical distribution function of the design points and that of the uniform distribution. unlike the previous three criteria, discrepancy does not depend on the distance between points. there are different ways to measure discrepancy, but in this study, we use the l2 norm. [19]. 𝐷𝑖𝑠𝑐 = ( 1 3 ) 𝑝 − 21−𝑝 𝑛 ∑ ∏ (1 − (𝑥𝑖 𝑗 ) 2 ) + 1 𝑛2 ∑ ∑ ∏ (1 − max(𝑥𝑖 𝑗 − 𝑥𝑘 𝑗 )) 𝑝 𝑗=1 𝑛 𝑘=1 𝑛 𝑖=1 𝑝 𝑗=1 𝑛 𝑖=1 in table 1, we present the results related to the discrepancy criterion for the proposed designs and compare them with sequences with low discrepancy (such as halton sequence [20], sobol sequence [21], and faure sequence [22]). it is interesting to note that the proposed designs have similarly low discrepancies as those of the low discrepancy sequences. table 1. discrepancy values for ccd designs: a low discrepancy halton sequence, a low discrepancy sobol sequence and a low discrepancy faure sequence. the presented results cover three, five, seven, and ten dimensions. the underlined values represent the lowest values for each dimension. dimension number of points ccd halton sequence sobol sequence faure sequence 3 5 7 10 20 40 50 60 0,001224 0,000634 0,000257 0.0000140 0,00173285 0,00040004 0,00012451 0,00005601 0,001382617 0,000260329 0,000074156 0,000007428 0,00113882 0,00028458 0,00012455 0,00001327 14 int. j. anal. appl. (2023), 21:51 the designs presented in this article are also compared with commonly used designs in computer experiments, excluding sequences with weak discrepancy. to ensure meaningful results, the criteria have been tested on 80 designs. the designs considered in this section are typically the following: • random designs (rd). • latin hypercube sampling (lhs) [23]. • maximin lhs designs (mlhs) [24]. • strauss designs (sd) [6]. • maximal entropy designs (dmax) [25]. • marked strauss designs (msd) ([7], [8]). • connected component designs (ccd). figures 4 and 5 present the criteria results in box plots for 80 designs in 10 and 5 dimensions, respectively. the plots clearly illustrate the distribution of the data and allow for easy comparison between designs. figure 4. representations of usual criteria calculated on 80 designs with 60 points for dimension 10. 15 int. j. anal. appl. (2023), 21:51 figure 5. representations of usual criteria calculated on 80 designs with 40 points for dimension 5. some remarks regarding the figures above are worth noting. maximal entropy designs, strauss designs, marked strauss designs, as well as connected component designs all scored well with regard to the discrepancy criterion. it is interesting to note that connected component designs are among the designs previously mentioned which also have very good distance criteria. the application of the recovery criterion helps to show that the proposed designs offer better results regarding this criterion. 6. conclusion the use of the connected component process and markov chain monte carlo (mcmc) method is instrumental in creating novel computer designs that are customized based on the distribution of a connected component model. this method is highly versatile, as it allows us to manipulate the distribution by representing it in a manner that makes it possible to impose certain properties such as filling. furthermore, mcmc methodology presents a fascinating alternative to the classical statistical approach that typically involves the independent realization of the same 16 int. j. anal. appl. (2023), 21:51 distribution. ultimately, the designs developed in this study were compared to those commonly used in computer experiments, and the outcomes were highly satisfactory. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] d.j. daley, d. vere-jones, an introduction to the theory of point processes. volume i: elementary theory and methods, 2nd edn, springer, berlin, 1988. [2] a.j. baddeley, j. møller, nearest neighbour markov point processes and random sets. int. stat. rev. 57 (1989), 89–121. https://doi.org/10.2307/1403381. [3] a.j. baddeley, m.n.m. van lieshout, area-interaction point processes, ann. inst. stat. math. 47 (1995), 601–619. https://doi.org/10.1007/bf01856536. [4] y.c. chin, a.j. baddeley, on connected component markov point processes, adv. appl. probab. 31 (1999), 279–282. https://doi.org/10.1239/aap/1029955135. [5] b.d. ripley, f.p. kelly, markov point processes, j. lond. math. soc. s2-15 (1977), 188–192. https://doi.org/10.1112/jlms/s2-15.1.188. [6] j. franco, planification d’expériences numériques en phase exploratoire pour des codes de calculs simulant des phénomènes complexes. doctoral thesis, l’ecole nationale supérieure des mines de saintetienne, france, 2008. [7] h. elmossaoui, n. oukid, f. hannane, construction of computer experiment designs using marked point processes, afr. mat. 31 (2020), 917–928. https://doi.org/10.1007/s13370-020-00770-9. [8] h. elmossaoui, contribution à la méthodologie de la recherche expérimentale, doctoral thesis, university saad dahleb, blida, algeria, 2020. [9] n. metropolis, a.w. rosenbluth, m.n. rosenbluth, a.h. teller, e. teller, equation of state calculations by fast computing machines, j. chem. phys. 21 (1953), 1087–1092. https://doi.org/10.1063/1.1699114. [10] w.k. hastings, monte carlo sampling methods using markov chains and their applications, biometrika. 57 (1970), 97–109. https://doi.org/10.1093/biomet/57.1.97. [11] a. baddeley, j. møller, a.g. pakes, properties of residuals for spatial point processes, ann. inst. stat. math. 60 (2007), 627–649. https://doi.org/10.1007/s10463-007-0116-6. [12] s. chib, e. greenberg, understanding the metropolis-hastings algorithm, amer. stat. 49 (1995), 327355. https://doi.org/10.2307/2684568. https://doi.org/10.2307/1403381 https://doi.org/10.1007/bf01856536 https://doi.org/10.1239/aap/1029955135 https://doi.org/10.1112/jlms/s2-15.1.188 https://doi.org/10.1007/s13370-020-00770-9 https://doi.org/10.1063/1.1699114 https://doi.org/10.1093/biomet/57.1.97 https://doi.org/10.1007/s10463-007-0116-6 https://doi.org/10.2307/2684568 17 int. j. anal. appl. (2023), 21:51 [13] s. chib, e. greenberg, markov chain monte carlo simulation methods in econometrics, econ. theory. 12 (1996), 409–431. https://doi.org/10.1017/s0266466600006794. [14] e. senata, non-negative matrices and markov chains, 2nd edition. springer, new york heidelberg berlin, 1981. [15] g. winkler, image analysis random fields and dynamic monte carlo methods, springer, berlin, 1995. [16] r.l. dobrushin, central limit theorem for nonstationary markov chains. i, theory probab. appl. 1 (1956), 65–80. https://doi.org/10.1137/1101006. [17] m. gunzburger, j. burkardt, uniformity measures for point samples in hypercubes. (2004). https://people.sc.fsu.edu/~jburkardt/publications/gb_2004.pdf. [18] m.e. johnson, l.m. moore, d. ylvisaker, minimax and maximin distance designs, j. stat. plan. inference. 26 (1990), 131–148. https://doi.org/10.1016/0378-3758(90)90122-b. [19] t.t. warnock, computational investigations of low-discrepancy point sets ii. in: niederreiter h., and shiue p.js. (eds) monte carlo and quasi-monte carlo methods in scientific computing. lecture notes in statistics, 106. springer, new york, 1995. [20] j.h. halton, on the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals, numer. math. 2 (1960), 84–90. https://doi.org/10.1007/bf01386213. [21] i.m. sobol, uniformly distributed sequences with an additional uniform property, ussr comput. math. math. phys. 16 (1976), 236–242. https://doi.org/10.1016/0041-5553(76)90154-3. [22] h. faure, discrépance de suites associées à un système de numération (en dimension s), acta arith. 41 (1982), 337–351. https://doi.org/10.4064/aa-41-4-337-351. [23] w.l. loh, on latin hypercube sampling, ann. stat. 24 (1996), 2058-2080. https://doi.org/10.1214/aos/1069362310. [24] m.d. morris, t.j. mitchell, exploratory designs for computational experiments, j. stat. plan. inference. 43 (1995), 381–402. https://doi.org/10.1016/0378-3758(94)00035-t. [25] m.c. shewry, h.p. wynn, maximum entropy sampling, j. appl. stat. 14 (1987), 165–170. https://doi.org/10.1080/02664768700000020. https://doi.org/10.1017/s0266466600006794 https://doi.org/10.1137/1101006 https://people.sc.fsu.edu/~jburkardt/publications/gb_2004.pdf https://doi.org/10.1016/0378-3758(90)90122-b https://doi.org/10.1007/bf01386213 https://doi.org/10.1016/0041-5553(76)90154-3 https://doi.org/10.4064/aa-41-4-337-351 https://doi.org/10.1214/aos/1069362310 https://doi.org/10.1016/0378-3758(94)00035-t https://doi.org/10.1080/02664768700000020 int. j. anal. appl. (2022), 20:14 some properties of controlled k-g-frames in hilbert c∗-modules rachid echarghaoui1, m’hamed ghiati1,∗, mohammed mouniane1, mohamed rossafi2 1laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, kenitra, morocco 2lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, b. p. 1796 fes atlas, morocco ∗corresponding author: mhamed.ghiati@uit.ac.ma abstract. this paper is devoted to studying the controlled k − g−frames in hilbert c∗−modules, some useful results are presented. also, the concept of controlled k−g−dual frames is given. finally, we discuss the stability problem for controlled k−g−frames in hilbert c∗−modules. 1. introduction and preliminaires frames for hilbert spaces were introduced by duffin and schaefer [2] in 1952 to study some deep problems in nonharmonic fourier series by abstracting the fundamental notion of gabor [4] for signal processing. many generalizations of the concept of frame have been defined in hilbert c∗-modules [3, 5, 6, 9, 11–15]. controlled frames in hilbert spaces have been introduced by p. balazs [1] to improve the numerical efficiency of iterative algorithms for inverting the frame operator. rashidi and rahimi [8] are introduced the concept of controlled frames in hilbert c∗−modules. let a be a unital c∗−algebra, let i be countable index set. throughout this paper h and l are countably generated hilbert a−modules and {hi}i∈i is a sequence of submodules of l. for each i ∈ i, end∗a(h,hi) is the collection of all adjointable a−linear maps from h to hi, and end ∗ a(h,h) received: jan. 24, 2022. 2010 mathematics subject classification. 42c15. key words and phrases. frame; g−frame; k−g−frame; controlled k−g−frames; hilbert c∗−modules. https://doi.org/10.28924/2291-8639-20-2022-14 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-14 2 int. j. anal. appl. (2022), 20:14 is denoted by end∗a(h). also let gl +(h) be the set of all positive bounded linear invertible operators on h with bounded inverse. definition 1.1. [10] let a be a unital c∗-algebra and h be a left a-module, such that the linear structures of a and h are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued in product 〈., .〉a:h ×h → a such that is sesquilinear, positive definite and respects the module action. in the other words, (i) 〈x,x〉a > 0 for all x ∈h and 〈x,x〉a = 0 if and only if x = 0. (ii) 〈ax + y,z〉a = a〈x,z〉a + 〈y,z〉a for all a ∈a and x,y,z ∈h. (iii) 〈x,y〉a = 〈y,x〉∗a for all x,y ∈h. for x ∈h we define ‖x‖= ‖〈x,x〉a‖ 1 2 . if h is complete with ‖.‖, it is called a hilbert a-module or a hilbert c∗-module over a. for every a in c∗-algebra a, we have |a| = (a∗a) 1 2 and the a-valued norm on h is defined by |x| = (x∗x) 1 2 for x ∈h. let h and k be tow hilbert a modules, a map t : h→k is said to be adjointable if there exists a map t∗ : k→h such that 〈tx,y〉a = 〈x,t∗y〉a for all y ∈k and x ∈h. lemma 1.1. [18] suppose that h1 and h2 two hilbert a-modules h and l1 ∈ end∗a(h1,h), l2 ∈ end∗a(h2,h). then the following assertions are equivalent: (i) r(l1) ⊆r(l2), (ii) l1l∗1 ≤ λ 2l2l ∗ 2 for some λ > 0, (iii) there exists a mapping u ∈ end∗a(h1,h2) such that l1 = l2u. moreover, if above conditions are valid, then there exists a unique operator u such that (i) ‖u‖2= inf{α > 0 l1l∗1 ≤ αl2l ∗ 2}, (ii) ker(l1) = ker(u), (iii) r(u) ⊆r(l∗2). if an operator u has a closed range, then there exists a right-inverse operator u†, (pseudo-inverse of u) in the following sense. lemma 1.2. [17] let u ∈ end∗a(h1,h2) be a bounded operator with closed range r(u). then there exists a bounded operator u† ∈ end∗a(h2,h1) for which uu†x = x, x ∈r(u). lemma 1.3. [10] let h and k two hilbert a-module and t ∈ end∗a(h,k). then, the following assertions are equivalent: (i) the operator t is bounded and a-linear, (ii) there exist k > 0 such that 〈tx,tx〉a ≤ k〈x,x〉a for all x ∈h. int. j. anal. appl. (2022), 20:14 3 definition 1.2. [7] a family λ := {λi ∈ end∗a(h,hi)}i∈i is called a g-frame in hilbert a module h with respect to {hi}i∈i if there exist constants 0 < a ≤ b < +∞ such that for each f ∈h, a〈f , f 〉a ≤ ∑ i∈i 〈λif , λif 〉a ≤ b〈f , f 〉a. theorem 1.1. [16] let λ := {λi ∈ end∗a(h,hi)}i∈i be a g-frame in hilbert a module h with respect to h{i∈ i} if and only if there exist constants a,b > 0 a‖〈f , f 〉a‖≤ ∥∥∥∥∥∑ i∈i 〈λif , λif 〉a ∥∥∥∥∥ ≤ b‖〈f , f 〉a‖ . (1.1) 2. some properties of controlled k-g-frames now, we define controlled k-g-frames in hilbert c∗-modules. definition 2.1. let c,c′ ∈gl+(h) and k ∈ end∗a(h), we say that λ := {λi ∈ end ∗ a(h,hi)}i∈i is a (c,c′)-controlled k-g-frame in hilbert a-module h if there exist constants 0 < acc′ < bcc′ < +∞ such that for each f ∈h, acc′〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λic′f , λicf 〉a ≤ bcc′〈f , f 〉a. (2.1) if the right hand of (2.1) holds, λ is called a (c,c′)-controlled k − g−bessel sequence in hilbert a-module h with bound bc. we call λ a parseval c,c′-controlled k-g-frame if 〈k∗f ,k∗f 〉a = ∑ i∈i 〈λic′f , λicf 〉a. if k = ih, then λ is c,c′-controlled g−frame. for simplicity, we will use a notation cc′ instead of c,c′. if λ is a cc′-controlled g-frame on hilbert a-module h, and c∗λ∗i λic ′ is positive for all i ∈ i, then for each f ∈h, acc′〈f , f 〉a ≤‖(c∗λ∗i λic ′) 1 2 f‖2≤ bcc′〈f , f 〉a. now, let r := {(c∗λ∗i λic ′) 1 2 f : f ∈h}i∈i ⊂ ( ∑ i∈i ⊕h)`2. it is easy to check that r is a closed subspace of ( ∑ i∈i ⊕h)`2. now, we can define the synthesis and analysis operators of the cc′-controlled g-frames as tcc′ : r→h, tcc′((c ∗λ∗i λic ′) 1 2 f )i∈i = ∑ i∈i (c∗λ∗i λic ′f ), and t∗cc′ : h→r, 4 int. j. anal. appl. (2022), 20:14 t∗cc′(f ) = ((c ∗λ∗i λic ′) 1 2 f )i∈i. thus, the cc′-controlled g-frame operator is given by scc′f = tcc′t ∗ cc′f = ∑ i∈i (c∗λ∗i λic ′f ). scc′ is positive, bounded, invertible and self-adjoint. moreover 〈scc′f , f 〉 = ∑ i∈i 〈λic′f , λicf 〉 and acc′ih < scc′ < bcc′ih. lemma 2.1. let c,c′ ∈ gl+(h). a sequence λ is a cc′-controlled g-bessel sequence in hilbert a-module with bound bcc′ if and only if the operator tcc′ : r→h, tcc′((c ∗λ∗i λic ′) 1 2 f )i∈i = ∑ i∈i (c∗λ∗i λic ′f ) is well defined and bounded with ‖tcc′‖≤ √ bcc′. proof. we only need to prove the sufficient condition. let tcc′ be a well-defined and bounded operator with ‖tcc′‖≤ √ bcc′. for each f ∈ h, we have∑ i∈i 〈λic′f , λicf 〉a = ∑ i∈i 〈c∗λ∗i λic ′f , f 〉a = 〈 ∑ i∈i c∗λ∗i λic ′f , f 〉a = 〈tcc′((c∗λ∗i λic ′) 1 2 f ))i∈i, f 〉a. hence, ‖〈tcc′((c∗λ∗i λic ′) 1 2 f ))i∈i, f 〉a‖≤‖tcc′((c∗λ∗i λic ′) 1 2 f ))i∈i‖‖f‖ ≤‖tcc′‖‖((c∗λ∗i λic ′) 1 2 f ))i∈i‖‖f‖. but ‖((c∗λ∗i λic ′) 1 2 f )‖2= ∑ i∈i 〈λic′f , λicf 〉a, ‖((c∗λ∗i λic ′) 1 2 f )‖≤‖tcc′‖‖f‖, ‖((c∗λ∗i λic ′) 1 2 f )‖2≤‖tcc′‖2‖f‖2. it follows that ∑ i∈i 〈λic′f , λicf 〉a ≤ bcc′‖〈f , f 〉a‖. int. j. anal. appl. (2022), 20:14 5 and this means that λ is a cc′-controlled g-bessel sequence. � lemma 2.2. let c,c′ ∈ gl+(h). a sequence λ is a cc′-controlled g-frame sequence in hilbert a-module if and only if the operator tcc′ : r→h, tcc′((c ∗λ∗i λic ′) 1 2 f ) = ∑ i∈i c∗λ∗i λic ′f is well defined, bounded and surjective. proof. suppose that λ is a cc′-controlled g-frame in hilbert a-module. since, scc′ is surjective operator, so tcc′. for the opposite implication, by lemma 2.1; tcc′ is a well-defined and bounded operator. so λ is a cc′-controlled g-bessel sequence. now, for each f ∈ h, we have f = tcc′t † cc′f . hence ‖f‖4 = ‖〈f , f 〉‖2 = ‖〈tcc′t † cc′f , f 〉‖ 2 = ‖〈t† cc′f ,t ∗ cc′f 〉‖ 2 ≤‖〈t† cc′f ,t † cc′f 〉‖ 2‖〈t∗cc′f ,t ∗ cc′f 〉‖ 2 ≤‖t† cc′f‖ 2‖‖t∗cc′f‖ 2 ≤‖t† cc′‖ 2‖f‖2 ∑ i∈i 〈λic′f , λicf 〉a. we conclude that (‖t† cc′‖ 2)−1‖〈f , f 〉‖≤ ∑ i∈i 〈λic′f , λicf 〉a. � proposition 2.1. let λ be a cc′-controlled k-g-frames in hilbert a-module h and k has a dense range. suppose that (c∗λ∗i λic ′) is positive and also vi = (c∗λ∗i λic ′) 1 2 for each i ∈ i. then ( ⋂ i∈i ker vi) ⊥ = h. proof. assume that acc′ and bcc′ are the frame bounds of λ. hence, acc′〈k∗f ,k∗f 〉a ≤‖(c∗λ∗i λic ′) 1 2‖2≤ bcc′〈f , f 〉a. (2.2) since ker k∗ = (r(k))⊥ and k has a dense range, k∗ injective. then from (2.2), for each i ∈ i, we get ⋂ i∈i ker vi ⊆ ker k∗ = {0}. 6 int. j. anal. appl. (2022), 20:14 remark 2.1. suppose that λ is a cc′-controlled k-g-frame in hilbert a with lower bound acc′. then, we have scc′ > acc ′kk∗, so by lemma 1.1, there exists an operator u ∈ end∗a(h,r) such that tcc′u = k. (2.3) now, we can obtain optimal frame bounds of λ by the operator u. indeed, it is obvious that bop = ‖scc′‖= ‖tcc′‖2. by lemma 1.1, the equation (2.3) has a unique solution as u0 such that ‖u0‖2 = inf{α > 0/kk∗ ≤ αtcc′t∗cc′} = inf{α > 0/〈kk∗f , f 〉≤ α〈tcc′t∗cc′f , f 〉, f ∈h} = inf{α > 0/〈k∗f ,k∗f 〉≤ α〈t∗cc′f ,t ∗ cc′f 〉, f ∈h} = inf{α > 0/ ‖〈k∗f ,k∗f 〉‖≤ α‖〈t∗cc′f ,t ∗ cc′f 〉‖, f ∈h} = inf{α > 0/‖k∗f‖2≤ α‖t∗cc′f‖ 2, f ∈h}. now, we have aop = sup{a > 0 \a‖k∗f‖2≤‖t∗cc′f‖ 2, f ∈h} = (inf{α > 0 \‖k∗f‖2≤ α‖t∗cc′f‖ 2, f ∈h})−1 = u−20 . � in the following, we consider some proper relations between the operators u,k ∈ end∗a(h) and c,c′ ∈ gl+(h) and investigate the cases that {λiu}i∈i, {λiu∗}i∈i can also cc′-controlled k-gframe. next, by putting connections between the operators sλ,k,c and c′, we reach to necessary and sufficient conditions that {λi}i∈i can be a parseval cc′-controlled k-g-frames. theorem 2.1. let λ be a cc′-controlled k-gframe in hilbert a module h. and u ∈ end∗a(h) such that r(u) ⊂r(k). then λ is a cc′-controlled u-g-frame in hilbert a-module h. proof. suppose that acc′ is a lower frame bound of λ. using lemma 1.1, there exists α > 0 such that uu∗ ≤ α2kk∗. now, for each f ∈h. we have 〈uu∗f , f 〉a ≤ α2〈kk∗f , f 〉a. we have acc′ (α2) 〈u∗f ,u∗f 〉a ≤ acc′〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λic′f , λicf 〉a ≤ bcc′〈f , f 〉a. � int. j. anal. appl. (2022), 20:14 7 theorem 2.2. let λ be a cc′-controlled k-gframe in hilbert amodule h. assume that k has a closed range and u ∈ end∗a(h) such that r(u ∗) ⊂ r(k) also suppose that u∗ commutes with c and c′. then {λiu∗}i∈i is a cc′-controlled k-gframe for r(u) if and only if there exists δ > 0 such that for each f ∈r(u), ‖u∗f‖> δ‖k∗f‖. proof. suppose that {λiu∗}i∈i is a cc′-controlled k-g-frame in hilbert a module h with a lower frame bound ecc′ > 0. if bcc′ is an upper frame bound of λ then for each f ∈r(u), we have ecc′〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a = ∑ i∈i 〈λic′u∗f , λicu∗f 〉a, thus ecc′〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λic′u∗f , λicu∗f 〉a ≤ bcc′〈u∗f ,u∗f 〉a, therefore ecc′‖〈k∗f ,k∗f 〉a‖≤‖ ∑ i∈i 〈λic′u∗f , λicu∗f 〉a‖≤ bcc′‖〈u∗f ,u∗f 〉a‖ thus ecc′‖k∗f‖2≤ bcc′‖u∗f‖2. so √ ecc′ bcc′‖k ∗f‖≤ ‖u∗f‖, for the opposite implication, for each f ∈ h, we have ‖u∗f‖= ‖(k†)∗k∗u∗f‖≤‖(k†)‖‖k∗u∗f‖. therefore, if acc′ is a lower frame bound of λ, we have acc′δ 2‖k†‖−2〈k∗f ,k∗f 〉≤ acc′‖k†‖−2〈u∗f ,u∗f 〉 ≤ acc′‖k∗u∗f‖2 ≤ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a. for the upper bound, it is clear that∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a ≤ bcc′〈u∗f ,u∗f 〉a ≤ bcc′‖u‖2〈f , f 〉a. so, (λiu∗)i∈i is a cc′-controlled k-g-frame in hilbert a-module h with frame bounds acc′δ2‖k†‖−2 and bcc′‖u‖2 . � theorem 2.3. let λ be a cc′-controlled k-g-frame in hilbert a-module h and the operator k has a dense rang. assume that u ∈ end∗a(h) has a closed range and u and u ∗ commute with c and c′. if {λiu∗}i∈i and {λiu}i∈i are cc′-controlled k-gframe in hilbert amodule h, then u is invertible. proof. suppose that {λiu∗}i∈i is a cc′-controlled k-g-frame in hilbert a module h with a lower frame bound a1, and b1. then for each f ∈h, a1〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a ≤ b1〈f , f 〉a. 8 int. j. anal. appl. (2022), 20:14 we have ‖a1〈k∗f ,k∗f 〉a‖≤‖ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a‖≤‖b1〈f , f 〉a‖, (2.4) hence, a1‖k∗f‖2≤‖ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉‖≤ b1‖f‖2. since k has a dense range, k∗ is injective. moreover, r(u) = (ker u∗)⊥ = h so u is surjective. suppose that {λiu∗}i∈i is a (cc′)-controlled k-g-frame in hilbert a module h with a lower frame bound a2 and b2. then, for each f ∈h, a2〈k∗f ,k∗f 〉a ≤ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a ≤ b2〈f , f 〉a ‖a2〈k∗f ,k∗f 〉a‖≤‖ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a‖≤‖b2〈f , f 〉a‖ a2‖k∗f‖2≤‖ ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a‖≤ b2‖f‖2. therefore u is injective, since ker u ⊆ ker k∗. thus, u is an invertible operator. � theorem 2.4. let λ be a cc′-controlled k-g-frame in hilbert amodule h and u ∈ end∗a(h) be a co-isometry (i.e. uu∗ = idh) such that uk = ku and u∗ commutes with c and c′. then {λiu∗}i∈i is a cc′-controlled k-g-frame in hilbert a-module h. proof. suppose λ be a cc′-controlled k-gframe in hilbert a-module h with a lower frame bound acc′. and bcc′ for each f ∈h, we have∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a = ∑ i∈i 〈λic′u∗f , λicu∗f 〉a ≤ bcc′〈u∗f ,u∗f 〉a hence, ∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a ≤ bcc′‖u∗‖2〈f , f 〉a. so, {λiu∗}i∈i is a cc′-controlled g-bessel sequence. for the lower bound, we can write∑ i∈i 〈λiu∗c′f , λiu∗cf 〉a = ∑ i∈i 〈λic′u∗f , λicu∗f 〉a > acc′〈k∗u∗f ,k∗u∗f 〉a = acc′〈(uk)∗f , (uk)∗f 〉a = acc′〈(ku)∗f , (ku)∗f 〉a = acc′〈u∗k∗f ,u∗k∗f 〉a = acc′〈uu∗k∗f ,u∗k∗f 〉a = acc′〈k∗f ,k∗f 〉a. � int. j. anal. appl. (2022), 20:14 9 theorem 2.5. let λ := {λi ∈ end∗a(h,hi)}i∈i and � := {�i ∈ end ∗ a(h,hi)}i∈i be tow cc ′controlled k − g− bessel sequences in hilbert amodule h with bounds bλ and b� respectively. suppose that tλ,c,c′ and t�,cc′ are their synthesis operators such that t�,cc′t ∗ λ,c,c′ = k ∗. then λ and � are cc′-controlled k and k∗-g-frames, respectively. proof. ‖k∗f‖4 = ‖〈k∗f ,k∗f 〉a‖2 = ‖〈t�,cc′t∗λ,c,c′f ,k ∗f 〉a‖2 ≤‖t∗λ,c,c′f‖ 2‖t∗�,cc′k ∗f‖2 = ∑ i∈i 〈λic′f , λicf 〉a ∑ i∈i 〈�ic′k∗f ,�ic′k∗f 〉a ≤ ∑ i∈i 〈λic′f , λicf 〉ab�‖〈k∗f ,k∗f 〉a‖. so, ‖〈k∗f ,k∗f 〉a‖≤ ∑ i∈i 〈λic′f , λicf 〉ab� � thus b−1� ‖〈k ∗f ,k∗f 〉a‖≤ ∑ i∈i 〈λic′f , λicf 〉a. this that λ is a cc′-controlled k-g-frame in hilbert a-module h with frame operator sλ. for each f ∈a, we have tλ,c,c′t∗�,cc′ = k ‖kf‖4 = ‖〈kf,kf 〉a‖2 = ‖〈tλ,c,c′t∗�,cc′f ,kf 〉a‖ 2 ≤‖t∗λ,c,c′kf‖ 2‖t∗�,cc′f‖ 2 = ∑ i∈i 〈λic′kf, λickf 〉a ∑ i∈i 〈�ic′f ,�ic′f 〉a ≤ ∑ i∈i 〈�ic′f ,�icf 〉abλ‖〈kf,kf 〉a‖. thus b−1λ ‖〈kf,kf 〉a‖≤ ∑ i∈i 〈�ic′f ,�icf 〉a. this that � is a cc′-controlled k-g-frame in hilbert amodule h. theorem 2.6. let λ be a g-frame in hilbert amodule h with frame operator sλ. also assume that λ is a cc′-controlled gbessel sequence with frame operator scc′. then λ is a parseval cc ′controlled k-gframe in hilbert a-module h if and only if c = (s−p λ )∗φ and c′ = (s−q λ )ψ where φ, ψ are two operators in hilbert amodule h such that φ∗ψ = kk∗ and p + q = 1 where p,q ∈r. 10 int. j. anal. appl. (2022), 20:14 proof. assume that λ is a parseval cc′-controlled k-g-frame in hilbert a-module h, ∑ i∈i 〈λic′f , λicf 〉a = 〈k∗f ,k∗f 〉a = ∑ i∈i 〈f ,c∗λ∗i λic ′f 〉a = 〈f , ∑ i∈i c∗λ∗i λic ′f 〉a = 〈f ,scc′f 〉a = 〈f ,kk∗f 〉a scc′(f ) = ∑ i∈i c∗λ∗i λic ′(f ) = c∗( ∑ i∈i λ∗i λic ′)(f ) = c∗sλc ′(f ). hence scc′ = c ∗sλc ′ and scc′ = kk ∗. therefore, for each p,q ∈r such that p +q = 1, we obtain kk∗ = c∗s p λ s q λ c′. we define φ = (sp λ )∗c and ψ = (sq λ )∗c′ so φ∗ψ = c∗s p λ s q λ c′ = kk∗. conversely, let φ and ψ be tow operators in hilbert amodule h such that φ∗ψ = kk∗. suppose that c = (s−p λ )∗φ and c′ = (s−q λ )∗ψ are tow operators on hilbert amodule h wherep,q ∈r and p + q = 1, since kk∗ = φ∗ψ = c∗s p λ s q λ c′ = c∗sλc ′ = scc′. so, for each f ∈h, 〈kk∗f , f 〉a = 〈k∗f ,k∗f 〉a = 〈 ∑ i∈i c∗λ∗i λic ′f , f 〉a. thus λ is parseval cc′-controlled k −g− frame on hilbert amodule h. � 3. duals of controlled k-g-frames in this section, by the concept of k-gdual pair, we present a bounded operator called dual operator and propose some known equalities and inequalities between dual operator cc′-controlled k-g-frame in hilbert amodule h. int. j. anal. appl. (2022), 20:14 11 definition 3.1. suppose that λ is cc′-controlled k-g-frame on hilbert amodule h with synthesis operator tλ,c,c′ then λ̃ := {λ̃i ∈ end∗a(h,hi)}i∈i is called a cc ′-controlled k −g− dual frame ( or brevitycc′ −kg− dual ) for λ if tλ,c,c′t ∗ λ̃,c,c′ = k, (3.1) and λ̃ is a cc′-controlled k − g− bessel sequence. in this cas, (λ, λ̃) is called a cc′-controlled k −g− dual pair. the following results presents equivalent conditions of the cc′-k-g-dual. proposition 3.1. let λ̃ be a cc′−k−g− dual for λ. then the following conditions are equivalent : (i) tλ,c,c′t ∗ λ̃,c,c′ = k (ii) t λ̃,c,c′ t∗λ,c,c′ = k ∗ (iii) for each f , f ′ ∈h, we have 〈kf ; f ′〉 = 〈t∗ λ̃,c,c′ f ,t∗ λ̃,c,c′ f 〉. theorem 3.1. if λ̃ be a cc′−k−g− dual for λ, then λ̃ is a cc′-controlled k∗−g− frame in hilbert amodule h. proof. we have ‖kf‖4 = ‖〈kf,kf 〉a‖2 = ‖〈tλ,c,c′t∗λ̃,c,c′f ,kf 〉‖ 2 = ‖〈t∗ λ̃,c,c′ f ,t∗λ,c,c′〉‖ 2 ≤‖t∗ λ̃,c,c′ f‖2‖t∗λ,c,c′kf‖ 2 ≤ ( ∑ i∈i 〈λ̃ic′f , λ̃icf 〉a)( ∑ i∈i 〈λic′kf, λickf 〉a) ≤ bc‖kf‖2( ∑ i∈i 〈λ̃ic′f , λ̃icf 〉a), it follows that b−1 c acc′‖〈kf,kf 〉a‖≤ ∑ i∈i 〈λ̃ic′f , λ̃icf 〉a ≤ bcc′‖〈f , f 〉a‖. therefore, λ̃ is a cc′-controlled k∗ −g− frame in hilbert amodule h. � theorem 3.2. assume that cop and dop are the optimal bounds of λ̃, respectively. then cop > b −1 op , dop > a −1 op , for which aop and bop are the optimal bounds of λ, respectively. assume (λ, λ̃) is called a cc′controlled k −g− dual pair and j ⊂i. we define sj f := ∑ i∈j (c∗λ∗i λic ′) 1 2 (c∗λ̃∗ i λ̃ic ′) 1 2 f , f ∈h, 12 int. j. anal. appl. (2022), 20:14 and we call it a dual operator. it is clear that sj ∈ end∗a(h) and sj + sj c = k where j c is the complement of j . if b1and b2 are the bounds of λ and λ̃ respectively, then, we have ‖sj f‖2 = ( sup ‖g‖=1 ‖〈sj f ,g‖〉)2 ≤ ( sup ‖g‖=1 ( ∑ i∈j ‖〈(c∗λ∗i λic ′) 1 2 (c∗λ̃∗ i λ̃ic ′) 1 2 f 〉‖)2 ≤ ( ∑ i∈i ‖(c∗λ∗i λic ′) 1 2 f‖2)(( sup ‖g‖=1 ( ∑ i∈j ‖c∗λ̃∗ i λ̃ic ′) 1 2‖)2 ≤ b1b2‖f‖2. so sj is bounded. now, by that operator sj we extend some well known equalities and inequalities for controlled k-gframes in the following theorems. theorem 3.3. if f ∈ h then ( ∑ i∈j〈(c ∗λ̃∗ i λ̃ic ′) 1 2 f , (c∗λ∗i λic ′) 1 2 kf 〉 − ‖sj f‖2= ( ∑ i∈j c 〈c∗λ̃ ∗ i λ̃ic ′)1/2f , (c∗λ∗ i λic ′)1/2kf 〉−‖sj cf‖2. proof. let f ∈h. we can write ( ∑ i∈j 〈(c∗λ̃∗ i λ̃ic ′) 1 2 f , (c∗λ∗i λic ′) 1 2 kf 〉−‖sj f‖2 = 〈k∗sj f , f 〉−‖sj f‖2 = 〈k∗sj f , f 〉−〈s∗jsj f , f 〉 = 〈(k −sj )∗sj f , f 〉 = 〈s∗j c (k −sj ), f 〉 = 〈s∗j ckf,f 〉−〈s ∗ j csj cf , f 〉 = 〈kf,sj cf 〉−〈sj cf ,sj cf 〉 = 〈sj cf ,kf 〉−‖sj cf‖2 = ( ∑ i∈j c 〈(c∗λ̃∗ i λ̃ic ′) 1 2 f , (c∗λ∗ i λic ′) 1 2 kf 〉 −‖sj cf‖2. � theorem 3.4. let λ be a parseval cc′-controlled k-g-frame in hilbert a-module h if j ⊆ i and e ⊆ jc, then for each f ∈h, ‖ ∑ i∈j∪e (c∗λ∗i λic ′)f‖2−‖ ∑ i∈jc\e (c∗λ∗i λic ′)f‖2 = ‖ ∑ i∈j (c∗λ∗i λic ′)f‖2−‖ ∑ i∈jc (c∗λ∗i λic ′)f‖2+2re( ∑ i∈e 〈λic′f , λic∗kk∗f 〉). int. j. anal. appl. (2022), 20:14 13 proof. let sλ,jf = ∑ i∈j (c∗λ∗i λic ′)f , therefore, sλ,i + sλ,ic = kk∗. hence s2λ,j −s 2 λ,jc = s 2 λ,j − (kk ∗ −sλ,j)2 = kk∗sλ,j + sλ,jkk ∗ − (kk∗)2 = kk∗sλ,j −sλ,jckk∗. now, for each f ∈ h, we obtain ‖s2λ,j‖ 2−‖s2λ,jc‖ 2= 〈kk∗sλ,jf , f 〉−〈sλ,jckk∗f , f 〉, consequently, for j ∪e instead of j: ‖ ∑ i∈j∪e (c∗λ∗i λic ′)f‖2−‖ ∑ i∈jc\e (c∗λ∗i λic ′)f‖2 = ( ∑ i∈j∪e 〈λic′f , λic∗kk∗f 〉) − ∑ i∈jc\e 〈λic′f , λic∗kk∗f 〉 = ( ∑ i∈j 〈λic′f , λic∗kk∗f 〉) − ∑ i∈jc 〈λic′f , λic∗kk∗f 〉 + 2re( ∑ i∈e 〈λic′f , λic∗kk∗f 〉) = ∑ i∈j (c∗λ∗i λic ′)f‖2−‖ ∑ i∈jc (c∗λ∗i λic ′)f‖2+2re( ∑ i∈e 〈λic′f , λic∗kk∗f 〉). � theorem 3.5. let λ be a parseval cc′-controlled k-g-frame in hilbert amodule h if j ⊆ i, then for each f ∈h, ‖ ∑ i∈j (c∗λ∗i λic ′)f‖2+re (∑ i∈jc 〈λic′f , λic∗kk∗f 〉 ) = ‖ ∑ i∈jc (c∗λ∗i λic ′)f‖2+re (∑ i∈j 〈λic′f , λic∗kk∗f 〉 ) > 3 4 ‖kk∗f‖2. proof. using the the proof of theorem 3.4, we have s2λ,j −s 2 λ,jc = kk ∗sλ,j −sλ,jckk∗. therefore s2λ,j + s 2 λ,jc = 2 ( kk∗ 2 −sλ,j )2 + (kk∗)2 2 > (kk∗)2 2 . 14 int. j. anal. appl. (2022), 20:14 thus kk∗sλ,j + s 2 λ,jc + (kk ∗sλ,j + s 2 λ,jc ) ∗ = kk∗sλ,j + s 2 λ,jc + sλ,jkk ∗ + s2λ,jc = kk∗(sλ,j + sλ,jc ) + s 2 λ,j + s 2 λ,jc > 3 4 (kk∗)2. now, for each f ∈ h, we obtain ‖ ∑ i∈j (c∗λ∗i λic ′)f‖2+re( ∑ i∈j 〈λic′f , λic∗kk∗f 〉) = 〈kk∗sλ,jf , f 〉 + 〈s2λ,jcf , f 〉 + 〈kk ∗ + s2λ,jcf , f 〉 + 〈f ,s 2 λ,jcf 〉> 3 4 (kk∗)2. � 4. the stability problem of controlled k −g−frames stability of the wavelet and gabor frames under perturbation is one of the important problems in frame theory. at first this problem was studied by paley and wienes for bases and then extended to frames.but the most important results are obtained by casazza and christensen. here we study the perturbation of cc′-controlled k-g-frames.in hilbert a-module h. theorem 4.1. let λ be a cc′-controlled k-gframe on hilbert amodule h with bounds acc′ and acc′. assume that � := {�i ∈ end∗a(h,hi)i∈i} is a sequence of operators such that for each f ∈ h and i ∈ i, ‖(c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖ ≤ λ1‖(c∗λ∗i λic ′)1/2f‖+λ2‖c∗ �∗i �ic ′)1/2f‖+ci〈k∗f ,k∗f 〉 1 2 where {ci}i∈i is a sequence of positive numbers such that η := ∑ i∈i c 2 i < ∞ and 0 ≤ λ1, λ2 ≤ 1. then � is a cc′-controlled k −g− frame on hilbert a-module h with bounds:( (1 −λ1) √ acc′ −η 1 + λ2 )2 , ( (1 + λ1) √ bcc′ + η‖k‖ 1 −λ2 )2 . proof. for each f ∈ h, we have ‖c∗ �∗i �ic ′)1/2f‖= ‖(c∗ �∗i �ic ′ −c∗λ∗i λic ′)1/2f + (c∗λ∗i λic ′)1/2f‖ ≤‖(c∗ �∗i �ic ′ −c∗λ∗i λic ′)1/2f‖+(c∗λ∗i λic ′)1/2f‖ ≤ λ1‖(c∗λ∗i λic ′)1/2f‖+λ2‖c∗ �∗i �ic ′)1/2f‖+ci〈k∗f ,k∗f 〉 1 2 + ‖+(c∗λ∗i λic ′)1/2f‖. hence (1 −λ2)‖(c∗ �∗i �ic ′)1/2f‖≤ (1 + λ1)‖(c∗λ∗i λic ′)1/2f‖+ci〈k∗f ,k∗f 〉 1 2 int. j. anal. appl. (2022), 20:14 15 since λ is a cc′-controlled k-g-frame, so ‖t∗cc′‖ 2 = ‖(c∗λ∗i λic ′)1/2f‖2 = ∑ i∈i 〈λic′f , λicf 〉a ≤ bcc′〈f , f 〉a. therefore ‖(c∗ �∗i �ic ′)1/2f‖≤ (1 + λ1)‖(c∗λ∗i λic ′)1/2f‖+ci〈k∗f ,k∗f 〉 1 2 1 −λ2 , ‖((c∗ �∗i �ic ′)1/2f‖2≤ ( (1 + λ1) √ bcc′ + η‖k‖ 1 −λ2 ))2〈f , f 〉a. now, for the lower bound we get ‖(c∗ �∗i �ic ′)1/2‖ = ‖c∗λ∗i λic ′)1/2f − (c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖ > ‖c∗λ∗i λic ′)1/2f‖−‖(c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖ > ‖c∗λ∗i λic ′)1/2f‖−λ1‖(c∗λ∗i λic ′)1/2f‖ −λ2‖c∗ �∗i �ic ′)1/2f‖−ci〈k∗f ,k∗f 〉 1 2 . therefore (1 + λ2)‖(c∗ �∗i �ic ′)1/2f‖> (1 −λ1)‖(c∗λ∗i λic ′)1/2f‖−ci〈k∗f ,k∗f 〉 1 2 or ‖(c∗ �∗i �ic ′)1/2f‖> (1 −λ1)‖(c∗λ∗i λic ′)1/2f‖−ci〈k∗f ,k∗f 〉 1 2 (1 + λ2) . since, ‖t∗cc′‖ 2= ‖(c∗λ∗i λic ′)1/2f‖2= ∑ i∈i 〈λic′f , λicf 〉a > acc′〈k∗f ,k∗f 〉a. thus ‖(c∗λ∗i λic ′)1/2f‖2> ( (1 −λ1) √ acc′ −η (1 + λ2) )2〈k∗f ,k∗f 〉a. � proposition 4.1. let λ be a cc′-controlled k−g− frame on hilbert amodule h with bounds acc′ and bcc′. assume that � := {�i ∈ end∗a(h,hi)i∈i} is a sequence of operators such that for each f ∈ h and i ∈ i, ‖(c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖≤ ci〈k∗f ,k∗f 〉 1 2 . where {ci}i∈i is a sequence of positive numbers such that η := ∑ i∈i c 2 i < ∞. then � is a cc ′controlled k −g− frame on hilbert amodule h with bounds : ( √ acc′ −η)2, ( √ bcc′ + η‖k‖)2. 16 int. j. anal. appl. (2022), 20:14 proof. for each f ∈ h, we have ‖(c∗ �∗i �ic ′)1/2f‖ = ‖c∗λ∗i λic ′)1/2f − (c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖ > ‖c∗λ∗i λic ′)1/2f‖−‖(c∗λ∗i λic ′ −c∗ �∗i �ic ′)1/2f‖ > √ acc′〈k∗f ,k∗f 〉 1 2 −η〈k∗f ,k∗f 〉 1 2 > ( √ acc′ −η)〈k∗f ,k∗f 〉 1 2 . thus ‖(c∗ �∗i �ic ′)1/2f‖2> ( √ acc′ −η)2〈k∗f ,k∗f 〉a. on the other hand ‖c∗ �∗i �ic ′)1/2f‖ = ‖(c∗ �∗i �ic ′ −c∗λ∗i λic ′)1/2f + (c∗λ∗i λic ′)1/2f‖ ≤‖(c∗ �∗i �ic ′ −c∗λ∗i λic ′)1/2f‖+‖(c∗λ∗i λic ′)1/2f‖ ≤ √ bcc′〈f , f 〉 1 2 + η〈k∗f ,k∗f 〉 1 2 a ≤ ( √ bcc′ + η‖k‖)〈f , f 〉 1 2 a. thus ‖(c∗ �∗i �ic ′)1/2f‖2≤ ( √ bcc′ + η‖k‖)2〈f , f 〉a. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. balazs, j.-p. antoine, a. gryboś, weighted and controlled frames: mutual relationship and first numerical properties, int. j. wavelets multiresolut inf. process. 08 (2010), 109–132. https://doi.org/10.1142/ s0219691310003377. [2] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341-366. [3] m. frank, d.r. larson, a-module frame concept for hilbert c∗-modules, funct. harmonic anal. wavelets contempt. math. 247 (2000), 207-233. [4] d. gabor, theory of communication, j. inst. elect. eng. 93 (1946), 429–457. [5] m. ghiati, s. kabbaj, h. labrigui, a. touri, m. rossafi, ∗-k-g-frames and their duals for hilbert a-modules, j. math. comput. sci. 12 (2022), 5. https://doi.org/10.28919/jmcs/6819. [6] s. kabbaj, m. rossafi, ∗-operator frame for end∗a(h), wavelet linear algebra, 5 (2018), 1-13. [7] a. khosravi, b. khosravi, fusion frames and g-frames in hilbert c*-modules, int. j. wavelets multiresolut. inform. process. 06 (2008), 433–446. https://doi.org/10.1142/s0219691308002458. [8] m. rashidi-kouchi, a. rahimi, on controlled frames in hilbert c*-modules, int. j. wavelets multiresolut. inform. process. 15 (2017), 1750038. https://doi.org/10.1142/s0219691317500382. [9] f. d. nhari, r. echarghaoui, m. rossafi, k − g−fusion frames in hilbert c∗−modules, int. j. anal. appl. 19 (2021), 836-857. https://doi.org/10.28924/2291-8639-19-2021-836. https://doi.org/10.1142/s0219691310003377 https://doi.org/10.1142/s0219691310003377 https://doi.org/10.28919/jmcs/6819 https://doi.org/10.1142/s0219691308002458 https://doi.org/10.1142/s0219691317500382 https://doi.org/10.28924/2291-8639-19-2021-836 int. j. anal. appl. (2022), 20:14 17 [10] w. l. paschke, inner product modules over b∗-algebras, trans. am. math. soc. 182 (1973), 443–468. https: //doi.org/10.1090/s0002-9947-1973-0355613-0. [11] m. rossafi, s. kabbaj, ∗-k-operator frame for end∗a(h), asian-eur. j. math. 13 (2020), 2050060. https: //doi.org/10.1142/s1793557120500606. [12] m. rossafi, s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra, 8 (2019), 85-95. [13] m. rossafi, s. kabbaj, ∗-k-g-frames in hilbert a-modules, j. linear topol. algebra, 7 (2018), 63-71. [14] m. rossafi, s. kabbaj, ∗-g-frames in tensor products of hilbert c∗-modules, ann. univ. paedagog. crac. stud. math. 17 (2018), 17-25. https://doi.org/10.2478/aupcsm-2018-0002. [15] m. rossafi, s. kabbaj, generalized frames for b(h,k), iran. j. math. sci. inform. accepted. [16] x.c. xiao, x.m. zeng, some properties of g-frames in hilbert c∗-modules, j. math. anal. appl. 363 (2010), 399–408. https://doi.org/10.1016/j.jmaa.2009.08.043. [17] q. xu, l. sheng, positive semi-definite matrices of adjointable operators on hilbert c*-modules, linear algebra appl. 428 (2008), 992–1000. https://doi.org/10.1016/j.laa.2007.08.035. [18] l.c. zhang, the factor decomposition theorem of bounded generalized inverse modules and their topological continuity, acta math. sinica. 23 (2007), 1413–1418. https://doi.org/10.1007/s10114-007-0867-2. https://doi.org/10.1090/s0002-9947-1973-0355613-0 https://doi.org/10.1090/s0002-9947-1973-0355613-0 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.2478/aupcsm-2018-0002 https://doi.org/10.1016/j.jmaa.2009.08.043 https://doi.org/10.1016/j.laa.2007.08.035 https://doi.org/10.1007/s10114-007-0867-2 1. introduction and preliminaires 2. some properties of controlled k-g-frames 3. duals of controlled k-g-frames 4. the stability problem of controlled k-g-frames references int. j. anal. appl. (2023), 21:19 the legitimacy of admission criteria in predicting the achievement of the students in the medical colleges at the university of hafr al-batin lulah alnaji∗ department of mathematics, university of hafr al batin, saudi arabia ∗corresponding author: laalnaji@uhb.edu.sa abstract. the purpose of this study is to show how the admission criteria can predict first-year college students’ performance. the study uses the data of the students’ high school gpa (hsgpa) and the scores of the prerequirement standardized tests in the kingdom of saudi arabia(achievement test score (act) and aptitude score (apt)), and the data of the first-year college gpa for correlations analysis and inferential statistics. as far as we are aware, no study has been conducted or has been made publicly accessible regarding the validity of admission criteria of the medical colleges at the university of hafr batin. the bayesian information criterion and the akaike information criterion are applied as model selection criteria in order to select the best model. actual data is used to establish the legitimacy of the admission criteria. 1. introduction making standardized tests a requirement for admission to universities has gained popularity in recent decades. the effectiveness of admission examinations in predicting students’ chances of success and predicting their college gpa based on entrance exams high school gpa (hsgpa), act score, and apt score has been the subject of several studies undertaken across the world. a significant number of research have focused on analyzing student populations from various universities. when examining the validity of the entrance test results for various institutes, it might be challenging to get an overall mean connection between the admission test scores and hsgpa since the data must be integrated in an attentive manner. for instance, [1] separately determined the correlations for each institution, adjusted the correlations using the [2] correction technique, and then calculated the average correlation received: dec. 30, 2022. 2020 mathematics subject classification. 62j05, 62j20, 62h20. key words and phrases. regression; bayesian information criterion; akaike information criterion; gpa; college admission. https://doi.org/10.28924/2291-8639-21-2023-19 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-19 2 int. j. anal. appl. (2023), 21:19 for numerous institutes. [1]’s report states that when comparing private and public organizations, the results of a thorough investigation into the validity of admissions exams were presented, and it was discovered that the correlation between the admission tests and hsgpa and (fygpa) is a little bit lower for public schools. [3]’s investigation of the numerous correlations between entrance test scores and fygpa of student populations across a variety of universities was part of a study that concentrated on several correlations of various institutions. the correlation between admission exam scores and fygpa of student populations in institutions with a greater range of extracurricular activities is less than that of those who stayed in the institutes’ housing, according to the conclusions of their study. the study highlighted on several aspects of the report, such as the higher correlation between entrance test scores and fygpa for liberal arts institutions than for the other universities included in the report. additionally, small towns’ institutes show a stronger correlation when compared to those in large cities [4] looked at student demographics across many university of california campuses for their research, and they got to the conclusion that there are differences between campuses in the correlation between entrance test scores and fygpa. although research including many institutions are helpful, they do not directly address the degree to which entrance exams correspond to student performance across multiple institutes or the related factors that contribute to that variability. it can be difficult to extrapolate the results of studies on the accuracy of standardized tests and hsgpa in forecasting students’ achievement across a variety of schools. each institute has a multi-layered clustering of its student body that takes into account factors like hsgpa, programs, colleges, and a variety of other factors. the requirements and testing processes for admission to various colleges within the same university vary as well. in order to give more comprehensive information on the greatest predictor of college students’ gpas, several institutes conducted regional research. other studies have examined the relationships between certain departments and entrance test results, while others have examined the relationships between the gpas of specific courses and the scores on those exams. the average achievement test score (ach), high school class rank, and the sat were determined to be the top predictors of student accomplishment at the university of pennsylvania, according to [5] research on the best predictors of admission requirements. at the university of california, san diego, researchers examined the gpas of more than 5,000 students [6]. the article came to the conclusion that both the hsgpa and sat greatly improve student performance predictions. using a sample of 521 students enrolled in a few classes of principles of economics courses, [7] investigated the extent to which the hsgpa and sat may predict performance at the university of south carolina. admission to saudi universities was based on the hsgpa prior to the saudi ministry of education turning toward the use of standardized exams as entry criteria for institutions. saudi colleges employ int. j. anal. appl. (2023), 21:19 3 high school grade point average and standardized examinations like the act and apt to assess admission. the act and apt exams are usually viewed by saudi colleges as being better and more accurate indicators of success than the hsgpa since all students are examined using the same materials and under the same conditions [8]. while the apt test’s outcome is focused on the verbal and quantitative components, the act test is a thorough review of high school courses in biology, chemistry, physics, mathematics, and english. for more details, see the following, [8]. since there are so many secondary school graduates and there aren’t enough slots in medical colleges ( college of medical sciences and college of pharmacy), students competing for places at the university of hafr al-batin are graded on standardized examinations and hsgpa, then the university of hafr al batin will make a choice based on the fygpa after initially admitting the students to the medical route for a year. students with good gpas have the option of enrolling in a department of one of the two schools of their choosing, while those with lower gpas are offered the chance to improve their grades or transfer to another college. to the best of our knowledge, no study nor results of research on the usefulness of the criteria for admission to medical colleges at the university of hafr batin have been published. the efficiency of standardized examinations and hsgpa in predicting fygpa for students in the medical colleges at the university of hafr al-batin is therefore examined for the first time in this study. the rest of this article is divided into the following sections. the descriptive statistics for the actual data are shown in section 2.the relationship between the inputs (hsgpa, apt, and act) and the outcome is demonstrated in section 3 (fygpa). the validity of the act, apt, and hsgpa in predicting fygpa is discussed in section 4 along with the results of the inferential statistics. section 5 of this report presents the investigation’s key conclusions. 2. descriptive statistics this study was conducted at the university of hafr al-batin, a public university with a predominance of female students. the study’s data were given by the university’s decision support and performance control section. the sample consisted of 197 male and 422 female students who were accepted in the fall of 2019. the university of hafr al-batin used a 4.00 scale to determine the gpa. the rstudio program was utilized for the data analysis. 2.1. dataset distribution. descriptive statistics are used in this part to characterize a dataset that was utilized in the study’s analysis in terms of its features and characteristics, tables (1) (3), for 422 female and 197 male students which represented graphically in figure (1). 4 int. j. anal. appl. (2023), 21:19 figure 1. pi-chart of the distribution of the dataset 2.2. central tendency. concentrating on the measures that characterize a variety of central measurements such as the mean and the median by looking at the usual central values within a dataset. the mean and median measures are clearly bigger for female students than male students for all scores. for instance, for female students, the median measurements for act, apt, hsgpa, and fygpa are m =85, m =82, m =98, and m =3.51, while for male students are m =77, m =79, m = 94, and m = 3.2, respectively. also, for female students, the mean measurements for act, apt, hsgpa, and fygpa are m =84.1, m =80.8, m =97.03, and m =3.45, meanwhile for male students are m =77.8, m =79.09, m =93.9, and m =3.2, respectively. it is noticeable the mean of hsgpa is higher than the mean of fygpa for both genders, which is an indication that the studies at the university are harder than the high schools. 2.3. variability. the dispersion or variability of a dataset is described in this section. considering that the term "variability" is used to describe a variety of metrics rather than just one single measure, such as standard deviation, range, kurtosis, skewness, minimum, and maximum values. the values of skewness and kurtosis of the act, apt and hsgpa do not exceed [-1, +1] and hence the distribution of the data is considered to be normal. • standard deviation: it is notable that for both genders, the amount of variation or dispersion for the standardized test is high while it is low for hsgpa which implies that most hsgpas are close to the mean of hsgpa. due to grading practices among high schools, high school grades are frequently seen as an unreliable criterion for college admissions, whereas standardized tests are seen as methodologically rigorous and offer a clear differentiation which is seen as more consistent and reliable for evaluating student ability and achievement. • minimum and maximum values: the minimum scores and gpas for male students are lower than the minimum scores and gpas for female students, moreover, the maximum values for male students are also lower than the maximum values for female students in most values. int. j. anal. appl. (2023), 21:19 5 table 1. descriptive statistic of the data for all students descriptive statistics act apt hsgpa nobs 619.000000 619.000000 619.000000 minimum 64.000000 60.000000 81.000000 maximum 98.000000 96.000000 100.000000 1. quartile 76.000000 75.000000 94.000000 3. quartile 89.000000 86.000000 99.000000 mean 82.095315 80.261712 96.024233 median 82.000000 80.000000 97.000000 se mean 0.300020 0.280580 0.132329 lcl mean 81.506133 79.710707 95.764364 ucl mean 82.684497 80.812718 96.284101 variance 55.717438 48.730749 10.839218 stdev 7.464411 6.980741 3.292297 skewness -0.000542 -0.134540 -1.153213 kurtosis -0.971225 -0.736097 1.380372 6 int. j. anal. appl. (2023), 21:19 table 2. descriptive statistic of the data for female students descriptive statistics act apt hsgpa nobs 422.000000 422.000000 422.000000 minimum 65.000000 60.000000 88.000000 maximum 98.000000 96.000000 100.000000 1. quartile 78.000000 75.000000 96.000000 3. quartile 90.000000 87.000000 99.000000 mean 84.097156 80.810427 97.030806 median 85.000000 82.000000 98.000000 se mean 0.346179 0.362663 0.121868 lcl mean 83.416702 80.097572 96.791260 ucl mean 84.777611 81.523281 97.270351 variance 50.572486 55.503169 6.267457 stdev 7.111433 7.450045 2.503489 skewness -0.183496 -0.199223 -1.052477 kurtosis -0.999609 -0.868899 0.953448 int. j. anal. appl. (2023), 21:19 7 table 3. descriptive statistic of the data for male students descriptive statistics act apt hsgpa nobs 197.000000 197.000000 197.000000 minimum 64.000000 64.000000 81.000000 maximum 92.000000 90.000000 100.000000 1. quartile 73.000000 75.000000 92.000000 3. quartile 82.000000 83.000000 97.000000 mean 77.807107 79.086294 93.868020 median 77.000000 79.000000 94.000000 se mean 0.450280 0.405518 0.265246 lcl mean 76.919090 78.286556 93.344917 ucl mean 78.695123 79.886032 94.391123 variance 39.942194 32.395577 13.860044 stdev 6.319984 5.691711 3.722908 skewness 0.255537 -0.258531 -0.723645 kurtosis -0.672641 -0.609607 0.200854 8 int. j. anal. appl. (2023), 21:19 figure 2. the plot of standardized tests and hsgpa for all students int. j. anal. appl. (2023), 21:19 9 figure 3. the plot of standardized test (act) for all students by gender figure 4. the plot of standardized test (apt) for all students by gender 10 int. j. anal. appl. (2023), 21:19 figure 5. the plot hsgpa for all students by gender 3. correlation the effectiveness of the entrance requirements and fygpa is evaluated quantitatively (via correlation) in this section. in general table (4) shows a moderate correlation between each of act, apt, hsgpa, and fygpa. at a glance, it can be seen that there is a correlation between each variable (act, apt, hsgpa) and fygpa. as each value increases, fygpa also tends to increase. figure (6) demonstrates the relationship between each variable and fygpa, each dot on the plots represents an individual student and her/his combination of each variable (say act) and fygpa. things get interesting when analyzing the correlation by gender as shown in tables (5) and (6). the correlation between each variable and fygpa remains but is stronger for females than males for every single variable. in table (5) r = 0.65 for act for females in compression to r = 0.29 for males. for apt and hsgpa r =0.56 and r =0.51, respectively, for female students, in contrast to r = 0.15 and r = 0.25. for both genders, the act is the best predictor for fygpa, and this needs to be confirmed by inferential statistics as well. table 4. correlation between the inputs (act, apt, hsgpa) and the outcome (fygpa) for male and female students correlation act apt hsgpa fygpa 0.5766857 fygpa 0.443804 fygpa 0.459829 int. j. anal. appl. (2023), 21:19 11 table 5. correlation between the inputs (act, apt, hsgpa) and the outcome (fygpa) for female students correlation act apt hsgpa fygpa 0.6504502 fygpa 0.5562798 fygpa 0.5073873 table 6. correlation between the inputs (act, apt, hsgpa) and the outcome (fygpa) for male students correlation act apt hsgpa fygpa 0.2907015 fygpa 0.1590556 fygpa 0.2510924 figure 6. the plot of the correlations between (act, apt, hsgpa) and fygpa for all students figure 7. the plot of the correlations between (act, apt, hsgpa) and fygpa for females 12 int. j. anal. appl. (2023), 21:19 figure 8. the plot of the correlations between (act, apt, hsgpa) and fygpa for males 4. inferential statistics section (2) established the descriptive statistics and concentrated on describing the important aspects of the study’s dataset. moreover, section (3) demonstrated that there exists a positive correlation between each variable and fygpa for both genders ranging from weak to relatively strong correlation, depending on the variable but act remains the best predictor for female and male students. regression analysis, which is presented in this section, is also necessary to validate that. the regression analysis for the study’s data is shown in table (7), and the analysis by gender is shown in table (8). using the models below, table (7) and table (8) assess the impacts of the predictors (act, apt, and hsgpa) on the first-year college performance: y = β0 +β1x1 + � (4.1) where y is the outcome (fygpa), x1 is the explanatory variable (act), and � is the random error. y = β0 +β1x1 + � (4.2) where y is the outcome (fygpa), x1 is the explanatory variable (apt), and � is the random error. y = β0 +β1x1 + � (4.3) where y is the outcome (fygpa), x1 is the explanatory variable (hsgpa), and � is the random error. y = β0 +β1x1 +β2x2 + � (4.4) where y is the outcome (fygpa), � is the random error, and x1 and x2 are the explanatory variables act and apt respectively. y = β0 +β1x1 +β2x2 +β3x3 + � (4.5) where y is the outcome (fygpa), � is the random error, and x1, x2, and x3 are the explanatory variables act, apt, and hsgpa, respectively. when the entire percentile score for the predictors is zero, the intercepts indicate the average fygpa. it is clear that no student has a score of zero, hence it is useless to evaluate the intercepts in these specific regression models (4.1) (4.5). as a result, it is safe to leave them out. the slopes int. j. anal. appl. (2023), 21:19 13 for single-variable models (i.e. (4.1) (4.3)) show that, on average, the fygpa tends to increase by 0.057, 0.032, and 0.026 for every increase of one in the act, apt, and hsgpa, respectively. model (4.1) explains 33% of the variation of fygpa which is the highest percentage among the single-variable models, moreover, (4.1) has the smallest value in compression to (4.2) and (4.3), but in compression to (4.1) and (4.5), clearly, the model that included all the variables (4.5) is the best. table 7. regression analysis for all students coeff model model 1 model 2 model 3 model 4 model 5 num.obs. 619 619 619 619 619 (intercept) 0.767 1.277 -2.137 0.466 -2.340 act 0.057 0.027 0.020 apt 0.032 0.009 0.010 hsgpa 0.026 0.034 r2 0.333 0.211 0.197 0.347 0.410 r2 adj. 0.331 0.196 0.210 0.345 0.407 aic 410.3 524.8 513.5 398.7 338.1 bic 423.6 538.1 526.8 416.4 360.3 in order to see how gender influences the variances in fygpa, the following models will be used: y = β0 +β1x1 +β2x2 + � (4.6) where y is the outcome (fygpa), x1 and x2 are the explanatory variable (act) and gender, and � is the random error. y = β0 +β1x1 +β2x2 + � (4.7) where y is the outcome (fygpa), x1 and x2 are the explanatory variable (apt) and gender, and � is the random error. y = β0 +β1x1 +β2x2 + � (4.8) where y is the outcome (fygpa), x1 and x2 are the explanatory variable (hsgpa) and gender, and � is the random error. y = β0 +β1x1 +β2x2 +β3x3 + � (4.9) 14 int. j. anal. appl. (2023), 21:19 where y is the outcome (fygpa), x1, x2 and x3 are the variables act, apt and gender, respectively, and � is the random error. y = β0 +β1x1 +β2x2 +β3x3 +β4x4 + � (4.10) where y is the outcome (fygpa), x1, x2, x3 and x4 are the variables act, apt, hsgpa and gender, respectively, and � is the random error. the intercepts show the average fygpa when all of the predictors’ percentile scores are 0. no student has a score of zero, hence it serves no purpose to interpret the intercepts in these particular regression models (4.6)-(4.10). as a result, it is safe to neglect them. when gender is taken into account. according to each model (4.6-4.8), an actscore change of one percentile correlates to a change in fygpa of 0.052 points on average and males’ fygpas drop on average by −0.055 points, while a one percentile change in the apt score correlates to a change in fygpa of 0.030 and males’ fygpas drop on average by −0.20 points, and finally, in the model (4.8), gender is taken into account, a one percentile change in hsgpa score is correlated with a 0.025 point change in fygpa, on average, and males’ fygpas drop on average by −0.081 points. model (4.10) explains greater differences in fygpa than models (4.64.9), according to r2adj model comparisons. additionally, model (4.10) is the best model, according to aic and bic model comparisons, as it has lower values than models (4.64.9). table 8. regression analysis by gender coeff model model 6 model 7 model 8 model 9 model 10 num.obs. 619 619 619 619 619 (intercept) 0.895 1.467 -1.620 0.610 -2.388 act 0.052 0.024 0.020 apt 0.030 0.010 0.009 hsgpa 0.025 0.035 gendermale -0.055 -0.204 -0.081 -0.07 6 0.008 r2 0.336 0.250 0.218 0.353 0.410 r2 adj. 0.334 0.247 0.216 0.350 0.406 aic 409.2 484.8 510.2 394.9 340.0 bic 427.0 502.5 527.9 417.0 366.6 int. j. anal. appl. (2023), 21:19 15 5. conclusion this article found that act is a better predictor of first-year gpa for medical students than apt and hsgpa since they have different effects on fygpa based on a student’s gender. in order to better understand the connections between academic ability and academic success, this article examines the academic components typically utilized in college entrance applications and used by the university to assess admission. future studies may take into account extracurricular aspects including family income, parents’ educational backgrounds, and environmental influences. investigating the relationship between the graduate gpa in compression and the fygpa is another option. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] j.l. kobrin, b.f. patterson, e.j. shaw, et al. validity of the sat® for predicting first-year college grade point average, research report no. 2008-5., college board (2008). [2] d. lawley, iv.—a note on karl pearson’s selection formulæ, proc. r. soc. edinburgh sect. a: math. 62 (1944), 28–30. https://doi.org/10.1017/s0080454100006385. [3] l.l. baird, predicting predictability: the influence of student and institutional characteristics on the prediction of grades, ets res. rep. ser. 1983 (1983), i–11. https://doi.org/10.1002/j.2330-8516.1983.tb00030.x. [4] r. zwick, t. brown, j.c. sklar, california and the sat: a reanalysis of university of california admissions data, research & occasional paper series: cshe.8.04. university of california, berkeley, (2004). [5] j. baron, m.f. norman, sats, achievement tests, and high-school class rank as predictors of college performance, educ. psychol. measure. 52 (1992), 1047–1055. https://doi.org/10.1177/0013164492052004029. [6] j.r. betts, d. morell, the determinants of undergraduate grade point average: the relative importance of family background, high school resources, and peer group effects, j. human resources. 34 (1999), 268-293. https://doi.org/10.2307/146346. [7] e. cohn, s. cohn, d.c. balch, et al. determinants of undergraduate gpas: sat scores, high-school gpa and highschool rank, econ. educ. rev. 23 (2004), 577–586. https://doi.org/10.1016/j.econedurev.2004.01.001. [8] national center for assessment in higher education, https://etec.gov.sa/en/pages/default.aspx. https://doi.org/10.1017/s0080454100006385 https://doi.org/10.1002/j.2330-8516.1983.tb00030.x https://doi.org/10.1177/0013164492052004029 https://doi.org/10.1016/j.econedurev.2004.01.001 https://etec.gov.sa/en/pages/default.aspx 1. introduction 2. descriptive statistics 2.1. dataset distribution 2.2. central tendency 2.3. variability 3. correlation 4. inferential statistics 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 18-27 http://www.etamaths.com a weak contraction principle in partially ordered cone metric space with three control functions binayak s. choudhury1, l. kumar2, t. som2,∗ and n. metiya3 abstract. in this paper we utilize three functions to define a weak contraction in a cone metric space with a partial order and establish that this contraction has necessarily a fixed point either under the continuity assumption or an order condition which we state here. the uniqueness of the fixed point is also derived under some additional assumptions. the result is supported with an example. the methodology used is a combination of order theoretic and analytic approaches. 1. introduction in this paper we consider a fixed point theorem in a space which is a generalization of metric space, namely, cone metric space. the space is introduced by allowing the metric to take up values in banach spaces. following the work of huang et al in [19], fixed point theory has experienced a rapid growth in cone metric spaces. a review of this development is given in [22]. weak contraction principle is a generalization of banach’s contraction principle which was first given by alber et al. in hilbert spaces [1]. it was subsequently extended to metric spaces by rhoades [30] and further generalized by several authors like dutta and choudhury [16], popescu [28], choudhury and kundu [9] etc. the weak contraction principle has been recently extended to cone metric spaces [6]. several weak contractive inequalities have been used in the fixed point theory in metric and cone metric spaces. references [7], [8], [10], [14], [32] are some examples of these works. in an attempt to blend the order theoretic and analytic aspects of fixed point theory, several authors have created a number of fixed point results in partially ordered metric spaces. some of these works are noted in [12], [18], [27], [29]. in cone metric spaces also, such efforts have been made in [2], [3], [11], [23] for examples. the purpose here is to establish weak contraction results in partially ordered cone metric spaces by using three control functions. control functions first appeared in fixed point theory in the work of khan et al. [25] and afterward this function and its generalizations have been used in a number of fixed point problems like [4], [5], [26], [31]. our result is illustrated with an example. 2. mathematical preliminaries 2010 mathematics subject classification. 54h10, 54h25. key words and phrases. partially ordered set; cone metric space; weak contraction; control function; fixed point. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 18 contraction principle in partially ordered cone metric space 19 definition 2.1 [19] let e always be a real banach space and p a subset of e. p is called a cone if and only if: (i) p is nonempty, closed, and p 6= {0}, (ii) a,b ∈ r, a,b ≥ 0, x,y ∈ p =⇒ ax + by ∈ p , (iii) x ∈ p and −x ∈ p =⇒ x = 0. given a cone p ⊂ e, a partial ordering ≤ with respect to p is naturally defined by x ≤ y if and only if y −x ∈ p , for x,y ∈ e. we shall write x < y to indicate that x ≤ y but x 6= y, while x � y will stand for y −x ∈ intp , where intp denotes the interior of p . the cone p is said to be normal if there exists a real number k ≥ 1 such that for all x,y ∈ e, 0 ≤ x ≤ y ⇒‖x‖≤ k‖y‖. the least positive number k satisfying the above statement is called the normal constant of p . the cone p is called regular if every increasing sequence which is bounded from above is convergent; that is, if {xn} is a sequence such that x1 ≤ x2 ≤ ... ≤ xn ≤ ... ≤ y, for some y ∈ e, then there is x ∈ e such that ‖xn − x‖ −→ 0 as n −→ ∞. equivalently, the cone p is regular if and only if every decreasing sequence which is bounded from below is convergent. it is well known that a regular cone is a normal cone. in the following we always suppose that e is a real banach space with cone p in e with intp 6= ∅ and ≤ is the partial ordering with respect to p. definition 2.2 [19] let x be a nonempty set. let the mapping d : x ×x −→ e satisfies (i) 0 ≤ d(x,y), for all x,y ∈ x and d(x, y) = 0 if and only if x = y, (ii) d(x,y) = d(y,x), for all x,y ∈ x, (iii) d(x,y) ≤ d(x,z) + d(z,y), for all x,y,z ∈ x. then d is called a cone metric on x and (x,d) is called a cone metric space. definition 2.3 [19] let (x,d) be a cone metric space, {xn} a sequence in x and x ∈ x. (i) if for every c ∈ e with 0 � c there exists n0 ∈ n such that for all n > n0, d(xn,x) � c, then {xn} is said to be convergent and {xn} converges to x, x is the limit of {xn}. we denote this by limnxn = x or xn −→ x as n −→∞. (ii) if for every c ∈ e with 0 � c there exists n0 ∈ n such that for all n,m > n0, d(xn,xm) � c, then {xn} is called a cauchy sequence in x. (iii) if every cauchy sequence in x is convergent in x, then x is called a complete cone metric space. if p is a normal cone, then {xn} converges to x if and only if d(xn,x) −→ 0 as n −→ ∞ and {xn} is a cauchy sequence if and only if d(xn,xm) −→ 0 as n,m −→∞ [19]. definition 2.4 let ψ : intp ∪{0}−→ intp ∪{0} be a function. 20 choudhury, kumar, som and metiya (i) we say ψ is strongly monotone increasing if for x, y ∈ intp ∪{0} x ≤ y ⇐⇒ ψ(x) ≤ ψ(y). (ii) ψ is said to be continuous at x0 ∈ intp ∪{0} if for any sequence {xn} in intp ∪{0}, xn −→ x0 =⇒ ψ(xn) −→ ψ(x0). the following is the definition of altering distance function in cone metric space. definition 2.5 a function ψ : intp ∪{0} −→ intp ∪{0} is called an altering distance function if the following properties are satisfied: (i) ψ is strongly monotone increasing and continuous, (ii) ψ(t) = 0 if and only if t = 0. definition 2.6 [20] let (x, d) be a cone metric space, t : x −→ x and x0 ∈ x. then the function t is continuous at x0 if for any sequence {xn} in x, xn −→ x0 implies txn −→ tx0. definition 2.7 [18] let (x,�) be a partially ordered set and t : x −→ x be a self map. we say that t is monotone non decreasing if x, y ∈ x, x � y =⇒ tx � ty. lemma 2.1. let e be a real banach space with cone p in e. then (i) if a ≤ b and b � c, then a � c [21], (ii) if a � b and b � c, then a � c [21], (iii) if 0 ≤ x ≤ y and a ≥ 0, where a is real number, then 0 ≤ ax ≤ ay [21], (iv) if 0 ≤ xn ≤ yn, for n ∈ n and limn xn = x, limn yn = y, then 0 ≤ x ≤ y [21], (v) p is normal if and only if xn ≤ yn ≤ zn and limn xn = limn zn = x imply limn yn = x [13]. lemma 2.2 [7] let (x, d) be a cone metric space with regular cone p such that d(x, y) ∈ intp, for x, y ∈ x with x 6= y. let φ : intp ∪{0} −→ intp ∪{0} be a function with the following properties: (i) φ(t) = 0 if and only if t = 0, (ii) φ(t) � t, for t ∈ intp and (iii) either φ(t) ≤ d(x, y) or d(x, y) ≤ φ(t), for t ∈ intp ∪{0} and x, y ∈ x. let {xn} be a sequence in x for which {d(xn, xn+1)} is monotonic decreasing. then {d(xn, xn+1)} is convergent to either r = 0 or r ∈ int p . lemma 2.3 [10] let (x, d) be a cone metric space. let φ : intp ∪ {0} −→ intp ∪{0} be a function such that (i) φ(t) = 0 if and only if t = 0 and (ii) φ(t) � t, for t ∈ intp. then a sequence {xn} in x is a cauchy sequence if and only if for every c ∈ e with 0 � c there exists n0 ∈ n such that d(xn,xm) � φ(c), for all n,m > n0. 3. main results lemma 3.1. let (x,d) be a cone metric space with regular cone p such that d(x, y) ∈ intp, for x, y ∈ x with x 6= y. let φ : intp ∪{0} −→ intp ∪{0} be a function with the following properties. (i) φ(t) = 0 if and only if t = 0, (ii) φ(t) � t, for t ∈ intp and contraction principle in partially ordered cone metric space 21 (iii) either φ(t) ≤ d(x, y) or d(x, y) ≤ φ(t), for t ∈ intp ∪{0} and x, y ∈ x. let {xn} be a sequence in x and x ∈ x for which {d(xn, x)} is monotonic decreasing. then {d(xn, x)} is convergent to either r = 0 or r ∈ int p . proof. let {xn} be a sequence in x and x ∈ x for which {d(xn, x)} is monotonic decreasing. since cone p is regular and 0 ≤ d(xn, x), for all n ∈ n, there exists r ∈ p such that d(xn, x) −→ r as n −→∞. if d(xn, x) = 0, for some n then trivially r = 0. hence we shall assume that d(xn, x) 6= 0, for all n ∈ n. then according to the conditions of the lemma, d(xn, x) ∈ int p , for all n ∈ n. let r 6= 0. since p is a regular cone, it is also a normal cone. let b = {t ∈ int p : ‖t‖ < ‖r‖ k }, where k is the normal constant of the cone p . for every positive real number a with a < ‖r‖ k and t ∈ int p , at‖t‖ ∈ b. therefore, b is non empty. now we claim that for every t ∈ b, φ(t) ≤ d(xn, x), for all n ∈ n. otherwise, there exists t0 ∈ b for which we can find a positive integer m such that d(xm, x) < φ(t0) (using the property (iii) of φ in the lemma). since {d(xn, x)} is monotonic decreasing, we have d(xn, x) ≤ d(xm, x) < φ(t0), for all n ≥ m, which implies d(xn, x) < φ(t0), for all n ≥ m. letting n −→∞ in the above inequality, by (iv) of lemma 2.1 and using a property of φ, we have r ≤ φ(t0) � t0, which implies ‖r‖≤ k‖t0‖, where k is the normal constant of cone p . that is, ‖t0‖≥ ‖r‖ k , which contradicts our assumption that t0 ∈ b. hence for every t ∈ b, φ(t) ≤ d(xn, x), for all n ∈ n. letting n −→∞ in the above inequality, we have φ(t) ≤ r. therefore, for every t ∈ b, r −φ(t) ∈ p ; that is, r = φ(t) + q, for some q ∈ p . now, 0 ≤ q � φ(t) + q (since φ(t) ∈ int p , for every t ∈ b). then by (i) of lemma 2.1, 0 � φ(t) + q; that is, 0 � r. therefore, r ∈ int p . hence the proof is completed. � theorem 3.1. let (x, �) be a partially ordered set and suppose that there exists a cone metric d in x for which the cone metric space (x, d) is complete with regular cone p such that d(x, y) ∈ intp , for x, y ∈ x with x 6= y. let t : x −→ x be a continuous and non decreasing mapping such that for all comparable x, y ∈ x ψ(d(tx,ty)) ≤ η(d(x,y)) −φ(d(x,y)), (3.1) where ψ, η, φ : intp ∪{0} −→ intp ∪{0} are such that ψ and η are continuous, φ is lower semi-continuous and also (i) ψ is strongly monotonic increasing, (ii) ψ(t) = η(t) = φ(t) = 0 if and only if t = 0, (iii) ψ(t) −η(t) + φ(t) > 0 for all t ∈ intp , (iv) φ(t) � t, for t ∈ intp and 22 choudhury, kumar, som and metiya (v) either φ(t) ≤ d(x, y) or d(x, y) � φ(t), for t ∈ intp ∪{0} and x, y ∈ x. if there exists x0 ∈ x such that x0 � tx0, then t has a fixed point in x. proof. let x0 ∈ x be such that x0 � tx0. since t is nondecreasing w.r.t. �, we construct the sequence {xn} such that xn = txn−1 = tnx0 and x0 � tx0 � t2x0 � ... � tnx0 � tn+1x0 � ...; that is x0 � x1 � x2 � ... � xn � xn+1 � .... (3.2) since x = xn and y = xn+1 are comparable, from (3.1), we have ψ(d(txn,txn+1)) ≤ η(d(xn,xn+1)) −φ(d(xn,xn+1)). (3.3) now, for all n ≥ 1, we have ψ(d(txn−1,txn)) −ψ(d(txn,txn+1)) ≥ ψ(d(txn−1,txn)) −η(d(xn,xn+1)) + φ(d(xn,xn+1)) = ψ(d(xn,xn+1)) −η(d(xn,xn+1)) + φ(d(xn,xn+1)) ≥ 0. (by (ii) and (iii)) this implies that ψ(d(txn,txn+1)) ≤ ψ(d(txn−1,txn)). then by (i), it follows that d(txn,txn+1) ≤ d(txn−1,txn), that is, d(xn+1,xn+2) ≤ d(xn,xn+1). therefore, {d(xn, xn+1)} is a monotone decreasing sequence. hence by lemma 2.2, there exists an r ∈ intp ∪{0} such that d(xn, xn+1) −→ r as n −→∞. (3.4) taking the limit as n → ∞ in (3.3) and using (3.4), continuities of ψ, η and the lower semi continuity of φ, we have ψ(r) ≤ η(r) −φ(r), that is, ψ(r) −η(r) + φ(r) ≤ 0, which by (ii) and (iii) implies that r = 0. hence, we have lim n→∞ d(xn+1, xn) −→ 0. (3.5) next we show that {xn} is a cauchy sequence. if {xn} is not a cauchy sequence, then by lemma 2.3, there exists a c ∈ e with 0 � c such that ∀ n0 ∈ n, ∃ n, m ∈ n with n > m > n0 such that d(xn, xm) <≮ φ(c). hence by a property of φ in (v) of the theorem, φ(c) ≤ d(xn, xm). therefore, there exist sequences {n(k)} and {m(k)} in n such that for all positive integers k, n(k) > m(k) > k and d(xn(k), xm(k)) ≥ φ(c). assuming that n(k) is the smallest such positive integer, we get d(xn(k), xm(k)) ≥ φ(c) and d(xn(k)−1, xm(k)) � φ(c). now, φ(c) ≤ d(xn(k), xm(k)) ≤ d(xn(k), xn(k)−1) + d(xn(k)−1, xm(k)), that is, φ(c) ≤ d(xn(k), xm(k)) ≤ d(xn(k), xn(k)−1) + φ(c). letting k −→∞ in the above inequality, using (3.5) and the property (v) of lemma 2.1, we have lim k→∞ d(xn(k), xm(k)) = φ(c). (3.6) again, d(xn(k), xm(k)) ≤ d(xn(k), xn(k)+1) + d(xn(k)+1, xm(k)+1) + d(xm(k)+1, xm(k)) and d(xn(k)+1, xm(k)+1) ≤ d(xn(k)+1, xn(k)) + d(xn(k), xm(k)) + d(xm(k), xm(k)+1). contraction principle in partially ordered cone metric space 23 letting k −→∞ in above inequalities, using (3.5) and (3.6), we have lim k→∞ d(xn(k)+1, xm(k)+1) = φ(c). (3.7) since for x = xn(k) and y = xm(k) are comparable, from (3.1), we have ψ(d(xn(k)+1, xm(k)+1)) = ψ(d(txn(k), txm(k))) ≤ η(d(xn(k), xm(k))) −φ(d(xn(k), xm(k))). letting k −→ ∞ in the above inequality, using (3.6), (3.7) and the continuities of ψ, η and the lower semi continuity of φ, we have ψ(φ(c)) ≤ η(φ(c)) −φ(φ(c)), that is, ψ(φ(c)) −η(φ(c)) + φ(φ(c)) ≤ 0, which by (ii) and (iii) implies that φ(c) = 0. now, by (ii), it follows that c = 0, which is a contradiction. hence, {xn} is a cauchy sequence. from the completeness of x, there exists u ∈ x such that xn −→ u as n −→∞. (3.8) since t is continuous and xn −→ u, we have u = lim n→∞ xn+1 = lim n→∞ txn = tu and this proves that u is a fixed point of t . � in our next theorem we relax the continuity assumption on t by imposing an order condition. theorem 3.2. let (x, �) be a partially ordered set and suppose that there exists a cone metric d in x for which the cone metric space (x, d) is complete with regular cone p such that d(x, y) ∈ intp , for x, y ∈ x with x 6= y. assume that if {xn} is a nondecreasing sequence in x such that xn −→ x then xn � x, for all n ∈ n. let t : x −→ x be a nondecreasing mapping such that for all comparable x, y ∈ x, (3.1) holds where the conditions upon ψ, η and φ are the same as in theorem 3.1. if there exists x0 ∈ x such that x0 � tx0, then t has a fixed point in x. proof. we take the same sequence {xn} as in the proof of theorem 3.1. then we have x0 � x1 � x2 � x3 � ... � xn � xn+1 � ..., that is, {xn} is nondecreasing sequence. also, this sequence converge to u. then xn � u, for all n ∈ n. therefore, we can use the condition (3.1) for x = u, y = xn and so we have ψ(d(tu, xn+1)) = ψ(d(tu, txn)) ≤ η(d(u, xn)) −φ(d(u, xn)). taking the limit as n −→ ∞ in the above inequality, using the properties of ψ, η and φ, we have ψ(d(tu, u)) ≤ 0. it follows by a property of ψ that d(tu, u) = 0; that is, tu = u, that is, u is a fixed point of t . � theorem 3.3 in addition to the hypotheses of theorem 3.1 and theorem 3.2, in both of the theorems, suppose that for every x, y ∈ x there exists a z ∈ x such that x � z and y � z. then t has a unique fixed point. proof. it follows from the theorem 3.1 or theorem 3.2, the set of fixed points of t is non-empty. if possible, let x, y ∈ x (x 6= y) be two fixed points of t . we distinguish two cases: case 1. suppose that x and y are comparable. without loss of generality we take y � x. 24 choudhury, kumar, som and metiya then tny = y � x = tnx, for n = 0, 1, 2, ... by the condition (3.1), we have for all n ≥ 1, ψ(d(x, y)) = ψ(d(tnx, tny)) ≤ η(d(tn−1x, tn−1y)) −φ(d(tn−1x, tn−1y)), that is, ψ(d(x, y)) ≤ η(d(x, y)) −φ(d(x, y)), that is, ψ(d(x, y)) −η(d(x, y)) + φ(d(x, y)) ≤ 0, which by (ii) and (iii) implies that d(x, y) = 0; that is, x = y. case 2. if x and y are not comparable, then there exists z ∈ x such that x � z and y � z. monotonicity of t implies that tnx = x � tnz and tny = y � tnz, for n = 0, 1, 2, ... by the condition (3.1), we have for all n ≥ 1, ψ(d(tnz, x)) = ψ(d(tnz, tnx)) ≤ η(d(tn−1z, tn−1x)) −φ(d(tn−1z, tn−1x)), that is, ψ(d(tnz, x)) ≤ η(d(tn−1z, x)) −φ(d(tn−1z, x)). (3.9) now, for all n ≥ 1, we have ψ(d(tn−1z, x)) −ψ(d(tnz, x)) = ψ(d(tn−1z, tn−1x)) −ψ(d(tnz, tnx)) = ψ(d(tn−1z, tn−1x)) −ψ(d(t(tn−1z), t(tn−1x))) ≥ ψ(d(tn−1z, tn−1x))−η(d(tn−1z, tn−1x)) +φ(d(tn−1z, tn−1x)) = ψ(d(tn−1z, x)) −η(d(tn−1z, x)) + φ(d(tn−1z, x)) ≥ 0. (by (ii) and (iii)) this implies that ψ(d(tnz, x)) ≤ ψ(d(tn−1z, x)). then by (i), it follows that d(tnz, x) ≤ d(tn−1z, x), therefore, {d(tnz, x)} is a monotone decreasing sequence. hence by lemma 3.1, there exists an r ∈ intp ∪{0} such that lim n→∞ d(tnz, x) = r. (3.10) taking the limit as n → ∞ in (3.9) and using (3.10), continuities of ψ, η and the lower semi continuity of φ, we have ψ(r) ≤ η(r) −φ(r), that is, ψ(r) −η(r) + φ(r) ≤ 0, which by (ii) and (iii) implies that r = 0. hence lim n→∞ d(tnz, x) = 0. analogously, it can proved that lim n→∞ d(tnz, y) = 0. finally, the uniqueness of the limit gives us x = y. from above two cases we have that fixed point of t is unique. � example 3.1 let x = [0, 1] with usual order � be a partially ordered set. let e = r2, with usual norm, be a real banach space. we define p = {(x,y) ∈ e : x,y ≥ 0}. the partial ordering ≤ with respect to the cone p be the partial ordering in e. then p is a regular cone. let d : x ×x −→ e be given as contraction principle in partially ordered cone metric space 25 d(x, y) = (| x−y |, | x−y |), for x, y ∈ x. then (x, d) is a complete cone metric space with the required properties of theorems 3.1 and 3.2. let ψ, η, φ : int p ∪{0}−→int p ∪{0} be defined respectively as follows: for t = (x, y) ∈ int p ∪{0}, ψ(t) =   0, if x = 0 and y = 0, (x, y), if 0 < x ≤ 1 and 0 < y ≤ 1, (x2, y), if x > 1 and 0 < y ≤ 1, (x, y2), if 0 < x ≤ 1 and y > 1, (x2, y2), if x > 1 and y > 1, and η(t) = (v, v) and φ(t) = ( v2 2 , v2 2 ), where v = min {x, y}. then ψ, η and φ have the properties mentioned in theorems 3.1 and 3.2. tx = x− x2 2 , for x ∈ x. then t has the required properties mentioned in theorems 3.1 and 3.2. without loss of generality we take x, y ∈ x with x > y. now, ψ(d(tx, ty)) = ψ(d(x− x2 2 , y − y2 2 )) = ψ(((x−y) − (x−y)(x + y) 2 , (x−y) − (x−y)(x + y) 2 )) [ since 0 ≤ (x−y)− (x−y)(x + y) 2 ≤ 1 ] = ((x−y) − (x−y)(x + y) 2 , (x−y) − (x−y)(x + y) 2 ) [ since (x−y) ≤ (x + y) ] ≤ ((x−y) − (x−y)2 2 , (x−y) − (x−y)2 2 ) = (x−y, x−y) − ( (x−y)2 2 , (x−y)2 2 ) = η((x−y, x−y)) −φ(( (x−y)2 2 , (x−y)2 2 )) = η(d(x, y)) −φ(d(x, y)). hence the conditions of theorems 3.1 and 3.2 are satisfied and it is seen that 0 is a fixed point of t . remark 3.1. it has been found that several fixed point problems in the cone metric spaces are reducible to problems of metric spaces ([15], [17], [22], [24]). this is not possible in general. particularly, weak contraction is not transferable to a corresponding weak contraction in the generated metric space and, therefore, is not derived from the results of weak contractions in metric spaces. in fact there is even no assurance that a cone metric space inequality will generate an inequality condition in metric spaces, although, it does in several important cases as has been pointed out in ([15], [17], [22], [24]). moreover, there is a problem when a partial ordering is defined on a cone metric space. in a partially ordered cone metric space with specific relations of the cone metric with the ordering, there is no natural way 26 choudhury, kumar, som and metiya of transferring these relations to metric spaces. our problem is outside the scope of those described in ([15], [17], [22], [24]). references [1] ya. i. alber and s. guerre-delabriere, principles of weakly contractive maps in hilbert spaces, oper. theory adv. appl. 98 (1997), 7-22. [2] i. altun, b. damjanović, d. djorić, fixed point and common fixed point theorems on ordered cone metric spaces, appl. math. lett. 23 (2010), 310-316. [3] i. altun, v. rakočević, ordered cone metric spaces and fixed point results, comput. math. appl. 60 (2010), 1145-1151. [4] b. s. choudhury, a common unique fixed point result in metric spaces involving generalised altering distances, math. commun. 10 (2005), 105-110. [5] b. s. choudhury, k. das, a coincidence point result in menger spaces using a control function, chaos solitons fractals 42 (2009), 3058-3063. [6] b. s. choudhury, n. metiya, fixed points of weak contractions in cone metric spaces, nonlinear anal. 72 (2010), 1589-1593. [7] b. s. choudhury, n. metiya, the point of coincidence and common fixed point for a pair of mappings in cone metric spaces, comput. math. appl. 60 (2010), 1686-1695. [8] b. s. choudhury, p. konar, b. e. rhoades, n. metiya, fixed point theorems for generalized weakly contractive mappings, nonlinear anal. 74 (2011), 2116-2126. [9] b. s. choudhury, a. kundu, (ψ, α, β) weak contractions in partially ordered metric spaces, appl. math. lett. 25 (2012), 6 10. [10] b. s. choudhury, n. metiya, coincidence point and fixed point theorems in ordered cone metric spaces, j. adv. math. stud. 5 (2012), 20-31. [11] b. s. choudhury, n. metiya, fixed point and common fixed point results in ordered cone metric spaces, an. st. univ. ovidius constanta 20 (2012), 55-72. [12] l. ćirić, m. abbas, r. saadati, n. hussain, common fixed points of almost generalized contractive mappings in ordered metric spaces, appl. math. comput. 217 (2011), 5784-5789. [13] k. deimling, nonlinear functional analysis, springer-verlage, 1985. [14] d. dorić, common fixed point for generalized (ψ, ϕ)-weak contractions, appl. math. lett. 22 (2009), 1896-1900. [15] w. s. du, a note on cone metric fixed point theory and its equivalence, nonlinear anal. 72 (2010), 2259-2261. [16] p. n. dutta, b. s. choudhury, a generalisation of contraction principle in metric spaces, fixed point theory appl. 2008 (2008), article id 406368. [17] a. a. harandi, m. fakhar, fixed point theory in cone metric spaces obtained via the scalarization method, comput. math. appl. 59 (2010), 3529-3534. [18] j. harjani, k. sadarangani, fixed point theorems for weakly contractive mappings in partially ordered sets, nonlinear anal. 71 (2009), 3403-3410. [19] l. g. huang, x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 332 (2007), 1468-1476. [20] d. ilić, v. rakoćević, common fixed point for maps on cone metric space, j. math. anal. appl. 341 (2008), 876-882. [21] s. janković, z. kadelburg, s. radenović, b. e. rhoades, assad-kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces, fixed point theory appl. 2009 (2009), article id 761086. [22] s. janković, z. kadelburg, s. radenović, on cone metric spaces: a survey, nonlinear anal. 74 (2011), 2591-2601. [23] z. kadelburg, m. pavlović, s. radenović, common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, comput. math. appl. 59 (2010), 3148-3159. [24] z. kadelburg, s. radenovic, v. rakocevic, a note on the equivalence of some metric and cone metric fixed point results, appl. math. lett. 24 (2011), 370-374. [25] m. s. khan, m. swaleh, s. sessa, fixed points theorems by altering distances between the points, bull. austral. math. soc. 30 (1984), 1-9. contraction principle in partially ordered cone metric space 27 [26] s. v. r. naidu, some fixed point theorems in metric spaces by altering distances, czechoslovak math. j. 53 (2003), 205-212. [27] j. j. nieto, r. rodroguez-lopez, existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, acta math. sinica 23 (2007), 22052212. [28] o. popescu, fixed points for (ψ, φ) weak contractions, appl. math. lett. 24 (2011), 1-4. [29] a. c. m. ran, m. c. b. reurings, a fixed point theorem in partially ordered sets and some applications to matrix equations, proc. amer. math. soc. 132 (2004), 1435-1443. [30] b. e. rhoades, some theorems on weakly contractive maps, nonlinear anal. 47 (200l), 26832693. [31] k. p. r. sastry, g. v. r. babu, some fixed point theorems by altering distances between the points, ind. j. pure. appl. math. 30 (1999), 641-647. [32] q. zhang, y. song, fixed point theory for generalized φ− weak contractions, appl. math. lett. 22 (2009), 75-78. 1department of mathematics, bengal engineering and science university, shibpur, howrah 711103, west bengal, india 2department of applied mathematics, indian institute of technology, (banaras hindu university), varanasi 221005, india 3department of mathematics, bengal institute of technology, kolkata 700150, west bengal, india ∗corresponding author international journal of analysis and applications volume 16, number 3 (2018), 306-316 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-306 solvability of multi-point value problems with integral condition at resonance rabah khaldi1,∗ and mohammed kouidri2 1laboratory of advanced materials, faculty of sciences, badji mokhtar-annaba university, p.o. box 12, 23000 annaba, algeria 2kasdi merbah ourgla university p.0.30000 ourgla, algeria ∗corresponding author: rkhadi@yahoo.fr abstract. in this paper, we study a boundary value problem at resonance with a multi-integral boundary conditions. by constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of mawhin. an example is given to show the effectiveness of our results. 1. introduction boundary value problem involves ordinary differential equation with non local condition appears in physical science and applied mathematics. moreover the theory of boundary value problems with integral condition is found in deferent areas like applied mathematics and applied physics for example plasma physics, heat conduction, themo-elasticity, underground water flew. in recent years, the boundary value problem at resonance for ordinary differential equations have been extensively studied and many results have been obtained, we refer to [1], [2], [5][7], [10][12] and the references therein. moreover, lots of works on multi-point boundary value problems have appeared, for examples, see [3][10]. received 2017-12-17; accepted 2018-02-07; published 2018-05-02. 2010 mathematics subject classification. 34b40, 34b15. key words and phrases. boundary value problem at resonance; existence of solution; coincidence degree; integral condition. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 306 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-306 int. j. anal. appl. 16 (3) (2018) 307 the goal of this paper is to provide sufficient conditions that ensure the existence of solutions for the following multi-point boundary value problem (bvp) x′′(t) = f(t,x(t),x′(t)), t ∈ (0, 1) (1.1) x(0) = 0,x(1) = m∑ k=1 λk ∫ ηk 0 x(t)dt, (1.2) where f : [0, 1] ×r2 → r is caratheodory function, ηk ∈ (0, 1) and m∑ k=1 λkη 2 k = 2. we say that the (bvp) (1.1)-(1.2) is a resonance problem if the dimension of the kernel of the linear operator lx = x′′ is not less than one under the boundary conditions 1.2. otherwise, we call them a problem at nonresonance. in the present work, if m∑ k=1 λkη 2 k = 2, then, (bvp) (1.1)-(1.2) is at resonance, since equation x′′(t) = 0, t ∈ (0, 1) with boundary condition 1.2 has nontrivial solutions x(t) = ct,c ∈ r, t ∈ [0, 1]. this paper is organized as follows, in section 2 we stated some definitions and lemmas needed in our proofs. in section 3 we treated the existence of solution by using the coincidence degree theory of mawhin [9], [10]. we ended by giving an example illustrating the previous results. 2. preliminaries for the convenience of the reader to understand the coincidence degree theory, we briefly recall some notations, definitions and theorems which will be used later. definition 2.1. let x,y , be reai banach spaces, the linear operator l : doml ⊂ x → y is said to be a fredholm map of index zero provided that ker l, the kernel of l, is of the same finite dimension as the y/ im l, where im l is the image of l. let l be a fredholm map of index zero, and p : x → x, q : y → y be continuous projectors, such that im p = ker l, kerq = im l. then x = ker l ⊕ ker p , y = im l ⊕ im q, thus l |dom l∩ker p : dom l∩ ker p → im l is invertible, denote its inverse by kp . definition 2.2. let l be a fredholm map of index zero and ω be an open bounded subset of x, such that doml ∩ ω 6= φ, the map n : x → y is said to be l − compact on ω, if qn(ω)is bounded and kp(i −q)n : ω → x is compact. int. j. anal. appl. 16 (3) (2018) 308 theorem 2.1. ( [13]) let l be a fredholm operator of index zero and let n be l−compact on ω. assume that the following conditions are satisfied. (i)lx 6= λnx, for every (x,λ) ∈ [(doml\kerl) ∩∂ω] × (0, 1). (ii) nx /∈ im l, for every x ∈ kerl∩∂ω . (iii) deg(qn|ker l, ker l∩ ω, 0) 6= 0, where q : y → y is a projection as above with im l = ker q. then, the equation lx = nx has at least one solution in doml∩ ω . in the following, we shall use the classical banach spaces x = c1[0, 1] and y = l1[0, 1] equipped respectively with the norm ‖x‖ = max{‖x‖∞ ,‖x ′‖∞}, ‖x‖∞ = max t∈[0,1] |x(t)| and ‖y‖1 = ∫ 1 0 |y(t)|dt. we will use the space ac2 [a,b] = { u ∈ c1 [a,b] ,u′ ∈ ac [a,b] } , where ac [a,b] is the space of absolutely continuous functions on [a,b] . 3. existence of solutions define the operator l : dom l ⊂ x → y by lx = x′′, where doml = { x ∈ w2,1(0, 1) : x(0) = 0,x(1) = m∑ k=1 λk ∫ ηk 0 x(t)dt; ηk ∈ (0, 1), m∑ k=1 λkη 2 k = 2 } . let n : x → y the operator nx(t) = f(t,x(t),x′(t)), t ∈ (0, 1). then, the equation 1.1, can be written as lx = nx. next, in order to apply theorem 2.1, we need the following lemma. lemma 3.1. (i) ker l = {x ∈ doml : x = ct,c ∈ r, t ∈ [0, 1]}; (ii) im l = {y ∈ y : ∫ 1 0 (1 −s)y(s)ds− 1 2 k=m∑ k=1 λk ∫η 0 (ηk −s)2y(s)ds = 0}; (iii) l : doml ⊂ x → y is a fredholm operator of index zero, and the linear continuous projector operator q : y → y can be defined as qy(t) = k.(ry).t where ry = ∫ 1 0 (1 −s)y(s)ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2y(s)ds and k−1 = 1 6 − 1 24 m∑ k=1 λkη 4 k. (iv) the linear operator kp : im l → doml∩ ker p can be written as kpy = ∫ t 0 (t−s)y(s)ds, int. j. anal. appl. 16 (3) (2018) 309 moreover, for all y ∈ im l we have ‖kpy‖≤‖y‖1 . (3.1) proof. (i) for x ∈ ker l, we have x′′ (t) = 0. then, we obtain x(t) = a + bt, where a,b ∈ r. from x(0) = 0, we have a = 0. again, since ∑m k=1 λkη 2 k = 2, then from x(1) = k=m∑ k=1 λk ∫ηk 0 x(t)dt, we get ker l = {x ∈ doml : x = ct,c ∈ r, t ∈ [0, 1]} (ii) to prove that im l = {y ∈ y : ∫ 1 0 (1 −s)y(s)ds− 1 2 k=m∑ k=1 λk ∫ ηk 0 (ηk −s) 2y(s)ds = 0}, we show that, the linear equation x′′ = y, (3.2) has a solution x(t) satisfied, x(0) = 0, x(1) = m∑ k=1 λk ∫ηk 0 x(t)dt, m∑ k=1 λkη 2 k = 2, if and only if ∫ 1 0 (1 −s)y(s)ds− 1 2 m∑ k=1 λk ∫ η 0 (ηk −s) 2y(s)ds = 0. in fact, by integrating equation 3.2 and tacking and account that x(0) = 0, we get x(t) = x′(0)t + ∫ t 0 (t−s)y(s)ds again from x(1) = k=m∑ k=1 λk ∫ηk 0 x(t)dt, we obtain x′(0) + ∫ 1 0 (1 −s)y(s)ds = m∑ k=1 λk ∫ ηk 0 [ x′(0)t + ∫ t 0 (t−s)y(s)ds ] dt = x′(0) + m∑ k=1 λk [∫ ηk 0 ∫ t 0 (t−s)y(s)dsdt ] = x′(0) + 1 2 m∑ k=1 λk [∫ ηk 0 (ηk −s) 2y(s)ds ] which implies ∫ 1 0 (1 −s)y(s)ds− 1 2 m∑ k=1 λk [∫ ηk 0 (ηk −s) 2y(s)ds ] = 0 (iii) for y ∈ y , we take the projector q as qy = k (∫ 1 0 (1 −s)y(s)ds− 1 2 m∑ k=1 λk [∫ ηk 0 (ηk −s) 2y(s)ds ]) t, int. j. anal. appl. 16 (3) (2018) 310 where k−1 = ∫ 1 0 (1 −s)sds− 1 2 m∑ k=1 λk [∫ ηk 0 (ηk −s) 2sds ] = 1 6 − 1 24 m∑ k=1 λkη 4 k. it follows from ker q = im l that y = im l⊕ im q, thus, co dim l = dim ker l = 1. hence, l is a fredholm operator of index zero. (iv) taking p : x → x as follows, px = x′(0)t then, the generalized inverse kp : im l → doml∩ ker p of l can be written as kpy = ∫ t 0 (t−s)y(s)ds in fact, for y ∈ im l, we have (lkp)y(t) = y(t) and, for x ∈ doml∩ ker p , we know (kpl)x(t) = ∫ t 0 (t−s)x′′(s)ds = x(t) this shows that kp = (l|doml∩ker p ) −1. finally from the definition of kp, we get ‖kpy‖∞ ≤ ∫ 1 0 (1 −s) |y(s)|ds ≤ ∫ 1 0 |y(s)|ds = ‖y‖1 . � now, we give the result on the existence of a solution for the problem (1.1)-(1.2). theorem 3.1. assume that the following conditions are satisfied : (h1) there exists nonnegative functions α,β,γ ∈ l1[0, 1], such that, for all (x,y) ∈ r2, t ∈ [0, 1], satisfying the following inequalities: |f(t,x,y)| ≤ α(t) |x| + β(t) |y| + γ(t) (3.3) (h2) there exists a constant m > 0, such that, for x ∈ doml, if |x′(t)| > m, for all t ∈ [0, 1], then, ∫ 1 0 (1 −s)f(s,x(s),x′(s))ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),x′(s))ds 6= 0 (h3) there exists a constant m ∗ > 0, such that, for any x(t) = ct ∈ ker l with |c| > m ∗either int. j. anal. appl. 16 (3) (2018) 311 c [∫ 1 0 (1 −s)f(s,x(s),c)ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),c)ds ] < 0 or c [∫ 1 0 (1 −s)f(s,x(s),c)ds− 1 2 k=m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),c)ds ] > 0 then bvp (1.1)-(1.2) has at least one solution in c1 [0, 1] , provided ‖α(t)‖ + ‖β(t)‖≤ 1 2 . next, in order to prove theorem 3.1, we need the following lemma. lemma 3.2. suppose that ω is an open bounded subset of x such that dom l∩ω 6= ∅. then n is l−compact on ω. proof. suppose that ω ⊂ x is a bounded set. without loss of generality, we may assume that ω = b (0,r) , then for any x ∈ ω, ‖x‖≤ r. for x ∈ ω, and by condition 3.3, we obtain |qnx| ≤ k ∫ 1 0 |f(s,x(s),x′(s))|ds + k 2 m∑ k=1 λk ∫ ηk 0 |f(s,x(s),x′(s))|ds ≤ k ∫ 1 0 |α(s)| |x(s)| + |β(s)| |x′(s)| + |γ(t)|ds + k 2 m∑ k=1 λk ∫ ηk 0 |α(s)| |x(s)| + |β(s)| |x′(s)| + |γ(t)|ds ≤ ( k + k 2 m∑ k=1 λk ) [r (‖α(t)‖1 + ‖β(t)‖1) + ‖γ(t)‖1] thus, ‖qnx‖1 ≤ ( k + k 2 m∑ k=1 λk ) [r (‖α(t)‖1 + ‖β(t)‖1) + ‖γ(t)‖1] , (3.4) which implies that qn ( ω ) is bounded. next, we show that kp (i −q) n ( ω ) is compact. for x ∈ ω, by condition 3.3 we have ‖nx‖1 = ∫ 1 0 |f(t,x(s),x′(s))|ds ≤ r (‖α(t)‖1 + ‖β(t)‖1) + ‖γ(t)‖1 . (3.5) on the other hand, from the definition of kp and together with (3.1), (3.4) and (3.5) one gets ‖kp (i −q) nx‖ ≤ ‖(i −q) nx‖1 ≤‖nx‖1 + ‖qnx‖1 ≤ ( 1 + k + k 2 m∑ k=1 λk ) (r (‖α(t)‖1 + ‖β(t)‖1) + ‖γ(t)‖1) . it follows that kp (i −q) n ( ω ) is uniformly bounded. let us prove that kp (i −q) n ( ω ) is equicontinuous. for any x ∈ ω, and any t1,t2 ∈ [0, 1] , t1 < t2, we have (kp (i −q) nx) (t1) − (kp (i −q) nx) (t2) = = ∣∣∣∣ ∫ t1 0 (t1 −s) (i −q) nx (s) ds int. j. anal. appl. 16 (3) (2018) 312 − ∫ t2 0 (t2 −s) (i −q) nx (s) ds ∣∣∣∣ ≤ [∫ t1 0 (t2 − t1) |(i −q) nx (s)|ds + ∫ t2 t1 (t2 −s) |(i −q) nx (s)|ds ] → 0, as t1 → t2. on the other hand we have ∣∣(kp (i −q) nx)′ (t1) − (kp (i −q) nx)′ (t2)∣∣ = ∣∣∣∣ ∫ t1 0 (i −q) nx (s) ds− ∫ t2 0 (i −q) nx (s) ds ∣∣∣∣ → 0, as t1 → t2. so kp (i −q) n ( ω ) is equicontinuous. so, the ascoli-arzela theorem ensure that kp(i−q)n : ω → x is compact . the proof is complete � now we give the proof of theorem 3.1 proof. firstly, we need to construct the set ω satisfying all the conditions in theorem 2.1, which is separated into the following three steps. step 1. first we show that the following set ω1 = {x ∈ doml\ker l : lx = λnx,forsomeλ ∈ [0, 1]}, is bounded. in fact, let x ∈ ω1, we have lx = λnx and lx 6= 0, so λ 6= 0, thus qnx = 0, from which it yields ∫ 1 0 (1 −s)f(s,x(s),x′(s))ds− 1 2 k=m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),x′(s))ds = 0, thus, from condition (h2) of theorem 3.1, there exists t0 ∈ [0, 1], such that |x′(t0)| ≤ m. in view of x′(0) = x′(t0) − t0∫ 0 x′′(t)dt then, ‖px‖ = |x′(0)| ≤ m + ‖x′′‖1 ≤ m + ‖nx‖1 (3.6) again for x ∈ ω1, x ∈ doml\ker l, then (i −p)x ∈ doml∩ ker p and lpx = 0, thus from lemma 3.1 , we know ‖(i −p)x‖ = ‖kpl(i −p)x‖≤‖l(i −p)x‖1 = ‖lx‖1 ≤‖nx‖1 (3.7) int. j. anal. appl. 16 (3) (2018) 313 from (3.6) and (3.7), we have ‖x‖≤‖px‖ + ‖(i −p)x‖≤ m + 2‖nx‖1 (3.8) if condition 3.3 holds, then from (3.8) , we obtain ‖x‖≤ 2 [ ‖α‖1 ‖x‖∞ + ‖β‖1 ‖x ′‖∞ + ‖γ‖1 + m 2 ] . (3.9) since ‖x‖∞ ≤‖x‖, then from (3.9) it yields ‖x‖∞ ≤ 2 1 − 2‖α‖1 [ ‖β‖1 ‖x ′‖∞ + ‖γ‖1 + m 2 ] (3.10) by using ‖x′‖∞ ≤‖x‖, (3.9) and (3.10) one has ‖x′‖∞ [ 1 − 2‖β‖1 1 − 2‖α‖1 ] ≤ 2 1 − 2‖α‖1 [ ‖γ‖1 + m 2 ] therefore, ‖x′‖∞ [ 1 − 2‖α‖1 − 2‖β‖1 1 − 2‖α‖1 ] ≤ 1 1 − 2‖α‖1 [2‖γ‖1 + m] i.e. ‖x′‖∞ ≤ 2 [ ‖γ‖1 + m 2 ] 1 − 2‖α‖1 − 2‖β‖1 , (3.11) thus, from (3.10) and (3.11), there exist m1 > 0, such that ‖x‖≤ m1 which proves that ω1 is bounded. step 2. we will show that the set ω2 = {x ∈ ker l : nx ∈ im l} is bounded. let x ∈ ω2, then x(t) = bt,b ∈ r, t ∈ [0, 1], and qnx = 0, therefor ∫ 1 0 (1 −s)f(s,bs,b)ds− 1 2 k=m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,bs,b)ds = 0 in view of condition (h2) of theorem 3.1, there exists t0 ∈ [0, 1], such that |x′(t0)| ≤ m that is |b| ≤ m, so ‖x′‖∞ = |b| ≤ m, from which, we get ‖x‖ = max{‖x‖∞ ,‖x ′‖∞} = |b| ≤ m, int. j. anal. appl. 16 (3) (2018) 314 which implies that ω2 is bounded. step 3. in the next, we show the boundedness of the following set ω3 = {x ∈ ker l : −λjx + (1 −λ)qnx = 0,λ ∈ [0, 1]}, where, j : ker l → im q is the linear isomorphism given by j(ct) = ct,∀c ∈ r, t ∈ [0, 1]. if the first part of condition (h3) of theorem 3.1 holds, that is, there exists m ∗ > 0, such that, for any c ∈ r, if |c| > m∗, then c [∫ 1 0 (1 −s)f(s,x(s),c)ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),c)ds ] < 0 ((3.12)) let x ∈ ω3, then x(t) = ct and λjx = (1 −λ)qnx, that is equivalently written as λc = (1 −λ)k [∫ 1 0 (1 −s)f(s,x(s),c)ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),c)ds ] , if λ = 1, then c = 0. otherwise, if |c| > m∗, then in view of (3.12) and together with the fact that k > 0, we get λc2 = (1 −λ)kc [∫ 1 0 (1 −s)f(s,x(s),c)ds− 1 2 m∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),c)ds ] < 0 which contradicts λc2 ≥ 0. thus, ω3 ⊂{x ∈ kerl : ‖x‖≤ m∗} is bounded. on the other hand, if the second part of condition (h3) of theorem 3.1 holds, by applying similar reasoning as above, we can prove that ω3 is bounded. finally, we shall prove that all conditions of theorem 2.1 are satisfied. let ω be a bounded open subset of x, such that ∪3i=1ωi ⊂ ω. it follows from lemma 3.1 that l is a fredholm operator of index zero. by lemma 3.2, we have n is l-compact on ω. by virtue of the definition of ω, the assumptions (i) and (ii) are satisfied. now we prove that condition (iii) of theorem 2.1 is satisfied. let h (x,λ) = ±λjx + (1 −λ) qnx. since ω3 ⊂ ω, then h (x,λ) 6= 0 for every x ∈ ker l∩∂ω. by the homotopy property of degree, we get deg (qn |ker l, ω ∩ ker l, 0) = deg ( h (·, 0) , ω ∩ ker l, 0) = deg ( h (·, 1) , ω ∩ ker l, 0) = deg ( ±j, ω ∩ ker l, 0) 6= 0. hence, the assumption (iii) of theorem 2.1 holds. since all hypothesis of theorem 2.1 are satisfied. therefore, equation lx = nx has at least one solution in dom l∩ω; i.e. boundary value problem (1.1)-(1.2) has at least one solution in x. the proof is completed. � int. j. anal. appl. 16 (3) (2018) 315 4. an illustrative example in this section we give an example to illustrate the usefulness of our main results. consider the multi-point boundary value problem (p)   x′′ (t) = f (t,x (t) ,x′(t)) , t ∈ (0, 1) , x (0) = 0, x(1) = 4 1 2∫ 0 x(t)dt + 16 9 2 3∫ 0 x(t)dt with f(t,x,y) = t 4 x + ( 1 − t2 4 ) y + t. since 2∑ k=1 λkη 2 k = 4 ( 1 2 )2 + 16 9 ( 3 4 )2 = 2, the problem (p) is at resonance. we have |f (t,x,y)| ≤ α (t) |x| + β (t) |y| + γ(t), where α (t) = t 4 ,β (t) = 1−t 2 4 and γ(t) = t, then α, β are nonnegative and belong to l1 [0, 1], so, hypothesis (h1) of theorem 3.1 is satisfied. we claim that condition (h2) of theorem 3.1 is satisfied, indeed, for m = 1. 821 4 > 0 and x ∈ doml, x(t) = ct, if |x′(t)| > m, for all t ∈ [0, 1], then, ∫ 1 0 (1 −s)f(s,x(s),x′(s))ds− 1 2 2∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),x′(s))ds = ∫ 1 0 (1 −s)( c 4 + s)ds− 1 2 ( 4 ∫ 1 2 0 ( 1 2 −s)2( c 4 + s)ds + 16 9 ∫ 3 4 0 ( 3 4 −s)2( c 4 + s)ds ) = 7 96 c + 17 128 6= 0. now, for m∗ = 2 > 0 and any x (t) = ct ∈ ker l with |c| > m∗, we have c [∫ 1 0 (1 −s)f(s,x(s),x′(s))ds− 1 2 2∑ k=1 λk ∫ ηk 0 (ηk −s) 2f(s,x(s),x′(s))ds ] = 7 96 c2 + 17 128 c > 0, consequently, condition (h3) of theorem 3.1 is satisfied. finally, a simple calculus gives ‖α‖1 +‖β‖1 = 1 8 + 1 6 ≤ 1 2 . we conclude from theorem 3.1 that the problem (p) has at least one solution in c1[0, 1]. references [1] a. guezane-lakoud and a. frioui, third order boundary value problem with integral condition at resonance, math. comput. sci. 3 (1) (2013) 56-64. int. j. anal. appl. 16 (3) (2018) 316 [2] a. guezane lakoud, r. khaldi and a. kılıçman, solvability of a boundary value problem at resonance, springer plus 5 (2016), art. id 1504. [3] a. guezane-lakoud, n. hamidane and r. khaldi, on a third-order three-point boundary value problem. int. j. math. math. sci. 2012 (2012), art. id 513189. [4] a. guezane-lakoud, r. khaldi, study of a third-order three-point boundary value problem, aip conf. proc., 1309(2010), 329-335. [5] c. p. gupta, solvability of multi-point boundary value problems at resonance, results math. 28(1995), 270-276. [6] c. p. gupta, a second order m-point boundary value problem at resonance, nonlinear anal. 24 (1995), 1483-1489. [7] c. p. gupta, existence theorems for a second order m-point boundary value problem at resonance, internat. j. math. math. sci. 18 (1995), no. 4, 705-710. [8] n. kosmatov, a multi-point boundary value problem with two critical conditions. nonlinear anal. 65 (2006), no. 3, 622–633. [9] s. k. ntouyas and p. ch. tsamatos, multi-point boundary value problems for ordinary differential equations. an. ştiinţ. univ. al. i. cuza iaşi. mat. (n.s.) 45 (1999), no. 1, 57–64 (2000). [10] lin, x., z. du. and f. meng, a note on a third-order multi-point boundary value problem at resonance. math. nachr. 284 (2011), 1690 – 1700. [11] r. ma, multiplicity results for a third order value problem at resonance, nonlinear anal. 32 (1998), no. 4, 493–499. [12] r. k. nagle and k. l. pothoven, on a third-order nonlinear boundary value problems at resonance, j. math. anal. appl. 195 (1995), no 1, 148-159. [13] j. mawhin, topological degree methods in nonlinear boundary value problems. expository lectures from the cbms regional conference held at harvey mudd college, claremont, calif., june 9–15, 1977. cbms regional conference series in mathematics, 40. american mathematical society, providence, r.i., 1979. [14] mawhin, j. topological degree and boundary value problems for nonlinear differential equations. topological methods for ordinary differential equations (montecatini terme, 1991), 74–142, lecture notes in math., 1537, springer, berlin, 1993. 1. introduction 2. preliminaries 3. existence of solutions 4. an illustrative example references int. j. anal. appl. (2022), 20:13 an extension of a variational inequality in the simader theorem to a variable exponent sobolev space and applications: the dirichlet case junichi aramaki∗ division of science, tokyo denki university, hatoyama-machi, saitama, 350-0394, japan ∗corresponding author: aramaki@hctv.ne.jp abstract. in this paper, we shall extend a fundamental variational inequality which is developed by simader in w1,p to a variable exponent sobolev space w1,p(·). the inequality is very useful for the existence theory to the poisson equation with the dirichlet boundary conditions in lp(·)-framework, where lp(·) denotes a variable exponent lebesgue space. furthermore, we can also derive the existence of weak solutions to the stokes problem in a variable exponent lebesgue space. 1. introduction in simader [25], the author derived a variational inequality of a bilinear form. more precisely, let g is a bounded domain of rd (d ≥ 2) with a c1-boundary ∂g and 1 < p < ∞. the author proved that there exists a positive constant c = c(p,g) > 0 such that ‖∇∇∇u‖lp(g) ≤ c sup 06=v∈w 1,p ′ 0 (g) |〈∇∇∇u,∇∇∇v〉g| ‖∇∇∇v‖ lp ′ (g) for all u ∈ w 1,p0 (g), (1.1) where 〈∇∇∇u,∇∇∇v〉g = ∫ g ∇∇∇u ·∇∇∇vdx, ∇∇∇ denotes the gradient operator and p′ is the conjugate exponent of p, that is, 1 p + 1 p′ = 1. he also considered the case where g is an exterior domain and got a variational inequality like as in (1.1). this inequality has many applications. for example, let v ∈lp(g), then it follows from (1.1) that the dirichlet problem for the poisson equation{ ∆u = divv in g, u = 0 on ∂g (1.2) received: jan. 5, 2022. 2010 mathematics subject classification. 35a15; 58e35; 35j25; 35d05; 35b30. key words and phrases. variational inequality; dirichlet problem for the poisson equation; the helmholtz decomposition; the stokes problem; variable exponent sobolev spaces. https://doi.org/10.28924/2291-8639-20-2022-13 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-13 2 int. j. anal. appl. (2022), 20:13 has a unique solution in a generalized sense. the equation (1.2) plays an essential role for the existence of a solution to the stokes problem (cf. fujiwara and morimoto [15] and kozono and yanagisawa [20]). it is also basic for the treatment of the navier-stokes equation, for example, see [15], miyakawa [22]. in this paper, we attempt to derive an improvement of the above variational inequality (1.1) in the sobolev opace w 1,p0 (g) to a variable exponent sobolev space w 1,p(·) 0 (g) (theorem 3.1). we restrict ourselves to the case where g is a bounded domain. though we follow the argument of [25], we have to proceed the analysis very carefully. the result brings about the existence theory of weak solutions to the dirichlet problem for the laplacian in the variable exponent sobolev space, that is, for given functions f ∈ w−1,p(·)(g) and g ∈ tr(w 1,p(·)(g)), where w 1,p(·)(g) is a variable exponent sobolev space, w−1,p(·)(g) is the dual space of w 1,p ′(·) 0 (g) and tr(w 1,p(·)(g)) denotes the trace space,{ −∆u = f in g, u = g on ∂g has a unique weak solution. according to our best knowledge, the result for the dirichlet problem in a variable exponent sobolev space is simplest. furthermore, we show that the stokes problem in a variable exponent sobolev space has a unique weak (strong) solution by a new approach which is an application of theorem 3.1. the study of differential equations with p(·)-growth conditions is a very interesting topic recently. studying such problem stimulated its application in mathematical physics, in particular, in elastic mechanics (zhikov [29]), in electrorheological fluids (diening [10], halsey [18], mihăilescu and rădulescu [21], růžička [23]). for the neumann case of the variational inequality, we gave a result in the previous work aramaki [5] (cf. simader and sohr [24] for the case p(·) = p = const.). the paper is organized as follows. in section 2, we give some preliminaries on variable exponent lebesgue-sobolev spaces. in section 3, we give main theorems (theorem 3.1) which is an extension of variational inequality of type (1.1) to one in a variable exponent sobolev space. section 4 is a preparation of a proof of the main theorem. in section 5, we give a proof of the main theorem. in section 6, we consider the dirichlet problem of the poisson equation. finally, section 7 is devoted to the existence of a weak (strong) solution for the stokes problem by a new approach. 2. preliminaries throughout this paper, we only consider vector spaces of real valued functions over r. for any space b, we denote bd by the boldface character b. hereafter, we use this character to denote vectors and vector-valued functions, and we denote the standard inner product of vectors a = (a1, . . . ,ad) and b = (b1, . . . ,bd) in rd by a ·b = ∑d i=1 aibi and |a| = (a ·a) 1/2. occasionally, we also use the same character for matrix values functions. moreover, for the dual space b′ of b (resp. b′ of b), we denote the duality bracket between b′ and b (resp. b′ and b) by 〈·, ·〉b′,b (resp. 〈·, ·〉b′,b). int. j. anal. appl. (2022), 20:13 3 in this section, we recall some well-known results on variable exponent lebesgue-sobolev spaces. see diening et al. [11], fan and zhao [14], fan and zhang [12], kovác̆ik and rácosnic [19] and references therein for more detail. let g be a (lebesgue) measurable subset of rd (d ≥ 2) with the measure |g| > 0. then we define a set of variable exponents by p(g) = {p; g → [1,∞); p is measurable in g} and for p ∈p(g), define p− = ess inf x∈g p(x) and p+ = ess sup x∈g p(x). for any real valued measurable function u on g and p ∈p(g), a modular ρp(·),g is defined by ρp(·),g(u) = ∫ g |u(x)|p(x)dx. the variable exponent lebesgue space is defined by lp(·)(g) = { u; u is a measurable function on g satisfying ρp(·),g(u) < ∞ } equipped with the luxemburg norm ‖u‖lp(·)(g) = inf { λ > 0; ρp(·),g (u λ ) ≤ 1 } . then lp(·)(g) is a banach space and it is separable if p+ < ∞ and reflexive if 1 < p− ≤ p+ < ∞. define p+(g) = {p ∈p(g); 1 < p− ≤ p+ < ∞}. the following proposition is well known (see fan et al. [13], wei and chen [26], [14], zhao et al. [28], yücedağ [27]). proposition 2.1. let g be a measurable set of rd, p ∈p+(g) and let u,un ∈ lp(·)(g) (n = 1, 2, . . .). then we have (i) ‖u‖lp(·)(g) < 1(= 1,> 1) ⇐⇒ ρp(·),g(u) < 1(= 1,> 1). (ii) ‖u‖lp(·)(g) > 1 =⇒‖u‖ p− lp(·)(g) ≤ ρp(·),g(u) ≤‖u‖ p+ lp(·)(g) . (iii) ‖u‖lp(·)(g) < 1 =⇒‖u‖ p+ lp(·)(g) ≤ ρp(·),g(u) ≤‖u‖ p− lp(·)(g) . (iv) limn→∞‖un −u‖lp(·)(g) = 0 ⇐⇒ limn→∞ρp(·),g(un −u) = 0. (v) ‖un‖lp(·)(g) →∞ as n →∞⇐⇒ ρp(·),g(un) →∞ as n →∞. the following proposition is a generalized hölder inequality. proposition 2.2. let g be a measurable set of rd and p ∈ p+(g). for any u ∈ lp(·)(g) and v ∈ lp ′(·)(g), we have∫ g |uv|dx ≤ ( 1 p− + 1 (p′)− ) ‖u‖lp(·)(g)‖v‖lp′(·)(g) ≤ 2‖u‖lp(·)(g)‖v‖lp′(·)(g), where p′(·) is the conjugate exponent of p(·), that is, 1 p(x) + 1 p′(x) = 1. 4 int. j. anal. appl. (2022), 20:13 when g is a domain (open and connected subset) of rd and p ∈p+(g), we can define a sobolev space, for any integer m ≥ 0, wm,p(·)(g) = {u ∈ lp(·)(g); ∂αu ∈ lp(·)(g) for |α| ≤ m}, where α = (α1, . . . ,αd) is a multi-index, |α| = ∑d i=1 αi, ∂ α = ∂ α1 1 · · ·∂ αd d and ∂i = ∂/∂xi, endowed with the norm ‖u‖wm,p(·)(g) = ∑ |α|≤m ‖∂αu‖lp(·)(g). of course, w 0,p(·)(g) = lp(·)(g). the local sobolev space is defined by w m,p(·) loc (g) = {u; for all open subset u b g,u ∈ wm,p(·)(u)}, where u b g means that the closure u of u is compact and u ⊂ g. for p ∈p+(g), define p∗(x) = { dp(x) d−p(x) if p(x) < d, ∞ if p(x) ≥ d and p∂(x) = { (d−1)p(x) d−p(x) if p(x) < d, ∞ if p(x) ≥ d. proposition 2.3. let p ∈p+(g) and m ≥ 0 be an integer. then we can see the following properties. (i) the space wm,p(·)(g) is a separable and reflexive banach space. (ii) let g be a bounded domain of rd. if q(·) ∈ p+(g) satisfies q(x) ≤ p(x) for all x ∈ g, then wm,p(·)(g) ↪→ wm,q(·)(g), where ↪→ means that the embedding is continuous. (iii) let g be a bounded domain of rd. if p,q ∈p+(g) ∩c(g) satisfies that q(x) < p∗(x) for all x ∈ g, then the embedding w 1,p(·)(g) ↪→ lq(·)(g) is compact. next we consider the trace. let g be a domain of rd with a lipschitz-continuous boundary ∂g and p ∈ p+(g). since wm,p(·)(g) ⊂ wm,1loc (g), the trace γ0(u) = u ∣∣ ∂g to ∂g of any function u in wm,p(·)(g) is well defined as a function in l1loc(∂g). we define tr(wm,p(·)(g)) = {f ; γ0(u) = f for some function u ∈ wm,p(·)(g)} equipped with the norm ‖f‖tr(wm,p(·)(g)) = inf{‖u‖wm,p(·)(g); u ∈ w m,p(·)(g) satisfying γ0(u) = f on ∂g} for f ∈ tr(wm,p(·)(g)). then tr(wm,p(·)(g)) is a banach space. more precisely, see [11, chapter 12]. we define a space ◦ w m,p(·)(g) = wm,p(·)(g) ∩wm,10 (g). int. j. anal. appl. (2022), 20:13 5 for a general measurable subset g of rd, we say that p ∈ plog(g) if p ∈ p+(g) and p has the globally log-hölder continuity and globally log-hölder decay condition in g, that is, p : g → r satisfies that there exist a constant clog(p) > 0 and p∞ ∈ r such that the following inequalities hold: |p(x) −p(y)| ≤ clog(p) log(e + 1/|x −y|) for all x,y ∈ g, and |p(x) −p∞| ≤ clog(p) log(e + |x|) for all x ∈ g, respectively. proposition 2.4. if g is a domain of rd and p ∈plog(g), then p has an extension q ∈plog(rd) with clog(q) = clog(p), q− = p− and q+ = p+. if g is unbounded, then additionally q∞ = p∞. for the proof, see [11, proposition 4.1.7]. we note that if g is bounded, the global log-hölder decay condition always holds. for a domain g, we write plog+ (g) = p log(g) ∩p+(g). let g be a domain of rd and p ∈p log + (g), define w m,p(·) 0 (g) = the closure of c ∞ 0 (g) in w m,p(·)(g). from definition, we can easily see that wm,p(·)0 (g) ⊂ ◦ w m,p(·)(g) and ◦ w m,p(·)(g) is a closed subspace of wm,p(·)(g). when p(·) = p is a constant, wm,p0 (g) = ◦ w m,p(g), however, in general, w m,p(·) 0 (g) ( ◦ w m,p(·)(g). theorem 2.5. if g is a bounded domain with a lipschitz-continuous boundary ∂g and p ∈plog+ (g), then (i) c∞(g) is dense in wm,p(·)(g). (ii) w m,p(·) 0 (g) = ◦ w m,p(·)(g), in addition, when m ≥ 3, assume that g has a cm,1-boundary. then we have w m,p(·) 0 (g) = {u ∈ w m,p(·)(ω); γ0(u) = · · · = γm−1(u) = 0 a.e. on ∂g}, where γj(u) = ∂ju ∂nj = ∑ |α|=j n α∂αu, n = (n1, . . . ,nd) is the outer unit normal vector to ∂g and nα = n α1 1 · · ·n αd d . for the proof, see [14, theorem 2.6] and galdi [16, theorem 3.2]. lemma 2.6. let g be a bounded domain of rd with a lipschitz-continuous boundary ∂ω and let p ∈ plog+ (g). assume that q ∈ p log + (∂ω) such that q(x) < p ∂(x) for all x ∈ ∂g. then the trace operator tr = γ0 : w 1,p(·)(g) → lq(·)(∂g) is compact, in particular, tr : w 1,p(·)(g) → lp(·)(∂g) is compact. for the proof, see deng [9, theorem 2.1]. frequently we use the following poincaré inequality later. 6 int. j. anal. appl. (2022), 20:13 theorem 2.7. (i) if g is a bounded domain of rd and p ∈ plog+ (g), then there exists a constant c depending only on d and clog(p) such that ‖u‖lp(·)(g) ≤ c diam(g)‖∇∇∇u‖lp(·)(g) for all u ∈ w 1,p(·) 0 (g), where diam(g) denotes the diameter of g. (ii) if g is a bounded domain of rd with a lipschitz-continuous boundary ∂g and p ∈ plog+ (g), then there exists a constant c depending only on d and clog(p) such that ‖u −〈u〉g‖lp(·)(g) ≤ c diam(g)‖∇∇∇u‖lp(·)(g) for all u ∈ w 1,p(·)(g), where 〈u〉g = 1|g| ∫ g udx. for the proof, see [11, theorem 8.2.4]. corollary 2.8. let g be a bounded domain of rd with a lipschitz-continuous boundary and let p ∈plog+ (g). furthermore, let a ⊂ g such that |a| ≈ |g|. then there exists a constant c depending only on d and clog(p) such that ‖u −〈u〉a‖lp(·)(g) ≤ c diam(g)‖∇∇∇u‖lp(·)(g) for all u ∈ l 1 loc(g) with ∇∇∇u ∈l p(·)(g). for the proof, see [11, corollary 8.2.6]. we introduce a generalized poincaré inequality which is found in ciarlet and dinca [8, theorem 4.1]. theorem 2.9. let g be a bounded domain of rd with a lipschitz-continuous boundary γ = ∂ω, g being locally on the same side of γ. moreover, let γ0 be a measurable subset of γ such that |γ0| > 0, and let p ∈plog+ (g). define u = {v ∈ w 1,p(·)(g); v ∣∣ γ0 = 0}. then there exists a constant c = c(p,d,u) such that ‖v‖lp(·)(g) ≤ cdiam(g)‖∇∇∇v‖lp(·)(g) for all v ∈ u. 3. the weak dirichlet problem for the laplacian ∆ in a variable exponent sobolev space in a bounded domain in this section, we state main theorems of this paper. let g be a bounded domain of rd (d ≥ 2) and p ∈ plog+ (g). then taking the poincaré inequality (theorem 2.7) into consideration, we may assume that the space w 1,p(·)0 (g) is equipped with the norm ‖∇∇∇v‖lp(·)(g). the first theorem is a variational inequality in w 1,p(·)0 (g). int. j. anal. appl. (2022), 20:13 7 theorem 3.1. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary and p ∈ plog+ (g). then there exists a constant cp = c(p,g) > 0 such that ‖∇∇∇u‖lp(·)(g) ≤ cp sup 06=v∈w 1,p ′(·) 0 (g) |〈∇∇∇u,∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(g) for all u ∈ w 1,p(·)0 (g). (3.1) the second theorem is a functional representation in w 1,p(·)0 (g) which shows existence of weak solution to the dirichlet problem of the poisson equation in a bounded domain g. theorem 3.2. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary and p ∈plog+ (g). for every f ′ ∈ (w 1,p ′(·) 0 (g)) ′ = w−1,p(·)(g), there exists a unique u ∈ w 1,p(·)0 (g) such that f ′(v) = 〈∇∇∇u,∇∇∇v〉g for all v ∈ w 1,p′(·) 0 (g). (3.2) furthermore, with cp > 0 in theorem 3.1, the following inequality holds. c−1p ‖∇∇∇u‖lp(·)(g) ≤‖f ′‖ (w 1,p′(·) 0 (g)) ′ ≤ 2‖∇∇∇u‖lp(·)(g), (3.3) where ‖f ′‖ (w 1,p′(·) 0 (g)) ′ = sup{|f ′(v)|; v ∈ w 1,p ′(·) 0 (g) and ‖∇∇∇v‖lp′(·)(g) ≤ 1}. before a proof of theorem 3.1, we show that theorem 3.1 implies theorem 3.2. for this purpose, we use the following proposition (cf. amrouche and seloula [1, theorem 4.2]). proposition 3.3. let x and m be two reflexive banach spaces, and a be a continuous bilinear form defined on x ×m. assume that a ∈l(x,m′) and a′ ∈l(m,x′) are operators defined by 〈au,v〉 = 〈u,a′v〉 = a(u,v) for u ∈ x,v ∈ m and put v = kera. then the following statements are equivalent. (i) there exists β > 0 such that inf 0 6=w∈m sup 0 6=v∈x a(v,w) ‖v‖x‖w‖m ≥ β. (3.4) (ii) a : x/v → m′ is an isomorphism and 1/β is the continuity constant of a−1. (iii) a′ : m → v ⊥ := {f ∈ x′;〈f ,v〉 = 0 for all v ∈ v} is also an isomorphism and 1/β is the continuity constant of (a′)−1. let x = (w 1,p(·)0 (g),‖∇∇∇·‖lp(·)(g)) and m = (w 1,p′(·) 0 (g),‖∇∇∇·‖lp′(·)(g)). since p,p ′ ∈ plog+ (g), x and m are reflexive banach spaces (proposition 2.3). define a(u,v) = 〈∇∇∇u,∇∇∇v〉g = ∫ g ∇∇∇u ·∇∇∇vdx for u ∈ x,v ∈ m. then clearly a is a bilinear form on x × m, and it follows from the generalized hölder inequality (proposition 2.2) that a is continuous. if u ∈ kera, then a(u,v) = 〈au,v〉 = 0 for all v ∈ w 1,p ′(·) 0 (g). from theorem 3.1, we have ∇∇∇u = 000 in lp(·)(g). from the poincaré inequality, we have u = 0 in 8 int. j. anal. appl. (2022), 20:13 lp(·)(g), so kera = {0}. from theorem 3.1, (i) in proposition 3.3 holds with β = 1/cp. thus for any f ′ ∈ (w 1,p ′(·) 0 (g)) ′ = m′, there exists uniquely u ∈ w 1,p(·)0 (g) such that f ′ = au, that is, f ′(v) = 〈au,v〉 = 〈∇∇∇u,∇∇∇v〉g for all v ∈ w 1,p ′(·)(g) and ‖∇∇∇u‖lp(·)(g) ≤ 1 β ‖f ′‖ (w 1,p′(·) 0 (g)) ′. therefore, c−1p ‖∇∇∇u‖lp(·)(g) ≤‖f ′‖ (w 1,p′(·) 0 (g)) ′. since by the generalized hölder inequality (proposition 2.2), ‖f ′‖ (w 1,p′(·) 0 (g)) ′ = sup{|f ′(v)|; v ∈ w 1,p ′(·) 0 (g),‖∇∇∇v‖lp′(·)(g) ≤ 1} = sup{|〈∇∇∇u,∇∇∇v〉g|; v ∈ w 1,p′(·) 0 (g),‖∇∇∇v‖lp′(·)(g) ≤ 1} ≤ 2‖∇∇∇u‖lp(·)(g), we can see that (3.3) holds. 4. preparation to a proof of theorem 3.1 we use the localization method for a proof of theorem 3.1. for any open set g ⊂ rd (d ≥ 2) (not necessarily bounded), we say that g satisfies (ga) if g has a c1-boundary and there exists a non-empty open set k in rd such that g = rd \k. definition 4.1. if g satisfies (ga) and p ∈plog+ (g), define ŵ 1,p(·) 0 (g) = {v : g → r; v is measurable in g,v ∈ l p(·)(gr) for each r > 0, ∇∇∇v ∈lp(·)(g), there exists a sequence {vi}∞i=1 ⊂ c ∞ 0 (g) such that ‖v −vi‖lp(·)(gr) → 0 for each r > 0 and ‖∇∇∇v −∇∇∇vi‖lp(·)(g) → 0 as i →∞}, where gr = g ∩br,br = {x ∈ rd; |x| < r}. definition 4.2. if g satisfies (ga) and p ∈plog+ (g), define ŵ 1,p(·) • (g) = {v : g → r; v is measurable in g,v ∈ lp(·)(gr) for each r > 0, ∇∇∇v ∈lp(·)(g) and for any η ∈ c∞0 (r d),ηv ∈ w 1,p(·)0 (g)}. we note that if g is bounded, then w 1,p(·) 0 (g) = ŵ 1,p(·) 0 (g) = ŵ 1,p(·) • (g). we examine the properties of the spaces ŵ 1,p(·)0 (g) and ŵ 1,p(·) • (g). int. j. anal. appl. (2022), 20:13 9 lemma 4.3. suppose (ga). let p ∈ plog+ (g) and let v ∈ ŵ 1,p(·) • (g). then for every r > 0, there exists a sequence {vi}⊂ c∞0 (g) possibly depending on r > 0 such that ‖v −vi‖lp(·)(gr) + ‖∇∇∇v −∇∇∇vi‖lp(·)(gr) → 0 as i →∞. proof. for given r > 0, choose η ∈ c∞0 (b2r) such that η = 1 on br. since ηv ∈ w 1,p(·) 0 (g), there exists a sequence {vi} ⊂ c∞0 (g) such that ‖ηv − vi‖w 1,p(·)(g) → 0 . since η = 1 on br, we have ‖v −vi‖w 1,p(·)(gr) → 0. � theorem 4.4. suppose (ga) and let p ∈plog+ (g). then the following properties hold. (i) w 1,p(·) 0 (g) ⊂ ŵ 1,p(·) 0 (g) ⊂ ŵ 1,p(·) • (g). (ii) for v ∈ ŵ 1,p(·)• (g), ‖∇∇∇v‖lp(·)(g) is a norm on ŵ 1,p(·) • (g). (iii) the space ŵ 1,p(·)• (g) equipped with the norm ‖∇∇∇·‖lp(·) is a reflexive banach space. (iv) the space ŵ 1,p(·)0 (g) is a closed subspace of ŵ 1,p(·) • (g) and ŵ 1,p(·) 0 (g) = the closure of c ∞ 0 (g) with respect to ‖∇∇∇·‖lp(·)(g)-norm. (v) the space w 1,p(·)0 (g) is dense in ŵ 1,p(·) 0 (g) with respect to ‖∇∇∇·‖lp(·)(g)-norm. (vi) if we define e∞0 (g) = {∇∇∇φ; φ ∈ c ∞ 0 (g)} and e p(·) 0 (g) = {∇∇∇v; v ∈ ŵ 1,p(·) 0 (g)}, then the closure of e∞0 (g) in l p(·)(g) is equal to ep(·)0 (g). proof. (i) it is trivial that w 1,p(·)0 (g) ⊂ ŵ 1,p(·) 0 (g). let v ∈ ŵ 1,p(·) 0 (g) and let η ∈ c ∞ 0 (r d). choose r > 0 so that supp η ⊂ br. by definition of ŵ 1,p(·) 0 (g), there exists a sequence {vi}⊂ c ∞ 0 (g) such that ‖v −vi‖lp(·)(gr) → 0 and ‖∇∇∇v −∇∇∇vi‖lp(·)(g) → 0 as i →∞. then ηvi ∈ c ∞ 0 (g) and ‖ηv −ηvi‖w 1,p(·)(g) ≤ c1(‖η‖l∞(g) + ‖∇∇∇η‖l∞(g))‖v −vi‖lp(·)(gr) + c2‖η‖l∞(g)‖∇∇∇v −∇∇∇vi‖lp(·)(g) → 0. thus ηv ∈ w 1,p(·)0 (g), so v ∈ ŵ 1,p(·) • (g). (ii) clearly ŵ 1,p(·)• (g) is a linear space. if v ∈ ŵ 1,p(·) • (g) satisfies ‖∇∇∇v‖lp(·)(g) = 0, then we show that v = 0 in g. to do so, it suffices to show that v = 0 in gr for every r > 0. choose η ∈ c∞0 (r d) such that η = 1 on br and supp η ⊂ b2r. since ηv ∈ w 1,p(·) 0 (g) by definition of ŵ 1,p(·) • (g), there exists a sequence {vi} ⊂ c∞0 (g) such that ‖ηv − vi‖w 1,p(·)(g) → 0. thus ηv ∈ w 1,p− 0 (g2r) and ‖ηv −vi‖w 1,p−(g2r) → 0. by [25, lemma 1.2], we have ‖vi‖lp−(g2r) ≤ cr d/p−+1−1/p−‖∇∇∇vi‖lp−(g2r). by the limit process, we have ‖v‖ lp − (gr) ≤ crd/p −+1−1/p−‖∇∇∇v‖ lp − (gr) . since lp(·)(gr) ↪→lp − (gr) and ∇∇∇v = 0 in lp − (gr), we have v = 0 in gr. the other properties of norm clearly hold. 10 int. j. anal. appl. (2022), 20:13 (iii) we already showed that ŵ 1,p(·)• (g) is a normed linear space. we derive the completeness of ŵ 1,p(·) • (g) equipped with the norm ‖∇∇∇·‖lp(·)(g). let {vi} ∞ i=1 ⊂ ŵ 1,p(·) • (g) be a cauchy sequence, that is, ‖∇∇∇vi −∇∇∇vj‖lp(·)(g) → 0 as i, j →∞. for k ∈ n, put gk = g∩bk. then there exists k0 ∈ n such that for k ≥ k0, gk has a portion γk of ∂g with |γk| > 0. since vi ∈ w 1,p(·)(gk) and vi ∣∣ γk = 0, it follows from a generalized poincaré inequality (theorem 2.9) that ‖vi −vj‖lp(·)(gk ) ≤ c(k)‖∇∇∇vi −∇∇∇vj‖lp(·)(gk ) ≤ c(k)‖∇∇∇vi −∇∇∇vj‖lp(·)(g). thus {vi ∣∣ gk } is a cauchy sequence in lp(·)(gk). therefore, there exists v(k) ∈ lp(·)(gk) such that vi ∣∣ gk → v(k) in lp(·)(gk). after choosing a subsequence, we may assume that vi ∣∣ gk → v(k) a.e. in gk. eventually after changing v(k+1) on a subset nk ⊂ gk with measure zero, we may assume that v(k+1) ∣∣ gk = v(k). define a unique measurable function v : g → r so that v(x) = v(k)(x) for x ∈ gk. hence for each r > 0 ‖vi −v‖lp(·)(gr) → 0 as i →∞. since {∇∇∇vi} is a cauchy sequence in l p(·)(g), there exists f = (f1, . . . , fd) ∈ lp(·)(g) such that ∇∇∇vi → f in lp(·)(g). let φ ∈ c∞0 (g). then supp φ ⊂ gk for some k ∈ n. for l = 1, . . . ,d, 〈v,∂lφ〉g = lim i→∞ 〈vi,∂lφ〉gk = − lim i→∞ 〈∂lvi,φ〉gk = −〈fl,φ〉g. hence ∂lv = fl ∈ lp(·)(g), so ∇∇∇v = f ∈ lp(·)(g). for η ∈ c∞0 (r d), choose r > 0 such that supp η ⊂ br. choose ζ ∈ c∞0 (b2r) so that 0 ≤ ζ ≤ 1 and ζ = 1 on br. since ζvi ∈ w 1,p(·) 0 (g) by definition of ŵ 1,p(·)• (g), there exists φi ∈ c∞0 (g) such that ‖ζvi − φi‖w 1,p(·)(g) ≤ 2 −i. since ηv = ηζv, we have ‖ηv −ηφi‖lp(·)(g) = ‖ηζv −ηφi‖lp(·)(g) ≤ ‖ηζv −ηζvi‖lp(·)(g) + ‖ηζvi −ηφi‖lp(·)(g) ≤ ‖η‖l∞(rd )(‖ζv −ζvi‖lp(·)(g) + ‖ζvi −φi‖lp(·)(g)) ≤ ‖η‖l∞(rd )(‖v −vi‖lp(·)(g) + 2 −i ) → 0 as i →∞ and ‖∇∇∇(ηv −ηφi )‖lp(·)(g) = ‖∇∇∇(ηζv −ηφi )‖lp(·)(g) ≤ ‖∇∇∇(ηζv −ηζvi )‖lp(·)(g) + ‖∇∇∇(ηζvi −ηφi )‖lp(·)(g) ≤ ‖ηζ‖l∞(rd )(‖∇∇∇(v −vi )‖lp(·)(g) +‖∇∇∇(ηζ)‖l∞(rd )‖vi −φi‖lp(·)(gr) +(‖η‖l∞(rd ) + ‖∇∇∇η‖l∞(rd ))‖ζvi −φi‖w 1,p(·)(g) → 0 as i → ∞. since ηφ ∈ c∞0 (g), we can see that ηv ∈ w 1,p(·) 0 (g), so v ∈ ŵ 1,p(·) • (g). hence ŵ 1,p(·) • (g) is complete. int. j. anal. appl. (2022), 20:13 11 we show the reflexivity of ŵ 1,p(·)• (g). if we define e p(·) • (g) = {∇∇∇v; v ∈ ŵ 1,p(·) • (g)}, then the gradient operator ∇∇∇ : ŵ 1,p(·)• (g) → e p(·) • (g) is isometric isomorphism. since ŵ 1,p(·) • (g) is complete, ep(·)• (g) is a closed subspace of a reflexive banach space lp(·)(g). therefore, e p(·) • (g) is reflexive, so ŵ 1,p(·)• (g) is also reflexive. (iv) let {vi} ⊂ ŵ 1,p(·) 0 (g) and v ∈ ŵ 1,p(·) • (g) such that ‖∇∇∇v −∇∇∇vi‖lp(·)(g) → 0 as i → ∞. by definition of ŵ 1,p(·)0 (g), there exists φi ∈ c ∞ 0 (g) such that ‖∇∇∇vi −∇∇∇φi‖lp(·)(g) ≤ 2 −i. hence ‖∇∇∇v −∇∇∇φi‖lp(·)(g) ≤‖∇∇∇v −∇∇∇vi‖lp(·)(g) + ‖∇∇∇vi −∇∇∇φi‖lp(·)(g) → 0 as i →∞. by the generalized poincaré inequliaty (theorem 2.9), for large r > 0, ‖v −φi‖lp(·)(gr) ≤ c(r)‖∇∇∇v −∇∇∇φi‖lp(·)(gr) ≤ c(r)‖∇∇∇v −∇∇∇φi‖lp(·)(g) → 0. from definition of ŵ 1,p(·)0 (g), we can see that v ∈ ŵ 1,p(·) 0 (g), so ŵ 1,p(·) 0 (g) is a closed subspace of ŵ 1,p(·) • (g). since c∞0 (g) ⊂ ŵ 1,p(·) 0 (g), the closure of c ∞ 0 (g) with respect to ‖∇∇∇ · ‖lp(·)(g) is contained in the closure of ŵ 1,p(·)0 (g) with respect to ‖∇∇∇·‖lp(·)(g) which is equal to ŵ 1,p(·) 0 (g). conversely, let v ∈ ŵ 1,p(·)0 (g). then there exists {φi} ⊂ c ∞ 0 (g) such that ‖∇∇∇v −∇∇∇φi‖lp(·)(g) → 0 as i → ∞. thereby v is contained in the closure of c∞0 (g) with respect to ‖∇∇∇·‖lp(·)(g)-norm. (v) by definition of ŵ 1,p(·)0 (g), the space c ∞ 0 (g) is dense in ŵ 1,p(·) 0 (g) with respect to ‖∇∇∇·‖lp(·)(g)norm. since c∞0 (g) ⊂ w 1,p(·) 0 (g) ⊂ ŵ 1,p(·) 0 (g), we see that w 1,p(·) 0 (g) is dense in ŵ 1,p(·) 0 (g) with respect to ‖∇∇∇·‖lp(·)(g)-norm. (vi) from (iv), it is clear that the closure of e∞0 (g) in l p(·)(g) is contained in ep(·)0 (g). let v ∈ ŵ 1,p(·)0 (g). then there exists a sequence {φi} ⊂ c ∞ 0 (g) such that ‖∇∇∇v −∇∇∇φi‖lp(·)(g) → 0 as i →∞. therefore, ∇∇∇v is contained in the closure of e∞0 (g) in l p(·)(g). � we can improve lemma 4.3. lemma 4.5. suppose (ga). let p ∈ plog+ (g) and let v ∈ ŵ 1,p(·) • (g). then there exists a sequence {vi}⊂ c∞0 (g) such that for every r > 0, ‖v −vi‖lp(·)(gr) + ‖∇∇∇v −∇∇∇vi‖lp(·)(gr) → 0 as i →∞, that is, we can choose {vi}⊂ c∞0 (g) independent of r > 0. proof. choose ζ ∈ c∞0 (g) such that 0 ≤ ζ ≤ 1, and ζ(x) = 1 for |x| ≤ 1 and ζ = 0 for |x| ≥ 2. put ζi (x) = ζ(i−1x). by definition of ŵ 1,p(·) • (g), we see that ζiv ∈ w 1,p(·) 0 (g). hence there exists {vi}⊂ c∞0 (g) such that ‖ζiv −vi‖w 1,p(·)(g) ≤ i −1. for each r > 0, let i ≥ r. then ‖v −vi‖lp(·)(gr) + ‖∇∇∇v −∇∇∇vi‖lp(·)(gr) ≤ i −1 → 0 as i →∞. � 12 int. j. anal. appl. (2022), 20:13 here we characterize of ŵ 1,p(·)• (g). theorem 4.6. suppose (ga) and let p ∈plog+ (g). then we have ŵ 1,p(·) • (g) = mp(·), where mp(·) = {v : g → r; v is measurable ,v ∈ l p(·)(gr) for each r > 0, ∇∇∇v ∈lp(·)(g) and there exists {vi}⊂ c∞0 (g) such that ‖v −vi‖lp(·)(gr) + ‖∇∇∇v −∇∇∇vi‖lp(·)(gr) → 0 for each r > 0 as i →∞}. proof. by lemma 4.5, ŵ 1,p(·)• (g) ⊂ mp(·). conversely, let v ∈ mp(·), and let η ∈ c∞0 (r d). choose r > 0 such that supp η ⊂ br. then ‖ηv −ηvi‖lp(·)(g) ≤‖η‖l∞(rd )‖v −vi‖lp(·)(gr) → 0 and ‖∇∇∇(ηv − ηvi )‖lp(·)(g) ≤ ‖η‖l∞(rd )‖∇∇∇v − ∇∇∇vi‖lp(·)(gr) + ‖∇∇∇η‖l∞(rd )‖v − vi‖lp(·)(gr) → 0. ] since ηvi ∈ c∞0 (g) and ηvi → ηv in w 1,p(·)(g), we see that ηv ∈ w 1,p(·)0 (g), so v ∈ ŵ 1,p(·) • (g). � definition 4.7. let g be a domain of rd (d ≥ 2) such that rd \g 6= ∅, and let s ∈plog+ (g). (a) we say that g has the property pa(s) if there exists a constant cs = c(s,g) > 0 such that ‖∇∇∇u‖ls(·)(g) ≤ cs sup 0 6=v∈ŵ 1,s ′(·) • (g) |〈∇∇∇u,∇∇∇v〉g| ‖∇∇∇v‖ ls ′(·)(g) for all u ∈ ŵ 1,s(·)• (g). (4.1) (b) let the bounded linear operator σs : ŵ 1,s(·) • (g) → (ŵ 1,s′(·) • (g)) ′ be defined by σs(u)(φ) = 〈∇∇∇u,∇∇∇φ〉g for u ∈ ŵ 1,s(·) • (g) and φ ∈ ŵ 1,s′(·) • (g). (4.2) we say that g has the property pb(s) if σs is a bijection and there exists a constant c̃s = c̃(s,g) > 0 such that c̃s‖∇∇∇u‖ls(·)(g) ≤‖σs(u)‖(ŵ 1,s′(·)• (g))′ ≤ 2‖∇∇∇u‖ls(·)(g) for all u ∈ ŵ 1,s(·) • (g). (4.3) theorem 4.8. let g be a domain of rd (d ≥ 2) such that rd \g 6= ∅ and let p ∈ plog+ (g). then g has the property pa(s) for s = p and s = p′ if and only if g has the property pb(s) for s = p and s = p′. proof. assume that g has the property pa(s) for s = p and s = p′. let u ∈ ŵ 1,s(·) • (g) and define ss(u) = sup{〈∇∇∇u,∇∇∇φ〉g; φ ∈ ŵ 1,s′(·) • (g),‖∇∇∇φ‖ls′(·)(g) ≤ 1}. by (4.1) and the hölder inequality (proposition 2.2), c−1s ‖∇∇∇u‖ls(·)(g) ≤ ss(u) = ‖σs(u)‖(ŵ 1,s′(·)• (g))′ ≤ 2‖∇∇∇u‖ls(·)(g) for all u ∈ ŵ 1,s(·) • (g). int. j. anal. appl. (2022), 20:13 13 hence (4.3) holds with c̃s = c−1s . from this, we see that σs(ŵ 1,s′(·) • (g)) is a closed subspace of (ŵ 1,s′(·) • (g)) ′. suppose σs(ŵ 1,s(·) • (g)) ( (ŵ 1,s′(·) • (g)) ′. by the hahn-banach theorem, there exists f ′′ ∈ (ŵ 1,s ′(·) • (g)) ′′ such that f ′′ 6= 0 and f ′′ ∣∣ σs (ŵ 1,s(·) • (g)) = 0. since ŵ 1,s ′(·) • (g) is reflexive, there exists uniquely φ ∈ ŵ 1,s′(·) • (g) such that f ′′(f ′) = f ′(φ) for all f ′ ∈ (ŵ 1,s ′(·) • (g)) ′ and ‖f ′′‖ (ŵ 1,s′(·) • (g)) ′′ = ‖∇∇∇φ‖ls′(·)(g) > 0. on the other hand, for all u ∈ ŵ 1,s(·) • (g), 0 = f ′′(σs(u)) = σs(u)(φ) = 〈∇∇∇u,∇∇∇φ〉g. by the property pa(s′), we have ‖∇∇∇φ‖ls′(·)(g) ≤ cs′ss′(φ) = 0. this is a contradiction. conversely, assume that g has the property pb(s) for s = p and s = p′. let u ∈ ŵ 1,s(·) • (g). since σs′ is a bijection, for f ′ ∈ (ŵ 1,s(·)• (g))′, there exists uniquely φ ∈ ŵ 1,s′(·) • (g) such that f ′ = σs′(φ). hence ‖∇∇∇u‖ls(·)(g) = sup { |f ′(u)| ‖f ′‖ (ŵ 1,s(·) • (g)) ′ ; 0 6= f ′ ∈ (ŵ 1,s(·)• (g))′ } ≤ sup { |〈∇∇∇u,∇∇∇φ〉g| c̃s′‖∇∇∇φ‖ls′(·)(g) ; 0 6= φ ∈ ŵ 1,s ′(·) • (g) } . thus (4.1) holds with cs = c̃ −1 s′ . � now we consider the case g = rd. lemma 4.9. if we define m := {∆v; v ∈d(rd) := c∞0 (r d)}, then m is dense in lp(·)(rd). proof. suppose that m lp(·)(rd), where m is the closure of m in lp(·)(rd). by the hahnbanach theorem, there exists f ′ ∈ (lp(·)(rd))′ with ‖f ′‖(lp(·)(rd ))′ > 0 and f ′ ∣∣ m = 0. since we can regard (lp(·)(rd))′ = lp ′(·)(rd) isometrically, there exists v ∈ lp ′(·)(rd) such that ‖v‖ lp ′(·)(rd ) = ‖f ′‖(lp(·)(rd ))′ > 0 and f ′(w) = 〈v,w〉rd := ∫ rd v(x)w(x)dx for all w ∈ l p(·)(rd). since f ′ ∣∣ m = 0, we can see that 〈v, ∆φ〉rd = 0 for all φ ∈d(rd), so ∆v = 0 in d′(rd). by the hypoellipticity of the laplacian, we can regard that v ∈ c∞(rd) (eventually after change of a set of measure zero), so v is harmonic in rd. for any x ∈ rd fixed, it follows from the second mean value theorem for harmonic functions that v(x) = 1 |br(x)| ∫ br(x) v(y)dy, where br(x) = {y ∈ rd : |y−x| < r} and |br(x)| denotes the volume of br(x). by the generalized hölder inequality (proposition 2.2), |v(x)| ≤ 2 |br(x)| ‖v‖ lp ′(·)(br(x)) ‖1‖lp(·)(br(x)). 14 int. j. anal. appl. (2022), 20:13 since ‖1‖lp(·)(br(x)) ≤ ρp(·),br(x)(1) 1/p− = |br(x)|1/p − for large r > 0 and p− > 1, we have |v(x)| ≤ 2|br(x)|−1+1/p − ‖v‖ lp ′(·)(rd ) → 0 as r →∞. therefore, we have v(x) = 0. since x ∈ rd is arbitrary, v ≡ 0 in rd. this is a contradiction. � define ∇∇∇2v = (∂i∂jv)i,j=1,...,d for v ∈d(rd). then there exists a constant c = c(p,d) > 0 such that c‖∇∇∇2v‖lp(·)(rd ) ≤‖∆v‖lp(·)(rd ) for all v ∈d(r d). (4.4) for the proof, see [11, corollary 14.1.7] (cf. when p(·) = p (constant), see gilbarg and trudinger [17, corollary 9.10]). for p ∈plog+ (r d), we have ep(·)(rd) = {∇∇∇u; u ∈ lp(·) loc (rd),∇∇∇u ∈lp(·)(rd)} by definition. lemma 4.10. let p,q ∈plog+ (r d). if u ∈ lq(·) loc (rd) with ∇∇∇u ∈lq(·)(rd) satisfies sup 0 6=v∈d(rd ) |〈∇∇∇u,∇∇∇v〉rd | ‖∇∇∇v‖ lp ′(·)(rd ) < ∞, (4.5) then u ∈ lp(·) loc (rd) with ∇∇∇u ∈ lp(·)(rd). furthermore, there exists a constant c1 = c1(p,d) > 0 such that ‖∇∇∇u‖lp(·)(rd ) ≤ c1 sup 0 6=v∈d(rd ) |〈∇∇∇u,∇∇∇v〉rd | ‖∇∇∇v‖ lp ′(·)(rd ) (4.6) for all u ∈ lp(·) loc (rd) with ∇∇∇u ∈lp(·)(rd). in particular, if u ∈ lp(·) loc (rd) with ∇∇∇u ∈lp(·)(rd), then ‖∇∇∇u‖lp(·)(rd ) ≤ c1 sup 0 6=v∈d(rd ) |〈∇∇∇u,∇∇∇v〉rd | ‖∇∇∇v‖ lp ′(·)(rd ) (4.7) proof. let u ∈ lq(·) loc (rd) with ∇∇∇u ∈lq(·)(rd). for every i = 1, . . . ,d, using (4.4), ∞ > sup 06=v∈d(rd ) |〈∇∇∇u,∇∇∇v〉rd | ‖∇∇∇v‖ lp ′(·)(rd ) ≥ sup 06=w∈d(rd ) |〈∇∇∇u,∇∇∇(∂iw)〉rd | ‖∇∇∇∂iw‖lp′(·)(rd ) ≥ sup 06=w∈d(rd ) |〈∂iu, ∆w〉rd | ‖∇∇∇2w‖ lp ′(·)(rd ) ≥ c sup 06=w∈d(rd ) |〈∂iu, ∆w〉rd | ‖∆w‖ lp ′(·)(rd ) , (4.8) where c is the constant in (4.4). define a bounded linear functional l∗ by l∗(∆w) = 〈∂iu, ∆w〉rd on the dense subspace m of l p′(·)(rd) int. j. anal. appl. (2022), 20:13 15 (cf. lemma 4.9). then the functional l∗ has a unique and norm-preserving extension as a continuous linear functional on lp ′(·)(rd). thus there exists g ∈ lp(·)(rd) such that 〈∂iu,v〉rd = 〈g,v〉rd for all v ∈ m, that is, 〈∂iu −g, ∆w〉rd = 0 for all w ∈d(r d). if we define w = ∂iu −g, then ∆w = 0 in d′(rd), so w is harmonic in rd. by the same argument as in the proof of lemma 4.9, we can regard w (x) ≡ 0, so ∂iu = g ∈ lp(·)(rd). hence we have ∇∇∇u ∈lp(·)(rd). since ∇∇∇u ∈lp(·)(rd) and u ∈ lq(·) loc (rd) ⊂ l1loc(r d), for any ball b, it follows from theorem 2.7 (ii) that ‖u −〈u〉b‖lp(·)(b) ≤ c(p,d,b)‖∇∇∇u‖lp(·)(b). this implies that u ∈ lp(·) loc (rd), that is, ∇∇∇u ∈ep(·)(rd). by continuity, we have sup 06=w∈d(rd ) |〈∂iu, ∆w〉rd | ‖∆w‖ lp ′(·)(rrd ) = sup 0 6=f∈lp′(·)(rd ) |〈∂iu,f 〉rd | ‖f‖ lp ′(·)(rrd ) = ‖∂iu‖lp(·)(rd ). from (4.8), ‖∂iu‖lp(·)(rd ) ≤ c −1 p′ sup 06=φ∈d(rd ) |〈∇∇∇u,∇∇∇φ〉rd | ‖∇∇∇φ‖ lp ′(·)(rd ) . therefore, we get (4.6). � remark 4.11. we can show that (4.6) implies (4.4). indeed, let φ ∈d(rd) and put u = ∂iφ. from (4.6) replaced p with p′, ‖∇∇∇(∂iφ)‖lp′(·)(rd ) ≤ c1(p) sup 0 6=v∈d(rd ) |〈∇∇∇(∂iφ),∇∇∇v〉rd | ‖∇∇∇v‖lp(·)(rd ) ≤ c1(p) sup 0 6=v∈d(rd ) |〈∆φ,∂iv〉rd | ‖∇∇∇v‖lp(·)(rd ) ≤ 2c1(p)‖∆φ‖lp′(·)(rd ). next we consider the case where g is a half-space or a bended half-space. let ω be a c1-function defined on rd−1, h = {x = (x′,xd); x′ = (x1, . . . ,xd−1) ∈ rd−1,xd < 0} and hω = {x = (x′,xd); xd < ω(x′)}. lemma 4.12. let ω be a c1-function defined on rd−1 with ‖∇∇∇′ω‖l∞(rd−1) < ∞, where ∇∇∇ ′ = (∂1, . . . ,∂d−1), and let p ∈p log + (hω). then we have ŵ 1,p(·) • (hω) = ŵ 1,p(·) 0 (hω). proof. step 1. let 0 < ρ < ∞, 0 < r < ∞ and put zωρ,r = {x = (x ′,xd) ∈ rd; |x′| < ρ,−r < xd < ω(x′)}. assume u ∈ ŵ 1,p(·)• (hω). then since u ∣∣ ∂hω = 0, it follows from the generalized poincaré ineequality (theorem 2.9) that there exists a constant c = c(d,clog(p)) such that ‖u‖lp(·)(zω ρr ) ≤ c(ρ + r)‖∇∇∇u‖lp(·)(zω ρ,r ). (4.9) 16 int. j. anal. appl. (2022), 20:13 since |ω(x′) −ω(0)| ≤ ‖∇∇∇′ω‖l∞(rd−1)|x′| ≤ ‖∇∇∇ ′ω‖l∞(rd−1)ρ for |x′| < ρ. hence |ω(x′)| ≤ ‖∇∇∇′ω‖l∞(rd−1)ρ + ω0, where ω0 = |ω(0)|. step 2. choose τ ∈ c∞0 (r d) such that 0 ≤ τ ≤ 1 and τ(x) = 1 for |x| ≤ 1 and τ(x) = 0 for |x| ≥ 2, and for k ∈ n, put τk(x) = τ(k−1x). then |∇∇∇τk(x)| ≤ k−1‖∇∇∇τ‖l∞(rd ), and supp(∇∇∇τk) ⊂ ak := {x ∈ rd; k < |x| < 2k}. put ρk = 2k,rk = 2k(‖∇∇∇′ω‖l∞(rd−1) + 1) + ω0 and zk = zωρk,rk. if x ∈ hω ∩b2k, then |x′| < 2k and −2k < xd < ω(x′). hence ω(x′) −rk ≤‖∇∇∇′ω‖l∞(rd−1)2k + ω0 −rk = −2k < xd < ω(x ′). therefore, hω ∩b2k ⊂ zk. from (4.9), we have ‖u∇∇∇τk‖lp(·)(hω) ≤ k −1‖∇∇∇′ω‖l∞(rd−1)‖u‖lp(·)(hω∩b2k ) (4.10) ′ ≤ k−1‖∇∇∇′ω‖l∞(rd−1)‖u‖lp(·)(zk ) ≤ ck−1‖∇∇∇′ω‖l∞(rd−1)(ρk + rk)‖∇∇∇u‖lp(·)(hω) ≤ c1‖∇∇∇u‖lp(·)(hω), where c1 is a constant independent of k. by definition of ŵ 1,p(·) • (hω), τku ∈ w 1,p(·) 0 (hω) ⊂ ŵ 1,p(·) 0 (hω). step 3. let f ′ ∈ (ŵ 1,p(·)• (hω))′, that is, |f ′(φ)| ≤ ‖f ′‖ (ŵ 1,p(·) • (hω)) ′‖∇∇∇φ‖lp(·)(hω) for all φ ∈ ŵ 1,p(·)(hω). since ŵ 1,p(·)• (hω) is complete and e p(·) 0 (hω) is a closed subspace of l p(·)(hω), we can regard f ′ as a continuous linear functional on ep(·)0 (hω). by the hahn-banach theorem, f ′ may be extended to a functional f̃ ′ ∈ (lp(·)(hω))′ which is norm-preserving. hence there exists f ∈lp ′(·)(hω) such that ‖f‖ lp ′(·)(hω) = ‖f̃ ′‖(lp(·)(hω))′ = ‖f‖(ŵ 1,p(·)• (hω))′ and f ′(φ) = f̃ ′(φ) = 〈f ,∇∇∇φ〉hω for all φ ∈ ŵ 1,p(·) • (hω). then f ′(u) −f ′(τku) = 〈(1 −τk)f ,∇∇∇u〉hω −〈f ,u∇∇∇τk〉hω. we have |〈(1 −τk)f ,∇∇∇u〉hω| ≤ 2‖(1 −τk)f‖lp′(·)(hω)‖∇∇∇u‖lp(·)(hω). by the lebesgue dominated convergence theorem, we have ρp′(·),hω ((1 −τk)f ) = ∫ hω |(1 −τk)f |p ′(x)dx → 0 as k →∞. so it follows from proposition 2.1 that ‖(1 − τk)f‖lp′(·)(hω) → 0 as k → ∞. by (4.10) and supp(∇∇∇τk) ⊂ ak, |〈f ,u∇∇∇τk〉hω| ≤ 2c1‖f‖lp′(·)(hω∩ak )‖∇∇∇u‖lp(·)(hω) → 0 as k →∞. int. j. anal. appl. (2022), 20:13 17 hence f ′(τku) → f ′(u) as k → ∞. since f ′ ∈ (ŵ 1,p(·) • (hω)) ′ is arbitrary, we see that τku → u weakly in ŵ 1,p(·)• (hω). by step 2, τku ∈ ŵ 1,p(·) 0 (hω). since ŵ 1,p(·) 0 (hω) is a closed subspace of ŵ 1,p(·) • (hω) (theorem 4.4), it is weakly closed. therefore u ∈ ŵ 1,p(·) 0 (hω). � lemma 4.13. let p ∈plog+ (h). for x ∈ r d, define p̃(x) = { p(x) for xd ≤ 0, p(x′,−xd) for xd > 0. then clearly p̃ ∈plog+ (r d). for u ∈ ŵ 1,p(·)• (h), define u1(x) =   u(x) for xd < 0, 0 for xd = 0, −u(x′,−xd) for xd > 0. then u1 ∈ w 1,p̃(·) loc (rd), ∇∇∇u1 ∈lp̃(·)(rd), and furthermore, ∂iu1(x) =   (∂iu)(x) for xd < 0, 0 for xd = 0, −(∂iu)(x′,−xd) for xd > 0 for i = 1, . . . ,d − 1 and ∂du1(x) = { (∂du)(x) for xd < 0, (∂du)(x ′,−xd) for xd > 0. in addition, ‖∇∇∇u‖lp(·)(h) ≤‖∇∇∇u1‖lp̃(rd ) ≤ 2‖∇∇∇u‖lp(·)(h). for φ ∈ d(rd), let (t1φ)(x) = φ(x) − φ(x′,−xd) for x ∈ h. then t1φ ∈ ŵ 1,p(·) 0 (h) ∩ c 1(h), (t1φ)(x ′, 0) = 0 and there exists r = r(φ) > 0 such that (t1φ)(x) = 0 for |x| > r and ‖∇∇∇(t1φ)‖lp(·)(h) ≤ 2‖∇∇∇φ‖lp̃(·)(rd ). furthermore, for u ∈ ŵ 1,p(·)0 (h) and φ ∈d(r d), 〈∇∇∇u1,∇∇∇φ〉rd = 〈∇∇∇u,∇∇∇(t1φ)〉h. since this lemma follows from elementary calculations (cf. [25, lemma 2.3]). we omit the proof. lemma 4.14. let p,q ∈plog+ (h). if u ∈ ŵ 1,q(·) 0 (h) satisfies sp(u) := sup 0 6=φ∈c∞0 (h) |〈∇∇∇u,∇∇∇φ〉h| ‖∇∇∇φ‖ lp ′(·)(h) < ∞, then u ∈ ŵ 1,p(·)0 (h) and ‖∇∇∇u‖lp(·)(h) ≤ c2(p)sp(u), where c2(p) = 2c1 > 0, c1 is a constant as in (4.6). 18 int. j. anal. appl. (2022), 20:13 proof. for any function φ ∈ d(rd), we consider t1φ. if supp φ ⊂ br, then supp t1φ ⊂ br, (t1φ)(x ′, 0) = 0 and ∇∇∇(t1φ) ∈ l∞(h). choose η ∈ c∞(rd) such that 0 ≤ η ≤ 1 and η(x) = 1 for |x| ≥ 1 and η(x) = 0 for |x| ≤ 1/2. for k ∈ n, put ηk(x) = η(kx) and φk(x) = ηk(x)(t1φ)(x). then for s = q′ and s = p′, ‖∇∇∇(t1φ) −∇∇∇φk‖ls(·)(h) ≤‖(1 −ηk(x))∇∇∇(t1φ)‖ls(·)(h) + ‖(∇∇∇ηk(x))t1φ‖ls(·)(h). here from the lebesgue dominated convergence theorem, ‖(1 −ηk(x))∇∇∇(t1φ)‖ls(·)(h) → 0 as k →∞. since supp ηk ⊂ {x ∈ rd; 1/(2k) < |x| < 1/k} =: ak, it follows from the poincaré inequality (theorem 2.9) that ‖(∇∇∇ηk(x))t1φ‖ls(·)(h) ≤ k‖∇∇∇η‖l∞(rd )‖t1φ‖ls(·)(h∩ak ) ≤ k‖∇∇∇η‖l∞(rd ) 1 k ‖∇∇∇(t1φ)‖ls(·)(h∩ak ) = ‖∇∇∇η‖l∞(rd )‖∇∇∇(t1φ)‖ls(·)(h∩ak ) → 0 as k →∞. therefore, since u ∈ ŵ 1,q(·)0 (h), we have sp(u) ≥ |〈∇∇∇u,∇∇∇φk〉h| ‖∇∇∇φk‖lp′(·)(h) → |〈∇∇∇u,∇∇∇(t1φ)〉h| ‖∇∇∇(t1φ)‖lp′(·)(h) . hence for 0 6= φ ∈ c∞0 (r d) such that t1φ 6= 0, by lemma 4.13 |〈∇∇∇u1,∇∇∇φk〉rd | ‖∇∇∇φ‖ lp̃ ′(·)(rd ) ≤ 2 |〈∇∇∇u,∇∇∇(t1φ)〉h| ‖∇∇∇t1φ‖lp′(·)(h) ≤ 2sp(u). by lemma 4.10, we see that ∇∇∇u1 ∈ lp̃(rd), so ∇∇∇u ∈ lp(·)(h). since u ∈ ŵ 1,q(·) 0 (h), we can see that u ∈ ŵ 1,p(·)0 (h) as in the proof of lemma 4.12, and ‖∇∇∇u‖lp(·)(h) ≤‖∇∇∇u1‖lp̃(·)(rd ) ≤ 2c1sp(u). � lemma 4.15. let ω be a c1-function on rd−1 such that there exists r = r(ω) > 0 such that ω(x′) = 0 for |x′| > r and let p ∈plog+ (hω). assume that there exists a constant kp = k(p,d) > 0 such that ‖∇∇∇′ω‖l∞(rd−1) ≤ kp. then there exists a constant c3(s) = c3(s,d,kp) > 0 such that for all u ∈ ŵ 1,s(·) 0 (hω), ‖∇∇∇u‖ls(·)(hω) ≤ c3(s) sup 0 6=φ∈c∞0 (hω) |〈∇∇∇u,∇∇∇φ〉hω| ‖∇∇∇φ‖ ls ′(·)(hω) (4.11) for s = p,p′. int. j. anal. appl. (2022), 20:13 19 proof. let y : rd → rd be a map defined by{ yi (x) = xi for i = 1, . . . ,d − 1, yd(x) = xd −ω(x′) . then y is a c1-map and bijective, y(hω) = h,y(∂hω) = ∂h and the jacobian j(y(x)) = 1 for x ∈ rd. the inverse map x : rd → rd is given by{ xi (y) = yi for i = 1, . . . ,d − 1, xd(y) = yd + ω(y ′). for s ∈ plog+ (hω), define s̃(y) = s(x(y)). then s̃ ∈ p log + (h). if u ∈ ŵ 1,s(·) 0 (hω) and define ũ(y) = u(x(y)) for y ∈ h, then ũ ∈ ŵ 1,s̃(·)0 (h). conversely, if ũ ∈ ŵ 1,s̃(·) 0 (h), then u(x) = ũ(y(x)) for x ∈ hω belongs to ŵ 1,s(·) 0 (hω). since{ ∂iu(x) = (∂iũ)(y(x)) − (∂dũ)(y(x))∂iω(x′) for i = 1, . . . ,d − 1, ∂du(x) = (∂dũ)(y(x)), there exists a constant d1(s) = d1(s,d) > 0 such that ‖∇∇∇u‖ls(·)(hω) ≤ d1(s)(1 + ‖∇∇∇ ′ω‖l∞(rd−1))‖∇∇∇ũ‖ls̃(·)(h), ‖∇∇∇ũ‖ls̃(·)(h) ≤ d1(s)(1 + ‖∇∇∇ ′ω‖l∞(rd−1))‖∇∇∇u‖ls(·)(hω). thus the map ŵ 1,s(·)0 (hω) 3 u 7→ ũ ∈ ŵ 1,s̃(·) 0 (h) is continuous, linear and bijective. let u ∈ ŵ 1,s(·)0 (hω) and φ ∈ ŵ 1,s′(·) 0 (hω). by elementary calculations, we have 〈∇∇∇u,∇∇∇φ〉hω = ∫ hω ∇∇∇u(x) ·∇∇∇φ(x)dx = ∫ h (∇∇∇ũ)(y) · (∇∇∇φ̃)(y)dy −bω[∇∇∇ũ,∇∇∇φ̃], where bω[∇∇∇ũ,∇∇∇φ̃] = − d−1∑ i=1 ∫ h ( (∂dũ)(∂iφ̃) + (∂iũ)(∂dφ̃)∂iω(y ′) ) dy + ∫ h (∂dũ)(∂dφ̃)|∇∇∇ω|2dy. by the generalized hölder inequality, there exists a constant d2(s) > 0 such that |bω[∇∇∇ũ,∇∇∇φ̃]| ≤ d2(s)‖∇∇∇′ω‖l∞(rd−1)(1 + ‖∇∇∇ ′ω‖l∞(rd−1))‖∇∇∇ũ‖ls̃(·)(h)‖∇∇∇φ̃‖ls̃′(·)(h). therefore, for 0 6= φ ∈ ŵ 1,s ′(·) 0 (hω), |〈∇∇∇u,∇∇∇φ〉hω| ‖∇∇∇φ‖ ls ′(·)(hω) ≥ ( d1(s ′)(1 + ‖∇∇∇′ω‖l∞(rd−1) ))−1{|〈∇∇∇ũ,∇∇∇φ̃〉h| ‖∇∇∇φ̃‖ ls̃ ′(·)(h) −d2(s)‖∇∇∇′ω‖l∞(rd−1)(1 + ‖∇∇∇ ′ω‖l∞(rd−1))‖∇∇∇ũ‖ls̃(·)(h) } . define kp = min { 1 2 , min { (4c2(s)d2(s)) −1; s = p,p′ }} , 20 int. j. anal. appl. (2022), 20:13 where c2(s) > 0 is a constant defined in lemma 4.14. if ‖∇∇∇′ω‖l∞(rd−1) ≤ kp, then sup 06=φ∈ŵ 1,s ′(·) 0 (hω) |〈∇∇∇u,∇∇∇φ〉hω| ‖∇∇∇φ‖ ls ′(·)(hω) ≥ (2d1(s′))−1 { sup 06=φ̃∈ŵ 1,s̃′(·)(h) |〈∇∇∇ũ,∇∇∇φ̃〉h| ‖∇∇∇φ̃‖ ls̃ ′(·)(h) − 2d2(s)((4c2(s)d2(s))−1‖∇∇∇ũ‖ls̃(·)(h) } ≥ (4ds(s′)cs(s))−1‖∇∇∇ũ‖ls̃(·)(h) ≥ c3(s)‖∇∇∇u‖ls(·)(hω), where c3(s) = (8ds(s)d1(s′)c2(s))−1. � lemma 4.16. suppose (ga). let x0 ∈ g and br(x0) b g, and let p ∈ p log + (r d). then for 0 < r′ < r, there exists a constant c3(p,r,r′) > 0 such that ‖∇∇∇(ηu)‖lp(·)(g) ≤ c3(p,r,r ′) sup 06=v∈c∞0 (br(x0)) |〈∇∇∇(ηu),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(br(x0)) for all u ∈ ŵ 1,p(·)• (g) and η ∈ c∞0 (br′(x0)). proof. let ρ ∈ d(br(x0)) such that 0 ≤ ρ ≤ 1 and ρ(x) = 1 for x ∈ br′(x0). if φ ∈ d(rd), put cφ = 1 |br(x0)| ∫ br(x0) φdx and v = ρ(φ−cφ). by the poincaré inequlity (theorem 2.7), ‖φ−cφ‖lp′(·)br′(x0)) ≤ cr‖∇∇∇v‖lp′(·)(br(x0)). here we have ‖∇∇∇v‖ lp ′(·)(br(x0)) ≤ (1 + cr‖∇∇∇ρ‖l∞(br(x0)))‖∇∇∇φ‖lp′(·)(rd ). since ∇∇∇ρ = 000 on br′(x0), ρ = 1 on br′(x0) and ∇∇∇v = (∇∇∇ρ)(φ−cφ) + ρ∇∇∇φ, we see that ∇∇∇v = ∇∇∇φ on br′(x0). if φ 6= 0 and v 6= 0, then we have |〈∇∇∇(ηu),∇∇∇φ〉rd | ‖∇∇∇φ‖ lp ′(·)(rd ) ≤ (1 + cr‖ρ‖l∞(br(x0))) |〈∇∇∇(ηu),∇∇∇φ〉g| ‖∇∇∇v‖ lp ′(·)(br(x0)) . by lemma 4.10, we can see that ‖∇∇∇(ηu)‖lp(·)(g) ≤ c1(p) sup 06=φ∈c∞0 (rd ) |〈∇∇∇(ηu),∇∇∇φ〉rd | ‖∇∇∇φ‖ lp ′(·)(rd ) ≤ c1(p)(1 + cr‖∇∇∇ρ‖l∞(rd )) sup 0 6=v∈c∞0 (br(x0)) |〈∇∇∇(ηu),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(br(x0)) . � lemma 4.17. suppose (ga) and p ∈plog+ (g). for each x0 ∈ ∂g, there exist r = r(p,x0,∂g) > 0 and a constant c5 = c5(r) > 0 such that ‖∇∇∇(ηu)‖lp(·)(g) ≤ c5 sup 0 6=v∈ŵ 1,p′(·)(gr(x0)) |〈∇∇∇(ηu),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(gr(x0)) (4.12) for all u ∈ ŵ 1,p(·)• (g) and η ∈ c∞0 (br/2(x0)). int. j. anal. appl. (2022), 20:13 21 proof. there exist ρ > 0 and a c1-function σ on bρ(x0) with (∇∇∇σ)(x0) 6= 000 such that g ∩bρ(x0) = {x ∈ bρ(x0); σ(x) < 0} and ∂g ∩ br(x0) = {x ∈ bρ(x0); σ(x) = 0}. then |∇∇∇σ(x0)|−1∇∇∇σ(x0) is the unit outer normal vector at x0. hence there exists an orthogonal matrix s such that s(|∇∇∇σ(x0)|−1∇∇∇σ(x0)) = ed = (0, . . . , 0, 1)t. define a transformation y = y(x) = s(x − x0). then y : bρ(x0) → b̂ρ(0) = {y ∈ rd; |y| < ρ} is a c1-bijective mapping and define σ̂(y) = σ(x0 + s −1y) = σ(x). hence (∇∇∇yσ̂)(0) = s(∇∇∇xσ)(x0) = |∇∇∇σ(x0)|ed, so (∇∇∇y ′σ̂)(0) 6= 000 and (∂ydσ̂)(0) = |∇∇∇σ(x0)| > 0. by the implicit function theorem, there exist 0 < ρ ′ < ρ, h > 0 and ψ ∈ c1(b′ ρ′), where b ′ ρ′ = {y ′ ∈ rd−1; |y ′| < ρ′}, such that z = zρ′,h = {y = (y ′,yd) ∈ rd; |y ′| < ρ′, |yd| < h}⊂ b̂ρ(0), (y ′,ψ(y ′)) ∈ z if y ′ ∈ b′ρ′ and σ̂(y ′,ψ(y ′)) = 0 for y ′ ∈ b′ρ′, conversely, if (y ′,yd) ∈ z and σ(y ′,yd) = 0, then yd = ψ(y ′), ψ(0) = 0 and ∇∇∇′y ′ψ(0) = 000. then clearly, ĝ ∩z = {y = (y ′,yd) ∈ z; yd < ψ(y ′)} and ∂ĝ ∩ z = {y = (y ′,yd) ∈ z; yd = ψ(y ′)}. ψ(0) = 0 and (∇∇∇′y ′ψ)(0) = 000. put gρ = g ∩ bρ(x0), ĝ = sg and ĝρ = ĝ ∩ b̂ρ(0). for p ∈ p log + (gρ), u ∈ ŵ 1,p(·) • (gρ) and v ∈ ŵ 1,p ′(·) 0 (gρ), define p̂(y) = p(x0 + s −1y), û(y) = u(x0 + s−1y) and v̂(y) = v(x0 + s−1y). then by the elementary calculations, we have 〈∇∇∇û,∇∇∇v̂〉 ĝρ = 〈∇∇∇u,∇∇∇v〉gρ and ‖∇∇∇û‖ lp̂(·)(ĝρ) = ‖∇∇∇u‖lp(·)(gρ). let η ∈ d(rd−1) such that η(y ′) = 1 for |y ′| ≤ 1 and η(y ′) = 0 for |y ′| ≥ 2. for 0 < λ < ρ′/2, put ηλ(y ′) = η(λ−1y ′), and define ωλ(y ′) = { ηλ(y ′)ψ(y ′) for |y ′| ≤ ρ′, 0 otherwise . then ∇∇∇′ωλ(y ′) = (∇∇∇′ηλ(y ′))ψ(y ′) + ηλ∇∇∇′ψ(y ′). since ψ(0) = 0 and ψ ∈ c1(b′ρ′), using the mean value theorem, |ψ(y ′)| = |ψ(y ′) −ψ(0)| ≤ |(∇∇∇′ψ)(θy ′)||y ′| ≤ 2λ|(∇∇∇′ψ)(θy ′)| for some 0 < θ < 1. hence |(∇∇∇′ηλ(y ′))ψ(y ′)| ≤ 2λ−1|(∇∇∇′η)(λ−1y ′)|λ|∇∇∇′ψ(θy ′)| = 2|(∇∇∇′η)(λ−1y ′)||∇∇∇′ψ(θy ′)|. therefore, we have sup |y ′|≤ρ |(∇∇∇ηλ(y ′))ψ(y ′)| ≤ ‖∇∇∇′η‖l∞(rd−1) sup |y ′|≤2λ |(∇∇∇′ψ)(y ′)|→ 0 as λ → 0 because ∇∇∇′ψ is continuous function and ∇∇∇′ψ(0) = 000. moreover, sup |y ′|≤ρ |ηλ(y ′)∇∇∇′ψ(y ′)| ≤ sup |y ′|≤2λ ‖η‖l∞(rd−1)‖(∇∇∇ ′ψ)(y ′)|→ 0 as λ → 0. 22 int. j. anal. appl. (2022), 20:13 thereby, if we choose λ > 0 small enough, then ‖∇∇∇′ωλ‖l∞(rd−1) ≤ kp, where kp is as in lemma 4.15. let r = r(p̂, 0,∂ĝ) = λ. then hωλ ∩br = ĝ∩br. if η ∈d(br/2) and u ∈ ŵ 1,p(·) • (g), then ηû ∈ w 1,p̂0 (ĝ) by definition of ŵ 1,p̂ • (ĝ) and ηû vanishes outside ĝ ∩br = hωλ ∩br. we extend ηû by zero to hωλ. then ηû ∈ w 1,p̂ 0 (hω) ⊂ ŵ 1,p̂ 0 (hω). by lemma 4.15, we have ‖∇∇∇(ηû)‖lp̂(·)(hω) ≤ c3(p) sup 0 6=v̂∈c∞0 (hω) |〈∇∇∇(ηû),∇∇∇v̂〉hω| ‖∇∇∇v̂‖lp̂(·)(hω) . (4.13) we show (4.12). let ρ ∈ c∞0 (br) such that 0 ≤ ρ ≤ 1 and ρ(x) = 1 on br/2. if v̂ ∈ ŵ 1,p̂(·) • (hω), then ρv̂ ∈ w 1,p̂(·)0 (hω) and by poincaré inequality, ‖∇∇∇(ρv̂)‖ lp̂ ′(·)(hωλ ) ≤‖∇∇∇ρ‖l∞(br)‖v̂‖lp̂′(·)(zωλ r,r ) + ‖∇∇∇v̂‖ lp̂ ′(·)(hωλ ) ≤ (‖∇∇∇ρ‖l∞(br)cr + 1)‖∇∇∇v̂‖lp̂′(·)(hωλ ). if v̂ 6= 0 and ρv̂ 6= 0, then we have |〈∇∇∇(ηû),∇∇∇v̂〉hωλ| ‖∇∇∇v̂‖ lp̂ ′(·)(hωλ ) ≤ (‖∇∇∇ρ‖l∞(br)cr + 1) |〈∇∇∇(ηû),∇∇∇(ρv̂)〉hωλ| ‖∇∇∇(ρv̂)‖ lp̂ ′(·)(hωλ ) . thus (4.12) follows from lemma 4.15 with c5 = c2(p)(‖∇∇∇ρ‖l∞(br)cr + 1). � 5. proof of theorem 3.1 first we derive the uniqueness. theorem 5.1. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary ∂g, and let p ∈plog+ (g). if u ∈ w 1,p(·)0 (g) satisfies 〈∇∇∇u,∇∇∇φ〉g = 0 for all φ ∈ w 1,p′(·) 0 (g), then we have u = 0 a.e. in g. proof. since p(x) ≥ p− for all x ∈ g and g is a bounded domain, we see that w 1,p(·)0 (g) ⊂ w 1,p− 0 (g). since d(g) ⊂ w 1,p ′(·) 0 (g), we have 〈∇∇∇u,∇∇∇φ〉g = 0 for all φ ∈d(g). hence it follows from the fact that d(g) is dense in w 1,(p −)′(g) that we can see that 〈∇∇∇u,∇∇∇φ〉g = 0 for all φ ∈ w 1,(p−)′(·) 0 (g) by continuity. therefore by [25, theorem 3.1], we have u = 0 a.e. in g. � we give a proof of theorem 3.1. suppose that (3.1) does not hold. then there exists {uk}∞k=1 ⊂ w 1,p(·) 0 (g) such that ‖∇∇∇uk‖lp(·)(g) = 1 (5.1) and εk = sup 06=φ∈w 1,p ′(·) 0 (g) |〈∇∇∇uk,∇∇∇φ〉g| ‖∇∇∇φ‖ lp ′(·)(g) → 0 as k →∞. (5.2) int. j. anal. appl. (2022), 20:13 23 by the poincaré inequality and (5.1), ‖uk‖lp(·)(g) ≤ c diam(g)‖∇∇∇uk‖lp(·)(g) = c diam(g). hence {uk}∞k=1 is bounded in a reflexive banach space w 1,p(·) 0 (g), so passing to a subsequence (still denoted by {uk}), we may assume that there exists u ∈ w 1,p(·) 0 (g) such that uk → u weakly in w 1,p(·) 0 (g). for each φ ∈ w 1,p′(·) 0 (g), from (5.2), we have 〈∇∇∇u,∇∇∇φ〉g = lim k→∞ 〈∇∇∇uk,∇∇∇φ〉g = 0. therefore it follows from theorem 5.1 that u = 0. since g is bounded, the embedding w 1,p(·)0 (g) ↪→ lp(·)(g) is compact (cf. [11, theorem 8.4.2], so uk → 0 strongly in lp(·)(g). by lemma 4.17, for each x0 ∈ ∂g, there exist r0 = r0(p,x0,∂g) > 0 and c5 = c5(r0) > 0 such that ‖∇∇∇(ηu)‖lp(·)(g) ≤ c5 sup 06=v∈w 1,p ′(·) 0 (g∩br0 (x0)) |〈∇∇∇(ηu),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(g∩br0 (x0)) (5.3) for all u ∈ w 1,p(·)0 (g) and η ∈ c ∞ 0 (br0/2)(x0)). since ∂g is compact, there exist finitely many xi ∈ ∂g (i = 1, . . . ,m), ri > 0 and ci > 0 such that ∂g ⊂ ∪mi=1bi, where bi = bri/4(xi ), and (5.3) holds with r0 = ri and c5 = ci. we note that g1 := g \ (∪mi=1bi ) is compact and g1 ⊂ g. according to lemma 4.16, for each x0 ∈ g1, thete exist r0 > 0 such that br0 (x0) ⊂ g and c3 > 0 such that ‖∇∇∇(ηu)‖lp(·)(g) ≤ c3 sup 0 6=v∈w 1,p ′(·) 0 (g∩br0 (x0)) |〈∇∇∇(ηu),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(g∩br0 (x0)) (5.4) for all u ∈ w 1,p(·)0 (g) and η ∈ c ∞ 0 (br0/2)(x0)). since g1 is compact, there exist finitely many xi ∈ g, ri > 0 and ci > 0 (i = m + 1, . . . ,n) such that g1 ⊂ ∪ni=m+1bi, where bi = bri/4(xi ) and (5.4) holds with r0 = ri and c3 = ci. for each i = 1, . . . ,n, choose ψi ∈ c∞0 (b ′ i ), where b ′ i = rri/2(xi ) such that 0 ≤ ψi ≤ 1, ψi = 1 on bi and denote gi = g ∩bri/4(xi ). then from (5.3) and (5.4), we have ‖∇∇∇uk‖lp(·)(gi ) ≤‖∇∇∇(ψiuk)‖lp(·)(g) ≤ c i sup 06=v∈w 1,p ′(·) 0 (gi ) |〈∇∇∇(ψiuk),∇∇∇v〉g| ‖∇∇∇v‖ lp ′(·)(gi ) =: d ik (5.5) fix i = 1, . . . ,n. for each k ∈ n, there exists vk ∈ w 1,p′(·) 0 (gi ) satisfying ‖∇∇∇vk‖lp′(·)(gi ) = 1 and 0 ≤ d ik −|〈∇∇∇(ψiuk),∇∇∇vk〉g| ≤ 1/k. therefore, 0 ≤ d ik ≤ 1 k + |〈∇∇∇uk,∇∇∇(ψivk)〉g| + |〈∇∇∇uk,vk∇∇∇ψi〉g| + |〈uk∇∇∇ψi,∇∇∇vk〉g| (5.6) ≤ 1 k + εk‖∇∇∇(ψkvk)‖lp′(·)(g) + |〈∇∇∇uk,vk∇∇∇ψi〉g| + |〈uk∇∇∇ψi,∇∇∇vk〉g|. using again the poincaré inequality, we can see that the sequence {vk}∞k=1 is bounded in w 1,p′(·) 0 (gi ). passing to a subsequence (still denoted by {vk}), there exists v ∈ w 1,p′(·) 0 (gi ) such that vk → v 24 int. j. anal. appl. (2022), 20:13 weakly in w 1,p ′(·) 0 (gi ), so vk → v strongly in l p′(·)(gi ). we estimate the right-hand side of (5.6). by the hölder inequality, |〈∇∇∇uk,vk∇∇∇ψi〉g| ≤ |〈∇∇∇uk, (vk −v)∇∇∇ψi〉g| + |〈∇∇∇uk,v∇∇∇ψi〉g| ≤ 2‖∇∇∇uk‖lp(·)(g)‖∇∇∇ψi‖l∞(gi )‖vk −v‖lp(·)(gi ) + |〈∇∇∇uk,v∇∇∇ψi〉g|→ 0 as k →∞ and |〈uk∇∇∇ψi,∇∇∇vk〉g| ≤ ‖uk∇∇∇ψi‖lp(·)(gi ) ≤‖∇∇∇ψi‖l∞(b′i )‖uk‖lp(·)(g) → 0 as k →∞. by the poincaré inequality, ‖vk‖lp′(·)(gi ) ≤ c diam(gi )‖∇∇∇vk‖lp′(·)(gi ) = c diam(gi ). hence ‖∇∇∇(ψivk)‖lp′(·)(g) ≤‖(∇∇∇ψi )vk‖lp′(·)(g) + ‖ψi∇∇∇vk‖lp′(·)(gi ) ≤‖∇∇∇ψi‖l∞(b′i )c diam(gi ) + 1. summing up the above, we see that d ik → 0 as k →∞ for every i = 1, . . . ,n. since g ⊂∪ n i=1bi, ‖∇∇∇uk‖lp(·)(g) ≤ n∑ i=1 ‖∇∇∇uk‖lp(·)(gi ) ≤ n∑ i=1 d ik → 0 as k →∞. this contradicts ‖∇∇∇uk‖lp(·)(g) = 1. this completes the proof of theorem 3.1. we can derive the lp(·)-regularity. theorem 5.2. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary. assume that p,q ∈plog+ (g) satisfies q(x) ≤ p(x) for all x ∈ g. if u ∈ w 1,q(·) 0 (g) satisfies sp(u) := sup 0 6=φ∈c∞0 (g) |〈∇∇∇u,∇∇∇φ〉g| ‖∇∇∇φ‖ lp ′(·)(g) < ∞, (5.7) then u ∈ w 1,p(·)0 (g) and ‖∇∇∇u‖lp(·)(g) ≤ cpsp(u), (5.8) where cp is the constant in theorem 3.1. proof. define a functional f ′ such that f ′(φ) = 〈∇∇∇u,∇∇∇φ〉g for φ ∈d(g). from (5.7), |f ′(φ)| ≤ ‖f ′‖ (w 1,p′(·) 0 (g)) ′‖∇∇∇φ‖lp′(·)(g) for φ ∈d(g), where ‖f ′‖ (w 1,p′(·) 0 (g)) ′ = sp(u). since d(g) is dense in w 1,p′(·) 0 (g) with respect to the norm ‖∇∇∇· ‖ lp ′(·)(g), f ′ has an extension f̃ ′ ∈ (w 1,p ′(·) 0 (g)) ′ which is unique and norm-preserving, by continuity. by theorem 3.2, there exists uniquely up ∈ w 1,p(·) 0 (g) such that 〈∇∇∇up,∇∇∇φ〉g = f̃ ′(φ) for all φ ∈ w 1,p′(·) 0 (g). hence 〈∇∇∇up,∇∇∇φ〉g = f̃ ′(φ) = f ′(φ) = 〈∇∇∇u,∇∇∇φ〉g int. j. anal. appl. (2022), 20:13 25 for all φ ∈d(g). since d(g) is dense in w 1,q ′(·) 0 (g) with respect to ‖∇∇∇·‖lq′(·)(g)-norm and q(x) ≤ p(x) for all x ∈ g, so u −up ∈ w 1,q(·) 0 (g), we have 〈∇∇∇(u −up),∇∇∇φ〉g = 0 for all φ ∈ w 1,q′(·) 0 (g). by theorem 5.1, u −up = 0, so u = up ∈ w 1,p(·) 0 (g) and (3.1) holds, so (5.8) follows. � corollary 5.3. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary. assume that p,q ∈ plog+ (g) satisfies q(x) ≤ p(x) for all x ∈ g. suppose that u ∈ w 1,q(·) 0 (g) and there exists f ∈l p(·)(g) such that 〈∇∇∇u,∇∇∇φ〉g = 〈f ,∇∇∇φ〉g for all φ ∈d(g). (5.9) then u ∈ w 1,p(·)0 (g) and satisfies (5.7). moreover, we have ‖∇∇∇u‖lp(·)(g) ≤ 2cp‖f‖lp(·)(g), where cpis the constant in theorem 3.1. proof. by the generalized hölder inequality, |〈∇∇∇u,∇∇∇φ〉g| = |〈f ,∇∇∇φ〉g| ≤ 2‖f‖lp(·)(g)‖∇∇∇φ‖lp′(·)(g) for all φ ∈d(g). hence (5.7) holds and sp(u) ≤ 2‖f‖lp(·)(g). hence we have ‖∇∇∇u‖lp(·)(g) ≤ cpsp(u) ≤ 2cp‖f‖lp(·)(g). � 6. dirichlet problem for the poisson equation let g be a bounded domain of rd (d ≥ 2) with a c1-boundary ∂g. we consider the following dirichlet problem for the poisson equation.{ −∆u = f in g, u = g on ∂g. (6.1) we are in a position to state the main theorem of this section. theorem 6.1. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary ∂g and let p ∈plog+ (g). assume that f ∈ w−1,p(·)(g) and g ∈ tr(w 1,p(·)(g)). then the system (6.1) has a unique weak solution u ∈ w 1,p(·)(g) in the sense that u ∣∣ ∂g = g and 〈∇∇∇u,∇∇∇v〉g = 〈f ,v〉 w−1,p(·)(g),w 1,p′(·) 0 (g) for all v ∈ w 1,p ′(·) 0 (g). (6.2) furthermore, there exists a constant c = c(p,d,g) > 0 such that ‖u‖w 1,p(·)(g) ≤ c(‖f‖w−1,p(·)(g) + ‖g‖tr(w 1,p(·)(g))). (6.3) 26 int. j. anal. appl. (2022), 20:13 proof. first we reduce the problem (6.1) to the homogeneous dirichlet problem. since g ∈ tr(w 1,p(·)(g)), there exists w ∈ w 1,p(·)(g) such that w ∣∣ ∂g = g and ‖w‖w 1,p(·)(g) = ‖g‖tr(w 1,p(·)(g)). (6.4) indeed, by definition of ‖g‖tr(w 1,p(·)(g)), there exists {wj}⊂ w 1,p(·)(g) with wj ∣∣ ∂g = g and ‖g‖tr(w 1,p(·)(g)) = lim j→∞ ‖wj‖w 1,p(·)(g)). hence {wj} is bounded in a reflexive banach space w 1,p(·)(g), so passing to a subsequence of {wj} (still denoted by {wj}) we may assume that wj → w weakly in w 1,p(·)(g). by lemma 2.6, wj ∣∣ ∂g → w ∣∣ ∂g in lp(·)(∂g), so w ∣∣ ∂g = g. therefore, ‖g‖tr(w 1,p(·)(g)) ≤‖w‖w 1,p(·)(g) ≤ lim inf j→∞ ‖wj‖w 1,p(·)(g) = ‖g‖tr(w 1,p(·)(g)). since ∆w ∈ w−1,p(·)(g), if we replace an unknown function u with v = u −w and a known function f with f = f + ∆w ∈ w−1,p(·)(g), the problem (6.1) is reduced the following problem.{ −∆v = f in g, v = 0 on ∂g. (6.5) therefore, we consider (6.5), that is, find v ∈ w 1,p(·)0 (g) such that 〈∇∇∇v,∇∇∇φ〉g = 〈f,φ〉 w−1,p(·)(g),w 1,p′(·) 0 (g) for all φ ∈ w 1,p ′(·) 0 (g). (6.6) according to theorem 3.2, there exists a unique v ∈ w 1,p(·)0 (g) such that (6.6) holds and c−1p ‖∇∇∇v‖lp(·)(g) ≤ ‖f‖w−1,p(·)(g) ≤ ‖f‖w−1,p(·)(g) + ‖∆w‖w−1,p(·)(g) ≤ ‖f‖w−1,p(·)(g) + c(p,g)‖w‖w 1,p(·)(g) ≤ ‖f‖w−1,p(·)(g) + c(p,g)‖g‖tr(w 1,p(·)(g)). by the poincaré inequality, ‖v‖w 1,p(·)(g) ≤ c1(p,g)‖∇∇∇v‖lp(·)(g). if we put u = v + w, (6.2) and the estimate (6.3) follows. � remark 6.2. the authors in [11] showed that if g is a bounded domain with a c1,1-boundary and f ∈ lp(·)(g),g ∈ tr(w 2,p(·)(g)), the system (6.1) has a unique strong solution u ∈ w 2,p(·)(g) and there exists a constant c depending only on p and g such that ‖u‖w 2,p(·)(g) ≤ c(‖f‖lp(·)(g) + ‖g‖tr(w 2,p(·)(g))). (6.7) they used the newton potential, and only announced the existence of a weak solution as in theorem 6.1. however, we can easily show that theorem 6.1 holds using theorem 3.1 and 3.2 under the weaker assumption of the regularity of boundary. int. j. anal. appl. (2022), 20:13 27 7. an approach to the stokes problem in this section, let g be a bounded domain of rd (d ≥ 2) with a c1-boundary γ = ∂g. we consider the following homogeneous stokes problem.  −∆u + ∇∇∇π = f in g, divu = 0 in ω, u = 000 on γ, π = 0 on γ. (7.1) we have the following theorem. theorem 7.1. let g be a bounded domain of rd (d ≥ 2) with a c1-boundary γ = ∂g and let p ∈ plog+ (g). assume that f ∈ l p(·)(g). then the problem (7.1) has a unique weak solution (u,π) ∈w1,p(·)0 (g) ×w 1,p(·) 0 (g), in the sense of 〈∇∇∇u,∇∇∇v〉g + 〈∇∇∇π,v〉g = 〈f ,v〉g for all v ∈w 1,p′(·) 0 (g), (7.2) and there exists a constant c = c(p,d,g) > 0 such that ‖u‖w1,p(·)(g) + ‖π‖w 1,p(·)(g) ≤ c‖f‖lp(·)(g). (7.3) furthermore, if g is of class c1,1, then u ∈ w2,p(·)(g) and (u,π) is a strong solution of (7.1). moreover, we have ‖u‖w2,p(·)(g) + ‖π‖w 1,p(·)(g) ≤ c ′‖f‖lp(·)(g) where c′ is a constant depending only on p,d and g. proof. first we consider the following dirichlet problem for the laplacian ∆.{ ∆π = div f in g, π = 0 on γ. (7.4) suppose that f ∈ lp(·)(g). since div f ∈ w−1,p(·)(g), if follows from theorem 6.1 that (7.4) has a unique weak solution π ∈ w 1,p(·)0 (g) and there exist positive constants c = c(p,d,g) and c1 = c1(p,d,g) such that ‖π‖w 1,p(·)(g) ≤ c‖div f‖w−1,p(·)(g) ≤ c1‖f‖lp(·)(g). (7.5) we note that f −∇∇∇π ∈lp(·)(g) and div (f −∇∇∇π) = 0 in the distribution sense. (7.6) we apply proposition 3.3. for this purpose, define xp(·)(g) = {u ∈w1,p(·)0 (g); divu = 0 in g}. 28 int. j. anal. appl. (2022), 20:13 then clearly xp(·)(g) is a closed subspace of a reflexive banach space w1,p(·)0 (g), so x p(·)(g) is also reflexive banach space. since xp(·)(g) ⊂w1,p(·)0 (g), if follows from theorem 3.1 0 < c−1p ≤ inf 0006=u∈w1,p(·)0 (g) sup 000 6=v∈w1,p ′(·) 0 (g) |〈∇∇∇v,∇∇∇u〉g| ‖∇∇∇u‖lp(·)(g)‖∇∇∇v‖lp′(·)(g) (7.7) ≤ inf 0006=u∈xp(·)(g) sup 0006=v∈w1,p ′(·) 0 (g) |〈∇∇∇v,∇∇∇u〉g| ‖∇∇∇u‖lp(·)(g)‖∇∇∇v‖lp′(·)(g) . taking the poincaré inequality into consideration, let x = w1,p ′(·) 0 (g) equipped with the norm ‖v‖x = ‖∇∇∇v‖ lp ′(·)(g) and m = x p(·)(g) equipped with the norm ‖u‖m = ‖∇∇∇u‖lp(·)(g). define a(v,u) = 〈∇∇∇v,∇∇∇u〉g for v ∈ x,u ∈ m. by the generalized hölder inequality, we have |a(v,u)| ≤ 2‖∇∇∇v‖ lp ′(·)(g)‖∇∇∇u‖lp(·)(g) = 2‖v‖x‖u‖m. thus a(v,u) is a continuous bilinear form on x×m. define bounded linear operators a : x → m′ and a′ : m → x′ by 〈av,u〉 = 〈v,a′u〉 = a(v,u). then |〈v,a′u〉| ≤ 2‖v‖x‖u‖m for all v ∈ x,u ∈ m. from (7.7), c−1p ≤ inf 000 6=u∈m sup 0006=v∈x |a(v,u)| ‖v‖x‖u‖m . put v = kera. we characterize v = kera. lemma 7.2. it follows that v = {v = (−∆)−1∇∇∇ϕ; ϕ ∈ lp ′(·)(g)}. here v = (−∆)−1g, g ∈ w−1,p ′(·)(g) means that v ∈w1,p ′(·) 0 (g) is a unique weak solution of the following problem.{ −∆v = g in g, v = 000 on γ. (7.8) proof. let v ∈ v . then 〈av,u〉m′,m = 0 for all u ∈ m, that is, 〈−∆v,u〉 w−1,p ′(·)(g),w 1,p(·) 0 (g) = 〈∇∇∇v,∇∇∇u〉g = 〈av,u〉m′,m = 0 for all u ∈ xp(·)(g). by the de rham theorem (cf. aramaki [3,4]), there exists ϕ ∈ lp ′(·)(g) such that −∆v = ∇∇∇ϕ in w−1,p ′(·)(g). since v = 000 on γ, we have v = (−∆)−1∇∇∇ϕ. thus we have v ⊂{v = (−∆)−1∇∇∇ϕ; ϕ ∈ lp ′(·)(g)}. conversely, let v = (−∆)−1∇∇∇ϕ for some ϕ ∈ lp ′(·)(g). then v is a unique weak solution of (7.8) with g = ∇∇∇ϕ. for any u ∈xp(·)(g), we have 〈av,u〉m′,m = 〈∇∇∇v,∇∇∇u〉g = 〈−∆v,u〉w−1,p′(·)(g),w1,p(·)0 (g) = 〈∇∇∇ϕ,u〉 w−1,p ′(·)(g),w 1,p(·) 0 (g) = −〈ϕ, divu〉g = 0. hence av = 000 in m′, that is, v ∈ kera = v . � denote that v ⊥ = {f ∈ x′ = w−1,p(·)(g);〈f ,v〉x′,x = 0 for all v ∈ v}. int. j. anal. appl. (2022), 20:13 29 lemma 7.3. if g ∈lp(·)(g) satisfies divg = 0, then g ∈ v ⊥. proof. according to lemma 7.2, for any v ∈ v , there exists ϕ ∈ lp ′(·)(g) such that v = (−∆)−1∇∇∇ϕ. thereby, we have 〈g,v〉x′,x = 〈g, (−∆)−1∇∇∇ϕ〉 w−1,p(·)(g),w 1,p′(·) 0 (g) = 〈(−∆)−1g,∇∇∇ϕ〉 w 1,p(·) 0 (g),w −1,p′(·)(g) = −〈div (−∆)−1g,ϕ〉g. if we put w = (−∆)−1g, we have −∆w = g in ω and w = 000 on γ. hence (−∆)divw = −div ∆w = divg = 0. therefore, (−∆)−1(−∆)divw = divw = 0, that is, div (−∆)−1g = 0. this implies that 〈g,v〉x′,x = 0 for all v ∈ v . thus g ∈ v ⊥. � we continue the proof of theorem 7.1. from (7.6), we know that f − ∇∇∇π ∈ lp(·)(g) and div (f −∇∇∇π) = 0. by proposition 3.3, a′ : m → v ⊥ is an isomorphism and cp is the continuity constant of (a′)−1. by lemma 7.3, we see that f −∇∇∇π ∈ v ⊥, so there exists a unique u ∈ m such that a′u = f −∇∇∇π, that is, divu = 0 in g, u = 000 on γ and 〈∇∇∇v,∇∇∇u〉g = 〈v, f −∇∇∇π〉x,x′ for all v ∈w 1,p′(·) 0 (g), so (7.2) holds. furthermore, we have ‖u‖w1,p(·)(g) ≤ cp‖f −∇∇∇π‖w−1,p(·)(g) ≤ c′p‖f −∇∇∇π‖lp(·)(g) ≤ c′p(‖f‖lp(·)(g) + ‖∇∇∇π‖lp(·)(g)) ≤ c′′p‖f‖lp(·)(g). summing up this inequality and (7.5), we get the estimate (7.3). if, in particular, g is of class c1,1, since −∆u = f −∇∇∇π ∈lp(·)(g) in g and u = 000 on γ, it follows from [11, theorem 14.1.2] that u ∈w2,p(·)(g) and ‖u‖w2,p(·)(g) ≤ c‖f −∇∇∇π‖lp(·)(g) ≤ c1‖f‖lp(·)(g). � now we consider the inhomogeneous stokes problem.  −∆u + ∇∇∇π = f in g, divu = ϕ in g, u = g on γ, π = π0 on γ. (7.9) 30 int. j. anal. appl. (2022), 20:13 theorem 7.4. let g be a bounded domain of rd with a c1,1-boundary γ and let p ∈ plog+ (g). assume that f ∈ lp(·)(g), π0 ∈ tr(w 1,p(·)(g)), ϕ ∈ w 1,p(·)(g) and g ∈ tr(w2,p(·)(g)) satisfy the compatibility condition ∫ g ϕdx = ∫ γ g ·ndσ, (7.10) where dσ is the surface measure on γ. then there exists a unique solution (u,π) ∈ w2,p(·)(g) × w 1,p(·)(g) of (7.9) and there exists a constant c = c(p,d,g) > 0 such that ‖u‖w2,p(·)(g) + ‖π‖w 1,p(·)(g) ≤ c(‖f‖lp(·)(g) + ‖π0‖tr(w 1,p(·)(g)) + ‖ϕ‖w 1,p(·)(g) + ‖g‖tr(w1,p(·)(g))). before the proof, it is necessary to prepare some arguments. proposition 7.5. let g be a bounded domain of rd with a c1,1-boundary γ and p ∈plog+ (g). if we assume that g ∈ tr(w 1,p(·)(g)), then there exists u ∈ w 2,p(·)(ω) such that γ1(u) = g and γ0(u) = 0. proof. we use the argument of boyer and fabrie [7, proof of theorem iii.2.23]. let δ(x) be the signed distance from x to γ, that is, δ(x) = { d(x, γ) if x ∈ g, −d(x, γ) if x /∈ g. then δ is lipschitz-continuous in rd with the lipschitz constant lip(δ) ≤ 1. let η be a standard mollifier, that is, η ∈ c∞0 (r d), supp η ⊂ b (the unit sphere of rd), η ≥ 0, ∫ rd ηdx = ∫ b ηdx = 1 and η(x) only depends on |x|. for x ∈ rd and τ ∈ r, define a function g(x,τ) = ∫ b δ ( x + τ 2 z ) η(z)dx. then we can clearly see that g ∈ c∞(rd × (r\{0}) and |g(x,τ1) −g(x,τ2)| ≤ 1 2 |τ1 −τ2|. therefore, by banach fixed-point theorem, for any x ∈ rd, there exists uniquely ρ(x) ∈ r such that ρ(x) = g(x,ρ(x)). we call ρ a regularized distance function of g. the regularized distance function ρ has the following properties. (i) ρ(x) = 0 ⇐⇒ δ(x) = 0 ⇐⇒ x ∈ γ, and there exists constants c1,c2 > 0 such that c1 ≤ δ(x)/ρ(x) ≤ c2 for x ∈ rd \ γ. (ii) ρ ∈ c1,1(rd) ∩c∞(rd \ γ). (iii) ∇∇∇ρ(x) = ∇∇∇δ(x) = −n(x) for all x ∈ γ, and there exists an open neighborhood u of γ such that infu |∇∇∇ρ| > 0. int. j. anal. appl. (2022), 20:13 31 for g ∈ tr(w 1,p(·)(g)), there exists v ∈ w 1,p(·)(g) such that γ0(v) = g and there exists a constant c > 0 such that ‖v‖w 1,p(·)(g) ≤ c‖g‖tr(w 1,p(·)(g)). we define rng(x) = −ρ(x) ∫ b v(x + αρ(x)z)η(z)dz, (7.11) where α > 0 so that α ≤ c1 and αlip(ρ) < 1, c1 is the constant of (i). for x ∈ g and z ∈ b, we have x + αρ(x)z ∈ g, so rng is well defined and rng ∈ c∞(g). we show that rng ∈ w 2,p(·)(g). by the calculations, for i, j = 1, . . . ,d, we have ∂rng ∂xi (x) = − ∂ρ ∂xi (x) ∫ b v(x + αρ(x)z)η(z)dz −ρ(x) ∫ b ∇∇∇v(x + αρ(x)z) · ( ei + α ∂ρ ∂xi (x)z ) η(z)dz, (7.12) ∂2rng ∂xi∂xj = − ∂2ρ ∂xi∂xj (x) ∫ b v(x + αρ(x)z)ψ(z)dz − ∂ρ ∂xi (x) ∫ b ∂v ∂xj (x + αρ(x)z)ψ(z)dz −α ∂ρ ∂xi (x) ∂ρ ∂xj (x) ∫ b ∇∇∇v(x + αρ(x)z) ·ψ(z)zdz +(d − 1) ∂ρ ∂xj (x) ∫ b ∂v ∂xi (x + αρ(x)z)η(z)dz + 1 α ∫ b ∂v ∂xi (x + αρ(x)z) ∂η ∂zj (z)dz + ∂ρ ∂xj (x) ∫ b ∂v ∂xi (x + αρ(x)z)∇∇∇η(z) ·zdz, where ψ(z) = η(z) − div z (η(z)z). since all the derivatives of ρ up to second-order are bounded, it suffices to show that the terms of the form f (x) = ∫ b f (x + αρ(x)z)ψ̃(z)dz for x ∈ g, where f ∈ lp(·)(g) and ψ̃ ∈ c∞0 (b), belong to l p(·)(g). we note that we can not use the jensen inequality in the case of variable exponent. however, applying a variant of the jensen inequality (cf. [11, theorem 4.2.4 and corollary 4.2.5]), there exists a constant c > 0 such that ρp(·),g(f ) ≤ c‖f‖ p+−p− lp(·)(g) ρp(·),g(f ) + c‖f‖ p+ lp(·)(g) . therefore, we see that f ∈ lp(·)(g), so u := rng ∈ w 2,p(·)(g). from (7.11) and property (i), we see that γ0(u) = 0. from (7.12) and property (iii), we can see that γ1(u) = γ0(∇∇∇rng) ·n = γ0(v) = g. � 32 int. j. anal. appl. (2022), 20:13 lemma 7.6. let g be a bounded domain of rd with a c1,1-boundary γ and let p ∈ plog+ (g). for (g0,g1) ∈ tr(w 2,p(·)(g)) × tr(w 1,p(·)(g)), there exists u ∈ w 2,p(·)(g) such that γ0(u) = g0 and γ1(u) = g1, moreover there exists a constant c > 0 such that ‖u‖w 2,p(·)(g) ≤ c(‖g0‖tr(w 2,p(·)(g)) + ‖g1‖tr(w 1,p(·)(g))). (7.13) proof. from theorem 2.5, we see that w 2,p(·) 0 (g) = {v ∈ w 2,p(·)(g); γ0(v) = γ1(v) = 0}. we consider the mapping γ : w 2,p(·)(g)/w 2,p(·) 0 (g) 3 [u] 7→ (γ0(u),γ1(u)) ∈ tr(w 2,p(·)(g)) × tr(w 1,p(·)(g)). since n is a lipschitz function on γ, we can extend n to a lipschitz function on g, so γ1(u) = γ0(∇∇∇u · n). thus γ is a linear continuous injection. we show that γ is surjective. let (g0,g1) ∈ tr(w 2,p(·)(g)) × tr(w 1,p(·)(g)). choose v0 ∈ w 2,p(·)(g) such that γ0(v0) = g0 and define v = v0 + rn(g1 −γ1(v0)) ∈ w 2,p(·)(g). then by proposition 7.5, γ0(v) = γ0(v0) = g0 and γ1(v) = g1. thereby γ is surjective. by the banach open mapping theorem, γ−1 is also linear and continuous. moreover there exists a constant c > 0 such that ‖[v]‖ w 2,p(·)(g)/w 2,p(·) 0 (g) ≤ c(‖g0‖tr(w 2,p(·)(g)) + ‖g1‖tr(w 1,p(·)(g))). we show that ‖[v]‖ w 2,p(·)(g)/w 2,p(·) 0 (g) = inf{‖v + w‖w 2,p(·)(g); w ∈ w 2,p(·) 0 (g)} is achieved. indeed, choose wj ∈ w 2,p(·) 0 (g) such that lim j→∞ ‖v + wj‖w 2,p(·)(g) = ‖[v]‖w 2,p(·)(g)/w 2,p(·)0 (g) . then {wj} is bounded in a reflexive banach space w 2,p(·) 0 (g). passing to a subsequence, we may assume that wj → w weakly in w 2,p(·)(g). hence ‖v + w‖w 2,p(·)(g) ≤ lim inf j→∞ ‖u + wj‖w 2,p(·)(g) = ‖[v]‖w 2,p(·)(g)/w 2,p(·)0 (g) . if we put u = v + w ∈ w 2,p(·)(g), then we have γ0(u) = g0, γ1(u) = g1 and the estimate (7.13) holds. � the following lemma is the celebrated héron formula (cf. amrouche and girault [2, lemma 3.5]). lemma 7.7. let g be a bounded domain of rd with a c1,1-boundary γ and let p ∈ plog+ (g). then for u ∈w2,p(·)(g), the following héron formula holds. γ0(divu) = div γ(γ0(u)t) + γ1(u) ·n− 2kγ0(u) ·n, where k denotes the mean curvature of γ, div γ is the surface divergence and vt = v−(v ·n)n is the tangent component of v. int. j. anal. appl. (2022), 20:13 33 proposition 7.8. let g be a bounded domain of rd with a c1,1-boundary γ and let p ∈ plog+ (g). then for every g ∈ tr(w2,p(·)(g)) and ϕ ∈ tr(w 1,p(·)(g)), there exists u ∈ w2,p(·)(g) such that γ0(divu) = ϕ and γ0(u) = g, and there exists a constant c > 0 depending only on p and g such that ‖u‖w2,p(·)(g) ≤ c(‖g‖tr(w2,p(·)(g)) + ‖ϕ‖tr(w 1,p(·)(g))). (7.14) proof. put g0 = g ∈ tr(w2,p(·)(g)),g1 = 2kg − ndiv γ(gt) + ϕn. by lemma 7.6, there exists u ∈ w2,p(·)(g) such that γ0(u) = g,γ1(u) = 2kg −ndiv γ(gt) + ϕn, and (7.14) holds. then by lemma 7.7, we have γ0(divu) = ϕ. � proposition 7.9. let g be a bounded domain of rd with a c1,1-boundary γ and let p ∈plog+ (g). for any g ∈ tr(w2,p(·)(g)) and any ϕ ∈ w 1,p(·)(g) satisfying the compatibility condition (7.10), there exists u0 ∈ w2,p(·)(g) such that divu0 = ϕ in g and γ0(u0) = g, moreover, there exists a constant c > 0 depending only on p,d and g such that ‖u0‖w2,p(·)(g) ≤ c(‖ϕ‖w 1,p(·)(g) + ‖g‖tr(w2,p(·)(g))). (7.15) proof. by proposition 7.8, there exists u ∈ w2,p(·)(g) such that γ0(divu) = γ0(ϕ), γ0(u) = g and (7.14) holds. then divu−ϕ ∈ w 1,p(·)(g). since γ0(divu−ϕ) = 0, we see that divu−ϕ ∈ w 1,p(·) 0 (g). since it follows from the compatibility condition (7.10) and the green theorem that∫ g (divu−ϕ)dx = ∫ γ g ·ndσ − ∫ g ϕdx = 0. by [3, 4, theorem 3.1] (e) (cf. aramaki [6] for the case p(·) = p = const.), there exists w ∈ w 2,p(·) 0 (g), unique up to an additive function of kerdiv := {v ∈ w 2,p(·) 0 (g); divv = 0 in g}, such that divw = divu−ϕ in g, and there exists a constant c > 0 such that ‖[w]‖w2,p(·)(g)/kerdiv ≤ c‖divu−ϕ‖w 1,p(·)(g) ≤ c1(‖g‖tr(w2,p(·)(g)) + ‖ϕ‖w 1,p(·)(g)). since we can easily see that ‖[w]‖w2,p(·)(g)/kerdiv = inf{‖w + v‖w2,p(·)0 (g) with divv = 0 in g} is achieved, there exists u1 ∈w 2,p(·) 0 (g) such that ‖w + u1‖w2,p(·)(g) = ‖[w]‖w2,p(·)(g)/kerdiv ≤ c1(‖g‖tr(w2,p(·)(g)) + ‖ϕ‖w 1,p(·)(g)). it suffices to put u0 = w + u1. � proof of theorem 7.4 by proposition 7.9, there exists u0 ∈w2,p(·)(g) such that divu0 = ϕ in g and γ0(u0) = g 34 int. j. anal. appl. (2022), 20:13 and the estimate (7.15) holds. moreover, there exists π̃ ∈ w 1,p(·)(g) such that γ0(π̃) = π0 and ‖π̃‖w 1,p(·)(g) ≤ c‖π0‖tr(w 1,p(·)(g)). (7.16) if we put v = u−u0 and π̂ = π−π̃ in problem (7.9), then the system (7.9) is reduced to the following problem   −∆v + ∇∇∇π̂ = f + ∆u0 + ∇∇∇π̃ in g, divv = 0 in g, v = 000 on γ, π̂ = 0 on γ. (7.17) from theorem 7.1, the estimates (7.15) and (7.16), the conclusion is clear. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] c. amrouche, n.h. seloula, lp-theory for vector potentials and sobolev’s inequalities for vector fields: application to the stokes equations with pressure boundary conditions, math. models methods appl. sci. 23 (2013), 37-92. https://doi.org/10.1142/s0218202512500455. [2] c. amrouche, v. girault, decomposition of vector spaces and application to the stokes problem in arbitrary dimension, czech. math. j. 44 (1994), 109-140. [3] j. aramaki, equivalent relations with the j. l. lions lemma in a variable exponent sobolev space and an application to the korn inequality, submitted. [4] j. aramaki, equivalent relations with the j. l. lions lemma in a variable exponent sobolev space and their applications, submitted. [5] j. aramaki, an extension of a variational inequality in the simader theorem to a variational exponent sobolev space and applications: the neumann case, submitted. [6] j. aramaki, on the j.l. lions lemma and its applications to the maxwell-stokes type problem and the korn inequality, commun. math. res. 37 (2021), 209–235. https://doi.org/10.4208/cmr.2020-0043. [7] f. boyer and p. fabrie, mathematical tools for the study of the incompressible navier-stokes equations and related models, springer, new york, 2010. [8] p.g. ciarlet, g. dinca, a poincaré inequality in a sobolev space with a variable exponent, chin. ann. math. ser. b. 32 (2011), 333–342. https://doi.org/10.1007/s11401-011-0648-1. [9] s.g. deng, eigenvalues of the p(x)-laplacian steklov problem, j. math. anal. appl. 339 (2008), 925–937. https: //doi.org/10.1016/j.jmaa.2007.07.028. [10] l. diening, theoretical and numerical results for electrorheological fluids, ph.d. thesis, university of freiburg, germany 2002. [11] l. diening, p. harjulehto, p. hästö and m. růžička, lebesgue and sobolev spaces with variable exponent. lecture notes in math., springer, berlin, 2017. [12] x.-l. fan, q.-h. zhang, existence of solutions for p(x)-laplacian dirichlet problem, nonlinear anal.: theory methods appl. 52 (2003), 1843–1852. https://doi.org/10.1016/s0362-546x(02)00150-5. [13] x. fan, q. zhang, d. zhao, eigenvalues of p(x)-laplacian dirichlet problem, j. math. anal. appl. 302 (2005), 306–317. https://doi.org/10.1016/j.jmaa.2003.11.020. https://doi.org/10.1142/s0218202512500455 https://doi.org/10.4208/cmr.2020-0043 https://doi.org/10.1007/s11401-011-0648-1 https://doi.org/10.1016/j.jmaa.2007.07.028 https://doi.org/10.1016/j.jmaa.2007.07.028 https://doi.org/10.1016/s0362-546x(02)00150-5 https://doi.org/10.1016/j.jmaa.2003.11.020 int. j. anal. appl. (2022), 20:13 35 [14] x. fan, d. zhao, on the spaces lp(x)(ω) and wm,p(x)(ω), j. math. anal. appl. 263 (2001), 424–446. https: //doi.org/10.1006/jmaa.2000.7617. [15] d. fujiwara, h. morimoto, an lr-theorem of the helmholtz decomposition of vector fields, j. fac. sci. univ. tokyo, sec. i, 24 (1977), 685-700. [16] g. galdi. an introduction to the mathematical theory of the navier-stokes equations, linearized steady problem, vol. 38 of tracts in natural philosophy, springer, new york, 1994. [17] d. gilbarg, n.s. trudinger, elliptic partial differential equations of second order, springer, berlin, heidelberg, new york, 1998. [18] t.c. halsey, electrorheological fluids, science. 258 (1992), 761–766. https://doi.org/10.1126/science.258. 5083.761. [19] o. kovác̆ik, j. rákosnic, on spaces lp(x)(ω) and wk,p(x)(ω), czechoslovak math. j. 41 (1991), 592-618. [20] h. kozono, t. yanagisawa, global div-curl lemma on bounded domains in r3, j. funct. anal. 256 (2009), 38473859. [21] m. mihăilescu, v. rădulescu, a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, proc. r. soc. a. 462 (2006), 2625–2641. https://doi.org/10.1098/rspa.2005.1633. [22] t. miyakawa, on nonstationary solutions of the navier-stokes equations in an exterior domain, hiroshima math. j. 12 (1982), 115-140. [23] m. růžička, electrotheological fluids: modeling and mathematical theory, lecture notes in mathematics, vol. 1784, berlin, springer-verlag, 2000. [24] c.g. simader, h. sohr, a new approach to the helmholtz decomposition and the neumann problem in lq-spaces for bounded and exterior domains, in: g.p. galdi (ed.), mathematical problem relating to the navier-stokes equations, in; ser. adv. math. appl. sci., world scientific, singapore, new jersey, london, hong kong, (1992), 1-35. [25] c.g. simader, the dirichlet problem for the laplacian in bounded and unbounded domains, pitman res. noes math. ser., vol. 360, longman, 1996, [26] z. wei, z. chen, existence results for the p(x)-laplacian with nonlinear boundary condition, isrn appl. math. 2012 (2012), 727398. [27] z. yücedağ, solutions of nonlinear problems involving p(x)-laplacian operator, adv. nonlinear anal. 4 (2015), 285–293. https://doi.org/10.1515/anona-2015-0044. [28] d. zhao, w.j. qiang, x.l. fan, on generalized orlicz space lp(x)(ω), j. gansu sci. 9 (1996), 1-7 (in chinese). [29] v.v. zhikov, averaging of functionals of the calculus of variations and elasticity theory, math. ussr izv. 29 (1987), 33–66. https://doi.org/10.1070/im1987v029n01abeh000958. https://doi.org/10.1006/jmaa.2000.7617 https://doi.org/10.1006/jmaa.2000.7617 https://doi.org/10.1126/science.258.5083.761 https://doi.org/10.1126/science.258.5083.761 https://doi.org/10.1098/rspa.2005.1633 https://doi.org/10.1515/anona-2015-0044 https://doi.org/10.1070/im1987v029n01abeh000958 1. introduction 2. preliminaries 3. the weak dirichlet problem for the laplacian in a variable exponent sobolev space in a bounded domain 4. preparation to a proof of theorem 3.1 5. proof of theorem 3.1 6. dirichlet problem for the poisson equation 7. an approach to the stokes problem references international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 83-89 http://www.etamaths.com quasi-compact perturbations of the weyl essential spectrum and application to singular transport operators leila mebarki1, mohammed benharrat2 and bekkai messirdi1,∗ abstract. this paper is devoted to the investigation of the stability of the weyl essential spectrum of closed densely defined linear operator a subjected to additive perturbation k such that (λ − a − k)−1k or k(λ − a − k)−1 is a quasi-compact operator. the obtained results are used to describe the weyl essential spectrum of singular neutron transport operator. 1. introduction and preliminaries let x and y be complex infinite dimensional banach spaces, and let c(x,y ) (resp. l(x,y )) denote the set of all closed, densely defined linear operators from x into y (resp. the banach algebra of all bounded operators), abbreviate c(x,x) (resp. l(x,x)) to c(x) (resp. l(x)). for a ∈c(x), write d(a) ⊂ x,n(a), r(a), σ(a) and ρ(a) respectively, the domain, the null space, the range, the spectrum and the resolvent set of a. the subset of all compact operators of l(x,y ) is noted by k(x,y ) and if x = y we write k(x). let i denote the identity operator in x. let a ∈c(x), we know that d(a) provided with the graph norm ‖x‖a = ‖x‖+‖ax‖ is a banach space denoted by xa. recall that an operator b is relatively bounded with respect to a or simply a-bounded if d(a) ⊆d(b) and b is bounded on xa. definition 1.1. an operator v ∈l(x) is said to be quasi-compact operator if there exists a compact operator k and an integer m such that ‖v m −k‖ < 1. we denote by qk(x) the set of all quasi-compact operators. if r = r(v ) is the spectral radius of a bounded linear operator v on x, another equivalent definition is given in [2], to quasi-compactness, is that if there exists m and n closed v -invariant subspaces of x such that x = m ⊕n with r(v/m) < r and dimn < ∞. we refer the reader to [2] for a detailed presentation of the quasi-compactness. for a ∈ c(x,y ), the nullity, α(a), of a is defined as the dimension of n(a) and the deficiency, β(a), of a is defined as the codimension of r(a) in y . the set of fredholm operators from x into y is defined by φ(x,y ) = {a ∈c(x,y ) : r(a) is closed in y,β(a) < ∞ and α(a) < ∞}, if a ∈ φ(x,y ), the number ind(a) = α(a) −β(a) is called the index of a. the operator a is weyl if it is fredholm of index zero. the fredholm essential spectrum σef (a) and the weyl essential spectrum σew(a) are defined by: σef (a) := {λ ∈ c : (λ−a) is not fredholm } and σew(a) := {λ ∈ c : (λ−a) is not weyl }. 2010 mathematics subject classification. 47a10. key words and phrases. quasi-compact operators; weyl essential spectrum; transport operators. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 83 84 mebarki, benharrat and messirdi for a ∈l(x), the fredholm essential spectrum of a is also equal to the spectrum of a+k(x) in the calkin algebra l(x)/k(x). accordingly, the essential spectral radius of a, denoted by re(a), is given by the formula re(a) = lim n→+∞ dist(an,k(x)) 1 n = inf n∈n dist(an,k(x)) 1 n where dist(a,k(x)) = infk∈k(x) ‖a−k‖. thus a is quasi-compact if and only if re(a) < 1. we say that a is a riesz operator if λ − a is fredholm for all non-zero complex numbers λ. thus a is riesz if and only if re(a) = 0. a useful property of a bounded linear operator a on a banach space is that each spectral element of a which lies in the unbounded component of the complement of the essential spectrum of a is an eigenvalue of finite multiplicity. further, if there are infinitely many of them, then they cluster only on the essential spectrum. certainly we have the implications: compact ⇒ riesz ⇒ quasi-compact. the following theorem gives an important and useful characterization of quasi-compact operators: theorem 1.2. [1, theorem i.6] if v ∈ qk(x) then for all for all complex number λ such that |λ| ≥ 1 then (λ−a) is a weyl operator. as a consequence of this theorem, v is quasi-compact if (and only if) the peripheral spectrum of v contains only many poles of the resolvent r(.,v ) of v , and if v is power-bounded operator the residue of r(.,v ) at each peripheral pole is of finite rank. however, the concept of quasicompact operator plays a crucial role and seems to be more appropriate because it evokes not only the special configuration of the spectrum of v , but also the fact that the spectral values of greatest modulus are eigenvalues associated with finite dimensional generalized eigenspaces. the purpose of this paper is to point out how, by means of the concept of the quasicompactness it is possible to improve the definition of the weyl essential spectrum. more precisely, we establish that σew(a) = σew(a + k) for all closed densely defined linear operator a, with k ∈ c(x) such that k is a-bounded and (λ−a−k)−1k or k(λ−a−k)−1 is a quasi-compact operator for all λ ∈ ρ(a + k). our results generalize many known ones in the literature. the next section is concerned with the definition of the weyl essential spectrum of closed densely defined linear operator a subjected to additive perturbation k, a-bounded, such that (λ − a − k)−1k or k(λ − a − k)−1 is a quasi-compact operator for some λ ∈ ρ(a + k) and we establish some properties of quasi-compact operators. in the last section we apply the results obtained in the second section to investigate the weyl essential spectrum of the following singular neutron transport operator aψ(x,ξ) = −ξ∇xψ(x,ξ) −σ(ξ)ψ(x,ξ) + ∫ rn κ(x,ξ′)ψ(x,ξ′) dξ′ (x,ξ) ∈ ω ×v = tψ(x,ξ) + kψ(x,ξ), with the vacum boundary conditions ψ|γ− (x,ξ) = 0 (x,ξ) ∈ γ− = {(x,ξ) ∈ ∂ω ×v ; ξ ·n(x) < 0}. where ω is a smooth open subset of rn (n ≥ 1), v is the support of a positive radon measure dµ on rn and n(x) stands for the outward normal unit at x ∈ ∂ω. the operator a describes the transport of particle (neutrons, photons, molecules of gas, etc.) in the domain ω. the function ψ ∈ lp(ω×v,dxdµ(ξ)) (1 ≤ p < ∞) represents the number (or probability) density of particles having the position x and the velocity ξ. the functions σ(.) and κ(., .) are called, respectively, the collision frequency and the scattering kernel and will be assumed to be unbounded. more precisely, we will assume that there exist a closed subset e ⊂ rn with zero dµ measure and a constant σ0 > 0 such that (1.1a) σ(.) ∈ l∞loc(r n \e), σ(ξ) > σ0; quasi-compact perturbations of the weyl essential spectrum 85 (1.1b) [∫ rn ( κ(.,ξ′) σ(ξ′ 1 p )qdµ(ξ′) ]1 q ∈ lp(rn). where q denotes the conjugate exponent of p. we describe here the weyl essential spectrum of the operator a subjected to assumptions (1.1a), (1.1b) and the scattering kernel taking the form κ(., .) = κ1(., .) + κ2(., .) with κi(., .) are non-negative i = 1, 2. theorem 3.3 assert that if conditions (1.1a) and (1.1b) are satisfied, the hyperplanes of rn have zero dµ-measure, (i.e. for each e ∈ sn−1,dµ{ξ ∈ rn,ξ.e = 0} = 0, where sn−1 denotes the unit ball of rn), the collision operator k1 : ψ(x,ξ) −→ ∫ rn κ1(x,ξ ′)ψ(x,ξ′)dξ′ is compact from lp(rn,σ(ξ)dµ(ξ)) into lp(rn,dµ(ξ)), and if further sup ξ∈rn σ(ξ) (reλ + σ(ξ))p ∥∥∥∥∥∥ [∫ rn ( κ2(.,ξ ′) σ(ξ′ 1 p )qdµ(ξ′) ]1 q ∥∥∥∥∥∥ lp(rn) < 1 for reλ > −σ0. then k(λ−a−k)−1 is a sum of a compact operator k1(µ−t)−1[i+(µ−λ+k)(λ−t−k)−1]+ k2(µ−t)−1k1(λ−t−k)−1 and a pure contraction k2(µ−t)−1[i +(µ−λ+k2)(λ−t−k)−1] on xp. hence k(λ − a − k)−1 is a quasi-compact operator on xp. now, by the knowledge of the weyl essential spectrum of the streaming operator t and theorem 2.3 we assert that σew(a) = {λ ∈ c : reλ ≤−σ0}. 2. invariance of the weyl essential spectrum if a ∈c(x) we define the sets ra(x) = {k ∈l(x) such that (λ−a−k)−1k ∈qk(x) for some λ ∈ ρ(a + k)}, la(x) = {k ∈l(x) such that k(λ−a−k)−1 ∈qk(x) for some λ ∈ ρ(a + k)}, theorem 2.1. let a ∈c(x) with nonempty resolvent set. then (2.1) σew(a) = ⋂ k∈ra(x) σ(a + k) = ⋂ k∈la(x) σ(a + k). proof. set σ1 = ⋂ k∈ra(x) σ(a + k) (resp. σ2 = ⋂ k∈la(x) σ(a + k)). we first claim that σew(a) ⊆ σ1 (resp. σew(a) ⊆ σ2). indeed, if λ /∈ σ1 (resp. λ /∈ σ2), then there exists k ∈ra(x) (resp. k ∈la(x)) such that λ ∈ ρ(a + k) and (λ−a−k)−1k ∈qk(x) (resp. k(λ−a−k)−1 ∈ qk(x)). therefore theorem 1.2 proves that i + (λ−a−k)−1k (resp. i + k(λ−a−k)−1) is a weyl operator. next, using the relation λ−a = (λ−a−k)(i + (λ−a−k)−1k) ( resp. λ−a = (i + k(λ−a−k)−1)(λ−a−k)) together with atkinson’s theorem we get (λ−a) is a weyl operator. this shows that λ /∈ σew(a). the opposite inclusion follows from k(x) ⊆ra(x) (resp. k(x) ⊆la(x)). � it follows, immediately, from theorem 2.1 that corollary 2.2. let u(x) a subset of l(x) (not necessarily an ideal). if k(x) ⊆ u(x) ⊆ ra(x) or k(x) ⊆ u(x) ⊆la(x). then σew(a) = σew(a + k) for all k ∈ u(x). proof. we have (2.2) ⋂ k∈ra(x) σ(a + k) ⊆ ⋂ k∈u(x) σ(a + k) ⊆ σew(a). and (2.3) ⋂ k∈la(x) σ(a + k) ⊆ ⋂ k∈u(x) σ(a + k) ⊆ σew(a). 86 mebarki, benharrat and messirdi now, the result follows from theorem 2.1. � let a ∈ c(x) and let j be an arbitrary a-bounded operator on x. hence we can regard a and j as operators from xa into x. they will be denoted by â and ĵ, respectively. these belong to l(xa,x). furthermore, we have the obvious relations (2.4)   α(â) = α(a), β(â) = β(a), r(â) = r(a) α(â + ĵ) = α(a + j) β(â + ĵ) = β(a + j), and r(â + ĵ) = r(a + j) theorem 2.3. let a ∈c(x) with nonempty resolvent set. then (2.5) σew(a) = ⋂ k∈∆a(x) σ(a + k). where ∆a(x) = {k ∈c(x),k is a-bounded and k(λ−a−k)−1 ∈qk(x) for all λ ∈ ρ(a + k)}. proof. since k(x) ⊂ ∆a(x), we refer that ⋂ k∈∆a(x) σ(a+k) ⊂ σew(a). conversely, suppose that there exists k ∈ ∆a(x), hence by using (2.4) and theorem 2.1, we infer that i + k(λ− a − k)−1 is a weyl operator for all λ ∈ ρ(a + k). the fact that λ − a = (i + k(λ − a − k)−1)(λ − a − k) and by using the atkinson′s theorem we get (λ − a) is a weyl operator. this shows that λ /∈ σew(a). � remark 2.4. since k(x) ⊂ ∆a(x), k(x) is the minimal subset of c(x) (in the sense of inclusion) which characterize the essential weyl spectrum . hence theorem 2.3 provides an improvement of the definition of σew(a) and is valid for a somewhat large variety of subsets of c(x). furthermore, σew(a + k) = σew(a), for all k ∈ ∆a(x). it follows, immediately, from theorem 2.3 that corollary 2.5. let a ∈c(x) and let m(x) be any subset of qk(x) (not necessarily an ideal) satisfying the condition (2.6) k(x) ⊆ m(x) ⊆qk(x). then σew(a) = σew(a + k) for all k ∈ ha(x). ha(x) = {k ∈c(x),k is a-bounded and k(λ−a−k)−1 ∈ m(x) for all λ ∈ ρ(a + k)}. proof. we have from (2.6) that k(x) ⊆ ha(x) ⊆ ∆a(x). from this we infer that (2.7) ⋂ k∈∆a(x) σ(a + k) ⊆ ⋂ k∈ha(x) σ(a + k) ⊆ σew(a). now, the result follows from theorem 2.3. � note that in applications (transport operators, operators arising in dynamic populations, etc.), we deal with operators a and b such that b = a + k where a ∈ c(x) and k is, in general, a closed (or closable) a-bounded linear operator. the operator k does not necessarily satisfy the hypotheses of the previous results. for some physical conditions on k, we have information about the operator (λ − a)−1 − (λ − b)−1 (λ ∈ ρ(a) ∩ ρ(b)). so we have the following useful stability result. theorem 2.6. let a,b ∈ c(x) such that ρ(a) ∩ρ(b) 6= ∅. if for some λ ∈ ρ(a) ∩ρ(b) the operator (λ−a)−1 − (λ−b)−1 ∈ ∆a(x), then σef (a) = σef (b) and σew(a) = σew(b). quasi-compact perturbations of the weyl essential spectrum 87 proof. without loss of generality, we suppose that λ = 0. hence 0 ∈ ρ(a) ∩ρ(b). therefore, we can write for µ 6= 0 µ−a = −µ(µ−1 −a−1)a. since, a is one to one and onto, then α(µ−a) = α(µ−1 −a−1) and β(µ−a) = β(µ−1 −a−1). this shows that (µ − a) is a fredholm operator if and only if (µ−1 − a−1) is a fredholm operator and ind(µ−a) = ind(µ−1 −a−1). now, assume that a−1 −b−1 ∈ ∆a(x). hence using theorem 2.3 we conclude that (µ − a) is a fredholm operator if and only if (µ − b) is a fredholm operator and ind(µ − a) = ind(µ − b) for each µ /∈ σef (a). this proves σef (a) = σef (b) and σew(a) = σew(b). � 3. application to transport operator in this section we are concerned with the weyl essential spectrum of singular transport operators (3.1a) aψ(x,ξ) = −ξ ·∇xψ(x,ξ) −σ(ξ)ψ(x,ξ) + ∫ rn κ(x,ξ′)ψ(x,ξ′) dξ′ (x,ξ) ∈ ω ×v, (3.1b) ψ|γ− (x,ξ) = 0 (x,ξ) ∈ γ−. where ω is a smooth open subset of rn (n ≥ 1), v is the support of a positive radon measure dµ on rn and ψ ∈ lp(ω×v,dxdµ(ξ)) (1 ≤ p < ∞). in (3.1b) γ− denotes the incoming part of the boundary of the phase space ω ×v γ− = {(x,ξ) ∈ ∂ω ×v ; ξ ·n(x) < 0}, where n(x) stands for the outward normal unit at x ∈ ∂ω. the operator a describes the transport of particle (neutrons, photons, molecules of gas, etc.) in the domain ω. the function ψ represents the number (or probability) density of particles having the position x and the velocity ξ. the functions σ(.) and κ(., .) are called, respectively, the collision frequency and the scattering kernel. let us first introduce the functional setting we shall use in the sequel. let xp = l p(ω ×v,dxdµ(ξ)), xσp = l p(ω ×v,σ(ξ)dxdµ(ξ)) 1 ≤ p < ∞, lpσ(r n) = lp(rn,σ(ξ)dµ(ξ)). we define the partial sobolev space wp = {ψ ∈ xp ; ξ ·∇xψ ∈ xp}. for any ψ ∈ wp, one can define the space traces ψ|γ− on γ−, w̃p = {ψ ∈ wp ; ψ|γ− = 0}. the streaming operator t associated with the boundary condition (3.1b) is{ t : d(t) ⊂ xp → xp ψ 7→ tψ(x,ξ) := −ξ ·∇xψ(x,ξ) −σ(ξ)ψ(x,ξ), with domain d(t) := w̃p ∩xσp . the transport operator (3.1) can be formulated as follows a = t + k, where k denotes the following collision operator k : xp → xp ψ 7→ ∫ rn κ(x,ξ′)ψ(x,ξ′) dξ′ . 88 mebarki, benharrat and messirdi we will assume that the scattering kernel κ(., .) = κ1(., .) + κ2(., .), κi(., .) are non-negative i = 1, 2 and there exist a closed subset e ⊂ rn with zero dµ measure and a constant σ0 > 0 such that (3.2a) σ(.) ∈ l∞loc(r n \e), σ(ξ) > σ0; (3.2b) [∫ rn ( κi(.,ξ ′) σ(ξ′ 1 p ) )qdµ(ξ′) ]1 q ∈ lp(rn), i = 1, 2. where q denotes the conjugate exponent of p. denote by kiψ(x,ξ) = ∫ rn κi(x,ξ ′)ψ(x,ξ′) dξ′ i = 1, 2. using boundedness of ω and the assumption (3.2b) we can fined that ki ∈l(xσp ,xp) with (3.3) ‖ki‖l(xσp ,xp) ≤ ∥∥∥∥∥∥ [∫ rn ( κi(.,ξ ′) σ(ξ′ 1 p ) )qdµ(ξ′) ]1 q ∥∥∥∥∥∥ lp(rn) i = 1, 2. note that a simple calculation using the assumption (3.2a) shows that xσp is a subset of xp and the the embedding xσp ↪→ xp is continuous. let us now consider the resolvent equation (3.4) (λ−t)ψ = ϕ, where ϕ is a given element of xp and the unknown ψ must be founded in d(t). for reλ > −σ0, the solution of (3.4) reads as follows (3.5) ψ(x,ξ) = ∫ t(x,ξ) 0 e−(λ+σ(ξ))sϕ(x−sξ,ξ)ds, where t(x,ξ) = sup{t > 0 ; x−sξ ∈ ω, ∀0 < s < t} = inf{s > 0 ; x−sξ /∈ ω}. an immediate consequence of these facts is that σ(t) ⊆{λ ∈ c : reλ ≤−σ0}, and in [8, 7] shows that σ(t) is reduced to the continuous spectrum σc(t) of t, that is (3.6) σ(t) = σc(t) = {λ ∈ c : reλ ≤−σ0}, since all essential spectra are enlargement of the continuous spectrum we infer that (3.7) σew(t) = σef (t) = {λ ∈ c : reλ ≤−σ0}. lemma 3.1. the collision operator k is t -bounded. proof. let λ ∈ c be such that reλ > −σ0 and consider ψ ∈ xp. it follows from (3.5) that∫ ω ∣∣(λ−t)−1ψ(x,ξ)∣∣p dx ≤ 1 (reλ + σ(ξ))p ∫ ω |ψ(x,ξ)|p dx therefore, (3.8) ∥∥(λ−t)−1ψ∥∥ xσp ≤ sup ξ∈rn σ(ξ) (reλ + σ(ξ))p ‖ψ‖xp hence, (λ−t)−1 ∈ l(xp,xσp ). using now the equation (3.3) to deduce that the operator k is t-bounded. � proposition 3.2 ([7, proposition 4.1]). let ω be a bounded subset of rn and 1 < p < ∞. if the hypotheses (3.2a) and (3.2b) are satisfied for κ1(., .), the measure dµ satisfies (3.9) { the hyperplanes have zero dµ-measure, i.e., for each, e ∈ sn−1,dµ{ξ ∈ rn,ξ.e = 0} = 0, where sn−1 denotes the unit ball of rn and the collision operator k1 : lpσ(r n) −→ lp(rn) is compact. then for any λ ∈ c such that reλ > −σ0, the operator k1(λ−t)−1 is compact on xp. quasi-compact perturbations of the weyl essential spectrum 89 now we are in position to state the main result of this section. theorem 3.3. assume that the hypotheses of proposition 3.2 are satisfied and (3.10) sup ξ∈rn σ(ξ) (reλ + σ(ξ))p ∥∥∥∥∥∥ [ ∫ rn ( κ2(.,ξ ′) σ(ξ′ 1 p ) )qdµ(ξ′) ]1 q ∥∥∥∥∥∥ lp(rn) < 1 for reλ > −σ0. then (1) for all λ ∈ ρ(a + k), we have k(λ−a−k)−1 is quasi-compact on xp, (2) σew(a) = σew(t) = {λ ∈ c : reλ ≤−σ0}. proof. (1) let λ ∈ ρ(a + k) and µ ∈ ρ(t) such that∥∥i + (µ−λ + k2)(λ−t −k)−1∥∥ < 1. by this estimate and (3.10) we can deduce that k2(µ−t)−1[i + (µ−λ + k2)(λ−t −k)−1] is a pure contraction. on other hand, we have k(λ−a−k)−1 = k(µ−t)−1[i + (µ−λ + k)(λ−t −k)−1] = k1(µ−t)−1[i + (µ−λ + k)(λ−t −k)−1] + k2(µ−t)−1k1(λ−t −k)−1 + k2(µ−t)−1[i + (µ−λ + k2)(λ−t −k)−1]. by this equation and proposition 3.2 we have k(λ−a−k)−1 is a sum of a compact operator and a pure contraction on xp. hence k(λ−a−k)−1 is a quasi-compact operator on xp. (2) the first item together with hypotheses on k implies that k ∈ ∆a(xp). now the second item follows from theorem 2.3, remark 2.4 and the relation (3.7). � references [1] a. brunel and d. revuz, quelques applications probabilistes de la quasi-compacité. ann. inst. henri et poincaré, b 10(3) (1974), 301-337. [2] h. hennion and l. hervé, limit theorems for markov chains and stochastic properties of dynamical systems by quasi-compactness. springer-verlag berlin heidelberg (2001). [3] k. latrach, essential spectra on spaces with the dunford-pettis property , j. math. anal. appl., 233, (1999), 607-622. [4] k. latrach, a. dehici, fredholm, semi-fredholm perturbations and essential spectra, j. math. anal. appl., 259 (2001), 277-301. [5] k. latrach, a. dehici, relatively strictly singular perturbations, essential spectra, and application to transport operators, j. math. anal. appl., 252(2001), 767-789. [6] k. latrach, a. jeribi, some results on fredholm operators, essential spectra, and application, j. math. anal. appl., 225, (1998), 461-485. [7] k. latrach, j. martin paoli, relatively compact-like perturbations, essensial spectra and application, j. aust. math. soc. 77 (2004), 73-89. [8] m. mokhtar-kharroubi, time asymptotic behaviour and compactness in neutron transport theory, euro. j. mech., b fluids, 11(1992), 39-68. [9] m. schechter, invariance of essential spectrum, bull. amer. math. soc. 71 (1965), 365-367. [10] k. yosida, quasi-completely continuous linear functional operations, jap. journ. of math., 15 (1939), 297– 301. [11] k. yosida, functional analysis, springer-verlag berlin heiderlberg, 1980. [12] k. yosida, s. kakutani, operator-theoretical treatment of markov’s process and mean ergodic theorem, ann. of math., 42, (1941), 188-228. 1département de mathématiques, université d’oran 1, algérie 2département de mathématiques et informatique, ecole nationale polytechnique d’oran (ex. enset d’oran); b.p. 1523 oran-el m’naouar, oran, algérie ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 49-61 http://www.etamaths.com dhage iteration method for nonlinear first order hybrid differential equations with a linear perturbation of second type b.c. dhage∗ abstract. in this paper the authors prove algorithms for the existence and approximation of the solutions for an initial and a periodic boundary value problem of nonlinear first order ordinary hybrid differential equations with a linear perturbation of second type via dhage iteration method. examples are furnished to illustrate the hypotheses and main abstract results of this paper. 1. introduction given a closed and bounded interval j = [0,t] in the real line r, consider the initial and periodic boundary value problems of first order nonlinear hybrid differential equation (in short hde), (1.1) d dt [ x(t) −f(t,x(t)) ] + λ [ x(t) −f(t,x(t)) ] = g(t,x(t)), t ∈ j, x(0) = α0,   and (1.2) d dt [ x(t) −f(t,x(t)) ] + λ [ x(t) −f(t,x(t)) ] = g(t,x(t)), t ∈ j, x(0) = x(t),   where λ ∈ r, λ > 0 and f,g : j ×r → r are continuous functions. by a solution of the hde (1.1) or (1.2) we mean a function x ∈ c(j,r) such that (i) the function t 7→ x−f(t,x) is differentiable for each x ∈ r, and (ii) x satisfies the equations in (1.1) or (1.2), where c(j,r) is the space of continuous real-valued functions defined on j. the hdes (1.1) and (1.2) are linear perturbations of the second type of the nonlinear differential equations (1.3) x′(t) = g(t,x(t)), t ∈ j, x(t0) = α0, } and (1.4) x′(t) = g(t,x(t)), t ∈ j, x(0) = x(t), } and a sharp classification of different types of perturbations of a differential equation appears in dhage [2] which can be treated with the hybrid fixed point theory (see dhage [2, 3] and dhage and lakshmikantham [14]). the hde (1.1) with λ = 0 has been thoroughly discussed in the literature for different basic aspects of the solutions such as existence theorem, differential inequalities, maximal and minimal solutions, comparison principle under some mixed lipschitz and compactness type conditions. 2010 mathematics subject classification. 34a12, 34h34, 47h07, 47h10. key words and phrases. hybrid differential equation; dhage iteration method; hybrid fixed point theorem; approximation theorem. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 49 50 dhage see for example dhage and jadhav [13], dhage [5, 6] and references therein. however, the hde (1.2) is new to the literature . in this paper we prove algorithms in terms of successive approximations for proving the existence and approximate solutions of the considered hybrid differential equations. we claim that the results of this paper are new basic and important contribution to the theory of nonlinear ordinary differential equations. 2. auxiliary result unless otherwise mentioned, throughout this paper that follows, let e denote a partially ordered real normed linear space with an order relation � and the norm ‖ · ‖ in which the addition and the scalar multiplication by positive real numbers are preserved by � . a few details of a partially ordered normed linear space appear in dhage [3], heikkilä and lakshmikantham [16] and the references therein. two elements x and y in e are said to be comparable if either the relation x � or y � x holds. a non-empty subset c of e is called a chain or totally ordered if all the elements of c are comparable. it is known that e is regular if {xn} is a nondecreasing (resp. nonincreasing) sequence in e such that xn → x∗ as n → ∞, then xn � x∗ (resp. xn � x∗) for all n ∈ n. the conditions guaranteeing the regularity of e may be found in heikkilä and lakshmikantham [16] and the references therein. we need the following definitions in the sequel. definition 2.1. a mapping t : e → e is called isotone or monotone nondecreasing if it preserves the order relation �, that is, if x � y implies t x � t y for all x,y ∈ e. similarly, t is called monotone nonincreasing if x � y implies t x � t y for all x,y ∈ e. finally, t is called monotonic or simply monotone if it is either monotone nondecreasing or monotone nonincreasing on e. definition 2.2 (dhage [4]). a mapping t : e → e is called partially continuous at a point a ∈ e if for � > 0 there exists a δ > 0 such that ‖t x−t a‖ < � whenever x is comparable to a and ‖x−a‖ < δ. t called partially continuous on e if it is partially continuous at every point of it. it is clear that if t is partially continuous on e, then it is continuous on every chain c contained in e. definition 2.3 (dhage [3, 4]). a non-empty subset s of the partially ordered banach space e is called partially bounded if every chain c in s is bounded. a nondecreasing operator t on a partially normed linear space e into itself is called partially bounded if t (c) is bounded for every chain c in e. t is called uniformly partially bounded if all chains c in e, t (c) are bounded by a unique constant. definition 2.4 (dhage [3, 4]). a non-empty subset s of the partially ordered banach space e is called partially compact if every chain c in s is compact. a nondecreasing mapping t : e → e is called partially compact if t (c) is a relatively compact subset of e for all totally ordered sets or chains c in e. t is called uniformly partially compact if t is a uniformly partially bounded and partially compact operator on e. t is called partially totally bounded if for any totally ordered and bounded subset c of e, t (c) is a relatively compact subset of e. if t is partially continuous and partially totally bounded, then it is called partially completely continuous on e. definition 2.5 (dhage [3]). the order relation � and the metric d on a non-empty set e are said to be compatible if {xn} is a monotone sequence, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk} of {xn} converges to x ∗ implies that the original sequence {xn} converges to x∗. similarly, given a partially ordered normed linear space (e,�,‖ · ‖), the order relation � and the norm ‖·‖ are said to be compatible if � and the metric d defined through the norm ‖·‖ are compatible. a subset s of e is called janhavi if the order relation � and the metric d or the norm ‖·‖ are compatible in it. in particular, if s = e, then e is called a janhavi metric or janhavi banach space. clearly, the set r of real numbers with usual order relation ≤ and the norm defined by the absolute value function | · | has this property. similarly, the finite dimensional euclidean space rn with usual componentwise order relation and the standard norm possesses the compatibility property and so is a janhavi banach space. dhage iteration method for hybrid differential equations 51 definition 2.6. an upper semi-continuous and monotone nondecreasing function ψ : r+ → r+ is called a d-function provided ψ(0) = 0. an operator t : e → e is called partially nonlinear dcontraction if there exists a d-function ψ such that (2.1) ‖t x−t y‖≤ ψ ( ‖x−y‖ ) for all comparable elements x,y ∈ e, where 0 < ψ(r) < r for r > 0. in particular, if ψ(r) = k r, k > 0, t is called a partial lipschitz operator with a lischitz constant k and moreover, if 0 < k < 1, t is called a partial linear contraction on e with a contraction constant k. the dhage iteration principle or method (in short dip or dim) developed in dhage [3, 4, 7] may be described as “ the sequence of successive approximations of a nonlinear equation beginning with a lower or an upper solution as its first or initial approximation converges monotonically to the solution.” the aforsaid convergence principle forms a basic and powerful tool in the study of nonlinear differential and integral equations. see dhage [7, 8], dhage and dhage [9, 10, 11, 12] and the references therein. the following applicable hybrid fixed point theorem of dhage [4] containing the dip is used as a key tool for the work contained in this paper. theorem 2.7 (dhage [4]). let ( e,�,‖·‖ ) be a regular partially ordered complete normed linear space such every compact chain c of e is janhavi. let a,b : e → e be two nondecreasing operators such that (a) a is partially bounded and partially nonlinear d-contraction, (b) b is partially continuous and partially compact, and (c) there exists an element x0 ∈ e such that x0 �ax0 + bx0 or x0 �ax0 + bx0. then the operator equation ax + bx = x has a solution x∗ in e and the sequence {xn} of successive iterations defined by xn+1 = axn + bxn, n=0,1,. . . , converges monotonically to x∗. remark 2.8. the condition that every compact chain of e is janhavi holds if every partially compact subset of e possesses the compatibility property with respect to the order relation � and the norm ‖ · ‖ in it. this simple fact is used to prove the desired characterization of the mild solution of the problem (1.1) on j. remark 2.9. we remark that hypothesis (a) of theorem 2.7 implies that the operator a is partially continuous and consequently both the operators a and b in the theorem are partially continuous on e. the regularity of e in above theorem 2.7 may be replaced with a stronger continuity condition of the operators a and b on e which is a result proved in dhage [3, 4]. 3. main results in this section, we prove an existence and approximation result for the hde (1.1) on a closed and bounded interval j = [0,t] under mixed partial lipschitz and partial compactness type conditions on the nonlinearities involved in it. we place the hde (1.1) in the function space c(j,r) of continuous real-valued functions defined on j. we define a norm ‖ · ‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) for all t ∈ j. clearly, c(j,r) is a banach space with respect to above supremum norm and also partially ordered w.r.t. the above partially order relation ≤. it is known that the partially ordered banach space c(j,r) is regular and lattice so that every pair of elements of e has a lower and an upper bound in it. the following lemma concerning the janhaviness of subsets of c(j,r) follows immediately form the arzelá-ascoli theorem for compactness. lemma 3.1. let ( c(j,r),≤,‖ · ‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2) respectively. then every partially compact subset of c(j,r) is janhavi. proof. the proof of the lemma appears in dhage and dhage [9] and so we omit the details. � 52 dhage we consider the following basic hypotheses in what follows. (a0) the mapping x 7→ x−f(t,x) is increasing in r for each t ∈ j. (a1) there exists a d-function ψ such that 0 ≤ f(t,x) −f(t,y) ≤ ψ(x−y) for all t ∈ j and x,y ∈ r with x ≥ y. moreover, 0 < ψ(r) < r for r > 0. (a2) there exists aconstant mf > 0 such that |f(t,x)| ≤ mf for all t ∈ j and x ∈ r. (b1) there exists a mg > 0 such that |g(t,x)| ≤ mg, for all t ∈ j and x ∈ r. (b2) g(t,x) is nondecreasing in x for each t ∈ j. remark 3.2. if the hypothesis (a0) holds, then the function x 7→ x−f(t,x) is injective in r for each t ∈ j. remark 3.3. the hypotheses (a1) and (a2) hold, in particular if f satisfies the inequality 0 ≤ f(t,x) −f(t,y) ≤ l(x−y) k + (x−y) for all x,y ∈ r with x ≥ y, where l > 0 and k > 0 are constants satisfying l ≤ k. 3.1. initial value problem. the following useful lemma follows from the theory of calculus and linear differential equations. lemma 3.4. assume that hypothesis (a0) holds. then for any continuous function h : j → r, the function x ∈ c(j,r) is a solution of the hde (3.3) d dt [x(t) −f(t,x(t))] + λ[x(t) −f(t,x(t))] = h(t), t ∈ j, x(0) = α0,   if and only if x satisfies the hybrid integral equation (hie) (3.4) x(t) = ce−λt + f(t,x(t)) + e−λt ∫ t 0 eλsh(s) ds, t ∈ j, where c = α0 −f(0,α0). proof. let h ∈ c(j,r). assume first that x is a solution of the hde (3.3) defined on j. by definition, the function t 7→ x(t) − f(t,x(t)) is continuous on j, and so, differentiable there, whence d dt [ x(t) − f(t,x(t)) ] is integrable on j. applying integration to (3.3) from 0 to t, we obtain the hie (3.4) on j. conversely, assume that x satisfies the hie (3.3). then by direct differentiation we obtain the first equation in (3.4). again, substituting t = 0 in (3.4) yields x(0) −f(0,x(0)) = α0 −f(0,α0). since the mapping x 7→ x−f(t,x) is increasing in r for all t ∈ j, the mapping x 7→ x−f(0,x) is injective in r, whence x(0) = α0. hence the proof of the lemma is complete. � we need the following definition in what follows. definition 3.5. a function u ∈ c(j,r) is called a lower solution of the hde (1.1) on j if the function t 7→ x−f(t,x) is differentiable and satisfies the inequalities d dt [ x(t) −f(t,x(t)) ] + λ [ x(t) −f(t,x(t)) ] ≤ g(t,x(t)), x(0) ≤ α0,   for all t ∈ j. similarly, an upper solution of the hde (1.1) on j is defined. (b3) the hde (1.1) has a lower solution u ∈ c(j,r). now we are in a position to prove the following existence and approximation theorem for the hde (1.1) on j. dhage iteration method for hybrid differential equations 53 theorem 3.6. assume that the hypotheses (a0) through (a2) and (b1) through (b3) hold. then the hde (1.1) has a solution x∗ defined on j and the sequence {xn}∞n=0 of successive approximations defined by (3.5) x0 = u, xn+1(t) = ce −λt + f(t,xn(t)) + e −λt ∫ t 0 eλsg(s,xn(s)) ds, t ∈ j, converges monotonically to x∗, where c = α0 −f(0,α0). proof. set e = c(j,r). then, in view of lemma 3.1, every compact chain c in e possesses the compatibility property with respect to the norm ‖ · ‖ and the order relation ≤ so that every compact chain c is janhavi in e. now, using the hypotheses (a0) and (a2), it can be shown by an application of lemma 3.4 that the hde (1.1) is equivalent to the nonlinear hie (3.6) x(t) = ce−λt + f(t,x(t)) + e−λt ∫ t 0 eλsg(s,x(s)) ds for t ∈ j. define two operators a,b : e → e by (3.7) ax(t) = f(t,x(t)), t ∈ j, and (3.8) bx(t) = ce−λt + e−λt ∫ t 0 eλsg(s,x(s)) ds, t ∈ j. then, the hie (3.6) is transformed into an operator equation as (3.9) ax(t) + bx(t) = x(t), t ∈ j. we shall show that the operators a and b satisfy all the conditions of theorem 2.7. firstly, we show that the operators a and b are nondecreasing on e. let x,y ∈ e be such that x ≥ y. then, by hypothesis (a1), ax(t) = f(t,x(t)) ≥ f(t,y(t)) = ay(t) for all t ∈ j. similarly, by hypothesis (a3), bx(t) = ce−λt + e−λt ∫ t 0 eλsg(s,x(s)) ds ≥ ce−λt + e−λt ∫ t 0 eλsg(s,y(s)) ds = by(t) for all t ∈ j. this shows that a and b are nondecreasing operators on e into e. from (a2) it follows that ‖ax‖≤ sup t∈j |ax(t)| ≤ sup t∈j |f(t,x)| ≤ mf for all x ∈ e. as a result, a is bounded and consequently partially bounded on e. next, we show that a is a partial nonlinear d-contraction on e with a d function ψ. let x,y ∈ e be such that x ≥ y. then, by hypothesis (a1), |ax(t) −ay(t)| = |f(t,x(t)) −f(t,y(t))| = f(t,x(t)) −f(t,y(t)) ≤ ψ(x(t) −y(t)) = ψ(|x(t) −y(t)|) ≤ ψ(‖x−y‖) 54 dhage for all t ∈ j. taking the supremum over t, we obtain ‖ax−ay‖≤ ψ(‖x−y‖) for all x,y ∈ e with x ≥ y. this shows that a is a partial nonlinear d-contraction on e with the d-function ψ. next, we show that b is a partially compact and partially continuous operator on e into e. first we show that b is a partially continuous on e. let {xn} be a sequence in a chain c of e converging to a point x ∈ c. then by the dominated convergence theorem for integration, we obtain lim n→∞ bxn(t) = lim n→∞ [ ce−λt + e−λt ∫ t 0 eλsg(s,xn(s)) ds ] = ce−λt + lim n→∞ e−λt ∫ t 0 eλsg(s,xn(s)) ds = ce−λt + e−λt ∫ t 0 eλs [ lim n→∞ g(s,xn(s)) ] ds = ce−λt + e−λt ∫ t 0 eλsg(s,x(s)) ds = bx(t) for all t ∈ j. moreover, it can be shown as below that {bxn} is an equicontinuous sequence of functions in e. now, following the arguments similar to that given in granas and dugundji [15], it is proved that b is a a partially continuous operator on e. next, we show that b is a partially compact operator on e. it is enough to show that b(c) is a uniformly bounded and equi-continuous set in e for every chain c in e. let x ∈ c be arbitrary. then by yhe hypothesis (a2), |bx(t)| ≤ ∣∣ce−λt∣∣ + ∣∣∣∣e−λt ∫ t 0 eλsg(s,x(s)) ds ∣∣∣∣ ≤ |α0 −f(t0,α0)| + ∫ t t0 eλsmg ds ≤ |α0 −f(t0,α0)| + (eλt − 1)mg λ for all t ∈ j. taking the supremum over t, we obtain ‖bx‖≤ |α0 −f(t0,α0)| + (eλt − 1)mg λ for all x ∈ c. this shows that b is uniformly bounded on c. dhage iteration method for hybrid differential equations 55 again, let t1, t2 ∈ j be arbitrary. then for any x ∈ c, one has |bx(t1) −bx(t2)| = |c| ∣∣e−λt1 −e−λt2∣∣ + ∣∣∣∣e−λt1 ∫ t1 0 eλsg(s,x(s)) ds−e−λt2 ∫ t2 0 eλsg(s,x(s)) ds ∣∣∣∣ ≤ |c| ∣∣e−λt1 −e−λt2∣∣ + ∣∣e−λt1 −e−λt2∣∣ ∣∣∣∣ ∫ t1 0 eλsg(s,x(s)) ds ∣∣∣∣ + ∣∣e−λt2∣∣ ∣∣∣∣ ∫ t1 0 eλsg(s,x(s)) ds− ∫ t2 0 eλsg(s,x(s)) ds ∣∣∣∣ ≤ |c| ∣∣e−λt1 −e−λt2∣∣ + [(eλt − 1)mg λ ]∣∣e−λt1 −e−λt2∣∣ + ∣∣∣∣ ∫ t1 t2 eλs|g(s,x(s))|ds ∣∣∣∣ ≤ [ |c| + (eλt − 1)mg λ ]∣∣e−λt1 −e−λt2∣∣ + |p(t1) −p(t2)| where, p(t) = ∫ t 0 eλsmg ds. since the functions t 7→ e−λt and t 7→ p(t) are continuous on compact j, they are uniformly continuous there. hence, for � > 0, there exists a δ > 0 such that |t1 − t2| < δ =⇒ |bx(t1) −bx(t2)| < � uniformly for all t1, t2 ∈ j and for all x ∈ s. this shows that b(c) is an equi-continuous set in e. now the set b(c) is uniformly bounded and equicontinuous in e, so it is compact by arzelá-ascoli theorem. as a result, b is a partially continuous and partially compact operator on e. thus, all the conditions of theorem 2.7 are satisfied and hence the operator equation ax +bx = x has a solution x∗ in e and the sequence of successive approximations {xn} defined by xn = axn−1 + bxn−1 converges monotonically to x∗. as a result, the hde (1.1) has a solution x∗ defined on j and the sequence of successive approximations {xn} defined by (3.5) converges monotonically to x∗. this completes the proof. � remark 3.7. we remark that theorem 3.6 also remains true if we replace the hypothesis (b3) with the following one: (b′3) the hde (1.1) has an upper solution v ∈ c(j,r). remark 3.8. we note that if the hde (1.1) has a lower solution u as well as an upper solution v such that u ≤ v, then under the given conditions of theorem 3.6 it has corresponding solutions x∗ and x∗ and these solutions satisfy x∗ ≤ x∗. hence they are the minimal and maximal solutions of the hde (1.1) in the vector segment [u,v] of the banach space e = c(j,r), where the vector segment [u,v] is a set of elements in c(j,r) defined by [u,v] = {x ∈ c(j,r) | u ≤ x ≤ v}. this is because the order relation ≤ defined by (3.2) is equivalent to the order relation defined by the order cone k = {x ∈ c(j,r) | x ≥ θ} which is a closed set in c(j,r). example 3.9. given a closed and bounded interval j = [0, 1] in r, consider the hde (3.10) d dt [ x(t) − tan−1 x(t) ] + [ x(t) − tan−1 x(t) ] = tanh x(t), t ∈ j, x(0) = 1.   here, λ = 1 and the functions f and g are given by f(t,x) = tan−1 x and g(t,x) = tanh x 56 dhage for all t ∈ j and x ∈ r. we show that the functions f and g satisfy all the hypotheses of theorem 3.6. first we show that f satisfies the hypotheses (a0)-(a2). now, ∂ ∂x [ x−f(t,x) ] = d dx [ x− tan−1 x ] = 1 − 1 1 + x2 > 0, for all x ∈ r and t ∈ j, so that the function x 7→ x−f(t,x) is increasing in r for each t ∈ j. therefore, hypothesis (a0) holds. next, let x,y ∈ r be such that x ≥ y. then, 0 ≤ f(t,x) −f(t,y) ≤ tan−1 x− tan−1 y = 1 1 + ξ2 (x−y) for all x > ξ > y, showing that f satisfies the hypothesis (a1) with d-function ψ given by ψ(r) = r 1 + ξ2 < r, r > 0, where ξ 6= 0. again, |f(t,x)| = |tan−1 x| ≤ π 2 , for all t ∈ j and x ∈ r. this shows that f satisfies hypothesis (a2) with mf = π 2 . furthermore, |g(t,x)| = |tanh x| ≤ 1, for all t ∈ j and x ∈ r, so that the hypothesis (b1) holds with mg = 1. again, since the function x 7→ tanh x is nondecreasing in r and so the hypothesis (b2) is satisfied. finally, the function u(t) = −(t + 3) is a lower solution of the hde (3.10) defined on j = [0, 1]. thus the functions f and g satisfy all the conditions of theorem 3.6. hence we apply and conclude that the hde (3.10) has a solution x∗ defined on j and the sequence {xn} of successive approximations defined by x0 = u, xn+1(t) = 1 − π 2 + tan−1 xn(t) + e −t ∫ t 0 es tanh xn(s) ds, for each t ∈ j, converges monotonically to x∗. a similar conclusion also remains true if we replace the lower solution u with the upper solution v(t) = t + 3, t ∈ j. 3.2. periodic boundary value problem. the following useful lemma is obvious and may be found in dhage [1] and nieto [17]. lemma 3.10. for any function σ ∈ l1(j,r), x is a solution to the differential equation (3.11) x′(t) + λx(t) = σ(t), t ∈ j, x(0) = x(t), } if and only if it is a solution of the integral equation (3.12) x(t) = ∫ t 0 gλ(t,s) σ(s) ds where, (3.13) gλ(t,s) =   eλs−λt+λt eλt − 1 , if 0 ≤ s ≤ t ≤ t, eλs−λt eλt − 1 , if 0 ≤ t < s ≤ t. notice that the green’s function gλ is continuous and nonnegative on j × j and therefore, the number kλ := max{|gλ(t,s)| : t,s ∈ [0,t]} exists for all λ ∈ r+. for the sake of convenience, we write gλ(t,s) = g(t,s) and kλ = k. dhage iteration method for hybrid differential equations 57 lemma 3.11. if there exists a differentiable function u ∈ c(j,r) such that (3.14) u′(t) + λu(t) ≤ σ(t), t ∈ j, u(0) ≤ u(t), } then (3.15) u(t) ≤ ∫ t 0 gλ(t,s) σ(s) ds, for all t ∈ j, where g(t,s) is a green’s function given by (3.5). proof. suppose that the function u ∈ c(j,r) satisfies the inequalities given in (3.14). multiplying the first inequality in (3.14) by eλt, ( eλtu(t) )′ ≤ eλtσ(t). a direct integration of above inequality from 0 to t yields (3.16) eλtu(t) ≤ u(0) + ∫ t 0 eλsσ(s) ds, for all t ∈ j. therefore, in particular, (3.17) eλtu(t) ≤ u(0) + ∫ t 0 eλsσ(s) ds. now u(0) ≤ u(t), so one has (3.18) u(0)eλt ≤ u(t)eλt . from (3.17) and 3.18) it follows that (3.19) eλtu(0) ≤ u(0) + ∫ t 0 eλsσ(s) ds which further yields (3.20) u(0) ≤ ∫ t 0 eλs (eλt − 1) σ(s) ds. substituting (3.20) in (3.16) we obtain u(t) ≤ ∫ t 0 g(t,s)σ(s) ds, for all t ∈ j. this completes the proof. � we need the following definition in what follows. definition 3.12. a function u ∈ c(j,r) is called a lower solution of the hde (1.1) if the function t 7→ x−f(t,x) is differentiable and satisfies the inequalities d dt [ u(t) −f(t,u(t)) ] + λ [ u(t) −f(t,u(t)) ] ≤ g(t,u(t)), u(0) ≤ u(t),   for all t ∈ j. similarly, an upper solution v ∈ c(j,r) of the hde (1.2) is defined. we need the following hypotheses in what follows. (b4) the function f(t,x) is periodic in t with period t for all x ∈ r, i.e., f(0,x) = f(t,x) for all x ∈ r. (b5) the hde (1.2) has a lower solution u ∈ c(j,r). 58 dhage lemma 3.13. assume that hypothesis (a0) holds. then for any λ ∈ r+, the function x ∈ c(j,r) is a solution of the hde (3.21) d dt [x(t) −f(t,x(t))] + λ[x(t) −f(t,x(t))] = g(t,x(t)), t ∈ j, x(0) = x(t),   if and only if x satisfies the hybrid integral equation (hie) (3.22) x(t) = f(t,x(t)) + ∫ t 0 g(t,s)g(s,x(s)) ds, t ∈ j. proof. assume first that x is a solution of the hde (3.22) defined on j. by definition, the function t 7→ x(t) − f(t,x(t)) is continuous on j, and so, differentiable there, whence d dt [ x(t) − f(t,x(t)) ] is integrable on j. again by hypothesis (b2), x(0) − f(0,x(0)) = x(t) − f(t,x(t)). so by a direct application of lemma 3.10, we obtain the hie (3.22) on j. conversely, assume that x satisfies the hie (3.21) on j. then by a direct differentiation of (3.22), we obtain the first equation in (3.21). again, substituting t = 0 in (3.2) yields x(0) −f(0,x(0)) = x(t) −f(0,x(t)). since the mapping x 7→ x−f(t,x) is increasing in r for all t ∈ j, the mapping x 7→ x−f(0,x) is injective in r, and so x(0) = x(t). hence the proof of the lemma is complete. � now we are in a position to prove the following existence and approximation theorem for the hde (1.2) on j. theorem 3.14. assume that the hypotheses (a0) (a2), (b1) (b2) and (b4) (b5) hold. then the hde (1.2) has a solution x∗ defined on j and the sequence {xn}∞n=0 of successive approximations defined by (3.23) x0 = u, xn+1(t) = f(t,xn(t)) + ∫ t 0 g(t,s)g(s,xn(s)) ds, t ∈ j, converges monotonically to x∗. proof. set e = c(j,r). then, in view of lemma 3.1, every compact chain c in e possesses the compatibility property with respect to the norm ‖ · ‖ and the order relation ≤ so that every compact chain c is janhavi set in e. now, using the hypotheses (a0), it can be shown by an application of lemma 3.4 that the hde (1.1) is equivalent to the nonlinear hie (3.24) x(t) = f(t,x(t)) + ∫ t 0 g(t,s)g(s,x(s)) ds for t ∈ j. define two operators a,b : e → e by (3.25) ax(t) = f(t,x(t)), t ∈ j, and (3.26) bx(t) = ∫ t 0 g(t,s)g(s,x(s)) ds, t ∈ j. then, the hie (3.24) is transformed into an operator equation as (3.27) ax(t) + bx(t) = x(t), t ∈ j. we shall show that the operators a and b satisfy all the conditions of theorem 3.6. now, it can be shown as in the proof of theorem 3.6 that a is a nondecreasing partially bounded and partially dhage iteration method for hybrid differential equations 59 nonlinear d-contraction on e with a d-function ψ. next, we show that the operator b is a nondecreasing, partially continuous and partially compact operator on e. let x,y ∈ e be such that x ≥ y. then, by hypothesis (b3), bx(t) = ∫ t 0 g(t,s)g(s,x(s)) ds ≥ ∫ t 0 g(t,s)g(s,y(s)) ds = by(t) for all t ∈ j. this shows that a and b are nondecreasing operators on e into e. next, let {xn} be a sequence in a chain c of e converging to a point x ∈ c. then by dominated convergence theorem for integration, we obtain lim n→∞ bxn(t) = lim n→∞ [∫ t 0 g(t,s)g(s,xn(s)) ds ] = ∫ t 0 g(t,s) [ lim n→∞ g(s,xn(s)) ] ds = ∫ t 0 g(t,s)g(s,x(s)) ds = bx(t) for all t ∈ j. moreover, it can be shown as below that {bxn} is an equicontinuous sequence of functions in e. now, following the arguments similar to that given in granas and dugundji [15], it is proved that b is a a partially continuous operator on e. next, we show that b is a partially compact operator on e. it is enough to show that b(c) is a uniformly bounded and equi-continuous set in e for every chain c in e. let x ∈ c be arbitrary. then by hypothesis (a2), |bx(t)| ≤ ∣∣∣∣∣ ∫ t 0 g(t,s)g(s,x(s)) ds ∣∣∣∣∣ ≤ ∫ t 0 g(t,s)|g(s,x(s))|ds ≤ ktmg for all t ∈ j. taking the supremum over t, ‖bx‖ ≤ ktmg for all x ∈ c. this shows that b is uniformly bounded on c. again, let t1, t2 ∈ j be arbitrary. then for any x ∈ c, one has |bxn(t2) −bxn(t1)| = ∣∣∣∣∣ ∫ t 0 g(t1,s)f̃(s,xn(s)) ds− ∫ t 0 g(t2,s)g(s,xn(s)) ds ∣∣∣∣∣ ≤ ∫ t 0 |g(t1,s) −g(t2,s)||g(s,xn(s))|ds ≤ mg ∫ t 0 |g(t1,s) −g(t2,s)|ds → 0 as t2 − t1 → 0. since the functions t 7→ g(t,s) is continuous on compact j, it is uniformly continuous there. hence, for � > 0, there exists a δ > 0 such that |t1 − t2| < δ =⇒ |bx(t1) −bx(t2)| < � uniformly for all t1, t2 ∈ j and for all x ∈ s. this shows that b(c) is an equi-continuous set in e. now the set b(c) is uniformly bounded and equicontinuous set in e, so it is compact by arzelá-ascoli theorem. as a result, b is a partially continuous and partially compact operator on e. finally, we show that u is a lower solution of the operator equation ax +bx = x. since u is a lower solution of the hde (1.2) on j, we have d dt [ u(t) −f(t,u(t)) ] + λ [ u(t) −f(t,u(t)) ] ≤ g(t,u(t)), u(0) ≤ u(t),   60 dhage for all t ∈ j. again, since the hypotheses (a0) and (b4) hold, one has d dt [ u(t) −f(t,u(t)) ] + λ [ u(t) −f(t,u(t)) ] ≤ g(t,u(t)),[ u(0) −f(0,u(0)) ] ≤ [ u(t) −f(t,u(t)) ] ,   for all t ∈ j. now an application lemma 3.11 yields that (3.28) u(t) ≤ f(t,u(t)) + ∫ t 0 g(t,s)g(s,u(s)) ds for t ∈ j. this further in view of definitions of the operators a and b implies that u ≤au + bu and that u is a lower solution of the operator equation ax + bx = x. thus, all the conditions of theorem 3.6 are satisfied by the operators a and b and hence the operator equation ax + bx = x has a solution x∗ in e and the sequence of successive approximations {xn} defined by xn = axn−1 + bxn−1 converges monotonically to x∗. consequently, the hde (1.1) has a solution x∗ defined on j and the sequence of successive approximations {xn} defined by (3.3) converges monotonically to x∗. this completes the proof. � remark 3.15. we remark that theorem 3.14 also remains true if we replace the hypothesis (b5) with the following one: (b′5) the hde (1.2) has an upper solution v ∈ c(j,r). remark 3.16. we note that if the hde (1.2) has a lower solution u as well as an upper solution v such that u ≤ v, then under the given conditions of theorem 3.14 it has corresponding solutions x∗ and x∗ and these solutions satisfy x∗ ≤ x∗. hence they are the minimal and maximal solutions of the hde (1.2) in the vector segment [u,v] of the banach space e = c(j,r), where the vector segment [u,v] is a set of elements in c(j,r) defined by [u,v] = {x ∈ c(j,r) | u ≤ x ≤ v}. this is because the order relation ≤ defined by (3.2) is equivalent to the order relation defined by the order cone k = {x ∈ c(j,r) | x ≥ θ} which is a closed set in c(j,r). example 3.17. given a closed and bounded interval j = [0, 1] in r, consider the hde (3.29) d dt [ x(t) − tan−1 x(t) ] + [ x(t) − tan−1 x(t) ] = tanh x(t), t ∈ j, x(0) = x(1).   here, λ = 1 and the functions f and g are given by f(t,x) = tan−1 x and g(t,x) = tanh x for all t ∈ j and x ∈ r. now, it can be shown that the functions f and g satisfy all the hypotheses of theorem 3.14 with u(t) = −4et, t ∈ [0, 1]. hence we conclude that the hde (3.29) has a solution x∗ defined on j and the sequence {xn} of successive approximations defined by x0 = u, xn+1(t) = tan −1 xn(t) + ∫ 1 0 g(t,s) tanh xn(s) ds, for each t ∈ j, converges monotonically to x∗, where g(t,s) is a green’s function associated with the homogeneous pbvp (3.30) x′(t) + x(t) = 0, t ∈ j, x(0) = x(1), } given by g(t,s) =   es−t+1 e− 1 , if 0 ≤ s ≤ t ≤ 1, es−t e− 1 , if 0 ≤ t < s ≤ 1. dhage iteration method for hybrid differential equations 61 again, a similar conclusion holds if we replace the lower solution u with the upper solution v(t) = 4et, t ∈ [0, 1] in view of remark 3.14. references [1] b.c. dhage, periodic boundary value problems of first order carathéodory and discontinuous differential equations, nonlinear funct. anal. & appl. 13(2) (2008), 323-352. [2] b.c. dhage, quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, differ. equ. appl. 2 (2010), 465–486. [3] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ appl. 5 (2013), 155-184. [4] b.c. dhage, partially condensing mappings in ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45 (2014), 397-426. [5] b.c. dhage, approximation methods in the theory of hybrid differential equations with linear perturbations of second type, tamkang j. math. 45 (2014), 39-61. [6] b.c. dhage, global existence and convergence analysis for hybrid differential equations, comm. appl. anal. 19 (2015), 439-452. [7] b.c. dhage, a new monotone iteration principle in the theory of nonlinear first order integro-differential equations, nonlinear studies 22 (3) (2015), 397-417. [8] b.c. dhage, operator theoretic techniques in the theory of nonlinear hybrid differential equations, nonlinear anal. forum 20 (2015), 15-31. [9] b.c. dhage, s.b. dhage, approximating solutions of nonlinear first order ordinary differential equations, gjms special issue for recent advances in mathematical sciences and applications-13, gjms vol. 2, no. 2, (2014), 25-35. [10] b.c. dhage, s.b. dhage, approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, indian j. math. 57(1) (2015), 103-119. [11] b.c. dhage, s.b. dhage, approximating positive solutions of nonlinear first order ordinary quadratic differential equations, cogent mathematics (2015), 2: 1023671. [12] b.c. dhage, s.b. dhage, approximating positive solutions of pbvps of nonlinear first order ordinary quadratic differential equations, appl. math. lett. 46 (2015), 133-142. [13] b.c. dhage and n. s. jadhav, basic results on hybrid differential equations with linear perturbation of second type, tamkang j. math. 44 (2) (2013), 171-186. [14] b.c. dhage, v. lakshmikantham, basic results on hybrid differential equations, nonlinear analysis: hybrid systems 4 (2010), 414-424. [15] a. granas, j. dugundji, fixed point theory, springer verlag, 2003. [16] s. heikkilä, v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. [17] j.j. nieto, basic theory for nonresonance impulsive periodic problems of first order, j. math. anal. appl. 205 (1997), 423–433. [18] e. zeidler, nonlinear functional analysis and its applications : part.i, springer-verlag, new york (1985). kasubai, gurukul colony, ahmedpur-413 515, dist: latur, maharashtra, india ∗corresponding author: bcdhage@email.email international journal of analysis and applications volume 16, number 1 (2018), 1-15 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-1 chebyshev rational approximations for the rosenau-kdv-rlw equation on the whole line mohammadreza foroutan∗ and ali ebadian department of mathematics, payame noor university, p.o.box 19395-3697, tehran, iran ∗corresponding author: foroutan−mohammadreza@yahoo.com abstract. in this paper, we consider the use of a modified chebyshev rational approximations for the rosenau-kdv-rlw equation on the whole line with initial-boundary values. it is shown that the proposed scheme leads to optimal error estimates. furthermore, the stability and convergence of the proposed schemes are proved. the fully discrete chebyshev pseudo-spectral scheme is constructed. numerical results confirm well with the theoretical results. the idea and techniques presented in this paper will be useful to solve many other problems. 1. introduction the application of spectral methods for approximating solutions of partial differential equations in unbounded domains has achieved great success and popularity in recent years. as a case in point, we can refer to the book by shen et al. [32] and a more recent research paper by foroutan et al. [18]. in general, spectral methods used for solving partial differential equations on unbounded domains can be classified into three families. the first family is to use spectral methods associated with some orthogonal systems such as the hermite spectral method and laguerre spectral method (see e.g. parand and taghavi [22], guo [12] and parand et al. [21]). the second family replaces infinite domain with [-l,l] and semi-infinite interval with [0,l] by choosing l, 2010 mathematics subject classification. 65m12, 41a20, 35q53. key words and phrases. error estimate; modified chebyshev rational approximation; spectral method; stability of the scheme. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-1 int. j. anal. appl. 16 (1) (2018) 2 sufficiently large. this method is named the domain truncation [5]. the third family, that is used in this paper too, is based on rational approximations. for example, boyd [6,7] and christov [8] developed some spectral methods on infinite intervals by using mutually orthogonal systems of rational functions. this family of spectral schemes is efficient specially for solving boundary value problems; see also [14–17, 34]. overall, rational spectral methods are highly flexible, but it is hard to obtain convergence results and error estimates for those rational spectral methods. to this end, we apply convergence and error estimates in the sense of guo [11, 13]. the purpose of this paper is to develop and analyze the modified chebyshev rational spectral methods for rosenau-kdv-rlw equation vt −vxxt + vxxxxt + vxxx + vx + vvx = 0, (x,t) ∈ ω × (0,t], (1.1) with the initial condition: v(x, 0) = v0(x), x ∈ ω, (1.2) and the boundary conditions: lim |x|→∞ v(x,t) = lim |x|→∞ vx(x,t) = lim |x|→∞ vxx(x,t) = 0, t ∈ [0,t], (1.3) where v0(x) is a known smooth function and ω = (−∞,∞). the nonlinear wave is one of the most widely researched areas. the dynamics of wave behaviors can be described by several models. some of these well-known models are korteweg-de vries (kdv) equation, regularized long wave (rlw) equation, and rosenau equation. in the following section, we address a short review of these important wave models. korteweg-de vries (kdv) equation as one of the well-known equations in mathematics and physics: vt + vxxx + 6vvx = 0. this equation has been applied in many various fields and its application for describing wave propagation and interaction has been studied widely. there are many numerical methods that can be used to solve kdv equation such as the modified legendre rational spectral methods applied on the semi-infinite interval [14], explicit scheme [20], petrov-galerkin method on the half line [36], finite-difference method [9], and solitary wave solution [1, 3]. the regularized long-wave (rlw) equation (also known as benjamin-bona-mahony equation) which was first introduced by peregrine [23] to describe the development of an undular bore is presented as follows: vt −vxxt + vx + vvx = 0. the rlw equation was well studied numerically and theoretically in the literature. for instance, biswas [2] has introduced an analytical solution of the rlw equation with power-law nonlinearity. on the other hand, islam et al. [19] investigated the meshfree method for the numerical solution of the rlw equation. int. j. anal. appl. 16 (1) (2018) 3 since the case of wave-wave and wave-wall interactions cannot be described by the kdv equation, rosenau [29, 30] proposed an equation known as the rosenau equation to over come this shortcoming of the kdv equation: vt + vxxxxt + vx + vvx = 0. the rosenau equation has been the subject of several analytical and numerical studies [4,24,27] and references therein. recently, the rosenau-kdv-rlw equation was proposed in [25] as a conjunction of rosenau-kdv and rosenau-rlw equations both of which are well studied and explained with regard to shallow water waves. also in this paper the results of rosenau-kdv-rlw equation have been reported without considering the effects of perturbation. for theoretical investigations, razborova et al. [26] explored the dynamics of perturbed soliton solutions to the rosenau-kdv-rlw equation with power-low nonlinearity. solutions of the perturbated rosenaukdvrlw equation are obtained [31]. soliton perturbation theory was applied to obtain the adiabatic parameter dynamics of these solitary waves [28]. the remainder of the paper is organized as follow. in section 2, we first review some basic results on chebyshev rational functions. some orthogonal projections with their properties are also given in this section as they play an important role in the error analysis. in section 3, we will discuss some basic techniques employed for stability of the spectral methods in infinite domains. in section 4, we use the results in the previous sections to validate the convergence of proposed scheme and derive error estimates. in section 5, we construct the fully-discrete chebyshev pseudo-spectral scheme, and obtain the optimum error estimate of approximation solutions. numerical results are shown in section 6. finally, the final section gives some concluding remarks. 2. modified chebyshev rational functions this section addresses the basic notions and working tools concerning orthogonal modified chebyshev rational functions. more specifically, it presents some properties of the modified chebyshev rational functions. the well-known chebyshev polynomials are orthogonal in the interval [-1,1] with respect to the weight function ρ(x) = 1√ 1−x2 and can be calculated through the following recurrence formula: t0(x) = 1, t1(x) = x, tn+1(x) = 2xtn(x) −tn−1(x), n = 1, 2, 3, .... the new basis functions denoted by rn(x), are defined by [14] in interval ω = (−∞,∞). rn(x) = 1 √ x2 + 1 tn( x √ x2 + 1 ), n = 0, 1, 2, ... . int. j. anal. appl. 16 (1) (2018) 4 rn(x) is the eigenfunction of the singular sturm-liouville problem (x2 + 1) 1 2 d dx ( (x2 + 1) d dx ((x2 + 1) 1 2 w(x) )) + n2w(x) = 0, x ∈ ω, n = 0, 1, 2, ... . (2.1) and satisfies the following recurrence relation: r0(x) = 1 √ x2 + 1 , r1(x) = x x2 + 1 , rn+1(x) = ( 2x √ x2 + 1 )rn(x) −rn−1(x), n = 1, 2, 3, ... . {rn(x)}n≥1 are orthogonal with respect to the weight function χ(x) = 1 in the interval (−∞,∞), with the orthogonality property: ∫ ω rn(x)rm(x)χ(x)dx = π 2 cnδn,m, where δn,m is the kronecker function and c0 = 2,cn = 1 for n ≥ 1. for 1 ≤ p ≤ ∞, we define the space lp(ω) and its norm ‖w‖lp(ω) as usual. in particular ‖w‖∞ = ‖w‖l∞(ω). for any nonnegative integer m, we define the sobolev space as follows: hm(ω) = {w : dkw dxk ∈ l2(ω), 0 ≤ k ≤ m}. equipped with the inner product, the semi-norm, and the norm are defined as follow: (v,w)m = m∑ k=0 ( dkv dxk , dkw dxk ), | w |m= ‖ dmw dxm ‖, ‖w‖m = (w,w) 1 2 m. for any real r > 0, we define the space hr(ω) and its norm |w|r by space interpolation. to describe approximation results, we introduce a sequence of hilbert spaces {hrzs}s≥1. for simplicity, let ∂xw(x) = ∂ ∂x w(x), etc. let a be the sturm-liouville operator in ( 2.1), namely aw(x) = −(x2 + 1) 1 2 d dx ( (x2 + 1) d dx ((x2 + 1) 1 2 w(x) )) . for any even integer r ≥ 0, hrzs (ω) = {w : w is a measurable on ω and ‖w‖r,zs < ∞}, where ‖w‖r,z0 = ‖a r 2 w‖, and for s ≥ 1, ‖w‖r,zs = ‖(x 2 + 1)∂x((x 2 + 1) 1 2 w))‖r−1,zs−1. int. j. anal. appl. 16 (1) (2018) 5 we define these spaces and their norms by space interpolation. now let n be any positive integer. in this case, we have: <n = span{r0,r1, ...,rn}. the l2(ω)-orthogonal projection ln : l 2(ω) −→<n is defined in the following way: (lnw −w,φ) = 0 , ∀φ ∈<n. the hm(ω)-orthogonal projection lmn : h m(ω) −→<n is defined as: (lmnw −w,φ)m = 0 , ∀φ ∈<n. let c denote a generic positive constant independent of any function and n. we have the following results: theorem 2.1. for any w ∈ hrzm (ω) and 0 ≤ m ≤ r, ‖lmnw −w‖m ≤ cn m−r‖w‖r,zm. theorem 2.2. let 0 ≤ µ ≤ m− 1 2 , with a positive integer m. then for any w ∈ hmzm (ω) ∩h m 0 (ω), ‖lmnw‖µ,∞ ≤ c‖w‖r,zm. the above two theorems were proved in [17]. in order to analyze the modified chebyshev rational approximation of the equation ( 1.1), we need another orthogonal projection. let hm0 (ω) = {w : w ∈ h m(ω) and ∂kxw(0) = 0, for 0 ≤ k ≤ m}, and <m,0n = <n ∩h m 0 (ω). we denote in particular <0n = < 1,0 n and define the orthogonal projection l m,0 n : h m 0 (ω) −→< m,0 n by (l m,0 n w −w,φ)m = 0 , ∀φ ∈< m,0 n . int. j. anal. appl. 16 (1) (2018) 6 theorem 2.3. for any w ∈ hrzm (ω) ∩h m 0 (ω) and 0 ≤ m ≤ r, ‖lm,0n w −w‖m ≤ cn m−r‖w‖r,zm. proof. let v(x) = ∫ x 0 (z2 + 1)∂z((z 2 + 1) 1 2 w(z) ) dz, and ϕ(x) = 1 √ x2 + 1 ∫ x 0 1 (z2 + 1) l m−1,0 n−1 ∂zv(z)dz. clearly ϕ ∈<m,0n . the desired result follows from the same argument as in the proof of theorem 4 [17]. � 3. stability of the rosenau-kdv-rlw equation this section discusses the application of modified chebyshev rational approximations for the rosenaukdv-rlw equation ( 1.1) and presents the stability results of the proposed approach. a weak form of equation ( 1.1) is to find v ∈ h20 (ω) such that for any w ∈ h20 (ω),  (vt,w) + (vxt,wx) + (vxxt,wxx) − (vxx,wx) + (vx,w) − 1 2 (v2,wx) = 0 v(x, 0) = vo(x), x ∈ ω. (3.1) the chebyshev rational spectral scheme of ( 3.1) is to find vn (t) ∈<0n such that for any φ ∈< 0 n ,  (vnt,φ) + (vnxt,φx) + (vnxxt,φxx) − (vnxx,φx) + (vnx,φ) − 1 2 (v2n,φx) = 0 vn (x, 0) = l 2 nvo(x), x ∈ ω. (3.2) since the initial value term cannot be exactly evaluated, here we consider how stable the numerical solution of ( 3.2) depending on the initial value term. we now analyze the stability of ( 3.2) in the sense of guo [11, 13]. to this end, we suppose that vn has the error ṽn . then for any φ ∈ <0n and t ∈ [0,t], we derive the following equation from ( 3.2) as follows:  (vnt(t) + ṽnt(t),φ) + (vnxt(t) + ṽnxt(t),φx) + (vnxxt(t) + ṽnxxt(t),φxx) −(vnxx(t) + ṽnxx(t),φx) + (vnx(t) + ṽnx(t),φ) − 12 ((vn (t) + ṽn (t)) 2,φx) = 0 ṽn (0) = ṽn,0. (3.3) it can be demonstrated that ṽn must satisfy the following relation:  (ṽnt(t),φ) + (ṽnxt(t),φx) + (ṽnxxt(t),φxx) − (ṽnxx(t),φx) + (ṽnx(t),φ) −1 2 (ṽn (t) + 2vn (t)ṽn (t),φx) = 0, ∀φ ∈<0n, 0 ≤ t ≤ t ṽn (0) = ṽn,0. by taking φ = 2ṽn in ( 3.4), it follows that d dt ‖ṽn (t)‖22 = 2a(t), (3.4) int. j. anal. appl. 16 (1) (2018) 7 where a(t) = (vn (t)ṽn (t), ṽnx(t)). then we have: |a(t)| ≤ ‖vn (t)‖∞‖ṽn (t)‖|ṽn (t)|1 ≤ 1 2 ‖vn (t)‖2∞‖ṽn (t)‖ 2 + 1 2 |ṽn (t)|21. (3.5) to describe the error, let ρ(ϑ,t) = ‖ϑn (t)‖22. also let |‖w|‖∞ = sup0≤t≤t‖w(t)‖∞, and m(w) = (1 + |‖w|‖∞). by substituting ( 3.5) into ( 3.4) and integrating the resulting inequality for t, we find that ‖ṽn (t)‖22 ≤ m(vn ) ∫ t 0 ‖ṽn (τ)‖22dτ + ρ(ṽn,0, t). (3.6) applying the gronwall lemma to the above inequality, we draw the following conclusion. theorem 3.1. let vn be the solution of ( 3.2), and ṽn be its error induced by ṽn,0. then for all 0 ≤ t ≤ t , ‖ṽn (t)‖22 ≤ ρ(ṽn,0, t)e m(vn )t. (3.7) 4. convergence analysis and error estimate for the rosenau-kdv-rlw equation in this section, we will discuss some basic techniques use to estimate error bounds for spectral methods in infinite domains. furthermore, to show that the convergence results of the spectral methods for the rosenaukdv-rlw equation ( 1.1), we use the modified chebyshev rational approximations. for this purpose, we first present an a priori estimate: lemma 4.1. if for r ≥ 2, v0 ∈ hrz2 (ω) ∩h 2 0 (ω) then for any t ∈ [0,t], ‖vn (t)‖22 ≤ 2‖v0‖ 2 2 + c0n 4−2r‖v0‖2r,z2. (4.1) proof. taking φ = vn in ( 3.2), we have 1 2 ∂t(‖vn‖2 + ‖vnx‖2 + ‖vnxx‖2) − (vnxx,vnx) + (vnx,vn ) − 1 2 (v2n,vnx) = 0. (4.2) clearly (vnxx,vnx) = 1 2 ∫ ω ∂xv 2 nx(x,t)dx = 0, (4.3) (vnx,vn ) = 1 2 ∫ ω ∂xvn (x,t)dx = 0, (4.4) (v2n,vnx) = 1 3 ∫ ω ∂xv 3 n (x,t)dx = 0. (4.5) hence ∂t(‖vn‖2 + ‖vnx‖2 + ‖vnxx‖2) = 0, (4.6) int. j. anal. appl. 16 (1) (2018) 8 which implies ‖vn (t)‖2 + ‖vnx(t)‖2 + ‖vnxx(t)‖2 = ‖vn (0)‖2 +‖vnx(0)‖2 + ‖vnxx(0)‖2 = ‖l2nv0‖ 2 2. (4.7) next, according to theorem 2.2 for v0 ∈ h20 (ω), we have ‖l2,0n v0‖ 2 2 ≤ 2‖v0‖ 2 2 + 2‖v0 −l 2,0 n v0‖ 2 2 ≤ 2‖v0‖ 2 2 + c0n 4−2r‖v0‖2r,z2. (4.8) finally, a combination of ( 4.7) and ( 4.8) leads to the desired result. � therefore, the modified chebyshev rational approximations are well suited for numerical approximation of the rosenau-kdv-rlw equation. indeed, we have the following result concerning the convergence and error estimate for ( 3.2). theorem 4.1. if for r ≥ 2, v ∈ l∞(0,t; w 1,∞(ω)) ∩h2(0,t; hrz2 (ω)), and hrz2 (ω), then we have ‖v −vn‖22 ≤ c2(v)n 2−r, ∀t ∈ (0,t], (4.9) where c2(v) is positive constant depending only on the norms of v and v0 in the spaces mentioned above. proof. for simplicity, let ξ = v −l2,0n v and η = vn −l 2,0 n v. (4.10) obviously, vn −v = η − ξ. hence, by ( 3.1) and ( 3.2) we obtain that for any φ ∈<0n , (ηt,φ) + (ηxt,φx) + (ηxxt,φxx) − (ηxx,φx) + (ηx,φ) + (vnvnx −vvx,φ) (4.11) = (ξt,φ) + (ξxt,φx) + (ξxxt,φxx) − (ξx,φxx) − (ξ,φx). take φ = η in ( 4.11). it can be shown that (ηxx,ηx) = 0 , (ηx,η) = 0, (4.12) and it is clear that (ηxxt,ηx) + (ηxt,ηx) + (ηt,η) = 1 2 ∂t‖η‖22. (4.13) int. j. anal. appl. 16 (1) (2018) 9 next, by theorems 2.2 and 2.3 we obtain (vnvnx −vvx,η) = (vn (ηx − ξx) + vx(η − ξ),η) ≤‖η‖2 + ‖vn (ηx − ξx)‖2 + ‖vx(η − ξ)‖2 ≤‖η‖2 + c‖vn‖2∞‖ηx − ξx‖ 2 + c‖vx‖2∞‖η − ξ‖ 2 ≤‖η‖2 + cn4−2r(‖v‖21,∞ + ‖v‖ 2 r,z2 ). (4.14) using theorem 2.3 again yields (ξt,η) + (ξxt,ηx) + (ξxxt,ηxx) ≤‖η‖22 + ‖ξt‖ 2 2 ≤‖η‖ 2 2 + cn 4−2r‖∂tv‖2r,z2, (4.15) (ξx,ηxx) ≤‖ηxx‖2 + ‖ξx‖2 ≤‖ηxx‖2 + cn4−2r‖v‖2r,z2, (4.16) (ξ,ηx) ≤‖ηx‖2 + ‖ξ‖2 ≤‖ηx‖2 + cn4−2r‖v‖2r,z2. (4.17) in addition, theorems 2.1 and 2.3 lead to ‖η(0)‖22 ≤‖v0 −l 2 nv0‖ 2 2 + ‖v0 −l 2,0 n v0‖ 2 2 ≤ cn 4−2r‖v0‖2r,z2. (4.18) hence, by inserting ( 4.12)–( 4.17) in to ( 4.11), we obtain ∂t‖η‖22 ≤‖η‖ 2 2 + c1(v)n 4−2r(‖v‖2r,z2 + ‖∂tv‖ 2 r,z2 ), (4.19) where c1(v) is a positive constant depending only on ‖v‖l∞(0,t,h2 z2 (ω)∩w1,∞(ω)). substituting ( 4.18) into ( 4.20) and integrating the result with respect to t, we obtain: ‖η‖22 ≤ ∫ t 0 ‖η(s)‖22ds + c2(v)n 4−2r, (4.20) where c2(v) is a positive constant depending only on c1(v), ‖v‖h2(0,t,hr z2 (ω)) and ‖v0‖hr z2 (ω)). the desired result follows from the gronwall inequality and theorem 2.3. � 5. fully-discretization scheme in this section, we describe the numerical implementation and present some numerical results. at first, we introduce the operator of interpolation at the chebyshevgauss rational nodes {xn,j = cot( π(2j + 1) 2n + 2 )}0≤j≤n, denoted by inv ∈<0n and inv(xn,j) = v(xn,j), 0 ≤ j ≤ n. the following theorem can be found in [17]. int. j. anal. appl. 16 (1) (2018) 10 theorem 5.1. for any v ∈ hrz1 (ω) and 0 ≤ µ ≤ 1 ≤ r, ‖inv −v‖µ ≤ cnµ−r+1‖w‖r,z1. let τ be the mesh size in t and set tk = kτ (k = 0, 1, ...,n + 1 = [ t τ ]). for simplicity, we denote vk(x) := v(x,tk) by v k and vk t̂ = vk+1 −vk−1 2τ , v̂kt = vk+1 + vk−1 2 . (5.1) the fully-discretization chebyshev pseudo-spectral method for ( 1.1)-( 1.3) is to find vkn ∈< 0 n , such that vk nt̂ − ∂2 ∂x2 vk nt̂ + ∂4 ∂x4 vk nt̂ + ∂3 ∂x3 v̂kn + ∂ ∂x v̂kn + in (v k n ∂ ∂x v̂kn ) = 0. (5.2) v0n (x) = inv0(x). (5.3) the equations ( 5.2) and ( 5.3) are satisfied at chebyshev-gauss rational nodes x`,` = 0, 1, ...,n. let vk −vkn = (v k −l2,0n v k) + (l 2,0 n v k −vkn ) = ξ k + ηk. note that (ξk,w) = 0, ∀w ∈ <0n . taking the inner product of ( 1.1) with w and subtracting ( 5.2) from ( 1.1), we get (ηk t̂ ,w) + ( ∂ηk t̂ ∂x , ∂w ∂x ) − ( ∂2ηk t̂ ∂x2 , ∂2w ∂x2 ) + ( ∂η̂k ∂x , ∂2w ∂x2 ) − (η̂k, ∂w ∂x ) (5.4) +(in (v k ∂v̂ k ∂x −vkn ∂v̂kn ∂x ),w) = (γk,w), where γk is truncation error γk = (vk t̂ − ∂vk ∂t ) + ∂2 ∂x2 ( ∂vk ∂t −vk t̂ ) + ∂4 ∂x4 (vk t̂ − ∂vk ∂t ) + ∂3 ∂x3 (vk − v̂k) (5.5) + ∂ ∂x (vk − v̂k) + (vk ∂v̂k ∂x − in (vkn ∂v̂kn ∂x )). by applying taylor’s theorem, theorems 2.3 and 5.1, we get γk = τ2 12 ( ∂3v ∂t3 (tk1 ) + ∂3v ∂t3 (tk2 )) + τ2 12 ∂2 ∂x2 ( ∂3v ∂t3 (tk3 ) + ∂3v ∂t3 (tk4 ) + τ2 12 ∂4 ∂x4 ( ∂3v ∂t3 (tk5 ) + ∂3v ∂t3 (tk6 ) + τ2 4 ∂3 ∂x3 ( ∂2v ∂t2 (tk7 ) + ∂2v ∂t2 (tk2 )) + τ2 4 ∂ ∂x ( ∂2v ∂t2 (tk9 ) + ∂2v ∂t2 (tk10)) +(vk ∂v̂k ∂x − in (vkn ∂v̂kn ∂x )). where tk−1 ≤ tk` ≤ t k+1,` = 1, ..., 10. taking w = η̂k in equation ( 5.5), we have (γk, η̂k) = ‖ηk+1‖2 −‖ηk−1‖2 4τ + ‖ηk+1x ‖2 −‖ηk−1x ‖2 4τ + ‖ηk+1xx ‖2 −‖ηk−1xx ‖2 4τ +( ∂η̂k ∂x , ∂2η̂k ∂x2 ) − (η̂k, ∂η̂k ∂x ) + fk. (5.6) int. j. anal. appl. 16 (1) (2018) 11 where fk = (in (v k ∂v̂ k ∂x −vkn ∂v̂kn ∂x ), η̂k). clearly, ( ∂η̂k ∂x , ∂2η̂k ∂x2 ) = 0, (η̂k, ∂η̂k ∂x ) = 0. (5.7) in the forthcoming discussions, we denote by c a positive constant independent of τ and n, and estimate fk of ( 5.6). by applying taylor’s theorem, cauchy-schwartz inequality and algebraic inequality, we deduce that |fk| ≤ c(n−2r + ‖ηk‖2 + ‖η̂k‖2 + ‖η̂k‖21). (5.8) furthermore, by theorem 2.3 and the cauchy-schwartz inequality, we obtain |(γk, η̂k)| ≤ c(n−2r + τ4 + ‖η̂k‖2). (5.9) inserting ( 5.7)-( 5.9) into ( 5.6), we get ‖ηk+1‖2 −‖ηk−1‖2 4τ + ‖ηk+1x ‖2 −‖ηk−1x ‖2 4τ + ‖ηk+1xx ‖2 −‖ηk−1xx ‖2 4τ ≤ c(n−2r + τ4 + ‖ηk‖2 + ‖η̂k‖2 + ‖η̂k‖21). (5.10) obviously, we can verify from ( 5.1) that ‖v̂k‖2 ≤ 1 2 (‖vk+1‖2 + ‖vk−1‖2). by summing up ( 5.10) for k = 1, 2, ...,n, we get ‖ηn‖22 ≤ c(‖η 0‖22 + τ 4 + n−2r) + cτ n−1∑ k=1 ‖ηn‖22. note that ‖η0‖22 = 0. tanks to the discrete gronwall inequality, we get c(τ4 + n−2r) ≤ mect . ‖ηn‖22 ≤ c(τ 4 + n−2r)ec(n+1)τ. where m is the positive constant. therefore, we have the following result. theorem 5.2. let v be the solution of ( 1.1) and v ∈ hrzm (ω) ∩h m 0 (ω), 0 ≤ m ≤ r. also, let vn be the solution of ( 5.2) in <0n and τ is sufficiently small. then there exist constant m, independent of τ and n such that for k = 1, 2, ...,n− 1, ‖vk+1 −vk+1n ‖2 ≤ m(τ 2 + n−r). int. j. anal. appl. 16 (1) (2018) 12 6. numerical experiments in this section, we will conduct some numerical experiments to verify the theoretical results obtained in the previous sections. we report ‖e‖∞ and the ‖e‖2 errors of the solution that are defined as ‖e‖∞ = max 0≤j≤n |v(xj, t) −vn (xj, t)|, and ‖e‖2 = ∑n j=0 |v(xj, t) −vn (xj, t)| 2∑n j=0 |v(xj, t)|2 , where vn (xj, t) is the solution of numerical scheme ( 5.2) and ( 5.3), while v(xj, t) is the exact solution of ( 1.1)-( 1.3). consider the initial boundary value problem of the rosenau-kdv-rlw equation ( 1.1)-( 1.3) with the exact solution as follow [35]: v(x,t) = 5 456 (25 − 13 √ 457)sech4[ 1 √ 288 √ −13 + √ 457(x− ( 241 + 13 √ 457 266 )t)], and the initial condition is set as v0(x) = 5 456 (25 − 13 √ 457)sech4[( 1 √ 288 √ −13 + √ 457)x]. ‖e‖∞ and ‖e‖2 errors for various of n with t = 20 and τ = 0.1, are reported in table 1. we can observe from the table, that the results from the present study are in good agreement with the exact solutions. n ‖e‖∞ ‖e‖2 10 6.008 × 10−2 9.863 × 10−2 20 4.439 × 10−3 8.746 × 10−3 30 3.540 × 10−5 5.298 × 10−5 40 3.516 × 10−5 5.255 × 10−5 50 3.510 × 10−5 5.140 × 10−5 table 1. norm infinity and norm relative of errors for τ = 0.1,t = 20 and several values of n table 2 gives the errors between numerical solutions and exact solutions. we can see that when we use smaller time and temporal mesh, numerical solutions are almost the same as the exact solutions. int. j. anal. appl. 16 (1) (2018) 13 τ ‖e‖∞ ‖e‖2 0.1 1.439 × 10−5 3.611 × 10−5 0.05 7.160 × 10−6 6.228 × 10−6 0.025 5.343 × 10−7 8.390 × 10−7 0.0125 1.905 × 10−7 3.110 × 10−7 0.00625 1.777 × 10−8 1.943 × 10−8 table 2. norm infinity and norm relative of errors for n = 35,t = 10 and several values of τ 7. conclusion in this paper, we proposed the modified chebyshev rational approximations for the rosenau-kdv-rlw equation on the infinite intervals. the corresponding spectral scheme was constructed and its convergence was proved. to show the performance of the modified chebyshev rational approximations, an a prior estimate was derived. we proved that the numerical solutions tend to weak solutions of ( 1.1) in a suitable sense. the numerical results demonstrate that the suggested method possesses high-order accuracy for the rosenau-kdv-rlw equation on the whole line with analytical solutions. references [1] a. biswas, singular solitons, shock waves, and other solutions to potential kdv equation, nonlinear dyn. 58 (2009), 345–348. [2] a. biswas , solitary waves for power-law regularized long-wave equation and r(m, n) equation, nonlinear dyn. 59 (2010), 423–426. [3] a. biswas, solitary wave solution for the generalized kdv equation with time-dependent damping and dispersion, commun. nonlinear sci. numer. simul. 14 (2009), 3503–3506. [4] a. biswas, h. triki, m. labidi, bright and dark solitons of the rosenaukawahara equation with power law nonlinearity, phys. wave phenom. 19 (2011), 24–29. [5] j.p. boyd, chebyshev and fourier spectral methods, second ed., dover, new york, 2000. [6] j.p. boyd, orthogonal rational functions on a semi-infinite interval, j. comput. phys. 70 (1978), 63–88. [7] j.p. boyd, spectral methods using rational basis functions on an infinite interval, j. comput. phys. 69 (1978), 112–142. [8] c.i. christov, a complete orthogonal system of functions in l2(−∞,∞) spaces, siam j. appl. math. 42 (1982), 1337– 1344. [9] m. dehghan, a.shokri, a numerical method for kdv equation using collocation and radial basis functios, nonlinear dyn. 50 (2007), 111–120. [10] d.gottlieb, m.y. hussaini, s.orszag, theory and applications of spectral methods in spectral methods for partial differential equations edited by r. voigt and d. gottlieb and m.y. hussaini, siam, philadelphia, 1984. [11] b.y. guo, a class of difference schemes of two-dimensional viscous fluid flow, acta math. sinica, 17 (1974), 242–258. [12] b.y. guo, error estimation of hermite spectral method for nonlinear partial differential equations, math. comput. 68 (1999), 1067–1078. int. j. anal. appl. 16 (1) (2018) 14 [13] b.y. guo, generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, contemp. math. 163 (1994), 33–54. [14] b.y. guo, j.shen, on spectral approximations using modified legendre rational functions: approximation to the korteweg-ele vries equation on the half line, indiana university mathematics journal, 50 (2001), 181–204. [15] b.y. guo, j. shen, z.q. wang, chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, int. j. numer. meth. engng. 53 (2002), 65–84. [16] b.y. guo, j. shen, z.q. wang, a rational approximation and its applications to differential equations on the half line, journal of scientific computing, 15 (2000), 117–148. [17] b.y. guo, z.q. wang, modified chebyshev rational spectral method for the whole line, discrete contin. dyn. syst. supplement volume (2003), 365–374. [18] m.r. foroutan, a. ebadian, s.najafzadeh, the use of generalized laguerre functions for solving the equation of magnetohydydinamic flow due to a stretching cylinder , sema journal 73(4) (2016), 335–346. [19] s.u. islam, s. haq, a. ali, a meshfree method for the numerical solution of the rlw equation, j. comput. appl. math. 223 (2009), 997–1012. [20] z.q. lv, m. xue, y.s. wang, a new multi-symplectic scheme for the kdv equation, chin. phys. lett. 28 (2011), 060205. [21] k. parand, m. dehghan, a.r. rezaei, s.m. ghaderi, an approximation algorithm for the solution of the nonlinear laneemden type equations arising in astrophysics using hermite functions collocation method, comput. phys. commun. 181 (2010), 1096–1108. [22] k. parand, a. taghavi, rational scaled generalized laguerre function collocation method for solving the blasius equation, j. comput. appl. math. 233(4) (2009), 980–989. [23] d.h. peregrine, calculations of the development of an undular bore, j. fluid mech. 25 (1966), 321–330. [24] j.l. ramos, explicit finite difference methods for the ew and rlw equations, appl. math. comput. 179(2) (2006), 622–638. [25] p. razborova, b. ahmed, a. biswas, solitons, shock waves and conservation laws of rosenau-kdv-rlw equation with power law nonlinearity, appl.math. inf. sci.8(2) (2014), 485–491. [26] p. razborova, l. moraru, a. biswas, perturbation of dispersive shallow water waves with rosenau-kdv-rlw equation and power law nonlinearity, rom. j. phys. 59(7-8) (2014), 658–676. [27] p. razborova, a.h. kara, a. biswas, additional conservation laws for rosenau-kdv-rlw equation with power law nonlinearity by lie symmetry, nonlinear dyn. 179 (2015), 743–748. [28] p. razborova, h. triki, a. biswas, perturbation of dispersive shallow water waves, cean eng. 63 (2013), 1–7. [29] p. rosenau, a quasi-continuous description of a nonlinear transmission line , phys. scr. 34 (1986), 827–829. [30] p. rosenau, dynamics of dense discrete systems, prog. theor. phys. 79 (1988), 1028–1042. [31] p. sanchez, g. ebadi, a. mojaver, m. mirzazadeh, m.eslami, a. biswas, solitons and other solutions to perturbed rosenau-kdv-rlw equation with power law nonlinearity, acta phys. pol. a, 127 (2015), 1577–1586. [32] j. shen, t. tang, l.l. wang, spectral methods: algorithms, analysis and applications, springer, first edition 2010. [33] t. tajvidi, m. razzaghi, m. dehghan, modified rational legendre approach to laminar viscous flow over a semi-infinite, caos, solitons fractals, 35 (2008), 59–66. [34] z.q. wang, b.y. guo, modified legendre rational spectral method for the whole line, j. comput. math. 22 (2004), 457–474. [35] b. wongsaijai, k. poochinapan: a three-level average implicit finite difference scheme to solve equation obtained by coupling the rosenau-kdv equation and the rosenau-rlw equation. appl. math. comput. 245 (2014), 289–304. int. j. anal. appl. 16 (1) (2018) 15 [36] z.q. zhang, h.p. ma, a rational spectral method for the kdv equation on the half line, j. comput. appl. math. 230 (2009), 614–625. 1. introduction 2. modified chebyshev rational functions 3. stability of the rosenau-kdv-rlw equation 4. convergence analysis and error estimate for the rosenau-kdv-rlw equation 5. fully-discretization scheme 6. numerical experiments 7. conclusion references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 57-61 http://www.etamaths.com about heinz mean inequalities daeshik choi∗ abstract. we present some inequalities related to heinz means. among them, we will provide an inequality involving heinz means and heron means, which is reverse to the one found by bhatia. 1. introduction throughout the paper, b stands for the set of all bounded linear operators on a hilbert space h and b+ denotes the subset of b consisting of positive invertible operators. for self-adjoint operators a,b in b, a ≥ b implies that a−b is positive semidefinite. for 0 ≤ v ≤ 1, the heinz mean hv(a,b) of positive numbers a,b is defined by hv(a,b) = 1 2 (a1−vbv + avb1−v). it is easy to see that hv(a,b), as a function of v, attains its minimum at v = 1/2 and its maximum at v = 0 or v = 1. thus √ ab ≤ hv(a,b) ≤ a + b 2 (1.1) holds for all ≤ v ≤ 1. for a,b ∈ b+ and 0 ≤ v ≤ 1, the v-weighted arithmetic mean a∇vb and geometric mean a]vb are defined, respectively, by a∇vb = (1 −v)a + vb, a]vb = a 1/2(a−1/2ba−1/2)va1/2. for convenience of notation, we write a∇1/2b as a∇b and a]1/2b as a]b. the heinz operator mean of a,b ∈b+ is defined by hv(a,b) = 1 2 (a]vb + a]1−vb) for 0 ≤ v ≤ 1. the operator mean inequalities corresponding to (1.1) are a]vb ≤ hv(a,b) ≤ a∇vb, (1.2) which are easily derived by the operator monotonicity of continuous functions, which states that if f is a real valued continuous function defined on the spectrum of a self-adjoint operator a, then f(t) ≥ 0 for every t in the spectrum of a implies that f(a) is a positive operator. we refer to [2–4] for more results related to heinz inequalities. using the taylor series of hyperbolic functions, bhatia [1] and liang and shi [5,6] derived interesting heinz operator inequalities. in this paper, we will improve their results using a simple but useful lemma. in particular, we note the following inequality [1]: hv(a,b) ≤ f(2v−1)2 (a,b), ∀a,b > 0, 0 ≤ v ≤ 1, (1.3) where fα(a,b) = (1 −α) √ ab + α a + b 2 received 8th may, 2017; accepted 11th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 47a63. key words and phrases. heinz mean; heron mean; positive definite operators; operator inequalities. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 57 58 choi are heron means of a,b. as mentioned in [1], there is no inequality reverse to (1.3) in the sense that fα(a,b) ≤ hv(a,b) for all a,b > 0, 0 < α < 1, and 0 < v < 1 2 . however, we will present a kind of reverse inequality to (1.3) (see theorem 2.2 or (2.9)). 2. improvements of heinz means the following is the main lemma in this paper. lemma 2.1. for c > 1 and ρ ∈ r, we define ϕ by ϕ(x) = cx − c−x xρ for x > 0. then, (1) if ρ ≤ 1, ϕ is increasing on (0,∞), and (2) if ρ > 1, then there exists xρ > 0 such that ϕ is decreasing on (0,xρ) and increasing on (xρ,∞). if t = tρ is the root of the equation t+1 2(t−1) ln t = ρ, then xρ = ln tρ 2 ln c and min x>0 ϕ(x) = ϕ(xρ) = ( ln c ρ · tρ + 1 tρ − 1 )ρ (t1/2ρ − t −1/2 ρ ). proof. let f(t) = t+1 2(t−1) ln t for t > 1. by direct computation, we have xρ+1cxϕ′(x) = x(c2x + 1) ln c−ρ(c2x − 1), (2.1) = 1 2 (s + 1) ln s−ρ(s− 1), = (s− 1)(f(s) −ρ), where s = c2x > 1. simple algebra shows that limt→1 f(t) = 1 and that f is strictly increasing on (1,∞). thus if ρ ≤ 1, xρ+1cxϕ′(x) = (s− 1) (f(s) −ρ) > (s− 1) (1 −ρ) ≥ 0 for any x > 0, which implies that ϕ is increasing on (0,∞). meanwhile, if ρ > 1, let tρ > 1 be the (unique) zero of f(t) = ρ. then for xρ = ln tρ 2 ln c , (2.1) says that ϕ′(x) < 0 on (0,xρ), ϕ ′(x) > 0 on (xρ,∞), and the minimum value of ϕ on (0,∞) is ϕ(xρ) = cxρ − c−xρ x ρ ρ = ( 2 ln c ln tρ )ρ (t1/2ρ − t −1/2 ρ ) = ( ln c ρ · tρ + 1 tρ − 1 )ρ (t1/2ρ − t −1/2 ρ ), where the last equation follows from 2 ln tρ = tρ+1 ρ(tρ−1) . � heinz means hv(a,b) or hv(a,b) were defined for 0 ≤ v ≤ 1, but we don’t restrict v to be in the interval [0, 1] in the following theorem. lemma 2.1 with ρ = 1 is used below. theorem 2.1. for a,b ∈b+ and r,s,t ∈ r with 0 < |1 − 2r| ≤ |1 − 2s| ≤ |1 − 2t|, we have hs(a,b) ≥ ( 1 − (1 − 2s)2 (1 − 2r)2 ) a]b + (1 − 2s)2 (1 − 2r)2 hr(a,b), hs(a,b) ≤ ( 1 − (1 − 2s)2 (1 − 2t)2 ) a]b + (1 − 2s)2 (1 − 2t)2 ht(a,b). proof. let a,b > 0 and f(x) = {( h(1−x)/2(a,b) − √ ab ) /x2, x ∈ r\{0} 1 8 ( ln a b )2 √ ab, x = 0 . letting c = (ab−1)1/4, we have f(x) = √ ab 2 · c2x + c−2x − 2 x2 = √ ab 2 ( cx − c−x x )2 . about heinz mean inequalities 59 without loss of generality, we assume c > 1. since f is even on (−∞,∞) and increasing on (0,∞) by lemma 2.1, we have f(1 − 2r) ≤ f(1 − 2s) ≤ f(1 − 2t) for r,s,t ∈ r with 0 < |1 − 2r| ≤ |1 − 2s| ≤ |1 − 2t|, which can be written as hr(a,b) − √ ab (1 − 2r)2 ≤ hs(a,b) − √ ab (1 − 2s)2 ≤ ht(a,b) − √ ab (1 − 2t)2 or equivalently, hs(a,b) ≥ ( 1 − (1 − 2s)2 (1 − 2r)2 )√ ab + (1 − 2s)2 (1 − 2r)2 hr(a,b), hs(a,b) ≤ ( 1 − (1 − 2s)2 (1 − 2t)2 )√ ab + (1 − 2s)2 (1 − 2t)2 ht(a,b). by the operator monotonicity of continuous functions, we get the desired operator inequalities. � remark 2.1. the second inequality in theorem 2.1 is shown in [5, theorem 2.1] with 0 ≤ s,t ≤ 1. now we use lemma 2.1 with ρ ≥ 1 below. theorem 2.2. for a,b ∈b+ and 0 ≤ s ≤ 1, we have hs(a,b) ≤ ( 1 − (1 − 2s)2 ) a]b + (1 − 2s)2a∇b. (2.2) for ρ > 1, let tρ > 1 be the root of the equation t+1 2(t−1) ln t = ρ . (1) if a > t2ρb or b > t 2 ρa, then hs(a,b) ≥ ( 1 + αρ|1 − 2s|2ρ(2 ln tρ)2ρ ) a]b (2.3) where αρ = ( tρ + 1 4ρ(tρ − 1) )2ρ (tρ + t−1ρ 2 − 1 ) . (2) if t−2ρ b ≤ a ≤ t2ρb, then hs(a,b) ≥ ( 1 −|1 − 2s|2ρ ) a]b + |1 − 2s|2ρa∇b. (2.4) proof. first, we will show the following: h(1−x)/2(a,b) ≤ ( 1 −x2 )√ ab + x2 a + b 2 , (2.5) h(1−x)/2(a,b) ≥ {√ ab ( 1 + αρ|x|2ρ| ln a− ln b|2ρ ) , if tρ < µa,b( 1 −|x|2ρ )√ ab + |x|2ρ a+b 2 , if tρ ≥ µa,b (2.6) for −1 ≤ x ≤ 1, where µa,b = max {√ a b , √ b a } . since h(1−x)/2(a,b) = h(1+x)/2(a,b) and h1/2(a,b) = √ ab, we may assume x > 0. for ρ ≥ 1, define fρ by fρ(x) = h(1−x)/2(a,b) − √ ab x2ρ for x > 0. letting c = (ab−1)1/4, we have fρ(x) = √ ab 2 · c2x + c−2x − 2 x2ρ = √ ab 2 ( cx − c−x xρ )2 . 60 choi without loss of generality, we assume c > 1. by lemma 2.1, f1(x) ≤ √ ab 2 ( c− c−1 )2 (2.7) = √ ab 2 (√ a b + √ b a − 2 ) = a + b 2 − √ ab, fρ(x) ≥ √ ab 2 (ln c)2ρ ( tρ + 1 ρ(tρ − 1) )2ρ (tρ + t −1 ρ − 2) (2.8) = √ ab(ln a− ln b)2ραρ for ρ > 1. (2.5) follows from (2.7). using the same notation as in lemma 2.1, we know ϕ(x) = cx − c−x xρ ≥ ϕ(xρ) for all x > 0. here we consider x with |x| ≤ 1. then we can bound ϕ as follows: ϕ(x) ≥ { ϕ(xρ), if xρ < 1 ϕ(1), if xρ ≥ 1 . since xρ < 1 ⇐⇒ tρ < c2 = √ a b and fρ(1) = a+b 2 − √ ab, we can improve (2.8) as fρ(x) ≥ {√ ab(ln a− ln b)2ραρ, if tρ < √ a b a+b 2 − √ ab, if tρ ≥ √ b a which implies (2.6). we get (2.2) from (2.5) by the operator monotonicity of continuous functions. meanwhile, since tρ < µa,b ⇐⇒ a > t2ρb or b > t 2 ρa, if tρ < µa,b, then | ln a− ln b| ≥ 2 ln tρ and hs(a,b) ≥ ( 1 + αρ|1 − 2s|2ρ|2 ln tρ|2ρ )√ ab from the first inequality of (2.6). on the other hand, if tρ ≥ µa,b, that is , if a ≤ t2ρb and b ≤ t2ρa , then hs(a,b) ≥ ( 1 −|1 − 2s|2ρ )√ ab + |1 − 2s|2ρ a + b 2 from the second inequality of (2.6). finally, (2.3) and (2.4) follow from the operator monotonicity of continuous functions. � remark 2.2. in the proof of theorem 2.2, we showed that hs(a,b) ≥ {( 1 + αρ|1 − 2s|2ρ| ln a− ln b|2ρ )√ ab, if tρ < µa,b( 1 −|1 − 2s|2ρ )√ ab + |1 − 2s|2ρ a+b 2 , if tρ ≥ µa,b (2.9) for any ρ > 1. the above inequality improves the known relation hs(a,b) ≥ √ ab considerably. note that the minimum value of the right hand side of (2.9), as a function in s, is √ ab (when s = 1/2). figure 1 shows the graphs of the both sides of (2.9) as functions in s ∈ [0, 1] for some values of a,b, where ρ = 1.1 and tρ = 3.0237. the following corollary also improves the heinz mean geometric mean inequality: hs(a,b) ≥ √ ab, a,b > 0 and hs(a,b) ≥ a]b, a,b ∈b+ under a condition. about heinz mean inequalities 61 figure 1. the graphs of hs(a,b) (solid curves) and the right hand side (dotted lines) of (2.9) as functions in s ∈ [0, 1], where ρ = 1.1 and tρ = 3.0237; a = 3.8390,b = 1.3615,µa,b = 1.6792 on the left figure and a = 0.9575,b = 96.4889,µa,b = 10.0385 on the right figure. the horizontal dotted lines denote √ ab which is the minimum value of the two functions. corollary 2.1. for 0 ≤ s ≤ 1 and a,b > 0, we have hs(a,b) ≥ ( 1 + 1 8 (1 − 2s)2(ln a b )2 )√ ab. (2.10) for 0 ≤ s ≤ 1 and a,b ∈b+ with either b ≥ αa or a ≥ αb for a real number α ≥ 1, we have hs(a,b) ≥ ( 1 + 1 8 (1 − 2s)2(ln α)2 ) a]b. (2.11) proof. it is easily shown that αρ → 18 and tρ → 1 as ρ → 1. thus (2.10) follows from the first inequality of (2.9). to show (2.11), it suffices to consider the case b ≥ αa, since hs(a,b) = hs(b,a) and a]b = b]a. letting a = 1 and assuming b ≥ α ≥ 1 in (2.10), we get 1 2 (bs + b1−s) ≥ ( 1 + 1 8 (1 − 2s)2(ln α)2 )√ b. (2.12) thus if b ≥ αa, then for x = a−1/2ba−1/2 we have 1 2 (xs + x1−s) ≥ ( 1 + 1 8 (1 − 2s)2(ln α)2 ) x1/2 from (2.12). multiplying each side of the above inequality by a1/2 on its leftand right-hand sides, we get (2.11). � references [1] r. bhatia, interpolating the arithmetic–geometric mean inequality and its operator version, lin. alg. and its appl. 413 (2006) 355–363. [2] r. bhatia, c. davis, more matrix forms of the arithmetic–geometric mean inequality, siam j. matrix anal. appl. 14 (1993) 132–136. [3] f. kittaneh, m. krnic, n. lovricevic, and j. pecaric, improved arithmetic-geometric and heinz means inequalities for hilbert space operators, publ. math. debrecen. 80 (3-4) (2012), 465– 478. [4] f. kittaneh and m. krnic, refined heinz operator inequalities, linear multilinear algebra, 61 (8) (2013), 1148–1157. [5] j. liang and g. shi, refinements of the heinz operator inequalities, linear multilinear algebra, 63 (7) (2015), 1337–1344. [6] j.liang and g. shi, some means inequalities for positive operators in hilbert spaces, j. inequal. appl. 2017 (2017), art. id 14. dept. of mathematics and statistics, southern illinois university edwardsville, box 1653, edwardsville, il 62026, usa ∗corresponding author: math.dchoi@gmail.com 1. introduction 2. improvements of heinz means references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 1-7 http://www.etamaths.com the essential spectrum of a sequence of linear operators in banach spaces aymen ammar1,∗, noui djaidja2 and aref jeribi1 abstract. in this work we introduce some essential spectra (σei, i = 1, ...,5) of a sequence of closed linear operators (tn)n∈n on banach space, we prove that if (tn)n∈n converges in the generalized sense to a closed linear operator t, then there exists n0 ∈ n such that, for every n ≥ n0, we have σei(λ0 − (tn + b)) ⊆ σei(λ0 − (t + b)), i = 1, ...,5, where b is a bounded linear operator, and λ0 ∈ c. the same treatment is made when (tn −t) converges to zero compactly. 1. introduction let x and y be two banach spaces. we denote by l(x,y ) (resp., c(x,y )) the set of all bounded (resp., closed, densely defined) linear operators from x into y while k(x,y ) designates the subspace of compact operators from x into y . if t ∈ c(x,y ), we write n(t) and r(t) for the null space and range of t, we set α(t)=dimn(t), β(t) = codimr(t). the classes of fredholm, upper semifredholm and lower semi-fredholm operators from x into y are, respectively, the following: φ(x,y ) := { t ∈c(x,y ) : α(t) < ∞ and β(t) < ∞ ,r(t) is closed iny } . φ+(x,y ) := { t ∈c(x,y ) : α(t) < ∞ and r(t) is closed in y } . φ−(x,y ) := { t ∈c(x,y ) : β(t) < ∞ and r(t) is closed in y } . the set of semi-fredholm operators from x into y is defined by φ±(x,y ) := φ+(x,y ) ∪ φ−(x,y ). the set of fredholm operators from x into y is defined by φ(x,y ) := φ+(x,y ) ∩ φ−(x,y ). for t ∈ φ±(x,y ), the number i(t) = α(t) −β(t) is called the index of t. definition 1.1. an operator f ∈l(x,y ) is called a fredholm perturbation if t +f ∈ φ(x,y ) whenever t ∈ φ(x,y ). f is called an upper (respectively, lower) fredholm perturbation if t +f ∈ φ+(x,y )( respectively, φ−(x,y ) ) whenever t ∈ φ+(x,y ) ( respectively, φ−(x,y ) ) . the sets of fredholm, upper semi-fredholm and lower semi-fredholm perturbations are denoted by f(x,y ), f+(x,y ) and f−(x,y ), respectively. let φb(x,y ), φb+(x,y ) and φ b −(x,y ) denote the set φ(x,y )∩l(x,y ), φ+(x,y )∩l(x,y ) and φ−(x,y ) ∩l(x,y ), respectively. definition 1.2. let a be a closable linear operator in a banach space x. the resolvent set and the spectrum of a are, respectively, defined as ρ(a) := { λ ∈ c, such that (λ−a) is injective and (λ−a)−1 ∈l(x) } , σ(a) := c\ρ(a). received 26th april, 2017; accepted 6th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 47a10. key words and phrases. essential spectra; convergence in the generalized sense; convergence to zero compactly. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 ammar, djaidja and jeribi definition 1.3. let a be a closed linear operator in a banach space x. we define the sets σe1(a) := { λ ∈ c, such that λ−a 6∈ φ+(x) } , σe2(a) := { λ ∈ c, such that λ−a 6∈ φ−(x) } , σe3(a) := { λ ∈ c, such that λ−a 6∈ φ−(x) ∪ φ+(x) } , σe4(a) := { λ ∈ c, such that λ−a 6∈ φ(x) } , σe5(a) := ⋂ k∈k(x) σ(t + k). σe1(.) and σe2(.) are the gustafson and weidman’s essential spectra. σe3(.) is the kato’s essential spectrum. σe4(.) is the wolf ’s essential spectrum, and σe5(.) is the schechter’s essential spectrum. proposition 1.1. [8, theorem 7.27, p.172] let t ∈c(x). then λ /∈ σe5(t) if, and only if, (λ−t) ∈ φ(x) and i(λ−t) = 0. definition 1.4. let x be a banach space and e,f be closed subspaces of x. let be be the unit sphere of e. let us define δ(e,f) : =   supx∈be dist(x,f), if e 6= {0},0, otherwise, and δ̂(e,f) := max { δ(e,f),δ(f,e) } . the quantity δ̂(e,f) is called the gap between the subspaces e and f . remark 1.1. (i) the gap measures the distance between two subspaces and it easily follows, from the definitions, (i1) δ(e,f) = δ(e,f) and δ̂(e,f) = δ̂(e,f). (i2) δ(e,f) = 0 if, and only if, e ⊂ f. (i3) δ̂(e,f) = 0 if, and only if, e = f. (ii) δ̂(·, ·) is a metric on the set v(x) of all linear closed subspaces of x and the convergence en → f in v(x) is obviously defined by δ̂(en,f) → 0. moreover, (v(x), δ̂) is a complete metric space. definition 1.5. (i) let x and y be two banach spaces, and let t, s be two closed linear operators acting from x to y . let us define δ ( g(t),g(s) ) = sup x ∈d(t) ‖x‖2 + ‖tx‖2 = 1 [ inf y∈d(s) ( ‖x−y‖2 + ‖tx−sy‖2 )1 2 ] . δ̂ ( t,s ) is called the gap between s and t . (ii) let t and s be two closable operators. we define the gap between t and s by δ(t,s) = δ(t,s) and δ̂(t,s) = δ̂(t,s). definition 1.6. a sequence (tn)n∈n of bounded linear operators mapping on x is said to converge to zero compactly if for all x ∈ x, tnx → 0 and (tnxn)n is relatively compact for every bounded sequence (xn)n ⊂ x. remark 1.2. clearly, tn converges to 0 implies that tn converges to zero compactly. definition 1.7. let (tn)n∈n be a sequence of closable linear operators from x into y and let t be a closable linear operator from x into y . (tn)n∈n is said to converge in the generalized sense to t if δ̂(tn,t) converges to 0 as, n →∞. 2. preliminaries theorem 2.1. [2, theorem 4] let an be a sequence of bounded linear operators converging to zero compactly and let t be a closed linear operator. if t is a semi-fredholm operator, there exists n0 ∈ n such that for all n ≥ n0, essential spectrum of linear operators 3 (i) (t + an) is semi-fredholm, (ii) α(t + an) < α(t), (iii) β(t + an) < β(t), and (iv) i(t + an) = i(t). proposition 2.1. [3, proposition 7.8.1 ]. let (tn)n∈n be a sequence of bounded linear operators and let t ∈l(x) such that tn −t converges to zero compactly. then, (i) if tn ∈fb(x), then t ∈fb(x), (ii) if tn ∈fb+(x), then t ∈fb+(x), and (iii) if tn ∈fb−(x), then t ∈fb−(x). theorem 2.2. [1, theorem 2.1] let t and s be two closed densely defined linear operators. then, we have: (i) δ(t,s) = δ(s∗,t∗) and δ̂(t,s) = δ̂(s∗,t∗). (ii) if s and t are one-to-one, then δ(s,t) = δ(s−1,t−1) and δ̂(s,t) = δ̂(s−1,t−1). (iii) let a ∈l(x,y ). then δ̂(a + s,a + t) ≤ 2(1 + ‖a‖2)δ̂(s,t). (iv) let t be fredholm operator (respectively semi-fredholm operator). if δ̂(t,s) < γ(t)(1+[γ(t)]2) −1 2 , then s is fredholm operator (respectively semi-fredholm operator ), α(s) ≤ α(t) and β(s) ≤ β(t). furthermore, there exists b > 0 such that δ̂(t,s) < b, which implies i(s) = i(t). (v) let t ∈l(x,y ). if s ∈c(x,y ) and δ̂(t,s) ≤ [ 1 +‖t‖2 ]−1 2 , then s is bounded operator (so that d(s) is closed). theorem 2.3. [1, theorem 2.3] let (tn)n∈n be a sequence of closable linear operators from x into y and let t be a closable linear operator from x into y . (i) the sequence tn converges in the generalized sense to t if, and only if, tn + s converges in the generalized sense to t + s, for all s ∈l(x,y ). (ii) let t ∈ l(x,y ). tn converges in the generalized sense to t if, and only if, tn ∈ l(x,y ) for sufficiently larger n and tn converges to t . (iii) let tn converges in the generalized sense to t . then, t −1 exists and t−1 ∈ l(y,x), if, and only if, t−1n exists and t −1 n ∈l(y,x) for sufficiently larger n and t−1n converges to t−1. 3. the main result in this section we investigate the essential spectra (σei, i = 1, . . . , 5) of the sequence of linear operators in a banach space x. theorem 3.1. let (tn)n∈n be a bounded linear operators mapping on x, and let t and b be two operators in l(x), λ0 ∈ c, and u ⊆ c is open . (a) if ((λ0 −tn−b)−(λ0 −t −b)) converges to zero compactly, and 0 ∈u, then there exists n0 ∈ n such that, for all n ≥ n0. σei(λ0 −tn −b) ⊆ σei(λ0 −t −b) + u. and, δ ( σei(λ0 −tn −b),σei(λ0 −t −b) ) = 0, i=1,. . . ,5 (b) if (λ0 −tn −b) converges to zero compactly then there exists n0 ∈ n such that for all n ≥ n0 σei((λ0 −t −b) + (λ0 −tn −b)) ⊆ σei(λ0 −t −b). and, δ ( σei((λ0 −t −b) + (λ0 −tn −b)),σei(λ0 −t −b) ) = 0, i = 1, . . . , 5. proof. (a) for i = 1. assume that the assertion fails. then by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈ σe1(λ0 −tn −b) such that λn 6∈ σe1(λ0 −t −b) + u. it is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe1(λ0 −t −b) + u. using 4 ammar, djaidja and jeribi the fact that 0 ∈u, hence we have λ 6∈ σe1(λ0 −t −b), and therefore, (λ− (λ0 −t −b)) ∈ φb+(x). let an = λn − λ + (λ0 −t −b) − (λ0 −tn −b). since an converges to zero compactly, writing λn−(λ0 −tn −b) = λ−(λ0 −t −b) +an and according to theorem 2.1, we infer that, there exists n0 ∈ n such that for all n ≥ n0 we have (λn − (λ0 −tn −b)) ∈ φ+(x) and i(λn − (λ0 −tn −b)) = i(λ− (λ0 −t −b) + an) = i(λ− (λ0 −t −b)). so, λn 6∈ σe1(λ0 −tn −b), which is a contradiction. then σe1(λ0 −tn −b) ⊆ σe1(λ0 −t −b) + u, for all n ≥ n0. since 0 ∈ u, we obtain σe1(λ0 −tn −b) ⊆ σe1(λ0 −t −b). hence by remark 1.1 (i2), we get δ ( σe1(λ0 −tn −b),σe1(λ0 −t −b) ) = 0, for all n ≥ n0. for i = 2, 3, 4, by using a similar proof as in (i = 1), by replacing σe1(.), and φ+(x) by σe2(.), σe3(.), σe4(.), and φ−(x), φ−(x) ∪ φ+(x), φ(x), respectively, we get if ((λ0 −tn −b) − (λ0 −t −b)) converges to zero compactly, and 0 ∈ u, then there exists n0 ∈ n such that, for all n ≥ n0. σei(λ0 −tn −b) ⊆ σei(λ0 −t −b) + u. and δ ( σei(λ0 −tn −b),σei(λ0 −t −b) ) = 0. for i = 5. assume that the assertion fails. then by passing to a subsequence, it may be deduced that, for each n, there exists λn ∈ σe5(λ0 −tn −b) such that λn 6∈ σe5(λ0 −t −b) + u. it is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe5(λ0 −t −b) + u. using the fact that 0 ∈ u, we have λ 6∈ σe5(λ0 −t −b) and therefore, λ − (λ0 −t −b) ∈ φb(x) and i(λ− (λ0 −t −b)) = 0. let an = λn −λ + (λ0 −t −b) − (λ0 −tn −b). since an converges to zero compactly, writing λn−(λ0 −tn −b) = λ−(λ0 −t −b)+an and according to theorem 2.1, we infer that, there exists n0 ∈ n such that for all n ≥ n0 we have λn−(λ0 −tn−b) ∈ φ(x) and i(λn− (λ0 −tn −b)) = i(λ− (λ0 −t −b) + an) = i(λ− (λ0 −t −b)) = 0. so, λn 6∈ σe5(λ0 −tn −b), which is a contradiction. then σe5(λ0 −tn −b) ⊆ σe5(λ0 −t −b) + u, for all n ≥ n0. since 0 ∈u, we have σe5(λ0 −tn −b) ⊆ σe5(λ0 −t −b). hence by remark 1.1 (i2), we have δ ( σe5(λ0 −tn −b),σe5(λ0 −t −b) ) = 0, for all n ≥ n0. (b) for i = 1. let λ 6∈ σe1(λ0 −t −b). then, (λ− (λ0 −t −b)) ∈ φb+(x). since (λ0 −tn −b) converges to zero compactly and applying [2, theorem 4] to the operators (λ0−t−b) and (λ0−tn−b), we prove that, there exists n0 ∈ n such that (λ−(λ0 −t −b)+(λ0 −tn −b)) ∈ φ+(x) for all n ≥ n0. hence λ 6∈ σe1((λ0 −t −b) + (λ0 −tn −b)). we conclude that σe1(λ0 −tn −b) ⊆ σe1(λ0 −t −b). now applying remark 1.1 (i2) we obtain δ ( σe1((λ0 −t −b) + (λ0 −tn −b)),σe1(λ0 −t −b) ) = 0, for all n ≥ n0. for i = 2, 3, 4, by using a similar proof as in (i = 1), by replacing σe1(.), and φ+(x) by σe2(.), σe3(.), σe4(.), and φ−(x), φ−(x) ∪ φ+(x), φ(x), respectively, we get if (λ0 −tn −b) converges to zero compactly then there exists n0 ∈ n such that for all n ≥ n0. σei((λ0 −t + b) + (λ0 −tn −b)) ⊆ σei(λ0 −t −b). and, δ ( σei((λ0 −t −b) + (λ0 −tn −b)),σei(λ0 −t −b) ) = 0, for all n ≥ n0. for i = 5. let λ 6∈ σe5(λ0 −t −b). then, (λ− (λ0 −t −b)) ∈ φb(x) and i(λ − (λ0 −t −b)) = 0. since (λ0 −tn −b) converges to zero compactly and by applying the [2, theorem 4] to the operators (λ0 −t −b) and (λ0 −tn −b), we prove that, there exists n0 ∈ n such that (λ− (λ0 −t −b) + (λ0 −tn −b)) ∈ φ(x) for all n ≥ n0. hence λ 6∈ σe5((λ0 −t −b) + (λ0 −tn −b)). we conclude that σe5(λ0 −tn −b) ⊆ σe5(λ0 −t −b) essential spectrum of linear operators 5 now applying remark 1.1 (i2) we have δ ( σe5((λ0 −t −b) + (λ0 −tn −b)),σe5(λ0 −t −b) ) = 0, for all n ≥ n0. � theorem 3.2. let (tn)n∈n be a sequence of closed linear operators mapping on banach spaces x and let t ∈ c(x),and let b and l be two operators in l(x), λ0 ∈ c such that tn converges in the generalized sense to t , and λ0 ∈ ρ(t + b), u ⊆ c is open. (a) if 0 ∈u, then there exists n0 ∈ n such that, for every n ≥ n0, we have σei(λ0 −tn −b) ⊆ σei(λ0 −t −b) + u. (3.1) and, δ ( σei(λ0 −tn −b),σei(λ0 −t −b) ) = 0, i = 1, . . . , 5. (b) there exist ε > 0 and n ∈ n such that, for all ‖l‖ < ε, we have σei(λ0 −tn −b + l) ⊆ σei(λ0 −t −b) + u, for all n ≥ n0. and, δ ( σei(λ0−tn−b +l),σei(λ0−t −b) ) = δ ( σei(λ0−t −b +l),σei(λ0−t −b) ) , i = 1, . . . , 5. ♦ proof. (a) for i = 1, since (b−λ0) be a bounded operator and λ0 ∈ ρ(t +b). according to theorem 2.3 (i) and (iii)the sequence (λ0 −tn −b) converges in the generalized sense to (λ0 −t −b), and λ0 ∈ ρ(tn + b) for a sufficiently large n and (λ0 −tn −b) −1 converges to (λ0 −t −b) −1 . now to prove such that the inclusion (3.1)holds it suffices to prove there exist n0 ∈ n, such that for all n ≥ n0, we have σe1(λ0 −tn −b)−1 ⊆ σe1(λ0 −t −b)−1 + u. (3.2) in first step by an indirect proof, we suppose that the (3.2) does not hold, and for each n ∈ n there exists λn ∈ σe1(λ0 −tn −b)−1 such that λn 6∈ σe1(λ0 −t −b)−1 + u. it is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe1(λ0 −t −b)−1 + u. using the fact that 0 ∈u hence we have λ 6∈ σe1(λ0 −t −b)−1. therefore (λ−(λ0 −t −b)−1) ∈ φb+(x) and applying theorem 2.3 (ii), we conclude that δ̂(λn − (λ0 −tn −b)−1,λ− (λ0 −t −b)−1) → 0, as n →∞. let γ(λ − (λ0 −t −b)−1) = δ > 0. then there exists n ∈ n such that, for all n ≥ n we have δ̂(λn − (λ0 −tn −b)−1,λ − (λ0 −t −b)−1) ≤ δ√1+δ2 . according theorem 2.2 (iv) we infer (λn − (λ0 −tn −b)−1) ∈ φb+(x). then we obtain λn /∈ σe1((λ0 −tn −b)−1), which this is a contradicts our assumption. hence (3.2) holds. now, if λ ∈ σe1(λ0 −tn −b) then 1λ ∈ σe1((λ0 −tn −b) −1). according then (3.1) we conclude that 1 λ ∈ σe1((λ0 −t −b)−1) + u. (3.3) since 0 ∈u, then (3.3) implies that 1 λ ∈ σe1((λ0 −t −b)−1). we have to prove λ ∈ σe1(λ0 −t −b) + u. (3.4) we will proceed by contradiction, we suppose that λ 6∈ σe1(λ0 −t −b) + u. the fact that 0 ∈ u implies that λ 6∈ σe1(λ0 −t −b) and so, 1λ 6∈ σe1((λ0 −t −b) −1) which this is a contradicts our assumption. so λ ∈ σe1(λ0 −t −b) +u. therefore (3.1) holds. since u is an arbitrary neighborhood of 0 and by using the relation (3.1) we have σe1(λ0 −tn −b) ⊆ σe1(t + b −λ0), for all n ≥ n0. hence by remark 1.1 (i2) δ ( σe1(λ0 −tn −b),σe1(λ0 −t −b) ) = δ ( σe1(λ0 −tn −b),σe1(λ0 −t + b) ) = 0 for all n ≥ n0.this ends the proof (i=1). for i = 2, 3, 4, by using a similar proof as in ((a) for i = 1), by replacing σe1(.), and φ+(x) by σe2(.), σe3(.), σe4(.), and φ−(x), φ−(x) ∪ φ+(x), φ(x), respectively, we get σei(λ0 −tn −b) ⊆ σei(λ0 −t −b) + u. and, δ ( σei(λ0 −tn −b),σei(λ0 −t −b) ) = 0, for all n ≥ n0. 6 ammar, djaidja and jeribi for i = 5, since (λ0 − b) be a bounded operator and λ0 ∈ ρ(t + b), according to theorem 2.3 (i) and (iii) the sequence (λ0 − tn − b) converges in the generalized sense to (λ0 −t −b), and λ0 ∈ ρ(tn + b) for a sufficiently large n and (λ0 −tn −b) −1 converges to (λ0 −t −b) −1 . now to prove that (3.1)holds it suffices to prove there exist n0 ∈ n, such that for all n ≥ n0, we have σe5(λ0 −tn −b)−1 ⊆ σe5(λ0 −t −b)−1 + u. (3.5) in first step by an indirect proof, we suppose that the inclusion (3.5) does not hold, and for each n ∈ n there exists λn ∈ σe5(λ0 −tn −b)−1 such that λn 6∈ σe5(λ0 −t −b)−1 + u. it is clear that lim n→+∞ λn = λ since (λn) is bounded, this implies that λ 6∈ σe5(λ0 −t −b)−1 + u. using the fact that 0 ∈ u, hence we have λ 6∈ σe5(λ0 −t −b)−1. therefore (λ − (λ0 −t −b)−1) ∈ φb(x) and i(λ− (λ0 −t −b)−1)=0, and applying theorem 2.3 (ii), we conclude that δ̂(λn − (λ0 −tn −b)−1,λ− (λ0 −t −b)−1) → 0 as n →∞. let γ(λ− (λ0 −t −b)−1) = δ > 0. then there exists n ∈ n such that, for all n ≥ n we have δ̂(λn − (λ0 −tn −b)−1,λ− (λ0 −t −b)−1) ≤ δ √ 1 + δ2 . according to theorem 2.2 (iv) we infer (λn − (λ0 −tn −b)−1) ∈ φb(x) and i(λn − (λ0 −tn −b)−1) = i(λ− (λ0 −t −b)−1)=0. then we obtain λn /∈ σe5((λ0 −tn −b)−1), which this is a contradicts our assumption. hence (3.1) holds. now, if λ ∈ σe5(λ0 − tn − b) then 1 λ ∈ σe5((λ0 −tn −b)−1). according then (3.1) we conclude that 1 λ ∈ σe5((λ0 −t −b)−1) + u. (3.6) since 0 ∈u, then (3.6) implies that 1 λ ∈ σe5(λ0 −t −b)−1. we have to prove λ ∈ σe5(λ0 −t −b) + u. (3.7) we will proceed by contradiction , we suppose that λ 6∈ σe5(λ0 − t − b) + u. the fact that 0 ∈ u implies that λ 6∈ σe5(λ0 − t − b) and so, 1λ 6∈ σe5((λ0 − t − b) −1) which this is a contradicts our assumption. so λ ∈ σe5(λ0 −t −b) + u. therefore (3.1)holds. since u is an arbitrary neighborhood of 0 and by using (3.1) we have σe5(λ0 −tn −b) ⊆ σe5(λ0 −t −b) for all n ≥ n0. hence by remark 1.1 (i2) δ ( σe5(λ0 −tn −b),σe5(λ0 −t −b) ) = δ ( σe5(λ0 −tn −b),σe5(λ0 −t −b) ) = 0 for all n ≥ n0.this ends the proof of, (a) . (b) for i = 1, since λ0 ∈ ρ(t +b), then (t +b−λ0)−1 exists and bounded. we put 1‖(λ0−t−b)−1‖ = ε1. let l ∈l(x) such that ‖l‖ < ε1 this implies ‖l (λ0 −t −b) −1 ‖ < 1. by according theorem 2.3 (i) the squence (λ0−tn−b +l) converges in the generalized sense to (λ0− t−b+l), and the neumann series ∑∞ k=0(−l (λ0 −t −b) −1 )k converges to (i+l (λ0 −t −b) −1 )−1 and ‖(i + l (λ0 −t −b) −1 )−1‖ < 1 1 −‖l‖‖(λ0 −t −b) −1 ‖ since (λ0 − t − b + l)−1 = (λ0 −t −b) −1 )(i + l (λ0 −t −b) −1 ))−1, then λ0 ∈ ρ(t + b + l). now applying ((a)for i = 1), we deduce that there exists n0 ∈ n such that σe1(λ0 −tn −b + l) ⊆ σe1(λ0 −t −b + l) +u, for all n ≥ n0. let λ 6∈ σe1(λ0−t −b). then (λ−(λ0−t −b)) ∈ φ+(x). by applying [8, theorem 7.9] there exists ε2 > 0 such that for ‖l‖ < ε2, one has (λ−(λ0−t−b)−l) ∈ φ+(x) and, this implies that λ 6∈ σe1(λ0 − t − b + l). from what has been mentioned and if we take ε = min(ε1,ε2) then for all ‖l‖ < ε, there exists n0 ∈ n such that σe1(λ0 −tn −b + l) ⊆ σe1(λ0 −t −b) + u, for all n ≥ n0. since 0 ∈u then we have δ ( σe1(λ0 −tn −b + l),σe1(λ0 −t −b) ) = 0 and δ ( σe1(λ0 −t −b + l),σe1(λ0 −t −b) ) = 0. essential spectrum of linear operators 7 therefore, (i = 1) holds. for i = 2, 3, 4. by using a similar proof as in ((b)for i = 1), by replacing σe1(.), and φ+(x) by σe2(.), σe3(.), σe4(.), and φ−(x), φ−(x) ∪ φ+(x), φ(x), respectively, we get there exist ε > 0 and n ∈ n such that, for all ‖l‖ < ε, we have σei(λ0 −tn −b + l) ⊆ σei(λ0 −t −b) + u, for all n ≥ n0. and, δ ( σei(λ0 −tn −b + l),σei(λ0 −t −b) ) = δ ( σei(λ0 −t −b + l),σei(λ0 −t −b) ) . for i = 5, since λ0 ∈ ρ(t + b), then (λ0 −t −b)−1 exists and bounded. we put 1‖(λ0−t−b)−1‖ = ε1. let l ∈l(x) such that ‖l‖ < ε1 this implies ‖l (λ0 −t −b) −1 ‖ < 1. by according theorem 2.3 (i) we have (λ0−tn−b +l) converges in the generalized sense to (λ0−t − b + l) , and the neumann series ∑∞ k=0(−l (λ0 −t −b) −1 )k converges to (i + l (λ0 −t −b) −1 )−1 and ‖(i + l (λ0 −t −b) −1 )−1‖ < 1 1 −‖l‖‖(λ0 −t −b) −1 ‖ . since (λ0 − t − b + l)−1 = (λ0 −t −b) −1 )(i + l (λ0 −t −b) −1 ))−1, then λ0 ∈ ρ(t + b + l). now applying ((a)for i = 5), we deduce that there exists n0 ∈ n such that σe5(λ0 −tn −b + l) ⊆ σe5(λ0−t −b +l) +u, for all n ≥ n0. let λ 6∈ σe5(λ0−t −b). then (λ−(λ0−t −b)) ∈ φ(x). by applying [8, theorem 7.9] there exists ε2 > 0 such that for ‖l‖ < ε2, one has (λ−(λ0 −t −b)−l) ∈ φ(x) and i(λ−(λ0−t −b−l)) = i(λ−(λ0−t −b) = 0. this implies that λ 6∈ σe5(λ0−t −b +l). from what has been mentioned and if we take ε = min(ε1,ε2) then for all ‖l‖ < ε, there exists n0 ∈ n such that σe5(λ0 −tn −b + l) ⊆ σe5(λ0 −t −b) + u, for all n ≥ n0. since 0 ∈u then we have δ ( σe5(λ0 −tn −b + l),σe5(λ0 −t −b) ) = 0 = δ ( σe5(λ0 −t −b + l),σe5(λ0 −t −b) ) . therefore, (i = 5) holds. � references [1] a. ammar and a. jeribi, the weyl essential spectrum of a sequence of linear operators in banach spaces, indag. math., new ser. 27 (1) (2016), 282-295. [2] s. goldberg, perturbations of semi-fredholm operators by operators converging to zero compactly, proc. amer. math. soc. 45 (1974), 93-98 . [3] a. jeribi, spectral theory and applications of linear operators and block operator matrices. springer-verlag, new york, 2015. [4] a. jeribi and n. moalla, a characterisation of some subsets of schechter’s essential spectrum and applications to singular transport equation, j. math. anal. appl. 358 (2) (2009), 434-444. [5] a.jeribi, une nouvelle caractrisation du spectre essentiel et application, comp rend.acad.paris serie i, 331 (2000),525-530. [6] t. kato, perturbation theory for linear operators, second edition. grundlehren der mathematischen wissenschaften, band 132. springer-verlag, berlin-new york, 1966. [7] k. latrach, a. jeribi, some results on fredholm operators, essential spectra, and application, j. math. anal. appl. 225 (1998), 461-485. [8] m. schechter, principles of functional analysis, grad. stud. math. vol. 36, amer. math. soc., providence, 2002. [9] m. schechter, riesz operators and fredholm perturbations, bull. amer. math. soc. 74 (1968), 1139-1144. 1department of mathematics, university of sfax, faculty of sciences of sfax, tunisia 2department of mathematics, university of mohamed boudiaf, m’sila, algeria ∗corresponding author: ammar aymen84@yahoo.fr 1. introduction 2. preliminaries 3. the main result references international journal of analysis and applications volume 16, number 3 (2018), 445-453 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-445 qk-type spaces of quaternion-valued functions m.a. bakhit∗ department of mathematics, faculty of science, jazan university, jazan, saudi arabia. ∗corresponding author: mabakhit@jazu.edu.sa abstract. in this paper we develop the necessary tools to generalize the qk -type function classes to the case of the monogenic functions defined in the unit ball of r3, some important basic properties of these functions are also considered. further, we show some relations between qk (p,q) and α-bloch spaces of quaternion-valued functions. 1. introduction 1.1. analytic function spaces. the so called qk-type spaces of analytic functions on d = {z ∈ c : |z| < 1} the unit open complex disk, were introduced by wulan and zhou in [12]. for k : [0,∞) → [0,∞) is a non-decreasing and non-negative function, and 0 < p < ∞,−2 < q < ∞, an analytic function f in d belongs to the qk(p,q) if sup a∈d ∫ d |f′(z)|p(1 −|z|2)qk(1 −|ϕa(z)|2)dxdy < ∞. moreover, if lim |a|→1 ∫ d |f′(z)|p(1 −|z|2)qk(1 −|ϕa(z)|2)dxdy = 0, then f ∈ qk,0(p,q), where ϕa(z) = (a− z)/(1 − āz) is the automorphism of the unit disk d that changes 0 and a. the qk(p,q) class is banach under the norm ‖f‖ = ‖f‖qk(p,q) + |f(0)|, when p ≥ 1. if k(t) = ts, 0 ≤ s < ∞, then qk(p,q) = f(p,q,s) see [3]. for more results of qk(p,q) classes see [3] and [7]. received 2017-10-27; accepted 2018-01-08; published 2018-05-02. 2010 mathematics subject classification. 46e15, 30g35. key words and phrases. clifford analysis; bloch-type classes of quaternion-valued functions; qk -type spaces. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 445 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-445 int. j. anal. appl. 16 (3) (2018) 446 1.2. quaternion function spaces. we will work throughout this paper in the field h (the skew field of quaternion-valued functions), i.e. each element a ∈ h with basis 1,e1,e2,e3, can be given in the form a := a0 + a1e1 + a2e2 + a3e3, ak ∈ r,k = 0, 1, 2, 3 the multiplication rules of these elements are given by e21 = e 2 2 = e 2 3 = −1, e1e2 = −e2e1 = e3, e2e3 = −e3e2 = e1, e3e1 = −e1e3 = e2. the conjugation element ā of an element a is ā = a0 −a1e1 −a2e2 −a3e3, with the property aā = āa = |a|2 = a20 + a 2 1 + a 2 2 + a 2 3. if a ∈ h \{0}, then a−1 := ā/|a|2 and |ab| = |a||b| for each a,b ∈ h. let x = (x0,x1,x2) ∈ r3 of the form x = x0 + x1e1 + x2e2 be a quaternion point. given ω ⊂ r3 a domain and let f : ω −→ h the quaternion-valued functions defined in ω. for p ∈ n∪{0}, thus the notation cp(ω; h) has the usual componentwise meaning. we consider d and d the generalization of a cauchy-riemann operator and it’s conjugate, respectively, and they are defined on c1(ω; h) by df = ∂f ∂x0 + e1 ∂f ∂x1 + e2 ∂f ∂x2 , df = ∂f ∂x0 −e1 ∂f ∂x1 −e2 ∂f ∂x2 . the equation df = 0 has the solutions for all x ∈ ω, are called left-hyperholomorphic functions and they are generalized of the analytic function classes from the functions in one complex variable theory. for more details about monogenic function classes and general clifford analysis, we refer to [2, 6, 11] and others. let b be the unit ball in ⊂ r3, with boundary s = ∂b. the class m(b) consists of all monogenic functions on b. for r > 0 and a ∈ r3, let b(a,r) denote by the ball with center a and radius r. also, for a ∈ b and 0 < r < 1, an euclidean ball u(a,r) = {x : |ϕa(x)| < r}, with center and radius, respectively, (1−r2)a 1−r2|a|2 and (1−|a|2)r 1−r2|a|2 , is called the pseudo-hyperbolic ball. where ϕa(x) : b → b is defined by ϕa(x) = (a−x)/(1− āx), for a ∈ b. let α > 0, the quaternion α-bloch space bα (see [4, 9]) defined by : bα = {f ∈m(b) : ‖f‖bα = sup x∈b |df(x)|(1 −|x|2)α < ∞}. int. j. anal. appl. 16 (3) (2018) 447 if α = 3 2 , we have the standard quaternion bloch space b. the space bα0 is called the quaternion little α-bloch, which consists of all f ∈bα such that lim |x|→1− |df(x)|(1 −|x|2)α = 0. for f ∈m(b), the weighted quaternion dirichlet space dp,q, (0 < p < ∞,−2 < q < ∞), is given by: dp,q = { f ∈m(b) : ‖f‖dp,q = ∫ b |df(x)|p(1 −|x|2)qdx < ∞ } . if q = 0, we have the space d2,0 (the quaternion dirichlet space d). through this work, we let k(t), 0 < t < ∞, be a non-decreasing and non-negative (righ-continuous) function, which is not equal to 0 identically. for 0 < p < ∞,−2 < q < ∞ and f ∈ m(b), define jk,p,qf : b → [0,∞) by j p,q k f(a) = ∫ b |df(x)|p(1 −|x|2)qk(1 −|ϕa(x)|2)dx, a ∈ b. the set qk(p,q) given by qk(p,q) := { f ∈m(b) : ‖f‖qk(p,q) = sup a∈b j p,q k f(a) < ∞ } , and the little quaternion qk,0(p,q) is defined by qk,0(p,q) := { f ∈m(b) : ‖f‖qk,0(p,q) = lim|a|→1− j p,q k f(a) = 0 } . remark 1.1. if we put s < 3 and k(t) = ts, then qk(p,q) = f(p,q,s) (see [8]). if p = 2,q = 0, then qk(2, 0) = qk (see [1]). also, if k(t) = 1, then qk(p,q) = dp,q, the quaternion dirichlit space. for 0 < p < ∞,−1 < q < ∞, define the dk(p,q) quaternion dirichlet-type space as the set of f ∈m(b) satisfying j p,q k f(0) < ∞. from the definition of qk(p,q) spaces the following lemma become immediate with a = 0. lemma 1.1. let 0 ≤ p < ∞,−1 < q < ∞, then qk(p,q) ⊂dk(p,q). from now, we assume that 1∫ 0 (1 −ρ2)qk ( 1 −ρ2 ) ρ2dρ < ∞. (1.1) otherwise, qk(p,q) contains only constant functions. fact 1 let 0 ≤ p < ∞,−1 < q < ∞, and let f ∈ m(b) be a non-constant function. if ( 1.1) does not hold, then int. j. anal. appl. 16 (3) (2018) 448 f /∈qk(p,q). proof. let f ∈ qk(p,q) be a non constant function. then, there is x0 ∈ b and 0 < r < 1 such that |df(x)| > 0 for each x ∈ b(x0,r). thus by lemma 1.1 and subharmonicity of |df|p where a(x0,r) = b\b(0, |x0|−r), we obtain ∞ > jp,qk f(0) ≥ ∫ a(x0,r) |df(x)|p(1 −|x|2)qk ( 1 −|x|2 ) dx ≥ 1∫ |x0| (1 −ρ2)qk ( 1 −ρ2 ) ρ2 ∫ s |df(ρζ)|pdσ(ζ)dρ ≥ ∫ s |df(|x0|ζ)|pdσ(ζ) 1∫ |x0| (1 −ρ2)qk ( 1 −ρ2 ) ρ2dρ = ∞, where dσ denotes the normalized surface element in s. this is a contradiction; therefor f is constant and the fact is proved. in this work, we introduce a classes of h-valued functions on r3. these classes are so called qk(p,q) spaces of monogenic function. we will study these classes and their relations to the quaternion α-bloch space. we shall prove some basic properties concerning qk(p,q) and qk,0(p,q) spaces in hyperholomorphic functions. our results in this work are extensions of our results in [1] and the results due to essén and wulan (see [3]) in hyperholomorphic functions case. for simplicity we restricted us tor3 the lowest noncommutative case and quaternion-valued functions. next, the hyperholomorphic function spaces were the aim of many works as [1, 4, 8] and [9]. in particular, we will need the following results for quaternion sense in the sequel: lemma 1.2. [5] let 1 ≤ p < ∞,f ∈m(b) and let 0 < r < 1. then, we have |df(0)|p ≤ 3 4πr2 ∫ u(a,r) |df(x)|pdx, for all a ∈ b. (1.2) lemma 1.3. [9] let 1 < p < ∞,f ∈m(b) and let 0 < r < 1. then, for every a ∈ b, we have |df(a)|p ≤ c4p r3(1 −r2)2p(1 −|a|2)3 ∫ u(a,r) |df(x)|pdx, for all a ∈ b, (1.3) where c = 48 π . int. j. anal. appl. 16 (3) (2018) 449 remark 1.2. the problem in quaternion sense is that, df(x) is monogenic, but df(ϕa(w)) is not monogenic. from [10] we know that 1−w̄a|1−āw|3 df(ϕa(w)) is hyperholomorphic. so, by the jacobian determinant( 1−|a|2 |1−āw|2 )3 , which has no singularities we can solve this problem. lemma 1.4. [8] let f ∈m(b),fa = f ◦ϕa and let ψfa : b → h given by ψfa(x) = 1 −xa |1 −ax|3 df(ϕa(x)). (1.4) then ψfa ∈m(b) and |ψfa| is a subharmonic function. 2. characterizations of qk(p,q) classes in this part, we prove some essential properties of quaternion qk(p,q) spaces as basic scale properties. proposition 2.1. let k satisfy (1.1) and let f ∈m(b), 1 ≤ p < ∞, and −2 < q < ∞. then, we have (1 −|a|2)q+3|df(a)|p ≤ 4q−p+3 c(r) j p,q k f(a), for 0 < r < 1. proof. since 0 < r < 1, by lemma 1.4 after the change of variable x = ϕa(w) then we deduce that j p,q k f(a) ≥ ∫ u(a,r) |df(x)|p(1 −|x|2)qk ( 1 −|ϕa(w)|2 ) dx ≥ (1 −|a|2)q+3 4q−p+3 ∫ b(a,r|1−a|) |ψfa(w)| p(1 −|w|2)qk ( 1 −|w|2 ) dw ≥ 1 4q−p+3 (1 −|a|2)q+3|df(a)|p ∫ r 0 (1 −ρ2)qk ( 1 −ρ2 ) ρ2dρ ≥ c(r) 4q−p+3 (1 −|a|2)q+3|df(a)|p, with ψfa(w) = 1−w̄a |1−āw|3 df(ϕa(w)). which implies that (1 −|a|2)q+3|df(a)|p ≤ 4q−p+3 c(r) j p,q k f(a). theorem 2.1. suppose that k satisfy (1.1), 1 ≤ p < ∞,−2 < q < ∞ and f ∈ m(b). when fn → f, assume that dfn → df uniformly on compact set m ⊂ b, as n → ∞. then, the space qk(p,q) under the norm ‖f‖k = |df(0)| + ‖f‖qk(p,q) is a banach space. proof. since 1 ≤ p < ∞, t is easy to prove that ‖.‖k is a norm. to show the completeness of (‖.‖k,qk(p,q)), fix 0 < r < 1. applying proposition 2.1, we obtain (1 −|a|2)q+3|df(a)|p ≤ 4q−p+3 c(r) j p,q k f(a), which gives ‖f‖ b q+3 p ≤ 4q−p+3 c(r) ‖f‖qk(p,q). int. j. anal. appl. 16 (3) (2018) 450 by the fact that (1 −|x|2)3 ≈ |u(a,r)|, from lemma 1.2 and lemma 1.3, we get |df(a) −df(0)| ≤ ( 4q−p+3 c(r) − ( 3 4πr3 )1 p )(∫ u(a,r) |df(x)|pdx )1 p ≤ c(r,p,q)‖f‖ b q+3 p (∫ u(a,r) 1 (1 −|x|2)q+3 dx )1 p ≤ c(r,p,q)‖f‖ b q+3 p 1 |u(a,r)| q+3 3p (∫ u(a,r) dx )1 p ≤ c1(r,p,q)‖f‖ b q+3 p , where a positive constant c1(r,p,q) is depending on r,p and q, which implies that |df(a)| ≤ |df(0)| + c1(r,p,q)‖f‖ b q+3 p . so, fore each compact set m ⊂ b, there is a constant c ∈ m such that |df(a)| ≤ |df(0)| + c1(r,p,q)‖f‖qk(p,q) ≤ c‖f‖k, (2.1) where the constant c = max { 1,c(m,c1(r,p,q), q+3 p ) } . now, we let {fn} be a cauchy sequence in qk(p,q) spaces. from (2.1) we deduce that {fn} is also a cauchy sequence in the topology of uniform convergence on compact sets. thus thereis a function f ∈ m(b) such that fn → f also, dfn → df uniformly on compact subsets of b, as n →∞. to show that ‖fn −f‖k → 0 as n → ∞, we give ε > 0. since, {fn} is a cauchy sequence, there is an n > 0 such that ‖fk −fn‖k < ε2 and |dfn(0) −df(0)| < ε2 for all n,k ∈≥ n. for each a ∈ b and n ≥ n, by applying fatou’s lemma, we obtain∫ b |df(x) −dfn(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx = ∫ b lim k→∞ |dfk(x) −dfn(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx ≤ lim k→∞ ∫ b |dfk(x) −dfn(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx = lim k→∞ ‖fk −fn‖ p qk(p,q) < ( ε 2 )p . thus, for all n ≥ n, ‖fn −f‖k = |dfn(0) −df(0)| + ( sup a∈b ∫ b |df(x) −dfn(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx )1 p ≤ ε, which implies that fn → f in qk(p,q). hence, the norm ‖.‖k is complete, therefore qk(p,q) spaces is a banach space in clifford setting. int. j. anal. appl. 16 (3) (2018) 451 3. the quaternion bloch and qk(p,q) spaces in this part of the paper, we consider the relations between qk(p,q) and α-bloch spaces in quaternion sense. we characterize the quaternion α-bloch spaces by the help of integral norms of quaternion qk(p,q) spaces. our results extend the results due to wulan and zhou [12] in quaternion sense. theorem 3.1. let f ∈m(b) and let 1 ≤ p < ∞,−2 < q < ∞. then (i): qk(p,q) ⊂b q+3 p , (ii): qk(p,q) = b q+3 p ; if ∫ 1 0 (1 −ρ2)−3k(1 −ρ2)ρ2dρ < ∞. (3.1) proof. (i) let 0 < r < 1 be fixed and a ∈ b. from proposition 2.1, we acquire (1 −|a|2)q+3|df(a)|p ≤ 4q−p+3 c(r) j p,q k f(a). if f ∈qk(p,q), then by estimate above we have f ∈b q+3 p . (ii) let f ∈b q+3 p be non constant. then, there is m > 0 constant such that (1 −|a|2) q+3 p |df(a)| ≤ m, for all x ∈ b. now we change the variable x = ϕa(w), then we acquire j p,q k f(a) ≤ ∫ b mp(1 −|x|2)−3k ( 1 −|ϕa(x)|2 ) dx ≤ mp ∫ b (1 −|ϕa(w)|2)−3k(1 −|w|2) (1 −|a|2)3 |1 − āw|6 dw ≤ mp ∫ b (1 −|w|2)−3k(1 −|w|2)dw ≤ mp ∫ 1 0 (1 −ρ2)−3k(1 −ρ2)ρ2dρ < ∞. thus, f ∈qk(p,q). this show that b q+3 p ⊂qk(p,q). combining theorem 3.1, we deduce the following corollary: corollary 3.1. let 1 ≤ p < ∞,−2 < q < ∞ and let f ∈m(b). then (i): qk,0(p,q) ⊂b q+3 p 0 , (ii): qk,0(p,q) = b q+3 p 0 ; if (3.1) holds. int. j. anal. appl. 16 (3) (2018) 452 proposition 3.1. let 1 ≤ p < ∞,−2 < q < ∞ and let f ∈m(b). then f ∈b q+3 p if and only if there is an 0 < r < 1, such that sup a∈b ∫ u(a,r) |df(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx < ∞. (3.2) proof. if f ∈b q+3 p , and a ∈ b. then for any 0 < r < 1, we deduce ∫ u(a,r) |df(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx ≤ ∫ b(0,r) |ψfa(w)| p(1 −|ϕa(w)|2)qk(1 −|w|2) (1 −|a|2)3 |1 − āw|6−2p dw ≤ ‖f‖ b q+3 p ∫ b(0,r) (1 −|w|2)−3k ( 1 −|w|2 ) dw ≤ c‖f‖ b q+3 p . conversely, let (3.2) holds then, we deduce ∫ u(a,r) |df(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx ≥ k ( 1 −r2 )∫ b(0,r) |df(x)|p(1 −|x|2)qdx ≥ c(r)k ( 1 −r2 ) 4q−p+3 (1 −|a|2)q+3|df(a)|p, which shows that f ∈b q+3 p . corollary 3.2. let k : (0,∞) → [0,∞) and f ∈m(b). then f ∈b q+3 p 0 if and only if there is an 0 < r < 1, such that lim |a|→1 ∫ u(a,r) |df(x)|p(1 −|x|2)qk ( 1 −|ϕa(x)|2 ) dx = 0. conclusion. our results in this work will be of important uses in the study of operator theory at the interface of monogenic function spaces. this work is a try to synthesize the achievements in the properties of monogenic qk(p,q) function spaces. the problem in quaternion sense is that, df(x) is monogenic, but df(φ(x)) is not monogenic, where φ : b → b is a monogenic function. the following question is open problem: what properties of operators act between this classes of monogenic functions, like f(p,q,s) and qk(p,q) classes? in quaternion case, several authors have studied function spaces and classes like qp,qk classes and f(p,q,s) spaces, see [1, 3, 8] and others. int. j. anal. appl. 16 (3) (2018) 453 references [1] m.a. bakhit, qk classes in clifford analysis, turk. j. anal. number theory, 4(3) (2016), 82-86. [2] f. brackx, r. delanghe and f. sommen, clifford analysis, pitman research notes in math. boston, london, melbourne, 1982. [3] m. essen and h. wulan, on analytic and meromorphic functions and spaces of qk -type, illinois j. math. 46 (2002), 1233–1258. [4] k. gürlebeck, u. kähler, m. shapiro, and l.m. tovar, on qp spaces of quaternion-valued functions, complex variables theory appl. 39 (1999), 115–135. [5] k. gürlebeck and h.r. malonek, on strict inclusions of weighted dirichlet spaces of monogenic functions, bull. austral. math. soc. 64 (2001), 33–50. [6] k. gürlebeck and w. sprössig, quaternion and clifford calculus for engineers and physicists, john wiley &. sons, chichester, 1997. [7] x. meng, some sufficient conditions for analytic functions to belong to qk,0(p,q) space, abstr appl. anal. 2008 (2008), article id 404636. [8] a.g. miss, l.f. resndis, l.m. tovar, quaternion f(p,q,s) function spaces, complex anal. oper. theory 9 (2015), 999– 1024. [9] l.f. reséndis and l.m. tovar, besov-type characterizations for quaternion bloch functions, in: le hung son et al (eds) finite or infinite complex analysis and its applications, adv. complex analysis and applications, boston ma: kluwer academic publishers (2004), 207–220. [10] j. ryan, conformally covariant operators in clifford analysis, z. anal. anwend. 4(4) (1995), 677–704. [11] a. sudbery, quaternion analysis, math. proc. cambridge philos. soc. 85 (1979), 199–225. [12] h. wulan and j. zhou, qk type spaces of analytic functions, j. funct. spaces appl. 4(1) (2006), 37–84. 1. introduction 1.1. ù’analytic function spaces 1.2. quaternion function spaces 2. characterizations of qk(p,q) classes 3. the quaternion bloch and qk(p,q) spaces references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 40-42 http://www.etamaths.com a new result on generalized absolute cesàro summability hüseyin bor1,∗ and ram n. mohapatra2 abstract. in [4], a main theorem dealing with an application of almost increasing sequences, has been proved. in this paper, we have extended that theorem by using a general class of quasi power increasing sequences, which is a wider class of sequences, instead of an almost increasing sequence. this theorem also includes some new and known results. 1. introduction a positive sequence (bn) is said to be an almost increasing sequence if there exists a positive increasing sequence (cn) and two positive constants m and n such that mcn ≤ bn ≤ ncn (see [1]). a sequence (dn) is said to be δ-quasi monotone, if dn → 0, dn > 0 ultimately, and ∆dn ≥ −δn, where ∆dn = dn −dn+1 and δ= (δn) is a sequence of positive numbers (see [2]). a positive sequence x = (xn) is said to be a quasi-f-power increasing sequence if there exists a constant k = k(x,f) ≥ 1 such that kfnxn ≥ fmxm for all n ≥ m ≥ 1, where f = {fn(σ,γ)} = {nσ(log n)γ, γ ≥ 0, 0 < σ < 1}(see [11]). if we take γ=0, then we get a quasi-σpower increasing sequence. every almost increasing sequence is a quasi-σ-power increasing sequence for any non-negative σ, but the converse is not true for σ > 0 (see [9]). let ∑ an be a given infinite series. we denote by tα,βn the nth cesàro mean of order (α,β), with α + β > −1, of the sequence (nan), that is (see [6]) (1) t α,β n = 1 a α+β n n∑ v=1 a α−1 n−va β vvav, where (2) a α+β n = o(n α+β ), a α+β 0 = 1 and a α+β −n = 0 for n > 0. let (θα,βn ) be a sequence defined by (see [3]) θ α,β n = { ∣∣tα,βn ∣∣ , α = 1,β > −1 max1≤v≤n ∣∣tα,βv ∣∣ , 0 < α < 1,β > −1.(3) the series ∑ an is said to be summable | c,α,β |k, k ≥ 1, if (see [7]) (4) ∞∑ n=1 1 n | tα,βn | k < ∞. if we take β = 0, then | c,α,β | k summability reduces to | c,α | k summability (see [8]). the first author has proved the following main theorem. theorem a ([4]). let (θα,βn ) be a sequence defined as in (3). let (xn) be an almost increasing sequence such that | ∆xn |= o(xn/n) and let λn → 0 as n → ∞. suppose that there exists a sequence of numbers (an) such that it is δ-quasi-monotone with ∑ nδnxn < ∞, ∑ anxn is convergent, and | ∆λn |≤ | an | for all n. if the condition m∑ n=1 (θα,βn ) k n = o(xm) as m →∞(5) satisfies, then the series ∑ anλn is summable | c,α,β |k, 0 < α ≤ 1, α + β > 0, and k ≥ 1. 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g05, 40g99. key words and phrases. cesàro mean; power increasing sequence; quasi-monotone sequence; summability factors; infinite series; hölder inequality; minkowski inequality. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 40 a new result on generalized absolute cesàro summability 41 2. the main result. the aim of this paper is to extent theorem a by using a quasi-f-power increasing sequence, which is a general class of quasi power increasing sequences, instead of an almost increasing sequence. we shall prove the following theorem. theorem. let (θα,βn ) be a sequence defined as in (3). let (xn) be a quasi-f-power increasing sequence and let λn → 0 as n →∞. suppose that there exists a sequence of numbers (an) such that it is δ-quasi-monotone with ∆an ≤ δn, ∑ nδnxn < ∞, ∑ anxn is convergent, and | ∆λn |≤ | an | for all n. if the condition (5) is satisfied, then the series ∑ anλn is summable | c,α,β |k, 0 < α ≤ 1, α + β > 0, and k ≥ 1. if we take (xn) as an almost increasing sequence such that | ∆xn |= o(xn/n), then we get theorem a, in this case condition ’∆an ≤ δn’ is not needed. we need the following lemmas for the proof of our theorem. lemma 1 ([3]). if 0 < α ≤ 1, β > −1, and 1 ≤ v ≤ n, then (6) | v∑ p=0 a α−1 n−pa β pap |≤ max 1≤m≤v | m∑ p=0 a α−1 m−pa β pap | . lemma 2 ([5]). let (xn) be a quasi-f-power increasing sequence. if (an) is a δ-quasi-monotone sequence with ∆an ≤ δn and ∑ nδnxn < ∞ , then ∞∑ n=1 nxn | ∆an |< ∞,(7) nanxn = o(1) as n →∞.(8) 3. proof of the theorem let (tα,βn ) be the nth (c,α,β) mean of the sequence (nanλn). then, by (1), we have t α,β n = 1 a α+β n n∑ v=1 a α−1 n−va β vvavλv. applying abel’s transformation first and then using lemma 1, we obtain that t α,β n = 1 a α+β n n−1∑ v=1 ∆λv v∑ p=1 a α−1 n−pa β ppap + λn a α+β n n∑ v=1 a α−1 n−va β vvav, | tα,βn | ≤ 1 a α+β n n−1∑ v=1 | ∆λv || v∑ p=1 a α−1 n−pa β ppap | + | λn | a α+β n | n∑ v=1 a α−1 n−va β vvav | ≤ 1 a α+β n n−1∑ v=1 a (α+β) v θ α,β v | ∆λv | + | λn | θ α,β n = t α,β n,1 + t α,β n,2 . to complete the proof of the theorem, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 1 n | tα,βn,r | k < ∞, for r = 1, 2. 42 bor and mohapatra when k > 1, we can apply hölder’s inequality with indices k and k′, where 1 k + 1 k′ = 1, we get that m+1∑ n=2 1 n | tα,βn,1 | k ≤ m+1∑ n=2 1 n | 1 a α+β n n−1∑ v=1 a (α+β) v θ α,β v ∆λv | k = o(1) m+1∑ n=2 1 n1+(α+β)k { n−1∑ v=1 v (α+β)k| av |(θα,βv ) k } × { n−1∑ v=1 | av | }k−1 = o(1) m∑ v=1 v (α+β)k| av |(θα,βv ) k m+1∑ n=v+1 1 n1+(α+β)k = o(1) m∑ v=1 v (α+β)k| av |(θα,βv ) k ∫ ∞ v dx x1+(α+β)k = o(1) m∑ v=1 v| av | (θα,βv ) k v = o(1) m−1∑ v=1 ∆(v| av |) v∑ p=1 (θα,βp ) k p + o(1)m| am | m∑ v=1 (θα,βv ) k v = o(1) m−1∑ v=1 | (v + 1)∆ | av | − | av || xv + o(1)m| am |xm = o(1) m−1∑ v=1 v | ∆av | xv + o(1) m−1∑ v=1 | av |xv + o(1)m| am |xm = o(1) as m →∞, in view of hypotheses of the theorem and lemma 2. similarly, we have that m∑ n=1 1 n | tα,βn,2 | k = o(1) m∑ n=1 | λn | n (θ α,β n ) k = o(1) m∑ n=1 (θα,βn ) k n ∞∑ v=n | ∆λv | = o(1) ∞∑ v=1 | ∆λv | v∑ n=1 (θα,βn ) k n = o(1) ∞∑ v=1 | ∆λv | xv = o(1) ∞∑ v=1 | av |xv < ∞. this completes the proof of the theorem. if we take β = 0, then we get a new result concerning the | c,α | k summability factors. if we set β = 0, α = 1, and xn= logn, then we obtain the result of mazhar dealing with | c, 1 | k summability factors (see [10]). finally, if we take γ=0, then we get a new result dealing with an application of quasi-σ-power increasing sequences. references [1] n. k. bari and s. b. stečkin, best approximation and differential properties of two conjugate functions, trudy. moskov. mat. obšč., 5 (1956), 483-522 (in russian). [2] r. p. boas, quasi positive sequences and trigonometric series, proc. london math. soc., 14a (1965), 38-46. [3] h. bor, on a new application of power increasing sequences, proc. est. acad. sci., 57 (2008), 205-209. [4] h. bor, on generalized absolute cesàro summability, an. ştiinţ. univ. al. i. cuza iaşi. mat. (n.s.), lvii (2011), 323-328. [5] h. bor, on the quasi monotone and generalized power increasing sequences and their new applications, j. classical anal., 2 (2013), 139-144. [6] d. borwein, theorems on some methods of summability, quart. j. math. oxford ser. (2), 9 (1958), 310-316. [7] g. das, a tauberian theorem for absolute summability, proc. camb. phil. soc., 67 (1970), 321-326. [8] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc., 7 (1957), 113-141. [9] l. leindler, a new application of quasi power increasing sequences, publ. math. debrecen, 58 (2001), 791-796. [10] s. m. mazhar, on generalized quasi-convex sequence and its applications, indian j. pure appl. math., 8 (1977), 784-790. [11] w. t. sulaiman, extension on absolute summability factors of infinite series, j. math. anal. appl., 322 (2006), 1224-1230. 1p. o. box 121, tr-06502 bahçelievler, ankara, turkey 2university of central florida, orlando, fl 32816, usa ∗corresponding author: hbor33@gmail.com international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 101-111 http://www.etamaths.com dhage iteration method for approximating positive solutions of pbvps of nonlinear quadratic differential equations with maxima shyam b. dhage, bapurao c. dhage∗ abstract. in this paper authors prove the existence as well as approximation of the positive solutions for a periodic boundary value problem of first order ordinary nonlinear quadratic differential equations with maxima. an algorithm for the solutions is developed and it is shown that certain sequence of successive approximations converges monotonically to the positive solution of considered quadratic differential equations under some suitable mixed hybrid conditions. our results rely on the dhage iteration principle embodied in a recent hybrid fixed point theorem of dhage (2014). a numerical example is also provided to illustrate the hypotheses and abstract theory developed in this paper. 1. introduction motivated by the linear differential equation with maxima that occurs in the automatic control of some technical system, several mathematicians started working in the field of differential equations with maxima for qualitative aspects of the solutions. see magomedov [17, 18], bainov and hristova [1], otrocol and rus [15] and the references therein for the details. again, mishkis [19] pointed out in his survey the necessity of the study of differential equations with maxima and gave an impetus in the study of such equations. similarly, the study of periodic boundary value problems of nonlinear quadratic differential equations is initiated in the works of dhage et.al. [13] and dhage [2]. very recently, dhage et.al. [13] initiated the study of initial value problems of nonlinear quadratic differential equations with maxima and proved the existence and approximation results for such equations. the purpose of the present paper is to blend these two ideas together and discuss the pbvps of first order quadratic differential equations with maxima for existence and numerical aspects of the solutions. it is well-known that the hybrid differential equations can be tackled using the dhage iteration method embodied in the hybrid fixed point theory initiated by dhage [3, 5, 7]) which also yields the algorithms for the solutions. therefore, it is of interest to establish algorithms for the quadratic differential equations with maxima (1.1) for existence and approximation of the solutions along similar lines. we claim that our method as well as our results of this paper are new to the literature in the theory of nonlinear differential equations with maxima. given a closed and bounded interval j = [0,t] in the real line r , consider the periodic boundary value problem (in short pbvp) of nonlinear first order ordinary quadratic differential equation (qde) with maxima, (1.1) d dt [ x(t) f(t,x(t)) ] + λ [ x(t) f(t,x(t))) ] = g(t,x(t),x(t))), t ∈ j, x(0) = x(t),   for λ ∈ r, λ > 0, where f : j ×r×r → r\{0} and g : j ×r×r → r are continuous functions and x(t) = max t0≤ξ≤t x(ξ) for t ∈ j. 2010 mathematics subject classification. 34a12, 34a38. key words and phrases. quadratic differential equation with maxima; periodic boundary value problem; dhage iteration principle; approximate positive solution. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 101 102 s.b. dhage and b.c. dhage by a solution of the qde (1.1) we mean a function x ∈ c1(j,r) that satisfies (i) t 7→ x f(t,x) is a continuously differentiable function for each x,y ∈ r, and (ii) x satisfies the equations in (1.1) on j, where c1(j,r) is the space of continuously differentiable real-valued functions defined on j. the qde (1.1) is a quadratic perturbation of second type of the nonlinear pbvp of first order differential equations, (1.2) x′(t) = f(t,x(t)), t ∈ j, x(0) = x(t), } the details of different types perturbations is given in dhage [3]. the qde (1.1) includes some known classes of nonlinear differential equations studied earlier in the literature. in the following we list a few special cases of (1.1) defined on j. 1. when f(t,x) = 1 for all t ∈ j and x,y ∈ r, the qde (1.1) reduces to the following known nonlinear differential equation with maxima (1.3) x′(t) + λx(t) = g(t,x(t),x(t)), t ∈ j, x(0) = x(t). } the above nonlinear differential equation with maxima (3.10) has already been discussed in the literature for existence and uniqueness of the solutions via classical methods of schauder and banach fixed point principles. see bainov and hristova [1] and the references therein. here our method is different and constructive. therefore, theorem 3.1 includes the existence and approximation theorem for the differential equation with maxima (3.10) as a special case under weak partial compactness type conditions. 2. again, when g(t,x,y) = g(t,x) for all t ∈ j and x,y ∈ r, the qde (1.1) reduces to the following qde without maxima, (1.4) d dt [ x(t) f (t,x(t)) ] + λ [ x(t) f (t,x(t)) ] = g (t,x(t)) , t ∈ j, x(0) = x(t).,   which has been discussed in dhage and dhage [11] via dhage iteration method and established the existence and approximation result for positive solutions. 3. if we take g(t,x,y) = py +f(t) for all t ∈ j and x,y ∈ r in (1.3), then it reduces to the standard linear differential equation of automatic regulation, (1.5) x′(t) + λx(t) = px(t) + f(t) x(0) = x(t), } for all t ∈ j, where λ > 0, p > 0 are constants and f : j → r is a continuous perturbation function. the differential equation with maxima (1.5) is the motivation for development of the subject of differential equations with maxima. therefore, our qde (1.1) is more general and the existence and approximation result of this problem includes the existence and approximation results for all the above differential equations with maxima as special cases. 2. auxiliary results in this section we give all the preliminaries and key tool that is used in subsequent part of the paper. unless otherwise mentioned, throughout this paper that follows, let e denote a partially ordered real normed linear space with an order relation � and the norm ‖ · ‖. it is known that e is regular if {xn}n∈n is a nondecreasing (resp. nonincreasing) sequence in e such that xn → x∗ as n → ∞, then xn � x∗ (resp. xn � x∗) for all n ∈ n. clearly, the partially ordered banach space c(j,r) is regular and the conditions guaranteeing the regularity of any partially ordered normed linear space e may be found in heikkilä and lakshmikantham [16] and the references therein. positive solutions of quadratic differential equations with maxima 103 we need the following preliminary definitions given in dhage [4, 5, 6] in what follows. a mapping t : e → e is called isotone or nondecreasing if it preserves the order relation �, that is, if x � y implies t x �t y for all x,y ∈ e. a mapping t : e → e is called partially continuous at a point a ∈ e if for � > 0 there exists a δ > 0 such that ‖t x−t a‖ < � whenever x is comparable to a and ‖x−a‖ < δ. t is called partially continuous on e if it is partially continuous at every point of it. it is clear that if t is partially continuous on e, then it is continuous on every chain c contained in e. a non-empty subset s of the partially ordered banach space e is called partially bounded if every chain c in s is bounded. an operator t : e → e is called partially bounded if every chain c in t(e) is bounded. t is called uniformly partially bounded if all chains c in t (e) are bounded by a unique constant. t is called bounded if t(e) is a bounded subset of e. a non-empty subset s of the partially ordered banach space e is called partially compact if every chain c in s is compact. an operator t : e → e is called partially compact if every chain or totally ordered set c in t (e) is a relatively compact subset of e. t is called uniformly partially compact if t (e) is a uniformly partially bounded and partially compact on e. t is called partially totally bounded if for any bounded subset s of e, t (s) is a relatively compact subset of e. if t is partially continuous and partially totally bounded, then it is called partially completely continuous on e. remark 2.1. suppose that t is a nondecreasing operator on e into itself. then t is a partially bounded or partially compact if t (c) is a bounded or relatively compact subset of e for each chain c in e. definition 2.1 (dhage [4]). the order relation � and the metric d on a non-empty set e are said to be compatible if {xn} is a monotone sequence, that is, monotone nondecreasing or monotone nonincreasing sequence in e and if a subsequence {xnk} of {xn} converges to x ∗ implies that the original sequence {xn} converges to x∗. similarly, given a partially ordered normed linear space (e,� ,‖ ·‖), the order relation � and the norm ‖ ·‖ are said to be compatible if � and the metric d defined through the norm ‖ · ‖ are compatible. a subset s of e is called janhavi if the order relation � and the metric d or the norm ‖·‖ are compatible in it. in particular, if s = e, then e is called a janhavi metric or janhavi banach space. clearly, the set r of real numbers with usual order relation ≤ and the norm defined by the absolute value function | · | has this property. similarly, the finite dimensional euclidean space rn with usual componentwise order relation and the standard norm possesses the compatibility property and so is a janhavi banach space. definition 2.2 (dhage [5]). an upper semi-continuous and nondecreasing function ψ : r+ → r+ is called a d-function provided ψ(0) = 0. let (e,�,‖ · ‖) be a partially ordered normed linear space. a mapping t : e → e is called partially nonlinear d-lipschitz if there exists a d-function ψ : r+ → r+ such that (2.1) ‖t x−t y‖≤ ψ(‖x−y‖) for all comparable elements x,y ∈ e. if ψ(r) = k r, k > 0, then t is called a partially lipschitz with a lipschitz constant k. let (e,�,‖ ·‖) be a partially ordered normed linear algebra. denote e+ = { x ∈ e | x � θ, where θ is the zero element of e } and (2.2) k = {e+ ⊂ e | uv ∈ e+ for all u,v ∈ e+}. the elements of the set k are called the positive vectors in e. the following lemma follows immediately from the definition of the set k which is often times used in the hybrid fixed point theory of banach algebras and applications to nonlinear differential and integral equations. lemma 2.1 (dhage [4]). if u1,u2,v1,v2 ∈k are such that u1 � v1 and u2 � v2, then u1u2 � v1v2. definition 2.3. an operator t : e → e is said to be positive if the range r(t ) of t is such that r (t ) ⊆k. 104 s.b. dhage and b.c. dhage the dhage iteration principle (in short dip) developed in dhage [5, 7] may be described as “the monotonic convergence of the sequence of successive approximations to the solutions of a nonlinear equation beginning with a lower or an upper solution of the equation as its initial or first approximation.” the aforesaid convergence principle is a useful tool in nonlinear analysis and is called dhage iteration method (in short dim) for nonlinear equations. dhage iteration method embodied in the following applicable hybrid fixed point theorem of dhage [7] is used as a key tool for our work contained in the present paper. a few other hybrid fixed point theorems containing the dhage iteration method along with applications appear in dhage [5, 7]. theorem 2.1 (dhage [5, 7]). let ( e,�,‖ · ‖ ) be a regular partially ordered complete normed linear algebra such that every compact chain of e is janhavi. let a,b : e → k be two nondecreasing operators such that (a) a is partially bounded and partially nonlinear d-lipschitz with d-function ψa, (b) b is partially continuous and uniformly partially compact, (c) 0 < mψa(r) < r, r > 0, where m = sup{‖b(c)‖ : c is a chain in e}, and (d) there exists an element x0 ∈ x such that x0 �ax0 bx0 or x0 �ax0 bx0. then the operator equation (2.3) axbx = x has a positive solution x∗ in e and the sequence {xn} of successive iterations defined by xn+1 = axnbxn, n = 0, 1, . . . ; converges monotonically to x∗. remark 2.2. the condition that every compact chain of e is janhavi holds if every partially compact subset of e possesses the compatibility property with respect to the order relation � and the norm ‖ ·‖ in it. 3. main results the qde (1.1) is considered in the function space c(j,r) of continuous real-valued functions defined on j. we define a norm ‖ · ‖ and the order relation ≤ in c(j,r) by (3.1) ‖x‖ = sup t∈j |x(t)| and (3.2) x ≤ y ⇐⇒ x(t) ≤ y(t) for all t ∈ j respectively. clearly, c(j,r) is a banach algebra with respect to above supremum norm and is also partially ordered w.r.t. the above partially order relation ≤. it is known that the partially ordered banach algebra c(j,r) has some nice properties w.r.t. the above order relation in it. the following lemma follows by an application of arzellá-ascolli theorem. lemma 3.1. let ( c(j,r),≤,‖ · ‖ ) be a partially ordered banach space with the norm ‖ · ‖ and the order relation ≤ defined by (3.1) and (3.2) respectively. then every partially compact subset s of c(j,r) is janhavi. proof. the proof of the lemma is given in dhage and dhage [9, 10, 11] and so we omit the details of it. � we need the following definition in what follows. definition 3.1. a function u ∈ c(j,r) is said to be a lower solution of the qde (1.1) if the function t 7→ u(t) f(t,u(t)) is differentiable and satisfies d dt [ u(t) f(t,u(t)) ] + λ [ u(t) f(t,u(t)) ] ≤ g(t,u(t),u(t)), u(0) ≤ u(t),   positive solutions of quadratic differential equations with maxima 105 for all t ∈ j, where u(t) = max 0≤ξ≤t x(ξ), t ∈ j. similarly, a function v ∈ c(j,r) is said to be an upper solution of the qde (1.1) if the function t 7→ v(t) f(t,v(t)) is differentiable and satisfies the above inequalities with reverse sign. we consider the following set of assumptions in what follows: (a0) the map x 7→ x f(t,x) is increasing for each t ∈ j. (a1) f defines a function f : j ×r → r+. (a2) there exists a constant mf > 0 such that 0 < f(t,x) ≤ mf for all t ∈ j and x ∈ r. (a3) there exists a d-function ϕ such that 0 ≤ f(t,x) −f(t,y) ≤ ϕ ( x−y ) , for all t ∈ j and x,y ∈ r, x1 ≥ y1. (a4) the function f(t,x) is periodic in t with period t for all x ∈ r, i.e., f(t,x) = f(t + t,x) for each x ∈ r. (b1) g defines a function g : j ×r×r → r+. (b2) there exists a constant mg > 0 such that g(t,x,y) ≤ mg for all t ∈ j and x,y ∈ r. (b3) g(t,x,y) is nondecreasing in x and y for all t ∈ j. (b4) the qde (1.1) has a lower solution u ∈ c1(j,r). remark 3.1. notice that if hypothesis (a0) holds, then the function x 7→ x f(t,x) is injective for each t ∈ j. the following useful lemma is obvious and may be found in nieto and lopez [20]. lemma 3.2. for any h ∈ l1(j,r+) and σ ∈ l1(j,r), x is a solution to the differential equation (3.3) x′(t) + h(t)x(t) = σ(t), t ∈ j, x(0) = x(t), } if and only if it is a solution of the integral equation (3.4) x(t) = ∫ t 0 gh(t,s) σ(s) ds where, (3.5) gh(t,s) =   eh(s)−h(t)+h(t) eh(t) − 1 , if 0 ≤ s ≤ t ≤ t, eh(s)−h(t) eh(t) − 1 , if 0 ≤ t < s ≤ t. and h(t) = ∫ t 0 h(s) ds. notice that the green’s function gh is continuous and nonnegative on j × j and therefore, the number kh := max{|gh(t,s)| : t,s ∈ [0,t]} exists for all h ∈ l1(j,r+). in particular, if h(t) = λ for all t ∈ j, then for the sake of convenience we write gλ(t,s) = g(t,s) and kλ = k. an application of above lemma 3.2 we obtain lemma 3.3. suppose that hypothesis (a1) holds. then a function u ∈ c(j,r) is a solution of the pbvp (1.1) if and only if it is a solution of the nonlinear integral equation, (3.6) x(t) = [ f(t,x(t)) ](∫ t 0 g(t,s)g(s,x(s),x(s)) ds ) 106 s.b. dhage and b.c. dhage for all t ∈ j, where (3.7) g(t,s) =   eλs−λt+λt eλt − 1 , if 0 ≤ s ≤ t ≤ t, eλs−λt eλt − 1 , if 0 ≤ t < s ≤ t. theorem 3.1. assume that hypotheses (a0)-(a4) and (b1)-(b3) hold. furthermore, assume that (3.8) k mg t ϕ(r) < r, r > 0, then the qde (1.1) has a positive solution x∗ defined on j and the sequence {xn}∞n=1 of successive approximations defined by (3.9) xn+1(t) = [ f(t,xn(t)) ](∫ t 0 g(t,s) g(s,xn(s),xn(s)) ds ) , t ∈ j, where x1 = u, converges monotonically to x ∗. proof. set e = c(j,r). then, by lemma 3.1, every compact chain in e possesses the compatibility property with respect to the norm ‖ ·‖ and the order relation ≤ in e. define two operators a and b on e by (3.10) ax(t) = f(t,x(t)), t ∈ j, and (3.11) bx(t) = ∫ t 0 g(t,s)g(s,x(s),x(t)) ds, t ∈ j. from the continuity of the integral, it follows that a and b define the maps a,b : e → e. now by lemma 3.3, the qde (1.1) is equivalent to the operator equation (3.12) ax(t)bx(t) = x(t), t ∈ j. we shall show that the operators a and b satisfy all the conditions of theorem 2.1. this is achieved in the series of following steps. step i: a and b are nondecreasing on e. let x,y ∈ e be such that x ≥ y. then x(t) ≥ y(t) for all t ∈ j. since y is continuous on [a,t], there exists a ξ∗ ∈ [a,t] such that y(ξ∗) = max a≤ξ≤t y(ξ). by definition of ≤, one has x(ξ∗) ≥ y(ξ∗). consequently, we obtain x(t) = max a≤ξ≤t x(ξ) ≥ x(ξ∗) ≥ y(ξ∗) = max a≤ξ≤t y(ξ) = y (t) for each t ∈ j. then by hypothesis (a3), we obtain ax(t) = f(t,x(t)) ≥ f(t,x(t)) = ay(t), for all t ∈ j. this shows that a is nondecreasing operator on e into e. similarly using hypothesis (b3), it is shown that the operator b is also nondecreasing on e into itself. thus, a and b are nondecreasing positive operators on e into itself. step ii: a is partially bounded and partially d-lipschitz on e. let x ∈ e be arbitrary. then by (a2), |ax(t)| ≤ ∣∣f(t,x(t))∣∣ ≤ mf, for all t ∈ j. taking the supremum over t, we obtain ‖ax‖≤ mf and so, a is bounded. this further implies that a is partially bounded on e. next, let x,y ∈ e be such that x ≥ y. then, |ax(t) −ay(t)| = ∣∣f(t,x(t)) −f(t,y(t))∣∣ ≤ ϕ(|x(t) −y(t)) positive solutions of quadratic differential equations with maxima 107 ≤ ϕ(‖x−y‖), for all t ∈ j. taking the supremum over t, we obtain ‖ax−ay‖≤ ϕ(‖x−y‖), for all x,y ∈ e with x ≥ y. hence, a is a partial nonlinear d-lipschitz on e which further also implies that a is a partially continuous on operator on e. step iii: b is a partially continuous operator on e. let {xn}n∈n be a sequence in a chain c of e such that xn → x for all n ∈ n. then, by dominated convergence theorem, we have lim n→∞ bxn(t) = ∫ t 0 g(t,s) [ lim n→∞ g(s,xn(s),xn(s)) ] ds = ∫ t 0 g(t,s)g(s,x(s),xn(s)) ds = bx(t), for all t ∈ j. this shows that bxn converges to bx pointwise on j. next, we will show that {bxn}n∈n is an equicontinuous sequence of functions in e. let t1, t2 ∈ j be arbitrary. then, by hypothesis (b2), |bxn(t2) −bxn(t1)| ≤ ∣∣∣∣ ∫ t 0 g(t2,s)g(s,xn(s),xn(s)) ds − ∫ t 0 g(t1,s)g(s,xn(s),xn(s)) ds ∣∣∣∣ ≤ ∫ t 0 ∣∣g(t2,s) −g(t1,s)∣∣ ∣∣g(s,xn(s),xn(s))∣∣ds ≤ mg ∫ t 0 ∣∣g(t2,s) −g(t1,s)∣∣ds → 0 as t2 − t1 → 0 uniformly for all n ∈ n. this shows that the convergence bxn → bx is uniform and hence b is a partially continuous on e. step iv: b is a uniformly partially compact operator on e. let c be an arbitrary chain in e. we show that b(c) is a uniformly bounded and equicontinuous set in e. first we show that b(c) is uniformly bounded. lety ∈b(c) be any element. then there is an element x ∈ c be such that y = bx. now, by hypothesis (b2), |y(t)| ≤ ∣∣∣∫ t 0 g(t,s)g(s,x(s),x(t)) ds ∣∣∣ ≤ k ∫ t 0 |g(s,x(s),x(t))|ds ≤ k mg t = m, for all t ∈ j. taking the supremum over t, we obtain ‖y‖ = ‖bx‖≤ m for all y ∈b(c). hence, b(c) is a uniformly bounded subset of e. moreover, ‖b(c)‖ ≤ m for all chains c in e. hence, b is a uniformly partially bounded operator on e. next, we will show that b(c) is an equicontinuous set in e. let t1, t2 ∈ j be arbitrary. then, for any y ∈b(c), there is a x ∈ c such that y(t) = bx(t). hence, proceeding with arguments as in step 108 s.b. dhage and b.c. dhage iii, |y(t2) −y(t1)| ≤ ∣∣∣∫ t 0 g(t2,s)g(s,x(s),x(s)) ds− ∫ t 0 g(t1,s)g(s,x(s),x(s)) ds ∣∣∣ ≤ mg ∫ t 0 |g(t2,s) −g(t1,s)|ds → 0 as t2 − t1 → 0 uniformly for all y ∈ b(c). hence b(c) is an equicontinuous subset of e. now, b(c) is a uniformly bounded and equicontinuous set of functions in e, so it is compact. consequently, b is a uniformly partially compact operator on e into itself. step v: u satisfies the operator inequality u ≤aubu. by hypothesis (b3), the qde (1.1) has a lower solution u defined on j. then, we have (3.13) d dt [ u(t) f(t,u(t)) ] + λ [ u(t) f(t,u(t)) ] ≤ g(t,u(t),u(t)), u(0)) ≤ x(t),   for all t ∈ j. multiplying the first inequality in (3.13) by the integrating factor eλt, we obtain (3.14) ( eλt u(t) f(t,u(t)) )′ ≤ eλtg(t,u(t),u(t)), for all t ∈ j. a direct integration of (3.14) from 0 to t yields (3.15) eλt u(t) f(t,u(t)) ≤ u(0) f(0,u(0)) + ∫ t 0 eλsg(s,u(s)) ds, for all t ∈ j. therefore, in particular, (3.16) eλt u(t) f(t,u(t)) ≤ u(0) f(0,u(0)) + ∫ t 0 eλsg(s,u(s),u(s)) ds. now u(0) ≤ u(t), so by hypothesis (a0) one has (3.17) u(0) f(0,u(0)) eλt ≤ u(t) f(t,u(t)) eλt . from (3.15) and (3.17) it follows that (3.18) eλt u(0) f(0,u(0)) ≤ u(0) f(0,u(0)) + ∫ t 0 eλsg(s,u(s),u(s)) ds which further yields (3.19) u(0) f(0,u(0)) ≤ ∫ t 0 eλs (eλt − 1) g(s,u(s),u(s)) ds substituting (3.19) in (3.15) we obtain u(t) ≤ [ f(t,u(t)) ](∫ t 0 g(t,s)g(s,u(s),u(s)) ds ) , t ∈ j. from the definitions of the operators a and b it follows that u(t) ≤au(t)bu(t) for all t ∈ j. hence u ≤aubu. step vi: d-function ϕ satisfies the growth condition mψa(r) < r, r > 0. finally, the d-function ϕ of the operator a satisfies the inequality given in hypothesis (d) of theorem 2.1. now from the estimate given in step iv, it follows that mψa(r) ≤ k mg t ϕ(r) < r for all r > 0. thus, a and b satisfy all the conditions of theorem 2.1 and we apply it to conclude that the operator equation axbx = x has a positive solution x∗. therefore, the integral equation (3.6) and consequently the qde (1.1) has a positive solution x∗ defined on j. furthermore, the sequence positive solutions of quadratic differential equations with maxima 109 {xn}∞n=1 of successive approximations defined by (3.9) converges monotonically to x∗. this completes the proof. � remark 3.2. the conclusion of theorem 3.1 also remains true if we replace the hypothesis (b3) with the following: (b′3) the qde (1.1) has an upper solution v ∈ c1(j,r). the proof under this new hypothesis is similar to the proof of theorem 3.1 with appropriate modifications. remark 3.3. we note that if the qde (1.1) has a lower solution u as well as an upper solution v such that u ≤ v, then under the given conditions of theorem 3.1 it has corresponding solutions x∗ and x∗ and these solutions satisfy x∗ ≤ x∗. hence they are the minimal and maximal solutions of the pbvp (1.1) in the vector segment [u,v] of the banach space e = c1(j,r), where the vector segment [u,v] is a set in c1(j,r) defined by [u,v] = {x ∈ c1(j,r) | u ≤ x ≤ v}. this is because the order relation ≤ defined by (3.2) is equivalent to the order relation defined by the order cone k = {x ∈ c(j,r) | x ≥ θ} which is a closed set in c(j,r). finally, we give an example to illustrate our hypotheses (a0)-(a4) and (b1)-(b3) and the abstract result formulated in theorem 3.1. example 3.1. given a closed and bounded interval j = [0, 1], consider the pbvp of qdes, (3.20) d dt [ x(t) f(t,x(t)) ] + [ x(t) f(t,x(t)) ] = 1 8 [ 2 + tanh x(t) + tanh x(t) ] , x(0) = x(1),   for all t ∈ j, where the functions f : j ×r → r\{0} and g : j ×r×r → r are defined as f(t,x,y) =   1, if x ≤ 0, 1 + x, if 0 < x < 3, 4, if x ≥ 3. and g(t,x,y) = 1 20 [ 3 + tanh x ] . clearly, the functions f and g are continuous nonnegative real-valued functions on j × r and j × r × r respectively. as ∂ ∂x ( x f(t,x) ) > 0 for all t ∈ j and x ∈ r, the function x 7→ x f(t,x) is increasing for each t ∈ j and so, the hypothesis (a0) is satisfied. next, f satisfies the hypothesis (a3) with ϕ(r) = r. to see this, we have 0 ≤ f(t,x) −f(t,y) ≤ x−y for all x,y ∈ r, x ≥ y. therefore, ϕ(r) = r. moreover, the function f(t,x) is periodic in t for each x ∈ r and is also bounded on j × r × r with bound mf = 4 and so, the hypotheses (a2) and(a3) are satisfied. again, since g is positive and bounded on j × r × r by mg = 14 , the hypothesis (b2) holds. furthermore, g(t,x,y) is nondecreasing in x and y for each t ∈ j, and thus hypothesis (b3) is satisfied. here, the green’s function g(t,s) associated with the homogeneous pbvp (3.21) x′(t) + x(t) = 0, t ∈ [0, 1], x(0) = x(t),   110 s.b. dhage and b.c. dhage is given by (3.22) 0 ≤ g(t,s) =   es−t+1 e− 1 , if 0 ≤ s ≤ t ≤ 1, es−t e− 1 , if 0 ≤ t ≤ s ≤ 1 ≤ e e− 1 ≤ 2 = k. also the condition (3.8) of theorem 3.1 holds. finally, after a simple computation, it is shown that the function u(t) = 1 20 ∫ 1 0 g(t,s) ds is a lower solution of the qde (3.20) defined on j. thus, all the hypotheses of theorem 3.1 are satisfied. hence we apply theorem 3.1 and conclude that the qde (3.20) has a positive solution x∗ defined on j and the sequence {xn}∞n=1 of successive approximations defined by x1(t) = 1 20 ∫ 1 0 g(t,s) ds, xn+1(t) = 1 20 [ f(t,xn(t)) ](∫ 1 0 g(t,s) [ 2 + tanh xn(s) + tanh xn(s) ] ds ) , for all t ∈ j, converges monotonically to x∗. references [1] d.d. bainov, s. hristova, differential equations with maxima, chapman & hall/crc pure and applied mathematics, 2011. [2] b.c. dhage, periodic boundary value problems of first order carathéodory and discontinuous differential equations, nonlinear funct. anal. & appl. 13(2) (2008), 323-352. [3] b.c. dhage, quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, differ. equ. appl. 2 (2010), 465–486. [4] b.c. dhage, hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, differ. equ. appl. 5 (2013), 155-184. [5] b.c. dhage, partially condensing mappings in partially ordered normed linear spaces and applications to functional integral equations, tamkang j. math. 45 (4) (2014), 397-426. [6] b.c. dhage, nonlinear d-set-contraction mappings in partially ordered normed linear spaces and applications to functional hybrid integral equations, malaya j. mat. 3(1) (2015), 62-85. [7] b.c. dhage, operator theoretic techniques in the theory of nonlinear hybrid differential equations, nonlinear anal. forum 20 (2015), 15-31. [8] b.c. dhage, a new monotone iteration principle in the theory of nonlinear first order integro-differential equations, nonlinear studies 22 (3) (2015), 397-417. [9] b.c. dhage, s.b. dhage, approximating solutions of nonlinear first order ordinary differential equations, gjms special issue for recent advances in mathematical sciences and applications-13, global journal of mathematical sciences, 2 (2014), 25-35. [10] b.c. dhage, s.b. dhage, approximating solutions of nonlinear pbvps of hybrid differential equations via hybrid fixed point theory, indian j. math. 57(1) (2015), 103-119. [11] b.c. dhage, s.b. dhage, approximating positive solutions of pbvps of nonlinear first order ordinary quadratic differential equations, appl. math. lett. 46 (2015), 133-142. [12] s.b. dhage, b.c. dhage, d. octrocol, dhage iteration method for approximating positive solutions of nonlinear first order ordinary quadratic differential equations with maxima, fixed point theory, in press. [13] b.c. dhage, j. henderson, s.k. ntouyas, periodic boundary value problems of first order differential equations in banach algebras, j. nonlinear funct. anal. & diff. equ. 1 (2007), 103-120. [14] b.c. dhage, v. lakshmikantham, basic results on hybrid differential equations, nonlinear analysis: hybrid systems 4 (2010), 414-424. [15] d. otrocol, i.a. rus, functional-differential equations with maxima of mixed type argument, fixed point theory, 9(2008), no. 1, pp. 207–220. [16] s. heikkilä, v. lakshmikantham, monotone iterative techniques for discontinuous nonlinear differential equations, marcel dekker inc., new york 1994. [17] a.r. magomedov, on some questions about differential equations with maxima, izv. akad. naukazerbaidzhan. ssr ser. fiz.-tekhn. mat. nauk, 1 (1977), 104–108 (in russian). [18] a.r. magomedov, theorem of existence and uniqueness of solutions of linear differential equations with maxima, izv. akad. naukazerbaidzhan. ssr ser. fiz. tekhn. mat. nauk, 5 (1979), 116-118 (in russian). [19] a.d. myshkis, on some problems of the theory of differential equations with deviating argument, russian math. surveys 32 (1977), 181-210. positive solutions of quadratic differential equations with maxima 111 [20] j.j. nieto, r. rodriguez-lopez, existence and approximation of solution for nonlinear differential equations with periodic bounday conditions, compt. math. appl. 40 (2000), 435-442. kasubai, gurukul colony, ahmedpur-413 515, dist: latur maharashtra, india ∗corresponding author: bcdhage@gmail.com international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 114-120 http://www.etamaths.com on the integral representation of strictly continuous set-valued maps anaté k. lakmon∗ and kenny k. siggini abstract. let t be a completely regular topological space and c(t ) be the space of bounded, continuous real-valued functions on t . c(t ) is endowed with the strict topology (the topology generated by seminorms determined by continuous functions vanishing at infinity). r. giles ([13], p. 472, theorem 4.6) proved in 1971 that the dual of c(t ) can be identified with the space of regular borel measures on t . we prove this result for positive, additive setvalued maps with values in the space of convex weakly compact non-empty subsets of a banach space and we deduce from this result the theorem of r. giles ([13], theorem 4.6, p.473). 1. introduction the strict topology β was for the first time introduced by r. c. buck ([1], [2]) on the space c(t) of all bounded continuous functions on a locally compact space t. he has proved among others that the dual space of (c(t),β) is the space of all finite signed regular borel measures on t . after a large number of papers have appeared in the literature concerned with extending the results contained in buck’s paper [1]( see e.g. [4], [5], [6], [7], [8], [12],[14], [15], [17], [18], [19], [22], [25] and [27]). r. giles has generalized this notion of the strict topology introduced by buck for completely regular space t and has proved buck’s results, particulary the theorem 2 in [1] for an arbitrary (not necessarily hausdorff) completely regular space t. in this paper we generalize giles’s result ([13], theorem 4.6, p.473) to additive, positive, positively homogeneous and strictly continuous set-valued maps defined on c+(t) with values in the space cc(e) of all convex weakly compact non-empty subsets of a banach space e. we deduce from this result the theorem of r. giles. 2. notations and definitions let t be a completely regular topological space and let b(t) be the borel σalgebra of t and let c(t) be the space of bounded continuous real-valued functions on t . let c0(t) be the subspace of c(t) consisting of functions f vanishing at infinity i.e. for any ε > 0 there is a compact set kε ⊂ t such that |f(x)| < ε for x ∈ t\kε. we denote by c+(t) the subspace of c(t) consisting of non-negative functions and by 1a the characteristic function of each a ⊂ t . for all f ∈ c(t), we put f+ = sup(f, 0),f− = sup(−f, 0) and ||f||∞ = sup{|f(t)|; t ∈ t}. we denote 2010 mathematics subject classification. primary 28b20, secondary 54c60. key words and phrases. set-valued measure; strict topology; regular set-valued measure; additive and positive set-valued map. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 114 on the integral representation 115 by r the set of real numbers. let e be a banach space, e′ its dual and cc(e) be the space of all non-empty, convex weakly compact subsets of e; we denote by ‖.‖ the norm on e and e′. if x and y are subsets of e we shall denote by x +y the family of all elements of the form x+y with x ∈ x and y ∈ y . the support function of x is the function δ∗(.|x) from e′ to [−∞; +∞] defined by δ∗(y|x) = sup{y(x),x ∈ x}. we endow cc(e) with a hausdorff distance, denoted by δ. for all k ∈ cc(e) and for all k′ ∈ cc(e),δ(k,k′) = sup{|δ∗(y|k) − δ∗(y|k′)|; y ∈ e′,‖y‖≤ 1}. recall that (cc(e),δ) is a complete metric space ([16], theorem 9, p.185) and ([21], theorem 15, p.2-2). definition 2.1. (1) let m : b(t) → r be a positive countable additive measure. we say that m is: (i) inner regular if for all a ∈b(t) and ε > 0, there exists a compact kε subset of t such that kε ⊂ a and m(a\kε) < ε. (ii) outer regular if for all a ∈b(t) and for all ε > 0, there exists an open subset oε of t such that oε ⊃ a and m(oε\a) < ε. (iii) regular if it is inner regular and outer regular. (2) a signed measure µ : b(t) → r is regular if and only if its total variation v(µ) is regular. note that v(µ) : b(t) → r+ (a 7→ v(µ)(a) = sup{ ∑ i |µ(ai)|; (ai) finite partition of a,ai ∈b(t)}). definition 2.2. a map m : b(t) → cc(e) is a set-valued measure if m(a ∪ b) = m(a) + m(b) for every pair of disjoint sets a,b in b(t),m(∅) = {0} and m( +∞⋃ n=1 an) = +∞∑ n=1 m(an) for every sequence (an) of mutually disjoint elements of b(t); which amounts to saying that for all y ∈ e′ the map δ∗(y|m(.)) : b(t) → r(a 7→ δ∗(y|m(a))) is a countably additive measure ([21], corollary p. 2-25). we say that a set-valued measure m is: (i) positive if for all a ∈b(t), 0 ∈ m(a) (ii) regular if for all y ∈ e′, the measure δ∗(y|m(.)) is regular. let ϕ ∈ c0(t), let k be a compact subset of t . we denote by pϕ and pk the semi-norms on c(t) defined by pϕ(f) = sup{|f(t)ϕ(t)|; t ∈ t} and pk(f) = sup{|f(t)|; t ∈ k} for every f ∈ c(t). definition 2.3. the topology determined by the set of semi-norms {pϕ; ϕ ∈ c0(t)} (resp. {pk; k belongs to the family of compact subsets of t}) is called the strict (resp. the compact convergence) topology. we say that a map defined on c(t) is strictly continuous if it is continuous for this topology. definition 2.4. a map l : c+(t) → cc(e) is: (i) additive set-valued map if for all f,g ∈ c+(t) l(f + g) = l(f) + l(g) (ii) positively homogeneous if for f ∈ c+(t) and for λ ≥ 0 l(λf) = λl(f). (iii) positive if for every f ∈ c+(t), 0 ∈ l(f). definition 2.5. ([24], p. 04) let m be a bounded linear functional on c(t), and let b(0, 1) be the unit ball of c(t). we say that m is tight if its restriction to b(0, 1) is continuous for the topology of compact convergence. 116 lakmon and siggini 3. main result lemma 3.1. let m be a bounded linear functional on c(t). if m is tight then for all ε > 0 there is a compact subset kε of t such that for all f ∈ c(t) and |f| ≤ 1t\kε , we have |m(f)| < ε. proof. assume that m is tight. then for every ε > 0 there is a compact subset kε of t and there is η > 0 such that for all f ∈ b(0, 1) and pkε (f) = sup{|f(t)|; t ∈ kε} < η. we have |m(f)| < ε. in particular for all f ∈ b(0, 1) such that |f| ≤ 1t\kε , one has |m(f)| < ε. � lemma 3.2. let m : b(t) −→ cc(e) be a positive, regular set-valued measure. then the real-valued measure δ∗(y|m(.)) are uniformly tight with respect to y ∈ e′,‖y‖ ≤ 1 ie for every a ∈ b(t) and for every ε > 0 there is a compact subset kε of t such that kε ⊂ a and sup{δ∗(y|m(a\kε)); y ∈ e′,‖y‖≤ 1}≤ ε. proof. let us consider the set {δ∗(y|m(.)),y ∈ e′,‖y‖ ≤ 1} of countably additive real-valued measures. it is uniformly countable additive (see [9], theorem 10, p. 88–89; [28], lemma 3.1, p. 275). according to ([10], p. 443, theorem 10.7) there is a sequence (cn) of real numbers and there is a sequence (δ ∗(yn|m(.))), |yn| ≤ 1 of measures such that µ(a) = +∞∑ n=1 cnδ ∗(yn|m(a)) exists for each a ∈ b(t) and such that the series ∑ |cn|δ∗(yn|m(a)) is uniformly convergent for a ∈b(t); moreover the countable additive measure ν : b(t) → r(a 7→ ν(a) = +∞∑ n=1 |cn|δ∗(yn|m(a))) verifies the following relation: lim ν(a)→0 [sup{δ∗(y|m(a)); y ∈ e′,‖y‖ ≤ 1}] = 0 (*). we deduce from the uniform convergence of the series ∑ |cn|δ∗(yn|m(a)) for a ∈ b(t), that ν is regular. indeed, given ε > 0 choose n0 ∈ n such that sup a∈b(t) ∣∣∣∣ν(a) − n0∑ k=1 |ck|δ∗(yk|m(a)) ∣∣∣∣ < ε/2. for a ∈ b(t), choose a compact subset k of t such that k ⊂ a and for every k ∈ {1, 2, ...,n0} δ∗(yk|m(a\k)) ≤ ε2(n0+1)r0 with r0 = sup{|ck|; k ∈ {1, 2, ...,n0}} then n0∑ k=1 |ck|δ∗(yk|m(a\k)) ≤ ε/2, therefore ν(a\k) ≤ ε. the relation (*) and the inner regularity of ν show that for each ε > 0 and each a ∈ b(t) there exists a compact subset k of t such that k ⊂ a and sup{δ∗(y|m(a\k)); y ∈ e′,‖y‖≤ 1}≤ ε. � let m be a positive set-valued measure defined on b(t). for the construction of the integral ∫ fm, with f ∈ c+(t) we refer to ([23], p. 17). lemma 3.3. let m : b(t) → cc(e) be a positive regular set-valued measure. then the set-valued map l : c+(t) → cc(e)(f 7→ l(f) = ∫ fm) is additive, positively homogeneous, positive and strictly continuous. proof. we only prove the strict continuity. the other properties follow from the construction of the integral ∫ fm,f ∈ c+(t). for each n ∈ n∗ there exists a compact subset kn of t such that sup{δ∗(y|m(t\kn)); y ∈ e′,‖y‖ ≤ 1} ≤ 2−2n (lemma 3.2). we then have a sequence (kn) of compact subsets of t that we may assume monotone increasing. we repeat here the proof of r. giles ([13], p. 471, on the integral representation 117 lemma 4.2). consider ϕ = +∞∑ n=1 2−n1kn , we have 2 −n−1 ≤ ϕ(x) ≤ 2−n for all x ∈ kn+1\kn. the function 1/ϕ is measurable and is δ∗(y|m(.)) integrable for each y ∈ e′,‖y‖ ≤ 1. we have ∫ 1/ϕδ∗(y|m(.)) = ∫ ∪+∞n=1(kn+1\kn) 1/ϕ δ∗(y|m(.)) = +∞∑ n=1 ∫ kn+1\kn 1/ϕ δ∗(y|m(.)) ≤ +∞∑ n=1 2n+1 [δ∗(y|m(kn+1)) −δ∗(y|m(kn))] ≤ +∞∑ n=1 2n+1.2−2n = 2. let ε > 0 and let ψn ∈ c0 such that ψn(x) = 2−n for x ∈ kn and 0 ≤ ψn ≤ 2−n1t . put ψ = +∞∑ n=1 ψn. then ψ ∈ c0 and ϕ ≤ ψ. for all f ∈ {g ∈ c+(t),p2ψ/ε(g) < 1} we have f < ε/2ϕ and ∫ fδ∗(y|m(.)) < ε for all y ∈ e′ with ‖y‖ ≤ 1. since δ∗(y| ∫ fm) = ∫ fδ∗(y|m(.)), one has δ( ∫ fm,{0}) < ε. therefore the map f →∫ fm is strictly continuous at 0. the equality δ∗(y| ∫ fm) = ∫ fδ∗(y|m(.)) for each f ∈ c+(t) and each y ∈ e′ enable us to prove the continuity on c+(t). � definition 3.4. a map s : e′ → r is said to be sublinear if for every y ∈ e′ and y′ ∈ e′ and for every λ ≥ 0 one has s(y + y′) ≤ s(y) + s(y′) and s(λy) = λs(y). the lemme below is a particular case of l. hörmander’s result ([16], theorem 5, p. 182). we give here an alternative proof. lemma 3.5. let e be a banach space, and let e′ its dual space endowed with the mackey topology τ(e′,e). let s : e′ → r be a sublinear map. then s is continuous if and only if there is c ∈ cc(e) such that s = δ∗(.|c). proof. assume that s is continuous. let ∇s = {l : e′ → r; linear and l ≤ s}. by the hahn-banach theorem ([11], theorem 10, p. 62), s(y) = sup{l(y); l ∈∇s} for each y ∈ e′. let l ∈∇s; then l is continuous for the mackey topology τ(e′,e). therefore l determines an element xl ∈ e that verifies l(y) = y(xl) for each y ∈ e′. let ∇es = {xl; l ∈ ∇s}. since ∇s is equicontinuous there is a neighborhood v of 0 in e′ such that ∇es ⊂ v ◦, where v ◦ is the polar of v in e. by the alaoglu-bourbaki’s theorem ([20], p. 248), one has v ◦ ∈ cc(e). since ∇es is convex , its closure is one of elements of cc(e) we want. the converse is obvious. note that if s is non-negative then 0 ∈∇es. � theorem 3.6. let t be a completely regular topological space and let c+(t) be the space of bounded continuous non-negative functions defined on t endowed with the strict topology. let e be a banach space and cc(e) be the space of convex weakly compact non-empty subsets of e endowed with the hausdorff distance. let l : c+(t) → cc(e) be a positive, additive, positively homogeneous and strictly continuous set-valued map. then there is a unique positive regular set-valued measure m defined on b(t) to cc(e) such that l(f) = ∫ fm for all f ∈ c+(t). conversely for all positive regular set-valued measure m : b(t) → cc(e), the setvalued map θ : c+(t) → cc(e) (f 7→ θ(f) = ∫ fm) is positive, additive, positively homogeneous and strictly continuous. proof. let y ∈ e′. the map δ∗(y|l(.)) : c+(t) → r (f 7→ δ∗(y|l(f))) is additive, positively homogeneous and continuous. then it can be extended to a continuous linear functional on c(t). this extension is unique. it is denoted by δ∗(y|l̄(.)). let f ∈ c(t), one has f = f+ − f− and δ∗(y|l̄(.)) is defined by δ∗(y|l̄(.))(f) = 118 lakmon and siggini δ∗(y|l(f+)) − δ∗(y|l(f−)). since δ∗(y|l̄(.)) is strictly continuous it is tight ([26], p. 41). by the lemma 3.1 and ([3], proposition 5, p.58) there exists a unique regular positive borel measure µy on t that verifies δ ∗(y|l̄(f)) = ∫ fµy for all f ∈ c(t). let 0 an open subset of t and let so the map defined on e′ to r by so(y) = µy(o) for each y ∈ e′. we have µy(o) = sup{ ∫ fµy; f ∈ c+(t),f ≤ 1o} = sup{δ∗(y|l(f)); f ∈ c+(t),f ≤ 1o}, therefore so is a sublinear map. let now a ∈b(t). we denote by sa the map defined on e′ to r by sa(y) = µy(a) for each y ∈ e′. since the measure µy is regular we have sa(y) = inf{µy(o); o ⊂ t,o open and o ⊃ a} = inf{so(y); o ⊂ t,o open and o ⊃ a}. let y,y′ ∈ e′ and let ε > 0, there exists two open subsets oε and o ′ ε of t containing a and such that sa(y) ≥ µy(oε) −ε/2, sa(y′) ≥ µy′ (o′ε) −ε/2. we have µy(oε) + µy′ (o′ε) ≤ sa(y) +sa(y ′) +ε, then µy(oε∩o′ε) +µy′ (oε∩o′ε) ≤ sa(y) +sa(y′) +ε, therefore µy+y′ (oε ∩ o′ε) ≤ sa(y) + sa(y′) + ε. we have µy+y′ (a) ≤ µy+y′ (oε ∩ o′ε) ≤ sa(y) + sa(y ′) + ε. it follows from this sa(y + y ′) ≤ sa(y) + sa(y′). it is obvious that for all λ ≥ 0 and for all y ∈ e′, sa(λy) = λsa(y). so sa is a nonnegative sublinear map. let us prove now that sa is continuous for the mackey topology τ(e′,e). we have sa(y) ≤ µy(t) = δ∗(y|l(1t )). let l̃(1t ) be the closed absolutely convex cover of l(1t ), one has l̃(1t ) ∈ cc(e) and sa(y) ≤ δ∗ ( y|l̃(1t ) ) for each y ∈ e′ and a ∈ b(t). we deduce that sa is continuous for the mackey topology for each a ∈ b(t). by the lemma 3.5 there is ca ∈ cc(e) such that sa(y) = δ ∗(y|ca) for all y ∈ e′. let m : b(t) → cc(e) (a 7→ m(a) = ca). we have δ∗(y|m(a)) = µy(a) for all y ∈ e′, hence the map δ∗(y|m(.)) : b(t) → r (a 7→ δ∗(y|m(a))) is a positive regular countably additive measure. then m is a regular set-valued measure. since sa is non-negative then m is positive. let f ∈ c+(t) and let y ∈ e′, ∫ fδ∗(y|m(.)) = ∫ fµy = δ ∗(y|l(f)). it follows that l(f) = ∫ fm for all f ∈ c+(t) because ∫ fδ∗(y|m(.)) = δ∗(y| ∫ fm). let us prove that m is unique. assume that there exist two regular positive set-valued measures m and m′ which verify ∫ fm = l(f) = ∫ fm′. let 0 be an open subset of t and let y ∈ e′. according to the inner regularity of δ∗(y|m(.)) and ([3] lemme 1 p. 55) we have δ∗(y|m(o)) = sup{δ∗(y|l(f)); f ∈ c+(t), f ≤ 1o} = δ∗(y|m′(o)). moreover the outer regularity of δ∗(y|m(.)) shows that δ∗(y|m(a)) = δ∗(y|m′(a)) for all a ∈ b(t) and y ∈ e′, hence m(a) = m′(a) for all a ∈ b(t). the second assertion of the theorem is justified by the lemma 3.3. � the following corollary is the result of r. giles. corollary 3.7. ([13], theorem 4.6 ) for any completely regular space t the dual of c(t) under the strict topology is the space of all bounded signed borel regular measures on t . proof. let l be a strictly continuous linear functional on c(t); l is bounded. therefore l is the difference of two non-negative linear functional. we may assume that l is non-negative. let k0 be an element of cc(e) that contains 0 and that is subset of the unit ball of e. consider the map l′ : c+(t) → cc(e) defined by l′(f) = l(f)k0 = {l(f)k; k ∈ k0} for all f ∈ c+(t). the map l′ is positive, positively homogeneous and strictly continuous. let us prove that l′ is additive. the inclusion l′(f + g) ⊂ l′(f) + l′(g) for all f,g ∈ c+(t) is trivial. let u ∈ k0 and each let v ∈ k0, l(f)u + l(g)v = l(f + g) [ l(f) l(f+g) u + l(g) l(f+g) v ] . since k0 is convex and l positive, l(f) l(f+g) u + l(g) l(f+g) v ∈ k0. then l′(f) + l′(g) ⊂ l′(f + g). on the integral representation 119 by the theorem 3.6, there is a unique positive regular set-valued measure m : b(t) → cc(e) that satisfies the condition ∫ fm = l′(f) for all f ∈ c+(t). let y0 ∈ e′ such that δ∗(y0|l′(.)) = l. since δ∗(y0| ∫ fm) = ∫ fδ∗(y0|m(.)) for all f ∈ c+(t) we then have ∫ fδ∗(y0|m(.)) = l(f) for all f ∈ c+(t) and therefore∫ fδ∗(y0|m(.)) = l(f) for all f ∈ c(t). the uniqueness of δ∗(y0|m(.)) follows from the regularity of m. taking the lemma 3.3 (for the scalar measures) into account we conclude that there is a bijection between the dual space of (c(t),β) and the space of all bounded signed regular borel measures on t . � references [1] r. c. buck, bounded continuous functions on locally compact space, michigan math. j. 5 (1958), 95–104. [2] r. c. buck, operator algebras and dual spaces, proc. amer. math. soc. 3.681– 687 (1952). [3] bourbaki, eléments de maths. livre vi intégration-chp. ix, ed. hermann, paris 1969. [4] a. choo, strict topology on spaces of continuous vector-valued functions, canad. j. math. 31 (1979), 890–896. [5] a. choo, separability in the strict topology, j. math. anal. appl. 75 (1980), 219–222. [6] h. s collins, on the space l∞(s), with the stict topology, math. zeitschr. 106, 361–373 (1968). [7] h. s. collins and j. r. dorroh, remarks on certain function spaces, math. ann., 176, (1968), 157–168 . [8] j. b. conway, the strict topology and compactness in the space of measures, bull. amer. math. soc. 72, (1966), 75–78 . [9] , j. diestel, sequences and series in banach spaces, graduate texts in math., vol.92, springer-verlag, 1984. [10] drewnowsky, topological rings of sets, continuous set functions, integration. iii, bull. acad. polon. sci., sér. sci. math., astronom. et phys., 20 (1972), 441–445. [11] n. dunford and j. schwartz, linear operators part i, new york: interscience 1958. [12] r. a. fontenot, strict topologies for vector-valued functions, canadian. j. math. 26 (1974), 841–853. [13] r. giles, a generalization of the strict topology, trans. amer. math. soc. 161(1971), 467–474. 120 lakmon and siggini [14] d. gulick, the σ-compact-open topology and its relatives, math. scand.. 30 (1972), 159–176. [15] j. hoffman-jörgenson, a generalization of strict topology, math. scand. 30 (1972), 313–323. [16] l. hörmander, sur la fonction d’appui des ensembles convexes dans un espace localement convexe, arkiv för matematik. 3 nr 12 (1954). [17] a. k. katsaras, on the strict topology in the non-locally convex setting ii, acta. math. hung. 41 (1-2) (1983), 77–88. [18] a. k. katsaras, some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals, trans. amer. math. soc. 216 (1979), 367–387. [19] l. a. khan, the strict topology on a space of vector-valued functions, proc. edinburgh math. soc., 22 (1979), 35–41. [20] g. köthe, topological vector spaces i second printing, springer-verlag, newyork, 1983. [21] r. pallu de la barriere, publications mathématiques de l’université pierre et marie curie no33. [22] f. d. sentilles, bounded continuous functions on a completely regular space, trans. amer. math. soc., 168 (1972), 311–336. [23] k. k. siggini, narrow convergence in spaces of set-valued measures, bull. of the polish acad. of sc. math. vol. 56, n◦1, (2008). [24] k. k. siggini, sur les propriétés de régularité des mesures vectorielles et multivoques sur les espaces topologiques généraux, thèse de doctorat de l’université de paris 6. [25] c. todd, stone-weierstrass theorems for the strict topolgy, proc. amer. math. soc. 16 (1965), 657–659. [26] a. c. m. van rooij, tight functionals and the strict topology, kyungpook math.j.7 (1967), 41–43. [27] j. wells, bounded continuous vector-valued functions on a locally compact space, michigan math. j. 12 (1965), 119–126. [28] x. xiaoping, c. lixin, l. goucheng, y. xiaobo, set valued measures and integral representation, comment.math.univ.carolin. 37,2 (1996)269–284 university of lomé, faculty of sciences, department of mathematics, bp 1515 lométogo ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 88-98 http://www.etamaths.com generalized steffensen inequalities for local fractional integrals mehmet zeki sarikaya1, tuba tunç1,∗ and samet erden2 abstract. firstly we give a important integral inequality which is generalized steffensen’s inequality. then, we establish weighted version of generalized steffensen’s inequality for local fractional integrals. finally, we obtain several inequalities related these inequalities using the local fractional integral. 1. introduction in [17], j. s. steffensen established the following result which is known as steffensen’s inequality in the literature. theorem 1.1. let a and b be real numbers such that a < b, f,g : [a,b] → r be integrable functions such that f is nonincreasing and for every x ∈ [a,b] , 0 ≤ g(x) ≤ 1. then b∫ b−λ f(x)dx ≤ b∫ a f(x)g(x)dx ≤ a+λ∫ a f(x)dx (1.1) where λ = b∫ a g(x)dx. the most basic inequality which deals with the comparison between integrals over a whole interval [a,b] and integrals over a subset of [a,b] is the following inequality. the inequality (1.1) has attracted considerable attention and interest from mathematicans and researchers. due to this, over the years, the interested reader is also refered to ( [1], [2], [4][7], [10], [11] and [18]) for integral inequalities. in [19], wu and srivastava proved the following inequality which is weighted version of the inequality (1.1). theorem 1.2. let f, g and h be integrable functions defined on [a,b] with f nonincreasin. also let 0 ≤ g(x) ≤ h(x) for all x ∈ [a,b] . then, the following inequalities hold: b∫ b−λ f(x)h(x)dx ≤ b∫ b−λ (f(x)h(x) − [f(x) −f(b−λ)] [h(x) −g(x)]) dx ≤ b∫ a f(x)g(x)dx ≤ a+λ∫ a (f(x)h(x) − [f(x) −f(a + λ)] [h(x) −g(x)]) dx ≤ a+λ∫ a f(x)h(x)dx 2010 mathematics subject classification. 26d15, 26a33. key words and phrases. steffensen’s inequality; local fractional integral; fractal space; generalized convex function. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 88 generalized steffensen inequalities 89 where λ is given by a+λ∫ a h(x)dx = b∫ a g(x)dx = b∫ b−λ h(x)dx. 2. preliminaries recall the set rα of real line numbers and use the gao-yang-kang’s idea to describe the definition of the local fractional derivative and local fractional integral, see [20, 21] and so on. recently, the theory of yang’s fractional sets [20] was introduced as follows. for 0 < α ≤ 1, we have the following α-type set of element sets: zα : the α-type set of integer is defined as the set {0α,±1α,±2α, ...,±nα, ...} . qα : the α-type set of the rational numbers is defined as the set {mα = ( p q )α : p,q ∈ z, q 6= 0}. jα : the α-type set of the irrational numbers is defined as the set {mα 6= ( p q )α : p,q ∈ z, q 6= 0}. rα : the α-type set of the real line numbers is defined as the set rα = qα ∪jα. if aα,bα and cα belongs the set rα of real line numbers, then (1) aα + bα and aαbα belongs the set rα; (2) aα + bα = bα + aα = (a + b) α = (b + a) α ; (3) aα + (bα + cα) = (a + b) α + cα; (4) aαbα = bαaα = (ab) α = (ba) α ; (5) aα (bαcα) = (aαbα) cα; (6) aα (bα + cα) = aαbα + aαcα; (7) aα + 0α = 0α + aα = aα and aα1α = 1αaα = aα. the definition of the local fractional derivative and local fractional integral can be given as follows. definition 2.1. [20] a non-differentiable function f : r → rα, x → f(x) is called to be local fractional continuous at x0, if for any ε > 0, there exists δ > 0, such that |f(x) −f(x0)| < εα holds for |x−x0| < δ, where ε,δ ∈ r. if f(x) is local continuous on the interval (a,b) , we denote f(x) ∈ cα(a,b). definition 2.2. [20] the local fractional derivative of f(x) of order α at x = x0 is defined by f(α)(x0) = dαf(x) dxα ∣∣∣∣ x=x0 = lim x→x0 ∆α (f(x) −f(x0)) (x−x0) α , where ∆α (f(x) −f(x0)) =̃γ(α + 1) (f(x) −f(x0)) . if there exists f(k+1)α(x) = k+1 times︷ ︸︸ ︷ dαx ...d α xf(x) for any x ∈ i ⊆ r, then we denoted f ∈ d(k+1)α(i), where k = 0, 1, 2, ... definition 2.3. [20] let f(x) ∈ cα [a,b] . then the local fractional integral is defined by, ai α b f(x) = 1 γ(α + 1) b∫ a f(t)(dt)α = 1 γ(α + 1) lim ∆t→0 n−1∑ j=0 f(tj)(∆tj) α, with ∆tj = tj+1 − tj and ∆t = max{∆t1, ∆t2, ..., ∆tn−1} , where [tj, tj+1] , j = 0, ...,n − 1 and a = t0 < t1 < ... < tn−1 < tn = b is partition of interval [a,b] . here, it follows that ai α b f(x) = 0 if a = b and ai α b f(x) = −bi α a f(x) if a < b. if for any x ∈ [a,b] , there exists ai α x f(x), then we denoted by f(x) ∈ iαx [a,b] . lemma 2.1. [20] we have i) dαxkα dxα = γ(1 + kα) γ(1 + (k − 1) α) x(k−1)α; ii) 1 γ(α + 1) b∫ a xkα(dx)α = γ(1 + kα) γ(1 + (k + 1) α) ( b(k+1)α −a(k+1)α ) , k ∈ r. 90 sarikaya the interested reader is able to look over the references [3], [8], [9], [12][16], [20][24] for local freactional theory. in this study, generalized steffensen’s inequality is established. then, some inequalities related generalized this inequality are given by using local fractional integrals. 3. main results we start the following important inequality for local fractional integrals: theorem 3.1 (generalized steffensen’s inequality). let f(x),g(x) ∈ iαx [a,b] such that f never increases and 0 ≤ g(x) ≤ 1 on [a,b] with a < b. then b−λi α b f(x) ≤ ai α b f(x)g(x) ≤ a i α a+λf(x) (3.1) where λα = γ(α + 1) ai α b g(x). (3.2) proof. for the proof of theorem, we give two different methods: first method: by direct computation, we get 1 γ(α + 1) a+λ∫ a f(x)(dx)α − aiαb f(x)g(x) (3.3) = 1 γ(α + 1) a+λ∫ a [f(x) −f (a + λ)] [1 −g(x)] (dx)α + 1 γ(α + 1) b∫ a+λ [f (a + λ) −f(x)] g(x)(dx)α. using the equality (3.3), because f is nonincreasing, we obtain the second inequality of (3.1). similarly, we have ai α b f(x)g(x) − 1 γ(α + 1) b∫ b−λ f(x)(dx)α (3.4) = 1 γ(α + 1) b−λ∫ a [f(x) −f (b−λ)] g(x)(dx)α + 1 γ(α + 1) b∫ b−λ [f (b−λ) −f(x)] [1 −g(x)] (dx)α. using the equality (3.4), because f is nonincreasing, we obtain the first inequality of (3.1). thus, the proof is completed. second method: now, we prove the same of above theorem in a deffirent way. generalized steffensen inequalities 91 because f is nonincreasing, the second inequality of (3.1) may be derived as follows: 1 γ(α + 1) a+λ∫ a f(x)(dx)α − aiαb f(x)g(x) = 1 γ(α + 1) a+λ∫ a f(x) [1 −g(x)] (dx)α − 1 γ(α + 1) b∫ a+λ f(x)g(x)(dx)α ≥ f(a + λ) γ(α + 1) a+λ∫ a [1 −g(x)] (dx)α − 1 γ(α + 1) b∫ a+λ f(x)g(x)(dx)α = f(a + λ) γ(α + 1)  λα − a+λ∫ a g(x)(dx)α  − 1 γ(α + 1) b∫ a+λ f(x)g(x)(dx)α = 1 γ(α + 1) b∫ a+λ [f(a + λ) −f(x)] g(x)(dx)α ≥ 0. the first inequality of (3.1) can be proved in a similar way. however, the second inequality implies the first. indeed, let g(x) = 1 −g(x) and λα = γ(α + 1) ai α b g(x). note that 0 ≤ g(x) ≤ 1 if 0 ≤ g(x) ≤ 1 in (a,b) . suppose the second inequality of (3.1) holds. then, we obtain 1 γ(α + 1) b∫ a f(x)g(x)(dx)α ≤ 1 γ(α + 1) a+λ∫ a f(x)(dx)α 1 γ(α + 1) b∫ a f(x)(dx)α − 1 γ(α + 1) a+λ∫ a f(x)(dx)α ≤ 1 γ(α + 1) b∫ a f(x)g(x)(dx)α 1 γ(α + 1) b∫ a+λ f(x)(dx)α ≤ 1 γ(α + 1) b∫ a f(x)g(x)(dx)α. because of λα = γ(α + 1) ai α b g(x) = b α −aα −λα we have the identity λ + a = b−λ. (3.5) from (3.5), we get the inequality 1 γ(α + 1) b∫ b−λ f(x)(dx)α ≤ aiαb f(x)g(x) which is the first inequality of (3.1). the proof is thus completed. � in order to prove weighted version of generalized steffensen’s inequality we need the following lemma: 92 sarikaya lemma 3.1. let f, g and h belong to iαx [a,b] . suppose also that λ is a real number such that ai α a+λh(x) = ai α b g(x) = b−λi α b h(x). then, we have ai α b f(x)g(x) (3.6) = 1 γ(α + 1) a+λ∫ a (f(x)h(x) − [f(x) −f(a + λ)] [h(x) −g(x)]) (dx)α + 1 γ(α + 1) b∫ a+λ [f(x) −f(a + λ)] g(x)(dx)α and ai α b f(x)g(x) (3.7) = 1 γ(α + 1) b∫ b−λ (f(x)h(x) − [f(x) −f(b−λ)] [h(x) −g(x)]) (dx)α + 1 γ(α + 1) b−λ∫ a [f(x) −f(b−λ)] g(x)(dx)α. proof. the essumptions of the lemma imply that a ≤ a + λ ≤ b and a ≤ b−λ ≤ b. firstly, we prove the validity of the equality (3.6). indeed, by direct computation, we find that ai α b f(x)g(x) (3.8) = 1 γ(α + 1) a+λ∫ a f(x)g(x)(dx)α + 1 γ(α + 1) b∫ a+λ f(x)g(x)(dx)α + f(a + λ) γ(α + 1)   b∫ a g(x)(dx)α − a+λ∫ a g(x)(dx)α − b∫ a+λ g(x)(dx)α   now, if we apply the following assumption of the lemma: ai α a+λh(x) = ai α b g(x) to (3.8), we obtain ai α b f(x)g(x) (3.9) = 1 γ(α + 1) a+λ∫ a (f(x)g(x) + f(a + λ) [h(x) −g(x)]) (dx)α + 1 γ(α + 1) b∫ a+λ [f(x) −f(a + λ)] g(x)(dx)α. if we add 1 γ(α + 1) a+λ∫ a f(x)h(x)(dx)α − 1 γ(α + 1) a+λ∫ a f(x)h(x)(dx)α to right side of (3.9) and also we use elementary analysis, then we easily get the equality (3.6). secondly, if we apply above the operations for the following assumption of the lemma ai α b g(x) = b−λi α b h(x) generalized steffensen inequalities 93 and also we consider the case a ≤ b − λ ≤ b, then we obtain the equality (3.7). thus, the proof is completed. � now, we prove weighted version generalized steffensen’s inequality using local fractional integrals. theorem 3.2. let f, g and h belong to iαx [a,b] with f nonincreasing. suppose also that 0 ≤ g(x) ≤ h(x) for all x ∈ [a,b] . then, we have the following inequalities b−λi α b f(x)h(x) (3.10) ≤ 1 γ(α + 1) b∫ b−λ (f(x)h(x) − [f(x) −f(b−λ)] [h(x) −g(x)]) (dx)α ≤ aiαb f(x)g(x) ≤ 1 γ(α + 1) a+λ∫ a (f(x)h(x) − [f(x) −f(a + λ)] [h(x) −g(x)]) (dx)α ≤ aiαa+λf(x)h(x) where λ is given by ai α a+λh(x) = ai α b g(x) = b−λi α b h(x). proof. in view of the assumptions that the function f is nonincreasing on [a,b] and that 0 ≤ g(x) ≤ h(x) for all x ∈ [a,b] , we find that 1 γ(α + 1) b−λ∫ a [f(x) −f(b−λ)] g(x)(dx)α ≥ 0, (3.11) 1 γ(α + 1) b∫ b−λ [f(b−λ) −f(x)] [h(x) −g(x)] (dx)α ≥ 0, (3.12) 1 γ(α + 1) b∫ a+λ [f(x) −f(a + λ)] g(x)(dx)α ≤ 0, (3.13) and 1 γ(α + 1) a+λ∫ a [f(a + λ) −f(x)] [h(x) −g(x)] (dx)α ≤ 0. (3.14) using the equality (3.7) together with the inequalities (3.11) and (3.12), we obtain that b−λi α b f(x)h(x) (3.15) ≤ 1 γ(α + 1) b∫ b−λ (f(x)h(x) − [f(x) −f(b−λ)] [h(x) −g(x)]) (dx)α ≤ aiαb f(x)g(x). using the equality (3.6) together with the inequalities (3.13) and (3.14) either, we get that ai α b f(x)g(x) (3.16) ≤ 1 γ(α + 1) a+λ∫ a (f(x)h(x) − [f(x) −f(a + λ)] [h(x) −g(x)]) (dx)α ≤ aiαa+λf(x)h(x). 94 sarikaya combining the inequalities (3.15) and (3.16), we easily deduce required inequalities. � in particular, if we chose h(t) = 1 in (3.10), we obtain the following refinement of generalized steffensen’s inequality. corollary 3.1. let f(x),g(x) ∈ iαx [a,b] such that f never increases and 0 ≤ g(x) ≤ 1 on [a,b] with a < b. then b−λi α b f(x) ≤ 1 γ(α + 1) b∫ b−λ (f(x) − [f(x) −f(b−λ)] [1 −g(x)]) (dx)α ≤ aiαb f(x)g(x) ≤ 1 γ(α + 1) a+λ∫ a (f(x) − [f(x) −f(a + λ)] [1 −g(x)]) (dx)α ≤ aiαa+λf(x) where λα = γ(α + 1) ai α b g(x). theorem 3.3. let f, g and h belong to iαx [a,b] with f nonincreasing. also let 0 ≤ ψ(x) ≤ g(x) ≤ h(x) −ψ(x) for all x ∈ [a,b] . then we have the inequalities b−λi α b f(x)h(x) + 1 γ(α + 1) b∫ a |[f(x) −f(b−λ)] ψ(x)|(dx)α ≤ aiαb f(x)g(x) ≤ aiαa+λf(x)h(x) − 1 γ(α + 1) b∫ a |[f(x) −f(a + λ)] ψ(x)|(dx)α where λ is given by ai α a+λh(x) = ai α b g(x) = b−λi α b h(x). proof. by the assumptions that the function f is nonincreasing on [a,b] and that 0 ≤ ψ(x) ≤ g(x) ≤ h(x) −ψ(x) generalized steffensen inequalities 95 for all x ∈ [a,b] , it follows that 1 γ(α + 1) a+λ∫ a [f(x) −f(a + λ)] [h(x) −g(x)] (dx)α (3.17) + 1 γ(α + 1) b∫ a+λ [f(a + λ) −f(x)] g(x)(dx)α = 1 γ(α + 1) a+λ∫ a |f(x) −f(a + λ)| [h(x) −g(x)] (dx)α + 1 γ(α + 1) b∫ a+λ |f(a + λ) −f(x)|g(x)(dx)α ≥ 1 γ(α + 1) a+λ∫ a |f(x) −f(a + λ)|ψ(x)(dx)α + 1 γ(α + 1) b∫ a+λ |f(a + λ) −f(x)|ψ(x)(dx)α = 1 γ(α + 1) b∫ a |[f(x) −f(a + λ)] ψ(x)|(dx)α. similarly, we find that 1 γ(α + 1) b∫ b−λ [f(b−λ) −f(x)] [h(x) −g(x)] (dx)α (3.18) + 1 γ(α + 1) b−λ∫ a [f(x) −f(b−λ)] g(x)(dx)α ≥ 1 γ(α + 1) b∫ a |[f(x) −f(b−λ)] ψ(x)|(dx)α. if we use the equalities (3.6) and (3.7) and the inequalities (3.17) and (3.18), we obtain required inequalities. � corollary 3.2. under the same assumptions of theorem 3.3 with h(x) = 1 and ψ(x) = mα, then the following inequalities hold: b−λi α b f(x) + mα γ(α + 1) b∫ a |[f(x) −f(b−λ)]|(dx)α ≤ aiαb f(x)g(x) ≤ a+λiαb f(x) − mα γ(α + 1) b∫ a |[f(x) −f(a + λ)]|(dx)α where mα ∈ rα+ ∪{0α} and λα = γ(α + 1) ai α b g(x). 96 sarikaya finally, we give a general result on a considerably improved version of generalized steffensen’s inequality by introducing the additional paramaters λ1 and λ2. theorem 3.4. let f(x),g(x) ∈ iαx [a,b] such that f never increases on [a,b] . also let 0α ≤ λα1 ≤ λ α = γ(α + 1) ai α b g(x) ≤ λ α 2 ≤ (b−a) α and 0 ≤ mα ≤ g(x) ≤ (1 −m)α for all x ∈ [a,b] . then, we have the inequalities (3.19) b−λ1i α b f(x) + f(b) γ(α + 1) (λ−λ1) α + mα γ(α + 1) b∫ a |[f(x) −f(b−λ)]|(dx)α ≤ aiαb f(x)g(x) ≤ aiαa+λ2f(x) + f(b) γ(α + 1) (λ2 −λ) α − mα γ(α + 1) b∫ a |[f(x) −f(a + λ)]|(dx)α. proof. by direct computation, we obtain ai α b f(x)g(x) −a i α a+λ2 f(x) + f(b) γ(α + 1)  λα2 − b∫ a g(x)(dx)α   (3.20) = 1 γ(α + 1)   b∫ a f(x)g(x)(dx)α − a+λ2∫ a f(x)(dx)α   + 1 γ(α + 1)   a+λ2∫ a f(b)(dx)α − b∫ a f(b)g(x)(dx)α   ≤ 1 γ(α + 1) b∫ a [f(x) −f(b)] g(x)(dx)α − 1 γ(α + 1) a+λ∫ a [f(x) −f(b)] (dx)α. because of the following assumption of the theorem 0α ≤ λα1 ≤ λ α ≤ λα2 ≤ (b−a) α , we find that aα ≤ aα + λα ≤ aα + λα2 ≤ b α that is a ≤ a + λ ≤ a + λ2 ≤ b. also, since f is nonincreasing, we have f(x) −f(b) ≥ 0 for all x ∈ [a.b] . onthe other hand, since the hypothesis of the theorem, we we conclude that the function f(x)−f(b) belong to iαx [a,b] and nonincreasing on [a,b] . thus, substituting f(x)−f(b) instead of f(x) in corollary generalized steffensen inequalities 97 3.2, we find that 1 γ(α + 1) b∫ a [f(x) −f(b)] g(x)(dx)α − 1 γ(α + 1) b∫ a+λ [f(x) −f(b)] (dx)α (3.21) ≤ − m γ(α + 1) b∫ a |[f(x) −f(a + λ)]|(dx)α. combining the inequalities (3.20) and (3.21), we obtain ai α b f(x)g(x) −a i α a+λ2 f(x) + f(b) γ(α + 1) (λα2 −λ α) ≤ − mα γ(α + 1) b∫ a |[f(x) −f(a + λ)]|(dx)α. which is the second inequality of (3.19). in a similar way, we can prove that ai α b f(x)g(x) −b−λ1 i α b f(x) − f(b) γ(α + 1)   b∫ a g(x)(dx)α −λα1   ≥ 1 γ(α + 1) b∫ a [f(x) −f(b)] g(x)(dx)α + 1 γ(α + 1) b∫ b−λ [f(b) −f(x)] (dx)α ≥ mα γ(α + 1) b∫ a |[f(x) −f(b−λ)]|(dx)α which is the first inequality of (3.19). the proof is thus completed. � references [1] s. abramovich, m.k. bakula, m. matic´, j.e. pečarić, a variant of jensen–steffensen’s inequality and quasiarithmetic means, j. math. anal. appl., 307 (2005), 370–386. [2] j. bergh, a generalization of steffensen’s inequality, j. math. anal. appl., 41 (1973), 187–191. [3] g-s. chen, generalizations of hölder’s and some related integral inequalities on fractal space, journal of function spaces and applications 2013 (2013), article id 198405. [4] h. gauchman, on a further generalization of steffensen’s inequality, j. inequal. appl., 5 (2000), 505–513. [5] f. qi, b.-n. guo, on steffensen pairs, j. math. anal. appl. 271 (2002) 534–541. [6] d. s. mitrinović, analytic inequalities, springer-verlang new-york, heidelberg, berlin, 1970, pp. 107-118. [7] d. s. mitrinović, j. e. pečarič, and a. m. fink, classical and new inequalities in analysis, ser. math. appl. (east european ser.). dordrecht: kluwer academic publishers group, 1993, vol. 61, pp. 311-331. [8] h. mo, x sui and d yu, generalized convex functions on fractal sets and two related inequalities, abstract and applied analysis, 2014 (2014), article id 636751. [9] h. mo, generalized hermite-hadamard inequalities involving local fractional integral, arxiv:1410.1062. [10] j. e. pečarić, on the bellman generalization of steffensen’s inequality. ii, j. math. anal. appl. 104 (1984) 432–434. [11] j. e. pečarić, on the bellman generalization of steffensen’s inequality, j. math. anal. appl. 88 (1982) 505–507. [12] m. z. sarikaya and h budak, generalized ostrowski type inequalities for local fractional integrals, rgmia res. rep. collect. 18 (2015), article id 62. [13] m. z. sarikaya, s.erden and h. budak, some generalized ostrowski type inequalities involving local fractional integrals and applications, rgmia res. rep. collect. 18 (2015), article id 63. [14] m. z. sarikaya h. budak, on generalized hermite-hadamard inequality for generalized convex function, rgmia res. rep. collect. 18 (2015), article id 64. [15] m. z. sarikaya, s.erden and h. budak, some integral inequalities for local fractional integrals, rgmia res. rep. collect. 18(2015), article id 65. [16] m. z. sarikaya, h. budak and s.erden, on new inequalities of simpson’s type for generalized convex functions, rgmia res. rep. collect. 18(2015), article id 66. [17] j. f. steffensen, on certain inequalities and methods of approximation, j. inst. actuaries, 51 (1919), 274–297. 98 sarikaya [18] u. m. ozkan and h. yildirim, steffensen’s integral inequality on time scales, j. inequal. appl. 2007 (2007), article id 46524. [19] s.-h. wu and h.m. srivastava, some improvements and generalizations of steffensen’s integral inequality, appl. math. comput. 192 (2007) 422-428. [20] x. j. yang, advanced local fractional calculus and its applications, world science publisher, new york, 2012. [21] j. yang, d. baleanu and x. j. yang, analysis of fractal wave equations by local fractional fourier series method, adv. math. phys. 2013 (2013), article id 632309. [22] x. j. yang, local fractional integral equations and their applications, adv. comput. sci. appl. 1 (4) 2012, 234-239. [23] x. j. yang, generalized local fractional taylor’s formula with local fractional derivative, journal of expert systems, 1(1) (2012), 26-30. [24] x. j. yang, local fractional fourier analysis, advances in mechanical engineering and its applications, 1 (1) 2012, 12-16. 1department of mathematics, faculty of science and arts, düzce university, düzce-turkey 2department of mathematics, faculty of science, bartın university, bartın-turkey ∗corresponding author: tubatunc03@gmail.com 1. introduction 2. preliminaries 3. main results references int. j. anal. appl. (2023), 21:86 numerical computation of spectral solutions for sturm-liouville eigenvalue problems sameh gana∗ department of basic sciences, deanship of preparatory year and supporting studies, imam abdulrahman bin faisal university, p.o. box 1982, dammam, 34212, saudi arabia ∗corresponding author:sbgana@iau.edu.sa abstract. this paper focuses on the study of sturm-liouville eigenvalue problems. in the classical chebyshev collocation method, the sturm-liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled chebyshev points. we are concerned with the computation of high-order eigenvalues of sturm-liouville problems using an effective method of discretization based on the chebfun software algorithms with domain truncation. we solve some numerical sturm-liouville eigenvalue problems and demonstrate the efficiency of computations. 1. introduction the sturm-liouville problem arises in many applied mathematics, science, physics and engineering areas. many biological, chemical and physical problems are described by using models based on sturmliouville equations. for example, problems with cylindrical symmetry, diffraction problems (astronomy) resolving power of optical instruments and heavy chains. in quantum mechanics, the solutions of the radial schrödinger equation describe the eigenvalues of the sturm-liouville problem. these solutions also define the bound state energies of the non-relativistic hydrogen atom. for more applications, see [1], [2] and [3]. in this paper, we consider the sturm-liouville problem − d dx [p(x) d dx ]y +q(x)y = λw(x)y,a ≤ x ≤ b, (1.1) cay(a)+day ′(a)=0, (1.2) received: may 30, 2023. 2020 mathematics subject classification. 34l05, 34l40, 81q20, 74s25. key words and phrases. sturm-liouville problems; spectral method; differential equations; chebyshef spectral collocation; chebfun; chebop; matlab. https://doi.org/10.28924/2291-8639-21-2023-86 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-86 2 int. j. anal. appl. (2023), 21:86 cby(b)+dby ′(b)=0, (1.3) where p(x) > 0 , w(x) > 0, ca, da, cb and db are constants. there is a great interest in developing accurate and efficient methods of solutions for sturm-liouville problems. the purpose of this paper is to determine the solution of some sturm-liouville problems using the chebfun package and to demonstrate the highest performance of the chebfun system compared with classical spectral methods in solving such problems. there are many different methods for the numerical solutions of differential equations, which include finite difference, finite element techniques, galerkin methods, taylor collocation method and chebyshef collocation method. spectral methods provide exponential convergence for several problems, generally with smooth solutions. the chebfun system provides greater flexibility in solving various differential problems than the classical spectral methods. many packages solve sturm-liouville problems such as matslise [12], sledge [13], sleign [14]. however, these numerical methods are not suitable for the approximation of the high-index eigenvalues for sturm-liouville problems. the main purpose of this paper is to assert that chebfun, along with the spectral collocation methods, can provide accuracy, robustness and simplicity of implementation. in addition, these methods can compute the whole set of eigenvectors and provide some details on the accuracy and numerical stability of the results provided. for more complete descriptions of the chebyshef collocation method and more details on the chebfun software system, we refer to [4], [5], [6], [8] and [9]. in this paper, we explain in section 2 the concept of the chebfun system and chebyshev spectral collocation methodology. then, in section 3, some numerical examples demonstrate the method’s accuracy. finally, we end up with the conclusion section. 2. chebfun system and chebyshev spectral collocation methodology the chebfun system, in object-oriented matlab, contains algorithms that amount to spectral collocation methods on chebyshev grids of automatically determined resolution. the chebops tools in the chebfun system for solving differential equations are summarized in [15] and [16]. the implementation of chebops combines the numerical analysis idea of spectral collocation with the computer science idea of the associated spectral discretization matrices. the chebfun system explained in [8] solves the eigenproblem by choosing a reference eigenvalue and checks the convergence of the process. the central principle of the chebfun, along with chebops, can accurately solve highly sturmliouville problems. int. j. anal. appl. (2023), 21:86 3 the spectral collocation method for solving differential equations consists of constructing weighted interpolants of the form [4]: y(x)≈ pn(x)= n∑ j=0 α(x) α(xj) φj(x)yj, (2.1) where xj for j =0, ....,n are interpolation nodes, α(x) is a weight function, yj = y(xj), and the interpolating functions φj(x) satisfy φj(xk)= δj,k and y(xk)= pn(xk) for k =0, ....,n. hence pn(x) is an interpolant of the function y(x). by taking l derivatives of 2.1 and evaluating the result at the nodes xj, we get: y(l)(xk)≈ n∑ j=0 d l dx l [ α(x) α(xj) φj(x) ] x=xk , k =0, ....,n. the entries define the differentiation matrix: d (l) k,j = d l dx l [ α(x) α(xj) φj(x) ] x=xk . the derivatives values y(l) are approximated at the nodes xk by d(l)y. the derivatives are converted to a differentiation matrix form and the differential equation problem is transformed into a matrix eigenvalue problem. our interest is to compute the solutions of sturm-liouville problems defined in 1.1 with high accuracy. first, we rewrite 1.1 in the following form: − d2y dx2 − p̃(x) dy dx + q̃(x)y = λw̃(x)y, (2.2) where p̃(x) = p ′(x) p(x) , q̃(x) = q(x) p(x) , w̃(x) = w(x) p(x) defined in the canonical interval [−1,1] . since the differential equation is posed on [a,b], it should be converted to [−1,1] through the change of variable x to 1 2 ((b−a)x +b+a). the eigenfunctions y(x) of the eigenvalue problem approximate finite terms of chebyshev polynomials as pn(x)= n∑ j=0 φj(x)yj, (2.3) 4 int. j. anal. appl. (2023), 21:86 where the weight function w̃(x)=1, φj(x) is the chebyshev polynomial of degree ≤ n and yj = y(xj), the chebyshev collocation points are defined by: xj = cos( jπ n ), j =0, ....,n. (2.4) a spectral differentiation matrix for the chebyshev collocation points is created by interpolating a polynomial through the collocation points, i.e., the polynomial pn(xk)= n∑ j=0 φj(xk)yj. the derivatives values of the interpolating polynomial 2.3 at the chebyshev collocation points 2.4 are: p (l) n (x)= n∑ j=0 φ (l) j (xk)yj. the differentiation matrix d(l) with entries d (l) k,j = φ (l) j (xk) is explicitly determined in [6] and [7]. if we rewrite equation 2.2 using the differentiation matrix form, we get (−d(2) n − p̃d(1) n + q̃)y = λw̃y, where p̃ = diag(p̃), q̃ = diag(q̃) and w̃ = diag(w̃). the boundary conditions 1.2 and 1.3 can be determined by: c1pn(1)+d1p ′ n(1)=0, c−1pn(−1)+d−1p ′n(−1)=0. then the sturm-liouville eigenvalue problem, defined as a block operator, is transformed into a discretization matrix diagram:  −d(2) n − p̃d(1) n + q̃ c1i +d1d (1) n c−1i +d−1d (1) n  y = λ   w̃i 0 0  y. (2.5) the approximate solutions of the sturm-liouville problem defined in 1.1 with boundary conditions 1.2 and 1.3 are determined by solving the generalized eigenvalue problem 2.5. for more details on convergence rates, the collocation differentiation matrices and the efficiency of the chebyshev collocation method, see [7]. int. j. anal. appl. (2023), 21:86 5 3. numerical computations in this section, we apply the chebyshev spectral collocation methodology outlined in the previous section, chebfun and chebop system described in [8] and [9] to some sturm-liouville problems. we examine the accuracy and efficiency of this methodology in a selected variety of examples. in each example, the relative error measures the technique’s efficiency. en = |λexactn −λn| |λexactn | where λexactn for n =0,1,2, ..... are the exact eigenvalues and λn are the numerical eigenvalues. example 1. we consider the sturm-liouville eigenvalue problem studied in [10] − d2y dx2 = λw(x)y, (3.1) where w(x) > 0 , y(0)= y(π)=0. the eigenvalue problem has an infinite number of non-trivial solutions: the eigenvalues λ1,λ2,λ3, ..... are discrete, positive real numbers and non-degenerate. the eigenfunctions yn(x) associated with different eigenvalues λn are orthogonal with respect to the weight function w(x). using the wkb theory, we approximate λn and yn(x) when n is large by the formulas: λn ∼ [ nπ∫π 0 √ w(t)dt ]2 , n −→∞. and yn(x)∼ [∫ π 0 √ w(t) 2 dt ]−1 2 w− 1 4(x)sin [ nπ ∫ x 0 √ w(t)dt∫π 0 √ w(t)dt ] , n −→∞. we choose the weight function w(x)= (x+π)4, then the sturm-liouville problem 3.1 is transformed to − d2y dx2 = λ(x +π)4y, y(0)= y(π)=0. (3.2) then, the approximate eigenvalues and eigenfunctions of the eigenvalue problem 3.2 are given by λn ∼ 9n2 49π4 , n −→∞ and yn(x)∼ √ 6 7π3 sin [ n(x3+3x2π+3π2x) 7π2 ] π +x , n −→∞. the chebyshev collocation approach to solve 3.2 consists of constructing the (n +1)× (n +1) second derivative matrix d(2) n associated with the nodes 2.4, but shifted from [−1,1] to [0,π]. the incorporation of the boundary conditions y(0) = y(π) = 0 requires that the first and last rows of the matrix d(2) n are removed, as well as its first and last columns, see [5]. 6 int. j. anal. appl. (2023), 21:86 the collocation approximation of the differential eigenvalue problem 3.2 is now represented by the (n −1)× (n −1) matrix eigenvalue problem −d(2) n y = λw̃y (3.3) where w̃ = diag(w)= diag((xj+π)4) and y is the vector of approximate eigenfunction at the interior nodes xj. the convergence rate can be estimated theoretically. fitting the regularity ellipse (defined in [17]) for chebyshev interpolation through the pole at x =−π indicates a convergence rate of o( 1 (3 √ 8)n )' o(0.17n). the typical rate of convergence in polynomial interpolation (and also differentiation) is exponential, where the decay rate determines the singularity’s location concerning the interval (see [5] and [17]). we approximate the solutions of the sturm-liouville problem 3.2 by solving the matrix eigenvalue problem 3.3 using a chebfun code: l = chebop(0,pi) ; l.op = @(x,u) -(pi+x)^-4*diff(u,2) ; l.bc ='dirichlet '; n = 40; [v, d] = eigs(l,n ) ; diag(d) in table 1, we compute the first forty eigenvalues and the related relative error between the numerical calculation and the exact solution. we consider the wkb approximations by bender and orzag [10] as the exact solutions for the calculations of errors since there is no explicit form of eigenvalues. in table 2, we compute the numerical values of some eigenvalues with a high index of the problem 3.2. it is clear that the eigenvalues as n increases are approximately calculated with an accuracy better than the low-order eigenvalues. the numerical results in table 1 and table 2 by chebfun algorithms closely match the exact eigenvalues of the sturm–liouville problem in example 1. figure 1 shows the numerical computations of some eigenfunctions for n = 1, 20, 50 and 100. int. j. anal. appl. (2023), 21:86 7 table 1. computations of the first forty eigenvalues λn and the relative error in example 1 n λn current work λwkbn relative error en = | λwkbn −λn λwkbn | λn( [10]) 1 0.001744014 0.001885589 0.075082675 0.00174401 2 0.007348655 0.007542354 0.025681583 0.734865 3 0.016752382 0.016970297 0.012840988 0.0167524 4 0.029938276 0.030169417 0.00766145 0.0299383 5 0.046900603 0.047139714 0.0050724 0.0469006 6 0.067636933 0.067881189 0.003598284 7 0.092146088 0.09239384 0.002681481 8 0.120427442 0.120677669 0.002073519 9 0.152480637 0.152732675 0.001650191 10 0.188305458 0.188558858 0.001343874 11 0.227901771 0.228156218 0.00111523 12 0.271269487 0.271524755 0.00094013 13 0.318408545 0.31866447 0.000803115 14 0.369318906 0.369575361 0.000693919 15 0.424000539 0.42425743 0.000605508 16 0.482453422 0.482710676 0.000532935 17 0.544677542 0.544935099 0.000472638 18 0.610672885 0.610930699 0.000422002 19 0.680439443 0.680697476 0.000379072 20 0.753977208 0.754235431 0.000342363 0.753977 21 0.831286176 0.831544563 0.00031073 22 0.912366343 0.912624871 0.00028328 23 0.997217704 0.997476357 0.000259308 24 1.085840256 1.086099021 0.000238251 25 1.178233999 1.178492861 0.000219655 26 1.274398929 1.274657878 0.000203152 27 1.374335046 1.374594073 0.000188439 28 1.478042348 1.478301445 0.000175266 29 1.585520834 1.585779994 0.000163427 30 1.696770504 1.69702972 0.000152747 31 1.811791355 1.812050623 0.00014308 32 1.930583389 1.930842703 0.000134301 33 2.053146604 2.053405961 0.000126306 34 2.179480999 2.179740395 0.000119003 35 2.309586575 2.309846007 0.000112316 36 2.443463331 2.443722796 0.000106176 37 2.581111267 2.581370762 0.000100526 38 2.722530382 2.722789906 9.53152e-05 39 2.867720677 2.867980226 9.0499e-05 40 3.016682151 3.016941724 8.60386e-05 3.01668 8 int. j. anal. appl. (2023), 21:86 table 2. computations of the high index eigenvalues λn and the relative error in example 1 n =500 n =1000 n λn λwkbn en n λn λ wkb n en 100 18.56689897 18.85588577 0.01532608 500 470.55992 471.3971443 0.001776049 150 41.93487514 42.42574299 0.011570047 600 677.1850808 678.8118879 0.002396551 200 75.01170433 75.4235431 0.005460348 700 921.8333917 923.9384029 0.002278303 250 117.0623718 117.8492861 0.006677293 750 1059.838653 1060.643575 0.0007589 300 169.030728 169.702972 0.003961298 800 1204.898974 1206.77669 0.001555976 350 230.1328596 230.9846007 0.003687437 850 1359.411329 1362.337747 0.002148085 400 300.5074522 301.6941724 0.00393352 900 1525.004757 1527.326748 0.001520297 450 380.7355745 381.8316869 0.002870669 950 1700.149371 1701.743691 0.000936875 figure 1. some eigenfunctions of the sturm liouville problem in example 1 with n = 1, 20, 50, 100. example 2: we consider the sturm-liouville eigenvalue problem − d2y dx2 +x4y = λy (3.4) int. j. anal. appl. (2023), 21:86 9 with the homogeneous boundary conditions y(±∞)=0. in the study of quantum mechanics, if the potential well v (x) rises monotonically as x −→±∞, the differential equation d2y dx2 =(v (x)−e)y, describes a particle of energy e confined to a potential well v (x). the eigenvalue e satisfies ∫ b a √ e −v (x)dx =(n+ 1 2 )π, where the turning points a and b are the two solutions to the equation v (x)−e =0. the wkb eigenfunctions yn(x) satisfy the formula yn(x)=2 √ πc( 3 2 s0) 1 6 [v (x)−e]− 1 4ai( 3 2 s0) 2 3 , where s0 = ∫ x 0 √ (v (t)−e)dt and ai is the airy function (see [10]). thus the eigenvalues of the problem 3.4 satisfy λn ∼ [ 3γ(3 4 )(n+ 1 2 ) √ π γ(1 4 ) ]4 3 , n −→∞ where γ is the gamma function. we use the chebyshev collocation method by discretizing 3.4 in the interval [−d,d] with the boundary conditions y(−d)= y(d)=0: − d2y dx2 +x4y = λy (3.5) with the boundary conditions y(−d)= y(d)=0. the collocation approximation to the differential eigenvalue problem 3.5 is represented by the (n − 1)× (n −1) matrix eigenvalue problem: −d(2) n y + q̃y = λy (3.6) where q̃ = diag(x4j ). we approximate the solutions of the sturm-liouville problem 3.4 by solving the matrix eigenvalue problems 3.5 and 3.6 using a chebfun code: d = 10; l = chebop(-d,d); l . op = @(x,u) diff(u,2)+(x^4)*u; l . bc ='dirichlet '; n = 100; [v, d] = eigs(l ,100); diag(d) 10 int. j. anal. appl. (2023), 21:86 in table 3, we list the numerical results of this method by computing the first thirty eigenvalues and the relative error between this technique and the wkb approximations by bender and orzag [10]. the numerical results in table 3 show the high performance of the current technique. figure 2 shows the numerical computations of relative errors. these results illustrate the high accuracy and efficiency of the algorithms. figure 3 shows some eigenfunctions of the sturm-liouville problem in example 2 for n = 10, 30, 50 and 100. table 3. computations of the first thirty eigenvalues λn of and the relative error in example 2 with d =10 n λn current work λwkbn relative error en = | λwkbn −λn λwkbn | 1 3.7996730297979 3.7519199235504 1.27276454e-02 2 7.4556979379858 7.4139882528108 5.62580945e-03 3 11.6447455113774 11.6115253451971 2.86096488e-03 4 16.2618260188517 16.2336146927052 1.73783391e-03 5 21.2383729182367 21.2136533590572 1.16526648e-03 6 26.5284711836832 26.5063355109631 8.35108750e-04 7 32.0985977109688 32.0784641156416 6.27635889e-04 8 37.9230010270330 37.9044718450677 4.88838943e-04 9 43.9811580972898 43.9639483585989 3.91451162e-04 10 50.2562545166843 50.2401523191723 3.20504552e-04 11 56.7342140551754 56.7190570966241 2.67228676e-04 12 63.4030469867205 63.3887079062501 2.26208751e-04 13 70.2523946286162 70.2387714452705 1.93955319e-04 14 77.2732004819871 77.2602101293507 1.68137682e-04 15 84.4574662749449 84.4450400943621 1.47151101e-04 16 91.7980668089950 91.7861473252516 1.29861467e-04 17 99.2886066604955 99.2771452225694 1.15448907e-04 18 106.923307381733 106.912262402219 1.03308819e-04 19 114.696917384982 114.686253003331 9.29874451e-05 20 122.604639001000 122.594324052793 8.41388726e-05 21 130.642068748629 130.632075959854 7.64956746e-05 22 138.805147911395 138.795453260716 6.98484745e-05 23 147.090121257603 147.080703465973 6.40314563e-05 24 155.493502268682 155.484342386656 5.89119257e-05 25 164.012043622866 164.003124693834 5.43826775e-05 26 172.642711962846 172.634018745858 5.03563379e-05 27 181.382666185766 181.374184925625 4.67611206e-05 28 190.229238652464 190.220956887619 4.35376047e-05 29 199.179918833747 199.171825234752 4.06362646e-05 30 208.232339005093 208.224423237910 3.80155558e-05 int. j. anal. appl. (2023), 21:86 11 figure 2. relative errors e(n) of some high index eigenvalues of the sturm-liouville problem in example 2 with d =10 12 int. j. anal. appl. (2023), 21:86 figure 3. some eigenfunctions of the sturm liouville problem for example 2 with n = 10, 30, 50, 100 and d =10 example 3: we consider the sturm-liouville eigenvalue problem − d2y dx2 = λ (1+x)2 y (3.7) with the boundary conditions y(0)= y(1)= 0. the exact eigenvalues of the problem 3.7 satisfy the explicit formula (see [11]): λn = 1 4 +( πn ln2 )2, n =1,2,3, ... the eigenfunctions yn(x) associated with different eigenvalues λn are given by: yn(x)= √ 1+x sin( πn ln2 ln(1+x)). the collocation approximation of the differential eigenvalue problem 3.7 is now represented by the (n −1)× (n −1) matrix eigenvalue problem −d(2) n y = λw̃y (3.8) int. j. anal. appl. (2023), 21:86 13 where w̃ = diag( 1 (1+xj) 2). now, the approximate eigenvalues of the sturm-liouville problem 3.7 are obtained by solving the matrix eigenvalue problem 3.8 using the chebyshev spectral collocation technique based on chebfun and chebop codes. in table 4, we compute the first thirty eigenvalues and the related absolute error between the numerical calculation and the exact solution en = |λexactn −λn| |λexactn | . the eigenvalues obtained are extremely close to the exact eigenvalues. the results show significant improvement in the convergence. in figure 4, we plot some eigenfunctions in example 3 for n =1,n =30,n =50 and n =80. table 4. computations of the first thirty eigenvalues λn and the relative error in example 3 n λn current work λexactn relative error en = |λexactn −λn| |λexactn | 1 20.79228845522 20.79228845517 2.40469e-12 2 82.4191538209 82.41915382087 3.63913e-13 3 185.13059609701 185.13059609709 4.32157e-13 4 328.92661528358 328.92661528388 9.11961e-13 5 513.8072113806 513.80721137895 3.21132e-12 6 739.77238438806 739.77238439069 3.55512e-12 7 1006.82213430597 1006.82213430587 9.9243e-14 8 1314.95646113432 1314.95646113354 5.93335e-13 9 1664.17536487313 1664.17536487301 7.20502e-14 10 2054.47884552238 2054.47884552193 2.18994e-13 11 2485.86690308208 2485.86690308175 1.32756e-13 12 2958.33953755223 2958.33953755204 6.41739e-14 13 3471.89674893283 3471.89674893313 8.6339e-14 14 4026.53853722387 4026.53853722418 7.70655e-14 15 4622.26490242536 4622.26490242571 7.578e-14 16 5259.0758445373 5259.07584453724 1.13997e-14 17 5936.97136355969 5936.97136355928 6.90036e-14 18 6655.95145949252 6655.95145949275 3.46947e-14 19 7416.0161323358 7416.01613233622 5.65889e-14 20 8217.16538208953 8217.16538208989 4.39108e-14 21 9059.39920875371 9059.3992087539 2.09559e-14 22 9942.71761232833 9942.71761232853 2.00991e-14 23 10867.1205928134 10867.1205928132 1.83894e-14 24 11832.6081502089 11832.6081502087 1.68889e-14 25 12839.1802845148 12839.1802845162 1.09127e-13 26 13886.8369957313 13886.8369957319 4.31718e-14 27 14975.5782838581 14975.5782838589 5.33776e-14 28 16105.4041488954 16105.4041488943 6.82455e-14 29 17276.3145908432 17276.3145908419 7.53159e-14 30 18488.3096097014 18488.3096097008 3.25471e-14 14 int. j. anal. appl. (2023), 21:86 figure 4. some eigenfunctions of the sturm liouville problem for example 3 with n = 1, 30, 50, 80. conclusion the numerical computations prove the efficiency of the technique based on the chebfun and chebop systems. this technique is unbeatable regarding the accuracy, computation speed, and information it provides on the accuracy of the computational process. chebfun provides greater flexibility compared to classical spectral methods. however, in the presence of various singularities, the maximum order of approximation can be reached and the chebfun issues a message that warns about the possible inaccuracy of the computations. the methodology can be used to obtain high-accurate solutions to other sturm-liouville problems, generalized differential equations involving higher order derivatives and non-linear partial differential equations in multiple space dimensions. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2023), 21:86 15 references [1] s. flügge, practical quantum mechanics, springer, berlin, 1994. https://doi.org/10.1007/ 978-3-642-61995-3. [2] j.d pryce, numerical solution of sturm–liouville problems, oxford university press, oxford, 1993. https://orca. cardiff.ac.uk/id/eprint/101057. [3] g.w. hanson, a.b. yakovlev, operator theory for electromagnetics, springer, new york, 2002. https://doi. org/10.1007/978-1-4757-3679-3. [4] c. canuto, m.y. hussaini, a. quarteroni, t.a. zang, spectral methods in fluid dynamics, springer, berlin, 1988. https://doi.org/10.1007/978-3-642-84108-8. [5] b. fornberg, a practical guide to pseudospectral methods, cambridge university press, 1996. https://doi.org/ 10.1017/cbo9780511626357. [6] l.n. trefethen, spectral methods in matlab, siam, 2000. https://doi.org/10.1137/1.9780898719598. [7] j.a. weideman, s.c. reddy, a matlab differentiation matrix suite, acm trans. math. softw. 26 (2000), 465-519. https://doi.org/10.1145/365723.365727. [8] t.a. driscoll, f. bornemann, l.n. trefethen, the chebop system for automatic solution of differential equations, bit numer. math. 48 (2008), 701-723. https://doi.org/10.1007/s10543-008-0198-4. [9] j.l. aurentz, l.n. trefethen, block operators and spectral discretizations, siam rev. 59 (2017), 423-446. https://doi.org/10.1137/16m1065975. [10] c.m. bender, s.a. orszag, advanced mathematical methods for scientists and engineers, mcgraw-hill, new york, 1978. [11] l.d. akulenko, s.v. nesterov, high-precision methods in eigenvalue problems and their applications, chapman and hall/crc, 2004. https://doi.org/10.4324/9780203401286. [12] v. ledoux, m.v. daele, g.v. berghe, matslise: a matlab package for the numerical solution of sturmliouville and schrödinger equations, acm trans. math. softw. 31 (2005), 532-554. https://doi.org/10.1145/ 1114268.1114273. [13] s. pruess, c.t. fulton, mathematical software for sturm-liouville problems, acm trans. math. softw. 19 (1993), 360-376. https://doi.org/10.1145/155743.155791. [14] p.b. bailey, m.k. gordon, l.f. shampine, automatic solution of the sturm-liouville problem, acm trans. math. softw. 4 (1978), 193-208. https://doi.org/10.1145/355791.355792. [15] l.n. trefethen, t.a. driscoll, n. hale, chebfun-numerical computing with functions. http://www.chebfun.org. [16] t.a. driscoll, n. hale, l.n. trefethen, chebfun guide, pafnuty publications, oxford, 2014. [17] e. tadmor, the exponential accuracy of fourier and chebyshev differencing methods, siam j. numer. anal. 23 (1986), 1-10. https://doi.org/10.1137/0723001. https://doi.org/10.1007/978-3-642-61995-3 https://doi.org/10.1007/978-3-642-61995-3 https://orca.cardiff.ac.uk/id/eprint/101057 https://orca.cardiff.ac.uk/id/eprint/101057 https://doi.org/10.1007/978-1-4757-3679-3 https://doi.org/10.1007/978-1-4757-3679-3 https://doi.org/10.1007/978-3-642-84108-8 https://doi.org/10.1017/cbo9780511626357 https://doi.org/10.1017/cbo9780511626357 https://doi.org/10.1137/1.9780898719598 https://doi.org/10.1145/365723.365727 https://doi.org/10.1007/s10543-008-0198-4 https://doi.org/10.1137/16m1065975 https://doi.org/10.4324/9780203401286 https://doi.org/10.1145/1114268.1114273 https://doi.org/10.1145/1114268.1114273 https://doi.org/10.1145/155743.155791 https://doi.org/10.1145/355791.355792 http://www.chebfun.org https://doi.org/10.1137/0723001 1. introduction 2. chebfun system and chebyshev spectral collocation methodology 3. numerical computations example 1 example 2: example 3: conclusion references international journal of analysis and applications volume 16, number 2 (2018), 264-275 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-264 existence and uniqueness of mild solutions for the damped burgers equation in weighted sobolev spaces on the half line mohammadreza foroutan∗ and ali ebadian department of mathematics, payame noor university, p.o.box 19395-3697, tehran, iran ∗corresponding author: foroutan mohammadreza@yahoo.com abstract. this paper addresses an initial boundary value problem for the damped burgers equation in weighted sobolev spaces on half line. first, it introduces two normed spaces and present relations between them, which in turn enables us to analysis the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. the paper also studies the well-posedness of this equation in a semi-infinite interval. 1. introduction nonlinear partial differential equations arise in a large number of mathematical and engineering problems [1, 2, 6, 7, 17, 25, 27]. a number of problems arising in science and engineering are set in semi-infinite domains, such as fluid flows in an infinite strip, nonlinear wave equations in quantum mechanics and so on. burgers equation is one of the well-known equations in mathematics and physics, which is used to describe various kinds of phenomena such as mathematical model of turbulence [3] and the approximate theory of flow through a shock wave traveling in a viscous fluid [4]. in this work, we consider the space variable x in ω = [0,∞), and the damped burgers equation [20, 23, 28] in the following form ϕt −ϕxx + ϕϕx + λϕ = 0, (x,t) ∈ ω × ω, (1.1) with the initial condition ϕ(x, 0) = ϕ0(x), x ∈ ω, (1.2) 2010 mathematics subject classification. 35q53, 35a01, 35a02. key words and phrases. damped burgers equation; hardy inequalities; mild solutions; semi-groups; strong solution. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 264 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-264 int. j. anal. appl. 16 (2) (2018) 265 and the boundary conditions ϕ(0, t) = lim x→∞ ϕ(x,t) = lim x→∞ ϕx(x,t) = 0, t ∈ ω, (1.3) where λ is positive constant. the analytical and numerical methods for solving (1.1) including the pseudospectral methods [24], mixed finite difference and galerkin methods [5], taylor-galerkin and taylor-collocation methods [10], chebyshev spectral collocation methods [15], modified extended backward differentiation formula [13] and modified legendre rational spectral methods among others [19] have received much attention. for more details see [11, 12, 16, 21, 26], among others. but, less attention has been paid to the study of existence and uniqueness of solution for burgers equation. moreover, the study of solutions for the partial differential equations in bounded domains have achieved great success and popularity in recent years, while the study of solutions for the partial differential equations in unbounded domains have only received limited attention. in [8], goubet and shen studied the third-order kdv equation in a framework based on the dual-petrov-galerkin method on finite interval. also in [14] khanal et al. studied the fifth-order kawahara equation in weighted sobolev spaces on finite interval. this work deals with the existence and uniqueness of local mild solutions and of global strong solutions for the even-order equation posed on semi-infinite domains. lu and li [19] employed an algebraic mapping of the form ω(x) = x−1 x+1 and proposed a set of legendre rational functions which are orthogonal in (0,∞). in this paper, two normed spaces with the weight function χ(x) = 1 2 d dx ω(x) = 1 (x+1)2 on the half line are introduced. then the weak formulation of the linearized equation is considered: −ϕxx + λϕ = g. (1.4) the existence and uniqueness of solutions to the above formulation is established with any g in these weighted spaces. furthermore, by applying fixed-point argument we present the uniqueness of a local mild solution of (1.1) in semi-infinite interval. the outline of this paper is as follows. section 2 addresses, the special case of (1.1) in the form (1.4), then two normed spaces are introduced and their properties are investigated. section 3 presents well-posedness results in the weighted spaces and describe the uniqueness of a local mild solution in these spaces. finally, section 4 addresses the existence of a global strong solution in weighted spaces. 2. hardy inequalities in weighted sobolev spaces and representation of bilinear forms in this section, a weak formulation of the boundary value problem for the linear equation is presented as follows:   −ϕxx + λϕ = g, x ∈ ω,ϕ(0, t) = limx→∞ϕ(x,t) = limx→∞ϕx(x,t) = 0, (2.1) int. j. anal. appl. 16 (2) (2018) 266 is presented. we first introduce some notations. let lp(ω) with p ∈ [1, +∞] denote the usual lebesgue space and hk(ω) the usual l2-based space. let hk0 (ω) denote the completion of c ∞ 0 (ω) under h k-norm. define φ(ω) = { ϕ ∈ h10 (ω) : ∫ ω ϕ2(x)χ(x)dx < ∞ } . (2.2) ψ(ω) = { ϕ ∈ φ(ω) : ∫ ω ϕ2x(x)χ(x)dx < ∞ } . (2.3) we define the bilinear form in ψ × ψ by b(ϕ,φ) = ∫ ω ϕx(x)(φ(x)χ(x))xdx + λ ∫ ω ϕ(x)φ(x)χ(x)dx, (2.4) for any ϕ and φ in ψ. thus, for any g ∈ p the weak form of (2.1) is defined by b(ϕ,φ) = (g,φ)p , (2.5) in addition, we write p(ω) for l2χ(ω) and denote the inner product in p by (., .)p . we present the existence and uniqueness results for (2.1). for this purpose, we first present the following two lemmas. lemma 2.1. the spaces φ and ψ endowed with norm ‖ϕ‖l2χ(ω) and norm ‖ϕx‖l2χ(ω) are hilbert spaces. the embedding relations c∞0 (ω) ↪→ ψ ↪→ φ are dense and continuous and the following hardy type inequality holds: ∫ ω ϕ2(x) (x + 1)4 dx ≤ 1 9 ∫ ω ϕ2x(x) (x + 1)2 dx, ∀ϕ ∈ φ. (2.6) ∫ ω ϕ2(x) (x + 1)2 dx ≤ 64 9 ∫ ω ϕ2x(x) (x + 1)2 dx, ∀ϕ ∈ ψ (2.7) proof. it is clear that ‖ϕ‖l2χ(ω) and ‖ϕx‖l2χ(ω) are norms in φ and ψ, respectively. to show that c ∞ 0 (ω) is dense in ψ(ω), it suffices to show c∞0 (ω) ⊥ = 0. let ϕ ∈ c∞0 (ω) ⊥ . in this case, for any test function φ, we have ∫ ∞ 0 ϕx(x)φx(x) 1 (x + 1)2 dx = 0. then, straightway computations lead to ϕx(x) = a(x + 1) 3. since ϕx ∈ l2χ(ω), it can be concluded that a = 0. integrating once again, we get ϕ(x) = b. then the boundary condition ϕ(0) = 0 leads to b = 0. this being so, c∞0 (ω) ⊥ = 0. in a similar manner, it can be shown that c∞0 (ω) is dense in φ(ω). we now prove the hardy inequality (2.6). let z ∈ i = (−1, 1),x = 1+z 1−z and ψ(z) = ϕ(x). to prove the first inequality, it suffices to show that∫ i ψ2(z)(1 −z)2dz ≤ 4 9 ∫ i (∂zψ(z)) 2(1 −z)4dz. (2.8) using ψ(−1) = 0, for any z ∈ i we have ψ2(z)(1 −z)3 = ∫ z −1 ∂z(ψ 2(z)(1 −z)3)dz. int. j. anal. appl. 16 (2) (2018) 267 therefore ψ2(z)(1 −z)3 + 3 ∫ z −1 ψ 2(z)(1 −z)2dz = 2 ∫ z −1 ψ(z)∂zψ(z)(1 −z) 3dz ≤ 2( ∫ z −1 ψ 2(z)(1 −z)2dz) 1 2 ( ∫ z −1 (∂zψ(z)) 2(1 −z)4dz) 1 2 . with letting z → 1, we get (2.8). moreover, we have ∫ ω ϕ2(x) (x + 1)4 dx = 1 8 ∫ i ψ2(z)(1 −z)2dz,∫ ω (∂xϕ(x)) 2 (x + 1)2 dx = 1 2 ∫ i (∂zψ(z)) 2(1 −z)4dz. (2.9) hence, by combining (2.8) and (2.9), we obtain the first result. to see that ψ ↪→ φ, one can apply (2.6), ‖ϕ‖2φ = ∫ ω ϕ2(x) 1 (x + 1)2 dx ≤‖ϕ‖φ sup x∈ω |ϕ(x) 1 x + 1 |. the bound for supx∈ω |ϕ(x) 1 x+1 | can be obtained as follows. ϕ2(x) 1 (x + 1)2 = 2 ∫ x 0 ϕ(x)ϕx(x) 1 (x + 1)2 dx− 2 ∫ x 0 ϕ2(x) 1 (x + 1)3 dx ≤ 2‖ϕ(x) 1 x + 1 ‖l2‖ϕx(x) 1 x + 1 ‖l2 + 2‖ϕ(x) 1 x + 1 ‖l2‖ϕ(x) 1 (x + 1)2 ‖l2 ≤ 8 3 ‖ϕ‖φ‖ϕ‖ψ. therefore ‖ϕ‖φ ≤ 8 3 ‖ϕ‖ψ. � lemma 2.2. (lax-milgram theorem) assume that w ⊆ v be two hilbert spaces with continuous and dense embedding. let b(v,w) denote a bilinear form v ×w and suppose that there exists m,m > 0 such that b(v,w) ≤ m‖v‖v‖w‖w , ∀v ∈ v,w ∈ w. (2.10) b(w,w) ≥ m‖w‖2w , ∀w ∈ w. (2.11) then for each g ∈ v ∗ (the dual space of v ), there exists v ∈ v such that b(v,w) = g(w), ∀w ∈ w. (2.12) this theorem is useful in showing the existence and uniqueness of a solution for a given differential equation. this general version of lax-milgram theorem is due to lions [18]; see also lemma 4.4.4.1 in [9]. int. j. anal. appl. 16 (2) (2018) 268 theorem 2.1. let λ ≥ 3. then, for each g ∈ p , there exists a unique solution ϕ ∈ ψ such that, for all φ ∈ ψ, b(ϕ,φ) = (g,φ)p . (2.13) therefore, we able define an operator s : d(s) → p given by sϕ = g, (2.14) where d(s) = {ϕ ∈ ψ, sϕ ∈ p}. proof. for ϕ ∈ ψ and φ ∈ ψ, we check that b(ϕ,φ) as defined in (2.4) satisfies the conditions set in lemma 2.2. hence, form (2.4) we can write b(ϕ,φ) = ∫ ω ϕx(x)φx(x)χ(x)dx + ∫ ω ϕx(x)φ(x)χ ′ (x)dx + λ ∫ ω ϕ(x)φ(x)χ(x)dx. (2.15) by using (2.6) and (2.7), the various terms of (2.15) on the right can be written in the following way.∫ ω ϕx(x)φx(x)χ(x)dx ≤ ( ∫ ω ϕ2x(x) (x + 1)2 dx) 1 2 ( ∫ ω φ2x(x) (x + 1)2 dx) 1 2 = ‖ϕ‖ψ‖φ‖ψ. ∫ ω ϕx(x)φ(x)χ ′ (x)dx ≤ 2( ∫ ω ϕ2x(x) (x + 1)2 dx) 1 2 ( ∫ ω φ2(x) (x + 1)4 dx) 1 2 ≤ 2 3 ( ∫ ω ϕ2x(x) (x + 1)2 dx) 1 2 ( ∫ ω φ2x(x) (x + 1)2 dx) 1 2 = 2 3 ‖ϕ‖ψ‖φ‖ψ. ∫ ω ϕ(x)φ(x)χ(x)dx ≤ ( ∫ ω ϕ2(x) (x + 1)2 dx) 1 2 ( ∫ ω φ2(x) (x + 1)2 dx) 1 2 ≤ 64 9 ( ∫ ω ϕ2x(x) (x + 1)2 dx) 1 2 ( ∫ ω φ2x(x) (x + 1)2 dx) 1 2 = 64 9 ‖ϕ‖ψ‖φ‖ψ. combining these three results, we have b(ϕ,φ) ≤ ( 5 3 + 64 9 λ)‖ϕ‖ψ‖φ‖ψ. (2.16) in order to prove the weak coercivity (2.11), for all ϕ ∈ ψ we have b(ϕ,ϕ) = ∫ ω ϕ2x(x)χ(x)dx + ∫ ω ϕx(x)ϕ(x)χ ′ (x)dx + λ ∫ ω ϕ2(x)χ(x)dx = ∫ ω ϕ2x(x) (x + 1)2 dx− 3 ∫ ω ϕ2(x) (x + 1)3 dx + λ ∫ ω ϕ2(x) (x + 1)2 dx ≥ ∫ ω ϕ2x(x) (x + 1)2 dx = ‖ϕ‖2ψ. thus, by the lemma 2.2, there exists ϕ ∈ ψ such that b(ϕ,φ) defines a linear functional g on ψ: g(φ) = b(ϕ,φ). (2.17) int. j. anal. appl. 16 (2) (2018) 269 from (2.16), ‖g‖ψ∗ = sup φ∈ψ ‖g(φ)‖ ‖φ‖ψ ≤ α‖ϕ‖ψ < ∞. (2.18) hence, the functional g is continuous. thanks to riesz representation theorem, there exists a unique g such that g(φ) = (g,φ)p . since, g depends on ϕ, so we can write this dependence as sϕ = g, where s : d(s) → φ is a linear operator. therefore g(φ) = b(ϕ,φ) = (sϕ,φ)p . (2.19) and ‖sϕ‖p ≤ m‖ϕ‖ψ which means that s is continuous. also we have ‖sϕ‖p = sup φ∈ψ ‖b(ϕ,φ)‖ ‖φ‖p ≥ sup φ∈ψ ‖b(ϕ,φ)‖ ‖φ‖ψ ≥ β1‖ϕ‖ψ. (2.20) hence, s is bounded below. we now show that the range r(s) of s is closed. indeed, if {sϕn} ∈ p is a cauchy sequence then so is {ϕn}∈ ψ and by (2.20) we obtain ‖ϕm −ϕn‖ψ ≤ β2‖s(ϕm −ϕn)‖p = β2‖sϕm −sϕn‖p . so that {ϕn} converges to some ϕ ∈ ψ. since s is continuous, {sϕn} converges to sϕ which proves that r(s) is closed. furthermore, we show that s is surjective. if this was not true, there exists a ϕ0 ∈ r(s)⊥ with ϕ0 6= 0 such that (sϕ,ϕ0)p = b(ϕ,ϕ0) = 0 for each ϕ ∈ ψ. but this is contradiction with (2.4). thus ϕ0 = 0 and r(s) = p . using (2.20) we get sup ϕ∈ψ ‖b(ϕ,φ)‖ ‖ϕ‖ψ = sup ϕ∈ψ ‖(sϕ,φ)p‖ ‖ϕ‖ψ ≥ sup ϕ∈ψ ‖(sϕ,φ)p‖ β−11 ‖sϕ‖p = β1 sup ψ∈ψ ‖(ψ,φ)p‖ ‖ψ‖p = β1‖φ‖p . (2.21) we now show that there is at most one element ϕ ∈ ψ satisfying (2.17). assume the contrary, i.e., that b(ϕ1,φ) = g(φ) and b(ϕ2,φ) = g(φ) for all φ ∈ ψ. then by linearity, b(ϕ1 −ϕ2,φ) = 0, ∀φ ∈ ψ. (2.22) according to lemma 2.1, as c∞0 (ω) is densely embedded in ψ, there is a sequence ϕnc ∞ 0 (ω) such that ϕn → ϕ1 −ϕ2, in ψ. from (2.21), ‖ϕn‖2ψ ≤ b(ϕn,ϕn) = b(ϕ1 −ϕ2,ϕn) + b(ϕn − (ϕ1 −ϕ2),ϕn) ≤ m‖ϕn − (ϕ1 −ϕ2)‖ψ‖ϕn‖ψ. letting n →∞, one has ϕn → 0 in ψ and consequently ‖ϕ1 −ϕ2‖ψ ≤‖ϕn‖ψ + ‖ϕn − (ϕ1 −ϕ2)‖ψ → 0 asn →∞. therefore, ϕ1 = ϕ2 in ψ. � int. j. anal. appl. 16 (2) (2018) 270 3. the damped burgers equation in this section, we consider the damped burgers equation (1.1) with initial and boundary values (1.2) and (1.3). we study the mild solutions of this equation. to this end, we show that −s is an infinitesimal generator of a semi-group where s is defined in theorem 2.1. using classical theory of linear semi-group and the hille-yosida theorem as presented in [22] we establish the following theorem for the operator s. theorem 3.1. let p,s and d(s) be defined as in the previous section. then operator −s in the infinitesimal generator of a semi-group of contraction e−ts in p . proof. according to the hille-yosida theorem it is sufficient to check that s is closed, d(s) is dense in p and ‖(λ−s)−1g‖p ≤ 1λ‖g‖p for any λ > 0. from (2.21) it follows that s −1 is one to one. thus, s is closed. d(s) is dense in p , since c∞0 (ω) ⊂ d(s). let ϕ = (λ + s)−1g where g ∈ p . then (λ + s)ϕ = g and (g,ϕ)p = ((λ + s)ϕ,ϕ)p = λ‖ϕ‖2p + (sϕ,ϕ)p . (3.1) from (2.20) it immediately follows that (sϕ,ϕ)p ≥ 0. then by (3.1) we have ‖ϕ‖p ≤ 1 λ ‖g‖p . � before studying the mild solution, lets define the bilinear form on ψ × ψ as a(ϕ,φ) = (ϕφ)x (ϕ,φ) ∈ ψ × ψ. (3.2) a mild solution of the initial boundary value problem (1.1) is a function ϕ ∈ h = c([0,t]; p) ∩l2(0,t; ψ) satisfying   dϕ dt + sϕ = −a(ϕ,ϕ) ϕ(0) = ϕ0 (3.3) where ϕ0 ∈ p and t > 0. let k(t) = e−ts where s is defined in the previous section. we plan to apply a fixed-point argument to the integral equation ϕ(t) = k(t)ϕ0 − 1 2 ∫ t 0 k(t− τ)a(ϕ,ϕ)(τ)dτ. (3.4) for this purpose, we show that if t is small enough, then ϕ(t) → f(ϕ(t)), (3.5) is a contraction h, where the right side of (3.4) is denoted by f(ϕ(t)). int. j. anal. appl. 16 (2) (2018) 271 theorem 3.2. for any given ϕ0 ∈ p , there exists t > 0 such that the problem (1.1) has a unique mild solution in h. furthermore, the following energy identity holds for all t ≥ 0: ‖ϕ(t)‖2p + ∫ t 0 ‖ϕ(τ)‖2ψdτ ≤‖ϕ0‖ 2 p + c ∫ t 0 ‖ϕ(τ)‖2l2‖ϕ(τ)‖p‖ϕ(τ)‖ψdτ. (3.6) proof. we first prove that for any (ϕ,φ) ∈ ψ × ψ, ‖a(ϕ,φ)‖ψ∗ ≤ c‖ϕ‖l2‖φ‖ 1 2 p‖φ‖ 1 2 ψ. (3.7) let ψ ∈ ψ, one obtains by integrating by parts (a(ϕ,φ),ψ)p = ∫ ω (ϕ(x)φ(x))xψ(x)χ(x)dx = − ∫ ω ϕ(x)φ(x)ψx(x)χ(x)dx− ∫ ω ϕ(x)φ(x)ψ(x)χ ′ (x)dx. (3.8) by using cauchy-schwarz inequality to bound the first term on the right-hand side of this equality, we have∫ ω ϕ(x)φ(x)ψx(x)χ(x)dx = ∫ ω ϕ(x)φ(x) 1 x + 1 ψx(x) 1 x + 1 dx ≤‖ϕ(x)‖l2‖ψ(x)‖ψ sup x∈ω |φ(x) 1 x + 1 |. to bound supx∈ω |φ(x) 1 x+1 |, apply (2.6) in lemma 2.1 to obtain φ2(x) 1 (x + 1)2 = 2 ∫ x 0 φ(x)φx(x) 1 (x + 1)2 dx− 2 ∫ x 0 φ2(x) 1 (x + 1)3 dx ≤ 2‖φ(x) 1 (x + 1) ‖l2‖φx(x) 1 (x + 1) ‖l2 + 2‖φ(x) 1 (x + 1) ‖l2‖φ(x) 1 (x + 1)2 ‖l2 ≤ c‖φ(x)‖p‖φ(x)‖ψ. (3.9) therefore ∫ ω ϕ(x)φ(x)ψx(x)χ(x)dx ≤ c‖ϕ(x)‖l2‖φ(x)‖ 1 2 p‖φ(x)‖ 1 2 ψ‖ψ(x)‖ψ. (3.10) by using (2.6) in lemma 2.1, the second term in (3.8) can be bounded similarly.∫ ω ϕ(x)φ(x)ψ(x)χ ′ (x)dx = −2 ∫ ω ϕ(x)φ(x) 1 x + 1 ψ(x) 1 (x + 1)2 dx ≤ 3‖ϕ(x)‖l2‖ψ(x) 1 (x + 1) ‖l2 sup x∈ω |φ(x) 1 x + 1 | ≤ c‖ϕ(x)‖l2‖φ(x)‖ 1 2 p‖φ(x)‖ 1 2 ψ‖ψ(x)‖ψ. (3.11) hence, by inserting (3.10) and (3.11) into (3.8), we obtain (3.7). similar process show that ‖a(ϕ,φ)‖ψ∗ ≤ c‖φ‖l2‖ϕ‖ 1 2 p‖ϕ‖ 1 2 ψ. (3.12) we now prove that ‖ϕ‖l2 ≤‖ϕ0‖l2. (3.13) int. j. anal. appl. 16 (2) (2018) 272 taking the l2-inner product of (1.1) with ϕ(x) over the interval (0,∞), we obtain∫ ω ϕ(x)ϕt(x)dx− ∫ ω ϕ(x)ϕxx(x)dx + ∫ ω ϕ2(x)ϕx(x)dx = 0. (3.14) using mainly integration by parts, we can write the various terms as∫ ω ϕ(x)ϕt(x)dx = 1 2 ∫ ω d dt ϕ2(x)dx = 1 2 d dt ∫ ω ϕ2(x)dx = 1 2 d dt ‖ϕ(x)‖2l2. (3.15) − ∫ ω ϕ(x)ϕxx(x)dx = ∫ ω ϕ2x(x)dx = ‖ϕx(x)‖ 2 l2. (3.16)∫ ω ϕ2(x)ϕx(x)dx = 0. (3.17) substituting (3.15)-(3.17) into (3.14), we obtain d dt ‖ϕ(x)‖2l2 = −2‖ϕx(x)‖ 2 l2. (3.18) using the poincare inequality on the right-hand side of inequality (3.18), we attain d dt ‖ϕ(x)‖2l2 = −2‖ϕx(x)‖ 2 l2 ≤−2‖ϕ(x)‖ 2 l2. (3.19) integrating inequality (3.19) with respect to time from 0 to t, we get ‖ϕ(x)‖l2 ≤ e−t‖ϕ0(x)‖l2. (3.20) since ϕ0 ∈ l2(ω) and t ∈ [0,t],then (3.13) is achieved. now, let us introduce the banach space p with norm ‖ϕ‖h = sup t∈[0,t] ‖ϕ‖p + ‖ϕ‖l2(0,t;ψ). (3.21) let d = 2‖ϕ0‖l2 and bd = {ϕ ∈ h : ‖ϕ‖h ≤ d}. we now prove that if t is small enough, then f maps bd into itself. let ϕ ∈ bd and f(ϕ) satisfies d dt f(ϕ) + sf(ϕ) = −a(ϕ,ϕ). hence d dt ‖f(ϕ)‖2p + 2b(f(ϕ),f(ϕ)) = −2(a(ϕ,ϕ),f(ϕ))p . according to the proof of theorem 2.1, we get 2b(f(ϕ),f(ϕ)) ≥ 2‖f(ϕ)‖2ψ. by using (3.7), 2|(a(ϕ,ϕ),f(ϕ))p | ≤ 2‖a(ϕ,ϕ)‖ψ∗‖f(ϕ)‖ψ ≤ 3‖f(ϕ)‖2ψ + c‖ϕ‖ 2 l2‖ϕ‖p‖ϕ‖ψ. int. j. anal. appl. 16 (2) (2018) 273 therefore ‖f(ϕ)‖2p + ∫ t 0 ‖f(ϕ)‖2ψdτ ≤‖ϕ0‖ 2 p + c ∫ t 0 ‖ϕ(τ)‖2l2‖ϕ(τ)‖p‖ϕ(τ)‖ψdτ. if we choose t > 0 such that d2 + 8c √ td4 < d2, then ‖f(ϕ)‖h < d. to show f is a contraction, first note that f(ϕ) −f(φ) = − ∫ t 0 k(t− τ)(a(ϕ−φ,ϕ) + a(φ,ϕ−φ))dτ. using once again (3.7) and (3.12) for a(ϕ−φ,ϕ) and a(φ,ϕ−φ), respectively ‖f(ϕ) −f(φ)‖2p + ∫ t 0 ‖f(ϕ) −f(φ)‖2ψ ≤ ∫ t 0 (‖ϕ‖2l2‖ϕ−φ‖p‖ϕ−φ‖ψ + ‖φ‖ 2 l2‖ϕ−φ‖p‖ϕ−φ‖ψ)dτ ≤ c √ t(‖ϕ‖2l2 + ‖φ‖ 2 l2 )‖ϕ−φ‖ 2 h. if t is further restricted to δ2 = 2c √ t < 1, then ‖f(ϕ) −f(φ)‖h ≤ δ‖ϕ−φ‖h. applying the contraction mapping principle completes the proof of the theorem. � 4. strong solutions in this section, we study solutions of the boundary initial value problem (1.1) in a stronger sense and establish the global existence and uniqueness of such solutions. theorem 4.1. let t0 > 0 and ϕ0 ∈ p(ω) be given. then there exists t ∈ (0,t0] such that it possesses a unique solution ϕ ∈ c([0,t]; p) ∩l2(0,t; ψ). moreover, if there exist a constant c = c(t) > 0 such that ‖ϕ0‖l2(ω) ≤ c, (4.1) then the problem (1.1) has a strong solution on [0,t]. proof. for ϕ0 ∈ p, theorem 3.2 assures that (1.1) admits the unique solution ϕ ∈ c([0,t]; p) ∩l2(0,t; ψ). (4.2) for the global existence, we only have to show that the solution cannot blow up at any finite time t . for this purpose, we apply (3.13) to show that, for t ∈ [0,t], ‖ϕ(t)‖p ≤ c(t)‖ϕ0‖p . (4.3) int. j. anal. appl. 16 (2) (2018) 274 taking the h-inner product of (1.1) with ϕ over the interval (0,∞), we get∫ ω ϕ(x)ϕt(x)χ(x)dx− ∫ ω ϕ(x)ϕxx(x)χ(x)dx + ∫ ω ϕ2(x)ϕx(x)χ(x)dx + λ ∫ ω ϕ2(x)χ(x)dx = 0. (4.4) the terms on the left can be written as − ∫ ω ϕ(x)ϕxx(x)χ(x)dx = ∫ ω ϕ2x(x)χ(x)dx + ∫ ω ϕ(x)ϕx(x)χ ′ (x)dx = ‖ϕ‖2ψ − 1 2 ∫ ω ϕ2(x)χ ′′ (x)dx = ‖ϕ‖2ψ − 3 ∫ ω ϕ2(x) 1 (x + 1)4 dx. (4.5) ∫ ω ϕ2(x)ϕx(x)χ(x)dx = − 1 3 ∫ ω ϕ3(x)χ ′ (x)dx = 2 3 ∫ ω ϕ3(x) 1 (x + 1)3 dx. (4.6) substituting (4.5) and (4.6) into (4.4), we obtain d dt ‖ϕ‖2p + ‖ϕ‖ 2 ψ = 3 ∫ ω ϕ2(x) 1 (x + 1)4 dx− 2 3 ∫ ω ϕ3(x) 1 (x + 1)3 dx−λ ∫ ω ϕ2(x) 1 (x + 1)2 dx. (4.7) the terms on the right-hand can be bounded as∫ ω ϕ2(x) 1 (x + 1)4 dx ≤ ∫ ω ϕ2(x) 1 (x + 1)2 dx = ‖ϕ‖2p . (4.8) ∫ ω ϕ3(x) 1 (x + 1)3 dx ≤ ∫ ω ϕ3(x) 1 (x + 1)2 dx = ∫ ω ϕ(x)ϕ2(x) 1 (x + 1)2 dx ≤ 1 2 ‖ϕ‖2p + 1 2 ‖ϕ‖2l2‖ϕ‖ 2 p ≤ 1 2 ‖ϕ‖2p + 1 2 ‖ϕ0‖2l2‖ϕ‖ 2 p . (4.9) inserting (4.8) and (4.9) into (4.7), we have d dt ‖ϕ‖2p + ‖ϕ‖ 2 ψ − ( 10 3 ‖ϕ‖2p + 1 3 ‖ϕ0‖2l2‖ϕ‖ 2 p + λ‖ϕ‖ 2 p ) ≤ 0. on the other hand, since ‖ϕ‖p ≤‖ϕ‖ψ, we conclude that d dt ‖ϕ‖2p − ( 7 3 + 1 3 ‖ϕ0‖2l2 + λ)‖ϕ‖ 2 p ) ≤ 0. applying the gronwall’s inequality completes the proof of this theorem. � references [1] r. abazari, application of extended tanh function method on kdv-burgers equation with forcing term, rom. j. phys. 59(1-2) (2014), 3-11. [2] a. biswas, s. kumar, e.v. krishnan, b. ahmed, a. strong, s. johnson, a. yildirim topological solitons and other solutions to potential kdv equation, rom. j. phys. 65(4) (2013), 1125–1137. [3] j.m. burgers, a mathematical model illustrating the theory of turbulence, adv. appl. mech. 1 (1948), 171–199. [4] j.d. cole, on a puasilinear parabolic equations occurring in aerodynamics, quart. appl. math. 9 (1951), 225–236. [5] m. dehghan, b.n. saray and m. lakestani, mixed finite difference and galerkin methods for solving burgers equations using interpolating scaling functions, math. meth. appl. sci. 37 (6) (2014), 894–912. [6] g. ebadi, a. mojaver, h. triki, a. yildirim, a. biswas, topological solitons and other solutions of the rosenauckdv equation with power law nonlinearity, rom. j. phys. 58 (2013), 3–14. int. j. anal. appl. 16 (2) (2018) 275 [7] g. ebadi, n. yousefzadeh, h. triki, a. yildirim, a. biswas, envolope solitons, peridic waves and other solutions to boussinesqcburgers equstion, rom. j. phys. 64 (2012), 915–932. [8] d. goubet and j. shen, on the dual petrov-galerkin formulation of the kdv equation on a finite interval, adv. diff. equs. 12 (2007), 221–239. [9] p. grisvard, el liptic problems in nonsmooth domains, volume 24 of monographs and studies in mathematics, pitman (advanced publishing program), boston, ma (1985). [10] d. idris, c.i. aynur and s. ali, taylor galerkin and taylor collocation methods for the numerical solutions of burgers equation using b-splines, commun. nonlinear sci. numer. simul. 16:7 (2011), 2696–2708. [11] i.e. inan, d. kaya and y. ugurlu, auto backlund transformation and similarity reductions for coupled burgers equation, appl. math. comput. 216 (2010), 2507–2511. [12] k. ismail and s. ibrahim, an efficient computational method for the optimal control problem for the burgers equation, math. comput. model. 44 (2006), 973–982. [13] m. javidi, a numerical solution of burgers equation based on modified extended bdf scheme, int. math. forum 1,(2006), 1565–1570. [14] n. khanal, j. wu and j.m. yuan, the kawahara equation in weighted sobolev spaces, nonlinearity 21 (2008), 1489–1505. [15] a.h. khater, r.s. temsah and m.m. hassan, a chebyshev spectral collocation method for solving burgers-type equation, j. comput. appl. math. 222 (2008), 333–350. [16] m. k. kunisch and s. volkwein, control of the burgers equation by a reduced-order approach using proper orthogonal decomposition, j. optim. theory appl. 102 (1999), 345–371. [17] jerome l.v. lewandowski, a marker method for the solution of the damped burgers equation, numer. methods partial differ. 22(1) (2005), 48–68. [18] j.l. lions, sur les problems aux limites du type derive oblique, ann. math. 64 (1956), 207–239. [19] s. lu and m. li, modified legendre rational spectral method for thr burgers equation on the half line, int. j. comput. math. 85:6 (2008), 865–875. [20] w. malfliet, approximate solution of the damped burgers equation, j. phys. a: math. gen. 26 (1993), l723–l728. [21] r.c. mittal and g. arora, numerical solution of the coupled viscous burgers equation, commun. nonlinear sci. numer. simul. 16 (2011), 1304–1313. [22] a. pazy, semigroups of linear operators and applications to partial differential equations, new york: springer, (1983). [23] y. peng, w. chen, a new similarity solution of the burgers equation with linear damping czech, j, phys. 56 (2008) 317-428. [24] z. sabeh, m. shamsi and m. dehghan, distributed optimal control of the viscous burgers equation via a legendre pseudospectral approach, math. meth. appl. sci. 39(12) (2015),3350–3360 [25] h. triki, a. yildirim, t. hayat, o.m. aldossary, a. biswas, topological and non-topological solitons of a generalized derivative nonlinear schrodingers equation with perturbation terms, rom. j. phys. 64 (2012), 672–684. [26] f. yilmaz and b. karasozen, solving optimal control problems for the unsteady burgers equation in comsol multiphysics, j. comput. appl. math. 235 (16) (2011), 4839–4850. [27] n.y. fard, m.r. foroutan, m. eslami, m. mirzazadeh, a. biswas, solitary waves and other solutions to kadomtsevpetviashvili equation with spatio-temporal dispersion, rom. j. phys. 60 (2015), 1337–1360. [28] b. m. vaganan, m. s. kumaran, kummer function solutions of damped burgers equations with time-dependent viscosity by exact linearization, nonlinear anal. real world appl. 4 (2003), 723–741. 1. introduction 2. hardy inequalities in weighted sobolev spaces and representation of bilinear forms 3. the damped burgers equation 4. strong solutions references int. j. anal. appl. (2022), 20:49 on tripolar fuzzy pure ideals in ordered semigroups nuttapong wattanasiripong1, jirapong mekwian2, hataikhan sanpan2, somsak lekkoksung2,∗ 1division of applied mathematics, faculty of science and technology, valaya alongkorn rajabhat university under the royal patronage, pathum thani 13180, thailand 2division of mathematics, faculty of engineering, rajamangala university of technology isan, khon kaen campus, khon kaen 40000, thailand ∗corresponding author: lekkoksung_somsak@hotmail.com abstract. tripolar fuzzy sets are a concept that deals with tripolar information. this idea is a generalization of bipolar and intuitionistic fuzzy sets. in this paper, the notions of tripolar fuzzy pure ideals in ordered semigroups are introduced, and some algebraic properties of tripolar fuzzy pure ideals are studied. we obtain some characterizations of weakly regular ordered semigroups in terms of tripolar fuzzy pure ideals. finally, we introduce the concepts of tripolar weakly pure ideals and prove that the tripolar fuzzy ideals are tripolar weakly pure ideals if such tripolar fuzzy ideals satisfy the idempotent property. 1. introduction the theory of fuzzy sets is the most appropriate theory for dealing with uncertainty was introduced by zadeh [14] in 1965. after the introduction of the concept of fuzzy sets by zadeh, several researchers researched the generalizations of the notions of fuzzy sets with huge applications in computer science, artificial intelligence, control engineering, robotics, automata theory, decision theory, finite state machine, graph theory, logic, operations research and many branches of pure and applied mathematics. for example, xie et al. applied fuzzy set theory to switching method [13]. there are many extensions of fuzzy sets, for example, intuitionistic fuzzy sets, hesitant fuzzy sets, interval-valued fuzzy sets, vague sets, type-2 fuzzy sets, fuzzy multi-sets, bipolar fuzzy sets, and cubic sets. the received: aug. 10, 2022. 2010 mathematics subject classification. 06f05, 08a72. key words and phrases. ordered semigroup; weakly regular ordered semigroup; tripolar fuzzy pure ideal; tripolar fuzzy weakly pure ideal. https://doi.org/10.28924/2291-8639-20-2022-49 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-49 2 int. j. anal. appl. (2022), 20:49 fuzzification of the algebraic structure was introduced by rosenfeld [11] and he introduced the notion of fuzzy subgroups in 1971. a bipolar fuzzy set is an extension of a fuzzy set whose membership degree range is [−1, 1]. in 1994, zhang [15] initiated the concept of the bipolar fuzzy set as a generalization of the fuzzy set. the notion of an intuitionistic fuzzy set was introduced by atanassov [2] as a generalization of the notion of a fuzzy set. the tripolar fuzzy set is a generalization of fuzzy set, bipolar fuzzy set, and intuitionistic fuzzy set. rao [7] introduced the notion of tripolar fuzzy set as a generalization of fuzzy set, bipolar fuzzy set, and intuitionistic fuzzy set, and studied tripolar fuzzy interior ideals of γ-semigroup. rao and venkateswarlu [8, 9] investigated tripolar fuzzy interior ideal, tripolar fuzzy soft ideal, and tripolar fuzzy soft interior ideal of γ-semigroup and γ-semiring. rao [10] introduced the notion of tripolar fuzzy interior ideal, tripolar fuzzy soft ideal, and tripolar fuzzy soft interior ideal over semigroup and studied some of their algebraic properties. ahsan and takahashi [1] introduced the notions of pure ideals and purely prime ideals in semigroups without order. bashir et al. [3] defined the concepts of pure ideals, weakly purely ideals, and purely prime ideals in ternary semigroups. in [4] changphas and sanborisoot introduced the concepts of pure ideals, weakly pure ideals, and purely prime ideals in ordered semigroups. siribute and sanborisoot [12] applied fuzzy theory to semigroup theory. they introduced the concepts of pure fuzzy and weakly pure fuzzy ideals in ordered semigroups and characterized weakly regular ordered semigroups by pure fuzzy ideals. linesawat et al. [6] introduced the concepts of anti-hybrid pure ideals in ordered semigroups and studied some algebraic properties of anti-hybrid pure ideals. they characterize weakly regular ordered semigroups in terms of anti-hybrid pure ideals. finally, they also gave the concept of antihybrid weakly pure ideals. they prove that the anti-hybrid ideals are anti-hybrid weakly pure ideals if such anti-hybrid ideals satisfy the idempotent property. based on the concept of the purity of fuzzy ideals considered by siribute and sanborisoot [12], we apply the concept of purity to tripolar fuzzy ideals in ordered semigroups. in this present paper, the notions of tripolar fuzzy pure ideals in ordered semigroups are introduced, and some algebraic properties of tripolar fuzzy pure ideals are studied. we obtain some characterizations of weakly regular ordered semigroups in terms of tripolar fuzzy pure ideals. finally, we introduce the concepts of tripolar weakly pure ideals and prove that the tripolar fuzzy ideals are tripolar weakly pure ideals if such tripolar fuzzy ideals satisfy the idempotent property. 2. preliminaries in this section, we will recall some of the fundamental concepts and definitions, which are necessary for this paper. an ordered semigroup is a structure (s; ·,≤) such that int. j. anal. appl. (2022), 20:49 3 (1) (s; ·) is a semigroup. (2) (s;≤) is a partially ordered set, and (3) x ≤ y implies u ·x ≤ v ·y and x ·u ≤ y ·v for all u,v,x,y ∈ s. for simplicity, we will be written xy instead of x · y, and an ordered semigroup (s; ·,≤), will be written in its universe set as a bold letter s. for k ⊆ s, we denote (k] := {a ∈ s | a ≤ k for some k ∈ k}. let a and b be two nonempty subsets of s. then we define ab := {ab | a ∈ a and b ∈ b}. definition 2.1. [5] let s be an ordered semigroup. a nonempty subset a of s is called a left (resp., right) ideal of s if (1) sa ⊆ a (resp., as ⊆ a). (2) for a ∈ s,b ∈ a, if a ≤ b, then a ∈ a. a nonempty subset i of s is called a two-side ideal (ideal) of s if it is both a left and a right ideal of s. definition 2.2. [4] let s be an ordered semigroup. an ideal i of s is called a right (resp., left) pure ideal of s if for each a ∈ i there exist x ∈ i such that a ≤ ax (resp., a ≤ xa). an ideal i of s is called a pure ideal of s if it is both a left and a right pure ideal of s. a fuzzy subset of a nonempty set x (or fuzzy set in a nonempty subset x) is a mapping f : x → [0, 1] from x to a unit closed interval, (see [14]). definition 2.3. [7] a fuzzy set f of a universe set x is said to be a tripolar fuzzy set, if f := {(x; f +(x), f ∗(x), f−(x)) | x ∈ x and 0 ≤ f +(x) + f ∗(x) ≤ 1}, where f + : x → [0, 1], f ∗ : x → [0, 1] and f− : x → [−1, 0]. the membership degree f +(x) characterizes the extent that the element x satisfies the property corresponding to tripolar fuzzy set f , f ∗(x) characterizes the extent that the element x satisfies the not property (irrelevant) corresponding to tripolar fuzzy set f , and f−(x) characterizes the extent that the element x satisfies the implicit counter property corresponding to tripolar fuzzy set f . for simplicity f := (f +, f ∗, f−) has been used for f := {(x; f +(x), f ∗(x), f−(x)) | x ∈ x and 0 ≤ f +(x) + f ∗(x) ≤ 1}. let a ∈ s. then, we set sa := {(x,y) ∈ s × s | a ≤ xy}. we denote tri(s) the set of all tripolar fuzzy subsets of s and define an operation on such set as follows: let f := (f +, f ∗, f−),g := 4 int. j. anal. appl. (2022), 20:49 (g+,g∗,g−) be elements in tri(s). then the product f ◦ g of f and g as the tripolar fuzzy set, denote by f ◦g := (f + ◦g+, f ∗ ◦g∗, f− ◦g−), of s defined as follows: for each x ∈ s. (f + ◦g+)(x) =   ∨ (a,b)∈sx {min{f +(a),g+(b)}} if sx 6= ∅ 0 otherwise, (f ∗ ◦g∗)(x) =   ∧ (a,b)∈sx {max{f ∗(a),g∗(b)}} if sx 6= ∅ 1 otherwise, and (f− ◦g−)(x) =   ∧ (a,b)∈sx {max{f−(a),g−(b)}} if sx 6= ∅ 0 otherwise. it is easy to verify that the structure (tri(s);◦) is a semigroup. in the set of all tripolar fuzzy subsets of s we define the order relation as follows: f ⊆ g if and only if f +(x) ≤ g+(x), f ∗(x) ≥ g∗(x) and f−(x) ≥ g−(x) for all x ∈ s and then such structure (tri(s);◦,⊆) is an ordered semigroup. finally for tripolar fuzzy subsets f and g of s we define the operation f ∩g as the tripolar fuzzy subset of s defined by: f ∩g := (f + ∩g+, f ∗ ∪g∗, f− ∪g−), where (f +∩g+)(x) := min{f +(x),g+(x)}, (f ∗∪g∗)(x) := max{f ∗(x),g∗(x)}, and (f−∪g−)(x) := max{f−(x),g−(x)} for all x ∈ s. the tripolar fuzzy subset 1 := (1+, 0, 1−) of s defined by 1+(x) := 1, 0(x) := 0 and 1−(x) = −1 for all x ∈ s. let a ⊆ s. we denoted by χa := (χ+a,χ ∗ a,χ − a ) the characteristic tripolar fuzzy subset of a in s and it is defined as follows: (χ+ a )(x) := { 1 if x ∈ a 0 otherwise, (χ∗a)(x) := { 0 if x ∈ a 1 otherwise, and (χ− a )(x) := { −1 if x ∈ a 0 otherwise. in the case of a = s, we define χa = 1. definition 2.4. let s be an ordered semigroup. a tripolar fuzzy subset f = (f +, f ∗, f−) of s is called a tripolar fuzzy right (resp., left) ideal of s if for every x,y ∈ s, (1) f +(xy) ≥ f +(x) (resp., f +(xy) ≥ f +(y)), (2) f ∗(xy) ≤ f ∗(x) (resp., f ∗(xy) ≤ f ∗(y)), (3) f−(xy) ≤ f−(x) (resp., f−(xy) ≤ f−(y)), int. j. anal. appl. (2022), 20:49 5 (4) if x ≤ y, then f +(x) ≥ f +(y), f ∗(x) ≤ f ∗(y), f−(x) ≤ f−(y). f is called a tripolar fuzzy two-side ideal (tripolar fuzzy ideal) of s if f is both a tripolar fuzzy right and a tripolar fuzzy left ideal of s. example 2.1. let s = {a,b,c}. define the binary operation ∗ on s by the followsing table: ∗ a b c a a a a b a a a c a b c and define an order on s as follows: ≤:= {(a,b)}∪ ∆s, where ∆s is an equality relation on s. that is, ∆s := {(x,x) ∈ s×s | x ∈ s}. then, s := (s;∗,≤) is an ordered semigroup. we define a tripolar fuzzy subset f = (f +, f ∗, f−) of s by: s f +(x) f ∗(x) f−(x) a 0.7 0.1 −0.6 b 0.5 0.2 −0.5 c 0.7 0.1 −0.6 then f is a tripolar fuzzy left ideal of s but, f is not a tripolar fuzzy right ideal of s since f +(c∗b) = f +(b) = 0.5 < 0.7 = f +(c). example 2.2. let s = {a,b,c}. define the binary operation ◦ on s by the following table: ◦ a b c a a a a b a a a c a c c and define an order on s as follows: ≤:= {(a,b), (a,c)}∪ ∆s, where ∆s is an equality relation on s. that is, ∆s := {(x,x) ∈ s×s | x ∈ s}. then, s := (s;◦,≤) is an ordered semigroup. we define a tripolar fuzzy subset f = (f +, f ∗, f−) of s by: s f +(x) f ∗(x) f−(x) a 0.8 0.1 −0.8 b 0.7 0.2 −0.7 c 0.7 0.2 −0.7 then f is a tripolar fuzzy ideal of s. 6 int. j. anal. appl. (2022), 20:49 3. main results in this main section, we introduce the concepts of tripolar fuzzy pure ideals in ordered semigroups and study some algebraic properties of such tripolar fuzzy pure ideals. we also characterize weakly regular ordered semigroups in terms of tripolar fuzzy pure ideals. finally, we introduce the concept of tripolar weakly pure ideals and prove that any tripolar fuzzy ideal is a tripolar fuzzy weakly pure ideal whenever it is idempotent. definition 3.1. let s be an ordered semigroup. a tripolar fuzzy ideal f of s is (1) left pure if f ∩g = f ◦g for every tripolar fuzzy left ideal g of s. (2) right pure g ∩ f = g ◦ f for every tripolar fuzzy right ideal g of s. a tripolar fuzzy ideal of s is called a tripolar fuzzy pure ideal of s it is both a tripolar fuzzy right pure and a tripolar fuzzy left pure ideal of s. the following lemmas are important in illustrating our first theorem. therefore, we give useful tools. lemma 3.1. let s be an ordered semigroup and a a nonempty subset of s. then the following conditions are equivalent: (1) a is a right (resp., left) ideal of s. (2) χa is a tripolar fuzzy right (resp., left) ideal of s. proof. (1)⇒(2). let a be a right ideal of an ordered semigroup s. first, let x,y ∈ s and x ∈ a. then xy ∈ a and, we obtain χ+ a (xy) = 1 = χ+ a (x), χ∗a(xy) = 0 = χ ∗ a(x), and χ − a (xy) = −1 = χ− a (x). if x /∈ a, we obtain χ+ a (xy) ≥ 0 = χ+ a (x), χ∗a(xy) ≤ 1 = χ ∗ a(x), and χ − a (xy) ≤ 0 = χ− a (x). secondly, let x,y ∈ s be such that x ≤ y. if y ∈ a, then x ∈ a and then χ+ a (x) = 1 = χ+ a (y), χ∗a(x) = 0 = χ ∗ a(y), and χ − a (x) = −1 = χ− a (y). if y /∈ a, we obtain χ+ a (x) ≥ 0 = χ+ a (y), χ∗a(x) ≤ 1 = χ ∗ a(y), and χ − a (x) ≤ 0 = χ− a (y). altogether, it is completed to prove that χa is a tripolar fuzzy right ideal of s. (2)⇒(1). let χa be a tripolar fuzzy right ideal of s. first, let x,y ∈ s and x ∈ a. we obtain 1 ≥ χ+ a (xy) ≥ χ+ a (x) = 1, which implies that χ+ a (xy) = 1 and then xy ∈ a, 0 ≤ χ∗a(xy) ≤ χ ∗ a(x) = 0, which implies that χ∗a(xy) = 0 and then xy ∈ a, similarly, −1 ≤ χ − a (xy) ≤ χ− a (x) = −1, which implies that χ− a (xy) = −1 and then xy ∈ a. secondly, let x,y ∈ s be such that x ≤ y and y ∈ a. we obtain int. j. anal. appl. (2022), 20:49 7 1 ≥ χ+ a (x) ≥ χ+ a (y) = 1, which implies that χ+ a (x) = 1 and then x ∈ a, 0 ≤ χ∗a(x) ≤ χ ∗ a(y) = 0, which implies that χ∗a(x) = 0 and then x ∈ a, similarly, −1 ≤ χ − a (x) ≤ χ− a (y) = −1, which implies that χ− a (x) = −1 and then x ∈ a. altogether, we obtain a is a right ideal of s. similarly, we can show that a is left ideal if and only if χa is tripolar fuzzy left ideal. � as a consequence of the above lemma, we have that a is an ideal of s if and only if χa is a tripolar fuzzy ideal of s. lemma 3.2. let s be an ordered semigroup and a, b nonempty subsets of a set s. then the following conditions are equivalent: (1) a ⊆ b if and only if χa ⊆ χb; (2) χa ∩χb = χa∩b; (3) χa ◦χb = χ(ab]. proof. we will give proof only (3). let x ∈ (ab]. then x ≤ ab for some a ∈ a and b ∈ b and then sx 6= ∅, we obtain 1 ≥ (χ+ a ◦χ+ b )(x) = ∨ (y,z)∈sx {min{χ+ a (y),χ+ b (z)}}≥ min{χ+ a (a),χ+ b (b)} = 1. this implies that (χ+ a ◦χ+ b )(x) = 1 = χ+ (ab] (x), 0 ≤ (χ∗a ◦χ ∗ b)(x) = ∧ (y,z)∈sx {max{χ∗a(y),χ ∗ b(z)}}≤ max{χ ∗ a(a),χ ∗ b(b)} = 0. this implies that (χ∗a ◦χ ∗ b)(x) = 0 = χ ∗ (ab] (x) and −1 ≤ (χ− a ◦χ− b )(x) = ∧ (y,z)∈sx {max{χ− a (y),χ− b (z)}}≤ max{χ− a (a),χ− b (b)} = −1. this implies that (χ− a ◦χ− b )(x) = −1 = χ− (ab] (x). therefore χa ◦χb = χ(ab]. � lemma 3.3. [4] let s be an ordered semigroup and a an ideal of s. then the following conditions are equivalent: (1) a is a right pure ideal of s. (2) b ∩a = (ba] for every right ideal b of s. lemma 3.4. [4] let s be an ordered semigroup and a an ideal of s. then the following conditions are equivalent: (1) a is a left pure ideal of s. (2) a∩b = (ab] for every left ideal b of s. the following theorem provides a characterization of right (resp., left) pure ideals in ordered semigroups using tripolar fuzzy right (resp., left) pure ideals. 8 int. j. anal. appl. (2022), 20:49 theorem 3.1. let a be an ideal of an ordered semigroup s. then the following conditions are equivalent: (1) a is a right (resp., left) pure ideal of s. (2) χa is a tripolar fuzzy right (resp., left) pure ideal of s. proof. (1)⇒(2). assume that a is a right pure ideal of s. let f be a tripolar fuzzy right ideal of s and a ∈ s. suppose that a /∈ a, we consider two cases as follows: if sa = ∅, then we have (f + ◦χ+ a )(a) = 0 = χ+ a (a) = min{f +(a),χ+ a (a)} = (f + ∩χ+ a )(a), (f ∗ ◦χ∗a)(a) = 1 = χ ∗ a(a) = max{f ∗(a),χ∗a(a)} = (f ∗ ∪χ∗a)(a), and (f− ◦χ− a )(a) = 0 = χ− a (a) = max{f−(a),χ− a (a)} = (f− ∪χ− a )(a). if sa 6= ∅, then, by a right purity of a, we have that v /∈ a for all (u,v) ∈sa. then (f + ◦χ+ a )(a) = ∨ (x,y)∈sa {min{f +(x),χ+ a (y)}} = 0 = min f +(a),χ+ a (a) = (f + ∩χ+ a )(a), (f ∗ ◦χ∗a)(a) = ∧ (x,y)∈sa {max{f ∗(x),χ∗a(y)}} = 1 = max{f ∗(a),χ∗a(a)} = (f ∗ ∪χ∗a)(a), and (f− ◦χ− a )(a) = ∧ (x,y)∈sa {max{f−(x),χ− a (y)}} = 0 = max{f−(a),χ− a (a)} = (f− ∪χ− a )(a). now, we assume that a ∈ a. by the right purity of a, we have sa /∈ ∅. more precisely, there exists (a,x) ∈sa such that x ∈ a. then, by the tripolar fuzzy right ideality of f and the tripolar fuzzy left int. j. anal. appl. (2022), 20:49 9 ideality of χa, we have that (f + ∩χ+ a )(a) = min{f +(a),χ+ a (a)} = f +(a) = min{f +(a),χ+ a (x)} ≤ ∨ (u,v)∈sa {min{f +(u),χ+ a (v)}}≤ ∨ (u,v)∈sa {min{f +(uv),χ+ a (uv)}} ≤ ∨ (u,v)∈sa {min{f +(a),χ+ a (a)}} = min{f +(a),χ+ a (a) = (f + ∩χ+ a )(a). this implies that (f + ∩χ+ a )(a) = ∨ (u,v)∈sa {min{f +(u),χ+ a (v)}} = (f + ◦χ+ a )(a), (f ∗ ∪χ∗a)(a) = max{f ∗(a),χ∗a(a)} = f ∗(a) = max{f ∗(a),χ∗a(x)} ≥ ∧ (u,v)∈sa {max{f ∗(u),χ∗a(v)}}≥ ∧ (u,v)∈sa {max{f ∗(uv),χ∗a(uv)}} ≥ ∧ (u,v)∈sa {max{f ∗(a),χ∗a(a)}} = max{f ∗(a),χ∗a(a) = (f ∗ ∪χ∗a)(a). this implies that (f ∗ ∪χ∗a)(a) = ∧ (u,v)∈sa {max{f ∗(u),χ∗a(v)}} = (f ∗ ◦χ∗a)(a), and (f− ∪χ− a )(a) = max{f−(a),χ− a (a)} = f−(a) = max{f−(a),χ− a (x)} ≥ ∧ (u,v)∈sa {max{f−(u),χ− a (v)}}≥ ∧ (u,v)∈sa {max{f−(uv),χ− a (uv)}} ≥ ∧ (u,v)∈sa {max{f−(a),χ− a (a)}} = max{f−(a),χ− a (a) = (f− ∪χ− a )(a). this implies that (f−∪χ− a )(a) = ∧ (u,v)∈sa {max{f−(u),χ− a (v)}} = (f−◦χ− a )(a). altogether, we have χa is a tripolar fuzzy right pure ideal of s. (2)⇒(1). assume that χa is a tripolar fuzzy right pure ideal of s. let b be a right ideal of s. by lemma 3.1, χb is a tripolar fuzzy right ideal of s, by assumption, we obtain χb∩a = χb ∪χa = χb ◦χa = χ(ba]. by lemma 3.2 (1), we have b ∩a = (ba] and, by lemma 3.3, we obtain a is a right pure ideal of s. similarly to prove a is a left pure ideal of s if and only if χa is a tripolar fuzzy left pure ideal of s. � by the above theorem, we obtain the following consequence. corollary 3.1. let a be an ideal of an ordered semigroup s. then the following conditions are equivalent: (1) a is a pure ideal of s. (2) χa is a tripolar fuzzy pure ideal of s. the following results illustrate some properties of tripolar fuzzy right (resp., left) pure ideals in an ordered semigroup. 10 int. j. anal. appl. (2022), 20:49 theorem 3.2. let f and g be tripolar fuzzy right pure ideals of an ordered semigroup s. then f ∩g is a tripolar fuzzy right pure ideal of s proof. let h be a tripolar fuzzy right ideal of s. we have h◦ (f ∩g) = (h◦ f ) ∩ (h◦g) = (h∩ f ) ∩ (h∩g) = h∩ (f ∩g). it is completed to prove that f ∩g is a tripolar fuzzy right pure ideal of s. � by similar method of theorem 3.2, we have the following theorem. theorem 3.3. let f and g be tripolar fuzzy left pure ideals of an ordered semigroup s. then f ∩g is a tripolar fuzzy left pure ideal of s combining theorem 3.2 and 3.3, we obtain the following result. corollary 3.2. let f and g be tripolar fuzzy pure ideals of an ordered semigroup s. then f ∩g is a tripolar fuzzy pure ideal of s let f and g be tripolar fuzzy subsets of s. then f dg := (f +∪g+, f ∗∩g∗, f−∩g−) is a tripolar fuzzy subset of s and is defined as follows: (f + ∪g+)(x) := max{f +(x),g+(x)}, (f ∗ ∩g∗)(x) := min{f ∗(x),g∗(x)} and (f− ∩g−)(x) := min{f−(x),g−(x)} for all x ∈ s. theorem 3.4. let f and g be tripolar fuzzy right pure ideals of an ordered semigroup s. then f dg is a tripolar fuzzy right pure ideal of s. proof. let h be a tripolar fuzzy right ideal of an ordered semigroup s and a ∈ s. if sa = ∅, then we obtain that (h+ ∩ (f + ∪g+))(a) = min{h+(a), (f + ∪g+)(a)} = min{h+(a), max{f +(a),g+(a)}} = max{min{h+(a), f +(a)}, min{h+(a),g+(a)}} = max{(h+ ∩ f +)(a), (h+ ∩g+)(a)} = max{(h+ ◦ f +)(a), (h+ ◦g+)(a)} = 0 = (h+ ◦ (f + ∪g+))(a), int. j. anal. appl. (2022), 20:49 11 (h∗ ∪ (f ∗ ∩g∗))(a) = max{h∗(a), (f ∗ ∩g∗)(a)} = max{h∗(a), min{f ∗(a),g∗(a)}} = min{max{h∗(a), f ∗(a)}, max{h∗(a),g∗(a)}} = min{(h∗ ∪ f ∗)(a), (h∗ ∪g∗)(a)} = min{(h∗ ◦ f ∗)(a), (h∗ ◦g∗)(a)} = 1 = (h∗ ◦ (f ∗ ∩g∗))(a), and (h− ∪ (f− ∩g−))(a) = max{h−(a), (f− ∩g−)(a)} = max{h−(a), min{f−(a),g−(a)}} = min{max{h−(a), f−(a)}, max{h−(a),g−(a)}} = min{(h− ∪ f−)(a), (h− ∪g−)(a)} = min{(h− ◦ f−)(a), (h− ◦g−)(a)} = 0 = (h− ◦ (f− ∩g−))(a). suppose that sa 6= ∅, and we obtain (h+ ∩ (f + ∪g+))(a) = min{h+(a), (f + ∪g+)(a)} = min{h+(a), max{f +(a),g+(a)}} = max{min{h+(a), f +(a)}, min{h+(a),g+(a)}} = max{(h+ ∩ f +)(a), (h+ ∩g+)(a)} = max{(h+ ◦ f +)(a), (h+ ◦g+)(a)} = max{ ∨ (x,y)∈sa {min{h+(x), f +(y)}, ∨ (x,y)∈sa {min{h+(x),g+(y)}}} = ∨ (x,y)∈sa {min{h+(x), max{f +(y),g+(y)}}} = ∨ (x,y)∈sa {min{h+(x), (f + ∪g+)(y)}} = (h+ ◦ (f + ∪g+))(a), 12 int. j. anal. appl. (2022), 20:49 (h∗ ∪ (f ∗ ∩g∗))(a) = max{h∗(a), (f ∗ ∩g∗)(a)} = max{h∗(a), min{f ∗(a),g∗(a)}} = min{max{h∗(a), f ∗(a)}, max{h∗(a),g∗(a)}} = min{(h∗ ∪ f ∗)(a), (h∗ ∪g∗)(a)} = min{(h∗ ◦ f ∗)(a), (h∗ ◦g∗)(a)} = min{ ∧ (x,y)∈sa {max{h∗(x), f ∗(y)}, ∧ (x,y)∈sa {max{h∗(x),g∗(y)}}} = ∧ (x,y)∈sa {max{h∗(x), min{f ∗(y),g∗(y)}}} = ∧ (x,y)∈sa {max{h∗(x), (f ∗ ∩g∗)(y)}} = (h∗ ◦ (f ∗ ∩g∗))(a), and (h− ∪ (f− ∩g−))(a) = max{h−(a), (f− ∩g−)(a)} = max{h−(a), min{f−(a),g−(a)}} = min{max{h−(a), f−(a)}, max{h−(a),g−(a)}} = min{(h− ∪ f−)(a), (h− ∪g−)(a)} = min{(h− ◦ f−)(a), (h− ◦g−)(a)} = min{ ∧ (x,y)∈sa {max{h−(x), f−(y)}, ∧ (x,y)∈sa {max{h−(x),g−(y)}}} = ∧ (x,y)∈sa {max{h−(x), min{f−(y),g−(y)}}} = ∧ (x,y)∈sa {max{h−(x), (f− ∩g−)(y)}} = (h− ◦ (f− ∩g−))(a). for any two cases, we obtain f dg is a tripolar fuzzy right pure ideal of s. � by similar method of theorem 3.4, we have the following theorem. theorem 3.5. let f and g be tripolar fuzzy left pure ideals of an ordered semigroup s. then f dg is a tripolar fuzzy left pure ideal of s. combining theorem 3.4 and 3.5, we obtain the following result. corollary 3.3. let f and g be tripolar fuzzy pure ideals of an ordered semigroup s. then adb is a tripolar fuzzy pure ideal of s. int. j. anal. appl. (2022), 20:49 13 an ordered semigroup s is said to be right (resp., left) weakly regular [4] if for any a ∈ s there exist x,y ∈ s such that a ≤ axay (resp., a ≤ xaya). an ordered semigroup s is called weakly regular if it is both a right and a left weakly regular ordered semigroup. lemma 3.5. [4] let s be an ordered semigroup. then the following statements are equivalent: (1) s is right (resp., left) weakly regular. (2) every ideal of s is a right (resp., left) pure ideal of s. lemma 3.6. [4] let s be an ordered semigroup. then the following statements are equivalent: (1) s is weakly regular. (2) every ideal of s is a pure ideal of s. we characterize right weakly regular ordered semigroups in terms of tripolar fuzzy right pure ideals as follows: theorem 3.6. let s be an ordered semigroup. then the following statements are equivalent: (1) s is right weakly regular. (2) every tripolar fuzzy ideal of s is right pure. proof. (1)⇒(2). let f be a tripolar fuzzy ideal of s and g a tripolar fuzzy right ideal of s. let a ∈ s. since s is right weakly regular, there exist x,y ∈ s such that a ≤ axay = (ax)(ay). this implies that sa 6= ∅ and then (g+ ◦ f +)(a) = ∨ (u,v)∈sa {min{g+(u), f +(v)}} ≤ ∨ (u,v)∈sa {min{g+(uv), f +(uv)}} ≤ ∨ (u,v)∈sa {min{g+(a), f +(a)}} = min{g+(a), f +(a)} = (g+ ∩ f +)(a). on other inclusion, we have (g+ ◦ f +)(a) = ∨ (u,v)∈sa {min{g+(u), f +(v)}} ≥ min{g+(ax), f +(ay)} ≥ min{g+(a), f +(a)} = (g+ ∩ f +)(a). 14 int. j. anal. appl. (2022), 20:49 therefore (g+ ◦ f +)(a) = (g+ ∩ f +)(a), (g∗ ◦ f ∗)(a) = ∧ (u,v)∈sa {max{g∗(u), f ∗(v)}} ≥ ∧ (u,v)∈sa {max{g∗(uv), f ∗(uv)}} ≥ ∧ (u,v)∈sa {max{g∗(a), f ∗(a)}} = max{g∗(a), f ∗(a)} = (g∗ ∪ f ∗)(a). on other inclusion, we have (g∗ ◦ f ∗)(a) = ∧ (u,v)∈sa {max{g∗(u), f ∗(v)}} ≤ max{g∗(ax), f ∗(ay)} ≤ max{g∗(a), f ∗(a)} = (g∗ ∪ f ∗)(a). therefore (g∗ ◦ f ∗)(a) = (g∗ ∪ f ∗)(a), and (g− ◦ f−)(a) = ∧ (u,v)∈sa {max{g−(u), f−(v)}} ≥ ∧ (u,v)∈sa {max{g−(uv), f−(uv)}} ≥ ∧ (u,v)∈sa {max{g−(a), f−(a)}} = max{g−(a), f−(a)} = (g− ∪ f−)(a). on other inclusion, we have (g− ◦ f−)(a) = ∧ (u,v)∈sa {max{g−(u), f−(v)}} ≤ max{g−(ax), f−(ay)} ≤ max{g−(a), f−(a)} = (g− ∪ f−)(a). therefore (g−◦ f−)(a) = (f−∪g−)(a), and then g◦ f = g∩ f . hence f is a tripolar fuzzy right pure ideal of s. int. j. anal. appl. (2022), 20:49 15 (2)⇒(1). let a and b be an ideal of s and a right ideal of s, respectively. by lemma 3.1, we obtain χa and χb is a tripolar fuzzy ideal of s and a tripolar fuzzy right ideal of s, respectively. by our assumption, χb is a tripolar fuzzy right pure ideal of s. then χ(ba] = χb ◦χa = χb ∩χa = χb∩a. by lemma 3.2 (1), we obtain b∩a = (ba]. this means that a is a right pure ideal of s. therefore, by lemma 3.5, s is right weakly regular. � by similar method of theorem 3.6, we have the following theorem. theorem 3.7. let s be an ordered semigroup. then the following statements are equivalent: (1) s is left weakly regular. (2) every tripolar fuzzy ideal of s is left pure. combining theorem 3.6 and 3.7, we obtain the following result. corollary 3.4. let s be an ordered semigroup. then the following statements are equivalent: (1) s is weakly regular. (2) every tripolar fuzzy ideal of s is pure. now, we present the concepts of right weakly purity and left weakly purity of tripolar fuzzy ideals. in our last main result, the coincidence of these two concepts is provided. definition 3.2. a tripolar fuzzy ideal f of s is called a tripolar fuzzy right (resp., left) weakly pure ideal if g ◦ f = g ∩ f (resp., f ◦g = f ∩g) for every tripolar fuzzy ideal g of s. a tripolar fuzzy ideal is called a tripolar weakly pure ideal of s if it is both a tripolar fuzzy right and a tripolar fuzzy left ideal of s. a tripolar fuzzy subset f of s is idempotent with respect to ◦ if f ◦ f = f . lemma 3.7. let s be an ordered semigroup and f ,g are tripolar fuzzy right ideals of s. then f ∩g is a tripolar fuzzy right ideal of s. proof. let f ,g be tripolar fuzzy right ideals of s and x,y ∈ s. then we obtain (f + ∩g+)(xy) = f +(xy) ∩g+(xy) ≥ f +(x) ∩g+(x) = (f + ∩g+)(x), (f ∗ ∪g∗)(xy) = f ∗(xy) ∪g∗(xy) ≤ f ∗(x) ∪g∗(x) = (f ∗ ∪g∗)(x), and (f− ∪g−)(xy) = f−(xy) ∪g−(xy) ≤ f−(x) ∪g−(x) = (f− ∪g−)(x). 16 int. j. anal. appl. (2022), 20:49 let x,y ∈ s be such that x ≤ y. then (f + ∩g+)(x) = f +(x) ∩g+(x) ≥ f +(y) ∩g+(y) = (f + ∩g+)(y), (f ∗ ∪g∗)(x) = f ∗(x) ∪g∗(x) ≤ f ∗(y) ∪g∗(y) = (f ∗ ∪g∗)(y), and (f− ∪g−)(x) = f−(x) ∪g−(x) ≤ f−(y) ∪g−(y) = (f− ∪g−)(y). therefore f ∩g is a tripolar fuzzy right ideal of s. � by similar method of lemma 3.7, we have the following lemma. lemma 3.8. let s be an ordered semigroup and f ,g are tripolar fuzzy left ideals of s. then f ∩g is a tripolar fuzzy left ideal of s. combining lemma 3.7 and 3.8, we obtain the following result. corollary 3.5. let s be an ordered semigroup and f ,g are tripolar fuzzy ideals of s. then f ∩g is a tripolar fuzzy ideal of s. our last result illustrates that the concepts of right weakly purity and left weakly purity of tripolar fuzzy ideals coincide. theorem 3.8. let s be an ordered semigroup and f a tripolar fuzzy ideal of s. then the following statements are equivalent. (1) f is tripolar fuzzy right weakly pure ideal. (2) f is idempotent with respect to ◦. (3) f is tripolar fuzzy left weakly pure ideal. proof. (1)⇒(2). let f be a tripolar fuzzy pure ideal of s. then, we obtain f ◦ f = f ∩ f = f . therefore f is idempotent with respect to ◦. (2)⇒(1). let g be a tripolar fuzzy ideal of s. by corollary 3.5, we obtain that g ∩ f is a tripolar fuzzy ideal of s. by our assumption, we have g ∩ f = (g ∩ f ) ◦ (g ∩ f ) ⊆ g ◦ f . on the other hand, let a ∈ s. if sa = ∅, then (g+ ◦ f +)(a) = 0 ≤ (g− ∩ f +)(a), (g∗ ◦ f ∗)(a) = 1 ≥ (g− ∪ f ∗)(a), and (g− ◦ f−)(a) = 0 ≥ (g− ∪ f−)(a). int. j. anal. appl. (2022), 20:49 17 suppose that sa 6= ∅. then, (g+ ◦ f +)(a) = ∨ (u,v)∈sa {min{g+(u), f +(v)}} ≤ ∨ (u,v)∈sa {min{g+(uv), f +(uv)}} ≤ ∨ (u,v)∈sa {min{g+(a), f +(a)}} = min{g+(a), f +(a)} = (g+ ∩ f +)(a), (g∗ ◦ f ∗)(a) = ∧ (u,v)∈sa {max{g∗(u), f ∗(v)}} ≥ ∧ (u,v)∈sa {max{g∗(uv), f ∗(uv)}} ≥ ∧ (u,v)∈sa {max{g∗(a), f ∗(a)}} = max{g∗(a), f ∗(a)} = (g∗ ∪ f ∗)(a), and (g− ◦ f−)(a) = ∧ (u,v)∈sa {max{g−(u), f−(v)}} ≥ ∧ (u,v)∈sa {max{g−(uv), f−(uv)}} ≥ ∧ (u,v)∈sa {max{g−(a), f−(a)}} = max{g−(a), f−(a)} = (g− ∪ f−)(a), thus g ◦ f ⊆ g ∩ f and altogether, we have g ◦ f = g ∩ f . this means that f is a tripolar fuzzy right weakly pure ideal of s. illustrating (2)⇔(3) can be done similarly. � by theorem 3.8, we obtain the following result. corollary 3.6. let s be an ordered semigroup and f a tripolar fuzzy ideal of s. then the following statements are equivalent. (1) f is idempotent with respect to ◦. (2) f is tripolar fuzzy weakly pure ideal. 18 int. j. anal. appl. (2022), 20:49 4. conclusion in this present paper, we introduced the concept of tripolar fuzzy pure ideals in ordered semigroups. some related properties of tripolar pure ideals are studied. we characterized weakly regular ordered semigroups in terms of tripolar fuzzy pure ideals. finally, we introduced the concept of tripolar weakly pure ideals. we proved that the tripolar fuzzy ideals are tripolar fuzzy weakly pure ideals if such tripolar fuzzy ideals satisfied idempotent property. in our future work, we will apply the notions of tripolar fuzzy ideals and tripolar fuzzy pure ideals to the theory of hyperstructures, ordered hyperstructures, semirings, groups, bci/bck algebras, etc. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. ahsan, m. takahashi, pure spectrum of a momoid with zero, kobe j. math. 6 (1989), 163-182. https: //cir.nii.ac.jp/crid/1571698601700142976. [2] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://dl.acm.org/doi/10.5555/ 1708507.1708520. [3] n. rehman, pure fuzzy ideals in ternary semigroups, int. j. algebra stat. 1 (2012), 1-7. [4] t. changphas, j. sanborisoot, pure ideals in ordered semigroups, kyungpook math. j. 54 (2014), 123–129. https://doi.org/10.5666/kmj.2014.54.1.123. [5] n. kehayopulu, m. tsingelis, fuzzy sets in ordered groupoids, semigroup forum. 65 (2001), 128–132. https: //doi.org/10.1007/s002330010079. [6] k. linesawat, n. lekkoksung, s. lekkoksung, anti-hybrid pure ideals in ordered semigroups, int. j. innov. comput. inform. control. 18 (2022), 1275-1290. https://doi.org/10.24507/ijicic.18.04.1275. [7] m.m.k. rao, tripolar fuzzy interior ideals of γ-semigroup, ann. fuzzy math. inform. 15 (2018), 199–206. https: //doi.org/10.30948/afmi.2018.15.2.199. [8] m. murali krishna rao, b. venkateswarlu, tripolar fuzzy ideals of γ-semirings, asia pac. j. math. 5 (2018), 192-207. https://doi.org/10.28924/apjm/5-2-192-207. [9] m. murali krishna rao, b. venkateswarlu, tripolar fuzzy interior ideals and tripolar fuzzy soft interior ideals over γ-semirings, facta univ. ser.: math. inform. 35 (2020), 29-42. https://doi.org/10.22190/fumi2001029k. [10] m. murali krishna rao, tripolar fuzzy interior ideals and tripolar fuzzy soft interior ideals over semigroups, ann. fuzzy math. inform. 20 (2020), 243-256. https://doi.org/10.30948/afmi.2020.20.3.243. [11] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 338-353. [12] k. siribute, j. sanborisoot, on pure fuzzy ideals in ordered semigroups, int. j. math. computer sci. 14 (2019), 867-877. [13] y. xie, j. liu, l. wang, further studies on h∞ filtering design for fuzzy system with known or unknown premise variables, ieee access. 7 (2019), 121975–121981. https://doi.org/10.1109/access.2019.2938797. [14] l.a. zadeh, fuzzy sets, inform. control, 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [15] wen-ran zhang, bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, in: nafips/ifis/nasa ’94. proceedings of the first international joint conference https://cir.nii.ac.jp/crid/1571698601700142976 https://cir.nii.ac.jp/crid/1571698601700142976 https://dl.acm.org/doi/10.5555/1708507.1708520 https://dl.acm.org/doi/10.5555/1708507.1708520 https://doi.org/10.5666/kmj.2014.54.1.123 https://doi.org/10.1007/s002330010079 https://doi.org/10.1007/s002330010079 https://doi.org/10.24507/ijicic.18.04.1275 https://doi.org/10.30948/afmi.2018.15.2.199 https://doi.org/10.30948/afmi.2018.15.2.199 https://doi.org/10.28924/apjm/5-2-192-207 https://doi.org/10.22190/fumi2001029k https://doi.org/10.30948/afmi.2020.20.3.243 https://doi.org/10.1109/access.2019.2938797 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x int. j. anal. appl. (2022), 20:49 19 of the north american fuzzy information processing society biannual conference. ieee, san antonio, tx, usa, 1994: pp. 305–309. https://doi.org/10.1109/ijcf.1994.375115. https://doi.org/10.1109/ijcf.1994.375115 1. introduction 2. preliminaries 3. main results 4. conclusion references international journal of analysis and applications volume 16, number 3 (2018), 400-413 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-400 on weakly 2-absorbing semi-primary submodules of modules over commutative rings pairote yiarayong∗ and manoj siripitukdet department of mathematics, faculty of science, naresuan university, phitsanuloke 65000, thailand ∗corresponding author: pairote0027@hotmail.com abstract. let r be a commutative ring with identity and let m be a unitary r-module. we say that a proper submodule n of m is a weakly 2-absorbing semi-primary submodule if a1,a2 ∈ r,m ∈ n with 0 6= a1a2m ∈ n, then a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. in this paper, we study weakly 2-absorbing semi-primary submodules and we prove some basic properties of these submodules. also, we give a characterization of weakly 2-absorbing semi-primary submodules and we investigate weakly 2-absorbing semi-primary submodules of some well-known modules. 1. introduction throughout this paper, we assume that all rings are commutative with 1 6= 0. let r be a commutative ring and let m be an r-module. we will denote by (n : m) a residual of n by m, that is, the set of all r ∈ r such that rm ⊆ n. clearly, √ i = {r ∈ r : rn ∈ i for some positive integer n} denotes the radical ideal of r. in 2003, anderson and smith [1] introduced the concept of a weakly prime ideal of a commutative ring. they said that a proper ideal p of the commutative ring r is weakly prime if a, b ∈ r and 0 6= ab ∈ p , then a ∈ p or b ∈ p . a weakly primary ideals were first introduced and studied by atani and farzalipour in [2]. recall that a proper ideal p of r is called a weakly primary ideal of r as in [2] if for a, b ∈ r with 0 6= ab ∈ p , then a ∈ p or bn ∈ p for some positive integer n. clearly, a weakly prime ideal of r is also a 2010 mathematics subject classification. 13a15,13f05. key words and phrases. weakly 2-absorbing semi-primary submodule; 2-absorbing semi-primary submodule; absorbing semiprimary triple-zero; weakly 2-absorbing primary ideal. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 400 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-400 int. j. anal. appl. 16 (3) (2018) 401 weakly primary ideal of r. the concept of weakly 2-absorbing ideals, which is a generalization of 2-absorbing ideals, was introduced by badawi and darani in [3]. recall from [3] that a proper ideal i of r is said to be a weakly 2-absorbing ideal of r if whenever a, b, c ∈ r with 0 6= abc ∈ i, then ab ∈ i or ac ∈ i or bc ∈ i. in [4], badawi et. al. defined a proper ideal i of a commutative ring r to be a weakly 2-absorbing primary ideal if whenever a, b, c ∈ r and 0 6= abc ∈ i, then ab ∈ i or ac ∈ √ i or bc ∈ √ i. the concept of weakly prime submodule was introduced and studied by behboodi and koohi [5]. we recall that a proper submodule n of m is called a weakly prime submodule, if 0 6= rm ∈ n, where r ∈ r, m ∈ m, then m ∈ n or r ∈ (n : m). the idea of decomposition of submodules into weakly primary submodules were introduced by atani and farzalipour in [2]. a weakly primary submodule n of m to be a proper submodule of m and if r ∈ r, m ∈ m and 0 6= rm ∈ n, then m ∈ n or rn ∈ (n : m) for some positive integer n. clearly, every primary submodule of a module is a weakly primary submodule. in [6], the concept of weakly 2-absorbing submodule generalized to 2-absorbing submodule of a module over a commutative ring. a proper submodule n of m is called a weakly 2-absorbing submodule, if whenever a, b ∈ r and m ∈ m with 0 6= abm ∈ n, then ab ∈ (n : m) or am ∈ n or bm ∈ n. in 2016, mostafanasab et al. [11] introduced the concept of weakly 2-absorbing primary submodules of modules over commutative rings with identities. recall that a proper submodule n of m is called a weakly 2-absorbing primary submodule of m as in [11] if whenever 0 6= abm ∈ n for some a, b ∈ r and m ∈ m, then ab ∈ (n : m) or am ∈ m −rad(n) or bm ∈ m − rad(n). the concept of weakly classical prime submodule, which is a generalization of classical prime submodule, was introduced by mostafanasab et al. in [10]. recall from [10] that a proper submodule n of m is said to be a weakly classical prime submodule of m if whenever a, b ∈ r and m ∈ m with 0 6= abm ∈ n, then am ∈ n or bm ∈ n. the concept of weakly classical primary submodule, a generalization of primary submodules was introduced and investigated in [9]. he weakly classical primary submodule n of m to be a proper submoduleof r and if a, b ∈ r and 0 6= abm ∈ n, then am ∈ n or mbn ∈ n for some positive integer n. motivated and inspired by the above works, the purposes of this paper are to introduce generalizations of weakly 2-absorbing primary submodule to the context of weakly 2-absorbing semi-primary submodule. a proper submodule n of m to be a weakly 2-absorbing semi-primary submodule of m if whenever 0 6= a1a2m ∈ n for a1, a2 ∈ r, m ∈ m, then a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. some characterizations of weakly 2-absorbing semi-primary submodules are obtained. moreover, we investigate relationships between 2-absorbing semi-primary and weakly 2-absorbing semi-primary submodules of modules over commutative rings. int. j. anal. appl. 16 (3) (2018) 402 2. properties of weakly 2-absorbing semiprimary submodules the results of the following theorems seem to play an important role to study weakly 2-absorbing semiprimary submodules of modules over commutative rings; these facts will be used frequently and normally we shall make no reference to this definition. definition 2.1. a proper submodule n of an r-module m is called a weakly 2-absorbing semi-primary (2-absorbing semi-primary) submodule, if for each m ∈ m and a1, a2 ∈ r, 0 6= a1a2m ∈ n(a1a2m ∈ n), then a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. remark 2.1. it is easy to see that every weakly 2-absorbing primary submodule (2-absorbing semi-primary) submodule is weakly 2-absorbing semi-primary submodule. the following example shows that the converse of definition 2.1 is not true. example 2.1. let r = z and m = z. consider the submodule n = 12z of m. it is easy to see that n is a 2-absorbing semi-primary submodule of m. notice that 2 · 2 · 3 ∈ n, but 2 · 3 6∈ n and (2 · 2)n 6∈ (n : m) for all positive integer n. therefore n is not a 2-absorbing primary submodule of m. example 2.2. let r = z and m = z30. consider the submodule n = {[0]} of m. it is easy to see that n is a weakly 2-absorbing semi-primary submodule of m. notice that (2·3)[5] ∈{[0]}, but 2·3 6∈ √ (n : m), 2[5] 6∈ {[0]} and 3n[5] 6∈ {[0]} for all positive integer n. therefore n is not a 2-absorbing semi-primary submodule of m. theorem 2.1. let n be a proper submodule of an r-module m. then the following statements hold: (1) if n is a weakly 2-absorbing semi-primary submodule of m, then (n : m) is a weakly 2-absorbing primary ideal of r for every m ∈ m −n. (2) for every m ∈ m − n if (n : m) is a weakly primary ideal of r, then n is a weakly 2-absorbing semi-primary submodule of m. proof. 1. let a1, a2, a3 ∈ r such that 0 6= a1a2a3 ∈ (n : m). clearly, 0 6= a1a3(a2m) ∈ n. by definition 2.1, a1a3 ∈ √ (n : m) ⊆ √ (n : m) or a1a2m ∈ n or an3 a2m ∈ n for some positive integer n. therefore a1a2 ∈ (n : m) or a2a3 ∈ √ (n : m) or a1a3 ∈ √ (n : m). hence (n : m) is a weakly 2-absorbing primary ideal of r. 2. let a1, a2 ∈ r such that 0 6= a1a2m ∈ n. then 0 6= a1a2 ∈ (n : m). by assumption, a1 ∈ (n : m) or an2 ∈ (n : m) for some positive integer n. therefore a1m ∈ n or an2 m ∈ n for some positive integer n. hence n is a weakly 2-absorbing semi-primary submodule of m. � but the converse of the above theorem is not true. for every m ∈ m − n, if (n : m) is weakly 2absorbing primary ideal, then n may not be weakly 2-absorbing semi-primary. let m = z × z × z be an int. j. anal. appl. 16 (3) (2018) 403 z-module. consider the submodule n = {0}× 6z × z of m. clearly, (n : (m1, m2, m3)) = {0} is a weakly 2-absorbing primary ideal of r, where (m1, m2, m3) ∈ m −n. notice that (0, 0, 0) 6= (2 ·3)(0, 1, 1) ∈ n, but 2 · 3 6∈ √ (n : m), 2(0, 1, 1) 6∈ n and 3n(0, 1, 1) 6∈ n for all positive integer n. therefore n is not a weakly 2-absorbing semi-primary submodule of m. theorem 2.2. if n is a weakly 2-absorbing semi-primary submodule of an r-module m, then (n : r) is a weakly 2-absorbing semi-primary submodule of m containing n for every r ∈ r − (n : m). proof. let a1, a2 ∈ r and m ∈ m such that 0 6= a1a2m ∈ (n : r). then 0 6= a1a2(rm) = ra1a2m ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) or a1rm ∈ n or an2 rm ∈ n for some positive integer n. therefore a1a2 ∈ √ (n : m) or a1m ∈ (n : r) or an2 ∈ (n : r) for some positive integer n. hence (n : r) is a weakly 2-absorbing semi-primary submodule of m. � theorem 2.3. let {0} be a 2-absorbing semi-primary submodule of an r-module m. then n is a weakly 2-absorbing semi-primary submodule of m if and only if n is a 2-absorbing semi-primary submodule of m. proof. suppose that n is a 2-absorbing semi-primary submodule of m. clearly, n is a weakly 2-absorbing semi-primary submodule of m. conversely, assume that n is a weakly 2-absorbing semi-primary submodule of m. let a1, a2 ∈ r and m ∈ m such that a1a2m ∈ n. if a1a2m 6∈ {0}, then 0 6= a1a2m ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. now if a1a2m ∈ {0}, then a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. hence n is a 2-absorbing semi-primary submodule of m. � theorem 2.4. let m and ḿ be two r-modules and f : m → ḿ be an epimorphism of an r-module. if n is a weakly 2-absorbing semi-primary submodule of m such that kerf ⊆ n, then f(n) is a weakly 2-absorbing semi-primary submodule of ḿ. proof. let a1, a2 ∈ r and ḿ ∈ ḿ such that 0 6= a1a2ḿ ∈ f(n). thus 0 6= a1a2ḿ = ḿ0 for some ḿ0 ∈ f(n). since f is an epimorphism, there exist m ∈ m and m0 ∈ n such that ḿ = f(m) and ḿ0 = f(m0). this implies that 0 6= a1a2f(m) = f(m0). therefore f(a1a2m−m0) = 0 and so a1a2m−m0 ∈ kerf ⊆ n. also, 0 6= a1a2m ∈ n, because if a1a2m = 0, then m0 ∈ kerf. it follows that f(m0) = 0, a contradiction. now, since n is a weakly 2-absorbing semi-primary, we have a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. therefore a1a2 ∈ √ (f(n) : ḿ) or a1ḿ ∈ f(n) or an2 ḿ ∈ f(n) for some positive integer n. hence f(n) is a 2-absorbing semi-primary submodule of ḿ. � theorem 2.5. let m be an r-module and n ⊆ k be two submodules of m. if k is a weakly 2-absorbing semi-primary submodule of m, then k/n is a weakly 2-absorbing semi-primary submodule of m/n. int. j. anal. appl. 16 (3) (2018) 404 proof. let a1, a2 ∈ r and m ∈ m such that n 6= a1a2(m + n) ∈ (k/n). then 0 6= a1a2m ∈ k. by definition 2.1, a1a2 ∈ √ (k : m) or a1m ∈ k or an2 m ∈ k for some positive integer n. therefore a1a2 ∈ √ (k/n : m/n) or a1(m + n) ∈ k/n or an2 (m + n) ∈ k/n for some positive integer n. hence k/n is a weakly 2-absorbing semi-primary submodule of m/n. � theorem 2.6. let m be an r-module and n ⊆ k be two submodules of m. suppose that n is a weakly 2absorbing semi-primary submodule of m. if k/n is a weakly 2-absorbing semi-primary submodule of m/n, then k is a weakly 2-absorbing semi-primary submodule of m. proof. let a1, a2 ∈ r and m ∈ m such that 0 6= a1a2m ∈ k. if a1a2m ∈ n, then 0 6= a1a2m ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) ⊆ √ (k : m) or a1m ∈ n ⊆ k or an2 m ∈ n ⊆ k for some positive integer n. if a1a2m 6∈ n, then n 6= a1a2(m + n) ∈ n. again, by definition 2.1, a1a2 ∈ √ (k/n : m/n) or a1(m + n) ∈ k/n or an2 (m + n) ∈ k/n for some positive integer n. thus a1a2 ∈ √ (k : m) or a1m ∈ k or an2 m ∈ k for some positive integer n. hence k is a weakly 2-absorbing semi-primary submodule of m. � corollary 2.1. then n is a weakly 2-absorbing semi-primary submodule of an r-module m if and only if n/{0} is a weakly 2-absorbing semi-primary submodule of an r-module m/{0}. proof. it is straightforward by theorem 2.5 and theorem 2.6. � theorem 2.7. let n be a submodule of an r-module m and s be a multiplicative subset of r. if n is a weakly 2-absorbing semi-primary submodule of m such that (n : m) ∩ s = ∅, then s−1n is a weakly 2-absorbing semi-primary submodule of s−1m. proof. clearly, s−1n is a proper submodule of s−1m. let a1, a2 ∈ r, s1, s2, s3 ∈ s and m ∈ m such that 0 6= a1 s1 a2 s2 m s3 ∈ s−1n. then there exists s ∈ s such that sa1a2m ∈ n. if sa1a2m = 0, then a1s1 a2 s2 m s3 = sa1 ss1 a2 s2 m s3 = 0 1 , a contradiction. if sa1a2m 6= 0, then 0 6= a1a2(sm) ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) or a1sm ∈ n or an2 sm ∈ n for some positive integer n. thus a1 s1 a2 s2 ∈ √ (s−1n : s−1m) or a1 s1 m s3 = a1sm s1s3s ∈ s−1n or ( a2 s2 )n m s3 = an2 sm sn2 s3s ∈ s−1n for some positive integer n. hence s−1n is a weakly 2-absorbing semi-primary submodule of s−1m. � theorem 2.8. let n be a submodule of an r-module m and s be a multiplicative subset of r. if s−1n is a weakly 2-absorbing semi-primary submodule of s−1m such that s ∩ zd(n) = ∅ and s ∩ zd(m/n) = ∅, then n is a weakly 2-absorbing semi-primary submodule of m. proof. let a1, a2 ∈ r and m ∈ m such that 0 6= a1a2m ∈ n. then a11 a2 1 m 1 ∈ s−1n. if a1 1 a2 1 m 1 = 0 1 , then there exists s ∈ s such that sa1a2m = 0 which is a contradiction. if a11 a2 1 m 1 6= 0 1 , then 0 1 6= a1 1 a2 1 m 1 ∈ s−1n. by definition 2.1, a1 1 a2 1 ∈ √ (s−1n : s−1m) or a1 1 m 1 ∈ s−1n or ( a2 1 )n m 1 ∈ s−1n for some positive integer int. j. anal. appl. 16 (3) (2018) 405 n. if a1 1 a2 1 ∈ √ (s−1n : s−1m), then ( a1 1 a2 1 )n ∈ (s−1n : s−1m) for some positive integer n. thus there exists s ∈ s such that s(a1a2)nm ⊆ n for some positive integer n. since s ∩ zd(m/n) = ∅, we have (a1a2) nm ⊆ n so a1a2 ∈ √ (n : m). if a1 1 m 1 ∈ s−1n, there exists s ∈ s such that sa1m ∈ n. thus s(a1m + n) = sa1m + n = n. but s ∩ zd(m/n) = ∅, a1m ∈ n. if ( a21 ) n am 1 ∈ n, there exists s ∈ s such that such that san1 m ∈ n for some positive integer n. thus s(an2 m + n) = san2 m + n = n for some positive integer n. since s ∩zd(m/n) = ∅, we have an2 m ∈ n for some positive integer n. therefore n is a weakly 2-absorbing semi-primary submodule of m. � theorem 2.9. let n be a proper submodule of an r-module m. the following conditions are equivalent: (1) n is a weakly 2-absorbing semi-primary submodule of m. (2) for every a1, a2 ∈ r − (n : m) if a1a2 ∈ r − √ (n : m), then (n : a1a2) ⊆ (0 : a1a2) ∪ (n : a1) ∪ (n : an2 ) for some positive integer n. (3) for every a1, a2 ∈ r − (n : m) if r is a u-ring and a1a2 ∈ r − √ (n : m), then (n : a1a2) ⊆ (0 : a1a2) or (n : a1a2) ⊆ (n : a1) or (n : a1a2) ⊆ (n : an2 ) for some positive integer n. proof. (1 ⇒ 2) let m ∈ (n : a1a2). then a1a2m ∈ n. if a1a2m = 0, then m ∈ (0 : a1a2) ⊆ (0 : a1a2) ∪ (n : a1) ∪ (n : an2 ) for some positive integer n. if a1a2m 6= 0, then 0 6= a1a2m ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. but a1a2 ∈ r − √ (n : m), m ∈ (n : a1) or m ∈ (n : an2 ) for some positive integer n. therefore m ∈ (n : a1) ∪ (n : an2 ) for some positive integer n. hence (n : a1a2) = (0 : a1a2) ∪ (n : a1) ∪ (n : an2 ) for some positive integer n. (2 ⇔ 3) it is obvious. (2 ⇒ 1) let a1, a2 ∈ r such that 0 6= a1a2m ∈ n. then m ∈ (n : a1a2) and m 6∈ (n : 0). by assumption, m ∈ (0 : a1a2) ∪ (n : a1) ∪ (n : an2 ) for some positive integer n. clearly, a1m ∈ n or an2 m ∈ n for some positive integer n. hence n is a weakly 2-absorbing semi-primary submodule of m. � corollary 2.2. let n be a proper submodule of an r-module m. the following conditions are equivalent: (1) n is a weakly 2-absorbing semi-primary submodule of m. (2) for every a ∈ r − (n : m) and every ideal i of r such that i 6⊆ (n : m), if ai 6⊆ √ (n : m), then (n : ai) ⊆ (0 : ai) ∪ (n : a) ∪ (n : in) for some positive integer n. (3) for every a ∈ r − (n : m) and every ideal i of r such that i 6⊆ (n : m), if r is a u-ring and ai 6⊆ √ (n : m), then (n : ai) ⊆ (0 : ai) or (n : ai) ⊆ (n : a) or (n : ai) ⊆ (n : in) for some positive integer n. (4) for every ideals i, j of r such that i, j 6⊆ (n : m), if ij 6⊆ √ (n : m), then (n : ij) ⊆ (0 : ij) ∪ (n : i) ∪ (n : jn) for some positive integer n. (5) for every ideals i, j of r such that i, j 6⊆ (n : m), if r is a u-ring and ij 6⊆ √ (n : m), then (n : ij) ⊆ (0 : ij) or (n : ij) ⊆ (n : i) or (n : ij) ⊆ (n : jn) for some positive integer n. int. j. anal. appl. 16 (3) (2018) 406 proof. it is clear from theorem 2.9. � theorem 2.10. let n be a proper submodule of an r-module m. the following conditions are equivalent: (1) n is a weakly 2-absorbing semi-primary submodule of m. (2) for every a ∈ r − (n : m) and m ∈ m, if am 6∈ n, then (n : am) ⊆ (0 : am) ∪ ( √ ((n : m) : a) ∪ √ (n : m). proof. (1 ⇒ 2) let a ∈ r − (n : m) and m ∈ m such that am 6∈ n. assume that r ∈ (n : am). then ram ∈ n. if ram 6= 0, then 0 6= ram ∈ n. by definition 2.1, ar ∈ √ (n : m) or am ∈ n or rnm ∈ n for some positive integer n. since am 6∈ n, we have r ∈ ( √ (n : m) : a) or r ∈ √ (n : m). this implies that r ∈ ( √ (n : m) : a) ∪ √ (n : m) ⊆ (0 : am) ∪ ( √ ((n : m) : a) ∪ √ (n : m). thus (n : am) ⊆ (0 : am)∪( √ ((n : m) : a)∪ √ (n : m). if ram = 0, then r ∈ (0 : am) ⊆ (0 : am)∪( √ ((n : m) : a)∪ √ (n : m). therefore (n : am) ⊆ (0 : am) ∪ ( √ ((n : m) : a) ∪ √ (n : m). (2 ⇒ 1) it is clear. � corollary 2.3. let n be a proper submodule of an r-module m. the following conditions are equivalent: (1) n is a weakly 2-absorbing semi-primary submodule of m. (2) for every ideal i of r such that i ⊆ r − (n : m) and m ∈ m, if im 6⊆ n, then (n : im) ⊆ (0 : im) ∪ ( √ (n : m) : i) ∪ √ (n : m). proof. it is clear from theorem 2.10. � definition 2.2. let n be a proper submodule of m. if n is a 2-absorbing semi-primary submodule and a1a2m = 0, a1a2 6∈ √ (n : m), a1m 6∈ n and an2 m 6∈ n for all positive integer n, then (a1, a2, m) is called a absorbing semi-primary triple-zero of n where a1, a2 ∈ r, m ∈ m. theorem 2.11. let n be a weakly 2-absorbing semi-primary submodule of an r-module m. suppose that k is a submodule of m and a1, a2 ∈ r such that n ⊆ k and a1a2k ⊆ n. if (a1, a2, m) is not a absorbing semi-primary triple-zero of n for every m ∈ k, then a1a2 ∈ √ (k : m) or a1k ⊆ n or an2 k ⊆ n for some positive integer n. proof. assume that a1a2 6∈ √ (k : m), a1k 6⊆ n and an2 k 6⊆ n for all positive integer n. then there are k1, k2 ∈ k such that a1k1 6∈ n and an2 k2 6∈ n for all positive integer n. if a1a2k1 6= 0, then 0 6= a1a2k1 ∈ n. by definition 2.1, an12 k1 ∈ n for some positive integer n1. so let a1a2k1 = 0. by definition 2.2, a n2 2 k1 ∈ n for some positive integer n2. now if a1a2k2 6= 0, then 0 6= a1a2k2 ∈ n. again, by definition 2.1, ak2 ∈ n. next let a1a2k2 = 0. now by definition 2.2, a1k2 ∈ n. let n0 = max{n1, n2}. then an02 k1, a1k2 ∈ n. since a1a2k ⊆ n, we have a1a2(k1 + k2) ∈ n. if a1a2(k1 + k2) 6= 0, then 0 6= a1a2(k1 + k2) ∈ n. thus by definition 2.1, a1(k1 + k2) ∈ n or an32 (k1 + k2) ∈ n for some positive integer n3. this implies that a1k1 ∈ n int. j. anal. appl. 16 (3) (2018) 407 or an42 k2 ∈ n where n4 = max{n0, n3} and we get a contradiction. assume that a1a2(k1 + k2) = 0. new since (a1, a2, k1 + k2) is not a absorbing semi-primary triple-zero of n, we have a1(k1 + k2) ∈ n or an52 (k1 + k2) ∈ n for some positive integer n5. clearly, a1k1 ∈ n or a n6 2 k2 ∈ n, where n6 = max{n0, n5}, which again is a contradiction. hence a1a2 ∈ √ (k : m) or a1k ⊆ n or an2 k ⊆ n for some positive integer n. � theorem 2.12. let n be a weakly 2-absorbing semi-primary submodule of an r-module m. suppose that (a1, a2, m) is a absorbing semi-primary triple-zero of n for some a1, a2 ∈ r and m ∈ m. then (1) a1a2n = {0}; (2) a1(n : m)m = {0}; (3) (n : m)a2m = {0}; (4) (n : m)2m = {0}; (5) a1(n : m)n = {0}; (6) (n : m)a2n = {0}. proof. 1. suppose that a1a2n 6= {0}. then there exists m0 ∈ n such that a1a2m0 6∈ {0}. thus a1a2m + a1a2m0 6= 0 so 0 6= a1a2(m + m0) ∈ n. by definition 2.1, a1a2 ∈ √ (n : m) or a1(m + m0) ∈ n or an2 (m + m0) ∈ n for some positive integer n. therefore a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. this is a contradiction. hence a1a2n = {0}. 2. suppose that a1(n : m)m 6= {0}. then there exists r ∈ (n : m) such that a1rm 6= 0. since rm ∈ n, we have 0 6= a1(a2 + r)m ∈ n. by definition 2.1, a1(a2 + r) ∈ √ (n : m) or a1m ∈ n or (a2 + r)nm ∈ n for some positive integer n. thus a1a2 ∈ √ (n : m) or a1m ∈ n or an2 ∈ n for some positive integer n. this is a contradiction. hence a1(n : m)m = {0}. 3. the proof is similar to part 2. 4. assume that (n : m)2m 6= {0}. then there exist r, s ∈ (n : m) such that rsm 6= 0. then by parts 1 and 2, (a1 + r)(a2 + s)m 6= 0. clearly, 0 6= (a1 + r)(a2 + s)m ∈ n. by definition 2.1, (a1 + r)(a2 + s) ∈ √ (n : m) or (a1 + r)m ∈ n or (a2 + s)nm ∈ n for some positive integer n. therefore a1a2 ∈ √ (n : m) or a1m ∈ n or an2 ∈ n for some positive integer n. this is a contradiction. hence (n : m)2m = {0}. 5. suppose that a1(n : m)n 6= {0}. then there exist r ∈ (n : m) and m0 ∈ n such that a1rm0 6= 0. therefore by parts 1 and 2 we conclude that a1(a2 + r)(m + m0) 6= 0. clearly, 0 6= a1(a2 + r)(m + m0) ∈ n. by definition 2.1, a1(a2 + r) ∈ √ (n : m) or a1(m + m0) ∈ n or (a2 + r)n(m + m0) ∈ n for some positive integer n. therefore a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n for some positive integer n. this is a contradiction. hence a1(n : m)n = {0}. 6. the proof is similar to part 5. � int. j. anal. appl. 16 (3) (2018) 408 theorem 2.13. let m be an r-module. if n is a weakly 2-absorbing semi-primary submodule of m that is not 2-absorbing semi-primary, then (n : m)2n = {0}. proof. suppose that n is a weakly 2-absorbing semi-primary submodule of m that is not 2-absorbing semi-primary submodule. then there exists a absorbing semi-primary triple-zero (a1, a2, m) of n for some a1, a2 ∈ r and m ∈ m. assume that (n : m)2n 6= {0}. then there exist r, s ∈ (n : m) and m0 ∈ n such that rsm0 6= 0. since (a1 + r)(a2 + s)(m + m0) 6= 0, we have 0 6= (a1 + r)(a2 + s)(m + m0) ∈ n. by definition 2.1, (a1 + r)(a2 + s) ∈ √ (n : m) or (a1 + r)(m + n) ∈ n or (a2 + s)n(m + n) ∈ n for some positive integer n. therefore a1a2 ∈ √ (n : m) or a1m ∈ n or an2 m ∈ n. this is a contradiction. hence (n : m)2n = {0}. � corollary 2.4. let m be a multiplication r-module. if n is a weakly 2-absorbing semi-primary submodule of m that is not 2-absorbing semi-primary submodule, then n3 = {0}. proof. suppose that n is a weakly 2-absorbing semi-primary submodule of m that is not 2-absorbing semiprimary submodule. by assumption, n = (n : m)m. then by theorem 2.13, n3 = (n : m)3m = (n : m)2((n : m)m) = (n : m)2n = {0}. � lemma 2.1. suppose that n is a weakly 2-absorbing semi-primary submodule of an r-module m and (0 : m2) is a 2-absorbing primary ideal of a ring r where m2 ∈ m − n. for all m1 ∈ m, if rs ∈ (n : m1) − √ (n : m2), then (n : rsm2) ⊆ (n : rm2) ∪ √ (n : snm2) for some positive integer n. proof. suppose that rs ∈ (n : m1) − (n : m2) where m1 ∈ m and m2 ∈ m − n. let a ∈ (n : rsm2). then (ars)m2 = a(rsm2) ∈ n so ars ∈ (n : m2). if arsm2 6= 0, then 0 6= ars ∈ (n : m2). by assumption, ar ∈ (n : m2) or as ∈ √ (n : m2) or rs ∈ √ (n : m2). by the assumption, ar ∈ (n : m2) or as ∈ √ (n : m2). thus a ∈ (n : rm2) or a ∈ √ (n : snm2) for some positive integer n. this implies that (n : rsm2) ⊆ (n : rm2)∪ √ (n : snm2) for some positive integer n. now if arsm2 = 0, then ars ∈ (0 : m2). thus ar ∈ (0 : m2) or as ∈ √ (n : m2) or rs ∈ √ (n : m2). therefore (n : rsm2) ⊆ (n : rm2)∪ √ (n : snm2) for some positive integer n. � proposition 2.1. let n be an irreducible submodule of an r-module m. for all r ∈ r if (n : r) = (n : r2), then n is a weakly 2-absorbing semi-primary submodule of m. proof. let a1, a2 ∈ r and m ∈ m such that 0 6= a1a2m ∈ n. suppose that a1a2 6∈ √ (n : m), a1m 6∈ n and an2 m 6∈ n for all positive integer n. clearly, n ⊆ (n + a1a2m)∩(n + ra1m)∩(n + ran2 m) for all positive integer n. let m0 ∈ (n + a1a2m) ∩ (n + ra1m) ∩ (n + ran2 m). this implies that m0 ∈ n + a1a2m, m0 ∈ n +ra1m and m0 ∈ n +ran2 m. then there exist r1, r2 ∈ r, m1 ∈ m and n1, n2 ∈ n such that n1+a1a2m1 = m0 = n2 + r1a1m = m0 = n3 + b n 2 m. since a1n1 + a 2 1a2m1 = a1m0 = a1n2 + r1a 2 1m = a1m0 = a1n3 + a1b n 2 m, int. j. anal. appl. 16 (3) (2018) 409 we have a21r1m ∈ n. it follows that r1m ∈ (n : a21). by the assumption, r1m ∈ (n : a1), so that r1a1m ∈ n. thus n = (n +a1a2m)∩(n +ra1m)∩(n +ran2 m). now since n is an irreducible, we have n +a1a2m ⊆ n or a1m ∈ n + ra1m ⊆ n or an2 m ∈ n + ran2 m ⊆ n, a contradiction. hence n is a weakly 2-absorbing semi-primary submodule of m. � theorem 2.14. let mi be an ri-module and ni be a proper submodule of mi, for i = 1, 2. if n1 × m2 is a weakly 2-absorbing semi-primary submodule of m1 × m2, then n1 is a weakly 2-absorbing semi-primary submodule of m1. proof. suppose that n1×m2 is a weakly 2-absorbing semi-primary submodule of m1×m2. let a1, a2 ∈ r1 and m ∈ m1 such that 0 6= a1a2m ∈ n1. then (0, 0) 6= (a1, 0)(a2, 0)(m, 0) = (a1a2m, 0) ∈ n1 × m2. by definition 2.1, (a1a2, 0) = (a1, 0)(a2, 0) ∈ √ (n1 ×m2 : m1 ×m2) or (a1m, 0) = (a1, 0)(m, 0) ∈ n1 × m2 or (an2 m, 0) = (a2, 0) n(m, 0) ∈ n1 × m2 for some positive integer n. this implies that a1a2 ∈ √ (n1 : m1) or a1m ∈ n1 or an2 m ∈ n1 for some positive integer n. hence n1 is a weakly 2-absorbing semi-primary submodule of m1. � corollary 2.5. let mi be an ri-module and ni be a proper submodule of mi, for i = 1, 2. if m1 × n2 is a weakly 2-absorbing semi-primary submodule of m1 × m2, then n2 is a weakly 2-absorbing semi-primary submodule of m2. proof. it is clear from theorem 2.14. � corollary 2.6. let mi be an ri-module and ni be a proper submodule of mi, for i = 1, 2, . . . , k. if m1 × m2×. . .×mj−1×nj×mj+1×. . .×mk is a weakly 2-absorbing semi-primary submodule of m1×m2×. . .×mk, then nj is a weakly 2-absorbing semi-primary submodule of mj. proof. it is clear from theorem 2.14 and corollary 2.5. � theorem 2.15. let mi be an r-module and let ni be a proper submodule of mi, for i = 1, 2. then the following conditions are equivalent: (1) n1 ×m2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2. (2) (a) n1 is a weakly 2-absorbing semi-primary submodule of m1. (b) for each a1, a2 ∈ r and m ∈ m1 such that a1a2m = 0, if a1a2 6∈ √ (n1 : m1) and a1m 6∈ n1, a n 2 m 6∈ n1 for all positive integer n, then a1a2 ∈ (0 : m2). proof. (1 ⇒ 2). (a). this follows from theorem 2.14. (b). let a1a2m = 0, a1m 6∈ n1 and an2 m 6∈ n1 for all positive integer n, where a1, a2 ∈ r and m ∈ m1. suppose that a1a2 6∈ (0 : m2). there exists m2 ∈ m2 such that a1a2m2 6= 0. thus (0, 0) 6= a1a2(m, m2) = int. j. anal. appl. 16 (3) (2018) 410 (a1a2m, a1a2m2) ∈ n1 × m2. by part 1, i.e., a1a2 ∈ √ (n1 ×m2 : m1 ×m2) or a1(m, m2) ∈ n1 × m2 or an2 (m, m2) ∈ n1 × m2 for some positive integer n. thus a1a2 ∈ √ (n1 : m1) or a1m ∈ n1 or an2 m ∈ n1 which is a contradiction. hence a1a2 ∈ (0 : m2). (2 ⇒ 1). let a1, a2 ∈ r and (m1, m2) ∈ m1 ×m2 such that (0, 0) 6= (a1a2m1, a1a2m2) = a1a2(m1, m2) ∈ n1 × m2. if a1a2m1 6= 0, then 0 6= a1a2m1 ∈ n1. by part (a), a1a2 ∈ √ (n1 : m1) or a1m1 ∈ n1 or an2 m1 ∈ n1 for some positive integer n. so a1a2 ∈ √ (n1 ×m2 : m1 ×m2) or a1(m1, m2) = (a1m1, a1m2) ∈ n1×m2 or an2 (m1, m2) = (an2 m1, an2 m2) ∈ n1×m2, and thus we are done. if a1a2m1 = 0, then a1a2m2 6= 0. therefore a1a2 6∈ (0 : m2). by part (b), a1a2 ∈ √ (n1 : m1) or a1m1 ∈ n1 or an2 m1 ∈ n1 for some positive integer n. thus a1a2 ∈ √ (n1 ×m2 : m1 ×m2) or a1(m1, m2) ∈ n1 ×m2 or an2 (m1, m2) ∈ n1 ×m2. hence n1 ×m2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2. � corollary 2.7. let mi be an r-module and let ni be a proper submodule of mi, for i = 1, 2. then the following conditions are equivalent: (1) m1 ×n2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2. (2) (a) n2 is a weakly 2-absorbing semi-primary submodule of m2. (b) for each a1, a2 ∈ r and m ∈ m2 such that a1a2m = 0, if a1a2 6∈ √ (n2 : m2), a1m 6∈ n2 and an2 m 6∈ n2 for all positive integer n, then a1a2 ∈ (0 : m1). proof. this follows from theorem 2.15. � corollary 2.8. let mi be an r-module and let ni be a proper submodule of mi, for i = 1, 2, . . . , k. then the following conditions are equivalent: (1) m1 × m2 × . . . × mi−1 × ni × mi+1 × mk is a weakly 2-absorbing semi-primary submodule of m1 ×m2 × . . .×mk. (2) (a) ni is a weakly 2-absorbing semi-primary submodule of mi. (b) for each a1, a2 ∈ r and m ∈ m2 such that a1a2m = 0, if a1a2 6∈ √ (n2 : m2), a1m 6∈ n2 and an2 m 6∈ n2 for all positive integer n, then there exists j ∈{1, 2, . . . , k} such that a1a2 ∈ (0 : mj). proof. this follows from theorem 2.15. � theorem 2.16. let ni be a proper submodule of an ri-module mi, for i = 1, 2. then the following conditions are equivalent: (1) n1 is a 2-absorbing semi-primary submodule of m1. (2) n1 ×m2 is a 2-absorbing semi-primary submodule of m1 ×m2. (3) n1 ×m2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2, where m2 6= {0}. proof. (1 ⇒ 2). this is clear, by theorem 2.15. int. j. anal. appl. 16 (3) (2018) 411 (2 ⇒ 3). the proof is clear. (3 ⇒ 1). suppose that n1 × m2 is a weakly 2-absorbing semi-primary submodule of m1 × m2, where m2 6= {0}. let a1, a2 ∈ r1 and m ∈ m1 such that a1a2m ∈ n1. by assumption, there exists m2 ∈ m2 such that m2 6= 0. since (a1, 1)(a2, 1)(m, m2) = (a1a2m, m2) 6= (0, 0) , we have (0, 0) 6= (a1, 1)(a2, 1)(m, m2) ∈ n1 × m2. by definition 2.1, (a1, 1)(a2, 1) ∈ √ (n1 ×m2 : m1 ×m2) or (a1, 1)(m, m2) ∈ n1 × m2 or (a2, 1) n(m, m2) ∈ n1 × m2 for some positive integer n. therefore a1a2 ∈ √ (n1 : m1) or a1m ∈ n1 or an2 m ∈ n1 for some positive integer n and hence n1 is a 2-absorbing semi-primary submodule of m1. � corollary 2.9. let ni be a proper submodule of an ri-module mi, for i = 1, 2. then the following conditions are equivalent: (1) n2 is a 2-absorbing semi-primary submodule of m1. (2) m1 ×n2 is a 2-absorbing semi-primary submodule of m1 ×m2. (3) m1 ×n2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2, where m1 6= {0}. proof. this follows from theorem 2.16. � corollary 2.10. let ni be a proper submodule of an ri-module mi, for i = 1, 2, . . . , k. then the following conditions are equivalent: (1) ni is a 2-absorbing semi-primary submodule of m1. (2) m1×m2×. . .×mi−1×ni×mi+1×mk is a 2-absorbing semi-primary submodule of m1×m2×. . .×mk. (3) m1 × m2 × . . . × mi−1 × ni × mi+1 × mk is a weakly 2-absorbing semi-primary submodule of m1 ×m2 × . . .×mk, where mj 6= {0}. proof. this follows from theorem 2.16 and corollary 2.9. � theorem 2.17. let ni be a proper submodule of an ri-module mi, for i = 1, 2. if n1 × n2 is a weakly 2-absorbing semi-primary submodule of m1 ×m2, then (1) n1 is a weakly 2-absorbing semi-primary submodule of m1. (2) n2 is a weakly 2-absorbing semi-primary submodule of m2. proof. (1). suppose that n1×n2 is a weakly 2-absorbing semi-primary submodule of m1×m2. let a1, a2 ∈ r1 and m ∈ m1 such that 0 6= a1a2m ∈ n1. clearly, (0, 0) 6= (a1, 1)(a2, 1)(m, m2) = (a1a2m, m2) ∈ n1×n2. by definition 2.1, (a1a2, 1) = (a1, 1)(a2, 1) ∈ √ (n1 ×n2 : m1 ×m2) or (a1m, m2) = (a1, 1)(m, m2) ∈ n1×n2 or (an2 m, m2) = (a2, 1)n(m, m2) ∈ n1×n2 for some positive integer n. therefore a1a2 ∈ √ (n1 : m1) or a1m ∈ n1 or an2 m ∈ n1 for some positive integer n. hence n1 is a weakly 2-absorbing semi-primary submodule of m1. (2). this follows from part 1. � int. j. anal. appl. 16 (3) (2018) 412 example 2.3. let m = z×z be an z-module. consider the submodule n = 5z×12z of m. it is easy to see that 5z and 12z are weakly 2-absorbing semi-primary submodule of m. notice that (0, 0) 6= 2 · 3(5, 2) ∈ n, but 2 · 3 6∈ √ (m : n), 2(5, 2) 6∈ n, and (2 · 3)n 6∈ (n : m) for all positive integer n. therefore n is not a weakly 2-absorbing semi-primary submodule of m. this example shows that the converse of theorem 2.17 is not true. theorem 2.18. let ni be a submodule of an ri-module mi, for i = 1, 2, 3. if n is a weakly 2-absorbing semi-primary submodule of m1×m2×m3, then n = {(0, 0, 0)} or n is a 2-absorbing semi-primary submodule of m1 ×m2 ×m3. proof. suppose that n is a weakly 2-absorbing semi-primary submodule of m1 × m2 × m3 that is not 2-absorbing semi-primary. we will show that n = {(0, 0, 0)}. now suppose that n1 × n2 × n3 = n 6= {0}×{0}×{0}. thus ni 6= {0}, for some i = 1, 2, 3. we claim that n1 6= {0}. there exists m1 ∈ n1 such that m1 6= 0. to show that n2 = m2 or n3 = m3. assume that n2 6= m2 and n3 6= m3. thus there exist m2 ∈ m2 and m3 ∈ m3 such that m2 6∈ n2 and m3 6∈ n3. since (1, 0, 1)(1, 1, 0)(m1, m2, m3) = (m1, 0, 0) 6= (0, 0, 0), we have (0, 0, 0) 6= (1, 0, 1)(1, 1, 0)(m1, m2, m3) ∈ n1 ×n2 ×n3. by definition 2.1, we get (1, 0, 1)(1, 1, 0) ∈√ (n1 ×n2 ×n3 : m1 ×m2 ×m3) or (1, 0, 1)(m1, m2, m3) ∈ n or (1, 1, 0)n(m1, m2, m3) ∈ n, for some positive integer n. so m2 ∈ n2 or m3 ∈ n3, a contradiction. therefore n = n1 × m2 × n3 or n = n1×n2×m3. if n = n1×m2×n3, then (0, 1, 0) ∈ (n : m1×m2×m3). by theorem 2.13, {0}×m2×{0} = (0, 1, 0)2n ⊆ (n : n1 ×m2 ×n3)2n = {(0, 0, 0)}, which is a contradiction. hence n = {(0, 0, 0)}. � theorem 2.19. let ni be a submodule of an ri-module mi, for i = 1, 2, 3. if n 6= {(0, 0, 0)} and n is a 2-absorbing semi-primary submodule of m1 × m2 × m3, then n is a weakly 2-absorbing semi-primary submodule of m1 ×m2 ×m3. proof. similar to the proof of theorem 2.18 � the above theorem shows the relationship between 2-absorbing semi-primary and weakly 2-absorbing semi-primary submodules in r1×r2×r3-modules. from the above theorem, we have the following corollary. corollary 2.11. let ni be a submodule of an ri-module mi, for i = 1, 2, 3 with n 6= {(0, 0, 0)}. then n is a weakly 2-absorbing semi-primary submodule of m1 ×m2 ×m3 if and only if n is a weakly 2-absorbing semi-primary submodule of m1 ×m2 ×m3. proof. this follows from theorem 2.18. � corollary 2.12. let ni be a submodule of an ri-module mi, for i = 1, 2, . . . , k ≥ 3 with n 6= {(0, 0, . . . , 0)}. then n is a weakly 2-absorbing semi-primary submodule of m1×m2× . . .×mk if and only if n is a weakly 2-absorbing semi-primary submodule of m1 ×m2 × . . .×mk. int. j. anal. appl. 16 (3) (2018) 413 proof. this follows from theorem 2.19. � references [1] d. anderson and e. smith, weakly prime ideals. houston j. math., 29 (2003), 831 – 840. [2] s.e. atani and f. farzalipour, on weakly primary ideals. georgian math. j., 12 (3) (2005), 423 – 429. [3] a. badawi and a.y. darani, on weakly 2-absorbing ideals of commutative rings. houston j. math., 39 (2013), 441 – 452. [4] a. badawi, u. tekir and e. yetkin, on weakly 2-absorbing primary ideals of commutative rings. j. korean math. soc., 52(1)(2015), 97 – 111. [5] m. behboodi and h. koohy, weakly prime modules. vietnam j. math., 32 2 (2004), 185 – 195. [6] a.y. darani and f. soheilnia, on 2-absorbing and weakly 2absorbing submodules. thai j. math., 9, (2011), 577 – 584. [7] z.a. el-bast and p.f. smith, multiplication modules. comm. algebra, 6(4)(1988), 755 – 779. [8] m. larsen and p. mccarthy, multiplicative theory of ideals. academic press, new york, london (1971). [9] h. mostafanasab, on weakly classical primary submodules. bull. belg. math. soc., 22(5)(2015), 743 – 760. [10] h. mostafanasab, u. tekir and k.h. oral, weakly classical prime submodules. math. j., 56 (2016), 1085 – 1101. [11] h. mostafanasab, e. yetkin, u. tekir and a.y. darani, on 2-absorbing primary submodules of modules over commutative rings. an. st. univ. ovidius constanta, 24(1)(2016), 335 – 351. [12] sh. payrovi and s. babaei, on the 2-absorbing submodules. iran. j. math. sci. inform., 10(1) (2015), 131 –137. [13] r. sharp, steps in commutative algebra. cambridge university press, cambridge-new york-sydney (2000). [14] n. zamani, φ-prime submodules. glasgow math. j., 52(2) (2010), 253 – 259. 1. introduction 2. properties of weakly 2-absorbing semiprimary submodules references int. j. anal. appl. (2023), 21:4 solution for a system of first-order linear fuzzy boundary value problems s. nagalakshmi, g. suresh kumar∗, b. madhavi department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, vaddeswaram, guntur, 522302, andhra pradesh, india ∗corresponding author: drgsk006@kluniversity.in abstract. in this paper, we consider homogeneous and non-homogeneous system of first order linear fuzzy boundary value problems (sfolbvps) under granular differentiability. using the concept of horizontal membership function, we introduced the notion of first order granular differentiability for n-dimensional fuzzy functions. we present granular integral and its properties. theorems on the existence and uniqueness of solutions for homogeneous and non-homogeneous sfolfbvps are proved. we develop an algorithm for solution of non-homogeneous sfolbvps under granular differentiability. we provide some examples to illustrate the validity of the proposed algorithm. 1. introduction mathematical models to deal with uncertainty are frequently used in fuzzy differential equations. the rich work on fuzzy differential equations (fde) is applied to population, bioinformatics, growth and decay, economics, quantum optics, and friction models. first-order linear fuzzy systems (folfs) are modeled by behaviors of many dynamical systems (ds) with uncertainty. buckley et al. [4] presented two types of solutions using the extension principle and standard interval arithmetic (sia) to the first-order system of equations with fuzzy initial conditions. gasilov et al. [6] proposed a geometric approach to solve the fuzzy system of differential equations (fsdes) with crisp real coefficients and fuzzy initial conditions. fard et al. [5] introduced an iterative technique to solve fsdes with fuzzy constant coefficients using the h-differentiability concept. hashemi et al. [7] developed the series solution to sfdes under h-differentiability. mondal et al [10] analyzed adaptive schemes to study received: dec. 15, 2022. 2020 mathematics subject classification. 34a07, 34b05. key words and phrases. n-dimensional horizontal membership function; n-dimensional granular metric; n-dimensional granular derivative; system of first order linear fuzzy boundary value problems. https://doi.org/10.28924/2291-8639-21-2023-4 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-4 2 int. j. anal. appl. (2023), 21:4 the fsdes in two types, fuzzy and in the crip sense. bara et al. [1] analyzed numerical solutions for fsdes using the variational iteration technique. keshavarz et al [8] proposed to get an analytical solution for fsdes under gh-differentiability. boukezzoula et al. [3] proposed a new technique to solve the fsdes with variables as fuzzy intervals. suhhiem and khwayyit [16] proposed to get a semi-analytical solution for autonomous fdes using the adomian decomposition method. but, these derivatives possess some drawbacks such as derivatives may not always exist, doubling property, a multiplicity of solutions, unnatural behavior in modeling (ubm) phenomenon, and monotonicity of the uncertainty. piegat et al. [14] presented a horizontal membership function (hmf) for fuzzy function (ff) and solved distinct granular problems. recently piegat et al. [15], provide a detailed comparison of hmfs and inverse hmfs, highlighting the key distinctions between the two kinds of functions. mazandarani et al. [9] established granular differentiability (gr-differentiability), a novel idea of ff differentiability based on rdm-ia and horizontal membership functions (hmf). najariyan et al. [12] investigated the solution of singular fdes with the concept of gr-differentiability. under the concept of gr-differentiability, najariyan et al. [11] effectively tune the fuzzy granular pid controller using a particle swarm optimization algorithm. these findings were made by studying fdes under the gr-differentiability to overcome all the shortcomings as discussed. in this present work, we consider sfolbvps under gr-differentiability. section 2, presents basic definitions and results related to hmfs, gr-metric, gr-differentiability, and gr-integration of n-dimensional ff. the existence and uniqueness of theorems for sfolfivps under gr-differentiability are established in section 3. section 4 presents a working method to solve sfofbvps under gr-differentiability and highlighted proposed results with suitable examples. concluding remarks and future works are discussed in section 5. 2. preliminaries this section presents some useful definitions, notations, and results that are useful to establish the main results. definition 2.1. a non-empty fuzzy subset p of r, with membership function, p : r → [0,1], is said to be a fuzzy number, if it is semi continuous, fuzzy convex, normal and compactly supported on r. here p(y) is the membership degree of y, for every y ∈ r. let rf denotes the space of fuzzy numbers(fns) in r. the β-level sets of p are defined by [p]β = {y ∈ r : p(y) ≥ β} = [pβ l ,p β r ], for 0 < β ≤ 1 and [p]0 = cl{y ∈r : p(y) > 0}. for notations, definitions and basic results related to hmf, gr-derivative and gr-integrations of fuzzy numbers refer to [9]. int. j. anal. appl. (2023), 21:4 3 definition 2.2. let rnf = rf ×rf ×rf ×···×rf︸ ︷︷ ︸ ntimes , be the space of n-tuple fuzzy numbers. then the addition and scalar multiplication defined component wise as follows: if u =(u1,u2, · · · ,un),v =(v1,v2, · · · ,vn)∈rnf , then (i) u +v =(u1 +v1,u2 +v2, · · · ,un +vn), (ii) ku =(ku1,ku2, · · · ,kun), where ui,vi ∈rf , i =1,2, · · ·n and k ∈r. definition 2.3. if g : [b,c] → rnf , is a ff, then it is called an n-dimensional vector of fn valued function on [b,c]. definition 2.4. if g : [b,c] → rnf is a n-dimensional ff, include mn ∈ n, distinct fns such that ui = (ui1,ui2, . . . ,uin), i = 1,2, · · ·m, then the hmf of g is indicated by h(g(x)) ≡ ggr(x,β,αg), and interpreted as ggr : [b,c] × [0,1] × [0,1]× [0,1]×···× [0,1]︸ ︷︷ ︸ mntimes → rn, in which αg ≡ (αi1,αi2, . . . ,αin), i = 1,2, · · ·m, where αi1,αi2, . . . ,αin, are the rdm variables related to ui1, ui2, . . ., uin. definition 2.5. let p and q be two n-dimensional fns. then h(p)= h(q), for all αp = αq ∈ [0,1] if and only if p and q are said to be equal. definition 2.6. let p,q ∈rnf . the function d n gr :r n f ×r n f →r + ∪{0}, defined by dngr(p,q)= sup β max αp,αq ‖pgr(β,αp)−qgr(β,αq)‖, which is called a n-dimensional granular distance between two n-dimentional fns p and q, where ‖.‖ represents euclidean norm in rn. definition 2.7. if g,h : [b,c]→rnf are n-dimensional ffs, then the granular distance is dgr(g(y),h(y))= sup β max αg,αh ‖ggr(y,β,αg)−hgr(y,β,αh)‖, where y ∈ [b,c]⊂r and β,αg,αh ∈ [0,1]. now, we define first order gr-differentiability for n-dimensional ff. definition 2.8. let g : [b,c]→rnf , be the n-dimensional ff. if there exists dgrg(y0) dy ∈rnf , such that lim h→0 g(y0 +h)−g(y0) h = dgrg(y0) dy =g′gr(y0), this limit is taken in the metric space (rnf ,d n gr). then g is said to be first order grdifferentiable at a point y0 ∈ [b,c]. 4 int. j. anal. appl. (2023), 21:4 theorem 2.1. let g : [b,c]→rf be a n-dimensional ff, then g is gr-differentiable if and only if its hmf is differentiable with respect to y ∈ [b,c]. moreover, h ( dgrg(y) dy ) = ∂ggr(y,β,αf ) ∂y . proof. suppose g is gr-differentiable and y ∈ (b,c). based on the definition 2.8, for all �1 > 0, there exits δ1 > 0 such that |h| < δ1 =⇒ dngr( g(y+h)−g(y) h , dgrg(y) dy ) < �1 =⇒ sup β max αg ‖ ggr(y +h,β,αg)−ggr(y,β,αg) h − dgrggr(y,β,αg) dy ‖ < �1 =⇒ ‖ ggr(y +h,β,αg)−ggr(y,β,αg) h − dgrggr(y,β,αg) dy ‖ < �1 =⇒ lim h→0 ggr(y +h,β,αg)−ggr(y,β,αg) h = dgrggr(y,β,αg) dy =⇒ ∂ggr(y,β,αg) ∂y = h ( dgrg(y) dy ) . � definition 2.9. suppose that g : [b,c] → rnf , is continuous and the hmf h(g(y)) = ggr(y,β,αg) is integrable on [b,c]. if there exists a m such that h(m) = ∫ c b h(g(y))dy, then m is called the gr-integral of g on [b,c] and m= ∫ c b g(y)dy. proposition 2.1. assume that f : [b,c] → rnf is gr-differentiable and g(y) = dgrf(y) dy is continuous on [b,c]. then, ∫ c b g(y)dy = f(c)−f(b). theorem 2.2. assume that g,h : [b,c] → rnf are gr-integrable n-dimensional ffs and l,m ∈ r. then the following properties hold: (i) ∫ c b [lg(y)+mh(y)]dy = l ∫ c b g(y)dy +m ∫ c b h(y)dy; (ii) dngr(g,h) is integrable; (iii) dngr (∫ c b g(y)dy, ∫ c b h(y)dy ) ≤ ∫ c b dngr(g(y),h(y))dy; (iv) ∫ c b g(y)dy = ∫ a b g(y)dy + ∫ c a g(y)dy, for each a ∈ (b,c). proof. (i) consider h (∫ c b [lg(y)+mh(y)]dy ) = ∫ c b h (lg(y)+mh(y))dy = h(l) ∫ c b h (g(y))dy +h(m) ∫ c b h (h(y))dy = h(l)h (∫ c b g(y)dy ) +h(m)h (∫ c b h(y)dy ) = h ( l ∫ c b g(y)dy ) +h ( m ∫ c b h(y)dy ) = h ( l ∫ c b g(y)dy +m ∫ c b h(y)dy ) . int. j. anal. appl. (2023), 21:4 5 by the definition 2.5 ∫ c b [lg(y)+mh(y)]dy = l ∫ c b g(y)dy +m ∫ c b h(y)dy. (ii) consider dngr([g(y)] β, [h(y)]β)= max αg,αh ‖ggr(y,β,αg)−hgr(y,β,αh)‖, β ∈ [0,1]. since f,g are integrable n-dimensional ffs on [b,c], so that ggr(y,β,αg),hgr(y,β,αh) are also integrable on [b,c] or all β,αg,αh ∈ [0,1]. let β ∈ [0,1] be fixed. then, dngr([g(y)]β, [h(y)]β) is measurable on [b,c]. by the definition of gr-distance, we have dngr(f (y),g(y))= sup β dngr([g(y)]β, [h(y)]β), β ∈ [0,1]. further more, we have dngr(g(y),h(y))≤d n gr(g(y),0)+d n gr(0,h(y)) ≤ g1(y)+h1(y), where g1, h1 are integrable bounded functions for g, h respectively. thus dngr(g(y),h(y)) is integrable. (iii) consider, dngr (∫ c b g(y)dy, ∫ c b h(y)dy ) =sup β max αg,αh ‖ ∫ c b ggr(y,β,αg)dy − ∫ c b hgr(y,β,αh)dy‖, for all β,αg,αh ∈ [0,1]. since the fact that ‖ ∫ c b ggr(y,β,αg)dy − ∫ c b hgr(y,β,αh)dy‖ = ‖ ∫ c b [ggr(y,β,αg)−hgr(y,β,αh)]dy‖ ≤ ∫ c b ‖ [ggr(y,β,αg)−hgr(y,β,αh)]‖dy. it follows that max αg,αh ‖ ∫ c b [ggr(y,β,αg)−hgr(y,β,αh)]dy‖≤ max αg,αh ∫ c b ‖ [ggr(y,β,αg)−hgr(y,β,αh)]‖dy. thus sup β max αg,αh ‖ ∫ c b [ggr(y,β,αg)−hgr(y,β,αh)]dy‖≤ sup β max αg,αh ∫ c b ‖ [ggr(y,β,αg)−hgr(y,β,αh)]‖dy, we get inequality (iii). the proof of (iv) is deduced directly by the definition 2.6 � definition 2.10. [13] let a be a square matrix of order n with real numbers. the exponential of a is represented by the notation exp(ay) and defined as exp(ay)= i +ay + a 2y2 2! + a3y3 3! + · · · , for all y ∈r and the following results hold: 6 int. j. anal. appl. (2023), 21:4 (i) exp(ay)|y=0 = i. (ii) dgr dy (exp(ay))= a(exp(ay))= (exp(ay))a. (iii) (exp(ay))−1 =(exp(−ay)). 3. main results 3.1. the fundamental theorem for sfolfbvps. theorem 3.1. let a be a square matrix of order n with real numbers. then for a given z(x0),z(x1)∈ rnf , the fbvp z ′ gr(x)= az(x), with mz(x0)+nz(x1)=0, has a unique trivial solution if [me ax0+ neax1] is non-singular. proof. consider the fbvp z′gr(x)= az(x), (3.1) with, mz(x0)+nz(x1)=0. (3.2) e−axz′gr(x)−e −axaz(x)=0 =⇒ dgr dx (e−axz(x))=0 =⇒ e−axz(x)= k, k ∈rnf =⇒ z(x)= eaxk. now mz(x0)+nz(x1)=0 =⇒ meax0k +neax1k =0 =⇒ [meax0 +neax1]k =0. thus (3.1) and (3.2), has a unique trivial solution if [meax0k +neax1] is non-singular. � 3.2. non-homogeneous sfolfbvps. consider the following sfolfbvp z′gr(x)= az(x)+f(x), (3.3) with, mz(x0)+nz(x1)= r. (3.4) theorem 3.2. the non-homogeneous sfolfbvps (3.3) and (3.4), has a unique solution if the corresponding homogeneous system (3.1) and (3.2) has only the trivial solution. if this condition holds then the solution of system (3.3) and (3.4) given by z(x) = z0(x)+ x1∫ x0 g(x,s)f(s)ds, where z0(x) is a solution of the homogeneous system (3.1) and (3.4) and g is the green’s matrix of non homogeneous system (3.3) and (3.4). int. j. anal. appl. (2023), 21:4 7 proof. let eax be the fundamental matrix of the system (3.1) and (3.2). then the general solution of non homogeneous system (3.3) is, z(x)= eaxk +eax x∫ x0 e−asf(s)ds. (3.5) using boundary condition (3.4), we have meax0k +n[eax1k +eax1 x1∫ x0 e−asf(s)ds] = r =⇒ [meax0 +neax1]k = r−neax1 x1∫ x0 e−asf(s)ds =⇒ k = d−1r−d−1neax1 x1∫ x0 e−asf(s)ds, where d = [meax0 +neax1]. z(x)= eax[d−1r−d−1neax1 x1∫ x0 e−asf(s)ds]+eax x∫ x0 e−asf(s)ds = eaxd−1r+ x1∫ x0 g(x,s)f(s)ds, where g(x,s)=  e axd−1meax0e−as if x0 ≤ s ≤ x ≤ x1 −eaxd−1neax1e−as if x0 ≤ x ≤ s ≤ x1, which is the greens matrix of sfolfbvp (3.1) and (3.2). � 4. an algorithm for solving system of first-order linear fuzzy boundary value problems under gr-differentiability consider a sfolfbvp, z′gr(x)= az(x)+f(x), withmz(x0)+nz(x1)= r. (4.1) the matrix form of (4.1) is, [ y ′gr(x) z′gr(x) ] = [ a b c d ][ y(x) z(x) ] + [ f(x) g(x) ] , (4.2) subject to, [ e 0 0 f ][ y(x0) z(x0) ] + [ g 0 0 h ][ y(x1) z(x1) ] = [ r1 r2 ] . (4.3) 8 int. j. anal. appl. (2023), 21:4 the following algorithm describes the procedure to compute β-cut solution of sfolfbvp (4.1) if it exists. step 1 : applying hmf on both sides of (4.2) and (4.3), we get[ ∂ygr(x,β,αy) ∂x ∂zgr(x,β,αz) ∂x ] = [ a b c d ][ ygr(x,β,αy) zgr(x,β,αz) ] + [ fgr(x,β,αf ) ggr(x,β,αg) ] , (4.4) [ e 0 0 f ][ ygr(x0,β,αy0) zgr(x0,β,αz0) ] + [ g 0 0 h ][ ygr(x1,β,αy1) zgr(x1,β,αz1) ] = [ r1(β,αr1) r2(β,αr2) ] , (4.5) where β,αy, αz,αf , αg, αr1, αr2,αy0, αz0, αy1, αz1 ∈ [0,1]. here, (4.4) is a system of partial differential equations with single independent variable x. therefore, (4.4) and (4.5) taken as a ordinary second order system of differential equations. step 2 : solving (4.4) and (4.5), we get h(y(x))= ygr(x,β,αy),and (4.6) h(z(x))= zgr(x,β,αz). (4.7) step 3 : applying inverse hmf on both sides of (4.6) and (4.7), we get [y(x)]β = [ inf β≤α≤1 min αy ygr(x,α,αy), sup β≤α≤1 max αy zgr(x,α,αy)], (4.8) [z(x)]β = [ inf β≤α≤1 min αz zgr(x,α,αz), sup β≤α≤1 max αz zgr(x,α,αz)], (4.9) which is the required β-cut solution of sfolfbvp (4.1). example 4.1. consider a homogeneous sfolfbvp with fuzzy boundary conditions, y ′gr(x)=3y(x)+2z(x), z′gr(x)= y(x)+4z(x), with fuzzy boundary values, y(0)+2y(1)= r1, z(0)+3z(1)= r2. suppose that the β-level sets of fuzzy boundary values are [r1]β = [1+β,3−β], [r2]β = [2+β,4−β]. then the matrix equation is, [ y ′gr(x) z′gr(x) ] = [ 3 2 1 4 ][ y(x) z(x) ] , (4.10) subject to, [ 1 0 0 1 ][ y(x0) z(x0) ] + [ 2 0 0 3 ][ y(x1) z(x1) ] = [ r1 r2 ] . (4.11) int. j. anal. appl. (2023), 21:4 9 taking hmf on both sides of (4.10) and (4.11), we have[ ∂ygr(x,β,αy) ∂x ∂zgr(x,β,αz) ∂x ] = [ 3 2 1 4 ][ ygr(x,β,αy) zgr(x,β,αz) ] , (4.12) subject to, [ 1 0 0 1 ][ ygr(x0) zgr(x0) ] + [ 2 0 0 3 ][ ygr(x1) zgr(x1) ] = [ r1(β,α1) r2(β,α2) ] , (4.13) where the granule of boundary values are r1gr(β,α1)= [1+β +2(1−β)α1], r2gr(β,α2)= [2+β + 2(1−β)α2], where β, α1, α2 ∈ [0,1]. the solution for system of equations (4.12) and (4.13) is ygr(x,β,α1,α2) and zgr(x,β,α1,α2). (4.14) applying inverse hmf on (4.14), we get [y(x)]β = [ inf β≤α≤1 min α1,α2 ygr(x,α,α1,α2), sup β≤α≤1 max α1,α2 ygr(x,α,α1,α2)], [z(x)]β = [ inf β≤α≤1 min α1,α2 zgr(x,α,α1,α2), sup β≤α≤1 max α1,α2 zgr(x,α,α1,α2)]. the β-cut solution is computed using matlab and is depicted in fig. 1. (a) the black curve represents y(x) at β = 1. (b) the black curve represents z(x) at β = 1. figure 1. the span of the information granule (β-level sets) of y(x) and z(x). example 4.2. consider a non-homogeneous sfolfbvp with fuzzy force functions, y ′gr(x)=3y(x)+2z(x)+ f (x), z′gr(x)= y(x)+4z(x)+g(x), with boundary values, y(0)−2y(1)= r1 =2, z(0)−3z(1)= r2 =3. 10 int. j. anal. appl. (2023), 21:4 suppose that the β-level sets of fuzzy boundary values are [f ]β = [1+β,3−β], [g]β = [2+β,4−β]. then the matrix equation is, [ y ′gr(x) z′gr(x) ] = [ 3 2 1 4 ][ y(x) z(x) ] + [ f (x) g(x) ] , (4.15) subject to, [ 1 0 0 1 ][ y(x0) z(x0) ] − [ 2 0 0 3 ][ y(x1) z(x1) ] = [ r1 r2 ] . (4.16) taking hmf on both sides of (4.15) and (4.16), we have[ ∂ygr(x,β,αy) ∂x ∂zgr(x,β,αz) ∂x ] = [ 3 2 1 4 ][ ygr(x,β,αy) zgr(x,β,αz) ] + [ fgr(x,β,αf ) ggr(x,β,αg) ] (4.17) subject to [ 1 0 0 1 ][ ygr(x0) zgr(x0) ] − [ 2 0 0 3 ][ ygr(x1) zgr(x1) ] = [ 2 3 ] , (4.18) where the granule of fuzzy force functions are fgr(β,α1) = [1 + β + 2(1 − β)α1], ggr(β,α2) = [2+β +2(1−β)α2], where β, α1, α2 ∈ [0,1]. the solution for system of equations (4.17) and (4.18) is ygr(x,β,α1,α2) and zgr(x,β,α1,α2). (4.19) applying inverse hmf on (4.19), we get [y(x)]β = [ inf β≤α≤1 min α1,α2 ygr(x,α,α1,α2), sup β≤α≤1 max α1,α2 ygr(x,α,α1,α2)], [z(x)]β = [ inf β≤α≤1 min α1,α2 zgr(x,α,α1,α2), sup β≤α≤1 max α1,α2 zgr(x,α,α1,α2)]. the β-cut solution is computed using matlab and is depicted in fig. 2. (a) the black curve represents y(x) at β = 1. (b) the black curve represents z(x) at β = 1. figure 2. the span of the information granule (β-level sets) of y(x) and z(x). int. j. anal. appl. (2023), 21:4 11 example 4.3. consider a non-homogeneous sfolfbvp with fuzzy boundary conditions and fuzzy force functions are, y ′gr(x)=3y(x)+2z(x)+ f (x), z′gr(x)= y(x)+4z(x)+g(x), with fuzzy boundary values, y(0)+2y(1)= r1, z(0)+3z(1)= r2. suppose that the β-level sets of fuzzy force functions are [f ]β = [1+β,3−β], [g]β = [2+β,4−β], [r1] β = [β,2−β], [r2]β = [1+β,3−β]. then the matrix equation is,[ y ′gr(x) z′gr(x) ] = [ 3 2 1 4 ][ y(x) z(x) ] + [ f (x) g(x) ] , (4.20) subject to, [ 1 0 0 1 ][ y(x0) z(x0) ] + [ 2 0 0 3 ][ y(x1) z(x1) ] = [ r1 r2 ] . (4.21) taking hmf on both sides of (4.20) and (4.21), we have [ ∂ygr(x,β,αy) ∂x ∂zgr(x,β,αz) ∂x ] = [ 3 2 1 4 ][ ygr(x,β,αy) zgr(x,β,αz) ] + [ fgr(x,β,α1) ggr(x,β,α2) ] , (4.22) subject to, [ 1 0 0 1 ][ ygr(x0) zgr(x0) ] + [ 2 0 0 3 ][ ygr(x1) zgr(x1) ] = [ r1(β,α3) r2(β,α1) ] , (4.23) where the granule of fuzzy boundary values and force functions are fgr(β,α1)= [1+β+2(1−β)α1], ggr(β,α2)= [2+β+2(1−β)α2] r1gr(β,α3)= [β+2(1−β)α3], r2gr(β,α1)= [1+β+2(1−β)α1], where β, α1, α2, α3 ∈ [0,1]. the solution for system of equations (4.17) and (4.18) is ygr(x,β,α1,α2,α3) and zgr(x,β,α1,α2,α3). (4.24) applying inverse hmf on (4.24), we get [y(x)]β = [ inf β≤α≤1 min α1,α2,α3 ygr(x,α,α1,α2,α3), sup β≤α≤1 max α1,α2,α3 ygr(x,α,α1,α2,α3)], [z(x)]β = [ inf β≤α≤1 min α1,α2,α3 zgr(x,α,α1,α2,α3), sup β≤α≤1 max α1,α2,α3 zgr(x,α,α1,α2,α3)]. the β-cut solution is computed using matlab and is depicted in fig. 3. 12 int. j. anal. appl. (2023), 21:4 (a) the black curve represents y(x) at β = 1. (b) the black curve represents z(x) at β = 1. figure 3. the span of the information granule (β-level sets) of y(x) and z(x). 5. conclusions the results proposed in this paper are useful for examining and determining solutions for sfolfbvps. the granular differentiability and integrability are extended to an n-dimensional fuzzy function. the sfolfbvps with fuzzy boundary conditions are researched under granular differentiability. we have established the existence and uniqueness of solutions for homogeneous and non-homogeneous sfolfbvps. the proposed algorithm is useful to determine the solution of the first-order fsdes with fuzzy boundary conditions. we provide various examples to demonstrate the effectiveness and applicability of our method. in the future, this work will be extended for higher-order fsdes with fuzzy boundary conditions and investigating applications in real-life. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. barazandeh, b. ghazanfari, approximate solution for systems of fuzzy differential equations by variational iteration method, punjab univ. j. math. 51 (2019), 13-33. [2] b. bede, mathematics of fuzzy sets and fuzzy logic, springer, berlin, heidelberg, 2013. https://doi.org/10. 1007/978-3-642-35221-8. [3] r. boukezzoula, l. jaulin, d. coquin, a new methodology for solving fuzzy systems of equations: thick fuzzy sets based approach, fuzzy sets syst. 435 (2022), 107–128. https://doi.org/10.1016/j.fss.2021.06.003. [4] j.j. buckley, t. feuring, y. hayashi, linear systems of first order ordinary differential equations: fuzzy initial conditions, soft comput. 6 (2002), 415–421. https://doi.org/10.1007/s005000100155. [5] o.s. fard, n. ghal-eh, numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients, inform. sci. 181 (2011), 4765–4779. https://doi.org/10.1016/j.ins.2011.06. 007. [6] n. gasilov, s.e. amrahov, a.g. fatullayev, a geometric approach to solve fuzzy linear systems of differential equations, appl. math. inform. sci. 5 (2011), 484-499. https://doi.org/10.48550/arxiv.0910.4307. https://doi.org/10.1007/978-3-642-35221-8 https://doi.org/10.1007/978-3-642-35221-8 https://doi.org/10.1016/j.fss.2021.06.003 https://doi.org/10.1007/s005000100155 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.48550/arxiv.0910.4307 int. j. anal. appl. (2023), 21:4 13 [7] m.s. hashemi, j. malekinagad, h.r. marasi, series solution of the system of fuzzy differential equations, adv. fuzzy syst. 2012 (2012), 407647. https://doi.org/10.1155/2012/407647. [8] m. keshavarz, t. allahviranloo, s. abbasbandy, m.h. modarressi, a study of fuzzy methods for solving system of fuzzy differential equations, new math. nat. comput. 17 (2021), 1–27. https://doi.org/10.1142/ s1793005721500010. [9] m. mazandarani, n. pariz, a.v. kamyad, granular differentiability of fuzzy-number-valued functions, ieee trans. fuzzy syst. 26 (2018), 310–323. https://doi.org/10.1109/tfuzz.2017.2659731. [10] s. prasad mondal, n. alam khan, o. abdul razzaq, t. kumar roy, adaptive strategies for system of fuzzy differential equation: application of arms race model, j. math. computer sci. 18 (2018), 192–205. https: //doi.org/10.22436/jmcs.018.02.07. [11] m. najariyan, y. zhao, granular fuzzy pid controller, expert syst. appl. 167 (2021), 114182. https://doi.org/ 10.1016/j.eswa.2020.114182. [12] m. najariyan, n. pariz, h. vu, fuzzy linear singular differential equations under granular differentiability concept, fuzzy sets syst. 429 (2022), 169–187. https://doi.org/10.1016/j.fss.2021.01.003. [13] l. perko, differential equations and dynamical systems, springer, new york, 2013. [14] a. piegat, m. landowski, solving different practical granular problems under the same system of equations, granul. comput. 3 (2017), 39–48. https://doi.org/10.1007/s41066-017-0054-5. [15] a. piegat, m. pluciński, the differences between the horizontal membership function used in multidimensional fuzzy arithmetic and the inverse membership function used in gradual arithmetic, granul. comput. 7 (2021), 751–760. https://doi.org/10.1007/s41066-021-00293-z. [16] m.h. suhhiem, r.i. khwayyit, semi analytical solution for fuzzy autonomous differential equations, int. j. anal. appl. 20 (2022), 61. https://doi.org/10.28924/2291-8639-20-2022-61. https://doi.org/10.1155/2012/407647 https://doi.org/10.1142/s1793005721500010 https://doi.org/10.1142/s1793005721500010 https://doi.org/10.1109/tfuzz.2017.2659731 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.1016/j.eswa.2020.114182 https://doi.org/10.1016/j.eswa.2020.114182 https://doi.org/10.1016/j.fss.2021.01.003 https://doi.org/10.1007/s41066-017-0054-5 https://doi.org/10.1007/s41066-021-00293-z https://doi.org/10.28924/2291-8639-20-2022-61 1. introduction 2. preliminaries 3. main results 3.1. the fundamental theorem for sfolfbvps 3.2. non-homogeneous sfolfbvps 4. an algorithm for solving system of first-order linear fuzzy boundary value problems under gr-differentiability 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 48-57 http://www.etamaths.com convergence theorem for generalized mixed equilibrium problem and common fixed point problem for a family of multivalued mappings j. n. ezeora∗ abstract. in this paper, a new hybrid iterative algorithm is constructed using the shrinking projection method introduced by takahashi. the sequence of the algorithm is proved to converge strongly to a common element of the set of solutions of generalized mixed equilibrium problem and the set of common fixed points of a finite family of multivalued strictly pseudocontractive mappings in real hilbert spaces. furthermore, we apply our main result to convex minimization problem. 1. introduction let h be a real hilbert space with inner product 〈., .〉 and norm || · ||, and let k be a nonempty closed convex subset of h. let b : k → h be a nonlinear mapping, ϕ : k → r ∪{+∞} be a function and f : k×k → r, be a bifunction where r is the set of real numbers. the generalized mixed equilibrium problem is defined as follows: (1.1) find x ∈ k : f(x,y) + ϕ(y) −ϕ(x) + 〈bx,y −x〉≥ 0 ∀ y ∈ k. the set of solutions of (1.1) is denoted by gmep(f,ϕ,b). if b = 0, problem (1.1) reduces to the following mixed equilibrium problem: (1.2) find x ∈ k : f(x,y) + ϕ(y) −ϕ(x) ≥ 0 ∀ y ∈ c. the set of solutions of (1.2) is denoted by mep(f,ϕ). if ϕ = 0, problem (1.1) becomes the following generalized equilibrium problem: (1.3) find x ∈ k : f(x,y) + 〈bx,y −x〉≥ 0 ∀ y ∈ c. the set of solutions of (1.3) is denoted by gep(f,b). if ϕ = 0 and b = 0, problem (1.1) becomes the following equilibrium problem: (1.4) find x ∈ k : f(x,y) ≥ 0 ∀ y ∈ k. the set of solutions of (1.4) is denoted by ep(f). if f(x,y) = 0 for all x,y ∈ k, problem (1.1) becomes the following generalized variational inequality problem: (1.5) find x ∈ k : ϕ(y) −ϕ(x) + 〈bx,y −x〉≥ 0 ∀ y ∈ k. if ϕ = 0 and f(x,y) = 0 for all x,y ∈ k, problem (1.1) becomes the following variational inequality problem: (1.6) find x ∈ k : 〈bx,y −x〉≥ 0 ∀ y ∈ k. if b = 0 and f(x,y) = 0 for all x,y ∈ k, problem (1.1) becomes the following convex minimization problem: (1.7) find x ∈ c : ϕ(y) ≥ ϕ(x) ∀ y ∈ k. 2010 mathematics subject classification. 47h04, 47h06, 47h15, 47h17, 47j25. key words and phrases. equilibrium problem; strict pseudo-contractive mappings; multivalued mappings; hausdorff metric. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 48 generalized mixed equilibrium problem and common fixed point problem 49 let x be a normed space. a subset k of x is called proximinal (see [36]) if for each x ∈ x, there exists an element k ∈ k such that d(x,k) = d(x,k), where d(x,k) = inf{||x−y|| ∀ y ∈ k} is the distance from the point x to the set k. let k be a nonempty closed convex subset of x. we denote by cb(k), the family of nonempty closed, bounded subsets of k, p(k) the family of nonempty proximinal bounded subsets of k. the hausdorff metric (see [29]) on cb(x) is defined by d(a,b) = max { sup x∈a d(x,b), sup y∈b d(y,a) } ∀a,b ∈ cb(x). let t : d(t) ⊂ x → cb(x). an element x ∈ d(t) is called a fixed point of t if x ∈ tx. the set of fixed points of t is denoted by f(t). for multivalued mappings t : k → p(k), the best approximation operator, pt (see [21]) is defined by pt (x) := {y ∈ t(x) : ||x−y|| = d(x,t(x))} ∀x ∈ k. a multi-valued mapping t : d(t) ⊆ x → cb(x) is called l-lipschitzian if there exists l > 0 such that (1.8) d(tx,ty) ≤ l||x−y|| ∀ x,y ∈ d(t). when l ∈ (0, 1) in (1.8), we say that t is a contraction, and t is called nonexpansive if l = 1. a multi-valued map t : d(t) ⊂ h → cb(h) is called k-strictly pseudo-contractive (see [10]) if there exists k ∈ (0, 1) such that for all x,y ∈ d(t), (1.9) (d(tx,ty))2 ≤ ||x−y||2 + k||x−y − (u−v)||2 ∀ u ∈ tx,v ∈ ty. it is well known that the generalized mixed equilibrium problem and indeed equilibrium problem include variational inequality problem, optimization problem, problems of nash equilibria, saddle point problems, fixed point problems and complementarity problems as special cases.( see [5, 19, 20, 28, 33] and the references therein). different iterative algorithms for solving generalized mixed equilibrium problems, mixed equilibrium problems and equilibrium problems have been developed and studied by many authors. see for instance, [6, 7, 12, 14, 17, 26, 34, 37, 39] and the references therein. for several years, the study of fixed point theory for multi-valued nonlinear mappings has attracted the interest of several well known mathematicians (see, for example, brouwer [4], chidume et al. [10], denavari [13], kakutani [22], nash [31, 32] geanakoplos [18], nadler [29], downing and kirk[15]). interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in realworld applications, such as in game theory, market economy, non-smooth differential equations and so on (see e.g., [10, 16]). game theory is perhaps the most successful area of application of fixed point theory for multi-valued mappings. however, it has been remarked that the applications of this theory to equilibrium problems in game theory are mostly static in the sense that while they enhance the understanding of conditions under which equilibrium may be achieved, they do not indicate how to construct a process starting from a non-equilibrium point that will converge to an equilibrium solution. iterative methods for fixed points of multivalued mappings are designed to address this problem. for more detail, see [8, 10, 16]. the classical mann iteration process has been employed successfully to approximate fixed points of nonlinear mappings (single valued or multi valued). however, it is known to yield only weak convergence even in hilbert spaces. to overcome this weakness, takahashi [40], introduced a method known as the shrinking projection method, and obtained strong convergence results of the method. the shrinking projection method has been studied extensively in the literature. see for instance, tada and takahashi [39], aoyama et al. [2], yao et al. [41], kang et. al. [23], kimura et. al. [25], cholamjiak and suantai [42] and the references contained therein. 50 ezeora motivated by the result of takahashi [40], bunyawat and suantai [6] used the shrinking projection method and defined a hybrid method for mixed equilibrium problem and fixed point problem for a family of nonexpansive multivalued mappings in real hilbert spaces. precisely, they proved the following result: theorem s (bunyawat and suantai [6]). let d be a nonempty closed and convex subset of a real hilbert space h. let f be a bifunction from d×d to r satisfying (a1) − (a4), and let ϕ be a proper, lower semi continuous and convex function from d to r∪{+∞} such that d∩domϕ 6= ∅. let ti : d → p(d) be multivalued nonexpansive mappings for all i ∈ n with ω := ⋂∞ i=1 f(ti)∩mep(f,ϕ) 6= ∅ such that all pti are nonexpansive. assume that either (b1) or (b2) holds and {αn,i} ⊂ (0, 1) satisfies the condition lim infn→∞αn,iαn,0 > 0 for all i ∈ n. define {xn} as follows: x1 ∈ d = c1, (1.10)   f(un,y) + ϕ(y) −ϕ(un) + 1rn〈y −un,un −xn〉≥ 0 ∀ y ∈ d, yn = αn,0un + ∑n i=1 αn,ixn,i, xn,i ∈ pti (un), cn+1 = {z ∈ cn : ||yn −z|| ≤ ||xn −z||}, xn+1 = pcn+1x0, n ≥ 0. where pcn is the metric projection of h onto cn. they proved that {xn} converges strongly to pωx0, ω = ∩∞i=1f(ti) ∩mep(f,ϕ). the class of strictly pseudocontractive mappings was introduced in 1967 by browder and petryshyn [3] as a generalization of the class of nonexpansive mappings. this class of operators have been studied by several authors under different assumptions. see for instance, [1, 9, 27, 35, 38, 43] and the references therein. in 2013, chidume et. al. [10] introduced and studied the class of multivalued strictly pseudocontractive mappings as a generalization of the class of multi valued nonexpansive mappings in real hilbert spaces. recently, chidume and ezeora [11] introduced a krasnoselskii-type sequence and proved that the sequence converges strongly to a common fixed point of a finite family of multivalued strictly pseudo contractive mappings in real hilbert spaces under some compactness assumption on the operators. motivated by the results of takahashi [40], bunyawat and suantai [6], chidume and ezeora [11], it is our purpose in this paper to introduce a new hybrid iterative algorithm based on the shrinking projection method and prove that the sequence of the scheme converges strongly to a common element of the set of solution of generalized mixed equilibrium problem and the set of common fixed points of a finite family of multivalued strictly pseudocontractive mappings in real hilbert spaces. our result extends that of bunyawat and suantai [6] from multivalued nonexpansive mappings to the more general class of multivalued strictly pseudocontractive mappings and many other important results. in proving our result, compactness assumption imposed on the operators by chidume and ezeora [11] was dispensed with. 2. preliminaries lemma 2.1. [30] let k be a nonempty closed convex subset of a real hilbert space h and pk : h → k be the metric projection from h onto k. then the following inequality holds: ‖y −pkx‖ 2 + ‖x−pkx‖ 2 ≤‖x−y‖2,∀ x ∈ h,y ∈ k. lemma 2.2. (see [11]) let h be a real hilbert space and {xi}mi=1 ⊂ h. for αi ∈ (0, 1), i = 1, . . . ,m such that ∑m i=1 αi = 1, the following identity holds:∣∣∣∣∣ ∣∣∣∣∣ m∑ i=1 αixi ∣∣∣∣∣ ∣∣∣∣∣ 2 = m∑ i=1 αi ||xi|| 2 − m∑ i,j=1,i6=j αiαj ||xi −xj|| 2 (2.1) lemma 2.3. (see [10]) let x be a reflexive real banach space and let a,b ∈ cb(x). assume that b is weakly closed. then, for every a ∈ a, there exists b ∈ b such that (2.1) ||a− b|| ≤ d(a,b) generalized mixed equilibrium problem and common fixed point problem 51 lemma 2.4. (see [10]) let k be a nonempty subset of a real hilbert space h and let t : k → cb(k) be a multivalued k-strictly pseudocontractive mapping. assume that for every x ∈ k, the set tx is weakly closed. then, t is lipschitzian. that is (2.2) d(tx,ty) ≤ l||x−y|| ∀ x,y ∈ k. lemma 2.5. [24] let d be a nonempty closed and convex subset of a real hilbert space h. given x,y,z ∈ h and also given a ∈ r, the set {v ∈ d : ‖y −v‖2 ≤‖x−v‖2 + 〈z,v〉 + a} is convex and closed. for solving the generalized mixed equilibrium problem, we assume the bifunction f, ϕ and the set k satisfy the following conditions: (a1) f(x,x) = 0 for all x ∈ k; (a2) f is monotone, that is, f(x,y) + f(y,x) ≤ 0 ∀ x,y ∈ k; (a3) for each y ∈ k,x 7→ f(x,y) is weakly upper semicontinuous (a4) for each x ∈ k,y 7→ f(x,y) is convex and lower semicontinuous; (b1) for each x ∈ h and r > 0, there exist a bounded subset kx ⊆ k and yx ∈ k ∩domϕ such that for any z ∈ k \kx, f(z,yx) + ϕ(yx) + 〈bz,yx −z〉 + 1 r 〈yx −z,z −x〉 < ϕ(z); (b2) k is a bounded set. lemma 2.6. [26] let k be a nonempty closed and convex subset of a real hilbert space h. let f : k×k → r be a bifunction satisfying conditions (a1)-(a4) and ϕ : k → r∪{+∞} be a paper lower semicontinuous and convex function such that k ∩ domϕ 6= ∅. for r > 0 and x ∈ k, define a mapping tr : h → k as follows: tr(x) = {z ∈ k : f(z,y) + ϕ(y) + 〈bz,yx −z〉 + 1 r 〈y −z,z −x〉≥ ϕ(z), ∀y ∈ k} for all x ∈ h. assume that either (b1)or (b2) holds. then the following conditions hold: (1) for all x ∈ h,tr(x) 6= ∅; (2) tr is singlevalued; (3) tr is firmly nonexpansive, that is, for any x,y ∈ h, ‖tr(x) −tr(y)‖ 2 ≤〈tr(x) −tr(y),x−y〉; (4) f(tr(i −rb)) = gmep(f,ϕ,b); (5) gmep(f,ϕ,b) is closed and convex. 3. main result let k be a nonempty closed convex subset of a real hilbert space h. in this section, we denote by cb(k), the family of nonempty, closed,convex and bounded subsets of k. theorem 3.1. let k be a nonempty closed convex subset of a real hilbert space h, f be a bifunction from k × k to r satisfying (a1) − (a4), and let ϕ be a proper lower semicontinuous and convex function from k to r ∪{+∞} such that k ∩ domϕ 6= ∅ and b an α-inverse strongly monotone mapping from k into h. let ti : k → cb(k) be multivalued ki-strictly pseudo-contractive mappings, ki ∈ (0, 1), i = 1, . . . ,m with ω := ∩mi=1f(ti) ∩ gmep(f,ϕ,b) 6= ∅. assume that for p ∈ ⋂m i=1 f(ti), tip = {p} and that either (b1) or (b2) holds with {αn,i} ⊂ (k, 1), i = 0, 1, · · · ,m. define the sequence {xn} as follows: x1 ∈ k = c1, (3.1)   f(un,y) + ϕ(y) −ϕ(un) + 〈bxn,y −un〉 + 1 rn 〈y −un,un −xn〉≥ 0,∀ y ∈ k, yn = αn,0un + ∑m i=1 αn,ix i n, x i n ∈ tiun, cn+1 = {z ∈ cn : ‖yn −z‖≤‖xn −z‖}, xn+1 = pcn+1x0, n ≥ 0, 52 ezeora where the sequence rn ∈ (0,∞) with lim inf n→∞ rn > 0 and ∑m i=0 αn,i = 1. then, the sequence {xn} converges strongly to pωx0. proof. we split the proof into steps. step 1. we show that pcn+1x0 is well defined for every x0 ∈ k. by lemma 2.6 and the condition on cb(k), we obtain that gmep (f,ϕ,b) and ∩mi=1f(ti) are closed and convex subsets of k. hence ω is a closed and convex subset of k. from lemma 2.5, we have that cn+1 is closed and convex for each n ≥ 0. let p ∈ ω, then ti(p) = {p}, i = 1, 2, · · · ,m. since un = trn (xn −rnxn), we have using lemma 2.6 that ‖un −p‖ = ‖trn (xn −rnbxn) −trn (p−rnbp)‖≤‖xn −p‖, ∀ n ≥ 0. using lemma 2.2 and lemma 2.3, we obtain the following estimates: ||yn −p||2 = αn,0||un −p||2 + m∑ i=1 αn,i||xin −p|| 2 − m∑ i=1 αn,iαn,0||un −xin|| 2 − m∑ i,j=1,i6=j αn,iαn,j||xin −x j n|| 2 ≤ αn,0||un −p||2 + m∑ i=1 αn,i||xin −p|| 2 − m∑ i=1 αn,iαn,0||un −xin|| 2 ≤ αn,0||un −p||2 + m∑ i=1 αn,i(d(tiun,tip)) 2 − m∑ i=1 αn,iαn,0||un −xin|| 2 ≤ αn,0||un −p||2 + m∑ i=1 αn,i||un −p||2 + m∑ i=1 αn,ik||un −xin|| 2 − m∑ i=1 αn,iαn,0||un −xin|| 2 = ||un −p||2 − m∑ i=1 αn,i(αn,0 −k)||un −xin|| 2.(3.2) since αn,i ∈ (k, 1), we obtain ||yn −p||2 ≤ ||un −p||2 so that ||yn −p|| ≤ ||un −p|| ≤ ||xn −p||. this implies that ||yn −p|| ≤ ||xn −p||.(3.3) hence p ∈ cn+1, and so ω ⊂ cn+1. therefore, pcn+1x0 is well defined. step 2. we show that lim n→∞ ‖xn −x0‖ exists. since ω is a nonempty closed convex subset of h, there exists a unique v ∈ ω such that v = pωx0. since xn = pcnx0 and xn+1 ∈ cn+1 ⊂ cn, ∀ n ≥ 0, we have ‖xn −x0‖≤‖xn+1 −x0‖, ∀ n ≥ 0. on the other hand, since v ∈ ω ⊂ cn, we obtain ‖xn −x0‖≤‖v −x0‖, ∀ n ≥ 0. it follows that the sequence {xn} is bounded and {||xn−x0||} is non decreasing and bounded. therefore, lim n→∞ ‖xn −x0‖ exists. generalized mixed equilibrium problem and common fixed point problem 53 step 3. we show that lim n→∞ xn exists in k. for m > n, by the definition of cn, we get xm = pcmx0 ∈ cm ⊂ cn. by applying lemma 2.1, we have ‖xm −xn‖ 2 ≤‖xm −x0‖ 2 −‖xn −x0‖ 2 . since lim n→∞ ‖xn − x0‖ exists, it follows that {xn} is a cauchy sequence. hence, there exists x∗ ∈ k such that lim n→∞ xn = x ∗. step 4. we show that ‖xin −xn‖→ 0 as n →∞, i = 1, 2, · · · ,m. from xn+1 ∈ cn+1, we have (3.4) ‖xn −yn‖≤‖xn −xn+1‖ + ‖xn+1 −yn‖≤ 2‖xn −xn+1‖→ 0 as n →∞. for p ∈ ω, using inequality (3.2), we get ‖yn −p‖ 2 ≤ αn,0||un −p||2 + m∑ i=1 αn,i||un −p||2 + m∑ i=1 αn,ik||un −xin|| 2 − m∑ i=1 αn,iαn,0||un −xin|| 2 = ||un −p||2 − m∑ i=1 αn,i(αn,0 −k)||un −xin|| 2 ≤ ||xn −p||2 − m∑ i=1 αn,i(αn,0 −k)||un −xin|| 2. thus, αn,iαn,0‖xin −un‖ 2 ≤ m∑ i=1 αn,iαn,0‖xin −un‖ 2 ≤ ‖xn −p‖ 2 −‖yn −p‖ 2 ≤ m‖xn −yn‖,(3.5) where m = supn≥0{‖xn −p‖ + ‖yn −p‖}. by the given condition on {αn,i} and (3.5), we get lim n→∞ ‖xin −un‖ = 0, i = 1, 2, · · · ,m. by lemma 2.6, we have ‖un −p‖ 2 = ‖trn (xn −rnbxn) −trn (p−rnbp)‖ 2 ≤〈trn (xn −rnbxn) −trn (p−rnbp),xn −p〉 = 〈un −p,xn −p〉 = 1 2 {||un −p||2 + ||xn −p||2 −||xn −un||2}. hence, ‖un −p‖ 2 ≤ ‖xn −p‖ 2 −‖xn −un‖ 2 . from inequality (3.2), we get ‖yn −p‖ 2 ≤ ‖xn −p‖ 2 −‖xn −un‖ 2 . ⇒‖xn −un‖ 2 ≤ ‖xn −p‖ 2 −‖yn −p‖ 2 ≤ m‖xn −yn‖, where m = sup n≥0 {‖xn −p‖ + ‖yn −p‖}. applying (3.4), we have ‖xn −un‖→ 0,n →∞. 54 ezeora hence, ‖xin −xn‖≤‖x i n −un‖ + ‖un −xn‖→ 0 as n →∞, i = 1, 2, · · · ,m. step 5. we show that x∗ ∈ ω. using the assumption that lim inf n→∞ rn > 0, we have (3.6) ∣∣∣∣xn −un rn ∣∣∣∣ = 1 rn ‖xn −un‖→ 0,n →∞. so, since lim n→∞ xn = x ∗, we obtain lim n→∞ un = x ∗. first, we show that x∗ ∈ gmep(f,ϕ,b). this proof follows as in the proof of theorem 3.1 of [26], we omit the proof. hence, x∗ ∈ gmep(f,ϕ,b). next, we have to show that x∗ ∈∩mi=1f(ti). for each i = 1, 2, ...,m, using lemma 2.4, we have d(x∗,tix ∗) ≤ d(x∗,xn) + d(xn,xin) + d(x i n,tix ∗) ≤ d(x∗,xn) + d(xn,xin) + d(tiun,tix ∗) ≤ d(x∗,xn) + d(xn,xin) + li||un −x ∗|| ≤ d(x∗,xn) + d(xn,xin) + l||un −x ∗||, where l = max1≤i≤m{li}. applying step 3-4, we have d(x∗,tix ∗) = 0. hence x∗ ∈ tix∗ for all i = 1, 2, ...,m. that is, x∗ ∈∩mi=1f(ti). step 6. we show that x∗ = pωx0. since xn = pcnx0, we get 〈ξ −xn,x0 −xn〉≤ 0,∀ ξ ∈ cn. since x∗ ∈ ω ⊂ cn, we have 〈ξ −x∗,x0 −x∗〉≤ 0,∀ ξ ∈ ω. thus, x∗ = pωx0, completing the proof. 2 if for each i = 1, 2, · · · ,m, ti : k → cb(k) is multivalued nonexpansive mappings, then we have the following result. corollary 3.2. let k be a nonempty closed and convex subset of a real hilbert space h, f be a bi-function from k × k to r satisfying (a1) − (a4), and let ϕ be a proper lower semicontinuous and convex function from k to r ∪ {+∞} such that k ∩ domϕ 6= ∅. let ti : k → cb(k) be multivalued nonexpansive mappings with ω := ∩mi=1f(ti) ∩ gmep(f,ϕ,b) 6= ∅. assume that for p ∈ ⋂m i=1 f(ti), tip = {p} and that either (b1) or (b2) holds with {αn,i} ⊂ [0, 1) satisfying the condition lim inf n→∞ αn,iαn,0 > 0 ∀ i = 1, 2, · · · ,m. define the sequence {xn} as follows: x1 ∈ k = c1, (3.7)   {f(un,y) + ϕ(y) −ϕ(un) + 1 rn 〈y −un,un −xn〉≥ 0,∀ y ∈ k, yn = αn,0un + ∑m i=1 αn,ix i n, x i n ∈ tiun, cn+1 = {z ∈ cn : ‖yn −z‖≤‖xn −z‖}, xn+1 = pcn+1x0, n ≥ 0, where the sequence rn ∈ (0,∞) with lim inf n→∞ rn > 0 and ∑m i=0 αn,i = 1. then the sequence {xn} converges strongly to pωx0. setting ϕ ≡ 0 in theorem 3.1, we have the following result. corollary 3.3. let k be a nonempty closed and convex subset of a real hilbert space h. let f be a bifunction from k × k → r satisfying (a1) − (a4). let ti : k → cb(k) be multivalued ki-strictly pseudo-contractive mappings, ki ∈ (0, 1), i = 1, . . . ,m with ω := ∩mi=1f(ti) ∩ep(f,ϕ) 6= ∅. assume generalized mixed equilibrium problem and common fixed point problem 55 that for p ∈ ⋂m i=1 f(ti), tip = {p} and that either (b1) or (b2) holds with {αn,i}⊂ [0, 1) satisfying the condition lim inf n→∞ αn,iαn,0 > 0 ∀ i = 1, 2, · · · ,m. define the sequence {xn} as follows: x1 ∈ k = c1, (3.8)   {f(un,y) + 1 rn 〈y −un,un −xn〉≥ 0,∀ y ∈ k, yn = αn,0un + ∑ i=1 m αn,ix i n, x i n ∈ tiun, cn+1 = {z ∈ cn : ‖yn −z‖≤‖xn −z‖}, xn+1 = pcn+1x0, n ≥ 0, where the sequence rn ∈ (0,∞) with lim inf n→∞ rn > 0 and ∑m i=0 αn,i = 1. then the sequence {xn} converges strongly to pωx0. setting f ≡ 0 in theorem 3.1, we have the following result. corollary 3.4. let k be a nonempty closed and convex subset of a real hilbert space h. let f be a bifunction from k × k → r satisfying (a1) − (a4). let ti : k → cb(k) be multivalued ki-strictly pseudo-contractive mappings, ki ∈ (0, 1), i = 1, . . . ,m with ω := ∩mi=1f(ti) ∩cmp(ϕ) 6= ∅. assume that for p ∈ ⋂m i=1 f(ti), tip = {p} and that either (b1) or (b2) holds with {αn,i}⊂ (k, 1). define the sequence {xn} as follows: x1 ∈ k = c1, (3.9)   {ϕ(y) −ϕ(un) + 1 rn 〈y −un,un −xn〉≥ 0,∀ y ∈ k, yn = αn,0un + ∑ i=1 m αn,ix i n, x i n ∈ tiun, cn+1 = {z ∈ cn : ‖yn −z‖≤‖xn −z‖}, xn+1 = pcn+1x0, n ≥ 0, where the sequence rn ∈ (0,∞) with lim inf n→∞ rn > 0 and ∑m i=0 αn,i = 1. then the sequence {xn} converges strongly to pωx0. setting f ≡ 0,ϕ ≡ 0 in theorem 3.1, we have the following result. corollary 3.5. let k be a nonempty closed and convex subset of a real hilbert space h. let f be a bifunction from k ×k → r satisfying (a1) − (a4). let ti : k → cb(k) be multi valued ki-strictly pseudo-contractive mappings, ki ∈ (0, 1), i = 1, · · · ,m with ω := ∩mi=1f(ti) 6= ∅. assume that for p ∈ ⋂m i=1 f(ti), tip = {p} and that {αn,i}⊂ (k, 1). define the sequence {xn} as follows: x1 ∈ k = c1, (3.10)   {yn = αn,0un + ∑ i=1 m αn,ix i n, x i n ∈ tiun, cn+1 = {z ∈ cn : ‖yn −z‖≤‖xn −z‖}, xn+1 = pcn+1x0, n ≥ 0, where the sequence ∑m i=0 αn,i = 1. then the sequence {xn} converges strongly to pωx0. remark 3.6. 1. theorem 3.1 is a significant improvement on theorem 3.1 of [11] for the following reasons; (a) it solves two major problems, generalized mixed equilibrium problem and common fixed point problem. (b) to prove theorem 3.1 of [11], compactness assumptions were placed on the operators ti, i = 1, 2, · · · ,n, this is dispensed with in the proof of theorem 3.1of this paper. 2. theorem 3.1 of this paper generalizes theorem 3.1 of [6] from multivalued nonexapnsive mappings to multivalued strictly pseudo contractive mapping and from mixed equilibrium problem to generalized mixed equilibrium problem. 56 ezeora 3. corollary 3.4 solves the convex minimization problem (1.7). references [1] acedo g. l., xu h. k., iterative methods for strict pseudo-contractions in hilbert spaces, nonlinear anal. 67(2007), 2258-2271. [2] aoyama k. , kohsaka f ., takahashi w. , shrinking projection methods for firmly nonexpansive mappings. nonlinear anal. 71(2009), 1626-1632. [3] browder f. e. , petryshyn w. v. , construction of fixed points of nonlinear mappings in hilbert spaces. j. math. anal. appl. 20(1967), 197-228. [4] brouwer l. e. j. , ober abbidung von manningfatigkeiten.math.ann.71(4)(1912), 598. [5] blum e. and oettli w., from optimization and variational inequalities to equilibrium problems, math. student, 63(1994), 123-145. [6] bunyawat a. and suantai s. , hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings. fixed point theory and appl. 2013(2013), article id 236. [7] ceng l. c. and yao j. c. , a hybrid iterative scheme for mixed equilibrium problems and fixed point problems, j. comput. appl. math., 214(2008),186-201. [8] chang k. c. , the obstacle problem and partial differential equations with discontinuous nonlinearities. commun. pure appl.math. 33(1980), 177-146. [9] chidume c. e. ,shahzad n., weak convergence theorems for a finite family of strict pseudocontractions, nonlinear anal. 72(2010), 1257-1265. [10] chidume c. e. , chidume c. o. , djitte n. , minjibir m. s. , convergence theorems for fixed points of multivalued strictly pseudo-contractive mappings in hilbert spaces. abstr. appl. anal. 2013. [11] chidume c. e. ,ezeora j. n. , krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings. fixed point theory and appl. 2014(2014), article id 111. [12] colao v. , marino g. and xu h.k., an iterative method for finding common solutions of equilibrium and fixed point problems, j. math. anal. appl., 344(2008), 340-352. [13] denavari t. , frigon m. , fixed point results for multivalued contractions on a metric space with a graph.j.math.anal.appl.405(2013), 507-517. [14] ding x. p. , iterative algorithm of solutions for generalized mixed implicit equilibriumlike problems, appl. math. comput., 162(2)(2005), 799-809. [15] downing d. ,kirk w. a., fixed point theorems for set-valued mappings in metric and banach spaces. math. jpn. 22(1)(1977), 99-112. [16] erbe l. , krawcewicz w. , existence of solutions to boundary value problems for impulsive second order differential inclusions. rocky mt.j.math.22(1992), 1-20. [17] ezeora j. n. , iterative solution of fixed points problem, system of generalized mixed equilibrium problems and variational inclusion problems. thai j. math. 12(1)(2014), 223-244. [18] geanakoplos j. , nash and walras equilibruim via brouwer, econ. theory 21(2003), 585-603. [19] giannessi f. , maugeri a. and pardalos m. , equilibrium problems, nonsmooth optimization and variational inequality models, kluwer academic, dordrecht, (2001). [20] huang c. ,ma x. on generalized equilibrium problems and strictly pseudocontractive mappings in hilbert spaces. fixed point theory and applications 2014(2014), article id 145. [21] hussain n. , khan a. r. , applications of the best approximation operator to ∗-nonexpansive maps in hilbert spaces. numer. funct. anal. optim. 24(2003), 327-338. [22] kakutani s. , a generalization of brouwer’s fixed point theorem. duke math.j. 8(3)(1941), 457-459. [23] kang j. l. , su y. f. ,zhang x., shrinking projection algorithm for fixed points of firmly nonexpansive mappings and its applications. fixed point theory 11(2)(2010), 301-310. [24] kim t. h. , xu h. k., strongly convergence of modified mann iterations for with asymptotically nonexpansive mappings and semigroups. nonlinear anal. 64(2006), 1140-1152. [25] kimura y. , takahashi w. ,yao j. c., strong convergenceof an iterative scheme by a new type projection method for a family of quasi nonexpansive mappings. j. optim.theory appl.149(2011), 239-253. [26] li s. ,li l. , cao l. , he x. and yue x. , hybrid extragradient method for generalized mixed equilibrium problems and fixed point problems in hilbert space. fixed point theory and appl. 2013(2013), article id 240. [27] marino g. xu h. k. , weak and strong convergence theorems for strict pseudo-contractions in hilbert spaces. j. math. anal. appl. 329(2007), 336-346. [28] moudafi a. , mixed equilibrium problems, sensitivity analysis and algorithmic aspects, comput. math. appl., 44(2002), 1099-1108. [29] nadler s. b. , multivalued contraction mappings. pac. j. math. 30, 475-488(1969) [30] nakajo k. ,takahashi w., strongly convergence theorems for nonexpansive mappings and nonexpansive semigroups. j. math. anal. appl. 279(2003), 372-379. [31] nash j. f., non-cooperative games.ann.math. 54(2), 286-295(1951) [32] nash j. f. , equilibrium points in n-person games. proc. natl. acad.sci.usa 36(1)(1950), 48-49. [33] noor m. a. , multivalued general equilibrium problems, j. math. anal. appl., 283(2003), 140-149. [34] noor m. a. , auxiliary principle technique for equilibrium problem, j. optim. theory appl., 122(2004), 371-386. generalized mixed equilibrium problem and common fixed point problem 57 [35] osilike m. o. , udomene a. , demiclosedness principle and convergence results for strictly pseudocontractive mappings of browder-petryshyn type, j. math. anal. appl. 256(2001), 431-445. [36] panyanak b. , mann and ishikawa iteration processes for multi-valued mappings in banach apces. comput.math.appl.4(2007), 72-87. [37] qin x. , cho s. y., kang s. m., an extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. j. glob. optim. 49(2011), 679-693. [38] sahu d. r. , petrusel a. , strong convergence of iterative methods by strictly pseudocontractive mappings in banach spaces, nonlinear anal. 74(2011), 6012-6023. [39] tada a. and takahashi w. , weak and strong convergence for a a nonexpansive mapping and an equilibrium problem, j. optim. theory appl., 133(2007), 359-370. [40] takahashi w. , takeuchi y., kubota r., strong convergence theorems by hybrid methods for families of nonexpansive mappings in hilbert spaces. j. math.anal. appl. 341(2008), 276-286. [41] yao y. h. , liou y. c. , marino g., a hybrid algorithm for pseudo-contractive mappings. nonlinear anal. 71(2009), 4997-5002. [42] cholamjiak w., suantai s. , a hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems. discrete dyn. nat.soc. 2010(2010), article id 349158. [43] chidume c. e., geometric properties of banach spaces and nonlinear iterations, springer verlag, lecture notes math. 1965 (2009). department of ind. mathematics and statistics, ebonyi state university, abakaliki, nigeria ∗corresponding author: jerryezeora@yahoo.com international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 146-154 doi: 10.28924/2291-8639-15-2017-146 fractional ostrowski type inequalities for functions whose first derivatives are s-preinvex in the second sense badreddine meftah∗ abstract. in this paper, we establish an fractional identity. using this new identity we derives some fractional ostrowski’s inequalities for functions whose first derivatives are s-preinvex in the second sense. 1. introduction in 1938, a.m. ostrowski proved an interesting integral inequality, given by the following theorem theorem 1.1. [10] let f : i → r, where i ⊆ r is an interval, be a mapping in the interior i◦of i, and a,b ∈ i◦, with a < b. if |f′| ≤ m for all x ∈ [a,b], then∣∣∣∣∣∣f(x) − 1b−a b∫ a f (t) dt ∣∣∣∣∣∣ ≤ m (b−a) [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] , ∀x ∈ [a,b] . (1.1) in the last decades, the inequality (1.1) has attracted much interest by many researchers, and considerable papers have been published concerning the generalizations, variants, and extensions of the inequality (1.1), for more detail we refer readers to [4, 7–9, 13, 16, 17] and references cited therein. recently, lot of efforts have been made by many mathematicians to generalize the classical convexity. hanson [3], introduced a new class of generalized convex functions, called invex functions. in [1], the authors gave the concept of preinvex functions which is special case of invexity, and many authors have study their basic properties, and their role in optimization, variational inequalities and equilibrium problems, we refer readers to [11, 12, 15, 20, 21]. i̇şcan [5] established the following ostrowski inequalities for functions whose derivatives are preinvex theorem 1.2. [5, theorem 2.2] let a ⊆ r be an open invex subset with respect to η : a×a → r and a,b ∈ a with a < a + η(b,a). suppose that f : a → r is a differentiable function and |f′| is preinvex function on a. if f′ is integrable on [a,a + η(b,a)], then the following inequality∣∣∣∣∣∣∣f(x) − 1η(b,a) a+η(b,a)∫ a f(u)du ∣∣∣∣∣∣∣ ≤ η(b,a) 6 × {( 3 ( x−a η(b,a) )2 − 2 ( x−a η(b,a) )3 + 2 ( a+η(b,a)−x η(b,a) )3) |f′(a)| + ( 1 − 3 ( x−a η(b,a) )2 + 4 ( x−a η(b,a) )3) |f′(b)| } holds for each x ∈ [a,a + η(b,a)]. theorem 1.3. [5, theorem 2.8] let a ⊆ r be an open invex subset with respect to η : a×a → r and a,b ∈ a with a < a + η(b,a). suppose that f : a → r is a differentiable function and |f′|q is received 21st july, 2017; accepted 30th september, 2017; published 1st november, 2017. 2010 mathematics subject classification. 26d10, 26d15, 26a51. key words and phrases. ostrowski inequality; midpoint inequality; hölder inequality; power mean inequality; spreinvex functions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 146 fractional ostrowski type inequalities 147 preinvex function on [a,a + η(b,a)] for some fixed q ≥ 1. if f′ is integrable on [a,a + η(b,a)], then the following inequality∣∣∣∣∣∣∣f(x) − 1η(b,a) a+η(b,a)∫ a f(u)du ∣∣∣∣∣∣∣ ≤ η (b,a) ( 1 2 )1−1 q × {( x−a η(b,a) )2(1−1q ) ((x−a)2(3η(b,a)−2x+2a) 6η3(b,a) |f′(a)|q + 1 3 ( x−a η(b,a) )3 |f′(b)|q )1 q + ( a+η(b,a)−x η(b,a) )2(1−1q ) (1 3 ( a+η(b,a)−x η(b,a) )3 |f′(a)|q + ( 1 6 + (x−a)2(2x−3η(b,a)−2a) 6η3(b,a) ) |f′(b)|q )1 q } holds for each x ∈ [a,a + η(b,a)]. kirmaci [7] established the following midpoint inequalities for differentiable convex functions theorem 1.4. [7, theorem 2.2] let f : i◦ ⊂ r → r be a differentiable mapping on i◦, a,b ∈ i◦ (i◦ is the interior of i) with a < b. if |f′| is convex on [a,b], then we have∣∣∣∣∣∣ 1b−a b∫ a f(x)dx−f ( a+b 2 )∣∣∣∣∣∣ ≤ b−a8 (|f′(a)| + |f′(b)|) . theorem 1.5. [7, theorem 2.3] let f : i◦ ⊂ r → r be a differentiable mapping on i◦, a,b ∈ i◦ (i◦ is the interior of i) with a < b and let p > 1. if |f′| p p−1 is convex on [a,b], then we have∣∣∣∣∣∣ 1b−a b∫ a f(x)dx−f ( a+b 2 )∣∣∣∣∣∣ ≤ b−a16 ( 4 p+1 )1 p (( 3 |f′(a)| p p−1 + |f′(b)| p p−1 )p−1 p + ( |f′(a)| p p−1 + 3 |f′(b)| p p−1 )p−1 p ) . wang et al. [18] established the following midpoint inequalities for functions whose the power of the absolute value of the first derivatives are preinvex theorem 1.6. [18, theorem 3.1] let a ⊆ r be an open invex subset with respect to η : a×a → r and let f : a → r be a differentiable function. if |f′|q is preinvex on a for q ≥ 1, then for every a,b ∈ a with η (b,a) 6= 0 we have∣∣∣∣∣∣∣ 1η(b,a) a+η(b,a)∫ a f(u)du−f ( 2a+η(b,a) 2 )∣∣∣∣∣∣∣ ≤ |η(b,a)| 8 (( |f′(a)|q+2|f′(b)|q 3 )1 q + ( 2|f′(a)|q+|f′(b)|q 3 )1 q ) . theorem 1.7. [18, corollary 3.2] let a ⊆ r be an open invex subset with respect to η : a×a → r and let f : a → r be a differentiable function. if |f′| is preinvex on a, then for every a,b ∈ a with η (b,a) 6= 0 we have∣∣∣∣∣∣∣ 1η(b,a) a+η(b,a)∫ a f(u)du−f ( 2a+η(b,a) 2 )∣∣∣∣∣∣∣ ≤ |η(b,a)| 8 (|f′(a)| + |f′(b)|) . motivated by these results, in this paper we establish an fractional identity, and then using this equality we derive some ostrowski’s inequalities for functions whose first derivatives in absolute value are s-preinvex in the second sense. 148 b. meftah 2. preliminaries in this section we recall some concepts of convexity that are well known in the literature. throughout this section i is an interval of r. definition 2.1. [14] a function f : i → r is said to be convex, if f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f(y) holds for all x,y ∈ i and all t ∈ [0, 1]. definition 2.2. [2] a nonnegative function f : i ⊂ [0,∞) → r is said to be s-convex in the second sense for some fixed s ∈ (0, 1], if the following inequality f(tx + (1 − t)y) ≤ tsf(x) + (1 − t)sf(y) holds for all x,y ∈ i and t ∈ [0, 1]. let k be a subset in rn and let f : k → r and η : k ×k → rn be continuous functions. definition 2.3. [20] a set k is said to be invex at x with respect to η, if x + tη (y,x) ∈ k holds for all x,y ∈ k and t ∈ [0, 1]. k is said to be an invex set with respect to η if k is invex at each x ∈ k. definition 2.4. [20] a function f on the invex set k is said to be preinvex with respect to η, if f (x + tη (y,x)) ≤ (1 − t) f (x) + tf(y) holds for all x,y ∈ k and t ∈ [0, 1]. definition 2.5. [19] a nonnegative function f on the invex set k ⊆ [0,∞) is said to be s-preinvex in the second sense with respect to η, for some fixed s ∈ (0, 1], if f (x + tη (y,x)) ≤ (1 − t)sf (x) + tsf(y) holds for all x,y ∈ k and t ∈ [0, 1]. definition 2.6. [6] let f ∈ l1[a,b]. the riemann-liouville integrals jαa+f and j α b− f of order α > 0 with a ≥ 0 are defined by jαa+f(x) = 1 γ (α) x∫ a (x− t)α−1 f(t)dt, x > a jαb−f(x) = 1 γ (α) b∫ x (t−x)α−1 f(t)dt, b > x respectively, where γ(α) = ∞∫ 0 e−ttα−1dt, is the gamma function and j0 a+ f(x) = j0 b− f(x) = f(x). 3. main results in what follows η : k ×k → r, and k ⊂ r an invex subset with respect to η, and a,b ∈ k◦ the interior of k such that [a,a + η(b,a)] ⊂ k. at first, we prove the following lemma. lemma 3.1. let f : [a,a + η(b,a)] → r be a differentiable function with a < a + η(b,a). if f′ ∈ l ([a,a + η(b,a)]), then the following equality for fractional integrals(( x−a η(b,a) )α + ( 1 − x−a η(b,a) )α) f(x) − γ(α+1) (η(b,a))α (jαx−f(a) + j α x+f(a + η (b,a))) = η (b,a)   x−a η(b,a)∫ 0 tαf′(a + tη (b,a))dt− 1∫ x−a η(b,a) (1 − t)α f′(a + tη (b,a))dt   (3.1) holds for all x ∈ [a,a + η(b,a)]. fractional ostrowski type inequalities 149 proof. integrating by parts right hand side of (3.1), we get η (b,a)   x−a η(b,a)∫ 0 tαf′(a + tη (b,a))dt− 1∫ x−a η(b,a) (1 − t)α f′(a + tη (b,a))dt   =  ( x−aη(b,a))α f(x) −α x−a η(b,a)∫ 0 tα−1f(a + tη (b,a))dt + ( 1 − x−a η(b,a) )α f(x) −α 1∫ x−a η(b,a) (1 − t)α−1 f(a + tη (b,a))dt   = (( x−a η(b,a) )α + ( 1 − x−a η(b,a) )α) f(x) −α   x−a η(b,a)∫ 0 tα−1f(a + tη (b,a))dt + 1∫ x−a η(b,a) (1 − t)α−1 f(a + tη (b,a))dt   . (3.2) using the change of variable u = a + tη (b,a), (3.2) becomes η (b,a)   x−a η(b,a)∫ 0 tαf′(a + tη (b,a))dt− 1∫ x−a η(b,a) (1 − t)α f′(a + tη (b,a))dt   = (( x−a η(b,a) )α + ( 1 − x−a η(b,a) )α) f(x) − α (η(b,a))α   x∫ a (u−a)α−1 f(u)du + a+η(b,a)∫ x (η (b,a) + a−u)α−1 f(u)du   = (( x−a η(b,a) )α + ( 1 − x−a η(b,a) )α) f(x) − γ(α+1) (η(b,a))α (jαx−f(a) + j α x+f(a + η (b,a))) , which is the desired result. � theorem 3.1. let f : [a,a + η (b,a)] → r be a differentiable function such that η(b,a) > 0 and f′ ∈ l ([a,a + η (b,a)]). if |f′| is s-preinvex in the second sense for some fixed s ∈ (0, 1], then the following inequality for fractional integrals∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η (b,a) (( b x−a η(b,a) (α + 1,s + 1) + 1 α+s+1 ( 1 − x−a η(b,a) )α+s+1) |f′(a)| + ( 1 α+s+1 ( x−a η(b,a) )α+s+1 + b (s + 1,α + 1) − b x−a η(b,a) (s + 1,α + 1) ) |f′(b)| ) (3.3) holds for all x ∈ [a,a + η(b,a)]. 150 b. meftah proof. from lemma 3.1, and properties of modulus, we have∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η (b,a)   x−a η(b,a)∫ 0 tα |f′(a + tη (b,a))|dt + 1∫ x−a η(b,a) (1 − t)α |f′(a + tη (b,a))|dt   . using the s-preinvexity of |f′|, we obtain∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η (b,a)   x−a η(b,a)∫ 0 ( tα (1 − t)s |f′(a)| + tα+s |f′(b)| ) dt + 1∫ x−a η(b,a) ( (1 − t)α+s |f′(a)| + (1 − t)α ts |f′(b)| ) dt   = η (b,a)     x−a η(b,a)∫ 0 tα (1 − t)s dt + 1∫ x−a η(b,a) (1 − t)α+s dt   |f′(a)| +   x−a η(b,a)∫ 0 tα+sdt + 1∫ x−a η(b,a) ts (1 − t)α dt   |f′(b)|   = η (b,a) (( b x−a η(b,a) (α + 1,s + 1) + 1 α+s+1 ( 1 − x−a η(b,a) )α+s+1) |f′(a)| + ( 1 α+s+1 ( x−a η(b,a) )α+s+1 + b (s + 1,α + 1) −b x−a η(b,a) (s + 1,α + 1) ) |f′(b)| ) , (3.4) where we use the facts that x−a η(b,a)∫ 0 tα (1 − t)s dt = b x−a η(b,a) (α + 1,s + 1) 1∫ x−a η(b,a) (1 − t)α+s dt = 1 α+s+1 ( 1 − x−a η(b,a) )α+s+1 x−a η(b,a)∫ 0 tα+sdt = 1 α+s+1 ( x−a η(b,a) )α+s+1 1∫ x−a η(b,a) ts (1 − t)α dt = b (s + 1,α + 1) −b x−a η(b,a) (s + 1,α + 1) . (3.5) the proof is completed. � remark 3.1. in theorem 3.1, if we put α = s = 1, we obtain theorem 2.2 from [5]. fractional ostrowski type inequalities 151 corollary 3.1. in theorem 3.1, if we choose x = 2a+η(b,a) 2 , then the following midpoint inequality holds for fractional integrals ∣∣∣∣f( 2a+η(b,a)2 ) − 2α−1γ(α+1)(η(b,a))α ( jα2a+η(b,a) 2 −f(a) + j α 2a+η(b,a) 2 +f(a + η (b,a)) )∣∣∣∣ ≤ η (b,a) (( b1 2 (α + 1,s + 1) + 1 (α+s+1)2α+s+1 ) |f′(a)| + ( 1 (α+s+1)2α+s+1 + b (s + 1,α + 1) −b1 2 (s + 1,α + 1) ) |f′(b)| ) remark 3.2. in corollary 3.1, if we put α = s = 1, we obtain corollary 3.2 from [18]. moreover if we take η (b,a) = b−a, we obtain theorem 2.2 from [7]. theorem 3.2. let f : [a,a + η (b,a)] → r be a differentiable function such that η(b,a) > 0 and f′ ∈ l ([a,a + η (b,a)]) and let q > 1 with 1 p + 1 q = 1. if |f′|q is s-preinvex in the second sense for some fixed s ∈ (0, 1], then the following inequality for fractional integrals ∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η(b,a) (s+1) 1 q (αp+1) 1 p (( x−a η(b,a) )α+ 1 p (( x−a η(b,a) )s+1 |f′(b)|q + ( 1 − ( 1 − x−a η(b,a) )s+1) |f′(a)|q )1 q + ( 1 − x−a η(b,a) )α+ 1 p × (( 1 − x−a η(b,a) )s+1 |f′(a)|q + ( 1 − ( x−a η(b,a) )s+1) |f′(b)|q )1 q ) (3.6) holds for all x ∈ [a,a + η(b,a)]. proof. from lemma 3.1, properties of modulus, and hölder’s inequality, we have ∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η (b,a)     x−a η(b,a)∫ 0 tαpdt   1 p   x−a η(b,a)∫ 0 |f′(a + tη (b,a))|q dt   1 q +   1∫ x−a η(b,a) (1 − t)αp dt   1 p   1∫ x−a η(b,a) |f′(a + tη (b,a))|q dt   1 q   = η(b,a) (αp+1) 1 p   ( x−a η(b,a) )α+ 1 p   x−a η(b,a)∫ 0 |f′(a + tη (b,a))|q dt   1 q + ( 1 − x−a η(b,a) )α+ 1 p   1∫ x−a η(b,a) |f′(a + tη (b,a))|q dt   1 q   . (3.7) 152 b. meftah since |f′|q is s-preinvex, we deduce∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η(b,a) (αp+1) 1 p   ( x−a η(b,a) )α+ 1 p   x−a η(b,a)∫ 0 (1 − t)s |f′(a)|q + ts |f′(b)|q dt   1 q + ( 1 − x−a η(b,a) )α+ 1 p   1∫ x−a η(b,a) (1 − t)s |f′(a)|q + ts |f′(b)|q dt   1 q   = η(b,a) (s+1) 1 q (αp+1) 1 p (( x−a η(b,a) )α+ 1 p (( x−a η(b,a) )s+1 |f′(b)|q + ( 1 − ( 1 − x−a η(b,a) )s+1) |f′(a)|q )1 q + ( 1 − x−a η(b,a) )α+ 1 p × (( 1 − x−a η(b,a) )s+1 |f′(a)|q + ( 1 − ( x−a η(b,a) )s+1) |f′(b)|q )1 q ) , which completes the proof. � corollary 3.2. in theorem 3.2, if we choose x = 2a+η(b,a) 2 , then the following midpoint inequality holds for fractional integrals∣∣∣∣f( 2a+η(b,a)2 ) − 2α−1γ(α+1)(η(b,a))α ( jα2a+η(b,a) 2 −f(a) + j α 2a+η(b,a) 2 +f(a + η (b,a)) )∣∣∣∣ ≤ η(b,a) 2 α+ 1 p (s+1) 1 q (αp+1) 1 p (( |f′(b)|q+(2s+1−1)|f′(a)|q 2s+1 )1 q + ( |f′(a)|q+(2s+1−1)|f′(b)|q 2s+1 )1 q ) . remark 3.3. in corollary 3.2, if we choose α = s = 1, and η (b,a) = b−a, we obtain theorem 2.3 from [7]. theorem 3.3. let f : [a,a + η (b,a)] → r be a differentiable function such that η(b,a) > 0 and f′ ∈ l ([a,a + η (b,a)]) and let q > 1. if |f′|q s-preinvex in the second sense for some fixed s ∈ (0, 1], then the following inequality for fractional integrals∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η(b,a) (α+1) 1− 1 q ((( x−a η(b,a) )(α+1)(1−1 q ))( b x−a η(b,a) (α + 1,s + 1) |f′(a)|q + 1 α+s+1 ( x−a η(b,a) )α+s+1 |f′(b)|q )1 q + ( 1 − x−a η(b,a) )(α+1)(1−1 q ) × ( 1 α+s+1 ( 1 − x−a η(b,a) )α+s+1 |f′(a)|q ? + ( b (s + 1,α + 1) −b x−a η(b,a) (s + 1,α + 1) ) |f′(b)|q )1 q ) (3.8) holds for all x ∈ [a,a + η(b,a)]. fractional ostrowski type inequalities 153 proof. from lemma 3.1, properties of modulus, and power mean inequality, we have∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η (b,a)     x−a η(b,a)∫ 0 tαdt   1− 1 q   x−a η(b,a)∫ 0 tα |f′(a + tη (b,a))|q dt   1 q +   1∫ x−a η(b,a) (1 − t)α dt   1− 1 q   1∫ x−a η(b,a) (1 − t)α |f′(a + tη (b,a))|q dt   1 q   = η(b,a) (α+1) 1− 1 q   ( x−a η(b,a) )(α+1)(1−1 q )  x−a η(b,a)∫ 0 tα |f′(a + tη (b,a))|q dt   1 q + ( 1 − x−a η(b,a) )(α+1)(1−1 q )  1∫ x−a η(b,a) (1 − t)α |f′(a + tη (b,a))|q dt   1 q   . (3.9) since |f′|q is s-preinvex, we deduce∣∣∣(( x−aη(b,a))α + (1 − x−aη(b,a))α)f(x) − γ(α+1)(η(b,a))α (jαx−f(a) + jαx+f(a + η (b,a)))∣∣∣ ≤ η(b,a) (α+1) 1− 1 q ((( x−a η(b,a) )(α+1)(1−1 q )) ×  |f′(a)|q x−a η(b,a)∫ 0 tα(1 − t)sdt + |f′(b)|q x−a η(b,a)∫ 0 tα+sdt   1 q + ( 1 − x−a η(b,a) )(α+1)(1−1 q ) ×  |f′(a)|q 1∫ x−a η(b,a) (1 − t)α+sdt + |f′(b)|q 1∫ x−a η(b,a) ts (1 − t)α dt   1 q   . (3.10) substituting (3.5) into (3.10), we obtain the desired result. � remark 3.4. in theorem 3.3, if we take α = s = 1, we obtain theorem 2.8 from [5]. corollary 3.3. in theorem 3.3, if we choose x = 2a+η(b,a) 2 , then the following midpoint inequality holds for fractional integrals∣∣∣∣∣ 12α−1 f( 2a+η(b,a)2 ) − 2α−1γ(α+1)(η(b,a))α ( jα 2a+η(b,a) 2 −f(a) + j α 2a+η(b,a) 2 +f(a + η (b,a)) )∣∣∣∣∣ ≤ η(b,a) 2 (α+1)(1− 1 q )(α+1)1− 1 q (( b1 2 (α + 1,s + 1) |f′(a)|q + 1 (α+s+1)2α+s+1 |f′(b)|q )1 q + ( 1 (α+s+1)2α+s+1 |f′(a)|q + ( b (s + 1,α + 1) −b1 2 (s + 1,α + 1) ) |f′(b)|q )1 q ) . remark 3.5. in corollary 3.3, if we put α = s = 1, we obtain theorem 3.1 from [18]. 154 b. meftah references [1] a. ben-israel and b. mond, what is invexity?, j. austral. math. soc., ser. b, 28(1986), no. 1, 1-9. [2] w. w. breckner, stetigkeitsaussagen für eine klasse verallgemeinerter konvexer funktionen in topologischen linearen räumen, publ. inst. math. (beograd), 23, (1978), 13–20. [3] m. a. hanson, on sufficiency of the kuhn-tucker conditions, j. math. anal. appl. 80 (1981) 545-550. [4] havva kavurmacı, m. emin özdemir and merve avcı, new ostrowski type inequalityes for m-convex functions and applications, hacettepe journal ofmathematics and statistics, volume 40 (2) (2011), 135–145. [5] i̇. i̇şcan, ostrowski type inequalities for functions whose derivatives are preinvex. bulletin of the iranian mathematical society. vol. 40 (2014), no. 2, pp. 373-386. [6] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractional differential equations. north-holland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [7] u. s. kirmaci, inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. appl. math. comput. 147 (2004), no. 1, 137–146. [8] b. meftah, some new ostrwoski’s inequalities for functions whose nth derivatives are r-convex. international journal of analysis, 2016, 7 pages [9] b. meftah, ostrowski inequalities for functions whose first derivatives are logarithmically preinvex. chin. j. math. (n.y.) 2016, art. id 5292603, 10 pp. [10] d. s. mitrinović, j. e. pečarić and a. m. fink, classical and new inequalities in analysis. mathematics and its applications (east european series), 61. kluwer academic publishers group, dordrecht, 1993. [11] m. a. noor, variational-like inequalities, optimization, 30 (1994), 323-330. [12] m. a. noor, invex equilibrium problems, j. math. anal. appl., 302 (2005), 463-475. [13] m. e. özdemir, h. kavurmaci, e. set, ostrowski’s type inequalities for (α,m)-convex functions, kyungpook math. j., 50(2010), 371-378. [14] j. pečarić, f. proschan and y. l. tong, convex functions, partial orderings, and statistical applications. mathematics in science and engineering, 187. academic press, inc., boston, ma, 1992. [15] r. pini, invexity and generalized convexity, optimization 22 (1991) 513-525. [16] m. z. sarikaya, on the ostrowski type integral inequality. acta math. univ. comenian. (n.s.) 79 (2010), no. 1, 129–134. [17] e. set, m. e. özdemir and m. z. sarıkaya, new inequalities of ostrowski’s type for s-convex functions in the second sense with applications. facta univ. ser. math. inform. 27 (2012), no. 1, 67–82. [18] y. wang, b. -y. xi and f. qi, hermite-hadamard type integral inequalities when the power of the absolute value of the first derivative of the integrand is preinvex. matematiche (catania) 69 (2014), no. 1, 89–96. [19] y. wang, s-h. wang and f. qi, simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is s-preinvex, facta univ. ser. math. inform. 28 (2) (2013), 151–159. [20] t. weir and b. mond, (1988). pre-invex functions in multiple objective optimization, j. math. anal.appl. 136, 29-38. [21] x. -m. yang and d. li, (2001). on properties of preinvex functions, j. math. anal. appl. 256,229-241. laboratoire des télécommunications, faculté des sciences et de la technologie, university of 8 may 1945 guelma, p.o. box 401, 24000 guelma, algeria. ∗corresponding author: badrimeftah@yahoo.fr 1. introduction 2. preliminaries 3. main results references int. j. anal. appl. (2022), 20:51 received: jul. 14, 2022. 2010 mathematics subject classification. 90c30. key words and phrases. nonlinear programming problem; fractional gradient-based system; residual power series method; optimal solution; caputo fractional derivative. https://doi.org/10.28924/2291-8639-20-2022-51 © 2022 the author(s) issn: 2291-8639 1 analytical solution of nonlinear fractional gradient-based system using fractional power series method radwan abu-gdairi* department of mathematics, faculty of science, zarqa university, zarqa 13132, jordan *corresponding author: rgdairi@zu.edu.jo abstract. this paper adapted a reliable treatment technique, called the fractional residual power series, to the fractional gradient-based system in solving a class of nonlinear programming model in caputo’s sense. the gradient-based system has been constructed to transform the nonlinear programming problem with equality constraints to unconstrained optimization problem, and then the fractional residual power series method is implemented to obtain the essential behavior of underlying problem. the proposed methods have been applied effectively to produce optimal solution in rapidly convergent fractional series representations without linearization, or any limitations. to confirm the performance of the proposed methods, some optimization problems are tested. further, numerical comparisons with other existing methods are also given. the results exhibit that the frps method is easy, simple, effective, and fully compatible with the complexity of such models. 1. introduction over the past years, the topic of optimization has had the interest of many scholars in various applications of science and technology and associated with several categories of optimization problems. besides, many effective methods have been developed to find the optimal solution to these problems. for more details see [1-6]. the gradient-based approach is one such method that is used to solve nonlinear programming (nlp) problems. transforming the optimization problem into a system of ordinary differential equations (odes) is the basic idea of this method, which is equipped https://doi.org/10.28924/2291-8639-20-2022-51 2 int. j. anal. appl. (2022), 20:51 with ideal conditions to reach the optimal solutions to this problem [7-13]. fractional calculus is a generalization of the derivatives and integrals of an arbitrary system. recently, the subject of fractional calculus has received the attention of scientists and engineers because of its important applications in various fields, whether science or engineering [14-19]. many real-life problems in various fields of applied science have been modeled using fractional differential equations (fdes), which are generalizations of odes. for describing the behavior of the unknowns of fdes, many researchers usually implement some numerical or numerical analytical techniques instead. in this regard, some recent techniques are proposed for solving fdes. among them decomposition technique, symmetric perturbation technique, variable frequency technique, and partial differential transformation technique [20-26]. further, more applications and promising approaches are utilized to treat the nonlinear fractional gradient-based systems of fdes could be founded in the references [27-30]. motivated by the existing techniques, the main contribution of this article is to transform equality constrained nlp problem to unconstrained optimization problem by identifying a penalty function, then construct a gradientbased system of fdes. besides the, the fractional residual power series (frps) technique is applied to provide us the accordance between the optimal solution of the nlp problem and the power series solution of the obtained fdes system. frps technique is one of the modern numeric-analytic techniques was initially proposed in [31] to investigate the sequential solution of fuzzy differential equations of both first and second degree. it has been used to generate accurate approximate solutions in terms of fractional series formulas for several kinds for linear and non-linear fdes, partial fdes and fuzzy fdes (see [3236]). this scheme is used basically the residual-error function and employed the fractional differentiation in each stage in determining the coefficient of the suggested series expansion without linearity, division, or perturbation (see [37-42]). it does not require any converting while switching from the lower order to the higher order; as a result, the method can be applied directly to the given problem by choosing an initial guess approximation. frps is quick and easy calculation to find series solutions via utilizing software package. also, different taylor series method, frps needs easy computation state with high reliability and less time. the organization of this paper was to present a brief presentation of some basic and necessary definitions and properties in fractional calculus in section 2, in addition to the fact that the central problem in this paper was presented in section 3. section 4 presents the details of the application of 3 int. j. anal. appl. (2022), 20:51 the proposed technique and provides a clear and simple algorithm for the basic steps of this technique. clarifying the applicability and efficiency of the proposed technique by comparing the results derived from it for some numerical examples with the results of the fourth degree rung-kutta method, done in section 5. section 6 is designed to provide some concluding observations. 2. preliminaries in this section, we present the definition of the riemann-liouville fractional integral operator, the caputo partial derivative, and some of their properties [43-49]. throughout this paper, the symbol 𝑅 denotes the set of real numbers, 𝑁 the set of integers, and γ(. ) is a gamma function. for more details, please see [45-49] and references therein. definition 1. let 𝛼 ≥ 0 and for each 𝛼, 𝑥 ∈ ℝ. the riemann-liouville fractional integral operator of order, for a function 𝑢(𝑥), is defined as follow 𝐼𝛼 𝑢(𝑥) = 1 γ(𝛼) ∫ (𝑥 − 𝑠)1−𝛼 𝑢(𝑠)𝑑𝑠 𝑥 0 , 𝑡 > 0, in particular, if 𝛼 = 0, then 𝐼𝛼 𝑢(𝑡) = 𝑢(𝑡). definition 2. the caputo fractional derivative of a function 𝑢(𝑥) with 𝑚 − 1 < 𝛼 ≤ 𝑚, 𝑚 ∈ ℕ, is defined as 𝐷𝛼 𝑢 = { 1 𝛤(𝑚 − 𝛼) ∫ (𝑥 − 𝑠)𝑚−𝛼−1𝑢(𝑚)(𝑠)𝑑𝑠 , 𝑚 − 1 < 𝛼 < 𝑚, 𝑥 > 0 𝑥 0 . 𝑢(𝑚)(𝑥) , 𝛼 = 𝑚 . on the other hand, the operator 𝐷𝛼 has some basic properties such as, for any real number 𝐴, then we have 𝐷𝛼 𝐴 = 0, and for 𝑢(𝑥) = 𝑥𝑘 , 𝑘 ≥ −1, we have 𝐷𝛼 𝑥𝑘 = 𝛤(𝑘+1) 𝛤(𝑘+1−𝛼) 𝑥 𝑘−𝛼. moreover, the caputo fractional derivative has the linearity property, this means, for each constant 𝛾 and 𝜇 we have 𝐷𝛼 [𝛾𝑢(𝑥) + 𝜇𝑣(𝑥)] = 𝛾𝐷𝛼 𝑢(𝑥) + 𝜇𝐷𝛼 𝑣(𝑥). lemma 1. for 𝑚 − 1 < 𝛼 ≤ 𝑚 and 𝑢(𝑥) ∈ 𝐶𝑚 [0, ∞), 𝑚 ∈ ℕ, then we have 1. 𝐷𝛼 𝐼𝛼 𝑢(𝑥) = 𝑢(𝑥), 2. 𝐼𝛼 𝐷𝛼 𝑢(𝑥) = 𝑢(𝑥) − ∑ 𝑢(𝑘)𝑚−1𝑖=0 (0 +) 𝑥𝑘 𝑘! , 𝑥 > 0. 3. optimization problem the second part presents the details of the optimization problem to be studied in this paper. we consider the following nlp problem with equality constrains 4 int. j. anal. appl. (2022), 20:51 min 𝑢(𝑥) 𝑠. 𝑡. 𝑤(𝑥) = 0, 𝑥 ∈ ℝ𝑛 , (2) where 𝑥 ∈ ℝ𝑛 is decision variable, 𝑢(𝑥) is a vector-valued function of a real variable, and 𝑤 = (𝑤1, 𝑤2, … , 𝑤𝑝) 𝑇 : ℝ𝑛 → ℝ𝑝 (𝑝 ≤ 𝑛) are twice continuously differentiable function such that whose gradient ∇𝑤(𝑥) has full rank. we assume that a feasible region of (2) is nonempty bounded set. the penalty function can be defined as 𝑃(𝑥) = 𝑢(𝑥) + 𝜙(𝑥), (3) where 𝜙(𝑥) is the penalty term defined on ℝ𝑛 and has the following property 𝜙(𝑥) = { 0, 𝑤(𝑥) = 0 𝑛𝑜𝑛𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒, 𝑤(𝑥) ≠ 0. (4) one can be defined the penalty term 𝜙(𝑥) as 𝜙(𝑥) = 1 𝜇 ‖𝑤(𝑥)‖𝜇 𝜇 = 1 𝜇 ∑ |𝑤𝑖 (𝑥)| 𝜇 𝑝 𝑖=1 , (5) where 𝜇 > 0 is a constant. a well-known penalty function for the problem (2) has been defined as 𝑃(𝑥, 𝜂𝑚) = 𝑢(𝑥) + 𝜂𝑚 𝜇 ∑ |𝑤𝑖 (𝑥)| 𝜇 𝑝 𝑖=1 , (6) where the penalty parameter 𝜂𝑚 satisfying the inequality 0 < 𝜂𝑚 < 𝜂𝑚+1 for all 𝑚, 𝜂𝑚 → ∞. it is worth mentioning that the penalty parameter 𝜂𝑚 can be chosen randomly. thus, 𝜂𝑚can be chosen in positive constant depends on the difficulty of minimizing the penalty function at every iteration. it can be shown that under some suitable conditions the solution of the equality constrains nlp problem (2) are solution of the following unconstrained optimization problem min 𝑃(𝑥, 𝜂𝑚) = 𝑢(𝑥) + 𝜂𝑚 𝜇 ∑ |𝑤𝑖 (𝑥)| 𝜇 𝑝 𝑖=1 𝑠. 𝑡 𝑥 ∈ ℝ𝑛 . (7) we assume that the unconstrained optimization problem (7) for each 𝑚, has a solution and we denote it by 𝑥𝑚. the main results that connect the solutions of the equality constrained nlp problem (2) and unconstrained problem (7) present in the following theorem. theorem 1. let {xm} be a sequence generated by the penalty method of the unconstrained problem (7). then any limit point of the sequence is a solution to the equality constrained nlp problem (2). the author in [12] showed that the unconstrained optimization problem (7) can be described by the following gradient based dynamic system of odes 𝑑𝑥(𝑡) 𝑑𝑡 = −∇𝑥 (𝑢(𝑥) + 𝜂𝑚 𝜇 ∑ |𝑤𝑖 (𝑥)| 𝜇 𝑝 𝑖=1 ) , 𝑥(𝑡0) = 𝑏 ∈ ℝ 𝑛 (8) 5 int. j. anal. appl. (2022), 20:51 where ∇𝑥 is the gradient vector with respect to the 𝑥 ∈ ℝ 𝑛. we can describe the system (8) by an approach based on fractional gradient based dynamic system by the following system of fdes 𝐷𝑡 𝛼 𝑥(𝑡) = −∇𝑥 (𝑢(𝑥) + 𝜂𝑚 𝜇 ∑ |𝑤𝑖 (𝑥)| 𝜇𝑝 𝑖=1 ) , 𝑥(𝑡0) = 𝑏 ∈ ℝ 𝑛, 0 < 𝛼 ≤ 1. (9) the system (9) has an equilibrium point 𝑥𝑒, if 𝑥𝑒 ∈ ℝ 𝑛 is satisfies the right-hand side of the system. a more convenient form of the system (9) can be expressed as follows 𝐷𝑡 𝛼 𝑥𝑖 (𝑡) = 𝑓𝑖 (𝑡, 𝜂𝑚, 𝑥1, 𝑥2, … , 𝑥𝑛), 𝑖 = 1,2, … , 𝑛, 𝑥(𝑡0) = 𝑏 ∈ ℝ 𝑛 , 0 < 𝛼 ≤ 1. (10) the stable equilibrium point of the fractional system (10) is acquired with the rps technique. 4. frps technique this subsection devoted to applying the rps method to derive analytic solution of the system of fdes (10). we begin by propose the definition of fractional power series. definition 3. a power series (ps) expansion at 𝑡 = 𝑡0 of the following form ∑ 𝑐𝑖 (𝑡 − 𝑡0) 𝑗𝛼 ∞ 𝑖=0 = 𝑐0 + 𝑐1(𝑡 − 𝑡0) 𝛼 + 𝑐2(𝑡 − 𝑡0) 2𝛼 + ⋯, (11) for 0 ≤ 𝑛 − 1 < 𝛼 ≤ 𝑛, 𝑛 ∈ ℕ, 𝑡 ≥ 𝑡0 where 𝑐𝑗’s are constants called the fractional power series (fps). theorem 2. suppose that 𝑥(𝑡) has a fps representation at 𝑡 = 𝑡0 of the form 𝑥(𝑡) = ∑ 𝑐𝑗 (𝑡 − 𝑡0) 𝑗𝛼 ∞ 𝑖=0 , 𝑡0 ≤ 𝑡 ≤ 𝑡0 + 𝑅. (12) if 𝑥(𝑡) ∈ 𝐶[𝑡0, 𝑡0 + 𝑅), and 𝐷𝑡 𝑖𝛼 𝑥(𝑡) ∈ 𝐶(𝑡0, 𝑡0 + 𝑅), for 𝑖 = 0,1,2, …, then the coefficients 𝑐𝑖 in eq. (6) will take the form 𝑐𝑖 = 𝐷𝑖𝛼𝑥(𝑡0) 𝛤(𝑖𝛼+1) , where 𝐷𝑖𝛼 = 𝐷𝛼 ∙ 𝐷𝛼 ∙∙∙ 𝐷𝛼 (𝑖-times) and 𝑅 is the convergent radius. theorem 3. if 𝐾 ∈ (0,1), such that ‖𝑥𝑖+1(𝑡)‖ ≤ 𝐾‖𝑥𝑖 (𝑡)‖ ∀𝑖 ∈ ℕ and 0 < 𝑡 < τ < 1, then the series of numerical solutions converges to an exact solution proof. we notice that ∀ 0 < 𝑡 < τ < 1, ‖𝑥(𝑡) − 𝑥𝑖 (𝑡)‖ = ‖ ∑ 𝑥𝑖 (𝑡) ∞ 𝑚=𝑖+1 ‖ ≤ ∑ ‖𝑥𝑖 (𝑡)‖ ∞ 𝑚=𝑖+1 ≤ ‖𝑏‖ ‖ ∑ 𝐾𝑚 ∞ 𝑚=𝑖+1 ‖ = 𝐾𝑘+1 1−𝐾 ‖𝑏‖ → 0 𝑎𝑠 𝑘 → ∞. 6 int. j. anal. appl. (2022), 20:51 now, to utilize the frps technique to solve the system (10), we substitute the fps expansion (12) among the recurrent fractional differentiation of truncation residual functions. suppose that the solution of the system (10) about the initial point 𝑡 = 𝑡0 takes the form 𝑥𝑖 (𝑡) = ∑ 𝑐𝑖𝑗 ∞ 𝑗=0 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑖 = 1,2, … , 𝑛. (13) differentiate the expansion form (13) for 𝑡 > 0 within the interval of convergence, we get 𝐷𝑡 𝛼 𝑥𝑖 (𝑡) = ∑ 𝑐𝑖𝑗 γ(𝑗𝛼 + 1) γ((𝑗 − 1)𝛼 + 1) ∞ 𝑗=1 (𝑡 − 𝑡0) (𝑗−1)𝛼 , 𝑖 = 1,2, … , 𝑛. (14) hence, 𝐷𝑡 2𝛼 𝑥𝑖 (𝑡) = ∑ 𝑐𝑖𝑗 γ(𝑗𝛼 + 1) γ((𝑗 − 2)𝛼 + 1) ∞ 𝑗=2 (𝑡 − 𝑡0) (𝑗−2)𝛼 , 𝑖 = 1,2, … , 𝑛, (15) where 𝐷𝑡 2𝛼 = 𝐷𝑡 𝛼 𝐷𝑡 𝛼. therefore, we suppose that the approximate solution of 𝑥𝑖 (𝑡) can be constructed by the following series 𝑥𝑖𝑘 (𝑡) = ∑ 𝑐𝑖𝑗 𝑘 𝑗=0 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑖 = 1,2, … , 𝑛. (16) since 𝑥𝑖 (𝑡) satisfy the initial condition 𝑥𝑖 (𝑡0) = 𝑏𝑖 , 𝑖 = 1,2, … , 𝑛, then 𝑐𝑖0 = 𝑏𝑖 and series (16) can be rewritten by 𝑥𝑖𝑘 (𝑡) = 𝑏𝑖 + ∑ 𝑐𝑖𝑗 𝑘 𝑗=1 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑘 = 1,2,3, . . . , 𝑖 = 1,2, … , 𝑛. (17) according the rps technique, we define the residual function, 𝑅𝑒𝑠𝑥𝑖 (𝑡), about 𝑡 = 𝑡0 for the system (10) as follows 𝑅𝑒𝑠𝑥𝑖 (𝑡) = 𝐷𝑡 𝛼 𝑥𝑖 (𝑡) + ∇𝑥𝑖 𝑃(𝑥𝑖 , 𝜂), 𝑖 = 1,2, … , 𝑛. (18) by assuming the penalty parameter 𝜂𝑚 = 𝜂 (constant). the 𝑘-th residual function, 𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡), can be defined by 𝑅𝑒𝑠𝑥𝑖,𝑘(𝑡) = 𝐷𝑡 𝛼 𝑥𝑖𝑘 (𝑡) + ∇𝑥𝑖𝑘 𝑃(𝑥𝑖𝑘 , 𝜂), 𝑘 = 1,2,3, … , 𝑖 = 1,2, … , 𝑛. (19) note that, 𝑅𝑒𝑠𝑥𝑖 (𝑡) = 0 and 𝐿𝑖𝑚𝑘→∞𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡) = 𝑅𝑒𝑠𝑥𝑖 (𝑡), 𝑖 = 1,2, … , 𝑛, for all 𝑡 ≥ 0. as a matter of fact, it yields the following algebraic system 𝐷𝑡 (𝑘−1)𝛼 𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡0) = 0, 𝑘 = 1,2,3, … , 𝑖 = 1,2, … , 𝑛. (20) 7 int. j. anal. appl. (2022), 20:51 through the algebraic system (20) we can obtained the coefficient 𝑐𝑗 . 𝑗 = 1,2,3, … , 𝑘 by applying the following steps. algorithm 1. algorithm of finding the coefficients 𝒄𝒊𝒋 of the k-th truncated series (16). step 1: substitute the 𝑘-th residual expansion (19) into (18), where (21) 𝑥𝑖𝑘 (𝑡) = 𝑐𝑖0 + ∑ 𝑐𝑖𝑗 𝑘 𝑗=1 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑐𝑖𝑗 = 𝐷𝑗𝛼 𝑏𝑖 𝛤(𝑗𝛼 + 1) . step 2: find the formula 𝐷𝑡 (𝑘−1)𝛼 (𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡0)) , 𝑖 = 1,2, … , 𝑛. step 3: the coefficient 𝑐𝑖𝑗 can be obtained by solve the algebraic system (22) 𝐷𝑡 (𝑘−1)𝛼 𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡0) = 0, 𝑘 = 1,2,3, … , 𝑖 = 1,2, … , 𝑛. step 4: substitute the obtained values of 𝑐𝑖𝑗 back into eq. (17). 5. numerical implementation and results this section is designed to apply the proposed technique, frps, and evaluate its performance and accuracy in deriving some numerical solutions for a number of examples and comparing the results obtained with the analytical solutions of the examples. we used the mathematica software package to perform numerical and symbolic calculations. example 1. consider the following nlp problem min 𝑢(𝑥) = 100(𝑥1 2 − 𝑥2) 2 + (𝑥1 − 1), subject to 𝑤(𝑥) = 𝑥1(𝑥1 − 4) − 2𝑥2 + 12 = 0. (23) according to (6), the correspond penalty function for the problem (23) at 𝜇 = 2 is given by 𝑃(𝑥1, 𝑥2, 𝜂) = 100(𝑥1 2 − 𝑥2) 2 + (𝑥1 − 1) + 1 2 𝜂(𝑥1 2 − 4𝑥1 − 2𝑥2 + 12) 2, (24) where the penalty variable 𝜂 ∈ ℝ+, 𝜂 → ∞. hence, we get the following correspond system of fdes 𝐷𝑡 𝛼 𝑥1(𝑡) = −400(𝑥1 2 − 𝑥2)𝑥1 − 2(𝑥1 − 1) −𝜂(2𝑥1 − 4)(𝑥1 2 − 4𝑥1 − 2𝑥2 + 12), 𝐷𝑡 𝛼 𝑥2(𝑡) = 200(𝑥1 2 − 𝑥2) + 2𝜂(𝑥1 2 − 4𝑥1 − 2𝑥2 + 12), 𝑥1(0) = 𝑥2(0) = 0, (25) 8 int. j. anal. appl. (2022), 20:51 where 0 < 𝛼 ≤ 1. to apply the frps technique to solve the system (25), we suppose that 𝑥1(𝑡) = ∑ 𝑐1𝑗 ∞ 𝑗=0 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑥2(𝑡) = ∑ 𝑐2𝑗 ∞ 𝑗=0 (𝑡 − 𝑡0) 𝑗𝛼 . (26) thus, the k-th truncated series is 𝑥1𝑘 (𝑡) = ∑ 𝑐1𝑗 𝑘 𝑗=1 (𝑡 − 𝑡0) 𝑗𝛼 , 𝑥2𝑘 (𝑡) = ∑ 𝑐2𝑗 𝑘 𝑗=1 (𝑡 − 𝑡0) 𝑗𝛼 . (27) consequently, the 𝑘-th residual function, 𝑅𝑒𝑠𝑥𝑖,𝑘 (𝑡), can be defined by 𝑅𝑒𝑠𝑥1,𝑘 (𝑡) = 𝐷𝑡 𝛼 𝑥1𝑘 (𝑡) + 400(𝑥1 2 − 𝑥2)𝑥1 + 2(𝑥1 − 1) +𝜂(2𝑥1 − 4) (𝑥1 2 − 4𝑥1 − 2𝑥2 + 12), 𝑅𝑒𝑠𝑥2,𝑘 (𝑡) = 𝐷𝑡 𝛼 𝑥2𝑘 (𝑡) − 200(𝑥1 2 − 𝑥2) − 2𝜂(𝑥1 2 − 4𝑥1 − 2𝑥2 + 12), (28) finally, algorithm 1 applied to find the coefficients 𝑐𝑖𝑗, then getting the approximate solution of the system (25). the efficiency of frps approach is introduced via establishing some numerical comparisons for the obtained results and the results obtained by runge-kutta approach and listed in table 1, for problem (23), from this table we noted that the frps solutions are compatible with the exact solution more that the rk4 solutions. the geometric behavior of the frps approximate solutions are shown against the exact and rk4 solutions as in figure 1. clearly, the figure indicates that the exact and frps approximate solutions are in good agreement for different values of 𝛼, comparing with the exact and rk4 approximate solutions over the domain of 𝛼. table 1. comparison of 𝑥1(𝑡) and 𝑥2(𝑡) for problem (23) between frps with rk4 solutions at 𝜂 = 200 and 𝛼 = 1. 𝑡 𝐹𝑅𝑃𝑆 𝑥1(𝑡) 𝐹𝑅𝑃𝑆 𝑥2(𝑡) 𝑅𝐾4 𝑥1(𝑡) 𝑅𝐾4 𝑥2(𝑡) absolute error 𝑥1(𝑡) absolute error 𝑥2(𝑡) 0.0000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.0005 1.971356 3.872061 1.970899 3.871887 0.000457 0.000174 0.0010 1.978467 3.908321 1.978274 3.907993 0.000193 0.000328 0.0013 1.98151 3.923481 1.981384 3.922554 0.000126 0.000927 0.0015 1.984216 3.932718 1.983132 3.930578 0.001084 0.00214 0.0020 1.9864 3.937884 1.984252 3.935654 0.002148 0.00223 9 int. j. anal. appl. (2022), 20:51 (a) (b) figure 1. comparison of 𝑥1(𝑡) and 𝑥2(𝑡) for problem (23) between frps with rk4 solutions at 𝜂 = 200 and 𝛼 = 1: (a) 𝑥1(𝑡), (b) 𝑥2(𝑡), solidline: frps, dashline: rk4. example 2. consider the following nlp problem: min 𝑢(𝑥) = (𝑥1 − 1) 2 + (𝑥1 − 𝑥2) 2 + (𝑥2 − 𝑥3) 2 + (𝑥3 − 𝑥4) 4 + (𝑥4 − 𝑥5) 4, subject to: 𝑤1(𝑥) = 𝑥1 + 𝑥2 2 + 𝑥3 3 − 2 − 3√2 = 0 𝑤2(𝑥) = 𝑥2 − 𝑥3 2 + 𝑥4 + 2 − 2√2 = 0, 𝑤3(𝑥) = 𝑥1𝑥5 − 2 = 0 (29) we define the penalty function for the problem (29) as 𝑃(𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝜂) = (𝑥1 − 1) 2 + (𝑥1 − 𝑥2) 2 + (𝑥2 − 𝑥3) 2 + (𝑥3 − 𝑥4) 4 + (𝑥4 − 𝑥5) 4 + 𝜂 2 [(𝑥1 + 𝑥2 2 + 𝑥3 3 − 2 − 3√2) 2 + (𝑥2 − 𝑥3 2 + 𝑥4 + 2 − 2√2) 2 + (𝑥1𝑥5 − 2) 2], (30) where the penalty variable 𝜂 ∈ ℝ+, 𝜂 → ∞. depending on this penalty function, the correspond fdes system can be written as 𝐷𝑡 𝛼 𝑥𝑖 (𝑡) = ∇𝑥 𝑖 𝑢(𝑥) + 𝜂 ∑ ∇𝑥 𝑖 𝑤𝑖 (𝑥) 5 𝑖=1 𝑤𝑖 (𝑥), 𝑥𝑖 (0) = 2, 𝑖 = 1,2,3,4,5, (31) where 0 < 𝛼 ≤ 1. we adapt the frps technique to the fdes system (31) with penalty variable 𝜂 = 300 at fractional order derivative 𝛼 = 0.9, we acquired solutions as shown in table 2. figure 2 present the obtained frps solutions 𝑥1(𝑡) and 𝑥4(𝑡) at various fractional derivative order 𝛼. obviously, from figure 2, the curves-frps approximate solutions consistent with each other and approach the exact curve with increasing fractional values to the integer-order value 𝛼 = 1. 10 int. j. anal. appl. (2022), 20:51 table 2. the obtained solution of nlp problem (29) by frps technique at 𝜂 = 300 and 𝛼 = 0.9. 𝑡 𝑥1(𝑡) 𝑥2(𝑡) 𝑥3(𝑡) 𝑥4(𝑡) 𝑥5(𝑡) 0.000 2.00000 2.00000 2.00000 2.00000 2.00000 1.000 1.19855 1.36387 1.47429 1.64587 1.67875 2.000 1.19951 1.36981 1.47237 1.63658 1.67801 3.000 1.20181 1.36473 1.48531 1.63782 1.67947 4.000 1.21414 1.36588 1.48431 1.63842 1.68371 5.000 1.26941 1.36947 1.48997 1.64889 1.68753 6.000 1.27542 1.37842 1.49568 1.64998 1.68947 (a) (b) figure 2. the frps solutions: (a) 𝑥1(𝑡), (b) 𝑥4(𝑡); 𝛼 = 0.9 solid, 𝛼 = 0.8 dashed, and 𝛼 = 0.7 dot-dashed, at 𝜂 = 300. example 3. consider the following nlp problem: 𝑚𝑖𝑛. 𝑢(𝑡) = (𝑥1 − 20) 2 + (𝑥2 + 20) 2, subject to: 𝑤(𝑡) = 𝑥1 2 100 + 𝑥2 2 4 − 1 = 0, (32) the penalty function of this problem can be written as 𝑃(𝑥1, 𝑥2, 𝜂) = (𝑥1 − 20) 2 + (𝑥2 + 20) 2 + 𝜂 2 ( 𝑥1 2 100 + 𝑥2 2 4 − 1) 2 , (33) where the penalty variable 𝜂 ∈ ℝ+, 𝜂 → ∞. therefore, we can write the correspond fdes system as 11 int. j. anal. appl. (2022), 20:51 𝐷𝑡 𝛼 𝑥1(𝑡) = 2𝑥1 − 40 + 𝜂 ( 𝑥1 3 5000 + 𝑥1𝑥2 2 200 − 𝑥1 50 ), 𝐷𝑡 𝛼 𝑥2(𝑡) = 2𝑥2 + 40 + 𝜂 ( 𝑥1 2𝑥2 200 + 𝑥2 3 8 − 𝑥2 2 ), 𝑥1(0) = 𝑥2(0) = 0, (34) where 0 < 𝛼 ≤ 1. the frps utilize to construct the solution of this system and we get the following numerical solutions of the nlp problem (32) as shown in table 3. it is very clear that the results obtained for example 3 indicated that the proposed method is simple, and its performance is very effective comparing with runge-kutta method. table 3. comparison of 𝑥1(𝑡) and 𝑥2(𝑡) for problem (32) between frps with rk4 solutions 𝜂 = 106 and 𝛼 = 1. 𝑡 𝐹𝑅𝑃𝑆 𝑥1(𝑡) 𝐹𝑅𝑃𝑆 𝑥2(𝑡) 𝑅𝐾 𝑥1(𝑡) 𝑅𝐾 𝑥2(𝑡) error 𝑥1(𝑡) error 𝑥2(𝑡) 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 1.0 5.850119 −0.47496 5.847465 −0.477547 0.002654 0.002584 2.0 6.744776 −0.53968 6.741904 −0.542553 0.002872 0.002873 4.0 8.460532 −0.58119 8.457551 −0.584133 0.002981 0.002948 6.0 8.735192 −0.61163 8.732010 −0.615305 0.003182 0.003675 7.0 8.980451 −0.63668 8.977187 −0.640466 0.003264 0.003782 8.0 9.092537 −0.663578 9.090838 −0.661673 0.001699 0.001905 10.0 9.304238 −0.691245 9.303232 −0.680066 0.001006 0.018393 5. conclusions in this a novel paper, the corresponding system of fdes for nlp was designed and analyzed under the meaning of caputo derivative and then solved the target system via a modern efficient technique, named frps technique. the benefit of utilizing the present technique is that it gives accurate convergence approximate solution to an exact solution with needs a small size of computation to the optimal solutions to the nlp problems. three attractive nlp are considered to validate the applicability and superiority of the frps scheme. simulation data and graphical 12 int. j. anal. appl. (2022), 20:51 representations are discussed and indicated that the obtained results by frps approach are in good agreement with each other and with exact solution at various values of fractional derivative order 𝛼, as well as from the figures the results emphasized that obtained frps solutions more accurate from the obtained runge-kutta solutions versus the exact solution. consequently, the presented technique has the ability to handle both linear and nonlinear fractional biological phenomena. in future works, the frps can be extended to generate accurate approximate solutions for systems of fractional partial differential equations. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] w. sun, y.x. yuan, optimization theory and methods: nonlinear programming, springer-verlag, new york (2006). [2] n. özdemir, f. evirgen, solving nlp problems with dynamic system approach based on smoothed penalty function, selcuk j. appl. math. 10 (2009), 63–73. https://hdl.handle.net/20.500.12462/4852. [3] h. yamashita, a differential equation approach to nonlinear programming, math. program. 18 (1980), 155–168. https://doi.org/10.1007/bf01588311. [4] m. al-smadi, o.a. arqub, d. zeidan, fuzzy fractional differential equations under the mittag-leffler kernel differential operator of the abc approach: theorems and applications, chaos solitons fractals. 146 (2021), 110891. https://doi.org/10.1016/j.chaos.2021.110891. [5] a.v. fiacco, g.p. mccormick. nonlinear programming: sequential unconstrained minimization techniques, john wiley, new york (1968). [6] c.a. botsaris, differential gradient methods, j. math. anal. appl. 63 (1978), 177–198. https://doi.org/10.1016/0022-247x(78)90114-2. [7] m. al‐smadi, fractional residual series for conformable time‐fractional sawada–kotera–ito, lax, and kaup–kupershmidt equations of seventh order, math methods appl. sci. (2021). https://doi.org/10.1002/mma.7507. [8] a.a. brown, m.c. bartholomew-biggs, ode versus sqp methods for constrained optimization, j. optim. theory appl. 62 (1989), 371–386. https://doi.org/10.1007/bf00939812. [9] j. schropp, i. singer, a dynamical systems approach to constrained minimization, numer. funct. anal. optim. 21 (2000), 537–551. https://doi.org/10.1080/01630560008816971. https://hdl.handle.net/20.500.12462/4852 https://doi.org/10.1007/bf01588311 https://doi.org/10.1016/j.chaos.2021.110891 https://doi.org/10.1016/0022-247x(78)90114-2 https://doi.org/10.1002/mma.7507 https://doi.org/10.1007/bf00939812 https://doi.org/10.1080/01630560008816971 13 int. j. anal. appl. (2022), 20:51 [10] s. wang, x.q. yang, k.l. teo, a unified gradient flow approach to constrained nonlinear optimization problems, comput. optim. appl. 25 (2003), 251–268. https://doi.org/10.1023/a:1022973608903. [11] m. al-smadi, o.a. arqub, computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates, appl. math. comput. 342 (2019), 280–294. https://doi.org/10.1016/j.amc.2018.09.020. [12] l. jin, l.w. zhang, x.t. xiao, two differential equation systems for equality-constrained optimization, appl. math. comput. 190 (2007), 1030–1039. https://doi.org/10.1016/j.amc.2006.11.041. [13] m. al-smadi, a. freihat, h. khalil, s. momani, r. ali khan, numerical multistep approach for solving fractional partial differential equations, int. j. comput. methods. 14 (2017), 1750029. https://doi.org/10.1142/s0219876217500293. [14] n. andrei, gradient flow algorithm for unconstrained optimization, ici technical report, 2004. [15] l. jin, a stable differential equation approach for inequality constrained optimization problems, appl. math. comput. 206 (2008), 186–192. https://doi.org/10.1016/j.amc.2008.09.007. [16] n. özdemir, f. evirgen, a dynamic system approach to quadratic programming problems with penalty method, bull. malays. math. sci. soc. (2) 33 (2010), 79–91. [17] s. hasan, m. al-smadi, a. el-ajou, et al. numerical approach in the hilbert space to solve a fuzzy atangana-baleanu fractional hybrid system, chaos solitons fractals. 143 (2021) 110506. https://doi.org/10.1016/j.chaos.2020.110506. [18] a. qazza, r. hatamleh, m. al-hawari, dirichlet problem in the simply connected domain, bounded by unicursal curve, int. j. appl. math. 22 (2009), 599-614. [19] a. qazza, r. hatamleh, dirichlet problem in the simply connected domain, bounded by the nontrivial kind, adv. differ. equ. control processes, 17 (2016), 177-188. [20] r. saadeh, m. alaroud, m. al-smadi, et al. application of fractional residual power series algorithm to solve newell–whitehead–segel equation of fractional order, symmetry. 11 (2019), 1431. https://doi.org/10.3390/sym11121431. [21] r. edwan, r. saadeh, s. hadid, m. al-smadi, s. momani, solving time-space-fractional cauchy problem with constant coefficients by finite-difference method, in: d. zeidan, s. padhi, a. burqan, p. ueberholz (eds.), computational mathematics and applications, springer singapore, singapore, 2020: pp. 25–46. https://doi.org/10.1007/978-981-15-8498-5_2. [22] a. burqan, a. el-ajou, r. saadeh, m. al-smadi, a new efficient technique using laplace transforms and smooth expansions to construct a series solution to the time-fractional navier-stokes equations, alexandria eng. j. 61 (2022), 1069–1077. https://doi.org/10.1016/j.aej.2021.07.020. [23] r. saadeh, a. qazza, a. burqan, a new integral transform: ara transform and its properties and applications, symmetry. 12 (2020), 925. https://doi.org/10.3390/sym12060925. https://doi.org/10.1023/a:1022973608903 https://doi.org/10.1016/j.amc.2018.09.020 https://doi.org/10.1016/j.amc.2006.11.041 https://doi.org/10.1142/s0219876217500293 https://doi.org/10.1016/j.amc.2008.09.007 https://doi.org/10.1016/j.chaos.2020.110506 https://doi.org/10.3390/sym11121431 https://doi.org/10.1007/978-981-15-8498-5_2 https://doi.org/10.1016/j.aej.2021.07.020 https://doi.org/10.3390/sym12060925 14 int. j. anal. appl. (2022), 20:51 [24] m. al-smadi, n. djeddi, s. momani, et al. an attractive numerical algorithm for solving nonlinear caputofabrizio fractional abel differential equation in a hilbert space, adv. differ. equ. 2021 (2021), 271. https://doi.org/10.1186/s13662-021-03428-3. [25] s.a. el-wakil, a. elhanbaly, m.a. abdou, adomian decomposition method for solving fractional nonlinear differential equations, appl. math. comput. 182 (2006), 313-324. https://doi.org/10.1016/j.amc.2006.02.055. [26] m. al-smadi, h. dutta, s. hasan, s. momani, on numerical approximation of atangana-baleanu-caputo fractional integro-differential equations under uncertainty in hilbert space, math. model. nat. phenom. 16 (2021), 41. https://doi.org/10.1051/mmnp/2021030. [27] a. khan, a. khan, m. sinan, ion temperature gradient modes driven soliton and shock by reduction perturbation method for electron-ion magneto-plasma, math. model. numer. simul. appl. 2 (2022), 112. https://doi.org/10.53391/mmnsa.2022.01.001. [28] f. evirgen, conformable fractional gradient based dynamic system for constrained optimization problem, acta phys. pol. a. 132 (2017), 1066–1069. https://doi.org/10.12693/aphyspola.132.1066. [29] f. evirgen, m. yavuz, an alternative approach for nonlinear optimization problem with caputo fabrizio derivative, itm web conf. 22 (2018), 01009. https://doi.org/10.1051/itmconf/20182201009. [30] a. alvarado-sánchez, e. rangel-cortes, a. hernández-hernández, bi-dimensional crime model based on anomalous diffusion with law enforcement effect, math. model. numer. simul. appl. 2 (2022), 26–40. https://doi.org/10.53391/mmnsa.2022.01.003. [31] o. abu arqub. series solution of fuzzy differential equations under strongly generalized differentiability, j. adv. res. appl. math. 5 (2013), 31-52. https://doi.org/10.5373/jaram.1447.051912. [32] m. alaroud, m. al-smadi, r.r. ahmad, et al. computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations, int. j. differ. equ. 2018 (2018), 8686502. https://doi.org/10.1155/2018/8686502. [33] s. hasan, m. al-smadi, h. dutta, et al. multi-step reproducing kernel algorithm for solving caputo– fabrizio fractional stiff models arising in electric circuits, soft comput. 26 (2022), 3713–3727. https://doi.org/10.1007/s00500-022-06885-4. [34] s. momani, n. djeddi, m. al-smadi, s. al-omari, numerical investigation for caputo-fabrizio fractional riccati and bernoulli equations using iterative reproducing kernel method, appl. numer. math. 170 (2021), 418-434. https://doi.org/10.1016/j.apnum.2021.08.005. [35] m. al-smadi, o. abu arqub, s. momani, numerical computations of coupled fractional resonant schrödinger equations arising in quantum mechanics under conformable fractional derivative sense, physica scripta, 95 (2020), 075218. https://doi.org/10.1088/1402-4896/ab96e0. [36] a. elsaid. fractional differential transform method combined with the adomian polynomials, appl. math. comput. 218 (2012), 6899-6911. https://doi.org/10.1016/j.amc.2011.12.066. https://doi.org/10.1186/s13662-021-03428-3 https://doi.org/10.1016/j.amc.2006.02.055 https://doi.org/10.1051/mmnp/2021030 https://doi.org/10.53391/mmnsa.2022.01.001 https://doi.org/10.12693/aphyspola.132.1066 https://doi.org/10.1051/itmconf/20182201009 https://doi.org/10.53391/mmnsa.2022.01.003 https://doi.org/10.5373/jaram.1447.051912 https://doi.org/10.1155/2018/8686502 https://doi.org/10.1007/s00500-022-06885-4 https://doi.org/10.1016/j.apnum.2021.08.005 https://doi.org/10.1088/1402-4896/ab96e0 https://doi.org/10.1016/j.amc.2011.12.066 15 int. j. anal. appl. (2022), 20:51 [37] a. freihet, s. hasan, m. al-smadi, et al. construction of fractional power series solutions to fractional stiff system using residual functions algorithm, adv. differ. equ. 2019 (2019), 95. https://doi.org/10.1186/s13662-019-2042-3. [38] m. al-smadi, o. abu arqub, s. hadid. an attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative, commun. theor. phys. 72 (2020), 085001. https://doi.org/10.1088/1572-9494/ab8a29. [39] m. al-smadi, o. abu arqub, s. hadid, approximate solutions of nonlinear fractional kundu-eckhaus and coupled fractional massive thirring equations emerging in quantum field theory using conformable residual power series method, physica scripta, 95 (2020), 105205. https://doi.org/10.1088/1402-4896/abb420. [40] m. al-smadi, a. freihat, o. abu arqub, et al. a novel multistep generalized differential transform method for solving fractional-order lü chaotic and hyperchaotic systems, j. comput. anal. appl. 19 (2015), 713–724. [41] m. alabedalhadi, m. al-smadi, s. al-omari, et al. new optical soliton solutions for coupled resonant davey-stewartson system with conformable operator, opt. quant. electron. 54 (2022), 392. https://doi.org/10.1007/s11082-022-03722-8. [42] i. podlubny, fractional differential equation, academic press, san diego, 1999. [43] m. alabedalhadi, m. al-smadi, s. al-omari, et al. structure of optical soliton solution for nonliear resonant space-time schrödinger equation in conformable sense with full nonlinearity term, physica scripta, 95 (2020), 105215. https://doi.org/10.1088/1402-4896/abb739. [44] z. altawallbeh, m. al-smadi, i. komashynska, a. ateiwi, numerical solutions of fractional systems of two-point bvps by using the iterative reproducing kernel algorithm, ukr. math j. 70 (2018), 687–701. https://doi.org/10.1007/s11253-018-1526-8. [45] a.a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier science b.v., amsterdam, 2006. [46] s. hasan, m. al-smadi, a. freihet, et al.two computational approaches for solving a fractional obstacle system in hilbert space, adv. differ. equ. 2019 (2019), 55. https://doi.org/10.1186/s13662-019-19965. [47] s. momani, o. abu arqub, a. freihat, et al. analytical approximations for fokker-planck equations of fractional order in multistep schemes, appl. comput. math. 15 (2016), 319-330. [48] m. al‐smadi, o. abu arqub, m. gaith, numerical simulation of telegraph and cattaneo fractional‐type models using adaptive reproducing kernel framework, math. meth. appl. sci. 44 (2020), 8472-8489. https://doi.org/10.1002/mma.6998. [49] r. abu-gdairi, m. al-smadi, g. gumah, an expansion iterative technique for handling fractional differential equations using fractional power series scheme, j. math. stat. 11 (2015), 29-38. https://doi.org/10.3844/jmssp.2015.29.38. https://doi.org/10.1186/s13662-019-2042-3 https://doi.org/10.1088/1572-9494/ab8a29 https://doi.org/10.1088/1402-4896/abb420 https://doi.org/10.1007/s11082-022-03722-8 https://doi.org/10.1088/1402-4896/abb739 https://doi.org/10.1007/s11253-018-1526-8 https://doi.org/10.1186/s13662-019-1996-5 https://doi.org/10.1186/s13662-019-1996-5 https://doi.org/10.1002/mma.6998 https://doi.org/10.3844/jmssp.2015.29.38 international journal of analysis and applications issn 2291-8639 volume 13, number 2 (2017), 161-169 http://www.etamaths.com on the composition and neutrix composition of the delta function and the function cosh−1(|x|1/r + 1) brian fisher1, emin ozcag2,∗ and fatma al-sirehy3 abstract. let f be a distribution in d′ and let f be a locally summable function. the composition f(f(x)) of f and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {fn(f(x))} is equal to h(x), where fn(x) = f(x) ∗ δn(x) for n = 1, 2, . . . and {δn(x)} is a certain regular sequence converging to the dirac delta function. it is proved that the neutrix composition δ(s)[cosh−1(x 1/r + + 1)] exists and δ(s)[cosh−1(x 1/r + + 1)] = − m−1∑ k=0 kr+r∑ i=0 (k i )(−1)i+krcr,s,k (kr + r)k! δ(k)(x), for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , where cr,s,k = i∑ j=0 (i j )(−1)kr+r−i(2j − i)s+1 2s+i+1 , m is the smallest integer for which s− 2r + 1 < 2mr and r ≤ s/(2m + 2). further results are also proved. 1. introduction let d be the space of infinitely differentiable functions with compact support, let d′ be the space of distributions defined on d. a sequence of functions {fn} is said to be regular if (i) fn is infinitely differentiable for all n, (ii) the sequence {〈fn,ϕ〉} converges to a limit 〈f,ϕ〉 for every ϕ ∈d, (iii) 〈f,ϕ〉 is continuous in ϕ in the sense that limn→∞〈fn,ϕ〉 = 0 for each sequence ϕn → 0 in d, see [24]. there are many ways to construct a sequence of regular functions which converges to δ(x). for instance let ρ be a fixed infinitely differentiable function having the properties: (i) ρ(x) = 0 for |x| ≥ 1, (ii) ρ(x) ≥ 0, (iii) ρ(x) = ρ(−x), (iv) ∫ 1 −1 ρ(x) dx = 1, putting δn(x) = nρ(nx) for n = 1, 2, . . . , it follows that {δn(x)} is a regular sequence of infinitely differentiable functions converging to the dirac delta-function δ(x). further, if f is a distribution in d′ and fn(x) = 〈f(x− t),δn(x)〉, then {fn(x)} is a regular sequence of infinitely differentiable functions converging to f(x). in the framework of the theory of distributions, no meaning can be generally given to expressions of the form f(f(x)) where f and f are arbitrary distributions. however, in elementary particle physics one finds the need to evaluate δ2(x) when calculating the transition rates of certain particle interactions, [14]. in addition, there are terms proportional to powers of the δ functions at the origin received 7th september, 2016; accepted 4th november, 2016; published 1st march, 2017. 2010 mathematics subject classification. 33b10, 46f30, 46f10, 41a30. key words and phrases. distribution; delta function; composition of distributions; neutrix composition of distributions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 161 162 fisher, ozcag and al-sirehy coming from the measure of path integration [10]. the composition of a distribution and an infinitely differentiable function is extended to distributions by continuity provided the derivative of the infinitely differentiable function is different from zero, [2]. the composition of a distribution and an infinitely differentiable function is extended to distributions by continuity provided the derivative of the infinitely differentiable function is different from zero, [2]. fisher [5] defined the composition of a distribution f and a summable function f which has a single simple root in the open interval (a,b), and it was recently generalized in [18] by allowing f to be a distribution. antosik [1] defined the composition g(f(x)) as the limit of the sequence {gn(fn)} providing the limit exists. by this definition he defined the compositions √ δ = 0, √ δ2 + 1 = 1 + δ, log(1 + δ) = 0, sin δ = 0, cos δ = 1 and 1 1+δ = 1. for many pairs of distributions, it is not possible to define their compositions by using the definition of antosik. using the neutrix calculus developed by van der corput [3], fisher gave a general principle for the discarding of unwanted infinite quantities from asymptotic expansions and this has been exploited in context of distributions, see [4, 5]. the technique of neglecting appropriately defined infinite quantities was devised by hadamard and the resulting finite value extracted from divergent integral is referred to as the hadamard finite part, see [16]. in fact his method can be regarded as a particular applications of the neutrix calculus. the following definition of the neutrix composition of distributions is a generalization of gel’fand and shilov’s definition of the composition involving the delta function [15], and was given in [5]. definition 1.1. let f be a distribution in d′ and let f be a locally summable function. we say that the neutrix composition f(f(x)) exists and is equal to h on the open interval (a,b), with −∞ < a < b < ∞, if n−lim n→∞ ∫ ∞ −∞ fn(f(x))ϕ(x)dx = 〈h(x),ϕ(x)〉 for all ϕ in d[a,b], where fn(x) = f(x) ∗ δn(x) for n = 1, 2, . . . and n is the neutrix, see [3], having domain n′ the positive and range n′′ the real numbers, with negligible functions which are finite linear sums of the functions nλ lnr−1 n, lnr n : λ > 0, r = 1, 2, . . . and all functions which converge to zero in the usual sense as n tends to infinity. in particular, we say that the composition f(f(x)) exists and is equal to h on the open interval (a,b) if lim n→∞ ∫ ∞ −∞ fn(f(x))ϕ(x)dx = 〈h(x),ϕ(x)〉 for all ϕ in d[a,b]. note that taking the neutrix limit of a function f(n), is equivalent to taking the usual limit of hadamard’s finite part of f(n), see [4, 6, 7, 16]. 2. main results by using fisher’s definition koh and li give meaning to δk and (δ′)k for k = 2, 3, . . . , see [17], and the more general form (δ(r))k was considered by kou and fisher in [18]. the meaning has been given to the symbol δk+ in [22] and the k-th powers of δ for negative integers were defined in [21]. recently, in [20] chenkuan li and changpin li used caputo fractional derivatives and definition 1.1 and chose the following δ−sequence δn(x) = (n π ) e−nx 2 (x ∈ r) to redefine powers of the distributions δk(x) and (δ′)k(x) for some values of k ∈ r. the following two theorems were proved in [6] and [7] respectively. theorem 2.1. the neutrix composition δ(s)(sgn x|x|λ) exists and δ(s)(sgn x|x|λ) = 0 composition δ(s)(cosh−1(|x|1/r + 1)) 163 for s = 0, 1, 2, . . . and (s + 1)λ = 1, 3, . . . and δ(s)(sgn x|x|λ) = (−1)(s+1)(λ+1)s! λ[(s + 1)λ− 1]! δ((s+1)λ−1)(x) for s = 0, 1, 2, . . . and (s + 1)λ = 2, 4, . . . . theorem 2.2. the compositions δ(2s−1)(sgn x|x|1/s) and δ(s−1)(|x|1/s) exist and δ(2s−1)(sgn x|x|1/s) = 1 2 (2s)!δ′(x), δ(s−1)(|x|1/s) = (−1)s−1δ(x) for s = 1, 2, . . . . the next theorem was proved in [9]. theorem 2.3. the neutrix composition δ(s)(sinh−1 x 1/r + ) exists and δ(s)(sinh−1 x 1/r + ) = m−1∑ k=0 kr+r−1∑ i=0 ( kr + r − 1 i ) (−1)i+kras,k,i 2kr+rk! δ(k)(x), for s = 0, 1, 2, . . . and r = 1, 2, . . . , where m is the smallest positive integer greater than (s−r + 1)/r and ar,s,k,i = (−1)s[(kr + r − 2i)s + (kr + r − 2i− 2)s] 2 . in particular, the neutrix composition δ(sinh−1 x 1/r + ) exists and δ(sinh−1 x 1/r + ) = 0, for r = 2, 3, . . . . in the following, we define the function δ(s)[cosh−1(x 1/r + + 1)] by δ(s)[cosh−1(x 1/r + + 1)] = { δ(s)[cosh−1(|x|1/r + 1)], x ≥ 0, 0, x < 0 and we define the function δ(s)[cosh−1(x 1/r − + 1)] by δ(s)[cosh−1(x 1/r − + 1)] = { δ(s)[cosh−1(|x|1/r + 1)], x ≤ 0, 0, x > 0 for r = 1, 2, . . . and s = 0, 1, 2, . . . . we also use the following easily proved lemma. lemma 2.1. ∫ 1 0 tiρ(s)(t) dt = { 0, 0 ≤ i < s, 1 2 (−1)ss!, i = s for s = 0, 1, 2, . . . . we now prove theorem 2.4. the neutrix composition δ(s)[cosh−1(x 1/r + + 1)] exists and δ(s)[cosh−1(x 1/r + + 1)] = m−1∑ k=0 kr+r∑ i=0 ( k i ) (−1)krcr,s,k (kr + r)k! δ(k)(x) (2.1) for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , where cr,s,k = i∑ j=0 ( i j ) (−1)kr+r+s−i(2j − i)s+1 2i+1 , m is the smallest integer for which s− 2r + 1 < 2mr and r ≤ s/(2m + 2). in particular, the neutrix composition δ[cosh−1(x+ + 1)] exists and δ[cosh−1(x+ + 1)] = 0 (2.2) 164 fisher, ozcag and al-sirehy for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(x+ + 1)] exists and δ′[cosh−1(x+ + 1)] = 1 4 δ(x). (2.3) proof. to prove equation (1), we first of all have to evaluate∫ 1 −1 δ(s)n [cosh −1(x 1/r + + 1)]x k dx = ns+1 ∫ 1 −1 ρ(s)[n cosh−1(x 1/r + + 1)]x k dx = ns+1 ∫ 1 0 ρ(s)[n cosh−1(x1/r + 1)]xk dx +ns+1 ∫ 0 −1 ρ(s)(0)xk dx = i1 + i2. (2.4) it is obvious that n−lim n→∞ i2 = n−lim n→∞ ns+1 ∫ 0 −1 ρ(s)(0)xk dx = 0, (2.5) for k = 0, 1, 2, . . . . making the substitution t = n cosh−1(x1/r + 1), we have for large enough n i1 = rn s ∫ 1 0 [cosh(t/n) − 1]kr+r−1 sinh(t/n)ρ(s)(t) dt = − rns+1 kr + r ∫ 1 0 [cosh(t/n) − 1]kr+rρ(s+1)(t) dt = − rns+1 kr + r kr+r∑ i=0 ( kr + r i ) (−1)kr+r−i ∫ 1 0 coshi(t/n)ρ(s+1)(t) dt = − rns+1 kr + r kr+r∑ i=0 ( kr + r i ) i∑ j=0 ( i j ) (−1)kr+r−i 2i ∫ 1 0 exp[(2j − i)t/n]ρ(s+1)(t) dt = − rns+1 kr + r kr+r∑ i=0 ( kr + r i ) i∑ j=0 ( i j ) ∞∑ m=0 (−1)kr+r−i(2j − i)m 2im!nm ∫ 1 0 tmρ(s+1)(t) dt. it follows that n−lim n→∞ i1 = r kr + r kr+r∑ i=0 ( kr + r i ) i∑ j=0 ( i j ) (−1)kr+r+s−i(2j − i)s+1 2i(s + 1)! ∫ 1 0 ts+1ρ(s+1)(t) dt = r kr + r kr+r∑ i=0 ( kr + r i ) i∑ j=0 ( i j ) (−1)kr+r+s−i(2j − i)s+1 2i+1 = r kr + r kr+r∑ i=0 ( kr + r i ) cr,s,k, (2.6) for k = 0, 1, 2, . . . . when k = m, we have |i1| ≤ rns+1 mr + r ∫ 1 0 ∣∣∣[cosh(t/n) − 1]mr+rρ(s+1)(t)∣∣∣dt ≤ rns+1 ∫ 1 0 [(t/n)2 + o(n−4)]mr+r|ρ(s+1)(t)|dt ≤ rns−2mr−2r+1 ∫ 1 0 [1 + o(n−4mr−4r)]|ρ(s+1)(t)|dt = o(ns−2mr−2r+1). composition δ(s)(cosh−1(|x|1/r + 1)) 165 thus, if ψ is an arbitrary continuous function, then lim n→∞ ∫ 1 0 δ(s)n [cosh −1(x 1/r + + 1)]x mψ(x) dx = 0, (2.7) since s− 2mr − 2r + 1 < 0. we also have ∫ 0 −1 δ(s)n [cosh −1(x 1/r + + 1)]ψ(x) dx = n s+1 ∫ 0 −1 ρ(s)(0)ψ(x) dx and it follows that n−lim n→∞ ∫ 0 −1 δ(s)n [(sinh −1 x+) 1/r]ψ(x) dx = 0. (2.8) if now ϕ is an arbitrary function in d[−1, 1], then by taylor’s theorem, we have ϕ(x) = m−1∑ k=0 ϕ(k)(0) k! xk + xm m! ϕ(m)(ξx), where 0 < ξ < 1, and so n−lim n→∞ 〈δ(s)n [cosh −1(x 1/r + + 1)],ϕ(x)〉 = n−lim n→∞ m−1∑ k=0 ϕ(k)(0) k! ∫ 1 0 δ(s)n [cosh −1(x 1/r + + 1)]x k dx + n−lim n→∞ m−1∑ k=0 ϕ(k)(0) k! ∫ 0 −1 δ(s)n [cosh −1(x 1/r + + 1)]x k dx + lim n→∞ 1 m! ∫ 1 0 δ(s)n [cosh −1(x 1/r + + 1)]x mϕ(m)(ξx) dx + lim n→∞ 1 m! ∫ 0 −1 δ(s)n [cosh −1(x 1/r + + 1)]x mϕ(m)(ξx) dx = m−1∑ k=0 kr+r∑ i=0 ( k i ) rcr,s,kϕ (k)(0) (kr + r)k! + 0 = m−1∑ k=0 kr+r∑ i=0 ( k i ) (−1)krcr,s,k (kr + r)k! 〈δ(k)(x),ϕ(x)〉, (2.9) on using equations (4) to (9). this proves equation (1) on the interval (−1, 1). it is clear that δ(s)[cosh−1(x 1/r + + 1)] = 0 for x > 0 and so equation (1) holds for x > 0. now suppose that ϕ is an arbitrary function in d[a,b], where a < b < 0. then∫ b a δ(s)n [cosh −1(x 1/r + + 1)]ϕ(x) dx = n s+1 ∫ b a ρ(s)(0)ϕ(x) dx and so n−lim n→∞ ∫ b a δ(s)n [cosh −1(x 1/r + + 1)]ϕ(x) dx = 0. it follows that δ(s)[cosh−1(x 1/r + + 1)] = 0 on the interval (a,b). since a and b are arbitrary, we see that equation (1) holds on the real line. to prove equation (2), we note that in this case s = 0 and so m = 0 for r = 1, 2, . . . . the sum in equation (1) is therefore empty and equation (2) follows. when r = s = 1 it follows that m = 1 and equation (3) then follows from equation (1). this completes the proof of the theorem. corollary 2.1. the neutrix composition δ(s)[cosh−1(x 1/r − + 1)] exists and δ(s)[cosh−1(x 1/r − + 1)] = m−1∑ k=0 kr+r∑ i=0 ( k i ) rcr,s,k (kr + r)k! δ(k)(x) (2.10) 166 fisher, ozcag and al-sirehy for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , in particular, the neutrix composition δ[cosh−1(x 1/r − + 1)] exists and δ[cosh−1(x 1/r − + 1)] = 0 (2.11) for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(x 1/r − + 1)] exists and δ′[cosh−1(x 1/r − + 1)] = 1 4 δ(x). (2.12) proof. equations (10) to (12) follow immediately on replacing x by −x in equations (1) to (3) respectively. corollary 2.2. the neutrix composition δ(s)[cosh−1(|x|1/r + 1)] exists and δ(s)[cosh−1(|x|1/r + 1)] = m−1∑ k=0 kr+r∑ i=0 ( k i ) [1 + (−1)k]rcr,s,k (kr + r)k! δ(k)(x) (2.13) for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , in particular, the neutrix composition δ[cosh−1(|x|1/r + 1)] exists and δ[cosh−1(|x|1/r + 1)] = 0 (2.14) for r = 1, 2, . . . and the neutrix composition δ′[cosh−1(|x|1/r + 1)] exists and δ′[cosh−1(|x|1/r + 1)] = 1 2 δ(x). (2.15) proof. equation (13) follows from equations (1) and (10) on noting that δ(s)[cosh−1(|x|1/r + 1)] = δ(s)[cosh−1(x1/r+ + 1)] + δ (s)[cosh−1(x 1/r − + 1)]. equations (14) to (15) follow similarly. theorem 2.5. the neutrix composition δ(s)[cosh−1(x+ + 1) 1/r] exists and δ(s)[cosh−1(x+ + 1) 1/r] = m−1∑ k=0 kr+r∑ i=0 ( kr + r i ) (−1)kbr,s,k k! δ(k)(x) (2.16) for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , where br,s,k = ri+r∑ j=0 ( ri + r j ) (−1)s+k−ir(2j −ri−r)s+1 2ri+r+1(ri + r) , m is the smallest integer for which s + 1 < 2mr and r ≤ (s + 1)/(2m). proof. to prove equation (16), we first of all have to evaluate∫ 1 −1 δ(s)n [cosh −1(x+ + 1) 1/r]xk dx = ns+1 ∫ 1 −1 ρ(s)[n cosh−1(x+ + 1) 1/r]xk dx = ns+1 ∫ 1 0 ρ(s)[n cosh−1(x+ + 1) 1/r]xk dx +ns+1 ∫ 0 −1 ρ(s)(0)xk dx = j1 + j2. (2.17) it is obvious that n−lim n→∞ j2 = n−lim n→∞ ns+1 ∫ 0 −1 ρ(s)(0)xk dx = 0, (2.18) for k = 0, 1, 2, . . . . composition δ(s)(cosh−1(|x|1/r + 1)) 167 making the substitution t = n cosh−1(x+ + 1) 1/r, we have for large enough n j1 = rn s ∫ 1 0 [coshr(t/n) − 1]k coshr−1(t/n) sinh(t/n)ρ(s)(t) dt = rns k∑ i=0 ( k i ) (−1)k−i ∫ 1 0 coshri+r−1(t/n) sinh(t/n)ρ(s)(t) dt = −rns+1 k∑ i=0 ( k i ) (−1)k−i ri + r ∫ 1 0 coshri+r(t/n)ρ(s+1)(t) dt = −rns+1 k∑ i=0 ( k i )ri+r∑ j=0 ( ri + r j ) (−1)k−i 2ri+r(ri + r) ∫ 1 0 exp[(2j −ri−r)t/n]ρ(s+1)(t) dt = −rns+1 kr+r∑ i=0 ( kr + r i )ri+r∑ j=0 ( ri + r j ) ∞∑ m=0 (−1)k−i(2j −ri−r)m 2ri+r(ri + r)m!nm ∫ 1 0 tmρ(s+1)(t) dt. it follows that n−lim n→∞ j1 = − kr+r∑ i=0 ( kr + r i )ri+r∑ j=0 ( ri + r j ) (−1)k−ir(2j −ri−r)s+1 2ri+r(ri + r)(s + 1)! ∫ 1 0 ts+1ρ(s+1)(t) dt = kr+r∑ i=0 ( kr + r i )ri+r∑ j=0 ( ri + r j ) (−1)s+k−ir(2j −ri−r)s+1 2ri+r+1(ri + r) = kr+r∑ i=0 ( kr + r i ) br,s,k, (2.19) for k = 0, 1, 2, . . . . when k = m, we have |j1| ≤ rns ∫ 1 0 ∣∣∣[coshr(t/n) − 1]m coshr−1(t/n) sinh(t/n)ρ(s)(t)∣∣∣ dt ≤ rns ∫ 1 0 ∣∣∣[(t/n)2r + o(n−4r)]m coshr−1(t/n) sinh(t/n)ρ(s)(t)∣∣∣ dt = o(ns−2mr−1). thus, if ψ is an arbitrary continuous function, then lim n→∞ ∫ 1 0 δ(s)n [cosh −1(x+ + 1) 1/r]xmψ(x) dx = 0, (2.20) since s− 2mr − 1 < 0. we also have ∫ 0 −1 δ(s)n [cosh −1(x+ + 1) 1/r]ψ(x) dx = ns+1 ∫ 0 −1 ρ(s)(0)ψ(x) dx and it follows that n−lim n→∞ ∫ 0 −1 δ(s)n [(sinh −1 x+) 1/r]ψ(x) dx = 0. (2.21) if now ϕ is an arbitrary function in d[−1, 1], then by taylor’s theorem, we have ϕ(x) = m−1∑ k=0 ϕ(k)(0) k! xk + xm m! ϕ(m)(ξx), 168 fisher, ozcag and al-sirehy where 0 < ξ < 1, and so n−lim n→∞ 〈δ(s)n [cosh −1(x+ + 1) 1/r],ϕ(x)〉 = = n−lim n→∞ m−1∑ k=0 ϕ(k)(0) k! ∫ 1 0 δ(s)n [cosh −1(x+ + 1) 1/r]xk dx + n−lim n→∞ m−1∑ k=0 ϕ(k)(0) k! ∫ 0 −1 δ(s)n [cosh −1(x+ + 1) 1/r]xk dx + lim n→∞ 1 m! ∫ 1 0 δ(s)n [cosh −1(x+ + 1) 1/r]xmϕ(m)(ξx) dx + lim n→∞ 1 m! ∫ 0 −1 δ(s)n [cosh −1(x+ + 1) 1/r]xmϕ(m)(ξx) dx = m−1∑ k=0 kr+r∑ i=0 ( kr + r i ) br,s,kϕ (k)(0) k! + 0 = m−1∑ k=0 kr+r∑ i=0 ( kr + r i ) (−1)kbr,s,k k! 〈δ(k)(x),ϕ(x)〉, (2.22) on using equations (17) to (22). this proves equation (16) on the interval (−1, 1). replacing x by −x in equation (16), we get corollary 2.3. the neutrix composition δ(s)[cosh−1(x− + 1) 1/r] exists and δ(s)[cosh−1(x− + 1) 1/r] = m−1∑ k=0 kr+r∑ i=0 ( kr + r i ) br,s,k k! δ(k)(x) (2.23) for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , corollary 2.4. the neutrix composition δ(s)[cosh−1(|x| + 1)1/r] exists and δ(s)[cosh−1(|x| + 1)1/r] = m−1∑ k=0 kr+r∑ i=0 ( kr + r i ) [1 + (−1)]kbr,s,k k! δ(k)(x) (2.24) for s = m − 1,m,m + 1, . . . and r = 1, 2, . . . , proof. equation (24) follows from equations (16) and (23) on noting that δ(s)[cosh−1(|x| + 1)1/r] = δ(s)[cosh−1(x+ + 1)1/r] + δ(s)[cosh−1(x− + 1)1/r]. for further related results on the neutrix composition of distributions, see [11], [12], [13], [19] and [23]. references [1] p. antosik, composition of distributions, technical report no.9 university of wisconsin, milwaukee, (1988-89), pp.1-30. [2] p. antosik, j. mikusinski and r. sikorski, theory of distributions, the sequential approach, pwn-elsevier, warszawa-amsterdam (1973). [3] j.g. van der corput, introduction to the neutrix calculus, j. analyse math., 7(1959), 291-398. [4] b. fisher, on defining the distribution δ(r)(f(x)), rostock. math. kolloq., 23(1983), 73-80. [5] b. fisher, on defining the change of variable in distributions, rostock. math. kolloq., 28(1985), 33-40. [6] b. fisher, the delta function and the composition of distributions, dem. math., 35(1)(2002), 117-123. [7] b. fisher, the composition and neutrix composition of distributions, in: kenan taş et al. (eds.), mathematical methods in engineering, springer, dordrecht, 2007, pp. 59-69. [8] b. fisher and b. jolevska-tuneska, two results on the composition of distributions, thai. j. math., 3(1)(2005), 17-26. [9] b. fisher and a. kılıçman, on the composition and neutrix composition of the delta function and powers of the inverse hyperbolic sine function, integral transforms spec. funct. 21(12)(2010), 935-944. [10] h. kleinert, a. chervyakov, rules for integrals over products of distributions from coordinate independence of path integrals, eur. phys. j. c part. fields 19(4)(2001), 743-747. composition δ(s)(cosh−1(|x|1/r + 1)) 169 [11] b. fisher, a. kananthai, g. sritanatana and k. nonlaopon, the composition of the distributions xms− ln x− and x r−p/m + , integral transforms spec. funct., 16(1)(2005), 13-20. [12] b. fisher and e. ozcag, some results on the neutrix composition of the delta function, filomat, 26(6)(2012), 1247-1256. [13] b. fisher and k. taş, on the composition of the distributions x−1 ln |x| and xr+, integral transforms spec. funct., 16(7)(2005), 533-543. [14] s. gasiorowics, elementery particle physics, john wiley and sons, new york, 1966. [15] i.m. gel’fand and g.e. shilov, generalized functions, volume i, academic press, new york and london, 1st edition, 1964. [16] d. s. jones, hadamard’s finite part., math. methods appl. sci. 19(13)(1996), 1017-1052. [17] e. l. koh and c. k. li, on distributions δk and (δ′)k, math. nachr., 157(1992), 243-248. [18] h. kou and b. fisher, on composition of distributions, publ. math. debrecen, 40(3-4)(1992), 279-290. [19] l. lazarova, b. jolevska-tuneska, i. akturk and e. ozcag note on the distribution composition (x µ +) λ. bull. malaysian math. soc., (2016). doi:10.1007/s40840-016-0342-2. [20] c. k. li and c. li, on defining the distributions δk and (δ′)k by fractional derivatives, appl. math. compt., 246(2014), 502-513. [21] e. ozcag, defining the k-th powers of the dirac-delta distribution for negative integers, appl. math. letters, 14(2001), 419-423. [22] e. ozcag, u. gulen and b. fisher, on the distribution δk+, integral transforms spec. funct., 9(2000), 57-64. [23] e. ozcag, l. lazarova and b. jolevska-tuneska, defining compositions of x µ +, |x| µ,x−s and x−s ln |x| as neutrix limit of regular sequences, commun. math. stat., 4(1)(2016), 63-80. [24] g. temple, the theory of generalized functions, proc. roy. soc. ser. a 28(1955), 175-190. 1department of mathematics and computer science, leicester university, england 2department of mathematics, hacettepe university, ankara, turkey 3department of mathematics, jeddah, king abdulaziz university, jeddah, saudi arabia ∗corresponding author: ozcag1@hacettepe.edu.tr 1. introduction 2. main results references int. j. anal. appl. (2022), 20:22 application of the f-expansion method for solving the fokas-lenells equation ohoud a. alshahrani∗ department of mathematics, faculty of sciences, university of tabuk, p.o.box 741, tabuk 71491, saudi arabia ∗corresponding author: ohoud1972@yahoo.com, ahoud.ksa@hotmail.com abstract. by the aid of traveling wave hypothesis, the f-expansion method has been implemented in this paper to obtain jacobian-elliptic function solutions for the optical fokas-lenells model. the hyperbolic-function solutions are derived as special cases from the jacobian-elliptic function solutions. the present approach is straightforward to determine the exact solutions for the fokas-lenells equation. the existence criteria of the obtained solutions are also reported. 1. introduction in the field of telecommunications engineering, the optical soliton perturbation and the pdes are the most active areas of research [1-15]. the dynamics of soliton molecules are administered by a variety of nonlinear evolution equations. the nonlinear schrödinger’s equation is the most studied model in this context. although the nonlinear schrödinger’s model has been extensively studied by many authors with different forms of nonlinearity, the present work analyzes the pulse propagation engineering through optical fibers and pcf with a newly established model known as the fokas– lenells equation (fle) [12, 13]. such model has been studied to obtain various kinds of soliton solutions by using the complex–amplitude ansatz and other approaches [12, 13]. the objective of this paper is to apply the f-expansion method by the aid of the traveling wave hypothesis to deduct more general solution as well as soliton solutions. the paper is organized as follows. in the next subsection, the fle is presented. in section 2, the traveling wave hypothesis is introduced. section 3 is devoted to the application of the f-expansion method on the current model. the exact solutions in terms of jacobian-elliptic functions are displayed in section 4. the hyperbolic-function solutions are derived received: feb. 4, 2022. 2010 mathematics subject classification. 78a60, 37k10, 35q51, 35q55. key words and phrases. optics; fokas–lenells equation; solitons. https://doi.org/10.28924/2291-8639-20-2022-22 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-22 2 int. j. anal. appl. (2022), 20:22 from the jacobian-elliptic function solutions in section 5. the paper is ended by a conclusion section 6 and two appendices. 1.1. governing model. the dimensionless form of perturbed fle that has been proposed takes the form [16, 17]: iqt +a1qxx +a2qxt + |q|2 (b1q + iσqx)= i [ αqx +λ ( |q|2m q ) x +µ ( |q|2m ) x q ] , (1) where the right hand side represents all the perturbation terms. in equation (1), the independent variables are x and t that represent spatial and temporal variables respectively while q(x,t) is the complex–valued wave profile representing the soliton profile. here, a1 is the coefficient of group velocity dispersion while a2 is the coefficient of spatio–temporal dispersion that was proposed to be included a few years ago. then, σ is the coefficient of nonlinear dispersion. on the right hand side, α is the coefficient of inter–modal dispersion while λ accounts for self–steepening effect and finally µ gives another form of nonlinear dispersion. the parameter b1 indicates the effect of self-phase modulation while the parameter m refers to the full nonlinearity. 2. traveling wave hypothesis the solutions of (1) may be supposed as q(x,t)= eiθ(x,t) u(ω), (2) where ω = x −γt and the phase θ(x,t) = −kx +βt + θ0, u(ω) is the amplitude component of the wave and γ is its speed. k is the soliton frequency, β is its wave-number and θ0 is the phase constant. equation (1) can be decomposing into real and imaginary parts yields a pair of relations. the real and imaginary parts of eq. (1) are respectively (a1 −a2)u′′ +(a2ωk −a1k2 −αk −ω)u +(b1 +kσ)u3 −ku[(2m+1)λ+2mµ]u2m =0, (3) and γ +2ka1 −a2(γk +ω)−σu2 +α+[(2m+1)λ+2mµ]u2m =0. (4) we notice from (4) that [λ(2m+1)+2mµ]u2m =−γ −2ka1 +a2(γk +ω)+σu2 −α. (5) setting (2m+1)λ+2mµ =0, σ =0, (6) in (5), then λ = −2mµ (2m+1) , (7) int. j. anal. appl. (2022), 20:22 3 and γ = 2ka1 −a2ω +α (a2k −1) . (8) accordingly, eq. (3) reduces to (a1 −a2γ)u′′ +(a1k2 −ω +kγ(1−a2k))u +b1u3 =0. (9) using n1 =(a1 −a2γ), n2 =(a1k2 −ω +kγ(1−a2k)), (10) hence (9) gives n1u ′′ +n2u +b1u 3 =0. (11) 3. application the f-expansion method to the fle assume that the solution of (11) is in the form [18] u(ω)= n∑ i=0 aif i(ω), (12) where ai, i = 0,1,2, . . . ,n, are constants to be determined, n is a positive integer which can be evaluated by balancing the highest-order linear term u ′′ and nonlinear term u3, this gives n = 1. moreover, f(ω) satisfies the following auxiliary equation f ′(ω)=± √ pf4(ω)+qf2(ω)+r, (13) where p, q, and r are constants. eq. (13) for f(ω) leads to  f ′′ =2pf3 +qf, f ′′′ =(6pf2 +q)f ′, f ′′′′ =24p2f5 +20pqf3 +(12pr+q2)f . . . (14) in appendix a, we present 46 forms of exact solutions for eq. (13), (see ref. [18] for details). in fact, these exact solutions can be used to construct more exact solutions for eq. (11). according to n =1, eq. (12) becomes u(ω)= a0 +a1f(ω). (15) 4 int. j. anal. appl. (2022), 20:22 substituting (15) into (11), we obtain the following system of algebraic equations  b1a 3 0 +a0n2 =0, 3b1a 2 0a1 +qa1n1 +a1n2 =0, 3b1a 2 1a0 =0, b1a 3 1 +2pa1n1 =0. (16) solving the last system (16), we have a0 =0, a1 =± √ −2pn1 b1 , q =− n2 n1 . (17) with these values of a0, a1 and q, the exact solution of (11) can be obtained by 46 forms depending on the values of p, q and r with the corresponding solution f(ω) in appendix a. however, some selected cases are presented in the following sections. 4. jacobian-elliptic function solutions case: 1: let us consider the inputs of case 1 in appendix a with implementing (15) and (17), thus  p = m2, q =−n2 n1 =−(1+m2), r =1, f(ω)= snω, u =± √ −2n1m2 b1 snω, n2 =(1+m2)n1, n1 < 0, q(x,t)=± √ −2n1m2 b1 ei (−kx+βt+θ0) sn(x −γt). (18) this general solution has not been reported in [16, 17]. moreover, it will be demonstrated in section 5 that the solution (18) and other solutions of this section reduce to hyperbolic forms as special cases. case: 2: the solution of this case (case 2 in appendix a) is expressed in terms of another kind of jacobianelliptic functions as   p = m2, q =−n2 n1 =−(1+m2), r =1, f(ω)= cdω, u =± √ −2n1m2 b1 cdω, n2 =(1+m2)n1, n1 < 0, q(x,t)=± √ −2n1m2 b1 ei (−kx+βt+θ0) cd(x −γt). (19) which was not also reported in refs. [16, 17]. int. j. anal. appl. (2022), 20:22 5 case: 3: the solution of this case (by using the inputs of case 26 in appendix a) is expressed as  p > 0, q =−n2 n1 < 0, r = m 2q2 (1+m2)2p , f(ω)= √ −m2q (1+m2)p sn (√ −q 1+m2 ω ) , u =± √ −2m2n2 (1+m2)b1 sn (√ n2 (1+m2)n1 ω ) , n2n1 > 0, , n2b1 < 0, q(x,t)=± √ −2m2n2 (1+m2)b1 ei (−kx+βt+θ0) sn (√ n2 (1+m2)n1 (x −γt) ) . (20) case: 4: the solution of this case (by using the inputs of case 27 in appendix a) is expressed as  p < 0, q =−n2 n1 > 0, r = (1−m2)q2 (m2−2)2p , f(ω)= √ −q (2−m2)p dn (√ q 2−m2 ω ) , u =± √ 2n2 (2−m2)b1 dn (√ −n2 (2−m2)n1 ω ) , n2n1 < 0, , n2b1 > 0, q(x,t)=± √ 2n2 (2−m2)b1 ei (−kx+βt+θ0) dn (√ −n2 (2−m2)n1 (x −γt) ) . (21) which was not reported in refs. [16, 17]. case: 5: on using the inputs of case 28 in appendix a, we have  p < 0, q =−n2 n1 > 0, r = m2(m2−1)q2 (2m2−1)2p , f(ω)= √ −m2q (2m2−1)p cn (√ q 2m2−1 ω ) , u =± √ 2m2n2 (2m2−1)b1 cn (√ −n2 (2m2−1)n1 ω ) , n2n1 < 0, , n2b1 > 0, q(x,t)=± √ 2m2n2 (2m2−1)b1 ei (−kx+βt+θ0) cn (√ −n2 (2m2−1)n1 (x −γt) ) . (22) 5. hyperbolic-function solutions some soliton−like solutions of eq. (11) can be obtained in the limited case when the modulus m → 1 (see appendix b), as: case: 1:   p =1, q =−n2 n1 =−2, r =1, f(ω)= tanhω, u =± √ −2n1 b1 tanhω, n2 =2n1, n1 < 0, q(x,t)=± √ −2n1 b1 ei (−kx+βt+θ0) tanh(x −γt). (23) 6 int. j. anal. appl. (2022), 20:22 case: 2:   p =1, q =−n2 n1 =−2, r =1, f(ω)=1, u =± √ −2n1 b1 , n2 =2n1, n1 < 0, q(x,t)=± √ −2n1 b1 ei (−kx+βt+θ0). (24) case: 3:   p > 0, q =−n2 n1 < 0, r = q 2 4p , f(ω)= √ −q 2p tanh (√ −q 2 ω ) , u =± √ −n2 b1 tanh (√ n2 2n1 ω ) , n2n1 > 0, , n2b1 < 0, q(x,t)=± √ −n2 b1 ei (−kx+βt+θ0) tanh (√ n2 2n1 (x −γt) ) . (25) cases: 4:   p < 0, q =−n2 n1 > 0, r =0, f(ω)= √ −q p sech (√ q ω ) , u =± √ 2n2 b1 sech (√ −n2 n1 ω ) , n2n1 < 0, , n2b1 > 0, q(x,t)=± √ 2n2 b1 ei (−kx+βt+θ0) sech (√ −n2 n1 (x −γt) ) . (26) case: 5: this case leads to the same hyperbolic-function solution given in (26). here, it should be noted that the solutions presented in the previous section in terms of the jacobianelliptic function are more general than those previously obtained in the relevant literature. in addition, the obtained hyperbolic-function solutions were derived as special cases from our jacobian-elliptic function solutions. moreover, some of the present solutions have not been reported in previous works [16, 17] which analyzed the same fokas-lenells equation. as a final observation is that all of the current solutions are obtained by using only one method, however, three different methods have been applied in [16, 17] to obtain only three solutions. finally, several kinds of soliton solutions such as singular soliton solution and dark-singular combo soliton solution can be derived by considering more inputs of the 46 cases in appendix a with the aid of appendix b. 6. conclusions this paper revealed new types of exact solutions for the perturbed fle, where the perturbation terms are of hamiltonian type and appeared with full nonlinearity. the f-expansion method was applied in this paper to obtain several kinds of jacobian-elliptic function solutions for the optical fokas-lenells model. in special cases, the solito-like solutions in terms of the hyperbolic-functions are int. j. anal. appl. (2022), 20:22 7 derived from the jacobian-elliptic function solutions. the results have not been reported in previous works in relevant literatures. several kinds of soliton solutions such as singular soliton solution and dark-singular combo soliton solution can be derived by further investigations of the suggested method. appendix a relations between values of (p, q, r) and corresponding f(ω) in eq. (13), where a, b and c are arbitrary constants and m1 = √ 1−m2. case p q r f(ω) 1 m2 −(1+m2) 1 snω 2 m2 −(1+m2) 1 cdω=cnω/dnω 3 −m2 2m2 −1 1−m2 cnω 4 −1 2−m2 m2 −1 dnω 5 1 −(1+m2) m2 nsω =(snω)−1 6 1 −(1+m2) m2 dcω = dnω/cnω 7 1−m2 2m2 −1 −m2 ncω =(cnω)−1 8 m2 −1 2−m2 −1 ndω =(dnω)−1 9 1−m2 2−m2 1 scω = snω/cnω 10 −m2(1−m2) 2m2 −1 1 sdω = snω/dnω 11 1 2−m2 1−m2 csω = cnω/snω 12 1 2m2 −1 −m2(1−m2) dsω = dnω/snω 13 1/4 (1−2m2)/2 1/4 nsω ±csω 14 (1−m2)/4 (1+m2)/2 (1−m2)/4 ncω ± scω 15 1/4 (m2 −2)/2 m2/4 nsω ±dsω 16 m2/4 (m2 −2)/2 m2/4 snω ± icnω 17 m2/4 (m2 −2)/2 m2/4 √ 1−m2sdω ±cdω 18 1/4 (1−m2)/2 1/4 m cdω ± i √ 1−m2ndω 19 1/4 (1−2m2)/2 1/4 m snω ± idnω 20 1/4 (1−m2)/2 1/4 √ 1−m2scω ±dcω 21 (m2 −1)/4 (m2 +1)/2 (m2 −1)/4 m sdω ±ndω 22 m2/4 (m2 −2)/2 1/4 snω 1±dnω 23 −1/4 (m2 +1)/2 (1−m2)2/4 m cnω ±dnω 24 (1−m2)2/4 (m2 +1)/2 1/4 dsω ±csω 25 m 4(1−m2) 2(2−m2) 2(1−m2) m2−2 1−m2 2(2−m2) dcω ± √ 1−m2ncω 26 p > 0 q < 0 m 2q2 (m2+1)2p √ −m2q (m2+1)p sn (√ −q m2+1 ω ) 27 p < 0 q > 0 (1−m 2)q2 (m2−2)2p √ −q (2−m2)p dn (√ q 2−m2ω ) 8 int. j. anal. appl. (2022), 20:22 28 p < 0 q > 0 m 2(m2−1)q2 (2m2−1)2p √ − m2q (2m2−1)p cn (√ q 2m2−1ω ) 29 1 2−4m2 1 snωdnωcnω 30 m4 2 1 snωcnωdnω 31 1 m2 +2 1−2m2 +m4 dnωcnωsnω 32 a 2(m−1)2 4 m2+1 2 +3m (m−1)2 4a2 dnωcnω a(1+snω)(1+msnω) 33 a 2(m+1)2 4 m2+1 2 −3m (m+1) 2 4a2 dnωcnω a(1+snω)(1−msnω) 34 − 4 m 6m−m2 −1 −2m3 +m4 +m2 mcnωdnω msn2ω+1 35 4 m −6m−m2 −1 2m3 +m4 +m2 mcnωdnω msn2ω−1 36 1/4 1−2m 2 2 1/4 sn 1±cnω 37 1−m 2 4 1+m2 2 1−m2 4 cnω 1±snω 38 4m1 2+6m1 −m2 2+2m1 −m2 m 2snωcnω m1−dn2ω 39 −4m1 2−6m1 −m2 2−2m1 −m2 −m 2snωcnω m1+dn2ω case p q r f(ω) 40 2−m 2−2m1 4 m2 2 −1−3m1 2−m 2−2m1 4 m2snωcnω sn2ω+(1+m1)dnω−1−m1 41 2−m 2+2m1 4 m2 2 −1+3m1 2−m 2+2m1 4 m2snωcnω sn2ω+(−1+m1)dnω−1+m1 42 c 2m4−(b2+c2)m2+b2 4 m2+1 2 m2−1 4(c2m2−b2) √ (b2−c2) (b2−c2m2) +snω b cnω+c dnω 43 b 2+c2m2 4 1 2 −m2 1 4(c2m2+b2) √ (c2m2+b2−c2) (b2+c2m2) +cnω b snω+c dnω 44 b 2+c2 4 m2 2 −1 m 4 4(c2+b2) √ (b2+c2−c2m2) (b2+c2) +dnω b snω+c cnω 45 −(m2 +2m+1)b2 2m2 +2 2m−m 2−1 b2 m sn2ω−1 b(m sn2ω+1) 46 −(m2 −2m+1)b2 2m2 +2 −2m+m 2+1 b2 m sn2ω+1 b(m sn2ω−1) appendix b the jacobi−elliptic functions degenerate into hyperbolic functions when m → 1 as: snω → tanhω, {cnω, dnω}→ sechω, {scω, sdω}→ sinhω, {dsω, csω}→ cschω, {ncω, ndω}→ coshω, nsω → cothω, {cdω, dcω}→ 1. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. biswas, 1-soliton solution of 1+2 dimensional nonlinear schrödinger’s equation in power law media, commun. nonlinear sci. numer. simul. 14 (2009), 1830–1833. https://doi.org/10.1016/j.cnsns.2008.08.003. [2] a. biswas, d. milovic, travelling wave solutions of the non-linear schrödinger’s equation in non-kerr law media, commun. nonlinear sci. numer. simul. 14 (2009), 1993–1998. https://doi.org/10.1016/j.cnsns.2008.04. 017. https://doi.org/10.1016/j.cnsns.2008.08.003 https://doi.org/10.1016/j.cnsns.2008.04.017 https://doi.org/10.1016/j.cnsns.2008.04.017 int. j. anal. appl. (2022), 20:22 9 [3] s.h. crutcher, a.j. osei, a. biswas, wobbling phenomena with logarithmic law nonlinear schrödinger equations for incoherent spatial gaussons, optik. 124 (2013), 4793–4797. https://doi.org/10.1016/j.ijleo.2013.01.081. [4] m. eslami, m. mirzazadeh, b. fathi vajargah, a. biswas, optical solitons for the resonant nonlinear schrödinger’s equation with time-dependent coefficients by the first integral method, optik. 125 (2014), 3107–3116. https: //doi.org/10.1016/j.ijleo.2014.01.013. [5] a.j. mohamad jawad, m.d. petković, a. biswas, modified simple equation method for nonlinear evolution equations, applied mathematics and computation. 217 (2010), 869–877. https://doi.org/10.1016/j.amc.2010.06.030. [6] a.a. gaber, a.f. aljohani, a. ebaid, j.t. machado, the generalized kudryashov method for nonlinear space–time fractional partial differential equations of burgers type, nonlinear dyn. 95 (2019), 361–368. https://doi.org/ 10.1007/s11071-018-4568-4. [7] a.a. alqarni, a. ebaid, a.a. alshaery, h.o. bakodah, a. biswas, s. khan, m. ekici, q. zhou, s.p. moshokoa, m.r. belic, optical solitons for lakshmanan–porsezian–daniel model by riccati equation approach, optik. 182 (2019), 922–929. https://doi.org/10.1016/j.ijleo.2019.01.057. [8] y.m. mahrous, s.m. khaled, a. ebaid, an internet traffic flow model via a conformable derivative: the exact soliton solutions, adv. differ. equ. control processes. 21 (2019), 227–237. https://doi.org/10.17654/de021020227. [9] d.a. lott, a. henriquez, b.j.m. sturdevant, a. biswas, a numerical study of optical soliton-like structures resulting from the nonlinear schrödinger’s equation with square-root law nonlinearity, appl. math. comput. 207 (2009), 319–326. https://doi.org/10.1016/j.amc.2008.10.038. [10] m. mirzazadeh, m. eslami, b.f. vajargah, a. biswas, optical solitons and optical rogons of generalized resonant dispersive nonlinear schrödinger’s equation with power law nonlinearity, optik. 125 (2014), 4246–4256. https: //doi.org/10.1016/j.ijleo.2014.04.014. [11] h.o. bakodah, m.a. banaja, b.a. alrigi, a. ebaid, r. rach, an efficient modification of the decomposition method with a convergence parameter for solving korteweg de vries equations, j. king saud univ. sci. 31 (2019), 1424–1430. https://doi.org/10.1016/j.jksus.2018.11.010. [12] h. triki, a.-m. wazwaz, combined optical solitary waves of the fokas—lenells equation, waves rand. complex media. 27 (2017), 587–593. https://doi.org/10.1080/17455030.2017.1285449. [13] h. triki, a.-m. wazwaz, new types of chirped soliton solutions for the fokas–lenells equation, int. j. numer. methods heat fluid flow. 27 (2017), 1596–1601. https://doi.org/10.1108/hff-06-2016-0252. [14] b. salah, e.r. el-zahar, a.f. aljohani, a. ebaid, m. krid, optical soliton solutions of the time-fractional perturbed fokas-lenells equation: riemann-liouville fractional derivative, optik. 183 (2019), 1114–1119. https://doi.org/ 10.1016/j.ijleo.2019.02.016. [15] a. ebaid, e.r. el-zahar, a.f. aljohani, b. salah, m. krid, j.t. machado, exact solutions of the generalized nonlinear fokas-lennells equation, results phys. 14 (2019), 102472. https://doi.org/10.1016/j.rinp.2019.102472. [16] a.j. mohamad jawad, a. biswas, q. zhou, s.p. moshokoa, m. belic, optical soliton perturbation of fokas–lenells equation with two integration schemes, optik. 165 (2018), 111–116. https://doi.org/10.1016/j.ijleo.2018. 03.104. [17] a.f. aljohani, e.r. el-zahar, a. ebaid, m. ekici, a. biswas, optical soliton perturbation with fokas-lenells model by riccati equation approach, optik. 172 (2018), 741–745. https://doi.org/10.1016/j.ijleo.2018.07.072. [18] a. ebaid, e.h. aly, exact solutions for the transformed reduced ostrovsky equation via the -expansion method in terms of weierstrass-elliptic and jacobian-elliptic functions, wave motion. 49 (2012), 296–308. https://doi. org/10.1016/j.wavemoti.2011.11.003. https://doi.org/10.1016/j.ijleo.2013.01.081 https://doi.org/10.1016/j.ijleo.2014.01.013 https://doi.org/10.1016/j.ijleo.2014.01.013 https://doi.org/10.1016/j.amc.2010.06.030 https://doi.org/10.1007/s11071-018-4568-4 https://doi.org/10.1007/s11071-018-4568-4 https://doi.org/10.1016/j.ijleo.2019.01.057 https://doi.org/10.17654/de021020227 https://doi.org/10.1016/j.amc.2008.10.038 https://doi.org/10.1016/j.ijleo.2014.04.014 https://doi.org/10.1016/j.ijleo.2014.04.014 https://doi.org/10.1016/j.jksus.2018.11.010 https://doi.org/10.1080/17455030.2017.1285449 https://doi.org/10.1108/hff-06-2016-0252 https://doi.org/10.1016/j.ijleo.2019.02.016 https://doi.org/10.1016/j.ijleo.2019.02.016 https://doi.org/10.1016/j.rinp.2019.102472 https://doi.org/10.1016/j.ijleo.2018.03.104 https://doi.org/10.1016/j.ijleo.2018.03.104 https://doi.org/10.1016/j.ijleo.2018.07.072 https://doi.org/10.1016/j.wavemoti.2011.11.003 https://doi.org/10.1016/j.wavemoti.2011.11.003 1. introduction 1.1. governing model 2. traveling wave hypothesis 3. application the f-expansion method to the fle 4. jacobian-elliptic function solutions 5. hyperbolic-function solutions 6. conclusions appendix a appendix b references international journal of analysis and applications volume 18, number 6 (2020), 1083-1107 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1083 coupled coincidence point for f(ψ,ϕ)−contractions via generalized α−admissible mappings with an application dhekra m. albaqeri1, hasanen a. hammad2,∗, manuel de la sen3 1department of mathematics, faculty of education, sanaa university, sanaa 1247, yemen 2department of mathematics, faculty of science, sohag university, sohag 82524, egypt 3institute of research and development of processes university of the basque country 48940leioa (bizkaia), spain ∗corresponding author: hassanein hamad@science.sohag.edu.eg abstract. the main objective of this manuscript is to discuss some coupled coincidence point (ccp) results for generalized α− admissible mappings which are f(ψ,ϕ)− contractions in the context of b−metric spaces (b-ms). also, an example to support the obtained theoretical theorems is derived. ultimately, an analytical solution for nonlinear integral equation (nie) is discussed as an application. 1. introduction and elementary discussions fixed point techniques plays an enormous role in many applications of mathematics. during the past thirty years various extension of a metric space have been discussed. the banach contraction principle is a popular tool helps to solve problems in nonlinear analysis. a number of publications are interested to the study and solutions of many practical and theoretical problems by using this principle [1–8]. bakhtin [9] in 1993 and czerwik [10] in 1998 introduced the concept of (b-ms). since then, several papers have been published on the fixed point theory of both classes of single-valued and multi-valued operators in (b-ms). [11], [12], [13–16]. received september 13th, 2020; accepted october 15th, 2020; published november 4th, 2020. 2010 mathematics subject classification. 46n40, 47h10, 46t99. key words and phrases. coupled coincidence point; generalized α− admissible mapping; b−metric spaces; nonlinear integral equations. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1083 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1083 int. j. anal. appl. 18 (6) (2020) 1084 definition 1.1. [10] let γ be a nonempty set and s ≥ 1 be a given real number. a function νb : γ × γ → [0,∞) is a b-metric (b-m) iff, for all e,r,ζ ∈ υ , the stipulations below are fulfilled: (b1) νb(e,r) = 0 ⇔ e = r; (b2) νb(e,r) = νb(r,e); (b3) νb(e,r) ≤ s[νb(e,ζ) + νb(ζ,r)]. the pair (γ,νb) is called a (b-ms) with a constant s ≥ 1. example 1.1. let γ = [0,∞). define the function νb : υ2 → [0,∞) by νb(e,r) = (e−r)2. then (γ,νb) is a (b-ms) with a constant s = 2. definition 1.2. [10] suppose that (γ,νb) is a (b-ms). so a sequence {en} in γ is called: (i) convergent if there is e ∈ γ so that νb(en,e) → 0 as n →∞. (ii) cauchy sequence iff limm,n→∞νb(em,en) → 0 as n,m →∞. (iii) the pair (γ,νb) is called a complete iff every cauchy sequence {en} in γ converges to e ∈ γ. lemma 1.1. [11] let (γ,νb) be a (b-ms) with a coefficient s ≥ 1, {en} and {rn} be a convergent to points e,r ∈ γ, respectively. then we have 1 s2 νb(e,r) ≤ lim inf n→∞ νb(en,rn) ≤ lim sup n→∞ νb(en,rn) ≤ s2νb(e,r). in particular, if e = r, then limn→∞νb(en,rn) = 0. moreover, for each δ ∈ γ, we have 1 s νb(e,δ) ≤ lim inf n→∞ νb(en,δ) ≤ lim sup n→∞ νb(en,δ) ≤ sνb(e,δ). lemma 1.2. [12] let {en} be a sequence in a (b-ms) (γ,νb) so that νb(en,en+1) ≤ λνb(en−1,en), for some λ, 0 < λ < 1 s , and for each n ∈ n. then {en} is a cauchy sequence in γ. the idea of coupled fixed point initiated and studied by guo and lakshmikantham [17]. after that, the monotone property is studied by bhaskar and lakshmikantham [18]. many works are made to generalized this concept in various spaces under certain conditions, the reader can shed light on [19–23, 25, 26]. definition 1.3. [18] an element (e,r) ∈ υ × υ is called a (ccp) of the mappings υ : γ × γ → γ and λ : γ → γ if υ(e,r) = λe and υ(r,e) = λr. definition 1.4. [27] an element (e,r) ∈ γ × γ is called a (ccp) of mappings υ, λ : γ × γ → γ if υ(e,r) = λ(e,r) and υ(r,e) = λ(r,e). example 1.2. let υ, λ : r×r → r be defined by υ(e,r) = λr and λ(e,r) = 2 3 (e + r) for all (e,r) ∈ γ×γ . note that (0, 0), (1, 2) and (2, 1) are (ccp) of υ and λ . int. j. anal. appl. 18 (6) (2020) 1085 definition 1.5. [27] let υ, λ : γ × γ → γ. we say that the pair (υ, λ) is generalized compatible if νb(υ(λ(en,rn), λ(rn,en)), λ(υ(en,rn), υ(rn,en))) → 0 as n →∞, νb(υ(λ(rn,en), λ(en,rn)), λ(υ(rn,en), υ(en,rn))) → 0 as n →∞, whenever {en} and {rn} are sequences in γ such that for all t1, t2 ∈ γ, we have lim n→∞ υ(en,rn) = lim n→∞ λ(en,rn) = t1, lim n→∞ υ(rn,en) = lim n→∞ λ(rn,en) = t2. definition 1.6. [28] let υ : γ → γ and α : γ×γ → [0, +∞). we say that υ is an α− admissible mapping if α(e,r) ≥ 1 implies α(υe, υr) ≥ 1 for all e,r ∈ υ. ansari [29] initiated the remarkable of c−class functions. this contribution covers a large class of contractive conditions. here, we denote c-class functions as c. definition 1.7. [29] a c-class function is a continuous mapping f : [0,∞)2 → r which fulfills the stipulations below: (1) f(e,r) ≤ e, (2) f(e,r) = e implies that either e = 0 or r = 0 for all e,r ∈ [0,∞). example 1.3. the functions below f : [0,∞)2 → r are elements of c, for all e,r ∈ [0,∞) : (1) f(e,r) = e−r; (2) f(e,r) = λe, 0 < λ < 1; (3) f(e,r) = e (1+r)σ ; σ ∈ (0,∞); (4) f(e,r) = log(r+ae) (1+r) ,a > 1. here in this manuscript, we refers to: • ψ = {ψ : ψ : [0,∞) → [0,∞) is a strictly nondecrasing and continuous function, ψ(t) = 0 ⇔ t = 0}. • an ultra altering distance function φ = {ϕ : ϕ : [0,∞) → [0,∞)} is a continuous, non-decreasing mapping such that ϕ(t) > 0 for t > 0 and ϕ(0) ≥ 0. the goal of this paper is to obtain some new (ccp) results for a certain class of f(ψ,ϕ)contractive via generalized α− admissible mappings in (b-ms). ultimately, to support our work we present an example and application to find an analytical solution to the (nie). 2. main results we begin this part with the definition below: int. j. anal. appl. 18 (6) (2020) 1086 definition 2.1. let υ, λ : γ2 → γ and α : γ2×γ2 → r+ be given mappings. we say that υ is a generalized α-admissible with respect to (w.r.t.) λ if α((λ(e,r), λ(r,e)), (λ(µ,κ), λ(κ,µ))) ≥ 1 implies α((υ(e,r), υ(r,e)), (υ(µ,κ), υ(κ,µ))) ≥ 1, for all (e,r), (µ,κ) ∈ γ2. now, we present the first main theorem: theorem 2.1. let (γ,νb) be a complete (b-ms) (with parameter s > 1), and υ, λ : γ 2 → γ be two generalized compatible mappings such that υ is a generalized α−admissible mapping w.r.t. λ and λ is continuous. let there is f ∈ c, ψ ∈ ψ,ϕ ∈ φ so that the stipulation below holds (ψ (sσνb(υ(e,r), υ(µ,κ)) + ρ) α   (λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))  α   (λ(µ,κ), λ(κ,µ)), (υ(µ,κ), υ(κ,µ))   ≤ f   ψ(νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ))2 ), ϕ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 )   +ρ,(2.1) for all e,r,µ,κ ∈ γ,σ,ρ > 0, and α : (γ2 × γ2) → [0,∞). assume that (i) υ(γ2) ⊆ λ(γ2), (ii) there is e0,r0 ∈ γ so that α((λ(e0,r0), λ(r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α((λ(r0,e0), λ(e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. also, suppose either (iv) υ is continuous, or (v) {en},{rn} are two sequences in γ so that α((en+1,rn+1), (en,rn)) ≥ 1 and α((rn,en), (rn+1,en+1)) ≥ 1. for all n ∈ n∪{0}, and en → e,rn → r as n →∞, e,r ∈ γ, we have α((e,r), (en,rn)) ≥ 1 and α((rn,en), (r,e)) ≥ 1. then υ and λ have a (ccp). proof. let e0,r0 ∈ γ, so by condition (ii), we have α((λ(e0,r0), λ(r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α((λ(r0,e0), λ(e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. int. j. anal. appl. 18 (6) (2020) 1087 according to (i), define two sequences {en},{rn} in γ by υ(en,rn) = λ(en+1,rn+1), υ(rn,en) = λ(rn+1,en+1),∀n = 0, 1, 2, ... . since υ(γ2) ⊆ λ(γ2), then we can write α((λ(e0,r0), λ(r0,e0)), (λ(e1,r1), λ(r1,e1))) = α((λ(e0,r0), λ(r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α((λ(r0,e0), λ(e0,r0)), (λ(r1,e1), λ(e1,r1))) = α((λ(r0,e0), λ(e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. again, since υ is a generalized α-admissible mapping w.r.t. λ, then we have that α((υ(e0,r0), υ(r0,e0)), (υ(e1,r1), υ(r1,e1))) ≥ 1, and α((υ(r0,e0), υ(e0,r0)), (υ(r1,e1), υ(e1,r1))) ≥ 1. by induction, we get for all n ∈ n∪{0}, α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))) = α((υ(en,rn), υ(rn,en)), (υ(en+1,rn+1), υ(rn+1,en+1))) ≥ 1, and α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1))) = α((υ(rn,en), υ(en,rn)), (υ(rn+1,en+1), υ(en+1,rn+1))) ≥ 1. (2.2) denote λn = νb(λ(en,rn), λ(en+1,rn+1)) + νb(λ(rn,en), λ(rn+1,en+1)),n ∈ n∪{0}. we suppose that λn > 0,∀n ∈ n because if not, (en,rn) will be a (ccp) and the proof is finished. we claim that ψ(sσλn+1) ≤ ψ(λn). using (2.2) and letting e = en,r = rn,µ = en+1, and κ = rn+1 in (2.1), int. j. anal. appl. 18 (6) (2020) 1088 we get ψ (sσνb(λ(en+1,rn+1), λ(en+2,rn+2)) + ρ = ψ (sσνb(υ(en,rn), υ(en+1,rn+1)) + ρ ≤ (ψ (sσνb(υ(en,rn), υ(en+1,rn+1)) + ρ) µn , ≤ f   ψ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 ) , ϕ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 )   + ρ = f ( ψ( λn 2 ),ϕ( λn 2 ) ) + ρ,(2.3) where µn = α((λ(en,rn), λ(rn,en)), (υ(en,rn), υ(rn,en))) × α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))). similarly, we have ψ (sσνb(λ(rn+2,en+2), λ(rn+1,en+1)) + ρ = ψ (sσνb(υ(rn+1,en+1), υ(rn,en)) + ρ ≤ (ψ (sσνb(υ(rn+1,en+1), υ(rn,en)) + ρ) µn , ≤ f   ψ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 ) , ϕ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 )   + ρ = f ( ψ( λn 2 ),ϕ( λn 2 ) ) + ρ,(2.4) where µn = α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1)) × α((λ(rn,en), λ(en,rn)), (υ(rn,en), υ(en,rn)). summing (2.3), (2.4), and since ψ is nondecreasing, we get (2.5) ψ (sσλn+1) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) ≤ ψ( λn 2 ), since ψ is nondecreasing. by inequality (2.5), one can write λn+1 ≤ 1 sσ λn. hence, by lemma 1.2, the sequence λn is b−cauchy in γ and then {λ(en,rn)} and {λ(rn,en)} are also cauchy sequences in γ. by the completeness of γ, there exist e,r ∈ γ so that (2.6) lim n→∞ λ(en,rn) = υ(en,rn) = λ, and lim n→∞ λ(rn,en) = υ(rn,en) = r. int. j. anal. appl. 18 (6) (2020) 1089 since the pair (υ, λ) satisfies the generalized compatible, by (2.6), we can write (2.7) lim n→∞ νb(υ(λ(en,rn), λ(rn,en)), λ(υ(en,rn), υ(rn,en))) = 0, (2.8) lim n→∞ νb(υ(λ(rn,en), λ(en,rn)), λ(υ(rn,en), υ(en,rn))) = 0. now, if (i) holds, that is υ is continuous and λ is already continuous from the hypothesis of the theorem, then in view of triangle inequality, we have νb(λ(e,r), υ(λ(en,rn), λ(rn,en))) ≤ s[νb(λ(e,r), λ(υ(en,rn), υ(rn,en))) + νb(λ(υ(en,rn), υ(rn,en)), υ(λ(en,rn), λ(rn,en)))]. passing n → ∞ in the above inequality and using (2.6), (2.7), and the continuity of υ and λ, we get λ(e,r) = υ(e,r). similarly, by (2.6), (2.8), we can show that λ(r,e) = υ(r,e). next, assume that (ii) holds. since the pair υ,λ satisfies the generalized compatible and λ is continuous, we have lim n→∞ λ(λ(en,rn), λ(rn,en)) = λ(e,r) = lim n→∞ λ(υ(en,rn), υ(rn,en)) = lim n→∞ υ(λ(en,rn), λ(rn,en)),(2.9) and lim n→∞ λ(λ(rn,en), λ(en,rn)) = λ(r,e) = lim n→∞ λ(υ(rn,en), υ(en,rn)) = lim n→∞ υ(λ(rn,en), λ(en,rn)).(2.10) then, we have α((λ(λ(en,rn),e(rn,en)), λ(λ(rn,en), λ(en,rn))), (λ(e,r), λ(r,e))) ≥ 1, and α((λ(λ(rn,en), λ(en,rn)), λ(λ(en,rn), λ(rn,en))), (λ(r,e), λ(e,r))) ≥ 1. int. j. anal. appl. 18 (6) (2020) 1090 applying (2.1), we get ψ(νb(υ(e,r),e(e,r))) = lim n→∞ ψ(νb(υ(e,r),e(υ(en,rn), υ(rn,en)))) ≤ lim n→∞ ψ(νb(υ(e,r), υ(e(en,rn), λ(rn,en)))) + ρ ≤ lim n→∞ (ψ(sσνb(υ(e,r), υ(e(en,rn), λ(rn,en)))) + ρ) µn, ≤ lim n→∞ f   ψ ( νb(λ(e,r),λ(λ(en,rn),λ(rn,en)))+νb(λ(r,e),λ(λ(rn,en),λ(enrn))) 2 ) , ϕ ( νb(λ(e,r),λ(λ(en,rn),λ(rn,en)))+νb(λ(r,e),λ(λ(rn,en),λ(enrn))) 2 )   + ρ, where µn = α(λ(e,r), λ(r,e)), (υ(e,r), υ(r,e)) × α   (λ(λ(rn,en), λ(en,rn)), λ(λ(en,rn), λ(rn,en))), (υ(λ(rn,en)), λ(en,rn), υ(λ(en,rn)), λ(rn,en))   . using (2.9),(2.10), we get ψ(νb(υ(e,r), λ(e,r))) = 0 implies that υ(e,r) = λ(e,r). similarly, we can prove that υ(r,e) = λ(r,e). � theorem 2.2. let (γ,νb) be a complete (b-ms) (with parameter s > 1), and υ, λ : γ 2 → γ be two generalized compatible mappings so that υ is a generalized α−admissible mapping w.r.t. λ and λ is continuous. let there is f ∈ c and ψ ∈ ψ,ϕ ∈ φ so that the stipulation below holds:  α   (λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))  α   (λ(µ,κ), λ(κ,µ)), (υ(µ,κ), υ(κ,µ))   +1   ψ(sσνb(υ(e,r),υ(µ,κ)) ≤ 2f ( ψ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 ),ϕ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 ) ) ,(2.11) for all e,r,µ,κ ∈ γ,ρ,σ > 0, and α : (γ2 × γ2) → [0,∞). assume that (i) υ(γ2) ⊆ λ(γ2), (ii) there is e0,r0 ∈ γ so that α((λ(e0,r0), λ(r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α((λ(r0,e0), λ(e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. also, suppose either (iv) υ is continuous, or (v) {en},{rn} are two sequences in γ such that α((en+1,rn+1), (en,rn)) ≥ 1 and α((rn,en), (rn+1,en+1)) ≥ 1. int. j. anal. appl. 18 (6) (2020) 1091 for all n ∈ n∪{0}, and en → e,rn → r as n →∞, e,r ∈ γ, we have α((e,r), (en,rn)) ≥ 1 and α((rn,en), (r,e)) ≥ 1. then υ and λ have a (ccp). proof. as in theorem (2.1), we can conclude that for all n ∈ n∪{0}, α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))) = α((υ(en,rn), υ(rn,en)), (υ(en+1,rn+1), υ(rn+1,en+1))) ≥ 1, and α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1))) = α((υ(rn,en), υ(en,rn)), (υ(rn+1,en+1), υ(en+1,rn+1))) ≥ 1. (2.12) denote λn = νb(λ(en,rn), λ(en+1,rn+1)) + νb(λ(rn,en), λ(rn+1,en+1)),n ∈ n∪{0}. we suppose that λn > 0,∀n ∈ n because if not, (en,rn) will be a (ccp) and the proof is finished. we claim that ψ(sσλn+1) ≤ ψ(λn). using (2.12), letting e = en,r = rn,µ = en+1, and κ = rn+1 in (2.11), we have 2ψ(s σνb(λ(en+1,rn+1),λ(en+2,rn+2)) = 2ψ(s σνb(υ(en,rn),υ(en+1,rn+1)) ≤ (µn + 1)ψ(s σνb(υ(en,rn),υ(en+1,rn+1)) ≤ 2 f   ψ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 ) , ϕ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 )   = 2f(ψ( λn 2 ),ϕ( λn 2 )),(2.13) where µn = α((λ(en,rn), λ(rn,en)), (υ(en,rn), υ(rn,en))) × α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))). thus, we get ψ (νb(λ(en+1,rn+1), λ(en+2,rn+2)) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) . int. j. anal. appl. 18 (6) (2020) 1092 similarly, we have 2ψ(s σνb(λ(rn+2,en+2),λ(rn+1,en+1)) = 2ψ(s σνb(υ(rn+1,en+1),υ(rn,en)) ≤ (µn + 1)ψ(s σνb(υ(rn+1,en+1),υ(rn,en)) ≤ 2 f   ψ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 ) , ϕ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 )   = 2f(ψ( λn 2 ),ϕ( λn 2 )),(2.14) where µn = α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1))) × α((λ(rn,en), λ(en,rn)), (υ(rn,en), υ(en,rn))). thus, we get (2.15) ψ (sσνb(λ(rn+2,en+2), λ(rn+1,en+1)) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) . summing both inequalities (2), (2.15), and since ψ is nondecreasing, we have (2.16) ψ (sσλn+1) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) ≤ ψ( λn 2 ), since ψ is nondecreasing and from inequality (2.16), one can get λn+1 ≤ 1 sσ λn. hence, by lemma 1.2, the sequence λn is b−cauchy in γ and then {λ(en,rn)} and {λ(rn,en)} are also cauchy sequences in γ. by the completeness of γ, there exist e,r ∈ γ such that (2.17) lim n→∞ e(en,rn) = υ(en,rn) = e, and lim n→∞ e(rn,en) = υ(rn,en) = r. since the pair (υ,e) satisfies the generalized compatible condition, then by (2.17), one can write (2.18) lim n→∞ νb(υ(λ(en,rn), λ(rn,en)), λ(υ(en,rn), υ(rn,en))) = 0, (2.19) lim n→∞ νb(υ(λ(rn,en), λ(en,rn)), λ(υ(rn,en), υ(en,rn))) = 0. now, if (i) holds, we apply the same steps of theorem (2.1), and we get λ(r,e) = υ(r,e). now, assume that (ii) holds. since the pair υ,λ satisfies the generalized compatible and λ is continuous, we int. j. anal. appl. 18 (6) (2020) 1093 have lim n→∞ λ(λ(en,rn),e(rn,en)) = λ(e,r) = lim n→∞ λ(υ(en,rn), υ(rn,en)) = lim n→∞ υ(λ(en,rn), λe(rn,en)),(2.20) and lim n→∞ λ(λ(rn,en), λ(en,rn)) = λ(r,e) = lim n→∞ λ(υ(rn,en), υ(en,rn)) = lim n→∞ υ(λ(rn,en), λ(en,rn)).(2.21) then, we can write α((λ(λ(en,rn), λ(rn,en)), λ(λ(rn,en), λ(en,rn))), (λ(e,r), λ(r,e))) ≥ 1, and α((λ(λ(rn,en), λ(en,rn)), λ(λ(en,rn), λ(rn,en))), (λ(r,e), λ(e,r))) ≥ 1. applying (2.11), we get 2ψ(νb(υ(e,r),λ(e,r))) = lim n→∞ 2ψ(νb(υ(e,r),e(υ(en,rn),υ(rn,en)))) = lim n→∞ 2ψ(νb(υ(e,r),υ(e(en,rn),λ(rn,en)))) ≤ lim n→∞ (µn + 1) ψ(sσνb(υ(e,r),υ(e(en,rn),λ(rn,en)))), ≤ lim n→∞ 2f(ψ(χn),ϕ(χn)), where µn = α(λ(e,r), λ(r,e)), (υ(e,r), υ(r,e)) × α   λ(λ(rn,en), λ(en,rn)), λ(λ(en,rn), λ(rn,en))), (υ(λ(rn,en), λ(en,rn)), υ(λ(en,rn), λ(rn,en)))   , and χn = νb(λ(e,r), λ(e(en,rn), λ(rn,en))) + νb(λ(r,e), λ(λ(rn,en), λ(enrn))) 2 , using (2.20),(2.21), we get ψ(νb(υ(e,r), λ(e,r))) = 0 this leads to υ(e,r) = λ(e,r). similarly, we can prove that υ(r,e) = λ(r,e). � int. j. anal. appl. 18 (6) (2020) 1094 theorem 2.3. let (γ,νb) be a complete (b-ms) (with parameter s > 1), and υ, λ : γ 2 → γ be two generalized compatible mappings such that υ is a generalizedα−admissible mapping w.r.t. λ and λ is continuous. let there is f ∈ c, ψ ∈ ψ,ϕ ∈ φ so that the stipulation below holds: α   (λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))    α   (λ(µ,κ), λ(κ,µ)), (υ(µ,κ), υ(κ,µ)))  ψ(sσνb(υ(e,r), υ(µ,κ)))   ≤ f   ψ(νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ))2 ), ϕ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 )   ,(2.22) for all e,r,µ,κ ∈ γ,ρ,σ > 0,, and α : (γ2 × γ2) → [0,∞). assume that (i) υ(γ2) ⊆ λ(γ2) (ii) there is e0,r0 ∈ γ so that α((λ(e0,r0), λ(r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α((λ(r0,e0), λ(e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. also, suppose either (iv) υ is continuous, or (v) {en},{rn} are two sequences in γ so that α((en+1,rn+1), (en,rn)) ≥ 1, α((rn,en), (rn+1,en+1)) ≥ 1. for all n ∈ n∪{0}, and en → e,rn → r as n →∞, e,r ∈ γ, we have α((e,r), (en,rn)) ≥ 1, α((rn,en), (r,e)) ≥ 1. then υ and λ have a (ccp). proof. again, as in theorem (2.1), we can conclude that for all n ∈ n∪{0}, α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))) = α((υ(en,rn), υ(rn,en)), (υ(en+1,rn+1), υ(rn+1,en+1))) ≥ 1,and α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1))) = α((υ(rn,en), υ(en,rn)), (υ(rn+1,en+1), υ(en+1,rn+1))) ≥ 1. (2.23) int. j. anal. appl. 18 (6) (2020) 1095 denote λn = νb(λ(en,rn), λ(en+1,rn+1)) + νb(λ(rn,en), λ(rn+1,en+1)),n ∈ n∪{0} . we suppose that λn > 0,∀n ∈ n because if not, (en,rn) will be a (ccp) and the proof is finished. we claim that ψ(sσλn+1) ≤ ψ(λn). using (2.12), letting e = en,r = rn,µ = en+1, and κ = rn+1 in (2.22), we have ψ (sσνb(λ(en+1,rn+1), λ(en+2,rn+2)) = ψ (sσνb(υ(en,rn), υ(en+1,rn+1)) ≤ µnψ (sσνb(υ(en,rn), υ(en+1,rn+1)) ≤ f   ψ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 ) , ϕ ( νb(λ(en,rn),λ(en+1,rn+1))+νb(λ(rn,en),λ(rn+1,en+1)) 2 )   = f ( ψ( λn 2 ),ϕ( λn 2 ) ) ,(2.24) where µn = α((λ(en,rn), λ(rn,en)), (υ(en,rn), υ(rn,en))) × α((λ(en+1,rn+1), λ(rn+1,en+1)), (υ(en+1,rn+1), υ(rn+1,en+1))). thus, we get ψ (sσνb(λ(en+1,rn+1), λ(en+2,rn+2)) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) .(2.25) similarly, we have ψ (sσνb(λ(rn+2,en+2), λ(rn+1,en+1)) = ψ (sσνb(υ(rn+1,en+1), υ(rn,en)) ≤ µnψ (sσνb(υ(rn+1,en+1), υ(rn,en)) ≤ f   ψ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 ) , ϕ ( νb(λ(rn+1,en+1),λ(rn,en))+νb(λ(en+1,rn+1),λ(en,rn)) 2 )   = f ( ψ( λn 2 ),ϕ( λn 2 ) ) ,(2.26) where µn = α((λ(rn+1,en+1), λ(en+1,rn+1)), (υ(rn+1,en+1), υ(en+1,rn+1))) × α((λ(rn,en), λ(en,rn)), (υ(rn,en), υ(en,rn))). int. j. anal. appl. 18 (6) (2020) 1096 thus, we get ψ (sσνb(λ(rn+2,en+2), λ(rn+1,en+1)) ≤ f ( ψ( λn 2 ),ϕ( λn 2 )) ) .(2.27) by inequalities (2.25), (2.27), and since ψ is non-decreasing, we have ψ (sσλn+1) ≤ f ( ψ( λn 2 ),ϕ( λn 2 ) ) ≤ ψ( λn 2 ).(2.28) since ψ is nondecreasing and by inequality (2.28), we get λn+1 ≤ 1 sσ λn. thus, by lemma 1.2, the sequence λn is b-cauchy in γ and then {λ(en,rn)} and {λ(rn,en)} are also cauchy sequences in γ. by the completeness of γ, there exist e,r ∈ γ such that lim n→∞ λ(en,rn) = υ(en,rn) = e, and lim n→∞ λ(rn,en) = υ(rn,en) = r.(2.29) since the pair (υ,e) satisfies the generalized compatible condition, then by (2.29), one can write lim n→∞ νb(υ(λ(en,rn), λ(rn,en)), λ(υ(en,rn), υ(rn,en))) = 0,(2.30) lim n→∞ νb(υ(λ(rn,en), λ(en,rn)), λ(υ(rn,en), υ(en,rn))) = 0.(2.31) now, if (i) holds, we apply the same steps of theorem (2.1), and we get λ(r,e) = υ(r,e). now, assume that (ii) holds. since the pair υ, λ satisfies the generalized compatible condition and λ is continuous, we have lim n→∞ λ(λ(en,rn), λ(rn,en)) = λ(e,r) = lim n→∞ λ(υ(en,rn), υ(rn,en)) = lim n→∞ υ(λ(en,rn), λ(rn,en)),(2.32) and lim n→∞ λ(λ(rn,en), λ(en,rn)) = λ(r,e) = lim n→∞ λ(υ(rn,en), υ(en,rn)) = lim n→∞ υ(λ(rn,en), λ(en,rn)).(2.33) then, we have α((λ(λ(en,rn), λ(rn,en)), λ(λ(rn,en), λ(en,rn))), (λ(e,r), λ(r,e))) ≥ 1, and α((λ(λ(rn,en), λ(en,rn)), λ(λ(en,rn), λ(rn,en))), (λ(r,e), λ(e,r))) ≥ 1. int. j. anal. appl. 18 (6) (2020) 1097 applying (2.22), we get ψ(νb(υ(e,r), λ(e,r))) = lim n→∞ ψ(νb(υ(e,r), λ(υ(en,rn), υ(rn,en)))) = lim n→∞ ψ(νb(υ(e,r), υ(λ(en,rn), λ(rn,en)))) ≤ lim n→∞ µnψ(νb(υ(e,r), υ(λ(en,rn), λ(rn,en)))), ≤ lim n→∞ f (ψ(χn),ϕ(χn)) , where µn = α(λ(e,r), λ(r,e)), (υ(e,r), υ(r,e)) × α   (λ(λ(en,rn), λ(rn,en)), λ(λ(rn,en), λ(en,rn))), (υ(λ(en,rn), λ(rn,en)), υ(λ(rn,en), λ(en,rn)))   . and χn = νb(λ(e,r), λ(λ(en,rn), λ(rn,en))) + νb(λ(r,e), λ(λ(rn,en), λ(enrn))) 2 . using (2.32),(2.33), we get ψ(νb(υ(e,r), λ(e,r))) = 0 implies that υ(e,r) = λ(e,r). similarly, we can prove that υ(r,e) = λ(r,e). � theorem 2.4. suppose that all requirements of theorems (2.1) or (2.2) or (2.3) are fulfilled. in addition, let the stipulation below holds: (vi) if λ(e,r) = υ(e,r) and λ(r,e) = υ(r,e) then α((λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))) ≥ 1, and α((λ(r,e), λ(e,r)), (υ(r,e), υ(e,r))) ≥ 1. then λ and υ have a unique (ccp). proof. from theorem (2.1) or (2.2) or (2.3), we know that the set of (ccp) of λ and υ is nonempty. let (e,r) and (e∗,r∗) are (ccp) of λ and υ, this yields λ(e,r) = υ(e,r), λ(r,e) = υ(r,e), and λ(e∗,r∗) = υ(e∗,r∗), λ(r∗,e∗) = υ(r∗,e∗). int. j. anal. appl. 18 (6) (2020) 1098 we will prove that λ(e,r) = λ(e∗,r∗) and λ(r,e) = λ(r∗,e∗). it follows from (vi) that α((λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))) ≥ 1 α((λ(r,e), λ(e,r)), (υ(r,e), υ(e,r))) ≥ 1, and α ((λ(e∗,r∗), λ(r∗,e∗)), (υ(e∗,r∗), υ(r∗,e∗))) ≥ 1 α((λ(r∗,e∗), λ(e∗,r∗)), (υ(r∗,e∗), υ(e∗,r∗))) ≥ 1, from theorem 2.1, using above inequalities, we have ψ (νb(λ(e,r), λ(e ∗,r∗)) + ρ ≤ ψ (sσνb(υ(e,r), υ(e∗,r∗)) + ρ ≤ (ψ (sσνb(υ(e,r), υ(e∗,r∗)) + ρ) µn ≤ f   ψ(νb(λ(e,r),λ(e∗,r∗))+νb(λ(r,e),λ(r∗,e∗))2 ), ϕ( νb(λ(e,r),λ(e ∗,r∗))+νb(λ(r,e),λ(r ∗,e∗)) 2 )   + ρ,(2.34) where µn = α((λ(e,r), λ(r,e)), (υ(e,r), υ(r,e))),α((λ(e ∗,r∗), λ(r∗,e∗)), (υ(e∗,r∗), υ(r∗,e∗))). from theorem 2.2, we have 2ψ(νb(λ(e,r),λ(e ∗,r∗))) ≤ 2ψ(s σνb(υ(e,r),υ(e ∗,r∗))) ≤ (µn + 1)ψ(s σνb(υ(e,r),υ(e ∗,r∗))) ≤ 2f ( ψ( νb(λ(e,r),λ(e ∗,r∗))+νb(λ(r,e),λ(r ∗,e∗)) 2 ),ϕ( νb(λ(e,r),λ(e ∗,r∗))+νb(λ(r,e),λ(r ∗,e∗)) 2 ) ) .(2.35) from theorem 2.3, we have ψ(νb(λ(e,r), λ(e ∗,r∗))) ≤ ψ(sσνb(υ(e,r), υ(e∗,r∗))) ≤ (µn + 1)ψ(sσνb(υ(e,r), υ(e∗,r∗))) ≤ f   ψ(νb(λ(e,r),λ(e∗,r∗))+νb(λ(r,e),λ(r∗,λ∗))2 ), ϕ( νb(λ(e,r),λ(e ∗,r∗))+νb(λ(r,e),λ(r ∗,e∗)) 2 )   .(2.36) from (2.34), (2.35) and (2.36), we have f (ψ(νb(λ(e,r), λ(e ∗,r∗)),ϕ(νb(λ(e,r), λ(e ∗,r∗)))) = ψ(νb(λ(e,r), λ(e ∗,r∗)). int. j. anal. appl. 18 (6) (2020) 1099 by the hypotheses of f,ψ,ϕ, we get either ψ(νb(λ(e,r), λ(e ∗,r∗)) = 0 or ϕ(νb(λ(e,r), λ(e ∗,r∗))) = 0. thus we have λ(e,r) = λ(e∗,r∗). similarly, we can prove that λ(r,e) = λ(r∗,e∗). � an example below support theorems 2.1, 2.2, 2.3, and 2.4. example 2.1. let f(e,r) = τe, 0 < τ < 1 and γ = [0,∞) endowed with νb(e,r) = (e−r) 2 for all e,r ∈ γ. it is clear that (γ,νb) is a complete b−ms with a coefficient s = 2. assume that υ, λ : γ×γ → γ by υ(e,r) =   e−r e ≥ r0 otherwise and λ(e,r) =   e + r e ≥ r0 otherwise , it is obvious that υ(γ2) ⊆ λ(γ2) and the mapping υ is continuous. define α : γ2 × γ2 → [0, +∞) by α ((e,r), (µ,κ)) =   2, e ≥ µ, r ≤ κ0, otherwise . then for each e◦,r◦ ∈ γ, we find that α [(λ (e◦,r◦) , λ (r◦,e◦)) , (υ (e◦,r◦) , υ (r◦,e◦))] = 2 > 1, α [(λ (r◦,e◦) , λ (e◦,r◦)) , (υ (r◦,e◦) , υ (e◦,r◦))] = 2 > 1. for all n ∈ n, let en = nn+1 and rn = 1 n be two sequences such that α [(λ (en+1,rn+1) , λ (rn,en))] ≥ 1, α [(λ (rn,en) , λ (en+1,rn+1))] ≥ 1. then, limn→∞en = 1 and limn→∞rn = 0. certainly, 0, 1 ∈ γ and α [(λ (0, 1) , λ (rn,en))] = 2 > 1, α [(λ (1, 0) , λ (en,rn))] = 2 > 1. under this sequences, we can write lim n→∞ νb (υ [λ (en,rn) , λ (rn,en)] , λ [υ (en,rn) , υ (rn,en)]) = νb (υ [λ (1, 0) , λ (0, 1)] , λ [υ (1, 0) , υ (0, 1)]) = νb (υ (1, 0) , λ (1, 0)) = νb (1, 1) = 0, similarly, one can prove that lim n→∞ νb (υ [λ (rn,en) , λ (en,rn)] , λ [υ (rn,en) , υ (en,rn)]) = 0. int. j. anal. appl. 18 (6) (2020) 1100 whenever en,rn ∈ γ, such that lim n→∞ υ (en,rn) = υ (1, 0) = 1 = λ (1, 0) = lim n→∞ λ (en,rn) , lim n→∞ υ (rn,en) = υ (0, 1) = 0 = λ (0, 1) = lim n→∞ λ (rn,en) . therefore, υ and λ are generalized compatible. now, for e = κ = 1 and r = µ = 0, if α [(λ (e,r) , λ (r,e)) , (λ (µ,κ) , λ (κ,µ))] > 1, this implies that α [(υ (e,r) , υ (r,e)) , (υ (µ,κ) , υ (κ,µ))] = α [(υ (1, 0) , υ (0, 1)) , (υ (0, 1) , υ (1, 0))] = α [(1, 0) , (0, 1)] = 2 > 1 for all (1, 0), (0, 1) ∈ γ2. this prove that υ is a generalized α−admissible w.r.t. λ. finally, we will try to verify the contractive conditions (2.1),(2.11) and (2.22) of theorems 2.1, 2.2, and 2.3 respectively. take ψ(θ) = θ 3 and φ(θ) = θ for all θ ∈ [0,∞). put σ = ρ = 1, τ = 1 2 , e = κ = 1 2 and r = µ = 1 4 , then we have α [(λ (e,r) , λ (r,e)) , (υ (e,r) , υ (r,e))] ×α [(λ (µ,κ) , λ (κ,µ)) , (υ (µ,κ) , υ (κ,µ))] = α [( λ ( 1 2 , 1 4 ) , λ ( 1 4 , 1 2 )) , ( υ ( 1 2 , 1 4 ) , υ ( 1 4 , 1 2 ))] ×α [( λ ( 1 4 , 1 2 ) , λ ( 1 2 , 1 4 )) , ( υ ( 1 4 , 1 2 ) , υ ( 1 2 , 1 4 ))] = α [( 3 4 , 0 ) , ( 0, 1 4 )] ×α [( 0, 3 4 ) , ( 0, 1 4 )] = 2 × 0 = 0,(2.37) ψ ( νb (λ (e,r) , λ (µ,κ)) + νb (λ (r,e) , λ (κ,µ)) 2 ) = ψ ( νb ( λ ( 1 2 , 1 4 ) , λ ( 1 4 , 1 2 )) + νb ( λ ( 1 4 , 1 2 ) , λ ( 1 2 , 1 4 )) 2 ) = ψ ( νb ( 3 4 , 0 ) + νb ( 0, 3 4 ) 2 ) = ψ ( 9 16 ) = 3 16 ,(2.38) (2.39) φ ( νb (λ (e,r) , λ (µ,κ)) + νb (λ (r,e) , λ (κ,µ)) 2 ) = φ ( 9 16 ) = 9 16 , int. j. anal. appl. 18 (6) (2020) 1101 (2.40) sσνb (υ (e,r) , υ (µ,κ)) = 2νb ( υ ( 1 2 , 1 4 ) , υ ( 1 4 , 1 2 )) = 2νb ( 1 4 , 0 ) = 1 8 . it follows from definition of f and (2.37)-(2.40) that (ψ (sσνb (υ (e,r) , υ (µ,κ))) + ρ)   α   (λ (e,r) , λ (r,e)) , (υ (e,r) , υ (r,e))   ×α   (λ (µ,κ) , λ (κ,µ)) , (υ (µ,κ) , υ (κ,µ))     = ( ψ ( 1 4 ) + 1 )0 = 1 ≤ 35 32 = 1 2 × 3 16 + 1 = f ( 3 16 , 9 16 ) + 1 = f   ψ ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 ) , φ ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 )   + ρ.(2.41) also, we can write   α   (λ (e,r) , λ (r,e)) , (υ (e,r) , υ (r,e))   ×α   (λ (µ,κ) , λ (κ,µ)) , (υ (µ,κ) , υ (κ,µ))   +1   ψ(sσνb(υ(e,r),υ(µ,κ))) = (0 + 1)ψ( 1 8 ) = 1 < 2 3 32 = 2 3 16 ×1 2 = 2f( 3 16 , 9 16 ) = 2 f ( ψ ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 ) ,φ ( νb(λ(e,r),λ(µ,κ))+νb(λ(r,e),λ(κ,µ)) 2 )) .(2.42) additionally, the contractive condition (2.22) of theorem 2.3 is directly hold. so, by (2.41)-(2.42) all hypotheses of theorems 2.1, 2.2, and 2.3 are fulfilled, so by theorem 2.4 the mappings υ and e have a unique (ccp), here it is (0, 0) ∈ γ2. if we put λ = i, (where i is the identity mapping) in theorem 2.1, we get the important result below. corollary 2.1. let (γ,νb) be a complete (b-ms) (with parameter s > 1), and υ,i : γ 2 → γ be two generalized compatible mappings such that υ is generalized α−admissible mapping w.r.t. i. let there is int. j. anal. appl. 18 (6) (2020) 1102 f ∈ c, ψ ∈ ψ,ϕ ∈ φ so that the stipulation below holds (ψ (sσνb(υ(e,r), υ(µ,κ)) + ρ) α   ((e,r), (r,e)), (υ(e,r), υ(r,e))  α   ((µ,κ), (κ,µ)) , (υ(µ,κ), υ(κ,µ)   ≤ f   ψ ( νb((e,r),(µ,κ))+νb((r,e),(κ,µ)) 2 ) , ϕ ( νb((e,r),(µ,κ))+νb((r,e),(κ,µ)) 2 )   +ρ,(2.43) for all e,r,µ,κ ∈ γ,ρ,σ > 0, and α : (γ2 × γ2) → [0,∞). assume that (i) υ(γ2) ⊆ γ2, (ii) there is e0,r0 ∈ γ so that α(((e0,r0), (r0,e0)), (υ(e0,r0), υ(r0,e0))) ≥ 1, α(((r0,e0), (e0,r0)), (υ(r0,e0), υ(e0,r0))) ≥ 1. also, suppose either (iv) υ is continuous, or (v) {en},{rn} are two sequences in γ so that α((en+1,rn+1), (en,rn)) ≥ 1, α((rn,en), (rn+1,en+1)) ≥ 1. for all n ∈ n∪{0}, and en → e,rn → r as n →∞, e,r ∈ γ, we have α((e,r), (en,rn)) ≥ 1, α((rn,en), (r,e)) ≥ 1. then υ has a (ccp). 3. an important application this part is very important in this paper, where the existence solution to a (nie) using corollary 2.1 is presented. here, we refers to χ by the class functions κ : [0,∞) → [0,∞) so that κ is an increasing function and there is ψ ∈ ψ, φ ∈ φ, and f ∈ c such that κ(κ) = 1 2 f(ψ(κ),φ(κ)) for all κ ∈ [0,∞). assume the problem below: (3.1) j($) = v∫ u (a1($,`) + a2($,`)) (k1(`,j(`)) + k2(`,j(`))) d`, for all $ ∈ [u,v]. suppose that a1,a2,k1,k2 are continuous functions which satisfy the hypotheses below: int. j. anal. appl. 18 (6) (2020) 1103 (i) for all $,` ∈ [u,v], a1($,`),a2($,`) ≥ 0, (ii) for all y,z ∈ r with y ≥ z, there is f,u so that 0 ≤ k1(`,y) −k1(`,z) ≤ fκ(y −z), 0 ≤ k2(`,y) −k2(`,z) ≤uκ(y −z), (iii) we get max{f2,u2}   sup$∈[u,v] v∫ u (a1($,`) + a2($,`))d`   2 ≤ 1. to discuss the existence of a unique solution of the problem (3.1), we formulate the theorem below: theorem 3.1. under the assumptions (i)-(iii) with a1,a2 ∈ c([u,v]× [u,v],r) and k1,k2 ∈ c([u,v]×r× r), the problem (3.1) has a solution in c([u,v],r). proof. let γ = c([u,v],r) be the set of real continuous functions on [u,v] endowed with the distance νb(e,r) = sup $∈[u,v] (|e($) −r($)|)2 , ∀e,r ∈ γ. it’s obvious that, the pair (γ,νb) is a complete b−ms with a coefficient s = 2. define mappings υ : γ × γ → γ and α : γ2 × γ2 → r+ by υ(e,r)($) = v∫ u a1($,`) (k1(`,e(`)) + k2(`,r(`))) d` + q∫ p a2($,`) (k1(`,e(`)) + k2(`,r(`))) d`, and α ((e,r), (µ,κ)) =   1, e ≥ µ, r ≤ κ0, otherwise . for all $ ∈ [u,v], (e,r), (µ,κ) ∈ γ2. if the mapping υ has a (ccp) in γ, then it is a solution of the problem (3.1). since, for each e,r,µ,κ ∈ γ, α [((e,r) , (r,e)) , (i (µ,κ) ,i (κ,µ))] = 1 and α [(υ (e,r) , υ (r,e)) , (υ (µ,κ) , υ (κ,µ))] = 1, we conclude that υ is a generalized α−admissible w.r.t. i and by the continuity of a1,a2,k1, and k2, we have υ is a continuous mapping. also, for any two sequences {en} and {rn} in γ, suppose that lim n→∞ νb (υ [i (en) ,i (rn)] ,i [υ (en,rn) , υ (rn,en)]) = 0, lim n→∞ νb (υ [i (rn) ,i (en)] ,i [υ (rn,en) , υ (en,rn)]) = 0. int. j. anal. appl. 18 (6) (2020) 1104 thus, we have lim n→∞ υ (en,rn) = lim n→∞ ien = lim n→∞ en, lim n→∞ υ (rn,en) = lim n→∞ rn. therefore, the pair (υ,i) is generalized compatible. again, it follows from the definition of α that if α [(i (e) ,i (r)) , (i (µ) ,i (κ))] = α (e,r,µ,κ) = 1, this implies that α [(υ (e,r) , υ (r,e)) , (υ (µ,κ) , υ (κ,µ))] = 1, and (3.2) α [(e,r) , (υ (e,r) , υ (r,e))] ×α [(µ,κ) , (υ (µ,κ) , υ (κ,µ))] = 1. now we are going to verify the hypothesis (2.43) of corollary 2.1, for all e,r,µ,κ ∈ γ, νb (υ(e,r), υ(µ,κ)) = sup $∈[u,v] (|υ(e,r)($) − υ(µ,κ)($)|)2 = sup $∈[u,v]   ∣∣∣∣ v∫ u a1($,`) (k1(`,e(`)) + k2(`,r(`))) d` + v∫ u a2($,`) (k1(`,e(`)) + k2(`,r(`))) d` − v∫ u a1($,`) (k1(`,µ(`)) + k2(`,κ(`))) d`− v∫ u a2($,`) (k1(`,µ(`)) + k2(`,κ(`))) d` ∣∣∣∣   2 = sup $∈[u,v]   ∣∣∣∣ v∫ u a1($,`) [(k1(`,e(`)) −k1(`,µ(`))) + (k2(`,r(`)) −k2(`,κ(`)))] d` + v∫ u a2($,`) [(k1(`,e(`)) −k1(`,µ(`))) + (k2(`,r(`)) −k2(`,κ(`)))] d` ∣∣∣∣   2 . applying assumption (ii), one can get νb (υ(e,r), υ(µ,κ)) ≤ sup $∈[u,v]   ∣∣∣∣ v∫ u a1($,`) [fκ(e(`) −µ(`))) + uκ(r(`) −κ(`))] d` + v∫ u a2($,`) [fκ(e(`) −µ(`))) + uκ(r(`) −κ(`))] d` ∣∣∣∣   2 ≤ max{f2,u2}× sup $∈[u,v]   ∣∣∣∣∣∣ v∫ u (a1($,`) + a2($,`)) [κ(|e(`) −µ(`)|) + κ(|r(`) −κ(`)|)] d` ∣∣∣∣∣∣  2 .(3.3) by using the definition of κ and the distance νb, we have (3.4) κ |e(`) −µ(`)|2 ≤ κνb(e,µ) and κ |r(`) −κ(`)| 2 ≤ κνb(r,κ), ∀$ ∈ [u,v]. int. j. anal. appl. 18 (6) (2020) 1105 it follows from (3.3), (3.4) and assumption (iii) that νb (υ(e,r), υ(µ,κ)) ≤ max{f2,u2}× [κ2νb(e,µ) + κ2νb(r,κ)] ×   sup$∈[u,v] v∫ u (a1($,`) + a2($,`)) d`   2 ≤ κ2νb(e,µ) + κ2νb(r,κ) = 2π2 ( νb(e,µ) + νb(r,κ) 2 ) ≤ 2 × 1 4 f ( ψ ( νb(e,µ) + νb(r,κ) 2 ) ,φ ( νb(e,µ) + νb(r,κ) 2 )) = 1 2 f ( ψ ( νb(e,µ) + νb(r,κ) 2 ) ,φ ( νb(e,µ) + νb(r,κ) 2 )) . thus, for all e,r,µ,κ ∈ γ, we 21νb (υ(e,r), υ(µ,κ)) ≤ f ( ψ ( νb(e,µ) + νb(r,κ) 2 ) ,φ ( νb(e,µ) + νb(r,κ) 2 )) . add ρ > 0 to the both sides, we have ( 21νb (υ(e,r), υ(µ,κ)) + ρ ) ≤ f ( ψ ( νb(e,µ) + νb(r,κ) 2 ) ,φ ( νb(e,µ) + νb(r,κ) 2 )) + ρ. put ψ(κ) = κ, for all κ ∈ [0,∞), s = 2, σ = 1, and using (3.2), we get ψ(sσνb (υ(e,r), υ(µ,κ)) + ρ) α[(e,r),(υ(e,r),υ(r,e))]×α[(µ,κ),(υ(µ,κ),υ(κ,µ))] ≤ f ( ψ ( νb(e,µ) + νb(r,κ) 2 ) ,φ ( νb(e,µ) + νb(r,κ) 2 )) + ρ. therefore all stipulations of corollary 2.1 are fulfilled. then the mapping υ has a (ccp) which is a solution of the system (3.1) in γ. � availability of data and material: not applicable. funding: this work was supported in part by the basque government under grant it1207-19. author contributions: all authors contributed equally and significantly in writing this article. all authors read and approved the final manuscript. acknowledgments: the authors are grateful to the spanish government and the european commission for grant it1207-19. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. berzig, s. chandok, m.s. khan, generalized krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem, appl. math. comput. 248 (2014), 323–327. int. j. anal. appl. 18 (6) (2020) 1106 [2] d. gopal, m. abbas, c. vetro, some new fixed point theorems in manger pm-spaces with application to volterra type integral equation, appl. math. comput. 232 (2014), 955–967. [3] z. liu, x. li, s.m. kang, s.y. cho, fixed point theorems for mappings satisfying contractive condition of integral type and applications, fixed point theory appl. 2011 (2011), 64. [4] h.k. pathak, m.s. khan, r. tiwari, a common fixed point theorem and its application to nonlinear integral equations, comput. math. appl. 53 (2007), 961–971. [5] h.a. hammad, m. de la sen, a solution of fredholm integral equation by using the cyclic r q s -rational contractive mappings technique in b−metric-like spaces, symmetry, 11 (2019), 1184. [6] h. a. hammad, m. de la sen, solution of nonlinear integral equation via fixed point of cyclic α q s−rational contraction mappings in metric-like spaces, bull. braz. math. soc. new ser. 51 (2020), 81-105. [7] n. shahzad, o. valero, m.a. alghamdi, a fixed point theorem in partial quasi-metric spaces and an application to software engineering, appl. math. comput. 268 (2015), 1292–1301. [8] n.hussain, m.a. taoudi, krasnoselskii-type fixed point theorems with applications to volterra integral equations, fixed point theory appl. 2013, (2013), 196. [9] i.a. bakhtin, the contraction principle in quasimetric spaces, funct. anal. 30 (1989) 26-37. [10] s. czerwik, contraction mappings in b-metric spaces, acta math. inf. univ. ostrav. 1 (1993), 5-11. [11] a. aghajani, m. abbas, j.r. roshan, common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, math. slovaca, 64 (4) (2014), 941-960. [12] r. miculescu, a. mihail, new fixed point theorems for set-valued contractions in b-metric spaces, j. fixed point theory appl. 19 (2017), 2153-2163. [13] m. shah, s. simic, n. hussain, a. sretenovic, s. radenović, common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces, j. comput. anal. appl. 14 (2012), 290-297. [14] n. hussain, m. shah, kkm mappings in cone b-metric spaces, comput. math. appl. 62 (2011), 1677-1684. [15] m. khamsi and n. hussain, kkm mappings in metric type spaces, nonlinear anal., theory meth. appl. 73 (2010), 3123-3129. [16] s. radenović, k. zoto, n. dedović, v. šešum-cavic, a. ansari, bhaskar-guo-lakshmikantam-ćirić type results via new functions with applications to integral equations, appl. math. comput. 357 (2019), 75-87. [17] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal., theory meth. appl. 11 (1987), 623-632. [18] t.g. bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal., theory meth. appl. 65 (2006), 1379-1393. [19] m. abbas, m.a. khan, s. radenović, common coupled fixed point theorems in cone metric spaces for w-compatible mappings, appl. math. comput. 217 (2010), 195-202. [20] h. aydi, m. postolache, w. shatanawi, coupled fixed point results for (ψ,φ)−weakly contractive mappings in ordered g-metric spaces, comput. math. appl. 63 (2012), 298-309. [21] v. berinde, coupled fixed point theorems for contractive mixed monotone mappings in partially ordered metric spaces, nonlinear anal., theory meth. appl. 75 (2012), 3218-3228. [22] b.s. choudhury, p. maity, coupled fixed point results in generalized metric spaces, math. comput. model. 54 (2011), 73-79. [23] h.a. hammad, m. de la sen, a coupled fixed point technique for solving coupled systems of functional and nonlinear integral equations, mathematics, 7 (2019), 634. int. j. anal. appl. 18 (6) (2020) 1107 [24] h.a. hammad, d.m. albaqeri, r.a. rashwan, coupled coincidence point technique and its application for solving nonlinear integral equations in rpocbml spaces, j. egypt. math. soc. 28 (2020), 8. [25] n. hussain, m. abbas, a. azam, j. ahmad, coupled coincidence point results for a generalized compatible pair with applications, fixed point theory appl. 2014 (2014), 62. [26] h.a. hammad, h. aydi, m. de la sen, generalized dynamic process for an extended multi-valued f-contraction in metriclike spaces with applications, alex. eng. j. 59 (2020), 3817–3825. [27] v. lakshmikantham, lj. b. ćirić, coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, nonlinear anal. theory methods appl. 70, (2009), 4341-4349. [28] h. alsulami, s. gulyaz, e. karapinar, i.m. erhan, fixed point theorems for a class of α− admissible contractions and applications to boundary value problem, abstr. appl. anal. 2014 (2014), article id 187031. [29] a.h. ansari, s. chandok, c. ionescu, fixed point theorems on b-metric spaces for weak contractions with auxiliary functions, j. inequal. appl. 2014 (2014), 429. 1. introduction and elementary discussions 2. main results 3. an important application references international journal of analysis and applications volume 16, number 3 (2018), 368-373 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-368 some remarks on the zeros of tribonacci polynomials öznur öztunç kaymak∗ balıkesir university, 10145 balıkesir, turkey ∗corresponding author: oztunc@balikesir.edu.tr abstract. in this paper, the zeros of tribonacci polynomials are studied. the bound containing the zeros of tribonacci polynomials has been numerically examined with comparisons. on the other hand, a new algorithm is given so that it can be used in other boundary problems. 1. introduction the features of polynomials have played an important role in many scientific areas such as control theory, signal processing, crytography and mathematical biology. if we have an accurate estimate of the region containing all the zeros of a polynomial, then the amount of work needed to find exact zeros can be considerably reduced in comparison with using the classical methods, and for this reason there is always a need for better and better estimates for the region containing all the zeros of a polynomial. so, there have been a number of theorems on computations for the roots of polynomials. we consider the generalization of well-known fibonacci polynomials called r−bonacci polynomials. the polynomial rn(x) is defined by the following recursive equation in [6] for any integer n and r ≥ 2 : rn(x) = [ (r−1)(n−1)r ]∑ j=0   n− j − 1 j   r x(r−1)(n−1)−rj, (1.1) received 2017-12-19; accepted 2018-02-12; published 2018-05-02. 2010 mathematics subject classification. 11b39, 12e10, 30c15. key words and phrases. tribonacci polynomials; r-bonacci polynomials; fibonacci numbers. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 368 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-368 int. j. anal. appl. 16 (3) (2018) 369 where an =   n r   r is the r−nomial coefficient. recently, some results have been given on the zeros of r−bonacci polynomials rn(x) when r = 2, 3 in [5]. m. x. he, p. e. ricci and d. simon found interesting curves composing of the zeros of r−bonacci polynomials in [12]. (if we set r = 2, 3, then the classical fibonacci polynomials and tribonacci polynomials are obtained respectively.) explicit forms have been given for the roots fibonacci polynomials in [7]. since these forms have been used in many practical areas eg. graph energy, stable polynomials, hand printed characters in [15], [16], [18] and [19], it is important to find the zeros of these polynomials. on the other hand, moore and prodinger examined the asymptotic behaviour of the maximal roots of fibonacci-type polynomials in [2] and [9], respectively. furthermore, the absolute values of complex zeros of fibonacci-like polynomials have been investigated by matyas in [8]. unlike fibonacci polynomials, explicit forms for the zeros of tribonacci polynomials have not been found yet. instead, zero attractors of these polynomials for r = 3 are determined by w. goh, m. x. he and p. e. ricci in [11]. the symmetric polynomials of the zeros of tribonacci polynomials are found by m. x. he, d. simon and p. e. ricci in [3]. these results are generalized for the zeros of r−bonacci polynomials and their derivatives in [17]. the aim of this paper is to determine the desired region on the complex plane containing the zeros of tribonacci polynomials. to support this region, we present numerical results for these polynomials to compare the results with the known regions. afterwards, we develop a new algorithm to use in numerical calculations. 2. bounds for polynomials there have been a number of theorems on computations for the roots of polynomials. these studies date back to the work of cauchy [1]. let p(z) = zn + n−1∑ k=0 ak z k, ak ∈ c (2.1) be a complex polynomial of degree n. all the zeros of the polynomial p(z) lie in the region b = {z : |z| < 1 + γ} , (2.2) where γ = max0≤k≤n−1 |ak|. recently, diaz-barrero have improved this region by describing two annuli containing all the zeros of a polynomial where the inner and outer radii are stated in terms of the well-known fibonacci numbers, respectively [13]. int. j. anal. appl. 16 (3) (2018) 370 theorem 2.1. [13] let p(z) = n∑ k=0 ak z k (ak 6= 0, 1 ≤ k ≤ n) be a non-consant polynomial with complex coefficients. then, all its zeros lie in the ring shaped region <1 = {z ∈ c : r1 ≤ |z| ≤ r2} , (2.3) where r1 = 3 2 min 1≤k≤n { 2nfk c(n,k) f4n ∣∣∣∣a0ak ∣∣∣∣ }1 k (2.4) and r2 = 2 3 max 1≤k≤n { f4n 2nfk c(n,k) ∣∣∣∣an−kan ∣∣∣∣ }1 k . (2.5) for another annulus, they considered lucas numbers which are defined by ln = ln−1+ln−2 for n ≥ 0 with the initial conditions l0 = 2, l1 = 1 (see [5] and [10] for the basic properties of these number sequences). theorem 2.2. [4] let p(z) = n∑ k=0 ak z k (ak 6= 0, 0 ≤ k ≤ n) be a non-consant polynomial with complex coefficients. then its all zeros of p(z) lie in the annulus <2 = {z ∈ c : r1 ≤ |z| ≤ r2} , (2.6) where r1 = min 1≤k≤n { lk ln+2 − 3 ∣∣∣∣a0ak ∣∣∣∣ }1 k (2.7) and r2 = max 1≤k≤n { ln+2 − 3 lk ∣∣∣∣an−kan ∣∣∣∣ }1 k . (2.8) 3. tribonacci polynomials for r = 3 in (1.1), r−bonacci polynomials are named tribonacci polynomials, defined by the recurrence relation tn+3 (x) = x 2tn+2 (x) + xtn+1 (x) + tn (x) (3.1) with t0 (x) = 0, t1 (x) = 1 and t2 (x) = x 2. despite its complicate form, the coefficients of these polynomials have many interesting properties. it is known that tribonacci zeros constitute 3-stars as shown below in figure 1. as seen in the above figure 1, the zeros of these polynomial being invariant with rotation 2π/3 are separated into 3 sets. so, it may be interesting to find a smallest disc that contains the zeros of these polynomials since the explicit expressions for the zeros of these polynomials are not known. in order to examine a new disc containing the zeros of a given polynomial in (3.1), we have used the following identity n∑ i=1 fi = fn+2 − 1, (3.2) int. j. anal. appl. 16 (3) (2018) 371 figure 1. the zeros of t25(x) where fn is the n−th fibonacci number in [14]. recall that fibonacci numbers are defined by fn = fn−1 + fn−2 for n ≥ 2 with the initial conditions f0 = 0, f1 = 1. this identity (3.2) can be found in [5]. now, we consider the following theorem. theorem 3.1. [14] let p(z) = n∑ k=0 ak z k (ak 6= 0, 0 ≤ k ≤ n) be a non-consant complex polynomial. then all its zeros lie in the annulus <3 = {z ∈ c : r1 ≤ |z| ≤ r2} , (3.3) where r1 = min 1≤k≤n { fk fn+2 − 1 ∣∣∣∣a0ak ∣∣∣∣ }1 k (3.4) and r2 = max 1≤k≤n { fn+2 − 1 fk ∣∣∣∣an−kan ∣∣∣∣ }1 k . (3.5) now we give the following theorem. theorem 3.2. let tn(x) be a tribonacci polynomial with n ≡ k(mod 3). then all its zeros lie <3 = { z ∈ c : r1 ≤ |z| ≤ { fn+2 − 1 f3 |an−3| }1 3 } , (3.6) where r1 = { f3 fn+2−1 ∣∣∣ 1a3 ∣∣∣}13 for k = 1 and r1 = 0 for k = 0, 2. proof. the result is obviously obtained when the r-nomial coefficients are written instead � int. j. anal. appl. 16 (3) (2018) 372 it will be immediately noticed that the areas containing the zeros of tribonacci polynomials will be a disc or an annulus. after reaching this result for the zeros of these polynomials, it is numerically examined by comparing with the existing boundaries. when comparisons are made, one of the oldest boundaries and the latest bounds are selected. below is a table giving inner, outer radius and the area for the zeros of these polynomials. polynomials <1 − area <2 − area cauchy′s bound <3 − area t3 (x) |z| ≤ 1.31 − 5.39 |z| ≤ 1.55 − 7.54 |z| ≤ 2 − 12.56 |z| ≤ 1.51 − 7.16 t4 (x) 0.57 ≤ |z| ≤ 2.20 − 14.18 0.35 ≤ |z| ≤ 2.80 − 24.24 |z| ≤ 3 − 28.27 0.36 ≤ |z| ≤ 2.71 − 22.66 t5 (x) |z| ≤ 4.07 − 52.04 |z| ≤ 4.48 − 63.05 |z| ≤ 4 − 50.26 |z| ≤ 4.32 − 58.62 t6 (x) |z| ≤ 7.90 − 196.06 |z| ≤ 6.83 / 146.55 |z| ≤ 7 − 153.93 |z| ≤ 6.58 − 136.02 t7 (x) 0.10 ≤ |z| ≤ 15.81 − 785.22 0.09 ≤ |z| ≤ 10.16 − 324.26 |z| ≤ 11 − 380.13 0.10 ≤ |z| ≤ 9.79 − 301.07 t8 (x) |z| ≤ 32.28 − 3273.53 |z| ≤ 14.89 − 696.52 |z| ≤ 17 − 907.92 |z| ≤ 14.35 − 646.92 t9 (x) |z| ≤ 66.92 − 14069 |z| ≤ 21.62 − 1468.46 |z| ≤ 31 − 3019.07 |z| ≤ 20.83 − 1363.1 t10 (x) 0.14 ≤ |z| ≤ 140.30 − 61839.3 0.03 ≤ |z| ≤ 31.15 − 3048.36 |z| ≤ 51 − 8171.28 0.03 ≤ |z| ≤ 30.02 − 2831.2 above, the upper bounds containing the zeros of the first ten tribonacci polynomials are shown numerically. (since the zero of t0 (x) ,t2 (x) is only 0 and t1 (x) is a fixed polynomial, they are not included in this table.) at first glance, it seems that diaz-barrero’s boundary(<1) is the most useful among others. but, it is clearly observed that actual area is gradually growing except for first few tribonacci polynomials. and this shows that this bound is unusable for these polynomials. on the other hand, it is clear that our boundary(<3) gives better results in terms of both r1, r2 and area, when compared to the bound of cauchy, one of the oldest boundaries and the bound of dalal and govil(<2) which have been recently obtained. although not included in the table, these calculations have been numerically verified for the first 20 tribonacci polynomials. hence, the disk or annulus obtained by our boundary and containing the zeros of each tribonacci polynomial will remain the smallest. this is the desired ideal situation. 4. application in this section, we develop an algorithm. at the beginning of the algorithm, all tribonacci polynomials are generated and then the coefficients are taken to be used at any boundary. the algorithm makes it easy for us to make this computation for the tribonacci polynomial as many as we would like to. furthermore, this algorithm can be developed and adapted for other boundary value problems. and, thanks to this algorithm, testing of the usability of any boundary can be done without needing long and complicated operations. int. j. anal. appl. 16 (3) (2018) 373 table 1. the algorithm t[0]=0; t[1]=1; t[2] = x2; for[i=2,i¡16,i++, t[n](x ):=t(i+1)=x2t(i) + xt(i− 1) + t(i− 2); a[n]:=coefficient[t(n)(x),x,2 i]; a[n-3]:=coefficient[t(n)(x),x,2 i-3]; s[n]:=coefficient[t(n)(x),x,0]; e[n]:=exponent[t(n)(x),x]; print [ max ({ n [ 3 √ |a(n−3)a(n) |(fe(n)+2−1) f3 ]})]] references [1] a. l. cauchy, exercises de mathématiques iv, anné de bure fréres, paris, 1829. [2] g. moore, the limit of the golden numbers is 3/2, fibonacci quart. 32 (1994), 211-217. [3] m. x. he, d. simon, p. e. ricci, dynamics of the zeros of fibonacci polynomials, fibonacci quart. 35 (1997), 160-168. [4] a. dalal, n. k. govil, on region containing all the zeros of a polynomial, appl. math. comput. 219 (2013), 9609-9614. [5] t. koshy, fibonacci and lucas numbers with applications, wiley-interscience, 2001. [6] v. e. hoggat, m. bicknell, generalized fibonacci polynomials, fibonacci quart. 11 (1973), 457-465. [7] v. e. hoggat, m. bicknell, roots of fibonacci polynomials, fibonacci quart. 11 (1973), 271-274. [8] f. matyas, bounds for the zeros of fibonacci-like polynomials, acta acad. paed. agriensis, sec. math. 25 (1998), 15-20. [9] h. prodinger, the asymptotic behavior of the golden numbers, fibonacci quart. 34 (1996), 224-225. [10] j. h. conway, r. guy, the book of numbers, copernicus, new york, 1996. [11] w. goh, m. x. he, p. e. ricci, on the universal zero attractor of the tribonacci-related polynomials, calcolo 46 (2009), 95-129. [12] m. x. he, p. e. ricci, d. simon, numerical results on the zeros of generalized fibonacci polynomials, calcolo 34 (1997), 25-40. [13] j.l. diaz-barrero, an annulus for the zeros of polynomials, j. math. anal. appl. 273 (2002), 349-352. [14] ö. öztunç, r−bonacci polynomials and their derivatives, ph. d. thesis, balıkesir university, (2014). [15] x. li, y. shi, i. gutman, graph energy, springer, new york, 2012. [16] s. fisk, polynomials, roots, and interlacing, arxiv mat/0612833v2, (2006). [17] n. y. özgür, ö. öztunç, on the zeros of r−bonaci polynomials, arxiv:1711.01150, (2017). [18] p. g. anderson, r. s. gaborski, the polynomial method augmented by supervised training for hand-printed character recognition, artificial neural networks and genetic algorithms, proceedings of the international conference, (1993). [19] s. ucar, n. y. ozgur, right circulant matrices with generalized fibonacci and lucas polynomials and coding theory, arxiv:1801.01766 [math.co]. 1. introduction 2. bounds for polynomials 3. tribonacci polynomials 4. application references int. j. anal. appl. (2023), 21:82 on existence and attractivity of ψ-hilfer hybrid fractional-order langevin differential equations savita rathee1, yogeeta narwal2,∗ 1department of mathematics, maharshi dayanand university, rohtak, india 2government college, baund kalan, charkhi dadri-127025, india ∗corresponding author: yogeetawork@gmail.com abstract. the work reported in this article studies the equivalence relationship between fractional integral equation and ψ-hilfer hybrid langevin differential equations of fractional order with nonlocal initial conditions, and then we use this relationship to establish the existence of the results by means of banach algebra and schauder’s fixed point theorem. we then demonstrate the uniform local attractiveness of all the solutions. 1. introduction odes are extended to include fractional differential equations (fdes), where the order of the derivative can be any positive number. for this reason, approaching the problem as an fde typically allows us to model an experimental dynamic more effectively. which fractional derivative (fd) is most appropriate at this point? the solution to this question typically depends on the problem and hence on the collected information. consider using a definition of fractional operators that is more broad to get around the multitude of definitions for fds. for better and more accurate simulations, we use ψ-hilfer fractional derivative(ψ-hfd) and fixed point theory as an important tool to derive existence criterion of solutions. kilbas et al. [1] introduced the notion of fd with respect to another function in the context of the rl fd. similar to this, almeida [15] proposed the ψcaputo fd and looked at a variety of intriguing aspects of this operator. fd operator with two parameters was presented by hilfer [16]. the hilfer derivative unifies the rl fd and caputo fd-based theories of fdes. sousa and oliveira received: jun. 15, 2023. 2020 mathematics subject classification. 26a33, 34a08, 34b15, 47h10. key words and phrases. fractional langevin differential equation; ψ-hilfer fractional derivative; schauder fixed point theorem. https://doi.org/10.28924/2291-8639-21-2023-82 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-82 2 int. j. anal. appl. (2023), 21:82 presented the hilfer fd with respect to another function in [9]; known as the ψ-hfd. the ψ-hfd’s significance stems from the fact that it uses a number of well-known fd operators as its specific cases, for example, rl [1], caputo [1], hadmard [1], riesz [1], erdeyl-kober [1], ψ-caputo [15], katugampola [21], hilfer [16, 17] and so on. with this approach, it is possible to examine a wide range of properties of fde solutions that employ several fd operators using a single fd operator. a number of researches have been conducted using ψ-hfd [6,7,10,12–14,17–20]. “in 2021, bachir et al. [8] proved the existence and attractivity of solutions for ψ-hilfer hybrid fdes: dλ,σ;ψ u(t) v(t,u(t)) = w(t,u(t)); a.e. t ∈r+, (1.1) (ψ(t) − ψ(0))1−ζu(t)|t=0 = u0; u0 ∈r, (1.2) where r+ = [0,∞), 0 < λ < 1, 0 ≤ σ ≤ 1, ζ = λ + σ(1 −λ), dλ,σ;ψ is the ψ-hfd of order λ and type σ, v : r+ ×r→r∗ and w : r+ ×r→r are given functions." “in 2022, kucche et al. [11] established the existence of solutions in the weighted space for the following ψhilfer hybrid fde: dµ,ν;ψ y(t) f (t,y(t)) = g(t,y(t)); a.e. t ∈ (0,t ], (1.3) (ψ(t) − ψ(0))1−ξy(t)|t=0 = y0; y0 ∈r, (1.4) where 0 < µ < 1, 0 ≤ ν ≤ 1, ξ = µ + ν(1 − µ), dµ,ν;ψ is the ψ-hfd of order µ and type ν, f ∈ c(i ×r,r−{0}) is bounded, i = [0,t ] and g ∈ c(i ×r,r)={h|the map w → h(t,w) is continuous for each t and the map t → h(t,w) is measurable for each w}." motivated by [11], [8], the following ivp of the ψ-hilfer type fractional-order langevin equation with nonlocal initial conditions is explored, and the existence and attractivity results are obtained: dν1,β1;ψ 0+ ( dν2,β2;ψ 0+ κ(t) g(t,κ(t)) + pκ(t) ) = f(t,κ(t)) (1.5) κ(t)|t=0 = 0, (ψ(t) − ψ(0))1−γ1−ν2κ(t)|t=0 = κ0 (1.6) where dνi,βi ;ψ 0+ , i = 1, 2 is the ψ-hfd of order νi, 0 < νi < 1 and type βi, 0 ≤ βi ≤ 1; 1 < ν1 + ν2 ≤ 2, f : i×r→r is a continuous function, p ∈r, t ∈ [0,ε] and γi = νi + βi (1 −νi ). special cases: (a) for β1,β2 = 0, ψ(t) = t; we get nonlinear hybrid rl langevin fde for a.e. t ∈ (0,ε) of the form rldν1 ( rldν2 κ(t) g(t,κ(t)) + pκ(t) ) = f(t,κ(t)) κ(t)|t=0 = 0 int. j. anal. appl. (2023), 21:82 3 for ν1 = 1,ν2 = 1, we generate hybrid differential equations of integer order d dt { d dt κ(t) g(t,κ(t)) + pκ(t) } = f(t,κ(t)) κ(t)|t=0 = 0 (b) for ν1 = 0, β2 = 0, p = 0, ψ(t) = t, κ0 = 0 and for a.e. t ∈ (0,ε) we obtain the nonlocal hybrid fdes of the form rldν2 [ κ(t) g(t,κ(t)) ] = f(t,κ(t)) κ(t)|t=0 = 0 the existence results for which are obtained in [24]. for ν2 = 1 and t ∈ (0,ε) a.e., the investigations of [4] regarding the hybrid differential equation of integer order are incorporated into the results of the current work d dt [ κ(t) g(t,κ(t)) ] = f(t,κ(t)) κ(t)|t=0 = 0 (c) the outcomes are relevant to the below mentioned nonlinear ψ-hfd for t ∈ (0,ε) and g = 1 dν1,β1;ψ 0+ ( dν2,β2;ψ 0+ + p ) κ(t) = f(t,κ(t)) κ(t)|t=0 = 0, (ψ(t) − ψ(0))1−γ1−ν2κ(t)|t=0 = κ0 when g is nothing but constant function with value equal to 1, β1,β2 = 0,ν1 = 0, ψ(t) = t, i.e. γ1 = 0 the investigation of nonlinear fdes involving rl fd is among the obtained outcomes of [22] for a.e. t ∈ (0,ε) rldν2κ(t) = f(t,κ(t)) t1−ν2κ(t)|t0 = κ0 ∈r. with β1 = 0,β2 = 1,ν1 = 0, ψ(t) = t, i.e.γ1 = 0, g = 1 the investigation of nonlinear fdes utilising the caputo fd is among the obtained outcomes [23] for a.e. t ∈ (0,ε) cdν2κ(t) = f(t,κ(t)) κ(t)|t=0 = κ0 the following is how the paper is organised: section 2 goes through a few crucial foundational concepts from fractional calculus. finding corresponding integral equations for the hybrid-hfde is the main topic of section 3, along with the existence of solutions to the ivp (1.5)-(1.6) using fpt and uniform local attractiveness of the solution. section 4 provides a summary of the results. 4 int. j. anal. appl. (2023), 21:82 2. auxiliary results let lp([0,ε],r) be the banach space of all lebesgue measurable functions from [0,ε] to r with ‖f‖lp[0,ε] < ∞. consider a function which is differentiable and increasing for all t ∈ [0,ε] = i, say ψ ∈ c1(i,r), where (0 < ε < ∞) and ψ ′ (t) 6= 0. we will be using, pσψ(t,s) = (ψ(t) − ψ(s)) σ, wνψ(t,s) = ψ ′ (s)(ψ(t) − ψ(s))ν and γ1 + ν2 = σ throughout this paper to reduce the length of the equations. here, we’ve listed some of the spaces used in this article: (a) c1−σ;ψ([0,ε],r), the weighted banach space of all partially ordered functions with ||.||c1−σ;ψ[0,ε] defined as: c1−σ;ψ[0,ε] = {h : (0,ε] →r|p1−σψ (t, 0)h(t) ∈c[0,ε]} where, ||h||c1−σ;ψ[0,ε] = max t∈[0,ε] |p1−σψ (t, 0)h(t)|. let ξ = ( c1−σ;ψ(i,r),‖.‖c1−σ;ψ(i,r) ) be a banach algebra where (κy)(t) = κ(t)y(t), t ∈ i is how the product of vectors is defined. (b) let bc = bc(r+) be the banach space of all functions φ : r+ to r which are bounded as well as continuous. (c) bc1−σ = bc1−σ(r+), denote the weighted space defined by bc1−σ = {φ : r+ →r : p1−σψ (t, 0)φ(t) ∈bc} of all bounded and continuous functions with the norm ‖φ‖bc1−σ = sup t∈r+ |p1−σψ (t, 0)φ(t)|. let’s revisit some fractional calculus definitions and characteristics. definition 2.1. [1] “let ν > 0, ν ∈ r, and g ∈ l1([0,ε],r).the ψ-r-l fractional integral of a function g with respect to ψ is defined by iν;ψ a+ = 1 γ(ν) ∫ t a ψ ′ (s)(ψ(t) − ψ(s))ν−1g(s)ds.” definition 2.2. [9] “let n − 1 < ν < n, n ∈ n and g ∈ cn([0,ε],r). the ψ-hfd hdν,β;ψ a+ (.) of a function g of order ν and type 0 ≤ β ≤ 1 is defined by hdν,β;ψ a+ g(t) = i β(n−ν);ψ a+ ( 1 ψ ′ (s) d dt )n i (1−β)(n−ν);ψ a+ g(t).” lemma 2.1. [9] “let ν > 0 and δ > 0. then (i) iν;ψ a+ iν;ψ a+ h(t) = iν+ν;ψ a+ h(t); (ii) iν;ψ a+ (ψ(t) − ψ(a))δ−1 = γ(ν) γ(ν+δ) (ψ(t) − ψ(a))ν+δ−1." and we observe that hdν,β;ψ a+ (ψ(t) − ψ(a))(γ−1) = 0. int. j. anal. appl. (2023), 21:82 5 lemma 2.2. [9] “let f ∈ l(0,ε),n−1 < ν ≤ n,n ∈n, 0 ≤ β ≤ 1, γ = ν + β(1−ν), i(1−β)(n−ν) a+ f ∈ ack[0,ε]. then( iν;ψ a+ hdν,β;ψ a+ f ) (t) = f (t) − n∑ k=1 (ψ(t) − ψ(s))γ−k γ(γ −k + 1) ( 1 ψ ′ (s) d dt )n lim t→a+ (i (1−β)(n−ν) a+ f )(t).” also, note that dν1,β;ψ a+ i ν2;ψ a+ f (t) = i ν2−ν1 a+ f (t), if ν2 > ν1 and dν1−ν2a+ f (t), if ν1 > ν2. for the readers’ convenience, we have included some of the fixed point theorems (fpts) that were utilised in this article. lemma 2.3. [3]“let s be a non-empty closed, convex and bounded subset of the banach algebra ξ and let a : ξ → ξ and b : s → ξ be two operators such that (i) a is lipschitzian with a lipschitz constant α; (ii) b is completely continuous; (iii) y = aybκ =⇒ y ∈ s for all κ ∈ s and (iv) αm < 1 where m = sup{‖bκ‖ : κ ∈ s}. then, the operator equation y = ayby has a solution in s." lemma 2.4. [8] “solution of equation (k(κ))(t) = t are locally attractive if there exists a ball b(κ0,µ) in the space bc such that, for any solutions y = y(t) and σ = σ(t) of above equations that belong to b(κ0,µ) ∩ λ, we can write lim t→∞ (y(t) −σ(t)) = 0. (2.1) if the limit (2.1) is uniform with respect to b(κ0,µ) ∩ λ, where φ 6= λ ⊂ bc, then the solutions are said to be uniformly locally attractive (or, equivalently, that the solutions are locally asymptotically stable)." lemma 2.5. [5] “let m ⊂ bc. then m is relatively compact in bc if the following conditions are satisfied: (i) m is uniformly bounded in bc; (ii) the functions belonging to m are almost equicontinuous in r+, i.e., equicontinuous on every compact set in r+; (iii) the functions from m are equiconvergent, i.e. given � > 0, there exists l(�) > 0 such that |κ(t) − lim t→∞ κ(t)| < �, for any t ≥ l(�) and κ ∈ m." theorem 2.1. [2] (schauder fixed-point theorem). “let f be a banach space, let u be a nonempty bounded convex and closed subset of f, and let k : u → u be a compact and continuous map. then, k has at least one fixed point in u." 6 int. j. anal. appl. (2023), 21:82 3. main results here, we develop an auxiliary lemma showing the relationship between the fractional ivp (1.5)-(1.6) and a corresponding fractional ie. lemma 3.1. the hybrid fractional ivp (1.5)-(1.6) for t ∈ [0,ε] is equivalent to the hybrid fractional ie κ(t) = g(t,κ(t)) { κ0 g(0,κ(0)) pσ−1ψ (t, 0) + i ν1+ν2;ψ 0+ f(t,κ(t)) −piν2;ψ 0+ κ(t) } (3.1) and thus a function κ ∈c1−σ(i,r) is a solution of (1.5)-(1.6) iff it is a solution of (3.1). proof. we shall establish that a solution of the ivp (1.5)-(1.6) is a solution of the fractional ie (3.1). using lemma (2.2) and the ψ-r-l fi of order ν1 on equation (1.5), we obtain dν2,β2;ψ 0+ κ(t) g(t,κ(t)) + pκ(t) = iν1;ψ 0+ f(t,κ(t)) + c0 γ(γ1) pγ1−1 ψ (t, 0). (3.2) using lemma (2.2) and the ψ-r-l fi of order ν2 on equation (3.2), we get κ(t) g(t,κ(t)) = i ν1+ν2;ψ 0+ f(t,κ(t)) −piν2;ψ 0+ κ(t) + c0 γ(σ) pσ−1ψ (t, 0) + c1 γ(γ2) pγ2−1 ψ (t, 0). (3.3) using κ(t)|t=0 = 0, we get c1 = 0 for g(0,κ(0)) = 0 thus, κ(t) = g(t,κ(t)) { c0 γ(σ) pσ−1ψ (t, 0) + i ν1+ν2;ψ 0+ f(t,κ(t)) −piν2;ψ 0+ κ(t) } . (3.4) multiplying p1−σ ψ (t, 0) on both sides of above equation, we get p1−σψ (t, 0)κ(t) = c0 γ(σ) g(t,κ(t)) + p1−σψ (t, 0)g(t,κ(t))i ν1+ν2;ψ 0+ f (t,κ(t)) −p g(t,κ(t))p1−σψ (t, 0)i ν2;ψ 0+ κ(t). applying initial condition (1.6) and substituting t = 0, we obtain c0 = κ0γ(σ) g(0,κ(0)) . replacing c0 in eqn. (3.4), we get κ(t) = g(t,κ(t)) { κ0 g(0,κ(0)) pσ−1ψ (t, 0) + i ν1+ν2;ψ 0+ f(t,κ(t)) −piν2;ψ 0+ κ(t) } . conversely, a solution of the fractional ie (3.1) is also a solution of the ivp (1.5)-(1.6). then, the aforementioned equation may be expressed as κ(t) g(t,κ(t)) = κ0 g(0,κ(0)) pσ−1ψ (t, 0) + i ν1+ν2;ψ 0+ f(t,κ(t)) −piν2;ψ 0+ κ(t). (3.5) operating the ψ-hd, dν2,β2;ψ on both sides and using the lemma (2.2) , we obtain dν2,β2;ψ κ(t) g(t,κ(t)) = κ0 g(0,κ(0)) pγ1−1 ψ (t, 0) + i ν1;ψ 0+ f(t,κ(t)) −pκ(t). int. j. anal. appl. (2023), 21:82 7 again applying dν1,β1;ψ on above equation , we get dν1,β1;ψ ( dν2,β2;ψ κ(t) g(t,κ(t)) + pκ(t) ) = κ0 g(0,κ(0)) dν1,β1;ψpγ1−1 ψ (t, 0) + f(t,κ(t)). now, using the lemma (2.1) dν1,β1;ψpγ1−1 ψ (t, 0) = 0, we get dν1,β1;ψ ( dν2,β2;ψ κ(t) g(t,κ(t)) + pκ(t) ) = f(t,κ(t)). at t = 0 and f(0,κ(0)) = 0, the given equation simplifies to κ(t)|t=0 = 0 and from equation (3.5) and lemma 2.1(ii), we get p1−σ ψ (t, 0)κ(t)|t=0 = κ0. � in the next theorem, we utilise banach algebra to demonstrate the existence of solution for (1.5)(1.6). we require the following hypotheses on g and f in order to establish our conclusion: (a) g is a bounded function in c ( i×r,r−{0} ) such that: (i) κ → κg(t,κ) for t ∈ i a.e. is an increasing map in r ; (ii) for all κ,y ∈r, t ∈ i, such that g satisfies lipchitz condition for second variable. (b) for all κ ∈r and t ∈ i a.e. ∃h1,h2 ∈c(i,r), such that |f(t,κ)| ≤ h1(t) and κ(t) ≤ h2(t). theorem 3.1. if (a)-(b) holds. then, ∃ a solution κ ∈c1−σ;ψ(i,r) of the hybrid fde (1.5)-(1.6) provided l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (ε, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (ε, 0) γ(ν2 + 1) } < 1. (3.6) proof. define, s = {κ ∈ ξ : ‖κ‖c1−σ;ψ(i,r) ≤ r} where r = k {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (t, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (t, 0) γ(ν2 + 1) } and k is bound on g. it is evident that s is a bounded subset of ξ which is closed and convex. define a : ξ → ξ and b : s → ξ as aκ(t) =g(t,κ(t)), t ∈ i, bκ(t) = κ0 g(0,κ(0)) pσ−1ψ (t, 0) + 1 γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds −p 1 γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds thus, equation (3.1) is nothing but κ = aκbκ, κ ∈ ξ. we shall demonstrate that a and b meet all of the criteria of lemma(2.3): firstly, we shall prove that ‖aκ −ay‖c1−σ,ψ(i,r) ≤l‖κ −y‖c1−σ,ψ(i,r) (3.7) 8 int. j. anal. appl. (2023), 21:82 i.e. a is an operator satisfying lipschitz condition. from assumption (a)(ii), we observe that |p1−σψ (t, 0)(aκ(t) −ay(t))| = ∣∣∣∣p1−σψ (t, 0) ( g(t,κ(t)) −g(t,y(t)) )∣∣∣∣ ≤l|p1−σψ (t, 0)(κ(t) −y(t))| ≤l‖κ −y‖c1−σ,ψ(i,r) next, we need to prove that b : s → ξ is completely continuous. for this we shall prove that b is continuous, uniformly bounded and equicontinuous. for continuity of b, consider a sequence κn →κ in s. then, ‖bκn −bκ‖c1−σ;ψ(i,r) = max t∈i |p1−σψ (t, 0) ( bκn(t) −bκ(t) ) | ≤ max t∈i { p1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κn(s)) −f(s,κ(s))|ds −p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κn(s) −κ(s)|ds } . as n →∞, ‖bκn −bκ‖c1−σ;ψ(i,r) → 0 by virtue of continuity of f and lebesgue dominated convergence theorem. for any t ∈ i and κ ∈s, we shall exhibit that b(s) = {bκ : κ ∈s} is uniformly bounded. |p1−σψ (t, 0)bκ(t)| ≤ ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s))|ds + p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s)|ds ≤ ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (t, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (t, 0) γ(ν2 + 1) . therefore, ‖bκ‖c1−σ;ψ(i,r) ≤ ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (t, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (t, 0) γ(ν2 + 1) . (3.8) now, for any κ ∈s and t1, t2 ∈ i with t1 < t2 we shall prove the equicontinuity of b(s). making use of assumption (b), we have |p1−σψ (t2, 0)bκ(t2) −p 1−σ ψ (t1, 0)bκ(t1)| = ∣∣∣∣ { κ0 g(0,κ(0)) + p1−σ ψ (t2, 0) γ(ν1 + ν2) ∫ t2 0 wν1+ν2−1 ψ (t2, 0)f(s,κ(s))ds −p p1−σ ψ (t2, 0) γ(ν2) ∫ t2 0 wν2−1 ψ (t2,s)|κ(s)|ds } − { κ0 g(0,κ(0)) + p1−σ ψ (t1, 0) γ(ν1 + ν2) ∫ t1 0 wν1+ν2−1 ψ (t1,s)f(s,κ(s))ds int. j. anal. appl. (2023), 21:82 9 −p p1−σ ψ (t1, 0) γ(ν2) ∫ t1 0 wν2−1 ψ (t1,s)|κ(s)|ds }∣∣∣∣ ≤ ∣∣∣∣p1−σψ (t2, 0)γ(ν1 + ν2) ‖h1‖∞ ∫ t2 0 wν1+ν2−1 ψ (t1, 0)ds− p1−σ ψ (t1, 0) γ(ν1 + ν2) ‖h1‖∞ ∫ t1 0 wν1+ν2−1 ψ (t1,s)ds −p p1−σ ψ (t2, 0) γ(ν2) ‖h2‖∞ ∫ t2 0 wν2−1 ψ (t2,s)ds + p p1−σ ψ (t1, 0) γ(ν2) ‖h2‖∞ ∫ t1 0 wν2−1 ψ (t2,s)ds ∣∣∣∣ ≤ ‖h1‖∞ γ(ν1 + ν2) { p1−γ1+ν1 ψ (t2, 0) −p 1−γ1+ν1 ψ (t1, 0) } + p ‖h2‖∞ γ(ν2) { p1−γ1 ψ (t1, 0) −p 1−γ1 ψ (t2, 0) } . thus, the continuity of ψ implies |p1−σ ψ (t2, 0)bκ(t2) −p1−σψ (t1, 0)bκ(t1)|→ 0 as |t1 − t2| → 0. thus, arzela-ascoli theorem implies b(s) is relatively compact and hence a compact operator as a result. it is completely continuous from the continuity and compactness of b : s → ξ. now, we shall show that for any u ∈ ξ, u = aubκ =⇒ u ∈s for all κ ∈s. let any u ∈ ξ and κ ∈ s such that u = aubκ. the function g being bounded and using the hypothesis (b), for any t ∈ i, we have |p1−σψ (t, 0)u(t)| = |p 1−σ ψ (t, 0)au(t)bκ(t)| ≤ ∣∣∣∣p1−σψ (t, 0)g(t,u(t)) { κ0 g(0,κ(0)) pσ−1ψ (t, 0) + 1 γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds−p 1 γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds }∣∣∣∣ ≤|g(t,u(t))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s))|ds + p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s)|ds } ≤k {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2 + 1)‖h1‖∞pν1+ν2ψ (t, 0) + pp 1−σ ψ (t, 0) γ(ν2 + 1) ‖h2‖∞pν2ψ (t, 0) } . i.e., ‖u‖c1−σ;ψ(i,r) ≤ k {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (t, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (t, 0) γ(ν2 + 1) } = r =⇒ u ∈s. in the end, we shall show that for m = sup{‖bu‖c1−σ;ψ(i,r) : u ∈ s}, we have αm < 1. utilising inequality (3.8), we obtain m = sup { ‖bκ‖c1−σ;ψ(i,r) : κ ∈s } ≤ {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (ε, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (ε, 0) γ(ν2 + 1) } 10 int. j. anal. appl. (2023), 21:82 making use of inequality (3.7), we get α = l. therefore, as a consequence of the condition (3.6), we get the required αm ≤l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ‖h1‖∞p1−γ1+ν1ψ (ε, 0)γ(ν1 + ν2 + 1) + p‖h2‖∞p 1−γ1 ψ (ε, 0) γ(ν2 + 1) } < 1. on applying lemma (2.3), the solution for equation κ = aκbκ in s is obtained and thus for hybrid fde (1.5)-(1.6). � using schauder’s fpt, we can now exhibit the existence and attractiveness of solutions. assume the following: (c) for each κ ∈ bc1−σ, t → f(t,κ(t)) is measurable on r+ ; for a.e. t ∈ r+ the mapping κ →f(t,κ(t)) is continuous on bc1−σ and κ →g(t,κ(t)) is continuous and bounded. (d) for each κ ∈r and a.e. t ∈r+, ∃ t : r+ →r+ such that t is a continuous function and f(t,κ(t)) ≤ t (t) 1 + p|κ| , lim t→∞ p1−σψ (t, 0)(i ν1+ν2;ψ 0+ + pi ν2;ψ 0+ )t (t) = 0. set t∗ = sup t∈r+ p1−σψ (t, 0)(i ν1+ν2;ψ 0+ + pi ν2;ψ 0+ )t (t) < ∞. theorem 3.2. if (c)(d) holds, then, ∃ at least one solution for problem (1.5) defined on r+ which is uniformly locally attractive. proof. define k for κ ∈bc1−σ (kκ)(t) = g(t,κ(t)) { κ0pσ−1ψ (t, 0) g(0,κ(0)) + 1 γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds −p 1 γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds } let function g be bounded by n . now, for κ ∈bc1−σ, t ∈r+ |p1−σψ (t, 0)(kκ)(t)| ≤|g(t,κ(t))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s))|ds + pp1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s)|ds } , ≤n {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + t∗ } = r∗. ‖k(κ)‖bc ≤r∗, (3.9) implies, k(κ) ∈bc1−σ and k(bc1−σ) ⊂bc1−σ as a result of the continuity of k(κ) on r+; for any κ ∈bc1−σ. consider br∗ = b(0,r∗) = {g ∈bc1−σ : ‖g‖bc1−σ ≤ r∗}. int. j. anal. appl. (2023), 21:82 11 equation (3.9) implies k transforms the ball br∗ into itself. from lemma (3.1) the solutions of problem (1.5)-(1.6) are nothing but the fixed points of k(κ). we shall show that the operator k satisfies all the assumptions of theorem (2.1). step 1. firstly, we shall prove the continuity of k. consider a convergent sequence {κn}n∈n in br∗ such that κn →κ . then, for each t ∈r+, we have |q1−σψ (t, 0)(kκn)(t) −q 1−σ ψ (t, 0)(kκ)(t)| ≤ ∣∣∣∣g(t,κn(t)) { κ0 g(0,κ(0)) + q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κn(s))ds−p q1−σ ψ (t, 0) γ(ν2)∫ t 0 wν2−1 ψ (t,s)κn(s)ds } −g(t,κ(t)) { κ0 g(0,κ(0)) + q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds −p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣g(t,κn(t)) { κ0 g(0,κ(0)) + q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κn(s))ds−p q1−σ ψ (t, 0) γ(ν2)∫ t 0 wν2−1 ψ (t,s)κn(s)ds } −g(t,κ(t)) { κ0 g(0,κ(0)) + q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κn(s))ds −p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κn(s)ds } + g(t,κ(t)) {∣∣∣∣ κ0g(0,κ(0)) + q 1−σ ψ (t, 0) γ(ν1 + ν2)∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κn(s))ds−p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κn(s)ds } −g(t,κ(t)) { κ0 g(0,κ(0)) + q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds−p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣g(t,κn(t)) −g(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0g(0,κ(0)) + q 1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κn(s))ds −p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κn(s)ds ∣∣∣∣ } + |g(t,κ(t))| {∣∣∣∣q1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)( f(s,κn(s)) −f(s,κ(s)) ) ds−p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)(κn(s) −κ(s))ds ∣∣∣∣ } ≤ ∣∣∣∣g(t,κn(t)) −g(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + q1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κn(s))|ds + p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κn(s)|ds } + n { q1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s) |f(s,κn(s)) −f(s,κ(s))|ds + p q1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κn(s) −κ(s)|ds } . case i. for t ∈ [0,ε], by applying lebesgue dominated convergence theorem and κn →κ as n →∞ on above equation along with the continuity of g and f, we get ‖k(κn)−k(κ)‖bc1−σ → 0 as n →∞. 12 int. j. anal. appl. (2023), 21:82 case ii. for t ∈ (ε,∞), then, from the hypotheses and above equation, we have |p1−σψ (t, 0)(kκn)(t) −p 1−σ ψ (t, 0)(kκ)(t)| ≤ ∣∣∣∣g(t,κn(t)) −g(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0) ( i ν1+ν2;ψ 0+ f(s,κn(s)) + piν2;ψ0+ κn(s) ) + n { p1−σψ (t, 0) ( i ν1+ν2;ψ 0+ |f(s,κn(s)) −f(s,κ(s))| + piν2;ψ 0+ |κn(s) −κ(s)| )} ≤ ∣∣∣∣g(t,κn(t)) −g(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0) ( i ν1+ν2;ψ 0+ + pi ν2;ψ 0+ ) t (t) } + 2np1−σψ (t, 0) ( i ν1+ν2;ψ 0+ + pi ν2;ψ 0+ ) t (t). since, κn → κ as n → ∞, g is continuous and p1−σψ (t, 0)(i ν1+ν2;ψ 0+ + pi ν2;ψ 0+ )t (t) → 0 as t → ∞, it follows from above equation that ‖k(κn) −k(κ)‖bc1−σ → 0 as n →∞. step 2. on every compact subset [0,ε] of r+,ε > 0, we need to prove the uniform boundedness and equi-continuity of l(br∗). since br∗ is bounded and l(br∗) ⊂ br∗, so l(br∗) is uniformly bounded. for each κ ∈ br∗ and t1, t2 ∈ [0,ε], t1 < t2, we have |p1−σψ (t2, 0)(kκ)(t2) −p 1−σ ψ (t1, 0)(kκ)(t1)| ≤ ∣∣∣∣g(t2,κ(t2)) { κ0 g(0,κ(0)) + p1−σψ (t2, 0) γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t1,s)f(s,κ(s))ds−p p1−σψ (t2, 0) γ(ν2)∫ t2 0 wν2−1ψ (t2,s)κ(s)ds } −g(t1,κ(t1)) { κ0 g(0,κ(0)) + p1−σψ (t1, 0) γ(ν1 + ν2) ∫ t1 0 wν1 +ν2−1ψ (t1,s) f(s,κ(s))ds−p p1−σψ (t1, 0) γ(ν2) ∫ t1 0 wν2−1ψ (t2,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣g(t2,κ(t2)) { κ0 g(0,κ(0)) + p1−σψ (t2, 0) γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t2,s)f(s,κ(s))ds−p p1−σψ (t2, 0) γ(ν2)∫ t2 0 wν2−1ψ (t2,s)κ(s)ds } −g(t1,κ(t1)) { κ0 g(0,κ(0)) + p1−σψ (t2, 0) γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t2,s) f(s,κ(s))ds−p p1−σψ (t2, 0) γ(ν2) ∫ t2 0 wν2−1ψ (t2,s)κ(s)ds } + g(t1,κ(t1)) { κ0 g(0,κ(0)) + p1−σψ (t2, 0) γ(ν1 + ν2)∫ t2 0 wν1 +ν2−1ψ (t2,s)f(s,κ(s))ds−p p1−σψ (t2, 0) γ(ν2) ∫ t2 0 wν2−1ψ (t2,s)κ(s)ds } −g(t1,κ(t1)){ κ0 g(0,κ(0)) + p1−σψ (t1, 0) γ(ν1 + ν2) ∫ t1 0 wν1 +ν2−1ψ (t1,s)f(s,κ(s))ds−p p1−σψ (t1, 0) γ(ν2) ∫ t1 0 wν2−1ψ (t1,s)κ(s)ds }∣∣∣∣ ≤|g(t2,κ(t2)) −g(t1,κ(t1))| {∣∣∣∣ κ0g(0,κ(0)) + p 1−σ ψ (t2, 0) γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t2,s)f(s,κ(s))ds + p p1−σψ (t2, 0) γ(ν2) ∫ t2 0 wν2−1ψ (t2,s)κ(s)ds ∣∣∣∣ } + |g(t1,κ(t1))| {∣∣∣∣p1−σψ (t2, 0)γ(ν1 + ν2) ∫ t1 0 wν1 +ν2−1ψ (t2,s)f(s,κ(s))ds + p1−σψ (t2, 0) γ(ν1 + ν2) ∫ t2 t1 wν1 +ν2−1ψ (t2,s)f(s,κ(s))ds− p1−σψ (t1, 0) γ(ν1 + ν2) ∫ t1 0 wν1 +ν2−1ψ (t1,s)f(s,κ(s))ds } int. j. anal. appl. (2023), 21:82 13 + p { p1−σψ (t2, 0) γ(ν2) ∫ t1 0 wν2−1ψ (t2,s)κ(s)ds + p1−σψ (t2, 0) γ(ν2) ∫ t2 t1 wν2−1ψ (t2,s)κ(s)ds− p1−σψ (t1, 0) γ(ν2)∫ t1 0 wν2−1ψ (t1,s)κ(s)ds ∣∣∣∣ } ≤|g(t2,κ(t2)) −g(t1,κ(t1))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t2, 0)γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t1,s)|f(s,κ(s))|ds + p p1−σψ (t2, 0) γ(ν2) ∫ t2 0 wν2−1ψ (t2,s)|κ(s)|ds } + l {(∫ t1 0 ∣∣∣∣p1−σψ (t2, 0)γ(ν1 + ν2) wν1 +ν2−1ψ (t2,s) − p1−σψ (t1, 0) γ(ν1 + ν2) wν1 +ν2−1ψ (t1,s) ∣∣∣∣|f(s,κ(s))|ds + p1−σψ (t2, 0)γ(ν1 + ν2) ∫ t2 t1 wν1 +ν2−1ψ (t2,s)|f(s,κ(s))|ds ) + p (∫ t1 0 ∣∣∣∣p1−σψ (t2, 0)γ(ν2) wν2−1ψ (t2,s) − p 1−σ ψ (t1, 0) γ(ν2) wν2−1ψ (t1,s) ∣∣∣∣|κ(s)|ds + p1−σψ (t2, 0) γ(ν2) ∫ t2 t1 wν2−1ψ (t2,s)|κ(s)|ds )} ≤|g(t2,κ(t2)) −g(t1,κ(t1))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t2, 0)γ(ν1 + ν2) ∫ t2 0 wν1 +ν2−1ψ (t1,s)t (s)ds } + l {(∫ t1 0 ∣∣∣∣p1−σψ (t2, 0)γ(ν1 + ν2) wν1 +ν2−1ψ (t2,s) − p 1−σ ψ (t1, 0) γ(ν1 + ν2) wν1 +ν2−1ψ (t1,s) ∣∣∣∣t (s)ds + p1−σψ (t2, 0) γ(ν1 + ν2) ∫ t2 t1 wν1 +ν2−1ψ (t1,s)t (s)ds )} . given that t, g are continuous and setting t∗ = supt∈[0,ε] t (t), we obtain |p1−σψ (t2, 0)(kκ)(t2) −p 1−σ ψ (t1, 0)(kκ)(t1)| ≤|g(t2,κ(t2)) −g(t1,κ(t1))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + t∗p1−σψ (t2, 0)γ(ν1 + ν2) ∫ t2 0 wν1+ν2−1 ψ (t1,s)ds } + lt∗(∫ t1 0 ∣∣∣∣p1−σψ (t2, 0)γ(ν1 + ν2) wν1+ν2−1ψ (t1,s) − p 1−σ ψ (t1, 0) γ(ν1 + ν2) wν1+ν2−1 ψ (t1,s) ∣∣∣∣ds + p1−σψ (t2, 0)γ(ν1 + ν2)∫ t2 t1 wν1+ν2−1 ψ (t1,s)ds ) . as t1 → t2, we have |p1−σψ (t2, 0)(kκ)(t2) −p 1−σ ψ (t1, 0)(kκ)(t1)|→ 0. step 3. to prove the equiconvergence of l(br). for any κ ∈ l(br∗), |p1−σψ (t, 0)(kκ)(t)| ≤|g(t,κ(t))| {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + ∣∣∣∣p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)f(s,κ(s))ds −p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)κ(s)ds ∣∣∣∣ } , t ∈r+ ≤l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)(iν1+ν2;ψ0+ + piν2;ψ0+ )t (t) } . 14 int. j. anal. appl. (2023), 21:82 since p1−σ ψ (t, 0)(i ν1+ν2;ψ 0+ + i ν2;ψ 0+ )t (t) → 0 as t →∞, we find |(kκ(t)| ≤l {∣∣∣∣ κ0p1−σ ψ (t, 0)g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)(iν1+ν2;ψ0+ t )(t)p1−σ ψ (t, 0) + p p1−σ ψ (t, 0)(i ν2;ψ 0+ t )(t) p1−σ ψ (t, 0) } . hence, |(kκ)(t) − (kκ)(∞)| → 0 as t → ∞. thus, k : br∗ → br∗ is compact and continuous using lemma (2.5). applying schauder fpt (2.1),the fixed point of k is a solution of (1.5) on r+. step 4. uniform local attractivity. let κ∗ be a solution of hybrid fde (1.5) and κ ∈ b ( κ∗, 2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2t∗ }) , we have |p1−σψ (t, 0)k(κ)(t) −p 1−σ ψ (t, 0)(κ∗)(t)| ≤|p1−σψ (t, 0)k(κ)(t) −p 1−σ ψ (t, 0)k(κ∗)(t)| ≤ ∣∣∣∣g(t,κ(t)) −g(t,κ∗(t)) ∣∣∣∣ {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s))|ds + p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s)|ds } + l { p1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s)) −f(s,κ∗(s))|ds + p p1−σ ψ (t, 0) γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s) −κ∗(s)|ds } ≤2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + p1−σψ (t, 0)γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|t (s)|ds } + 2l p1−σ ψ (t, 0) γ(ν1 + ν2) ∫ t 0 wν1+ν2−1 ψ (t,s)|t (s)|ds ≤2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2t∗ } . thus, we get ‖k(κ) −κ∗‖bc1−σ ≤ 2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2t∗ } . this implies the continuity of k such that k ( b ( κ∗, 2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2t∗ })) ⊂ ( b ( κ∗, 2l {∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2t∗ })) . moreover, if κ is a solution of problem (1.5), then |κ(t) −κ∗(t)| = |kκ(t) −kκ∗(t)| ≤ |g(t,κ(t)) −g(t,κ∗(t))| { pσ−1ψ (t, 0) ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + { 1 γ(ν1 + ν2)∫ t 0 wν1+ν2−1 ψ (t,s)|f(s,κ(s)) −f(s,κ∗(s))|ds− p γ(ν2) ∫ t 0 wν2−1 ψ (t,s)|κ(s) −κ∗(s)|ds } ≤ 2l { pσ−1ψ (t, 0) ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2p1−σψ (t, 0)(iν1+ν2;ψ0+ t )(t)p1−σ ψ (t, 0) + 2p p1−σ ψ (t, 0)(i ν2;ψ 0+ t )(t) p1−σ ψ (t, 0) } . int. j. anal. appl. (2023), 21:82 15 therefore, |κ(t) −κ∗(t)| =|(k(κ(t)) − (k(κ∗(t))| ≤ 2l { pσ−1ψ (t, 0) ∣∣∣∣ κ0g(0,κ(0)) ∣∣∣∣ + 2 p1−σ ψ (t, 0)(i ν1+ν2;ψ 0+ t )(t) p1−σ ψ (t, 0) + 2p p1−σ ψ (t, 0)(i ν2;ψ 0+ t )(t) p1−σ ψ (t, 0) } . (3.10) by using (3.10) and lim t→∞ p1−σψ (t, 0)(i ν1+ν2;ψ 0+ + i ν2;ψ 0+ )t (t) = 0, we conclude lim t→∞ |κ(t) −κ∗(t)| = 0. the lemma (2.4) indicates that solutions of ivp (1.5) has uniform local attractiveness. � 4. conclusion the criteria presented in this work ensured the existence and uniform local attractivity of solutions for some hybrid fdes with ψ-hilfer fd. the methodology is predicated on banach algebras and schauder’s fpt. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, vol. 204. elsevier, amsterdam (2006). [2] a. granas, j. dugundji, elementary fixed point theorems, in: fixed point theory, springer, new york, 2003: pp. 9-84. https://doi.org/10.1007/978-0-387-21593-8_2. [3] b.c. dhage, on α-condensing mappings in banach algebras, math. stud. 63 (1994), 146-152. [4] b.c. dhage, v. lakshmikantham, basic results on hybrid differential equations, nonlinear anal.: hybrid syst. 4 (2010), 414-424. https://doi.org/10.1016/j.nahs.2009.10.005. [5] c. corduneanu, integral equations and stability of feedback systems, academic press, new york, 1973. [6] c. derbazi, h. hammouche, a. salim, m. benchohra, weak solutions for fractional langevin equations involving two fractional orders in banach spaces, afr. mat. 34 (2022), 1. https://doi.org/10.1007/ s13370-022-01035-3. [7] c. nuchpong, s.k. ntouyas, d. vivek, j. tariboon, nonlocal boundary value problems for ψ-hilfer fractional-order langevin equations, bound. value probl. 2021 (2021), 34. https://doi.org/10.1186/s13661-021-01511-y. [8] f. si bachir, s. abbas, m. benbachir, m. benchohra, g.m. n’guérékata, existence and attractivity results for ψ-hilfer hybrid fractional differential equations, cubo. 23 (2021), 145-159. https://doi.org/10.4067/ s0719-06462021000100145. [9] j. vanterler da c. sousa, e. capelas de oliveira, on the ψ-hilfer fractional derivative, commun. nonlinear sci. numer. simul. 60 (2018), 72-91. https://doi.org/10.1016/j.cnsns.2018.01.005. [10] k.d. kucche, a.d. mali, j.v. da c. sousa, on the nonlinear ψ-hilfer fractional differential equations, comput. appl. math. 38 (2019), 73. https://doi.org/10.1007/s40314-019-0833-5. https://doi.org/10.1007/978-0-387-21593-8_2 https://doi.org/10.1016/j.nahs.2009.10.005 https://doi.org/10.1007/s13370-022-01035-3 https://doi.org/10.1007/s13370-022-01035-3 https://doi.org/10.1186/s13661-021-01511-y https://doi.org/10.4067/s0719-06462021000100145 https://doi.org/10.4067/s0719-06462021000100145 https://doi.org/10.1016/j.cnsns.2018.01.005 https://doi.org/10.1007/s40314-019-0833-5 16 int. j. anal. appl. (2023), 21:82 [11] k.d. kucche, a.d. mali, on the nonlinear ψ-hilfer hybrid fractional differential equations, comput. appl. math. 41 (2022), 86. https://doi.org/10.1007/s40314-022-01800-x. [12] k. guida, l. ibnelazyz, k. hilal, s. melliani, existence and uniqueness results for sequential ψ-hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions, aims math. 6 (2021), 8239-8255. https://doi.org/10.3934/math.2021477. [13] k.m. furati, m.d. kassim, n. tatar, existence and uniqueness for a problem involving hilfer fractional derivative, computers math. appl. 64 (2012), 1616-1626. https://doi.org/10.1016/j.camwa.2012.01.009. [14] m. houas, a. devi, a. kumar, existence and stability results for fractional-order pantograph differential equations involving riemann-liouville and caputo fractional operators, int. j. dynam. control. 11 (2022), 1386-1395. https://doi.org/10.1007/s40435-022-01005-4. [15] r. almeida, a caputo fractional derivative of a function with respect to another function, commun. nonlinear sci. numer. simul. 44 (2017), 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006. [16] r. hilfer, ed., applications of fractional calculus in physics, world scientific, singapore (2000). [17] r. hilfer, experimental evidence for fractional time evolution in glass forming materials, chem. phys. 284 (2002), 399-408. https://doi.org/10.1016/s0301-0104(02)00670-5. [18] r.p. agarwal, a. assolami, a. alsaedi, b. ahmad, existence results and ulam–hyers stability for a fully coupled system of nonlinear sequential hilfer fractional differential equations and integro-multistrip-multipoint boundary conditions, qual. theory dyn. syst. 21 (2022), 125. https://doi.org/10.1007/s12346-022-00650-6. [19] s. asawasamrit, a. kijjathanakorn, s.k. ntouyas, j. tariboon, nonlocal boundary value problems for hilfer fractional differential equations, bull. korean math. soc. 55 (2018), 1639-1657. https://doi.org/10.4134/bkms. b170887. [20] s.k. ntouyas, d. vivek, existence and uniqueness results for sequential ψ-hilfer fractional differential equations with multi-point boundary conditions, acta math. univ. comen. 90 (2021), 171-185. [21] u.n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218 (2011), 860-865. https://doi.org/10.1016/j.amc.2011.03.062. [22] v. lakshmikantham, s. leela, j. vasundhara devi, theory of fractional dynamic systems, cambridge scientific publications, cambridge, (2009). [23] v. lakshmikantham, a.s. vatsala, theory of fractional differential inequalities and applications, commun. appl. anal. 11 (2007), 395-402. [24] y. zhao, s. sun, z. han, q. li, theory of fractional hybrid differential equations, computers math. appl. 62 (2011), 1312-1324. https://doi.org/10.1016/j.camwa.2011.03.041. https://doi.org/10.1007/s40314-022-01800-x https://doi.org/10.3934/math.2021477 https://doi.org/10.1016/j.camwa.2012.01.009 https://doi.org/10.1007/s40435-022-01005-4 https://doi.org/10.1016/j.cnsns.2016.09.006 https://doi.org/10.1016/s0301-0104(02)00670-5 https://doi.org/10.1007/s12346-022-00650-6 https://doi.org/10.4134/bkms.b170887 https://doi.org/10.4134/bkms.b170887 https://doi.org/10.1016/j.amc.2011.03.062 https://doi.org/10.1016/j.camwa.2011.03.041 1. introduction 2. auxiliary results 3. main results 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 107-121 http://www.etamaths.com global uniqueness result for functional differential equations driven by a wiener process and fractional brownian motion toufik guendouzi∗ and soumia idrissi abstract. we prove a global existence and uniqueness result for the solution of a mixed stochastic functional differential equation driven by a wiener process and fractional brownian motion with hurst index h > 1/2. we also study the dependence of the solution on the initial condition. 1. introduction fractional brownian motion (fbm) with a hurst parameter h ∈ (0, 1) is defined formally as a continuous centered gaussian process bht = {bht , t ≥ 0} with the covariance (1) rh = 1 2 (t2h + s2h −|t−s|2h). for h > 1/2 it exhibits a property of long-range dependence, which makes it a popular model for long-range dependence in natural sciences, financial mathematics etc. for this reason, equations driven by fractional brownian motion have been an object of intensive study during the last decade. from (1) we deduce that ie(|bht −bhs |2) = |t−s|2h and, as a consequence, the trajectories of bh are almost surely locally α-hölder continuous for all α ∈ (0,h). since bh is not a semimartingale if h 6= 1/2 (see [7]), we cannot use the classical itô theory to construct a stochastic calculus with respect to the fbm. over the last years some new techniques have been developed in order to define stochastic integrals with respect to fbm. essentially two different types of integrals can be defined: one possibility is skorokhod, or divergence integral introduced in the fractional brownian setting in [2]. however this definition is not very practical: it is based on wick rather than usual products, and unlike brownian case, in the fractional brownian case this makes difference when integrating non-anticipating functions because of dependence of increments. this makes this definition worthless for most applications (most notably, those in financial mathematics). moreover, it is impossible to solve stochastic differential equations with such integral except the cases of additive or multiplicative noise; the latter case was considered in [10]. another approach is a pathwise integral, defined first in [13] for fbm with 2010 mathematics subject classification. 60g15, 60g22, 60h10. key words and phrases. fractional brownian motion, wiener process, mixed stochastic functional differential equation, fractional integrals and derivatives. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 107 108 toufik guendouzi and soumia idrissi h > 1/2 as a young integral. the papers [6, 11] were the first to prove existence and uniqueness of stochastic differential equations involving such integrals. later the pathwise approach was extended with the help of lyons?rough path theory to the case of arbitrary h in [1] where also unique solvability of equations with h > 1/4 was proved. very recently, the stochastic differential equations driven simultaneously by a fractional brownian motion and standard brownian motion have been studied by several authors. in [5] guerra and nualart have proved an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven by a multidimensional fractional brownian motion with hurst parameter h > 1/2 and a multidimensional standard brownian motion using a techniques of the classical fractional calculus and the classical itô stochastic calculus. their (existence) result is based on the yamada-watanabe theorem. in [8] the existence and uniqueness of solutions is proved by mishura and shevchenko for differential equations driven by a fractional brownian motion with parameter h > 1/2 and a wiener process in one dimensional case, under mild regularity assumptions on the coefficients. for the same equation, with nonhomogeneous coefficients and random initial condition, the convergence in besov space of the solutions depending on a parameter has been studied in [9] by mishura and posashkova. in this paper we focus on the following mixed stochastic functional differential equation involving wiener process and fractional brownian motion, with nonconstant delay (2) x(t) = φ(0) + ∫ t 0 b(s,xs)ds + ∫ t 0 σw (s,xs)dw(s) + ∫ t 0 σh(s,xs)db h(s), t ≥ 0 x0 = φ ∈cr, where bh = {bh(t); t ∈ [0,t]} is a fractional brownian motion with hurst index h ∈ ( 1 2 , 1), w = {w(t); t ∈ [0,t]} is a wiener process and cr is the space of all continuous functions f from [−r, 0] to ir endowed by the uniform norm ‖ · ‖. here, xt ∈cr denote the function defined by xt(u) = x(t + u), ∀u ∈ [−r, 0] and the coefficients b,σw ,σh : [0,t] ×cr → ir are appropriate functions. the stochastic integral w.r.t. wiener process in (2) is the standard itô integral, and the integral w.r.t. fbm is pathwise generalized lebesgue-stieltjes, or young integral. our goal in this paper is to prove the existence and uniqueness of the solution for equation (2). then we will study the dependance of the solution on the initial condition. we first prove our results for deterministic equations and we will easily apply them pathwise to the wiener processus and fractional brownian motion. the paper is organized as follows. in section 2, we state the problem and list our assumptions on the coefficients of eq. (2). section 3, contains some basic facts about extended stieltjes integrals. in section 4, we derive some precise estimates for the integrals involved in eq. (2). section 5 is devoted to obtain the existence, uniqueness and dependence on the initial data for the solution of the deterministic equations. in section 6, we apply the results of the previous sections to stochastic equations driven by both wiener process and fractional brownian motion and we give the proofs of our main theorems. global uniqueness result 109 2. main result let (ω,f, (ft, t ∈ [0,t]),ip) be a complete probability space with a filtration satisfying the standard conditions. denote by {w(t),ft, t ∈ [0,t]} the standard wiener process adapted to this filtration. suppose that b = {b(t); t ∈ [0,t]} is an ft-fbm with hurst index h ∈ ( 12, 1). consider the mixed stochastic functional differential equation (2) and let us consider the following assumptions on the coefficients. (hb) the function b(t,y) is continuous. moreover, it is lipschitz continuous in the variable y and has linear growth in the same variable, uniformly in t, that is, there exist constants l1 and l2 such that |b(t,y) − b(t,z)| ≤ l1‖y −z‖, |b(t,y)| ≤ l2(1 + ‖y‖), for all y,z ∈cr and t ∈ [0,t]. (hσw ) the function σw (t,y) is continuous. moreover, it is lipschitz continuous in y and has linear growth in the same variable, uniformly in t, that is, there exist constants l3 and l4 such that |σw (t,y) −σw (t,z)| ≤ l3‖y −z‖, |σw (t,y)| ≤ l4(1 + ‖y‖), for all y,z ∈cr and t ∈ [0,t]. (hσh) the function σh(t,y) is continuous and fréchet differentiable in the variable y. moreover, there exist constants l5, l6 and l7 such that |∇yσh(t,y)|l(cr,ir) ≤ l5, |∇yσh(t,y) −∇yσh(t,z)|l(cr,ir) ≤ l6‖y −z‖, |σh(t,y) −σh(s,y)| + |∇yσh(t,y) −∇yσh(s,y)|l(cr,ir) ≤ l7|t−s|, for all y,z ∈cr and t ∈ [0,t]. note that (hσh) implies the linear growth property, i. e., there exists a constant l such that |σh(t,y)| ≤ l(1 + ‖y‖), for all y ∈cr and t ∈ [0,t]. let us define for λ ∈ (0, 1] the space cλ of λ-hölder continuous functions f : [0,t] → ir, equipped with the norm ‖f‖λ := ‖f‖∞ + sup 0≤s<t≤t |f(t) −f(s)| (t−s)λ < ∞, where ‖f‖∞ := sup t∈[0,t] |f(t)|. our main results are the following theorems on the uniqueness, existence and dependence of the solution of eq. (2) on the initial condition. theorem 2.1. let the assumptions (hb), (hσw ) and (hσh) be satisfied, and c be a generic constant which depends on the constants li, 1 ≤ i ≤ 7. (1) if 1 − h < α < h and φ is a stochastic process whose trajectories belong to the space c1−α([−r, 0]) ip-a.s, then there exists a unique solution x of mixed equation (2) with paths in c1−α([−r, 0]) ip-a.s. 110 toufik guendouzi and soumia idrissi (2) if in addition α + h > 3 2 , c is independent of ω and the process φ satisfies ie‖φ‖p1−α < ∞ for p ≥ 1, then the solution x satisfies ie‖x‖ p 1−α < ∞ for p ≥ 1. theorem 2.2. let the assumptions (hb), (hσw ) and (hσh) be satisfied, φ,φ n ∈ c1−α([−r, 0]) and c be a generic constant which depends on the constants li, 1 ≤ i ≤ 7. let x be a solution of the mixed equation (2) and xn the solution of the same equation with φn in place of φ. we assume that 1 −h < α < h. (1) if lim n ‖φn −φ‖1−α = 0, a.s., then we have, for ip-almost all ω ∈ ω, lim n ‖xn(ω,.) −x(ω,.)‖1−α = 0. (2) if in addition α+h > 3 2 , c is independent of ω and φ,φn are deterministic functions, then lim n ie‖xn −x‖p1−α = 0 for p ≥ 1. remark 2.3. we note that the regularity and absolute continuity results for the above mixed equation in d-dimensional case, but without delay, was studied in [5] by guerra and nualart. for the equations driven only by fbm, and the constant delay situation, we refer the reader to [4]. 3. generalized stieltjes integral let α ∈ (0, 1 2 ). for any measurable function f : [0,t] → ir we introduce the following notation (3) ‖f(t)‖α := |f(t)| + ∫ t 0 |f(t) −f(s)| (t−s)α+1 ds. denote by wα,∞ the space of measurable functions f : [0,t] → ir such that (4) ‖f(t)‖α,∞ := sup t∈[0,t] ‖f(t)‖α < ∞. a equivalent norm can be defined by (5) ‖f‖α,µ = sup t∈[0,t] e−µt ( |f(t)| + ∫ t 0 |f(t) −f(s)| (t−s)α+1 ds ) ; µ ≥ 0. note that for any �, (0 < � < α), we have the inclusions cα+�([0,t]; ir) ⊂ wα,∞([0,t]; ir) ⊂cα−�([0,t]; ir) (for more details, see [7]). in particular, both the fractional brownian motion bh, with h > 1 2 , and the standard brownian motion w , have their trajectories in wα,∞. we refer the reader to [7, 5] for further details on this topics. we denote by w 1−α,∞ t ([0,t]; ir) the space of continuous functions g : [0,t] → ir such that ‖g‖1−α,∞,t := sup 0<s<t<t ( |g(t) −g(s)| (t−s)1−α + ∫ t s |g(y) −g(s)| (y −s)2−α dy ) < ∞. clearly, for all � > 0 we have c1−α+�([0,t]; ir) ⊂ w1−α,∞t ([0,t]; ir) ⊂c 1−α([0,t]; ir). denoting λα(g; [0,t]) = 1 γ(1 −α) sup 0<s<t<t |(d1−α t− gt− )(s)|, global uniqueness result 111 where γ(α) = ∫ ∞ 0 rα−1e−rdr is the euler function and (d1−α t− gt− )(s) = eiπ(1−α) γ(α) ( g(s) −g(t) (t−s)1−α + (1 −α) ∫ t s g(s) −g(y) (y −s)2−α dy ) 1(0,t)(s). we also define the space wα,1([0,t]; ir) of measurable functions f on [0,t] such that ‖f‖α,1;[0,t] = ∫ t 0 [ |f(t)| tα + ∫ t 0 |f(t) −f(y)| (t−y)α+1 dy ] dt < ∞. we have wα,∞([0,t]; ir) ⊂ wα,1([0,t]; ir) and ‖f‖α,1;[0,t] ≤ ( t + t 1−α 1−α ) ‖f‖α,∞;[0,t]. in [13], zähle introduced the generalized stieltjes integral (6) ∫ t 0 f(t)dg(t) = (−1)α ∫ t 0 (dα0+f)(t)(d 1−α t− gt−)(t)dt, defined in terms of the fractional derivative operators (dα0+f)(t) = 1 γ(1 −α) ( f(t) tα + α ∫ t 0 f(t) −f(y) (t−y)α+1 dy ) 1(0,t)(t), and (d1−αt− gt−)(t) = e−iπα γ(α) ( gt−(t) (t − t)1−α + (1 −α) ∫ t t gt−(t) −gt−(y) (y − t)2−α dy ) 1(0,t)(t). the following proposition is the estimate of the generalized stieltjes integral. proposition 3.1 ([7]). fix 0 < α < 1 2 . given two functions g ∈ w1−α,∞t (0,t) and f ∈ wα,1(0,t) we set gts(f) = ∫ t s frdgr. then for all r < t ≤ t we have (7) ∣∣∣∣ ∫ t s frdgr ∣∣∣∣ ≤ sup s≤r<τ≤t |(d1−ατ− gτ−)(r)| ∫ t s |(dαs+f)(τ)|dτ ≤ λα(g; [s,t])‖f‖α,1;[0,t] ≤ cα,t λα(g; [s,t])‖f‖α,∞, cα,t = ( t + t1−α 1 −α ) . 4. a priori estimates we will first deduce useful estimates for the integrals involved in equation (2). let λ ∈ ( 1 2 , 1) be fixed and α ∈ (1 −λ,λ). for h ∈ cα(0,t,ir), g ∈ cλ(0,t,ir) and x ∈c1−α([−r,t]), we denote fbt (x) := ∫ t 0 b(s,xs)ds, g σh t (x) := ∫ t 0 σh(s,xs)dhs and g σg t (x) := ∫ t 0 σg(s,xs)dgs. proposition 4.1. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, h ∈ cα(0,t,ir), g ∈ cλ(0,t,ir) and x ∈c1−α([−r,t]). then (1) fbt (x) ∈c1−α(0,t,ir). 112 toufik guendouzi and soumia idrissi (2) g σh t (x) ∈c1−α(0,t,ir). (3) g σg t (x) ∈c1−α(0,t,ir). proof. (1) it is easy to see that fbt (x) ∈c1(0,t,ir) and for 0 ≤ s ≤ t ≤ t |fbt (x) −fbs (x)| = ∣∣∣∣∣ ∫ t 0 b(u,xu)du− ∫ s 0 b(u,xu)du ∣∣∣∣∣ = ∣∣∣∣∣ ∫ t s b(u,xu)du ∣∣∣∣∣ ≤ c(1 + ‖x‖)(t−s) ≤ c(t + tα)(1 + ‖x‖) where c is defined as in above theorems. hence, ‖fbt (x) −f b s (x)‖1−α ≤ c1(1 + ‖x‖1−α), where c1 constants depending only on c,α and t . (2) it follows from the assumption (hσw ) and the garsia-rodemich-rumsey inequality (see, theorem 2.1.3 of [12]) that for any α ∈ (1 −λ,λ) and any t ∈ [0,t] there exists a continuous random variable ζ(α,t,h) in t with finite moments of any order such that |gσht (x) −gσhs (x)| = ∣∣∣∣∣ ∫ t s σh(u,xu)dhu ∣∣∣∣∣ ≤ c(1 + ‖x‖)ζ(α,t,h)(t−s) 1 2 −α ≤ ct 1 2 −αζ(1 + ‖x‖), where ζ is defined as ζ(α,t,h) = cα (∫ t 0 ∫ t 0 |hτ −hθ|2/α |τ −θ|1/α dτdθ )α/2 , cα constants depending only on α. hence, ‖gσht (x) −g σh s (x)‖1−α ≤ c2ζ(1 + ‖x‖1−α), c2 depend only on c,α and t . (3) let 0 ≤ s < t ≤ t . using the proposition 3.1 we have for any α ∈ (1−λ,λ) and λ ∈ ( 1 2 , 1) (8) |gσgt (x) −g σg s (x)| = ∣∣∣∣∣ ∫ t s σg(u,xu)dgu ∣∣∣∣∣ ≤ λα(g) ∫ t s ( |σg(θ,xθ)| (θ −s)α + α ∫ θ s |σg(θ,xθ) −σg(v,xv)| (θ −v)α+1 dv ) dθ. global uniqueness result 113 hence, by condition (hσh), (9) ∫ t s |σg(θ,xθ)| (θ −s)α dθ ≤ ctα (t−s)1−α 1 −α (1 + ‖x‖1−α), and (10) ∫ t s ∫ θ s |σg(θ,xθ) −σg(v,xv)| (θ −v)α+1 dvdθ ≤ ∫ t s ∫ θ s |σg(θ,xθ) −σg(θ,xv)| + |σg(θ,xv) −σg(v,xv)| (θ −v)α+1 dvdθ ≤ ∫ t s ∫ θ s c(θ −v)1−α(‖x‖1−α + (θ −v)α) (θ −v)α+1 dvdθ ≤ c(t−s)1−α tα 1 −α [ ‖x‖1−α α + t ] . then the above estimates lead to the third assertion. � consider the following equivalent norm in the space c1−α defined for any ν ≥ 0 by ‖x‖1−α,ν = sup t e−νt|x(t)| + sup s<t e−νt |x(t) −x(s)| (t−s)1−α . we now present some estimates in order to be able to use the banach fixed point theorem. proposition 4.2. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, h ∈ cα(0,t,ir), g ∈ cλ(0,t,ir) and x ∈c1−α([−r,t]). then there exist c∗i , 1 ≤ i ≤ 3, such that (1) ‖fb(x)‖1−α,ν ≤ c∗1(ν) ( 1 + ‖x‖1−α,ν ) (2) ‖gσh(x)‖1−α,ν ≤ c∗2(ν) ( 1 + ‖x‖1−α,ν ) (3) ‖gσg (x)‖1−α,ν ≤ c∗3(ν) ( 1 + ‖x‖1−α,ν ) where c∗i (ν) → 0 as ν →∞. proof. (1) we remark that e−νt(t−s)α−1|fbt (x) −fbs (x)| ≤ e−νt(t−s)α−1 ∫ t s |b(u,xu)|du ≤ e−νt(t−s)α−1 ∫ t s c(1 + ‖x‖u)du ≤ c sup t e−νt(t−s)α−1 ∫ t s (1 + ‖x‖u)du ≤ c sup t ∫ t s e−ν(t−u) 1 + e−νu‖xu‖ (t−u)1−α du ≤ cνα−1 t2α−1 2α− 1 ( 1 + ‖x‖ν1−α,ν ) , 114 toufik guendouzi and soumia idrissi and therefore ‖fb(x)‖1−α,ν = sup t e−νt|fbt (x)| + sup t e−νt(t−s)α−1|fbt (x) −f b s (x)| ≤ c ν1−α(2α− 1) (t1−α + t2α−1) ( 1 + ‖x‖ν1−α,ν ) . then the above inequalities yield that there exist c∗1(ν) such that ‖fb(x)‖1−α,ν ≤ c∗1(ν) ( 1 + ‖x‖1−α,ν ) , and c∗1(ν) → 0 as ν →∞. (2) we have |gσht (x) −gσhs (x)|e−νt(t−s)α−1 ≤ e−νt(t−s)α−1 ∫ t s |σh(u,xu)|dhu ≤ e−νt(t−s)α−1ct 1 2 −αζ(α,t,h) ∫ t s (1 + ‖xu‖)du ≤ ct 1 2 −αζ(α,t,h) sup t ∫ t s e−ν(t−u) 1 + e−νu‖xu‖ (t−u)1−α du, and therefore ‖gσh(x)‖1−α,ν ≤ c∗2(ν) ( 1 + ‖x‖1−α,ν ) , where c∗2(ν) = c tα−1 (2α− 1)ν1−α ζ(α,t,h), and c∗2(ν) → 0 as ν →∞. (3) using proposition 3.1 we have for s,t ∈ [0,t] with s < t (11) |gσgt (x) −g σg s (x)|e−νt(t−s)α−1 ≤ e−νt(t−s)α−1λα(g) ∫ t s ( |σg(θ,xθ)| (θ −s)α + α ∫ θ s |σg(θ,xθ) −σg(v,xv)| (θ −v)α+1 dv ) dθ. by the assumption (hσh) we have, (12) e−νt(t−s)α−1 ∫ t s |σg(θ,xθ)| (θ −s)α dθ ≤ e−νt(t−s)α−1 ∫ t s c 1 + ‖xθ‖ (θ −s)α dθ ≤ ctα−1 sup t ∫ t s e−ν(t−θ) 1 + e−νθ‖xθ‖ (θ −s)α dθ ≤ c tα(1 − 2α)να ( 1 + ‖x‖1−α,ν ) , and (13) e−νt(t−s)α−1 ∫ t s ∫ θ s |σg(θ,xθ) −σg(v,xv)| (θ −v)α+1 dvdθ ≤ e−νt(t−s)α−1 ∫ t s ∫ θ s c (θ −v)1−α(‖xθ −xv‖ + 1) (θ −v)α+1 dvdθ ≤ ctα−1 sup t ∫ t s ∫ θ s e−ν(θ−v) 1 + e−νv‖xθ −xv‖ (θ −v)2α dvdθ ≤ c t2α−1 (2α− 1)ν2α−1 ( 1 + ‖x‖1−α,ν ) . global uniqueness result 115 thanks to (11)-(12) and (13) one easily remarks that there exist c∗3(ν) such that ‖gσg (x)‖1−α,ν ≤ c∗3(ν) ( 1 + ‖x‖1−α,ν ) , c∗3(ν) → 0 as ν →∞. � proposition 4.3. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, h ∈ cα(0,t,ir), g ∈ cλ(0,t,ir) and x,y ∈c1−α([−r,t]). then there exist c∗i , 1 ≤ i ≤ 3, such that (1) ‖fb(x) −fb(y)‖1−α,ν ≤ c∗1(ν)||x−y||1−α,ν (2) ‖gσh(x) −gσh(y)‖1−α,ν ≤ c∗2(ν)‖x−y‖1−α,ν (3) ‖gσg (x) −gσg (y)‖1−α,ν ≤ c∗3(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν where c∗i (ν) → 0 as ν →∞ for 1 ≤ i ≤ 3. proof. (1) using the lipschitz property of b, we see that for 0 ≤ s < t ≤ t e−νt(t−s)α−1 ∣∣∣∣∣ ( fbt (x)−f b t (y) ) − ( fbs (x)−f b s (y) )∣∣∣∣∣ ≤ cνα−1(2α+1)t2α+1‖x−y‖1−α,ν which imply the first claim. (2) let s,t ∈ [0,t] with s < t, we have by the lipschitz property of σh ≤ e−νt(t−s)α−1 ∣∣∣∣∣ ( g σh t (x) −g σh t (y) ) − ( gσhs (x) −gσhs (x) )∣∣∣∣∣ ≤ e−νt(t−s)α−1 ∫ t s |σh(u,xu) −σh(u,yu)|du ≤ ct 1 2 −αζ(α,t,h) sup t ∫ t s e−νu‖xu −yu‖e−ν(t−u)(t−u)α−1du. hence, ‖gσh(x) −gσh(y)‖1−α,ν ≤ c tα−1 α(α− 1)να−1 ζ(α,t,h)‖x−y‖1−α,ν, which give the result of the second assertion. (3) we have for s,t ∈ [0,t] with s < t∣∣∣(gσgt (x) −gσgt (y))−(gσgs (x) −gσgs (y))∣∣∣e−νt(t−s)α−1 ≤ e−νt(t−s)α−1λα(g)×∫ t s ( |σg(θ,xθ) −σg(θ,yθ)| (θ −s)α + α ∫ θ s |σg(θ,xθ) −σg(θ,yθ) −σg(v,xv) + σg(v,yv)| (θ −v)α+1 dv ) dθ. by the lipschitz property of σg we obtain (14) e−νt(t−s)α−1 ∫ t s |σg(θ,xθ) −σg(θ,yθ)| (θ −s)α dθ ≤ c t1−α α(1 −α)να ‖x−y|‖1−α,ν. 116 toufik guendouzi and soumia idrissi remark that for all u,v ∈ [0,t]∣∣∣(σg(θ,xθ) −σg(θ,yθ))−(σg(v,xv) −σg(v,yv))∣∣∣ ≤ ∣∣∣∫ 1 0 ∇σg(v,vxv + yv −vyv) ( (xθ −yθ) − (xv −yv) ) dv ∣∣∣ + ∣∣∣∫ 1 0 [ ∇σg(θ,vxθ + yθ −vyθ) −∇σg(v,vxv + yv −vyv) ] (xθ −yθ)dv ∣∣∣ ≤ c [ ‖(xθ −yθ) − (xv −yv)‖ + ‖(xθ −yθ‖ ( ‖xθ −xv‖ + ‖yθ −yv‖ + (θ −v) )] , and therefore e−νt(t−s)α−1 ∫ t s ∫ θ s |(σg(θ,xθ) −σg(θ,yθ)) − (σg(v,xv) −σg(v,yv))| (θ −v)α+1 dvdθ ≤ e−νt(t−s)α−1c ∫ t s ∫ θ s ‖(xθ −yθ) − (xv −yv)‖ (θ −v)α+1 dvdθ + e−νt(t−s)α−1c ∫ t s ∫ θ s ‖xθ −yθ‖ (θ −v)α dvdθ + e−νt(t−s)α−1c ∫ t s ∫ θ s ‖xθ −yθ‖(‖xθ −xv‖ + ||yθ −yv‖) (θ −v)α+1 dvdθ ≤ (t−s)α−1c ∫ t s ∫ θ s e−ν(t−θ)‖x−y‖1−α,ν (θ −v)2α dvdθ + (t−s)α−1c ∫ t s ∫ θ s e−ν(t−θ)‖x−y‖1−α,ν (θ −v)α dvdθ + e−νt(t−s)α−1c ∫ t s ∫ θ s ‖xθ −yθ‖(‖x‖1−α + ‖y‖1−α) (θ −v)2α dvdθ. thus (15) e−νt(t−s)α−1 ∫ t s ∫ θ s |(σg(θ,xθ) −σg(θ,yθ)) − (σg(v,xv) −σg(v,yv))| (θ −v)α+1 dvdθ ≤ ct2α−1 ν1−α(1 − 2α) ‖x−y‖1−α,ν + ctα−1 ν(1 −α) ‖x−y‖1−α,ν + ct2α−1 ν1−α(1 −α) ‖x−y‖1−α,ν ( ‖x‖1−α + ‖y‖1−α ) from (14) and (15) we can derive the estimate: sup s<t ∣∣∣(gσgt (x) −gσgt (y))−(gσgs (x) −gσgs (y))∣∣∣e−νt(t−s)α−1 ≤ c∗3(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν, c∗3(ν) → 0 as ν →∞, which implies the third claim. � consider the equation on ir ψ(x)(t) = x(0) + ∫ t 0 b(s,xs)ds + ∫ t 0 σh(s,xs)dh(s) + ∫ t 0 σg(s,xs)dg(s), t ≥ 0, where the function ψ is defined from c1−α([−r,t]) into c1−α([−r,t]) by ψ(x)(t) = x(t), t ∈ [−r, 0]. proposition 4.2 and proposition 4.3 combined together give the following result. global uniqueness result 117 proposition 4.4. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, h ∈ cα(0,t,ir), g ∈ cλ(0,t,ir) and x,y ∈c1−α([−r,t]). then there exist c̃i, i = 1, 2 such that (1) ‖ψ(x)‖1−α,ν ≤‖x0‖1−α,ν + c̃1(ν) ( 1 + ‖x‖1−α,ν ) , (2) ‖ψ(x) −ψ(y)‖1−α,ν ≤ ‖x0 −y0‖1−α,ν + c̃1(ν) ( 1 + ‖x‖1−α + ‖y‖1−α ) ‖x− y‖1−α,ν, c̃i(ν) → 0, i = 1, 2, as ν →∞. 5. deterministic functional equation in this section we fix the parameters λ and α such that 1 2 < λ < 1, 1 −λ < α < λ. consider the deterministic functional equation (16) x(t) = ϕ(0) + ∫ t 0 b(s,xs)ds + ∫ t 0 σh(s,xs)dh(s) + ∫ t 0 σg(s,xs)dg(s), x0 = ϕ ∈cr, for g ∈cλ(0,tir), h ∈cα(0,t,ir) and t ≥ 0. the following theorem is the main result of this section. theorem 5.1. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, ϕ ∈c1−α([−r, 0]). then eq. (16) has unique solution x. moreover the solution is (1 −α)-hölder continuous on [−r,t]. proof. existence. we shall prove the existence of the solution by a fixed point argument. we first define c1−α([−r,t],ϕ) as the space of all x ∈ c1−α([−r,t]) such that x = ϕ on [−r, 0]. let γ be an operator defined from c1−α([−r,t],ϕ) into itself by γ(x)(t) = ϕ(t) for t ∈ [−r, 0] and γ(x)(t) = ϕ(0) + ∫ t 0 b(s,xs)ds + ∫ t 0 σh(s,xs)dh(s) + ∫ t 0 σg(s,xs)dg(s), t ≥ 0. from proposition 4.4. we remark that ‖γ(x)‖1−α,ν ≤‖ϕ‖1−α,ν + c̃(ν) ( 1 + ‖x‖1−α,ν ) , where c̃(ν) → 0 as ν →∞. let ν = ν0 be sufficiently large such that c̃(ν0) ≤ 12 . if ||x||1−α,ν0 ≤ 2(1 + ||ϕ||1−α,ν0 ), then ‖γ(x)‖1−α,ν0 ≤ 2(1 + ‖ϕ‖1−α,ν0 ) and hence γ(bν0 ) ⊂ bν0 where bν0 = { x ∈c1−α([−r,t],ϕ) : ‖x‖1−α,ν0 ≤ 2(1 + ‖ϕ‖1−α,ν0 ) } . as consequence, γ maps bν0 into itself. we now show that there exists ν > ν0 such that the operator γ is a contraction on bν0 under the norm ‖ · ‖1−α,ν. using proposition 4.4., we have for all x,y ∈ c1−α([−r,t],ϕ) ‖γ(x) − γ(y)‖1−α,ν ≤ c̃(ν)(1 + ‖x‖1−α + ‖y‖1−α)‖x−y‖1−α,ν. 118 toufik guendouzi and soumia idrissi if l0 = sup x∈bν0 ‖x‖1−α, then for all x,y ∈ bν0 we have ‖γ(x) − γ(y)‖1−α,ν ≤ c̃(ν)(1 + 2l0)‖x−y‖1−α,ν. let ν > ν0 be sufficiently large such that c̃(ν)(1+2l0) < 1/2. then for all x,y ∈ bν0 we have ‖γ(x) − γ(y)‖1−α,ν ≤ 1 2 ‖x−y‖1−α,ν. consequently, the operator γ is a contraction on the closed subset bν0 of the complete metric space c1−α([−r,t]) which implies that it has a unique fixed point x in bν0 . so from the definition of γ it follows that x is a solution of eq. (16) in c1−α([−r,t]). uniqueness. assume that x,y are two solutions of (16) in the space c1−α([−r,t]) and using proposition 4.4., with ν sufficiently large, we get ||x−y||ν ≤ 1 2 ||x−y||1−α,ν, and, therefore, x = y. � theorem 5.2. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, ϕ ∈c1−α([−r, 0]). then the solution x of eq. (16) satisfies ‖x‖1−α ≤ ĉ1 ( 1 + ‖ϕ‖1−α ) exp ( ĉ2λ 1/α α (g) ) , where ĉ1, ĉ2 are constants depending only on α,t and c. proof. set j(t) = sup s∈[−r,t] |x(s)| + sup −r≤s<u≤t |x(u) −x(s)| (u−s)1−α , t ≥ 0. we have for 0 ≤ s < u ≤ t, |x(u) −x(s)| (u−s)1−α ≤ (u−s)α−1 (∣∣∣∫ u s b(v,xv)dv ∣∣∣+∣∣∣∫ u s σh(v,xv)dh ∣∣∣+∣∣∣∫ u s σg(v,xv)dg ∣∣∣). by assumption (hb) we have (17) (u−s)α−1 (∣∣∣∫ u s b(v,xv)dv ∣∣∣) ≤ c ∫ t 0 (t−v)α−1(1 + j(v))dv. by assumption (hσw ) we get (18) (u−s)α−1 (∣∣∣∫ u s σh(v,xv)dh ∣∣∣) ≤ ct 12−αζ(α,t,h) ∫ u s 1 + ‖xv‖ (u−v)1−α dv ≤ ct 1 2 −αζ(α,t,h) ∫ t 0 (t−v)α−1(1 + j(v))dv. global uniqueness result 119 using proposition 3.1., we have for 1 −λ < α < λ and 1 2 < λ < 1 (u−s)α−1 (∣∣∣∣∣ ∫ u s σg(v,xv)dg ∣∣∣∣∣ ) ≤ (u−s)α−1λα(g) ∫ u s ( |σg(v,xv)| (v −s)α + α ∫ v s |σg(v,xv) −σg(τ,xτ )| (v − τ)α+1 dτ ) dv. by the lipschitz property of σg, we have (19) (u−s)α−1λα(g) ∫ u s |σg(v,xv)| (v −s)α dv ≤ (u−s)α−1λα(g) ∫ u s |σg(v,xv) −σg(v,xs)| + |σg(v,xs)| (v −s)α dv ≤ λα(g)c t2α 1 −α + λα(g)c(t 1−2α + 2α) ∫ t 0 (t−s)α−1j(v)dv, and (20) (u−s)α−1λα(g) ∫ u s ∫ v s |σg(v,xv) −σg(τ,xτ )| (v − τ)α+1 dτdv ≤ (u−s)α−1λα(g) ∫ u s ∫ v s c (v − τ)1−α(1 + ‖xv −xτ‖) (v − τ)α+1 dτdv ≤ c t2α−1 (2α− 1) λα(g) ∫ t 0 (t−v)α−1j(v)dv. inequalities (17), (18), (19) and (20) together imply that sup s<u |x(u) −x(s)| (u−s)1−α ≤ ĉ0(1 + λα(g)) [ 1 + ∫ t 0 (t−s)α−1j(v)dv ] , and j(t) ≤‖ϕ‖1−α + ĉ0(1 + λα(g)) [ 1 + ∫ t 0 (t−s)α−1j(s)ds ] . using the gronwall lemma (see [11]), we have since j(t) ≤‖ϕ‖1−α + ĉ0(1 + λα(g)) [ 1 + ∫ t 0 j(s)(t−s)α−1t1−αs−(1−α)ds ] then ‖x‖1−α ≤ ĉ1(1 + ‖ϕ‖1−α) exp ( ĉ2λ 1/α α (g) ) . � the following result shows the dependance of the solution of eq.(16) on the initial condition. lemma 5.3. let the assumptions (hb), (hσw ) and (hσh) be satisfied for the coefficients b, σh and σg respectively, ϕ,ϕ n ∈ c1−α([−r, 0]). let x be the solution of eq.(16) and xn be the solution of the same equation with ϕn in place of ϕ. then for 1 −λ < α < λ and 1 2 < λ < 1 we have ‖x−xn‖1−α ≤ ĉ1‖ϕ−ϕn‖1−α exp ( ĉ2(‖x‖ 1/α 1−α + ‖x n‖1/α1−α) ) exp ( ĉ3λ 1/α α (g) ) , where ĉ1, ĉ2 and ĉ3 depend only on α,t and c. 120 toufik guendouzi and soumia idrissi proof. for t ≤ 0, let jn(t) = sup −r≤s≤t |x(s) −xn(s)| + sup s<u |x(u) −xn(u) −x(s) + xn(s)| (u−s)1−α . following the same lines as in proof of theorem 5.2. we obtain for t ≤ 0 jn(t) ≤‖ϕ−ϕn‖1−α + ĉ0 [ 1 + λα(g)(1 +‖x‖1−α +‖xn‖1−α) ]∫ t 0 jn(s)(t−s)α−1ds. therefore, jn(t) ≤‖ϕ−ϕn‖1−α+ĉ0 [ 1+λα(g)(1+‖x‖1−α+‖xn‖1−α) ]∫ t 0 j(s)(t−s)α−1t1−αs−(1−α)ds. by the gronwall lemma ([11]) we get ‖x−xn‖1−α ≤ ĉ1‖ϕ−ϕn‖1−α exp ( ĉ3λ 1/α α (g) ) exp ( ĉ2(‖x‖ 1/α 1−α + ‖x n‖1/α1−α) ) . � 6. functional equation driven by a wiener process and fbm in this section we apply the deterministic results in order to prove the main theorems of this article. proof.(theorem 2.1) the existence and uniqueness of the solution can be established following the same argument as in the deterministic theorem 5.1. using theorem 5.2., we get for α > 1 −h ‖x‖1−α ≤ ĉ1(1 + ‖φ‖1−α) exp ( ĉ2λ 1/α α (b) ) , ĉ1, ĉ1 depend only on α,t and c. therefore, for all p ≥ 1 we have (21) ie‖x‖p1−α ≤ 1 2 ĉ 2p 1 ie(1 + ‖φ‖1−α) 2p + 1 2 ie exp ( 2pĉ2λ 1/α α (b) ) . hence for any 0 < γ < 2 we have by fernique’s theorem ([3]) ie [ exp λα(b) γ ] < ∞. as consequence ie||x||p1−α < ∞, ∀p ≥ 1 such that 1 α < 2 with h should be greater than 3 4 and α + h > 3 2 . � proof.(theorem 2.2) the almost-sure convergence can be obtained using lemma 5.3. the ilp-convergence can also be obtained by a dominated convergence argument since we have for any n ≥ 0 ‖xn −x‖1−α ≤ ‖xn‖1−α + ‖x‖1−α ≤ ĉ1(2 + ‖φ‖1−α + ‖φn‖1−α) exp ( ĉ2λ 1/α α (b) ) and ||φn||1−α is bounded, the we can write |‖xn −x‖≤ ĉ4 exp ( ĉ2λ 1/α α (b) ) := y global uniqueness result 121 and iey p < ∞, ∀p ≥ 1. � references [1] l. coutin, z. qian, stochastic analysis, rough path analysis and fractional brownian motions. probab. theory related fields 122 (2002) 108-140. [2] l. decreusefond, a. s. üstünel, stochastic analysis of the fractional brownian motion. potential anal. 10 (1998) 177-214. [3] x. fernique, regularité des trajectoires des fonctions aléatoires gaussiennes, in: ecole d’été de probabilités de saint-flour, iv-1974, in: lecture notes in math. 480 (1975) 1-96. [4] m. ferrante, c. rovira, stochastic delay differential equations driven by fractional brownian motion with hurst parameter h > 1/2. bernoulli 12 (1) (2006) 85-100. [5] j. guerra, d. nualart, stochastic differential equations driven by fractional brownian motion and standard brownian motion. stoch. anal. app. 26 (2008) 1053-1075. [6] k. kubilius, the existence and uniqueness of the solution of the integral equation driven by fractional brownian motion. liet. mat. rink. 40, spec. iss. (2000) 104-110. [7] y. mishura, stochastic calculus for fractional brownian motion and related processes. springer-verlag berlin heidelberg 2008. [8] y. mishura, g. shevchenko, existence and uniqueness of the solution of stochastic differential equation involving wiener process and fractional brownian motion with hurst index h > 1/2. comm. stat. theo. meth., 40 (2011) 3492-3508. [9] y. mishura, s.v. posashkova, stochastic differential equations driven by a wiener process and fractional brownian motion: convergence in besov space with respect to a parameter. j.camwa 62 (2011) 1166-1180. [10] y. mishura, quasi-linear stochastic differential equations with fractional brownian component. theory probab. math. stat. 68 (2004) 103-116. [11] d. nualart, a. răşcanu, differential equations driven by fractional brownian motion. collect. math. 53 (2002) 55-81. [12] d.w. stroock, s.r.s. varadhan, multidimensional diffusion processes. grundlehren der mathematischen wissenschaften, 233. springer-verlag, berlin-new york, 1979. [13] m. zähle, integration with respect to fractal functions and stochastic calculus i. prob. theory relat. fields 111 (1998) 333-374. laboratory of stochastic models, statistic and applications, tahar moulay university po.box 138 en-nasr, 20000 saida, algeria ∗corresponding author int. j. anal. appl. (2023), 21:62 strong and ∆-convergence of a new iteration for common fixed points of two asymptotically nonexpansive mappings j. robert dhiliban1,∗, a. anthony eldred2 1department of mathematics, arul anandar college (autonomous), madurai-625514, tamilnadu, india 2department of mathematics, st. joseph’s college (autonomous), affiliated to bharathidasan university, tiruchirappalli-620002, tamilnadu, india ∗corresponding author: jrdhiliban@gmail.com abstract. the purpose of this paper is to study strong and ∆ convergence of a newly defined iteration to a common fixed point of two asymptotically nonexpansive self mappings in a hyperbolic space framework. we provide an example and a comparison table to support our assertions. 1. introduction globel and kirk [1] introduced the concept of asymptotically nonexpansive mappings and proved that every asymptotically nonexpansive self mapping on a non empty closed subset k of a uniformly convex banach space x posseses a fixed point. ever since, many authors (see, [2], [3], [4] and [5]) have established strong and weak convergence theorems for asymptotically nonexpansive mappings based on the modified mann [6] and ishikawa [7] iterations. tan and xu [8] studied the modified ishikawa iteration scheme:   x1 ∈ k xn+1 = (1 −αn)xn + αntnyn yn = (1 −βn)xn + βntnxn, n ≥ 1 (1.1) where {αn} and {βn} are sequences in (0, 1) bounded away from 0 and 1. received: apr. 28, 2023. 2020 mathematics subject classification. 47h10, 47j25. key words and phrases. asymptotically nonexpansive mapping; common fixed points; ∆-convergence; hyperbolic space. https://doi.org/10.28924/2291-8639-21-2023-62 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-62 2 int. j. anal. appl. (2023), 21:62 aggarwal et al [9] in an attempt to obtain a faster rate of convergence, modified the above iteration process (1.1) as following:   x1 ∈ k xn+1 = (1 −αn)tnxn + αntnyn yn = (1 −βn)xn + βntnxn, n ≥ 1 (1.2) this iteration is called the modified s-iteration process. for further results on ishikawa iteration process, (refer, [10], [11], [12] and [13]). recently, iterative approximations are defined and investigated in the framework of hyperbolic spaces. several authors (refer, [14], [15] and [16]) have put forward different notions of hyperbolic spaces in order to blend convexity and metric structure. the following definition given by kohlenbach [17] is widely used. definition 1.1. [17] a hyperbolic space is a triplet (x,d,w ), where (x,d) is a metric space and w : x2 → [0, 1] is a mapping that satisfies the following conditions: (1) d(u,w (x,y,α)) ≤ (1 −α)d(u,x) + αd(u,y) (2) d(w (x,y,α),w (x,y,β)) = |α−β|d(x,y) (3) w (x,y,α) = w (y,x, (1 −α)) (4) d(w (x,z,α),w (y,v,α)) ≤ (1 −α)d(x,y) + αd(z,v) for all x,y,z,u,v ∈ x and α,β ∈ [0, 1]. 2. preliminaries we recall some definitions and basic concepts which will be useful for our work. definition 2.1. [1] let (x,d) be a metric space and let k be a closed convex subset of x. a mapping t : k → k is said to be asymptotically nonexpansive, if there is a sequence of real numbers {kn}∈ [1,∞) such that lim n→∞ kn = 1 and d(tnx,tny) ≤ knd(x,y) for all x,y ∈ x and ∀ n ∈n. the concept of an asymptotically nonexpansive mapping is a natural generalization of a nonexpansive mapping (d(tx,ty) ≤ d(x,y)). the set f (t ) = {tx = x : x ∈ k} shall denote the set of all fixed points of any mapping t. definition 2.2. [23] a subset k of a hyperbolic space (x,d,w ) is convex if w (x,y,α) ∈ k for all x,y ∈ k and ∀ α ∈ [0, 1]. definition 2.3. [24] a hyperbolic space (x,d,w ) is said to be uniformly convex if for any x,y,z ∈ x, r > 0 and � ∈ (0, 2], there is a δ ∈ (0, 1] so that d(w (x,y, 1 2 ),z) ≤ (1 − δ)r whenever d(x,z) ≤ r, d(y,z) ≤ r and d(x,y) ≥ �r. definition 2.4. [25], [26] consider a bounded sequence {xn} in a hyperbolic space (x,d,w ). for any x ∈ x, define, r(x,{xn}) = lim n→∞ sup d(xn,x) and r({xn}) = inf{r(x,{xn})/x ∈ x}. the int. j. anal. appl. (2023), 21:62 3 asymptotic center a({xn}) of a bounded sequence {xn} is defined as a({xn}) = {x ∈ x/r(x,{xn}) ≤ r(y,{xn}),∀ y ∈ x}. it is well known that in uniformly convex banach spaces, bounded sequences have unique asymptotic centers. the following lemma proved by leustean [27] guarantees that complete uniformly convex hyperbolic spaces also enjoy this property. lemma 2.1. [27] let (x,d,w ) be a complete uniformly convex hyperbolic space. then every bounded sequence {xn} in x has a unique asymptotic center. definition 2.5. [28] a sequence {xn} in a hyperbolic space (x,d,w ) is said to ∆-converge to a point x ∈ x, if every subsequence {zn} of {xn} has x as its unique asymptotic center. lemma 2.2. [29] let k be a nonempty closed convex subset of a uniformly convex hyperbolic space (x,d,w ) and let {xn} be a bounded sequence in k such that a({xn}) = {z} and r({xn}) = ω. if {zm} is a sequence in k such that lim m→∞ r(zm,{xn}) = ω, then lim m→∞ zm = z. lemma 2.3. [29] let (x,d,w ) be a uniformly convex hyperbolic space. let x ∈ x and let {tn} be a sequence in (0, 1) such that δ ≤ tn ≤ 1 −δ for all n ∈n and for some δ > 0. if {xn} and {yn} are sequences in x such that lim n→∞ sup d(xn,x) ≤ c, lim n→∞ sup d(yn,x) ≤ c and lim n→∞ d(w (xn,yn,tn),x) = c for some c ≥ 0, then lim n→∞ d(xn,yn) = 0. lemma 2.4. [3] let {αn}, {βn} and {δn} be sequences of nonnegative numbers such that δn+1 ≤ αnδn + βn ∀ n ∈n. if αn ≥ 1 ∀ n ∈n and ∑∞ n=1(αn − 1) < ∞ and βn < ∞, then limn→∞δn exists. uniformly convex banach spaces and cat(0) spaces are some of the known examples of hyperbolic spaces. sahin and basarir [18] studied the following iterative process in a hyperbolic space setting and established some convergence results under suitable conditions:  x1 ∈ k xn+1 = w (t nxn,t nyn,αn) yn = w (xn,t nxn,βn), n ≥ 1 (2.1) ishikawa type iteration is also employed to study the convergence of common fixed points of two asymptotically nonexpansive mappings. in a banach space framework, das and debata [19] initiated the study of two mapping iterative procedure. the authors in [20] and [21] have studied the following iteration for the convergence of common fixed points:  x1 ∈ k xn+1 = w (xn,s nyn,αn) yn = w (xn,t nxn,βn), n ≥ 1 (2.2) 4 int. j. anal. appl. (2023), 21:62 where s and t are asymptotically nonexpansive mappings with atleast one common fixed point and {αn} and {βn} are sequences in (0, 1). recently, saluja [22] modified the iterative procedure introduced by khan et al [13] in hyperbolic spaces to obtain a faster iterative procedure:  x1 ∈ k xn+1 = w (t nxn,s nyn,αn) yn = w (xn,t nxn,βn), n ≥ 1 (2.3) where s and t are asymptotically nonexpansive mappings with atleast one common fixed point and {αn} and {βn} are sequences in (0, 1). the purpose of this paper is to introduce and study a new iterative procedure (3.1) even in banach spaces to approximate the common fixed points of two asymptotically nonexpansive mappings. we prove strong and ∆ convergence of such an iteration in the general nonlinear framework of hyperbolic spaces. 3. main results in this section, we introduce and study a new iterative scheme to approximate common fixed points of two asymptotically nonexpansive mappings in a hyperbolic space. let (x,d,w ) be a uniformly convex hyperbolic space. let k be a non-empty subset of x. let s and t be two asymptotically nonexpansive self mappings on k. let {αn} and {βn} be sequences in (0, 1) such that, δ ≤ αn, βn ≤ 1 −δ, for all n ∈n and for some δ > 0. we define the following iteration:  x1 = x ∈ k xn+1 = w (s nxn,t nyn,αn) yn = w (xn,s n(tnxn),βn), n ≥ 1 (3.1) lemma 3.1. let k be a non-empty subset of a uniformly convex hyperbolic space x. let s and t be asymptotically nonexpansive self mappings on k with a common sequence of real numbers kn ≥ 1 satisfying ∑ (k2n − 1) < ∞. let f denote the set of all common fixed points of s and t. i.e., f = f (s) ∩f (t ). let p ∈ f. if {xn} and {yn} are sequences as defined in (3.1), then lim n→∞ d(xn,p) and lim n→∞ d(yn,p) exist and lim n→∞ d(xn,p) = lim n→∞ d(yn,p). proof. since p ∈ f (s) ∩f (t ), d(xn+1,p) = d(w (s nxn,t nyn,αn),p) ≤ (1 −αn)d(snxn,p) + αnd(tnyn,p) int. j. anal. appl. (2023), 21:62 5 ≤ (1 −αn)knd(xn,p) + αnknd(yn,p) (3.2) d(yn,p) = d(w (xn,s n(tnxn),βn),p) ≤ (1 −βn)d(xn,p) + βnd(sn(tnxn),p) = (1 −βn)d(xn,p) + βnd(sn(tnxn),sn(tnp)) ≤ (1 −βn)d(xn,p) + βnknd(tnxn,tnp) ≤ (1 −βn)d(xn,p) + βnk2nd(xn,p) = d(xn,p) [ (1 −βn) + βnk2n ] (3.3) by substituting (3.3) in (3.2), we get, d(xn+1,p) ≤ (1 −αn)knd(xn,p) + αnkn [ (1 −βn) + βnk2n ] d(xn,p) = [ (1 −αn)kn + αnkn((1 −βn) + βnk2n) ] d(xn,p) = [ kn −αnknβn + αnβnk3n ] d(xn,p) = [ 1 + (kn − 1) −αnβnkn + αnβnk3n ] d(xn,p) = [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) (3.4) hence, d(xn+1,p) ≤ [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) by lemma 2.4, lim n→∞ d(xn,p) exists. let lim n→∞ d(xn,p) = c. (3.5) from (3.2), we have, d(yn,p) ≤ [ (1 −βn) + βnk2n ] d(xn,p) hence, lim n→∞ sup d(yn,p) ≤ lim n→∞ sup d(xn,p) i.e., lim n→∞ sup d(yn,p) ≤ c. (3.6) now consider, d(xn+1,p) = d(w (s nxn,t nyn,αn),p) ≤ (1 −αn)knd(xn,p) + αnknd(yn,p) = [ 1 + (kn − 1) + (k2n − 1)αnβnkn ] d(xn,p) by (3.5), we have, lim n→∞ sup d(xn+1,p) = c and lim n→∞ sup d(xn,p) = c. hence, from (3.2) and (3.4), lim n→∞ sup [ (1 −αn)knd(xn,p) + αnknd(yn,p) ] = c. 6 int. j. anal. appl. (2023), 21:62 i.e, lim n→∞ sup [ knd(xn,p) −knαnd(xn,p) + αnknd(yn,p) ] = c. since, lim n→∞ sup kn = 1, we have, c + lim n→∞ sup αnkn [ d(yn,p) −d(xn,p) ] = c =⇒ lim n→∞ sup αnkn [ d(yn,p) −d(xn,p) ] = 0. since, lim n→∞ sup αnkn > 0, this will imply that, lim n→∞ sup [ d(yn,p) −d(xn,p) ] = 0. therefore, lim n→∞ sup d(yn,p) = c. similarly, we can show that, lim n→∞ inf d(yn,p) = c. hence, lim n→∞ d(yn,p) = c (3.7) � lemma 3.2. let k be a non-empty subset of a uniformly convex hyperbolic space x. let s and t be asymptotically nonexpansive self mappings on k with a common sequence of real numbers kn ≥ 1 satisfying ∑ (k2n − 1) < ∞. if {xn} is a sequence as defined in (3.1) and d(xn,xn+1) → 0 as n →∞, then lim n→∞ d(xn,sxn) = 0 and lim n→∞ d(xn,txn) = 0. proof. let f denote the set of all common fixed points of s and t . i.e., f = f (s) ∩f (t ). let p ∈ f. now since, lim n→∞ kn = 1, from (3.5), we have, lim n→∞ sup d(tnyn,p) ≤ lim n→∞ sup d(xn,p) = c. similarly, lim n→∞ sup d(snxn,p) ≤ c. (3.8) now, d(xn+1,p) = d(w (s nxn,t nyn,αn),p) ≤ [1 + (kn − 1) + (k2n − 1)αnβnkn]d(xn,p). from (3.5), we have, d(w (snxn,tnyn,αn),p) = c. by lemma 2.3, we have, lim n→∞ d(snxn,t nyn) = 0. (3.9) int. j. anal. appl. (2023), 21:62 7 now consider, d(yn,p) = d(w (xn,s n(tnxn),βn),p) ≤ (1 −βn)d(xn,p) + βnd(sn(tnxn),p) = d(xn,p) [ (1 −βn) + βnk2n ] . since, lim n→∞ sup d(yn,p) = c and lim n→∞ sup d(xn,p) = c, we have, d(w (xn,s n(tnxn),βn),p) → c further, lim n→∞ sup d(sn(tnxn),p) ≤ c. (3.10) so, using lemma 2.3, we conclude that, lim n→∞ d(xn,s n(tnxn)) = 0. (3.11) now, d(yn,xn) = d(w (xn,s n(tnxn),βn),xn) ≤ (1 −βn)d(xn,xn) + βnd(sn(tnxn),xn). using (3.11), lim n→∞ d(xn,yn) = 0. (3.12) from d(yn,s n(tnxn)) ≤ d(yn,xn) + d(xn,sn(tnxn)), we have, lim n→∞ d(yn,s n(tnxn)) = 0. (3.13) now, d(xn+1,s nxn) = d(w (s nxn,t nyn,αn),s nxn) ≤ (1 −αn)d(snxn,snxn) + αnd(tnyn,snxn) ≤ (1 −αn)knd(xn,xn) + αnd(tnyn,snxn). so, lim n→∞ d(xn+1,s nxn) = 0. (3.14) further, d(xn+1,t nyn) = d(w (s nxn,t nyn,αn),t nyn) ≤ (1 −αn)d(snxn,tnyn) + αnd(tnyn,tnyn) ≤ (1 −αn)d(snxn,tnyn) + αnknd(yn,yn) yields, lim n→∞ d(xn+1,t nyn) = 0. (3.15) now, d(xn,s nxn) ≤ d(xn,xn+1) + d(xn+1,snxn) so, lim n→∞ d(xn,s nxn) = 0. (3.16) 8 int. j. anal. appl. (2023), 21:62 now consider, d(xn,sxn) ≤ d(xn,xn+1) + d(xn+1,sn+1xn+1) + d(sn+1xn+1,sn+1xn) + d(sn+1xn,sxn) ≤ d(xn,xn+1) + d(xn+1,sn+1xn+1) + kn+1d(xn+1,xn) + k1d(snxn,xn). thus, we conclude that, lim n→∞ d(xn,sxn) = 0. (3.17) and from, d(xn,t nyn) ≤ d(xn,xn+1) + d(xn+1,tnyn) we obtain, lim n→∞ d(xn,t nyn) = 0 (3.18) and therefore, lim n→∞ d(yn,t nyn) = 0. (3.19) also, d(yn,yn+1) ≤ d(yn,xn) + d(xn,xn+1) + d(xn+1,yn+1). thus, lim n→∞ d(yn,yn+1) = 0. (3.20) now consider, d(yn,tyn) ≤ d(yn,yn+1) + d(yn+1,tn+1yn+1) + d(tn+1yn+1,t n+1yn) + d(t n+1yn,tyn) ≤ d(yn,yn+1) + d(yn+1,tn+1yn+1) + kn+1d(yn+1,yn) + k1d(tnyn,yn). therefore, lim n→∞ d(yn,tyn) = 0. (3.21) by the asymptotic nonexpansive property of t, d(txn,tyn) ≤ k1d(xn,yn). hence, lim n→∞ d(txn,tyn) = 0. (3.22) from, d(xn,txn) ≤ d(xn,yn) + d(yn,tyn) + d(tyn,txn), we conclude that, lim n→∞) d(xn,txn) = 0. this completes the proof. (3.23) � theorem 3.1. let k be a non-empty closed convex subset of a uniformly convex hyperbolic space (x,d,w ). let t : k → k and s : k → k be asymptotically nonexpansive mappings with f (t ) 6= φ and f (s) 6= φ and kn ≥ 1 satisfying ∑∞ n=1(k 2 n − 1) < ∞. for any initial point x1 ∈ k, define the sequence {xn} iteratively by (3.1). suppose d(xn,xn+1) → 0 as n → ∞, then, {xn} ∆-converges to an element of f (t ) ∩f (s). int. j. anal. appl. (2023), 21:62 9 proof. from lemma 3.2, d(xn,txn) → 0 and d(xn,sxn) → 0 as n →∞. lemma 2.1 ensures that any bounded sequence has a unique asymptotic center. let {zn} be a subsequence of {xn}. since {xn} is bounded, {zn} is also bounded and suppose that a({xn}) = x and a({zn}) = z. using the asymptotic nonexpansive property of t, we have, lim n→∞ d(tkzn,t k+1zn) = 0, where k = 1, 2, 3, ... our purpose is to show that, z = x and z ∈ f (t ) ∩f (s). let m and n be positive integers. now, d(tmz,zn) ≤ d(tmz,tmzn) + d(tmzn,tm−1zn) + ... + d(tzn,zn) ≤ kmd(z,zn) + m−1∑ k=0 d(tkzn,t k+1zn). taking lim sup as n →∞, for any fixed m, we have, r(tmz,{zn}) = lim n→∞ sup d(tmz,{zn}) ≤ km lim n→∞ sup d(z,{zn}) = kmr(z,{zn}). now, taking lim sup as m →∞, we obtain, lim m→∞ sup r(tmz,{zn}) ≤ r(z,{zn}). since a({zn}) = z, we have, r(z,{zn}) ≤ r(tmz,{zn}), for any fixed m ∈ n, which implies that, lim m→∞ r(tmz,{zn}) = r(z,{zn}). using lemma 2.2, we conclude that, tmz → z and z ∈ f (t ). by a similar argument, we can show that z ∈ f (s). we now claim that, z is the unique asymptotic center for each subsequence {zn} of {xn}. suppose x 6= z. since z ∈ f (t ) ∩f (s), by lemma 3.1, lim n→∞ d(xn,z) exists and therefore by the uniqueness of asymptotic centers, we have, lim n→∞ sup d(zn,z) < lim n→∞ sup d(zn,x) ≤ lim n→∞ sup d(xn,x) < lim n→∞ sup d(xn,z) = lim n→∞ sup d(zn,z). this contradiction proves that z must be equal to x. since the choice of the subsequence {zn} is arbitrary, we have, a({zn}) = {x}, for all subsequences {zn} of {xn}. thus, we conclude that, {xn} ∆-converges to a common fixed point of t and s. � theorem 3.2. let k be a non-empty subset of a uniformly convex hyperbolic space x. let s and t be asymptotically nonexpansive self mappings on k. let {xn} and {yn} be sequences as defined in 10 int. j. anal. appl. (2023), 21:62 (3.1) and d(xn,xn+1) → 0 as n → ∞. if either of the mappings t or s is demi-compact, then {xn} and {yn} converge strongly to an element of f (t ) ∩f (s). proof. assume t is demi-compact. by theorem 3.1, we have, d(xn,txn) → 0 as n → ∞. then, there exists a subsequence {xnp} of {xn} such that txnp → z∗. now, d(xnp,z ∗) ≤ d(xnp,txnp) + d(txnp,z∗) → 0 as p →∞. since, lim n→∞ d(xn,txn) → 0, we have z∗ ∈ f (t ). also, lim n→∞ d(xn,z ∗) exists. hence, xn → z∗ and d(xn,yn) → 0 implies that lim n→∞ d(yn,z ∗) exists. further, d(xn,sxn) → 0 implies that z∗ ∈ f (s). hence, {xn} and {yn} converges strongly to z∗ ∈ f (t ) ∩f (s). � as an illusration, we consider the following example in a banach space setting. example 3.1. consider k = b(0; 0.9) , the ball centred at 0 and radius 0.9 in r2. let s and t be self mappings on k defined by s(x1,x2) = (x21,x 2 2) and t (x1,x2) = (sin x1, sin x2). let x,y ∈ k, so that x = (x1,x2) and y = (y1,y2). assume that y1 < x1 and y2 < x2. now, d(snx,sny) = ‖snx −sny‖ = ∥∥(x2n1 ,x2n2 )−(y2n1 ,y2n2 )∥∥ = [( x2n1 −y 2n 1 )2 + ( x2n2 −y 2n 2 )2]12 = [ |x1 −y1|2 { x2n−11 + y1x 2n−2 1 + ... + y 2n−1 1 }2 + |x2 −y2|2 { x2n−12 + y2x 2n−2 2 + ... + y 2n−1 2 }2]12 ≤ [ |x1 −y1|2 { 2nx2n−11 }2 + |x2 −y2|2 { 2nx2n−12 }2]12 take ln = max { 1, 2nx2n−11 } and mn = max { 1, 2nx2n−12 } . let kn = max{ln,mn}. then clearly kn → 1 as n →∞. so, d(snx,sny) ≤ kn [ |x1 −y1|2 + |x2 −y2|2 ]1 2 = kn‖x −y‖. hence s is an asymptotically nonexpansive mapping on k. also t is a nonexpansive mapping on k and (0, 0) is a common fixed point of t and s. the following table shows that our new iterative scheme has a comparitively better rate of convergence than some of the existing iterative schemes. here, we take x1 = ( 3 4 , 3 4 ) and αn = βn = 1 2 ,∀n ∈n. int. j. anal. appl. (2023), 21:62 11 iterations new iteration defined as in (3.1) iteration defined as in (2.3) iteration defined as in (2.2) i y1 = (0.607316, 0.607316) y1 = (0.715819, 0.715819) y1 = (0.715819, 0.715819) x2 = (0.566583, 0.566583) x2 = (0.597018, 0.597018) x2 = (0.631199, 0.631199) ii y2 = (0.286736, 0.286736) y2 = (0.456532, 0.456532) y2 = (0.489716, 0.489716) x3 = (0.091520, 0.091520) x3 = (0.179742, 0.179742) x3 = (0.344357, 0.344357) iii y3 = (0.045760, 0.045760) y3 = (0.092728, 0.092728) y3 = (0.191416, 0.191416) x4 = (0.000048, 0.000048) x4 = (0.002857, 0.002857) x4 = (0.172179, 0.172179) iv y4 = (0.000024, 0.000024) y4 = (0.001428, 0.001428) y4 = (0.086520, 0.086520) x5 = (0.000000, 0.000000) x5 = (0.000000, 0.000000) x5 = (0.086090, 0.086090) v y5 = (0.000000, 0.000000) y5 = (0.000000, 0.000000) y5 = (0.043047, 0.043047) x6 = (0.000000, 0.000000) x6 = (0.000000, 0.000000) x6 = (0.043045, 0.043045) v i y6 = (0.021522, 0.021522) x7 = (0.021522, 0.021522) v ii y7 = (0.010761, 0.010761) x8 = (0.010761, 0.010761) v iii y8 = (0.005381, 0.005381) x9 = (0.005381, 0.005381) ix y9 = (0.002690, 0.002690) x10 = (0.002690, 0.002690) x y10 = (0.001345, 0.001345) x11 = (0.001345, 0.001345) xi y11 = (0.000673, 0.000673) x12 = (0.000673, 0.000673) xii y12 = (0.000336, 0.000336) x13 = (0.000336, 0.000336) xiii y13 = (0.000168, 0.000168) x14 = (0.000168, 0.000168) xiv y14 = (0.000084, 0.000084) x15 = (0.000084, 0.000084) xv y15 = (0.000042, 0.000042) x16 = (0.000042, 0.000042) xv i y16 = (0.000021, 0.000021) x17 = (0.000021, 0.000021) xv ii y17 = (0.000011, 0.000011) x18 = (0.000011, 0.000011) xv iii y18 = (0.000005, 0.000005) x19 = (0.000005, 0.000005) xix y19 = (0.000003, 0.000003) x20 = (0.000003, 0.000003) xx y20 = (0.000001, 0.000001) x21 = (0.000001, 0.000001) xxi y21 = (0.000001, 0.000001) x22 = (0.000001, 0.000001) xxii y22 = (0.000000, 0.000000) x23 = (0.000000, 0.000000) 12 int. j. anal. appl. (2023), 21:62 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] k. goebel, w. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35 (1972), 171-174. [2] d.r. sahu, d. o’regan, r.p. agarwal, fixed point theory for lipschitzian-type mappings with applications, springer, new york, 2009. https://doi.org/10.1007/978-0-387-75818-3. [3] m.o. osilike, s.c. aniagbosor, weak and strong convergence theorems for fixed points of asymptotically nonexpensive mappings, math. computer model. 32 (2000), 1181-1191. https://doi.org/10.1016/s0895-7177(00) 00199-0. [4] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, j. math. anal. appl. 158 (1991), 407-413. https://doi.org/10.1016/0022-247x(91)90245-u. [5] k.k. tan, h.k. xu, approximating fixed points of nonexpansive mappings by the ishikawa iteration process, j. math. anal. appl. 178 (1993), 301-308. https://doi.org/10.1006/jmaa.1993.1309. [6] w.r. mann, mean value methods in iteration, proc. amer. math. soc. 4 (1953), 506-510. [7] s. ishikawa, fixed points by a new iteration method, proc. amer. math. soc. 44 (1974), 147-150. [8] k.k. tan, h.k. xu, fixed point iteration processes for asymptotically nonexpansive mappings, proc. amer. math. soc. 122 (1994), 733-739. [9] r. agarwal, d.o. regan, d. sahu, iterative construction of fixed points of nearly asymptotically nonexpansive mappings, j. nonlinear convex anal. 8 (2007), 61-79. [10] m. abbas, b.s. thakur, d. thakur, fixed points of asymptotically nonexpansive mappings in the intermediate sense in cat(0) spaces, commun. korean math. soc. 28 (2013), 107-121. https://doi.org/10.4134/ckms. 2013.28.1.107. [11] s. chang, l. wang, h. j. lee, c. k. chan, and l. yang, demiclosed principle and δ-convergence theorems for total asymptotically nonexpansive mappings in cat(0) spaces, appl. math. comput. 219 (2012), 2611-2617. https://doi.org/10.1016/j.amc.2012.08.095. [12] s. dhompongsa, b. panyanak, on δ-convergence theorems in cat(0) spaces, computers math. appl. 56 (2008), 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036. [13] s.h. khan, m. abbas, strong and ∆-convergence of some iterative schemes in cat(0) spaces, computers math. appl. 61 (2011), 109-116. https://doi.org/10.1016/j.camwa.2010.10.037. [14] k. goebel, iteration processes for nonexpansive mappings. in: s.p. singh, s. thomeier, (eds.) topological methods in nonlinear functional analysis, contemporary mathematics, vol. 21, pp. 115–123. american math. soc, providence, 1983. [15] w.a. kirk, krasnoselskii’s iteration process in hyperbolic space, numer. funct. anal. optim. 4 (1982), 371-381. https://doi.org/10.1080/01630568208816123. [16] s. reich, i. shafrir, nonexpansive iterations in hyperbolic spaces, nonlinear anal.: theory meth. appl. 15 (1990), 537-558. https://doi.org/10.1016/0362-546x(90)90058-o. [17] u. kohlenbach, some logical metatheorems with applications in functional analysis, trans. amer. math. soc. 357 (2005), 89-128. [18] a. şahin, m. başarır, on the strong convergence of a modified s-iteration process for asymptotically quasinonexpansive mappings in a cat(0) space, fixed point theory appl. 2013 (2013), 12. https://doi.org/10. 1186/1687-1812-2013-12. [19] g. das, fixed points of quasinonexpansive mappings, indian j. pure appl. math. 17 (1986), 1263-1269. https://doi.org/10.1007/978-0-387-75818-3 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/s0895-7177(00)00199-0 https://doi.org/10.1016/0022-247x(91)90245-u https://doi.org/10.1006/jmaa.1993.1309 https://doi.org/10.4134/ckms.2013.28.1.107 https://doi.org/10.4134/ckms.2013.28.1.107 https://doi.org/10.1016/j.amc.2012.08.095 https://doi.org/10.1016/j.camwa.2008.05.036 https://doi.org/10.1016/j.camwa.2010.10.037 https://doi.org/10.1080/01630568208816123 https://doi.org/10.1016/0362-546x(90)90058-o https://doi.org/10.1186/1687-1812-2013-12 https://doi.org/10.1186/1687-1812-2013-12 int. j. anal. appl. (2023), 21:62 13 [20] s. h. khan and w. takahashi, approximating common fixed points of two asymptotically nonexpansive mappings, sci. math. japon. 53 (2001), 143-148. [21] w. takahashi, t. tamura, limit theorems of operators by convex combinations of nonexpansive retractions in banach spaces, j. approx. theory. 91 (1997), 386-397. https://doi.org/10.1006/jath.1996.3093. [22] g.s. saluja, convergence of modified s-iteration process for two asymptotically quasi-nonexpansive type mappings in cat(0) spaces, demonstr. math. 49 (2016), 107-118. https://doi.org/10.1515/dema-2016-0010. [23] w. takahashi, a convexity in metric space and nonexpansive mappings. i., kodai math. j. 22 (1970), 142-149. https://doi.org/10.2996/kmj/1138846111. [24] t. shimizu, w. takahashi, fixed points of multivalued mappings in certain convex metric spaces, topol. meth. nonlinear anal. 8 (1996), 197-203. [25] m. edelstein, the construction of an asymptotic center with a fixed-point property, bull. amer. math. soc. 78 (1972), 206-208. [26] k. goebel, w.a. kirk, topics in metric fixed point theory, 1st ed., cambridge university press, 1990. https: //doi.org/10.1017/cbo9780511526152. [27] l. leustean, nonexpansive iterations in uniformly convex w-hyperbolic spaces, in: nonlinear analysis and optimization i: nonlinear analysis, vol. 513, pp. 193-209, 2010. [28] t.c. lim, remarks on some fixed point theorems, proc. amer. math. soc. 60 (1976), 179-182. [29] a.r. khan, h. fukhar-ud-din, m.a. ahmad khan, an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, fixed point theory appl. 2012 (2012), 54. https://doi.org/10.1186/ 1687-1812-2012-54. https://doi.org/10.1006/jath.1996.3093 https://doi.org/10.1515/dema-2016-0010 https://doi.org/10.2996/kmj/1138846111 https://doi.org/10.1017/cbo9780511526152 https://doi.org/10.1017/cbo9780511526152 https://doi.org/10.1186/1687-1812-2012-54 https://doi.org/10.1186/1687-1812-2012-54 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 122-129 http://www.etamaths.com two step modified ishikawa iteration scheme for multi-valued mappings in cat(0) space pankaj kumar jhade1,∗ and a. s. saluja2 abstract. the aim of this paper is to prove some strong convergence theorems for the modified ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings in the framework of cat(0) spaces. 1. introduction let k be a nonempty convex subset of a banach space x = (x,‖ · ‖). the set k is called proximal if for each x ∈ x, there exists an element y ∈ k such that ‖x−y‖ = d(x,k), where d(x,k) = inf{‖x− z‖ : z ∈ k}. let cb(k),k(k) and p(k) denote the family of nonempty closed bounded subsets, nonempty compact subsets and nonempty proximal bounded subsets of k respectively. the hausdorff metric on cb(k) is defined by h(a,b) = max{sup x∈a d(x,b), sup y∈b d(y,a)} for a,b ∈ cb(k). a single-valued mapping t : k → k is called nonexpansive if ‖t(x) −t(y)‖ ≤ ‖x−y‖ for x,y ∈ k. a multi-valued mapping t : k → cb(k) is said to be nonexpansive if h(t(x),t(y)) ≤‖x−y‖ for all x,y ∈ k. an element p ∈ k is called a fixed point of t : k → k (respectively, t : k → cb(k)) if p = t(p) (respectively, p ∈ t(p)). the set of fixed points of t is denoted by f(t). the mapping t : k → cb(k) is called quasi-nonexpansive [27] if f(t) 6= φ and h(t(x),t(p)) ≤ ‖x − p‖ for all x ∈ k and all p ∈ f(t). it is clear that every nonexpansive multi-valued mapping t with f(t) 6= φ is quasi-nonexpansive. but there exists quasi-nonexpansive mappings that are not nonexpansive. example 1.1. let k = [0,∞) with the usual metric and t : k → cb(k) be defined by t(x) =   {0}, if x ≤ 1;[ x− 3 4 ,x− 1 3 ] , if x > 1 then clearly f(t) = {0} and for any x we have h(t(x),t(0)) ≤‖x−0‖, hence t is quasi-nonexpansive. however, if x = 2, y = 1 we get h(t(x),t(y)) > |x−y| = 1 and hence not nonexpansive. 2010 mathematics subject classification. primary: 47h10; secondary: 54h25, 54e40. key words and phrases. quasi-nonexpansive multi-valued maps, modified ishikawa iteration scheme, fixed points, cat(0) space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 122 two step modified ishikawa iteration scheme 123 the mapping t : k → cb(k) is called hemi-compact if, for any sequence {xnk} of {xn} such that xnk → p ∈ k. we note that if k is compact, then every multivalued mapping t : k → cb(x) is hemi-compact . t : k → cb(k) is said to satisfy condition (i )[24], if there is a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for r ∈ (0,∞) such that d(x,t(x)) ≥ f(d(x,f(t))) for all x ∈ k. iterative techniques for approximating fixed points of nonexpansive single-valued mappings have been studied by various authors (see; [24],[30],[11],[22]) using the mann iteration scheme or the ishikawa iteration scheme. for details on the subject, we refer the reader to berinde [2]. sastry and babu [23] studied the mann and ishikawa iteration schemes for multivalued mappings and proved that these schemes for a multi-valued map t with a fixed point p converges to a fixed point q of t under certain conditions. they also claimed that the fixed point q may be different from p. panyanak [21] extended the result of sastry and babu [23] by modifying the iteration schemes of sastry and babu [23] in the setting of uniformly convex banach spaces but the domain of t remains compact. song and wang [28,29] noted that there was a gap in the proof of theorem 3.1 of [21] and theorem 5 of [23]. because the iteration xn depends on a fixed point p ∈ f(t) as well as t . if q ∈ f(t) and q 6= p, then the iteration xn defined by q is different from the one defined by p. therefore, one cannot derive the monotonicity of sequence {‖xn −q‖} from the monotonicity of {‖xn −p‖}. so the conclusion of theorem 3.1 [21] and theorem 5 [23] are very dubious. they further solved/revised the gap and also gave the affirmative answer to the above question using the following ishikawa iteration scheme. yn = βnzn + (1 −βn)xn, xn+1 = αnz ′ n + (1 −αn)xn where ‖zn−z ′ n‖≤ h(t(xn),t(yn))+γn and ‖zn+1−z ′ n‖≤ h(t(xn+1),t(yn))+γn for zn ∈ t(xn) and z ′ n ∈ t(yn). recently, shahzad and zegeye [26] introduced the modified ishikawa iteration schemes as follows: (sz1): let k be a nonempty convex subset of a banach space x and αn,βn ∈ [0, 1]. the sequence of ishikawa iterates is defined by x0 ∈ k, yn = βnzn + (1 −βn)xn, n ≥ 0 where zn ∈ t(xn), and xn+1 = αnz ′ n + (1 −αn)xn, n ≥ 0 where z ′ n ∈ t(yn) they also proved some interesting results on the strong convergence of the sequence defined by (sz1). motivated and inspired by the above work, we introduced the following modified ishikawa iteration schemes and prove some strong convergence theorems for these schemes in the setting of cat(0) space. 124 jhade and saluja modified ishikawa iteration scheme: (ps1): let k be a nonempty convex subset of a complete cat(0) space x and αn,βn ∈ [0, 1]. the sequence of ishikawa iterates is defined by x0 ∈ k, yn = βnzn + (1 −βn)xn, n ≥ 0 where zn ∈ t(xn), and xn+1 = αnz ′ n + (1 −αn)zn, n ≥ 0 where z ′ n ∈ t(yn) the aim of this paper, is to prove strong convergence theorems of the modified ishikawa iteration scheme (ps1) in the setting of cat(0) space. 2. preliminaries a metric space x is a cat(0) space if it is geodesically connected, and if every geodesic triangle in x is at least as ’thin’ as its comparison triangle in the euclidean plane. it is well-known that any complete, simply connected riemannian manifold having nonpositive sectional curvature is a cat(0) space. other examples include pre-hilbert spaces, r-trees (see [3]), euclidean buildings (see [4]), the complex hilbert ball with a hyperbolic metric (see [10]), and many others. for a thorough discussion of these spaces and of the fundamental role they play in geometry see bridson and haefliger [3]. fixed point theory in a cat(0) space was first studied by kirk (see [12] and [15]). he showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete cat(0) space always has a fixed point. since then the fixed point theory for single-valued and multi-valued mappings in cat(0) spaces has been rapidly developed and much papers have appeared (see, e.g., [9],[17],[25],[26],[6]-[8], [12]-[13]). let (x,d) be a metric space. a geodesic path joining x ∈ x to y ∈ x (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ r to x such that c(0) = x, c(l) = y, and d(c(t),c(t′)) = |t − t′| for all t,t′ ∈ [0, l].in particular, c is an isometry and d(x,y) = l. the image α of c is called a geodesic (or metric) segment joining x and y. when it is unique this geodesic segment is denoted by [x,y]. the space (x,d) is said to be a geodesic space if every two points of x are joined by a geodesic, and x is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,y ∈ x. a subset y ⊆ x is said to be convex if y includes every geodesic segment joining any two of its points. a geodesic triangle ∆(x1,x2,x3) in a geodesic metric space (x,d) consists of three points x1,x2,x3 in x (the vertices of ∆) and a geodesic segment between each pair of vertices (the edges of ∆). a comparison triangle for the geodesic triangle ∆(x1,x2,x3) in (x,d) is a triangle ∆(x1,x2,x3) := ∆(x̄1, x̄2, x̄3) in the euclidean plane e2 such that de2 (x̄i, x̄j) = d(xi,xj) for i,j ∈{1, 2, 3}. a geodesic space is said to be a cat(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. cat(0): let ∆ be a geodesic triangle in x and let ∆ be a comparison triangle for ∆. then ∆ is said to satisfy the cat(0) inequality if for all x,y ∈ ∆ and all comparison points x̄, ȳ ∈ ∆̄, d(x,y) ≤ de2 (x̄, ȳ) two step modified ishikawa iteration scheme 125 if x,y1,y2 are points in a cat(0) space and if y0 is the midpoint of the segment [y1,y2], then the cat(0) inequality implies (2.1) d(x,y0) 2 ≤ 1 2 d(x,y1) 2 + 1 2 d(x,y2) 2 − 1 4 d(y1,y2) 2 this is the (cn) inequality of bruhat and tits [5]. in fact (cf. [11], p.163), a geodesic space is a cat(0) space if and only if it satisfies the (cn) inequality. in the sequel, we need the following lemmas which will be used frequently in the proofs of our main results. lemma 2.1. ([21]) let {αn},{βn} be two real sequences such that (1) 0 ≤ αn,βn < 1; (2) βn → 0 as n →∞; (3) ∑ αnβn = ∞ let {γn} be a nonnegative real sequence such that ∑ αnβn(1 − βn)γn is bounded. then {γn} has a subsequence which converges to zero. lemma 2.2. ([3, proposition 2.4]) let (x,d) be a cat(0) space.let k be a subset of x which is complete in the induced metric. then, for every x ∈ x, there exists a unique point p(x) ∈ k such that d(x,p(x)) = inf{d(x,y) : y ∈ k}. moreover, the map x 7→ p(x) is a nonexpansive retract from x into k. lemma 2.3. ([8, lemma 2.1(iv)]) let (x,d) be a cat(0) space. for x,y ∈ x and t ∈ [0, 1], there exists a unique point z ∈ [x,y] such that (2.2) d(x,z) = td(x,y) and d(y,z) = (1 − t)d(x,y) we use the notation (1 − t)x⊕ ty for the unique point z satisfying (2.2). lemma 2.4. ([8, lemma 2.4]) let let (x,d) be a cat(0) space. for x,y,z ∈ x and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z) ≤ (1 − t)d(x,z) + td(y,z) lemma 2.5. ([8, lemma 2.5]) let let (x,d) be a cat(0) space. for x,y,z ∈ x and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z)2 ≤ (1 − t)d(x,z)2 + td(y,z)2 − t(1 − t)d(x,y)2 3. main results lemma 3.1. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a quasi-nonexpansive multi-valued mapping. suppose that lim n→∞ d(xn,t(xn)) = 0 for some sequence {xn} in k. then t has a fixed point. moreover, if {d(xn,y)} converges for each y ∈ f(t), then {xn} strongly converges to a fixed point of t . proof. by the compactness of k, there exists a subsequence {xnk} of {xn} such that xnk → q ∈ k. thus d(q,tq) ≤ d(q,xnk ) + d(xnk,txnk ) + h(txnk,q) → 0 as,k →∞ 126 jhade and saluja this implies that q is a fixed point of t. since the limit of {d(xn,q)} exists and limk→∞ d(xnk,q) = 0, we have limk→∞ d(xn,q) = 0. this shows that the sequence {xn} converges strongly to a fixed point of q ∈ k. � theorem 3.2. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a quasi-nonexpansive multi-valued mapping and f(t) 6= φ satisfying t(p) = {p} for any fixed point p ∈ f(t). let {xn} be the sequence of ishikawa iterates defined by ps1. assume that (1) αn,βn ∈ [0, 1); (2) limn→∞ βn = 0; (3) ∑∞ n=0 αnβn = ∞. then the sequence {xn} strongly converges to a fixed point of t . proof. let p ∈ f(t). then by lemma 2.5, we have d(xn+1,p) 2 = d((1 −αn)zn ⊕αnz ′ n,p) 2 ≤ (1 −αn)d(zn,p)2 + αnd(z ′ n,p) 2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)(h(txn,tp))2 + αn(h(tyn,tp))2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)d(xn,p)2 + αnd(yn,p)2 −αn(1 −αn)d(zn,z ′ n) 2 ≤ (1 −αn)d(xn,p)2 + αnd(yn,p)2 also d(yn,p) = d(βnzn ⊕ (1 −βn)xn,p)2 ≤ (1 −βn)d(xn,p)2 + βnd(zn,p)2 −βn(1 −βn)d(xn,zn)2 ≤ (1 −βn)d(xn,p)2 + βn(h(txn,tp))2 −βn(1 −βn)d(xn,zn)2 ≤ (1 −βn)d(xn,p)2 + βnd(xn,p)2 −βn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −βn(1 −βn)d(xn,zn)2 so d(xn+1,p) 2 ≤ (1 −αn)d(xn,p)2 + αnd(xn,p)2 −αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −αnβn(1 −βn)d(xn,zn)2 this implies (3.1) d(xn+1,p) 2 ≤ d(xn,p)2 (3.2) αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 it follows from (3.1) that the sequence {d(xn,p)} is decreasing and hence limn→∞ d(xn,p) exists. on the other hand (3.2) implies ∞∑ n=0 αnβn(1 −βnd(xn,zn)2 ≤ d(x1,p)2 ≤∞ then by lemma 2.1, there exists a subsequence {d(xnk,znk )} of d(xn,zn) such that lim k→∞ d(xnk,znk ) = 0 two step modified ishikawa iteration scheme 127 this implies lim k→∞ d(xnk,txnk ) = 0 by lemma 3.1, {xnk} converges to a point q ∈ f(t). since the limit of {d(xn,q)} exists, it must be the case that limn→∞ d(xn,q) = 0 and hence the conclusion follows. � theorem 3.3. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a quasi-nonexpansive multi-valued mapping that satisfying condition (i). let {xn} be the sequence of ishikawa iterates defined by ps1. assume that f(t) 6= φ satisfying t(p) = {p} for any fixed point p ∈ f(t) and αn,βn ∈ [a,b] ⊂ (0, 1). then the sequence {xn} converges strongly to a fixed point of t . proof. similar to the proof of theorem 3.2, we obtain limn→∞ d(xn,p) exists for all p ∈ f(t) and (3.3) αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 then (3.4) a2(1 − b)d(xn,zn)2 ≤ αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 this implies (3.5) ∞∑ n=0 a2(1 − b)d(xn,zn)2 ≤ d(x1,p)2 < ∞ thus, limn→∞ d(xn,zn) 2. since zn ∈ t(xn), (3.6) d(xn,t(xn)) ≤ d(xn,zn) therefore limn→∞ d(xn,t(xn)) = 0. furthermore, (3.7) lim n→∞ d(xn,f(t)) = 0 the proof of remaining part closely follows the proof of [21, theorem 3.8], simply replacing ‖ ·‖ with d(·, ·). � corollary 3.4. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a nonexpansive multi-valued mapping that satisfying condition (i). let {xn} be the sequence of ishikawa iterates defined by ps1. assume that f(t) 6= φ satisfying t(p) = {p} for any fixed point p ∈ f(t) and αn,βn ∈ [a,b] ⊂ (0, 1). then the sequence {xn} converges strongly to a fixed point of t . theorem 3.5. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a quasi-nonexpansive multi-valued mapping and f(t) 6= φ satisfying t(p) = {p} for any fixed point p ∈ f(t). let {xn} be the sequence of ishikawa iterates defined by ps1. assume that t is hemicompact and continuous, and (1) αn,βn ∈ [0, 1); (2) limn→∞ βn = 0; (3) ∑∞ n=0 αnβn = ∞. 128 jhade and saluja then the sequence {xn} strongly converges to a fixed point of t . proof. let p ∈ f(t). then, from 3.2 αnβn(1 −βn)d(xn,zn)2 ≤ d(xn,p)2 −d(xn+1,p)2 which implies that ∞∑ n=0 αnβn(1 −βnd(xn,zn)2 ≤ d(x1,p)2 < ∞ thus, limn→∞ d(xn,zn) = 0. since d(xn,t(xn)) ≤ d(xn,zn) → 0 as n →∞ and t is hemicompact, there is a subsequence {xnk} of {xn} such that xnk → q for some q ∈ k. since t is continuous, we have d(xnk,t(xnk )) → d(q,t(q)). thus, we have d(q,t(q)) = 0 and so q ∈ f(t). by theorem 3.2 limn→∞ d(xn,p) exists for each p ∈ f(t), it follows that {xn} converges strongly to q. this completes the proof of the theorem. � corollary 3.6. let k be a nonempty compact convex subset of a complete cat(0) space x, and let t : k → cb(k) be a nonexpansive multi-valued mapping and f(t) 6= φ satisfying t(p) = {p} for any fixed point p ∈ f(t). let {xn} be the sequence of ishikawa iterates defined by ps1. assume that t is hemicompact and continuous, and (1) αn,βn ∈ [0, 1); (2) limn→∞ βn = 0; (3) ∑∞ n=0 αnβn = ∞. then the sequence {xn} strongly converges to a fixed point of t . references [1] n.a. assad and w.a. kirk, fixed point theorems for set-valued mappings of contractive type, pacific j. math., 43(3)(1972),553-562. [2] v. berinde, iterative approximation of fixed points, in: lecture notes in mathematics, vol. 1912, springer, berlin, 2007. [3] m. bridson and a. haefliger, metric spaces of non-positive curvature, vol. 319 of fundamental principles of mathematical sciences, springer-verlag, berlin, germany, 1999. [4] k. s. brown, buildings, springer-verlag, new york, 1989. [5] f. bruhat and j. tits, groupes réductifs sur un corps local, publications mathematiques de lthes,41(1)1972,5-251. [6] p. chaoha and a. phon-on, a note on fixed sets in cat(0) spaces, j. math. anal. appl., 320(2)(2006), 983-987. [7] s. dhompongsa, a. kaewkhao and b. panyanak, lim’s theorems for multi-valued mappings in cat(0) spaces, j. math. anal. appl., 312(2)(2005), 478-487. [8] s. dhompongsa and b. panyanak, on ∆−convergence theorems in cat(0) spaces, comput. math. appl., 56(2008), 2572-2579. [9] h. fujiwara, k. nagano and t. shioya, fixed point sets of parabolic isometries of cat(0) spaces, commentarii mathematici helvetici, 81(2)(2006), 305-335. two step modified ishikawa iteration scheme 129 [10] k. goebel and r. reich, uniform convexity, hyperbolic geometry and nonexpansive mappings, vol. 83 of monographs and textbooks in pure & applied mathematics, marcel dekker inc., new york, 1984. [11] s. ishikawa, fixed point and iteration of a nonexpansive mapping in a banach space, proc. amer. math. soc., 59(1976), 65-71. [12] w. a. kirk, geodesic geometry and fixed point theory, in seminar of mathematical analysis (malaga/seville, 2002/2003), vol. 64(2003) of colecc. abierta, pp. 195-225, univ. sevilla serc. publ., seville. [13] w.a. kirk, fixed point theorems in cat(0) spaces and r-trees, fixed point theory & application, vol. 2004(4)(2004),309-316. [14] w.a. kirk and b. panyanak, a concept of convergence in geodesic spaces, nonlinear analysis: tma, 68(12), 3689-3696. [15] w.a. kirk, geodesic geometry and fixed point theory ii, international conference on fixed point theory & applications, yokohama publ. japan, (2004), 113-142. [16] w. laowang and b. panyanak, srong and ∆−convergence theorems for multi-valued mappings in cat(0) spaces, j.inequal. appl., vol. 2009, article id 730132, 16 pages. [17] l. leustean, a quadratic rate of asymptotic regularity for cat(0) spaces, j. math. anal. appl., 325(1)(2007), 386-399. [18] w. r. mann, mean value methods in iteration, proc. amer. math. soc., 4(1953), 506-510. [19] s. b. nadler jr., multi-valued contraction mappings, pacific j. math., 30(1969), 475-487. [20] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc.,73(1967), 591-597. [21] b. panyanak, mann and ishikawa iterative processes for multi-valued mappings in banach spaces, comput. math. appl., 54(6)(2007), 872-877. [22] s. reich, weak convergence theorems for nonexpansive mappings in banach spaces, j. math. anal. appl., 67(1979), 274-276. [23] k.p.r. sastry and g.v.r. babu, convergence of ishikawa iterates for a multivalued mapping with fixed point, czechoslovak math. j., 55(1)(2005), 817-399. [24] h.f. senter and w.g. dotson, approximating fixed points of nonexpansive mappings, proc. amer. math. soc., 44(1974), 375-380. [25] n. shahzad and j. markin, invariant approximations for commuting mappings in cat(0) and hyperconvex spaces, j. math. anal. appl., 337(2)(2008), 1457-1464. [26] n. shahzad and h. zegeye, on mann and ishikawa iteration schemes for multi-valued maps in banach spaces, nonlinear analysis, 71(2009), no. 3-4, 838-844. [27] c. shiau, k.k. tan and c.s. wang, quasi-nonexpansive multi-valued maps and selections, fund. math., 87(1975), 109-119. [28] y. song and h. wang, convergence of iterative algorithms for multivalued mappings in banach spaces, nonlinear analysis: theory, method & applications, 70(4)(2009), 15471556. [29] y. song and h. wang, erratum to ”mann and ishikawa iterative processes for multi-valued mappings in banach spaces”[comput. math. appl. 54(2007),872-877], comput. math. appl., 55(12)(2008), 2999-3002. [30] k.k. tan and h.k. xu, approximating fixed points of nonexpansive mappings by ishikawa iteration process, j. math. anal. appl., 178(1993), 301-308. 1department of mathematics, nri institute of information science & technology, bhopal-462021 , india 2department of mathematics, j. h. government (pg) college, betul 460001, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 129-141 http://www.etamaths.com iterative solutions of nonlinear integral equations of hammerstein type abebe r. tufa, h. zegeye∗ and m. thuto abstract. let h be a real hilbert space. let f, k : h → h be lipschitz monotone mappings with lipschtiz constants l1 and l2, respectively. suppose that the hammerstein type equation u + kfu = 0 has a solution in h. it is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized hammerstein type equation. the results obtained in this paper improve and extend known results in the literature. 1. introduction let h be a real hilbert space. a mapping a : d(a) ⊂ h → h is said to be l−lipschitz if there exists l ≥ 0 such that ||ax−ay|| ≤ l||x−y||, for all x,y ∈ d(a).(1.1) a is called nonexpansive mapping if l = 1 and it is called contraction mapping if l < 1. it is easy to observe that the class of lipschitz mappings includes the class of nonexpansive and hence the class of contraction mappings. a mapping a : d(a) ⊂ h → h is said to be γ− inverse strongly monotone if there exists a positive real number γ such that 〈x−y,ax−ay〉≥ γ||ax−ay||2, for all x,y ∈ d(a).(1.2) if a is γ−inverse strongly monotone, then it is lipschitz continuous with lipschitz constant 1 γ . a is said to be α-strongly monotone if for each x,y ∈ d(a) there exists α > 0 such that 〈x−y,ax−ay〉≥ α||x−y||2.(1.3) a mapping a : d(a) ⊂ h → h is called monotone if for each x,y ∈ d(a), the following inequality holds: 〈x−y,ax−ay〉≥ 0.(1.4) evidently the set of γ-inverse strongly monotone and the set of α-strongly monotone mappings are included in the set of monotone mappings. 2010 mathematics subject classification. 47h30, 47h05, 26a16. key words and phrases. hammerstein type equation; lipschitz mapping; monotone mapping. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 129 130 tufa, zegeye and thuto a monotone mapping a : h → h is said to be maximal monotone if r(i + λa), the range of (i + λa), is h for every λ > 0, where i is the identity mapping on h. this is equivalent to saying that, a monotone mapping a is said to be maximal monotone if it is not properly contained in any other monotone mapping. for a maximal monotone mapping a and r > 0, a mapping jr : r(i +ra) → d(a) given by jr = (i + ra) −1 is called the resolvent of a. it is well known that the resolvent operator, jr, is single valued and nonexpansive mapping. the class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. interests in monotone mappings stems mainly from the fact that many physically significant problems (see e.g [20]) can be modelled by initial value problems of the form: x′(t) + ax(t) = 0,x(0) = x0,(1.5) where a is a monotone mapping in an hilbert space h. such evolution equation can be found in the heat, wave and schrödinger equations. if x(t) is independent of t, the equation (1.5) reduces to au = 0,(1.6) whose solutions correspond to the equilibrium points of the system (1.5). a variety of problems, for example, convex optimization, linear programming, and elliptic differential equations can be formulated as finding a zero of maximal monotone mappings. consequently, many research efforts (see, e.g., zarantonello [16], minty [11], kacurovskii [9] and vainberg and kacurovskii [14]) have been devoted to methods of finding appropriate solutions, if it exists, of equation (1.6) and then u + au = 0.(1.7) one important generalization of equation (1.7) is the so-called equation of hammerstein type (see e.g., [8]), where a nonlinear integral equation of hammerstein type is one of the form: (1.8) u(x) + ∫ ω k(x,y)f(y,u(y))dy = h(x), where dy is a σ-finite measure on the measure space ω, the real kernel k is defined on ω×ω, f is a real-valued function defined on ω×r and is, in general, nonlinear and h is a given function on ω. if we now define a mapping k by kv(x) := ∫ ω k(x,y)v(y)dy; x ∈ ω, and the so-called superposition or nemytskii mapping by fu(y) := f(y,u(y)) then, the integral equation (1.8) can be put in operator theoretic form as follows: (1.9) u + kfu = 0, where, without loss of generality, we have taken h ≡ 0. given h in the function space h, the integral equation then asks for some u in h such that (i +kf)(u) = h. we note that if k and f are monotone, then a := i + kf need not be necessarily monotone. solution of hammerstein typ equation 131 equations of hammerstein type play a crucial role in the theory that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess greens functions can, as a rule, be transformed into the form (1.9) (see e.g., [12], chapter iv). several existence and uniqueness theorems have been proved for equations of hammerstein type (see e.g., [1, 2, 3, 4, 7]). in general, equations of hammerstein type (1.9) are nonlinear and there is no known standard method to find solutions for them. consequently, methods of approximating solutions of such equations are of interest. in 2004, chidume and zegeye [6] used an auxiliary operator (in their proof), defined in a real hilbert space in terms of k and f that is monotone whenever k and f are, and constructed an iterative procedure that converges strongly to the solution of equation (1.9). in fact, they proved the following theorem. theorem 1.1. ([6]) let h be a real hilbert space. let f : d(f) ⊂ h → h,k : d(k) ⊂ h → h be bounded monotone mappings with r(f) ⊆ d(k) where d(f) and d(k) are closed convex subsets of h satisfying certain condition. suppose the equation 0 = u+kfu has a solution in d(f). let {λn} and {θn} be real sequences in (0, 1] satisfying the following conditions: (i) lim n→∞ θn = 0, (ii) ∑∞ n=1 λnθn = ∞, limn→∞ λn θn = 0, (iii) lim n→∞ ( θn−1 θn − 1 ) λnθn = 0. let sequences {un} ⊆ d(f) and {vn} ⊆ d(k) be generated from u0 ∈ d(f) and v0 ∈ d(k), respectively by{ un+1 = pd(f) ( un −λn(fun −vn + θn(un −w1)) ) , vn+1 = pd(k) ( vn −λn(kvn + un + θn(vn −w2)) )(1.10) where w1 ∈ d(f) and w2 ∈ d(k) are arbitrary but fixed. then, there exists d > 0 such that if λn ≤ d and λnθn ≤ d 2 for all n ≥ 0, the sequences {un} and {vn} converge strongly to u∗ and v∗, respectively in h, where u∗ is a solution of the equation 0 = u + kfu and v∗ = fu∗. in 2012, chidume and djitte [5] introduced an iterative scheme and proved the following theorem. theorem 1.2. ([5]) let h be a real hilbert space. let f,k : h → h be a bounded, monotone mapping and satisfy the range condition. let {un} and {vn} be sequences in h defined iteratively from arbitrary u1,v1 ∈ h by{ un+1 = un −λn(fun −vn) −λnθn(un −u1),n ≥ 1 vn+1 = vn −λn(kvn + un) −λnθn(vn −v1),n ≥ 1 (1.11) where {λn} and {θn} are sequences in (0, 1) satisfying the following conditions. (i) lim n→∞ θn = 0, (ii) ∑∞ n=1 λnθn = ∞, λn = o(θn), (iii) limn→∞ ( θn−1 θn − 1 ) λnθn = 0. suppose that u+kfu = 0 has a solution in h. then, there exists a constant d0 > 0 such that if λn ≤ d0θn, for all n ≥ n0 for some n0 ≥ 1, then the sequence {un} converges to u∗, a solution of u + kfu = 0. 132 tufa, zegeye and thuto more recently, zegeye and malonza [17] introduced a method which contains an auxiliary operator, defined in an hilbert space in terms of k and f which, under certain conditions, is monotone whenever k and f are, and whose zeros are solutions of equation (1.9). they proved the following theorem. theorem 1.3. ([17]) let h be a real hilbert space. let f : h → h and k : h → h be continuous and bounded monotone operators. let e := h × h with norm ||z||2e = ||u|| 2 h + ||v|| 2 h, for z = (u,v) ∈ e and let a map t : e → e defined by tz = t(u,v) := (fu−v,u + kv) be γ−inverse strongly monotone. let a sequence {xn} be generated by:   x0 = w ∈ e chosen arbitrarily, wn = xn −γntxn, xn+1 = αnw + βnxn + λnwn, (1.12) where αn,βn,γn,λn ∈ (0, 1) satisfy αn + βn + λn = 1 and limn→∞αn = 0,∑∞ n=1 αn = ∞; 0 < β ≤ βn,λn, for all n ≥ 0 and 0 < a0 ≤ γn ≤ γ, for some a0,β ∈ r. then the sequence {xn} converges strongly to x∗ = [u∗,v∗] ∈ e, where u∗ is a solution of the equation 0 = u + kfu and v∗ = fu∗. we observe that in theorem 1.1 and theorem 1.2, the convergence of the schemes to the solution of the equation u+kfu = 0 is granted by the existence of a constant which is not clear how it is calculated. in theorem 1.3, the auxiliary operator t is used in the iteration scheme and the condition imposed on t , which is γ−inverse strongly monotone, is strong. these lead us to the following question. question: is it possible to construct an iterative scheme which converges strongly to a solution of hammerstein type equation (1.9) which does not require the existence of a constant and does not involve an auxiliary operator? it is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized hammerstein type equation (1.9). our theorems provide an affirmative answer to the above question in hilbert spaces. the results obtained in this paper improve and extend the results in this direction. 2. preliminaries let h be a real hilbert space and c be a nonempty, closed and convex subset of h. it is well known that for every point x ∈ h, there exists a unique nearest point in c, denoted by pcx, i.e, ||x−pcx|| ≤ ||x−y|| for all y ∈ c.(2.1) the mapping pc is called the metric projection of h onto c and characterized by the following property (see, e.g., [13]): pcx ∈ c and 〈x−pcx,pcx−y〉≥ 0, for all x ∈ h,y ∈ c.(2.2) in the sequel we shall make use of the following lemmas. solution of hammerstein typ equation 133 lemma 2.1. [13] let h be a real hilbert space and a : h → h be a monotone mapping. then, a is maximal monotone if and only if r(i + ra) = h for some r > 0. lemma 2.2. [20] let h be a real hilbert space. if a : h → h is monotone and continuous, then a is maximal monotone. lemma 2.3. [18] let h be a real hilbert space. then for all xi ∈ h and αi ∈ [0, 1] for i = 0, 1, 2, 3, ...,n such that α0 + α1 + α2 + ... + αn = 1 the following equality holds: ||α0x0 + α1x1 + ... + αnxn||2 = n∑ i=1 αi||xi||2 − ∑ 1≤i,j≤n αiαj||xi −xj||2. lemma 2.4. let h be a real hilbert space. then, for any given x,y ∈ h, the following inequality holds: ||x + y||2 ≤ ||x||2 + 2〈y,x + y〉. lemma 2.5. [15] let {an} be a sequence of nonnegative real numbers satisfying the following relation: an+1 ≤ (1 −αn)an + αnδn,n ≥ n0, where {αn}⊂ (0, 1) and {δn}⊂ r satisfying the following conditions: lim n→∞ αn = 0, ∞∑ n=1 αn = ∞, and lim sup n→∞ δn ≤ 0. then, lim n→∞ an = 0. lemma 2.6. [19] let h be a real hilbert space and let a : h → h be a continuous monotone mapping. then, n(a) = {x ∈ h : ax = 0} is closed and convex. lemma 2.7. [10] let {an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani < ani+1, for all i ∈ n. then, there exists a nondecreasing sequence {mk}⊂ n such that mk →∞ and the following properties are satisfied by all (sufficiently large) numbers k ∈ n: amk ≤ amk+1 and ak ≤ amk+1. in fact, mk = max{j ≤ k : aj < aj+1}. lemma 2.8. [6] let h be a real hilbert space. let e = h ×h with norm ||z||2e = ||u|| 2 h + ||v|| 2 h for z = (u,v) ∈ e. then, e is a real hilbert space and for w1 = (u1,v1),w2 = (u2,v2) ∈ e, we have that 〈w1,w2〉 = 〈u1,u2〉 + 〈v1,v2〉. lemma 2.9. [6] let c and d be nonempty subsets of a real hilbert space h. let f : c → h, k : d → h be monotone mappings. let e = h × h with norm ||z||2e = ||u|| 2 h + ||v|| 2 h for z = (u,v) ∈ e. define a mapping t : c × d → e by tz = t(u,v) := (fu−v,kv + u). then, t is monotone mapping. 3. main result we first prove the following lemma which will be used in the sequel. lemma 3.1. let c and d be nonempty subsets of a real hilbert space h. let f : c → h, k : d → h be monotone mappings. let e = h × h with norm ||z||2e = ||u|| 2 h + ||v|| 2 h for z = (u,v) ∈ e. define a mapping t : c × d → e by tz = t(u,v) := (fu−v,kv + u). then we have the following. 134 tufa, zegeye and thuto (a) if f and k are lipschitz, then t is lipschitz. (b) if f and k are maximal monotone, then t is maximal monotone. proof. since f and k are monotone, by lemma 2.9, t is monotone mapping. (a) let z1 = (u1,v1),z2 = (u2,v2) ∈ c × d and let l1 and l2 be lipschitz constants of f and k, respectively. then, we have ||tz1 −tz2||2 = ||(fu1 −v1,kv1 + u1) − (fu2 −v2,kv2 + u2)||2 = ||fu1 −fu2 − (v1 −v2)||2 + ||kv1 −kv2 + (u1 −u2)||2 ≤ ||fu1 −fu2||2 + 2||fu1 −fu2||||v1 −v2|| + ||v1 −v2||2 +||kv1 −kv2||2 + 2||kv1 −kv2||||u1 −u2|| + ||u1 −u2||2 ≤ ||fu1 −fu2||2 + ||fu1 −fu2||2 + ||v1 −v2||2 + ||v1 −v2||2 +||kv1 −kv2||2 + ||kv1 −kv2||2 + ||u1 −u2||2 + ||u1 −u2||2 ≤ 2||fu1 −fu2||2 + 2||v1 −v2||2 + 2||kv1 −kv2||2 + 2||u1 −u2||2 ≤ 2l21||u1 −u2|| 2 + 2||v1 −v2||2 + 2l22||v1 −v2|| 2 + 2||u1 −u2||2 ≤ 2(l21 + 1)||u1 −u2|| 2 + 2(l22 + 1)||v1 −v2|| 2 ≤ l2(||u1 −u2||2 + ||v1 −v2||2), where l = √ 2 max{ √ l21 + 1, √ l22 + 1}. thus ||tz1 −tz2|| ≤ l||z1 −z2|| and hence t is lipschitz mapping. (b) let 0 < r < 1. then, since f and k are maximal monotone we have that r(i + rf) = h and r(i + rk) = h. moreover, the resolvent jfr = (i + rf)−1 of f and jkr = (i + rk) −1 of k are nonexpansive. now, let h = (h1,h2) ∈ e. define g := e → e by gw = (jfr (h1 +rv),jkr (h2−ru)) for all w = (u,v) ∈ e. by the nonexpansiveness of jfr and jkr , we have ||gw1 −gw2|| ≤ r||w1 −w2|| , for all w1,w2 ∈ e. thus, g is a contraction mapping. then, by the banach contraction principle, g has a unique fixed point say w∗ = (u∗,v∗) ∈ e. that is, gw∗ = w∗, where u∗ = jfr (h1 + rv∗) and v∗ = jkr (h2 − ru∗). thus, for every h = (h1,h2) ∈ e, there exists w∗ = (u∗,v∗) ∈ e such that (i + rt)(w∗) = h. hence, r(i + rt) = e. therefore, by lemma 2.1, t is maximal monotone. � now, consider the sequences {un}, {vn} ⊂ h and let u′n = f(un − γn(fun − vn)),v ′ n = k(vn −γn(kvn + un)). then through out the rest of the paper, we use the following notations. i) tn = un −γn [ u′n −vn + γn(kvn + un) ] , ii) sn = vn −γn [ v′n + un −γn(fun −vn) ] . we now prove the following theorem. theorem 3.2. let h be a real hilbert space. let f,k : h → h be lipschitz monotone mappings with lipschtiz constants l1 and l2, respectively. suppose that the equation 0 = u + kfu has a solution in h. let ū, v̄ ∈ h and the sequences {un}, {vn}⊂ h be generated from arbitrary u0, v0 ∈ h by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)(anvn + (1 −an)sn), (3.1) solution of hammerstein typ equation 135 where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2 max{ √ l21 + 1, √ l22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn}⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, in h, where u∗ is the solution of 0 = u + kfu and v∗ = fu∗. proof. f and k are maximal monotone by lemma 2.2. now, let e := h × h be endowed with the norm ||z||2e = ||u|| 2 h + ||v|| 2 h, for z = (u,v) ∈ e. define t : e → e by t(z) = t(u,v) := (fu − v,kv + u). then, by lemma 3.1, t is lipschtiz and maximal monotone mapping. we also observe that u∗ is the solution of 0 = u + kfu if and only if z∗ = (u∗,v∗) is a solution of 0 = tz for v∗ = fu∗. thus, n(t) = {z ∈ e : tz = 0} 6= ∅. now, for initial point z0 = (u0,v0) ∈ e, define the sequence {zn} by { xn = zn −γntzn, zn+1 = αnw + (1 −αn)[anzn + (1 −an)(zn −γntxn)], (3.2) where w = (ū, v̄). observe that we have zn = [un,vn], where {un} and {vn} are sequences in (3.1). let yn = zn−γntxn and p ∈ n(t). then, by the monotonicity of t, we have ||yn −p||2 = ||zn −γntxn −p||2 −||zn −γntxn −yn||2 = ||zn −p||2 −||zn −yn||2 + 2γn〈txn,p−yn〉 = ||zn −p||2 −||zn −yn||2 + 2γn ( 〈txn −tp,p−xn〉 +〈tp,p−xn〉 + 〈txn,xn −yn〉 ) ≤ ||zn −p||2 −||zn −yn||2 + 2γn〈txn,xn −yn〉 = ||zn −p||2 −||zn −xn||2 − 2〈zn −xn,xn −yn〉 −||xn −yn||2 + 2γn〈txn,xn −yn〉 = ||zn −p||2 −||zn −xn||2 −||xn −yn||2 +2〈zn −γntxn −xn,yn −xn〉.(3.3) but since xn = zn −γntzn and t is lipschitzian we obtain 〈zn −γntxn −xn,yn −xn〉 = 〈zn −γntzn −xn,yn −xn〉 + 〈γntzn −γntxn,yn −xn〉 ≤ 〈γntzn −γntxn,yn −xn〉≤ γnl||zn −xn||||yn −xn||.(3.4) thus, from (3.3) and (3.4) we have that ||yn −p||2 ≤ ||zn −p||2 −||zn −xn||2 −||xn −yn||2 + 2lγn||zn −xn||||yn −xn|| ≤ ||zn −p||2 −||zn −xn||2 −||xn −yn||2 +γnl(||zn −xn||2 + ||xn −yn||2) ≤ ||zn −p||2 + (γnl− 1)||zn −xn||2 + (γnl− 1)||xn −yn||2.(3.5) 136 tufa, zegeye and thuto thus, from (3.2), lemma 2.3, and (3.5) we have the following: ||zn+1 −p||2 = ||αnw + (1 −αn)[anzn + (1 −an)yn] −p||2 ≤ αn||w −p||2 + (1 −αn)||an(zn −p) + (1 −an)(yn −p)||2 ≤ αn||w −p||2 + (1 −αn) [ an||zn −p||2 + (1 −an)||yn −p||2 ] ≤ αn||w −p||2 + (1 −αn)an||zn −p||2 + (1 −αn)(1 −an) [ ||zn −p||2 +(γnl− 1)||zn −xn||2 + (γnl− 1)||xn −yn||2 ] . = αn||w −p||2 + (1 −αn)||zn −p||2 + (1 −αn)(1 −an)(3.6) ×(γnl− 1) [ ||zn −xn||2 + ||xn −yn||2 ] now, since from the hypotheses, we have γn < 1 l for all n ≥ 1, the inequality (3.6) implies that ||zn+1 −p||2 ≤ αn||w −p||2 + (1 −αn)||zn −p||2.(3.7) therefore, by induction we get that ||zn+1 −p||2 ≤ max{||z0 −p||2, ||w −p||2},∀n ≥ 0, which implies that {zn}, {xn}, and {yn} are bounded. let z∗ = pn(t)w. then, using (3.2), lemma 2.4, lemma 2.3, (3.5), (3.6) and the fact that γn < 1 l , we obtain the following: ||zn+1 −z∗||2 = ||αn(w −z∗) + (1 −αn) [ anzn + (1 −an)yn −z∗ ] ||2 ≤ (1 −αn)||anzn + (1 −an)yn −z∗||2 +2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)an||zn −z∗||2 + (1 −αn)(1 −an)||yn −z∗||2 +2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn) ( an||zn −z∗||2 + (1 −an)[||zn −z∗||2 + (γnl− 1)(||zn −xn||2 +||xn −yn||2)] ) + 2αn〈w −z∗,zn+1 −z∗〉 = (1 −αn)||zn −z∗||2 + (1 −αn)(1 −an)(γnl− 1)(||zn −xn||2(3.8) +||xn −yn||2) + 2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)||zn −z∗||2 + 2αn〈w −z∗,zn+1 −z∗〉 ≤ (1 −αn)||zn −z∗||2 + 2αn〈w −z∗,zn −z∗〉 + 2αn||zn+1 −zn||||w −z∗||.(3.9) now, we consider two cases. case 1. suppose that there exists n0 ∈ n such that {||zn −z∗||} is decreasing for all n ≥ n0. then, we get that, {||zn − z∗||)} is convergent. thus, from (3.8), the fact that γn < b < 1 l for all n ≥ 0 and αn → 0 as n →∞, we have that yn −xn → 0,zn −xn → 0 as n →∞.(3.10) moreover, from (3.2), (3.10) and letting n →∞, we get zn+1 −zn = αn(w −zn) + (1 −αn)(1 −an)(yn −zn) → 0.(3.11) furthermore, since {zn} is bounded subset of h, which is reflexive, we can choose a subsequence {znj} of {zn} such that znj ⇀ ẑ and lim sup n→∞ 〈w − z∗,zn − z∗〉 = solution of hammerstein typ equation 137 lim j→∞ 〈w−z∗,znj −z ∗〉. this together with (3.10) implies that ynj ⇀ ẑ and xnj ⇀ ẑ. now, we show that ẑ ∈ n(t). but, since t is lipschitz continuous, we have ||tynj −txnj||→ 0 as j →∞. let (s,t) ∈ g(t). then, we have t−ts = 0 and hence we get 〈s−z,t−ts〉 = 0, for all z ∈ e. on the other hand, since ynj = znj −γnjtxnj , we have 〈znj −γnjtxnj − ynj,ynj −s〉 = 0, and hence, 〈s−ynj, (ynj −znj )/γnj + txnj〉 = 0. thus, we get 〈s−ynj, t〉 = 〈s−ynj,ts〉 = 〈s−ynj,ts〉−〈s−ynj, (ynj −znj )/γnj + txnj〉 = 〈s−ynj,ts−tynj〉 + 〈s−ynj,tynj −txnj〉 −〈s−ynj, (ynj −znj )/γnj〉 ≥ 〈s−ynj,tynj −txnj〉−〈s−ynj, (ynj −znj )/γnj〉. this implies that 〈s − ẑ, t〉 ≥ 0, as j → ∞. then, maximality of t gives that ẑ ∈ n(t). thus, from (2.2), we immediately obtain that lim sup n→∞ 〈w −z∗,zn −z∗〉 = lim j→∞ 〈w −z∗,znj −z ∗〉 = 〈w −z∗, ẑ −z∗〉≤ 0.(3.12) hence, it follows from (3.9), (3.11), (3.12) and lemma 2.5 that ||zn − z∗|| → 0 as n →∞. consequently, zn → z∗ = (u∗,v∗) = pn(t)w. case 2. suppose that there exists a subsequence {ni} of {n} such that ||zni −z ∗|| < ||zni+1 −z ∗||, for all i ∈ n. then, by lemma 2.7, there exist a nondecreasing sequence {mk}⊂ n such that mk →∞, and ||zmk −z ∗|| ≤ ||zmk+1 −z ∗|| and ||zk −z∗|| ≤ ||zmk+1 −z ∗||,(3.13) for all k ∈ n. now, from (3.8), the fact that γn < 1l for all n ≥ 0 and αn → 0 as n → ∞, we get that ymk −xmk → 0,zmk −xmk → 0 as k → ∞. thus, following the method in case 1, we obtain lim sup k→∞ 〈w −z∗,zmk −z ∗〉≤ 0.(3.14) now, replacing zn by zmk in (3.9), we have that ||zmk+1 −z ∗||2 ≤ (1 −αmk||zmk −z ∗||2 + 2αmk〈w −z ∗,zmk −z ∗〉, +2αmk||zmk+1 −zmk||.||w −z ∗||,(3.15) and hence (3.13) and (3.15) imply that αmk||zmk −x ∗||2 ≤ 2αmk〈w −z ∗,zmk −z ∗〉 + 2αmk||zmk+1 −zmk||.||w −z ∗||. but the fact that αmk > 0 implies that ||zmk −z ∗||2 ≤ 2〈w −z∗,zmk −z ∗〉 + 2||zmk+1 −zmk||.||w −z ∗||. thus, using (3.14) and (3.11) we get that ||zmk −z ∗||→ 0 as k →∞. this together with (3.15) implies that ||zmk+1−z ∗||→ 0 as k →∞. but ||zk−z∗|| ≤ ||zmk+1−z ∗|| for all k ∈ n gives that xk → z∗. therefore, from the above two cases, we can conclude that {zn} converges strongly to a point z∗ = (u∗,v∗) = pn(t)w, where u∗ is the solution of 0 = u + kfu and v∗ = fu∗. the proof is complete. � 138 tufa, zegeye and thuto if, in theorem 3.2, we assume that f is γ1-inverse strongly monotone and k is γ2-inverse strongly monotone, then both f and k are lipschitz with lipschitz constant l′ = max{ 1 γ1 , 1 γ2 } and hence we get the following corollary. corollary 3.3. let h be a real hilbert space. let f : h → h be γ1-inverse strongly monotone and k : h → h be γ2-inverse strongly monotone mappings. suppose that the equation 0 = u + kfu has a solution in h. let ū, v̄ ∈ h and the sequences {un},{vn}⊂ h be generated from arbitrary u0 and v0 in h by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)sn), (3.16) where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2((l′)2 + 1), {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, where u∗ is the solution of the equation 0 = u + kfu and v∗ = fu∗. if, in theorem 3.2, we assume that f is lipschitz α1-strongly monotone with lipschitz constant l1 and k is lipschitz α2-strongly monotone with lipschitz constant l2, then one can show that f is α1 l21 -inverse strongly monotone and k is α2 l22 -inverse strongly monotone and hence we get the following corollary. corollary 3.4. let h be a real hilbert space. let f : h → h be lipschitz α1strongly monotone and k : h → h be lipschitz α2-strongly monotone mappings. suppose that the equation 0 = u + kfu has a solution in h. let ū, v̄ ∈ h and the sequences {un},{vn}⊂ h be generated from arbitrary u0 and v0 in h by{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)sn), (3.17) where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2((l′′)2 + 1) and l′′ = max{l 2 1 α1 , l22 α2 }, {an}⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and∑ αn = ∞. then, the sequences {un} and {vn} converge strongly to u∗ and v∗ respectively, where u∗ is the solution of the equation 0 = u + kfu and v∗ = fu∗. if, in theorem 3.2, we assume that f = i, an identity mapping on h, then f is lipschitz monotone with lipschitz constant l1 = 1 and the sequences {tn} and {sn} reduce to: i) t′n = (1 −γn)un + (1 −γn)γnvn −γ2nkvn, ii) s′n = (1 −γ2n)vn + (γn − 1)γnun −γnv′n, where v′n = k(vn −γn(kvn + un)) and hence we get the following corollary. corollary 3.5. let h be a real hilbert space. let k : h → h be lipschitz monotone mapping with lipschtiz constant l2. suppose that the equation 0 = u + ku has a solution in h. let ū, v̄ ∈ h and the sequences {un},{vn} ⊂ h be generated from arbitrary u0 and v0 in h by{ un+1 = αnū + (1 −αn)(anun + (1 −an)t′n), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)s′n), (3.18) solution of hammerstein typ equation 139 where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2 max{ √ 2, √ l22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. then, the sequences {un} and {vn} both converge strongly to u∗, where u∗ is the solution of the equation 0 = u + ku. if, in theorem 3.2, we assume that k = i, an identity mapping on h, then k is lipschitz monotone with lipschitz constant l2 = 1 and the sequences {tn} and {sn} reduce to: i) t′′n = (1 −γ2n)un + (1 −γn)γnvn −γnu′n, ii) s′′n = (1 −γn)vn + (γn − 1)γnun + γ2nfun, where u′n = f(un −γn(fun −vn)) and hence we get the following corollary. corollary 3.6. let h be a real hilbert space. let f : h → h be lipschitz monotone mapping with lipschtiz constant l1. suppose that the equation 0 = u+fu has a solution in h. let ū, v̄ ∈ h and the sequences {un}, {vn} ⊂ h be generated from arbitrary u0 and v0 in h by{ un+1 = αnū + (1 −αn)(anun + (1 −an)t′′n), vn+1 = αnv̄ + (1 −αn)[anvn + (1 −an)s′′n), (3.19) where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2 max{ √ 2, √ l21 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn} ⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. then, the sequences {un} and {vn} converge strongly to u∗ and −u∗, respectively, where u∗ is the solution of the equation 0 = u + ku. we note that the method of proof of theorem 3.2 provides the following theorem for approximating the unique minimum norm point of solution of the hammerstein type equation. theorem 3.7. let h be a real hilbert space. let f,k : h → h be lipschitz monotone mappings with lipschtiz constants l1 and l2, respectively. suppose that the equation 0 = u + kfu has a solution in h. let the sequences {un},{vn} ⊂ h be generated from arbitrary u0 and v0 in h by{ un+1 = (1 −αn)(anun + (1 −an)tn), vn+1 = (1 −αn)(anvn + (1 −an)sn), (3.20) where γn ⊂ [a,b] ⊂ (0, 1l), for l := √ 2 max{ √ l21 + 1, √ l22 + 1}, {an} ⊂ (0,r] ⊂ (0, 1) and {αn}⊂ (0,c] ⊂ (0, 1) for all n ≥ 0 satisfies lim n→∞ αn = 0 and ∑ αn = ∞. then, the sequence {zn} = {(un,vn)} converges strongly to the unique minimum norm point z∗ = (u∗,v∗) in h × h, where u∗ is a solution of 0 = u + kfu and v∗ = fu∗. remark 3.8. theorem 3.2 improves theorem 3.4 of chidume and zegeye [6] and theorem 3.1 of chidume and djitte [5] in the sense that the convergence of our scheme does not require the existence of a constant number. remark 3.9. theorem 3.2 extends theorem 3.4 of zegeye and malonza [17] in the sense that our scheme, which does not involve the auxiliary mapping, provides strong convergence to a solution of hammerstein type equation for a more general class of monotone mappings. our theorems provide an affirmative answer to the above question in hilbert spaces. 140 tufa, zegeye and thuto 4. numerical example now, we give an example of lipschitz monotone mappings satisfying conditions of theorem 3.2 and some numerical experiment result to explain the conclusion of the theorem. let h = r with absolute value norm. let f,k : r → r be defined by fx = 3x and kx = 2x− 14.(4.1) clearly, f and k are lipschitz maximal monotone mappings with constants 3 and 2, respectively. furthermore, we observe that u∗ = 2 is the solution of u+kfu = 0. now if we take, αn = 1 n+100 , γn = 1 n+200 + 0.01, an = 1 n+100 + 0.01, and w = (ū, v̄) = (1, 0), we observe that the conditions of theorem 3.2 are satisfied and scheme (3.1) reduces to{ un+1 = αnū + (1 −αn)(anun + (1 −an)tn), vn+1 = αnv̄ + (1 −αn)(anvn + (1 −an)sn), (4.2) where tn = (1 − 3γn + 8γ2n)un + (1 − 5γn)γnvn + 14γ2n, and sn = (3γ 2 n − 2γn + 1)vn + (5γn − 1)γnun − 28γ2n + 14γn. thus, for (u0,v0) = (1, 3), (un,vn) converges strongly to (u ∗,v∗) = (2, 6) = pn(t)(w), where 2 is the solution of u + kfu = 0 and 6 = f(2). see the following table and figure. n 1 101 1001 2001 3001 4001 5001 6001 7001 7901 un 1.0000 1.5552 1.9025 1.9470 1.9636 1.9723 1.9776 1.9812 1.9838 1.9856 vn 3.0000 5.0504 5.7888 5.8853 5.9213 5.9400 5.9516 5.9594 5.9650 5.9689 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1 2 3 4 5 6 iterations, n u n, v n u 0 =1, u n v 0 =3, v n figure 1 solution of hammerstein typ equation 141 references [1] h. brezis and f. browder, nonlinear integral equations and systems of hammerstein type, advances in math., 18 (1975), 115-147. [2] h. brezis and f. browder, existence theorems for nonlinear integral equations of hammerstein type, bull. amer. math. soc. 81 (1975), 73-78. [3] f. e. browder, d. g. de figueiredo and p. gupta, maximal monotone operators and a nonlinear integral equations of hammerstein type, bull. amer. math. soc. 76 (1970), 700705. [4] felix e. browder and chaitan p. gupta, monotone operators and nonlinear integral equations of hammerstein type, bull. amer. math. soc. 75 (1969), 1347-1353. [5] c.e. chidume and n. djitte, approximation of solutions of nonlinear integral equations of hammerstein type, isrn mathematical analysis, 2012(2012), article id 169751, 12 pages. [6] c.e. chidume and h. zegeye, approximation of solutions of nonlinear equations of hammerstein type in hilbert space, proc. amer. math. soc. 133(2004), 851-858. [7] c. l. dolph, nonlinear integral equations of the hammerstein type,trans. amer. math. soc. 66 (1949), 289-307. [8] a. hammerstein, nichtlineare integralgleichungen nebst anwendungen, acta math. soc. 54 (1930), 117-176. [9] kacurovskii, on monotone operators and convex functionals, uspekhi mat. nauk, 15 (1960), 213-215. [10] p. e. mainge, strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, set-valued anal. 16 (2008), 899-912. [11] g. j. minty, monotone operators in hilbert spaces. duke math. j., 29 (1962), 341-346. [12] d. pascali and sburlan, nonlinear mappings of monotone type, editura academiai, bucuresti, romania (1978). [13] w. takahashi, nonlinear functional analysis. fixed point theory and its applications,yokohama publishers, yokohama, japan (2000). [14] m. m. vainberg and r. i. kacurovskii, on the variational theory of nonlinear operators and equations, dokl. akad. nauk 129(1959), 1199-1202. [15] h.k.xu, another control condition in an iterative method for nonexpansive mappings, bull. austral. math. soc., 65(2002),109-113. [16] e. h. zarantonello, solving functional equations by contractive averaging, mathematics research center rep, #160, mathematics research centre,univesity of wisconsin, madison, 1960. [17] h. zegeye and david m. malonza, habrid approximation of solutions of integral equations of the hammerstein type, arab.j.math., 2(2013),221-232. [18] h. zegeye and n. shahzad, convergence of mann’s type iteration method for generalized asymptotically nonexpansive mappings, comput. math. appl. 62(2011), 4007-4014. [19] h. zegeye and n. shahzad, approximating common solution of variational inequality problems for two monotone mappings in banach spaces, optim. lett. 5(2011), 691-704. [20] e. zeidler, nonlinear functional analysis and its applications, part ii: monotone operators, springer-verlag, berlin(1985). department of mathematics, university of botswana, pvt. bag 00704, gaborone, botswana ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 45-53 http://www.etamaths.com applications of some classes of sequences on approximation of functions (signals) by almost generalized nörlund means of their fourier series xhevat z. krasniqi abstract. in this paper, using rest bounded variation sequences and head bounded variation sequences, some new results on approximation of functions (signals) by almost generalized nörlund means of their fourier series are obtained. to our best knowledge this the first time to use such classes of sequences on approximations of the type treated in this paper. in addition, several corollaries are derived from our results as well as those obtained previously by others. 1. introduction and preliminaries given two sequences p := (pn) and q := (qn) the convolution (p∗ q)n is defined by rn := (p∗q)n := n∑ m=0 pmqn−m, and we also write pn := (p∗ 1)n = ∑n m=0 pm and qn := (1 ∗ q)n = ∑n m=0 qm =∑n m=0 qn−m. let (sn) be a sequence. when rn 6= 0 for all n, the generalized nörlund transform of the sequence (sn) is the sequence {tp,qn } obtained by putting tp,qn = 1 rn n∑ m=0 pn−mqmsm. if sn −→ s(n −→∞) induces tp,qn −→ s(n −→∞) then the method (n,pn,qn) is called to be regular. the necessary and sufficient condition for (n,pn,qn) method to be regular is ∑n m=0 |pn−mqm| = o(|(p∗q)n|) and pn−m = o(|(p∗q)n|) as n −→∞ for every fixed m ≥ 0 (see borwein [1]). the method (n,pn,qn) reduces to nörlund method (n,pn) if qn = 1 for all n and to riesz method (n,qn) if pn = 1 for all n. it is well-known that (n,pn) mean or (n,qn) mean includes as a special case cesàro and harmonic means or logarithmic mean, respectively. 2010 mathematics subject classification. 40c99, 40g99, 41a25, 42a16. key words and phrases. fourier series; almost generalized nörlund means; degree of approximation. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 45 46 krasniqi let f be a 2π periodic function (signal) and lebesgue integrable i.e. f ∈ l[0, 2π]. then the fourier series of the function (signal) f at the point x is given by (1.1) f(x) ∼ a0 2 + ∞∑ m=1 (am cos mx + bm sin mx), with its partial sums sn(f; x) being a trigonometric polynomial of order n with n + 1 terms. a function (signal) f ∈ lip α if |f(x + t) −f(x)| = o(|t|α) for 0 < α ≤ 1. a function (signal) f ∈ lip (α,r) for a ≤ x ≤ b if (1.2) wr(t; f) = {∫ b a |f(x + t) −f(x)|r }1/r ≤ m(|t|α) for r ≥ 1 and 0 < α ≤ 1, where m is an absolute positive constant not necessarily the same at each occurrence (see mcfadden [5]). it should be noted that if r −→ ∞ in lip(p,r) class then this class reduces to lipα. according to lorentz [3] a bounded sequence (sk) of k-th sums of the fourier series (1.1) is said to be almost convergent to s, if (1.3) lim n→∞ sn,r = lim n→∞ sr + sr+1 + · · · + sr+n n + 1 = lim n→∞ 1 n + 1 n+r∑ k=r sk = s. it is said (see [9]) that the the fourier series (1.1) is said to be almost riesz summable to the finite number s, if τn,r = 1 pn n∑ m=0 pmsm,r −→ s as n −→∞ uniformly with respect to r, where sm,r = 1 m + 1 m+r∑ j=r sj. it is a well-known fact that a convergent sequence is almost convergent and the limits are the same. a bounded sequence (sn) is said to be almost riesz summable to s if the riesz transform of (sn) is almost convergent to s (see [2]). the theory of approximation which is originated from a well-known theorem of weierstrass has been an excitatory interdisciplinary field of study till nowadays. the approximations of the functions have a wide applications in signal analysis, digital communications, theory of machines in mechanical engineering and in particular in digital signal processing see [7] and [8] (also the interested reader could find several new results on these approximations and their applications into references given in [6]). very recently mishra et al [6] determined the degree of approximation of a signal f ∈ lip(α,r), (r ≥ 1) by almost riesz summability means of its fourier series. before we recall their results we need first some known definitions given below. the lr-norm of an function f : r −→ r is defined by ‖f‖r = (∫ 2π 0 |f(x)|rdx )1/r , r ≥ 1. some classes of sequences on approximation of functions 47 the l∞-norm of an function f : r −→ r is defined by ‖f‖∞ = sup{|f(x)| : x ∈ r}. a signal (function) f is approximated by trigonometric polynomial τn(f; x) of order n and the degree of approximation en(f) of a function f ∈ lr is given by en(f) = min n ‖f(x) − τn(f; x)‖r, in terms of n. the degree of approximation of a function f : r −→ r by a trigonometric polynomial τn(f; x) of order n under sup norm ‖ ·‖∞ is defined by ‖f(x) − τn(f; x)‖∞ = sup{|f(x) − τn(f; x)| : x ∈ r}. throughout this paper we will write ψ(t) = f(x + t) −f(x− t) − 2f(x). now we are able to formulate the result obtained in [6]: theorem 1.1. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lip(α,r), (r ≥ 1) class, then the degree of approximation of the function f by almost riesz means of its fourier series is given by (1.4) ‖f(t) − τn(f(t); x)‖r = o ( p1/r−αn ) ,∀n, and ψ(t) satisfies the following conditions (1.5) [∫ π/pn 0 ( t|ψ(t)| tα )r dt ]1/r = o ( p−1n ) , (1.6) [∫ π π/pn ( t−δ|ψ(t)| tα )r dt ]1/r = o ( pδn ) , where δ is a finite quantity, riesz means are regular and r + s = rs such that 1 ≤ r ≤∞. note that in this theorem is not mentioned explicitly that the sequence (pn) is a non-decreasing one but in its proof it is used this property. we say that the the fourier series (1.1) is said to be almost generalized nörlund summable to the finite number s ([1]), if tp,qn,r = 1 rn n∑ m=0 pmqn−msm,r −→ s as n −→∞ uniformly with respect to r, where sm,r = 1 m + 1 m+r∑ j=r sj. now we give definitions of two classes of sequences (see [4]). a sequence c := {cn} of nonnegative numbers tending to zero is called of rest bounded variation, or briefly c ∈ rbv s, if it has the property (1.7) ∞∑ n=m |cn − cn+1| ≤ k(c)cm 48 krasniqi for all natural numbers m, where k(c) is a constant depending only on c. a sequence c := {cn} of nonnegative numbers will be called of head bounded variation, or briefly c ∈ hbv s, if it has the property (1.8) m−1∑ n=0 |cn − cn+1| ≤ k(c)cm for all natural numbers m, or only for all m ≤ n if the sequence c has only finite nonzero terms, and the last nonzero term is cn . the purpose of this paper is to determine the degree of approximation of a function (signal) f ∈ lip(α,r), (r ≥ 1) by almost generalized nörlund means of its fourier series under conditions that (pn) ∈ hbv s and (qn) ∈ rbv s. as is pointed out in figure 2 constructed in [10] the class of sequences rbv s is a wider one than that of monotone sequences. this fact shows that in some way our results are very extensive results. 2. main results we prove the following main result. theorem 2.1. let (pn) ∈ hbv s and (qn) ∈ rbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lip(α,r), (r ≥ 1) class, then the degree of approximation of the function f by almost generalized nörlund means of its fourier series tp,qn,r(f(t); x) is given by (2.1) ‖f(t) − tp,qn,r(f(t); x)‖r = o ( r1/r−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.2) [∫ π/rn 0 ( t|ψ(t)| tα )r dt ]1/r = o ( r−1n ) , (2.3) [∫ π π/rn ( t−δ|ψ(t)| tα )r dt ]1/r = o ( rδn ) , where δ is a finite quantity, generalized nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞. proof. it is almost a routine that for partial sums sk(f(t); x) of the fourier series (1.1) the equality sk,r(f(t); x) −f(t) = 1 2π(k + 1) ∫ π 0 ψ(t) cos(rt) − cos(k + r + 1)t 2 sin2 t 2 dt holds true. some classes of sequences on approximation of functions 49 whence, for almost generalized nörlund means of sk,r(f(t); x) we have tp,qn,r(f(t); x) −f(t) = 1 rn n∑ m=0 pmqn−m{sm,r(f(t); x) −f(t)} = 1 2πrn ∫ π 0 ψ(t) n∑ m=0 pmqn−m m + 1 · cos(rt) − cos(m + r + 1)t 2 sin2 t 2 dt = 1 2πrn (∫ π/rn 0 + ∫ π π/rn ) ψ(t) n∑ m=0 pmqn−m m + 1 · sin(m + 2r + 1) t 2 · sin(m + 1) t 2 2 sin2 t 2 dt := l1 + l2.(2.4) applying hölder’s inequality, f(t) ∈ lip(α,s) =⇒ ψ(t) ∈ lip(α,s) on [0,π] (see [5]), condition (2.2), the well-known inequalities (2.5) sin u ≥ 2 π u, for u ∈ (0,π/2], (2.6) |sin(mu)| ≤ m|sin u| for all u ∈ r,m ∈ n, r + s = rs such that 1 ≤ r ≤∞, we obtain |l1| ≤ 1 2πrn [∫ π/rn 0 ( t|ψ(t)| tα )r dt ]1/r × × [∫ π/rn 0 ( 1 t1−α ∣∣∣∣∣ n∑ m=0 pmqn−m m + 1 · sin(m + 2r + 1) t 2 · sin(m + 1) t 2 2 sin2 t 2 ∣∣∣∣∣ )s dt ]1/s = o ( r−2n )[∫ π/rn 0 ( 1 t1−α ∣∣∣∣∣ n∑ m=0 pmqn−m · 1 t ∣∣∣∣∣ )s dt ]1/s = o ( r−2n )[∫ π/rn 0 rsn · t (α−2)s dt ]1/s = o ( r−1n ) ·o ( 1 r α−2+ 1 s n ) = o ( 1 r α−1 r n ) .(2.7) to estimate |l2| from above we again apply hölder’s inequality to obtain |l2| ≤ 1 2πrn [∫ π π/rn ( t−δ|ψ(t)| tα )r dt ]1/r × × [∫ π π/rn ( tδ+α ∣∣∣∣∣ n∑ m=0 pmqn−m m + 1 · sin(m + 2r + 1) t 2 · sin(m + 1) t 2 2 sin2 t 2 ∣∣∣∣∣ )s dt ]1/s . 50 krasniqi next, using again the fact that f(t) ∈ lip(α,s) =⇒ ψ(t) ∈ lip(α,s) on [0,π] (see [5]), conditions (2.3), (2.5), (2.6), and r + s = rs such that 1 ≤ r ≤∞, we get |l2| = o ( r−1n ) × × [∫ π π/rn ( tδ+α n∑ m=0 ∣∣pmqn−m sin(m + 2r + 1) t2∣∣ · (m + 1) ∣∣sin t2∣∣ 2(m + 1) sin2 t 2 )s dt ]1/s = o ( r−1n ) o ( rδn )[∫ π π/rn ( tδ+α sin t 2 n∑ m=0 ∣∣∣∣pmqn−m sin(m + 2r + 1) t2 ∣∣∣∣ )s dt ]1/s . since (pk) ∈ hbv s, then by (1.8) we have pm −pn ≤ |pm −pn| ≤ n−1∑ k=m |pk −pk+1| ≤ n−1∑ k=0 |pk −pk+1| ≤ k(p)pn which implies (2.8) pm ≤ (k(p) + 1)pn,∀m ∈ [0,n]. also, since (qk) ∈ rbv s, then by (1.7) we have qn−m ≤ ∞∑ k=m |qn−k −qn−k−1| ≤ ∞∑ k=0 |qn−k −qn−k−1| ≤ k(q)qn which implies (2.9) qn−m ≤ k(q)qn,∀m ∈ [0,n]. using the well-known fact d∑ `=j e−i`t = o ( t−1 ) , 0 ≤ j ≤ d, (2.8) and (2.9) we find that n∑ m=0 ∣∣∣∣pmqn−m sin(m + 2r + 1) t2 ∣∣∣∣ ≤ (k(p) + 1)k(q)pnqn max 0≤j≤n j∑ m=0 sin(m + 2r + 1) t 2 = o ( rnt −1) . subsequently, we obtain |l2| = o ( rδ−1n )[∫ π π/rn ( rnt δ+α−2)s dt ]1/s = o ( 1 r α−1 r n ) .(2.10) inserting (2.7) and (2.10) into (2.4) we immediately obtain |f(t) − tp,qn,r(f(t); x)| = o ( r1/r−αn ) . some classes of sequences on approximation of functions 51 finaly, using lr-norm and the lastest estimate we find that ‖f(t) − tp,qn,r(f(t); x)‖r = [∫ 2π 0 |f(t) − tp,qn,r(f(t); x)| r dt ]1/r = [∫ 2π 0 o ( r1/r−αn )r dt ]1/r = o ( r1/r−αn ) . the proof of the theorem is completed. � if we take qn = 1 for all n ≥ 0 then we obtain: corollary 2.1. let (pn) ∈ hbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lip(α,r), (r ≥ 1) class, then the degree of approximation of the function f by almost riesz means tpn,r(f(t); x) of its fourier series is given by ‖f(t) − tpn,r(f(t); x)‖r = o ( p1/r−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.11) [∫ π/pn 0 ( t|ψ(t)| tα )r dt ]1/r = o ( p−1n ) , (2.12) [∫ π π/pn ( t−δ|ψ(t)| tα )r dt ]1/r = o ( pδn ) , where δ is a finite quantity, riesz means are regular and r + s = rs such that 1 ≤ r ≤∞. if we take pn = 1 for all n ≥ 0 then we obtain: corollary 2.2. let (qn) ∈ rbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lip(α,r), (r ≥ 1) class, then the degree of approximation of the function f by almost nörlund means tqn,r(f(t); x) of its fourier series is given by ‖f(t) − tqn,r(f(t); x)‖r = o ( q1/r−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.13) [∫ π/qn 0 ( t|ψ(t)| tα )r dt ]1/r = o ( q−1n ) , (2.14) [∫ π π/qn ( t−δ|ψ(t)| tα )r dt ]1/r = o ( qδn ) , where δ is a finite quantity, nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞. if we take r →∞ then lip(α,r) ≡ lipα and we derive the following. 52 krasniqi corollary 2.3. let (pn) ∈ hbv s and (qn) ∈ rbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lipα class, then the degree of approximation of the function f by almost generalized nörlund means of its fourier series is given by |f(t) − tp,qn,r(f(t); x)| = o ( r−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.2) and (2.3), where δ is a finite quantity, generalized nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞. proof. for r →∞ and theorem 2.1 we have |f(t) − tp,qn,r(f(t); x)|∞ = sup 0≤x≤2π |f(t) − tp,qn,r(f(t); x)| = o ( r−αn ) . subsequently, we find that |f(t) − tp,qn,r(f(t); x)| ≤ |f(t) − t p,q n,r(f(t); x)|∞ = o ( r−αn ) , which completes the proof. � finally, if for all n ≥ 0 we take qn = 1 or pn = 1 in corollary 2.3 respectively, then we obtain the following two corollaries. corollary 2.4. let (pn) ∈ hbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lipα class, then the degree of approximation of the function f by almost riesz means of its fourier series is given by |f(t) − tpn,r(f(t); x)| = o ( p−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.11) and (2.12), where δ is a finite quantity, riesz means are regular and r + s = rs such that 1 ≤ r ≤∞. corollary 2.5. let (qn) ∈ rbv s. if f : r −→ r is a 2π periodic function, lebesgue integrable and belonging to the lipα class, then the degree of approximation of the function f by almost nörlund means of its fourier series is given by |f(t) − tqn,r(f(t); x)| = o ( q−αn ) , ∀n, and ψ(t) satisfies the following conditions (2.13) and (2.14), where δ is a finite quantity, nörlund means are regular and r + s = rs such that 1 ≤ r ≤∞. remark 2.1. if we had assumed in theorem 2.1 that (pn) is a non-decreasing sequence and (qn) is a non-increasing one, then it would also hold true. thus, taking qn = 1 for all n ≥ 0, then all results obtained in [6] are immediate results of ours. references [1] d. borwein, on product of sequences, j. london math. soc., 33 (1958), 352-357. [2] j. p. king, almost summable sequence, proc. amer. math. soc. 17 (1966), 1219–1225. [3] g. g. lorentz, a contribution to the theory of divergent series, acta math. 80 (1948), 167190. [4] l. leindler, on the uniform convergence and boundedness of a certain class of sine series, anal. math. 27 (2001), 279–285. [5] l. mcfadden, absolute nörlund summability, duke math. j. 9 (1942), 168-207. [6] v. n. mishra et al , on the degree of approximation of signals lip(α,r), (r ≥ 1) class by almost riesz means of its fourier series, journal of classical analysis 4 (2014), 79–87. [7] j. g. proakis, digital communications, mcgraw–hill, new york, 1995. some classes of sequences on approximation of functions 53 [8] e. z. psariks, g. v. moustakids, an l2 -based method for the design of 1 d zero phase fir digital filters, stockticker, ieee transactions on circuit and systems, fundamental theory & applications, 4, 7 (1997), 591–601. [9] p. l. sharma and k. qureshi, on the degree of approximation of a periodic function f by almost riesz means, ranchi univ. math. j. 11 (1980), 29–33 (1982). [10] s. p. zhou et al, ultimate generalization to monotonicity for uniform convergence of trigonometric series, arxiv:math/0611805v1 [math.ca] 27 nov 2006. university of prishtina ”hasan prishtina”, faculty of education, department of mathematics and informatics, avenue ”mother theresa” 5, 10000 prishtina, kosovo international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 155-161 http://www.etamaths.com graph quasicontinuous functions and densely continuous forms ľubica holá1,∗ and dušan holý2 abstract. let x, y be topological spaces. a function f : x → y is said to be graph quasicontinuous if there is a quasicontinuous function g : x → y with the graph of g contained in the closure of the graph of f. there is a close relation between the notions of graph quasicontinuous functions and minimal usco maps as well as the notions of graph quasicontinuous functions and densely continuous forms. every function with values in a compact hausdorff space is graph quasicontinuous; more generally every locally compact function is graph quasicontinuous. 1. definitions and preliminaries in what follows let x,y be topological spaces and r be the space of real numbers with the usual metric. in the paper [16] kempisty introduced a notion of quasicontinuity for real-valued functions defined in r. for general topological spaces this notion can be given the following equivalent formulation [22]. a function f : x → y is called quasicontinuous at x ∈ x if for every open set v ⊂ y , f(x) ∈ v and open set u ⊂ x, x ∈ u there is a nonempty open set w ⊂ u such that f(w) ⊂ v . if f is quasicontinuous at every point of x, we say that f is quasicontinuous. quasicontinuous functions found applications in the study of minimal usco and minimal cusco maps [12, 13], in the study of densely continuous forms [12], in the study of topological groups [4, 20, 21], in the study of dynamical systems [6], in proofs of some generalizations of michael’s selection theorem [10] and in other areas. the notion of graph quasicontinuity was introduced in [19]. a function f : x → y is said to be graph quasicontinuous if there is a quasicontinuous function g : x → y with the graph g contained in the closure f of the graph f. in sections 2 and 3 we will show that there is a close relation between notions of graph quasicontinuous functions and minimal usco maps as well as between notions of graph quasicontinuous functions and densely continuous forms. in section 4 we will prove that the uniform limit of graph quasicontinuous functions is graph quasicontinuous if the range space is a boundedly compact metric space. a set-valued map, or a multifunction, from x to y is a function that assigns to each element of x a subset of y . if f is a set-valued map from x to y , then its graph is the set {(x,y) ∈ x×y : y ∈ f(x)}. conversely, if f is a subset of x ×y and x ∈ x, define f(x) = {y ∈ y : (x,y) ∈ f}. then we can assign to each subset f of x ×y a set-valued map which takes the value f(x) at each point x ∈ x and which graph is f. in this way, we identify set-valued maps with their graphs. following [5] the term map is reserved for a set-valued map. notice that if f : x → y is a single-valued function, we will use the symbol f also for the graph of f. received 2nd april, 2017; accepted 31st may, 2017; published 3rd july, 2017. 2010 mathematics subject classification. primary 54c08, 54c60; secondary 54e35. key words and phrases. graph quasicontinuity; quasicontinuity, usco map; densely continuous form. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 155 156 holá and holý given two maps f,g : x → y , we write g ⊂ f and say that g is contained in f if g(x) ⊂ f(x) for every x ∈ x. a set-valued map f : x → y is upper semi-continuous at a point x ∈ x, if for every open set v containing f(x), there exists an open neighbourhood u of x such that f(u) = ⋃ {f(u) : u ∈ u}⊂ v. f is upper semi-continuous if it is upper semicontinuous at each point of x. a set-valued map f : x → y is upper quasicontinuous at a point x ∈ x [22] if for every open set v containing f(x) and every open set u containing x, there is a nonempty open set w ⊂ u such that f(w) ⊂ v . f is upper quasicontinuous if it is upper quasicontinuous at each point of x. following christensen [7] we say that a set-valued map f is usco if it is upper semi-continuous and takes nonempty compact values. finally, a set-valued map f is said to be minimal usco [5] if it is a minimal element in the family of all usco maps (with domain x and range y ); that is if it is usco and does not contain properly any other usco map. densely continuous forms were introduced by hammer and mccoy in [14]. densely continuous forms can be considered as set-valued maps from a topological space x into a topological space y which have a kind of minimality property found in the theory of minimal usco maps. in particular, every minimal usco map from a baire space into a metric space is a densely continuous form. there is also a connection between differentiability properties of convex functions and densely continuous forms as expressed via the subdifferentials of convex functions, which are a kind of convexification of minimal usco maps [14]. a function f : x → y is subcontinuous at x ∈ x [9] if for every net (xi) convergent to x, there is a convergent subnet of (f(xi)). if f is subcontinuous at every x ∈ x, we say that f is subcontinuous. a very useful characterization of minimal usco maps using quasicontinuous selections was given in [12] and it will be important also for our analysis. theorem 1.1. let x,y be topological spaces and y be a regular t1-space. let f : x → y be a set-valued map. the following are equivalent: (1) f is a minimal usco map; (2) every selection f of f is quasicontinuous, subcontinuous and f = f ; (3) there exists a quasicontinuous, subcontinuous selection f of f with f = f . notice that the notion of subcontinuity can be extended for so-called densely defined functions. let a be a dense subset of a topological space x and y be a topological space. let f : a → y be a function. we say that f is densely defined. let a be a dense subset of a topological space x. we say that densely defined function f : a → y is subcontinuous at x ∈ x [17] if for every net (xi) ⊂ a converging to x, there is a convergent subnet of (f(xi)). we say that f : a → y is subcontinuous if it is subcontinuous at every x ∈ x. let a be a dense subset of a topological space x. we say that a densely defined function f : a → y is a densely defined quasicontinuous function if f : a → y is quasicontinuous with respect to the induced topology on a. let x,y be topological spaces and f : x → y be a map. we say that a densely defined function f is a densely defined selection of a set-valued map f, if f(x) ∈ f(x) for every x ∈ domf, (domf denotes the domain of f). in [13] a characterization of minimal usco maps using densely defined quasicontinuous subcontinuous selections is given. graph quasicontinuous functions and densely continuous forms 157 theorem 1.2. let x,y be topological spaces and y be a t1 regular space. let f : x → y be a map. the following are equivalent: (1) f is minimal usco; (2) there is a densely defined quasicontinuous subcontinuous selection f of f such that f = f . 2. graph quasicontinuous functions and usco maps theorem 2.1. let x,y be topological spaces and y be a regular t1-space. let f : x → y be a function such that f contains a graph of a minimal usco map. then f is graph quasicontinuous. proof. by theorem 1.1 every selection of minimal usco map is quasicontinuous. thus we are done. (let f : x → y be a minimal usco map such that f ⊂ f. let g : x → y be a selection of f . then g ⊂ f.) � corollary 2.1. let x,y be topological spaces and y be a regular t1-space. let f : x → y be a function such that f contains a graph of a usco map. then f is graph quasicontinuous. proof. by an easy application of kuratowski-zorn principle we can guarantee that every usco map from x to y contains a minimal usco map. by theorem 2.1 f is graph quasicontinuous. � the following corollary is a generalization of corollary 1 in [11]. corollary 2.2. let x,y be topological spaces and y be a compact hausdorff space. then every function f : x → y is graph quasicontinuous. proof. it is the well-known fact that every set-valued map with a closed graph and with values in a compact hausdorff space is a usco map (see [2]). thus f is the graph of a usco map. by corollary 2.1 f is graph quasicontinuous. � we say that a set-valued mapping f : x → y is locally compact at x ∈ x [15] if there are an open neighbourhood u of x and a compact set k ⊂ y such that f(u) ⊂ k. if f is locally compact at every x ∈ x, we say that f is locally compact. if f is a (single-valued) function, we have a definition of a locally compact function. notice that if y = r, the notions of local boundedness and local compactness coincide. we have the following generalization of corollary 2 in [11]. corollary 2.3. let x,y be topological spaces and y be a regular t1-space. let f : x → y be a locally compact function. then f is graph quasicontinuous. proof. let f : x → y be a locally compact function. it is very easy to verify that f is the graph of a locally compact set-valued map. it is the well-known fact that a locally compact set-valued map with a closed graph is usco ( see [2]). thus by corollary 2.1 f is graph quasicontinuous. � the folowing example shows that the condition (concerning the existence of a minimal usco map) given in theorem 2.1 is only a sufficient condition and not necessary. example 2.1. let x = [0, 1] with the usual topology and let y = r also with the usual topology. let f : x → y be a function defined as follows: f(x) = { n, x ∈ ⋃ n≥2( 1 2n−1, 1 2n−2 ]; 0, x ∈ ⋃ n∈n ( 1 2n , 1 2n−1 ] ∪{0}. it is easy to verify that f is a quasicontinuous function, however f does not contain any graph of a usco map. 158 holá and holý notice that there is a graph quasicontinuous function f and quasicontinuous function g such that g ⊂ f but f ∩ g = ∅. it is easy to define a function f : [0, 1] → [0, 1] such that f(x) 6= 0 for every x ∈ [0, 1] and with the property that f = [0, 1] × [0, 1]. let g : [0, 1] → [0, 1] be the constant function equal to zero. then f and g have the above property. theorem 2.2. let x,y be topological spaces, y be a regular t1-space and a be a dense subspace of x. let f : a → y be a densely defined quasicontinuous subcontinuous function. then the function f has a quasicontinuous extension over x. proof. by theorem 1.2 f is minimal usco. for every x ∈ x\a we choose a point yx ∈{y ∈ y ; (x,y) ∈ f}. define a function g : x → y as follows: g(x) = { f(x), x ∈ a; yx, x ∈ x \a. by theorem 1.1 the function g is quasicontinuous function from x to y and so g is a quasicontinuous extension of f over x. � theorem 2.3. let x,y be topological spaces, y be a regular t1-space and a be a dense subspace of x. let f be a quasicontinuous function from a to y and for each point x ∈ x \a there is an open neighborhood u(x) of x such that the set v (x) = f(a∩u(x)) is compact. then the function f has a quasicontinuous extension g : x → y such that g ⊂ f. proof. denote by u(x) a base of open neighborhoods of x ∈ x \ a such that u ⊂ u(x) for every u ∈u(x). put b(x) = ⋂ {f(a∩u(x)) : u ∈u(x)}. since v (x) is compact and {f(a∩u(x)) : u ∈ u(x)} has the finite intersection property, b(x) is nonempty for every x ∈ x \a. for every x ∈ x \a choose a point yx ∈ b(x). define a function g : x → y as follows: g(x) = { f(x), x ∈ a; yx, x ∈ x \a. it is easy to verify that g ⊂ f. we show that g is quasicontinuous. let x ∈ a. let g be an open set with g(x) ∈ g and h be an open set with x ∈ h. let g1 be an open set with g(x) ∈ g1 such that g1 ⊂ g. since f is quasicontinuous at x there is an open set o ⊂ a ∩ h in a such that f(o) ⊂ g1. there is an open set ox ⊂ h in x such that o = ox ∩ a. we show that g(ox) ⊂ g. if z ∈ ox ∩ a then g(z) = f(z) ∈ g1 ⊂ g. let z ∈ ox \a. then g(z) ∈ f(ox ∩a) ⊂ g1 ⊂ g. let now x ∈ x \a. let g be an open set with g(x) ∈ g and h be an open set with x ∈ h. since g(x) ∈ b(x) there is a point (z,f(z)) ∈ h ×g. then the proof can continue as above. � now we give a generalization of theorem 1 in [11]. theorem 2.4. let x,y be topological spaces, y be a regular t1-space and f : x → y be a function. if there is a dense subset a ⊂ x such that the restricted function f � a is quasicontinuous and for each point x ∈ x \a there is an open neighborhood u(x) of x such that the set v (x) = f(a∩u(x)) is compact, then the function f is graph quasicontinuous. proof. by theorem 2.3 there is a quasicontinuous function g : x → y such that g ⊂ f � a ⊂ f. thus f is graph quasicontinuous. � we have the following characterization of graph quasicontinuous functions with values in locally compact hausdorff spaces. graph quasicontinuous functions and densely continuous forms 159 theorem 2.5. let x,y be topological spaces and y be a locally compact hausdorff space. the following are equivalent: (1) f : x → y is graph quasicontinuous; (2) there is a set-valued map g : x → y such that g ⊂ f, g is usco at every x in some open dense set a ⊂ x and g is single-valued and upper quasicontinuous at every x /∈ a. proof. (1) ⇒ (2) let g : x → y be a quasicontinuous function such that g ⊂ f. let τ be the topology on x. define the following set a = {x ∈ x : ∃u ∈ τ,x ∈ u,∃ compact k ⊂ y,g(u) ⊂ k}. it is easy to verify that the set a is open. now we prove that a is a dense set. let v be an open set in x and let x ∈ v . let k be a compact set such that g(x) ∈ intk. the quasicontinuity of g at x implies that there is a nonempty open set h ⊂ x such that h ⊂ v and g(h) ⊂ intk ⊂ k. thus h ⊂ a∩v . let g : x → y be the following set-valued map: g(x) = { {g(x)}, x ∈ a; {g(x)}, x ∈ x \a. it is easy to verify that for every x ∈ a, g is locally compact at x and thus g is usco at every x ∈ a. now g is single-valued for every x /∈ a by the definition; we prove that g is upper quasicontinuous at every x /∈ a. let x /∈ a. let u be an open set in x such that x ∈ u and v be an open set in y such that g(x) ⊂ v . let o be an open set in y such that g(x) ∈ o ⊂ o ⊂ v and o is compact in y . the quasicontinuity of g at x implies that there is a nonempty open set h in x such that g(h) ⊂ o. thus g(h) ⊂ o ⊂ v ; i.e. g is upper quasicontinuous at x. (2) ⇒ (1) let f : a → y be the restriction g � a of g to a. then f is usco and thus by [12] there must exist a quasicontinuous selection h : a → y of f. define now the following function: g(x) = { h(x), x ∈ a; g(x), x ∈ x \a. then g : x → y is single-valued, g ⊂ g ⊂ f. obviously, g is quasicontinuous at every x ∈ a. the upper quasicontinuity of g at x /∈ a, implies that g is quasicontinuous at every x ∈ x. � 3. graph quasicontinuous functions and densely continuous forms to define a densely continuous form from x to y [14], denote by dc(x,y ) the set of all functions f : x → y such that the set c(f) of points of continuity of f is dense in x. we call such functions densely continuous. of course dc(x,y ) contains the set c(x,y ) of all continuous functions from x to y . if y = r and x is a baire space, then all upper and lower semicontinuous functions on x belongs to dc(x,y ) and if x is a baire space and y is a metric space then every quasicontinuous function f : x → y has a dense gδ-set c(f) of the points of continuity of f [22]. notice that points of continuity and quasicontinuity of functions are studied in [3]. for every f ∈ dc(x,y ) we denote by f � c(f) the closure of the graph of f � c(f) in x ×y . we define the set d(x,y ) of densely continuous forms by d(x,y ) = {f � c(f) : f ∈ dc(x,y )}. densely continuous forms from x to y may be considered as set-valued maps, where for each x ∈ x and f ∈ d(x,y ), f(x) = {y ∈ y : (x,y) ∈ f}. 160 holá and holý theorem 3.1. let x be a topological space and y be a regular t1 space. let f : x → y be a function such that f contains a graph of a densely continuous form with nonempty values. then f is graph quasicontinuous. proof. the proof follows from proposition 3.2 in [12]. � we have the following characterizations of elements of d(x,y ) with nonempty values [12]. theorem 3.2. let x be a baire space and y be a metric space. let f be a set-valued map from x to y such that f(x) 6= ∅ for every x ∈ x. the following are equivalent: (1) f ∈ d(x,y ); (2) there is a quasicontinuous function f : x → y such that f = f ; (3) every selection f of f is quasicontinuous and f = f . corollary 3.1. let x be a baire space and y be a metric space. let f : x → y be a function. the following are equivalent: (1) f is graph quasicontinuous; (2) f contains a graph of a densely continuous form with nonempty values. notice that closures of graphs of quasicontinuous functions were studied also in [18]. 4. topology of uniform convergence on graph quasicontinuous functions we say that a metric space (y,d) is boundedly compact ( [1]) if every closed bounded subset is compact. therefore (y,d) is a locally compact, separable metric space and d is complete. in fact, any locally compact, separable metric space has a compatible metric d such that (y,d) is a boundedly compact space ( [23]). the following result is an improvement of theorem 2 in [11] for boundedly compact metric spaces. notice that we use entirely different ideas in our proof. theorem 4.1. let x,y be topological spaces and (y,d) be a boundedly compact metric space. let f : x → y be a graph quasicontinuous function. if for a function h : x → y there is a real m > 0 such that d(h(x),f(x)) ≤ m for every x ∈ x, then h is a graph quasicontinuous function. proof. let f : x → y be a graph quasicontinuous function. let g : x → y be a quasicontinuous function such that g ⊂ f. let g be a maximal family of pairwise disjoint open sets such that diam[g(g)] < 1 2 for every g ∈g. of course ⋃ g is dense in x. for every g ∈g exists a set dg ⊂ g, dense in g such that (*) g � g ⊂ f � dg and diam[f(dg)] ≤ 1. thus for every g ∈g diam[h(dg)] ≤ 2m + 1, i.e. h(dg) is compact. for every g ∈g the map h � dg ∩ (g×y ) is usco. there exists a quasicontinuous selection lg of h � dg ∩ (g×y ). the quasicontinuity of g and the property (*) imply that g ⊂ g � ⋃ g∈g g = ⋃ g∈g g � g ⊂ ⋃ g∈g f � dg = f � ⋃ g∈g dg. define the function h : x → y as follows: if x ∈ ⋃ g∈g g, then there exists g ∈g such that x ∈ g. put h(x) = lg(x). now let x ∈ x \ ⋃ g∈g g. then (x,g(x)) ∈ f � ⋃ g∈g dg ; i.e. there exists a net {xσ; σ ∈ σ}⊂ ⋃ g∈g dg such that g(x) = lim f(xσ). without loss of generality we can suppose that b(g(x),m + 1) contains the net {h(xσ) : σ ∈ σ}, where b(y,r) = {z ∈ y : d(y,z) ≤ r}. thus b(g(x), 3m + 2) is compact and contains {h(xσ) : σ ∈ σ}, i.e. there exists a cluster point r(x) of {h(xσ) : σ ∈ σ}. put h(x) = r(x). we claim that h is quasicontinuous and h ⊂ h. if x ∈ g, then (x,h(x)) ⊂ h � dg ⊂ h. if x /∈ ⋃ g∈g g, (x,h(x)) ∈ ⋃ g∈g h � dg ⊂ h � ⋃ g∈g dg ⊂ h. � graph quasicontinuous functions and densely continuous forms 161 let x be a topological space and (y,d) be a metric space. denote by f(x,y ) the space of all functions from x to y , by g(x,y ) the space of all graph quasicontinuous functions from x to y and by τu the topology of uniform convergence on f(x,y ). we have the following corollary of the above theorem: corollary 4.1. let x be a topological space and (y,d) be a boundedly compact metric space. then g(x,y ) is clopen set in (f(x,y ),τu ). acknowledgement. authors would like to thank to grant vega 2/0006/16. references [1] g. beer: topologies on closed and closed convex sets, kluwer academic publisher 1993. [2] c. berge: topological spaces, oliver and boyd, edinburgh 1963. [3] j. borśık: points of continuity, quasicontinuity and cliquishness, rend. ist. math. univ. trieste 26 (1994), 5–20. [4] a. bouziad: every čech-analytic baire semitopological group is a topological group, proc. amer. math. soc. 124 (1996), 953–959. [5] l. drewnowski and i. labuda: on minimal upper semicontinuous compact valued maps, rocky mountain j. math. 20 (1990), 737–752. [6] a. crannell, m. frantz and m. lemasurier: closed relations and equivalence classes of quasicontinuous functions, real anal. exch. 31 (2006/2007), 409-424. [7] j.p.r. christensen: theorems of namioka and r.e. johnson type for upper semicontinuous and compact valued mappings, proc. amer. math. soc. 86 (1982), 649–655. [8] r. engelking: general topology, pwn 1977. [9] r.v. fuller: set of points of discontinuity, proc. amer. math. soc. 38 (1973), 193–197. [10] j.r. giles and m.o. bartlett, modified continuity and a generalization of michael’s selection theorem, set-valued anal. 1 (1993), 247-268. [11] z. grande: a note on the graph quasicontinuity, demonstr. math. 39 (2006), 515–518. [12] ľ. holá and d. holý: minimal usco maps, densely continuous forms and upper semicontinuous functions, rocky mountain j. math. 39 (2009), 545–562. [13] ľ. holá and d. holý: new characterization of minimal cusco maps, rocky mount. math. j. 44 (2014), 1851– 1866. [14] s.t. hammer, r.a. mccoy: spaces of densely continuous forms, set-valued anal. 5 (1997), 247–266. [15] ľ. holá and b. novotný: subcontinuity of multifunctions, math. slovaca 62 (2012), 345-362. [16] s. kempisty: sur les fonctions quasi-continues, fund. math. 19 (1932), 184–197. [17] a. lechicki and s. levi: extensions of semicontinuous multifunctions, forum math. 2 (1990), 341–360. [18] m. matejdes, minimality of multifunctions, real anal. exch. 32 (2007), 519–526. [19] a. mikucka: graph quasi-continuity, demonstr. math. 36 (2003), 483–494. [20] w.b. moors: any semitopological group that is homeomorphic to a product of čech-complete spaces is a topological group, set-valued var. anal. 21 (2013), 627-633. [21] w.b. moors: semitopological groups, bouziad spaces and topological groups, topology appl. 160 (2013), 20382048. [22] t. neubrunn: quasi-continuity, real anal. exch. 14 (1988), 259–306. [23] h. vaughan: on locally compact metrizable spaces, bull. amer. math. soc. 43 (1937), 532–535. 1academy of sciences, institute of mathematics štefánikova 49, 81473 bratislava, slovakia 2department of mathematics and computer science, faculty of education, trnava university, priemyselná 4, 918 43 trnava, slovakia ∗corresponding author: hola@mat.savba.sk 1. definitions and preliminaries 2. graph quasicontinuous functions and usco maps 3. graph quasicontinuous functions and densely continuous forms 4. topology of uniform convergence on graph quasicontinuous functions references international journal of analysis and applications volume 16, number 2 (2018), 162-177 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-162 periodic and nonnegative periodic solutions of nonlinear neutral dynamic equations on a time scale manel gouasmia1, abdelouaheb ardjouni1,2,∗ and ahcene djoudi1 1applied mathematics lab, faculty of sciences, department of mathematics, univ annaba, p.o. box 12, annaba 23000, algeria 2faculty of sciences and technology, department of mathematics and informatics, univ souk ahras, p.o. box 1553, souk ahras, 41000, algeria ∗corresponding author: abd ardjouni@yahoo.fr abstract. let t be a periodic time scale. we use krasnoselskii–burton’s fixed point theorem to show new results on the existence of periodic and nonnegative periodic solutions of nonlinear neutral dynamic equation with variable delay of the form x∆(t) = −a(t)h(xσ(t)) + q(t,x(t− τ(t)))∆ + g(t,x(t),x(t− τ(t))), t ∈ t. we invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a completely continuous map. the caratheodory condition is used for the functions q and g. the results obtained here extend the work of mesmouli, ardjouni and djoudi [16]. 1. introduction in 1988, stephan hilger [11] introduced the theory of time scales (measure chains) as a means of unifying discrete and continuum calculi. since hilger’s initial work there has been significant growth in the theory 2010 mathematics subject classification. 34a37, 34a12, 35b09, 35b10, 45j05. key words and phrases. krasnoselskii-burton’s theorem; large contraction; neutral differential equation; integral equation; periodic solution; time scales. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 162 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-162 int. j. anal. appl. 16 (2) (2018) 163 of dynamic equations on time scales, covering a variety of different problems; see [8, 9, 14] and references therein. let t be a periodic time scale such that 0 ∈ t. in this article, we are interested in the analysis of qualitative theory of periodic and positive periodic solutions of neutral dynamic equations. motivated by the papers [1][7], [10], [12], [13], [15], [16] and the references therein, we consider the following nonlinear neutral dynamic equation x∆(t) = −a(t)h(xσ(t)) + q(t,x(t− τ(t)))∆ + g(t,x(t),x(t− τ(t))), t ∈ t. (1.1) throughout this paper we assume that a and τ are positive rd-continuous functions, id − τ : t → t is increasing so that the function x (t− τ (t)) is well defined over t. the function h is continuous, q and g satisfying the caratheodory condition. to reach our desired end we have to transform (1.1) into an integral equation written as a sum of two mapping, one is a contraction and the other is continuous and compact. after that, we use krasnoselskii-burton’s fixed point theorem, to show the existence of periodic and nonnegative periodic solutions. the organization of this paper is as follows. in section 2, we introduce some notations and definitions, and state some preliminary material needed in later sections. we will state some facts about the exponential function on a time scale as well as the fixed point theorems. for details on fixed point theorems we refer the reader to [10, 17]. in section 3, we establish the existence of periodic solutions. in section 4, we give sufficient conditions to ensure the existence of nonnegative periodic solutions. the results presented in this paper extend the main results in [16]. 2. preliminaries in this section, we consider some advanced topics in the theory of dynamic equations on a time scales. again, we remind that for a review of this topic we direct the reader to the monographs of bohner and peterson [8] and [9]. a time scale t is a closed nonempty subset of r. for t ∈ t the forward jump operator σ, and the backward jump operator ρ, respectively, are defined as σ(t) = inf {s ∈ t : s > t} and ρ(t) = sup{s ∈ t : s < t}. these operators allow elements in the time scale to be classified as follows. we say t is right scattered if σ(t) > t and right dense if σ(t) = t. we say t is left scattered if ρ(t) < t and left dense if ρ(t) = t. the graininess function µ : t → [0,∞), is defined by µ(t) = σ(t) − t and gives the distance between an element and its successor. we set inf ∅ = sup t and sup ∅ = inf t. if t has a left scattered maximum m, we define tk = t\{m}. otherwise, we define tk = t. if t has a right scattered minimum m, we define tk = t\{m}. otherwise, we define tk = t. int. j. anal. appl. 16 (2) (2018) 164 let t ∈ tk and let f : t → r. the delta derivative of f(t), denoted f∆ (t), is defined to be the number (when it exists), with the property that, for each ε > 0, there is a neighborhood u of t such that ∣∣f (σ (t)) −f (s) −f∆ (t) [σ (t) −s]∣∣ ≤ ε |σ (t) −s| , for all s ∈ u. if t = r then f∆ (t) = f′ (t) is the usual derivative. if t = z then f∆ (t) = ∆f(t) = f (t + 1) −f (t) is the forward difference of f at t. a function is right dense continuous (rd-continuous), f ∈ crd = crd(t,r), if it is continuous at every right dense point t ∈ t and its left-hand limits exist at each left dense point t ∈ t. the function f : t → r is differentiable on tk provided f∆ (t) exists for all t ∈ tk. we are now ready to state some properties of the delta-derivative of f. note fσ(t) = f(σ(t)). theorem 2.1 ( [8, theorem 1.20]). assume f,g : t → r are differentiable at t ∈ tk and let α be a scalar. (i) (f + g) ∆ (t) = f∆(t) + g∆(t). (ii) (αf) ∆ (t) = αf∆(t). (iii) the product rules (fg) ∆ (t) = f∆ (t) g (t) + fσ (t) g∆ (t) , (fg) ∆ (t) = f (t) g∆ (t) + f∆ (t) gσ (t) . (iv) if g (t) gσ (t) 6= 0 then ( f g )∆ (t) = f∆ (t) g (t) −f (t) g∆ (t) g (t) gσ (t) . definition 2.1 ( [12]). we say that a time scale t is periodic if there exist a w > 0 such that if t ∈ t then t±w ∈ t. for t 6= r, the smallest positive w is called the period of the time scale. definition 2.2 ( [12]). let t 6= r be a periodic time scales with the period w. we say that the function f : t → r is periodic with period t if there exists a natural number n such that t = nw, f(t±t) = f(t) for all t ∈ t and t is the smallest number such that f(t±t) = f(t). if t = r, we say that f is periodic with period t > 0 if t is the smallest positive number such that f(t±t) = f(t) for all t ∈ t. remark 2.1 ( [12]). if t is a periodic time scale with period w, then σ(t±nw) = σ(t)±nw. consequently, the graininess function µ satisfies µ(t±nw) = σ(t±nw) − (t±nw) = σ(t) − t = µ(t) and so, is a periodic function with period w. the next theorem is the chain rule on time scales ( [8, theorem 1.93]). theorem 2.2 (chain rule). assume v : t → r is strictly increasing and t̃ := v(t) is a time scale. let w : t̃ → r. if v∆(t) and w∆̃ (v(t)) exist for t ∈ tk, then (w ◦v)∆ = (w∆̃ ◦v)v∆. int. j. anal. appl. 16 (2) (2018) 165 in the sequel we will need to differentiate and integrate functions of the form f(t−τ(t)) = f (v (t)) where, v(t) := t− τ(t). our next theorem is the substitution rule ( [8, theorem 1.98]). theorem 2.3. assume v : t → r is strictly increasing and t̃ := v(t) is a time scale. if f : t → r is rd-continuous function and v is differentiable with rd-continuous derivative, then for a,b ∈ t ,∫ b a f(t)v∆(t)∆t = ∫ v(b) v(a) ( f ◦v−1 ) (s) ∆̃s. a function p : t → r is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ tk. the set of all regressive rd-continuous function f : t → r is denoted by r. the set of all positively regressive functions r+, is given by r+ = {f ∈r : 1 + µ (t) f (t) > 0 for all t ∈ t}. let p ∈r and µ (t) 6= 0 for all t ∈ t. the exponential function on t is defined by ep(t,s) = exp (∫ t s 1 µ (z) log (1 + µ (z) p (z)) ∆z ) . it is well known that if p ∈r+ , then ep(t,s) > 0 for all t ∈ t. also, the exponential function y(t) = ep(t,s) is the solution to the initial value problem y∆ = p (t) y, y (s) = 1. other properties of the exponential function are given by the following lemma. lemma 2.1 ( [8, theorem 2.36]). let p,q ∈r. then (i) e0 (t,s) = 1 and ep (t,t) = 1, (ii) ep (σ (t) ,s) = (1 + µ (t) p (t)) ep (t,s), (iii) 1 ep(t,s) = e p (t,s), where e p (t,s) = − p(t) 1+µ(t)p(t) , (iv) ep (t,s) = 1 ep(s,t) = e p (t,s), (v) ep (t,s) ep (s,r) = ep (t,r), (vi) e∆p (.,s) = pep (.,s) and ( 1 ep(.,s) )∆ = − p(t) eσp (.,s) . theorem 2.4 ( [7, theorem 2.1]). let t be a periodic time scale with period w > 0. if p ∈ crd(t) is a periodic function with the period t = nw, then∫ b+t a+t p(u)∆u = ∫ b a p(u)∆u, ep(b + t,a + t) = ep(b,a) if p ∈r, and ep(t + t,t) is independent of t ∈ t whenever p ∈r. lemma 2.2 ( [1]). if p ∈r+, then 0 < ep (t,s) ≤ exp (∫ t s p (u) ∆u ) , ∀t ∈ t. corollary 2.1 ( [1]). if p ∈r+ and p (t) < 0 for all t ∈ t, then for all s ∈ t with s ≤ t we have 0 < ep (t,s) ≤ exp (∫ t s p (u) ∆u ) < 1. int. j. anal. appl. 16 (2) (2018) 166 now, we give some definitions which we are going to use in what follows. definition 2.3. a function f : [0,t]×rn → r is an l1∆−caratheodory function if it satisfies the following conditions (i) for each z ∈ rn, the mapping t 7→ f(t,z) is ∆−measurable. (ii) for almost all t ∈ [0,t], the mapping z 7→ f(t,z) is continuous on rn. (iii) for each r > 0, there exists αr ∈ l1∆ ([0,t] ,r) such that for almost all t ∈ [0,t] and for all z such that |z| < r, we have |f(t,z)| ≤ αr(t). burton observed that krasnoselskii’s result (see [17]) can be more attractive in applications with certain changes and formulated theorem 2.5 below (see [10] for the proof). definition 2.4. let (m,d) be a metric space and assume that b : m 7→ m. b is said to be a large contraction, if for ϕ,ψ ∈m, with ϕ 6= ψ, we have d(bϕ,bψ) < d(ϕ,ψ), and if ∀ε > 0, ∃δ < 1 such that [ϕ,ψ ∈m, d(ϕ,ψ) ≥ ε] ⇒ d(bϕ,bψ) < δd(ϕ,ψ). it is proved in [10] that a large contraction defined on a closed bounded and complete metric space has a unique fixed point. theorem 2.5 (krasnoselskii-burton). let m be a closed bounded convex nonempty subset of a banach space (b,‖.‖). suppose that a and b map m into m such that (i) a is completely continuous, (ii) b is large contraction, (iii) x,y ∈m, implies ax + by ∈m. then there exists z ∈m with z = az + bz. 3. existence of periodic solutions let t > 0, t ∈ t be fixed and if t 6= r, t = nw for some n ∈ n. by the notation [a,b] we mean [a,b] = {t ∈ t : a ≤ t ≤ b}, unless otherwise specified. the intervals [a,b), (a,b] and (a,b) are defined similarly. define pt = {φ ∈ c(t,r), φ(t + t) = φ(t)} , where c(t,r) is the space of all real valued rd-continuous functions. then (pt ,‖.‖) is a banach space when it is endowed with the supremum norm ‖φ‖ = sup t∈[0,t] |φ(t)| . we will need the following lemma whose proof can be found in [12]. int. j. anal. appl. 16 (2) (2018) 167 lemma 3.1. let x ∈ pt . then ‖xσ‖ = ‖x◦σ‖ exists and ‖xσ‖ = ‖x‖. in this paper we assume that h is continuous, a ∈r+ is rd-continuous and a(t−t) = a(t), τ(t−t) = τ(t), τ(t) ≥ τ∗ > 0, (3.1) with τ continuously and τ∗ is positive constant, a is positive function and 1 −e�a(t,t−t) ≡ 1 η 6= 0. (3.2) the functions q(t,x) and g(t,x,y) are periodic in t of period t . that is q(t−t,x) = q(t,x), g(t−t,x,y) = g(t,x,y). (3.3) the following lemma is fundamental to our results. lemma 3.2. suppose (3.1)–(3.3) hold. if x ∈ pt , then x is a solution of equation (1.1) if and only if x(t) = η ∫ t t−t k(t,u)a(u)[xσ(u) −h(xσ(u))]∆u + q(t,x(t− τ(t))) + η ∫ t t−t k(t,u) [−a(u)qσ(u,x(u− τ(u))) + g(u,x(u),x(u− τ(u)))] ∆u, (3.4) where k(t,u) = e�a(t,u). (3.5) proof. let x ∈ pt be a solution of (1.1). rewrite the equation (1.1) as (x(t) −q(t,x(t− τ(t))))∆ + a(t)[xσ(t) −qσ(t,x(t− τ(t)))] = a(t) [xσ(t) −h(xσ(t))] −a(t)qσ(t,x(t− τ(t))) + g(t,x(t),x(t− τ(t))). multiply both sides of the above equation by ea(t, 0) and then integrate from t−t to t to obtain ∫ t t−t [(x(u) −q(u,x(u− τ(u))))ea(u, 0)] ∆ ∆u = ∫ t t−t a(u)[xσ(u) −h(xσ(u))]ea(u, 0)∆u + ∫ t t−t [−a(u)qσ(u,x(u− τ(u))) + g(u,x(u),x(u− τ(u)))]ea(u, 0)∆u. int. j. anal. appl. 16 (2) (2018) 168 as a consequence, we arrive at (x(t) −q(t,x(t− τ(t))))ea(t, 0) − (x(t−t) −q(t−t,x(t−t − τ(t−t))))ea(t−t, 0) = ∫ t t−t a(u)[xσ(u) −h(xσ(u))]ea(u, 0)∆u + ∫ t t−t [−a(u)qσ(u,x(u− τ(u))) + g(u,x(u),x(u− τ(u)))]ea(u, 0)∆u. by dividing both sides of the above equation by ea(t, 0) and using the fact that x(t) = x(t−t), we obtain x(t) −q(t,x(t− τ(t))) = η ∫ t t−t a(u)[xσ(u) −h(xσ(u))]e�a(t,u)∆u + η ∫ t t−t [−a(u)qσ(u,x(u− τ(u))) + g(u,x(u),x(u− τ(u)))]e�a(t,u)∆u. (3.6) the converse implication is easily obtained and the proof is complete. � to apply theorem 2.5, we need to define a banach space b, a closed bounded convex subset m of b and construct two mappings; one is a completely continuous and the other is large contraction. so, we let (b,‖.‖) = (pt ,‖.‖) and m = {ϕ ∈ pt , ‖ϕ‖≤ l} , (3.7) with l ∈ (0, 1]. for x ∈m, let the mapping h be defined by h(x) = xσ −h(xσ), (3.8) and by (3.4), define the mapping s : pt → pt by (sϕ) (t) = η ∫ t t−t k(t,u)a(u)h(ϕ(u))∆u + q(t,ϕ(t− τ(t))) + η ∫ t t−t k(t,u)[−a(u)qσ(u,ϕ(u− τ(u))) + g(u,ϕ(u),ϕ(u− τ(u)))]∆u. (3.9) therefore, we express the above equation as (sϕ) (t) = (aϕ) (t) + (bϕ) (t), int. j. anal. appl. 16 (2) (2018) 169 where a,b : pt → pt are given by (aϕ) (t) = q(t,ϕ(t− τ(t))) + η ∫ t t−t k(t,u) [−a(u)qσ(u,ϕ(u− τ(u))) + g(u,ϕ(u),ϕ(u− τ(u)))] ∆u. (3.10) and (bϕ) (t) = η ∫ t t−t k(t,u)a(u)h(ϕ(u))∆u. (3.11) we will assume that the following conditions hold. (h1) a ∈ l1∆ [0,t] is bounded. (h2) q and g satisfy caratheodory conditions with respect to l1∆ [0,t]. (h3) there exist periodic functions q1,q2 ∈ l1∆ [0,t], with period t, such that |q(t,x)| ≤ q1(t) |x| + q2(t). (h4) there exist periodic functions g1,g2,g3 ∈ l1∆ [0,t], with period t , such that |g(t,x,y)| ≤ g1(t) |x| + g2(t) |y| + g3(t). now, we need the following assumptions q1(t)l + q2(t) ≤ γ1 2 l, (3.12) g1(t)l + g2(t)l + g3(t) ≤ γ2la(t), (3.13) and j (γ1 + γ2) ≤ 1, (3.14) where γ1, γ2 and j are positive constants with j ≥ 3. lemma 3.3. for a defined in (3.10), suppose that (3.1)–(3.3), (3.12)–(3.14 ) and (h1)–(h4) hold. then a : m→m. proof. let a be defined by (3.10). obviously, aϕ is rd-continuous. first by (3.1) and (3.3), a change of variable in (3.10) shows that (aϕ)(t + t) = (aϕ)(t). that is, if ϕ ∈ pt then aϕ is periodic with period t. int. j. anal. appl. 16 (2) (2018) 170 next, let ϕ ∈m, by (3.12)–(3.14) and (h1)–(h4) we have |(aϕ) (t)| ≤ |q(t,ϕ(t− τ(t)))| + η ∫ t t−t k(t,u) [a(u) |qσ(u,ϕ(u− τ(u)))| + |g(u,ϕ(u),ϕ(u− τ(u)))|] ∆u ≤ q1(t) |ϕ(t− τ(t))| + q2(t) + η ∫ t t−t k(t,u)a(u)[q1(u) |ϕ(u− τ(u))| + q2(u)]∆u + η ∫ t t−t k(t,u)[g1(u) |ϕ(u)| + g2(u) |ϕ(u− τ(u))| + g3(u)]∆u ≤ γ1l + γ2l ≤ l j ≤ l. that is aϕ ∈m. � lemma 3.4. for a : m → m defined in (3.10), suppose that (3.1)–(3.3), (3.12)–(3.14) and (h1)–(h4) hold. then a is completely continuous. proof. we show that a is continuous in the supremum norm, let ϕn ∈m where n is a positive integer such that ϕn → ϕ as n →∞. then |(aϕn) (t) − (aϕ) (t)| ≤ |q(t,ϕn(t− τ(t)) −q(t,ϕ(t− τ(t))| + η ∫ t t−t k(t,u)a(u) |qσ(u,ϕn(u− τ(u))) −qσ(u,ϕ(u− τ(u)))|∆u + η ∫ t t−t k(t,u) |g(u,ϕn(u),ϕn(u− τ(u))) −g(u,ϕ(u),ϕ(u− τ(u)))|∆u. by the dominated convergence theorem, limn→∞ |(aϕn) (t) − (aϕ) (t)| = 0. then a is continuous. we next show that a is completely continuous. let ϕ ∈m, then, by lemma 3.3, we see that ‖aϕ‖≤ l. int. j. anal. appl. 16 (2) (2018) 171 and so the family of functions aϕ is uniformly bounded. again, let ϕ ∈m. without loss of generality, we can pick ω < t such that t−ω < t . then |(aϕ) (t) − (aϕ) (ω)| ≤ |q(t,ϕ(t− τ(t))) −q(ω,ϕ(ω − τ(ω)))| + η ∣∣∣∣ ∫ t t−t k(t,u)a(u)qσ(u,ϕ(u− τ(u)))∆u − ∫ ω ω−t k(ω,u)a(u)qσ(u,ϕ(u− τ(u)))∆u ∣∣∣∣ + η ∣∣∣∣ ∫ t t−t k(t,u)g(u,ϕ(u),ϕ(u− τ(u)))∆u − ∫ ω ω−t k(ω,u)g(u,ϕ(u),ϕ(u− τ(u)))∆u ∣∣∣∣ ≤ |q(t,ϕ(t− τ(t))) −q(ω,ϕ(ω − τ(ω)))| + 2ηk0 ∫ t−t ω−t [ a(u)ql(u) + g√2l(u) ] ∆u + η ∫ ω ω−t |k(t,u) −k(ω,u)| [ a(u)ql(u) + g√2l(u) ] ∆u ≤ |q(t,ϕ(t− τ(t))) −q(ω,ϕ(ω − τ(ω)))| + 2ηk0 ∫ t ω [ a(u)ql(u) + g√2l(u) ] ∆u + η ∫ t 0 |k(t,u) −k(ω,u)| [ a(u)ql(u) + g√2l(u) ] ∆u, where k0 = max u∈[t−t,t] {k(t,u)}, then by the dominated convergence theorem |(aϕ)(t) − (aϕ)(ω)| → 0 as t−ω → 0 independently of ϕ ∈ m. thus (aϕ) is equicontinuous. hence by ascoli-arzela’s theorem a is completely continuous. � now, we state an important result see [1] and for convenience we present below its proof, we deduce by this theorem that the following are sufficient conditions implying that the mapping h given by (3.8) is a large contraction on the set m. (h5) h : r → r is continuous on [−l,l] and differentiable on (−l,l), (h6) the function h is strictly increasing on [−l,l], (h7) sup t∈(−l,l) h′(t) ≤ 1. theorem 3.1. let h : r → r be a function satisfying (h5)–(h7). then the mapping h in (3.8) is a large contraction on the set m. int. j. anal. appl. 16 (2) (2018) 172 proof. let ϕσ,ψσ ∈m with ϕσ 6= ψσ. then ϕσ(t) 6= ψσ(t) for some t ∈ t. let us denote the set of all such t by d(ϕ,ψ), i.e., d(ϕ,ψ) = {t ∈ t : ϕσ(t) 6= ψσ(t)} . for all t ∈ d(ϕ,ψ), we have |(hϕ)(t) − (hψ)(t)| ≤ |ϕσ(t) −ψσ(t) −h(ϕσ(t)) + h(ψσ(t))| ≤ |ϕσ(t) −ψσ(t)| ∣∣∣∣1 − h(ϕσ(t)) −h(ψσ(t))ϕσ(t) −ψσ(t) ∣∣∣∣ . (3.15) since h is a strictly increasing function we have h(ϕσ(t)) −h(ψσ(t)) ϕσ(t) −ψσ(t) > 0 for all t ∈ d(ϕ,ψ). (3.16) for each fixed t ∈ d(ϕ,ψ) define the interval it ⊂ [−l,l] by it =   (ϕ σ(t),ψσ(t)) if ϕσ(t) < ψσ(t), (ψσ(t),ϕσ(t)) if ψσ(t) < ϕσ(t). the mean value theorem implies that for each fixed t ∈ d(ϕ,ψ) there exists a real number ct ∈ it such that h(ϕσ(t)) −h(ψσ(t)) ϕσ(t) −ψσ(t) = h′(ct). by (h6) and (h7) we have 0 ≤ inf u∈(−l,l) h′(u) ≤ inf u∈it h′(u) ≤ h′(ct) ≤ sup u∈it h′(u) ≤ sup h′ u∈(−l,l) (u) ≤ 1. (3.17) hence, by (3.15)–(3.17) we obtain |(hϕ)(t) − (hψ)(t)| ≤ |ϕσ(t) −ψσ(t)| ∣∣∣∣1 − inf u∈(−l,l) h′(u) ∣∣∣∣ , (3.18) for all t ∈ d(ϕ,ψ). this implies a large contraction in the supremum norm. to see this, choose a fixed ε ∈ (0, 1) and assume that ϕ and ψ are two functions in m satisfying ε ≤ sup t∈(−l,l) |ϕ(t) −ψ(t)| = ‖ϕ−ψ‖ . if |ϕσ(t) −ψσ(t)| ≤ ε 2 for some t ∈ d(ϕ,ψ), then we get by (3.17) and (3.18) that |(hϕ)(t) − (hψ)(t)| ≤ |ϕσ(t) −ψσ(t)| ≤ 1 2 ‖ϕ−ψ‖ . (3.19) since h is continuous and strictly increasing, the function h(u + ε 2 )−h(u) attains its minimum on the closed and bounded interval [−l,l]. thus, if ε 2 ≤ |ϕσ(t) −ψσ(t)| for some t ∈ d(ϕ,ψ), then by (h6) and (h7) we conclude that 1 ≥ h(ϕσ(t)) −h(ψσ(t)) ϕσ(t) −ψσ(t) > λ, int. j. anal. appl. 16 (2) (2018) 173 where λ := 1 2l min { h(u + ε 2 ) −h(u) : u ∈ [−l,l] } > 0. hence, (3.15) implies |(hϕ)(t) − (hψ)(t)| ≤ (1 −λ)‖ϕ−ψ‖ . (3.20) consequently, combining (3.19) and (3.20) we obtain |(hϕ)(t) − (hψ)(t)| ≤ δ‖ϕ−ψ‖ , (3.21) where δ = max { 1 2 , 1 −λ } . the proof is complete. � the next result shows the relationship between the mappings h and b in the sense of large contractions. assume that max{|h(−l)| , |h(l)|}≤ 2l j . (3.22) lemma 3.5. let b be defined by (3.11), suppose (h5)–(h7) hold. then b : m→m is a large contraction. proof. let b be defined by (3.11). obviously, bϕ is continuous and it is easy to show that (bϕ)(t + t) = (bϕ)(t). let ϕ ∈m |(bϕ)(t)| ≤ η ∫ t t−t k(t,u)a(u) max{|h(−l)| , |h(l)|}∆u ≤ 2l j < l, which implies b : m→m. by theorem 3.1, h is large contraction on m, then for any ϕ,ψ ∈ m, with ϕ 6= ψ and for any ε > 0, from the proof of that theorem, we have found a δ < 1, such that |(bϕ)(t) − (bψ)(t)| = ∣∣∣∣η ∫ t t−t k(t,u)a(u)[h(ϕ(u)) −h(ψ(u))]∆u ∣∣∣∣ ≤ δ‖ϕ−ψ‖η ∫ t t−t k(t,u)a(u)∆u ≤ δ‖ϕ−ψ‖ . the proof is complete. � theorem 3.2. suppose the hypothesis of lemmas 3.3, 3.4 and 3.5 hold. let m defined by (3.7). then the equation (1.1) has a t -periodic solution in m. int. j. anal. appl. 16 (2) (2018) 174 proof. by lemma 3.3, 3.4, a is continuous and a(m) is contained in a compact set. also, from lemma 3.5, the mapping b is a large contraction. next, we show that if ϕ,ψ ∈ m, we have ‖aψ + bϕ‖ ≤ l. let ϕ,ψ ∈m with ‖ϕ‖ ,‖ψ‖≤ l. by (3.12)–(3.14) ‖aψ + bϕ‖≤ (γ1 + γ2)l + 2l j ≤ l j + 2l j ≤ l. clearly, all the hypotheses of the krasnoselskii-burton’s theorem are satisfied. thus there exists a fixed point z ∈m such that z = az + bz. by lemma 3.2 this fixed point is a solution of (1.1). hence (1.1) has a t-periodic solution. � 4. existence of nonnegative periodic solutions in this section we obtain the existence of a nonnegative periodic solution of (1.1). by applying theorem 2.5, we need to define a closed, convex, and bounded subset m of pt . so, let m = {φ ∈ pt : 0 ≤ φ ≤ k} , (4.1) where k is positive constant. to simplify notation, we let m = min u∈[t−t,t] e�a(t,u), m = max u∈[t−t,t] e�a(t,u). (4.2) it is easy to see that for all (t,u) ∈ [0, 2t]2, m ≤ k(t,u) ≤ m. (4.3) then we obtain the existence of a nonnegative periodic solution of (1.1) by considering the two cases; 1) q(t,y) ≥ 0, ∀t ∈ [0,t] , y ∈ m. 2) q(t,y) ≤ 0, ∀t ∈ [0,t] , y ∈ m. in the case one, we assume for all t ∈ [0,t], x,y ∈ m, that there exists a positive constant c1 such that 0 ≤ q(t,y) ≤ c1y, (4.4) c1 < 1, (4.5) 0 ≤−a(t)qσ(t,y) + g(t,x,y), (4.6) a(t)h(ϕ(t)) −a (t) qσ (t,y) + g(t,x,y) ≤ k (1 − c1) mηt . (4.7) lemma 4.1. let a and b given by (3.10) and (3.11) respectively, assume that (4.4)–(4.7) hold. then a,b : m → m. int. j. anal. appl. 16 (2) (2018) 175 proof. let a defined by (3.11). so, for any ϕ ∈ m, we have 0 ≤ (aϕ) (t) ≤ q(t,ϕ(t− τ(t))) + η ∫ t t−t k(t,u) [−a(u)qσ(u,ϕ(u− τ(u))) + g(u,ϕ(u),ϕ(u− τ(u)))] ∆u ≤ η ∫ t t−t m k (1 − c1) mηt ∆u + c1k = k, that is aϕ ∈ m. now, let b defined by (3.11). so, for any ϕ ∈ m, we have 0 ≤ (bϕ)(t) ≤ η ∫ t t−t m k (1 − c1) mηt ∆u ≤ mηt k mηt = k. that is bϕ ∈ m. � theorem 4.1. suppose the hypothesis of lemmas 3.4, 3.5 and 4.1 hold. then equation (1.1) has a nonnegative t -periodic solution x in the subset m. proof. by lemma 3.4, a is completely continuous. also, from lemma 3.5, the mapping b is a large contraction. by lemma 4.1, a,b : m → m. next, we show that if ϕ,ψ ∈ m, we have 0 ≤ aψ + bϕ ≤ k. let ϕ,ψ ∈ m with 0 ≤ ϕ,ψ ≤ k. by (4.4)–(4.7) (aψ)(t) + (bϕ)(t) = η ∫ t t−t k(t,u)a(u)h(ϕ(u))∆u + q(t,ψ(t− τ(t))) + η ∫ t t−t k(t,u) [−a(u)qσ(u,ψ(u− τ(u))) + g(u,ψ(u),ψ(u− τ(u)))] ∆u ≤ η ∫ t t−t k(t,u) k (1 − c1) mηt ∆u + c1k ≤ η ∫ t t−t m k (1 − c1) mηt ∆u + c1k = k. on the other hand, (aψ)(t) + (bϕ)(t) ≥ 0. clearly, all the hypotheses of the krasnoselskii-burton’s theorem are satisfied. thus there exists a fixed point z ∈ m such that z = az + bz. by lemma 1 this fixed point is a solution of (1.1) and the proof is complete. � in the case two, we substitute conditions (4.4)–(4.7) with the following conditions respectively. we assume that there exist a negative constant c2 such that c2y ≤ q(t,y) ≤ 0, (4.8) int. j. anal. appl. 16 (2) (2018) 176 − c2 < 1, (4.9) −c2k mηt ≤ a(t)h(ϕ(t)) −a(t)q(t,y) + g(t,x,y), (4.10) a(t)h(ϕ(t)) −a (t) q (t,y) + g(t,x,y) ≤ k mηt . (4.11) theorem 4.2. suppose (4.8)–(4.11) and the hypothesis of lemmas 3.3, 3.4 and 3.5 hold. then equation (1.1) has a nonnegative t -periodic solution x in the subset m. proof. by lemma 3.3, 3.4, a is completely continuous. also, from lemma 3.5, the mapping b is a large contraction. to see that, it is easy to show as in lemma 4.1, a,b : m → m. next, we show that if ϕ,ψ ∈ m, we have 0 ≤ aψ + bϕ ≤ k. let ϕ,ψ ∈ m, with 0 ≤ ϕ,ψ ≤ k. by (4.8)– (4.11) (aψ) (t) + (bϕ) (t) = η ∫ t t−t k(t,u)a(u)h(ϕ(u))∆u + q(t,ψ(t− τ(t))) + η ∫ t t−t k(t,u) [−a(u)qσ(u,ψ(u− τ(u))) + g(u,ψ(u),ψ(u− τ(u)))] ∆u ≤ η ∫ t t−t k(t,u) k mηt ∆u = η ∫ t t−t m k mηt ∆u = k. on the other hand, (aψ)(t) + (bϕ) (t) ≥ η ∫ t t−t m −c2k mηt ∆u + c2k = 0. clearly, all the hypotheses of the krasnoselskii-burton’s theorem are satisfied. thus there exists a fixed point z ∈ m such that z = az + bz. by lemma 3.2 this fixed point is a solution of (1.1) and the proof is complete. � references [1] m. adivar and y. n. raffoul, existence of periodic solutions in totally nonlinear delay dynamic equations, electronic journal of qualitative theory of differential equations, 2009, no. 1, 1–20. [2] a. ardjouni and a. djoudi, existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. commun. nonlinear sci. numer. simulat., 17 (2012), 3061–3069. [3] a. ardjouni and a. djoudi, existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, malaya j. mat. 2(1) (2013) 60–67. [4] a. ardjouni and a. djoudi, existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale, acta univ. palacki. olomnc., fac. rer. nat., mathematica 52, 1 (2013) 5–19. [5] a. ardjouni and a. djoudi, a. existence, uniqueness and positivity of solutions for a neutral nonlinear periodic dynamic equation on a time scale, j. nonlinear anal. optim. 6 (2) (2015), 19–29. [6] m. belaid, a. ardjouni and a.djoudi, stability in totally nonlinear neutral dynamic equations on time scales, int. j. anal. appl. 11 (2) (2016), 110–123. int. j. anal. appl. 16 (2) (2018) 177 [7] l. bi, m. bohner and m. fan, periodic solutions of functional dynamic equations with infinite delay, nonlinear anal. 68 (2008), 1226–1245. [8] m. bohner, a. peterson, dynamic equations on time scales, an introduction with applications, birkhäuser, boston, 2001. [9] m. bohner, a. peterson, advances in dynamic equations on time scales, birkhäuser, boston, 2003. [10] t. a. burton, stability by fixed point theory for functional differential equations, dover publications, new york, 2006. [11] s. hilger, ein masskettenkalkül mit anwendung auf zentrumsmanningfaltigkeiten. phd thesis, universität würzburg, 1988. [12] e. r. kaufmann and y. n. raffoul, periodic solutions for a neutral nonlinear dynamical equation on a time scale, j. math. anal. appl. 319 (2006), no. 1, 315–325. [13] e. r. kaufmann and y. n. raffoul, periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, electron. j. differential equations, 2007 (2007), no. 27, 1–12. [14] v. lakshmikantham, s. sivasundaram, b. kaymarkcalan, dynamic systems on measure chains, kluwer academic publishers, dordrecht, 1996. [15] m. b. mesmouli, a. ardjouni, a. djoudi, existence and stability of periodic solutions for nonlinear neutral differential equations with variable delay using fixed point technique, acta univ. palacki. olomuc., fac. rer. nat., mathematica 54 (1) (2015), 95–108. [16] m. b. mesmouli, a. ardjouni and a. djoudi, study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points, acta univ. sapientiae, mathematica, 8 (2) (2016), 255–270. [17] d. r. smart, fixed point theorems, cambridge tracts in mathematics, no. 66. cambridge university press, london-new york, 1974. 1. introduction 2. preliminaries 3. existence of periodic solutions 4. existence of nonnegative periodic solutions references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 125-137 doi: 10.28924/2291-8639-15-2017-125 some integral inequalities using quantum calculus approach muhammad uzair awan1,∗, muhammad aslam noor2,3 and khalida inayat noor3 abstract. the aim of this paper is to introduce a new class of preinvex functions which is called as generalized beta preinvex functions. we show that this class includes some other new classes of preinvex functions. we derive some new integral inequalities using the approach of quantum calculus. these integral inequalities involve generalized preinvex functions and q-euler-beta functions. our results can be viewed as new quantum estimates for trapezoidal like inequalities. some new special cases are also discussed which can be deduced from the main results of the paper. 1. introduction and preliminaries the property of convexity of a function has attracted several researchers over the years and consequently this property has been generalized in different dimensions according to need, for some useful details, interested readers are referred to [3, 4, 6, 9, 14, 15, 17, 18, 24, 25, 28, 34] and the references therein. varošanec [33] introduced the notion of h-convex functions which not only generalizes the class of classical convex function but also generalizes several other classes of convex functions, such as breckner type of s-convex functions, godunova-levin-dragomir type of s-convex functions, godunova-levin functions and p-functions. so naturally the class of h-convex functions is quite unifying one. in recent years several authors have investigated the class of h-convex functions with respect to integral inequalities. recently tunç et al. [32] introduced the notion of so-called tgs-convex functions as: definition 1.1 ( [32]). let f : i ⊂ r → r be a non-negative function. we say that f is tgs-convex function on i, if f((1 − t)u + tv) ≤ t(1 − t)[f(u) + f(v)], ∀u,v ∈ i,t ∈]0, 1[. (1.1) note that f is tgs-concave function if −f is tgs-convex function. also if t = 0, 1, then, according to the hypothesis the function is equal to zero. remark 1.1. it has been noticed that the class of tgs-convex functions is also contained in the class of h-convex functions by taking suitable choice of the function h(·). hanson [9] introduced the notion of invex functions while studying mathematical programming. ben israel and mond introduced the class of the invex sets and then using invex sets as domain they defined the notion of preinvex functions. they have shown that the differentiable preinvex functions imply invex functions. under certain suitable conditions, one can show that these two classes of are equivalent. noor [17] had shown that the minimum of the differentiable preinvex functions on the invex sets can be characterized by a class of variational inequalities, which is known as variational-like inequalities. recently noor et al. [25] extended the classes of preinvex functions and h-convex functions and introduced the notion of h-preinvex functions. this class not only contains the classes of preinvex functions and h-convex functions but also other classes of convex functions. inequalities play pivotal role in mathematical analysis. convexity property of functions has also a close relationship with theory of inequalities. this relationship has attracted several authors and resultantly numerous new inequalities have been obtained via convex functions and also for its other generalizations. for some more information, see [2, 5, 6, 8, 10, 14, 16, 20–22, 24, 27]. received 24th june, 2017; accepted 12th august, 2017; published 1st november, 2017. 2010 mathematics subject classification. 26a51; 26d15. key words and phrases. convex; generalized preinvex; functions; quantum; q-differentiable; integral inequalities. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 125 126 awan, noor and noor in this paper, we define a new class of preinvex functions which is called as generalized preinvex functions. we obtain some new quantum bounds which involve primarily the property of generalized preinvexity. some special cases are also discussed which can be deduced from the main results of the paper. it is expected that interested readers may find some novel applications of generalized preinvex functions and its related inequalities in other fields of pure and applied sciences. this is the main motivation of this paper. 2. preliminaries let kη be a nonempty closed set in rn. let f : kη → r be a continuous function and let η(., .) : kη × kη → rn be a continuous bifunction. first of all, we recall some known results and concepts. definition 2.1 ( [1]). a set kη is said to be invex set with respect to η(., .), if u + tη(v,u) ∈ kη, ∀u,v ∈ kη, t ∈ [0, 1]. (2.1) the invex set kη is also called η-connected set. if η(v,u) = v −u, then we have classical convex set (1 − t)u + tv ∈ k . definition 2.2 ( [34]). a function f : kη → r is said to be preinvex with respect to arbitrary bifunction η(., .), if f(u + tη(v,u)) ≤ (1 − t)f(u) + tf(v), ∀u,v ∈ kη, t ∈ [0, 1]. (2.2) for η(v,u) = v −u in (2.2), the preinvex functions reduces to classical convex functions. definition 2.3. a function f : k → r is said to be convex in the classical sense, if f(u + t(v −u)) ≤ (1 − t)f(u) + tf(v), ∀u,v ∈ kη, t ∈ [0, 1]. noor et al. [25] have defined the class of h-preinvex functions as: definition 2.4 ( [25]). let h : j → r where (0, 1) ⊆ j be an interval in r, and let kη be an invex set with respect to η(., .). a function f : kη → r is called h-preinvex with respect to η(., .), if f(u + tη(v,u)) ≤ h(1 − t)f(u) + h(t)f(v), u,v ∈ kη, t ∈ (0, 1). if above inequality is reversed, then f is said to be h-preincave with respect to bifunction η(., .). noor et al. [25] have shown that the class of h-preinvex functions includes several other classes of preinvex functions as special cases. remark 2.1. in this paper function η(., .) : r×r → r is supposed to have the following property: η(v + t1η(u,v),v + t2η(u,v)) = (t1 − t2)η(u,v), ∀t1, t2 ∈ [0, 1], t1 ≤ t2. (2.3) in this case the following consequences hold: (1) if t1 = t2 = 0 then (2.3) implies that η(v,v) = 0 for all v ∈ r. (2) if t1 = 0 and t2 = t > 0 then η(v,v + tη(u,v)) = −tη(u,v) for all u,v ∈ r. this is the first requirement of condition c introduced in [13]. (3) if η(u,v) > 0 for some (u,v) ∈ r then η(v,v + tη(u,v)) ≤ 0 for all t ∈ [0, 1]. it means that property (2.3) implies that function η has not constant sign on r×r. now we define the class of so-called generalized beta preinvex functions. definition 2.5. let kη be an invex set with respect to bifunction η(., .) and g,h : (0, 1) → r be real functions. a function f : kη → r is said to be generalized beta preinvex with respect to bifunction η(., .), if f(u + tη(v,u)) ≤ g(t)h(1 − t)f(u) + g(1 − t)h(t)f(v), ∀u,v ∈ kη, t ∈]0, 1[. we now discuss some special cases of definition 2.5. i. if we take g(t) = tα and h(t) = tγ, where α,γ ∈ [0, 1], then we have following definition of (α,γ)preinvex functions. quantum calculus 127 definition 2.6. let kη be an invex set with respect to bifunction η(., .). a function f : kη → r is said to be (α,γ)-preinvex with respect to bifunction η(., .), if f(u + tη(v,u)) ≤ tα(1 − t)γf(u) + (1 − t)αtγf(v), ∀u,v ∈ kη, t ∈]0, 1[,α,γ ∈ [0, 1]. ii. if we take g(t) = tα and h(t) = tα, where α ∈ [0, 1], then we have following definition of α-preinvex functions. definition 2.7. let kη be an invex set with respect to bifunction η(., .). a function f : kη → r is said to be α-preinvex with respect to bifunction η(., .), if f(u + tη(v,u)) ≤ [tα(1 − t)α][f(u) + f(v)], ∀u,v ∈ kη, t ∈]0, 1[,α ∈ [0, 1]. iii. if we take g(t) ≡ 1, then definition 2.5 reduces to the definition of h-preinvex functions [25]. we now define another new class of generalized preinvex functions. this class is not included in the class of generalized beta preinvex functions. definition 2.8. let kη be an invex set with respect to bifunction η(., .). a function f : kη → r is said to be generalized preinvex with respect to bifunction η(., .), if f(u + tη(v,u)) ≤ [tα(1 − t)γ][f(u) + f(v)], ∀u,v ∈ kη, t ∈]0, 1[,α,γ ∈ [0, 1]. note that the class of generalized preinvex functions is included in the class of h-preinvex functions. also if η(v,u) = v −u and α = 1 = γ, then we have classical tgs-convex functions. we now recall some previously known concepts and results of quantum calculus. these results will be helpful in the development of our main results. these results are mainly due to tariboon et al. [30, 31]. let j = [a,b] ⊆ r be an interval and 0 < q < 1 be a constant. the q-derivative of a function f : j → r at a point x ∈ j on [a,b] is defined as follows. definition 2.9. let f : j → r be a continuous function and let x ∈ j. then q-derivative of f on j at x is defined as dqf(x) = f(x) −f(qx + (1 −q)a) (1 −q)(x−a) , x 6= a. (2.4) a function f is q-differentiable on j if dqf(x) exists for all x ∈ j. definition 2.10. let f : j → r is a continuous function. a second-order q-derivative on j, which is denoted as d2qf, provided dqf is q-differentiable on j is defined as d 2 qf = dq(dqf) : j → r. similarly higher order q-derivative on j is defined by dnq f := j → r. lemma 2.1. let α ∈ r, then dq(x−a)α = (1 −qα 1 −q ) (x−a)α−1. tariboon et al. [30, 31] defined the q-integral as: definition 2.11. let f : i ⊂ r → r be a continuous function. then q-integral on i is defined as x∫ a f(t)dqt = (1 −q)(x−a) ∞∑ n=0 qnf(qnx + (1 −qn)a), (2.5) for x ∈ j. 128 awan, noor and noor these integrals can be viewed as riemann-type q-integral. moreover, if c ∈ (a,x), then the definite q-integral on j is defined by x∫ c f(t)dqt = x∫ a f(t)dqt− c∫ a f(t)dqt = (1 −q)(x−a) ∞∑ n=0 qnf(qnx + (1 −qn)a) − (1 −q)(c−a) ∞∑ n=0 qnf(qnc + (1 −qn)a). theorem 2.1. let f : i → r be a continuous function, then (1) dq x∫ a f(t) dqt = f(x) (2) x∫ c dqf(t)dqt = f(x) −f(c) for x ∈ (c,x). theorem 2.2. let f,g : i → r be a continuous functions, α ∈ r, then x ∈ j (1) x∫ a [f(t) + g(t)] dqt = x∫ a f(t) dqt + x∫ a g(t) dqt (2) x∫ a (αf(t))(t) dqt = α x∫ a f(t) dqt (3) x∫ a f(t) adqg(t) dqt = (fg)|xc − x∫ c g(qt + (1 −q)a)dqf(t)dqt for c ∈ (a,x). lemma 2.2. let α ∈ r\{−1}, then x∫ a (t−a)αdqt = ( 1 −q 1 −qα+1 ) (x−a)α+1. we now recall the definitions of q-gamma (γq(.)) and q-beta (bq(., .)) functions. definition 2.12 ( [12]). for α > 0, the γq(.) function is defined as: γq(α) = 1 1−q∫ 0 tα−1e−qtq dqt, where exq is one of the following q-analogues of the exponential function: etq := ∞∑ n=0 q n(n−1) 2 tn [n]! = (1 + (1 −q)t)∞q = ∞∏ j=0 (1 + qj(1 −q)t); etq := ∞∑ n=0 tn [n]! = 1 (1 − (1 −q)t)∞q = 1 ∞∏ j=0 (1 −qj(1 −q)t) . definition 2.13 ( [12]). for α > 0,γ > 0, the bq(., .) function is defined as: bq(α,γ) = 1∫ 0 tα−1(1 −qt)γ−1q dqt, where (1 −qt)γ−1q = (1 −qt)∞q (1 −qγt)∞q . quantum calculus 129 q-gamma and q-beta functions are related by the following relation: bq(α,γ) = γq(α)γq(γ) γq(α + γ) . also we have (1 − t)γ ≤ (1 −qt)γ ≤ (1 −qt)γq , for 0 ≤ t ≤ 1, γ > 0 and 0 < q < 1. for more details one can consult [12]. noor et al. [23] established a new quantum integral identity for first order q-differentiable functions, which reads as follows: lemma 2.3. let f : iη → r be a continuous function and 0 < q < 1. if adqf is an integrable function on i0η, then r ′ f (a,a + η(b,a); q; η) = qη(b,a) 1 + q 1∫ 0 (1 − (1 + q)t) adqf(a + tη(b,a)) 0dqt, where r ′ f (a,a + η(b,a); q; η) = 1 η(b,a) a+η(b,a)∫ a f(x) adqx− qf(a) + f(a + η(b,a)) 1 + q noor et al. [19] also established the following integral identity for twice q-differentiable functions. lemma 2.4. let f : iη → r be a twice q-differentiable function on i◦η such that d2qf be continuous and integrable on iη where 0 < q < 1, then r ′′ f (a,a + η(b,a); q; η) = q2η2(b,a) 1 + q 1∫ 0 t(1 −qt)d2qf(a + tη(b,a))dqt, where r ′′ f (a,a + η(b,a); q; η) = qf(a) + f(a + η(b,a)) 1 + q − 1 η(b,a) a+η(b,a)∫ a f(x)dqx. for some useful information on quantum calculus interested readers are referred to [7, 11, 12, 29]. from now onwards i = [a,a + η(b,a)] will be the interval unless otherwise specified. 3. some q-hermite-hadamard type inequalities via generalized preinvex functions in this section, we discuss our main results. theorem 3.1. let f : i ⊂ r → r be generalized preinvex function, where η(b,a) > 0 and moreover η(., .) satisfies condition c, then 2α+γ−1f ( a + η(b,a) 2 ) ≤ 1 η(b,a) a+η(b,a)∫ a f(x)dqx ≤ bq(α + 1,γ + 1)[f(a) + f(b)]. proof. since it is given that f is a generalized preinvex function and η(., .) satisfies condition c, then f ( 2a + η(b,a) 2 ) ≤ 1 2α+γ [f(a + (1 − t)η(b,a)) + f(a + tη(b,a))]. q-integrating above inequality with respect to t on [0, 1], we have 2α+γ−1f ( 2a + η(b,a) 2 ) ≤ 1 η(b,a) a+η(b,a)∫ a f(x)dqx. (3.1) 130 awan, noor and noor also f(a + tη(b,a)) ≤ tα(1 − t)γ[f(a) + f(b)]. q-integrating above inequality with respect to t on [0, 1], we have 1 η(b,a) b∫ a f(x)dqx ≤ bq(α + 1,γ + 1)[f(a) + f(b)]. (3.2) on summation of inequalities (3.1) and (3.2), we get the required result. � if q → 1, α = γ = 1 and η(b,a) = b−a in theorem 3.1, we get theorem 2.1 [32]. if q → 1 and α = γ = 1 in theorem 3.1, we have following new result for generalized preinvex functions corollary 3.1. under the assumptions of theorem 3.1, if q → 1 and α = γ = 1, then we have 2f ( a + η(b,a) 2 ) ≤ 1 η(b,a) a+η(b,a)∫ a f(x)dx ≤ f(a) + f(b) 6 . if α = γ = 1 in theorem 3.1, we have following new result for tgs-preinvex functions corollary 3.2. under the assumptions of theorem 3.1, if α = γ = 1, then we have 2f ( a + η(b,a) 2 ) ≤ 1 η(b,a) a+η(b,a)∫ a f(x)dqx ≤ ( 1 1 + q − 1 1 + q + q2 ) [f(a) + f(b)]. our next result is q-hermite-hadamard’s inequality via product of two generalized preinvex functions. theorem 3.2. let f,g : i ⊂ r → r be two generalized preinvex functions. moreover η(., .) satisfies condition c and η(b,a) > 0, then i. the left side of the inequality reads as: 22(α+γ)−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) −bq(α + γ + 1,α + γ + 1)[m(a,b) + n(a,b)] ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx, ii. the right side of the inequality reads as: 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx ≤ [m(a,b) + n(a,b)]bq(α + γ + 1,α + γ + 1), where m(a,b) = f(a)g(a) + f(b)g(b); (3.3) n(a,b) = f(a)g(b) + f(b)g(a). (3.4) proof. i. since it is given that f and g are generalized preinvex functions, then f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22(α+γ) [f(a + (1 − t)η(b,a)) + f(a + tη(b,a))][g(a + (1 − t)η(b,a)) + g(a + tη(b,a))] = 1 22(α+γ) {f(a + (1 − t)η(b,a))g(a + (1 − t)η(b,a)) + f(a + tη(b,a))g(a + tη(b,a)) +f(a + tη(b,a))g(a + (1 − t)η(b,a)) + f(a + (1 − t)η(b,a))g(a + tη(b,a))} . quantum calculus 131 q-integrating both sides of above inequality with respect to t on [0, 1], we have f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22(α+γ)   1∫ 0 f(a + (1 − t)η(b,a))g(a + (1 − t)η(b,a))dqt + 1∫ 0 f(a + tη(b,a))g(a + tη(b,a))dqt + 1∫ 0 f(a + tη(b,a))g(a + (1 − t)η(b,a))dqt + 1∫ 0 f(a + (1 − t)η(b,a))g(a + tη(b,a))dqt   = 1 22(α+γ)   2η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + 2 1∫ 0 tα+γ(1 − t)α+γ[f(a) + f(b)][g(a) + g(b)]dqt   = 1 22(α+γ)−1 { 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + bq(α + γ + 1,α + γ + 1) × [f(a) + f(b)][g(a) + g(b)] } = 1 22(α+γ)−1 { 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + bq(α + γ + 1,α + γ + 1)[m(a,b) + n(a,b)] } . this completes the proof of first part. ii. now we prove second part of the theorem. using the hypothesis of the theorem that f and g are generalized preinvex functions, we have f(a + tη(b,a))g(a + tη(b,a)) ≤ tα+γ(1 − t)α+γ[f(a) + f(b)][g(a) + g(b)]. q-integrating both sides of above inequality with respect to t on [0, 1], we have 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx = 1∫ 0 f(a + tη(b,a))g(a + tη(b,a))dqt ≤ [f(a) + f(b)][g(a) + g(b)] 1∫ 0 tα+γ(1 − t)α+γdqt = [m(a,b) + n(a,b)]bq(α + γ + 1,α + γ + 1). this completes the proof of second part. � note that when q → 1, α = γ = 1 and η(b,a) = b − a theorem 3.2 reduces to previously known results, see [32]. when q → 1 and α = γ = 1 theorem 3.2 reduces to new result for tgs-preinvexity. corollary 3.3. under the assumptions of theorem 3.2, if q → 1 and α = γ = 1, then, we have i. 8f ( 2a+η(b,a) 2 ) g ( 2a+η(b,a) 2 ) − 1 30 [m(a,b) + n(a,b)] ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dx, ii. 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dx ≤ 1 30 [m(a,b) + n(a,b)], where m(a,b) and n(a,b) are given by (3.3) and (3.4) respectively. 132 awan, noor and noor when α = γ = 1 theorem 3.2 reduces to new result for tgs-preinvexity. corollary 3.4. under the assumptions of theorem 3.2, if α = γ = 1, then, we have i. 8f ( 2a+η(b,a) 2 ) g ( 2a+η(b,a) 2 ) −ψ1(q)[m(a,b) + n(a,b)] ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx, ii. 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx ≤ ψ1(q)[m(a,b) + n(a,b)], where m(a,b) and n(a,b) are given by (3.3), (3.4) and ψ1(q) := 1 1 + q + q2 + 1 1 + q + q2 + q3 + q4 − 2 1 + q + q2 + q3 , (3.5) respectively. theorem 3.3. let f,g : i ⊂ r → r be two generalized preinvex functions, then 1 2η2(b,a)bq(α + γ + 1,α + γ + 1) a+η(b,a)∫ a a+η(b,a)∫ a 1∫ 0 f(a + tη(b,a))g(a + tη(b,a))dqtdqydqx ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + b2q(α + 1,γ + 1)[m(a,b) + n(a,b)], where m(a,b) and n(a,b) are given by (3.3) and (3.4) respectively. proof. since it is given that f and g are generalized preinvex functions, then f(x + tη(y,x))g(x + tη(y,x)) ≤ tα+γ(1 − t)α+γ[f(x) + f(y)][g(x) + g(y)]. q-integrating both sides of above inequality with respect to t on the interval [0, 1], we have 1∫ 0 f(x + tη(y,x))g(x + tη(y,x))dqt ≤ 1∫ 0 tα+γ(1 − t)α+γ[f(x) + f(y)][g(x) + g(y)]dqt = bq(α + γ + 1,α + γ + 1)[f(x) + f(y)][g(x) + g(y)]. quantum calculus 133 now again q-integrating both sides of above inequality on [a,a + η(b,a)] × [a,a + η(b,a)], we have a+η(b,a)∫ a a+η(b,a)∫ a 1∫ 0 f(x + tη(y,x))g(x + tη(y,x))dqtdqydqx ≤ bq(α + γ + 1,α + γ + 1) a+η(b,a)∫ a a+η(b,a)∫ a [f(x) + f(y)][g(x) + g(y)]dqydqx = bq(α + γ + 1,α + γ + 1) × a+η(b,a)∫ a a+η(b,a)∫ a [f(x)g(x) + f(y)g(y) + f(x)g(y) + f(y)g(x)]dqydqx = bq(α + γ + 1,α + γ + 1) ×   a+η(b,a)∫ a a+η(b,a)∫ a [f(x)g(x) + f(y)g(y)]dqydqx + a+η(b,a)∫ a f(x)dqx a+η(b,a)∫ a g(y)dqy + a+η(b,a)∫ a f(y)dqy a+η(b,a)∫ a g(x)dqx   . using theorem 3.1, we have a+η(b,a)∫ a a+η(b,a)∫ a 1∫ 0 f(x + tη(y,x))g(x + tη(y,x))dqtdqydqx ≤ bq(α + γ + 1,α + γ + 1) ×  2η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + 2η 2(b,a)b2q(α + 1,γ + 1)[m(a,b) + n(a,b)]   = 2bq(α + γ + 1,α + γ + 1) ×  η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + η 2(b,a)b2q(α + 1,γ + 1)[m(a,b) + n(a,b)]   . multiplying both sides of of above inequality by 1 η2(b,a) completes the proof. � note that when q → 1, α = γ = 1 and η(b,a) = b−a in theorem 3.3, we get theorem 2.4 [32]. if we take q → 1 and α = γ = 1 in theorem 3.3, we get a new result for tgs-preinvexity. corollary 3.5. under the assumptions of theorem 3.3, if q → 1 and α = γ = 1, then, we have 15 η2(b,a) a+η(b,a)∫ a a+η(b,a)∫ a 1∫ 0 f(a + tη(b,a))g(a + tη(b,a))dtdydx ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dx + 1 36 [m(a,b) + n(a,b)], where m(a,b), n(a,b) are given by (3.3) and (3.4), respectively. if we take α = γ = 1 in theorem 3.3, we get a new result for tgs-preinvexity. 134 awan, noor and noor corollary 3.6. under the assumptions of theorem 3.3, if α = γ = 1, then, we have ψ−11 (q) 2η2(b,a) a+η(b,a)∫ a a+η(b,a)∫ a 1∫ 0 f(a + tη(b,a))g(a + tη(b,a))dqtdqydqx ≤ 1 η(b,a) a+η(b,a)∫ a f(x)g(x)dqx + ( 1 1 + q − 1 1 + q + q2 ) [m(a,b) + n(a,b)], where m(a,b), n(a,b) and ψ1(q) are given by (3.3), (3.4) and (3.5), respectively. now we prove some q-hermite-hadamard type inequalities via q-differentiable generalized preinvex functions. theorem 3.4. let f : i = [a,a+η(b,a)] ⊂ r → r be a q-differentiable function on i◦ (the interior of i) with dq be continuous and integrable on i where 0 < q < 1. if |dq|r, r ≥ 1 is generalized preinvex function, then ∣∣r′f (a,a + η(b,a); q; η)∣∣ ≤ qη(b,a)1 + q ( 2q (1 + q)2 )1−1 r (ψ2(q){|dqf(a)| r + |dqf(b)|r}) 1 r , where ψ2(q) = bq(α + 1,γ + 1) − (1 + q)bq(α + 2,γ + 1). (3.6) proof. utilizing lemma 2.3, property of modulus, power mean inequality and the hypothesis of the theorem, we have∣∣r′f (a,a + η(b,a); q; η∣∣ = ∣∣∣∣∣∣qη(b,a)1 + q 1∫ 0 (1 − (1 + q)t)dqf(a + tη(b,a)) dqt ∣∣∣∣∣∣ ≤ qη(b,a) 1 + q   1∫ 0 |1 − (1 + q)t|dq  1− 1 r   1∫ 0 (1 − (1 + q)t)|dqf(a + η(b,a))|rdqt   1 r ≤ qη(b,a) 1 + q ( 2q (1 + q)2 )1−1 r   1∫ 0 (1 − (1 + q)t)[tα(1 − t)γ{|dqf(a)|r + |dqf(b)|r}]dqt   1 r = qη(b,a) 1 + q ( 2q (1 + q)2 )1−1 r {[bq(α + 1,γ + 1) − (1 + q)bq(α + 2,γ + 1)]{|dqf(a)|r + |dqf(b)|r}} 1 r = qη(b,a) 1 + q ( 2q (1 + q)2 )1−1 r (ψ2(q){|dqf(a)| r + |dqf(b)|r}) 1 r . this completes the proof. � now using lemma 2.4, we drive some more quantum estimates for hermite-hadamard type inequalities via twice q-differentiable generalized preinvex functions. theorem 3.5. let f : iη → r be a twice q-differentiable function on i◦η such that d2qf be continuous and integrable on iη where 0 < q < 1. if |d2qf(x)| is generalized preinvex function, then∣∣r′′f (a,a + η(b,a); q; η)∣∣ ≤ q2η2(b,a)1 + q ωq(α,γ){|d2qf(a)| + |d2qf(b)|}, where ωq(α,γ) := bq(α + 2,γ + 1) −qbq(α + 3,γ + 1). (3.7) quantum calculus 135 proof. using lemma 2.4, property of modulus and the fact that |d2qf(x)| is generalized preinvex function, we have∣∣r′′f (a,a + η(b,a); q; η)∣∣ = ∣∣∣∣∣∣q 2η2(b,a) 1 + q 1∫ 0 t(1 −qt)d2qf(a + tη(b,a))dqt ∣∣∣∣∣∣ ≤ q2η2(b,a) 1 + q 1∫ 0 t(1 −qt)|d2qf(a + tη(b,a))|dqt ≤ q2η2(b,a) 1 + q 1∫ 0 t(1 −qt)[tα(1 − t)γ{|d2qf(a)| + |d 2 qf(b)|}]dqt = q2η2(b,a) 1 + q (bq(α + 2,γ + 1) −qbq(α + 3,γ + 1)){|d2qf(a)| + |d 2 qf(b)|}. this completes the proof. � theorem 3.6. let f : iη → r be a twice q-differentiable function on i◦η such that d2qf be continuous and integrable on iη where 0 < q < 1. if |d2qf(x)|r is generalized preinvex function, then, for 1 p + 1 r = 1, r > 1, we have∣∣r′′f (a,a + η(b,a); q; η)∣∣ ≤ q2η2(b,a)1 + q φ 1p (α,q) [bq(α + 1,γ + 1){|d2qf(a)|r + |d2qf(b)|r}]1r , where φ(α,q) = (1 −q) ∞∑ n=0 (qn)p+1(1 −qn+1)p. (3.8) proof. using lemma 2.4, holder’s inequality and the fact that |d2qf(x)|r is generalized preinvex function, we have∣∣r′′f (a,a + η(b,a); q; η)∣∣ = ∣∣∣∣∣∣q 2η2(b,a) 1 + q 1∫ 0 t(1 −qt)d2qf(a + tη(b,a))dqt ∣∣∣∣∣∣ ≤ q2η2(b,a) 1 + q 1∫ 0 t(1 −qt)|d2qf(a + tη(b,a))|dqt ≤ q2η2(b,a) 1 + q   1∫ 0 (t−qt2)pdqt   1 p  {|d2qf(a)|r + |d2qf(b)|r} 1∫ 0 tα(1 − t)γdqt   1 r = q2η2(b,a) 1 + q ( (1 −q) ∞∑ n=0 (qn)p+1(1 −qn+1)p )1 p [ bq(α + 1,γ + 1){|d2qf(a)| r + |d2qf(b)| r} ]1 r . this completes the proof. � theorem 3.7. let f : iη → r be a twice q-differentiable function on i◦η such that d2qf be continuous and integrable on iη where 0 < q < 1. if |d2qf(x)|r is generalized preinvex function, then, for r ≥ 1, we have ∣∣r′′f (a,a + η(b,a); q; η)∣∣ ≤ q2η2(b,a)1 + q θ1−1rq ω 1rq (α,γ) [|d2qf(a)|r + |d2qf(b)|r]1r , 136 awan, noor and noor where ωq(α,γ) is given by (3.7) and θq = 1 (1 + q)(1 + q + q2) . (3.9) respectively. proof. using lemma 2.4, power means inequality, and the fact that |d2qf(x)|r is generalized preinvex function, we have∣∣r′′f (a,a + η(b,a); q; η)∣∣ = ∣∣∣∣∣∣q 2η2(b,a) 1 + q 1∫ 0 t(1 −qt)d2qf(a + tη(b,a))dqt ∣∣∣∣∣∣ ≤ q2η2(b,a) 1 + q   1∫ 0 t(1 −qt)dqt  1− 1 r   1∫ 0 t(1 −qt)|d2qf(a + tη(b,a))| rdqt   1 r ≤ q2η2(b,a) 1 + q ( 1 (1 + q)(1 + q + q2) )1−1 r ×   1∫ 0 tα+1(1 −qt)(1 − t)γ[|d2qf(a)| r + |d2qf(b)| r]dqt   1 r = q2η2(b,a) 1 + q ( 1 (1 + q)(1 + q + q2) )1−1 r [ ωq(α,γ){|d2qf(a)| r + |d2qf(b)| r} ]1 r . this completes the proof. � acknowledgements the authors are thankful to anonymous referees for their helpful comments and suggestions. authors are pleased to acknowledge the support of ”the support of distinguished scientist fellowship program(dsfp), king saud university, riyadh, saudi arabia”. references [1] a. ben-israel, b. mond, what is invexity? j. austral. math. soc. ser., b, 28 (1986) 1-9. [2] a. barani, a. g. ghazanfari, s. s. dragomir, hermite-hadamard inequality for functions whose derivatives absolute values are preinvex, j. inequal. appl. (2012) (2012), art. id. 247. [3] g. cristescu, l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrecht, holland, 2002. [4] g. cristescu, m. a. noor, m. u. awan, bounds of the second degree cumulative frontier gaps of functions with generalized convexity, carpath. j. math, 31(2) (2015), 173-180. [5] s. s. dragomir, r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11(5), 91–95, (1998). [6] s. s. dragomir, c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university australia, (2000). [7] t. ernst, a comprehensive treatment of q-calculus, springer basel heidelberg new york dordrecht london (2014). [8] h. gauchman, integral inequalities in q-calculus. comput. math. appl. 47 (2004), 281-300. [9] m. a. hanson, on sufficiency of the kuhn-tucker conditions, j. math. anal. appl., 80 (1981), 545-550. [10] d. a. ion, some estimates on the hermite-hadamard inequality through quasi-convex functions, annals of university of craiova, math. comp. sci. ser., 34 (2007), 82–87. [11] f. h. jackson, on a q-definite integrals, q. j. pure appl. math. 41 (1910), 193-203. [12] v. kac, p. cheung, quantum calculus. springer, new york (2002). [13] s. r. mohan, s. k. neogy, on invex sets and preinvex functions, j. math. anal. appl. 189 (1995), 901-908. [14] b. b.-mohsin, m. u. awan, m. a. noor, k. i. noor, s. iftikhar, a. g. khan, generalized beta-convex functions and integral inequalities, int. j. anal. appl., 14 (2) (2017), 180-192. [15] c. p. niculescu, l.-e. persson , convex functions and their applications. a contemporary approach, cms books in mathematics vol. 23, springer-verlag, new york, (2006). [16] m. a. noor, hermite-hadamard integral inequalities for log-preinvex functions, j. math. anal. approx. theory, 2 (2007), 126-131. quantum calculus 137 [17] m. a. noor, invex equilibrium problems, j. math. anal. appl, 302 (2005) 463-475. [18] m. a. noor, advanced convex analysis, lecture notes, comsats institute of information technology, islamabad, pakistan, 2008-2015. [19] m. a. noor, m. u. awan, k. i. noor, new quantum estimates of integral inequalities via generalized preinvex functions, magnt research report, 4(1) (2016), 1-23. [20] m. a. noor, m. u. awan, k. i. noor some new q-estimates for certain integral inequalities, facta universitatis (nis) ser. math. inform. 31(4) (2016), 801-813. [21] m. a. noor, m. u. awan, k. i. noor, quantum ostrowski inequalities for q-differentiable convex functions, j. math. inequal., 10(4) (2016), 1013-1018. [22] m. a. noor, k. i. noor, m. u. awan, some quantum estimates for hermite-hadamard inequalities, appl. math. comput. 251 (2015), 675-679. [23] m. a. noor, k. i. noor, m. u. awan, some quantum integral inequalities via preinvex functions, appl. math. comput., 269 (2015), 242-251. [24] m. a. noor, k. i. noor, m. u. awan, quantum analogues of hermite-hadamard type inequalities for generalized convexity, in: n. daras and m.t. rassias (ed.), computation, cryptography and network security (2015). [25] m. a. noor, k. i. noor, m. u. awan, j. li, on hermite-hadamard inequalities for h-preinvex functions, filomat, 28(7), (2014), 1463-1474. [26] ozdemir m. e., on iyengar-type inequalities via quasi-convexity and quasi-concavity, arxiv:1209.2574v1 [math.fa] (2012). [27] pearce c. e. m., pecaric j. e: inequalities for differentiable mappings with application to special means and quadrature formulae, appl. math. lett. 13 (2000), 51-55. [28] pecaric j. e., prosch f., tong y. l., : convex functions, partial orderings, and statistical applications, academic press, new york, (1992). [29] w. sudsutad, s. k. ntouyas, j. tariboon, quantum integral inequalities for convex functions, j. math. inequal. 9(3) (2015), 781-793. [30] j. tariboon, s. k. ntouyas, quantum calculus on finite intervals and applications to impulsive difference equations. adv. differ. equ. 2013 (2013), art. id 282. [31] j. tariboon, s. k. ntouyas, quantum integral inequalities on finite intervals, j. inequal. app. 2014 (2014), art. id 121. [32] m. tunç, e. gov, u. sanal, on tgs-convex function and their inequalities, facta universitatis (nis) ser. math. inform. 30(5) (2015), 679-691. [33] s. varošanec, on h-convexity, j. math. anal. appl. 326(2007), 303-311. [34] t. weir, b. mond, preinvex functions in multiobjective optimization, j. math. anal. appl. 136(1988), 29-38. 1department of mathematics, gc university, faisalabad, pakistan 2mathematics department, king saud university, riyadh, saudi arabia 3department of mathematics, comsats institute of information technology, islamabad, pakistan ∗corresponding author: awan.uzair@gmail.com 1. introduction and preliminaries 2. preliminaries 3. some q-hermite-hadamard type inequalities via generalized preinvex functions acknowledgements references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 167-174 http://www.etamaths.com generalized stabilities of euler-lagrange-jensen (a,b)-sextic functional equations in quasi-β-normed spaces john michael rassias1,∗, krishnan ravi2 and beri venkatachalapathy senthil kumar3 abstract. the aim of this paper is to investigate generalized ulam-hyers stabilities of the following euler-lagrange-jensen-(a,b)-sextic functional equation f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 ) [ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 ) ( a4 − b4 ) [f(x) + f(y)] where a 6= b, such that k ∈ r; k = a+b 6= 0,±1 and λ = 1+(a−b)6−2 ( a6 + b6 ) −62(ab)2 ( a2 + b2 ) 6= 0, in quasi-β-normed spaces by using fixed point method. in particular, we prove generalized stabilities involving the sum of powers of norms, product of powers of norms and the mixed product-sum of powers of norms of the above functional equation in quasi-β-normed spaces by using fixed point method. a counter-example for a singular case is also indicated. 1. introduction the classical theory of stability of functional equations was instigated by the question of ulam [42] in the year 1940. in the subsequent year 1941, hyers [16] was the foremost mathematician to establish the pioneering result connected with the stability of functional equations. the result obtained by hyers is called as hyers-ulam stability of functional equation. later in the year 1950, aoki [4] made a further simplification to the theorem of hyers. in the year 1978, th.m. rassias [41] took a broad view in the hyers result by taking the upper bound as a sum of powers of norms. the result obtained by th.m. rassias is recognized as hyers-ulam-rassias stability of functional equation. john m. rassias ( [29], [30], [31]) provided a further generalization of the result of hyers by using weaker conditions controlled by a product of different powers of norms. the result proved by john m. rassias is termed as ulam-gavruta-rassias stability of functional equation. further, in the year 1994, gavruta [14] provided a generalization of th.m. rassias theorem by replacing a general control function as an upper bound. the stability result ascertained by gavruta is celebrated as generalized ulam-hyers stability of functional equation. in the year 2008, ravi et al. [39] investigated the stability of the following quadratic functional equation q(`x + y) + q(`x−y) = 2q(x + y) + 2q(x−y) + 2 ( `2 − 2 ) q(x) − 2q(y) for any arbitrary but fixed real constant ` with ` 6= 0; ` 6= ±1; ` 6= ± √ 2 using mixed product-sum of powers of norms. this stability result acquired by ravi et al. is known as j.m. rassias stability involving mixed product-sum of powers of norms. several stability results have recently been obtained for various functional equations and functional inequalities, also for mappings with more general domains and ranges (see [6], [7], [8], [11], [13], [22], [23], [24], [40]). many research monographs are also available on functional equations, one can see ( [1], [2], [3], [10], [17], [20], [21]). received 6th march, 2017; accepted 27th april, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 39b82, 39b72. key words and phrases. quasi-β-normed spaces; sextic mapping; (β,p)-banach spaces; generalized ulam-hyers stabilities. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 167 168 rassias, ravi and kumar in 1996, isac and th.m. rassias [18] were the first to provide applications of the stability theory of functional equations for the proof of new fixed point theorems with applications. the stability problems of several various functional equations have been extensively investigated by a number of authors using fixed point methods (see [5], [26], [43], [45]). john m. rassias [32] introduced euler-lagrange type quadratic functional equation of the form f(ax + by) + f(bx−ay) = ( a2 + b2 ) (f(x) + f(y)) (1.1) motivated from the following pertinent algebraic equation |ax + by|2 + |bx−ay|2 = ( a2 + b2 )( |x|2 + |y|2 ) . (1.2) the solution of the functional equation (1.1) is called an euler-lagrange quadratic type mapping. in addition, john m. rassias ( [32], [33], [34], [35], [36]) generalized the standard quadratic equation to the following quadratic equation m1m2 |a1x1 + a2x2| 2 + |m2a2x1 −m1a1x2| 2 = ( m1 |a1| 2 + m2 |a2| 2 )( m2 |x1| 2 + m1 |x2| 2 ) . he introduced and investigated the general pertinent euler-lagrange quadratic mappings. these euler-lagrange mappings are named euler-lagrange-rassias mappings, and the corresponding eulerlagrange equations are called euler-lagrange-rassias equations (see [15], [25], [27], [28]). these notions provide a cornerstone in analysis, because of their particular interest in probability theory and stochastic analysis in marrying these fields of research to functional equations via the pioneering introduction of the euler-lagrange-rassias quadratic weighted means and fundamental mean equations ( [15], [34], [35]). john m. rassias [38] introduced the cubic functional equation, as follows: f(x + 2y) − 3f(x + y) + 3f(x) −f(x−y) = 6f(y). (1.3) this inspiring cubic functional equation was the transition from the following famous euler-lagrangerassias quadratic functional equation: f(x + y) − 2f(x) + f(x−y) = 2f(y) to the cubic functional equations. john m. rassias [37] introduced also the following quartic functional equation: f(x + 2y) + f(x− 2y) + 6f(x) = 4f(x + y) + 4f(x−y) + 24f(y). (1.4) it is easy to see that f(x) = x4 is a solution of equation (1.4). for this reason, the equation (1.4) is called a quartic functional equation. the general solution of (1.4) is determined without assuming any regularity conditions on the unknown function (refer [9]). since the solution of equation (1.4) is even, we can rewrite (1.4) as f(2x + y) + f(2x−y) = 4f(x + y) + 4f(x−y) + 24f(x) − 6f(y). (1.5) in 2010, xu et al. [44] achieved the general solution and proved the stability of the quintic functional equation f(x + 3y) − 5f(x + 2y) + 10f(x + y) − 10f(x) + 5f(x−y) −f(x− 2y) = 120f(y) (1.6) and the sextic functional equation f(x + 3y) − 6f(x + 2y) + 15f(x + y) − 20f(x) + 15f(x−y) − 6f(x + 2y) + f(x− 3y) = 720f(y) (1.7) in quasi-β-normed spaces using fixed point method. euler-lagrange-jensen (a,b)-sextic functional equations 169 in this paper, the first author of this paper introduces a new euler-lagrange-jensen (a,b; k = a+b)sextic functional equation f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 )[ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 )( a4 − b4 ) [f(x) + f(y)] (1.8) where a 6= b, such that k ∈ r; k = a+b 6= 0,±1 and λ = 1+(a−b)6−2 ( a6 + b6 ) −62(ab)2 ( a2 + b2 ) 6= 0. then we investigate the generalized ulam-hyers stability of the equation (1.8) in quasi-β-normed spaces using fixed point method. we extend the stability results involving sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms of the above functional equation. we also provide a counter-example to show that the functional equation (1.8) is not stable for singular case. it is easy to see that the function f(x) = kx6 is a solution of the equation (1.8). hence we say that it is a sextic functional equation. 2. preliminaries in this section, we recall some fundamental notions in association with quasi-β-normed spaces and m-additive symmetric mappings. let β be a fixed real number with 0 < β ≤ 1 and let k denote either r or c. definition 2.1. let x be a linear space over k. a quasi-β-norm ‖·‖ is a real-valued function on x satisfying the following conditions: (i) ‖a‖≥ 0 for all a ∈x and ‖a‖ = 0 if and only if a = 0. (ii) ‖ηa‖ = |η|β · ‖a‖ for all η ∈ k and all a ∈x. (iii) there is a constant k ≥ 1 such that ‖a + b‖≤ k (‖a‖ + ‖b‖) for all a,b ∈x . the pair (x ,‖·‖) is called quasi-β-normed space if ‖·‖ is a quasi-β-norm on x. the smallest possible k is called the modulus of concavity of ‖·‖. definition 2.2. a complete quasi-β-normed space is called a quasi-β-banach space. definition 2.3. a quasi-β-norm ‖·‖ is called a (β,p)-norm (0 < p < 1) if ‖x + y‖p ≤‖x‖p + ‖y‖p for all x,y ∈x. in this case, a quasi-β-banach space is called a (β,p)-banach space. 3. generalized ulam-hyers stability of equation (1.8) throughout this section, we assume that x is a linear space and y is a (β,p)-banach space with (β,p)-norm ‖·‖y. let k be the modulus of concavity of ‖·‖y. for notational convenience, we define the difference operator for a given mapping f : x →y as dsf(x,y) = f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2 ( a2 + b2 )[ f ( x + y 2 ) + f ( x−y 2 )] + 2 ( a2 − b2 )( a4 − b4 ) [f(x) + f(y)] for all x,y ∈x . lemma 3.1. (see [44]). let j ∈ {−1, 1} be fixed, m,b ∈ n with b ≥ 2 and φ : x → [0,∞) be a function such that there exists an l < 1 with φ ( bjx ) ≤ bjmβlφ(x) for all x ∈x. let g : x →y be a mapping satisfying ‖g(bx) − bmg(x)‖y ≤ φ(x) (3.1) 170 rassias, ravi and kumar for all x ∈x, then there exists a uniquely determined mapping g : x →y such that g(bx) = bmg(x) and ‖g(x) −g(x)‖y ≤ 1 bmβ |1 −lj| φ(x) (3.2) for all x ∈x. theorem 3.1. let i ∈ {−1, 1} be fixed. let φ : x ×x → [0,∞) be a function such that there exists an l < 1 with φ ( kix,kiy ) ≤ k6iβlφ(x,y) for all x,y ∈x. let f : x →y be a mapping satisfying ‖dsf(x,y)‖y ≤ φ(x,y) (3.3) for all x,y ∈x. then there exists a unique sextic mapping s : x →y such that ‖f(x) −s(x)‖y ≤ 1 k6β |1 −li| ψ(x) (3.4) for all x ∈x, where ψ(x) = k 2β [ φ(x,x) + 32β(ab)2β ( a2 + b2 )β λβ φ(0, 0) ] . proof. plugging (x,y) into (0, 0) in (3.3), we obtain ‖f(0)‖y ≤ 1 2βλβ φ(0, 0). (3.5) switching (x,y) to (x,x) in (3.3), one finds∥∥f(kx) −k6f(x) − 32(ab)2 (a2 + b2)f(0)∥∥y ≤ 12β φ(x,x) (3.6) for all x ∈x . using (3.5) and (3.6), we arrive∥∥f(kx) −k6f(x)∥∥y ≤ ψ(x) (3.7) for all x ∈ x . by lemma 3.1, there exists a unique mapping s : x → y such that s(kx) = k6s(x) and ‖f(x) −s(x)‖y ≤ 1 k6β |1 −li| ψ(x) for all x ∈x . it remains to show that s is a sextic map. by (3.3), we have∥∥∥∥ 1k6indsf (kinx,kiny) ∥∥∥∥ y ≤ k−6inβφ ( kinx,kiny ) ≤ k−6inβ ( k6iβl )n φ(x,y) = lnφ(x,y) for all x,y ∈ x and n ∈ n. so ‖dss(x,y)‖y = 0 for all x,y ∈ x . thus the mapping s : x → y is sextic, which completes the proof of theorem. � corollary 3.1. let x be a quasi-α-normed space with quasi-α-norm ‖·‖x, and let y be a (β,p)banach space with (β,p)-norm ‖·‖y. let k1,p be positive numbers with p 6= 6β α and f : x → y be a mapping satisfying ‖dsf(x,y)‖y ≤ k1 (‖x‖ p x + ‖y‖ p x) for all x,y ∈x. then there exists a unique sextic mapping s : x →y such that ‖f(x) −s(x)‖y ≤   k1k 2β(k6β−kpα) ‖x‖ p x , p ∈ ( 0, 6β α ) kpαk1k k6β2β(kpα−k6β) ‖x‖ p x , p ∈ ( 6β α ,∞ ) for all x ∈x. proof. the proof is obtained by taking φ(x,y) = k1 (‖x‖ p x + ‖y‖ p x), for all x,y ∈ x and l = kpα k6β in theorem 3.1. � euler-lagrange-jensen (a,b)-sextic functional equations 171 corollary 3.2. let x be a quasi-α-normed space with quasi-α-norm ‖·‖x, and let y be a (β,p)-banach space with (β,p)-norm ‖·‖y. let k2,p,q be positive numbers with ρ = p + q 6= 6β α and f : x →y be a mapping satisfying ‖dsf(x,y)‖y ≤ k2 ‖x‖ p x ‖y‖ q x for all x,y ∈ x. then there exists a unique sextic mapping s : x →y such that ‖f(x) −s(x)‖y ≤   k2k 2β(k6β−kρα) ‖x‖ ρ x , ρ ∈ ( 0, 6β α ) kραk2k k6β2β(kρα−k6β) ‖x‖ ρ x , ρ ∈ ( 6β α ,∞ ) for all x ∈x. proof. letting φ(x,y) = k2 ‖x‖ p x ‖y‖ q x , for all x,y ∈ x and l = kρα k6β in theorem 3.1, we obtain the required results. � corollary 3.3. let x be a quasi-α-normed space with quasi-α-norm ‖·‖x, and let y be a (β,p)banach space with (β,p)-norm ‖·‖y. let k3,r be positive numbers r 6= 3β α and f : x → y be a mapping satisfying ‖dsf(x,y)‖y ≤ k3 [ ‖x‖rx ‖y‖ r x + ( ‖x‖2rx + ‖y‖ 2r x )] for all x,y ∈x. then there exists a unique sextic mapping s : x →y such that ‖f(x) −s(x)‖y ≤   3k3k 2β(k6β−k2rα) ‖x‖ 2r x , r ∈ ( 0, 3β α ) 3k2rαk3k k6β2β(k2rα−k6β) ‖x‖ 2r x , r ∈ ( 3β α ,∞ ) for all x ∈x. proof. by taking ϕ(x,y) = k3 [ ‖x‖rx ‖y‖ r x + ( ‖x‖2rx + ‖y‖ 2r x )] , for all x,y ∈ x and l = k 2rα k6β in theorem 3.1, we arrive at the desired results. � 4. counter-example in this section, using the idea of the well-known counter-example provided by z. gajda [12], we illustrate a counter-example that the functional equation (1.8) is not stable for p = 6β α in corollary 3.1. we consider the function ϕ(x) = { x6, for |x| < 1 1, for |x| ≥ 1. (4.1) where ϕ : r → r. let f : r → r be defined by f(x) = ∞∑ n=0 2−6nϕ(2nx) (4.2) for all x ∈ r. the function f serves as a counter-example for the fact that the functional equation (1.8) is not stable for p = 6β α in corollary 3.1 in the following theorem. theorem 4.1. if the function f defined in (4.2) satisfies the functional inequality |dsf(x,y)| ≤ 643δ 63 ( |x|6 + |y|6 ) (4.3) where δ = 2 [ 1 + (a− b)6 − 2 ( a6 + b6 ) − 62(ab)2 ( a2 + b2 )] > 0, for all x,y ∈ r, then there do not exist a sextic mapping s : r → r and a constant � > 0 such that |f(x) −s(x)| ≤ � |x|6 , for all x ∈ r. proof. first, we are going to show that f satisfies (4.3). |f(x)| = ∣∣∣∣∣ ∞∑ n=0 2−6nϕ(2nx) ∣∣∣∣∣ ≤ ∞∑ n=0 1 26n = 64 63 . 172 rassias, ravi and kumar therefore, we see that f is bounded by 64 63 on r. if |x|6 + |y|6 = 0 or |x|6 + |y|6 ≥ 1 64 , then |dsf(x,y)| ≤ 64δ 63 ≤ 642δ 63 ( |x|6 + |y|6 ) . now, suppose that 0 < |x|6 + |y|6 < 1 64 . then there exists a non-negative integer k such that 1 64k+1 ≤ |x|6 + |y|6 < 1 64k . (4.4) hence 64k |x|6 < 1, 64k |y|6 < 1 and 2n(ax + by), 2n(bx + ay), 2n ( ax−by a−b ) , 2n ( bx−ay b−a ) , 2n ( x+y 2 ) , 2n ( x−y 2 ) , 2nx, 2ny ∈ (−1, 1) for all n = 0, 1, 2, . . . ,k − 1. hence for n = 0, 1, 2, . . . ,k − 1, ϕ (2n(ax + by)) + ϕ (2n(bx + ay)) + (a− b)6 [ ϕ ( 2n ( ax− by a− b )) + ϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ϕ ( 2n ( x + y 2 )) + ϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 ) [ϕ (2nx) + ϕ (2ny)] = 0. (4.5) from the definition of f and the inequality (4.4), we obtain that |dsf(x,y)| = ∣∣∣ ∞∑ n=0 2−6nϕ (2n(ax + by)) + ∞∑ n=0 2−6nϕ (2n(bx + ay)) + (a− b)6 [ ∞∑ n=0 2−6nϕ ( 2n ( ax− by a− b )) + ∞∑ n=0 2−6nϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ∞∑ n=0 2−6nϕ ( 2n ( x + y 2 )) + ∞∑ n=0 2−6nϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 )[ ∞∑ n=0 2−6nϕ (2nx) + ∞∑ n=0 2−6nϕ (2ny) ]∣∣∣ ≤ ∞∑ n=0 2−6n ∣∣∣ϕ (2n(ax + by)) + ϕ (2n(bx + ay)) + (a− b)6 [ ϕ ( 2n ( ax− by a− b )) + ϕ ( 2n ( bx−ay b−a ))] − 64(ab)2 ( a2 + b2 )[ ϕ ( 2n ( x + y 2 )) + ϕ ( 2n ( x−y 2 ))] − 2 ( a2 − b2 )( a4 − b4 ) [ϕ (2nx) + ϕ (2ny)] ∣∣∣ ≤ ∞∑ n=0 2−6nδ = 26(1−k)δ 63 ≤ 643δ 63 ( |x|6 + |y|6 ) . (4.6) therefore, f satisfies (4.3) for all x,y ∈ r. now, we claim that the functional equation (1.8) is not stable for p = 6β α in corollary 3.1. suppose on the contrary that there exists a sexticc mapping s : r → r and a constant � > 0 such that |f(x) −s(x)| ≤ � |x|6 , for all x ∈ r. then there exists a constant c ∈ r such that s(x) = cx6 for all rational numbers x (see [19]). so we obtain that |f(x)| ≤ (� + |c|) |x|6 (4.7) euler-lagrange-jensen (a,b)-sextic functional equations 173 for all x ∈ q. let m ∈ n with m + 1 > � + |c|. if x is a rational number in (0, 2−m), then 2nx ∈ (0, 1) for all n = 0, 1, 2, . . . ,m, and for this x, we get f(x) = ∞∑ n=0 2−6nϕ(2nx) ≥ m∑ n=0 2−6n (2nx) 6 = (m + 1)x6 > (� + |c|)x6 (4.8) which contradicts (4.7). hence the functional equation (1.8) is not stable for p = 6β α in corollary 3.1. � references [1] j. aczel, lectures on functional equations and their applications, vol. 19, academic press, new york, 1966. [2] j. aczel, functional equations, history, applications and theory, d. reidel publ. company, 1984. [3] c. alsina, on the stability of a functional equation, general inequalities, vol. 5, oberwolfach, birkhauser, basel, (1987), 263-271. [4] t. aoki, on the stability of the linear transformation in banach spaces, j. math. soc. japan, 2 (1950), 64-66. [5] l. cadariu and v. radu, fixed points and stability for functional equations in probabilistic metric and random normed spaces, fixed point theory appl. 2009 (2009), art. id 589143, 18 pages. [6] b. bouikhalene and e. elquorachi, ulam-gavruta-rassias stability of the pexider functional equation, int. j. appl. math. stat., 7 (2007), 7-39. [7] i. s. chang and h. m. kim, on the hyers-ulam stability of quadratic functional equations, j. ineq. appl. math. 33 (2002), 1-12. [8] i. s. chang and y. s. jung, stability of functional equations deriving from cubic and quadratic functions, j. math. anal. appl., 283 (2003), 491-500. [9] j. k. chung and p. k. sahoo, on the general solution of a quartic functional equation, bull. korean math. soc., 40 (4) (2003), 565-576. [10] s. czerwik, functional equations and inequalities in several variables, world scientific publishing company, new jersey, london, singapore and hong kong, 2002. [11] m. eshaghi gordji, s. zolfaghari, j. m. rassias and m. b. savadkouhi, solution and stability of a mixed type cubic and quartic functional equation in quasi-banach spaces, abst. appl. anal., 2009 (2009), art. id 417473. [12] z. gajda, on stability of additive mappings, int. j. math. math. sci. 14 (3) (1991), 431-434. [13] n. ghobadipour and c. park, cubic-quartic functional equations in fuzzy normed spaces, int. j. nonlinear anal. appl., 1 (2010), 12-21. [14] p. gǎvrutǎ, a generalization of the hyers-ulam-rassias stability of approximately additive mappings, j. math. anal. appl., 184 (1994), 431-436. [15] heejeong koh and dongseung kang, solution and stability of euler-lagrange-rassias quartic functional equations in various quasi-normed spaces, abstr. appl. anal., 2013 (2013), art. id 908168, 8 pages. [16] d. h. hyers, on the stability of the linear functional equation, proc. nat. acad. sci., u.s.a., 27 (1941), 222-224. [17] d. h. hyers, g. isac and th. m. rassias, stability of functional equations in several variables, birkhauser, basel, 1998. [18] g. isac and th. m. rassias, stability of ψ-additive mappings: applications to nonlinear analysis, int. j. math. math. sci., 19(2) (1996), 219-228. [19] k. w. jun and h. m. kim, on the stability of euler-lagrange type cubic mappings in quasi-banach spaces, j. math. anal. appl. 332(2) (2007), 1335-1350. [20] s. m. jung, hyers-ulam-rassias stability of functional equations in mathematical analysis, hardonic press, palm harbor, 2001. [21] pl. kannappan, quadratic functional equation and inner product spaces, results math. 27(3-4) (1995), 368-372. [22] j. r. lee, d. y. shin and c. park, hyers-ulam stability of functional equations in matrix normed spaces, j. inequal. appl. 2013 (2013), art. id 22. [23] e. movahednia, fixed point and generalized hyers-ulam-rassias stability of a quadratic functional equation, j. math. comput. sci., 6 (2013), 72-78. [24] a. najati and c. park, cauchy-jensen additive mappings in quasi-banach algebras and its applications, j. nonlinear anal. appl., 2013 (2013), art. id jnaa-00191. [25] p. nakmahachalasint, hyers-ulam-rassias and ulam-gavruta-rassias stabilities of additive functional equation in several variables, int. j. math. math. sci. 2007 (2007) art. id 13437, 6 pages. [26] c. park, fixed points and the stability of an aqcq-functional equation in non-archimedean normed spaces, abstr. appl. anal., 2010 (2010) art. id 849543, 15 pages. [27] c. g. park, stability of an euler-lagrange-rassias type additive mapping, int. j. appl. math. stat., 7 (2007), 101-111. [28] a. pietrzyk, stability of the euler-lagrange-rassias functional equation, demonstr. math., 39(3) (2006), 523 530. [29] j. m. rassias, on approximately of approximately linear mappings by linear mappings, j. funct. anal. 46 (1982), 126-130. [30] j. m. rassias, on approximately of approximately linear mappings by linear mappings, bull. sci. math., 108 (4) (1984), 445-446. 174 rassias, ravi and kumar [31] j. m. rassias, on a new approximation of approximately linear mappings by linear mappings, discuss. math., 7 (1985), 193-196. [32] j. m. rassias, on the stability of the euler-lagrange functional equation, chinese j. math., 20 (1992), 185-190. [33] j. m. rassias, on the stability of the non-linear euler-lagrange functional equation in real normed linear spaces, j. math. phys. sci., 28 (1994), 231-235. [34] j .m. rassias, on the stability of the general euler-lagrange functional equation, demonstr. math., 29 (1996), 755-766. [35] j. m. rassias, solution of the ulam stability problem for euler-lagrange quadratic mappings, j. math. anal. appl., 220 (1998), 613-639. [36] j. m. rassias, on the stability of the multi-dimensional euler-lagrange functional equation, j. indian math. soc., 66 (1999), 1-9. [37] j. m. rassias, solution of the ulam stability problem for quartic mappings, glasnic matematicki. serija iii, 34(2) (1999), 243-252. [38] j. m. rassias, solution of the ulam stablility problem for cubic mappings, glasnik matematicki. serija iii, 36(1) (2001), 63-72. [39] k. ravi, m. arunkumar and j. m. rassias, ulam stability for the orthogonally general euler-lagrange type functional equation, int. j. math. stat. 3(a08) (2008), 36-46. [40] k. ravi, j. m. rassias, m. arunkumar and r. kodandan, stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, j. inequ. pure appl. math., 10(4) (2009), 1-29. [41] th. m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc., 72 (1978), 297-300. [42] s. m. ulam, problems in modern mathematics, rend. chap. vi, wiley, new york, 1960. [43] t. z. xu, j. m. rassias and w. x. xu, a fixed point approach to the stability of a general mixed aqcq-functional equation in non-archimedean normed spaces, discrete dyn. nat. soc. 2010 (2010) art. id 812545, 24 pages. [44] t. z. xu, j. m. rassias, m. j. rassias and w. x. xu, a fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces,j. inequal. appl., 2010 (2010), art. id 423231. [45] t. z. xu, j. m. rassias and w. x. xu, a fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, int. j. phys. sci. 6(2) (2011), 313-324. 1pedagogical department e.e., section of mathematics and informatics, national and capodistrian university of athens, 4, agamemnonos str., aghia paraskevi, athens, attikis 15342, greece 2department of mathematics, sacred heart college, tirupattur-635 601, tamil nadu, india 3department of mathematics, c. abdul hakeem college of engineering and technology, melvisharam 632 509, tamil nadu, india ∗corresponding author: jrassias@primedu.uoa.gr 1. introduction 2. preliminaries 3. generalized ulam-hyers stability of equation (1.8) 4. counter-example references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 70-81 http://www.etamaths.com a generalized and refined perturbed version of ostrowski type inequalities m. z. sarikaya1, h. budak1,∗, s. erden2 and a. qayyum3 abstract. in this paper, we first obtain a new identity for twice differentiable mappings. then, we establish generalized and improved perturbed version of ostrowski type inequalities for functions whose derivatives are of bounded variation or second derivatives are either bounded or lipschitzian. 1. introduction in 1938, ostrowski first declared his inequality for different differentiable mappings. ostrowski inequalities appear in most of the domains of mathematics. its importance has increased remarkably during the past few years and it is now cosidered as an independent branch of mathematics. the development of the theory of ostrowski inequality was initiated by dragomir. in [6], dragomir et al. obtained ostrowski type inequalities for functions whose second derivatives are bounded. during the time, the growing interest for the ostrowski inequalities led to the apparition of several research papers in the area. in this sense, we mention ( [6], [8], [16], [17], [19][21]). in recent years, modern theory of inequalities is used at large and many efforts devoted to establish several generalizations of the ostrowski’s inequalities for mappings of bounded variation ( [1][5], [7], [9][13], [15], [18]). in this study, we establish some perturbed version of ostrowski type inequalities for twice differentiable functions whose derivatives are of bounded variation or second derivatives are either bounded or lipschitzian. theorem 1.1. [14] let f : [a,b] → r be a differentiable mapping on (a,b) whose derivative f′ : (a,b) → r is bounded on (a,b) , i.e. ‖f′‖∞ := sup t∈(a,b) |f′(t)| < ∞. then, we have the inequality ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ , (1.1) for all x ∈ [a,b]. the constant 1 4 is the best possible. in [9], dragomir proved the following ostrowski type inequalitiesfor functions of bounded variation: theorem 1.2. let f : [a,b] → r be a mapping of bounded variation on [a,b] . then∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) f(x) ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f) (1.2) holds for all x ∈ [a,b] . the constant 1 2 is the best possible. the following lemma is required to prove the main theorem. received 30th july, 2016; accepted 6th october, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d07, 26d10, 26d15. key words and phrases. ostrowski inequality; function of bounded variation; lipschitzian mappings. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 70 a refined ostrowski type inequalities 71 lemma 1.1. let f : [a,b] → c be a twice differantiable function on (a,b) . then for any λi(x), i = 1, 2, ..5 complex number the following identity holds (1.3) 1 2 (b−a)   a+x 2∫ a (t−a)2 [f ′′ (t) −λ1(x)] dt + x∫ a+x 2 ( t− 3a + b 4 )2 [f ′′ (t) −λ2(x)] dt + a+b−x∫ x ( t− a + b 2 )2 [f ′′ (t) −λ3(x)] dt + a+2b−x 2∫ a+b−x ( t− a + 3b 4 )2 [f ′′ (t) −λ4(x)] dt + b∫ a+2b−x 2 (t− b)2 [f ′′ (t) −λ5(x)] dt   = a + 1 48 (b−a) {( x− a + b 2 )3 [λ2(x) + 16λ3(x) + λ4(x)] −(x−a)3 [λ1(x) + λ5(x)] − 8 ( x− 3a + b 4 )3 [λ2(x) + λ4(x)] } , for all x ∈ [ a, a+b 2 ] , where a is defined by a (1.4) = 1 b−a b∫ a f (t) dt− 1 4 [ f (x) + f (a + b−x) + f ( a + x 2 ) + f ( a + 2b−x 2 ) + ( x− 5a + 3b 8 ) {f ′ (a + b−x) −f ′ (x)} + 1 2 ( x− 3a + b 4 ){ f ′ ( a + 2b−x 2 ) −f ′ ( a + x 2 )}] . proof. integrating the by parts for each integral, we can easily obtain the required result (1.3). � now with the help of above lemma, we will prove the following inequalities. 2. inequalities for functions whose second derivatives are bounded recall the sets of complex-valued functions: u[a,b] (γ, γ) : = { f : [a,b] → c|re [ (γ −f(t)) ( f(t) ) −γ ] ≥ 0 for almost every t ∈ [a,b] } and ∆[a,b] (γ, γ) := { f : [a,b] → c| ∣∣∣∣f(t) − γ + γ2 ∣∣∣∣ ≤ 12 |γ −γ| for a.e. t ∈ [a,b] } . proposition 2.1. for any γ, γ ∈ c, γ 6= γ, we have that u[a,b] (γ, γ) and ∆[a,b] (γ, γ) are nonempty and closed sets and u[a,b] (γ, γ) = ∆[a,b] (γ, γ) . let i1 = [ a, a+x 2 ] , i2 = [ a+x 2 ,x ] i3 = [x,a + b−x] i4 = [ a + b−x, a+2b−x 2 ] and i5 = [ a+2b−x 2 ,b ] . theorem 2.1. let f : [a,b] → c be a twice differantiable function on (a,b) and x ∈ (a,b) . suppose that γi(x), γi(x) ∈ c, γi(x) 6= γi(x), i = 1, 2, 3, 4, 5 and f′′ ∈ 5⋂ i=1 uii (γi, γi) 72 sarikaya, budak, erden and qayyum then we have the inequality ∣∣∣∣∣a + 196 (b−a) [( x− a + b 2 )3 × [γ2(x) + γ2(x) + 16 (γ3(x) + γ3(x)) + γ4(x) + γ4(x)] −(x−a)3 [γ1(x) + γ1(x) + γ5(x) + γ5(x)] −8 ( x− 3a + b 4 )3 [γ2(x) + γ2(x) + γ4(x) + γ4(x)] ]∣∣∣∣∣ ≤ 1 96 (b−a) { (x−a)3 |γ1(x) −γ1(x)| + [ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] |γ2(x) −γ2(x)| +16 ( a + b 2 −x )3 |γ3(x) −γ3(x)| + [ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] |γ4(x) −γ4(x)| + (x−a)3 |γ5(x) −γ5(x)| } , where a is defined as in (1.4). proof. taking the modulus identity (1.3) for λi(x) = γi(x)+γi(x) 2 , i = 1, 2, ..., 5, since f′′ ∈ 5⋂ i=1 uii (γi, γi), we have ∣∣∣∣∣a + 196 (b−a) [( x− a + b 2 )3 × [γ2(x) + γ2(x) + 16 (γ3(x) + γ3(x)) + γ4(x) + γ4(x)] −(x−a)3 [γ1(x) + γ1(x) + γ5(x) + γ5(x)] −8 ( x− 3a + b 4 )3 [γ2(x) + γ2(x) + γ4(x) + γ4(x)] ]∣∣∣∣∣ ≤ 1 2 (b−a)   a+x 2∫ a (t−a)2 ∣∣∣∣f ′′ (t) − γ1(x) + γ1(x)2 ∣∣∣∣dt + x∫ a+x 2 ( t− 3a + b 4 )2 ∣∣∣∣f ′′ (t) − γ2(x) + γ2(x)2 ∣∣∣∣dt a refined ostrowski type inequalities 73 + a+b−x∫ x ( t− a + b 2 )2 ∣∣∣∣f ′′ (t) − γ3(x) + γ3(x)2 ∣∣∣∣dt + a+2b−x 2∫ a+b−x ( t− a + 3b 4 )2 ∣∣∣∣f ′′ (t) − γ4(x) + γ4(x)2 ∣∣∣∣dt + b∫ a+2b−x 2 (t− b)2 ∣∣∣∣f ′′ (t) − γ5(x) + γ5(x)2 ∣∣∣∣dt   ≤ 1 96 (b−a) { (x−a)3 |γ1(x) −γ1(x)| + [ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] |γ2(x) −γ2(x)| +16 ( a + b 2 −x )2 |γ3(x) −γ3(x)|[ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] |γ4(x) −γ4(x)| + (x−a)3 |γ5(x) −γ5(x)| } . this completes the proof. � remark 2.1. if we choose x = a in theorem 2.1, we obtain the inequality∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− f (a) + f (b) 2 −(b−a) f ′ (b) −f ′ (a) 8 − (b−a)2 48 (γ3(x) + γ3(x)) ∣∣∣∣∣ ≤ (b−a) 48 |γ3(x) −γ3(x)| which was given by sarikaya et al. in [15]. corollary 2.1. under assumption of theorem 2.1 with x = a+b 2 , we have∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− 1 4 [ f ( 3a + b 4 ) + 2f ( a + b 2 ) + f ( a + 3b 4 ) + 1 8 (b−a) { f ′ ( a + 3b 4 ) −f ′ ( 3a + b 4 )}] − (b−a)2 768 [γ1(x) + γ1(x) + γ2(x) + γ2(x) +γ4(x) + γ4(x) + γ5(x) + γ5(x)]| ≤ (b−a)2 768 [|γ1(x) −γ1(x)| + |γ2(x) −γ2(x)| + |γ4(x) −γ4(x)| + |γ5(x) −γ5(x)|] . 74 sarikaya, budak, erden and qayyum corollary 2.2. under assumption of theorem 2.1 with x = 3a+b 4 , we have∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− 1 4 [ f ( 3a + b 4 ) + f ( a + 3b 4 ) +f ( 7a + b 8 ) + f ( a + 7b 8 ) − 1 8 (b−a) { f ′ ( a + 3b 4 ) −f ′ ( 3a + b 4 )}] + (b−a)2 6144 [γ1(x) + γ1(x) + γ2(x) + γ2(x) +16 (γ3(x) + γ3(x)) + γ4(x) + γ4(x) + γ5(x) + γ5(x)]| ≤ (b−a)2 6144 [|γ1(x) −γ1(x)| + 8 |γ2(x) −γ2(x)| + 16 |γ4(x) −γ4(x)| +8 |γ4(x) −γ4(x)| + |γ5(x) −γ5(x)|] . 3. inequalities for mappings of bounded variation in this section, we establish some inequalities for function whose second derivatives are of bounded variation. let f :[a,b] → c be a twice differentiable function on i◦(i◦ is the interior of i) and [a,b] ⊂ i◦.then, from (1.3), we have for λ1(x) = f ′′ (a) , λ2(x) = f ′′ ( a+x 2 ) + f ′′ (x) 2 , λ3(x) = f ′′ (x) + f ′′ (a + b−x) 2 , λ4(x) = f ′′ (a + b−x) + f ′′ ( a+2b−x 2 ) 2 , λ5(x) = f ′′ (b) , 1 2 (b−a)   a+x 2∫ a (t−a)2 [f ′′ (t) −f ′′ (a)] dt + x∫ a+x 2 ( t− 3a + b 4 )2 × [ f ′′ (t) − f ′′ ( a+x 2 ) + f ′′ (x) 2 ] dt + a+b−x∫ x ( t− a + b 2 )2 [ f ′′ (t) − f ′′ (x) + f ′′ (a + b−x) 2 ] dt + a+2b−x 2∫ a+b−x ( t− a + 3b 4 )2 [ f ′′ (t) − f ′′ (a + b−x) + f ′′ ( a+2b−x 2 ) 2 ] dt + b∫ a+2b−x 2 (t− b)2 [f ′′ (t) −f ′′ (b)] dt   a refined ostrowski type inequalities 75 = a + 1 48 (b−a) [ 1 2 ( x− a + b 2 )3 (3.1) × { f ′′ ( a + x 2 ) + 17 (f ′′ (x) + f ′′ (a + b−x)) + f ′′ ( a + 2b−x 2 )} −(x−a)3 [f ′′ (a) + f ′′ (b)] − 4 ( x− 3a + b 4 )3 × { f ′′ ( a + x 2 ) + f ′′ (x) + f ′′ (a + b−x) + f ′′ ( a + 2b−x 2 )}] for any x ∈ [ a, a+b 2 ] , where a is defined as in (1.4). theorem 3.1. let f : [a,b] → c be a twice differentiable function on i◦(i◦ is the interior of i) and [a,b] ⊂ i◦. if the second derivative f ′′ is of bounded variation on [a,b] , then we have∣∣∣∣∣a + 148 (b−a) [ 1 2 ( x− a + b 2 )3 (3.2) × { f ′′ ( a + x 2 ) + 17 (f ′′ (x) + f ′′ (a + b−x)) + f ′′ ( a + 2b−x 2 )} −(x−a)3 [f ′′ (a) + f ′′ (b)] − 4 ( x− 3a + b 4 )3 × { f ′′ ( a + x 2 ) + f ′′ (x) + f ′′ (a + b−x) + f ′′ ( a + 2b−x 2 )}]∣∣∣∣ ≤ 1 48 (b−a)  (x−a)3 a+x 2∨ a (f ′′) + [ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] x∨ a+x 2 (f ′′) +8 ( a + b 2 −x )3 a+b−x∨ x (f ′′) + [ 8 ( x− 3a + b 4 )3 − ( x− a + b 2 )3] a+2b−x2∨ a+b−x (f ′′) + (x−a)3 b∨ a+2b−x 2 (f ′′)   , for all x ∈ [ a, a+b 2 ] , where a is defined as in (1.4). proof. from (3.1), we find that∣∣∣∣∣a + 148 (b−a) [ 1 2 ( x− a + b 2 )3 × { f ′′ ( a + x 2 ) + 17 (f ′′ (x) + f ′′ (a + b−x)) + f ′′ ( a + 2b−x 2 )} −(x−a)3 [f ′′ (a) + f ′′ (b)] − 4 ( x− 3a + b 4 )3 × { f ′′ ( a + x 2 ) + f ′′ (x) + f ′′ (a + b−x) + f ′′ ( a + 2b−x 2 )}]∣∣∣∣ 76 sarikaya, budak, erden and qayyum ≤ 1 2 (b−a)   a+x 2∫ a (t−a)2 |f ′′ (t) −f ′′ (a)|dt + x∫ a+x 2 ( t− 3a + b 4 )2 [ f ′′ (t) − f ′′ ( a+x 2 ) + f ′′ (x) 2 ] dt + a+b−x∫ x ( t− a + b 2 )2 [∣∣∣∣f ′′ (t) − f ′′ (x) + f ′′ (a + b−x)2 ∣∣∣∣ ] dt + a+2b−x 2∫ a+b−x ( t− a + 3b 4 )2 ∣∣∣∣∣f ′′ (t) − f ′′ (a + b−x) + f ′′ ( a+2b−x 2 ) 2 ∣∣∣∣∣dt + b∫ a+2b−x 2 (t− b)2 |f ′′ (t) −f ′′ (b)|dt   . since f ′′ is of bounded variation on [a,b] , we get |f ′′ (t) −f ′′ (a)| ≤ t∨ a (f ′′) for t ∈ [ a, a+x 2 ] ∣∣∣∣∣f ′′ (t) − f ′′ ( a+x 2 ) + f ′′ (x) 2 ∣∣∣∣∣ ≤ 12 x∨ a+x 2 (f ′′) < x∨ a+x 2 (f ′′) for t ∈ [ a+x 2 ,x ] ∣∣∣∣f ′′ (t) − f ′′ (x) + f ′′ (a + b−x)2 ∣∣∣∣ ≤ 12 a+b−x∨ x (f ′′) for t ∈ [x,a + b−x] ∣∣∣∣∣f ′′ (t) − f ′′ (a + b−x) + f ′′ ( a+2b−x 2 ) 2 ∣∣∣∣∣ ≤ 12 a+2b−x 2∨ a+b−x (f ′′) < a+2b−x 2∨ a+b−x (f ′′) for t ∈ [ a + b−x, a+2b−x 2 ] |f ′′ (t) −f ′′ (b)| ≤ b∨ t (f ′′) for t ∈ [ a+2b−x 2 ,b ] . thus, using the elementary analysis operations, we deduce desired inequality (3.2) which completes the proof. � remark 3.1. if we choose x = a in (3.2), then we get the result proved by sarikaya et al. [15]. a refined ostrowski type inequalities 77 corollary 3.1. under assumption of theorem 3.1 with x = a+b 2 , we have the inequality ∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− 1 4 [ f ( 3a + b 4 ) + 2f ( a + b 2 ) + f ( a + 3b 4 ) + 1 8 (b−a) { f ′ ( a + 3b 4 ) −f ′ ( 3a + b 4 )}] − (b−a) 384 [ f′′(a) + f′′(b) + f′′ ( a + b 2 ) + 1 2 [ f′′ ( a + 3b 4 ) + f′′ ( 3a + b 4 )]]∣∣∣∣ ≤ 1 384 b∨ a (f ′′) . 4. inequalities for lipschitzian mappings in this section we obtain some inequalities for function whose second derivatives are lipschitzian. we say that the function g : [a,b] → c is lipschitzian with the constant l > 0 if |g(t) −g(s)| ≤ l |t−s| for any t,s ∈ [a,b] . theorem 4.1. let f : [a,b] → c be a twice differantiable function on (a,b) . if the second derivative f′′ is a lipschitzian mapping with the constant l > 0,then we have the inequality ∣∣∣∣∣a + 148 (b−a) [( x− a + b 2 )3 (4.1) × [ f ′′ ( 3a + b 4 ) + 16f ′′ ( a + b 2 ) + f ′′ ( a + 3b 4 )] −(x−a)3 [f ′′ (a) + f ′′ (b)] −8 ( x− 3a + b 4 )3 [ f ′′ ( 3a + b 4 ) + f ′′ ( a + 3b 4 )]]∣∣∣∣∣ ≤ l 128 (b−a) { 2 (x−a)4 + sgn ( 3a + b 4 −x ) × [ 16 ( x− 3a + b 4 )4 − ( x− a + b 2 )4] +31 ( x− a + b 2 )4 + 16 ( x− 3a + b 4 )4] , for all x ∈ [ a, a+b 2 ] , where a is defined as in (1.4). 78 sarikaya, budak, erden and qayyum proof. if we take the λ1 = f ′′ (a) , λ2 = f ′′ ( 3a+b 4 ) , λ3 = f ′′ ( a+b 2 ) , λ4 = f ′′ ( a+3b 4 ) and λ5 = f ′′ (b) in equality (1.3), we have 1 2 (b−a)   a+x 2∫ a (t−a)2 [f ′′ (t) −f ′′ (a)] dt+ (4.2) x∫ a+x 2 ( t− 3a + b 4 )2 [ f ′′ (t) −f ′′ ( 3a + b 4 )] dt + a+b−x∫ x ( t− a + b 2 )2 [ f ′′ (t) −f ′′ ( a + b 2 )] dt + a+2b−x 2∫ a+b−x ( t− a + 3b 4 )2 [ f ′′ (t) −f ′′ ( a + 3b 4 )] dt + b∫ a+2b−x 2 (t− b)2 [f ′′ (t) −f ′′ (b)] dt   = 1 b−a b∫ a f (t) dt− 1 4 [ f (x) + f (a + b−x) + f ( a + x 2 ) + f ( a + 2b−x 2 ) + ( x− 5a + 3b 8 ) {f ′ (a + b−x) −f ′ (x)} + 1 2 ( x− 3a + b 4 ){ f ′ ( a + 2b−x 2 ) −f ′ ( a + x 2 )}] + 1 48 (b−a) [( x− a + b 2 )3 × [ f ′′ ( 3a + b 4 ) + 16f ′′ ( a + b 2 ) + f ′′ ( a + 3b 4 )] −(x−a)3 [f ′′ (a) + f ′′ (b)] −8 ( x− 3a + b 4 )3 [ f ′′ ( 3a + b 4 ) + f ′′ ( a + 3b 4 )]] for all x ∈ [ a, a+b 2 ] . since f′′ is lipschitzian, taking the madulus in (4.2), we have ∣∣∣∣∣a + 148 (b−a) [( x− a + b 2 )3 × [ f ′′ ( 3a + b 4 ) + 16f ′′ ( a + b 2 ) + f ′′ ( a + 3b 4 )] −(x−a)3 [f ′′ (a) + f ′′ (b)] a refined ostrowski type inequalities 79 −8 ( x− 3a + b 4 )3 [ f ′′ ( 3a + b 4 ) + f ′′ ( a + 3b 4 )]]∣∣∣∣∣ ≤ l 128 (b−a) { 2 (x−a)4 + sgn ( 3a + b 4 −x ) × [ 16 ( x− 3a + b 4 )4 − ( x− a + b 2 )4] +31 ( x− a + b 2 )4 + 16 ( x− 3a + b 4 )4] ≤ l 2 (b−a)   a+x 2∫ a (t−a)3 dt + x∫ a+x 2 ∣∣∣∣t− 3a + b4 ∣∣∣∣3 dt + a+b−x∫ x ∣∣∣∣a + b2 − t ∣∣∣∣3 dt + a+2b−x 2∫ a+b−x ( a + 3b 4 − t )3 dt + b∫ a+2b−x 2 (b− t)3 dt   . if we calculate the above five integrals, then we obtain the inequality (4.1). thus proof is completed. � corollary 4.1. under assumption of theorem 4.1 with x = a, we get the inequality ∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− f (a) + f (b) 2 −(b−a) f ′ (b) −f ′ (a) 8 − (b−a)2 24 f ′′ ( a + b 2 )∣∣∣∣∣ ≤ 1 64 (b−a)3 l. corollary 4.2. under assumption of theorem 4.1 with x = a+b 2 , we get the inequality ∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− 1 4 [ f ( 3a + b 4 ) + 2f ( a + b 2 ) + f ( a + 3b 4 ) + 1 8 (b−a) { f ′ ( a + 3b 4 ) −f ′ ( 3a + b 4 )}] + (b−a)2 384 [ f ′′ (a) + f ′′ (b) + f ′′ ( 3a + b 4 ) + f ′′ ( a + 3b 4 )] ≤ 1 512 (b−a)3 l. 80 sarikaya, budak, erden and qayyum corollary 4.3. under assumption of theorem 4.1 with x = 3a+b 4 , we get the inequality∣∣∣∣∣∣ 1b−a b∫ a f (t) dt− 1 4 [ f ( 3a + b 4 ) + f ( a + 3b 4 ) + f ( 7a + b 8 ) + f ( a + 7b 8 ) − 1 8 (b−a) { f ′ ( a + 3b 4 ) −f ′ ( 3a + b 4 )}] − 1 3072 (b−a)2 [ f ′′ (a) + f ′′ ( 3a + b 4 ) + 16f ′′ ( a + b 2 ) +f ′′ ( a + 3b 4 ) + f ′′ (b) ]∣∣∣∣ ≤ 17 214 (b−a)3 l. references [1] h. budak and m. z. sarikaya, a new ostrowski type inequality for functions whose first derivatives are of bounded variation, moroccan j. pure appl. anal., 2(1)(2016), 1–11. [2] h. budak and m.z. sarikaya, a companion of ostrowski type inequalities for mappings of bounded variation and some applications, rgmia research report collection, 19(2016), article id 24. [3] h. budak, m.z. sarikaya and a. qayyum, improvement in companion of ostrowski type inequalities for mappings whose first derivatives are of bounded variation and application, rgmia research report collection, 19(2016), article id 25. [4] h. budak, m.z. sarikaya and s.s. dragomir, some perturbed ostrowski type inequality for twice differentiable functions, rgmia research report collection, 19(2016), article id 47. [5] h. budak and m. z. sarikaya, some perturbed ostrowski type inequality for functions whose first derivatives are of bounded variation, rgmia research report collection, 19 (2016), article id 54. [6] s. s. dragomir and n.s. barnett, an ostrowski type inequality for mappings whose second derivatives are bounded and applications, rgmia research report collection, 1(2)(1998) , 69 − 77. [7] s. s. dragomir, the ostrowski integral inequality for mappings of bounded variation, bulletin of the australian mathematical society, 60(1) (1999), 495-508. [8] s. s. dragomir and a. sofo, an integral inequality for twice differentiable mappings and application, tamkang j. math., 31(4) 2000, 257-266. [9] s. s. dragomir, on the ostrowski’s integral inequality for mappings with bounded variation and applications, mathematical inequalities & applications, 4(1) (2001), 59–66. [10] s. s. dragomir, a companion of ostrowski’s inequality for functions of bounded variation and applications, international journal of nonlinear analysis and applications, 5(1) (2014), 89-97. [11] s. s. dragomir, some perturbed ostrowski type inequalities for functions of bounded variation, asian-european journal of mathematics, 8(4)(2015, ), article id 1550069. doi:10.1142/s1793557115500692 [12] s. s. dragomir, perturbed companions of ostrowski’s inequality for functions of bounded variation, rgmia research report collection, 17(2014), article id 1. [13] w. liu and y. sun, a refinement of the companion of ostrowski inequality for functions of bounded variation and applications, arxiv:1207.3861v1, (2012). [14] a. m. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, comment. math. helv. 10(1938), 226-227. [15] m. z. sarikaya, h. budak, t. tunc, s. erden and h. yaldiz, perturbed companion of ostrowski type inequality for twice differentiable functions, rgmia research report collection, 19 (2016), article id 59. [16] e. set and m. z. sarikaya, on a new ostrowski-type inequality and related results, kyungpook mathematical journal, 54(2014), 545-554. [17] j. park, some companions of an ostrowski-like type inequality for twice differentiable functions, applied mathematical sciences, 8 (47) (2014), 2339 2351. [18] m. liu, y. zhu and j. park, some companions of perturbed ostrowski-type inequalities based on the quadratic kernel function with three sections and applications, j. of ineq. and applications, 2013 (2013), article id 226. [19] a. qayyum, m. shoaib and i. faye, companion of ostrowski-type inequality based on 5-step quadratic kernel and applications, journal of nonlinear science and applications, 9 (2016), 537–552. [20] a. qayyum, i. faye and m. shoaib, a companion of ostrowski type integral inequality using a 5-step kernel with some applications, filomat, in press. [21] a. qayyum, m. shoaib and i. faye, derivation and applications of inequalities of ostrowski type for n-times differentiable mappings for cumulative distribution function and some quadrature rules, journal of nonlinear sciences and applications, 9 (2016), 1844-1857. 1department of mathematics, faculty of science and arts, düzce university, düzce, turkey. a refined ostrowski type inequalities 81 2department of mathematics, faculty of science, bartin university, bartin,turkey. 3department of mathematics, university of hail, p. o. box 2440, saudi arabia. ∗corresponding author: hsyn.budak@gmail.com 1. introduction 2. inequalities for functions whose second derivatives are bounded 3. inequalities for mappings of bounded variation 4. inequalities for lipschitzian mappings references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 162-166 http://www.etamaths.com some improvements of conformable fractional integral inequalities fuat usta∗ and mehmet zeki sarikaya abstract. in this study, we wish to set up and present some new conformable fractional integral inequalities of the gronwall type which have a great variety of implementation area in differential and integral equations. 1. introduction & preliminaries in light of recent events in theory of differential and integral equations, it is becoming extremely difficult to ignore the existence of integral inequalities which help to determine of bounds on unknown functions. for example, gronwall and pachpatte have great contribution in the literature [19], [20], [5], [6]. together with this contributions, gronwall inequality has been extended and applied in a number of context. however, in non-integer order of models the bound provided by the above authors are not feasible. additionally non-integer order calculus called fractional calculus has a number of fields of application such as control theory, computational analysis and engineering [12], see also [13]. thus a number of new definitions have been introduced in academia to provide the best method for fractional calculus. for instance in more recent times a new local, limit-based definition of a conformable derivative has been introduced in [1], [4], [10], with several follow-up papers [2], [3], [7][9], [11], [14][18]. in this research, we presented conformable fractional version of some significant integral inequalities with the help of the katugampola conformable fractional calculus. in detail, katugampola conformable derivatives for α ∈ (0, 1] and t ∈ [0,∞) given by dα (f) (t) = lim ε→0 f ( teεt −α ) −f (t) ε , dα (f) (0) = lim t→0 dα (f) (t) , (1.1) provided the limits exist (for detail see, [10]). if f is fully differentiable at t, then dα (f) (t) = t1−α df dt (t) . (1.2) a function f is α−differentiable at a point t ≥ 0 if the limit in (1.1) exists and is finite. this definition yields the following results; theorem 1.1. let α ∈ (0, 1] and f,g be α−differentiable at a point t > 0. then i. dα (af + bg) = adα (f) + bdα (g) , for all a,b ∈ r, ii. dα (λ) = 0, for all constant functions f (t) = λ, iii. dα (fg) = fdα (g) + gdα (f) , iv. dα ( f g ) = fdα (g) −gdα (f) g2 v. dα (tn) = ntn−α for all n ∈ r vi. dα (f ◦g) (t) = f′ (g (t)) dα (g) (t) for f is differentiable at g(t). 2010 mathematics subject classification. 26d15, 26a51, 26a33, 26a42. key words and phrases. gronwall integral inequality; conformable fractional differential equation; global existence. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 162 some improvements of conformable fractional integral inequalities 163 definition 1.1 (conformable fractional integral). let α ∈ (0, 1] and 0 ≤ a < b. a function f : [a,b] → r is α-fractional integrable on [a,b] if the integral∫ b a f (x) dαx := ∫ b a f (x) xα−1dx exists and is finite. all α-fractional integrable on [a,b] is indicated by l1α ([a,b]) remark 1.1. iaα (f) (t) = i a 1 ( tα−1f ) = ∫ t a f (x) x1−α dx, where the integral is the usual riemann improper integral, and α ∈ (0, 1]. we will also use the following important results, which can be derived from the results above. lemma 1.1. let the conformable differential operator dα be given as in (1.1), where α ∈ (0, 1] and t ≥ 0, and assume the functions f and g are α-differentiable as needed. then i. dα (ln t) = t−α for t > 0 ii. dα [∫ t a f (t,s) dαs ] = f(t,t) + ∫ t a dα [f (t,s)] dαs iii. ∫ b a f (x) dα (g) (x) dαx = fg| b a − ∫ b a g (x) dα (f) (x) dαx. in this paper, by using the katugampola type conformable fractional calculus, we introduced retarded gronwall-bellman and bihari like conformable fractional integrals inequalities. 2. main findings & cumulative results in this article, all the functions which appear in the inequalities are assumed to be real-valued and all the integrals involved exist on the respective domains of their definitions, and c (m,s) and c1 (m,s) denote the class of all continuous functions and the first order conformable derivative, respectively, defined on set m with range in the set s. additionally, r denotes the set of real numbers such that r+ = [0,∞), r1 = [1,∞) and q = [0,t) are the given subset of r. theorem 2.1. [14] let k,y,x ∈ c (r+,r+) , r ∈ c1 (r+,r+) and assume that r is nondecreasing with r(t) ≤ t for t ≥ 0. if u ∈ c (r+,r+) satisfies u(t) ≤ k(t) + y(t) ∫ r(t) 0 x(s)u(s)dαs, t ≥ 0, (2.1) then u(t) ≤ k(t) + y(t) ∫ t 0 e ∫ r(t) r(τ) x(s)y(s)dαsx (r(τ)) k(r(τ))dαr(τ)dατ, t ≥ 0. (2.2) theorem 2.2. let u,c,x,h,y ∈ c(r+,r+), r ∈ c1(r+,r+) and assume that r is non-decreasing with r(t) ≤ t for t ≥ 0. let w(t,u) be a positive , continuous, monotonic, non-decreasing, sub-additive and sub-multiplicative function for u > 0 for each fixed t. let the function k(t) > 0 and ψ(t) ≥ 0 be a non-decreasing in t and continuous on [0,∞). ψ(0) = 0 and suppose further that the inequality u(t) ≤ k(t) + c(t) ∫ r(t) 0 x(s)u(s)dαs + h(t)ψ [∫ r(t) 0 y(s)w(s,u(s))dαs ] (2.3) is satisfied for all t > 0. then u(t) ≤ [ k(t) + h(t)ψ ( g−1 ( g [∫ t 0 y(s)w(s,k(s)m(s))dαr(s)dαs ] + ∫ t 0 [y(s)w(s,h(s)m(s))dαr(s)]dαs ))] m(t) (2.4) where m(t) = 1 + c(t) ∫ t 0 e ∫ r(t) r(τ) x(s)c(s)dαsx(r(τ))dαr(τ)dατ (2.5) 164 usta and sarikaya and g−1 is inverse of g such that g(ξ) =: ∫ ξ 1 1 w(s, ψ(s)) dαs, ξ ≥ 0, and g [∫ t 0 y(s)w(s,k(s)m(s))dαr(s)dαs ] + ∫ t 0 [y(s)w(s,h(s)m(s))dαr(s)]dαs ∈ dom(g−1), ∀t ≥ 0. proof. let define z(t) = ∫ r(t) 0 y(s)w(s,u(s))dαs. (2.6) so z(0) = 0, then u(t) ≤ [k(t) + h(t)ψ(z(t))] + c(t) ∫ r(t) 0 x(s)u(s)dαs. (2.7) as [k(t) + h(t)ψ(z(t))] is positive, monotonic, non-decreasing, continuous function over [0,∞), we can apply the theorem 2.1, that is u(t) ≤ [k(t) + h(t)ψ(z(t))] m(t) (2.8) where m(t) defined in 2.5. then if we take the conformable fractional derivative of equation 2.6, we obtain dαz(t) = y(r(t))w(r(t),u(r(t)))dαr(t) ≤ y(t)w(t,u(t))dαr(t) ≤ y(t)w(t, [k(t) + h(t)ψ(z(t))] m(t))dαr(t) ≤ y(t)w(t,k(t)m(t))dαr(t) + y(t)w(t,h(t)m(t))w(t, ψ(z(t)))dαr(t) hence dαz(t) w(t, ψ(z(t))) ≤ y(t)w(t,k(t)m(t)) w(t, ψ(z(t))) dαr(t) + y(t)w(t,h(t)m(t))dαr(t). (2.9) then using the definition of g, we get g(z(t)) ≤g [∫ t 0 y(s)w(s,k(s)m(s))dαr(s)dαs ] + ∫ t 0 [y(s)w(s,h(s)m(s))dαr(s)]dαs. (2.10) hence z(t) ≤g−1 ( g [∫ t 0 y(s)w(s,k(s)m(s))dαr(s)dαs ] + ∫ t 0 [y(s)w(s,h(s)m(s))dαr(s)]dαs ) . (2.11) if we combine the equation 2.8 and 2.11, we get the desired bound. � theorem 2.3. let u,x ∈ c(r+,r+), r ∈ c1(r+,r+) and assume that r is non-decreasing with r(t) ≤ t for t ≥ 0, for which the inequality dαu(t) ≤ p + ∫ r(t) 0 x(s)dαuq(s)[u(s) + dαu(s)]dαs (2.12) holds, where p is a positive constant and 0 < q < 1. if [1 −q(p + u(0))q ∫ r(t) 0 x(s)eqsdαs] > 0, t ≥ 0, (2.13) then dαu(t) ≤ (pβ + β ∫ t 0 x(s)ω(s)dαs) 1/β, t ≥ 0. (2.14) where q + β = 1, ω(t) = (u(0) + p)et [1 −q(u(0) + p)q ∫ t 0 x(s)eqsdαs]1/q . (2.15) some improvements of conformable fractional integral inequalities 165 proof. let denote the right hand side of equation 2.12 by z(t), that is z(t) = p + ∫ r(t) 0 x(s)dαuq(s)[u(s) + dαu(s)]dαs. (2.16) here z(0) = p and dαu(t) ≤ z(t). if we integrate both sides of dαu(t) ≤ z(t) according to rules of conformable fractional calculus, we get u(t) ≤ u(0) + ∫ t 0 z(s)dαs. (2.17) then if we take conformable fractional derivative of equation 2.16, we obtain dαz(t) ≤ x(r(t))dαuq(r(t))[u(r(t)) + dαu(r(t))]dαr(t). (2.18) after simple manipulation, we get dαz(t) ≤ x(t)zq(t)[u(0) + z(t) + ∫ t 0 z(s)dαs]d αr(t) (2.19) let define w(t) = u(0) + z(t) + ∫ t 0 z(s)dαs. (2.20) here w(0) = u(0) + p. then by taking both sides of conformable fractional derivative of above expression and using dαz(t) ≤ x(t)zq(t)w(t)dαr(t) and z(t) < w(t), we get dαw(t) = dαz(t) + z(t) ≤ x(t)zq(t)w(t)dαr(t) + w(t) ≤ x(t)wq+1(t)dαr(t) + w(t). so we have w(t) ≤ ω(t), t ≥ 0, (2.21) where ω(t) = (u(0) + p)et [1 −q(u(0) + p)q ∫ t 0 x(s)eqsdαs]1/q (2.22) if we substitute 2.22 into dαz(t) ≤ x(t)zq(t)w(t)dαr(t), we get dαz(t) ≤ x(t)zq(t)ω(t)dαr(t). (2.23) which implies the estimation for z(t) such that, z(t) ≤ (pβ + β ∫ t 0 x(s)ω(s)dαs) 1/β, t ≥ 0. (2.24) if we combine the equation 2.24 and dαu(t) ≤ z(t), we get the desired result. � 3. concluding remark in this study we established the explicit bounds on retarded integral inequalities with the help of conformable fractional calculus. we take the advantage of katugampola type conformable fractional derivatives and integrals. references [1] t. abdeljawad, on conformable fractional calculus, j. comput. appl. math. 279 (2015) 57–66. [2] d. r. anderson and d. j. ulness, results for conformable differential equations, preprint, 2016. [3] a. atangana, d. baleanu, and a. alsaedi, new properties of conformable derivative, open math. 2015; 13: 889–898. [4] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65-70. [5] s. s. dragomir, some gronwall type inequalities and applications, rgmia monographs, victoria university, australia, 2002. [6] s. s. dragomir, on volterra integral equations with kernels of (l)-type, ann. univ. timisoara facult de math. infor., 25 (1987), 21-41. 166 usta and sarikaya [7] o.s. iyiola and e.r.nwaeze, some new results on the new conformable fractional calculus with application using d’alambert approach, progr. fract. differ. appl., 2 (2) (2016), 115-122. [8] m. a. hammad, r. khalil, conformable fractional heat differential equations, int. j. differ. equ. appl. 13 (3) (2014), 177-183. [9] m. a. hammad, r. khalil, abel’s formula and wronskian for conformable fractional differential equations, int. j. differ. equ. appl. 13(3) (2014), 177-183. [10] u. katugampola, a new fractional derivative with classical properties, arxiv:1410.6535 [math.ca]. [11] a. zheng, y. feng and w. wang, the hyers-ulam stability of the conformable fractional differential equation, math. aeterna, 5 (3) (2015), 485-492. [12] a. a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier b.v., amsterdam, netherlands, 2006. [13] s.g. samko, a.a. kilbas and o.i. marichev, fractional integrals and derivatives: theory and applications, gordonand breach, yverdon et alibi, 1993. [14] m. z. sarikaya, gronwall type inequality for conformable fractional integrals, konuralp j. math. 4(2) (2016), 217222. [15] m. z. sarikaya and huseyin budak, new inequalities of opial type for conformable fractional integrals, turkish j. math. in press. [16] f. usta, explicit bounds on certain integral inequalities via conformable fractional calculus, cogent math. 4 (1) (2017), art. id 1277505. [17] f. usta and m.z. sarikaya , on generalization conformable fractional integral inequalities, rgmia res. rep. collection, 19 (2016), article 123. [18] f. usta and m.z. sarikaya , a retarded conformable fractional integrals inequalities and its application, in press. [19] b. g. pachpatte, on some new inequalities related to certain inequalities in the theory of differential equations, j. math. anal. appl. 189 (1995), 128-144. [20] t.h. gronwall, note on derivatives with respect to a parameter of the solutions of a system of differential equations, ann. math. 20 (4) (1919), 292-296. department of mathematics, faculty of science and arts, düzce university, düzce-turkey ∗corresponding author: fuatusta@duzce.edu.tr 1. introduction & preliminaries 2. main findings & cumulative results 3. concluding remark references int. j. anal. appl. (2022), 20:65 generalization of fixed point approximation of contraction and suzuki generalized non-expansive mappings in banach domain n. muhammad, a. asghar, m. aslam, s. irum, m. iftikhar, m. m. abbas, a. qayyum∗ department of mathematics, institute of southern punjab multan, pakistan ∗corresponding author: atherqayyum@isp.edu.pk abstract. by principal motivation from the results of the new iterative scheme that produces faster results than k-iteration. in this article, we study generalized results by a new iteration scheme to approximate fixed points of generalized contraction and suzuki non-expansive mappings. we establish strong convergence results of generalized contraction mappings of closed convex banach space and also deduce data dependent results. furthermore, we prove some weak and strong convergence theorems in the sense of generalized suzuki non-expansive mapping by applying condition (c). 1. introduction mappings play a vital role in the field of inequalities (see of example [17–20]). the mappings which have lipschitzís constant equal to 1 are called as non-expansive mappings. let z be a non empty bounded closed convex subset of k. a banach space z has the fixed point property (fpp) for non expansive mapping if for every non-empty bounded closed convex subset of z contains a fixed point. meanwhile, in 1965 major struggle has been proposed to study the theory of fixed point of non-expansive mappings in the setting of reflexive and non-reflexive banach domain. since then, a number of generalizations and extensions of non-expansive mappings and their results have been obtained by many authors. we can say that fpp provides basis of physical appearance of the banach space. when k is a weakly compact convex subset of z, a non-expansive self-mapping of k requires not have fixed point. however, if the norm of z has suitable ordered properties (i.e., uniform convex and some others) each non-expansive self-mapping of every weakly closed convex subset of z has a fixed point. in this case, k is called a weak fixed point property. all over in this article we assume that received: oct. 2, 2022. 2010 mathematics subject classification. 47h09, 47h10, 47j25. key words and phrases. fixed point, contraction mapping, suzuki generalized non expansive mapping, condition (c). https://doi.org/10.28924/2291-8639-20-2022-65 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-65 2 int. j. anal. appl. (2022), 20:65 k is a non-empty subset of a banach space z and q(t ), the set of all fixed points of the mapping t over k. a mapping t : k → k is called to be a non-expansive if ‖txo −tyo‖ ≤ ‖xo −yo‖, of all xo, yo ∈ k. this is also called quasi non-expansive if q(t ) 6= φ and ‖txo −p‖ ≤ ‖xo −p‖, of all xo ∈ k and of all p ∈ q(t ). it is known as q(t ) is non-empty while z is uniformly convex, k be a bounded closed convex subset of x and t be a non-expansive mapping [2]. in 2008, japanese mathematician suzuki [3] presented idea of generalized non-expansive mappings which is also called condition (c) and defined as a self-mapping t on k is said to be condition (c) that, 1 2 ‖xo −txo‖≤‖xo −yo‖ =⇒‖txo −tyo‖≤‖xo −yo‖ ,∀xo,yo ∈ k of such mappings, suzuki also obtained the existence of fixed point and convergence results. in [4], he proved that condition(c) have faster results as compared to non-expansive mappings. for a selfmapping t be defined over [0, 3] t (xo) = [0, if xo 6= 0 1, if xo = 0]. by this, we claim that t satisfy the condition (c), but t is not a non-expansive mapping. in extension, picard’s iterative scheme is approximate the fixed point of contraction mappings in the banach contraction principle. over time, many mathematicians [6–10] played a fundamental role in the development of the current literature. sahu v. k [5] and many other tried their best as compared to the previous one and added outstanding work. inspired by the above, now we generalize some results by a new iteration scheme to approximate fixed point of generalized contraction and suzuki’s non-expansive mappings. also we discuss strong convergence theorems of generalized contraction mappings with closed convex banach space and some data dependence results are also deduce. moreover, we prove some weak and strong convergence results in type of generalized suzuki non-expansive mappings by using condition (c). 2. preliminaries (2.1) [12] opial property if for each sequence {$n} in x, (where x be a banach space) converging weakly to xo ∈ x take lim n→∞ sup‖$n −xo‖ < lim n→∞ sup‖$n −yo‖∀ yo ∈ x such that yo 6= xo. (2.2) let k be a non-empty bounded sequence convex subset of a banach space z and consider {$n} be in z of x0 ∈ z, that d(z,{$n} = lim n→∞ sup‖$n −xo‖ . the asymptotic radius of {$n} comparative to k is given that d (k,{$n}) = inf {d(xo,{$n}) : xo ∈ k} . int. j. anal. appl. (2022), 20:65 3 the asymptotic center of {$n} relative to k is the set b(k,$n}) = {xo ∈ k : d(xo,{$n}) = d(k,{$n})}. (2.3) a uniformly convex banach space, x and {$n} be a real sequence such that 0 < s ≤ $n ≤ t < 1 ,∀ n ≥ 1. consider {$n} and {ωn} be two sequences of k given that limn→∞ sup‖$n‖≤ d, limn→∞ sup‖ωn‖ ≤ d and limn→∞ sup‖$n + (1 −$n)ωn‖ = d holds of some d ≥ 0. then, limn→∞‖$n −ωn‖ = 0. it is called a uniformly convex banach space, b(k,{$n}) contains exactly one point. (2.4) a mapping t : k → k is called demi-closed with respect to yo ∈ k if of each sequence {$n} in k and k be a closed, convex and non-empty subset of a banach space k of each xo ∈ k,{$n} converges weakly at xo and {t$n} converges strongly at yo =⇒ txo = yo. (2.5) [16] let{un}∞n=0 and {vn} ∞ n=0 are two fixed points iteration sequences that converges to the same fixed point q. if ‖un −q‖ ≤ an and ‖vn −q‖ ≤ bn, of all n ≥ 0, wherever {an}∞n=0 and {bn} ∞ n=0 be two real convergent sequences. then we say that {un}∞n=0 converge faster than {vn} ∞ n=0 to q if {an}∞n=0 converges faster as compare to {bn} ∞ n=0. (2.6) let k be a non-empty subset of a banach space z. consider that a mapping t : k → k is said to be generalized contraction when ∃ 0 ≤ h ≤ 1 such that ‖ts −tt‖ ≤ h max[‖s − t‖ ,‖s −ts‖ ,‖t −tt‖ ,‖s −tt‖ + ‖t −ts‖] ∀ s,t ∈ k. (2.7) a banach space k is known as uniformly convex if of each � belongs to (0, 2] there is a δ > 0 such that of s,t ∈ k ‖s‖ ≤ 1, ‖t‖ ≤ 1, ‖s − t‖ > �, implies that ‖s + t‖ 2 ≤ δ. [1] let x be a non-empty set and ϕ is collection of x, then (1) x belongs to τ. (2) absolute union of number of τ belongs to τ. (3) limited intersection of τ belongs to τ. than τ be a topology over x so, (x,τ) is called topological space. topology also helpful in different properties like convergence, existence, convex and many other. 4 int. j. anal. appl. (2022), 20:65 3. some basic results proposition (3.1) ( [3]) let z be a non-empty subset k of a banach space k and t be a self mappings. (a1) if t be non-expansive mapping then t satisfies condition (c). (a2) every mapping satisfying condition (c) with a fixed point is quasi non-expansive. (a3) if t satisfies condition (c) ‖xo −tyo‖≤ 3‖txo −yo‖ =⇒‖txo −tyo‖ + ‖xo −yo‖∀xo,yo ∈ c. lemma (3.2) let {λm}∞m=0 and {µm} ∞ m=0 be a non negative real sequences satisfying the given inequality λm+1 ≤ (1 − ξm)λm + µm, also ξm ∈ (0, 1) ∀ m ∈ n, σ∞m=0 ξm = ∞ and µm ξm → 0 as m →∞, then limm→∞λm = 0. lemma (3.3) ( [15]) let{λn}∞n=0 be a non-negative real sequence for which assume that ∃ n0 ∈ n such that ∀ n ≥ n0, the given inequalities satisfies λn+1 ≤ (1 − νn)λn + νnµn, also νn ∈ (0, 1) ∀ n ∈ n, σ∞n=0νn = ∞ and µn ≥ 0 ∀, n ∈ n, so 0 ≤ limn→∞ sup λn ≤ limn→∞ sup µn. lemma (3.4) [13] let k be a uniformly convex banach space and t be a self-mapping over a weakly compact convex subset k. consider that t fulfil condition (c), then t has a fixed point. lemma (3.5) suppose that x be a subset k of a banach space with the opial’s property [12] and t be a self mapping over x. suppose t satisfies the criteria of condition (c). if {$n} converges weakly to τ and limn→∞‖$n−t$n = 0‖ , then tτ = τ. it is i −t demi-closed at 0. here we define our new iterative process, it has better approximations and have faster rate of convergence then previous all (for further details see [11]). now we generalized our results by this faster iterative scheme u0 ∈ k zn = t [(1 −δn)un + δntun] yn = t [(1 −αn)tun + αntzn] un+1 = tyn. (1) 4. main results in this section we generalize results via new faster iterative scheme to approximate fixed point of generalized contraction and suzuki’s non-expansive mappings. we generalize strong convergence results in closed convex banach space and some data dependence results are also deduce. theorem 4.1. suppose k be a non-empty closed convex subset of a banach space z and t : k → k be generalized contraction mapping. assume {hn}∞n=0 be an iterative sequence which is generated by (1), with the real sequence {ηn} ∞ n=0 and {κn} ∞ n=0 in [0, 1] satisfying σ ∞ n=0ηnγn = 0 then, {hn} ∞ n=0 converges strongly to a unique fixed point of t. int. j. anal. appl. (2022), 20:65 5 proof. the well-known banach principle has guarantees of existence and uniqueness of fixed point g. we prove that {hn} converges to a fixed point g, by using (1) we get ‖zn −g‖ = ‖t [(1 −γn) hn + γnthn] −g‖ ≤ h max[‖(1 −γn)h + γnthn −g‖ , ‖((1 −γn)hn + γnthn) −t{(1 −γn)hn + hnγn}‖ , ‖g −tg‖ ,‖(1 −γn)hn + γnthn −tg‖ + ‖g −t ((1 −γn)hn + γnhn)‖] ≤ h max[‖(1 −γn) hn + γnthn −tg‖ ,‖((1 −γn)hn + γnthn) −zn‖ , ‖(1 −γn)hn + γnthn −g‖ + ‖g −zn‖]. (2) case#1 let ‖zn −g‖≤ h[‖(1 −γn) hn + γnthn −g‖]. (3) case#2 let ‖zn −g‖ ≤ h[‖(1 −γn) hn + γnthn −zn‖] = h[‖(1 −γn) hn + γnthn −g + g −zn‖] ≤ [‖(1 −γn) hn + γnthn −g‖ + ‖zn −g‖] ‖zn −g‖ ≤ h 1 −h [‖(1 −γn) hn + γnhn −g‖]. (4) case#3 let ‖zn −g‖ ≤ h[‖(1 −γn) hn + γnthn −g‖ + ‖zn −g‖] ‖zn −g‖ ≤ h 1 −h [‖(1 −γn) hn + γnthn −g‖]. let η = max[h, h 1−h ] ∈ (0, 1) ‖zn −g‖ ≤ η‖(1 −γn) hn + γnhn −g‖ ≤ η‖(1 −γn) hn − (1 −γn)g + γnthn −γng‖ ≤ η [(1 −γn)‖hn −g‖ + γn‖thn −g‖ . now ‖thn −g‖ ≤ h max[‖hn −g‖ ,‖th−g‖ ,‖tg −g‖ + ‖g −thn‖] = h max[‖hn −g‖ ,‖thn −g‖ ,‖thn −g‖ + ‖hn −g‖] = h max[‖hn −g‖ ,‖thn −g‖ + ‖hn −g‖] ≤ η‖hn −g‖ . 6 int. j. anal. appl. (2022), 20:65 ‖zn −g‖ ≤ η [(1 −γn)‖hn −g‖ + γnη‖hn −g‖ ≤ η [(1 −γn + γnη)]‖hn −g‖ ≤ η [(1 −γn)(1 −α)]‖hn −g‖ . (5) similarly ∥∥h′n −g∥∥ = ‖t (1 −ηn)thn + ηntzn −tg‖ ≤ h max[‖(1 −ηn)thn + ηntzn −g‖ , ‖(1 −ηn)thn + ηntzn −t ((1 −ηn)thn + ηntzn)‖ , ‖g −tg‖ ,‖((1 −ηn)thn + ηntzn) −tg‖ + ‖g −t ((1 −ηn)thn + ηntzn)‖ . case#1 ∥∥h′n −g∥∥ ≤ h[‖(1 −ηn)thn + ηntzn −g‖]. (6) case#2 ∥∥h′n −g∥∥ ≤ h[∥∥(1 −ηn)thn + ηntzn −h′n∥∥] = h[ ∥∥(1 −ηn)thn + ηntzn −g + g −h′n∥∥]∥∥h′n −g∥∥ ≤ h1 −h[‖(1 −ηn)thn + ηntzn −g‖]. (7) case#3 ∥∥h′n −g∥∥ ≤ h[‖(1 −ηn)tqn + ηntzn −q‖ + ∥∥q −h′n∥∥]∥∥h′n −g∥∥ ≤ h1 −h[‖(1 −ηn)thn + ηntzn −q‖]. let η = max{h, h 1−h}∈ (0, 1)∥∥h′n −g∥∥ ≤ η [‖(1 −ηn)thn + ηntzn −g‖] = η [(1 −ηn)‖thn −g‖ + ηn‖tzn −g‖] ≤ η [(1 −ηn)‖thn −g‖ + ηηn‖zn −g‖] ≤ η [η (1 −ηn)‖hn −g‖ + η 2ηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn)‖hn −g‖ + ηηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn)‖hn −g‖ + ηηn(1 −γn(1 −η))‖hn −g‖] ≤ η2[(1 −ηn + ηηn −ηηnγn(1 −η)]‖hn −g‖ ≤ η2[(1 −ηn(1 −η) −ηηnγn(1 −ηn))]‖hn −g‖ ∥∥h′n −g∥∥ ≤ η2[(1 − (1 −η) ηn(1 + ηγn)]‖hn −g‖ (8) int. j. anal. appl. (2022), 20:65 7 ‖hn+1 −g‖ = ∥∥th′n −g∥∥ = ‖t [t (1 −ηn) thn + ηntzn] −g‖ ≤ max[ ∥∥h′n −g∥∥ ,∥∥h′n −th′n∥∥ ,‖g −tg‖ ,∥∥h′n −tg∥∥ + ∥∥g −th′n∥∥ ≤ h max‖t (1 −ηn) thn + ηntzn −g‖ , ‖t ((1 −ηn)thn + ηntzn) −t (t (1 −ηn)thn + ηntzn))‖ , ‖tg −g‖ ,‖t (t (1 −ηn)thn −ηntzn)) −g‖ +‖t ((1 −ηn)thn + ηntzn) −g‖] = h max[‖hn −g‖ , ∥∥h′n −th′n∥∥ , 0,∥∥th′n −g∥∥ + ∥∥h′n −g∥∥ = h max[ ∥∥h′n −g∥∥ ,∥∥hn+1 −h′n∥∥ ,‖hn+1 −g‖ + ∥∥h′n −g∥∥ . case#1 ‖hn+1 −g‖≤ h ∥∥h′n −g∥∥ . case#2 ‖hn+1 −g‖≤ h 1 −h ∥∥h′n −g∥∥ . case#3 ‖hn+1 −g‖≤ h 1 −h ∥∥h′n −g∥∥ η = max{h, h 1−h}∈ (0, 1) ‖hn+1 −g‖ ≤ η ∥∥h′n −g∥∥ ≤ η[η2(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖] ≤ η3(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖]. (9) repetition of above scheme gives the following inequalities ‖hn+1 −g‖ ≤ η3(1 −ηn(1 + ηγn) (1 −η))‖hn −g‖ ‖hn −g‖ ≤ η3(1 −ηn−1(1 + ηγn−1) (1 −η))‖hn−1 −g‖ ‖hn−1 −g‖ ≤ η3(1 −ηn−2(1 + ηγn−2) (1 −η))‖hn−2 −g‖ ... ‖h1 −g‖ ≤ η3(1 −η0(1 + ηγ0) (1 −η))‖h0 −g‖ . (10) from (10) we can easily derive ‖hn+1 −g‖≤‖h0 −g‖η3(n+1)πnk=01 −ηk(1 + ηγk) (1 −η) (11) 8 int. j. anal. appl. (2022), 20:65 where 1 −ηk(1 + ηγk) (1 −η) < 1 because η ∈ (0, 1) and ηnγn ∈ (0, 1) ∀ n ∈ n. we know that 1 −h ≤ %−h ∀ x ∈ (0, 1) then by (11), we have ‖hn+1 −g‖≤ ‖h0 −q‖η3(n+1) %(1−η)σn k=0 ηk(1 + ηγk) (12) taking limit of both sides in (12), we get limn→∞‖hn −g‖ i.e hn → g of n →∞ as required. � theorem 4.2. suppose that k be a non-empty closed convex subset of a banach space z and t : k → k be a generalized contraction mappings. consider {xn}∞n=0 be an iterative sequence that is generated from (1) with real sequences {αn}∞n=0 and {βn} ∞ n=o in [0, 1] satisfying the criteria of σ∞n=0αnβn = ∞. then, iteration scheme (1) be tstable. proof. let {un}∞n=0 ⊂ z be arbitrary sequence in k. suppose that the sequence generated from (1) be xn+1 = f (t,xn) converging to a unique fixed point q (by theorem 4.1) and �n = ||un+1−f (t,un)|| we prove that limn→∞ �n = 0 ⇐⇒ limn→∞un = q. assume limn→∞ �n = 0 we take ‖un+1 −q‖ ≤ ‖un+1 − f (t,un)‖ + ‖f (t,un) −q‖ = �n + ‖t (t ((1 −βn)tun + βnt ((1 −αn)un + αntun))) −q‖ ≤ α3(1 − (1 −α))αn(1 + βnα)‖un −q‖ + �n. since α ∈ (0, 1) and αn,βn ∈ [0, 1] ∀n ∈ n and limn→∞�n = 0. so, by above inequality and lemma 3.2 which leads limn→∞‖un −q‖ = 0. hence limn→∞un = q. conversely consider that limn→∞un = q we get �n = ‖un+1 − f (t,un)‖ ≤ ‖un+1 −q‖ + ‖f (t,un) −q‖ ≤ ‖un+1 −q‖ + α3(1 − (1 −α)αn(1 + αβn)‖un −q‖ ⇐⇒ limn→∞�n = 0. hence, (1) is stable w.r.t t . � theorem 4.3. suppose that k be a non-empty closed convex subset of a banach space z and t : k → k be a generalized contraction mapping of a fixed point p . it is given that u0 = x0 ∈ c, let {un}∞n=0 and {xn} ∞ n=0 be iterative sequences generated by (1) respectively, with real sequences {αn}∞n=0 and {βn} ∞ n=0 in [0, 1] satisfying (s1) α ≤ αn < 1 and β ≤ βn < 1, for some results like α,β > 0 and ∀ n ∈ n. then, {xn}∞n=0 converges to p faster as than {un}∞n=0 does. int. j. anal. appl. (2022), 20:65 9 proof. by (11) we get ‖xn+1 −p‖≤‖x0 −p‖α3(n+1)πnk=0(1 − (1 −α))αk(1 + αβk). (13) the following inequality is due to (9) and lemma (3.2) which is obtained from (1), also converges to a unique fixed point p. ‖un+1 −p‖≤‖u0 −p‖α2(n+1)πnk=0(1 − (1 −α))αk(1 + αβk) (14) together with supposition (s1) and (13) implies that ‖xn+1 −p‖ ≤ ‖x0 −p‖α3(n+1)πnk=0[(1 − (1 −α))α(1 + αβ)] = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1. (15) similarly (15) and supposition (s1) ‖un+1 −p‖ = ‖u0 −p‖α2(n+1)πnk=0(1 − (1 −α))α(1 + βα) = ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + βα)]n+1 (16) define an = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 bn = ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 then ψn = an bn = ‖x0 −p‖α3(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 ‖u0 −p‖α2(n+1)[(1 − (1 −α))α(1 + αβ)]n+1 = ∥∥αn+1∥∥ (17) since lim n→∞ ψn+1 ψn = lim n→∞ αn+2 αn+1 = α < 1 by applying the ratio test we get σ∞n=0ψn < ∞ hence from (17), we have lim n→∞ an bn = lim n→∞ ψn = 0 implies that {xn}∞n=0 is faster than {un} ∞ n=0. now we prove following data dependence results. � 10 int. j. anal. appl. (2022), 20:65 theorem 4.4. suppose that t̃ be an approximate operator of a generalized contraction mapping t. consider that {hn}∞n=0 be an iterative sequence generated from equation (1) for t and we define an iterative sequence {h̃n}∞n=0 which is given as h̃0 ∈ k z̃n = t [(1 −γn)h̃n + γnt̃ h̃n] h̃′n = t̃ [(1 −αn)t̃ h̃n + αnt̃ z̃n] h̃n+1 = t̃ h̃ ′ n (18) with real sequences {αn}∞n=0 and {γn} ∞ n=0 in [0, 1] which satisfying the (i) 1 2 ≤ αnγn ∀ n ∈ n (ii) σ∞n=0αnγn = ∞ if tq = q and t̃ q̃ =q̃ such that limn→∞ h̃n = q̃, then we get ‖q − q̃‖≤ 7� 1 −α where � ≥ 0 is a fixed number. proof. it follows from (1) and (18) ‖zn − z̃n‖ = ∥∥∥t (1 −γn)hn + γnthn) − t̃ ((1 −γn)h̃n −γnt̃ h̃n)∥∥∥ ≤ ∥∥∥t (1 −γn)hn + γnthn) −t (1 −γn)h̃n + γnt̃ h̃n)∥∥∥ + ∥∥∥t ((1 −γn)h̃n + γnt̃ h̃n) − t̃ ((1 −γn)h̃n + γnt̃ h̃n)∥∥∥ ≤ α ∥∥∥(1 −γn)hn + γnthn − (1 −γn)h̃n −γnt̃ h̃n)∥∥∥ + � ≤ α[(1 −γn) ∥∥∥hn + h̃n∥∥∥ + γn ∥∥∥thn − t̃ h̃n∥∥∥ + � ≤ α[(1 −γn) ∥∥∥hn − h̃n∥∥∥ + γn{∥∥∥thn − t̃ h̃n∥∥∥ + ∥∥∥th̃n − t̃ h̃n∥∥∥} + � ≤ α[(1 −γn) ∥∥∥hn − h̃n∥∥∥ + γnα∥∥∥hn − h̃n∥∥∥ + γn�] + � ≤ α[1 −γn (1 −α) ∥∥∥hn − h̃n∥∥∥ + γn�] + �. (19) using (19), we have∥∥∥h′n − h̃′n∥∥∥ = ∥∥∥t ((1 −αn)thn + αntzn) − t̃ ((1 −αn)t̃ h̃n + t̃ z̃n)∥∥∥ ≤ t ((1 −αn)thn + αntzn) −t ((1 −αn)t̃ h̃n + αnt̃ z̃n) +t ((1 −αn)t̃ h̃n + αnt̃ z̃n) − t̃ ((1 −αn)t̃ h̃n + αnt̃ z̃n) ≤ α ∥∥∥(1 −αn)thn + αntzn − (1 −αn)t̃ h̃n −αnt̃ z̃n)∥∥∥ + � ≤ α[(1 −αn) ∥∥∥thn − t̃ h̃n∥∥∥ + αn ∥∥∥tzn − t̃ z̃n∥∥∥ + � int. j. anal. appl. (2022), 20:65 11 ≤ α[(1 −αn) ∥∥∥thn −th̃n∥∥∥ + ∥∥∥th̃n − t̃ h̃n∥∥∥ +αn[‖tzn −tz̃n‖ + ∥∥∥tz̃n − t̃ z̃n∥∥∥] + � ≤ α[(1 −αn)α ∥∥∥xn − h̃n∥∥∥ +αnα[α (1 −γn) (1 −α) ||hn − h̃n|| + γn� + �] + � ≤ α2[(1 −αn) ∥∥∥xn − h̃n∥∥∥ + α3αn[1 −γn(1 −α) ∥∥∥hn − h̃n∥∥∥ +α3αnγn + α 2αn]� ≤ α2[1 −αn + αnα + α (1 −α) αnγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + � ≤ α2[1 − (1 −α)αn −α (1 −α) αnγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + � ≤ α2[1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +α�(1 + ααnγn) + �. (20) by using (20), we have∥∥∥h′n+1 − h̃′n−1∥∥∥ = ∥∥∥th′n − t̃ h̃′n∥∥∥ ≤ ∥∥∥h′n − h̃′n∥∥∥ + � ≤ α3[1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +α2�(1 + ααnγn)] + �α + � ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +�(1 + ααnγn)] + � + � ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn� + 3� ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn� + 3 (1 −αnγn + αnγn) � (21) by supposition (i) we have 1 −αnγn ≤ αnγn∥∥∥hn+1 − h̃n+1∥∥∥ ≤ [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +7αnγn� = [1 − (1 −α) αn(1 + αγn)] ∥∥∥hn − h̃n∥∥∥ +αnγn (1 −α) 7� 1 −α . (22) 12 int. j. anal. appl. (2022), 20:65 let ψn = ∥∥∥hn − h̃n∥∥∥ , φn = (1 −α) αn(1 + αγn), φn = 7�1−α, then from lemma 3.2 together with (22) we get 0 ≤ lim n→∞ sup ∥∥∥hn − h̃n∥∥∥ ≤ lim n→∞ sup 7� 1 −α (23) since by theorem 4.1 we have limn→∞hn = q and from supposition i and ii we get limn→∞h̃n = q̃ by using these together with (23) and we get ‖q − q̃‖≤ 7� 1−α as required. � 5. convergence results of suzuki generalized non-expansive mappings of condition (c) in this section, we prove some weak and strong convergence theorems of a sequence generated from new iterative scheme of suzuki generalized non-expansive mappings with condition (c) by uniformly convex banach spaces. lemma 5.1. suppose that k be a non-empty uniformly closed convex subset of a banach space z. let t : k → k be a mapping satisfying condition (c) with q(t ) 6= 0. for arbitrary chosen h0 ∈ k, consider that a sequence {hn} is generated from (1), then limn→∞‖hn −q‖ exists for any g ∈ q(t ). proof. consider that g ∈ q(t ) and z ∈ k. so t satisfies condition (c) ≤ 0 1 2 ‖g −tg‖ = 0 ≤‖g −z‖⇒‖tg −tz‖≤‖g −z‖ so by proposition (a2) we get ‖zn −g‖ = ‖t [(1 −γn)hn + γnδtn] −g‖ = ‖t [(1 −γn)hn + γntn] −tg‖ ≤ ||(1 −γn)hn + γntn −g|| ≤ (1 −γn)‖hn −g‖ + γn‖txn −g‖ ≤ (1 −γn)‖hn −g‖ + γn‖hn −g‖ ≤ ‖hn −g‖ . (24) by using (24) we have ∥∥h′n −g∥∥ = ‖t ((1 −δn)thn + δntzn) −g‖ ≤ ‖(1 −δn)thn + δntzn −g‖ ≤ (1 −δn)‖txn −g‖ + δn‖tzn −g‖ ≤ (1 −δn)‖hn −g‖ + δn‖zn −g‖ ≤ (1 −δn)‖hn −g‖ + δn‖hn −g‖ = ‖hn −g‖ . (25) int. j. anal. appl. (2022), 20:65 13 similarly by using (25) we have ‖hn+1 −g‖ = ∥∥th′n −g∥∥ ≤ ∥∥h′n −g∥∥ ≤ ‖hn −g‖ (26) ⇒‖hn −g‖ be a bounded and decreasing ∀ g ∈ q (t ) so, limn→∞‖hn −g‖ exist as required. � theorem 5.1. suppose that k is a non-empty and uniformly closed convex of subset of a banach space z. and let t : k → k be a mapping satisfying condition (c). for arbitrary chosen h0 ∈ k, consider that the sequence {hn} be generated from (1) ∀ n ≥ 1, where {αn} and {βn} are two sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1. so, q(t ) 6= θ ⇐⇒ {hn} is bounded sequence and limn→∞‖thn −hn‖ = 0. proof. suppose that q(t ) 6= φ and let g ∈ q(t ). then, from lemma 5.1, limn→∞‖hn −g‖ exists and {hn} is bounded. lim n→∞ ‖hn −g‖ = r (27) from (24) and (27), we have lim n→∞ sup‖zn −g‖≤ lim n→∞ sup‖hn −g‖ = r (28) so by proposition 3.1 (a2) lim n→∞ sup‖thn −g‖≤ lim n→∞ sup‖hn −g‖ = r (29) also ‖hn+1 −g‖ = ∥∥th′n −g∥∥ ≤ ∥∥h′n −g∥∥ = ‖t ((1 −αn)thn + αntzn) −g‖ ≤ ‖(1 −αn)thn + αntzn −g‖ ≤ (1 −αn)‖thn −g‖ + αn‖tzn −g‖ ≤ (1 −αn)‖hn −g‖ + αn‖zn −g‖ ≤ ‖hn −g‖−αn‖hn −g‖ + αn‖zn −g‖ (30) this implies ‖hn+1 −g‖−‖hn −g‖ αn ≤ ‖zn −g‖−‖yn −g‖ 14 int. j. anal. appl. (2022), 20:65 ‖hn+1 −g‖−‖hn −g‖ ≤ ‖hn+1 −g‖−‖hn −g‖ αn ≤ ‖zn −g‖−‖hn −g‖ =⇒ ‖xn+1 −p‖≤‖zn −p‖ therefore r ≤ limn→∞ inf ‖zn −g‖ from (28) and (30), we have r = ||zn −g|| = lim n→∞ ‖t ((1 −γn)hn + γntn)g‖ ≤ lim n→∞ ‖γn(thn −g) + (1 −γn)hn −g‖ (31) from (27) , (29) and (31) together with lemma 3.3 we get limn→∞‖thn −hn‖ = 0. conversely suppose that {hn} is bounded and lim n→∞ ‖thn −hn‖ = 0 consider that g ∈ (c,{hn}) by proposition 3.1 we get r (tg,{hn}) = lim n→∞ sup‖hn −tg‖ ≤ lim n→∞ sup 3‖hn −tg‖ + ‖hn −g‖ ≤ lim n→∞ sup‖hn −g‖ = r (g,{hn}) . this implies that tg ∈ b(k,{hn}). since z is uniformly convex, b(k,{hn}) singleton and we get tg = g. hence q(t ) 6= φ. we are able to prove weak convergence theorem. � theorem 5.2. let k be a non-empty closed convex subset of a uniformly convex banach space z, with opial property, and consider that t : k → k be a mapping satisfying condition (c). for arbitrary chosen x0 ∈ c, let the sequence {xn} is generated from (1) of all n ≥ 1, where {αn} and {βn} are sequences of a real numbers in [l,m] for some l,m with 0 < l ≤ m < 1 such that q(t ) 6= φ. then {xn} converges weakly to a fixed point of t. proof. from theorem 5.1 we get {xm} be bounded and limn→∞‖txm −xm‖ = 0. since, x be a uniformly convex and reflexive thereof, from eberlin’s theorem ∃ a subsequence {xmu} of {xm} which converges weakly to some points q1 ∈ z. since, k is closed and convex from mazur’s theorem q1 ∈ k. from lemma 3.4, q1 ∈ q(t ). now, we prove that {xm} converges weakly to q1. in fact if this is false than there must exist a subsequence {xmv} of {xm} such that {xmv} converges weakly to int. j. anal. appl. (2022), 20:65 15 q2 ∈ k and q2 6= q1. from lemma 3.5 q2 ∈ q(t ). since, limn→∞‖xm −p‖ exists ∃, p ∈ q(t ). by theorem 5.1 and by opial’s property, we get lim m→∞ inf ‖xm −q1‖ = lim u→∞ inf ‖xmu −q1‖ < lim u→∞ inf ‖xmu −q2‖ = lim m→∞ inf ‖xm −q2‖ = lim v→∞ inf ‖xmv −q2‖ < lim v→∞ inf ‖xmv −q1‖ = lim m→∞ inf ‖xm −q1‖ which is contradiction so q1 = q2. this implies {xm} converges weakly to a fixed point of t . now we establish strong convergence results. � theorem 5.3. suppose that k be a non-empty compact convex subset of a uniformly convex banach space z. let t : k → k be a mapping satisfying condition (c). for arbitrary chosen l0 ∈ k, consider that a sequence {ln} is generated from (1) ∀, n ≥ 1, where {αn} and {βn} are two sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1. then {xn} converges strongly to a fixed point for t. proof. by lemma 3.4, q(t ) 6= φ and by theorem 5.1 we have limn→∞‖tln − ln‖ = 0. since k is compact and ∃ a subsequence {lnk} of {ln} such as {lnk} converges strongly to p for some p ∈ k. from proposition a3 we get ‖lnk −tp‖≤ 3‖tlnk − lnk‖ + ‖lnk −p‖ for all n ≥ 1. suppose that k → ∞, than we get tp = p, i.e., p ∈ q(t ). since, from lemma 5.1, limn→∞‖ln −p‖ exists for all p ∈ q(t ), since xn converges strongly to p. senter and dotson [14] both mathematicians discovered notion of mapping which satisfying condition (i) as. a mapping t : k → k is called to satisfy condition (i), if ∃ an increasing function f : [0,∞) → [0,∞) in f (0) = 0 and f (r ′) > 0 for all r ′ > 0 such as ‖l −tl‖ ≥ f (d(l,q(t ))) for all l ∈ k, and d(l, q(t )) = infp ∈ q(t )‖l −p‖. � theorem 5.4. suppose that k be a non-empty uniformly closed convex subset of a banach space z, and consider that t : k → k be a mapping which satisfying condition (c). for arbitrary chosen y0 ∈ k, let the sequence {yn} be generated from (1) for all n ≥ 1. since {αn} and {βn} are sequences of real numbers in [u,v] for some u,v with 0 < u ≤ v < 1 such that q(t ) 6= φ. if t fulfil condition (i), so {yn} converges strongly to a fixed point of t . 16 int. j. anal. appl. (2022), 20:65 proof. by lemma 5.1, we get limn→∞‖yn −p‖ exist ∀ p ∈ q(t ) and limn→∞d(yn,q(t )) exists. consider that limn→∞‖yn −q‖ = s′ for some s′ ≥ 0. if s′ = 0 then results follows. suppose that s′ > 0, from proposition (3.1) and condition (i), f (d(yn,q(t ))) ≤‖tyn −yn‖ (32) since, q(t ) 6= φ, from theorem 5.2 with (32) ⇐⇒ limn→∞‖tyn −yn‖ = 0 lim n→∞ f (d(yn,q(t ))) = 0 (33) since f is an increasing function and by (33), we have limn→∞d(yn,q(t )) = 0. thus we get a subsequence {ynk} of {yn} and a sequence {y ′k}⊂ q(t ) such that∥∥ynk −y ′k∥∥ < 12k ∀, k ∈ n than by applying (26), we have∥∥ynk+1 −y ′k∥∥ ≤ ∥∥ynk −y ′k∥∥ < 12k ≤ 1 2k+1 + 1 2k 1 2k−1 → 0 as k →∞ this shows that {y ′k} is a cauchy sequence with q(t ) and so it converges to a point p. since q(t ) is closed, therefore p ∈ q(t ) and then {ynk} converges strongly to p. since limn→∞‖yn −p‖ exists, we get yn → p ∈ q(t ). � 6. conclusion in this article we discussed generalized results by using new iterative scheme to approximate fixed point of generalized contraction and suzuki non-expansive mappings. here we developed new strongly convergence results of generalized contraction mappings of closed convex banach space and also produced some new data dependence results. in addition, we proved some weak and strong convergence results in sense of generalized suzuki non expansive mapping by applying condition (c). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. asghar, a. qayyum, n. muhammad, different types of topological structures by graphs, eur. j. math. anal. 3 (2022), 3. https://doi.org/10.28924/ada/ma.3.3. [2] f.e. browder, nonexpansive nonlinear operators in a banach space, proc. natl. acad. sci. u.s.a. 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041. [3] t. suzuki, fixed point theorems and convergence theorems for some generalized nonexpansive mappings, j. math. anal. appl. 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023. https://doi.org/10.28924/ada/ma.3.3 https://doi.org/10.1073/pnas.54.4.1041 https://doi.org/10.1016/j.jmaa.2007.09.023 int. j. anal. appl. (2022), 20:65 17 [4] j. garcia-falset, e. llorens-fuster, t. suzuki, fixed point theory for a class of generalized nonexpansive mappings, j. math. anal. appl. 375 (2011), 185–195. https://doi.org/10.1016/j.jmaa.2010.08.069. [5] v.k. sahu, h.k. pathak, r. tiwari, convergence theorems for new iteration scheme and comparison results, aligarh bull. math. 35 (2016), 19-42. [6] w. kassab, t. turcanu, numerical reckoning fixed points of (%e)-type mappings in modular vector spaces, mathematics, 7 (2019), 390. https://doi.org/10.3390/math7050390. [7] s. dhompongsa, w. inthakon, a. kaewkhao, edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, j. math. anal. appl. 350 (2009), 12–17. https://doi.org/10.1016/j.jmaa.2008.08. 045. [8] j. garcia-falset, e. llorens-fuster, t. suzuki, fixed point theory for a class of generalized nonexpansive mappings, j. math. anal. appl. 375 (2011), 185–195. https://doi.org/10.1016/j.jmaa.2010.08.069. [9] i. uddin, m. imdad, j. ali, convergence theorems for a hybrid pair of generalized nonexpansive mappings in banach spaces, bull. malays. math. sci. soc. 38 (2014), 695–705. https://doi.org/10.1007/s40840-014-0044-6. [10] m. de la sen, m. abbas, on best proximity results for a generalized modified ishikawa’s iterative scheme driven by perturbed 2-cyclic like-contractive self-maps in uniformly convex banach spaces, j. math. 2019 (2019), 1356918. https://doi.org/10.1155/2019/1356918. [11] n. muhammad, a. asghar, s. irum, a. akgül, e.m. khalil, m. inc, approximation of fixed point of generalized non-expansive mapping via new faster iterative scheme in metric domain, aims math. 8 (2023), 2856–2870. https://doi.org/10.3934/math.2023149. [12] z. opial, weak convergence of the sequence of successive approximations of non-expansive mappings, bull. amer. math. soc. 73 (1967), 595-597. [13] j. schu, weak and strong convergence to fixed points of asymptotically nonexpansive mappings, bull. austral. math. soc. 43 (1991), 153–159. https://doi.org/10.1017/s0004972700028884. [14] h.f. senter, w.g. dotson, approximating fixed points of nonexpansive mappings, proc. amer. math. soc. 44 (1974), 375–380. https://doi.org/10.1090/s0002-9939-1974-0346608-8. [15] s.m. soltuz, data dependence of mann iteration, octogon math. mag. 9 (2001), 825-828. https://dl.acm. org/doi/10.5555/605858.605878. [16] v. berinde, iterative approximation of fixed points, springer, berlin, (2007). [17] a. qayyum, a weighted ostrowski-grüss type inequality of twice differentiable mappings and applications, int. j. math. comput. 1 (2008), 63-71. [18] m.m. saleem, z. ullah, t. abbas, m.b. raza, a. qayyum, a new ostrowski’s type inequality for quadratic kernel, int. j. anal. appl. 20 (2022), 28. https://doi.org/10.28924/2291-8639-20-2022-28. [19] t. hussain, m.a. mustafa, a. qayyum, a new version of integral inequalities of a linear function of bounded variation, turk. j. inequal. 6 (2022), 7-16. [20] j. amjad, a. qayyum, s. fahad, m. arslan, some new generalized ostrowski type inequalities with new error bounds, innov. j. math. 1 (2022), 30–43. https://doi.org/10.55059/ijm.2022.1.2/23. https://doi.org/10.1016/j.jmaa.2010.08.069 https://doi.org/10.3390/math7050390 https://doi.org/10.1016/j.jmaa.2008.08.045 https://doi.org/10.1016/j.jmaa.2008.08.045 https://doi.org/10.1016/j.jmaa.2010.08.069 https://doi.org/10.1007/s40840-014-0044-6 https://doi.org/10.1155/2019/1356918 https://doi.org/10.3934/math.2023149 https://doi.org/10.1017/s0004972700028884 https://doi.org/10.1090/s0002-9939-1974-0346608-8 https://dl.acm.org/doi/10.5555/605858.605878 https://dl.acm.org/doi/10.5555/605858.605878 https://doi.org/10.28924/2291-8639-20-2022-28 https://doi.org/10.55059/ijm.2022.1.2/23 1. introduction 2. preliminaries 3. some basic results 4. main results 5. convergence results of suzuki generalized non-expansive mappings of condition ( c) 6. conclusion references int. j. anal. appl. (2023), 21:16 pairwise semiregular properties on generalized pairwise lindelöf spaces zabidin salleh∗ department of mathematics, faculty of ocean engineering technology and informatics, universiti malaysia terengganu, 21030 kuala nerus, terengganu, malaysia ∗corresponding author: zabidin@umt.edu.my abstract. let (x,τ1,τ2) be a bitopological space and ( x,τs(1,2),τ s (2,1) ) its pairwise semiregularization. then a bitopological property p is called pairwise semiregular provided that (x,τ1,τ2) has the property p if and only if ( x,τs(1,2),τ s (2,1) ) has the same property. in this work we study pairwise semiregular property of (i, j)-nearly lindelöf, pairwise nearly lindelöf, (i, j)-almost lindelöf, pairwise almost lindelöf, (i, j)-weakly lindelöf and pairwise weakly lindelöf spaces. we prove that (i, j)-almost lindelöf, pairwise almost lindelöf, (i, j)-weakly lindelöf and pairwise weakly lindelöf are pairwise semiregular properties, on the contrary of each type of pairwise lindelöf space which are not pairwise semiregular properties. 1. introduction semiregular properties in topological spaces have been studied by many topologist. some of them related to this research studied by mrsevic et al. [14, 15] and fawakhreh and kılıçman [3]. but in bitopological space, the study of this topic is still open for investigation. the purpose of this paper is to study pairwise semiregular properties on generalized pairwise lindelöf spaces, that we have studied in [9,13,16,17], namely, (i, j)-nearly lindelöf, pairwise nearly lindelöf, (i, j)-almost lindelöf, pairwise almost lindelöf, (i, j)-weakly lindelöf and pairwise weakly lindelöf spaces. in 2010, salleh and kılıçman [19] studied the pairwise semiregular properties of (i, j)-almost regularlindelöf, pairwise almost regular-lindelöf, (i, j)-weakly regular-lindelöf and pairwise weakly regularlindelöf spaces. they also show that the (i, j)-nearly regular-lindelöf and pairwise nearly-regularlindelöf spaces are pairwise semiregular invariant properties. received: jan. 4, 2023. 2020 mathematics subject classification. 54a05, 54a10, 54d20, 54e55. key words and phrases. bitopological space; (i, j)-nearly lindelöf; (i, j)-almost lindelöf; (i, j)-weakly lindelöf; pairwise semiregular property. https://doi.org/10.28924/2291-8639-21-2023-16 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-16 2 int. j. anal. appl. (2023), 21:16 the main results is that the lindelöf, b-lindelöf, s-lindelöf and p-lindelöf spaces are not pairwise semiregular properties. while (i, j)-almost lindelöf, pairwise almost lindelöf, (i, j)-weakly lindelöf and pairwise weakly lindelöf spaces are pairwise semiregular properties. we also show that (i, j)-nearly lindelöf and pairwise nearly lindelöf spaces are satisfying pairwise semiregular invariant properties. 2. preliminaries throughout this paper, all spaces (x,τ) and (x,τ1,τ2) (or simply x) are always mean topological spaces and bitopological spaces, respectively unless explicitly stated. if p is a topological property, then ( τ i,τ j ) -p denotes an analogue of this property for τ i has property p with respect to τ j, and p-p denotes the conjunction (τ1,τ2)-p∧(τ2,τ1)-p, i.e., p-p denotes an absolute bitopological analogue of p. also note that (x,τ i ) has a property p ⇐⇒ (x,τ1,τ2) has a property τ i-p. sometimes the prefixes ( τ i,τ j ) or τ iwill be replaced by (i, j)or irespectively, if there is no chance for confusion. by i-open cover of x, we mean that the cover of x by i-open sets in x; similar for the (i, j)-regular open cover of x etc. by i-int (a) and i-cl (a), we shall mean the interior and the closure of a subset a of x with respect to topology τ i, respectively. in this paper always i, j ∈ {1, 2} and i 6= j. the reader may consult [2] for the detail notations. the following are some basic concepts. definition 2.1. [6,20] a subset s of a bitopological space (x,τ1,τ2) is said to be (i, j)-regular open (resp. (i, j)-regular closed) if i-int (jcl (s)) = s (resp. i-cl (jint (s)) = s), where i, j ∈{1, 2} , i 6= j. s is called pairwise regular open (resp. pairwise regular closed) if it is both (1, 2)-regular open and (2, 1)-regular open (resp. (1, 2)-regular closed and (2, 1)-regular closed). definition 2.2. [6,21] a bitopological space (x,τ1,τ2) is said to be (i, j)-almost regular if for each point x ∈ x and for each (i, j)-regular open set v containing x, there exists an (i, j)-regular open set u such that x ∈ u ⊆ j-cl (u) ⊆ v . x is called pairwise almost regular if it is both (1, 2)-almost regular and (2, 1)-almost regular. in any bitopological space (x,τ1,τ2), the family of all (i, j)-regular open sets is closed under finite intersections. thus the family of (i, j)-regular open sets in any bitopological space (x,τ1,τ2) forms a base for a coarser topology called (i, j)-semiregularization of (x,τ1,τ2), which is defined as follows. definition 2.3. [16] the topology generated by the (i, j)-regular open subsets of (x,τ1,τ2) is denoted by τs (i,j) and it is called (i, j)-semiregularization of x. the topologies is pairwise semiregularization of x if the first topology is (1, 2)-semiregularization of x and the second topology is (2, 1)-semiregularization of x. if τ i ≡ τs(i,j), then x is said to be (i, j)-semiregular. (x,τ1,τ2) is called pairwise semiregular if it is both (1, 2)-semiregular and (2, 1)-semiregular, that is, whenever int. j. anal. appl. (2023), 21:16 3 τ i ≡ τs(i,j) for each i, j ∈{1, 2} and i 6= j. in other words, (x,τ1,τ2) is (i, j)-semiregular if the family of (i, j)-regular open sets form a base for the topology τ i. it is very clear that τs (i,j) ⊆ τ i, but it is not necessary τ i ⊆ τs(i,j). thus with every given bitopological space (x,τ1,τ2) there is associated another bitopological space ( x,τs (1,2) ,τs (2,1) ) in the manner described above (see [20]). we provide the following example in order to understand the concept of pairwise semiregular spaces clearly. example 2.1. for the set of all real numbers r, let τu denotes the usual topology and τs denote the sorgenfrey topology, i.e., topology generated by right half-open intervals (see [22]). then (r,τu,τs) is (τu,τs)-semiregular since τu = τs(τu,τs), i.e., τu generated by (τu,τs)-regular open subsets of r. (r,τu,τs) is also (τs,τu)-semiregular since τs = τs(τs,τu) because any set e ∈ τs is the union of a collection of (τs,τu)-regular open sets in r. thus (r,τu,τs) is pairwise semiregular. khedr and alshibani [6] defined the equivalent definition of (i, j)-semiregular spaces as follows. definition 2.4. a bitopological space x is said to be (i, j)-semiregular if for each x ∈ x and for each i-open subset v of x containing x, there is an i-open set u such that x ∈ u ⊆ i-int (jcl (u)) ⊆ v . x is called pairwise semiregular if it is both (1, 2)-semiregular and (2, 1)-semiregular. definition 2.5. [1] a bitopological space (x,τ1,τ2) is said to be (i, j)-extremally disconnected if the i-closure of every j-open set is j-open. x is called pairwise extremally disconnected if it is both (1, 2)-extremally disconnected and (2, 1)-extremally disconnected. recall that a property p will be called bitopological property (resp. p-topological property) if whenever (x,τ1,τ2) has property p, then every space homeomorphic (resp. p-homeomorphic) to (x,τ1,τ2) also has property p (see [8]). if a bitopological space x has bitopological (or p-topological) property p, one may ask, does the pairwise semiregularization of x satisfies the property p also? now we arrive to the concept of pairwise semiregular property. definition 2.6. let (x,τ1,τ2) be a bitopological space and let ( x,τs (1,2) ,τs (2,1) ) its pairwise semiregularization. a bitopological property p is called pairwise semiregular provided that (x,τ1,τ2) has the property p if and only if ( x,τs (1,2) ,τs (2,1) ) has the property p. lemma 2.1. [16] let (x,τ1,τ2) be a bitopological space and let ( x,τs (1,2) ,τs (2,1) ) its pairwise semiregularization. then (a) τ i-int (c) = τs(i,j)-int (c) for every τ j-closed set c; (b) τ i-cl (a) = τs(i,j)-cl (a) for every a ∈ τ j; (c) the family of ( τ i,τ j ) -regular open sets of (x,τ1,τ2) are the same as the family of ( τs (i,j) ,τs (j,i) ) regular open sets of ( x,τs (1,2) ,τs (2,1) ) ; 4 int. j. anal. appl. (2023), 21:16 (d) the family of ( τ i,τ j ) -regular closed sets of (x,τ1,τ2) are the same as the family of ( τs (i,j) ,τs (j,i) ) regular closed sets of ( x,τs (1,2) ,τs (2,1) ) ; (e) ( τs (i,j) )s (i,j) = τs (i,j) . 3. pairwise semiregularization of pairwise lindelöf spaces definition 3.1. [5, 7]. a bitopological space (x,τ1,τ2) is said to be i-lindelöf if the topological space (x,τ i ) is lindelöf. x is called lindelöf if it is i-lindelöf for each i = 1, 2. in other words, (x,τ1,τ2) is called lindelöf if the topological space (x,τ1) and (x,τ2) are both lindelöf. note that i-lindelöf property as well as lindelöf property is not a pairwise semiregular property by the following example. example 3.1. let x be a set with cardinality 2c, where c = card (r). let τ1 be a co-c topology on x consisting of ∅ and all subsets of x whose complements have cardinality at most c and let τ2 be a cofinite topology on x. then (x,τ1,τ2) is τ2-lindelöf but not τ1-lindelöf and hence not lindelöf. observe that ( x,τs (1,2) ,τs (2,1) ) is τs (1,2) -lindelöf and τs (2,1) -lindelöf since τs (1,2) and τs (2,1) are indiscrete topologies. hence ( x,τs (1,2) ,τs (2,1) ) is lindelöf. definition 3.2. a bitopological space (x,τ1,τ2) is called (i, j)-lindelöf [5,7] if for every i-open cover of x there is a countable j-open subcover. x is called b-lindelöf [5] or p1-lindelöf [7] if it is both (1, 2)-lindelöf and (2, 1)-lindelöf. an (i, j)-lindelöf property as well as b-lindelöf property is not pairwise semiregular property by the following example. example 3.2. let (x,τ1,τ2) be a bitopological space as in example 3.1. then (x,τ1,τ2) is not (τ1,τ2)-lindelöf but it is (τ2,τ1)-lindelöf and hence not b-lindelöf. observe that ( x,τs (1,2) ,τs (2,1) ) is ( τs (1,2) ,τs (2,1) ) -lindelöf and ( τs (2,1) ,τs (1,2) ) -lindelöf since τs (1,2) and τs (2,1) are indiscrete topologies. hence ( x,τs (1,2) ,τs (2,1) ) is b-lindelöf. definition 3.3. a cover u of a bitopological space (x,τ1,τ2) is called τ1τ2-open [23] if u ⊆ τ1∪τ2. if, in addition, u contains at least one nonempty member of τ1 and at least one nonempty member of τ2, it is called p-open [4]. definition 3.4. [5] a bitopological space (x,τ1,τ2) is called s-lindelöf (resp. p-lindelöf) if every τ1τ2-open (resp. p-open) cover of x has a countable subcover. a p-lindelöf property is not pairwise semiregular property by the following example. thus the s-lindelöf property is also not pairwise semiregular property. int. j. anal. appl. (2023), 21:16 5 example 3.3. let (x,τ1,τ2) be a bitopological space as in example 3.1. then (x,τ1,τ2) is not p-lindelöf and hence not s-lindelöf. observe that ( x,τs (1,2) ,τs (2,1) ) is p-lindelöf and s-lindelöf since τs (1,2) and τs (2,1) are indiscrete topologies. 4. pairwise semiregularization of generalized pairwise lindelöf spaces definition 4.1. [9, 13, 16] a bitopological space x is said to be (i, j)-nearly lindelöf (resp. (i, j)-almost lindelöf, (i, j)-weakly lindelöf) if for every i-open cover {uα : α ∈ ∆} of x, there exists a countable subset {αn : n ∈n} of ∆ such that x = ⋃ n∈n i-int (jcl (uαn ))( resp. x = ⋃ n∈n jcl (uαn ) , x = jcl (⋃ n∈n (uαn ) )) . x is called pairwise nearly lindelöf (resp. pairwise almost lindelöf, pairwise weakly lindelöf) if it is both (1, 2)-nearly lindelöf (resp. (1, 2)-almost lindelöf, (1, 2)-weakly lindelöf) and (2, 1)-nearly lindelöf (resp. (2, 1)-almost lindelöf, (2, 1)-weakly lindelöf). our first result is analogue with the result of mršević et al. [15, theorem 1]. theorem 4.1. a bitopological space (x,τ1,τ2) is ( τ i,τ j ) -nearly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. proof. let (x,τ1,τ2) be a ( τ i,τ j ) -nearly lindelöf and let {uα : α ∈ ∆} be a τs(i,j)-open cover of( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx and since for each αx ∈ ∆,uαx ∈τs(i,j), there exists a ( τ i,τ j ) -regular open set vαx in (x,τ1,τ2) such that x ∈ vαx ⊆ uαx . so x = ⋃ x∈x vαx and hence {vαx : x ∈ x} is a ( τ i,τ j ) -regular open cover of x. since (x,τ1,τ2) is ( τ i,τ j ) -nearly lindelöf, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = ⋃ n∈n vαxn ⊆ ⋃ n∈n uαxn . this shows that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. conversely, suppose that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf and let {vα : α ∈ ∆} be a ( τ i,τ j ) regular open cover of (x,τ1,τ2). since vα ∈ τs(i,j) for each α ∈ ∆,{vα : α ∈ ∆} is a τ s (i,j) -open cover of ( x,τs (1,2) ,τs (2,1) ) . since ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf, there exists a countable subcover such that x = ⋃ n∈n vαn. this implies that (x,τ1,τ2) is ( τ i,τ j ) -nearly lindelöf. � corollary 4.1. a bitopological space (x,τ1,τ2) is pairwise nearly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is lindelöf. proposition 4.1. a bitopological space ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -nearly lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. proof. the sufficient condition is obvious by the definitions. so we need only to prove necessary condition. suppose that {uα : α ∈ ∆} is a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -semiregular, there exists a 6 int. j. anal. appl. (2023), 21:16 τs (i,j) -open set vαx in ( x,τs (1,2) ,τs (2,1) ) such that x ∈ vαx ⊆ τs(i,j)-int ( τs (j,i) cl (vαx ) ) ⊆ uαx . hence x = ⋃ x∈x vαx and thus the family {vαx : x ∈ x} forms a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -nearly lindelöf, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = ⋃ n∈n τs (i,j) -int ( τs (j,i) cl ( vαxn )) ⊆ ⋃ n∈n uαxn . this shows that( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. � corollary 4.2. a bitopological space ( x,τs (1,2) ,τs (2,1) ) is pairwise nearly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is lindelöf. from the definition 2.6, if the property p is not bitopological property but it satisfies the condition (x,τ1,τ2) has the property p if and only if ( x,τs (1,2) ,τs (2,1) ) has the property p, then the property p will be called pairwise semiregular invariant property. the following theorem prove that (i, j)nearly lindelöf as well as pairwise nearly lindelöf property satisfying the pairwise semiregular invariant property since (i, j)-nearly lindelöf and pairwise nearly lindelöf are not i-topological property [8] and bitopological property, respectively. this is because the i-continuity and (i, j)-δ-continuity (resp. continuity and p-δ-continuity) are independent notions (see [12]). theorem 4.2. a bitopological space (x,τ1,τ2) is ( τ i,τ j ) -nearly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -nearly lindelöf. proof. it is obvious by theorem 4.1 and proposition 4.1. � corollary 4.3. a bitopological space (x,τ1,τ2) is pairwise nearly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is pairwise nearly lindelöf. theorem 4.3. [20] if (x,τ1,τ2) is pairwise semiregular, then (x,τ1,τ2) = ( x,τs (1,2) ,τs (2,1) ) . the converse of theorem 4.3 is also true by the definitions. proposition 4.2. let (x,τ1,τ2) be a pairwise semiregular space. then (x,τ1,τ2) is (i, j)-nearly lindelöf if and only if it is i-lindelöf. proof. by theorem 4.3, (x,τ1,τ2) = ( x,τs (1,2) ,τs (2,1) ) . the result follows immediately by proposition 4.1. � corollary 4.4. let (x,τ1,τ2) be a pairwise semiregular space. then (x,τ1,τ2) is pairwise nearly lindelöf if and only if it is lindelöf. unlike all types of pairwise lindelöf properties, the (i, j)-almost lindelöf, pairwise almost lindelöf, (i, j)-weakly lindelöf and pairwise weakly lindelöf properties are pairwise semiregular properties as we prove in the following theorems. int. j. anal. appl. (2023), 21:16 7 theorem 4.4. a bitopological space (x,τ1,τ2) is ( τ i,τ j ) -almost lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -almost lindelöf. proof. let (x,τ1,τ2) be a ( τ i,τ j ) -almost lindelöf and let {uα : α ∈ ∆} be a τs(i,j)-open cover of( x,τs (1,2) ,τs (2,1) ) . since τs (i,j) ⊆ τ i, {uα : α ∈ ∆} is a τ i-open cover of the ( τ i,τ j ) -almost lindelöf space (x,τ1,τ2). then there is a countable subset {αn : n ∈n} of ∆ such that x = ⋃ n∈n τ jcl (uαn ). by lemma 2.1, we have x = ⋃ n∈n τs (j,i) -cl (uαn ), which implies ( x,τs (1,2) ,τs (2,1) ) is( τs (i,j) ,τs (j,i) ) -almost lindelöf. conversely suppose that ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -almost lindelöf and let {vα : α ∈ ∆} be a τ i-open cover of (x,τ1,τ2). since vα ⊆ τ i-int ( τ jcl (vα) ) and τ i-int ( τ jcl (vα) ) ∈ τs (i,j) , we have { τ iint ( τ jcl (vα) ) : α ∈ ∆ } is a τs (i,j) -open cover of the ( τs (i,j) ,τs (j,i) ) -almost lindelöf space( x,τs (1,2) ,τs (2,1) ) . so there is a countable subset {αn : n ∈n} of ∆ such that x = ⋃ n∈n τs (j,i) cl ( τ iint ( τ jcl (vαn ) )) . by lemma 2.1, we have x = ⋃ n∈n τ j-cl ( τ iint ( τ jcl (vαn ) )) ⊆ ⋃ n∈n τ j-cl (vαn ). this implies that (x,τ1,τ2) is ( τ i,τ j ) -almost lindelöf. � corollary 4.5. a bitopological space (x,τ1,τ2) is pairwise almost lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is pairwise almost lindelöf. note that, the (i, j)-almost lindelöf property and the pairwise almost lindelöf property are both bitopological properties (see [18]). utilizing this fact, theorem 4.4 and corollary 4.5, we easily obtain the following corollary. corollary 4.6. the (i, j)-almost lindelöf property and the pairwise almost lindelöf property are both pairwise semiregular properties. proposition 4.3. let (x,τ1,τ2) be a ( τ i,τ j ) -almost regular space. then (x,τ1,τ2) is ( τ i,τ j ) almost lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. proof. let (x,τ1,τ2) be a ( τ i,τ j ) -almost lindelöf and let {uα : α ∈ ∆} be a τs(i,j)-open cover of( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx and since uαx ∈ τs(i,j), there exists a ( τ i,τ j ) -regular open set vαx in (x,τ1,τ2) such that x ∈ vαx ⊆ uαx . since (x,τ1,τ2) is( τ i,τ j ) -almost regular, there is a ( τ i,τ j ) -regular open set cαx in (x,τ1,τ2) such that x ∈ cαx ⊆ τ jcl (cαx ) ⊆ vαx . hence x = ⋃ x∈x cαx and thus the family {cαx : x ∈ x} forms a ( τ i,τ j ) -regular open cover of (x,τ1,τ2). since (x,τ1,τ2) is ( τ i,τ j ) -almost lindelöf, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = ⋃ n∈n τ j-cl ( cαxn ) ⊆ ⋃ n∈n vαxn ⊆ ⋃ n∈n uαxn . this shows that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. conversely, let ( x,τs (1,2) ,τs (2,1) ) be a τs (i,j) -lindelöf and let {uα : α ∈ ∆} be a τ i-open cover of (x,τ1,τ2). since uα ⊆ τ i-int ( τ jcl (uα) ) and τ iint ( τ jcl (uα) ) ∈ τs (i,j) , { τ iint ( τ jcl (uα) ) : α ∈ ∆ } is τs (i,j) -open cover of the τs (i,j) -lindelöf space 8 int. j. anal. appl. (2023), 21:16( x,τs (1,2) ,τs (2,1) ) . then there exists a countable subset {αn : n ∈n} of ∆ such that x = ⋃ n∈n τ iint ( τ jcl (uαn ) ) ⊆ ⋃ n∈n τ j-cl (uαn ). this implies that (x,τ1,τ2) is ( τ i,τ j ) -almost lindelöf. � corollary 4.7. let (x,τ1,τ2) be a pairwise almost regular space. then (x,τ1,τ2) is pairwise almost lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is lindelöf. proposition 4.4. let ( x,τs (1,2) ,τs (2,1) ) be a ( τs (j,i) ,τs (i,j) ) -extremally disconnected space. then( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -almost lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. proof. the sufficient condition is obvious by the definitions. so we need only to prove necessary condition. suppose that {uα : α ∈ ∆} is a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) semiregular, there exists a τs (i,j) -open set vαx in ( x,τs (1,2) ,τs (2,1) ) such that x ∈ vαx ⊆ τs(i,j)int ( τs (j,i) cl (vαx ) ) ⊆ uαx . hence x = ⋃ x∈x vαx and thus the family {vαx : x ∈ x} forms a τs (i,j) -open cover of ( x,τs (1,2) ,τs (2,1) ) . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -almost lindelöf and( τs (j,i) ,τs (i,j) ) -extremally disconnected, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = ⋃ n∈n τs (j,i) -cl ( vαxn ) = ⋃ n∈n τs (i,j) -int ( τs (j,i) cl ( vαxn )) ⊆ ⋃ n∈n uαxn . this shows that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. � corollary 4.8. let ( x,τs (1,2) ,τs (2,1) ) be a pairwise extremally disconnected space. then( x,τs (1,2) ,τs (2,1) ) is pairwise almost lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is lindelöf proposition 4.5. let (x,τ1,τ2) be a pairwise semiregular and (j, i)-extremally disconnected space. then (x,τ1,τ2) is (i, j)-almost lindelöf if and only if it is i-lindelöf. proof. by theorem 4.3, (x,τ1,τ2) = ( x,τs (1,2) ,τs (2,1) ) . the result follows immediately by proposition 4.4. � corollary 4.9. let (x,τ1,τ2) be a pairwise semiregular and pairwise extremally disconnected space. then (x,τ1,τ2) is pairwise almost lindelöf if and only if it is lindelöf. theorem 4.5. a bitopological space (x,τ1,τ2) is ( τ i,τ j ) -weakly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -weakly lindelöf. proof. the proof is similar to the proof of theorem 4.4 by using the fact that τs(j,i)cl (⋃ n∈n τ iint ( τ jcl (vαn ) )) = τ jcl (⋃ n∈n τ iint ( τ jcl (vαn ) )) ⊆ τ jcl (⋃ n∈n τ jcl (vαn ) ) ⊆ τ jcl (⋃ n∈n vαn ) . int. j. anal. appl. (2023), 21:16 9 thus we choose to omit the details. � corollary 4.10. a bitopological space (x,τ1,τ2) is pairwise weakly lindelöf if and only if( x,τs (1,2) ,τs (2,1) ) is pairwise weakly lindelöf. note that, the (i, j)-weakly lindelöf property and the pairwise weakly lindelöf property are both bitopological properties (see [18]). utilizing this fact, theorem 4.5 and corollary 4.10, we easily obtain the following corollary. corollary 4.11. the (i, j)-weakly lindelöf property and the pairwise weakly lindelöf property are both pairwise semiregular properties. recall that, a bitopological space x is called (i, j)-weak p-space [13] if for each countable family {un : n ∈n} of i-open sets in x, we have j-cl (⋃ n∈n un ) = ⋃ n∈n j-cl (un). x is called pairwise weak p-space if it is both (1, 2)-weak p-space and (2, 1)-weak p-space. proposition 4.6. let (x,τ1,τ2) be a ( τ i,τ j ) -almost regular and ( τ i,τ j ) -weak p-space. then (x,τ1,τ2) is ( τ i,τ j ) -weakly lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. proof. necessity: let {uα : α ∈ ∆} be a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx and since uαx ∈ τs(i,j) for each αx ∈ ∆, there exists a( τ i,τ j ) -regular open set vαx in (x,τ1,τ2) such that x ∈ vαx ⊆ uαx . since (x,τ1,τ2) is ( τ i,τ j ) almost regular, there is a ( τ i,τ j ) -regular open set cαx in (x,τ1,τ2) such that x ∈ cαx ⊆ τ jcl (cαx ) ⊆ vαx . hence x = ⋃ x∈x cαx and thus the family {cαx : x ∈ x} forms a ( τ i,τ j ) -regular open cover of (x,τ1,τ2). since (x,τ1,τ2) is ( τ i,τ j ) -weakly lindelöf and ( τ i,τ j ) -weak p-space, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = τ j-cl (⋃ n∈n cαxn ) = ⋃ n∈n τ jcl ( cαxn ) ⊆ ⋃ n∈n vαxn ⊆ ⋃ n∈n uαxn . this shows that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. sufficiency: let {uα : α ∈ ∆} be a τ i-open cover of (x,τ1,τ2). since uα ⊆ τ i-int ( τ jcl (uα) ) and τ i-int ( τ jcl (uα) ) ∈ τs (i,j) , { τ iint ( τ jcl (uα) ) : α ∈ ∆ } is τs (i,j) -open cover of the τs (i,j) -lindelöf space ( x,τs (1,2) ,τs (2,1) ) . then there exists a countable subset {αn : n ∈n} of ∆ such that x =⋃ n∈n τ i-int ( τ jcl (uαn ) ) ⊆ ⋃ n∈n τ j-cl (uαn ) = τ j-cl (⋃ n∈n uαn ) . this implies that (x,τ1,τ2) is( τ i,τ j ) -weakly lindelöf. � corollary 4.12. let (x,τ1,τ2) be a pairwise almost regular and pairwise weak p-space. then (x,τ1,τ2) is pairwise weakly lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is lindelöf. proposition 4.7. let ( x,τs (1,2) ,τs (2,1) ) be a ( τs (j,i) ,τs (i,j) ) -extremally disconnected and ( τs (i,j) ,τs (j,i) ) weak p-space. then ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -weakly lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. 10 int. j. anal. appl. (2023), 21:16 proof. the sufficient condition is obvious by the definitions. so we need only to prove necessary condition. suppose that {uα : α ∈ ∆} is a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . for each x ∈ x, there exists αx ∈ ∆ such that x ∈ uαx . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -semiregular, there exists a τs (i,j) -open set vαx in ( x,τs (1,2) ,τs (2,1) ) such that x ∈ vαx ⊆ τs(i,j)-int ( τs (j,i) cl (vαx ) ) ⊆ uαx . hence x = ⋃ x∈x vαx and thus the family {vαx : x ∈ x} forms a τs(i,j)-open cover of ( x,τs (1,2) ,τs (2,1) ) . since ( x,τs (1,2) ,τs (2,1) ) is ( τs (i,j) ,τs (j,i) ) -weakly lindelöf, ( τs (j,i) ,τs (i,j) ) -extremally disconnected and( τs (i,j) ,τs (j,i) ) -weak p-space, there exists a countable subset of points x1, . . . ,xn, . . . of x such that x = τs (j,i) -cl (⋃ n∈n vαxn ) = ⋃ n∈n τs (j,i) -cl ( vαxn ) = ⋃ n∈n τs (i,j) -int ( τs (j,i) cl ( vαxn )) ⊆ ⋃ n∈n uαxn . this shows that ( x,τs (1,2) ,τs (2,1) ) is τs (i,j) -lindelöf. � corollary 4.13. let ( x,τs (1,2) ,τs (2,1) ) be a pairwise extremally disconnected and pairwise weak pspace. then ( x,τs (1,2) ,τs (2,1) ) is pairwise weakly lindelöf if and only if ( x,τs (1,2) ,τs (2,1) ) is lindelöf. proposition 4.8. let (x,τ1,τ2) be a pairwise semiregular, (j, i)-extremally disconnected and (i, j)weak p-space. then (x,τ1,τ2) is (i, j)-weakly lindelöf if and only if it is i-lindelöf. proof. by theorem 4.3, (x,τ1,τ2) = ( x,τs (1,2) ,τs (2,1) ) . the result follows immediately by proposition 4.7. � corollary 4.14. let (x,τ1,τ2) be a pairwise semiregular, pairwise extremally disconnected and pairwise weak p-space. then (x,τ1,τ2) is pairwise weakly lindelöf if and only if it is lindelöf. 5. acknowledgement the authors are gratefully acknowledge the ministry of higher education malaysia and universiti malaysia terengganu that this research was partially supported under the fundamental research grant scheme (frgs) project code frgs/1/2021/stg06/umt/02/1 and vote no. 59659. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] m.c. datta, projective bitopological spaces, j. aust. math. soc. 13 (1972), 327–334. https://doi.org/10. 1017/s1446788700013744. [2] b.p. dvalishvili, bitopological spaces: theory, relations with generalized algebraic structures, and applications, north-holland math. stud. 199, elsevier, 2005. [3] a.j. fawakhreh, a. kılıçman, semiregular properties and generalized lindelöf spaces, mat. vesnik. 56 (2004), 77-80. [4] p. fletcher, h.b. hoyle, iii, c.w. patty, the comparison of topologies, duke math. j. 36 (1969), 325-331. https://doi.org/10.1215/s0012-7094-69-03641-2. [5] a.a. fora, h.z. hdeib, on pairwise lindelöf spaces, rev. colombiana mat. 17 (1983), 37-57. https://doi.org/10.1017/s1446788700013744 https://doi.org/10.1017/s1446788700013744 https://doi.org/10.1215/s0012-7094-69-03641-2 int. j. anal. appl. (2023), 21:16 11 [6] f.h. khedr, a.m. al-shibani, on pairwise super continuous mappings in bitopological spaces, int. j. math. math. sci. 14 (1991), 715–722. https://doi.org/10.1155/s0161171291000960. [7] a. kılıçman, z. salleh, on pairwise lindelöf bitopological spaces, topol. appl. 154 (2007), 1600-1607. https: //doi.org/10.1016/j.topol.2006.12.007. [8] a. kılıçman and z. salleh, mappings and pairwise continuity on pairwise lindelöf bitopological spaces, albanian j. math.1(2) (2007), 115-120. [9] a. kılıçman, z. salleh, pairwise almost lindelöf bitopological spaces ii, malaysian j. math. sci. 1 (2007), 227-238. [10] a. kılıçman, z. salleh, pairwise weakly regular-lindelöf spaces, abstr. appl. anal. 2008 (2008), 184243. https: //doi.org/10.1155/2008/184243. [11] a. kılıçman, z. salleh, on pairwise almost regular-lindelöf spaces, sci. math. japon. 70 (2009), 285-298. [12] a. kılıçman, z. salleh, mappings and decompositions of pairwise continuity on pairwise nearly lindelöf spaces, albanian j. math. 4 (2010), 31-47. [13] a. kılıçman, z. salleh, on pairwise weakly lindelöf bitopological spaces, bull. iran. math. soc. 39 (2013), 469486. [14] m. mršević, i. l. reilly, m. k. vamanamurthy, on semi-regularization topologies, j. aust. math. soc. a. 38 (1985), 40–54. https://doi.org/10.1017/s1446788700022588. [15] m. mršević, i.l. reilly, m. k. vamanamurthy, on nearly lindelöf spaces, glasnik math. 21 (1986), 407-414. [16] z. salleh, a. kılıçman, pairwise nearly lindelöf bitopological spaces, far east j. math. sci. 77 (2013), 147-171. [17] z. salleh, a. kılıçman, some results on pairwise almost lindelöf spaces, jp j. geometry topol. 15 (2014), 81-98. [18] z. salleh, a. kılıçman, mappings and decompositions of pairwise continuity on (i, j)-almost lindelöf and (i, j)-weakly lindelöf spaces, proyecciones j. math. 40 (2021), 815-836. https://doi.org/10.22199/issn. 0717-6279-3253. [19] z. salleh and a. kılıçman, pairwise semiregular properties on generalized pairwise regular-lindelöf spaces, istanb. univ., sci. fac., j. math. phys. astron. 3 (2010), 127-136. [20] a.r. singal, s.p. arya, on pairwise almost regular spaces, glasnik math. 6 (1971), 335-343. [21] m.k. singal, a.r. singal, some more separation axioms in bitopological spaces, ann. son. sci. bruxelles. 84 (1970), 207-230. [22] l.a. steen, j.a. seebach jr., counterexamples in topology, 2nd edition, springer-verlag, new york, 1978. [23] j. swart, total disconnectedness in bitopological spaces and product bitopological spaces, nederl. akad. wetensch., proc. ser. a 74 = indag math. 33 (1971), 135-145. https://doi.org/10.1155/s0161171291000960 https://doi.org/10.1016/j.topol.2006.12.007 https://doi.org/10.1016/j.topol.2006.12.007 https://doi.org/10.1155/2008/184243 https://doi.org/10.1155/2008/184243 https://doi.org/10.1017/s1446788700022588 https://doi.org/10.22199/issn.0717-6279-3253 https://doi.org/10.22199/issn.0717-6279-3253 1. introduction 2. preliminaries 3. pairwise semiregularization of pairwise lindelöf spaces 4. pairwise semiregularization of generalized pairwise lindelöf spaces 5. acknowledgement references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 18-22 http://www.etamaths.com positive solutions for multi-order nonlinear fractional systems a. guezane-lakoud and r. khaldi∗ abstract. in this paper, we study the existence of positive solutions for a class of multi-order systems of fractional differential equations with nonlocal conditions. the main tool used is schauder fixed point theorem and upper and lower solutions method. the results obtained are illustrated by a numerical example. 1. introduction recently, the investigation of fractional differential equations attracted more attention since it has many applications in several fields of sciences such as in engineering, physics, chemistry, biology, etc ... [8], [10]. in this work, we use the method of upper and lower solutions to prove the existence of positive solutions for a system of multi-order fractional differential equations with nonlocal boundary conditions, where each equation has an order that may be different from the order of the other equations, that is : (p) { dα 0+ u (t) + f (t,u (t)) = 0, 0 < t < 1, u (0) = u′ (0) = 0, au (1) = bu′ (1) , where the function u = (u1,u2, ...,un), ui : [0, 1] → r, dα0+u (t) = ( dα1 0+ u1 (t) ,d α2 0+ u2 (t) , ...,d αn 0+ un (t) ) , dαi 0+ denotes the reimann-liouville fractional derivative of order αi, 2 < αi < 3, i ∈ {1, ..,n} , n ≥ 2, the function f is such that f (t,u) = (f1 (t,u) , ...,fn (t,u)) , u = (u1,u2, ...,un) , fi ∈ c ([0, 1] ×rn,r+) , a = (a1, ...,an) , b = (b1, ...,bn) ∈ rn. fractional differential systems can arise from sciences problems such population problems, dielectric polarization, electromagnetic waves,...see [3]. many methods are used for the investigation of fractional differential equations, such fixed point theory, lower and upper solutions method, mawhin theory,...see [1], [2], [4], [5], [6], [7], [9], [11]. this paper is organized as follows: in the second section, we state some preliminary materials that will be used later. in section three, we use the upper and lower solutions method to prove the existence of positive solutions for problem (p). finally, we give an example illustrating the obtained results. 2. preliminaries in this section, we recall the basic definitions and lemmas from fractional calculus theory and the details can be found in [7], [10]. received 30th april, 2017; accepted 11th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 34b10, 26a33, 34b15. key words and phrases. fractional rieman-liouville derivative; fractional differential equation; upper and lower solutions method. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 18 positive solutions for multi-order nonlinear fractional systems 19 definition 2.1. the riemann-liouville fractional integrals of order α of a function h is defined as iα0+h (t) = 1 γ (α) ∫ t 0 h (s) (t−s)1−α ds, where γ (α) = ∫∞ 0 e−ttα−1dt is the gamma function, α > 0. definition 2.2. the riemann-liouville derivative of fractional order α > 0 for a function h is defined as dα0+h (t) = 1 γ (n−α) ( d dt )n ∫ t 0 (t−s)n−α−1 h (s) ds, where n = [α] + 1 ([α] denotes the integer part of the real number α). lemma 2.1. for α > 0, the solution of the homogeneous equation dα0+h (t) = 0, is given by h(t) = c1t α−1 + c2t α−2 + ··· + cntα−n, where ci, i = 1, 2, ...,n, are real constants. lemma 2.2. let p, q ≥ 0, h ∈ l1 [0, 1]. then i p 0+ i q 0+ h (t) = i p+q 0+ h (t) = i q 0+ i p 0+ h (t) . 3. main results lemma 3.1. let yi ∈ c ([0, 1] ,r) , i ∈{1, ..,n}. assume that ai > 0 and bi < 0, then for i ∈{1, ..,n}, the linear nonhomogeneous problem (si) =   dαi 0+ ui (t) = −yi (t) , 0 < t < 1, ui (0) = u ′ i (0) = 0, aiui (1) = biu ′ i (1) , (3.1) has the following solution ui (t) = ∫ 1 0 gi (t,s) yi (s) ds, 0 ≤ t ≤ 1,∀i ∈{1, ..,n} (3.2) where gi (t,s) =   −(t−s)αi−1 γ (αi) + tαi−1 (1 −s)αi−2 (ai − bi (αi − 1)) γ (αi − 1) ( ai (1 −s) αi − 1 − bi ) , s ≤ t, tαi−1 (1 −s)αi−2 (ai − bi (αi − 1)) γ (αi − 1) ( ai (1 −s) αi − 1 − bi ) ,s ≥ t. proof. by assuming that ui is a solution of the fractional boundary value problem (p) and using lemma 2.1, we obtain ui (t) = −iαi0+yi (t) + c1t αi−1 + c2t αi−2 + c3t αi−3, (3.3) according to conditions ui (0) = 0 and u ′ i (0) = 0, we obtain c2 = c3 = 0. using the nonlocal condition aiui (1) = biu ′ i (1), it yields c1 = 1 ai − bi (αi − 1) ( aii αi 0+ y (1) − biiαi−10+ y (1) ) . (3.4) substituting c1 in equation 3.3, we get what follows ui (t) = ∫ 1 0 gi (t,s) yi (s) ds, (3.5) where gi is given above. � lemma 3.2. if ai > 0 and bi < 0, i ∈ {1, ..,n} , then the functions gi are nonnegative, continuous and 0 ≤ gi (t,s) ≤ 1 γ (αi) , 0 ≤ s,t ≤ 1,∀i ∈{1, ..,n} , 20 guezane-lakoud, khaldi proof. the proof is direct, we omit it. � let x be the banach space c ([0, 1] ,r) × ...×c ([0, 1] ,r)︸ ︷︷ ︸ n times , equipped with the norm ‖u‖ = ∑i=n i=1 maxt∈[0,1] |ui (t)| . define the integral operator t : x → x by tu = (t1u,t2u,...,tnu) where tiu (t) = ∫ 1 0 gi (t,s) fi (s,u (s)) ds, ∀i ∈{1, ..,n} (3.6) let c = (c1, ...,cn) , d = (d1, ...,dn) ∈ rn+ such that d > c. we recall that for x = (x1, ...,xn) , y = (y1, ...,yn) then x ≤ y means xi ≤ yi, for all i ∈{1, ..,n} and [c,d] = {x = (x1, ...,xn) ,ci ≤ xi ≤ di,∀i ∈{1, ..,n}} . we define the upper and lower control operators u = ( u1, ...,un ) , u = (u1, ...,un) respectively by ui (t,x) = sup{fi (t,y) ,c ≤ y ≤ x} , ui (t,x) = inf {fi (t,y) ,x ≤ y ≤ d} , 0 ≤ t ≤ 1. from the definition of ui and ui we have ui (t,x) ≤ fi (t,x) ≤ ui (t,x) , x ∈ [c,d] , 0 ≤ t ≤ 1, i ∈{1, ..,n} . lemma 3.3. the function u ∈ x is a solution of the system (p) if and only if tiu (t) = ui (t), for all t ∈ [0, 1] , ∀i ∈{1, ...,n} . consequently, the existence of solutions for system (p) can be turned into a fixed point problem in x for the operator t. define the cone k = {u ∈ x,u(t) ≥ 0, 0 ≤ t ≤ 1} . let us make the following hypothesis: (h) there exist two functions θ = ( θ1, ...,θn ) , θ = (θ1, ...,θn) ∈ k, such that c ≤ θ (t) ≤ θ (t) ≤ d, 0 ≤ t ≤ 1 and { θi (t) ≥ ∫ 1 0 gi (t,s) ui ( s,θ (s) ) ds,i ∈{1, ..,n} θi(t) ≤ ∫ 1 0 gi (t,s) ui (s,θ (s)) ds,i ∈{1, ..,n} the functions θ and θ are called respectively upper and lower solutions for problem (p). now we are ready to give the main result for problem (p). theorem 3.1. assume that hypothesis (h) holds and f (t, 0) 6= 0, 0 ≤ t ≤ 1, then the fractional boundary value problem (p) has at least one positive solution u ∈ k satisfying θ (t) ≤ u (t) ≤ θ (t) , 0 ≤ t ≤ 1. proof. clearly, the continuity of the operator t follows from the continuity of f. set ω = { u ∈ k : θ (t) ≤ u (t) ≤ θ (t) , 0 ≤ t ≤ 1 } , then ω is a nonempty, closed and convex subset of x. firstly, we show that t (ω) ⊂ ω. in fact, let u ∈ ω, then by the definition of the control functions and hypothesis (h), it yields tiu (t) = ∫ 1 0 gi (t,s) fi (s,u (s)) ds ≤ ∫ 1 0 gi (t,s) ui ( s,θ (s) ) ds ≤ θi (t) , i ∈{1, ..,n} thus tu (t) ≤ θ (t) , 0 ≤ t ≤ 1. similarly, we get tiu (t) = ∫ 1 0 gi (t,s) fi (s,u (s)) ds ≥ ∫ 1 0 gi (t,s) ui (s,θ (s)) ds ≥ θi (t) , i ∈{1, ..,n} positive solutions for multi-order nonlinear fractional systems 21 from which follows tu (t) ≥ θ (t) , 0 ≤ t ≤ 1 thus t (ω) ⊂ ω. now, we prove that t : ω → x is completely continuous operator. set mi = max{fi (t,u(t)) , 0 ≤ t ≤ 1,u ∈ ω} , then we have |tiu (t)| ≤ ∫ 1 0 gi (t,s) fi (s,u (s)) ds ≤ mi γ (αi) . taking the supremum over [0, 1], then summing the obtained inequalities according to i from 1 to n, we get ‖tu‖≤ n∑ i=1 mi γ (αi) , which implies that t (ω) is uniformly bounded. let us show that (tu) is equicontinuous. indeed, let u ∈ ω and 0 ≤ t1 < t2 ≤ 1, then |tiu (t1) −tiu (t2)| ≤ ∫ 1 0 |gi (t1,s) −gi (t2,s)|fi (s,u (s)) ds ≤ mi [∫ t1 0 |gi (t1,s) −gi (t2,s)|ds + ∫ t2 t1 |gi (t1,s) −gi (t2,s)|ds + ∫ 1 t2 |gi (t1,s) −gi (t2,s)|ds ] by computation, we get |tiu (t1) −tiu (t2)| ≤ mi ( (t2 − t1) (αi − 1) γ (αi) + (t2 − t1) αi−1 γ (αi) + 3 ( tαi−12 − t αi−1 1 ) ai − bi (αi − 1) ( ai γ (αi) + bi γ (αi − 1) )) . as t1 → t2, the right-hand side of the above inequality tends to zero. by ascoli-arzela theorem, we conclude that the operator t : ω → ω is completely continuous. finally, schauder fixed point theorem implies that t has at least one fixed point u ∈ ω and then problem (p) has at least one positive solution in ω. as direct consequence of theorem 3.1, we get the following corollary. � corollary 3.1. assume that fi are continuous, nonnegative, fi (t, 0) 6= 0, 0 ≤ t ≤ 1 and there exist two positive constants li and li such that 0 < li ≤ fi (t,x) ≤ li, x ≥ 0, 0 ≤ t ≤ 1, i ∈{1, ..,n} , (3.7) then problem (p) has at at least one positive solution u ∈ x. furthermore the solution satisfies 0 < li ∫ 1 0 gi (t,s) ds ≤ ui (t) ≤ li ∫ 1 0 gi (t,s) ds, 0 ≤ t ≤ 1, ∀i ∈{1, ..,n} . 22 guezane-lakoud, khaldi proof. from equation 3.7 we have ui (t,x) ≤ li, ui (t,x) ≥ li, 0 ≤ t ≤ 1, x ≥ 0. let us choose   θi (t) = li ∫ 1 0 gi (t,s) ds = li tαi−1 αiγ (αi) ( 1 + 1 + 1 αi ) ≥ ∫ 1 0 gi (t,s) ui ( s,θ (s) ) ds, i ∈{1, ..,n} θi(t) = li ∫ 1 0 gi (t,s) ds = li tαi−1 αiγ (αi) ( 1 + 1 + 1 αi ) ≤ ∫ 1 0 gi (t,s) ui (s,θ (s)) ds, i ∈{1, ..,n} , then the conclusion follows from theorem 3.1. � now, we give an examples to illustrate the usefulness of our main results. example 3.1. consider the following two-dimensional fractional order system (s) =   d 5 2 u1 (t) + ( 1 + 1 1+u1+u2 ) = 0, d 8 3 u2 (t) + (1 + e −u1 ) = 0, u1 (0) = 0, u ′ 1 (0) = 0, u1 (1) −u′1 (0) = 0, u2 (0) = 0, u ′ 2 (0) = 0, u2 (1) −u′2 (0) = 0. we have α = ( 5 2 , 8 3 ) , a1 = a2 = 1, b1 = b2 = −1, f1 (t,u1,u2) = 1 + 11+u1+u2 , f2 (t,u1,u2) = 1 + e −u1 , fi ∈ c ( [0, 1] ×r2,r+ ) , fi (t, 0) 6= 0, and 1 ≤ fi (t,u1,u2) ≤ 2. from corollary 3.1, we conclude the system (s) has at at least one positive solution u ∈ x. furthermore, the solution u satisfies 0.722 15t 3 2 ≤ u1 (t) ≤ 1.444 3t 3 2 0.591 95t 5 3 ≤ u2 (t) ≤ 1.183 9t 5 3 . references [1] b. ahmad, a. alsaedi, existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. fixed point theory appl. 2010 (2010), art. id 364560. [2] b. ahmad, juan j. nieto, riemann–liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. bound value probl. 2011 (2011), art. id 36. [3] y. chai, l. chen, r. wu, inverse projective synchronization between two different hyperchaotic systems with fractional order. j. appl. math. 2012 (2012), article id 762807. [4] m. feng, x. zhang, w. ge, new existence results for higher-order nonlinear fractional differential equation with integral boundary conditions . bound. value probl. 2011 (2011), art. id 720702. [5] a. guezane-lakoud, a. ashyralyev, positive solutions for a system of fractional differential equations with nonlocal integral boundary conditions. differ. equ. dyn. syst., doi: 10.1007/s12591-015-0255-9. [6] j. henderson, s. k. ntouyas, i.k. purnaras, positive solutions for systems of generalized three-point nonlinear boundary value problems. comment. math. univ. carolin. 49 (2008), 79-91. [7] r. khaldi, a. guezane-lakoud, upper and lower solutions method for higher order boundary value problems, progress in fractional differentiation and applications, progr. fract. differ. appl. 3 (1) (2017), 53-57. [8] a. a. kilbas, h. m. srivastava, j. j. trujillo, theory and applications of fractional differential equations. elsevier, amsterdam 2006. [9] s. k. ntouyas, m. obaid, a coupled system of fractional differential equations with nonlocal integral boundary conditions. adv. differ. equ. 2012 (2012), article id 130. [10] i. podlubny, fractional differential equations mathematics in sciences and engineering. academic press, new york 1999. [11] m. rehman, r. khan, a note on boundary value problems for a coupled system of fractional differential equations. comput. math. appl. 61 (2011), 2630-2637. laboratory of advanced materials, faculty of sciences, badji mokhtar-annaba university, p.o. box 12, 23000 annaba, algeria ∗corresponding author: rkhadi@yahoo.fr 1. introduction 2. preliminaries 3. main results references int. j. anal. appl. (2023), 21:13 some results on conditionally sequential absorbing maps in multiplicative metric space t. thirupathi1,∗, v. srinivas2 1department of mathematics, sreenidhi institute of science and technology, hyderabad, telangana, india 2department of mathematics, osmania university, hyderabad, telangana, india ∗corresponding author: thotathirupathi1986@gmail.com abstract. this paper aims to prove two general fixed point theorems in multiplicative metric space (mms) by using reciprocally continuous mappings and conditionally sequential absorbing mappings. further our outcomes are validated by discussing two appropriate examples. 1. introduction one of the most exciting areas of contemporary mathematics is fixed point theory, which is also interesting topic of the analysis. further this topic has became a platform due to its wide applications in pure and applied mathematics. in this connection s. young cho et al [1] proved a common fixed point theorem over a complete metric space. later,many researchers generated results in diffeent spaces. in this process monika verma et al [2] generalized [1] for multiplicative metric space.furthermore some results can be witnessed like [3], [4], [5], [6], [7] [8] and [9] in mms. using the conditions conditionally sequential absorption and reciprocally continuous mappings, the goal of this research is to derive two common fixed point theorems for mms. further two suitable examples are discussed to validate our theorems. 2. preliminaries definition 2.1 let x be a non empty set and d : x × x → r+ then (x,d) is said to be mms if satisfying the following conditions: received: nov. 28, 2022. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. multiplicative metric space; fixed point; reciprocally continuous mapping; conditionally sequential absorbing mapping. https://doi.org/10.28924/2291-8639-21-2023-13 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-13 2 int. j. anal. appl. (2023), 21:13 (i) d(ζ,η)≥ 1 for all ζ,η ∈ x and d(ζ,η)=1 if and only if x = y (ii) d(ζ,η)= d(ζ,η) for all ζ,η ∈ x (iii) d(ζ,η)≤ d(ζ,β).d(β,η) for all ζ,η,β ∈ x (multiplicative triangle inequality). then (x,d) is called mms. the pair of mapping (g,j) of a mms (x,d) is said to be definition 2.2 compatible if lim j→∞ d(gjηj,jgηj) = 1, whenever ηj is a sequence in x such that gηj = jηj = ζ for some ζ ∈ x. definition 2.3 weakly compatible if gζ = jζ for some ζ ∈ x such that ijζ = jiζ. definition 2.4 if there is a coincidence point where the mappings commute then it is said to be occasionally weakly compatible (owc). example 2.4.1 let (x,d) be a mms and ∀η,ζ ∈ x we have (.η,ζ)= e |η−ζ|. now the self mappings g,j are defined on x = [0,∞) and given below g(η)= η+1 2 and j(η)= η 2+1 2 for all η ∈ x. from above η =0,1 are coincidence points for the mappings g,j. at η =0 g(0)= j(0)= 1 2 , gj(0)= g(1 2 )= 3 4 , jg(0)= j(1 2 )= 5 8 . therefore gj(0) 6= jg(0). and also gj(1)= jg(1)=1. resulting that the maps g, j are owc but not weakly compatible. definition 2.5 conditionally sequentially absorbing if whenever a sequence (ζj) satisfying {(ζj): lim j→∞ gζj = lim j→∞ jζj} 6= ∅ then there exists another sequence (ηj) in x with lim j→∞ gηj= lim j→∞ jηj = u for some u ∈ x such that lim j→∞ d(gηj,gjηj)=1 and lim j→∞ d(jηj,jgηj)=1. example 2.5.1 lt (x,d) be an mms and ∀η,ζ ∈ x we have d(η,ζ)= e|η−ζ|. now the self mappings g,j are defined on x = [0,∞) and given below g(η)= { sinη if 0≤ η < π 2 2η2 if 0π 2 ≤ η ≤ π; j(η)= { cosη if 0≤ η < π 2 πη if 0π 2 ≤ η ≤ π; from above η =0, π 2 are coincidence points for the mappings g,j. at η =0 g(0)= j(0)=0, gj(0)= g(0)=0, jg(0)= j(0)=0. int. j. anal. appl. (2023), 21:13 3 therefore gj(0)= jg(0). and also gj(π 2 )= g(π 2 2 )= π 4 2 , jg(π 2 )= g(π2)=2π4, therefore gj(π 2 ) 6= jg(π 2 ). resulting that the maps g, j are not weakly compatible. let (pj)= √ 4 j , for all j ≥ 1. then lim j→∞ gpj = lim j→∞ g( √ 4 j )= lim j→∞ sin( √ 4 j )=0 (2.1) and lim j→∞ jpj = lim j→∞ j( √ 4 j )= lim j→∞ 1−cos( √ 4 j )=1−1=0. (2.2) from (2.1) and (2.2), we get lim j→∞ gpj = lim j→∞ jpj (2.3) from (2.3) implies {(pj) : lim j→∞ gpj = lim j→∞ jpj} 6= ∅. then ∃ another sequence qj = π2 + 5 j , for all j ≥ 1. lim j→∞ gqj = lim j→∞ g( π 2 + 5 j )= lim j→∞ =2( π 2 + 5 j )2 = π2 2 (2.4) and lim j→∞ jqj = lim j→∞ j( π 2 + 5 j )= lim j→∞ π( π 2 + 5 j )= π2 2 . (2.5) from (2.4) and (2.5), we get lim j→∞ gpj = lim j→∞ jpj = π2 2 . (2.6) now lim j→∞ gj(qj)= gj( π 2 + 5 j ) = lim j→∞ g(π(π 2 + 5 j )) = lim j→∞ g(π 2 2 + 5π j )= π 3 2 and lim j→∞ jg(qj)= jg( π 2 + 5 j ) = lim j→∞ j(2((π 2 + 5 j )2)) =π 3 2 . therefore lim j→∞ d(gqj,gjqj)=1 and lim j→∞ d(jqj,jgqj)=1. hence the pair(g,j) is conditionally sequentially absorbing but not weakly compatible. definition 2.6 reciprocally continuous whenever (ηj) is a sequence in x such that lim j→∞ gηj = lim j→∞ jηj = ζ for some ζ ∈ x such that lim j→∞ d(gζ,gjηj)=1 and lim j→∞ d(jζ,jgηj)=1. example 2.5.1 lt (x,d) be a multiplicative metric space and ∀η,ζ ∈ x we have d(η,ζ)= e|η−ζ|. now the self mappings g,j are defined on x = [0,∞) and given below g(η)= { πcosη if 0≤ η < π 2 η2 if 0π 2 ≤ η ≤ π; 4 int. j. anal. appl. (2023), 21:13 j(η)= { πsecη if 0≤ η < π 2 πη if 0π 2 ≤ η ≤ π; from above η =0,π are coincidence points for the mappings g,j. from above at η =0 g(0)= j(0)= π, gj(0)= g(π)= π3, jg(0)= j(π)= π3. therefore gj(0)= jg(0). and also gj(π)= g(π3)= π7, jg(π)= j(π3)= π9. therefore gj(π) 6= jg(π) resulting that the maps g, j are not weakly compatible. let (rj)= π − 3j3 for all j ≥ 1. then lim j→∞ grj = lim j→∞ g(π − 3 j3 )= lim j→∞ (π − 3 j3 )2 = π2 (2.7) and lim j→∞ jrj = lim j→∞ j(π − 3 j3 )= lim j→∞ π(π − 3 j3 )= π2. (2.8) from (2.7) and (2.8), we get lim j→∞ grj = lim j→∞ jrj (2.9) now lim j→∞ gj(rj)= gj(π − 3j3) = limj→∞ g(π(π − 3 j3 )) = π3 and lim j→∞ jg(rj)= jg(π − 3j3) = limj→∞ j(π(π − 3 j3 )) =π4. therefore lim j→∞ d(g(π2),gjrj)=1 and lim j→∞ d(j(π2),jgrj)=1. hence the maps g, j are reciprocally continuous. in [1], the following theorem was established. theorem 2.7 assume that (x,d) is an mms which is complete and the mappings b, s, a, and t are defined on x such that (b1) b(x)⊆ s(x) and a(x)⊆ t(x) (b2) d(au,bv)≤ (max{d(au,su),d(bv,tv), √ [d(au,tv).d(bv,su)],d(su,tv)})p. (max{d(au,su),d(bv,tv)})q.(max{d(au,tv),d(bv,su)})r for all u,v ∈ x, where 0 < h = p+q +2r < 1 (p,q and r are non-ve real numbers). (b3) among the subspaces ax or bx or sx or tx is complete (b4) both the pairs (a,s) and (b,t) are weakly compatible. then the four maps a,b,s and t above share a common single fixed point. now we generalize the above theorem 2.7 as below. int. j. anal. appl. (2023), 21:13 5 3. main result theorem 3.1 assume that (x,d) is an mms which is complete and the mappings a, b, s, and t are defined on x such that (d1) a(x)⊆ t(x) and b(x)⊆ s(x) (d2) d(au,bv)≤ (max{d(au,su),d(bv,tv), √ [d(au,tv).d(bv,su)],d(su,tv)})p. (max{d(au,su),d(bv,tv)})q.(max{d(au,tv),d(bv,su)})r for all u,v ∈ x, where h = p+q +2r and o < h < 1 (p,qandrarenon−verealnumbers). (d3) the pairs (a,s) reciprocally continuous and conditionally sequentially absorbing and (b,t) is occasionally weakly compatible. then the four mappings share a single fixed point which is common in x. proof: by (d1), there is a point here u0 ∈ x such that au0 = tu1 = y1. for this point u1 ∈ x there exists a point u2 in x such that bu1 = su2 = y2 and so on. similarly, we can inductively define bu2j−1 = su2j = y2j;au2j = tu2j+1 = y2j+1 for n =0,1,2, ... we can now show that the sequence {vj} is a cauchy in x. put u = u2j and v = u2j+1 in (d2) then d(v2j+1,v2n+2)= d(au2j,bu2j+1)≤ (max{d(au2j,su2j),d(bu2j+1,tu2j+1), √ [d(au2j,tu2j+1).d(bu2j+1,su2j)],d(su2j,tu2j+1)})p. (max{d(au2j,su2j),d(bu2j+1,tu2j+1)})q.(max{d(au2j,tu2j+1),d(bu2j+1,su2j)})r d(v2j+1,v2n+2)≤ (max{d(v2j+1,v2j),d(v2j+2,v2j+1), √ [d(v2j+1,v2j+1).d(v2j+2,v2j)],d(v2j,v2j+1)})p. (max{d(v2j+1,v2j),d(v2j+2,v2j+1)})q.(max{d(v2j+1,v2j+1),d(v2j+2,v2j)})r d(v2j+1,v2n+2)≤ (max{d(v2j+1,v2j),d(v2j+2,v2j+1), √ [d(v2j+1,v2j+1).d(v2j+1,v2j).d(v2j+1,v2j+2)],d(v2j,v2j+1)})p. (max{d(v2j+1,v2j),d(v2j+2,v2j+1)})q.(max{d(v2j+1,v2j+1),d(v2j+1,v2j).d(v2j+1,v2j+2)})r in the above equation, if d(v2j+2,v2j+1) > d(v2j+1,v2j) for some +ve integer j, then we have d(v2j+1,v2j+2)≤ d(v2j+1,v2j+2)h, where o < h = p+q +2r < 1, a contradiction. therefore we have d(v2j+2,v2j+1)≤ d(v2j,v2j+1)h. likewise, we have d(vj,v2j+1)≤ (d(vj−1,vj)h)≤ (d(vj−2,vj −1)h 2 ≤ .... ≤ (d(v0,v1))h n . let l, j lnn such that l > j, we get d(vl,vj)≤ d(vl,vl−1)....d(vj+1,vj) ≤ (d(v1,v0))h l−1+....hj ≤ (d(v1,v0)) hj 1−h → 1 as l, j →∞. as a result, the sequence {vj} is a cauchy. 6 int. j. anal. appl. (2023), 21:13 by the completeness of x ∃ w ∈ x such that vj → w as j →∞. accordingly, the sequences au2j,su2j → z,tu2j+1,bu2j+1 → z (3.1) as j →∞. use the notion l{a,s}= {(uj) : lim j→∞ auj = lim j→∞ suj}. by (d3) the pair of mapping (a,s) is conditionally sequentially absorbing from (3.1) l{a,s} 6= ∅⇒∃(vj) such that lim j→∞ avj = lim j→∞ svj = ψ (3.2) =⇒ d(avj,asvj)=1andd(svj,savj)=1 (3.3) by the reciprocally continuous of the pair (a,s) implies whenever lim j→∞ avj = lim j→∞ svj = ψ (3.4) =⇒ d(aψ,asvj)=1andd(sψ,savj)=1. (3.5) using (3.2) and (3.5) in (3.3), we get aψ = sψ = ψ. since aψ is an element in a(x) by (d1) there exists ϕ such that ψ = sψ = aψ = tϕ. (3.6) claim bϕ = tϕ. putting u = ψ , v = ϕ in (d2) d(aψ,bϕ)≤ (max{d(aψ,sψ),d(bϕ,tϕ), √ [d(aψ,tϕ).d(bϕ,sψ)],d(sψ,tϕ)})p. (max{d(aψ,sψ),d(bϕ,tϕ)})q.(max{d(aψ,tϕ),d(bϕ,sψ)})r. letting n →∞ we get, d(tϕ,bϕ)≤ (max{d(ψ,ψ),d(bϕ,tϕ), √ [d(ψ,ψ).d(bϕ,tϕ)],d(ψ,ψ)})p. (max{d(ψ,ψ),d(bϕ,tϕ)})q.(max{d(ψ,ψ),d(bϕ,tϕ)})r d(tϕ,bϕ)≤ d(tϕ,bϕ)p+q+r, which is a contradiction. hence tϕ = bϕ. which gives ψ = sψ = aψ = tϕ = bϕ. (3.7) from (d3) we have the pair (b,t) is occasionally weakly compatible which gives btϕ = tbϕ implies that bψ = tψ from (3.7). int. j. anal. appl. (2023), 21:13 7 claim ψ = bψ. putting u = v = ψ in (d2) d(ψ,bψ)≤ (max{d(ψ,ψ),d(tψ,tψ), √ [d(ψ,bψ).d(bψ,ψ)],d(ψ,bψ)})p. (max{d(ψ,ψ),d(tψ,tψ)})q.(max{d(ψ,bψ),d(bψ,ψ)})r d(ψ,bψ)≤ d(ψ,bψ)p+r, a contradiction which impies ψ = bψ. therefore ψ = sψ = aψ = tψ = bψ. which implies that ψ is the required common fixed point. for uniqueness: assume that ρ be the another fixed point then ρ = sρ = aρ = tρ = bρ. putting u = ψ and v = ρ in (d2), we get d(aψ,bρ)≤ (max{d(aψ,sψ),d(bρ,tρ), √ [d(aψ,tρ).d(bρ,sψ)],d(sψ,tρ)})p. (max{d(aψ,sψ),d(bρ,tρ)})q.(max{d(aψ,tρ),d(bρ,sψ)})r d(ψ,ρ)≤ (max{d(ψψ),d(ρ,ρ), √ [d(ψ,ρ).d(ρ,ψ)],d(ψ,ρ)})p. (max{d(ψ,ψ),d(ρ,ρ)})q.(max{d(ψ,ρ),d(ρ,ψ)})r d(ψ,ρ)≤ d(ψ,ρ)p+q+r, a contradiction which implies ψ = ρ. this proves the uniqueness. now we discuss an example. example 3.2 assume that (x,d) is an mms space with d(u,v)= e|u−v| for all u,v ∈ x. a, b, s, and t are the self maps that are defined on x = [0,1] as follows: a(η)= { η2+1 2 if 0≤ η < 1 5 η if 1 5 ≤ η ≤ 1; s(η)= { η2+η+1 2 if 0≤ η < 1 5 η2 if 1 5 ≤ η ≤ 1; b(η)= { η2+4η+1 2 if 0≤ η < 1 5 1 5 if 1 5 ≤ η ≤ 1; t(η)= { η2+3η+1 2 if 0≤ η < 1 5 η if 1 5 ≤ η ≤ 1; now a(x) = [1 2 ,0.52]∪ (1 5 ,1], s(x) = [1 2 ,0.9)∪{1 5 } , b(x) = [1 2 ,0.62]∪{1 5 } and t(x) = [1 2 ,0.52]∪ (1 5 ,1]. clearly a(x)⊆ t(x) and b(x)⊆ s(x) so that (d1) is satisfied. for the pair of mappings (a,s) and (b,t), it is evident that 0 and 1 are coincidence points. at η =0 ⇒ a(0)= s(0)= 1 2 . 8 int. j. anal. appl. (2023), 21:13 but as(0)= a(1 2 )= 1 2 and sa(0)= s(1 2 )= 1 5 . therefore as(0) 6= sa(0). also at η =0 ⇒ b(0)= t(0)= 1 2 . but bt(0)= b(1 2 )= 1 8 and tb(0)= t(1 2 )= 1 2 . therefore bt(0) 6= tb(0). as a result, the mappings (a,s) and (b,t) are not weakly compatible. take a sequence ηk = 3 2k for all k ≥ 0. then lim k→∞ aηk = a( 3 2k )= ( 3 2k )2+1 2 = 1 2 and lim n→∞ sηk = s( 3 2k )= 1 2 . implies lim k→∞ aηk = lim k→∞ sηk = 1 2 . now lim k→∞ asηk= lim k→∞ as( 3 2k )= lim k→∞ a(1 2 + 9+6k 8k2 )= 1 2 and lim k→∞ saηk= lim k→∞ sa( 3 2k2 + 1 2 )= (1 2 + 9 8k2 )2 = 1 4 . therefore the pair (a,s) is non-compatible so that ∃ another sequence βk = 15 + 2 3k for all k ≥ 1. lim k→∞ aβk = lim k→∞ sβk = 1 5 . also we have lim k→∞ asβk= lim k→∞ as(1 5 + 2 3m )= lim k→∞ a(1 5 )= 1 5 and lim k→∞ saβk= lim k→∞ sa(1 5 + 2 3k )= lim k→∞ s(1 5 + 2 3k )= 1 5 . thus from above lim k→∞ d(aβk,asβk)= d( 1 5 , 1 5 )= e| 1 5 −1 5 | =1 and lim k→∞ d(sβk,saβk)= d( 1 5 , 1 5 )= e| 1 5 −1 5 | =1. further lim k→∞ d(asβk,a( 1 5 ))= d(1 5 , 1 5 )= e| 1 5 −1 5 | =1. from the above we can conclude that the pairs (a,s) and (b,t) are non-compatible reciprocally continuous and conditionally sequential absorbing mappings. also it is observed that a(1 5 )= s(1 5 )= b(1 5 )= t(1 5 )= 1 5 . it is found that the only common fixed point shared by the four self-maps is 1 5 . now we prove another generalization of theorem 2.7, as given below. theorem 3.3 assume that (x,d) is an mms which is complete and the mappings a,b,s, and t are defined on x such that (e1) a(x)⊆ t(x) and b(x)⊆ s(x) (e2) d(au,bv)≤ (max{d(au,su),d(bv,tv), √ [d(au,tv).d(bv,su)],d(su,tv)})p. (max{d(au,su),d(bv,tv)})q.(max{d(au,tv),d(bv,su)})r for all u,v ∈ x, where h = p+q +2r and 0 < h < 1 (p,q and r are non-ve real numbers). (e3) the mappings for the pairs (a,s) and (b,t) are non-compatible reciprocally continuous and conditionally sequential absorbing mappings. then the four mappings share a single fixed point which is common in x. int. j. anal. appl. (2023), 21:13 9 proof: by (e3) we have the pair (a,s) non-compatible =⇒ there is a sequence (uj) with lim j→∞ auj = lim j→∞ suj = ψ (3.8) for some ψ ∈ x. =⇒ limj→∞ d(asuj,sauj) not exist or limj→∞ d(asuj,sauj) 6=1. considering that the pair (a,s) is conditionally sequentially absorbing from (3.8) we have l{a,s} 6= ∅=⇒∃(vj) such that lim j→∞ avj = lim j→∞ svj = ψ (say) =⇒ limj→∞ d(avj,asvj)=1 and limj→∞ d(svj,savj)=1. also from (e3) we have (a,s) is reciprocally continuous means whenever lim j→∞ avj = lim j→∞ svj = ψ. (3.9) =⇒ limj→∞ d(aψ,asvj)=1 and limj→∞ d(sψ,savj)=1. using the above equations, we get aψ = sψ = ψ. (3.10) since the pair (b,t) is non compatible implies there is sequence (uj) with lim j→∞ buj = lim j→∞ tuj = ϕ (3.11) for some ϕ ∈ x. =⇒ limj→∞ d(btuj,tbuj)1 not exist or limj→∞ d(btuj,tbuj) 6=1. from (e3) the pair (b,t) is conditionally sequential absorbing from (3.11) l{b,t} 6= ∅=⇒∃(vj) such that lim j→∞ bvj = lim j→∞ tvj = β (say) =⇒ limj→∞ d(bvj,btvj)=1 and limj→∞ d(tvj,tbvj)=1. also the pair (b,t) is reciprocally continuous implies whenever lim j→∞ bvj = lim j→∞ tvj = β (say) =⇒ limj→∞ d(bβ,btvj)=1 and limj→∞ d(tβ,tbvj)=1. using the above equation, we get bβ = tβ = β. (3.12) claim β = ψ. assume that β 6= ψ. putting u = ψ and v = β in (e2) d(aψ,bβ)≤ (max{d(aψ,sψ),d(bβ,tβ), √ [d(aψ,tβ).d(bβ,sψ)],d(sψ,tβ)})p. (max{d(aψ,sψ),d(bβ,tβ)})q.(max{d(aψ,tβ),d(bβ,sψ)})r. letting as n →∞ 10 int. j. anal. appl. (2023), 21:13 d(ψ,β)≤ (max{d(ψ,ψ),d(β,β), √ [d(ψ,β).d(β,ψ)],d(ψ,β)})p. (max{d(ψ,ψ),d(β,β)})q.(max{d(ψ,β),d(β,ψ)})r. d(ψ,β)≤ d(ψ,β)p+r. a contradiction hence ψ = β. therefore aψ = sψ = bψ = tψ = ψ. which implies that ψ is the required unique common fixed point. from (e2) uniqueness follows easily. now we give an example to support theorem 3.3. example 3.4 assume that (x,d) is a mms space with d(u,v)= e|u−v| for all u,v ∈ x. a, b, s, and t are the self maps that are defined on x = [0,12] as follows: a(η)= { η4 if 0≤ η ≤ 1 2 if 1 < η ≤ 12; s(η)= { η2 if 0≤ η ≤ 1 2logη if 1 < η ≤ 12; b(η)= { η5 if 0≤ η ≤ 1 4 if 1 < η ≤ 12; t(η)= { η3 if 0≤ η ≤ 1 4logη if 1 < η ≤ 12. now a(x)= [0,1]∪{2}, s(x)= [0,4.96], b(x)= [0,1]∪4 and t(x)= [0,5.45]. we have from above maps η = e and 1 are coincidence points for (a,s) and (b,t). at η = e , a(e)= s(e)=2 and b(e)= t(e)=4. as(e)= a(2)=2,sa(e)= s(2)=2log2 and bt(e)= b(1)=1,tb(e)= t(4)=4log4. clearly as(e) 6= sa(e) and bt(e) 6= tb(e). as a result, the mappings (a,s) and (b,t) are not weakly comparable. now take a sequence ηj = e + 3 4j , for all j ≥ 1. then lim j→∞ aηj = lim j→∞ a(e + 3 4j )= lim j→∞ 2=2 (3.13) and lim j→∞ sηj = lim j→∞ s(e + 3 4j )= lim j→∞ 2log(e + 3 4j )=2. (3.14) implies lim j→∞ aηj = lim j→∞ sηj = 2. now lim j→∞ as(e + 3 4j )= lim j→∞ a(2log(e + 3 4m ))= lim j→∞ 2=2 and lim j→∞ sa(e + 3 4j )= lim j→∞ s(2)= 2log2. therefore the pair (a,s) is non-compatible. further from (3.13) and (3.14) we get int. j. anal. appl. (2023), 21:13 11 {(ηj) : lim j→∞ aηj = lim j→∞ sηj} 6= ∅. further there exists another sequence βj =1− 54j for all j ≥ 1. lim j→∞ aβj = lim j→∞ a(1− 5 4j )= lim j→∞ (1− 5 4j )4 =1 (3.15) and lim j→∞ sβj = lim j→∞ s(1− 5 4j )= lim j→∞ (1− 5 4j )2 =1. (3.16) now lim j→∞ as(βj) = lim j→∞ as(1− 5 4j )= lim j→∞ a(1− 5 4m )2= lim j→∞ a(1− 15 16j )= lim j→∞ (1− 5 4j )4= 1 and lim j→∞ sa(βj) = lim j→∞ sa(1− 5 4j )= lim j→∞ s(1− 5 4j )4= lim j→∞ (1− 5 4j )8= 1. therefore lim j→∞ d(asβj,aβj) =d(1,1)=e|1−1| =1 and lim j→∞ d(saβj,sβj) =d(1,1)= e|1−1| =1. further lim j→∞ d(asβj,a(1)) =d(1,1)=e|1−1| =1 and lim j→∞ d(saβj,s(1)) = d(1,1) e|1−1| =1. it follows that the pairs (a,s) and (b,t) have unique fixed point η = 1 and are non-compatible reciprocally continuous and conditionally sequentially absorbing mappings. further the maps a,s,t and b are discontinuous at η =1.moreover the pairs (a,s) and (b,t) are not weakly compatible and hence all the conditions of theorem 3.3 are satisfied. 4. conclusion in this paper we generalized theorem 2.7 using (i) conditionally sequential absorbing, reciprocally continuous and owc by removing weakly compatible mappings in theorem 3.1. (ii) further the weakly compatible mappings are replaced by non-compatible reciprocally continuous and conditionally sequential absorbing mappings in theorem 3.3. moreover the above two results are substantiated by two suitable examples. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s.y. cho, m.j. yoo, common fixed points of generalized contractive mappings, east asian math. j. 25 (2009), 1-10. [2] m. verma, p. kumar, n. hooda, common fixed point theorems using weakly compatible maps in multiplicative metric spaces, electron. j. math. anal. appl. 9 (2021), 284-292. 12 int. j. anal. appl. (2023), 21:13 [3] v. srinivas, t. thirupathi, an affirmative result on banach space, int. j. anal. appl. 20 (2022) 2. https: //doi.org/10.28924/2291-8639-20-2022-2. [4] k. mallaiah, v. srinivas, common fixed point of four maps in sm-metric space, int. j. anal. appl. 19 (2021), 915–928. https://doi.org/10.28924/2291-8639-19-2021-915. [5] v. srinivas, k. satyanna, a result on probabilistic 2-metric space using strong semi compatible mappings, indian j. sci. technol. 14 (2021), 3467–3474. https://doi.org/10.17485/ijst/v14i47.1513. [6] v. srinivas, t. tirupathi, k. mallaiah, a fixed point theorem using e.a property on multiplicative metric space, j. math. comput. sci. 10 (2020), 1788-1800. https://doi.org/10.28919/jmcs/4752. [7] v. srinivas, k. mallaiah, a result on multiplicative metric space, j. math. comput. sci. 10 (2020), 1384-1398. https://doi.org/10.28919/jmcs/4628. [8] b. vijayabaskerreddy, some results in multiplicative metric space using absorbing mappings, indian j. sci. technol. 13 (2020), 4161–4167. https://doi.org/10.17485/ijst/v13i39.1629. [9] a.e. bashirov, e.m. kurpınar, a. ozyapıcı, multiplicative calculus and its applications, j. math. anal. appl. 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081. https://doi.org/10.28924/2291-8639-20-2022-2 https://doi.org/10.28924/2291-8639-20-2022-2 https://doi.org/10.28924/2291-8639-19-2021-915 https://doi.org/10.17485/ijst/v14i47.1513 https://doi.org/10.28919/jmcs/4752 https://doi.org/10.28919/jmcs/4628 https://doi.org/10.17485/ijst/v13i39.1629 https://doi.org/10.1016/j.jmaa.2007.03.081 1. introduction 2. preliminaries 3. main result 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 11-18 http://www.etamaths.com growth and zeros of meromorphic solutions to second-order linear differential equations maamar andasmas and benharrat belaïdi∗ abstract. the main purpose of this article is to investigate the growth of meromorphic solutions to homogeneous and non-homogeneous second order linear differential equations f′′ + af′ + bf = f, where a (z) , b (z) and f (z) are meromorphic functions with finite order having only finitely many poles. we show that, if there exist a positive constants σ > 0, α > 0 such that |a (z)| ≥ eα|z| σ as |z| → +∞, z ∈ h, where dens{|z| : z ∈ h} > 0 and ρ = max{ρ (b) , ρ (f)} < σ, then every transcendental meromorphic solution f has an infinite order. further, we give some estimates of their hyper-order, exponent and hyper-exponent of convergence of distinct zeros. 1. introduction and statement of results we will assume that the reader is familiar with the fundamental results and the standard notations of nevanlinna theory of meromorphic functions (see [11], [14], [16]). in addition, for a meromorphic function f in the complex plane c, we will use the notations λ (f) and λ (f) to denote respectively the exponent of convergence of the zeros and the distinct zeros of a meromorphic function f, ρ (f) to denote the order of growth of f. in order to estimate the rate of growth of meromorphic function of infinite order more precisely, we recall the following definition. definition 1.1 ([13, 16]). let f be a meromorphic function. then the hyper-order ρ2 (f) of f (z) is defined by ρ2 (f) = lim sup r→+∞ log log t (r,f) log r , where t (r,f) is the nevanlinna characteristic function of f. if f is an entire function, then the hyper-order ρ2 (f) of f (z) is defined by ρ2 (f) = lim sup r→+∞ log log t (r,f) log r = lim sup r→+∞ log log log m (r,f) log r , where m (r,f) = max|z|=r |f (z)|. definition 1.2 ([7]). let f be a meromorphic function. then the hyper-exponent of convergence of the sequence of zeros of f (z) is defined by λ2 (f) = lim sup r→+∞ log log n ( r, 1 f ) log r , 2010 mathematics subject classification. 34m10, 34m05, 30d35. key words and phrases. linear differential equation; meromorphic function; order of growth; hyper order; exponent of convergence of zeros; hyper-exponent of convergence of zeros. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 11 12 andasmas and belaïdi where n ( r, 1 f ) is the integrated counting function of zeros of f (z) in {z : |z| ≤ r}. similarly, the hyper-exponent of convergence of the sequence of distinct zeros of f (z) is defined by λ2 (f) = lim sup r→+∞ log log n ( r, 1 f ) log r , where n ( r, 1 f ) is the integrated counting function of distinct zeros of f (z) in {z : |z| ≤ r}. the linear measure of a set e ⊂ (0, +∞) is defined as m (e) = ∫ +∞ 0 χe (t) dt. the logarithmic measure of a set e ⊂ (1, +∞) is defined by lm (e) = ∫ +∞ 1 χe(t) t dt, where χe (t) is the characteristic function of the set e. the upper density of a set e ⊂ (0, +∞) is defined by dense = lim sup r−→ +∞ m (e ∩ [0,r]) r . consider the second-order linear differential equation (1.1) f′′ + a (z) f′ + b (z) f = f, where a (z) , b (z) and f (z) are meromorphic functions of finite order having only finitely many poles. several authors have investigated the growth of solutions of the corresponding homogeneous equation (1.2) f′′ + a (z) f′ + b (z) f = 0. from the works of gundersen (see [10]) and hellerstein et al. (see [12]), we know that if a (z) and b (z) are entire functions with ρ (a) < ρ (b), or a (z) is a polynomial, and b (z) is transcendental, or ρ (a) < ρ (b) ≤ 1 2 , then every solution f 6≡ 0 of (1.2) is of infinite order. for entire solutions of infinite order more precise estimates for their rate of growth would be an important achievement. kwon (see [13]) and chen and yang (see [7]) have investigated the hyper-order ρ2 (f) of solutions of (1.2), and obtained the following results. theorem a ([13]) . let h be a set of complex numbers satisfying dens{|z| : z ∈ h} > 0, and let a (z) and b (z) be entire functions such that for real constants α (> 0) , β (> 0) , |a (z)| ≤ exp { o (1) |z|β } and |b (z)| ≥ exp { (1 + o (1)) α |z|β } as z → +∞ for z ∈ h. then every solution f 6≡ 0 of equation (1.2) has infinite order and ρ2 (f) ≥ β. theorem b ([7]) . let h be a set of complex numbers satisfying dens{|z| : z ∈ h} > 0, and let a (z) and b (z) be entire functions with ρ (a) ≤ ρ (b) = ρ < +∞ such that for real constant c (> 0) and for any given ε > 0, |a (z)| ≤ exp { o (1) |z|ρ−ε } and |b (z)| ≥ exp { (1 + o (1)) c |z|ρ−ε } as z → ∞ for z ∈ h. then every solution f 6≡ 0 of equation (1.2) has infinite order and ρ2 (f) = ρ (b) . these results were improved by beläıdi in [2, 3] by considering more general conditions to higher order linear differential equations with entire coefficients. recently in [8] chen extended the previous results by studying the zeros and the growth of meromorphic solutions of equation (1.1) when a (z) , b (z) , f (z) are meromorphic functions. there exists a natural question: how about the growth of (1.1) when a (z) ,b (z) and f (z) are meromorphic functions of finite order having only finitely many poles and the dominant coefficient is a (z) instead of b (z)? second-order linear differential equations 13 in this paper, we answer the above question and obtain the following results. theorem 1.1 let h ⊂ [0, +∞) be a set with a positive upper density, and let a (z) , b (z) and f (z) be meromorphic functions of finite order having only finitely many poles. suppose there exist positive constants σ > 0, α > 0 such that |a (z)| ≥ eαr σ as |z| = r ∈ h, r → +∞, and ρ = max{ρ (b) , ρ (f)} < σ. then every transcendental meromorphic solution f of equation (1.1) satisfies ρ (f) = +∞ and ρ2 (f) ≤ ρ (a) . furthermore, if f (z) 6≡ 0 then every transcendental meromorphic solution f of equation (1.1) satisfies λ (f) = λ (f) = ρ (f) = +∞ and λ2 (f) = λ2 (f) = ρ2 (f) ≤ ρ (a) . remark 1.1 it is clear that ρ (a) = β ≥ σ in theorem 1.1. indeed, suppose that ρ (a) = β < σ. then, by using lemma 2.2 of this paper, there exists a set e2 ⊂ (1, +∞) that has finite linear measure such that when |z| = r /∈ [0, 1] ∪e2, r −→ +∞, we have for any given ε (0 < ε < σ −β) (1.3) |a (z)| ≤ er β+ε . on the other hand, by the hypotheses of theorem 1.1, there exist positive constants σ > 0, α > 0 such that (1.4) |a (z)| ≥ eαr σ as |z| = r ∈ h, r → +∞, where h is a set with m (h) = ∞. from (1.3) and (1.4) , we obtain for |z| = r ∈ h\ [0, 1] ∪e1, r → +∞ eαr σ ≤ |a (z)| ≤ er β+ε and by ε (0 < ε < σ −β) this is a contradiction as r → +∞. hence ρ (a) = β ≥ σ. corollary 1.1 let a (z) , b (z) , f (z) be meromorphic functions of finite order having only finitely many poles such that ρ = max{ρ (b) , ρ (f)} < ρ (a) = σ < 1 2 . then every transcendental meromorphic solution f of equation (1.1) satisfies ρ (f) = +∞ and ρ2 (f) ≤ ρ (a) = σ. furthermore, if f (z) 6≡ 0 then every transcendental meromorphic solution f of equation (1.1) satisfies λ (f) = λ (f) = ρ (f) = +∞ and λ2 (f) = λ2 (f) = ρ2 (f) ≤ ρ (a) = σ. 2. lemmas for the proofs of theorems our results depend mainly on the following lemmas. lemma 2.1 ([9]) . let f (z) be a transcendental meromorphic function of finite order ρ, and let ε > 0 be a given constant. then, there exists a set e0 ⊂ (1, +∞) that has finite logarithmic measure, such that for all z satisfying |z| /∈ e0 ∪ [0, 1] , and for all k, j, 0 ≤ j < k, we have (2.1) ∣∣∣∣f(k) (z)f(j) (z) ∣∣∣∣ ≤ |z|(k−j)(ρ−1+ε) . similarly, there exists a set e1 ⊂ [0, 2π) that has linear measure zero such that for all z = reiθ with |z| sufficiently large and θ ∈ [0, 2π) \e1, and for all k, j, 0 ≤ j < k, the inequality (2.1) holds. lemma 2.2 ([6]) . let f (z) be a meromorphic function of order ρ (f) = ρ < +∞. then for any given ε > 0, there exists a set e2 ⊂ (1, +∞) that has finite linear measure and finite logarithmic measure such that when |z| = r /∈ [0, 1] ∪e2, r −→ +∞, we have |f (z)| ≤ exp { rρ+ε } . 14 andasmas and belaïdi lemma 2.3 ([14]) . let p (z) = anz n + an−1z n−1 + · · · + a1z + a0 with an 6= 0 (n ≥ 1 is an integer ) be a non constant polynomial. then for every ε > 0, there exists r = r (ε) > 0 such that for all z, |z| = r > r, we have (1 −ε) |an|rn ≤ |p (z)| ≤ (1 + ε) |an|rn. lemma 2.4 ([1]) . suppose that k ≥ 2 and a0,a1,a2, · · · ,ak−1 ( for at least as 6≡ 0, s ∈ {0, 1, · · · ,k − 1}) are meromorphic functions that have finitely many poles. let ρ = max{ρ (aj) (j = 0, 1, · · · ,k − 1) , ρ (f)} < +∞ and let f (z) be a meromorphic solution of infinite order of equation f(k) + ak−1 f (k−1) + · · · + a1f′ + a0f = f. then ρ2 (f) ≤ ρ. lemma 2.5 ([1]) . let f (z) be a meromorphic function having only finitely many poles, and suppose that g (z) := log+ ∣∣f(s) (z)∣∣ |z|ρ , (s ≥ 1 is an integer ) is unbounded on some ray arg z = θ with constant ρ > 0. then there exists an infinite sequence of points zn = rne iθ (n = 1, 2, · · ·) tending to infinity such that g (zn) →∞ and∣∣∣∣f(j) (zn)f(s) (zn) ∣∣∣∣ ≤ 1(s− j)! (1 + o (1)) |zn|s−j (j = 0, · · · ,s− 1) as n → +∞. lemma 2.6 ([15]) . let f (z) be an entire function with ρ (f) < +∞. suppose that there exists a set e3 ⊂ [0, 2π] which has linear measure zero, such that log+ |f ( reiθ ) | ≤ mrσ for any ray arg (z) = θ ∈ [0, 2π]\e3, where m is a positive constant depending on θ, while σ is a positive constant independent of θ. then ρ (f) ≤ σ. lemma 2.7 ([4]) . let f (z) be an entire function of order ρ where 0 < ρ (f) = ρ < 1 2 , and let ε > 0 be a given constant. then there exists a set h ⊂ [0, +∞) with densh ≥ 1 − 2ρ such that for all z satisfying |z| = r ∈ h, we have |f (z)| ≥ exp { rρ−ε } . lemma 2.8 ([5]) . let aj (j = 0, 1, · · · ,k − 1) , f 6≡ 0 be finite order meromorphic functions. if f (z) is an infinite order meromorphic solution of the equation f(k) + ak−1f (k−1) + · · · + a1f′ + a0f = f, then f satisfies λ (f) = λ (f) = ρ (f) = +∞. 3. proof of theorem 1.1 assume that f is a transcendental (f′ 6≡ 0) meromorphic solution of (1.1) with ρ (f) < σ. it follows from (1.1) that (3.1) − f′′ f′ −b (z) f f′ + f (z) f′ = a (z) . since ρ = max{ρ (b) , ρ (f)} < σ, then the order of growth of the left side of equation (3.1) is ρ1 = max{ρ (b) , ρ (f) ,ρ (f)} < σ, hence ρ (a) ≤ ρ1. by lemma 2.2, for any given ε (0 < ε < σ −ρ1) , there exists a set e2 ⊂ (1, +∞) with a finite linear measure and finite logarithmic measure such that (3.2) |a (z)| ≤ er ρ1+ε holds for all z satisfying |z| = r /∈ [0, 1] ∪e2, r → +∞. from hypotheses of theorem 1.1, there exist a set h with densh > 0 and positive constants σ > 0, α > 0 such that (3.3) |a (z)| ≥ eαr σ holds for all z satisfying |z| = r ∈ h, r → +∞. by (3.2) and (3.3), we conclude that eαr σ ≤ er ρ1+ε second-order linear differential equations 15 that is, e(1−o(1))αr σ ≤ 1 for all z satisfying |z| = r ∈ h \ [0, 1] ∪ e2, r → +∞, this contradicts the fact e(1−o(1))αr σ → +∞. consequently, any transcendental meromorphic solution f of (1.1) is of ρ (f) ≥ σ. now, we prove that ρ (f) = +∞. let f be a transcendental meromorphic solution of (1.1). we assume that f is of finite order and ρ (f) = δ. then, we have ρ (f) = δ ≥ σ. it follows from (1.1) that (3.4) |a| ≤ ∣∣∣∣f′′f′ ∣∣∣∣ + |b| ∣∣∣∣ ff′ ∣∣∣∣ + ∣∣∣∣ff′ ∣∣∣∣ . by lemma 2.1, there exists a set e1 ⊂ [0, 2π) that has linear measure zero such that if θ ∈ [0, 2π)\e1, then there is a constant r0 = r0 (θ) > 1 such that for all z satisfying arg z = θ and |z| = r ≥ r0, we have (3.5) ∣∣∣∣f′′ (z)f′ (z) ∣∣∣∣ ≤ r2δ. we now proceed to show that g (z) = log+ |f′ (z)| |z|ρ+ε is bounded on the ray arg z = θ. supposing that this is not the case, then by lemma 2.5, there exists an infinite sequence of points zm = rme iθ (m = 1, 2, · · ·) tending to infinity such that (3.6) ∣∣∣∣ f (zm)f′ (zm) ∣∣∣∣ ≤ (1 + o (1)) |zm| as m → +∞ and (3.7) log+ |f′ (zm)| |zm| ρ+ε →∞. from (3.7) for any positive constant number m > 0, we have (3.8) |f′ (zm)| > em|zm| ρ+ε as m → +∞. since f (z) is a meromorphic function with only finitely many poles, then by hadamard factorization theorem, we can write f (z) = h(z) π(z) where π (z) is a polynomial, h (z) is an entire function with ρ (h) = ρ (f). from (3.8), for m sufficiently large (rm →+∞), we have∣∣∣∣f (zm)f′ (zm) ∣∣∣∣ = ∣∣∣∣ h (zm)π (zm) f′ (zm) ∣∣∣∣ ≤ ∣∣∣∣ h (zm)crsmem|zm|ρ+ε ∣∣∣∣ ≤ |h (zm)|em|zm|ρ+ε , where c > 0 is a constant and s = deg π ≥ 1 is an integer. since ρ (h) = ρ (f) ≤ ρ, then we have (3.9) ∣∣∣∣ h (zm)π (zm) f′ (zm) ∣∣∣∣ ≤ |h (zm)|em|zm|ρ+ε → 0 as m → +∞. by lemma 2.2, for any given ε (0 < ε < σ −ρ) , there exists a set e2 ⊂ (1, +∞) with a finite linear measure and a finite logarithmic measure such that (3.10) |b (z)| ≤ er ρ+ε holds for all z satisfying |z| = r /∈ [0, 1] ∪e2, r → +∞. also by the hypotheses of theorem 1.1, there exists a set h with densh > 0, such that for all z satisfying |z| = r ∈ h, r → +∞, we have (3.11) |a (z)| ≥ eαr σ . using (3.5), (3.6), (3.9), (3.10) and (3.11), we conclude from (3.4) that for all zm = rme iθ satisfying θ ∈ [0, 2π) \e1 and rm ∈ h \ [0, 1] ∪e2, rm → +∞, we have eαr σ m ≤ r2δm + e rρ+εm rm (1 + o (1)) + o (1) ≤ 3r2δ+1m e rρ+εm , that is, eα(1−o(1))r σ m ≤ 3r2δ+1m 16 andasmas and belaïdi which is a contradiction for m is large enough. therefore, log+|f′(z)| |z|ρ+ε is bounded on the ray arg (z) = θ, then there exists a bounded constant m1 > 0 such that |f′ (z)| ≤ em1|z| ρ+ε on the ray arg (z) = θ. then (3.12) |f (z)| ≤ (1 + o (1)) r |f′ (z)| ≤ em1r ρ+2ε on the ray arg (z) = θ. since a ,b and f are meromorphic functions having only finitely many poles and the poles of f can only occur at the poles of a, b and f, then f (z) must have only finitely many poles. therefore, by hadamard factorization theorem, we can write f as f (z) = g(z) d(z) where d (z) is a polynomial and g (z) is an entire function with ρ (g) = ρ (f) ≥ σ. from (3.12), we have∣∣∣∣g (z)d (z) ∣∣∣∣ ≤ em1rρ+2ε on the ray arg (z) = θ. then |g (z)| ≤ |d (z)|em1r ρ+2ε ≤ arkem1r ρ+2ε on the ray arg (z) = θ, where a > 0 is a constant and k = deg d ≥ 1 is an integer. hence (3.13) |g (z)| ≤ em1r ρ+3ε on the ray arg (z) = θ. therefore, for any given θ ∈ [0, 2π)\e1, where e1 ⊂ [0, 2π) is a set of linear measure zero, we have (3.13) holds, for sufficiently large |z| = r. then by lemma 2.6, we get ρ (g) ≤ ρ + 3ε < σ for a small positive ε, a contradiction with ρ (g) ≥ σ. hence, every transcendental meromorphic solution f of (1.1) must be of infinite order. by remark 1.1 we have ρ (a) ≥ σ and since ρ = max{ρ (b) , ρ (f)} < σ, then by using lemma 2.4, we obtain (3.14) ρ2 (f) ≤ ρ (a) . suppose that f 6≡ 0. then, by lemma 2.8, we obtain λ (f) = λ (f) = ρ (f) = +∞. we know that if f has a zero at z0 of order l (l > 2) , and a (z) , b (z) are analytic at z0, then f (z) must have a zero at z0 of order l− 2. therefore, we get by f 6≡ 0 that (3.15) n ( r, 1 f ) ≤ 2n ( r, 1 f ) + n ( r, 1 f ) + n (r,a) + n (r,b) . on the other hand, (1.1) may be rewritten as follows 1 f = 1 f [ f′′ f + a f′ f + b ] . so (3.16) m ( r, 1 f ) ≤ m ( r, 1 f ) + m (r,a) + m (r,b) + 2∑ j=1 m ( r, f(j) f ) + o (1) . hence, by the lemma of logarithmic derivative [11], there exists a set e having finite linear measure such that for all r /∈ e, we have (3.17) m ( r, f(j) f ) = o (log (rt (r,f))) (j = 1, 2) . by (3.15), (3.16) and (3.17), we obtain t (r,f) = t ( r, 1 f ) + o (1) ≤ 2n ( r, 1 f ) + n ( r, 1 f ) + n (r,a) + n (r,b) +m ( r, 1 f ) + m (r,a) + m (r,b) + 2∑ j=1 m ( r, f(j) f ) + o (1) ≤ 2n ( r, 1 f ) (3.18) +t (r,f) + t (r,a) + t (r,b) + c log (rt (r,f)) , second-order linear differential equations 17 where c is a positive constant. set β = ρ (a) = max{ρ, ρ (a)} . then, for any given ε > 0 and sufficiently large r, we have (3.19) c log (rt (r,f)) ≤ 1 2 t (r,f) , t (r,a) ≤ rβ+ε, t (r,b) ≤ rβ+ε, t (r,f) ≤ rβ+ε. then for r /∈ e and r sufficiently large, by using (3.18) and (3.19), we conclude that t (r,f) ≤ 2n ( r, 1 f ) + 3rβ+ε + 1 2 t (r,f) , that is, (3.20) t (r,f) ≤ 4n ( r, 1 f ) + 6rβ+ε. hence, by (3.20), we get ρ2 (f) ≤ λ2 (f) . it follows that (3.21) λ2 (f) ≥ λ2 (f) ≥ ρ2 (f) . we have n (r,f) ≤ n (r,f) ≤ t (r,f) , then (3.22) λ2 (f) ≤ λ2 (f) ≤ ρ2 (f) . therefore, by (3.21) and (3.22) , we obtain λ2 (f) = λ2 (f) = ρ2 (f) . from (3.14), we get λ2 (f) = λ2 (f) = ρ2 (f) ≤ ρ (a) . 4. proof of corollary 1.1 since a is a meromorphic function having only finitely many poles and ρ(a) = σ, then by hadamard factorization theorem, we can write a to a(z) = k(z) p(z) , where k(z) is an entire function with ρ(a) = ρ(k) = σ and p(z) is a polynomial. hence, by lemma 2.7, for any ε (0 < ε < σ), there exists a set h ⊂ [0, +∞) with densh ≥ 1 − 2σ > 0 such that (4.1) |k (z)| ≥ er σ−ε holds for all z, |z| = r ∈ h and r → +∞. also, by lemma 2.3, there exist positive constants c > 0, m ≥ 1 such that (4.2) |p (z)| ≤ crm. hence from (4.1) and (4.2) , we have (4.3) |a (z)| = ∣∣∣∣k (z)p (z) ∣∣∣∣ ≥ er σ−ε crm ≥ er σ−2ε . since ρ = max{ρ (b) , ρ (f)} < σ, then for any given ε with 0 < 2ε < σ −ρ, we have (4.3) and (4.4) ρ = max{ρ (b) , ρ (f)} < σ − 2ε. by using theorem 1.1 for equation (1.1), we find that every transcendental meromorphic solution f of equation (1.1) satisfies (4.5) ρ (f) = +∞ and ρ2 (f) ≤ ρ (a) = σ. furthermore, by using (4.5) and the fact f 6≡ 0, we conclude from theorem 1.1 that every transcendental meromorphic solution f of equation (1.1) with f 6≡ 0 satisfies λ (f) = λ (f) = ρ (f) = +∞ and λ2 (f) = λ2 (f) = ρ2 (f) ≤ ρ (a) = σ. 18 andasmas and belaïdi references [1] m. andasmas and b. beläıdi, on the growth and fixed points of meromorphic solutions of second order nonhomogeneous linear differential equations, int. j. math. comput. 18(2013), no. 1, 28-45. [2] b. beläıdi, estimation of the hyper-order of entire solutions of complex linear ordinary differential equations whose coefficients are entire functions, electron. j. qual. theory differ. equ. 2002 (2002), article id 5. [3] b. beläıdi, growth of solutions of certain non-homogeneous linear differential equations with entire coefficients, jipam. j. inequal. pure appl. math., 5 (2004), no. 2, article id 40. [4] a. besicovitch, on integral functions of order < 1, math. ann. 97 (1927), no. 1, 677-695. [5] z. x. chen, zeros of meromorphic solutions of higher order linear differential equations, analysis, 14 (1994), no. 4, 425–438. [6] z. x. chen, the zero, pole and order of meromorphic solutions of differential equations with meromorphic coefficients, kodai math. j., 19 (1996), no. 3, 341–354. [7] z. x. chen and c. c. yang, some further results on the zeros and growths of entire solutions of second order linear differential equations, kodai math. j., 22 (1999), no. 2, 273–285. [8] y. chen, estimates of the zeros and growths of meromorphic solutions of homogeneous and non-homogeneous second order linear differential equations, math. appl. (wuhan) 23 (2010), no. 1, 18-26. [9] g. g. gundersen, estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, j. london math. soc. 37 (1988), no. 1, 88-104. [10] g. g. gundersen, finite order solutions of second order linear differential equations, trans. amer. math. soc. 305 (1988), no. 1, 415-429. [11] w. k. hayman, meromorphic functions, clarendon press, oxford, 1964. [12] s. hellerstein, j. miles, j. rossi, on the growth of solutions of f′′ + gf′ + hf = 0, trans. amer. math. soc., 324 (1991), no. 2, 693–706. [13] k. h. kwon, on the growth of entire functions satisfying second order linear differential equations, bull. korean math. soc., 33 (1996), no. 3, 487-496. [14] i. laine, nevanlinna theory and complex differential equations, de gruyter studies in mathematics, 15. walter de gruyter & co., berlin, 1993. [15] j. wang, i. laine, growth of solutions of nonhomogeneous linear differential equations, abstr. appl. anal. 2009 (2009), article id 363927. [16] c. c. yang and h. x. yi, uniqueness theory of meromorphic functions, mathematics and its applications, 557. kluwer academic publishers group, dordrecht, 2003. department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem, algeria ∗corresponding author: belaidibenharrat@yahoo.fr international journal of analysis and applications volume 18, number 6 (2020), 1048-1055 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-1048 generalized spectrum and numerical rang of matrix the lorentzian oscillator group of dimension four rafik derkaoui∗, abderrahmane smail university of oran 1, laboratory geanlab, p.o. box 1524, oran, 31000, algeria ∗corresponding author: rafikderkaoui27@yahoo.com abstract. in this paper, we find the spectrum, pseudo-spectrum and numerical rang of matrix of the metric ga. 1. introduction connected lie groups that admit a bi-invariant lorentzian metric were determined by the first of the authors in [14]. among them, those that are solvable, non-commutative, and simply connected are called oscillator groups. this group has many properties useful both in geometry and physics. we study here the geometry of these groups and their networks, i.e their discrete sub-groups co-compact. if g is an oscillator group, its networks determine compact homogeneous lorentz manifolds, on which g acts by isometries. let h2k+1 = r × ck be the heisenberg group and let λ = (λ1,λ2, . . . ,λk) k be strictly positive real numbers. let the additive group r act on h2k+1 by the action: ρ(t)(u, (zj)) = (u, (e iλjtzj)). the group gk(λ), a semi-direct product of r by h2k+1 following ρ, is provided with a bi-invariant lorentz metric. here is how it is built: g = r×r×r2k received september 18th, 2020; accepted october 9th, 2020; published october 28th, 2020. 2010 mathematics subject classification. 53b30, 47a10, 47a12. key words and phrases. oscillator group; spectrum; pseudo-spectrum; numerical rang. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1048 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-1048 int. j. anal. appl. 18 (6) (2020) 1049 is the tangent space at the origin. let us extend the usual scalar product of r2k into a symmetric bilinear form over g so that the plane r×r is hyperbolic and orthogonal to r2k. this form defines an invariant lorentz metric on the left on gk(λ), it is also invariant on the right because the adjoint operators on g are antisymmetric [15]. groups gk(λ) are characterized [14] by: theorem 1.1. the groups gk(λ) are the only lie group simply connected , resolvable and noncommutative which admit a bi-invariant lorentz metric. remark 1.1. it is easy to see that the groups g1(λ) are isomorphous; the group g1 = g1(1) is usually known as the oscillator group [20]. since [1], [2] et [3] the oscillator group has been generalized to a dimension equal to an even number 2n with n ≥ 2, plus this provides a known example of homogeneous space-time [6]. for n = 2, the oscillator group of dimension 4 admits a lorentzian metric invariant on the left and on the right (bi-invariant). this bi-invariant metric has been generalized a family ga, −1 < a < 1, invariant lorentzian metrics on the left. for a = 0, the metric g0 become or the only example of lorentzian bi-invariant metric [7] the researchers giovani and zaeim extracted three vectors feilds from the oscillator group, which are: killing vector feild, affine vector feild, parallel vector feild (see [4]). and also giovani and zaeim classified the totally geodesic and parallel hypersurfaces of four-dimensional groups (see [3]). varah published an article entitled ”on the separation of two matrices” in which he defined with standard 2 the pseudospectrum using the smallest singular value σmin(zi−a) under the notion λ�(a) see [23]. in the 1960s the pseudospectrum was studied in several by l. n. trefethen [19], [21]. in recent years the study of the pseudospectrum has been very active, many contributions related to the pseudospectrum have been made by various researchers, for example, j. s. baggett, a. bottcher, m. embree, l. n. trefethen, l. reichel, s.c.reddy, t.a. driscoll. the pseudospectrum of a normal matrix a consists of circles of radius � around each eigenvalue. for nonnirmal matrices, the pseudospectrum takes different forms in the complex plane. in [19] the pseudospectrum of thirteen highly non-normal matrices is presented. 2. preliminaries at the moment we consider on gλ a family parametre of left-invariant lorentzian metrics ga. with respect to coordinates (x1,x2,x3,x4), this metric ga is explicitly given by ga = adx 2 1 + 2ax3dx1dx2 + (1 + ax 2 3)dx 2 2 + dx 2 3 + 2dx1dx4 + 2x3dx2dx4 + adx 2 4, int. j. anal. appl. 18 (6) (2020) 1050 with −1 < a < 1. note that for a = 0 and λ = 1 we have the bi-invariant metric on the oscillator group g1 [7]. in all other cases, ga is only invariant on the left. the matrix of the metric ga is given by aa =   a ax3 0 1 ax3 1 + ax 2 3 0 x3 0 0 1 0 1 x3 0 a   . numerical rang definition 2.1. let a be an n×n complex matrix. then the numerical rang of a, w(a), is defined to be w(a) = { x∗ax x∗x , x ∈ cn, x 6= 0 } . where x∗ denotes the conjugate transpose of the vector x. proposition 2.1. based on the definition of the numerical range, one can now fairly easily deduce the following basic properties; for details see primarily [ [9], chapter 1] but also [8]. 1− for any a ∈ mn(c) and for any a,b ∈ c, w(aa + bin) = aw(a) + b. 2− for any a,b ∈ mn(c), w(a + b) ⊆ w(a) + w(b). 3− for any a ∈ mn(c), w(a) contains the convex hull of the eigenvalues of a. if a is normal, i.e., a∗a = aa∗, then w(a) equals the convex hull of σ(a). 4− for any a ∈ mn(c), w(a) ⊂ r if and only if a is hermitian, i.e., a∗ = a, in this case, the endpoints of w(a) coincide with the minimum and the maximum eigenvalues of a. furthermore, w(a) is a line segment in the complex plane if and only if the matrix a is normal and has collinear eigenvalues; or equivalently, if and only if a = ah + bi for some a,b ∈ c and an hermitian matrix h. 3. eigenvalues and pseudo-spectrum of matrix aa proposition 3.1. the eigenvalues of the matrix aa are: λ1 = 1, λ2 = 2 3 a + 1 3 ax23 − 1 2 s + c 2s + 1 3 − √ 3 2 i ( s + c s ) , λ3 = λ2, λ4 = 2 3 a + 1 3 ax23 + s − c s + 1 3 , with s = 3 √ m + √ n − 8 27 , int. j. anal. appl. 18 (6) (2020) 1051 and m = 2 9 a + 1 9 a2 − 1 27 a3 + 1 6 x23 + 11 18 ax23 + 1 6 ax43 − 1 18 a2x23 − 1 18 a3x23 + 1 9 a2x43 + 1 18 a3x43 + 1 27 a3x63 n = 4 27 a3 − 4 27 a2 − 1 27 a4 − 8 27 x23 − 13 108 x43 − 1 27 x63 − 2 9 ax23 − 1 54 ax43 − 1 54 ax63 + 7 27 a2x23 + 4 27 a3x23 + 7 36 a2x43 − 1 9 a4x23 + 1 18 a2x63 − 11 108 a4x43 + 1 27 a3x63 + 1 54 a5x43 − 1 108 a2x83 − 1 108 a6x43 − 1 54 a5x63 + 1 54 a4x83 − 1 54 a6x63 − 1 108 a6x83 c = 2 9 a− 1 9 a2 − 1 3 x23 − 2 9 ax23 − 1 9 a2x23 − 1 9 a2x43 − 4 9 proof. we have det(aa −λi4) = (1 −λ)(−λ3 + lλ2 + kλ + (a2 − 1)), with l = ( 1 + 2a + ax23 ) , k = (−a2 − 2a−a2x23 + x 2 3 + 1), so, det(a−λi4) = 0, if and only if either λ1 = 1 or −λ3 + lλ2 + kλ + (a2 − 1) = 0. according to the cardan method we find, z3 + pz + q = 0, such as (3.1) z = λ− l 3 , z ∈ c, int. j. anal. appl. 18 (6) (2020) 1052 and p = −( 1 3 l2 + k) = −1 3 (4 + a2x43 + ax 2 3 + a 2 − 2a + a2x23 + 3x23) , q = − 1 27 (−16 + 2a3x63 + 6a2x43 + 33ax23 − 2a3 + 6a2 − 3a3x23 + 12a + 3a3x43 − 3a2x23 + 9x23) . then the cardan method he says that the 3 solutions are: zk = j k 3 √√√√1 2 ( −q + √ −∆ 27 ) + j−k 3 √√√√1 2 ( −q − √ −∆ 27 ) , 0 ≤ k ≤ 2 such as , ∆ = −4p3 − 27q2, j = ei2 π 3 . so, according to (3.1) we find, λk = zk + l 3 , 0 ≤ k ≤ 2 � pseudo-spectrum of aa: since a is symmetrical therefore aa is normal, therefore pseudo-spectrum noted by λ�(aa) given by: λ�(aa) = {z ∈ c : |z −λi| ≤ �} with i ∈{1, . . . , 4} . 3.1. numerical rang of matrix aa. proposition 3.2. the numerical rang of matrix aa check the following relation:∣∣∣∣x∗aaxx∗x ∣∣∣∣ ≤ (1 + |a|)(1 + |x3|) + ∣∣ax23∣∣ proof. we have w(a) = { x∗ax x∗x , x ∈ c4, x 6= 0 } we put x =   z1 z2 z3 z4   , with zi = rie iθi. we have x∗aax = a |z1| 2 + a |z4| 2 + |z2| 2 + |z3| 2 + ax3(z1z2 + z2z1) + x3(z2z4 + z4z2) + (z1z4 + z4z1) + a |z2| 2 x23, int. j. anal. appl. 18 (6) (2020) 1053 so, x∗ax x∗x = 1 + (a− 1)(|z1| 2 + |z4| 2 ) 4∑ i=1 |zi| 2 + ax3 z1z2 + z2z1 4∑ i=1 |zi| 2 + x3 z2z4 + z4z2 4∑ i=1 |zi| 2 + z1z4 + z4z1 4∑ i=1 |zi| 2 + ax23 |z2| 2 4∑ i=1 |zi| 2 . we have (3.2) |zj| 2 4∑ i=1 |zi| 2 ≤ 1, ∀j ∈{1, . . . , 4} . and (3.3) zizj + zjzi 4∑ i=1 |zi| 2 ≤ 1, ∀i,j ∈{1, . . . , 4} , so from (3.2) and (3.3) we find∣∣∣∣x∗aaxx∗x ∣∣∣∣ ≤ 1 + |ax3| + |x3| + |a| + ∣∣ax23∣∣ . it had to be proven. � example 3.1. 1) for a = 0 and x3 = 0, a00 =   0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0   , so g00(x ∗,x) x∗x = 1 − r21 + r 2 4 − 2r1r4 cos(θ1 −θ4) r21 + r 2 2 + r 2 3 + r 2 4 ≤ 1, moreover 1 ∈ w(a00) on the other hand, we have − r21 + r 2 4 − 2r1r4 cos(θ1 −θ4) r21 + r 2 2 + r 2 3 + r 2 4 ≥−2, therefore g00(x ∗,x) x∗x ≥−1, moreover −1 ∈ w(a00). so w(a00) = [−1, 1] 2) for a = 0 and x3 = 0.5, a0.50 =   0 0 0 1 0 1 0 0.5 0 0 1 0 1 0.5 0 0   , int. j. anal. appl. 18 (6) (2020) 1054 so g0.50 (x ∗,x) x∗x = g00(x ∗,x) x∗x + r2r4 cos(θ2 −θ4) r21 + r 2 2 + r 2 3 + r 2 4 . we have, r2r4 cos(θ2 −θ4) r21 + r 2 2 + r 2 3 + r 2 4 ≤ 1 2 , and g0.50 (x ∗,x) x∗x ≥− 5 4 so − 5 4 ≤ g0.50 (x ∗,x) x∗x ≤ 3 2 , but −5 4 and 3 2 does not belong to w(a0.50 ), so we get w(a 0.5 0 ) ⊂ ] −5 4 , 3 2 [ . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. boucetta, a. medina. solutions of the yang-baxter equations on orthogonal groups: the case of oscillator groups, j. geom. phys. 61 (2011), 2309–2320. [2] s. bromberg, a. medina, geometry of oscillator groups and locally symmetric manifolds, geom. dedicata. 106 (2004), 97–111. [3] g. calvaruso, j. van der veken: totally geodesic and parallel hypersurfaces of four-dimensional oscillator groups. results math. 64 (2013), 135–153. [4] g. calvaruso, a. zaeim, on the symmetries of the lorentzian oscillator group, collect. math. 68 (2017), 51-67. [5] r. duran diaz, p.m. gadea, j.a. oubiña: reductive decompositions and einstein-yang-mills equationsassociated to the oscillator group. j. math. phys. 40 (1999), 3490–3498. [6] r. duran diaz, p.m. gadea, j.a. oubiña: the oscillator group as a homogeneous spacetime. lib. math. 19 (1999), 9–18. [7] p.m. gadea, j.a. oubiña: homogeneous lorentzian structures on the oscillator groups. arch. math. 73 (1999), 311–320. [8] k.e. gustafson and d.k.m. rao, numerical range, springer, new york, 1997. [9] r.a. horn and c.r. johnson, topics in matrix analysis, cambridge university press, cambridge, 1991. [10] v. khiem ngo, an approach of eigenvalue perturbation theory. appl. numer. anal. comput. math. 2 (2005), 108-125. [11] a. v. levichev, chronogeometry of an electromagnetic wave given by a biinvariant metric on the oscillator group, siberian math. j. 27 (1986), 237–245. [12] a. medina, groupes de lie munis de métriques bi-invariantes, tohoku math. j. 37 (1985), 405–421. [13] a. medina, p. revoy, les groupes oscillateurs et leurs réseaux, manuscripta math. 52 (1985), 81–95. [14] a. medina-perea, groupes de lie munis de pseudo-métriques de riemann bi-invariantes. séminaire de géométrie différentielle 1981–1982, exposé 4, institut de mathématiques, université des sciences et techniques du languedoc, montpellier [15] j. milnor, curvatures of left invariant metrics on lie groups, adv. math. 21 (1976), 283–329. [16] d. müller, f. ricci. on the laplace-beltrami operator on the oscillator group. j. reine angew. math. 390 (1988), 193–207. [17] t. nomura, the paley-wiener theorem for the oscillator group, j. math. kyoto univ. 22 (1982/83), 71–96. int. j. anal. appl. 18 (6) (2020) 1055 [18] b. o’neill, semi-riemannian geometry, academic press, new york, (1983). [19] l. reichel, l.n. trefethen, eigenvalues and pseudo-eigenvalues of toeplitz matrice, linear algebra appl. 162-164 (1992), 153-185. [20] r.f. streater, the representations of the oscillator group, commun. math. phys. 4 (1967), 217–236. [21] l. trefethen, m. embree, spectra and pseudospectra: the behavior of non-normal matrices and operators, princeton university press, princeton, 2005. [22] l.n. trefethen, pseudospectra of matrices, in d.f. griffiths and g.a. watson, eds., numerical analysis 1991, longman, harlow, uk, 1992. [23] c. van loan, a study of the matrix exponential, numerical analysis report no. 10, university of manchester, uk, august, 1975, reissued as mims eprint, manchester institute for mathematical sciences, the university of manchester, uk, november 2006. 1. introduction 2. preliminaries 3. eigenvalues and pseudo-spectrum of matrix aa 3.1. numerical rang of matrix aa references international journal of analysis and applications issn 2291-8639 volume 10, number 1 (2016), 9-16 http://www.etamaths.com homotopy perturbation method for solving the fractional fisher’s equation mountassir hamdi cherif∗, kacem belghaba and djelloul ziane abstract. in this paper, we apply the modified hpm suggested by momani and al. [23] for solving the time-fractional fisher’s equation and we use the classical hpm to derive numerical solutions of the space-fractional fisher’s equation. we compared our solution with the exact solution. the results show that the hpm modified is an appropriate method for solving nonlinear fractional derivative equations. 1. introduction fractional analysis is a branch of mathematics, was the first debut in 1695 with the question posed by leibnitz as follows: what could be the derivative of order (half) of a function x? from that date to today, the evolution of this branch of mathematics where he became a major development has many uses, among them, for example: fractional derivatives have been widely used in the mathematical model of the visco-elasticity of the material [1]. the electromagnetic problems can be described using the fractional integro-differential equations [2]. in biology, it was deduced that the membranes of biological organism cells have the electrical conductance of fractional order [3], and then is classified into groups of non-integer order models. in economics, some finance systems can display a dynamic fractional order [4]. in addition to the above, we find that the development of this branch has also led to the emergence of linear and nonlinear differential equations of fractional order, which became requires researchers to use conventional methods to solve them. among these methods there is the homotopy perturbation method (hpm). this method was established in 1998 by he ([5]-[9]) and applied by many researchers to solve various linear and nonlinear problems (see [10][16]). the method is a powerful and efficient technique to find the solutions of nonlinear equations. the coupling of the perturbation and homotopy method is called the homotopy perturbation method. this method can take the advantages of the conventional perturbation method while eliminating its restrictions [16]. our concern in this work is to consider the numerical solution of the nonlinear fisher’s equation with time and space fractional derivatives of the form (1) cdαt u = cdβxu + γu(1 −u), 0 < α 6 1, 1 < β 6 2, where cdαt u = ∂α ∂tα , cd β t u = ∂β ∂tβ . in the case α = 1 and β = 2, this equation become (2) ut = uxx + γu(1 −u), which is a fisher’s partial differential equation. we will extend the application of the modified hpm in order to derive analytical approximate solutions to nonlinear time-fractional fisher’s equation and we use the classical hpm to resolve the 2010 mathematics subject classification. 35r11, 26a33, 47j35. key words and phrases. caputo fractional derivative; homotopy perturbation method; fisher’s equation; fractional partial diferential equation. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 9 10 cherif, belghaba and ziane nonlinear space-fractional fisher’s equation. precisely, we use the modified homotopy perturbation method described in [23] for handling an iterative formula easy-to-use for computation. observing the numerical results, and comparing with the exact solution, the proposed method reveals to be very close to the exact solution and consequently, an efficient way to solve the nonlinear time-fractional fisher’s equation. this is the raison why we try to use it in this work. 2. basic definitions we give some basic definitions and properties of the fractional calculus and the laplace transform theory which are used further in this paper. (see [18]-[20]). definition 1. let ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis r. the riemann–liouville fractional integrals iαa+f of order α ∈ c (re(α) > 0) is defined by (3) (iαa+f)(t) = 1 γ (α) ∫ t a f(τ)dτ (t− τ)1−α , t > 0,re(α) > 0, here γ(α) is the gamma function. theorem 2. let re(α) > 0 and let n = [re(α)] + 1. if f(t) ∈ acn [a,b] , then the caputo fractional derivatives (cdαt f)(t) exist almost evrywhere on [a,b] . if α /∈ n, (cdαt f)(t) is represented by: (4) (cdαt f)(t) = 1 γ (n−α) ∫ t a f(n)(τ)dτ (t− τ)α−n+1 , and if α ∈ n, we obtain (cdαt f)(t) = f(n)(t). definition 3. let u ∈ cn−1, n ∈ n∗. then the (left sided) caputo fractional derivative of u is defined (for t > 0) as (5) cdαt u(x,t) = ∂αu(x,t) ∂tα = { 1 γ(n−α) ∫ t 0 (t− τ)n−α−1 ∂ nu(x,t) ∂tn dτ,n− 1 < α < n,n ∈ n∗ ∂nu(x,t) ∂tn ,α = n ∈ n. according to (5), we can obtain: cdαk = 0, k is a constant, and cdαtβ = { γ(β+1) γ(β−α+1)t β−α,β > α− 1 0,β ≤ α− 1. 3. the homotopy perturbation method to illustrate the basic ideas of this method, we consider the following nonlinear differential equation (6) a(u) −f(r) = 0, r ∈ ω, with the following boundary conditions (7) b(u, ∂u ∂n ) = 0, r ∈ γ, where a is a general differential operator, f(r) is a known analytic function, b is a boundary operator, u is the unknown function, and γ is the boundary of the domain ω. the operator a can be generally divided into two operators, l and n, where l is a linear, and n a nonlinear operator. therefore, equation (6) can be written as follows: (8) l(u) + n(u) −f(r) = 0. using the homotopy technique, we construct a homotopy v(r,p) : ω × [0, 1] −→ r, which satisfies (9) h(v,p) = (1 −p)[l(υ) −l(u0)] + p[a(υ) −f(r)] = 0, or (10) h(v,p) = l(v) −l(u0) + pl(u0) + p[n(v) −f(r)] = 0, fractional fisher’s equation 11 where p ∈ [0, 1] is an embedding parameter, and u0 is the initial approximation of equation (6) which satisfies the boundary conditions. clearly, from eq. (9) and (10) we will have h(v, 0) = l(v) −l(u0) = 0,(11) h(v, 1) = a(v) −f(r) = 0.(12) the changing process of p from zero to unity is just that of v(r,p) changing from u0(r) to u(r). in topology, this is called deformation and l(v) − l(u0) and a(v) − f(r) are called homotopic. if the embedding parameter p, (0 6 p 6 1) is considered as a “small parameter”, applying the classical perturbation technique, we can assume that the solution of equation (9) or (10) can be given as a power series in p (13) v = v0 + pv1 + p 2v2 + · · · . setting p = 1, results in the approximate solution of eq. (6) (14) u = lim p→1 v = v0 + v1 + v2 + · · · . the convergence of the series (14) has been proved in ([21], [22]). 3.1. new modification of the hpm . momani and al. [23] introduce an algorithm to handle in a realistic and efficient way the nonlinear pdes of fractional order. they consider the nonlinear partial differential equations with time fractional derivative of the form (15) { cdαt u(x,t) = f(u,ux,uxx) = l(u,ux,uxx) + n(u,ux,uxx) + h(x,t), t > 0 uk(x, 0) = gk(x), k = 0, 1, 2, ....m− 1, where l is a linear operator, n is a nonlinear operator which also might include other fractional derivatives of order less than α. the function h is considered to be a known analytic function and cdα, m− 1 < α 6 m, is the caputo fractional derivative of order α. in view of the homotopy technique, we can construct the following homotopy (16) ∂um ∂tm −l(u,ux,uxx) −h(x,t) = p[ ∂um ∂tm + n(u,ux,uxx) −c dαt u], or (17) ∂um ∂tm −h(x,t) = p[ ∂um ∂tm + l(u,ux,uxx) + n(u,ux,uxx) −c dαt u], where p ∈ [0, 1]. the homotopy parameter p always changes from zero to unity. in case p = 0, eq. (16) becomes the linearized equation (18) ∂um ∂tm = l(u,ux,uxx) + h(x,t), or in the second form, eq. (17) becomes the linearized equation (19) ∂um ∂tm = h(x,t). when it is one, eq. (16) or eq. (17) turns out to be the original fractional differential equation (15). the basic assumption is that the solution of eq. (16) or eq. (17) can be written as a power series in p (20) u = u0 + pu1 + p 2u2 + p 3u3 · · · . finally, we approximate the solution by 12 cherif, belghaba and ziane (21) u(x,t) = ∞∑ n=0 un(x,t). 4. numerical applications in this section, we apply the modified homotopy perturbation method for solving fisher’s equation with time-fractional derivative and we use the classical hpm to obtain analytical solution for fisher’s equation with space-fractional derivative. 4.1. numerical solutions of time-fractional fisher’s equation. if β = 2, we obtain the following form of the time-fractional fisher’s equation (22) cdαt u = uxx + γu(1 −u), 0 < α 6 1, with the initial condition u(x, 0) = f(x). application of the new modification of the hpm in view of eq. (17), the homotopy of eq. (22) can be constructed as (23) ∂u ∂t = p[ ∂u ∂t + uxx + γu−γu2 − cdαt u]. substituting (20) into (23) and equating the terms of the same power p, one obtains the following set of linear partial differential equations (24) p0 : ∂u0 ∂t = 0, p1 : ∂u1 ∂t = ∂u0 ∂t + u0xx + γu0 −γu20 − cdαt u0, p2 : ∂u2 ∂t = ∂u1 ∂t + u1xx + γu1 −γ(2u0u1) − cdαt u1, p3 : ∂u3 ∂t = ∂u2 ∂t + u2xx + γu2 −γ(2u0u2 + u21) − cdαt u2, ... with the following conditions (25) u0(x, 0) = f(x), (26) ui(x, 0) = 0 for i = 1, 2, .... case 1: consider the following form of the time-fractional equation (for γ = 1) (27) cdαt u = uxx + u(1 −u), 0 < α 6 1, with the initial condition (28) u(x, 0) = λ. using the initial condition (28) and solving the above eqs. (24) yields u0 (x,t) = λ, u1 (x,t) = λ(1 −λ)t, u2 (x,t) = λ(1 −λ)t + λ(1 −λ)(1 − 2λ)t 2 2! −λ(1 −λ) t 2−α γ(3−α), u3 (x,t) = λ(1 −λ)t + 2λ(1 −λ)(1 − 2λ)t 2 2! + λ(1 −λ)(1 − 6λ + 6λ2)t 3 3! − 2λ(1 −λ) t 2−α γ(3−α) +λ(1 −λ) t 3−2α γ(4−2α) − 2λ(1 −λ)(1 − 2λ) t3−α γ(4−α), ... and so on. the first four terms of the decomposition series solution for eq. (27) is given as fractional fisher’s equation 13 u(x,t) = λ + 3λ(1 −λ)t + 3λ(1 −λ)(1 − 2λ) t2 2! + λ(1 −λ)(1 − 6λ + 6λ2) t3 3! (29) −3λ(1 −λ) t2−α γ(3 −α) + λ(1 −λ) t3−2α γ(4 − 2α) − 2λ(1 −λ)(1 − 2λ) t3−α γ(4 −α) . substituting α = 1 into (30), we get the same approximate solution of nonlinear partial diferential fisher’s equation obtained in[24] as u(x,t) = λ + λ(1 −λ)t + λ(1 −λ)(1 − 2λ) t2 2! + λ(1 −λ)(1 − 6λ + 6λ2) t3 3! + · · ·(30) = λet 1 −λ + λet . case 2 next we consider the following form of the time-fractional fisher’s equation (for γ = 6) (31) cdαt u = uxx + 6u(1 −u), 0 < α 6 1, subject to the initial condition (32) u(x, 0) = 1 (1 + ex)2 . the use of the initial condition (32) and solving the above eq (24), we obtain u0 (x,t) = 1 (1+ex)2 , u1 (x,t) = 10ex (1+ex)3 t, u2 (x,t) = 10ex (1+ex)3 t + 50ex(2ex−1) (1+ex)4 t2 2! − 10e x (1+ex)3 t2−α γ(3−α), u3 (x,t) = 10ex (1+ex)3 t + 50ex(2ex−1) (1+ex)4 t2 2! + 250ex(4e2x−7ex+1) (1+ex)4 t3 3! − 20e x (1+ex)3 t2−α γ(3−α) + 10ex (1+ex)3 t3−2α γ(4−2α) − 50ex(2ex−1) (1+ex)4 t3−α γ(4−α), ... and so on. the first three terms of the decomposition series solution for eq. (31) is given as u(x,t) = 1 (1 + ex)2 + 30ex (1 + ex)3 t + 100ex(2ex − 1) (1 + ex)4 t2 2! + 250ex(4e2x − 7ex + 1) (1 + ex)4 t3 3! (33) − 30ex (1 + ex)3 t2−α γ(3 −α) + 10ex (1 + ex)3 t3−2α γ(4 − 2α) − 50ex(2ex − 1) (1 + ex)4 t3−α γ(4 −α) . substituting α = 1 into (34), we obtain: u(x,t) = 1 (1 + ex)2 + 10ex (1 + ex)3 t + 50ex(2ex − 1) (1 + ex)4 t2 2! + 250ex(4e2x − 7ex + 1) (1 + ex)4 t3 3! + · · ·(34) = 1 (1 + ex−5t)2 the same solution as presented in [24]. 4.2. numerical solutions of space-fractional fisher’s equation. now we consider the following form of the space-fractional fisher’s equation with initial condition (35) ut = c d β t uxx + γu(1 −u), 1 < β 6 2, (36) u(x, 0) = x2. 14 cherif, belghaba and ziane solution.png figure 1. (left): exact solution for eq. (31)-(32); (right): approximative solution of eq. (31)-(32) with four terms for α = 1. figure 2. (left): approximative solution of eq. (31)-(32) for α = 0.5; (right): approximative solution of eq. (31)-(32) for α = 0.9. the initial condition has been taken as the above polynomial to avoid heavy calculation of fractional differentiation. application of the hpm according to the hpm, we construct the following homotopy (37) ut −v0t + p [ −cdβx −u(1 −u) + v0t ] , 1 < β 6 2, where p ∈ [0; 1], v0 = u(x; 0) = x2 and γ = 1. in view of the hpm, substituting eq. (20) into eq. (37) and equating the coefficients of like powers of p, we get the following set of differential equations (38) p0 : ∂u0 ∂t = v0t, p1 : ∂u1 ∂t =c dβxu0 + u0 −u20 −v0t, p2 : ∂u2 ∂t =c dβxu1 + u1 − 2u0u1, p3 : ∂u3 ∂t =c dβxu2 + u2 − (2u0u2 + u21), ...pn : ∂un ∂t =c dβxun−1 + un−1 − (∑n−1 i=0 uiun−i−1 ) , n > 1, with the following conditions (39) u0(x, 0) = x 2,ui(x, 0) = 0 for i = 1, 2, .... using the initial conditions (39) and solving the above eqs. (38) yields fractional fisher’s equation 15 (40) u0 (x,t) = x 2, u1 (x,t) = (a1x 2−β + x2 −x4)t, u2 (x,t) = (a2x 2−2β + a3x 2−β + a4x 4−β + x2 − 3x4 + 2x6)t 2 2! , u3 (x,t) = (a5x 2−3β + a6x 2−2β + a7x 2−β + a8x 4−2β + a9x 4−β + a10x 6−β + x2 − 6x4 + 10x6 − 5x8)t 3 3! , ... where a1 = 2 γ(3−β), a2 = 2 γ(3−2β), a3 = 4 γ(3−β), a4 = − 24 γ(5−β) − 4 γ(3−β), a5 = 2 γ(3−3β), a6 = 6 γ(3−2β), a7 = 6 γ(3−β), a8 = − 24 γ(5−2β) − 4γ(5−β) γ(3−β)γ(5−2β) − 4 γ(3−2β) − 4 γ2(3−β), a9 = − 96 γ(5−β) − 16 γ(3−β), a10 = 2 6! γ(7−β) + 48 γ(5−β) + 12 γ(3−β). here, setting p = 1, we have the following solution for three iterations u(x,t) = x2 + (a1x 2−β + x2 −x4)t(41) + (a2x 2−2β + a3x 2−β + a4x 4−β + x2 − 3x4 + 2x6) t2 2! + (a5x 2−3β + a6x 2−2β + a7x 2−β + a8x 4−2β + a9x 4−β + a10x 6−β + x2 − 6x4 + 10x6 − 5x8) t3 3! . case 1: substituting β = 2 into (42), we obtain u(x,t) = x2 + (2 + x2 −x4)t + (4 − 15x2 − 3x4 + 2x6) t2 2! (42) + (−30 − 63x2 − 90x4 + 10x6 − 5x8) t3 3! . case 2: substituting β = 3 2 into (42), we get u(x,t) = x2 + ( 4 √ π x 1 2 + x2 −x4)t + ( 8 √ π x 1 2 + x2 − 104 5 √ π x 5 2 − 3x4 + 2x6) t2 2! (43) + ( 12 √ π x 1 2 − 39π + 16 π x + x2 − 416 5 √ π x 5 2 − 6x4 + 7328 105 √ π x 9 2 + 10x6 − 5x8) t3 3! . in the same manner, we can obtain the approximate solution of higher order of eq. (35) by using the iteration formulas (38) and maple. figure 3. (left): approximative solution of eq. (35)-(36) with β = 2; (right): approximative solution of eq. (35)-(36) with β = 3/2. 16 cherif, belghaba and ziane 5. conclusion in this work, the modified homotopy perturbation method was successfully used for solving fisher’s equation with time-fractional derivative, and the classical hpm has been used for solving fisher’s equation with space-fractional derivative. the final results obtained from modified hpm and compared with the exact solution shown that there is a similarity between the exact and the approximate solutions. calculations show that the exact solution can be obtained from the third term. that’s why we say that modified hpm is an alternative analytical method for solving the nonlinear time-fractional equations. references [1] a. hanyga, fractional-order relaxation laws in non-linear viscoelasticity, continuum mechanics and thermodynamics, 19 (2007), 25-36. [2] v. e. tarasov, fractional integro-differential equations for electromagnetic waves in dielectric media, theoretical and math. phys, 158 (2009), 355-359. [3] g. chen and g. friedman, an rlc interconnect model based on fourier analysis, comput. aided des. integr. circuits syst, 24 (2005), 170-183. [4] t. j. anastasio, the fractional-order dynamics of brainstem vestibule-oculumotor neurons, biol. cybern, 72 (1994), 69-79. [5] j. h. he, homotopy perturbation technique, comput. meth. appl. mech. eng, 178 (1999), 257-262. [6] j. h. he, application of homotopy perturbation method to nonlinear wave equations, chaos solitons fractals, 26 (2005), 695-700. [7] j. h. he, a coupling method of homotopy technique and perturbation technique for nonlinear problems, int. j. of nonlinear mech, 35 (2000), 37-43. [8] j. h. he, some asymptotic methods for strongly nonlinear equations, int. j. modern phys, b 20 (2006), 1141-1199. [9] j. h. he, a new perturbation technique which is also valid for large parameters, j. sound vib, 229 (2000), 1257-1263. [10] a. j. khaleel, homotopy perturbation method for solving special types of nonlinear fredholm integro-differentiel equations, j. al-nahrain uni, 13 (2010), 219-224. [11] d. d. ganji, h. babazadeh, f. noori, m. m. pirouz and m. janipour, an application of homotopy perturbation method for non-linear blasius equation to boundary layer flow over a flat plate, int. j. nonlinear sci, 7 (2009), 399-404. [12] r. taghipour, application of homotopy perturbation method on some linear and nonlnear parabolique equations, ijrras, 6 (2011), 55-59. [13] w. asghar khan, homotopy perturbation techniques for the solution of certain nonlinear equations, appl. math. sci, 6 (2012), 6487-6499. [14] s. m. mirzaei, homotopy perturbation method for solving the second kind of non-linear integral equations, int. math. forum, 5 (2010), 1149-1154. [15] h. el qarnia, application of homotopy perturbation method to non-homogeneous parabolic partial and non linear differential equations, w. j. mod. simul, 5 (2009), 225-231. [16] a. j. al-saif and d. a. abood, the homotopy perturbation method for solving k(2,2) equation, j. basrah researches ((sciences)), 37 (2011). [17] m. matinfar, m. mahdavi and z. raeisy, the implementation of variational homotopy perturbation method for fisher’s equation, int. j. of nonlinear sci, 9 (2010), 188-194. [18] i. podlubny, fractional differential equations, academic press, new york, 1999. [19] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [20] k. diethelm, the analysis fractional differential equations, springer-verlag berlin heidelberg, 2010. [21] j. biazar and h. ghazvini, convergence of the homotopy perturbation method for partial dfferential equations, nonlinear analysis: real world applications, 10 (2009), 2633-2640. [22] j. biazar and h. aminikhah, study of convergence of homotopy perturbation method for systems of partial differential equations, comput. math. appli, 58 (2009), 2221-2230. [23] s. momani and z. odibat, homotopy perturbation method for nonlinear partial differential equations of fractional order, phys. lett. a 365 (2007), 345-350. [24] m. matinfar, z. raeisi and m. mahdavi, variational homotopy perturbation method for the fisher’s equation, int. j. of nonlinear sci, 9 (2010), 374-378. [25] a. bouhassoun and m. hamdi cherif, homotopy perturbation method for solving the fractional cahn-hilliard equation, journal of interdisciplinary mathematics, 18 (2015), 513-524. laboratory of mathematics and its applications (lamap), university of oran1, p.o. box 1524, oran, 31000, algeria ∗corresponding author: mountassir27@yahoo.fr int. j. anal. appl. (2022), 20:20 payne-sperb-stakgold type inequality for a wedge-like membrane abir sboui1,3,5,∗, abdelhalim hasnaoui2,4 1department of mathematics, faculty of arts and science (turaif), northern border university, ksa 2faculty of arts and science (rafha), northern border university, ksa 3department of mathematics, issatm, university of carthage, tunisia 4department of mathematics, fst, university of tunis el manar, tunisia 5laboratory of partial differential equations and applications (lr03es04), faculty of sciences of tunis, university of tunis el manar, 1068 tunis, tunisia ∗corresponding author: abir.sboui@nbu.edu.sa, abirsboui@yahoo.fr abstract. for a bounded domain contained in a wedge, we give a new payne-sperb-stakgold type inequality for the solution of a semi-linear equation. the result is isoperimetric in the sense that the sector is the unique extremal domain. 1. introduction for a two-dimensional bounded domain d, payne and rayner proved [9,10] that the eigenfunction u of the dirichlet laplacian corresponding to the fondamental eigenvalue λ(d) satisfies the following inequality ∫ d u2da ≤ λ(d) 4π (∫ d u da )2 , (1.1) where da denotes the lebesgue measure. equality is achieved if, and only if, d is a disk. the importance of this inequality is that it is a reverse cauchy-schwarz type inequality for the first eigenfunction . this inequality was extended to higher dimension by kohler kohler-jobin [5,6]. her inequality states that received: feb. 23, 2022. 2010 mathematics subject classification. 35p15, 45a12, 58e30. key words and phrases. payne-sperb-stakgold inequality; semi-linear equation; isoperimetric inequality. https://doi.org/10.28924/2291-8639-20-2022-20 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-20 2 int. j. anal. appl. (2022), 20:20 ∫ d u2 da ≤ λd/2 2d cd j d−2 d/2−1,1 (∫ d u da )2 (1.2) where d is a bounded domain in rd, cd denotes the volume of the unit ball in rd, and jd/2−1,1 is the first positive zero of the bessel function jd/2−1. using the comparison method due to giorgio talenti, chiti [1] proved that(∫ d uq da )1 q ≤ k(p,q,d) λ d 2 ( 1 p −1 q ) (∫ d up da )1 p for q ≥ p > 0. (1.3) here k(p,q,d) = (d cd) 1 q −1 p j d ( 1 q −1 p ) d 2 −1,1 (∫ 1 0 rd−1+q(1− d 2 )j q d 2 −1 (jd 2 −1,1 r)dr )1 q (∫ 1 0 rd−1+p(1− d 2 )j p d 2 −1 (jd 2 −1,1 r)dr )1 p . equality holds if and only if d is a ball. a more interesting inequality in the spirit of the above has been proved by payne, sperb and stakgold [11] for the following nonlinear problem ∆ u + f (u) = 0 in ω ⊂r2, (1.4) u > 0 in ω ⊂r2, (1.5) u = 0 on ∂ω, for a given continuous function f (t), with f (0) = 0. this includes dirichlet eigenvalue problem for the laplace operator when f (t) = λt. for this problem, the payne-rayner inequality takes the form(∫ ω f (u) dx )2 ≥ 8π ∫ ω f (u) dx (1.6) where f (u) = ∫u 0 f (t)dt. finally, mossino [7] prove a generalization of the latest inequality for the p-laplacian and the case of equality was discussed by kesavan and pacella [4]. our aims is to give a version of payne-sperb stakgold inequality for the case of wedge like domains. 2. preliminary tools and main result before stating our result, we give some notation . let α ≥ 1 and w be the wedge defined in polar coordinates (r,θ) by w = { (r,θ) ∣∣ r > 0, 0 < θ < π α } . (2.1) whenever pertinent, the arc length element will be denoted by ds2 = dr2 + r2dθ2 while the element of area is denoted by da = rdrdθ, and we let v(r,θ) = rα sin αθ. (2.2) then, v is a positive harmonic function in w which is zero on the boundary ∂w. int. j. anal. appl. (2022), 20:20 3 we are interested in the solution u of the following quasi-linear problem: p1 :   ∆u + f (u v )v = 0 in d u > 0 in d u = 0 on ∂d, where d is a sufficiently smooth bounded domain completely contained in the wedge w and the g((r,θ),t) = f ( t v(r,θ) )v(r,θ) is locally hölder continuous and satisfies the following hypotheses. (h1) there exists a ∈ l1(d) and c > 0 such that |g((r,θ),t)| ≤ a(r,θ) + c|t|p,∀((r,θ),t) ∈ d×r, where p > 0. (h2) for t > 0, we have g((r,θ),t) > 0. the role of hypothesis (h1)is to ensure that every weak solution of the problem (p1) is a c2-solution of (p1). notice that, the problem(p1) includes the eigenvalue problem for the laplace operator with dirichlet boundary condition, when we take f (u v ) = λu v . now, if we write the solution of (p1) as u = vw, then the problem above transforms to p2 :   −div(v2∇w) = f (w)v2 in d v > 0 in d v = 0 on ∂d∩w. the solution w may be interpreted as a solution of the nonlinear classical problem (p1) for the 4-dimensional domain symmetric about the x2-axis when α = 1 and for the 6-dimensional domain bi-axially symmetric about the x1-axis and the x2-axis when α = 2, see [8] and [2]. now, we need to introduce some notations and definitions. let µ denoted measure defined by dµ = v2da. then, the weighted unidimensional decreasing rearrangement of the function w with respect to measure µ is the function w∗ : [0,µ(d)] → [0, +∞) defined by w∗(0) = sup w, w∗(ξ) = inf { t ≥ 0; mw (t) < ξ } , ∀ξ ∈ (0,µ(d)], where mw (t) = µ ({ (r,θ) ∈ d; w(r,θ) > t }) , ∀t ∈ [0, sup w]. (2.3) the main result is given in the following theorem. theorem 2.1. let d be a smooth bounded domain completely contained in the wedge. assume that (h1) and (h2) are satisfied. let f be the primitive of f such that f (0) = 0. then the solution u of the problem (p1) satisfies the inequality 4(2α + 2)(2α + 1) ( π 2α(2α + 2) ) 1 α+1 ∫ µ(d) 0 ξ α α+1 f ( ( u v )∗(ξ) ) dξ ≤ ∫ d f ( u v )v2 da. 4 int. j. anal. appl. (2022), 20:20 equality holds if and only if d is a perfect sector sr. the proof of this inequality and the equality case will be discussed in the next section . 3. the weighted version of payne-sperb-stackgold inequality to beginning, we introduce the space w (d,dµ) of measurable functions ϕ that possess weak gradient denoted by |∇ϕ| and satisfy the following conditions (i) ∫ d |∇ϕ|2dµ + ∫ d |ϕ|2dµ < +∞ (ii) there exists a sequence of functions ϕn ∈ c1(d) such that ϕn(r,θ) = 0 on ∂d∩w and lim n→+∞ ∫ d |∇(ϕ−ϕn)|2dµ + ∫ d |ϕ−ϕn|2dµ = 0. (3.1) using the fact that v is harmonic and the divergence theorem , we see ∫ d |∇u|2da = ∫ d |∇(wv)|2da = ∫ d |∇w|2v2da = ∫ d |∇w|2dµ. thus w satisfies the first condition (i). since u is a smooth solution of the problem p1, then w is also smooth and by the boundary condition in p2,we conclude that w satisfies the second condition (ii). then w is in the space w (d,dµ). we introduce now the function φ(t) = ∫ dt f (w)dµ. (3.2) since w and w∗ are equimeaserable then we have φ(t) = ∫ dt f (w)dµ = ∫ m(t) 0 f (w∗)dξ. (3.3) to proceed further, we need to show that m(t) is absolutely continuous on (0,m). indeed, assume that µ({w = t}) is positive. recall that w ∈ w (d,dµ) and proceeding as in the proof of stampacchia’s theorem [3] to conclude that ∇w = 0 almost everywhere on the set {w = t}. substitute this into p2, we obtain f (w) = 0 on and so g((r,θ),u) = f (u v )v = 0 on this set, which contradicts the hypothesis h2. thus, w is continuous on (0,m) and by the fact that w∗ is the left inverse of m(t),we get φ′(t) = f (w∗(m(t))m′(t) = f (t)m′(t). (3.4) by a weak solution to the problem p2 we mean a function w belong to w (d,dµ) and satisfies the equality ∫ d ∇w ·∇ϕdµ = ∫ d f (w)ϕdµ, (3.5) for every ϕ in c1(d), such that ϕ = 0 on ∂d∩w. choose the test function ϕ defined by ϕ(r,θ) = { ( w(r,θ) − t ) , if w(r,θ) > t 0, otherwise , (3.6) int. j. anal. appl. (2022), 20:20 5 where 0 ≤ t < m. plugging (3.6) into (3.5) we get∫ w>t |∇w|2dµ = ∫ w>t f (w)(w − t)dµ. (3.7) then, for � > 0, we have 1 � (∫ w>t |∇w|2dµ− ∫ w>t+� |∇w|2dµ ) = ∫ w>t f (w)dµ+ ∫ t<w≤t+� f (w)( w − t − � � )dµ, (3.8) which, on letting � go to zero, gives, lim �→0 1 � (∫ w>t |∇w|2dµ− ∫ w>t+� |∇w|2dµ ) = ∫ w>t f (w)dµ. (3.9) the same computation for −� gives the same value of the limit. thus − d dt ∫ w>t |∇w|2dµ = ∫ w>t f (w)dµ. (3.10) now, applying the cauchy schwarz inequality( 1 � ∫ t<w≤t+� |∇w|dµ )2 ≤ ( 1 � ∫ t<w≤t+� |∇w|2dµ )( 1 � ∫ t<w≤t+� dµ ) (3.11) and letting � go to zero, we get( − d dt ∫ w>t |∇w|dµ )2 ≤−m′(t)φ(t). (3.12) from the coarea formula, we have − d dt ∫ w>t |∇w|dµ = ∫ ∂{w>t} v2ds. (3.13) then, an application of the payne-weinberger isoperimetric inequality for the wedge-like membrane [12] leads to ( π 2α )2 ( 4α(α + 1) π m(t) )2α+1 α+1 ≤ (∫ ∂{w>t} v2ds )2 ≤−m′(t)φ(t). (3.14) by appealing to (3.13), we obtain ( π 2α ) 1 α+1 (2α + 2) 2α+1 α+1 (m(t)) 2α+1 α+1 f (t) ≤−φ′(t)φ(t). (3.15) integrating both sides of the last relation from 0 to m, then we have ( π 2α ) 1 α+1 (2α + 2) 2α+1 α+1 ∫ m 0 (m(t)) 2α+1 α+1 f (t) ≤ 1 2 φ2(0), (3.16) 6 int. j. anal. appl. (2022), 20:20 since φ(m) = 0. but on the left hand side we have ∫ m 0 (m(t)) 2α+1 α+1 f (t)dt = ∫ m 0 2α + 1 α + 1 f (t) ∫ m(t) 0 ξ α α+1 dξdt (3.17) = 2α + 1 α + 1 ∫ m 0 f (t) ∫ µ(d) 0 ξ α α+1 χ{w∗>t}(ξ)dξdt = 2α + 1 α + 1 ∫ µ(d) 0 ∫ w∗(ξ) 0 f (t)ξ α α+1 dtdξ = 2α + 1 α + 1 ∫ µ(d) 0 f (w∗(ξ))ξ α α+1 dtdξ. substituting the last result into (3.16), the desired inequality in theorem 2.1 follows. moreover, if equality is achieved in theorem 2.1, then obviously inequality (3.15) reduces to equality. since φ′(t) = f (t)m′(t) and f (t) > 0, then equality in (3.15) implies equality in (3.14) and so payneweinberger lemma [12] implies that almost all level sets dt are concentric sectors with fixed angle π α . since d = {w > 0} is the increasing union of such sectors then d is a sector as well. acknowledgement: the authors gratefully acknowledge the approval and the support of this research study by the grant number 7912-sat-2018-3-9-f from the deanship of scientific research at northern border university, arar, ksa. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] g. chiti, an isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, boll. unione mat. ital., vi. ser., a. 1 (1982), 145-151. [2] a. hasnaoui, l. hermi, isoperimetric inequalities for a wedge-like membrane, ann. henri poincaré. 15 (2014), 369–406. https://doi.org/10.1007/s00023-013-0243-y. [3] s. kesavan, topics in functional analysis and applications, new age international, 1989. [4] s. kesavan, p. filomena, symmetry of positive solutions of a quasilinear elliptic equation via isoperimetric inequalities, appl. anal. 54 (1994), 27–37. https://doi.org/10.1080/00036819408840266. [5] m.-t. kohler-jobin, sur la première fonction propre d’une membrane: une extension àn dimensions de l’inégalité isopérimétrique de payne-rayner, j. appl. math. phys. (zamp). 28 (1977), 1137–1140. https://doi.org/10. 1007/bf01601680. [6] m.-t. kohler-jobin, isoperimetric monotonicity and isoperimetric inequalities of payne-rayner type for the first eigenfunction of the helmholtz problem, z. angew. math. phys. 32 (1981), 625–646. https://doi.org/10.1007/ bf00946975. [7] j. mossino, a generalization of the payne-rayner isoperimetric inequality, boll. unione mat. ital., vi. ser., a. 2 (1983), 335–342. https://doi.org/10.1007/s00023-013-0243-y https://doi.org/10.1080/00036819408840266 https://doi.org/10.1007/bf01601680 https://doi.org/10.1007/bf01601680 https://doi.org/10.1007/bf00946975 https://doi.org/10.1007/bf00946975 int. j. anal. appl. (2022), 20:20 7 [8] l.e. payne, isoperimetric inequalities for eigenvalue and their applications, autovalori e autosoluzioni: lectures given at a summer school of the centro internazionale matematico estivo (c.i.m.e.) held in chieti, italy, 1962, g. fichera (ed.), c.i.m.e. summer schools, 27 (1962), 1-58. [9] l.e. payne, m.e. rayner, an isoperimetric inequality for the first eigenfunction in the fixed membrane problem, j. appl. math. phys. (zamp). 23 (1972), 13–15. https://doi.org/10.1007/bf01593198. [10] l.e. payne, m.e. rayner, some isoperimetric norm bounds for solutions of the helmholtz equation, j. appl. math. phys. (zamp). 24 (1973), 105–110. https://doi.org/10.1007/bf01594001. [11] l.e. payne, r. sperb, i. stakgold, on hopf type maximum principles for convex domains, nonlinear anal., theory methods appl. 1 (1977), 547–559. https://doi.org/10.1016/0362-546x(77)90016-5. [12] l.e. payne, h.f. weinberger, a faber-krahn inequality for wedge-like membranes, j. math. phys. 39 (1960), 182–188. https://doi.org/10.1002/sapm1960391182. https://doi.org/10.1007/bf01593198 https://doi.org/10.1007/bf01594001 https://doi.org/10.1016/0362-546x(77)90016-5 https://doi.org/10.1002/sapm1960391182 1. introduction 2. preliminary tools and main result 3. the weighted version of payne-sperb-stackgold inequality references international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 77-84 http://www.etamaths.com characterizations of p-wavelets on positive half line using the walsh-fourier transform abdullah∗ abstract. in this paper, we study the characterization of wavelets on positive half line by means of two basic equations in the fourier domain. we also give another characterization of wavelets. 1. introduction the characterization of wavelets of l2(r) was obtained by gripenberg [7] in terms of two basic equations involving the fourier transform of the wavelets (see also [8]). this result was generalized by calogero [3] for wavelets associated with a general dilation matrix. bownik [2] provided a new approach to characterizing multiwavelets in l2(rn). this characterization was obtained by using the result about shift invariant systems and quasi-affine systems in [4] and [9]. farkov [5] has given general construction of compactly supported orthogonal p-wavelets in l2(r+). farkov et al. [6] gave an algorithm for biorthogonal wavelets related to walsh functions on positive half line. on the other hand, shah and debnath [10] have constructed dyadic wavelet frames on the positive half-line r+ using the walsh-fourier transform and have established a necessary condition and a sufficient condition for the system {ψj,k(x) = 2j/2ψ(2jx k) : j ∈ z,k ∈ z+} to be a frame for l2(r+). further, a constructive procedure for constructing tight wavelet frames on positive halfline using extension principles was recently considered by shah in [11], in which he has pointed out a method for constructing affine frames in l2(r+). moreover, the author has established sufficient conditions for a finite number of functions to form a tight affine frames for l2(r+). in the present paper, we study characterization of wavelet on positive half line by using the results on affine and quasi-affine frames on positive half-line. the paper is structured as follows. in section 2, we introduce some notations and preliminaries related to the operations on positive half-line r+ including the definition of the walsh-fourier transform. in section 3, some results on the affine and quasi-affine systems on positive half-line is given and use them to provide a characterization of wavelets. 2. notations and preliminaries on walsh-fourier analysis let p be a fixed natural number greater than 1. as usual, let r+ = [0,∞) and z+ = {0, 1, ...}. denote by [x] the integer part of x. for x ∈ r+ and for any positive integer j, we set xj = [p jx](mod p), x−j = [p 1−jx](mod p), (2.1) where xj, x−j ∈{0, 1, ...,p− 1}. consider the addition defined on r+ as follows: x⊕y = ∑ j<0 ξjp −j−1 + ∑ j>0 ξjp −j (2.2) 2010 mathematics subject classification. 42c15, 40a30. key words and phrases. wavelets; affine frame; quasi-affine frame; wash-fourier transform. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 77 78 abdullah with ξj = xj + yj(mod p), j ∈ z\{0}, (2.3) where ξj ∈ {0, 1, 2, ...,p − 1} and xj, yj are calculated by (2.1). moreover, we write z = x y if z ⊕y = x. for x ∈ [0, 1), let r0(x) be given by r0(x) =   1, x ∈ [ 0, 1 p ) , εjp, x ∈ [ jp−1, (j + 1)p−1 ) , j = 1, 2, ...,p− 1, (2.4) where εp = exp ( 2πi p ) . the extension of the function r0 to r+ is defined by the equality r0(x + 1) = r0(x), x ∈ r+. then the generalized walsh functions {ωm(x)}m∈z+ are defined by ω0(x) = 1, ωm(x) = k∏ j=0 ( r0 ( pjx ))µj , where m = ∑k j=0 µjp j, µj ∈{0, 1, 2, ...,p− 1}, µk 6= 0. for x, ω ∈ r+, let χ(x,ω) = exp  2πi p ∞∑ j=1 (xjω−j + x−jωj)   , (2.5) where xj and ωj are calculated by (2.1). we observe that χ ( x, m pn−1 ) = χ ( x pn−1 , m ) = ωm ( x pn−1 ) ∀x ∈ [0, pn−1), m ∈ z+. the walsh-fourier transform of a function f ∈ l1(r+) is defined by f̃(ω) = ∫ r+ f(x)χ(x,ω)dx, (2.6) where χ(x,ω) is given by (2.5). if f ∈ l2(r+) and jaf(ω) = ∫ a 0 f(x)χ(x, ω)dx (a < 0), (2.7) then f̃ is defined as limit of jaf in l 2(r+) as a →∞. the properties of walsh-fourier transform are quite similar to the classical fourier transform. it is known that systems {χ(α,.)}∞α=0 and {χ(.,α)}∞α=0 are orthonormal bases in l2(0, 1). let us denote by {ω} the fractional part of ω. for l ∈ z+, we have χ(l,ω) = χ(l,{ω}). if x,y,ω ∈ r+ and x⊕y is p-adic irrational, then χ(x⊕y,ω) = χ(x,ω)χ(y,ω), χ(x y,ω) = χ(x,ω)χ(y,ω), (2.8) characterizations of p-wavelets 79 3. characterization of p-wavelets definition 3.1. let ψ = {ψ1,ψ2, ...,ψl} be a finite family of functions in l2(r+). the affine system generated by ψ is the collection x(ψ) = {ψlj,k : 1 ≤ l ≤ l,j ∈ z,k ∈ z +} where ψlj,k(x) = p j/2ψl(pjx k). the quasi-affine system generated by ψ is x̃(ψ) = {ψ̃lj,k : 1 ≤ l ≤ l,j ∈ z,k ∈ z +}, where ψ̃lj,k(x) = { pj/2ψl(pjx k), j ≥ 0, k ∈ z+ pjψl(pj(x k)), j < 0, k ∈ z+. (3.1) we say that ψ is a set of basic wavelets of l2(r+) if the affine system x(ψ) forms an orthonormal basis for l2(r+). definition 3.2. x ⊂ l2(r+) is a bessel family if there exists b > 0 so that∑ η∈x | < f,η > |2 ≤ b‖f‖2 for f ∈ l2(r+). (3.2) if, in addition, there exists a constant a > 0, a ≤ b such that a‖f‖2 ≤ ∑ η∈x | < f,η > |2 ≤ b‖f‖2 for all f ∈ l2(r+), (3.3) then x is called a frame. the frame is tight if we can choose a and b such that a = b. the (quasi) affine system x(ψ) (resp. xq(ψ)) is a (quasi) affine frame if (3.3) holds for x = x(ψ) (x = xq(ψ)). in [9], chui, shi and stöckler have obsereved the relationship between affine and quasi-affine frame in rn. in [1], we have extended their result to positive half line. theorem 3.3. let ψ be a finite subset of l2(r+). then (a) x(ψ) is a bessel family if and only if x̃(ψ) is a bessel family. furthermore, their exact upper bounds are equal. (b) x(ψ) is an affine frame if and only if x̃(ψ) is a quasi-affine frame. furthermore, their lower and upper exact bounds are equal. definition 3.4. given {ti : i ∈ n}⊂ l2(z+), define the operator h : l2(z+) → l2(n) by h(v) = (< v,ti >)i∈n . if h is bounded then g̃ = h ∗h : l2(z+) → l2(n) is called the dual gramian of {ti : i ∈ n}. observe that g̃ is a non negative definite operator on l2(z+). also, note that for r,s ∈ z+, we have < g̃er,es >=< her,hes >= ∑ i∈n ti(r)ti(s), where {ei : i ∈ z+} is the standard basis of l2(z+). the following result characterizes when the system of translates of a given family of functions is a frame in terms of the dual gramian. theorem 3.5. let {ϕi : i ∈ n}⊂ l2(r+). then for a.e. ξ ∈ [0, 1), let g̃(ξ) denote the dual gramian of {ti = (ϕ̂i(ξ⊕k))k∈z+ : i ∈ n}⊂ l2(z+). the system of translates {tkϕi : k ∈ z+, i ∈ n} is a frame for l2(r+) with constants a,b if and only if g̃(ξ) is bounded for a.e. ξ ∈ [0, 1/2) and 80 abdullah a‖v‖2 ≤〈g̃(ξ)v,v〉≤ b‖v‖2 for v ∈ l2(z+), a.e. ξ ∈ [0, 1/2), that is, the spectrum of g̃(ξ) is contained in [a,b] for a.e. ξ ∈ [0, 1/2) we first prove a lemma which gives necessary and sufficient conditions for the orthonormality of an affine system. lemma 3.6. suppose ψ = {ψ1,ψ2, ...,ψl} ⊂ l2(r+). the affine system x(ψ) is orthonormal in l2(r+) if and only if∑ k∈z+ ψ̂l(ξ ⊕k)ψ̂m(pj(ξ ⊕k)) = δj,0δl,m for a.e. ξ ∈ r+, 1 ≤ l,m ≤ l, j ≥ 0. (3.4) proof. by a simple change of variables 〈ψlj,k,ψ l′ j′,k′〉 = δl,l′δj,j′δk,k′, j,j ′ ∈ z, k,k′ ∈ z+, 1 ≤ l, l′ ≤ l is equivalent to 〈ψlj,k,ψ l′ 0,0〉 = δl,l′δj,0δk,0, j ≥ 0, k ∈ z +, 1 ≤ l, l′ ≤ l. now, let 1 ≤ l, l′ ≤ l, j ≥ 0, k ∈ z+. then δl,l′δj,0δk,0 = 〈ψ̂lj,k, ψ̂ l′ 0,0〉 = ∫ r+ p−j/2ψ̂l(p−jξ)χ(k,p−jξ)ψ̂l ′ (ξ)dξ = ∫ r+ pj/2ψ̂l(ξ)χ(k,ξ)ψ̂l ′ (pjξ)dξ = ∑ n∈z+ pj/2 ∫ n+[0,1/2) ψ̂l(ξ)ψ̂l ′ (pjξ)χ(k,ξ)dξ = pj/2 ∫ [0,1/2) [ ∑ n∈z+ ψ̂l(ξ ⊕n)ψ̂l′(pj(ξ ⊕n)) ] χ(k,ξ)dξ = pj/2 ∫ [0,1/2) k(ξ)χ(k,ξ)dξ, where k(ξ) = [∑ n∈z+ ψ̂ l(ξ ⊕n)ψ̂l′(pj(ξ ⊕n)) ] . the interchange of summation and integration is justified by ∫ [0,1/2) ∑ n∈z+ ∣∣∣ψ̂l(ξ ⊕n)ψ̂l′(pj(ξ ⊕n))∣∣∣dξ = ∫ r+ ∣∣∣ψ̂l(ξ)∣∣∣ ∣∣∣ψ̂l′(pjξ)∣∣∣dξ ≤ p−j/2‖ψl‖2‖ψl ′ ‖2 < ∞. the above computation shows that all fourier coefficients of k(ξ) ∈ l1([0, 1/2)) are zero except for the coefficient corresponding to k = 0 which is 1 if j = 0 and l = l′. therefore, k(ξ) = δj,0δl,l′ for a.e. ξ ∈ [0, 1/2). suppose we have ψ = {ψ1, ...,ψl}⊂ l2(r+). define dj as follows: characterizations of p-wavelets 81 dj = { {0, 1, ...,pj − 1}, j ≥ 0, 0, j < 0. since the quasi affine system xq(ψ) is invariant under shifts by k ∈ z+, we have xq(ψ) = {tkϕ : k ∈ z+, ϕ ∈a}, a = {ψ̃lj,d : j ∈ z, d ∈dj, l = 1, ...,l}. the dual gramian g̃(ξ) of the quasi affine system xq(ψ) at ξ ∈ [0, 1/2) is defined as the dual gramian of {(ϕ̂(ξ ⊕k))k∈z+ : ϕ ∈a}⊂ l2(z+). for s ∈ z+\pz+, define the function ts(ξ) = l∑ l=1 ∞∑ j=0 ψ̂l(pjξ)ψ̂l(pj(ξ ⊕s)). in the following lemma we compute the dual gramian ĝ(ξ) of the quasi-affine system xq(ψ) at ξ ∈ [0, 1/2) in terms of the fourier transforms of the functions in ψ. lemma 3.7. let ψ = {ψ1,ψ2, ...,ψl}⊆ l2(r+) and g̃ be the dual gramian of xq(ψ) at ξ ∈ [0, 1/2). then < g̃(ξ)ek,ek >= l∑ l=1 ∑ j∈z |ψ̂l(pj(ξ ⊕k))|2 for k ∈ z+ (3.7) and < g̃(ξ)ek,ek′ >= tp−m(k′ k)(p −mξ ⊕p−mξ) for k 6= k′ ∈ z+, (3.8) where m = max{j ≥ 0 : p−j(k′ k) ∈ z+}, and functions ts,s ∈ z+\pz+ are given by (3.6). proof. for k,k′ ∈ z+, we have < g̃(ξ)ek,ek′ >= ∑ ϕ∈a ϕ̂(ξ ⊕k)ϕ̂(ξ ⊕k′) = l∑ l=1 ∑ j<0 ψ̂l(p−j(ξ ⊕k))ψ̂l(p−j(ξ ⊕k′)) + l∑ l=1 ∑ j≥0 ψ̂l(p−j(ξ ⊕k))ψ̂l(p−j(ξ ⊕k′))   ∑ d∈dj p−jχ(k,pjd)χ(k′,pjd)   . the expression in the bracket is equal to ∑ d∈dj p−jχ(k,pjd)χ(k′,pjd) = ∑ d∈dj p−jχ((k k′),pjd) = { 1 if k k′ ∈ pjz+ 0 otherwise. therefore, if k = k′, then < g̃(ξ)ek,ek >= l∑ l=1 ∑ j∈z |ψ̂l(pj(ξ ⊕k))|2 if k 6= k′, let m = max{j ≥ 0 : k k′ ∈ pjz+}. then 82 abdullah < g̃(ξ)ek,ek′ >= l∑ l=1 m∑ j=−∞ ψ̂l(p−j(ξ ⊕k))ψ̂l(p−j(ξ ⊕k′)) = l∑ l=1 ∞∑ j=−m ψ̂l(pj(ξ ⊕k))ψ̂l(pj(ξ ⊕k′)) = l∑ l=1 ∑ j≥0 ψ̂l(pj−m(ξ ⊕k))ψ̂l(pj−m(ξ ⊕k′)) = l∑ l=1 ∑ j≥0 ψ̂l(pj(p−mξ ⊕p−mk))ψ̂l(pj(p−mξ ⊕p−mk ⊕p−m(k′ k))) = tp−m(k′ k)(p −mξ ⊕p−mk)). in the following theorem, we provide a characterization of wavelets in terms of two basic equations. theorem 3.8. suppose ψ = {ψ1,ψ2, ...,ψl} ⊂ l2(r+). the affine system x(ψ) is a tight frame with constant 1 for l2(r+), i.e., ‖f‖22 = l∑ l=1 ∑ j∈z ∑ k∈z+ | < f,ψl,j,k > |2 for all f ∈ l2(r+) if and only if l∑ l=1 ∑ j∈z |ψ̂l(pjξ)|2 = 1 for a. e. ξ ∈ r+, (3.9) and ts(ξ) = 0 for a. e. ξ ∈ r+ and for all s ∈ z+\pz+. (3.10) in particular, ψ is a set of basic wavelets of l2(r+) if and only if ‖ψl‖2 = 1 for l = 1, 2, ...,l and (3.9) and (3.10)hold. proof. by theorem 3.3, x(ψ) is a tight frame with constant 1 if and only if xq(ψ) is a tight frame with constant 1. by theorem 3.5, this is equivalent to the spectrum g̃(ξ) consisting of single point 1 i.e. g̃(ξ) is identity on l2(z+) for a.e. ξ ∈ [0, 1/2). by lemma 3.7, this is equivalent to (3.9) and (3.10). by theorem 1.8, section 7.1 in [8], a tight frame x(ψ) is an orthonormal basis if and only if ‖ψl‖2 = 1 for l = 1, 2, ...,l. theorem 3.9. suppose ψ = {ψ1,ψ2, ...,ψl}⊆ l2(r+). then the following are equivalent: (i) x(ψ) is a tight frame with constant 1. (ii) ψ satisfies (3.9) (iii) ψ satisfies l∑ l=1 ∫ r+ |ψ̂l(ξ)|2 dξ |ξ| = ∫ d dξ |ξ| (3.11) where d ⊂ r+ is such that {pjd : j ∈ z} is a partition of r+. proof. it is obvious from theorem 3.8 that (i) ⇒ (ii). to show (ii) implies (iii), assume that (3.9) holds, then characterizations of p-wavelets 83 l∑ l=1 ∫ r+ |ψ̂l(ξ)|2 dξ |ξ| = l∑ l=1 ∑ j∈z ∫ pjd |ψ̂l(ξ)|2 dξ |ξ| = l∑ l=1 ∫ d ∑ j∈z |ψ̂l(pjξ)|2 dξ |ξ| = ∫ d dξ |ξ| to prove (iii) ⇒ (i), we assume that (3.11) holds. since x(ψ) is a bessel family with costant 1, then xq(ψ) is also a bessel family with constant 1 by theorem 3.3 (a). let g̃(ξ) be the dual gramian of xq(ψ) at ξ ∈ [ 0, 1 2 ) . by theorem 3.5, we have ‖g̃(ξ)‖ ≤ 1 for a.e. ξ ∈ [ 0, 1 2 ) . in particular, ‖g̃(ξ)ek‖≤ 1. hence, 1 ≥‖g̃(ξ)ek‖2 = ∑ k′∈z+ | < g̃(ξ)ek,ek′ > |2 = | < g̃(ξ)ek,ek > |2 + ∑ k′∈z+,k 6=k′ | < g̃(ξ)ek,ek′ > |2. (3.12) by lemma 3.7, we have l∑ l=1 ∑ j∈z ∣∣∣ψ̂l(pj(ξ ⊕k))∣∣∣2 ≤ 1 for k ∈ z+,a.e.ξ ∈ [0, 1/2). hence, ∫ d dξ |ξ| = l∑ l=1 ∫ r+ |ψ̂l(ξ)|2 dξ |ξ| = ∫ d   l∑ l=1 ∑ j∈z ∣∣∣ψ̂l(pjξ)∣∣∣2   dξ |ξ| ≤ ∫ d dξ |ξ| , we have ∑l l=1 ∑ j∈z ∣∣∣ψ̂l(pjξ)∣∣∣2 = 1 for a.e. ξ ∈ d and hence for a.e. ξ ∈ r+, i.e., equation (3.9) holds. by lemma 3.7 and (3.9), | < g̃(ξ)ek,ek > |2 = 1 for all k ∈ z+. hence by (3.12), it follows that < g̃(ξ)ek,ek′ >= 0 for k 6= k′ so that g̃(ξ) is the identity operator on l2(z+). hence, by theorem 3.5, xq(ψ) is a tight frame with constant 1. so is x(ψ) by theorem 3.3. theorem 3.10. suppose ψ = {ψ1,ψ2, ...,ψl}⊆ l2(r+). then the following are equivalent: (a) ψ is a set of basic wavelets of l2(r+). (b) ψ satisfies (3.4) and (3.9). (c) ψ satisfies (3.4) and (3.11). proof. it follows from theorem 3.9 and lemma 3.7 that (a) ⇒ (b) ⇒ (c). we now prove that (c) implies (a). assume that ψ satisfies (3.4) and (3.11). the equation (3.4) implies that x(ψ) is an orthonormal system, hence it is a bessel family with constant 1. by theorem 3.9 and (3.11), x(ψ) is a tight frame with constant 1. since each ψl has l2 norm 1, it follows that x(ψ) is an orthonormal basis for l2(k). that is, ψ is a set of basic wavelets of l2(k). references [1] abdullah, affine and quasi-affine frames on positive half line, j. math. ext., in press. [2] m. bownik, on characterizations of multiwavelets in l2(rn), proc. amer. math. soc., 129 (2001), 3265-3274. [3] a. calogero, a characterization of wavelets on general lattices, j. geom. anal. 10 (2000), 597-622. [4] c. k. chui, x. shi and j. stöckler, affine frames, quasi-affine frames, and their duals, adv. comput. math., 8 (1998), 1-17. 84 abdullah [5] y. a. farkov, orthogonal p-wavelets on r+, in proceedings of international conference wavelets and splines, st. petersberg state university, st. petersberg (2005), 4-6. [6] y. a. farkov, a. y. maksimov and s. a. stroganov, on biorthogonal wavelets related to the walsh functions, int. j. wavelets, multiresolut. inf. process. 9(3) (2011), 485-499. [7] g. gripenberg, a necessary and sufficient condition for the existence of a father wavelet, stud. math. 114 (1995), 207-226. [8] e. hernández and g. weiss, a first course on wavelets, crc press, new york, 1996. [9] a. ron and z. shen, frames and stable bases for shift invariant subspaces of l2(rd), canad. j. math., 47 (1995), 1051-1094. [10] f a shah and l. debnath, dyadic wavelet frames on a half-line using the walsh-fourier transform, integ. trans. special funct., 22(2011), 477-486. [11] f. a. shah, tight wavelet frames generated by the walsh polynomials, int. j. wavelets, multiresolut. inf. process., 11 (2013), article id 1350042. department of mathematics, zhdc, university of delhi, jln marg, new delhi-110 002, india ∗corresponding author: abd.zhc.du@gmail.com international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 1-10 http://www.etamaths.com existence results for some nonlinear functional-integral equations in banach algebra with applications lakshmi narayan mishra1,2,∗ h. m. srivastava3,4 and mausumi sen1 abstract. in the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. by utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as darbo’s theorem in banach algebra concerning the estimate on the solutions. the results obtained in this paper extend and improve essentially some known results in the recent literature. we also provide an example of nonlinear functional-integral equation to show the ability of our main result. 1. introduction measures of noncompactness and fixed point theorems are the most valuable and effective implements in the framework of nonlinear analysis, which act as principal role for the solvability of linear and nonlinear integral equations. recently, the theory of such integral equations is developed effectively and emerge in the fields of mathematical analysis, engineering, mathematical physics and nonlinear functional analysis (see [2, 1, 33, 4, 13, 23, 27, 26, 34, 35, 36, 37, 19, 8, 15, 18, 7] and some references therein). in connection with some of the integro-differential equations, the paper should be further motivated by somehow connecting the work with the works [25, 17, 3, 12, 16, 31, 32]. maleknejad et al. [29, 30] examined the existence of solutions for the nonlinear functional-integral equations (for short nlfie) of the form x(t) = g(t,x(t)) + f  t, t∫ 0 u(t,s,x(s))ds,x(α(t))   ,(1.1) and x(t) = f(t,x(α(t)) t∫ 0 u(t,s,x(s))ds,(1.2) respectively, by availing the darbo fixed-point theorem with suitable combination of measure of noncompactness defined in [5]. banaś and sadarangani [11] as well as maleknejad et al. [28] discussed the existence of solutions for nlfie (1.3) f  t, t∫ 0 v(t,s,x(s))ds,x(α(t))   ·g  t, a∫ 0 u(t,s,x(s))ds,x(β(t))   . 2010 mathematics subject classification. 45g10, 47h08, 47h10. key words and phrases. measures of noncompactness; nonlinear functional-integral equation; fixed point theorem; banach algebra. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 mishra, srivastava and sen banaś and rzepka [9, 10] dealt the existence of solutions of nlfie and nonlinear quadratic volterra integral equation x(t) = f(t,x(t)) t∫ 0 u(t,s,x(s))ds,(1.4) x(t) = p(t) + f(t,x(t)) t∫ 0 v(t,s,x(s))ds,(1.5) respectively. the popular nonlinear volterra integral equation and urysohn integral equation are given as follows x(t) = a(t) + t∫ 0 u(t,s,x(s))ds,(1.6) x(t) = b(t) + a∫ 0 v(t,s,x(s))ds,(1.7) respectively. dhage [20] discussed the following nonlinear integral equation x(t) = a(t) a∫ 0 v(t,s,x(s))ds +   t∫ 0 u(t,s,x(s))ds   ·   a∫ 0 v(t,s,x(s))ds   .(1.8) moreover, the familiar quadratic integral equation of chandrasekhar type [14] has the form x(t) = 1 + x(t) a∫ 0 t t + s φ(s)x(s)ds,(1.9) which is applicable in the theories of radiative transfer, neutron transport and kinetic energy of gases (see [14, 22, 24]). in this paper, we study the existence of solutions of nlfie x(t) =  q(t) + f(t,x(t),x(θ(t))) + f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))     ×g  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   ,(1.10) for t ∈ [0,a]. it is worthwhile mentioning that up to now equations (1.1)-(1.9) are a particular case of equation (1.10). moreover, nlfie (1.10) also involve with the functional equation of the first order having the form x(t) = f(t,x(t),x(θ(t))). this paper investigates existence of solutions of nlfie (1.10) under some relevant results of fixed point theorem for the product of two operators which satisfies the darbo condition with suitable combination of a measure of noncompactness in the banach algebra of continuous functions in the interval [0,a]. the existence results are interesting in themselves although their solutions are continuous and stable. 2. definitions and preliminaries this section is devoted to revise some data which will be required in our further circumstances. let e is a real banach space with the norm ‖ ·‖ and zero element θ ′ . symbolically b(x,r) represents the closed ball centered at x and with radius r, as well as we indicates by br the ball b(θ ′ ,r). the notation me appears for the family of all nonempty and bounded subsets of e and notation ne also appears for its subfamily consisting of all relatively compact subsets. additionally, if x(6= φ) ⊂ e nonlinear functional-integral equations 3 then the symbols x̄,convx in consideration of the closure and convex closure of x, respectively. we exercise the definition on the concept of a measure of noncompactness [5] as follows. definition 2.1. let x ∈me and µ(x) = inf { δ > 0 : x = m⋃ i=1 xi with diam(xi) ≤ δ, i = 1, 2, ...m } , where for a fixed number t ∈ [0,a], we denote diam x(t) = sup{|x(t) −y(t)| : x,y ∈ x}. clearly, 0 ≤ µ(x) < ∞. µ(x) is called the kuratowski measure of noncompactness. theorem 2.1. let x,y ∈me and λ ∈ r. then (i) µ(x) = 0 if and only if x ∈ne; (ii) x ⊆ y ⇒ µ(x) ≤ µ(y ); (iii) µ(x̄) = µ(convx) = µ(x); (iv) µ(x ∪y ) = max{µ(x),µ(y )}; (v) µ(λx) = |λ|µ(x), where λx = {λx : x ∈ x}; (vi) µ(x + y ) ≤ µ(x) + µ(y ), where x + y = {x + y : x ∈ x,y ∈ y}; (vii) |µ(x) −µ(y )| ≤ 2dh(x,y ), where dh(x,y ) denotes the hausdorff metric of x and y , i.e. dh(x,y ) = max { sup y∈y d(y,x), sup x∈x d(x,y ) } , where d(., .) is the distance from an element of e to a set of e. furthermore, every function µ : me → [0,∞), satisfying conditions (i)-(vi) of theorem 2.1, will be called a regular measure of noncompactness in the banach space e (cf. [9]). now let us theorize that ω is a nonempty subset of a banach space e and s : ω → e is a continuous operator, which transforms bounded subsets of ω to bounded ones. additionally, let µ be a regular measure of noncompactness in e. definition 2.2. (see [5]) the continuous operator s satisfies the darbo condition with a constant k ′ with respect to measure µ provided µ(sx) ≤ k ′ µ(x) for each x ∈me such that x ⊂ ω. if k ′ < 1, then s is called a contraction with respect to µ. in the continuation, consider the space c[0,a] is consisting of all real functions defined and continuous on the interval [0,a]. the space c[0,a] is equipped with standard norm ‖x‖ = sup{|x(t)| : t ∈ [0,a]}. evidently, the space c[0,a] has also the structure of banach algebra. taking into our considerations, we will utilize a regular measure of noncompactness defined in [6] (cf. also [5]). let us fix a set x ∈ mc[0,a]. for x ∈ x and for a given � > 0 denote by w(x,�) the modulus of continuity of x, i.e., w(x,�) = sup{|x(t) −x(s)| : t,s ∈ [0,a]; |t−s| ≤ �}. further, put w(x,�) = sup{w(x,�) : x ∈ x}, w0(x) = lim �→0 w(x,�). the function w0(x) is a regular measure of noncompactness in the space c[0,a], which can be shown in [6]. for our purposes we will require the following lemma and theorem [21, 6]. lemma 2.1. let d be a bounded, closed and convex subset of e. if operator s : d → d is a strict set contraction, then s has a fixed point in d. 4 mishra, srivastava and sen theorem 2.2. let us suppose that ω is a nonempty, bounded, convex and closed subset of c[0,a] and the operators p and t transform continuously the set ω into c[0,a], just like that p(ω) and t(ω) are bounded. furthermore, let the operator s = p ·t transform ω into itself. if the each operators p and t satisfies the darbo condition on the set ω with the constants k1 and k2, respectively, then the operator s satisfies the darbo condition on ω with the constant ‖p(ω)‖k2 + ‖t(ω)‖k1. remark 2.1. in theorem 2.2, if ‖p(ω)‖k2 + ‖t(ω)‖k1 < 1, then s is a contraction with respect to the measure w0 and has at least one fixed point in the set ω. now we will identify solutions of the integral equation (1.10). 3. main result in this section, we will study the solvability of nlfie (1.10) for x ∈ c[0,a], under the following hypotheses. (a1) the function q : [0,a] → r is continuous and bounded with k = supt∈[0,a] |q(t)|. (a2) the functions f : [0,a] × r × r → r; f,g : [0,a] × r × r × r → r are continuous and there exists nonnegative constants l,m such that |f(t, 0, 0)| ≤ l, |f(t, 0, 0, 0)| ≤ m, |g(t, 0, 0, 0)| ≤ m. (a3) there exists the continuous functions aj : [0,a] → [0,a], for j = 1, 2, ...8 such that |f(t,x1,x2) −f(t,y1,y2)| ≤ a1(t)|x1 −y1| + a2(t)|x2 −y2|, |f(t,x1,y1,x2) −f(t,x3,y2,x4)| ≤ a3(t)|x1 −x3| + a4(t)|y1 −y2| + a5(t)|x2 −x4|, |g(t,x1,y1,x2) −g(t,x3,y2,x4)| ≤ a6(t)|x1 −x3| + a7(t)|y1 −y2| + a8(t)|x2 −x4|, for all t ∈ [0,a] and x1,x2,x3,x4,y1,y2 ∈ r. (a4) the functions u = u(t,s,x(á(s))) and v = v(t,s,x(c(s))) act continuously from the set [0,a]× [0,a] × r into r. moreover, the functions θ, á,b,c and d transform continuously the interval [0,a] into itself. (a5) there exists a nonnegative constant k such that k = max j {aj(t) : t ∈ [0,a]}, for j = 1, 2, ...8. (a6) (sublinear condition) there exists the constants ξ and η such that |u(t,s,x(á(s)))| ≤ ξ + η|x|, |v(t,s,x(c(s)))| ≤ ξ + η|x|, for all t,s ∈ [0,a] and x ∈ r. (a7) 4στ < 1, for σ = 4k + kaη and τ = k + l + kaξ + m. now we can formulate the main result of this paper. theorem 3.1. under the assumptions (a1) − (a7), nlfie (1.10) has at least one solution in the banach algebra c = c[0,a]. proof. to prove this result using theorem 2.2, we consider the operators p and t on the banach algebra c[0,a] in the following way: (px)(t) = q(t) + f(t,x(t),x(θ(t))) + f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))   , (tx)(t) = g  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   , nonlinear functional-integral equations 5 for t ∈ [0,a]. now, taking into account the assumptions (a1), (a2) and (a4), it is clear that p and t transforms the banach algebra c[0,a] into itself. now, the operator s defined on the algebra c[0,a] as follows sx = (px) · (tx). definitely, s transform c[0,a] into itself. next, let us fix x ∈ c[0,a], then using our imposed assumptions for t ∈ [0,a], we obtain |(sx)(t)| = |(px)(t)|× |(tx)(t)| = ∣∣∣∣∣∣q(t) + f(t,x(t),x(θ(t))) + f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))   ∣∣∣∣∣∣ × ∣∣∣∣∣∣g  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   ∣∣∣∣∣∣ ≤ { k + |f(t,x(t),x(θ(t))) −f(t, 0, 0)| + |f(t, 0, 0)| + ∣∣∣∣∣∣f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −f(t, 0, 0, 0) ∣∣∣∣∣∣ + |f(t, 0, 0, 0)| } ×   ∣∣∣∣∣∣g  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))  −g(t, 0, 0, 0) ∣∣∣∣∣∣ + |g(t, 0, 0, 0)|   ≤  k + a1(t)|x(t)| + a2(t)|x(θ(t))| + l + a3(t)|x(t)| + a4(t) t∫ 0 |u(t,s,x(á(s)))|ds + a5(t)|x(b(t))| + m   ×  a6(t)|x(t)| + a7(t) a∫ 0 |v(t,s,x(c(s)))|ds + a8(t)|x(d(t))| + m   ≤{k + 4k‖x‖ + l + ka(ξ + η‖x‖) + m} ·{2k‖x‖ + ka(ξ + η‖x‖) + m} ≤{(4k + kaη)‖x‖ + k + l + kaξ + m}2. let σ = 4k + kaη and τ = k + l + kaξ + m, then from the above estimate, it follows that ‖px‖≤ σ‖x‖ + τ,(3.1) ‖tx‖≤ σ‖x‖ + τ,(3.2) ‖sx‖≤ (σ‖x‖ + τ)2,(3.3) for x ∈ c[0,a]. from estimate (3.3), we conclude that the operator s maps the ball br ⊂ c[0,a] into itself for r1 ≤ r ≤ r2, where r1 = 1 − 2στ − √ 1 − 4στ 2σ2 , r2 = 1 − 2στ + √ 1 − 4στ 2σ2 . in the following, we will assume that r = r1. moreover, let us observe that from estimates (3.1) and (3.2), we obtain ‖pbr‖≤ σr + τ,(3.4) ‖tbr‖≤ σr + τ.(3.5) 6 mishra, srivastava and sen now, we have to prove that the operator p is continuous on the ball br. to do this, fix � > 0 and take arbitrary x,y ∈ br such that ‖x−y‖≤ �. then for t ∈ [0,a], we have |(px)(t) − (py)(t)| ≤ |f(t,x(t),x(θ(t))) −f(t,y(t),y(θ(t)))| + ∣∣∣∣∣∣f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −f  t,y(t), t∫ 0 u(t,s,y(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ ≤ a1(t)|x(t) −y(t)| + a2(t)|x(θ(t)) −y(θ(t))| + ∣∣∣∣∣∣f  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −f  t,y(t), t∫ 0 u(t,s,x(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ + ∣∣∣∣∣∣f  t,y(t), t∫ 0 u(t,s,x(á(s)))ds,y(b(t))  −f  t,y(t), t∫ 0 u(t,s,y(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ ≤ a1(t)|x(t) −y(t)| + a2(t)|x(θ(t)) −y(θ(t))| + a3(t)|x(t) −y(t)| + a5(t)|x(θ(t)) −y(θ(t))| + a4(t) t∫ 0 |u(t,s,x(á(s))) −u(t,s,y(á(s)))|ds ≤ 4k‖x−y‖ + ka w(u,�) ≤ 4k� + ka w(u,�), where w(u,�) = sup{|u(t,s,x) −u(t,s,y)| : t,s ∈ [0,a]; x,y ∈ [−r,r];‖x−y‖≤ �}. in view of uniformly continuous of the function u = u(t,s,x) on the bounded subset [0,a]×[0,a]×[−r,r], we have that w(u,�) → 0 as � → 0. thus, from the above inequality the operator p is continuous on br. similarly, the operator t is also continuous on br. hence, we conclude that s is continuous operator on br. next, we prove that the operators p and t satisfies the darbo condition with respect to the measure w0, defined in section 2, in the ball br. to do this, we take a nonempty subset x of br and x ∈ x. let � > 0 be fixed and t1, t2 ∈ [0,a] with t2 − t1 ≤ � and we can assume that t1 ≤ t2. then, taking into account our assumptions, it follows |(px)(t2) − (px)(t1)| ≤ |q(t2) −q(t1)| + |f(t2,x(t2),x(θ(t2))) −f(t1,x(t1),x(θ(t1)))| + ∣∣∣∣∣∣f  t2,x(t2), t2∫ 0 u(t2,s,x(á(s)))ds,x(b(t2))   − f  t1,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ ≤ w(q,�) + |f(t2,x(t2),x(θ(t2))) −f(t2,x(t1),x(θ(t1)))| + |f(t2,x(t1),x(θ(t1))) −f(t1,x(t1),x(θ(t1)))| + ∣∣∣∣∣∣f  t2,x(t2), t2∫ 0 u(t2,s,x(á(s)))ds,x(b(t2))   (3.6) nonlinear functional-integral equations 7 − f  t2,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ + ∣∣∣∣∣∣f  t2,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   − f  t1,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ ≤ w(q,�) + a1(t)|x(t2) −x(t1)| + a2(t)|x(θ(t2)) −x(θ(t1))| + wf (�, ., .) + a3(t)|x(t2) −x(t1)| + a4(t) ∣∣∣∣∣∣ t2∫ 0 u(t2,s,x(á(s)))ds− t1∫ 0 u(t1,s,x(á(s)))ds ∣∣∣∣∣∣ + a5(t)|x(b(t2)) −x(b(t1))| + wf (�, ., ., .) ≤ w(q,�) + 2kw(x,�) + kw(x,w(θ,�)) + wf (�, ., .) + k   t1∫ 0 |u(t2,s,x(á(s))) −u(t1,s,x(á(s)))|ds + t2∫ t1 |u(t2,s,x(á(s)))|ds   + kw(x,w(b,�)) + wf (�, ., ., .) w(px,�) ≤ w(q,�) + 2kw(x,�) + kw(x,w(θ,�)) + wf (�, ., .) + k{wu(�, ., .)a + k′�} + kw(x,w(b,�)) + wf (�, ., ., .)(3.7) where wf (�, ., .) = sup{|f(t,x1,x2) −f(t′,x1,x2)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x1,x2 ∈ [−r,r]}, wu(�, ., .) = sup{|u(t,s,x) −u(t′,s,x)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x ∈ [−r,r]}, wf (�, ., ., .) = sup{|f(t,x1,y1,x2) −f(t′,x1,y1,x2)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x1,x2 ∈ [−r,r]; y1 ∈ [−k′a,k′a]}, k′ = sup{|u(t,s,x)| : t,s ∈ [0,a]; x ∈ [−r,r]}. since, the functions q = q(t), f = f(t,x1,x2) and f = f(t,x1,y1,x2) are uniformly continuous on the set [0,a], [0,a]×r×r and [0,a]×r×r×r, respectively, and the function u = u(t,s,x) is also uniformly continuous on the set [0,a]×[0,a]×r. hence, we deduce that w(q,�) → 0,wf (�, ., .) → 0,wu(�, ., .) → 0 and wf (�, ., ., .) → 0 as � → 0. thus, from the above estimate (3.6) we conclude (3.8) w0(px) ≤ 4kw0(x). similarly, we can show that (3.9) w0(tx) ≤ 2kw0(x). finally, from the estimates (3.4), (3.5), (3.7), (3.8) and keeping in mind theorem 2.2, we conclude that the operator s satisfies the darbo condition on br with respect to the measure w0 with constant 4k(σr + τ) + 2k(σr + τ). thus, we have 6k(σr + τ) = 6k(σr1 + τ) = 6k { σ ( (1 − 2στ) − √ 1 − 4στ 2σ2 ) + τ } = 3k σ (1 − √ 1 − 4στ). 8 mishra, srivastava and sen taking into account the assumption (a7), since 1 − √ 1 − 4στ < 1 and 3k σ = 3k 4k + kaη < 1. therefore, the operator s is a contraction on br with respect to measure w0. thus, s has at least one fixed point in the ball br, by applying theorem 2.2 and remark 2.1. consequently, the nlfie (1.10) has at least one solution in the ball br. � 4. an example now, we begin with an example of a nlfie and to illustrate the existence of its solutions by using theorem 3.1. example 4.1. consider the following nlfie: x(t) = [ te−(t+3) + t 7(1 + t) arctan |x(t)| + t 16 ln(1 + |x(1 − t)|) + 1 12 t∫ 0 { cos(x(1 −s)) 3 + 2t arctan ( |x(1 −s)| 1 + |x(1 −s)| )} ds ] × [ 1 17 1∫ 0 { t sin x( √ s) 3 + (1 + t) ln(1 + |x( √ s)|) } ds ] ,(4.1) where t ∈ [0, 1]. observe that equation (4.1) is a particular case of equation (1.10). let us take q : [0, 1] → r; f : [0, 1] ×r×r → r; f,g : [0, 1] ×r×r×r → r and u,v : [0, 1] × [0, 1] ×r → r and comparing (4.1) with equation (1.10), we get q(t) = te−(t+3),f(t,x1,x2) = t 7(1 + t) arctan |x1| + t 16 ln(1 + |x2|), f(t,x1,y1,x2) = 1 12 y1,g(t,x1,y1,x2) = 1 17 y1, u(t,s,x) = cos x 3 + 2t arctan ( |x| 1 + |x| ) ,v(t,s,x) = t sin x 3 + (1 + t) ln(1 + |x|), then we can easily test that the assumptions of theorem 3.1 are satisfied. in fact, we have that the function q(t) is continuous and bounded on [0, 1] with k = e−4 = 0.0183156... . thus, the assumption (a1) is satisfied. moreover, these functions are continuous and satisfies the assumption (a3) with a1 = 1 14 ,a2 = 1 16 ,a3 = a5 = a6 = a8 = 0,a4 = 1 12 ,a7 = 1 17 . in this case, we have k = max { 1 14 , 1 16 , 0, 1 12 , 1 17 } = 1 12 . further, |f(t, 0, 0)| = 0, |f(t, 0, 0, 0)| = 0, |g(t, 0, 0, 0)| = 0, |u(t,s,x)| ≤ 1 3 + 2|x|, |v(t,s,x)| ≤ 1 3 + 2|x|. it is observed that l = m = 0,ξ = 1 3 ,η = 2 and a = 1. finally, we see that 4στ = 4(4k + kaη)(k + l + kaξ + m) < 1. hence, all the assumptions from (a1) to (a7) are satisfied. now, on the basis of result obtained in theorem 3.1, we deduce that nlfie (4.1) has at least one solution in banach algebra c[0, 1]. nonlinear functional-integral equations 9 acknowledgments the authors wishes to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. the authors are also grateful to all the editorial board members and reviewers of this esteemed journal. the first author lakshmi narayan mishra is thankful to the ministry of human resource development, new delhi, india and department of mathematics, national institute of technology, silchar, india for supporting this research article. references [1] r.p. agarwal, n. hussain, m.a. taoudi, fixed point theorems in ordered banach spaces and applications to nonlinear integral equations, abstr. appl. anal. 2012 (2012), article id 245872. [2] r.p. agarwal, d. o’regan, p.j.y. wong, positive solutions of differential, difference and integral equations, kluwer academic, dordrecht, 1999. [3] g. anichini, g. conti, existence of solutions of some quadratic integral equations, opuscula math. 28 (4) (2008), 433-440. [4] j. banaś, a. chlebowicz, on existence of integrable solutions of a functional integral equation under carathéodory conditions, nonlinear anal. 70 (9) (2009), 3172-3179. [5] j. banaś, k. goebel, measures of noncompactness in banach spaces, lecture notes in pure and applied mathematics, vol. 60, marcel dekker, new york, 1980. [6] j. banaś, m. lecko, fixed points of the product of operators in banach algebra, panamer. math. j. 12 (2) (2002), 101-109. [7] j. banaś, m. mursaleen, sequence spaces and measures of noncompactness with applications to differential and integral equations, springer, new york, 2014. [8] j. banaś, b. rzepka, an application of a measure of noncompactness in the study of asymptotic stability, appl. math. lett. 16 (1) (2003), 1-6. [9] j. banaś, b. rzepka, on existence and asymptotic stability of solutions of a nonlinear integral equation, j. math. anal. appl. 284 (1) (2003), 165-173. [10] j. banaś, b. rzepka, on local attractivity and asymptotic stability of solutions of a quadratic volterra integral equation, appl. math. comput. 213 (1) (2009), 102-111. [11] j. banaś, k. sadarangani, solutions of some functional-integral equations in banach algebra, math. comput. modelling 38 (2003), 245-250. [12] v.c. boffi, g. spiga, an equation of hammerstein type arising in particle transport theory, j. math. phys. 24 (6) (1983), 1625-1629. [13] t.a. burton, b. zhang, fixed point and stability of an integral equation: nonuniqueness, appl. math. lett. 17 (7) (2004), 839-846. [14] s. chandrasekhar, radiative transfer, oxford univ press, london, 1950. [15] c. corduneanu, integral equations and applications, cambridge university press, new york, 1990. [16] g. darbo, punti uniti in trasformazioni a codominio non compatto, rend. sem. mat. univ. padova 24 (1955), 84-92. [17] m.a. darwish, k. sadarangani, nondecreasing solutions of a quadratic abel equation with supremum in the kernel, appl. math. comput. 219 (14) (2013), 7830-7836. [18] deepmala, h.k. pathak, a study on some problems on existence of solutions for nonlinear functional-integral equations, acta math. sci. 33 (5) (2013), 1305-1313. [19] k. deimling, nonlinear functional analysis, springer-verlag, new york, 1985. [20] b. c. dhage, on α-condensing mappings in banach algebras, math. student 63 (1994), 146-152. [21] d. guo, v. lakshmikantham, x.z. liu, nonlinear integral equations in abstract spaces, kluwer, dordrecht, 1996. [22] s. hu, m. khavani, w. zhuang, integral equations arising in the kinetic theory of gases, appl. anal. 34 (1989), 261-266. [23] x.l. hu, j.r. yan, the global attractivity and asymptotic stability of solution of a nonlinear integral equation, j. math. anal. appl. 321 (1) (2006), 147-156. [24] c.t. kelly, approximation of solutions of some quadratic integral equations in transport theory, j. integral equations 4 (3) (1982), 221-237. [25] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematical studies, vol. 204, elsevier (north-holland) science publishers, amsterdam, london and new york, 2006. [26] z. liu, s.m. kang, existence and asymptotic stability of solutions to a functional-integral equation, taiwanese j. math. 11 (1) (2007), 187-196. [27] z. liu, s.m. kang, existence of monotone solutions for a nonlinear quadratic integral equation of volterra type, rocky mountain j. math. 37 (6) (2007), 1971-1980. [28] k. maleknejad, r. mollapourasl, k. nouri, study on existence of solutions for some nonlinear functional integral equations, nonlinear anal. 69 (8) (2008), 2582-2588. 10 mishra, srivastava and sen [29] k. maleknejad, k. nouri, r. mollapourasl, existence of solutions for some nonlinear integral equations, commun. nonlinear sci. numer. simul. 14 (6) (2009), 2559-2564. [30] k. maleknejad, k. nouri, r. mollapourasl, investigation on the existence of solutions for some nonlinear functionalintegral equations, nonlinear anal. 71 (12) (2009), e1575-e1578. [31] l.n. mishra, r.p. agarwal, m. sen, solvability and asymptotic behavior for some nonlinear quadratic integral equation involving erdélyi-kober fractional integrals on the unbounded interval, progr. fract. differ. appl. in press. [32] l.n. mishra, m. sen, on the concept of existence and local attractivity of solutions for some quadratic volterra integral equation of fractional order, appl. math. comput. 285 (2016) 174-183. [33] l.n. mishra, m. sen, r.n. mohapatra, on existence theorems for some generalized nonlinear functional-integral equations with applications, filomat, in press. [34] d. o’regan, existence results for nonlinear integral equations, j. math. anal. appl. 192 (3) (1995), 705-726. [35] d. o’regan, m. meehan, existence theory for nonlinear integral and integrodifferential equations, kluwer, dordrecht, 1998. [36] h.k. pathak, deepmala, remarks on some fixed point theorems of dhage, appl. math. lett. 25 (11) (2012), 1969-1975. [37] p.p. zabrejko, a.i. koshelev, m.a. krasnosel’skii, s.g. mikhlin, l.s. rakovshchik, v.j. stetsenko, integral equations, noordhoff, leyden, 1975. 1department of mathematics, national institute of technology, silchar 788 010, cachar, assam, india 2l. 1627 awadh puri colony, phase iii, beniganj, opposite industrial training institute (i.t.i.), ayodhya main road, faizabad 224 001, uttar pradesh, india 3department of mathematics and statistics, university of victoria, victoria, british columbia v8w 3r4, canada 4china medical university, taichung 40402, taiwan, republic of china ∗corresponding author: lakshminarayanmishra04@gmail.com international journal of analysis and applications issn 2291-8639 volume 9, number 2 (2015), 90-95 http://www.etamaths.com titchmarsh’s theorem for the cherednik-opdam transform in the space l2α,β(r) s. el ouadih∗ and r. daher abstract. in this paper, we prove the generalization of titchmarshs theorem for the cherednikopdam transform for functions satisfying the (ψ, 2)-cherednik-opdam lipschitz condition in the space l2 α,β (r). 1. introduction and preliminaries in [3], e. c. titchmarsh’s characterizes the set of functions in l2(r) satisfying the cauchylipschitz condition by means of an asymptotic estimate growth of the norm of their fourier transform, namely we have theorem 1.1 [3] let δ ∈ (0, 1) and assume that f ∈ l2(r). then the following are equivalents (i) ‖f(t + h) −f(t)‖ = o(hδ), as h → 0, (ii) ∫ |λ|≥r |f̂(λ)|2dλ = o(r−2δ) as r →∞, where f̂ stands for the fourier transform of f. in this paper, we prove the generalization of theorem 1.1 for the cherednik-opdam transform for functions satisfying the (ψ, 2)-cherednik-opdam lipschitz condition in the space l2α,β(r). for this purpose, we use the generalized translation operator. in this section, we develop some results from harmonic analysis related to the differentialdifference operator t(α,β). further details can be found in [1] and [2]. in the following we fix parameters α, β subject to the constraints α ≥ β ≥−1 2 and α > −1 2 . let ρ = α + β + 1 and λ ∈ c. the opdam hypergeometric functions g(α,β)λ on r are eigenfunctions t(α,β)g (α,β) λ (x) = iλg (α,β) λ (x) of the differential-difference operator t(α,β)f(x) = f′(x) + [(2α + 1) coth x + (2β + 1) tanh x] f(x) −f(−x) 2 −ρf(−x), that are normalized such that g (α,β) λ (0) = 1. in the notation of cherednik one would write t(α,β) as t(k1 + k2)f(x) = f ′(x) + { 2k1 1 + e−2x + 4k2 1 −e−4x } (f(x) −f(−x)) − (k1 + 2k2)f(x), with α = k1 +k2− 12 and β = k2− 1 2 . here k1 is the multiplicity of a simply positive root and k2 the (possibly vanishing) multiplicity of a multiple of this root. by [1] or [2], the eigenfunction 2010 mathematics subject classification. 46l08. key words and phrases. cherednik-opdam operator; cherednik-opdam transform; generalized translation. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 90 titchmarsh’s theorem for the cherednik-opdam transforml spectrum 91 g (α,β) λ is given by g (α,β) λ (x) = ϕ α,β λ (x) − 1 ρ− iλ ∂ ∂x ϕ α,β λ (x) = ϕ α,β λ (x) + ρ 4(α + 1) sinh(2x)ϕ α+1,β+1 λ (x), where ϕ α,β λ (x) =2 f1( ρ+iλ 2 ; ρ−iλ 2 ; α + 1;−sinh2 x) is the classical jacobi function. lemma 1.2. [4] the following inequalities are valids for jacobi functions ϕ α,β λ (x) (i) |ϕα,βλ (x)| ≤ 1. (ii) 1 −ϕα,βλ (x) ≤ x 2(λ2 + ρ2). (iii) there is a constant c > 0 such that 1 −ϕα,βλ (x) ≥ c, for λx ≥ 1. denote l2α,β(r), the space of measurable functions f on r such that ‖f‖2,α,β = (∫ r |f(x)|2aα,β(x)dx )1/2 < +∞, where aα,β(x) = (sinh |x|)2α+1(cosh |x|)2β+1. the cherednik-opdam transform of f ∈ cc(r) is defined by hf(λ) = ∫ r f(x)g (α,β) λ (−x)aα,β(x)dx for all λ ∈ c. the inverse transform is given as h−1g(x) = ∫ r g(λ)g (α,β) λ (x) ( 1 − ρ iλ ) dλ 8π|cα,β(λ)|2 , here cα,β(λ) = 2ρ−iλγ(α + 1)γ(iλ) γ( 1 2 (ρ + iλ))γ( 1 2 (α−β + 1 + iλ)) . the corresponding plancherel formula was established in [1], to the effect that∫ r |f(x)|2aα,β(x)dx = ∫ +∞ 0 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ), where f̌(x) := f(−x) and dσ is the measure given by dσ(λ) = dλ 16π|cα,β(λ)|2 . according to [2] there exists a family of signed measures µ (α,β) x,y such that the product formula g (α,β) λ (x)g (α,β) λ (y) = ∫ r g (α,β) λ (z)dµ (α,β) x,y (z), holds for all x,y ∈ r and λ ∈ c, where dµ(α,β)x,y (z) =   kα,β(x,y,z)aα,β(z)dz if xy 6= 0 dδx(z) if y = 0 dδy(z) if x = 0 and kα,β(x,y,z) = mα,β|sinh x. sinh y. sinh z|−2α ∫ π 0 g(x,y,z,χ) α−β−1 + × [1 −σχx,y,z + σ χ x,z,y + σ χ z,y,x + ρ β + 1 2 coth x. coth y. coth z(sin χ)2] × (sin χ)2βdχ 92 ouadih and daher if x,y,z ∈ r\{0} satisfy the triangular inequality ||x|−y|| < |z| < |x|+|y|, and kα,β(x,y,z) = 0 otherwise. here ∀x,y,z ∈ r,χ ∈ [0, 1],σχx,y,z =   cosh x+cosh y−cosh z cos χ sinh x sinh y if xy 6= 0 0 if xy = 0 and g(x,y,z,χ) = 1 − cosh2 x− cosh2 y. cosh2 z + 2 cosh x. cosh y. cosh z. cos χ. lemma 1.3. [2] for all x,y ∈ r, we have (i) kα,β(x,y,z) = kα,β(y,x,z). (ii) kα,β(x,y,z) = kα,β(−x,z,y). (iii) kα,β(x,y,z) = kα,β(−z,y,−x). the product formula is used to obtain explicit estimates for the generalized translation operators τ(α,β)x f(y) = ∫ r f(z)dµ(α,β)x,y (z). it is known from [2] that hτ(α,β)x f(λ) = g (α,β) λ (x)hf(λ),(1.1) for f ∈ cc(r). 2. main result in this section we give the main result of this paper. we need first to define (ψ, 2)-cherednikopdam lipschitz class. denote nh by nh = τ (α,β) h + τ (α,β) −h − 2i, where i is the unit operator in the space l2α,β(r). definition 2.1. a function f ∈ l2α,β(r) is said to be in the (ψ, 2)-cherednik-opdam lipschitz class, denoted by lip(ψ, 2), if ‖nhf(x)‖2,α,β = o(ψ(h)) as h → 0, where ψ is a continuous increasing function on [0,∞),ψ(0) = 0 , ψ(ts) = ψ(t)ψ(s) for all t,s ∈ [0,∞) and this function verify∫ 1/h 0 sψ(s−2)ds = o(h−2ψ(h2)), h → 0. lemma 2.2. if f ∈ cc(r), then hτ̌(α,β)x f(λ) = g (α,β) λ (−x)hf̌(λ).(2.1) proof. for f ∈ cc(r), we have hτ̌(α,β)x f(λ) = ∫ r τ(α,β)x f(−y)g (α,β) λ (−y)aα,β(y)dy = ∫ r τ(α,β)x f(y)g (α,β) λ (y)aα,β(y)dy = ∫ r [∫ r f(z)kα,β(x,y,z)aα,β(z)dz ] g (α,β) λ (y)aα,β(y)dy = ∫ r f(z) [∫ r g (α,β) λ (y)kα,β(x,y,z)aα,β(y)dy ] aα,β(z)dz. titchmarsh’s theorem for the cherednik-opdam transforml spectrum 93 since kα,β(x,y,z) = kα,β(−x,z,y), it follows from the product formula that hτ̌(α,β)x f(λ) = g (α,β) λ (−x) ∫ r f(z)g (α,β) λ (z)aα,β(z)dz = g (α,β) λ (−x) ∫ r f(−z)g(α,β)λ (−z)aα,β(z)dz = g (α,β) λ (−x)hf̌(λ). lemma 2.3. for f ∈ l2α,β(r), then ‖nhf(x)‖22,α,β = 4 ∫ +∞ 0 |ϕα,βλ (h) − 1| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ). proof. from formulas (1) and (2), we have h(nhf)(λ) = (g (α,β) λ (h) + g (α,β) λ (−h) − 2)h(f)(λ), and h(ňhf)(λ) = (g (α,β) λ (−h) + g (α,β) λ (h) − 2)h(f̌)(λ). since g (α,β) λ (h) = ϕ α,β λ (h) + ρ 4(α + 1) sinh(2h)ϕ α+1,β+1 λ (h), and ϕ α,β λ is even, then h(nhf)(λ) = 2(ϕ α,β λ (h) − 1)h(f)(λ) and h(ňhf)(λ) = 2(ϕ α,β λ (h) − 1)h(f̌)(λ). now by plancherel theorem, we have the result. theorem 2.4. let f ∈ l2α,β(r). then the following are equivalents (a) f ∈ lip(ψ, 2), (b) ∫ +∞ r ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) = o(ψ(r−2)), as r →∞. proof. (a) ⇒ (b) let f ∈ lip(ψ, 2). then we have ‖nhf(x)‖2,α,β = o(ψ(h)) as h → 0. from lemma 2.2, we have ‖nhf(x)‖22,α,β = 4 ∫ +∞ 0 |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ. if λ ∈ [ 1 h , 2 h ], then λh ≥ 1 and (iii) of lemma 1.2 implies that 1 ≤ 1 c2 |1 −ϕα,βλ (h)| 2. then∫ 2 h 1 h ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) ≤ 1 c2 ∫ 2 h 1 h |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) ≤ 1 c2 ∫ +∞ 0 |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) ≤ 1 4c2 ‖nhf(x)‖22,α,β = o(ψ(h2)). 94 ouadih and daher we obtain ∫ 2r r ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) ≤ cψ(r−2), r →∞, where c is a positive constant. now,∫ +∞ r ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) = ∞∑ i=0 ∫ 2i+1r 2ir ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) ≤ cψ(r−2) ∞∑ i=0 (ψ(2−2))i ≤ ccδψ(r−2), where cδ = (1 −ψ(2−2))−1 since ψ(2−2) < 1. consequently ∫ +∞ r ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) = o(ψ(r−2)), as r →∞. (b) ⇒ (a). suppose now that∫ +∞ r ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) = o(ψ(r−2)), as r →∞, and write ‖nhf(x)‖22,α,β = 4(i1 + i2), where i1 = ∫ 1 h 0 |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ, and i2 = ∫ +∞ 1 h |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ. firstly, we use the formula |ϕα,βλ (h)| ≤ 1 and i2 ≤ 4 ∫ +∞ 1 h ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ) = o(ψ(h2)), as h → 0. to estimate i1, we use the inequalities (i) and (ii) of lemma 1.2 i1 = ∫ 1 h 0 |1 −ϕα,βλ (h)| 2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ ≤ 2 ∫ 1 h 0 |1 −ϕα,βλ (h)| ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ ≤ 2h2 ∫ 1 h 0 (λ2 + ρ2) ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ = i3 + i4, where i3 = 2h 2ρ2 ∫ 1 h 0 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ, and i4 = 2h 2 ∫ 1 h 0 λ2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ. note that i3 ≤ 2h2ρ2 ∫ +∞ 0 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ = 2h2ρ2‖f‖22,α,β = o(ψ(h 2)), as h → 0. titchmarsh’s theorem for the cherednik-opdam transforml spectrum 95 for a while, we put φ(s) = ∫ +∞ s ( |hf(λ)|2 + |hf̌(λ)|2 ) dσ(λ). using integration by parts, we find that h2 ∫ 1/h 0 λ2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ = h2 ∫ 1/h 0 −s2φ′(s)ds = h2 ( − 1 h2 φ( 1 h ) + 2 ∫ 1/h 0 sφ(s)ds ) = −φ( 1 h ) + 2h2 ∫ 1/h 0 sφ(s)ds. since φ(s) = o(ψ(s−2)), we have sφ(s) = o(sψ(s−2)) and∫ 1/h 0 sφ(s)ds = o (∫ 1/h 0 sψ(s−2)ds ) = o(h−2ψ(h2)). then h2 ∫ 1/h 0 λ2 ( |hf(λ)|2 + |hf̌(λ)|2 ) dσλ ≤ 2c1h2h−2ψ(h2), where c1 is a positive constant. finally i4 = o(ψ(h 2)), which completes the proof of the theorem. � references [1] e. m. opdam, harmonic analysis for certain representations of graded hecke algebras, acta math. 175 (1995), no. 1, 75c121. [2] j. p. anker, f. ayadi, and m. sifi, opdams hypergeometric functions: product formula and convolution structure in dimension 1, adv. pure appl. math. 3 (2012), no. 1, 11c44. [3] e. c. titchmarsh , introduction to the theory of fourier integrals . claredon , oxford, 1948, komkniga.moxow.2005. [4] s. s. platonov, approximation of functions in l2-metric on noncompact rank 1 symmetric space . algebra analiz .11(1) (1999), 244-270. department of mathematics, faculty of sciences äın chock, university hassan ii, casablanca, morocco ∗corresponding author international journal of analysis and applications volume 16, number 2 (2018), 290-305 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-290 an approximation of fuzzy numbers based on polynomial form fuzzy numbers sh. yeganehmanesh and m. amirfakhrian∗ department of mathematics, central tehran branch, islamic azad university, tehran, iran ∗corresponding author: amirfakhrian@iauctb.ac.ir abstract. in this paper, we approximate an arbitrary fuzzy number by a polynomial fuzzy number through minimizing the distance between them. throughout this work, we used a distance that is a meter on the set of all fuzzy numbers with continuous left and right spread functions. to support our claims analytically, we have proven some theorems and given supplementary corollaries. 1. introduction comparison of fuzzy numbers is an indispensable part of most systems using such numbers. to this end, many researchers active in the fuzzy theory domain have tried to make fuzzy numbers comparable. some authors have approximated a fuzzy number by a single crisp number. this method which is called ranking suffers from loss of some useful information. some authors such as [8] convert a given fuzzy number into an interval and solve an interval arithmetic problem instead of a more complicated fuzzy computation. however, the fuzzy central concept fades here. finding the nearest triangular or trapezoidal fuzzy number associated to an arbitrary given fuzzy number is another method on which some authors such as [2], [4], [6], [10] and [11] have concentrated. however, this method fails to guarantee the same modal value (or interval). also, some authors such as [12] and [13] have made a considerable contribution to the coefficients of polynomial, the concept that we have used in this research. received 2017-10-28; accepted 2018-01-08; published 2018-03-07. 2010 mathematics subject classification. 00a00. key words and phrases. fuzzy number; parametric form; distance; polynomial form. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 290 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-290 int. j. anal. appl. 16 (2) (2018) 291 in this paper, we propose two methods for approximating a given arbitrary fuzzy number with a polynomial fuzzy number to a great degree of accuracy. the first method splits the approximation problem into two sub-problems and solves them separately whereas the second one solves the problem in a general form. 2. basic concepts in this section, the basic concepts used throughout the paper are given. let f(r) be the set of all fuzzy numbers (the set of all normal and convex fuzzy sets) on the real line. definition 2.1. a generalized lr fuzzy number ũ with the membership function µũ(x), x ∈ r can be defined as [1]: µũ(x) =   lũ(x), a ≤ x ≤ b, 1, b ≤ x ≤ c, rũ(x), c ≤ x ≤ d, 0, otherwise, (2.1) where lũ is the left membership function and rũ is the right membership function. it is assumed that lũ is increasing in [a,b] and rũ is decreasing in [a,b], and that lũ(a) = rũ(d) = 0 and lũ(b) = rũ(c) = 1. in addition, if lũ and rũ are linear, then ũ is a trapezoidal fuzzy number, which is denoted by ũ = (a,b,c,d). if b = c, we denoted it by ũ = (a,c,d), which is a triangular fuzzy number. the parametric form of a fuzzy number is given by ũ = (u,u), where u and u are functions defined over [0, 1] and satisfy the following requirements: (1) u is a monotonically increasing left continuous function. (2) u is a monotonically decreasing left continuous function. (3) u ≤ u, in [0, 1]. we name u and u, left and right spread functions, respectively. if a is a crisp number, then u(r) = u(r) = a, for ∀r ∈ [0, 1]. definition 2.2. we say that a fuzzy number ṽ has an m−degree polynomial form, if there exist two polynomials p and q of degree at most m such that ṽ = (p,q) [3]. let ṽ ∈ fm(r) be the set of all m−degree polynomial form fuzzy numbers. for 0 < α ≤ 1, α-cut of a fuzzy number ũ is defined by [5] as follows: [ũ]α = {t ∈ r | µũ(t) ≥ α}. (2.2) int. j. anal. appl. 16 (2) (2018) 292 the core of a fuzzy number is defined by [5] as follows: core(ũ) = {t ∈ r | µũ(t) = 1}. (2.3) let fc(r) be the set of all fuzzy numbers with continuous left and right spread functions and let fm(r) be the set of all m−degree polynomial form fuzzy numbers [3]. we also consider πm as the set of all polynomials of degree at most m. we can write a fuzzy number ũ ∈fm(r) as follows: ũ = (u,u), (2.4) where u, u ∈ πm. 3. a parametric distance in order to measure the distance between two fuzzy numbers, here, we propose a new definition. definition 3.1. for ũ, ṽ ∈f(r), the distance of ũ and ṽ is defined by dp,q(ũ, ṽ) = (∫ 1 0 q|u(r) −v(r)|p dr + ∫ 1 0 (1 −q)|u(r) −v(r)|p dr )1 p , (3.1) where q ∈ [0, 1] and p > 0. theorem 3.1. dp,q is a metric on fc(r). proof. it can be found in [7]. � as the q changes in (3.1), the distance dp,q gets biased towards either the left spread function or the right one. 4. the best polynomial fuzzy number to an arbitrary fuzzy number knowing the fact that a fuzzy number can be approximated in terms of an m-degree polynomial, it is now aimed at finding the nearest m-degree polynomial to a given fuzzy number. to this end, the proposed parametric distance defined in section 3 is used. assume that ũ is an arbitrary fuzzy number and ṽ is an approximated m-degree polynomial form fuzzy number. for p = 2 in (3.1), the distance becomes as: d2,q(ũ, ṽ) = (∫ 1 0 q|u(r) −v(r)|2 dr + ∫ 1 0 (1 −q)|u(r) −v(r)|2 dr )1 2 , (4.1) int. j. anal. appl. 16 (2) (2018) 293 where q ∈ [0, 1]. now, the approximation problem becomes as:  min ṽ∈fm d2,q(ũ, ṽ), s.t. ṽ(1) = ũ(1), (4.2) which can be expanded as follows:   min v,v∈πm ∫ 1 0 q|u(r) −v(r)|2 dr + ∫ 1 0 (1 −q)|u(r) −v(r)|2 dr, s.t. v(1) = u(1), v(1) = u(1). (4.3) before solving this problem, let’s present the lemma 4.1 which will come in handy in our approximation method. lemma 4.1. let f and g be two arbitrary functions defined on a domain d ⊆ r. then over this domain we have: min(f(x) + g(x)) ≥ min f(x) + min g(x). (4.4) proof. straightforward. � in the following, we propose our two new approximation methods which minimize the distance first based on splitting the problem and second based on general form. 4.1. minimization by splitting the problem. from lemma 4.1, it is clear that splitting the problem (4.3) into two sub-problems will lead us to have a less objective value. since q is constant and both terms of the objective functions in (4.3) are non-negative, by lemma 4.1 the problem is divided into two independent sub-problems:   min v∈πm ∫ 1 0 |u(r) −v(r)|2 dr, s.t. v(1) = u(1). (4.5) and   min v∈πm ∫ 1 0 |u(r) −v(r)|2 dr, s.t. v(1) = u(1). (4.6) int. j. anal. appl. 16 (2) (2018) 294 assume that v(r) = ∑m j=0 ajr j, for solving the problem (4.5) with lagrangian method we define the following function: f(a,λ) = ∫ 1 0 |u(r) −v(r)|2 dr −λ(u(1) −v(1)), (4.7) where a = (a0,a1, ...,am) t. the necessary condition to minimize the function f is that the gradient of function should be zero. the gradient of function f can be shown as follows: 5f(a,λ) =   2ha + λ1 − 2m 1ta −u(1)   , (4.8) where 1 = (1, ..., 1)t, h is the m + 1 hermitian matrix which its elements defined as hij = (i + j + 1) −1 and m is the momentum u : m = [∫ 1 0 riu(r) dr ]t i=0,...,m . (4.9) 5f = 0 gives following:   ha + 1 2 λ1 − m = 0, v(1) = u(1). (4.10) thus, we define qf : rm+2 −→ rm+2 as follows: qf (a,λ) =   ha + 1 2 λ1 − m 1ta −u(1)   . (4.11) hence, we try to solve qf (x) = 0 which is a linear system such that: x = (a,λ)t. (4.12) to solve this system, we have: qf (a,λ) = a x − r, (4.13) where r =   m u(1)   , (4.14) and a is as follows: a =   h 1 2 1 1t 0   . (4.15) int. j. anal. appl. 16 (2) (2018) 295 hence, we have: x = a−1 r, (4.16) theorem 4.1. the inverse matrix of a has the following form: a−1 =   h−1 − vvt 1 m + 1 v 2 m + 1 vt − 2 (m + 1)2   , (4.17) where v = 1 m+1 h−11. proof. it is straightforward. � we consider the solution of minimization problem (4.7) as x∗ = (a∗,λ∗)t such that:  a∗ = (h−1 − vvt)m + u(1) m + 1 v, λ∗ = 2 m + 1 vtm − 2u(1) (m + 1)2 . (4.18) now, in the same way, we solve the problem (10). let v(r) = m∑ j=0 bjr j, b = (b0,b1, ...,bm) t. for solving with lagrangian method , we continue by defining g as follows: g(b,µ) = ∫ 1 0 |u(r) −v(r)|2 dr −µ(u(1) −v(1)). (4.19) let define the momentum vector of u as: m = [∫ 1 0 riu(r) dr ]t i=0,...,m , (4.20) thus, we define qg : rm+2 −→ rm+2 as follows: qg(b,µ) =   hb + 1 2 µ1 − m 1tb −u(1)   . (4.21) as we did it before we have a linear system az follows: qg(b,µ) = a x − r, (4.22) where x = (b,µ)t, (4.23) thus: x = a−1 r, (4.24) int. j. anal. appl. 16 (2) (2018) 296 considering the solution of minimization problem (4.7) as x∗ = (b∗,µ∗)t such that:  b∗ = (h−1 − vvt)m + u(1) m + 1 v, µ∗ = 2 m + 1 vtm − 2u(1) (m + 1)2 . (4.25) in summary, assuming ũ ∈f(r) be an arbitrary fuzzy number, we find the best approximation of ũ out of fm for a fixed integer m. in this case, ũ∗m is the best approximation of ũ, such that: u∗(r) = m∑ j=0 a∗jr j and u∗(r) = m∑ j=0 b∗jr j, where a∗ = (a∗0,a ∗ 1, ...,a ∗ m) t and b∗ = (b∗0,b ∗ 1, ...,b ∗ m) t. we denote the best approximation of ũ ∈ f out of fm by ũ∗m. in following theorem we show that the best approximation of an arbitrary polynomial fuzzy number is itself. theorem 4.2. if ũ ∈fm then its best approximation ũ∗m, out of fm(r) with respect to distance (3.1) exists and ũ∗m = ũ. proof. straightforward. � corollary 4.2. best approximation of an arbitrary trapezoidal fuzzy number is itself. proof. it can obtained by theorem 4.2. � 4.2. minimization of the problem in general form. in this section, we try to solve problem (4.3) in general form. assume that v(r) = m∑ j=0 ajr j and v(r) = m∑ j=0 bjr j. to this end, for solving the problem (4.3) with lagrangian method we define the following function: e(a, b,λ,µ) = ∫ 1 0 q(u(r) −v(r))2 dr + ∫ 1 0 (1 −q)(u(r) −v(r))2 dr −λ(u(1) −v(1)) −µ(u(1) −v(1)), (4.26) where a = (a0,a1, ...,am) t, b = (b0,b1, ...,bm) t. the necessary condition to minimize the function e is that the gradient of function should be zero. the gradient of function e can be shown as follows: 5e(a, b,λ,µ) =   2qha + λ1 − 2qm 2(1 −q)hb + µ1 − 2(1 −q)m 1ta −u(1) 1tb −u(1)   , (4.27) int. j. anal. appl. 16 (2) (2018) 297 where h is the m + 1 hermitian matrix, 1 = (1, ..., 1)t, m and m respectively are the momentum vectors of u and u defined in (4.9) and (4.20). 5e = 0 gives following:   qha + 1 2 λ11 −qm = 0, (1 −q)hb + 1 2 µ11 − (1 −q)m = 0, v(1) = u(1), v(1) = u(1). (4.28) now, we define qe as follows: qe(a,λ, b,µ) =   qha + 1 2 λ1 −qm 1ta −u(1) (1 −q)hb + 1 2 µ1 − (1 −q)m 1tb −u(1)   . (4.29) hence, we try to solve qe(t) = 0 which is a linear system such that: t = (a,λ, b,µ)t. (4.30) to solve this system, we let qe(t) = ae t−z = 0 where: ae =   qh 1 2 1 0 0 1t 0 0 0 0 0 (1 −q)h 1 2 1 0 0 1t 0   4×4 , (4.31) int. j. anal. appl. 16 (2) (2018) 298 and z =   qm u(1) (1 −q)m u(1)   . (4.32) now we countinue with finding the invers of coefficient matrix, ae. by considering ae,γ as ae,γ =   γh 1 2 1 1t 0   , (4.33) where γ ∈ [0, 1] and we have ae =   ae,q 0 0 ae,(1−q)   . (4.34) lemma 4.3. a−1e,γ has the following form: a−1e,γ =   1 γ (h−1 − vvt) 1 m + 1 v 2 m + 1 vt −γ 2 (m + 1)2   , (4.35) such that v = 1 m+1 h−11. proof. straightforward. � theorem 4.3. the inverse matrix of ae has the following form: a−1e =   a−1e,q 0 0 a−1 e,(1−q)   , (4.36) proof. it is straightforward. � to do this end, with theorem 4.3 we have: t = a−1e z, (4.37) int. j. anal. appl. 16 (2) (2018) 299 we consider the solution of minimization problem (4.26) as xe ∗ = (a∗,λ∗, b∗,µ∗)t such that:  a∗ = (h−1 − vvt)m + u(1) m + 1 v, b∗ = (h−1 − vvt)m + u(1) m + 1 v, λ∗ = 2q m + 1 vtm − 2qu(1) (m + 1)2 , µ∗ = 2(1 −q) m + 1 vtm − 2(1 −q)u(1) (m + 1)2 . (4.38) analogous to the theorem 4.2, in following theorem we again show that the best approximation of an arbitrary polynomial fuzzy number is itself. theorem 4.4. if ũ ∈fm then its best approximation ũ∗m, out of fm(r) with respect to distance (3.1) exists and ũ∗m = ũ. proof. it can be proved by (4.38). � corollary 4.4. if ũ ∈fl where (l ≤ m), then ũ∗m = ũ . proof. straightforward. � note that if the obtained approximated coefficients yield a polynomial form fuzzy number, this polynomial is the best approximation of the given fuzzy number. 5. convergence of approximation in this section, the convergence of the proposed approximation methods are shown. lemma 5.1. let m ∈ n lim m→∞ 1 m m∑ j=1 1 j = 0. proof. it is trivial. � theorem 5.1. if u and u are integrable functions in [0, 1] and ũ∗m is the best approximation of ũ by splitting the problem in subsection 4.1 out of fm, then lim m→∞ ũ∗m = ũ. int. j. anal. appl. 16 (2) (2018) 300 proof. from (4.18), we have lim m→∞ λ∗ = lim m→∞ ( 2 m + 1 vtm − 2u(1) (m + 1)2 ) = 2 lim m→∞ 1 m + 1 m∑ j=0 1∫ 0 rju(r) dr (5.1) since rj is nonnegative in [0, 1], according to midpoint theorem for integrals there exists θj ∈ (0, 1), such that 1∫ 0 rju(r) dr = u(θj) j + 1 , j = 0, ...,m (5.2) therefore, from (5.1) and lemma 5.1 we have lim m→∞ λ∗ = 2 lim m→∞ 1 m + 1 m∑ j=0 u(θj) j + 1 ≤‖u‖∞2 lim m→∞ 1 m + 1 m+1∑ j=1 1 j = 0. (5.3) from (4.15), (4.18) and (5.3), when m →∞, a∗ is the solution of ha = m, where h is a hermitian matrix. in this case, a∗ is the solution of a common crisp problem and for this solution we have the convergence. similarly from (4.25) , we have lim m→∞ µ∗ = 0 (5.4) and these claims hold for b∗ in hb = m. since a∗ and b∗ are both convergent, therefore, u∗ and u∗ are also convergent and this completes the proof. � theorem 5.2. if u and u are integrable functions in [0, 1] and ũ∗m is the best approximation of ũ of general form in subsection 4.2 out of fm, then lim m→∞ ũ∗m = ũ. proof. it was obtained by (4.38) and lemma 5.1. � corollary 5.2. if ũ ∈fm, the approximation sequence converges to the exact solution in the first iteration. proof. straightforward. � considering (4.18), (4.25) and (4.38) for an arbitrary fuzzy number, the best approximation regarding both methods are identical. according to following lemma we present an explicit formula to approximate an arbitrary fuzzy number with a trapezoidal fuzzy number. int. j. anal. appl. 16 (2) (2018) 301 theorem 5.3. if ũ = (u,u) is an arbitrary fuzzy number, then its best linear approximation ũ∗1 regarding the distance (3.1) is ũ∗1 = (a0 + a1r,b0 + b1r) where: a0 = 1 2 (6 ∫ 1 0 u(r) dr − 6 ∫ 1 0 ru(r) dr −u(1)), (5.5) a1 = − 3 2 (2 ∫ 1 0 u(r) dr − 2 ∫ 1 0 ru(r) dr −u(1)), (5.6) b0 = 1 2 (6 ∫ 1 0 u(r) dr − 6 ∫ 1 0 ru(r) dr −u(1)), (5.7) b1 = − 3 2 (2 ∫ 1 0 u(r) dr − 2 ∫ 1 0 ru(r) dr −u(1)). (5.8) proof. straightforward. � in the following, let’s present the lemma 5.3 which will come in handy in showing our best linear approximation of an arbitrary fuzzy number is a trapezoidal fuzzy number. lemma 5.3. for any arbitrary function g, if g is a monotonically increasing left continuous function then:∫ 1 0 xg(x) dx− ∫ 1 0 g(x) dx + 1 2 g(1) ≥ 0, and if g is a monotonically decreasing left continuous function then:∫ 1 0 xg(x) dx− ∫ 1 0 g(x) dx + 1 2 g(1) ≤ 0, proof. straightforward. � lemma 5.4. the best linear approximation of an arbitrary fuzzy number ũ = (u,u) regarding the distance (3.1) is a trapezoidal fuzzy number. proof. as regards to distance (3.1) and by lemma 5.3 and 5.3 it was obtained. � due to the theorem 5.3 and lemma 5.4 for any arbitrary fuzzy number, the nearest trapezoidal fuzzy number regarding the distance (3.1) can be obtained from equations (5.5) (5.8). 6. numerical examples in this section we present some examples which have been solved by mathematica software using 10 decimal digits. example 6.1. let ũ = (2r2 + 1, 5 − r2). by assuming m = 2 and each q ∈ [0, 1] the best approximation of ũ is itself. according to theorem 4.2 it could be foretold. example 6.2. let ũ = (r2 + 1, 3 −r2). by assuming m = 1 and each q ∈ [0, 1] the best approximation of ũ can be found by lemma 5.3. the best approximation is ( 3 4 + 5 4 r, 13 4 − 5 4 r). int. j. anal. appl. 16 (2) (2018) 302 example 6.3. let ũ = (er,e2−r). for m = 1 and m = 3, the best approximations of ũ are ũ∗1 and ũ ∗ 3, where:   u∗1(r) = 1 2 (−12 + 5e) + 3 2 (4 −e)r u∗1(r) = 5 2 e− 3 2 er  u∗3(r) = 1 4 (−4560 + 1679e) − 15 4 (−3216 + 1183e)r + 15 4 (−7168 + 2637e)r2 − 35 4 (−1824 + 671e)r3 u∗3(r) = 1 4 (3599e− 1320e2) + 15 4 (−2719e + 1000e2)r − 15 4 (−6285e + 2312e2)r2 + 35 4 (−1631e + 600e2)r3 and for q = 0.5 the distance (4.1) between ũ and ũ∗1 is d(ũ, ũ ∗ 1) = 0.185451 and the distance between ũ and ũ∗3 is d(ũ, ũ ∗ 3) = 0.000835893. example 6.4. let ũ = (ln [(e− 1)r + 1] , 2−ln [(e− 1)r + 1]), pi(r) = ri and an arbitrary q . the distances (4.1) between ũ and ũ∗m, for m = 1, · · · , 7, are shown in table 1. m d(ũ, ũ∗m) 1 4.85342 × 10−2 2 7.2679 × 10−3 3 1.27813 × 10−3 4 2.44005 × 10−4 5 4.89405 × 10−5 6 1.04687 × 10−5 7 5.88816 × 10−6 table 1: distances for different values of m regarding theorem 5.1, it was predictable that increasing the variable m would reduce the associated error. since ũ is a symmetric fuzzy number and by (4.38) the best approximation of it is independent of q, the distance (4.1) between ũ and ũ∗m is independent of q. example 6.5. in this example, we approximate a fuzzy number with m = 1 by a trapezoidal one and compare the results from our method with the results obtained from other four methods proposed in [2, 9, 11, 14] in a tabular format in table 2. int. j. anal. appl. 16 (2) (2018) 303 u(r) / u(r) (1) (2) (3) (4) (5) 1 − 0.3 √ − ln r 2 + 0.7 √ − ln r 0.484391 1 2 3.20309 0.50052 0.96775 2.07526 3.16546 0.06790 1.67725 1.67725 3.2866 0.52195 0.89256 2.10743 2.9722 0.50052 0.96775 2.07526 3.16546 3r 7 − 3r 0.740495 3 4 6.2595 0.84003 2.80092 4.19908 6.15997 0.4905 3.5 3.5 6.5095 0.48633 2.97779 4.02221 6.51367 0.84003 2.80092 4.19908 6.15997 1 2 1 1 2 2 1 1 2 2 0.75 1.5 1.5 2.25 1 1 1 2 1 1 2 2 r + 1 5 − 3r 1 2 2 5 1 2 2 5 −0.5 2 2 4.5 1 2 2 2.5 1 2 2 5 1 3 −r 1 1 2 3 1 1 2 3 0.5 1.75 1.75 3 −0.33333 0.66667 2.33333 2.33333 1 1 2 3 table 2. numerical results of examples as it is obvious, our method yields trapezoidal fuzzy numbers with closer cores in comparison with the ones obtained from the other four methods. while the other four methods fail in approximating the m-degree polynomial form fuzzy numbers, our method can approximate all the trapezoidal, triangular and m-degree polynomial form fuzzy numbers. example 6.6. let ũ = (2 + er−1, 4 − ln [(e− 1)r + 1]), pi(r) = ri. the distances between ũ and ũ∗m, for m = 0, · · · , 4 and q = {0, 0.25, 0.5, 0.75, 1}, are shown in table 3. d(ũ, ũ∗m) q = 0 q = 0.25 q = 0.5 q = 0.75 q = 1 m = 0 0.504053 0.482261 0.459435 0.435415 0.409989 m = 1 0.0485342 0.0458563 0.0430119 0.0399657 0.0366673 m = 2 0.0072679 0.00643446 0.0054756 0.00430839 0.00267248 m = 3 0.00127813 0.00110962 0.000910436 0.000653097 0.000155493 m = 4 0.000244005 0.000211315 0.000172538 0.000122003 0.000000000 table 3: distances for different values of m and q int. j. anal. appl. 16 (2) (2018) 304 this table shows that as the variable m increases, the distance (4.1) between exact given fuzzy number and our approximated polynomial form fuzzy number reduces(it can be predicted by theorem 5.1). in this example ũ is not a symmetric fuzzy number. hence, the distance between ũ and ũ∗m depends on q. as we can see in the table, whenever q increases from 0 to 1, the distance decreases. considering the distance equation (4.1), it can be deduced that the approximation of right spread is more precise than the approximation of the left one. 7. conclusion in this paper, a new distance metric was proposed on the set of all fuzzy numbers with continuous left and right spread functions. using this metric, a given fuzzy number can be approximated through finding the nearest polynomial form fuzzy number out of the set of all m-degree polynomial form fuzzy numbers. hence, two methods were proposed to solve the approximation problem. we showed that both of the methods not only yield the same results, but also are convergent. finally, we investigated our theorems in some numerical examples. references [1] s. abbasbandy, m. amirfakhrian, the nearest trapezoidal form of a generalized left right fuzzy number, int. j. approx. reason. 43 (2006), 166-178. [2] s. abbasbandy, b. asady, the nearest trapezoidal fuzzy number to a fuzzy quantity, appl. math. comput. 156 (2004), 381-386. [3] m. amirfakhrian, numerical solution of a system of polynomial parametric form fuzzy linear equations, book chapter of ferroelectrics, intech publisher, austria, (2010). [4] a. i. ban, l. coroianu, existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition, fuzzy sets syst. 3 (2014) 3-22. [5] d. dubois, h. prade, fuzzy sets and systems: theory and application, academic press, new york, (1980). [6] l. coroianu, m. gagolewski, p. grzegorzewski, nearest piecewise linear approximation of fuzzy numbers, fuzzy sets syst. 233 (2013), 26-51. [7] p. grzegorzewski, metrics and orders in space of fuzzy numbers, fuzzy sets syst. 97 (1998), 83-94. [8] p. grzegorzewski, nearest interval approximation of a fuzzy number, fuzzy sets syst. 130 (2002), 321-330. [9] p. grzegorzewski, p. mró wka, trapezoidal approximations of fuzzy numbers, fuzzy sets syst. 153 (2005), 115-135. [10] p. grzegorzewski, k. pasternak-winiarska,natural trapezoidal approximations of fuzzy numbers, fuzzy sets syst. 250 (2014), 90-109. [11] m. ma and a. kandel and m. friedman, correction to ”a new approach for defuzzification”, fuzzy sets syst. 128 (2002), 133-134. [12] v. powers, b. reznick, polynomials that are positive on an interval, trans. amer. math. soc. 352 (10) (2000), 4677c4692. [13] j. stolfi, m.v.a. andrade, j.l.d. comba, and r.van iwaarden. affine arithmetic: a correlation-sensitive variant of interval arithmetic, accessed january 17, (2008). [14] w. voxman, some remarks on distance between fuzzy numbers, fuzzy sets syst. 100 (1998), 353-365. int. j. anal. appl. 16 (2) (2018) 305 [15] h. j. zimmermann, fuzzy set theory and its applications, 2nd edition, kluwer academic, boston, (1991). 1. introduction 2. basic concepts 3. a parametric distance 4. the best polynomial fuzzy number to an arbitrary fuzzy number 4.1. minimization by splitting the problem 4.2. minimization of the problem in general form 5. convergence of approximation 6. numerical examples 7. conclusion references international journal of analysis and applications volume 16, number 3 (2018), 317-327 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-317 lacunary i2-invariant convergence and some properties uǧur ulusu, erdi̇nç dündar∗ and fati̇h nuray department of mathematics, faculty of science and literature, afyon kocatepe university, 03200, afyonkarahisar, turkey ∗corresponding author: edundar@aku.edu.tr abstract. in this paper, the concept of lacunary invariant uniform density of any subset a of the set n×n is defined. associate with this, the concept of lacunary i2-invariant convergence for double sequences is given. also, we examine relationships between this new type convergence concept and the concepts of lacunary invariant convergence and p-strongly lacunary invariant convergence of double sequences. finally, introducing lacunary i∗2 -invariant convergence concept and lacunary i2-invariant cauchy concepts, we give the relationships among these concepts and relationships with lacunary i2-invariant convergence concept. 1. introduction several authors have studied invariant convergent sequences (see, [8–10, 13, 15–17, 19]). let σ be a mapping of the positive integers into themselves. a continuous linear functional φ on `∞, the space of real bounded sequences, is said to be an invariant mean or a σ-mean if it satisfies following conditions: (1) φ(x) ≥ 0, when the sequence x = (xn) has xn ≥ 0 for all n, (2) φ(e) = 1, where e = (1, 1, 1, ...) and (3) φ(xσ(n)) = φ(xn) for all x ∈ `∞. the mappings σ are assumed to be one-to-one and such that σm(n) 6= n for all positive integers n and m, where σm(n) denotes the m th iterate of the mapping σ at n. thus, φ extends the limit functional on c, 2010 mathematics subject classification. 40a05, 40a35. key words and phrases. double sequence; i-convergence; lacunary sequence; invariant convergence; i-cauchy sequence. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 317 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-317 int. j. anal. appl. 16 (3) (2018) 318 the space of convergent sequences, in the sense that φ(x) = lim x for all x ∈ c. in the case σ is translation mappings σ(n) = n + 1, the σ-mean is often called a banach limit. by a lacunary sequence we mean an increasing integer sequence θ = {kr} such that k0 = 0 and hr = kr −kr−1 →∞ as r →∞. the intervals determined by θ is denoted by ir = (kr−1,kr] (see, [4]). the concept of lacunary strongly σ-convergence was introduced by savaş [17] as below: lθ = { x = (xk) : lim r→∞ 1 hr ∑ k∈ir |xσk(m) −l| = 0, uniformly in m } . pancaroǧlu and nuray [13] defined the concept of lacunary invariant summability and the space [vσθ]q as follows: a sequence x = (xk) is said to be lacunary invariant summable to l if lim r→∞ 1 hr ∑ m∈ir xσm(n) = l, uniformly in n. a sequence x = (xk) is said to be strongly lacunary q-invariant convergent (0 < q < ∞) to l if lim r→∞ 1 hr ∑ m∈ir |xσm(n) −l|q = 0, uniformly in n and it is denoted by xk → l ( [vσθ]q ) . the idea of i-convergence was introduced by kostyrko et al. [5] as a generalization of statistical convergence which is based on the structure of the ideal i of subset of the set of natural numbers n. a family of sets i ⊆ 2n is called an ideal if and only if (i) ∅∈i, (ii) for each a,b ∈i we have a∪b ∈i, (iii) for each a ∈i and each b ⊆ a we have b ∈i. an ideal is called non-trivial if n /∈i and non-trivial ideal is called admissible if {n}∈i for each n ∈ n. a family of sets f ⊆ 2n is called a filter if and only if (i) ∅ /∈f, (ii) for each a,b ∈f we have a∩b ∈f, (iii) for each a ∈f and each b ⊇ a we have b ∈f. for any ideal there is a filter f(i) corresponding with i, given by f(i) = { m ⊂ n : (∃a ∈i)(m = n\a) } . recently, the concepts of lacunary σ-uniform density of the set a ⊆ n, lacunary iσ-convergence, lacunary i∗σ-convergence, lacunary iσ-cauchy and i∗σ-cauchy sequences of real numbers were defined by ulusu and int. j. anal. appl. 16 (3) (2018) 319 nuray [20] and similar concepts can be seen in [12]. let θ = {kr} be a lacunary sequence, a ⊆ n and sr := min n {∣∣a∩{σm(n) : m ∈ ir} ∣∣} and sr := max n {∣∣a∩{σm(n) : m ∈ ir}∣∣}. if the following limits exist vθ(a) := lim r→∞ sr hr , vθ(a) := lim r→∞ sr hr then they are called a lower lacunary σ-uniform (lower σθ-uniform) density and an upper lacunary σ-uniform (upper σθ-uniform) density of the set a, respectively. if vθ(a) = vθ(a), then vθ(a) = vθ(a) = vθ(a) is called the lacunary σ-uniform density or σθ-uniform density of a. denote by iσθ the class of all a ⊆ n with vθ(a) = 0. let iσθ ⊂ 2n be an admissible ideal. a sequence (xk) is said to be lacunary iσ-convergent or iσθconvergent to the number l if for every ε > 0 aε := { k : |xk −l| ≥ ε } belongs to iσθ; i.e., vθ(aε) = 0. in this case we write iσθ − lim xk = l. the set of all iσθ-convergent sequences will be denoted by iσθ. let iσθ ⊂ 2n be an admissible ideal. a sequence x = (xk) is said to be i∗σθ-convergent to the number l if there exists a set m = {m1 < m2 < ...} ∈ f(iσθ) such that lim k→∞ xmk = l. in this case we write i∗σθ − lim xk = l. a sequence (xk) is said to be lacunary iσ-cauchy sequence or iσθ-cauchy sequence if for every ε > 0, there exists a number n = n(ε) ∈ n such that a(ε) = { k : |xk −xn| ≥ ε } belongs to iσθ; i.e., vθ ( a(ε) ) = 0. a sequence x = (xk) is said to be i∗σθ-cauchy sequences if there exists a set m = {m1 < m2 < ... < mk < ...}∈f(iσθ) such that lim k,p→∞ |xmk −xmp| = 0. int. j. anal. appl. 16 (3) (2018) 320 convergence and i-convergence of double sequences in a metric space and some properties of this convergence, and similar concepts which are noted following can be seen in [1, 2, 6, 7, 14, 18]. a double sequence x = (xkj)k,j∈n of real numbers is said to be convergent to l ∈ r in pringsheim’s sense if for any ε > 0, there exists nε ∈ n such that |xkj − l| < ε, whenever k,j > nε. in this case, we write p − lim k,j→∞ xkj = l or lim k,j→∞ xkj = l. a double sequence x = (xkj) is said to be bounded if supk,j xkj < ∞. the set of all bounded double sequences of sets will be denoted by `2∞. a nontrivial ideal i2 of n × n is called strongly admissible ideal if {i}× n and n ×{i} belong to i2 for each i ∈ n. it is evident that a strongly admissible ideal is admissible also. throughout the paper we take i2 as a strongly admissible ideal in n×n. i02 = { a ⊂ n × n : (∃m(a) ∈ n)(i,j ≥ m(a) ⇒ (i,j) 6∈ a) } . then i02 is a strongly admissible ideal and clearly an ideal i2 is strongly admissible if and only if i02 ⊂i2. an admissible ideal i2 ⊂ 2n×n satisfies the property (ap2) if for every countable family of mutually disjoint sets {e1,e2, ...} belonging to i2, there exists a countable family of sets {f1,f2, ...} such that ej∆fj ∈ i02 , i.e., ej∆fj is included in the finite union of rows and columns in n × n for each j ∈ n and f = ⋃∞ j=1 fj ∈i2 (hence fj ∈i2 for each j ∈ n). let (x,ρ) be a metric space. a sequence x = (xmn) in x is said to be i2-convergent to l ∈ x, if for any ε > 0 a(ε) = { (m,n) ∈ n×n : ρ(xmn,l) ≥ ε } ∈i2. in this case, we write i2 − lim m,n→∞ xmn = l. the double sequence θ = {(kr,ju)} is called double lacunary sequence if there exist two increasing sequence of integers such that k0 = 0, hr = kr −kr−1 →∞ and j0 = 0, h̄u = ju − ju−1 →∞ as r,u →∞. we use the following notations in the sequel: kru = krju, hru = hrh̄u, iru = {(k,j) : kr−1 < k ≤ kr and ju−1 < j ≤ ju}, qr = kr kr−1 and qu = ju ju−1 . int. j. anal. appl. 16 (3) (2018) 321 also, the idea of i2-invariant convergence concepts and i2-invariant cauchy concepts of double sequences were defined by dündar and ulusu (see [3]). 2. lacunary i2-invariant convergence in this section, firstly, the concepts of lacunary invariant convergence of double sequence and lacunary invariant uniform density of any subset a of the set n×n are defined. associate with this uniform density, the concept of lacunary i2-invariant convergence for double sequences is given. also, we examine relationships between this new type convergence concept and the concepts of lacunary invariant convergence, p-strongly lacunary invariant convergence for double sequences. definition 2.1. a double sequence x = (xkj) is said to be lacunary invariant convergent to l if lim r,u→∞ 1 hru ∑ k,j∈iru xσk(m),σj(n) = l, uniformly in m,n and it is denoted by xkj → l ( v σθ2 ) . definition 2.2. let θ = {(kr,ju)} be a double lacunary sequence, a ⊆ n×n and sru := min m,n ∣∣∣a∩{(σk(m),σj(n)) : (k,j) ∈ iru}∣∣∣ and sru := max m,n ∣∣∣a∩{(σk(m),σj(n)) : (k,j) ∈ iru}∣∣∣. if the following limits exist v θ2 (a) := lim r,u→∞ sru hru , v θ2 (a) := lim r,u→∞ sru hru , then they are called a lower lacunary σ-uniform density and an upper lacunary σ-uniform density of the set a, respectively. if v θ2 (a) = v θ 2 (a), then v θ 2 (a) = v θ 2 (a) = v θ 2 (a) is called the lacunary σ-uniform density of a. denote by iσθ2 the class of all a ⊆ n×n with v θ2 (a) = 0. throughout the paper we take iσθ2 as a strongly admissible ideal in n×n. definition 2.3. a double sequence x = (xkj) is said to be lacunary i2-invariant convergent or iσθ2 -convergent to the l if for every ε > 0, the set aε := { (k,j) ∈ iru : |xkj −l| ≥ ε } belongs to iσθ2 ; i.e., v θ2 (aε) = 0. in this case, we write iσθ2 − lim xkj = l or xkj → l ( iσθ2 ) . int. j. anal. appl. 16 (3) (2018) 322 the set of all iσθ2 -convergent sequences will be denoted by iσθ2 . theorem 2.1. if iσθ2 − lim xkj = l1 and iσθ2 − lim ykj = l2, then (i) iσθ2 − lim(xkj + ykj) = l1 + l2 (ii) iσθ2 − lim αxkj = αl1 (α is a constant). proof. the proof is clear so we omit it. � theorem 2.2. suppose that x = (xkj) is a bounded double sequence. if (xkj) is lacunary i2-invariant convergent to l, then (xkj) is lacunary invariant convergent to l. proof. let θ = {(kr,ju)} be a double lacunary sequence, m,n ∈ n be an arbitrary and ε > 0. now, we calculate t(k,j,r,u) := ∣∣∣∣∣∣ 1hru ∑ k,j∈iru xσk(m),σj(n) −l ∣∣∣∣∣∣ . we have t(k,j,r,u) ≤ t(1)(k,j,r,u) + t(2)(k,j,r,u), where t(1)(k,j,r,u) := 1 hru ∑ k,j∈iru |x σk(m),σj(n) −l|≥ε |xσk(m),σj(n) −l| and t(2)(k,j,r,u) := 1 hru ∑ k,j∈iru |x σk(m),σj(n) −l|<ε |xσk(m),σj(n) −l|. we get t(2)(k,j,r,u) < ε, for every m,n = 1, 2, . . . . the boundedness of x = (xkj) implies that there exists a k > 0 such that |xσk(m),σj(n) −l| ≤ k, ((k,j) ∈ iru; m,n = 1, 2, ...). then, this implies that t(1)(k,j,r,u) ≤ k hru ∣∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣∣ ≤ k max m,n ∣∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣∣ hru = k sru hru , hence (xkj) is lacunary invariant summable to l. � int. j. anal. appl. 16 (3) (2018) 323 the converse of theorem 2.2 does not hold. for example, x = (xkj) is the double sequence defined by following; xkj :=   1 , if kr−1 < k < kr−1 + [ √ hr], jr−1 < j < jr−1 + [ √ h̄u], and k + j is an even integer. 0 , if kr−1 < k < kr−1 + [ √ hr], jr−1 < j < jr−1 + [ √ h̄u], and k + j is an odd integer. when σ(m) = m + 1 and σ(n) = n + 1, this sequence is lacunary invariant convergent to 1 2 but it is not lacunary i2-invariant convergent. in [20], ulusu and nuray gave some inclusion relations between [vσθ]q-convergence and lacunary iinvariant convergence, and showed that these are equivalent for bounded sequences. now, we shall give analogous theorems which states inclusion relations between [v σθ2 ]p-convergence and lacunary i2-invariant convergence, and show that these are equivalent for bounded double sequences. definition 2.4. a double sequence x = (xkj) is said to be strongly lacunary invariant convergent to l if lim r,u→∞ 1 hru ∑ k,j∈iru |xσk(m),σj(n) −l|, uniformly in m,n and it is denoted by xkj → l ( [v σθ2 ] ) . definition 2.5. a double sequence x = (xkj) is said to be p-strongly lacunary invariant convergent (0 < p < ∞) to l if lim r,u→∞ 1 hru ∑ k,j∈iru |xσk(m),σj(n) −l| p = 0, uniformly in m,n and it is denoted by xkj → l ( [v σθ2 ]p ) . theorem 2.3. if a double sequence x = (xkj) is p-strongly lacunary invariant convergent to l, then (xkj) is lacunary i2-invariant convergent to l. int. j. anal. appl. 16 (3) (2018) 324 proof. assume that xkj → l ( [v σθ2 ]p ) and given ε > 0. then, for every double lacunary sequence θ = {(kr,ju)} and for every m,n ∈ n, we have∑ k,j∈iru ∣∣xσk(m),σj(n) −l∣∣p ≥ ∑ (k,j)∈iru |x σk(m),σj(n) −l|≥ε |xσk(m),σj(n) −l| p ≥ εp ∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣ ≥ εp max m,n ∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣ and 1 hru ∑ k,j∈iru ∣∣xσk(m),σj(n) −l∣∣p ≥ εp maxm,n ∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣ hru = εp sru hru . this implies lim r,u→∞ sru hru = 0 and so (xkj) is iσθ2 -convergent to l. � theorem 2.4. if a double sequence x = (xkj) ∈ `2∞ and (xkj) is lacunary i2-invariant convergent to l, then (xkj) is p-strongly lacunary invariant convergent to l (0 < p < ∞). proof. suppose that x = (xkj) ∈ `2∞ and xkj → l ( iσθ2 ) . let 0 < p < ∞ and ε > 0. by assumption we have v θ2 ( aε ) = 0. the boundedness of (xkj) implies that there exists k > 0 such that |xσk(m),σj(n) −l| ≤ k, ((k,j) ∈ ir,u; m,n = 1, 2, . . . ). observe that, for every m,n ∈ n we have 1 hru ∑ k,j∈iru ∣∣xσk(m),σj(n) −l∣∣p = 1 hru ∑ k,j∈iru |x σk(m),σj(n) −l|≥ε |xσk(m),σj(n) −l| p + 1 hru ∑ k,j∈iru |x σk(m),σj(n) −l|<ε |xσk(m),σj(n) −l| p ≤ k max m,n ∣∣{(k,j) ∈ iru : |xσk(m),σj(n) −l| ≥ ε}∣∣ hru + εp ≤ k sru hru + εp. int. j. anal. appl. 16 (3) (2018) 325 hence, we obtain lim r,u→∞ 1 hru ∑ k,j∈iru ∣∣xσk(m),σj(n) −l∣∣p = 0, uniformly in m,n. � theorem 2.5. a double sequence x = (xkj) ∈ `2∞ and (xkj) is lacunary i2-invariant convergent to l if and only if (xkj) is p-strongly lacunary invariant convergent to l (0 < p < ∞.) proof. this is an immediate consequence of theorem 2.3 and theorem 2.4. � now, introducing lacunary i∗2 -invariant convergence concept, lacunary iσ2 -cauchy double sequence and iσθ2∗ -cauchy double sequence concepts, we give the relationships among these concepts and relationships with lacunary i2-invariant convergence concept. definition 2.6. a double sequence x = (xkj) is lacunary i∗2 -invariant convergent or iσθ2∗ -convergent to l if and only if there exists a set m2 ∈f(iσθ2 ) (n×n\m2 = h ∈iσθ2 ) such that lim k,j→∞ (k,j)∈m2 xkj = l. (2.1) in this case, we write iσθ2∗ − lim xkj = l or xkj → l ( iσθ2∗ ) . theorem 2.6. if a double sequence x = (xkj) is lacunary i∗2 -invariant convergent to l, then this sequence is lacunary i2-invariant convergent to l. proof. since iσθ2∗ − lim k,j→∞ xkj = l, there exists a set m2 ∈f(iσθ2 ) (n×n\m2 = h ∈iσθ2 ) such that lim k,j→∞ (k,j)∈m2 xkj = l. given ε > 0. by (2.1), there exists k0,j0 ∈ n such that |xkj −l| < ε, for all (k,j) ∈ m2 and k ≥ k0,j ≥ j0. hence, for every ε > 0, we have t(ε) = { (k,j) ∈ n×n : |xkj −l| ≥ ε } ⊂ h ∪ ( m2 ∩ ( ({1, 2, ..., (k0 − 1)}×n) ∪ (n×{1, 2, ..., (k0 − 1)}) )) . since iσθ2 ⊂ 2n×n is a strongly admissible ideal, h ∪ ( m2 ∩ ( ({1, 2, ..., (k0 − 1)}×n) ∪ (n×{1, 2, ..., (k0 − 1)}) )) ∈iσθ2 , so we have t(ε) ∈iσθ2 that is v θ2 ( t(ε) ) = 0. hence, iσθ2 − lim k,j→∞ xkj = l. � the converse of theorem 2.6, which it’s proof is similar to the proof of theorems in [1–3], holds if iσθ2 has property (ap2). int. j. anal. appl. 16 (3) (2018) 326 theorem 2.7. let iσθ2 has property (ap2). if a double sequence x = (xkj) is lacunary i2-invariant convergent to l, then this sequence is lacunary i∗2 -invariant convergent to l. finally, we define the concepts of lacunary i2-invariant cauchy and lacunary i∗2 -invariant cauchy double sequences. definition 2.7. a double sequence (xkj) is said to be lacunary i2-invariant cauchy sequence or iσθ2 -cauchy sequence, if for every ε > 0, there exist numbers s = s(ε), t = t(ε) ∈ n such that a(ε) = { (k,j), (s,t) ∈ iru : |xkj −xst| ≥ ε } ∈iσθ2 , that is, v θ2 ( a(ε) ) = 0. definition 2.8. a double sequence (xkj) is lacunary i∗2 -invariant cauchy sequence or iσθ2∗ -cauchy sequence if there exists a set m2 ∈ f(iσθ2 ) (i.e., n × n\m2 = h ∈ iσθ2 ) such that for every (k,j), (s,t) ∈ m2 lim k,j,s,t→∞ |xkj −xst| = 0. the proof of the following theorems are similar to the proof of theorems in [2, 3, 11], so we omit them. theorem 2.8. if a double sequence x = (xkj) is iσθ2 -convergent, then (xkj) is an iσθ2 -cauchy sequence. theorem 2.9. if a double sequence x = (xkj) is iσθ2∗ -cauchy sequence, then (xkj) is iσθ2 -cauchy sequence. theorem 2.10. let iσθ2 has property (ap2). if a double sequence x = (xkj) is iσθ2 -cauchy sequence then, (xkj) is iσθ2∗ -cauchy sequence. acknowledgements: this study supported by afyon kocatepe university scientific research coordination unit with the project number 17.kariyer.21. references [1] p. das, p. kostyrko, w. wilczyński and p. malik, i and i∗-convergence of double sequences, math. slovaca, 58(5) (2008), 605-620. [2] e. dündar and b. altay, i2-convergence and i2-cauchy of double sequences, acta math. sci., 34b(2) (2014), 343-353. [3] e. dündar, u. ulusu and f. nuray, on ideal invariant convergence of double sequences and some properties, creat. math. inf., 27(2) (2018), (in press). [4] j. a. fridy and c. orhan, lacunary statistical convergence, pacific j. math., 160(1) (1993), 43–51. [5] p. kostyrko, t. šalát and w. wilczyński, i-convergence, real anal. exchange, 26(2) (2000), 669–686. [6] v. kumar, on i and i∗-convergence of double sequences, math. commun. 12 (2007), 171–181. [7] s. a. mohiuddine and e. savaş, lacunary statistically convergent double sequences in probabilistic normed spaces, ann univ. ferrara, 58 (2012), 331–339. [8] m. mursaleen, matrix transformation between some new sequence spaces, houston j. math., 9 (1983), 505–509. int. j. anal. appl. 16 (3) (2018) 327 [9] m. mursaleen, on finite matrices and invariant means, indian j. pure appl. math., 10 (1979), 457–460. [10] m. mursaleen and o. h. h. edely, on the invariant mean and statistical convergence, appl. math. lett., 22(11) (2009), 1700–1704. [11] a. nabiev, s. pehlivan and m. gürdal, on i-cauchy sequences, taiwanese j. math., 11(2) (2007), 569–576. [12] f. nuray, h. gök and u. ulusu, iσ-convergence, math. commun. 16 (2011) 531–538. [13] n. pancaroǧlu and f. nuray, statistical lacunary invariant summability, theor. math. appl., 3(2) (2013), 71–78. [14] a. pringsheim, zur theorie der zweifach unendlichen zahlenfolgen, math. ann., 53 (1900), 289?21. [15] r. a. raimi, invariant means and invariant matrix methods of summability, duke math. j., 30(1) (1963), 81–94. [16] e. savaş, some sequence spaces involving invariant means, indian j. math., 31 (1989), 1–8. [17] e. savaş, strongly σ-convergent sequences, bull. calcutta math., 81 (1989), 295–300. [18] e. savaş and r. patterson, double σ-convergence lacunary statistical sequences, j. comput. anal. appl., 11(4) (2009). [19] p. schaefer, infinite matrices and invariant means, proc. amer. math. soc., 36 (1972), 104–110. [20] u. ulusu and f. nuray, lacunary iσ-convergence, (under review). 1. introduction 2. lacunary i2-invariant convergence references international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 1-8 http://www.etamaths.com univalent biharmonic mappings and linearly connected domains z. abdulhadi1 and l. el hajj2,∗ abstract. a four times continuously differentiable complex-valued function f = u + iv in a simply connected domain ω is biharmonic if the laplacian of f is harmonic. every biharmonic mapping f in ω has the representation f = |z|2g + k, where g and k are harmonic in ω. this paper investigates the relationship between the univalence of f and of k using the concept of linearly connected domains. 1. introduction a planar harmonic mapping in a simply connected domain ω ⊂ c is a complexvalued harmonic function f(z) defined on ω, where z = x+iy. the mapping f has a canonical decomposition f = h + g, where h and g are analytic (holomorphic) in ω (see [13, 14]). we say that f is locally univalent and sense preserving if and only if its jacobian jf (z) is positive, where jf (z) is given by jf (z) = |fz(z)|2 −|fz(z)|2 = |h′(z)|2 −|g′(z)|2, (see lewy [11]). clunie and sheil-small made the following important observation : f is locally univalent and orientation -preserving in d if and only if |g′(z)| < |h′(z)| in ω; or equivalently if h′(z) 6= 0 and the dilatation ω(z) = g ′(z) h′(z) has the property |ω(z)| < 1. a four times continuously differentiable complex-valued function f = u + iv in a simply connected domain ω is biharmonic if the laplacian of f is harmonic. note that 4f is harmonic in ω, if 4f satisfies laplace’s equation 4(4f) = 0, where 4 = 4 ∂2 ∂z∂z := ∂2 ∂x2 + ∂2 ∂y2 . every harmonic function is biharmonic but not necessarily the converse. moreover, it is easy to see that every biharmonic mapping f in ω has the representation (1.1) f = |z|2g + k, where g and k are harmonic in ω and they can be expressed as, g = g1 + g2,(1.2) k = k1 + k2, 2010 mathematics subject classification. 30c35, 30c45, 35q30. key words and phrases. biharmonic mappings; univalent; linearly connected domains. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 abdulhadi and hajj where g1, g2, k1 and k2 are analytic in ω (for details see [2]). note that the composition f ◦φ of a harmonic function f with analytic function φ is harmonic, while this is not true when f is biharmonic. biharmonic mappings arise in a lot of physical situations, particularly in fluid dynamics and elasticity problems, and have many important applications in engineering and biology. most important applications of the theory of functions of a complex variable were obtained in the plane theory of elasticity and in the approximate theory of plates subject to normal loading. that is, in cases when the solutions are biharmonic functions or functions associated with them ( see [15, 16 ]). moreover, biharmonic mapping are closely related to the theory of laguerre minimal surfaces (for details see [5, 7, 8, 9, 17, 18]). investigation of biharmonic mappings in the context of geometric function theory started only recently (for details see[ 1, 2, 3, 4, 10 ]). for example, in [2], abdulhadi, abumuhanna and khuri analyze the univalence of the solutions of the biharmonic equations. throughout we consider harmonic and biharmonic functions defined on the unit disk d = {z : |z| < 1}. definition 1. a domain ω ⊂ c is linearly connected if there exists a constant m < ∞ such that any two points w1,w2 ∈ ω are joined by a path γ, γ ⊂ ω, of length `(γ) ≤ m|w1 −w2|. such a domain is necessarily a jordan domain, and for piecewise smoothly bounded domains, linear connectivity is equivalent to the boundary having no inwardpointing cusps. in [12], chuaqui and hermandez, considered the relationship between the harmonic mapping f = h + g and its analytic factor h on linearly connected domains. they show that if h is an analytic univalent function, then every harmonic mapping f = h + g with dilatation |ω| < c is univalent if and only if h(d) is linearly connected. in this paper, we scrutinize the relationship between the univalence of the biharmonic function f = |z|2g + k and the univalence of the harmonic function k. 2. main results in our first results, we deduce the univalence of f(z) from the univalence of k(z). we first consider subclasses, where g,k are assumed to be analytic or antianalytic. theorem 1. let f(z) = |z|2g(z) +k(z) be a biharmonic function in the unit disk d, where g,k are analytic. if k is univalent and k(d) is a linearly connected domain with constant m, and if 2|g| + |g′| |k′| < 1 m , then f(z) is univalent. proof. let h(z) = |z|2g(z). we define ϕ = h ◦ k−1. given w � k(d), we claim w+ϕ(w) is univalent. assume w+ϕ(w) is not univalent, then there exists w1 6= w2 such that ϕ(w2) −ϕ(w1) = w1 −w2. let γ be a path in k(d) joining w1,w2 such that l(γ) ≤ m|w2 −w1|. biharmonic mappings 3 then |ϕ(w2) −ϕ(w1)| ≤ ∣∣∣∣ ∫ γ ϕwdw + ϕwdw ∣∣∣∣ . but ϕw = hz(k −1)w + hz(k−1)w = hz k′ = zg + |z|2g′ k′ , ϕw = hz(k −1)w + hz(k−1)w = hz k ′ = zg k ′ , where z = k−1(w) ∈ d. therefore, |ϕ(w2) −ϕ(w1)| ≤ ∫ γ sup d |2g| + |g′| |k′| |dw| < 1 m l(γ) < |w2 −w1| which is a contradiction. hence f(z) is univalent. � remark 1. in the above proof , if k(d) is convex we may take m = 1, and thus f will be univalent as long as 2|g|+|g′| |k′| < 1. the special case m = 1 when k is convex, is an important special case and we will state it separately as a corollary. corollary 1. let f(z) = |z|2g(z) + k(z) be a biharmonic function in the unit disk d, where g,k are analytic. if k is univalent and convex with 2|g| + |g′| |k′| < 1, then f(z) is univalent. as a consequence of theorem 1, we have the following corollary : corollary 2. let f(z) = |z|2g(z) +k(z) be a biharmonic function in the unit disk d, where g,k are antianalytic. if k is univalent and k(d) is a linearly connected domain with constant m, and 2|g| + |gz| |kz| < 1 m , then f(z) is univalent. our next result is the general case, where g ,k are harmonic in the unit disk d : theorem 2. let f(z) = |z|2g(z) +k(z) be a biharmonic function in the unit disk d, where g,k are harmonic. if k is univalent and k(d) is a linearly connected domain with constant m, and if 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) < 1 m , then f(z) is univalent. in the above ωk,ωg denotes the dilations ωk = k′2 k′1 , ωg = g′2 g′1 . 4 abdulhadi and hajj proof. let h(z) = |z|2g(z). we define ϕ = h ◦k−1. given w � k(d), we claim w + ϕ(w) is univalent. assume w + ϕ(w) is not univalent, then there exists w1 6= w2 such that ϕ(w2) −ϕ(w1) = w1 −w2. let γ be a path in k(d) joining w1,w2 such that l(γ) ≤ m|w2 −w1|.then |ϕ(w2) −ϕ(w1)| ≤ ∣∣∣∣ ∫ γ ϕwdw + ϕwdw ∣∣∣∣ ≤ ∫ γ (|ϕw| + |ϕw|) |dw|. but ϕw = hz(k −1)w + hz(k−1)w ϕw = hz(k −1)w + hz(k−1)w. differentiating k−1(k(z)) = z, we show that (k−1)w = k′1 |k′1|2 −|k′2|2 , (k−1)w = −k′2 |k′1|2 −|k′2|2 . it follows ϕw = hz(k −1)w + hz(k−1)w = (zg + |z|2g′1) k′1 |k′1|2 −|k′2|2 + (zg + |z|2g′2) −k′2 |k′1|2 −|k′2|2 , ϕw = hz(k −1)w + hz(k−1)w = (zg + |z|2g′1) −k′2 |k′1|2 −|k′2|2 + (zg + |z|2g′2) k′1 |k′1|2 −|k′2|2 . therefore, |ϕw| + |ϕw| ≤ 2|z||g|(|k′1| + |k′2|) |k′1|2 −|k′2|2 + |z|2(|g′1||k′2| + |g′2||k′1|) |k′1|2 −|k′2|2 = 2|z||g| + |z|2 (|g′1| + |g′2|) |k′1|− |k′2| = 2|z||g| + |z|2|g′1|(1 + |ωg|) |k′1|(1 −|ωk|) , where z = k−1(w) ∈ d. then we have |ϕ(w2) −ϕ(w1)| ≤ ∫ γ sup d 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) |dw| < 1 m l(γ) < |w2 −w1| which is a contradiction. hence f(z) is univalent. � corollary 3. let f(z) = |z|2g(z) + k(z) be a biharmonic function in the unit disk d, where g,k are harmonic. if k is univalent and convex with 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) < 1, then f(z) is univalent. biharmonic mappings 5 the following corollary follows immediately from theorem 2, for the case when g is analytic. corollary 4. let f(z) = |z|2g(z) + k(z) be a biharmonic function in the unit disk d, where g is analytic, k is harmonic. if k is univalent and k(d)is a linearly connected domain with constant m, and if 2|g| + |g′| |k′1|(1 −|ωk|) < 1 m , then f(z) is univalent. in the above ωk denotes the dilation ωk = k′2 k′1 . in each of the previous results, it follows under the same conditions that f(d) will also be linearly connected. we will prove it for the general case and the proof is along the same lines for the special cases. proposition 1. let f(z) = |z|2g(z) + k(z) be a biharmonic function in the unit disk d, where g ,k are harmonic. if k is univalent and k(d) is a linearly connected domain with constant m, and if 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) ≤ c, where c < 1 m , then f(d) is linearly connected. proof. given w ∈ ω = k(d), we let ψ(w) = w + ϕ(w), where ϕ = h ◦ k−1, and h = |z|2g. since k is univalent, we may look at r = f(d), as the image of ω = k(d) under the mapping ψ, and we show ψ(ω) is linearly connected. let ς1 = ψ(w1), ς2 = ψ(w2), w1,w2 ∈ ω. since k(d) is a linearly connected domain, then there exists a curve γ ⊂ ω satisfying l(γ) ≤ m|w2 −w1|. let γ = ψ(γ). in the proof of theorem 2, we have showed that |ϕw| + |ϕw| ≤ 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) < c, it follows |ψw| + |ψw| ≤ 1 + |ϕw| + |ϕw| < 1 + c. hence we have, l(γ) = ∫ γ dς ≤ ∫ γ (|ψw| + |ψw|) dw < (1 + c)l(γ) ≤ (1 + c)m|w2 −w1|. but, |ς1 − ς2| = |w1 −w2 + ϕ(w1) −ϕ(w2)| ≥ |w1 −w2|− |ϕ(w1) −ϕ(w2)| ≥ |w1 −w2|− ∫ γ (|ϕw| + |ϕw|) dw > |w1 −w2|−cl(γ) ≥ (1 −cm)|w1 −w2|. then we get, l(γ) ≤ (1 + c)m 1 −cm |ς1 − ς2| and so r is linearly connected with constant (1+c)m 1−cm . � in our next result, we deduce the univalence of k from the univalence of f. 6 abdulhadi and hajj theorem 3. let f(z) = |z|2g(z) +k(z) be a biharmonic function in the unit disk d. suppose f is univalent and f(d) is a linearly connected domain with constant m and satisfies 2|g| + |gz| + |gz| ||fz|− |fz|| < 1 m , then k(z) is univalent. proof. let h(z) = |z|2g(z). aiming for a contradiction assume k(z) is not univalent, then there exists z1 6= z2 , such that k(z1) = k(z2). hence we get f(z1) −f(z2) = h(z1) −h(z2). given w = f(z)� f(d), the above equation is equivalent to w1 −w2 = ϕ(w2) −ϕ(w1), where ϕ = h ◦f−1. let γ be a path in f(d) joining w1,w2 such that l(γ) ≤ m|w2 −w1|. then |ϕ(w2) −ϕ(w1)| ≤ ∣∣∣∣ ∫ γ ϕwdw + ϕwdw ∣∣∣∣ ≤ ∫ γ (|ϕw| + |ϕw|) |dw|. but ϕw = hz(f −1)w + hz(f−1)w ϕw = hz(f −1)w + hz(f−1)w. differentiating f−1(f(z)) = z, we get the following two equations (f−1)wfz + (f −1)wfz = 1 (f−1)wfz + (f −1)wfz = 0. solving the above system we get (f−1)w = fz jf , (k−1)w = −fz jf , where jf denotes the jacobian jf = |fz|2 −|fz|2. it follows, ϕw = hz(f −1)w + hz(f−1)w = hz fz jf −hz fz jf ϕw = hz(f −1)w + hz(f−1)w = −hz fz jf + hz fz jf . therefore, |ϕw| + |ϕw| ≤ |hz||fz| + |hz||fz| + |hz||fz| + |hz||fz| |jf | = (|hz| + |hz|)(|fz| + |fz|) |jf | = |hz| + |hz| ||fz|− |fz|| biharmonic mappings 7 ≤ 2|g| + |gz| + |gz| ||fz|− |fz|| . hence |ϕ(w2) −ϕ(w1)| ≤ ∫ γ sup d 2|g| + |gz| + |gz| ||fz|− |fz|| |dw| < 1 m l(γ) < |w2 −w1|, which is a contradiction. thus k(z) is univalent. � corollary 5. let f(z) = |z|2g(z) + k(z) be a biharmonic function in the unit disk d, where g ,k are harmonic. if k is univalent and convex with 2|g| + |g′1|(1 + |ωg|) |k′1|(1 −|ωk|) < 1, then k(z) is univalent. references [1] z. abdulhadi and y.abumuhanna , “ landau’s theorem for biharmonic mappings,” journal of mathematical analysis and applications, 338(2008), 705-709. [2] z. abdulhadi, y.abumuhanna and s. khoury, “ on univalent solutions of the biharmonic equations,” journal of inequalities and applications, 2005(2005), 469-478. [3] z. abdulhadi, y.abumuhanna and s. khoury, “on the univalence of the log-biharmonic mappings” journal of mathematical analysis and applications, 289(2004), 629-638. [4] z. abdulhadi, y.abumuhanna and s. khoury, “ on some properties of solutions of the biharmonic equations,” appl. math. comput. 177(2006), 346-351. [5] y.abu-muhanna and r. m. ali“ biharmonic maps and laguerre minimal surfaces,” journal of abstract and applied analysis, 2013(2013), article id 843156, 9 pages. [6] y.abu-muhanna and g. schober, harmonic mappings onto convex mapping domains, can. j. math, 39(1987), 1489-1530. [7] a. bobenko and u. pinkall, “discrete isothermic surfaces,”journal fur die reine und angewandte mathematik, 475(1996), 187-208. [8] w. blaschke, “uber die geometrie von laguerre iii: beitra ge ¨zur fl achentheorie,” abhandlungen aus dem mathematischen seminar der universitat hamburg ¨ , 4(1925), 1-12. [9] w. blaschke, “uber die geometrie von laguerre ii: fla chen¨theorie in ebenenkoordinaten,” abhandlungen aus dem mathematischenseminar der universitat hamburg ¨, 3(1924), 195212. [10] s. chen, s. ponnusamy, and x. wang, “ landau’s theorem for certain biharmonic mappings,” applied mathematics and computation, 208(2009), 427-433. [11] g. choquet, sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, bull. sci. math. 69(1945), 156–165. [12] m.chuaqui, r.hermandez,univalent harmonic mappings and linearly connected domains, j. math.anal.appl., 332(2007), 1189-1197. [13] j. clunie and t. sheil-small, harmonic univalent functions, annales acad. sci. fenn. series a. mathematica, 9(1984), 3-25. [14] p. duren, harmonic mappings in the plane, cambridge university press, 2004. [15] j. happel and h. brenner, low reynolds number hydrodynamics, pretice-hall, 1965. [16] w.e. langlois, slow viscous flow, macmillan company, 1964. [17] m. peternell and h. pottmann, “a laguerre geometric approachto rational offsets,” computer aided geometric design, 15(1998), 223-249. [18] h. pottmann, p. grohs, and n. j. mitra, “laguerre minimal surfaces, isotropic geometry and linear elasticity,” advances in computational mathematics, 31(2009), 391-419. 8 abdulhadi and hajj 1department of mathematics and statistics, aus, p.o.box 26666, sharjah, uae 2mathematics division , aud, p.o.box 28282 , dubai, uae ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 140-146 http://www.etamaths.com identities on genocchi polynomials and genocchi numbers concerning binomial coefficients qing zou∗ abstract. in this paper, the author gives some new identities on genocchi polynomials and genocchi numbers. 1. introduction the researches on genocchi numbers and genocchi polynomials have a long history. it can be traced back to angelo genocchi (1817–1889). nowadays, genocchi numbers and kinds of genocchi polynomials have become a popular research topic. during these very recent years, some researchers such as araci [1–7] did many researches on this interesting topic. they studied genocchi numbers and genocchi polynomials extensively in many branches of mathematics, such as elementary number theory, analytic number theory, theory of modular forms, p-adic analytic number theory and etc.. now, let us show the definitions of genocchi numbers and genocchi polynomials. the genocchi numbers are a sequence of integers that satisfy the following exponential generating function 2t et + 1 = ∞∑ n=0 gn tn n! , |t| < π, with the convention that replacing gn by gn. the first few genocchi numbers are g0 = 0, g1 = 1, g2 = −1, g3 = 0, g4 = 1, g5 = 0, g6 = −3, g7 = 0, g8 = 17. the classic genocchi polynomials are usually defined by mean of the following exponential generating function 2t et + 1 ·ext = ∞∑ n=0 gn(x) tn n! , |t| < π, with the convention that replacing gn(x) by gn(x). it is clear that gn(0) = gn. according to the classic genocchi polynomials, some mathematicians introduced several new polynomials that extended the classic genocchi polynomials. araci [6] and kim et al. [8] did some researches on the so-called genocchi polynomials of order k, which were defined by ( 2t et + 1 )k ·ext = ∞∑ n=0 g(k)n (x) tn n! . araci [6] and he [9, 10] introduced the apostol–genocchi polynomials defined by 2t λet + 1 ·ext = ∞∑ n=0 gn(x,λ) tn n! . received 9th march, 2017; accepted 25th may, 2017; published 3rd july, 2017. 2010 mathematics subject classification. primary 11b68; secondary 05a19. key words and phrases. genocchi polynomial; genocchi number; binomial inverse formula. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 140 identities on genocchi polynomials and genocchi numbers 141 based on which, araci [7] introduced the high order apostol–genocchi polynomials which can be called the generalized apostol–genocchi polynomials of order k ∈ c,( 2t λet + 1 )k ·ext = ∞∑ n=0 g(k)n (x,λ) tn n! . in [11], lim defined the degenerated genocchi polynomials g(k)n (x,λ) of order k to be( 2t (1 + λt)1/λ + 1 )k (1 + λt)x/λ = ∞∑ n=0 g(k)n (x,λ) tn n! . besides these generalizations, araci [1], duran et al. [12] and agyuz et al. [13] also introduced the q-alanogue of the genocchi polynomials as follows, ∞∑ n=0 gn,q(x) tn n! = t ∫ zp q−yet[y+x]qdµ−q(y), where [x]q = 1 −qx 1 −q , [x]−q = 1 − (−q)x 1 + q . this definition used p-adic fermionic q-integral on zp with respect to µ−q. it can also be defined by ∞∑ n=0 gn,q(x) tn n! = [2]qt ∞∑ m=0 (−1)met[m+x]q. in which when we take x = 0, it becomes gn,q(0) := gn,q, which we call it the n-th q-genocchi number. when it comes to genocchi numbers, the most common thing comes to our mind is to research the relations between genocchi numbers, bernoulli numbers [14–16] and euler numbers [14, 17]. indeed, most researches on genocchi numbers concern the relations between these three kinds of numbers (see for example [2–4, 18, 19]). in other words, there are many literatures that provide identities on these three kinds of numbers. similarly, when it comes to genocchi polynomials, the most common thing is to research on the relations between genocchi polynomials, bernoulli polynomials and euler polynomials (see for example [2–4, 9, 18–21]). even though when it comes to the generalized genocchi numbers and generalized genocchi polynomials, it is unavoidable to research the relations as above. in this paper, we do not want to find relations between the three kinds of numbers or the three kinds of polynomials. we will focus only on genocchi numbers themselves and genocchi polynomials themselves. in other words, in this paper, we will give some identities only concern genocchi numbers and genocchi polynomials. actually, by these identities combining with the identities between genocchi numbers (polynomials), bernoulli numbers (polynomials) and euler numbers (polynomials), one can obtain some other identities. while we do not want to show them here since the process of combining two identities is not very novel. 2. identities on genocchi numbers and genocchi polynomials let us start this section with some straightforward derived identities on genocchi numbers and genocchi polynomials. differentiating both sides of the exponential generating function for gn with respect to x yields d dx gn(x) = ngn−1(x), deg gn+1(x) = n. by which we can get ∫ b a gn(x)dx = gn+1(b) −gn+1(a) n + 1 . thanks to [3, 18], we have gn(x) = n∑ k=0 ( n k ) gkx n−k. 142 q. zou combining the above two identities and the relation (2.7) below shows∫ 1 0 gn(x)dx = { 0 n = 0 −2gn+1 n+1 n ≥ 1 . besides these classical identities, one can also find more identities concerning genocchi numbers and genocchi polynomials in [4]. next, we show some new identities on genocchi numbers and genocchi polynomials. theorem 2.1. for n ≥ 2, we have 1 2 n∑ k=0 ( n k ) gk(x) ·gn−k+1(1) n−k + 1 = x ·gn(x) − n n + 1 gn+1(x). (2.1) 1 2 n∑ k=0 ( n k ) gk+1(x) ·gn−k(1) k + 1 = x ·gn(x) − n n + 1 gn+1(x). (2.2) proof. let us recall the generating function of genocchi polynomials first, 2t et + 1 ·ext = ∞∑ n=0 gn(x) tn n! . taking the partial derivative with respect to t on the right hand side, we deduce that ∂ ∂t ∞∑ n=0 gn(x) tn n! = ∂ ∂t ( g0(x) + g1(x)t + g2(x) t2 2! + g3(x) t3 3! + · · · ) =g1(x) + g2(x)t + g3(x) t2 2! + · · · = ∞∑ n=0 gn+1(x) tn n! . (2.3) now, let us look at the left hand side. ∂ ∂t ( 2t et + 1 ·ext ) = (2ext + xext · 2t)(et + 1) − (2text ·et) (et + 1)2 = 1 t 2t ·ext et + 1 + x 2t ·ext et + 1 − 1 2t 2t ·ext et + 1 2t ·et et + 1 = 1 t ∞∑ n=0 gn(x) tn n! + x ∞∑ n=0 gn(x) tn n! − 1 2t ∞∑ n=0 gn(x) tn n! ∞∑ n=0 gn(1) tn n! = ∞∑ n=0 gn+1(x) n + 1 tn n! + ∞∑ n=0 xgn(x) tn n! − 1 2 ∞∑ n=0 gn(x) tn n! ∞∑ n=0 gn+1(1) n + 1 tn n! = ∞∑ n=0 [ gn+1(x) n + 1 + x ·gn(x) − 1 2 n∑ k=0 ( n k ) gk(x) gn−k+1(1) n−k + 1 ] tn n! . (2.4) in the second to last step, we used the fact that g0(x) = 0. comparing the coefficients of t n n! in (2.3) and (2.4) yields gn+1(x) n + 1 + x ·gn(x) − 1 2 n∑ k=0 ( n k ) gk(x) gn−k+1(1) n−k + 1 = gn+1(x). then (2.1) follows from rearranging the terms in this identity. note that the second to last step can also be written as ∂ ∂t ( 2t et + 1 ·ext ) = ∞∑ n=0 gn+1(x) n + 1 tn n! + ∞∑ n=0 xgn(x) tn n! − 1 2 ∞∑ n=0 gn+1(x) n + 1 tn n! ∞∑ n=0 gn(1) tn n! , identities on genocchi polynomials and genocchi numbers 143 which gives us gn+1(x) n + 1 + x ·gn(x) − 1 2 n∑ k=0 ( n k ) gk+1(x) k + 1 gn−k(1) = gn+1(x). rearranging the terms above yeilds (2.2). this completes the proof. � remark 2.1. according to the process of the proof above, one can also obtain that 1 2 n∑ k=0 ( n k ) gn−k(x) ·gk+1(1) k + 1 = x ·gn(x) − n n + 1 gn+1(x), (2.5) and 1 2 n∑ k=0 ( n k ) gn−k+1(x) ·gk(1) n−k + 1 = x ·gn(x) − n n + 1 gn+1(x). (2.6) but we should notice that (2.5) and (2.6) are equivalent to (2.1) and (2.2), respectively. this is because when k goes from 0 to n, n−k also goes from 0 to k. hence if we replace k by n−k in (2.1) and (2.2), we can then get (2.5) and (2.6) respectively. from this point of view, we do not regard (2.5) and (2.6) as new identities. corollary 2.1. for n ≥ 2, we have 1 2 n∑ k=0 ( n k ) gk ·gn−k+1(1) n−k + 1 = − n n + 1 gn+1. 1 2 n∑ k=0 ( n k ) gk+1 ·gn−k(1) k + 1 = − n n + 1 gn+1. proof. this lemma follows from taking x = 0 in theorem 2.1. � having developed to this point, it is necessary to say something about gn(1). since 2t et + 1 ·et = ∞∑ n=0 gn(1) tn n! . then ∞∑ n=0 (gn + gn(1)) tn n! = ∞∑ n=0 gn tn n! + ∞∑ n=0 gn(1) tn n! = 2t. thus, g1 + g1(1) = 2 and for n ≥ 2, gn + gn(1) = 0, which means gn(1) = { 1, n = 1, −gn, n ≥ 2. (2.7) so, in this sense, we can call the integer sequence gn(1) the negative genocchi numbers. with this fact, we can obtain corollary 2.2. for n ≥ 2, we have 1 2 n−1∑ k=0 ( n k ) gk(x) ·gn−k+1 n−k + 1 = ( 1 2 −x) ·gn(x) + n n + 1 gn+1(x). (2.8) 1 2 n−2∑ k=0 ( n k ) gk+1(x) ·gn−k k + 1 = ( 1 2 −x) ·gn(x) + n n + 1 gn+1(x). (2.9) proof. since gn(1) = −gn except for n = 1. then we can replace gn−k+1(1) by −gn−k+1 except for k = n. this gives us 1 2 gn(x) − 1 2 n−1∑ k=0 ( n k ) gk(x) ·gn−k+1 n−k + 1 = x ·gn(x) − n n + 1 gn+1(x), which means (2.8) holds true. 144 q. zou similarly, we can show (2.9) through (2.2). � if we take x = 0 in corollary 2.2, we can arrive at the following conclusion. corollary 2.3. for n ≥ 2, we have 1 2 n−1∑ k=0 ( n k ) gk ·gn−k+1 n−k + 1 = 1 2 gn + n n + 1 gn+1. 1 2 n−2∑ k=0 ( n k ) gk+1 ·gn−k k + 1 = 1 2 gn + n n + 1 gn+1. next, let us talk about gn(x + y) which is given by 2t et + 1 ·e(x+y)t = ∞∑ n=0 gn(x + y) tn n! . as the basic properties we have mentioned for gn(x), gn(x + y) has the same properties, such as∫ d c ∫ b a gn(x + y)dxdy = gn+2(a + c) −gn+2(b + c) (n + 2)(n + 1) − gn+2(a + d) −gn+2(b + d) (n + 2)(n + 1) . now, we would like to show some identities on gn(x + y). theorem 2.2. for y 6= 0, gn(x + y) = n∑ k=0 ( n k ) gk(x)y n−k. (2.10) conversely, we have gn(x) = (−1)n n∑ k=0 (−1)k ( n k ) gk(x + y)y n−k. (2.11) symmetrically, when x 6= 0, we have gn(x + y) = n∑ k=0 ( n k ) gk(y)x n−k, (2.12) and gn(y) = (−1)n n∑ k=0 (−1)k ( n k ) gk(x + y)x n−k. (2.13) proof. since ∞∑ n=0 gn(x + y) tn n! = 2t et + 1 e(x+y)t = 2t et + 1 ext ·eyt = ∞∑ n=0 gn(x) tn n! ∞∑ n=0 yn tn n! = ∞∑ n=0 ( n∑ k=0 ( n k ) gn−k(x)y n−k ) tn n! . comparing the coefficients of t n n! shows (2.10) holds true. the binomial inverse formula [22, pp.192, (5.48)] reads as an = n∑ k=0 ( n k ) (−1)kbk ⇔ bn = n∑ k=0 ( n k ) (−1)kak. equation (2.10) can be rewritten as gn(x + y) yn = n∑ k=0 ( n k ) gk(x) yk . identities on genocchi polynomials and genocchi numbers 145 taking ak = gk(x+y) yk and bk = (−1)k gk(x) yk gives us (−1)n gn(x) yn = n∑ k=0 (−1)k ( n k ) gk(x + y) yk , which shows (2.11) holds. since x and y are symmetric, then we can obtain (2.12) and (2.13) by changing the position of x and y. � remark 2.2. if we want (2.11) and (2.13) to be more beautiful, we can replace k by n−k. then we can have gn(x) = n∑ k=0 (−1)k ( n k ) gn−k(x + y)y k, and gn(y) = n∑ k=0 (−1)k ( n k ) gn−k(x + y)x k. corollary 2.4. n∑ k=0 (−1)k+1 ( n k ) gk = (−1)ngn + 2n. proof. thanks to [3, 18], we have gn(x + 1) + gn(x) n = 2xn−1. (2.14) taking x = −1 in (2.14) shows gn + gn(−1) = 2n · (−1)n−1. (2.15) let x = 0 and y = −1 in (2.10), we deduce that gn(−1) = n∑ k=0 (−1)n−k ( n k ) gk. (2.16) plugging (2.16) in (2.15) shows gn + n∑ k=0 (−1)n−k ( n k ) gk = 2n · (−1)n−1. then multiplying (−1)n−1 on the both sides proves this corollary. � corollary 2.5. for n ≥ 2, we have gn + n∑ k=0 ( n k ) gk = 0. proof. let x = 1 and y = 0 in (2.12), we can get that gn(1) = n∑ k=0 ( n k ) gk. since we have mentioned above that gn(1) + gn = 0 when n ≥ 2. then the conclusion follows. � 146 q. zou references [1] s. araci, m. acikgoz, h. jolany and j. j. seo, a unified generating function of the q-genocchi polynomials with their interpolation functions, proc. jangjeon math. soc. 15 (2) (2012), 227–233. [2] s. araci, novel identities for q-genocchi numbers and polynomials, j. funct. spaces appl. 2012 (2012), article id 214961. [3] s. araci, m. acikgoz and e. sen, some new identities of genocchi numbers and polynomials involving bernoulli and euler polynomials, arxiv:1209.0628 [math.nt]. [4] s. araci, novel identities involving genocchi numbers and polynomials arising from applications of umbral calculus, appl. math. comput. 233 (2014), 599–607. [5] s. araci, m. acikgoz and e. sen, on the von staudt–clausen’s theorem associated with q-genocchi numbers, appl. math. comput. 247 (2014), 780–785. [6] s. araci, e. sen and m. acikgoz, theorems on genocchi polynomials of higher order arising from genocchi basis, taiwanese j. math. 18 (2) (2014), 473–482. [7] s. araci, w. a. khan, m. acikgoz, c. ozel and p. kumam, a new generalization of apostol type hermite–genocchi polynomials and its applications, springerplus, 5 (2016), art. id 860. [8] t. kim, s. h. rim, d. v. dolgy and s. h. lee, some identities of genocchi polynomials arising from genocchi basis, j. ineq. appl. 2013 (2013), article id 43. [9] y. he, s. araci, h. m. srivastava and m. acikgoz, some new identities for the apostol–bernoulli polynomials and the apostol–genocchi polynomials, appl. math. comput. 262 (2015), 31–41. [10] y. he, some new results on products of the apostol–genocchi polynomials, j. comput. anal. appl. 22 (4) (2017), 591–600. [11] d. lim, some identities of degenerate genocchi polynomials, bull. korean math. soc. 53 (2) (2016), 569–579. [12] u. duran, m. acikgoz and s. araci, symmetric identities involving weighted q-genocchi polynomials under s4, proc. jangjeon math. soc. 18 (4) (2015), 455–465. [13] e. agyuz, m. acikgoz and s. araci, a symmetric identity on the q-genocchi polynomials of higher-order under third dihedral group d3, proc. jangjeon math. soc. 18 (2) (2015), 177–187. [14] l. carlitz, q-bernoulli numbers and polynomials, duke math. j. 15 (1948), 987–1000. [15] j. choi and y.-h. kim, a note on high order bernoulli numbers and polynomials using differential equations, appl. math. comput. 249 (2014), 480–486. [16] d. s. kim, t. kim and d. v. dolgy, a note on degenerate bernoulli numbers and polynomials associated with p-adic invariant integral on zp, appl. math. comput. 259 (2015), 198–204. [17] j. choi, p. j. anderson and h. m. srivastava, carlitzs q-bernoulli and q-euler numbers and polynomials and a class of generalized q-hurwitz zeta functions, appl. math. comput. 215 (2009), 1185–1208. [18] s. araci, m. acikgoz and e. sen, some new formulae for genocchi numbers and polynomials involving bernoulli and euler polynomials, int. j. math. math. sci. 2014 (2014), article id 760613. [19] t. kim, some identities for the bernoulli, the euler and the genocchi numbers and polynomials, adv. stud. contemp. math. 20 (1) (2010), 23–28. [20] y. he and t. kim, general convolution identities for apostol-bernoulli, euler and genocchi polynomials, j. nonlinear sci. appl. 9 (2016), 4780–4797. [21] t. agoh, convolution identities for bernoulli and genocchi polynomials, electronic j. combin. 21 (2014), article id p1.65. [22] r. l. graham. d. e. knuth and o. patashnik, concrete mathematics — a foundation for computer science, 2nd edn. addison-wesley publishing company, reading, 1994. department of mathematics, the university of iowa, iowa city, ia 52242-1419, usa ∗corresponding author: zou-qing@uiowa.edu 1. introduction 2. identities on genocchi numbers and genocchi polynomials references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 147-154 http://www.etamaths.com oscillation of nonlinear delay differential equation with non-monotone arguments özkan öcalan1,∗, nurten kiliç2, sermin şahi̇n3 and umut mutlu özkan3 abstract. consider the first-order nonlinear retarded differential equation x′(t) + p(t)f (x (τ(t))) = 0, t ≥ t0 where p(t) and τ(t) are function of positive real numbers such that τ(t) ≤ t for t ≥ t0, and limt→∞ τ(t) = ∞. under the assumption that the retarded argument is non-monotone, new oscillation results are given. an example illustrating the result is also given. keywords: delay differential equation; non-monotone argument; oscillatory solutions; nonoscillatory solutions. 2010 mathematics subject classification: 34k11, 34k06. 1. introduction consider the nonlinear retarded differential equation x′(t) + p(t)f (x (τ(t))) = 0, t ≥ t0 (1.1) where p(t) and τ(t) are functions of nonnegative real numbers, and τ(t) is non-monotone or nondecreasing such that τ(t) ≤ t for t ≥ t0, and lim t→∞ τ(t) = ∞, (1.2) and f ∈ c(r,r) and xf(x) > 0 for x 6= 0. (1.3) by a solution of (1.1) we mean a continuously differentiable function defined on [τ(t0),∞] for some t0 ≥ t0 and such that (1.1) is satisfied for t ≥ t0. such a solution is called oscillatory if it has arbitrarily large zeros. otherwise, it is called nonoscillatory. recently there has been an increasing interest in the study of the oscillatory behavior of the following special form of (1.1) x′(t) + p(t)x (τ(t)) = 0, t ≥ t0. (1.4) see, for example, [1−19] and the references cited therein. the first systematic study for the oscillation of all solutions of equation (1.4) was made by myshkis. in 1950 [17] he proved that every solution of (1.4) oscillates if lim sup t→∞ [t− τ(t)] < ∞ and lim inf t→∞ [t− τ(t)] lim inf t→∞ p(t) > 1 e . in 1972, ladas, lakshmikantham and papadakis [16] proved that the same conclusion holds if, in addition, τ is a non-decreasing function and lim sup t→∞ t∫ τ(t) p(s)ds > 1. (1.5) in 1982, koplatadze and canturija [14] established the following result. received 12th april, 2017; accepted 5th june, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 34k11, 34k06. key words and phrases. delay differential equation; non-monotone argument; oscillatory solutions; nonoscillatory solutions. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 147 148 öcalan, kiliç, şahi̇n and özkan if τ(t) is a non-monotone or nondecreasing and lim inf t→∞ t∫ τ(t) p(s)ds > 1 e , (1.6) then all solutions of eq.(1.4) oscillate, while if lim sup t→∞ t∫ τ(t) p(s)ds < 1 e , (1.7) then the equation (1.4) has a nonoscillatory solution. to the best of our knowledge, there are few papers dealing with the oscillatory behavior of solutions of (1.1), see, for example, [9, 17]. the following theorem was given by ladde et al. in [17]. theorem a. assume that the f, p and τ in eq.(1.1) satisfy the following conditions: i) the condition (1.2) holds and let τ(t) be strictly increasing on r+, ii) p(t) is locally integrable and p(t) ≥ 0, a.e.; iii) the condition (1.3) holds and let f be nondecreasing, and lim x→0 x f(x) = n < +∞. assume further that lim sup t→∞ t∫ τ(t) p(s)ds > n, or lim inf t→∞ t∫ τ(t) p(s)ds > n e . then every solution of eq.(1.1) is oscillatory. the following theorem was given by fukagai and kusano in [9]. theorem b. suppose that the conditions (1.2) and (1.3) hold. suppose moreover that lim sup x→0 |x| |f(x)| = λ < ∞. if lim inf t→∞ t∫ τ(t) p(s)ds > λ e , then every solution of eq.(1.1) is oscillatory. thus, in this paper, our aim is to obtain some oscillation criteria for all solutions of eq.(1.1) under the assumption that τ(t) is non-monotone. 2. main results in this section, we present a new sufficient conditions for the oscillation of all solutions of eq.(1.1), under the assumption that the argument τ(t) is non-monotone or nondecreasing. set h(t) := sup s≤t τ(s), t ≥ 0. (2.1) clearly, h(t) is nondecreasing, and τ(t) ≤ h(t) for all t ≥ 0. assume that the f in eq.(1.1) satisfy the following condition: lim sup x→0 x f(x) = m, 0 ≤ m < ∞. (2.2) oscillation of nonlinear delay differential 149 theorem 2.1. assume that (1.2), (1.3) and (2.2) holds. if τ(t) is non-monotone or nondecreasing, and lim inf t→∞ t∫ τ(t) p(s)ds > m e , (2.3) then all solutions of eq.(1.1) oscillate. proof. assume, for the sake of contradiction, that there exists a nonoscillatory solution x(t) of (1.1). since −x(t) is also a solution of (1.1), we can confine our discussion only to the case where the solution x(t) is eventually positive. then there exists t1 > t0 such that x(t), x (τ(t)) > 0, for all t ≥ t1. thus, from (1.1) we have x′(t) = −p(t)f (x (τ(t))) ≤ 0, for all t ≥ t1. thus x(t) is nonincreasing and has a limit l ≥ 0 as t →∞. now, we claim that l = 0. condition (2.3) implies that ∞∫ a p(t)dt = ∞. (2.4) in view of (2.4) and by the theorem 3.1.5 in [17] that limt→∞x (t) = 0. suppose m > 0. then, in view of (2.2) we can choose t2 > t1 so large that f(x (t)) ≥ 1 2m x(t) for t ≥ t2. (2.5) on the other hand, we know from lemma 2.1.1 [7] that lim inf t→∞ t∫ h(t) p(s)ds = lim inf t→∞ t∫ τ(t) p(s)ds. (2.6) since h(t) ≥ τ(t) and x(t) is nonincreasing , by (1.1) and (2.5) we have x′(t) + 1 2m p(t)x(h(t)) ≤ 0, t ≥ t3. (2.7) also, from (2.3) and (2.6) it follows that there exists a constant c > 0 such that t∫ h(t) p(s)ds ≥ c > m e , t ≥ t3 ≥ t2. (2.8) so, from (2.8), there exists a real number t∗ ∈ (h(t), t), for all t ≥ t3 such that t∗∫ h(t) p(s)ds > m 2e and t∫ t∗ p(s)ds > m 2e . (2.9) integrating (2.7) from h(t) to t∗ and using x(t) is nonincreasing then we have x(t∗) −x (h(t)) + 1 2m t∗∫ h(t) p(s)x (h(s)) ds ≤ 0, or x(t∗) −x (h(t)) + 1 2m x (h(t∗)) t∗∫ h(t) p(s)ds ≤ 0. thus, by (2.9), we have −x (h(t)) + 1 2m x (h(t∗)) m 2e < 0. (2.10) 150 öcalan, kiliç, şahi̇n and özkan integrating (2.7) from t∗ to t and using the same facts , we get x(t) −x (t∗) + 1 2m t∫ t∗ p(s)x (h(s)) ds ≤ 0. thus, by (2.9), we have −x (t∗) + 1 2m x (h(t)) m 2e < 0. (2.11) combining the inequalities (2.10) and (2.11), we obtain x(t∗) > x (h(t)) 1 4e > x (h(t∗)) ( 1 4e )2 , and hence we have x (h(t∗)) x(t∗) < (4e) 2 for t ≥ t4. let w = x(h(t∗)) x(t∗) ≥ 1, and because of 1 ≤ w < (4e)2 , w is finite. now dividing (1.1) with x(t) and then integrating from h(t) to t we obtain t∫ h(t) x′(s) x(s) ds + t∫ h(t) p(s) f(x(τ(s))) x(s) ds = 0 and ln x(t) x(h(t)) + t∫ h(t) p(s) f(x(τ(s))) x(τ(s)) x(τ(s)) x(s) ds = 0 since x(t) is nonincreasing, we get ln x(t) x(h(t)) + t∫ h(t) p(s) f(x(τ(s))) x(τ(s)) x(h(s)) x(s) ds ≤ 0 and ln x(h(t)) x(t) ≥ f(x(τ(ξ))) x(τ(ξ)) x(h(ξ)) x(ξ) t∫ h(t) p(s)ds, (2.12) where ξ is defined with h(t) < ξ < t, while t −→ ∞, ξ −→ ∞ and because of this h(t) −→ ∞. then taking lower limit on both side of (2.12), we obtain ln w ≥ w e . but this is impossible since ln x ≤ x e for all x > 0. the case where m = 0 can be discussed similarly. the proof of the theorem is completed. � theorem 2.2. assume that (1.2), (1.3), (2.2) and (2.4) holds. if τ(t) is non-monotone, and lim sup t→∞ t∫ h(t) p(s)ds > 2m (2.13) where h(t) is defined by (2.1), then all solutions of eq.(1.1) oscillate. oscillation of nonlinear delay differential 151 proof. assume, for the sake of contradiction, that there exist a nonoscillatory solution x(t) of (1.1). in view of (2.4), we know from theorem 2.1 that lim t→∞ x(t) = 0, for t ≥ t1. considering equation (1.1) x′(t) + p(t)f(x(τ(t))) = 0 by (2.5) we get x′(t) + 1 2m p(t)x(τ(t)) ≤ 0 since h(t) ≥ τ(t) and x(t) is nonincreasing x′(t) + 1 2m p(t)x(h(t)) ≤ 0 (2.14) integrating (2.14) from h(t) to t, and using the fact that the function x(t) is nonincreasing and the function h(t) is nondecreasing x(t) −x(h(t)) + 1 2m t∫ h(t) p(s)x(h(s))ds ≤ 0 or x(t) −x(h(t)) + 1 2m x(h(t)) t∫ h(t) p(s)ds ≤ 0 this implies x(t) −x(h(t)) +  1 − 1 2m t∫ h(t) p(s)ds   ≤ 0 and hence t∫ h(t) p(s)ds < 2m for sufficiently t. therefore, lim sup t→∞ t∫ h(t) p(s)ds ≤ 2m this is a contradiction to (2.13). the proof is completed. � now, assume that f is nondecreasing function then we have the following result. theorem 2.3. assume that (1.2), (1.3), (2.2) and (2.4) hold. if τ(t) is non-monotone, f is nondecreasing function and lim sup t→∞ t∫ τ(t) p(s)ds > m (2.15) where h(t) is defined by (2.1), then all solutions of eq.(1.1) oscillate. 152 öcalan, kiliç, şahi̇n and özkan proof. assume, for the sake of contradiction, that there exist a nonoscillatory solution x(t) of (1.1). in view of (2.4), we know from theorem 2.1 that lim t→∞ x(t) = 0, for t ≥ t1. considering equation (1.1) x′(t) + p(t)f(x(τ(t))) = 0 since τ(t) ≤ h(t), x(t) is nonincreasing and f is nondecreasing we have x′(t) + p(t)f(x(h(t))) ≤ 0 (2.16) integrating (2.16) from h(t) to t and using the fact that x(t) is nonincreasing and f, h(t) are nondecreasing x(t) −x(h(t)) + t∫ h(t) p(s)f(x(h(s)))ds ≤ 0 or x(t) −x(h(t)) + f(x(h(t))) t∫ h(t) p(s)ds ≤ 0 and so x(t) −x(h(t))  1 − f(x(h(t))) x(h(t)) t∫ h(t) p(s)ds   ≤ 0 therefore 1 > f(x(h(t))) x(h(t)) t∫ h(t) p(s)ds ≥ 1 m lim sup t→∞ t∫ h(t) p(s)ds that is a contradiction. the proof is completed. � we remark that if τ(t) is nondecreasing, then we have τ(t) = h(t) for all t, and the condition (2.13) and (2.15), respectively, reduce to lim sup t→∞ t∫ τ(t) p(s)ds > 2m (2.16) and lim sup t→∞ t∫ τ(t) p(s)ds > m (2.17) now, we have the following example. example 2.1. consider the nonlinear delay differential equation x′(t) + 1 e x (τ(t)) ln (10 + |x (τ(t))|) = 0, t > 0, (2.18) oscillation of nonlinear delay differential 153 where τ(t) =   t− 1, if t ∈ [3k, 3k + 1] −3t + 12k + 3, if t ∈ [3k + 1, 3k + 2] 5t− 12k − 13, if t ∈ [3k + 2, 3k + 3] , k ∈ n0. by (2.1), we see that h(t) := sup s≤t τ(s) =   t− 1, if t ∈ [3k, 3k + 1] 3k, if t ∈ [3k + 1, 3k + 2.6] 5t− 12k − 13, if t ∈ [3k + 2.6, 3k + 3] , k ∈ n0. if we put p(t) = 1 e and f(x) = x ln(10 + |x|). then, we have m = lim sup x→0 x f(x) = lim sup x→0 x x ln(10 + |x|) = 1 ln 10 and lim inf t→∞ ∫ t τ(t) p(s)ds = 1 e > m e = 1 e ln 10 that is, all conditions of theorem 2.1 are satisfied and therefore all solutions of (2.18) oscillate. references [1] o. arino, i. győri and a. jawhari, oscillation criteria in delay equations, j. differential equations 53 (1984), 115-123. [2] l. berezansky and e. braverman, on some constants for oscillation and stability of delay equations, proc. amer. math. soc. 139 (11) (2011), 4017-4026. [3] e. braverman, b. karpuz, on oscillation of differential and difference equations with non-monotone delays, appl. math. comput. 218 (2011) 3880-3887. [4] george e. chatzarakis and özkan öcalan, oscillations of differential equations with non-monotone retarded arguments, lms j. comput. math., 19 (1) (2016) 98–104. [5] a. elbert and i. p. stavroulakis, oscillations of first order differential equations with deviating arguments, univ of ioannina t. r. no 172 (1990), recent trends in differential equations, 163-178, world sci. ser. appl. anal., 1, world sci. publishing co. (1992). [6] a. elbert and i. p. stavroulakis, oscillation and non-oscillation criteria for delay differential equations, proc. amer. math. soc., 123 (1995), 1503-1510. [7] l. h. erbe, qingkai kong and b.g. zhang, oscillation theory for functional differential equations, marcel dekker, new york, 1995. [8] l. h. erbe and b. g. zhang, oscillation of first order linear differential equations with deviating arguments, differential integral equations, 1 (1988), 305-314. [9] n. fukagai and t. kusano, oscillation theory of first order functional differential equations with deviating arguments, ann. mat. pura appl.,136 (1984), 95-117. [10] k.gopalsamy, stability and oscillations in delay differential equations of population dynamics, kluwer academic publishers, 1992. [11] m. k. grammatikopoulos, r. g. koplatadze and i. p. stavroulakis, on the oscillation of solutions of first order differential equations with retarded arguments, georgian math. j., 10 (2003), 63-76. [12] i. győri and g. ladas, oscillation theory of delay differential equations with applications, clarendon press, oxford, 1991. [13] b. r. hunt and j. a. yorke, when all solutions of x′ = ∑ qi(t)x(t−ti(t)) oscillate, j. differential equations 53 (1984), 139-145. [14] r. g. koplatadze and t. a. chanturija, oscillating and monotone solutions of first-order differential equations with deviating arguments, (russian), differentsial’nye uravneniya, 8 (1982), 1463-1465. [15] g. ladas, sharp conditions for oscillations caused by delay, applicable anal., 9 (1979), 93-98. [16] g. ladas, v. laskhmikantham and j.s. papadakis, oscillations of higher-order retarded differential equations generated by retarded arguments, delay and functional differential equations and their applications, academic press, new york, 1972, 219–231. [17] g.s. ladde, v. lakshmikantham, b.g. zhang, oscillation theory of differential equations with deviating arguments, monographs and textbooks in pure and applied mathematics, vol. 110, marcel dekker, inc., new york, 1987. [18] a. d. myshkis, linear homogeneous differential equations of first order with deviating arguments, uspekhi mat. nauk, 5 (1950), 160-162 (russian). [19] x.h. tang, oscillation of first order delay differential equations with distributed delay, j. math. anal. appl. 289 (2004), 367-378. 154 öcalan, kiliç, şahi̇n and özkan 1akdeniz university, faculty of science, department of mathematics, 07058, antalya, turkey 2dumlupınar university, faculty of science and arts, department of mathematics, 43000, kütahya, turkey 3afyon kocatepe university, faculty of science and arts, department of mathematics, ans campus, 03200, afyon, turkey ∗corresponding author: ozkanocalan@akdeniz.edu.tr 1. introduction 2. main results references international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 123-135 http://www.etamaths.com on some isomorphisms between bounded linear maps and non-commutative lp-spaces e. j. atto1,∗, v.s.k. assiamoua1 and y. mensah1,2 abstract. we define a particular space of bounded linear maps using a von neumann algebra and some operator spaces. by this, we prove some isomorphisms, and using interpolation in some particular cases, we get analogue of non-commutative lp spaces. 1. introduction in the fifties, many authors have studied on non-commutative lp-spaces like segal [13], kunze [9], dixmier [4], stinespring [14]. but the recent emergency of the theory of operator spaces from the late 80’s to the early 90’s in the works of effros and ruan [5],[6], [7],[12], blecher and paulsen [2],[3] allowed gilles pisier since the mid 90’s to expose the general theory of non-commutative lp-spaces [11], using for instance a von neumann algebra m equipped with a particular type of trace ϕ. as the complex interpolation method contribute to define a new banach space using a compatible pair of banach spaces, this method was also used to define non-commutative lp-spaces. our aim in this paper is to define for each 1 ≤ p ≤ ∞ a particular space of bounded linear maps denoted lp(m,ϕ,e; f), using some operator spaces e, f and the non-commutative spaces l1(m,ϕ), l∞(m,ϕ), such that those particular spaces have some properties with the non-commutative lp spaces like the isomorphism. we firstly view those types of spaces as banach spaces and secondly give them an operator space structure. before stating our results, we shall recall the concept of operator space and the complex interpolation method to make the paper more comprehensive. 2. preliminary notes 2.1. operator spaces. h being an hilbert space, we denote by b(h) the banach space of all bounded operators from h into h, endowed with the operator norm ‖t‖∞ = sup{‖tξ‖h,ξ ∈ h,‖ξ‖≤ 1}. a closed subset e ⊂ b(h) is called an operator space. but there exist an abstract characterization of an operator space given by ruan (see [7] and [12] for more details): 2010 mathematics subject classification. 46b03; 46l07. key words and phrases. operator spaces, banach spaces, linear maps, non-commutative lpspaces. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 123 124 atto, assiamoua and mensah theorem 2.1 (ruan theorem). a complex vector space e is an operator space if and only if for each n ≥ 1, there is a complete norm ‖.‖n on mn(e), the space of n×n matrices with entries in e, such that the following properties are satisfied: (1) ‖αxβ‖n ≤‖α‖‖x‖n‖β‖ (2) ‖x⊕y‖n+m = max{‖x‖n,‖y‖m} for x ∈ mn(e),y ∈ mm(e),α,β ∈ mn. let m be a von neumann algebra on the hilbert space h, that is, m ⊂ b(h) is a c∗-algebra closed in the weak operator topology and contains the identity operator. we denote by m+ the set of all positive elements of m. then, we recall the following definitions concerning the notion of trace: definition 2.2. a trace ϕ on m+ is a function ϕ : m+ →]0, +∞], such that (1) ϕ(x + y) = ϕ(x) + ϕ(y) for any x,y ∈ m+. (2) ϕ(λx) = λϕ(x) for any 0 ≤ λ ≤∞ and x ∈ m+ with the usual convention 0.∞ = 0. (3) ϕ(xy) = ϕ(yx) for any x,y ∈ m+. definition 2.3. a trace ϕ is called: i) faithful if x ∈ m+,ϕ(x) = 0 =⇒ x = 0. ii) finite if ϕ(x) < ∞ for any x ∈ m+. iii) normal if for any x ∈ m+ and any increasing net (xα) converging to x in the strong operator topology, ϕ(xα) → ϕ(x). iv) semi-finite if for any x ∈ m+ , there exist y ∈ m+such that ϕ(y) < ∞ and y ≤ x. in the following, the von neumann algebra m is assumed to be equipped with a faithful, normal and semi-finite trace ϕ. 2.2. the complex interpolation method and the non-commutative lp spaces. a couple of (complex) banach spaces (x0; x1) is said to be compatible if they are both embedded by continuous injective linear maps into a hausdorff topological vector space x. in this case, x0 and x1 are viewed as vector subspaces of x. then their intersection x0 ∩x1 is equipped with the norm ‖x‖x0∩x1 = max{‖x‖x0,‖x‖x1} and their sum is defined by x0 + x1 = {x0 + x1 : xk ∈ xk, k = 0, 1} with the norm ‖x‖x0+x1 = inf{‖x0‖x0 + ‖x1‖x1 : x = x0 + x1,xk ∈ xk, k = 0, 1}. it is easy to check that x0 ∩x1 and x0 + x1 are again banach spaces. then we have x0 ∩x1 ⊂ x0, x1 ⊂ x0 + x1, contractiveinjections. let b = {z ∈ c : 0 ≤ re(z) ≤ 1}. let f(x0,x1) be the family of all functions f : b → x0 + x1 satisfying the following conditions: (1) f is continuous on b and analytic in the interior of b; on some isomorphisms 125 (2) f(k + it) ∈ xk for t ∈ r and the function t 7→ f(k + it) is continuous from r into xk, k = 0, 1; (3) lim |t|→∞ ‖f(k + it)‖xk = 0, k = 0, 1. we equip f(x0,x1) with the norm: ‖f‖f(x0,x1) = {sup t∈r ‖f(it)‖x0, sup t∈r ‖f(1 + it)‖x1}. then it is easy to check that f(x0,x1) becomes a banach space. let 0 < θ < 1, the complex interpolation space (x0,x1)θ is defined as the space of all those x ∈ x0 + x1 for which there exists f ∈f(x0,x1) such that f(θ) = x. equipped with ‖x‖θ = inf{‖f‖f(x0,x1) : f(θ) = x, f ∈f(x0,x1)}, (x0,x1)θ becomes a banach space. indeed, by the maximum principle, the map f 7→ f(θ) is a contraction from f(x0,x1) to x0 + x1. then (x0,x1)θ can be isometrically identified with the quotient of f(x0,x1) by the kernel of this map. remark 2.1. the following properties are easy: i) (x0,x1)θ = (x1,x0)1−θ isometrically. ii) x0 ∩x1 is dense in (x0,x1)θ. iii) let (x0,x1) and (y0,y1) be two compatible couples. let t : x0 + x1 → y0 + y1 be a linear map which is bounded from xk to yk for k = 0 and k = 1. then t is bounded from (x0,x1)θ to (y0,y1)θ for any 0 < θ < 1; moreover ‖t : (x0,x1)θ → (y0,y1)θ‖≤‖t : x0 → y0‖1−θ‖t : x1 → y1‖θ. this statement is usually called interpolation theorem. note that by tradition in interpolation theory, the assumption on t in the statement (iii) above means that t maps xk into yk and its restriction to xk belongs to b(xk,yk) (k = 0, 1). set ip = {x ∈ m : ϕ(|x|p) < ∞}, (1 ≤ p < ∞), equipped with the norm ‖x‖p = (ϕ(|x|p)1/p. the completion of ip under this norm is a banach space and is denoted lp(m,ϕ) by g. pisier in [11] where it is called non-commutative lp space. since all lp(m,ϕ), 1 ≤ p ≤∞, are injected into l1(m,ϕ), (lp0 (m,ϕ),lp1 (m,ϕ)) is a compatible couple for any p0,p1 ∈ [1;∞]. the following is the complex interpolation theorem on non-commutative lp-spaces. theorem 2.4. let 1 ≤ p0 < p1 ≤ ∞, 0 < θ < 1 and let p be determined by 1 p = 1−θ p0 + θ p1 . then (lp0 (m,ϕ),lp1 (m,ϕ))θ = lp(m,ϕ) with equal norms. for more details, see [1] and [8]. particularly, for p0 = 1 and p1 = ∞ we get (l1(m,ϕ),l∞(m,ϕ))θ = lp(m,ϕ) whith θ = 1 p 126 atto, assiamoua and mensah remark 2.2. (i) e ⊂ b(h) being an operator space, the non-commutative vector valued lp spaces for 1 ≤ p ≤∞ are defined as follow: l1(m,ϕ,e) = l1(m,ϕ)⊗̂e l0∞(m,ϕ,e) = m ⊗min e lp(m,ϕ,e) = (l1(m,ϕ,e),l 0 ∞(m,ϕ,e))θ 1 < p < ∞ whith θ = 1 p . (ii) the dual space of lp(m,ϕ) is lq(m,ϕ) for 1 ≤ p < ∞( 1p + 1 q = 1) with respect to the following duality 〈x,y〉 = ϕ(xy), x ∈ lp(m,ϕ), y ∈ lq(m,ϕ). in over words (lp(m,ϕ)) ′ = lq(m,ϕ) isometrically. (iii) in the lebesgue-bochner theory, if e is a banach space, it is well known that the dual of lp(ω,µ; e) is not in general lq(ω,µ; e ′) unless e′ possesses the radon nikodym property (in short the rnp). in [11], it was introduced an operator space analogue of the rnp which is called the ornp. thus, if e is an operator space such that is dual e′ has the ornp, then the dual theorem is confirmed. namely the dual of lp(m,ϕ,e) is completely isometric to lq(m,ϕ,e ′). one must also note that the ornp of an operator space implies the rnp of the underlying banach space, but the converse is not true. the following theorem has been proved by g. pisier in [11] pages 49,50. theorem 2.5 (pisier). let (m,ϕ) be any non-commutative probability space (in short n.c.p. space). let e be an operator space. if e′ has the ornpp′ with 1 < p < ∞ and q = p/(p− 1), then we have a completely isometric identity lp(m,ϕ,e) ′ = lq(m,ϕ,e ′). 3. main results 3.1. the spaces l1(m,ϕ,e; f) and l∞(m,ϕ,e; f).. let e,f ⊂ b(h) two operator spaces. we denote by ‖.‖e and ‖.‖f the operator norm on e and f inherited from b(h) and l(e,f) the space of all bounded linear maps from e into f equipped with the norm ‖f‖l(e,f) = sup{‖f(x)‖f : ‖x‖e ≤ 1}. set l1(m,ϕ,e; f) the space of all bounded linear maps from l1(m,ϕ) into l(e,f). theorem 3.1. the mapping u 7−→‖u‖ϕ,1 = sup{‖u(x)‖l(e,f) : ϕ(|x|) ≤ 1} is a norm on l1(m,ϕ,e; f) . l1(m,ϕ,e; f) equipped with this norm is a banach space. proof. it is obvious that ‖.‖ϕ,1 is a norm. since l1(m,ϕ), (e,‖.‖e) and (f,‖.‖f ) are banach spaces, then (l1(m,ϕ,e; f),‖.‖ϕ,1) is a banach space. � on some isomorphisms 127 moreover, l1(m,ϕ,e; f) is a banach algebra if endowed with the inner product denoted · as follow: for all u,v ∈ l1(m,ϕ,e; f), u ·v = w such that w(x)(y) = u(x)(y) ◦ v(x)(y), with x ∈ l1(m,ϕ) and y ∈ e. here, the product between the two elements u(x)(y),v(x)(y) of f is the natural product of operator inherited from the banach algebra b(h). more precisely, ∀t1,t2 ∈ b(h),t1 ◦t2(ξ) = t1(t2(ξ)) with ξ ∈ h. and we obtain ‖u ·v‖ϕ,1 = sup{‖(u ·v)(x)‖l(e,f) : ϕ(|x|) ≤ 1} = sup{sup{‖(u ·v)(x)(y)‖f : ‖y‖e ≤ 1} : ϕ(|x|) ≤ 1} = sup{sup{‖u(x)(y) ◦v(x)(y)‖f : ‖y‖e ≤ 1} : ϕ(|x|) ≤ 1} ≤ sup{sup{‖u(x)(y)‖f‖v(x)(y)‖f : ‖y‖e ≤ 1} : ϕ(|x|) ≤ 1} ≤ sup{ sup ‖y‖e≤1 {‖u(x)(y)‖f‖v(x)(y)‖f} : ϕ(|x|) ≤ 1} ≤ sup{( sup ‖y‖e≤1 ‖u(x)(y)‖f )( sup ‖y‖e≤1 ‖v(x)(y)‖f ) : ϕ(|x|) ≤ 1} ≤ sup{ ( ‖u(x)‖l(e,f) )( ‖v(x)‖l(e,f) ) : ϕ(|x|) ≤ 1} ≤ sup ϕ(|x|)≤1 { ( ‖u(x)‖l(e,f) )( ‖v(x)‖l(e,f) ) } ≤ ( sup ϕ(|x|)≤1 ‖u(x)‖l(e,f))( sup ϕ(|x|)≤1 ‖v(x)‖l(e,f)) ≤ ‖u‖ϕ,1‖v‖ϕ,1 definition 3.2. set l∞(m,ϕ,e; f) the space of all bounded linear maps from m into l(e,f) equipped with the norm ‖u‖ϕ,∞ = sup{‖u(x)‖l(e,f) : ‖x‖∞ ≤ 1}, where ‖.‖∞ is the operator norm on m ⊂ b(h). theorem 3.3. l∞(m,ϕ,e; f) is a banach space. proof. the proof of this theorem is in the same spirit of the one of the previous theorem . � it is also easy to check that l∞(m,ϕ,e; f) endowed with the same inner product used for l1(m,ϕ,e; f) is a banach algebra. theorem 3.4 (duality). the topological dual of l1(m,ϕ,e; f) is isometrically isomorph to l∞(m,ϕ,e; f ′) where f ′ is the dual of f : (l1(m,ϕ,e; f)) ′ ' l∞(m,ϕ,e; f ′) proof. let us consider the mapping t : l∞(m,ϕ,e; f ′) −→ (l1(m,ϕ,e; f))′ u 7−→ tu, such that 〈tu,v〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉|, where u ∈ l∞(m,ϕ,e,f ′), v ∈ (l1(m,ϕ,e; f))′, and so: ∀x ∈ l1(m,ϕ),y ∈ e we have u(x)(y) ∈ f ′, v(x)(y) ∈ f. 128 atto, assiamoua and mensah (1) linearity and boundedness of t: the linearity of t is trivial. let us prove t is bounded. we have: |〈tu,v〉| = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉| ≤ sup ϕ(|x|)≤1,‖y‖e≤1 ‖u(x)(y)‖f′‖v(x)(y)‖f ≤ sup ϕ(|x|)≤1,‖y‖e≤1 ‖u(x)(y)‖f′ sup ϕ(|x|)≤1,‖y‖e≤1 ‖v(x)(y)‖f ≤ sup ‖x‖∞≤1,‖y‖e≤1 ‖u(x,y)‖f′ sup ϕ(|x|)≤1,‖y‖e≤1 ‖v(x)(y)‖f ≤ ‖u‖ϕ,∞‖v‖ϕ,1 thus, ‖tu‖≤‖u‖ϕ,∞ and t is bounded with ‖t‖≤ 1. (2) we prove now that ‖t‖ = 1. since ‖t‖ ≤ 1, all we have to do is to prove ‖t‖≥ 1. let a ∈ f such that ‖a‖f = 1. since a 6= 0, there exist b∗ ∈ f ′ such that ‖b∗‖ = 1 and 〈b∗,a〉 = ‖a‖f = 1. for (x0,y0) fixed in l1(m,ϕ) ×e, one define u ∈ l∞(m,ϕ,e; f ′) as follow: u(x)(y) = { b∗ if (x,y) = (x0,y0) 0 if not and v ∈ l1(m,ϕ,e; f) by v(x)(y) = { a if (x,y) = (x0,y0) 0 if not afterwards, 〈tu,v〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉| = 〈u(x0)(y0),v(x0)(y0)〉 = 〈b∗,a〉 = 1 since 〈tu,v〉 ≤ ‖t‖‖u‖‖v‖, with ‖u‖ = ‖v‖ = 1, then ‖t‖ ≥ 1 and ‖t‖ = 1. (3) subjectivity of t suppose f ∈ (l1(m,ϕ,e; f))′ and (x0,y0) fixed in l1(m,ϕ) ×e. let φ ∈ l1(m,ϕ,e; f) such that φ(x)(y) = 0 if (x,y) 6= (x0,y0). we set a(x0,y0) = φ(x0)(y0) ∈ f, then there exist a linear functional b(x0,y0) ∈ f ′ such that 〈f,φ〉 = 〈b(x0,y0),a(x0,y0〉 > 0 set φ ∈ l∞(m,ϕ,e; f) as follow: φ(x)(y) = { b(x0,y0) if (x,y) = (x0,y0) 0 if not so we have: 〈f,φ〉 = 〈φ(x0)(y0),φ(x0)(y0)〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈φ(x)(y),φ(x)(y)〉| = 〈tφ,φ〉. hence the linear functional f and tφ coincide on l∞(m,ϕ,e; f ′). in over words f = tφ and t is subjective. on some isomorphisms 129 in conclusion, t is isometric. � 3.2. the spaces l 0p (m,ϕ,e; f) with 1 < p < ∞. definition 3.5. let l 0p (m,ϕ,e; f) be the space of all bounded linear maps from lp(m,ϕ) into l(e,f) for 1 < p < ∞. theorem 3.6. for each 1 < p < ∞,the space l 0p (m,ϕ,e; f) endowed with the norm ‖u‖ϕ,p = sup{‖u(x)‖l(e,f) : ϕ(|x|p) ≤ 1} is a banach space. proof. in the same spirit as in theorem 3.1 � theorem 3.7 (duality). the topological dual of l 0p (m,ϕ,e; f) is isometrically isomorph to l 0q (m,ϕ,e; f ′), ( 1 p + 1 q = 1) where f ′ is the dual of f : (l 0p (m,ϕ,e; f)) ′ ' l 0q (m,ϕ,e,f ′) proof. let us consider the mapping t : lq(m,ϕ,e; f ′) −→ (lp(m,ϕ,e; f))′ u 7−→ tu, such that 〈tu,v〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉|, where u ∈ lq(m,ϕ,e,f ′), v ∈ (lp(m,ϕ,e; f))′, and so: ∀x ∈ l1(m,ϕ),y ∈ e we have u(x)(y) ∈ f ′, v(x)(y) ∈ f. (1) linearity and boundedness of t: the linearity of t is trivial. let us prove t is bounded. we have: |〈tu,v〉| = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉| ≤ sup ϕ(|x|)≤1,‖y‖e≤1 ‖u(x)(y)‖f′‖v(x)(y)‖f ≤ sup ϕ(|x|)≤1,‖y‖e≤1 ‖u(x)(y)‖f′ sup ϕ(|x|)≤1,‖y‖e≤1 ‖v(x)(y)‖f ≤ sup ϕ(|x|q)≤1,‖y‖e≤1 ‖u(x)(y)‖f′ sup ϕ(|x|p)≤1,‖y‖e≤1 ‖v(x)(y)‖f ≤ ( sup ϕ(|x|q)≤1 ‖u(x)‖l(e,f′) )( sup ϕ(|x|p)≤1 ‖v(x)‖l(e,f ) ≤ ‖u‖ϕ,q‖v‖ϕ,p thus, ‖tu‖≤‖u‖ϕ,q and t is bounded with ‖t‖≤ 1. (2) we prove now that ‖t‖ = 1. since ‖t‖ ≤ 1, all we have to do is to prove ‖t‖≥ 1. let a ∈ f such that ‖a‖f = 1. since a 6= 0, there exist b∗ ∈ f ′ such that ‖b∗‖ = 1 and 〈b∗,a〉 = ‖a‖f = 1. for (x0,y0) fixed in lp(m,ϕ) ×e, one define u ∈ lq(m,ϕ,e; f ′) as follow: u(x)(y) = { b∗ if (x,y) = (x0,y0) 0 if not 130 atto, assiamoua and mensah and v ∈ lp(m,ϕ,e; f) by v(x)(y) = { a if (x,y) = (x0,y0) 0 if not afterwards, 〈tu,v〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈u(x)(y),v(x)(y)〉| = 〈u(x0)(y0),v(x0)(y0)〉 = 〈b∗,a〉 = 1 since 〈tu,v〉 ≤ ‖t‖‖u‖‖v‖, with ‖u‖ = ‖v‖ = 1, then ‖t‖ ≥ 1 and ‖t‖ = 1. (3) subjectivity of t suppose f ∈ (lp(m,ϕ,e; f))′ and (x0,y0) fixed in lp(m,ϕ) ×e. let φ ∈ lp(m,ϕ,e; f) such that φ(x)(y) = 0 if (x,y) 6= (x0,y0). we set a(x0,y0) = φ(x0)(y0) ∈ f , then there exist a linear functional b(x0,y0) ∈ f ′ such that 〈f,φ〉 = 〈b(x0,y0),a(x0,y0〉 > 0 set φ ∈ lq(m,ϕ,e; f) as follow: φ(x)(y) = { b(x0,y0) if (x,y) = (x0,y0) 0 if not so we have: 〈f,φ〉 = 〈φ(x0)(y0),φ(x0)(y0)〉 = sup ϕ(|x|)≤1,‖y‖e≤1 |〈φ(x)(y),φ(x)(y)〉| = 〈tφ,φ〉. hence the linear functional f and tφ coincide on lq(m,ϕ,e; f ′). in over words f = tφ and t is subjective. in conclusion, t is isometric. � corollary 3.8. for 1 ≤ p < ∞ l 0p (m,ϕ,c; c) is isomorphic to lq(m,ϕ): l 0p (m,ϕ,c; c) ' lq(m,ϕ) where q is such that 1 p + 1 q = 1 (called the conjugate of p). proof. l 0p (m,ϕ,c; c) = l(lp(m,ϕ),l(c,c)) ' l(lp(m,ϕ),c) ' (lp(m,ϕ))′ ' lq(m,ϕ) � on some isomorphisms 131 3.3. the spaces lp(m,ϕ,e; f) by using the interpolation method. theorem 3.9. there is a contractive injection from l∞(m,ϕ,e; f) into l1(m,ϕ,e; f) proof. since l1(m,ϕ,e; f) denotes the space of all bounded linear maps from l1(m,ϕ) into l(e,f) and l∞(m,ϕ,e; f) the space of all bounded linear maps from m into l(e,f), with m ⊂ l1(m,ϕ), it is obvious to claim that l∞(m,ϕ,e; f) ⊂ l1(m,ϕ,e; f). let z0 be an element of f. now, we consider the injection: f : l∞(m,ϕ,e; f) −→ l1(m,ϕ,e; f) u 7−→ f(u) such that ∀(x,y) ∈ l∞(m,ϕ) ×e , f(u)(x)(y) = { u(x)(y) if x ∈ l1(m,ϕ) z0 if not then we have: ‖f(u)‖ϕ,1 = sup{‖f(u)(x)(y)‖f : ϕ(|x|) ≤ 1,‖y‖e ≤ 1} ≥ sup{‖f(u)(x)(y)‖f : ‖x‖∞ ≤ 1,‖y‖e ≤ 1} ≥ sup{‖u(x)(y)‖f : ‖x‖∞ ≤ 1,‖y‖e ≤ 1} ≥ ‖u‖ϕ,∞ � this theorem allows as to view the pair (l∞(m,ϕ,e; f), l1(m,ϕ,e; f)) as a compatible couple of banach spaces and so we can apply the complex interpolation method to define a new banach space. definition 3.10. let ϕ be a semi-finite normal faithful trace on an injective von neumann algebra m ⊂ b(h) and let e,f ⊂ b(h) two operators spaces. if 1 < p < ∞, we define lp(m,ϕ,e; f) = (l∞(m,ϕ,e; f), l1(m,ϕ,e; f))θ where θ = 1 p . theorem 3.11. l1(m,ϕ,e; f) is isomorphic to l(l1(m,ϕ)⊗e,f) and l∞(m,ϕ,e; f) is isomorphic to l(m ⊗e,f): l1(m,ϕ,e; f) 'l(l1(m,ϕ) ⊗e,f), l∞(m,ϕ,e; f) 'l(m ⊗e,f). proof. we want to prove firstly that l1(m,ϕ,e; f) 'l(l1(m,ϕ) ⊗e,f). let us consider the map: f : l1(m,ϕ,e; f) −→l(l1(m,ϕ) ⊗e,f) u 7−→f(u) such that f(u)(x⊗y) = u(x)(y) 132 atto, assiamoua and mensah i) linearity: we have f(λu)(x⊗y) = (λu)(x)(y) = λu(x)(y) = λfu(x⊗y) and f(u + v)(x⊗y) = (u + v)(x)(y) = u(x)(y) + v(x)(y) = f(u)(x⊗y) + f(v)(x⊗y) = [f(u) + f(v)](x⊗y) ii) f is bijective for all v ∈l(l1(m,ϕ) ⊗e,f) and x ∈ l1(m,ϕ), set: g(v)(x) the mapping y 7→ v(x ⊗ y), where y ∈ e. this mapping is an element of l(e,f). in fact for x fixed, y 7→ v(x ⊗ y) is linear, and is bounded since v is a bounded linear map. thus, we get a map: g(v) : l1(m,ϕ) −→l(e,f). now we want to prove that this map is linear. for all x,x′ ∈ l1(m,ϕ), y ∈ e, g(v)(x + λx′)(y) = v((x + λx′) ⊗y) = v(x⊗y + λ(x′ ⊗y)) = v(x⊗y) + λv(x′ ⊗y) = g(v)(x)(y) + λg(v)(x′)(y) = [g(v)(x) + λg(v)(x′)](y) since ∀y,g(v)(x + λx′)(y) = [g(v)(x) + λg(v)(x′)](y), then g(v)(x + λx′) = g(v)(x) + λg(v)(x′), and g(v) is linear. by this, we claim that g is a map from l(l1(m,ϕ) ⊗e,f) into l1(m,ϕ,e; f). finally we prove that g is the inverse of f: (g ◦f)(u)(x⊗y) = g[f(u)(x⊗y)] = g(u(x)(y)) = u(x⊗y) since (g ◦f)(u)(x⊗y) = u(x⊗y), then (g ◦f)(u) = u (f ◦g)(v)(x)(y) = f[g(v)(x)(y)] = f(v(x⊗y)) = v(x)(y) since (f ◦g)(v)(x)(y) = v(x)(y), then (g ◦f)(v) = v. so l1(m,ϕ,e; f) 'l(l1(m,ϕ) ⊗e,f). the proof of the second isomorphism use the same method by replacing for instance l1(m,ϕ) by m and l1(m,ϕ,e; f) by l∞(m,ϕ,e; f). � on some isomorphisms 133 corollary 3.12. l1(m,ϕ,e; f) 'l(l1(m,ϕ),e),f) and l∞(m,ϕ,e; f) 'l(l∞(m,ϕ,e),f) proof. l1(m,ϕ,e; f) ' l(l1(m,ϕ) ⊗e,f) ' l(l1(m,ϕ)⊗̂e,f) ' l(l1(m,ϕ,e),f) and l∞(m,ϕ,e; f) ' l(m ⊗e,f) ' l(m ⊗min e,f) ' l(l∞(m,ϕ,e),f) � corollary 3.13. for all 1 < p < ∞, lp(m,ϕ,e; f) is isomorphic to (l(l∞(m,ϕ,e),f),l(l1(m,ϕ,e),f))θ: lp(m,ϕ,e; f) ' (l(l∞(m,ϕ,e),f),l(l1(m,ϕ,e),f))θ with θ = 1 p proof. this is a direct consequence of definition 3.10 and theorem 3.11. � corollary 3.14. for 1 ≤ p < ∞ lp(m,ϕ,c; c) is isomorphic to lq(m,ϕ): lp(m,ϕ,c; c) ' lq(m,ϕ) where q is such that 1 p + 1 q = 1 (called the conjugate of p). proof. lp(m,ϕ,c; c) = (l∞(m,ϕ,c; c), l1(m,ϕ,c; c))θ = (l(l∞(m,ϕ),l(c; c)),l(l1(m,ϕ),l(c; c)))θ ' (l(l∞(m,ϕ),c)),l(l1(m,ϕ),c)))θ ' ((l∞(m,ϕ))′, (l1(m,ϕ))′)θ ' ((l1(m,ϕ)), (l∞(m,ϕ)))θ ' ((l∞(m,ϕ)), (l1(m,ϕ)))1−θ ' ((l∞(m,ϕ)), (l1(m,ϕ)))1/q ' lq(m,ϕ) � 3.4. operator space structure. for any integer n ∈ n∗, we identify mn(l1(m,ϕ,e; f)) with l1(m,ϕ,e; mn(f)): mn(l1(m,ϕ,e; f)) ≈ l1(m,ϕ,e; mn(f)), via the correspondence mn(l1(m,ϕ,e; f)) → l1(m,ϕ,e; mn(f)) (ϕkl)1≤k,l≤n 7→ ϕn where ϕn is an element of l1(m,ϕ,e; mn(f)) defined by ∀(x,y) ∈ l∞(m,ϕ) ×e, ϕn(x,y) = (ϕkl(x,y))1≤k,l≤n . 134 atto, assiamoua and mensah now, by identifying mn(l1(m,ϕ,e; f)) with l1(m,ϕ,e; mn(f)), we can set an operator space structure on l1(m,ϕ,e; f). thus, the norm ‖·‖n in mn(l1(m,ϕ,e; f)) is the one defined on l1(m,ϕ,e; mn(f)). in fact, mn(f) being an operator space, l1(m,ϕ,e; mn(f)) is well-defined as a banach space and so the norm ‖·‖n is complete for all n ∈ n∗ . let us prove that the two conditions of ruan theorem are realised: let n,m ∈ n∗, u ∈ mn(l1(m,ϕ,e; f)), v ∈ mm(l1(m,ϕ,e; f)) and α,β ∈ mn. first condition: ‖u⊕v‖n+m = ‖u⊕v‖l1(m,ϕ,e;mn+m(f)) = sup{‖(u⊕v)(x,y)‖mn+m(f) : ϕ(|x|) ≤ 1,‖y‖e ≤ 1} = sup{max { ‖u(x,y)‖mn(f),‖v(x,y)‖mm(f) } : ϕ(|x|) ≤ 1,‖y‖e ≤ 1} = sup ϕ(|x|)≤1,‖y‖e≤1 { max { ‖u(x,y)‖mn(f),‖v(x,y)‖mm(f) }} = max { sup ϕ(|x|)≤1,‖y‖e≤1 { ‖u(x,y)‖mn(f) } , sup ϕ(|x|)≤1,‖y‖e≤1 { ‖v(x,y)‖mm(f) }} = max { ‖u‖l1(m,ϕ,e;mn(f)),‖v‖l1(m,ϕ,e;mm(f)) } second condition: ‖αuβ‖n = ‖αuβ‖l1(m,ϕ,e;mn(f)) = sup ϕ(|x|)≤1,‖y‖e≤1 ‖(αuβ)(x,y)‖mn(f) = sup ϕ(|x|)≤1,‖y‖e≤1 ‖α(u(x,y))β‖mn(f) u(x,y) being in mn(f), where f is an operator space, according to the second condition of ruan theorem, we have obviously ‖α(u(x,y))β‖mn(f) ≤‖α‖‖u(x,y)‖mn(f)‖β‖. so ‖αuβ‖ ≤ sup ϕ(|x|)≤1,‖y‖e≤1 ‖α‖‖u(x,y)‖mn(f)‖β‖ ≤ ‖α‖ ( sup ϕ(|x|)≤1,‖y‖e≤1 ‖u(x,y)‖mn(f) ) ‖β‖ ≤ ‖α‖‖u‖n‖β‖. using the same method, we give to l∞(m,ϕ,e; f) an operator space structure by identifying mn(l∞(m,ϕ,e; f)) with l∞(m,ϕ,e; mn(f)). in the following, l1(m,ϕ,e; f) and l∞(m,ϕ,e; f) are viewed as operator spaces and by interpolation, we define for 1 < p < ∞, the operator space: lp(m,ϕ,e; f) = (l∞(m,ϕ,e; f), l1(m,ϕ,e; f))θ where θ = 1 p . the following theorem is the analogous of pisier’s theorem that we’ve recalled in the previous sequence (theorem 2.5). theorem 3.15. let (m,ϕ) be any n.c.p. space. let e be an operator space. if e′ has the ornpq with 1 < p < ∞ and q = pp−1 , then we have a completely isometric identity lp(m,ϕ,e; c) = lq(m,ϕ,e′). on some isomorphisms 135 proof. lp(m,ϕ,e; c) = l(lp(m,ϕ,e),c) = (lp(m,ϕ,e)) ′ = (lq(m,ϕ,e ′) � references [1] j. bergh and j. lfstrm, interpolation spaces. springer-verlag, berlin, 1976. [2] d. blecher and v. paulsen, tensor products of operators spaces. j. funct. anal. 99 (1991) 262-292. [3] d. blecher, the standard dual of an operator space. pacific j. math. 153 (1992) 15-30. [4] j. dixmier, formes linaires sur un anneau d’oprateurs. bull. soc. math. france 81 (1953) 9-39. [5] e. effros and z. j. ruan, a new approach to operator spaces. canadian math. bull. 34 (1991) 329-337. [6] e. effros and z. j. ruan, recent development in operator spaces. current topics in operator algebras. proceedings of the icm-90 satelite conference held in nara (august 1990). world sci. publishing, river edge, n. j., 1991, p146-164. [7] e. effros and z. j. ruan, on the abstract characterization of operator spaces. proc. amer. math. soc. 119 (1993) 579-584. [8] h. kosaki, applications of the complex interpolation method to a von neumann algebra: non-commutative lp-spaces. j. funct. anal. 56 (1984) 29-78. [9] r. kunze, lp fourier transforms on locally compact unimodular groups. trans. amer. math. soc. 89 (1958) 519-540. [10] g. pisier, the operator hilbert space oh, complex interpolation and tensor norms. memoirs amer. math. soc. vol. 122, 585 (1996) 1-103. [11] g. pisier, non-commutative vector valued lp-spaces and completely p-summing maps. astérisque (soc. math. france) 247 (1998), 1-131. [12] z. j. ruan, subspaces of c∗−algebras. j. funct. anal. 76 (1988)217-230. [13] i. segal, a non-commutative extension of abstract extension. ann. of math. 57 (1953) 401457. [14] w. stinespring, integration theorem for gages and duality for unimodular groups. trans. amer. math. soc. 90 (1959) 15-26. [15] q. xu, operator spaces and non-commutative lp. the part on non-commutative lp-spaces. lectures in the summer school on banach spaces and operator spaces, nankai university -china, july 16 july 20, 2007. 1department of mathematics, university of lomé, pobox 1515 lomé, togo 2international chair in mathematical physics and applications (icmpa-unesco chair),university of abomey-calavi, benin ∗corresponding author international journal of analysis and applications volume 16, number 6 (2018), 809-821 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-809 algebraic structure of graph operations in terms of degree sequences vishnu narayan mishra1,2∗, sadik delen3 and ismail naci cangul3 1department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur 484 887, madhya pradesh, india 2l. 1627 awadh puri colony beniganj, (phase-iii), opposite i.t.i., ayodhya main road, faizabad 224 001, uttar pradesh, india 3dept. of mathematics, uludag university, gorukle 16059 bursa, turkey ∗corresponding author vishnunarayanmishra@gmail.com abstract. in this paper, by means of the degree sequences (ds) of graphs and some graph theoretical and combinatorial methods, we determine the algebraic structure of the set of simple connected graphs according to two graph operations, namely join and corona product. we shall conclude that in the case of join product, the set of graphs forms an abelian monoid whereas in the case of corona product, this set is not even associative, it only satisfies two conditions, closeness and identity element. we also give a result on distributive law related to these two operations. 1. introduction 1,2 let g = (v (g),e(g)) be a simple and connected graph with | v (g) |= n vertices and | e(g) |= m edges. usually we use the notations v and e instead of v (g) and e(g), respectively. here, by the word ”simple”, we mean that the graphs we consider do not have loops or multiple edges. similar studies can be received 2017-11-27; accepted 2018-02-01; published 2018-11-02. 2010 mathematics subject classification. 05c07, 05c10, 05c30, 05c76. key words and phrases. graph; degree sequence; join; corona product; graph operation. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 809 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-809 int. j. anal. appl. 16 (6) (2018) 810 done for non-simple graphs as well. for a vertex v ∈ v , we denote the degree of v by dg(v), which is defined as the number of edges of g meeting at v. a vertex with degree one is called a pendant vertex. the notion of degree of a graph provides us an area to study various structural properties of graphs and hence attracts the attention of many graph theorists and also other scientists including chemists. if di, 1 ≤ i ≤ n, are the degrees of the vertices vi of a graph g in any order, then the degree sequence (ds) of g is the sequence {d1, d2, · · · , dn}. also, in many papers, the ds is taken to be a non-decreasing sequence, whenever possible. conversely, a non-negative sequence {d1, d2, · · · , dn} will be called realizable if it is the ds of any graph. it is clear from the definition that for a realizable ds, there is at least one graph having this ds. for example, the completely different two graphs in figure 1 have the same ds. fig. 1 graphs with the same ds for convenience and brevity, we shall denote the ds having repeated degrees with a shorter ds. for example, if the degree di of the vertex vi appears zi times in the ds of a graph g, then we use { d (z1) 1 , d (z2) 2 , · · · , d (z`) l } instead of {d1, d2, · · · , dn} where ` ≤ n. here the members zi are called the frequencies of the degrees. when ` = n, that is, when all degrees are different, the ds is called perfect. it is an open problem to determine that which dss are realizable and there are several algorithms to determine that. as usual, we denote by pn, cn, sn, kn, tr,s and kr,s the path, cycle, star, complete, complete bipartite and tadpole graphs, respectively, which are the most used graph examples in literature, see figure 2. int. j. anal. appl. 16 (6) (2018) 811 fig. 2 p5, c6, s7, k6, t3,2, k2,5 the number of vertices and edges of these well-known graph classes are given in table 1. table 1. the number of vertices and edges of some graphs g ] vertices ] edges pn n n-1 cn n n sn n n-1 kn n ( n 2 ) kr,s r+s rs tr,s r+s r+s another important reason to study the dss of graphs is topological indices. a topological index (or a graph invariant) is a fixed invariant number for two isomorphic graphs and gives some information about the graph under consideration. these indices are especially useful in the study of molecular graphs. some of the topological indices are defined by means of the vertex degrees: first and second zagreb indices, first and second multiplicative zagreb indices, atom-bond connectivity index, narumi-katayama index, geometricarithmetic index, harmonic index and sum-connectivity index etc. therefore to know about the ds of the graph will help to obtain information about, e.g., the chemical properties of the graph. there are many papers on degree based topological indices, see e.g. [2][3]. int. j. anal. appl. 16 (6) (2018) 812 the modern study of dss started in 1981 by bollobas, [1]. the same year, tyshkevich et.al. established a correspondence between ds of a graph and some structural properties of this graph, [8]. in 1987, tychkevich et.al. written a survey on the same correspondence, [9]. in [10], the authors gave a new version of the erdös-gallai theorem on the realizability of a given ds. in 2008, a new criterion on the same problem is given by triphati and tyagi, [7]. the same year, kim et.al. gave a necessary and sufficient condition for the same problem, [5]. ivanyi et.al, [4], gave an enumeration of dss of simple graphs. miller, [6] also gave a criteria for the realizability of given dss. there are several graph operations used in calculating some chemical invariants of graphs. amongst these the join, cartesian, corona product, union, disjunction, and symmetric difference are well-known. in this paper, after recalling two of these operations, join and corona product, we shall determine the ds of these new product graphs and by means of these calculations, we shall study the algebraic properties of the join and corona product of two graphs. let g1 and g2 be two graphs with n1 and n2 vertices and m1 and m2 edges, respectively. the join g1∨g2 of graphs g1 and g2 with disjoint vertex sets v (g1) and v (g2) and edge sets e(g1) and e(g2) is the graph union g1 ∪g2 together with all the edges joining v (g1) and v (g2). thus, for example, kp ∨kq = kp,q, the complete bipartite graph. we have |v (g1 ∨g2)| = n1 + n2 and |e(g1 ∨g2)| = m1 + m2 + n1n2. the corona product g1 ◦g2 of two graphs g1 and g2 is defined to be the graph γ obtained by taking one copy of g1 (which has n1 vertices) and n1 copies of g2, and then joining the i−th vertex of g1 to every vertex in the i−th copy of g2, for i = 1, 2, · · · , n1. let g1 = (v1,e1) and g2 = (v2,e2) be two graphs such that v (g1) = {u1, u2, · · · , un1}, |e(g1)| = m1 and v (g2) = {v1, v2, · · ·vn2}, |e(g2)| = m2. then it follows from the definition of the corona product that g1 ◦g2 has n1(1 + n2) vertices and m1 + n1m2 + n1n2 edges, where v (g1 ◦g2) = {(ui,vj), i = 1, 2, ...,n1; j = 0, 1, 2, ...,n2} and e(g1 ◦g2) = {((ui,v0), (uk,v0)), (ui,uk) ∈ e(g1)} ∪{((ui,vj), (ui,v`)), (vj,v`) ∈ e(g2), i = 1, 2, ...,n1} ∪{((ui,v0), (ui,v`)),` = 1, 2, ...,n2, i = 1, 2, ...,n1} . int. j. anal. appl. 16 (6) (2018) 813 it is clear that if g1 is connected, then g1 ◦ g2 is connected, and in general g1 ◦ g2 is not isomorphic to g2 ◦g1. 2. algebraic properties of join in this section, we deal with some algebraic properties of the join of two graphs. we shall try to determine the abstract algebraic structure of this new graph and also give the ds of the join graph g1 ∨ g2 of two graphs g1 and g2 where g1 and g2 are choosen from pn, cn, sn, kn, tr,s and kr,s. in particular, in the case of join operation, the set of graphs forms an abelian monoid whereas in the case of corona product, the set of graphs is not even associative, it only satisfies two conditions, closedness and identity element. it is clear to see that there are no zero divisors for both products. theorem 2.1. the dss of all possible joins of the path, cycle, star, complete, tadpole and complete bipartite graphs are given in table 2. proof. we make the proof only for pr ∨ps and sr ∨cs. let pr = { 1(2), 2(r−2) } and ps = { 1(2), 2(s−2) } . to visualize the situation, see figure 3. there are two types of vertices in each of pr and ps. therefore there are 2 + 2 = 4 types of vertices in pr ∨ ps. the first type is the two end vertices of pr (red ones) which are connected with the next green vertex in pr and s vertices in ps. each of these two vertices add s + 1 to the ds of pr ∨ps. therefore they add (s + 1)(2). the second type of vertices are the mid ones in pr (green ones) each of which is connected to two neighboring vertices in pr and s vertices in ps. each of these r − 2 vertices adds s + 2 to the ds of pr ∨ ps. therefore (s + 2)(r−2) is added. the third type of vertices are the two end vertices of ps (black ones) each of which is connected to one vertex in ps and r vertices in pr. they add (r + 1) 2 to the ds of pr ∨ ps. the fourth and last type of vertices are the mid-vertices (blue ones) in ps. their number is s−2 and each of which similarly adds r + 2 to the ds of pr ∨ps. so their contribution is (r + 2)(s−2). therefore the required ds is pr ∨ps = {(s + 1)(2), (s + 2)(r−2), (r + 1)(2), (r + 2)(s−2)}. now we recall that sr = {1(r−1),r − 1} and cs = { 2(s) } . in figure 4, the join of these two graphs is drawn for r = s = 5. int. j. anal. appl. 16 (6) (2018) 814 table 2. the dss of the join of some well-known graph types g1 g2 g1 ∨ g2 pr ps { (s + 1)(2), (s + 2)(r−2), (r + 1)(2), (r + 2)(s−2) } pr cs { (s + 1)(2), (s + 2)(r−2), (r + 2)(s) } pr ss { (s + 1)(2), (s + 2)(r−2), (r + 1)(s−1), r + s − 1 } pr ks { (s + 1)(2), (s + 2)(r−2), (r + s − 1)(s) } pr ts,t { (s + t + 1)(2), (s + t + 2)(r−2), r + 1, (r + 2)(s+t−2), r + 3 } pr ks,t { (s + t + 1)(2), (s + t + 2)(r−2), (r + s)(t), (r + t)(s) } cr ps { (s + 2)(r), (r + 1)(2), (r + 2)(s−2) } cr cs { (s + 2)(r), (r + 2)(s) } cr ss { (s + 2)(r), r + s − 1, (r + 1)(s−1) } cr ks { (s + 2)(r), (r + s − 1)(s) } cr ts,t { (s + t + 2)(r), r + 1, (r + 2)(s+t−2), r + 3 } cr ks,t { (s + t + 2)(r), (r + s)(t), (r + t)(s) } sr ps { (s + 1)(r−1), r + s − 1, (r + 1)(2), (r + 2)(s−2) } sr cs { (s + 1)(r−1), r + s − 1, (r + 2)(s) } sr ss { (s + 1)(r−1), (r + s − 1)(2), (r + 1)(s−1) } sr ks { (s + 1)(r−1), s + r − 1, (r + s − 1)(s) } sr ts,t { (s + t + 1)(r−1), r + s + t − 1, r + 1, (r + 2)(s+t−2), r + 3 } sr ks,t { (s + t + 1)(r−1), r + s + t − 1, (r + s)(t), (r + t)(s) } kr ps { (r + s − 1)(r), (r + 1)(2), (r + 2)(s−2) } kr cs { (r + s − 1)(r), (r + 2)(s) } kr ss { (r + s − 1)(r), (r + 1)(s−1), r + s − 1 } kr ks { (r + s − 1)(r+s) } kr ts,t { (r + s + t − 1)(r), r + 1, (r + 2)(s+t−2), r + 3 } kr ks,t { (r + s + t − 1)(r), (r + s)(t), (r + t)(s) } tr,s pt { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 1)(2), (r + s + 2)(t−2) } tr,s ct { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 2)(t) } tr,s st { (t + 2)(r+s−2), t + 1, t + 3, (r + s + 1)(t−1), r + s + t − 1 } tr,s kt { (t + 2)(r+s−2), t + 1, t + 3, (r + s + t − 1)(t) } tr,s tt,m   (t + m + 2)(r+s−2), t + m + 1, t + m + 3, (r + s + 2)(t+m−2), r + s + 1, r + s + 3   tr,s kt,m { (t + m + 2)(r+s−2), t + m + 1, t + m + 3, (r + s + t)(m), (r + s + m)(t) } kr,s pt { (r + t)(s), (s + t)(r), (r + s + 1)(2), (r + s + 2)(t−2) } kr,s ct { (r + t)(s), (s + t)(r), (r + s + 2)(t) } kr,s st { (r + t)(s), (s + t)(r), (r + s + 1)(t−1), r + s + t − 1 } kr,s kt { (r + t)(s), (s + t)(r), (r + s + t − 1)(t) } kr,s tt,m { (r + t + m)(s), (s + t + m)(r), (r + s + 2)(t+m−2), r + s + 1, r + s + 3 } kr,s kt,m { (r + t + m)(s), (s + t + m)(r), (r + s + m)(t), (r + s + t)(m) } int. j. anal. appl. 16 (6) (2018) 815 there are two types of vertices in star graph sr and only one type in cycle cs. therefore there are 2 + 1 = 3 types of vertices in sr∨cs. the first type of those is the end vertices (black ones) of the star graph sr. each of these is connected to the central vertex (blue coloured) by an edge in sr and to all s vertices in cs. therefore each of these vertices adds s + 1, and in total, (s + 1) (r−1) is added to the ds. the second type is the unique central vertex in sr which is connected to r − 1 end vertices in sr and also to all of s vertices in cs. it adds a total of r + s− 1 to the ds. the third and final type of vertices is the s vertices in cs (green ones). each of them adds r + 2 to the ds. therefore a total of (r + 2) (s) is added. hence the ds of the required join graph is sr ∨cs = {(s + 1)(r−1), r + s− 1, (r + 2)(s)}. � now we can study the algebraic structure of the set of graphs according to join operation: theorem 2.2. let g be the set of all simple connected graphs. then g is an abelian monoid with the join operation. proof. let us have three graphs g1 = {α (β11) 11 , · · · ,α (β1`) 1` }, g2 = {α (β21) 21 , · · · ,α (β2m) 2m } and g3 = {α (β31) 31 , · · · ,α (β3n) 3n } with the number of vertices n1,n2,n3, respectively. we shall show that g with the join operation is closed, associative, commutative, has identity element but no inverse elements: first of all, the join of two simple connected graphs, by definition, is another simple connected graph, so g is closed. for associativeness, note that (g1 ∨g2) ∨g3 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21) (β21), · · · , (n1 + α2m)(β2m)}∨{α (β31) 31 ,α (β32) 32 ,α (β33) 33 } = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} and g1 ∨ (g2 ∨g3) = {α (β11) 11 , · · · ,α (β1`) 1` }∨{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (n2 + α31) (β31), · · · , (n2 + α3n)(β3n)} = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} int. j. anal. appl. 16 (6) (2018) 816 = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (n1 + n2 + α31) (β31), · · · , (n1 + n2 + α3n)(β3n)} therefore g is associative. as g1 ∨g2 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21)(β21), · · · , (n1 + α2m)(β2m)} = {(n1 + α21)(β21), · · · , (n1 + α2m)(β2m), (n2 + α11)(β11), · · · , (n2 + α1`)(β1`)} = g2 ∨g1, the operation is commutative. therefore, to find the identity element, one needs to find a graph z with the property that g1 ∨ z = g1. let z = {ab11 , · · · ,a bk k } and let the graph z have c vertices. then {α(β11)11 , · · · ,α (β1`) 1` }∨{a (b1) 1 , · · · ,a (bk) k } = {α (β11) 11 , · · · ,α (β1`) 1` } implies that {(c + α11)(β11), · · · , (c + α1`)(β1`), (n1 + a1) (b1) , · · · , (n1 + ak) (bk)} = {α(β11)11 , · · · ,α (β1`) 1` } and this is only possible when c = 0. for c = 0, we have {α(β11)11 , · · · ,α (β1`) 1` , (n1 + a1) (b1) , · · · , (n1 + ak) (bk)} = {α(β11)11 , · · · ,α (β1`) 1` }. to have this equality, we must have (n1 + a1) (b1) = 0, · · · , (n1 + ak) (bk) = 0. hence we must have no terms (n1 + a1) (b1) , · · · , (n1 + ak) (bk) in the ds of the identity element z. this implies that b1 = · · · = bk = 0. therefore we can symbolically take z = {1(0)} as the identity element. finally, let the inverse element of the graph g1 = {α (β11) 11 , · · · ,α (β1`) 1` } be denoted by {c (d1) 1 , ...,c (dk) k }. let the number of vertices of the inverse element be e. then g1 ∨{c (d1) 1 , · · · ,c (dk) k } = z = {1 (0)} implies that {α(β11)11 , · · · ,α (β1`) 1` }∨{c (d1) 1 , · · · ,c (dk) k } = {1 (0)} and therefore {(e + α11)(β11), · · · , (e + α1`)(β1`), (n1 + c1) (d1) , · · · , (n1 + ck) (dk)} = {1(0)}. as there is no solution to that equation, we conclude that there is no inverse element for the join operation. therefore the result follows. � int. j. anal. appl. 16 (6) (2018) 817 3. algebraic properties of corona product theorem 3.1. the dss of all possible corona products of the path, cycle, star, complete, tadpole and complete bipartite graphs are given in table 3. theorem 3.2. let g be the set of all simple connected graphs. then g with corona product operation is closed with identity. proof. first, by the definition of the operation, g is closed. secondly, for associativeness, we should note that (g1 ◦g2) ◦g3 = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (1 + α21) (n1β21), · · · , (1 + α2m)(n1β2m)}◦{α (β31) 31 , · · · ,α (β3n) 3n } = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (1 + α31) (n1(n1+n2)β31), · · · , (1 + α3n)(n1(n1+n2)β3n)} and g1 ◦ (g2 ◦g3) = { α (β11) 11 , · · · ,α (β1`) 1` } ◦{(n3 + α21)(β21), · · · , (n3 + α2m)β2m, (1 + α31) (n2β31), · · · , (1 + α3n)(n2β3n)} = {(n2 + n2n3 + α11)(β11), · · · , (n2 + n2n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (2 + α31) (n1n2β31), · · · , (2 + α3n)(n1n2β3n)}. that is, g is not associative. for the identity element, we should find a graph z such that g1 ◦ z = g1. let z = {a (b1) 1 , · · · ,a (bk) k } and let the number of vertices of z be c. we have {α(β11)11 , · · · ,α (β1`) 1` }◦{a (b1) 1 , · · · ,a (bk) k } = {α (β11) 11 , · · · ,α (β1`) 1` } and hence, we get {(c + α11)(β11), · · · , (c + α1`)(β1`), (1 + a1) (n1b1) , · · · , (1 + ak) (n1bk)} = {α(β11)11 , · · · ,α (β1`) 1` }. for this equation to have a solution, we must have c = 0 and also b1, · · · ,bk = 0. for the sake of brevity, if we take 1 instead of 1 + a1, · · · , 1 + ak, we conclude that z = {1(0)} is the required identity element. let int. j. anal. appl. 16 (6) (2018) 818 table 3. the dss of the corona product of well-known graph types g1 g2 g1 ◦ g2 g1 g2 g1 ◦ g2 pr ps { 2(2r), 3(r(s−2)), (s + 1)(2), (s + 2)(r−2) } pr cs { 3(rs), (s + 1)(2), (s + 2)(r−2) } pr ss { 2(r(s−1)), s(r), (s + 1)(2), (s + 2)(r−2) } pr ks { s(rs), (s + 1)(2), (s + 2)(r−2) } pr ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 1)(2), (s + t + 2)(r−2) } pr ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 1)(2), (s + t + 2)(r−2) } cr ps { 2(2r), 3(r(s−2)), (s + 2)(r) } cr cs { 3(rs), (s + 2)(r) } cr ss { 2(r(s−1)), s(r), (s + 2)(r) } cr ks { s(rs), (s + 2)(r) } cr ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 2)(r) } cr ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 2)(r) } sr ps { 2(2r), 3(r(s−2)), (s + 1)(r−1), r + s − 1 } sr cs { 3(rs), (s + 1)(r−1), r + s − 1 } sr ss { 2(r(s−1)), s(r), (s + 1)(r−1), r + s − 1 } sr ks { s(rs), (s + 1)(r−1), r + s − 1 } sr ts,t { 2(r), 3(r(s+t−2)), 4(r), (s + t + 1)(r−1), r + s + t − 1 } sr ks,t { (s + 1)(rt), (t + 1)(rs), (s + t + 1)(r−1), r + s + t − 1 } kr ps { 2(2r), 3(r(s−2)), (r + s − 1)(r) } kr cs { 3(rs), (r + s − 1)(r) } kr ss { 2(r(s−1)), s(r), (r + s − 1)(r) } kr ks { s(rs), (r + s − 1)(r) } kr ts,t { 2(r), 3(r(s+t−2)), 4(r), (r + s + t − 1)(r) } kr ks,t { (s + 1)(rt), (t + 1)(rs), (r + s + t − 1)(r) } tr,s pt { 2(2(r+s)), 3((r+s)(t−2)), t + 1, (t + 2)(r+s−2), t + 3 } tr,s ct { 3((r+s)t), t + 1, (t + 2)(r+s−2), t + 3 } tr,s st { 2((r+s)(t−1)), t(r+s), t + 1, (t + 2)(r+s−2), t + 3 } tr,s kt { t((r+s)t), t + 1, (t + 2)(r+s−2), t + 3 } tr,s tt,m   2(r+s), 3((r+s)(t+m−2)), 4(r+s), t + m + 1, (t + m + 2)(r+s−2), t + m + 3   tr,s kt,m   (t + 1)((r+s)m), (m + 1)((r+s)t), t + m + 1, (t + m + 2)(r+s−2), t + m + 3   kr,s pt { 2(2(r+s)), 3((r+s)(t−2)), (r + t)(s), (s + t)(r) } kr,s ct { 3((r+s)t), (r + t)(s), (s + t)(r) } kr,s st { 2((r+s)(t−1)), t(r+s), (r + t)(s), (s + t)(r) } kr,s kt { t((r+s)t), (r + t)(s), (s + t)(r) } kr,s tt,m { 2(r+s), 3((r+s)(t+m−2)), 4(r+s), (r + t + m)(s), (s + t + m)(r) } kr,s kt,m { (t + 1)((r+s)m), (m + 1)((r+s)t), (r + t + m)(s), (s + t + m)(r) } int. j. anal. appl. 16 (6) (2018) 819 t = {c(d1)1 , · · · , c (dk) k } be the inverse element of a graph g1 = {α (β11) 11 , · · · , α (β1`) 1` }. then they must satisfy the equation g1 ◦{c (d1) 1 , · · · ,c (dk) k } = z = {1 (0)}. if the number of vertices of t is e, then we have {α(β11)11 , · · · ,α (β1`) 1` }◦{c (d1) 1 , · · · ,c (dk) k } = {1 (0)}. in this case, this equation cannot be hold, implying that there is no inverse element in g. finally, as {α(β11)11 , · · · ,α (β1`) 1` }◦{α (β21) 21 , · · · ,α β2m 2m } = {(n2 + α11) (β11), · · · , (n2 + α1`)(β1`), (1 + α21) (n1β21), · · · , (1 + α2m)(n1β2m)} and {α(β21)21 , · · · ,α (β2m) 2m }◦{α (β11) 11 , · · · ,α (β1`) 1` } = {(n1 + α21) (β21), · · · , (n1 + α2m)(β2m), (1 + α11) (n2β11), · · · , (1 + α1`)(n2β1`)}, g can only be commutative when g1 = g2. in general, g is not commutative. � finally, we check whether the distributive law holds when we have join and corona in place of · and +, or vice versa: theorem 3.3. neither join nor corona operation is not distributive on each other. that is (i) g1 ∨ (g2 ◦g3) 6= (g1 ∨g2) ◦ (g1 ∨g3) , (ii) g1 ◦ (g2 ∨g3) 6= (g1 ◦g2) ∨ (g1 ◦g3). proof. both claims follow after the following calculations: (i) g1 ∨ (g2 ◦g3) = {α (β11) 11 , · · · ,α (β1`) 1` }∨{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (1 + α31) (n2β31), · · · , (1 + α3n)(n2β3n)} = {(n2 + n2n3 + α11)(β11), · · · , (n2 + n2n3 + α1`)(β1`), (n1 + n3 + α21) (β21), · · · , (n1 + n3 + α2m)(β2m), (1 + n1 + α31) (n2β31), · · · , (1 + n1 + α3n)(n2β3n)} int. j. anal. appl. 16 (6) (2018) 820 and (g1 ∨g2) ◦ (g1 ∨g3) = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (n1 + α21)(β21), · · · , (n1 + α2m) (β2m)}◦{(n3 + α11)(β11), · · · , (n3 + α1`)(β1`), (n1 + α31) (β31), · · · , (n1 + α3n)(β3n)} = {(n1 + n2 + n3 + α11)(β11), · · · , (n1 + n2 + n3 + α1`)(β1`), (2n1 + n3 + α21) (β21), · · · , (2n1 + n3 + α2m)(β2m), (1 + n3 + α11) (n1+n2)β11, · · · , (1 + n3 + α1`)((n1+n2)β1`), (1 + n1 + α31) ((n1+n2)β31), · · · , (1 + n1 + α3n)((n1+n2)β3n)}. (ii)g1 ◦ (g2 ∨g3) = {α (β11) 11 , · · · , α (β1`) 1` }◦{(n3 + α21) (β21), · · · , (n3 + α2m)(β2m), (n2 + α31) (β31), · · · , (n2 + α3n)(β3n)} = {(n2 + n3 + α11)(β11), · · · , (n2 + n3 + α1`)(β1`), (1 + n3 + α21) (n1β21), · · · , (1 + n3 + α2m)(n1β2m), (1 + n2 + α31) (n1β31), · · · , (1 + n2 + α3n)(n1β3n)} and also (g1 ◦g2) ∨ (g1 ◦g3) = {(n2 + α11)(β11), · · · , (n2 + α1`)(β1`), (1 + α21)(n1β21), · · · , (1 + α2m) (n1β2m)}∨{(n3 + α11)(β11), · · · , (n3 + α1`)(β1`), (1 + α31) (n1β31), · · · , (1 + α3n)(n1β3n)} = {(n1 + n1n3 + n2 + α11)β11, · · · , (n1 + n1n3 + n2 + α1`)(β1`), (1 + n1 + n1n3 + α21) (n1β21), · · · , (1 + n1 + n1n3 + α2m)(n1β2m), (n1 + n1n2 + n3 + α11) β11, · · · , (n1 + n1n2 + n3 + α1`)(β1`), (1 + n1 + n1n2 + α31) (n1β31), · · · , (1 + n1 + n1n2 + α3n)(n1β3n)}. � int. j. anal. appl. 16 (6) (2018) 821 references [1] b. bollobas, degree sequences of random graphs, discrete math. 33 (1981), 1-19. [2] k. c. das, n. akgunes, m. togan, a. yurttas, i. n. cangul, a. s. cevik, on the first zagreb index and multiplicative zagreb coindices of graphs, an. tiin. univ. ovidius constana, ser. mat. 24 (1) (2016), 153-176. [3] k. c. das, a. yurttas, m. togan, i. n. cangul, a. s. cevik, the multiplicative zagreb indices of graph operations, j. inequal. appl. 90 (2013), 1-14. [4] a. ivanyi, l. lucz, g. gombos, t. matuszka, parallel enumeration of degree sequences of simple graphs ii, acta univ. sapientiae, informatica 5 (2) (2013), 245-270. [5] h. kim, z. toroczkai, i. miklos, p. l. erdös, l. a. szekely, on realizing all simple graphs with a given degree sequence, j. phys. a: math. theor. 42 (2009), 1-6. [6] j. w. miller, reduced criteria for degree sequences, discrete math. 313 (2013), 550-562. [7] a. triphati, h. tyagi, a simple criterion on degree sequences of graphs, discrete appl. math. 156 (2008), 3513-3517. [8] r. i. tyshkevich, o. i. mel’nikov, v. m. kotov, on graphs and degree sequences: canonical decomposition, kibernetika 6 (1981), 5-8. [9] r. i. tyshkevich, a. a. chernyak, zh. a. chernyak, graphs and degree sequences i, cybernetics 23 (6) (1987), 734-745. [10] i. e. zverovich, v. e. zverovich, contributions to the theory of graphic sequences, discrete math. 105 (1992), 293-303. 1. introduction 2. algebraic properties of join 3. algebraic properties of corona product references international journal of analysis and applications volume 16, number 3 (2018), 340-352 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-340 harmonic m-preinvex functions and inequalities muhammad aslam noor1, khalida inayat noor1, sabah iftikhar1, awais gul khan2 1department of mathematics, comsats institute of information technology, islamabad, pakistan 2department of mathematics, gc university, faisalabad, pakistan ∗corresponding author: noormaslam@gmail.com abstract. in this paper, we introduce a new class of harmonic functions, which is called harmonic mpreinvex functions for a fixed m. some hermite-hadamard inequality for harmonic m-preinvex functions are derived. several special cases are discussed as applications of the main results. the ideas and techniques of this paper may be starting point for further research. 1. introduction convex functions and their variant forms are being used to study a wide class of problems which arises in various branches of pure and applied sciences. the concept of convexity have been generalize by many researchers using novel and innovative techniques and ideas. it is know that the convex function can be characterized by some integral inequalities, which are known as hermite-haramard inequalities. hanson [11] introduced the concept of invex functions. ben-israel and mond [5] introduced the concept of invex sets and peinvex functions. for the applications, properties and other aspects of the preinvex functions, see [2,3,17–24] and the references therein. varosanec [36] introduced the class of h-convex functions. this class of functions unifies various classes of convex functions and is being used to discuss several concepts in a unified manners. received 2017-11-03; accepted 2018-01-16; published 2018-05-02. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. harmonic preinvex convex function; relative harmonic preinvex function; m-preinvex function; hermite-hadamard type inequality. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 340 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-340 int. j. anal. appl. 16 (3) (2018) 341 anderson et al. [1] and iscan [13] have investigated various properties of harmonic convex function. iscan [13] has obtained several hermite-hadamard inequalities for harmonic convex functions. noor et al. [26] introduced and investigated another class of harmonic convex functions, which is called harmonic preinvex functions and can be viewed as significant generalization of both the harmonic convex functions and preinvex functions. for recent developments and other aspects of harmonic convex functions, see [14, 25–30]. toader [34] define the concept of m-convexity, an intermediate between usual convexity and star shape functions. noor et. al [30] have introduced the concept of harmonic m-convex function on a harmonic convex set. in particular, a function f : i = [a,b] ⊂ r \ {0} −→ r is said to be harmonic m-convex function with respect to an arbitrary nonnegative function h, where m ∈ (0, 1], if 1 h ( 1 2 )f( 2ab a + b ) ≤ ab b−a ∫ b a f(x) + mf(xm) x2 dx ≤ 1 2 { f(a) + f(b) + 2m[f(am) + f(bm)] +m2[f(am2) + f(bm2)] }∫ 1 0 h(t)dt, (1.1) which is known as hermite-hadamard inequality for harmonic m-convex function with respect to an arbitrary nonnegative function h. we would like to emphasize that m-convex functions and harmonic h-preinvex functions are two distinct classes of convex functions. it is natural to introduce a new class of convex functions, which unifies these concepts. motivated and inspired by the on going research in the convexity theory, we introduce harmonic m-preinvex functions on a harmonic m-invex set. it is shown that several new classes of harmonic convex functions and harmonic preinvex functions can be obtained as special case. we have obtained several new hermite-hadamard inequality and related inequalities for harmonic m-preinvex functions. one can easily show that this new class includes harmonic m-convex functions, harmonic m-preinvex functions and harmonic beta m-preinvex functions as special cases. the techniques and the ideas of this paper may stimulate further research. 2. preliminaries let i be a nonempty closed set in r . let f : iη ⊆ r −→ r be a continuous function and η(·, ·) : iη×iη −→ r be a continuous bifunction. definition 2.1. [7]. a set i ⊂ r is said to be m-convex set with respect to a fixed constant m ∈ [0, 1], if (1 − t)x + mty ∈ i, ∀x,y ∈ i, t ∈ [0, 1]. int. j. anal. appl. 16 (3) (2018) 342 the m-convex set contains the line segment between points x and my for every pair of points x and y of i. definition 2.2. [34]. a function f : i ⊂ r → r is said to be m-convex function, where m ∈ [0, 1], if f((1 − t)x + mty) ≤ (1 − t)f(x) + mf(y), ∀x,y ∈ i, t ∈ [0, 1]. if we take m = 1, then we recapture the concept of convex functions and if we take t = 1, then f(my) ≤ mf(y) ∀t ∈ [0, 1], y ∈ i. this shows that the function f is sub-homogeneous. definition 2.3. a set iη ⊂ r \ {0} is said to be a harmonic m-invex set with respect to an arbitrary bifunction η : iη × iη → r, if x(x + η(my,x)) x + (1 − t)η(my,x) ∈iη, ∀x,y ∈iη, t ∈ [0, 1]. the harmonic m-invex set contains the path between points x and x + η(my,x) for every pair of points x and y of iη. every harmonic m-invex set is harmonic invex respecting the mapping η(my,x) = my −x. we now introduce the concept of harmonic m-preinvex function as. definition 2.4. let h : j = [0, 1] → r be a nonnegative function. a function f : iη ⊂ r \{0}→ r is said to be harmonic m-preinvex function, where m ∈ (0, 1], if f ( x(x + η(my,x)) x + (1 − t)η(my,x) ) ≤ h(1 − t)f(x) + h(t)mf(y), ∀x,y ∈iη, t ∈ (0, 1). if t = 1 2 , then we have f ( 2x(x + η(my,x)) 2x + η(my,x) ) ≤ h ( 1 2 ) [f(x) + mf(y)], ∀x,y ∈iη. the function f is called jensen type harmonic m-preinvex function. now we discuss some special cases of harmonic m-preinvex functions, which appears to be new ones. i. if η(my,x) = my−x in definition 2.4, then it reduces to the definition of harmonic m-convex function. definition 2.5. a function f : i ⊂ r\{0}→ r is said to be harmonic m-convex, where m ∈ (0, 1], if f ( mxy tx + (1 − t)my ) ≤ h(1 − t)f(x) + h(t)mf(y), ∀x,y ∈i, t ∈ [0, 1]. ii. if h(t) = ts in definition 2.4, then it reduces to the definition of harmonic (s,m)-preinvex function in the second sense. int. j. anal. appl. 16 (3) (2018) 343 definition 2.6. a function f : iη ⊂ r\{0}→ r is said to be harmonic (s,m)-preinvex, where s,m ∈ (0, 1], if f ( x(x + η(my,x)) x + (1 − t)η(my,x) ) ≤ (1 − t)sf(x) + tsmf(y), ∀x,y ∈iη, t ∈ [0, 1]. iii. if h(t) = tp(1 − t)q in definition 2.4, then it reduces to the definition of harmonic beta-m-preinvex function. definition 2.7. a function f : iη ⊂ r\{0}→ r is said to be harmonic beta-m-preinvex, where m ∈ (0, 1] and p,q ≥−1, if f ( x(x + η(my,x)) x + (1 − t)η(my,x) ) ≤ m(1 − t)ptqf(xm) + tp(1 − t)qf(y), ∀x,y ∈iη, t ∈ (0, 1). in brief, for suitable and appropriate choice of the functions, one can obtain several new and known classes of harmonic, preinvex and convex functions as special cases of harmonic m-preinvex functions. this shows that the class of harmonic m-preinvex functions is very general and unifying one. definition 2.8. [31]. two functions f,g are said to be similarly ordered (f is g-monotone), if and only if, 〈f(x) −f(y),g(x) −g(y)〉≥ 0, ∀x,y ∈ rn. the euler beta function is a special function defined by b(x,y) = ∫ 1 0 tx−1(1 − t)y−1dt = γ(x)γ(y) γ(x + y) , ∀x,y > 0. where γ(·) is a gamma function. we now show that the product of two harmonic m-preinvex functions is again a harmonic m-preinvex function under certain condition, which is the main motivation of our next result. lemma 2.1. let f and g be two similarly ordered harmonic m-preinvex functions. if h(1−t) + mh(t) ≤ 1, then the product fg is again a harmonic m-preinvex function. int. j. anal. appl. 16 (3) (2018) 344 proof. let f and g be harmonic m-preinvex functions. then f ( x(x + η(my,x)) x + (1 − t)η(my,x) ) g ( x(x + η(my,x)) x + (1 − t)η(my,x) ) ≤ [h(1 − t)f(x) + h(t)mf(y)][h(1 − t)g(x) + h(t)mg(y))] = [h(1 − t)]2f(x)g(x) + mh(t)h(1 − t)[f(x)g(y) + f(y)g(x)] +m2[h(t)]2f(y)g(y) ≤ [h(1 − t)]2f(x)g(x) + mh(t)h(1 − t)[f(x)g(x) + f(y)g(y)] +m2[h(t)]2f(y)g(y) = [h(1 − t)f(x)g(x) + mh(t)f(y)g(y)][h(1 − t) + mh(t)] ≤ h(1 − t)f(x)g(x) + h(t)mf(y)g(y), where we have used the fact that h(1 − t) + mh(t) ≤ 1. this shows that product of two similarly ordered harmonic m-preinvex functions is again a harmonic m-preinvex function. � 3. main results in this section, we obtain hermite-hadamard inequalities for harmonic m-preinvex function. throughout this section, we take iη = [a,a + η(mb,a)] unless otherwise specified, where a < a + η(mb,a). theorem 3.1. let f : iη ⊂ r \ {0} −→ r be harmonic m-preinvex function, where m ∈ (0, 1]. if f ∈ l[a,a + η(mb,a)], then a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x) x2 dx ≤ [f(a) + mf(b)] ∫ 1 0 h(t)dt. proof. let f be harmonic m-preinvex function. then we have f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ h(1 − t)f(a) + h(t)mf(b). integrating over t ∈ [0, 1], we obtain∫ 1 0 f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt ≤ [f(a) + mf(b)] ∫ 1 0 h(t)dt. this implies a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x) x2 dx ≤ [f(a) + mf(b)] ∫ 1 0 h(t)dt, which is the required result. � corollary 3.1. if η(my,x) = my −x, then theorem 3.1 reduces to: mab mb−a ∫ mb a f(x) x2 dx ≤ [f(a) + mf(b)] ∫ 1 0 h(t)dt, int. j. anal. appl. 16 (3) (2018) 345 corollary 3.2. under the assumptions of theorem 3.1 and h(t) = ts, we have a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x) x2 dx ≤ f(a) + mf(b) s + 1 . corollary 3.3. under the assumptions of theorem 3.1 and h(t) = tp(1 − t)q, we have a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x) x2 dx ≤ [f(a) + mf(b)]β(p + 1,q + 1). theorem 3.2. let f,g : iη ⊂ r \ {0} −→ r be harmonic m-preinvex functions, where m ∈ (0, 1]. if f ∈ l[a,a + η(mb,a)], then a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ m(a,b), where m(a,b) = [ f(a)g(a) + mf(b)mg(b) ]∫ 1 0 [h(t)]2dt + [ f(a)mg(b) + mf(b)g(a) ]∫ 1 0 h(t)h(1 − t)dt. (3.1) proof. let f,g be harmonic m-preinvex functions, we have f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ h(1 − t)f(a) + h(t)mf(b) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ h(1 − t)g(a) + h(t)mg(b). now consider f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ [ h(1 − t)f(a) + h(t)mf(b) ][ h(1 − t)g(a) + h(t)mg(b) ] = [h(1 − t)]2f(a)g(a) + h(t)h(1 − t)[f(a)mg(b) + mf(b)g(a)] +[h(t)]2mf(b)mg(b) int. j. anal. appl. 16 (3) (2018) 346 integrating over [0, 1], we have∫ 1 0 f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) g ( a(a + η(mb,a) a + (1 − t)tη(mb,a) ) dt ≤ f(a)g(a) ∫ 1 0 [h(1 − t)]2dt +[f(a)mg(b) + mf(b)g(a)] ∫ 1 0 h(t)h(1 − t)dt +mf(b)mg(b) ∫ 1 0 [h(t)]2dt = [ f(a)g(a) + mf(b)mg(b) ]∫ 1 0 [h(t)]2dt + [ f(a)mg(b) + mf(b)g(a) ]∫ 1 0 h(t)h(1 − t)dt. this implies a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ [ f(a)g(a) + mf(b)mg(b) ]∫ 1 0 [h(t)]2dt + [ f(a)mg(b) + mf(b)g(a) ]∫ 1 0 h(t)h(1 − t)dt, which is the required result. � corollary 3.4. if η(my,x) = my −x, then theorem 3.2 reduces to: mab mb−a ∫ mb a f(x)g(x) x2 dx ≤ [ f(a)g(a) + mf(b)mg(b) ]∫ 1 0 [h(t)]2dt + [ f(a)mg(b) + mf(b)g(a) ]∫ 1 0 h(t)h(1 − t)dt, corollary 3.5. under the assumptions of theorem 3.2 and h(t) = ts, we have a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ f(a)g(a) + mf(b)mg(b) 2s + 1 +β(s + 1,s + 1)[f(a)mg(b) + mf(b)g(a)]. corollary 3.6. under the assumptions of theorem 3.2 and h(t) = tp(1 − t)q, we have a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ m(a,b), where m(a,b) = [f(a)g(a) + mf(b)mg(b)]β(2p + 1, 2q + 1) +[f(a)mg(b) + mf(b)g(a)]β(p + q + 1,p + q + 1). int. j. anal. appl. 16 (3) (2018) 347 theorem 3.3. let f,g : iη ⊂ r \ {0} −→ r be harmonic m-preinvex functions, where m ∈ (0, 1]. if f ∈ l[a,a + η(mb,a)], then ( a(a + η(mb,a)) η(mb,a) )2 ∫ 1 a 1 a+η(mb,a) h ( x− 1 a + η(mb,a) )[ f(a)g ( 1 x ) + g(a)f ( 1 x )] dx ( a(a + η(mb,a)) η(mb,a) )2 ∫ 1 a 1 a+η(mb,a) h ( 1 a −x )[ mf(b)g ( 1 x ) + mg(b)f ( 1 x )] dx ≤ m(a,b) + a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx, where m(a,b) is given by (3.1). proof. let f,g be harmonic m-preinvex functions, we have f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ h(1 − t)f(a) + h(t)mf(b) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ h(1 − t)g(a) + h(t)mg(b). now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ r) and x1 < x2, x3 < x4, we have f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) )[ h(1 − t)g(a) + h(t)mg(b) ] +g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) )[ h(1 − t)f(a) + h(t)mf(b) ] ≤ [ h(1 − t)f(a) + h(t)mf(b) ][ h(1 − t)g(a) + h(t)mg(b) ] +f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) thus g(a)h(1 − t)f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) + mg(b)h(t)f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) +f(a)h(1 − t)g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) + mf(b))h(t)g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) ≤ [h(1 − t)]2f(a)g(a) + h(t)h(1 − t)[f(a)mg(b) + mf(b)g(a)] +m2[h(t)]2f(b)g(b) +f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) int. j. anal. appl. 16 (3) (2018) 348 integrating the above inequality with respect to t over [0, 1], we have g(a) ∫ 1 0 h(1 − t)f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt +mg(b) ∫ 1 0 h(t)f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt +f(a) ∫ 1 0 h(1 − t)g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt +mf(b) ∫ 1 0 h(t)g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt ≤ [ f(a)g(a) + mmf(b)g(b) ]∫ 1 0 [h(t)]2dt + [ f(a)mg(b) + mf(b)g(a) ]∫ 1 0 h(t)h(1 − t)dt + ∫ 1 0 f ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) g ( a(a + η(mb,a)) a + (1 − t)η(mb,a) ) dt this implies ( a(a + η(mb,a)) η(mb,a) )2 ∫ 1 a 1 a+η(mb,a) h ( x− 1 a + η(mb,a) )[ f(a)g ( 1 x ) + g(a)f ( 1 x )] dx ( a(a + η(mb,a)) η(mb,a) )2 ∫ 1 a 1 a+η(mb,a) h ( 1 a −x ) m [ f(b)g ( 1 x ) + g(b)f ( 1 x )] dx ≤ m(a,b) + a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x)g(x) x2 dx, which is the required result. � corollary 3.7. if η(my,x) = my −x, then theorem 3.3 reduces to: ( mab mb−a )2 ∫ 1 a 1 mb h ( x− 1 mb )[ f(a)g ( 1 x ) + g(a)f ( 1 x )] dx ( mab mb−a )2 ∫ 1 a 1 mb h ( 1 a −x ) m [ f(b)g ( 1 x ) + g(b)f ( 1 x )] dx ≤ m(a,b) + mab mb−a ∫ mb a f(x)g(x) x2 dx, lemma 3.1. let f : iη ⊂ r\{0}−→ r be harmonic m-preinvex function, where m ∈ (0, 1]. then f ( a(a + η(mb,a)x (2a + η(mb,a))x−a(a + η(mb,a)) ) ≤ [ f(a) + mf(b) ] [h(1 − t) + h(t)] −f(x). int. j. anal. appl. 16 (3) (2018) 349 proof. as we know that x ∈ [a,a + η(mb,a)], can be represented as x = a(a+η(mb,a)) a+(1−t)η(mb,a) , ∀t ∈ [0, 1]. thus f ( a(a + η(mb,a)x (2a + η(mb,a))x−a(a + η(mb,a)) ) = f ( a(a + η(mb,a) a + tη(mb,a) ) ≤ h(t)f(a) + mh(1 − t)f(b) = h(1 − t) [ f(a) + mf(b) ] + h(t)[f(a) + mf(b)] −[h(1 − t)f(a) + h(t)mf(b)] ≤ [ f(a) + mf(b) ] [h(1 − t) + h(t)] −f(x). � theorem 3.4. let f : iη ⊂ r \ {0} −→ r be harmonic m-preinvex function, where m ∈ (0, 1]. if f ∈ l[a,a + η(mb,a)], then 1 2h( 1 2 ) f ( 2a(a + η(mb,a)) 2a + η(mb,a) )∫ a+η(mb,a) a g(x) x2 dx ≤ 1 2 ∫ a+η(mb,a) a (m + 1)f(x)g(x) x2 dx ≤ [f(a) + mf(b)] 2 ∫ b a [ h ( (a + η(mb,a))(x−a) x(η(mb,a)) ) +h ( a((a + η(mb,a)) −x) x(η(mb,a)) )] g(x) x2 dx + m− 1 2 ∫ a+η(mb,a) a f(x)g(x) x2 dx, where g : [a,a + η(mb,a)] ⊂ r\{0} is nonnegative, integrable and satisfies g(x) = g ( a(a + η(mb,a))x [2a + η(mb,a)]x−a(a + η(mb,a)) ) , for all x ∈ [a,a + η(mb,a)]. proof. using the given fact, we have 1 2h( 1 2 ) f ( 2a(a + η(mb,a)) 2a + η(mb,a) )∫ a+η(mb,a) a g(x) x2 dx = 1 2h( 1 2 ) ∫ b a f ( 2a(a + η(mb,a))x (2a + η(mb,a))x−a(a + η(mb,a)) + a(a + η(mb,a)) ) g(x) x2 dx ≤ 1 2h( 1 2 ) ∫ b a h ( 1 2 )[ f ( a(a + η(mb,a))x (2a + η(mb,a))x−a(a + η(mb,a)) ) + mf(x) ] g(x) x2 dx int. j. anal. appl. 16 (3) (2018) 350 = 1 2 ∫ a+η(mb,a) a f ( a(a + η(mb,a))x (2a + η(mb,a))x−a(a + η(mb,a)) ) g(x) x2 dx + m 2 ∫ a+η(mb,a) a f(x)g(x) x2 dx = 1 2 ∫ a+η(mb,a) a (m + 1)f(x)g(x) x2 dx to prove the other part of the inequality, we consider 1 2 ∫ a+η(mb,a) a (m + 1)f(x)g(x) x2 dx = 1 2 ∫ b a f ( a(a + η(mb,a))x (2a + η(mb,a))x−a(a + η(mb,a)) ) g(x) x2 dx + m 2 ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ 1 2 ∫ a+η(mb,a) a [[ f(a) + mf(b) ] [h(1 − t) + h(t)] −f(x) ]g(x) x2 dx + m 2 ∫ a+η(mb,a) a f(x)g(x) x2 dx ≤ [f(a) + mf(b)] 2 ∫ b a [ h ( (a + η(mb,a))(x−a) x(η(mb,a)) ) +h ( a((a + η(mb,a)) −x) x(η(mb,a)) )] g(x) x2 dx + m− 1 2 ∫ a+η(mb,a) a f(x)g(x) x2 dx. this completes the proof. � corollary 3.8. if η(my,x) = my −x, then theorem 3.4 reduces to: 1 2h( 1 2 ) f ( 2mab a + mb )∫ mb a g(x) x2 dx ≤ 1 2 ∫ mb a (m + 1)f(x)g(x) x2 dx ≤ [f(a) + mf(b)] 2 ∫ b a [ h ( mb(x−a) x(mb−a) ) +h ( a(mb−x) x(mb−a) )] g(x) x2 dx + m− 1 2 ∫ mb a f(x)g(x) x2 dx. corollary 3.9. under the assumptions of theorem 3.4 with g(x) = 1, we have 1 2h( 1 2 ) f ( 2a(a + η(mb,a)) 2a + η(mb,a) ) ≤ a(a + η(mb,a)) η(mb,a) ∫ a+η(mb,a) a f(x) x2 dx ≤ [f(a) + mf(b)] ∫ 1 0 h(t)dt. int. j. anal. appl. 16 (3) (2018) 351 4. conclusion in this paper we have introduced and studied a new class of harmonic preinvex functions with respect to an arbitrary non-negative function h and the parameter m. it is shown that this class of harmonic m-preinvex functions is quite general, flexible and unifying one. new hermite-hadamard type inequalities are obtained. special cases of the main results are discussed. acknowledgment. authors are grateful to the rector, comsats institute of information technology, pakistan, for providing the excellent academic and research environment. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] m. u. awan, m. a. noor, v. n. mishra and k. i. noor, some characterizations of general preinvex functions, int. j. anal. appl. 15(1), 46-56. [3] m. u. awan, m. a. noor and k. i. noor, some integral inequalities using quantum calculus approach, int. j. anal. appl. 15(2)(2017), 125-137. [4] m. k. bakula, m. e. ozdemir and j. pecaric, hadamard type inequalities for m-convex and (α,m)-convex functions, j. ineq. pure appl. math., 9(4)(2008), art. 96, 12 pages. [5] a. ben-isreal and b. mond, what is invexity? j. australian math. soc., ser. b, 28(1)(1986), 1-9. [6] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publisher, dordrechet, holland, (2002). [7] s.s. dragomir and g. toader, some inequalities for m-convex functions, studia univ babes-bolyai math., 38(1993), 21-28. [8] s.s. dragomir, on some inequalities of hermite-hadamard type for m-convex functions,tamkang j. math., 33(1)(2002). [9] j. hadamard, etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann. j. math. pure appl., 58(1893), 171-215. [10] c. y. he, y. wang, b. y. xi and f. qi, hermite-hadamard type inequalities for (α,m)-ha and strongly (α,m)-ha convex functions, j. nonlinear sci. appl., 10(2017), 205c214. [11] m. a. hanson. on sufficiency of the kuhn-tucker conditions, j. math. anal. appl., 80(1981), 545-550. [12] c. hermite, sur deux limites d’une integrale difinie. mathesis, 3(1883), 82. [13] i. iscan, hermite-hadamard type inequalities for harmonically convex functions. hacet, j. math. stats., 43(6)(2014), 935-942. [14] i. a. baloch and i. iscan, some hermite-hadamard type inequalities for harmonically (s,m)-convex functions in second sense, arxiv:1604.08445 [math.ca], (2016). [15] s. r. mohan and s. k. neogy, on invex sets and preinvex functions, j. math. anal. appl., 189(1995), 901-908. [16] c. p. niculescu and l. e. persson, convex functions and their applications, springer-verlag, new york, (2006). [17] m. a. noor, variational-like inequalities, optimization, 30(1994), 323-330. [18] m. a. noor, hermite-hadamard integral inequalities for log-preinvex functions, j. math. anal. approx. theory, 2(207), 126-131. [19] m.a. noor, hadamard integral inequalities for product of two preinvex function, nonlinear anal. forum, 14(2009), 167-173. int. j. anal. appl. 16 (3) (2018) 352 [20] m.a. noor, on hadamard integral inequalities involving two log-preinvex functions, j. inequal. pure appl. math., 8(3)(2007), 1-14. [21] m. a. noor and k. i. noor, some characterization of strongly preinvex functions, j. math. anal. appl., 316(2006), 697-706. [22] m. a. noor, k. i. noor and s. iftikhar, harmonic mt-preinvex functions and integral inequalities, rad hazu math. znan. 20(2016), 51-70. [23] some new bounds of the quadrature formula of gauss-jacobi type via(p,q)-prinvex functions, appl. math. inofrm. sci. letters, 5(2)(2017), 51-56. [24] m. a. noor, th. m. rassias, k. i. noor and s. iftikhar, inequalities for coordinated harmonic preinvex functions, proceed. jangjeon math. soc. 20(4)(2017), 647-658. [25] m. a. noor, k. i. noor, m. u. awan and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b. sci. bull. series a, 77(1)(2015), 5-16. [26] m. a. noor, k. i. noor and s. iftikhar, hermite-hadamard inequalities for harmonic preinvex functions, saussurea, 6(1)(2016), 34-53. [27] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions, twms j. pure math., 7(1)(2016), 3-19. [28] m. a. noor, k. i. noor and s. iftikhar,on harmonic (h,r)-convex functions, proc. jangjeon math. soc., 19(2016). [29] m. a. noor, k. i. noor and s. iftikhar, fractional ostrowski inequalities for harmonic h-preinvex functions, facta(nis), ser. math. inform. 31(2)(2016), 417-445 [30] m. a. noor, k. i. noor and s. iftikhar, relative harmonic m-convex functions and integral inequalities, j. adv. math. stud. 10(2)(2017), 231-242. [31] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [32] r. pini, invexity and generalized convexity, optimization, 22(1991), 513-525. [33] h. n. shi and zhang, some new judgement theorems of schur geometric and schur harmonic convexities for a class of symmetric functions, j. inequal. appl., 2013(2013), art. id 527. [34] g. h. toader, some generalizations of convexity, proc. colloq. approx. optim, cluj napoca (romania), (1984), 329-338. [35] t. weir and b. mond, preinvex functions in multiple objective optimization, j. math. anal. appl., 136(1998), 29-38. [36] s. varosanec: on h-convexity, j. math. anal. appl., 326(2007), 303-311. 1. introduction 2. preliminaries 3. main results 4. conclusion references int. j. anal. appl. (2023), 21:12 received: dec. 21, 2022. 2020 mathematics subject classification. 65n06, 65n38. key words and phrases. finite cloud method; uniformly distributed clouds; boundary conditions. https://doi.org/10.28924/2291-8639-21-2023-12 © 2023 the author(s) issn: 2291-8639 1 an improved finite cloud method with uniformly distributed clouds and enhanced boundary conditions miew leng oh, see pheng hang* department of mathematical sciences, faculty of science, universiti teknologi malaysia, 81300 johor bahru, johor, malaysia *corresponding author: sphang@utm.my abstract. finite cloud method (fcm) employs the fixed kernel reproducing technique to construct the interpolation function and point collocation approach is adopted for the discretization. in this study, an improved fcm is proposed such that a node of interest is approximated with its nearest cloud. this feature enables a set of uniformly distributed clouds of various densities such that all the information in the problem domain is captured and stored in the clouds. additionally, the instability of fcm near the boundaries is treated by having the boundary nodes also satisfy the governing differential equation. besides, a splitting mechanism is suggested for the node refinement to improve the accuracy of solution. parameters are introduced to control the density of clouds and the singularity of the moment matrices associated with the clouds. thus, a more controllable numerical simulation is developed. numerical examples are presented and the results have shown that the improved fcm produces a stable and better accuracy of solution. 1. introduction meshfree methods have been studied intensively by researchers in the field of engineering. meshfree methods ([1], [2], [3]) become one of the hottest topics in researchers’ eyes, owing to the fact that meshfree methods possess an attractive advantage over conventional mesh-based methods, which is that the mesh generation is not required in the formulation procedure. hence, handling https://doi.org/10.28924/2291-8639-21-2023-12 2 int. j. anal. appl. (2023), 21:12 problems such as fracture, large deformation and moving boundaries is easier for meshfree methods since adding nodes to the affected area is much simpler, compared to mesh-based methods. there are numerous meshfree methods available in the literature. the element-free galerkin method (efg) ([4], [5]) employs the moving least square (mls) approximation to construct the interpolation function based on nodes in a local domain. for meshless local petrov-galerkin method (mlpg) ([6], [7], [8]), a method that solves local weak form of pdes, the weighted residual method is implemented in integral form. the integration is confined to a small local subdomain of a particular node. the random differential quadrature method (rdq) ([9]) is a meshfree method that combined fixed reproducing kernel particle method (fixed rkpm) and differential quadrature method (dqm) ([10], [11], [12]) motivated by having dqm applied to irregular domain with randomly distributed field nodes. recently, the improved interpolating element-free galerkin (iiefg) method has emerged to be another outstanding meshfree method that has made improvements to the interpolation accuracy. iiefg employs the improved interpolating moving least-square technique to construct the interpolation functions which possess the kronecker delta property and the related studies ([13], [14]) have shown that this method can achieve better accuracy of solution compared to efg method. on the other hand, researchers and scientists also find ways to improve the existing meshbased methods in meshfree directions. the hybrid finite element-meshfree method ([15], [16]), the meshfree finite volume method (fvm) ([17]) and the meshfree finite difference method ([18]). there is also an innovative approach being suggested to solve the mesh nodes and the meshfree points which are arbitrarily mixed in the computational domain using fvm ([19], [20]). meshfree methods allow a more flexible way of adding new nodes while keeping the existing field nodes because the connection information among the field nodes is not needed. hence, this would be an added advantage for node refinement since new field nodes can be added to the critical region where the detailed analysis is required. therefore, some researchers have devoted their effort to coupling a meshfree method with a mesh-based method ([21]) such that node refinement is carried out in the critical region by using the meshfree method whereas the mesh-based method is adopted in the smooth region ([22]). fcm is a meshfree method that employs fixed rkpm for the construction of interpolation functions and then adopts the point collocation approach to discretize the governing differential equation. fcm has been applied in various fields, like computer-aided design ([23], [24], [25]) and 3 int. j. anal. appl. (2023), 21:12 the simulation of the behavior of hydrogel ([26]). several papers related to some improvements to fcm are found in the literature ([27], [28], [29], [30]). however, the moment matrix associated with a cloud may become singular if the cloud size is not large enough to have a sufficient number of field nodes or the cloud center is too near the boundary such that it becomes incomplete and consequently have fewer field nodes for its interpolation. in order to have a more controllable numerical simulation, parameters are introduced to cater for the cloud distribution. more importantly, the cloud density can be adjusted when necessary or even during the node refinement phase to obtain the desired accuracy of solutions. in this paper, parameters are introduced to control the density of clouds as well as the distance of the clouds, which are adjacent to the boundaries, from the boundaries. a study is carried out to investigate the effect of the distance of the cloud centers from the boundaries to the singularity of the moment matrices associated with these clouds. apart from that, the relationship between the density of clouds and the accuracy of solution is also examined. 2. finite cloud method (fcm) fcm is a meshfree method that employs fixed reproducing kernel technique to construct the interpolation functions using a set of overlapping clouds defined over the problem domain. each cloud consists of a set of field nodes. as clouds are overlapping, a field node may belong to more than one cloud. with the appropriate order of polynomial basis, a moment matrix is created for each cloud defined in the domain. when the moment matrices are ready, the interpolation functions will then be constructed for the clouds. for the discretization of the governing partial differential equations, the diffuse derivative approach is adopted to approximate the terms of the derivatives. since the moment matrices in fixed rpkm are constants, the derivatives of the interpolation functions are straightforward, that is the differentiations involve only the polynomial basis of the interpolation functions. 2.1. fixed reproducing kernel particle method rkpm is inherited from smoothed particle hydrodynamics (sph) method where an extra term, the correction function, is added to the interpolation function ([31]) to achieve a higher order of reproducibility of the field variables. the rkpm is given as 4 int. j. anal. appl. (2023), 21:12 𝑢𝑎(𝑥,𝑦) = ∫ 𝐶(𝑥,𝑦,𝑠,𝑡) 𝜓(𝑥 − 𝑠,𝑦 − 𝑡) 𝑢(𝑠,𝑡) 𝑑𝑠 𝑑𝑡 ω . (2.1) ω ∈ ℝ2. 𝐶(𝑥,𝑦,𝑠,𝑡) is the correction function whereas 𝜓(𝑥,𝑦,𝑠,𝑡) is the kernel function or window function which acts as the low pass filter that reduces noise in the solution. the discrete form of rkpm is given as 𝑢𝑎 (𝑥,𝑦) = ∑𝑁𝐼(𝑥,𝑦) 𝑢𝐼 𝑁𝑃 𝐼=1 , (2.2) where 𝑁𝐼(𝑥,𝑦) = 𝐶(𝑥,𝑦,𝑥𝐼,𝑦𝐼) 𝜓(𝑥 − 𝑥𝐼,𝑦 − 𝑦𝐼) 𝑑𝑉i, 𝑢𝐼 is the nodal unknown for field node i and np is the total number of field nodes in the domain ω. fixed rkpm is a special case of rkpm where the kernel is fixed at the central node (𝑥𝑘,𝑦𝑘) and hence produces a constant moment matrix, unlike classical rpkm, moving rpkm and multiple rpkm which generate moment matrices with entries comprised of functions of x and y. the approximate solution for the fixed rkpm is as follows: 𝑢𝑎(𝑥,𝑦) = ∫ 𝐶(𝑥,𝑦,𝑠,𝑡) 𝜓(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑢(𝑠,𝑡) 𝑑𝑠 𝑑𝑡 ω . (2.3) (𝑥𝑘,𝑦𝑘) is the centre of the kernel 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) where the local interpolation takes place. 2.2. polynomial basis function several types of basis functions are available in the literature, like polynomial basis, fourier bases and b-spline basis function. for fcm, the relatively simpler basis functions, i.e. polynomial basis function, is employed (as shown in equation 2.4). 𝑃𝑇(𝑠,𝑡) = {𝑝1,𝑝2,…,𝑝𝑚} (2.4) 𝑃𝑇(𝑠,𝑡) is the basis function of basis degree d. the two-dimension basis function of order m can be computed by equation 2.5. 𝑚 = (𝐷 + 1) ∗ (𝐷 + 2) 2 (2.5) if d = 1, then m = 3 and the corresponding linear basis is 𝑃𝑇(𝑠,𝑡) = [1,𝑠,𝑡]. (2.6) if d = 2, then m = 6. this leads to a quadratic basis given by 5 int. j. anal. appl. (2023), 21:12 𝑃𝑇(𝑠,𝑡) = [1,𝑠,𝑡,𝑠2,𝑠𝑡,𝑡2]. (2.7) 3. formulation of fcm in fcm, the interpolation functions are constructed by fixed rkpm where its derivatives are easy to derive due to the constant moment matrix associated with each cloud ([32]). then, jin et al. ([28]) have investigated the fcm and proposed the use of shifted polynomial basis in the construction of interpolation function and have proven that the improved fcm produces superior convergence of solutions. the approximate equation derived by the fcm is presented in this section. the fixed rkpm interpolation function is first constructed followed by the discretization accomplished by point collocation approach. 3.1. formulation of fixed rkpm interpolation functions with shifted polynomial basis the approximate solution for the fixed rkpm is defined as 𝑢𝑎(𝑥,𝑦) = ∫ 𝐶(𝑥,𝑦,𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) ψ(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑢(𝑠,𝑡) 𝑑𝑠 𝑑𝑡 ω . (3.1) 𝐶(𝑥,𝑦,𝑥𝐾 − 𝑠,𝑦𝐾 − 𝑡) is the correction function as follow: 𝐶(𝑥,𝑦,𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) = 𝑃 𝑇(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑐(𝑥,𝑦), (3.2) where 𝑃𝑇(𝑠,𝑡) is the basis function as shown in equation 2.4 and 𝑐(𝑥,𝑦) is the unknown correction function coefficients. from equation 3.2, equation 2.3 becomes 𝑢𝑎(𝑥,𝑦) = ∫ 𝑃𝑇(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑐(𝑥,𝑦)𝜓(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑢(𝑠,𝑡) 𝑑𝑠 𝑑𝑡 ω . (3.3) since every monomial of the polynomial defined in equation 2.4 has to satisfy the consistency condition of the approximate function, the following equation is obtained: 𝑝𝑖(𝑥,𝑦) = ∫ 𝑃 𝑇(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑐(𝑥,𝑦) 𝜓(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑝𝑖(𝑠,𝑡)𝑑𝑠 𝑑𝑡, 𝑖 = 1,2, . . ,𝑚 ω (3.4) and the relevant discretized form is rewritten as 6 int. j. anal. appl. (2023), 21:12 𝑝𝑖(𝑥,𝑦) = ∑ 𝑃 𝑇 (𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) 𝑐(𝑥,𝑦) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) 𝑝𝑖 (𝑥𝐼,𝑦𝐼) 𝑑𝑉𝐼 𝑁𝑃 𝐼=1 (3.5) where 𝑖 = 1,2, . . ,𝑚. np denotes the total field nodes in the cloud and 𝑑𝑉𝐼 is the nodal volume at the 𝐼𝑡ℎfield node. let m be the m x m moment matrix given as 𝑀𝑖𝑗 = ∑ 𝑝𝑗(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) 𝑝𝑖(𝑥𝐼,𝑦𝐼) 𝑑𝑉𝐼 𝑁𝑃 𝐼=1 , 𝑖, 𝑗 = 1,2,…,𝑚. (3.6) then, the equation 3.6 can be written in a matrix form as 𝑀𝑐(𝑥,𝑦) = 𝑃(𝑥,𝑦). (3.7) note that the moment matrix m is not function of x and y. in another words, m is a constant matrix. from equation 3.7, the unknown correction function coefficients can be determined as 𝑐(𝑐,𝑦) = 𝑀−1 𝑃(𝑥,𝑦). (3.8) substituting equation 3.8 into equation 2.3 gives 𝑢𝑎(𝑥,𝑦) = ∫ 𝑃𝑇(𝑥,𝑦) 𝑀−𝑇 𝑃(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡)𝜓(𝑥𝑘 − 𝑠,𝑦𝑘 − 𝑡) 𝑢(𝑠,𝑡) 𝑑𝑠 𝑑𝑡 ω (3.9) and the corresponding discrete form is 𝑢𝑎(𝑥,𝑦) = ∑𝑁𝐼(𝑥,𝑦)𝑢𝐼 𝑁𝑃 𝐼=1 , (3.10) where the interpolation function for node i is 𝑁𝐼(𝑥,𝑦) = 𝑃 𝑇(𝑥,𝑦) 𝑀−𝑇 𝑃(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼. (3.11) 3.2. derivatives of interpolation functions due to the fact that the moment matrix (as in equation 3.6) is constant, the derivatives of the interpolation functions involve only the differentiations of the polynomial basis. for quadratic polynomial basis in two dimensions, m = 6, 𝑃𝑇(𝑥,𝑦) = [1 𝑥 𝑦 𝑥2 𝑥𝑦 𝑦2], then the derivatives of the interpolation functions (equation 3.11) are 𝑁𝐼, 𝑥 (𝑥,𝑦) = [0 1 0 2𝑥 𝑦 0] 𝑀 −1𝑃(𝑥𝐼,𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼 (3.12) 7 int. j. anal. appl. (2023), 21:12 𝑁𝐼, 𝑥𝑥 (𝑥,𝑦) = [0 0 0 2 0 0] 𝑀 −1𝑃(𝑥𝐼,𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼 (3.13) 𝑁𝐼, 𝑦 (𝑥,𝑦) = [0 0 1 0 𝑥 2𝑦] 𝑀 −1𝑃(𝑥𝐼,𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼 (3.14) 𝑁𝐼, 𝑦𝑦 (𝑥,𝑦) = [0 0 0 0 0 2] 𝑀 −1𝑃(𝑥𝐼,𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼 (3.15) 𝑁𝐼, 𝑥𝑦 (𝑥,𝑦) = [0 0 0 0 1 0] 𝑀 −1𝑃(𝑥𝐼,𝑦𝐼) 𝜓(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) ∆𝑉𝐼. (3.16) 3.3. formulation of point collocation method (pcm) in fcm, the point collocation approach is adopted to discretize the governing equation with a set of collocation nodes, then the approximate solution is determined from the conditions that the governing equation is satisfied at the collocation nodes. in addition, the point collocation technique is capable of enforcing the boundary conditions exactly. 4. the development of the improved fcm the development of the improved fcm starts with the discretization of the problem domain with a set of field nodes. then, clouds with their associated center node are also defined for the problem domain. when the two sets of nodes are ready, a moment matrix is constructed for each cloud with the field nodes located in the vicinity of the cloud. with the moment matrices, the interpolation functions for all field nodes are ready for the next stage, which is the discretization of the governing equation. in this study, the set of nodes that are used to generate the set of linear system of equations, namely collocation nodes, includes the internal nodes and the boundary nodes. by point collocation scheme with boundary conditions, a set of linear system of equations is established and the nodal parameters are computed by solving the system of equations, and thus the approximate solutions are obtained. 4.1. discretisation of problem domain consider a simple problem model of size 𝐿𝑥 × 𝐿𝑦. the problem domain is divided into 𝑁𝑖𝑛𝑡 cells where 𝑁𝑖𝑛𝑡 = 𝐼𝑛𝑡𝑋 × 𝐼𝑛𝑡𝑌. 𝐼𝑛𝑡𝑋 is the number of intervals along the x-axis whereas 𝐼𝑛𝑡𝑌 is the number of intervals along the y-axis. a field node is placed at the center of each cell (figure 1(a)). alternatively, field nodes can be placed randomly in the domain as depicted in figure 1(b). note that the field nodes in this study do not include boundary nodes. 8 int. j. anal. appl. (2023), 21:12 4.2. cloud definition and cloud distribution in this work, the cloud centers are independent of collocation nodes and distributed uniformly over the problem domain as shown in figure 1(c). these clouds are overlapping and can be of a degree of density. however, the denser the cloud distribution, the computation time required will be increased as the total number of clouds will increase too. since a collocation node (𝑥𝐶,𝑦𝐶) ≠ (𝑥𝑘,𝑦𝑘) where (𝑥𝑘,𝑦𝑘) is a cloud center, the computation work to determine the nearest cloud center for each collocation node has to be carried out ahead of time. under this circumstance, a moment matrix associated with a cloud may be used more than once for the approximation of solutions. thus, the computation time required can be optimized by adjusting the number of clouds defined in the problem domain. in figure 1(d), the cloud, 𝐶𝑙𝑑1, is centred at (𝑥𝑘,𝑦𝑘) and the field nodes covered by the 𝐶𝑙𝑑1 are 𝐷𝑃1,𝐷𝑃2,𝐷𝑃3,𝐷𝑃4,𝐷𝑃5,𝐷𝑃6 and 𝐷𝑃7. note that 𝐷𝑃1is the field node that is shared among 𝐶𝑙𝑑1, 𝐶𝑙𝑑2 and 𝐶𝑙𝑑3. (a) (b) (c) (d) figure 1. (a) & (b) discretization of problem domain (c) cloud distribution (d) definition of cloud 9 int. j. anal. appl. (2023), 21:12 having overlapping clouds is one of the features or advantages of fcm, due to their localized interpolation domain as well as the independence of the moment matrix of one cloud from the others. note that a moment matrix is generated for one cloud. besides, the interpolation functions constructed by fixed reproducing kernel technique are satisfying the sum to unity property. considering the consistency condition again. when ℓ = 1, 𝑝1(𝑥,𝑦) = 1.0, then substituting them into equation 3.5 gives ∑𝑃𝑇(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼)∁(𝑥,𝑦) 𝜙(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) δ𝑉𝐼 = 1.0 𝑁𝑃 i=1 . (4.1) compare with the interpolation function defined in equation 3.11 and we conclude that ∑𝑁𝐼(𝑥,𝑦) 𝑁𝑃 𝐼=1 = 1.0 (4.2) theoretically, since the node of interest (𝑥,𝑦) can be at any point in the domain, the clouds can be centered at any location, as long as the node of interest is in the vicinity of the cloud. however, this may lead to two minor problems: • multivalued interpolation functions • singularity of moment matrices for the clouds which are near the boundaries 4.3. multivalued interpolation functions according to aluru and li ([32]), the multivalued interpolation functions can be avoided by centering the kernel at the node of interest. in another word, the node of interest (𝑥,𝑦) is also the cloud center node (𝑥𝑘,𝑦𝑘). this means that a list of nodes of interest or collocation nodes should be defined ahead of time before constructing the interpolation functions. the more flexible way of defining cloud centers is by employing the improved fcm ([22]) so that with the adjustments to the interpolation functions, the cloud center nodes can be predefined and distributed uniformly over the problem domain. the approximate solution of any node of interest is computed with the adjusted interpolation functions of its nearest cloud. as a result, after the approximate solution is obtained, any new node of interest does not involve the generation of the 10 int. j. anal. appl. (2023), 21:12 moment matrix and the construction of the interpolation functions for the new cloud, one only needs to find the nearest cloud to the node of interest for the construction of the corresponding interpolation functions to compute the approximate solution. 4.4. singularity of moment matrices a moment matrix may become singular if its associated cloud does not have a sufficient number of local support nodes. there are two reasons why a cloud might become singular. one of them is the size of the cloud is not large enough to enclose a sufficient number of field nodes. another reason is the cloud is located near the boundaries where the cloud becomes incomplete. hence, parameters are introduced to control the density of the clouds as well as the distance of clouds away from the boundaries. for a better illustration, clouds of circle shape are used (as shown in figure 2(a)). 𝛽𝑅𝑥 and 𝛽𝑅𝑦 are the distance between the boundaries of two adjacent overlapping clouds in 𝑥-direction and 𝑦-direction respectively, where 𝛽 is the coefficient of overlapping of clouds and 𝑅𝑥 and 𝑅𝑦 are the radius of clouds in 𝑥 and 𝑦 directions. 𝜆𝑥 and 𝜆𝑦 are the distances between a boundary and the center node of the cloud which are near the boundary in 𝑥 and 𝑦 direction. 𝜇𝑥 and 𝜇𝑦 represent the distances between two adjacent clouds in 𝑥 and 𝑦 direction and are given by 𝜇𝑥 = 2𝑅𝑥 − 𝛽𝑅𝑥 𝜇𝑦 = 2𝑅𝑦 − 𝛽𝑅𝑦. (4.3) let 𝑁𝐶𝑙𝑑𝑥 and 𝑁𝐶𝑙𝑑𝑦 be the unknown number of clouds defined in both directions. with the given 𝜆𝑥, 𝜆𝑦, 𝛽𝑥, 𝛽𝑦, 𝑁𝐶𝑙𝑑𝑥 and 𝑁𝐶𝑙𝑑𝑦 can be computed as follow: 𝑁𝐶𝑙𝑑𝑥 = 𝐿𝑥 − 2𝜆𝑥 𝑅𝑥(2 − 𝛽𝑥) + 1 𝑁𝐶𝑙𝑑𝑦 = 𝐿𝑦 − 2𝜆𝑦 𝑅𝑦(2 − 𝛽𝑦) + 1. (4.4) 𝑁𝐶𝑙𝑑𝑥 and 𝑁𝐶𝑙𝑑𝑦 are the smallest integers of the rhs of equation 4.4. thus, the total number of clouds defined in the domain is 𝑁𝑐𝑙𝑑 = 𝑁𝐶𝑙𝑑𝑥 ∗ 𝑁𝐶𝑙𝑑𝑦 (4.5) 11 int. j. anal. appl. (2023), 21:12 and the clouds are distributed as depicted in figure 2(b). (a) (b) figure 2. (a) cloud density by parameters (b) cloud distribution note that clouds, generally called local domains in other meshfree methods, are independent of nodes of interest or solution points. since each cloud has its own set of interpolation functions, thus any node of interest, which is in the vicinity of a cloud, can utilize the corresponding interpolation functions to compute its approximate solution. however, in this study, the nearest cloud will be chosen for its interpolation for the reason of only the neighbouring field nodes are included for a better approximation of solution. additionally, by having uniformly distributed clouds, the information carried by all field nodes is captured and stored in the clouds and these clouds are then be used to establish the set of system of equations 4.5. discretization with point collocation approach and enhancement at the boundary in this study, a two-dimensional model problem is considered. the model consists of the governing differential equation that acts on the problem domain ω surrounded by the boundary γ𝐷 and γ𝑁 which satisfy the dirichlet boundary condition and the neumann boundary condition respectively (as in equation 4.6-4.8): ℒ𝑢𝑎(𝑥𝐼,𝑦𝐼) = 𝑓(𝑥𝐼,𝑦𝐼), 𝐼 = 1,2,…,𝑁𝑖𝑛𝑡 in 𝛺 (4.6) 𝑢𝑎(𝑥𝐼,𝑦𝐼) = 𝑔(𝑥𝐼,𝑦𝐼), 𝐼 = 1,2,…,𝑁𝐷 on 𝛤𝐷 (4.7) 𝜕𝑢𝑎 𝜕𝑛 (𝑥𝐼,𝑦𝐼) = ℎ(𝑥𝐼,𝑦𝐼), 𝐼 = 1,2,…,𝑁𝑁 on 𝛤𝑁. (4.8) ℒ is the differential operator and 𝑢𝑎 is the approximate solution. 12 int. j. anal. appl. (2023), 21:12 in point collocation method, the collocation nodes are comprised of 𝑁𝑖𝑛𝑡 internal field nodes, 𝑁𝐷 dirichlet boundary nodes and 𝑁𝑁 neumann boundary nodes. note that only the 𝑁𝑖𝑛𝑡 internal field nodes are satisfying the governing equation. the 𝑁𝐷 and 𝑁𝑁 boundary nodes are satisfying their relevant boundary condition only. hence, an approach to strengthen the ties between the governing equation and the 𝑁𝐷 + 𝑁𝑁 boundary nodes is suggested, that is to have additional equations for boundary nodes to ensure that the boundary nodes are also satisfying the governing differential equation (equation 4.6). ℒ𝑢𝑎(𝑥𝐼,𝑦𝐼) = 𝑓(𝑥𝐼,𝑦𝐼), 𝐼 = 1,2,…,𝑁𝐷 + 𝑁𝑁. (4.9) as a result, the modified system of equations established by equation 4.6-4.9 in matrix form may consist of 𝑁𝑖𝑛𝑡 + 2(𝑁𝐷 + 𝑁𝑁) rows and 𝑁𝑖𝑛𝑡 columns, which is an over-determined system of linear equations as shown below: 𝑅𝑀×𝑁 ′ 𝑢𝑁×1 ′ = 𝑏𝑀×1 ′ , (4.10) where m is the total number of rows and n is the total number of columns of the final stiffness matrix. by applying the least square approach to the system of linear equations as shown in equation 4.10, the following is obtained: (𝑅′)𝑁×𝑀 𝑇 𝑅𝑀×𝑁 ′ 𝑢𝑁×1 ′ = (𝑅′)𝑁×𝑀 𝑇 𝑏𝑀×1 ′ (4.11) after solving the final stiffness matrix, the unknown 𝑢′ is determined. at this point, the approximate solution 𝑢𝑎 for any node (𝑥,𝑦) in the problem domain can be computed with equation 3.10 by using the moment matrix of the nearest cloud in the problem domain. 4.6. node refinement the accuracy of solution for fcm can be improved further with node refinement. the node refinement mechanism adopted in this work is by splitting a field node as presented in figure 4. 13 int. j. anal. appl. (2023), 21:12 figure 4. the splitting of a field node. (a) before the split. (b) after the split. the relative error for each collocation node is computed after the first approximate solution is obtained. if the relative error exceeds the given threshold, then the corresponding field nodes are marked. each of the marked field nodes is then split into 4 field nodes. the newly added field nodes are used for the numerical analysis of fcm at the node refinement stage. by generating the affected moment matrices with the new sets of field nodes and deriving the new final stiffness matrix, the new approximate solution is obtained. 5. numerical examples the fcm with proposed improvements is applied to a problem domain ω = {(𝑥,𝑦)| 0 < 𝑥 < 1, 0 < 𝑦 < 1} with the poisson equation as the governing differential equation and the given boundary conditions (as expressed in equation 5.1-5.2). 𝜕2𝑢 𝜕𝑥2 + 𝜕2𝑢 𝜕𝑦2 = 4 − 2𝜔𝛼2 𝑒 [– 𝛼 2(𝑦−𝑐)2] + 4𝜔𝛼4(𝑦 − 𝑐)2 𝑒−𝛼 2(𝑦−𝑐)2 (5.1) { 𝑢 |𝑥=0 = 𝑦 2 + 𝜔𝑒 [−𝛼 2(𝑦−𝑐)2] 𝑢|𝑥=1.0 = 1.0 + 𝑦 2 + 𝜔𝑒 [−𝛼 2(𝑦−𝑐)2] 𝑢𝑦|𝑦=0 = 2𝜔𝛼2𝑐𝑒−𝛼 2𝑐2 𝑢𝑦|𝑦=1.0 = 2.0 − 2𝜔𝛼2(1.0 − 𝑐)𝑒−𝛼 2(1.0−𝑐)2 . (5.2) then, the cubic spline kernel function is defined as 𝜙(𝑥𝑘 − 𝑥𝐼,𝑦𝑘 − 𝑦𝐼) = 1 𝑑𝑥 𝜔( 𝑥𝑘 − 𝑥𝐼 𝑑𝑥 ) 1 𝑑𝑦 𝜔( 𝑦𝑘 − 𝑦𝐼 𝑑𝑦 ), (5.3) 14 int. j. anal. appl. (2023), 21:12 where 𝜔(𝓏) = { 0, 𝓏 < −2 1 6 (𝓏 + 2)3, −2 ≤ 𝓏 ≤ −1, 2 3 − 𝓏2 (1 + 𝓏 2 ) , −1 ≤ 𝓏 ≤ 0, 2 3 − 𝓏2 (1 − 𝓏 2 ) , 0 ≤ 𝓏 ≤ 1, − 1 6 (𝓏 − 2)3, 1 ≤ 𝓏 ≤ 2, 0, 𝓏 > 2 (5.4) and 𝑧𝐼 = 𝑥𝑘−𝑥𝐼 𝑑𝑥 or 𝑦𝑘−𝑦𝐼 𝑑𝑦 and 𝑑𝑥 and 𝑑𝑦 is the cloud size in x and y direction respectively. the exact solution for the poisson equation is given by 𝑢(𝑥,𝑦) = 𝑥2 + 𝑦2 + 𝜔𝑒 [−𝛼 2(𝑦−𝑐)2] (5.5) and the global error ([32]) are computed as 𝜀 = 1 |𝑢𝑒|𝑚𝑎𝑥 √ 1 𝑁𝐶 ∑ [𝑢𝐼 𝑒 − 𝑢𝐼 𝑎]2𝑁𝐶𝐼=1 , (5.6) where 𝑁𝐶 is the total number of collocation nodes, 𝑢𝑒 is the exact solution and 𝑢𝑎 is the approximate solution. in this work, we have two models. model a is of 𝛼 = 10, 𝜔 = 10 and 𝑐 = 0.5 and model b is having 𝛼 = 5, 𝜔 = 2 and 𝑐 = 0.5 . the cloud size is 2.4 ∆𝑥 and 2.4 ∆𝑦. the two models use four sets of field nodes for their analysis, i.e. 𝑁𝑃 = 144,196,256 and 324. in addition, the clouds are defined with various 𝜆 and 𝛽 and the results obtained are observed. the distance of a cloud center adjacent to a boundary from the boundary in 𝑥 and 𝑦 directions, 𝜆𝑥 and 𝜆𝑦, are used to observe the singularity of moment matrices and the results are presented in table 1. table 1. the relationship between 𝜆𝑥 or 𝜆𝑦 and the number of singular matrices observed with 𝛽𝑥 = 𝛽𝑦 = 1.5 and the cloud size = 2.4 δ𝑥. distance, λ (in ∆𝑥 𝑜𝑟 δ𝑦) 0.025 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 no. of singular moment matrix/total clouds 31/ 256 31/ 256 31/ 256 21/ 256 24/ 256 19/ 256 25/ 256 15/ 225 15/ 225 8/ 225 5/ 225 6/ 225 4/ 225 1/ 196 0/ 196 15 int. j. anal. appl. (2023), 21:12 and the relationship of clouds versus the singularity of their moment matrices are depicted in figure 5(a) and (b) as shown below: (a) (b) figure 5. clouds versus the singularity of moment matrices. (a) 𝜆𝑥 = 0.1δ𝑥 and 𝜆𝑦 = 0.1δ𝑦. (b) 𝜆𝑥 = 1.0δ𝑥 and 𝜆𝑦 = 1.0δ𝑦. from figure 5, it is clearly shown that the larger the distance the cloud center, which is adjacent to the boundary, away from the boundary, its associated moment matrix is less likely to be singular. from table 1, we can conclude that when the value of 𝜆 increases, the number of singular matrices will be reduced due to the fact that the cloud adjacent to the boundary could include more field nodes causing its associated moment matrix becomes less likely to be singular. however, if a cloud center which is adjacent to a boundary is too far away from a boundary could cause the information of field nodes which are near the boundaries are not captured into the system of equations and hence lead to less accurate solution. therefore, it is important to choose a suitable 𝜆 for our numerical model. in this work, we have 𝜆𝑥 = 0.1δ𝑥 and 𝜆𝑦 = 0.1δ𝑦. for cloud distribution, we adopted several 𝛽 values and make comparison of the computed solutions. the cloud distribution of 𝛽 = 1.19, 1.51 and 1.80 and the number of clouds defined are 100, 256 and 1444 (as shown in figure 6). numerical results are computed for several 𝛽s and global errors, 𝜖, are obtained using equation 5.6. graphs are plotted to describe the relationships between the global errors and the cloud density, 𝛽 (as shown in figure 7(a) for model a and figure 7(b) and 7(c) are for model b). 16 int. j. anal. appl. (2023), 21:12 (a) (b) (c) figure 6. cloud distribution. (a) 𝛽 = 1.19 (b) 𝛽 = 1.51 (c) 𝛽 = 1.80 (a) (b) (c) figure 7. the global error for fcm with various β. (a) model a (b) & (c) model b 17 int. j. anal. appl. (2023), 21:12 figure 7(c) is the enlarged version of figure 7(b) for model b. from the numerical results, it is obvious that the computed solutions are more stable and converge slowly when 𝛽 increases with more clouds being defined in the domain. for model a which exhibits high gradient solution, the global error tends to become stable as the number of field nodes increases and the density of cloud may not play a significant role in the performance of the model. in this case, we may consider increasing the number of field nodes. then, we may have larger size of clouds than 2.4∆𝑥 or 2.4∆𝑦 or denser clouds to capture more details from the high gradient region. on the contrary, modal b has a smoother gradient of solution and the numerical results have shown less oscillation and converged slowly when 𝛽 approaches 1.9. the numerical results prove that the more clouds are defined, the better the accuracy of solutions. this is because a collocation node could find the nearest cloud which contains the information of its nearest field nodes while retaining the reusability of moment matrices. note that a cloud can be used more than once for approximating solutions. this feature provides the flexibility of allocating available resources while obtaining the desired accuracy of solution. in addition, the results also have shown that the shorter the distance between the cloud center and the node of interest or collocation node, the better the accuracy of solution as the longest distance between a node of interest and its associated nearest cloud center for a 𝛽 cloud distribution, will decrease as 𝛽 increases. additionally, the numerical model is modified to adopt the enhancement at the boundaries. for model b with 𝛽 = 1.8, the numerical results obtained are analyzed and presented in figure 8. figure 8 has shown that the numerical model with enhancement at the boundaries produces a smoother approximate solution compared with the solution computed without enhancement at the boundary. the numerical results have proven that the enhancement at the boundaries can lead to a less oscillated solution. 18 int. j. anal. appl. (2023), 21:12 figure 8. the global errors for fcm with and without the enhancement at the boundary (np=144, 196, 256, 324 and 400) besides the enhancement at the boundaries, the approximate solution can be improved further with node refinement process. by employing the node refinement mechanism suggested in section 4.6, after going through the splitting process at the threshold = |𝑢𝑎−𝑢𝑒| 𝑢𝑒 , where 𝑢𝑎 is the approximate solution and 𝑢𝑒 is the exact solution, the initial set of field nodes and the new set of field nodes during the node refinement stage are depicted in figure 9(a) and 9(b) respectively. (a) (b) figure 9. (a) field nodes at initial stage (𝑁𝑃 = 256). (b) field nodes at node refinement stage (𝑁𝑃 = 604) the numerical model with enhancement to the boundaries proceeds to node refinement stage and the results obtained are analyzed and presented in table 2. 19 int. j. anal. appl. (2023), 21:12 table 2. the global errors for the boundary enhancement model b (with 𝛼 = 8.0,𝜔 = 2.0) before and after the node refinement process. global error after the enhancement at the boundaries but before the node refinement global error after the enhancement at the boundaries and node refinement 𝑁𝑃 = 144 , 𝜀 = 0.3731 𝑁𝑃 = 321 , 𝜀 = 0.3570 𝑁𝑃 = 196 , 𝜀 = 0.3197 𝑁𝑃 = 370 , 𝜀 = 0.3429 𝑁𝑃 = 256 , 𝜀 = 0.3494 𝑁𝑃 = 604 , 𝜀 = 0.2320 𝑁𝑃 = 324 , 𝜀 = 0.2971 𝑁𝑃 = 609 , 𝜀 = 0.2946 𝑁𝑃 = 400 , 𝜀 = 0.2819 𝑁𝑃 = 763 , 𝜀 = 0.2499 the numerical results have shown improvement in the accuracy of the approximate solution obtained after the node refinement stage (except for 𝑁𝑃 = 196). hence, the improvements for fcm suggested in this research are effective in producing a better accuracy of solution. 6. conclusion in this paper, we have introduced a way of defining uniformly distributed cloud using parameters 𝜆 and 𝛽. each collocation node is assigned to its nearest cloud for the computation of its approximate solution. we have studied the role of 𝜆 and 𝛽 in detail by using a numerical example. the numerical results reveal the relationships of 𝜆 and 𝛽 and the accuracy of solution. in addition, the suggested enhancement at the boundaries and the node refinement mechanism are also proven can produce a more stable and accurate of approximate solution. acknowledgement: the authors wish to thank the malaysian ministry of education (moe), universiti teknologi malaysia (utm) and research management centre (rmc) for financial sponsorship to this work through grant funding number r.j130000.2654.17j37 utm tier 2. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.s. chen, m. hillman, s.w. chi, meshfree methods: progress made after 20 years, j. eng. mech. 143 (2017), 04017001. https://doi.org/10.1061/(asce)em.1943-7889.0001176. https://doi.org/10.1061/(asce)em.1943-7889.0001176 20 int. j. anal. appl. (2023), 21:12 [2] s.d. daxini, j.m. prajapati, a review on recent contribution of meshfree methods to structure and fracture mechanics applications, sci. world j. 2014 (2014), 247172. https://doi.org/10.1155/2014/247172. [3] s. garg, m. pant, meshfree methods: a comprehensive review of applications, int. j. comput. methods. 15 (2018), 1830001. https://doi.org/10.1142/s0219876218300015. [4] q. duan, x. gao, b. wang, x. li, h. zhang, t. belytschko, y. shao, consistent element‐free galerkin method, int. j. numer. methods eng. 99 (2014), 79-101. https://doi.org/10.1002/nme.4661. [5] q. duan, t. belytschko, gradient and dilatational stabilizations for stress‐point integration in the element‐free galerkin method, int. j. numer. methods eng. 77 (2009), 776-798. https://doi.org/10.1002/nme.2432. [6] m. najafi, v. enjilela, natural convection heat transfer at high rayleigh numbers–extended meshless local petrov–galerkin (mlpg) primitive variable method, eng. anal. bound. elements. 44 (2014), 170184. https://doi.org/10.1016/j.enganabound.2014.01.022. [7] r. singh, k.m. singh, interpolating meshless local petrov-galerkin method for steady state heat conduction problem, eng. anal. bound. elements. 101 (2019), 56-66. https://doi.org/10.1016/j.enganabound.2018.12.012. [8] r. divya, v. sriram, k. murali, wave-vegetation interaction using improved meshless local petrov galerkin method, appl. ocean res. 101 (2020), 102116. https://doi.org/10.1016/j.apor.2020.102116. [9] s.s. mulay, h. li, s. see, on the development of adaptive random differential quadrature method with an error recovery technique and its application in the locally high gradient problems, comput. mech. 45 (2010), 467-493. https://doi.org/10.1007/s00466-010-0468-2. [10] c.w. bert, m. malik, differential quadrature method in computational mechanics: a review, appl. mech. rev. 49 (1996), 1–28. https://doi.org/10.1115/1.3101882. [11] x. liang, t. wang, d. huang, z. liu, r. zhu, c. wang, an improved rbf based differential quadrature method, eng. anal. bound. elements. 135 (2022), 299–314. https://doi.org/10.1016/j.enganabound.2021.11.023. [12] t. liu, j. yu ding, x. yu xu, differential quadrature method for partial differential dynamic equations of beam–ring structure, aiaa j. 60 (2022), 2542-2554. https://doi.org/10.2514/1.j061113. [13] f.x. sun, j.f. wang, y.m. cheng, an improved interpolating element-free galerkin method for elasticity, chin. phys. b. 22 (2013), 43–50. https://doi.org/10.1088/1674-1056/22/12/120203. [14] s. wu, y. xiang, b. liu, g. li, a weak-form interpolation meshfree method for computing underwater acoustic radiation, ocean eng. 233 (2021), 109105. https://doi.org/10.1016/j.oceaneng.2021.109105. [15] s. r. idelsohn, e. onate, n. calvo, f. del pin, the meshless finite element method, int. j. numer. methods eng. 58 (2003), 893–912. https://doi.org/10.1002/nme.798. https://doi.org/10.1155/2014/247172 https://doi.org/10.1142/s0219876218300015 https://doi.org/10.1002/nme.4661 https://doi.org/10.1002/nme.2432 https://doi.org/10.1016/j.enganabound.2014.01.022 https://doi.org/10.1016/j.enganabound.2018.12.012 https://doi.org/10.1016/j.apor.2020.102116 https://doi.org/10.1007/s00466-010-0468-2 https://doi.org/10.1115/1.3101882 https://doi.org/10.1016/j.enganabound.2021.11.023 https://doi.org/10.2514/1.j061113 https://doi.org/10.1088/1674-1056/22/12/120203 https://doi.org/10.1016/j.oceaneng.2021.109105 https://doi.org/10.1002/nme.798 21 int. j. anal. appl. (2023), 21:12 [16] y. chai, c. cheng, w. li, y. huang, a hybrid finite element-meshfree method based on partition of unity for transient wave propagation problems in homogeneous and inhomogeneous media, appl. math. model. 85 (2020), 192–209. https://doi.org/10.1016/j.apm.2020.03.026. [17] h. li, q. zhang, a meshfree finite volume method with optimal numerical integration and direct imposition of essential boundary conditions, appl. numer. math. 153 (2020), 98-113. https://doi.org/10.1016/j.apnum.2020.02.005. [18] i. jaworska and s. milewski, on two-scale analysis of heterogeneous materials by means of the meshless finite difference method, int. j. multiscale comput. eng. 14 (2016), 113–134. https://doi.org/10.1615/intjmultcompeng.2016014435. [19] y. jiang, algebraic-volume meshfree method for application in finite volume solver, comput. methods appl. mech. eng. 355 (2019), 44–66. https://doi.org/10.1016/j.cma.2019.05.048. [20] y. jiang, general mesh method: a unified numerical scheme, comput. methods appl. mech. eng. 369 (2020), 1–28. https://doi.org/10.1016/j.cma.2020.113049. [21] y. gu, l.c. zhang, coupling of the meshfree and finite element methods for determination of the crack tip fields, eng. fracture mech. 75 (2008), 986-1004. https://doi.org/10.1016/j.engfracmech.2007.05.003. [22] m. l. oh and s. h. yeak, a hybrid multiscale finite cloud method and finite volume method in solving high gradient problem, int. j. comput. methods, 19 (2022), 2250002. https://doi.org/10.1142/s0219876222500025. [23] d.r. burke, t.j. smy, optical mode solving for complex waveguides using a finite cloud method, optics express. 20 (2012), 17783-17796. https://doi.org/10.1364/oe.20.017783. [24] d.r. burke, t.j. smy, thermal models for optical circuit simulation using a finite cloud method and model reduction techniques, ieee trans. computer-aided design integrated circuits syst. 32 (2013), 1177-1186. https://doi.org/10.1109/tcad.2013.2253835. [25] d.r. burke, a meshless approach to solving partial differential equations using the finite cloud method for the purposes of computer aided design, doctoral dissertation, carleton university, 2013. [26] s.k. de, n.r. aluru, a chemo-electro-mechanical mathematical model for simulation of ph sensitive hydrogels, mech. mater. 36 (2004), 395–410. https://doi.org/10.1016/s0167-6636(03)00067-x. [27] x.z. jin, g. li, n.r. aluru, on the equivalence between least-squares and kernel approximations in meshless methods, computer model. eng. sci. 2 (2001), 447-462. [28] x.z. jin, g. li, n.r. aluru, positivity conditions in meshless collocation methods, computer methods appl. mech. eng. 193 (2004), 1171-1202. https://doi.org/10.1016/j.cma.2003.12.013. [29] x.z. jin, g. li, n.r. aluru, new approximations and collocation schemes in the finite cloud method, comput. struct. 83 (2005), 1366-85. https://doi.org/10.1016/j.compstruc.2004.08.030. https://doi.org/10.1016/j.apm.2020.03.026 https://doi.org/10.1016/j.apnum.2020.02.005 https://doi.org/10.1615/intjmultcompeng.2016014435 https://doi.org/10.1016/j.cma.2019.05.048 https://doi.org/10.1016/j.cma.2020.113049 https://doi.org/10.1016/j.engfracmech.2007.05.003 https://doi.org/10.1142/s0219876222500025 https://doi.org/10.1364/oe.20.017783 https://doi.org/10.1109/tcad.2013.2253835 https://doi.org/10.1016/s0167-6636(03)00067-x https://doi.org/10.1016/j.cma.2003.12.013 https://doi.org/10.1016/j.compstruc.2004.08.030 22 int. j. anal. appl. (2023), 21:12 [30] w.x. chan, h. son, y.j. yoon, computational efficiency of meshfree methods with local-coordinates algorithm, int. j. precis. eng. manuf. 16 (2015), 547–556. https://doi.org/10.1007/s12541-015-00745. [31] g.r. liu, y.t. gu, a point interpolation method, in: proceedings of the fourth asia-pacific conference on computational mechanics, singapore. (1999), 1009-1014. [32] n.r. aluru, g. li, finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation, int. j. numer. meth. eng. 50 (2001), 2373-2410. https://doi.org/10.1002/nme.124. https://doi.org/10.1007/s12541-015-0074-5 https://doi.org/10.1007/s12541-015-0074-5 https://doi.org/10.1002/nme.124 int. j. anal. appl. (2022), 20:45 bipolar fuzzy sublattices and ideals u. venkata kalyani1,∗, t. eswarlal1, j. kavikumar2, a. iampan3 1department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, vaddeswaram, ap, india 2department of mathematics and statistics, faculty of applied sciences and technology, universiti tun hussein onn malaysia campus, pagoh 84600, johor, malaysia 3department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand ∗corresponding author: u.v.kalyani@gmail.com abstract. in this article, we introduce and study the theory of bipolar fuzzy sublattices (bfls) and bipolar fuzzy ideals (bfis) of a lattice, and some interesting properties of these bfls and bfis are established. moreover, we study the properties of bfis under lattice homomorphisms and also an application of bfls. 1. introduction zadeh [4] introduced the concept of fuzzy sets (fss) in 1965, and it has become a thriving area of research in a variety of fields. following that, several researchers applied this concept to various algebraic structures. the fuzzy set theory has various expansions, such as vague sets (vss), intervalvalued fuzzy sets (ivfss), intuitionistic fuzzy sets (ifs) and so on. the ifs was introduced by atanassov [2] in 1986 as a generalization of the fs. in both the fs and ifs, the membership value range is in [0,1]. later, ajmal and thomas [5] specifically applied the concept of fss in lattice theory and developed the theory of fuzzy sublattices (fsls). thereafter, thomas and nair [3] introduced the concept of intuitionistic fuzzy sublattices (ifsls) in 2011. characterization of intuitionistic fuzzy ideals and filters based on lattice operations were studied by milles [11] in 2017. later, rough vague lattices were studied by rao [13] in 2019. vague lattices were introduced by rao [12] in 2020. in received: jul. 29, 2022. 2010 mathematics subject classification. 06d50, 06d72, 03e72. key words and phrases. bipolar fuzzy set; bipolar fuzzy sublattice; bipolar fuzzy ideal; homomorphism. https://doi.org/10.28924/2291-8639-20-2022-45 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-45 2 int. j. anal. appl. (2022), 20:45 2020, milles [8, 9] researched on the principal intuitionistic fuzzy ideals and filters on a lattice and the lattice of intuitionistic fuzzy topologies generated by intuitionistic fuzzy relations. zhang [10] studied intuitionistic fuzzy filters on residuated lattices. nowadays, bipolarity is playing a vital role in many areas. this has become a thriving area of research in many fields like artificial intelligence (ai), machine learning (ml) etc. lee [1] introduced the concept of bipolar fuzzy sets (bfss) in 2000, with membership values ranging from [-1,1]. eswarlal and kalyani [6,7] investigated bipolar vague cosets, homomorphism, and anti homomorphism in bipolar vague normal groups (bvngs), and used bipolar vague sets (bvss) to solve mcdm problems. in this paper, we introduce the concepts of bfls and bfis of a lattice. some interesting characterizations and properties of these bfls and bfis are established. in addition, we study the properties of bfis under lattice homomorphisms and also an application of bfls. 2. preliminaries throughout this paper, unless otherwise stated, l always represents a lattice (l,∨,∧) and l1 represents a lattice (l1,∨,∧). here, we will review a few standard definitions that are relevant to this work. definition 2.1. [4] a mapping δ : z → [0, 1] is represented as a fuzzy set (fs) in a non-empty set z. definition 2.2. [3] let δ be a fs in l. then δ is called a fuzzy sublattice (fl) of l if for all t ,k ∈ l, (i) δ(t ∨k) ≥ min{δ(t ),δ(k)}, (ii) δ(t ∧k) ≥ min{δ(t ),δ(k)}. definition 2.3. [1] suppose x is a universal set. a bipolar fuzzy set (bfs) bδ in x is an object having the form bδ = {< t ,bpδ (t ),b n δ (t ) >| t ∈ x} determined by a positive and a negative membership function, respectively, where bpδ : x → [0, 1] and b n δ : x → [−1, 0]. for convenience, the bfs bδ is denoted by bδ = (bpδ ,b n δ ). definition 2.4. [1] let bδ and bω be bfss in a non-empty set x. (i) bδ is a subset of bω, denoted by bδ ⊆ bω, if for each t ∈ x, bpδ (t ) ≤ b p ω (t ) and bnδ (t ) ≥ bnω (t ). (ii) the complement of bδ, denoted by bcδ = ((b c δ ) p , (bcδ ) n), is a bfs in x defined as: for each t ∈ x, bcδ (t ) = (1−b p δ (t ),−1−b n δ (t )), i.e., (b c δ ) p (t ) = 1−bpδ (t ) and (b c δ ) n(t ) = −1−bnδ (t ). (iii) the intersection of bδ and bω, denoted by bδ ∩ bω, is a bfs in x defined as: for each t ∈ x, (bδ ∩bω)(t ) = (bpδ (t ) ∧b p ω (t ),bnδ (t ) ∨b n ω (t )). (iv) the union of bδ and bω, denoted by bδ ∪bω, is a bfs in x defined as: for each t ∈ x, (bδ ∪ bω)(t ) = (bpδ (t ) ∨b p ω (t ),bnδ (t ) ∧b n ω (t )). int. j. anal. appl. (2022), 20:45 3 3. bipolar fuzzy sublattices and ideals in this section, we introduce and study bfls and bfis and their characterizations. theorem 3.1. let bδ = (bpδ ,b n δ ) be a bfs in l. then for all t ,k ∈ l, the following conditions are equivalent: (i) t ≤ k ⇒ (bpδ (t ) ≥ b p δ (k),b n δ (t ) ≤ b n δ (k)), (ii) bpδ (t ∧k) ≥ max{b p δ (t ),b p δ (k)},b n δ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}, (iii) bpδ (t ∨k) ≤ min{b p δ (t ),b p δ (k)},b n δ (t ∨k) ≥ max{b n δ (t ),b n δ (k)}. proof. for any t ,k ∈ l, we have t ∧k ≤t and t ∧k ≤ k. then from (i), we have bpδ (t ∧k) ≥ b p δ (t ),b n δ (t ∧k) ≤ b n δ (t ), bpδ (t ∧k) ≥ b p δ (k), and b n δ (t ∧k) ≤ b n δ (k). thus bpδ (t ∧k) ≥ max{b p δ (t ),b p δ (k)} and b n δ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}. now for any t ,k ∈ l, we have t ≤t ∨k and k ≤t ∨k, using (i) we have bpδ (t ) ≥ b p δ (t ∨ k),b p δ (k) ≥ b p δ (t ∨ k),b n δ (t ) ≤ b n δ (t ∨ k), and b n δ (k) ≤ bnδ (t ∨k). thus bpδ (t ∨k) ≤ min{b p δ (t ),b p δ (k)} and b n δ (t ∨k) ≥ max{b n δ (t ),b n δ (k)}. hence, (ii) and (iii) are valid. suppose that (ii) is true. let t ,k ∈ l be such that t ≤ k. then t ∧ k = t ⇒ bpδ (t ) = b p δ (t ∧ k) ≥ max{b p δ (t ),b p δ (k)} and b n δ (t ) = b n δ (t ∧ k) ≤ min{bnδ (t ),b n δ (k)}. thus bpδ (t ) ≥ b p δ (k) and b n δ (t ) ≤ b n δ (k). finally, suppose (iii) holds. let t ,k ∈ l be such that t ≤ k. then t ∨ k = k ⇒ bpδ (k) = b p δ (t ∧ k) ≤ min{b p δ (t ),b p δ (k)} and b n δ (t ) = b n δ (t ∨ k) ≥ max{bnδ (t ),b n δ (k)}. thus bpδ (t ) ≥ b p δ (k) and b n δ (t ) ≤ b n δ (k). hence, the proof is completed. � similar to theorem 3.1, we get the following theorem. theorem 3.2. let bδ = (bpδ ,b n δ ) be a bfs in l. then for all t ,k ∈ l, the following conditions are equivalent: (i) t ≤ k ⇒ bpδ (t ) ≤ b p δ (k),b n δ (t ) ≥ b n δ (k), (ii) bpδ (t ∧k) ≤ min{b p δ (t ),b p δ (k)},b n δ (t ∧k) ≥ max{b n δ (t ),b n δ (k)}, (iii) bpδ (t ∨k) ≥ max{b p δ (t ),b p δ (k)},b n δ (t ∨k) ≤ min{b n δ (t ),b n δ (k)}. definition 3.1. let bδ = (bpδ ,b n δ ) be a bfs in l. then bδ is called a bipolar fuzzy sublattice (bfl) of l if the following conditions are satisfied for all t ,k ∈ l, (i) bpδ (t ∨k) ≥ min{b p δ (t ),b p δ (k)}, 4 int. j. anal. appl. (2022), 20:45 (ii) bpδ (t ∧k) ≥ min{b p δ (t ),b p δ (k)}, (iii) bnδ (t ∨k) ≤ max{b n δ (t ),b n δ (k)}, (iv) bnδ (t ∧k) ≤ max{b n δ (t ),b n δ (k)}. example 3.1. consider the lattice l of "divisors of 10". then l = {1, 2, 5, 10}. let bδ = (bpδ ,b n δ ) be given by < 1, 0.6,−0.4 >,< 2, 0.1,−0.5 >,< 5, 0.3,−0.4 >,< 10, 0.2,−0.7 > . we can routinely prove that bδ is a bfl of l. definition 3.2. let bδ = (bpδ ,b n δ ) be a bfs in l. then bδ is called a bipolar fuzzy ideal (bfi) of l if the following conditions are satisfied for all t ,k ∈ l, (i) bpδ (t ∨ k) ≥ min{b p δ (t ),b p δ (k)}, (ii) b p δ (t ∧ k) ≥ max{b p δ (t ),b p δ (k)}, (iii) b n δ (t ∨ k) ≤ max{bnδ (t ),b n δ (k)}, (iv) b n δ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}. example 3.2. consider the lattice l in example 3.1. let bδ = (bpδ ,b n δ ) be given by < 1, 0.7,−0.5 >,< 2, 0.5,−0.3 >,< 5, 0.6,−0.5 >,< 10, 0.4,−0.2 > . we can routinely prove that bδ is a bfi of l. theorem 3.3. if j and m are two bfls (bfis) of a lattice l, then j ∩m is a bfl (bfi) of l. proof. let j = (δpj ,δ n j ) and m = (δ p m,δ n m) be two bfls of l. now, δpj∩m(t ∨k) = min{δ p j (t ∨k),δ p m(t ∨k)} ≥ min{min{δpj (t ),δ p m(k)}, min{δ p j (t ),δ p m(k)} = min{min{δpj (t ),δ p m(t )}, min{δ p j (k),δ p m(k)} = min{δpj∩m(t ),δ p j∩m(k)}. thus δpj∩m(t ∨k) ≥ min{δ p j∩m(t ),δ p j∩m(k)} for all t ,k ∈ l. similarly, we get δpj∩m(t ∧k) ≥ min{δ p j∩m(t ),δ p j∩m(k)} for all t ,k ∈ l. now, δnj∩m(t ∨k) = max{δ n j (t ∨k),δ n m(t ∨k)} ≤ max{max{δnj (t ),δ n m(k)}, max{δ n j (t ),δ n m(k)} = max{max{δnj (t ),δ n m(t )}, max{δ n j (k),δ n m(k)} = max{δnj∩m(t ),δ n j∩m(k)}. int. j. anal. appl. (2022), 20:45 5 thus δnj∩m(t ∨k) ≤ max{δ n j∩m(t ),δ n j∩m(k)} for all t ,k ∈ l. similarly, we get δnj∩m(t ∧k) ≤ max{δ n j∩m(t ),δ n j∩m(k)} for all t ,k ∈ l. hence, j ∩m is a bfl of l. similarly, we can prove that j ∩m is a bfi of l if j and m are bfis of l. � example 3.3. consider the lattice l given in example 3.1. if j = {< 1, 0.7,−0.3 >,< 2, 0.4,−0.4 >,< 5, 0.1,−0.3 >,< 10, 0.2,−0.5 >} and m = {< 1, 0.6,−0.4 >,< 2, 0.1,−0.5 >,< 5, 0.3,−0.4 >,< 10, 0.2,−0.7 >} are two bfls of l, then j ∩m = {< 1, 0.6,−0.3 >,< 2, 0.1,−0.4 >,< 5, 0.1,−0.3 >,< 10, 0.2,−0.5 >} is a bfl of l. example 3.4. consider the lattice l given in example 3.1. if j = {< 1, 0.7,−0.5 >,< 2, 0.5,−0.6 >,< 5, 0.6,−0.5 >,< 10, 0.5,−0.6 >} and m = {< 1, 0.7,−0.3 >,< 2, 0.4,−0.2 >,< 5, 0.2,−0.1 >,< 10, 0.2,−0.1 >} are two bfis of l, then j ∩m = {< 1, 0.7,−0.3 >,< 2, 0.4,−0.2 >,< 5, 0.2,−0.1 >,< 10, 0.2,−0.1 >} is a bfi of l. remark 3.1. the union of two bfls of a lattice l need not be a bfl. consider the lattice l given in example 3.1. if j = {< 1, 0.7,−0.3 >,< 2, 0.4,−0.4 >,< 5, 0.1,−0.3 >,< 10, 0.2,−0.5 >} and m = {< 1, 0.6,−0.4 >,< 2, 0.1,−0.5 >,< 5, 0.3,−0.4 >,< 10, 0.2,−0.7 >} are two bfls of l, then j ∪m = {< 1, 0.7,−0.4 >,< 2, 0.4,−0.5 >,< 5, 0.3,−0.4 >,< 10, 0.2,−0.7 >}. here, δpj∪m(2∨5) = δ p j∪m(10) = 0.2 � 0.3 = min{0.4, 0.3} = min{δ p j∪m(2),δ p j∪m(5)}. hence, j∪m is not a bfl of l. 6 int. j. anal. appl. (2022), 20:45 remark 3.2. every bfi of l is a bfl, but the converse need not be true. consider the lattice l given in example 3.1. then the bfi given by j = {< 1, 0.7,−0.5 >,< 2, 0.5,−0.6 >,< 5, 0.6,−0.5 >,< 10, 0.5,−0.6 >} is a bfl of l. but the bfl given by m = {< 1, 0.5,−0.3 >,< 2, 0.4,−0.4 >,< 5, 0.4,−0.3 >,< 10, 0.7,−0.5 >} is not a bfi of l as δpm(2 ∧ 10) = 0.4 � 0.7 = max{δ p m(2),δ p b(10)}. remark 3.3. the union of two bfis of a lattice l need not be a bfi. consider the lattice l given in example 3.1. if j = {< 1, 0.7,−0.5 >,< 2, 0.5,−0.6 >,< 5, 0.6,−0.5 >,< 10, 0.5,−0.6 >} and m = {< 1, 0.7,−0.3 >,< 2, 0.4,−0.2 >,< 5, 0.2,−0.1 >,< 10, 0.2,−0.1 >} are two bfis of l, then j ∪m = {< 1, 0.7,−0.5 >,< 2, 0.5,−0.6 >,< 5, 0.6,−0.5 >,< 10, 0.5,−0.6 >}. here, δnj∪m(2 ∧ 5) = δ n j∪m(1) = −0.5 −0.6 = {−0.6,−0.5} = min{δ n j∪m(2),δ n j∪m(5)}. hence, j ∪m is not a bfi of l. theorem 3.4. let bδ = (bpδ ,b n δ ) be a bfl of l. then for all t ,k ∈ l, the following four statements hold: (i) bpδ (t ∧k) ≥ max{b p δ (t ),b p δ (k)}⇔ (t ≤ k ⇒ b p δ (t ) ≥ b p δ (k)), (ii) bpδ (t ∨k) ≥ max{b p δ (t ),b p δ (k)}⇔ (t ≤ k ⇒ b p δ (t ) ≤ b p δ (k)), (iii) bnδ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}⇔ (t ≤ k ⇒ b n δ (t ) ≤ b n δ (k)), (iv) bnδ (t ∨k) ≤ min{b n δ (t ),b n δ (k)}⇔ (t ≤ k ⇒ b n δ (t ) ≥ b n δ (k)). proof. let t ,k ∈ l. (i) suppose bpδ (t ∧k) ≥ max{b p δ (t ),b p δ (k)}. if t ≤ k, then t ∧k = t . since b p δ (t ∧k) ≥ max{bpδ (t ),b p δ (k)}, we have b p δ (t ) = b p δ (t ∧ k) ≥ max{b p δ (t ),b p δ (k)}. hence, b p δ (t ) ≥ bpδ (k). conversely, suppose (t ≤ k ⇒ bpδ (t ) ≥ b p δ (k)). then b p δ (t ∧k) ≥ b p δ (t ) and b p δ (t ∧k) ≥ bpδ (k). hence, b p δ (t ∧k) ≥ max{b p δ (t ),b p δ (k)}. (ii) suppose bpδ (t ∨k) ≥ max{b p δ (t ),b p δ (k)}. if t ≤ k, then t ∨k = k. since b p δ (t ∨k) ≥ max{bpδ (t ),b p δ (k)}, we have b p δ (k) = b p δ (t ∨ k) ≥ max{b p δ (t ),b p δ (k)}. hence, b p δ (t ) ≤ bpδ (k). conversely, suppose (t ≤ k ⇒ bpδ (t ) ≤ b p δ (k)). then b p δ (t ) ≤ b p δ (t ∨ k) and b p δ (k) ≤ bpδ (t ∨k). hence, b p δ (t ∨k) ≥ max{b p δ (t ),b p δ (k)}. int. j. anal. appl. (2022), 20:45 7 (iii) suppose bnδ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}. if t ≤ k, then t ∧k = t . since b n δ (t ∧k) ≤ min{bnδ (t ),b n δ (k)}, we have b n δ (t ) = b n δ (t ∧ k) ≤ min{b n δ (t ),b n δ (k)}. hence, b n δ (t ) ≤ bnδ (k). conversely, suppose (t ≤ k ⇒ bnδ (t ) ≤ b n δ (k)). then b n δ (t ∧k) ≤ b n δ (t ) and (b n δ (t ∧k) ≤ bnδ (k). hence, b n δ (t ∧k) ≤ min{b n δ (t ),b n δ (k)}. (iv) suppose bnδ (t ∨k) ≤ min{b n δ (t ),b n δ (k)}. if t ≤ k, then t ∨k = k. since (b n δ (t ∨k) ≤ min{bnδ (t ),b n δ (k)), we have b n δ (k) = b n δ (t∨k) ≤ min{b n δ (t ),b n δ (k)}. hence, b n δ (t ) ≥ b n δ (k). conversely, suppose (t ≤ k ⇒ bnδ (t ) ≥ b n δ (k)). then b n δ (t ) ≥ (b n δ (t ∨ k) and (b n δ (k) ≥ bnδ (t ∨k) ≥ b n δ (k). hence, b n δ (t ∨k) ≥ min{b n δ (t ),b n δ (k)}. � theorem 3.5. let bη = (bpη ,b n η ) be a bfl of l. then bη is a bfi of l if and only if the following two conditions are satisfied for all t ,k ∈ l, (i) bpη (t ∨k) = min{bpη (t ),bpη (k)}, (ii) bnη (t ∨k) = max{bnη (t ),bnη (k)}. proof. suppose that bη is a bfi of l. let t ,k ∈ l. then bpη (t ∨k) ≥ min{bpη (t ),bpη (k)} and bnη (t ∨k) ≤ max{bnη (t ),bnη (k)}. since t ≤t ∨k and k ≤t ∨k, then by theorem 3.4, we have bpη (t ) ≥ bpη (t ∨ k) and bpη (k) ≥ bpη (t ∨ k). hence, min{bpη (t ),bpη (k)} ≥ bpη (t ∨ k). thus bpη (t ∨k) = min{bpη (t ),bpη (k)}. now, since t ≤t ∨k and k ≤t ∨k, then by theorem 3.4, we have bnη (t ) ≤ bnη (t ∨ k) and bnη (k) ≤ bnη (t ∨ k). hence, max{bnη (t ),bnη (k)} ≤ bnη (t ∨ k). thus bnη (t ∨k) = max{bnη (t ),bnη (k)}. conversely, suppose that bpη (t∨k) = min{bpη (t ),bpη (k)} and bnη (t∨k) = max{bnη (t ),bnη (k)} for any t ,k ∈ l. then it is clear that bpη (t ∨ k) ≥ min{bpη (t ),bpη (k)} and bnη (t ∨ k) ≤ max{bnη (t ),bnη (k)} for any t ,k ∈ l. next, we shall show that bpη (t ∧k) ≥ max{bpη (t ),bpη (k)} and bnη (t ∧k) ≤ min{bnη (t ),bnη (k)} for any t ,k ∈ l. let t ,k ∈ l. since t ∨ (t ∧k) = t and k ∨ (t ∧k) = k, we have bpη (t ∨ (t ∧k)) = bpη (t ) and bpη (k ∨ (t ∧k)) = bpη (k). thus min{bpη (t ),bpη (t ∧k)} = bpη (t ) and min{bpη (k),bpη (t ∧ k)} = bpη (k), hence, bpη (t ∧ k) ≥ bpη (t ) and bpη (t ∧ k) ≥ bpη (k). therefore, bpη (t ∧ k) ≥ max{bpη (t ),bpη (k)} for any t ,k ∈ l. let t ,k ∈ l. since t ∨ (t ∧k) = t and k ∨ (t ∧k) = k, we have bnη (t ∨ (t ∧k)) = bnη (t ) and bnη (k ∨ (t ∧k)) = bnη (k). thus max{bnη (t ),bnη (t ∧k)} = bnη (t ) and max{bnη (k),bnη (t ∧ k)} = bnη (k). hence, bnη (t ∧ k) ≤ bnη (t ) and bnη (t ∧ k) ≥ bnη (k). therefore, bnη (t ∧ k) ≤ min{bnη (t ),bnη (k)} for any t ,k ∈ l. hence, bη is a bfi of l. � 8 int. j. anal. appl. (2022), 20:45 4. bipolar fuzzy ideals under lattice homomorphisms definition 4.1. let θ : l → l1 be a mapping and bδ = (bpδ ,b n δ ) be a bfs in l. then the image θ(bδ) is defined as θ(bδ) = {< k,θ(bpδ )(k),θ(b n δ )(k) >| k ∈ l 1}, θ(bpδ )(k) = { sup{bpδ (t ) | t ∈ θ −1(k)} if θ−1(k) 6= ∅, 0 if otherwise and θ(bnδ )(k) = { inf{bnδ (t ) | t ∈ θ −1(k)} if θ−1(k) 6= ∅, 0 if otherwise. similarly, if bη = (bpη ,b n η ) be a bfs in l 1, then θ−1(bη) = {< t ,θ−1(bpη (t )),θ−1(bnη (t )) >| t ∈ l}, where θ−1(bpη (t )) = bpη (θ(t )) and θ−1(bnη (t )) = bnη (θ(t )). theorem 4.1. let θ : l → l1 be an epimorphism. if bδ is a bfi of l, then θ(bδ) is a bfi of l1. proof. let bδ = (bpδ ,b n δ ) be a bfi of l. let s,w ∈ l 1. then θ(bpδ )(s ∨w) = sup{b p δ (t ) : t ∈ θ −1(s ∨w)} ≥ sup{bpδ (h∨k) | h ∈ θ −1(s),k ∈ θ−1(w)} ≥ sup{min{bpδ (h),b p δ (k)} | h ∈ θ −1(s),k ∈ θ−1(w)} = min{sup{bpδ (h) | h ∈ θ −1(s)}, sup{bpδ (k) | k ∈ θ −1(w)}} = min{θ(bpδ )(s),θ(b p δ )(w)}, θ(bpδ )(s ∧w) = sup{b p δ (t ) | t ∈ θ −1(s ∧w)} ≥ sup{bpδ (h∧k) | h ∈ θ −1(s),k ∈ θ−1(w)} ≥ sup{max{bpδ (h),b p δ (k)} | h ∈ θ −1(s),k ∈ θ−1(w)} = max{sup{bpδ (h) | h ∈ θ −1(s)}, sup{bpδ (k) | k ∈ θ −1(w)}} = max{θ(bpδ )(s),θ(b p δ )(w)}, θ(bnδ )(s ∨w) = inf{b n δ (t ) | t ∈ θ −1(s ∨w)} ≤ inf{bnδ (h∨k) | h ∈ θ −1(s),k ∈ θ−1(w)} ≤ inf{max{bnδ (h),b n δ (k)} | h ∈ θ −1(s),k ∈ θ−1(w)} = max{inf{bnδ (h) | h ∈ θ −1(s)}, inf{bnδ (k) | k ∈ θ −1(w)}} = max{θ(bnδ )(s),θ(b n δ )(w)}, int. j. anal. appl. (2022), 20:45 9 and θ(bnδ )(s ∧w) = inf{b n δ (t ) | t ∈ θ −1(s ∧w)} ≤ inf{bnδ (h∧k) | h ∈ θ −1(s),k ∈ θ−1(w)} ≤ inf{min{bnδ (h),b n δ (k)} | h ∈ θ −1(s),k ∈ θ−1(w)} = min{inf{bnδ (h) | h ∈ θ −1(s)}, inf{bnδ (k) | k ∈ θ −1(w)}} = min{θ(bnδ )(s),θ(b n δ )(w)}. hence, θ(bδ) is a bfi of l1. � . theorem 4.2. let θ : l → l1 be a homomorphism. if bη is a bfi of l1, then θ−1(bη) is a bfi of l. proof. let bη = (bpη ,b n η ) be a bfi of l 1. let t ,k ∈ l. then θ−1(bpη )(t ∨k) = b p η (θ(t ∨k)) = bpη {(θ(t ) ∨θ(k)} ≥ min{bpη (θ(t )),b p η (θ(k))} = min{θ−1(bpη )(t ),θ −1(bpη )(k)}, θ−1(bpη )(t ∧k) = b p η (θ(t ∧k)) = bpη {(θ(t ) ∧θ(k)} ≥ max{bpη (θ(t )),b p η (θ(k))} = max{θ−1(bpη )(t ),θ −1(bpη )(k)}, θ−1(bnη )(t ∨k) = b n η (θ(t ∨k)) = bnη {(θ(t ) ∨θ(k)}} ≤ max{bnη (θ(t )),b n η (θ(k)) = max{θ−1(bnη )(t ),θ −1(bnη )(k)}, and θ−1(bnη )(t ∧k) = b n η (θ(t ∧k)) = bnη {(θ(t ) ∧θ(k)} ≤ min{bnη (θ(t )),b n η (θ(k))} = min{θ−1(bnη )(t ),θ −1(bnη )(k)}. 10 int. j. anal. appl. (2022), 20:45 hence, θ−1(bη) is a bfi of l. � theorem 4.3. let θ : l → l1 be a homomorphism and let µ and η be bfls of l and l1, respectively. then (i)θ(µ) is a bfl of l1, (ii) θ−1(η) is a bfl l. proof. the proof is omitted since it follows the same proof of theorems 4.1 and 4.2. � theorem 4.4. if θ : l → l1 is an surjection and bη,bδ are bfss of l and l1, respectively, then (i) θ[θ−1(bδ)] = bδ, (ii) bη ⊆ θ−1[θ(bη)]. proof. (i) let α ∈ l1. then θ[θ−1(bpδ )](α) = sup{θ −1(bpδ )(γ) | γ ∈ θ −1(α)} = sup{bpδ (θ(γ)) | γ ∈ l,θ(γ) = α} = bpδ (α) because θ is onto, for every α ∈ l 1, there exists γ in l such that θ(γ) = α. similarly, θ[θ−1(bnδ )](α) = b n δ (α). hence, θ[θ −1(bδ)] = bδ. (ii) let γ ∈ l. then θ−1[θ(bpη )](γ) = θ(bpη )(θ(γ)) = sup{bpη (γ) | γ ∈ θ−1[θ(γ)]} ≥ bpη (γ) and θ−1[θ(bnη )](γ) = θ(b n η )(θ(γ)) = inf{bnη (γ) | γ ∈ θ−1[θ(γ)]} ≤ bnη (γ). hence, bη ⊆ θ−1[θ(bη)]. � definition 4.2. let f : l → l1 be a surjection and bδ = (bpδ ,b n δ ) be a bfs in l. then bδ is said to be f -invariant if for any w,s ∈ l1 such that f (w) = f (s) implies bpδ (w) = b p δ (s) and bnδ (w) = b n δ (s). from theorem 4.4 and definition 4.2, we have the following theorem. theorem 4.5. let f : l → l1 be a surjection and bδ = (bpδ ,b n δ ) be a bfs in l. if a bfs bδ is f -invariant, then f−1(f (bδ)) = bδ. theorem 4.6. let f : l → l1 be a surjection, bδ and bη be bfss of l, and b1δ and b 1 η be bfss of l1. then (i) bδ ⊆ bη ⇒ f (bδ) ⊆ f (bη), (ii) b1δ ⊆ b 1 η ⇒ f−1(b1δ ) ⊆ f −1(b1η). proof. let bδ = (bpδ ,b n δ ) and bη = (b p η ,b n η ) be bfss in l such that bδ ⊆ bη. then bpδ ≤ b p η and b n δ ≥ b n η . also, f (bδ) = {< t,f (bpδ )(t), f (b n δ )(t) >| t ∈ l 1} and f (bη) = {< t,f (bpη )(t), f (bnη )(t) >| t ∈ l1}. now, for any t ∈ l, we have f (bpδ )(t) = sup{(b p δ (k) | k ∈ f−1(t))} ≤ sup{(bpη (k) | k ∈ f−1(t))} = f (bpη )(t) and f (bnδ )(t) = inf{(b n δ (k) | k ∈ f −1(t))} ≤ sup{(bnη (k) | k ∈ f−1(t))} = f (bnη )(t). hence, f (bδ) ⊆ f (bη). similarly, we can prove that b1δ ⊆ b 1 η ⇒ f−1(b1δ ) ⊆ f −1(b1η). � theorem 4.7. if f : l → l1 is an epimorphism, then there is one to one order preserving correspondence between the bfis of l1 and those of l which are f -invariant. int. j. anal. appl. (2022), 20:45 11 proof. let b(l1) denote the set of all bfis of l1 and b(l) denote the set of all bfis of l which are f -invariant. define ς : b(l) → b(l1) and ψ : b(l1) → b(l) such that ς(bδ) = f (bδ) and ψ(b1δ ) = f −1(b1δ ). by theorems 4.1 and 4.2, we have ς and ψ are well-defined. also by theorems 4.4 and 4.5, we have ς and ψ are the inverse to each other which gives that the oneto-one correspondence. also by theorem 4.6, we get bδ ⊆ bη ⇒ f (bδ) ⊆ f (bη). hence, the correspondence is order preserving. � 5. an application of bipolar fuzzy sublattices the single pattern: the one-minute microwave [15]. the one-minute microwave is a simple system with the following requirements: 1. there is a single button available for the user. 2. if the door is closed and the button is pushed, the oven will be energized for one minute. 3. if the button is pushed while the oven is energized, the cooking time is increased by one minute. 4. if the door is open, pushing the button has no effect. 5. the oven has a light that is turned on when the door is open, and also when the oven is cooking. otherwise, the light is off. 6. opening the door stops the cooking and clears the timer (i.e., the remaining cooking time is set to zero). 7. when the cooking is complete (oven times out) a beeper sounds and the light is turned off. here in this application, we consider the one-minute microwave as a lattice l = {button, timer, oven-door} with the operations on and off. the final output will be cooking the food or not cooking the food. when we consider the button there may be two cases that is the button may be pressed or unpressed. when we consider the timer the cases will be the timer may be initiated or uninitiated. when we consider the oven-door then there may be a chance that the door is closed or open. to check whether it forms a bipolar fuzzy lattice first let us know what the possible cases arise. the cases will be like, oven-door closed and button pressed, oven-door closed and button unpressed, button pressed and timer initiated, timer initiated and oven-door opened, timer initiated and oven-door closed, oven-door closed and button unpressed etc. we shall represent b for the button, t for timer, d for oven-door, and join operator as ‘on’ and meet operator as ‘off’. if we take the operator ‘on’ between b and t then it is considered as a button pressed and timer initiated. then cooking will be done. so, the important thing here in this case is pushing the button. so b ∨t = b. if we take the operator ‘off’ between b and t then it is considered the button is unpressed. in this case, there is no question about whether the timer is initiated or not. so b∧t = t. similarly, if we take the operator ‘on’ between b and d then it is considered the button is pressed and the door is closed. then cooking will be done. so, b ∨d = b. if we take the operator ‘off’ 12 int. j. anal. appl. (2022), 20:45 between b and d then it is considered the button is unpressed. in this case, there is no question about whether the timer is initiated or not. so, b ∧d = d. similarly, if we take the operator ‘on’ between t and d then it is considered the timer is initiated and oven-door is closed. then cooking will be done. so, t ∨d = t . if we take the operator ‘off’ between t and d then it is considered the timer is uninitiated and door is open. so, t ∧d = b (here the minimum considered to be b, because the possibility that the timer is uninitiated is that the button is unpressed). let us consider a bfs in l, as bδ = {(b, 0.5,−0.5), (t, 0.4,−0.6)(d, 0.3,−0.5)}. in this set the positive value shows the button pressed, timer initiated, door closed, and the negative values show the button unpressed, timer uninitiated, and door open. now, we check bδ forms a bf-lattice or not. we can routinely prove that bδ is a bfl of l. 6. conclusion and future work in this article, we have introduced the concepts of bfls and bfis of a lattice. interesting properties of these bfls and bfis are developed. moreover, we investigated the properties of bfis under lattice homomorphism and an application of bfls is given. our future work is to develop the bipolar fuzzy prime ideals, bipolar fuzzy principal ideals, quotient ideals, bipolar fuzzy filters, and bipolar fuzzy prime filters of a lattice. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] k.m. lee, bipolar-valued fuzzy sets and their operations, in: proc. int. conf. on intelligent technologies, bangkok, thailand, (2000), 307-312. [2] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [3] k.v. thomas, l.s. nair, intuitionistic fuzzy sublattices and ideals, fuzzy inform. eng. 3 (2011), 321-331. https: //doi.org/10.1007/s12543-011-0086-5. [4] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [5] n. ajmal, k.v. thomas, fuzzy lattices, inform. sci. 79 (1994), 271-291. https://doi.org/10.1016/ 0020-0255(94)90124-4. [6] u. venkata kalyani, t. eswarlal, homomorphism on bipolar vague normal groups, adv. math., sci. j. 9 (2020), 3315-3324. https://doi.org/10.37418/amsj.9.6.11. [7] u. venkata kalyani, t. eswarlal, bipolar vague cosets, adv. math., sci. j. 9 (2020), 6777-6787. https://doi. org/10.37418/amsj.9.9.36. [8] s. boudaoud, s. milles, l. zedam, principal intuitionistic fuzzy ideals and filters on a lattice, discuss. math. gen. algebra appl. 40 (2020), 75-88. https://doi.org/10.7151/dmgaa.1325. https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1007/s12543-011-0086-5 https://doi.org/10.1007/s12543-011-0086-5 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/0020-0255(94)90124-4 https://doi.org/10.1016/0020-0255(94)90124-4 https://doi.org/10.37418/amsj.9.6.11 https://doi.org/10.37418/amsj.9.9.36 https://doi.org/10.37418/amsj.9.9.36 https://doi.org/10.7151/dmgaa.1325 int. j. anal. appl. (2022), 20:45 13 [9] s. milles, the lattice of intuitionistic fuzzy topologies generated by intuitionistic fuzzy relations, appl. appl. math. 15 (2020), 942-956. https://digitalcommons.pvamu.edu/aam/vol15/iss2/13. [10] h. zhang, q. li, intuitionistic fuzzy filter theory on residuated lattices, soft comput. 23 (2018), 6777-6783. https://doi.org/10.1007/s00500-018-3647-2. [11] s. milles, l. zedam, e. rak, characterizations of intuitionistic fuzzy ideals and filters based on lattice operations, j. fuzzy set valued anal. 2017 (2017), 143-159. https://doi.org/10.5899/2017/jfsva-00399. [12] b. nageswararao, n. ramakrishna, t. eswarlal, vague lattices, studia rosenthaliana, 12 (2020), 191-202. [13] r.p. rao, v.s. kumar, a.p. kumar, rough vague lattices, j. xi’an univ. architect. technol. 9 (2019), 115-124. [14] m. gorjanac ranitovic, a. tepavcevic, a lattice-theoretical characterization of the family of cut sets of intervalvalued fuzzy sets, fuzzy sets syst. 333 (2018), 1-10. https://doi.org/10.1016/j.fss.2016.11.014. [15] k. jacob, s.p. tiwari, n. shamsidah ah, et al. restricted cascade and wreath products of fuzzy finite switchboard state machines, iran. j. fuzzy syst. 16 (2019), 75-88. https://doi.org/10.22111/ijfs.2019.4485. https://digitalcommons.pvamu.edu/aam/vol15/iss2/13 https://doi.org/10.1007/s00500-018-3647-2 https://doi.org/10.5899/2017/jfsva-00399 https://doi.org/10.1016/j.fss.2016.11.014 https://doi.org/10.22111/ijfs.2019.4485 1. introduction 2. preliminaries 3. bipolar fuzzy sublattices and ideals 4. bipolar fuzzy ideals under lattice homomorphisms 5. an application of bipolar fuzzy sublattices 6. conclusion and future work references int. j. anal. appl. (2023), 21:14 estimation of finite population mean by utilizing the auxiliary and square of the auxiliary information saddam hussain1, anum iftikhar2, kleem ullah3, gulnaz atta4, usman ali5, ulfat parveen5, muhammad yasir arif5, ather qayyum5,∗ 1department of statistics, university of mianwali, pakistan 2school of statistics, shanxi university of finance and economics taiyuan, china 3foundation university medical college, foundation university, islamabad, pakistan 4department of mathematics, university of education lahore, dgk campus, pakistan 5department of mathematics, institute of southern punjab multan, pakistan ∗corresponding author: atherqayyum@isp.edu.pk abstract. this article fundamentally aims at the proposition of new family of estimators using auxiliary information to assist the estimation of finite population mean of the study variable. the objectives are achieved by devising dual use of supplementary information through straightforward manner. the additional information is injected in mean estimating procedure by considering squared values of auxiliary variable. the utility of the proposed scheme is substantiated by providing rigorous comparative account of the newly materialized structure with the well celebrated existing family of grover and kaur (2014). the contemporary advents of the new family are documented throughout the article. 1. introduction the utility of auxiliary information to improve the effectiveness of estimation procedures estimating the attributes of population under study is well cherished. the documented realizations of the advents of employing auxiliary information while estimating study parameters can be traced back in eighteenth century france. a keen review of literature substantiate that pierre-simon laplace documented the early advocacy of using supplementary information to assist the estimation of population of country. received: oct. 8, 2022. 2020 mathematics subject classification. 60k35. key words and phrases. auxiliary variable; bias; mean squared error; percentage relative efficiency; second raw moment of auxiliary variable. https://doi.org/10.28924/2291-8639-21-2023-14 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-14 2 int. j. anal. appl. (2023), 21:14 he advised “the register of births, which are kept with care in order to assure the condition of the citizens, can serve to determine the population of great empire without resorting a census of its inhabitants. but for this it is necessary to know the ratio of population to annual the birth.” [16] later on, the applicability of the additional information to enhance the efficiency of underlying estimation procedures was materialized by exploiting the correlation structure existent to govern both study variable and auxiliary variable, [3]. over the time, streams of propositions devising novel mechanism expounding efficient estimation of study parameter can be in available literature. for a comprehensive review of ongoing research efforts, one may consult to [4], [5], [8], [10], [11], [12], [13], [14], [21], and [23]. it is noteworthy that these efforts comprehends the advances on two fronts, (i) – competent use of auxiliary information and (ii) – introducing novel functional forms incorporating additional information in estimation synergy. in adjacent past, [5] instigated the use of exponent-based formation to inject supplementary information and thus proposed generalized family of estimators encapsulating numerous exiting specifications. the optimality of the proposed scheme was delineated through rigorous empirical evaluation. motivated by the ongoing proceedings, this research targets the aforementioned both fronts, simultaneously. we propose new family of estimators capable of entertaining dual use of auxiliary information by the application of efficient exponent-based functional formation. the comparative performance of the newly developed mechanism is documented with respect to promising novel family of [5] estimators. the empirical evaluations are conducted by the consideration of numerous data sets from statistics and allied research fields. this article is mainly divided into seven major parts. section 2 briefs about the preliminaries extensively used in this research whereas section 3 summarizes the contemporary methods. section 4 is dedicated to expound the proposed scheme, whereas section 5 documents the efficiency conditions. section 6 explore the empirical performance of the competing techniques. lastly, section 7 comprehends the investigation by offering highlights of the research along with few prospective research venues. 2. notation and symbols let v be a finite population of n units, such as v = {v1,v2, . . . ,vn}. we draw a sample of size n from the population through simple random sampling with out replacement scheme. let yi, xi, and ui are study, auxiliary variable, and squared values of auxiliary variable, respectively, for the ith (i = 1, 2, . . . ,n) unit of the population. let, ȳ = 1 n n∑ i=1 yi, x̄ = 1 n n∑ i=1 xi, and ū = 1 n n∑ i=1 ui are sample means of the study, auxiliary variable, and squared values of the auxiliary variable respectively. ȳ = 1 n n∑ i=1 yi, x̄ = 1 n n∑ i=1 xi, and ū = 1 n n∑ i=1 ui, are population means of the study, auxiliary variable, and squared values of the auxiliary variable respectively. on these grounds sample variances of study, int. j. anal. appl. (2023), 21:14 3 auxiliary variable, and squared values of auxiliary variable are defined as s2y = 1 n−1 n∑ i=1 (yi − ȳ)2, s2x = 1 n−1 n∑ i=1 (xi − x̄)2, and s2u = 1 n−1 n∑ i=1 (ui − ū)2. further more, let us define coefficients of variation of x, y , and u as cx, cy, cu, where cy = sy/ȳ , cx = sx/x̄, and cu = su/ū. we now define error terms as e0 = (ȳ − ȳ )/ȳ , e1 = (x̄ − x̄)/x̄, e2 = (ū − ū)/ū, such that e(ei ) = 0, i = 0, 1, 2. e(e20 ) = λc 2 y , e(e 2 1 ) = λc 2 x , e(e 2 2 ) = λc 2 u, where λ = ( 1 n − 1 n ) commonly known as sample fraction. in procession the error covariances are derived as e(e0e1) = λcycxρyx, e(e0e2) = λcycuρyu, e(e1e2) = λcxcuρxu, where ρyx, ρyu, and ρxu, represents sample correlation coefficients defined as ρyx = syx sysx , ρyu = syu sysu , and ρxu = sxu sxsu . 3. some existing estimators (i) the typically independent suggest estimator is ȳ with variance var(ȳ) = λȳ 2c2y . (3.1) (ii) [3] and [17] proposed traditional ratio type and product type estimators, ˆ̄yr and ˆ̄yp , respectively, given by ˆ̄yr = ȳ( x̄ x̄ ), (3.2) ˆ̄yp = ȳ( x̄ x̄ ). (3.3) it is a famous truth that, irrespective of the biasedness, the classical ratio and product estimator, ˆ̄yr and ˆ̄yp , is more accurate than the mean per unit estimator ȳ if there exist a high positive and negative correlation between y and x, i.e ρyx > cx/2cy and ρyx < −cx/2cy. the mses of ˆ̄yr and ˆ̄yp , respectively, given by mse( ˆ̄yr) ∼= λȳ 2(c2y + c 2 x − 2cycxρyx ) (3.4) mse( ˆ̄yp ) ∼= λȳ 2(c2y + c 2 x + 2cycxρyx ). (3.5) several authors have proposed a few converted ratio-kind estimators for estimating the finite population mean with the aid of using the use of auxiliary information. some suitable research on this path include [2], [23], [24], [25], and many others authors. in a latest study, [15] proposed a standard elegance of estimators ˆ̄yk that consists of a few tailored ratio-kind estimators, given by ˆ̄yk ∼= ȳ { ax̄ + b α(ax̄ + b)) + (1 −α)(ax̄ + b) }g , . (3.6) 4 int. j. anal. appl. (2023), 21:14 ˆ̄yk ∼= ȳ { ax̄ + b α(ax̄ + b)) + (1 −α)(ax̄ + b) }g , (3.7) in which a( 6= 0) and b are known recognized elements of any recognized population parameters, such as cx the coefficient of variation; ρyx the correlation between y and x; β1(x) the coefficient of skewness and so on. the minimal mse of ˆ̄yk on the top of the line cost of (αϑg), where ϑ = ax̄ ax̄+b , is given by msemin( ˆ̄yk) ∼= λȳ 2c2y (1 −ρ 2 yx ) (3.8) (iii) the usual difference estimator ˆ̄yd is ˆ̄yd = ȳ + t1(x̄ − x̄), (3.9) in which t1 is an unknown elements. it is straightforward to expose that ˆ̄yd is unbiased. the minimal variance of ˆ̄yd on the top of the line fee of t1, that is, t1(opt) = ρyx (sy/sx ), is given by varmin( ˆ̄yd) ∼= λȳ 2c2y (1 −ρ 2 yx ), (3.10) which is same to the variance of the classical regression estimator ˆ̄ylr = ȳ + b(x̄ − x̄), where b is the slope estimator of the population regression coefficient β = t1(opt). the difference estimator ˆ̄yd is always better perform than the ratio type ˆ̄yr and product type ˆ̄yp estimators when estimating ȳ . (iv) [19] proposed an improved difference type estimatorof ˆ̄yd, given by ˆ̄yr,d = t2ȳ + t3(x̄ − x̄), (3.11) where t2 and t3 are selected quantities. the minimum mse of ˆ̄yr,d at the optimum values, t2(opt) = 1 1 + λc2y (1 −ρ2yx ) , (3.12) and t3(opt) = ȳ cyρyx x̄cx [1 + λc 2 y (1 −ρ2yx )] , (3.13) is given by msemin( ˆ̄yr,d) ∼= λȳ 2c2y (1 −ρ2yx ) 1 + λc2y (1 −ρ2yx ) . (3.14) the above (11), it is shown that the ˆ̄yr,d is better perform than ˆ̄yd, i.e, msemin( ˆ̄yr,d) ∼= varmin( ˆ̄yd) − λȳ 2c4y (1 −ρ2yx )2 1 + λc2y (1 −ρ2yx ) . (3.15) (v) [1] proposed a ratio and product type exponential estimators, given by ˆ̄ybt,r = ȳ exp (x̄ − x̄ x̄ + x̄ ) , (3.16) ˆ̄ybt,p = ȳ exp (x̄ − x̄ x̄ + x̄ ) . (3.17) int. j. anal. appl. (2023), 21:14 5 the mses of ˆ̄ybt,r and ˆ̄ybt,p , respectively, are given by msemin( ˆ̄ybt,r) ∼= λȳ 2 4 ( 4c2y + c 2 x − 4ρyxcycx ) , (3.18) msemin( ˆ̄ybt,p ) ∼= λȳ 2 4 ( 4c2y + c 2 x + 4ρyxcycx ) . (3.19) following the work in [1], [22] proposed a generalized ratio type exponential estimator, ˆ̄ys = ȳ exp( a(x̄ − x̄) a(x̄ + x̄) + 2b ). (3.20) the minimum mse of ˆ̄ys turns out to equivalent to varmin( ˆ̄yd), i.e, msemin( ˆ̄ys) ∼= λȳ c2y (1 −ρ2yx ). (vi) based on the estimator [1], [19], [20], [22] proposed a estimator, given by ˆ̄ysg = {t4ȳ + t5(x̄ − x̄)}exp ( x̄ − x̄ x̄ + x̄ + 2nx̄ ) , (3.21) where t4 and t5 are suitably chosen constant. following these work, [4] proposed related estimator by combining [1] and [19] ˆ̄ysg = {t6ȳ + t7(x̄ − x̄)}exp( x̄ − x̄ x̄ + x̄ ), (3.22) in which t6 and t7 are elements. in a recant study, proposed a estimators, given by ˆ̄ygk,g = {t8ȳ + t9(x̄ − x̄)}exp( a(x̄ − x̄) a(x̄ + x̄) + 2b ), (3.23) in which t8 and t9 are elements. note that ˆ̄ygk,g contains the estimators given known in [4] and [9]. the minimum mse of ˆ̄ygk,g at the optimum values, t8(opt) = 8 −λϑ2c2x 8{1 + λc2y (1 −ρ2yx )} , and t9(opt) = ȳ [λϑ3c3x + 8cyρyx −λϑ2c2xc2yρyx − 4ϑcx { 1 −λc2y (1 −ρ2yx )}] 8x̄cx { 1 + λc2y (1 −ρ2yx ) } . is given by msemin( ˆ̄ygk,g) ∼= λȳ 2 { 64c2y (1 −ρ2yx ) −λϑ4c4x − 16λϑ2c2xc2y (1 −ρ2yx ) } 64 { 1 + λc2y (1 −ρ2yx ) } . (3.24) the above equation can be written as, given by msemin( ˆ̄ygk,g) ∼= varmin( ˆ̄yd) −v1. (3.25) where v1 = λ2ȳ 2{ϑ2c2x +8c2y (1−ρ2yx )}2 64{1+λc2y (1−ρ2yx )} 6 int. j. anal. appl. (2023), 21:14 4. proposed estimator it is well-recognized that the usage of auxiliary variables increase the precision of an estimator each the estimation level and at the designing level. in many surveys, the auxiliary records is broadly speaking to be had text-color the sampling design or frame. the concept is that if there exists the ideal amount of correlation among the examine and auxiliary variables, the squared values of the auxiliary variables also are correlated with the values of the examine variable. thus, the squared auxiliary variable (this is the squared values of auxiliary variable) may be taken into consideration a brand new auxiliary variable, and this greater records may also assist us to growth the performance of an estimator. by those notions, we endorse an advanced estimator of the finite populace mean. the proposed estimator consists of the greater records withinside the shape of an auxiliary variable and withinside the shape of the squared cost of the auxiliary variable. following [4], [5] and [20], we suggest a exponential type estimator ˆ̄ypr, given by ˆ̄ypr = {t10ȳ + t11(x̄ − x̄) + t12(ū − ū)}exp ( a(x̄ − x̄) a(x̄ + x̄) + 2b ) , (4.1) where t10, t11, and t12 are suitably constants, which will be determined later. where a and b are explained in table1. ˆ̄ypr also rewriting as ˆ̄ypr = {t10ȳ (1 + e0) − t11x̄e1 − t12ūe2} { 1 − ϑe1 2 + 3ϑ2e21 8 + . . . } . (4.2) by expending (25) and upto two degree of approximation in ei, we can write ( ˆ̄ypr − ȳ ) = −ȳ + t10ȳ + t10ȳ e0 − 1 2 t10ϑȳ e1 − t11x̄e1 − t12ūe2 −− 1 2 t10ϑȳ e0e1 + 3 8 t10ϑ 2ȳ e21 + 1 2 t11ϑx̄e 2 1 + 1 2 t12ϑūe2. from (26), the bias and mse of ˆ̄ypr up to first degree of approximation are, respectively, given by bias( ˆ̄ypr ) = 1 8 [ −8ȳ + 4λϑcx (t11x̄cx + t12ūcuρxu) + t10ȳ{8 + λϑcx (3ϑcx − 4cyρyx )} ] , (4.3) msemin( ˆ̄ypr ) ∼= ȳ 2 + t11λx̄c2x (−ȳ ϑ + t11x̄) + t 2 12λū 2c2u + t12λūcxcuρxu(−ȳ ϑ + 2t11x̄) + ȳ 2t210 [ 1 + λ { c2y + ϑcx (ϑcx − 2cyρyx ) }] + 1 4 t10ȳ { − 8ȳ + λcx { ϑcx (−3ϑȳ + 8t11x̄) + 8t12ūcuρxu + 4cyρyx (ȳ ϑ− 2t11x̄) } − 8t12λūcycuρyu } . the values of t10, t11 and t12 obtained from (28) are, respectively, given by t10(0pt) = 8 −λϑ2c2x 8{1 + λc2y (1 −ϕ2yxu)} t11(opt) = ȳ  λϑc3x (−1 + ρ2xu) + (−8cy + λϑ2c2xcy )(ρyx −ρyuρxu) +4ϑcx (−1 + ρ2xu) − 1 + λc 2 y (1 −ϕ 2 yxu)   8x̄cx (−1 + ρ2xu){1 + λc2y (1 −ϕ2yxu)} int. j. anal. appl. (2023), 21:14 7 and t12(0pt) = ȳ (8 −λϑ2c2x )cy (ρyxρxu −ρyu) 8ūcu(−1 + ρ2xu){1 + λc2y (1 −ϕ2yxu)} , where ϕ2yxu = ρ2yx +ρ 2 yu−2ρyxρyuρxu 1−ρ2xu . putting the above obtained values of t10, t11, and t12 in (28), and after a few simplifications, we get the minimal mse of ˆ̄ypr, given by msemin( ˆ̄ypr ) ∼= λȳ 2 { 64c2y (1 −ϕ2yxu) −λϑ4c4x − 16λϑc2yc2x (1 −ϕ2yxu) } 64{1 + λc2y (1 −ϕ2yxu)} . (4.4) it can be shown that proposed estimator msemin( ˆ̄ypr ) is always better perform then the difference estimator varmin( ˆ̄yd), i.e., msemin( ˆ̄ypr ) ∼= varmin( ˆ̄yd) −{v1 + v2}, (4.5) where v1 is defined as before and v2, is given by v2 = λȳ 2c2y (ρyu −ρyxρxu)2(−8 + λϑ2c2x )2 64(1 −ρ2xu){1 + λc2y (1 −ρ2yx )}{1 + λc2y (1 −ϕ2yxu)} . table 1. some possible members of the suggested family of estimators s.no a b ˆ̄ygk,g ˆ̄ypr 1 1 cx ˆ̄ygk,g ˆ̄ypr 2 1 β2(x) ˆ̄ygk,g ˆ̄ypr 3 β2(x) cx ˆ̄ygk,g ˆ̄ypr 4 cx β2(x) ˆ̄ygk,g ˆ̄ypr 5 1 ρyx ˆ̄ygk,g ˆ̄ypr 6 cx ρyx ˆ̄ygk,g ˆ̄ypr 7 ρyx cx ˆ̄ygk,g ˆ̄ypr 8 β2(x) ρyx ˆ̄ygk,g ˆ̄ypr 9 ρyx β2(x) ˆ̄ygk,g ˆ̄ypr 10 1 nx̄ ˆ̄ygk,g ˆ̄ypr 8 int. j. anal. appl. (2023), 21:14 table 2. some possible members of the suggested family of estimators s.no a b ˆ̄ygk,g ˆ̄ypr 1 1 cx ˆ̄ygk,g ˆ̄ypr 2 1 β2(x) ˆ̄ygk,g ˆ̄ypr 3 β2(x) cx ˆ̄ygk,g ˆ̄ypr 4 cx β2(x) ˆ̄ygk,g ˆ̄ypr 5 1 ρyx ˆ̄ygk,g ˆ̄ypr 6 cx ρyx ˆ̄ygk,g ˆ̄ypr 7 ρyx cx ˆ̄ygk,g ˆ̄ypr 8 β2(x) ρyx ˆ̄ygk,g ˆ̄ypr 9 ρyx β2(x) ˆ̄ygk,g ˆ̄ypr 10 1 nx̄ ˆ̄ygk,g ˆ̄ypr 5. conclusion this research persuaded the enhancement of efficiency of estimation procedures involving the estimation of finite population mean. the objectives are aimed by devising new class of estimators facilitating the launch of dual use of auxiliary information through exponent-based formation. the duality of the supplementary information is executed by considering the squares of information available on supplementary variable. the comparative performance of the proposed approach is enumerated by the application of well acknowledged data sets from multi-disciplinary research literature. furthermore, rigorous evaluation analysis is conducted in comparison to the leading family of grover and kaur (2014). the tedious comparative performance evaluation of contemporary methods reveals that every instant of newly proposed family out performs every member of existing family. moreover, this distinction is witnessed with respect to all considered data sets. in future, it will be interesting to extend the proposed scheme for more complicated study designs. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. bahl, r.k. tuteja, ratio and product type exponential estimators, j. inform. optim. sci. 12 (1991), 159–164. https://doi.org/10.1080/02522667.1991.10699058. [2] p.k. bedi, efficient utilization of auxiliary information at estimation stage, biometrical j. 38 (1996), 973–976. https://doi.org/10.1002/bimj.4710380809. [3] w.g. cochran, the estimation of the yields of cereal experiments by sampling for the ratio of grain to total produce, j. agric. sci. 30 (1940), 262–275. https://doi.org/10.1017/s0021859600048012. [4] l.k. grover, p. kaur, an improved estimator of the finite population mean in simple random sampling, model assisted stat. appl. 6 (2011), 47–55. https://doi.org/10.3233/mas-2011-0163. https://doi.org/10.1080/02522667.1991.10699058 https://doi.org/10.1002/bimj.4710380809 https://doi.org/10.1017/s0021859600048012 https://doi.org/10.3233/mas-2011-0163 int. j. anal. appl. (2023), 21:14 9 [5] l.k. grover, p. kaur, a generalized class of ratio type exponential estimators of population mean under linear transformation of auxiliary variable, commun. stat. simul. comput. 43 (2014), 1552–1574. https://doi.org/ 10.1080/03610918.2012.736579. [6] d.n. gujarati, basic econometrics, tata mcgraw-hill education, new delhi, (2009). [7] v.v. kulish, j.l. lage, application of fractional calculus to fluid mechanics, j. fluids eng. 124 (2002), 803–806. https://doi.org/10.1115/1.1478062. [8] s. gupta, j. shabbir, on improvement in estimating the population mean in simple random sampling, j. appl. stat. 35 (2008), 559–566. https://doi.org/10.1080/02664760701835839. [9] s. gupta, j. shabbir, s. sehra, mean and sensitivity estimation in optional randomized response models, j. stat. plan. inference. 140 (2010), 2870–2874. https://doi.org/10.1016/j.jspi.2010.03.010. [10] a. haq, m. khan, z. hussain, a new estimator of finite population mean based on the dual use of the auxiliary information, commun. stat. theory methods. 46 (2016), 4425–4436. https://doi.org/10.1080/03610926. 2015.1083112. [11] a. haq, j. shabbir, improved family of ratio estimators in simple and stratified random sampling, commun. stat. theory methods. 42 (2013), 782–799. https://doi.org/10.1080/03610926.2011.579377. [12] c. kadilar, h. cingi, ratio estimators in simple random sampling, appl. math. comput. 151 (2004), 893–902. https://doi.org/10.1016/s0096-3003(03)00803-8. [13] c. kadilar, h. cingi, an improvement in estimating the population mean by using the correlation coefficient, hacettepe j. math. stat. 35 (2006), 103-109. [14] c. kadilar, h. cingi, improvement in estimating the population mean in simple random sampling, appl. math. lett. 19 (2006), 75–79. https://doi.org/10.1016/j.aml.2005.02.039. [15] m. khoshnevisan, r. singh, p. chauhan, et al. a general family of estimators for estimating population mean using known value of some population parameter(s), far east j. theor. stat. 22 (2007), 181–191. [16] s.l. lohr, sampling: design and analysis, duxbury press, (1999). [17] m.n. murthy, product method of estimation, sankhya a. 26 (1964), 69–74. [18] m.n. murthy, sampling theory and methods, statistical publishing society, (1967). [19] t.j. rao, on certail methods of improving ratio and regression estimators, commun. stat. theory methods. 20 (1991), 3325–3340. [20] j. shabbir, s. gupta, some estimators of finite population variance of stratified sample mean, commun. stat. theory methods. 39 (2010), 3001–3008. https://doi.org/10.1080/03610920903170384. [21] j. shabbir, a. haq, s. gupta, a new difference-cum-exponential type estimator of finite population mean in simple random sampling, rev. colomb. estad. 37 (2014), 199-211. https://doi.org/10.15446/rce.v37n1.44366. [22] r. singh, p. chauhan, n. sawan, et al. improvement in estimating the population mean using exponential estimator in simple random sampling, int. j. stat. econ. 3 (2009), 13–18 [23] h.p. singh, r.s. solanki, efficient ratio and product estimators in stratified random sampling, commun. stat. theory methods. 42 (2013), 1008–1023. https://doi.org/10.1080/03610926.2011.592257. [24] b. sisodia, v. dwivedi, modified ratio estimator using coefficient of variation of auxiliary variable, j.-indian soc. agric. stat. 33 (1981), 13–18. [25] l.n. upadhyaya, h.p. singh, use of transformed auxiliary variable in estimating the finite population mean, biometrical j. 41 (1999), 627–636. https://doi.org/10.1002/(sici)1521-4036(199909)41:5<627::aid-bimj627>3. 0.co;2-w. [26] a. iftikhar, h. shi, s. hussain, a. qayyum, m. el-morshedy, s. al-marzouki, estimation of finite population mean in presence of maximum and minimum values under systematic sampling scheme, aims math. 7 (2022), 9825–9834. https://doi.org/10.3934/math.2022547. https://doi.org/10.1080/03610918.2012.736579 https://doi.org/10.1080/03610918.2012.736579 https://doi.org/10.1115/1.1478062 https://doi.org/10.1080/02664760701835839 https://doi.org/10.1016/j.jspi.2010.03.010 https://doi.org/10.1080/03610926.2015.1083112 https://doi.org/10.1080/03610926.2015.1083112 https://doi.org/10.1080/03610926.2011.579377 https://doi.org/10.1016/s0096-3003(03)00803-8 https://doi.org/10.1016/j.aml.2005.02.039 https://doi.org/10.1080/03610920903170384 https://doi.org/10.15446/rce.v37n1.44366 https://doi.org/10.1080/03610926.2011.592257 https://doi.org/10.1002/(sici)1521-4036(199909)41:5<627::aid-bimj627>3.0.co;2-w https://doi.org/10.1002/(sici)1521-4036(199909)41:5<627::aid-bimj627>3.0.co;2-w https://doi.org/10.3934/math.2022547 1. introduction 2. notation and symbols 3. some existing estimators 4. proposed estimator 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 62-74 http://www.etamaths.com characterizations of abel grassmann’s groupoids by the properties of their double-framed soft ideals asghar khan1,∗, muhammad izhar1 and aslihan sezgin2 abstract. in this paper, we introduce the concept of double-framed soft ideals and investigate properties of these ideals in regular, intra-regular, right regular and left regular ag-groupoids. we also characterize intra-regular ag-groupoids in terms of the double-framed soft ideals. 1. introduction the uncertainty which appeared in economics, engineering, environmental science, medical science and social science and so many other applied sciences is too complicated to be solved within traditional mathematical framework. molodtsov [1] introduced the concept of soft set which can be used as a generic mathematical tool for dealing with uncertainties. molodtsov pointed out several directions for the applications of soft sets. worldwide, there has been a rapid growth in interest in soft set theory and its applications. evidence of this can be found in the increasing number of quality articles on soft sets and related topics that have been published in recent years. maji et al. [2] described the application of soft set to a decision making problem. maji et al. [3] also studied several operations on soft sets. jun et al. [4] introduced the notion of soft ordered semigroup. at present, soft set theory is applied to different algebraic structure. we refer the reader to the papers [5–13]. the idea of generalization of a commutative semigroup, (known as left almost semigroup) was introduced by m. a. kazim and m. naseeruddin in 1972 (see [15]). some other names have also been used in literature for left almost semigroups. cho et al. [16] studied this structure under the name of right modular groupoid. holgate [17] studied it as left invertive groupoid. similarly, stevanovic and protic [18] called this structure an abel-grassmann groupoid (or simply ag-groupoid), which is the primary name under which this structure in known nowadays. there are many important applications of ag-groupoids in the theory of flocks [19]. recently, jun et al. extended the notions of union and intersectional soft sets into double-framed soft sets and defined double-framed soft subalgebra of a bck/bci-algebra and studied the related properties in [21]. in [14], jun et al. also defined the concept of a double-framed soft ideal (briefly, dfs ideal) of a bck/bci-algebra and gave many valuable results. in the present paper, we apply the idea given by jun et al. in [21], to ag-groupoids and introduce the concept of double-framed soft ideals in ag-groupoids and investigate their related properties. the respective examples of these notions are investigated. intra-regular ag-groupoids are characterized using the dfs ideals of ag-groupoids. 2. preliminaries a groupoid (s, ·) is called an ag-groupoid if it satisfies the left invertive law: (ab)c = (cb)a for all a,b,c ∈ s. this structure is closely related with a commutative semigroup because if an ag-groupoid contains right identity then it becomes a commutative monoid. an ag-groupoid may or may not contain a left identity. if there exist a left identity in an ag-groupoid then it is unique. received 10th may, 2017; accepted 14th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 03c05, 06d72. key words and phrases. dfs-set; dfs ag-groupoid; dfs ideal; dfs int-uni product; dfs including set. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 62 characterizations of abel grassmann’s groupoids 63 every ag-groupoid s satisfies the medial law: (ab)(cd) = (ac)(bd) for all a,b,c,d ∈ s. every aggroupoid s with left identity satisfies the paramedial law: (ab)(cd) = (db)(ca) for all a,b,c,d ∈ s. in an ag-groupoid s with left identity, using the paramedial law, it is easy to prove that (ab)(cd) = (dc)(ba) for all a,b,c,d ∈ s. in an ag-groupoid s with left identity, we have a(bc) = b(ac) for all a,b,c ∈ s. an ag-groupoid s is called ag∗∗-groupoid if x(yz) = y(xz) for all x,y,z ∈ s. throughout this paper, s will represent an ag-groupoid unless otherwise stated. for nonempty subsets a and b of s we have ab := {ab|a ∈ a and b ∈ b}. a nonempty subset a of an ag-groupoid s is called sub ag-groupoid of s if a2 ⊆ a. a nonempty subset a of an ag-groupoid s is called left ( resp. right ) ideal of s if sa ⊆ a (resp. as ⊆ a). if a is both a left and a right ideal of s then it is called a two-sided ideal or simply an ideal of s. we denote by l[a2],r[a2] and j[a2], the priciple left ideal, right ideal, two sided ideal of an aggroupoid s generated by a2 ∈ s. note that the principal left ideal, right ideal, two sided ideals of an ag-groupoid s generated by a2 are equal. that is l[a2] = r[a2] = j[a2] = sa2 = a2s = sa2s = {sa2 : s ∈ s}. the reader is invited to read [25, 26] an ag-groupoid s is called; i) right regular if for all a ∈ s, there exist x ∈ s such that a = (aa)x. ii) left regular if for all a ∈ s, there exist x ∈ s such that a = x(aa). iii) regular if for all a ∈ s, there exist x ∈ s such that a = (ax)a. iv) intra-regular if for all a ∈ s, there exist x,y ∈ s such that a = (xa2)y. 3. soft set (basic operations) in [6], atagun and sezgin introduced some new operations on soft set theory and defined soft sets in the following way: let u be an initial universe, e a set of parameters, p(u) the power set of u and a ⊆ e. then soft set fa over u is a function defined by: fa : e −→ p(u) such that fa(x) = ∅ if x /∈ a. here fa is called an approximate function. a soft set over u can be represented by the set of ordered pairs fa := {(x,fa(x)) : x ∈ e, fa(x) ∈ p(u)} . it is clear that a soft set is a parameterized family of subsets of u. the set of all soft sets over u is denoted by s(u). definition 3.1. let fa,fb ∈ s(u). then fa is a soft subset of fb, denoted by fa⊆̃fb if fa(x) ⊆ fb(x) for all x ∈ e. two soft sets fa,fb are said to be equal soft sets if fa⊆̃fb and fb⊆̃fa and is denoted by fa=̃fb. definition 3.2. let fa,fb ∈ s(u). then the union of fa and fb, denoted by fa∪̃fb, is defined by fa∪̃fb = fa∪b, where fa∪b(x) = fa(x) ∪fb(x), for all x ∈ e. definition 3.3. let fa,fb ∈ s(u). then the intersection of fa and fb, denoted by fa∩̃fb, is defined by fa∩̃fb = fa∩b, where fa∩b(x) = fa(x) ∩fb(x), for all x ∈ e. definition 3.4. [22] let fa,fb ∈ s(u). then the soft product of fa and fb, denoted by fa◦̃fb, is defined by (fa◦̃fb) (x) :=   ⋃ x=yz {fa(y) ∩fb(z)} if ∃ y,z ∈ s such that x = yz ∅ otherwise. throughout this paper, let e = s, where s is an ag-groupoid and a,b,c, · · · are sub aggroupoids, unless otherwise stated. definition 3.5. [21] a double-framed soft pair 〈(α,β) ; a〉 is called a double-framed soft set of a over u (briefly, dfs -set of a), where α and β are mappings from a to p(u). the set of all dfs-sets of s over u will be denoted by dfs(u). for a dfs-set 〈(α,β) ; a〉 of a and two subsets γ and δ of u, the γ-inclusive set and the δ-exclusive set of 〈(α,β) ; a〉, denoted by ia(α; γ) and ea(β; δ), respectively, are defined as follows: ia(α; γ) := {x ∈ a|α (x) ⊇ γ} 64 khan, izhar and sezer and ea(β; δ) := {x ∈ a|β (x) ⊆ δ} respectively. the set dfa (α,β)(γ,δ) := {x ∈ a|α (x) ⊇ γ,β (x) ⊆ δ} is called a double framed soft including set [21] of 〈(α,β) ; a〉 . it is clear that dfa (α,β)(γ,δ) := ia(α; γ) ∩ea(β; δ). let 〈(α,β) ; a〉 and 〈(f,g) ; b〉 be two double-framed soft sets of a over u. then the int-uni soft product [23] is denoted by 〈(α,β) ; a〉♦〈(f,g) ; b〉 and is defined as a double framed soft set 〈(α◦̃f,β◦̃g) ; s〉 defined to be a double-framed soft set over u, in which α◦̃f, and β◦̃g are soft mappings from s to p(u), given as follows: α◦̃f : s −→ p(u),x 7−→   ⋃ x=yz {α (y) ∩f (z)} if ∃ y,z ∈ s such that x = yz, ∅ otherwise, β◦̃g : s −→ p(u),x 7−→   ⋂ x=yz {β (y) ∪g (z)} if ∃ y,z ∈ s such that x = yz, u otherwise. one can easily see that the operation “♦” is well-defined. let 〈(α,β) ; a〉 and 〈(f,g) ; b〉 be two double-framed soft sets of a and b respectively over a common universe u. then 〈(α,β) ; a〉 is called a double-framed soft subset (briefly, dfs subset) of 〈(f,g) ; b〉, denoted by 〈(α,β) ; a〉v 〈(f,g) ; b〉, if i) a ⊆ b, ii) (∀e ∈ a) ( α and f are identical approximations. i.e. α(e) ⊆ f(e) βc and gc are identical approximations. i.e. β(e) ⊇ g(e) ) . for any two dfs sets 〈(α,β) ; a〉 and 〈(f,g) ; a〉 of a over u, the dfs int-uni set [21] of 〈(α,β) ; a〉 and 〈(f,g) ; a〉 , is defined to be a dfs set 〈( α∩̃f,β∪̃g ) ; a 〉 whereα∩̃f, and β∪̃g are mappings given by α∩̃f : a → p (u) ,x → α (x) ∩f (x), β∪̃g : a → p (u) ,x → β (x) ∪g (x). it is denoted by 〈(α,β) ; a〉u〈(f,g) ; a〉 = 〈( α∩̃f,β∪̃g ) ; a 〉 . for a non-empty subset a of s, the dfs set xa=(χa,χ c a; a) is called the double framed characteristic soft set where χa : s → p(u),x → { u if x ∈ a ∅ if x /∈ a , χca : s → p(u),x → { ∅ if x ∈ a u if x /∈ a . we have the following lemmas. lemma 3.1. (cf. [24]) if s is an ag-groupoid then the set (dfs(u),♦) is an ag-groupoid. lemma 3.2. (cf. [24]) if s is an ag-groupoid then the medial law holds in dfs(u). that is for 〈(α,β) ; s〉 ,〈(f,g) ; s〉 , 〈(h,k) ; s〉 and 〈(p,q) ; s〉 ∈ dfs(u), we have (α◦̃f) ◦̃(h◦̃p) = ((h◦̃p) ◦̃f) ◦̃α and (β◦̃g) ◦̃(k◦̃q) = ((k◦̃q) ◦̃g) ◦̃β. lemma 3.3. (cf. [24]) if s is an ag-groupoid with left identity then the paramedial law holds in dfs(u). that is for all 〈(α,β) ; s〉 ,〈(f,g) ; s〉 , 〈(h,k) ; s〉 and 〈(p,q) ; s〉 ∈ dfs(u), (α◦̃f)◦̃(h◦̃p) = (p◦̃f)◦̃(h◦̃α) and (β◦̃g)◦̃(k◦̃q) = (q◦̃g)◦̃(k◦̃β) lemma 3.4. let 〈(α,β) ; s〉 ,〈(f,g) ; s〉 ,〈(h,k); s〉 and 〈(p,q) ; s〉 ∈ dfs(u) then, i) 〈(α,β) ; s〉� (〈(f,g) ; s〉u〈(h,k); s〉) = (〈(α,β) ; s〉� 〈(f,g) ; s〉) u (〈(α,β) ; s〉� 〈(h,k); s〉) . ii) if 〈(f,g) ; s〉v 〈(h,k); s〉 then 〈(α,β) ; s〉� 〈(f,g) ; s〉v 〈(α,β) ; s〉� 〈(h,k); s〉 . iii) if 〈(α,β) ; s〉v 〈(f,g) ; s〉 and 〈(h,k); s〉v 〈(p,q) ; s〉 then 〈(α,β) ; s〉�〈(h,k); s〉v 〈(f,g) ; s〉� 〈(p,q) ; s〉 . proof. straightforward. � characterizations of abel grassmann’s groupoids 65 lemma 3.5. let a and b be two non empty subsets of an ag-groupoid s then the following properties hold: i) if a ⊆ b then xav xb. ii) xa u xb = xa∩b. iii) xa � xb = xab. proof. straightforward. � 4. double-framed soft ideals in this section, we define double-framed soft ag-groupoids, double-framed soft left (resp. right ) ideal of s over u and discuss their properties in regular, intra-regular, right regular and left regular ag-groupoids. definition 4.1. [24] let s be an ag-groupoid and 〈(α,β); a〉 be a dfs-set of a over u. then 〈(α,β) ; a〉 is called a double-framed soft ag-groupoid (briefly, dfs ag-groupoid) of a over u if it satisfies α (xy) ⊇ α (x) ∩α (y) and β (xy) ⊆ β (x) ∪β (y) for all x,y ∈ a. example 4.1. consider an ag-groupoid s = {0, 1, 2, 3, 4} with the following multiplication table: · 0 1 2 3 4 0 4 1 1 2 4 1 1 1 1 1 1 2 1 1 1 1 1 3 1 1 1 2 1 4 4 1 1 1 4 consider a double-framed soft 〈(α,β) ; s〉 of s over u = z− defined by: α (0) = {−1} ,α (1) = {−1,−2,−3,−4,−5,−6} ,α (2) = {−1,−2,−3} ,α (3) = {−1,−3} , α(4) = {−1,−3,−5}. β (0) = {−1,−2,−3,−4,−5} ,β (1) = {−1,−2} ,β(2) = {−1,−2,−3,−4} , β (3) = {−1,−2,−3,−4,−6} ,β(4) = {−2,−4}. by routine checking it is easy to verify that 〈(α,β); s〉 is a double-framed soft ag-groupoid of s over u. again consider u = {[ x x 0 0 ] |x ∈ z3 } , the set of all 2 × 2 matrices with entries from z3 be the universal set. define a double-framed soft set 〈(f,g) ; b〉 of s over u as follows: f (0) = {[ 0 0 0 0 ] , [ 2 2 0 0 ]} = f(2), f (1) = f(3) = {[ 0 0 0 0 ] , [ 1 1 0 0 ]} , f (4) = {[ 1 1 0 0 ] , [ 2 2 0 0 ]} . g (0) = {[ 0 0 0 0 ] , [ 1 1 0 0 ]} = g (2) ,g(1) = g(3) {[ 0 0 0 0 ] , [ 2 2 0 0 ]} , g (4) = {[ 0 0 0 0 ] , [ 1 1 0 0 ] , [ 2 2 0 0 ]} . since f (3 · 3) = f (2) + f (3) ∩f(3) and/or g (0 · 0) = g (4) * g (0) ∪g(0). hence, 〈(f,g) ; b〉 is not a dfs ag-groupoid of s over u. theorem 4.1. let 〈(α,β) ; a〉 be a dfs-set over u. then 〈(α,β) ; a〉 is a dfs ag-groupoid over u if and only if 〈(α,β) ; a〉♦〈(α,β) ; a〉v 〈(α,β) ; a〉 . proof. assume that 〈(α,β) ; a〉 is a dfs ag-groupoid over u. let a ∈ a ⊆ s. if (α◦̃α) (a) = ∅ and (β◦̃β) (a) = u, then obviously, (α◦̃α) (a) ⊆ α (a) and (β◦̃β) (a) ⊇ β (a) . suppose that there exist 66 khan, izhar and sezer x,y ∈ s such that a = xy. then (α◦̃α) (a) = ⋃ a=xy {α (x) ∩α (y)}⊆ ⋃ a=xy α (xy) = ⋃ a=xy α (a) = α (a) , and (β◦̃β) (a) = ⋂ a=xy {β (x) ∪β (y)}⊇ ⋂ a=xy β (xy) = ⋂ a=xy β (a) = β (a) . thus, (α◦̃α) (a) ⊆ α (a) and (β◦̃β) (a) ⊇ β (a) .hence 〈(α,β) ; a〉♦〈(α,β) ; a〉v 〈(α,β) ; a〉 . conversely, assume that 〈(α,β) ; a〉♦〈(α,β) ; a〉 v 〈(α,β) ; a〉 . hence α◦̃α ⊆ α and β◦̃β ⊇ β. let x,y ∈ a ⊆ s and a = xy, then we have α (xy) = α (a) ⊇ (α◦̃α) (a) = ⋃ a=xy {α (x) ∩α (y)}⊇ α (x) ∩α (y) and β (xy) = β (a) ⊆ (β◦̃β) (a) = ⋂ a=xy {β (x) ∪β (y)}⊆ β (x) ∪β (y) . hence, 〈(α,β) ; a〉 is a dfs ag-groupoid over u. � theorem 4.2. for a dfs-set 〈(α,β) ; a〉 of a, the following are equivalent: (1) 〈(α,β) ; a〉 is a dfs ag-groupoid of a. (2) the non-empty γ-inclusive set and δ-exclusive set of 〈(α,β) ; a〉 are sub ag-groupoids of s for any subsets γ and δ of u. proof. suppose that 〈(α,β) ; a〉 is dfs ag-groupoid of a. let γ and δ be subsets of u such that ia(α; γ) 6= ∅ 6= ea(β; δ). then there exist x,a ∈ a such that α(x) ⊇ γ and β(a) ⊆ δ. let p,q ∈ ia(α; γ) then α(p) ⊇ γ and α(q) ⊇ γ. since 〈(α,β) ; a〉 is dfs ag-groupoid of a, hence α(pq) ⊇ α(p) ∩α(q) ⊇ γ ∩γ = γ. thus pq ∈ ia(α; γ) and so ia(α; γ) is sub ag-groupoid of s. now suppose v,u ∈ ea(β; δ) then β(v) ⊆ δ and β(u) ⊆ δ. since 〈(α,β) ; a〉 is dfs ag-groupoid of a, hence β(vu) ⊆ β(v) ∪β(u) ⊆ δ ∪δ = δ. thus vu ∈ ea(β; δ) and so ea(β; δ) is sub ag-groupoid of s. conversely, suppose the non-empty γ-inclusive set and δ-exclusive set of 〈(α,β) ; a〉 are sub aggroupoids of s for any subsets γ and δ of u. let x,y ∈ a such that α(x) = γ1, α(y) = γ2, β(x) = δ1, β(y) = δ2. let us take γ = γ1 ∩γ2 and δ = δ1 ∪δ2..now α(x) = γ1 ⊇ γ1 ∩γ2 = γ and so x ∈ ia(α; γ). similarly y ∈ ia(α; γ). by hypothesis, ia(α; γ) is a sub ag-groupoid of s, hence xy ∈ ia(α; γ) and so α(xy) ⊇ γ = γ1 ∩ γ2 = α(x) ∩ α(y). also as β(x) = δ1 ⊆ δ1 ∪ δ2 = δ then x ∈ ea(β; δ). similarly y ∈ ea(β; δ). by hypothesis, ea(β; δ) is a sub ag-groupoid of s, hence xy ∈ ea(β; δ) and so β(xy) ⊆ δ = δ1 ∪ δ2 = β(x) ∪β(y). therefore 〈(α,β; a)〉 is a dfs ag-groupoid of a. � for any dfs-set 〈(α,β) ; e〉 of e, let 〈(α∗,β∗); e〉 be a dfs-set of e defined by α∗ : e → p(u), x → { α(x) if x ∈ ie(α; γ) η otherwise β∗ : e → p(u), x → { β(x) if x ∈ ee(β; δ) ρ otherwise where γ,δ,η and ρ are subsets of u with η ( α(x) and ρ ) β(x). theorem 4.3. if 〈(α,β) ; a〉 is a dfs ag-groupoid of a over u then so is 〈(α∗,β∗); a〉. characterizations of abel grassmann’s groupoids 67 proof. suppose that 〈(α,β) ; a〉 is a dfs ag-groupoid of a over u then non-empty γ-inclusive set and δ-exclusive set of 〈(α,β) ; a〉 are sub ag-groupoids of s for any subsets γ and δ of u. let x,y ∈ a. if x,y ∈ ia(α; γ) then xy ∈ ia(α; γ) and hence α∗(xy) = α(xy) ⊇ α(x) ∩ α(y) = α∗(x) ∩ α∗(y). if x /∈ ia(α; γ) or y /∈ ia(α; γ) then α∗(x) = η or α∗(y) = η. hence α∗(xy) ⊇ η = α∗(x) ∩α∗(y). now if x,y ∈ ea(β; δ) then xy ∈ ea(β; δ) and hence β∗(xy) = β(xy) ⊆ β(x)∪β(y) = β∗(x)∪β∗(y). if x /∈ ea(β; δ) or y /∈ ea(β; δ) then β∗(x) = ρ or β∗(y) = ρ. hence β∗(xy) ⊆ ρ = β∗(x) ∪ β∗(y). therefore 〈(α∗,β∗); a〉 is a dfs ag-groupoid of a. � the converse of this theorem is not true in general. example 4.2. suppose there are ten patients in the initial universe u given by: u = {p1,p2,p3,p4,p5,p6,p7,p8,p9,p10}. let e = {e1,e2,e3,e4} be set of parameters showing status of patients in which e1 stands for the parameter “chest pain” e2 stands for the parameter “head ache” e3 stands for the parameter “tooth ache” e4 stands for the parameter “back pain” with the following multiplication table · e1 e2 e3 e4 e1 e3 e3 e3 e4 e2 e4 e4 e3 e3 e3 e4 e4 e4 e4 e4 e4 e4 e4 e4 define a dfs set 〈(α,β); e〉 by α : e −→ p(u), x −→   {p1,p2,p3,p5} if x = e1 {p1,p2,p3,p4,p5} if x = e2 {p1,p3,p5,p7} if x = e3 {p1,p3,p5,p7,p9} if x = e4 β : e −→ p(u), x −→   {p1,p3} if x = e1 {p1,p3,p5} if x = e2 {p1,p2} if x = e3 {p1,p2} if x = e4 then ie(α; γ) = {e3,e4} for γ = {p1,p3,p5,p7} and ee(β; δ) = {e3,e4} for δ = {p1,p2}. according to the definition, we have 〈(α∗,β∗); e〉 is defined as α∗ : e −→ p(u), x −→   {p1,p3} if x = e1 {p1,p3} if x = e2 {p1,p3,p5,p7} if x = e3 {p1,p3,p5,p7,p9} if x = e4 β∗ : e −→ p(u), x −→   {p1,p2,p3,p4,p5} if x = e1 {p1,p2,p3,p4,p5} if x = e2 {p1,p2} if x = e3 {p1,p2} if x = e4 by routine checking, we have 〈(α∗,β∗); e〉 is dfs ag-groupoid. but 〈(α,β); e〉 is not dfs aggroupoid because α(e3) = α(e1e1) + α(e1) ∩α(e2) or β(e3) = β(e1e1) * β(e1) ∪β(e1). theorem 4.4. let a be a nonempty subset of an ag-groupoid s. then a is a sub ag-groupoid of s if and only if the dfs-set xa = 〈(χa,χca) ; a〉 is a dfs ag-groupoid of s over u. proof. straightforward. � let 〈(α,β) ; a〉 and 〈(α,β) ; b〉 be two dfs-sets over u then (α∧,β∨)-product of 〈(α,β) ; a〉 and 〈(α,β) ; b〉 is defined [21] to be a dfs-set 〈(αa∧b,βa∨b); a×b〉 over u in which 68 khan, izhar and sezer αa∧b : a×b → p(u), (x,y) → α(x) ∩α(y) βa∨b : a×b → p(u), (x,y) → β(x) ∪β(y) theorem 4.5. for any ag-groupoids e and f as set of parameters, let 〈(α,β); e〉 and 〈(α,β); f〉 be dfs ag-groupoids of e and f respectively. then (α∧,β∨)-product of 〈(α,β); e〉 and 〈(α,β); f〉 is a dfs ag-groupoid of e ×f . proof. we note that e ×f is also an ag-groupoid with the operation (a,b) ∗ (c,d) = (ac,bd) for all (a,b), (c,d) ∈ e ×f . let (u,v), (s,t) ∈ e ×f, we have αe∧f ((u,v) ∗ (s,t)) = αe∧f (us,vt) = α(us) ∩α(vt) ⊇ α(u) ∩α(s) ∩α(v) ∩α(t) = α(u) ∩α(v) ∩α(s) ∩α(t) = αe∧f (u,v) ∩αe∧f (s,t), and βe∨f ((u,v) ∗ (s,t)) = βe∨f (us,vt) = β(us) ∪β(vt) ⊆ β(u) ∪β(s) ∪β(v) ∪β(t) = β(u) ∪β(v) ∪β(s) ∪β(t) = βe∨f (u,v) ∪βe∨f (s,t). hence (α∧,β∨)-product of 〈(α,β); e〉 and 〈(α,β); f〉 is a dfs ag-groupoid of e ×f. � definition 4.2. a dfs-set 〈(α,β) ; a〉 of a over u is called a double-framed soft left (resp. right) ideal (briefly, dfs left (right) ideal) of a over u if it satisfies: α (ab) ⊇ α (b) ( resp. α (ab) ⊇ α (a)) and β (ab) ⊆ β (b) (resp. β (ab) ⊆ β (a)) for all a,b ∈ a. a dfs-set 〈(α,β) ; a〉 of a over u is called a double-framed soft two-sided ideal (briefly, dfs two-sided ideal) of a over u if it is both a dfs left and a dfs right ideal of a over u. example 4.3. there are six women patients in the initial universe set u given by u := {p1,p2,p3,p4,p5,p6}. let s := {e0,e1,e2} be the set of status of each patient in u with the following type of disease e0 stands for the parameter “headache”, e1 stands for the parameter “chest pain”, e2 stands for the parameter “mental depression”, with the following binary operation ∗ given in the cayley table: ∗ e0 e1 e2 e0 e0 e0 e0 e1 e2 e2 e2 e2 e0 e0 e0 then (s,∗) is an ag-groupoid. consider a dfs-set 〈(α,β) ; s〉 over u as follows: α : s −→ p(u),x 7−→   {p1,p2,p3} if x = e0, {p2,p3} if x = e1, {p1,p2,p3} if x = e2, and β : s −→ p(u),x 7−→   {p2,p4} if x = e0, {p1,p2,p3,p4} if x = e1, {p1,p2,p4} if x = e2. then one can easily show that 〈(α,β) ; s〉 is a dfs ideal over u. however, if we define another double-framed soft set 〈(f,g) ; s〉 such that f : s −→ p(u),x 7−→   {p1,p2,p6} if x = e0 {p1} if x = e1 {p2,p4,p6} if x = e2 and characterizations of abel grassmann’s groupoids 69 g : s −→ p(u),x 7−→   {p2,p4,p6} if x = e0 {p1,p6} if x = e1 {p1,p2,p3} if x = e2 . then 〈(f,g) ; s〉 is not dfs ideal of s over u, because f (e2 ∗e0) = f (e0) = {p1,p2,p6} + {p2,p4,p6} = f (e2) and/or g (e2 ∗e0) = f (e0) = {p2,p4,p6} * {p1,p2,p3,p4} = g (e2) . proposition 4.1. let 〈(α,β) ; a〉 be a dfs-set over u. then 〈(α,β) ; a〉 is a dfs ideal of s over u if and only if α (xy) ⊇ α (x) ∪α (y) and β (xy) ⊆ β (x) ∩β (y) for all x,y ∈ s. proof. let 〈(α,β) ; a〉 be a dfs ideal of s over u. then α (xy) ⊇ α (y) ,α (xy) ⊇ α (x) and β (xy) ⊆ β (y) ,β (xy) ⊆ β (x) for all x,y ∈ s. thus, α (xy) ⊇ α (x) ∪ α (y) and β (xy) ⊆ β (x) ∩ β (y) for all x,y ∈ s. conversely, suppose that α (xy) ⊇ α (x) ∪ α (y) and β (xy) ⊆ β (x) ∩ β (y) for all x,y ∈ s. then α (xy) ⊇ α (x) ∪α (y) ⊇ α (x) ,α (y) and β (xy) ⊆ β (x) ∩β (y) ⊆ β (x) ,β (y) . hence 〈(α,β) ; a〉 is a dfs ideal of s over u. � proposition 4.2. let 〈(α,β) ; a〉 be a dfs-set over u. if 〈(α,β) ; a〉 is a dfs left (resp., right or two-sided) ideal over u. then 〈(α,β) ; a〉 is a dfs ag-groupoid over u. proof. straightforward. � proposition 4.3. if s is an ag-groupoid with left identity e then every dfs right ideal is dfs ideal. proof. let 〈(α,β) ; a〉 be a dfs right ideal of a over u. now let x,y ∈ a, then α(xy) = α((ex)y) = α((yx)e) ⊇ α(yx) ⊇ α(y) and β(xy) = β((ex)y) = β((yx)e) ⊆ β(yx) ⊆ β(y). hence α(xy) ⊇ α(y) and β(xy) ⊆ β(y) for all x,y ∈ a. thus 〈(α,β) ; a〉 is dfs left ideal and hence 〈(α,β) ; a〉 is dfs ideal of a over u. � the converse of the above theorem is not true in general. example 4.4. let s = {1, 2, 3, 4} with the following multiplication table: · 1 2 3 4 1 2 2 4 4 2 2 2 2 2 3 1 2 3 4 4 1 2 1 2 it is easy to see that 3 is left identity in s. consider a dfs-set 〈(α,β) ; s〉 over u = z as follows: α : s −→ p(u),x 7−→   4z if x = 1, z if x = 2, 8z if x = 3, 2z if x = 4 and β : s −→ p(u),x 7−→   8z if x = 1, 16z if x = 2, z if x = 3, 4z if x = 4 then one can easily show that 〈(α,β) ; s〉 is a dfs left ideal over u. however, 〈(α,β); s〉 is not dfs right ideal over u because α(1) = α(41) + α(4) and/or β(4) = β(14) * β(1). 70 khan, izhar and sezer proposition 4.4. (cf. [24]) let a be a nonempty subset of an ag-groupoid s. then a is an ideal of s if and only if the dfs-set xa = 〈(χa,χca) ; a〉 is a dfs ideal of s over u. proof. straightforward � theorem 4.6. (cf. [24]) a dfs set 〈(α,β); a〉 is dfs left (resp. right ) ideal of a over u if and only if xa �〈(α,β); a〉v 〈(α,β); a〉 (resp. 〈(α,β); a〉� xa v〈(α,β); a〉). proof. straightforward � theorem 4.7. if 〈(α,β) ; s〉 is a dfs left (resp. right) ideal of s over u then so is 〈(α∗,β∗); s〉. proof. suppose that 〈(α,β) ; s〉 is a dfs left ideal of s over u then non-empty γ-inclusive set and δexclusive set of 〈(α,β) ; s〉 are left ideals of s for any subsets γ and δ of u. let a,b ∈ s. if b ∈ is(α; γ) then ab ∈ is(α; γ). thus α∗(ab) = α(ab) ⊇ α(b) = α∗(b). if b /∈ is(α; γ) then ab ∈ is(α; γ) or ab /∈ is(α; γ). if ab ∈ is(α; γ) then α∗(ab) = α(ab) ⊃ η = α∗(b). if ab /∈ is(α; γ) then α∗(ab) = η = α∗(b). in either case α∗(ab) ⊇ α∗(b) for all a,b ∈ s. now if b ∈ es(β; δ) then ab ∈ es(β; δ) and hence β∗(ab) = β(ab) ⊆ β(b) = β∗(b). if b /∈ es(β; δ) then ab ∈ es(β; δ) or ab /∈ es(β; δ). if ab ∈ es(β; δ) then β∗(ab) = β(ab) ⊂ ρ = β∗(b). if ab /∈ es(β; δ) then β∗(ab) = ρ = β∗(b). in either case β∗(ab) ⊆ β∗(b). therefore 〈(α∗,β∗); s〉 is a dfs left ideal of s. over u. in a similar fashion, we can prove the result for dfs right ideal. � the converse of the above theorem is not true in general. example 4.5. suppose u = z and s = {0, 1, 2} with the following multiplication table · 0 1 2 0 0 0 0 1 2 2 2 2 0 0 0 then (s, ·) an ag-groupoid. consider a dfs 〈(α,β); s〉 over u as follows: α : s → p(u), x 7−→   4z if x = 0 6z if x = 1 4z if x = 2 and β : s → p(u), x 7−→   16z if x = 0 6zif x = 1 16z if x = 2 then for γ = δ = 4z we have is(α; γ) = es(β; δ) = {0, 2}. now define 〈(α∗,β∗); s〉 as follows: α∗ : s → p(u), x 7−→   4z if x = 0 12z if x = 1 4z if x = 2 and β∗ : s → p(u), x 7−→   16z if x = 0 z if x = 1 16z if x = 2 routine calculations shows that 〈(α∗,β∗); s〉 is a dfs left ideal over u. but 〈(α,β); s〉 is not dfs left ideal over u since α(0) = α(01) + α(1) and/or β(0) = β(01) * β(1). characterizations of abel grassmann’s groupoids 71 theorem 4.8. for any ag-groupoids e and f as set of parameters, let 〈(α,β); e〉 and 〈(α,β); f〉 be dfs left (resp. right) ideals of e and f respectively. then (α∧,β∨)-product of 〈(α,β); e〉 and 〈(α,β); f〉 is a dfs left (resp. right) ideal of e ×f . proof. by definition, the (α∧,β∨)-product of 〈(α,β); e〉 and 〈(α,β); f〉 is a dfs 〈(αe∧f ,βe∨f ); e ×f〉 in which αe∧f : e ×f → p(u), (x,y) → α(x) ∩α(y) and βe∨f : e ×f → p(u), (x,y) → β(x) ∪β(y). we note that e × f is also an ag-groupoid with the operation (a,b) ∗ (c,d) = (ac,bd) for all (a,b), (c,d) ∈ e ×f . let (u,v), (s,t) ∈ e×f, we have αe∧f ((u,v)∗(s,t)) = αe∧f (us,vt) = α(us)∩α(vt) ⊇ α(s)∩α(t) = αe∧f (s,t), and βe∨f ((u,v) ∗ (s,t)) = βe∨f (us,vt) = β(us) ∪β(vt) ⊆ β(s) ∪β(t) = βe∨f (s,t). hence (α∧,β∨)product of 〈(α,β); e〉 and 〈(α,β); f〉 is a dfs left ideal of e ×f. � theorem 4.9. a dfs-set of a right regular ag-groupoid s is dfs left ideal iff it is a dfs right ideal of s over u. proof. let s be a right regular ag-groupoid and let 〈(α,β); a〉 be a dfs-left ideal of a over u. now let a,b ∈ a ⊆ s, so a ∈ s. but since s is right regular so there exist an element x such that a = (aa)x. now α(ab) = α(((aa)x)b) = α((bx)(aa)) ⊇ α(aa) ⊇ α(a) and β(ab) = β(((aa)x)b) = β((bx)(aa)) ⊆ β(aa) ⊆ β(a). hence 〈(α,β); a〉 is dfs-right ideal of a over u. conversely, let 〈(α,β); a〉 be a dfs-right ideal of a over u. take a,b ∈ a ⊆ s, so a ∈ s. but since s is right regular so there exist an element x such that a = (aa)x. now α(ab) = α(((aa)x)b) = α((bx)(aa)) = α((ba)(xa)) ⊇ α(ba) ⊇ α(b) and β(ab) = β(((aa)x)b) = β((bx)(aa)) = β((ba)(xa)) ⊆ β(ba) ⊆ β(b). hence 〈(α,β); a〉 is dfs-left ideal of a over u. � proposition 4.5. a dfs-set of an intra-regular ag-groupoid s is a dfs right ideal if and only if it is a dfs left ideal. proof. let 〈(α,β) ; a〉 be a dfs right ideal of a over u. let a,b ∈ a. since a ∈ s and s is intra-regular ag-groupoid so there exists x,y ∈ s such that a = (xa2)y. then we have α(ab) = α(((xa2)y)b) = α((by)(xa2)) ⊇ α(by) ⊇ α(b). also β(ab) = β(((xa2)y)b) = β((by)(xa2)) ⊆ β(by) ⊆ β(b). hence 〈(α,β) ; a〉 is dfs left ideal of a over u. conversely, assume that 〈(α,β) ; a〉 is a dfs left ideal of a over u. now α(ab) = α(((xa2)y)b) = α((by)(xa2)) ⊇ α(xa2) ⊇ α(a2) ⊇ α(a). also β(ab) = β(((xa2)y)b) = β((by)(xa2)) ⊆ β(xa2) ⊆ β(a2) ⊆ β(a). hence 〈(α,β) ; a〉 is dfs right ideal of a over u. � proposition 4.6. a dfs right ideal of a regular ag-groupoid s is a dfs left ideal of s. proof. let 〈(α,β); s〉 be a dfs right ideal of a regular ag-groupoid s. let x,y ∈ s. since x ∈ s and s is regular so there exist a ∈ s such that x = (xa)x. thus, α(xy) = α(((xa)x)y) = α((yx)(xa)) ⊇ α(yx) ⊇ α(y) and β(xy) = β(((xa)x)y) = β((yx)(xa)) ⊆ β(yx) ⊆ β(y). hence 〈(α,β); s〉 is dfs left ideal. � proposition 4.7. every dfs right ideal of a regular ag-groupoid s is idempotent. proof. let 〈(α,β); s〉 be a dfs right ideal of a regular ag-groupoid s. then 〈(α,β); s〉�〈(α,β); s〉v 〈(α,β); s〉� xs v〈(α,β); s〉 . now we show that 〈(α,β); s〉v 〈(α,β); s〉�〈(α,β); s〉 . since s is regular, so for any x ∈ s, there exist an element y ∈ s such that x = (xy)x. we have (α◦̃α)(x) = ⋃ x=ab {α(a) ∩α(b)}⊇ α(xy) ∩α(x) ⊇ α(x) ∩α(x) = α(x) and (β◦̃β)(x) = ⋂ x=ab {β(a) ∪β(b)}⊆ β(xy) ∪β(x) ⊆ β(x) ∪β(x) = β(x). hence 〈(α,β); s〉 v 〈(α,β); s〉 � 〈(α,β); s〉 and so 〈(α,β); s〉 = 〈(α,β); s〉 � 〈(α,β); s〉, which is the desired result. � proposition 4.8. let 〈(α,β) ; s〉 be a dfs right ideal and 〈(f,g) ; s〉 a dfs left ideal of s over u, respectively. then 〈(α,β) ; s〉♦〈(f,g) ; s〉v 〈(α,β) ; s〉u〈(f,g) ; s〉. 72 khan, izhar and sezer proof. let 〈(α,β) ; s〉 be a dfs right ideal and 〈(f,g) ; s〉 be dfs left ideal of s over u. then 〈(α,β); s〉v xs and 〈(f,g) ; s〉v xs always true. we have 〈(α,β); s〉� 〈(f,g) ; s〉v 〈(α,β); s〉� xs v 〈(α,β); s〉 and 〈(α,β); s〉 � 〈(f,g) ; s〉 v xs♦〈(f,g) ; s〉 v 〈(f,g) ; s〉 . it follows that 〈(α,β); s〉 � 〈(f,g) ; s〉 v 〈(α,β); s〉u〈(f,g) ; s〉 . � definition 4.3. a dfs 〈(α,β); a〉 of a over u is called dfs semiprime if α(a) ⊇ α(a2) and β(a) ⊆ β(a2) for all a ∈ a. theorem 4.10. for a non empty subset a of an ag-groupoid s, the following conditions are equivalent: i) a is semiprime. ii) the dfs characteristics function xa is dfs semiprime. proof. (i)⇒(ii). assume that a is semiprime. let a ∈ a. if a2 ∈ a then a ∈ a since a is semiprime. thus χa(a) = u = χa(a 2) and χca(a) = ∅ = χ c a(a 2). if a2 /∈ a then χa(a2) = ∅⊆ χa(a) and χca(a 2) = u ⊇ χca(a). hence xa is dfs semiprime. (ii)⇒(i). assume xa is dfs semiprime. let a2 ∈ a. then u = χa(a2). but χa(a) ⊇ χa(a2) = u. hence χa(a) = u and so a ∈ a. also χca(a) ⊆ χ c a(a 2) = ∅, so χca(a) = ∅. thus a ∈ a. hence a is semiprime. � proposition 4.9. for any dfs ag-groupoid 〈(α,β); a〉 of a over u, the following conditions are equivalent: i) 〈(α,β); a〉 is dfs semiprime. ii) α(a) = α(a2) and β(a) = β(a2) for all a ∈ a. proof. (i)⇒(ii). assume 〈(α,β); a〉 is dfs semiprime and let a ∈ a. now α(a) ⊇ α(a2) = α(aa) ⊇ α(a)∩α(a) ⊇ α(a), so α(a) = α(a2). also β(a) ⊆ β(a2) = β(aa) ⊆ β(a)∪β(a) = β(a), so β(a) = β(a2). (ii)⇒(i). it is obvious. � proposition 4.10. for an ag-groupoid s with left identity e, the following conditions are equivalent: i) s is left regular. ii) every dfs left ideal of s is dfs semiprime. proof. (i)⇒(ii). assume that s is left regular. let 〈(α,β); s〉 is dfs left ideal of s. let a ∈ s. since s is left regular so there exist x ∈ s such that a = x(aa). now α(a) = α(x(aa)) ⊇ α(aa) = α(a2) and β(a) = β(x(aa)) ⊆ β(aa) = β(a2). thus 〈(α,β); s〉 is dfs semiprime. (ii)⇒(i). assume (ii) holds. since sa2 is left ideal so xsa2 = 〈 (χsa2,χ c sa2 ); sa2 〉 is dfs left ideal and so by hypothesis xsa2 = 〈 (χsa2,χ c sa2 ); sa2 〉 is dfs semiprime. since s is ag-groupoid with left identity so a2 ∈ sa2 and hence u = χsa2 (a2) ⊇ χsa2 (a) ⊇ χsa2 (a2). thus χsa2 (a) = u hence a ∈ sa2. in the other case ∅ = χc sa2 (a2) ⊆ χc sa2 (a) ⊆ χc sa2 (a2). so χc sa2 (a) = ∅which imply a ∈ sa2. hence in any case a ∈ sa2 and so s is left regular. � 5. characterizations of intra-regular ag-groupoids in terms of dfs ideals in this section, we give some characterizations of intra-regular ag-groupoids using their dfs ideals. proposition 5.1. for an ag groupoid s, the following conditions are equivalent: i) s is intra-regular. ii) every dfs ideal 〈(α,β); a〉 is dfs soft semiprime. iii) α(a) = α(a2) and β(a) = β(a2) for every dfs ideal 〈(α,β); a〉 for all a ∈ a. proof. (i)⇒(iii). suppose that s is intra-regular. let 〈(α,β); a〉 is a dfs ideal which is semiprime. take a ∈ a ⊆ s, so there exist x,y ∈ s such that a = (xa2)y. thus, α(a) = α((xa2)y) ⊇ α(xa2) = α(x(aa)) ⊇ α(aa) ⊇ α(a) and so α(a) = α(a2). now β(a) = β((xa2)y) ⊆ β(xa2) = β(x(aa)) ⊆ β(aa) ⊆ β(a) and so β(a) = β(a2). (iii)⇒(i). assume that for every dfs ideal 〈(α,β); a〉 of a over u, we have α(a) = α(a2) and characterizations of abel grassmann’s groupoids 73 β(a) = β(a2) for all a ∈ a. since j[a2] is an ideal of s, so xj[a2] is dfs ideal of s. since a 2 ∈ j[a2], we have χj[a](a) = χj[a2](a 2) = u. thus a ∈ j[a2] = (sa2)s. also χc j[a2] (a) = χc j[a2] (a2) = ∅. in this case, too, a ∈ j[a2] = (sa2)s. hence s is intra-regular. (iii)⇒(ii). obvious. (ii)⇒(iii). let 〈(α,β); a〉 is a dfs ideal which is semiprime. now α(a) ⊇ α(a2) = α(aa) ⊇ α(a). thus α(a) = α(a2). also β(a) ⊆ β(a2) = β(aa) ⊆ β(a). thus β(a) = β(a2). this completes the proof. � theorem 5.1. for an ag-groupoid s with left identity e, the following conditions are equivalent. i) s is intra-regular. ii) l∩r ⊆ lr for every left ideal l and every right ideal r of s and r is semiprime. iii) 〈(α,β); a〉u 〈(f,g); b〉 v 〈(α,β); a〉 � 〈(f,g); b〉 .for every dfs left ideal 〈(α,β); a〉 and every dfs right ideal 〈(f,g); b〉 and 〈(f,g); b〉 is dfs semiprime. proof. (i)⇒(iii). assume that s is intra-regular. let 〈(α,β); a〉 is dfs left ideal and 〈(f,g); b〉 is dfs right ideal over u. since s is intra-regular, so for a ∈ s, there exist elements x,y in s such that a = (xa2)y = ((x(aa))y) = ((a(xa))y) = (y(xa))a = (y(xa))(ea) = (ye)((xa)a) = (xa)((ye)a) = (xa)((ae)y). now (α◦̃f)(a) = ⋃ a=pq {α(p) ∩f(q)}⊇ α(xa) ∩f((ae)y) ⊇ α(a) ∩f(ae) ⊇ α(a) ∩f(a) and (β◦̃g)(a) = ⋂ a=pq {β(p) ∪g(q)}⊆ β(xa) ∪g((ae)y) ⊆ β(a) ∪g(ae) ⊆ β(a) ∪g(a). hence 〈(α,β); a〉u〈(f,g); b〉v 〈(α,β); a〉� 〈(f,g); b〉 . now f(a) = f((xa2)y) = f((xa2)(ey)) = f((ye)(a2x)) = f(a2((ye)x)) ⊇ f(a2) and g(a) = g((xa2)y) = g((xa2)(ey)) = g((ye)(a2x)) = g(a2((ye)x)) ⊆ g(a2). thus 〈(f,g); b〉 is dfs semiprime. (iii)⇒(ii). assume that (iii) holds. let l and r be left ideal and right ideal of s respectively then xl is dfs left ideal and xr is dfs right ideal. thus by hypothesis xl u xr v xlr and xr is dfs semiprime. let a ∈ l ∩ r then a ∈ l and a ∈ r. hence u = χl∩r(a) = (χl ∩ χr)(a) ⊆ (χl ◦χr)(a) = χlr(a). that is χlr(a) = u and so a ∈ lr. in the other case ∅ = χcl∩r(a) = (χ c l ∪ χ c r)(a) ⊇ (χ c l ◦ χ c r)(a) = χ c lr(a). that is χ c lr(a) = ∅ and so a ∈ lr. hence in any case l ∩ r ⊆ lr. by theorem 4.10, since xr is dfs semiprime, so r is semiprime. (ii)⇒(i). assume that (ii) holds. we prove s is intra-regular. let a ∈ s. then a = ea ∈ sa, where sa is left ideal of s and a2 ∈ a2s and a2s is right ideal of s. by hypothesis a2s is semiprime and so a ∈ a2s. thus a ∈ sa ∩ a2s ⊆ (sa)(a2s) = (sa2)(as) ⊆ (sa2)(ss) ⊆ (sa2)s. hence s is intra-regular. � lemma 5.1. for an ag groupoid s with left identity, the following conditions are equivalent: i) s is intra-regular. ii) r∩l = rl for every left ideal l and right ideal r of s and r is semiprime. proof. proof is available in [25]. � theorem 5.2. for an ag-groupoid s with left identity e, the following conditions are equivalent: i) s is intra-regular. ii) 〈(α,β); a〉u〈(f,g); b〉 = 〈(α,β); a〉 � 〈(f,g); b〉 .for every dfs right ideal 〈(α,β); a〉 and every dfs left ideal 〈(f,g); b〉 and 〈(α,β); a〉 is dfs semiprime. proof. (i)⇒(ii). let 〈(α,β); a〉 is dfs right ideal and 〈(f,g); b〉 dfs left ideal of s over u. by proposition 4.8, 〈(α,β); a〉� 〈(f,g); b〉v 〈(α,β); a〉u〈(f,g); b〉. next we have since s is intra-regular, so for each a ∈ s, there exist x,y ∈ s such that a = (xa2)y. thus a = (xa2)y = (x(aa))y = (a(xa))y = (y(xa))a = ((ey)(xa))a = ((ax)(ye))a. hence (α◦̃f)(a) = ⋃ a=pq {α(p) ∩β(q)}⊇ α((ax)(ye)) ∩β(a) ⊇ α(ax) ∩β(a) ⊇ α(a) ∩β(a) 74 khan, izhar and sezer and (β◦̃g)(a) = ⋂ a=pq {β(p) ∪g(q)}⊆ β((ax)(ye)) ∪g(a) ⊆ β(ax) ∪g(a) ⊆ β(a) ∪g(a), and so 〈(α,β); a〉u 〈(f,g); b〉 v 〈(α,β); a〉 � 〈(f,g); b〉. thus 〈(α,β); a〉u 〈(f,g); b〉 = 〈(α,β); a〉 � 〈(f,g); b〉. also α(a) = α((xa2)y) = α((xa2)(ey)) = α((ye)(a2x)) = α(a2((ye)x)) ⊇ α(a2), and β(a) = β((xa2)y) = β((xa2)(ey)) = β((ye)(a2x)) = β(a2((ye)x)) ⊆ β(a2). thus 〈(α,β); a〉 is dfs semiprime. (ii)⇒(i). assume (ii) holds. let l be a left ideal and r be a right ideal of s. then xl is dfs left ideal and xr is dfs right ideal. by hypothesis, xr u xl = xr � xl and xr is dfs semiprime. since xr is dfs semiprime, so by theorem 4.10, r is semiprime. let a ∈ r∩l, then a ∈ r and a ∈ l. hence u = χr∩l(a) = χr(a) ∩χl(a) = (χr◦̃χl)(a) = χrl(a), so a ∈ rl. also ∅ = χcr∩l(a) = χ c r(a) ∩ χ c l(a) = (χ c r◦̃χ c l)(a) = χ c rl(a), so a ∈ rl. in any case r∩l ⊆ rl. the other inclusion rl ⊆ r∩l is obvious, since s is intra-regular. thus r∩l = rl. this along with r is semiprime implies that s is intra-regular. � references [1] d. molodtsov, soft set theory first results, comput. math. appl., 37 (1999) 19-31. [2] p. k. maji, r. biswas and a. r. roy, an application of soft sets in a decision making problem, comput. math. appl., 44 (2002) 1077-1083. [3] p. k. maji, r. biswas and a. r. roy, soft set theory, comput. math. appl., 45 (2003) 555-562. [4] y. b. jun, k. j. lee and a. khan, soft ordered semigroups, math. logic q., 56, (2010) 42–50. [5] u. acar, f. koyuncu and b. tanay, soft sets and soft rings, comput. math. appl., 59, (2010) 3458–3463. [6] a. o. atagun and a. sezgin, soft substructures of rings, fields and modules, comput. math. appl., 61, (2011) 592–601. [7] n. cagman and s. enginoglu, fp-soft set theory and its applications, ann. fuzzy math. inform., 2, (2011) 219–226. [8] f. feng, soft rough sets applied to multicriteria group decision making, ann. fuzzy math. inform., 2, (2011) 69–80. [9] f. feng, y. b. jun and x. zhao, soft semirings, comput. math. appl., 56, (2008) 2621–2628. [10] y. b. jun, soft bck/bci-algebras, comput. math. appl., 56, (2008) 1408–1413. [11] y. b. jun, k. j. lee and c. h. park, soft set theory applied to ideals in d-algebras, comput. math. appl., 57, (2009) 367–378. [12] y. b. jun, k. j. lee and j. zhan, soft p-ideals of soft bci-algebras, comput. math. appl., 58, (2009) 2060–2068. [13] y. b. jun and c. h. park, applications of soft sets in ideal theory of bck/bci-algebras, inform. sci., 178, (2008) 2466–2475. [14] y. b. jun, g. muhiuddin, a. m. al-roqi, ideal theory of bck/bci-algebras based on double-framed soft set, appl. math. inf. sci., 7 (5) (2013) 1879-1887. [15] m. a. kazim and m. naseeruddin, on almost semigroups, aligarh bull. math. 2 (1972) 1-7. [16] j. r. cho, pusan, j. jezek and t. kepka, paramedial groupoids, czechoslovak math. j., 49 (124) (1996) 277-290. [17] p. holgate, groupoids satisfying a simple invertive law, math. stud., 61, (1992) 101-106. [18] p. v. protic and n. stevanovic, on abel-grassmann’s groupoids (review), proc. math. conf. pristina, (1999) 31-38. [19] m. naseeruddin, some studies on almost semigroups and flocks, ph.d. thesis, the aligarh muslim university india, 1970. [20] q. mushtaq and s. m. yusuf, on ag-groupoids, aligarh bull. math., 8 (1978), 65-70. [21] y. b. jun and s. s. ahn, double-framed soft sets with applications in bck/bci-algebras, j. appl. math., 2012 (2012), article id 178159. [22] a. s. sezer, a new approach to la-semigroup theory via the soft sets, j. intell. fuzzy syst. 26 (2014), 2483-2495. [23] a. khan, t. asif and y. b. jun, double-framed soft ordered semigroups, to appear. [24] a. khan, m. izhar, double framed soft la-semigroups, to appear. [25] m. khan, t. asif, characterizations of intra-regular left almost semigroups by their fuzzy ideals, j. math. research 2 (3) (2010), 87-96. [26] f. yousafzai, a. khan, v. amjad, a. zeb, on fuzzy fully regular ag-groupoids, j. intell. fuzzy syst. 26 (2014), 2973-2982. 1department of mathematics, abdul wali khan university mardan, kp, pakistan. 2department of elementary education, amasya university, amasya, turkey. ∗corresponding author: azhar4set@yahoo.com 1. introduction 2. preliminaries 3. soft set (basic operations) 4. double-framed soft ideals 5. characterizations of intra-regular ag-groupoids in terms of dfs ideals references international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 130-147 http://www.etamaths.com general stability of a reciprocal type functional equation in three variables k. ravi1,∗, j.m. rassias2 and b.v. senthil kumar3 abstract. in this paper, we obtain the solution of a reciprocal type functional equation in three variables of the form g (2(k− 1)x1 + x2 + x3) = g((k− 1)x1 + x2)g((k− 1)x1 + x3) g((k− 1)x1 + x2) + g((k− 1)x1 + x3) and investigate its generalized hyers-ulam stability where k ≥ 2 is a positive integer, g : x → r is a mapping with x as the space of non-zero real numbers and 2(k − 1)x1 + x2 + x3 6= 0, g((k − 1)x1 + x2) + g((k − 1)x1 + x2) 6= 0, for all x1,x2,x3 ∈ x. we also provide counter-examples for non-stability. 1. introduction an inquisitive question that was given a serious thought by s.m. ulam [42] concerning the stability of group homomorphisms gave rise to the stability problem of functional equations. the laborious intellectual strivings of d.h. hyers [15] did not go in vain because he was the first to come out with a partial answer to solve the question posed by ulam on banach spaces. in course of time, the theorem formulated by hyers was generalized by t. aoki [4] for additive mappings and by th.m. rassias [40] for linear mappings by taking into consideration an unbounded cauchy difference. the findings of th.m. rassias have exercised a delectable influence on the development of what is addressed as the generalized hyers-ulam-rassias stability of functional equations. a generalized and modified form of the theorem evolved by th.m. rassias was advocated by p. gavruta [13] who replaced the unbounded cauchy difference by driving into study a general control function within the viable approach designed by th.m. rassias. in 1982-1989, a generalization of the result of d.h. hyers was proved by j.m. rassias using weaker conditions controlled by a product of different powers of norms ([31], [32], [33]). the investigation of stability of functional equations involving with the mixed type product-sum of powers of norms is introduced by j.m. rassias [34]. a further research materialized by f. skof [41] found solution to hyers-ulamrassias stability problem for quadratic functional equation (1.1) f(x + y) + f(x−y) = 2f(x) + 2f(y) 2010 mathematics subject classification. 39b22, 39b52, 39b72. key words and phrases. reciprocal function; reciprocal type functional equation; generalized hyers-ulam stability. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 130 stability of a reciprocal functional equation in three variables 131 for a class of functions f : a → b, where a is a normed space and b is a banach space. the functional equation (1.1) is used to characterize inner product spaces ([1], [2], [24]). the stability problems of several functional equations have been extensively investigated by a number of mathematicians, posed with creative thinking and critical dissent who have arrived at interesting results (see [5], [8], [9], [11], [14], [16], [25], [35]). functional equations find a lot of application in information theory, information science, measure of information, coding theory, computer graphics, spatial filtering in image processing, behavioral and social sciences, astronomy, number theory, fuzzy system models, economics, stochastic processes, mechanics, cryptography and physics. in 1998, s.m. jung [18] investigated the hyers-ulam-rassias stability for the jensen functional equation (1.2) 2f ( x + y 2 ) = f(x) + f(y) and applied the stability result to the study of an asymptotic behaviour of the additive mappings. in 2008, w.g. park and j.h. bae [29] obtained the general solution and the stability of the functional equation f(x + y + z,u + v + w) + f(x + y −z,u + v + w) + 2f(x,u,−w) + 2f(y,v,−w) = f(x + y,u + w) + f(x + y,v + w) + f(x + z,u + w) + f(x−z,u + v −w) + f(y + z,v + w) + f(y −z,u + v −w).(1.3) the function f : r × r → r given by f(x,y) = x3 + ax + b − y2 having level curves as elliptic curves is a solution of (1.3). the stability result of the functional equation (1.3) is related with the canonical height function of the elliptic curves. in the year 1996, isac and th.m. rassias [17] were the first to provide applications of stability theorem of functional equations for the proof of new fixed point theorems with applications. usually, the stability problem for functional equations is solved by direct method in which the exact solution of the functional equation is explicitly constructed as a limit of a (hyers) sequence, starting from the given approximate solution of f (see [3], [10], [16], [18], [19]). in the year 2003, radu [30] proposed a new method for obtaining the existence of exact solutions and error estimations, based on the fixed point alternative. this method was recently been used by many authors (see [20], [21], [22], [23], [27], [28]). cadariu and radu ([6], [7]) applied a fixed point method to investigate the jensen’s and cauchy additive functional equations. in the year 2010, k. ravi and b.v. senthil kumar [36] investigated the generalized hyers-ulam stability for the reciprocal functional equation (1.4) g(x + y) = g(x)g(y) g(x) + g(y) where g : x → y is a mapping on the spaces of non-zero real numbers. the reciprocal function g(x) = c x is a solution of the functional equation (1.4). the functional equation (1.4) holds good for the “reciprocal formula” of any electric 132 ravi, rassias and kumar circuit with two resistors connected in parallel. s.m. jung [22] applied fixed point method for proving hyers-ulam stability for the reciprocal functional equation (1.4). k. ravi, j.m. rassias and b.v. senthil kumar [37] disucssed the generalized hyers-ulam stability for the reciprocal functional equation in several variables of the form (1.5) g ( m∑ i=1 αixi ) = ∏m i=1 g(xi)∑m i=1 [ αi (∏m j=1,j 6=i g(xj) )] for arbitrary but fixed real numbers (α1,α2, . . . ,αm) 6= (0, 0, . . . , 0), so that 0 < α = α1 + α2 + · · · + αm = ∑m i=1 αi 6= 1 and g : x → y with x and y are the spaces of non-zero real numbers. later, j.m. rassias and et.al., [38] introduced the reciprocal difference functional equation (1.6) r ( x + y 2 ) −r(x + y) = r(x)r(y) r(x) + r(y) and the reciprocal adjoint functional equation (1.7) r ( x + y 2 ) + r(x + y) = 3r(x)r(y) r(x) + r(y) and investigated the generalized hyers-ulam stability of the equations (1.6) and (1.7). soon after, j.m. rassias and et.al., [39] applied fixed point method to investigate the generalized hyers-ulam stability of the equations (1.6) and (1.7). g.l. forti [12] obtained the stability of the functional equation ψ ◦f ◦a = f, by proving the following theorem. theorem 1.1. assume that (y,d) is a complete metric space, k is a nonempty set, f : k → y , ψ : y → y , a : k → k, h : k → [0,∞), λ ∈ [0,∞), d(ψ ◦ f ◦ a(x),f(x)) ≤ h(x) for x ∈ k, d(ψ(x), ψ(y)) ≤ λd(x,y) for x,y ∈ y, and h(x) = ∞∑ i=0 λih ( ai(x) ) < ∞ for x ∈ k. then, for every x ∈ k, the limit r(x) = lim n→∞ ψn ◦f ◦an(x) exists and r : k → y is a unique function such that ψ ◦r ◦a = r and d(f(x),r(x)) ≤ h(x), for x ∈ k. in this paper, we obtain the general solution and investigate the generalized hyers-ulam stability of the reciprocal type functional equation in three variables of the form (1.8) g (2(k − 1)x1 + x2 + x3) = g((k − 1)x1 + x2)g((k − 1)x1 + x3) g((k − 1)x1 + x2) + g((k − 1)x1 + x3) where k ≥ 2 is a positive integer, g : x → r is a mapping with x as the space of non-zero real numbers and 2(k − 1)x1 + x2 + x3 6= 0, g((k − 1)x1 + x2) + g((k − 1)x1 + x2) 6= 0, for all x1,x2,x3 ∈ x using direct method, fixed point method and theorem 1.1. we also provide counter-examples for non-stability. throughout this paper, let x be the space of non-zero real numbers. we also assume that 2(k − 1)x1 + x2 + x3 6= 0, g(x) 6= 0, g((k − 1)x1 + x2) 6= 0, stability of a reciprocal functional equation in three variables 133 g((k − 1)x1 + x3) 6= 0 and g((k − 1)x1 + x2) + g((k − 1)x1 + x3) 6= 0 for all x,x1,x2,x3 ∈ x to prove our main results. for the sake of convenience, let us define d3g(x1,x2,x3) = g (2(k − 1)x1 + x2 + x3) − g((k − 1)x1 + x2)g((k − 1)x1 + x3) g((k − 1)x1 + x2) + g((k − 1)x1 + x3) (1.9) for all x1,x2,x3 ∈ x. the paper is organised as follows. in section 2, we present preliminaries required for proving our main results. in section 3, we obtain the general solution of the functional equation (1.8). in section 4, we investigate the generalized hyersulam stability of the equation (1.8) using direct method. in section 5, we apply fixed point method to gain the generalized hyers-ulam stability of the equation (1.8) and in section 6, we find the generalized hyers-ulam stability of the equation (1.8) in the sense of g.l. forti. in section 7, we illustrate counter-examples for singular cases. 2. preliminaries in this section, we present the definition of generalized metric and fundamental result of fixed point theory. let a be a set. a function d : a×a → [0,∞] is called a generalized metric on a if d satisfies the following conditions: 1. d(x,y) = 0 if and only if x = y; 2. d(x,y) = d(y,x) for all x,y ∈ a; 3. d(x,z) ≤ d(x,y) + d(y,z) for al x,y,z ∈ a. we note that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include infinity. the following theorem is very useful for proving our results in section 5, which is due to margolis and diaz [26]. theorem 2.1. let (a,d) be a complete generalized metric space and let j : a → a be a strictly contractive mapping with lipschitz constant l < 1. then for each given element x ∈ a, either (2.1) d ( jnx,jn+1x ) = ∞ for all nonnegative integers n or there exists a positive integer n0 such that 1. d ( jnx,jn+1x ) < ∞ for all n ≥ n0; 2. the sequence {jnx} converges to a fixed point y∗ of j; 3. y∗ is the unique fixed point of j in the set y = {y ∈ a|d (jn0x,y) < ∞}; 4. d(y,y∗) ≤ 1 1−ld(y,jy) for all y ∈ y . 134 ravi, rassias and kumar 3. general solution of functional equation (1.8) theorem 3.1. a mapping g : x → r satisfies the functional equation (1.8) for all x1,x2,x3 ∈ x if and only if there exists a reciprocal mapping g : x → r satisfying the reciprocal functional equation (1.4) for all x,y ∈ x. hence, every solution of functional equation (1.8) is also a reciprocal function. proof. let the mapping g : x → r satisfy the functional equation (1.8). replacing (x1,x2,x3) by ( x+y k−1 ,−y,−x ) in (1.8), we obtain the equation (1.4). conversely, let the mapping g : x → r satisfy the functional equation (1.4). replacing (x,y) by ((k−1)x1 + x2, (k−1)x1 + x3) in (1.4), we obtain the equation (1.8), which completes the proof of theorem 3.1. � 4. generalized hyers-ulam stability of equation (1.8) using direct method theorem 4.1. let g : x → r be a mapping satisfying (1.9) and (4.1) |d3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ x, where φ : x ×x ×x → r be a given function such that (4.2) φ(x) = ∞∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) with the condition (4.3) lim n→∞ 2nφ ( 2nx k , 2nx k , 2nx k ) = 0 holds for every x ∈ x. then there exists a unique reciprocal mapping r : x → r which satisfies (1.8) and the inequality (4.4) |g(x) −r(x)| ≤ φ(x) for all x ∈ x. proof. replacing (x1,x2,x3) by ( x k , x k , x k ) in (4.1) and multiplying by 2, we get (4.5) |2g(2x) −g(x)| ≤ 2φ (x k , x k , x k ) for all x ∈ x. again replacing x by 2x in (4.5), multiplying by 2 and summing the resulting inequality with (4.5), we get ∣∣22g(22x) −g(x)∣∣ ≤ ∑1i=0 2i+1φ(2ixk , 2ixk , 2ixk ) for all x ∈ x. proceeding further and using induction on a positive integer n, we stability of a reciprocal functional equation in three variables 135 obtain |2ng(2nx) −g(x)| ≤ n−1∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) ≤ ∞∑ i=0 2i+1φ ( 2ix k , 2ix k , 2ix k ) (4.6) for all x ∈ x. in order to prove the convergence of the sequence {2ng (2nx)}, replace x by 2px in (4.6) and multiply by 2p, we find that for n > p > 0∣∣2pg (2px) − 2n+pg(2n+px)∣∣ = 2p ∣∣g (2px) − 2ng(2n+px)∣∣ ≤ ∞∑ i=0 2p+i+1φ ( 2p+ix k , 2p+ix k , 2p+ix k ) → 0 as p →∞. thus the sequence {2ng (2nx)} is a cauchy sequence. allowing n →∞ in (4.6), we arrive (4.4) with r(x) = lim n→∞ 2ng (2nx) . to show that r satisfies (1.8), replacing (x,y) by (2nx, 2ny) in (4.1) and multiplying by 2n, we obtain (4.7) 2n |d3g (2nx1, 2nx2, 2nx3)| ≤ 2nφ (2nx1, 2nx2, 2nx3) . allowing n → ∞ in (4.7), we see that r satisfies (1.8) for all x1,x2,x3 ∈ x. to prove r is a unique reciprocal mapping satisfying (1.8), let r1 : x → r be another reciprocal mapping which satisfies (1.8) and the inequality (4.4). clearly r1 (2 nx) = 2−nr1(x), r (2 nx) = 2−nr(x) and using (4.4), we arrive |r1(x) −r(x)| ≤ 2n |r1 (2nx) −r (2nx)| ≤ 2n (|r1 (2nx) −g (2nx)| + |g (2nx) −r (2nx)|) ≤ 2 ∞∑ i=0 2n+i+1φ ( 2n+ix k , 2n+ix k , 2n+ix k ) → 0 as n →∞ which implies that r is unique. this completes the proof of theorem 4.1. � theorem 4.2. let g : x → r be a mapping satisfying (1.9) and (4.1), for all x1,x2,x3 ∈ x, where φ : x ×x ×x → r be a given function such that (4.8) φ(x) = ∞∑ i=0 1 2i φ ( x 2i+1k , x 2i+1k , x 2i+1k ) with the condition (4.9) lim n→∞ 1 2n φ ( x 2n+1k , x 2n+1k , x 2n+1k ) = 0 136 ravi, rassias and kumar holds for every x ∈ x. then there exists a unique reciprocal mapping r : x → r which satisfies (1.8) and the inequality (4.10) |g(x) −r(x)| ≤ φ(x) for all x ∈ x. proof. the proof is obtained by replacing (x1,x2,x3) by ( x 2k , x 2k , x 2k ) in (4.1) and proceeding by similar arguments as in theorem 4.1. � corollary 4.3. let g : x → r be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) |d3g(x1,x2,x3)| ≤ θ1 ( 3∑ i=1 |xi|q ) for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.12) |g(x) −r(x)| ≤   6θ1 kq(1−2q+1)|x| q, for q < −1 6θ1 kq(2q+1−1)|x| q, for q > −1 for every x ∈ x. proof. the proof follows immediately by taking φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ x in theorems 4.1 and 4.2 respectively. � corollary 4.4. let g : x → r be a mapping and let there exist a real number q 6= −1. let there exist θ2 ≥ 0 such that (4.13) |d3g(x1,x2,x3)| ≤ θ2 ( 3∏ i=1 |xi| q 3 ) for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.14) |g(x) −r(x)| ≤   2θ2 kq(1−2q+1)|x| q, for q < −1 2θ2 kq(2q+1−1)|x| q, for q > −1 for every x ∈ x. proof. the required results in corollary 4.4 can be easily derived by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ x in theorems 4.1 and 4.2 respectively. � stability of a reciprocal functional equation in three variables 137 corollary 4.5. let θ3 ≥ 0 and q 6= −1 be real numbers, and g : x → r be a mapping satisfying the functional inequality (4.15) |d3g(x1,x2,x3)| ≤ θ3 ( 3∏ i=1 |xi| q 3 + ( 3∑ i=1 |xi|q )) for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.16) |g(x) −r(x)| ≤   8θ3 kq(1−2q+1)|x| q, for q < −1 8θ3 kq(2q+1−1)|x| q, for q > −1 for every x ∈ x. proof. by choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q 3 + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ x in theorems 4.1 and 4.2 respectively, the proof of corollary 4.5 is complete. � 5. generalized hyers-ulam stability of equation (1.8) using fixed point method theorem 5.1. suppose that the mapping g : x → r satisfies the inequality (5.1) |d3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ x, where φ : x×x×x → r is a given function. if there exists l < 1 such that the mapping x → φ(x) = 2φ (x k , x k , x k ) has the property φ(2x) ≤ 1 2 lφ(x), for all x ∈ x and the mapping φ has the property (5.2) lim n→∞ 2nφ (2nx1, 2 nx2, 2 nx3) = 0 for all x1,x2,x3 ∈ x, then there exists a unique reciprocal mapping r : x → r such that (5.3) |g(x) −r(x)| ≤ 1 1 −l φ(x) for all x ∈ x. proof. define a set s by s = {h : x → r|h is a function} 138 ravi, rassias and kumar and introduce the generalized metric d on s as follows: (5.4) d(g,h) = dφ(g,h) = inf{c ∈ r+ : |g(x) −h(x)| ≤ cφ(x), for all x ∈ x}. now, we show that (s,d) is complete. using the idea from [21], we prove the completeness of (s,d). let {hn} be a cauchy sequence in (s,d). then for any � > 0, there exists an integer n� > 0 such that d(hm,hn) ≤ �, for all m,n ≥ n�. from (5.4), we arrive (5.5) ∀� > 0,∃n� ∈ n,∀m,n ≥ n�,∀x ∈ x : |hm(x) −hn(x)| ≤ �φ(x). if x is a fixed number, (5.5) implies that {hn(x)} is a cauchy sequence in (r, |.|). since (r, |.|) is complete, {hn(x)} converges for all x ∈ x. therefore, we can define a function h : x → r by h(x) = lim n→∞ hn(x) and hence h ∈ s. letting m →∞ in (5.5), we have ∀� > 0,∃n� ∈ n,∀n ≥ n�,∀x ∈ x : |h(x) −hn(x)| ≤ �φ(x). by considering (5.4), we arrive ∀� > 0,∃n� ∈ n,∀n ≥ n� : d(h,hn) ≤ �, which implies that the cauchy sequence {hn} converges to h in (s,d). hence (s,d) is complete. define a mapping σ : s → s by (5.6) σh(x) = 2h(2x) (x ∈ x) for all h ∈ s. we claim that σ is strictly contractive on s. for any given g,h ∈ s, let cgh ∈ [0,∞] be an arbitrary constant with d(g,h) ≤ cgh. hence d(g,h) < cgh ⇒|g(x) −h(x)| ≤ cghφ(x), for all x ∈ x ⇒|2g(2x) − 2h(2x)| ≤ 2cghφ(2x), for all x ∈ x ⇒|2g(2x) − 2h(2x)| ≤ lcghφ(x), for all x ∈ x ⇒ d(σg,σh) ≤ lcgh. therefore, we see that d(σg,σh) ≤ ld(g,h), for all g,h ∈ s. that is, σ is strictly contractive mapping of s, with the lipschitz constant l. now, replacing (x1,x2,x3) by ( x k , x k , x k ) in (5.1) and multiplying by 2, we get |2g(2x) −g(x)| ≤ 2φ (x k , x k , x k ) = φ(x) stability of a reciprocal functional equation in three variables 139 for all x ∈ x. hence (5.4) implies that d(σf,f) ≤ 1. hence by applying the fixed point alternative theorem 2.1, there exists a function r : x → r satisfying the following: (1) r is a fixed point of σ, that is (5.7) r(2x) = 1 2 r(x) for all x ∈ x. the mapping r is the unique fixed point of σ in the set µ = {f ∈ s : d(f,g) < ∞}. this implies that r is the unique mapping satisfying (5.7) such that there exists c ∈ (0,∞) satisfying |r(x) −g(x)| ≤ cφ(x), ∀x ∈ x. (2) d(σng,r) → 0 as n →∞. thus, we have (5.8) lim n→∞ 2ng (2nx) = r(x) for all x ∈ x. (3) d(g,r) ≤ 1 1−ld(σg,r), which implies d(g,r) ≤ 1 1 −l . thus the inequality (5.3) holds. hence from (5.1), (5.2) and (5.8), we have |d3g(x1,x2,x3)| = lim n→∞ 2n|d3g(2nx1, 2nx2, 2nx3)| ≤ lim n→∞ 2nφ(2nx1, 2 nx2, 2 nx3) = 0 for all x1,x2,x3 ∈ x. hence r is a solution of the functional equation (1.8). by theorem 3.1, r : x → r is a reciprocal mapping. next, we show that r is the unique reciprocal mapping satisfying (1.8) and (5.3). suppose, let r1 : x → r be another reciprocal mapping satisfying (1.8) and (5.3). then from (1.8), we have that r1 is a fixed point of σ. since d(g,r1) < ∞, we have r1 ∈ s∗ = {f ∈ s|d(f,g) < ∞}. from theorem 2.1 (3) and since both r and r1 are fixed points of σ, we have r = r1. therefore, r is unique. hence, there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (5.3), which completes the proof of theorem 5.1. � 140 ravi, rassias and kumar theorem 5.2. suppose that the mapping g : x → r satisfies the inequality (5.1) for all x1,x2,x3 ∈ x, where φ : x×x×x → r is a given function. if there exists l < 1 such that the mapping x → φ(x) = φ ( x 2k , x 2k , x 2k ) has the property φ (x 2 ) ≤ 2lφ(x), for all x ∈ x and the mapping φ has the property (5.9) lim n→∞ 2−nφ ( 2−nx1, 2 −nx2, 2 −nx3 ) = 0 for all x1,x2,x3 ∈ x, then there exists a unique reciprocal mapping r : x → r such that (5.10) |g(x) −r(x)| ≤ 1 1 −l φ(x) for all x ∈ x. proof. the proof of theorem 5.2 goes through the same way as in theorem 5.1. � corollary 5.3. let g : x → r be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) holds for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.12) for every x ∈ x. proof. the proof is obtained by assuming φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ x and l = 2−q−1,l = 2q+1 in theorems 5.1 and 5.2 respectively. � corollary 5.4. let g : x → r be a mapping and let there exist a real number q 6= −1. let there exist θ2 ≥ 0 such that (4.13) holds for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.14) for every x ∈ x. proof. it is easy to derive the required results in corollary 5.4 by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ x and l = 2−q−1,l = 2q+1 in theorems 5.1 and 5.2 respectively. � corollary 5.5. let θ3 ≥ 0 and q 6= −1 be real numbers, and g : x → r be a mapping satisfying the functional inequality (4.15) for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.16) for every x ∈ x. stability of a reciprocal functional equation in three variables 141 proof. the proof is complete by choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q e + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ x and l = 2−q−1,l = 2q+1 in theorems 5.1 and 5.2 respectively. � 6. generalized hyers-ulam stability of equation (1.8) in the sense of g.l. forti theorem 6.1. suppose that the mapping g : x → r satisfies the inequality (6.1) |d3g(x1,x2,x3)| ≤ φ(x1,x2,x3) for all x1,x2,x3 ∈ x, where φ : x×x×x → r is a given function. suppose there exists β ∈ (0,∞) such that 2β < 1, (6.2) φ ( 2x1 k , 2x2 k , 2x3 k ) ≤ βφ (x1 k , x2 k , x3 k ) for all x1,x2,x3 ∈ x and k(> 1) ∈ z+. then there exists a unique reciprocal mapping r : x → r such that (6.3) |r(x) −g(x)| ≤ 2 1 − 2β φ (x k , x k , x k ) for all x ∈ x. proof. replacing (x1,x2,x3) by ( x k , x k , x k ) in (6.1), we get |g(2x) − 1 2 g(x)| ≤ φ (x k , x k , x k ) , for all x ∈ x. hence, we obtain |2g(2x) −g(x)| = 2 ∣∣∣∣g(2x) − 12g(x) ∣∣∣∣ ≤ 2φ (x k , x k , x k ) , for x ∈ x. considering f = 1 2 g, ψ(z) = 2z, λ = 2, h(x) = 2φ ( x k , x k , x k ) , a(x) = 2x and d(x,y) = |x−y|, for all x,y ∈ x in theorem 1.1, we see that the limit r(x) exists and |r(x) −g(x)| ≤ h(x), for all x ∈ x. using (6.1), we obtain (6.4) 2n|d3g (2nx1, 2nx2, 2nx3) | ≤ (2β)nφ(x1,x2,x3) for all x1,x2,x3 ∈ x and n ∈ n. allowing n → ∞ in (6.4), we see that r satisfies (1.8). next we show that r is the unique reciprocal mapping satisfying (1.8) and (6.3). suppose, let r1 : x → r be another reciprocal mapping satisfying (1.8) and (6.3), and |r1(x) −g(x)| ≤ h(x), for all x ∈ x. then ψ ◦r1 ◦a = r1 and hence by theorem 1.1, r = r1, which proves that r is unique. � 142 ravi, rassias and kumar theorem 6.2. suppose that the mapping g : x → r satisfies (6.1), for all x1,x2,x3 ∈ x, where φ is a function defined as in theorem 6.1. suppose there exists β ∈ (0,∞) such that β 2 < 1, (6.5) φ (x1 2k , x2 2k , x3 2k ) ≤ βφ (x1 k , x2 k , x3 k ) for all x1,x2,x3 ∈ x and k(> 1) ∈ z+. then there exists a unique reciprocal mapping r : x → r such that (6.6) |r(x) −g(x)| ≤ 2β 2 −β φ (x k , x k , x k ) for all x ∈ x. proof. replacing (x1,x2,x3) by ( x 2k , x 2k , x 2k ) in (6.1), we get∣∣∣∣g(x) − 12g (x 2 )∣∣∣∣ ≤ φ( x2k, x2k, x2k ) , for all x ∈ x. the rest of the proof is obtained by taking f = g, ψ(z) = 1 2 z, λ = 1 2 , h(x) = φ( x 2k , x 2k , x 2k ), a(x) = 1 2 x and d(x,y) = |x−y|, for all x,y ∈ x in theorem 1.1 and using similar arguments as in theorem 6.1. � corollary 6.3. let g : x → r be a mapping and let there exist real numbers q 6= −1 and θ1 ≥ 0 such that (4.11) holds for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.12) for every x ∈ x. proof. the proof is obtained by assuming φ(x1,x2,x3) = θ1 (∑3 i=1 |xi| q ) , for all x1,x2,x3 ∈ x and β = 2q,β = 12q in theorems 6.1 and 6.2 respectively. � corollary 6.4. let g : x → r be a mapping and let there exist a real number q 6= −1. let there exist θ2 ≥ 0 such that (4.13) holds for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.14) for every x ∈ x. proof. it is easy to derive the required results in corollary 6.4 by considering φ(x1,x2,x3) = θ2 (∏3 i=1 |xi| q 3 ) , for all x1,x2,x3 ∈ x and β = 2q,β = 12q in theorems 6.1 and 6.2 respectively. � corollary 6.5. let θ3 ≥ 0 and q 6= −1 be real numbers, and g : x → r be a mapping satisfying the functional inequality (4.15) for all x1,x2,x3 ∈ x. then there exists a unique reciprocal mapping r : x → r satisfying (1.8) and (4.16) for every x ∈ x. stability of a reciprocal functional equation in three variables 143 proof. the proof is complete by choosing φ(x1,x2,x3) = θ3 (∏3 i=1 |xi| q e + (∑3 i=1 |xi| q )) , for all x1,x2,x3 ∈ x and β = 2q,β = 12q in theorems 6.1 and 6.2 respectively. � remark 6.6. from section 4, section 5 and section 6, we observe that the results obtained in corollaries 4.3, 5.3 and 6.3 are the same. similarly the upper bounds in the corollaries 4.4, 5.4 and 6.4 are identical. also, we find that the stability results in corollaries 4.5, 5.5 and 6.5 are similar. therefore, from the above results, we conclude that the method of g.l. forti is the easiest method in comparison with the direct method and fixed point method. 7. counter-examples the following example illustrates the fact that the functional equation (1.8) is not stable for q = −1 in corollary 4.3. example 7.1. let ϕ : x → r be a mapping defined by ϕ(x) = { c1 x for x ∈ (1,∞) c1 otherwise where c1 > 0 is a constant, and define a mapping g : x → x by g(x) = ∞∑ n=0 ϕ(2−nx) 2n , for all x ∈ x. then the mapping g satisfies the inequality (7.1) |d3g(x1,x2,x3)| ≤ 3c1 ( 3∑ i=1 |xi|−1 ) for all xi ∈ x,i = 1, 2, 3. therefore there do not exist a reciprocal mapping r : x → r and a constant δ > 0 such that (7.2) |g(x) −r(x)| ≤ δ|x|−1 for all x ∈ x. proof. |g(x)| ≤ ∑∞ n=0 |ϕ(2−nx)| |2n| ≤ ∑∞ n=0 c1 2n = 2c1. hence g is bounded by 2c1. if(∑3 i=1 |xi| −1 ) ≥ 1, then the left hand side of (4.1) is less than 3c1. now, suppose that 0 < (∑3 i=1 |xi| −1 ) < 1. then there exists a positive integer m such that (7.3) 1 2m+1 < ( 3∑ i=1 |xi|−1 ) < 1 2m . hence (∑3 i=1 |xi| −1 ) < 1 2m implies 2m|xi|−1 < 1, for i = 1, 2, 3 or xi 2m > 1, for i = 1, 2, 3 or xi 2m−1 > 2 > 1, for i = 1, 2, 3 144 ravi, rassias and kumar and consequently 1 2m−1 (2(k − 1)x1 + x2 + x3), 1 2m−1 ((k − 1)x1 + x2), 1 2m−1 ((k − 1)x1 + x3) > 1. therefore, for each value of n = 0, 1, 2, . . . ,m− 1, we obtain 1 2n (2(k − 1)x1 + x2 + x3), 1 2n ((k − 1)x1 + x2), 1 2n ((k − 1)x1 + x3) > 1 and d3ϕ ( 1 2n x1, 1 2n x2, 1 2n x3 ) = 0, for n = 0, 1, 2, . . . ,m − 1. using (7.3) and the definition of g, we obtain |d3g(x1,x2,x3)| (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ n=m ∣∣∣∣ϕ (2−n(2(k − 1)x1 + x2 + x3)) − ϕ(2−n((k−1)x1+x2))ϕ(2−n((k−1)x1+x3))ϕ(2−n((k−1)x1+x2))+ϕ(2−n((k−1)x1+x3)) ∣∣∣∣ 2n (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ k=0 3 2 c1 2k2m (|x1|−1 + |x2|−1 + |x3|−1) ≤ ∞∑ k=0 3 2 c1 2k = 3 2 c1 ( 1 − 1 2 )−1 = 3c1, for all x,y ∈ x. that is, the inequality (7.1) holds true. now, assume that there exists a reciprocal mapping r : x → r satisfying (7.2). therefore, we have (7.4) |g(x)| ≤ (δ + 1)|x|−1. however, we can choose a positive integer p with pc1 > δ + 1. if x ∈ (1, 2p−1), then 2−nx ∈ (1,∞) for all n = 0, 1, 2, . . . ,p− 1 and therefore |g(x)| = ∞∑ n=0 ϕ(2−nx) 2n ≥ m−1∑ n=0 c1 2−nx 2n = pc1 x > (δ + 1)x−1 which contradicts (7.4). therefore, the reciprocal type of functional equation (1.8) is not stable for q = −1 in corollary 4.3. � the following example illustrates the fact that the functional equation (1.8) is not stable for q = −1 in corollary 4.5. example 7.2. let ψ : x → r be a mapping defined by ψ(x) = { c2 x for x ∈ (1,∞) c2 otherwise where c2 > 0 is a constant, and define a mapping g : x → r by g(x) = ∞∑ n=0 ψ(2−nx) 2n , for all x ∈ x. then the mapping g satisfies the inequality |d3g(x1,x2,x3)| ≤ 3c2 ( 3∏ i=1 |xi|− 1 3 + ( 3∑ i=1 |xi|−1 )) stability of a reciprocal functional equation in three variables 145 for all xi ∈ x,i = 1, 2, 3. therefore there do not exist a reciprocal mapping r : x → r and a constant δ > 0 such that |g(x) −r(x)| ≤ δ|x|−1, for all x ∈ x. proof. the proof is analogous to the proof of example 7.1. � references [1] j. aczel, lectures on functional equations and their applications, vol. 19, academic press, new york, 1966. [2] j. aczel, functional equations, history, applications and theory, d. reidel publ. company, 1984. [3] c. alsina, on the stability of a functional equation, general inequalities, vol. 5, oberwolfach, birkhauser, basel, (1987), 263-271. [4] t. aoki, on the stability of the linear transformation in banach spaces, j. math.soc. japan, 2(1950), 64-66. [5] c. baak and m.s. moslehian, on the stability of j∗-homomorphisms, nonlinear analysis-tma 63 (2005), 42-48. [6] l. cadariu and v. radu, fixed points and the stability of jensen’s functional equation, j. inequal.pure and appl. math., 4 (2003), no.1, art. 4. [7] l. cadariu and v. radu, on the stability of the cauchy functional equation: a fixed point approach , grazer math. ber., 346(2004), 43-52. [8] i.s. chang and h.m. kim, on the hyers-ulam stability of quadratic functional equations, j. ineq. appl. math. 33(2002), 1-12. [9] i.s. chang and y.s. jung, stability of functional equations deriving from cubic and quadratic functions, j. math. anal. appl. 283(2003), 491-500. [10] s. czerwik, functional equations and inequalities in several variables, world scientific publishing co., singapore, new jersey, london, 2002. [11] m. eshaghi gordji, s. zolfaghari, j.m. rassias and m.b. savadkouhi, solution and stability of a mixed type cubic and quartic functional equation in quasi-banach spaces, abst. appl. anal. vol. 2009, article id 417473(2009), 1-14. [12] g.l. forti, comments on the core of the direct method for proving hyers-ulam stability of functional equations, j. math. anal. appl. 295(2004), 127-133. [13] p. gavruta, a generalization of the hyers-ulam-rassias stability of approximately additive mapppings, j. math. anal. appl. 184(1994), 431-436. [14] n. ghobadipour and c. park, cubic-quartic functional equations in fuzzy normed spaces, int. j. nonlinear anal. appl. 1(2010), 12-21. [15] d.h. hyers, on the stability of the linear functional equation, proc. nat. acad. sci. u.s.a. 27(1941), 222-224. [16] d.h. hyers, g. isac and th.m. rassias, stability of functional equations in several variables, birkhauser, basel, 1998. [17] g. isac and th.m. rassias, stability of ψ-additive mappings: applications to nonlinear analysis, int. j. math. math. sci., 19(2)(1996), 219-228. 146 ravi, rassias and kumar [18] s.m. jung, hyers-ulam-rassias stability of jensen’s equation and its application, proc. amer. math. soc. 126(11)(1998), 3137-3143. [19] s.m. jung, hyers-ulam-rassias stability of functional equations in mathematical analysis, hardonic press, palm harbor, 2001. [20] s.m. jung, a fixed point approach to the stability of isometries, j. math. anal. appl., 329 (2007), 879-890. [21] s.m. jung, a fixed point approach to the stability of a volterra integral equation, fixed point theory and applications, vol. 2007 (2007), article id 57064, 9 pages. [22] s.m. jung, a fixed point approach to the stability of the equation f(x + y) = f(x)f(y) f(x)+f(y) , the australian j. math. anal. appl., 6(8) (2009), 1-6. [23] y.s. jung and i.s. chang, the stability of a cubic type functional equation with the fixed point alternative, j. math. ana.l appl., 306(2)(2005), 752-760. [24] pl. kannappan, quadratic functional equation inner product spaces, results math. 27(3-4)(1995), 368-372. [25] h. khodaei and th.m. rassias, approximately generalized additive functions in several variables, int. j. nonlinear anal. appl. 1(2010), 22-41. [26] b. margolis and j. diaz, a fixed point theorem of the alternative for contractions on a generalized complete metric space, bull. amer. math. soc., 74 (1968), 305-309. [27] m. mirzavaziri and m.s. moslehian, a fixed point approach to stability of a quadratic equation, bull. braz. math. soc., 37(3)(2006), 361-376. [28] c. park, fixed points and hyers-ulam-rassias stability of cauchy-jensen functional equations in banach algebras, fixed point theory and applications, vol. 2007 (2007), article id 13437, 6 pages. [29] w.g. park and j.h. bae, a functional equation originating from elliptic curves, abst. appl. anal. vol. 2008, article id 135237, 10 pages. [30] v. radu, the fixed point alternative and the stability of functional equations, fixed point theory, 4 (2003), 91-96. [31] j.m. rassias, on approximation of approximately linear mappings by linear mappings, j. funct. anal. 46(1982), 126-130. [32] j.m. rassias, on approximation of approximately linear mappings by linear mappings, bull. sci. math. 108(1984), 445-446. [33] j.m. rassias, solution of a problem of ulam, j. approx. theory, 57(1989), 268-273. [34] k. ravi, m. arunkumar and j.m. rassias, ulam stability for the orthogonally general euler-lagrange type functional equation, int. j. math. stat. 3(a08)(2008), 36-46. [35] k. ravi, j.m. rassias, m. arunkumar and r. kodandan, stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, j. ineq. pure & appl. math. 10(4)(2009), 1-29. [36] k. ravi and b.v. senthil kumar, ulam-gavruta-rassias stability of rassias reciprocal functional equation, global j. of appl. math. and math. sci. 3(1-2) (jan-dec 2010), 57-79. [37] k. ravi, j.m. rassias and b.v. senthil kumar, ulam stability of generalized reciprocal functional equation in several variables, int. j. app. math. stat. 19(d10) 2010, 1-19. stability of a reciprocal functional equation in three variables 147 [38] k. ravi, j.m. rassias and b.v. senthil kumar, ulam stability of reciprocal difference and adjoint functional equations, the australian j. math. anal. appl., 8(1), art. 13 (2011), 1-18. [39] k. ravi, j.m. rassias and b.v. senthil kumar, a fixed point approach to the generalized hyers-ulam stability of reciprocal difference and adjoint functional equations, thai j. math., 8(3) (2010), 469-481. [40] th.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72(1978), 297-300. [41] f. skof, proprieta locali e approssimazione di operatori, rend. sem. mat. fis. milano 53(1983), 113-129. [42] s.m. ulam, problems in modern mathematics, chapter vi, wiley-interscience, new york, 1964. 1pg & research department of mathematics, sacred heart college, tirupattur 635 601, tamilnadu, india 2pedagogical department e.e., section of mathematics and informatics, national and capodistrian university of athens, 4, agamemnonos str., aghia paraskevi, athens, attikis 15342, greece 3department of mathematics, c. abdul hakeem college of engg. and tech., melvisharam 632 509, tamilnadu, india ∗corresponding author int. j. anal. appl. (2022), 20:61 received: sep. 13, 2022. 2010 mathematics subject classification. 34a07. key words and phrases. fuzzy adomian decomposition method; fuzzy autonomous differential equation; fuzzy series solution. https://doi.org/10.28924/2291-8639-20-2022-61 © 2022 the author(s) issn: 2291-8639 1 semi analytical solution for fuzzy autonomous differential equations mazin h. suhhiem1,*, raad i. khwayyit2 1university of sumer, iraq 2ministry of education, iraq *corresponding author: mazin.suhhiem@yahoo.com abstract. in this work, we have used fuzzy adomian decomposition method to find the fuzzy semi analytical solution of the fuzzy autonomous differential equations with fuzzy initial conditions. this method allows for the solution of the fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. the fuzzy series solutions that we have obtained are accurate solutions and very close to the fuzzy exact analytical solutions. some numerical results have been given to illustrate the efficiency of the used method. 1. introduction the topic of fuzzy semi analytical methods (fuzzy series method) for solving fuzzy initial value problems (fivps) has been rapidly growing in recent years. several fuzzy semi analytical methods have been proposed to obtain the fuzzy series solution of the linear and non-linear fivb. fuzzy adomian decomposition method is one of the fuzzy semi analytical methods used to obtain the fuzzy series solution of the fivbs. researchers and scientists are continuing to develop this method for solving various types of the fivbs because it represents an efficient and effective technique (for more details, see [1, 2, 3, 4, 6, 7, 8, 10]). https://doi.org/10.28924/2291-8639-20-2022-61 2 int. j. anal. appl. (2022), 20:61 in this work, we will need many basic concepts in the fuzzy theory. these concepts can be found in detail in [5, 8, 11]. 2. fuzzy autonomous differential equations a fuzzy ordinary differential equation is called autonomous if it is independent of its independent crisp variable x. this implies that the nth order fuzzy autonomous differential equation is of the form [11]: 𝑢(𝑛)(x) = f ( 𝑢 (x) , 𝑢′(𝑥) , 𝑢′′(𝑥) , … . , 𝑢(𝑛−1)(𝑥)) , 𝑥 ∈ [𝑥0 , ℎ] (2.1) with the fuzzy initial conditions: u(𝑥0) = 𝑢0 𝑢′(𝑥0) = 𝑢0 ′ 𝑢′′(𝑥0) = 𝑢0 ′′ . . . 𝑢(𝑛−1)(𝑥0) = 𝑢0 (𝑛−1) where: 𝑢 is a fuzzy function of the crisp variable 𝑥, f (𝑢 (x) , 𝑢′(𝑥) , 𝑢′′(𝑥) , … . , 𝑢(𝑛−1)(𝑥)) is a fuzzy function of the crisp variable 𝑥 and the fuzzy variable 𝑢, 𝑢(𝑛)(x) is the fuzzy derivative of 𝑢 (x) , 𝑢′(𝑥) , 𝑢′′(𝑥) , … ., 𝑢(𝑛−1)(𝑥), and u(𝑥0) , 𝑢 ′(𝑥0) , 𝑢 ′′(𝑥0) , … , 𝑢 (𝑛−1)(𝑥0) are fuzzy numbers. the general idea of solving the fuzzy differential equation is based on transforming this equation into a system of non-fuzzy (crisp) differential equations. thus, problem (2.1) can be written as: 𝑢(𝑛)(x) = 𝑓 ( 𝑢 , 𝑢′, 𝑢′′ , … , 𝑢(𝑛−1)) (2.2) = h( 𝑢 , 𝑢′ , 𝑢′′ , … , 𝑢(𝑛−1) , 𝑢 , 𝑢 ′ , 𝑢 ′′ , … , 𝑢 (𝑛−1) ) with the initial conditions: u(𝑥0) = 𝑢 0 𝑢′(𝑥0) = 𝑢 0 ′ 𝑢′′(𝑥0) = 𝑢 0 ′′ 3 int. j. anal. appl. (2022), 20:61 . . . 𝑢(𝑛−1)(𝑥0) = 𝑢 0 (𝑛−1) 𝑢 (𝑛) (x) = 𝑓 ( 𝑢 , 𝑢′, 𝑢′′ , … , 𝑢(𝑛−1)) (2.3) = g( 𝑢 , 𝑢′ , 𝑢′′ , … , 𝑢(𝑛−1) , 𝑢 , 𝑢 ′ , 𝑢 ′′ , … , 𝑢 (𝑛−1) ) with the initial conditions: 𝑢 (𝑥0) = 𝑢0 𝑢 ′ (𝑥0) = 𝑢0 ′ 𝑢 ′′ (𝑥0) = 𝑢0 ′′ . . . 𝑢 (𝑛−1) (𝑥0) = 𝑢0 (𝑛−1) where h( 𝑢 , 𝑢′ , 𝑢′′ , … , 𝑢(𝑛−1) , 𝑢 , 𝑢 ′ , 𝑢 ′′ , … , 𝑢 (𝑛−1) ) (2.4) =min{ 𝑓 (𝑥 , 𝑧) ∶ 𝑧 ∈ [𝑢 , 𝑢′ , 𝑢′′ , … , 𝑢(𝑛−1) , 𝑢 , 𝑢 ′ , 𝑢 ′′ , … , 𝑢 (𝑛−1) ] }, g( 𝑥 , 𝑥 ′ , 𝑥′′ , … , 𝑥 (𝑛−1) , 𝑥 , 𝑥 ′ , 𝑥 ′′ , … , 𝑥 (𝑛−1) ) (2.5) =max{ 𝑓 (𝑥 , 𝑧) ∶ 𝑧 ∈ [𝑢 , 𝑢′ , 𝑢′′ , … , 𝑢(𝑛−1) , 𝑢 , 𝑢 ′ , 𝑢 ′′ , … , 𝑢 (𝑛−1) ] } , the parametric form of system (2.4 2.5) is given by: 𝑢(𝑛)(x , r) =h( 𝑢(𝑥 , r), 𝑢′(𝑥 , r) , … , 𝑢(𝑛−1)(𝑥 , r) , 𝑢 (𝑥 , r) , 𝑢 ′ (𝑥 , r) , … , 𝑢 (𝑛−1) (𝑥 , r) ) (2.6) with the initial conditions: u(𝑥0 , r)=𝑢 0(r), 𝑢′(𝑥0 , r) = 𝑢 0 ′ (r) 𝑢′′(𝑥0 , r) = 𝑢 0 ′′ (r) . . . 𝑢(𝑛−1)(𝑥0 , r) = 𝑢 0 (𝑛−1) (r) 4 int. j. anal. appl. (2022), 20:61 𝑢 (𝑛) (x , r) =g( 𝑢(𝑥 , r) ,𝑢′(𝑥 , r) , … , 𝑢(𝑛−1)(𝑥 , r) , 𝑢 (𝑥 , r) , 𝑢 ′ (𝑥 , r) , … , 𝑢 (𝑛−1) (𝑥 , r) ) (2.7) with the initial conditions: 𝑢 (𝑥0 , r)= 𝑢0(r) 𝑢 ′ (𝑥0 , r) = 𝑢0 ′ (r) 𝑢 ′′ (𝑥0 , r) = 𝑢0 ′′ (𝑟) . . . 𝑢 (𝑛−1) (𝑥0 ,r) = 𝑢0 (𝑛−1) (r) both equation (2.6) and equation (2.7) have only one solution on the interval [𝑥0 , ℎ]. therefore, equation (2.1) has a unique fuzzy solution on [𝑥0 , ℎ], where 𝑟 ∈ [0 ,1] (for more details, see [11]). in order to illustrate the above, we give the following example: if we consider the second order fuzzy autonomous differential equation 𝑢´´ (𝑥) = 4 𝑢´(𝑥) − 4 𝑢(𝑥), x ∈ [0 , 1] (2.8) with the fuzzy initial conditions: 𝑢(0) = [2 + r , 4 − r ], 𝑢ˊ(0) = [5 + r , 7 − r] and 𝑟 ∈ [0 ,1]. to convert problem (2.8) into a system of the second order crisp ordinary differential equations, we apply the following steps: [ 𝑢´´(𝑥) ]r = [4 𝑢´(𝑥) − 4 𝑢(𝑥) ]r, for all 𝑟 ∈ [0 ,1] (2.9) with the fuzzy initial conditions: [𝑢(0)]r = [ 2 + r , 4 − r ], [𝑢ˊ(0) ]r = [ 5 + r , 7 − r ] then we get [𝑢´´(𝑥)]r = 4[𝑢´(𝑥)]r − 4[𝑢(𝑥)]r, for all 𝑟 ∈ [0 ,1] (2.10) with the fuzzy initial conditions: [𝑢(0)]r = [ 2 + r , 4 − r ], [𝑢ˊ(0) ]r = [ 5 + r , 7 − r ] 5 int. j. anal. appl. (2022), 20:61 then we have [ [𝑢´´(𝑥)]r 𝐿 , [𝑢´´(𝑥)]r 𝑈 ] = [ 4[𝑢´(𝑥)]r 𝐿 − 4[𝑢(𝑥)]r 𝐿 , 4[𝑢´(𝑥)]r 𝑈 − 4[𝑢(𝑥)]r 𝑈 ] (2.11) with the fuzzy initial conditions: [ [𝑢(0)]r 𝐿 , [𝑢(0)]r 𝑈 ] = [ 2 + r , 4 − r] [ [𝑢ˊ(0)]r 𝐿 , [𝑢ˊ(0)]r 𝑈 ] = [ 5 + r , 7 − r ] then we get the following system of second order crisp ordinary differential equations: [𝑢´´(𝑥)]r 𝐿 = 4[𝑢´(𝑥)]r 𝐿 − 4[𝑢(𝑥)]r 𝐿 ; (2.12) with the initial conditions: [𝑢(0)]r 𝐿 = 2 + r [𝑢ˊ(0)]r 𝐿 = 5 + r [𝑢´´(𝑥)]r 𝑈 = 4[𝑢´(𝑥)]α 𝑈 − 4[𝑢(𝑥)]r 𝑈; (2.13) with the initial conditions: [𝑢(0)]r 𝑈 = 4 − r [𝑢ˊ(0)]r 𝑈 = 7 − r which gives the unique crisp solutions [𝑢(𝑥)]r 𝐿 = (2 + 𝑟) 𝑒2𝑥 + (1 − 𝑟 ) 𝑥 𝑒2𝑥 (2.14) [𝑢(𝑥)]r 𝑈 = (4 − 𝑟) 𝑒2𝑥 + (r − 1 ) 𝑥 𝑒2𝑥 (2.15) then the unique fuzzy solution of problem (8) is [𝑢(𝑥)]r = [ (2 + 𝑟) 𝑒 2𝑥 + (1 − 𝑟 ) 𝑥 𝑒2𝑥 , (4 − 𝑟) 𝑒2𝑥 + (r − 1 ) 𝑥 𝑒2𝑥 ] (2.16) 3. fuzzy adomian decomposition method to understand the fuzzy adomian decomposition method, we consider the nth order fuzzy differential equation [2,3,7]: [𝐹(𝑢(𝑥))]𝑟 = [𝑔(𝑥) ]𝑟 (3.1) where f represents a general nonlinear fuzzy ordinary (or fuzzy partial) differential operator including both linear and nonlinear terms, x denotes the independent crisp variable, 𝑢(𝑥) and 𝑔(𝑥) are unknown fuzzy functions. from section (2), we can conclude that: 6 int. j. anal. appl. (2022), 20:61 [𝐹(𝑢(𝑥))]𝑟 = [ 𝐹(𝑢(𝑥))]𝑟 𝐿 , [𝐹(𝑢(𝑥))]𝑟 𝑈 (3.2) [𝑔(𝑥)]𝑟 = [ 𝑔(𝑥)]𝑟 𝐿 , [𝑔(𝑥)]𝑟 𝑈 ] (3.3) where [𝐹(𝑢(𝑥))]𝑟 𝐿 = [𝑔(𝑥)]𝑟 𝐿 (3.4) [𝐹(𝑢(𝑥))]𝑟 𝑈 = 𝑔(𝑥)]𝑟 𝑈 (3.5) the fuzzy linear terms of (3.1) are decomposed into [𝐿]𝑟 + [𝑅]𝑟, where: [𝐿]𝑟 is invertible and is taken as the highest order fuzzy derivative. this implies that: 𝐿(∗) = 𝑑𝑚 𝑑𝑥𝑚 (∗) , 𝑚 = 1,2,3, … (3.6) 𝐿−1(∗) = ∫ ∫ …………… 𝑚−𝑡𝑖𝑚𝑒𝑠 𝑥 0 𝑥 0 ∫ (∗)𝑑𝑥𝑑𝑥 𝑥 0 …………… 𝑚−𝑡𝑖𝑚𝑒𝑠 𝑑𝑥 (3.7) and [𝑅]𝑟 is the remainder of the fuzzy linear operator. thus the equation (3.1) may be written as: [𝐿(𝑢)]𝑟 + [𝑅(𝑢)]𝑟 + [𝑁(𝑢)]𝑟 = [𝑔]𝑟 (3.8) where [𝑁(𝑢)]𝑟 represents the fuzzy nonlinear terms. by the concepts of section (2), we get: [𝐿(𝑢)]𝑟 𝐿 + [𝑅(𝑢)]𝑟 𝐿 + [𝑁(𝑢)]𝑟 𝐿 = [𝑔]𝑟 𝐿 (3.9) [𝐿(𝑢)]𝑟 𝑈 + [𝑅(𝑢)]𝑟 𝑈 + [𝑁(𝑢)]𝑟 𝑈 = [𝑔]𝑟 𝑈 (3.10) the fuzzy adomian decomposition method represents the fuzzy solution [𝑢(𝑥)]𝑟 of problem (3.1) as a fuzzy series of the form: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] (3.11) where [𝑢(𝑥)]𝑟 𝐿 = ∑ [𝑢𝑛 (𝑥)]𝑟 𝐿∞ 𝑛=0 = [𝑢0(𝑥)]𝑟 𝐿 + [𝑢1(𝑥)]𝑟 𝐿 + [𝑢2(𝑥)]𝑟 𝐿 + [𝑢3(𝑥)]𝑟 𝐿 + ⋯ (3.12) [𝑢(𝑥)]𝑟 𝑈 = ∑ [𝑢𝑛 (𝑥)]𝑟 𝑈∞ 𝑛=0 = [𝑢0(𝑥)]𝑟 𝑈 + [𝑢1(𝑥)]𝑟 𝑈 + [𝑢2(𝑥)]𝑟 𝑈 + [𝑢3(𝑥)]𝑟 𝑈 + ⋯ (3.13) such that: ●) [𝑢0(𝑥)]𝑟 = [ 𝑢0(𝑥)]𝑟 𝐿 , [𝑢0(𝑥)]𝑟 𝑈 ] (3.14) where [𝑢0(𝑥)]𝑟 𝐿 = [𝜃0]𝑟 𝐿 + 𝐿−1([𝑔(𝑥)]𝑟 𝐿 ) (3.15) [𝑢0(𝑥)]𝑟 𝑈 = [𝜃0]𝑟 𝑈 + 𝐿−1([𝑔(𝑥)]𝑟 𝑈 ) (3.16) 7 int. j. anal. appl. (2022), 20:61 ●) [𝑢1(𝑥)]𝑟 = [ 𝑢1(𝑥)]𝑟 𝐿 , [𝑢1(𝑥)]𝑟 𝑈 ] (3.17) where [𝑢1(𝑥)]𝑟 𝐿 = −𝐿−1([𝑅(𝑢0)]𝑟 𝐿 ) −𝐿−1([𝐴0]𝑟 𝐿 ) (3.18) [𝑢1(𝑥)]𝑟 𝑈 = −𝐿−1([𝑅(𝑢0)]𝑟 𝑈 ) −𝐿−1([𝐴0]𝑟 𝑈 ) (3.19) ●) [𝑢2(𝑥)]𝑟 = [ 𝑢2(𝑥)]𝑟 𝐿 , [𝑢2(𝑥)]𝑟 𝑈 ] (3.20) where [𝑢2(𝑥)]𝑟 𝐿 = −𝐿−1([𝑅(𝑢1)]𝑟 𝐿 ) −𝐿−1([𝐴1]𝑟 𝐿 ) (3.21) [𝑢2(𝑥)]𝑟 𝑈 = −𝐿−1([𝑅(𝑢1)]𝑟 𝑈 ) −𝐿−1([𝐴1]𝑟 𝑈 ) (3.22) . . . ●) [𝑢𝑛+1(𝑥)]𝑟 = [ 𝑢𝑛+1(𝑥)]𝑟 𝐿 , [𝑢𝑛+1(𝑥)]𝑟 𝑈 ] , 𝑛 ≥ 0 (3.23) where [𝑢𝑛+1(𝑥)]𝑟 𝐿 = −𝐿−1([𝑅(𝑢𝑛)]𝑟 𝐿 ) −𝐿−1([𝐴𝑛]𝑟 𝐿 ) (3.24) [𝑢𝑛+1(𝑥)]𝑟 𝑈 = −𝐿−1([𝑅(𝑢𝑛)]𝑟 𝑈 ) −𝐿−1([𝐴𝑛]𝑟 𝑈 ) (3.25) note that: [𝜃0]𝑟 = [ [𝜃0]𝑟 𝐿 , [𝜃0]𝑟 𝑈 ] (3.26) and it can be calculated as follows: ●) if 𝐿 = 𝑑 𝑑𝑥 , then we have: [𝜃0]𝑟 𝐿 = [𝑢(0)]𝑟 𝐿 (3.27) [𝜃0]𝑟 𝑈 = [𝑢(0)]𝑟 𝑈 (3.28) ●) if 𝐿 = 𝑑2 𝑑𝑥2 , then we have: [𝜃0]𝑟 𝐿 = [𝑢(0)]𝑟 𝐿 + 𝑥[𝑢′(0)]𝑟 𝐿 (3.29) [𝜃0]𝑟 𝑈 = [𝑢(0)]𝑟 𝑈 + 𝑥[𝑢′(0)]𝑟 𝑈 (3.30) ●) if 𝐿 = 𝑑3 𝑑𝑥3 , then we have: [𝜃0]𝑟 𝐿 = [𝑢(0)]𝑟 𝐿 + 𝑥[𝑢′(0)]𝑟 𝐿 + 𝑥2 2! [𝑢′′(0)]𝑟 𝐿 (3.31) [𝜃0]𝑟 𝑈 = [𝑢(0)]𝑟 𝑈 + 𝑥[𝑢′(0)]𝑟 𝑈 + 𝑥2 2! [𝑢′′(0)]𝑟 𝑈 (3.32) 8 int. j. anal. appl. (2022), 20:61 . . . ●) if 𝐿 = 𝑑𝑛+1 𝑑𝑥𝑛+1 , then we have: [𝜃0]𝑟 𝐿 = [𝑢(0)]𝑟 𝐿 + 𝑥[𝑢′(0)]𝑟 𝐿 + 𝑥2 2! [𝑢′′(0)]𝑟 𝐿 + ⋯ + 𝑥𝑛 𝑛! [𝑢(𝑛)(0)]𝑟 𝐿 (3.33) [𝜃0]𝑟 𝑈 = [𝑢(0)]𝑟 𝑈 + 𝑥[𝑢′(0)]𝑟 𝑈 + 𝑥2 2! [𝑢′′(0)]𝑟 𝑈 + ⋯ + 𝑥𝑛 𝑛! [𝑢(𝑛)(0)]𝑟 𝑈 (3.34) also, note that: [𝐴0]𝑟 , [𝐴1]𝑟 , [𝐴2]𝑟 , … , [𝐴𝑛 ]𝑟 are the fuzzy adomian polynomials, which can be found as follows: ●) [𝐴0]𝑟 = [ 𝐴0]𝑟 𝐿 , [𝐴0]𝑟 𝑈 ] (3.35) where [𝐴0]𝑟 𝐿 = [𝑁(𝑢0)]𝑟 𝐿 (3.36) [𝐴0]𝑟 𝑈 = [𝑁(𝑢0)]𝑟 𝑈 (3.37) ●) [𝐴1]𝑟 = [ 𝐴1]𝑟 𝐿 , [𝐴1]𝑟 𝑈 ] (3.38) where [𝐴1]𝑟 𝐿 = [𝑢1]𝑟 𝐿 [𝑁′(𝑢0)]𝑟 𝐿 (3.39) [𝐴1]𝑟 𝑈 = [𝑢1]𝑟 𝑈 [𝑁′(𝑢0)]𝑟 𝑈 (3.40) ●) [𝐴2]𝑟 = [ 𝐴2]𝑟 𝐿 , [𝐴2]𝑟 𝑈 ] (3.41) where [𝐴2]𝑟 𝐿 = [𝑢2]𝑟 𝐿 [𝑁′(𝑢0)]𝑟 𝐿 + 1 2! ([𝑢1]𝑟 𝐿 )2[𝑁′′(𝑢0)]𝑟 𝐿 (3.42) [𝐴2]𝑟 𝑈 = [𝑢2]𝑟 𝑈 [𝑁′(𝑢0)]𝑟 𝑈 + 1 2! ([𝑢1]𝑟 𝑈 )2[𝑁′′(𝑢0)]𝑟 𝑈 (3.43) . . . ●)[𝐴𝑛 ]𝑟 = [ 𝐴𝑛 ]𝑟 𝐿 , [𝐴𝑛]𝑟 𝑈 ], 𝑛 = 0,1,2, … (3.44) where [𝐴𝑛 ]𝑟 𝐿 = 1 𝑛! 𝑑𝑛 𝑑𝛽𝑛 ( [𝑁( ∑ 𝛽𝑛𝑢𝑛 ∞ 𝑛=0 )]𝑟 𝐿 )|𝛽=0 (3.45) [𝐴𝑛 ]𝑟 𝑈 = 1 𝑛! 𝑑𝑛 𝑑𝛽𝑛 ( [𝑁( ∑ 𝛽𝑛𝑢𝑛 ∞ 𝑛=0 )]𝑟 𝑈 )|𝛽=0 (3.46) 9 int. j. anal. appl. (2022), 20:61 4. applied examples in this section, we will solve three fuzzy problems to illustrate the accuracy of the fuzzy adomian decomposition method. example 1: consider the first order fuzzy autonomous differential equation 𝑢′(𝑥) = 𝑢(𝑥) + 𝑢2(𝑥) , 𝑥 ∈ [0 , 0.1] , with the fuzzy initial condition: [𝑢(0)]𝑟 = [0.96 + 0.04𝑟 , 1.01 − 0.01𝑟] , 𝑟 ∈ [0,1]. solution: we define: 𝐿(𝑢) = 𝑑 𝑑𝑥 (𝑢) 𝑅(𝑢) = [ 𝑅(𝑢)]𝑟 𝐿 = [ 𝑅(𝑢)]𝑟 𝑈 = −𝑢 𝑁(𝑢) = [ 𝑁(𝑢)]𝑟 𝐿 = [ 𝑁(𝑢)]𝑟 𝑈 = −𝑢2 𝑔(𝑥) = [ 𝑔(𝑥)]𝑟 𝐿 = [ 𝑔(𝑥)]𝑟 𝑈 = 0 from section (3), we can find: [𝜃0]𝑟 𝐿 = 0.96 + 0.04𝑟 [𝑢0]𝑟 𝐿 = (0.96 + 0.04𝑟) [𝐴0]𝑟 𝐿 = −(0.96 + 0.04𝑟 )2 [𝑢1]𝑟 𝐿 = ((0.96 + 0.04𝑟) + (0.96 + 0.04𝑟 )2)𝑥 [𝐴1]𝑟 𝐿 = (−2(0.96 + 0.04𝑟 )2 − 2(0.96 + 0.04𝑟 )3)𝑥 [𝑢2]𝑟 𝐿 = ( 1 2 (0.96 + 0.04𝑟) + 3 2 (0.96 + 0.04𝑟 )2 + (0.96 + 0.04𝑟 )3) 𝑥2 [𝐴2]𝑟 𝐿 = (−2(0.96 + 0.04𝑟 )2 − 5(0.96 + 0.04𝑟 )3 − 3(0.96 + 0.04𝑟 )4)𝑥2 [𝑢3]𝑟 𝐿 = ( 1 6 (0.96 + 0.04𝑟) + 7 6 (0.96 + 0.04𝑟 )2 + 2(0.96 + 0.04𝑟 )3 + (0.96 + 0.04𝑟 )4) 𝑥3 . . . also, we find: [𝜃0]𝑟 𝑈 = 1.01 − 0.01𝑟 [𝑢0]𝑟 𝑈 = (1.01 − 0.01𝑟) [𝐴0]𝑟 𝑈 = −(1.01 − 0.01𝑟 )2 10 int. j. anal. appl. (2022), 20:61 [𝑢1]𝑟 𝑈 = ((1.01 − 0.01𝑟) + (1.01 − 0.01𝑟 )2)𝑥 [𝐴1]𝑟 𝑈 = (−2(1.01 − 0.01𝑟 )2 − 2(1.01 − 0.01𝑟 )3)𝑥 [𝑢2]𝑟 𝑈 = ( 1 2 (1.01 − 0.01𝑟) + 3 2 (1.01 − 0.01𝑟 )2 + (1.01 − 0.01𝑟 )3) 𝑥2 [𝐴2]𝑟 𝑈 = (−2(1.01 − 0.01𝑟 )2 − 5(1.01 − 0.01𝑟 )3 − 3(1.01 − 0.01𝑟 )4)𝑥2 [𝑢3]𝑟 𝑈 = ( 1 6 (1.01 − 0.01𝑟) + 7 6 (1.01 − 0.01𝑟 )2 + 2(1.01 − 0.01𝑟 )3 + (1.01 − 0.01𝑟 )4) 𝑥3 . . . therefore, the fuzzy semi analytical solution is: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] where [𝑢(𝑥)]𝑟 𝐿 = (0.96 + 0.04𝑟) + ((0.96 + 0.04𝑟) + (0.96 + 0.04𝑟 )2)𝑥 +( 1 2 (0.96 + 0.04𝑟) + 3 2 (0.96 + 0.04𝑟 )2 + (0.96 + 0.04𝑟 )3) 𝑥2 + ( 1 6 (0.96 + 0.04𝑟) + 7 6 (0.96 + 0.04𝑟 )2 + 2(0.96 + 0.04𝑟 )3 + (0.96 + 0.04𝑟 )4) 𝑥3 + ⋯ [𝑢(𝑥)]𝑟 𝑈 = (1.01 − 0.01𝑟) + ((1.01 − 0.01𝑟) + (1.01 − 0.01𝑟 )2)𝑥 +( 1 2 (1.01 − 0.01𝑟) + 3 2 (1.01 − 0.01𝑟 )2 + (1.01 − 0.01𝑟 )3) 𝑥2 + ( 1 6 (1.01 − 0.01𝑟) + 7 6 (1.01 − 0.01𝑟 )2 + 2(1.01 − 0.01𝑟 )3 + (1.01 − 0.01𝑟 )4) 𝑥3 + ⋯ example 2: consider the second order fuzzy autonomous differential equation 𝑢′′(𝑥) + 𝑢(𝑥) = 5 , 𝑥 ∈ [0 , 1], with the fuzzy initial conditions: [𝑢(0)]𝑟 = [𝑟 , 2 − 𝑟] [𝑢′(0)]𝑟 = [1 + 𝑟 , 3 − 𝑟] , 𝑟 ∈ [0,1]. solution: we define: 𝐿(𝑢) = 𝑑2 𝑑𝑥2 (𝑢) 𝑅(𝑢) = [ 𝑅(𝑢)]𝑟 𝐿 = [ 𝑅(𝑢)]𝑟 𝑈 = 𝑢 11 int. j. anal. appl. (2022), 20:61 𝑁(𝑢) = [ 𝑁(𝑢)]𝑟 𝐿 = [ 𝑁(𝑢)]𝑟 𝑈 = 0 𝑔(𝑥) = [ 𝑔(𝑥)]𝑟 𝐿 = [ 𝑔(𝑥)]𝑟 𝑈 = 5 from section (3), we can find: [𝜃0]𝑟 𝐿 = 𝑟 + (1 + 𝑟)𝑥 [𝑢0]𝑟 𝐿 = 𝑟 + (1 + 𝑟)𝑥 + 5 2 𝑥2 [𝐴0]𝑟 𝐿 = 0 [𝑢1]𝑟 𝐿 = − 𝑟 2 𝑥2 − (𝑟+1) 6 𝑥3 − 5 24 𝑥4 [𝐴1]𝑟 𝐿 = 0 [𝑢2]𝑟 𝐿 = 𝑟 24 𝑥4 + (𝑟+1) 120 𝑥5 + 5 720 𝑥6 [𝐴2]𝑟 𝐿 = 0 [𝑢3]𝑟 𝐿 = − 𝑟 720 𝑥6 − (𝑟+1) 5040 𝑥7 − 5 40320 𝑥8 . . . also, we find: [𝜃0]𝑟 𝑈 = (2 − 𝑟) + (3 − 𝑟)𝑥 [𝑢0]𝑟 𝑈 = (2 − 𝑟) + (3 − 𝑟)𝑥 + 5 2 𝑥2 [𝐴0]𝑟 𝑈 = 0 [𝑢1]𝑟 𝑈 = − (2−𝑟) 2 𝑥2 − (3−𝑟) 6 𝑥3 − 5 24 𝑥4 [𝐴1]𝑟 𝑈 = 0 [𝑢2]𝑟 𝑈 = (2−𝑟) 24 𝑥4 + (3−𝑟) 120 𝑥5 + 5 720 𝑥6 [𝐴2]𝑟 𝑈 = 0 [𝑢3]𝑟 𝑈 = − (2−𝑟) 720 𝑥6 − (3−𝑟) 5040 𝑥7 − 5 40320 𝑥8 . . . therefore, the fuzzy semi analytical solution is: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] where 12 int. j. anal. appl. (2022), 20:61 [𝑢(𝑥)]𝑟 𝐿 = 𝑟 + (𝑟 + 1)𝑥 + ( 5−𝑟 2 ) 𝑥2 − ( 𝑟+1 6 ) 𝑥3 + ( 𝑟−5 24 ) 𝑥4 + ( 𝑟+1 120 ) 𝑥5 + ( 5−𝑟 720 ) 𝑥6 − ( 𝑟+1 5040 ) 𝑥7 − ( 5 40320 ) 𝑥8 + ⋯ = 5 + (𝑟 − 5)𝑐𝑜𝑠𝑥 + (𝑟 + 1)𝑠𝑖𝑛𝑥 [𝑢(𝑥)]𝑟 𝑈 = (2 − 𝑟) + (3 − 𝑟)𝑥 + ( 3+𝑟 2 ) 𝑥2 − ( 3−𝑟 6 ) 𝑥3 − ( 3+𝑟 24 ) 𝑥4 + ( 3−𝑟 120 ) 𝑥5 + ( 3+𝑟 720 ) 𝑥6 − ( 3−𝑟 5040 ) 𝑥7 − ( 5 40320 ) 𝑥8 + ⋯ = 5 − (3 + 𝑟)𝑐𝑜𝑠𝑥 + (3 − 𝑟)𝑠𝑖𝑛𝑥 which is the fuzzy exact analytical solution. example 3: consider the second order fuzzy autonomous differential equation 𝑢′′(𝑥) + [2𝑟 + 1, −2𝑟 + 5]𝑢(𝑥) + √𝑢(𝑥) = 0 , 𝑥 ∈ [0 , 1] , with the fuzzy initial conditions: [𝑢(0)]𝑟 = [0 , 0] [𝑢′(0)]𝑟 = [𝑟 + 4 , −𝑟 + 6] , 𝑟 ∈ [0,1]. solution: we define: 𝐿(𝑢) = 𝑑2 𝑑𝑥2 (𝑢) [𝑅(𝑢)]𝑟 𝐿 = (2𝑟 + 1)𝑢 [𝑅(𝑢)]𝑟 𝑈 = (−2𝑟 + 5)𝑢 𝑁(𝑢) = [ 𝑁(𝑢)]𝑟 𝐿 = [ 𝑁(𝑢)]𝑟 𝑈 = √𝑢 𝑔(𝑥) = [ 𝑔(𝑥)]𝑟 𝐿 = [ 𝑔(𝑥)]𝑟 𝑈 = 0 from section (3), we can find: [𝜃0]𝑟 𝐿 = (𝑟 + 4)𝑥 [𝑢0]𝑟 𝐿 = (𝑟 + 4)𝑥 [𝐴0]𝑟 𝐿 = ((𝑟 + 4)𝑥) 1 2 [𝑢1]𝑟 𝐿 = − (2𝑟+1) 6(𝑟+4)2 ((𝑟 + 4)𝑥)3 − 4 15(𝑟+4)2 ((𝑟 + 4)𝑥) 5 2 [𝐴1]𝑟 𝐿 = − (2𝑟+1) 12(𝑟+4)2 ((𝑟 + 4)𝑥) 5 2 − 2 15(𝑟+4)2 ((𝑟 + 4)𝑥)2 13 int. j. anal. appl. (2022), 20:61 [𝑢2]𝑟 𝐿 = 1 90(𝑟+4)4 ((𝑟 + 4)𝑥) 4 + (2𝑟+1)2 120(𝑟+4)4 ((𝑟 + 4)𝑥) 5 + (2𝑟+1) 90(𝑟+4)4 ((𝑟 + 4)𝑥) 9 2 [𝐴2]𝑟 𝐿 = (2𝑟+1) 60(𝑟+4)4 ((𝑟 + 4)𝑥) 4 − 1 300(𝑟+4)4 ((𝑟 + 4)𝑥) 7 2 + (2𝑟+1)2 1440(𝑟+4)4 ((𝑟 + 4)𝑥) 9 2 [𝑢3]𝑟 𝐿 = − (2𝑟+1) 1080(𝑟+4)6 ((𝑟 + 4)𝑥) 6 − (2𝑟+1)3 5040(𝑟+4)6 ((𝑟 + 4)𝑥) 7 + 1 7425(𝑟+4)6 ((𝑟 + 4)𝑥) 11 2 − 17(2𝑟+1)2 51480(𝑟+4)6 ((𝑟 + 4)𝑥) 13 2 . . . also, we find: [𝜃0]𝑟 𝑈 = (6 − 𝑟)𝑥 [𝑢0]𝑟 𝑈 = (6 − 𝑟)𝑥 [𝐴0]𝑟 𝑈 = ((6 − 𝑟)𝑥) 1 2 [𝑢1]𝑟 𝑈 = − (5−2𝑟) 6(6−𝑟)2 ((6 − 𝑟)𝑥)3 − 4 15(6−𝑟)2 ((6 − 𝑟)𝑥) 5 2 [𝐴1]𝑟 𝑈 = − (5−2𝑟) 12(6−𝑟)2 ((𝑟 + 4)𝑥) 5 2 − 2 15(6−𝑟)2 ((6 − 𝑟)𝑥)2 [𝑢2]𝑟 𝑈 = 1 90(6−𝑟)4 ((6 − 𝑟)𝑥) 4 + (5−2𝑟)2 120(6−𝑟)4 ((6 − 𝑟)𝑥) 5 + (5−2𝑟) 90(6−𝑟)4 ((6 − 𝑟)𝑥) 9 2 [𝐴2]𝑟 𝑈 = (2𝑟+1) 60(6−𝑟)4 ((6 − 𝑟)𝑥) 4 − 1 300(6−𝑟)4 ((6 − 𝑟)𝑥) 7 2 + (2𝑟+1)2 1440(6−𝑟)4 ((6 − 𝑟)𝑥) 9 2 [𝑢3]𝑟 𝑈 = − (5−2𝑟) 1080(6−𝑟)6 ((𝑟 + 4)𝑥) 6 − (5−2𝑟)3 5040(6−𝑟)6 ((6 − 𝑟)𝑥) 7 + 1 7425(6−𝑟)6 ((6 − 𝑟)𝑥) 11 2 − 17(5−2𝑟)2 51480(6−𝑟)6 ((6 − 𝑟)𝑥) 13 2 . . . therefore, the fuzzy semi analytical solution is: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] where [𝑢(𝑥)]𝑟 𝐿 = (𝑟 + 4) [ 𝑥 − (2𝑟 + 1) 𝑥3 3! + (2𝑟 + 1)2 𝑥5 5! − (2𝑟 + 1)3 𝑥7 7! ] + (𝑟 + 4) 1 2 [ − 4 15 𝑥 5 2 + 1 90 (2𝑟 + 1) 𝑥 9 2 − 17 51480 (2𝑟 + 1)2 𝑥 13 2 ] + [ 1 90 𝑥4 − 1 1080 (2𝑟 + 1) 𝑥6 + 1 7425 (𝑟 + 4) − 1 2 𝑥 11 2 ] + ⋯ 14 int. j. anal. appl. (2022), 20:61 [𝑢(𝑥)]𝑟 𝑈 = (6 − 𝑟) [ 𝑥 − (5 − 2𝑟) 𝑥3 3! + (5 − 2𝑟)2 𝑥5 5! − (5 − 2𝑟)3 𝑥7 7! ] + (6 − 𝑟) 1 2 [ − 4 15 𝑥 5 2 + 1 90 (5 − 2𝑟) 𝑥 9 2 − 17 51480 (5 − 2𝑟)2 𝑥 13 2 ] + [ 1 90 𝑥4 − 1 1080 (5 − 2𝑟) 𝑥6 + 1 7425 (6 − 𝑟) − 1 2 𝑥 11 2 ] + ⋯ 5. discussion to show the accuracy of the results, we will give a numerical comparison between the exact analytical solution and the semi analytical solution. if we go back to example 1: 𝑢′(𝑥) = 𝑢(𝑥) + 𝑢2(𝑥) , 𝑥 ∈ [0 , 0.1] , the fuzzy exact analytical solution for this problem is: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] where [𝑢(𝑥)]𝑟 𝐿 = (1.96+0.04𝑟) (1.96+0.04𝑟)−(0.96+0.04𝑟)𝑒 𝑥 − 1 [𝑢(𝑥)]𝑟 𝑈 = (2.01−0.01𝑟) (2.01−0.01𝑟)−(1.01−0.01𝑟)𝑒 𝑥 − 1 while the fuzzy semi analytical solution that we got (at 𝑟 = 0.5 ) is: [𝑢(𝑥)]𝑟 = [ 𝑢(𝑥)]𝑟 𝐿 , [𝑢(𝑥)]𝑟 𝑈 ] where [𝑢(𝑥)]𝑟 𝐿 = 0.98 + 1.9404𝑥 + 2.871792 𝑥2 + 4.08855216𝑥3 + ⋯ [𝑢(𝑥)]𝑟 𝑈 = 1.005 + 2.015025𝑥 + 3.032612625 𝑥2 + 4.396163251𝑥3 + ⋯ we test the accuracy by computing the absolute errors: [𝑒𝑟𝑟𝑜𝑟]𝑟 𝐿 = | [𝑢𝑒𝑥𝑎𝑐𝑡(x)]𝑟 𝐿 − [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝐿 | [𝑒𝑟𝑟𝑜𝑟]𝑟 𝑈 = | [𝑢𝑒𝑥𝑎𝑐𝑡(x)]𝑟 𝑈 − [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝑈 | 15 int. j. anal. appl. (2022), 20:61 table 1: numerical results for example 1. 𝑥 [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝐿 [𝑒𝑟𝑟𝑜𝑟]𝑟 𝐿 [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝑈 [𝑒𝑟𝑟𝑜𝑟]𝑟 𝑈 0 0.980000000000000 0 1.005000000000000 0 0.01 0.999695267752160 5.90 e-8 1.025457907425751 6.46 e-8 0.02 1.019989425217280 9.57 e-7 1.046548714356008 1.05 e-6 0.03 1.040907003708320 4.92 e-6 1.068298797770277 5.39 e-6 0.04 1.062472534538240 1.58 e-5 1.090734534648064 1.73 e-5 0.05 1.084710549020000 3.91 e-5 1.113882301968875 4.29 e-5 0.06 1.107645578466560 8.23 e-5 1.137768476712216 9.03 e-5 0.07 1.131302154190880 1.55 e-4 1.162419435857593 1.70 e-4 0.08 1.155704807505920 2.69 e-4 1.187861556384512 2.95 e-4 0.09 1.180878069724640 4.37 e-4 1.214121215272479 4.80 e-4 0.10 1.206846472160000 6.78 e-4 1.241224789501000 7.44 e-4 16 int. j. anal. appl. (2022), 20:61 table 2: numerical results for example 1. 𝑥 [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝐿 [𝑒𝑟𝑟𝑜𝑟]𝑟 𝐿 [𝑢𝑠𝑒𝑟𝑖𝑒𝑠(x)]𝑟 𝑈 [𝑒𝑟𝑟𝑜𝑟]𝑟 𝑈 0.001 0.981943275880552 5.82 e-12 1.007018062008788 6.37 e-12 0.003 0.985847156518908 4.73 e-10 1.011072487210033 5.18 e-10 0.005 0.989774305869020 3.66 e-9 1.015151489836031 4.01 e-9 0.007 0.993724920181391 1.41 e-8 1.019255280902620 1.54 e-8 0.009 0.997699195706525 3.86 e-8 1.023384071425635 4.23 e-8 0.011 1.001697328694925 8.64 e-8 1.027538072420912 9.47 e-8 0.013 1.005719515397095 1.69 e-7 1.031717494904287 1.85 e-7 0.015 1.009765952063540 3.01 e-7 1.035922549891597 3.29 e-7 0.017 1.013836834944762 4.97 e-7 1.040153448398677 5.45 e-7 0.019 1.017932360291266 7.78 e-7 1.044410401441363 8.53 e-7 0.021 1.022052724353554 1.17 e-6 1.048693620035492 1.28 e-6 17 int. j. anal. appl. (2022), 20:61 6. conclusion in this work, we have studied the fuzzy semi analytical solutions of the fuzzy autonomous differential equations. we have used the fuzzy adomian decomposition method to find these solutions. based on the numerical results we obtained, the fuzzy adomian decomposition method is a highly efficient method in solving and gives accurate results, and in some cases, this method gives us the exact analytical solution. the accuracy of this method varies from one fuzzy differential equation to another, and this depends on the type of fuzzy differential equation, whether it is of the first order or the highest order, and also depends on the nature of the fuzzy differential equation, whether it is linear or non-linear. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] t. allahviranloo, l. jamshidi, solution of fuzzy differential equations under generalized differentiability by adomian decomposition method, iran. j. optim. 1 (2009), 57-75. [2] l. wang, s. guo, adomian method for second-order fuzzy differential equation, int. j. math. comput. sci. 5 (2011), 613-616. [3] s. narayanamoorthy, t. yookesh, an adomian decomposition method to solve linear fuzzy differential equations, in: proceeding of the international conference on mathematical methods and computation, india, 13-14 february 2014. [4] m. paripour, e. hajilou, a. hajilou, h. heidari, application of adomian decomposition method to solve hybrid fuzzy differential equations, j. taibah univ. sci. 9 (2015), 95–103. https://doi.org/10.1016/j.jtusci.2014.06.002. [5] a. jameel, numerical and approximate – analytical solutions of fuzzy initial value problems, ph.d. thesis, school of quantitative sciences, university utara malaysia, malaysia, 2015. [6] s. biswas, s. banerjee, t. roy, solving intuitionistic fuzzy differential equations with linear differential operator by adomian decomposition method, notes ifs, 22 (2016), 25-41. http://ifigenia.org/wiki/issue:nifs/22/4/25-41. [7] s. biswas, t. roy, adomian decomposition method for fuzzy differential equations with linear differential operator, j. inform. comput. sci. 11 4 (2016), 243-250. https://doi.org/10.1016/j.jtusci.2014.06.002 http://ifigenia.org/wiki/issue:nifs/22/4/25-41 18 int. j. anal. appl. (2022), 20:61 [8] m. suhhiem, fuzzy artificial neural network for solving fuzzy and non-fuzzy differential equations, ph.d. thesis, college of sciences, university of al-mustansiriyah, iraq, 2016. [9] a.k. ateeah, approximate solution for fuzzy differential algebraic equations of fractional order using adomian decomposition method, ibn al-haitham j. pure appl. sci. 30 (2017), 202-213. [10] s. askari, t. allahviranloo, s. abbasbandy, solving fuzzy fractional differential equations by adomian decomposition method used in optimal control theory, int. trans. j. eng. manage. appl. sci. technol. 10 (2019), 10a12p. [11] h.a. sabr, b.n. abood, m. suhhiem, fuzzy homotopy analysis method for solving fuzzy autonomous differential equation, ratio math. 40 (2021), 191-212. https://doi.org/10.23755/rm.v40i1.589. https://doi.org/10.23755/rm.v40i1.589 international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 198-210 doi: 10.28924/2291-8639-15-2017-198 countably infinitely many positive solutions for even order boundary value problems with sturm-liouville type integral boundary conditions on time scales k. r. prasad∗ and md. khuddush abstract. in this paper, we establish the existence of countably infinitely many positive solutions for a certain even order two-point boundary value problem with integral boundary conditions on time scales by using hölder’s inequality and krasnoselskii’s fixed point theorem for operators on a cone. 1. introduction the study of dynamic equations on time scales unifies existing results in differential and finite difference equations, and provides powerful new tools for exploring connections between the traditionally separated fields. for details refer to the books by bohner and peterson [6], [7], lakshmikantham et al. [23] and the papers [1], [3], [19]. the boundary value problems with integral boundary conditions occur in the study of nonlocal phenomena in many different areas of applied mathematics, physics and engineering, in particular, in heat conduction, chemical engineering, underground waterflow, thermo-elasticity, plasma physics, [2], [10], [11], [21], [22], [27], [33], [36] and reference therin. recently, authors established the existence of positive solutions to boundary value problems with integral boundary conditions on time scales; for details, see [9], [12], [13], [18], [26], [28], [32], [34] and reference therein. however, to the best of our knowledge, little work has been done on the existence of positive solutions for higher order boundary value problems with integral boundary conditions on time scales. we would like to mention some results of karasa and tokmak [20], li and wang [25], li and sun [24], cetin and topal [8], and sreedhar et al [31] which motivate us to consider the problem (1.6)-(1.7). in 2013, karasa and tokmak [20] established the existence of a positive solution of the following third order boundary value problem with integral boundary conditions,( φ(−u∆∆(t)) )∆ + q(t)f(t,u(t),u∆(t)) = 0, t ∈ [0, 1]t , au(0) − bu∆(0) = ∫ 1 0 g1(s)u(s)∆s, cu(1) + du ∆(1) = ∫ 1 0 g2(s)u(s)∆s, u ∆∆(1) = 0,   (1.1) by using four functionals fixed point theorem. in the same year, li and wang [25] studied the existence of a positive solution of the following nonlinear third order boundary value problem with integral boundary conditions,( φ(−u∆∆(t)) )∆ + q(t)f(t,u(t),u∆(t)) = 0, t ∈ [0, 1]t , au(0) − bu∆(0) = ∫ 1 0 g1(s,u(s))∆s, cu(1) + du ∆(1) = ∫ 1 0 g2(s)u(s)∆s, u ∆∆(1) = 0,   (1.2) by applying a generalization of the leggett-williams fixed point theorem. in [24], li and sun studied the following boundary value problem on time scales, x∆(t) + p(t)xσ(t) = f ( t,xσ(t) ) , t ∈ (0,t)t, x(0) −βxσ(t) = α ∫ σ(t) 0 xσ(s)∆g(s),   (1.3) received 14th august, 2017; accepted 18th october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 34b18, 34n05. key words and phrases. green’s function; boundary value problem; positive solution; cone. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 198 countable many positive sol. for eobvps with ibcs on time scales 199 where xσ = x ◦ σ, and by using fixed point index theory the existence of infinitely many positive solutions for (1.3) are obtained. in [8], cetin and topal investigated the existence of solutions for integral boundary value problems on time scales, [p(t)x∆(t)]∇ + q(t)x(t) = f(t,x(t)), t ∈ [a,b]t , αx(ρ(a)) −βx[∆](ρ(a)) = ∫ b ρ(a) h1(x(s))∇s,γx(b) + δx[∆](b) = ∫ b ρ(a) h2(x(s))∇s,   (1.4) by using schauder fixed point theorem in a cone and by the method of upper and lower solutions. in 2017, sreedhar et al [31] considered the 2nth order boundary value problem with integral boundary conditions on time scales, (−1)nu∆ 2n (t) = f ( t,u(t) ) , t ∈ (0, 1)t, u∆ 2i (0) = u∆ 2i (1) = ∫ 1 0 ai+1(x)u ∆2i(x)∆x, 0 ≤ i ≤ n− 1,   (1.5) where n act as positive. by using avery-henderson fixed point theorem, the authors established the existence of even number of positive solutions for (1.5). motivated by the work mentioned above, in this paper we investigate the existence of infinitely many positive solutions for the even order boundary value problem on time scales given by (−1)nu(∆∇) n (t) = ω(t)f ( u(t) ) , t ∈ [0, 1]t, (1.6) satisfying the sturm-liouville type integral boundary conditions αi+1u (∆∇)i(0) −βi+1u(∆∇) i∆(0) = ∫ 1 0 ai+1(s)u (∆∇)i(s)∇s, 0 ≤ i ≤ n− 1, γi+1u (∆∇)i(1) + δi+1u (∆∇)i∆(1) = ∫ 1 0 bi+1(s)u (∆∇)i(s)∇s, 0 ≤ i ≤ n− 1,   (1.7) where n ≥ 1,t is a time scale, f ∈ c ( [0, +∞), [0, +∞) ) ,ω(t) ∈ lp∇[0, 1] for some p ≥ 1 and has countably many singularities in (0, 1 2 )t. we show that the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions by imposing suitable conditions on ω and f. the key tool in our approach is the hölder’s inequality and krasnoselskii’s fixed point theorem for operators on a cone. 2. preliminaries in this section, we provide some definitions and lemmas which are useful for our later discussions; for details, see [2], [4], [5], [6], [16], [30], [35]. definition 2.1. a time scale t is a nonempty closed subset of the real numbers r. t has the topology that it inherits from the real numbers with the standard topology. it follows that the jump operators σ,ρ : t → t, σ(t) = inf{r ∈ t : r > t}, ρ(t) = sup{r ∈ t : r < t} (supplemented by inf ∅ := sup t and sup∅ := inf t) are well defined. the point t ∈ t is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) < t, respectively. by an interval time scale, we mean the intersection of a real interval with a given time scale. i.e., [a,b]t = [a,b] ∩t other intervals can be defined similarly. definition 2.2. let µ∆ and µ∇ be the lebesgue ∆− measure and the lebesgue ∇−measure on t, respectively. if a ⊂ t satisfies µ∆(a) = µ∇(a), then we call a is measurable on t, denoted µ(a) and this value is called the lebesgue measure of a. let p denote a proposition with respect to t ∈ t. (i) if there exists e1 ⊂ a with µ∆(e1) = 0 such that p holds on a\e1, then p is said to hold ∆–a.e. on a. (ii) if there exists e2 ⊂ a with µ∇(e2) = 0 such that p holds on a\e2, then p is said to hold ∇–a.e. on a. 200 prasad and khuddush definition 2.3. let e ⊂ t be a ∇–measurable set and p ∈ r̄ ≡ r∪{−∞, +∞} be such that p ≥ 1 and let f : e → r̄ be ∇–measurable function. we say that f belongs to lp∇(e) provided that either∫ e |f|p(s)∇s < ∞ if p ∈ r, or there exists a constant m ∈ r such that |f| ≤ m, ∇−a.e. on e if p = +∞. lemma 2.1. let e ⊂ t be a ∇–measurable set. if f : t → r is a ∇–integrable on e, then∫ e f(s)∇s = ∫ e f(s)ds + ∑ i∈ie ( ti −ρ(ti) ) f(ti), where ie := {i ∈ i : ti ∈ e} and {ti}i∈i,i ⊂ n, is the set of all left-scattered points of t. for convenience, we introduce the following notation throughout the paper: for τ ∈ (0, 1 2 )t, φi(t) := γi + δi −γit, ψi(t) := βi + αit, di := γiβi + αiδi + αiγi, gi(t,s) := 1 di { φi(s)ψi(t), t ≤ s, φi(t)ψi(s), s ≤ t, a := ∫ 1 0 [∫ 1 0 gi(s,r)ai(s)∇s ] ω(r)∇r, b := ∫ 1 0 [∫ 1 0 gi(s,r)bi(s)∇s ] ω(r)∇r, ui := 1 di ∫ 1 0 ai(t)φi(t)∇t, u∗i := 1 di ∫ 1 0 ai(t)ψi(t)∇t, vi := 1 di ∫ 1 0 bi(t)ψi(t)∇t, v∗i := 1 di ∫ 1 0 bi(t)φi(t)∇t, θi(t) := (1 −vi)φi(t) + v∗i ψi(t) di ( (1 −ui)(1 −vi) −u∗iv ∗ i ), ζi(t) := (1 −ui)ψi(t) + u∗iφi(t) di ( (1 −ui)(1 −vi) −u∗iv ∗ i ), θ∗i := max t∈j0 θi(t), θ ∗∗ i := max t∈[τ, 1−τ]t θi(t), ζ ∗ i := max t∈j0 ζi(t), ζ ∗∗ i := max t∈[τ, 1−τ]t ζi(t), a∗i := ∫ 1 0 ai(t)∇t, ai(τ) := ∫ 1−τ τ ai(t)∇t, b∗i := ∫ 1 0 bi(t)∇t, bi(τ) := ∫ 1−τ τ bi(t)∇t, gi := ∫ 1 0 gi(t,t)∇t, gi(τ) := ∫ 1−τ τ gi(t,t)∇t, and j0 := [0, 1]t. we make the following assumptions throughout the paper: (h1) there exists a sequence {tk}∞k=1 (k ∈ n), t1 < 12, limk→∞tk = t ∗ ≥ 0 and lim t→tk ω(t) = +∞ for all k = 1, 2, 3, · · · . (h2) ω ∈ lp∇(j0) for some 1 ≤ p ≤ +∞ and there exists m > 0 such that ω(t) ≥ m for all [t∗, 1 − t∗]t, (h3) f : j0 × [0, +∞) → [0, +∞) is continuous, (h4) ai,bi ∈ l1∇(j0) for all 1 ≤ i ≤ n, are nonnegative and (1 − ui)(1 − vi) − v ∗ i u ∗ i > 0 for all 1 ≤ i ≤ n, on j0, (h5) αi,βi,γi,δi ≥ 0 such that di := γiβi + αiδi + αiγi > 0 for each 1 ≤ i ≤ n. the rest of the paper is organized in the following fashion. in section 2, we provide some definitions and lemmas that provide us with some useful information concerning the behavior of solution of the boundary value problem (1.6)-(1.7). in section 3, we construct the green’s function for the homogeneous problem corresponding to (1.6)-(1.7), estimate bounds for the green’s function, and some lemmas which are needed in establishing our main results are provided. in section 4, we establish a criteria for the existence of countable number of positive solutions for the boundary value problem (1.6)-(1.7) by applying krasnoselskii’s fixed point theorem in cones. finally, we provide an example of a family of functions ω(t) that satisfy conditions (h1) − (h3). countable many positive sol. for eobvps with ibcs on time scales 201 3. green’s function and bounds in this section, we construct the green’s function for the homogeneous problem corresponding to (1.6)-(1.7) and estimate bounds for the green’s function. lemma 3.1. let (h4), (h5) hold. then for any h(t) ∈ c(j0), the boundary value problem, −u∆∇(t) = h(t), t ∈ j0, (3.1) αiu(0) −βiu∆(0) = ∫ 1 0 ai(s)u(s)∇s, 1 ≤ i ≤ n, (3.2) γiu(1) + δiu ∆(1) = ∫ 1 0 bi(s)u(s)∇s, 1 ≤ i ≤ n, (3.3) has a unique solution u(t) = ∫ 1 0 ki(t,s)h(s)∇s, for 1 ≤ i ≤ n, (3.4) where, ki(t,s) = gi(t,s) + θi(t) ∫ 1 0 gi(r,s)aj(r)∇r + ζi(t) ∫ 1 0 gi(r,s)bi(r)∇r, (3.5) for 1 ≤ i ≤ n, proof. suppose that u is a solution of (3.1), then, we have u(t) = − ∫ t 0 ∫ s 0 h(r)∇r∆s + at + b = − ∫ t 0 (t−s)h(s)∇s + at + b where a = lim t→0+ u∆(t) and b = u(0). using the boundary conditions (3.1), (3.2) we can determined a and b as a = 1 di ∫ 1 0 [αibi(s) −γiai(s)]u(s)∇s + αi di ∫ 1 0 [γi(1 −s) + δi]h(s)∇s b = 1 di ∫ 1 0 [(γi + δi)ai(s) + βibi(s)]u(s)∇s + βi di ∫ 1 0 [γi(1 −s) + δi]h(s)∇s thus, we have u(t) = 1 di [∫ t 0 (γi + δi −γit)(βi + αis)h(s)∇s + ∫ 1 t (γi + δi −γis)(βi + αit)h(s)∇s ] + 1 di [∫ 1 0 [(γi + δi −γit)ai(s) + (βi + αit)bi(s)]u(s)∇s ] = 1 di [∫ t 0 φi(t)ψi(s)h(s)∇s + ∫ 1 t φi(s)ψi(t)h(s)∇s ] + 1 di ∫ 1 0 [φi(t)ai(s) + ψi(t)bi(s)]u(s)∇s, from which, we obtain u(t) = ∫ 1 0 gi(t,s)h(s)∇s + 1 di φi(t) ∫ 1 0 ai(s)u(s)∇s + 1 di ψi(t) ∫ 1 0 bi(s)u(s)∇s. (3.6) after certain computations we can determined,∫ 1 0 ai(s)u(s)∇s = (1 −vi)ai + u∗ibi (1 −ui)(1 −vi) −v∗i u ∗ i . (3.7) ∫ 1 0 bi(s)u(s)∇s = v∗i ai + (1 −ui)bi (1 −ui)(1 −vi) −v∗i u ∗ i . (3.8) 202 prasad and khuddush from (3.7) and (3.8), (3.6) can be written as u(t) = ∫ 1 0 gi(t,s)h(s)∇s + 1 di φi(t) (1 −vi)ai + u∗ibi (1 −ui)(1 −vi) −v∗i u ∗ i + 1 di ψi(t) v∗i ai + (1 −ui)bi (1 −ui)(1 −vi) −v∗i u ∗ i = ∫ 1 0 gi(t,s)h(s)∇s + (1 −vi)φi(t) + v∗i ψi(t) di ( (1 −ui)(1 −vi) −v∗i u ∗ i )ai + (1 −ui)ψi(t) + u∗iφi(t) di ( (1 −ui)(1 −vi) −v∗i u ∗ i )bi = ∫ 1 0 gi(t,s)h(s)∇s + θi(t)ai + ζi(t)bi = ∫ 1 0 gi(t,s)h(s)∇s + θi(t) ∫ 1 0 (∫ 1 0 gi(r,s)ai(r)∇r ) h(s)∇s + ζi(t) ∫ 1 0 (∫ 1 0 gi(r,s)bi(r)∇r ) h(s)∇s = ∫ 1 0 [ gi(t,s) + θi(t) ∫ 1 0 gi(r,s)ai(r)∇r + ζi(t) ∫ 1 0 gi(r,s)bi(r)∇r ] h(s)∇s = ∫ 1 0 ki(t,s)h(s)∇s, where ki(t,s) is defined in (3.5). the proof is complete. � lemma 3.2. assume that (h4), (h5) hold and for τ ∈ (0, 1 2 )t define ηi(τ) = min { αiτ + βi αi + βi , γiτ + δi γi + δi } < 1. then gi(t,s) for 1 ≤ i ≤ n, satisfies the following properties: (i) 0 ≤ gi(t,s) ≤ gi(s,s) for all t,s ∈ j0, (ii) 0 ≤ ηi(τ)gi(s,s) ≤ gi(t,s) for all t ∈ [τ, 1 − τ]t and s ∈ j0, lemma 3.3. assume that (h4), (h5) holds. then ki(t,s) for 1 ≤ i ≤ n, have the following properties: (i) 0 ≤ ki(t,s) ≤ ( 1 + θ∗i a ∗ i + ζ ∗ i b ∗ i ) gi(s,s) for all t,s ∈ j0, (ii) 0 ≤ ηi(τ) ( 1 + θ∗∗i ai(τ) + ζ ∗∗ i bi(τ) ) gi(s,s) ≤ ki(t,s) for all t ∈ [τ, 1 − τ]t and s ∈ j0 lemma 3.4. assume that (h4), (h5) hold and kj(t,s) for 1 ≤ j ≤ n, is given in (3.5). let h1(t,s) = k1(t,s) and recursively define hj(t,s) = ∫ 1 0 hj−1(t,r)kj(r,s)∇r, for 2 ≤ j ≤ n. (3.9) then hn(t,s) is the green’s function for the homogeneous boundary value problem (−1)nu(∆∇) n (t) = 0, t ∈ j0, αi+1u (∆∇)i(0) −βi+1u(∆∇) i∆(0) = ∫ 1 0 ai+1(s)u (∆∇)i(s)∇s, 0 ≤ i ≤ n− 1, γi+1u (∆∇)i(1) + δi+1u (∆∇)i∆(1) = ∫ 1 0 bi+1(s)u (∆∇)i(s)∇s, 0 ≤ i ≤ n− 1. countable many positive sol. for eobvps with ibcs on time scales 203 lemma 3.5. assume that (h4), (h5) hold. define g = n−1∏ j=1 gj, φ = n∏ j=1 ( 1 + θ∗ja ∗ j + ζ ∗ j b ∗ j ) , ητ = n∏ j=1 ηj(τ) ( 1 + θ∗∗j aj(τ) + ζ ∗∗ j bj(τ) ) , gτ = n−1∏ j=1 gj(τ), then the green’s function hn(t,s) satisfies the following inequalities: (i) 0 ≤ hn(t,s) ≤ gφgn(s,s), for all t,s ∈ j0 and (ii) hn(t,s) ≥ ητgτgn(s,s), for all t ∈ [τ, 1 − τ]t and s ∈ j0, proof. it is clear that green’s function hn(t,s) ≥ 0, for all t,s ∈ j0. now we prove the inequality by induction on n and denote the statement by p(n). from (3.5) we have h1(t,s) = g1(t,s) + θ1(t) ∫ 1 0 g1(r,s)a1(r)∇r + ζ1(t) ∫ 1 0 g1(r,s)b1(r)∇r ≤ g1(s,s) + θ1(t) ∫ 1 0 g1(s,s)a1(r)∇r + ζ1(t) ∫ 1 0 g1(s,s)b1(r)∇r ≤ ( 1 + θ1(t) ∫ 1 0 a1(r)∇r + ζ1(r) ∫ 1 0 b1(r)∇r ) g1(s,s) ≤ ( 1 + θ∗1a ∗ 1 + ζ ∗ 1 b ∗ 1 ) g1(s,s) and for t ∈ [τ, 1 − τ]t, h1(t,s) = g1(t,s) + θ1(t) ∫ 1 0 g1(r,s)a1(r)∇r + ζ1(t) ∫ 1 0 g1(r,s)b1(r)∇r ≥ g1(t,s) + θ1(t) ∫ 1−τ τ g1(r,s)a1(r)∇r + ζ1(t) ∫ 1−τ τ g1(r,s)b1(r)∇r ≥ η1(τ)g1(s,s) + θ1(t)η1(τ)g1(s,s) ∫ 1−τ τ a1(r)∇r + ζ1(t)η1(τ)g1(s,s) ∫ 1−τ τ b1(r)∇r ≥ ( 1 + θ1(t) ∫ 1−τ τ a1(r)∇r + ζ1(r) ∫ 1−τ τ b1(r)∇r ) η1(τ)g1(s,s) ≥ ( 1 + θ∗∗1 a1(τ) + ζ ∗∗ 1 b1(τ) ) η1(τ)g1(s,s) hence, p(1) is true. from (3.9), we have hm+1(t,s) = ∫ 1 0 hm(t,r)km+1(r,s)∇r ≤ ∫ 1 0 gm(r,r)gφ ( 1 + θ∗m+1a ∗ m+1 + ζ ∗ m+1b ∗ m+1 ) gm+1(s,s)∇r ≤ (∫ 1 0 gm(r,r)∇r ) gφ ( 1 + θ∗m+1a ∗ m+1 + ζ ∗ m+1b ∗ 1 ) gm+1(s,s) ≤ ( m∏ i=1 gi )m+1∏ i=1 ( 1 + θ∗i a ∗ i + ζ ∗ i b ∗ i ) gm+1(s,s) 204 prasad and khuddush and for t ∈ [τ, 1 − τ]t, hm+1(t,s) = ∫ 1 0 hm(t,r)km+1(r,s)∇r ≥ ∫ 1−τ τ ητgτgm(r,r)ηm+1(τ) ( 1 + θ∗∗m+1am+1(τ) + ζ ∗∗ m+1bm+1(τ) ) gm+1(s,s)∇r ≥ gτ (∫ 1−τ τ gm(r,r)∇r ) ητηm+1(τ) ( 1 + θ∗∗m+1am+1(τ) + ζ ∗∗ m+1bm+1(τ) ) gm+1(s,s) ≥ ( m∏ i=1 gi(τ) )m+1∏ i=1 ηm+1(τ) ( 1 + θ∗∗i ai(τ) + ζ ∗∗ i bi(τ) ) gm+1(s,s) so, p(m + 1) holds. thus p(n) is true by induction � let x denotes the banach space cld(j0,r) with norm ‖u‖ = max t∈j0 |u(t)|. for τ ∈ (0, 1 2 )t, define the cone pτ ⊂ x by pτ = { u ∈ x : u(t) ≥ 0 and min t∈[τ, 1−τ]t u(t) ≥ ξτ‖u(t)‖ } , where ξτ = ητgτ gφ . for any u ∈ pτ, define an operator t : pτ → x by (tu)(t) = ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s. (3.10) lemma 3.6. assume that (h1)-(h3) hold. then t(pτ ) ⊂ pτ and t : pτ → pτ is completely continuous for each τ ∈ (0, 1 2 )t. proof. fix τ ∈ (0, 1 2 ). since ω(s)f(u(s)) ≥ 0 for all s ∈ j0,u ∈ pτ and since hn(t,s) ≥ 0 for all t,s ∈ j0, then t(u(t)) ≥ 0 for all t ∈ j0,u ∈ pτ. on the other hand, by lemma 3.5 we obtain (tu)(t) = ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≤ gφ ∫ 1 0 gn(s,s)ω(s)f ( u(s) ) ∇s, min t∈[τ,1−τ]t u(t) = min t∈[τ,1−τ]t ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≥ ∫ 1 0 min t∈[τ,1−τ]t hn(t,s)ω(s)f ( u(s) ) ∇s = ητgτ ∫ 1 0 gn(s,s)ω(s)f ( u(s) ) ∇s ≥ ξτtu(t) for all t ∈ j0. thus min t∈[τ,1−τ]t u(t) ≥ ξτ‖tu‖. so, tu ∈ pτ and then t(pτ ) ⊂ pτ. next, by standard methods and the arzela-ascoli theorem, one can easily prove that the operator t is completely continuous. the proof is complete. � 4. main results in this section, we establish the existence of countably infinitely many positive solutions for the boundary value problem (1.6)-(1.7) by applying krasnoselskii’s fixed point theorem in cones. theorem 4.1. [14] let b be a banach space and let p ⊂b be a cone in b. assume that ω1, ω2 are open with 0 ∈ ω1, ω̄1 ⊂ ω2, and let t : p ∩ (ω̄2\ω1) → p be a completely continuous operator such that either (i) ‖tu‖≤‖u‖,u ∈ p ∩∂ω1, and ‖tu‖≥‖u‖,u ∈ p ∩∂ω2, or countable many positive sol. for eobvps with ibcs on time scales 205 (ii) ‖tu‖≥‖u‖,u ∈ p ∩∂ω1, and ‖tu‖≤‖u‖,u ∈ p ∩∂ω2. then t has a fixed point in p ∩ (ω̄2\ω1). theorem 4.2. [4, 29] let f ∈ lp∇(j) with p > 1,g ∈ l q ∇(j) with q > 1, and 1 p + 1 q = 1. then fg ∈ l1∇(j) and ‖fg‖l1∇ ≤‖f‖lp∇‖g‖lq∇. where ‖f‖lp∇ :=   [∫ j |f|p(s)∇s ]1 p , p ∈ r, inf { m ∈ r/ |f| ≤ m ∇−a.e., onj } , p = ∞, and j = (a,b]. moreover, if f ∈ l1∇(j) and g ∈ l ∞ ∇ (j). then fg ∈ l 1 ∇(j) and ‖fg‖l1∇ ≤‖f‖l1∇‖g‖l∞∇ . we consider the following three cases for ω ∈ lp∇(j0) : p > 1,p = 1,p = ∞. case p > 1 is treated in the following theorem. theorem 4.3. assume that (h1)−(h5) hold, let {τk}∞k=1 be such that tk+1 < τk < tk, k = 1, 2, 3, · · · . let {rk}∞k=1 and {rk} ∞ k=1 be such that rk+1 < ξτkrk < crk < rk, k ∈ n, where c = max { 1 ητ1gτ1m ∫ 1−τ1 τ1 gn(s,s)∇s , 1 } . assume that f satisfies (a1) f(u) ≤ m1rk for all t ∈ j0, 0 ≤ u ≤ rk, where m1 < 1 gφ‖gn(s,s)‖lq∇‖ω‖lp∇ , (a2) f(u) ≥ crk for all t ∈ [τk, 1 − τk]t, ξτrk ≤ u ≤ rk. then the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions {uk}∞k=1. furthermore, rk ≤‖uk‖≤ rk for each k ∈ n. proof. consider the sequences {ω1,k}∞k=1 and {ω2,k} ∞ k=1 of open subsets of e defined by ω1,k = {u ∈b : ‖u‖ < rk}, ω2,k = {u ∈b : ‖u‖ < rk}. let {τk}∞k=1 be as in the hypothesis and note that t ∗ < tk+1 < τk < tk < 1 2 , for all k ∈ n. for each k ∈ n, define the cone pτk by pτk = { u ∈ x : u(t) ≥ 0 and min t∈[τk, 1−τk]t u(t) ≥ ξτk‖u(t)‖ } . let u ∈ pτk ∩∂ω1,k. then, u(s) ≤ rk = ‖u‖ for all s ∈ j0. by (a1), ‖tu‖ = max t∈j0 ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≤ gφ ∫ 1 0 gn(s,s)ω(s)∇sm1rk ≤ gφ‖gn(s,s)‖lq∇‖ω‖lp∇m1rk ≤ rk. since ‖u‖ = rk for all u ∈ pτk ∩∂ω1,k, then ‖tu‖≤‖u‖. (4.1) let s ∈ [τk, 1 − τk]t. then, rk = ‖u‖≥ u(s) ≥ min s∈[τk, 1−τk]t u(s) ≥ ξτ‖u‖≥ ξτkrk. 206 prasad and khuddush by (a2), ‖tu‖ = max t∈j0 ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≥ max t∈j0 ∫ 1−τk τk hn(t,s)ω(s)f ( u(s) ) ∇s ≥ max t∈j0 ∫ 1−τk τk hn(t,s)ω(s)∇scrk ≥ crkm max t∈[τ1, 1−τ1]t ∫ 1−τ1 τ1 hn(t,s)∇s ≥ ητ1gτ1mcrk max t∈[τ1, 1−τ1]t ∫ 1−τ1 τ1 gn(s,s)∇s ≥ rk = ‖u‖. thus, if u ∈ pτ ∩∂ω2,k, then ‖tu‖≥‖u‖. (4.2) it is obvious that 0 ∈ ω2,k ⊂ ω̄2,k ⊂ ω1,k. by (4.1),(4.2), it follows from theorem 4.1 that the operator t has a fixed point uk ∈ pτk ∩ ( ω̄1,k\ω2,k ) such that rk ≤‖uk‖≤ rk. since k ∈ n was arbitrary, the proof is complete. � now we deal with the case p = 1. theorem 4.4. assume that (h1)−(h5) hold, let {τk}∞k=1 be such that tk+1 < τk < tk, k = 1, 2, 3, · · · . let {rk}∞k=1 and {rk} ∞ k=1 be such that rk+1 < ξτkrk < crk < rk, k ∈ n, assume that f satisfies (b1) f(u) ≤ m2rk for all t ∈ j0, 0 ≤ u ≤ rk, where m2 < min { 1 gφ‖gn(s,s)‖l∞∇‖ω‖l1∇ , c } and (a2). then the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions {uk}∞k=1. furthermore, rk ≤‖uk‖≤ rk for each k ∈ n. proof. for a fixed k, let ω1,k be as in the proof of theorem 4.3 and let u ∈ pτk ∩∂ω2,k. again u(s) ≤ ak = ‖u‖, for all s ∈ j0. by (b1) and theorem 4.3, ‖tu‖ = max t∈j0 ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≤ gφ ∫ 1 0 gn(s,s)ω(s)∇sm2rk ≤ gφ‖gn(s,s)‖l∞∇‖ω‖l1∇m2rk ≤ rk. thus, ‖tu‖≤‖u‖, for u ∈ pτk∩∂ω1,k. now define ω2,k = {u ∈b : ‖u‖ < rk}. let u ∈ pτk∩∂ω2,k and let s ∈ [τk, 1−τk]t. then, the argument leading to (4.2) carries over to the present case and completes the proof. � finally we consider the case of p = ∞. countable many positive sol. for eobvps with ibcs on time scales 207 theorem 4.5. assume that (h1) − (h5) hold. let {rk}∞k=1 and {rk} ∞ k=1 be such that rk+1 < ξτrk < crk < rk, k ∈ n, assume that f satisfies (e1) f(u) ≤ m3rk for all t ∈ j0, 0 ≤ u ≤ rk, where m3 < min { 1 gφ‖gn(s,s)‖l1∇‖ω‖l∞∇ , c } and (a2). then the boundary value problem (1.6)-(1.7) has countably infinitely many positive solutions {uk}∞k=1. furthermore, rk ≤‖uk‖≤ rk for each k ∈ n. proof. by (e1), ‖tu‖ = max t∈j0 ∫ 1 0 hn(t,s)ω(s)f ( u(s) ) ∇s ≤ gφ ∫ 1 0 gn(s,s)ω(s)∇sm3rk ≤ gφ‖gn(s,s)‖l1∇‖ω‖l∞∇ m3rk ≤ rk. this shows that if u ∈ pτk ∩∂ω1,k, where ω1,k = {u ∈b : ‖u‖ < rk}, then, ‖tu‖≤‖u‖. define ω2,k = {u ∈b : ‖u‖ < rk} and let u ∈ pτk ∩∂ω2,k. then, the argument employed in the proof of theorem 4.3 applies directly to yield ‖tu‖≥‖u‖. by the theorem 4.1, completes the proof. � 5. example in this section, we provide an example of a family of functions ω(t) that satisfy conditions (h1), (h2) corresponding to the cases p = 1 and p = 2. let t = [0, 1 2 ] ∪{3 5 , 3 4 , 4 5 }∪ [ 5 6 , 1] and consider the family of functions ω(t,�) : t → (0, +∞] given by ω(t,�) =   ∞∑ k=1 χ[νk, νk−1] |t− tk|� if 0 ≤ t ≤ 1 2 , 1 |t− 1 2 |� if 1 2 < t < 5 6 , 1 |t− 4 5 |� if 5 6 ≤ t ≤ 1, where t0 = 5 16 , tk = t0 − k−1∑ i=0 1 (i + 2)4 , k = 1, 2, 3, · · · , ν0 = 1, νk = 1 2 (tk + tk+1), k = 1, 2, 3, · · · . at first, it is easily seen that ω(t,�) ≥ ω(1,�) = 1|1−4 5 |� = 5 �, t1 = 1 4 < 1 2 , tk − tk+1 = 1(k+2)4 , k = 1, 2, 3, · · · , and note that ∑∞ k=1 1 k4 = π 4 90 . t∗ = lim k→∞ tk = 5 16 − ∞∑ i=0 1 (i + 2)4 = 5 16 − ( π4 90 − 1 ) = 21 16 − π4 90 > 1 5 . 208 prasad and khuddush we claim that if � = 1 2 , then ω(t,�) ∈ l1∇[0, 1]. note that ∑∞ k=1 1 k2 = π 2 6 , we have∫ 1 0 ω(t,�)∇t = ∫ 1 2 0 ω(t,�)∇t + ∫ 1 1 2 ω(t,�)∇t = ∫ 1 2 0 ∞∑ k=1 χ[νk, νk−1] |t− tk|� ∇t + ∫ 1 5 6 1 |t− 4 5 |� ∇t + [( 3 5 − 1 2 ) ω ( 3 5 ,� ) + ( 3 4 − 3 5 ) ω ( 3 4 ,� ) + ( 4 5 − 3 4 ) ω ( 4 5 ,� )( 5 6 − 4 5 ) ω ( 5 6 ,� )] = ∞∑ k=1 ∫ νk−1 νk 1 |t− tk|� ∇t + ∫ 1 5 6 1( t− 4 5 )�∇t + [ 1 10 × 10� + 3 20 × 4� + 1 20 × ( 10 3 )� + 1 30 × 3� ] = ∞∑ k=1 [∫ tk νk 1 (tk − t)� ∇t + ∫ νk−1 tk 1 (t− tk)� ∇t ] + 1 1 − � [ 1 51−� − 1 301−� ] + [ 1 10 × 10� + 3 20 × 4� + 1 20 × ( 10 3 )� + 1 30 × 3� ] = ∞∑ k=1 [∫ tk tk+tk+1 2 1 (tk − t)� ∇t + ∫ tk−1+tk 2 tk 1 (t− tk)� ∇t ] + 1 1 − � [ 1 51−� − 1 301−� ] + [ 10�−1 + 3 5 × 4�−1 + 1 20 × ( 10 3 )� + 1 10 × 3�−1 ] = 1 1 − � ∞∑ k=1 [( tk − tk+1 2 )1−� + ( tk−1 − tk 2 )1−� + 1 1 − � [ 1 51−� − 1 301−� ] + [ 10�−1 + 3 5 × 4�−1 + 1 20 × ( 10 3 )� + 1 10 × 3�−1 ] = 1 21−�(1 − �) ∞∑ k=1 [ 1 (k + 2)4(1−�) + 1 (k + 1)4(1−�) ] + 1 1 − � [ 1 51−� − 1 301−� ] + [ 10�−1 + 3 5 × 4�−1 + 1 20 × ( 10 3 )� + 1 10 × 3�−1 ] = √ 2 ∞∑ k=1 [ 1 (k + 1)2 + 1 (k + 1)2 ] + 1 15 (6 √ 5 − √ 30) + 1 60 [6( √ 10 + 3) + √ 3( √ 10 + 2)] = √ 2 ( π2 3 − 9 4 ) + 1 60 [24 √ 5 − 3 √ 30 + 6 √ 10 + 2 √ 3 + 18], which implies that ω(t,�) ∈ l1∇[0, 1]. next, we claim that if � = 1 4 , then ω(t,�) ∈ l2∇[0, 1]. in this case, we need the cauchy product, ∞∑ k=1 ak · ∞∑ k=1 bk = ∞∑ k=1 ck, (5.1) where ck = k∑ n=1 anbk−n+1. (5.2) note that ∫ 1 0 ω2(t,�)∇t = ∫ 1 2 0 [ ∞∑ k=1 χ[νk, νk−1] |t− tk|� ]2 ∇t + ∫ 1 1 2 ω2(t,�)∇t, (5.3) countable many positive sol. for eobvps with ibcs on time scales 209 we use (5.1) and (5.2) and the fact that, if x ∩y = ∅, then χ[x] ·χ[y ] = 0 to simplify the integrand,[ ∞∑ k=1 χ[νk, νk−1] |t− tk|� ]2 = ∞∑ k=1 k∑ n=1 χ[νn, νk−1] |t− tn|� χ[νk−n+1, νk−n] |t− tk−n+1|� = ∞∑ k=1 χ[νk, νk−1] |t− tk|2� a.e., and so (5.3) may be written as∫ 1 0 ω2(t,�)∇t = ∞∑ k=1 ∫ 1 2 0 χ[νk, νk−1] |t− tk|2� ∇t + ∫ 1 1 2 ω2(t,�)∇t = ∞∑ k=1 ∫ νk−1 νk 1 |t− tk|2� ∇t + ∫ 1 5 6 ω2(t,�)∇t + + [( 3 5 − 1 2 ) ω2 ( 3 5 ,� ) + ( 3 4 − 3 5 ) ω2 ( 3 4 ,� ) + ( 4 5 − 3 4 ) ω2 ( 4 5 ,� )( 5 6 − 4 5 ) ω2 ( 5 6 ,� )] = ∞∑ k=1 [∫ tk νk 1 (tk − t)2� ∇t + ∫ νk−1 tk 1 (t− tk)2� ∇t ] + ∫ 1 5 6 1( t− 4 5 )2�∇t + [ 1 10 × 102� + 3 20 × 42� + 1 20 × ( 10 3 )2� + 1 30 × 32� ] = ∞∑ k=1 [∫ tk tk+tk+1 2 1 (tk − t)2� ∇t + ∫ tk−1+tk 2 tk 1 (t− tk)2� ∇t ] + 1 1 − 2� [ 1 51−2� − 1 301−2� ] + [ 102�−1 + 3 5 × 42�−1 + 1 20 × ( 10 3 )2� + 1 10 × 32�−1 ] = 1 1 − 2� ∞∑ k=1 [( tk − tk+1 2 )1−2� + ( tk−1 − tk 2 )1−2� + 1 1 − 2� [ 1 51−2� − 1 301−2� ] + [ 102�−1 + 3 5 × 42�−1 + 1 20 × ( 10 3 )2� + 1 10 × 32�−1 ] = 1 21−2�(1 − 2�) ∞∑ k=1 [ 1 (k + 2)4(1−2�) + 1 (k + 1)4(1−2�) ] + 1 1 − 2� [ 1 51−2� − 1 301−2� ] + [ 102�−1 + 3 5 × 42�−1 + 1 20 × ( 10 3 )2� + 1 10 × 32�−1 ] = √ 2 ∞∑ k=1 [ 1 (k + 1)2 + 1 (k + 1)2 ] + 1 15 (6 √ 5 − 5 √ 6) + 1 60 [6( √ 10 + 3) + √ 3( √ 10 + 2)] = √ 2 ( π2 3 − 9 4 ) + 1 60 [24 √ 5 − 20 √ 6 + 6( √ 10 + 3) + √ 3( √ 10 + 2)], which implies ω(t,�) ∈ l2∇[0, 1]. references [1] r. p. agarwal, m. bohner, basic calculus on time scales and some of its applications. result math, 35(1999), 3–22. [2] r. p. agarwal, v. otero-espinar, k. perera and d.r. vivero, basic properties of sobolev’s spaces on time scales. advan. diff. eqns., 2006(2006), art. id 038121. [3] d. r. anderson, i. y. karaca, higher-order three-point boundary value problem on time scales. comput. math. appl.56(2008), 2429–2443. [4] g. a. anastassiou, intelligent mathematics: computational analysis. vol. 5. heidelberg: springer, 2011. [5] m. bohner and h. luo, singular second-order multipoint dynamic boundary value problems with mixed derivatives, adv. diff. eqns., 2006(2006), art. id 054989. [6] m. bohner, and a. peterson, dynamic equations on time scales: an introduction with applications. birkhauser, boston, (2001). [7] m. bohner, and a. peterson, advances in dynamic equations on time scales. birkhauser, boston, (2003). [8] e. cetin, f. s. topal, existence results for solutions of integral boundary value problems on time scales, abstr. appl. anal., 2013(2013), art. id 708734. 210 prasad and khuddush [9] f. t. fen, and i. y. karaca, existence of positive solutions for nonlinear second-order impulsive boundary value problems on time scales, med. j. math., 13(2016), 191-204. [10] m. feng, existence of symmetric positive solutions for boundary value problem with integral boundary conditions, appl. math. lett., 24(2011), 1419-1427. [11] j. m. gallardo, second-order differential operators with integral boundary conditions and generation of analytic semigroups, rocky mountain j. math., 30(2000), no. 4, 1265-1292. [12] c. s. goodrich, existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale, comment. math. univ. carol., 54(2013), no. 4, 509-525. [13] c. s. goodrich, on a first-order semipositone boundary value problem on a time scale, appl. anal. disc. math., 8 (2014), 269-287. [14] d. guo and v. lakshmikantham, nonlinear problems in abstract cones, academic press, new york, 1988. [15] y. guo, y. liu and y. liang, positive solutions for the third order boundary value problems with the second derivatives, bound. value probl., 2012(2012), art. id 34. [16] g. s. guseinov, integration on time scales. j. math. anal. app., 285(2003), no. 1, 107–127. [17] n. a. hamal and f. yoruk, symmetric positive solutions of fourth order integral bvp for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales, comput. math. appl., 59(2010), no. 11, 3603-3611. [18] m. hu and l. wang, triple positive solutions for an impulsive dynamic equation with integral boundary condition on time scales, inter. j. app. math. stat., 31(2013), no. 1, 67-78. [19] i. y. karaca, positive solutions for boundary value problems of second-order functional dynamic equations on time scales, adv. difference equ. 21(2009) art. id 829735. [20] i. y. karaca, f. tokmak, existence of positive solutions for third-order boundary value problems with integral boundary conditions on time scales, j. ineq. app., 2013(2013), no. 1, 1-12. [21] g. l. karakostas and p. ch. tsamatos, multiple positive solutions of some fredholm integral equations arisen from nonlocal boundary-value problems, electron. j. differential equations, 30(2002), no. 30, 1-17. [22] r. a. khan, the generalized method of quasi-linearization and nonlinear boundary value problems with integral boundary conditions, electron. j. qual. theory differential equations, 19(2003), no. 15, 1-15. [23] v. lakshmikantham, s. sivasundaram, b. kaymakcalan, dynamic systems on measure chains, kluwer, dordrecht, (1996). [24] y. li and l. sun, infinite many positive solutions for nonlinear first-order bvps with integral boundary conditions on time scales, top. meth. nonl. anal., 41(2013), no. 2, 305-321. [25] y. li and l. wang, multiple positive solutions of nonlinear third-order boundary value problems with integral boundary conditions on time scales, adv. diff. eqns., 2015(2015), art. id 90. [26] y. li and t. zhang, multiple positive solutions for second-order p-laplacian dynamic equations with integral boundary conditions, boun. val. prob., 2011(2010), art. id 19. [27] a. lomtatidze and l. malaguti, on a nonlocal boundary-value problems for second order nonlinear singular differential equations, georgian math. j., 7(2000), 133-154. [28] a. d. oguz and f. s. topal, symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales, j. appl. anal and comp., 6(2016), no. 2, 531–542. [29] u. m. ozkan and m. z. sarikaya and h. yildirim, extensions of certain integral inequalities on time scales. appl. math. let., 21(2008), no. 10, 993–1000. [30] b. p. rynne, l2 spaces and boundary value problems on time-scales. j. math. anal. app., 328(2007), no. 2, 1217-1236. [31] n. sreedhar, v. v. r. r. b. raju and y. narasimhulu, existence of positive solutions for higher order boundary value problems with integral boundary conditions on time scales. j. nonlinear funct. anal., 2017(2017), article id 5, 1-13. [32] p . thiramanus, and t. jessada, positive solutions of m-point integral boundary value problems for second-order p-laplacian dynamic equations on time scales, adv. diff. eqns., 2013(2013), art. id 206. [33] s. p. timoshenko and j. m. gere, theory of elastic stability, mcgraw-hill, new york(1961). [34] s. g. topal and a. denk, existence of symmetric positive solutions for a semipositone problem on time scales, hacet. j. math. stat., 45(2016), no. 1, 23–31. [35] p. a. williams, unifying fractional calculus with time scales [ph.d. thesis], university of melbourne, (2012). [36] x. zhang and m. feng, w. ge, existence of solutions of boundary value problems with integral boundary conditions for second order impulsive integro-differential equations in banach spaces, comput. appl. math., 233(2010), 19151926. department of applied mathematics, andhra university, visakhapatnam, 530 003, india ∗corresponding author: rajendra92@rediffmail.com 1. introduction 2. preliminaries 3. green's function and bounds 4. main results 5. example references international journal of analysis and applications volume 16, number 4 (2018), 484-502 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-484 a study of non-associative ordered semigroups in terms of semilattices via smallest (double-framed soft) ideals faisal yousufzai1, tauseef asif2, asghar khan2,∗ and bijan davvaz3 1military college of engineering national university of sciences and technology (nust), islamabad, pakistan 2department of mathematics, abdul wali khan university, mardan, kpk, pakistan 3department of mathematics, yazd university, yazd, iran ∗corresponding author: asghar@awkum.edu.pk abstract. soft set theory, introduced by molodtsov has been considered as a successful mathematical tool for modeling uncertainties. a double-framed soft set is a generalization of a soft set, consisting of union soft sets and intersectional soft sets. an ordered ag-groupoid can be referred to as a non-associative ordered semigroup, as the main difference between an ordered semigroup and an ordered ag-groupoid is the switching of an associative law. in this paper, we define the smallest left (right) ideals in an ordered ag-groupoid and use them to characterize a strongly regular class of a unitary ordered ag-groupoid along with its semilattices and double-framed soft (briefly dfs) l-ideals (r-ideals). we also give the concept of an ordered a*g**-groupoid and investigate its structural properties by using the generated ideals and dfs l-ideals (r-ideals). these concepts will verify the existing characterizations and will help in achieving more generalized results in future works. received 2018-01-16; accepted 2018-04-06; published 2018-07-02. 2010 mathematics subject classification. 00a00. key words and phrases. dfs-sets; ordered ag-groupoid, pseudo-inverses; ordered a*g**-groupoid, left invertive law; smallest ideals and dfs ideals. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 484 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-484 int. j. anal. appl. 16 (4) (2018) 485 1. introduction the concept of soft set theory was introduced by molodtsov in [17]. this theory can be used as a generic mathematical tool for dealing with uncertainties. in soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields [1, 2, 5–9]. at present, the research work on soft set theory in algebraic fields is progressing rapidly [20, 22–24]. a soft set is a parameterized family of subsets of the universe set. in the real world, the parameters of this family arise from the view point of fuzzy set theory. most of the researchers of algebraic structures have worked on the fuzzy aspect of soft sets. soft set theory is applied in the field of optimization by kovkov in [13]. several similarity measures have been discussed in [16], decision making problems have been studied in [22], reduction of fuzzy soft sets and its applications in decision making problems have been analyzed in [14]. the notions of soft numbers, soft derivatives, soft integrals and many more have been formulated in [15]. this concept have been used for forecasting the export and import volumes in international trade [26]. recently, jun et al. further extended the notion of softs set into double-framed soft sets and defined double-framed soft subalgebra of bck/bci algebra and studied the related properties in [8]. jun et al. also defined the concept of a double-framed soft ideal (briefly, dfs ideal) of a bck/bci-algebra and gave many valuable results for this theory. in [12], khan et al. have applied the idea of double-framed soft set to ordered semigroups and defined prime and irreducible dfs ideals of an ordered semigroup over a universe set u. khan et al. have also characterized different classes of an ordered semigroup by using different dfs ideals. in the present paper, we apply the idea given by jun et al. in [8], to ordered ag-groupoids. we introduce and investigate the notions of dfs l-ideals and dfs r-ideals, and study the relationship between these dfs ideals in detail. as an application of our results we get characterizations of a strongly regular class of a unitary ordered ag-groupoid (an ordered a*g**-groupoid) in terms of its semilattices, one-sided (two-sided) ideals based on dfs-sets and generated commutative monoids. 2. preliminaries an ag-groupoid is a non-associative and a non-commutative algebraic structure lying in a grey area between a groupoid and a commutative semigroup. commutative law is given by abc = cba in ternary operations. by putting brackets on the left of this equation, i.e. (ab)c = (cb)a, in 1972, m. a. kazim and m. naseeruddin introduced a new algebraic structure called a left almost semigroup abbreviated as an la-semigroup [10]. this identity is called the left invertive law. p. v. protic and n. stevanovic called the same structure an abel-grassmann’s groupoid abbreviated as an ag-groupoid [21]. this structure is closely related to a commutative semigroup because a commutative ag-groupoid is a semigroup [18]. it was proved in [10] that an ag-groupoid s is medial, that is, ab · cd = ac · bd holds for all int. j. anal. appl. 16 (4) (2018) 486 a,b,c,d ∈ s. an ag-groupoid may or may not contain a left identity. the left identity of an ag-groupoid permits the inverses of elements in the structure. if an ag-groupoid contains a left identity, then this left identity is unique [18]. in an ag-groupoid s with left identity, the paramedial law ab · cd = dc · ba holds for all a,b,c,d ∈ s. by using medial law with left identity, we get a · bc = b ·ac for all a,b,c ∈ s. we should genuinely acknowledge that much of the ground work has been done by m. a. kazim, m. naseeruddin, q. mushtaq, m. s. kamran, p. v. protic, n. stevanovic, m. khan, w. a. dudek and r. s. gigon. one can be referred to [3, 4, 11, 18, 19, 21, 25] in this regard. an ag-groupoid (s, ·) together with a partial order ≤ on s that is compatible with an ag-groupoid operation, meaning that for x,y,z ∈ s, x ≤ y ⇒ zx ≤ zy and xz ≤ yz, is called an ordered ag-groupoid [28]. let us define a binary operation ”◦e” (e-sandwich operation) on an ordered ag-groupoid (s, ·,≤) with left identity e as follows: a◦e b = ae · b, ∀ a,b ∈ s. then (s, ◦e,≤) becomes an ordered semigroup [28]. note that an ordered ag-groupoid is the generalization of an ordered semigroup because if an ordered ag-groupoid has a right identity then it becomes an ordered semigroup. let ∅ 6= a ⊆ s, we denote (a] by (a] := {x ∈ s/x ≤ a for some a ∈ a}. if a = {a}, then we write ({a}]. for ∅ 6= a,b ⊆ s, we denote ab =: {ab/a ∈ a,b ∈ b}. • let s be an ordered ag-groupoid. by an ordered ag-subgroupoid of s, we means a nonempty subset a of s such that (a2] ⊆ a. • a nonempty subset a of an ordered ag-groupoid s is called a left (right) ideal of s if: (i ) sa ⊆ a (as ⊆ a); (ii) if a ∈ a and b ∈ s such that b ≤ a, then b ∈ a. equivalently: a nonempty subset a of an ordered ag-groupoid s is called a left (right) ideal of s if (sa] ⊆ a ((as] ⊆ a). • by two-sided ideal or simply ideal, we mean a nonempty subset of an ordered ag-groupoid s which is both left and right ideal of s. lemma 2.1. [28] let s be an ordered ag-groupoid and ∅ 6= a,b ⊆ s. then the following hold: (i) a ⊆ (a] ; (ii) if a ⊆ b, then (a] ⊆ (b] ; (iii) (a] (b] ⊆ (ab] ; (iv) (a] = ((a]] ; (vi) ((a] (b]] = (ab] ; (vii) (t] = t, for every ideal t of s; int. j. anal. appl. 16 (4) (2018) 487 (viii) (ss] = s = ss, if s has a left identity. 3. soft sets in [24], sezgin and atagun introduced some new operations on soft set theory and defined soft sets in the following way. let u be an initial universe set, e a set of parameters, p(u) the power set of u and a ⊆ e. then a soft set fa over u is a function defined by: fa : e → p(u) such that fa(x) = ∅, if x /∈ a. here fa is called an approximate function. a soft set over u can be represented by the set of ordered pairs fa = {(x, fa(x)) : x ∈ e, fa(x) ∈ p(u)} . it is clear that a soft set is a parameterized family of subsets of u. the set of all soft sets is denoted by s(u). • let fa, fb ∈ s(u). then fa is a soft subset of fb, denoted by fa ∼ ⊆ fb if fa(x) ⊆ fb(x) for all x ∈ s. two soft sets fa, fb are said to be equal soft sets if fa ∼ ⊆ fb and ∼ fb ⊆ fa and is denoted by fa ∼ = fb. the union of fa and fb, denoted by fa ∼ ∪fb, is defined by fa ∼ ∪fb = fa∪b, where fa∪b(x) = fa(x) ∪fb(x), ∀ x ∈ e. in a similar way, we can define the intersection of fa and fb. • let s be an ordered ag-groupoid, let fa, fb ∈ s(u). then the soft product [24] of fa and fb, denoted by fa ∼ ◦ fb, is defined as follows: (fa ∼ ◦ fb)(x) =   ⋃ (y,z)∈ax {fa(y) ∩gb (z)} if ax 6= ∅ ∅ if ax = ∅ where ax = {(y,z) ∈ s ×s/x ≤ yz}. • a double-framed soft pair 〈 (f+a , f − a ; a 〉 is called a double-framed soft set (briefly, dfs-set of a) [8] of a over u, where f+a and f − a are mappings from a to p(u). the set of all dfs-sets of a over u will be denoted by dfs(u). • let fa = 〈 (f+a , f − a ); a 〉 and ga = 〈 (g+a, g − a ); a 〉 be two double-framed soft sets of an ordered aggroupoid s over u. then the uni-int soft product [12], denoted by fa � ga = 〈 (f+a ∼ ◦ g+a, f − a ∼ ? g−a ); a 〉 is defined to be a double-framed soft set of s over u, in which f+a ∼ ◦ g+a and f − a ∼ ? g−a are mapping from s to p(u), given as follows: int. j. anal. appl. 16 (4) (2018) 488 f+a ∼ ◦ g+a : s −→ p(u),x 7−→   ⋃ (y,z)∈ax {f+a (y) ∩g + a (z)} if ax 6= ∅ ∅ if ax = ∅, f−a ∼ ? g−a : s −→ p(u),x 7−→   ⋂ (y,z)∈ax {f−a (y) ∪g − a (z)} if ax 6= ∅ u if ax = ∅. • let fa = 〈 (f+a , f − a ); a 〉 and ga = 〈 (g+a, g − a ); a 〉 be two double-framed soft sets over a common universe set u. then 〈 (f+a , f − a ); a 〉 is called a double-framed soft subset (briefly, dfs-subset ) [12] of 〈 (g+a, g − a ); a 〉 , denote by 〈 (f+a , f − a ); a 〉 v 〈 (g+a, g − a ); a 〉 if: (i) a ⊆ b; (ii) (∀e ∈ a)   f+a and g+a are identical approximations (f+a (e) ⊆ g+a (e)) f−a and g − a are identical approximations (f − a (e) ⊇ g − a (e))   . • for two dfs-sets fa = 〈 (f+a , f − a ); a 〉 and ga = 〈 (g+a, g − a ); a 〉 over u are said to be equal, denoted by〈 (f+a , f − a ); a 〉 = 〈 (g+a, g − a ); a 〉 , if 〈 (f+a , f − a ); a 〉 v 〈 (g+a, g − a ); a 〉 and 〈 (g+a, g − a ); a 〉 v 〈 (f+a , f − a ); a 〉 . • for two dfs-sets fa = 〈 (f+a , f − a ); a 〉 and ga = 〈 (g+a, g − a ); a 〉 over u, the dfs int-uni set [12] of〈 (f+a , f − a ); a 〉 and 〈 (g+a, g − a ); a 〉 , is defined to be a dfs-set 〈 (f+a ∩g + a, f − a ∪g − a ); a 〉 , where f+a ∩g + a and f−a ∪g − a are mapping given as follows: f+a ∩g + a : a −→ p(u),x 7−→ f + a (x) ∩g + a (x); f−a ∪g − a : a −→ p(u),x 7−→ f − a (x) ∪g − a (x). it is denoted by 〈 (f+a , f − a ); a 〉 u 〈 (g+a, g − a ); a 〉 = 〈 (f+a ∩g + a, f − a ∪g − a ); a 〉 . • a double-framed soft set fa = 〈 (f+a , f − a ); a 〉 of s over u is called a double-framed soft ag-subgroupoid (briefly, dfs ag-subgroupoid) of s over u if it satisfies f+a (xy) ⊇ f + a (x)∩f + a (y), f − a (xy) ⊆ f − a (x)∪f − a (y), ∀ x, y ∈ s. • a double-framed soft set fa = 〈 (f+a , f − a ); a 〉 of s over u is called (i) a double-framed soft left ideal (briefly, dfs l-ideal) of s over u if it satisfies: (a) f+a (xy) ⊇ f + a (y) and f − a (xy) ⊆ f − a (y); (b) x ≤ y =⇒ f+a (x) ⊇ f + a (y) and f − a (x) ⊆ f − a (y), ∀ x, y ∈ s. (ii) a double-framed soft right ideal (briefly, dfs r-ideal) of s over u if it satisfies: (a) f+a (xy) ⊇ f + a (x) and f − a (xy) ⊆ f − a (x); (b) x ≤ y =⇒ f+a (x) ⊇ f + a (y) and f − a (x) ⊆ f − a (y), ∀ x, y ∈ s. (iii) a double-framed soft ideal (briefly, dfs ideal) of s over u, if it is both dfs l-ideal and dfs r-ideal of s over u. • let a be a nonempty subset of s. then the characteristic double-framed soft mapping of a, denoted by 〈 (x+a , x − a ); a 〉 = xa is defined to be a double-framed soft set, in which x+a and x − a are soft mappings int. j. anal. appl. 16 (4) (2018) 489 over u, given as follows: x+a : s −→ p(u),x 7−→   u if x ∈ a∅ if x /∈ a, x−a : s −→ p(u),x 7−→   ∅ if x ∈ au if x /∈ a. note that the characteristic mapping of the whole set s, denoted by xs = 〈 (x+s , x − s ); s 〉 , is called the identity double-framed soft mapping, where x+s (x) = u and x − s (x) = ∅, ∀ x ∈ s. the following result holds for an ordered semigroup [6] just because of the closure property which makes very clear for an ordered ag-groupoid to hold the same lemma. lemma 3.1. for a nonempty subset a of an ordered ag-groupoid s, the following conditions are equivalent: (i) a is a left ideal (right ideal ) of s; (ii) the dfs set xa of s over u is a dfs l-ideal (dfs r-ideal ) of s over u. the following result holds for an ordered semigroup [12] just because of the closure property which makes very clear for an ordered ag-groupoid to hold the same lemma. lemma 3.2. let fa = 〈(f+a ,f − a ); a〉 be any dfs-set of an ordered ag-groupoid s over u. then fa is a dfs r-ideal (l-ideal) of s over u if and only if fa �xs v fa (xs �fa v fa). • a double-framed soft set fa = 〈 (f+a , f − a ); a 〉 of s over u is called dfs semiprime if fa(x) w fa(x2), ∀ x ∈ a. lemma 3.3. let a be any right (left) ideal of an ordered ag-groupoid s. then a is semiprime if and only if xa is dfs semiprime. proof. let a be a right (left) ideal of s, then by lemma 3.1, xa is a dfs r-ideal (dfs l-ideal) of s over u. let a2 ∈ a, then x+a (a) ⊇ x + a (a 2), therefore x+a (a 2) = u ⊆ x+a (a), this implies x + a (a) = u and similarly x−a (a) = ∅. thus a ∈ a and therefore a is semiprime. converse is simple. � remark 3.1. the set (dfs(u),�,v) forms an ordered ag-groupoid and satisfies all the basic laws. remark 3.2. if s is an ordered ag-groupoid, then xs �xs = xs. the following result also holds for an ordered semigroup [12] just because of the closure property which is very trivial for an ordered ag-groupoid to hold the same lemma. lemma 3.4. let s be an ordered ag-groupoid. for ∅ 6= a,b ⊆ s, the following assertions hold: int. j. anal. appl. 16 (4) (2018) 490 (i) a ⊆ b ⇔ xa vxb; (ii) xa uxb = xa∩b; (iii) xa txb = xa∪b; (iv) xa �xb = x(ab]. 4. on dfs strongly regular ordered ag-groupoids throughout this paper, let e = s, where s is a unitary ordered ag-groupoid, unless otherwise stated. by a unitary ordered ag-groupoid, we shall mean an ordered ag-groupoid with left identity. 4.1. basic results. this section contains some examples and basic results which will be essential for up coming section. theorem 4.1. let s be an ordered ag-groupoid. a nonempty subset a of s is a left (resp. right) ideal of s if and only if the dfs-set 〈 (g+b, g − b ); b 〉 , defined by g+b (x) =   γ1 if x ∈ aγ2 if x ∈ s\a   and g−b (x) =   δ1 if x ∈ aδ2 if x ∈ s\a   , is a dfs l-ideal (resp. dfs r-ideal) of s over u, where γ1,γ2,δ1,δ2 ⊆ u such that γ2 ⊆ γ1 and δ1 ⊆ δ2. proof. necessity. let x,y ∈ s be such that x ≤ y. if y /∈ a, then g+b (y) = γ2 ⊆ g + b (x) and g − b (y) = δ2 ⊇ g−b (x). if y ∈ a, then γ1 = g + b (y) and δ2 = g − b (y). since x ≤ y ∈ a, and a is a left ideal of s, we have x ∈ a. then g+b (x) = γ1 = g + b (y) and g − b (x) = δ1 = g − b (y). for x,y ∈ s, we discuss the following two cases. case 1. if x ∈ s and y ∈ a, then xy ∈ a and we have g+b (x) = γ1 = g + b (y) and g − b (x) = δ1 = g − b (y). case 2. if x ∈ s and y /∈ a, then g+b (y) = γ2 ⊆ g + b (xy) and g − b (y) = δ2 ⊇ g − b (xy). therefore 〈 (g+b, g − b ); b 〉 is a dfs l-ideal of s over u. similarly we can prove the result for a dfs r-ideal of s over u. sufficiency. assume that 〈 (g+b, g − b ); b 〉 is a dfs l-ideal of s over u. let x,y ∈ s be such that x ≤ y. if y ∈ a, then g+b (y) ⊇ γ1 and g − b (y) ⊆ δ1. since g + b (x) ⊇ g + b (y) ⊇ γ1 and g − b (x) ⊆ g + b (y) ⊆ δ1, we have x ∈ a. let x ∈ s and y ∈ a, then g+b (y) = γ1 and g − b (y) = δ1. by hypothesis, g + b (xy) ⊇ g + b (y) = γ1 and g−b (xy) ⊆ g + b (y) = δ1. hence xy ∈ a. thus a is a left ideal of s. similarly, we can show that a is a right ideal of s. � example 4.1. there are six different chemicals which have been used in an experiment. take a collection of chemicals as the initial universe set u given by u = {γ1,γ2,γ3,γ4,γ5,γ6}. let a set of parameters e = {1, 2, 3, 4, 5} be a set of particular properties of each chemical in u with the following type of natures: int. j. anal. appl. 16 (4) (2018) 491 1 stands for the parameter ”density ”, 2 stands for the parameter ”melting point ”, 3 stands for the parameter ”combustion”, 4 stands for the parameter ”enthalpy ”, 5 stands for the parameter ”toxicity ”. let us define the following binary operation and order on a set of parameters e as follows. ∗ 1 2 3 4 5 1 2 2 4 4 5 2 2 2 2 2 5 3 1 2 3 4 5 4 1 2 1 2 5 5 1 5 5 5 5 ≤= {(1, 1), (1, 2), (3, 3), (1, 3), (4, 4), (1, 5), (5, 5), (2, 2)}. it is easy to observe that (e,∗,≤) is a unitary ordered ag-groupoid. let a = {1, 2, 5} and define a dfs-set 〈 (f+a , f − a ); a 〉 of s over u as follows: f+a (x) =   {γ1,γ2} if x = 1 {γ1,γ2,γ3} if x = 2 {γ5} if x = 3 {γ5} if x = 4 {γ1,γ2,γ3,γ4} if x = 5   and f−a (x) =   {γ1,γ2,γ3} if x = 1 {γ1,γ2} if x = 2 { } if x = 3 { } if x = 4 {γ2} if x = 5   . then it is easy to verify that 〈 (f+a , f − a ); a 〉 is a dfs l-ideal of s over u. let b = {1, 2, 4} and define a dfs-set 〈 (g+b, g − b ); b 〉 of s over u as follows: g+b (x) =   {γ4} if x = 1 {γ2,γ3,γ4} if x = 2 {γ1,γ2,γ5} if x = 3 {γ1,γ2,γ5} if x = 4 {γ2,γ4} if x = 5   and g−b (x) =   {γ1,γ2,γ3,γ4} if x = 1 {γ2,γ3} if x = 2 {γ1,γ2, ...,γ5} if x = 3 {γ1,γ2,γ3,γ4} if x = 4 {γ2,γ3} if x = 5   . then it is easy to verify that 〈 (g+b, g − b ); b 〉 is a dfs r-ideal of s over u. remark 4.1. every dfs r-ideal of a unitary ordered ag-groupoid s over u is a dfs l-ideal of s over u but the converse inclusion is not true in general which can be followed from above example. int. j. anal. appl. 16 (4) (2018) 492 lemma 4.1. let r be a right ideal and l be a left ideal of a unitary ordered ag-groupoid s. then (rl] is a left ideal of s. proof. let r be a right ideal and l be a left ideal of s. then by using lemma 2.1, we get s(rl] = (ss](rl] ⊆ (ss ·rl] = (sr ·sl] ⊆ (sr · (sl]] = (sr ·l] = ((ss]r ·l] ⊆ ((ss)r ·l] = ((rs)s ·l] ⊆ ((rs]s ·l] ⊆ (rl],which shows that (rl] is a left ideal of s. � lemma 4.2. let s be a unitary ordered ag-groupoid. if a = a2 for all a ∈ s, then ra = (sa∪sa2] is the smallest right ideal of s containing a. proof. assume that a = a2 for all a ∈ s. then by using lemma 2.1, we have (sa∪sa2]s = (sa∪sa2](s] ⊆ ((sa∪sa2)s] = (sa ·s ∪sa2 ·s] = (sa ·ss ∪sa2 ·ss] = (s ·as ∪s ·a2s] = (a ·ss ∪a2 ·ss] = (a2 ·ss ∪a2 ·ss] = (ss ·a2 ∪ss ·a2] = (sa∪sa2], which shows that (sa∪sa2] is a right ideal of s. it is easy to see that a ∈ (sa∪sa2]. let r be another right ideal of s containing a. since (sa∪sa2] = (ss ·a∪a ·sa] = (as ·s ∪a ·sa] ⊆ (rs ·s ∪rs] ⊆ r, hence (sa∪sa2] is the smallest right ideal of s containing a. � lemma 4.3. let s be a unitary ordered ag-groupoid and a = a2 for all a ∈ s. then s becomes a commutative monoid. proof. it is simple. � corollary 4.1. ra = (sa∪sa2] is the smallest right ideal of an ordered commutative monoid s containing a. lemma 4.4. let s be a unitary ordered ag-groupoid and a ∈ s. then la = (sa] is the smallest left ideal of s containing a. proof. it is simple. � theorem 4.2. let s be a unitary ordered ag-groupoid and ∅ 6= e ⊆ s . then the following assertions hold: (i) e forms a semilattice, where e = {x ∈ s : x = x2}; (ii) e is a singleton set, if a = ax ·a, ∀ a,x ∈ s. int. j. anal. appl. 16 (4) (2018) 493 proof. (i). it is simple. (ii). let y,z ∈ e. then by using (i), we get y = yz ·y = zy ·y = yy ·z = yz = zy = zz ·y = yz ·z = zy ·z = z. � • recall that an ordered ag**-groupoid is an ordered ag-groupoid in which a · bc = b ·ac, ∀ a,b,c ∈ s. note that an ordered ag**-groupoid also satisfies the paramedial law as well. now let us introduce the concept of an ordered a*g**-groupoid as follows: • an ordered ag**-groupoid s is called an ordered a*g**-groupoid if s = (s2]. corollary 4.2. let s be an ordered a*g**-groupoid and ∅ 6= e ⊆ s . then the following assertions hold: (i) e forms a semilattice, where e = {x ∈ s : x = x2}; (ii) e is a singleton set if a = ax ·a, ∀ a,x ∈ s. lemma 4.5. let s be an ordered a*g**-groupoid. then 〈r〉a2 = (sa 2 ∪a2] (〈l〉a = (sa∪a]) is the right (resp. left) ideal of s. proof. let a ∈ s, then by using lemma 2.1, we get (sa2 ∪a2]s = (sa2 ∪a2](s] = ((sa2 ∪a2)s] = (sa2 ·s ∪a2s] = (ss ·a2s ∪ss ·aa] = (s ·a2s ∪sa2] = (a2 ·ss ∪sa2] = (sa2] ⊆ (sa2 ∪a2], which is what we set out to prove. similarly we can prove that s(sa∪a] ⊆ (sa∪a]. � • an element a of an ordered ag-groupoid s is called a strongly regular element of s, if there exists some x in s such that a ≤ ax ·a and ax = xa, where x is called a pseudo-inverse of a. s is called strongly regular ordered ag-groupoid if all elements of s are strongly regular. theorem 4.3. let s be an ordered ag-groupoid (an ordered a*g**-groupoid) with left identity. an element a of s is strongly regular if and only if a ≤ ax ·ay for some x,y ∈ s. proof. necessity. let a ∈ s is strongly regular, then a ≤ ax ·a ≤ (ax) · (xa)(ax ·a) = (ax) · (a ·ax)(ax) = (ax) ·a((a ·ax)x) = ax ·ay, where (a ·ax)x = y ∈ s. thus a ≤ ax ·ay for some x,y ∈ s. sufficiency. let a ∈ s such that a ≤ ax·ay for some x,y ∈ s, then a ≤ ax·ay = (ay·x)a = (xy·a)a = ua·a, where xy = u ∈ s. thus au ≤ (ua ·a)u = ua ·ua = u(ua ·a) ≤ ua, and a ≤ ua ·a = au ·a. thus s is strongly regular. � int. j. anal. appl. 16 (4) (2018) 494 lemma 4.6. let fa = 〈 (f+a , f − a ); a 〉 be any dfs r-ideal (dfs l-ideal) of a strongly regular ordered a*g**-groupoid s over u. then the following assertions hold: (i) fa = fa �xs (fa = xs �fa); (ii) fa is dfs semiprime. proof. it is simple. � 4.2. characterization problems. in this section, we generalize the results of an ordered semigroup and get some interesting characterizations which we usually do not find in an ordered semigroup. from now onward, r (resp. l) will denote any right (resp. left) ideal of an ordered ag-groupoid s; ra (resp. la) will denote any smallest right (resp. smallest left) ideal of s containing a. any dfs r-ideal of an ordered ag-groupoid s (resp. dfs l-ideal of s) over u will be denoted by fa (resp. gb) unless otherwise specified. theorem 4.4. let fa,gb be any dfs l-ideals of a unitary ordered ag-groupoid s. then the following conditions are equivalent: (i) s is strongly regular; (ii) s is strongly regular commutative monoid; (iii) (rala] ∩la = ((ra ·rala)la ·la], (a = a2, ∀ a ∈ s); (iv) (rl] ∩l = ((r ·rl)l ·l]; (v) fa ugb = (fa �gb) �fa; (vi) s is strongly regular and |e|= 1, (a = ax ·a, ∀ a,x ∈ e); (vii) s is strongly regular and ∅ 6= e ⊆ s is semilattice. proof. (i) =⇒ (vii) : it can be followed from theorem 4.2 (i). (vii) =⇒ (vi) : it can be followed from theorem 4.2 (ii). (vi) =⇒ (v) : let fa and gb be any dfs l-ideals of a strongly regular s over u. now for a ∈ s, there exist some x,y ∈ s such that a ≤ ax·ay = ya·xa ≤ y(ax·ay)·xa = (ax)(y·ay)·xa = (ay·y)(xa)·xa = (y2a·xa)(xa). thus (y2a ·xa,xa) ∈ aa. therefore ((f+a ∼ ◦ g+b ) ∼ ◦ f+a )(a) = ⋃ (y2a·xa,xa)∈aa { (f+a ∼ ◦ g+b )(y 2a ·xa) ∩f+a (xa) } ⊇ ⋃ y2a·xa≤y2a·xa {f+a (y 2a) ∩g+b (xa)}∩f + a (xa) ⊇ f+a (y 2a) ∩g+b (xa) ∩f + a (xa) ⊇ f + a (a) ∩g + b (a), and similarly, we get int. j. anal. appl. 16 (4) (2018) 495 ((f−a ∼ ? g−b ) ∼ ? f−a )(a) = ⋂ (y2a·xa,xa)∈aa { (f−a ∼ ? g−b )(y 2a ·xa) ∪f−a (xa) } ⊆ ⋂ y2a·xa≤y2a·xa {f−a (y 2a) ∪g−b (xa)}∪f − a (xa) ⊆ f−a (y 2a) ∪g−b (xa) ∪f − a (xa) ⊆ f − a (a) ∪g − b (a), which shows that (fa � gb) � fa w fa u gb. by using lemmas 3.2 and 4.6, it is easy to show that (fa �gb) �fa v fa ugb. thus fa ugb = (fa �gb) �fa. (v) =⇒ (iv) : let r and l be any right and left ideals of s respectively. then by using lemmas 3.1 and 4.1, x(rl] and xl are the dfs l-ideals of s over u. now by using lemma 3.4, we get x(rl]∩l = x(rl] uxl = (x(rl] �xl) �x(rl] = x((rl]l·(rl]], which give us (rl] ∩l = ((rl]l · (rl]]. now by using lemma 2.1, we get ((rl]l · (rl]] = ((rl)l ·rl] = (l2r ·rl] = (lr ·rl2] = (r(lr ·l2)] = (r(l2 ·rl)] = (r(r ·l2l)] = (r ·rl3] = (r(r ·l2l)] = (r(l2 ·rl)] = ((r ·rl)l ·l]. (iv) =⇒ (iii) : it is simple. (iii) =⇒ (ii) : since (sa∪sa2] is the smallest right ideal of s containing a and (sa] is the smallest left ideal of s containing a, where a = a2, ∀ a ∈ s. thus by using given assumption and lemma 2.1, we get a ∈ ((sa∪sa2](sa]] ∩ (sa] = (((sa∪sa2] · (sa∪sa2](sa])(sa] · (sa]] = (((sa∪sa2) · (sa∪sa2)(sa))(sa) · (sa)] ⊆ (s(sa) · (sa)] = (s2a ·sa] = (sa ·sa] = (as ·as]. hence by using lemma 4.2, s is strongly regular commutative monoid. (ii) =⇒ (i) : it is obvious. � theorem 4.5. let s be an ordered ag-groupoid. then the following conditions are equivalent: (i) s is strongly regular ; (ii) s is strongly regular commutative monoid ; (iii) ra ∩la = (ra(lara ·ra)], (a = a2, ∀ a ∈ s); (iv) r∩l = (r(lr ·r)]; (v) fa ugb = f3a �gb; int. j. anal. appl. 16 (4) (2018) 496 (vi) s is strongly regular and |e|= 1, (a = ax ·a, ∀ a,x ∈ e); (vii) s is strongly regular and ∅ 6= e ⊆ s is semilattice. proof. (i) =⇒ (vii) : it can be followed from theorem 4.2 (i). (vii) =⇒ (vi) : it can be followed from theorem 4.2 (ii). (vi) =⇒ (v) : let fa and gb be any dfs r-ideal and dfs l-ideal of a strongly regular s over u respectively. from lemma 3.2, it is easy to show that f+3a �g + b v f + a ug + b . now for a ∈ s, there exist some x,y ∈ s such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = y(ax ·ay) ·x(ax ·ay) = (ax)(y ·ay) · (ax)(x ·ay) = (ax)(ay2) · (ax)(a ·xy) = (y2a)(xa) · (ax)(a ·xy) = ((ax)(a ·xy))(xa) ·y2a = ((ax)(a ·xy))(ex ·a) ·y2a = ((ax)(a ·xy))(ax ·e) ·y2a = bc ·y2a = d ·y2a, where d = bc = ((ax)(a ·xy))(ax ·e). thus ((f+a ∼ ◦ f+a ) ∼ ◦ f+a )(d) = ⋃ d≤bc {(f+a ∼ ◦ f+a )(b) ∩f + a (c)}⊇ (f + a ∼ ◦ f+a )(b) ∩f + a (c) = ⋃ b≤(ax)(a·xy) {f+a (ax) ∩f + a (a ·xy)}∩f + a (ax ·e) ⊇ f+a (ax) ∩f + a (a ·xy) ∩f + a (ax ·e) ⊇ f + a (a). therefore (f+3a ∼ ◦ g+b )(a) = ⋃ a≤d·y2a {((f+a ∼ ◦ f+a ) ∼ ◦ f+a )(d) ∩g + b (y 2a)}⊇ f+a (a) ∩g + b (a), which shows that f+a ∩g + b ⊆ f +3 a ∼ ◦ g+b , and similarly f − a ∪g − b ⊇ f −3 a ∼ ? g−b . thus fa ugb = f 3 a �gb. (v) =⇒ (iv): let r and l be any right and left ideals of s respectively. then by using lemma 3.1, xr and xl are the dfs r-ideal and dfs l-ideal of s over u respectively. now by using lemma 3.4, we get xr∩l = xr uxl = ((xr �xr) �xr) �xl = x(r3] �xl = x((r3]l], which implies that r ∩ l = ((r3]l]. now by using lemma 2.1, we get r ∩ l = ((r3]l] = (r3l] = (r2r ·l] = (lr ·r2] = (r2 ·rl] = (r ·r2l] = (r(lr ·r)]. (iv) =⇒ (iii) : it is simple. int. j. anal. appl. 16 (4) (2018) 497 (iii) =⇒ (ii) : since (sa∪sa2] is the smallest right ideal of s containing a and (sa] is the smallest left ideal of s containing a. thus by using given assumption and lemma 2.1, we get a ∈ (sa∪sa2] ∩ (sa] = ((sa∪sa2]((sa](sa∪sa2] · (sa∪sa2])] = ((sa∪sa2)((sa)(sa∪sa2) · (sa∪sa2))] ⊆ (s(s(sa∪sa2) · (sa∪sa2))] = (s((s2a∪s2a2)(sa∪sa2))] = ((s2a∪s2a2)(s(sa∪sa2))] = ((s2a∪s2a2)(s2a∪s2a2)] = ((sa∪a2s2)(sa∪a2s2)] = ((sa∪s2a ·a)(sa∪s2a ·a)] ⊆ ((sa∪sa)(sa∪sa)] = (sa ·sa] = (as ·as]. hence by using lemma 4.2, s is strongly regular commutative monoid. (ii) =⇒ (i) : it is obvious. � let s be an ordered a*g**-groupoid. from now onward, r (resp. l) will denote any right (resp. left) ideal of s; 〈r〉a2 will denote a right ideal (sa 2 ∪ a2] of s containing a2 and 〈l〉a will denote a left ideal (sa∪a] of s containing a; fa (resp. gb) will denote any dfs r-ideal over u (resp. dfs l-ideal over u) of s unless otherwise specified. theorem 4.6. let s be an ordered a*g**-groupoid. then s is strongly regular if and only if 〈r〉a2 ∩〈l〉a = (〈r〉2a2 〈l〉 2 a] and 〈r〉a2 is semiprime. proof. necessity: let s be strongly regular. it is easy to see that (〈r〉2a2 〈l〉 2 a] ⊆ 〈r〉a2 ∩ 〈l〉a . let a ∈ 〈r〉a2 ∩〈l〉a . then there exist some x,y ∈ s such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = (x ·ay)(ax) · (y ·ay)(ax) = (a ·xy)(ax) · (ay2)(ax) = (a ·xy)(ax) · (xa)(y2a) ∈ (〈r〉a2 s · 〈r〉a2 s)(s 〈l〉a ·s 〈l〉a) ⊆〈r〉 2 a2 〈l〉 2 a , which shows that 〈r〉a2 ∩〈l〉a = (〈r〉 2 a2 〈l〉 2 a]. it is easy to see that 〈r〉a2 is semiprime. int. j. anal. appl. 16 (4) (2018) 498 sufficiency: since (sa2∪a2] and (sa∪a] are the right and left ideals of s containing a2 and a respectively. thus by using given assumption and lemma 2.1, we get a ∈ (sa2 ∪a2] ∩ (sa∪a] = ((sa2 ∪a2]2(sa∪a]2] = ((sa2 ∪a2)(sa2 ∪a) · (sa∪a)(sa∪a)] ⊆ (s(sa2 ∪a) ·s(sa∪a)] = ((s ·sa2 ∪sa)(s ·sa∪sa)] = ((a2s ·s ∪sa)(as ·s ∪sa)] = ((a2s ·s ∪sa)(as ·s ∪sa)] = ((sa2 ∪sa)(sa∪sa)] = ((a2s ∪sa)(sa∪sa)] = ((sa ·a∪sa)(sa∪sa)] ⊆ ((sa∪sa)(sa∪sa)] = (sa ·sa] = (as ·as]. this implies that s is strongly regular. � corollary 4.3. let s be an ordered a*g**-groupoid. then s is strongly regular if and only if 〈r〉a2∩〈l〉a = (〈l〉2a 〈r〉 2 a2 ] and 〈r〉a2 is semiprime. theorem 4.7. let s be an ordered a*g**-groupoid. then the following conditions are equivalent: (i) s is strongly regular ; (ii) 〈r〉a2 ∩〈l〉a = (〈l〉 2 a 〈r〉 2 a2 ] and 〈r〉a2 is semiprime; (iii) r∩l = (l2r2] and r semiprime; (iv) fa ugb = (fa �gb) � (fa �gb) and fa is dfs semiprime; (v) s is strongly regular and |e|= 1, (a = ax ·a, ∀ a,x ∈ e); (vi) s is strongly regular and ∅ 6= e ⊆ s is semilattice. proof. (i) =⇒ (vi) : it can be followed from corollary 4.2 (i). (vi) =⇒ (v) : it can be followed from corollary 4.2 (ii). (v) =⇒ (iv) : let fa and gb be any dfs r-ideal and dfs l-ideal of a strongly regular s over u respectively. from lemma 3.2, it is easy to show that (fa �gb) � (fa �gb) v fa ugb. now for a ∈ s, there exist some x,y ∈ s such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = (ax ·ay) · ((ax ·ay)x)y = (ax ·ay) · (yx)(ax ·ay) = (ax ·ay) · (ax)(yx ·ay) = (ax ·ay) · (ay ·yx)(xa) = (ax ·ay) · ((yx ·y)a)(xa) = (ax)((yx ·y)a) · (ay)(xa) = (ax)(ba) · (ay)(xa), where yx ·y = b. int. j. anal. appl. 16 (4) (2018) 499 thus (ax · ba,ay ·xa) ∈ aa. therefore ((f+a ∼ ◦ g+b ) ∼ ◦ (f+a ∼ ◦ g+b ))(a) = ⋃ (ax·ba,ay·xa)∈aa {(f+a ∼ ◦ g+b )(ax · ba) ∩ (f + a ∼ ◦ g+b )(ay ·xa)} ⊇ ⋃ ax·ba≤ax·ba {f+a (ax) ∩g + b (ba)}∩ ⋃ ay·xa=ay·xa {f+a (ay) ∩g + b (xa)} ⊇ f+a (ax) ∩g + b (ba) ∩f + a (ay) ∩g + b (xa) ⊇ f + a (a) ∩g + b (a), which shows that (f+a ∼ ◦g+b ) ∼ ◦(f+a ∼ ◦g+b ) ⊇ f + a∩g + b . similarly we can show that (f − a ∼ ?g−b ) ∼ ?(f−a ∼ ?g−b ) ⊆ f − a∪g − b. thus fa ugb v (fa �gb) � (fa �gb). hence fa ugb = (fa �gb) � (fa �gb). also by using lemma 4.6, fa is dfs semiprime. (iv) =⇒ (iii) : let r and l be any left and right ideals of s. then by using lemma 3.1, xr and xl are the dfs r-ideal and dfs l-ideal of s over u respectively. now by using lemma 3.4, we get xr∩l = xr uxl = (xr �xl) � (xr �xl) = (xr �xr) � (xl �xl) = x(r2] �x(l2] = x(r2l2] = x(l2r2], which implies that r∩l = (l2r2]. (iii) =⇒ (ii) : it is simple. (ii) =⇒ (i) : it can be followed from corollary 4.3. � theorem 4.8. let s be an ordered a*g**-groupoid. then the following conditions are equivalent: (i) s is strongly regular ; (ii) 〈r〉a2 ∩〈l〉a = (〈r〉a2 〈l〉a · 〈r〉a2 ] and 〈r〉a2 is semiprime; (iii) r∩l = (rl ·r] and r is semiprime; (iv) fa ugb = (fa �gb) �fa and fa is dfs semiprime; (v) s is strongly regular and |e|= 1, (a = ax ·a, ∀ a,x ∈ e); (vi) s is strongly regular and ∅ 6= e ⊆ s is semilattice. proof. (i) =⇒ (vi) : it can be followed from corollary 4.2 (i). (vi) =⇒ (v) : it can be followed from corollary 4.2 (ii). (v) =⇒ (iv) : let fa and gb be any dfs l-ideals of a strongly regular s over u. now for a ∈ s, there exist some x,y ∈ s such that a ≤ ax ·ay ≤ ax ·(ax ·ay)y = ((ax ·ay)y ·x)a = (xy ·(ax ·ay))a = (ax ·(xy ·ay))a = (ax · (a · (xy)y))a. int. j. anal. appl. 16 (4) (2018) 500 thus (ax · (a · (xy)y),a) ∈ aa. therefore ((f+a ∼ ◦ g+b ) ∼ ◦ f+a )(a) = ⋃ (ax·(a·(xy)y),a)∈aa {(f+a ∼ ◦ g+b )(ax · (a · (xy)y)) ∩g + b (a)} ⊇ ⋃ ax·(a·(xy)y≤ax·(a·(xy)y {f+a (ax) ∩g + b (a · (xy)y)}∩g + b (a) ⊇ f+a (ax) ∩g + b (a · (xy)y) ∩g + b (a) ⊇ f + a (a) ∩g + b (a), which shows that (f+a ∼ ◦g+b ) ∼ ◦f+a ⊇ f + a ∩g + b . similarly we can show that (f − a ∼ ? g−b ) ∼ ? f−a ⊆ f − a ∪g − b. thus (fa �gb) �fa w fa ugb. by using lemmas 3.2 and 4.6, it is easy to show that (fa �gb) �fa v fa ugb. thus fa ugb = (fa �gb) �fa. also by using lemma 4.6, fa is dfs semiprime. (iv) =⇒ (iii) : let r and l be any left and right ideals of s. then by lemma 3.1, xr and xl are the dfs r-ideal and dfs l-ideal of s over u respectively. now by using lemmas 3.4, 4.1 and 2.1, we get xr∩l = xr uxl = (xr �xl) �xl = x((rl]·r] = x(rl·r], which shows that r∩l = (rl ·r]. also by using lemma 3.3, r is semiprime. (iii) =⇒ (ii) : it is simple. (ii) =⇒ (i) : since (sa2∪a2] and (sa∪a] are the right and left ideals of s containing a2 and a respectively. thus by using given assumption and lemma 2.1, we get a ∈ (sa2 ∪a2] ∩ (sa∪a] = ((sa2 ∪a2](sa∪a] · (sa2 ∪a2]] = ((sa2 ∪a2)(sa∪a) · (sa2 ∪a2)] ⊆ (s(sa∪a) · (sa2 ∪a2)] = ((s2a∪sa)(sa2 ∪a2)] = ((s2a ·sa2) ∪ (s2a ·a2) ∪ (sa ·sa2) ∪ (s2a ·a2)] ⊆ ((sa ·a2s) ∪ (sa ·sa) ∪ (sa ·a2s) ∪ (sa ·sa)] ⊆ ((sa ·sa) ∪ (sa ·sa) ∪ (sa ·sa) ∪ (sa ·sa)] = (sa ·sa] = (as ·as]. hence s is strongly regular. � 5. conclusions we have got some interesting and new characterizations which we usually do not find in other algebraic structures. we have considered the following problems in detail: i) define and compare dfs left/right ideals of an ordered ag-groupoid and respective examples are provided. ii) introduce the concept of an ordered a*g**-groupoid and characterize it by using dfs left/right ideals. int. j. anal. appl. 16 (4) (2018) 501 iii) study the structural properties of a unitary ordered ag-groupoid and ordered a*g**-groupoid in terms of its semilattices, strongly regular classes and generated commutative monoids. this paper generalized the theory of an ag-groupoid in the following ways: i) in an ag-groupoid (without order) by using the dfs-sets. ii) in an ag-groupoid (with and without order) by using fuzzy sets instead of dfs-sets. some important issues for future work are: i) to develop strategies for obtaining more valuable results in related areas. ii) to apply these notions and results for studying dfs expert sets and applications in decision making problems. references [1] m. i. ali, f. feng, x. liu, w. k. mine and m. shabir, on some new operations in soft set theory, comput math appl., 57 (2009), 1547-1553. [2] n. cagman and s. enginoglu, fp-soft set theory and its applications, ann. fuzzy math. inform., 2 (2011), 219-226. [3] w. a. dudek and r. s. gigon, congruences on completely inverse ag**-groupoids, quasigroups and related systems, 20 (2012), 203-209. [4] w. a. dudek and r. s. gigon, completely inverse ag**-groupoids. semigroup forum, 87 (2013), 201-229. [5] f. feng, soft rough sets applied to multicriteria group decision making, ann. fuzzy math. inform., 2 (2011), 69-80. [6] f. feng, y. b. jun and x. zhao, soft semirings, comput. math. appl., 56 (2008), 2621-2628. [7] y. b. jun, soft bck/bci-algebras, comput. math. appl., 56 (2008), 1408-1413. [8] y. b. jun and s. s. ahn, double-framed soft sets with applications in bck/bci-algebras, j. appl. math., 2012, pp. 15. [9] y. b. jun, k. j. lee and a. khan, soft ordered semigroups, math. logic q., 56 (2010), 42-50. [10] m. a. kazim and m. naseeruddin, on almost semigroups, the alig. bull. math., 2(1972), 1-7. [11] m. khan, some studies in ag*-groupoids, ph. d thesis, quaid-i-azam university, pakistan, 2008. [12] a. khan, t. asif and y. b. jun, double-framed soft ordered semigroups, submitted. [13] d. v. kovkov, v. m. kolbanov and d. a. molodtsov, soft sets theory based optimization, j. comput. syst. sci. int., 46 (6) (2007), 872-880. [14] p. k. maji a. r. roy and r. biswas, an application of soft sets in a decision making problem, comput. math. appl., 44 (2002), 1077-1083. [15] d. molodtsov, v. y. leonov and d. v. kovkov, soft sets technique and its application, nechetkie sistemy i myagkie vychisleniya, 1 (1) (2006), 8-39. [16] p. majumdar and s. k. samanta, similarity measures of soft sets, new math. neutral comput., 4 (1) (2008), 1-12. [17] d. molodtsov, soft set theory, comput math appl., 37 (1999), 19-31. [18] q. mushtaq and s. m. yusuf, on la-semigroups, alig. bull. math., 8(1978), 65-70. [19] q. mushtaq and s. m. yusuf, on locally associative left almost semigroups, j. nat. sci. math., 19(1979), 57-62. [20] d. w. pei and d. miao, from soft sets to information systems, ieee international conference on granular computing, (2005), 617-621. [21] p. v. protić and n. stevanović, ag-test and some general properties of abel-grassmann’s groupoids, pu. m. a., 4, 6 (1995), 371-383. int. j. anal. appl. 16 (4) (2018) 502 [22] a. r. roy and p. k. maji, a fuzzy soft set theoretic approach to decision making problems, j. comput. appl. math., 203 (2007), 412-418. [23] a. sezgin, a. o. atagun and n. cagman, soft intersection nearrings with applications, neural comput. and appl., 21 (2011), 221-229. [24] a. seizgin and a. o. atagun, on operations of soft sets, comput. math. appl., 61 (2011), 1457-1467. [25] n. stevanović and p. v. protić, composition of abel-grassmann’s 3-bands, novi sad, j. math., 2, 34 (2004), 175-182. [26] x. yang, d. yu, j. yang, c. wu, generalization of soft set theory from crisp to fuzzy case, fuzzy inf. eng., 40 (2007), 345-355. [27] f. yousafzai, n. yaqoob and k. hila, on fuzzy (2,2)-regular ordered γ-ag**-groupoids, upb sci. bull., ser. a, 74 (2012), 87-104. [28] f. yousafzai, a. khan, v. amjid and a. zeb, on fuzzy fully regular ordered ag-groupoids, j. intell. fuzzy syst., 26 (2014), 2973-2982. 1. introduction 2. preliminaries 3. soft sets 4. on dfs strongly regular ordered ag-groupoids 4.1. basic results 4.2. characterization problems 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 10, number 2 (2016), 85-89 http://www.etamaths.com beurling’s theorem and lp −lq morgan’s theorem for the generalized bessel-struve transform a. abouelaz, a. achak, r. daher, n. safouane∗ abstract. the generalized bessel-struve transform satisfies some uncertainty principles similar to the euclidean fourier transform. a generalization of beurling’s theorem and lp − lq morgan’s theorem obtained for the generalized bessel-struve transform. 1. introduction and preliminaries there are many theorems known which state that a function and its classical fourier transform on r cannot both be sharply localized. that is, it is impossible for a nonzero function and its fourier transform to be simultaneously small. here a concept of the smallness had taken different interpretations in different contexts. morgan [5] and beurling [3] for example interpreted the smallness as sharp pointwise estimates or integrable decay of functions. in particular beurling’s theorem, which was found by beurling and his proof was published much later by hörmander [4], says that theorem 1. if f ∈ l2(r) satisfies that∫ r ∫ r |f(x)||f̂(y)|e|x||y|dxdy < ∞, then f = 0 a.e. morgan [5] has established a famous theorem stating that for γ > 2 and η = γ γ−1, if (aγ) 1 γ (bη) 1 η > (sin(π 2 (η− 1)) 1 η , ea|x| γ f ∈ l∞(r) and eb|x| η f(f) ∈ l∞(r). then f is null almost everywhere. s. ben farah and k. mokni [2] have generalized morgan’s theorem to an lp −lq−version where 1 ≤ p,q ≤ +∞. the outline of the content of this paper is as follows. in section 2 we give an analogue of beurling’s theorem for fb,sα,n . section 3 is devoted to lp −lq-morgan’s theorem for fb,sα,n . let us now be more precise and describe our results. to do so, we need to introduce some notations. throughout this paper, the letter c indicates a positive constant not necessarily the same in each occurrence. we denote by • (1) aα = 2γ (α + 1) √ πγ ( α + 1 2 ) where α > −1 2 . • mn the map defined by mn(f(x)) = x2nf(x). • lpα(r) the class of measurable functions f on r for which ‖f‖p,α < ∞, where ‖f‖p,α = (∫ r |f(x)|p|x|2α+1dx )1 p , ifp < ∞, and ‖f‖∞,α = ‖f‖∞ = ess supx≥0|f(x)|. • lpα,n(r) the class of measurable functions f on r for which ‖f‖p,α,n = ‖m−1n f‖p,α+2n < ∞. 2010 mathematics subject classification. 42a38, 44a35, 34b30. key words and phrases. generalized bessel-struve transform; uncertainty principles. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 85 86 abouelaz, achak, daher and safouane • k0 the space of functions f infinitely differentiable on r∗ with bounded support verifying for all n ∈ n, lim y→0− ynf(n)(y) and lim y→0+ ynf(n)(y) exist. • d dx2 = 1 2x d dx , where d dx is the first derivative operator. in this section we recall some facts about harmonic analysis related to the generalized bessel-struve operator fα,nb,s. we cite here, as briefly as possible, only some properties. for more details we refer to [1]. for λ ∈ c and x ∈ r, put ψλ,α,n(x) = aα+2nx 2n ∫ 1 0 (1 − t2)α+2n− 1 2 eλxtdt. ψλ,α,n satisfies ∀ξ ∈ r, ∀ζ ∈ r,∀x ∈ r |ψλ,α,n(i(ξ + iζ)x)| ≤ x2ne|x||ζ|.(2) ∀n ∈ n, ∀λ ∈ r, ∀x ∈ r, | dn dxn (x−2nψiλ,α,n(x))| ≤ |λ|n.(3) definition 1. the generalized bessel-struve transform is defined on l1α,n(r) by ∀λ ∈ r, fα,nb,s(f)(λ) = ∫ r f(x)ψ−iλ,α,n(x)|x|2α+1dx. definition 2. for f ∈ l1α,n(r) with bounded support, the integral transform wα,n, given by wα,n(f(x)) = aα+2n ∫ +∞ |x| (y2 −x2)α+2n− 1 2 y1−2nf(sgn(x)y)dy, x ∈ r\{0} is called the generalized weyl integral transform associated with bessel-struve operator. proposition 1. wα,n is a bounded operator from l1α,n(r) to l 1(r), where l1(r) is the space of lebesgue-integrable functions. remark 1. from proposition 1 we can find a constant c such that∫ r |wα,n(f)(x)|dx ≤ c‖f‖α,n,1 proposition 2. if f ∈ l1α,n(r) then (4) fα,nb,s = f ◦wα,n, where f is the classical fourier transform defined on l1(r) by f(g)(λ) = ∫ r g(x)e−iλxdx. definition 3. let α = k + 1 2 where k ∈ n. we define the operator vα,n on k0 as follows vα,nf(x) = (−1)k+1 22k+4n+1(k + 2n)! (2k + 4n + 1)! x2n( d dx2 )k+2n+1(f(x)), x ∈ r∗. theorem 2. let f ∈ k0, vα,n and wα,n are related by the following relation vα,n(wα,n(f)) = f. beurling’s theorem and lp −lq morgan’s theorem 87 2. beurling’s theorem for the generalized bessel-struve transform in this section we will prove beurling’s theorem for the generalized bessel-struve transform. theorem 3. let k ∈ n, α = k + 1 2 and f ∈ l2α,n(r) satisfy (5) ∫ r ∫ r |f(x)||fα,nb,s(f)(y)|e |x||y||x|2(α+n)+1dxdy < ∞, then f = 0 almost everywhere. proof. we start with the following lemma. lemma 1. we suppose that f ∈ l2α,n(r) satisfies (5), then f ∈ l1α,n(r). proof. we may assume that f 6= 0 in l2α,n(r). (5) and the fubini theorem for almost every y ∈ r |fα,nb,s(f)(y)| ∫ r |f(x)|e|x||y||x|2(α+n)+1dx < ∞, since fα,nb,s(f) 6= 0, there exist y0 ∈ r,y0 6= 0 such that f α,n b,s(f)(y0) 6= 0, therefore ∫ r |f(x)|e|x||y0||x|2(α+n)+1dx < ∞ ∫ r |f(x)| x2n e|x||y0||x|2(α+2n)+1dx < ∞∫ r |m−1n f(x)|e |x||y0||x|2(α+2n)+1dx < ∞, since e|x||y0| > 1 for large |x| it follows ∫ r |m −1 n f(x)||x|2(α+2n)+1dx < ∞. this lemma and proposition 1 imply that wα,n(f) is well-defined a.e on r. by remark 1 we can find a positif constant c such that∫ r |wα,n(f)(x)|dx ≤ c‖f‖α,n,1 ≤ c‖m−1n f‖α+2n,1 ≤ c ∫ r |m−1n f(x)||x| 2(α+2n)+1dx ≤ c ∫ r | f(x) x2n ||x|2(α+2n)+1dx ≤ c ∫ r |f(x)||x|2(α+n)+1dx. thus∫ r ∫ r |wα,n(f(x))||f α,n b,s(f)(y)|e |x||y|dxdy ≤ c ∫ r ∫ r |f(x)||fα,nb,s(f)(y)|e |x||y||x|2(α+n)+1dxdy < ∞. it follows from proposition 2 that∫ r ∫ r |wα,n(f(x))||f ◦wα,n(f)(y)|e|x||y|dxdy < ∞. according to theorem 1, we can deduce that wα,n(f) = 0, applying lemma 1 we obtain f = vα,n ◦wα,n(f) = 0. 88 abouelaz, achak, daher and safouane corollaire 1. (gelfand-shilov) if f ∈ l2α,n(r) such that∫ r |f(x)|e |x|p p |x|2(α+n)+1dx < ∞, ∫ r |fα,nb,s(f)(y)|e |y|q q dy < ∞ then f = 0. proof. let m and m∗ be functions satisfying xy ≤ m(x) + m∗(y).(6) if ∫ r |f(x)|em(x)|x|2(α+n)+1dx < ∞, ∫ r |fαb,s(f)(y)|e m∗(y)dy < ∞ then ∫ r ∫ r |f(x)||fα,nb,s(f)(y)|e |x||y||x|2(α+n)+1dxdy ≤ ∫ r ∫ r |f(x)||fα,nb,s(f)(y)|e m(x)+m∗(y)|x|2(α+n)+1dxdy = ∫ r |f(x)|em(x)|x|2(α+n)+1dx ∫ r |fα,nb,s(f)(y)|e m∗(y)dy < ∞. consequently, beurling’s theorem implies that f(y) = 0. in particular, if m(x) = |x| p p and m∗(y) = |y|q q , where p,q are conjugate exponents p−1 + q−1 = 1, then the pair (m, m∗) satisfies the condition (6). thus, we obtain an analogue of the gelfand-shilov uncertainty principle for the bessel-struve transform. 3. lp −lq morgan’s theorem for the generalized bessel-struve transform in this section, we prove lp −lq morgan’s theorem for the generalized bessel-struve transform. lemma 2. we assume that ρ ∈]1, 2[, q ∈ [1,∞], σ > 0 and b > σ sin(π 2 (ρ − 1)). if g is an entire function on c verifying |g(x + iy)| ≤ ceσ|y| ρ ∈ lpα+2n(r) eb|x| ρ g|r ∈ l q α+2n(r) for all x, y ∈ r, then g = 0. proof. see [2]. lemma 3. let p ∈ [1,∞] , γ > 2 and f a measurable function on r verifying (7) ∀a > 0, ea|x| γ f ∈ lpα,n(r). then the function defined on c by (8) fα,nb,s(f)(z) = ∫ r f(x)ψ−iz,α,n(x)|x|2α+1dx is well defined and entire on c. moreover, we have (9) ∀ξ,ζ ∈ r, |fα,nb,s(f)(ξ + iζ)| ≤ ∫ r |f(x)|e|x||ζ||x|2α+1dx. proof. relation (7) assert that fα,nb,s(f) is well defined. applying again relation (7), the analytic theorem on (8) and the fact that ψ−iλ,α,n(x) verifies (9), we deduce that z →f α,n b,s(f)(z) is an entire on c. the relation (10) is obtained from relation (2). theorem 4. let p,q ∈ [1,∞], a > 0, b > 0, γ > 2 and η = γ γ−1. suppose that f a measurable function on r such that ea|x| γ f ∈ lpα,n(r) and e b|x|ηfα,nb,s(f) ∈ l q α+2n(r).(10) if (aγ) 1 γ (bη) 1 η > (sin(π 2 (η − 1)) 1 η , then f is null almost everywhere. beurling’s theorem and lp −lq morgan’s theorem 89 proof. we notice that ea|x| γ f ∈ lpα,n ⇔ ea|x| γ m−1n f ∈ l p α+2n. first case: 1 < p < ∞. applying hölder inequality, we get |fα,nb,s(f)(ξ + iζ)| ≤ ‖f‖α,n,p (∫ r e−ap ′|x|γe|x||ζ|p ′ |x|2α+1dx ) 1 p′ where p′ verifies 1 p′ + 1 p = 1. now, we take c ∈](bη)−η sin(π 2 (η − 1)) 1 η ), (aγ) 1 γ [. using a convexity’s inequality, we obtain |x||ζ| ≤ cγ γ |x|γ + 1 ηcη |ζ|η(11) and the following relation holds∫ r e−ap ′|x|γe|x||ζ|p ′ |x|2α+1dx ≤ e p′|ζ|η ηcη ∫ r e−p ′(a−c γ γ |x|γ)|x|2α+1dx. so we get ∀ξ,ζ ∈ r, |fα,nb,s(f)(ξ + iζ)| ≤ const e 1 ηcη |ζ|η. second case: p = 1 or p = +∞. from relations (2) and (11), we get |fb,sα,n (f)(ξ + iζ)| ≤ e 1 ηcη |ξ|η ∫ r ea|x| γ |f(x)|e−(a− cγ γ )|x|γ|x|2α+1dx. therefore |fb,sα,n (f)(ξ + iζ)| ≤ const e 1 ηcη |ξ|η. hence (12) ∀p ∈ [1,∞], ∀ξ,ζ ∈ r, |fα,nb,s(f)(ξ + iζ)| ≤ const e 1 ηcη |ζ|η. by virtue of relations (10), (12) and lemma 2, we obtain that fα,nb,sf = 0. the injectivity of the generalized bessel-struve transform implies that f = 0 almost everywhere. references [1] a. abouelaz, a. achak, r. daher and n. safouane, harmonic analysis associated with the generalized bessel-struve operator on the real line, int. refereed j. eng. sci., 4 (2015), 72-84. [2] s. ben farah, k. mokni, uncertainty principle and the lp − lq-version of morgan’s theorem on semi-simple lie groups, integrals transform spec. funct. 16 (2005), 281-289. [3] a. beurling, the collect works of arne beurling, birkhäuser. boston, 1989, 1-2. [4] l. hörmander, a uniqueness theorem of beurling for fourier transform pairs, ark. för math., 2(1991), 237-240. [5] g.w. morgan, a note on fourier transforms, j. london math. soc., 9 (1934), 188-192. department of mathematics, faculty of sciences aïn chock, university of hassan ii, casablanca 20100, morocco ∗corresponding author: safouanenajat@gmail.com int. j. anal. appl. (2023), 21:74 received: may 12, 2023. 2020 mathematics subject classification. 47h10, 54h25. key words and phrases. common fixed point; coincident point; graphical structures; neutrosophic metric space; unique solution; non-linear fractional differential equations. https://doi.org/10.28924/2291-8639-21-2023-74 © 2023 the author(s) issn: 2291-8639 1 certain fixed-point results via ds-weak commutativity condition in neutrosophic metric spaces with application to non-linear fractional differential equations umar ishtiaq1,*, muhammad saeed2, khaleel ahmad2, ilsa shokat3, manuel de la sen4 1office of research, innovation and commercialization, university of management and technology, lahore, pakistan 2department of mathematics, university of management and technology, lahore, pakistan 3department of mathematics, bahauddin zakariya university multan, sub campus, vehari, pakistan 4department of electricity and electronics, institute of research and development of processes, university of the basque country, campus of leioa, 48940 leioa, bizkaia, spain *corresponding author: umarishtiaq000@gmail.com abstract. this study demonstrates that, for the non-linear contractive conditions in neutrosophic metric spaces, a common fixed-point theorem may be proved without requiring the continuity of any mappings. a novel commutativity requirement for mappings weaker than the compatibility of mappings is used to demonstrate the conclusion. we provide several examples to illustrate our major idea. also, we provide an application to the non-linear fractional differential equation to show the validity of our main result. 1. introduction and prelimainaries there is still one significant deficiency despite the development of the computer industry and its incredible achievement in changing several fields of research. computers are not intended to process phrases with uncertainty. logic must be developed to handle uncertainty using methods other than the classical ones. one approach to uncertainty is fuzzy set theory, where topological structures are https://doi.org/10.28924/2291-8639-21-2023-74 2 int. j. anal. appl. (2023), 21:74 fundamental building blocks for creating mathematical models that are applicable to real-world scenarios. el naschie's [1] established a relation between topology, algebra and geometry, he showed that the approach of dimension is based on topological structure. el naschie [2] established a pure mathematical derivation of the structure of real spacetime from quantum set theory that is exceedingly straightforward and simple to understand. this is done by combining the von neumannconnes dimensional function of the klein-penrose modular holographic boundary of the e8e8 exceptional lie group bulk of our universe with components of the menger-urysohn dimensional theory and the topological theory of cobordism. the end result of a paper is a clear, concise mental image: quantum spacetime is just the border or surface of the quantum wave empty set, while the quantum wave itself is an empty set that represents the surface, or boundary, of the zero-set quantum particle. the fundamental distinction between quantum spacetime and a quantum wave is that the latter is a multi-fractal type of infinitely many empty sets with varying degrees of emptiness, whilst the former is a simple empty set. the notion of intuitionistic fuzzy metric spaces (ifmss), which introduced and discussed by park in [3], is useful in modeling. it is based on the notion of the intuitionistic fuzzy set (ifs) given by atanssov [4] and the concept of fuzzy metric space (fms) given by george and veeramani [5]. using the concept of ifss, alaca et al. [6] established the concept of fms as it was introduced by kramosil and michalek [7]. additionally, they developed the concept of cauchy sequences in ifms and used the notion introduced by grabiec [8] to demonstrate the extension of the banach contraction principle given in [9] to fms and edelstein [10] to ifms. the common fixed-point theorem of jungck [11] was extended to ifms by turkoglu et al. in [12]. the idea of ifms and their applications were researched by gregory et al. [13], sadati and park [14], and rodriguez-lopez and ramagurea [15]. sessa [16] developed the idea of weak commuting(w-commuting) mappings in metric spaces. broader commutativity, or compatibility, was introduced by jungck [11]. mishra et al. [17] established the notion of suitable maps in the context of fmss. vasuki [18] established the notion of an r-weak commutativity in the context of fmss. r-wcommuting mappings of type (a) in metric spaces were established by pathak et al. [19], who demonstrated that these mappings are incompatible and that r-w-commuting mappings are not always r-w-commuting of type (a). r-wcommuting mappings on ifms were defined by jesic and babacev [20]. sharma and deshpande [21] described (ds)-weak commutativity in fmss and 3 int. j. anal. appl. (2023), 21:74 established the notion of r-w-commuting mappings of type (a) in the context of fmss. recently, boyd and wong [22] and jesic and babacev [20] derived a common fixed-point theorem by utilizing nonlinear contractive conditions and assuming continuity for a pair of mappings on ifms. the method of neutrosophic metric spaces (nmss), which deals with membership, non-membership, and naturalness functions, was proposed by kirişci and simsek [23]. in the case of nmss, sowndrarajan, et al. [24] demonstrated some fixed-point results. schweizer and sklar worked on statistical metric spaces and deshpande [26] derived several fixed-point results and under some interesting conditions in the context of an ifms. in this manuscript, we will study some interesting circumstances in which continuity may be not necessary to find an existence and uniqueness of a solution. also, we establish an application to nonlinear fractional differential equations to show the validity of our main result. now, we discuss some important notions that will be helpful for readers to understand main section. definition 1.1: [25] a binary operation ∗:[0,1] → [0,1] is said to be continuous t-norm (ctn) if ∗ is fulfill the aforementioned requirements: (tn1) 𝜎 ∗ 𝜍 = 𝜍 ∗ 𝜎, (tn2) 𝜎 ∗ ( 𝜍 ∗ 𝜐) = (𝜎 ∗ 𝜍) ∗ 𝜐, (tn3) ∗ is continuous, (tn4) 𝜎 ∗ 1 = 1 for all 𝜎 ∈ [0,1], (tn5) 𝜎 ∗ 𝜍 ≤ 𝜐 ∗ 𝛥 whenever 𝜎 ≤ 𝜐 and 𝜍 ≤ 𝛥,for all 𝜎,𝜍,𝜐,𝛥 ∈ [0,1]. definition 1.2: [25] a binary operation ♢:[0,1] × [0,1] → [0,1] is said to be continuous t-conorm (ctcn) if ♢ satisfies (tn1)-(tn3), (tn5) and the following condition: (tn4)* 𝜎♢0 = 𝜎 for all 𝜎 ∈ [0,1]. definition 1.3: [6] a 5-tuple (𝛯,𝔅,𝔒,∗,♢) is called an ifms if 𝛯 is an arbitrary set ∗ is a ctn, ♢ is ctcn and 𝔅,𝔒 are fuzzy sets on 𝛯2 × [0,∞) verifies the below conditions: (ifm1) 𝔅(𝜛,𝜔,𝜏) + 𝔒(𝜛,𝜔,𝜏) ≤ 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (ifm2) 𝔅(𝜛,𝜔,0) = 0 for all 𝜛,𝜔 ∈ 𝛯, (ifm3) 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 if and only if 𝜛 = 𝜔, (ifm4) 𝔅(𝜛,𝜔,𝜏) = 𝔅(𝜔,𝜛,𝜏) for all 𝜛,𝜔 ∈ 𝛯,𝜏 > 0, (ifm5) 𝔅(𝜛,𝜔,𝜏) ∗ 𝔅(𝜔,𝜇,𝑠) ≤ 𝔅(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (ifm6) for all 𝜛,𝜔 ∈ 𝛯,𝔅(𝜛,𝜔, .): [0,∞) → (0,1] is left-continuous, 4 int. j. anal. appl. (2023), 21:74 (ifm7) lim 𝜏→∞ 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (ifm8) 𝔒(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (ifm9) 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (ifm10) 𝔒(𝜛,𝜔,𝜏) = 𝔒 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (ifm11) 𝔒(𝜛,𝜔,𝜏)♢ 𝔒(𝜔,𝜇,𝑠) ≥ 𝔒(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (ifm12) for all 𝜛,𝜔 ∈ 𝛯,𝔒(𝜛,𝜔, .) ∶ [0,∞) → [0,1] is a right-continuous, (ifm13) lim 𝜏→∞ 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. then (𝔅,𝔒) is called an intuitionistic fuzzy metric on 𝛯. example 1.1: suppose (𝛯,δ) be a metric space. define ctn by 𝜎 ∗ 𝜍 = min{𝜎,𝜍} and ctcn by 𝜎♢𝜍 = max{𝜎,𝜍} for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, 𝔅𝛥 (𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝔒𝛥 (𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) . then (𝛯,𝔅,𝔒,∗,♢) is an ifms. definition 1.4: [23] a 6-tuple (ξ,𝔅,𝔒,𝔖,∗,♢) is called an nms if 𝛯 is an arbitrary set, ∗ is a ctn, ♢ is ctcn and 𝔅,𝔒,𝔖 are neutrosophic sets on 𝛯2 × [0,∞) verifies the following conditions: (nms1) 𝔅(𝜛,𝜔,𝜏) + 𝔒(𝜛,𝜔,𝜏) + 𝔖(𝜛,𝜔,𝜏) ≤ 3 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (nms2) 𝔅(𝜛,𝜔,0) = 0 for all 𝜛,𝜔 ∈ 𝛯, (nms3) 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 if and only if 𝜛 = 𝜔, (nms4) 𝔅(𝜛,𝜔,𝜏) = 𝔅(𝜔,𝜛,𝜏) for all 𝜛,𝜔 ∈ 𝛯,𝜏 > 0, (nms5) 𝔅(𝜛,𝜔,𝜏) ∗ 𝔅(𝜔,𝜇,𝑠) ≤ 𝔅(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (nms6) for all 𝜛,𝜔 ∈ 𝛯,𝔅(𝜛,𝜔, .):[0,∞) → (0,1] is left-continuous, (nms7) lim 𝜏→∞ 𝔅(𝜛,𝜔,𝜏) = 1 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (nms8) 𝔒(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (nms9) 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (nms10) 𝔒(𝜛,𝜔,𝜏) = 𝔒 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, (nms11) 𝔒(𝜛,𝜔,𝜏)♢ 𝔒(𝜔,𝜇,𝑠) ≥ 𝔒(𝜛,𝜇,𝜏 + 𝑠)for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (nms12) for all 𝜛,𝜔 ∈ 𝛯,𝔒(𝜛,𝜔, . ) ∶ [0,∞) → [0,1] is a right-continuous, (nms13) lim 𝜏→∞ 𝔒(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. (nms14) 𝔖(𝜛,𝜔,0) = 1 for all 𝜛,𝜔 ∈ 𝛯, (nms15) 𝔖(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0 iff 𝜛 = 𝜔, (nms16) 𝔖(𝜛,𝜔,𝜏) = 𝔖 (𝜔,𝜛,𝜏)for all 𝜛,𝜔 ∈ 𝛯 and 𝜏 > 0, 5 int. j. anal. appl. (2023), 21:74 (nms17) 𝔖(𝜛,𝜔,𝜏)♢ 𝔖(𝜔,𝜇,𝑠) ≥ 𝔖(𝜛,𝜇,𝜏 + 𝑠) for all 𝜛,𝜔,𝜇 ∈ 𝛯 and 𝑠,𝜏 > 0, (nms18) for all 𝜛,𝜔 ∈ 𝛯,𝔖(𝜛,𝜔, . ) ∶ [0,∞) → [0,1] is a right-continuous, (nms19) lim 𝜏→∞ 𝔖(𝜛,𝜔,𝜏) = 0 for all 𝜛,𝜔 ∈ 𝛯. then (𝔅,𝔒,𝔖) is called a neutrosophic metric on 𝛯. definition 1.5: [23] a sequence {𝜛𝑛} in nms (𝛯,𝔅,𝔒,𝔖,∗,♢) is called a convergent to a point 𝜛 ∈ 𝛯 if and only if lim 𝑛→∞ 𝔅(𝜛𝑛,𝜛,𝜏) = 1, lim 𝑛→∞ 𝔒(𝜛𝑛,𝜛,𝜏) = 0 and lim 𝑛→∞ 𝔖(𝜛𝑛,𝜛,𝜏) = 0 for each 𝜏 > 0. definition 1.6: [23] a sequence in nms (𝛯,𝔅,𝔒,𝔖,∗,♢) is called a cauchy if for each > 0 and for each 𝜏 > 0,there exist 𝑛° ∈ ℕ such that 𝔅(𝜛𝑛,𝜛𝑚,𝜏) > 1 − ,𝔒(𝜛𝑛,𝜛𝑚,𝜏) < and 𝔖(𝜛𝑛,𝜛𝑚,𝜏) < for all 𝑛,𝑚 ≥ 𝑛𝑜, (𝛯,𝔅,𝔒,𝔖,∗,♢) is considered to be complete if and only if each cauchy sequence is convergent. 2. main results in this section, we show the existence and uniqueness of a fixed point by utilizing some interesting conditions with contractions. lemma 2.1: if (𝛯,𝔅,𝔒,𝔖 ∗,♢) is an nms and lim 𝑛→∞ 𝜛𝑛 = 𝜛, lim 𝑛→∞ 𝜔𝑛 = 𝜔 then lim 𝑛→∞ 𝔅(𝜛𝑛,𝜔𝑛,𝜏) = 𝔅(𝜛,𝜔,𝜏) , lim 𝑛→∞ 𝔒(𝜛𝑛,𝜔𝑛,𝜏) = 𝔒(𝜛,𝜔,𝜏), lim 𝑛→∞ 𝔖(𝜛𝑛,𝜔𝑛,𝜏) = 𝔖(𝜛,𝜔,𝜏). definition 2.1: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms and 𝑄 ⊆ 𝛯. the smallest closed set that contains 𝑄 is known as closure of the set 𝑄 and is indicated by �̅�. remark 2.1: an element 𝜛 ∈ �̅� if and only if there exist a sequence {𝜛𝑛} in 𝑄 such that 𝜛𝑛 → 𝜛. definition 2.2: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms. let for all 𝑟 ∈ (0,1) then a collection {𝐹𝑛}𝑛∈ℕ have neutrosophic diameter zero (nd-zero) if for each 𝜏 > 0 there exists 𝑛𝑜 ∈ ℕ such that 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝑛𝑜. theorem 2.1: an nms (𝛯,𝔅,𝔒,𝔖,∗,♢) is said to be complete if and only if each nested sequence {𝐹𝑛}𝑛∈ℕ of non-empty closed sets with (nd-zero) have non-empty intersection. remark 2.2: an element 𝜛 is unique if 𝜛 ∈∩𝑛∈ℕ 𝐹𝑛. definition 2.3: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms. a subset 𝑄 of 𝛯 is called a neutrosophic bounded if there exists 𝜏 > 0 and 𝑟 ∈ (0,1) such that 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝑄. 6 int. j. anal. appl. (2023), 21:74 definition 2.4: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms. let the mapping 𝛿𝑄(𝜏) ∶ (0,∞) → [0,1], 𝜌𝑄(𝜏):(0,∞) → [0,1] and 𝛾𝑄(𝜏):(0,∞) → [0,1] be define as 𝛿𝑄 (𝜏) = inf 𝜛,𝜔∈𝑄 sup 𝜀<𝜏 𝔅(𝜛,𝜔, ) , 𝜌𝑄(𝜏) = sup 𝜛,𝜔∈𝑄 inf 𝜀<𝜏 𝔒(𝜛,𝜔, ) , 𝛾𝑄(𝜏) = sup 𝜛,𝜔∈𝑄 inf 𝜀<𝜏 𝔖(𝜛,𝜔, ) . the constants 𝛿𝑄 = sup𝜏>0𝛿𝑄(𝜏),𝜌𝑄 = inf𝜏>0𝜌𝑄(𝜏) and 𝛾𝑄 = inf𝜏>0𝛾𝑄(𝜏), we will call neutrosophic diameter of nearness, non-nearness and naturalness of set 𝑄. remark 2.3: the inequalities 𝛿𝑄 ≥ 1 − 𝑟,𝜌𝑄 ≤ 𝑟 and 𝛾𝑄 ≤ 𝑟 are fulfilled, if 𝑄 is a neutrosophic bounded set. definition 2.5: the set 𝑄 is called neutrosophic strongly bounded set (nsbs), if 𝛿𝑄 = 1, 𝜌𝑄 = 0 and 𝛾𝑄 = 0. lemma 2.2: a mapping 𝜙:(0,∞) → (0,∞) is continuous, and non-decreasing, if it verifies 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. then lim 𝑛→∞ 𝜙𝑛 (𝜏) = 0, for all 𝜏 > 0, where 𝜙𝑛(𝜏) shows the nth iteration of 𝜙. lemma 2.3: suppose (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms. a mapping 𝜙:(0,∞) → (0,∞) be a continuous, non-decreasing, if it verifies the condition 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. then the below circumstances are satisfied: (a) if for all 𝜛,𝜔 ∈ 𝛯 it satisfies that 𝔅(𝜛,𝜔,𝜙(𝜏)) ≥ 𝔅(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. (b) if for all 𝜛,𝜔 ∈ 𝛯 it holds that 𝔒(𝜛,𝜔,𝜙(𝜏)) ≤ 𝔒(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. (c) if for all 𝜛,𝜔 ∈ 𝛯 it holds that 𝔖(𝜛,𝜔,𝜙(𝜏)) ≤ 𝔖(𝜛,𝜔,𝜏) then 𝜛 = 𝜔. definition 2.6: let 𝑄 and 𝑈 are the self-mappings of 𝛯 in nms (𝛯,𝔅,𝔒,𝔖,∗,♢). these maps are said to be compatible if for all 𝜏 > 0, lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 0, lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 0, whenever {𝜛𝑛} is a sequence in 𝛯 such that, lim 𝑛→∞ 𝑄𝜛𝑛 = lim 𝑛→∞ 𝑈𝜛𝑛 = 𝜇 for some 𝜇 ∈ 𝛯. definition 2.7: let 𝑄 and 𝑈 are the self-mappings of 𝛯 in nms (𝛯,𝔅,𝔒,𝔖,∗,♢). if there exist a positive real number 𝑅 then the mappings are called (𝐷𝑆𝑄)-w-commuting at 𝜛 ∈ 𝛯 such that 𝔅(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≥ 𝔅(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔒(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≤ 𝔒(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 7 int. j. anal. appl. (2023), 21:74 𝔖(𝑄𝑈𝜛,𝑈𝑈𝜛,𝜏) ≤ 𝔖(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ). then, 𝑄 and 𝑈 are (𝐷𝑆𝑄)-w-commuting on 𝛯 for all 𝜛 ∈ 𝛯. definition 2.8: suppose (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms and 𝑄 and 𝑈 self-mappings of 𝛯. if there exists a positive real number 𝑅, then the mappings 𝑄 and 𝑈 are called (𝐷𝑆𝑈)-w-commuting at 𝜛 ∈ 𝛯 such that 𝔅(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≥ 𝔅(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔒(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≤ 𝔒(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ), 𝔖(𝑈𝑄𝜛,𝑄𝑄𝜛,𝜏) ≤ 𝔖(𝑄𝜛,𝑈𝜛, 𝜏 𝑅 ). if these inequalities are fulfilled then 𝑄 and 𝑈 are (𝐷𝑆𝑈)-w-commuting on 𝛯 for all 𝜛 ∈ 𝛯. if the self– mappings 𝑄 and 𝑈 holds definitions 2.7 and 2.8 then 𝑄 and 𝑈 are called (𝐷𝑆)-w-commuting mappings. example 2.1: suppose 𝛯 = [1,5] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔|. for each 𝜏 ∈ (0,∞). define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔒(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 , 𝜛,𝜔 ∈ 𝛯. clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms, where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{1,𝜎 + 𝜍}. define 𝑄,𝑈:𝛯 → 𝛯 by 𝑄(𝜛) = 𝜛 if 𝜛 ∈ [0,1), 𝑄(𝜛) = 1 4 if 𝜛 ∈ [1,5), 𝑈(𝜛) = 1 1 + 𝜛 for all 𝜛 ∈ [0,5]. suppose the sequence {𝜛𝑛 = 3 + 1 𝑛 ∶ 𝑛 ≥ 1} in 𝛯. then lim 𝑛→∞ 𝑄𝜛𝑛 = 1 4 , lim 𝑛→∞ 𝑈𝜛𝑛 = 1 4 but lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = 𝜏 𝜏 + |1 4 − 4 5 | ≠ 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = |1 4 + 4 5 | 𝜏 + |1 4 − 4 5 | ≠ 0, 8 int. j. anal. appl. (2023), 21:74 lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = |1 4 + 4 5 | 𝜏 ≠ 0. which is shows that 𝑄 and 𝑈 are non-compatible. but we can obtain for 𝑅 ≥ 2 3 and 𝑄 and 𝑈 are (𝐷𝑆𝑄)-w-commuting at 𝜛 = 1. example 2.2: suppose 𝛯 = [1,10] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔| for each 𝜏 ∈ (0,∞). define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯 𝔒(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 + 𝛥(𝜛,𝜔) , 𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = 𝛥(𝜛,𝜔) 𝜏 , 𝜛,𝜔 ∈ 𝛯, clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be an nms where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{1,𝜎 + 𝜍}. define 𝑄,𝑈:𝛯 → 𝛯 by 𝑄𝜛 = 𝜛 if 1 ≤ 𝜛 ≤ 5,𝑄𝜛 = 𝜛 − 3 if 5 < 𝜛 ≤ 10, 𝑈𝜛 = 2 if 1 ≤ 𝜛 ≤ 5,𝑈𝜛 = 𝜛 − 1 2 if 5 < 𝜛 ≤ 1, consider the sequence {𝜛𝑛 = 3 + 1 𝑛 ∶ 𝑛 ≥ 1} in 𝛯. then lim 𝑛→∞ 𝑄𝑈𝜛 = 2, lim 𝑛→∞ 𝑈𝜛𝑛 = 2. lim 𝑛→∞ 𝔅(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim 𝜏 𝜏 + |2 + 1 2𝑛 − 2| = 1, lim 𝑛→∞ 𝔒(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim |2 + 1 2𝑛 − 2| 𝜏 + |2 + 1 2𝑛 −2| = 0, lim 𝑛→∞ 𝔖(𝑄𝑈𝜛𝑛,𝑈𝑄𝜛𝑛,𝜏) = lim |2 + 1 2𝑛 − 2| 𝜏 = 0. which shows that 𝑄 and 𝑈 are compatible and (𝐷𝑆𝑄)-w-commuting for 𝑅 ≥ 1, at 𝜛 = 7. theorem 2.2: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete nms. let 𝜉,𝛶:𝛯 → 𝛯. suppose 𝛶(𝛯) is a nsbs and 𝛶(𝛯) ⊆ 𝜉(𝜛) satisfying the following conditions: 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏),≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏) 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏) 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏) ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏) }. (2.1) a mapping 𝜙:(0,∞) → (0,∞), is non-decreasing that verifies 𝜙(𝜏) < 𝜏 for all 𝜏 > 0. then 𝜉 and 𝛶 have a coincidence point. if 𝜉 and 𝛶 are ds-w-commuting at coincidence point then both mapping 𝜉 and 𝛶 have a unique common fixed point. 9 int. j. anal. appl. (2023), 21:74 proof: suppose 𝜛 ∈ 𝛯 be any element of 𝛯. because 𝛶(𝛯) ⊆ 𝜉(𝛯), so there exist an element 𝜛 ∈ 𝛯 such that 𝛶(𝜛𝑜) = 𝜉(𝜛1). by induction {𝜛𝑛} be a sequence such that 𝛶(𝜛𝑛) = 𝜉(𝜛𝑛+1). assume a nested sequence of non-empty closed sets defined by 𝐹 = {𝛶𝜛𝑛,𝛶𝜛𝑛+1 ,…}̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅, 𝑛 ∈ ℕ. we shall prove that the ndzero of the family {𝐹𝑛}𝑛∈ℕ. moreover, suppose 𝑟 ∈ (0,1) and 𝜏 > 0 be randomly picked. by 𝐹𝜚 ∈ g(𝜛̅̅ ̅̅ ̅) it follows that 𝐹𝜚 is a nsbs for any 𝜚 ∈ ℕ. this implies that there exists 𝜏o > 0 such that 𝔅(𝜛,𝜔,𝜏𝑜) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏𝑜) < 𝑟 and 𝔖(𝜛,𝜔,𝜏𝑜) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝜚. (1) from lim 𝑛→∞ 𝜙𝑛(𝜏𝑜) = 0 we deduce that there exists 𝑚 ∈ ℕ such that 𝜙 𝑚(𝜏𝑜) < 𝜏. let 𝑛 = 𝑚 + 𝜚 and 𝜛,𝜔 ∈ 𝐹𝑛 be arbitrary. there exists sequence {𝛶𝜛𝑛(𝑖)},{𝛶𝜛𝑛(𝑗)} in 𝐹𝑛(𝑛(𝑖),𝑛(𝑗) ≥ 𝑛,𝑖, 𝑗 ∈ ℕ) such that lim 𝑖→∞ 𝛶𝜛𝑛(𝑖) = 𝜛 and lim 𝑗→∞ 𝛶𝜛𝑛(𝑗) = 𝜔. from (1), we have 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≥ 𝔅(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔅(𝛶𝜛𝑛(𝑖)−1,𝜏) 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≤ 𝔒(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔒(𝛶𝜛𝑛(𝑖)−1,𝛶𝜛𝑛(𝑗)−1,𝜏), and 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙(𝜏)) ≤ 𝔖(𝜉𝜛𝑛(𝑖),𝜉𝜛𝑛(𝑗),𝜏) = 𝔖(𝛶𝜛𝑛(𝑖)−1,𝛶𝜛𝑛(𝑗)−1,𝜏). thus, by induction, we get 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏)) ≥ 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏), 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏) ≤ 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏), and 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏) ≤ 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏). since 𝜙𝑚(𝜏𝑜) < 𝜏 and because 𝔅(𝜛,𝜔, .) is non-decreasing and 𝔒(𝜛,𝜔, .) is non-increasing function, 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≥ 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜) ) ≥ 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜), 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≤ 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜)) ≤ 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜), 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) ≤ 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜙 𝑚(𝜏𝑜)) ≤ 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜)} . (2) as {𝛶𝜛𝑛(𝑖)−𝑚} and {𝛶𝜛𝑛(𝑗)−𝑚} are sequence in 𝐹𝜚 from (1), it follows that 𝔅(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) > 1 − 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ, 𝔒(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ, 𝔖(𝛶𝜛𝑛(𝑖)−𝑚,𝛶𝜛𝑛(𝑗)−𝑚,𝜏𝑜) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ }. (3) 10 int. j. anal. appl. (2023), 21:74 by implying (1) to (3), we examine 𝔅(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) > 1 − 𝑟, 𝔒(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) < 𝑟, 𝔖(𝛶𝜛𝑛(𝑖),𝛶𝜛𝑛(𝑗),𝜏) < 𝑟 ∀ 𝑖, 𝑗 ∈ ℕ. by applying the limits as 𝑖, 𝑗 → ∞ and the lemma 2.1. we conclude 𝔅(𝜛,𝜔,𝜏) > 1 − 𝑟, 𝔒(𝜛,𝜔,𝜏) < 𝑟 and 𝔖(𝜛,𝜔,𝜏) < 𝑟 for all 𝜛,𝜔 ∈ 𝐹𝑛. then, {𝐹𝑛}𝑛∈ℕ has ndzero. by using the theorem 2.1 we examine that {𝐹𝑛}𝑛∈ℕ has non-empty intersection, which consists of exactly one point 𝜇. since {𝐹𝑛}𝑛∈ℕ has ndzero and 𝜇 ∈ 𝐹𝑛 for all 𝑛 ∈ ℕ then for each 𝑟 ∈ (0,1) and each 𝜏 > 0 there exists 𝑛𝑜 ∈ ℕ such that for all 𝑛 ≥ 𝑛𝑜 𝔅(𝛶𝜛𝑛,𝜇,𝜏) > 1 − 𝑟, 𝔒(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟 and 𝔖(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟. for all 𝑟 ∈ (0,1), it’s obviously satisfied lim 𝑛→∞ 𝔅(𝛶𝜛𝑛,𝜇,𝜏) > 1 − 𝑟 and lim 𝑛→∞ 𝔒(𝛶𝜛𝑛,𝜇,𝜏) < 𝑟. by using the limit that 𝑟 → 0, we get lim 𝑛→∞ 𝔅(𝛶𝜛𝑛,𝜇,𝜏) = 1, lim 𝑛→∞ 𝔒(𝛶𝜛𝑛,𝜇,𝜏) = 0 and lim 𝑛→∞ 𝔖(𝛶𝜛𝑛,𝜇,𝜏) = 0. that is, lim 𝑛→∞ 𝛶𝜛𝑛 = 𝜇. sequence {𝜉𝜛𝑛} follows the condition that lim 𝑛→∞ 𝜉𝜛𝑛 = 𝜇. since 𝛶(𝛯) ⊆ 𝜉(𝛯), there exists 𝑢 ∈ 𝛯 such that 𝜇 = 𝜉(𝑢). then utilizing (1), we get 𝔅(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≥ 𝔅(𝜉(𝑢),𝜉(𝜛𝑛),𝜏), 𝔒(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≤ 𝔒(𝜉(𝑢),𝜉(𝜛𝑛),𝜏), 𝔖(𝛶(𝑢),𝛶(𝜛𝑛),𝜙(𝜏)) ≤ 𝔖(𝜉(𝑢),𝜉(𝜛𝑛),𝜏). letting 𝑛 → ∞, we get 𝔅(𝛶(𝑢),𝜇,𝜙(𝜏)) ≥ 𝔅(𝜇,𝜇,𝜏) = 1, 𝔒(𝛶(𝑢),𝜇,𝜙(𝜏)) ≤ 𝔒(𝜇,𝜇,𝜏) = 0, 𝔖(𝛶(𝑢),𝜇,𝜙(𝜏)) ≤ 𝔖(𝜇,𝜇,𝜏) = 0. since 𝔅(𝛶(𝑢),𝜇,𝜏 ) ≥ 𝔅(𝛶(𝑢),𝜇,𝜙(𝜏)), 𝔒(𝛶(𝑢),𝜇,𝜏) ≤ 𝔒(𝛶(𝑢),𝜇,𝜙(𝜏)), 𝔖(𝛶(𝑢),𝜇,𝜏) ≤ 𝔖(𝛶(𝑢),𝜇,𝜙(𝜏)). we get, 𝔅(𝛶(𝑢),𝜇,𝜏) = 1, 𝔒(𝛶(𝑢),𝜇,𝜏) = 0 and 𝔖(𝛶(𝑢),𝜇,𝜏) = 0 for all 𝜏 > 0. thus 𝛶(𝑢) = 𝜇. therefore 𝜉(𝑢) = 𝛶(𝑢) = 𝜇 which is shows that 𝑢 is coincidence point of 𝜉 and 𝛶. since 𝜉 and 𝛶 are (𝐷𝑆)-w-commuting at coincidence point, so 𝜉 and 𝛶 are (𝐷𝑆𝜉)-w-commuting at coincidence points, such that 11 int. j. anal. appl. (2023), 21:74 𝔅(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≥ 𝜉((𝑢),𝛶(𝑢), 𝜏 𝑅 ), 𝔒(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≤ 𝔒(𝜉(𝑢),𝛶(𝑢), 𝜏 𝑅 ) , 𝔖(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) ≤ 𝔖(𝜉(𝑢),𝛶(𝑢), 𝜏 𝑅 ). thus, 𝔅(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 1, 𝔒(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 0 and 𝔖(𝜉𝛶𝑢,𝛶𝛶𝑢,𝜏) = 0. so, 𝜉𝛶𝑢 = 𝛶𝛶𝑢 that is 𝜉𝜇 = 𝛶𝜇. again using (1), we have 𝔅(𝛶(𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ), 𝔒((𝛶𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ), 𝔖((𝛶𝜛𝑛),𝛶(𝜇),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜛𝑛),𝜉(𝜇),𝜏 ). letting 𝑛 → ∞, we have 𝔅(𝜇,𝛶(𝜇),𝜙(𝜏)) ≥ 𝔅(𝜇,𝛶(𝜏),𝜏 ), 𝔒(𝜇,𝛶(𝜏),𝜙(𝜏)) ≤ 𝔒(𝜇,𝛶(𝜇),𝜏 ), 𝔖(𝜇,𝛶(𝜏),𝜙(𝜏)) ≤ 𝔖(𝜇,𝛶(𝜇),𝜏 ) for all 𝜏 > 0. utilizing the lemma 2.3, we have 𝛶(𝜇) = 𝜇. thus 𝜉(𝜇) = 𝛶(𝜇) = 𝜇. now, we show the uniqueness of 𝜇. suppose 𝜇 ≠ 𝑤 be another fixed point, then utilizing (2.1), we get 𝔅(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜇),𝜉(𝑤),𝜏), 𝔒(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜇),𝜉(𝑤),𝜏 ), 𝔖(𝛶(𝜇),𝛶(𝑤),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜇),𝜉(𝑤),𝜏 ) ∀ 𝜏 > 0. that is 𝔅(𝜇,𝑢,𝜙(𝜏)) ≥ 𝔅(𝜇,𝑢,𝜏), 𝔒(𝜇,𝑢,𝜙(𝜏)) ≤ 𝔒(𝜇,𝑢,𝜏), 𝔖(𝜇,𝑢,𝜙(𝜏)) ≤ 𝔖(𝜇,𝑢,𝜏) for all 𝜏 > 0. by using the lemma 2.3, it follows that 𝜇 = 𝑤. example 2.3: consider 𝛯 = [0,2] with the usual metric space 𝛥(𝜛,𝜔) = |𝜛 − 𝜔| and for each 𝜏 ∈ [0,1], define 𝔅(𝜛,𝜔,𝜏) = 𝜏 𝜏 + |𝜛 − 𝜔| ,𝜛,𝜔 ∈ 𝛯, 𝔒(𝜛,𝜔,𝜏) = |𝜛 − 𝜔| 𝜏 + |𝜛 − 𝜔| ,𝜛,𝜔 ∈ 𝛯, 𝔖(𝜛,𝜔,𝜏) = |𝜛 − 𝜔| 𝜏 ,𝜛,𝜔 ∈ 𝛯. clearly, (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete nms where 𝜎 ∗ 𝜍 = 𝜎𝜍 and 𝜎♢𝜍 = min{𝜎,𝜍}. 12 int. j. anal. appl. (2023), 21:74 define 𝜉,𝛶:𝛯 → 𝛯 by 𝜉(𝜛) = { 1 if 𝜛 = 1 𝜛 2 + 1 if 𝜛 ≠ 1, 𝛶(𝜛) = { 1 if 𝜛 = 1 1 1 + 𝜛 if 𝜛 ≠ 1, 𝜙(𝜏) = 𝜏 2 ,𝜏 > 0. then 𝛶(𝛯) ⊆ 𝜉(𝛯). we see that {𝜛𝑛} is decreasing sequence, so lim 𝑛→∞ 𝜉(𝜛𝑛) = lim 𝑛→∞ 𝛶(𝜛𝑛). thus, mappings are not compatible. but both are (𝐷𝑆)-w-commuting at coincidence point. we will demonstrate that axiom (1) is also met. clearly, ∀ 𝜛,𝜔 ∈ 𝛯. we get 1 (1 + 𝜛)(1 + 𝜔) ≤ 1. we get 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 𝜏 𝜏 + 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) ≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏), as easy to see in figure 1. figure 1 shows the graphical behavior of the contraction 𝔅(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≥ 𝔅(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. now for second 𝔒, we have 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) 𝜏 + 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏), 13 int. j. anal. appl. (2023), 21:74 as easy to see in figure 2. figure 2 shows the graphical behavior of the contraction 𝔒(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≤ 𝔒(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. now for third function 𝔖, we have 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) = 2( |𝜛−𝜔| (1+𝜛)(1+𝜔) ) 𝜏 ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏), as easy to see in figure 3. figure 3 shows the graphical behavior of the contraction 𝔖(𝛶(𝜛),𝛶(𝜔),𝜙(𝜏)) ≤ 𝔖(𝜉(𝜛),𝜉(𝜔),𝜏) when 𝜏 = 1. 14 int. j. anal. appl. (2023), 21:74 all the conditions of theorem 2.2 are fulfill. so, both mappings 𝜉(𝜛) and 𝛶(𝜛) have a unique common fixed point, which is 𝜛 = 1. theorem 2.3: let (𝛯,𝔅,𝔒,𝔖,∗,♢) be a complete nms. let 𝛶:𝛯 → 𝛯. suppose 𝛶(𝛯) is nsbs verifying the axiom (2) with 𝔅(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≥ 𝔅(𝜛,𝜔,𝜏), 𝔒(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≤ 𝔒(𝜛,𝜔,𝜏), 𝔖(𝛶(𝜛),𝛶(𝜔),𝜚𝜏) ≤ 𝔖(𝜛,𝜔,𝜏) for some 𝜚 ∈ (0,1). then 𝛶 has a unique fixed point. remark 2.4: the hypothesis of theorem 2.2 that is 𝛶(𝛯) is a nsbs, can be exchanged with the following one: there exists an element 𝜛𝑜 ∈ 𝛯 such that the orbit of an element 𝜛𝑜 described by 𝜙(𝜛𝑜,𝛶) = {𝜛𝑜,𝛶𝜛𝑜,𝛶 2𝜛𝑜,…} is a nsbs. this tells us the sequence used in the start of the proof of the theorem 2.2 for 𝜉 = 1 is a sequence of picard iterates defined by 𝜛𝑛+1 = 𝛶𝜛𝑛. 3. application now, we aim to apply theorem 2.3 to obtain the existence of solution to a nonlinear fractional differential equation (nfde) 𝐷0+ 𝑝 ℎ(𝜛) = 𝜉(𝜛,ℎ(𝜛)), 0 < 𝜛 < 1 (4) with the boundary conditions ℎ(0) + ℎ′(0) = 0,ℎ(1) + ℎ′(1) = 0, where 1 < 𝑝 ≤ 2 is a number, 𝐷0+ 𝑝 is the caputo fractional derivative and 𝜉 ∶ [0,1] × [0,∞) → [0,∞) is a continuous function. let 𝛯 = 𝐶([0,1],(0,1]) denote the space of all continuous functions defined on [0,1]. let 𝜎 ∗ 𝜍 = min{𝜎,𝜍} for all 𝜎,𝜍 ∈ [0,1] and define an nms as follows: 𝔅(ℎ(t),𝛿(𝜛),𝜏) = { 1 if ℎ(𝜏) = 𝛿(𝜛), 𝛼𝜏 𝛼𝜏 + 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} ,otherwise, 𝔒(ℎ(t),𝛿(𝜛),𝜏) = { 0 if ℎ(𝜏) = 𝛿(𝜛), 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} 𝛼𝜏 + 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} ,otherwise, and 𝔖(ℎ(t),𝛿(𝜛),𝜏) = { 0 if ℎ(𝜏) = 𝛿(𝜛), 𝛾 max{sup𝜛𝜖[0,1] ℎ(𝜛),sup𝜛𝜖[0,1] 𝛿(𝜛)} 𝛼𝜏 ,otherwise. observe that ℎ ∈ 𝛯 solves (4) whenever ℎ ∈ 𝛯 solves the below integral equation: 15 int. j. anal. appl. (2023), 21:74 ℎ(𝜛) = 1 г(𝑝) ∫ (1 − 𝑠 1 0 )𝑝−1(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 г(𝑝 − 1) ∫ (1 − 𝑠 1 0 )𝑝−2(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1𝜉(𝑠,ℎ(𝑠))𝛥𝑠. 𝜛 0 the graphical behavior of ℎ(𝜛) for different values of 𝑝 is shown in figure 4. figure 4 shows the graphical behavior of ℎ(𝜛) for 𝑝 = 1.1, 𝑝 = 1.4, 𝑝 = 1.7 and 𝑝 = 2. theorem 4.2 the integral operator 𝛶:𝛯 → 𝛯 is given by 𝛶ℎ(𝜛) = 1 г(𝑝) ∫ (1 − 𝑠 1 0 )𝑝−1(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 г(𝑝 − 1) ∫ (1 − 𝑠 1 0 )𝑝−2(1 − 𝜛)𝜉(𝑠,ℎ(𝑠))𝛥𝑠 + 1 г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1𝜉(𝑠,ℎ(𝑠))𝛥𝑠, 𝜛 0 where 𝜉:[0,1] × [0,∞) → [0,∞) fulfilling the following criteria: max{sup𝑠∈[0,1] 𝜉 (𝑠,ℎ(𝑠)), sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))} ≤ 1 4 max{ sup𝑠∈[0,1] ℎ(𝑠), sup𝑠∈[0,1] 𝛿(𝑠)} , for all ℎ,𝛿 ∈ 𝛯. also, suppose that sup𝜛∈(0,1) 1 4 [ 1 − 𝜛 г(𝑝 + 1) + 1 − 𝜛 г(𝑝) + 𝜛𝑝 г(𝑝 + 1) ] ≤ 𝜚 < 1. 16 int. j. anal. appl. (2023), 21:74 then nfde has a unique solution in 𝛯. proof: let max{𝛶ℎ(𝜛),𝛶𝛿(𝜛)} = 1 − 𝜛 г(𝑝) ∫ (1 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))}𝛥𝑠 1 0 + 1 − 𝜛 г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2 1 0 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))}𝛥𝑠 + 1 г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]𝜉(𝑠,ℎ(𝑠)),sup𝑠∈[0,1] 𝜉(𝑠,𝛿(𝑠))} 𝜛 0 ≤ 1 − 𝜛 г(𝑝) ∫ (1 − 𝑠)𝑝−1 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} 4 𝛥𝑠 1 0 + 1 − 𝜛 г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2 max{sup𝑠∈[0,1]ℎ(𝑠),sup𝑠∈[0,1]𝛿(𝑠)} 4 𝛥𝑠 1 0 + 1 г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 max{sup𝑠∈[0,1]ℎ(𝑠),sup𝑠∈[0,1]𝛿(𝑠)} 4 𝛥𝑠 𝜛 0 ≤ 1 4 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} ( 1 − 𝜛 г(𝑝) ∫ (1 − 𝑠)𝑝−1 1 0 𝛥𝑠 + 1 − 𝜛 г(𝑝 − 1) ∫ (1 − 𝑠)𝑝−2𝛥𝑠 1 0 + 1 г(𝑝) ∫ (𝜛 − 𝑠)𝑝−1 𝜛 0 𝛥𝑠) ≤ 1 4 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)} sup𝜛∈[0,1] [ 1 − 𝜛 г(𝑝 + 1) + 1 − 𝜛 г(𝑝) + 𝜛𝑝 г(𝑝 + 1) ] = 𝜚 max{sup𝑠∈[0,1] ℎ(𝑠),sup𝑠∈[0,1] 𝛿(𝑠)}, where, 𝜚 = sup𝜛𝜖(0,1) 1 4 [ 1 − 𝜛 г(𝑝 + 1) + 1 − 𝜛 г(𝑝) + 𝜛𝑝 г(𝑝 + 1) ]. therefore, the above equation max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≤ 𝜚max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝛼𝜏 + 𝛾 𝜚 max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≤ 𝛼𝜏 + 𝛾max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝛼(𝜚𝜏) 𝛼(𝜚𝜏) + 𝛾max{sup𝜛∈[0,1] 𝛶ℎ(𝜛),sup𝜛∈[0,1] 𝛶𝛿(𝜛)} ≥ 𝛼𝜏 𝛼𝜏 + 𝛾max{sup𝜛∈[0,1] ℎ(𝜛),sup𝜛∈[0,1] 𝛿(𝜛)} ⇒ 𝔅(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≥ 𝔅(ℎ,𝛿,𝜏), 17 int. j. anal. appl. (2023), 21:74 and on the same technique, it is easy to deduce that 𝔒(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≤ 𝔒(ℎ,𝛿,𝜏), 𝔖(𝛶ℎ,𝛶𝛿,𝜚𝜏) ≤ 𝔖(ℎ,𝛿,𝜏), for some 𝛼,𝛾 > 0. observe that the conditions of the theorem 2.3 are fulfilled. resultantly, 𝛶 has a fixed-point fixed point; accordingly, the specified nfde has a solution. 4. conclusion in this manuscript, we established numerous interesting conditions in the context of nms. we provided numerous non-trivial examples and their graphical views via computational techniques. also, we derived several coincident points and common fixed-point results for contraction mappings in the context of nms, as well, we presented a graphical view of defined contractions. at the end, we provide a novel application to support the validity of our main result. funding: basque government, grant it1155-22 contributions: all the authors contributed equally. all authors read and approved the manuscript. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.s. el naschie, wild topology, hyperbolic geometry and fusion algebra of high energy particle physics, chaos, solitons fractals. 13 (2002), 1935-1945. https://doi.org/10.1016/s0960-0779(01)00242-9. [2] m.s. el naschie, from experimental quantum optics to quantum gravity via a fuzzy kähler manifold, chaos solitons fractals. 25 (2005), 969-977. https://doi.org/10.1016/j.chaos.2005.02.028. [3] j.h. park, intuitionistic fuzzy metric spaces, chaos solitons fractals. 22 (2004), 1039–1046. https://doi.org/10.1016/j.chaos.2004.02.051. [4] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/s01650114(86)80034-3. [5] a. george, p. veeramani, on some results in fuzzy metric spaces, fuzzy sets syst. 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7. [6] c. alaca, d. turkoglu, c. yildiz, fixed points in intuitionistic fuzzy metric spaces, chaos solitons fractals. 29 (2006), 1073–1078. https://doi.org/10.1016/j.chaos.2005.08.066. [7] ivan kramosil; jiří michálek, fuzzy metrics and statistical metric spaces, kybernetika, 11 (1975), 336-344. http://dml.cz/dmlcz/125556. [8] m. grabiec, fixed points in fuzzy metric spaces, fuzzy sets syst. 27 (1988), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4. https://doi.org/10.1016/s0960-0779(01)00242-9 https://doi.org/10.1016/j.chaos.2005.02.028 https://doi.org/10.1016/j.chaos.2004.02.051 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/0165-0114(94)90162-7 https://doi.org/10.1016/j.chaos.2005.08.066 http://dml.cz/dmlcz/125556 https://doi.org/10.1016/0165-0114(88)90064-4 18 int. j. anal. appl. (2023), 21:74 [9] s. banach, theoriles operations. linearies manograie mathematyezne, warsaw, poland, 1932. [10] m. edelstein, on fixed and periodic points under contractive mappings, j. lond. math. soc. s1-37 (1962), 74–79. https://doi.org/10.1112/jlms/s1-37.1.74. [11] g. jungck, compatible mappings and common fixed points, int. j. math. math. sci. 9 (1986), 771–779. https://doi.org/10.1155/s0161171286000935. [12] d. turkoglu, c. alaca, y.j. cho, c. yildiz, common fixed point theorems in intuitionistic fuzzy metric spaces, j. appl. math. comput. 22 (2006), 411–424. https://doi.org/10.1007/bf02896489. [13] v. gregori, s. romaguera, p. veeramani, a note on intuitionistic fuzzy metric spaces, chaos solitons fractals. 28 (2006), 902–905. https://doi.org/10.1016/j.chaos.2005.08.113. [14] r. sadati, j.h. park, on the intuitionistic topological spaces, chaos solutions fractals, 27 (2006), 331-344. [15] j. rodrı ́guez-lópez, s. romaguera, the hausdorff fuzzy metric on compact sets, fuzzy sets syst. 147 (2004), 273–283. https://doi.org/10.1016/j.fss.2003.09.007. [16] s. sessa, on a weak commutativity condition of mappings in fixed point considerations, publ. inst. math.32 (1982), 149–153. [17] s.n. mishra, n. sharma, s.l. singh, common fixed points of maps on fuzzy metric spaces, int. j. math. math. sci. 17 (1994), 253–258. [18] n. simsek, m. kirişci, fixed point theorems in neutrosophic metric spaces, sigma j. eng. nat. sci. 10 (2019), 221230. [19] h.k. pathak, y.j. cho, s.m. kang, remarks on r-weakly commuting mappings and common fixed point theorems, bull. korean math. soc. 34 (1997), 247–257. [20] s.n. ješić, n.a. babačev, common fixed point theorems in intuitionistic fuzzy metric spaces and l-fuzzy metric spaces with nonlinear contractive condition, chaos solitons fractals. 37 (2008), 675–687. https://doi.org/10.1016/j.chaos.2006.09.048. [21] s. sharma, b. deshpande, common fixed point theorems for non-compatible mappings and meir–keeler type contractive condition in fuzzy metric spaces, int. rev. fuzzy math. 1(2006), 147–159. [22] d.w. boyd, j.s.w. wong, on nonlinear contractions, proc. amer. math. soc. 20 (1969), 458–464. [23] m. kirişci, n. şimşek, neutrosophic metric spaces, math. sci. 14 (2020), 241–248. https://doi.org/10.1007/s40096020-00335-8. [24] s. sowndrarajan, m. jeyarama, f. smarandache, fixed point results for contraction theorems in neutrosophic metric spaces, neutrosophic sets syst. 36 (2020), 23. [25] b. schweizer, a. sklar, statistical metric spaces, pac. j. math. 10 (1960), 314–334. [26] b. deshpande, fixed point and (ds)-weak commutativity condition in intuitionistic fuzzy metric spaces, chaos solitons fractals. 42 (2009), 2722–2728. https://doi.org/10.1016/j.chaos.2009.03.178. https://doi.org/10.1112/jlms/s1-37.1.74 https://doi.org/10.1155/s0161171286000935 https://doi.org/10.1007/bf02896489 https://doi.org/10.1016/j.chaos.2005.08.113 https://doi.org/10.1016/j.fss.2003.09.007 https://doi.org/10.1016/j.chaos.2006.09.048 https://doi.org/10.1007/s40096-020-00335-8 https://doi.org/10.1007/s40096-020-00335-8 https://doi.org/10.1016/j.chaos.2009.03.178 international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 23-30 http://www.etamaths.com steffensen’s integral inequality for conformable fractional integrals mehmet zeki sarikaya, hatice yaldiz∗ and hüseyin budak abstract. the aim of this paper is to establish some steffensen’s type inequalities for conformable fractional integral. the results presented here would provide generalizations of those given in earlier works. 1. introduction the most basic inequality which deals with the comparison between integrals over a whole interval [a,b] and integrals over a subset of [a,b] is the following inequality, which was estab-lished by j.f. steffensen in 1919,(see [10]). theorem 1.1 (steffensen’s inequality). let a and b be real numbers such that a < b, f and g be integrable functions from [a,b] into r such that f is nonincreasing and for every x ∈ [a,b], 0 ≤ g (x) ≤ 1. then b∫ b−λ f (x) dx ≤ b∫ a f (x) g (x) dx ≤ a+λ∫ a f (x) dx, (1.1) where λ = b∫ a g (x) dx. a comprehensive survey on this inequality can be found in [9]. steffensen’s inequality plays an important role in the study of integral inequalities. for more results concerning new proofs, generalizations, weaker hypothesis or different forms were emerging one after another see [6]– [11], and the references therein. 2. definitions and properties of conformable fractional derivative and integral the following definitions and theorems with respect to conformable fractional derivative and integral were referred in (see, [1][5]). definition 2.1 (conformable fractional derivative). given a function f : [0,∞) → r. then the “conformable fractional derivative” of f of order α is defined by dα (f) (t) = lim �→0 f ( t + �t1−α ) −f (t) � (2.1) for all t > 0, α ∈ (0, 1) . if f is α−differentiable in some (0,a) , α > 0, lim t→0+ f(α) (t) exist, then define f(α) (0) = lim t→0+ f(α) (t) . (2.2) we can write f(α) (t) for dα (f) (t) to denote the conformable fractional derivatives of f of order α. in addition, if the conformable fractional derivative of f of order α exists, then we simply say f is α−differentiable. theorem 2.1. let α ∈ (0, 1] and f,g be α−differentiable at a point t > 0. then received 1st may, 2017; accepted 9th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 26d15. key words and phrases. steffensen inequality; conformable fractional integral. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 23 24 sarikaya, yaldiz and budak i. dα (af + bg) = adα (f) + bdα (g) , for all a,b ∈ r, ii. dα (λ) = 0, for all constant functions f (t) = λ, iii. dα (fg) = fdα (g) + gdα (f) , iv. dα ( f g ) = fdα (g) −gdα (f) g2 . if f is differentiable, then dα (f) (t) = t 1−αdf dt (t) . (2.3) definition 2.2 (conformable fractional integral). let α ∈ (0, 1] and 0 ≤ a < b. a function f : [a,b] → r is α-fractional integrable on [a,b] if the integral∫ b a f (x) dαx := ∫ b a f (x) xα−1dx (2.4) exists and is finite. all α-fractional integrable on [a,b] is indicated by l1α ([a,b]) . remark 2.1. iaα (f) (t) = i a 1 ( tα−1f ) = ∫ t a f (x) x1−α dx, where the integral is the usual riemann improper integral, and α ∈ (0, 1]. theorem 2.2. let f : (a,b) → r be differentiable and 0 < α ≤ 1. then, for all t > a we have iaαd a αf (t) = f (t) −f (a) . (2.5) theorem 2.3 (integration by parts). let f,g : [a,b] → r be two functions such that fg is differentiable. then ∫ b a f (x) daα (g) (x) dαx = fg| b a − ∫ b a g (x) daα (f) (x) dαx. (2.6) theorem 2.4. assume that f : [a,∞) → r such that f(n)(t) is continuous and α ∈ (n,n + 1]. then, for all t > a we have daαf (t) i a α = f (t) . theorem 2.5 (fractional steffensen’s inequality). ( [4]) let α ∈ (0, 1] and a and b be real numbers such that 0 ≤ a < b. let f : [a,b] → [0,∞) and g : [a,b] → [0, 1] be α-fractional integrable functions on [a,b] with f is decreasing. then b∫ b−` f (x) dαx ≤ b∫ a f (x) g (x) dαx ≤ a+`∫ a f (x) dαx, (2.7) where ` := α(b−a) bα−aα b∫ a g (x) dαx. the aim of this paper is to establish some steffensen’s type inequalities for conformable fractional integral. the results presented here would provide generalizations of those given in earlier works. steffensen’s integral inequality 25 3. steffensen’s type inequalities for conformable fractional integrals lemma 3.1. let α ∈ (0, 1] and a,b ∈ r with 0 ≤ a < b, g and h be α−fractional integrable function on [a,b], 0 ≤ g (t) ≤ h (t) all t ∈ [a,b], and define l := (b−a) b∫ a h (t) dα (t) b∫ a g (t) dα (t) ∈ [0,b−a] . (3.1) then, we have b∫ b−l h (t) dα (t) ≤ b∫ a g (t) dα (t) ≤ a+l∫ a h (t) dα (t) . (3.2) proof. since 0 ≤ g (t) ≤ h (t) for all t ∈ [a,b], l given in (3.1) satisfies, 0 ≤ l = (b−a) b∫ a h (t) dα (t) b∫ a g (t) dα (t) ≤ (b−a) b∫ a h (t) dα (t) b∫ a h (t) dα (t) = b−a, and by average values, we get the following inequalities 1 l b∫ b−l h (t) dα (t) ≤ 1 b−a b∫ a h (t) dα (t) ≤ 1 l a+l∫ a h (t) dα (t) and then b∫ b−l h (t) dα (t) ≤ l b−a b∫ a h (t) dα (t) ≤ a+l∫ a h (t) dα (t) . by (3.1), we obtain the following inequalities b∫ b−l h (t) dα (t) ≤ b∫ a g (t) dα (t) ≤ a+l∫ a h (t) dα (t) . this completes the proof. � remark 3.1. if we take h(t) = 1 in lemma 3.1, then lemma 3.1 reduces to the lemma 2.1 in [4]. theorem 3.1. let α ∈ (0, 1] and a,b ∈ r with 0 ≤ a < b, f,g,h : [a,b] → [0,∞) be α−fractional integrable function on [a,b], 0 ≤ g (t) ≤ h (t) all t ∈ [a,b], with f decreasing function. then b∫ b−l h (t) f (t) dα (t) ≤ b∫ a f (t) g (t) dα (t) ≤ a+l∫ a h (t) f (t) dα (t) (3.3) where l is given by (3.1). proof. we will prove only the case in (3.3) for right inequality; the proof for the left inequality is similar, and relies on (3.2). by definition of l and the conditions on g,h the inequality (3.2) holds. 26 sarikaya, yaldiz and budak since f is decreasing function, we obtain that a+l∫ a h (t) f (t) dα (t) − b∫ a f (t) g (t) dα (t) = a+l∫ a f (t) [h (t) −g (t)] dα (t) − b∫ a+l f (t) g (t) dα (t) ≥ f (a + l) a+l∫ a [h (t) −g (t)] dα (t) − b∫ a+l f (t) g (t) dα (t) = f (a + l)   a+l∫ a h (t) dα (t) − a+l∫ a g (t) dα (t)  − b∫ a+l f (t) g (t) dα (t) ≥ f (a + l)   b∫ a g (t) dα (t) − a+l∫ a g (t) dα (t)  − b∫ a+l f (t) g (t) dα (t) = f (a + l) b∫ a+l g (t) dα (t) − b∫ a+l f (t) g (t) dα (t) = b∫ a+l [f (a + l) −f (t)] g (t) dα (t) ≥ 0. this completes the proof. � remark 3.2. if we take h(t) = 1 in theorem 3.1, then the inequality (3.3) reduces to the inequality (2.7). remark 3.3. if we take h(t) = 1 and α = 1 in theorem 3.1, then the inequality (3.3) reduces to the inequality (1.1). in order to obtain our other results, we need the following lemma. lemma 3.2. under the assumptions of lemma 3.1 and l is defined by a+l∫ a h (t) dα (t) = b∫ a g (t) dα (t) = b∫ b−l h (t) dα (t) . (3.4) then, we have b∫ a f (t) g (t) dα (t) = a+l∫ a (f (t) h (t) − [f (t) −f (a + l)] [h (t) −g (t)]) dα (t) (3.5) + b∫ a+l [f (t) −f (a + l)] g (t) dα (t) , steffensen’s integral inequality 27 and b∫ a f (t) g (t) dα (t) = b∫ b−l (f (t) h (t) − [f (t) −f (b− l)] [h (t) −g (t)]) dα (t) (3.6) + b−l∫ a [f (t) −f (b− l)] g (t) dα (t) . proof. we know that a ≤ a + l ≤ b, a ≤ b − l ≤ b. firstly, we calculate identity (3.5). by direct computation, we have a+l∫ a (f (t) h (t) − [f (t) −f (a + l)] [h (t) −g (t)]) dα (t) − b∫ a f (t) g (t) dα (t) = a+l∫ a (f (t) h (t) −f(t)g(t) − [f (t) −f (a + l)] [h (t) −g (t)]) dα (t) + a+l∫ a f (t) g (t) dα (t) − b∫ a f (t) g (t) dα (t) = a+l∫ a f (a + l) [h (t) −g (t)] dα (t) − b∫ a+l f (t) g (t) dα (t) = f (a + l)   a+l∫ a h (t) dα (t) − a+l∫ a g (t) dα (t)  − b∫ a+l f (t) g (t) dα (t) = f (a + l)   b∫ a g (t) dα (t) − a+l∫ a g (t) dα (t)  − b∫ a+l f (t) g (t) dα (t) = f (a + l) b∫ a+l g (t) dα (t) − b∫ a+l f (t) g (t) dα (t) . which completes the proof. similarly, the second part is obtained. the proof of the lemma is completed. � 28 sarikaya, yaldiz and budak theorem 3.2. under the assumptions of theorem 3.1. then b∫ b−l f (t) h (t) dα (t) ≤ b∫ b−l (f (t) h (t) − [f (t) −f (b− l)] [h (t) −g (t)]) dα (t) ≤ b∫ a f (t) g (t) dα (t) ≤ a+l∫ a (f (t) h (t) − [f (t) −f (a + l)] [h (t) −g (t)]) dα (t) ≤ a+l∫ a f (t) h (t) dα (t) where l is given by (3.4). proof. from 0 ≤ g (t) ≤ h (t) and f is decreasing function on [a,b], then we have b−l∫ a [f (t) −f (b− l)] g (t) dα (t) ≥ 0 (3.7) and b∫ b−l [f (b− l) −f (t)] [h (t) −g (t)] dα (t) ≥ 0. (3.8) using the identity (3.6) together with the inequalities (3.7) and (3.8), we obtain b∫ b−l f (t) h (t) dα (t) ≤ b∫ b−l (f (t) h (t) − [f (t) −f (b− l)] [h (t) −g (t)]) dα (t) ≤ b∫ a f (t) g (t) dα (t) . in the same way as above, we can prove that b∫ a f (t) g (t) dα (t) ≤ a+l∫ a (f (t) h (t) − [f (t) −f (a + l)] [h (t) −g (t)]) dα (t) ≤ a+l∫ a f (t) h (t) dα (t) . this completes the proof. � steffensen’s integral inequality 29 theorem 3.3. let α ∈ (0, 1] and g ∈ l1 ([0, 1]) such that 0 ≤ g(x) ≤ 1 for all x ∈ [0, 1]. if ϕ : [0, 1] → [0,∞) is a convex, α-fractional differentiable function with ϕ(0) = 0, then ϕ  α 1∫ 0 g (x) dαx   ≤ 1∫ 0 g (x) dαϕ (x) dαx. (3.9) proof. the function ϕ is convex and α-fractional differentiable on [0, 1] and dαϕ is nondecreasing for all x ∈ [0, 1]. then −dαϕ is decreasing and we take f(x) = −dαϕ, a = 0 and b = 1 in the fractional steffensen’s inequality (2.7) it follows that `∫ 0 dαϕ(x)dαx ≤ 1∫ 0 g (x) dαϕ(x)dαx ≤ 1∫ 1−` dαϕ(x)dαx. by simple computation, we have ϕ(`) −ϕ(0) ≤ 1∫ 0 g (x) dαϕ(x)dαx ≤ ϕ(1) −ϕ(1 − `). since ` := α b∫ a g (x) dαx and ϕ(0) = 0, we obtain the desired result (3.9). � now, we give the new inequality for functions g ∈ l1α ([0, 1]) as follows: theorem 3.4. let α ∈ (0, 1] and g ∈ l1α ([0, 1]) such that 0 ≤ g(x) ≤ 1 for all x ∈ [0, 1] . if ϕ : [0, 1] → [0,∞) is a convex, α-fractional differentiable function with ϕ(0) = 0, then ϕ  α 1∫ 0 g (x) dαx   ≤ 1∫ 0 g (x) dαϕ (x) dαx for all x ∈ [0, 1]. proof. let g ∈ l1α ([0, 1]) and ε = 1 n > 0, there exists a sequence (gn)n∈n of functions which are continuous on [0, 1] such that ‖gn −g‖α,1 < 1 n . since gn is continuous, then by theorem 3.3, we obtain that ϕ  α 1∫ 0 gn (x) dαx   ≤ 1∫ 0 gn (x) dαϕ (x) dαx = 1∫ 0 g (x) dαϕ (x) dαx + 1∫ 0 [gn (x) −g(x)] dαϕ (x) dαx. since ∣∣∣∣∣∣ 1∫ 0 gn (x) dαx− 1∫ 0 g (x) dαx ∣∣∣∣∣∣ ≤ 1∫ 0 |gn (x) −g(x)|dαx < 1 αn → 0 as n →∞, it follows that ϕ  α 1∫ 0 g (x) dαx   ≤ 1∫ 0 g (x) dαϕ (x) dαx which is completed the proof. � 30 sarikaya, yaldiz and budak references [1] t. abdeljawad, on conformable fractional calculus, j. comput. appl. math. 279 (2015) 57–66. [2] m. abu hammad, r. khalil, conformable fractional heat differential equations, int. j. pure appl. math. 94(2) (2014), 215-221. [3] m. abu hammad, r. khalil, abel’s formula and wronskian for conformable fractional differential equations, int. j. differ. equ. appl. 13( 3) 2014, 177-183. [4] d. r. anderson, taylor’s formula and integral inequalities for conformable fractional derivatives, contrib. math. eng. springer, (2016). [5] r. khalil, m. al horani, a. yousef, m. sababheh, a new definition of fractional derivative, j. comput. appl. math. 264 (2014), 65-70. [6] p. cerone, on some generalizations of steffensen’s inequality and related results, j. ineq. pure appl. math. 3 (2) (2001), art. id 28. [7] z. liu, more on steffensen type inequalities, soochow j. math., 31 (3) (2005), 429–439. [8] z. liu, on steffensen type inequalities, j. nanjing univ. math. biquart. 19 (2) (2002), 25-30. [9] d.s. mitrinovic, j.e. pecaric and a.m. fink, classical and new inequalities in analysis, kluwer, dordrecht (1993). [10] j. f. steffensen, on certain inequalities and methods of approximation, j. inst. actuaries 51(1919), 274–297. [11] s.-h. wu and h. m. srivastava, some improvements and generalizations of steffensen’s integral inequality, appl. math. comput. 192 (2007), 422-428. department of mathematics, faculty of science and arts, düzce university, düzce, turkey ∗corresponding author: yaldizhatice@gmail.com 1. introduction 2. definitions and properties of conformable fractional derivative and integral 3. steffensen's type inequalities for conformable fractional integrals references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 229-237 doi: 10.28924/2291-8639-15-2017-229 quasi-slowly oscillating sequences in locally normal riesz spaces bipan hazarika1 and ayhan esi2,∗ abstract. in this paper, we study quasi-slowly oscillating sequences, on quasi-slowly oscillating compactness and quasi-slowly oscillating continuous functions in locally normal riesz space. 1. introduction the concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics especially in computer science, information theory, biological science, speech analysis, bioinformatics. a real valued function is continuous on the set of real numbers if and only if it preserves cauchy sequences. using the idea of continuity of a real function and the idea of compactness in terms of sequences, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: forward continuity [6], slowly oscillating continuity [9, 12, 14, 15, 35], statistical ward continuity [7], δ-ward continuity [11], ideal ward continuity [5, 22]. the concept of a cauchy sequence involves far more than that the distance between successive terms is tending to zero. nevertheless, sequences which satisfy this weaker property are interesting in their own right. a sequence (xn) of points in r is called quasi-cauchy if (∆xn) is a null sequence where ∆xn = xn+1 −xn. in [4] burton and coleman named these sequences as ”quasi-cauchy” and in [8] çakallı used the term ”ward convergent to 0” sequences. in terms of quasi-cauchy we restate the definitions of ward compactness and ward continuity as follows: a function f is ward continuous if it preserves quasi-cauchy sequences, i.e. (f(xn)) is quasi-cauchy whenever (xn) is, and a subset e of r is ward compact if any sequence x = (xn) of points in e has a quasi-cauchy subsequence z = (zk) = (xnk) of the sequence x. a riesz space is an ordered vector space which is a lattice at the same time. it was first introduced by f. riesz [32] in 1928. riesz spaces have many applications in measure theory, operator theory and optimization. they have also some applications in economics (see [2]), and we refer to [1, 3, 21, 23, 25, 27, 28, 30, 31, 33, 36] for more details. 2. preliminaries and notations it is known that a sequence x = (xn) of points in r, the set of real numbers, is slowly oscillating, denoted by x ∈ so, if lim λ→1+ limn max n+1≤k≤[λn] |xk −xn| = 0 where [λn] denotes the integer part of λn. this is equivalent to the following if (xm−xn) → 0 whenever 1 ≤ m n → 1 as m,n →∞. using ε > 0 and δ this is also equivalent to the case when for any given ε > 0, there exists δ = δ(ε) > 0 and n = n(ε) such that |xm −xn| < ε if n ≥ n(ε) and n ≤ m ≤ (1 + δ)n (see [9]). a function defined on a subset e of r is called slowly oscillating continuous if it preserves slowly oscillating sequences, i.e. (f(xn)) is slowly oscillating whenever (xn) is. connor and grosse-erdman [13] gave sequential definitions of continuity for real functions calling g-continuity instead of a-continuity and their results covers the earlier works related to a-continuity received 2nd july, 2017; accepted 26th august, 2017; published 1st november, 2017. 2010 mathematics subject classification. 46a19; 40a05; 40g15; 46a50; 54e35. key words and phrases. riesz space; continuity; quasi-cauchy sequence; quasi-slowly oscillating sequences. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 229 230 hazarika and esi where a method of sequential convergence, or briefly a method, is a linear function g defined on a linear subspace of s, space of all sequences, denoted by cg, into r. a sequence x = (xn) is said to be g-convergent to ` if x ∈ cg and g(x) = `. in particular, lim denotes the limit function lim x = limn xn on the linear space c. a method g is called regular if every convergent sequence x = (xn) is g-convergent with g(x) = lim x. a method is called subsequential if whenever x is g-convergent with g(x) = `, then there is a subsequence (xnk) of x with limk xnk = ` (for details see [10]). let x be a real vector space and ≤ be a partial order on this space. then x is said to be an ordered vector space if it satisfies the following properties: (i) if x,y ∈ x and y ≤ x, then y + z ≤ x + z for each z ∈ x. (ii) if x,y ∈ x and y ≤ x, then ay ≤ ax for each a ≥ 0. if, in addition, x is a lattice with respect to the partial ordered, then x is said to be a riesz space (or a vector lattice)(see [36]). for an element x of a riesz space x, the positive part of x is defined by x+ = x∨ 0 = sup{x, 0}, the negative part of x by x− = −x∨ 0 and the absolute value of x by |x| = x∨ (−x), where 0 is the zero element of x. a subset s of a riesz space x is said to be normal if y ∈ s and |x| ≤ |y| implies x ∈ s. a topological vector space (x,τ) is a vector space x which has a topology (linear) τ, such that the algebraic operations of addition and scalar multiplication in x are continuous. continuity of addition means that the function f : x × x → x defined by f(x,y) = x + y is continuous on x × x, and continuity of scalar multiplication means that the function f : r ×x → x defined by f(a,x) = ax is continuous on r×x. every linear topology τ on a vector space x has a base n for the neighborhoods of θ satisfying the following properties: (1) each y ∈ n is a balanced set, that is, ax ∈ y holds for all x ∈ y and for every a ∈ r with |a| ≤ 1. (2) each y ∈ n is an absorbing set , that is , for every x ∈ x, there exists a > 0 such that ax ∈ y. (3) for each y ∈ n there exists some e ∈ n with e + e ⊆ y. a linear topology τ on a riesz space x is said to be locally normal or solid if τ has a base at zero consisting of normal sets. a locally normal (solid) riesz space (x,τ) is a riesz space equipped with a locally normal (solid) topology τ. recall that a first countable space is a topological space satisfying the ”first axiom of countability”. specifically, a space x is said to be first countable if each point has a countable neighborhood basis (local base). that is, for each point x in x there exists a sequence v1,v2, · · · of open neighborhoods of x such that for any open neighborhood v of x there exists an integer j with vj contained in v. the idea of statistical convergence first appeared, under the name of almost convergence, in the first edition zygmund [37] of celebrated monograph [38] of zygmund. later, this idea was introduced by fast [16] and steinhaus [34] and many authors. actually, this concept is based on the natural density of subsets of n of positive integers. a subset e of n is said to have natural or asymptotic density δ(e), if δ(e) = lim n→∞ |e(n)| n exists, where e(n) = {k ≤ n : k ∈ e} and |e| denotes the cardinality of the set e. a sequence x = (xn) of points in x is said to be statistically convergent (see [1]) to an element l in x if for each τneighborhood v of zero, δ({n ∈ n : xn −l /∈ v}) = 0, i.e. lim k→∞ 1 k |{n ≤ k : xn −l /∈ v}| = 0. kostyrko et al. [26] introduced the notion of ideal convergence which is a generalization of statistical convergence (see [16,18]) based on the structure of the admissible ideal i of subsets of natural numbers n. a family of sets i ⊂ p(n) (the power sets of n) is said to be an ideal on n if and only if φ ∈ i for each a,b ∈ i, we have a∪b ∈ i for each a ∈ i and each b ⊂ a, we have b ∈ i. a non-empty family of sets f ⊂ p(n) is said to be a filter on n if and only if φ /∈ f for each a,b ∈ f, we have a∩b ∈ f each a ∈ f and each b ⊃ a, we have b ∈ f. an ideal i is called non-trivial ideal if i 6= φ quasi-slowly oscillating sequences in locally normal riesz spaces 231 and n /∈ i. clearly i ⊂ p(n) is a non-trivial ideal if and only if f = f(i) = {n−a : a ∈ i} is a filter on n. a non-trivial ideal i ⊂ p(n) is called admissible if and only if {{n} : n ∈ n} ⊂ i. throughout we assume i is a non-trivial admissible ideal in n. a sequence x = (xn) of points in a locally normal riesz space x is said to be ideally convergent to x0 ∈ x if for every τ-neighborhood v of zero, the set {n ∈ n : xn−x0 /∈ v}∈ i. in this case we write xn iτ→ ` i.e. iτ -lim xn = ` (for details see [21]). the notion of lacunary statistical convergence was introduced by fridy and orhan [19] and has been investigated for the real case in [20]. a sequence x = (xn) in x is called lacunary statistically convergent to an element l in x (see [29]) if for every τ-neighborhood v of zero, lim r→∞ 1 hr |{n ∈ jr : xn −l /∈ v}| = 0 where j = (kr−1,kr] and k0 = 0,hr := kr − kr−1 → ∞ as r → ∞ and (θ) = (kr) is an increasing sequence of positive integers. throughout the article, the symbol nnor we will denote any base at zero consisting of normal sets and satisfying the conditions (1), (2) and (3) in a locally normal topology. also (x,τ) a locally normal riesz space (in short lnrs) and n and r will denote the set of all positive integers, and the set of all real numbers, respectively. we will use boldface letters x, y, z, . . . for sequences x = (xn), y = (yn), z = (zn), . . . of points in x. 3. quasi-slowly oscillating sequences in lnrs in this section we introduce the concepts of quasi-slowly oscillating continuity and quasi-slowly oscillating compactness in lnrs and establish some interesting results related to these notions. a sequence x = (xn) of points in x is called quasi-cauchy if for each τ-neighborhood v of zero, there exists an m0 ∈ n such that xn+1 − xn ∈ v for n ≥ m0. it is clear that cauchy sequences are slowly oscillating not only the real case but also in the lnrs setting. it is easy to see that any slowly oscillating sequence of points in x is quasi-cauchy and therefore cauchy sequence is quasi-cauchy. the converses are not always true. there are quasi-cauchy sequences which are not cauchy. there are quasi-cauchy sequences which are not slowly oscillating. any subsequence of cauchy sequence is cauchy. the analogous property fails for quasi-cauchy sequences and slowly oscillating sequences as well. a sequence x = (xn) of points in x is said to be slowly oscillating, denoted by x ∈ so(x) if (xn) is a slowly oscillating sequences, i.e. for each τ-neighborhood v of zero, there exist δ = δ(v ) > 0 and m = m(v ) such that xk −xn ∈ v for n ≥ m(v ) and n ≤ k ≤ (1 + δ)n. a sequence x = (xn) of points in x is called ideally quasi-cauchy if for each τ-neighborhood v of zero, the set {n ∈ n : xn+1 −xn /∈ v}∈ i. it is clear that slowly oscillating sequence of points in x is ideally quasi-cauchy. now we introduce the notion of quasi-slowly oscillating sequences and quasi-slowly oscillating continuity in lnrs. definition 3.1. a sequence x = (xn) of points in x is said to be quasi-slowly oscillating, denoted by x ∈ qso(x) if (∆xn) is a slowly oscillating sequences, i.e. for each τ-neighborhood v of zero, there exist δ = δ(v ) > 0 and m = m(v ) such that ∆xk − ∆xn ∈ v for n ≥ m(v ) and n ≤ k ≤ (1 + δ)n. it is clear that a convergent sequence is slowly oscillating, since every convergent sequence is a cauchy sequence, and any slowly oscillating sequence is quasi-cauchy, but the converse need not to be true in general. for examples, if x = r, then ( ∑∞ n=1 1 n ), (ln n), (ln ln n) are slowly oscillating, but not cauchy. the sequence ( ∑k n=1 1 n ) is quasi-cauchy, but not slowly oscillating. theorem 3.1. if a sequence is slowly oscillating then it is a quasi-slowly oscillating. 232 hazarika and esi proof. let (xn) be a slowly oscillating sequence. for each τ-neighborhood v of zero, there exists a y ∈ nnor such that y ⊆ v. choose w ∈ nnor such that w −w ⊆ y. since (xn) is a slowly oscillating sequence, there exist δ = δ(w) > 0 and a positive integer n1 = n1(w) such that xk −xn ∈ w for all n ≥ n1 and n ≤ k ≤ (1 + δ)n. hence for all n ≥ n1(w) and n ≤ k ≤ (1 + δ)n we have ∆xk − ∆xn = (xk −xk+1) − (xn −xn+1) = (xk −xn) − (xk+1 −xn+1) ∈ w −w ⊆ y ⊆ v. it implies that (xn) is a quasi-slowly oscillating sequence. � definition 3.2. a function f defined on a subset e of x is called quasi-slowly oscillating continuous if it transforms quasi-slowly oscillating sequences to quasi-slowly oscillating sequences of points in e, that is, (f(xn)) is quasi-slowly oscillating whenever (xn) is quasi-slowly oscillating sequences of points in e. we note that sum of two quasi-slowly oscillating continuous functions is quasi-slowly oscillating continuous and the composite of two quasi-slowly oscillating continuous functions is quasi-slowly oscillating continuous in lnrs. in connection with slowly oscillating sequences, quasi-slowly oscillating sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on x. (qso-qso): : (xn) ∈ qso(x) ⇒ (f(xn)) ∈ qso(x) (qso-c): : (xn) ∈ qso(x) ⇒ (f(xn)) ∈ c(x) (c-c): : (xn) ∈ c(x) ⇒ (f(xn)) ∈ c(x) (c-qso): : (xn) ∈ c(x) ⇒ (f(xn)) ∈ qso(x) (qso-so): : (xn) ∈ qso(x) ⇒ (f(xn)) ∈ so(x) (so-qso): : (xn) ∈ so(x) ⇒ (f(xn)) ∈ qso(x) (u): : uniform continuity of f. it is clear that (qso-qso) implies (so-qso), but (so-qso) need not imply (qso-qso). also (qso-c) implies (c-qso) and (qso-c) implies (c-c) and we see that (c-c) need not imply (qso-c), because identity function is an example for it. we also see that (u) implies (so-qso). theorem 3.2. if f is quasi-slowly oscillating continuous on a subset e of x then it is continuous on e in the ordinary sense. proof. suppose that f is quasi-slowly oscillating continuous on e and let (xn) be any convergent sequence of points in e with limn→∞xn = x0. then the sequence (x1,x1,x0,x0,x2,x2,x0,x0, ...,xn−1,xn−1,x0,x0,xn,xn,x0,x0, ...) is also convergent to x0 and hence (yn) is quasi-slowly oscillating. since f is quasi-slowly oscillating continuous, the sequence (f(x1),f(x1),f(x0),f(x0),f(x2),f(x2),f(x0),f(x0), ..., f(xn−1),f(xn−1),f(x0),f(x0),f(xn),f(xn),f(x0),f(x0), ...) is also quasi-slowly oscillating. from the definition of quasi-slowly oscillation, we have that the sequence (0,f(x1) −f(x0), 0,f(x0) −f(x2), 0,f(x2) −f(x0), 0,f(x0) −f(x3), 0,f(x3) −f(x0), 0, ..., 0,f(x0) −f(xn−1), 0,f(xn−1) −f(x0),f(x0) −f(xn), 0,f(xn) −f(x0), ...) is a slowly oscillating. since any slowly oscillating sequence is quasi-cauchy, then the sequence (f(x0) −f(x1),f(x1) −f(x0),f(x0) −f(x2),f(x2) −f(x0),f(x0) −f(x3),f(x3) −f(x0), ..., f(x0) −f(xn−1),f(xn−1) −f(x0),f(x0) −f(xn),f(xn) −f(x0), ...) is a null sequence. now it follows that if for each τ-neighborhood v of zero, there exists m = m(v ) such that f(xn) −f(x0) ∈ v for n ≥ m. this completes the proof of theorem. � quasi-slowly oscillating sequences in locally normal riesz spaces 233 in general the converse is not true. if x = r. then, it follows from the function f(x) = x2 + 1 and the sequence (xn) = ( √ n). corollary 3.1. any quasi-slowly oscillating continuous function is g-continuous for any regular subsequential method g. corollary 3.2. if f is quasi-slowly oscillating continuous on a subset e of x, then it is ideally continuous on e. corollary 3.3. if f is quasi-slowly oscillating continuous on a subset e of x, then it is statistically continuous on e. corollary 3.4. if f is quasi-slowly oscillating continuous on a subset e of x, then it is lacunary statistically continuous on e. theorem 3.3. if f is a uniformly continuous function defined on a subset e of x, then it is quasislowly oscillating continuous on e. proof. let f be uniformly continuous function and x = (xn) be any quasi-slowly oscillating sequence in e. let w be a τ-neighborhood of zero. since f is uniformly continuous on e, then there exists a τ-neighborhood v of zero such that f(x) − f(y) ∈ w whenever x − y ∈ v. since (xn) is slowly oscillating, for the same τ-neighborhood w of zero, there exist m = m(v ) and δ = δ(v ) > 0 such that ∆xk −∆xn ∈ v for n ≥ m(v ) and n ≤ k ≤ (1 + δ)n. hence we have ∆f(xk)−∆f(xn) ∈ w whenever n ≥ m(v ) and n ≤ k ≤ (1 + δ)n. it follows that (f(xn)) is quasi-slowly oscilatting. this completes the proof of theorem. � definition 3.3. a sequence (xn) of points in x is called cesáro quasi-slowly oscillating if (tn) is -quasi-slowly oscillating, where tn = 1 n ∑n k=1 xk, is the cesáro means (see [17]) of the sequence (xn). also a function f defined on a subset e of x is called cesáro quasi-slowly oscillating continuous if it preserves cesáro quasi-slowly oscillating sequences of points in e. by using the similar argument used in proof of theorem 3.3, we immediately have the following result. theorem 3.4. if f is a uniformly continuous on a subset e of x and (xn) is a quasi-slowly oscillating sequence in e, then (f(xn)) is cesáro quasi-slowly oscillating. definition 3.4. a sequence of functions (fn) defined on a subset e of x is said to be uniformly convergent to a function f if for each τ-neighborhood v of zero, there exists an integer n0 = n0(v ) such that fn(x) −f(x) ∈ v for all n ≥ n0 and x ∈ e. theorem 3.5. if (fn) is a sequence of quasi-slowly oscillating continuous functions defined on a subset e of x and (fn) is uniformly convergent to a function f on e, then f is quasi-slowly oscillating continuous on e. proof. let (xn) be any quasi-slowly oscillating sequence of points in e. by uniform convergence of (fn), if for each τ-neighborhood v of zero, there exists a y ∈ nnor such that y ⊆ v. choose w ∈ nnor such that −w + w + w −w + w ⊆ y. then there exists n1 = n1(w) such that fn(x) −f(x) ∈ w for each x ∈ e and for all n ≥ n1(w). also since fn1 is quasi-slowly oscillating continuous, there exist n2 = n2 > n1 and δ = δ(w) > 0 such that ∆fn1 (xk) − ∆fn1 (xn) ∈ w whenever n ≥ n2(w) and n ≤ k ≤ (1 + δ)n. therefore if n ≥ n1(w) and n ≤ k ≤ (1 + δ)n we have ∆f(xk) − ∆f(xn) = [f(xk) −f(xk+1)] − [f(xn) −f(xn+1)] = [f(xk) −fn1 (xk)] + [fn1 (xk+1) −f(xk+1)] + [fn1 (xn) −f(xn)] + [f(xn+1) −fn1 (xn+1)] +[fn1 (xk) −fn1 (xk+1) −fn1 (xn) + fn1 (xn+1)] ∈−w + w + w −w + w ⊆ y ⊆ v. thus it implies that ∆f(xk) − ∆f(xn) ∈ v if n ≥ n1 and n ≤ k ≤ (1 + δ)n. it follows that (f(xn)) is a quasi-slowly oscillating sequences of points in e which completes the proof of theorem. � 234 hazarika and esi using the same techniques as in the theorem 3.5, the following result can be obtained easily. theorem 3.6. if (fn) is a sequence of cesáro quasi-slowly oscillating continuous functions defined on a subset e of x and (fn) is uniformly convergent to a function f on e, then f is cesáro quasi-slowly oscillating continuous on e. theorem 3.7. the set of all quasi-slowly oscillating continuous functions defined on a subset e of x is a closed subset of all continuous functions on e, that is qso(e) = qso(e), where qso(e) is the set of all quasi-slowly oscillating continuous functions defined on e and qso(e) denotes the set of all cluster points of qso(e). proof. let f be any element of qso(e). then there exists a sequence of points (fn) in qso(e) such that limn→∞fn = f. to show that f is quasi-slowly oscillating sequence on e. now let (xn) be any quasi-slowly oscillating sequence in e. let v be an arbitrary τ-neighborhood of zero. there exists a y ∈ nnor such that y ⊆ v. choose w ∈ nnor such that w +w +w +w +w ⊆ y. since (fn) converges to f, there exists a positive integer n1 such that for all x ∈ e and for all n ≥ n1, fn(x) −f(x) ∈ w. also since fn1 is quasi-slowly oscillating continuous, there exist an integer n2 = n2 > n1 and δ > 0 such that ∆fn1 (xk) − ∆fn1 (xn) ∈ w whenever n ≥ n2 and n ≤ k ≤ (1 + δ)n. hence, for all n ≥ n1 and n ≤ k ≤ (1 + δ)n we have ∆f(xk) − ∆f(xn) = [∆f(xk) − ∆fn1 (xn)] + [∆fn1 (xn) − ∆fn1 (xk)] + [∆fn1 (xk) − ∆f(xn)] = [f(xk) −f(xk+1)] − [fn1 (xn) −fn1 (xn+1)] + [fn1 (xn) −fn1 (xn+1)] − [fn1 (xk) −fn1 (xk+1)] +[fn1 (xk) −fn1 (xk+1)] − [f(xn) −f(xn+1)] ∈ w + w + w + w + w ⊆ y ⊆ v. thus it implies that f(xk)−f(xn) ∈ v for all n ≥ n1 and n ≤ k ≤ (1 +δ)n. thus f is slowly oscillating continuous function on e and this completes the proof of theorem. � corollary 3.5. the set of all quasi-slowly oscillating continuous functions defined on a subset e of x is a complete subspace of the space of all continuous functions on e. an element x0 in x is called an ideal limit point of a subset e of x if there is an e-valued sequence of points with ideal limit x0. it follows that the set of all ideal limit points of e is equal to the set of all limit points of e in the ordinary sense. an element x0 in x is called an ideal accumulation point of a subset e if it is an ideal limit point of the set e −{x0}. the set of all ideal accumulation points of e is equal to the set of all accumulation points of e in the ordinary sense. a function f on x is said to have an ideally sequential limit at a point x0 of x if the image sequence (f(xn)) is ideally convergent to x0 for any ideally convergent sequence x = (xn) with ideal limit x0 and a function f is to be ideally sequentially continuous at a point x0 of x if the sequence (f(xn)) is ideally convergent to f(x0) for any ideally convergent sequence x = (xn) with ideal limit x0 (for details see [5]). lemma 3.1. a function f on x has an ideally sequential limit at a point x0 of x if and only if it has an ideal limit at a point x0 of x in ordinary sense. proof. the proof follows from the fact that any ideally convergent sequence has a convergent subsequence (also see [5]). � next we define the concept of quasi-slowly oscillating compactness in lnrs. definition 3.5. a subset e of x is called quasi-slowly oscillating compact if any sequence of points in e has a quasi-slowly oscillating subsequence. we see that any compact subset of x is quasi-slowly oscilatting compact, union of two quasislowly oscillating compact subsets of x is quasi-slowly oscillating compact. any subset of quasi-slowly oscillating compact set is also quasi-slowly oscillating compact and so intersection of any quasi-slowly oscillating compact subsets of x is quasi-slowly oscillating compact. theorem 3.8. a quasi-slowly oscillating continuous image of a quasi-slowly oscillating compact subset of x is quasi-slowly oscillating compact. quasi-slowly oscillating sequences in locally normal riesz spaces 235 proof. let f be a quasi-slowly oscillating continuous function on x and e be a quasi-slowly oscillating compact subset of x. let y = (yn) be a sequence of points in f(e). then we can write yn = f(xn) where (xn) is sequence of points in e for each n ∈ n. since e is quasi-slowly oscillating compact, there is a quasi-slowly oscillating subsequence z = (zk) = (xnk) of (xn). then, quasi-slowly oscillating continuity of f implies that f(zk) is a quasi-slowly oscillating subsequence of f(xn). hence f(e) is quasi-slowly oscillating compact. � corollary 3.6. for any regular subsequential method g, if e is g-sequentially compact subset of x, then it is quasi-slowly oscillating compact. proof. the proof of the result follows from the regularity and subsequence property of g. � corollary 3.7. a real valued function defined on a bounded subset of r is uniformly continuous if and only if it is slowly oscillating continuous. proof. the proof of the result follows from the fact that totally boundedness coincides with slowly oscillating compactness and boundedness coincides with totally boundedness in r. � corollary 3.8. any totally bounded subset of x is quasi-slowly oscillating compact. proof. the proof of the result follows from the fact that any sequence of points in a totally bounded subset of x has a cauchy subsequence, which is quasi-slowly oscillating. � now we give the definition on ideal continuous function in lnrs. definition 3.6. let (x,τ1) and (y,τ2) be lnr spaces and e ⊂ y. a function f : e → y is called ideally continuous at a point x0 ∈ e if xn iτ1→ x0 in e implies f(xn) iτ2→ f(x0) in y. theorem 3.9. let (x,τ1) and (y,τ2) be lnr spaces. if a function f : x → y is uniformly continuous, then f is ideally continuous. proof. let f : x → y be uniformly continuous and xn iτ1→ x0 in x. let θ1 and θ2 be denote the zeros in x and y, respectively. let w be an arbitrary τ2-neighborhood of θ2. since f is uniformly continuous, there exists some τ1-neighborhood v of θ1 such that x−y ∈ v ⇒ f(x) −f(y) ∈ w. (3.1) since xn iτ1→ x0, we put k = {n ∈ n : xn −x0 /∈ v}, so k ∈ i. then from (3.1) we have f(xn) −f(x0) /∈ w for all n ∈ k. therefore we have {n ∈ n : f(xn) −f(x0) /∈ w}⊆ k and hence {n ∈ n : f(xn) −f(x0) /∈ w}∈ i. i.e. we have f(xn) iτ2→ f(x0), which shows that f is ideally continuous. � theorem 3.10. a function f on x is ideally sequentially continuous at a point x0 of x if and only if it is continuous at a point x0 in ordinary sense. proof. the proof follows from the fact that any ideally convergent sequence has a convergent subsequence and from the above lemma 3.1. � theorem 3.11. let f : x → x be any function and (xn) be a sequence of points in x such that iτ − limn→∞xn = x0 implies limn→∞f(xn) = f(x0), then it is a constant function. proof. for the proof of the theorem follows form theorem 3 in [12]. � theorem 3.12. if a function is quasi-slowly oscillating continuous on a subset e of x, then it is ideally sequentially continuous on e. 236 hazarika and esi proof. let f be any quasi-slowly oscillating continuous on e. by theorem 3.2, we have f is continuous on e. also from theorem 3.10, we see that f is ideally sequentially continuous on e. this completes the proof. � theorem 3.13. if a function is δ-ward continuous on a subset e of x, then it is ideally sequentially continuous on e. proof. let f be any δ-ward continuous function on e. it follows from corollary 2 in [11] that f is continuous. by theorem 3.10 we obtain that f is ideally sequentially continuous on e. this completes the proof of the theorem. � references [1] h. albayrak and s. pehlivan, statistical convergence and statistical continuity on locally solid riesz spaces, topology appl. 159 (2012), 1887-1893. [2] c. d. aliprantis and o. burkinshaw, locally solid riesz spaces with applications to economics, second ed., amer. math. soc., 2003. [3] a. alotaibi, b. hazarika and s. a. mohiuddine, on the ideal convergence of double sequences in locally solid riesz spaces, abst. appl. anal. 2014 (2004), article id 396254, 6 pages. [4] d. burton and j. coleman, quasi-cauchy sequences, amer. math. monthly 117(4)(2010), 328-333. [5] h. çakallı and b. hazarika, ideal-quasi-cauchy sequences, j. inequal. appl. 2012 (2012), art. id 234, 11 pages. [6] h. çakallı forward continuity, j. comput. anal. appl. 13(2)(2011), 225-230. [7] h. çakallı, statistical ward continuity, appl. math. lett. 24(10)(2011), 1724-1728. [8] h. çakallı statistical-quasi-cauchy sequences, math. comput. modelling 54(5-6)(2011), 1620-1624. [9] h. çakallı slowly oscillating continuity, abst. appl. anal. 2008(2008), article id 485706, 5 pages. [10] h. çakallı on g-continuity, comput. math. appl. 61(2011), 313-318. [11] h. çakallı delta quasi-cauchy sequences, math. comput. modelling 53(2011), 397-401. [12] h. çakallı and a. sönmez, slowly oscillating continuity in abstract metric spaces, filomat 27(5)(2013), 925-930. [13] j. connor and k. g. grosse-erdmann, sequential definitions of continuity for real functions, rocky mountain j. math. 33(1)(2003), 93-121. [14] h. çakallı i. çanak and m. dik, ∆-quasi-slowly oscillating continuity. appl. math. comput. 216(10)(2010), 2865– 2868. [15] m. dik and i. çanak, new types of continuities, abst. appl. anal. 2010(2010), article id 258980, 6 pages. [16] h. fast, sur la convergence statistique, colloq. math. 2(1951), 241-244. [17] a. r. freedman, j. j. sember and m. raphael, some cesaro-type summability spaces, proc. london math. soc. 3(37)(1978), 508-520. [18] j. a. fridy, on statistical convergence, analysis 5(1985), 301-313. [19] j. a. fridy and c. orhan, lacunary statistical convergence, pacific j. math. 160(1)(1993), 43-51. [20] j. a. fridy and c. orhan, lacunary statistical summability, j. math. anal. appl. 173(1993), 497-504. [21] b. hazarika, ideal convergence in locally solid riesz spaces, filomat 28(4)(2014), 797-809. [22] b. hazarika, second order ideal-ward continuity, int. j. anal. 2014(2014), article id 480918, 4 pages. [23] b. hazarika, s. a. mohiuddine and m. mursaleen, lacunary density and some inclusion results in locally solid riesz spaces, iranian j. sci. tech., transactions-a, 38a1 (2014), 61-68 [24] b. hazarika, m. kemal ozdemir and a. esi, slowly oscillating sequences in locally normal riesz spaces, int. j. adv. appl. sci. 4(1)(2017), 96-101. [25] l. v. kantorovich, lineare halbgeordnete raume, rec. math. 2 (1937), 121-168. [26] p. kostyrko, t. s̆alát and w. wilczyński, i-convergence, real anal. exch. 26(2)(2000-2001), 669-686. [27] w. a. j. luxemburg and a. c. zaanen, riesz spaces i, north-holland, amsterdam, 1971. [28] s. a. mohiuddine, a. alotaibi and m. mursaleen, statistical convergence of double sequences in locally solid riesz spaces, abst. appl. anal. 2012(2012), article id 719729, 9 pages. [29] s. a. mohiuddine and m. a. alghamdi, statistical summability through lacunary sequence in locally solid riesz spaces, j. inequal. appl. 2012(2012), article id 225, 9 pages. [30] s. a. mohiuddine, a. alotaibi and m. mursaleen, statistical convergence of double sequences in locally solid riesz spaces, abst. appl. anal. 2012(2012), article id 719729, 9 pages. [31] s. a. mohiuddine, b. hazarika and a. alotaibi, double lacunary density and some inclusion results in locally solid riesz spaces, abst. appl. anal. 2013(2013), article id 507962, 8 pages. [32] f. riesz, sur la décomposition des opérations fonctionelles linéaires, in: atti del congr. internaz. dei mat., 3, bologna, 1928, zanichelli, 1930, 143-148. [33] g. t. roberts, topologies in vector lattices, math. proc. camb. phil. soc. 48 (1952), 533-546. [34] h. steinhaus, sur la convergence ordinaire et la convergence asymptotique, colloq. math. 2 (1951), 73-74. [35] r. w. vallin, creating slowly oscillating sequences and slowly oscillating continuous functions, with an appendix by vallin and cakalli, acta math. univ. comenian(n.s.) 80(1)(2011), 71-78. [36] a. c. zannen, introduction to operator theory in riesz spaces, springer-verlag, 1997. [37] a. zygmund, trigonometrical series, vol. 5 of monografýas de matemáticas, warszawa-lwow, 1935. [38] a. zygmund, trigonometric series, cambridge university press, cambridge, uk, 2nd edition, 1979. quasi-slowly oscillating sequences in locally normal riesz spaces 237 1department of mathematics, rajiv gandhi university, rono hills, doimukh-791112, arunachal pradesh, india 2adıyaman university, science and art faculty, department of mathematics, 02040, adıyaman, turkey ∗corresponding author: aesi23@hotmail.com 1. introduction 2. preliminaries and notations 3. quasi-slowly oscillating sequences in lnrs references int. j. anal. appl. (2022), 20:17 received: feb. 6, 2022. 2010 mathematics subject classification. 37m10, 93a30. key words and phrases. daily temperature forecasting; seasonal autoregressive integrated moving-average model; hybrid model; non-stationarity measure; decomposition methods. https://doi.org/10.28924/2291-8639-20-2022-17 ©2022 the author(s) issn: 2291-8639 1 comparative study of wavelet-sarima and emd-sarima for forecasting daily temperature series luwam ghide, siyuan wei, yiming ding* center for mathematical sciences and department of mathematics, wuhan university of technology, wuhan 430070, china *corresponding author: dingym@whut.edu.cn abstract. this paper aims to find a forecasting model based on the comparative study of wavelet arima and emd-arima models and residual-based model selection technique for temperature time series. time series analysis is essential in studying temperature data for investigating the variation and predicting the future trend, in which we can control the changes and make good decisions. and most important is to understand the trend in the series with time. this study applied hybridized models of wavelet transform and empirical mode decomposition with seasonal autoregressive integrated moving average (sarima), which combines two models to get better accuracy, for forecasting daily average temperature time series data in the central region of eritrea, asmara. daily data was collected for 30 years, from january 1, 1991, to december 31, 2020. the study compares wt-sarima and emdsarima models to find a well fit and better forecasting model. model selection techniques are essential for time series analysis to determine which model best fits our data. aic and bic are the most used methods in model selection. this paper uses an additional method based on the residual series. in estimating accurate parameters, the structure of the residual sequence had a lot of connection, in which a stationary residual depict an accurate estimation. from this perspective, a nonstationarity measurement of the residual series is used for model selection. the relative performance is based on the predictive capability of sample forecasts assessed. the results indicate that the hybridized wavelet-sarima model is more effective than the other models, and matlab soft-wire is used for this analysis. https://doi.org/10.28924/2291-8639-20-2022-17 2 int. j. anal. appl. (2022), 20:17 1.introduction climate change is one of the most critical issues in our time. earth's climate has changed throughout history and threatens the lives and livelihood of billions of people. earth-orbiting satellites and additional technological advances have enabled scientists to see the big picture by collecting different information about our planet and its climate on a global scale. the collected data over many years show the signals of climate change. research conducted in 2018 [1] says that climate change is the most severe environmental problem. it directly affects human activities and property. precipitation, melting ice, drought, poverty, river drying, and reducing the number and distribution of groundwater resources are some of the climate change signals. temperature is one of the main climate change elements. increasing or decreasing in temperature will affect the weather pattern in the lives of plants and animals. understanding and predicting the future courses of temperature quantities based on the historical time series data is fundamental in climate change to know the pattern and increase in the frequency and magnitude of extreme events [2]–[5]. in addition, a good forecast with minimum error support for future preparation and good decisionmaking. most time series forecasting methods are based on analyzing the past observed data by assuming that the past patterns in the data can be used to forecast future events. the autoregressive integrated moving average (arima) model is one of the most popular time series modelling methods. the main aim of the arima model is to cautiously and rigorously study the past observations of a time series to develop a suitable model which can predict future value. over the past years, the arima model has been widely used in temperature time series forecasting. for annual temperature prediction in libya, [6] study showed that the linear arima model and the quadratic arima model had the best performance in making short-term predictions. the study [32] used the arima model for 50 year time period (1955-2005) in the south of iran, and the arima model was selected as the optimal model for temperature data. furthermore, [7] used the seasonal autoregressive integrated moving average (sarima) model to forecast the monthly mean of the maximum surface air temperature of india, and [8] used sarima for monthly mean temperature in nanjing, china. in all these studies, the selected models had higher prediction accuracy. but a disadvantage of the arima model, it assumes that data are stationary and has a limited ability to capture nonstationarity and nonlinear data. arima model cannot handle noisy time series data without preprocessing it. so removing the noise before applying the arima model makes it easier in the modelling process to capture the features of the actual series. different int. j. anal. appl. (2022), 20:17 3 papers have introduced the hybridized approach (combination of other methods to model to get more accurate results) of decomposition models and arima model to get a well-fitted model with good prediction performance of nonstationary series, [9]–[13]. decomposition methods decompose the main time series into stationary subcomponents, which help us to see the detailed structure of the series and improve the model's prediction performance by removing noisy components. wavelet transform (wt) and empirical mode decomposition (emd) based arima model achieve better prediction accuracy than direct applying arima model to our time series. a comparative study in bangladesh [9] used wavelet decomposition with the arima and artificial neural network models for temperature prediction. in the study, the hybrid wt-arima model result was effective. this technique supports the arima model to fit the data structure, but it also has progressed in forecasting performance [10]–[12][34]. wavelet analysis has beautiful and deep mathematical properties, making them a well-adapted tool for different data types. it has much attention in signal processing since its theoretical development in 1984 by grossman & morlet. the studies of climate changes using wavelet analysis have received much consideration and applied in detecting climate signals [13]–[15]. in theory and applications, wavelet transforms enormously relates to fourier transform. still, the advantage of using wt is that it offers a simultaneous localization of output in frequency and time domain with faster computation. the most significant benefit is separating fine details in the signal using small wavelets [16]. as a result, several applied fields are making use of wavelets such as[17], [18], [33]. empirical mode decomposition (emd) is a data-adaptive time-frequency representation technique proposed by huang. it requires only that the decomposed component consists of a simple intrinsic mode of oscillations [19], which makes it quite different from wt. this method is most suitable for nonlinear and nonstationary time series data. the emd methodology is based on a sifting process, which identifies local maxima and minima and results in intrinsic mode functions (imfs). in detail, it decomposes the time series into a finite sum of intrinsic mode functions [20]. emd decomposes a nonstationary signal into several stationary series components. and its combination with the arima model improves prediction performance than traditional [21]. the hybrid emd-arima model shows effective long-term and short-term prediction [22], [23]. after denoising the original series with emd and wavelet analyzer for the applied arima model, we must get a suitable parameters estimation technique to build a well-fitted model, which is the main problem in model selection. a good model selecting process will balance the goodness of 4 int. j. anal. appl. (2022), 20:17 model fit with simplicity. in most studies, aic and bic are used to estimate the parameters of the arima (p, d, q) model. another effective method of finding a well fit model is based on the residual structure of the model [24]. accurate parameter estimation makes its residual sequence stationary, which means a well fit model makes a stationary residual series. from this perspective, [25] used the stationarity of estimated residual series to calibrate inaccurate estimation in the case of data with complex noise. a nonstationary residual series shows a trend in the sequence that the model does not capture, which lead to under-fitting. and the stationarity of residual series can be measured by the nonstationary measure (ns) technique proposed by ding et al. [26], [27], [28]. ns is constructed based on shannon entropy and ergodic theory, the level of nonstationarity of a data series and the value of ns is in the interval [0, 1]. but the more the ns value is smaller, the more stationary the series is. from this perspective, the nonstationary measure is used in this paper to check the ns value of residual in both the denoising and model selection process. this method has been used in different studies, such as [29] using ns measure to select forecasting models for irrigation water consumption and [30] using decomposition of noise and trend of a series based emd and nonstationary measure. this paper aims to find a model that can predict all data features. and the main problems are the limitation of the arima model to nonstationary data and model selection techniques. this paper examines the daily mean temperature's statistical properties. we constructed a predictive model to forecast the daily mean temperature for long term prediction, using a comparative study of the seasonal arima model based on decomposition methods. wavelet transforms (wt) and empirical mode decomposition (emd) remove noise from the temperature series and seasonal autoregressive integrated moving-average (sarima) model applied to the denoised series. and nonstationary measures are used in model construction in all processes. model performance is tested in three years and one year and a half ahead forecasting. 2. study area and data collection temperature time-series data is used in this study. our study aims to build a well fit model representing the series and forecasting the future temperature trend. the study area is in eritrea, and one city is chosen, asmara. we chose asmara as the study location to represent eritrea's high land climatic condition as it is the central area of the region. daily average temperature data of 30 years collected from january 1, 1991, to december 31, 2020, from the website int. j. anal. appl. (2022), 20:17 5 https://power.larc.nasa.gov/data-access-viewer/. collected time series data has strong seasonality, irregular patterns and trends. we do not have any missing data. eritrea found in east africa or the horn of africa, between 12° and 18° north, and 36° and 44° east. it borders the red sea, ethiopia, djibouti, and sudan. asmara is the capital city of eritrea, home to almost 1 million people, and it is the largest and most populated city. asmara has a high temperature in the spring from april to june and again in september reaches an averagely of 27.8°c. it went to the lowest temperature from november to february, on averagely, to 21.6°c. the horn africa region has an unpredictable change in climate, which causes frequent droughts. so as eritrea is part of this region, the climate signals and future prediction are essential. for a daily time series t y with a sample size of n, the statistical description such as mean, standard deviation, minimum and maximum observation and trend analysis with mann–kendall (m–k) test are used in this study, table 1. 3. method 3.1. autoregressive integrated moving-average the main point of time series analysis using the arima model is to cautiously analyze and rigorously process the past observation to develop a suitable model that describes the series's inherent structure. it brings to light the possibility of explaining the data to facilitate prediction, monitoring, and control of the data. autoregressive integrated moving average (arima) models are one approach for this process. arima is a prevalent model for time series prediction; it has been widely used since box and jenkins initially proposed it. arima model is a combination of three different models, the auto-regressive (ar), integrated (i), and moving average (ma). the general equation for arima (p, d, q) model is: 𝝓𝒑(𝑳)(𝟏 − 𝑳) 𝒅𝒀𝒕 = 𝒄 + 𝜽𝒒(𝑳)𝜺𝒕 (3. 1) where 𝜙𝑝(𝐿) represent the ar and 𝜃𝑞 (𝐿) the ma part of the model with 𝜀𝑡 error[2], [9]. the ar model is predicted 𝑌𝑡 as the linear combination of past values of the variable. ba uses the linear combination of past forecasting error of a variable. arima model combines these models with differencing in nonstationary time series. for a series of data with seasonal behavior seasonal arima, sarima(p, d, q)(p, d, q)s models are appropriate to use to handle seasonality. it extends the arima model, explicitly supporting data analysis with a seasonal component. 𝜱𝑷(𝑳 𝒔)𝝓𝒑(𝑳)(𝟏 − 𝑳 𝒔)𝑫(𝟏 − 𝑳)𝒅𝒀𝒕 = 𝒄 + 𝜣𝑸(𝑳 𝒔)𝜽𝒒(𝑳) 𝜺𝒕 (3. 2) https://power.larc.nasa.gov/data-access-viewer/ 6 int. j. anal. appl. (2022), 20:17 3.2. wavelet analysis wavelet analysis is a mathematical model that reveals nonstationary time series data information in the time and frequency domain. that is why it is suitable for the nonstationary and seasonal temperature series. the main advantage of wavelet transform is that it decomposes a signal into different frequencies at a different resolution and is used as an alternative to fourier transform. the basic wavelet function called "mother wavelet" is defined as: 𝝍𝒂,𝒃(𝒚) = 𝟏 √|𝒂| 𝝍 ( 𝒕−𝒃 𝒂 ) for 𝒂, 𝒃 𝝐𝑹; 𝒂 ≠ 𝟎 (3. 3) the parameter a is called the scaling parameter and measure the degree of compression, while b the translation parameter moves the function in the time location to analyze the signal. the transformation process is implemented using a multi-resolution decomposition technique. it divides the original series 𝑌𝑡 into different domain components, the detail (high frequency) series 𝑑𝑗,𝑡 and the smooth (approximate) series 𝐴𝐿,𝑡 using a high pass filter and low pass filter, respectively. there exist various choices of wavelet basis functions such as haar, daubechies, symlet, meyer, biorthogonal wavelet, etc. a detailed explanation of wavelet transform functions is in [11], [13]. in our study, the wavelet family was selected based on the ns measure of the residual between denoised series and original series. when conducting wavelet analysis, selecting the optimal number of decomposition levels is one of the keys to determining the performance of a model in the wavelet domain. based on [9], the number of decomposition levels is selected using the following formula: 𝑳 = 𝒊𝒏𝒕[𝒍𝒐𝒈(𝑵)] (3. 4) where l is the number of decomposition levels, n is time-series length. the original time-series data 𝑌𝑡 will be decomposed into its detailed 𝑑𝑗,𝑡 and approximate (smooth) series 𝐴𝐿,𝑡. and with the wavelet de-nosing method, the series was reconstructed by removing the high-frequency component. 𝒀𝒕 = 𝑨𝑳,𝒕 + ∑ 𝒅𝒋,𝒕 𝑳 𝒋=𝟏 for t=1,2,3….n. (3. 5) 3.3. empirical mode decomposition emd decomposes time series into components in which all are in a time domain and has the same length as the original series, where varying frequency in time is preserved, which is generally hidden in fourier transform. the overall result of the decomposition is to remove the highest frequencies or noise from a series successively. emd can extract valuable information from the original data series and has received int. j. anal. appl. (2022), 20:17 7 more attention in signal denoising. but it also has a disadvantage of mode mixed in the resulting imfs, which means similar time scales are broken down into different components. generally, emd decomposes a noisy signal into different imfs plus residuals. the decomposed signal is written as: 𝒀𝒕 = ∑ 𝒉𝒊,𝒕 𝒍 𝒊=𝟏 + 𝒓𝒕 (3. 6) where ℎ𝑖,𝑡 indicates the i th imf and 𝑟𝑡 represents the residual of the signal 𝑌𝑡 . in the emd, each imf function must satisfy the following conditions: 1. the number of maximum and minimum extrema and the number of zero crossings must either be equal or differ at most by one. 2. at any point in the imf series, the mean of the envelope defined by the local maxima and the local minima is zero. from the second condition, we can understand that imf is stationary, which simplifies the analysis, but imf can have variable amplitude and frequency. an iterative procedure of emd called the "sifting process" is adopted to extract the separate imf components. the detailed guidelines of the process can refer it in [31]. the residual signal contains information about the lower frequency components. the sifting process continues until the final residue is constant or a monotonic function, a function with only one maximum and minima, in which imf cannot be obtained. the denoising series is constructed by choosing a low-frequency component that has more relation with the actual series. selecting components was based on the ns value of the error between the denoised and original data. after adding the low-frequency component to reconstruct the new series, find the error by subtracting it from the training set. step1: let 𝑌𝑡 be the original time series, decompose the series with emd decomposition procedure to 𝑌𝑡 → 𝑖𝑚𝑓1, 𝑖𝑚𝑓2, . … , 𝑖𝑚𝑓𝑛 , 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 step2: construct the denoised series 𝑌 𝑡 𝑑 by adding low-frequency components 𝒀 𝒕 𝒅 = 𝒓𝒆𝒔𝒊𝒅𝒖𝒂𝒍 + ∑ 𝒊𝒎𝒇𝒏 , 𝒏−𝒊 𝒏 for i=1,2,….,n-1. (3. 7) step3: calculate the difference series between original and denoised data 𝛆𝐭 = 𝐘𝐭 − 𝐘 𝐭 𝐝, t=1, 2,…, n (3. 8) step4: if the ns value of the difference series is 𝑁𝑆(𝜀𝑡 ) ≤ 0.05 then 𝑌 𝑡 𝑑 is the new denoised series. if the value is bigger, go to step 2. and i=i+1. continue the process until we get a stationary residual series. 8 int. j. anal. appl. (2022), 20:17 3.4. hybrid decomposition models with arima model the noise component affects the forecasting results' accuracy significantly because arima model cannot handle nonstationary data. to solve this problem, wavelet and emd were chosen as denoising methods and used the seasonal arima model to control the series's seasonal behavior. the first step is to de-noise the series with emd and wavelet analysis. in each method, the ns value of residual between the denoised series and the actual series is checked. the removed component must be stationary to consider noise or a series without trend patterns. the advantage of using ns measurement is to check if there is any trend in the removed component, which makes the ns value bigger. the threshold we select is 0.05, and a higher value is considered a nonstationary residual. figure 1: model construction flow chart note: 𝑌𝑡 and 𝑌𝑡 𝑑 donate the actual and denoised time series respectively with time t = 1, 2, …, n 3.5. model selection and forecasting performance evaluation model selection is an essential task of selecting a well-fitted model from a candidate set of models. a good model selection technique finds a model that shows the data's features for data points collected under random noise and predicts future points without any available future information. aic and bic are model selection criteria’s, which are the most used techniques in arima model selection. test how well the model fits the time series. another more effective method can be used based on the residual characteristics. by checking if the estimated residual is int. j. anal. appl. (2022), 20:17 9 close to noise? let 𝑌𝑡 been a time series data collected somewhere in the past, and the relation we need between the collected series of data and the actual model f which represents it is written as: 𝒀𝒕 = 𝒇(𝒕) + 𝜺𝒕 (3. 9) where 𝜀𝑡 is identically and independently distributed noise. if 𝑓 is the estimated function of f, then the estimated residual 𝜀�̂� of the model at each time t={t=1, 2, …., n} is: �̂�𝒕 = (𝒚𝒕 − 𝒇(𝒕)) + (𝒇(𝒕) − �̂�(𝒕)) (3. 10) where 𝑦𝑡 denote the observation at time t. let's rewrite the equation as: 𝜀�̂� = 𝑦𝑡 − 𝑓(𝑡) (3. 11) the estimated residual will be close to random noise if the estimated model correctly fits the actual time series. but if any trend appears in the residual series, it shows that the model fitted poorly [24][25]. from this perspective, model selection based on the residual structure is more effective. for checking residual series behaving random, a nonstationary measure can be taken to the residual series. an accurate estimation had a stationary residual sequence. the ns measure proposed by ding et al. is constructed based on ergodic theory and shannon entropy and is sensitive to nonstationary signals. the value of ns is from [0,1]. the bigger ns value represents the higher level of nonstationarity. in this study, 0.05 is taken as the threshold to check if the residual series t  is stationary. but measuring the residual stationarity is not enough. because even the residual is stationary, the model may over-fit the data and read the noisy component with actual patterns of the series. we use aic, bic, and ns comparatively in model selection and choose models that give better performance to solve such problems. in the forecasting process, for each selected model the forecasting performance was assessed using the mean absolute error (mae), the root means square error (rmse), and the mean absolute scaled error. 𝑴𝑨𝑬 = 𝟏 𝒏 ∑ |𝒀𝒕 − �̂�𝒕| 𝒏 𝒕=𝟏 (3. 12) 𝑹𝑴𝑺𝑬 = √ 𝟏 𝒏 ∑ (𝒀𝒕 − �̂�𝒕) 𝟐𝒏 𝒕=𝟏 (3. 13) where 𝑌𝑡 testing set and �̂�𝑡 forecasted data series with n length series. mae is the mean absolute difference between the testing set and forecasted data. rmse, the square root of the average square difference of test set and forecasted value. in both, the minimum value is a better result. the formula gives for mase: 10 int. j. anal. appl. (2022), 20:17 𝑴𝑨𝑺𝑬 = 𝑴𝑨𝑬 𝑸 (3. 14) where q is the scaling statistic computed on the training set, for non-seasonal time series, the q value is the mean absolute error of the one-step naïve forecasting method. 𝑸 = 𝟏 𝒏−𝟏 ∑ |𝒀𝒋 − 𝒀𝒋−𝟏| 𝒏 𝒋=𝟏 (3. 15) and for seasonal series: 𝑸 = 𝟏 𝒏−𝒎 ∑ |𝒀𝒋 − 𝒀𝒋−𝒎| 𝒏 𝒋=𝒎+𝟏 (3. 16) where m is the season period and n the size of the training set, if the mase result is less than one, it means a better forecasting value than the naïve forecast, and if the result is greater than one, it shows worse forecasting. a model with a minimum value in all tests is considered better performance. 4. result and discussion 4.1. data division the average temperature time series data used in this study were daily collected data of 30 years in asmara, eritrea. a nonparametric trend analysis is used in our research. the seasonal mannkendall (m-k) test shows an increasing trend in the series as the p-value is <0.0001. table 1. demonstrates the statistical description of the data and the type of trend in the series. sen’s slope refers to the slope of the trend, which is 0.013. as we see the structure of our data next step was to find a model that captures the trend and features of the series. and categorizing the series into groups is essential because different sets are needed for training data, while another set is used as testing data. training data is manipulated to train and develop the forecasting model. in contrast, testing data is treated as future data that must be used to compare the fitted model's predictability accuracy. it gives the ability to compare the effectiveness of different models of prediction. the partitioning way of the series is essential, which could affect the performance of the forecasting result of the models, and several aspects such as the data type and the size of the available data should be taken into consideration. in this research, the training set was chosen from january 1, 1991december 31, 2017, 90% of the data, and the test set from january 1, 2018 december 31, 2020, 10%. from the fig.2 of the actual data graph, it is noticeable that the temperature series had seasonality and instabilities. int. j. anal. appl. (2022), 20:17 11 table 1: descriptive statistics and trend analysis of the daily temperature data observations missing data minimum maximum mean std. deviation 10957 0 13.795 °c 29.320 °c 22.707 °c 2.285 mann-kendall trend test kendall's tau sen's slope p-value alpha, α there is a trend in the series 0.140 0.031 <0.0001 0.05 figure 2: original series graph 4.2. data decomposition two decomposition methods are used in denoising the temperature time series, and the first step is to decompose the noisy series into different components. in the wavelet transform procedure, the decomposition level is chosen based on the log value of the time series size de. based on eqn. (3.4). the optimal number of decomposition for this series is four levels. many researchers have used this method to choose the decomposition level and stated that the data series could be expressed helpfully and more meticulously. good forecasting results can be achieved for nonstationary time series [10]. in this study, the haar family, with decomposition level 4, was chosen compared to other families of wavelet transformation because it presented relevant smooth 12 int. j. anal. appl. (2022), 20:17 and depending on the residual series between the actual series and denoised series data. residual series tested for ns measurement and haar family denoising series were suitable methods. applying to the time series 𝑌𝑡 results in 5 series, 𝑎4 the approximate series and𝑑4, 𝑑3, 𝑑2 and 𝑑1 are the detail, with residual ns value of the training set 𝑌𝑡 and denoised series𝑌 𝑡 𝑑: 𝜀𝑡 = 𝑌𝑡 − 𝑌 𝑡 𝑑 ; 𝑁𝑆(𝜀𝑡 ) = 0.02 figure 3: wavelet decomposed component note: d1, d2, d3 and d4 are the detail component and a4 approximate component the second method used for denoising is emd. the training set 𝑌𝑡 decomposed into residual and 7 imf components, as shown in fig.4. the original series is decomposed into relatively stationary series components. the high-frequency component is extracted in the first few imfs, such as imf1, imf2 showing the high variation in the series. and the low-frequency components are extracted in the last imfs reveals the seasonal behavior and the trend of the series. a denoised series was built by adding residual components and the low-frequency imfs, starting from residual and imf7 to higher imf step-by-step. each time we add one imf, considering it as the new series, check the int. j. anal. appl. (2022), 20:17 13 ns value of the error t  between the new series and training set data and the ns value less or equal to 0.05 consider being stationary. continue the process until we get a stationary series 𝜀𝑡and a smaller ns value. in this study, the reconstructed series was: 𝑌 𝑡 𝑑 = {𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 + 𝑖𝑚𝑓7 + 𝑖𝑚𝑓6 + 𝑖𝑚𝑓5 + 𝑖𝑚𝑓4}, with smallest ns value 𝜀𝑡 = 𝑌𝑡 − 𝑌 𝑡 𝑑 ; 𝑁𝑆(𝜀𝑡 ) = 0.035 . figure 4: emd decomposed component the denoised series from both wt and emd with the original series are given in fig 5. denoised series must have same trend and seasonality as the original series. the wavelet denoised series except the noise component it had similar trend and seasonal patterns as the original series. but in the case of emd denoised series there is showing some different patterns which is out of the original series structure. 14 int. j. anal. appl. (2022), 20:17 figure 5: denoised series from each decomposition method vs training set data 4.3. arima model construction the time-series data is stationary if the acf graph cuts off rapidly, but if the acf graph dies down too slow, the series is considered nonstationary. in fig.6, it is noticeable that the acf graph drops gradually; thus, the training set of the actual data is considered nonstationary and nonlinear. in this study, decomposition methods were added to remove the noise component of the series, and as the decomposed components illustrate, the original series has seasonal behavior. therefore, the arima model will use seasonal differencing to handle the seasonal behavior of the series. sarima applied to the reconstructed denoised time series. we have two different methods; the emd based seasonal arima (emd-sarima) and wavelet-based seasonal arima (wt-sarima) are the hybrids models. and seasonal arima model was applied directly to the original series to compare the improvement of the hybrid methods. int. j. anal. appl. (2022), 20:17 15 figure 6: acf and pacf graph of temperature series data (training set) as the proposed procedure, the original and denoised data will be treated as input to develop their respective models. as mentioned in section ii, all procedures will be applied to the series with the assistance of matlab software. the model parameters are chosen based on the aic, bic, and ns values residual series between the training set and model predicted data. in aic and bic criteria, a model with the minimum result shows it fits the data well. on the other hand, measuring the ns value to the residual, the difference between model prediction and actual observation (training set), will tell us if there is ignored or not captured information of the original data. it will handle the under-fitting problem. so a well fit model has a smaller value in those three methods. but in some models, the result of three model selection criteria may not reduce proportionally. in this situation model is selected by balancing these three criteria to find the minimum value of aic and bic and stationary residual series where the residual ns value must be less or equal to 0.05. results of aic, bic and ns of sample models from wt and emd based sarima are listed in table 2. the table shows that aic, bic result and ns value for wt-sarima hybrid models do not have proportional change. in the model with smaller aic, bic has a bigger residual ns value and vice versa. comparatively, sarima(3,0,2)(1,1,1)365 has a better result. sarima(2,0,1)(1,1,1)365 and sarima(3,0,2)(1,1,1)365 selected to test the forecasting performance examine the effect of residual structure in model performance. 16 int. j. anal. appl. (2022), 20:17 sarima(2,0,1)(1,1,1)365 has the minimum aic and bic value, but as shown in table 2., the ns value of its residual is greater than 0.05, which implies its residual series is not stationary. while sarima(3,0,2)(1,1,1)365 was chosen based on the minimum value of the three test. similarly sarima(3,1,2)(1,1,1) and sarima(3,1,1)(0,1,1) selected from direct applied models on training set without decomposition methods. but still, for all series, the ns value is bigger than 0.05. applying the sarima model directly to the nonstationary series had a weak performance. a bigger ns value, as we say it early, there is a trend in the residual series because the model has not captured every structure of the series. the other hybrid models were emd based seasonal arima (emd-sarima). unlike wt-sarima in the emd denoised series, the result of three model selection methods was proportional, so we chose one model with a minimum value in all the tests, sarima(7,0,0)(1,1,0)365. table 3. illustrate all the result of the selected models, and all hybrid models have stationary residual series, which show how well they fit the training set. table 2: sample models for aic, bic and ns value of wt-sarima models and emd-sarima model aic bic ns wt-sarima sarima(2,0,1)(1,1,1)365 3.2844e+03 3.327e+03 0.066 sarima(3,0,2)(0,1,1)365 3.3928e+03 3.4432e+03 0.057 sarima(3,0,2)(1,1,1)365 3.3589e+03 3.4165e+03 0.039 sarima(2,0,1)(0,1,1)365 3.4762e+03 3.5122e+03 0.024 sarima(2,0,1)(1,1,0)365 5.7783e+03 5.8143e+03 0.015 emd-sarima sarima(7,0,0)(1,1,0)365 -1.4393e+05 -1.4387e+05 0.038 sarima(8,0,0)(1,1,0)365 -1.4394e+05 -1.4386e+05 0.038 sarima(7,0,0)(0,1,0)365 -1.4108 e+05 -1.4102e+05 0.038 int. j. anal. appl. (2022), 20:17 17 table 3: selected models from wt-sarima, emd-sarima and sarima applied to original data model ns aic bic actual data sarima(3,1,2)(1,1,1) 0.72 2.4365e+04 2.4422e+04 sarima(3,1,1)(0,1,1) 0.54 2.4533e+04 2.4570e+04 wavelet sarima(2,0,1)(1,1,1)365 0.066 3.2844e+03 3.327e+03 sarima(3,0,2)(1,1,1)365 0.039 3.3589e+03 3.4165e+03 emd sarima(7,0,0)(1,1,0)365 0.038 -1.4393e+05 -1.4387e+05 4.4. effectiveness evaluation result the mae, mase, and rmse results of testing data for all hybrid forecasting models and sarima are exhibited in fig 7, table 4 and 5. depict the graph of the forecasted result of chosen three models. smaller error testing result indicates a better forecasting accuracy approach. and tables 4 and 5. showing the wt-sarima(3,0,2)(1,1,1)365 model has the best performance in both 1095 and 546 days prediction. it gives a smooth series and captures the original series seasonal behavior. however, emdsarima model has weak performance. referring to both tables, it is clear that the mae, mase and rmse result of emd based sarima model is bigger than the results of all other methods. and fig 7. depicts the graph also shows that it weakly fit the original time series data. on the other hand, in wt-sarima and directly applied models, we chose two models based on minimum aic and bic values with bigger ns measures of residual series and models with smaller ns values of the residual. the forecasting performance improved in both sarima and wtsarima models selected based on residual test, which has smaller ns results. in the modelling process, sometimes the model over-fit the series by coping with the noisy part or under-fits the series when the model ignores some information or features of the actual series. that gives a wrong picture of the original time series structure. for this reason, we have to choose appropriate model selection methods. aic and bic are most used methods in model selection, but as we can see the results models chosen base on minimum value of both criteria are having nonstationary residual series. in other word, there is some information which is not captured by the model, using ns measure in the residual help to control this problem. and applying these three criteria improves the accuracy of the prediction. following all performance evaluation outcomes, we can deduce that the proposed hybrid model, wavelet transformation and sarima models can give a superior and improved prediction outcome than the straightforward application of the arima model and others. 18 int. j. anal. appl. (2022), 20:17 table 4: forecasting performance of models for 1095 days model mase rank rmse rank mae ran k actual data sarima(3,1,2)(1,1,1) 0.844633 4 1.51558 4 1.201135 4 sarima(3,1,1)(0,1,1) 0.822906 3 1.47866 3 1.170238 3 wavelet sarima(2,0,1)(1,1,1)365 0.761913 2 1.359894 2 1.0835 2 sarima(3,0,2)(1,1,1)365 0.735675 1 1.302188 1 1.046187 1 emd sarima(7,0,0)(1,1,0)365 0.890858 5 1.569255 5 1.266871 5 table 5: 546 days forecasting test model mase rank rmse rank mae rank actual data sarima(3,1,2)(1,1,1) 0.906336 4 1.579976 4 1.288881 4 sarima(3,1,1)(0,1,1) 0.872747 3 1.526071 3 1.241115 3 wavelet sarima(2,0,1)(1,1,1)365 0.791927 2 1.382071 2 1.126183 2 sarima(3,0,2)(1,1,1)365 0.743171 1 1.314283 1 1.056848 1 emd sarima(7,0,0)(1,1,0)365 0.982199 5 1.704065 5 1.396765 5 in this work, forecasting performance is tested for long-term predictions. frist evaluation was for 10% of the original data, which is 1095days, and the performance of all models was checked with residual test table 4. the prediction performance was evaluated for 546 days (year and a half) results are in table 5. and the hybrid models of wavelet transformation and sarima had the smallest value in all tests. in general, the wt-sarima model has the highest performance in this study and captures all critical features of the original series. emdsarima has weak performance in this study. in our opinion, this could be because emd has a weak ability to read large size data and mix information in different components. in the emd denoising process, all the original series features were not captured. as we can see in fig 7. the emd-sarima predicted series graph has some different patterns from the original series. int. j. anal. appl. (2022), 20:17 19 figure 7: 1095 days forecasting series vs testing set table 6: parameter estimation of wt-sarima (3,0,2)(1,1,1)365 coef. std. error t-statistic p-value ar{1} -0.45845 0.020271 -22.616 2.98e-113 ar{2} 0.43318 0.011512 37.629 7.1269e-310 ar{3} 0.90625 0.018038 50.241 0 sar{365} -0.1488 0.006316 -23.575 6.88e-123 ma{1} 1.4175 0.023856 59.417 0 ma{2} 0.91483 0.022793 40.136 0 sma{365} -0.8159 0.003962 -205.93 0 from eqn.(3. 2). the equation of wtsarima(3,0,2)(1,1,1)365 model with estimated parameters, l lag operator and t  the error at t is defined as: (1 − 𝛷365𝐿 365)(1 − 𝜙1𝐿 − 𝜙2𝐿 2 − 𝜙3𝐿 3)(1 − 𝐿365)𝑌𝑡 = (1 + 𝛩365𝐿 365)(1 + 𝜃1𝐿 + 𝜃2𝐿 2)𝜀𝑡 the estimated coefficients value, standard error, t-statistic and p-value of the model are given in table 6. since all p-values are less than 0.05, the results are statistically significant. 20 int. j. anal. appl. (2022), 20:17 5. conclusion eritrea is located in the area where climate change is a major issue. and the climate of this country is affected by topography, which divide the country into high land and low land. most of the population lives in the high land region and more than 70% of the population life depends on agriculture. therefore, studies conducting on climate change are important to understand the trend and future prediction. as temperature is one element of climate change in this study we focus on finding a model for long term prediction of temperature in the city asmara. this study assessed the capability of prediction of sarima based two different decomposition methods and the characteristics of the temperature data. the result from mann-kendall test show increasing trend in the series with kendall’s tau 0.014 and sen’s slope 0.031. the wavelet transform and empirical mode decomposition were used to denoise the temperature series and sarima applied to the denoised series. and wavelet transform base sarima model give better prediction results. the selected wt-sarima model can now forecast future daily temperature values. due to the fundamental importance of forecast accuracy, a test should be performed to verify the forecasting accuracy by comparing the forecast values with observational values in the testing set. the test methods can also support avoiding under-fitting or over-fitting. and as we see in the 1095 and 546 days forecasting result tables, it had a good performance. the temperature time series had seasonal behaviors, trends, and irregularities. appling arima models directly without preprocessing will not give accurate results. as we see in this paper and much other research, using decomposition methods with the arima model provides better performance. using the hybrid models of wt with sarima models had shown improvement in the accuracy of the results we had. our study supports using hybrid models instead of directly applying the arima model to nonstationary and nonlinear data. in this study, wavelet transformation with the seasonal autoregressive integrated moving average model was best to choose for our time series data. but as the result and graph of the forecasting test elucidate the hybrid model base, emd has weak performance. fig 8. shows that the model has some different seasonal patterns. the other important point we focused on is the model selection problem. using of nonstationary measurement in model selection shows good performance compare to aic and bic in this study. ns measure helps us check if the residual series of the model has a trend or behaves nonrandom, which indicate if there is removed information or trend from the series and treated as noise. if there is any trend in the residual, the ns value gets bigger. ns measure results can show how well int. j. anal. appl. (2022), 20:17 21 the models capture all the information and structure of the series. as we see in the model construction and forecasting evaluation, models chosen based on the ns value has higher performance. as we are going for minimum aic and bic, the model's residual is showing nonstationarity. and this is one drawback of aic and bic we see in this study. measuring the ns value of the residual is practical to find a well-fitted model. acknowledgement this work was supported by the national key research and development program of china (no. 2020yfa0714200). conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.g. ghebrezgabher, x. yang, spatio-temporal assessment of climate change in eritrea based on precipitation and temperature variables, world wide j. multidiscip. res. develop. 4 (2018), 1–10. [2] ]m. murat, i. malinowska, m. gos, j. krzyszczak, forecasting daily meteorological time series using arima and regression models, int. agrophys. 32 (2018), 253–264. https://doi.org/10.1515/intag2017-0007. [3] d.b. lobell, a. sibley, j. ivan ortiz-monasterio, extreme heat effects on wheat senescence in india, nat. clim. change. 2 (2012), 186–189. https://doi.org/10.1038/nclimate1356. [4] m.a. semenov, p.r. shewry, modelling predicts that heat stress, not drought, will increase vulnerability of wheat in europe, sci. rep. 1 (2011), 66. https://doi.org/10.1038/srep00066. [5] n. subash, a.k. sikka, trend analysis of rainfall and temperature and its relationship over india, theor. appl. climatol. 117 (2014), 449–462. https://doi.org/10.1007/s00704-013-1015-9. [6] e. el-mallah, s. elsharkawy, time-series modeling and short term prediction of annual temperature trend on coast libya using the box-jenkins arima model, adv. res. 6 (2016), 1–11. https://doi.org/10.9734/air/2016/24175. [7] k. anitha kumari, n. kumar boiroju, p.r. reddy, forecasting of monthly mean of maximum surface air temperature in india, int. j. stat. math. 9 (2014), 14–19. [8] p. chen, a. niu, d. liu, w. jiang, b. ma, time series forecasting of temperatures using sarima: an example from nanjing, iop conf. ser.: mater. sci. eng. 394 (2018), 052024. https://doi.org/10.1088/1757-899x/394/5/052024. https://doi.org/10.1515/intag-2017-0007 https://doi.org/10.1515/intag-2017-0007 https://doi.org/10.1038/nclimate1356 https://doi.org/10.1038/srep00066 https://doi.org/10.1007/s00704-013-1015-9 https://doi.org/10.9734/air/2016/24175 https://doi.org/10.1088/1757-899x/394/5/052024 22 int. j. anal. appl. (2022), 20:17 [9] a.h. nury, k. hasan, md.j.b. alam, comparative study of wavelet-arima and wavelet-ann models for temperature time series data in northeastern bangladesh, j. king saud univ. sci. 29 (2017), 47–61. https://doi.org/10.1016/j.jksus.2015.12.002. [10] n.q.n. md-khair, r. samsudin, a. shabri, wavelet transform and autoregressive integrated moving average combination approach in crude oil prices forecasting, int. j. innov. comput. 8 (2018), 41-48. https://doi.org/10.11113/ijic.v8n2.177. [11] a. ben mabrouk, n. ben abdallah, z. dhifaoui, wavelet decomposition and autoregressive model for time series prediction, appl. math. comput. 199 (2008), 334–340. https://doi.org/10.1016/j.amc.2007.09.067. [12] t. kriechbaumer, a. angus, d. parsons, m. rivas casado, an improved wavelet–arima approach for forecasting metal prices, resources policy. 39 (2014), 32–41. https://doi.org/10.1016/j.resourpol.2013.10.005. [13] s.a.a. karim, m.t. ismail, m.k. hasan, j. sulaiman, denoising the temperature data using wavelet transform, appl. math. sci. 7 (2013), 5821–5830. https://doi.org/10.12988/ams.2013.38450. [14] a. araghi, m. mousavi baygi, j. adamowski, j. malard, d. nalley, s.m. hasheminia, using wavelet transforms to estimate surface temperature trends and dominant periodicities in iran based on gridded reanalysis data, atmospheric res. 155 (2015), 52–72. https://doi.org/10.1016/j.atmosres.2014.11.016. [15] k.-m. lau, h. weng, climate signal detection using wavelet transform: how to make a time series sing, bull. amer. meteor. soc. 76 (1995), 2391–2402. https://doi.org/10.1175/15200477(1995)076<2391:csduwt>2.0.co;2. [16] m. sifuzzaman, m. islam, m. ali, application of wavelet transform and its advantages compared to fourier transform, j. phys. sci. 13 (2009), 121–134. [17] n. al bassam, v. ramachandran, s. eratt parameswaran, wavelet theory and application in communication and signal processing, in: s. mohammady (ed.), wavelet theory, intechopen, 2021. https://doi.org/10.5772/intechopen.95047. [18] g. bousaleh, f. hassoun, t. ibrahim, application of wavelet transform in the field of electromagnetic compatibility and power quality of industrial systems, in: 2009 international conference on advances in computational tools for engineering applications, ieee, beirut, lebanon, 2009: pp. 284–289. https://doi.org/10.1109/actea.2009.5227896. [19] n.e. huang, z. shen, s.r. long, m.c. wu, h.h. shih, q. zheng, n.-c. yen, c.c. tung, h.h. liu, the empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis, proc. r. soc. lond. a. 454 (1998), 903–995. https://doi.org/10.1098/rspa.1998.0193. [20] m. kedadouche, m. thomas, a. tahan, a comparative study between empirical wavelet transforms and empirical mode decomposition methods: application to bearing defect diagnosis, mech. syst. signal https://doi.org/10.1016/j.jksus.2015.12.002 https://doi.org/10.11113/ijic.v8n2.177 https://doi.org/10.1016/j.amc.2007.09.067 https://doi.org/10.1016/j.resourpol.2013.10.005 https://doi.org/10.12988/ams.2013.38450 https://doi.org/10.1016/j.atmosres.2014.11.016 https://doi.org/10.1175/1520-0477(1995)076%3c2391:csduwt%3e2.0.co;2 https://doi.org/10.1175/1520-0477(1995)076%3c2391:csduwt%3e2.0.co;2 https://doi.org/10.5772/intechopen.95047 https://doi.org/10.1109/actea.2009.5227896 https://doi.org/10.1098/rspa.1998.0193 int. j. anal. appl. (2022), 20:17 23 process. 81 (2016), 88–107. https://doi.org/10.1016/j.ymssp.2016.02.049. [21] h. wang, l. liu, z. (sean) qian, h. wei, s. dong, empirical mode decomposition–autoregressive integrated moving average: hybrid short-term traffic speed prediction model, transport. res. record. 2460 (2014), 66–76. https://doi.org/10.3141/2460-08. [22] z.y. wang, j. qiu, f.f. li, hybrid models combining emd/eemd and arima for long-term streamflow forecasting, water. 10 (2018), 853. https://doi.org/10.3390/w10070853. [23] y. zhou, m. huang, lithium-ion batteries remaining useful life prediction based on a mixture of empirical mode decomposition and arima model, microelectron. reliab. 65 (2016), 265–273. https://doi.org/10.1016/j.microrel.2016.07.151. [24] q. tan, h. jiang, y. ding, model selection method based on maximal information coefficient of residuals, acta math. sci. 34 (2014), 579–592. https://doi.org/10.1016/s0252-9602(14)60031-x. [25] z. zhou, z. zhou, l. wu, calibration for parameter estimation of signals with complex noise via nonstationarity measure, complexity. 2021 (2021), 8840757. https://doi.org/10.1155/2021/8840757. [26] y. ding, w. fan, q. tan, k. wu, y. zou, nonstationarity measure of data stream (in chinese), acta math. sci. 30a (2010), 1364–1376. [27] k. wu, nonstationarity of stock returns, in: m. bohner, y. ding, o. došlý (eds.), difference equations, discrete dynamical systems and applications, springer international publishing, cham, 2015: pp. 153–165. https://doi.org/10.1007/978-3-319-24747-2_12. [28] q. tan, the nonstationarity measure of time series and its application (in chinese), ph.d. thesis, university of chinese academy of sciences, beijing, china, 2013. [29] j. lan, q. tan, and q. dong, selection of forecasting models for irrigation water consumption based on nonstationarity measure (in chinese), eng. j. wuhan univ. 47 (2014), 721–725. [30] q. tan, l. wu, b. li, decomposition of noise and trend based on emd and nonstationarity measure (in chinese), acta math. sci. 36a (2016), 783–794. [31] s. maheshwari and a. kumar, empirical mode decomposition: theory & applications, int. j. electron. electric. eng. 7 (2014), 873–878. [32] b. yadollah, f. gharib, b. ali, a study and prediction of annual temperature in shiraz using arima model, geograph. space, 12 (2012), 127-144. [33] n. al bassam, v. ramachandran, s. eratt parameswaran, wavelet theory and application in communication and signal processing, in: s. mohammady (ed.), wavelet theory, intechopen, 2021. https://doi.org/10.5772/intechopen.95047. [34] y. wei, j. wang, c. wang, network traffic prediction based on wavelet transform and season arima model, in: d. liu, h. zhang, m. polycarpou, c. alippi, h. he (eds.), advances in neural networks – isnn 2011, springer berlin heidelberg, berlin, heidelberg, 2011: pp. 152–159. https://doi.org/10.1007/978-3-642-21111-9_17. https://doi.org/10.1016/j.ymssp.2016.02.049 https://doi.org/10.3141/2460-08 https://doi.org/10.3390/w10070853 https://doi.org/10.1016/j.microrel.2016.07.151 https://doi.org/10.1016/s0252-9602(14)60031-x https://doi.org/10.1155/2021/8840757 https://doi.org/10.1007/978-3-319-24747-2_12 https://doi.org/10.5772/intechopen.95047 https://doi.org/10.1007/978-3-642-21111-9_17 int. j. anal. appl. (2023), 21:21 the möbius invariant qth spaces munirah aljuaid∗ department of mathematics, northern border university, arar 73222, saudi arabia ∗corresponding author: moneera.mutlak@nbu.edu.sa abstract. in this article, we introduce a new space of harmonic mappings that is an extension of the well known space qt in the unit disk d in term of non decreasing function. several characterizations of the space qth are investigated. we also define the little subspace of q t h. finally, the boundedness of the composition operators cϕ mapping into the space qth and q t h,0 are considered. 1. introduction a harmonic mapping on a simply connected domain ψ is a complex-valued function k such that the laplace’s equation satisfied ∆k := 4kηη ≡ 0, on ψ, where kηη represents the mixed complex derivative of k. the harmonic mapping k admits a representaion of the form f + g, where f and g are analytic functions. this representaion is unique up to an additive constant. in this work, we consider all the functions defined on the open unit disk d := {η ∈ c : |η| < 1} so, the representaion of k is given by k = f + g and g(0) = 0. let h(d) denotes the collection of all analytic functions on d and h(d) be the collection of harmonic mappings on d. the operator theory of spaces of analytic functions on a various settings on the unit disk has been completely analyzed and a enormous amount of research papers on this matter have appeared in the literature, but the study of a similarly coverage in the harmonic setting is still limited. received: jan. 27, 2023. 2020 mathematics subject classification. 47b33. key words and phrases. qt space; harmonic mapping; composition operators. https://doi.org/10.28924/2291-8639-21-2023-21 issn: 2291-8639 © 2023 the author(s). https://orcid.org/0000-0002-5748-9738 https://doi.org/10.28924/2291-8639-21-2023-21 2 int. j. anal. appl. (2023), 21:21 in recent years, some papers have concentrated on the study of harmonic mappings. besides [2], for characterization of bloch type spaces of harmonic mapping, see [6], for harmonic zygmund spaces. in [18], the authors investigate the compactness and boundedness of cϕ mapping into weighted banach spaces of harmonic mappings. we also encourage the reader to see the additional references related to the harmonic mappings such as [ [21] [5], [16], [14], [15], [17], [13], [7], [8], [10], [11], [12], [17], [9]]. the results carried out in [19] bring the interesting question for whether we can extend the space qt to the harmonic setting and study the operator theoretic properties of cϕ. 2. preliminaries and background we start this section with several preliminaries facts on the spaces that will be used in this work. harmonic bloch space bh can be seen as the collection of k ∈ h(d) and the a semi-norm bk satisfies the following condition bk := sup η∈d (1 −|η|2)(|f ′(η)| + |g′(η)|) < ∞. (2.1) bh is a banach space when it is equipped with the harmonic bloch norm defined as ‖k‖bh := |k(0)| + bk. bh space extends the well known bloch space b. an analytic function f ∈b if and only if bf = sup η∈d (1 −|η|2)|f ′(η)| < ∞, (2.2) with norm ‖f‖b = |f (0)| + bf . in [3], the author obtains that the bloch constant of k can be written as follows bk := sup η∈d (1 −|η|2)(|kη(η)| + |kη̄(η)|) < ∞. (2.3) and max{bf ,bg}≤ bk ≤ bf + bg. (2.4) consequently, a harmonic mapping k belongs to the harmonic bloch space if and only if the functions f ,g ∈ h(d) such that k = f + ḡ with g(0) = 0 are in the classical bloch space. for more details, see [2]. the little harmonic bloch space bh,0 is the subspace of bh such that bh,0 := {k ∈bh : lim |η|→1 (1 −|η|2) ( |kη(η)| + |kη̄(η)| ) = 0}. int. j. anal. appl. (2023), 21:21 3 and the little bloch spaces b0 defined as b0 := {f ∈b : lim |η|→1 (1 −|η|2)|f ′(η)| = 0}. consider nondecreasing function t : [0, +∞) → [0, +∞). the logarithmic order of t (r) is given by λ = lim r→∞ log∗ log∗t (r) log r , where log∗γ = max{0, log γ} if λ > 0, the logarithmic type of the function t (r) is given by γ = lim r→∞ log∗t (r) rλ , the space qt is the collection of analytic functions f defined on d and qt (f ) = sup ν∈d (∫ d (|f ′(η)|2t (g(η,ν))da(η) )1 2 < ∞, where da(η) represents the area measure on the unit disk and g(η,ν) = − log |σν(η)| is the green function of d with pole at ν ∈ d and σν(η) = (ν −η) (1 − ν̄η) be a möbius transformation of d. 3. the möbius invariant qth spaces we now introduce the harmonic qth space of harmonic mapping by a nondecreasing function t (r) on r ∈ [0,∞). definition 3.1. for nondecreasing function t : [0, +∞) → [0, +∞). a harmonic mapping k ∈h(d) is said to be in the class qth if [qt (k)]2 = sup ν∈d ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) < ∞, and the norm of qth is defined as: ‖k‖qt h := |k(0)| + qt (k). (3.1) the little harmonic qth,0 is the subspace of q t h such that qth,0 := { k ∈h(d) : lim |η|→1 ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) = 0 } . remark 3.1. as a special case when k ∈ h(d), the functions f ,g in the canonical decomosition of k are given by k = f and g ≡ 0. moreover, the collections of analytic function on the unit disk in the qth is just the space q t . 4 int. j. anal. appl. (2023), 21:21 corollary 3.1. for t : [0, +∞) → [0, +∞) be non-decreasing function. let f ∈ h(d), if k ∈h(d) be the real part of f or imaginary part of f then qt (k) = qt (f ) proof. assume f = re(k). then we have, k = 1 2 (f + f̄ ). therefore, qt (k) = ( sup ν∈d ∫ d ( 1 2 |f ′(η)| + 1 2 |f ′(η)|)2t (g(η,ν))da(η) )1 2 = ( sup ν∈d ∫ d |f ′(η)|2t (g(η,ν))da(η) )1 2 = qt (f ) in a similar way, assume f = im(k), then we have k = 1 2i f − 1 2i f̄ . thus, qt (k) = (sup ν∈d ∫ d ( 1 2 |f ′(η)| + 1 2 |f ′(η)|)2t (g(η,ν))da(η)) 1 2 = (sup ν∈d ∫ d |f ′(η)|2t (g(η,ν))da(η)) 1 2 = qt (f ) theorem 3.1. for t : [0, +∞) → [0, +∞) be non-decreasing function. let k = f +ḡ ∈h(d) where f ,g ∈ h(d).then f ,g ∈qt if and only if k ∈qth. moreover, if g(0) = 0, then 1 2 (‖f‖qt + ‖g‖qt ) ≤‖k‖qt h ≤ 2((‖f‖qt + ‖g‖qt )). proof. consider f ,g ∈qt and let k = f + ḡ. then f ′ = kη and g ′ = kη̄. therefore, (|kη(η)| + |hη̄(η)|)2 < 22(|kη(η)|2 + |kη̄(η)|2) the above inequality follows from the fact that for c1,c2 ≥ 0,( c1 + c2 2 )2 ≤ [max{c1,c2}]2 = max{c21 ,c 2 2}≤ c 2 1 + c 2 2 , int. j. anal. appl. (2023), 21:21 5 we have qt (k)2 = sup ν∈d ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) ≤ 22 [ sup ν∈d ∫ d (|kη(η)|)2t (g(η,ν))da(η) + sup ν∈d ∫ d (|kη̄(η)|)2t (g(η,ν))da(η) ] < ∞. therefore k ∈qth and, qt (f + ḡ)2 ≤ 4(qt (f )2 + qt (g)2). (3.2) taking the square root, we get qt (k) ≤ 2 √( qt (f )2 + qt (g)2 ) < 2 ( qt (f ) + qt (g) ) . moreover, using |k(0)| ≤ |f (0)| + |g(0)|, the upper estimate holds conversely, let k ∈qth and note that |f ′(η)|2 + |g′(η)|2 ≤ (|f ′(η)| + |g′(η)|)2, thus sup ν∈d ∫ d (|kη(η)|)2t (g(η,ν))da(η) + sup ν∈d ∫ d (|kη̄(η)|)2t (g(η,ν))da(η)) ≤ sup ν∈d ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) < ∞. therefore, both f and g are in the space qt and qt (f )2 + qt (g)2 ≤ qt (k)2. hence, by 3.2 1 2 [qt (f ) + qt (g)] ≤ √ qt (f )2 + qt (g)2. then, we combine these two inequalities to get 1 2 [qt (f ) + qt (g)] ≤ qt (k). by the assumption g(0) = 0, we have 1 2 |f (0)| ≤ |f (0)| = |k(0)|. therefore, 1 2 [‖f‖qt + ‖g‖qt ] ≤‖k‖qt h , we deduce the lower estimate. 6 int. j. anal. appl. (2023), 21:21 lemma 3.1. for t : [0, +∞) → [0, +∞) be non-decreasing function. then k ∈qth if and only if sup ν∈d (∫ d (|kη(η)| + |kη̄(η)|)2t (1 −|σν(η)|2)da(η) )1 2 < ∞, (3.3) proof. recall that for s ∈ (0, 1], we have −2 log s ≥ 1 − s2 and for s ∈ ( 1 4 , 1) we have − log s ≤ 4(1 − s2) assume k ∈qth then we have, qt (k) = sup ν∈d (∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) )1 2 (3.4) ≤ sup ν∈d (∫ d (|kη(η)| + |kη̄(η)|)2t (1 −|σν(η)|2)da(η) )1 2 (3.5) since ∫ d (|kη(η)| + |kη̄(η)|)2|dη| is increasing function on δ ∈ (0, 1), we have ∫ d (|kη(η)| + |kη̄(η)|)2|dη| ≤ ∫ d/d(0, 1 4 ) (|kη(η)| + |kη̄(η)|)2t (1 −|σν(η)|2)da(η) ≤ (qt (k))2. this inequality with 3.4, prove the theorem. � we now study the relationship between k ∈qth and the associated real and imaginary parts. proposition 3.1. for t : [0, +∞) → [0, +∞) be non-decreasing function. let k ∈h(d) and assume that τ be the real part of k and θ is the imaginary part of k such that τ = re(k) and θ = im(k). then k ∈qth, if and only if τ,θ ∈q t h. moreover 1 4 ( ‖τ‖qt h + ‖θ‖qt h ) ≤‖k‖qt h ≤‖τ‖qt h + ‖θ‖qt h . proof. assume τ,θ ∈ qth . due to linearity, k ∈ q t h and the upper estimate hold directly by the property of the norm (triangle inequality ) . let k ∈qth and recall that j(τ,θ) = τxθy −θxτy we have 2|j(τ,θ)| ≤ ‖∇τ‖2 + ‖∇θ‖2, (3.6) int. j. anal. appl. (2023), 21:21 7 where ∇τ = (τx,τy ) , and ∇θ = (θx,θy ). from this, we get (‖∇τ‖2 + ‖∇θ‖2 + 2j(τ,θ)) 1 2 + (‖∇τ‖2 + ‖∇θ‖2 − 2j(τ,θ)) 1 2 ≥ √ 2(‖∇τ‖2 + ‖∇θ‖2) 1 2 (3.7) by squaring (3.7), the left-hand side becomes ‖∇τ‖2 + ‖∇θ‖2 + 2j(τ,θ) + ‖∇τ‖2 + ‖∇θ‖2 − 2j(τ,θ) + 2 ( ‖∇τ‖2 + ‖∇θ‖2)2 − 4(j(τ,θ)2 )1 2 , thus, by neglecting the last term and simple calculation, we obtain 2(‖∇τ‖2 + ‖∇θ‖2). now, we may find |kη| + |kη̄| with respect to τ and θ by using the partials with respect to η and η̄, then calculating the modulus, after that applying (3.7) |kη| + |kη̄| = |τη + iθη| + |τη̄ + iθη̄| = 1 2 ∣∣τx + θy + i(θx −τy )∣∣ + 1 2 ∣∣τx −θy + i(θx + τy )∣∣ = 1 2 √(( τx + θy )2 + ( θx −τy )2) + 1 2 √(( τx −θy )2 + ( θx + τy )2) = 1 2 √( ‖∇τ‖2 + ‖∇θ‖2 + 2j(τ,θ) ) + 1 2 √( ‖∇τ‖2 + ‖∇θ‖2 − 2j(τ,θ) ) ≥ 1 √ 2 √ ‖∇τ‖2 + ‖∇θ‖2 ≥ 1 2 ( ‖∇τ‖ + ‖∇θ‖ ) , in the last step, we apply the following inequality ‖(η1,η2)‖≥ |η1| + |η2|√ 2 f or η1,η2 ∈ c. (3.8) therefore, (qt (k))2 ≥ 1 2 sup η∈d ∫ d (‖∇τ(η)‖ + ‖∇θη‖)2t (g(η,ν)da(η) ≥ 1 2 max{qtτ ,q t θ } ≥ 1 4 (qtτ + q t θ ) (3.9) therefore, by using inequality (3.8) one more time, we obtain |k(0)| ≥ 1 √ 2 (|τ(0)| + |θ(0)|) (3.10) now, combine (3.9) and (3.10) to get ‖k‖qt h ≥ 1 4 (‖τ‖qt h + ‖θ‖qt h ) 8 int. j. anal. appl. (2023), 21:21 thus, τ and θ are in qth, and that the other estimate is hold. theorem 3.2. (qth,‖ ·‖qt h ) is a banach space. proof. obviously, qth is a normed linear space, we only wish to show completeness. for each n ∈ n, let {kn} be a cauchy sequence in qth . by theorem 3.1, the analytic functions {fn} and {gn} such that kn = fn + ḡn with gn(0) = 0 are in qt and {fn} and {gn} are cauchy sequence in qt . by proposition 2.2 in [4], qt is complete. thus, {fn} and {gn} converge to f and g, respectively in the qt norm. define k = f + ḡ. then, k ∈qth by the estimates in theorem 3.1, and ‖kn −k‖qt h ≤ 2(‖fn − f‖qt + ‖gn −g‖qt ) → 0, as n →∞. we ends up with kn → k in qth. theorem 3.3. for nondecreasing function t : [0, +∞) → [0, +∞). the space qth is a subset of bh. moreover, for k ∈qth we have ‖k‖bh ≤ m‖k‖qt h , for some constant m > 0. proof. assume k ∈qth and let sup ν∈d ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) = m < ∞, for δ ∈ (0, 1) define d(�,�) := {η ∈ d : |σν(η)| < δ}. since t is nondecreasing function and by the change of variable w = σν(η) we have m ≥ ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) ≥ ∫ d(�,�) (|kη(η)| + |kη̄(η)|)2t ( log 1 σν(η) ) da(η) ≥ t ( log 1 δ ) ∫ d(�,�) (|kη(η)| + |kη̄(η)|)2da(η) = t ( log 1 δ ) ∫ |w|<δ (|(k ◦σν)w (w)| + |(k ◦σν)w̄ (w)|)2da(w) ≥ πδ2t ( log 1 δ ) (|(k ◦σν)ν(0)| + |(k ◦σν)ν̄(0)|)2 = πδ2t ( log 1 δ ) (|(kν(ν)| + |(kν̄(ν)|)2(1 −|ν|2)2 int. j. anal. appl. (2023), 21:21 9 fix δ0 ∈ (0, 1). thus sup ν∈d (1 −|ν|2)[|(kν(ν)| + |(kν̄(ν)|] ≤ √ m πδ20t ( log 1 δ0 ) therefore, bk ≤ qt (k) δ0 √ πt ( log 1 δ0 ) (3.11) we obtained that k ∈bh and qth ⊂bh. � theorem 3.4. if the logarithmic type γ and the logarithmic order λ of t (r) satisfying one of the following cases, (1) λ > 1, (2) γ > 2 and λ = 1, then the space qth has only constant functions(trivial space). proof. by theorem 3.3, it is sufficient to prove that for each non constant harmonic bloch function k can not be in the space qth. indeed, if either λ > 1 or γ > 2 and λ = 1, there is a sequence {rj} as j →∞, the sequence {rj}→∞ as follows lim j→∞ log∗ log∗t (rj) log rj = λ > 1, (3.12) or lim j→∞ log∗t (rj) rj = γ > 2, (3.13) in the case 3.12 or 3.13, we get lim j→∞ t (rj) e2rj = ∞. (3.14) set hj = e−rj , for j ∈ n, then lim j→∞ h2j t ( log 1 hj ) = ∞. (3.15) assume k ∈bh be a non-constant. then it is clear that the semi-norm bk 6= 0. however, by 3.11, and 3.15, as j →∞ we obtain sup ν∈d ∫ d (|kη(η)| + |kη̄(η)|)2t (g(η,ν))da(η) ≥ πb2k h 2 j t (log 1 hj ) →∞. that implies k /∈qth which proves the theorem. � the next theorem shows that the möbius invariance of qt space extends to the harmonic setting. 10 int. j. anal. appl. (2023), 21:21 theorem 3.5. for t : [0, +∞) → [0, +∞) be non-decreasing function. qth is a möbius invariant space. proof. it is obvious that rotations have no effect on the semi-norm qt (k). we wish to show qt (k ◦ ϕν) = q t (k), for ν ∈ d and k ∈qth. for ν ∈ d, and since ϕν is its own inverse, we have (1 −|η|2)|ϕ ′ (η)| = 1 −|ϕν(η)|2 and ϕ ′ ν(ϕν(η)) = 1 ϕ ′ ν(η) by change of variables ξ = ϕν(η), we get qt (k ◦ϕν)2 = sup ν∈d ∫ d t (1 −|ϕν(η)|2)[|(k ◦ϕν)η(η)| + |(k ◦ϕν)η̄(η)|]2da(η) = sup ν∈d ∫ d t (1 −|ϕν(η)|2)[|kη(ϕν(η))ϕ ′ ν(η)| + |(kη̄(ϕν(η))ϕ ′ ν(η))|] 2da(η) = sup ν∈d ∫ d t (1 −|ϕν(η)|2)|ϕ ′ ν(η)| 2[|kη(ϕν(η))| + |kη̄(ϕν(η))|]2da(η) = sup ν∈d ∫ d t (1 −|ξ|2)|ϕ ′ ν(ϕν(ξ))| 2[|kη(ξ)| + |(kη̄(ξ))|]2|ϕ ′ ν(ξ)| 2da(ξ) = sup ν∈d ∫ d t (1 −|ξ|2) 1 |ϕ′ν(ξ)|2 [|hη(ξ)| + |hη̄(η)|]2|ϕ ′ ν(ξ)| 2da(ξ) = sup ν∈d ∫ d t (1 −|ξ|2)[|kη(ξ)| + |kη̄(ξ)|]2da(ξ) = qt (k)2 as desired. finally, we move our attention to study the boundedness of composition operator cϕ from the harmonic bloch space bh to qth and q t h,0. 4. boundedness due to the representation of the harmonic mapping, the composition operator cϕ induced by analytic or a conjugate analytic self-maps of d is given by cϕk = k ◦ϕ, for all k belonging to a class of harmonic mappings. the following is a basic property of the harmonic bloch space was introduced in [20]. int. j. anal. appl. (2023), 21:21 11 lemma 4.1. for η ∈ d. if k1 , k2 ∈bh we have (1 −|η|2)−1 ≤ (k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)|. the next result which will be used in the proof of the main theorem of this section is a special case of theorem 3.6 in [1] lemma 4.2. for k ∈bh and ϕ : d → d, |k(ϕ(0))| ≤ |k(0)| + 1 2 log 1 + |ϕ(0)| 1 −|ϕ(0)| bk. theorem 4.1. for t : [0, +∞) → [0, +∞) be non-decreasing function. let ϕ be analytic function such that ϕ : d → d. then cϕ : bh →qth is bounded operator if and only if sup ν∈d ∫ d |ϕ′(η)|2 (1 −|ϕ(η)|2)2 t (g(η,ν))da(η) < ∞. (4.1) proof. let us assume 4.1 holds and let ρ21 be the supremum in 4.1. let η ∈ d and k ∈bh, then∫ d t (g(η,ν))[|(k ◦ϕ)η(η)| + |(k ◦ϕ)η̄(η)|]2da(η) = ∫ d t (g(η,ν))|ϕ ′ (η)|2[|kη(ϕ(η))| + |kη̄(ϕ(η))|]2da(η) ≤ b2k ∫ d t (g(η,ν)) |ϕ ′ z (ξ)|2 (1 −|ϕ(η)|2)2 da(η) ≤ ρ21b 2 k. therefore, qt (k ◦ϕ) ≤ ρ1 bk. since k ∈bh we have ‖cϕk‖2qt h = ( |k ◦ϕ(0)| + qt (cϕk) )2 ≤ ( |k(0)| + 1 2 log 1 + |ϕ(0)| 1 −|ϕ(0)| bk + ρ1 bk) )2 ≤ ρ2 ( |k(0)| + bk )2 = ρ2‖k‖2bh. where ρ = max{1,ρ1 + 1 2 log 1 + |ϕ(0)| 1 −|ϕ(0)| }. therefore, ‖cϕk‖qt h ≤ ρ‖k‖bh which implies that cϕ : bh →q t h is bounded. conversely, assume the boundedness of cϕ : bh → qth holds, then there is a positive constant ρ > 0 for all k ∈ bh, we have ‖cϕk‖qt h ≤ ρ‖k‖bh. on the other hand, by lemma 4.1 for all η ∈ d, there exist k1 , k2 ∈bh such that (1 −|η|2)−1 ≤ |(k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)| 12 int. j. anal. appl. (2023), 21:21 therefore, |ϕ(η)′|2[ 1 −|ϕ(η)|2 ]2 ≤ 2|(k1 ◦ϕ)η(η)|2 + 2|(k1 ◦ϕ)η̄(η)|2 + 2|(k2 ◦ϕ)η(η)|2 + 2|(k2 ◦ϕ)η̄(η)|2 ≤ 2[|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + 2[|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2 where the last inequity follows from the fact that for c1,c2 ≥ 0 and m > 1 we have cm1 + c m 2 ≤ (c1 + c2) m moreover,∫ d t (g(η,ν)) |ϕ(η)′|2( 1 −|ϕ(η)|2 )2 da(η) ≤ 2 ∫ d [ [|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + [|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2 ] t (g(η,ν))da(η) ≤ 2ρ2 ( ‖k1‖2bh + ‖k2‖ 2 bh ) , thus, take the supremum over all η ∈ d, the quantity 4.1 holds since ρ is a constant and k ∈bh. � theorem 4.2. for nondecreasing function t : [0, +∞) → [0, +∞). let ϕ be analytic function such that ϕ : d → d. then cϕ : bh →qth,0 is bounded operator if and only if lim |ν|→1 ∫ d |ϕ′(η)|2 (1 −|ϕ(η)|2)2 t (g(η,ν))da(η) = 0. (4.2) proof. by theorem 4.1, we know that cϕ : bh →qth is bounded since the condition 4.2 implies the following sup ν∈d ∫ d |ϕ′(η)|2 (1 −|ϕ(η)|2)2 t (g(η,ν))da(η) < ∞. we only wish to show that cϕk ∈qth,0 for each k ∈bh and this comes from the inequality∫ d t (g(η,ν))[|(k ◦ϕ)η(η)| + |(k ◦ϕ)η̄(η)|]2da(η) = ∫ d t (g(η,ν))|ϕ ′ (η)|2[|kη(ϕ(η))| + |kη̄(ϕz (η))|]2da(η) ≤ b2k ∫ d t (g(η,ν)) |ϕ ′ z (η)|2 (1 −|ϕ(η)|2)2 da(η) thus, cϕk ∈qth,0. conversely, consider cϕ : bh → qth,0 is bounded. by lemma 4.1 there exist k1 , k2 ∈ bh such that (1 −|η|2)−1 ≤ |(k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)| int. j. anal. appl. (2023), 21:21 13 then cϕk1 , cϕk2 ∈qth,0. therefore, lim |ν|→1 ∫ d t (g(η,ν)) |ϕ(η)′|2[ 1 −|ϕ(η)|2 ]2 da(η) ≤ 2 lim |ν|→1 ∫ d t (g(η,ν)) ( [|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + [|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2 ) da(η) = 0 then 4.2 holds and this complete the proof. � conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] m. aljuaid, the operator theory on some spaces of harmonic mappings, doctoral dissertation, george mason university, 2019. [2] m. aljuaid, f. colonna, characterizations of bloch-type spaces of harmonic mappings, j. funct. spaces. 2019 (2019), 5687343. https://doi.org/10.1155/2019/5687343. [3] f. colonna, the bloch constant of bounded harmonic mappings, indiana univ. math. j. 38 (1989), 829–840. https://www.jstor.org/stable/24895370. [4] p. wu, h. wulan, composition operators from the bloch space into the spaces qt , int. j. math. math. sci. 2003 (2003), 1973–1979. https://doi.org/10.1155/s0161171203207122. [5] m. aljuaid, f. colonna, composition operators on some banach spaces of harmonic mappings, j. funct. spaces. 2020 (2020), 9034387. https://doi.org/10.1155/2020/9034387. [6] m. aljuaid, f. colonna, on the harmonic zygmund spaces, bull. aust. math. soc. 101 (2020), 466–476. https: //doi.org/10.1017/s0004972720000180. [7] c. boyd, p. rueda, isometries of weighted spaces of harmonic functions, potential anal. 29 (2008), 37–48. https://doi.org/10.1007/s11118-008-9086-4. [8] s. chen, s. ponnusamy, a. rasila, lengths, areas and lipschitz-type spaces of planar harmonic mappings, nonlinear anal.: theory methods appl. 115 (2015), 62–70. https://doi.org/10.1016/j.na.2014.12.005. [9] sh. chen, s. ponnusamy, x. wang, landau’s theorem and marden constant for harmonic ν-bloch mappings, bull. aust. math. soc. 84 (2011), 19–32. https://doi.org/10.1017/s0004972711002140. [10] sh. chen, s. ponnusamy, x. wang, on planar harmonic lipschitz and planar harmonic hardy classes, ann. acad. sci. fen. math. 36 (2011), 567–576. [11] sh. chen, x. wang, on harmonic bloch spaces in the unit ball of cn, bull. aust. math. soc. 84 (2011), 67–78. https://doi.org/10.1017/s0004972711002164. [12] x. fu, x. liu, on characterizations of bloch spaces and besov spaces of pluriharmonic mappings, j. inequal. appl. 2015 (2015), 360. https://doi.org/10.1186/s13660-015-0884-0. [13] j. laitila, h.o. tylli, composition operators on vector-valued harmonic functions and cauchy transforms, indiana univ. math. j. 55 (2006), 719–746. https://www.jstor.org/stable/24902369. [14] w. lusky, on weighted spaces of harmonic and holomorphic functions, j. lond. math. soc. 51 (1995), 309–320. https://doi.org/10.1112/jlms/51.2.309. [15] w. lusky, on the isomorphism classes of weighted spaces of harmonic and holomorphic functions, stud. math. 175 (2006), 19–45. https://doi.org/10.1155/2019/5687343 https://www.jstor.org/stable/24895370 https://doi.org/10.1155/s0161171203207122 https://doi.org/10.1155/2020/9034387 https://doi.org/10.1017/s0004972720000180 https://doi.org/10.1017/s0004972720000180 https://doi.org/10.1007/s11118-008-9086-4 https://doi.org/10.1016/j.na.2014.12.005 https://doi.org/10.1017/s0004972711002140 https://doi.org/10.1017/s0004972711002164 https://doi.org/10.1186/s13660-015-0884-0 https://www.jstor.org/stable/24902369 https://doi.org/10.1112/jlms/51.2.309 14 int. j. anal. appl. (2023), 21:21 [16] a.l. shields, d.l. williams, bounded projections, duality, and multipliers in spaces of harmonic functions, j. reine angew. math. 299-300 (1978), 256–279. https://doi.org/10.1515/crll.1978.299-300.256. [17] r. yoneda, a characterization of the harmonic bloch space and the harmonic besov spaces by an oscillation, proc. edinburgh math. soc. 45 (2002), 229–239. https://doi.org/10.1017/s001309159900142x. [18] m. aljuaid, f. colonna, norm and essential norm of composition operators mapping into weighted banach spaces of harmonic mappings, preprint. [19] h. wulan, p. wu, characterizations of qt spaces, j. math. anal. appl. 254 (2001), 484–497. [20] a. kamal, q-type spaces of harmonic mappings, j. math. 2022 (2022), 1342051. https://doi.org/10.1155/ 2022/1342051. [21] m. aljuaid, m.a. bakhit, on characterizations of weighted harmonic bloch mappings and its carleson measure criteria, j. funct. spaces. 2023 (2023), 8500633. https://doi.org/10.1155/2023/8500633. https://doi.org/10.1515/crll.1978.299-300.256 https://doi.org/10.1017/s001309159900142x https://doi.org/10.1155/2022/1342051 https://doi.org/10.1155/2022/1342051 https://doi.org/10.1155/2023/8500633 1. introduction 2. preliminaries and background 3. the möbius invariant qth spaces 4. boundedness references int. j. anal. appl. (2022), 20:24 k −g−duals in hilbert c∗−modules mohamed rossafi1, fakhr-dine nhari2,∗ 1lasma laboratory department of mathematics, faculty of sciences dhar el mahraz, university sidi mohamed ben abdellah, p. o. box 1796 fez atlas, morocco 2laboratory analysis, geometry and applications department of mathematics, faculty of sciences, university of ibn tofail, p. o. box 133 kenitra, morocco ∗corresponding author: nharidoc@gmail.com abstract. generalized frames with adjointable operators called k−g−frame is a generalization of a g−frame. in this paper, we give some results of dual k−g−bessel sequence, finally we obtain a new properties of approximate k−g−duals in hilbert c∗−module. 1. introduction frames, introduced by duffin and schaefer [1] in 1952 to analyse some deep problems in nonharmonic fourier series by abstracting the fundamental notion of gabor [2] for signal processing. in 2000, frank-larson [4] introduced the concept of frames in hilbet c∗−modules as a generalization of frames in hilbert spaces. the basic idea was to consider modules over c∗−algebras of linear spaces and to allow the inner product to take values in the c∗−algebras [5]. many generalizations of the concept of frame have been defined in hilbert c∗-modules [3,6,8–10,12–15]. throughout this paper, h is considered to be a countably generated hilbert a−module. let {hj}j∈j be the collection of submodules of h where i is a finite or countable index set. end∗a(h,hj) is the set of all adjointable operator from h to hj. in particular end∗a(h) denote the set of all adjointable operators on h. received: feb. 28, 2022. 2010 mathematics subject classification. primary 41a58, secondary 42c15. key words and phrases. g-frame, k-g-frame, c∗-algebra, hilbert a-modules. https://doi.org/10.28924/2291-8639-20-2022-24 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-24 2 int. j. anal. appl. (2022), 20:24 define the module l2({hj}j∈j) = {{xj}j∈j : xj ∈ hj,‖ ∑ j∈j 〈xj,xj〉‖ < ∞} with a−valued inner product 〈x,y〉 = ∑ j∈j〈xj,yj〉, where x = {xj}j∈j and y = {yj}j∈j, clearly l2({hj}j∈j) is a hilbert a−module. in the following we briefly recall some definitions and basic properties. for a c∗-algebra a if a ∈a is positive we write a ≥ 0 and a+ denotes the set of positive elements of a. definition 1.1. [7]. let a be a unital c∗-algebra and h be a left a-module, such that the linear structures of a and u are compatible. h is a pre-hilbert a-module if h is equipped with an a-valued inner product 〈., .〉 : h ×h → a, such that is sesquilinear, positive definite and respects the module action. in the other words, (i) 〈x,x〉≥ 0 for all x ∈ h and 〈x,x〉 = 0 if and only if x = 0. (ii) 〈ax + y,z〉 = a〈x,z〉 + 〈y,z〉 for all a ∈a and x,y,z ∈ h. (iii) 〈x,y〉 = 〈y,x〉∗ for all x,y ∈ h. for x ∈ h, we define ||x|| = ||〈x,x〉|| 1 2 . if h is complete with ||.||, it is called a hilbert a-module or a hilbert c∗-module over a. for every a in c∗-algebra a, we have |a| = (a∗a) 1 2 and the a-valued norm on h is defined by |x| = 〈x,x〉 1 2 for x ∈ h. definition 1.2. suppose that x and y are hilbert a−modules and t ∈ end∗a(x,y ). the moorepenrose inverse of t (if it exists) is an element t + of end∗a(y,x) satisfying tt +t = t, t +tt + = t +, (tt +)∗ = tt +, (t +t )∗ = t +t lemma 1.1. let x and y be two hilbert a−modules and t ∈ end∗a(x,y ) then the moore-penrose inverse t + exixts if and only if t has a closed range. definition 1.3. suppose that k ∈ end∗a(h) and {λj}j∈j is a k − g−frame for h. a g−bessel sequence {γj}j∈j for h is said to be a dual k −g−bessel sequence of {λj}j∈j if kx = ∑ j∈j λ∗j γjx, ∀x ∈ h. definition 1.4. [3] a sequence {λj ∈ end∗a(h,hj), j ∈ j} is called a g−frame with respect to {hj}j∈j if there exist constants c,d > 0 such that for every x ∈ h c〈x,x〉≤ ∑ j∈j 〈λjx, λjx〉≤ d〈x,x〉. (1.1) as usual c and d are g−frame bounds of {λj}j∈j. if only upper inequality of (1.1) holds, then {λj}j∈j is called g−bessel sequence for h. int. j. anal. appl. (2022), 20:24 3 definition 1.5. let k ∈ end∗a(h) and {λj ∈ end ∗ a(h,hj), j ∈ j}. a sequence {λj}j∈j is called k −g−frame for h with respect to {hj}j∈j, if there exist constants 0 < a ≤ b < ∞ such that a〈k∗x,k∗x〉≤ ∑ j∈j 〈λjx, λjx〉≤ b〈x,x〉,∀x ∈ h. a k −g−frame {λj}j∈j is said to be tight if there exists a constant a > 0 such that ∑ j∈j 〈λjx, λjx〉 = a〈k∗x,k∗x〉,∀x ∈ h. 2. main results theorem 2.1. suppose k ∈ end∗a(h) has closed range and {λj ∈ end ∗ a(h,hj)} is a k −g−frame for h. for each j ∈ j, let γj ∈ end∗a(h,hj). then the following statement are equivalent (1) the sequence {γj}j∈j is a dual k −g−bessel sequence of {λj}j∈j. (2) for each a ∈ ker(k), ∑ j∈j λ ∗ j γja = 0. the sequence {γj}j∈j is a g−bessel sequence for h, which can naturally generate a g−bessel sequence {θj}j∈j for h in the forme θj = γjk +prange(k) for each j ∈ j such that h = ∑ j∈j λ∗j θjh = ∑ j∈j θ∗j λjh, ∀h ∈ range(k) (2.1) where prange(k) denotes the orthogonal projection onto range(k). (1) =⇒ (2) assume that {γj}j∈j is a dual k − g−bessel sequence of {λj}j∈j, then for each a ∈ ker(k), ∑ j∈j λ ∗ j γja = ka = 0 and we have for any h ∈ range(k) h = kk+h = ∑ j∈j λ∗j γjk +h = ∑ j∈j λ∗j γjk +prange(k)h, put θj = γjk+prange(k), for each j ∈ j so for any x ∈ h ∑ j∈j 〈θjx,θjx〉 = ∑ j∈j 〈γjk+prange(k)x, γjk +prange(k)x〉 ≤ dγ〈k+x,k+x〉 ≤ dγ‖k+‖2〈x,x〉, hence {θj}j∈j is g−bessel sequence for h. 4 int. j. anal. appl. (2022), 20:24 let g,h ∈ range(k) 〈g,h〉 = 〈 ∑ j∈j λ∗j θjg,h〉 = ∑ j∈j 〈θjg, λjh〉 = ∑ j∈j 〈γjk+prange(k)g, λjh〉 = ∑ j∈j 〈g, (γjk+prange(k)) ∗λjh〉 = 〈g, ∑ j∈j θ∗j λjh〉, therefore, ∑ j∈j θ ∗ j λjh = h (2) =⇒ (1) we have range(k +) = (ker(k))⊥ then each x ∈ h can be expressed as x = x1 + x2, where x1 ∈ range(k+) and x2 ∈ (range(k+))⊥ = ker(k), so kx1 = ∑ j∈j λ∗j γjk +prange(k)kx1 = ∑ j∈j λ∗j γjk +kx1 = ∑ j∈j λ∗j γjx1. and we have, kx2 = ∑ j∈j λ ∗ j γjx2 = 0, we obtain kx = k(x1 + x2) = kx1 = ∑ j∈j λ∗j γjx1 = ∑ j∈j λ∗j γj(x1 + x2) = ∑ j∈j λ∗j γjx. hence, {γj}j∈j is a dual k −g−bessel sequence of {λj}j∈j. theorem 2.2. suppose that k ∈ end∗a(h), λj, λ ′ j ∈ end ∗ a(h,hj) for each j ∈ j and that {γj ∈ end∗a(hj,wji )}i∈ij is a g−frame for hj with bounds cj,dj such that 0 < c = infj∈j cj ≤ supj∈j dj = d < ∞. let {γ ′ ji ∈ end ∗ a(hj,wji )}i∈ij be a dual g−frame of {γji}i∈ij for each j ∈ j. then the following conditions are equivalent (1) the pair {λj}j∈j and {λ ′ j}j∈j are a dual k −g−frame pair. (2) the pair {γji λj}j∈j,i∈ij and {γ ′ ji λ ′ j}j∈j,i∈ij are a dual k −g−frame pair. int. j. anal. appl. (2022), 20:24 5 proof. suppose that {λj}j∈j is a k −g−frame for h with bounds dλ and cλ. for each x ∈ h, we have ∑ j∈j ∑ i∈ij 〈γji λjx, γji λjx〉≤ ∑ j∈j dj〈λjx, λjx〉 ≤ ddλ〈x,x〉. on the other hand ∑ j∈j ∑ i∈ij 〈γji λjx, γji λjx〉≥ ∑ j∈j cjλjx, λjx〉 ≥ ccλ〈k∗x,k∗x〉. assume now that {γji λj}j∈j,i∈ij is a k −g−frame for h with bounds a,b. for each x ∈ h cj〈λjx, λjx〉≤ ∑ i∈ij 〈γji λjx, γji λjx〉≤ dj〈λjx, λjx〉, then ∑ j∈j 〈λjx, λjx〉≤ ∑ j∈j 1 cj ∑ i∈ij 〈γji λjx, γji λjx〉 ≤ 1 c ∑ j∈j ∑ i∈ij 〈γji λjx, γji λjx〉 ≤ b c 〈x,x〉. on the other hand ∑ j∈j 〈λjx, λjx〉≥ ∑ j∈j 1 dj ∑ i∈ij 〈γji λjx, γji λjx〉 ≥ 1 d ∑ j∈j ∑ i∈ij 〈γji λjx, γji λjx〉 ≥ a d 〈k∗x,k∗x〉. therefore {λj}j∈j being a k −g−frame for h is equivalent to {γji λj}j∈j,i∈ij being a k −g−frame for h, it remains only to prove the duality, then for each x ∈ h, we have∑ j∈j λ∗j λ ′ jx = ∑ j∈j λ∗j (∑ i∈ij γ∗ji γ ′ ji λ ′ jx ) = ∑ j∈j ∑ i∈ij λ∗j γ ∗ ji γ ′ ji λ ′ jx = ∑ j∈j ∑ i∈ij (γji λj) ∗γ ′ ji λ ′ jx, 6 int. j. anal. appl. (2022), 20:24 so {λ ′ j}j∈j is a dual k−g−bessel sequence of {λj}j∈j if and only if {γ ′ ji λ ′ j}j∈j,i∈ij is a dual k−g−bessel sequence of {γji λj}j∈j,i∈ij . � theorem 2.3. let {λj}j∈j be a parseval k−g−frame for h where k ∈ end∗a(h) has closed range. then {λj(k+)∗}j∈j is a dual k −g−bessel sequence of {λj}j∈j. proof. for each x ∈ h, we have∑ j∈j 〈λj(k+)∗x, λj(k+)∗x〉 = 〈k∗(k+)∗x,k∗(k+)∗x〉 ≤ ‖k‖2‖k+‖2〈x,x〉, then {λj(k+)∗}j∈j is g−bessel sequence for h. and we have for each g ∈ range(k∗), g = k∗(k∗)+g = k∗(k+)∗g, so kg = kk∗(k+)∗g = ∑ j∈j λ∗j λj(k +)∗g. if h ∈ (range(k∗))⊥ = ker(k) we obtain h ∈ ker((k+)∗) implying that ∑ j∈j λ ∗ j λj(k +)∗h = 0 = kh, altogether we have kf = ∑ j∈j λ ∗ j λj(k +)∗f for each f ∈ h. � theorem 2.4. suppose that k ∈ end∗a(h) has closed range and that {λj ∈ end ∗ a(h,hj)}j∈j is a parseval k −g−frame for h. then the following condition hold. (1) for any dual k −g−bessel sequence {θj}j∈j of {λj}j∈j we have t∗λ̃tλ̃ = t ∗ λ̃ tθ, where tλ̃ is the analysis operator of {λj(k+)∗}j∈j. (2) if {γj ∈ end∗a(h,hj)}j∈j is also a parseval k − g−frame for h such that t ∗ λtγ = 0, then {λj}j∈j and {γj}j∈j admit a common dual k−g−bessel sequence {λj(k+)∗ + γj(k+)∗}j∈j. proof. assume that {θj}j∈j is a dual k −g−bessel sequence of {λj}j∈j. so t∗λ (tλ̃x −tθx) = ∑ j∈j λ∗j λj(k +)∗x − ∑ j∈j λ∗j θjx = kx −kx = 0, then, 〈t∗ λ̃ (t λ̃ −tθ)x,y〉 = 〈k+t∗λ (tλ̃ −tθ)x,y〉 = 0, hence, t∗ λ̃ (t λ̃ −tθ)x = 0, so t∗λ̃tλ̃ = t ∗ λ̃ tθ. (2) since t∗λtγ = 0, then∑ j∈j λ∗j (λj(k +)∗ + γj(k +)∗)x = ∑ j∈j λ∗j λj(k +)∗x = kx = ∑ j∈j γ∗j γj(k +)∗x = ∑ j∈j γ∗j (λj(k +)∗ + γj(k +)∗)x int. j. anal. appl. (2022), 20:24 7 � theorem 2.5. let {λj ∈ end∗a(h,hj); j ∈ j} be a k −g−frame for h with bound a and b. then {λj}j∈j is a g−frame for h if k∗ is bounded below. proof. since k∗ is bounded below, then there exists c > 0 such that 〈k∗x,k∗x〉≤ c〈x,x〉,∀x ∈ h. and we have a〈k∗x,k∗x〉≤ ∑ j∈j 〈λjx, λjx〉≤ b〈x,x〉,∀x ∈ h. so, ac〈x,x〉≤ a〈k∗x,k∗x〉≤ ∑ j∈j 〈λjx, λjx〉≤ b〈x,x〉,∀x ∈ h. hence, {λj}j∈j is a g−frame for h. � theorem 2.6. if {λj ∈ end∗a(h,hj); j ∈ j} is a tight k−g−frame with bounded a, then {λj}j∈j is a tight g−frame with bounded b if and only if the right inverse of the operator k is a b k∗. proof. now if {λj}j∈j is a tight g−frame with bounded b, then∑ j∈j 〈λjx, λjx〉 = b〈x,x〉,∀x ∈ h. since {λj}j∈j is a tight k −g−frame with bounded a then, a〈k∗x,k∗x〉 = b〈x,x〉,∀x ∈ h, hence, 〈kk∗x,x〉 = 〈 b a x,x〉, therefore, kk∗ = b a ih, so, a b k∗ is the right inverse of the operator k. conversely, assume that a b k∗ is the right inverse of the operator k, then kk∗ = b a ih so, a〈kk∗x,x〉 = b〈x,x〉, hence, a〈k∗x,k∗x〉 = b〈x,x〉, since, {λj}j∈j is tight k −g−frame with bound a, then b〈x,x〉 = ∑ j∈j 〈λjx, λjx〉,∀x ∈ h. 8 int. j. anal. appl. (2022), 20:24 � theorem 2.7. let i ⊂ j be given. suppose that {λj ∈ end∗a(h,hj), j ∈ j} is a k −g−frame with bounds a,b and k −g−frame operator sλ,j. then the following statements are equivalent: (1) irange(k) −s−1λ,jsλ,i is boundedly invertible on range(k), (2) the sequence {λj}j∈j−i is a k − g−frame for h with lower k − g−frame bound b−1 ‖s−1 λ,j ‖2‖k∗(irange(k)−s −1 λ,j sλ,i) −1‖2 . proof. denote the frame operator of the k −g−frame {λj}j∈j−i by sλ,j−i. we have sλ,j−i = sλ,j −sλ,i = sλ,j(irange(k) −s −1 λ,jsλ,i), then, {λj}j∈j−i is a k −g−frame if and only if irange(k) −s−1λ,jsλ,i is boundedly invertible. suppose that irange(k) −s−1λ,jsλ,i is invertible. for any x ∈ h x = s−1λ,jsλ,jx = s−1λ,j (∑ j∈j λ∗j λjx + ∑ j∈j−i λ∗j λjx ) = s−1λ,jsλ,ix + ∑ j∈j−i s−1λ,jλ ∗ j λjx. so, (irange(k) −s −1 λ,jsλ,i)x = ∑ j∈j−i s−1λ,jλ ∗ j λjx. hence, ‖(irange(k) −s −1 λ,jsλ,i)x‖ = ‖ ∑ j∈j−i s−1λ,jλ ∗ j λjx‖ = sup ‖y‖=1 ∣∣∣∣ ∣∣∣∣〈 ∑ j∈j−i s−1λ,jλ ∗ j λjx,y〉 ∣∣∣∣ ∣∣∣∣ = sup ‖y‖=1 ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjx, λjs−1λ,jy〉 ∣∣∣∣ ∣∣∣∣ ≤ sup ‖y‖=1 ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjx, λjx〉 ∣∣∣∣ ∣∣∣∣ 1 2 ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjs−1λ,jy, λjs −1 λ,jy〉 ∣∣∣∣ ∣∣∣∣ 1 2 ≤ √ b‖s−1λ,j‖ ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjx, λjx〉 ∣∣∣∣ ∣∣∣∣ 1 2 , therefore, ‖(irange(k) −s −1 λ,jsλ,i)x‖≤ √ b‖s−1λ,j‖ ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjx, λjx〉 ∣∣∣∣ ∣∣∣∣ 1 2 , int. j. anal. appl. (2022), 20:24 9 then irange(k) −s−1λ,jsλ,i is well defined. if irange(k) −s −1 λ,j sλ,i is invertible on h, then for any x ∈ h we have ‖k∗x‖≤‖k∗(irange(k) −s −1 λ,jsλ,i) −1‖‖(irange(k) −s −1 λ,jsλ,i)x‖ hence, b−1 ‖s−1 λ,j ‖2‖k∗(irange(k) −s−1λ,jsλ,i)−1‖2 ‖k∗x‖2 ≤‖ ∣∣∣∣ ∣∣∣∣ ∑ j∈j−i 〈λjx, λjx〉 ∣∣∣∣ ∣∣∣∣. � definition 2.1. consider two g−bessel sequences {λj ∈ end∗a(h,hj), j ∈ j} and {θj ∈ end∗a(h,hj), j ∈ j}. the sequence {λj}j∈j and {θj}j∈j are said to be approximately k − g−dual frames if ‖irange(k) −tλt∗θ‖ < 1. in this case, we say that {θj}j∈j is an approximate k −g−dual of {λj}j∈j. theorem 2.8. if {θj}j∈j is an approximate k −g−dual of {λj}j∈j, then {θj(tλt∗θ ) −1}j∈j is a k − g−dual of {λj}j∈j. proof. it is easy to see that {θj(tλt∗θ ) −1}j∈j is a g−bessel sequence and x = (tλt ∗ θ )(tλt ∗ θ ) −1x = ∞∑ j=0 λ∗j θj(tλt ∗ θ ) −1x = ∞∑ j=0 λ∗j ( θj ∞∑ j=0 (irange(k) −−tλt ∗ θ ) nx ) . then, θj(tλt∗θ ) −1 = {θj ∑∞ j=0(irange(k) −−tλt ∗ θ ) n}j∈j is a k −g−dual of {λj}j∈j. � conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.j. duffin, a.c. schaeffer, a class of nonharmonic fourier series, trans. amer. math. soc. 72 (1952), 341–366. https://doi.org/10.1090/s0002-9947-1952-0047179-6. [2] d. gabor, theory of communication, j. inst. electric. eng. 93 (1946), 429-457. [3] a. khosravi, b. khosravi, fusion frames and g-frames in hilbert c*-modules, int. j. wavelets multiresolut. inf. process. 06 (2008), 433–446. https://doi.org/10.1142/s0219691308002458. [4] m. frank, d.r. larson, a module frame concept for hilbert c*-modules, in: l.w. baggett, d.r. larson (eds.), contemporary mathematics, american mathematical society, providence, rhode island, 1999: pp. 207–233. https://doi.org/10.1090/conm/247/03803. [5] e.c. lance, hilbert c*-modules: a toolkit for operator algebraists, london mathematical society lecture note series, 210, cambridge university press, cambridge, 1995. [6] s. kabbaj, m. rossafi, ∗-operator frame for end∗a(h), wavelet linear algebra, 5 (2018), 1-13. [7] i. kaplansky, modules over operator algebras, amer. j. math. 75 (1953), 839. https://doi.org/10.2307/ 2372552. https://doi.org/10.1090/s0002-9947-1952-0047179-6 https://doi.org/10.1142/s0219691308002458 https://doi.org/10.1090/conm/247/03803 https://doi.org/10.2307/2372552 https://doi.org/10.2307/2372552 10 int. j. anal. appl. (2022), 20:24 [8] f.d. nhari, r. echarghaoui, m. rossafi, k−g−fusion frames in hilbert c∗−modules, int. j. anal. appl. 19 (6) (2021), 836-857. https://doi.org/10.28924/2291-8639-19-2021-836. [9] m. rossafi, f.d. nhari, c. park, s. kabbaj, continuous g-frames with c∗-valued bounds and their properties, complex anal. oper. theory. 16 (2022), 44. https://doi.org/10.1007/s11785-022-01229-4. [10] m. rossafi, f.d. nhari, controlled k−g−fusion frames in hilbert c∗−modules, int. j. anal. appl. 20 (2022), 1. https://doi.org/10.28924/2291-8639-20-2022-1. [11] m. rossafi, s. kabbaj, ∗-k-operator frame for end∗a(h), asian-eur. j. math. 13 (2020), 2050060. https: //doi.org/10.1142/s1793557120500606. [12] m. rossafi, s. kabbaj, operator frame for end∗a(h), j. linear topol. algebra, 8 (2019), 85-95. [13] m. rossafi, s. kabbaj, ∗-k-g-frames in hilbert a-modules, j. linear topol. algebra, 7 (2018), 63-71. [14] m. rossafi, s. kabbaj, ∗-g-frames in tensor products of hilbert c∗-modules, ann. univ. paedagog. crac. stud. math. 17 (2018), 17-25. https://doi.org/10.2478/aupcsm-2018-0002. [15] m. rossafi, s. kabbaj, generalized frames for b(h,k), iran. j. math. sci. inf. accepted. https://doi.org/10.28924/2291-8639-19-2021-836 https://doi.org/10.1007/s11785-022-01229-4 https://doi.org/10.28924/2291-8639-20-2022-1 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.1142/s1793557120500606 https://doi.org/10.2478/aupcsm-2018-0002 1. introduction 2. main results references int. j. anal. appl. (2023), 21:8 some invariant point results using simulation function venkatesh∗, naga raju department of mathematics, osmania university, hyderabad, telangana-500007, india ∗corresponding author: venkat409151@gmail.com abstract. through this article, we establish an invariant point theorem by defining generalized zscontractions in relation to the simulation function in s-metric space. in this article, we generalized the results of nihal tas, nihal yilmaz ozgur and n.mlaiki. in addition to that, we bestow an example which supports our results. 1. introduction fixed point is also known as an invariant point. banach principle of contraction [2] on metric space plays very important role in the field of invariant point theory and non linear analysis. in 1922, stefan banach initiated the concept of contraction and established well known banach contraction theorem. in the year 2006, b sims and mustafa [9], established theory on g-metric spaces, that is an extension of metric spaces and established some properties. later, a.aliouche, s.sedghi and n.shobe [13] initiated s-metric spaces, it is a generalization of g-metric spaces in the year 2012. in 2014, s.radojevic, n.v.dung and n.t.hieu [4] proved by examples that s-metric space is not a generalization of gmetric space and vice versa. invariant points of various contractive maps on s-metric spaces were studied in [ [1], [3], [6][8], [11]]. in 2015, f.khajasteh, satish shukla and s.radenovic [5] introduced simulation function and the concept of z-contration in relation to simulation function and proved an invariant point theorem which generalizes the banach contraction principle. very recently, murat olgun, o.bicer and t.alyildiz [10] defined generalized z-contraction in relation to the simulation function and proved an invariant point theorem. received: dec. 3, 2022. 2020 mathematics subject classification. 54h25, 47h09, 47h10. key words and phrases. simulation function; z-contraction; fixed point; s-metric space. https://doi.org/10.28924/2291-8639-21-2023-8 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-8 2 int. j. anal. appl. (2023), 21:8 in the year 2019, nihal tas, nihal ylimaz ozgur and nabil mlaiki [8] proved an invariant point theorem by employing the collection of simulation mappings on s-metric spaces. in this article, we generalized the results of nihal tas , nihal yilmaz ozgur and n.mlaiki. 2. preliminaries definition 2.1. [13] let x 6= ∅, then a mapping s:x3 → [0,∞) is said to be an s-metric on x if: (s1) s(ξ,ϑ,w) > 0 for all ξ,ϑ,w ∈ x with ξ 6= ϑ 6= w. (s2) s(ξ,ϑ,w) = 0 if ξ = ϑ = w. (s3) s(ξ,ϑ,w)≤ [s(ξ,ξ,a)+s(ϑ,ϑ,a)+s(w,w,a)] ∀ξ,ϑ,w,a ∈ x. then we call (x,s) is an s-metric space. example 2.1. [13] define s:x3 → [0,∞) by s(ξ,ϑ,w) = d(ξ,ϑ) + d(ξ,w) + d(ϑ,w) for any ξ,ϑ,w ∈ x, where (x,d) be a metric space. then (x,s) is an s-metric space. example 2.2. [4] suppose x=r, collection of all real numbers and let s(ξ,ϑ,w)= |ϑ+w −2ξ|+ |ϑ−w| for all ξ,ϑ,w ∈ x. then (x,s) is an s-metric space. example 2.3. [12] suppose x=r, collection of all real numbers and let s(ξ,ϑ,w)= |ξ−w|+|ϑ−w| for all ξ,ϑ,w ∈ x. then (x,s) is an s-metric space. example 2.4. suppose x=[0,1] and s:x3 → [0,∞) be defined by s(ξ,ϑ,w)=  0 if ξ = ϑ = w max{ξ,ϑ,w} otherwise . then (x,s) is an s-metric space. lemma 2.1. [13] in the s-metric space, we observe s(ξ,ξ,ϑ)= s(ϑ,ϑ,ξ). lemma 2.2. [4] in the s-metric space, we observe (i) s(ξ,ξ,ϑ)≤ 2s(ξ,ξ,w)+s(ϑ,ϑ,w) and (ii) s(ξ,ξ,ϑ)≤ 2s(ξ,ξ,w)+s(w,w,ϑ) definition 2.2. [13] let (x,s) be a s-metric space. we have: (i) if s(ξn,ξn,ξ)→ 0 as n →∞. ,then we say sequence {ξn}∈ x converges to ξ ∈ x. i.e., for every � > 0, it can be found a natural number n0 so that to each n≥ n0, s(ξn,ξn,ξ) < � and we indicate it by limn→∞ξn = ξ. (ii) a sequence {ξn} ∈ x is known as cauchy sequence if to each � > 0, it can be found n0 ∈ n so that s(ξn,ξn,ξm) < � for every n,m≥ n0. (iii)if each cauchy sequence of x is convergent, then say x is complete. definition 2.3. [13] a self map h is defined on s-metric space (x,s) is known as an s-contraction if we get a constant 0≤ τ < 1 so that s(h(ξ),h(ξ),h(ϑ))≤ τs(ξ,ξ,ϑ) for all ξ,ϑ ∈ x. int. j. anal. appl. (2023), 21:8 3 definition 2.4. [5] we say that a mapping γ : [0,∞)× [0,∞)→r is a simulation mapping if: (γ1) γ(0,0) = 0 (γ2) γ(p,q) < q −p for p,q > 0 (γ3) if {pn},{qn} are sequences of (0,∞) so that limn→∞pn = limn→∞qn > 0, then limn→∞sup γ(pn,qn) < 0. we indicate z as the collection of all simulation mappings. for example, γ(p,q) = τq − p for 0≤ τ <1 belongning to z. definition 2.5. [5] let h be a self map on a metric space (x,d) and γ ∈z. then h is known as a z-contraction in relation to γ if: γ(d(hξ,hϑ),d(ξ,ϑ))≥ 0 for all ξ,ϑ ∈ x. by considering the definition 2.5. it is concluded that each banach contraction becomes zcontraction in relation to γ(p,q) = τq − p with 0 ≤ τ < 1. further, it can be established from the definition of the simulation mapping that γ(p,q) < 0 for each p ≥ q > 0. hence, assume that h is a z-contraction in relation to γ ∈ z then d(hξ,hϑ) < d(ξ,ϑ) for all distinct ξ,ϑ ∈ x. theorem 2.1. [5] in complete metric space (x,d), each z-contraction has a unique invariant point and furthermore the invariant point is the limit of every picard’s sequence. 3. main results definition 3.1. [13] let h be a self map on an s-metric space x and γ ∈z. we say that h is a contraction if we find a constant 0≤ l < 1 such that s(hξ,hξ,hϑ)≤ ls(ξ,ξ,ϑ) for all ξ,ϑ ∈ x. nihal tas, n.y.ozgur and nabil mlaiki [8] defined the zs-contraction as follows. definition 3.2. [8] let h be a self map on an s-metric space (x, s) and γ ∈z. then h is said to be a zs-contraction in relation to γ if γ(s(hξ,hξ,hϑ),s(ξ,ξ,ϑ))≥ 0 for all ξ,ϑ ∈ x nihal tas, n.y.ozgur and nabil mlaiki [8] proved the following theorem. theorem 3.1. [8] let h be a self map on an s-metric space (x, s). then h has a unique invariant point a∈ x and the invariant point is the limit of the picard sequence {ξn}, whenever h is a zs-contraction in relation to γ. 4 int. j. anal. appl. (2023), 21:8 definition 3.3. let h be a self map on an s-metric space (x, s) and γ ∈ z. then h is said to be generalized zs-contraction in relation to γ if γ(s(hξ,hξ,hϑ),m(ξ,ξ,ϑ))≥ 0 f or all ξ,ϑ ∈ x (3.1) where m(ξ,ξ,ϑ) = max{s(ξ,ξ,ϑ),s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ), 1 2 [s(ξ,ξ,hϑ)+s(ϑ,ϑ,hξ)]} example 3.1. let h be a contraction on (x,s). if we take l∈[0,1) and γ(p,q) = lq-p for all 0 ≤ p,q < ∞, then a contraction h is a zs-contraction in relation to γ. in fact, consider p = s(hξ,hξ,hϑ) and q = m(ξ,ξ,ϑ). since h is a contraction, we obtain : s(hξ,hξ,hϑ)≤ ls(ξ,ξ,ϑ)≤ lm(ξ,ξ,ϑ) =⇒ lm(ξ,ξ,ϑ)−s(hξ,hξ,hϑ)≥ 0 =⇒ γ(s(hξ,hξ,hϑ),m(ξ,ξ,ϑ))≥ 0. for all ξ,ϑ ∈ x. therefore, h is a generalized zs-contraction in relation to γ. example 3.2. consider a complete s-metric space (x,s), where x = [0,1] and s : x3 → [0,∞) by s(ξ,ϑ,w)= |ξ−w|+ |ϑ−w|. define h:x → x by hξ =   2 5 , for ξ ∈ [0, 2 3 ) 1 5 , for ξ ∈ [2 3 ,1) now we prove that h be a generalized zs-contraction in relation to γ, where γ is defined by γ(p,q)= 6 7 q −p. now we get s(hξ,hξ,hϑ)≤ 3 7 [s(ξ,ξ,hξ)+s(ϑ,ϑ,hϑ)] ≤ 6 7 max{s(ξ,ξ,hξ),s(ϑ,ϑ,hϑ)} ≤ 6 7 m(ξ,ξ,ϑ) for all ξ,ϑ ∈ x. that is, we have γ(s(hξ,hξ,hϑ),m(ξ,ξ,ϑ))= 6 7 m(ξ,ξ,ϑ)−d(hξ,hξ,hϑ)≥ 0. for all ξ,ϑ ∈ x. definition 3.4. let (x,s) be an s-metric space. then we say that a mapping h:x → x is asymptotically regular at ξ ∈ x if limn→∞s(hnξ,hnξ,hn+1ξ)=0 by the following lemma, we can conclude that a generalized zs-contraction is asymptotically regular at each point of x. lemma 3.1. if h : x → x is a generalized zs-contraction in relation to γ, then h is an asymptotically regular at each point ξ ∈ x. int. j. anal. appl. (2023), 21:8 5 proof. let ξ ∈ x. if for some m∈n, we have hmξ = hm−1ξ, that is, hϑ = ϑ, where ϑ = hm−1ξ, then hnϑ = hn−1hϑ = hn−1ϑ = ... = hϑ = ϑ for each n∈n. therefore, we have: s(hnξ,hnξ,hn+1ξ)= s(hn−m+1hm−1ξ,hn−m+1hm−1ξ,hn−m+2hm−1ξ) = s(hn−m+1ϑ,hn−m+1ϑ,hn−m+2ϑ) = s(ϑ,ϑ,ϑ) =0 hence lim n→∞ s(hnξ,hnξ,hn+1ξ)=0 now, we assume that hnξ 6= hn+1ξ, for each n∈n. from the condition(γ2) and the generalized zs-contraction property, we get: 0≤ γ(s(hn+1ξ,hn+1ξ,hnξ),m(hnξ,hnξ,hn−1ξ)) (3.2) where m(hnξ,hnξ,hn−1ξ)= max{s(hnξ,hnξ,hn−1ξ),s(hnξ,hnξ,hhnξ),s(hn−1ξ,hn−1ξ,hhn−1ξ), 1 2 [s(hnξ,hnξ,hhn−1ξ)+s(hn−1ξ,hn−1ξ,hhnξ)]} = max{s(hnξ,hnξ,hn−1ξ),s(hnξ,hnξ,hn+1ξ),s(hn−1ξ,hn−1ξ,hnξ), 1 2 [s(hnξ,hnξ,hnξ)+s(hn−1ξ,hn−1ξ,hn+1ξ)} = max{s(hnξ,hnξ,hn−1ξ),s(hn+1ξ,hn+1ξ,hnξ)} if s(hn+1ξ,hn+1ξ,hnξ) > s(hnξ,hnξ,hn−1ξ) then, we get m(hnξ,hnξ,hn−1ξ) = s(hn+1ξ,hn+1ξ,hnξ) from equation (3.2) we have, 0≤ γ(s(hn+1ξ,hn+1ξ,hnξ),s(hn+1ξ,hn+1ξ,hnξ)) < s(hn+1ξ,hn+1ξ,hnξ)−s(hn+1ξ,hn+1ξ,hnξ)=0 which is a contradiction. hence m(hnξ,hnξ,hn−1ξ) = s(hnξ,hnξ,hn−1ξ). using generalized zs-contractive property, we get 0≤ γ(s(hn+1ξ,hn+1ξ,hnξ),m(hnξ,hnξ,hn−1ξ)) = γ(s(hn+1ξ,hn+1ξ,hnξ),s(hnξ,hnξ,hn−1ξ)) < s(hnξ,hnξ,hn−1ξ)−s(hn+1ξ,hn+1ξ,hnξ) 6 int. j. anal. appl. (2023), 21:8 i.e., s(hn+1ξ,hn+1ξ,hnξ) < s(hnξ,hnξ,hn−1ξ) for all n∈n. then {s(hnξ,hnξ,hn−1ξ)} is a nonnegative reals of decreasing sequence and so it should be convergent. suppose limn→∞s(hnξ,hnξ,hn+1ξ) = η ≥ 0. if η > 0, then from the condition (γ3) and the generalized zs-contraction property, we get 0≤ lim n→∞ sup γ(s(hn+1ξ,hn+1ξ,hnξ),m(hnξ,hnξ,hn−1ξ) = lim n→∞ sup γ(s(hn+1ξ,hn+1ξ,hnξ),s(hnξ,hnξ,hn−1ξ) < 0 which is a contradiction. it should be η = 0. therefore limn→∞s(hnξ,hnξ,hn+1ξ)=0. hence, h is asymptotically regular at each point ξ ∈ x. � lemma 3.2. the picard sequence {ξn} so that hξn−1 = ξn, to each n∈n the initial point ξ0 ∈ x is a bounded sequence, whenever h is a generalized zs-contraction in relation to γ. proof. consider {ξn} be the picard sequence in x with initial value ξ0. now we claim that {ξn} is a bounded sequence. assume that {ξn} is unbounded. let ξn+m 6= ξn, for each m,n∈n. since {ξn} is unbounded, we can find a subsequence {ξnk} of {ξn} so that n1 =1 and to each k∈n, nk+1 is the smallest integer so that s(ξnk+1,ξnk+1,ξnk) > 1 and s(ξm,ξm,ξnk)≤ 1 for nk ≤ m ≤ nk+1 −1 hence, from the lemma (2.2), we obtain 1 < s(ξnk+1,ξnk+1,ξnk) ≤ 2s(ξnk+1,ξnk+1,ξnk+1−1)+s(ξnk,ξnk,ξnk+1−1) ≤ 2s(ξnk+1,ξnk+1,ξnk+1−1)+1 letting k→∞ and using lemma (3.1), we have lim n→∞ s(ξnk+1,ξnk+1,ξnk)=1 1 < s(ξnk+1,ξnk+1,ξnk)≤ m(ξnk+1−1,ξnk+1−1,ξnk−1) = max{s(ξnk+1−1,ξnk+1−1,ξnk−1),s(ξnk+1−1,ξnk+1−1,ξnk+1),s(ξnk−1,ξnk−1,ξnk), 1 2 [s(ξnk+1−1,ξnk+1−1,ξnk)+s(ξnk−1,ξnk−1,ξnk+1)]} = max{s(ξnk−1,ξnk−1,ξnk+1−1),s(ξnk+1−1,ξnk+1−1,ξnk+1),s(ξnk−1,ξnk−1,ξnk), 1 2 [s(ξnk+1−1,ξnk+1−1,ξnk)+s(ξnk−1,ξnk−1,ξnk+1)]} int. j. anal. appl. (2023), 21:8 7 ≤ max{2s(ξnk−1,ξnk−1,ξnk)+s(ξnk+1−1,ξnk+1−1,ξnk),s(ξnk+1−1,ξnk+1−1,ξnk+1), s(ξnk−1,ξnk−1,ξnk), 1 2 [s(ξnk+1−1,ξnk+1−1,ξnk)+s(ξnk−1,ξnk−1,ξnk+1)]} ≤ max{2s(ξnk−1,ξnk−1,ξnk)+1,s(ξnk+1−1,ξnk+1−1,ξnk+1), s(ξnk−1,ξnk−1,ξnk), 1 2 [1+2s(ξnk−1,ξnk−1,ξnk)+s(ξnk,ξnk,ξnk+1)]} letting n→ ∞, we get 1≤ lim k→∞ m(ξnk+1−1,ξnk+1−1,ξnk−1)≤ 1. that is limk→∞m(ξnk+1−1,ξnk+1−1,ξnk−1)=1 from the condition(γ3) and the generalized zs-contraction property, we obtain 0≤ lim k→∞ sup γ(s(ξnk+1,ξnk+1,ξnk),m(ξnk+1−1,ξnk+1−1,ξnk−1)) = lim k→∞ sup γ(s(ξnk+1,ξnk+1,ξnk),s(ξnk+1−1,ξnk+1−1,ξnk−1)) < 0 which is a contradiction. hence our assumption is wrong. therefore {ξn} is bounded. � theorem 3.2. let h be a self map defined on complete s-metric space (x, s). then h has a unique invariant point a ∈ x and picard sequence {ξn} converges to the invariant element a, whenever h is a generalized zs-contraction in relation to γ. proof. let the picard sequence {ξn} be defined as hξn−1 = ξn, ∀n ∈n and ξ0 ∈ x. now, we claim that {ξn} be a cauchy sequence. to get this, consider tn = sup{s(ξi,ξi,ξj) : i, j ≥ n}. clearly {tn} be a nonnegative reals of decreasing sequence. hence, we can find τ ≥ 0 so that limn→∞tn = τ. now we prove that τ = 0. if possible suppose that τ > 0. from the definition of tn, for each k∈n, we can find mk,nk so that k ≤ nk < mk and tk − 1 k < s(ξmk,ξmk,ξnk)≤ tk therefore, we get limn→∞s(ξmk,ξmk,ξnk)= τ. from the lemma (2.2), lemma (3.1) and generalized zs-contraction property, we get s(ξmk,ξmk,ξnk)≤ s(ξmk−1,ξmk−1,ξnk−1) ≤ 2s(ξmk−1,ξmk−1,ξmk)+s(ξnk−1,ξnk−1,ξmk) ≤ 2s(ξmk−1,ξmk−1,ξmk)+2s(ξnk−1,ξnk−1,ξnk)+s(ξmk,ξmk,ξnk) letting as k→ ∞, we have lim k→∞ s(ξmk−1,ξmk−1,ξnk−1)= τ 8 int. j. anal. appl. (2023), 21:8 s(ξmk−1,ξmk−1,ξnk−1)≤ m(ξmk−1,ξmk−1,ξnk−1) = max{s(ξmk−1,ξmk−1,ξnk−1),s(ξmk−1,ξmk−1,hξmk−1),s(ξnk−1,ξnk−1,hξnk−1), 1 2 [s(ξmk−1,ξmk−1,hξnk−1)+s(ξnk−1,ξnk−1,hξmk−1)]} = max{s(ξmk−1,ξmk−1,ξnk−1),s(ξmk−1,ξmk−1,ξmk),s(ξnk−1,ξnk−1,ξnk), 1 2 [s(ξmk−1,ξmk−1,ξnk)+s(ξnk−1,ξnk−1,ξmk)]} ≤ max{s(ξmk−1,ξmk−1,ξnk−1),s(ξmk−1,ξmk−1,ξmk),s(ξnk−1,ξnk−1,ξnk), 1 2 [2s(ξmk−1,ξmk−1,ξmk)+s(ξmk,ξmk,ξnk)+ 2s(ξnk−1,ξnk−1,ξnk)+s(ξnk,ξnk,ξmk)]} letting k → ∞, we get lim k→∞ m(ξmk−1,ξmk−1,ξnk−1)= τ. from the condition (γ3) and the generalized zs-contraction property, we have 0≤ lim k→∞ sup γ(s(ξmk,ξmk,ξnk),m(ξmk−1,ξmk−1,ξnk−1)) < 0 this is a contraction, hence, τ = 0. that is {ξn} is a cauchy sequence in the complete s-metric space x, we can find η ∈ x so that limn→∞ξn = η. now we verify that, η is an invariant point of h. if suppose hη 6= η, then s(η,η,hη)= s(hη,hη,η) > 0. now, m(ξn,ξn,η)=max{s(ξn,ξn,η),s(ξn,ξn,hξn),s(η,η,hη), 1 2 [s(ξn,ξn,hη)+s(η,η,hξn)]} lim n→∞ m(ξn,ξn,η)= max{s(η,η,η),s(η,η,η),s(η,η,hη), 1 2 [s(η,η,hη)+s(η,η,η)]} = s(η,η,hη) from the conditions (γ2),(γ3) and zs-contraction property, we get 0≤ lim n→∞ sup γ(s(hξn,hξn,hη),m(ξn,ξn,η)) < 0 this is contradiction. hence s(η,η,hη)=0 =⇒ hη = η. hence, η is a invariant point of h. now we claim that η is unique. suppose α is an element in x such that α 6= η and hα = α. int. j. anal. appl. (2023), 21:8 9 now, m(η,η,α)= max{s(η,η,α),s(η,η,hη),s(α,α,hα), 1 2 [s(η,η,hα)+s(α,α,hη]} = max{s(η,η,α),s(η,η,η),s(α,α,α), 1 2 [s(η,η,α)+s(α,α,η)]} = s(η,η,α) from the condition (γ2) and zs-contraction property, we get 0≤ γ(s(hη,hη,hα),m(η,η,α))= γ(s(hη,hη,hα),s(η,η,α)) < s(η,η,α)−s(η,η,α)=0, this is a contradiction. it should be η = α. � example 3.3. consider a complete s-metric space (x, s), where x = [0, 1 4 ] and s : x3 → [0,∞) by s(ξ,ϑ,w) = |ξ −w|+ |ξ −2ϑ+w|. define h: x → x by hξ = ξ 1+ξ . from example 2.9 in [5], we have h be a z-contraction in relation to γ ∈ z, where γ(p,q)= q q+1 4 −p, for any p,q∈ [0,∞) therefore for all ξ,ϑ ∈ x, we get 0≤ γ(s(hξ,hξ,hϑ),s(ξ,ξ,ϑ)) = s(ξ,ξ,ϑ) s(ξ,ξ,ϑ)+ 1 4 −s(hξ,hξ,hϑ) ≤ m(ξ,ξ,ϑ) m(ξ,ξ,ϑ)+ 1 4 −s(hξ,hξ,hϑ) = γ(s(hξ,hξ,hϑ),m(ξ,ξ,ϑ)) thus, h is generalized zs-contraction in relation to γ, for some γ ∈ z. so, by using theorem 3.2, h has a unique invariant point a=0. references [1] v.r.b. guttia, l.b. kumssa, fixed points of (α,ψ,φ)-generalized weakly contractive maps and property(p) in s-metric spaces, filomat. 31 (2017,) 4469–4481. https://doi.org/10.2298/fil1714469b. [2] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund. math. 3 (1922), 133-181. http://eudml.org/doc/213289. [3] t. dosenovic, s. radenovic, a. rezvani, s. sedghi, coincidence point theorems in s-metric spaces using integral type of contractions, u.p.b. sci. bull., ser. a, 79 (2017), 145-158. [4] d.v. nguyen, h.t. nguyen, s. radojevic, fixed point theorems for g-monotone maps on partially ordered smetric spaces, filomat. 28 (2014), 1885–1898. https://doi.org/10.2298/fil1409885d. [5] f. khojasteh, s. shukla, s. radenovic, a new approach to the study of fixed point theory for simulation functions, filomat. 29 (2015), 1189–1194. https://doi.org/10.2298/fil1506189k. [6] a. gupta, cyclic contraction on smetric space, int. j. anal. appl. 3 (2013), 119-130. [7] j.k. kim, s. sedghi, a. gholidahneh, m.m. rezaee, fixed point theorems in s-metric spaces, east asian math. j. 32 (2016), 677–684. https://doi.org/10.7858/eamj.2016.047. https://doi.org/10.2298/fil1714469b http://eudml.org/doc/213289 https://doi.org/10.2298/fil1409885d https://doi.org/10.2298/fil1506189k https://doi.org/10.7858/eamj.2016.047 10 int. j. anal. appl. (2023), 21:8 [8] n. mlaiki, n.y. özgür, n. taş, new fixed-point theorems on an s-metric space via simulation functions, mathematics. 7 (2019), 583. https://doi.org/10.3390/math7070583. [9] z. mustafa, b. sims, a new approach to generalized metric spaces, j. nonlinear convex anal. 7 (2006), 289-297. [10] m. olgun, o. bicer, t. alyildiz, a new aspect to picard operators with simulation functions, turk. j. math. 40 (2016), 832–837. https://doi.org/10.3906/mat-1505-26. [11] n.y. ozgur, n. tas, some fixed point theorems on s-metric spaces, mat. vesnik, 69 (2017), 39-52. https: //hdl.handle.net/20.500.12462/6682. [12] s. sedghi, n.v. dung, fixed point theorems on s-metric spaces, math. vesnik, 66 (2014), 113-124. [13] s. sedghi, n. shobe, a. aliouche, a generalization of fixed point theorem in s-metric spaces, math. vesnik, 64 (2012), 258-266. https://doi.org/10.3390/math7070583. https://doi.org/10.3906/mat-1505-26 https://hdl.handle.net/20.500.12462/6682 https://hdl.handle.net/20.500.12462/6682 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 36-44 http://www.etamaths.com on the behavior near the origin of a sine series with coefficients of monotone type xhevat z. krasniqi abstract. in this paper we have obtained some asymptotic equalities of the sum function of a trigonometric sine series expressed in terms of its special type of coefficients. 1. introduction let us consider the sine series (1.1) ∞∑ m=1 am sin mx with coefficients tending to zero and such that the sequence {am} satifsies condition 4am = am−am+1 ≥ 0 or 42am = 4am−4am+1 ≥ 0 for all m. it is a well-known fact that under such conditions the series (1.1) converges for all x (see [12], page 95). we denote by g(x) its sum. as usually we write g(u) ∼ h(u),u → 0 if there exist absolute positive constants a and b such that ah(u) ≤ g(u) ≤ bh(u) is in a neighborhood of the point u = 0, and write g(u) ≈ h(u) if limu→0 g(u) h(u) = 1. likewise, throughout this paper the constants in the o-expression denote positive absolute constants and they may be different in different relations. several authors have investigated the behavior of the sum g(x) near the origin expressed in terms of the coefficients am. seemingly, the first was young [11] who consider this problem, and he was concerned solely about estimates of |g(x)| from above. then salem ([3], [4], theorem 1) proved that if the sequence {mam} is monotone decreasing, then the following order equality holds g(x) ∼ ∑̀ m=1 mamx, where x ∈ i` := ( π `+1 , π ` ] ,` = 1, 2, . . . , x → 0. later on, aljančić, bojanić and tomić ([5], theorem 2) give asymptotic expression for g(x) as x → 0, when the coefficients am are convex (42am ≥ 0) and can be represent as the values a(m) of a slowly varying (in karamata’s sense) function 2010 mathematics subject classification. 42a20, 42a32. key words and phrases. sine series, (k, s)-monotone, convex sequence, asymptotic equality. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 36 on the behavior near the origin of a sine series 37 a(z), i.e. for each t > 0 (1.2) lim z→∞ a(tz) a(z) = 1. their result is equivalent to the following statement which can be deduce from one result given by telyakovskĭı ([6], theorem 2) and it is formulated as a corollary in this form: corollary 1.1. suppose that the coefficients am of the series (1.1) are convex and that am = a(m), for a slowly varying function a(z). then the following asymptotic equality holds true: g(x) ≈ a` 1 x , x ∈ i`, x → 0. telyakovskĭı deduced this result after the proof, in the same paper, of the following two theorems: theorem 1.1. assume that am ↓ 0. then for x ∈ i` the following estimate is valid g(x) = ∑̀ m=1 mamx + o ( 1 `3 ∑̀ m=1 m3am ) . theorem 1.2. let am → 0 and let the sequence {am} be convex. if x ∈ i`, where ` ≥ 11, then the following estimate holds true a` 2 cot x 2 + 1 2` `−1∑ m=1 m24am ≤ g(x) ≤ a` 2 cot x 2 + 6 ` `−1∑ m=1 m24am. note also that the above theorems as well as some of [1] are generalized and extended in [7]-[10]. for an integer k ≥ 0 and a real sequence {am}∞m=0 denote 4kam = k∑ i=0 (−1)icikam+i (40am = am), {4}k am = k∑ i=0 cikam+i ({4}0 am = am). definition 1.1 ([2]). a sequence {am}∞m=0 is said to be (k,s)-monotone if am → 0 as m →∞ and 4k ({4}s am) ≥ 0, for some k ≥ 0,s ≥ 0 and all m. it is easy to see that that if a sequence {am} (am → 0 as m → ∞) is nonincreasing, then it is (1,s)-monotone for all s = 0, 1, 2, . . . . the converse statement is not always true. for example, if we consider the sequence {am} such that am → 0 as m →∞ and a2m = 0, a2m+1 ≥ a2m+3 for m = 0, 1, 2, . . . , then this sequence is not non-increasing but it is (1, 1)-monotone. chronologically this definition arises the following question: what is the behavior near the origin of the series (1.1) with (k,s)-monotone coefficients? the answer to this question is the main goal of this paper. precisely, we shall answer to this question only considering the cases when the series (1.1) has: (1, 1)-monotone, or (1, 2)-monotone, or (2, 1)-monotone, or (2, 2)-monotone coefficients. for the proof of our results we need the following two lemmas proved in [2]. 38 xhevat z. krasniqi lemma 1.1. let {am}∞m=0 be a sequence such that am → 0 as m → ∞ and 4kam ≥ 0 for some k ≥ 1 and all m. then for all r = 0, 1, . . . ,k−1 and all m the following inequality 4ram ≥ 0 holds. lemma 1.2. let {am}∞m=0 be a (k,s)-monotone sequence. if k = 1,s = 1 or s = 2, then g(x) = a0 2 ( 1 − tan x 2 ) + 1( 2 cos x 2 )s ∞∑ m=1 {4}sam−1 sin (ms− 2 + s) x 2 , allmost everywhere. lemma 1.3. let bm(x) = ∑m i=0 sin (i− 1) x 2 . then the following estimates hold:∣∣bm(x)∣∣ ≤ 2π x , 0 < x ≤ π. proof. after some elementary calculation we have∣∣bm(x)∣∣ = ∣∣∣∣ 12 sin x 2 m∑ i=0 [ cos (i− 2) x 2 − cos ix 2 ]∣∣∣∣ = ∣∣∣∣cos x2 + cos x− cos (m− 1) x2 − cos mx22 sin x 2 ∣∣∣∣ ≤ 2∣∣ sin x 2 ∣∣ ≤ 2πx , 0 < x ≤ π. � 2. main results the following theorem considers sine series with (1, 1)-monotone sequence. theorem 2.1. assume that {am}∞m=1 is a (1, 1)-monotone sequence. then for x ∈ i` the following estimate is valid g(x) = 1 2 cos x 2 { 1 2 ∑̀ m=1 m{4}1amx + o ( 1 `3 ∑̀ m=1 m3{4}1am )} .(2.1) proof. by the lemma 1.2 (a0 = 0) we have g(x) = 1 2 cos x 2 ∞∑ m=1 {4}1am−1 sin (m− 1) x 2 .(2.2) then the use of abel’s transformation gives h(x) = lim p→∞ { p−1∑ m=1 4({4}1am−1) bm(x) + {4}1ap−1bp(x) + {4}1a0 sin x 2 } = ∞∑ m=1 4({4}1am−1) bm(x) + {4}1a0 sin x 2 := h (1) ` (x) + h (2) ` (x),(2.3) where h (1) ` (x) = `+1∑ m=1 4({4}1am−1) bm(x) + {4}1a0 sin x 2 , on the behavior near the origin of a sine series 39 and h (2) ` (x) = ∞∑ m=`+2 4({4}1am−1) bm(x). let us estimate first h (1) ` (x). based on lemma 1.3, our assumption 4({4}1am) ≥ 0 for all m, the well-known relation sin t = t + o(t3), as t → 0, and x ∈ i` we have h (1) ` (x) = `+1∑ m=1 ({4}1am−1 −{4}1am) bm(x) + {4}1a0 sin x 2 = ∑̀ m=0 {4}1am [ bm+1(x) −bm(x) ] −{4}1a`+1b`+1(x) = ∑̀ m=1 {4}1am sin mx 2 + 2π x {4}1a`+1 = 1 2 ∑̀ m=1 m{4}1amx + o ( 1 `3 ∑̀ m=1 m3{4}1am ) + o (`{4}1a`) . by virtue of monotonicity of {4}1am we obtain `{4}1a` ≤ 4 `3 { `(` + 1) 2 }2 {4}1a` ≤ 4 `3 ∑̀ m=1 m3{4}1am. thus, h (1) ` (x) = 1 2 ∑̀ m=1 m{4}1amx + o ( 1 `3 ∑̀ m=1 m3{4}1am ) .(2.4) furthermore, since x ∈ i` and |bm(x)| = o ( 1 x ) by the lemma 1.2, we notice that h (2) ` (x) = o ( 1 x ∞∑ m=`+2 ({4}1am−1 −{4}1am) ) = o ((` + 1){4}1a`+1) = o (`{4}1a`) = o ( 1 `3 ∑̀ m=1 m3{4}1am ) .(2.5) finally, relations (2.2)-(2.5) prove completely estimation (2.1). � corollary 2.1. let {am}∞m=1 be a (1, 1)-monotone sequence and the series ∞∑ m=1 m (am + am+1) converges. then the following asymptotic equality lim x→0 g(x) x = 1 4 ∞∑ m=1 m (am + am+1) holds true. 40 xhevat z. krasniqi proof. in accordance with theorem 2.1 it is enough to prove that 1 `2 ∑̀ m=1 m3{4}1am → 0, as ` →∞. indeed, for an arbitrary natural number m we can write 1 `2 ∑̀ m=1 m3{4}1am ≤ 1 `2 m∑ m=1 m3{4}1am + ∞∑ m=m+1 m{4}1am. if a number ε > 0 be chosen, then by hypotesis a number m = m(ε) exists, such that ∞∑ m=m+1 m{4}1am < ε 2 . likewise, for all sufficiently large ` 1 `2 m∑ m=1 m3{4}1am < ε 2 . then obviously, for such ` we have 1 `2 ∑̀ m=1 m3{4}1am < ε 2 + ε 2 = ε. � the following statements can be proved similarly therefore we will skip their proofs. theorem 2.2. assume that {am}∞m=1 is a (1, 2)-monotone sequence. then for x ∈ i` the following estimate is valid g(x) = 1( 2 cos x 2 )2 {∑̀ m=0 (m + 1){4}2amx + o ( 1 `3 ∑̀ m=0 (m + 1)3{4}2am )} . corollary 2.2. suppose that {am}∞m=1 is a (1, 2)-monotone sequence and the series ∞∑ m=0 (m + 1) (am + 2am+1 + am+2) converges. then the following asymptotic equality lim x→0 g(x) x = 1 4 ∞∑ m=0 (m + 1) (am + 2am+1 + am+2) holds true. the proof of the next statement is more complicated and that is why we will sketch it in more details. on the behavior near the origin of a sine series 41 theorem 2.3. assume that {am}∞m=1 is a (2, 2)-monotone sequence. then for x ∈ i`, ` ≥ 11 the following estimate is valid {4}2a`−1 2 cot x 2 + 1 2` `−1∑ m=1 m24({4}2am−1) ≤ g(x) ( 2 cos x 2 )2 ≤ {4}2a`−1 2 cot x 2 + 6 ` `−1∑ m=1 m24({4}2am−1) . proof. by the lemma 1.1 the condition 42 ({4}2am) ≥ 0 implies 4({4}2am) ≥ 0. therefore by the lemma 1.2 we have g(x) = 1( 2 cos x 2 )2 ∞∑ m=1 {4}2am−1 sin mx. applying abel’s transformation we obtain (2.6) g(x) = 1( 2 cos x 2 )2 ∞∑ m=1 4({4}2am−1) d̃m(x), where d̃m(x) = ∑m i=1 sin ix is the conjugate dirichlet kernel. for x ∈ (0,π] and m = 0, 1, 2, . . . , introduce the functions ϕm(x) := − cos (m + 1/2) x 2 sin x/2 and ψm(x) := m∑ i=0 ϕi(x) = − sin (m + 1) x 4 sin2(x/2) . denoting h(x) := ∑∞ m=1 4({4}2am−1) d̃m(x) one can write h(x) = `−1∑ m=1 4({4}2am−1) d̃m(x) + ∞∑ m=` 4({4}2am−1) ( 1 2 cot x 2 + ϕm(x) ) = {4}2a` 2 cot x 2 + `−1∑ m=1 4({4}2am−1) d̃m(x) + ∞∑ m=` 4({4}2am−1) ϕm(x) = a`−1 + 2a` + a`+1 2 cot x 2 + e`(x) + f`(x).(2.7) we shall make use of the representation (2.7) for x ∈ i`, and from now and till the end of the proof of our theorem we supose that x ∈ i` but we shall not remind of it. the following estimate is true in view of the monotonous decay of 4({4}2am−1) and the positivity of d̃m(x) for m ≤ `: e`(x) ≥ 4({4}2a`−1) `−1∑ m=1 ( 1 2 cot x 2 + ϕm(x) ) = 4({4}2a`−1) ( ` 2 cot x 2 + ψ`−1(x) ) = 4({4}2a`−1) 4 sin2(x/2) (` sin x− sin `x) .(2.8) 42 xhevat z. krasniqi let us estimate f`(x) from above. applying abel’s transformation we have |f`(x)| = ∣∣∣∣∣ limn→∞ { n−1∑ m=` 42 ({4}2am−1) ψm(x) +4({4}2an−1) ψn(x) −4({4}2a`−1) ψ`−1(x) }∣∣∣∣∣ ≤ ∞∑ m=` 42 ({4}2am−1) |ψm(x) −ψ`−1(x)| ≤ 4({4}2a`−1) 4 sin2(x/2) (1 + sin `x) .(2.9) from (2.8) and (2.9), in a similiar way as telyakovskĭı did [6], for ` ≥ 11 we can show that 1 2 e`(x) + f`(x) > 0. further, if m < `, then d̃m(x) ≥ m∑ i=1 2 π ix ≥ m(m + 1) ` + 1 > m2 ` . therefore, (2.10) 1 2 e`(x) ≥ 1 2` `−1∑ m=1 m24({4}2am−1) . from (2.10), (2.7), and (2.6) we obtain the estimate of g(x) from below g(x) ≥ 1( 2 cos x 2 )2 ( a`−1 + 2a` + a`+1 2 cot x 2 + 1 2` `−1∑ m=1 m24({4}2am−1) ) . since d̃m(x) ≤ m2x ≤ πm2 ` , then (2.11) e`(x) ≤ π ` `−1∑ m=1 m24({4}2am−1) . for the estimate (2.9) we can write |f`(x)| ≤ 4({4}2a`−1) 2 sin2(x/2) ≤4({4}2a`−1) π2 2x2 ≤ (` + 1)2 2 4({4}2a`−1) , and for ` ≥ 11 (` + 1)2 2 < 2, 4 ` `−1∑ m=1 m2, hence, by reason of the monotonicity of 4({4}2a`−1) we get |f`(x)| ≤ 2, 4 ` `−1∑ m=1 m24({4}2am−1) .(2.12) on the behavior near the origin of a sine series 43 estimates (2.12), (2.13), and (2.7) give the estimate of g(x) from above g(x) ≤ 1( 2 cos x 2 )2 ( a`−1 + 2a` + a`+1 2 cot x 2 + 6 ` `−1∑ m=1 m24({4}2am−1) ) . the proof is completed. � it follows from theorem 2.3 that for x ∈ i` in a sufficiently small neighbourhood of the origin we have g(x) = 1 2 (1 + cos x) ( {4}2a`−1 2 cot x 2 + o ( 1 ` `−1∑ m=1 m24({4}2am−1) )) .(2.13) corollary 2.3. assume that {am}∞m=1 is a (2, 2)-monotone sequence. then the following order equality is true g(x) ∼ (`− 1){4}2a`−1 + 1 ` `−1∑ m=1 m{4}2am−1. proof. since limx→0 x cot x = 1, then it is enough to prove that 1 ` `−1∑ m=1 (2m− 1){4}2am−1 − (`− 1){4}2a`−1 ≤ 1 ` `−1∑ m=1 m24({4}2am−1) and 1 ` `−1∑ m=1 m24({4}2am−1) ≤ 1 ` `−1∑ m=1 (2m− 1){4}2am−1. indeed, putting {4}2am−1 := bm−1, we can write 1 ` `−1∑ m=1 m24bm−1 = 1 ` [ b0 + 3b1 + 5b2 + · · · + (2`− 3)b`−2 − (`− 1)2b`−1 ] ≤ 1 ` `−1∑ m=1 (2m− 1)bm−1 ≤ 1 ` `−1∑ m=1 (2m− 1){4}2am−1,(2.14) because by the lemma 1.1, bm−1 ≥ 0 holds true. on the other hand we get (`− 1)2b`−1 ≤ `(`− 1)b`−1, therefore the proof of the corollary is completed. � remark 2.1. similar statement with theorem 2.3 holds true for the series (1.1) with (2, 1)-monotone coefficients. references [1] a. yu. popov, estimates for the sums of sine series with monotone coefficients of certain classes, (russian) mat. zametki 74 (2003), no. 6, 877–888; translation in math. notes 74 (2003), no. 5-6, 829–840. [2] b. v. simonov, on trigonometric series with (k, s)-monotone coefficients in weighted spaces, izv. vyssh. uchebn. zaved., mat. 2003, no. 5, 42-54. [3] r. salem, détermination de l’ordre de grandeur á l’origine de certains séries trigonometriques, c. r. acad. sci. paris 186 (1928), 1804-1806. [4] r. salem, essais sur les series trigonometriques, paris, 1940. 44 xhevat z. krasniqi [5] s. aljančić, r. bojanić and m. tomić, sur le comportement asymtotique au voisinage de zéro des séries trigonométriques de sinus á coefficients monotones, publ. inst. math. acad. serie sci., 10 (1956), 101-120. [6] s. a. telyakovskĭı, on the behavior near the origin of the sine series with convex coefficients, publ. inst. math. nouvelle serie, 58 (72) (1995), 43-50. [7] xh. z. krasniqi, certain estimates for double sine series with multiple–monotone coefficients, acta math. acad. paedagog. nyházi. (n.s.) 27 (2011), no. 2, 233–243. [8] xh. z. krasniqi, on the behavior near the origin of double sine series with monotone coefficients. math. bohem. 134 (2009), no. 3, 255–273. [9] xh. z. krasniqi, some estimates of r-th derivative of the sums of sine series with monotone coefficients of higher order near the origin. int. j. math. anal. (ruse) 3 (2009), no. 1-4, 59–69. [10] xh. z. krasniqi, n. l. braha, on the behavior of r-derivative near the origin of sine series with convex coefficients. jipam. j. inequal. pure appl. math. 8 (2007), no. 1, article 22, 6 pp. (electronic). [11] w. h. young, on the mode of oscillation of a fourier series and of its allied series, proc. london math. soc. 12 (1913), 433-452. [12] n. bary, trigonometric series, moscow, 1961 (in russian). university of prishtina ”hasan prishtina”, faculty of education, department of mathematics and informatics, avenue ”mother theresa” 5, 10000 prishtina, kosova international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 175-179 http://www.etamaths.com factors for absolute weighted arithmetic mean summability of infinite series hüseyi̇n bor∗ abstract. in this paper, we proved a general theorem dealing with absolute weighted arithmetic mean summability factors of infinite series under weaker conditions. we have also obtained some known results. 1. introduction let ∑ an be a given infinite series with partial sums (sn). we denote by u α n the nth cesàro mean of order α, with α > −1, of the sequence (sn), that is (see [4]) uαn = 1 aαn n∑ v=0 aα−1n−vsv, (1.1) where aαn = (α + 1)(α + 2)....(α + n) n! = o(nα), aα−n = 0 for n > 0. (1.2) a series ∑ an is said to be summable | c,α |k, k ≥ 1, if (see [5]) ∞∑ n=1 nk−1 | uαn −u α n−1 | k< ∞. (1.3) if we take α=1, then we obtain | c, 1 |k summability. let (pn) be a sequence of positive numbers such that pn = ∑n v=0 pv → ∞ as n → ∞, (p−i = p−i = 0, i ≥ 1). the sequence-to-sequence transformation wn = 1 pn n∑ v=0 pvsv (1.4) defines the sequence (wn) of the weighted arithmetic mean or simply the (n̄,pn) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [6]). the series ∑ an is said to be summable | n̄,pn |k, k ≥ 1, if (see [1]) ∞∑ n=1 (pn/pn) k−1 | wn −wn−1 |k< ∞. (1.5) if we take pn = 1 for all values of n, then we obtain | c, 1 |k summability. also if we take k = 1, then we obtain | n̄,pn | summability (see [11]). for any sequence (λn) we write that ∆λn = λn −λn+1. 2. known result the following theorem is known dealing with | n̄,pn |k summability factors of infinite series. 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g99. key words and phrases. weighted arithmetic mean; absolute summability; summability factors; infinite series; hölder inequality; minkowski inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 175 176 bor theorem 2.1. [2] let (xn) be a positive non-decreasing sequence and suppose that there exists sequences (βn) and (λn) such that | ∆λn |≤ βn, (2.1) βn → 0 as n →∞, (2.2) ∞∑ n=1 n | ∆βn | xn < ∞, (2.3) | λn | xn = o(1). (2.4) if m∑ n=1 | sn |k n = o(xm) as m →∞, (2.5) and (pn) is a sequence such that pn = o(npn), (2.6) pn∆pn = o(pnpn+1), (2.7) then the series ∑∞ n=1 an pnλn npn is summable | n̄,pn |k, k ≥ 1. remark 2.1. it should be noted that, under the conditions on the sequence (λn) we have that (λn) is bounded and ∆λn = o(1/n) [2]. 3. main result the aim of this paper is to prove theorem 2.1 under weaker conditions. now, we shall prove the following theorem. theorem 3.1. let (xn) be a positive non-decreasing sequence. if the sequences (xn), (βn), (λn), and (pn) satisfy the conditions (2.1)-(2.4), (2.6)-(2.7), and m∑ n=1 | sn |k nxk−1n = o(xm) as m →∞, (3.1) then the series ∑∞ n=1 an pnλn npn is summable | n̄,pn |k, k ≥ 1. remark 3.1. it should be noted that condition (3.1) is the same as condition (2.5) when k=1. when k > 1, condition (3.1) is weaker than condition (2.5) but the converse is not true. as in [10], we can show that if (2.5) is satisfied, then we get m∑ n=1 | sn |k nxk−1n = o( 1 xk−11 ) m∑ n=1 | sn |k n = o(xm) as m →∞. to show that the converse is false when k > 1, as in [3], the following example is sufficient. we can take xn = n δ, 0 < δ < 1, and then construct a sequence (un) such that un = |sn|k nxn k−1 = xn −xn−1, hence m∑ n=1 |sn|k nxn k−1 = xm = m δ, and so m∑ n=1 |sn|k n = m∑ n=1 (xn −xn−1)xk−1n = m∑ n=1 (nδ − (n− 1)δ)nδ(k−1) ≥ δ m∑ n=1 nδ−1nδ(k−1) = δ m∑ n=1 nδk−1 ∼ mδk k as m →∞. absolute weighted arithmetic mean summability 177 it follows that 1 xm m∑ n=1 |sn|k n →∞ as m →∞ provided k > 1. this shows that (2.5) implies (3.1) but not conversely. we require the following lemmas for the proof of theorem 3.1. lemma 3.1. [7] under the conditions on (xn), (βn) and (λn) as as expressed in the statement of the theorem, we have the following; nxnβn = o(1), (3.2) ∞∑ n=1 βnxn < ∞. (3.3) lemma 3.2. [9] if the conditions (2.6) and (2.7) are satisfied, then ∆ ( pn npn ) = o ( 1 n ) . 4. proof of theorem 3.1 proof. let (tn) be the sequence of (n̄,pn) mean of the series ∑∞ n=1 anpnλn npn . then, by definition, we have tn = 1 pn n∑ v=1 pv v∑ r=1 arprλr rpr = 1 pn n∑ v=1 (pn −pv−1) avpvλv vpv . then we get that tn −tn−1 = pn pnpn−1 n∑ v=1 pv−1pvavλv vpv , n ≥ 1, (p−1 = 0). by using abel’s transformation, we have that tn −tn−1 = pn pnpn−1 n−1∑ v=1 sv∆ ( pv−1pvλv vpv ) + λnsn n = snλn n + pn pnpn−1 n−1∑ v=1 sv pv+1pv∆λv (v + 1)pv+1 + pn pnpn−1 n−1∑ v=1 pvsvλv∆ ( pv vpv ) − pn pnpn−1 n−1∑ v=1 svpvλv 1 v = tn,1 + tn,2 + tn,3 + tn,4. to complete the proof of the theorem 3.1, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 ( pn pn )k−1 | tn,r |k< ∞, for r = 1, 2, 3, 4. (4.1) applying abel’s transformation, we have that m∑ n=1 ( pn pn )k−1 | tn,1 |k= m∑ n=1 ( pn npn )k−1 | λn |k−1| λn | | sn |k n = o(1) m∑ n=1 | sn |k n ( 1 xn )k−1 | λn | = o(1) m−1∑ n=1 ∆ | λn | n∑ v=1 | sv |k vxv k−1 + o(1) | λm | m∑ n=1 | sn |k nxn k−1 = o(1) m−1∑ n=1 | ∆λn | xn + o(1) | λm | xm = o(1) m−1∑ n=1 βnxn + o(1) | λm | xm = o(1), as m →∞, 178 bor by the hypotheses of theorem 3.1 and lemma 3.1. now, by using (2.6) and applying hölder’s inequality, we obtain that m+1∑ n=2 ( pn pn )k−1 | tn,2 |k= o(1) m+1∑ n=2 pn pnp k n−1 | n−1∑ v=1 pvsv∆λv |k= o(1) m+1∑ n=2 pn pnp k n−1 { n−1∑ v=1 pv pv | sv | pv | ∆λv | }k = o(1) m+1∑ n=2 pn pnpn−1 n−1∑ v=1 ( pv pv )k | sv |k pvβvk × ( 1 pn−1 n−1∑ v=1 pv )k−1 = o(1) m∑ v=1 ( pv pv )k | sv |k pvβvk m+1∑ n=v+1 pn pnpn−1 = o(1) m∑ v=1 ( pv pv )k−1 βv k−1βv | sv |k = o(1) m∑ v=1 (vβv) k−1βv | sv |k = o(1) m∑ v=1 ( 1 xv )k−1 βv | sv |k= o(1) m∑ v=1 vβv | sv |k vxv k−1 = o(1) m−1∑ v=1 ∆(vβv) v∑ r=1 | sr |k rxr k−1 + o(1)mβm m∑ v=1 | sv |k vxv k−1 = o(1) m−1∑ v=1 | ∆(vβv) | xv + o(1)mβmxm = o(1) m−1∑ v=1 | (v + 1)∆βv −βv | xv + o(1)mβmxm = o(1) m−1∑ v=1 v | ∆βv | xv + o(1) m−1∑ v=1 xvβv + o(1)mβmxm = o(1), as m →∞, by the hypotheses of the theorem 3.1 and lemma 3.1. again, as in tn,1, we have that m+1∑ n=2 ( pn pn )k−1 | tn,3 |k= m+1∑ n=2 ( pn pn )k−1 | pn pnpn−1 n−1∑ v=1 pvsvλv∆ ( pv vpv ) |k = o(1) m+1∑ n=2 pn pnp k n−1 { n−1∑ v=1 pv | sv || λv | 1 v }k = o(1) m+1∑ n=2 pn pnp k n−1 { n−1∑ v=1 ( pv pv ) pv | sv || λv | 1 v }k = o(1) m+1∑ n=2 pn pnpn−1 n−1∑ v=1 ( pv vpv )k pv | sv |k| λv |k × { 1 pn−1 n−1∑ v=1 pv }k−1 = o(1) m∑ v=1 ( pv vpv )k | sv |k pv | λv |k m+1∑ n=v+1 pn pnpn−1 = o(1) m∑ v=1 ( pv vpv )k pv | sv |k| λv |k 1 pv . v v = o(1) m∑ v=1 ( pv vpv )k−1 | λv |k−1| λv | | sv |k v = o(1) m∑ v=1 ( 1 xv )k−1 | λv | | sv |k v = o(1) m∑ v=1 | λv | | sv |k vxv k−1 = o(1) m−1∑ v=1 xvβv + o(1)xm | λm |= o(1), as m →∞, by the hypotheses of the theorem 3.1, lemma 3.1 and lemma 3.2. finally, using hölder’s inequality, as in tn,3, we have get m+1∑ n=2 ( pn pn )k−1 | tn,4 |k= m+1∑ n=2 pn pnp k n−1 | n−1∑ v=1 sv pv v λv |k = m+1∑ n=2 pn pnp k n−1 | n−1∑ v=1 sv pv vpv pvλv |k≤ m+1∑ n=2 pn pnpn−1 n−1∑ v=1 | sv |k ( pv vpv )k pv | λv |k × ( 1 pn−1 n−1∑ v=1 pv )k−1 absolute weighted arithmetic mean summability 179 = o(1) m∑ v=1 ( pv vpv )k | sv |k pv | λv |k 1 pv . v v = o(1) m∑ v=1 ( pv vpv )k−1 | λv |k−1| λv | | sv |k v = o(1) m∑ v=1 ( 1 xv )k−1 | λv | | sv |k v = o(1) m∑ v=1 | λv | | sv |k vxv k−1 = o(1) m−1∑ v=1 xvβv + o(1)xm | λm |= o(1), as m →∞. this completes the proof of theorem 3.1. � 5. conclusions it should be noted that if we take pn = 1 for all n, then we obtain a known result of mishra and srivastava dealing with | c, 1 |k summability factors of infinite series (see [8]). also, if we set k = 1, then we have a known result of mishra and srivastava concerning the | n̄,pn | summability factors of infinite series (see [9]). references [1] h. bor, on two summability methods, math. proc. camb. philos soc. 97 (1985), 147-149. [2] h. bor, a note on | n̄, pn |k summability factors of infinite series, indian j. pure appl. math. 18 (1987), 330-336. [3] h. bor, quasi-monotone and almost increasing sequences and their new applications, abstr. appl. anal. 2012, art. id 793548, 6 pp. [4] e. cesàro, sur la multiplication des séries, bull. sci. math. 14 (1890), 114-120. [5] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc., 7 (1957), 113-141. [6] g. h. hardy, divergent series, clarendon press, oxford, (1949). [7] k. n. mishra, on the absolute nörlund summability factors of infinite series, indian j. pure appl. math. 14 (1983), 40-43. [8] k. n. mishra and r. s. l. srivastava, on the absolute cesàro summability factors of infinite series, portugal. math. 42 (1983/84), 53-61. [9] k. n. mishra and r. s. l. srivastava, on | n̄, pn | summability factors of infinite series, indian j. pure appl. math. 15 (1984), 651-656. [10] w. t. sulaiman, a note on |a|k summability factors of infinite series, appl. math. comput. 216 (2010), 2645-2648. [11] g. sunouchi, notes on fourier analysis. xviii. absolute summability of series with constant terms, tôhoku math. j. (2), 1 (1949), 57-65. p. o. box 121, tr-06502 bahçelievler, ankara, turkey ∗corresponding author: hbor33@gmail.com 1. introduction 2. known result 3. main result 4. proof of theorem 3.1 5. conclusions references international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 1-9 http://www.etamaths.com common best proximity points for cyclic ϕ−contraction maps m. ahmadi baseri1,∗, h. mazaheri1 and t. d. narang2 abstract. the purpose of this paper is to introduce new types of contraction condition for a pair of maps (s, t ) in metric spaces. we give convergence and existence results of best proximity points of such maps in the setting of uniformly convex banach spaces. moreover, we obtain existence theorems of best proximity points for such contraction pairs in reflexive banach spaces. our results generalize, extend and improve results on the topic in the literature. 1. introduction and preliminaries let a and b be nonempty subsets of a metric space x := (x,d) and t a cyclic map on a∪b i.e. t(a) ⊆ b, t(b) ⊆ a. an element x ∈ a ∪ b is called a best proximity point of the mapping t if d(x,tx) = d(a,b), where d(a,b) is distance of a and b i.e. d(a,b) = inf{d(x,y) : x ∈ a,y ∈ b}. the map t is called a cyclic contraction [2] if d(tx,ty) ≤ kd(x,y) + (1 −k)d(a,b), for some k ∈ (0, 1) and for all x ∈ a and y ∈ b. if ϕ : [0,∞) → [0,∞) is a strictly increasing map then the cyclic map t is called cyclic ϕ−contraction map [1] if d(tx,ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(d(a,b)), for every x ∈ a and y ∈ b. given two self maps s and t on a∪b, a common best proximity point of the pair (s,t) is a point x ∈ a∪b satisfying d(x,sx) = d(x,tx) = d(a,b). the pair (s,t) is called a semi-cyclic contraction [3] if: (i) s(a) ⊆ b, t(b) ⊆ a (ii) ∃α ∈ (0, 1), such that d(sx,ty) ≤ αd(x,y) + (1 −α)d(a,b), for every x ∈ a and y ∈ b. let s(a) ⊆ b, t(b) ⊆ a. if ϕ : [0,∞) → [0,∞) is a strictly increasing map and d(sx,ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(d(a,b)), for every x ∈ a and y ∈ b, then the pair (s,t) is called semi-cyclic ϕ−contraction [8]. clearly, if s = t then a semi-cyclic ϕ−contraction pair reduces to a cyclic ϕ− contraction map. the best proximity point theorems emerge as a natural generalization of fixed point theorems, because a best proximity point reduces to a fixed point if a∩b 6= ∅. a fundamental result in fixed point theory is the banach contraction principle. one of the interesting extentions of this result was given by kirk, srinivasan and veermani [5]. eldred and veeramani [2] gave existence and convergence results of best proximity point for cyclic contraction maps in uniformly convex banach spaces and metric spaces, to include the case a ∩ b = ∅. al-thagafi and shahzad [1] obtained some such results for cyclic ϕcontraction maps . also rezapour, derafshpour and shahzad [7] gave best proximity point of cyclic ϕcontraction map on reflexive banach spaces. in 2011, gabeleh and abkar [3] proved theorems on the existence and convergence of best proximity point for semi-cyclic contraction pair in banach spaces. in 2014, thakur and sharma [8] obtaind best 2010 mathematics subject classification. 41a65, 41a52, 46n10. key words and phrases. best proximity point; generalized semi-cyclic ϕ-contraction pair; proximal property. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 baseri, mazaheri and narang proximity point for semi-cyclic ϕ−contraction pair in uniformly convex banach spaces. inspired by these results, we introduce a generalized semi-cyclic ϕ-contraction pair in metric spaces, which contain the general contractive pair of maps, and prove the existence and convergence of best proximity point for such pair of maps in metric spaces, uniformly convex banach spaces and in reflexive banach spaces. the following, definitions are needed for our resuls. definition 1.1. [1] let a and b be nonempty subsets of a normed linear space x, t : a∪b → a∪b, t(a) ⊆ b and t(b) ⊆ a. we say that t satisfies the proximal property if xn w→ x ∈ a∪b, ‖xn −txn‖→ d(a,b) =⇒‖x−tx‖ = d(a,b), for {xn}n≥0 ∈ a∪b. definition 1.2. a banach space x is said to be (i) uniformly convex if there exists a stricly increasing function δ : (0, 2] → [0, 1] such that the following implication holds for all x1,x2,p ∈ x,r > 0 and r ∈ [0, 2r] : ‖xi −p‖≤ r, i = 1, 2 and ‖x1 −x2‖≥ r ⇒‖(x1 + x2)/2 −p‖≤ (1 − δ(r/r))r (ii) strictly convex if the following impilication holds for all x1,x2,p ∈ x and r > 0 ‖xi −p‖≤ r, i = 1, 2 and x1 6= x2 ⇒‖(x1 + x2)/2 −p‖ < r. 2. main results we introduce generalized semi-cyclic ϕ−contraction pair in metric spaces as under: definition 2.1. let a and b be nonempty subsets of a metric space x and s,t : a ∪ b → a ∪ b be such that s(a) ⊆ b and t(b) ⊆ a. then the pair (s,t) is said to be generalized semi-cyclic ϕ-contraction if ϕ : [0, +∞) → [0, +∞) is a strictly increasing map and d(sx,ty) ≤ (1/3){d(x,y) + d(sx,x) + d(ty,y)} − ϕ(d(x,y) + d(sx,x) + d(ty,y)) + ϕ(3d(a,b)), for all x ∈ a and y ∈ b. example 2.1. take ϕ(t) = (1 −k)(t/3) for t ≥ 0 and 0 < k < 1, we obtain d(sx,ty) ≤ (k/3){d(x,y) + d(x,sx) + d(y,ty)} + (1 −k)d(a,b), for all x ∈ a and y ∈ b, which is generalization of semi-cyclic contraction. note that, s = t then we obtain generalized cyclic contraction map [4]. example 2.2. let x = r with the usual metric. for a = [0, 1], b = [−1, 0], define s,t : a∪b → a∪b by s(x) =   −x 2 x ∈ a x 2 x ∈ b, t(x) =   x 2 x ∈ a −x 2 x ∈ b. clearly s(a) ⊆ b and t(b) ⊆ a. with a ∈ a, b ∈ b and ϕ(t) = t 2 1+8t for t ≥ 0, (s,t) is a generalized semi-cyclic ϕ-contraction. let (s,t) be a generalized semi-cyclic ϕ-contraction. consider x0 ∈ a, then sx0 ∈ b, so there exists y0 ∈ b such that y0 = sx0. now ty0 ∈ a, so there exists x1 ∈ a such that x1 = ty0. inductively, we define sequences {xn} and {yn} in a and b, respectively by (1) xn+1 = tyn, yn = sxn. lemma 2.1. let a and b be nonempty subsets of a metric space x and s,t : a ∪ b → a ∪ b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction . for x0 ∈ a∪b, the sequences {xn} and {yn} are generated by (1) then for all x ∈ a, y ∈ b, and n ≥ 1, we have (a) −ϕ(d(x,y) + d(x,sx) + d(y,ty)) + ϕ(3d(a,b)) ≤ 0, (b) d(sx,ty) ≤ (1/3){d(x,y) + d(x,sx) + d(y,ty)}, (c) d(xn,sxn) ≤ d(xn−1,sxn−1), (d) d(xn+1,yn) ≤ d(yn,tyn−1), common best proximity points 3 (e) d(yn+1,tyn) ≤ d(yn,tyn−1). proof. we have 3d(a,b) ≤ d(x,y) + d(tx,x) + d(ty,y). hence ϕ is a strictly increasing map, (a) and (b) are obtained. since d(xn,sxn) ≤ (1/3){d(yn−1,xn) + d(xn,sxn) + d(yn−1,xn)}, so (2) d(xn,sxn) ≤ d(yn−1,xn). also, since d(yn−1,xn) ≤ (1/3){d(yn−1,xn−1) + d(xn−1,sxn−1) + d(yn−1,xn)}, we have (3) d(yn−1,xn) ≤ d(xn−1,sxn−1). from (2) and (3), inequality (c) is obtained. since d(xn+1,yn) ≤ (1/3){d(xn,yn) + d(xn+1,yn) + d(xn,yn)}, so d(xn+1,yn) ≤ d(yn,tyn−1), that is inequlity (d). now, since d(yn+1,tyn) ≤ (1/3){d(xn+1,yn) + d(yn+1,tyn) + d(xn+1,yn)}, we have d(yn+1,tyn) ≤ d(xn+1,yn). by using (d), inequality (e) is obtained. following result will be needed in what follows. proposition 2.1. let a and b be nonempty subsets of a metric space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction map. for x0 ∈ a∪b, the sequences {xn} and {yn} are generated by (1). then d(xn,sxn) → d(a,b) and d(yn,tyn−1) → d(a,b). proof. let dn = d(xn,sxn). it follows from lemma 2.1(c), that {dn} is decreasing and bounded. so limn→∞dn = t0. since (s,t) is a generalized semi-cyclic ϕ-contraction pair, we obtain dn+1 ≤ d(sxn,tyn) ≤ (1/3){2dn + d(yn,tyn)}−ϕ(2dn + d(yn,tyn)) + ϕ(3d(a,b)) ≤ dn −ϕ(2dn + d(yn,tyn)) + ϕ(3d(a,b)). hence, ϕ(3d(a,b)) ≤ ϕ(2dn + d(yn,tyn)) ≤ dn −dn+1 + ϕ(3d(a,b)). thus (4) lim n→∞ ϕ(2dn + d(yn,tyn)) = ϕ(3d(a,b)). since ϕ is strictly increasing and dn ≥ d(yn,tyn) ≥ dn+1 ≥ t0 ≥ d(a,b), we have lim n→∞ ϕ(2dn + d(yn,tyn)) ≥ ϕ(3t0) ≥ ϕ(3d(a,b)). from (4), ϕ(3t0) = ϕ(3d(a,b)). as ϕ is strictly increasing, we have t0 = d(a,b). theorem 2.1. let a and b be nonempty subsets of a metric space x and s,t : a∪b → a∪b be such that the pair (s,t) is a generalized semi-cyclic ϕ-contraction map. for x0 ∈ a∪b, the sequences {xn} and {yn} are generated by (1). if {xn} and {yn} have convergent subsequences in a and b, then there exists x ∈ a and y ∈ b such that d(x,sx) = d(a,b) = d(y,ty). 4 baseri, mazaheri and narang proof. let {ynk} be a subsequence of {yn} such that ynk → y. the relation d(a,b) ≤ d(tynk,y) ≤ d(ynk,y) + d(ynk,tynk ) holds for each k ≥ 1. letting k →∞, by proposion 2.1 and lemma 2.1(d), we obtain lim k→∞ d(tynk,y) = d(a,b). from lemma2.1(b), d(ty,ynk ) ≤ (1/3){d(y,xnk ) + d(y,ty) + d(xnk,sxnk )} ≤ (1/3){d(y,ynk ) + d(ynk,xnk ) + d(y,ynk ) + d(ty,ynk ) + d(xnk,sxnk )}. letting k →∞, by proposion 2.1, we get (2/3)d(a,b) ≤ (2/3) lim k→∞ d(ty,ynk ) ≤ (2/3)d(a,b). so d(ty,y) = d(a,b). similary, it can be proved that d(x,sx) = d(a,b). proposition 2.2. let a and b be nonempty subsets of a metric space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction.then the sequences {xn} and {yn} generated by (1) are bounded. proof. by proposition 2.1, we have d(xn,sxn) → d(a,b) as n → ∞. it is sufficient to show that {sxn} is bounded. for the unbounded map ϕ, take m > 0 such that ϕ(m) > (4/3)d(x0,sx0) + ϕ(3d(a,b)). if {sxn} is not bounded, then there exists a natural number n ∈ n, such that d(x1,sxn ) > m, d(x1,sxn−1) ≤ m. then m < d(x1,sxn ) ≤ d(y0,xn ) ≤ (1/3){d(x0,yn−1) + d(x0,sx0) + d(yn−1,tyn−1)} −ϕ(d(x0,yn−1) + d(x0,sx0) + d(yn−1,tyn−1)) + ϕ(3d(a,b)) ≤ (1/3){d(x0,x1) + d(x1,yn−1) + d(x0,sx0) + d(xn−1,yn−1)} −ϕ(d(x0,yn−1)) + ϕ(3d(a,b)) ≤ (1/3){d(x0,y0) + d(y0,x1) + m + d(x0,sx0) + d(xn−1,yn−2)} −ϕ(d(x0,yn−1)) + ϕ(3d(a,b)) ≤ (1/3){3d(x0,sx0) + m + d(xn−2,yn−2)} −ϕ(d(x0,yn−1)) + ϕ(3d(a,b)) + ϕ(3d(a,b)) < (4/3)d(x0,sx0) + m −ϕ(d(x0,yn−1)) + ϕ(3d(a,b)). hence ϕ(d(x0,yn−1)) < (4/3)d(x0,sx0) + ϕ(3d(a,b)). therefore ϕ(m) < ϕ(d(x1,sxn )) ≤ ϕ(d(y0,xn )) ≤ ϕ(d(x0,yn−1)) < (4/3)d(x0,sx0) + ϕ(3d(a,b)), which is a contradiction. hence {sxn} is bounded, therefore {xn} is bounded. corollary 2.1. let a and b be nonempty subsets of a metric space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. if a and b are boundedly compact then there exists x ∈ a and y ∈ b such that d(x,sx) = d(a,b) = d(y,ty). proof. the result is an immediately consequence of proposition 2.2 and theorem 2.1. now, define a sequence {zn} in a∪b as: (5) zn = { tyk n = 2k sxk n = 2k − 1. common best proximity points 5 in the following, we consider a uniformly convex banach space x and give best proximity point for generalized semi-cyclic ϕ-contraction pair (s,t). lemma 2.2. let a and b be nonempty convex subsets of a uniformly convex banach space x and s,t : a ∪ b → a ∪ b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. for x0 ∈ a ∪ b, if the sequences {xn} and {yn} are generated by (1) and sequence {zn} is generated by (5), then ‖z2n+2 −z2n‖→ 0 and ‖z2n+3 −z2n+1‖→ 0 as n →∞. proof. to show ‖z2n+2 −z2n‖→ 0 as n →∞, assume the contrary. then there exists �0 > 0 such that for each k ≥ 1, there exists nk ≥ k such that ‖z2nk+2 −z2nk‖≥ �0. (6) choose � > 0 such that ( 1 − δ ( �0 d(a,b)+� )) (d(a,b) + �) < d(a,b). by proposition 2.1, there exists n1 such that ‖z2nk+2 −z2nk+1‖≤ d(a,b) + �, (7) for every nk ≥ n1. also, ‖z2nk −z2nk+1‖≤‖ynk −xnk+1‖≤‖xnk −ynk‖→ d(a,b), so, there exists n2 such that ‖z2nk −z2nk+1‖≤ d(a,b) + �, (8) for all nk ≥ n2. let n = max{n1,n2}. from (6)-(8) and the uniform convexity of x, we get∥∥∥z2nk+2+z2nk2 −z2nk+1∥∥∥ ≤ (1 − δ( �0d(a,b)+�)) (d(a,b) + �), for all nk ≥ n. as (z2nk+2 + z2nk )/2 ∈ a, the choice of � implies that∥∥∥z2nk+2+z2nk2 −z2nk+1∥∥∥ < d(a,b), for all nk ≥ n, a contradiction. by a similar argument we can show that ‖z2n+3 − z2n+1‖ → 0 as n →∞. proposition 2.3. let a and b be nonempty convex subsets of a uniformly convex banach space x and s,t : a ∪ b → a ∪ b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. for x0 ∈ a ∪ b, if the sequences {xn} and {yn} are generated by (1) and sequence {zn} is generated by (5), then for each � > 0, there exists a positive integer n0 such that for all m > n ≥ n0, ‖z2m −z2n+1‖ < d(a,b) + �. proof. suppose the contrary. so there exists �0 > 0 such that for each k ≥ 1, there is mk > nk ≥ k satisfying ‖z2mk −z2nk+1‖≥ d(a,b) + �0 (9) and ‖z2(mk−1) −z2nk+1‖ < d(a,b) + �0. (10) now from (9) and (10), we get d(a,b) + �0 ≤‖z2mk −z2nk+1‖≤‖z2mk −z2(mk−1)‖ + ‖z2(mk−1) −z2nk+1‖ < ‖z2mk −z2(mk−1)‖ + d(a,b) + �0. letting k →∞, lemma 2.2 implies lim k→∞ ‖z2mk −z2nk+1‖ = d(a,b) + �0. (11) by lemma 2.1(b) and (d), 6 baseri, mazaheri and narang ‖z2mk −z2nk+1‖≤‖z2mk −z2mk+2‖ + ‖z2mk+2 −z2nk+3‖ + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + (1/3){‖ymk+1 −xnk+2‖ +‖ymk+1 −xmk+2‖ + ‖xnk+2 −ynk+2‖} + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + (1/9){‖xmk+1 −ynk+1‖ +‖xmk+1 −ymk+1‖ + ‖ynk+1 −xnk+1‖} +(1/3){‖ymk+1 −xmk+1‖ + ‖xnk+2 −ynk+2‖} + ‖z2nk+3 −z2nk+1‖. letting k →∞, from (11), lemma 2.2 and proposition 2.1 we get d(a,b) + �0 ≤ (1/9)(d(a,b) + �0) + (2/9)d(a,b) + (2/3)d(a,b), so d(a,b) + �0 ≤ d(a,b) + (1/9)�0, this is a contradiction. theorem 2.2. let a and b be nonempty closed and convex subsets of a uniformly convex banach space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. for x0 ∈ a∪b, if the sequences {xn} and {yn} are generated by (1) and sequence {zn} is generated by (5), then there exist unique x ∈ a and y ∈ b such that z2n → x, z2n+1 → y and ‖x − sx‖ = d(a,b) = ‖y −ty‖. proof. first, we show that {z2n} is a cauchy sequence in a. if d(a,b) = 0, then let �0 > 0 be given. by lemma 2.1(d) and proposition 2.1, ‖z2n −z2n+1‖ = ‖tyn −sxn+1‖≤‖xn −sxn‖→ d(a,b) = 0. so, there exists a positive integer n1 such that ‖z2n −z2n+1‖ < �, for every n ≥ n1. by proposition 2.2, there exists a positive integer n2 such that ‖z2m −z2n+1‖ < �, for every m > n ≥ n2. let n = max{n1,n2}. it follows that ‖z2m −z2n‖≤‖z2m −z2n+1‖ + ‖z2n −z2n+1‖ < 2�, for all m > n ≥ n. therefore {z2n} is a cauchy sequence in a. now, we assume that d(a,b) > 0. to show that {z2n} is a cauchy sequence in a, we assume the contrary. then there exists �0 > 0 such that for each k ≥ 1 there exists mk > nk ≥ k so that ‖z2mk −z2nk‖≥ �0. (12) choose � > 0 such that ( 1 − δ ( �0 d(a,b) + � )) (d(a,b) + �) < d(a,b). by lemma 2.1(d) and proposition 2.1, ‖z2nk −z2nk+1‖ = ‖tynk −sxnk+1‖≤‖xnk −sxnk‖→ d(a,b). hence, there exists a positive integer n1 such that ‖z2nk −z2nk+1‖≤ d(a,b) + �, (13) for all nk ≥ n1. by proposition 2.2 there exists a positive integer n2 such that ‖z2mk −z2nk+1‖≤ d(a,b) + �, (14) for all mk > nk ≥ n2. let n = max{n1,n2}. from (12)-(14) and the uniform convexity of x, we get∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ ≤ ( 1 −δ ( �0 d(a,b) + � )) (d(a,b) + �), for all mk > nk ≥ n. as (z2mk + z2nk )/2 ∈ a, the choice of � implies that∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ < d(a,b), common best proximity points 7 for all mk > nk ≥ n, a contradiction. thus {z2n} cauchy sequence in a. by a similar argument, we can show that {z2n+1} is a cauchy sequence in b. the completeness of x and the closedness of a implies that z2n → x as n →∞. by theorem 2.1, ‖x−sx‖ = d(a,b). also, it follows from closedness of b and theorem 2.1 that ‖y−ty‖ = d(a,b). to prove uniqueness, assume that there is a ∈ a such that a 6= x and ‖a−sa‖ = d(a,b). by lemma 2.1 (b), ‖tsx−sx‖≤ (1/3){2‖sx−x‖ + ‖tsx−sx‖}, hence (2/3)d(a,b) ≤ (2/3)‖tsx−sx‖≤ (2/3)‖sx−x‖ = (2/3)d(a,b). therefore, ‖tsx−sx‖ = d(a,b), it follows that tsx = x. now ‖sx−a‖ = ‖sx−tsa‖≤ (1/3){‖sa−x‖ + ‖x−sx‖ + ‖sa−a‖} ≤ (1/9){‖sx−a‖ + ‖sa−a‖ + ‖x−sx‖} +(1/3){‖x−sx‖ + ‖sa−a‖}. hence (8/9)d(a,b) ≤ (8/9)‖sx−a‖≤ (8/9)d(a,b), which implies that, ‖sx−a‖ = d(a,b). from convexity of a and strict convexity of x, we get∥∥∥∥x + a2 −sx ∥∥∥∥ = ∥∥∥∥x−sx2 + a−sx2 ∥∥∥∥ < d(a,b), a contradiction. thus x = a. similarly, we show the uniqueness of y ∈ b. now, we show the existence of a best proximity point for generalized semi-cyclic ϕ-contraction pair (s,t) in reflexive banach spaces. first, we prove the following theorem. theorem 2.3. let a and b be nonempty weakly closed subsets of a reflexive banach space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. then there exists (x,y) ∈ a×b such that ‖x−y‖ = d(a,b). proof. if d(a,b) = 0, the result follows by theorem 3.1(i) of [6]. so we assume that d(a,b) > 0. for x0 ∈ a, if the sequences {xn} and {yn} are generated by (1) and sequence {zn} is generated by (5) then from proposition 2.3, the sequences {z2n} and {z2n+1} are bounded. since x is reflexive and a is weakly closed, the sequence {z2n} has a subsequence {z2nk} such that z2nk w→ x ∈ a. also b is weakly closed, hence z2nk+1 w→ y ∈ b as k → ∞. since z2nk − z2nk+1 w→ x−y 6= 0 as k → ∞, there exists a bounded liner functional f : x → [0,∞) such that ‖f‖ = 1 and f(x−y) = ‖x−y‖. for all k ≥ 1, we have |f(z2nk −z2nk+1)| ≤ ‖f‖‖z2nk −z2nk+1‖ = ‖z2nk −z2nk+1‖. since limk→∞ |f(z2nk −z2nk+1)| = ‖x−y‖, by lemma 2.1(d) and proposition 2.1, we get ‖x−y‖ = limk→∞ |f(z2nk −z2nk+1)| ≤ limk→∞‖z2nk −z2nk+1‖ ≤ limk→∞‖xnk −sxnk‖ = d(a,b). so, ‖x−y‖ = d(a,b). theorem 2.4. let a and b be nonempty weakly closed subsets of a reflexive banach space x and s,t : a∪b → a∪b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. then there exists x ∈ a and y ∈ b such that ‖x−sx‖ = d(a,b) = ‖ty −y‖, provided that one of the following conditions is satisfied (i) s is weakly continuous on a and t is weakly continuous on b. (ii) t, s satisfy the proximal property. 8 baseri, mazaheri and narang proof. if d(a,b) = 0, the result follows from theorem 3.1(i) of [6]. so we assume that d(a,b) > 0. for x0 ∈ a, if the sequences {xn} and {yn} are generated by (1) and sequence {zn} is generated by (5) then by proposition 2.3 the sequences {z2n} and {z2n+1} are bounded. since a is weakly closed, the sequence {z2n} has a subsequence {z2nk} such that z2nk w→ x ∈ a. from (i), z2nk+1 w→ sx ∈ b as k → ∞. so z2nk − z2nk+1 w→ x − sx 6= 0 as k → ∞. now the proof continues similar to that of theorem 2.3. also b is weakly closed, so z2nk+1 w→ y ∈ b as k → ∞. since, (i) holds, z2nk+2 w→ ty, as k →∞. next the proof continues similar to that of theorem 2.3. from (ii), by lemma 2.1(d) and proposition 2.1, ‖z2nk+1 − tz2nk+1‖ → d(a,b) as k → ∞. so ‖y −ty‖ = d(a,b). also, ‖z2nk −sz2nk‖→ d(a,b) as k →∞. thus ‖x−sx‖ = d(a,b). next, we consider reflexive and strictly convex banach spaces and give best proximity point for generalized semi-cyclic ϕ-contraction pair. theorem 2.5. let a and b be nonempty closed and convex subsets of a reflexive and strictly convex banach space x and s,t : a ∪ b → a ∪ b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. if (a−a) ∩ (b−b) = {0}, then there exists a unique x ∈ a and a unique y ∈ b such that ‖x−sx‖ = d(a,b) = ‖ty −y‖. proof. if d(a,b) = 0, the result follows from theorem 3.1(i) of [6]. so we assume that d(a,b) > 0. since a and b are closed and convex, they are weakly closed. by theorem 2.3, there exists (x,y) ∈ a×b with ‖x−y‖ = d(a,b). suppose that there exists (a,b) ∈ a×b with ‖a−b‖ = d(a,b). since (a−a) ∩ (b −b) = {0}, x−y 6= a− b. by the strict convexity of x, and convexity of a and b, we have ‖(x + a)/2 − (y + b)/2‖ = ‖(x−y)/2 + (a− b)/2‖ < d(a,b), which is a contraction. this shows that (x,y) is unique. theorem 2.6. let a and b be nonempty closed and convex subsets of a reflexive and strictly convex banach space x and s,t : a ∪ b → a ∪ b be such that the pair (s,t) is generalized semi-cyclic ϕ-contraction. then there exist unique x ∈ a and y ∈ b such that ‖x−sx‖ = d(a,b) = ‖ty −y‖, provided that one of the following conditions is satisfied (i) s is weakly continuous on a and t is weakly continuous on b. (ii) t, s satisfy the proximal property. proof. if d(a,b) = 0, the result follows from theorem 3.1(i) of [6]. so we assume that d(a,b) > 0. since a and b are closed and convex, they are weakly closed. by theorem 2.4, there exists x ∈ a and y ∈ b such that ‖x−sx‖ = d(a,b) = ‖ty −y‖. for the uniqueness of x, suppose that there exists a ∈ a such that ‖a−sa‖ = d(a,b). by the strict convexity of x, and convexity of a and b, we have ‖(x + a)/2 − (sx + sa)/2‖ = ‖(x−sx)/2 + (a−sa)/2‖ < d(a,b), which is a contraction. now, for uniqueness of y, suppose that there exists b ∈ b such that ‖tb− b‖ = d(a,b). since ‖(y + b)/2 − (ty + tb)/2‖ = ‖(y −ty)/2 + (b−tb)/2‖ < d(a,b), which is a contraction. references [1] m. a. al-thagafi, n. shahzad, convergence and existence result for best proximity points, nonlinear analysis, theory, methods and applications, 70(2009), 3665-3671. [2] a. anthony eldred, p. veeramani, existence and convergence of best proximity points, j. math. anal. appl, 323(2006), 1001-1006. common best proximity points 9 [3] m. gabeleh, a. abkar, best proximity points for semi-cyclic contractive pairs in banach spaces, int. math. forum, 6(2011), 2179 2186. [4] e. karapinar, best proximity points of cyclic mappings, appl. math. letters, 25(2012), 1761-1766. [5] w. a. kirk, p. s. srinivasan and p. veeramani, fixed points for mapping cyclic contractions, fixed point theory, 4(2003), 79-89. [6] b. prasad, a best proximity theorem for some general contractive pair of maps, proc. of int. conf. on emerging trends in engineering and technology, 2013. [7] sh. rezapour, m. derafshpour and n. shahzad, best proximty points of cyclic ϕ− contractions on reflexive banach spaces, fixed point theory and application, 2010(2010), art. id 946178. [8] b. s. thakur, a. sharma, existence and convergence of best proximity points for semi-cyclic contraction pairs, international journal of analysis and applications, 5(2014), 33-44. 1faculty of mathematics, yazd university, yazd, iran 2guru nanak dev university, amristar, india ∗corresponding author: m.ahmadi@stu.yazd.ac.ir international journal of analysis and applications volume 16, number 4 (2018), 569-593 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-569 study of solution for a parabolic integrodifferential equation with the second kind integral condition dehilis sofiane, bouziani abdelfatah and oussaeif taki-eddine∗ department of mathematics and informatics, the larbi ben m’hidi university, oum el bouaghi, algeria ∗corresponding author: taki maths@live.fr abstract. in this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. the existence, uniqueness of a strong solution for the linear problem based on a priori estimate ”energy inequality” and transformation of the linear problem to linear first-order ordinary differential equation with second member. then by using a priori estimate and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the integrodifferential problem. also we have developed an efficient numerical scheme, which uses temporary problems with standard boundary conditions. a suitable combination of the auxiliary solutions defines an approximate solution to the original nonlocal problem, the algebraic matrices obtained after the full discretization are tridiagonal, then the solution is obtained by using the thomas algorithm. some numerical results are reported to show the efficiency and accuracy of the scheme. 1. introduction the topic of integro-differential equations which are combination of differential and integral has attracted many scientists and researchers due to their applications in many areas; see, for example, [16, 17] . many mathematical formulation of physical phenomena contain integro-differential equations, and these equations may arise in fluid dynamics, biological models, and chemical kinetics; for more details, see [20, 40] . received 2017-10-23; accepted 2018-01-04; published 2018-07-02. 2010 mathematics subject classification. 35r09, 65m20. key words and phrases. parabolic integrodifferential equation; nonlocal problem; energy inequality. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 569 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-569 int. j. anal. appl. 16 (4) (2018) 570 integro-differential equations are usually difficult to solve analytically, so it is required to obtain an efficient approximate solution. nowadays various nonlocal problems for partial differential equations have been actively studied and one can find a lot of papers dealing with them (see [13][29] , [12][21] and references therein). afterwards, the nonlocal problems for integro-differential equation with integral conditions was studied by many authors, see a. merad and a. bouziani [23] , [26]. motivated by this we study a parabolic integrodifferential equation with nonlocal second kind integral condition. 2. preliminaries and functional spaces in the rectangular domain ω = (0, 1) × (0,t), with t < ∞,we consider the equation: lu = ∂u ∂t − ∂2u ∂x2 = ∫ t 0 a (t−s) g (s,u) ds + f(x,t), (2.1) with the initial data `u = u(x, 0) = ϕ (x) , x ∈ (0, 1) , (2.2) with the second kind integral conditions ux (0, t) = ∫ 1 0 k0 (x,t) u (x,t) dx, (2.3) ux (1, t) = ∫ 1 0 k1 (x,t) u (x,t) dx, (2.4) where f, ϕ, k0, k1 ansd g are known functions. note that a is bounded function where |a (t−s)| < a0, a0 is a positive constant. and the function g verify the following inequality ‖g (s,u)‖l2(ω) 6 c1 ‖u‖l2(ω) + c2, c1,c2 are positive constants. we shall assume that the function ϕ satisfies a compatibility conditions with (2.3) and (2.4) , i.e., ϕx (0) = ∫ 1 0 k0 (x, 0) ϕ (x) dx, ϕx (1) = ∫ 1 0 k1 (x, 0) ϕ (x) dx. some problems of modern physics and technology can be described in terms of partial differential equations with nonlocal conditions. the integral term of our problem (that is, ∫ t 0 a (t−s) g (s,u) ds appears,because in some fields such as the heat transfer, nuclear reactor dynamics and thermoelasticity, we need to reflect the effects of the memory of the system in model, but describing such a system as a function at a given space and time ignores the effect of past history. therefore, the way of remedy this difficulty is including an integral term in the basic partial differential equation that leads to a partial integro-differential equations( int. j. anal. appl. 16 (4) (2018) 571 pide) [39]. the study of the problem (2.1)-(2.2)with some special types of boundary conditions of the form ux(0, t) = α(t) and ∫ 1 0 u (x,t) dx = e(t) motivated by the works of dabas and bahuguna [15],and guezanelakoud et al. [18]. bouziani and mechri [8], studied , problem (2.1)-(2.2) with purely nonlocal (integral) conditions ∫ 1 0 u(x,t) = e(t) and ∫ 1 0 xu(x,t) = g(t) . for other models, we refer the reader, for instance, to [6][38], and references therein. most of the previous studies, the authors used the rothe method (see [10], [8], [15]), the laplace transform of the problem and then used numerical technique for the inverse laplace transform to obtain the numerical solution (see [1]). it is well known that the classical methods used widely to prove solvability of initial-boundary problems break down when applied to nonlocal problems. nowadays some methods have been advanced for overcoming difficulties arising from nonlocal conditions. these methods are different and the choice of a concrete one depends on a form of a nonlocal condition. in this article, we focus on spatial nonlocal integral conditions like [30], of which we give three examples:∫ 1 0 k (x,t) u (x,t) dx = 0, (2.5) ux (0, t) = ∫ 1 0 k (x,t) u (x,t) dx, (2.6) a (t) u (0, t) = ∫ 1 0 k (x,t) u (x,t) dx, (2.7) condition (2.5) is a nonlocal first kind condition, (2.6) and (2.7) are second kind nonlocal conditions. the kind of a nonlocal integral condition depends on the presence or lack of a term containing a trace of the required solution or its derivative outside the integral [30] . problems with nonlocal conditions of the forms (2.5) and (2.7) are investigated in [30], [11],and [36]. we pay attention on the second one, (2.6) which has not been studied so far with this class of integro-differential problems. this paper is organized as follows. in section 3, we establish the uniquness of solution by using a priori estimate method or the energy-integral method. in sect 4, we first establish the existence of solutions of the linear problem by using the density of the range of the operator generated by the abstract formulation of the stated linear problem; secondly reformulating the integro-differential problem to a semi-linear problem, and after that we prove the slovability of semi-linear problem by using a priori estimate and applying an iterative process based on results obtained for the linear problem (see [34]), we prove the existence, uniqueness of the weak generalized solution of the integrodifferential problem.section 5 is devoted to the construction of approximate solutions of problem (2.1)-(2.4), we discretize the problem by backward euler in time and finite differences in space. the main numerical difficulty become visible after the discretization,the presence of an integral operator in the boundary conditions gives rise to rows/collumns, which are full. to avoid the problems with special solvers for algebraic systems, we design a very easy numerical algorithm, based on int. j. anal. appl. 16 (4) (2018) 572 superposition principle, this technique lead to a linear systems have a tridiagonal coefficient matrix, so they can be solved very efficiently by fast gauss elimination (which is also known as the thomas algorithm). finally, in section 6 we presents two numerical examples to illustrate the performance and efficiency of the proposed algorithm. 3. an energy estimate and uniqueness of solution the method used here is one of the most efficient functional analysis methods and important techniques for solving partial differential equations with integral conditions, which has been successfully used in investigating the existence, uniqueness, and continuous dependence of the solutions of pde’s, the so-called a priori estimate method or the energy-integral method. this method is essentially based on the construction of multiplicators for each specific given problem, which provides the a priori estimate from which it is possible to establish the solvability of the posed problem. more precisely, the proof is based on an energy inequality and the density of the range of the operator generated by the abstract formulation of the stated problem, so to investigated the posed problem, we introduce the needed function spaces. in this paper, we prove the existence and the uniqueness for solution of the problem (2.1) − (2.4) as a solution of the operator equation lu = z. (3.1) where l = (l,`), with domain of difinition e consisting of functions u ∈ l2 (0,t,l2 (0, 1)) := l2 (ω) such that ux ∈ l2 (ω) and u satisfies condition (2.3) and (2.4) ; the operator l is considered from e to f, where e is the banach space consisting of all functions u(x,t) having a finite norm ‖u‖2e = ‖u‖ 2 l2(ω) + ‖ux‖ 2 l2(ω) , and f is the hilbert space consisting of all elements z = (f,ϕ) for which the norm ‖z‖2f = ‖f‖ 2 l2(ω) + ‖ϕ‖ 2 l2(0,1) is finite. theorem 3.1. if ε > 0, where ε << 1 2 . then for any function u ∈ e and we have the inequality ‖u‖e ≤ c‖lu‖f (3.2) where c is a positive constant independent of u. int. j. anal. appl. 16 (4) (2018) 573 proof. assume that a solution of the problem (2.1) − (2.4) exists. we multiply the equation (2.1) by u and integrating over ωτ, where ωτ = (0, 1) × (0,τ), we get ∫ ωτ u ·mu dxdt = ∫ ωτ ut ·u dxdt− ∫ ωτ uxx ·u dxdt = ∫ ωτ [∫ 1 0 a (t−s) g (s,u) ] ·u dxdt + ∫ ωτ f (x,t) ·u dxdt (3.3) integrating by parts each term of the left-hand side of (3.3) over ωτ, 0 < τ < t , we obtain 1 2 ∫ 1 0 u (x,τ) 2 dx + ∫ ωτ u2x dxdt = ∫ τ 0 ux (1, t) u (1, t) dt− ∫ τ 0 ux (0, t) u (0, t) dt + 1 2 ∫ 1 0 ϕ2 dx + ∫ ωτ [∫ 1 0 a (t−s) g (s,u) ] ·u dxdt + ∫ ωτ f ·u dxdt (3.4) our next aim is to derive estimates of the right-hand side part of (3.4) . by using the cauchy inequality with ε; we have ∫ τ 0 ux (1, t) u (1, t) dt < ε 2 ∫ τ 0 u2 (1, t) dt + 1 2ε ∫ τ 0 u2x (1, t) dt. (3.5)∫ τ 0 ux (0, t) u (0, t) dt < ε 2 ∫ τ 0 u2 (0, t) dt + 1 2ε ∫ τ 0 u2x (0, t) dt. (3.6) to obtain the estimate, we need the inequalities u2 (ξ,t) 6 2 ∫ ξ x u2x dx + 2u 2 which easily follow from the equalities u (ξ,t) = ∫ ξ x ux (x,t) dx + u (x,t) ξ = 0 or 1. also by (2.3) and (2.4), we obtain ∫ τ 0 ux (1, t) u (1, t) dt− ∫ τ 0 ux (0, t) u (0, t) dt 6 ε 2 ∫ τ 0 u2 (1, t) dt + 1 2ε ∫ τ 0 u2x (1, t) dt + ε 2 ∫ τ 0 u2 (0, t) dt + 1 2ε ∫ τ 0 u2x (0, t) dt 6 ε 2 ∫ τ 0 [ 2 ∫ 0 x u2x dx + 2u 2 ] dt + 1 2ε ∫ τ 0 [∫ 1 0 k0 (x,t) u (x,t) dx ]2 dt + ε 2 ∫ τ 0 [ 2 ∫ 1 x u2x dx + 2u 2 ] dt + 1 2ε ∫ τ 0 [∫ 1 0 k1 (x,t) u (x,t) dx ]2 dt int. j. anal. appl. 16 (4) (2018) 574 so, by using holder inequality, we have∫ τ 0 ux (1, t) u (1, t) dt− ∫ τ 0 ux (0, t) u (0, t) dt (3.7) 6 2ε ∫ ωτ u2x dxdt + 2ε ∫ τ 0 u2 dt + k ε ∫ ωτ u2 dxdt; where the constant k = maxi=0,1 ∫ ωτ k2i (x,t) dxdt. now, we estimate ∫ ωτ [∫ 1 0 a (t−s) g (s,u) ] ·u dxdt, first we can find an constant c verify ‖g (s,u)‖l2(ω) 6 c1 ‖u‖l2(ω) + c2 < c‖u‖l2(ω) , c > 0. then, we get ∫ ωτ [∫ 1 0 a (t−s) g (s,u) ] ·u dxdt 6 ε 2 ∫ ωτ u2dxdt + ta0 2ε ∫ ωτ g2dxdt 6 ( ε 2 + tca0 2ε )∫ ωτ u2dxdt. (3.8) remains apply the inequality the cauchy inequality with ε to the end terms of the right-hand side part of (3.4) and using (3.7) and (3.8) , we get 1 2 ∫ 1 0 u (x,τ) 2 dx + ∫ ωτ u2x dxdt (3.9) = 2ε ∫ ωτ u2x dxdt + 2ε ∫ τ 0 u2 dt + k ε ∫ ωτ u2 dxdt + 1 2 ∫ 1 0 ϕ2 dx + ( ε 2 + tca0 2ε )∫ ωτ u2dxdt + ε 2 ∫ ωτ u2dxdt + 1 2ε ∫ ωτ f2dxdt then, we obtain 1 2 ∫ 1 0 u (x,τ) 2 dx + ∫ ωτ u2x dxdt = 2ε ∫ ωτ u2x dxdt + ( 3ε + tca0 2ε + k ε )∫ ωτ u2dxdt + 1 2ε ∫ ωτ f2dxdt + 1 2 ∫ 1 0 ϕ2 dx using lemma 1 of gronwall in [32] , we have∫ 1 0 u (x,τ) 2 dx + ∫ ωτ u2x dxdt (3.10) ≤ d (∫ ω f2dxdt + ∫ 1 0 ϕ2dx ) , where d = 1 2 exp ( 3ε + tca0 2ε + k ε ) . int. j. anal. appl. 16 (4) (2018) 575 by integrating the inequality (3.10) over (0,t) , we obtain the desired inequality, where c = (td) 1 2 . so, we get ‖u‖2l2(ω) + ‖ux‖ 2 l2(ω) 6 c ( ‖f‖2l2(ω) + ‖ϕ‖ 2 l2(0,1) ) . (3.11) � 4. existence of solution of the integrodifferential problem this section is consecrated to the proof of the existence of the solution on the data of the problem (2.1) − (2.4). we can reformulating the integro-differential problem to a semi-linear problem by putting∫ t 0 a (t−s) g (s,u) ds + f(x,t) = h(x,t,u) where exists a positive constant δ such that |h(x,t,u1) −h(x,t,u2)| ≤ δ ( ‖u1 −u2‖l2(q) ) , (c∗) ∀u1,u2 ∈ l2(ω), (x,t) ∈ ω. therefore to study the existence of solution of previous problem (2.1)−(2.4), is enough to study the following semi-linear problem: lu = ∂u ∂t − ∂2u ∂x2 = h(x,t,u), (4.1) with the initial data `u = u(x, 0) = ϕ (x) , x ∈ (0, 1) , (4.2) with the second kind integral conditions ux (0, t) = ∫ 1 0 k0 (x,t) u (x,t) dx, (4.3) ux (1, t) = ∫ 1 0 k1 (x,t) u (x,t) dx, (4.4) let us consider the following auxiliary problem with homogeneous equation lw = ∂w ∂t − ∂2w ∂x2 = 0, (4.5) `w = w(x, 0) = ϕ(x), (4.6) wx (0, t) = ∫ 1 0 k0 (x,t) w (x,t) dx, (4.7) wx (1, t) = ∫ 1 0 k1 (x,t) w (x,t) dx, (4.8) int. j. anal. appl. 16 (4) (2018) 576 if u is a solution of problem (4.1)−(4.4) and w is a solution of problem (4.5)−(4.8), then y = u−w satisfies ly = ∂y ∂t − ∂2y ∂x2 = g (x,t,y) , (4.9) `y = y(x, 0) = 0, (4.10) yx (0, t) = 0, (4.11) yx (1, t) = 0. (4.12) where g (x,t,y) = h (x,t,y + w) , as the function h, the function g satisfies the condition (c∗) , that is there exists a positive constant δ such that |g(x,t,y1) −g(x,t,y2)| ≤ δ ( ‖y1 −y2‖l2(q) ) (c∗∗) ∀y1,y2 ∈ l2(ω), (x,t) ∈ ω. to show the existence of solutions of the problem (4.5)−(4.8), it is enough to transform the problem to the linear first-order ordinary differential equation with second member. for that we integrate the equation (4.5) over [0, 1] and using (4.7) − (4.8), we get∫ 1 0 ∂w ∂t dx = ∫ 1 0 (k1 (x,t) −k0 (x,t)) w (x,t) dx, ∀x ∈ [0, 1] ; then, we obtain ∫ 1 0 ( ∂w ∂t −k (x,t) w (x,t) ) dx = 0, where k1 (x,t) −k0 (x,t) = k (x,t) . (4.13) so, we can prove that there existe a function ψ verify that ∂w ∂t −k (x,t) w (x,t) = ψ (x,t) , where ∫ 1 0 ψ (x,t) dx = 0. (4.14) clearly, that the solution of (4.5) by using (4.6) is given by w (x,t) = ϕ (x) exp 1 exp (∫ t 0 k (x,θ) dθ ) + exp (∫ t 0 k (x,θ) dθ )∫ t 0 [ ψ(x,τ) exp ( − ∫ τ 0 k (x,θ) dθ )] dτ. therefore, the existence of solution is guaranteed. according to this results, we deduce that problem (4.5) − (4.8) admits a unique solution. therefore it remains to solve and prove that the problem (4.9) − (4.12) has a unique weak solution. let us construct an iteration sequence in the following way: starting with y(0) = 0, the sequence { y(n) } n∈n is defined as follows: given the element y(n−1), then for n = 1, 2, ... solve the problem: ∂y(n) ∂t − ∂2y(n) ∂x2 = g ( x,t,y(n−1) ) , (4.15) y(n)(x, 0) = 0, (4.16) int. j. anal. appl. 16 (4) (2018) 577 y(n)x (0, t) = 0, (4.17) y(n)x (1, t) = 0. (4.18) clearly, for fixed n, each problem (4.15) − (4.18) has a unique solution y(n) (x,t). if we set z(n)(x,t) = y(n+1)(x,t) −y(n)(x,t), then we have the new problem ∂z(n) ∂t − ∂2z(n) ∂x2 = p (n−1) (x,t) , (4.19) z(n)(x, 0) = 0, (4.20) z(n)x (0, t) = 0, (4.21) z(n)x (1, t) = 0. (4.22) where p (n−1) (x,t) = g ( x,t,y(n) ) −g ( x,t,y(n−1) ) . lemma 4.1. assume that condition (c∗∗) holds, then for the linearized problem (4.19) − (4.22), we have the a priori estimate ∥∥∥z(n)∥∥∥ l2(0,t; h1(0,1)) ≤ m ∥∥∥z(n−1)∥∥∥ l2(0,t; h1(0,1)) , (4.23) where m is a positive constant given by m = √ t ε δ2 min ( 1−εt 2 ,t ). proof. multiplying the equation (4.19) by z(n) and integrating over ωτ, where ωτ = (0, 1) × (0,τ), we get∫ ωτ ∂z(n) ∂t ·z(n)dxdt− ∫ ωτ ∂2z(n) ∂x2 ·z(n)dxdt = ∫ ωτ p (n−1) ·z(n)dxdt. (4.24) integrating by parts the second term of the left-hand side in (4.24) and taking into account conditions (4.20) , (4.21) and (4.22), we obtain 1 2 ∫ 1 0 ( z(n) (x,τ) )2 dx + ∫ ωτ ( ∂z(n) ∂x )2 dxdt = ∫ ωτ p (n−1) ·z(n)dxdt. (4.25) using the cauchy inequality to the right-hand side of (4.25), we get 1 2 ∫ 1 0 ( z(n) (x,τ) )2 dx + ∫ ωτ ( ∂z(n) ∂x )2 dxdt ≤ 1 2 ∫ ωτ ( p (n−1) )2 dxdt + 1 2 ∫ ωτ ( z(n) )2 dxdt. using lemma of gronwall, we obtain∫ 1 0 ( z(n) (x,τ) )2 dx + ∫ ωτ ( ∂z(n) ∂x )2 dxdt ≤ exp (t) ∫ ωτ ( p (n−1) )2 dxdt. (4.26) int. j. anal. appl. 16 (4) (2018) 578 on the other hand., by virtue of condition (c∗∗) , we have(∫ ωτ ( p (n−1) )2 dxdt ) (4.27) ≤ δ2 ∫ ωτ (∣∣∣z(n−1) (x,t)∣∣∣ + ∣∣∣∣∂z(n−1) (x,t)∂x ∣∣∣∣ )2 dxdt ≤ 2δ2 ∫ τ 0 (∥∥∥z(n−1) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n−1) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt. substituting (4.27) into (4.26), we get∫ 1 0 ( z(n) (x,τ) )2 dx + ∫ ωτ ( ∂z(n) ∂x )2 dxdt ≤ 2δ2 exp (t) ∫ t 0 (∥∥∥z(n−1) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n−1) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt. the right hand side here is independent of τ; hence, replacing the left hand side by the upper bound with respect to τ, we obtain∫ 1 0 ( z(n) (x,τ) )2 dx + ∫ ωt ( ∂z(n) ∂x )2 dxdt ≤ 2δ2 exp (t) ∫ t 0 (∥∥∥z(n−1) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n−1) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt. now by integrating over (0,t), we get∫ ωt ( z(n) )2 dx + t ∫ ωt ( ∂z(n) ∂x )2 dxdt ≤ 2tδ2 exp (t) ∫ t 0 (∥∥∥z(n−1) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n−1) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt. so, we obtain ∫ t 0 (∥∥∥z(n) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt ≤ 2tδ2 exp (t) min (1,t) ∫ t 0 (∥∥∥z(n−1) (•, t)∥∥∥2 l2(0,1) + ∥∥∥∥∂z(n−1) (•, t)∂x ∥∥∥∥2 l2(0,1) ) dt finally, we find ∥∥∥z(n)∥∥∥2 l2(0,t; h1(0,1)) ≤ m ∥∥∥z(n−1)∥∥∥2 l2(0,t; h1(0,1)) , (4.28) where m = 2tδ2 exp (t) min (1,t) . from the criteria of convergence of series, we see that the series ∑∞ n=1 z (n) converges if m < 1, that is if δ < √ min (1,t) 2t exp (t) . int. j. anal. appl. 16 (4) (2018) 579 since z(n)(x,t) = y(n+1)(x,t) −y(n)(x,t), then it follows that the sequence (y(n))n∈n defined by y(n)(x,t) = n−1∑ i=0 z(i) + y(0)(x,t), converges to an element y ∈ l2 ( 0,t; h1(0, 1) ) . � remains to precise the concept of the solution we are considering. let v = v(x,t) be any function from c1 (ω) . we shall compute the integral ∫ ω gvdxdt, for this we assume vx (0, t) = vx (1, t) = 0. by using conditions on y, we have − ∫ ω ∂2y ∂x vdxdt = ∫ ω ∂v ∂x ∂y ∂x dxdt. then we put a (y,v) = ∫ ω ∂y ∂t vdxdt + ∫ ω ∂v ∂x ∂y ∂x dxdt = ∫ ω vgdxdt, (4.29) definition 4.1. for every v ∈ c1 (ω), a function y ∈ l2(0,t; h1(0, 1)) is called a weak solution of problem (4.9) − (4.12) if (4.30) holds under the conditions of y. now, we must show that the limit function y is a solution of the problem under study. to do this, we will show that y verifies (4.30) as mentioned in definition 1. so, we consider the weak formulation of problem (4.9) − (4.12) : a (y,v) = ∫ ω vgdxdt. (4.30) from (4.30) , we have a ( y(n),v ) = a ( y(n) −y,v ) + a (y,v) = ∫ ω v [ g ( x,t,y(n−1) ) −g (x,t,y, ) ] dxdt∫ ω vg (x,t,y) dxdt. (4.31) however, we apply holder inequality, we get a ( y(n) −y,v ) = ∫ ω v [ g ( x,t,y(n−1) ) −g (x,t,y) ] dxdt ≤ δ 2 ‖v‖l2(ω) ∥∥∥y(n) −y∥∥∥ l2(ω) . (4.32) so by passing to the limit in (4.33) as n →∞, (4.31) become a ( y(n),v ) = ∫ ω vg (x,t,y) dxdt. (4.33) int. j. anal. appl. 16 (4) (2018) 580 again passing to the limit in (4.31) as n →∞, we obtain a (y,v) = ∫ ω vg (x,t,y) dxdt. therefore, we have established the following result: theorem 4.1. assume that condition (h2) holds and δ < √ min ( 1−εt 2 ,t ) t ε then the problem (4.9) − (4.12) admits a weak solution in l2 ( 0,t; h1(0, 1) ) . it remains to prove that problem (4.9) − (4.12) admits a unique solution. theorem 4.2. under the condition (c∗∗) , the solution of the problem (4.9) − (4.12) is unique. proof. suppose that y1 and y2 in l 2 ( 0,t; h1(0, 1) ) are two solution of (4.9) − (4.12), then h = y1 − y2 satisfies h ∈ l2 ( 0,t; h1(0, 1) ) and ∂h ∂t − ∂2h ∂x2 = ψ (x,t) (x,t) ∈ ω, (4.34) h(x, 0) = 0, (4.35) hx (0, t) = 0, (4.36) hx (1, t) = 0, (4.37) ψ (x,t) = g (x,t,y1) −g (x,t,y2) . following the same procedure done in establishing the proof of lemma 1, then for the problem (4.35)−(4.38), we get ‖h‖l2(0,t; h1(0,1)) ≤ m ‖h‖l2(0,t; h1(0,1)) . (4.38) since m < 1, then from (4.39) that (1 −m)‖h‖l2(0,t; h1(0,1) ≤ 0, from which we conclude that y1 = y2 in l 2 ( 0,t; h1(0, 1) ) . � int. j. anal. appl. 16 (4) (2018) 581 5. construction of approximate solutions in order to solve the problem (2.1)−(2.4) , first we divide the time interval [0,t] into n ∈ n equidistant subintervals (tj−1, tj) for tj = jτ , where τ = t n we introduce the following notation uj = uj(x) = u(x,tj), after replacing the derivative ∂u ∂t by backward finite difference approximations uj −uj−1 τ and the integral by rectangular rule. then problem (2.1) − (2.4) reduced to the solutions of recurrent system of ode problems at each successive time point tj for j = 1, ...,n find, successively for j = 1, ...,n ; functions uj : (0, 1) → r such that: uj −uj−1 τ − d2uj d2x = τ j−1∑ k=0 a(tj − tk)g(tk,uk) + f(x,tj) x ∈ (0, 1) (5.1) duj dx (0) = ∫ 1 0 k0(x,tj)u(x,tj)dx (5.2) duj dx (1) = ∫ 1 0 k1(x,tj)u(x,tj)dx (5.3) u0(x) = ϕ(x) x ∈ (0, 1) (5.4) the main numerical difficulty become visible after the full discretization of these nonlocal problem , the presence of an integral bc in the problem gives rise to rows, which are full(see algorithm 1). 5.1. algorithm 1 :(a1). for the space discretization we use the finite differences scheme . we divide the space interval [0, 1] into m ∈ n equidistant subintervals of equal lengths h = 1 m second-order difference is used to approximate the second order spatial derivative : ∂2ui,j ∂x2 = ui−1,j − 2ui,j + ui+1,j h2 + o(h2), where ui,j = u(xi, tj), and employing central-differences to approximat the first order spatial derivative in the boundary condition : ∂ui,j ∂x = ui+1,j −ui−1,j 2h + o(h2), we construct a difference scheme for the problem (5.1)-(5.4): ui,j − τ ui−1,j − 2ui,j + ui+1,j h2 =ui,j−1 + τ 2 j−1∑ k=0 a(tj − tk)g(tk,ui,k) (5.5) + τfi,j ,i = 0, ...,m int. j. anal. appl. 16 (4) (2018) 582 u1,j −u−1,j 2h = ∫ 1 0 k0(x,tj)u(x,tj)dx (5.6) um+1,j −um−1,j 2h = ∫ 1 0 k1(x,tj)u(x,tj)dx (5.7) ui,0 = ϕi i = 0, ...,m after some rearrangement, the equation (5.5) becomes : −rui−1,j + (1 + 2r)ui,j −rui+1,j =ui,j−1 + τ2 j−1∑ k=0 a(tj − tk)g(tk,ui,k) (5.8) + τfi,j , i = 0, ...,m where r = τ h2 . we approximate the integral in (5.6)-(5.7) numerically by the trapezoidal numerical integration rule: u1,j −u−1,j 2h = ∫ 1 0 k0(x,tj)u(x,tj)dx (5.9) = h 2 (k0(x0, tj)u0,j + 2 m−1∑ k=1 k0(xk, tj)uk,j + k0(xm, tj)um,j) um+1,j −um−1,j 2h = ∫ 1 0 k1(x,tj)u(x,tj)dx (5.10) = h 2 (k1(x0, tj)u0,j + 2 m−1∑ k=1 k1(xk, tj)uk,j + k1(xm, tj)um,j) which is the same second-order of accuracy in space as the methods used for spatial derivative . equation (5.9) presents m + 1 linear equations in m + 3 unknowns u−1,u0, ...,um+1. eliminating of the ”fictitious” value u−1,j beteween (5.8)i=0 and (5.9) gives : (1 + 2r + τk0(x0, tj))u0,j + (−2r + 2τk0(x1, tj))u1,j +2τk0(x2, tj)u2,j + ... + 2τk0(xm−1, tj)um−1,j + τk0(xm, tj)um,j =u0,j−1 + τ 2 j−1∑ k=0 a(tj − tk)g(tk,u0,k) + τf0,j (5.11) similarly, eliminating um+1,j beteween (5.8)i=m and (5.10) gives : − τk1(x0, tj)u0,j − 2τk1(x1, tj))u1,j − ...− 2τk1(xm−2, tj)um−2,j (5.12) + (−2r − 2τk1(xm−1, tj))um−1,j + (1 + 2r − τk1(xm, tj))um,j = um,j−1 + τ 2 j−1∑ k=0 a(tj − tk)g(tk,um,k) + τfm,j int. j. anal. appl. 16 (4) (2018) 583 combining (5.11), (5.9), with (5.12) yields an (m + 1)×(m + 1) linear system of equations whose coefficient matrix aj has the form: aj =   a00 a01 a02 ...... a0m −r 1 + 2r −r ...... 0 . . . ...... . 0 ...... −2r 1 + 2r −2r am0 am1 am2 ...... amm   where a00,a01, ...,a0m and am0,am1, ...,amm are the coefficients in (5.11) and (5.12), respectively. we will denote the right-side of the system by bj = (b0,b1, ..bm ) t , with bi = ui,j−1 + τ 2 ∑j−1 k=0 a(tj −tk)g(tk,ui,k) + τfi,j, i = 0, ...,m . we write the system in the matrix form : ajuj = bj (5.13) which have to be solved successively with increasing time step j = 1, ..,n. the main numerical problem is the special character of the algebraic matrix obtained, tridiagonal except that their first and last rows are full,this needs a special solver to get a result. but there exist a simple way how to avoid this complication, we explain it in algorithm 2. 5.2. algorithm 2:(a2). to get rid of the nonlocal bc,we make use of a slightly modified idea of [37], for any given j we introduce three auxiliary problems. the first one with an unknown function vi is given as:  vj − τ d2vj d2x = uj−1 + τ 2 ∑j−1 k=0 a(tj − tk)g(tk,uk) + τf(x,tj) x ∈ (0, 1) dvj dx (0) = 0 dvj dx (1) = 0 (5.14) and the initial condition v0(x) = ϕ(x) ,x ∈ [0, 1] . the second one with the unknown z reads as:   z − τ d2z d2x = 0 x ∈ (0, 1) dz dx (0) = 1 dz dx (1) = 0 (5.15) the third one with the unknown w reads as int. j. anal. appl. 16 (4) (2018) 584   w − τ d2w d2x = 0 x ∈ (0, 1) dw dx (0) = 0 dw dx (1) = 1 (5.16) let us note that the temporary problems are standard problems. let αj and βj be any real number, the principle of linear superposition gives that ωj := vj + αjz + βjw is the solution to the following bvp  ωj − τ ∂2ωj ∂2x = uj−1 + τ 2 ∑j−1 k=0 a(tj − tk)g(tk,uk) +τf(x,tj) x ∈ (0, 1) dωj dx (0) = αj dωj dx (1) = βj (5.17) and the initial condition ω0(x) = ϕ(x) ,x ∈ [0, 1] . we have to pick up the appropriate value of the free parameter αj and βj for which the function ωj be a solution to problem (5.1)-(5.4). we are looking for an αj and βj such that αj = ∫ 1 0 k0(x,tj)u(x,tj)dx = ∫ 1 0 k0(x,tj) ( vj + αjz(x) + βjw(x) ) dx βj = ∫ 1 0 k1(x,tj)u(x,tj)dx = ∫ 1 0 k1(x,tj) ( vj + αjz(x) + βjw(x) ) dx then ωj will be a solution to problem (5.1)−(5.3) if and only if the pair (αj, βj) is a solution of the following system of equations   αj(1 − ∫ 1 0 k0(x,tj)zdx) −βj ∫ 1 0 k0(x,tj)wdx = ∫ 1 0 k0(x,tj)vjdx −αj ∫ 1 0 k1(x,tj)zdx + βj(1 − ∫ 1 0 k1(x,tj)wdx) = ∫ 1 0 k1(x,tj)vjdx (5.18) we have to check if the determinant d = (1 − ∫ 1 0 k0(x,tj)zdx)(1 − ∫ 1 0 k1(x,tj)wdx) − ∫ 1 0 k0(x,tj)wdx ∫ 1 0 k1(x,tj)zdx (5.19) of system (5.18) is different from zero. if d 6= 0 then we easily deduce :   αj = (1 − ∫ 1 0 k1(x,tj)wdx) ∫ 1 0 k0(x,tj)vjdx + ∫ 1 0 k1(x,tj)vjdx ∫ 1 0 k0(x,tj)wdx d βj = αj(1 − ∫ 1 0 k0(x,tj)zdx) − ∫ 1 0 k0(x,tj)vjdx∫ 1 0 k0(x,tj)wdx (5.20) int. j. anal. appl. 16 (4) (2018) 585 lemma 5.1. let d = (1 − ∫ 1 0 k0(x,tj)zdx)(1 − ∫ 1 0 k1(x,tj)widx) − ∫ 1 0 k0(x,tj)wdx ∫ 1 0 k1(x,tj)zdx there exists τ0 > 0 such that d > 1 2 proof. one can see that the solution of the second auxiliary problem is : z(x) = − √ τch(x−1√ τ ) sh( 1√ τ ) we have lim τ→0 z(x) = lim τ→0 − √ τch(x−1√ τ ) sh( 1√ τ ) = lim τ→0 − √ τ(e x−2√ τ + e −x√ τ ) e 2√ τ − 1 = 0 and the solution of the third auxiliary problem is : w(x) = √ τch( x√ τ ) sh( 1√ τ ) and also lim τ→0 w(x) = lim τ→0 √ τch( x√ τ ) sh( 1√ τ ) = lim τ→0 √ τ(e x−1√ τ + e −x−1√ τ ) e 2√ τ − 1 = 0 the variational formulations of temporary problems are: (z, φ) + τ( dz dx , dφ dx ) = −τφ(0), for any φ ∈ h1(0, 1) (5.21) we set φ = z into (5.21) and we get ‖z‖2 + τ‖ dz dx ‖2 = −τφ(0) = τ √ τcoth( 1 √ τ ) ≤ c analogously for w (w, φ) + τ( dw dx , dφ dx ) = τφ(1), for any φ ∈ h1(0, 1) (5.22) we set φ = w into (5.22) and we get ‖w‖2 + τ‖ dw dx ‖2 = τφ(1) = τ √ τcoth( 1 √ τ ) ≤ c lebegue domineted theorem says : lim τ→0 ‖z‖2 = lim τ→0 ∫ 1 0 z2 = ∫ 1 0 lim τ→0 z2 = 0 analogously for w lim τ→0 ‖w‖2 = 0 cauchy inequality says | ∫ 1 0 k0z| ≤ ‖k0‖‖z‖→ 0 when τ → 0 | ∫ 1 0 k1w| ≤ ‖k1‖‖w‖→ 0 when τ → 0 therefor lim τ→0 d = 1 int. j. anal. appl. 16 (4) (2018) 586 from the definition of a limit we easily arrive at for ε = 1/2 there exists a τ0 such that: for any 0 < τ < τ0 we haved > 1/2 . � for the space discretization we use the same scheme in algorithm 1 for a better comparison. we construct a difference scheme for the first auxiliary problem (5.14) : vi,j − τ vi−1,j − 2vi,j + vi+1,j h2 =ui,j−1 + τ 2 j−1∑ k=0 a(tj − tk)g(tk,ui,k) (5.23) + τfi,j ,i = 0, ...,m v1,j −v−1,j 2h = 0 (5.24) vm+1,j −vm−1,j 2h = 0 (5.25) vi,0 = ϕi i = 0, ...,m after some rearrangement, the equation (5.23) becomes : −rvi−1,j + (1 + 2r)vi,j −rvi+1,j = ui,j−1 + τ2 j−1∑ k=0 a(tj − tk)g(tk,ui,k) + τfi,j , i = 0,m (5.26) where r = τ h 2 .there are m + 1 linear equations in m + 3 unknowns v−1,j,v0,j, ...,vm+1;j eliminating of the ”fictitious” value v−1,j beteween (5.23)i=0 and (5.24) gives : (1 + 2r)v0,j − 2rv1,j = u0,j−1 + τ2 j−1∑ k=0 a(tj − tk)g(tk,u0,k) + τf0,j, (5.27) eliminating vm+1,j beteween (5.23)i=m and (5.25) gives : − 2rvm−1,j − (1 + 2r)vm,j = um,j−1 + τ2 j−1∑ k=0 a(tj − tk)g(tk,um,k) + τfm,j, (5.28) combining (5.27), (5.26), with (5.27) yields an (m + 1) × (m + 1) linear system of equations, we write the system in the matrix form : ajv j = bj j = 1,n (5.29) where aj =   1 + 2r −2r 0 ...... 0 −r 1 + 2r −r ...... 0 . . . ...... . 0 0 ..... −2r 1 + 2r   int. j. anal. appl. 16 (4) (2018) 587 bj =   u0,j−1 + τ 2 ∑j−1 k=0 a(tj − tk)g(tk,u0,k) + τf0,j u1,j−1 + τ 2 ∑j−1 k=0 a(tj − tk)g(tk,u1,k) + τf1,j . um,j−1 + τ 2 ∑j−1 k=0 a(tj − tk)g(tk,um,k) + τfm,j   and v j =   v0,j v1,j . vm,j   then at each time level, the difference scheme can be written as systems of m + 1 tridiagonal linear algebraic equations, which is solved by thomas’ algorithm. after that computing the value of αj and βj from equation (5.20) . the integrals are approximated by the composite trapezoidal rule: ∫ xm x0 f(x)dx = h 2 [f(x0) + f(xm ) + 2 m−1∑ i=1 f(xi)] + o(h 2) then the approximative solution of (2.1)-(2.4) is obtained by: ui,j = vi,j + αjzi + βjwi, i = 0...,m,j = 1, ...,n. 6. numerical experiment to test the above algorithms , we use two examples as follows: example 1. consider (2.1) − (2.4) in ω = (0, 1) × (0, 1) , with a(t−s) = (t−s)2 g(t,u(x,t)) = 2u(x,t) f(x,t) = −(x(x− 1) − 2)(−3e−t − 4t + 2t2 + 4) − 2e−t k0(x,t) = 6 13 k1(x,t) = − 6 13 ϕ(x) = x(x− 1) − 2 it is easy to check that the exact solution of this test problem is u∗(x,t) = (x(x− 1) − 2)e−t int. j. anal. appl. 16 (4) (2018) 588 algorithm a1 a2 a1 a2 a1 a2 h h h h h h hh m (x,t) (0.2,0.5) (0.2,0.5) (0.6,0.5) (0.6,0.5) (1,0.5) (1,0.5) 20 -1.3206835 -1.3206835 -1.3695993 -1.3695993 -1.2228648 -1.2228648 40 -1.3153454 -1.3153454 -1.3640651 -1.3640651 -1.2179125 -1.2179125 80 -1.3127135 -1.3127135 -1.3613348 -1.3613348 -1.2154743 -1.2154743 160 -1.3114068 -1.3114068 -1.3599787 -1.3599787 -1.2142647 -1.2142647 320 -1.3107557 -1.3107557 -1.3593029 -1.3593029 -1.2136622 -1.2136622 640 -1.3104308 -1.3104308 -1.3589656 -1.3589656 -1.2133615 -1.2133615 1280 -1.3102684 -1.3102684 -1.3587971 -1.3587971 -1.2132114 -1.2132114 u∗(x, 0.05) -1.3101060 -1.3101060 -1.3586290 -1.3586290 -1.2130610 -1.2130610 table 1. some numerical results at t = 0.5 with τ = h 2 for example 1. algorithm a1 and a2 a1 and a2 a1 and a2 a1 a2 h h h h h h hh m (x,t) (0.2,0.5) (0.6,0.5) (1.0,0.5) cpu time (s) cpu time (s) 20 0.0105773 0.0109706 0.0098035 0.265 0.218 40 0.0052392 0.0054364 0.0048512 0.733 0.53 80 0.0026073 0.0027061 0.0024130 1.982 1.7 160 0.0013006 0.00135003 0.0012034 6.896 6.631 320 0.0006495 0.0006742 0.0006009 30.152 25.787 640 0.0003245 0.0003369 0.0003002 183.41 114.48 1280 0.0001500 0.0001684 0.0001622 1309.21 438.62 table 2. the absolute errors of some numerical solutions at t = 0.5 with τ = h 2 and cpu-times for example 1. m 10 20 40 80 160 n 40 160 640 2560 10240 ‖u−uhτ‖∞ 2.7724e-2 6.8285e-3 1.7008e-3 4.2479e-04 1.0617e-04 table 3. the maximum errors of the numerical solutions for example 1. int. j. anal. appl. 16 (4) (2018) 589 (a) (b) figure 1. the errors of the numerical solutions at t=0.5 for example1. example 2. now, consider problem (2.1) − (2.4) inω = (0, 1) × (0, 1) with a(t−s) = e(t−s) g(t,u(x,t)) = u(x,t) + 3 f(x,t) = et(π2cos(πx) + (cos(πx) + x)(1 − t) − 3) + 3 k0(x,t) = 1 + π2 π2 −e ex k1(x,t) = 1 −π2 −sin(1)π2 + (1 −π2)(cos(1) − 1) cos(x) ϕ(x) = cos(πx) + x it is easy to check that the exact solution of this test problem is u∗(x,t) = (cos(πx) + x)et (a) (b) figure 2. the errors of the numerical solutions at t=0.5 for example2. int. j. anal. appl. 16 (4) (2018) 590 algorithm a1 a2 a1 a2 a1 a2 h h h h h h hh m (x,t) (0.2,0.5) (0.2,0.5) (0.6,0.5) (0.6,0.5) (1,0.5) (1,0.5) 20 1.6716326 1.6716326 0.4887113 0.4887113 0.0112091 0.0112091 40 1.6668822 1.6668822 0.4843230 0.4843230 0.0061441 0.0061441 80 1.6650529 1.6650529 0.4820596 0.4820596 0.0032070 0.0032070 160 1.6642748 1.6642748 0.4809105 0.4809105 0.0016372 0.0016372 320 1.6639199 1.6639199 0.4803316 0.4803316 0.0008270 0.0008270 640 1.6637510 1.6637510 0.4800411 0.4800411 0.0004156 0.0004156 1280 1.6636686 1.6636686 0.4798955 0.4798955 0.0002083 0.0002083 u∗(x, 0.05) 1.6635880 1.6635880 0.4797498 0.4797498 0.0000000 0.0000000 table 4. some numerical results at t = 0.5 with τ = h 2 a for example 2. algorithm a1 and a2 a1 and a2 a1 and a2 a1 a2 h h h h h h hh m (x,t) (0.2,0.5) (0.6,0.5) (1.0,0.5) cpu time (s) cpu time (s) 20 0.0080448 0.0089614 0.0112091 0.22 0.19 40 0.0032945 0.0045731 0.0061441 0.54 0.49 80 0.0026073 0.0027061 0.0024130 1.63 1.57 160 0.0013006 0.0013500 0.0012034 6.80 6.15 320 0.0006495 0.0006742 0.0006009 29.53 24.87 640 0.0001632 0.0002912 0.0004156 169.47 109.80 1280 0.0000808 0.0001457 0.0002083 1271.96 464.24 table 5. the absolute errors of some numerical solutions at t = 0.5 with τ = h 2 and cpu-times for example 2. m 10 20 40 80 160 n 40 160 640 2560 10240 ‖u−uhτ‖∞ 3.8541e-2 9.5205e-3 2.3728e-3 5.9274e-04 1.4815e-04 table 6. the maximum errors of the numerical solutions for example 2. int. j. anal. appl. 16 (4) (2018) 591 our numerical experiment are performed using matlab and we used an intel core i3 with 2.1 ghz . table 1 and table 4 gives some numerical results and exact values at some points at the time t = 0.5. table 2 and table 5 gives the absolute errors of the numerical solutions at some points at the time t = 0.5, and this is also shown in figure 1 and figure 2. table 3 and table 6 gives the maximum errors of the numerical solutions. the maximum error is defined as follows: e(h,τ) = ‖u−uhτ‖∞ = max 0≤i≤m { max 0≤j≤n u(xi, tj) −uij} the results obtained using algorithm 1 and algorithm 2 have the same accuracy . it is also noted that the algorithm 2 will require less cpu time than algorithm 1 (see table 2 and table 5). from table 3 and table 6,we may see the errors decrease about by a factor of 4 as the spatial mesh size is reduced by a factor of 2 and the time mesh size is reduced by a factor of 4. conclusion it is important to note that, for non-local problems, there is not yet a general theory analogous to that of classical problems. this is due to the relative novelty of this topic on the one hand and to the complexity of the questions it raises on the other hand. each problem then requires a specific treatment, which highlights the topicality of the subject tackled in this article. especially, when combined a parabolic integrodifferential equation with the second kind integral condition. so in this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. for the theoretical studies we use the energy inequality and fixed point theorem methods. also we construct a new numerical scheme to solve parabolic integrodifferential equation with the second kind integral condition, which has the following advantage: the coefficient matrices of the scheme is tridiagonal,to solve the linear system of equations by thomas algorithm the cost is about 8m − 7 (m the order of the coefficient matrices ) , will save remarkable cpu time. references [1] w.t. ang, a method of solution for the one-dimentional heat equation subject to nonlocal conditions, southeast asian bull. math. 26 (2002), 185-191. [2] k. balachandran and j.y. park, existence of mild solution of a functional integrodifferential equation with nonlocal condition, bull. korean math. soc. 38 (2001), 175-182. [3] k. balachandran and k. uchiyama, existence of solutions of quasilinear integrodifferential equations with nonlocal condition, tokyo j. math. 23 (2000), 203-210. [4] k. balachandran and d.g. park, existence of solutions of quasilinear integrodifferential evolution equations in banach spaces. bull. korean math. soc. 46 (2009), 691-700. [5] k. balachandran and f.p. samuel, existence of solutions for quasilinear delay integrodifferential equations with nonlocal condition, electron. j. differ. equ. 6 (2009), 1-7. int. j. anal. appl. 16 (4) (2018) 592 [6] s.a. beilin, existence of solutions for one-dimentional wave nonlocal conditions, electron. j. differ. equ. 6(2001), 1-8. [7] a. bouziani and t-e. oussaeif and l. ben aoua, a mixed problem with an integral two-space-variables condition for parabolic equation with the bessel operator, j. math. 2013 (2013), art. id 457631. [8] a. bouziani and r. mechri, rothe’s method to semilinear parabolic integrodifferential equation with a nonclassical boundary conditions, int. j. stoch. anal. (2010), 519-684. [9] a. bouziani, mixed problem with boundary integral conditions for a certain parabolic equation, j. appl. math. stochastic anal. 9(1996), 323-330. [10] a. bouziani r. mechri, the rothe method to a parabolic integro-differential equation with a nonclassical boundary conditions,int. j. stoch. anal. (2010), art. id 519684. [11] a. bouziani, solution forte d’un problème mixte avec une condition non locale pour une classe d’équations hyperboliques ,bull. cl. sci., vi. sr., acad. r. belg.8 (1997), 53-70. [12] a. bouziani, on the solvability of parabolic and hyperbolic problems with a boundary integral condition, int. j. math. math. sci. 31(2002), 201-213. [13] l. byszewski, theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal cauchy problem, j. math. anal. appl. 162 (1991), 494-505. [14] l. byszewski and h. acka, existence of solutions of semilinear functional differential evolution nonlocal problems, nonlinear anal. 34 (1998), 65-72 [15] j. dabas and d. bahuguna, an integro-differential parabolic problem with an integral boundary condition, math. comput. modelling 50 (2009), 123-131. [16] a.a. elbeleze and a. kilicman and b.m. taib, application of homotopy perturbation and variational iteration method for fredholm integro-differential equation of fractional order, abstr. appl. anal. 2012(2012), art. id 763139. [17] a.a. elbeleze and a. kilicman and b.m. taib, homotopy perturbation method for fractional black-scholes european option pricing equations using sumudu transform,math. probl. eng. 2013 (2013). [18] a. guezane-lakoud and m.s. jasmati and a. chaoui, rothe’s method for an integrodifferential equation with integral conditions, nonlinear anal. 72 (2010), 1522-1530. [19] a.i. kozhanov, on solvability of certain spatially nonlocal boundary problems for linear parabolic equations, vestnik of samara state university 3(2008), 165-174. [20] p.k. kythe and p. puri, computational methods for linear integral equation, birkhauser, boston, 2002. [21] a.i. kozhanov, on the solvability of certain spatially nonlocal boundary-value problems for linear hyperbolic equations of second order, math. notes 90 (2011), 238-249. [22] j. liu and z. sun, finite difference method for reaction-diffusion equation with nonlocal boundary conditions, numer. math., j. chin. univ. 16 (2007), 1491-1496. [23] a. merad and a. bouziani and o. cenap and a. kilicman, on solvability of the integrodifferential hyperbolic equation with purely nonlocal conditions, acta math. sci. 35 (2015), 601-609. [24] a. merad and a. bouziani, a method of solution for integro-differential parabolic equation with purely integral conditions, springer proc. math. stat. 41 (2013), 317-327. [25] a.merad, and j. mart́ın-vaquero, a galerkin method for two-dimensional hyperbolic integro-differential equation with purely integral conditions,appl. math. comput. 291 (2016), 386-394. [26] a. merad and a. bouziani and c. ozel, inversion laplace transform for integrodifferential parabolic equation with purely nonlocal conditions; hacet. j. math. stat.44 (2015), 1087-1097. int. j. anal. appl. 16 (4) (2018) 593 [27] n.merazga and a. bouziani, rothe time-discretization method for a nonlocal problem arising in thermoelasticity, j. appl. math. stochastic anal. 1 (2005), 13-28. [28] s. mesloub and a. bouziani, mixed problem with integral conditions for a certain class of hyperbolic equations, j. appl. math. 1(2001), 107-116. [29] g.m. n’guérékata, a cauchy problem for some fractional abstract differential equation with non local conditions,nonlinear anal., theory methods appl.70( 2009), 1873-1876. [30] l.s. pulkina, nonlocal problems for hyperbolic equations with degenerate integral conditions, electron. j. differ. equ. 193 (2016), 1-12. [31] t-e. oussaeif and a. bouziani, mixed problem with an integral two-space-variables condition for a parabolic equation, int. j. evol. equ. 9( 2014 ), 181-198. [32] t-e. oussaeif and a. bouziani, mixed problem with an integral two-space-variables condition for a third order parabolic equation, int. j. anal. appl. 12 (2016), 98-117. [33] t-e. oussaeif and a. bouziani, existence and uniqueness of solutions to parabolic fractional differential equations with integral conditions, electron. j. differ. equ. 179 (2014), 1-10. [34] t-e. oussaeif and a. bouziani, solvability of nonlinear viscosity equation with a boundary integral condition, j. nonlinear evol. equ. appl. 3 (2015), 31-45. [35] l.s. pulkina, a mixed problem with integral condition for the hyperbolic equation, math. notes 74(2003), 411-421. [36] l.s. pulkina, boundary value problems for a hyperbolic equation with nonlocal conditions of the i and ii kind, russ. math. 56 (2012), 62-69. [37] m. slodička and s. dehilis, a numerical approach for a semilinear parabolic equation with a nonlocal boundary condition, j. comput. appl. math. 231(2009), 715-724. [38] h.stehfest, numerical inversion of the laplace transform, commun. acm 13(1970), 47-49. [39] e.g. yanik and g.fairweather, finite element methods for parabolic and hyperbolic partial integro-differential equations, nonlinear anal., theory methods appl. 12 (1988), 785-809. [40] a.m. wazwaz, a comparison study between the modified decomposition method and the traditional methods for solving nonlinear integral equation. appl. math. comput. 181 (2006), 1703-1712. 1. introduction 2. preliminaries and functional spaces 3. an energy estimate and uniqueness of solution 4. existence of solution of the integrodifferential problem 5. construction of approximate solutions 5.1. algorithm 1 :(a1) 5.2. algorithm 2:(a2) 6. numerical experiment conclusion references int. j. anal. appl. (2023), 21:43 tri-endomorphisms on bch-algebras patchara muangkarn1, cholatis suanoom1, jirayu phuto2, aiyared iampan3,∗ 1science and applied science center, program of mathematics, kamphaeng phet rajabhat university, kamphaeng phet 62000, thailand 2department of mathematics, faculty of science, naresuan university, phitsanulok 65000, thailand 3fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. in this paper, we use the concept of endomorphisms and bi-endomorphisms as a model to create tri-endomorphisms on of bch-algebras. we introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of bch-algebras and provide some properties. in addition, we obtain the properties between those tri-endomorphisms and some subsets of bch-algebras. 1. introduction the algebraic structures of bck-algebras and bci-algebras were studied by iséki and his colleague [4,5]. in 1983, hu and li [3] generalized a new class of algebras from bci-algebras, namely, a bchalgebra. next, bandru and rafi [1] introduced a new algebra, called a g-algebra. bch-algebras are also being studied extensively later, [2,3]. in this paper, we use the concept of endomorphisms and bi-endomorphisms as a model to create tri-endomorphisms. we introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of bch-algebras and provide some properties. before studying, we will review the definitions and well-known results. received: jul. 31, 2022. 2020 mathematics subject classification. 06f35, 03g25. key words and phrases. bch-algebra; left tri-endomorphism; central tri-endomorphism; right tri-endomorphism; complete tri-endomorphism. https://doi.org/10.28924/2291-8639-21-2023-43 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-43 2 int. j. anal. appl. (2023), 21:43 definition 1.1. [3] a bch-algebra is a non-empty set x with an element 0 and a binary operation ∗ satisfying the following conditions: (bch1) (∀x ∈x)(x ∗x =0), (bch2) (∀x,y ∈x)(x ∗y =0,y ∗x =0⇒ x = y), (bch3) (∀x,y,z ∈x)((x ∗y)∗z =(x ∗z)∗y). in a bch-algebra x =(x,∗,0), the binary relation ≤ on x is defined as follows: (∀x,y ∈x)(x ≤ y ⇔ x ∗y =0). example 1.1. let x = {0,a,b,c} with the following cayley table as follows: ∗ 0 a b c 0 0 0 0 0 a a 0 c c b b 0 0 b c c 0 0 0 then x =(x,∗,0) is a bch-algebra. proposition 1.1. [2,3] let x =(x,∗,0) be a bch-algebra. then the following hold: for all x,y ∈x, (bch4) (∀x,y ∈x)(x ∗ (x ∗y)≤ y), (bch5) (∀x ∈x)(x ∗0=0⇒ x =0), (bch6) (∀x,y ∈x)(0∗ (x ∗y)= (0∗x)∗ (0∗y)), (bch7) (∀x ∈x)(x ∗0= x), (bch8) (∀x,y ∈x)((x ∗y)∗x =0∗y), (bch9) (∀x,y ∈x)(x ≤ y ⇒ 0∗x =0∗y). for a bch-algebra x = (x,∗,0), some interesting subsets of x play a significant rule in the investigation of its properties described below. definition 1.2. a non-empty subset y of a bch-algebra x =(x,∗,0) is called a subalgebra of x if x ∗y ∈ y for all x,y ∈ y . a non-empty subset i of a bch-algebra x =(x,∗,0) is called an ideal of x if (1) 0∈ i, (2) (∀x,y ∈x)(x ∗y ∈ i,x ∈ i ⇒ y ∈ i). 2. main results in this section, we introduce the concepts of left tri-endomorphisms, central tri-endomorphisms, right tri-endomorphisms, and complete tri-endomorphisms of bch-algebras as follows. definition 2.1. let x =(x,∗,0) be a bch-algebra. a mapping f :x3 →x is called int. j. anal. appl. (2023), 21:43 3 (1) a left tri-endomorphism on x if (∀w,x,y,z ∈x)(f (x ∗w,y,z)= f (x,y,z)∗ f (w,y,z)), (2) a central tri-endomorphism on x if (∀w,x,y,z ∈x)(f (x,y ∗w,z)= f (x,y,z)∗ f (x,w,z)), (3) a right tri-endomorphism on x if (∀w,x,y,z ∈x)(f (x,y,z ∗w)= f (x,y,z)∗ f (x,y,w)), (4) a complete tri-endomorphism on x if (∀a,b,c,x,y,z ∈x)(f (x ∗a,y ∗b,z ∗c)= f (x,y,z)∗ f (a,b,c)). example 2.1. in example 1.1, we define fl :x3 →x by fl(x,y,z)=  x if y = z =0, 0 otherwise. then fl is a left tri-endomorphism on x. proposition 2.1. let x =(x,∗,0) be a bch-algebra and fl be a left tri-endomorphism on x. then (1) (∀y,z ∈x)(fl(0,y,z)=0), (2) (∀w,x,y,z ∈x)(x ≤w ⇒ fl(x,y,z)≤ fl(w,y,z)). proof. (1) let y,z ∈x. then, by bch1, we have fl(0,y,z)= fl(0∗0,y,z)= fl(0,y,z)∗fl(0,y,z)= 0. (2) let w,x,y,z ∈x be such that x ≤w. then, by (1), we have 0= fl(0,y,z)= fl(x∗w,y,z)= fl(x,y,z)∗ fl(w,y,z). hence, fl(x,y,z)≤ fl(w,y,z). � similarly, the properties of central and right tri-endomorphisms are easily obtained. proposition 2.2. let x = (x,∗,0) be a bch-algebra and fc be a central tri-endomorphism on x. then (1) (∀x,z ∈x)(fc(x,0,z)=0), (2) (∀w,x,y,z ∈x)(y ≤w ⇒ fc(x,y,z)≤ fc(x,w,z)). proposition 2.3. let x =(x,∗,0) be a bch-algebra and fr be a right tri-endomorphism on x. then (1) (∀x,y ∈x)(fr(x,y,0)=0), (2) (∀w,x,y,z ∈x)(z ≤w ⇒ fr(x,y,z)≤ fr(x,y,w)). theorem 2.1. let x = (x,∗,0) be a bch-algebra and f be a complete tri-endomorphism on x. then (1) f (0,0,0)=0, (2) if s is a subalgebra of x, then f (s3) is also a subalgebra of x, (3) if s is an ideal of x and f is bijective, then f (s3) is also an ideal of x, (4) if f is a left tri-endomorphism on x, then f (x,y,z)∗ f (x,0,0)=0 for any x,y,z ∈x, (5) if f is a central tri-endomorphism on x, then f (x,y,z)∗ f (0,y,0)=0 for any x,y,z ∈x, (6) if f is a right tri-endomorphism on x, then f (x,y,z)∗ f (0,0,z)=0 for any x,y,z ∈x, 4 int. j. anal. appl. (2023), 21:43 (7) if f is a left and right (central and right, left and central) tri-endomorphism on x, then f (x,y,z)=0 for any x,y,z ∈x, i.e., f is the zero map. proof. (1) by bch1, we have f (0,0,0)= f (0∗0,0∗0,0∗0)= f (0,0,0)∗ f (0,0,0)=0. (2) suppose that s is a subalgebra of x. let a,b ∈ f (s3). then there exist (x1,y1,z1),(x2,y2,z2) ∈ s3 such that a = f (x1,y1,z1) and b = f (x2,y2,z2). thus a ∗ b = f (x1,y1,z1) ∗ f (x2,y2,z2) = f (x1 ∗ x2,y1 ∗ y2,z1 ∗ z2) ∈ f (s3). hence, f (s3) is a subalgebra of x. (3) suppose that s is an ideal of x and f is bijective. since 0 ∈ s and by (1), we have 0 = f (0,0,0)∈ f (s3). assume that x∗y ∈ f (s3) and x ∈ f (s3). there exist (x1,y1,z1),(x2,y2,z2)∈s3 such that x ∗ y = f (x1,y1,z1) and x = f (x2,y2,z2). since f is surjective, there exists (a,b,c)∈x3 such that y = f (a,b,c). thus f (s3)3 f (x1,y1,z1)= x ∗y = f (x2,y2,z2)∗ f (a,b,c)= f (x2∗a,y2∗ b,z2 ∗ c). since f is injective, we have x2 ∗ a,y2 ∗ b,z2 ∗ c ∈ s. since s is an ideal of x, we get a,b,c ∈s. thus y = f (a,b,c)∈ f (s3). hence, f (s3) is an ideal of x. (4)-(6) it is obvious from propositions 2.1-2.3. (7) suppose that f is a left and right tri-endomorphism onx. letx,y,z ∈x. then, by propositions 2.1 and 2.3, bch1, bch7 0= f (0,y,z)= f (x∗x,y∗0,z∗0)= f (x,y,z)∗f (x,0,0)= f (x,y,z)∗0= f (x,y,z). hence, f is the zero map on x. � let tl(x) (resp., tc(x), tr(x) and t(x)) be the set of all left tri-endomorphisms (resp., right, central and complete tri-endomorphisms) on a bch-algebra x =(x,∗,0). we define an operation ? on tl(x) by (∀x,y,z ∈ x)((f ? g)(x,y,z) = f (x,y,z)∗g(x,y,z)). let f ∈ tl(x) and x,y,z ∈ x. then (f ?f )(x,y,z)= f (x,y,z)∗f (x,y,z)=0. this means that f ?f =0x, where 0x :x3 →x is a function that maps all members to 0. let f ,g ∈tl(x) be such that f ?g =0x and g?f =0x. then for all x,y,z ∈x, 0= (f ?g)(x,y,z)= f (x,y,z)∗g(x,y,z) and 0= (g ? f )(x,y,z)= g(x,y,z)∗ f (x,y,z). since g(x,y,z), f (x,y,z) ∈ x, we have f (x,y,z) = g(x,y,z) for all x,y,z ∈ x. hence, f = g. let f ,g,h ∈tl(x) and x,y,z ∈x. then ((f ?g)?h)(x,y,z)= (f ?g)(x,y,z)∗h(x,y,z)=( f (x,y,z)∗g(x,y,z) ) ∗h(x,y,z)= ( f (x,y,z)∗h(x,y,z) ) ∗g(x,y,z)= (f ?h)(x,y,z)∗g(x,y,z)= ((f ?h)?g)(x,y,z). hence, (f ?g)?h =(f ?h)?g. theorem 2.2. (tl(x),?,0x),(tc(x),?,0x),(tr(x),?,0x), and (t(x),?,0x) are bch-algebras. let x = (x,∗,0) be a bch-algebra. we define the binary operation � on x3 as follows: (∀(a,b,c),(x,y,z) ∈ x3)((a,b,c) � (x,y,z) = (a ∗ x,b ∗ y,c ∗ z)). then x3 = (x,�,(0,0,0)) is a bch-algebra. theorem 2.3. let x =(x,∗,0) be a bch-algebra and s1,s2,s3 be subsets of x. then (1) s1 ×s2 ×s3 is a subalgebra of x3 if and only if s1,s2 and s3 are subsets of x, (2) s1 ×s2 ×s3 is an ideal of x3 if and only if s1,s2 and s3 are ideals of x. int. j. anal. appl. (2023), 21:43 5 proof. (1) suppose that s1 × s2 × s3 is a subalgebra of x3. firstly, we will show that s1 is a subalgebra of x. let a,b ∈ s1. let x ∈ s2 and u ∈ s3. then (a,x,u),(b,x,u) ∈ s1 ×s2 ×s3. thus (a∗b,0,0)= (a∗b,x∗x,u∗u)= (a,x,u)�(b,x,u)∈s1×s2×s3, that is, a∗b ∈s1. hence, s1 is a subalgebra of x. on the other hand, we can show that s2 and s3 are subalgebras of x. conversely, let (x,y,z),(a,b,c) ∈ s1 ×s2 ×s3. then x ∗a ∈ s1,y ∗b ∈ s2, and z ∗c ∈ s3, so (x,y,z)� (a,b,c)= (x ∗a,y ∗b,z ∗c)∈s1 ×s2 ×s3. hence, s1 ×s2 ×s3 is a subalgebra of x3. (2) suppose that s1×s2×s3 is an ideal of x3. since (0,0,0)∈s1×s2×s3, we have 0∈si for all i =1,2,3. assume that a∗x ∈s1 and a∈s1. let b ∈s2 and c ∈s3. then (a,b,c)∈s1×s2×s3 and (x,b,c)∈x3. thus (a,b,c)�(x,b,c)= (a∗x,b∗b,c ∗c)= (a∗x,0,0)∈s1×s2×s3. since s1 ×s2 ×s3 is an ideal of x3, we have (x,b,c) ∈ s1 ×s2 ×s3, that is, x ∈ s1. hence, s1 is an ideal of x. similarly, we can show that s2 and s3 are ideals of x. conversely, suppose that s1, s2 and s3 are ideals of x. since 0 ∈ si for all i = 1,2,3, we have (0,0,0)∈s1×s2×s3. assume that (a,b,c)∗(x,y,z)∈s1×s2×s3 and (a,b,c)∈s1×s2×s3. we get (a ∗ x,b ∗ y,c ∗ z) ∈ s1 ×s2 ×s3. since a ∗ x,a ∈ s1, we have x ∈ s1. moreover, we can obtain that y ∈ s2 and z ∈ s3. this implies that (x,y,z) ∈ s1 ×s2 ×s3. hence, s1 ×s2 ×s3 is an ideal of x3. � acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.k. bandru, n. rafi, on g-algebras, sci. magna, 8 (2012), 1-7. [2] w.a. dudek, j. thomys, on decompositions of bch-algebras, math. japon. 35 (1990), 1131-1138. https: //cir.nii.ac.jp/crid/1572261549626076288. [3] q.p. hu, x. li, on bch-algebras, math. seminar notes, 11 (1983), 313-320. [4] k. iséki, an algebra related with a propositional calculus, proc. japan acad. ser. a math. sci. 42 (1966), 26-29. https://doi.org/10.3792/pja/1195522171. [5] k. iséki, on bci-algebras, math. seminar notes, 8 (1980), 125-130. https://cir.nii.ac.jp/crid/1572261549626076288 https://cir.nii.ac.jp/crid/1572261549626076288 https://doi.org/10.3792/pja/1195522171 1. introduction 2. main results references international journal of analysis and applications issn 2291-8639 volume 9, number 1 (2015), 9-18 http://www.etamaths.com fixed points of α-admissible mappings in cone metric spaces with banach algebra s.k. malhotra1 j.b. sharma2 and satish shukla3,∗ abstract. in this paper, we introduce the α-admissible mappings in the setting of cone metric spaces equipped with banach algebra and solid cones. our results generalize and extend several known results of metric and cone metric spaces. an example is presented which illustrates and shows the significance of results proved herein. keywords. cone metric space; α-admissible mapping; solid cone; banach algebra; fixed point. amc 2000. 47h10; 54h25. 1. introduction huang and zhang [7] introduced the notion of cone metric spaces as a generalization of metric spaces. they defined the distance of two points of space in terms of a vector lying in a particular subset of a banach space called cone. they also defined the cauchy sequence and convergence of a sequence in such spaces in terms of interior points of the underlying cone. moreover, they proved the banach contraction principle in the setting of cone metric spaces with the assumption that the cone is normal. later, the assumption of normality of cone was removed by rezapour and hamlbarani [10]. huang and zhang [7] also gave an example and showed the dependency of contractive nature of mappings on the cone metric spaces. although, some authors (see, e.g., [3, 16, 13, 14]) showed that the fixed point results proved on cone metric spaces are the simple consequences of corresponding results of usual metric spaces. liu and xu [4] used the cones over a banach algebra and proved some fixed point theorems on cone metric spaces. they improved the contractive condition on self-maps of cone metric spaces by replacing the contractive constant by a vector of cone. they also gave an example which shows that their fixed point results cannot be obtained by the corresponding results on usual metric spaces with an approach used, e.g., in [3, 16, 13, 14]. the results proved by liu and xu [4] demands the normality of the underlying cone. later on, xu and radenović [9] showed that the condition of normality of cone can be removed, and so, the results of liu and xu [4] are also true in case of a non-normal cone. on the other hand, samet et al. [2] introduced the study of fixed points for the α-admissible mappings and generalized several known results of metric spaces. in this paper, we use the concept of α-admissibility of mappings defined on cone 2010 mathematics subject classification. 47h10. key words and phrases. fixed points; univalent; α-admissible mappings. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 9 10 malhotra, sharma and shukla metric spaces with banach algebra and define the generalized lipschitz contractions on such spaces. our results extend and generalize several known results of metric and cone metric spaces. an example is also provided which verifies the usability and significance of our results. 2. preliminaries first, we recall some definitions and results about the banach algebras and cone metric spaces. let a be a real banach algebra, i.e., a is a real banach space in which an operation of multiplication is defined, subject to the following properties: for all x,y,z ∈ a,a ∈ r (1) x(yz) = (xy)z; (2) x(y + z) = xy + xz and (x + y)z = xz + yz; (3) a(xy) = (ax)y = x(ay); (4) ‖xy‖≤‖x‖‖y‖. in this paper, we shall assume that the banach algebra a has a unit, i.e., a multiplicative identity e such that ex = xe = x for all x ∈ a. an element x ∈ a is said to be invertible if there is an inverse element y ∈ a such that xy = yx = e. the inverse of x is denoted by x−1. for more details we refer to [11]. the following proposition is well known [11]. proposition 2.1. let a be a real banach algebra with a unit e and x ∈ a. if the spectral radius ρ(x) of x is less than one., ρ(x) = lim n→∞ ‖xn‖ 1 n = inf n≥1 ‖xn‖ 1 n < 1 then e−x is invertible. actually, (e−x)−1 = ∞∑ i=0 xi. a subset p of a is called a cone if (1) p is non-empty, closed and {θ,e}⊂ p, where θ is the zero vector of a; (2) a1p + a2p ⊂ p for all non-negative real numbers a1,a2; (3) p2 = pp ⊂ p (4) p ⋂ (−p) = {θ}. for a given cone p ⊂ a, we can define a partial ordering � with respect to p by x � y if and only if y − x ∈ p . the notation x � y will stand for y − x ∈ p◦, where p◦ denotes the interior of p. the cone p is called normal if there exists a number k > 0 such that for all a,b ∈ a, a � b implies ‖a‖≤ k‖b‖. the least positive value of k satisfying the above inequality is called the normal constant (see [7]). note that, for any normal cone p we have k ≥ 1 (see [10]). in the following we always assume that p is a cone in a real banach algebra a with p◦ 6= φ (i.e., the cone p is a solid cone) and � is the partial ordering with respect to p . the following lemmas and remark will be useful in the sequel. fixed points of α-admissible mappings 11 lemma 2.2 (see [15]). if e is a real banach space with a cone p and if a � λa with a ∈ p and 0 ≤ λ < 1, then a = θ. lemma 2.3 (see [8]). if e is a real banach space with a solid cone p and if θ � u � c for each θ � c, then u = θ. lemma 2.4 (see [8]). if e is a real banach space with a solid cone p and if ‖xn‖→ 0 as n →∞, then for any θ � c, there exists n0 ∈ n such that, xn � c for all n < n0. remark 2.5 (see [9]). if ρ(x) < 1 then ‖xn‖→ 0 as n →∞. definition 2.6 (see [4, 5, 7]). let x be a non-empty set. suppose that the mapping d: x ×x → a satisfies: (1) θ � d(x,y) for all x,y ∈ x and d(x,y) = θ if and only if x = y. (2) d(x,y) = d(y,x) for all x,y ∈ x. (3) d(x,y) � d(x,z) + d(z,y) for all x,y,z ∈ x. then d is called a cone metric on x, and (x,d) is called a cone metric space over the banach algebra a. definition 2.7 (see [7]). let (x,d) be a cone metric space, x ∈ x and {xn} be a sequence in x. then: (1) the sequence {xn} converges to x whenever for each c ∈ a with θ � c, there is n0 ∈ n such that d(xn,x) � c for all n > n0. we denote this by lim n→∞ xn = x or xn → x as n →∞. (2) the sequence {xn} is a cauchy sequence whenever for each c ∈ a with θ � c, there is n0 ∈ n such that d(xn,xm) � c for all n,m > n0. (3) (x,d) is a complete cone metric space if every cauchy sequence is convergent in x. it is obvious that the limit of a convergent sequence in a cone metric space is unique. a mapping t : x → x is called continuous at x ∈ x, if for every sequence {xn} in x such that xn → x as n →∞, we have txn → tx as n →∞. definition 2.8 (see [2]). let x be a nonempty set and α: x × x → [0,∞) be a function. we say that t is α-admissible if (x,y) ∈ x, α(x,y) ≥ 1 =⇒ α(tx,ty) ≥ 1. now, we define the generalized lipschitz contractions on the cone metric spaces with a banach algebra (see also, [4]). definition 2.9. let (x,d) be a complete cone metric space over a banach algebra a and p be the underlying solid cone. then the mapping t : x → x is said to be generalized lipschitz contraction if there exists k ∈ p such that ρ(k) < 1 and, d(tx,ty) � kd(x,y) for all x,y ∈ x with α(x,y) ≥ 1. here, the vector k is called the lipschitz vector of t. now we can state our main results. 12 malhotra, sharma and shukla 3. main results first, we prove an existence theorem for a generalized lipschitz contraction on cone metric space over banach algebras. theorem 3.1. let (x,d) be a complete cone metric space over a banach algebra a and p be the underlying solid cone. suppose, t : x → x be a generalized lipschitz contraction with lipschitz vector k and the following conditions are satisfied: (i) t is α-admissible; (ii) there exists x0 ∈ x such that α(x0,tx0) ≥ 1; (iii) t is continuous. then t has a fixed point x∗ ∈ x. proof. let x0 ∈ x such that α(x0,tx0) ≥ 1 and define a sequence {xn} in x such that xn = txn−1 for all n ∈ n. if xn = xn+1 for some n ∈ n, then x∗ = xn is a fixed point for t. assume that xn 6= xn+1 for all n ∈ n. since t is α-admissible we have α(x0,x1) = α(x0,tx0) ≥ 1 =⇒ α(tx0,t2x0) = α(x1,x2) ≥ 1. by induction, we get (1) α(xn,xn+1) ≥ 1 for all n ∈ n. since t is generalized lipscitz contraction, then d(xn,xn+1) = d(txn−1,txn) � kd(xn−1,xn) ... � knd(x0,x1). thus, for n < m we have d(xn,xm) � d(xn,xn+1) + d(xn+1,xn+2) + · · · + d(xm−1,xm) � knd(x0,x1) + kn+1d(x0,x1) + · · · + km−1d(x0,x1) = (e + k + ... + km−n−1)knd(x0,x1) � ( ∞∑ i=0 ki ) knd(x0,x1) = (e−k)−1knd(x0,x1). since ρ(k) < 1, by remark 2.5 we have ‖kn‖→ 0 as n →∞. therefore, by lemma 2.4 it follows that: for every c ∈ a with θ � c there exists n0 ∈ n such that d(xn,xm) � (e−k)−1knd(x0,x1) � c for all n > n0. it implies that {xn} is a cauchy sequence. by completeness of x, there exists x∗ ∈ x such that xn → x∗ as n →∞. since t is continuous, it follows that xn+1 = txn → tx∗ as n → ∞. by the uniqueness of limit we get x∗ = tx∗, that is x∗ is a fixed point of t. � in the above theorem, we use the continuity of the mapping t. now, we show that the assumption of continuity can be replaced by another condition. fixed points of α-admissible mappings 13 theorem 3.2. let (x,d) be a complete cone metric space over a banach algebra a and p be the underlying solid cone. suppose, t : x → x be a generalized lipschitz contraction with lipschitz vector k and the following conditions are satisfied: (i) t is α-admissible; (ii) there exists x0 ∈ x such that α(x0,tx0) ≥ 1; (iii) if xn is a sequence in x such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ x as n →∞, then α(xn,x) ≥ 1 for all n ∈ n. then t has a fixed point x∗ ∈ x. proof. by proof of theorem 3.1, we know that the sequence {xn} is a cauchy sequence in complete cone metric space (x,d). then, there exists x∗ ∈ x such that xn → x∗ as n →∞. on the other hand, from (1) and hypothesis (iii), we have (2) α(xn,x ∗) ≥ 1, for all n ∈ n. since t is a generalized lipschitz contraction, using (2) we obtain d(x∗,tx∗) � d(x∗,xn+1) + d(xn+1,tx∗) = d(x∗,xn+1) + d(txn,tx ∗) � d(x∗,xn+1) + kd(xn,x∗). as xn → x∗ as n →∞, for every c ∈ p with θ � c and for every m ∈ n there exists n(m) such that d(xn,x ∗) � c 2m for all n > n(m). therefore, kd(xn,x ∗) � kc 2m and it follows from the above inequality that d(x∗,tx∗) � c 2m + kc 2m = c 2m (e + k) for all n > n(m),m ∈ n. it implies that c 2m (e + k)−d(x∗,tx∗) ∈ p for all m ∈ n. since p is closed, letting m →∞ we obtain θ−d(x∗,tx∗) ∈ p . by definition, we must have d(x∗,tx∗) = θ, i.e., tx∗ = x∗. thus, x∗ is a fixed point of t. � next, we give an example which illustrate the above result. example 3.3. let a = c1r[0, 1] ×c 1 r[0, 1] with the norm ‖(x1,x2)‖ = ‖x1‖∞ + ‖x2‖∞ + ‖x′1‖∞ + ‖x ′ 2‖∞. define the multiplication on x by xy = (x1y1,x1y2 + x2y1) for all x = (x1,x2),y = (y1,y2) ∈ x. then, a is a banach algebra with usual sum of functions and scalar product on cartesian product c1r[0, 1] × c 1 r[0, 1] and with unit e = (0, 1). let p = {(x1(t),x2(t)) ∈ a: x1(t),x2(t) ≥ 0, t ∈ [0, 1]}. then p is a cone which is not normal. let x = r2 and define the cone metric d: x ×x → p by d((x1,x2), (y1,y2)) = (|x1 −y1|, |x2 −y2|) et ∈ p. then, (x,d) is a complete cone metric space. for a constant a ∈ q, define the mappings t : x → x and α: x ×x → [0,∞) by: t(x1,x2) = { (x1 2 , x2 3 + ax1 ) , if (x1,x2) ∈ q×q; (x1,x2), otherwise 14 malhotra, sharma and shukla and α((x1,x2), (y1,y2)) = { 1, if (x1,x2), (y1,y2) ∈ q×q; 0, otherwise. then, t is a generalized lipschitz contraction with lipschitz vector k = ( 1 2 ,a ) , where ρ(k) = 1 2 < 1. indeed, α(x1,x2) ≥ 1 implies that (x1,x2), (y1,y2) ∈ q × q. therefore, d(t(x1,x2),t(y1,y2)) = ( 1 2 |x1 −y1|, 1 3 |x2 −y2| + a|x1 −y1| ) et � ( 1 2 |x1 −y1|, 1 2 |x2 −y2| + a|x1 −y1| ) et = ( 1 2 ,a ) d((x1,x2), (y1,y2)). since a ∈ q, the mapping t is an α-admissible mapping, and for every (x1,x2) ∈ q × q we have α((x1,x2),t(x1,x2)) = 1. therefore, the conditions (i) and (ii) of theorem 3.2 are satisfied. finally, since q is complete, the condition (iii) of theorem 3.2 is satisfied. thus, all the conditions of theorem 3.2 are satisfied and we conclude the existence of at least one fixed point of t. indeed, (0, 0) and all the points of (x ×x) \ (q×q) are the fixed points of t. remark 3.4. in the above example, the mapping t is not a continuous mapping on the space x. also, ( 1 2 ,a ) 6� (1, 0) = e and ∥∥(1 2 ,a )∥∥ = 1+2a 2 > 1 (for a > 1). for large enough a one can see that the mapping is not a contraction in the sense of samet et al. [2] with respect to euclidian metric on x. again, it is easy to see that the mapping t is not a contraction in the sense of liu and xu [4], and so, we can not apply these known results on the mapping t. moreover, following similar arguments to those in the remark 2.3 of the paper [4] we can say that our results are actual generalization of the known results. in the example 3.3 we can see that the mapping t may have more than one fixed points. let us denote the set of all fixed points of t by fix(t). next, to assure the uniqueness of fixed point of a generalized lipschitz mapping we use the following property (see [2]): (h) ∀ x,y ∈ fix(t) ∃ z ∈ x : α(x,z) ≥ 1,α(y,z) ≥ 1. theorem 3.5. adding condition (h) to the hypothesis of theorem 3.1 (resp. theorem 3.2) we obtain the uniqueness of the fixed point of t. proof. following similar arguments to those in the proof of theorem 3.1 (resp. theorem 3.2) we obtain the existence of fixed point. let the condition (h) is satisfied and x∗,y∗ ∈ fix(t) and x∗ 6= y∗. by (h) there exists z ∈ x such that (3) α(x∗,z) ≥ 1 and α(y∗,z) ≥ 1. since t is α-admissible and x∗,y∗ ∈ fix(t), therefore from (3) we obtain (4) α(x∗,tnz) ≥ 1 and α(y∗,tnz) ≥ 1. for all n ∈ n. fixed points of α-admissible mappings 15 since t is generalized lipschitz contraction, using (4), we have d(x∗,tnz) = d(tx∗,t(tn−1z)) � kd(x∗,tn−1z) ... � knd(x∗,z) for all n ∈ n. since ρ(k) < 1, by remark 2.5 we have ‖kn‖→ 0 as n →∞, and so, ‖knd(x∗,z)‖≤‖kn‖‖d(x∗,z)‖→ 0 as n →∞. therefore, by lemma 2.4 it follows that: for every c ∈ a with θ � c there exists n0 ∈ n such that d(x∗,tnz) � knd(x∗,z) � c. it implies that tnz → x∗ as n →∞. similarly we get tnz → y∗ as n →∞. therefore, by uniqueness of the limit we obtain x∗ = y∗. this finishes the proof. � 4. some consequences in this section, we give some consequences of the results of previous section. the following result is an improved version of theorem 2.1 of liu and xu [4]. theorem 4.1 (xu and radenović [9]). let (x,d) be a complete cone metric space over a banach algebra a and p be the underlying solid cone with k ∈ p where ρ(k) < 1. suppose the mapping t : x → x satisfies generalized lipschitz condition : d(tx,ty) � kd(x,y) for all x,y ∈ x. then t has a unique fixed point in x. moreover, for any x ∈ x, the iterative sequence {tnx} converges to the fixed point of x. proof. define the function α: x × x → [0,∞) by α(x,y) = 1 for all x,y ∈ x. then, all the conditions of theorem 3.5 are satisfied, and so, the mapping t has a unique fixed point in x. � next, we derive the ordered and cyclic versions of banach contraction principle. in the next theorems, we generalize and unify the results of ran and reurings [1], liu and xu [4] and nieto, rodŕıguez-lópez [6] and kirk et al. [12]. the following theorem is the cone metric version of ran and reurings [1] when the cone metric is endowed with a banach algebra. theorem 4.2. let (x,v) be a partially ordered set and suppose that (x,d) be a complete cone metric space (x,d) over a banach algebra a with p the underlying solid cone. let t : x → x be a continuous nondecreasing mapping with respect to v . suppose that the following two assumptions hold: (i) there exists k ∈ p such that ρ(k) < 1 and d(tx,ty) � kd(x,y) for all x,y ∈ x with x v y; (ii) there exists x0 ∈ x such that x0 v tx0. then, t has a fixed point in x. 16 malhotra, sharma and shukla proof. define the mapping αr : x ×x → [0,∞) by αr(x,y) = { 1, if x v y; 0, otherwise. note that, the condition (i) implies that the mapping t a generalized lipschitz contraction with lipschitz vector k, where ρ(k) < 1. since t is nondecreasing it is an αr-admissible mapping. the condition (ii) implies that, there exists x0 ∈ x such that αr(x0,tx0) = 1. therefore, all the conditions of theorem 3.1 are satisfied, and so, the mapping t has a fixed point in x. � the following theorem is the cone metric version of nieto, rodŕıguez-lópez [6] when the cone metric is endowed with a banach algebra. theorem 4.3. let (x,v) be a partially ordered set and suppose that (x,d) be a complete cone metric space (x,d) over a banach algebra a with p the underlying solid cone. let t : x → x be a nondecreasing mapping with respect to v . suppose that the following three assumptions hold: (i) there exists k ∈ p such that ρ(k) < 1 and d(tx,ty) � kd(x,y) for all x,y ∈ x with x v y; (ii) there exists x0 ∈ x such that x0 v tx0; (iii) if {xn} is a nondecreasing sequence in x such that xn → x ∈ x as n → ∞, then xn v x for all n ∈ n. then, t has a fixed point in x. proof. define the mapping αr : x ×x → [0,∞) similar to that as in the proof of theorem 4.2. now, the proof follows from the theorem 3.2. � next, we define the cyclic contractions (see [12]) in cone metric spaces. let x be a nonempty set, t : x → x a mapping and a1,a2, . . . ,am be subsets of x. then x = m⋃ i=1 ai is a cyclic representation of x with respect to t if (1) ai, i = 1, 2, . . . ,m are nonempty sets; (2) t(a1) ⊂ a2, . . . ,t(am−1) ⊂ t(am),t(am) ⊂ t(a1). remark 4.4. (see [12]) if x = m⋃ i=1 ai is a cyclic representation of x with respect to t , then fix(t) ⊂ m⋂ i=1 ai. a cyclic contraction on a cone metric space is defined as follows. definition 4.5. let (x,d) be a complete cone metric space over a banach algebra a and p be the underlying solid cone. suppose, a1,a2, . . . ,am be subsets of x and y = m⋃ i=1 ai. a mapping t : y → y is called a generalized cyclic lipschitz contraction with lipschitz vector k if following conditions hold: (1) y = m⋃ i=1 ai is a cyclic representation of y with respect to t ; (2) there exists k ∈ p such that ρ(k) < 1 and (5) d(tx,ty) � kd(x,y) for any x ∈ ai,y ∈ ai+1 (i = 1, 2, . . . ,m where am+1 = a1). fixed points of α-admissible mappings 17 the following theorem is the cone metric version of kirk et al. [12] when the cone metric is endowed with a banach algebra. theorem 4.6. let (x,v) be a partially ordered set and suppose that (x,d) be a complete cone metric space (x,d) over a banach algebra a with p the underlying solid cone. suppose, a1,a2, . . . ,am be closed subsets of x and y = m⋃ i=1 ai and t : y → y be a a generalized cyclic lipschitz contraction with lipschitz vector k. then, t has a unique fixed point in x. proof. define the mapping αc : x ×x → [0,∞) by: αc(x,y) = { 1, if (x,y) ∈ ai ×ai+1 (i = 1, 2, . . . ,m where am+1 = a1); 0, otherwise. first, by definition of the function α and the cyclic representation, t is αc-admissible. again, by definition of the function αc, t is a generalized cyclic lipschitz contraction with lipschitz vector k. suppose, for a sequence {xn} we have α(xn,xn+1) ≥ 1 for all n and xn → x ∈ x as n →∞. then, as y = m⋃ i=1 ai is a cyclic representation with respect to t, we must have x ∈ m⋂ i=1 ai. therefore, αc(xn,x) ≥ 1 for all n ∈ n. now, the proof of existence of fixed point of t follows from theorem 3.2. for uniqueness, if x∗,y∗ ∈ fix(t), then by remark 4.4 we have x∗,y∗ ∈ m⋂ i=1 ai. since each ai, i ∈{1, 2, . . . ,m} is nonempty, there exists z ∈ y such that x∗,y∗ ∈ ai,z ∈ ai+1 for some i ∈ {1, 2, . . . ,m}, and so αc(x∗,z) = αc(y∗,z) = 1. thus, the condition (h) is satisfied and the uniqueness of fixed point follows from theorem 3.5. � references [1] a.c.m. ran, m.c.b. reurings, a fixed point theorem in partially ordered sets and some applications to matrix equations, proc. amer. math. soc., 132 (2003) 1435-1443. [2] b. samet, c. vetro, and p. vetro, fixed point theorems for α-ψ-contractive type mappings, nonlinear analysis, 75 (2012) 2154-2165. [3] h. çakallı, a. sönmez, ç. genç, on an equivalence of topological vector space valued cone metric spaces and metric spaces, appl. math. lett., 25, (2012) 429-433. [4] h. liu and s.-y. xu, cone metric spaces with banach algebras and fixed point theorems of generalized lipschitz mappings, fixed point theory appl., 2013, 2013:320. [5] h. liu and s.-y. xu, fixed point theorems of quasi-contractions on cone metric spaces with banach algebras, abstarct and applied analysis, volume 2013, article id 187348, 5 pages. [6] j.j. nieto, r. rodŕıguez-lópez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22 (2005) 223-239. [7] l.-g. huang and x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl., 332 (2007) 1468-1476. [8] s. radenović, b.e. rhoades, fixed point theorem for two non-self mappings in cone metric spaces, comput. math. appl., 57, 1701-1707 (2009) [9] s. xu, s. radenović, fixed point theorems of generalized lipschitz mappings on cone metric spaces over banach algebras without assumption of normality, fixed point theory appl., 2014, 2014:102. [10] sh. rezapour and r. hamlbarani, some notes on the paper cone metric spaces and fixed point theorems of contractive mappings, math. anal. appl., 345 (2008), 719-724. [11] w. rudin, functional analysis, 2nd ed., mcgraw-hill, 1991. [12] w.a. kirk, p.s. srinivasan, p. veeramani, fixed points for mappings satisfying cyclical contractive conditions, fixed point theory 4(1)(2003), 79-89. 18 malhotra, sharma and shukla [13] w.s. du, a note on cone metric fixed point theory and its equivalence, nonlinear anal., 72(5), (2010) 2259-2261. [14] y. feng, w. mao, the equivalence of cone metric spaces and metric spaces, fixed point theory, 11(2), (2010) 259-264. [15] z. kadelburg, m. pavlović, s. radenović, common fixed point theorems for ordered contractions and quasi-contractions in ordered cone metric spaces, comput. math. appl. 59, 3148-3159 (2010) [16] z. kadelburg, s. radenović, v. rakočević, a note on the equivalence of some metric and cone metric fixed point results, appl. math. lett., 24, (2011) 370-374. 1department of mathematics, govt. s.g.s.p.g. college ganj basoda, distt vidisha m.p., india 2department of mathematics, choithram college of professional studies, dhar road, indore (m.p.) 453001, india 3department of applied mathematics, shri vaishnav institute of technology & science, gram baroli, sanwer road, indore (m.p.) 453331, india ∗corresponding author int. j. anal. appl. (2023), 21:2 remarks on some higher dimensional hardy inequalities zraiqat amjad1, jebril iqbal1,∗, hawawsheh laith2, abudayah mohmammad2 1alzaytoonah university of jordan, jordan 2school of basic sciences and humanities, german jordanian university, p.o. box 35247, amman 11180, jordan ∗corresponding author: i.jebril@zuj.edu.jo abstract. in this note, we give an elementary proof of hardy inequality in higher dimensions introduced by christ and grafakos. the advantage of our approach is that it uses the one-dimensional hardy inequality to obtain higher dimensional versions. we go further and get some well-known weighted estimates using the same approach. 1. introduction let rn be the n-dimensional euclidean space. let h(f )(x) be the average of |f | ∈ lp(rn) over the euclidean ball b(0, |x|), that is, h(f )(x)= 1 |b(0, |x|)| ∫ b(0,|x|) |f (x)|dx. christ and grafakos [1] introduced the operator h in order to get a higher dimensional version of the classical hardy inequality in one dimension [2]. in fact, they proved the following sharp estimate: ||h(f )||lp(rn) ≤ p p−1 ||f ||lp(rn), (1.1) for f ∈ lp(rn) and 1 < p < ∞. the estimate 1.1 was obtained by using minkowski’s convolution inequality over the space lp(r+, dt t ), where r+ is the multiplicative topological group (0,∞). it received: oct. 12, 2022. 2020 mathematics subject classification. 26d15, 47a63. key words and phrases. hardy inequality; weight functions; weighted estimates. https://doi.org/10.28924/2291-8639-21-2023-2 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-2 2 int. j. anal. appl. (2023), 21:2 should be mentioned that the idea of their proof was very elegent, although they did not use the classical one dimensional estimate(∫ ∞ 0 ( 1 x ∫ x 0 |f(t)|dy )p dx )1 p ≤ p p−1 (∫ ∞ 0 |f(t)|p dt )1 p (1.2) or any of its variants at any stage of the proof. so it is natural to ask whether we can attain the sharp estimate 1.1 by invoking the old bottles of hardy inequalities in [2]. a humble attempt towards answering this question can be done by re-examining the left-hand side of 1.1. more precisely, using polar coordinates and applying hölder’s inequality give ||h(f )||lp(rn) = (∫ rn ( 1 |b(0, |x|)| ∫ b(0,|x|) |f (y)|dy )p dx )1 p ≤ ωn−1 νn (∫ ∞ 0 ( 1 r ∫ r 0 (∫ sn−1 |f (tθ)|ptn−1dθ )1 p dt )p dr )1 p , (1.3) where ωn−1 is the surface area of the unit sphere sn−1 and νn is the volume of the unit ball b(0,1). now, let f(t) := (∫ sn−1 |f (tθ)|ptn−1dθ )1 p and use 1.2 with the fact that ωn−1 = nνn to obtain ||h(f )||lp(rn) ≤ np p−1 ||f ||lp(rn). (1.4) the dependence on the dimension (n) in estimate 1.4 is natural and indicates that our inquiry makes sense. based upon this observation, we give a simple and direct proof of 1.1 which reveals that this inequality is one dimensional in spirit. let w and z be two weight functions on rn, that is, nonnegative and locally integrable on rn, and denote the conjugate exponent of p > 1 by p′ = p p−1 . in [3], the authors introduced a weighted version of hardy inequality in higher dimensions which generalizes 1.1. more precisely, they obtained the following result. theorem 1.1. let w and z be weight functions on rn. for 1 < p ≤ q < ∞, the inequality (∫ rn w(x) (∫ b(0,|x|) |f (y)|dy )q dx )1 q ≤ c (∫ rn z(x)|f (x)|p dx )1 p (1.5) holds if and only if a := sup α>0 (∫ |x|≥α w(x)dx )1 q (∫ |x|≤α z1−p ′ (x)dx ) 1 p′ < ∞. (1.6) moreover, if c is the smallest constant for which 1.5 holds, then a ≤ c ≤ ap′ 1 p′ p 1 q . int. j. anal. appl. (2023), 21:2 3 we remark here that the condition 1.6 extends the one dimensional condition which can be found in [4]. as a special and important case of theorem 1.1, we have, for 1 < p = q < ∞, s > 0, w(x)= |b(0,x)|−s−1 and u(x)= |b(0,x)|p−s−1, that (∫ rn |b(0, |x|)|−s−1 (∫ b(0,|x|) |f (y)|dy )p dx )1 p ≤ p s (∫ rn |b(0, |x|)|p−s−1|f (x)|p dx )1 p , (1.7) and the constant p s being best possible. we notice here that by taking s = p − 1 in 1.7 one can obtain 1.1. similarly, it is natural to ask whether we can deduce the estimates 1.5 and 1.7 using their one dimensional versions. in the following, we introduce a new and simple proofs of the higher dimensional hardy inequalities 1.1, 1.5 and 1.7. we show also in theorem 2.4 that our technique can be used to produce nice estimates by using the classical hardy inequalities. it should also mention that many authors worked on the equivalence between the higher dimensional and one-dimensional hardy’s inequalities. for instance, gord [7] studies hardy inequalities in higher dimensions where the averages are taken over appropriate dilates of a given star-shaped regions. we refer the readers to ( [5], [6], [8]) for more background information and relevant work. 2. proofs and further results in this section we introduce new and simple proofs of the estimates 1.1, 1.5 and 1.7 using some elementary tools and based on hardy’s inequality in one dimension. we start by recalling the following result. theorem 2.1. [2, theorem 330] let f be a measurable function. then (∫ ∞ 0 ( 1 x ∫ x 0 |f(t)|dt )p xη dx )1 p ≤ p p−1−η (∫ ∞ 0 |f(t)|p tη dt )1 p (2.1) holds for 1 < p < ∞ and η < p−1. now, we present our proofs of 1.1, 1.7. proof of estimate 1.1. applying polar coordinates and höldr’s inequality yield ||h(f )||lp(rn) = (∫ rn ( 1 |b(0, |x|)| ∫ b(0,|x|) |f (y)|dy )p dx )1 p ≤ n (∫ ∞ 0 ( 1 r ∫ r 0 (∫ sn−1 |f (tθ)|ptn−1dθ )1 p t n−1 p′ dt )p r (1−n)p p′ dr )1 p . (2.2) 4 int. j. anal. appl. (2023), 21:2 using 2.1 with η = (1−n)p p′ and f(t) := (∫ sn−1 |f (tθ)|ptn−1dθ )1 p t n−1 p′ , we obtain ||h(f )||lp(rn) ≤ n ( p p−1+ (n−1)p p′ )(∫ ∞ 0 (∫ sn−1 |f (tθ)|ptn−1dθ ) t (n−1)p p′ t (1−n)p p′ dt )1 p = p p−1 ||f ||lp(rn). proof of estimate 1.7. let f(t) := (∫ sn−1 |f (tθ)|ptn−1dθ )1 p t n−1 p′ dt, η = p−sn−1 and proceeding as above, we get (∫ rn |b(0, |x|)|−s−1 (∫ b(0,|x|) f (y)dy )p dx )1 p ≤ ν −s−1 p n ωn−1 (∫ ∞ 0 (∫ r 0 f(t)dt )p r−ns−1dr )1 p ≤ ν −s+p−1 p n (p s )(∫ ∞ 0 ∫ sn−1 |f (tθ)|ptnp−ns−1dθdt )1 p = ν −s+p−1 p n (p s )(∫ rn |f (x)|p |x|n(p−s−1)dx )1 p = (p s )(∫ rn |f (x)|p |b(0, |x|)|p−s−1dx )1 p . before introducing the proof of theorem 1.1, we need the following result. theorem 2.2. let u and v be nonnegative measurable functions on (0,∞). if f is a measurable function, then (∫ ∞ 0 (∫ x 0 f (t)dt )q u(x)dx )1 q ≤ c (∫ ∞ 0 f p(x)v(x)dx )1 p (2.3) holds for 1 < p ≤ q < ∞ if and only if a := sup x>0 (∫ ∞ x u(t)dt )1 q (∫ x 0 v1−p ′ (t)dt ) 1 p′ < ∞. (2.4) now, we are ready to prove theorem 1.1. in fact, our proof is simpler than the proof in [4] and it depends primarily on the appropriate choice of the one dimensional weights. theorem 2.3. in order to apply theorem 2.2 we carefully define the weight functions u, v and the function f. let int. j. anal. appl. (2023), 21:2 5 u(t) := (∫ sn−1 w(rϕ)dϕ ) rn−1, v(t) := t(1−n)(p−1) (∫ sn−1 z −p ′ p (tθ)dθ )− p p′ , f(t) := (∫ sn−1 |f (tθ)|pz(tθ)dθ )1 p (∫ sn−1 z −p ′ p (tθ)dθ ) 1 p′ tn−1. now, consider (∫ rn w(x) (∫ b(0,|x|) |f (y)|dy )q dx )1 q = (∫ ∞ 0 (∫ r 0 ∫ sn−1 |f (tθ)|tn−1dθdt )q u(r)dr )1 q = (∫ ∞ 0 (∫ r 0 ∫ sn−1 |f (tθ)|z 1 p(tθ)z −1 p(tθ)tn−1dθdt )q u(r)dr )1 q ≤ (∫ ∞ 0 (∫ r 0 f(t)dt )q u(r)dr )1 q . (2.5) then invoking theorem 2.2 we have that (∫ rn w(x) (∫ b(0,|x|) |f (y)|dy )q dx )1 q ≤ (∫ ∞ 0 (∫ r 0 f(t)dt )q u(r)dr )1 q ≤ c (∫ ∞ 0 fp(t)v(t)dt )1 p = c (∫ ∞ 0 (∫ sn−1 |f (tθ)|pz(tθ)dθ )(∫ sn−1 z −p ′ p (tθ)dθ ) p p′ t(n−1)pv(t)dt )1 p = c (∫ ∞ 0 (∫ sn−1 |f (tθ)|pz(tθ)dθ ) tn−1dt )1 p = c (∫ rn |f (y)|pz(y)dy )1 p (2.6) if and only if a := sup α>0 (∫ ∞ α u(t)dt )1 q (∫ α 0 v1−p ′ (t)dt ) 1 p′ = sup α>0 (∫ ∞ α (∫ sn−1 w(tϕ)dϕ ) tn−1dt )1 q (∫ α 0 (∫ sn−1 z1−p ′ (tθ) ) tn−1dt ) 1 p′ 6 int. j. anal. appl. (2023), 21:2 = sup α>0 (∫ |x|≥α w(x)dx )1 q (∫ |x|≤α z1−p ′ (x)dx ) 1 p′ < ∞. (2.7) next, we use the main scheme of the previous proofs to introduce the following result. theorem 2.4. let f be a measurable function and − n p′ < α < 1 p′ . then (∫ rn ( 1 |b(0, |x|)| ∫ b(0,|x|) |f (y)||x ·y|−α dy )p dx )1 p ≤ c (∫ rn |f (y) |y|−2α|p )1 p for 1 < p < ∞, where c =  cα,p,n ω1pn−1 νn  ( p np+αn−n ) . proof. let g(ϕ,t)= ∫ sn−1 |f (tθ)|t n−1 p |ϕ ·θ|−α dθ. then applying holder’s inequality and using the fact that∫ sn−1 |ϕ ·θ|−αp ′ dθ = ωn−2b ( 1−αp′ 2 , n−1 2 ) := cp ′ α,p,n for α < 1 p′ , we get g(ϕ,t)≤ (∫ sn−1 |f (tθ)|tn−1dθ )1 p cα,p,n (2.8) now set dα,p,n = n−1−(α+n−1)p, f(t)= (∫ sn−1 |f (tθ)|t n−1dθ )1 p t n−1 p′ −α and use 2.8 to obtain (∫ rn ( 1 |b(0, |x|)| ∫ b(0,|x|) |f (y)||x ·y|−α dy )p dx )1 p = 1 νn (∫ ∞ 0 ∫ sn−1 ( 1 r ∫ r 0 g(ϕ,t)t n−1 p′ −α dt )p rdα,p,n dϕdr )1 p ≤ cα,p,n ω 1 p n−1 νn (∫ ∞ 0 ( 1 r ∫ r 0 f(t)dt )p rdα,p,n dr )1 p . (2.9) finally, applying theorem 2.1 with η = dα,p,n and α > − np′ we get the desired result. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2023), 21:2 7 references [1] m. christ, l. grafakos, best constants for two nonconvolution inequalities, proc. amer. math. soc. 123 (1995), 1687–1693. https://doi.org/10.1090/s0002-9939-1995-1239796-6. [2] g.h. hardy, j.e. littlewood, g. polya, inequalities, cambridge university press, cambridge, 1952. [3] p. drábek, h.p. heinig, a. kufner, higher dimensional hardy inequality, in: c. bandle, w.n. everitt, l. losonczi, w. walter (eds.), general inequalities 7, birkhäuser basel, basel, 1997: pp. 3–16. https://doi.org/10.1007/ 978-3-0348-8942-1_1. [4] b. opic, a. kufner, hardy-type inequalities, volume 219, pitman research notes in mathematics series, longman scientific & technical, harlow, 1990. [5] a. kufner, l. maligranda, l.e. persson, the hardy inequality: about its history and some related results, vydavatelsky servis publishing house, pilsen, 2007. [6] a. kufner, l.e. persson, weighted inequalities of hardy type, world scientific publishing co. inc., river edge, 2003. [7] g. sinnamon, one-dimensional hardy-type inequalities in many dimensions, proc. royal soc. edinburgh: sect. a math. 128 (1998), 833–848. https://doi.org/10.1017/s0308210500021818. [8] w.g. alshanti, inequality of ostrowski type for mappings with bounded fourth order partial derivatives, abstr. appl. anal. 2019 (2019), 5648095. https://doi.org/10.1155/2019/5648095. https://doi.org/10.1090/s0002-9939-1995-1239796-6 https://doi.org/10.1007/978-3-0348-8942-1_1 https://doi.org/10.1007/978-3-0348-8942-1_1 https://doi.org/10.1017/s0308210500021818 https://doi.org/10.1155/2019/5648095 1. introduction 2. proofs and further results references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 43-53 http://www.etamaths.com exponential stability of the heat equation with boundary time-varying delays mouataz billah mesmouli1, abdelouaheb ardjouni1,2,∗ and ahcene djoudi1 abstract. in this paper, we consider the heat equation with a time-varying delays term in the boundary condition in a bounded domain of rn, the boundary γ is a class c2 such that γ = γd ∪γn , with γd ∩ γn = ∅, γd 6= ∅ and γn 6= ∅. well-posedness of the problems is analyzed by using semigroup theory. the exponential stability of the problem is proved. this paper extends in ndimensional the results of the heat equation obtained in [11]. 1. introduction time-delay often appears in many biological, electrical engineering systems and mechanical applications, and in many cases, delay is a source of instability [3]. in the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. [1, 2, 8, 9, 10, 14]). the stability issue of systems with delay is, therefore, of theoretical and practical importance. in present paper, we are interested in the effect of a time-varying delays in boundary stabilization of the heat equation in domains of rn. let ω ⊂ rn be an open bounded set with boundary γ of class c2. we assume that γ is divided into two parts γn and γd; i.e., γ = γd ∪ γn with γd ∩ γn = ∅, γd 6= ∅ and γn 6= ∅. in this domain ω, we consider the initial boundary value problem ut (x,t) − ∆u (x,t) = 0 in ω × (0,∞) ,(1.1) u (x,t) = 0 on γd × (0,∞) ,(1.2) ∂u ∂ν (x,t) = −µ1u (x,t) −µ2u (x,t− τ (t)) on γn × (0,∞) ,(1.3) u (x, 0) = u0 (x) in ω,(1.4) u (x,t− τ (0)) = f0 (x,t− τ (0)) on γn × (0,τ (0)) ,(1.5) where ν (x) denotes the outer unit normal vector to the point x ∈ γ and ∂u ∂ν is the normal derivative. moreover, τ (t) > 0, µ1,µ2 ≥ 0 are fixed nonnegative real numbers, the initial datum (u0,f0) belongs to a suitable space. on the functions τ (·) we assume that there exists a positive constants τ, such that (1.6) 0 < τ0 ≤ τ (t) ≤ τ, ∀t > 0, moreover, we assume (1.7) τ′ (t) < 1, ∀t > 0, and (1.8) τ ∈ w 2,∞ ([0,t]) , ∀t > 0. note that , if t < τ (t), then u (x,t− τ (t)) is in the past and we need an initial value in the past. moreover, by (1.7) and the mean value theorem, we have τ (t) − τ (0) < t, 2010 mathematics subject classification. 35l05, 93d15. key words and phrases. heat equation; delay feedbacks; stabilization; lyapunov method. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 43 44 mesmouli, ardjouni and djoudi which implies t− τ (t) > −τ (0) , we thus obtain the initial condition (1.5). the last boundary-value problem describes the propagation of heat in a homogeneous n-dimensional rod. here a stands for the heat conduction coefficient, u(x,t) is the value of the temperature field of the plant at time moment t and location x along the rod. in the sequel, the state dependence on time t and spatial variable x is suppressed whenever possible. the above problem, with both µ1,µ2 > 0 and a time-varying delay, has been studied in one space dimension by nicaise, valein and fridman [12]. in [12] an exponential stability result is given, under the condition (1.9) µ2 < √ 1 −dµ1, where d is a constant such that (1.10) τ′ (t) ≤ d < 1, ∀t > 0, we are interested in giving an exponential stability result for such a problem. let us denote by 〈v,w〉 the euclidean inner product between two vectors (v,w) ∈ rn. under a suitable relation between the above coefficients we can give a well-posedness result and an exponential stability estimate for problem (1.1)–(1.5). 2. well-posedness of the problem using semigroup theory we can give the well-posedness of problem (1.1)-(1.5). let us stand z (x,ρ,t) = u (x,t− τ (t) ρ) , x ∈ γn, ρ ∈ (0, 1) , t > 0. then, the problem (1.1)-(1.5) is equivalent to ut (x,t) − ∆u (x,t) = 0 in ω × (0,∞) ,(2.1) τ (t) zt (x,ρ,t) + (1 − τ′ (t) ρ) zρ (x,ρ,t) = 0 in γn × (0, 1) × (0,∞) ,(2.2) u (x,t) = 0 on γd × (0,∞) ,(2.3) ∂u ∂ν (x,t) = −µ1u (x,t) −µ2z (x, 1, t) on γn × (0,∞) ,(2.4) z (x, 0, t) = u (x,t) , x ∈ γn, t > 0,(2.5) u (x, 0) = u0 (x) in ω,(2.6) z (x,ρ, 0) = f0 (x,−τ (0) ρ) , x ∈ γn, ρ ∈ (0, 1) .(2.7) if we denote by u := (u,z) t , then u′ = ( ut zt ) = ( ∆u (τ′(t)ρ−1) τ(t) zρ ) . therefore, problem (2.1)–(2.7) can be rewritten as (2.8) { u′ = a(t) u, u (0) = (u0,f0 (·,−· τ (0))) t , in the hilbert space h defined by (2.9) h := l2(ω) ×l2(γn × (0, 1)), equipped with the standard inner product〈( u z ) , ( ũ z̃ )〉 h := ∫ ω u(x)ũ(x)dx + ∫ γn ∫ 1 0 z(x,ρ)z̃(x,ρ)dρdγ. exponential stability of the heat equation 45 the time varying operator a(t) is defined by a(t) ( u z ) := ( ∆u (τ′(t)ρ−1) τ(t) zρ ) , with domain d (a(t)) : = { (u,z) t ∈ ( e(∆,l2(ω)) ∩v ) ×l2 ( γn,h 1 (0, 1) ) : ∂u ∂ν = −µ1u−µ2z (·, 1) on γn, u = z (·, 0) on γn } , where, v = h1γd = { u ∈ h1 (ω) , u = 0 on γd } , and e(∆,l2(ω)) = {u ∈ h1(ω) : ∆u ∈ l2(ω)}. recall that for a function u ∈ e(∆,l2(ω)), ∂u ∂ν belongs to h−1/2(γn ) and the next green formula is valid (see section 1.5 of [4]) (2.10) ∫ ω ∇u∇ϕdx = − ∫ ω ∆uϕdx + 〈 ∂u ∂ν ,ϕ〉γn , ∀ϕ ∈ h 1 γd (ω), where 〈·; ·〉γn means the duality pairing between h−1/2(γn ) and h1/2(γn ). observe that the domain of a(t) is independent of the time t, i.e., (2.11) d(a(t)) = d(a(0)), t > 0. note further that for (u,z) t ∈d(a(t)), ∂u/∂ν belongs to l2(γn ), since z (x, 1) is in l2(γn ). a general theory for equations of type (2.8) has been developed using semigroup theory [6, 7, 13]. the simplest way to prove existence and uniqueness results is to show that the triplet {a,h,d(a(0))}, with a = {a(t) : t ∈ [0,t]}, for some fixed t > 0, forms a cd-system (or constant domain system, see [6, 7]). more precisely, we can obtain a well-posedness result using semigroup arguments by kato [5, 6, 13]. the following result is proved in [5, theorem 1.9]. theorem 1. assume that (i) d(a(0)) is a dense subset of h, (ii) d(a(t)) = d(a(0)) for all t > 0, (iii) for all t ∈ [0,t], a(t) generates a strongly continuous semigroup on h and the family a = {a(t) : t ∈ [0,t]} is stable with stability constants c and m independent of t (i.e. the semigroup (st(s))s≥0 generated by a(t) satisfies ‖st(s)u‖h ≤ cems‖u‖h, for all u ∈h and s ≥ 0), (iv) ∂ta belongs to l∞∗ ([0,t],b(d(a(0)),h)), the space of equivalent classes of essentially bounded, strongly measurable functions from [0,t] into the set b(d(a(0)),h) of bounded operators from d(a(0)) into h. then, problem (2.8) has a unique solution u ∈ c([0,t],d(a(0))) ∩ c1([0,t],h) for any initial datum in d(a(0)). our goal is then to check the above assumptions for problem (2.8). lemma 1. d(a(0)) is dense in h. proof. let (f,h)t ∈h be orthogonal to all elements of d(a(0)), that is, 0 = 〈( u z ) , ( g h )〉 h = ∫ ω u(x)g(x)dx + ∫ γn ∫ 1 0 z(x,ρ)h(x,ρ)dρdγ, for all (u,z)t ∈d(a(0)). we first take u = 0 and z ∈d(γn ×(0, 1)). as (0,z)t ∈ d(a(0)), we obtain∫ γn ∫ 1 0 z(x,ρ)h(x,ρ)dρdγ = 0. since d(γn × (0, 1)) is dense in l2(γn × (0, 1), we deduce that h = 0. in the same way, by taking z = 0 and u ∈d(ω) we see that g = 0. � 46 mesmouli, ardjouni and djoudi assuming (1.9) and (1.10) hold. let ξ be a positive constant that satisfies (2.12) µ2√ 1 −d ≤ ξ ≤ 2µ1 − µ2√ 1 −d . note that this choice of ξ is possible from assumption (1.9). we define on the hilbert space h the time dependent inner product (2.13) 〈( u z ) , ( ũ z̃ )〉 t := ∫ ω u (x) ũ (x) dx + ξτ (t) ∫ γn ∫ 1 0 z (x,ρ) z̃ (x,ρ) dpdγ. using this time dependent inner product and theorem 1, we can deduce a well-posedness result. theorem 2. for any initial datum u0 ∈d(a(0)) there exists a unique solution u ∈ c([0, +∞),d(a(0))) ∩c1([0, +∞),h), of system (2.8). proof. we first observe that (2.14) ‖φ‖t ‖φ‖s ≤ e c 2τ0 |t−s| , ∀t,s ∈ [0,t], where φ = (u,z)t and c is a positive constant. indeed, for all s,t ∈ [0,t], we have ‖φ‖2t −‖φ‖ 2 se c τ0 |t−s| = ( 1 −e c τ0 |t−s| )∫ ω u2dx + ξ ( τ(t) − τ(s)e c τ0 |t−s| )∫ γn ∫ 1 0 z2(x,ρ)dρdγ. we notice that 1 − e c τ0 |t−s| ≤ 0. moreover τ(t) − τ(s)e c τ0 |t−s| ≤ 0 for some c > 0. indeed, τ(t) = τ(s) + τ′(a)(t−s), where a ∈ (s,t), and thus, τ(t) τ(s) ≤ 1 + |τ′(a)| τ(s) |t−s|. by (1.8), τ′ is bounded on [0,t] and therefore, recalling also (1.7), τ(t) τ(s) ≤ 1 + c τ0 |t−s| ≤ e c τ0 |t−s| , which proves (2.14). now we calculate 〈a(t)u,u〉t for a fixed t. take u = (u,z)t ∈d(a(t)). then, 〈a(t)u,u〉t = 〈( ∆u τ′(t)ρ−1 τ(t) zρ ) , ( u z )〉 t = ∫ ω u(x)∆u(x)dx− ξ ∫ γn ∫ 1 0 (1 − τ′(t)ρ) zρ (x,ρ) z (x,ρ) dρdγ. so, by green’s formula, 〈a(t)u,u〉t = ∫ γn ∂u (x) ∂ν u(x)dγ − ∫ ω |∇u(x)|2dx − ξ ∫ γn ∫ 1 0 (1 − τ′(t)ρ)zρ(x,ρ)z(x,ρ)dρdγ.(2.15) integrating by parts in ρ, we obtain∫ γn ∫ 1 0 zρ(x,ρ)z(x,ρ)(1 − τ′(t)ρ) dρdγ = ∫ γn ∫ 1 0 1 2 ∂ ∂ρ z2(x,ρ)(1 − τ′(t)ρ)dρdγ = τ′(t) 2 ∫ γn ∫ 1 0 z2(x,ρ)dρdγ + 1 2 ∫ γn {z2(x, 1) (1 − τ′ (t)) −z2(x, 0)}dγ.(2.16) exponential stability of the heat equation 47 therefore, from (2.15) and (2.16), 〈a(t)u,u〉t = ∫ γn ∂u (x) ∂ν u(x)dγ − ∫ ω |∇u(x)|2dx − ξ 2 ∫ γn {z2(x, 1) (1 − τ′ (t)) −z2(x, 0)}dγ − ξτ′(t) 2 ∫ γn ∫ 1 0 z2(x,ρ)dρdγ = − ∫ γn [µ1u (x) + µ2z (x, 1)] u(x)dγ − ∫ ω |∇u(x)|2dx + ξ 2 ∫ γn u2(x)dγ − ξ 2 ∫ γn {z2(x, 1) (1 − τ′ (t)) dγ − ξτ′(t) 2 ∫ γ1 ∫ 1 0 z2(x,ρ)dρdγ = − ( µ1 − ξ 2 )∫ γn u2(x)dγ −µ2 ∫ γn z (x, 1) u(x)dγ − ∫ ω |∇u(x)|2dx − ξ 2 ∫ γn {z2(x, 1) (1 − τ′ (t)) dγ − ξτ′(t) 2 ∫ γn ∫ 1 0 z2(x,ρ)dρdγ, from which, using cauchy-schwarz’s, poincaré’s inequality and (1.10), it follows that 〈a(t)u,u〉t ≤ ( −µ1 + ξ 2 + µ2 2 √ 1 −d − 1 cp )∫ γn u2(x)dγ + ( µ2 √ 1 −d 2 − ξ 2 (1 −d) )∫ γn z2(x, 1)dγ + κ(t)〈u,u〉t,(2.17) where (2.18) κ(t) = (τ′2 (t) + 1) 1 2 2τ(t) . now, observe that from (2.12), −µ1 + ξ 2 + µ2 2 √ 1 −d ≤ 0, µ2 √ 1 −d 2 − ξ 2 (1 −d) ≤ 0. then (2.19) 〈a(t)u,u〉t −κ(t)〈u,u〉t ≤ 0, which means that the operator ã(t) = a(t) −κ(t)i is dissipative. moreover, κ′(t) = τ′′(t)τ′(t) 2τ(t)(τ′2(t) + 1) 1 2 − τ′(t)(τ′2(t) + 1) 1 2 2τ(t)2 , is bounded on [0,t] for all t > 0 (by (1.6) and (1.7)) and we have d dt a(t)u = ( 0 τ′′(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1) τ(t)2 zρ ) , with τ′′(t)τ(t)ρ−τ′(t)(τ′(t)ρ−1) τ(t)2 bounded on [0,t]. thus (2.20) d dt ã(t) ∈ l∞∗ ([0,t],b(d(a(0)),h)), the space of equivalence classes of essentially bounded, strongly measurable functions from [0,t] into b(d(a(0)),h). now, we show that λi −a(t) is surjective for fixed t > 0 and λ > 0. given (g,h)t ∈ h, we seek u = (u,z)t ∈d(a(t)) solution of (λi −a(t)) ( u z ) = ( g h ) , 48 mesmouli, ardjouni and djoudi that is verifying (2.21) { λu− ∆u = g, λz + 1−τ′(t)ρ τ(t) zρ = h. suppose that we have found u with the appropriate regularity. we can then determine z, indeed z satisfies the differential equation, λz(x,ρ) + 1 − τ′ (t) ρ τ (t) zρ(x,ρ) = h(x,ρ), for x ∈ γ,ρ ∈ (0, 1) , and the boundary condition (2.22) z(x, 0) = u(x), for x ∈ γn. therefore z is explicitly given by z(x,ρ) = u(x)e−λρτ(t) + τ(t)e−λρτ(t) ∫ ρ 0 h(x,σ)eλστ(t)dσ, if τ′(t) = 0, and z(x,ρ) = u(x)e λ τ(t) τ′(t) ln(1−τ ′(t)ρ) + e λ τ(t) τ′(t) ln(1−τ ′(t)ρ) ∫ ρ 0 h(x,σ)τ(t) 1 − τ′(t)σ e −λ τ(t) τ′(t) ln(1−τ ′(t)σ) dσ, otherwise. this means that once u is found with the appropriate properties, we can find z. in particular, if τ′(t) = 0, (2.23) z(x, 1) = u(x)e−λτ(t) + z0(x), x ∈ γn, with z0 ∈ l2(γn ) defined by (2.24) z0(x) = τ(t)e −λτ(t) ∫ 1 0 h(x,σ)eλστ(t)dσ, x ∈ γn, and, if τ′(t) 6= 0, (2.25) z(x, 1) = u(x)e λ τ(t) τ′(t) ln(1−τ ′(t)) + z0(x), x ∈ γn, with z0 ∈ l2(γn ) defined by (2.26) z0(x) = e λ τ(t) τ′(t) ln(1−τ ′(t)) ∫ 1 0 h(x,σ)τ(t) 1 − τ′(t)σ e −λ τ(t) τ′(t) ln(1−τ ′(t)σ) dσ, for x ∈ γn . then, we have to find u. in view of the equation (2.27) λu− ∆u = g. multiplying this identity by a test function φ and integrating in space (2.28) ∫ ω (λuφ− ∆uφ) dx = ∫ ω gφdx, ∀φ ∈ h1γd, using green’s formula, we obtain∫ ω (λuφ− ∆uφ) dx = ∫ ω (λuφ + ∇u∇φ) dx− ∫ γn ∂u ∂ν φdγ = ∫ ω (λuφ + ∇u∇φ) dx + ∫ γn (µ1u + µ2z (x, 1)) φdγ. by (2.23), we obtain ∫ ω (λuφ− ∆uφ) dx = ∫ ω (λuφ + ∇u∇φ) dx + ∫ γn ( µ1u + µ2 ( ue−λτ(t) + z0 )) φdγ, exponential stability of the heat equation 49 if τ′(t) = 0, and by (2.25)∫ ω (λuφ− ∆uφ) dx = ∫ ω (λuφ + ∇u∇φ) dx + ∫ γn ( µ1u + µ2 ( ue λ τ(t) τ′(t) ln(1−τ ′(t)) + z0 )) φdγ, otherwise. therefore, (2.28) can be rewritten as (2.29) ∫ ω (λuφ + ∇u∇φ) dx + ∫ γn ( µ1u + µ2 ( ue−λτ(t) + z0 )) φdγ = ∫ ω gφdx, if τ′(t) = 0, and∫ ω (λuφ + ∇u∇φ) dx + ∫ γn ( µ1u + µ2 ( ue λ τ(t) τ′(t) ln(1−τ ′(t)) + z0 )) φdγ = ∫ ω gφdx,(2.30) otherwise. as the left-hand side of (2.29) or (2.30) is coercive on h1γd (ω), the lax-milgram lemma guarantees the existence and uniqueness of a solution u ∈ h1γd (ω) of (2.29), (2.30). if we consider φ ∈ d(ω) in (2.29), (2.30), we have that u solves (2.27) in d′(ω) and thus u ∈ e(∆,l2(ω)). using green’s formula (2.10) in (2.29) and using (2.27), we obtain, if τ′(t) = 0∫ γn ( µ1 + µ2e −λτ(t) ) uφdγ + 〈 ∂u ∂ν ,φ〉γn = −µ2 ∫ γn z0φdγ, from which follows ∂u ∂ν + ( µ1 + µ2e −λτ(t) ) u = −µ2z0 on γn, which imply that ∂u ∂ν = −µ1u−µ2z (·, 1) on γn, where we have used (2.23) and (2.27). we find the same result if τ′(t) 6= 0. in conclusion, we have found (u,z)t ∈d(a), which verifies (2.21), and thus λi −a(t) is surjective for some λ > 0 and t > 0. again as κ(t) > 0, this proves that (2.31) λi −ã(t) = (λ + κ(t))i −a(t) is surjective, for any λ > 0 and t > 0. then, (2.14), (2.19) and (2.31) imply that the family ã = {ã(t) : t ∈ [0,t]} is a stable family of generators in h with stability constants independent of t, by [6, proposition 1.1]. therefore, the assumptions (i)-(iv) of theorem 1 are satisfied by (2.11), (2.14), (2.19), (2.31), (2.20) and lemma 1, and thus, the problem { ũ′ = ã(t)ũ, ũ(0) = u0, has a unique solution ũ ∈ c([0, +∞),d(a(0))) ∩ c1([0, +∞),h) for u0 ∈ d(a(0)). the requested solution of (2.8) is then given by u(t) = eβ(t)ũ(t), with β(t) = ∫ t 0 κ(s)ds, because u′eβ(t)ũ(t) + eβ(t)ũ′(t) = κ(t)eβ(t)ũ(t) + eβ(t)ã(t)ũ(t) = eβ(t)(κ(t)ũ(t) + ã(t)ũ(t)) = eβ(t)a(t)ũ(t) = a(t)eβ(t)ũ(t) = a(t)u(t). 50 mesmouli, ardjouni and djoudi this concludes the proof. � 3. the decay of the energy let us choose the following energy (3.1) e (t) = 1 2 ∫ ω u2 (x,t) dx + ξτ (t) 2 ∫ γn ∫ 1 0 u2 (x,t− τ (t) ρ) dpdγ, where ξ is a suitable positive constant. proposition 1. let (1.9) and (1.10) be satisfied. then for all regular solution of problem (2.8), the energy is decreasing and satisfies (3.2) e′ (t) ≤−c (∫ γn u2 (x,t) dγ + ∫ γn u2 (x,t− τ (t)) dγ ) . proof. differentiating (3.1), we get e′ (t) = ∫ ω uutdx + ξτ′ (t) 2 ∫ γn ∫ 1 0 u2 (x,t− τ (t) ρ) dpdγ + ξτ (t) ∫ γn ∫ 1 0 (1 − τ′ (t) ρ) u (x,t− τ (t) ρ) ut (x,t− τ (t) ρ) dpdγ, then e′ (t) = ∫ ω u∆udx + ξτ′ (t) 2 ∫ γn ∫ 1 0 u2 (x,t− τ (t) ρ) dpdγ + ξτ (t) ∫ γn ∫ 1 0 (1 − τ′ (t) ρ) u (x,t− τ (t) ρ) ut (x,t− τ (t) ρ) dpdγ. by green’s formula and integrating by parts in ρ, we obtain e′ (t) = − ∫ ω |∇u|2 dx + ∫ γn u ∂u ∂ν dγ − ξ 2 ∫ γn u2 (x,t− τ (t)) (1 − τ′ (t)) dγ + ξ 2 ∫ γn u2 (x,t) dγ, and by (1.3), we obtain e′ (t) = − ∫ ω |∇u|2 dx− ∫ γn [ µ1u 2 (x,t) + µ2u (x,t) u (x,t− τ (t)) ] dγ − ξ 2 ∫ γn u2 (x,t− τ (t)) (1 − τ′ (t)) dγ + ξ 2 ∫ γn u2 (x,t) dγ. by cauchy-schwarz’s and poincaré’s inequality, we get, e′ (t) ≤ ( − 1 cp −µ1 + ξ 2 + µ2 2 √ 1 −d )∫ γn u2 (x,t) dγ − ( ξ (1 −d) 2 + µ2 √ 1 −d 2 )∫ γn u2 (x,t− τ (t)) dγ. since the condition (2.12), we deduce that − 1 cp −µ1 + ξ 2 + µ2 2 √ 1 −d ≤ 0. which concludes the proof. � exponential stability of the heat equation 51 4. exponential stability in this section, we will give an exponential stability result for the problem (1.1)–(1.5) by using the following lyapunov functional (4.1) e (t) = e (t) + γê (t) , where γ > 0 is a parameter that will be fixed small enough later on, e is the standard energy defined by (3.1) and ê is defined by (4.2) ê (t) = ξτ (t) ∫ γn ∫ 1 0 e−2τ(t)ρu2 (x,t− τ (t) ρ) dpdγ. note that, the functional ê is equivalent to the energy e, that is there exist two positive constant d1, d2 such that (4.3) d1e (t) ≤e (t) ≤ d2e (t) . theorem 3. assume (1.6) and (1.7). then, there exist positive constants c1, c2 such that for any solution of problem (1.1)–(1.5), e(t) ≤ c1e(0)e−c2t, ∀t ≥ 0. proof. first, we differentiate ê (t) to have d dt ê (t) = τ′ (t) τ (t) ê (t) + ξτ (t) ∫ γn ∫ 1 0 (−2τ′ (t) ρ) e−2τ(t)ρu2 (x,t− τ (t) ρ) dpdγ + j, where j = 2ξτ (t) ∫ γn ∫ 1 0 e−2τ(t)ρ (1 − τ′ (t) ρ) ut (x,t− τ (t) ρ) u (x,t− τ (t) ρ) dpdγ. moreover, by noticing one more time that z (x,ρ,t) = u (x,t− τ (t) ρ) , x ∈ γn, ρ ∈ (0, 1) , t > 0, and by integrating by parts in ρ, we have j = −ξ ∫ γn ∫ 1 0 e−2τ(t)ρ (1 − τ′ (t) ρ) ∂ ∂ρ (z (x,ρ,t)) 2 dpdγ = ξ ∫ γn ∫ 1 0 e−2τ(t)ρ [−2τ (t) (1 − τ′ (t) ρ) − τ′ (t)] z2 (x,ρ,t) dpdγ −ξ ∫ γn e−2τ(t) (1 − τ′ (t)) z2 (x, 1, t) dγ + ξ ∫ γn z2 (x, 0, t) dγ = ξ ∫ γn ∫ 1 0 e−2τ(t)ρ [−2τ (t) (1 − τ′ (t) ρ) − τ′ (t)] u2 (x,t− τ (t) ρ) dpdγ −ξ ∫ γn e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dγ + ξ ∫ γn u2 (x,t) dγ. therefore, we have d dt ê (t) = τ′ (t) τ (t) ê (t) + ξ ∫ γn ∫ 1 0 e−2τ(t)ρ [−2τ (t) − τ′ (t)] u2 (x,t− τ (t) ρ) dpdγ − ξ ∫ γn e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dγ + ξ ∫ γn u2 (x,t) dγ = −2ê (t) − ξ ∫ γn e−2τ(t) (1 − τ′ (t)) u2 (x,t− τ (t)) dγ + ξ ∫ γn u2 (x,t) dγ. 52 mesmouli, ardjouni and djoudi as τ′ (t) < 1, we obtain (4.4) d dt ê (t) ≤−2ê (t) + ξ ∫ γn u2 (x,t) dγ. consequently, gathering (3.2), (4.1) and (4.4), we obtain d dt e (t) = d dt e (t) + γ d dt ê (t) ≤−2γê (t) + γξ ∫ γn u2 (x,t) dγ −c ∫ γn ( u2 (x,t) + u2 (x,t− τ (t)) ) dγ. then, for γ sufficiently small, we can estimate (4.5) d dt e (t) ≤−2γê (t) −c ∫ γn ( u2 (x,t) + u2 (x,t− τ (t)) ) dγ. now, observe that by assumption (1.6) on τ (t), we can deduce ê (t) ≥ ξτ (t) ∫ γn ∫ 1 0 e−2τρu2 (x,t− τ (t) ρ) dpdγ ≥ kξτ (t) 2 ∫ γn ∫ 1 0 u2 (x,t− τ (t) ρ) dpdγ,(4.6) for some positive constant k. therefore, from (4.5) and (4.6), d dt e (t) ≤ −2γê (t) −c ∫ γn ( u2 (x,t) + u2 (x,t− τ (t)) ) dγ ≤ −ke (t) ≤−ke (t) . for suitable positive constants k, k; where we used also the first inequality in (4.3). this clearly implies e (t) ≤ e−kte (0) , and so, using (4.3), e (t) ≤ c1e−c2te (0) , for suitable constants c1,c2 > 0. � references [1] f. ali mehmeti, nonlinear waves in networks, mathematical research, 80, akademie verlag, berlin, 1994. [2] r. datko, not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, siam j. control optim., 26 (1988), 697–713. [3] j. hale and s. verduyn lunel, introduction to functional differential equations, volume 99 of applied mathematical sciences, springer-verlag, new york, 1993. [4] p. grisvard, elliptic problems in nonsmooth domains, monographs and studies in mathematics, 21, pitman, boston-london-melbourne, 1985. [5] t. kato, nonlinear semigroups and evolution equations, 19 (1967), 508–520. [6] t. kato, linear and quasilinear equations of evolution of hyperbolic type, ii ciclo:125–191, 1976. [7] t. kato, abstract differential equations and nonlinear mixed problems, lezioni fermiane. [fermi lectures]. scuola normale superiore, pisa, 1985. [8] j. l. lions and e. magenes, problèmes aux limites non homogènes et applications. vol. 1, travaux et recherches math ematiques, 17, dunod, paris, 1968. [9] h. logemann, r. rebarber and g. weiss, conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop, siam j. control optim., 34 (1996), 572–600. [10] s. nicaise and c. pignotti, stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, siam j. control optim., 45 (2006), 1561–1585. [11] s. nicaise, j. valein and e. fridman, stability of the heat and of the wave equations with boundary time-varying delays, discrete contin. dyn. syst. ser. s. 2 (2009), 559–581. [12] s. nicaise, j. valein and e. fridman, stability of the heat and of the wave equations with boundary time-varying delays, discrete contin. dyn. syst. ser. s. 4 (2011), 693-722. [13] a. pazy, semigroups of linear operators and applications to partial differential equations, applied math. sciences. springer-verlag, new york, 1983. exponential stability of the heat equation 53 [14] r. rebarber and s. townley, robustness with respect to delays for exponential stability of distributed parameter systems, siam j. control optim., 37 (1999), 230–244. 1applied mathematics lab, faculty of sciences, department of mathematics, univ annaba, p.o. box 12, annaba 23000, algeria 2department of mathematics and informatics, univ souk ahras, p.o. box 1553, souk ahras, 41000, algeria ∗corresponding author: abd ardjouni@yahoo.fr international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 193-202 http://www.etamaths.com int-soft interior hyperideals of ordered semihypergroups asghar khan1,∗, muhammad farooq1 and bijan davvaz2 abstract. the main theme of this paper is to study ordered semihypergroups in the context of int-soft interior hyperideals. in this paper, the notion of int-soft interior hyperideals are studied and their related properties are discussed. we present characterizations of interior hyperideals in terms of int-soft interior hyperideals. the concepts of int-soft hyperideals and int-soft interior hyperideals coincide in a regular as well as in intra-regular ordered semihypergroups. we prove that every int-soft hyperideal is an int-soft interior hyperideal but the converse is not true which is shown with help of an example. furthermore we characterize simple ordered semihypergroups by means of int-soft hyperideals and int-soft interior hyperideals. 1. introduction the real world is inherently uncertain, imprecise, and vague. various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [19]. in response to this situation, zadeh [20], introduced fuzzy set theory as an alternative to probability theory. uncertainty is an attribute of information. in order to suggest a more general framework, the approach to uncertainty is outlined by zadeh [21]. to solve a complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. there are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. but all these theories have their own difficulties. uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of intuitionistic fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. however, all of these theories have their own difficulties which are pointed out in [6]. maji et al. [22] and molodtsov [6], suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. to overcome these difficulties, molodtsov [6], introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. molodtsov pointed out several directions for the applications of soft sets. worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years see [1–5, 9, 10, 16]. the concept of hyperstructure was first introduced by marty [7], at the 8th congress of scandinavian mathematicians in 1934, when he defined hypergroups and started to analyze its properties. now, the theory of algebraic hyperstructures has become a well-established branch in algebraic theory and it has extensive applications in many branches of mathematics and applied science. later on, people have developed the semihypergroups, which are the simplest algebraic hyperstructures having closure and associative properties. a comprehensive review of the theory of hyperstructures can be found in [11–15, 17]. in this paper, we study the concept of int-soft interior hyperideals in ordered semihypergroups and present some related examples of this concept. we show that int-soft hyperideals and int-soft interior hyperideals coincide in regular ordered semihypergroups and intra-regular ordered semihypergroups. we characterize ordered semihypergroups in terms of int-soft hyperideals and int-soft interior hyperideals. simple ordered received 13th february, 2017; accepted 4th april, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 20n20. key words and phrases. ordered semihypergroup; interior hyperideal; int-soft interior hyperideal; simple ordered semihypergroup; regular and intra-regular ordered semihypergroup . c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 193 194 khan, farooq and davvaz semihypergroups are characterized by using the notions of int-soft hyperideals and int-soft interior hyperideals. 2. preliminaries 2.1. basic results on ordered semihypergroups. a hypergroupoid is a nonempty set s equipped with a hyperoperation ◦, that is a map ◦ : s × s −→ p∗ (s), where p∗ (s) denotes the set of all nonempty subsets of s (see [7]). we shall denote by x◦y, the hyperproduct of elements x,y of s. a hypergroupoid (s,◦) is called a semihypergroup if (x◦y) ◦z = x◦ (y ◦z) for all x,y,z ∈ s. let a, b be the nonempty subsets of s. then the hyperproduct of a and b is defined as a◦b = ⋃ a∈a,b∈b a◦b. we shall write a◦x instead of a◦{x} and x◦a for {x}◦a. definition 2.1. (see [11]). an algebraic hyperstructure (s,◦,≤) is called an ordered semihypergroup (also called po-semihypergroup) if (s,◦) is a semihypergroup and (s,≤) is a partially ordered set such that the monotone condition holds as follows: a ≤ b implies that x◦a ≤ x◦ b and a◦x ≤ b◦x for all x,a,b ∈ s, where, if a,b ∈ p∗ (s) then we say that a � b if for every a ∈ a there exists b ∈ b such that a ≤ b. if a = {a} then we write a � b instead of {a}� b. definition 2.2. (see [13]). a nonempty subset a of an ordered semihypergroup (s,◦,≤) is called a subsemihypergroup of s if for all x,y ∈ a, implies that x◦y ⊆ a. equivalently a nonempty subset a of an ordered semihypergroup (s,◦,≤) is called a subsemihypergroup of s if a◦a ⊆ a. definition 2.3. (see [11]). let (s,◦,≤) be an ordered semihypergroup and a be a nonempty subset of s. then a is called a left (resp., right) hyperideal of s if: (1) s ◦a ⊆ a (resp., a◦s ⊆ a). (2) if a ∈ a and s 3 b ≤ a then b ∈ a. if a is both a right hyperideal and a left hyperideal of s, then it is called a hyperideal (or two-sided hyperideal) of s. definition 2.4. (see [18]) let (s,◦,≤) be an ordered semihypergroup. a subsemihypergroup a of s is called an interior hyperideal of s if: (1) s ◦a◦s ⊆ a. (2) if a ∈ a and s 3 b ≤ a then b ∈ a. for a ⊆ s, we denote (a] = {t ∈ s | t ≤ h for some h ∈ a} . lemma 2.1. (see [11]). let (s,◦,≤) be an ordered semihypergroup and a,b are the nonempty subsets of s. then the following statements hold: (1) a ⊆ (a] . (2) a ⊆ b implies that (a] ⊆ (b] . (3) (a] ◦ (b] ⊆ (a◦b] . (4) ((a] ◦ (b]] = (a◦b] . (5) ((a]] = (a] . definition 2.5. (see [18]). an ordered semihypergroup (s,◦,≤) is called regular if for each a ∈ s there exists x ∈ s such that a ≤ a◦x◦a. definition 2.6. (see [18]). an ordered semihypergroup (s,◦,≤) is called intra-regular if for each a ∈ s there exist x,y ∈ s such that a ≤ x◦a◦a◦y. 2.2. basic concepts of soft sets. in what follows, we take e = s as the set of parameters, which is an ordered semihypergroup, unless otherwise specified. from now on, u is an initial universe set, e is a set of parameters, p(u) is the power set of u and a,b,c... ⊆ e. definition 2.7. (see [6]). a soft set fa over u is defined as fa : e −→ p(u) such that fa(x) = ∅ if x /∈ a. hence fa is also called an approximation function. a soft set fa over u can be represented by the set of ordered pairs fa = {(x,fa(x))|x ∈ e, fa(x) ∈ p(u)} . it is clear from definition 2.7, that a soft set is a parameterized family of subsets of u. note that the set of all soft sets over u will be denoted by s(u). definition 2.8. (see [6]) (i) let fa,fb ∈ s(u). then fa is called a soft subset of fb, denoted by fa ⊆ fb if fa(x) ⊆ fb(x) for all x ∈ e. two soft sets fa and fb are said to be equal soft sets if fa ⊆ fb and fb ⊆ fa and is int-soft interior hyperideals of ordered semihypergroups 195 denoted by fa = fb. (ii) let fa,fb ∈ s(u). then the soft union of fa and fb, denoted by fa ∪fb = fa∪b, is defined by (fa ∪fb) (x) = fa(x) ∪fb(x) for all x ∈ e. (iii) let fa,fb ∈ s(u). then the soft intersection of fa and fb, denoted by fa∩fb = fa∩b, is defined by (fa ∩fb) (x) = fa(x) ∩fb(x) for all x ∈ e. for x ∈ s, we define ax = {(y,z) ∈ s ×s | x ≤ y ◦z}. definition 2.9. (see [8]). let fa and gb be two soft sets of an ordered semihypergroup s over u. then, the int-soft product, denoted by fa∗̃gb, is defined by fa∗̃gb : s −→ p(u),x 7−→ (fa∗̃gb) (x) = { ⋃ (y,z)∈ax {fa(y) ∩gb(z)} , if ax 6= ∅, ∅, if ax = ∅, for all x ∈ s. definition 2.10. (see [8]). for a nonempty subset a of s the characteristic soft set is defined to be the soft set sa of a over u in which sa is given as follows sa : s 7−→ p(u). x 7−→ { u, if x ∈ a ∅, otherwise for an ordered semihypergroup s, the soft set ”ss ” of s over u is defined as follows: ss : s −→ p(u),x 7−→ss(x) = u for all x ∈ s. the soft set ”ss” of an ordered semihypergroup s over u is called the whole soft set of s over u. definition 2.11. (see [8]). let fa be a soft set of an ordered semihypergroup s over u a subset δ such that δ ∈ p (u). the δ-inclusive set of fa is denoted by ea(fa,δ) and defined to be the set ea(fa,δ) = {x ∈ s | fa (x) ⊇ δ} . definition 2.12. (see [8]). a soft set fa of an ordered semihypergroup s over u is called an int-soft subsemihypergroup of s over u if: (∀x,y ∈ s) ⋂ α∈x◦y fa(α) ⊇ fa(x) ∩fa(y). definition 2.13. (see [8]). let fa be a soft set of an ordered semihypergroup s over u. then fa is called an int-soft left (resp., right) hyperideal of s over u if it satisfies the following conditions: (1) (∀x,y ∈ s) ⋂ α∈x◦y fa(α) ⊇ fa(y) ( resp., ⋂ α∈x◦y fa(α) ⊇ fa(x) ) . (2) (∀x,y ∈ s) x ≤ y =⇒ fa(x) ⊇ fa(y). a soft set fa over u is called an int-soft hyperideal (or int-soft two-sided hyperideal) of s over u if it is both an int-soft left hyperideal and an int-soft right hyperideal of s over u. definition 2.14. an int-soft subsemihypergroup fa of an ordered semihypergroup s over u is called an int-soft interior hyperideal of s over u if it satisfies the following conditions: (1) (∀x,y,a ∈ s) ⋂ α∈x◦a◦y fa(α) ⊇ fa(a). (2) (∀x,y ∈ s) x ≤ y =⇒ fa(x) ⊇ fa(y). example 2.1. let (s,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a} {a} c {a} {a} {a} {a,b} d {a} {a} {a,b} {a,b,c} ≤:= {(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (b,d), (c,d)}. suppose u = {p,q,r,s} and a = {a,b,c} . let us define fa (a) = {p,q,r,s} , fa (b) = {p} , fa (c) = {p,q,r} and fa (d) = ∅. then fa is an int-soft interior hyperideal of s over u. 196 khan, farooq and davvaz example 2.2. let (s,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d e a {a,b} {a,b} {a,b} {a,b} {a,b} b {a,b} {a,b} {a,b} {a,b} {a,b} c {a,b} {a,b} {c} {c} {e} d {a,b} {a,b} {c} {d} {e} e {a,b} {a,b} {c} {c} {e} ≤:= {(a,a), (b,b), (c,c), (d,d), (e,e) , (a,c), (a,d), (a,e) , (b,c) , (b,d), (b,e) , (c,d), (c,e)}. let u = {1, 2, 3, 4} and a = {a,b,c,e} . let us define fa (a) = {1, 2, 3, 4} , fa (b) = {1, 2, 3, 4} , fa (c) = {3, 4} , fa (d) = ∅ and fa (e) = {3, 4} . then fa is an int-soft interior hyperideal of s over u. proposition 2.1. let (s,◦,≤) be an ordered semihypergroup and a be a nonempty subset of s. then a is an interior hyperideal of s if and only if the characteristic function sa is an int-soft interior hyperideal of s over u. proof. suppose that a is an interior hyperideal of s. let x,y and a be any elements of s. if a ∈ a, then sa (a) = u. since a is an interior hyperideal of s, we have α ∈ x◦a◦y ⊆ s◦a◦s ⊆ a we have sa (α) = u. thus ⋂ α∈x◦a◦y sa (α) = u = sa (a) . if a /∈ a then sa (a) = ∅. since sa (α) ⊇∅ = sa (a) . thus ⋂ α∈x◦a◦y sa (α) ⊇sa (a) . let x,y ∈ s with x ≤ y. if y /∈ a then sa (y) = ∅ and so sa (x) ⊇∅ = sa (y) . if y ∈ a then sa (y) = u. since x ≤ y and a is an interior hyperideal of s, we have x ∈ a and thus sa (x) = u = sa (y) . since a is an interior hyperideal of s. therefore a is a subsemihypergroup of s. let x,y ∈ s. then we have, ⋂ α∈x◦y sa (α) ⊇sa (x)∩sa (y) for every α ∈ x◦y. indeed: if x◦y * a, then there exists α ∈ x ◦ y such that α /∈ a, and we have ⋂ α∈x◦y sa (α) = ∅. besides that x ◦ y * a implies that x /∈ a or y /∈ a. then sa (x) = ∅ or sa (y) = ∅ and hence ⋂ α∈x◦y sa (α) = sa (x)∩sa (y) . let x◦ y ⊆ a. then sa (α) = u for any α ∈ x◦ y. it implies that ⋂ α∈x◦y sa (α) = u. since we have sa (x) ⊆ u for any x ∈ a. thus ⋂ α∈x◦y sa (α) ⊇ sa (x) ∩sa (y) . therefore sa is an int-soft interior hyperideal of s over u. conversely, let ∅ 6= a ⊆ s such that sa is an int-soft interior hyperideal of s over u. we claim that a◦a ⊆ a. to prove the claim, let x,y ∈ a. by hypothesis, ⋂ α∈x◦y sa (α) ⊇sa (x)∩sa (y) = u which implies that sa (α) ⊇ u for any α ∈ x◦y. on the other hand sa (x) ⊆ u for all x ∈ s. thus for any α ∈ x◦y, sa (α) = u implies that α ∈ a. it thus follows that a◦a ⊆ a. let α ∈ s◦a◦s, then there exist x,y ∈ s and a ∈ a such that α ∈ x◦a◦y. since ⋂ α∈x◦a◦y sa (α) ⊇ sa (a) , and a ∈ a we have sa (a) = u. hence for each α ∈ s◦a◦s, we have sa (α) = u, and so α ∈ a. thus s◦a◦s ⊆ a. let x ∈ s and y ∈ a be such that x ≤ y. then sa (x) ⊇ sa (y) = u, and thus x ∈ a. therefore a is an interior hyperideal of s. proposition 2.2. let (s,◦,≤) be an ordered semihypergroup and fa be an int-soft hyperideal of s over u. then fa is an int-soft interior hyperideal of s over u. proof. let x,a,y ∈ s. since fa is an int-soft hyperideal of s over u. then for any α ∈ x◦a◦y, we have ⋂ α∈x◦a◦y fa (α) = ⋂ α∈x◦β β∈a◦y fa (α) ⊇ fa (β) ⊇ ⋂ β∈a◦y fa (β) ⊇ fa (a) . thus ⋂ α∈x◦a◦y fa (α) ⊇ fa (a) . therefore fa is an int-soft interior hyperideal of s over u. the converse of proposition 2.2, is not true in general. we can illustrate it by the following example. example 2.3. let (s,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a} {a} c {a} {a} {a,b} {a,b} d {a} {a} {a,b} {a} int-soft interior hyperideals of ordered semihypergroups 197 ≤:= {(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (d,b), (d,c)}. suppose u = {p,q,r} and a = {a,b,d} . let us define fa (a) = {p,q,r} , fa (b) = {p} , fa (c) = ∅ and fa (d) = {p,r} . then fa is an int-soft interior hyperideal of s over u. this is not an int-soft left hyperideal as ⋂ α∈c◦d={a,b} fa (a) ∩fa (b) = {p} + {p,r} = fa (d) . proposition 2.3. let (s,◦,≤) be a regular ordered semihypergroup and fa is an int-soft interior hyperideal of s over u. then fa is an int-soft hyperideal of s over u. proof. let x,y ∈ s. since fa is an int-soft interior hyperideal of s over u. then ⋂ α∈x◦y fa (α) ⊇ fa (x) . indeed: since s is regular and x ∈ s, then there exists z ∈ s such that x ≤ x ◦ z ◦ x. then we have x ◦ y ≤ (x◦z ◦x) ◦ y = (x◦z) ◦ (x◦y) . so there exist α ∈ x ◦ y, v ∈ x ◦ z and β ∈ v ◦ x ◦ y such that α ≤ β. since fa is an int-soft interior hyperideal of s over u, we have fa (α) ⊇ fa (β) ⊇ ⋂ β∈v◦x◦y fa (β) ⊇ fa (x) . thus ⋂ α∈x◦y fa (α) ⊇ fa (x) . therefore fa is an int-soft right hyperideal of s over u. in a similar way we prove that fa is an int-soft left hyperideal of s over u. by propositions 2.2 and 2.3 we have the following: theorem 2.1. in regular ordered semihypergroups the concepts of int-soft hyperideals and int-soft interior hyperideals coincide. proposition 2.4. let (s,◦,≤) be an intra-regular ordered semihypergroup and fa is an int-soft interior hyperideal of s over u. then fa is an int-soft hyperideal of s over u. proof. let a,b ∈ s. then ⋂ u∈a◦b fa (u) ⊇ fa (a) . indeed: since s is intra-regular and a ∈ s, there exist x,y ∈ s such that a ≤ x◦a◦a◦y. then a◦ b ≤ (x◦a◦a◦y) ◦ b = x◦a◦ (a◦y ◦ b). so there exist u ∈ a◦b, v ∈ a◦y◦b and α ∈ x◦a◦v such that u ≤ α. since fa is an int-soft interior hyperideal of s over u, we have fa (u) ⊇ fa (α) ⊇ ⋂ α∈x◦a◦v fa (α) ⊇ fa (a) . thus ⋂ u∈a◦b fa (u) ⊇ fa (a) . hence fa is an int-soft right hyperideal of s over u. similarly we can prove that fa is an int-soft left hyperideal of s over u. therefore fa is an int-soft hyperideal of s over u. by propositions 2.2 and 2.4 we have the following: theorem 2.2. in intra-regular ordered semihypergroups the concepts of int-soft hyperideals and intsoft interior hyperideals coincide. theorem 2.3. let fa be a soft set of an ordered semihypergroup s over u and δ ∈ p (u) . then fa is an int-soft interior hyperideal of s over u if and only if each nonempty δ-inclusive set ea(fa,δ) is an interior hyperideal of s. proof. assume that fa is an int-soft interior hyperideal of s over u. let δ ∈ p (u) such that ea(fa,δ) 6= ∅. let x,y ∈ ea(fa,δ). then fa (x) ⊇ δ and fa (y) ⊇ δ. by hypothesis, we have⋂ α∈x◦y fa (α) ⊇ fa (x) ∩ fa (y) ⊇ δ ∩ δ = δ. thus for any α ∈ x ◦ y, we have fa (α) ⊇ δ, implies that α ∈ ea(fa,δ). it follows that x◦y ⊆ ea(fa,δ). hence ea(fa,δ) is a subsemihypergroup of s. let y ∈ ea(fa,δ) and x,z ∈ s. then fa (y) ⊇ δ. since fa is an int-soft interior hyperideal of s over u. thus ⋂ w∈x◦y◦z fa (w) ⊇ fa (y) ⊇ δ. hence fa (w) ⊇ δ for any w ∈ x◦y ◦z implies that w ∈ ea(fa,δ). thus s◦ea(fa,δ)◦s ⊆ ea(fa,δ). let x ∈ ea(fa,δ) and y ∈ s with y ≤ x. then δ ⊆ fa (x) ⊆ fa (y) , we get y ∈ ea(fa,δ). therefore ea(fa,δ) is an interior hyperideal of s. conversely, suppose that ea(fa,δ) 6= ∅ is an interior hyperideal of s. if ⋂ α∈x◦y fa (α) ⊂ fa (x)∩fa (y) for some x,y ∈ s, then there exists δ ∈ p (u) such that ⋂ α∈x◦y fa (α) ⊂ δ ⊆ fa (x) ∩ fa (y) , which implies that x,y ∈ ea(fa,δ) and x◦y * ea(fa,δ). it contradicts the fact that ea(fa,δ) is an interior hyperideal of s. consequently, ⋂ α∈x◦y fa (α) ⊇ fa (x) ∩ fa (y) for all x,y ∈ s. next we show that⋂ α∈x◦a◦y fa (α) ⊇ fa (a) for all x,a,y ∈ s. choose fa (a) = δ, then a ∈ ea(fa,δ). since ea(fa,δ) is an interior hyperideal of s, we get x◦a◦y ⊆ ea(fa,δ). then for every α ∈ x◦a◦y, we have fa (α) ⊇ δ and so fa (a) = δ ⊆ ⋂ α∈x◦a◦y fa (α) . let x,y ∈ s such that x ≤ y. lf fa(y) = δ then y ∈ ea(fa,δ). since ea(fa,δ) is an interior hyperideal of s, we get x ∈ ea(fa,δ). so fa(x) ⊇ δ = fa(y). therefore fa is an int-soft interior hyperideal of s over u. 198 khan, farooq and davvaz example 2.4. let (s,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a,d} {a} c {a} {a} {a} {a} d {a} {a} {a} {a} ≤:= {(a,a), (b,b), (c,c), (d,d), (a,d)}. then the interior hyperideals of s are {a} , {a,b} , {a,c} , {a,d} , {a,b,d} , {a,c,d} and s. suppose u = {e1,e2,e3,e4} and a = {a,b,d} . let us define fa (a) = {e1,e2,e3,e4} , fa (b) = {e1,e3} , fa (c) = ∅ and fa (d) = {e1,e4} . then ea(fa,δ) =   {a,b,d} if δ = {e1} {a} if δ = {e2} {a,b} if δ = {e3} {a,d} if δ = {e4} {a} if δ = {e1,e2} {a,b} if δ = {e1,e3} {a,d} if δ = {e1,e4} {a} if δ = {e2,e3} {a} if δ = {e2,e4} {a} if δ = {e3,e4} {a} if δ = {e1,e2,e3} {a} if δ = {e1,e2,e4} {a} if δ = {e1,e3,e4} {a} if δ = {e2,e3,e4} {a} if δ = u so by theorem 2.3, fa is an int-soft interior hyperideal of s over u. theorem 2.4. let {fai | i ∈ i} be a family of int-soft interior hyperideals of an ordered semihypergroup s over u. then fa = ⋂ i∈i fai is an int-soft interior hyperideal of s over u where(⋂ i∈i fai ) (x) = ⋂ i∈i (fai (x)) . proof. let x,y ∈ s. then, since each fai (i ∈ i) is an int-soft interior hyperideals of s over u, so ⋂ α∈x◦y fai (α) ⊇ fai (x) ∩fai (y) . thus for any α ∈ x◦y, fai (α) ⊇ fai (x) ∩fai (y) , and we have fa (α) = (⋂ i∈i fai ) (α) = ⋂ i∈i (fai (α)) ⊇ ⋂ i∈i (fai (x) ∩fai (y)) = (⋂ i∈i (fai (x)) ) ∩ (⋂ i∈i (fai (y)) ) =(⋂ i∈i fai ) (x) ∩ (⋂ i∈i fai ) (y) = fa (x) ∩ fa (y) , which implies that ⋂ α∈x◦y fa (α) ⊇ fa (x) ∩ fa (y) . let a,x,y ∈ s and ⋂ β∈x◦a◦y fai (β) ⊇ fai (a) . thus for any β ∈ x ◦ a ◦ y, fai (β) ⊇ fai (a) . then fa (β) = (⋂ i∈i fai ) (β) = ⋂ i∈i (fai (β)) ⊇ ⋂ i∈i (fai (a)) = (⋂ i∈i fai ) (a) = fa (a) . thus⋂ β∈x◦a◦y fa (β) ⊇ fa (a). furthermore, if x ≤ y, then fa (x) ⊇ fa (y) . indeed: since every fai (i ∈ i) is an int-soft interior hyperideal of s over u, it can be obtained that fai (x) ⊇ fai (y) for all i ∈ i. thus fa (x) = (⋂ i∈i fai ) (x) = ⋂ i∈i (fai (x)) ⊇ ⋂ i∈i (fai (y)) = (⋂ i∈i fai ) (y) = fa (y) . thus fa is an int-soft interior hyperideals of s over u. lemma 2.2. let s be an ordered semihypergroup and fa is a soft set of s over u. if fa is an int-soft subsemihypergroup of s over u such that x ≤ y =⇒ fa(x) ⊇ fa(y), ∀x,y ∈ s, int-soft interior hyperideals of ordered semihypergroups 199 then fa∗̃fa⊆̃fa. conversely if fa∗̃fa⊆̃fa, then fa is an int-soft subsemihypergroup of s over u. proof. let x ∈ s. if ax = ∅, then (fa∗̃fa) (x) = ∅ ⊆ fa (x) . if ax 6= ∅, then (b,c) ∈ ax such that x ≤ b◦ c. this means that there exists α ∈ b◦ c such that x ≤ α. (fa∗̃fa) (x) = ⋃ (b,c)∈ax {fa (b) ∩fa (c)} ⊆ ⋃ (b,c)∈ax fa (α) ⊆ ⋃ (b,c)∈ax fa (x) = fa (x) . thus fa∗̃fa ⊆ fa. conversely, if fa∗̃fa ⊆ fa, then for all x,y ∈ s and α ∈ x◦y. we have fa (α) ⊇ (fa∗̃fa) (α) = ⋃ (x,y)∈aα {fa (x) ∩fa (y)} ⊇ {fa (x) ∩fa (y)} fa (α) ⊇ {fa (x) ∩fa (y)} . hence ⋂ α∈x◦y fa (α) ⊇{fa (x) ∩fa (y)} . thus fa is an int-soft subsemihypergroup of s over u. theorem 2.5. let (s,◦,≤) be an ordered semihypergroup and fa be a soft set of s over u. then fa is an int-soft interior hyperideal of s over u if and only if fa∗̃fa ⊆ fa and ss∗̃fa∗̃ss ⊆ fa. proof. let fa be an int-soft interior hyperideal of s over u. then (ss∗̃fa∗̃ss) (a) ⊆ fa (a) for all a ∈ s. indeed: if (ss∗̃fa∗̃ss) (a) = ∅, clearly, (ss∗̃fa∗̃ss) (a) ⊆ fa (a) . let (ss∗̃fa∗̃ss) (a) 6= ∅. then we can prove that (ss∗̃fa∗̃ss) (a) ⊆ fa (a) . in fact, let (x,y) ∈ aa and (p,q) ∈ ax, i.e., a ≤ x◦y and x ≤ p◦q. then a ≤ p◦q◦y, and there exists u ∈ p◦q◦y such that a ≤ u. since fa is an int-soft interior hyperideal of s over u. then fa (a) ⊇ fa (u) ⊇ ⋂ u∈p◦q◦y fa (u) ⊇ fa (q) . thus (ss∗̃fa∗̃ss) (a) = ⋃ (x,y)∈aa {(ss∗̃fa) (x) ∩ss (y)} = ⋃ (x,y)∈aa {(ss∗̃fa) (x) ∩u} = ⋃ (x,y)∈aa (ss∗̃fa) (x) = ⋃ (x,y)∈aa   ⋃ (p,q)∈ax (ss (p) ∩fa (q))   = ⋃ (x,y)∈aa   ⋃ (p,q)∈ax (u ∩fa (q))   = ⋃ (x,y)∈aa ⋃ (p,q)∈ax (fa (q)) ⊆ fa (a) . thus ss∗̃fa∗̃ss ⊆ fa. conversely, for any x,y,z ∈ s, let α ∈ x ◦ y ◦ z. then, there exists u ∈ x ◦ y ⊆ (x◦y] such that 200 khan, farooq and davvaz α ∈ u◦z ⊆ (u◦z] , and we have (x,y) ∈ au, (u,z) ∈ aα. since ss∗̃fa∗̃ss ⊆ fa, we have fa (α) ⊇ (ss∗̃fa∗̃ss) (α) = ⋃ (p,q)∈aα [{ss∗̃fa}(p) ∩ss (q)] ⊇ {(ss∗̃fa) (u) ∩ss (z)} = {(ss∗̃fa) (u) ∩u} = (ss∗̃fa) (u) = ⋃ (s,t)∈au [ss (s) ∩fa (t)] ⊇ {ss (x) ∩fa (y)} = {u ∩fa (y)} = fa (y) . it thus follows that ⋂ α∈x◦y◦z fa (α) ⊇ fa (y) . the rest of the proof is a consequence of the lemma 2.2. 3. characterizations of simple ordered semihypergroups in terms of int-soft hyperideals and int-soft interior hyperideals definition 3.1. (see [18]). an ordered semihypergroup (s,◦,≤) is called simple if it has no a proper hyperideal. lemma 3.1. (see [18]). an ordered semihypergroup (s,◦,≤) is a simple ordered semihypergroup if and only if for every a ∈ s, (s ◦a◦s] = s. let (s,◦,≤) is an ordered semihypergroup and a ∈ s, and fa be a soft set of s over u we denote by ia the subset of s defines as follows: ia = {b ∈ s | fa (b) ⊇ fa (a)} . proposition 3.1. let (s,◦,≤) be an ordered semihypergroup and fa is an int-soft right hyperideals of s over u. then the set ia is a right hyperideal of s for every a ∈ s. proof. let a ∈ s. first of all ∅ 6= ia ⊆ s. since a ∈ ia. let b ∈ ia and s ∈ s. then b◦s ⊆ ia. indeed: since fa is an int-soft right hyperideals of s over u and b,s ∈ s, we have ⋂ α∈b◦s fa (α) ⊇ fa (b) . since b ∈ ia, we have fa (b) ⊇ fa (a) . thus ⋂ α∈b◦s fa (α) ⊇ fa (a) , implies that fa (α) ⊇ fa (a) , so α ∈ ia and hence b◦ s ⊆ ia. let b ∈ ia and s 3 s ≤ b. then s ∈ ia. indeed: since fa is an int-soft right hyperideals of s over u, b,s ∈ s and s ≤ b, we have fa (s) ⊇ fa (b) . since b ∈ ia, we have fa (b) ⊇ fa (a) . then fa (s) ⊇ fa (a) , so s ∈ ia. in a similar way we prove the following: proposition 3.2. let (s,◦,≤) be an ordered semihypergroup and fa is an int-soft left hyperideals of s over u. then the set ia is a left hyperideal of s for every a ∈ s. by propositions 3.1 and 3.2 we have the following: proposition 3.3. let (s,◦,≤) be an ordered semihypergroup and fa is an int-soft hyperideals of s over u. then the set ia is a hyperideal of s for every a ∈ s. theorem 3.1. (see [8]). let (s,◦,≤) be an ordered semihypergroup and ∅ 6= i ⊆ s. then i is a hyperideal of s if and only if the characteristic function si is an int-soft hyperideals of s over u. theorem 3.2. an ordered semihypergroup (s,◦,≤) is a simple ordered semihypergroup if and only if every int-soft hyperideal of s over u is a constant function. proof. assume that s is a simple ordered semihypergroup. let fa is an int-soft hyperideal of s over u and a,b ∈ s. by proposition 3.3, we obtain ia is a hyperideal of s. by assumption, this implies that ia = s. then b ∈ ia, that is fa (b) ⊇ fa (a) . by symmetry we get fa (a) ⊇ fa (b) . therefore fa (a) = fa (b) . conversely, we assume that for every int-soft hyperideal of s over u is a constant function. let i be a hyperideal of s and x ∈ s. by theorem 3.1, we obtain the characteristic function si is an int-soft hyperideal of s over u. by assumption, si is a constant function, that is si (x) = si (b) for every int-soft interior hyperideals of ordered semihypergroups 201 b ∈ s. let a ∈ i. then si (x) = si (a) = u, and so x ∈ i. therefore s ⊆ i. theorem 3.3. let (s,◦,≤) be an ordered semihypergroup. then s is a simple ordered semihypergroup if and only if every int-soft interior hyperideal of s over u is a constant function. proof. assume that s is a simple ordered semihypergroup. let fa be an int-soft interior hyperideal of s over u and a,b ∈ s. by lemma 3.1, we have s = (s ◦ b◦s] . since a ∈ s, we have a ∈ (s ◦ b◦s] . then there exist x,y ∈ s such that a ≤ x◦ b◦y, i.e., there exists α ∈ x◦ b◦y such that a ≤ α. since fa is an int-soft interior hyperideal of s over u, we have fa (a) ⊇ fa (α) ⊇ ⋂ α∈x◦b◦y fa (α) ⊇ fa (b) . hence fa (a) ⊇ fa (b) . by symmetry we can prove that fa (b) ⊇ fa (a) . therefore fa (a) = fa (b) . conversely, assume that every int-soft interior hyperideal of s over u is a constant function. let fa is an int-soft hyperideal of s over u. then fa is an int-soft interior hyperideal of s over u. by assumption fa is a constant function. by theorem 3.2, s is a simple ordered semihypergroup. corollary 3.1. let (s,◦,≤) be an intra-regular ordered semihypergroup. then every int-soft interior hyperideal of s over u is a constant function. as a consequence of lemma 3.1, theorem 3.2, and theorem 3.3, we present characterizations of a simple ordered semihypergroup as the following theorem. theorem 3.4. let (s,◦,≤) be an ordered semihypergroup. then the following statements are equivalent: (1) s is a simple ordered semihypergroup. (2) s = (s ◦a◦s] for every a ∈ s. (3) every int-soft hyperideal of s over u is a constant function. (4) every int-soft interior hyperideal of s over u is a constant function. proposition 3.4. let (s,◦,≤) be an intra-regular ordered semihypergroup. then for every interior hyperideals a and b of s we have (1) (a◦a] = a. (2) (a◦b] = (b ◦a] . proof. (1) . let s be an intra-regular ordered semihypergroup and a, b are the interior hyperideals of s. let a ∈ a. since s is intra-regular, there exist x,y ∈ s such that a ≤ x ◦ a ◦ a ◦ y = (x◦a)◦(a◦y) ≤ x◦(x◦a◦a◦y)◦(x◦a◦a◦y)◦y = ((x◦x◦a) ◦a◦ (y))◦((x◦a) ◦a◦ (a◦y ◦y)) ⊆ (s ◦a◦s) ◦ (s ◦a◦s) ⊆ a◦a =⇒ a ∈ (a◦a] =⇒ a ⊆ (a◦a] . for the reverse inclusion, let a ∈ (a◦a] , then a ≤ a1 ◦ a2 for some a1,a2 ∈ a. then a ≤ x◦a◦a◦y = (x◦a)◦(a◦y) ≤ x◦(a1 ◦a2)◦(a1 ◦a2)◦y = (x◦a1 ◦a2)◦a1 ◦(a2 ◦y) ⊆ s◦a◦s ⊆ a =⇒ a ∈ (a] = a =⇒ (a◦a] ⊆ a. thus (a◦a] = a. (2) . let a and b be interior hyperideals of s. then (a◦b] = (b ◦a] . indeed: by (1) we have (a◦b] = ((a◦b] ◦ (a◦b]] = (((a◦b] ◦ (a◦b]) ◦ ((a◦b] ◦ (a◦b])] ⊆ (((a◦b) ◦ (a◦b)] ◦ ((a◦b) ◦ (a◦b)]] = (((a) ◦b ◦ (a◦b)] ◦ ((a◦b) ◦a◦ (b)]] ⊆ ((s ◦b ◦s] ◦ (s ◦a◦s]] ⊆ ((b] ◦ (a]] = (b ◦a] =⇒ (a◦b] ⊆ (b ◦a] . by symmetry we have (b ◦a] ⊆ (a◦b] . thus (a◦b] = (b ◦a] . proposition 3.5. let (s,◦,≤) be an intra-regular ordered semihypergroup and fa is an int-soft interior hyperideal of s over u. then for every a ∈ s such that a◦a ≤ a, we have the following (1) ⋂ v∈a◦a fa (v) = fa (a) . (2) ⋂ α∈a◦b fa (α) = ⋂ β∈b◦a fa (β) . proof. (1) . let s be an intra-regular ordered semihypergroup and fa is an int-soft interior hyperideal of s over u and a ∈ s. then ⋂ v∈a◦a fa (v) = fa (a) . indeed: since s is intra-regular and a ∈ s, there exist x,y ∈ s such that a ≤ x◦a◦a◦y for some x,y ∈ s. so there exist v ∈ a◦a and z ∈ x◦v ◦y such that a ≤ z. then fa (a) ⊇ fa (z) ⊇ ⋂ z∈x◦v◦y fa (z) ⊇ fa (v) . hence fa (a) ⊇ ⋂ v∈a◦a fa (v) . since a◦a ≤ a so there is v ∈ a◦a such that v ≤ a. then we have fa (v) ⊇ fa (a) . thus ⋂ v∈a◦a fa (v) ⊇ fa (a) . therefore ⋂ v∈a◦a fa (v) = fa (a) . 202 khan, farooq and davvaz (2) . suppose a,b ∈ s. let α ∈ a ◦ b and β ∈ b ◦ a. then we have ⋂ α∈a◦b fa (α) = ⋂ β∈b◦a fa (β) . indeed: by (1) we have fa (α) = ⋂ u∈α◦α fa (u) ⊇ ⋂ u∈a◦b◦a◦b fa (u) = ⋂ u∈a◦(b◦a)◦b fa (u) = ⋂ u∈a◦β◦b β∈b◦a fa (u) ⊇ fa (β) ⊇ ⋂ β∈b◦a fa (β) . it follows that ⋂ α∈a◦b fa (α) ⊇ ⋂ β∈b◦a fa (β) . by symmetry it can be shown that⋂ β∈b◦a fa (β) ⊇ ⋂ α∈a◦b fa (α) . hence ⋂ α∈a◦b fa (α) = ⋂ β∈b◦a fa (β) . references [1] h. aktas and n. cagman, soft sets and soft groups, inf. sci. 177 (13) (2007), 2726-2735. [2] f. feng, y. b. jun, and x. zhao, soft semirings, comput. math. appl. 56 (10) (2008), 2621–2628. [3] f. feng, m. i. ali, and m. shabir, soft relations applied to semigroups, filomat, 27 (7) (2013), 1183–1196. [4] f. feng and y.m. li, soft subsets and soft product operations, inf. sci. 232 (2013), 44–57. [5] y. b. jun, s. z. song, and g. muhiuddin, concave soft sets, critical soft points, and union-soft ideals of ordered semigroups, sci. world j. 2014 (2014), article id 467968. [6] d. molodtsov, soft set theory—-first results, comput. math. appl. 37 (4-5) (1999), 19–31. [7] f .marty, sur une generalization de la notion de group, 8iemcongress, math. scandenaves stockholm (1934), 45-49. [8] a. khan, m. farooq and b. davvaz, int-soft left (right) hyperideals of ordered semihypergroups, submitted. [9] s. naz and m. shabir, on soft semihypergroups, j. intell. fuzzy syst. 26 (2014). 2203-2213. [10] s. naz and m. shabir, on prime soft bi-hyperideals of semihypergroups, j. intell. fuzzy syst. 26 (2014). 1539-1546. [11] b. pibaljommee and b. davvaz, characterizations of (fuzzy) bi-hyperideals in ordered semihypergroups, j. intell. fuzzy syst. 28 2015. 2141-2148. [12] d. heidari and b. davvaz, on ordered hyperstructures, u.p.b. sci. bull. series a, 73 (2) 2011. 85-96. [13] t. changphas and b. davvaz, bi-hyperideals and quasi-hyperideals in ordered semihypergroups, ital. j. pure appl. math.-n, 35 (2015), 493-508. [14] p. corsini and v. leoreanu-fotea, applications of hyperstructure theory, advances in mathematics, kluwer academic publisher, (2003). [15] b. davvaz, fuzzy hyperideals in semihypergroups, ital. j. pure appl. math.-n, 8 (2000), 67-74. [16] d. molodtsov, the theory of soft sets, urss publishers, moscow, 2004 (in russian). [17] j. tang, a. khan and y. f. luo, characterization of semisimple ordered semihypergroups in terms of fuzzy hyperideals, j. intell. fuzzy syst. 30 (2016), 1735-1753. [18] n. tipachot and b. pibaljommee, fuzzy interior hyperideals in ordered semihypergroups, ital. j. pure appl. math.n, 36 (2016), 859-870. [19] l. a. zadeh, from circuit theory to system theory, proc. inst. radio eng. 50 (1962), 856–865. [20] l. a. zadeh, fuzzy sets, inf. comput. 8 (1965), 338–353. [21] l. a. zadeh, toward a generalized theory of uncertainty gtu —an outline, inf. sci. 172 (1-2) (2005), 1–40. [22] p. k. maji, a. r. roy, and r. biswas, an application of soft sets in a decision making problem, comput. math. appl. 44 (8-9) (2002), 1077–1083. 1department of mathematics, abdul wali khan university mardan, kp, pakistan 2department of mathematics, yazd university, yazd, iran ∗corresponding author: azhar4set@yahoo.com 1. introduction 2. preliminaries 2.1. basic results on ordered semihypergroups. 2.2. basic concepts of soft sets 3. characterizations of simple ordered semihypergroups in terms of int-soft hyperideals and int-soft interior hyperideals references international journal of analysis and applications volume 16, number 5 (2018), 654-672 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-654 robust stability and design of state feedback controller for straightforward active queue management mehrdad nozohour yazdi and ali delavarkhalafi∗ department of mathematics, yazd university, yazd, iran ∗corresponding author: delavarkh@yazd.ac.ir abstract. the straightforward active queue management (aqm), which is based on the prediction of arrival rate is investigated by means of state-space approach. we formulate the feedback control design problem for linearized system of additive increase multiplicative decrease (aimd) dynamic models as state-space model. then the lyapunov-krasovskii method is provided to achieve the robust stability and sufficient stabilization condition and afterwards the term of linear inequality matrix (lmi) is used to show the results. we present the simulation results and show the superiority of our proposed method to other control mechanisms. 1. introduction nowadays, computer networks develop rapidly and the growth of the amount of data communicated in computer networks become an important issue in this field. since the available capacity of the resource is usually less than the demand, congestion happens which is the main concerns to deal with in computer networks. two effective strategies to control congestion are transmission control protocol (tcp) and active queue management (aqm) which they control the congestion at the end and at the hosts, respectively. tcp recognize the congestion by receiving acknowledgements and moderate the tcp window sizes of senders(ref in congestion control paper). among the categories in control systems, networked control systems (ncs) are a category in which sensors, actuators, and controllers are connected though a network. different forms received 2017-10-29; accepted 2018-01-10; published 2018-09-05. 2010 mathematics subject classification. 93b05, 93e15. key words and phrases. network congestion control; active queue management(aqm); packet arrival rate; prediction; asymptotical stability. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 654 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-654 int. j. anal. appl. 16 (5) (2018) 655 and investigations in ncs have been exposed in [11]. for designing the control systems, parameter effects such as delay [14] and [10], quality of service (qos), and co-design of plant controller [7], [18], [26] and [30] are considered. the limitation on the amount of delay is assumed and there is a lot of attention on packet lost [16]. dai et al. [8] improved the quality of service (qos) and propagate the control performance of the system. in tcp/aqm, dynamical models for average tcp window size and the queue size by including the tcp networks and aqm strategies , are provided [12], [13] and [20]. in tcp/aqm models after detecting the congestion, they convey the information to end host, in which the routers drop or mark, when explicit congestion notification (ecn) is activated, packets and they do all these things before the buffer overflows. also, end hosts after redacting their congestion window size, reconcile the sending rates. since packet-dropping causes flow-synchronization and performance depreciation, red scheme was represented to allow to routers to facilitate tcp ’s management of network performance [9]. hollot et al. [13] derived proportional and proportional-integral control for aqm schemes in which, the dynamical model developed by misra et al. [20]. the aqm algorithms based on control theory is complicated to implement and some times parameter setting effects the system design [1] and [23]. to overcome with this, aqm algorithm with simple implementation based on prediction of arrival rate was presented in [27] and this prediction was derived from the analysis of the network congestion control. since these dynamical models are nonlinear, not only they are linearized at the operating point, but also this linearization is depend on a time delay which play an important role in models and their stability. time delay can be an uncertainties for the system and robust controller such as h∞ could be designed for them [21]. in order to rectify the delays and the impact on system performance, the state-space models developed to formulate the control design problem [4]. wang et al. [24] use the state-space model to derive model predictive controller and use the motion equation to linearized the system. the queue length in data buffer was predicted there and then packets at router were dropped. when the linear system obtained, stability analysis and stabilization of linear time delay systems must be considered. the stability of linear system may be required to consider the poles of closed-loop transfer function. the routh-hurwitz stability of a linear system, provides the stability by finding the roots of characteristic equation of the system [5], [6] and [15]. the nyquist criterion is a graphical method and deals with the loop gain transfer function, i.e., the open loop transfer function. hollot et al. [13] find the stability of pi and pid control and also represented that red’s queue-averaging is not beneficial by nyquist stability. nyquist plot specifies the factor in which the system magnitude and makes the system marginally stable [15]. that factor is called gain margin. phase margin is obtained by phase shift which determined by nyquist plot or bode plot [12]. stability theory influences the system theory and engineering and also the stability implies the behavior of a system over a long time period. the asymptotic behavior of a state of the system near the int. j. anal. appl. 16 (5) (2018) 656 steady-state solution like equilibrium points or operating points, helps to derive the stability of systems. the stability of equilibrium points, characterized in the sense of lyapunov. stability and asymptotically stability of system, were defined in [15]. the contribution of this investigation is to model state-space of the active queue management in which the drop probability policy is outspoken. the reason to go through to this designing is finding the asymptotical stability of the system which is the significant basis of the affectivity of the system. lyapunov method is used to locate the stability conditions of an equilibrium point without solving the state equation. the remain of this paper is organized into description of the straightforward active queue management in section 2. the control model of this system and stability condition for proposed method are derived in section 3. in section 4, the illustrative examples are provided to represented the effectiveness of the method. conclusion and suggestion of future work are located in section 5. 2. prediction of the change of packet arrival rate in order to formulate the system we first provide some relations. the system is a single congested router with a capacity c. let n, tcp flows labeled i = 1, ...,n, wi(t), vi(t) and ri(t) represent the congestion window size, packet sending rate and round trip time(rtt)of flow tcpi(i = 1, . . . ,n) at time t > 0, respectively. the packet arrival rate at time t > 0 is λ(t), then vi(t) = wi(t) ri(t) , (2.1) λ(t) = n∑ i=1 vi(t), ri(t) = ri + q(t) c , (2.2) where ri is the propagation of rrt that we called rtpt of tcpi, q(t) is the queue length, and the queueing delay of models is q(t) c . additive increase and multiplicative decrease (aimd) is the famous strategy is used here and the full description of this strategy can be found in [19]. the expectation of the increment of the arrival rate is as follows: ∆λ(t) = 1 wi(t)ri(t) (1 −p(t− τ)) − vi(t) 2 p(t− τ). (2.3) the moment that the packets are dropped or acknowledged to the moment that host receives the information. here in this system wi(t), vi(t) and ri(t) are unknown variables and also we assume that they have same fixed propagation delay ri(t) and we approximated to their expectations. the other assumptions here, are the desired value of queue occupation which is shown as qref and the average value of rtpt of all tcp sessions which is r. hence the rrt could be derived as: ri(t) ≈ r + qref c = r, (2.4) int. j. anal. appl. 16 (5) (2018) 657 where r is a rrt approximation. afterward the expectation of the sending rate vi(t) and window size wi(t) will be expressed as follows whilst vi(t) λ(t) is proportion of the packets that are generated by tcpi, v (t) = n∑ i=1 vi(t) vi(t) λ(t) ≥ ( ∑n i=1 vi(t)) 2 nλ(t) = λ(t) n , (2.5) w(t) = n∑ i=1 wi(t) vi(t) λ(t) ≥ ( ∑n i=1 vi(t)) 2r nλ(t) = λ(t)r n . (2.6) since the system is based on packet dropping probability, so we assume it updated at every time interval and represented as p(t). also the number of arriving packets at the router is m(t). now by combing the equations (2.4-2.6), the expectation of arrival at the router is obtained below, is the objective of analysis which is provided by ( [19] and [27]) and they shown that there is a mismatch between the operating value of the queue length and desired one by their method, actually it’s smaller than the desired one. ∆λ(t + rrt) = ∑m(t) n(1−p(t)) λ(t+rrt)r2 i (t+rrt) − λ(t+rrt)p(t)m(t) 2n ≈ n(1−p(t)) λ(t+rrt)r2 i − λ(t+rrt)p(t)m(t) 2n , (2.7) xu and sun [19] proposed the statistical method which is related to this dropping probability and they named it straightforward active queue management (sfaqm) . the arrival rate when the congestion information arrive at the hosts is λ(t + rrt) and the desired arrival rate is λref (t + rrt). so in order to reach to the desired queue length qref we obtain: λ(t + rrt) + ∆λ(t + rrt) = λref (t + rrt), (2.8) λref (t + rrt) = c + (qref−q(t)) α . (2.9) here, α is a parameter to control queue length to desired value. hence, p(t) = λ(t + rrt) + m(t)n λ(t+rrt)r2 −λref (t + rrt) ( λ(t+rrt) 2n + n λ(t+rrt)r2 )m(t) (2.10) by counting the number of arriving packets m(t) to estimate the arrival rate λ(t) will be expressed as: λ(t) = m(t) δ , (2.11) int. j. anal. appl. 16 (5) (2018) 658 figure 1. the flowchart of sfaqm [27]. where δ is the length of the sampling period. substituting (2.11) in (2.10) we can conclude: p(t) = m(t + rrt) + m(t)nδ2 m(t+rrt)r2 −mref (t + rrt) ( m(t+rrt) 2n + nδ 2 m(t+rrt)r2 )m(t) , (2.12) where mref (t + rrt) shows as follows: mref (t + rrt) = cδ + (qref −q(t))δ α . (2.13) the exponential weighted moving average (ewma) was used to predict m(t) by noticing that m(t + rrt) is predicted as a value between m(t) and mref (t + rrt), we have: m(t) = (1 −ω1)m(t− δ) + ω1m0, (2.14) m(t + rrt) = (1 −ω2)m(t) + ω2mref (t + rrt), (2.15) where • ω1, ω2: weight parameter, • m0: number of arriving packets during the previous interval, • m(t− δ): value of m(t) of last sample. the algorithm of (sfaqm) will be depicted as in figure 1. int. j. anal. appl. 16 (5) (2018) 659 misra et al. [12], [20] and also low et al. [19] provided the fluid-flow models to demonstrate the tcp and queue dynamics. xu and sun [19] claimed that their model has the same value dropping probability at the operating point. the following model as we mentioned before this is a network with one bottleneck link, with only one tcp flow from each end host.   ẇi(t) = wi(t−r) r (1 −p(t−r)) 1 wi(t) − wi(t−r) r wi(t) 2 p(t−r), q̇(t) = −c + ∑n i=1 wi(t) r , (2.16) where r, wi(t), p(t) are like pervious section. using (2.1) in this model, (2.16) changed to:   v̇ (t) = v (t−r) r2v (t) (1 −p(t−r)) − 1 2 v (t−r)v (t)p(t−r), q̇(t) = −c + nv (t). (2.17) the operating point will be derived by v̇ (t) = 0 and q̇(t) = 0 and considering that (v,q) as a state and p as the input, therefore, v̇ (t) = 0 =⇒ p0 = 1 v 20 r 2 2 + 1 , (2.18) q̇(t) = 0 =⇒ v0 = c n . (2.19) now equation (2.12) is concluded as follows by substituting equation (2.11), (2.13) and (2.14): p(t) = [(1 −ω2)λ(t) + ω2λref (t + τ)]2 + nδλ(t) r2 −λ(t)λref (t + τ)( [(1−ω2)λ(t)+ω2λref(t+τ)]2 2n + n r2 ) λ(t)δ . (2.20) then the packet dropping probability will be as follows when the system is at the steady state, and the queue occupation, arrival rate and desired arrival rate are stabilized at qref and c (link capacity), respectively: p0 = 1 c2r2 2n2 + 1 . (2.21) as we mentioned this value is like the same value as low et al. [19] calculated in their model. for linearizing the above equation the right hand side of equation (2.17) can be written as follows:   f(v,vτ,pτ ) = vτ r2v (1 −pτ ) − 12vτv pτ, g(v ) = −c + nv, h(q̇,q) = [(1−ω2)λ+ω2λref]2+nδλ r2 −λλref( [(1−ω2)λ+ω2λref ] 2 2n +n/r2 ) λδ , (2.22) int. j. anal. appl. 16 (5) (2018) 660 where vτ (t) = v (t− τ) and λ, λref can be shown as:   λ(t) = ∑n i=1 vi(t) = q̇(t) + c, λref = c + (qref−q(t)) α . (2.23) deriving partial derivatives at the operating point in (2.23) we have: ∂f ∂v = − v0(1 −p0) r2v 20 − 1 2 v0p0 = − 1 −p0 r2v0 − 1 2 v0p0 = − n r2c c2r2 2n2 c2r2 2n2 + 1 − 1 2 c n 1 c2r2 2n2 + 1 = − c n 1 c2r2 2n2 + 1 , (2.24) ∂f ∂vτ = (1 −p0) r2v0 − 1 2 v0p0 = n r2c c2r2 2n2 c2r2 2n2 + 1 − 1 2 c n 1 c2r2 2n2 + 1 = 0, (2.25) ∂f ∂pτ = − 1 2 v 2 = − c2 2n2 , (2.26) ∂g ∂v = n, (2.27) ∂h ∂q̇ = ∂h ∂λ ∂λ ∂q̇ = ∂h ∂λ , (2.28) ∂h ∂q = ∂h ∂λref ∂λref ∂q = − 1 α ∂h ∂λref . (2.29) in order to obtain the two last equations, let a and b be the numerator and denominator of h, respectively. so,   ∂a ∂λ = 2(1 −ω2)c + nδr2 −c, ∂b ∂λ = (3c2 − 2ω22c2) δ 2n + nδ r2 , ∂a ∂λref = 2ω2c −c, ∂b ∂λref = ω2c 2 n . (2.30) now we have:   ∂h ∂λ = ∂a ∂λ b−∂b ∂λ a b2 , ∂h ∂λref = ∂a ∂λref b− ∂b ∂λref a b2 . (2.31) int. j. anal. appl. 16 (5) (2018) 661 thus the linearizing form is as follows:   δv̇ (t) = − c n ( c2r2 2n2 +1 )δv (t) − c2 2n2 δp(t−r), δq̇(t) = nδv (t), δp(t) = ∂p ∂λ δq̇(t) − 1 α ∂p∂λrefδq(t). (2.32) 3. designing control model of tcp/aqm model (2.22) can be formulated as:  ẋp = apx(t) + bpu(t−r) + bdu(t),yp = cpx(t), (3.1) in which x(t) = [δv (t) δq(t)], δp(t) = u(t), y(t) = δq(t) and ap =   − cn(c2r22n2 +1) 0 n 1 α d2 d1   , bp =   c22n2 0   , bd =   0 1 d1   , where d1 = ∂p ∂λ = ∂h ∂λ and d2 = ∂p ∂λref = ∂h ∂λref and also, δq̇ = ( ∂p ∂λ )−1δp(t) + 1 α ( ∂p ∂λref )−1δq(t) = 1 d1 δp(t) + 1 α d2 d1 δq. we can write the state-space as:   ẋp = apx(t) + bpu(t−r), yp = cpx(t), u(t) = dpẋ(t) + dcx(t), (3.2) where dp = [0 uλ], dc = [0 −1 αuref ] and ap =   − cn(c2r22n2 +1) 0 n 0   , bp =   c22n2 0   . int. j. anal. appl. 16 (5) (2018) 662 as we can see the model (3.1) is a state-dependent delay differential equation with delay in control. assuming this model uses a state feedback controller un(t) = knxp(t) i.e., un(t) = kn1δv (t) + kn2δq(t), this equation will changed to:   ẋp = apxp(t) + bpkpxp(t−r) + bdknxp(t),yp = cpxp(t), (3.3)   ẋp = (ap + bdkn)xp(t) + bpkpxp(t−r),yp = cpxp(t), (3.4) a =   − cn(c2r22n2 +1) 0 n + 1 d1 1 α d2 d1   , ad =   − c22n2 0   . for proofing the stability of state-dependent system which is describe by (3.4), the following theorems are applied whilst stability conditions will be constructed to insure the stability and give the variables of pervious system. first the lemma which is proposed by lin [17] is provided as follows and will be used in the proof of aforementioned theorems. lemma 3.1. if there is exist arbitrary matrices x11, x12, x13, x22, x23 and x33 such that x =   x11 x12 x13 xt12 x22 x23 xt13 x t 23 x33   ≥ 0, (3.5) then we obtain − ∫ t t−rrt(t) ẋt (s)x33ẋ(s)ds ≤ (3.6) ∫ t t−rrt(t) [xt (t) xt (t−rrt(t)) ẋt (s)]   x11 x12 x13 xt12 x22 x23 xt13 x t 23 0     x(t) x(t−rrt(t)) ẋ(s)  ds. theorem 3.1. let the scalars h > 0 and µ < 1, the state-dependent delay system (3.3) is asymptotically stable, if the condition (∇rrt.ẋ(t) = λ(t) < µ < 1) is satisfied for every t and there exist positive-definite int. j. anal. appl. 16 (5) (2018) 663 symmetric matrices p , q and r, and a semi positive-definite matrix x =   x11 x12 x13 x21 x22 x23 x31 x32 x33   , such that the following linear matrix inequality(lmi) holds ap =   e11 e12 ha tr ∗ e22 hatd r ∗ ∗ −hr   < 0, r−x33 ≥ 0, where e11 = a tp + pa + q + (1 −µ)(x13 + xt13 + hx11), e12 = pad + (1 −µ)(−x13 + xt23 + hx12), e22 = (1 −µ)(−q−x23 −xt23 + hx22). lyapunov stability theory are concept that applied to investigate the ability of these theorems. shevitz and paden were proceeded the lyapunov stability for nonsmooth systems [22] and also, zhang et al. [28] utilized feedback control system and provided the stability for networked control systems where their approach is used to establish the stability . also, lyapunov-krasovskii functional and schur complement are using to proof of theorem. proof: the lyapunov-krasovskii that we use here, is as follows: v (xt) = x t (t)px(t) + ∫ t t−rrt x t (s)qx(s)ds + ∫ 0 −rrt ∫ t t+θ ẋt (s)rẋ(s)dsdθ. (3.7) obviously, this lyapunov functional candidate is positive definite. now we calculate the derivative of this functional: v̇ (xt) = ẋ t (t)px(t) + xtpẋ(t) + xt (t)qx(t) − (1 − ˙rrt(x))qx(t−rrt(x)) +rrt(x)ẋt (t)rẋ(t) − (1 − ˙rrt(x)) ∫ t t−rrt(x) ẋt (s)rẋ(s)ds. (3.8) int. j. anal. appl. 16 (5) (2018) 664 by substituting (3.7) to (3.8) yields v̇ (xt) = x t (t)[atp + pa + q]x(t) + xt (t)padx(t−rrt(x)) + (1 −λ(t))xt (t−rrt(x))qx(t−rrt(x)) + rrt(x)ẋt (t)rẋ(t) − (1 − ˙rrt(x)) ∫ t t−rrt(x) ẋt (s)(r−x33)ẋ(s)ds − (1 − ˙rrt(x)) ∫ t t−rrt(x) ẋt (s)x33ẋ(s)ds. (3.9) the leibniz-newton formula used here and the assumption on ˙rrt(x) < µ and rrt(x) < h. so we have −(1 −µ) ∫ t t−rrt(x) ẋt (s)x33ẋ(s)ds ≤ (1 −µ) ∫ t t−rrt(x) [xt xt (t−rrt(x)) ẋ(s)]   x11 x12 x13 x21 x22 x23 x31 x32 x33     xt xt (t−rrt(x)) ẋ(s)   ≤ (1 −µ)hxt (t)x11x(t) + xthx12x(t−rrt(x)) + xtx13 ∫ t t−rrt(x) ẋ(s)ds + hxt (t−rrt(x))xt12x(t) + hx t (t−rrt(x))x22x(t−rrt(x)) + xt (t−rrt(x))x23 ∫ t t−rrt(x) ẋ(s)ds + xt13x(t) ∫ t t−rrt(x) ẋ(s)ds + xt23x(t−rrt(x)) ∫ t t−rrt(x) ẋ(s)ds = (1 −µ)[xt (t)hx11x(t) + xt (t)hx12x(t−rrt(x)) + xt (t)x13x(t) −x(t−rrt(x))xt13x(t) + x t (t−rrt(x))hxt12x(t) + xt (t−rrt(x))hx22x(t−rrt(x)) + xt (t−rrt(x))x23x(t) −xt (t−rrt(x))x23x(t−rrt(x)) + xt (t)xt13x(t) −x t (t)xt13x(t−rrt(x)) + x(t)x t 23x(t−rrt(x)) −x(t−rrt(x))x t 23x(t−rrt(x))]. by substituting in (3.9), following equation will be concluded: v̇ (xt) < ξ t (t)zξ(t) − (1 −µ) ∫ t t−rrt(x) ẋ t (s)(r−x33)ẋ(s)ds, (3.10) where ξt (t) = [xt xt (t−rrt(x))] and   z11 z12 0 z22   , int. j. anal. appl. 16 (5) (2018) 665 z11 = a tp + pa + q + (1 −µ)(x13 + xt13 + hx11) + ha tra, z12 = pad + (1 −µ)(−x13 + xt23 + hx12) + ha trad, z22 = (1 −µ)(−q−x23 −xt23 + hx22) + ha t d rad. in order to achieve to negative derivative of lyapunov functional, we use the schur complement, (i.e. z/z11 = z22 −z12z−111 < 0) and lmi conditionals. so the asymptotically stability will be derived. using the proof of this theorem and substituting aqm model on stability condition yields: v̇ (x) = xt (t) [ − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   t p + p   − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   + q]x(t) + xtp   c22n2 kp 0  x(t−rrt(x)) + xt (t−rrt(x))[ c2 2n2 kp 0 ] qx(t−rrt(x)) × rrt(x)ẋt (t)rẋ(t) − (1 − ˙rrt(x)) ∫ t t−rrt(x) ẋt (s)(r−x33)ẋ(s)ds − (1 − ˙rrt(x)) ∫ t t−rrt(x) ẋt (s)x33x(s)ds. so the stability of lyapunov functional, we obtain: v̇ (x) < [xt (t) xt (t−rrt(x))] ×   m n 0 f  × [xt xt (t−rrt(x))]t −(1 −µ) ∫ t t−rrt(x) ẋt (s)(r−x33)ẋ(s)ds < 0,   − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   t p + p   − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   + (1 −µ)(x13 + xt13 + hx11) +h   − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   t r   − cn(c2r22n2 +1) 0 (n + 1 d1 )kn 1 α d2 d1   < 0, where, m = atp + pa + (1 −µ)(x13 + xt13 + hx11) + ha tra, n = pad + (1 −µ)(x13 + xt23 + hx12) + harad, f = (1 −µ)(x23 + xt23 + hx22) + ha t d rad. int. j. anal. appl. 16 (5) (2018) 666 simple calculation leads the following equation: (n + 1 d1 )knp21 + 2( 1 α d2 d1 p22) + h( 1 α d2 d1 )2r22 < 2( c n(c 2r2 2n + 1) )p11 −h( c n(c 2r2 2n + 1) )2r11. (3.11) theorem 3.2. for any given h and µ < 1, the state dependent delay is asymptotically stabilized via the state feedback controller u(t) = k̃x(t), if the condition (∇rrt.ẋ(t) = λ(t) < µ < 1) satisfies for every t and there exists positive definite symmetric matrices w , u, g and a matrix y with appropriate dimensions and a semi-positive definite matrix, t =   t11 t12 t13 t21 t22 t23 t31 t32 t33   ≥ 0, such that the following lmi holds, γ =   γ11 γ12 hw   − cn(c2r2/2n2+1) 0 (n + 1 d1 )kn 1 α d2 d1   ∗ γ22   c22n2 kp 0   + hy btd ∗ ∗ −hg   < 0, and w −t33 ≥ 0, where, γ11 = w   − cn(c2r2/2n2+1) 0 (n + 1 d1 )kn 1 α d2 d1  t +   − cn(c2r2/2n2+1) 0 (n + 1 d1 )kn 1 α d2 d1  w + (1 −µ)(t13 + tt13 + ht11), γ12 =   c22n2 kp 0  w +   0 1 d1  y + (1 −µ)(−t13 + tt23 + ht12), γ22 = (1 −µ)(−u −t23 −tt23 + ht22), and a stabilizing gain will be driven by:   kp 0 0 kn   = y w−1. int. j. anal. appl. 16 (5) (2018) 667 proof: since matrix ad including controller gains, and its needs to be disassembled, so it should be calculate and we will be factorized matrix ad as,  c22n2 0 kp   . replacing in lmi condition and multiplying to the both sides of that by diagp−1,r−1, defining w = p−1, g = r−1, p−1qp−1 = u, k̃w = y , p−1xijp −1 = tij and [r −1 p−1][r −x33]t , p−1 = w −t33 lead to the equation in theorem and the state feedback controller gain then can be found from k̃ = y w−1 the rate at which the queue length grows when the buffer is nonempty in which athuraliya et al. [2] mentioned as mismatch. also, the aqm method uses state feedback, but vk −v states might be not available at routers and it makes some difficulties in real network. an approximation uses here to conquer this problem. the control signal for network is un(t) = knxn(t) which yields: δp(t) = knxn(t) = [kn1 kn2][δq δv ] t , (3.12) (3.13) as [29], the second term of the state vector is investigated: x2 = v −v0 = v − τ0c n = τ0 n ( v n τ0 −c), (3.14) τ0 n [flowrate−c] = τ0 n ∗rate of mismatch. (3.15) this mismatch can be approximated by q̇(t) and it turns to: δp(t) = kn1.(q −q0) + kn2. τ0 n q̇(t). (3.16) 4. numerical simulation some examples of control systems have been simulated here.the parameters ω1, ω2, α and δ will be calculated according to stability condition that we provided in pervious section. [27] determined that qref as reference queue length can be set 500 packets which is the desired value. also, they set r = 0.1889s and rtpt = 0.1s. following examples consider the different network condition, actually we alter the network condition . we will see in following examples that all these parameters satisfy the stability condition corresponding to these network systems. the network topology that we use here is dumb-bell which the link between router b and router c is the int. j. anal. appl. 16 (5) (2018) 668 figure 2. the network topology [27]. bottleneck link and it’s depicted in figure (2). tcp/reno is the policy that the sources use. all routers use droptail except router b where it exploit sfaqm to control queue length. we use the network control system with following features: • propagation delay is set to 200ms, • nominal load is 1000 sessions, • desired queue length is set 100 packets, • buffer capacity is set 300 packets. in following examples the parameters corresponding to sfaqm policy will be derived according to the stability condition proposed in this paper. it must mentioned that δ is set to 0.00625s. the results explain the effectiveness and good performance of the presented method and also show the these parameters speed of convergence of sfaqm under this stability condition. example 4.1. consider the following system which is proposed by huang and ngnuang [14] ap =   0 1 0 0.1   ,bp =   0 0.1   ,bd =   0 0.01   . applying control algorithm proposed in [3] and also considering the theorem (3.2) and relation (3.16) , the controller gains are derived as: up(t) = [−1.4176 − 5.6137]xp(t), un(t) = [0.0002 0.0072]xn(t). the state trajectories of the system which is shown in figure (3), is derived by applying the obtained control into the aforementioned network control system with initial values x0 = [1 1] t . also the queue length int. j. anal. appl. 16 (5) (2018) 669 figure 3. the state trajectories of the proposed system . figure 4. the queue length of network control system. figure 5. the drop probability as a control law. figure 6. the state dependent delay. is represented in figure (4). figure (5), shows the drop probability of system and state dependent delay is depicted in figure (6). the stability of the network control system by proposed method is noticeable by this figures. as shown, the queue length and state dependent delay keep stayed around value. the result is coincide with the result of azadegan et al. [3], which is represent the affectivity of our proposed method. we must mentioned that we use lmi toolbox at the matlab to obtain this control law. int. j. anal. appl. 16 (5) (2018) 670 example 4.2. extending the unstable system where proposed in [25], as follows: ẋ(t) =   −1 0 −0.5 1 −0.5 0 0 0 0.5  x(t) +   0 0 0.1  x(t−r) +   0 0 1   . , and by applying the theorem (3.2), the controlled gain is obtained: up(t) = [0 0 − 0.8762]xp(t), un(t) = [0.0003 0.0018]xn(t), with initial state x0 = [−5 0 5]t . same as [3] and [25], the state components convergence to zero in faster time as expected. it must be noticed that the asymptotic stability of the aforementioned system is provided by [14] and it’s consider the effect of transmission delay will affect the system. 5. conclusion the stabilizing feedback control based on the state-space method for the straightforward aqm system has been investigated in this paper. we provided the state-feedback control and to regulate the sfaqm dynamical system, we derived the explicit delay compensation structures. the robust stability was obtained by lyapunov-krasovskii method and the stabilization condition has been presented in the form of lmis. using ns2 software and matlab toolbox, simulation results have been verified the effectiveness and superiority of our method compared with other aqm schemes on queue stability. in the future work we will use optimal control theory and m −matrix form to induce more robust aqm schemes. references [1] a. e. abharian, h. khaloozadeh, and r. amjadifard, genetic-sigmoid random early detection covariance control as a jitter controller, iet control theory appl. 6 (2012), 327 334,. [2] s. athuraliya, s., s. low, v. li, and q. yin, rem: active queue management, ieee network magazine, 15 (2001), 48-53,. [3] m. azadegan, m. t. beheshti, and b. tavassoli, using aqm for performance improvement of networked control systems, int. j. control autom. syst. 13 (2015), 764-772. [4] t. azuma, t. fujita, and m. fujita, congestion control for tcp/aqm networks using state predictive control, electr. eng. japan, 156 (2006), 41-47. [5] h. w. bode, network analysis and feedback amplifier design, van nostrand, new york (1945). [6] r. v. churchill, brown, j. v., and verhy, r. f., complex variable and applications, mcgraw-hill, new york, (1976). [7] d. b. dacic, and d. nesic, quadratic stabilization of linear networked control systems via simultaneous protocol and controller design, automatica, 43 (2007), 1145-1155. int. j. anal. appl. 16 (5) (2018) 671 [8] s. l. dai, h. lin, and s. s. ge, scheduling and control codesign for a collection of networked control systems with uncertain delays, ieee trans.control syst. technol., 18 (2010), 66-78,. [9] s. floyd, and v. jacobson, random early detection gateways for congestion avoidance, ieee/acm trans. networking, 1 (1993), 397-413. [10] y. ge, j. wang, and c. li, robust stability conditions for dmc controller with uncertain time delay, int. j. control autom. syst., 12 (2) (2014), 241-250. [11] r. a. gupta, and m. y. chow, networked control system: overview and research trends, ieee trans.ind. electron., 57 (7) (2010), 2527-2535. [12] c. v. hollot, v. misra, d. towsley, and w. gong on designing improved controllers for aqm routers supporting tcp flows, proc. ieee infocom, 3 (2001), 1726-34. [13] c. v. hollot, v. misra, d. towsley, and w. gong analysis and design of controllers for aqm routers supporting tcp flows, ieee trans. autom. control, 47 (6) (2002), 945-959. [14] d. huang, and s. k. nguang, state feedback control of uncertain networked control systems with random time delays, ieee trans. autom. control, 53 (3) (2008), 829-834. [15] w. s. levine, the control handbook, crc press, ieee press, boca raton, new york , (1996). [16] h. li, and y. shi, network-based predictive control for constrained nonlinear systems with two-channel packet dropouts, ieee trans. ind. electron., 61 (10) (2014), 1574-1582,. [17] p. liu, robust exponential stability for uncertain time-varying delay systems with delay dependence, j. franklin inst. 346 (3) (2009), 958-968. [18] x. liu, and a. goldsmith, wireless network design for distributed control, 43rd ieee conf. on decision and control, pp. 2823-2829, (2004). [19] s. h. low, f. paganini, and j. c. doyle, internet congestion control, 43rd ieee control syst. pp. 28-43, (2002). [20] v. misra, w. gong, and d. towsley fluid-based analysis of network of aqm routers supporting tcp flows with an application to red, proc. acm/sigcomm, 30 (2000), 151-160. [21] h. naito, t. azuma, a. nishimura, and m.fujita, experimental verification of congestion controllers for tcp/aqm networks, ieej trans. eis, 124 (2004), 2093-2100. [22] shevitz, d. and b. paden, lyapunov stability theory of nonsmooth systems, ieee trans. autom. control, 39 (9) (1994), 1910-1914. [23] h. wang, c. liao, and z. tian, effective adaptive virtual queue: a stabilising active queue management algorithm for improving responsiveness and robustness, iet commun. 5 (2011), 99-109. [24] p. wang, h. chen, x. yang, and y. ma, design and analysis of a model predictive controller for active queue management, isa trans. 51 (2012), 120-131. [25] j. xiong, and j. lam, stabilization of networked control systems with a logic zoh, ieee trans. autom. control, 54 (2009), 358-363. [26] h. xu, a. sahoo, and s. jagannathan, stochastic adaptive event-triggered control and network scheduling protocol co-design for distributed networked systems, iet contr. theory appl. 8 (18) (2014), 2253-2265. [27] q. xu, and j. sun, a simple active queue management based on the prediction of the packet arrival rate, j. network comput. appl. 42 (2014), 12-20. [28] w. zhang, m. s. branicky, and s. m. phillips, stability of networked control systems, ieee control syst. mag. 21 (1) (2001), 84-99. int. j. anal. appl. 16 (5) (2018) 672 [29] p. zhang, c. ye, x. ma, y. chen, and x. li, using lyapunov function to design optimal controller, j. zhejiang univ. sci. a, 8 (2007), 113-118. [30] l. zhanga, and d. h. varsakelis, communication and control co-design for networked control systems, automatica, 42 (2006), 953-958. 1. introduction 2. prediction of the change of packet arrival rate 3. designing control model of tcp/aqm 4. numerical simulation 5. conclusion references int. j. anal. appl. (2023), 21:71 l∞-convergence analysis of a finite element linear schwarz alternating method for a class of semi-linear elliptic pdes qais al farei, messaoud boulbrachene∗ department of mathematics, sultan qaboos university, p.o. box 36, muscat 123, oman ∗corresponding author: boulbrac@squ.edu.om abstract. in this paper, we prove uniform convergence of the standard finite element method for a schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the context of linear iterations and non-matching grids. more precisely, making use of the subsolutionbased concept, we prove that finite element schwarz iterations converge, in the maximum norm, to the true solution of the pde. we also give numerical results to validate the theory. this work introduces a new approach and generalizes the one in [14] as it encompasses a larger class of problems. 1. introduction the schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. the solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. the literature in this area is huge and one can refer to [2], [3] and to proceedings of the annual international symposium on domain decomposition for partial differential equations, starting from [1]. the mathematical analysis of schwarz alternating method for nonlinear elliptic boundary value problems has been extensively studied in the last three decades (c.f., e.g., [2], [3], [5], [6] and the references therein). on the numerical analysis side and, more specifically, non-matching grid discretizations, to the best of our knowledge, only few works are known in the literature regarding the convergence and error estimates analysis for discrete schwarz procedures (c.f. [7], [8], [9], [10], [12], [15]). received: jan. 3, 2023. 2020 mathematics subject classification. 65n30, 65n15. key words and phrases. schwarz method; finite elements; non-matching grids; subsolutions; uniform convergence. https://doi.org/10.28924/2291-8639-21-2023-71 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-71 2 int. j. anal. appl. (2023), 21:71 the main motivation in using non-matching grid discretizations resides in their flexibility as they can be applied to solve many practical problems which cannot be handled by global discretizations because of the complexity of the domain’s geometry. they allow the choice of different mesh sizes and different orders of approximate polynomials in different subdomains according to the different properties of the solution and different requirements of the practical problems. in the present paper, we are interested in a non-matching grid finite element approximation method for the class of pdes   −∆u = f (x,u) in ωu = g on ∂ω (1.1) where ω ⊂rd,d = 2, 3 is a bounded domain with boundary ∂ω, ∆ is the laplace operator, f (.) is a smooth nonlinearity, and g is a regular function defined on ∂ω. to be more specific, let ω = ω1 ∪ ω2 such that ω1 ∩ ω2 6= ∅, γi = ∂ωi∩ ωj, γi = ∂ωi ∩∂ω and ∂ωi ; i = 1, 2, the boundary of ωi. let also c(x) be a positive smooth function. then following the work of s.h.lui [6], given initial smooth guesses u01 and u 0 2, we approximate the solution of problem (1.1) by schwarz sequences ( uni ) such that un1 ∈ c 2 ( ω̄1 ) , n ≥ 1 solves the linear subproblem   −∆un1 + cu n 1 = f (u n−1 1 ) + cu n−1 1 in ω1 un1 = u n−1 2 on γ1 un1 = g on γ1 (1.2) on ω1,and un2 ∈ c 2 ( ω̄2 ) , n ≥ 1 solves the linear subproblem  −∆un2 + cu n 2 = f (u n−1 2 ) + cu n−1 2 in ω2 un2 = u n 1 on γ2 un2 = g on γ2 (1.3) on ω2. in this paper, motivated by the uniform convergence result [6], lim n→∞ ‖uni −u‖l∞(ωi ) = 0, i = 1, 2, we prove that the corresponding finite elements schwarz sequences (un1h1 ) and (u n 2h2 ), generated in the context of non-matching grids, converge, in the maximum norm, to the exact solution of problem (1.1). that is, lim n→∞ ∥∥u −unihi∥∥l∞(ωi ) = 0, i = 1, 2, where, hi is the mesh-size on ωi, and ui = u/ωi . int. j. anal. appl. (2023), 21:71 3 to that end, by means of the concept of subsolutions, we establish a fundamental lemma which consists of estimating the error, at each iteration, between the continuous and the discrete schwarz iterations, on each subdomain. this work introduces a new approach and uses weaker assumptions on the nonlinearity than the one developed in [14] to derive the convergence result. the layout of the paper is as follows. in section 2, we recall some standard results related to linear elliptic boundary problems. in section 3, we recall the existence of a solution for the nonlinear pde, and define both the continuous and discrete variational formulations of subproblems (1.2) and (1.3). in section 4, we prove the main results of this paper. finally, in section 5, we give some numerical results to validate the theory. 2. preliminaries we begin by recalling some definitions and classical results related to linear elliptic equations. 2.1. linear elliptic equations. we introduce the bilinear form a(ξ,v) = ∫ ω (∇ξ.∇v + cξv)dx ∀v ∈ h1 (ω) , (2.1) the linear form (f ,v) = ∫ ω f (x).v(x)dx ∀v ∈ h1 (ω) , (2.2) where the right hand side f is a regular function, (2.3) and the space v(g) = {v ∈ h1 (ω) such that v = g on ∂ω}, (2.4) where g is a regular function defined on ∂ω. note that v̊ = h10 (ω). we consider the linear elliptic equation: find ξ ∈v(g) such that a (ξ,v) = (f ,v) , ∀v ∈ v̊(ω) (2.5) lemma 2.1. [6] (weak maximum principle) let w ∈ h1 (ω) ∩ c(ω̄) satisfy a(w,φ) ≥ 0 ∀ nonnegative φ ∈ v̊, and w ≥ 0 on ∂ω. then w ≥ 0 on ω̄. definition 2.1. a function ξ̌ ∈ h1 (ω) is a subsolution of (2.5) if{ a(ξ̌,v) ≤ (f ,v) ∀v ≥ 0,v ∈ v̊(ω) ξ̌ ≤ g (2.6) definition 2.2. a function ξ̂ ∈ h1 (ω) is a supersolution of (2.5) if{ a(ξ̂,v) ≥ (f ,v) ∀v ≥ 0,v ∈ v̊(ω) ξ̂ ≥ g (2.7) 4 int. j. anal. appl. (2023), 21:71 lemma 2.2. the solution ξ of (2.5) is the least upper bound of the set of subsolutions. proof. (2.6) can be re-written as a(−ξ̌,v) ≥ (−f ,v) ∀v ≥ 0,v ∈ v̊. subtracting this result from (2.5) yields a(ξ− ξ̌,v) ≥ 0 ∀v ≥ 0,v ∈ v̊. since ξ− ξ̌ ≥ 0 on ∂ω, it follows from lemma 2.1 that ξ̌ ≤ ξ on ω̄, which completes the proof. � the proposition below establishes a continuous lipschitz property of the solution with respect to the data. notation 2.1. let (f ,g) and (f̃ , g̃) be a pair of data, and ξ = ∂ (f ,g) and ξ̃ = ∂( f̃ , g̃) be the corresponding solutions to (2.5). proposition 2.1. [9] let β be a positive constant such that c/β ≥ 1. let also lemma 2.1 hold. then, ∥∥ξ− ξ̃∥∥ l∞(ω) ≤ max { 1 β ∥∥f − f̃∥∥ l∞(ω) ;‖g − g̃‖l∞(∂ω) } (2.8) let vh be the space of finite elements consisting of continuous piece-wise linear functions, φs, s = 1, 2, ...,m(h) be the basis functions of vh. let also v̊h be the subspace of vh defined by v̊h = {v ∈vh such that v = 0 on ∂ω} (2.9) the discrete counterpart of (2.5) consists of finding ξh ∈v (g) h such that a(ξh,v) = (f ,v) ∀v ∈ v̊h (2.10) where v(g) h = {v ∈vh such that v = πhg on ∂ω }, (2.11) and πh is the linear lagrange interpolation operator on ∂ω. the discrete version of lemma 2.1 stays true provided a discrete maximum principle (d.m.p) holds (the matrix resulting from the finite element discretization is an m-matrix). see [16]. lemma 2.3. let wh ∈vh satisfy a(wh,φs) ≥ 0 ∀s = 1, 2, ...,m(h) and wh ≥ 0 on ∂ω. then, under the d.m.p, we have wh ≥ 0 on ω̄. definition 2.3. a function ξ̌h ∈vh is a subsolution of (2.10) if  a(ξ̌h,φs) ≤ (f ,φs) ∀φs ≥ 0,∀s = 1, 2, ...,m(h) ξ̌h ≤ πhg (2.12) int. j. anal. appl. (2023), 21:71 5 definition 2.4. a function ξ̂h ∈vh is a supersolution of (2.10) if  a(ξ̂h,φs) ≥ (f ,φs) ∀φs ≥ 0,∀s = 1, 2, ...,m(h)ξ̂h ≥ πhg (2.13) lemma 2.4. let the d.m.p hold. then the solution ξh of (2.10) is the least upper bound of the set of subsolutions. proof. the proof is similar to that of lemma 2.2. indeed, as φs ≥ 0 are non-negative, it suffices to make use of lemma 2.3. � now we give the finite element counterpart of proposition 2.1. notation 2.2. let (f ,g) and (f̃ , g̃) be a pair of data, with ξh = ∂h(f ,g) and ξ̃h = ∂h(f̃ , g̃) be the corresponding discrete solutions to (2.10). proposition 2.2. [9] let β be a positive constant such that c/β ≥ 1. then, under the d.m.p and conditions of lemma 2.3, we have∥∥ξh − ξ̃h∥∥l∞(ω) ≤ max { 1 β ∥∥f − f̃∥∥ l∞(ω) ;‖g − g̃‖l∞(∂ω) } (2.14) finally, we recall a standard maximum norm error estimate [18]. theorem 2.1. [18] under suitable regularity of the solution of problem (2.5), there exists a constant c independent of h such that ‖ξ−ξh‖l∞(ω) ≤ ch 2 |ln h| 3. schwarz method for nonlinear pdes we first recall the following classical existence result due to pao [4]. 3.1. the nonlinear pde. we shall consider the following nonlinear pde: find u ∈ c2(ω) such that   −∆u = f (x,u) in ω u = g on ∂ω (3.1) for the sake of convenience, we will suppress the dependence of the space variable x. definition 3.1. [4] a function ǔ ∈ c2(ω) is a subsolution of (3.1) if  −∆ǔ ≤ f (ǔ) in ωǔ ≤ g on ∂ω (3.2) 6 int. j. anal. appl. (2023), 21:71 definition 3.2. [4] a function û ∈ c2(ω) is a supersolution of (3.1) if   −∆û ≥ f (û) in ω û ≥ g on ∂ω (3.3) suppose that (3.1) has a subsolution ǔ and a supersolution û such that ǔ ≤ û on ω. define the sector a = {u ∈ c2(ω̄); ǔ ≤ u ≤ û on ω̄}. (3.4) assume that −c (u −v) ≤ f (u) − f (v) ∀v ≤ u ∈a (3.5) then, thanks to pao [4], (3.1) has a solution (not necessarily unique) in a. theorem 3.1. [6] let u02 = ǔ on ω̄; i = 1, 2, with ǔ = 0 on ∂ω. define the linear schwarz sequences generated by the subproblems (1.2) and (1.3).then uni → u in c 2(ω̄i ), where u is a solution of (3.1) in a. similarly, if u02 = û on ω̄ with û = 0 on ∂ω instead, then the same conclusion holds. 3.2. continuous variational schwarz subproblems. the weak form of (1.2) and (1.3) read as follows: find un1 ∈ h 1 (ω) such that:   a1(u n 1,v) = (f (u n−1 1 ),v) ∀v ∈ v̊1 un1 = u n−1 2 on γ1 un1 = g on γ1, (3.6) and un2 ∈ h 1 (ω) such that   a2(u n 2,v) = (f (u n−1 2 ),v) ∀v ∈ v̊2 un2 = u n 1 on γ2 un2 = g on γ2 (3.7) respectively, where ai (ui,v) = ∫ ωi (∇ui∇v + cuiv)dx, (3.8) and (f (ui ),v) = ∫ ωi (f (ui ) + cui )vdx ; i = 1, 2. (3.9) int. j. anal. appl. (2023), 21:71 7 3.3. finite element discretization. let τhi ; i = 1, 2 be a standard quasi-uniform regular finite element triangulation on ωi ; hi being its mesh size. we introduce the finite element spaces vhi and v̊hi as follows: vhi = {v ∈ c 0(ω̄i ) : v/k ∈ p1 ∀k ∈τhi}, (3.10) and v̊hi = {v ∈vhi : v = 0 on γi}, (3.11) where p1 denotes the space of linear polynomials on k ∈τhi, with degree ≤ 1. the two meshes are also assumed to be overlaping and non-matching in the sense that they are mutually independent on the overlap region. the discrete maximum principle (d.m.p). we assume that the meshing on each subdomain satisfies the discrete maximum principle. in other words, the matrices resulting from the discretization of (3.6) and (3.7) are m-matrices. 3.4. discrete variational schwarz subproblems. let u0ihi be the discrete analog of u 0 i , i.e.; u 0 ihi = rhi (u 0 i ), where rhi denotes the finite element lagrange interpolation operator on ωi. now, we define the discrete schwarz sequences (un1h1 ) such that u n 1h1 ∈vh1 solves   a1(u n 1h1 ,v) = (f (un−1 1h1 ),v) ∀v ∈ v̊h1 un1h1 = πh1 (u n−1 2h2 ) on γ1 un1h1 = πhg on γ1, (3.12) and ( un2h2 ) such that un2h2 ∈vh2 solves   a2(u n 2h2 ,v) = (f (un−1 2h2 ),v) ∀v ∈ v̊h2 un2h2 = πh2 (u n 1h1 ) on γ2 un2h2 = πhg on γ2, (3.13) where πhi denotes the lagrange interpolation operator on γi. below, we construct a finite element discretization of subproblems (3.12) and (3.13), as in figure 1, using a quasi-uniform regular finite element triangulation on both subdomains as stated before. 8 int. j. anal. appl. (2023), 21:71 figure 1. a sample of two overlapping nonmatching grids. 4. l∞convergence analysis this section is devoted to the proof of the main results of the present paper. we first introduce two continuous and two discrete auxiliary schwarz sequences and prove a fundamental lemma. 4.1. continuous auxiliary schwarz subproblems. for ũ0i = u 0 i ; i = 1, 2, we define the continuous auxiliary schwarz sequence (ũn1 ) such that ũ n 1 ∈v1 solves  a1(ũ n 1,v) = (f (u n−1 1h1 ),v) ∀v ∈ v̊1 ũn1 = πh1 (u n−1 2h2 ) on γ1 ũn1 = πhg on γ1 (4.1) and ( ũn2 ) such that ũn2 ∈v2 solves   a2(ũ n 2,v) = (f (u n−1 2h2 ),v) ∀v ∈ v̊2 ũn2 = πh2 (u n 1h1 ) on γ2 ũn2 = πhg on γ2 (4.2) where un1h1 and u n 2h2 are the schwarz iterates defined in (3.12) and (3.13), respectively. int. j. anal. appl. (2023), 21:71 9 4.2. discrete auxiliary schwarz subproblems. likewise, for ũ0ihi = u 0 ihi ; i = 1, 2, we define the discrete auxiliary schwarz sequences (ũn1h1 ) such that ũ n 1h1 ∈vh1 solves  a1(ũ n 1h1 ,v) = (f (un−11 ),v) ∀v ∈ v̊h1 ũn1h1 = πh1 (u n−1 2 ) on γ1 ũn1h1 = πhg on γ1 (4.3) and ( ũn2h2 ) such that ũn2h2 ∈vh2 solves  a2(ũ n 2h2 ,v) = (f (un−12 ),v) ∀v ∈ v̊h2 ũn2h2 = πh2 (u n 1 ) on γ2 ũn2h2 = πhg on γ2 (4.4) where un1 and u n 2 are the schwarz iterates defined in (3.6) and (3.7), respectively. notation 4.1. from now onward, we shall adopt the following notations: c is ageneric constant independent of h and n, ‖.‖1 = ‖.‖l∞(ω1) ; |.|1 = ‖.‖l∞(γ1) , ‖.‖2 = ‖.‖l∞(ω2) ; |.|2 = ‖.‖l∞(γ2) , πh1 = πh2 = πh, and h = max i=1,2 hi. lemma 4.1. assume that max { ‖ũni ‖w 2,p(ωi ) , ‖u n i ‖w 2,p(ωi ) } ≤ c. then, we have ∥∥ũni −unihi∥∥l∞(ωi ) ≤ ch2 |ln h| , (4.5) ∥∥uni − ũnihi∥∥l∞(ωi ) ≤ ch2 |ln h| . (4.6) where c is a constant independent of both hi ; i = 1, 2 and n. proof. it is clear that unihi and ũ n ihi are the discrete counterparts of ũni and u n i , respectively. so, as the latter are both uniformly bounded in w 2,p(ωi ), the desired error estimates follows from theorem 2.1. � 10 int. j. anal. appl. (2023), 21:71 4.3. the main results. the following lemma plays a crucial role in deriving the main result of this paper. lemma 4.2. assume that f (.) is a lipschitz continuous function, i.e., there is a constant k > 0 such that |f (x) − f (y)| ≤ k |x −y| ∀x,y ∈r. (4.7) then, ‖un1 −u n 1h‖1 ≤ (2n)ch 2 |ln h| (4.8) and ‖un2 −u n 2h‖2 ≤ (2n + 1)ch 2 |ln h| . (4.9) remark 4.1. note that the assumption k/β ≤ 1 used in [14] is no longer needed in this paper. proof. the proof will be carried out by induction. also, for the sake of simplicity, we shall ignore the boundary condition on γi ; i = 1, 2. indeed, on ω1, problem (4.1) for n = 1 reads as follows  a1(ũ 1 1,v) = (f (u 0 1h),v) ∀v ∈ v̊1 ũ11 = πh(u 0 2h) on γ1. (4.10) as ũ11 is also a subsolution for (4.10), we have  a1(ũ 1 1,v) ≤ (f (u 0 1h),v) ∀v ∈ v̊1,v ≥ 0 ũ11 ≤ πh(u 0 2h) on γ1. but   a1(ũ 1 1,v) ≤ (f (u 0 1h) −f (u 0 1 ) + f (u 0 1 ),v) ∀v ∈ v̊1,v ≥ 0 ũ11 ≤ πh(u 0 2h) −πh(u 0 2 ) + πh(u 0 2 ) on γ1, then, since f (.) is lipschitz continuous and γ1 ⊂ ω2, this implies  a1(ũ 1 1,v) ≤ (c ∥∥u01 −u01h∥∥1 + f (u01 ),v) ∀v ∈ v̊1,v ≥ 0 ũ11 ≤ ∥∥u02 −u02h∥∥2 + πh(u02 ) on γ1. then, making use of standard uniform estimate, we have∥∥u0i − rh(u0i )∥∥i ≤ ch2 |ln h| ; i = 1, 2. (4.11) hence   a1(ũ 1 1,v) ≤ (f (u 0 1 ) + ch 2 |ln h| ,v) ∀v ∈ v̊1,v ≥ 0 ũ11 ≤ πh(u 0 2 ) + ch 2 |ln h| on γ1 int. j. anal. appl. (2023), 21:71 11 let ũ11 be the solution of the problem with source term f (u 0 1 ) +ch 2 |ln h| and boundary data πh(u02 ) + ch2 |ln h| . that is, ũ11 = ∂(f (u 0 1 ) + ch 2 |ln h| , πh(u02 ) + ch 2 |ln h|) then, as u11 = ∂(f (u 0 1 ) , u 0 2 ), making use of proposition 2.1, yields∥∥ũ11 −u11∥∥1 ≤ max {ch2 |ln h| ; ch2 |ln h|} ≤ ch2 |ln h| . hence, due to lemma 2.2, we have ũ11 ≤ ũ 1 1 ≤ u 1 1 + ch 2 |ln h| . putting α11 = ũ 1 1 −ch 2 |ln h| , we get α11 ≤ u 1 1. (4.12) and using (4.5), for n = 1, we also get∥∥ũ11 −u11h∥∥1 ≤ ch2 |ln h| . thus, ∥∥α11 −u11h∥∥1 = ∥∥ũ11 −ch2 |ln h|−u11h∥∥1 (4.13) ≤ ch2 |ln h| + ch2 |ln h| ≤ 2ch2 |ln h| . now, consider problem (4.3) for n = 1:  a1(ũ 1 1h,v) = (f (u 0 1 ),v) ∀v ∈ v̊1h ũ11h = πh(u 0 2 ) on γ1. (4.14) as ũ11h is also a subsolution for (4.14), we have  a1(ũ 1 1h,φs) ≤ (f (u 0 1 ),φs) ∀φs ≥ 0,∀s ũ11h ≤ πh(u 0 2 ) on γ1, which implies   a1(ũ 1 1h,φs) ≤ (f (u 0 1 ) −f (u 0 1h) + f (u 0 1h),φs) ∀φs ≥ 0,∀s ũ11h ≤ πh(u 0 2 ) −πh(u 0 2h) + πh(u 0 2h) on γ1. 12 int. j. anal. appl. (2023), 21:71 since f (.) and πh are lipschitz, we get  a1(ũ 1 1h,φs) ≤ (c ∥∥u01 −u01h∥∥1 + f (u01h),φs) ∀φs ≥ 0,∀s ũ11h ≤ ∥∥ u02 −u02h∥∥2 + πh(u02h) on γ1. hence, using (4.11), yields  a1(ũ 1 1h,φs) ≤ (f (u 0 1h) + ch 2 |ln h| ,φs) ∀φs ≥ 0,∀s ũ11h ≤ πh(u 0 2h) + ch 2 |ln h| on γ1. let ũ11h be the solution of the problem with source term f (u 0 1h) + ch 2 |ln h| and boundary data πh(u 0 2h) + ch 2 |ln h|, that is, ũ11h = ∂h(f (u 0 1h) + ch 2 |ln h| , πh(u02h) + ch 2 |ln h|) then, as u11h = ∂h(f (u 0 1h) , πh(u 0 2h)), making use of proposition 2.2, yields ∥∥ũ11h −u11h∥∥1 ≤ max {ch2 |ln h| ; ch2 |ln h|} ≤ ch2 |ln h| , and due to lemma 2.4, we have ũ11h ≤ ũ 1 1h ≤ u 1 1h + ch 2 |ln h| . now, putting α11h = ũ 1 1h −ch 2 |ln h| , it follows that α11h ≤ u 1 1h. (4.15) and making use of (4.6) for n = 1, we get ∥∥u11 − ũ11h∥∥1 ≤ ch2 |ln h| . thus, ∥∥α11h −u11∥∥1 = ∥∥ũ11h −ch2 |ln h|−u11∥∥1 . (4.16) ≤ ch2 |ln h| + ch2 |ln h| ≤ 2ch2 |ln h| int. j. anal. appl. (2023), 21:71 13 now, combining (4.12), (4.13), (4.15) and (4.16), we get u11 ≤ α 1 1h + 2ch 2 |ln h| ≤ u11h + 2ch 2 |ln h| ≤ α11 + 4ch 2 |ln h| ≤ u11 + 4ch 2 |ln h| . that is, ∥∥u11 −u11h∥∥1 ≤ 2ch2 |ln h| . (4.17) similarly on ω2, for n = 1 in (4.2), we have  a2(ũ 1 2,v) = (f (u 0 2h),v) ∀v ∈ v̊2 ũ12 = πh(u 1 1h) on γ2. (4.18) the solution ũ12 is also a subsolution for (4.18). that is,  a2(ũ 1 2,v) ≤ (f (u 0 2h),v) ∀v ∈ v̊2,v ≥ 0 ũ12 ≤ πh(u 1 1h) on γ2, or   a2(ũ 1 2,v) ≤ (f (u 0 2h,v) −f (u 0 2,v) + f (u 0 2 ),v) ∀v ∈ v̊2,v ≥ 0 ũ12 ≤ πh(u 1 1h) −πh(u 1 1 ) + πh(u 1 1 ) on γ2. as f (.) is lipschitz continuous function and γ2 ⊂ ω1, this implies  a2(ũ 1 2,v) ≤ ( ∥∥u02 −u02h∥∥2 + f (u02 ),v) ∀v ∈ v̊2,v ≥ 0 ũ12 ≤ ∥∥u11 −u11h∥∥1 + πh(u11 ) on γ2 using (4.11) and the resulting estimate (4.17), we obtain  a2(ũ 1 2,v) ≤ (f (u 0 2 ) + ch 2 |ln h| ,v) ∀v ∈ v̊2,v ≥ 0 ũ12 ≤ πh(u 1 1 ) + 2ch 2 |ln h| on γ2. let ũ12 be the solution of the equation with source term f (u 0 2 ) +ch 2 |ln h| and boundary data πh(u11 ) + 2ch2 |ln h| , that is, ũ12 = ∂(f (u 0 2 ) + ch 2 |ln h| , πh(u11 ) + 2ch 2 |ln h|). then, as u12 = ∂(f (u 0 2 ) , u 1 1 ), making use of proposition 2.1, we get∥∥ũ12 −u12∥∥2 ≤ max {ch2 |ln h| ; 2ch2 |ln h|} ≤ 2ch2 |ln h| . 14 int. j. anal. appl. (2023), 21:71 also, due to lemma 2.2, we have ũ12 ≤ ũ 1 2 ≤ u 1 2 + 2ch 2 |ln h| now, putting α12 = ũ 1 2 − 2ch 2 |ln h| , yields α12 ≤ u 1 2. (4.19) and due to (4.5) for n = 1, we have ∥∥ũ12 −u12h∥∥2 ≤ ch2 |ln h| . thus, it follows that ∥∥α12 −u12h∥∥2 = ∥∥ũ12 − 2ch2 |ln h|−u12h∥∥2 (4.20) ≤ ∥∥ũ12 −u12h∥∥1 + 2ch2 |ln h| ≤ 3ch2 |ln h| . again, on ω2, for n = 1 in (4.4), we have  a2(ũ 1 2h,v) = (f (u 0 2 ),v) ∀v ∈ v̊2h ũ12h = πh(u 1 1 ) on γ2. (4.21) the solution ũ12h being also a subsolution, we have  a1(ũ 1 2h,φs) ≤ (f (u 0 2 ),φs) ∀φs ≥ 0,∀s ũ12h ≤ πh(u 1 1 ) on γ2. then, as f (.) and πh are lipschitz, we get  a1(ũ 1 2h,φs) ≤ (c ∥∥u02 −u02h∥∥2 + f (u02h),φs) ∀φs ≥ 0,∀s ũ12h ≤ ∥∥u11 −u11h∥∥1 + πh(u11h) on γ2, or   a1(ũ 1 2h,φs) ≤ (f (u 0 2h) + ch 2 |ln h| ,φs) ∀φs ≥ 0 ũ12h ≤ πh(u 1 1h) + 2ch 2 |ln h| on γ2. hence, ũ12h is a subsolution for the problem with source term f (u 0 2h) + ch 2 |ln h| and boundary term πh(u 1 1h) + 2ch 2 |ln h|. let ũ12h be the solution of such a problem, that is, ũ12h = ∂h(f (u 0 2h) + ch 2 |ln h| , πh(u11h) + 2ch 2 |ln h|) then, we have ũ12h ≤ ũ 1 2h. int. j. anal. appl. (2023), 21:71 15 as u12h = ∂h(f (u 0 2h) , πh(u 1 1h)), making use of proposition 2.2, we get∥∥ũ12h −u12h∥∥2 ≤ max {ch2 |ln h| ; 2ch2 |ln h|} ≤ 2ch2 |ln h| . so, due to lemma 2.4, we have ũ12h ≤ ũ 1 2h ≤ u 1 2h + 2ch 2 |ln h| now, putting α12h = ũ 1 2h − 2ch 2 |ln h| , yields α12h ≤ u 1 2h. (4.22) and making use of (4.6) for n = 1, we get∥∥α12h −u12∥∥2 = ∥∥ũ12h − 2ch2 |ln h|−u12∥∥2 (4.23) (4.24) ≤ 3ch2 |ln h| . now, combining statements (4.19), (4.20), (4.22) and (4.23), we obtain u12 ≤ α 1 2h + 3ch 2 |ln h| ≤ u12h + 3ch 2 |ln h| ≤ α12 + 6ch 2 |ln h| ≤ u12 + 6ch 2 |ln h| . that is, ∥∥u12 −u12h∥∥2 ≤ 3ch2 |ln h| . (4.25) now, for n = 2 on ω1, (4.1) reads  a1(ũ 2 1,v) = (f (u 1 1h),v) ∀v ∈ v̊1 ũ21 = πh(u 1 2h) on γ1. (4.26) as ũ21 is also a subsolution, we have  a1(ũ 2 1,v) ≤ (f (u 1 1h),v) ∀v ∈ v̊1,v ≥ 0 ũ21 ≤ πh(u 1 2h) on γ1. 16 int. j. anal. appl. (2023), 21:71 and, since f (.) is lipschitz, we get  a1(ũ 2 1,v) ≤ (f (u 1 1 ) + 2ch 2 |ln h| ,v) ∀v ∈ v̊1 ũ21 ≤ πh(u 1 2 ) + 3ch 2 |ln h| on γ1. so, ũ21 is a subsolution for the problem with source term f (u 1 1 ) + 2ch 2 |ln h| and boundary term πh(u 1 2 ) + 3ch 2 |ln h|. let ũ21 be the solution of such a problem. that is, ũ21 = ∂(f (u 1 1 ) + 2ch 2 |ln h| , πh(u12 ) + 3ch 2 |ln h|). then, due to lemma 2.2, we have ũ21 ≤ ũ 2 1. furthermore, as u21 = ∂(f (u 1 1 ) , u 1 2 ), making use of proposition 2.1, we get∥∥ũ21 −u21∥∥1 ≤ max {2ch2 |ln h| ; 3ch2 |ln h|} ≤ 3ch2 |ln h| . hence, ũ21 ≤ ũ 2 1 ≤ u 2 1 + 3ch 2 |ln h| . putting α21 = ũ 2 1 − 3ch 2 |ln h| , yields α21 ≤ u 2 1. (4.27) making use of (4.5) for n = 2, we get∥∥α21 −u21h∥∥1 = ∥∥ũ21 − 3ch2 |ln h|−u21h∥∥1 (4.28) ≤ ch2 |ln h| + 3ch2 |ln h| ≤ 4ch2 |ln h| . now for n = 2 on ω1, (4.3) reads  a1(ũ 2 1h,v) = (f (u 1 1 ),v) ∀v ∈ v̊1h ũ21h = πh(u 1 2 ) on γ1. (4.29) as ũ21h is also a subsolution, we have  a1(ũ 2 1h,φs) ≤ (f (u 1 1 ),φs) ∀φs ≥ 0,∀s ũ21h ≤ πh(u 1 2 ) on γ1. int. j. anal. appl. (2023), 21:71 17 similarly, as above, this implies  a1(ũ 2 1h,φs) ≤ (f (u 1 1h) + 2ch 2 |ln h| ,φs) ∀φs ≥ 0,∀s ũ21h ≤ πh(u 1 2h) + 3ch 2 |ln h| on γ1. and, due to lemma 2.4, ũ21h ≤ ũ 2 1h = ∂h(f (u 1 1h) + 2ch 2 |ln h| , πh(u12h) + 3ch 2 |ln h|) but u21h = ∂h(f (u 1 1h) , πh(u 1 2h)) then, using proposition 2.2, yields the estimate∥∥ũ21h −u21h∥∥1 ≤ max {2ch2 |ln h| ; 3ch2 |ln h|} ≤ 3ch2 |ln h| . hence ũ21h ≤ ũ 2 1h ≤ u 2 1h + 3ch 2 |ln h| . putting α21h = ũ 2 1h − 3ch 2 |ln h| , it follows that α21h ≤ u 2 1h, (4.30) and, (4.6) for n = 2, implies that∥∥α21h −u21∥∥1 = ∥∥ũ21h − 3ch2 |ln h|−u21∥∥1 (4.31) ≤ 4ch2 |ln h| . combining (4.27), (4.28), (4.30) and (4.31), we obtain u21 ≤ α 2 1h + 4ch 2 |ln h| ≤ u21h + 4ch 2 |ln h| ≤ α21 + 8ch 2 |ln h| ≤ u21 + 8ch 2 |ln h| . thus, ∥∥u21 −u21h∥∥1 ≤ 4ch2 |ln h| . (4.32) similarly, for n = 2 on ω2,(4.2) we have  a2(ũ 2 2,v) = (f (u 1 2h),v) ∀v ∈ v̊2 ũ22 = πh(u 2 1h) on γ2. 18 int. j. anal. appl. (2023), 21:71 using a similar argument as above, one can prove that ũ22 is a subsolution for the problem with source term f (u12 ) + 3ch 2 |ln h| and boundary condition πh(u21 ) + 4ch 2 |ln h|. let ũ22 be the solution of such a problem, that is ũ22 = ∂(f (u 1 2 ) + 3ch 2 |ln h| , πh(u21 ) + 4ch 2 |ln h|), then, as u22 = ∂(f (u 1 2 ) , u 2 1 ), making use of proposition 2.1, we get∥∥ũ22 −u22∥∥2 ≤ 4ch2 |ln h| . putting α22 = ũ 2 2 − 4ch 2 |ln h| , we obtain α22 ≤ u 2 2, (4.33) and, making use of (4.5) for n = 2, we get∥∥α22 −u22h∥∥2 = ∥∥ũ22 − 4ch2 |ln h|−u22h∥∥1 (4.34) ≤ 5ch2 |ln h| . likewise, for n = 2 on ω2, we can also establish that α22h ≤ u 2 2h (4.35) and ∥∥α22h −u22∥∥2 ≤ 5ch2 |ln h| (4.36) hence, combining (4.33), (4.34), (4.35) and (4.36), we obtain∥∥u22 −u22h∥∥2 ≤ 5ch2 |ln h| . (4.37) now, let us assume that (4.8) and (4.9) hold. we need to prove it for the (n + 1)th step. indeed, consider the problem   a1(ũ n+1 1 ,v) = (f (u n 1h),v) ∀v ∈ v̊1 ũn+11 = πh(u n 2h) on γ1. then, we also have   a1(ũ n+1 1 ,v) ≤ (f (u n 1h),v) ∀v ∈ v̊1,v ≥ 0 ũn+11 ≤ πh(u n 2h) on γ1, which can be rewritten as  a1(ũ n+1 1 ,v) ≤ (f (u n 1h) −f (u n 1 ) + f (u n 1 ),v) ∀v ∈ v̊1,v ≥ 0 ũn+11 ≤ πh(u n 2h) −πh(u n 2 ) + πh(u n 2 ) on γ1. int. j. anal. appl. (2023), 21:71 19 since f (.) is lipschitz continuous, γ1 ⊂ ω2, this implies that  a1(ũ n+1 1 ,v) ≤ (f (u n 1 ) + (2n)ch 2 |ln h| ,v) ∀v ∈ v̊1,v ≥ 0 ũn+11 ≤ πh(u n 2 ) + (2n + 1)ch 2 |ln h| on γ1. this means that ũn+11 is a subsolution for the problem with source term f (u n 1 ) + (2n)ch 2 |ln h| and boundary term πh(un2 ) + (2n + 1)ch 2 |ln h| . let ũn+11 be the solution of such a problem. that is, ũn+11 = ∂ ( f (un1 ) + (2n)ch 2 |ln h| , πh(un2 ) + (2n + 1)ch 2 |ln h| ) . then, making use of lemma 2.2, we have ũn+11 ≤ ũ n+1 1 = ∂(f (u n 1 ) + (2n)ch 2 |ln h| , πh(un2 ) + (2n + 1)ch 2 |ln h|) but un+11 = ∂(f (u n 1 ) , u n 2 ), then, making use of proposition 2.1, we have∥∥ũn+11 −un+11 ∥∥1 ≤ (2n + 1)ch2 |ln h| , and due to lemma 2.2, ũn+11 ≤ ũ n+1 1 ≤ u n+1 1 + (2n + 1)ch 2 |ln h| . putting αn+11 = ũ n+1 1 − (2n + 1)ch 2 |ln h| , (4.38) yields αn+11 ≤ u n+1 1 . and, using (4.5), we get ∥∥αn+11 −un+11h ∥∥1 ≤ (2n + 1)ch2 |ln h| . (4.39) the solution of (4.3) is also a subsolution:  a1(ũ n+1 1h ,φs) ≤ (f (un1 ),φs) ∀φs ≥ 0,∀s ũn+1 1h ≤ πh(un2 ) on γ1, which, in turn, can be rewritten as  a1(ũ n+1 1h ,φs) ≤ (f (un1 ) −f (u n 1h) + f (u n 1h),φs) ∀φs ≥ 0,∀s ũn+1 1h ≤ πh(un2 ) −πh(u n 2h) + πh(u n 2h) on γ1. since f (.) is lipschitz continuous, γ1 ⊂ ω2, this implies  a1(ũ n+1 1h ,φs) ≤ (f (un1h) + (2n)ch 2 |ln h| ,φs) ∀φs ≥ 0,∀s ũn+1 1h ≤ πh(un2h) + (2n + 1)ch 2 |ln h| on γ1. 20 int. j. anal. appl. (2023), 21:71 in other words, ũn+1 1h is a subsolution for the problem with data f (un1h) + (2n)ch 2 |ln h| and πh(un2h) + (2n + 1)ch2 |ln h| . let ũn+1 1h be the solution of such a problem. that is, ũn+11h = ∂h(f (u n 1h) + (2n)ch 2 |ln h| , πh(un2h) + (2n + 1)ch 2 |ln h|). but, as un+11h = ∂h(f (u n 1h) , πh(u n 2h)), making of proposition 2.2, we get∥∥ũn+11h −un+11h ∥∥1 ≤ (2n + 1)ch2 |ln h| . and so, due to lemma 2.4, we have ũn+11h ≤ ũ n+1 1h ≤ u n+1 1h + (2n + 1)ch 2 |ln h| . now putting αn+11h = ũ n+1 1h − (2n + 1)ch 2 |ln h| , (4.40) and using (4.6), we obtain ∥∥αn+11h −un+11 ∥∥1 ≤ 2(n + 1)ch2 |ln h| . (4.41) hence, similarly to above, combining (4.38), (4.39), (4.40) and (4.41), we obtain∥∥un+11 −un+11h ∥∥1 ≤ 2(n + 1)ch2 |ln h| . (4.42) which is the desired result in ω1. likewise, the estimate for the iterate n + 1 in ω2 can be proved using similar arguments as above, which yields ∥∥un+12 −un+12h ∥∥1 ≤ (2n + 3)ch2 |ln h| . � corollary 4.1. ∀n ≥ 1 fixed, we have lim h→0 ‖uni −u n ih‖i = 0, i = 1, 2. proof. the proof is straightforward. for fixed n ≥ 1, passing to the limit as h → 0 to both (4.8) and (4.9), the corollary follows on both subdomains. � now, we are in position to prove the following convergence result: corollary 4.2. there exists hn > 0 with hn → 0, such that lim n→∞ ∥∥ui −unihn∥∥i = 0; i = 1, 2. (4.43) int. j. anal. appl. (2023), 21:71 21 proof. let us give the proof of (4.43) on ω1, the one on ω2 is similar. we know that ‖u1 −un1h‖1 ≤‖u1 −u n 1‖1 + ‖u n 1 −u n 1h‖1 letting � > 0, theorem 3.1 implies that there exists n ∈n such that ‖u1 −un1‖1 ≤ � 2 ∀n > n hence, due to (4.8), we have ‖un1 −u n 1h‖1 ≤ (2n)ch 2 |ln h| , thus, the convergence result follows by choosing hn > 0 such that h2n |ln hn| ≤ � 4cn ∀n > n. � 5. numerical experiments in this section, we conduct numerical experiments on two model problems to validate the theory. the first model is chosen so that it does not have an exact solution, while we know an exact solution for the second one. for both models, we adopt the following notations: • hi ; i = 1, 2, are the mesh sizes of the triangulations in ωi. • δ is the size of overlap between both subdomains. • errorhi = ∥∥∥u −unihi∥∥∥i is the maximum error between the exact solution u and the discrete schwarz iterate on each subdomain. we shall conduct the two tests to investigate the behavior of errorhi as follows: (1) we fix the mesh sizes hi and vary the number of schwarz iterations n, (2) we fix the number of iterations n and vary the mesh sizes hi, (3) we consider the sequence of mesh sizes hi,n as n varies. for all the experiments, we consider ω = [0, 1] × [0, 1] . the "freefem++" software, see [19], is adapted to obtain the numerical results for both models. 5.1. first example: in this example, we consider the boundary value problem  −∆u = −σu 1 + au + bu2 in ω u = x + y on ∂ω (5.1) where σ = 1,a = 0 and b = 0.25. this problem describes the enzyme kinetics model with inhibition. the value of the constant c is defined by determining suitable lower and upper solutions to the problem 22 int. j. anal. appl. (2023), 21:71 (5.1) satisfying definitions (3.2) and (3.3), respectively. the sector 〈ǔ, û〉 = 〈0, 12〉 is taken and c is evaluated by [4]: c = max { − ∂f ∂u ; x ∈ ω̄, ǔ ≤ u ≤ û } . (5.2) for c = 1, f (u) satisfies the one-sided lipshitz condition f (u1) − f (u2) ≥−(u1 −u2) for ǔ ≤ u2 ≤ u1 ≤ û. (5.3) a unique positive solution for the above model is also ensured in the sector 〈0, 2〉, (see [4]). since it is difficult to obtain the exact solution for the problem, we use a p2− finite element approximation of the exact solution of the problem on ω, instead. figure 2 represents the solution u of the above problem using a uniform fixed mesh size of 1 30 . figure 2. p2−approximate solution we divide ω into two overlapping non-matching subdomains ω1 and ω2 such that each subdomain is independently discretized into a quasi-uniform mesh with p1 triangular elements and different mesh size hi ; i = 1, 2. in order for the maximum principle to be satisfied here, we construct a triangulation with acute angles for every k ∈τhi ; i = 1, 2, using a delaunay triangulation algorithm. when both mesh sizes are fixed to be h1 = 1 32 and h2 = 1 24 with two sizes of the overlap δ = 1 8 and δ = 1 4 , the approximated solution of the 35th iterate are represented in figure 3, respectively. this shows the convergence of schwarz sequences unihi to u. the same result can be easily shown for different mesh refinements. furthermore, tables 1 and 2 represent the approximate solution values at some points in the domain with a stopping criterion ε = e−5 for both subdomains. the obtained information of the tables show the monotone convergence of the schwarz sequences unihi , where u1h1 and u2h2 are the p2approximation of the nonlinear pde problem on ω1 and ω2, respectively. it is also seen that the number of schwarz iterations decreases as the overlap size increases. int. j. anal. appl. (2023), 21:71 23 table 1. approximate solution values at some points when δ = 1 8 n u1h1 ( 1 4 , 1 2 ) u1h1 ( 1 2 , 1 2 ) u2h2 ( 2 3 , 1 4 ) u2h2 ( 3 4 , 2 3 ) 0 0 0 0 0 1 0.345889 0.0817873 0.642487 1.1308 2 0.572373 0.536428 0.814077 1.32854 3 0.689899 0.809101 0.890129 1.40256 4 0.745107 0.943227 0.926039 1.43526 5 0.770443 1.00566 0.942779 1.44996 table 2. approximate solution values at some points when δ = 1 4 n u1h1 ( 1 4 , 1 2 ) u1h1 ( 1 2 , 1 2 ) u2h2 ( 2 3 , 1 4 ) u2h2 ( 3 4 , 2 3 ) 0 0 0 0 0 1 0.390914 0.16399 0.741398 1.23655 2 0.677421 0.800115 0.902063 1.41263 3 0.762939 0.992884 0.943188 1.45016 4 0.784635 1.04217 0.953611 1.45921 (a) (b) (c) (d) figure 3. iterative process of the first example on both ω1 and ω2. approximate solution at 35th iteration when (a)δ = 1 8 and (b) δ = 1 4 . maximum errors versus number of iterations (c) and versus meshsizes (d). 24 int. j. anal. appl. (2023), 21:71 in the first place, when we put h1 = 1 32 , h2 = 1 24 with both δ = 1 8 and δ = 1 4 as before, and vary the number of schwarz iterations over 1 ≤ n ≤ 35, one can observe how the maximum errors decrease as the number of schwarz iterations and the overlap size increase. next, we fix the number of iterations to be n = 8 when δ = 1 8 , n = 5 when δ = 1 4 and vary the mesh sizes to be h1 = 1 4×2n , h2 = 1 3×2n when 1 ≤ n ≤ 5, instead. one can notice that the maximum errors decrease as the mesh sizes get smaller. also, the bigger overlap size is, the smaller errors and the closer the curves are. figure 3 shows both plots of errorhi . 5.2. second example: we consider the following problem  −∆u = σup in ω u = 12 (x+y+1) 2 on ∂ω (5.4) where σ = −1 and p = 2. this problem describes the concentration of free atoms in the dissociation process. one can verify that the function f (u) satisfies the one-sided lipschitz condition (5.3). also, the value of c satisfying (5.2) is 24. hence, there exists a positive solution for the model in the sector 〈0, 12〉 . furthermore, one can verify that the exact solution of the model is given by u = 12 (x + y + 1) 2 the exact solution u is represented in figure 4 using a uniform fixed mesh size of 1 30 . figure 4. exact solution in this example, we build the same triangulation as in the first experiment in order to satisfy the maximum principle. we also examine the performance of the iterative approach for different values of the number of iterations, with only overlap size of δ = 1 8 , by doing a similar analysis to the one made in the first example. the numerical results are shown in figure 5, where the first and second figures represent the approximate solution at the initial and 35th iteration, while the third and fourth ones display the relationship of maximum errors with number of iterations and mesh sizes, respectively. int. j. anal. appl. (2023), 21:71 25 moreover, the approximated solution values at some certain points in the domain with the same stopping criterion as in the first example for both subdomains are represented in table 3. we notice the monotone convergence of the schwarz sequences unihi . (a) (b) (c) (d) figure 5. iterative process of the second example on both ω1 and ω2. (a) approximate solution at first iteration. (b) approximate solution at 35th iteration. (c) maximum errors versus number of iterations with fixed mesh sizes. (d)maximum errors versus meshsizes with fixed number of iterations. table 3. approximate solution values at some points when δ = 1 8 n u1h1 ( 1 4 , 1 2 ) u1h1 ( 1 2 , 1 2 ) u2h2 ( 2 3 , 1 4 ) u2h2 ( 3 4 , 2 3 ) 0 0 0 0 0 1 3.31901 0.709984 2.71408 1.64465 2 3.685 2.14238 3.13503 1.93498 3 3.86836 2.7503 3.23898 2.02499 4 3.90591 2.9386 3.26161 2.04795 5 3.9165 2.98647 3.2672 2.05378 6 3.91889 2.99829 3.26853 2.0552 7 3.9195 3.00117 3.26885 2.05555 26 int. j. anal. appl. (2023), 21:71 we conclude this section by validating the convergence result (corollary 4.2). applying the context of it to both examples with mesh sizes of h1,n = 1 2n+1 and h2,n = 1 3n+1 on both subdomains when 1 ≤ n ≤ 35, we see in figure 6 that ∥∥∥ui −unihi,n∥∥∥i ≤ 6n2 ; i = 1, 2,∀n in particular, the asymptotic behavior of our iterative approach is indicated to be at least o( 1 n2 ). this proves that our numerical results are in agreement with our theory. (a) (b) (c) (d) figure 6. plots of the maximum errors for both examples by considering the meshsize sequences hi,n for 1 ≤ n ≤ 35. (a) first maximum error for example 1. (b) second maximum error for example 1. (c) first maximum error for example 2. (d) second maximum error for example 2. int. j. anal. appl. (2023), 21:71 27 6. conclusion in this paper, we have proved the convergence of the standard finite element approximation of monotone linear schwarz alternating procedure for a class of semilinear elliptic pdes, in the context of non-matching grids. in order to prove the main result, we used the concept of subsolutions to estimate, at each schwarz iteration, the gap between the continuous and approximated schwarz sequences. we have also conducted numerical experiments to show the agreement with the theory. we believe that the availability of a rate of convergence of the schwarz procedure will help to derive an error estimate between the discrete schwarz sequence and the exact solution of the semilinear pde on each subdomain. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. glowinski, g.h. golub, g.a. meurant, j. periaux, domain decomposition methods for partial differential equations, in: proceedings of 1st international symposium on domain decomposition methods for partial differential equations, siam, philadelphia, pa, usa, (1988). [2] p.l. lions, on the schwarz alternating method i, in: proceedings of 1st international symposium on domain decomposition methods for partial differential equations, siam, philadelphia, pa, usa, pp. 1-40, (1988). [3] p.l. lions, on the schwarz alternating method ii, in: proceedings of the 2nd international symposium on domain decomposition methods for partial differential equations, siam, philadelphia, pa, usa, pp. 47-70, (1989). [4] c.v. pao, nonlinear parabolic and elliptic equations, pienium press, new york, (1992). [5] s.h. lui, on schwarz alternating methods for nonlinear elliptic pdes, siam j. sci. comput. 21 (1999), 15061523. https://doi.org/10.1137/s1064827597327553. [6] s.h. lui, on linear monotone iteration and schwarz methods for nonlinear elliptic pdes, numer. math. 93 (2002), 109-129. https://doi.org/10.1007/bf02679439. [7] x.c. cai, t.p. mathew, m.v. sarkis, maximum norm analysis of overlapping nonmatching grid discretizations of elliptic equations, siam j. numer. anal. 37 (2000), 1709-1728. https://doi.org/10.1137/s0036142998348741. [8] m. boulbrachene, s. saadi, maximum norm analysis of an overlapping non-matching grids method for the obstacle problem, adv. differ. equ. 2006 (2006), 85807. [9] a. harbi, m. boulbrachene, maximum norm analysis of a nonmatching grids method for nonlinear elliptic pdes, j. appl. math. 2011 (2011), 605140. https://doi.org/10.1155/2011/605140. [10] a. harbi, maximum norm analysis of an arbitrary number of nonmatching grids method for nonlinears elliptic pdes, j. appl. math. 2013 (2013), 893182. https://doi.org/10.1155/2013/893182. [11] m. boulbrachene, q. al farei, maximum norm error analysis of a nonmatching grids finite element method for linear elliptic pdes, appl. math. comput. 238 (2014), 21-29. https://doi.org/10.1016/j.amc.2014.03.146. [12] a. harbi, maximum norm analysis of a nonmatching grids method for a class of variational inequalities with nonlinear source terms, j. inequal. appl. 2016 (2016), 181. https://doi.org/10.1186/s13660-016-1110-4. [13] s. boulaaras, m.s. touati brahim, s. bouzenada, a posteriori error estimates for the generalized schwarz method of a new class of advection-diffusion equation with mixed boundary condition, math. meth. appl. sci. 41 (2018), 5493-5505. https://doi.org/10.1002/mma.5092. https://doi.org/10.1137/s1064827597327553 https://doi.org/10.1007/bf02679439 https://doi.org/10.1137/s0036142998348741 https://doi.org/10.1155/2011/605140 https://doi.org/10.1155/2013/893182 https://doi.org/10.1016/j.amc.2014.03.146 https://doi.org/10.1186/s13660-016-1110-4 https://doi.org/10.1002/mma.5092 28 int. j. anal. appl. (2023), 21:71 [14] m. boulbrachene, finite element convergence analysis of a schwarz alternating method for nonlinear elliptic pdes, squ journal for science. 24 (2020) 109-121. https://doi.org/10.24200/squjs.vol24iss2pp109-121. [15] q. al farei, m. boulbrachene, mixing finite elements and finite differences in nonlinear schwarz iterations for nonlinear elliptic pdes, comput. math. model. 33 (2022), 77-94. https://doi.org/10.1007/s10598-022-09558-x. [16] p.g. ciarlet, discrete maximum principle for finite-difference operators, service de mathematiques, l.c.p.c, and case institute of technology, (1969). [17] p.g. ciarlet, p.a. raviart, maximum principle and uniform convergence for the finite element method, computer methods appl. mech. eng. 2 (1973), 17-31. https://doi.org/10.1016/0045-7825(73)90019-4. [18] j. nitsche, l∞-convergence of finite element approximations, in: proceedings of the symposium on mathematical aspects of finite element methods, vol 606 of lecture notes in mathematics, pp. 261-274, (1977). [19] f. hecht, new development in freefem++, j. numer. math. 20 (2012), 251-265. https://doi.org/10.24200/squjs.vol24iss2pp109-121 https://doi.org/10.1007/s10598-022-09558-x https://doi.org/10.1016/0045-7825(73)90019-4 1. introduction 2. preliminaries 2.1. linear elliptic equations 3. schwarz method for nonlinear pdes 3.1. the nonlinear pde 3.2. continuous variational schwarz subproblems 3.3. finite element discretization 3.4. discrete variational schwarz subproblems 4. lconvergence analysis 4.1. continuous auxiliary schwarz subproblems 4.2. discrete auxiliary schwarz subproblems 4.3. the main results 5. numerical experiments 5.1. first example: 5.2. second example: 6. conclusion references int. j. anal. appl. (2023), 21:28 quasi-ideals and bi-ideals of near left almost rings thiti gaketem, tanaphong prommai∗ fuzzy algebras and decision-making problems research unit, department of mathematics school of science, university of phayao, phayao 56000, thailand ∗corresponding author: tanaphong.pr@up.ac.th abstract. in this paper, we define quasi-ideal, bi-ideal, and weak bi-ideal of nla-ring, and investigate it properties. 1. introduction m.a. kazim and md. naseeruddin defined la-semigroup as the following; a groupoid s is called a left almost semigroup, abbreviated as la-semigroup if (ab)c =(cb)a, ∀a,b,c ∈s m.a. kazim and md. naseeruddin [2] asserted that, in every la-semigroups g a medial law hold (a ·b) · (c ·d)= (a ·c) · (b ·d), ∀a,b,c,d ∈g. q. mushtaq and m. khan [4] asserted that, in every la-semigroups g with left identity (a ·b) · (c ·d)= (d ·b) · (c ·a), ∀a,b,c,d ∈g. further m. khan, faisal, and v. amjid [3], asserted that, if an la-semigroup g with left identity the following law holds a · (b ·c)= b · (a ·c), ∀a,b,c ∈g. m. sarwar (kamran) [6] defined la-group as the following; a groupoid g is called a left almost group, abbreviated as la-group, if (i) there exists e ∈g such that ea= a for all a∈g, (ii) for every a∈g there exists a′ ∈g such that, a′a= e, (iii) (ab)c =(cb)a for every a,b,c ∈g. received: jan. 30, 2023. 2010 mathematics subject classification. 16y30. key words and phrases. nla-ring; quasi-ideal bi-ideal; weak bi-ideal. https://doi.org/10.28924/2291-8639-21-2023-28 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-28 2 int. j. anal. appl. (2023), 21:28 a non-empty subset a of an la-group g is called an la-subgroup of g if a is itself an la-group under the same operation as defined in g. s.m. yusuf in [8] introduced the concept of a left almost ring (la-ring). that is, a non-empty set r with two binary operations “+” and “·” is called a left almost ring, if 〈r,+〉 is an la-group, 〈r, ·〉 is an la-semigroup and distributive laws of “·” over “+” holds. t. shah and i. rehman [8] asserted that a commutative ring 〈r,+, ·〉, we can always obtain an la-ring 〈r,⊕, ·〉 by defining, for a,b,c ∈ r, a⊕b = b−a and a ·b is same as in the ring. we can not assume the addition to be commutative in an la-ring. an la-ring 〈r,+, ·〉 is said to be an la-integral domain if for all a,b ∈r, with a ·b =0, then a = 0 or b = 0. let 〈r,+, ·〉 be an la-ring and s be a non-empty subset of r and s is itself and la-ring under the binary operation induced by r, then s is called an la-subring of 〈r,+, ·〉. if s is an la-subring of an la-ring 〈r,+, ·〉, then s is called a left ideal of r if rs ⊆ s. right and two-sided ideals are defined in the usual manner. by [5] a near-ring is a non-empty set n together with two binary operations “+” and “·” such that 〈n,+〉 is a group (not necessarily abelian), 〈n, ·〉 is a semigroup and one sided distributive (left or right) of “·” over “+” holds. by [1] if a subgroup q of 〈n,+〉 has the property qn∩nq⊆q, then it is called a quasi-ideal of n. by [9] if a subgroup b of 〈n,+〉 is said to be a bi-ideal of n if bnb∩ (bn)∗b ⊆b. if n has a zero symmetric near-ring a subgroup b of 〈n,+〉 is a bi-ideal if and only if bnb ⊆b. by [10] a subgroup b of 〈n,+〉 is said to be a weak bi-ideal of n if b3 ⊆b. in this paper we will define bi-ideal of near-ring has a zero symmetric. 2. near left almost rings t. shah, f. rehman and m. raees [7] introduces the concept of a near left almost ring (nla-ring). definition 2.1. [7]. a non-empty set n with two binary operation “+” and “·” is called a near left almost ring (or simply an nla-ring) if and only if (1) 〈n,+〉 is an la-group. (2) 〈n, ·〉 is an la-semigroup. (3) left distributive property of · over + holds, that is a ·(b+c)= a ·b+a ·c for all a,b,c ∈n. definition 2.2. [7]. an nla-ring 〈n,+〉 with left identity 1, such that 1·a= a for all a∈n, is called an nla-ring with left identity. definition 2.3. [7]. a non-empty subset s of an nla-ring n is said to be an nla-subring if and only if s is itself an nla-ring under the same binary operations as in n. int. j. anal. appl. (2023), 21:28 3 definition 2.4. [7]. an nla-subring i of an nla-ring n is called a left ideal of n if ni ⊆ i, and i is called a right ideal if for all n,m ∈n and i ∈ i such that (i +n)m−nm ∈ i, and is called two sided ideal or simply ideal if it is both left and right ideal. definition 2.5. [7]. let 〈n,+, ·〉 be an nla-ring. a non-zero element a of n is called a left zero divisor if there exists 0 6= b ∈n such that a ·b =0. similarly a is a right zero divisor if b ·a=0. if a is both a left and a right zero divisor, then a is called a zero divisor. definition 2.6. [7]. an nla-ring 〈d,+, ·〉 with left identity 1, is called an nla-ring integral domain if it has no left zero divisor. definition 2.7. [7]. an nla-ring 〈f,+, ·〉 with left identity 1, is called a near almost field (n-almost field) if and only if each non-zero element of f has inverse under “·” 3. quasi-ideals of near left almost rings definition 3.1. if an la-subgroup q of 〈n,+〉 has the property qn ∩nq ⊆ q, then it is called a quasi-ideal of n. lemma 3.1. let n be a nla-ring and q1,q2 are quasi-ideals of n. then q1∩q2 is a quasi-ideal of n. proof. since q1,q2 are la-subgroups of 〈n,+〉 we have q1 ∩q2 is a la-subgroup of 〈n,+〉. we must show that (q1 ∩q2)n∩n(q1 ∩q2)⊆q1 ∩q2. then (q1 ∩q2)n∩n(q1 ∩q2) ⊆ q1n∩q2n∩nq1 ∩nq2 = (q1n∩nq1)∩ (q2n∩nq2) ⊆ q1 ∩q2. thus q1 ∩q2 is a quasi-ideal of n. � theorem 3.1. each quasi-ideal of an nla-ring n is an nla-subring. proof. let q be a quasi-ideal an nla-ring n. then q is a nla-subring of 〈n,+〉. let a,b ∈q⊆n. then ab ∈nq⊆nq and ab ∈qn ⊆qn. thus ab ∈nq∩qn ⊂q, since q is a quasi-ideal of n. hence ab ∈q. therefore q is a nla-subring of n. � theorem 3.2. the set of all quasi-ideal of nla-ring. proof. let {qi}i∈i be a set of quasi-ideal in n and q=∩i∈iqi. then qn∩nq⊆ ⋂ i∈i qin∩n ⋂ i∈i qi ⊆qi for every i ∈ i. thus q is a quasi-ideal of n. � 4 int. j. anal. appl. (2023), 21:28 4. bi-ideals and weak bi-ideals of near left almost rings next we defined of a bi-ideal and weak bi-ideal in nla-ring is defines the same as a bi-ideal and weak bi-ideal in near-ring in [9] and [10]. definition 4.1. let n be an nla-ring. an la-subgroup b of 〈n,+〉 is a bi-ideal if (bn)b ⊆b. theorem 4.1. if b be a bi-ideal of a nla-ring n and s is an nla-subring of n. then b ∩s is a bi-ideal of s. proof. since b is a bi-ideal of n we have (bn)b ⊆ b. assume that c := b∩s. then (cs)c ⊆ (ss)s ⊆s, since s is a nla-subring of n and c ⊆s. on the other hand (cs)c ⊆ (bs)b ⊆ (bn)b ⊆b. hence (cs)c ⊆b∩s =c. therefore c is a bi-ideal of s. � theorem 4.2. let n be an nla-ring and a,b be bi-ideals of an nla-ring n. then a∩b is a bi-ideal of n. proof. since a,b is bi-ideals of an nla-ring n, we have a∩b is an la-subgroup of 〈n,+〉. thus [(a ∩ b)n](a ∩ b) ⊆ (an)(a ∩ b) = [(a ∩ b)n]a ⊆ (an)a ⊆ a and [(a ∩ b)n](a ∩ b) ⊆ (bn)(a∩b)= [(a∩b)n]b ⊆ (bn)b ⊆b. it following that a∩b is a bi-ideal of n. � theorem 4.3. the set of all bi-ideal of nla-ring. proof. let {bi}i∈i be a set of bi-ideal in n and b :=∩i∈ibi. then (bn)b ⊆ ( ⋂ i∈i bin) ⋂ i∈i bi ⊆bi for every i ∈ i. thus b is a bi-ideal of n. � definition 4.2. let n be an nla-ring. an element d of n is called distributive if (n+n′)d = nd+n′d for all n,n′ ∈n. theorem 4.4. let n be an nla-ring. if b is a bi-ideal of n then bn and n′b are bi-ideals of n where n,n′ ∈n and n′ is a distributive element in n. proof. since b is a bi-ideal we have bn and n′b are la-subgroup 〈n,+〉. thus ((bn)n)(bn)⊆ (bn)(bn)= ((bn)b)n ⊆bn. hence bn is a bi-ideal of n. again ((n′b)n)(n′b)⊆ ((n′b)n)b =(n′bn)b ⊆ n′b. thus n′b are bi-ideal of n. � corollary 4.1. let b be a bi-ideal of nla-ring. for b,c ∈b, if b is a distributive element in n, then bbc is a bi-ideal of n. int. j. anal. appl. (2023), 21:28 5 proof. let b be a bi-ideal of nla-ring and b is a distributive element in n. then (n+n′)b = nb+n′b for all n,n′ ∈n. since b is a bi-ideal we have bbc is an la-subgroup 〈n,+〉 then ((bbc)n)(bbc)⊆ (bn)b ⊆b. � definition 4.3. an nla-ring n is said to be b-simple if it has no proper bi-ideals. theorem 4.5. let n be an nla-ring with more than one element. if n is a near almost field. then n is a b-simple. proof. let n be a near almost field then {0} and n are the only bi-ideals of n. for if 0 6= b is a bi-ideal of n, then for 0 6= b ∈b we get n =nb and n = bn. now n = n2 = (bn)(nb) ⊆ bnb ⊆ b, since b is a bi-ideal of n. then n = b. thus n is a b-simple. � the following we defined weak bi-ideal and study properties it. definition 4.4. an la-subgroup b of 〈n,+〉 is said to be a weak bi-ideal of n if b3 ⊆b. theorem 4.6. every bi-ideal of an nla-ring is a weak bi-ideal. proof. since b3 =(bb)b ⊆ (bn)b ⊆b we have every bi-ideal is a weak bi-ideal. � theorem 4.7. if b is a weak bi-ideal of a nla-ring n and s is a nla-subring of n. then b∩s is a weak bi-ideal of n. proof. assume that c :=b∩s. then c3 = ((b∩s)(b∩s))(b∩s) = ((b∩s)(b∩s))b∩ ((b∩s)(b∩s))s ⊆ (bb)b∩sss = b3 ∩sss ⊆ b3 ∩ss ⊆ b3 ∩s ⊆ b∩s = c. thus c3 ⊆c. hence c is a weak bi-ideal of n. � theorem 4.8. let n be an nla-ring. if b is a weak bi-ideal of n then bn and n′b are weak bi-ideal of n where n,n′ ∈n and n′ is a distributive element in n. proof. since b is a weak bi-ideal we have bn and n′b an la-subgroup of 〈n,+〉. thus (bn)3 =(bnbn)bn ⊆ (bb)bn ⊆b3n ⊆bn. hence bn is a weak bi-ideal of n. 6 int. j. anal. appl. (2023), 21:28 again (n′b)3 =(n′bn′b)n′b ⊆ (n′bb)b = n′b3 ⊆ n′b. thus n′b is a weak bi-ideal of n. � corollary 4.2. let b be a weak bi-ideal of nla-ring. for b,c ∈b, if b is a distributive element in n, then bbc is a weak bi-ideal of n. 5. conclusion in this article, we give the concept of a quasi-ideals and biideals in nla-rings. we study properties of quasi-ideals and biideals. in the future we study primary and quasi-primary in nla-ring. acknowledgements: this research project was supported by the thailand science research and innovation fund and the department of mathematics, school of science, university of phayao, phayao 56000, thailand. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. dheena, s. manivasan, quasiideals of a p-regular near-rings, int. j. algebra, 5 (2011), 1005-1010. [2] m.a. kazim, md. naseeruddin, on almost semigroup, portugaliae math. 36 (1977), 41-47. https://eudml.org/ doc/115301. [3] m. khan, faisal, v. amjid, on some classes of abel-grassmann’s groupoids, (2010). http://arxiv.org/abs/ 1010.5965. [4] q. mushtaq, m. khan, m-system in la-semigroups, southeast asian bull. math. 33 (2009), 321-327. [5] g. pilz, near-rings, north holland, amesterdam-new york, 1997. [6] m. sarwar (kamran), conditions for la-semigroup to resemble associative structures, phd thesis, quaid-i-azam university, 1993. [7] t. shah, f. rehman, m. raees, on near left alomst rings, int. math. forum, 6 (2011), 1103-1111. [8] t. shah, i. rehman, on la-rings of finitely nonzero function, int. j. contemp. math. sci. 5 (2010), 209-222. [9] t.t. chelvam, n. ganesan, on bi-ideal of near-ring, indian j. pure appl. math. 18 (1987), 1002-1005. [10] y.u. cho, t.t. chelvam, s. jayalakshmi, weak bi-ideal of near-ring, j. korea soc. math. 14 (2007), 153-159. https://eudml.org/doc/115301 https://eudml.org/doc/115301 http://arxiv.org/abs/1010.5965 http://arxiv.org/abs/1010.5965 1. introduction 2. near left almost rings 3. quasi-ideals of near left almost rings 4. bi-ideals and weak bi-ideals of near left almost rings 5. conclusion references int. j. anal. appl. (2022), 20:59 geometry of warped product cr and semi-slant submanifolds in quasi-para-sasakian manifolds shamsur rahman1, abdul haseeb2,∗, nargis jamal3 1department of mathematics, maulana azad national urdu university, polytechnic, satellite campus darbhanga, bihar 846001, india 2department of mathematics, college of science, jazan university, jazan-45142, kingdom of saudi arabia 3department of mathematics, college of science (girls campus mehliya), jazan university, jazan-45142, kingdom of saudi arabia ∗corresponding author: malikhaseeb80@gmail.com, haseeb@jazanu.edu.sa abstract. in the present paper we study the existence or non-existence of warped product semi-slant submanifolds in quasi-para-sasakian manifolds and prove that there are no proper warped product semislant submanifolds in a quasi-para-sasakian manifold such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively. 1. introduction the concept of warped product manifolds was introduced by bishop and o’neill for constructing manifolds of non-positive curvature, as one of the most effective generalization of riemannian product manifold [15]. about two decades ago, chen extended the work of bishop and o’neill and studied the warped product cr-submanifold of kaehler manifolds [3,4], this study was also extended by many geometers in different settings [2,13,14]. the existence or non-existence of warped product manifolds plays an important role in differential geometry as well as in physics. in [6], blair introduced the notion of quasi-sasakian manifolds that unifies sasakian and cosymplectic manifolds. tanno [19] also contributed some remarkable results on quasi-sasakian structure. recently, quasi-sasakian structure have been studied in [1, 17, 18]). the geometry of almost paracontact manifold was studied by received: sep. 16, 2022. 2010 mathematics subject classification. 53c40, 53c42, 53b26. key words and phrases. warped product; semi-slant submanifolds; quasi-para-sasakian manifolds. https://doi.org/10.28924/2291-8639-20-2022-59 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-59 2 int. j. anal. appl. (2022), 20:59 kaneyuki and williams in [16] as a natural generalization of natural odd-dimensional analogue to almost para-hermitian structures. the study of almost paracontact metric manifolds was carried out in one of zamkovoy’s papers [20]. in [21], olszak studied normal almost contact metric manifolds of dimension 3. in 2009, welyczko [10] investigated curvature and torsion of frenet-legendre curves in 3-dimensional normal almost paracontact metric manifolds. recently, 3-dimensional normal almost paracontact metric manifolds were studied in [5,7,8]. 2. preliminaries let m̄ be a (2n + 1)-dimensional almost paracontact manifold with structure tensor (f ,ξ,υ,<,>), where f , ξ and υ be a tensor field of type (1, 1), a vector field, and a 1-form, respectively on m̄ satisfying f ξ = 0, f 2 = i −υ ⊗ξ, υ ◦ f = 0, (2.1) υ(ξ) = 1, υ(x) =< x ,ξ >, < f ·, f · >= − <,> +υ ⊗υ, (2.2) where i is the identity on the tangent bundle tm̄ of m̄. we say that m̄ is a paracontact metric manifold if there exists a one-form υ such that < x , fy >= dυ(x ,y) = 1 2 (xυ(y) −yυ(x) −υ([x ,y]), for all x ,y ∈ x(m̄), where x(m̄) denotes the lie algebra of vector fields on m̄, and < fx ,y > + < x , fy >= 0 (2.3) for all vector fields x and y on m̄. further, an almost paracontact metric manifold is called a quasi-para-sasakian manifold if (∇̄xf )y = υ(y)fx− < fx ,y > ξ, (2.4) and ∇̄xξ = −ffx , ffx = ffx , < fx ,y >= − < x ,fy >, (2.5) where ∇̄ denotes the levi-civita connection with respect to the metric tensor <,> and f is a tensor field of type (1, 1). by applying f to (2.5) and using (2.1), we obtain fx = υ(fx)ξ− f (∇̄xξ). (2.6) also by replacing x by ξ in (2.5) it follows that ∇̄ξξ = 0. (2.7) using (2.4), (2.6) and (2.7) we infer fξ = υ(fξ)ξ, (2.8) int. j. anal. appl. (2022), 20:59 3 and (∇̄ξf )x = 0 (2.9) for any x ∈ γ(tm̄). if m is a contact cr-submanifold of m̄ and the projections on d and d⊥ are denoted by p and q, respectively; then for all vector field x tangent to m, we infer x = px + qx + υ(x)ξ. (2.10) now we put bλ + cλ = f λ, (2.11) where bλ and cλ are tangential and normal part of f λ on m. next we define the tensor field of type (1, 1) on m by fx = f px , (2.12) and the γ(tm⊥)-valued 2-form ω by ωx = f qx . (2.13) since d is invariant by f , then it follows from (2.11) and (2.12) that b is γ(d⊥)-valued and t is γ(d)-valued, respectively. by using (2.1), (2.10), (2.12) and (2.13), we obtain ωx + tx = fx , (2.14) and t3 + t = 0; c3 + c = 0. (2.15) then by (2.15) we conclude that t and c are f -structure in sense of yano [11] on tm and tm⊥, respectively. now suppose <,> be the induced metric and ξ be tangent to m. further, we suppose ∇ and ∇⊥ be the induced connections on the tangent bundle tm and the normal bundle t⊥m of m, respectively. then the gauss and weingarten formulas are given respectively by ∇̄xy = σ(x ,y) + ∇xy, (2.16) ∇̄xλ = −λλx + ∇⊥xλ (2.17) for all vector fields x ,y tangent to m and any vector field λ normal to m, where σ and λλ are the second fundamental form and the shape operator for the immersion of m into m̄. the second fundamental form σ and shape operator λλ are related by < σ(x ,y),λ >=< λλx ,y > (2.18) for all vector fields x ,y tangent to m and vector field λ normal to m. 4 int. j. anal. appl. (2022), 20:59 furthermore, for any z ∈ γ(tm̄), we put fz = αz + βz, (2.19) where αz and βz are the tangent part and the normal part of fz, respectively. from (2.3) we have < tx ,y > + < x ,ty >= 0. (2.20) in account of (2.6), (2.11), (2.12) and (2.16) we obtain αx = υ(x)υ(fx)ξ− t(∇xξ) −bσ(x ,ξ), (2.21) and βx = −ω(∇xξ) −cσ(x ,ξ). (2.22) proposition 2.1. if m is a contact cr-submanifold of a quasi-para-sasakian manifold m̄, then γ(tm) is invariant with respect to the action of f if and only if we have ω(∇xξ) = 0, (2.23) and cσ(x ,ξ) = 0. (2.24) proof. from (2.22) it follows that f is a tensor field of type (1, 1) on m if and only if ω(∇xξ) + cσ(x ,ξ) = 0. (2.25) then (2.23) and (2.24) follows from (2.25) (since < ωy,cλ >= 0 for any y ∈ γ(tm)). corollary 2.1. if m is a contact cr-submanifold of a quasi-para-sasakian manifold m̄ such that γ(tm) is invariant with respect to the action of f, then both the distributions d and d⊥ are invariant with respect to the action of f. proof. let x ∈ γ(d), then by using the third relation of (2.5) and (2.8) we obtain < fx ,ξ >= − < x ,fξ >= υ(fξ) < x ,ξ >= 0. on the other hand, by using (2.2), the second relation of (2.5) and the invariace of d with respect to the action of f we infer < fx ,z >=< ffx ′,z >= − < fx ′, fz >= 0, where x ′ ∈ γ(d) and z ∈ γ(d⊥). hence d is invariant by f. in a similar way it follows that d⊥ is invariant by the action of f. the riemannian connections ∇ and ∇⊥ allow us to define the usual covariant derivatives as (∇xt)y = ∇xty − t∇xy, (2.26) int. j. anal. appl. (2022), 20:59 5 and (∇xω)y = ∇⊥xωy −ω∇xy. (2.27) now, the canonical structures t and ω on a submanifold m are said to be parallel if ∇t = 0 and ∇ω = 0, respectively. on a cr-submanifold of a quasi-para-sasakian manifold, it follows from (2.5) and (2.16) that ∇xξ = −ffx , (2.28) and σ(x ,ξ) = 0 (2.29) for each x ∈ tm . furthermore, from (2.29) we obtain λωξ = 0; υ(λω)x = 0. (2.30) lemma 2.1. for a contact cr-submanifold m of a quasi-para-sasakian manifold m̄, we infer (∇xt)y = λωyx + bσ(x ,y) + υ(y)αx− < fx ,y > ξ, (2.31) (∇xω)y = cσ(x ,y) −σ(x ,ty) + υ(y)βx . (2.32) proof. by using (2.4), (2.16)-(2.19), (2.26) and (2.27), we obtain (αx + βx)υ(y)− < fx ,y > ξ = (∇xt)y + (∇xω)y − λωyx −bσ(x ,y) −cσ(x ,y) + σ(x ,ty) for any x ,y ∈ γ(tm). by equating the tangential and the normal parts in above relation, (2.31) and (2.32), respectively follows. the covariant derivatives of b and c are given respectively by (∇xb)λ = ∇xbλ−b(∇⊥xλ), (2.33) and (∇⊥xc)λ = ∇ ⊥ xcλ−c(∇ ⊥ xλ) (2.34) for any x ∈ γ(tm) and ≥∈ γ(tm⊥). lemma 2.2. for a contact cr-submanifold m of a quasi-para-sasakian manifold m̄, we infer (∇xb)λ = λcλx − t(λλx)− < fx ,λ > ξ, (2.35) and (∇⊥xc)λ = −σ(x ,bλ) −ω(λλx) (2.36) for any x ∈ γ(tm) and λ ∈ γ(tm⊥). 6 int. j. anal. appl. (2022), 20:59 lemma 2.3. for a contact cr-submanifold m of a quasi-para-sasakian manifold m̄, we infer λfxy = λfyx , (2.37) and < σ(u,v), fz >=< ∇uz, fv > (2.38) for all u ∈ γ(tm),v ∈ γ(d) and x ,y,z ∈ γ(d⊥). proof. by using (2.2), (2.4) and (2.16)-(2.18), we have < λfxy,u >=< σ(y,u), fx >=< ∇̄uy, fx > − < ∇uy, fx > =< ∇uy, fx >= − < f (∇uy),x >= − < −(∇̄uf )y + ∇̄ufy,x > + < υ(y)fu− < fu,y > ξ,x > − < ∇̄ufy,x > − < −λfyu + ∇⊥u fy,x >=< λfyu,x >=< λfyx ,u > . since υ(y) = υ(x) = 0, therefore we find (2.37). next, by using (2.2), (2.4) and (2.16), we obtain < σ(u,v), fz >=< ∇̄uv, fz > − < v,∇̄ufz > − < v, (∇̄uf )z + f (∇̄uz) > − < v,υ(z)fu− < fu,z > ξ > − < v, f (∇̄uz) >=< fv,∇̄uz >=< fv,∇uz > which leads to (2.38). a submanifold m of an almost para contact metric manifold m̄ is said to be invariant if f is identically zero, that is, fx ∈ tm and anti-invariant if t is identically zero, that is, fx ∈ t⊥m, for any x ∈ tm. for each non-zero vector x tangent to m at any point x such that x is not proportional to ξ, we denote by θ(x), the angle between fx and txm for all x ∈m. definition 2.1. a submanifold n is said to be slant if the angle θ(x) is constant for all x ∈ txn−{ξ} and x ∈ n. the angle θ is called a slant angle or wirtinger angle. obviously, if θ = 0, then n is invariant; and if θ = π/2, then m is an anti-invariant submanifold. if the slant angle of n is different from 0 and π/2 then it is called proper slant. a characterization of slant submanifolds is given by the following theorem: theorem 2.1. [9] let n be slant submanifold of a quasi-para-sasakian manifold m̄ such that ξ is tangent to n. then n is slant submanifold if and only if there exists a constant λ ∈ [0, 1] such that t2x = µ(x −υ(x))ξ. (2.39) furthermore, if θ is the slant angle of n, then µ = cos2θ. int. j. anal. appl. (2022), 20:59 7 corollary 2.2. let n be a slant submanifold with slant angle θ of a quasi-para-sasakian manifold m̄ such that ξ is tangent to n. then we have < tz,tw >= cos2θ{− < z,w > +υ(z)υ(w)}, (2.40) < ωz,ωw >= sin2θ{− < z,w > +υ(z)υ(w)} (2.41) for any z,w tangent to n. 3. warped product semi-slant submanifolds a quasi-para-sasakian manifold for two riemannian manifolds (n1,<,>1) and (n2,<,>2) and a positive differentiable function δ on n1, the warped product of n1 and n2 is the riemannian manifold n1×δn2 = (n1×n2,<,>), where <,>=<,>1 +δ 2 <,>2 . (3.1) more explicitly, if the vector fields x and y are tangent to n1×δn2 at (x,y), then < x ,y >=<,>1 (π1 ∗x ,π1 ∗y) + δ2(x) <,>2 (π2 ∗x ,π2 ∗y), (3.2) where πi (i = 1, 2) are the canonical projections of n1×δn2 onto n1 and n2, respectively, and ∗ stands for derivative map. if m̃ = n1×δn2 is a warped product manifold, this means that n1 and n2 are totally geodesic and totally umbilical submanifolds of m̃, respectively. for warped product manifolds, we have the following proposition [12,15]: proposition 3.1. on a warped product manifold m̃ = n1×δn2, we have (1) ∇xy ∈ γ(tn1) is the lift of ∇xy on n1, (2) ∇ux = ∇xu = x(lnδ)u, (3) ∇uv = ∇ ′ uv− < u,v > ∇lnδ for any x,y ∈ γ(tn1) and u,v ∈ γ(tn2), where ∇ and ∇ ′ denote the levi-civita connections on m and n2, respectively. let us suppose that m̄ be a quasi-para-sasakian manifold and n1×δn2 be a warped product semislant submanifold of a quasi-para-sasakian manifold m̄. such submanifolds are always tangent to the structure vector field ξ. if the manifolds nθ and nt (resp., n⊥) are slant and invariant (resp., antiinvariant) submanifolds of a quasi-para-sasakian manifold m̄, then their warped product semi-slant submanifolds may be given by one of the following forms: (i) nθ×δnt , (ii) nθ×δn⊥, (iii) nt×δnθ, (iv) n⊥×δnθ. here, we are concerned with cases (i) and (ii). theorem 3.1. if m̄ is a quasi-para-sasakian manifold, then there do not exist proper warped product semi-slant submanifolds nθ×δnt such that nθ is a proper slant submanifold, nt is an invariant submanifold of m̄ and ξ is tangent to n . 8 int. j. anal. appl. (2022), 20:59 proof. let nθ×δnt be a proper warped product semi-slant submanifold of a quasi-para-sasakian manifold m̄. for any x ,y ∈ γ(tnθ) and u,v ∈ γ(tnt ), we have (∇̄xf )u = ∇̄xfu − f (∇̄xu). (3.3) thus, from (2.4), (2.11), (2.14) and (2.16) we obtain υ(u)fx− < fx ,u > ξ = σ(x ,tu) −bσ(x ,u) −cσ(x ,u). this means that bσ(x ,u) = 0, (3.4) and cσ(x ,u) −σ(x ,tu) = 0. (3.5) on the other hand, by interchanging roles of u and x in (3.3), we conclude tx log(δ)u = λωxu + x log(δ)tu + bσ(u,x), (3.6) and ∇⊥uωx + σ(u,tx) −cσ(u,x) = 0. (3.7) from (3.6), we arrive at tx log(δ) < u,u > = < λωxu,u > + < bσ(u,x),u > (3.8) = < σ(u,u),ωx > + < bσ(u,x),u > = < σ(u,u),ωx > − < σ(x ,u), fu > = < σ(u,u),ωx > . on the other hand, since the ambient space m̄ is a quasi-para-sasakian manifold, then by using (3.5) and (3.7) we get ch(z,ξ) = 0 (3.9) for any z ∈ γ(tn). by using (3.5) and (3.7), we get ωx = cσ(x ,ξ) = 0. thus we have tx log(δ) < u,u >= 0, this implies that tx log(δ) = 0, that is, the warping function δ is constant on nθ. � theorem 3.2. if m̄ is a quasi-para-sasakian manifold, then there do not exist proper warped product semi-slant submanifolds nθ×δn⊥ such that nθ is a proper slant submanifold, n⊥ is an invariant submanifold of m̄ and ξ is tangent to n . proof. let nθ×δn⊥ be a proper warped product semi-slant submanifold of a quasi-para-sasakian manifold m̄ such that ξ is tangent to n. for any x ,y ∈ γ(tnθ) and u,v ∈ γ(tn⊥), we have (∇̄xf )u = ∇̄xfu − f (∇̄xu). int. j. anal. appl. (2022), 20:59 9 using (2.4), (2.14), (2.16), (2.17) and proposition 3.1, the above equation takes the form υ(u)fx −g(fx ,u)ξ = −λωux + ∇⊥xωu −x(logδ)ωu (3.10) −f σ(x ,u). this means that λωux + bσ(x ,u) = 0, (3.11) and ∇⊥xωu −x(logδ)ωu −cσ(x ,u) = 0. (3.12) by interchanging roles of x and u in (3.10), we arrive at υ(u)fx− < fx ,u > ξ = tx log(δ)u + σ(u,tx) − λωxu (3.13) +∇⊥uωx −x log(δ)ωu −bσ(u,x) −cσ(u,x). equating the tangential and normal components in (3.13), we find tx log(δ)u = λωxu + bσ(u,x), (3.14) and σ(u,tx) + ∇⊥uωx −x log(δ)ωu −cσ(u,x) = 0, (3.15) respectively. from (3.14), we find < λωxu,ty > + < bσ(u,x),ty >= 0. (3.16) since the ambient space m̄ is a quasi-para-sasakian manifold, ξ is tangent to n and using (2.2), we obtain < bσ(x ,u),ty > = < f σ(x ,u), fy > = − < σ(x ,u),y > +υ(y)υ(σ(x ,u)) = 0. this implies that < bσ(x ,u),ty >=< σ(u,ty),ωx >= 0. (3.17) thus we have < σ(u,ty), fx >= 0 (3.18) for any x ,y ∈ γ(tnθ). moreover, making use of (3.11) and (3.18), we get < σ(x ,ty), fu >= 0. (3.19) 10 int. j. anal. appl. (2022), 20:59 by using the gauss-weingarten formulas and considering that nθ is totally geodesic in n, we arrive at < σ(x ,ty), fu > = < ∇̄tyx , fu) = − < f (∇̄tyx),u > (3.20) = − < ∇̄tyfx − (∇̄tyf )x ,u > = − < ∇̄tytx ,u > − < ∇̄tyωx ,u > + < υ(x)fty,u > − < fty,x >< ξ,u > = < λωxty,u > −υ(u) < fty,x > = < σ(ty,u),ωx > −υ(u) < fty,x > = υ(u) < ty,fx > . thus from (3.19) and (3.20), we conclude υ(u) < ty,fx >=< σ(x ,ty, fu >= 0. (3.21) here, if υ(u) = 0, then by using (2.32) and (3.12), we leads to x log(δ)ωu = υ(∇xu) = − < −ffx ,u >= 0. this is impossible. because u is a non-zero vector field and n⊥ 6= 0. thus < tx ,ty >= cos2θ{− < x ,y > +υ(x)υ(y)} = 0, this implies that the slant angle θ is either identically π/2 or the warping function δ is constant on nθ . this completes the proof. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a. haseeb, s. pandey, r. prasad, some results on η-ricci solitons in quasi-sasakian 3-manifolds, commun. korean math. soc. 36 (2021), 377–387. https://doi.org/10.4134/ckms.c200196. [2] b. sahin, non-existence of warped product semi-slant submanifolds of kaehler manifolds, geom. dedicata. 117 (2006), 195–202. https://doi.org/10.1007/s10711-005-9023-2. [3] b.y. chen, geometry of warped product cr-submanifolds in kaehler manifolds, monatsh. math. 133 (2001), 177–195. https://doi.org/10.1007/s006050170019. [4] b.y. chen, geometry of warped product cr-submanifolds in kaehler manifolds, ii, monatsh. math. 134 (2001), 103–119. https://doi.org/10.1007/s006050170002. [5] c.l. bejan, m. crasmareanu, second order parallel tensors and ricci solitons in 3-dimensional normal paracontact geometry, ann. glob. anal. geom. 46 (2014), 117–127. https://doi.org/10.1007/s10455-014-9414-4. [6] d.e. blair, contact manifolds in riemannian geometry, springer berlin heidelberg, 1976. https://doi.org/10. 1007/bfb0079307. [7] i.k. erken, some classes of 3-dimensional normal almost paracontact metric manifolds, honam math. j. 37 (2015), 457–468. https://doi.org/10.5831/hmj.2015.37.4.457. [8] i.k. erken, on normal almost paracontact metric manifolds of dimension 3, facta univ. ser. math. inform. 30 (2015), 777-788. https://doi.org/10.4134/ckms.c200196 https://doi.org/10.1007/s10711-005-9023-2 https://doi.org/10.1007/s006050170019 https://doi.org/10.1007/s006050170002 https://doi.org/10.1007/s10455-014-9414-4 https://doi.org/10.1007/bfb0079307 https://doi.org/10.1007/bfb0079307 https://doi.org/10.5831/hmj.2015.37.4.457 int. j. anal. appl. (2022), 20:59 11 [9] j.l. cabrerizo, a. carriazo, l.m. fernández, m. fernández, slant submanifolds in sasakian manifolds, glasgow math. j. 42 (2000), 125–138. https://doi.org/10.1017/s0017089500010156. [10] j. wełyczko, on legendre curves in 3-dimensional normal almost paracontact metric manifolds, results. math. 54 (2009), 377–387. https://doi.org/10.1007/s00025-009-0364-2. [11] k. yano, on structure defined by a tensor field f of type (1, 1), satisfying f 3+f =0, tensor (n. s.), 14 (1963), 99-109. [12] m. atceken, warped product semi-slant submanifolds in kenmotsu manifolds, turk. j. math. 34 (2010), 425-432. https://doi.org/10.3906/mat-0901-6. [13] m. a. khan and c. ozel, ricci curvature of contact cr-warped product submanifolds in generalized sasakian space forms admitting a trans-sasakian structure, filomat. 35 (2021), 125–146. https://doi.org/10.2298/ fil2101125k. [14] m.i. munteanu, a note on doubly warped product contact c r-submanifolds in trans-sasakian manifolds, acta math. hung. 116 (2007), 121–126. https://doi.org/10.1007/s10474-007-6013-x. [15] r.l. bishop, b. o’neill, manifolds of negative curvature, trans. amer. math. soc. 145 (1969), 1–49. https: //doi.org/10.1090/s0002-9947-1969-0251664-4. [16] s. kaneyuki, f.l. williams, almost paracontact and parahodge structures on manifolds, nagoya math. j. 99 (1985), 173–187. https://doi.org/10.1017/s0027763000021565. [17] s. rahman, some results on the geometry of warped product cr-submanifolds in quasi-sasakian manifold, cubo, math. j. 24 (2022), 105–114. https://doi.org/10.4067/s0719-06462022000100105. [18] s. rahman, m. s. khan, a. horaira, warped product semi slant submanifold of nearly quasi sasakian manifold, boson j. modern phys. 5 (2019), 443-453. [19] s. tanno, quasi-sasakian structure of rank 2p +1, j. differ. geom. 5 (1971), 317-324. [20] s. zamkovoy, canonical connections on paracontact manifolds, ann. glob. anal. geom. 36 (2009), 37–60. https: //doi.org/10.1007/s10455-008-9147-3. [21] z. olszak, normal almost contact metric manifolds of dimension three, ann. polon. math. 47 (1986), 41–50. https://doi.org/10.4064/ap-47-1-41-50. https://doi.org/10.1017/s0017089500010156 https://doi.org/10.1007/s00025-009-0364-2 https://doi.org/10.3906/mat-0901-6 https://doi.org/10.2298/fil2101125k https://doi.org/10.2298/fil2101125k https://doi.org/10.1007/s10474-007-6013-x https://doi.org/10.1090/s0002-9947-1969-0251664-4 https://doi.org/10.1090/s0002-9947-1969-0251664-4 https://doi.org/10.1017/s0027763000021565 https://doi.org/10.4067/s0719-06462022000100105 https://doi.org/10.1007/s10455-008-9147-3 https://doi.org/10.1007/s10455-008-9147-3 https://doi.org/10.4064/ap-47-1-41-50 1. introduction 2. preliminaries 3. warped product semi-slant submanifolds a quasi-para-sasakian manifold references int. j. anal. appl. (2023), 21:41 received: feb. 11, 2023. 2020 mathematics subject classification. 62p05. key words and phrases. neutrosophic statistics; classical statistics; simulation; robust type estimators; indeterminacy intervals. https://doi.org/10.28924/2291-8639-21-2023-41 © 2023 the author(s) issn: 2291-8639 1 neutrosophic generalized exponential robust ratio type estimators yashpal singh raghav* department of mathematics, faculty of science, jazan university, jazan, saudi arabia *corresponding author: yraghav@jazanu.edu.sa abstract. estimators proposed under classical statistics fail if data are vague or indeterminate. neutrosophic statistics are the only alternative because its deal with indeterminacy. extensive reserch has been conducted in this field because of its wide applicability. this study aimed to further develop the theory of neutosophic simple random sampling without replacement. in this study, a generalized neutrosophic exponential robust ratio-type estimator was proposed, and five of its member neutrosophic estimators were developed. derivations of the bias and mean square error were provided up to the first-order approximation. to demonstrate the high efficiency of the proposed neutrosophic estimators an empirical study on the stock price of moderna and four simulation studies have been conducted, and the results show that the proposed neutrosophic estimators are more efficient than similar existing ratio type estimators discussed in this paper in neutrosophic as well as classical forms. 1. introduction classical statistics and its methods deal with randomness but there are cases where the data at hand is indeterminate or vague or ambiguous or imprecise rather than random. in such situations estimation using classical statistical methods does not yield promising results. fuzzy logic [1, 2] is one solution to tackle such a problem but still, it ignores indeterminacy. in such cases, neutrosophic methods are much more reliable. they deal with both randomness and more importantly with indeterminacy. neutrosophic statistics refers to a set of data such that the data or a part of it is indeterminate and methods to analyze such a data [3]. neutrosophic statistics is an extension of classical statistics and when the indeterminacy is zero, neutrosophic statistics coincides with classical statistics [3]. estimation through neutrosophic https://doi.org/10.28924/2291-8639-21-2023-41 2 int. j. anal. appl. (2023), 21:41 methods is a new field and therefore it is unexplored unlike estimation problems in classical probability sampling designs where the data is determinate [4-8]. but, due to its wide applicability, it has gained much more importance than classical statistics and as a results it is being applied in various fileds for instance in decision making [9]. [10] developed a new sampling plan using neutrosophic process. [11] proposed neutrosophic analysis of variance. [12] used neutrosophic statistics in analyzing road traffic accidents. [13] proposed goodness of fit test in neutrosophic statistics. as a result filed of neutrosophic sampling has been developed and some neutrosophic ratio-type estimators has been proposed [14] and this paper is the second paper aimed at further developing the theory of neutrosophic srswor or nsrswor sampling. it has been observed in some sample surveys that the data collected containes some vagueness due to many factors like methodolgy used (observing blood pressure multiple times within an interval in nfhs 4 [16]) , observing daily stock price [15, 19] or daily temperature of a city [14]. all these are examples where the data contains some indeterminacy and clssical statistical measures like mean, median or standard deviation might not give results which are useful for decision making. thus the aim of this paper is to further develop neutrosophic probability sampling theory particulary nsrswor by developing various generalized neutrosophic exponential robust ratio type estimators. in section 2, the paper presents the terminologies of neutrosophic statistics for new readers. in section 3, existing related neutrosophic ratio-type estimators have been presented. in section 4, the proposed generalized neutrosophic exponential robust ratio type estimator and the five developed estimators along with their derivations of biases and mses are presented. in order to demonstrate the high efficiecny of the developed neutrosophic generalized neutrosophic exponential robust ratio type estimators four simulation studies have been conducted in section 5. the results are compared with their classical mse values as well. results and concluding remarks on this paper are provided in section 6 along with some future fruitful areas of research. 2. terminology a simple random neutrosophic sample of size n from a classical or neutrosophic population is a sample of n individuals such that at least one of them has some indeterminacy [3, 14]. as presented in [14], a neutrosophic observation is of the form 𝑍𝑁 = 𝑍𝐿 + 𝑍𝑈 𝐼𝑁, where 𝐼𝑁 ∈ [𝐼𝐿 , 𝐼𝑈 ] and 𝑍𝑁 ∈ [𝑍𝑙 , 𝑍𝑢 ]. now consider a simple random neutrosophic sample of size 𝑛𝑁 ∈ [𝑛𝐿 , 𝑛𝑈 ] drawn from a finite population of size n and 𝑦𝑁 (𝑖) ∈ [𝑦𝐿 , 𝑦𝑈 ] and 𝑥𝑁 (𝑖) are 𝑖 𝑡ℎ ∈ [𝑥𝐿 , 𝑥𝑈 ] neutrosophic sample 3 int. j. anal. appl. (2023), 21:41 observation. here the population mean of neutrosophic survey and auxiliary variable are �̅�𝑁 ∈ [𝑌𝐿 , 𝑌𝑈 ] and �̅�𝑁 ∈ [𝑋𝐿 , 𝑋𝑈 ] respectively. 𝐶𝑦𝑁 ∈ [𝐶𝑦𝑁𝐿 , 𝐶𝑦𝑁𝑈 ] and 𝐶𝑥𝑁 ∈ [𝐶𝑥𝑁𝐿 , 𝐶𝑥𝑁𝑈 ] are population coefficient of variation of neutrosophic survey and auxiliary variables respectively. in addition, 𝜌𝑥𝑦𝑁 ∈ [𝜌𝑥𝑦𝑁𝐿 , 𝜌𝑥𝑦𝑁𝑈 ], 𝛽1(𝑥𝑁 ) ∈ [𝛽1(𝑥𝑁𝐿 ), 𝛽1(𝑥𝑁𝑈 )] and 𝛽2(𝑥𝑁 ) ∈ [𝛽2(𝑥𝑁𝐿 ), 𝛽2(𝑥𝑁𝑈 )] are the correlation coefficient between the neutrosophic survey and auxiliary variables, coefficient of skewness and coefficient of kurtosis of the neutrosophic auxiliary variable respectively. the mse of a neutrosophic estimator is of the form, 𝑀𝑆𝐸(�̅�𝑁 ) ∈ [𝑀𝑆𝐸𝐿 , 𝑀𝑆𝐸𝑈 ]. the error terms in neutrosophic statistics are: �̅�𝑦𝑁 = �̅�𝑁 − �̅�𝑁, �̅�𝑥𝑁 = �̅�𝑁 − �̅�𝑁, 𝐸(�̅�𝑦𝑁 ) = 𝐸(�̅�𝑥𝑁) = 0, 𝐸(�̅�𝑦𝑁 2 ) = 𝑁−𝑛 𝑁𝑛 𝑆𝑦𝑁 2 �̅�𝑁 2 = ξ20 𝐸(�̅�𝑥𝑁 2 ) = 𝑁−𝑛 𝑁𝑛 𝑆𝑥𝑁 2 �̅�𝑁 2 = 𝜉02 𝐸(�̅�𝑥𝑁 �̅�𝑦𝑁) = 𝑁−𝑛 𝑁𝑛 𝑆𝑦𝑁𝑆𝑥𝑁 �̅�𝑁�̅�𝑁 = 𝜉11, where �̅�𝑦𝑁 ∈ [�̅�𝑦𝑁𝐿 , �̅�𝑦𝑁𝑈 ], �̅�𝑥𝑁 ∈ [�̅�𝑥𝑁𝐿 , �̅�𝑥𝑁𝑈 ], �̅�𝑦𝑁 2 ∈ [�̅�𝑦𝑁𝐿 2 , �̅�𝑦𝑁𝑈 2 ], �̅�𝑥𝑁 2 ∈ [�̅�𝑥𝑁𝐿 2 , �̅�𝑥𝑁𝑈 2 ]. 3. some related neutrosophic estimators since neutrosophic probability sampling is a new area of research handful of ratio type estimators are proposed in this neutrosophic simple random sampling without replacement (nsrswor). tahir et al. [14] proposed the following ratio-type estimators given by �̅�𝑅𝑁 = �̅�𝑁 �̅�𝑁 �̅�𝑁 , (3.1) �̅�𝑆𝐷𝑟𝑁 = �̅�𝑁 �̅�𝑁 +𝐶𝑥𝑁 �̅�𝑁+𝐶𝑥𝑁 , (3.2) �̅�𝑆𝐾𝑟𝑁 = �̅�𝑁 �̅�𝑁+𝛽2(𝑥𝑁) �̅�𝑁+𝛽2(𝑥𝑁) , (3.3) �̅�𝑈𝑆𝑟𝑁 = �̅�𝑁 �̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁 �̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁 , (3.4) 4 int. j. anal. appl. (2023), 21:41 where �̅�𝑁 ∈ [�̅�𝑁𝐿 , �̅�𝑁𝑈 ] and 𝑦𝑅𝑁 ∈ [𝑦𝑅𝐿 , 𝑦𝑅𝑈 ], �̅�𝑈𝑆𝑟𝑁 ∈ [�̅�𝑆𝐷𝑟𝐿 , �̅�𝑆𝐷𝑟𝑈 ], �̅�𝑆𝐾𝑟𝑁 ∈ [�̅�𝑆𝐾𝑟𝐿 , �̅�𝑆𝐾𝑟𝑈 ], and �̅�𝑈𝑆𝑟𝑁 ∈ [�̅�𝑈𝑆𝑟𝐿 , �̅�𝑈𝑆𝑟𝑈 ]. their expressions of mses are: 𝑀𝑆𝐸(�̅�𝑅 ) = 𝑁−𝑛 𝑁𝑛 �̅�𝑁 2[𝐶𝑦𝑁 2 + 𝐶𝑥𝑁 2 − 2𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ], (3.5) 𝑀𝑆𝐸(�̅�𝑆𝐷𝑟𝑁 ) = 𝑁−𝑛 𝑁𝑛 �̅�𝑁 2 [𝐶𝑦𝑁 2 + ( �̅�𝑁 �̅�𝑁 +𝐶𝑥𝑁 ) 𝐶𝑥𝑁 2 − 2 ( �̅�𝑁 �̅�𝑁+𝐶𝑥𝑁 ) 𝐶𝑥𝑁𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ], (3.6) 𝑀𝑆𝐸(�̅�𝑆𝐾𝑟𝑁 ) = 𝑁−𝑛 𝑁𝑛 �̅�𝑁 2 [𝐶𝑦𝑁 2 + ( �̅�𝑁 �̅�𝑁+𝛽2(𝑥𝑁) ) 𝐶𝑥𝑁 2 − 2 ( �̅�𝑁 �̅�𝑁+𝛽2(𝑥𝑁) ) 𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ] (3.7) and 𝑀𝑆𝐸(�̅�𝑈𝑆𝑟 ) = 𝑁−𝑛 𝑁𝑛 �̅�𝑁 2 [𝐶𝑦𝑁 2 + ( �̅�𝑁𝛽2(𝑥𝑁) �̅�𝑁 𝛽2(𝑥𝑁)+𝐶𝑥𝑁 ) 𝐶𝑥𝑁 2 − 2 ( �̅�𝑁𝛽2(𝑥𝑁) �̅�𝑁𝛽2(𝑥𝑁)+𝐶𝑥𝑁 ) 𝐶𝑥𝑁 𝐶𝑦𝑁 𝜌𝑥𝑦𝑁 ] (3.8) where 𝐶𝑦𝑁 2 ∈ [𝐶𝑦𝑁𝐿 2 , 𝐶𝑦𝑁𝑈 2 ], 𝐶𝑥𝑁 2 ∈ [𝐶𝑥𝑁𝐿 2 , 𝐶𝑥𝑁𝑈 2 ] and 𝜌𝑥𝑦𝑁 ∈ [𝜌𝑥𝑦𝑁𝐿 , 𝜌𝑥𝑦𝑁𝑈 ]. 4. proposed neutrosophic generalized estimators the aim of this article is to propose a generalized neutrosophic exponential robust ratio type estimator of finite neutrosophic population mean. motivated by [14], [17] and [18] we propose the following generalized neutrosophic exponential robust ratio type estimator 𝑡𝑝𝑁 g = (o1�̅�𝑁 + 𝑜2(x̅n − x̅n))𝑒𝑥𝑝( �̅�𝑁ω +ψ 𝛼(�̅�𝑁ω+ψ)+(1−𝛼)(�̅�𝑁 ω+ψ) − 1), (4.1) where, 𝑜1 and 𝑜2 are scalars which minimizes the mse of the proposed generalizedneutrosophic estimator 𝑡𝑝 𝐺 . further, ω and ψ are scalaras which would assume different known population parameter values of neutrosophic auxiliary variable precisely hodges lehmann, tri-mea, mid range and coefficient of variation. it should be noted that 𝑡𝑝𝑁 g ∈ [𝑡𝑝 g 𝑁 𝐿 , 𝑡𝑝 g 𝑁 𝑈 ], 𝑜1 ∈ [𝑜1l , 𝑜1u ], 𝑜2 ∈ [𝑜2l , 𝑜2u ], �̅�𝑁 ∈ [�̅�𝑁𝐿 , �̅�𝑁𝑈 ]. in order to obtain the expression of bias and mean squared error of the proposed generalized neutrosophic estimator 𝑡𝑝 𝐺 , we re-write it usingerror terms defined in section 2 and using taylor series obtain the expression as follows 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺 ) = −�̅�𝑁 + �̅�𝑁 θ𝑜2𝜉02 + 𝑜1(�̅�𝑁 + 3 2 �̅�𝑁 𝜃 2𝜉02 − �̅�𝑁 𝜃𝜉11), (4.2) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝐺 ) = −�̅�𝑁 2 + �̅�𝑁 𝑜2(−2�̅�𝑁 𝜃 + �̅�𝑁 𝑜2)𝜉02 + �̅�𝑁 𝑜1(−2�̅�𝑁 + 𝜃(−3�̅�𝑁 𝜃 + 4�̅�𝑁 𝑜2)𝜉02 + 2(�̅�𝑁 𝜃 + �̅�𝑁 𝑜2)𝜉11) + �̅�𝑁 2𝑜1 2(1+4𝜃2𝜉2 02 − 4𝜃𝜉11 + ξ20). (4.3) partially differentiating 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝐺 ) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑜1 and 𝑜2to find their optimum values we get 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) (4.4) 5 int. j. anal. appl. (2023), 21:41 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3𝜉202−2𝜉11(−1+𝜃𝜉11)−𝜃𝜉02(2+𝜃𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) (4.5) using these optimum values we get 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺 ) = �̅�𝑁 2{4𝜉211+𝜉02{𝜃 4𝜉202−4𝜃 2𝜉211+4(−1+𝜃 2𝜉02)𝜉20}} 4{𝜉211−𝜉02(1+𝜉20)} , (4.6) where 𝜃 = α �̅�𝑁 ω �̅�𝑁ω+ψ . from the proposed generalized neutrosophic exponential robust ratio type estimator 𝑡𝑝 𝐺 we have developed five generalized neutrosophic exponential robust ratio type estimators. (i) 𝑡𝑝𝑁 𝐺1 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝( �̅�𝑁hl +tm �̅�𝑁hl+tm − 1) (4.7) the bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺1 ) = −�̅�𝑁 + �̅�𝑁 𝜃1𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 + 3 2 �̅�𝑁 𝜃1 2𝜉02 − �̅�𝑁 𝜃1𝜉11), (4.8) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺1 ) = �̅�𝑁 24𝜉211+𝜉02𝜃1 4𝜉202−4𝜃1 2𝜉211+4(−1+𝜃1 2𝜉02)𝜉20 4𝜉211−𝜉02(1+𝜉20) , (4.9) where 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃1 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) , (4.10) 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3𝜉202−2𝜉11(−1+𝜃1𝜉11)−𝜃1𝜉02(2+𝜃1𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) , (4.11) and 𝜃1 = �̅�𝑁hl �̅�𝑁hl+tm , where, 𝑡𝑝𝑁 𝐺1 ∈ [𝑡𝑝 𝐺1 𝐿 , 𝑡𝑝 𝐺1 𝑈 ], 𝜃1 ∈ [𝜃1𝐿 , 𝜃1𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿 , 𝑜1𝑜𝑝𝑡 𝑈 ] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿 , 𝑜2𝑜𝑝𝑡 𝑈 ]. (ii) 𝑡𝑝𝑁 𝐺2 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝( �̅�𝑁 tm +mr �̅�𝑁tm+mr − 1) (4.12) the bias and mse_opt are 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺2 ) = −�̅�𝑁 + �̅�𝑁 𝜃2𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 + 3 2 �̅�𝑁 𝜃2 2𝜉02 − �̅�𝑁 𝜃2𝜉11), (4.13) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺2 ) = �̅�𝑁 24𝜉211+𝜉02𝜃2 4𝜉202−4𝜃2 2𝜉211+4(−1+𝜃2 2𝜉02)𝜉20 4𝜉211−𝜉02(1+𝜉20) , (4.14) where 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃2 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) , (4.15) 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3 𝜉202−2𝜉11(−1+𝜃2𝜉11)−𝜃1𝜉02(2+𝜃2𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) , (4.16) and 𝜃2 = �̅�𝑁tm �̅�𝑁tm+mr where, 𝑡𝑝𝑁 𝐺2 ∈ [𝑡𝑝 𝐺2 𝐿 , 𝑡𝑝 𝐺2 𝑈 ], 𝜃2 ∈ [𝜃2𝐿 , 𝜃2𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿 , 𝑜1𝑜𝑝𝑡 𝑈 ] and 𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿 , 𝑜2𝑜𝑝𝑡 𝑈 ]. 6 int. j. anal. appl. (2023), 21:41 (iii) 𝑡𝑝𝑁 𝐺3 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝( �̅�𝑁hl +mr �̅�𝑁hl+mr − 1) (4.17) the bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺3 ) = −�̅�𝑁 + �̅�𝑁 𝜃3𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 + 3 2 �̅�𝑁 𝜃3 2𝜉02 − �̅�𝑁 𝜃3𝜉11), (4.18) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺3 ) = �̅�𝑁 24𝜉211+𝜉02𝜃3 4𝜉202−4𝜃3 2𝜉211+4(−1+𝜃3 2𝜉02)𝜉20 4𝜉211−𝜉02(1+𝜉20) , (4.19) where 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃3 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) , (4.20) 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3 𝜉202−2𝜉11(−1+𝜃3𝜉11)−𝜃3𝜉02 (2+𝜃3𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) , (4.21) and 𝜃3 = �̅�𝑁 hl �̅�𝑁 hl+mr where, 𝑡𝑝𝑁 𝐺3 ∈ [𝑡𝑝 𝐺3 𝐿 , 𝑡𝑝 𝐺3 𝑈 ], 𝜃3 ∈ [𝜃3𝐿 , 𝜃3𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿 , 𝑜1𝑜𝑝𝑡 𝑈 ] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2 𝑜𝑝𝑡 𝐿 , 𝑜2𝑜𝑝𝑡 𝑈 ]. (iv) 𝑡𝑝𝑁 𝐺4 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝( �̅�𝑁 cxn +𝐻𝐿 �̅�𝑁cxn +hl − 1) (4.22) the bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺4 ) = −�̅�𝑁 + �̅�𝑁 𝜃4𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 + 3 2 �̅�𝑁 𝜃4 2𝜉02 − �̅�𝑁 𝜃4𝜉11), (4.23) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺4 ) = �̅�𝑁 24𝜉211+𝜉02𝜃4 4𝜉202−4𝜃4 2𝜉211+4(−1+𝜃4 2𝜉02)𝜉20 4𝜉211−𝜉02(1+𝜉20) (4.24) where, 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃4 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) , (4.25) 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3 𝜉202−2𝜉11(−1+𝜃4𝜉11)−𝜃4𝜉02(2+𝜃4𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) , (4.26) and 𝜃4 = �̅�𝑁cxn �̅�𝑁cxn +hl where, 𝑡𝑝𝑁 𝐺4 ∈ [𝑡𝑝 𝐺4 𝐿 , 𝑡𝑝 𝐺4 𝑈 ], 𝜃4 ∈ [𝜃4𝐿 , 𝜃4𝑈], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿 , 𝑜1𝑜𝑝𝑡 𝑈 ] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2 𝑜𝑝𝑡 𝐿 , 𝑜2𝑜𝑝𝑡 𝑈 ]. (v) 𝑡𝑝𝑁 𝐺5 = (𝑜1�̅�𝑁 + 𝑜2(�̅�𝑁 − �̅�𝑁 ))𝑒𝑥𝑝( �̅�𝑁cxn +𝑇𝑀 �̅�𝑁cxn +tm − 1) (4.27) the bias and 𝑀𝑆𝐸𝑜𝑝𝑡 are 𝐵𝑖𝑎𝑠(𝑡𝑝𝑁 𝐺5 ) = −�̅�𝑁 + �̅�𝑁 𝜃5𝑜2𝑜𝑝𝑡 𝜉02 + 𝑜1𝑜𝑝𝑡 (�̅�𝑁 + 3 2 �̅�𝑁 𝜃5 2𝜉02 − �̅�𝑁 𝜃5𝜉11), (4.28) 𝑀𝑆𝐸 (𝑡𝑝𝑁 𝑜𝑝𝑡 𝐺5 ) = �̅�𝑁 24𝜉211+𝜉02𝜃5 4𝜉202−4𝜃5 2𝜉211+4(−1+𝜃5 2𝜉02)𝜉20 4𝜉211−𝜉02(1+𝜉20) (4.29) 7 int. j. anal. appl. (2023), 21:41 𝑜1𝑜𝑝𝑡 = 𝜉02(2−𝜃5 2𝜉02) 2(−𝜉211+𝜉02(1+𝜉20)) , (4.30) 𝑜2𝑜𝑝𝑡 = �̅�𝑁{2𝜃 3 𝜉202−2𝜉11(−1+𝜃5𝜉11)−𝜃5𝜉02(2+𝜃5𝜉11−2𝜉20)) 2�̅�𝑁(−𝜉 2 11+𝜉02(1+𝜉20)) , (4.31) and 𝜃5 = �̅�𝑁cxn �̅�𝑁 cxn +tm where, 𝑡𝑝𝑁 𝐺5 ∈ [𝑡𝑝 𝐺5 𝐿 , 𝑡𝑝 𝐺5 𝑈 ], 𝜃5 ∈ [𝜃5𝐿 , 𝜃5𝑈 ], 𝑜1𝑜𝑝𝑡 ∈ [𝑜1𝑜𝑝𝑡 𝐿 , 𝑜1𝑜𝑝𝑡 𝑈 ] 𝑎𝑛𝑑 𝑜2𝑜𝑝𝑡 ∈ [𝑜2𝑜𝑝𝑡 𝐿 , 𝑜2𝑜𝑝𝑡 𝑈 ]. 5. empirical study in this section we have conducted an empirical study to demonstrate the high efficiency of the developed estimators. this study, is conducted using daily stock price of moderna. the rationale behind taking the stock price as a neutrosophic data is the fact that the daily stock price ranges between a high and a low values each day. pin pointing the point estimate of the daily stock price will not give a reliable estimate. thus, we have taken it as a neutrosophic dataset. in this empirical study, daily stock price of moderna has been considered form 1-september-2020 to 1-september2021 [20] (n=253). the neutrosophic survey variable 𝑦𝑁 i.e., varying price of the stock on each day where 𝑦𝑁 ∈ [𝑦𝑙 , 𝑦𝑢 ] ( 𝑦𝑢 is the highest price of the stock on each day and 𝑦𝑙 is the lowest price of the stock each day). 6. simulation study in this section we have conducted four simulation studies to demonstrate the high efficiency of the proposed generalized neutrosophic robust type exponential ratio estimator over similar existing ratio estimators discussed in this article. the comparison has been made on the basis of neutrosophic mses and neutrosophic res. 6.1 simulation study-1 the following algorithm is used in r language to perform the simulation study: (i) nutrosophic auxiliary variable 𝑥𝑁 has been generated from neutrosophic normal distribution nn([0.7, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single indetermincay where population mean 𝜇𝑋 is indeterminate. thus 𝑥𝑁 ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. (ii) neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 7𝑒 such that 𝑦𝑁 ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). 8 int. j. anal. appl. (2023), 21:41 (iii) for sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various values of neutrosophic estimates are obtained with 20000 iterations. (iv) for each neutrosophic sample size used, neutrosophic mses and res have been obtained and presented in tables. (v) values of estimates have been calculated under classical statistics as well and their mses and res are tabulated in tables 1-4. table 1: data statistics for empirical study 𝑆𝑦𝑈 2 = 9624, 𝑆𝑦𝐿 2 = 8111, 𝐶𝑦𝑈 2 = 0.3124, 𝐶𝑦𝐿 2 = 0.3022, 𝑆𝑥𝑈 2 = 8743, 𝑆𝑥𝐿 2 = 8965, 𝐶𝑥𝑈 2 = 0.3055, 𝐶𝑥𝐿 2 = 0.3092, 𝑁 = 253, 𝑛 = 160, 𝑇𝑀𝑈 = 150, 𝐻𝐿𝑈 = 153.44, 𝑀𝑅𝑈 = 270.76, β2𝑥𝑈 = 1.06, 𝑇𝑀𝐿 = 153, 𝐻𝐿𝐿 = 153.96, 𝑀𝑅𝐿 = 269.4, 𝛽2(𝑥𝐿 ) = 1.01 ρ𝑦𝑈𝑥𝑈 = 0.99, 𝜌𝑦𝐿𝑥𝐿 = 0.99. table 2: neutrosophic mse of the estimators estimators mse 𝑴𝑺𝑬[�̅�∗𝐿 , �̅�∗𝑈 ] relative eficiency 𝑹𝑬[�̅�∗𝐿 , �̅�∗𝑈 ] �̅�𝑅 𝑁 0.1038 0.1209 1 1 �̅�𝑆𝐷𝑟𝑁 0.1021 0.1223 1.01665 0.988553 �̅�𝑆𝐾𝑟𝑁 0.1009 0.124 1.028741 0.975 �̅�𝑈𝑆𝑟𝑁 0.1222 0.1021 0.849427 1.184133 𝑡𝑝𝑁 𝐺2 0.0835 0.116 1.243114 1.042241 𝑡𝑝𝑁 𝐺2 0.0836 0.116 1.241627 1.042241 𝑡𝑝𝑁 𝐺3 0.0836 0.1156 1.241627 1.045848 𝑡𝑝𝑁 𝐺4 0.0868 0.1193 1.195853 1.013412 𝑡𝑝𝑁 𝐺5 0.0869 0.119 1.194476 1.015966 *denotes appropriate estimator 9 int. j. anal. appl. (2023), 21:41 table 3: neutrosophic mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑀𝑆𝐸(𝐿, 𝑈) [60, 60] 0.79376 0.71468 0.68268 0.69029 0.92986 0.75644 0.67749 0.71476 0.36048 0.49557 0.37457 0.49494 0.49492 0.49492 0.38379 0.49608 0.38321 0.49603 [65, 65] 0.70738 0.64959 0.62434 0.63017 0.82481 0.68538 0.62051 0.65438 0.33174 0.46315 0.34235 0.46272 0.46271 0.46271 0.34892 0.46352 0.34851 0.46349 [70, 70] 0.6408 0.58546 0.56655 0.57157 1.02672 0.61103 0.56208 0.59378 0.30331 0.43304 0.31307 0.43287 0.43287 0.43287 0.31948 0.43323 0.31906 0.43321 [75, 75] 0.58023 0.53854 0.52302 0.52725 0.64351 0.55999 0.52060 0.54858 0.28057 0.40533 0.28759 0.40525 0.40525 0.40525 0.29178 0.40546 0.29152 0.40545 table 4: neutrosophic res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑅𝐸(𝐿, 𝑈) [60, 60] 1 1 1.162712 1.035319 0.085363 0.944794 1.171619 0.999888 2.201953 1.442137 2.119123 1.443973 2.121842 1.444031 2.068214 1.440655 2.071345 1.440800 [65, 65] 1 1 1.133004 1.030817 0.857628 0.947781 1.139998 0.99268 2.132333 1.402548 2.066248 1.403851 2.068363 1.403881 2.027342 1.401428 2.029727 1.401519 [70, 70] 1 1 1.131056 1.024301 0.624123 0.958153 1.140051 0.985988 2.11269 1.351977 2.046827 1.352508 2.048921 1.352508 2.005759 1.351384 2.0084 1.351446 [75, 75] 1 1 1.109384 1.021413 0.901664 0.961696 1.114541 0.981698 2.06804 1.328646 2.01756 1.328908 2.019104 1.328908 1.988587 1.32822 1.990361 1.328253 10 int. j. anal. appl. (2023), 21:41 table 5: classical mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 0.70002 0.67756 0.68675 0.67328 0.39037 0.38957 0.38955 0.39044 0.39042 65 0.63624 0.61897 0.62596 0.61597 0.34903 0.34853 0.34853 0.34907 0.34906 70 0.57653 0.56183 0.56783 0.55915 0.32683 0.32636 0.32635 0.32688 0.57341 75 0.53054 0.51869 0.52345 0.5168 0.31015 0.30982 0.30982 0.31018 0.31017 table 6: classical res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 1 1.03314 1.01932 1.03971 1.79322 1.79690 1.79699 1.79290 1.79299 65 1 1.02790 1.01642 1.03290 1.82288 1.82549 1.82549 1.82267 1.82272 70 1 1.02616 1.01532 1.03108 1.76400 1.76654 1.76660 1.76373 1.00544 75 1 1.02284 1.01354 1.02658 1.71059 1.71241 1.71241 1.71043 1.71048 6.2 simulation study-2 the following algorithm is used in r language to perform the simulation study: (i) nutrosophic auxiliary variable 𝑥𝑁 has been generated from neutrosophic normal distribution nn([0.7, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single indetermincay where population mean 𝜇𝑋 is indeterminate. thus 𝑥𝑁 ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. (ii) neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 6𝑒 such that 𝑦𝑁 ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). (iii) for sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various values of neutrosophic estimates are obtained with 20000 iterations. (iv) for each neutrosophic sample size used, neutrosophic mses and res have been obtained and presented in tables. (v) values of estimates have been calculated under classical statistics as well and their mses and res are tabulated in tables 5-8. 11 int. j. anal. appl. (2023), 21:41 table 7: neutrosophic mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑀𝑆𝐸(𝐿, 𝑈) [60, 60] 0.58317 0.52507 0.50408 0.50940 7.06050 0.55537 0.50234 0.53718 0.28861 0.42083 0.29898 0.42056 0.29863 0.42056 0.30594 0.42109 0.30549 0.42107 [65, 65] 0.51971 0.47725 0.4612 0.46516 0.60191 0.50290 0.46063 0.49209 0.26721 0.39468 0.27514 0.39456 0.27487 0.39455 0.28036 0.39484 0.28003 0.39483 [70, 70] 0.47079 0.43013 0.41825 0.42180 0.74261 0.44856 0.41663 0.44617 0.24573 0.37008 0.25288 0.37014 0.25264 0.37014 0.25790 0.70110 0.25756 0.37011 [75, 75] 0.42629 0.39566 0.38630 0.38914 0.47267 0.41108 0.38624 0.41245 0.22828 0.34735 0.23318 0.34747 0.23302 0.34748 0.23623 0.34735 0.23604 0.34734 table 8: neutrosophic res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑅𝐸(𝐿, 𝑈) [60, 60] 1 1 1.156900 1.030760 0.082596 0.945442 1.160907 0.977456 2.020616 1.247701 1.950532 1.248502 1.952818 1.248502 1.906158 1.246931 1.908966 1.24699 [65, 65] 1 1 1.126865 1.025991 0.863435 0.948996 1.128259 0.969843 1.94495 1.209207 1.888893 1.209575 1.890748 1.209606 1.853724 1.208717 1.855908 1.208748 [70, 70] 1 1 1.125619 1.019749 0.633967 0.958913 1.129995 0.964050 1.915883 1.162262 1.861713 1.162074 1.863482 1.16e-05 1.825475 0.613507 1.827885 1.162168 [75, 75] 1 1 1.103521 1.016755 0.901877 0.962489 1.103692 0.959292 1.867400 1.139082 1.828159 1.138688 1.829414 1.138655 1.804555 1.139082 1.806007 1.139114 12 int. j. anal. appl. (2023), 21:41 table 9: classical mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 0.51430 0.49854 0.50479 0.49612 0.33024 0.32978 0.32978 0.33028 0.33027 65 0.46744 0.45557 0.46018 0.45409 0.30102 0.30078 0.30077 0.30104 0.30103 70 0.42357 0.41335 0.41736 0.41196 0.2837 0.28345 0.28345 0.28373 0.28372 75 0.38978 0.38173 0.3848 0.38095 0.26904 0.26891 0.26891 0.26905 0.26905 table 10: classical res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 1 1.031612 1.01884 1.036644 1.557352 1.559525 1.559525 1.557164 1.557211 65 1 1.026055 1.015776 1.029399 1.552854 1.554093 1.554144 1.55275 1.552802 70 1 1.024725 1.014879 1.028182 1.493021 1.494338 1.494338 1.492863 1.492916 75 1 1.021088 1.012942 1.023179 1.448781 1.449481 1.449481 1.448727 1.448727 6.3 simulation study-3 the following algorithm is used in r language to perform the simulation study: (i) nutrosophic auxiliary variable 𝑥𝑁 has been generated from neutrosophic normal distribution nn([0.75, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single indetermincay where population mean 𝜇𝑋 is indeterminate. thus 𝑥𝑁 ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. (ii) neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 7𝑒 such that 𝑦𝑁 ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). (iii) for sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various values of neutrosophic estimates are obtained with 20000 iterations. (iv) for each neutrosophic sample size used, neutrosophic mses and res have been obtained and presented in tables . (v) values of estimates have been calculated under classical statistics as well and their mses and res are tabulated in tables 9-1. 13 int. j. anal. appl. (2023), 21:41 table 11: neutrosophic mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑀𝑆𝐸(𝐿, 𝑈) [60, 60] 0.56715 0.52507 0.50377 0.5094 0.64524 0.55537 0.50262 0.53718 0.30052 0.42083 0.30811 0.42056 0.30785 0.42056 0.31202 0.42109 0.31173 0.42107 [65, 65] 0.50815 0.47725 0.46082 0.46516 0.55396 0.5029 0.46063 0.49209 0.2788 0.39468 0.28427 0.39456 0.28405 0.39455 0.28699 0.39484 0.28679 0.39483 [70, 70] 0.46026 0.43013 0.41799 0.4218 5.3525 0.44856 0.41683 0.44617 0.25763 0.37008 0.26279 0.37014 0.26262 37014 0.26538 0.37011 0.26519 0.37011 [75, 75] 0.41836 0.39566 0.38598 0.38914 0.5705 0.41108 0.38646 0.41245 0.23982 0.34735 0.24343 0.34747 0.24331 0.34748 0.24543 0.34735 0.2451 0.34734 table 12: neutrosophic res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑅𝐸(𝐿, 𝑈) [60, 60] 1 1 1.125811 1.030762 0.878975 0.945442 1.128387 0.977456 1.887229 1.247701 1.840739 1.248502 1.842293 1.248502 1.817672 1.246931 1.819363 1.24699 [65, 65] 1 1 1.102708 1.025991 0.917304 0.948996 1.103163 0.969843 1.822633 1.209207 1.787561 1.209575 1.788946 1.209606 1.770619 1.208717 1.771854 1.208748 [70, 70] 1 1 1.101127 1.019749 0.08599 0.958913 1.104191 0.96405 1.786516 1.162262 1.751437 1.162074 1.75257 1.16e-05 1.734343 1.162168 1.735586 1.162168 [75, 75] 1 1 1.08389 1.016755 0.733322 0.962489 1.082544 0.959292 1.744475 1.139082 1.718605 1.138688 1.719453 1.138655 1.7046 1.139082 1.706895 1.139114 14 int. j. anal. appl. (2023), 21:41 table 13: classical mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 0.51298 0.4984 0.50418 49604 0.33895 0.33853 0.33852 0.33888 0.33887 65 0.46636 0.45541 0.45966 0.454 0.30994 0.30979 0.30972 0.3099 0.30989 70 0.4227 0.41324 0.41695 0.41189 0.2924 0.29217 0.29216 0.29236 0.29235 75 0.38903 0.38161 0.38443 0.38087 0.27718 0.27707 0.27707 0.27716 0.27716 table 14: classical res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 1 1.029254 1.017454 1.03e-05 1.513439 1.515316 1.515361 1.513751 1.513796 65 1 1.024044 1.014576 1.027225 1.504678 1.505407 1.505747 1.504873 1.504921 70 1 1.022892 1.013791 1.026245 1.445622 1.44676 1.44681 1.44582 1.44587 75 1 1.019444 1.011966 1.021425 1.403528 1.404086 1.404086 1.40363 1.40363 6.4 simulation study-4 the following algorithm is used in r language to perform the simulation study: (i) nutrosophic auxiliary variable 𝑥𝑁 has been generated from neutrosophic normal distribution nn([0.65, 1.1], 1.2) i.e., the neutrosophic auxiliary variable x has single indetermincay where population mean 𝜇𝑋 is indeterminate. thus 𝑥𝑁 ∈ [𝑥𝑁 𝐿 , 𝑥𝑁 𝑈 ]. (ii) neutrosophic survey variable is generated using the model 𝑦𝑁 = 𝑥𝑁 − 6𝑒 such that 𝑦𝑁 ∈ [𝑦𝑁 𝐿 , 𝑦𝑁 𝑈 ] where 𝑒 ~𝑁(0, 1). (iii) for sample sizes 𝑛1 ∈ [60, 60], 𝑛2 ∈ [65, 65], 𝑛3 ∈ [70, 70] and 𝑛4 ∈ [75, 75] various values of neutrosophic estimates are obtained with 20000 iterations. (iv) for each neutrosophic sample size used, neutrosophic mses and res have been obtained and presented in tables. (v) values of estimates have been calculated under classical statistics as well and their mses and res are tabulated in tables 13-16 15 int. j. anal. appl. (2023), 21:41 table 15: neutrosophic mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑀𝑆𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑀𝑆𝐸(𝐿, 𝑈) [60, 60] 0.60649 0.52507 0.50434 0.50940 160621 0.55537 0.50209 0.53718 0.28066 0.42083 0.29939 0.42056 0.29870 0.42056 0.31687 0.42109 0.31575 0.42107 [65, 65] 0.53563 0.47725 0.46151 0.46516 0.95151 0.50290 0.46012 0.49209 0.26131 0.39468 0.28742 0.39456 0.28619 0.39455 0.32693 0.39484 0.32387 0.39483 [70, 70] 0.48835 0.43013 0.41848 0.42180 0.58047 0.44856 0.41644 0.44617 0.23605 0.37008 0.25087 0.37014 0.25018 0.37014 0.27917 0.37011 0.27641 0.37011 [75, 75] 0.43770 0.39566 0.38660 0.38914 3.32296 0.41108 0.38605 0.41245 0.21814 0.34735 0.22565 0.34747 0.22539 0.34748 0.23187 0.34735 0.23148 0.34734 table 16: neutrosophic res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐷𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑆𝐾𝑟𝑁 𝑅𝐸(𝐿, 𝑈) �̅�𝑈𝑆𝑟𝑁 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺1 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺2 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺3 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺4 𝑅𝐸(𝐿, 𝑈) 𝑡𝑝𝑁 𝐺5 𝑅𝐸(𝐿, 𝑈) [60, 60] 1 1 1.202542 1.030762 3.78e-06 0.945442 1.207931 0.977456 2.160942 1.247701 2.025752 1.248502 2.030432 1.248502 1.914003 1.246931 1.920792 1.24699 [65, 65] 1 1 1.160603 1.025991 0.562926 0.948996 1.164109 0.969843 2.049788 1.209207 1.863579 1.209575 1.871589 1.209606 1.638363 1.208717 1.653843 1.208748 [70, 70] 1 1 1.166961 1.019749 0.841301 0.958913 1.172678 0.96405 2.068841 1.162262 1.946626 1.162074 1.951995 1.162074 1.749293 1.162168 1.76676 1.162168 [75, 75] 1 1 1.132178 1.016755 0.13172 0.962489 1.133791 0.959292 2.00651 1.139082 1.93973 1.138688 1.941967 1.138655 1.887696 1.139082 1.890876 1.139114 16 int. j. anal. appl. (2023), 21:41 table 17: classical mses of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 0.51574 0.49868 0.50544 0.4962 0.3217 0.32121 0.3212 0.32189 0.32188 65 0.46836 0.45572 0.46072 0.45418 0.29226 0.292 0.29199 0.29238 0.29237 70 0.42453 0.41346 0.4187 0.41203 0.27515 0.27487 0.27486 0.27526 0.27525 75 0.3906 0.38186 0.3852 0.38103 0.26101 0.26086 0.26085 0.26108 0.26107 table 18: classical res of all the neutrosophic estimators sample size �̅�𝑅 𝑁 �̅�𝑆𝐷𝑟𝑁 �̅�𝑆𝐾𝑟𝑁 �̅�𝑈𝑆𝑟𝑁 𝑡𝑝𝑁 𝐺1 𝑡𝑝𝑁 𝐺2 𝑡𝑝𝑁 𝐺3 𝑡𝑝𝑁 𝐺4 𝑡𝑝𝑁 𝐺5 60 1 1.03421 1.020378 1.039379 1.603171 1.605616 1.605666 1.602224 1.602274 65 1 1.027736 1.016583 1.031221 1.602546 1.603973 1.604028 1.601888 1.601943 70 1 1.026774 1.013924 1.030338 1.542904 1.544476 1.544532 1.542287 1.542343 75 1 1.022888 1.014019 1.025116 1.496494 1.497355 1.497412 1.496093 1.49615 7. discussion and conclusion vagueness or indetermincacy is usually observed in the collected data. instead of using fuzzy logic to deal with such a data set it would be more easier and cost resource efficient to use neutrosophic statistical tools. due to its need and wide applicability research in neutrosophic statistics has been regorously carried out. this paper aims at further developing the existing theory of neutrosophic simple random sampling without replacement (nsrswor). in this paper, a generalized neutrosophic exponential robust ratio type estimator 𝑡𝑝𝑁 g has been presented using some known population parameters of neutrosophic auxiliary variables. from the proposed generalized neutrosophic exponential robust ratio type estimator, five generalized neutrosophic exponential robust ratio type estimators 𝑡𝑝𝑁 𝐺1 -𝑡𝑝𝑁 𝐺5 have been developed using known population parameter values of auxiliary variables viz., hodges lehmann, tri mean, mid range and coefficient of variation. the high efficiency of the developed neutrosophic estimators have been demonstrated using an empirical and four simulation studies the results of which are presented in tables 1-18. 17 int. j. anal. appl. (2023), 21:41 in the empirical study on daily stock price, we can see that the proposed estimators provide a lower mse indicating high efficiency that the similar existing neutrosophic estimators. in simulation studies, it is clear from the results that the developed neutrosophic estimators 𝑡𝑝𝑁 𝐺1 -𝑡𝑝𝑁 𝐺5 𝑓𝑟𝑜𝑚 the proposed generalized neutrosphic estimator 𝑡𝑝𝑁 g provide a much lower mse as compared to the similar existing neutrosophic ratio type estimators discussed in this paper (table [3-4], table [7-8], table [11-12] and table [15-16]). the results of the neutrosophic estimators estimators have also been compared with their classical values (table [5-6], table [9-10], table [13-14] and table [17-18]). it can be seen that the classical values of mse falls in the indetermancy neutrosophic mse intervals implying that when the data contains some indetermincay, neutrosophic estimators should be used. further, it can be seen that, proposed neutrosophic estimators 𝑡𝑝𝑁 𝐺1 -𝑡𝑝𝑁 𝐺5 provide lowest mse in neutrosophic as well as classical form and thus it is advised to use the proposed neutrosophic estimators 𝑡𝑝𝑁 𝐺1 -𝑡𝑝𝑁 𝐺5 when the data at hand is neutrosophic. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. jan, l. zedam, t. mahmood, k. ullah, z. ali, multiple attribute decision making method under linguistic cubic information, j. intell. fuzzy syst. 36 (2019), 253–269. https://doi.org/10.3233/jifs181253. [2] d.f. li, t. mahmood, z. ali, y. dong, decision making based on interval-valued complex single-valued neutrosophic hesitant fuzzy generalized hybrid weighted averaging operators, j. intell. fuzzy syst. 38 (2020), 4359–4401. https://doi.org/10.3233/jifs-191005. [3] f. smarandache, introduction to neutrosophic statistics, arxiv. (2014). https://doi.org/10.48550/arxiv.1406.2000. [4] r. varshney, a. pal, mradula, i. ali, optimum allocation in the multivariate cluster sampling design under gamma cost function, j. stat. comput. simul. 93 (2022), 312–323. https://doi.org/10.1080/00949655.2022.2104845. [5] n. gupta, i. ali, shafiullah, a. bari, a fuzzy goal programming approach in stochastic multivariate stratified sample surveys, south pac. j. nat. app. sci. 31 (2013), 80-88. https://doi.org/10.1071/sp13009. https://doi.org/10.3233/jifs-181253 https://doi.org/10.3233/jifs-181253 https://doi.org/10.3233/jifs-191005 https://doi.org/10.48550/arxiv.1406.2000 https://doi.org/10.1080/00949655.2022.2104845 https://doi.org/10.1071/sp13009 18 int. j. anal. appl. (2023), 21:41 [6] n. kumar adichwal, a. ali h. ahmadini, y. singh raghav, r. singh, i. ali, estimation of general parameters using auxiliary information in simple random sampling without replacement, j. king saud univ. – sci. 34 (2022), 101754. https://doi.org/10.1016/j.jksus.2021.101754. [7] r. singh, r. mishra, ratio-cum-product type estimators for rare and hidden clustered population, sankhya b. (2022). https://doi.org/10.1007/s13571-022-00298-x. [8] a. haq, j. shabbir, improved family of ratio estimators in simple and stratified random sampling, commun. stat. – theory methods. 42 (2013), 782–799. https://doi.org/10.1080/03610926.2011.579377. [9] z. ali, t. mahmood, complex neutrosophic generalised dice similarity measures and their application to decision making, caai trans. intell. technol. 5 (2020), 78–87. https://doi.org/10.1049/trit.2019.0084. [10] m. aslam, a new sampling plan using neutrosophic process loss consideration, symmetry. 10 (2018), 132. https://doi.org/10.3390/sym10050132. [11] m. aslam, neutrosophic analysis of variance: application to university students, complex intell. syst. 5 (2019), 403–407. https://doi.org/10.1007/s40747-019-0107-2. [12] m. aslam, monitoring the road traffic crashes using newma chart and repetitive sampling, int. j. injury control safe. promotion. 28 (2020), 39–45. https://doi.org/10.1080/17457300.2020.1835990. [13] m. aslam, a new goodness of fit test in the presence of uncertain parameters, complex intell. syst. 7 (2020), 359–365. https://doi.org/10.1007/s40747-020-00214-8. [14] z. tahir, h. khan, m. aslam, j. shabbir, y. mahmood, f. smarandache, neutrosophic ratio-type estimators for estimating the population mean, complex intell. syst. 7 (2021), 2991–3001. https://doi.org/10.1007/s40747-021-00439-1. [15] yahoo finance: tesla. https://finance.yahoo.com/quote/tsla/history/. accessed 2021-09-13. [16] national family health survey (nfhs-4), (2015-2016). http://rchiips.org/nfhs/factsheet_nfhs-4.shtml. [17] r. singh, r. mishra, improved exponential ratio estimators in adaptive cluster sampling, j. stat. appl. probab. lett. 9 (2022), 19–29. https://doi.org/10.18576/jsapl/090103. [18] z. yan, b. tian, ratio method to the mean estimation using coefficient of skewness of auxiliary variable, in: r. zhu, y. zhang, b. liu, c. liu (eds.), information computing and applications, springer berlin heidelberg, berlin, heidelberg, 2010: pp. 103–110. https://doi.org/10.1007/978-3-642-163395_14. [19] r. mishra, b. ram, portfolio selection using r, yugoslav j. oper. res. 30 (2020), 137–146. https://doi.org/10.2298/yjor181115002m. [20] yahoo finance: mrna. https://finance.yahoo.com/quote/mrna/history/. accessed 2021-09-13. https://doi.org/10.1016/j.jksus.2021.101754 https://doi.org/10.1007/s13571-022-00298-x https://doi.org/10.1080/03610926.2011.579377 https://doi.org/10.1049/trit.2019.0084 https://doi.org/10.3390/sym10050132 https://doi.org/10.1007/s40747-019-0107-2 https://doi.org/10.1080/17457300.2020.1835990 https://doi.org/10.1007/s40747-020-00214-8 https://doi.org/10.1007/s40747-021-00439-1 https://finance.yahoo.com/quote/tsla/history/ http://rchiips.org/nfhs/factsheet_nfhs-4.shtml https://doi.org/10.18576/jsapl/090103 https://doi.org/10.1007/978-3-642-16339-5_14 https://doi.org/10.1007/978-3-642-16339-5_14 https://doi.org/10.2298/yjor181115002m https://finance.yahoo.com/quote/mrna/history/ international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 68-77 http://www.etamaths.com coefficient estimates of meromorphic bistarlike functions of complex order t. janani and g. murugusundaramoorthy∗ abstract. in the present investigation, we define a new subclass of meromorphic bi-univalent functions class σ′ of complex order γ ∈ c\{0}, and obtain the estimates for the coefficients |b0| and |b1|. further we pointed out several new or known consequences of our result. 1. introduction and definitions denote by a the class of analytic functions of the form (1.1) f(z) = z + ∞∑ n=2 anz n which are univalent in the open unit disc ∆ = {z : |z| < 1}. also denote by s the class of all functions in a which are univalent and normalized by the conditions f(0) = 0 = f′(0) − 1 in ∆. some of the important and well-investigated subclasses of the univalent function class s includes the class s∗(α)(0 ≤ α < 1) of starlike functions of order α in ∆ and the class k(α)(0 ≤ α < 1) of convex functions of order α < ( z f′(z) f(z) ) > α or < ( 1 + z f′′(z) f′(z) ) > α, (z ∈ ∆) respectively.further a function f(z) ∈a is said to be in the class s(γ) of univalent function of complex order γ(γ ∈ c\{0}) if and only if f(z) z 6= 0 and < ( 1 + 1 γ [ zf′(z) f(z) − 1 ]) > 0,z ∈ ∆. by taking γ = (1 − α)cosβ e−iβ, |β| < π 2 and 0 ≤ α < 1, the class s((1 − α)cosβ e−iβ) ≡ s(α,β) called the generalized class of β-spiral-like functions of order α(0 ≤ α < 1). an analytic function ϕ is subordinate to an analytic function ψ, written by ϕ(z) ≺ ψ(z), 2000 mathematics subject classification. 30c45 , 30c50. key words and phrases. analytic functions; univalent functions; meromorphic functions; biunivalent functions; bi-starlike and bi-convex functions of complex order; coefficient bounds. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 68 coefficient estimates of meromorphic bistarlike functions 69 provided there is an analytic function ω defined on ∆ with ω(0) = 0 and |ω(z)| < 1 satisfying ϕ(z) = ψ(ω(z)). ma and minda [9] unified various subclasses of starlike and convex functions for which either of the quantity z f′(z) f(z) or 1 + z f′′(z) f′(z) is subordinate to a more general superordinate function. for this purpose, they considered an analytic function φ with positive real part in the unit disk ∆,φ(0) = 1,φ′(0) > 0 and φ maps ∆ onto a region starlike with respect to 1 and symmetric with respect to the real axis. the class of ma-minda starlike functions consists of functions f ∈ a satisfying the subordination z f′(z) f(z) ≺ φ(z). similarly, the class of ma-minda convex functions consists of functions f ∈ a satisfying the subordination 1 + z f′′(z) f′(z) ≺ φ(z). it is well known that every function f ∈s has an inverse f−1, defined by f−1(f(z)) = z, (z ∈ ∆) and f(f−1(w)) = w, (|w| < r0(f); r0(f) ≥ 1/4) where (1.2) f−1(w) = w −a2w2 + (2a22 −a3)w 3 − (5a32 − 5a2a3 + a4)w 4 + · · · . a function f ∈a given by (1.1), is said to be bi-univalent in ∆ if both f(z) and f−1(z) are univalent in ∆, these classes are denoted by σ. earlier, brannan and taha [2] introduced certain subclasses of bi-univalent function class σ, namely bistarlike functions s∗σ(α) and bi-convex function kς(α) of order α corresponding to the function classes s∗(α) and k(α) respectively. for each of the function classes s∗σ(α) and kς(α), non-sharp estimates on the first two taylor-maclaurin coefficients |a2| and |a3| were found [2, 17]. but the coefficient problem for each of the following taylor-maclaurin coefficients: |an| (n ∈ n\{1, 2}; n := {1, 2, 3, · · ·}) is still an open problem(see[1, 2, 8, 10, 17]). recently several interesting subclasses of the bi-univalent function class σ have been introduced and studied in the literature(see[15, 18, 19]). a function f is bi-starlike of ma-minda type or bi-convex of ma-minda type if both f and f−1 are respectively ma-minda starlike or convex. these classes are denoted respectively by s∗σ(φ) and kς(φ).in the sequel, it is assumed that φ is an analytic function with positive real part in the unit disk ∆, satisfying 70 janani and murugusundaramoorthy φ(0) = 1,φ′(0) > 0 and φ(∆) is symmetric with respect to the real axis. such a function has a series expansion of the form (1.3) φ(z) = 1 + b1z + b2z 2 + b3z 3 + · · · , (b1 > 0). let σ′ denote the class of meromorphic univalent functions g of the form (1.4) g(z) = z + b0 + ∞∑ n=1 bn zn defined on the domain ∆∗ = {z : 1 < |z| < ∞}. since g ∈ σ′ is univalent, it has an inverse g−1 = h that satisfy g−1(g(z)) = z, (z ∈ ∆∗) and g(g−1(w)) = w, (m < |w| < ∞,m > 0) where (1.5) g−1(w) = h(w) = w + ∞∑ n=0 cn wn , (m < |w| < ∞). analogous to the bi-univalent analytic functions, a function g ∈ σ′ is said to be meromorphic bi-univalent if g−1 ∈ σ′. we denote the class of all meromorphic bi-univalent functions by mς′. estimates on the coefficients of meromorphic univalent functions were widely investigated in the literature, for example, schiffer[13] obtained the estimate |b2| ≤ 23 for meromorphic univalent functions g ∈ σ ′ with b0 = 0 and duren [3] gave an elementary proof of the inequality |bn| ≤ 2(n+1) on the coefficient of meromorphic univalent functions g ∈ σ′ with bk = 0 for 1 ≤ k < n2 . for the coefficient of the inverse of meromorphic univalent functions h ∈ mς′ , springer [14] proved that |c3| ≤ 1 and |c3 + 12c 2 1| ≤ 1 2 and conjectured that |c2n−1| ≤ (2n−1)! n!(n−1)!, (n = 1, 2, ...). in 1977, kubota [7] has proved that the springer conjecture is true for n = 3, 4, 5 and subsequently schober [12] obtained a sharp bounds for the coefficients c2n−1, 1 ≤ n ≤ 7 of the inverse of meromorphic univalent functions in ∆∗. recently, kapoor and mishra [6] (see [16]) found the coefficient estimates for a class consisting of inverses of meromorphic starlike univalent functions of order α in ∆∗. motivated by the earlier work of [4, 5, 6, 20], in the present investigation, a new subclass of meromorphic bi-univalent functions class σ′ of complex order γ ∈ c\{0}, is introduced and estimates for the coefficients |b0| and |b1| of functions in the newly introduced subclass are obtained. several new consequences of the results are also pointed out. definition 1.1. for 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ a function g(z) ∈ σ′ given by (1.4) is said to be in the class mγς′ (λ,µ,φ) if the following conditions are satisfied: (1.6) 1 + 1 γ [ (1 −λ) ( g(z) z )µ + λg′(z) ( g(z) z )µ−1 − 1 ] ≺ φ(z) and (1.7) 1 + 1 γ [ (1 −λ) ( h(w) w )µ + λh′(w) ( h(w) w )µ−1 − 1 ] ≺ φ(w) where z,w ∈ ∆∗, γ ∈ c\{0} and the function h is given by (1.5). coefficient estimates of meromorphic bistarlike functions 71 by suitably specializing the parameters λ and µ, we state the new subclasses of the class meromorphic bi-univalent functions of complex order mγς′ (λ,µ,φ) as illustrated in the following examples. example 1.1. for 0 ≤ λ < 1,µ = 1 a function g ∈ σ′ given by (1.4) is said to be in the class mγς′ (λ, 1,φ) ≡ f γ σ′ (λ,φ) if it satisfies the following conditions respectively: 1 + 1 γ [ (1 −λ) ( g(z) z ) + λg′(z) − 1 ] ≺ φ(z) and 1 + 1 γ [ (1 −λ) ( h(w) w ) + λh′(w) − 1 ] ≺ φ(w) where z,w ∈ ∆∗, γ ∈ c\{0} and the function h is given by (1.5). example 1.2. for λ = 1, 0 ≤ µ < 1 a function g ∈ σ′ given by (1.4) is said to be in the class mγς′ (1,µ,φ) ≡ b γ σ′ (µ,φ) if it satisfies the following conditions respectively: 1 + 1 γ [ g′(z) ( g(z) z )µ−1 − 1 ] ≺ φ(z) and 1 + 1 γ [ h′(w) ( h(w) w )µ−1 − 1 ] ≺ φ(w) where z,w ∈ ∆∗, γ ∈ c\{0} and the function h is given by (1.5). example 1.3. for λ = 1,µ = 0, a function g ∈ σ′ given by (1.4) is said to be in the class mγς′ (1, 0,φ) ≡s γ σ′ (φ) if it satisfies the following conditions respectively: 1 + 1 γ ( zg′(z) g(z) − 1 ) ≺ φ(z) and 1 + 1 γ ( wh′(w) h(w) − 1 ) ≺ φ(w) where z,w ∈ ∆∗, γ ∈ c\{0} and the function h is given by (1.5). 2. coefficient estimates for the function class mγς′ (λ,µ,φ) in this section we obtain the coefficients |b0| and |b1| for g ∈m γ σ′ (λ,µ,φ) associating the given functions with the functions having positive real part. in order to prove our result we recall the following lemma. lemma 2.1. [11] if φ ∈ p, the class of all functions with <(φ(z)) > 0, (z ∈ ∆) then |ck| ≤ 2, for each k, where φ(z) = 1 + c1z + c2z 2 + · · · for z ∈ ∆. 72 janani and murugusundaramoorthy define the functions p and q in p given by p(z) = 1 + u(z) 1 −u(z) = 1 + p1 z + p2 z2 + · · · and q(z) = 1 + v(z) 1 −v(z) = 1 + q1 z + q2 z2 + · · · . it follows that u(z) = p(z) − 1 p(z) + 1 = 1 2 [ p1 z + ( p2 − p21 2 ) 1 z2 + · · · ] and v(z) = q(z) − 1 q(z) + 1 = 1 2 [ q1 z + ( q2 − q21 2 ) 1 z2 + · · · ] . note that for the functions p(z),q(z) ∈p, we have |pi| ≤ 2 and |qi| ≤ 2 for each i. theorem 2.1. let g is given by (1.4) be in the class mγς′ (λ,µ,φ). then (2.1) |b0| ≤ ∣∣∣∣ γb1µ−λ ∣∣∣∣ and (2.2) |b1| ≤ ∣∣∣∣∣∣ γ √( (µ− 1)γb21 2(µ−λ)2 )2 + ( b2 µ− 2λ )2 ∣∣∣∣∣∣ where γ ∈ c\{0}, 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗. proof. it follows from (1.6) and (1.7) that (2.3) 1 + 1 γ [ (1 −λ) ( g(z) z )µ + λg′(z) ( g(z) z )µ−1 − 1 ] = φ(u(z)) and (2.4) 1 + 1 γ [ (1 −λ) ( h(w) w )µ + λh′(w) ( h(w) w )µ−1 − 1 ] = φ(v(w)). in light of (1.4), (1.5), (1.6) and (1.7), we have (2.5) 1 + 1 γ [ (1 −λ) ( g(z) z )µ + λg′(z) ( g(z) z )µ−1 − 1 ] = 1 + 1 γ [ (µ−λ) b0 z + (µ− 2λ)[ (µ− 1) 2 b20 + b1] 1 z2 + ... ] = 1 + b1p1 1 2z + [ 1 2 b1(p2 − p21 2 ) + 1 4 b2p 2 1 ] 1 z2 + ... coefficient estimates of meromorphic bistarlike functions 73 and (2.6) 1 + 1 γ [ (1 −λ) ( h(w) w )µ + λh′(w) ( h(w) w )µ−1 − 1 ] = 1 + 1 γ [ −(µ−λ) b0 z + (µ− 2λ)[ (µ− 1) 2 b20 − b1] 1 z2 + ... ] = 1 + b1q1 1 2w + [ 1 2 b1(q2 − q21 2 ) + 1 4 b2q 2 1 ] 1 w2 + ... now, equating the coefficients in (2.5) and (2.6), we get (2.7) 1 γ (µ−λ)b0 = 1 2 b1p1, (2.8) 1 γ (µ− 2λ) [ (µ− 1) b20 2 + b1 ] = 1 2 b1(p2 − p21 2 ) + 1 4 b2p 2 1, (2.9) − 1 γ (µ−λ)b0 = 1 2 b1q1, and (2.10) 1 γ (µ− 2λ) [ (µ− 1) b20 2 − b1 ] = 1 2 b1(q2 − q21 2 ) + 1 4 b2q 2 1. from (2.7) and (2.9), we get (2.11) p1 = −q1 and 8(µ−λ)2b20 = γ 2b21 (p 2 1 + q 2 1 ). hence, (2.12) b20 = γ2b21 (p 2 1 + q 2 1 ) 8(µ−λ)2 . applying lemma (2.1) for the coefficients p1 and q1, we have |b0| ≤ ∣∣∣∣ γb1µ−λ ∣∣∣∣ . next, in order to find the bound on |b1| from (2.8), (2.10) and (2.11), we obtain (2.13) (µ− 2λ)2 b21 = (µ− 2λ) 2(µ− 1)2 b40 4 −γ2 ( b21 4 p2q2 + (b2 −b1)b1(p2 + q2) p21 8 + (b1 −b2)2 p41 16 ) . using (2.12) and applying lemma (2.1) once again for the coefficients p1,p2 and q2, we get |b1| ≤ ∣∣∣∣∣∣ γ √( (µ− 1)γb21 2(µ−λ)2 )2 + ( b2 µ− 2λ )2 ∣∣∣∣∣∣ . � 74 janani and murugusundaramoorthy corollary 2.1. let g(z) is given by (1.4) be in the class fγς′ (λ,φ). then (2.14) |b0| ≤ ∣∣∣∣ γb11 −λ ∣∣∣∣ and (2.15) |b1| ≤ ∣∣∣∣ γb22λ− 1 ∣∣∣∣ where γ ∈ c\{0}, 0 ≤ λ < 1 and z,w ∈ ∆∗. corollary 2.2. let g(z) is given by (1.4) be in the class bγς′ (µ,φ). then (2.16) |b0| ≤ ∣∣∣∣ γb1µ− 1 ∣∣∣∣ and (2.17) |b1| ≤ ∣∣∣∣∣∣ γ √( γb21 2(µ− 1) )2 + ( b2 µ− 2 )2 ∣∣∣∣∣∣ where γ ∈ c\{0}, 0 ≤ µ < 1 and z,w ∈ ∆∗. corollary 2.3. let g(z) is given by (1.4) be in the class sγς′ (φ). then (2.18) |b0| ≤ |γ b1| and (2.19) |b1| ≤ ∣∣∣∣γ2 √ γ2b41 + b 2 2 ∣∣∣∣ where γ ∈ c\{0} and z,w ∈ ∆∗. 3. corollaries and concluding remarks analogous to (1.3), by setting φ(z) as given below: (3.1) φ(z) = ( 1 + z 1 −z )α = 1 + 2αz + 2α2z2 + · · · (0 < α ≤ 1), we have b1 = 2α, b2 = 2α 2. for γ = 1 and φ(z)is given by (3.1) we state the following corollaries: corollary 3.1. let g is given by (1.4) be in the class m1σ′ (λ,µ, ( 1+z 1−z )α ) ≡ mς′ (λ,α). then |b0| ≤ 2α |µ−λ| and |b1| ≤ ∣∣∣∣∣ 2α2 √ (µ− 1)2 (µ−λ)4 + 1 (µ− 2λ)2 ∣∣∣∣∣ where 0 < λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗. coefficient estimates of meromorphic bistarlike functions 75 corollary 3.2. let g(z) is given by (1.4) be in the class f1σ′ (λ, ( 1+z 1−z )α ) ≡fς′ (λ,α), then |b0| ≤ 2α |1 −λ| and |b1| ≤ 2α2 |1 − 2λ| where 0 ≤ λ < 1 and z,w ∈ ∆∗. corollary 3.3. let g(z) is given by (1.4) be in the class b1σ′ (λ, ( 1+z 1−z )α ) ≡bς′ (µ,α), then |b0| ≤ 2α |µ− 1| and |b1| ≤ ∣∣∣∣∣ 2α2 √ 1 (µ− 1)2 + 1 (µ− 2)2 ∣∣∣∣∣ where 0 ≤ µ < 1 and z,w ∈ ∆∗. corollary 3.4. let g(z) is given by (1.4) be in the class s1σ′ ([ 1+z 1−z ]α) ≡ sς′ (α) then |b0| ≤ 2α and |b1| ≤ α2 √ 5 where z,w ∈ ∆∗. on the other hand if we take (3.2) φ(z) = 1 + (1 − 2β)z 1 −z = 1 + 2(1 −β)z + 2(1 −β)z2 + · · · (0 ≤ β < 1), then b1 = b2 = 2(1 −β). for γ = 1 and φ(z)is given by (3.2) we state the following corollaries: corollary 3.5. let g is given by (1.4) be in the class m1σ′ ( λ,µ, 1+(1−2β)z 1−z ) ≡ mς′ (λ,µ,β). then |b0| ≤ 2(1 −β) |µ−λ| and |b1| ≤ ∣∣∣∣∣ 2(1 −β) √ (µ− 1)2(1 −β)2 (µ−λ)4 + 1 (µ− 2λ)2 ∣∣∣∣∣ where 0 ≤ λ ≤ 1,µ ≥ 0,µ > λ and z,w ∈ ∆∗. remark 3.1. we obtain the estimates |b0| and |b1| as obtained in the corollaries 3.2 to 3.4 for function g given by (1.4) are in the subclasses defined in examples 1.1 to 1.3. 76 janani and murugusundaramoorthy concluding remarks: let a function g ∈ σ′ given by (1.4). by taking γ = (1 − α)cosβ e−iβ, |β| < π 2 , 0 ≤ α < 1 the class mγς′ (λ,µ,φ) ≡ m β σ′ (α,λ,µ,φ) called the generalized class of β− bi spiral-like functions of order α(0 ≤ α < 1) satisfying the following conditions. eiβ [ (1 −λ) ( g(z) z )µ + λg′(z) ( g(z) z )µ−1] ≺ [φ(z)(1 −α) + α]cos β + isinβ and eiβ [ (1 −λ) ( h(w) w )µ + λh′(w) ( h(w) w )µ−1] ≺ [φ(w)(1 −α) + α]cos β + isinβ where 0 ≤ λ ≤ 1,µ ≥ 0 and z,w ∈ ∆∗ and the function h is given by (1.5). for function g ∈mβς′ (α,λ,µ,φ) given by (1.4),by choosing φ(z) = ( 1+z 1−z ), (or φ(z) = 1+az 1+bz ,−1 ≤ b < a ≤ 1), we obtain the estimates |b0| and |b1| by routine procedure (as in theorem2.1) and so we omit the details. references [1] d.a. brannan, j.g. clunie (eds.), aspects of contemporary complex analysis (proceedings of the nato advanced study institute held at the university of durham, durham; july 1–20, 1979), academic press, new york and london, 1980. [2] d.a. brannan, t.s. taha, on some classes of bi-univalent functions, in: s.m. mazhar, a. hamoui, n.s. faour (eds.), mathematical analysis and its applications, kuwait; february 18–21, 1985, in: kfas proceedings series, vol. 3, pergamon press, elsevier science limited, oxford, 1988, 53–60. see also studia univ. babeś-bolyai math. 31 (2) (1986) 70–77. [3] p. l. duren, coefficients of meromorphic schlicht functions, proceedings of the american mathematical society, vol. 28, 169–172, 1971. [4] e. deniz, certain subclasses of bi-univalent functions satisfying subordinate conditions, journal of classical analysis 2(1) (2013), 49–60. [5] s. g. hamidi,s. a. halim ,jay m. jahangiri,coefficent estimates for bi-univalent strongly starlike and bazilevic functions, international journal of mathematics research. vol. 5, no. 1 (2013), 87–96. [6] g. p. kapoor and a. k. mishra, coefficient estimates for inverses of starlike functions of positive order,journal of mathematical analysis and applications, vol. 329, no. 2, 922–934, 2007. [7] y. kubota, coefficients of meromorphic univalent functions, kōdai math. sem. rep. 28 (1976/77), no. 2-3, 253–261. [8] m. lewin, on a coefficient problem for bi-univalent functions, proc. amer. math. soc. 18 (1967) 63–68. [9] w.c. ma, d. minda, a unified treatment of some special classes of functions, in: proceedings of the conference on complex analysis, tianjin, 1992, 157 – 169, conf. proc.lecture notes anal. 1. int. press, cambridge, ma, 1994. [10] e. netanyahu, the minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, arch. rational mech. anal. 32 (1969) 100–112. [11] c. pommerenke, univalent functions, vandenhoeck & ruprecht, göttingen, 1975. [12] g. schober, coefficients of inverses of meromorphic univalent functions, proc. amer. math. soc. 67 (1977), no. 1, 111–116. [13] m. schiffer, sur un probléme dextrémum de la représentation conforme, bull. soc. math. france 66 (1938), 48–55. [14] g. springer, the coefficient problem for schlicht mappings of the exterior of the unit circle, trans. amer. math. soc. 70 (1951), 421–450. [15] h.m. srivastava, a.k. mishra, p. gochhayat, certain subclasses of analytic and bi-univalent functions, appl. math. lett. 23 (2010) 1188–1192. coefficient estimates of meromorphic bistarlike functions 77 [16] h. m. srivastava, a. k. mishra, and s. n. kund, coefficient estimates for the inverses of starlike functions represented by symmetric gap series, panamerican mathematical journal, vol. 21, no. 4, 105–123, 2011. [17] t.s. taha, topics in univalent function theory, ph.d. thesis, university of london, 1981. [18] q.-h. xu, y.-c. gui and h. m. srivastava, coefficient estimates for a certain subclass of analytic and bi-univalent functions, appl. math. lett. 25 (2012), 990–994. [19] q.-h. xu, h.-g. xiao and h. m. srivastava, a certain general subclass of analytic and biunivalent functions and associated coefficient estimate problems, appl. math. comput. 218 (2012), 11461–11465. [20] q-h. xu , chun-bo lv , h.m. srivastava coefficient estimates for the inverses of a certain general class of spirallike functions appl. math. comput. 219 (2013) 7000–7011. school of advanced sciences, vit university, vellore 632 014, india ∗corresponding author int. j. anal. appl. (2023), 21:17 on the behavior of the nonlinear difference equation yn+1 = ayn−1 +byn−3 + cyn−1+dyn−3 fyn−3−e turki d. alharbi1,2,∗, elsayed m. elsayed1,3 1department of mathematics, faculty of science, king abdulaziz university, jeddah, saudi arabia 2department of mathematics, university college in al-leeth, umm al-qura university, makkah, saudi arabia 3department of mathematics, faculty of science, mansoura university, mansoura, egypt ∗corresponding author: tdharbi@uqu.edu.sa abstract. the theory of difference equations got a significant position in the applicable analysis. therefore, studying the qualitative behavior of the difference equations is a fruitful area of research that has increasingly attracted many researchers. in this paper, we demonstrate the stability and the existence of periodic solutions of the nonlinear difference equation. moreover, we provide some numerical simulations to confirm our results. 1. introduction the major purpose of this study is to provide a substantial analysis on periodicity of solution, local asymptotic stability and global behavior of the following difference equations yn+1 = ayn−1 +byn−3 + cyn−1 +dyn−3 fyn−3 −e , n =0,1, ... (1.1) where the parameters a, b, c , and d are positive real numbers and the initial conditions y−3,y−2,y−1, and y0 are positive real. the study of difference equations is of utmost importance in mathematical applications. these equations also naturally appear as discrete analogs and as numerical solutions of some dynamical systems of differential equations that illustrate several phenomena in physics, biology, ecology, engineering, economics, etc. [1–10]. the theory of difference equations occupied a central position in received: jan. 18, 2023. 2020 mathematics subject classification. 39a11. key words and phrases. difference equation; recursive sequences; stability; periodicity; boundedness. https://doi.org/10.28924/2291-8639-21-2023-17 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-17 2 int. j. anal. appl. (2023), 21:17 the applicable analysis. so, there is no doubt that the theory of discrete time equations will persist in playing an important role in mathematics. therefore, it has been developing in terms of analysing the behavior and solving these equations. this progress can obviously be seen in the published studies, take for instance, alharbi et al. [11] analysed the stability and the periodicity of solutions and explored the form of solution for a special case of the rational difference equation zn+1 = azn−5 − bzn−5 czn−5 −dxn−11 , n =0,1, ... el-dessoky [12] obtained the local and global stability of the positive solutions, the periodic behavior, and the boundedness character of the following difference equation xn+1 = βxn−l +αxn−k + axn−t bxn−t +c , n =0,1, ... elsayed et al. [13] investigated the stability and periodicity as well as obtaining the solutions of a higher-order difference equation un+1 = un−9un−5un−1 un−7un−3(±1±un−9un−5un−1) , n =0,1, ... in [14], zayed et al. studied some qualitative properties of the solutions for the non-linear difference equation xn+1 = axn +bxn−k +cxn−l +dxn−σ + bxn−k +hxn−l dxn−k +exn−l , n =0,1, ... the boundedness solution, local stability, and global attractivity of the following second-order fractional equation xn+1 = α+γxn−1 bxn +dxnxn−1 +xn−1 , n =0,1, ... are demonstrated in [15] by kostrov et al. avotina [16] presented the periodic solution of three special cases of the rational difference equation: xn+1 = α+βxn +γxn−1 a+bxn +cxn−1 , n =0,1, ... for more recent studies, we refer the reader to [17-43] and references cited therein. 2. preliminaries and notation in this section, we introduce some definitions and theorems of the theory of difference equations that be utilized in our analysis. assume that s be a continuously differentiable function such that s : [a,b]k+1 → [a,b], where [a,b] is a real numbers interval and k is a positive integer. then the difference equation tn+1 = s(tn,tn−1, ...,tn−k), n =0,1,2, ... (2.1) has a unique solution {tn}∞n=−k for all set of initial values t−k,t−k+1, ...,t0 ∈ [a,b]. (kocic and ladas [5]) int. j. anal. appl. (2023), 21:17 3 definition 2.1. (equilibrium point) a point t∗ ∈ [a,b] is called an equilibrium point of equation (2.1) if t∗ = s(t∗,t∗, ...,t∗). that is, tn = t∗ for all n ≥ 0, is a solution of equation (2.1), or equivalently, t∗ is a fixed point of s. definition 2.2. (stability) the equilibrium point t∗ of equation 2.1 is said to be • locally stable if, for every α > 0, there exists β > 0 such that for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] with |t−k − t∗|+ |t−k+1 − t∗|+ ...+ |t0 − t∗| < β, we have |tn − t∗| < α ∀n ≥−k. • locally asymptotically stable if t∗ is locally stable solution of equation 2.1 and there exists µ > 0 such that for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] with |t−k − t∗|+ |t−k+1 − t∗|+ ...+ |t0 − t∗| < µ, we have lim n→∞ tn = t ∗. • global attractor if, for every t−k,t−k+1, ...,t−1,t0 ∈ [a,b] we have lim n→∞ tn = t ∗. • globally asymptotically stable if t∗ is locally stable, and also a global attractor of equation 2.1 • unstable if t∗ is not locally stable of equation 2.1. definition 2.3. (periodicity) a sequence {tn}∞n=−k is a periodic solution with period q if tn+q = tn for all n ≥−k. definition 2.4. (linearised equation) the linearized equation of the difference equation (2.1) about the equilibrium t∗ is the linear difference equation xn+1 = k−1∑ i=0 ∂s(t∗,t∗, ...,t∗) ∂tn−i xn−i (2.2) now, suppose that the characteristic equation associated with (2.2) is q(ζ)= q0ζ k +q1ζ k−1 + ...+qk−1ζ +qk =0 . (2.3) 4 int. j. anal. appl. (2023), 21:17 theorem a [8] assume that qi ∈ r, where i =1,2,3, ...,k and k ∈ {0,1,2,3,..}. then k∑ i=1 |qi| < 1 . is a sufficient condition for the asymptotic stability of the following the difference equation xn+k +q1xn+k−1 + ...+qkxn =0 . theorem b [9] assume that h is a continuous function such that h : [α,β]s+1 → [α,β], where k is a positive integer and [α,β] is a real numbers interval.and consider the difference equation tn+1 = h(tn,tn−1, ...,tn−k), n =0,1,2, ... (2.4) now, let h satisfies the following (1) for all 1≤ i ≤ k+1 where i is an integer , the function h(z1,z2, ...,zk+1) is weakly monotonic in zi for each z1,z2, ...,zk+1 . (2) assume (m,m) is a solution of the the system m = h(m1,m2, ...,mk+1) , m = h(m1,m2, ...,mk+1) . then m=m, fer each (i =1,2, ...,k +1) we set mi =  m, if h is non-decreasing in zi m, if h is non-inceasing in zi, and mi =  m, if h is non-decreasing in zi m, if h is non-inceasing in zi, m = m . so, there exists a unique fixed point t∗ of the equation (2.4) and any solution of (2.4) converges to t∗ 3. the local stability analysis in this section, we calculate the equilibrium points of equation (1.1). moreover, the local stability of these equilibrium points will be investigated. theorem 3.1. the non-linear difference equation (1.1) has two equilibrium points y∗1 = 0 and y ∗ 2 = c+d f(1−a−b) + e f . int. j. anal. appl. (2023), 21:17 5 proof. equation (1.1) can be written as y∗(1−a−b)= cy∗ +dy∗ fy∗ −e or fy2∗(1−a−b)−ey∗(1−a−b)−cy∗ −dy∗ =0, then, fy2∗(1−a−b)−y∗(e(1−a−b)+c +d)=0. so, the difference equation (1.1) has two equilibrium points y∗1 =0, y ∗ 2 = c +d f(1−a−b) + e f . theorem 3.2. the first equilibrium point y∗1 =0 of the difference equation (1.1) is locally asymptotically stable if |−c −d| < e(1−a−b). proof. suppose that g(0,∞)3 → (0,∞) is a function defined as follows g(u,w)= au +bw + cu +dw fw −e . (3.1) differentiating g(u,w) with respect to u and w. we get gu = a+ c fw −e , gw = b − (fcu +de) (fw −e)2 , substituting y∗1 =0 into gu, and gw. we get gu(y ∗ 1,y ∗ 1)= a− c e =−p1, gw(y∗1,y ∗ 1)= b − d e =−p2. hence, the linearized equation of (1.1) about the equilibrium point y∗1 is zn+1 +p1zn−1 +p2zn−3 =0 . (3.2) it follows by theorem a that the fixed point ,y∗1, of equation (1.1) is locally asymptotically stable if |p1|+ |p2| < 1 . so, |a− c e |+ |b − d e | < 1, this implies, |ae −c +be −d| < e. thus, the first equilibrium point y∗1 is locally asymptotically stable if |−c −d| < e(1−a−b). the proof is completed. 6 int. j. anal. appl. (2023), 21:17 theorem 3.3. suppose that |cα− (c +eα)α| < c +d−a−b. where α =(1−a−b), then the second equilibrium point y∗2 of equation (1.1) is locally asymptotically stable. proof. substituting y∗2 = c+d fα + e f into gu, and gw. we get gu(y ∗ 2,y ∗ 2)= a+ cα c +d =−q1, gw(y ∗ 2,y ∗ 2)= b − (c +eα)α c +d =−q2. where α =(1−a−b). so, the linearized equation of (1.1) about the equilibrium point y∗2 is zn+1 +q1zn−1 +q2zn−3 =0 . (3.3) it can be shown by theorem a that the fixed point y∗2 of the difference equation (1.1) is locally asymptotically stable if |q1|+ |q2| < 1 . so, |a+ cα c +d |+ |b − (c +eα)α c +d | < 1, thus, |a+cα+b − (c +eα)α| < c +d. therefore, the second equilibrium y∗2 is locally asymptotically stable if |cα− (c +eα)α| < c +d−a−b. the proof is completed. 4. global behaviour analysis we dedicate this section to showing the case under which the equilibrium points y∗ of equation (1.1) are asymptotically globally stable. theorem 4.1. the equilibrium points y∗ of the difference equation (1.1) is globally asymptotically stable if i ae +bf +d > c +be +e ii e +c +d > f int. j. anal. appl. (2023), 21:17 7 proof. suppose that k and r be real numbers and assume g(k,r)2 → (k,r) is a function that defined by g(u,w)= au +bw + cu +dw fw −e . (4.1) now, we consider two cases. case i. suppose that g(u,w) is increasing in u and w. then, assume (ζ,ρ) is a solution of the following system ζ = g(ζ,ζ) , ρ = g(ρ,ρ) . so, ζ = aζ +bζ + cζ +dζ fζ −e , ρ = aρ+bρ+ cρ+dρ fρ−e , this gives, fζ2(1−a−b)−eζ(1−a−b)= ζ(c +d), (4.2) fρ2(1−a−b)−eρ(1−a−b)= ρ(c +d), (4.3) after subtracting (4.3) from (4.2). we get (ζ2 −ρ2)f(1−a−b)− (ζ −ρ)e(1−a−b)− (ζ −ρ)(c +d)=0, (4.4) this implies, (ζ −ρ){(ζ +ρ)f(1−a−b)−e(1−a−b)− (c +d)}=0. (4.5) thus, when e +c +d > f, ζ = ρ . it follows by theorem b that y∗ is globally asymptotically stable. the proof is completed. case ii. suppose that g(u,w) is increasing in u and it is decreasing in w. then, assume (ζ,ρ) is a solution of the following system ζ = g(ζ,ρ) , ρ = g(ρ,ζ) . so, ζ = aζ +bρ+ cζ +dρ fρ−e , ρ = aρ+bζ + cρ+dζ fζ −e , this implies, ζ(1−a)(fρ−e)−bρ(fρ−e)−cζ −dρ =0, (4.6) 8 int. j. anal. appl. (2023), 21:17 ρ(1−a)(fζ −e)−bζ(fζ −e)−cρ−dζ =0. (4.7) now, subtracting (4.7) from (4.6). we get (ζ −ρ){ae −e +bf(ζ +ρ)−be −c +d}=0. (4.8) therefore, when ae +bf +d > c +be +e ζ = ρ . it can be shown by theorem b that y∗ is globally asymptotically stable. the proof is completed. 5. existence of periodic solutions this section discusses the existence of periodic behavior of the nonlinear difference equation (1.1). the following theorem states the necessary and sufficient conditions that assure eq.(1.1) has periodic behavior of prime period two. theorem 5.1. the difference equation (1.1) has solution of period two if and only if e(1−a−b)+c +d 6=0 (5.1) proof. assume that equation (1.1) has a solution of period two ...,α,β,α,β,... with α 6= β α = aα+bα+ cα+dα fα−e , β = aβ +bβ + cβ +dβ fβ −e . so, fα2(1−a−b)−eα(1−a−b)= α(c +d) , (5.2) fβ2(1−a−b)−eβ(1−a−b)= β(c +d) . (5.3) subtracting (5.3) from (5.2) gives f(1−a−b)(α2 −β2)−e(1−a−b)(α−β)= (c +d)(α−β) , this implies, f(1−a−b)(α+β)−eα(1−a−b)= (c +d) . consequently, α+β = e(1−a−b)+c +d f(1−a−b) . (5.4) int. j. anal. appl. (2023), 21:17 9 again, adding (5.2) and (5.3). we get f(1−a−b)(α2 +β2)= {e(1−a−b)+(c +d)}(α+β) . (5.5) by using (5.4), (5.5) , and the relation (α+β)2 = α2 +2αβ +β2, we obtain f(1−a−b){(α+β)2 −2αβ}= {e(1−a−b)+(c +d)}(α+β) , then, 2f(1−a−b)αβ = f(1−a−b)(α+β)2 −{e(1−a−b)+(c +d)}(α+β) , 2f(1−a−b)αβ = (e(1−a−b)+c +d)2 f(1−a−b) −{e(1−a−b)+c+d}( e(1−a−b)+c +d f(1−a−b) ) . thus, αβ =0. (5.6) therefore, it follows from equations (20) and (22) that α and β are the two distinct roots of the quadratic equation x2 − (α+β)x +αβ =0. (5.7) that is, x2 − ( e(1−a−b)+c +d f(1−a−b) )x =0, then, f(1−a−b)x2 − (e(1−a−b)+c +d)x =0, so, (e(1−a−b)+c +d)2 > 0. for (e(1−a−b)+c +d) 6=0, the condition (5.1) holds. on the other side, suppose that condition (5.1) is true. we will demonstrate that equation (1.1) has a prime period two solution. set y−3 = y−1 = p = e(1−a−b)+c +d f(1−a−b) and y−2 = y0 = q =0. now, we want to show that y1 = p, and y2 =0. it follows from equation (1.1) that y1 = ap+bp+ cp+dp fp−e , so, y1 =(a+b) (e(1−a−b)+c +d f(1−a−b) ) + (c +d)( e(1−a−b)+c+d f(1−a−b) ) f( e(1−a−b)+c+d f(1−a−b) )−e , 10 int. j. anal. appl. (2023), 21:17 =(a+b) (e(1−a−b)+c +d f(1−a−b) ) + (c +d)(e(1−a−b)+c +d f(c +d) , =(a+b) (e(1−a−b)+c +d f(1−a−b) ) + (e(1−a−b)+c +d f , = (e(1−a−b)+c +d f )( 1+ a+b (1−a−b) ) = e(1−a−b)+c +d f(1−a−b) = p, y2 = aq +bq + cq +dq fq −e =0= q. so, by induction we get y2n = q and y2n+1 = p for all n ≥−3. hence, equation (1.1) has the prime period two solution p and q. where p and q are the distinct roots of the quadratic equation (5.7). 6. numerical examples in this part, we provide some examples that verify our analytical results. matlab programming is used to show numerically the behavior of the nonlinear difference equation(1.1). example 6.1. figure 1 shows the behavior of eq.(1.1) tends to the first equilibrium point y∗1 = 0 when the parameters and the initial values are a = 0.1, b = 0.2, c = 1, d = 2, f = 4, e = 6, y−3 =−3, y−2 =2, y−1 =−0.5, and y0 =1. example 6.2. figure 2 presents the behavior of eq.(1) approaches to the second equilibrium point figure 1. the behaviour of equation (1.1) y∗2 =0 when we assume the parameters and the initial values that a =0.6, b =0.2, c =3, d =4, f =6, e =5, y−3 =5, y−2 =−3, y−1 =1, and y0 =4. int. j. anal. appl. (2023), 21:17 11 figure 2. the behaviour of equation (1.1) example 6.3. the unstable behavior of eq.(1.1) is shown in figure 3. we assume the parameters and the initial values that a = 0.1, b = 0.2, c = 5, d = 8, f = 0.4, e = 6, y−3 = 1, y−2 = −6, y−1 =3, and y0 =−4. figure 3. the behaviour of equation (1.1) example 6.4. in figure 4, the global stability behavior of eq.(1.1) is shown. it is clear that the behavior of eq.(1.1) tends to the fixed point y∗1 as n goes to ∞ under the following the initial conditions and the parameters a =0.1, b =0.2, c =1, d =2, f =4, e =8, y−3 =1, y−2 =−6, y−1 =3, and y0 =−4. 12 int. j. anal. appl. (2023), 21:17 figure 4. the behaviour of equation (1.1) example 6.5. figure 5 demonstrates the global stability behavior of the fixed point y∗2 when the initial conditions and the parameters are a = 0.2, b = 0.16, c = 0.123, d = 14, f = 0.5, e = 5, y−3 =5, y−2 =−3, y−1 =1, and y0 =−4. figure 5. the behaviour of equation (1.1) example 6.6. figure 6 shows that eq.(1.1) has a prime period two solution when the initial conditions and the parameters are a = 0.2, b = 0.1, c = 0.2, d = 3, f = 1, e = 0.6, y−3 = p, y−2 = q, y−1 = p, and y0 = q where p and q satisfied theorem 5.1 int. j. anal. appl. (2023), 21:17 13 figure 6. the behaviour of equation (1.1) 7. conclusion this study discusses the dynamics of the nonlinear difference equation (1.1). in section 3 we illustrated that when the local stability condition in theorem 3.2 is satisfied, the behavior tends to the stability state of the equilibrium point y∗1 =0. while, the equilibrium y ∗ 2 will be locally asymptotically stable when |cα − (c + eα)α| < c + d − a − b. the global solution of the equilibrium points conditions is shown in section 4. section 5 discussed the necessary and sufficient conditions to obtain the periodic solution of equation (1). for confirmation of our theoretical analysis, we presented some numerical examples in section 6, and figures 1-6 verified the results. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.e. mickens, difference equations: theory and applications, 2nd ed, chapman and hall, new york, (1990). [2] h.f. huo, w.t. li, permanence and global stability of positive solutions of a nonautonomous discrete ratiodependent predator-prey model, discr. dyn. nat. soc. 2005 (2005), 135–144. https://doi.org/10.1155/ddns. 2005.135. [3] g. ladas, g. tzanetopoulos, a. tovbis, on may’s host parasitoid model, j. differ. equ. appl. 2 (1996), 195–204. https://doi.org/10.1080/10236199608808054. [4] s. stevic, a global convergence results with applications to periodic solutions, indian j. pure appl. math. 33 (2002), 45-53. [5] v.l. kocic, g. ladas, global behavior of nonlinear difference equations of higher order with applications, springer netherlands, dordrecht, 1993. https://doi.org/10.1007/978-94-017-1703-8. [6] h. sedaghat, nonlinear difference equations, springer netherlands, dordrecht, 2003. https://doi.org/10.1007/ 978-94-017-0417-5. [7] e.c. pielou, population and community ecology, gordon and breach, new york, (1974). https://doi.org/10.1155/ddns.2005.135 https://doi.org/10.1155/ddns.2005.135 https://doi.org/10.1080/10236199608808054 https://doi.org/10.1007/978-94-017-1703-8 https://doi.org/10.1007/978-94-017-0417-5 https://doi.org/10.1007/978-94-017-0417-5 14 int. j. anal. appl. (2023), 21:17 [8] m. r. s. kulenovic and g. ladas, dynamics of second order rational difference equations with open problems and conjectures, chapman & hall/ crc press, new york, (2001). [9] ea. grove and g. ladas, periodicities in nonlinear difference equations. 1st ed, chapman & hall/ crc press, new york, (2004). [10] p. cull, m.e. flahive, r.o. robson, difference equations: from rabbits to chaos, springer, new york, (2005). [11] t.d. alharbi, e.m. elsayed, forms of solution and qualitative behavior of twelfth-order rational difference equation, int. j. differ. equ. 17 (2022), 281-292. [12] m.m. el-dessoky, studies on the higher order difference equation xn+1 = βxn−l + αxn−k + axn−t bxn−t+c , j. comput. anal. appl. 29 (2021), 116-131. [13] e.m. elsayed, b.s. alofi, a.q. khan, qualitative behavior of solutions of tenth-order recursive sequence equation, math. probl. eng. 2022 (2022), 5242325. https://doi.org/10.1155/2022/5242325. [14] m.a. el-moneam, e.m.e. zayed, dynamics of the rational difference equation, inform. sci. lett. 3 (2014), 45–53. https://doi.org/10.12785/isl/030202. [15] y. kostrov, z. kudlak, on a second-order rational difference equation with a quadratic term, int. j. differ. equ. 11 (2016), 179-202. [16] m. avotina, on three second-order rational difference equations with period-two solutions, int. j. differ. equ. 9 (2014), 23-35. [17] a. asiri, m.m. el-dessoky, e.m. elsayed, solution of a third order fractional system of difference equations, j. comput. anal. appl., 24 (2018), 444-453. [18] s. moranjkic, z. nurkanovic, local and global dynamics of certain second-order rational difference equations containing quadratic terms, adv. dyn. syst. appl. 12 (2017), 123-157. [19] m.n. phong, a note on a system of two nonlinear difference equations, electron. j. math. anal. appl. 3 (2015), 170-179. [20] w. wang, j. tian, difference equations involving causal operators with nonlinear boundary conditions, j. nonlinear sci. appl. 8 (2015), 267-274. [21] h.s. alayachi, m.s.m. noorani, a.q. khan, m.b. almatrafi, analytic solutions and stability of sixth order difference equations, math. probl. eng. 2020 (2020), 1230979. https://doi.org/10.1155/2020/1230979. [22] a.m. alotaibi, m.a. el-moneam, on the dynamics of the nonlinear rational difference equation xn+1 = axn−m+δxn β+γxn−kxn−1(xn−k+xn−1) , aims math. 7 (2022), 7374-7384. [23] j. bektesevic, m. mehuljic, v. hadziabdic, global asymptotic behavior of some quadratic rational second-order difference equations, int. j. differ. equ. 20 (2017), 169-183. [24] e. m. elsayed, k. n. alshabi and f. alzahrani, qualitative study of solution of some higher order difference equations, j. comput. anal. appl. 26 (2019), 1179-1191. [25] e.m. elsayed, k.n. alharbi, the expressions and behavior of solutions for nonlinear systems of rational difference equations, j. innov. appl. math. comput. sci. 2 (2022), 78–91. [26] e.m. elsayed, a. alshareef, qualitative behavior of a system of second order difference equations, eur. j. math. appl. 1 (2021), 15. https://doi.org/10.28919/ejma.2021.1.15. [27] e.m. elsayed, n.h. alotaibi, the form of the solutions and behavior of some systems of nonlinear difference equations, dyn. contin. discr. impuls. syst. ser. a: math. anal. 27 (2020), 283-297. [28] e.m. elsayed, h.s. gafel, some systems of three nonlinear difference equations, j. comput. anal. appl. 29 (2021), 86-108. [29] e.m. elsayed, j.g. al-juaid, h. malaikah, on the dynamical behaviors of a quadratic difference equation of order three, eur. j. math. appl. 3 (2023), 1. https://doi.org/10.28919/ejma.2023.3.1. https://doi.org/10.1155/2022/5242325 https://doi.org/10.12785/isl/030202 https://doi.org/10.1155/2020/1230979 https://doi.org/10.28919/ejma.2021.1.15 https://doi.org/10.28919/ejma.2023.3.1 int. j. anal. appl. (2023), 21:17 15 [30] e.m. elsayed, j.g. al-juaid, the form of solutions and periodic nature for some system of difference equations, fundam. j. math. appl. 6 (2023), 24-34. https://doi.org/10.33401/fujma.1166022. [31] e.m. elsayed, m. m. alzubaidi, on a higher-order systems of difference equations, pure appl. anal. 2023 (2023), 2. [32] e.m. elsayed, b. alofi, stability analysis and periodictly properties of a class of rational difference equations, manas j. eng. 10 (2022), 203-210. https://doi.org/10.51354/mjen.1027797. [33] e.m. elasyed, m.t. alharthi, the form of the solutions of fourth order rational systems of difference equations, ann. commun. math. 5 (2022), 161-180. [34] e.m. elsayed, a. alghamdi. dynamics and global stability of higher order nonlinear difference equation, j. comput. anal. appl. 21 (2016), 493-503. [35] e.m. elsayed, a. alshareef, qualitative behavior of a system of second order difference equations, eur. j. math. appl. 1 (2021), 15. https://doi.org/10.28919/ejma.2021.1.15. [36] m. garic-demirovic, m. nurkanovic, z. nurkanovic, stability, periodicity and neimark-sacker bifurcation of certain homogeneous fractional difference equations, int. j. differ. equ. 12 (2017), 27-53. [37] m. gümüş, r. abo-zeid, qualitative study of a third order rational system of difference equations, math. morav. 25 (2021), 81-97. [38] s. kalabusic, m. nurkanovic, z. nurkanovic, global dynamics of certain mix monotone difference equation, mathematics, 6 (2018), 10. https://doi.org/10.3390/math6010010. [39] a. khaliq, e. elsayed, the dynamics and solution of some difference equations, j. nonlinear sci. appl. 9 (2016), 1052-1063. [40] w.x. ma, global behavior of a higher-order nonlinear difference equation with many arbitrary multivariate functions, east asian j. appl. math. 9 (2019), 643–650. https://doi.org/10.4208/eajam.140219.070519. [41] s. moranjkic, z. nurkanovic, local and global dynamics of certain second-order rational difference equations containing quadratic terms, adv. dyn. syst. appl. 12 (2017), 123-157. [42] m. saleh, a. farhat, global asymptotic stability of the higher order equation xn+1 = axn+bxn−k a+bxn−k , j. appl. math. comput. 55 (2017), 135-148. [43] e.m.e. zayed, on the dynamics of a new nonlinear rational difference, dyn. contin. discr. impuls. syst. ser. a. math. anal., 27 (2020), 153-165. https://doi.org/10.33401/fujma.1166022 https://doi.org/10.51354/mjen.1027797 https://doi.org/10.28919/ejma.2021.1.15 https://doi.org/10.3390/math6010010 https://doi.org/10.4208/eajam.140219.070519 1. introduction 2. preliminaries and notation 3. the local stability analysis 4. global behaviour analysis 5. existence of periodic solutions 6. numerical examples 7. conclusion references international journal of analysis and applications volume 16, number 2 (2018), 276-289 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-276 on fuzzy ordered la-semihypergroups muhammad azhar1, muhammad gulistan1,∗, naveed yaqoob2, seifedine kadry3 1department of mathematics, hazara university, mansehra, kp, pakistan 2department of mathematics, college of science al-zulfi, majmaah university, al-zulfi, saudi arabia 3department of mathematics & computer science, beirut arab university, lebanon ∗corresponding author: gulistanmath@hu.edu.pk abstract. we introduce the notion of fuzzy ordered la-semihypergroups and provide different examples. we also discuss some results related with fuzzy left and right hyperideals. 1. introduction the theory of algebraic hyperstructure was introduced by marty in 1934, when marty [1] defined hypergroups. since then many hyperstructures were being studied by several authors, for instance, bonansinga and corsini [2], davvaz [3], freni [4], hila et al. [5], leoreanu [6], salvo et al. [7] and many others. the concept of ordered semihypergroup was studied by heidari and davvaz in [8], where they used a binary relation ” ≤ ” on semihypergroup (h,◦) such that the binary relation is a partial order and the structure (h,◦,≤) is known as ordered semihypergroup. there are several authors who study the ordering of hyperstructures, for instance, bakhshi and borzooei [9], chvalina [10], hoskova [11], kondo and lekkoksung [12] and novak [13]. another non-associative algebraic hyperstructure known as la-semihypergroup which is a useful generalization of semigroup, semihypergroups and la-semigroups was introduced by hilla and dine [14] in 2011 based on left invertive law given by kazim and naseerudin [15] in 1972. yaqoob et al. [16] extended the work received 2017-12-07; accepted 2018-02-01; published 2018-03-07. 2000 mathematics subject classification. 20n20. key words and phrases. la-semihypergroups; ordered la-semihypergroups; fuzzy hyperideals. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 276 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-276 int. j. anal. appl. 16 (2) (2018) 277 of hila and dine and characterized intra-regular left almost semihypergroups by their hyperideals using pure left identity. the ordering in la-semihypergroups was introduced by yaqoob and gulistan [17]. the concept of fuzzy set was introduced by zadeh in 1965 [18]. rosenfeld [19] introduced fuzzy sets in the context of group theory and formulated the concept of fuzzy subgroup of a group in 1971. later many researcher are engaged in extending the concept of abstract algebra to the frame work of fuzzy setting. fuzzy hyperstructures have been already considered by many researchers, for instance, corsini et al. [20, 21], davvaz [22, 23], hila and abdullah [24], khan et al. [25], pibaljommee et al. [26, 27], tang et al. [28–31], tipachot and pibaljommee [32] and zhan et al. [33, 34]. as a further study of ordered la-semihypergroups, we attempt in the present paper to study the fuzzy ordered la-semihypergroups in detail. 2. preliminaries let h be a non-empty set. then the map ◦ : h ×h →p∗(h) is called hyperoperation or join operation on the set h, where p∗(h) = p(h)\{∅} denotes the set of all non-empty subsets of h. a hypergroupoid is a set h together with a (binary) hyperoperation. for any non-empty subsets a,b of h, we denote a◦b = ⋃ a∈a,b∈b a◦ b instead of {a}◦a and b ◦{a} , we write a◦a and b ◦a, respectively. recently, in [14, 16] authors introduced the notion of la-semihypergroups as a generalization of semigroups, semihypergroups, and la-semigroups. a hypergroupoid (h,◦) is called an la-semihypergroup if for every x,y,z ∈ h, we have (x ◦ y) ◦ z = (z ◦ y) ◦ x. the law (x ◦ y) ◦ z = (z ◦ y) ◦ x is called a left invertive law. an element e ∈ h is called a left identity (resp., pure left identity) if for all x ∈ h, x ∈ e◦x (resp., x = e◦x). in an la-semihypergroup, the medial law (x◦y) ◦ (z ◦w) = (x◦z) ◦ (y ◦w) holds for all x,y,z,w ∈ h. an la-semihypergroup may or may not contains a left identity and pure left identity. in an la-semihypergroup h with pure left identity, the paramedial law (x◦y) ◦ (z ◦w) = (w ◦ z) ◦ (y ◦x) holds for all x,y,z,w ∈ h. if an la-semihypergroup contains a pure left identity, then by using medial law, we get x◦ (y ◦z) = y ◦ (x◦z) for all x,y,z ∈ h. definition 2.1. [17] let h be non-empty set and ≤ be an ordered relation on h. the triplet (h,◦,≤) is called an ordered la-semihypergroup if the following conditions are satisfied. (1) (h,◦) is an la-semihypergroup, (2) (h,≤) is a partially ordered set, (3) for every a,b,c ∈ h, a ≤ b implies a◦ c ≤ b◦ c and c◦a ≤ c◦ b, where a◦ c ≤ b◦ c means that for x ∈ a◦ c there exist y ∈ b◦ c such that x ≤ y. int. j. anal. appl. 16 (2) (2018) 278 definition 2.2. [17] if (h,◦,≤) is an ordered la-semihypergroup and a ⊆ h, then (a] is the subset of h defined as follows: (a] = {t ∈ h : t ≤ a, for some a ∈ a}. definition 2.3. [17] a non-empty subset a of an ordered la-semihypergroup (h,◦,≤) is called an lasubsemihypergroup of h if (a◦a] ⊆ (a]. definition 2.4. [17] a non-empty subset a of an ordered la-semihypergroup (h,◦,≤) is called a right (resp., left) hyperideal of h if (1) a◦h ⊆ a (resp., h ◦a ⊆ a), (2) for every a ∈ h, b ∈ a and a ≤ b implies a ∈ a. if a is both right hyperideal and left hyperideal of h, then a is called a hyperideal (or two sided hyperideal) of h. 3. fuzzy ordered la-semihypergroups let x ∈ h, then ax = {(y,z) ∈ h ◦h : x ≤ y ◦z} . let f and g be two fuzzy subsets of an ordered la-semihypergroup h, then f ∗g is defined as (f ∗g) (x) =   ∨ (y,z)∈ax {f (y) ∧g (z)} if x ≤ y ◦z, for some y,z ∈ h 0 otherwise. let f(h) denote the set of all fuzzy subsets of an ordered la-semihypergroup. theorem 3.1. let h be an ordered la-semihypergroup. then the set (f(h),∗,⊆) is an ordered lasemihypergroup. proof. clearly f(h) is closed. let f, g and h be in f(h) and let x be any element of h such that it is not expressible as product of two elements in h. then we have, ((f ∗g) ∗h) (x) = 0 = ((h∗g) ∗f) (x). int. j. anal. appl. 16 (2) (2018) 279 let ax 6= ∅. then there exist y and z in h such that (y,z) ∈ ax. therefore by using left invertive law, we have ((f ∗g) ∗h)(x) = ∨ (y,z)∈ax {(f ∗g) (y) ∧h(z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay {f(p) ∧g(q)}∧h(z)   = ∨ x≤(p◦q)◦z {f(p) ∧g(q) ∧h(z)} = ∨ x≤(z◦q)◦p {h(z) ∧g(q) ∧f(p)} = ∨ (w,p)∈ax   ∨ (z,q)∈aw (h(z) ∧g(q) ∧f(p))   = ∨ (w,p)∈ax {(h∗g) (w) ∧f(p)} = ((h∗g) ∗f)(x). hence (f(h),◦) is an la-semihypergroup. assume that f ⊆ g and let ax = ∅ for any x ∈ h, then (f ∗h)(x) = 0 = (g ∗h)(x) =⇒ f ∗h ⊆ g ∗h. similarly we can show that f ∗h ⊇ g ∗h. let ax 6= ∅. then there exist y and z in h such that (y,z) ∈ ax, therefore (f ∗h)(x) = ∨ (y,z)∈ax {f (y) ∧h(z)}≤ ∨ (y,z)∈ax {g (y) ∧h(z)} = (g ∗h)(x), similarly we can show that f ∗ h ⊇ g ∗ h. it is easy to see that f(h) is a poset. thus (f(h),∗,⊆) is an ordered la-semihypergroup. � theorem 3.2. let h be an ordered la-semihypergroup. then the property (f ∗g) ∗ (h∗k) = (f ∗h) ∗ (g ∗k) holds in f(h), for all f,g,h and k in f(h). proof. straightforward. � theorem 3.3. if an ordered la-semihypergroup h has a pure left identity, then the following properties hold in f(h). (i) (f ∗g) ∗ (h∗k) = (k ∗h) ∗ (g ∗f) , (ii) f ∗ (g ∗h) = g ∗ (f ∗h) , for all f,g,h and k in f(h). proof. straightforward. � int. j. anal. appl. 16 (2) (2018) 280 proposition 3.1. an ordered la-semihypergroup (f(h),∗,⊆) with f(h) = (f(h))2 is a commutative ordered semihypergroup if and only if (f ∗g) ∗h = f ∗ (h∗g) holds for all fuzzy subsets f,g,h ∈ f(h). proof. let an ordered la-semihypergroup f(h) be a commutative ordered semihypergroup. for any fuzzy subsets f,g,h ∈ f(h), if ax = ∅ for any x ∈ h, then ((f ∗ g) ∗ h)(x) = 0 = (f ∗ (h ∗ g))(x). let ax 6= ∅, then there exist s and t in h such that (s,t) ∈ ax, therefore by use of left invertive law and commutative law, we get ((f ∗g) ∗h)(x) = ∨ (s,t)∈ax {(f ∗g)(s) ∧h(t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as f(m) ∧g(n) ∧h(t)   = ∨ x≤(m◦n)◦t {f(m) ∧h(t) ∧g(n)} = ∨ x≤(t◦n)◦m {f(m) ∧h(t) ∧g(n)} = ∨ x≤m◦(t◦n) {f(m) ∧h(t) ∧g(n)} = ∨ (m,p)∈ax   ∨ (t,n)∈ap f(m) ∧h(t) ∧g(n)   = ∨ (m,p)∈ax {f(m) ∧ (h∗g)(p)} = (f ∗ (h∗g))(x). thus (f ∗g)∗h = f ∗(h∗g). conversely, let (f ∗g)∗h = f ∗(h∗g) holds for all fuzzy subsets f,g,h ∈ f(h). we have to show that h is a commutative ordered semihypergroup. let f and g be any arbitrary fuzzy subsets of h, if ax = ∅ for any x ∈ h, then (f ∗g)(x) = 0 = (g∗f)(x). let ax 6= ∅, then there exist s and t in h such that (s,t) ∈ ax. since f(h) = (f(h))2, so f = h∗k, where h and k are any fuzzy subsets of h. now by left invertive law, we have (f ∗g)(x) = ((h∗k) ∗g)(x) = ∨ (s,t)∈ax {(h∗k)(s) ∧g(t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as h(m) ∧k(n) ∧g(t)   = ∨ x≤(m◦n)◦t {g(t) ∧k(n) ∧h(m)} = ∨ x≤(t◦n)◦m {g(t) ∧k(n) ∧h(m)} int. j. anal. appl. 16 (2) (2018) 281 = ∨ (p,m)∈ax   ∨ (t,n)∈ap g(t) ∧k(n) ∧h(m)   = ∨ (p,m)∈ax {(g ∗k)(p) ∧h(m)} = (g ∗ (h∗k))(x) = (g ∗f)(x). this shows that f ∗g = g ∗ (h∗k) = g ∗f. therefore commutative law holds in f(h). now if ax = ∅ for any x ∈ h, then ((f ∗g) ∗k)(x) = 0 = (f ∗ (g ∗k))(x). let ax 6= ∅, then there exist s and t in h such that (s,t) ∈ ax, therefore by using left invertive law and commutative law, we get ((f ∗g) ∗k)(x) = ∨ (s,t)∈ax {(f ∗g)(s) ∧k(t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as f(m) ∧g(n) ∧k(t)   = ∨ x≤(m◦n)◦t {f(m) ∧g(n) ∧k(t)} = ∨ x≤(t◦n)◦m {f(m) ∧g(n) ∧k(t)} = ∨ x≤m◦(t◦n) {f(m) ∧g(n) ∧k(t)} = ∨ x≤m◦(n◦t) {f(m) ∧g(n) ∧k(t)} = ∨ (m,p)∈ax   ∨ (n,t)∈ap f(m) ∧g(n) ∧k(t)   = ∨ (m,p)∈ax {f(m) ∧ (g ∗k)(p)} = (f ∗ (g ∗k)(x). therefore associative law holds in f(h). thus f(h) is commutative ordered semihypergroup. � theorem 3.4. cm = {f | f ∈ f(h), f ∗α = f, where α = α∗α} is a commutative monoid in h. proof. the fuzzy subset cm of h is non-empty since α ∗ α = α, which implies that α is in cm. let f and γ be fuzzy subsets of h in cm, then f ∗ α = f and γ ∗ α = γ. if ax = ∅ for x ∈ h, then (f ∗γ) (x) = 0 = ((f ∗γ) ∗α) (x). let ax 6= ∅, then there exist y and z in h such that (y,z) ∈ ax. therefore int. j. anal. appl. 16 (2) (2018) 282 by using medial law, we have (f ∗γ)(x) = ∨ (y,z)∈ax {(f ∗α) (y) ∧ (γ ∗α) (z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay {f(p) ∧α(q)}∧ ∨ (u,v)∈az {γ(u) ∧α(v)}   = ∨ x≤(p◦q)◦(u◦v) {f(p) ∧α(q) ∧γ(u) ∧α(v)} = ∨ x≤(p◦u)◦(q◦v) {f(p) ∧γ(u) ∧α(q) ∧α(v)} = ∨ (m,n)∈ax   ∨ (p,u)∈am {f(p) ∧γ(u)}∧ ∨ (q,v)∈an {α(q) ∧α(v)}   = ∨ (m,n)∈ax {(f ∗γ)(m) ∧ (α∗α)(n)} = ((f ∗γ) ∗ (α∗α))(x). thus f ∗γ = (f ∗γ) ∗ (α∗α) = (f ∗γ) ∗α, which implies that cm is closed. now if ax = ∅ for x ∈ h, then (f ∗γ) (x) = 0 = (γ ∗f) (x). let ax 6= ∅, then there exist y and z in h such that (y,z) ∈ ax. therefore by using left invertive law, we have (f ∗γ)(x) = ∨ (y,z)∈ax {(f ∗α) (y) ∧γ(z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay f (p) ∧α(q) ∧γ(z)   = ∨ x≤(p◦q)◦z {γ(z) ∧α(q) ∧f (p)} = ∨ x≤(z◦q)◦p {γ(z) ∧α(q) ∧f (p)} = ∨ (t,p)∈ax   ∨ (z,q)∈at γ(z) ∧α(q) ∧f (p)   = ∨ (t,p)∈ax {(γ ∗α)(t) ∧f (p)} = ((γ ∗α) ∗f)(x). thus f∗γ = (γ∗α)∗f = γ∗f, which implies that commutative law holds in cm and associative law holds in cm due to commutativity. since for any fuzzy subset f in cm, we have f ∗α = f (where α is fixed) implies that α is a right identity in h and hence an identity. � for an ordered la-semihypergroup h, the fuzzy subset h of h is defined as follows: h : h → [0, 1]|x → h(x) := 1. lemma 3.1. in an ordered la-semihypergroup with a left identity h ∗ h = h. int. j. anal. appl. 16 (2) (2018) 283 proof. let x ∈ h. then x ≤ e◦x, that is (e,x) ∈ ax, where e is the left identity of h. therefore (h ∗ h) (x) = ∨ (e,x)∈ax (h (e) ∧ h (x)) = 1 = h (x) . so h ∗ h = h. � 4. fuzzy hyperideals in ordered la-semihypergroups in this section, we define the concept of a fuzzy right (resp., left) hyperideal and give relationships between them. definition 4.1. let (h,◦,≤) be an ordered la-semihypergroup. a fuzzy subset f : h → [0, 1] is called fuzzy la-subsemihypergroup of h if the following assertion are satisfied; (i) ∧ z≤a◦b f (z) ≥ min{f (a) ,f (b)} , (ii) if a ≤ b implies f (a) ≥ f (b) , for every a,b ∈ h. example 4.1. we consider a set h = {x,y,z} with the following hyperoperation ”◦” and the order ” ≤ ” : ◦ x y z x {x,y} {x,y} z y {x,z} {x,z} z z z z z ≤:= {(x,x) , (y,y) , (z,x) , (z,y) , (z,z)}. we give the covering relation ” ≺ ” and the figure of h as follows: ≺= {(z,x) , (z,y)}. then (h,◦,≤) is an ordered la-semihypergroup. now let f be a fuzzy subset of h such that f (x) = 0.5,f (y) = 0.3,f (z) = 0.9. clearly f is a fuzzy la-subsemihypergroup of h. int. j. anal. appl. 16 (2) (2018) 284 theorem 4.1. a fuzzy subset f of an ordered la-semihypergroup h is a fuzzy la-subsemihypergroup of h if and only if (i) f ∗f ⊆ f, (ii) if a ≤ b implies f (a) ≥ f (b) , for every a,b ∈ h. proof. consider that f is a fuzzy la-subsemihypergroup of h. let ax = ∅ for any x ∈ h. then (f∗f)(x) = 0 = f(x). let ax 6= ∅. then for x,y,a ∈ h, we have (f ∗f) (a) = ∨ a≤x◦y {f (x) ∧f (y)}≤ ∨ a≤x◦y f(a) = f(a). thus f ∗f ⊆ f. conversely, assume that f ∗f ⊆ f. let x,y,a ∈ h with a ∈ x◦y. we have f(a) ≥ (f ∗f)(a) = ∨ a≤x◦y {f (x) ∧f (y)}≥ f (x) ∧f (y) . thus ∧ a≤x◦y f (a) ≥ min{f (x) ,f (y)} . thus f is a fuzzy la-subsemihypergroup of h. � definition 4.2. let (h,◦,≤) be an ordered la-semihypergroup. a fuzzy subset f : h → [0, 1] is called a fuzzy right (resp., left) hyperideal of h if (1) ∧ z≤a◦b f (z) ≥ f (a) (resp., ∧ z≤a◦b f (z) ≥ f (b)), (2) a ≤ b implies f (a) ≥ f (b) , for every a,b ∈ h. if f is both fuzzy right hyperideal and fuzzy left hyperideal of h, then f is called a fuzzy hyperideal of h. definition 4.3. we consider a set h = {x,y,z} with the following hyperoperation ”◦” and the order ” ≤ ” : ◦ x y z w x x {x,w} {x,w} w y {x,w} {y,z} {y,z} w z {x,w} y y w w w w w w ≤:= {(x,x) , (x,y), (x,z), (y,y) , (z,z) , (w,x), (w,y), (w,z), (w,w)} . we give the covering relation ” ≺ ” and the figure of h as follows: ≺= {(x,y), (x,z), (w,x)} int. j. anal. appl. 16 (2) (2018) 285 then (h,◦,≤) is an ordered la-semihypergroup. now let f be a fuzzy subset of h such that f (a) =   0.6 if a = x 0.4 if a = y 0.2 if a = z 0.9 if a = w then f is a fuzzy two sided hyperideal of h. theorem 4.2. a fuzzy subset f of an ordered la-semihypergroup h is a fuzzy left (resp., right) hyperideal of h if and only if (i) h ∗f ⊆ f (resp., f ∗ h ⊆ f) (ii) if a ≤ b implies f (a) ≥ f (b) , for every a,b ∈ h. proof. the proof is similar to the proof of the theorem 4.1. � definition 4.4. let h be an ordered la-semihypergroup and f be a fuzzy subset of h. then for every t ∈ [0, 1] the set ft = {x : x ∈ h, f(x) ≥ t} is called the level set of h. definition 4.5. let h be an ordered la-semihypergroup and ∅ 6= a ⊆ h. then the characteristic function χa of a is defined as: χa : h → [0, 1] : x → χa(x) =   1 if x ∈ a0 if x /∈ a. theorem 4.3. let h be an ordered la-semihypergroup and f be a fuzzy subset of h. then f is a fuzzy la-subsemihypergroup (resp., right hyperideal, left hyperideal) of h if and only if for every t ∈ [0, 1], the non-empty level subset ft is a fuzzy la-subsemihypergroup (resp., right hyperideal, left hyperideal) of h. proof. assume that f is a fuzzy right hyperideal of h. let t ∈ [0, 1] with ft 6= φ. let a ∈ ft ◦h. we have a ∈ b◦h for some b ∈ ft and h ∈ h. by assumption, t ≤ f (b) ≤ ∧ a∈b◦h f (a) , we have f (a) ≥ t. this implies int. j. anal. appl. 16 (2) (2018) 286 ft ◦h ⊆ ft. let x ∈ ft and y ∈ h with y ≤ x. since t ≤ f (x) ≤ f (y) , we obtain y ∈ ft. therefore, ft is right hyperideal of h. conversely, we assume that for every t ∈ [0, 1] , ft is a right hyperideal of h. we show that fa ≤ ∧ c∈a◦b f (c) for a,b ∈ h. we put t◦ = f (a) . by assumption ft◦ is a right hyperideal of h. since a ∈ ft◦, a ◦ b ⊆ ft◦. then, for every c ∈ a◦ b, we obtain t◦ ≤ f (c) and hence, f (a) = t◦ ≤ ∧ c∈a◦b f (c) . let a,b ∈ h with a ≤ b. since a ≤ b, b ∈ ff(b) and ff(b) is a right hyperideal of h. we have a ∈ ffb. so, f (b) ≤ f (a) . therefore f is a fuzzy right hyperideal of h. � corollary 4.1. let h be an ordered la-semihypergroup and χi be the characteristic function of i. then, then i is an la-subsemihypergroup (resp., right hyperideal, left hyperideal) of h if and only if χi is a fuzzy la-subsemihypergroup (resp., right hyperideal, left hyperideal) of h. theorem 4.4. if {fi}i∈j is a family of fuzzy left hyperideals (resp., right hyperideals) of an ordered lasemihypergroup h, then ⋂ i∈j fi is a fuzzy left hyperideal (resp., right hyperideal) of h, where ⋂ i∈j fi = ∧ i∈j fi and ∧ i∈j fi (x) = inf {fi (x) : i ∈ j, x ∈ h} . proof. straightforward. � proposition 4.1. the fuzzy product of two fuzzy right (resp., left) hyperideals of an ordered la-semihypergroup h is again a fuzzy right (resp., left) hyperideal of h. proof. let f1 and f2 be two fuzzy right hyperideals of an ordered la-semihypergroup h. let x,y ∈ h such that x ≤ y. let (a,b) ∈ ay then y ≤ a◦b. since x ≤ y, so x ≤ a◦b implies (a,b) ∈ ax. hence ay ⊆ ax. now (f1 ∗f2)(y) = ∨ (a,b)∈ay {f1 (a) ∧f2 (b)} = ∨ (a,b)∈ay⊆ax {f1 (a) ∧f2 (b)} ≤ ∨ (a,b)∈ax {f1 (a) ∧f2 (b)} = (f1 ∗f2)(x) thus (f1 ∗f2)(x) ≥ (f1 ∗f2)(y). if ax = ∅, for any x ∈ h. then ∧ x≤a◦b (f1 ∗f2) (x) = 0 = (f1 ∗f2)(a). int. j. anal. appl. 16 (2) (2018) 287 if ax 6= ∅. since f1 and f2 are a fuzzy right hyperideals of h, then (f1 ∗f2)(x) = ∨ (a,b)∈ax {f1 (a) ∧f2 (b)} ≤ ∨ (a◦y,b◦y)∈ax◦y     ∧ m≤a◦y f1 (m)  ∧   ∧ n≤b◦y f2 (n)     = ∧ q≤x◦y (f1 ∗f2)(q). thus f1 ∗f2 is a fuzzy right hyperideal of h. � theorem 4.5. let f1 be a fuzzy right hyperideal and f2 a fuzzy left hyperideal of h. then f1 ∗f2 ⊆ f1 ∩f2. proof. let ax = ∅ for any x ∈ h. then (f1 ∗ f2)(x) = 0 = (f1 ∩ f2)(x). given that f1 is a fuzzy right hyperideal of h, i.e. ∧ a≤x◦y f1 (a) ≥ f1 (x) also given that f2 is fuzzy left hyperideal of h, i.e. ∧ a≤x◦y f2 (a) ≥ f2 (y) . let ax 6= ∅. then for x,y,a ∈ h, we have (f1 ∗f2) (a) = ∨ a≤x◦y {f1 (x) ∧f2 (y)}≤ ∨ a≤x◦y   ∧ a≤x◦y f1 (a) ∧ ∧ a≤x◦y f2 (a)   = (f1 ∩f2) (a) . thus f1 ∗f2 ⊆ f1 ∩f2. � lemma 4.1. let h be an ordered la-semihypergroup with left identity. then every fuzzy right hyperideal of h is fuzzy left hyperideal of h. proof. let h be an la-semihypergroup with pure left identity e, and f be a fuzzy right hyperideal of h. since f is a fuzzy right hyperideal of h, so f ∗ h ⊆ f. thus by lemma 3.1, and left invertive law, we have h ∗f = (h ∗ h) ∗f = (f ∗ h) ∗ h ⊆ f ∗ h ⊆ f. thus h ∗f ⊆ f. hence f is a fuzzy left hyperideal of h. � theorem 4.6. if f is a fuzzy left hyperideal of h with left identity, then f ∪ (f ∗ h) is a fuzzy hyperideal of h. proof. let f be a fuzzy left hyperideal of h. we have to show that f ∪ (f ∗ h) is fuzzy hyperideal. let (f ∪ (f ∗ h)) ∗ h = (f ∗ h) ∪ (f ∗ h) ∗ h = (f ∗ h) ∪ (h ∗ h) ∗f = (f ∗ h) ∪ (h ∗f) ⊆ (f ∗ h) ∪f = f ∪ (f ∗ h) . int. j. anal. appl. 16 (2) (2018) 288 hence f ∪ (f ∗ h) is fuzzy right hyperideal of h. since every fuzzy right hyperideal of an ordered lasemihypergroup with left identity is a fuzzy left hyperideal of h, so f ∪ (f ∗ h) is a fuzzy hyperideal of h. � definition 4.6. a fuzzy hyperideal f of an ordered la-semihypergroup h is called idempotent if f ∗f = f. proposition 4.2. every idempotent fuzzy left hyperideal of an ordered la-semihypergroup h is a fuzzy hyperideal of h. proof. let f be a fuzzy left hyperideal of h which is idempotent. then f ∗ h = (f ∗f) ∗ h = (h ∗f) ∗f ⊆ f ∗f = f. hence f is a fuzzy right hyperideal of h and so f is a fuzzy hyperideal of h. � proposition 4.3. if f is an idempotent fuzzy set in an ordered la-semihypergroup h with left identity. then h ∗f and f ∗ h are idempotents. proof. let f be an idempotent element in an ordered la-semihypergroup h with left identity. then by using medial law, we have (h ∗f) ∗ (h ∗f) = (h ∗ h) ∗ (f ∗f) = h ∗f. thus (h ∗f) ∗ (h ∗f) = h ∗f. the case for f ∗ h can be seen in a similar way. � 5. conclusion fuzzy set theory is a mathematical tools for dealing with uncertainties. this paper is devoted to the discussion of the combinations of fuzzy set in ordered la-semihypergroup. we combined these concepts to introduce fuzzy left (resp., right) hyperideals and discussed some interesting results. references [1] f. marty, sur uni generalization de la notion de groupe, 8th congress math. scandinaves, stockholm (1934) 45-49. [2] p. bonansinga and p. corsini, on semihypergroup and hypergroup homomorphisms, boll. un. mat. ital. b., 1(2) (1982) 717-727. [3] b. davvaz, some results on congruences in semihypergroups, bull. malyas. math. sci. so. 23 (2000) 53-58. [4] d. freni, minimal order semihypergroups of type u on the right, ii, j. algebra, 340 (2011) 77-80. [5] k. hilla, b. davvaz and k. naka, on quasi-hyperideals in semihypergroups, commun. algebra, 39 (2011) 4183. [6] v. leoreanu, about the simplifiable cyclic semihypergroups, italian j. pure appl. math., 7 (2000) 69-76. [7] m. d. salvo, d. freni and g. lo faro, fully simple semihypergroups, j. algebra, 399 (2014) 358-377. [8] d. heidari and b. davvaz, on ordered hyperstructures, u. p. b. sci. bull. series a, 73 (2011) 85-96. int. j. anal. appl. 16 (2) (2018) 289 [9] m. bakhshi and r.a. borzooei, ordered polygroups, ratio math, 24 (2013) 31-40. [10] j. chvalina, commutative hypergroups in the sense of marty and ordered sets, in: proc. summer school, gen. algebra ordered sets, olomouc (czech republic), (1994) 19-30. [11] s. hoskova, upper order hypergroups as a reflective subcategory of subquasiorder hypergroups, ital. j. pure appl. math., 20 (2006) 2015-222. [12] m. kondo and n. lekkoksung, on intra-regular ordered γ-semihypergroups, int. j. math. anal., 7(28) (2013) 1379-1386. [13] m. novak, el-hyperstructures, ratio math., 23 (2012) 65-80. [14] k. hila and j. dine, on hyperideals on left almost semihypergroups, isrn algebra, (2011) article id 953124 8pages. [15] m. a. kazim and m. naseerudin, on almost semigroup, aligarh bull. math., 2 (1972) 1-7. [16] n. yaqoob, p. corsini and f. yousafzai, on intra-regular left almost semihypergroups with pure left identity, j. math., article id 510790 (2013) 10 pages. [17] n. yaqoob and m. gulistan, partially ordered left almost semihypergroups, j. egyptian math. soc., 23(2) (2015) 231-235. [18] l. a. zadeh, fuzzy sets, inform. control., 8 (1965) 338-353. [19] a. rosenfeld, fuzzy groups, j. math. anal. appl., 35 (1971) 512-517. [20] p. corsini, m. shabir and t. mahmood, semisimple semihypergroups in terms of hyperideals, iran j. fuzzy. syst., 8 (2011) 95-111. [21] p. corsini, fuzzy multiset hyperstructures, european j. comb., 44 (2015) 198-207. [22] b. davvaz, characterization of subsemihypergroups by various triangular norms, czech. math. j., 55(4) (2005) 923-932. [23] b. davvaz, fuzzy hyperideals in semihypergroups, italian j. pure appl. math., 8 (200) 67-74. [24] k. hila and s. abdullah, a study on intuitionistic fuzzy sets in γ-semihypergroups, j. intell. fuzzy syst., 26 (2014) 1695-1710. [25] a. khan, m. farooq, m. izhar and b. davvaz, fuzzy hyperideals of left almost semihypergroups, int. j. anal. appl., 15(2) (2017) 155-171. [26] b. pibaljommee, k. wannatong and b. davvaz, an investigation on fuzzy hyperideals in ordered in semihypergroups, quasigroups and relat. syst., 23 (2015) 297-308. [27] b. pibaljommee and b. davvaz, characterizations of (fuzzy) bi-hyperideals in ordered semihypergroups, j. intell. fuzzy syst., 28 (2015) 2141-2148. [28] j. tang, a. khan and y. luo, characterizations of semisimple ordered semihypergroups in term of fuzzy hyperideals, j. intell. fuzzy syst., 30(3) (2016) 1735-1753. [29] j. tang, b. davvaz and x. xie, a study on (fuzzy) quasi-γ-hyperideals in ordered γ-semihypergroups, j. intell. fuzzy syst., 32(6) (2017) 3821-3838. [30] j. tang, b. davvaz and y. luo, a study on fuzzy hyperideals in ordered semihypergroups, italian j. pure appl. math., 36 (2016) 125-146. [31] j. tang, b. davvaz and y.f. luo, hyperfilters and fuzzy hyperfilters of ordered semihypergroups, j. intell. fuzzy syst., 29(1) (2015) 75-84. [32] n. tipachot and b. pibaljommee, fuzzy interior hyperideasl in ordered semihypergroups, italian j. pure appl. math., 36 (2016) 859-870. [33] j. zhan and b. davvaz, study of fuzzy algebraic hypersystems from a general view point, int. j. fuzzy syst., 12 (2010) 73-79. [34] j. zhan, b. davvaz and k.p. shum, a new view of fuzzy hypernear-rings, inform sci., 178 (2008) 425-438. 1. introduction 2. preliminaries 3. fuzzy ordered la-semihypergroups 4. fuzzy hyperideals in ordered la-semihypergroups 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 1, number 2 (2013), 71-78 http://www.etamaths.com on hyers-ulam stability for nonlinear differential equations of nth order maher nazmi qarawani abstract. this paper considers the stability of nonlinear differential equations of nth order in the sense of hyers and ulam. it also considers the hyersulam stability for superlinear emden-fowler differential equation of nth order. some illustrative examples are given. 1. introduction in 1940, ulam [1] posed the stability problem of of functional equations: given a group g1and a metric group g2 with metric ρ(., .). given ε > 0, does there exist a δ > 0 such that if f : g1 → g2 satisfies ρ(f(x),h(x)) < δ for all x,y ∈ g1,then a homomorphism h : g1 → g2 exists with ρ(f(x),h(x)) < ε,for all x,y ∈ g1? the problem for approximately additive mappings, on banach spaces, was solved by hyers [2]. the result obtained by hyers was generalized by rassias [3]. during the last two decades many mathematicians have extensively investigated the stability problems of functional equations (see [4-11]). alsina and ger [12] were the first mathematicians who investigated the hyersulam stability of the differential equation g′ = g.they proved that if a differentiable function y : i → r satisfies |y′ −y| ≤ ε for all t ∈ i,then there exists a differentiable function g : i → r satisfying g′(t) = g(t) for any t ∈ i such that |g −y| ≤ 3ε,for all t ∈ i. this result of alsina and ger has been generalized by takahasi et al [13] to the case of the complex banach space valued differential equation y′ = λy. furthermore, the results of hyers-ulam stability of differential equations of first order were also generalized by miura et al. [14], jung [15] and wang et al. [16]. li [17] established the stability of linear differential equation of second order in the sense of the hyers and ulam y′′ = λy. li and shen [18] proved the stability of nonhomogeneous linear differential equation of second order in the sense of the hyers and ulam y′′ + p(x)y′ + q(x)y + r(x) = 0, while gavruta et al. [19] proved the hyers-ulam stability of the equation y′′ + β(x)y = 0 with boundary and initial conditions. the author in his study [20] estabilshed the hyers-ulam stability of the equations of the second order z′′ = f(x,z) with the initial conditions z(x0) = 0 = z ′(x0). in this paper we investigate the hyers-ulam stability of the following nonlinear differential equation of nth order 2010 mathematics subject classification. 39a99, 39a30. key words and phrases. hyers-ulam stability, superlinear, emden-fowler differential equation. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 71 72 qarawani (1) y(n) = f(t,y,y′,y′′, ...,y(n−1)) with the initial conditions (2) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ c(n)(i),i = [t0, t1], (t, [y]) ≡ (t,y,y′,y′′, ...,y(n−1)) ∈ d, t ∈ i, −∞ < t0 < t1 < ∞, and f(t,y,y′,y′′, ...,y(n−1)) is defined on a closed bounded set d ⊂ rn+1 that satisfies the condition (3) |f(t, [y]) −f(t, [z])| ≤ g(t) |y(t) −z(t)| (t1 − t0) n−1 where g(t) : i → (0,∞) is integrable function. moreover we establish the hyers-ulam stability of the problem (1),(2) for f satisfying the lipschitz condition (4) |f(t, [y]) −f(t, [z])| ≤ a0 n−1∑ i=0 ∣∣y(i)(t) −z(i)(t)∣∣ where a0 > 0. definition 1 we will say that the equation (1) has the hyers -ulam stability if there exists a positive constant k > 0 with the following property: for every ε > 0, y ∈ c(n)(i), if (5) |y(n) −f(t, [y])| ≤ ε with the initial condition (2), then there exists a solution z(t) ∈ c(n)(i) of the equation (1), such that |y(t) − z(t)| ≤ kε, where k is a constant that does not depend on ε nor on y(t). 2. main results on hyers-ulam stability theorem 1 if y ∈ c(n)(i) and f(t,y,y′,y′′, ...,y(n−1)) satisfies condition (3) on a closed bounded set d ⊂ rn+1, then the initial value problem(1),(2) is stable in the sense of hyers and ulam. proof. let ε > 0 and y(t) be an approximate solution of the initial value problem (1),(2).we will show that there exists a function z(t) ∈ c(n)(i) satisfying the equation (1) and the initial condition (2) such that |y(t) −z(t)| ≤ kε from the inequality (5) we have (6) −ε ≤ y(n) −f(t, [y]) ≤ ε integrating the last inequality n times, we obtain (7) ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ ≤ (t− t0) nε n! on hyers-ulam stability 73 it is clear that z(t) = n−1∑ k=0 (t−t0)kyk k! + t∫ t0 f(s, [z]) (t−s)n−1 (n−1)! ds satisfies equation (1) and the initial condition (2). consider the difference |y(t) −z(t)| ≤ ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ + ∣∣∣∣∣∣ t∫ t0 f(s, [y]) −f(s, [z])) (t−s)n−1 (n− 1)! ds ∣∣∣∣∣∣ ≤ (t− t0)nε n! + 1 (n− 1)! t∫ t0 g(s) |y(s) −z(s)| (t1 − t0) n−1 (t1 − t0) n−1 ds(8) applying gronwall’s inequality, we obtain from inequalities (7) and (8) |y(t) −z(t)| ≤ (t1 − t0)nε n! exp   1 (n− 1)! t∫ t0 g(t)ds   whence max t0≤t≤t1 |y(t) −z(t)| ≤ kε hence the initial value problem (1),(2) is stable in the sense of hyers and ulam. theorem 2 if y ∈ c(n)(i) and f(t,y,y′,y′′, ...,y(n−1)) satisfies the lipschitz condition (4) on a closed bounded set d ⊂ rn+1, then the initial value problem(1),(2) is stable in the sense of hyers and ulam. proof. given ε > 0, assume that y is an approximate solution of eq. (1). we will show that there exists a function z(t) ∈ c(n)(i) satisfying equation (1) such that |y(t) −z(t)| ≤ kε from the inequality (5) we have (9) −ε ≤ y(n) −f(t, [y]) ≤ ε by integrating the inequality (9) k times, we obtain (10) ∣∣∣∣∣y(n−k)(t) − k−1∑j=0 (t−t0) jyn−k+j j! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ ≤ (t− t0) kε k! where 1 ≤ k ≤ n. we can easily verify that the function z(t) z(n−k)(t) = k−1∑ j=0 (t−t0)jyn−k+j j! + t∫ t0 f(s, [z]) (t−s)k−1 (k−1)! ds must satisfy the initial value problem (1),(2) 74 qarawani now let ∆(n−k) ≡ ∣∣y(n−k) − z(n−k)∣∣ . then, using the inequalities (4),(10), we get the estimation ∆(n−k) ≤ ∣∣∣∣∣y(n−k)(t) − k−1∑j=0 (t−t0) jyn−k+j j! − 1 (n−1)! t∫ t0 (t−s)n−1f(s, [y])ds ∣∣∣∣∣ + 1 (n− 1)! t∫ t0 |f(s, [y]) −f(s, [z])|(t−s)n−1ds ≤ (t1 − t0)kε k! + a0n (n− 1)! t∫ t0 ∣∣∣y(n−k)(s) −z(n−k)(s)∣∣∣ (t−s)n−1ds(11) thus, according to (4),(10) and (11), from gronwall’s inequality it follows that ∣∣∣y(n−k)(t) −z(n−k)(t)∣∣∣ ≤ (t1 − t0)kε k! exp ( a0(t1 − t0)n (n− 1)! ) consequently for k = n, we have max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( a0(t1 − t0)n (n)! ) hence the initial value problem (1),(2) is stable in the sense of hyers and ulam. remark 1 suppose that y ∈ c(n)(i) satisfies the inequality (9) with the zero initial condition y(t0) = 0, y ′(t0) = 0, ... , y (n−1)(t0) = 0. if f(t, [z]) satisfies lipschitz condition (4) and f(t, 0, , ..., 0) ≡ 0, then one can similarly show that the zero solution z0 ≡ 0 of equation (1) is stable in the sense of hyers and ulam. 3. hyers-ulam stability for superlinear nth order differential equation in this section we investigate the hyers ulam stability of solutions for superlinear nth order differential equation (12) y(n) = h(t) |y|α sgny , α > 1 with the initial condition (13) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ c(n)(i),i = [t0, t1], −∞ < t0 < t1 < ∞, and h(t) : i → r is continuous. theorem 3 if y ∈ c(n)(i),and h(t) : i → r is continuous, then the initial value problem (12),(13) is stable in the sense of hyers and ulam. proof. given ε > 0,suppose y(t) is an approximate solution of the initial value problem (12),(13).we show that there exists an exact solution z(t) ∈ c(n)(i) satisfying the equation (12) such that |y(t) −z(t)| ≤ kε where k is a constant that does not explicitly depend on ε nor on y(t). from the inequality (5) we have (14) −ε ≤ y(n) −h(t) |y|α sgny ≤ ε by integrating the last inequality n times, we obtain on hyers-ulam stability 75 (15) ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 h(s) |y|α sgny.(t−s)n−1ds ∣∣∣∣∣ ≤ (t− t0) nε n! where 1 ≤ k ≤ n. we can easily verify that the function z(t) z(t) = n−1∑ k=0 (t−t0)kyk k! + 1 (n−1)! t∫ t0 h(s) |z|α sgnz.(t−s)n−1ds must satisfy the initial value problem (12),(13). now since the derivative ∣∣∣∂(h(t)yα)∂y ∣∣∣ is bounded on s, then the function f(t,y) = h(t) |y|α sgny satisfies lipschitz condition |f(t,y) −f(t,z)| ≤ l |y(t) −z(t)| , (t,y), (t,z) ∈ s where s = [t0, t1] × [−m,m] ⊂ r2, and m = max t0≤t≤t1 |y(t)| . since h is continuous on i, then ∃ b0 > 0, |h(t)| ≤ b0, and from the inequality (15), we get the estimation |y(t) −z(t)| ≤ (t1 − t0)nε n! + b0l (n− 1)! t∫ t0 |y(s) −z(s)|(t−s)n−1ds from gronwall’s inequality it follows that |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( b0l(t1 − t0)n n! ) consequently, we have max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( (t1 − t0)n (n)! ) hence the initial value problem (12),(13) is stable in the sense of hyers and ulam. remark 2 suppose that y ∈ c(n)(i) satisfies the inequality (6) with the zero initial condition y(t0) = 0, y ′(t0) = 0, ... , y (n−1)(t0) = 0. if the function h : i → r is continuous, then one can similarly establish the hyers-ulam stability of zero solution z0 ≡ 0 of (12). example1 consider the problem y(5) = 8y2 sin t + et(16) y(k)(t0) = 0, k = 0, 4(17) and the inequality (18) −ε ≤ y(5) −y2 sin t + et ≤ ε where (t,y) ∈ [t0, t1] × [−m1,m1] , m1 = max t0≤t≤t1 |y(t)| . 76 qarawani integrating the inequality (18) five times and using the initial condition (17), we get that ∣∣∣∣∣∣y(t) − 13 t∫ t0 (y3 sin t + et)(t−s)4ds ∣∣∣∣∣∣ ≤ (t− t0) 5ε 5! one can easily show that z(t) z(t) = 1 3 t∫ t0 (z3 sin t + et)(t−s)4ds has to satisfy the initial value problem (16),(17). now let us estimate the difference: |y(t) −z(t)| ≤ ∣∣∣∣∣∣y(t) − 13 t∫ t0 (t−s)4(y3 sin t + et)ds ∣∣∣∣∣∣ + 1 3 t∫ t0 (t−s)4 ∣∣y3 −z3∣∣ |sin t|ds ≤ (t1 − t0)5ε 5! + m21 t∫ t0 (t−s)4 |y −z|ds therefore, we obtain max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)5ε 5! exp ( m21 (t1 − t0)5 5! ) hence the initial value problem (16),(17) is stable in the sense of hyers and ulam. 4. a special case of equation (11) consider the equation (19) y(n) = h(t)y with the initial conditions (20) y(t0) = y0 , y ′(t0) = y1, ... , y (n−1)(t0) = yn−1 where y ∈ c(n)(i),i = [t0, t1], −∞ < t0 < t1 < ∞, and h(t) : i → r is continuous. theorem 4 if y ∈ c(n)(i),and h(t) : i → r is continuous, then the initial value problem(19),(20) is stable in the sense of hyers and ulam. proof. assume that ε > 0 and that y is n times continuously differentiable realvalued function on i = [t0, t1]. we will show that there exists a function z(t) ∈ c2(i) satisfying equation (19) such that |y(t) −z(t)| ≤ kε we have (21) −ε ≤ y(n) −h(t)y ≤ ε by integrating the inequality (21) n times, we obtain on hyers-ulam stability 77 ∣∣∣∣∣y(t) −n−1∑k=0 (t−t0) kyk k! − 1 (n−1)! t∫ t0 (t−s)n−1h(s)yds ∣∣∣∣∣ ≤ (t− t0) nε n! where 1 ≤ k ≤ n. it is easily to verify that the function z(t) z(t) = n−1∑ k=0 (t−t0)kyk k! + 1 (n−1)! t∫ t0 (t−s)n−1h(s)zds satisfies the initial value problem (19),(20). one can get the estimation (22) |y(t) −z(t)| ≤ (t1 − t0)nε n! + b0 (n− 1)! t∫ t0 |y(t) −z(t)|(t−s)n−1ds using gronwall’s inequality we have hence max t0≤t≤t1 |y(t) −z(t)| ≤ (t1 − t0)nε n! exp ( b0(t1 − t0)n (n)! ) therefore, the initial value problem (19),(20) is stable in the sense of hyers and ulam. example 2 consider the equation (23) y(4) − (1 + cos t) y = 0 (24) y(0) = 0, y′(0) = 1, y′′(0) = −1, y′′′(0) = 0 and the inequality ∣∣∣ y(4) − (1 + cos t) y ∣∣∣ ≤ ε where 0 ≤ t ≤ b, b ∈ r. integrating the last inequality four times, we get∣∣∣∣∣∣y(t) − t + t 2 2 − 1 6 t∫ 0 (t−s)3 (1 + cos t) yds ∣∣∣∣∣∣ ≤ t 3ε 6 one can easily find that z(t) z(t) = t− t 2 2 + 1 6 t∫ 0 (t−s)3 (1 + cos t) zds satisfies the equation (23) and initial condition (24) then, we obtain an estimation |y(t) −z(t)| ≤ b3ε 6 exp(b4/12) hence eq. (23) has the hyers -ulam stability. 78 qarawani references [1] ulam s.m., (1964), problems in modern mathematics, john wiley & sons, new york, ny, usa, science edition. [2] hyers d. h., (1941), on the stability of the linear functional equation, proceedings of the national academy of sciences of the united states of america, vol. 27, pp. 222–224. [3] rassias t. m. (1978), on the stability of the linear mapping in banach spaces, proceedings of the american mathemaical society, vol. 72, no. 2, pp. 297–300. [4] miura t., takahasi s.-e., choda h., (2001), on the hyers-ulam stability of real continuous function valued differentiable map, tokyo journal of mathematics 24, pp. 467-476. [5] jung s. m., (1996), on the hyers-ulam-rassias stability of approximately additive mappings, journal of mathematics analysis and application 204, pp. 221-226. [6] park c. g., (2002), on the stability of the linear mapping in banach modules,journal of mathematics analysis and application 275, pp. 711-720. [7] gavruta p., (1994), a generalization of t he hyers-ulam-rassias stability of approximately additive mappings, journal of mathematical analysis and applications, vol. 184, no. 3 , pp. 431–436. [8] jun k .-w., lee y. -h., (2004), a generalization of the hyers-ulam-rassias stability of the pexiderized quadratice quations, journal of mathematcal analysis and applications , vol. 297, no. 1, pp. 70– 86. [9] jung s.-m. (2001), hyers-ulam-rassias stability of functional equations in mathematical analysis, hadronic press, palm harbor, fla, u sa. [10] park c. , (2005), homomorphisms between poisson jc*-algebras, bulletin of the brazilian mathematical society, vol. 36, no. 1 , pp. 79–97. [11] c. park, cho y.-s. and han m., (2007), functional inequalities associated with jordan-von neumanntype additive functional equations, journal of inequalities and applications , vol. 2007, article id 41820, 13 pages. [12] alsina c., ger r., (1998), on some inequalities and stability results related to the exponential function, journal of inequalities and application 2, pp 373-380. [13] takahasi e., miura t., and miyajima s., (2002), on the hyers-ulam stability of the banach space-valued differential equation y′ = λy, bulletin of the korean mathematical society, vol. 39, no. 2, pp 309–315. [14] miura t., miyajima s., takahasi s.-e., (2007), a characterization of hyers-ulam stability of first order linear differential operators, journal of mathematics analysis and application 286. pp. 136-146. [15] jung s. m., (2005), hyers-ulam stability of linear differential equations of first order, journal of mathematics analysis and application 311, pp. 139-146. [16] wang g., zhou m. and sun l. (2008), hyers-ulam stability of linear differential equations of first order, applied mathematics letters 21, pp 1024-1028. [17] li y., (2010), hyers-ulam stability of linear differential equations,thai journal of mathematics,vol. 8 no 2, pp 215–219. [18] li y. and shen y., (2009), hyers-ulam stability of nonhomogeneous linear differential equations of second order, international journal of mathematics and mathematical sciences, vol. 2009, article id 576852, pp. 7. [19] gavruta p., jung s. and li y. , (2011), hyers-ulam stability for second-order linear differential equations with boundary conditions, ejde, vol.2011, no. 80, pp1-7, .http://ejde.math.txstate.edu/volumes/2011/80/gavruta.pdf. [20] qarawani m. n., (2012), hyers-ulam stability of a generalized secondorder nonlinear differential equation, applied mathematics,vol. 3, no. 12, pp. 1857-1861. doi: 10.4236/am.2012.312252. department of mathematics, alquds open university, palestine, west-bank, salfit p.o.box 37 int. j. anal. appl. (2022), 20:67 existence suzuki type fixed point results in ab-metric spaces with application p. naresh1,2,∗, g. upender reddy2, b. srinuvasa rao3 1department of mathematics, sreenidhi institute of science and technology, ghatkesar, hyderabad-501301, telangana, india 2department of mathematics, mahatma gandhi university, nalgonda, telangana, india 3department of mathematics, dr. b.r. ambedkar university, srikakulam, etcherla-532410, andhra pradesh, india ∗corresponding author: parakala2@gmail.com abstract. in this paper, we give some applications to integral equations as well as homotopy theory via suzuki contractive type common coupled fixed point results in complete ab-metric space. we also furnish an example which supports our main result. 1. introduction the study of fixed points is aexquisite synthesis of analysis, topology, and geometry. its numerous applications in areas such as homotopy theory, integral, integro-differential, and impulsive differential equations, finding solutions to optimization problems, approximation theory, non-linear analysis, biomechanics, and algorithms have made it an essential tool. suzuki [1] recently established expanded versions of edelstein’s and banach’s fundamental conclusions, sparking a great deal of research in this area (see [2–6]). the b-metric space was first introduced by i.a. bakhtin [7] in 1989. numerous generalizations of metric spaces were created as a result of the development of b-metric space.the n-tuple metric space was first introduced and its topological features were examined by m.abbas et al. in 2015 [8]. ab-metric spaces were first described by m. ughade et al. [9] as a generalized version of n-tuple metric space. then, in partially ordered ab-metric received: oct. 21, 2022. 2010 mathematics subject classification. 47h10, 54h25, 58c30, 58j20. key words and phrases. suzuki type contraction; ω-compatible; ab-completeness; coupled common fixed points. https://doi.org/10.28924/2291-8639-20-2022-67 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-67 2 int. j. anal. appl. (2022), 20:67 spaces, n.mlaiki et al. [19] and k.ravibabu et al. [11,12] discovered original coupled common fixed point theorems. the idea of coupled fixed point was first suggested in 1987 by guo and lakshmikantham [13]. later, employing a weak contractivity type assumption, bhaskar and lakshmikantham [14] derived a novel fixed point theorem for a mixed monotone mapping on a metric space powered with partial ordering. see study findings in [15–20] and related sources for additional results on coupled fixed point outcomes. in the context of ab-metric spaces, the purpose of the current research is to establish original common coupled fixed point theorems for suzuki contractive type mapping. we can also provide examples that are appropriate and relevant applications to homotopy and integral equations. before we can demonstrate the primary findings, we need certain fundamental definitions, results and examples from the literature. 2. preliminaries definition 2.1. [9] let p be a non-empty set and b ≥ 1 be given real number. a mapping ab : pn → [0,∞) is called an ab-metric on p if and only if for all λi,ν ∈ p i = 1,2,3, ..n; the following conditions hold : (ab1) ab(λ1,λ2, ........,λn−1,λn)≥ 0, (ab2) ab(λ1,λ2, ........,λn−1,λn)=0⇔ λ1 = λ2 = · · · · · ·= λn−1 = λn, (ab3) ab(λ1,λ2, ........,λn−1,λn)≤ b   ab (λ1,λ1, ........,(λ1)n−1,ν) +ab (λ2,λ2, ........,(λ2)n−1,ν) + · · · · · ·+ab (λn−1,λn−1, ........,(λn−1)n−1,ν) +ab (λn,λn, ........,(λn)n−1,ν)   then the pair (p,ab) is called an ab-metric space. remark 2.1. [9] it should be noted that, the class of ab-metric spaces is effectively larger than that of a-metric spaces. indeed each a-metric space is a ab-metric space with b = 1.however, the converse is not always true. also ab-metric space is an "n-dimensional sb-metric space. therefore the sb-metric are special cases of an ab-metric with n =3. following example shows that a ab-metric on p need not be a a-metric on p. example 2.1. [9] let p = [0,+∞), define ab :pn → [0,+∞) as ab (λ1,λ2, ........,λn−1,λn) = ∑n i=1 ∑ i<j |λi −λj| 2 for all λi ∈ p, i = 1,2, · · · . then (p,ab) is an ab-metric space with b =2 > 1. definition 2.2. [9] a abmetric space (p,ab) is said to be symmetric if ab (λ,λ, · · · ,(λ)n−1,%)= ab (%,%, · · · ,(%)n−1,λ) for all λ,% ∈p. int. j. anal. appl. (2022), 20:67 3 definition 2.3. [9] let (p,ab) be a ab-metric space. then, for λ ∈p, r > 0 we defined the open ball bab(λ,r) and closed ball bab[λ,r] with center λ and radius r as follows respectively: bab(λ,r)= {% ∈p : ab(%,%, · · · ,(%)n−1,λ) < r}, and bab[λ,r] = {% ∈p : ab(%,%, · · · ,(%)n−1,λ)≤ r}. lemma 2.1. [9] in a ab-metric space, we have (1) ab(λ,λ, · · · ,(λ)n−1,%)≤ bab(%,%, · · · ,(%)n−1,λ); (2) ab(λ,λ, · · · ,(λ)n−1,ζ)≤ b(n−1)ab(λ,λ, · · · ,(λ)n−1,%)+b2ab(%,%, · · · ,(%)n−1,ζ). definition 2.4. [9] if (p,ab) be a ab-metric space. a sequence {λk} in p is said to be: (1) ab-cauchy sequence if, for each � > 0, there exists n0 ∈ n such that ab(λk,λk, · · · · · ·(λk)n−1,λm) < � for each m,k ≥ n0. (2) ab-convergent to a point λ ∈p if, for each � > 0, there exists a positive integer n0 such that ab(λk,λk, · · · · · ·(λk)n−1,λ) < � for all n ≥ n0 and we denote by lim k→∞ λk = λ. (3) a ab-metric space (p,ab) is called complete if every ab-cauchy sequence is ab-convergent in p. lemma 2.2. [9] if (p,ab) be a ab-metric space with b ≥ 1 and suppose that {λk} is a ab-convergent to λ and {ζk} is a ab-convergent to ζ, then we have (i) 1 b2 ab(λ,λ, · · · ,(λ)n−1,ζ) ≤ lim k→∞ inf ab(λk,λk, · · · ,(λk)n−1,ζk) ≤ lim k→∞ supab(λk,λk, · · · ,(λk)n−1,ζk) ≤ b2ab(λ,λ, · · · ,(λ)n−1,ζ). in particular, if ζk = ζ is constant, then (ii) 1 b2 ab(λ,λ, · · · ,(λ)n−1,ζ) ≤ lim k→∞ inf ab(λk,λk, · · · ,(λk)n−1,ζ) ≤ lim k→∞ supab(λk,λk, · · · ,(λk)n−1,ζ) ≤ b2ab(λ,λ, · · · ,(λ)n−1,ζ). theorem 2.1. [1] let (p;d) be a complete metric space, let t :p →p be a mapping and define a non increasing function θ : [0;1)→ (0;1] by θ(t)=   1, 0≤ t ≤ √ 5−1 2 (1− t)t−2, √ 5−1 2 ≤ t ≤ 1√ 2 (1+ t)−1, 1√ 2 < t ≤ 1 . 4 int. j. anal. appl. (2022), 20:67 assume that there exists t ∈ [0;1) such that θ(t)d(ρ,tρ)≤ d(ρ;%) implies d(tρ;t%)≤ td(ρ;%) for all ρ,% ∈p. then there exists a unique fixed point a of t . moreover, lim n→∞ tnρ = a for all ρ ∈p. in order to obtain our results we need to consider the followings. 3. main results definition 3.1. let (p,ab) be a ab-metric spaces and suppose f : p2 → p be a mapping. if f (ρ,%)= ρ, f (%,ρ)= % for ρ,% ∈p then (ρ,%) is called a coupled fixed point of f. definition 3.2. let (p,ab) be a ab-metric spaces and suppose f :p2 →p and f :p →p be two mappings. an element (ρ,%) is said to be a coupled coincident point of f and f if f (ρ,%) = f ρ, f (%,ρ)= f %. definition 3.3. let (p,ab) be a ab-metric spaces and suppose f : p2 → p, f : p → p be two mappings. an element (ρ,%) is said to be a coupled common point of f and f if f (ρ,%) = f ρ = ρ, f (%,ρ)= f % = %, definition 3.4. let (p,ab) be a ab-metric space. a pair (f,f ) is called weakly compatible if f (f(ρ,%))= f(f ρ,f %) whenever for all ρ,% ∈p such that f (ρ,%)= f ρ, f (%,ρ)= f %. theorem 3.1. let (p,ab) be a ab-metric space. suppose that t : p2 → p and f : p → p be a two mappings satisfying the following: η(θ)ab (f λ,f λ, · · · ,(f λ)n−1,t(λ,ζ))≤max   ab (f λ,f λ, · · · ,(f λ)n−1, f ρ) , ab (f ζ, f ζ, · · · ,(f ζ)n−1, f %) ab (f λ,f λ, · · · ,(f λ)n−1,t(λ,ζ)) , ab (f ζ, f ζ, · · · ,(f ζ)n−1,t(ζ,λ)) ,   implies ab (t(λ,ζ),t(λ,ζ), · · · ,(t(λ,ζ))n−1,t(ρ,%))≤ θmax   ab (f λ,f λ, · · · ,(f λ)n−1, f ρ) , ab (f ζ, f ζ, · · · ,(f ζ)n−1f %) , ab (f λ,f λ, · · · ,(f λ)n−1,t(λ,ζ)) , ab (f ζ, f ζ, · · · ,(f ζ)n−1,t(ζ,λ)) , ab (f ρ,f ρ, · · · ,(f ρ)n−1,t(ρ,%)) , ab (f %,f %, · · · ,(f %)n−1,t(%,ρ)) , ab (f ρ,f ρ, · · · ,(f ρ)n−1,t(λ,ζ)) , ab (f %,f %, · · · ,(f %)n−1t(ζ,λ))   . (3.1) int. j. anal. appl. (2022), 20:67 5 for all λ,ζ,ρ,% ∈p, where θ ∈ [0,1) and η : [0,1)→ (0,1] defined as η(θ)= 1 b2((n−1)+θ)is a strictly decreasing function, a) t(p2)⊆ f (p) and f (p) is complete, b) pair (t,f ) is ω-compatible. then there is a unique common coupled fixed point of t and f in p. proof. let λ0,ζ0 ∈p be arbitrary, and from (a), we construct the sequences {λp} ,{ζp} , in p as t (λp,ζp)= f λp+1, t (ζp,λp)= f ζp+1, where p =0,1,2, . . . . case (i): assume that f λp 6= f λp+1 or f ζp 6= f ζp+1∀ p. (3.2) since η(θ)ab (f λ0, f λ0, · · · ,t(λ0,ζ0)) = η(θ)ab (f λ0, f λ0, · · · , f λ1) ≤ ab (f λ0, f λ0, · · · , f λ1) ≤ max   ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) , ab (f λ0, f λ0, · · · ,t(λ0,ζ0)) , ab (f ζ0, f ζ0, · · · ,t(ζ0,λ0))   . then from (3.1), we can get ab(f λ1, f λ1, · · · , f λ2) = ab (t(λ0,ζ0),t(λ0,ζ0), · · · ,t(λ1,ζ1)) ≤ θmax   ab (f λ0, f λ0, · · · , f λ1) ,ab (f ζ0, f ζ0, · · · , f ζ1) , ab (f λ0, f λ0, · · · ,t(λ0,ζ0)) ,ab (f ζ0, f ζ0, · · · ,t(ζ0,λ0)) , ab (f λ1, f λ1, · · · ,t(λ1,ζ1)) ,ab (f ζ1, f ζ1, · · · ,t(ζ1,λ1)) , ab (f λ1, f λ1, · · · ,t(λ0,ζ0)) ,ab (f ζ1, f ζ1, · · · ,t(ζ0,λ0))   ≤ θmax { ab (f λ0, f λ0, · · · , f λ1) ,ab (f ζ0, f ζ0, · · · , f ζ1) , ab (f λ1, f λ1, · · · , f λ2) ,ab (f ζ1, f ζ1, · · · , f ζ2) } (3.3) similarly, we can prove that ab(f ζ1, f ζ1, · · · , f ζ2)≤ θmax   ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) , ab (f λ1, f λ1, · · · , f λ2) , ab (f ζ1, f ζ1, · · · , f ζ2)   . (3.4) 6 int. j. anal. appl. (2022), 20:67 due to (3.3)− (3.4), we conclude that max { ab(f λ1, f λ1, · · · , f λ2), ab(f ζ1, f ζ1, · · · , f ζ2) } ≤ θmax   ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) , ab (f λ1, f λ1, · · · , f λ2) , ab (f ζ1, f ζ1, · · · , f ζ2)   . (3.5) if max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } ≤max { ab (f λ1, f λ1, · · · , f λ2) , ab (f ζ1, f ζ1, · · · , f ζ2) } . then from (3.5), we have f λ1 = f λ2 or f ζ1 = f ζ2. it is contradiction to (3.2). hence from (3.5), we have max { ab (f λ1, f λ1, · · · , f λ2) , ab (f ζ1, f ζ1, · · · , f ζ2) } ≤ θmax { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } . continuing in this way, we get max { ab (f λp, f λp, · · · , f λp+1) , ab (f ζp, f ζp, · · · , f ζp+1) } ≤ θmax { ab (f λp−1, f λp−1, · · · , f λp) , ab (f ζp−1, f ζp−1, · · · , f ζp) } ≤ θ2max { ab (f λp−2, f λp−2, · · · , f λp−1) , ab (f ζp−2, f ζp−2, · · · , f ζp−1) } ... ≤ θp max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } . thus ab (f λp, f λp, · · · , f λp+1)≤ θp max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } , and ab (f ζp, f ζp, · · · , f ζp+1)≤ θp max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } now for q > p, by use of (ab3), we have ab (f λp, f λp, · · · ,(f λp)n−1, f λq)≤ b   ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +ab (f λp, f λp, ........,(f λp)n−1, f λp+1) + · · · · · ·+ab (f λp,λp, ........,(f λp)n−1, f λp+1) +ab (f λq, f λq, ........,(f λq)n−1, f λp+1)   ≤ b(n−1)ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +bab (f λq, f λq, ........,(f λq)n−1, f λp+1) int. j. anal. appl. (2022), 20:67 7 ≤ b(n−1)ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b2ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λq) ≤ b(n−1)ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b3(n−1)ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λp+2) +b4ab (f λp+2, f λp+2, ........,(f λp+2)n−1, f λq) ≤ b(n−1)ab (f λp, f λp, ........,(f λp)n−1, f λp+1) +b3(n−1)ab (f λp+1, f λp+1, ........,(f λp+1)n−1, f λp+2) +b5(n−1)ab (f λp+2, f λp+2, ........,(f λp+2)n−1, f λp+3) +b7(n−1)ab (f λp+3, f λp+3, ........,(f λp+3)n−1, f λp+4) + . . .+b2q−2p−2(n−1)ab (f ζq−2, f ζq−2, ........,(f ζq−2)n−1, f λq−1) +b2q−2p−3ab (f λq−1, f λq−1, ........,(f λq−1)n−1, f λq) ≤ (n−1) ( bθp +b3θp+1 +b5θp+2 + . . .+b2q−2p−2θq−2 ) max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } +b2q−2p−3θq−1max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp ( 1+b2θ+b4θ2 + . . .+b2q−2p−4θq−p−2 ) max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } +b2q−2p−3θq−p−1max { ab (f λ0, f λ0, · · · , f λ1) ,ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp ( 1+b2θ+b4θ2 +b6θ3 + . . . ) max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } ≤ (n−1)bθp 1−b2θ max { ab (f λ0, f λ0, · · · , f λ1) , ab (f ζ0, f ζ0, · · · , f ζ1) } → 0 as p,q →∞. hence {f λp} is a cauchy sequence in f (p) . similarly we can show that {f ζp}, is cauchy sequence in f (p). since f (p) is complete, there exist α,β and a,b in p such that lim p→∞ f λp = α = f a lim p→∞ f ζp = β = f b since f λp → α, f ζp → β, we may assume that f λp 6= α, f ζp 6= β for infinitely many p. we claim that max { ab (f a,f a, · · · ,t(λ,ζ)) , ab (f b,f b, · · · ,t(ζ,λ)) , } ≤ θmax   ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) , ab (f ζ, f ζ, · · · ,t(ζ,λ))   8 int. j. anal. appl. (2022), 20:67 for all λ,ζ ∈p with f a 6= f λ,f b 6= f ζ. let λ,ζ ∈p with f a 6= f λ,f b 6= f ζ. then there exists a positive integer p0 such that for p ≥ p0, we have ab (f a,f a, · · · , f λp)≤ 12b2(n−1)ab (f a,f a, · · · , f λ), ab (f b,f b, · · · , f ζp)≤ 12b2(n−1)ab (f b,f b, · · · , f ζ). now for p ≥ p0, η(θ)ab (f λp, f λp, · · · ,(f λp)n−1,t(λp,ζp)) ≤ ab (f λp, f λp, · · · ,(f λp)n−1,t(λp,ζp)) = ab (f λp, f λp, · · · ,(f λp)n−1, f λp+1) ≤ b(n−1)ab (f λp, f λp, · · ·(f λp)n−1, f a) +b2ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ b2(n−1)ab (f a,f a · · · ,(f a)n−1, f λp) +b2ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ b2(n−1)ab (f a,f a · · · ,(f a)n−1, f λp) +b2(n−1)ab (f a,f a, · · · ,(f a)n−1, f λp+1) ≤ 1 2 ab (f a,f a · · · ,(f a)n−1, f λ) + 1 2 ab (f a,f a, · · · ,(f a)n−1, f λ) ≤ ab (f a,f a · · · ,(f a)n−1, f λ) ≤ ab (f λ,f λ, · · · , f λp) ≤ max { ab (f λ,f λ, · · · , f λp) ,ab (f ζ, f ζ, · · · , f ζp) , ab (f λp, f λp, · · · ,t(λp,ζp)) ,ab (f ζp, f ζp, · · · ,t(ζp,λp)) } . from (3.1), we have ab (t(λp,ζp),t(λp,ζp), · · · ,t(λ,ζ)) ≤ θmax   ab (f λp, f λp, f λ) ,ab (f ζp, f ζp, f ζ) , ab (f λp, f λp, f λp+1) ,ab (f ζp, f ζp, f ζp+1) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) , ab (f λ,f λ, · · · , f λp+1) ,ab (f ζ, f ζ, · · · , f ζp+1)   . letting p →∞, we get ab (f a,f a, · · · ,t(λ,ζ))≤ θmax { ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) } . int. j. anal. appl. (2022), 20:67 9 similarly we can show that ab (f b,f b, · · · ,t(ζ,λ))≤ θmax { ab (f b,f b, · · · , f ζ) ,ab (f a,f a, · · · , f λ) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) } . thus max { ab (f a,f a, · · · ,t(λ,ζ)) , ab (f b,f b, · · · ,t(ζ,λ)) , } ≤ θmax   ab (f a,f a, · · · .f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) , ab (f ζ, f ζ, · · · ,t(ζ,λ))   . hence the claim. now consider ab (f λ,f λ, · · · ,t(λ,ζ)) ≤ (n−1)bab (f λ,f λ, · · · , f a)+b2ab (f a,f a, · · · ,t(λ,ζ)) ≤ (n−1)b2ab (f a,f a, · · · , f λ) +b2θmax { ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) , } ≤ b2 ((n−1)+θ)max   ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) , ab (f ζ, f ζ, · · · ,t(ζ,λ))   . thus η(θ)ab (f λ,f λ, · · · ,t(λ,ζ))≤max { ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) } . hence from (3.1), we have ab (t(λ,ζ),t(λ,ζ), · · · ,t(a,b)) ≤ θmax   ab (f λ,f λ, · · · , f a) ,ab (f ζ, f ζ, · · · , f b) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) , ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) , ab (f a,f a, · · · ,t(λ,ζ)) ,ab (f b,f b, · · · ,t(ζ,λ))   . now ab (f a,f a, · · · ,t(a,b))= lim p→∞ ab (f λp+1, f λp+1, · · · ,t(a,b)) = lim p→∞ ab (t(λp,yp),t(λp,ζp), · · · ,t(a,b)) ≤ lim p→∞ θmax   ab (f λp, f λp, · · · , f a) ,ab (f ζp, f ζp, · · · f b) , ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) , ab (f λp, f λp, · · · ,t(λp,ζp)) ,ab (f ζp, f ζp, · · · ,t(ζp,λp)) , ab (f a,f a, · · · ,t(λp,ζp)) ,ab (f b,f b, · · · ,t(ζp,λp)) ,   ≤ θmax { ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) } . 10 int. j. anal. appl. (2022), 20:67 similarly, we can have ab (f b,f b, · · · ,t(b,a))≤ θmax { ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) } thus max { ab (f a,f a, · · · ,t(a,b)) , ab (f b,f b, · · · ,t(b,a)) , } ≤ θmax { ab (f a,f a, · · · ,t(a,b)) , ab (f b,f b, · · · ,t(b,a)) } . so that t(a,b)= f a and t(b,a)= f b. thus (a,b) is a coupled coincidence point of t and f . since the pair (t,f ) is ω-compatible, we have f α = f 2a = f (t(a,b))= t(f a,f b)= t (α,β) f β = f 2b = f (t(b,a))= t(f b,f a)= t (β,α) (3.6) now η(θ)ab (f α,f α, · · · ,t (α,β))=0≤max { ab (f a,f a, · · · , f α) ,ab (f b,f b, · · · , f β) , ab (f α,f α, · · · ,t (α,β)) ,ab (f β,f β, · · · ,t (β,α)) } . hence from (3.1), we have ab(f α,f α, · · · , f a)= ab (t(α,β),t(α,β), · · · ,t(a,b)) ≤ θmax   ab (f α,f α, · · · , f a) ,ab (f β,f β, · · · , f b) , ab (f α,f α, · · · ,t(α,β)) ,ab (f β,f β, · · · ,t(β,α)) , ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) , ab (f a,f a, · · · ,t(α,β)) ,ab (f b,f b, · · · ,t(β,α))   . ≤ θmax { ab (f α,f α, · · · , f a) ,ab (f β,f β, · · · , f b) } . similarly, we have ab(f β,f β, · · · , f b)≤ θmax { ab (f α,f α, · · · , f a) ,ab (f β,f β, · · · , f b) } . thus max { ab (f α,f α, · · · , f a) ,ab (f β,f β, · · · , f b) , } ≤ θmax { ab (f α,f α, · · · , f a) , ab (f β,f β, · · · , f b) } . hence α = f a = f α and β = f b = f β,. hence from (3.6), we have (α,β) is a common coupled fixed point of t and f . in the following we will show the uniqueness of common coupled fixed point in p. for this purpose, assume that there is another coupled fixed point (α′,β′) of t,f . now consider, η(θ)ab (f α,f α, · · · ,t (α,β))=0≤max { ab (f α,f α, · · · , f α′) ,ab (f β,f β, · · · , f β′) , ab (f α,f α, · · · ,t (α,β)) ,ab (f β,f β, · · · ,t (β,α)) } int. j. anal. appl. (2022), 20:67 11 by (3.1), we have ab(α,α, · · · ,α′) = ab ( t(α,β),t(α,β), · · · ,t(α′,β′) ) ≤ θmax { ab (α,α, · · · ,α′) ,ab (β,β, · · · ,β′) } . similarly, we can show that ab(β,β, · · · ,β′) ≤ θmax { ab (α,α, · · · ,α′) ,ab (β,β, · · · ,β′) } . thus max { ab (α,α, · · · ,α′) ,ab (β,β, · · · ,β′) } ≤ θmax { ab (α,α, · · · ,α′) ,ab (β,β, · · · ,β′) } . hence α = α′,β = β′. thus (α,β) is the unique common coupled fixed point of t and f . case(ii): if f λp = f λp+1, f ζp = f ζp+1 for some p then f λp = t (λp,ζp), f ζp = t (ζp,λp) so that (λp,ζp) is a coupled coincidence point of t and f . now proceeding as in case (i) with f λp = α, f ζp = β, we can show that (α,β) is the unique common coupled fixed point of t and f . � example 3.1. let p = [0,+∞), define ab :pn → [0,+∞) as ab (λ1,λ2, ........,λn−1,λn) = ∑n i=1 ∑ i<j |λi −λj| 2 for all λi ∈ p, i = 1,2, · · · . then (p,ab) is an complete ab-metric space with b =2. let t :p2 →p and f :p →p be given by t(λ,ζ)= 4λ−2ζ+16n−2 16n and f (λ)= 6λ+32n−6 32n . then obviously, t(p2)⊆ f (p) and the pair (t,f ) are ω-compatible and clearly for all a,b ∈p 4 5 ab (f a,f a, · · · ,t(a,b)) ≤ ab (f a,f a, · · · ,t(a,b)) ≤ max { ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) , } now we have ab (t(a,b),t(a,b), · · · ,t(λ,ζ)) = (n−1)|t(a,b)−t(λ,ζ)|2 = (n−1)| 4a−2b+16n−2 16n − 4λ−2ζ +16n−2 16n )|2 = (n−1)| 4a−2b−4λ+2ζ 16n |2 ≤ (n−1) 2 | 6a+32n−6 32n − 4λ−2ζ +16n−2 16n |2 ≤ (n−1) 2 |f a−t(λ,ζ)|2 = 1 2 ab (f a,f a, · · · ,t(λ,ζ)) 12 int. j. anal. appl. (2022), 20:67 ≤ 1 2 max   ab (f a,f a, · · · , f λ) ,ab (f b,f b, · · · , f ζ) , ab (f a,f a, · · · ,t(a,b)) ,ab (f b,f b, · · · ,t(b,a)) , ab (f λ,f λ, · · · ,t(λ,ζ)) ,ab (f ζ, f ζ, · · · ,t(ζ,λ)) , ab (f λ,f λ, · · · ,t(a,b)) ,ab (f ζ, f ζ, · · · ,t(b,a))   . put n =2 and b =2, since θ = 1 2 < 1.thus all the conditions of the theorem 3.1 are satisfied and (1,1) is unique common coupled fixed point of t and f . corollary 3.1. let (p,ab) be a complete ab-metric space. suppose that t :p2 →p be a mapping satisfying: η(θ)ab (λ,λ, · · · ,t(λ,ζ))≤max { ab (λ,λ, · · · ,ρ) ,ab (ζ,ζ, · · · ,%) , ab (λ,λ, · · · ,t(λ,ζ)) ,ab (ζ,ζ, · · · ,t(ζ,λ)) } implies ab (t(λ,ζ),t(λ,ζ), · · · ,t(ρ,%))≤ θmax   ab (λ,λ, · · · ,ρ) ,ab (ζ,ζ, · · · ,%) , ab (λ,λ, · · · ,t(λ,ζ)) ,ab (ζ,ζ, · · · ,t(ζ,λ)) , ab (ρ,ρ, · · · ,t(ρ,%)) ,ab (%,%, · · · ,t(%,ρ)) , ab (ρ,ρ, · · · ,t(λ,ζ)) ,ab (%,%, · · · ,t(ζ,λ))   . for all λ,ζ,ρ,% ∈p, where θ ∈ [0,1) and η : [0,1)→ (0,1] defined as η(θ) = 1 b2((n−1)+θ) is a strictly decreasing function. then there is a unique coupled fixed point of t in p. 4. application to integral equations in this section, we study the existence of an unique solution to an initial value problem, as an application to corollary 3.1. theorem 4.1. consider the initial value problem λ′(t)= t(t,(λ,ζ)(t)), t ∈ i = [0,1], (λ,ζ)(0)= (λ0,ζ0) (4.1) where t : i × r → r with t∫ 0 t(s,(λ,ζ)(s))ds = max { t∫ 0 t(s,λ(s))ds, t∫ 0 t(s,ζ(s))ds } and λ0,ζ0 ∈r. then there exists unique solution in c (i,r) for the initial value problem (4.1). proof. the integral equation corresponding to initial value problem (4.1) is λ(t)= λ0 + √ n−1 2 b2 t∫ 0 t(s,(λ,ζ)(s))ds. let p = c (i,r) and ab (λ1,λ2, ........,λn−1,λn)= ∑n i=1 ∑ i<j |λi −λj| 2 for all λi ∈p, i =1,2, · · · , define r :p2 →p by r(λ,ζ)(t) = 2λ0√ (n−1)b2 + t∫ 0 t(s,(λ,ζ)(s))ds. int. j. anal. appl. (2022), 20:67 13 clearly, for all λ,ζ ∈p, we have 3 5 ab (λ,λ, · · · ,r(λ,ζ))≤max { ab (λ,λ, · · · ,a) ,ab (ζ,ζ, · · · ,b) , ab (λ,λ, · · · ,r(λ,ζ)) ,ab (ζ,ζ, · · · ,r(ζ,λ)) } . now ab (r(λ,ζ)(t),r(λ,ζ)(t), · · · ,r(a,b)(t)) = (n−1)|r(λ,ζ)(t)−r(a,b)(t)|2 = | 2λ0√ (n−1)b2 + t∫ 0 t(s,(λ,ζ)(s))ds − 2a0√ (n−1)b2 − t∫ 0 t(s,(a,b)(s))ds|2 = 4 b4 |λ(t)−a(t)|2 ≤ 1 2 (n−1)|λ(t)−a(t)|2 ≤ 1 2 ab(λ,λ, · · · ,a) ≤ 1 2 max   ab (λ,λ, · · · ,a) ,ab (ζ,ζ, · · · ,b) , ab (λ,λ, · · · ,r(λ,ζ)) ,ab (ζ,ζ, · · · ,r(ζ,λ)) , ab (a,a, · · · ,r(a,b)) ,ab (b,b, · · · ,r(b,a)) , ab (a,a, · · · ,r(λ,ζ)) ,ab (b,b, · · · ,r(ζ,λ))   . it follows from corollary 3.1, we conclude that r has a unique fixed point in p. � 5. application to homotopy in this section, we study the existence of a unique solution to homotopy theory. theorem 5.1. let (p,ab) be complete ab-metric space, u and u be an open and closed subset of p such that u ⊆ u. suppose h : u2×[0,1]→p be an operator with following conditions are satisfying, τ0) λ 6= h(λ,ζ,κ), ζ 6= h(ζ,λ,κ), for each λ,ζ ∈ ∂u and κ ∈ [0,1] (here ∂u is boundary of u in p); τ1) for all λ,ζ,ρ,% ∈ u and κ ∈ [0,1] such that η(θ)ab (λ,λ, · · · ,h(λ,ζ,κ))≤max { ab (λ,λ, · · · ,ρ) ,ab (ζ,ζ, · · · ,%) , ab (λ,λ, · · · ,h(λ,ζ,κ)) ,ab (ζ,ζ, · · · ,h(ζ,λ,κ)) } implies ab ( h(λ,ζ,κ),h(λ,ζ,κ), · · · ,h(ρ,%,κ) ) ≤ θmax   ab (λ,λ, · · · ,ρ) ,ab (ζ,ζ, · · · ,%) , ab (λ,λ, · · · ,h(λ,ζ,κ)) , ab (ζ,ζ, · · · ,h(ζ,λ,κ)) , ab (ρ,ρ, · · · ,h(ρ,%,κ)) , ab (%,%, · · · ,h(%,ρ,κ)) , ab (ρ,ρ, · · · ,h(λ,ζ,κ)) , ab (%,%, · · · ,h(ζ,λ,κ))   . (5.1) where θ ∈ [0,1) and η : [0,1)→ (0,1] defined as η(θ)= 1 b2((n−1)+θ)is a strictly decreasing function. τ2) ∃ m ≥ 03 ab(h(λ,ζ,κ),h(λ,ζ,κ), · · · ,h(λ,ζ,ζ))≤ m|κ−ζ| 14 int. j. anal. appl. (2022), 20:67 for every λ,ζ ∈ uand κ,ζ ∈ [0,1]. then h(.,0) has a coupled fixed point ⇐⇒ h(.,1) has a coupled fixed point. proof. let the set p = { κ ∈ [0,1] : h(λ,ζ,κ)= λ,h(ζ,λ,κ)= ζ for some λ,ζ ∈ u } . suppose that h(.,0) has a coupled fixed point in u2, we have that (0,0) ∈ p2. so that p is non-empty set. now we show that p is both closed and open in [0,1] and hence by the connectedness p = [0,1]. as a result, h(.,1) has a coupled fixed point in u2. first we show that p closed in [0,1]. to see this, let {κp}∞p=1 ⊆ p with κp → κ ∈ [0,1] as p → ∞. we must show that κ ∈ p. since κp ∈ p for p = 0,1,2,3, · · · , there exists sequences {λp} ,{ζp} with λp+1 = h(λp,ζp,κp), ζp+1 = h(ζp,λp,κp). consider ab(λp,λp, · · · ,(λp)n−1,h(λp,ζp,κp)) = ab (h(λp−1,ζp−1,κp−1),h(λp−1,ζp−1,κp−1), · · · ,(h(λp−1,ζp−1,κp−1))n−1,h(λp,ζp,κp)) ≤ b(n−1)ab ( h(λp−1,ζp−1,κp−1),h(λp−1,ζp−1,κp−1), · · · ,h(λp−1,ζp−1,κp) ) +b2ab ( h(λp−1,ζp−1,κp),h(λp−1,ζp−1,κp), · · · ,h(λp,ζp,κp) ) ≤ b(n−1)ab ( h(λp−1,ζp−1,κp−1),h(λp−1,ζp−1,κp−1), · · · ,h(λp−1,ζp−1,κp) ) +b2θmax   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , ab (λp,λp, · · · ,h(λp−1,ζp−1,κp)) , ab (ζp,ζp, · · · ,h(ζp−1,λp−1,κp))   ≤ b2((n−1)+θ)max   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , ab (λp,λp, · · · ,h(λp−1,ζp−1,κp)) , ab (ζp,ζp, · · · ,h(ζp−1,λp−1,κp))   ≤ b2((n−1)+θ)max   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , ab (h(λp,ζp,κp),h(λp,ζp,κp), · · · ,h(λp,ζp,κp+1)) , ab (h(ζp,λp.κp),h(ζp,λp.κp), · · · ,h(ζp,λp,κp+1))   ≤ b2((n−1)+θ)max { ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , m|κp −κp+1| } . thus, η(θ)ab(λp,λp, · · · ,(λp)n−1,h(λp,ζp,κp)) ≤ max   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , ab (h(λp,ζp,κp),h(λp,ζp,κp), · · · ,h(λp,ζp,κp+1)) , ab (h(ζp,λp.κp),h(ζp,λp.κp), · · · ,h(ζp,λp,κp+1))   . int. j. anal. appl. (2022), 20:67 15 from (5.1), we have that ab(h(λp,ζp,κp)h(λp,ζp,κp), · · · ,h(λp+1,ζp+1,κp+1)) ≤ θmax   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζq+1) , ab (λp,λp, · · · ,h(λp,ζp,κp)) , ab (ζp,ζp, · · · ,h(ζp,λp,κp)) , ab (λp+1,λp+1, · · · ,h(λp+1,ζp+1,κp+1)) , ab (ζp+1,ζp+1, · · · ,h(ζp+1,λp+1,κp+1)) , ab (λp+1,λp+1, · · · ,h(λp,ζp,κp)) , ab (ζp+1,ζp+1, · · · ,h(ζp,λp,κp))   ≤ θmax   ab (λp,λp, · · · ,λp+1) ,ab (ζp,ζp, · · · ,ζp+1) , ab (λp+1,λp+1, · · · ,λp+2) , ab (ζp+1,ζp+1, · · · ,yp+2)   . thus it follows that ab(h(λp,ζp,κp),h(λp,ζp,κp), · · · ,h(λp+1,ζp+1,κp+1)) ≤ θmax { ab (λp,λp, · · · ,λp+1) , ab (ζp,ζp, · · · ,ζp+1) } ... ≤ θp max { ab (λ0,λ0, · · · ,λ1) , ab (ζ0,ζ0, · · · ,ζ1) } . similarly, we can show that ab(h(ζp,λp,κp),h(ζp,λp,κp), · · · ,h(ζp+1,λp+1,κp+1)) ≤ θp max { ab (λ0,λ0, · · · ,λ1) , ab (ζ0,ζ0, · · · ,ζ1) } . thus max { ab(h(λp,ζp,κp),h(λp,ζp,κp), · · · ,h(λp+1,ζp+1,κp+1)), ab(h(ζp,λp,κp),h(ζp,λp,κp), · · · ,h(ζp+1,λp+1,κp+1)) } ≤ θp max { ab (λ0,λ0, · · · ,λ1) , ab (ζ0,ζ0, · · · ,ζ1) } . → o as p →∞. now for q > p, by use of (ab3), we have ab (λp,λp, · · · ,λq)≤ b(n−1)ab (λp,λp, · · · ,λp+1)+b2ab (λp+1,λp+1, · · · ,λq) letting p →∞, we get lim p→∞ ab (λp,λp, · · · ,λq) ≤ lim p→∞ b2ab (h(λp,ζp,κp),h(λp,ζp,κp), · · · ,h(λq−1,ζq−1,κq−1)) ≤ lim p→∞ b2q−2p−3ab (h(ζq−2,ζq−2,κq−2),h(ζq−2,ζq−2,κq−2), · · · ,h(λq−1,ζq−1,κq−1)) 16 int. j. anal. appl. (2022), 20:67 ≤ lim p→∞ b2q−2p−3θmax   ab (ζq−2,ζq−2, · · · ,λq−1) ,ab (ζq−2,ζq−2, · · · ,ζq−1) , ab (ζq−2,ζq−2, · · · ,h(ζq−2,ζq−2,κq−2)) , ab (ζq−2,ζq−2, · · · ,h(ζq−2,ζq−2,κq−2)) , ab (λq−1,λq−1, · · · ,h(λq−1,yq−1,κq−1)) , ab (ζq−1,ζq−1, · · · ,h(ζq−1,λq−1,κq−1)) , ab (λq−1,λq−1, · · · ,h(ζq−2,ζq−2,κq−2)) , ab (ζq−1,ζq−1, · · · ,h(ζq−2,ζq−2,κq−2))   . ≤ lim p→∞ b2q−2p−3θq−p−1max { ab (λ0,λ0, · · · ,λ1) ,ab (ζ0,ζ0, · · · ,ζ1) } → 0 as q →∞. hence {λp} is a cauchy sequence in ab metric spaces (p,ab). similarly we can show that {ζp}, is cauchy sequence in (p,ab) and by the completeness of (p,ab), there exist a,b ∈p with lim p→∞ λp+1 = a lim p→∞ ζp+1 = b (5.2) since η(θ)ab (a,a, · · · ,h(a,b,κ))≤max { ab (a,a, · · · ,λp) ,ab (b,b, · · · ,ζp) , ab (a,a, · · · ,h(a,b,κ)) ,ab (b,b, · · · ,h(b,a,κ)) } we have ab (a,a, · · · ,h(a,b,κ)) ≤ lim p→∞ ab (h(λp,ζp,κ),h(λp,ζp,κ), · · · ,h(a,b,κ)) ≤ lim n→∞ θmax   ab (λp,λp, · · · ,a) ,ab (ζp,ζp, · · · ,b) , ab (a,a, · · · ,h(a,b,κ)) ,ab (b,b, · · · ,h(b,a,κ)) , ab (λp,λp, · · · ,h(λp,ζp,κ)) ,ab (ζp,ζp, · · · ,h(ζp,λp,κ)) , ab (a,a, · · · ,h(λp,ζp,κ)) ,ab (b,b, · · · ,h(ζp,λp,κ))   ≤ θmax { ab (a,a, · · · ,h(a,b,κ)) ,ab (b,b, · · · ,h(b,a,κ)) } . therefore, max { ab (a,a, · · · ,h(a,b,κ)) , ab (b,b, · · · ,h(b,a,κ)) } ≤ θmax { ab (a,a, · · · ,h(a,b,κ)) , ab (b,b, · · · ,h(b,a,κ)) } . it follows that h(a,b,κ) = a,h(b,a,κ) = b. thus κ ∈ p. hence p is closed in [0,1]. let κ0 ∈ p, then there exist λ0,ζ0 ∈ u with λ0 = h(λ0,ζ0,κ0), ζ0 = h(ζ0,λ0,κ0). since u is open, then there exist r > 0 such that bab(λ0, r)⊆ u. choose κ ∈ (κ0 − �,κ0 + �) such that |κ−κ0| ≤ 1 mp < � 2 , then for λ ∈ bab(λ0, r)= {λ ∈p/ab(λ,λ, · · · ,λ0)≤ r +ab(λ0,λ0, · · · ,λ0)}. also η(θ)ab (λ,λ, · · · ,h(λ0,ζ0,κ))≤max { ab (λ,λ, · · · ,λ0) ,ab (ζ,ζ, · · · ,ζ0) , ab (λ,λ, · · · ,h(λ0,ζ0,κ)) ,ab (ζ,ζ, · · · ,h(ζ0,λ0,κ)) } int. j. anal. appl. (2022), 20:67 17 now we have ab (h(λ,ζ,κ),h(λ,ζ,κ), · · · ,λ0) = ab (h(λ,ζ,κ),h(λ,ζ,κ), · · · ,h(λ0,ζ0,κ0)) ≤ (n−1)bab (h(λ,ζ,κ),h(λ,ζ,κ), · · · ,h(λ,ζ,κ0)) +b2ab (h(λ,ζ,κ0),h(λ,ζ,κ0), · · · ,h(λ0,ζ0,κ0)) ≤ b(n−1)m|κ−κ0|+b2ab (h(λ,ζ,κ0),h(λ,ζ,κ0), · · · ,h(λ0,ζ0,κ0)) ≤ b(n−1) 1 mp−1 +b2ab (h(λ,ζ,κ0),h(λ,ζ,κ0), · · · ,h(λ0,ζ0,κ0)) . letting p →∞, we obtain ab (h(λ,ζ,κ),h(λ,ζ,κ), · · · ,λ0) ≤ b2ab (h(λ,ζ,κ0),h(λ,ζ,κ0), · · · ,h(λ0,ζ0,κ0)) . ≤ b2θmax   ab (λ,λ, · · · ,λ0) ,ab (ζ,ζ, · · · ,ζ0) , ab (λ,λ, · · · ,h(λ,ζ,κ)) ,ab (ζ,ζ, · · · ,h(ζ,λ,κ)) , ab (λ0,λ0, · · · ,h(λ0,ζ0,κ)) ,ab (ζ0,ζ0, · · · ,h(ζ0,λ0,κ)) , ab (λ0,λ0, · · · ,h(λ,ζ,κ)) ,ab (ζ0,ζ0, · · · ,h(ζ,λ,κ))   . ≤ b2θmax { ab (λ,λ, · · · ,λ0) ,ab (ζ,ζ, · · · ,ζ0) } . therefore, we have max { ab (h(λ,ζ,κ),h(λ,ζ,κ), · · · ,λ0) ab (h(ζ,λ,κ),h(ζ,λ,κ), · · · ,ζ0) } ≤ b2θmax { ab (λ,λ, · · · ,λ0) ,ab (ζ,ζ, · · · ,ζ0) } ≤ b2θmax { r +ab (λ0,λ0, · · · ,λ0) , r +ab (ζ0,ζ0, · · · ,ζ0) } . thus for each fixed κ ∈ (κ0 − �,κ0 + �), h(.,κ) : bab(λ0, r)→ bab(λ0, r), h(.,κ) : bab(ζ0, r)→ bab(ζ0, r). then all conditions of theorem 5.1 are satisfied. thus we conclude that h(.,κ) has a coupled fixed point in u 2 . but this must be in u2. since (τ0) holds. thus, κ ∈p for any κ ∈ (κ0−�,κ0+�). hence (κ0−�,κ0+�)⊆p. clearly p is open in [0, 1]. for the reverse implication, we use the same strategy. � 6. conclusion in this paper we conclude some applications to homotopy theory and integral equations by using suzuki contractive type fixed point theorems in the set up of ab-metric spaces. acknowledgements: authors are thankful to referees for their valuable suggestions. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 18 int. j. anal. appl. (2022), 20:67 references [1] t. suzuki, a generalized banach contraction principle that characterizes metric completeness, proc. amer. math. soc. 136 (2008), 1861-1869. https://www.jstor.org/stable/20535364. [2] i. altun, a. erduran, a suzuki type fixed-point theorem, int. j. math. math. sci. 2011 (2011), 736063. https: //doi.org/10.1155/2011/736063. [3] m. aggarwal, r. chugh, r. kamal, suzuki-type fixed point results in g-metric spaces and applications, int. j. computer appl. 47 (2012), 14–17. https://doi.org/10.5120/7239-0073. [4] n. hussain, d. ðorić, z. kadelburg, s. radenović, suzuki-type fixed point results in metric type spaces, fixed point theory appl. 2012 (2012), 126. https://doi.org/10.1186/1687-1812-2012-126. [5] k.p.r. rao, k.r.k. rao, v.c.c. raju, a suzuki type unique common coupled fixed point theorem in metric spaces, int. j. innov. res. sci. eng. technol. 2 (2013), 5187-5192. [6] t. suzuki, a new type of fixed point theorem in metric spaces, nonlinear anal.: theory methods appl. 71 (2009), 5313–5317. https://doi.org/10.1016/j.na.2009.04.017. [7] i.a. bakhtin, the contraction mapping principle in quasimetric spaces, funct. anal. unianowsk gos. ped. inst. 30 (1989), 26–37. [8] m. abbas, b. ali, y.i. suleiman, generalized coupled common fixed point results in partially ordered a-metric spaces, fixed point theory appl. 2015 (2015), 64. https://doi.org/10.1186/s13663-015-0309-2. [9] m. ughade, d. turkoglu, s.r. singh, r.d. daheriya, some fixed point theorems in ab-metric space, br. j. math. computer sci. 19 (2016), 1-24. [10] n. mlaiki, y. rohen, some coupled fixed point theorems in partially ordered ab-metric space, j. nonlinear sci. appl. 10 (2017), 1731–1743. https://doi.org/10.22436/jnsa.010.04.35. [11] r. konchada, s.r. chindirala, r.n. chappa, a novel coupled fixed point results pertinent to ab-metric spaces with application to integral equations, math. anal. contemp. appl. 4 (2022), 63–83. https://doi.org/10. 30495/maca.2022.1949822.1046. [12] k. ravibabu, c.s. rao, c.r. naidu, applications to integral equations with coupled fixed point theorems in ab-metric space, thai j. math. spec. iss. (2018), 148-167. [13] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal.: theory methods appl. 11 (1987), 623–632. https://doi.org/10.1016/0362-546x(87)90077-0. [14] t.g. bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal.: theory methods appl. 65 (2006), 1379–1393. https://doi.org/10.1016/j.na.2005.10.017. [15] m. abbas, m. ali khan, s. radenović, common coupled fixed point theorems in cone metric spaces for ωcompatible mappings, appl. math. comput. 217 (2010), 195–202. https://doi.org/10.1016/j.amc.2010.05. 042. [16] a. aghajani, m. abbas, e.p. kallehbasti, coupled fixed point theorems in partially ordered metric spaces and application, math. commun. 17 (2012), 497-509. https://hrcak.srce.hr/93280. [17] e. karapinar, coupled fixed point on cone metric spaces, gazi univ. j. sci. 1 (2011), 51-58. https://dergipark. org.tr/en/download/article-file/83133. [18] w. long, b.e. rhoades, m. rajović, coupled coincidence points for two mappings in metric spaces and cone metric spaces, fixed point theory appl. 2012 (2012), 66. https://doi.org/10.1186/1687-1812-2012-66. [19] j.g. mehta, m.l. joshi, on coupled fixed point theorem in partially ordered complete metric space, int. j. pure appl. sci. technol. 1 (2010), 87-92. [20] w. shantanawi, some common coupled fixed point results in cone metric spaces, int. j. math. anal. 4 (2010), 2381-2388. https://www.jstor.org/stable/20535364 https://doi.org/10.1155/2011/736063 https://doi.org/10.1155/2011/736063 https://doi.org/10.5120/7239-0073 https://doi.org/10.1186/1687-1812-2012-126 https://doi.org/10.1016/j.na.2009.04.017 https://doi.org/10.1186/s13663-015-0309-2 https://doi.org/10.22436/jnsa.010.04.35 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.1016/0362-546x(87)90077-0 https://doi.org/10.1016/j.na.2005.10.017 https://doi.org/10.1016/j.amc.2010.05.042 https://doi.org/10.1016/j.amc.2010.05.042 https://hrcak.srce.hr/93280 https://dergipark.org.tr/en/download/article-file/83133 https://dergipark.org.tr/en/download/article-file/83133 https://doi.org/10.1186/1687-1812-2012-66 1. introduction 2. preliminaries 3. main results 4. application to integral equations 5. application to homotopy 6. conclusion references int. j. anal. appl. (2022), 20:19 approximating the mode of the non-central chi-squared distribution v. ananiev∗, a. l. read department of physics, university of oslo, norway ∗corresponding author: victor.ananyev@gmail.com abstract. in this paper we consider the probability density function (pdf) of the non-central χ2 distribution with arbitrary number of degrees of freedom and non-centrality. for this function we find the approximate location of the maximum and discuss related edge cases of 1 and 2 degrees of freedom. we also use this expression to demonstrate the improved performance of the c++ boost’s implementation of the non-central χ2 and extend the domain of its applicability. 1. introduction properties of the non-central χ2 distribution were described before in literature [6–8]. however, the topic of the mode of the non-central χ2 was significantly underrepresented. we would like to focus on the mode specifically in this paper. let x1,x2, ...,xn be normally distributed random variables with unit variance and means µ1,µ2, ...,µn. the sum x21 + x 2 2 + ... + x 2 n follows the non-central χ 2 distribution with k = n degrees of freedom and non-centrality λ = µ21 + µ 2 2 + ... + µ 2 n. the probability density function of this distribution has a closed form expression: fk,λ(x) = 1 2 exp− x+λ 2 (x λ )k−2 4 ik−2 2 ( √ λx) , (1.1) where iν(x) is a modified bessel function of the first kind. we are interested in the value of xmode that maximizes fk,λ(x). typical shapes of the pdf of the non-central χ2 distribution are shown in fig. 1. received: feb. 8, 2022. 2010 mathematics subject classification. 41-02, 33c10, 62-04. key words and phrases. non-central chi-squared; mode; linear approximation; boost c++; performance. https://doi.org/10.28924/2291-8639-20-2022-19 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-19 2 int. j. anal. appl. (2022), 20:19 0 20 40 60 80 100 x 0.00 0.01 0.02 0.03 0.04 no nce nt ra l χ 2 de ns ity k=3, λ=20 k=10, λ=20 k=20, λ=20 k=50, λ=20 (a) dependency on the number of d.o.f. k. 0 20 40 60 80 100 x 0.00 0.02 0.04 0.06 0.08 no nce nt ra l χ 2 de ns ity k=3, λ=5 k=3, λ=10 k=3, λ=30 k=3, λ=50 (b) dependency on the noncentrality λ. figure 1. non-central χ2 distributions and behavior of the mode. when the number of degrees of freedom k is fixed, we can plot the dependency of the maximum of the pdf as a function of the non-centrality parameter λ, see fig. 2. 0 2 4 6 8 10 non-centrality, λ 2 4 6 8 10 12 14 16 18 m od e ncχ2, k=10 ncχ2, k=5 ncχ2, k=3 figure 2. mode of the non-central χ2 as a function of the non-centrality parameter λ. we observe that the bigger λ is the better the mode appears to be approximated with a straight line. the derivation of the line parameters together with the analysis of the edge cases of small number of degrees of freedom, where the mode does not exist, constitute the main results of the paper. 2. derivation 2.1. master equation. in this section we obtain the transcendental equation (eq. 2.2) that determines the mode of the non-central χ2 distribution. we reduce it to the ordinary differential equation (eq. 2.5), where the non-centrality parameter λ is the argument, and the number of degrees of freedom k is a parameter. finally, we solve the ode approximately with a taylor expansion (eq. 2.10) and investigate edge cases of 1 and 2 degrees of freedom (sec. 2.3). int. j. anal. appl. (2022), 20:19 3 we start by setting the derivative of the density of the non-central χ2 (eq. 2.1) to zero. this leads us to the transcendental equation (eq. 2.2) that determines the mode of the distribution: d dx χ2k,λ(x) = 1 2 χ2d,λ(x) ·  −1 + k − 2 2x + √ λ x i′k−2 2 ( √ λx) ik−2 2 ( √ λx)   , (2.1) d dx χ2k,λ(x) = 0 ⇒ √ λxi′k−2 2 ( √ λx) = (x − k − 2 2 )ik−2 2 ( √ λx) . (2.2) we can eliminate the derivative in eq. 2.2 by using the differential equation for the modified bessel function [1, eq. 10.25.1]: t2 d2 dt2 iν(t) + t d dt iν(t) − (t2 + ν2)id,λ(t) = 0 . (2.3) to make use of eq. 2.3, we need the expression for i′′k−2 2 , therefore, we differentiate eq. 2.2 by λ. since the mode depends on the non-centrality λ, we should remember that x = x(λ), thus dx dλ = x′. the resulting expression for i′′k−2 2 is as follows: √ λxi′′k−2 2 ( √ λx) = (x − k 2 )i′k−2 2 ( √ λx) + 2 √ λxx′ x + λx′ ik−2 2 ( √ λx) . (2.4) we substitute i′k−2 2 (eq. 2.2) and i′′k−2 2 (eq. 2.4) into the differential equation for the modified bessel function (eq. 2.3). we then use the property [1, eq. 10.29.4] to decrease the order of the derivatives of the modified bessel functions. assuming that the bessel function itself is non-zero at the mode, we arrive to the following differential equation for the mode as a function of the non-centrality parameter λ: λx′(x −k −λ + 4) + x(x −k −λ + 2) = 0 . (2.5) 2.2. approximate solution. we observed that the linear approximation works better with growing λ, thus we introduce the asymptotic parameter t = k λ << 1 to build the expansion. we expect the solution to be linear in λ, however the asymptotic expansion of x(t) = c0 + c1t + ... won’t provide us with a solution linear in λ. therefore, we reparametrize x(t) with a new function y(t) = tx(t): t = k λ , (2.6) y(t) = tx(t) . (2.7) we obtain the following equation after the reparametrization: − (y ′t −y)(y −kt −k + 4t) + y(y −kt −k + 2t) = 0 . (2.8) to solve eq. 2.8, we expand y(t) into the taylor series by the scale parameter t = k λ . we would like to find the linear solution and one extra term that estimates the error. thus, we cut the series at 4 int. j. anal. appl. (2022), 20:19 the third power of t in order to account for the derivative. after solving algebraic equations for the coefficients near each power of t, we arrive to the resulting approximate expression for the mode: y(t) = c0 + c1t + c2t 2 + c3t 3 + o(t4) , (2.9) c0 = k, c1 = k − 3, c2 = k − 3 2k , (2.10) xmode = λ + k − 3 + k − 3 2λ + o ( k2 λ2 ) . (2.11) we plot the linear approximation eq. 2.11 together with the precise numerical solution fig. 2 in order to verify the approximation is correct, see fig. 3. 0 2 4 6 8 10 non-centrality, λ 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 m od e k + λ 3 ncχ2, k=10 ncχ2, k=5 ncχ2, k=3 figure 3. linear approximation to the mode of the non-central χ2 compared to the more precise numerical solution as a function of the non-centrality parameter λ. 2.3. small number of degrees of freedom. 2.3.1. case k < 2. the asymptotic behavior of the modified bessel function at x → 0 [1, eq. 10.30.1] shows that the pdf of the non-central χ2 diverges, thus it doesn’t have a mode: χ2k,λ(x) → 1 2γ(k 2 ) 1 (2λ) k−2 2 e− λ 2 (√ λx )k−2 , x → 0 . (2.12) 2.3.2. case k = 2. in this case, the pdf at x = 0 is finite. if the derivative at x = 0 is positive, then the maximum is not there. the expression for the derivative (eq. 2.13) and its asymptotic behavior at x → 0 (eq. 2.14) are shown below: d dx χ2k,λ(x) = 1 2 χ2d,λ(x) · [ −1 + √ λ x i−1( √ λx) i0( √ λx) ] , (2.13) d dx χ2k,λ(x) → 1 2 χ2d,λ(x) · [ −1 + λ 2 ] , x → 0 . (2.14) we observe that when λ > 2, the pdf of the non-central χ2 doesn’t have its maximum at x = 0. in the region λ < 2, the asymptotic scale t = k λ > 1, hence our approximation is inapplicable in this region and we refrain from analysing it. int. j. anal. appl. (2022), 20:19 5 3. application there exist a number of numerical procedures for finding the mode of a distribution [2, ch. 10]. some of them require the search region to be specified. for example, the widely used c++ library boost [3] identifies the search region based on an initial guess for the mode x0. boost iteratively checks regions of the form [x0/2, 2x0], [x0/22, 22x0], etc. when the value of the pdf at both ends of the region becomes smaller than the value at the initial guess point x0, the algorithm initiates the search for the maximum inside of the region. at the time of writing, boost used x0 = k + 1 as the initial guess. we already know, based on the approximate solution (eq. 2.11), that the chosen guess will undershoot at large non-centrality values λ. let’s estimate λ above which the method will require the second iteration for the region to cover the mode. for this we compare the linear estimate for the location of the mode (eq. 2.11) to the initial guess x0 used by boost: k + λ− 3 > 2 · (k + 1) , (3.1) λ > k + 5 . (3.2) with eq. 3.2, for any number of d.o.f. k we are able to specify the threshold α, defined by λ = αk, at which the original initial guess starts undershooting: α > 5 k + 1 . (3.3) we see that large k corresponds to small thresholds α. the most conservative estimate for the threshold would be at the smallest k possible: k = 2. thus, α = 3.5 is the threshold that approximately works for k = 2 and is the overestimated threshold for bigger values of k. the threshold α (eq. 3.3) is closely related to the asymptotic scale t (eq. 2.6) that we used for finding the approximate solution, specifically: t = k λ = 1 α . for example, the conservative threshold α ≈ 3.5 corresponds to the asymptotic scale t ≈ 0.25 < 1. it means that the region where the original guess of boost undershoots, is, at the same time, the region where our approximate solution for the mode becomes applicable and can be used as a corrected initial guess. however, the fact that we use the conservative threshold may lead to the situation where the original method has already started undershooting but λ is not yet big enough to turn on the corrected regime. 3.1. dependency on λ. we fix the threshold to the conservative value k λ = 0.25. we then plot the dependency of the run time on the non-centrality λ for a set of d.o.f. k: 2, 15, 50, see fig. 4. for benchmarking we use the google benchmark library [4]. the benchmarking script itself became a part of the boost.math [5]. using this script we measure the run time 100 times and use the mean as a central value. the error bar is computed as a standard deviation. we add noise with standard deviation σ = 10−6 to parameters k and λ to avoid caching effects. the vertical line on the plots shows the threshold where the original initial guess for smaller λ is switched to the corrected value at 6 int. j. anal. appl. (2022), 20:19 bigger λ. therefore, we expect that both lines coincide below the threshold and the improved solution would lie lower above the threshold. one can notice missing values on the curve representing the original initial guess. the reason for this is the numerical instability of the algorithm in boost, that has been resolved after we corrected the initial guess. 101 102 103 λ 60 80 100 120 140 160 180 200 tim e, μ s baseline new (a) 101 102 103 λ 40 60 80 100 120 140 160 tim e, μ s baseline new (b) 101 102 103 λ 60 80 100 120 140 160 tim e, μ s baseline new (c) figure 4. run time as a function of the non-centrality λ for d.o.f. k = 2 (4a), k = 15 (4b), k = 50 (4c). vertical line shows the threshold at which the corrected expression replaces the original initial guess. 3.2. dependency on d.o.f. (k). in the set of plots in fig. 5, we fix the asymptotic scale to values k λ = 0.25, 0.15, 0.05 and investigate the dependency of the run time on the number of d.o.f. since the threshold is fixed, the difference in the run time is caused by the actual position where the original initial guess starts to undershoot, the non-conservative threshold. the farther the fixed threshold is from the non-conservative threshold, the more significant the effect of undershooting at the test point will be. therefore, we expect the difference in the run time to grow with number of d.o.f, as follows from eq. 3.3. for each value of the asymptotic scale, in addition to the full plot, we also show a zoomed version that shows the region where both original and improved methods were able to converge (fig. 5). 4. conclusion in this paper we present an approximate expression for the mode of the non-central χ2 distribution: xmode ≈ k +λ−3, where k is the number of degrees of freedom and λ is the non-centrality parameter. the approximation is based on an asymptotic expansion and is valid in the region where the scale parameter k λ << 1 and where the mode exists k > 2. the approximate formula can be used as the initial guess for iterative procedures searching for a precise solution. run time performance and the domain of applicability of the boost implementation of the mode search was improved using the presented approximate expression. the improvement became a part of the boost.math library [5]. int. j. anal. appl. (2022), 20:19 7 101 102 0 50 100 150 200 250 tim e, μ s baseline new 101 102 10μ k 100 200 μ00 400 500 tim e, μ s baseline new (a) 101 102 0 50 100 150 200 250 tim e, μ s baseline new 101 102 10μ k 200 400 600 tim e, μ s baseline new (b) 101 102 0 50 100 150 200 tim e, μ s baseline new 101 102 10μ k 200 400 600 800 1000 tim e, μ s baseline new (c) figure 5. run time as a function of the number of d.o.f. (k) for the asymptotic scale values k λ = 0.25 (5a), k λ = 0.15 (5b), k λ = 0.05 (5c). the upper plot in each pair shows the zoomed version, focused on the region where both original and improved methods were able to converge. 5. acknowledgements we would like to thank mykola semenyakin for the numerous fruitful and motivating discussions. we also would like to acknowledge the support of the boost community that allowed the contribution to become a part of the boost library. this research was supported by the european unions framework programme for research and innovation horizon 2020 (2014-2021) under the marie sklodowska-curie grant agreement no.765710. references [1] f.w.j. olver et al., nist digital library of mathematical functions, (2021). http://dlmf.nist.gov/. [2] w.h. press, s.a. teukolsky, w.t. vetterling, b.p. flannery, numerical recipes 3rd edition, (2007). http:// numerical.recipes/. [3] boost c++ libraries, v1.76.0, (2021). https://www.boost.org/. [4] google benchmark, (2021). https://github.com/google/benchmark. [5] boost math, (2021). https://github.com/boostorg/math/pull/645. [6] s. andrás, á. baricz, properties of the probability density function of the non-central chi-squared distribution, j. math. anal. appl. 346 (2008), 395–402. https://doi.org/10.1016/j.jmaa.2008.05.074. [7] d. horgan, c.c. murphy, on the convergence of the chi square and noncentral chi square distributions to the normal distribution, ieee commun. lett. 17 (2013), 2233–2236. https://doi.org/10.1109/lcomm.2013. 111113.131879. [8] l. saulis, asymptotic expansion for the distribution and density functions of the quadratic form of a stationary gaussian process in the large deviation cramer zone, nonlinear anal.: model. control. 6 (2001), 87–101. https: //doi.org/10.15388/na.2001.6.1.15218. [9] s.s. sawant, d.a. levin, v. theofilis, analytical prediction of low-frequency fluctuations inside a onedimensional shock, arxiv:physics.flu-dyn (2021). https://arxiv.org/abs/2101.00664. [10] v. pereyra, iterated deferred corrections for nonlinear operator equations, numer. math. 10 (1967), 316–323. https://doi.org/10.1007/bf02162030. http://dlmf.nist.gov/ http://numerical.recipes/ http://numerical.recipes/ https://www.boost.org/ https://github.com/google/benchmark https://github.com/boostorg/math/pull/645 https://doi.org/10.1016/j.jmaa.2008.05.074 https://doi.org/10.1109/lcomm.2013.111113.131879 https://doi.org/10.1109/lcomm.2013.111113.131879 https://doi.org/10.15388/na.2001.6.1.15218 https://doi.org/10.15388/na.2001.6.1.15218 https://arxiv.org/abs/2101.00664 https://doi.org/10.1007/bf02162030 1. introduction 2. derivation 2.1. master equation 2.2. approximate solution 2.3. small number of degrees of freedom 3. application 3.1. dependency on 3.2. dependency on d.o.f. (k) 4. conclusion 5. acknowledgements references int. j. anal. appl. (2023), 21:33 received: feb. 26, 2023. 2020 mathematics subject classification. 80a20. key words and phrases. fractional calculus; reduced differential transform method; diffusion equations. https://doi.org/10.28924/2291-8639-21-2023-33 © 2023 the author(s) issn: 2291-8639 1 fractional reduced differential transform method for solving mutualism model with fractional diffusion mohamed ahmed abdallah1,2,*, khaled abdalla ishag3 1department of mathematics, faculty of sciences, university of tabuk, tabuk, ksa 2department of basic science, faculty of engineering, university of sinnar, sinnar, sudan 3department of basic science, faculty of engineering science, omdurman islamic university, omdurman, sudan *corresponding author: mohamed.ah.abd@hotmail.com abstract. this study presents the fractional reduced differential transform method for a nonlinear mutualism model with fractional diffusion. the fractional derivatives are described by caputo's fractional operator. in this method, the solution is considered as the sum of an infinite series. which converges rapidly to the exact solution. the method eliminates the need to use adomian's polynomials to calculate the nonlinear terms. to show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with the ordinary derivative order index α=1 for the nonlinear mutualism model with fractional diffusion. approximate solutions for different values of the fractional derivatives together with non-fractional derivatives and absolute errors are represented graphically in two and three dimensions. from all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods. 1. introduction recently, it has turned out that many phenomena in engineering and other sciences can be described by models using mathematical tools from fractional calculus [1], fractional calculus owes its origin to a question of whether the meaning of a derivative to an integer order could be extended to still be valid when n is not an integer. diffusion phenomena is one the most important topic in heat transfer, especially in mechanics engineering and biological population. in the earlier literature most of the https://doi.org/10.28924/2291-8639-21-2023-33 2 int. j. anal. appl. (2023), 21:33 discussions are devoted to coupled systems of two equations. in the recent years, attention has been given to reaction-diffusion systems with three population species, the densities of three populations 𝑢,𝑣,𝑤 are governed by the following coupled equations: ([2], [3]) 𝐷𝑡 𝛼𝑢 − d1∇ 2u = 𝑢(𝑎1(𝑡,𝑥) − 𝑏1(𝑡,𝑥)𝑢 + 𝑐1(𝑡,𝑥)𝑣) 𝐷𝑡 𝛼𝑣 − d2∇ 2𝑣 = 𝑣(𝑎2(𝑡,𝑥) − 𝑏2(𝑡,𝑥)𝑣 + 𝑐2(𝑡,𝑥)𝑢 + 𝑒(𝑡,𝑥)𝑤) (1) 𝐷𝑡 𝛼𝑤 − d3∇ 2𝑤 = 𝑤(𝑎3(𝑡,𝑥) − 𝑏3(𝑡,𝑥)𝑤 + 𝑐3(𝑡,𝑥)𝑣) with initial conditions 𝑢(𝑥,0) = 𝑢0, 𝑣(𝑥,0) = 𝑣0, 𝑤(𝑥,0) = 𝑤0 where 𝑛 − 1 < 𝛼 ≤ 𝑛, for each 𝑖 = 1,2,3, 𝑑𝑖 is constant and 𝑎𝑖,𝑏𝑖,𝑐𝑖, 𝑒 are smooth functions [2], ∇ 2 denotes laplacian with respect to the variables 𝑥 = (𝑥1,𝑥2,𝑥3) and 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡),𝑤(𝑥,𝑡) is solution of eq. (1). if 𝑑𝑖 = 0 for each 𝑖 = 1,2,3 in eq. (1) we obtain a model of lotka volterra for preypredator. 2. preliminaries and fractional calculus in this section, gives some important definitions, such as the gamma function and basic definitions of the fractional derivatives. 2.1. gamma function gamma function γ(𝑛) is simply the generalization of factorial to complex and real arguments. the gamma function can be defined as ([5], [6]) γ(𝑛) = ∫ 𝑡𝑛−1𝑒−𝑡𝑑𝑡 = (𝑛 − 1)!, 𝑛 ∈ 𝐼𝑁 ∞ 0 (2) which is convergent for 𝑛 > 0. a recurrence formula for gamma function is ([5], [6]) γ(𝑛 + 1) = 𝑛γ(𝑛) 𝑓𝑜𝑟 𝑛 ∈ 𝐼𝑅+ (3) γ(𝑛) = γ(𝑛+1) n 𝑓𝑜𝑟 𝑛 ∈ 𝐼𝑅− (4) 2.2. fractional derivatives definition (1): riemann-liouville fractional integral operator suppose that 𝛼 > 0, 𝑛 − 1 < 𝛼 ≤ 𝑛, the riemann-lioville fractional integral define as [5] 𝐷𝑡 −𝛼 𝑎 𝑅𝐿 (𝑓(𝑡)) = 1 γ(𝛼) ∫(𝑡 − 𝑢)𝛼−1𝑓(𝑢)𝑑𝑢 𝑡 𝑎 (5) note: riemann-liouville fractional differential operator define as 𝐷𝛼𝑅𝑙 𝑓(𝑡) = 𝐷𝑛𝐷𝛼−𝑛𝑓(𝑡), 𝛼 < 𝑛 (6) definition (2): caputo fractional differential operator suppose that 𝛼 > 0, 𝑛 − 1 < 𝛼 ≤ 𝑛, the caputo fractional differential define as [5] 3 int. j. anal. appl. (2023), 21:33 𝐷𝑡 𝛼 𝑎 𝐶 (𝑓(𝑡)) = { 1 γ(𝑛 − 𝛼) ∫ 𝑓𝑛(𝑢) (𝑡 − 𝑢)𝛼−𝑛+1 𝑑𝑢, 𝑛 − 1 < 𝛼 < 𝑛 𝑡 𝑎 𝑑𝑛 𝑑𝑡𝑛 𝑓(𝑡) 𝛼 = 𝑛 ∈ 𝑁 (7) riemann-liouville and caputo fractional integral operator for polynomial is [5] 𝐷𝑡 −𝛼 0 𝑅𝐿 (𝑡𝑛) = 𝐷𝑡 −𝛼 0 𝐶 (𝑓(𝑡)) = γ(𝑛 + 1) γ(𝛼 + 𝑛 + 1) 𝑡𝛼+𝑛 (8) definition (3): the mittagleffler function suppose 𝛼 > 0,𝛽 > 0, then the mittag-leffler function define by [5] 𝐸𝛼,𝛽(𝑡) = ∑ 𝑡𝑘 γ(𝛼𝑘 + 𝛽) (9) ∞ 𝑘=0 3. fractional reduced differential transform method fractional reduced differential transform method (frdtm) is iteration method, suppose 𝑢(𝑡,𝑥1,𝑥2,…,𝑥𝑛) be analytical and continuously differentiable with respect to 𝑛 + 1 variables 𝑡,𝑥1,𝑥2, . . ,𝑥𝑛 in the domain of interest; then frdtm in 𝑛 dimensions for the following differential equation 𝐷𝑡 𝛼𝑢 + 𝐿𝑢 + 𝑁(𝑢) = 0 (10) where 𝐷𝑡 𝛼 is differential operator with respect time, 𝐿 differential operator with respect variables 𝑥1,𝑥2, . . ,𝑥𝑛 and 𝑁(𝑢) is nonlinear term ([7]-[10]). 𝑢𝑘(𝑥1,𝑥2,…,𝑥𝑛) = γ(𝑘𝛼 + 1) γ(𝛼(𝑘 + 1) + 1) [−𝐿(𝑢𝑘)− ∑𝑁(𝑢𝑟)𝑁(𝑢𝑘−𝑟) 𝑘 𝑟=0 ] (11) the approximate solution is given by ([7], [8]). 𝑢(𝑡,𝑥1,𝑥2,…,𝑥𝑛) = ∑ 𝑢𝑘𝑡 𝛼𝑘 = 𝑢0 + 𝑢1𝑡 𝛼 + 𝑢2𝑡 2𝛼 + ⋯ (12) ∞ 𝑘=0 4. numerical results in this section, we assume 𝑑𝑖 = 1,𝑎𝑖 = 1 𝑓𝑜𝑟 𝑖 = 1,2,3, 𝑏1 = 𝑏3 = 1, 𝑏2 = 3, 𝑐1 = 𝑐2 = 𝑐3 = 0.5, 𝑒 = 0.5 in eq. (1) 𝐷𝑡 𝛼𝑢 = uxx + 𝑢 − 𝑢 2 + 0.5𝑢𝑣 𝐷𝑡 𝛼𝑣 = vxx + 𝑣 − 3𝑣 2 + 0.5𝑢𝑣 + 0.5𝑤𝑣 𝐷𝑡 𝛼𝑤 = wxx + 𝑤 − 𝑤 2 + 0.5𝑣𝑤 4 int. j. anal. appl. (2023), 21:33 with initial conditions 𝑢(𝑥,0) = 𝑒𝑥, 𝑣(𝑥,0) = 𝑥, 𝑤(𝑥,0) = 𝑥 − 𝜋, 0 ≤ 𝑥 ≤ 10 applied frdtm 𝑢𝑘+1 = γ(𝑘𝛼 + 1) γ(𝛼(𝑘 + 1) + 1) ((𝑢𝑘)𝑥𝑥 + 𝑢𝑘 − ∑𝑢𝑟𝑢𝑘−𝑟 𝑘 𝑟=0 + 1 2 ∑𝑢𝑟𝑣𝑘−𝑟 𝑘 𝑟=0 ) 𝑣𝑘+1 = γ(𝑘𝛼 + 1) γ(𝛼(𝑘 + 1) + 1) ((𝑣𝑘)𝑥𝑥 + 𝑣𝑘 − 3∑𝑣𝑟𝑣𝑘−𝑟 𝑘 𝑟=0 + 1 2 ∑𝑣𝑟𝑢𝑘−𝑟 𝑘 𝑟=0 + 1 2 ∑𝑣𝑟𝑤𝑘−𝑟 𝑘 𝑟=0 ) 𝑤𝑘+1 = γ(𝑘𝛼 + 1) γ(𝛼(𝑘 + 1) + 1) ((𝑤𝑘)𝑥𝑥 + 𝑤𝑘 − ∑𝑤𝑟𝑤𝑘−𝑟 𝑘 𝑟=0 + 1 2 ∑𝑤𝑟𝑣𝑘−𝑟 𝑘 𝑟=0 ) given 𝑢0 = 𝑒 𝑥, 𝑣0 = 𝑥, 𝑤0 = 𝑥 − 𝜋 when 𝑘 = 0 𝑢1 = 1 γ(𝛼 + 1) ((𝑢0)𝑥𝑥 + 𝑢0 − 𝑢0 2 + 1 2 𝑢0𝑣0) 𝑢1 = (2 − 𝑒𝑥 + 0.5𝑥)𝑒𝑥 γ(𝛼 + 1) 𝑣1 = 1 γ(𝛼 + 1) ((𝑣0)𝑥𝑥 + 𝑣0 − 3𝑣0 2 + 1 2 𝑣0𝑢0 + 1 2 𝑣0𝑤0) 𝑣1 = 𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥 𝛤(𝛼 + 1) 𝑤1 = 1 γ(𝛼 + 1) ((𝑤0)𝑥𝑥 + 𝑤0 − 𝑤0 2 + 1 2 𝑤0𝑣0) 𝑤1 = (x − π)(1 + 𝜋 − 0.5𝑥) γ(𝛼 + 1) when 𝑘 = 1 𝑢2 = 1 γ(2𝛼 + 1) ((𝑢1)𝑥𝑥 + 𝑢1 − 2𝑢0𝑢1 + 1 2 𝑢0𝑣1 + 1 2 𝑢1𝑣0) 𝑢2 = (5 − 9𝑒𝑥 + 2𝑒2𝑥 + 0.25(7 − 𝜋)𝑥 − 0.5𝑥𝑒𝑥 − 𝑥2)𝑒𝑥 γ(2𝛼 + 1) 𝑣2 = 1 γ(2𝛼 + 1) ((𝑣1)𝑥𝑥 + 𝑣1 − 6𝑣0𝑣1 + 1 2 𝑣0𝑢1 + 1 2 𝑣1𝑢0 + 1 2 𝑣0𝑤1 + 1 2 𝑣1𝑤0) 𝑣2 = (1 − 0.5π − 5.5𝑥 + 0.5𝑒𝑥)(𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥)− 𝑥3 γ(2𝛼 + 1) + (0.25𝑒𝑥 − 0.5𝜋2 + 0.25𝜋 + 0.5)𝑥2 + (1.5𝑒𝑥 − 0.5𝑒2𝑥)𝑥 + 𝑒𝑥 − 5 γ(2𝛼 + 1) 5 int. j. anal. appl. (2023), 21:33 𝑤2 = 1 γ(2𝛼 + 1) ((𝑤1)𝑥𝑥 + 𝑤1 − 2𝑤0𝑤1 + 1 2 𝑤0𝑣1 + 1 2 𝑤1𝑣0) 𝑤2 = (𝑥 − 𝜋)(π + 1 − 0.5𝑥)(2π + 1 − 1.5𝑥)+ 0.5𝑥(𝑥 − 𝜋)(𝑥 − 0.5𝜋𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥) − 1 γ(2𝛼 + 1) when 𝑘 = 2 𝑢3 = 1 γ(3𝛼 + 1) ((𝑢2)𝑥𝑥 + 𝑢2 − 2𝑢0𝑢2 − 𝑢1 2 + 1 2 𝑢0𝑣2 + 1 2 𝑢1𝑣1 + 1 2 𝑢2𝑣0) 𝑣3 = 1 γ(3𝛼 + 1) ((𝑣2)𝑥𝑥 + 𝑣2 − 6𝑣0𝑣2 − 3𝑣1 2 + 1 2 𝑢0𝑣2 + 1 2 𝑢1𝑣1 + 1 2 𝑢2𝑣0 + 1 2 𝑣0𝑤2 + 1 2 𝑣1𝑤1 + 1 2 𝑣2𝑤0) 𝑤3 = 1 γ(3𝛼 + 1) ((𝑤2)𝑥𝑥 + 𝑤2 − 2𝑤0𝑤2 − 𝑤1 2 + 1 2 𝑤0𝑣2 + 1 2 𝑤1𝑣1 + 1 2 𝑤2𝑣0) . . . 𝑢(𝑥,𝑡) = ∑ 𝑢𝑘𝑡 𝛼𝑘 = 𝑢0 + 𝑢1𝑡 𝛼 + 𝑢2𝑡 2𝛼 + ⋯ ∞ 𝑘=0 𝑢(𝑥,𝑡) ≅ 𝑒𝑥 + (2 − 𝑒𝑥 + 0.5𝑥)𝑒𝑥 γ(𝛼 + 1) 𝑡𝛼 + (5 − 9𝑒𝑥 + 2𝑒2𝑥 + 0.25(7 − 𝜋)𝑥 − 0.5𝑥𝑒𝑥 − 𝑥2)𝑒𝑥 γ(2𝛼 + 1) 𝑡2𝛼 𝑣(𝑥,𝑡) = ∑ 𝑣𝑘𝑡 𝛼𝑘 = 𝑣0 + 𝑣1𝑡 𝛼 + 𝑣2𝑡 2𝛼 + ⋯ ∞ 𝑘=0 𝑣(𝑥,𝑡) ≅ 𝑥 + 𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥 𝛤(𝛼 + 1) 𝑡𝛼 + (1 − 0.5π − 5.5𝑥 + 0.5𝑒𝑥)(𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥 + 0.5𝜋𝑥) − 𝑥3 γ(2𝛼 + 1) 𝑡2𝛼 + (0.25𝑒𝑥 − 0.5𝜋2 + 0.25𝜋 + 0.5)𝑥2 + (1.5𝑒𝑥 − 0.5𝑒2𝑥)𝑥 + 𝑒𝑥 − 5 γ(2𝛼 + 1) 𝑡2𝛼 𝑤(𝑥,𝑡) = ∑ 𝑤𝑘𝑡 𝛼𝑘 = 𝑤0 + 𝑤1𝑡 𝛼 + 𝑤2𝑡 2𝛼 + ⋯ ∞ 𝑘=0 𝑤(𝑥,𝑡) ≅ (𝑥 − 𝜋)+ (x − π)(1 + 𝜋 − 0.5𝑥) γ(𝛼 + 1) 𝑡𝛼 + (𝑥 − 𝜋)(π + 1 − 0.5𝑥)(2π+ 1 − 1.5𝑥) + 0.5𝑥(𝑥 − 𝜋)(𝑥 − 0.5𝜋𝑥 − 2.5𝑥2 + 0.5𝑥𝑒𝑥)− 1 γ(2𝛼 + 1) 𝑡2𝛼 6 int. j. anal. appl. (2023), 21:33 table 1. numerical results of variable 𝑢(𝑥,𝑡) 𝒙 𝜶 = 𝟏 𝜶 = 𝟎.𝟖 𝜶 = 𝟎.𝟓 𝜶 = 𝟎.𝟐 1.0e+014 * 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0002 0.0001 0.0001 0.0000 0.0042 0.0034 0.0021 0.0010 0.0000 0.0851 0.0683 0.0425 0.0209 0.0000 1.7093 1.3734 0.8546 0.4193 7 int. j. anal. appl. (2023), 21:33 figure 1. graphical presentation of variable 𝑢(𝑥,𝑡) figure 2. compression derivatives order between non-fractional order with fractional orders of variable 𝑢(𝑥,𝑡). -10 0 10 -10 0 10 -5 0 5 x 10 8 x axis alpha=1 t axis u a x is -10 0 10 -10 0 10 -2 0 2 x 10 8 x axis alpha=0.8 t axis u a x is -10 0 10 -10 0 10 -5 0 5 x 10 7 x axis alpha=0.5 t axis u a x is -10 0 10 -10 0 10 -1 0 1 x 10 8 x axis alpha=0.2 t axis u a x is 0 1 2 3 4 5 6 7 8 9 10 -2 0 2 4 6 8 10 12 14 16 18 x 10 13 x axis u a x is comparison between derivative orders alpha=1 alpha=0.8 alpha=0.5 alpha=0.2 8 int. j. anal. appl. (2023), 21:33 table 1 shows the approximate solution of fractional diffusion of variable 𝑢(𝑥,𝑡), it is noted that only the second order of the frdtm. figure 1: the surface of diffusion variable 𝑢(𝑥,𝑡) is convergence between fractional order and ordinary order, in figure 2: we get small difference between ordinary order with multiple fractional orders. table 2. numerical results of variable 𝑣(𝑥,𝑡) 𝒙 𝜶 = 𝟏 𝜶 = 𝟎.𝟖 𝜶 = 𝟎.𝟓 𝜶 = 𝟎.𝟐 1.0e+009 * 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0003 -0.0002 -0.0002 -0.0001 0.0000 -0.0023 -0.0019 -0.0012 -0.0006 0.0000 -0.0182 -0.0147 -0.0091 -0.0045 0.0000 -0.1473 -0.1184 -0.0737 -0.0361 0.0000 -1.1998 -0.9641 -0.5999 -0.2943 0.0000 -9.7643 -7.8456 -4.8821 -2.3950 9 int. j. anal. appl. (2023), 21:33 figure 3. graphical presentation of variable 𝑣(𝑥,𝑡) figure 4. compression derivatives order between non-fractional order with fractional orders of variable 𝑣(𝑥,𝑡). -10 0 10 -10 0 10 -2 0 2 x 10 6 x axis alpha=1 t axis v a x is -10 0 10 -10 0 10 -1 0 1 x 10 6 x axis alpha=0.8 t axis v a x is -10 0 10 -10 0 10 -2 0 2 x 10 5 x axis alpha=0.5 t axis v a x is -10 0 10 -10 0 10 -2 -1 0 x 10 6 x axis alpha=0.2 t axis v a x is 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x 10 9 x axis v a x is comparison between derivative orders alpha=1 alpha=0.8 alpha=0.5 alpha=0.2 10 int. j. anal. appl. (2023), 21:33 table 2 shows the approximate solution of fractional diffusion of variable 𝑣(𝑥,𝑡), it is noted that only the second order of the frdtm. figure 3: the surface of diffusion of variable 𝑣(𝑥,𝑡) is convergence between fractional order and ordinary order, in figure 4: we get small difference between ordinary order with multiple fractional orders. table 3. numerical results of variable 𝑤(𝑥,𝑡) 𝒙 𝜶 = 𝟏 𝜶 = 𝟎.𝟖 𝜶 = 𝟎.𝟓 𝜶 = 𝟎.𝟐 1.0e+007 * 0.0000 -0.0001 -0.0001 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0001 0.0001 0.0000 0.0000 0.0000 0.0011 0.0009 0.0006 0.0003 0.0000 0.0077 0.0062 0.0038 0.0019 0.0000 0.0401 0.0322 0.0201 0.0098 0.0000 0.1828 0.1469 0.0914 0.0448 0.0000 0.7647 0.6144 0.3823 0.1876 0.0000 3.0144 2.4220 1.5072 0.7394 11 int. j. anal. appl. (2023), 21:33 figure 5. graphical presentation of variable 𝑤(𝑥,𝑡) figure 6. compression derivatives order between non-fractional order with fractional orders of variable 𝑤(𝑥,𝑡). -10 0 10 -10 0 10 -5 0 5 x 10 5 x axis alpha=1 t axis w a x is -10 0 10 -10 0 10 -5 0 5 x 10 5 x axis alpha=0.8 t axis w a x is -10 0 10 -10 0 10 -1 0 1 x 10 5 x axis alpha=0.5 t axis w a x is -10 0 10 -10 0 10 -5 0 5 x 10 4 x axis alpha=0.2 t axis w a x is 0 1 2 3 4 5 6 7 8 9 10 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 x 10 7 x axis w a x is comparison between derivative orders alpha=1 alpha=0.8 alpha=0.5 alpha=0.2 12 int. j. anal. appl. (2023), 21:33 table 3 shows the approximate solution of fractional diffusion of variable 𝑤(𝑥,𝑡), it is noted that only the second order of the frdtm. figure 5: the surface of diffusion of variable 𝑤(𝑥,𝑡) is convergence between fractional order and ordinary order, in figure 6: we get small difference between ordinary order with multiple fractional orders. 5. conclusions the fractional reduced differential transform method has been successfully applied to obtain an analytical approximate solution for the mutualism model with fractional diffusion. it is easy to recognize that frdtm is powerful mathematical tool for solving different kinds of linear and/or nonlinear fractional partial differential equations the frdtm is no need to use adomian's polynomials to calculate the nonlinear terms. we have concluded that the fractional derivative of diffusion mutualism model is more accurate than ordinary derivative order. from all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations so we recommended researchers would use fractional reduced differential transform method when derivation the mathematical models (biological phenomena) for fractional derivatives. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. liu, b. xin, numerical solutions of a fractional predator-prey system, adv. differ. equ. 2011 (2011), 190475. https://doi.org/10.1155/2011/190475. [2] c. wang, s. wang, x. yan, global asymptotic stability of 3-species mutualism models with diffusion and delay effects, discr. dyn. nat. soc. 2009 (2009), 317298. https://doi.org/10.1155/2009/317298. [3] m.a. abdallah, asymptotic behavior of solution of a periodic mutualistic system, int. j. sci. res. 5 (2016), 543-547. [4] s. abuasad, i. hashim, homotopy decomposition method for solving higher-order timefractional diffusion equation via modified beta derivative, sains malays. 47 (2018), 2899–2905. https://doi.org/10.17576/jsm-2018-4711-33. [5] k. abdalla ishag, the techniques for solving fractional burger's equations, asian j. pure appl. math. 4 (2022), 60-73. [6] l. debnath, nonlinear partial differential equations for scientists and engineers, birkhäuser boston, https://doi.org/10.1155/2011/190475 https://doi.org/10.1155/2009/317298 https://doi.org/10.17576/jsm-2018-4711-33 13 int. j. anal. appl. (2023), 21:33 boston, ma, 2005. https://doi.org/10.1007/b138648. [7] s. abuasad, i. hashim, s.a. abdul karim, modified fractional reduced differential transform method for the solution of multiterm time-fractional diffusion equations, adv. math. phys. 2019 (2019), 5703916. https://doi.org/10.1155/2019/5703916. [8] s. mukhtar, s. abuasad, i. hashim, s.a. abdul karim, effective method for solving different types of nonlinear fractional burgers’ equations, mathematics. 8 (2020), 729. https://doi.org/10.3390/math8050729. [9] s. abuasad, a. yildirim, i. hashim, s. abdul karim, j.f. gómez-aguilar, fractional multi-step differential transformed method for approximating a fractional stochastic sis epidemic model with imperfect vaccination, int. j. environ. res. public health. 16 (2019) 973. https://doi.org/10.3390/ijerph16060973. [10] d. lu, j. wang, m. arshad, abdullah, a. ali, fractional reduced differential transform method for space-time fractional order heat-like and wave-like partial differential equations, j. adv. phys. 6 (2017), 598–607. https://doi.org/10.1166/jap.2017.1383. https://doi.org/10.1007/b138648 https://doi.org/10.1155/2019/5703916 https://doi.org/10.3390/math8050729 https://doi.org/10.3390/ijerph16060973 https://doi.org/10.1166/jap.2017.1383 international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 191-197 http://www.etamaths.com some results on the drazin inverse of a modified matrix with new conditions abdul shakoor∗, hu yang and ilyas ali abstract. in this article, we consider representations of the drazin inverse of a modified matrix m = a − cddb with the generalized schur complement z = d − badc under different conditions given in recent articles on the subject. numerical example is given to illustrate our result. 1. introduction the importance of the drazin inverse and its applications to singular differential equations and difference equations, to morkov chains and iterative methods, to cryptography, to numerical analysis, to structured matrices and to perturbation bounds for the relative eigenvalue problems can be found in [1-3]. let cm×n represent the set of m × n complex matrices. let a ∈ cn×n, then there exist a unique matrix ad ∈ cn×n satisfying the following equations: ak+1ad = ak, adaad = ad, aad = ada,(1.1) ad is called the drazin inverse of a, where k = ind(a) is the index of a, the smallest nonnegative integer for which rank(ak+1) = rank(ak) (see[1-3]). in particular, when ind(a) = 1, the drazin inverse of a is called the group inverse of a. if a is nonsingular, it is clearly ind(a) = 0 and ad = a−1. throughout this article, we denote by aπ = i − aad and define a0 = i, where i is the identity matrix with proper sizes. in 1975, shoaf [4] derived the result of the drazin inverse of a modified square matrix, in 1994, kala et al. [5] gave an explicit representation for the generalized inverse of a modified matrix, and in 2002, wei [6] have discussed the expression of the drazin inverse of a modified square matrix a − cb. recently, in 2013, dopazo et al. [7], mosić [8] and shakoor et al. [9] presented some new results for the drazin inverse of a modified matrix m = a−cddb in terms of the drazin inverse of the matrix a and the generalized schur complement z = d − badc under the following conditions: (1) aπc = 0, cdπ = 0, dπb = 0, zπb = 0, czπ = 0 (see [7]); (2) baπ = 0, cdπ = 0, dπb = 0, zπb = 0, czπ = 0 (see [7]); (3) aπc = cdπ, dπb = 0, dzπ = 0 (see [8]); (4) baπ = dπb, cdπ = 0, zπd = 0 (see [8]); (5) aπc = 0, cdπzdb = 0, cddzπb = 0, czddπb = 0, czπddb = 0 (see [9]); 2010 mathematics subject classification. 15a09, 15a10, 65f20. key words and phrases. drazin inverse; generalized schur complement; modified matrix. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 191 192 shakoor, yang and ali (6) baπ = 0, cdπzdb = 0, cddzπb = 0, czddπb = 0 czπddb = 0 (see [9]). moreover, shakoor et al. [9] gave some new results for the drazin inverse of the modified matrix m = a − cddb, when the generalized schur complement z = 0 under the following conditions: (7) aπc = 0, cdπγdb = 0, cddγπb = 0, cγddπb = 0 cγπddb = 0 (see [9]); (8) baπ = 0, cdπγdb = 0, cddγπb = 0, cγddπb = 0 cγπddb = 0 (see [9]). in this article, we consider the drazin inverse of a modified matrix m = a − cddb in terms of the drazin inverse of the matrix a and the generalized schur complement z = d−badc under conditions weaker than conditions (5) and (6) in [9], which extends some results in [7,8]. furthermore, we consider some results for the drazin inverse of the modified matrix m = a − cddb, when the generalized schur complement z = 0 under different conditions in [9]. finally, we give a numerical example to illustrate our result. 2. the drazin inverse of a modified matrix in this section, we consider the drazin inverse of a modified matrix m = a − cddb in terms of the drazin inverse of the matrix a and the generalized schur complement z = d−badc is not necessarily invertible under different conditions presented in [7,8,9]. let a, b, c, d ∈ cn×n. throughout this section, we use the following notations: m = a−cddb, z = d −badc(2.1) and k = adc, h = bad, γ = hk.(2.2) first, we present the following theorem. theorem 2.1. let a, b, c, and d be complex matrices, where ind(a) = k. if aπc = cdπ, cddzπb = 0, czddπb = 0 and czπddb = 0, then md = ad + kzdh − k−1∑ i=0 (ad + kzdh)i+1kzdbaiaπ(2.3) and ind(m) ≤ ind(a). proof. let x = ad + kzdh. the assumption aπc = cdπ implies that aadc = cddd. firstly, we note the facts: mx = aad + aadczdbad −cddbad −cdd(d −z)zdbad = aad + cdddzdbad −cdddzdbad −cddzπbad = aad(2.4) and xm = ada + adczdbada−adcddb −adczd(d −z)ddb = ada−adczdbaπ −adczπddb = ada−kzdbaπ.(2.5) on the drazin inverse of a modified matrix 193 from (2.4), we have mmd = mx −mx k−1∑ i=0 xikzdbaiaπ = aad − k−1∑ i=0 xikzdbaiaπ(2.6) and using (2.5), we get mdm = xm − k−1∑ i=0 xi+1kzdbaiaπm = ada−kzdbaπ − k−1∑ i=0 xi+1kzdbai+1aπ = ada− k−1∑ i=0 xikzdbaiaπ.(2.7) thus mmd = mdm. secondly, from (2.7) and aπmd = 0, we obtain mdmmd = (ada− k−1∑ i=0 xikzdbaiaπ)md = adamd = md. finally, we shall prove that m −m2md is a nilpotent matrix. from (2.6), we get m −m2md = [i −mmd]m = (i −aad + k−1∑ i=0 xikzdbaiaπ)m = aaπ + k−1∑ i=0 xikzdbai+1aπ. by induction on integer n ≥ 1, we have (m −m2md)n = anaπ + k−1∑ i=0 xikzdbai+naπ.(2.8) from (2.8), it gives that (m − m2md)k = 0, where k = ind(a). therefore, we conclude that mk+1md = mk and ind(m) ≤ ind(a), that completes the proof. � from theorem 2.1, we obtain the following corollaries. 194 shakoor, yang and ali corollary 2.2 ([8]). let a, b, c, and d be complex matrices, where ind(a) = k. if aπc = cdπ, zπb = 0, dπb = 0 and czπ = 0, then md = ad + kzdh − k−1∑ i=0 (ad + kzdh)i+1kzdbaiaπ and ind(m) ≤ ind(a). corollary 2.3 ([8]). let a, b, c, and d be complex matrices, where ind(a) = k. if aπc = cdπ, dzπ = 0 and dπb = 0, then md = ad + kzdh − k−1∑ i=0 (ad + kzdh)i+1kzdbaiaπ and ind(m) ≤ ind(a). in the same way, we give a new theorem. theorem 2.4. let a, b, c, and d be complex matrices, where ind(a) = k. if baπ = dπb, cdπzdb = 0, cddzπb = 0 and czπddb = 0, then md = ad + kzdh − k−1∑ i=0 aπaiczdh(ad + kzdh)i+1 and ind(m) ≤ ind(a). similarly, from theorem 2.2. we have the following corollaries. corollary 2.5 ([8]). let a, b, c, and d be complex matrices, where ind(a) = k. if baπ = dπb, cdπ = 0, zπb = 0 and czπ = 0, then md = ad + kzdh − k−1∑ i=0 aπaiczdh(ad + kzdh)i+1 and ind(m) ≤ ind(a). corollary 2.6 ([8]). let a, b, c, and d be complex matrices, where ind(a) = k. if baπ = dπb, cdπ = 0 and zπd = 0, then md = ad + kzdh − k−1∑ i=0 aπaiczdh(ad + kzdh)i+1 and ind(m) ≤ ind(a). now, we consider some results for the drazin inverse of the modified matrix m = a − cddb, when the generalized schur complement z = 0 under different conditions in [9]. on the drazin inverse of a modified matrix 195 theorem 2.7. let a, b, c, and d be complex matrices, where ind(a) = k. if z = 0, aπc = cdπ, dπb = 0 and dγπ = 0, then md = (i −kγdh)ad(i −kγdh) + k−1∑ i=0 [(i −kγdh)ad]i+2kγdbaiaπ (2.9) and ind(m) ≤ ind(a). proof. let x = (i − kγdh)ad(i − kγdh). the assumptions aπc = cdπ and dγπ = 0 imply that aadc = cddd and ddγπ = 0. firstly, we note the facts: mx = (a−aadcγdbad −cddb + cdddγdbad)ad(i −kγdh) = (a−cddb)(ad − (ad)2cγdbad) = aad −adcγdbad −cddbad + cddγγdbad = aad −kγdh −cddγπbad = aad −kγdh.(2.10) since dddb = b, adc = adcddd and dγπ = 0, then xm = (i −kγdh)ad(a−cddb −adcγdbada + adcγddddb) = (i −kγdh)ad(a−cddb −adcγdbada + adcγdb) = (ad −adcγdb(ad)2)(a−cddb + adcγdbaπ) = ada−adcddb + (ad)2cγdbaπ −adcγdbad + adcγdγddb −adcγdb(ad)3cγdbaπ = ada−kγdh + (i −adcγdbad)(ad)2cγdbaπ −adcdddγπddb = ada−kγdh + (i −kγdh)adkγdbaπ.(2.11) from (2.10), we have mmd = mx + m(i −kγdh)ad k−1∑ i=0 [(i −kγdh)ad]i+1kγdbaiaπ = aad −kγdh + (aad −cddbad) k−1∑ i=0 [(i −kγdh)ad]i+1kγdbaiaπ = aad −kγdh + k−1∑ i=0 [(i −kγdh)ad]i+1kγdbaiaπ(2.12) and using (2.11), we get mdm = xm + k−1∑ i=0 [(i −kγdh)ad]i+2kγdbaiaπm = ada−kγdh + (i −kγdh)adkγdbaπ + k−1∑ i=0 [(i −kγdh)ad]i+2 ×kγdbai+1aπ = ada−kγdh + k−1∑ i=0 [(i −kγdh)ad]i+1kγdbaiaπ.(2.13) 196 shakoor, yang and ali thus mmd = mdm. from (2.13) and aπmd = 0, we obtain mdmmd = (ada−kγdh + k−1∑ i=0 [(i −kγdh)ad]i+1kγdbaiaπ)md = (ada−kγdh)md = md. by using (2.12) and dπb = 0, we get m −m2md = aaπ − k−1∑ i=0 [(i −kγdh)ad]ikγdbai+1aπ. by induction on integer n ≥ 1, we have (m −m2md)n = anaπ − k−1∑ i=0 [(i −kγdh)ad]ikγdbai+naπ. from above expression, it follows that (m − m2md)k = 0, where k = ind(a). therefore, we obtain that mk+1md = mk and ind(m) ≤ ind(a), which completes the proof. � in similar way, we present another result of this paper. theorem 2.8. let a, b, c, and d be complex matrices, where ind(a) = k. if z = 0, baπ = dπb, cdπ = 0 and γπd = 0, then md = (i −kγdh)ad(i −kγdh) + k−1∑ i=0 aπaicγdh[ad(i −kγdh)]i+2 and ind(m) ≤ ind(a). in the end of this section, we give a numerical example to demonstrate our result of a modified matrix. this numerical example describes matrices a, b, c and d which do not satisfy the conditions of [7, theorem 2.1] nor the conditions of [8, theorem 1] but they satisfy the conditions of theorem 2.1. therefore, we can apply the formula given in theorem 2.1 to obtain the drazin inverse of a modified matrix m. numerical example 2.9. consider the matrices a =   1 0 00 0 0 0 0 0  , b = ( 2 1 1 0 0 0 ) , c =   −1 −10 0 0 0  , d = ( 1 1 0 0 ) . note that ind(a) = 1 and ind(d) = 1, then we obtain ad =   1 0 00 0 0 0 0 0   , aπ =   0 0 00 1 0 0 0 1   , dd = ( 1 1 0 0 ) , dπ = ( 0 −1 0 1 ) . on the drazin inverse of a modified matrix 197 now we have m = a−cddb =   3 1 10 0 0 0 0 0   and z = d −badc = ( 3 1 0 0 ) , zd = 1 9 ( 3 1 0 0 ) , zπ = 1 3 ( 0 −1 0 3 ) . it can be calculated that: (i) czπ 6= 0, so the conditions given in [7, theorem 2.1] are not satisfied. (ii) dzπ 6= 0, thus the conditions given in [8, theorem 1] are not satisfied. on the other hand, we can observe that aπc = cdπ, cddzπb = 0, czddπb = 0 and czπddb = 0. then applying theorem 2.1, we obtain md = 1 9   3 1 10 0 0 0 0 0   . 3. acknowledgment this work was supported by the ph.d. programs foundation of ministry of education of china (grant no. 20110191110033). references [1] a. ben-israel, t.n.e. greville, generalized inverses: theory and applications, second ed., springer, new york, 2003. [2] s.l. campbell, c.d. meyer, generalized inverse of linear transformations, dover, new york, 1991. [3] g. wang, y. wei, s. qiao, generalized inverses: theory and computations, science press, beijing/new york, 2004. [4] j.m. shoaf, the drazin inverse of a rank-one modification of a square matrix, ph.d. dissertation, north carolina state university, 1975. [5] r. kala, k. klaczyński, generalized inverses of a sum of matrices, sankhya ser. a 56 (1994) 458-464. [6] y. wei, the drazin inverse of a modified matrix, appl. math. comput. 125 (2002) 295-301. [7] e. dopazo, m.f. mart́ınez-serrano, on deriving the drazin inverse of a modified matrix, linear algebra appl. 438 (2013) 1678-1687. [8] d. mosić, some results on the drazin inverse of a modified matrix, calcolo. 50 (2013) 305?11. [9] a. shakoor, h. yang, i. ali, some representations for the drazin inverse of a modified matrix, calcolo. doi 10.1007/s10092-013-0098-0 (2013). college of mathematics and statistics, chongqing university, chongqing, 401331, china ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 188-197 doi: 10.28924/2291-8639-15-2017-188 a new numerical technique for solving systems of nonlinear fractional partial differential equations mountassir hamdi cherif1∗ and djelloul ziane2 abstract. in this paper, we apply an efficient method called the aboodh decomposition method to solve systems of nonlinear fractional partial differential equations. this method is a combined form of aboodh transform with adomian decomposition method. the theoretical analysis of this investigated for systems of nonlinear fractional partial differential equations is calculated in the explicit form of a power series with easily computable terms. some examples are given to shows that this method is very efficient and accurate. this method can be applied to solve others nonlinear systems problems. 1. introduction over the last three decades, fractional calculus has been enormously developed and taken on in many fields of scientific research. one of the main reasons is its application in many scientific disciplines. the fractional calculus by its tools remains a very suitable means for the resolution of the differential systems. these generally translate mathematical or physical models of the many natural phenomena that surround us. integro-differential equations and fractional differential systems have recently proved to be very useful in the field of physics, engineering, control processing for visco-elastic systems, diffusion, .... we find that many researchers have been interested in solving this kind of linear and nonlinear differential equations also systems of fractional differential equations. consequently, the investigation of the exact solutions to nonlinear equations play an important role in the study of nonlinear physical phenomena, although the nonlinear differential equations are the most complex in the solution compared with linear differential equations. integral transformations such as laplace, sumudu, natural, elzaki and aboodh are unable to solve the nonlinear differential equations. so, we find some researchers are working on the combined of these transformations with many methods, among them we find the adomian decomposition method. this method was introduced in the 1980s by george adomian (1923-1996), and it was applied to many problems ( [1][4]). the adomian decomposition method was coupled with laplace transform method [14], with sumudu transform method [9], with elzaki transform method [10], with natural transform method [11] and with aboodh transform method. aboodh transform is derived from the classical fourier integral. based on the mathematical simplicity of the aboodh transform and its fundamental properties, aboodh transform was introduced by khalid aboodh in 2013, to facilitate the process of solving ordinary and partial differential equations in the time domain. this transformation has deeper connection with the laplace and elzaki transform [8]. the coupling of adomian decomposition method with aboodh transform method has been applied for solving linear and nonlinear equations [17], to solve system of linear and nonlinear partial differential equations [13] and for solving time-fractional diffusion equation [16]. the objective of this study is coupling the adomian decompositionmethod (adm) with aboodh transform in the sense of fractional derivative, then we apply this modified method to solve some examples related with systems of nolinear fractional partial differential equations. received 21st july, 2017; accepted 19th october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 26a33, 44a05, 34k37, 35f61. key words and phrases. aboodh transform; caputo fractional derivative; adomain decomposition method; system of nonlinear fractional differential equations. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 188 nonlinear fractional partial differential equations 189 2. basic definitions in this section, we give some basic notions about fractional calculus, aboodh transform and aboudh transform of fractional derivatives which are used further in this paper. 2.1. fractional calculus. we give some basic definitions and properties of the fractional calculus theory as the riemann–liouville fractional integrals and caputo fractional derivative (see [6], [7]). definition 2.1. let ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis r. the riemann–liouville fractional integral iα0+f of order α ∈ r (α > 0) is defined by (iα0+f)(t) = 1 γ (α) ∫ t 0 f(τ)dτ (t− τ)1−α , t > 0, α > 0 (i00+f)(t) = f(t) here γ(·) is the gamma function. theorem 2.1. let α > 0 and let n = [α]+1. if f(t) ∈ acn [a,b] , then the caputo fractional derivative (cdα0+f)(t) exist almost evrywhere on [a,b] . if α /∈ n, (cdα0+f)(t) is represented by (cdα0+f)(t) = 1 γ (n−α) ∫ t 0 f(n)(τ)dτ (t− τ)α−n+1 , (2.1) where d = d dx and n = [α] + 1. remark 2.1. in this paper, we consider the time-fractional derivative in the caputo’s sense. when α ∈ r+, the time-fractional derivative is defined as (cdαt u)(x,t) = ∂αu(x,t) ∂tα = { 1 γ(m−α) ∫ t 0 (t− τ)m−α−1 ∂ mu(x,τ) ∂τm , m− 1 < α < m, ∂mu(x,t) ∂tm , α = m, where m ∈ n∗. 2.2. definitions and properties of the aboodh transform. the aboodh transform was defined by k. s. aboodh [12] in 2013. in this section, we give some basic definitions and properties of this transform (see [12], [15], [16]). the aboodh transform is defined for functions of exponential order. we consider functions belonging to a class b, where b defined by b = { u(t) : |u(t)| < mekj|t|, if t ∈ (−1)j × [0, ∞,j = 1, 2; m, k1,k2 > 0 } . definition 2.2. the aboodh integral transform of the function u in b is defined by the integral equation a [u(t)] = u(v) = 1 v ∫ ∞ 0 u(t)e−vtdt; t ≥ o, v ∈ (k1,k2). (2.2) the variable v in this transform is used to factor the variable t in the argument of the function u. proposition 2.1. the aboodh transform of the time-fractional derivative in the caputo’s sense is defined as a [ (cdα0+u)(t); v ] = vαa[u(t)] − n−1∑ k=0 u(k)(0) v2−α+k , n− 1 < α ≤ n, n = 1, 2, . . . (2.3) and the aboodh transform of the function u(x,t) with caputo fractional derivative of order α is given by 190 cherif and ziane a [ (cdα0+u)(x,t); v ] = vαa[u(x,t)] − n−1∑ k=0 u(k)(x, 0) v2−α+k , n− 1 < α ≤ n, n = 1, 2, . . . (2.4) 2.2.1. somme properties of the aboodh transform . 1. the aboodh transform of the nth derivative of u(t) is given by a[u(n)(t)] = un(v) = v na[u(t)] − n−1∑ k=0 u(k)(0) v2−n+k (2.5) 2. some elementary functions and their transformations u(t) a [u(t)] 1 1 v2 t 1 v3 tn n! vn+2 , n = 0, 1, 2, . . . tα γ(α+1) vα+2 , α > 0. 3. analysis of fractional aboodh decomposition method to illustrate the basic idea of this method, we consider the general system of nonlinear fractional partial differential equations of the form cdαt u(x,t) + rw(x,t) + nu(x,t) = h1(x,t), cd β t w(x,t) + ru(x,t) + nw(x,t) = h2(x,t), (3.1) where n− 1 < α,β 6 n, n = 1, 2, ... and the initial conditions[ ∂n−1u(x,t) ∂tn−1 ] t=0 = fn−1(x), n = 1, 2, ...[ ∂n−1w(x,t) ∂tn−1 ] t=0 = gn−1(x), n = 1, 2, ... (3.2) cdαt u(x,t), cd β t w(x,t) are the caputo fractional derivatives of the functions u(x,t), w(x,t) respectively, r is the linear differential operator, n represent the general nonlinear differential operator, and h1(x,t), h2(x,t) are the source terms. applying the aboodh transform on both sides of (3.1) and using the differentiation property of this transform (2.3), we obtain a [u(x,t)] = 1 vα ∑n−1 k=0 u(k)(x,0) v2−α+k + 1 vα a [h1(x,t)] − 1vα a [rw(x,t) + nu(x,t)] , a [w(x,t)] = 1 vβ ∑n−1 k=0 w(k)(x,0) v2−β+k + 1 vβ a [h2(x,t)] − 1vβ a [ru(x,t) + nw(x,t)] . (3.3) taking the inverse aboodh transform on both sides of equations in system (3.3) and then by using initial conditions (3.2), we have v(x,t) = g(x,t) −a−1 ( 1 vα a[rw(x,t) + nu(x,t)] ) , w(x,t) = h(x,t) −a−1 ( 1 vβ a[ru(x,t) + nw(x,t)] ) , (3.4) where g(x,t) and h(x,t) are represents the terms arising from the nonhomogeneous terms and the prescribed initial conditions. now, we represent solutions as the following infinite series u(x,t) = ∞∑ n=0 un(x,t) ,w(x,t) = ∞∑ m=0 wn(x,t) (3.5) and the nonlinear terms can be decomposed as nu(x,t) = ∞∑ n=0 cn, nw(x,t) = ∞∑ n=0 dn (3.6) where cn and dn are adomian polynomials [5], and they can be calculated by the formulas given below nonlinear fractional partial differential equations 191 cn = 1 n! ∂n ∂λn [ n ( ∞∑ i=0 λiui )] λ=0 , dn = 1 n! ∂n ∂λn [ n ( ∞∑ i=0 λiwi )] λ=0 , i = 0, 1, 2, · · · (3.7) using (3.5) and (3.6), we can rewrite (3.4) as∑∞ n=0 un(x,t) = g(x,t) −a −1 [ 1 vα a [r ∑∞ n=0 wn + ∑∞ n=0 cn] ]∑∞ n=0 wn(x,t) = h(x,t) −a −1 [ 1 vβ a [r ∑∞ n=0 un + ∑∞ n=0 dn] ] . (3.8) by comparing both sides of (3.8), we get u0(x,t) = g(x,t), u1(x,t) = −a−1 [ 1 vα a [rw0(x,t) + c0] ] , u2(x,t) = −a−1 [ 1 vα a [rw1(x,t) + c1] ] . ... (3.9) and w0(x,t) = h(x,t), w1(x,t) = −a−1 [ 1 vβ a [ru0(x,t) + d0] ] , w2(x,t) = −a−1 [ 1 vβ a [ru1(x,t) + d1] ] , ... (3.10) we continue in this manner to obtain the general recursive relations un+1(x,t) = −a−1 [ 1 vα a [rwn(x,t) + cn] ] , n > 1 wn+1(x,t) = −a−1 [ 1 vβ a [run(x,t) + dn] ] , n > 1 . (3.11) finally, the approximate solution is calculated by u(x,t) = lim n→∞ n∑ n=0 un(x,t) w(x,t) = lim n→∞ n∑ n=0 wn(x,t). (3.12) 4. applications in this section, we apply the aboodh decomposition transform method for solving systems of nonlinear fractional partial differential equations. example 4.1. consider the system of nonlinear partial differential equations with time-fractional derivatives cdαt u(x,t) + w(x,t)ux(x,t) + u(x,t) = 1, 0 ≤ α < 1 cd β t w(x,t) −u(x,t)wx(x,t) −w(x,t) = 1, 0 ≤ β < 1 (4.1) with the initial conditions u(x, 0) = ex w(x, 0) = e−x (4.2) for α = β = 1, the exact solution of (4.1) is given by u(x,t) = ex−t w(x,t) = et−x (4.3) applying the aboodh transform on both sides of (4.1) and using its differentiation property, we get a [u(x,t)] = e x v2 + 1 vα a [1 −w(x,t)ux(x,t) −u(x,t)] , a [w(x,t)] = e −x v2 + 1 vβ a [1 + u(x,t)wx(x,t) + w(x,t)] . (4.4) taking the inverse aboodh transform on both sides of (4.4), we obtain 192 cherif and ziane u(x,t) = ex + t α γ(α+1) −a−1 ( 1 vα a [w(x,t)ux(x,t) + u(x,t)] ) , w(x,t) = e−x + t β γ(β+1) + a−1 ( 1 vβ a [u(x,t)wx(x,t) + w(x,t)] ) . (4.5) we represent the approximate solution as the following infinite series u(x,t) = ∞∑ n=0 un(x,t), w(x,t) = ∞∑ m=0 wn(x,t) (4.6) note that these nonlinear terms wux = ∞∑ n=0 cn , uwx = ∞∑ n=0 dn (4.7) are the adomian polynomials (see [5]). the first few components of cn and dn polynomials are given by c0 = w0u0x, c1 = w0u1x + w1u0x, c2 = w0u2x + w2u0x + w1u1x, ... and d0 = u0w0x, d1 = u0w1x + u1w0x, d2 = u0w2x + u2w0x + u1w1x, ... substituting (4.6) and (4.7) in (4.5), we have∑∞ n=0 un(x,t) = e x + t α γ(α+1) −a−1 ( 1 vα a [ ∑∞ n=0 cn + ∑∞ n=0 un(x,t)] )∑∞ n=0 wn(x,t) = e −x + t β γ(β+1) + a−1 ( 1 vβ a [ ∑∞ n=0 dn + ∑∞ m=0 wn(x,t)] ) . (4.8) by comparing both sides of (4.8), we can easily generate the recursive relations u0(x,t) = e x + t α γ(α+1) u1(x,t) = −a−1 ( 1 vα a [c0 + u0(x,t)] ) u2(x,t) = −a−1 ( 1 vα a [c1 + u1(x,t)] ) u3(x,t) = −a−1 ( 1 vα a [c2 + u2(x,t)] ) ... un+1(x,t) = −a−1 ( 1 vα a [cn + un(x,t)] ) , n ≥ 0. (4.9) and w0(x,t) = e −x + t β γ(β+1) w1(x,t) = a −1 ( 1 vβ a [d0 + w0(x,t)] ) w2(x,t) = a −1 ( 1 vβ a [d1 + w1(x,t)] ) w3(x,t) = a −1 ( 1 vβ a [d2 + w2(x,t)] ) ... wn+1(x,t) = a −1 ( 1 vβ a [dn + wn(x,t)] ) , n ≥ 0. (4.10) the first few components of the unknown functions un(x,t) and wn(x,t) are given as follows nonlinear fractional partial differential equations 193 u1(x,t) = − 1+e x γ(α+1) tα − e x γ(α+β+1) tα+β − t 2α γ(2α+1) w1(x,t) = −1+e−x γ(β+1) tβ − e −x γ(α+β+1) tα+β + t 2β γ(2β+1) . and u2(x,t) = 2 + ex γ(2α + 1) t2α + ex − 1 γ(α + β + 1) tα+β + ( 1 + 2ex + γ(α + β + 1)ex γ(α + 1)γ(β + 1) ) t2α+β γ(2α + β + 1) + γ(α + 2β + 1)ex γ(β + 1)γ(α + β + 1)γ(2α + 2β + 1) t2α+2β − ex γ(α + 2β + 1) tα+2β + t3α γ(3α + 1) w2(x,t) = −2 + e−x γ(2β + 1) t2β + 1 + e−x γ(α + β + 1) tα+β + ( 2 −e−x + γ(α + β + 1)e−x γ(α + 1)γ(β + 1) ) tα+2β γ(α + 2β + 1) + γ(2α + β + 1)e−x γ(α + 1)γ(α + β + 1)γ(2α + 2β + 1) t2α+2β + e−x γ(2α + β + 1) t2α+β + t3β γ(3β + 1) in the same manner, we can find the other components. finally , the series solution of the unknown functions u(x,t) and w(x,t) of (4.1) are given by u(x,t) = ex − ex γ(α + 1) tα + 1 + ex γ(2α + 1) t2α − 1 γ(α + β + 1) tα+β +( 1 + 2ex + γ(α + β + 1)ex γ(α + 1)γ(β + 1) ) t2α+β γ(2α + β + 1) + γ(α + 2β + 1)ex γ(β + 1)γ(α + β + 1)γ(2α + 2β + 1) t2α+2β − ex γ(α + 2β + 1) tα+2β + t3α γ(3α + 1) + ... w(x,t) = e−x + e−x γ(β + 1) tβ + −1 + e−x γ(2β + 1) t2β + e−x γ(α + β + 1) tα+β +( 2 −e−x + γ(α + β + 1)e−x γ(α + 1)γ(β + 1) ) tα+2β γ(α + 2β + 1) + γ(2α + β + 1)e−x γ(α + 1)γ(α + β + 1)γ(2α + 2β + 1) t2α+2β + e−x γ(2α + β + 1) t2α+β + t3β γ(3β + 1) + ... when α = 1 and β = 1, the series solutions of (4.1) are u(x,t) = u0(x,t) + u1(x,t) + u2(x,t) + ... = e x ( 1 − t + t 2 2! − t 3 3! + ... ) = ex−t w(x,t) = w0(x,t) + w1(x,t) + w2(x,t) + ... = e −x ( 1 + t + t 2 2! + t 3 3! + ... ) = e−x+t which is the exact solution of nonlinear system given in (4.3). 194 cherif and ziane example 4.2. we consider the following system of nonlinear partial differential equations with timefractional derivatives cdαt u = −u−hxwy + hywx, 0 ≤ α < 1 cd β t h = h, 0 ≤ β < 1 cd γ t w = w −uxwx −uywy, 0 ≤ γ < 1 (4.11) with the initial conditions u(x,y, 0) = x + y ; h(x,y, 0) = 1 + x−y; w(x,y, 0) = −x + y (4.12) taking aboodh transform with its differentiation property of (4.11) subject to the initial conditions (4.12), we have a [u] = x+y v2 + 1 vα a [−u−hxwy + hywx] , a [h] = 1+x−y v2 + 1 vβ a [h] , a [w] = −x+y v2 + 1 vγ a [w −uxwx −uywy] . (4.13) now, we apply the inverse aboodh transform on both sides of (4.13), we obtain u(x,y,t) = x + y + a−1 ( 1 vα a [−u−hxwy + hywx] ) , h(x,y,t) = 1 + x−y + a−1 ( 1 vβ a [h] ) , w(x,y,t) = −x + y + a−1 ( 1 vγ a [w −uxwx −uywy] ) . (4.14) we represent solutions as the following infinite series u(x,y,t) = ∞∑ n=0 un(x,y,t), h(x,y,t) = ∞∑ n=0 hn(x,y,t),w(x,y,t) = ∞∑ n=0 wn(x,y,t). (4.15) note that these nonlinear terms hxwy = ∞∑ n=0 an ; hywx = ∞∑ m=0 bn ; uxwx = ∞∑ n=0 cn ; uywy = ∞∑ m=0 dn (4.16) are the adomian polynomials [5]. substituting (4.15) and (4.16) in (4.14), we have ∑∞ n=0 un(x,y,t) = x + y + a −1 ( 1 vα a [− ∑∞ n=0 un − ∑∞ n=0 an + ∑∞ m=0 bn] ) ,∑∞ m=0 hn(x,y,t) = 1 + x−y + a −1 ( 1 vβ a [ ∑∞ n=0 hn] ) ,∑∞ n=0 wn(x,y,t) = −x + y + a −1 ( 1 vγ a [ ∑∞ n=0 wn − ∑∞ m=0 cn − ∑∞ n=0 dn] ) . (4.17) by comparing both sides of equations (4.17), we can easily generate the recursive relations u0(x,y,t) = x + y u1(x,y,t) = a −1 ( 1 vα a [−u0 −a0 + b0] ) u2(x,y,t) = a −1 ( 1 vα a [−u1 −a1 + b1] ) u3(x,y,t) = a −1 ( 1 vα a [−u2 −a2 + b2] ) ... un+1(x,y,t) = a −1 ( 1 vα a [−un −an + bn] ) , n ≥ 0. (4.18) and nonlinear fractional partial differential equations 195 h0(x,y,t) = 1 + x−y h1(x,y,t) = a −1 ( 1 vβ a [h0] ) h2(x,y,t) = a −1 ( 1 vβ a [h1] ) h3(x,y,t) = a −1 ( 1 vβ a [h2] ) ... hn+1(x,y,t) = a −1 ( 1 vβ a [hn] ) , n ≥ 0. (4.19) finally w0(x,y,t) = −x + y w1(x,y,t) = a −1 ( 1 vγ a [w0 −c0 −d0] ) w2(x,y,t) = a −1 ( 1 vγ a [w1 −c1 −d1] ) w3(x,y,t) = a −1 ( 1 vγ a [w2 −c2 −d2] ) ... wn+1(x,y,t) = a −1 ( 1 vγ a [wn −cn −dn] ) , n ≥ 0. (4.20) the first few components of the unknown functions un(x,y,t),hn(x,y,t) and wn(x,y,t) are given as follows u1(x,y,t) = a −1 ( 1 vα a [−u0 −h0xw0y + h0yw0x] ) = −(x + y) t α γ(α+1) h1(x,y,t) = a −1 ( 1 vβ a [h0] ) = (1 + x−y) t β γ(β+1) w1(x,y,t) = a −1 ( 1 vγ a [w0 −u0xw0x −u0yw0y] ) = (−x + y) t λ γ(λ+1) the second component of each solution series is given by the following formulas u2(x,y,t) = a −1 ( 1 vα a [−u1 −h0xw1y −h1xw0y + h0yw1x + h1yw0x] ) = (x + y) t 2α γ(2α+1) h2(x,y,t) = a −1 ( 1 vβ a [h1] ) = (1 + x−y) t 2β γ(2β+1) w2(x,y,t) = a −1 ( 1 vγ a [w1 −u0xw1x −u1xw0x −u0yw1y −u1yw0y] ) = (−x + y) t 2λ γ(2λ+1) we continue the calculations to find u3(x,y,t) = a −1 ( 1 vα a [−u2 −a2 + b2] ) = −(x + y) t 2α γ(2α+1) ... un(x,y,t) = (−1)n(x + y) t nα γ(nα+1) . (4.21) h3(x,y,t) = a −1 ( 1 vβ a [h2] ) = (1 + x−y) t 3β γ(3β+1) ... hn(x,y,t) = (1 + x−y) t nβ γ(nβ+1) . (4.22) 196 cherif and ziane w3(x,y,t) = a −1 ( 1 vγ a [w2 −c2 −d2] ) = (−x + y) t 3λ γ(3λ+1) ... wn(x,y,t) = (−x + y) t nλ γ(nλ+1) . (4.23) finally , the series solutions of the system (4.11) are given by u(x,y,t) = ∑∞ n=0 un(x,y,t) = (x + y) ( 1 − t α γ(α+1) + t 2α γ(2α+1) − t 3α γ(3α+1) + ...± t nα γ(nα+1) ± ... ) = (x + y) ∑∞ n=0 (−tα)n γ(nα+1) = (x + y)eα(−tα). (4.24) h(x,y,t) = ∑∞ n=0 hn(x,y,t) = (1 + x−y) ( 1 + t β γ(β+1) + t 2β γ(2β+1) + t 3β γ(3β+1) + ... + t nβ γ(nβ+1) + ... ) = (1 + x−y) ∑∞ n=0 (tβ)n γ(nβ+1) = (1 + x−y)eβ(tβ). (4.25) w(x,y,t) = ∑∞ n=0 wn(x,y,t) = (−x + y) ( 1 + t λ γ(λ+1) + (tλ)2 γ(2λ+1) + (tλ)3 γ(3λ+1) + ... + (tλ)n γ(nλ+1) + ... ) = (−x + y) ∑∞ n=0 (tλ)n γ(nλ+1) = (−x + y)eλ(tλ). (4.26) where, eα , eβ and eλ are the mittag-leffler functions. when α = 1, β = 1 and λ = 1, we get u(x,y,t) = (x + y)eα(−t) = (x + y)e−t. h(x,y,t) = (1 + x−y)eβ(t) = (1 + x−y)et. w(x,y,t) = (−x + y)eλ(t) = (−x + y)et. (4.27) these are the exact solutions of nonlinear system (4.11). 5. conclusion in this work, the aboodh transform method combined with adomian decomposition method has been successfully applied to solve systems of nonliner fractional partial differential equations. the approximate solutions obtained by this method are compared with the exact solutions. thus, the results show that this method is a powerful mathematical tool for solving systems of nonlinear fractional partial differential equations in other areas of science. references [1] g. adomian, nonlinear stochastic systems theory and applications to physics, kluwer academic publishers, netherlands, (1989). [2] g. adomian and r. rach, equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, comput. math. appl., 10 (1990), 9-12. [3] g. adomian, solving frontier problems of physics: the decomposition method, kluwer academic publishers, boston, (1994). [4] g. adomian, solutions of nonlinear p.d.e, appl. math. lett., 11 (1998), 121-123. [5] y. zhu, q. chang and s. wu, a new algorithm for calculating adomian polynomials, appl. math. comp., 169 (2005), 402–416. [6] k. diethelm, the analysis fractional differential equations, springer-verlag berlin heidelberg (2010). [7] i. podlubny, fractional differential equations, academic press, san diego, ca, (1999). [8] a. k. h. sedeeg and m. m. a. mahgoub, comparison of new integral transform aboodh transform and adomian decomposition method, int. j. math. and app., 4 (2–b) (2016), 127-135. [9] d. kumar, j. singh and s. rathore, sumudu decomposition method for nonlinear equations, int. math. for., 7 (2012), 515 521. [10] d. ziane and m. hamdi cherif, resolution of nonlinear partial differential equations by elzaki transform decomposition method, j. appr. theor. app. math., 5 (2015), 17-30. [11] h. m. baskonus, h. bulut and y. pandir, the natural transform decomposition method for linear and nonlinear partial differential equations, math. engineer. scie. aerospace., 5 (1) (2014), 111-126. nonlinear fractional partial differential equations 197 [12] k. s. aboodh, the new integral transform ”aboodh transform” , glob. j. pure. app. math., 9 (1) (2013), 35-43. [13] m. m. a. mahgoub and a. k. h. sedeeg , an efficient method for solving linear and nonlinear system of partial differential equations, british. j. math. comp. sci., 20 (1) (2017), 1-10. [14] j. ahmad, z. bibi and k. noor, laplace decomposition method using he’s polynomial to burgers equation, j. scie. arts., 2 (27) (2014), 131-138. [15] m. m. a. mahgoub and a. k. h. sedeeg , a comparative study of homotopy perturbation aboodh transform method and homotopy decomposition method for solving nonlinear fractional partial differential equations, int. j. theor. app. math., 2 (2) (2016), 45-51. [16] r. i. nuruddeen and k. s. aboodh, analytical solution for time-fractional diffusion equation by aboodh decomposition method, int. j. math. and app., 5 (1–a) (2017), 115-122. [17] r. i. nuruddeen and a. m. nass, aboodh decomposition method and its application in solving linear and nonlinear heat equations, europ. j. adv. eng. tech., 3 (7) (2016), 34-37. 1department of mathematics and informatics, electrical and energy engineering graduate school (esgee), oran, algeria., laboratory of mathematics and its applications (lamap), university of oran1 ahmed ben bella, oran, algeria. 2laboratory of mathematics and its applications (lamap), university of oran1 ahmed ben bella, oran, 31000, algeria. ∗corresponding author: mountassir27@yahoo.fr 1. introduction 2. basic definitions 2.1. fractional calculus 2.2. definitions and properties of the aboodh transform 3. analysis of fractional aboodh decomposition method 4. applications 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 75-85 http://www.etamaths.com optimality and duality defined by the concept of tempered fractional univex functions in multi-objective optimization rabha w. ibrahim abstract. in this paper, we purpose the concept of tempered univex functions by utilizing a tempered fractional difference-differential operator type caputo. this instruction indicates a new class of these functions in some optimal problems by exemplifying the settings on the modified formula. we call it the class of tempered fractional univex functions. our study is based on the strong, weak, converse, and strict converse duality propositions. a multi-objective optimal problem includes the new process is disentangled. 1. introduction in 1989, dunkl imposed a difference-differential operator [1] setting on some euclidean space and realizing the commutative law for a differentiable function on rn. this operator can be employed in various parts in pure mathematics, such as lee algebra, clifford algebra and complex analysis. in 1998, rosler and voit acquired into consideration this operator to adapt the tool of the markov processes [2]. unevenly, these operators can be expected as a simplification of the partial derivatives and various constructions of operators like the laplace operator, the fourier transform, and the hermite polynomials. also, these operators convoluted in famous processes such as the brownian motion and the cauchy processes. recently, this operator and its some simplifications have improved significant care in many fields of mathematics and physics. they shield a helpful method in the study of special functions and they are closely related with definite demonstrations of degenerate affine hecke algebras. furthermore, dunkl operator is obviously convoluted in the algebraic explanation of definite devotedly resolvable quantum multi-body systems. it can be used to identify the generalized method of the heat equation, which is called the dunkl heat equation. it can be recommended to adapt the idea of moments of probability measures on rn. our goal is to generalize the dunkl operator in view of the tempered fractional calculus and propose it to simplify the class of non-linear univex functions. this class typically appears in many non-linear multi-objective problems. the benefit of exploiting the dunkle operator is that this operator deals with multi-dimensional spaces. moreover, the author extend it to the complex plane and provided a modified differential-difference dunkl operator in the open unit disk. the study was in the field of geometric function theory [3]. fractional calculus is the most important branch of mathematical analysis, because it refers to the non-linearity studies in all science. the most famous operators are the riemann-liouville, caputo (continuous operators) and grunwald-letnikov (discrete operator) (see [4], [5]). it has been presented the fractional calculus discoveries usage in many categories of science and engineering, containing fluid flow, diffusive transport theory, electrical networks, electromagnetic theory, probability and statistics. the tempered fractional diffusion idea was established in statistics. this idea has demonstrated useful applications in geophysics and finance [6]. moreover, it applied to introduce a fractional multi-objective function in optimal control [7]. received 15th may, 2017; accepted 28th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 34a08, 26a33, 49j35. key words and phrases. fractional calculus; fractional differential operator; fractional differential equation; univex function. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 75 76 rabha w. ibrahim in this study, we aim to generalize the concept of tempered univex functions by utilizing a tempered fractional differential-difference operator (dunkl operator), based on different types of fractional calculus. this study gives us a new class of these functions in some optimal problems by illustrating conditions on the generalized functions. we call it the class of tempered fractional univex functions. the strong, weak, converse, and strict converse duality theorems are proposed. the main tool employed in the analysis is based on the tempered caputo operator. 2. handling we need the following concepts in the sequel of the article: 2.1. dunkl operator. suppose the two column vectors x = (x1, ...,xn) and v = (v1, ...,vn) ∈ rn with their dot product x.v = xt .v = ∑n i=1 vixi. the operator through the hyper-plane is defined by σv x = x− 2 v.x v.v v. in a matrix form, we have σv = i − 2 vvt vtv . note that σv can be represented by a symmetric matrix. the dunkl operator is formulated by: diφ(x) = ∂ ∂xi φ(x) + ∑ v∈r+ kv (1 −σv)φ(x) v.x vi, i = 1, ...,n, where vi is the i-th component of v, 1 ≤ i ≤ n, x ∈ rn. , and φ smooth function on rn. when kv = 0, then we have diφ(x) = ∂ ∂xi φ(x). one of the outcomes of these operators is satisfying di(djφ(x)) = dj(diφ(x)). (2.1) moreover, the operator achieves the product di[φ(x)ψ(x)] = ψ(x)diφ(x) + φ(x)diψ(x). note that, if φ is a polynomial of degree n, then (1 − σv)φ(x)/v.x is a polynomial of degree n − 1. moreover, the path of the dunkl process onto a subset of rn is collected by the set s = {x ∈ rn : x.v > 0, ∀v ∈ r+}. finally, dunkl processes are formulated as the markov processes which achieve the dunkl heat equation ∂ ∂t − 1 2 n∑ i=1 d2i = 0. 2.2. fractional calculus. the cauchy formula for frequent integration, to be specific as follows: (inφ)(χ) = 1 (n− 1)! ∫ χ 0 (χ− t)n−1φ(t) dt, drives in an explicit way to a generalization for real n. utilizing the gamma function to take off the discrete nature of the factorial function allows us a natural candidate for fractional usage of the integral operator. (iαφ)(χ) = 1 γ(α) ∫ χ 0 (χ− t)α−1φ(t) dt this operator is well-defined and it is represented to the classical fractional calculus, which is called the riemann-liouville fractional integral operator. it is straightforward to show that the integral operator achieves the semi-group property of fractional differ-integral operators (iα)(iβφ)(χ) = (iβ)(iαφ)(χ) = (iα+βφ)(χ) = 1 γ(α + β) ∫ χ 0 (χ− t)α+β−1φ(t) dt. optimality and duality 77 corresponding to the above fractional integral operator and for a general function φ(χ) and 0 < α < 1, the complete fractional derivative is defined as follows: dαφ(χ) = 1 γ(1 −α) d dχ ∫ χ 0 φ(t) (χ− t)α dt. for the fractional power α < 1, since the gamma function is sloppy for values whose real part is a negative integer with the imaginary part is equal to zero, it is important to employ the fractional derivative after the integer derivative has been accurate. for example, d5/4φ(χ) = d1/4 ( d1φ(χ) ) = d1/4 ( d dχ φ(χ) ) . in general, calculating n-th order derivative over the integral of order (n−α), is given by the formula ad α χφ(χ) = dn dχn ad −(n−α) χ φ(χ) = dn dχn ai n−α χ φ(χ). the riemann-liouville calculus admits a fast converge, historical property, natural generalization and wide applications in almost all science. one of the most property of this calculus is as follows: dαχm = γ(m + 1) γ(m−α + 1) χm−α, m ≥ 0. there is another fractional differential operator called the caputo operator, which is defined as follows: c a d α χφ(χ) = 1 γ(n−α) ∫ χ a φ(n)(τ) dτ (χ− τ)α+1−n . this type of fractional differential operator is applied to find the solutions of fractional differential equations with initial conditions. 2.2.1. tempered fractional calculus. the riemann-liouville tempered fractional derivative is formed as follows [9] d̂α,λφ(χ) = e−λχdα ( eλχφ(χ) ) −λαφ(χ), λ ≥ 0. and the the caputo tempered fractional derivative is formed as follows c a d̂ α,λφ(χ) = d̂α,λ[φ(χ) − n−1∑ i=0 χi i! φ(i)(0)], λ ≥ 0, where φ(i) is the derivative of order i, and n is the upper integer value less than α. when λ = 0, the above equation reduces to the usual formula of the caputo operator. 2.3. tempered-dunkl operator. based on the caputo tempered fractional calculus, assume that the fractional partial derivative is denoted by ca d̂ α,λ. then for x ∈ rnwe receive the tempered dunkl operator as follows: d α,λ i φ(x) = c a d̂ α,λ i φ(x) + n∑ i=1,v∈r+ kv (1 −σv)αφ(x) v.x vi, (2.2) ( i = 1, ...,n, 0 < α < 1 ) , where ca d̂ α,λ i is denoted the fractional derivative with respect the component xi, r+ is a positive subsystem, satisfying for all u ∈ r+, u.v > 0. dunkl operators in the direction of y ∈ rn is defined as follows: dα,λφ(x) = n∑ i=1 yid α,λ i φ(x). our aim is to include the generalized tempered dunkl operator in a class of fractional stochastic differential equations and to study the behavior of the solutions. we have the following properties of the new operator: 78 rabha w. ibrahim proposition 2.1 let φ(x) be analytic function converging in the interval (0,ρ] with the approximate form φ(x) = ∞∑ m=0 am x m+λ, λ > −1. then d α,λ i ( d β,λ j φ(x) ) = d β,λ j ( d α,λ i φ(x) ) , α,β ∈ (0, 1), i = 1, ...,n. proof. in view of theorem 3 in [3] and (2.1) , we have d α,λ i ( d β,λ j φ(x) ) = d α,λ i ( ∂β ∂x β j φ(x) + ∑ v∈r+ kv (1 −σv)βφ(x) v.x vj ) = d β,λ j ( ∂α ∂xαi φ(x) + ∑ v∈r+ kv (1 −σv)αφ(x) v.x vi ) = d β,λ j ( d α,λ i φ(x) ) . proposition 2.2 let φ and ψ be power functions in x. then dαi [φ(x)ψ(x)] = ψ(x)d α i φ(x) + φ(x)d α i ψ(x). proof. by applying the fractional generalization of the leibniz rule of the caputo derivative [8] ∂α ∂xα [φ(x)ψ(x)] = ∞∑ k=0 γ(α + 1) γ(α−k + 1)γ(k + 1) ∂α−kφ(x)∂kψ(x), we conclude the desire result. 2.4. tempered univex function. in this subsection, we generalize the concept of tempered univex function, by using the fractional calculus. let ω be a nonempty subset of rn,η : ω × ω → rm, ξ be an arbitrary point of ω and h : ω → rm,φ : rm → r. definition 2.1 a differential function h is said to be a tempered fractional univex function of order α ∈ (0, 1) in the direction of ξ ∈ ω if for all x ∈ ω, we have η(x,ξ).dα,λ h(x) ≤ φ ( h(x) −h(ξ) ) , where dα,λ h(x) = n∑ i=1 ξid α,λ i h(x), ξ = (ξ1, ...,ξn). the advantage of using the tempered fractional dunkl operator, is that can be acted on multidimensional euclidean spaces as well as it can be defined a parametric family of deformations of the polynomial . therefor, it can be employed in non-linear multi-objective problem minimize ψ(x) = ( ψ1(x), ...,ψm(x) ) subjectto θ(x) ≤ 0, (2.3) where ψ : ω → rm and θ : ω → rp and 0 is the zero vector in rp. the function ψ(x) can be applied in various studies. it can be considered as a utility function over some set of needs (goods), cost function of production presented a fixed quantity produced, growth function and others. definition 2.2 a point ξ ∈ λ := {x ∈ ω : θ(x) ≤ 0} is said to be an efficient solution of (2.3), if there exists no x ∈ λ, such that ψ(x) ≤ ψ(ξ). and it is called a weak efficient solution if ψ(x) < ψ(ξ). next, we define a new class of fractional univex function for the problem (2.3), we denote this class by : (α,ρ,η,ϑ) as follows: optimality and duality 79 definition 2.3 the couple (ψ, θ) is called (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω if for all x ∈ λ such that η1(x,ξ).d α,λ ψ(x) + ρ1‖ϑ(x,ξ)‖2 ≤ φ1 ( ψ(x) − ψ(ξ) ) and η2(x,ξ).d α,λ θ(x) + ρ2‖ϑ(x,ξ)‖2 ≤ −φ2 ( θ(x) − θ(ξ) ) , where η1 : ω × ω → rm, η2 : ω × ω → rp, ϑ : ω × λ → r,φ1 : rm → r, φ2 : rp → r and ρ1,ρ2 ∈ r. we have the following facts: remark 2.1 • if φ1 ( ψ(x) − ψ(ξ) ) ≤ 0 ⇒ η1(x,ξ).dα,λ ψ(x) ≤ −ρ1‖ϑ(x,ξ)‖2 and φ2 ( ψ(x) − ψ(ξ) ) ≥ 0 ⇒ η2(x,ξ).dα,λ ψ(x) ≤ −ρ2‖ϑ(x,ξ)‖2. then the couple (ψ, θ) is called weak pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω. • if φ1 ( ψ(x) − ψ(ξ) ) ≤ 0 ⇒ η1(x,ξ).dα,λ ψ(x) < −ρ1‖ϑ(x,ξ)‖2 and φ2 ( ψ(x) − ψ(ξ) ) ≥ 0 ⇒ η2(x,ξ).dα,λ ψ(x) < −ρ2‖ϑ(x,ξ)‖2. then the couple (ψ, θ) is called strong pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω. 3. results in this section, we investigate some sufficient optimality conditions for a point to be an efficient solution of (2.3) under the tempered (α,λ,ρ,η,ϑ)-type univex. theorem 3.1. let ξ be an initial solution of the multi-objective problem (2.3) and c1 and c2 be two non-negative constants such that (a) θ(ξ) = 0; (b) c1 ( η1(x,ξ).d α,λ ψ(x) ) + c2 ( η2(x,ξ).d α,λ θ(x) ) ≥ 0; (c) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω; (d) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (e) c1ρ1 + c2ρ2 ≥ 0. then ξ is an efficient solution of (2.3). proof. suppose that ξ is not an efficient solution of (2.3), then there exists x ∈ λ such that ψ(x) ≤ ψ(ξ). by the assumptions (a) and (d), we have φ1(ψ(x) − ψ(ξ)) ≤ 0, and φ2(θ(ξ)) ≥ 0. (3.1) in view of the assumption (c), we get c1 ( η1(x,ξ).d α,λ ψ(x) ) < −c1ρ1‖ϑ(x,ξ)‖2 (3.2) and c2 ( η2(x,ξ).d α,λ θ(x) ) ≤−c2ρ2‖ϑ(x,ξ)‖2. (3.3) summing the above inequalities and utilizing (e), we conclude that c1 ( η1(x,ξ).d α,λ ψ(x) ) + c2 ( η2(x,ξ).d α,λ θ(x) ) < − ( c1ρ1 + c2ρ2 ) ‖ϑ(x,ξ)‖2 ≤ 0, which contradicts the assumption (b). hence, ξ is an efficient solution of (2.3). this completes the proof. theorem 3.2. if the following conditions are satisfied: (a) ξ is a weakly efficient solution of (2.3); (b) θ is continuous in ξ; (c) the functions ψ and θ are fractional tempered univex functions of order α ∈ (0, 1), λ ≥ 0 in the direction of ξ ∈ λ. moreover, for some x̄ ∈ λ, we have θ(x̄) < 0. 80 rabha w. ibrahim then there are two constants c1 ≥ 0 and c2 ≥ 0 such that c1 ( η1(x,ξ).d α,λ ψ(x) ) + c2 ( η2(x,ξ).d α,λ θ(x) ) ≥ 0,( x ∈ ω, c2θ(ξ) = 0, η1 : ω × ω → rm, η2 : ω × ω → rp ) . proof. our aim is to show that the system η1(x,ξ).d α,λ ψ(x) < 0, η2(x,ξ).d α,λ θ(x) < 0, has no solution for x ∈ ω. let the system has a solution y ∈ ω. by the assumption (a), we have ψ(ξ + �1y) < ψ(ξ) and θ(ξ + �2y) < θ(ξ), for sufficient small arbitrary constants �1, �2 > 0. now, we let x̄ := ξ + �2y; which implies that x̄ ∈ λ ∩n�2 (ξ) thus by (b) and (c), we have θ(ξ + �2y) = θ(x̄) < 0; which contradicts (a), where ξ is a weak solution. therefore, the above inequalities are non-negative. hence, in view of (c) these are two constants c1 and c2 satisfy the inequality c1 ( η1(x,ξ).d α,λ ψ(x) ) + c2 ( η2(x,ξ).d α,λ θ(x) ) ≥ 0, with the property c2θ(ξ) = 0. this completes the proof. next, we consider the dual problem of (2.3) as follows: max ψ(χ) = ( ψ1(χ), ...,ψm(χ) ) subjectto c1 ( η1(x,χ).d α,λ ψ(x) ) + c2 ( η2(x,χ).d α,λ θ(x) ) ≥ 0, c2θ(χ) ≥ 0, (3.4) where χ ∈ ω, c1 and c2 be two non negative constants. theorem 3.3. let x,χ be initial solutions of the multi-objective problems (2.3) and (3.4) respectively. if (a) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω; (b) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (c) c1ρ1 + ρ2 ≥ 0; then ψ(x) � ψ(χ). proof. suppose that ψ(x) ≤ ψ(χ). since c1ρ1 + ρ2 ≥ 0 then by (b), we obtain φ1(ψ(x) − ψ(χ)) ≤ 0 φ2(θ(χ)) ≥ 0. in virtue of the assumption (a) the above inequalities yield( η1(x,χ).d α,λ ψ(χ) ) < −ρ1‖ϑ(x,χ)‖2( η2(x,χ).d α,λ θ(χ) ) ≤−ρ2‖ϑ(x,χ)‖2, consequently, we obtain c1 ( η1(x,ξ).d α,λ ψ(x) ) < −c1ρ1‖ϑ(x,χ)‖2 and c2 ( η2(x,χ).d α,λ θ(x) ) ≤−ρ2‖ϑ(x,χ)‖2. summing the above inequalities and utilizing (c), we conclude that c1 ( η1(x,χ).d α,λ ψ(χ) ) + c2 ( η2(x,χ).d α,λ θ(χ) ) < − ( c1ρ1 + ρ2 ) ‖ϑ(x,χ)‖2 ≤ 0, which contradicts the assumption (c). this completes the proof. theorem 3.4. let x0 and χ0 be initial solution for the problems (2.3) and (3.4) respectively. if ψ(x0) = ψ(χ0) then the (weak or strong) duality problems (2.3) and (3.4) has efficient solutions x0 and χ0 respectively. optimality and duality 81 proof. suppose that x0 is not efficient for (2.3), then for some x ∈ λ ψ(x) ≤ ψ(x0) = ψ(χ0), which contradicts weak (strong) duality theorems as χ0 is initial solution for (3.4). therefore, x0 is efficient for (2.3). similarly χ0 is efficient solution for (3.4). hence the proof. theorem 3.5. let χ0 be an initial solution of the multi-objective problem (3.4) and c1 and c2 be two non negative constants such that (a) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω; (b) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (c) c1ρ1 + ρ2 ≥ 0. then χ0 is an efficient solution of (3.4). proof. suppose that χ0 is not an efficient solution of (3.4), then there exists x0 ∈ λ such that ψ(x0) ≤ ψ(χ0). now going on as in theorem 3.3, we have a contradiction. hence, χ0 is an efficient solution of (3.4). theorem 3.6. let x0,χ0 be initial solutions of the multi-objective problems (2.3) and (3.4) respectively. if (a) ψ(x0) ≤ ψ(χ0); (b) the couple (ψ, θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ ω; (c) u ≤ 0 ∈ rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ rp ⇒ φ2(v) ≥ 0; (d) c1ρ1 + ρ2 ≥ 0; then x0 = χ0. proof. suppose that x0 6= χ0. since χ0 is an initial solution for (3.4) then by (a) and (c), we have φ1(ψ(x0) − ψ(χ0)) ≤ 0 φ2(θ(χ0)) ≥ 0. in virtue of the assumption (b) the above inequalities imply that( η1(x0,χ0).d α,λ ψ(χ0) ) < −ρ1‖ϑ(x0,χ0)‖2( η2(x0,χ0).d α,λ θ(χ0) ) ≤−ρ2‖ϑ(x0,χ0)‖2, which on summing yields c1 ( η1(x0,χ0).d α,λ ψ(χ0) ) + c2 ( η2(x0,χ0).d α,λ θ(χ0) ) < − ( c1ρ1 + ρ2 ) ‖ϑ(x0,χ0)‖2 ≤ 0, which contradicts to initially of χ0. then we obtain x0 = χ0. this completes the proof. 4. simulation in this section, we illustrate a simulation to show how the tempered fractional calculus is effected on the multi-objective functions. let ψ, θ : r → r2 such that ψ(x) = ( x2,x3 ) ; θ(x) = ( x,x2 ) . our aim is to show that the couple (ψ, θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. to determine the fractional dunkl operator on these functions, we shall introduce three cases depending on the value of kv for v = 1. 82 rabha w. ibrahim 4.1. case (i) λ = 0, kv = 0. the tempered fractional dunkl operator acts on the functions ψ and θ as follows: dα,λψ(x) = ( γ(3) γ(3 −α) x2−α, γ(4) γ(4 −α) x3−α ) ; dα,λθ(x) = ( γ(2) γ(2 −α) x1−α, γ(3) γ(3 −α) x2−α ) . now, by letting η1,2(x,ξ) = (x− ξ 2 , x− ξ 2 ) , ξ = 0, we have η1(x,ξ).d α,λψ(x) = x3−α γ(3 −α) + 3x4−α γ(4 −α) ; η2(x,ξ).d α,λθ(x) = x2−α 2γ(2 −α) + x3−α γ(3 −α) . consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain ‖ϑ(x,ξ)‖2 = x4, ξ = 0. it is clear that ψ(ξ) = ψ(0) = (0, 0); θ(ξ) = θ(0) = (0, 0), then by assuming φ1 ( ψ(x) − ψ(ξ) ) = 5x, φ2 ( θ(x) − θ(ξ) ) = −5x, x ∈ [0, 1], we conclude that η1(x,ξ).d α ψ(x) + ρ1‖ϑ(x,ξ)‖2 = x3−α γ(3 −α) + 3x4−α γ(4 −α) + x4 < 5x, x ∈ [0, 1] = φ1 ( ψ(x) − ψ(ξ) ) (4.1) and η2(x,ξ).d α,λ θ(x) + ρ2‖ϑ(x,ξ)‖2 = x2−α 2γ(2 −α) + x3−α γ(3 −α) + x4 < 5x, x ∈ [0, 1] = −φ2 ( θ(x) − θ(ξ) ) (4.2) hence, the couple (ψ, θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. table 1 shows that for various values of α ∈ (0, 1), the outcomes yield the tempered fractional univexty of the couple (ψ, θ). table 1. fractional multi-objective function, kv = 0 (α) eq. (4.1) eq. (4.2) 0.25 1.6 1.9 0.5 2.6 2.4 0.75 3.1 3.2 to apply the conditions of theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 +c2ρ2 = 2 > 0 with the inequalities (4.1) and (4.2). this leads that all the conditions of theorem 3.1 are achieved and hence, ξ = 0 is an efficient solution. note that if we let φ1(y ) = 3y and φ2(y ) = −3y, the couple (ψ, θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. 4.2. case (ii) λ = 0, kv = 1. to evaluate the tempered fractional dunkl operator, a calculation implies that σx2 = x 2 − 2 v.x2 v.v = −x2, σx3 = −x3. therefore, one can attain η1(x,ξ).d α,λψ(x) = x3−α γ(3 −α) + x(2x2)α 2 + 3x4−α γ(4 −α) + x(2x3)α 2 optimality and duality 83 and η2(x,ξ).d αθ(x) = x2−α 2γ(2 −α) + x(2x)α 2 + x3−α γ(3 −α) + x(2x2)α 2 . table 2 shows the evaluation of the tempered fractional multi-objective functions for different values of α. table 2. fractional multi-objective function, kv = 1 (α) eq. (4.1) eq. (4.2) 0.25 2.7 2.9 0.5 5 3.8 0.75 4.7 4.8 thus, we conclude that the conditions of theorem 3.1 are satisfied when c1 = c2 = 1; such that c1ρ1 + c2ρ2 = 2 > 0 with the inequalities (4.1) and (4.2). consequently, we obtain ξ = 0 is an efficient solution. 4.3. case (iii) λ = 0, kv = 2. by applying (2.2), we have η1(x,ξ).d α,λψ(x) = x3−α γ(3 −α) + x(2x2)α + 3x4−α γ(4 −α) + x(2x3)α and η2(x,ξ).d α,λθ(x) = x2−α 2γ(2 −α) + x(2x)α + x3−α γ(3 −α) + x(2x2)α. table 3 shows the evaluation of the tempered fractional multi-objective functions for different values of α. it is clear that the couple (ψ, θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. it is of (α,λ,ρ,η,ϑ)type univex at ξ ∈ [0, 1], when α ∈ (0, 0.25]. hence, theorem 3.1 can be applied only for this value of α. table 3. fractional multi-objective function, kv = 2 (α) eq. (4.1) eq. (4.2) 0.25 3.5 4.1 0.5 5.4 5.2 0.75 6.4 5.5 4.4. case (iv) λ = 1, kv = 0. the tempered fractional dunkl operator acts on the functions ψ and θ as follows: dα,λψ(x) = ( γ(3) γ(3 −α) x2−α + x2ex, γ(4) γ(4 −α) x3−α + x3ex ) ; dα,λθ(x) = ( γ(2) γ(2 −α) x1−α + xex, γ(3) γ(3 −α) x2−α + x2ex ) . now, by letting η1,2(x,ξ) = (x− ξ 2 , x− ξ 2 ) , ξ = 0, we have η1(x,ξ).d α,λψ(x) = x3−α γ(3 −α) + x3ex 2 + 3x4−α γ(4 −α) + x4ex 2 ; η2(x,ξ).d α,λθ(x) = x2−α 2γ(2 −α) + x2ex 2 + x3−α γ(3 −α) + x3ex 2 . consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain ‖ϑ(x,ξ)‖2 = x4, ξ = 0. it is clear that ψ(ξ) = ψ(0) = (0, 0); θ(ξ) = θ(0) = (0, 0), 84 rabha w. ibrahim then by assuming φ1 ( ψ(x) − ψ(ξ) ) = 7x, φ2 ( θ(x) − θ(ξ) ) = −7x, x ∈ [0, 1], we conclude that η1(x,ξ).d α ψ(x) + ρ1‖ϑ(x,ξ)‖2 = x3−α γ(3 −α) + x3ex 2 + 3x4−α γ(4 −α) + x4ex 2 + x4 < 7x, x ∈ [0, 1] = φ1 ( ψ(x) − ψ(ξ) ) (4.3) and η2(x,ξ).d α,λ θ(x) + ρ2‖ϑ(x,ξ)‖2 = x2−α 2γ(2 −α) + x2ex 2 + x3−α γ(3 −α) + x3ex 2 + x4 < 7x, x ∈ [0, 1] = −φ2 ( θ(x) − θ(ξ) ) (4.4) hence, the couple (ψ, θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. table 1 shows that for various values of α ∈ (0, 1), the outcomes yield the tempered fractional univexty of the couple (ψ, θ). table 4. , kv = 0,λ = 1 (α, 1) eq. (4.3) eq. (4.4) 0.25 4.3 4.6 0.5 5.4 5.1 0.75 5.8 5.9 to apply the conditions of theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 +c2ρ2 = 2 > 0 with the inequalities (4.3) and (4.4). this leads that all the conditions of theorem 3.1 are achieved and hence, ξ = 0 is an efficient solution. note that if we let φ1(y ) = 5y and φ2(y ) = −5y, the couple (ψ, θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. also, the case λ = 1 and kv = 1 does not imply the univex function when φ1(y ) = 7y and φ2(y ) = −7y. 5. conclusion this effort is generalized, for the first time, two important concepts in science. the dunkl tempered fractional operator and the tempered univex function, by utilizing the caputo tempered fractional differential operator. these two generalizations are combined to deliver the fractional multi-objective problems. we studied the duality cases by minimize and maximize the desired function in the rn. simulation is provided to apply the existing solutions. it has been found that the fractional case converges to the ordinary case. these problems can be employed in many studies not only in mathematics, but also in the economy; such as the utility function the cost function and the entropy function. references [1] c.f. dunkl, differential-difference operators associated to reflection groups, trans. amer. math. soc. 311 (1) (1989), 167–183. [2] m. rosler and m. voit, markov processes related with dunkl operators. adv. appl. math. 21(4)(1998) 575–643. [3] r. w. ibrahim, new classes of analytic functions determined by a modified differential-difference operator in a complex domain, karbala int. j. modern sci. 3 (2017) 53–58. [4] k.s. miller, b. ross, an introduction to the fractional calculus and fractional differential equations, john wily& sons (1993). [5] a.a. kilbas, h.m. srivastava, j.j. trujiilo, theory and applications of fractional differential equations. amsterdam, netherlands: elsevier (2006). [6] b. baeumer, m.m. meerschaert, tempered stable levy motion and transient super-diffusion, j. comput. appl. math. 233 (2010) 243–2448. [7] r. w. ibrahim, fractional calculus of multi-objective functions & multi-agent systems. lambert academic publishing, saarbrcken, germany 2017. [8] v.e. tarasov, leibniz rule and fractional derivatives of power functions. j. comput. nonlinear dyn. 11 (3) (2016), art. id 031014–4. optimality and duality 85 [9] m.m. meerschaert, a. sikorskii, stochastic models for fractional calculus, de gruyter, berlin, 2012. faculty of computer science and information technology, university of malaya, malaysia corresponding author: rabhaibrahim@yahoo.com 1. introduction 2. handling 2.1. dunkl operator 2.2. fractional calculus 2.3. tempered-dunkl operator 2.4. tempered univex function 3. results 4. simulation 4.1. case (i) =0, kv=0. 4.2. case (ii) =0, kv=1. 4.3. case (iii) =0, kv=2. 4.4. case (iv) =1, kv=0. 5. conclusion references international journal of analysis and applications volume 16, number 6 (2018), 793-808 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-793 various kinds of freeness in the categories of krasner hypermodules hossein shojaei∗ and reza ameri school of mathematics, statistics and computer science, college of science, university of tehran, p.o. box 14155-6455, tehran, iran ∗corresponding author: h shojaei@ut.ac.ir abstract. the purpose of this paper is to study the concept of freeness in the categories of krasner hypermodules over a krasner hyperring. in this regards first we construct various kinds of categories of hypermodules based on various kinds of homomorphisms of hypermodules, such as homomorphisms, good homomorphisms, multivalued homomorphisms and etc. then we investigate the notion of free hypermodule in these categories. this leads us to introduce different types of free, week free, �∗-free and fundamental free hypermodules and obtain the relationship among them. 1. introduction the concept of hyperstructure is the generalization of the concept of algebraic structure. as a matter of fact, the hyperstructures are more natural and general than the algebraic structures. for the first time, hypergroups, as a suitable generalization of groups, were defined by marty in 1934 [11]. recently, many hyperstructures, for example, hypergroups, hyperrings, hyperfield, hypermodules and hypervector spaces, have been introduced and studied by many authors, e.g., [1], [3], [4], [5], [6], [7], [9], and [13]. for the first time, the concepts of hyperring and hyperfield were introduced by krasner in connection with his work on valued fields. one of the most important hyperstructures satisfying the module-like axioms as a generalization of module is a type of hypermodule over a krasner hyperring that we call it krasner hypermodule (see [14]). received 2017-11-07; accepted 2018-01-17; published 2018-11-02. 2010 mathematics subject classification. 18d35. key words and phrases. krasner hyperring; krasner hypermodule; free hypermodule. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 793 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-793 int. j. anal. appl. 16 (6) (2018) 794 interested readers can find a wide generalization of krasner hypermodules in [15]. in this paper, we study these hyperstructures and focus on various freeness for a krasner hypermodule. next section is a summary and reminder of [14]. 2. preliminaries we start this section with some basic and fundamental concepts of category theory, and then we proceed to recall some requirements from hyperstructures theory. definition 2.1. a category denoted by c consists of (1) a class of objects: a,b,c,. . . (2) a class of morphisms or arrows: f,g,h, . . . with the following data: • given morphisms f : a −→ b and g : b −→ c, that is, with: cod(f) = dom(g) there is given a morphism: g ◦f : a −→ c called the composition of morphisms f and g. • associativity: h◦ (g ◦f) = (h◦g) ◦f for all f : a −→ b, g : b −→ c and h : a −→ c. • identity: f ◦ ida = f = idb ◦f for all f : a −→ b. notation 2.1. the classes of objects and morphisms of a category are denoted by ob(c) and mor(c), respectively. the class of all morphisms from a to b of category c is denoted by morc(a,b). note that a morphism f : a −→ b in 2.1 is not necessarily a function from a to b. recall that a zero object in an arbitrary category c is an object denoted by 0 such that |morc(m, 0)| = |morc(0,m)| = 1 for every m ∈ ob(c). the morphism a −→ 0 −→ b of morc(a,b) in a category c is called the zero morphism (see [2] or [8]). throughout this paper, p(x) denotes the set of all subsets of x and p∗(x) = p(x) \ {∅}. here, sets denotes the category of sets as objects with functions between sets as morphisms. now we state some basic definitions related to hyperstructures theory. let h be a non-empty set. then h together with the map · : h ×h −→ p∗(h) (a,b) 7→ a · b denoted by (h, ·) is called a hypergroupoid and · is called a hyperproduct or hyperoperation on h. let a,b ⊆ h. the hyperproduct a ·b is defined as a ·b = ⋃ (a,b)∈a×b a · b. int. j. anal. appl. 16 (6) (2018) 795 if there is no confusion, for simplicity {a}, a ·{b} and {a}·b are denoted by a, a ·b and a ·b, respectively. also we use ab instead of a · b for a,b ∈ h. a non-empty set s together with the hyperoperation ·, denoted by (s, ·) is called a semihypergroup if for all x,y,z ∈ s, (x ·y) ·z = x · (y ·z). a semihypergroup (h, ·) satisfying x ·h = h ·x = h for every x ∈ h, is called a hypergroup. let x be an element of semihypergroup (h, +) (resp., (h, ·)) such that e+y = y+e = y (resp., e·y = y·e = y). then x is called a scalar identity (resp., unit). every scalar identity or scalar unit in a semihypergroup h is unique. we denote the scalar identity (resp., unit) of h by 0h (resp., 1h). let 0h (resp., 1h) be the scalar identity (resp., unit) of hypergroup (h, +) (resp., (h, ·)) and x ∈ h. an element x′ ∈ h is called an inverse of x in (h, +) (resp., (h, ·)) if 0h ∈ x+x′∩x′+x (resp., 1h ∈ x·x′∩x′·x). a semihypergroup with a scalar identity is called a hypermonoid. a non-empty set h together with the hyperoperation + is called a canonical hypergroup if the following axioms hold: (1) (h, +) is a semihypergroup (associativity); (2) (h, +) is commutative (commutativity); (3) there is a scalar identity 0h (existence of scalar identity); (4) for every x ∈ h, there is a unique inverse denoted by −x such that 0h ∈ x + (−x), which for simplicity we write 0h ∈ x−x (existence of inverse); (5) ∀x,y,z ∈ h : x ∈ y + z =⇒ y ∈ x−z (reversibility). definition 2.2. a non-empty set r together with the hyperoperation + and the operation · is called a krasner hyperring if the following axioms hold: (1) (r, +) is a canonical hypergroup; (2) (r, ·) is a semigroup including 0r as a bilaterally absorbing element, that is 0r ·x = x · 0r = 0r for all x ∈ r; (3) (y + z) ·x = (y ·x) + (z ·x) and x · (y + z) = x ·y + x ·z for all x,y,z ∈ r. we say r has a unit (element) 1r when 1r ·r = r · 1r = r for all r ∈ r. example 2.1. let (r, +, ·) be a ring and n a normal subgroup of semigroup (r\{0}, ·). let r′ = r n be the set of classes of the form x̄ = x ·n. if for all x̄, ȳ ∈ r̄, we define x̄ +′ ȳ = {z̄| z ∈ x̄ + ȳ}, and x̄ ·′ ȳ = x ·y, then (r, +′, ·′) is a krasner hyperring. int. j. anal. appl. 16 (6) (2018) 796 now we state the concept of hypermodule over a hyperring. one of the most important and well-behaved classes of hypermodules is a class induced by the structure of a krasner hyperring. to study such hyperstructures, we start with the following concept. definition 2.3. let x and y be two non-empty sets. a map ∗ : x ×y −→ y sending (x,y) to x∗y ∈ y is called a left external multiplication on y . if u ⊆ x and v ⊆ y , then we define u∗v := ∪v∈v u∗v and u ∗v := ∪u∈uu∗v. analogously, a right external multiplication on y is defined by ∗ : y ×x −→ y sending (y,x) to y∗x ∈ y . definition 2.4. let (r, +, ·) be a krasner hyperring. a canonical hypergroup (a, +) together with the left external multiplication ∗ : r×a −→ a on a is called a left krasner hypermodule over r if for all r1,r2 ∈ r and for all a1,a2 ∈ a the following axioms hold: (1) r1 ∗ (a1 + a2) = r1 ∗a1 + r1 ∗a2; (2) (r1 + r2) ∗a1 = r1 ∗a1 + r2 ∗a1; (3) (r1 ·r2) ∗a1 = r1 ∗ (r2 ∗a1); (4) 0r ∗a1 = 0a. remark 2.1. (i) if a is a left krasner hypermodule over a krasner hyperring r, then we say that a is a left krasner r-hypermodule. clearly, a right krasner r-hypermodule is defined with the map ∗ : a×r −→ a possessing the smilar properties. (ii) if r is a krasner hyperring with 1r and a is a krasner r-hypermodule satisfying 1r ∗a = a (resp. a∗ 1r = a) for all a ∈ a, then a is said a unitary left (resp. right) krasner r-hypermodule. (iii) throughout the paper, for convenience, by hyperring r we mean a krasner hyperring with 1r and by r-hypermodule a we mean a unitary left krasner r-hypermodule unless otherwise stated. definition 2.5. a non-empty subset b of an r-hypermodule a is said to be an r-subhypermodule of a if b is an r-hypermodule itself, that is for all x,y ∈ b and all r ∈ r, x−y ⊆ b and r ∗x ∈ b. proposition 2.1. [14, remark 3.2] (i) in every (unitary) r-hypermodule a, (−1r) ∗a = −a for every a ∈ a. (ii) in every hyperring r, (−1r) ·r = −r for every r ∈ r. remark 2.2. let a be an r-hypermodule, r,s ∈ r and a ∈ a. in the sequel, when there is no confusion, we use rs and ra instead of r ·s and r ∗a, respectively. unlike the category of modules, there are various types of homomorphisms in the categories of hypermodules. int. j. anal. appl. 16 (6) (2018) 797 definition 2.6. let a and b be two r-hypermodules. a function f : a −→ b that satisfies the conditions: (i) f(x + y) ⊆ f(x) + f(y); (ii) f(rx) = rf(x); for all r ∈ r and all x,y ∈ a, is said to be an (inclusion) r-homomorphism from a into b. remark 2.3. if in (i) of definition 2.6 the equality holds, then f is called a strong (or good) r-homomorphism. the category whose objects are all r-hypermodules and whose morphisms are all r-homomorphisms is denoted by rhmod. the class of all r-homomorphisms from a into b is denoted by homr(a,b). also, rshmod is the category of all r-hypermodules whose morphisms are strong r-homomorphisms. the class of all strong r-homomorphisms from a into b is denoted by homsr(a,b). it is easy to see that rshmod is a subcategory of rhmod, and we write rshmod �rhmod and read rshmod is a subcategory of rhmod. so far we have considered the morphisms or arrows, as usual, the functions between objects. but one can consider a morphism from a to b as a function from a into p∗(b) called a multivalued function from a to b. considering multivalued functions between sets, we have the following definition: definition 2.7. category of hypersets denoted by hsets is a category with the following data: (1) ob(hsets) = ob(sets), (2) mor(hsets) = the class of all multivalued functions between objects, that the composition g ◦f is defined as the following: (g ◦f)(a) = ⋃ b∈f(a) g(b), ∀a ∈ a, (2.1) and an identity morphism for an object a is ida(x) = {x} for all x ∈ a. now we are ready to define a generalization of usual morphisms of rhmod. definition 2.8. if a and b are two r-hypermodules, then multivalued function f from a into b is a mapping f : a −→ p∗(b) satisfying the following conditions: (i) f(x + y) ⊆ f(x) + f(y); (ii) f(rx) = rf(x); for all r ∈ r and all x,y ∈ a, is said to be a multivalued r-homomorphism, for short rmv-homomorphism. remark 2.4. in (i) of definition 2.8, if the equality holds, then f is called a strong (or good) multivalued r-homomorphism, for short an rsmv-homomorphism. notation 2.2. the class of all rmv-homomorphisms (resp., rsmv-homomorphisms) from a into b is denoted by homr(a,b) (resp., hom s r(a,b)). int. j. anal. appl. 16 (6) (2018) 798 proposition 2.2. [14, remark 3.9] (i) for every f ∈ homr(a,b), f(0a) = 0b. (ii) for every f ∈ homr(a,b), f(−x) = −f(x) and f(x−y) = f(x) −f(y). let f ∈ homr(a,b) and h ∈ homr(b,c). the composition h◦f is defined as equation 2.1. also, for every r-hypermodule a, the r-homomorphism ida with definition ida(x) = {x} for all x ∈ a is the identity morphism as before. hereafter, rhmod (resp., rshmod) denotes the category whose objects are all r-hypermodules and whose morphisms from a to b are all rmv-homomorphisms (resp., rsmvhomomorphisms) from a into b. clearly, rshmod is a subcategory of rhmod, i.e., rshmod �rhmod. remark 2.5. (i) hereafter, we identify a singleton x = {a} by its element a. also, we sometimes write f(a) = b instead of f(a) = {b}. so every single-valued morphism f ∈ homr(a,b) (resp., f ∈ homsr(a,b)) is an element of homr(a,b) (resp., hom s r(a,b)), and conversely, every element of homr(a,b) (resp., hom s r(a,b)) can be considered as an element of homr(a,b) (resp., hom s r(a,b)), so rhmod �rhmod (resp., rshmod �rshmod). (ii) let f,g ∈ homr(a,b). define the relation ≤ on homr(a,b) in which f ≤ g means f(x) ⊆ g(x) for all x ∈ a. clearly (homr(a,b),≤) is a poset. for convnience and distinguishing, we call rhmod and rshmod primary categories of krasner rhypermodules. also, rhmod and rshmod are called secondary categories of krasner r-hypermodules. 3. freeness of hypermodules as it is well-known free objects play an important role in the study of modules theory. in [12] it was shown that free object does not exist in the category of hypergroups. also, in [10] the notion of free hypermodules in the category of krasner hypermodules was introduced. however, it is not clear that whether this definition is suitable in view point of category theory. here we give various types of freeness in the categories of r-hypermodules and investigate the relationship between them. fix a hyperring (r, +, ·). let u(r) denote the set of all expressions of the form ∑ i∈i ( ∏ j∈ji rj) in which rj ∈ r where i and all ji’s are finite. the relation γ is defined on r is defined as follows: for all x,y ∈ r, xγy ⇐⇒∃u ∈u(r) : x,y ∈ u. the transitive closure of the relation γ is called the fundamental relation of r denoted by γ∗. let γ∗(r) denote the equivalence class containing r ∈ r. then it is shown that r γ∗ with the sum ⊕ and product ⊗ is a ring as follows: int. j. anal. appl. 16 (6) (2018) 799 for all x,y ∈ r, γ∗(x) ⊕γ∗(y) = γ∗(z) ∀z ∈ γ∗(x) + γ∗(y); γ∗(x) ⊗γ∗(y) = γ∗(x ·y). the fundamental relation γ∗ is the smallest equivalence relation such that r γ∗ is a ring. the ring r γ∗ is called the fundamental ring of r. also, the fundamental relation of an r-hypermodule a can be defined similar to above denoted by �∗a that a �∗ a is a fundamental module over the ring r γ∗ with operations: �∗a(x) ⊕ � ∗ a(y) = � ∗ a(z) ∀z ∈ � ∗ a(x) + � ∗ a(y); γ∗(r) � �∗a(x) = � ∗ a(r ∗x), for all x,y ∈ a and r ∈ r. the fundamental relation �∗a is the smallest equivalence relation such that a �∗ a is a module over the ring r γ∗ . (for more details, see [16] and [17]). now we introduce the following concept: definition 3.1. an r-hypermodule f is said to be free on x ⊆ f if for every r-hypermodule a and for any morphism f : x −→ a in hsets, there exists a unique f̄ ∈ homr(f,a) such that f̄ ◦ i = f in which i = idf |x, i.e., the following diagram commutes: x i // f �� f ∃!f̄~~ a remark 3.1. in [10] ch. g. massouros defined a free r-hypermodule differently. in the sense of massouros, an r-hypermodule f is said to be free on x ⊆ f if x generates f (see definition 3.5) and for every rhypermodule a and for an arbitrary morphism f : x −→ a in sets, there exists f̄ ∈ homr(f,a) such that f̄(x) = {f(x)} for every x ∈ x, i.e., the following diagram commutes: x i // f �� f ∃f̄~~ a by this definition, the morphism f̄ is not necessarily unique rmv-homomorphism, but in [10], it was shown that f̄ is a maximum rmv-homomorphism, that is f̄ is a maximum element in the poset (homr(f,a),≤)) such that f̄ ◦ i = f. clearly, a free r-hypermodule on x ⊆ f based on this definition, is not really free on x in rhmod or rhmod. in the following, we introduce the concept of weak freeness over hsets or sets motivated by massouros definition. int. j. anal. appl. 16 (6) (2018) 800 definition 3.2. an r-hypermodule f is said to be weak free on x ⊆ f over hsets (resp., sets) if for every r-hypermodule a and for every morphism f : x −→ a in hsets (resp., sets), there exists a maximum f̄ ∈ homr(f,a) such that f̄ ◦ i = f in which i = idf |x, i.e., the diagram x i // f �� f ∃f̄~~ a commutes. remark 3.2. (i) every free r-hypermodule on x is weak free on x over hsets. (ii) every free r-hypermodule on x in the sense of massouros is really weak free on x over sets. notation 3.1. let a be an r-hypermodule and x ⊆ a and set �∗a(x) := ∪x∈x� ∗ a(x). for x = {x}, we write �∗a(x) instead of � ∗ a({x}). definition 3.3. two morphisms f and g in homr(a,b) is said to be � ∗-equivalent, and write f ∼�∗ g if and only if �∗b(f(x)) = � ∗ b(g(x)) for every x ∈ a. clearly ∼�∗ is an equivalence relation on homr(a,b). we denote the equivalence class of f with respect to ∼�∗ by [f]. definition 3.4. an r-hypermodule f is said to be �∗-free on x ⊆ f over hsets (resp., sets) if for every r-hypermodule b and for every morphism f : x −→ b in hsets (resp., sets), there exists an �∗-unique f̄ ∈ homr(f,b) such that f̄ ◦ i = f in which i = idf |x, i.e., if there exists f̄′ ∈ homr(f,b) such that f̄′ ◦ i = f, then [f] = [g]. now we recall the notion of a generating set. definition 3.5. let r be a hyperring not necessarily with 1r and a be an r-hypermodule and x ⊆ a. 〈x〉 denotes the smallest r-subhypermodule of a containing x or the intersection of all r-subhypermodules of a containing x. notation 3.2. let a be an r-hypermodule. then for x ∈ a and m ∈ z, mx =   x + x + · · · + x︸ ︷︷ ︸ m times x if n > 0 0a if n = 0 −x−x−···−x︸ ︷︷ ︸ −m times x if n < 0. int. j. anal. appl. 16 (6) (2018) 801 proposition 3.1. for a subset x of a not necessarily unitary r-hypermodule a, 〈x〉 is the set {a ∈ m∑ i=1 rixi + n∑ i=1 nixi + k∑ i=1 ki(xi − xi) | ri ∈ r, xi ∈ x, m,n,k,ki ∈ n, ni ∈ z}. proof. it is clear to straightforward. � definition 3.6. the set x is said to be a generating set for an r-hypermodule a, or x generates a, if a = 〈x〉. here, a is called finitely generated if it has a finite generating set. let x = {x}. for simplicity, we use 〈x〉 instead of 〈x〉. it is easy to see that 〈x〉 = {a ∈ rx + mx + n∑ i=1 ni(x−x) | r ∈ r, m ∈ z, n,ni ∈ n}. let rx = {rx | r ∈ r, x ∈ a}. remark 3.3. let r be with identity 1r and a be a unitary r-hypermodule. then (i) 〈x〉 = rx. indeed, since mx = x + x + · · · + x︸ ︷︷ ︸ m times x = 1r ∗x + 1r ∗x + · · · + 1r ∗x︸ ︷︷ ︸ m times x = (1r + 1r + · · · + 1r︸ ︷︷ ︸ m times 1r ) ∗x ⊆ rx and (by proposition 2.1 (i)) x−x = x + (−x) = 1r ∗x + ((−1r) ∗x) = (1r + (−1r)) ∗x = (1r − 1r) ∗x ⊆ rx, we have 〈x〉 = rx. (ii) letting x = {xi}i∈i ⊆ a, a = 〈x〉 if and only if for every a ∈ a, there exists a finite j ⊆ i such that a ∈ ∑ j∈j rjxj which rj ∈ r and xj ∈ x. definition 3.7. let a be an r-hypermodule and x ⊆ a. x is said linearly independent if for all n ∈ n and all x1,x2, . . . ,xn ∈ x, 0a ∈ n∑ i=1 rixi implies r1 = r2 = · · · = rn = 0r. definition 3.8. let f be an r-hypermodule and x be a generating set which is linearly independent. then x is called a basis for f . remark 3.4. the empty set is linearly independent and is a basis for the trivial r-hypermodule 0 = {0} (based on definition 3.5). if x is a basis for f, then for every a ∈ f there are x1,x2, . . . ,xn ∈ x and unique r1,r2, . . . ,rn ∈ r such that a ∈ n∑ i=1 rixi. in order to show the uniqueness of ri’s, let a ∈ m∑ i=1 r′ixi for some m ∈ n. without loss of generality, assume m = n. then 0f ∈ a−a ∈ n∑ i=1 rixi − n∑ i=1 r′ixi int. j. anal. appl. 16 (6) (2018) 802 =⇒ 0f ∈ n∑ i=1 (ri −r′)xi =⇒∃ci ∈ ri −r′i : 0f ∈ n∑ i=1 cixi since x is a basis, ci = 0r. so 0r ∈ ri −r′i implies ri = r ′ i. every ri (i = 1, 2, . . . ,n) is called the ith coordinate of a in r. in fact, every coordinate of a can be considered as a function from f into r mapping a to an appropriate ri denoted by fi. indeed, fi(xj) = 1r if i = j, fi(xj) = 0r if i 6= j and fi(a) = ri for a ∈ n∑ i=1 rixi. every fi is called ith coordinating function of a. clearly for all a,b ∈ f and all r ∈ r, we have fi(a + b) ⊆ fi(a) + fi(b) and fi(ra) = rfi(a). for an arbitrary morphism f : x −→ b in hsets, define the morphism f̄ : f −→ b f̄(a) = ∑ i fi(a) ∗f(xi) (3.1) that the summation is indeed taken finite for an appropriate n, and fi(a) = ri is the ith coordinate of a. since x is a basis and the ith coordinate of a is uniquely determined, so f̄ is well-defined. it is easy to see that f̄(xi) = f(xi) for all xi ∈ x. f̄(a + b) = ⋃ c∈a+b f̄(c) = ⋃ c∈a+b (∑ i (fi(c) ∗f(xi)) ) (by equation 3.1) ⊆ ∑ i ( ⋃ c∈a+b {fi(c)} ) ∗f(xi) ⊆ ∑ i fi(a + b) ∗f(xi) ⊆ ∑ i fi(a) ∗f(xi) + ∑ i fi(b) ∗f(xi) = f̄(a) + f̄(b) (by equation 3.1). note that the first inclusion is obtained from fi(a + b) ⊆ fi(a) + fi(b). clearly f̄(ra) = rf̄(a) for all r ∈ r. so f is an rmv-homomorphism and thus f̄ ∈ homr(f,b). now let f̄′ ∈ homr(f,b) be another rmvhomomorphism with f̄′(xi) = f̄(xi) for every xi ∈ x. then for every a ∈ f, we have a ∈ ∑ i fi(a) ∗xi and thus f̄′(a) ⊆ f̄′ (∑ i (fi(a) ∗xi) ) = ∑ i fi(a) ∗ f̄′(xi) = ∑ i fi(a) ∗f(xi) = f̄(a). (3.2) hence f̄′ ≤ f̄. thus we have the following statement: int. j. anal. appl. 16 (6) (2018) 803 theorem 3.1. let f be an r-hypermodule with basis x. then f is weak free on x over hsets. theorem 3.2. let f be an r-hypermodule with basis x. if f is weak free on x over sets, then it is �∗-free on x over sets. proof. following the proof of theorem 3.1, consider equation 3.2. if f is a morphism of set, then f̄′(a) and f̄(a) are contained in the finite linear combination ∑ i fi(a) ∗f(xi) in which fi(a) ∈ r and f(xi) ∈ b. thus �∗b(f̄ ′(a)) = �∗b(f̄(a)). hence f is � ∗-free on x. � definition 3.9. if a = ∑ i∈i ai in which every ai is an r-subhypermodule of a and aj ∩ ∑ i 6=j ai = {0a} for every j (j ∈ j), then we write a = ⊕i∈iai, and a is said to be the direct sum of {ai}i∈i. proposition 3.2. let x = {xi}i∈i be a subset of an r-hypermodule f which i is an index set. if for every a ∈ f , there exist a finite set i0, unique rj ∈ r and xj ∈ x such that a ∈ ∑ j∈i0 rjxj, then f = ⊕i∈irxi. proof. suppose every element of f is contained in a uniquely expressed linear combination of the form n∑ i=1 rixi in which ri ∈ r and xi ∈ x for an appropriate n ∈ n. consequently, rxi = 0f for r ∈ r implies r = 0r. also, for every element a ∈ f , a ∈ n∑ i=1 rxi for some n ∈ n. so f = ∑ i∈i rxi. suppose a ∈ rxj and a ∈ ∑ i 6=j rxi that the summation is taken finite. so assume a = rjxj and a ∈ n∑ i=1,i6=j rixi in which r1,r2, . . . ,rn ∈ r for an appropriate n ∈ n by a new indexing. thus 0f ∈ a−a ∈   n∑ i=1,i6=j rixi  − (rjxj). but −(rj ∗xj) = (−1r) ∗ (rj ∗xj) = (−1r ·rj) ∗xj = (−rj) ∗xj, from proposition 2.1. so 0f ∈ ( n∑ i=1,i6=j rixi ) + (−rj)xj. since x is a basis for f , we obtain −rj = 0r = ri for all 1 ≤ i ≤ n and i 6= j. consequently a = 0f . � corollary 3.1. let f be an r-hypermodule with basis x. then f = ⊕i∈irxi. proof. it is clear from proposition 3.2. � the following result shows that the converse of proposition 3.2 holds: indeed, theorem 3.3. let x = {xi}i∈i be a subset of an r-hypermodule f which i is an index set. for every a ∈ f , there exist a finite set i0, unique rj ∈ r and xj ∈ x such that a ∈ ∑ j∈i0 rjxj if and only if f = ⊕i∈irxi. int. j. anal. appl. 16 (6) (2018) 804 proof. according to proposition 3.2, we must suppose f = ⊕i∈irxi and prove for every a ∈ f, there exist a finite set i0, unique rj ∈ r and xj ∈ x such that a ∈ ∑ j∈i0 rjxj. clearly a ∈ f = ⊕i∈irxi implies that there exist n ∈ n, rj ∈ r and xj ∈ x such that a ∈ n∑ j=1 rjxj. to show the uniqueness of rj ∈ r, let a ∈ m∑ j=1 r′jxj for some m ∈ n. without loss of generality, n = m. then 0f ∈ a−a ∈ n∑ j=1 rjxj − n∑ j=1 r′jxj. thus 0f ∈ n∑ j=1 djxj where dj ∈ rj − r′j. if dj = 0r for every 1 ≤ j ≤ n, then by the reversibility of r, rj = r ′ j for every 1 ≤ j ≤ n. without loss of generality, suppose d1 6= 0r and d1x1 6= 0f . then 0f ∈ n∑ j=1 djxj, and thus d1x1 ∈ n∑ j=2 djxj by the reversibility of f. so d1x1 ∈ rxi∩ n∑ j=2 rxj that is a contradiction. hence the proof is complete. � theorem 3.4. given a set x, there exists an r-hypermodule f and some y ⊆ f with |x| = |y | such that f is weak free on y over hsets. proof. clearly, r can be regarded as an r-hypermodule, and so, one can form the direct sum f = ⊕i∈xri, where for all i ∈ x, ri = r. define l: x −→ f as follows: l(x) = (ri,x)i∈x where ri,x = δi,x. it can be easily shown that {(ri,x)i∈x | x ∈ x} is a basis for f. denote (ri,x)i∈x as ex. then we can write f = ⊕x∈xrex and every element of fis contained in a unique finite linear combination ∑ x∈x rxex where rx ∈ r. indeed, every element of f has the form (rx)x∈x in which all but only a finitely many rx’s are zero. so the subset {ex}x∈x is a basis for f . thus by theorem 3.1, f is weak free on {ex}x∈x. consequently, considering the injective map l: x −→ f with x 7→ ex and letting y = l(x), f is weak free on y . � theorem 3.5. for every r-hypermodule a, there is some surjective f̄ ∈ homr(f,a) in which f is weak free r-hypermodule on some y ⊆ f over hsets. proof. let a = 〈x〉. also, let f, y ⊆ f and l: x −→ y as in the proof of theorem 3.4. so f is a weak free r-hypermodule on y = l(x) over hsets. let a ∈ a = 〈x〉. acording to proposition 3.1, suppose a ∈ m∑ i=1 rixi + n∑ i=1 nixi + k∑ i=1 ki(xi − xi) : ri ∈ r, xi ∈ x, m,n,k,ki ∈ n, ni ∈ z}. define f̄(z) = ∑m i=1 ril −1(exi ) or f̄(z) = ∑m i=1 rixi. the surjectivity of f̄ is clear. y i // l−1∼= �� f ∃f̄ �� x i �� a (note that since y is a basis for f, we have z ∈ ∑ x∈x rxex for every z ∈ f where rx ∈ r as in the proof of theorem 3.4.) � int. j. anal. appl. 16 (6) (2018) 805 now we state a new notion by using the fundamental module of an r-hypermodule. definition 3.10. an r-hypermodule f is called fundamental free if its fundamental module, f �∗ f , is free r γ∗ -module. example 3.1. [13] let (g, ·) be a group with |g| ≥ 4, and define a hyperaddition and a multiplication on r = g∪{0}, by: a + 0 = 0 + a = a for all a ∈ r; a + a = {a, 0} for all a ∈ g; a + b = b + a = g\{a,b} for all a,b ∈ g : a 6= b; a� 0 = 0 �a = 0 for all a ∈ r; a� b = a · b for all a,b ∈ g. then (r, +,�) is a hyperring. clearly every hyperring r is an r-hypermodule and �∗r = γ ∗. on the other hand, for every x,y ∈ r we have xγ0γy, since x + x = {x, 0} and y + y = {y, 0}. so xγ∗y, and indeed we have only one equivalence class. hence r′ = r γ∗ is the trivial ring 0 = {0r′}. clearly r′ = rγ∗ is a free r′-module. thus r is fundamental free as an r-hypermodule. remark 3.5. note that in 3.1, every r-hypermodule is fundamental free, since every r′-module is free. indeed, every module over the trivial ring r′ = 0 is free. in general, we state the following proposition: proposition 3.3. for every hyperring r with trivial fundamental ring, all objects of rhmod (or rhmod) are fundamental free. proof. the proof is clear, since every r γ∗ -module is free. � definition 3.11. an r-homomorphism f ∈ homr(a,b) is �∗-inverse of g ∈ homr(b,a) if �∗b((f ◦ g)(b)) = �∗b(b) and � ∗ a((g ◦ f)(a)) = � ∗ a(a) for all (a,b) ∈ a × b, or equivalently [f ◦ g] = [idb] and [g◦f] = [ida]. in this case, we say f is an �∗-isomorphism and a is �∗-isomorphic to b denoted by a �∗∼= b. theorem 3.6. if f and f ′ are two �∗-free r-hypermodules on the sets x and x′ over hsets, respectively, and |x| = |x′|, then f �∗∼= f ′. int. j. anal. appl. 16 (6) (2018) 806 proof. since |x| = |x′|, we have a bijection h : x −→ x′ in sets. now consider the inclusion i′ : x′ −→ f ′ and set f = i′ ◦ h as a morphism in hsets. since f is �∗-free on x over hsets, we have an rmvhomomorphism f̄ such that f̄ ◦ i = f as the diagram x i // h �� f f̄ �� x′ i′ // f ′ commutes. also, consider the inclusion i : x −→ f and set g = i◦h−1 as a morphism in hsets. since f ′ is �∗-free on x′ over hsets, we have an rmv-homomorphism ḡ such that ḡ ◦ i′ = g as the diagram x′ i′ // h−1 �� f ′ ḡ �� x i // f commutes. thus we have the following commutative diagram: x i // h �� f f̄ �� x′ i′ // h−1 �� f ′ ḡ �� x i // f so ḡ ◦ f̄ ◦ i = i◦h−1 ◦h = i◦ idx = i. on the other hand, idf ◦ i = i. since f is �∗-free on x over hsets, we have [ḡ ◦ f̄] = [idf ]. similarly, one can check that [f̄ ◦ ḡ] = [idf′]. hence f �∗∼= f ′. � according to theorem 3.2, we have the following result: corollary 3.2. let x ⊆ f and x′ ⊆ f ′ be two bases for r-hypermodules f and f ′, respectively. if f and f ′ are weak free on x and x′ over sets, respectively, and |x| = |x′|, then f �∗∼= f ′. lemma 3.1. let x be a basis for an r-hypermodule. if every summation n∑ i=1 rixi for ri ∈ r and xi ∈ x, which xi’s are not necessarily distinct, is a singleton, then r is a ring and f is an r-module . proof. clearly, for all ri,rj ∈ r and xi,xj ∈ x, rixi + rjxj is a singleton. also, (ri + rj)xi = rixi + rjxi is a singleton. without loss of generality, assume x ∈ n∑ i=1 rixi and y ∈ n∑ i=1 r′ixi for ri,r ′ i ∈ r and xi ∈ x. then we can write x + y ⊆ n∑ i=1 tixi in which ti = ri + r ′ i. according to the assumption, if we prove ti is a singleton, then x+y is a singleton and f is an r-module. let r,s ∈ ti. since tixi = rixi +r′ixi is a singleton and rxi,sxi ∈ tixi, we have rxi = sxi = tixi. clearly, 0f ∈ rxi − sxi = (r − s)xi. on the other hand, int. j. anal. appl. 16 (6) (2018) 807 0f ∈ 0r ∗xi. so 0r ∈ r −s. hence r = s. consequently, ti is a singleton. now we prove r is ring. let r,s ∈ r and t,t′ ∈ r + s. clearly, txi, t ′xi ∈ (r + s)xi = rxi + sxi every xi ∈ x. then txi = t′xi. then 0f ∈ txi − t′xi = (t − t′)xi. on the other hand, 0f ∈ 0r ∗xi. so 0r ∈ t− t′. hence t = t′. consequently, r + s is a singleton. � proposition 3.4. let f be a free r-hypermodule on basis x ⊆ f such that for all r,s ∈ r and all x ∈ x, we have |rx + sx| = 1. then r is a ring and f is a free r-module on x in the category rhmod. proof. note that every free r-hypermodule on x ⊆ f is a weak free r-hypermodule on x over sets. thus if i = idf |x, then x i // i �� f ∃!f~~ f i.e., f ◦ i = i. since x is a basis, for an arbitrary x ∈ f, we have f(x) = ∑ xi∈x rxi xi in which every r x i ∈ r depends on x (by equation 3.1). note that ∑ xi∈x rxi xi is the unique presentation of x by the elements of the basis x, i.e., x ∈ ∑ xi∈x rxi xi and rxi ’s are unique. on the other hand, x i // i �� f idf~~ f and indeed, idf ◦i = i. so f = idf and thus f(x) = idf (x). this implies ∑ xi∈x rxi xi = x. consequently, every unique presentation of every element of f is a singleton. according to the assumption, every summation n∑ i=1 rixi, which xi’s are not necessarily distinct, is a singleton. thus the result is followed by lemma 3.1. � references [1] r. ameri, on categories of hypergroups and hypermodules, j. discrete math. sci. cryptogr. 6(2-3) (2003), 121–132. [2] s. awodey, category theory, second ed., oxford university press, inc. new york, 2010. [3] p. corsini, prolegomena of hypergroup theory, second ed., aviani editore, tricesimo, 1993. [4] p. corsini and v. leoreanu-fotea, applications of hyperstructure theory, advances in mathematics, kluwer academic publication, dordrecht, 2003. [5] b. davvaz, a brief survey of the theory of hv-structures, 8 th aha, greece, spanidis (2003), 39–70. [6] b. davvaz, polygroup theory and related systems, world scientific publishing co. pte. ltd., hackensack, nj, 2013. [7] b. davvaz and v. leoreanu-fotea, hyperring theory and applications, international academic press, palm harbor, usa, 2007. [8] h. herrlich and g. e. strecker, category theory, vol. 2. boston: allyn and bacon, 1973. [9] m. krasner, a class of hyperrings and hyperfelds, internat. j. math. math. sci. 6 (1983), 307–311. int. j. anal. appl. 16 (6) (2018) 808 [10] ch g. massouros, free and cyclic hypermodules, ann. math. pura appl. 150 (1) (1988), 153–166. [11] f. marty, sur uni generalization de la notion de group, in: 8th congress math. scandenaves, stockholm, (1934), 45–49. [12] s. sh. mousavi and m. jafarpour, on free and weak free (semi) hypergroups, algebra colloq. 18 (2011), 873–880. [13] a. nakassis, expository and survey article of recent results in hyperring and hyperfield theory, internat. j. math. math. sci. 11 (1988) 209–220. [14] h. shojaei and r. ameri, some results on categories of krasner hypermodules, j. fundam. appl. sci. 8 (3s) (2016), 2298–2306. [15] h. shojaei, r. ameri and s. hoskova-mayerova, on properties of various morphisms in the categories of general krasner hypermodules, italian j. pure appl. math. 39 (2018), 475-484. [16] t. vougiouklis, the fundamental relation in hyperrings, the general hyperfeld, 4th aha, xanthi 1990, world scientifc (1991), 203–211. [17] t. vougiouklis, hyperstructures and their representations, hadronic press, inc., 115, palm harber, usa, 1994. 1. introduction 2. preliminaries 3. freeness of hypermodules references int. j. anal. appl. (2023), 21:37 asymptotic behavior of solution for coupled reaction diffusion system by order m mebarki maroua1, barrouk nabila2,∗ 1faculty of science and technology, departement of mathematics and computer science, amine elokkal el hadj moussa ag akhamouk university, p.o.box 10034, tamanrasset 11000, algeria 2faculty of science and technology, department of mathematics and computer science, mohamed cherif messaadia university, p.o.box 1553, souk ahras 41000, algeria ∗corresponding author: n.barrouk@univ-soukahras.dz abstract. the aim of this paper is to prove that asymptotic behavior in the time of solutions for the weakly coupled reaction diffusion system:  ∂ui ∂t −di ∆ui = fi (u1,u2, . . . ,um) in ω ×r+, ∂ui ∂η = 0 in ∂ω ×r+, ui (.,0) = u 0 i (.) in ω, (0.1) where ω is an open bounded domain of class c1 in rn, ui (t,x), i = 1,m, t ≥ 0, x ∈ ω are real valued functions. we treat the system (0.1) as a dynamical system in c ( ω ) ×c ( ω ) × ...×c ( ω ) and apply lyapunov type stability techniques. a key ingredient in this analysis is a result which establishes that the orbits of the dynamical system are precompact in c ( ω ) ×c ( ω ) × ...×c ( ω ) . as a consequence of arzela-ascoli theorem, this will be satisfied if the orbits are, for example, uniformly bounded in c1 ( ω ) ×c1 ( ω ) × ...×c1 ( ω ) for t > 0. 1. introduction the existence, uniqueness, and asymptotic behavior of the solution of a balanced two component reaction diffusion system have been investigated. it was shown that a global and unique solution existed and it’s second component can be estimated using the lyapunov functional see [1, 14, 15]. received: mar. 8, 2023. 2020 mathematics subject classification. 35k57, 35b40, 35b45. key words and phrases. reaction diffusion system; semigroup; global existence; asymptotic behavior. https://doi.org/10.28924/2291-8639-21-2023-37 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-37 2 int. j. anal. appl. (2023), 21:37 it was, also, demonstrated that each component of the solution converged, at infinity, to a constant which can be used in terms of the reacting function and the initial data. the results of the current research can be used in several areas of applied mathematics, especially when the system equations originate from mathematical models of real systems such as in biology, chemistry, population dynamics, and other disciplines. we know that the problem (0.1) has a unique global solution see [5, 7, 17]. the main question we want to address is asymptotic behavior the solutions for system (0.1). in fact the subject of the asymptotic behavior of reaction diffusion systems has received a lot of attention in the last decades and several outstanding results have been proved by some of the major experts in the field. this question has been investigated by many authors by considering special forms of the nonlinear terms fi. in the case where i = 1, 2 :   ∂u1 ∂t −d1∆u1 = f1 (u1,u2) in ω ×r+, ∂u2 ∂t −d2∆u2 = f2 (u1,u2) in ω ×r+, λi ∂ui ∂η + (1 −λi ) ui = 0, in ∂ω ×r+, (1.1) when d1 6= d2, and nonnegative initial data arise, for example, as models for the diffusion of substances which at the same time react with each other chemically (cf. [8, 16]). also (1.1) is related to the rosenzweig-mac arthur equation in ecology (cf. [2]). in the case where f1 (u1,u2) = −f2 (u1,u2) = −u1uσ2 , note that, the behavior of non-negative total solutions (1.1) is treated in the paper of alikakos [2] obtained l∞-bounds of solutions global existence when 1 < σ < n + 2 n , and masuda [14] who showed that solutions exist globally for every σ ≥ 1 and, in addition, showed that the solutions converge as t goes to +∞. recently, haraux and youkana [13] established a global existence result of a system (1.1) for a large class of the function f1 and f2. more precisely, they showed that for f1 (u1,u2) = f2 (u1,u2) = −u1ψ (u2) , the problem (1.1) admits a global solution provided that the following condition holds: lim u2→+∞ [log (1 + ψ (u2))] u2 = 0. in the case where d2 ≥ 0, systems of the type (1.1) occur in many applications (cf. [8]). for triangular diffusion matrix, global bounds were proved by kirane in [11] if u02 (x) ≥ d2 d1−d3 u 0 1 (x) ≥ 0, x ∈ ω,. the author proved also that the solution (u1,u2) converges to a constant vector k = (k1,k2) as t →∞, uniformly in ω̄. furthermore, k1 ≥ 0, k2 ≥ 0 and k1ψ (k2) = 0. int. j. anal. appl. (2023), 21:37 3 in this paper we shall generalize the results obtained in [11]. we prove the asymptotic behavior of solutions of m-component reaction-diffusion systems with diagonal matrix and homogeneous neumann conditions. the reaction terms are assumed to be of polynomial growth. we consider the following m-equations of reaction-diffusion system, with m ≥ 2: ∂u ∂t −am∆u = f (u) in ω × (0, +∞), (1.2) where ω is an open bounded domain of class c1 in rn, the vectors u and f and the matrix am are defined as: u = (u1, ...,um) t , f = (f1, ..., fm) t , am =   d1 0 0 · · · 0 0 d2 0 ... ... 0 0 d3 ... 0 ... ... ... ... 0 0 · · · 0 0 dm   . the constants (di ) m i=1 , are supposed to be strictly positive which reflects the parabolicity of the system and implies at the same time that the diffusion matrix am is positive defnite. the boundary conditions and initial data (respectively) for the proposed system are assumed to satisfy: ∂ηu = 0 on ∂ω × (0, +∞), and u(0,x) = u0(x) = (u 0 1, ...,u 0 m) t on ω, where ∂ ∂η denotes the outward normal derivative on the boundary ∂ω, the vectors u0 are defined as: u0 = (u 0 1, ...,u 0 m) t . 2. notations and preliminary in the following we denote by ‖.‖p the norm in l p (ω) for 1 ≤ p ≤ +∞, ‖.‖∞ the norm in c (ω), and ‖.‖1,∞ the norm in c 1 (ω). for 1 < p < ∞, set   d (a) = { u : u ∈ w 2,p (ω) : ∂u ∂η = 0 on ∂ω } , au = ∆u for u ∈ d (a) . it is well known (cf. for example, [6]) that a is m-dissipative in lp (ω) for 1 < p < ∞. moreover, the restriction of a to c ( ω̄ ) is m-dissipative. let us now recall an overview of the asymptotic behavior of the solution for coupled reaction diffusion systems. this will pave the way to introduce our findings later on. 4 int. j. anal. appl. (2023), 21:37 consider the initial value problem { ut (t) = lu (t) + f (u (t)) u (0) = u0, (ivp) where l is the infinitesimal generator of a c0-semigroup s (t) on a real banach space x with norm ‖.‖, f : x →r is a given function, and u0 ∈ x is a given initial datum. theorem 2.1. [19] let t > 0. a function u : [0,t ] → x is a weak solution of (ivp) on [0,t ] if and only if f (u (t)) ∈ l1 (0,t,x) and u satisfies the variation of constants formula u (t) = s (t) u0 + ∫ t 0 s (t − s) f (u (s)) ds, for all s ∈ [0,t ] . definition 2.1. a function u : [0,t ] → x is called a strong solution of (ivp) if u (t) is strongly continuously differentiable in the interval 0 < t < t, u (t) ∈ d (l) for 0 < t < t, that equation (ivp) is satisfied for 0 < t < t and u (t) → u0 as t → 0. theorem 2.2. [18] let f : x → x be locally lipschitz continuous. then for u0 ∈ x, (ivp) has a unique weak solution defined in a maximal interval of existence [0,tmax), tmax > 0, u ∈ c ([0,tmax) ,x). moreover, if tmax < ∞, then lim t→tmax ‖u (t)‖ = +∞. now, let us recall the following definition. definition 2.2. let {s (t)}t≥0 be a nonlinear semigroup on a compact metric space x. if( u01,u 0 2, . . . ,u 0 m ) ∈ x, o ( u01,u 0 2, . . . ,u 0 m ) = { s (t) ( u01,u 0 2, . . . ,u 0 m )} t≥0 is the orbit through( u01,u 0 2, . . . ,u 0 m ) , then the w-limite set for ( u01,u 0 2, . . . ,u 0 m ) is defined by w ( u01,u 0 2, . . . ,u 0 m ) = {(u1,u2, . . . ,um) ∈ x : ∃tn →∞ : s (tn) ( u01,u 0 2, . . . ,u 0 m ) → (u1,u2, . . . ,um)}. 3. the main result in this section, we state the main result. theorem 3.1. the solution w = (u1,u2, . . . ,um) of the system (1.2) converges a constant vector of the form ξ = (ξ1,ξ2, . . . ,ξi ) as t →∞, uniformly in ω i.e ( ui → t→∞ ξi ) for i = 1,m. furthermore, we have: ξi ≥ 0, i = 1,m, fi (ξ1,ξ2, . . . ,ξm) = 0, and m∑ i=1 ξi = 1 ω ∫ ω m∑ i=1 u0i (x) dx. the following lemma is a useful tool in the proof of the theorem 3.1. int. j. anal. appl. (2023), 21:37 5 lemma 3.1. let (u1,u2, . . . ,um) be a solution of (1.2). we have∫ qt |∇ui| 2 dxdt < ∞ for i = 1,m, here qt = ω × [0,t ] and 0 < t < ∞. proof. we have for i = 1,m. ∂ui ∂t −di ∆ui = fi (u1,u2, . . . ,um) . (3.1) by integrating over (0,t ) is obtained∫ t 0 ∂ui ∂t (x,t) dt = di ∫ t 0 ∆uidt + ∫ t 0 fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dt, ui (x,t ) −ui (x, 0) = di ∫ t 0 ∆uidt + ∫ t 0 fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dt, and integrating a second time is collected over ω∫ ω ui (x,t ) dx − ∫ ω ui (x, 0) dx = di ∫ ω ∫ t 0 ∆uidtdx + ∫ ω ∫ t 0 fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx. green’s formula is applied to ∫ ω ∆uidx, we gain∫ ω ∆uidx = ∫ ∂ω ∂ui ∂η dσ − ∫ ω ∇ui∇1dx, therefore ∫ ω ∆uidx = 0, thus ∫ ω ∫ t 0 fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx = ∫ ω ui (x,t ) dx − ∫ ω u0i (x) dx < ∞, as a result of ui (t ) ∈ c ( ω ) we have∫ qt fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx < ∞, for i = 1,m. multiply now the ith equation of (1.2) by ui, for i = 1,m, and integrating over qt , we attain∫ ω ∫ t 0 ui ∂ui ∂t (x,t) dtdx = di ∫ ω ∫ t 0 ui ∆uidtdx + ∫ ω ∫ t 0 uifi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dtdx, by using the green formula∫ ω ui ∆uidx = ∫ ∂ω ui ∂ui ∂η dσ − ∫ ω |∇ui| 2 dx, therefore ∫ ω ui ∆uidx = − ∫ ω |∇ui| 2 dx, and a simple calculation, it becomes 1 2 ∫ ω [ u2i (x,t) ]∣∣t 0 dx = −di ∫ t 0 ∫ ω |∇ui| 2 dxdt + ∫ t 0 ∫ ω ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt, 6 int. j. anal. appl. (2023), 21:37 then ∫ ω u2i (x,t ) + 2di ∫ qt |∇ui| 2 dxdt = ∫ ω ( u0i (x) )2 dx +2 ∫ qt ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt, consequently 2di ∫ qt |∇ui| 2 dxdt ≤ ∫ ω ( u0i (x) )2 dx +2 ∫ qt ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt, (3.2) since ∫ ω ( u0i (x) )2 dx < ∞, for i = 1,m. and ∫ qt ui (x,t) fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt ≤‖ui‖l∞(qt ) ∫ qt fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dxdt < ∞, for i = 1,m. hence 2di ∫ qt |∇ui| 2 dxdt < ∞, for i = 1,m. consequently ∫ qt |∇ui| 2 dxdt < ∞, for i = 1,m. ∀t > 0. � 4. proof of the main result (theorem 3.1) we are now ready to prove the main result of this work: proof of theorem 3.1. first, if we integrate the ith equation of (1.2) over ω we have∫ ω ∂ui ∂t (x,t) dx = di ∫ ω ∆uidx + ∫ ω fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx, use green theorem to transform the terms ∆ui in the light of boundary conditions we observe that∫ ω ∂ui ∂t (x,t) dx = ∫ ω fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx, if we add this equations imliying∫ ω m∑ i=1 ∂ui ∂t (x,t) dx = ∫ ω m∑ i=1 fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx, for i = 1,m, if we assume that m∑ i=1 ∫ ω fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx = 0, int. j. anal. appl. (2023), 21:37 7 we get ∫ ω m∑ i=1 ∂ui ∂t (x,t) dx = 0, for i = 1,m, as ∫ t 0 ∫ ω m∑ i=1 ∂ui ∂t (x,t) dxdt = ∫ ω ∫ t 0 m∑ i=1 ∂ui ∂t (x,t) dtdx = ∫ ω m∑ i=1 ui (x,t) ∣∣∣∣∣ t 0 dx = ∫ ω m∑ i=1 ui (x,t) dx − ∫ ω m∑ i=1 u0i (x) dx = 0, we deduce that ∫ ω m∑ i=1 ui (x,t) dx = ∫ ω m∑ i=1 u0i (x) dx. (4.1) integrating the ith equation of the system (1.2) in ω, for i = 1,m, we have:∫ ω ∂ui ∂t (x,t) dx = ∫ ω fi (u1 (x,t) ,u2 (x,t) , . . . ,um (x,t)) dx > 0, as a means that d dt ∫ ω ui (x,t) dx > 0. then the fonction t → ∫ ω ui (x,t) dx is increasing and ω is bounded. then t → 1|ω| ∫ ω ui (x,t) dx is increasing and according to the positivity of ui was 1 |ω| ∫ ω ui (x,t) dx ≥ 0. therefore 1|ω| ∫ ω ui (x,t) dx is bounded below and increasing. formerly ∃ lim t→∞ 1 |ω| ∫ ω ui (x,t) dx = li, for i = 1,m. on the other hand, since sets ∪ t≥0 {ui (t)}t≥0 , for i = 1,m are precompacts in c ( ω ) . there exists a sequence (tn)n≥0, tn →∞ such that lim n→∞ ui (tn) = u s i , for i = 1,m in c ( ω ) , now, denote by w ( u01,u 0 2, . . . ,u 0 m ) the w-limite set for ( u01,u 0 2, . . . ,u 0 m ) and φ the set of the solution of the elliptic system{ −di ∆usi = fi ( us1 (x,t) ,u s 2 (x,t) , . . . ,u s m (x,t) ) in ω, ∂us i ∂η = 0 in ∂ω, (4.2) and prove φ = {(ξ1,ξ2, . . . ,ξm)} where ξ1,ξ2, . . . ,ξm are constants, in fact, multipliying the ith equation of the problem (4.2) by usi for i = 1,m and integrating over ω yields: −di ∫ ω usi ∆u s i dx = ∫ ω usi fi (u s 1 (x,t) ,u s 2 (x,t) , . . . ,u s m (x,t)) dx. apply green formular: di ∫ ω |∇usi | 2 dx = ∫ ω usi fi (u s 1 (x,t) ,u s 2 (x,t) , . . . ,u s m (x,t)) dx. 8 int. j. anal. appl. (2023), 21:37 adding the ith equations yields m∑ i=i di ∫ ω |∇usi | 2 dx = m∑ i=i ∫ ω usi fi (u s 1 (x,t) ,u s 2 (x,t) , . . . ,u s m (x,t)) dx. supposing m∑ i=i usi fi ( us1 (x,t) ,u s 2 (x,t) , . . . ,u s m (x,t) ) ≤ 0 for i = 1,m, then 0 ≤ m∑ i=i di ∫ ω |∇usi | 2 dx ≤ 0, therefore m∑ i=i di ∫ ω |∇usi | 2 dx = 0. we deduce that ∫ ω |∇usi | 2 dx = 0 ⇒∇usi = 0 ⇒ u s i = ξi. (4.3) replacing usi = ξi, for i = 1,m in the i th equation (4.2). it is clear that fi (ξ1,ξ2, . . . ,ξm) = 0. hereafter, we are going to show that w ( u01,u 0 2, . . . ,u 0 m ) 6= φ. now, ∀x ∈ ω, σ ∈ ]−1, 1[ and let pni (x,σ) = ui (x,tn + σ) , for i = 1,m, multiply the ith equation of the poblem (1.2) by ∂ui ∂t ∂ui ∂t ∂ui ∂t −di ∂u ∂t ∆ui = ∂ui ∂t fi (u1,u2, . . . ,um) , and integrate over ω we get: ∫ ω ( ∂ui ∂t )2 dx −di ∫ ω ∂ui ∂t ∆uidx = ∫ ω ∂ui ∂t fi (u1,u2, . . . ,um) dx, as ∥∥∥∥∂ui∂t ∥∥∥∥2 l2(ω) = di ∫ ω ∂ui ∂t ∆udx + ∫ ω ∂ui ∂t fi (u1,u2, . . . ,um) dx, intégrating result over (t0, +∞) , we have: ∫ +∞ t0 ∥∥∥∥∂ui∂t ∥∥∥∥2 l2(ω) dt = di ∫ +∞ t0 ∫ ω ∂ui ∂t ∆uidxdt + ∫ +∞ t0 ∫ ω ∂ui ∂t fi (u1,u2, . . . ,um) dxdt < ∞, int. j. anal. appl. (2023), 21:37 9 thus ∂ui ∂t ∈ l2 ( t0, +∞,l2 (ω) ) , ∀σ ∈ ]−1, 1[ we get: pni (x,σ) −ui (x,tn) = ui (x,tn + σ) −ui (x,tn) = ∫ tn+σ tn ∂ui ∂t (x,t) dt ≤ ∫ tn+1 tn−1 ∂ui ∂t (x,t) dt by reason of (tn − 1 < tn, σ < 1, tn + σ < tn + 1) ≤ (∫ tn+1 tn−1 (1) 2 dt )1 2 (∫ tn+1 tn−1 ( ∂ui ∂t (x,t) )2 dt )1 2 , ≤ √ 2 (∫ tn+1 tn−1 ( ∂ui ∂t (x,t) )2 dt )1 2 , as follows |pni (x,σ) −ui (x,tn)| 2 = 2 ∫ tn+1 tn−1 ( ∂ui ∂t (x,t) )2 dt, integrating the latter inequality ω yields∫ ω |pni (x,σ) −ui (x,tn)| 2 dx ≤ 2 ∫ ω ∫ tn+1 tn−1 ( ∂ui ∂t (x,t) )2 dtdx, we pass to the limit as n →∞, we have: ‖pni (x,σ) −u s i ‖ 2 l2(ω) ≤ 2 limn→∞ ∫ ω ∫ tn+1 tn−1 ( ∂ui ∂t (x,t) )2 dtdx = 0, so ‖pni (x,σ) −u s i ‖ 2 l2(ω) → 0n→∞ . as a result, we will all σ ∈ ]−1, 1[ , ‖pni (x,σ) −u s i ‖ 2 l2(ω) → 0n→∞, hence sup−1<σ<1 ‖pni (x,σ) −u s i ‖ 2 l2(ω) → 0n→∞, and by the same mode are obtained: sup −1<σ<1 ‖pni (x,σ) −u s i ‖ 2 l2(ω) → 0n→∞, for i = 1,m. also, we can have: sup −1<σ<1 ‖∇pni (x,σ) −∇u s i ‖ 2 l2(ω) →n→∞ 0 for i = 1,m, through positivity and boundedness of the solution was: 0 ≤ ui (x,tn + σ) ≤ mi, remarkably fi ∈ c∞ (rn), we can conclude, using lebesgue’s theorem, that fi (p1 (x,σ) ,p2 (x,σ) , . . . ,pm (x,σ)) → fi (us1,u s 2, . . . ,u s m) in l 2 (ω × (−1, 1)) weak. 10 int. j. anal. appl. (2023), 21:37 now, let %i ∈ c1 ( ω ) such that %i = 0 on ∂ω where i = 1,m, and let γ ∈ c1 ( ω ) such that suppγ ⊂ [−1, 1] , ∫ 1 −1 γ (s) ds = 1 and γ (−1) = γ (1) . we multiply the ith equation of problem (1.2) by γ (t − tn) %i and integrate over ω×(tn − 1,tn + 1) , we obtain ∫ tn+1 tn−1 ∫ ω γ (t − tn) %i ∂ui ∂t dxdt −di ∫ tn+1 tn−1 ∫ ω γ (t − tn) %i ∆uidxdt = ∫ tn+1 tn−1 ∫ ω γ (t − tn) %ifi (u1,u2, . . . ,um) dxdt. (4.4) forecast the integral ∫ tn+1 tn−1 γ (t − tn) %i ∂ui ∂t dt by part, we find ∫ tn+1 tn−1 γ (t − tn) %i ∂ui ∂t dt = − ∫ tn+1 tn−1 γ′ (t − tn) %iui (x,t) dt, (4.5) to appraise ∫ ω γ (t − tn) %i ∆uidx applying green’s formula∫ ω γ (t − tn) %i ∆uidx = ∫ ∂ω γ (t − tn) %i ∂ui ∂η dσ − ∫ ω ∇γ (t − tn) %i∇uidx = − ∫ ω ∇γ (t − tn) %i∇uidx, extremely ∫ ω γ (t − tn) %i ∆uidx = − ∫ ω ∇γ (t − tn) %i∇uidx, (4.6) injected (4.5) and (4.6) in (4.4) is accessed − ∫ tn+1 tn−1 ∫ ω γ′ (t − tn) %iui (x,t) dxdt + di ∫ tn+1 tn−1 ∫ ω ∇γ (t − tn) %i∇uidxdt (4.7) + ∫ tn+1 tn−1 ∫ ω γ (t − tn) %ifi (u1,u2, . . . ,um) dxdt = 0 for i = 1,m. by making the following change of variable σ = t − tn → dσ = dt if { t = tn − 1 t = tn + 1 → { σ = −1 σ = 1 accordingly the integral (4.7) becomes∫ +1 −1 ∫ ω γ′ (σ) %ip n i (x,σ) dxdσ −di ∫ +1 −1 ∫ ω ∇γ (σ) %i∇pni (x,σ) dxdσ ∫ +1 −1 ∫ ω γ (σ) %ifi (p1 (x,σ) ,p2 (x,σ) , . . . ,pm (x,σ)) dxdσ = 0, for i = 1,m. (4.8) int. j. anal. appl. (2023), 21:37 11 applying lesbegue’s theorem we gain: for i = 1,m lim n→∞ ∫ +1 −1 ∫ ω γ′ (σ) %ip n i (x,σ) dxdt = ∫ +1 −1 ∫ ω γ′ (σ) %iu s i dxdσ = ∫ +1 −1 γ′ (σ) dσ ∫ ω %iu s i dx = γ (σ)|+1−1 ∫ ω %iu s i dx = 0 by virtue of γ (1) = γ (−1) , from inequality (4.8), we make: for i = 1,m −di ∫ ω ∇%i∇usi + ∫ ω %ifi (u s 1,u s 2, . . . ,u s m) dx = 0, which is the variational formulation of (4.2), hence w = φ. finally, combining (4.2) with (4.1) yields∫ ω m∑ i=1 ξidx = ∫ ω m∑ i=1 u0i dx, |ω| m∑ i=1 ξi = ∫ ω m∑ i=1 u0i dx, m∑ i=1 ξi = 1 |ω| ∫ ω m∑ i=1 u0i dx, the proof of the theorem is complete. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. abdelmalek, s. kouachi, proof of existence of global solutions form-component reaction–diffusion systems with mixed boundary conditions via the lyapunov functional method, j. phys. a: math. theor. 40 (2007), 12335–12350. https://doi.org/10.1088/1751-8113/40/41/005. [2] n.d. alikakos, lp-bounds of solutions of reaction-diffusion equations, commun. part. differ. equ. 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113. [3] j.d. avrin, qualitative theory for a model of laminar flames with arbitrary nonnegative initial data, j. differ. equ. 84 (1990), 290-308. https://doi.org/10.1016/0022-0396(90)90080-9. [4] a. barabanova, on the global existence of solutions of a reaction diffusion equation with exponential nonlinearity, proc. amer. math. soc. 122 (1994), 827-831. [5] w. bouarifi, n.e. alaa, s. mesbahi, global existence of weak solutions for parabolic triangular reaction diffusion systems applied to a climate model, ann. univ. craiova, math. computer sci. ser. 42 (2015), 80-97. [6] m.g. crandall, a. pazy, l. tartar, remarks on generators of analytic semigroups, israel j. math. 32 (1979), 363–374. https://doi.org/10.1007/bf02760465. [7] p.v. danckwerts, gas-liquid reactions, mcgraw-hill, new york, (1970). [8] s.r. de groot, p. mazur, nonequilibrium thermodynamics, north-holland, amsterdam, (1962). https://doi.org/10.1088/1751-8113/40/41/005 https://doi.org/10.1080/03605307908820113 https://doi.org/10.1016/0022-0396(90)90080-9 https://doi.org/10.1007/bf02760465 12 int. j. anal. appl. (2023), 21:37 [9] a. haraux, m. kirane, estimations c1 pour des problèmes paraboliques semi-lineaires, ann. fac. sci. toulouse math. 5 (1983), 265-280. [10] a. haraux, a. youkana, on a result of k. masuda concerning reaction-diffusion equations, tohoku math. j. 40 (1988), 159-163. https://doi.org/10.2748/tmj/1178228084. [11] m. kirane, global bounds and asysmptotics for a system of reaction-diffusion equations, j. math. anal. appl. 138 (1989), 328-342. [12] s. kouachi, global existence of solutions to reaction diffusion systems via a lyapunov functional, electron. j. differ. equ. 2001 (2001), 68. [13] s. kouachi, a. youkana, global existence and asymptotics for a class of reaction diffusion systems, bull. polish acad. sci. math. 49 (2001). [14] k. masuda, on the global existence and asymptotic behavior of solutions of reaction-diffusion equations, hokkaido math. j. 12 (1983), 360-370. https://doi.org/10.14492/hokmj/1470081012. [15] m. mebarki, a. moumeni, global solution of system reaction diffusion with full matrix, glob. j. math. anal. (2015), 04-25. [16] a. moumeni, m. dehimi, global existence solutions of a system for reaction diffusion, int. j. math. arch. 4 (2013), 122-129. [17] b. rebiai, s. benachour, global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth, j. evol. equ. 10 (2010), 511–527. https://doi.org/10.1007/s00028-010-0059-x. [18] f. rothe, global existence of reaction-diffusion systems, lecture notes in mathematics, 1072, springer, berlin, (1984). [19] a. pazy, semi-groups of linear operators and applications to partial differential equations, springer, new york, (1983). https://doi.org/10.2748/tmj/1178228084 https://doi.org/10.14492/hokmj/1470081012 https://doi.org/10.1007/s00028-010-0059-x 1. introduction 2. notations and preliminary 3. the main result 4. proof of the main result (theorem 3.1) references international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 203-208 http://www.etamaths.com inequalities for the modified k-bessel function saiful rahman mondal1 and kottakkaran sooppy nisar2,∗ abstract. the article considers the generalized k-bessel functions and represents it as wright functions. then we study the monotonicity properties of the ratio of two different orders kbessel functions, and the ratio of the k-bessel and the k-bessel functions. the log-convexity with respect to the order of the k-bessel also given. an investigation regarding the monotonicity of the ratio of the k-bessel and k-confluent hypergeometric functions are discussed. 1. introduction one of the generalization of the classical gamma function γ studied in [4] is defined by the limit formula γk(x) := lim n→∞ n! kn(nk) x k −1 (x)n,k , k > 0, (1.1) where (x)n,k := x(x+k)(x+2k) . . . (x+(n−1)k) is called k-pochhammer symbol. the above k−gamma function also have an integral representation as γk(x) = ∫ ∞ 0 tx−1e− tk k dt, <(x) > 0. (1.2) properties of the k-gamma functions have been studies by many researchers [6, 8–11]. following properties are required in sequel: (i) γk (x + k) = xγk (x) (ii) γk (x) = k x k −1γ ( x k ) (iii) γk (k) = 1 (iv) γk (x + nk) = γk(x)(x)n,k motivated with the above generalization of the k-gamma functions, romero et. al. [1] introduced the k−bessel function of the first kind defined by the series j γ,λ k,ν (x) := ∞∑ n=0 (γ)n, k γk (λn + υ + 1) (−1)n (x/2)n (n!) 2 , (1.3) where k ∈ r+; α,λ,γ,υ ∈ c; <(λ) > 0 and <(υ) > 0. they also established two recurrence relations for j γ,λ k,ν . in this article, we are considering the following function: i γ,λ k,ν (x) := ∞∑ n=0 (γ)n, k γk (λn + υ + 1) (x/2) n (n!) 2 , (1.4) since lim k,λ,γ→1 i γ,λ k,ν (x) = ∞∑ n=0 1 γ (n + υ + 1) (x/2) n n! = ( 2 x )ν 2 iν( √ 2x), received 18th march, 2017; accepted 25th may, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 33c10, 26d07. key words and phrases. generalized k-bessel functions; monotonicity; log-convexity; turán type inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 203 204 mondal and nisar the classical modified bessel functions of first kind. in this sense, we can call i γ,λ k,ν as the modified k-bessel functions of first kind. in fact, we can express both j γ,λ k,ν and i γ,λ k,ν together in w γ,λ k,ν,c(x) := ∞∑ n=0 (γ)n, k γk(λn + ν + 1) (−c)n(x/2)n (n!) 2 , c ∈ r. (1.5) we can termed w γ,λ k,ν as the generalized k-bessel function. first we study the representation formulas for w γ,λ k,ν in term of the classical wright functions. then we will study about the monotonicity and log-convexity properties of i γ,λ k,ν . 2. representation formula for the generalized k-bessel function the generalized hypergeometric function pfq(a1, . . . ,ap; c1, . . . ,cq; x), is given by the power series pfq(a1, . . . ,ap; c1, . . . ,cq; z) = ∞∑ k=0 (a1)k · · ·(ap)k (c1)k · · ·(cq)k(1)k zk, |z| < 1, (2.1) where the ci can not be zero or a negative integer. here p or q or both are allowed to be zero. the series (2.1) is absolutely convergent for all finite z if p ≤ q and for |z| < 1 if p = q + 1. when p > q + 1, then the series diverge for z 6= 0 and the series does not terminate. the generalized wright hypergeometric function pψq(z) is given by the series pψq(z) = pψq [ (ai,αi)1,p (bj,βj)1,q ∣∣∣∣z ] = ∞∑ k=0 ∏p i=1 γ(ai + αik)∏q j=1 γ(bj + βjk) zk k! , (2.2) where ai,bj ∈ c, and real αi,βj ∈ r (i = 1, 2, . . . ,p; j = 1, 2, . . . ,q). the asymptotic behavior of this function for large values of argument of z ∈ c were studied in [13, 14] and under the condition q∑ j=1 βj − p∑ i=1 αi > −1 (2.3) in literature [18, 19]. the more properties of the wright function are investigated in [14–16]. now we will give the representation of the generalized k-bessel functions in terms of the wright and generalized hypergeometric functions. proposition 2.1. let, k ∈ r and λ,γ,ν ∈ c such that <(λ) > 0,<(ν) > 0. then w γ,λ k,ν,c(x) = 1 k ν+k+1 k γ ( γ k )1ψ2 [ (γk , 1)(ν+1 k , γ k ) (1, 1) ∣∣∣∣− cx 2k λ k −1 ] proof. using the relations γk (x) = k x k −1γ ( x k ) and γk (x + nk) = γk(x)(x)n,k, the generalized k-bessel functions defined in (1.5) can be rewrite as w γ,λ k,ν,c(x) = ∞∑ n=0 γk(γ + nk) γk(λn + ν + 1)γk(γ) (−c)n (n!)2 (x 2 )n (2.4) = 1 k ν+k+1 k γ ( γ k ) ∞∑ n=0 γ ( γ k + n ) γ ( λ k n + ν+1 k ) γ ( γ k ) (−c)n γ(n + 1)γ(n + 1) ( x 2k λ k −1 )n (2.5) = 1 k ν+k+1 k γ ( γ k )1ψ2 [ (γk , 1)(ν+1 k , γ k ) (1, 1) ∣∣∣∣− cx 2k λ k −1 ] (2.6) hence the result follows. � 3. monotonicity and log-convexity properties this section discuss the monotonicity and log-convexity properties for the modified k-bessel functions w γ,λ k,ν,−1(x) = i γ,λ k,ν (x). following lemma due to biernacki and krzyż [7] will be required. inequalities for the modified k-bessel function 205 lemma 3.1. [7] consider the power series f(x) = ∑∞ k=0 akx k and g(x) = ∑∞ k=0 bkx k, where ak ∈ r and bk > 0 for all k. further suppose that both series converge on |x| < r. if the sequence {ak/bk}k≥0 is increasing (or decreasing), then the function x 7→ f(x)/g(x) is also increasing (or decreasing) on (0,r). the above lemma still holds when both f and g are even, or both are odd functions. theorem 3.1. the following results holds true for the modified k-bessel functions. (1) for µ ≥ ν > −1, the function x 7→ iγ,λk,µ(x)/i γ,λ k,ν (x) is increasing on (0,∞) for some fixed k > 0. (2) if k ≥ λ ≥ m > 0, the function x 7→ iγ,λk,ν (x)/i γ,λ m,ν(x) is increasing on (0,∞) for some fixed ν > −1 and γ ≥ ν + 1. (3) the function ν 7→ iγ,λk,ν (x) is log-convex on (0,∞) for some fixed k,γ > 0 and x > 0. here, iγ,λk,ν (x) := γk(ν + 1)i γ,λ k,ν (x). (4) suppose that λ ≥ k > 0 and ν > −1. then (a) the function x 7→ iγ,λk,ν (x)/φk (a,c; x) is decreasing on (0,∞) for a ≥ c > 0 and 0 < γ ≤ ν + 1. here, φk (a; c; x) is the k-confluent hypergeometric functions. (b) the function x 7→ iγ,λk,ν (x)/φk (γ; λ; x/2) is decreasing on (0, 1) for γ > 0 and 0 < k ≤ λ ≤ ν + 1. (c) the function x 7→ iγ,λk,ν (x)/φk (γ; λ; x/2) is decreasing on [1,∞) for γ > 0 and 0 < k ≤ min{λ,ν + 1}. proof. (1) form (1.4) it follows that i γ,λ k,ν (x) = ∞∑ n=0 an(ν)x n and i γ,λ k,ν (x) = ∞∑ n=0 an(µ)x n, where an(ν) = (γ)n,k γk(λn + ν + 1)(n!)22n and an(µ) = (γ)n,k γk(λn + µ + 1)(n!)22n consider the function f(t) := γk(λt + µ + 1) γk(λt + ν + 1) . then the logarithmic differentiation yields f′(t) f(t) = λ(ψk(λt + µ + 1) − ψk(λt + ν + 1)). here, ψk = γ ′ k/γk is the k-digamma functions studied in [5] and defined by ψk(t) = log(k) −γ1 k − 1 t + ∞∑ n=1 t nk(nk + t) (3.1) where γ1 is the euler-mascheronis constant. a calculation yields ψ′k(t) = ∞∑ n=0 1 (nk + t)2 , k > 0 and t > 0. (3.2) clearly, ψk is increasing on (0,∞) and hence f′(t) > 0 for all t ≥ 0 if µ ≥ ν > −1. this, in particular, implies that the sequence {dn}n≥0 = {an(ν)/an(µ)}n≥0 is increasing and hence the conclusion follows from lemma 3.1. (2). this result also follows from lemma 3.1 if the sequence {dn}n≥0 = {akn(ν)/amn (µ)}n≥0 is increasing for k ≥ m > 0. here, akn (ν) = (γ)n,k γk (λn + ν + 1) (n!) 2 and amn (ν) = (γ)n,m γm (λn + ν + 1) (n!) 2 , 206 mondal and nisar which together with the identity γk (x + nk) = γk(x)(x)n,k gives dn = (γ)n,k (γ)n,m γm (λn + ν + 1) γk (λn + ν + 1) = γk (γ + nk) γm (λn + ν + 1) γk (γ + nm) γk (λn + ν + 1) . now to show that {dn} is increase, consider the function f(y) := γk (γ + yk) γm (λy + ν + 1) γk (γ + ym) γk (λy + ν + 1) the logarithmic differentiation of f yields f′(y) f(y) = kψk(γ + yk) + λψm (λy + ν + 1) −mψm(γ + ym) −λψk (λy + ν + 1) (3.3) if γ ≥ ν + 1 and k ≥ λ ≥ m, then (3.3) can be rewrite as f′(y) f(y) ≥ λ ( ψk(ν + 1 + yk) − ψk (λy + ν + 1) ) + m ( ψm (λy + ν + 1) − ψm(ν + 1 + ym) ) ≥ 0. (3.4) this conclude that f, and consequently the sequence {dn}n≥0, is increasing. finally the result follows from the lemma 3.1. (3). it is known that sum of the log-convex functions is log-convex. thus, to prove the result it is enough to show that ν 7→ akn (ν) := (γ)n,k γk (ν + 1) γk (λn + ν + 1) (n!) 2 is log-convex. a logarithmic differentiation of an(ν) with respect to ν yields ∂ ∂ν log ( akn (ν) ) = ψk (ν + 1) − ψk (λn + ν + 1) . this along with (3.2) gives ∂2 ∂ν2 log ( akn (ν) ) = ψ′k (ν + 1) − ψ ′ k (λn + ν + 1) = ∞∑ r=0 1 (rk + ν + 1)2 − ∞∑ r=0 1 (rk + λn + ν + 1)2 = ∞∑ r=0 λn(2rk + λn + 2ν + 2) (rk + ν + 1)2(rk + λn + ν + 1)2 > 0, for all n ≥ 0, k > 0 and ν > −1. thus, ν 7→ akn (ν) is log-convex and hence the conclusion. (4). denote φk (a,c; x) = ∑∞ n=0 cn,k(a,c)x n and i γ,λ k,ν (x) = ∑∞ n=0 an(ν)x n, where an(ν) = (γ)n,k γk(λn + ν + 1)(n!)22n and dn,k (a,c) = (a)n,k (c)n,k n! with v > −1 and a,c,λ,γ,k > 0. to apply lemma 3.1, consider the sequence {wn}n≥0 defined by wn = an (ν) dn,k (a,c) = γk (γ + nk) 2nγk (γ) γk (λn + α + 1) (n!) 2 . γk (a) γk (c + nk) n! γk (a + nk) γk (c) = γk (a) γk (γ) γk (c) ρk (n) where ρk (x) = γk (γ + xk) γk (c + xk) γk (λx + ν + 1) γk (a + xk) 2xγ(x + 1) . inequalities for the modified k-bessel function 207 in view of the increasing properties of ψk on (0,∞), and ρ′ (x) ρ (x) = kψk (γ + xk) + kψk (c + xk) −λψk (λx + α + 1) −kψk (a + xk) , it follows that for a ≥ c > 0, λ ≥ k and ν + 1 ≥ γ, the function ρ is decreasing on (0,∞) and thus the sequence {wn}n≥0 also decreasing. finally the conclusion for (a) follows from the lemma 3.1. in the case (b) and (c), the sequence {wn} reduces to wn = an (ν) dn,k (γ,λ) = ρk (n) γk (λ) where ρk (x) = γk (λ + xk) γk(ν + 1 + λx)γ(x + 1) . now as in the proof of part (a) ρ′k (x) ρk (x) = kψk(λ + xk) −λψk(ν + 1 + xk) − ψ(x + 1) > 0, if ν + 1 + λx ≥ λ + xk. now for x ∈ (0, 1), this inequality holds if 0 < k ≤ λ ≤ ν + 1, while for x ≥ 1, it is required that k ≤ min{λ,ν + 1}. � references [1] l.g. romero, g.a.dorrego and r.a. cerutti, the k-bessel function of first kind, int. math. forum, 38(7)(2012), 1859–1854. [2] gn. watson, a treatise on the theory of bessel functions, cambridge mathematical library edition, cambridge university press, cambridge (1995). reprinted (1996) [3] a. erdélyi, w. magnus, f. oberhettinger and f.g. tricomi, higher transcendental functions, i, ii, mcgraw-hill book company, inc., new york, 1953. new york, toronto, london, 1953. [4] r. diaz and e. pariguan, on hypergeometric functions and k-pochhammer symbol, divulg. mat. 15(2) (2007), 179–192. [5] k. nantomah, e. prempeh, some inequalities for the k-digamma function, math. aeterna, 4(5) (2014), 521–525. [6] s. mubeen, m. naz and g. rahman, a note on k-hypergemetric differential equations, j. inequal. spec. funct. 4(3) (2013), 8–43. [7] m. biernacki and j. krzyż, on the monotonicity of certain functionals in the theory of analytic functions, ann. univ. mariae curie-sk lodowska. sect. a. 9 (1957), 135–147. [8] c. g. kokologiannaki, properties and inequalities of generalized k-gamma, beta and zeta functions, int. j. contemp. math. sci. 5(13-16) (2010), 653–660. [9] c. g. kokologiannaki and v. krasniqi, some properties of the k-gamma function, matematiche (catania), 68(1) (2013), 13–22. [10] v. krasniqi, a limit for the k-gamma and k-beta function, int. math. forum, 5(33-36) (2010), 1613–1617. [11] m. mansour, determining the k-generalized gamma function γk(x) by functional equations, int. j. contemp. math. sci., 4(21-24) (2009), 1037–1042. [12] g. e. andrews, r. askey and r. roy, special functions, cambridge univ. press, cambridge, 1999. [13] c. fox, the asymptotic expansion of generalized hypergeometric functions, proc. london math. soc., 27(4) (1928), 389–400. [14] a. a. kilbas, m. saigo and j. j. trujillo, on the generalized wright function, fract. calc. appl. anal. 5(4) (2002), 437–460. [15] a. a. kilbas and n. sebastian, generalized fractional integration of bessel function of the first kind, integral transforms spec. funct. 19 (11-12) (2008), 869–883. [16] a. a. kilbas, h. m. srivastava, and j. j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies 204, elsevier, amsterdam, 2006. [17] e. d. rainville, special functions, macmillan, new york, 1960. [18] e. m. wright, the asymptotic expansion of integral functions defined by taylor series, philos. trans. roy. soc. london, ser. a. 238 (1940), 423–451. [19] e. m. wright, the asymptotic expansion of the generalized hypergeometric function, proc. london math. soc. (2) 46(1940), 389–408. 1 department of mathematics and statistics, college of science, king faisal university, hofuf, kingdom of saudi arabia 2 department of mathematics, college of arts & science-wadi al-dawaser, prince sattam bin abdulaziz university, alkharj, kingdom of saudi arabia 208 mondal and nisar ∗corresponding author: n.sooppy@psau.edu.sa, ksnisar1@gmail.com 1. introduction 2. representation formula for the generalized k-bessel function 3. monotonicity and log-convexity properties references int. j. anal. appl. (2023), 21:34 solvability of nonlinear wave equation with nonlinear integral neumann conditions iqbal m. batiha1,2,∗, zainouba chebana3, taki-eddine oussaeif3, adel ouannas3, iqbal h. jebril1, mutaz shatnawi4 1department of mathematics, al-zaytoonah university of jordan, amman 11733, jordan 2nonlinear dynamics research center (ndrc), ajman university, ajman 346, uae 3department of mathematics and computer science, university of larbi ben m’hidi, oum el bouaghi, algeria 4department of mathematics, irbid national university, irbid 21110, jordan ∗corresponding author: i.batiha@zuj.edu.jo abstract. in this paper, we examine a nonlinear hyperbolic equation with a nonlinear integral condition. in particular, we prove the existence and the uniqueness of the linear problem by the fadeo galerkin method, and by applying an iterative process to some significant results obtained for the linear problem, the existence and the uniqueness of the weak solution for the nonlinear problem are additionally examined. 1. introduction the nonlinear hyperbolic equations describes important processes in the nonlinear evolution equation basis of mathematical models of diverse phenomena and processes in mechanics, physics, technology, biophysics, biology, ecology, and many other areas [1–4]. such ubiquitous occurrence of nonlinear hyperbolic equations is to be explained, first of all, by the fact that they are derived from fundamental laws in the real world [5,6]. let us remark only that for broad classes of equations, the fundamental questions of solvability and uniqueness of solutions of various boundary value problems have been solved and that the differentiability properties of the solutions have been studied in detail [7–9]. general results of the solvability and uniqueness were inferred by different methods such as the energy method, upper lower method received: dec. 25, 2022. 2010 mathematics subject classification. 35d30, 35l05, 35l70. key words and phrases. hyperbolic equation; nonlinear integral condition; weak existence. https://doi.org/10.28924/2291-8639-21-2023-34 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-34 2 int. j. anal. appl. (2023), 21:34 and the fadeo galerkin methods. the later one is regarded one of the most important methods that were mainly developed in the 1960s, but they are still powerful tools today to deal with nonlinear evolution equations, especially those who are modeled by non-classical boundary conditions that consist of integral conditions [10–13]. non-local and integral partial differential equations are used to solve a vast range of current physics and technology challenges [14–19]. when it is hard to directly measure the minimum and maximum values on the border, the overall value or average is known. this method might be utilized for modeling where we can model more complicated domain with nonlinear integral condition. motivated by the above perspective, we trait in this work to discuss a nonlinear evolution equation with a nonlinear integral condition. in particular, we aim to focus on the solvability of the solution of nonlinear hyperbolic problems with the integral condition of the second type by the method of fadeo-galerkin. in the following sections, we present first the existence of the linear problem, and then by applying an iterative process based on the results obtained for the linear problem, we prove the existence and the uniqueness of the weak solution of the nonlinear problem. 2. the statement of the main problem in this section, we let q = { (x,t) ∈r2, x ∈ ω = ]0, l[ and 0 < t < t } , besides we consider the main following initial boundary value problem for a nonlinear hyperbolic equation:  ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t,u,ux ) u(x, 0) = ϕ(x) ut(x, 0) = ψ(x) ∂u ∂x (0,t) = ∫ l 0 k(x,t)g(ut)(x,t)dx ∂u ∂x (l, t) = ∫ l 0 k(x,t)h(ut)(x,t)dx. (p1) assuming that f ∈ l2 (q) and ϕ,ψ ∈ l2 (ω). the nonlinear hyperbolic equation is given as follows: lu = ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t,u,ux ), (2.1) which satisfies the following identities: • the initial conditions `u = { u(x, 0) = ϕ (x) ut (x, 0) = ψ (x) , x ∈ (0, l) . • the boundary conditions are integral conditions of the second type defined as: ∂u ∂x (0,t) = ∫ l 0 k(x,t)g(ut)(x,t)dx, t ∈ (0,t ) , ∂u ∂x (l, t) = ∫ l 0 k(x,t)h(ut)(x,t)dx, t ∈ (0,t ) , int. j. anal. appl. (2023), 21:34 3 where k(x,t) ≥ 0 ∀(x,t) ∈ qand g(ut)(x,t) ≤ h(ut)(x,t) ∀(x,t) ∈ q, and for all v ∈ l2(q), we have: ‖g(v)‖l2(q) ≤ c‖v‖l∞(0,t,l2(ω)) , in which we define the space v by v = h1 (ω). actually, the space v is provided with the norm ‖v‖v = ‖v‖h1(ω), and hence it is a hilbert space. from this point of view, we are now able to formulate problem (p2) in order to precisely study it. from this fact, we will need to the following hypothesis: (h) : { f ∈ l2 ( 0,t ; l2 (ω) ) (h.1) ϕ ∈ h1 (ω) (h.2) . 3. position of problem (p1) in the rectangular area q = ω×(0,t ), and t < ∞, we consider the following linear problem (p2):  ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t) ∀(x,t) ∈ q u (x, 0) = ϕ (x) ∀x ∈ (0, l) ut (x, 0) = ψ (x) ∀x ∈ (0, l) ∂u ∂x (0,t) = ∫ l 0 k(x,t)g(ut)(x,t)dx ∀t ∈ (0,t ) ∂u ∂x (l, t) = ∫ l 0 k(x,t)g(ut)(x,t)dx ∀t ∈ (0,t ) (p2) in which the hyperbolic equation is given as follows: lu = ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t), (3.1) with the initial conditions: `u = { u(x, 0) = ϕ (x) ut (x, 0) = ψ (x) , x ∈ (0, l) , and with the integral condition of the second type: ∂u ∂x (0,t) = ∫ l 0 k(x,t)g(ut)(x,t)dx, t ∈ (0,t ) , ∂u ∂x (l, t) = ∫ l 0 k(x,t)h(ut)(x,t)dx, t ∈ (0,t ) , where k(x,t) ≥ 0 ∀(x,t) ∈ q and 0 ≤ g(ut)(x,t) ≤ h(ut)(x,t) ∀(x,t) ∈ q, and ‖g(v)‖l2(q) ≤ c‖v‖l∞(0,t,l2(q)) , 4 int. j. anal. appl. (2023), 21:34 such that the space v = h1 (ω) provided with the norm ‖v‖v = ‖v‖h1(ω) is a hilbert space. we are now able to formulate the problem (p2), precisely to study it, according to the following hypothesis: (h) : { f ∈ l2 ( 0,t ; l2 (ω) ) (h.1) ϕ ∈ h1 (ω) (h.2) . definition 3.1. the weak solution of problem (p2) is a function that satisfies: • u ∈ l2 ( 0,t ; h1 (ω) ) ∩l∞ ( 0,t ; h1 (ω) ) . • u admits a strong derivative ∂u ∂t ∈ l2 ( 0,t ; l2 (ω) ) . • u (0) = ϕ,ut (0) = ψ. • the following identity: (utt,v) + a (ux,vx ) = (f ,v) + ux (l, t)v(l) −ux (0,t)v(0) ∀v ∈ v, ∀t ∈ [0,t ] . 3.1. variational formulation. by multiplying the equation: ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t) (3.2) by an element v ∈ v , and the by integrating the result over ω, we obtain:∫ ω ∂2u ∂t2 ·vdx −a ∫ ω ∂2u ∂x2 ·vdx = ∫ ω f ·vdx. (3.3) by using the boundary conditions and using green’s formula, ( 3.3) becomes (utt,v) + a (ux,vx ) = (f ,vt) + ux (l, t)v(l) −ux (0,t)v(0), ∀v ∈ v, (3.4) where (·, ·) denotes the scalar product l2 (ω). 3.2. study of the existence of weak solution of problem (p2). the demonstration of the existence of the solution of problem (p2) can be discussed based on the faedo-galerkin method which consists of carrying out the next three steps. 3.2.1. step 1: construction of the approximate solutions. as the space v is separable, then there exists a sequence w1,w2, · · · ,wm having the following properties:  wi ∈ v, ∀i, ∀m,w1,w2, ...,wm are linearly independent, vm = 〈{w1,w2, ...,wm}〉 is dense in v. (3.5) in particular, we can say: ∀ϕ ∈ v =⇒ ∃(αkm)m ∈ in∗, ϕm = m∑ k=1 αkmwk −→ ϕ when m −→ +∞. (3.6) ∀ψ ∈ v =⇒ ∃(βkm)m ∈ in∗, ψm = m∑ k=1 βkmwk −→ ϕ when m −→ +∞. (3.7) int. j. anal. appl. (2023), 21:34 5 now, the faedo galerkin’s approximation aims to search about a function in which t 7→ um (x,t) = m∑ i=1 gim (t) wi (x) verifies { um (t) ∈ vm, ∀t ∈ [0,t ] ((um(t))tt ,wk) + a (um(t),wk) = (f (t),wk) ∀k = 1,m , (p3) for any integer m ≥ 1, where ((um(t))tt ,wk) = (( m∑ i=1 gim (t) wi ) tt ,wk ) = ( m∑ i=1 ∂2gim ∂t (t) wi (x) ,wk ) = m∑ i=1 (wi,wk) ∂2gim ∂t2 (t) , (3.8) and a (um(t),wk) = a ( m∑ i=1 gim (t) wi,wk ) = a m∑ i=1 gim (t)  ∫ ω ∂wi ∂x ∂wk ∂x dx − ∂wi ∂x (l) wk(l) + ∂wi ∂x (0) wk(0)   = a m∑ i=1 gim (t) ∫ ω ∂wi (x) ∂x ∂wk(x) ∂x dx −a m∑ i=1 gim (t) ∂wi ∂x (l) wk(l) + a m∑ i=1 gim (t) ∂wi ∂x (0) wk(0) = m∑ i=1 a (wi,wk) gim (t) . (3.9) in addition, we have um(0) = m∑ i=1 gim (0) wi (x) = ϕm = m∑ i=1 αimwi (x). and u′m(0) = m∑ i=1 g′im (0) wi (x) = βm = m∑ i=1 βimwi (x). we obtain consequently a system of first-order nonlinear differential equations:  m∑ i=1 (wi,wk) ∂2gim ∂t2 (t) + a m∑ i=1 ( ∂wi ∂x , ∂wk ∂x ) gim (t) = (f (t),wk) + a m∑ i=1 gim (t) ∂wi ∂x (l) wk(l) −a m∑ i=1 gim (t) ∂wi ∂x (0) wk(0) gim (0) = αim ∀i = 1,m. g′im (0) = βim ∀i = 1,m. , (p4) from this view, we consider the vector: gm = (g1m(t), · · · ,gmm(t)) , fm = ((f ,w1) , · · · , (f ,wm)) , 6 int. j. anal. appl. (2023), 21:34 coupled with the matrices: bm = (( wi,wj )) 1≤i≤m 1≤j≤m ,am = (( ∂wi ∂x , ∂wj ∂x )) 1≤i≤m 1≤j≤m , and cm = ( ∂wi ∂x (l) ·wj(l) ) 1≤i≤m 1≤j≤m ,dm = ( ∂wi ∂x (0) ·wj(0) ) 1≤i≤m 1≤j≤m . now, we can immediately write problem (p4) in the matrix form as:  bm ∂gm ∂t (t) + aamgm + admgm = fm + acmgm gm (0) = (αim)1≤i≤m g′m (0) = (βim)1≤i≤m , as the matrix entries bm are linearly independent (because it is a diagonal matrix), then det bm 6= 0. so, it is invertible, and then gm is the solution of the following states:  ∂2gm ∂t2 (t) + ( ab−1m am + bb −1 m dm −ab−1m cm ) gm = b −1 m fm gm (0) = (αim)1≤i≤m . g′m (0) = (βim)1≤i≤m . (p5) now, it is easy to verify that this ordinary differential system has a solution where the matrix: ( ab−1m am + bb −1 m bm −ab −1 m cm ) is of constant coefficients and the vector b−1m fm are continuous functions and majorized by integrable functions on (0,t ). consequently, we can conclude that there exists a tm that depends only on |αim| and |βim|. 3.2.2. step 2: a priori estimate. herein, we intend to begin this step with state and prove the next result. lemma 3.1. for all m ∈n∗, if 1 8a > k2, the solution um ∈ l2 (0,t ; vm) of problem (p2) satisfies: ‖um‖l2(0,t ; h1(ω)) ≤ c1,∥∥∥∥∂um∂t ∥∥∥∥ l2(0,t ; l2(ω)) ≤ c2, where c1 and c2 are two positive constants independent of m. int. j. anal. appl. (2023), 21:34 7 proof. by multiplying the equation of (p3) by gkm(t), and then by summing the result over k, we obtain: m∑ k=1 ((um(t))tt ,wk) g ′ km(t) + a m∑ k=1 ( ∂um ∂x (t), ∂wk ∂x∂t ) g′km(t) = m∑ k=1 (f (t),wk) ·g′km(t) + a m∑ i=1 gim (t) ∂wi ∂x (l) m∑ k=1 g′km(t)wk(l) −a m∑ i=1 gim (t) ∂wi ∂x (0) m∑ k=1 g′km(t)wk(0). so, we obtain ((um(t))tt , (um(t))t) + a ( ∂um ∂x (t), ∂um ∂x∂t (t) ) = (f (t), (um(t))t) + a m∑ i=1 gim (t) ∂wi ∂x (l) m∑ k=1 g′km(t)wk(l) −a m∑ i=1 gim (t) ∂wi ∂x (0) m∑ k=1 g′km(t)wk(0). thus, we get ((um(t))tt , (um(t))t) + a ∥∥∥∥∂um∂x ∥∥∥∥2 l2(ω) = (f (t),um(t)) + a m∑ i=1 gim (t) ∂wi ∂x (l) m∑ k=1 g′km(t)wk(l) −a m∑ i=1 gim (t) ∂wi ∂x (0) m∑ k=1 g′km(t)wk(0). integrating the above equality over 0 to t coupled wit using the cauchy inequality with ε, i.e. |ab| ≤ a2 2ε + εb2 2 , we get 1 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(ω) + a 2 ‖∇um‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + a m∑ i=1 gim (t) ∂wi ∂x (l) m∑ k=1 g′km(t)wk(l) −a m∑ i=1 gim (t) ∂wi ∂x (0) m∑ k=1 g′km(t)wk(0) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) , ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + a ∂um ∂x (l, t) ∂um ∂t (l, t) −a ∂um ∂x (0,t) ∂um ∂t (0,t), ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) + a ∫ τ 0 [(∫ ω k(x,t)g ((ut)m) (x,t)dx ) ∂um ∂t (l, t) 8 int. j. anal. appl. (2023), 21:34 − (∫ ω k(x,t)h ((ut)m) (x,t)dx ) ∂um ∂t (0,t) ] ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) + a ∫ τ 0 [(∫ ω k(x,t)g ((ut)m) (x,t)dx ) ∂um ∂t (l, t) − (∫ ω k(x,t)h ((ut)m) (x,t)dx ) ∂um ∂t (0,t) ] . this means 1 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(ω) + a 2 ‖∇um‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) + a ∫ τ 0 [(∫ ω k(x,t)g ((ut)m) (x,t)dx ) ∂um ∂t (l, t) − (∫ ω k(x,t)g ((ut)m) (x,t)dx ) ∂um ∂t (0,t) ] ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) + ack ∥∥∥∥∂um∂t ∥∥∥∥ l∞(0,t ;l2(ω)) [∫ τ 0 ∫ ω ∂2um ∂t∂x dxdt ] ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) + akc ∥∥∥∥∂um∂t ∥∥∥∥ l∞(0,t ;l2(ω)) [∫ ω ∂um ∂x dx − ∫ ω ∂ψm ∂x dx ] , which yields 1 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(ω) + a 2 ‖∇um‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) + [(ack)2 2δ ∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + δ [ ‖∇um‖2l∞(0,t ;l2(ω)) + ‖∇ϕm‖ 2 l∞(0,t ;l2(ω)) ]] + 1 2 ‖ϕm‖2l2(ω) + a 2 ‖∇ψm‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ε 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(q) (ack) 2 2δ ∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + δ‖∇um‖2l∞(0,t ;l2(ω)) + a 2 ‖∇ψm‖2l2(ω) + max( 1 2 ,δ)‖ϕm‖2h1(ω) , int. j. anal. appl. (2023), 21:34 9 or 1 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(ω) + a 2 ‖∇um‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ( εct 2 + (ack) 2 2δ )∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + δ‖∇um‖2l∞(0,t ;l2(ω)) + a 2 ‖∇ψm‖2l2(ω) + max( 1 2 ,ackδ)‖ϕm‖2h1(ω) , (3.10) where k = max ∫ q k2 (x,t) dxdt. consequently, we obtain 1 2 ∥∥∥∥∂um∂t ∥∥∥∥2 l2(ω) + a 2 ‖∇um‖2l2(ω) ≤ 1 2ε ‖f‖2l2(q) + ( εct 2 + (ack) 2 2δ )∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + δ‖∇um‖2l∞(0,t ;l2(ω)) + a 2 ‖∇ψm‖2l2(ω) + max( 1 2 ,δ)‖ϕm‖2h1(ω) , which gives:( 1 2 − ( εct 2 + (ack) 2 2δ ))∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + (a 2 −δ ) ‖∇um‖2l∞(0,t ;l2(ω)) ≤ 1 2ε ‖f‖2l2(q) + a 2 ‖∇ψm‖2l2(ω) + max( 1 2 ,δ)‖ϕm‖2h1(ω) by putting ε = 1 4ct and δ = 4 (ack)2, we get∥∥∥∥∂um∂t ∥∥∥∥2 l∞(0,t ;l2(ω)) + ‖∇um‖2l∞(0,t ;l2(ω)) ≤ c1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ] , (3.11) or c1 = max ( 2ct , a 2 , max( 1 2 , (ack) 2 ) ) min { 1 4 , ( a 2 − 4 (ack)2) )} . from (3.11), we can also get:∥∥∥∥∂um∂t ∥∥∥∥ l2(ω) ≤ √ c1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ]1 2 . (3.12) by integrating (3.12) over [0,t ], we obtain:∥∥∥∥∥∥ τ∫ 0 ∂um ∂t ∥∥∥∥∥∥ l2(ω) ≤ τ∫ 0 ∥∥∥∥∂um∂t ∥∥∥∥ l2(ω) ≤ t √ c1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ]1 2 , 10 int. j. anal. appl. (2023), 21:34 or ∥∥∥∥∥∥ τ∫ 0 ∂um ∂t ∥∥∥∥∥∥ l2(ω) ≤ t √ c1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ]1 2 , which implies: ‖um −ϕm‖l2(ω) ≤ t √ c1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ]1 2 , i.e., ‖um‖2l2(ω) + ‖ϕm‖ 2 l2(ω) ≤ tc1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ] + 2‖um‖l2(ω) ‖ϕm‖l2(ω) . by applying cauchy inequality with γ, we get: ‖um‖2l2(ω) + ‖ϕm‖ 2 l2(ω) ≤ tc1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ] + 1 γ ‖um‖2l2(ω) + γ‖ϕm‖ 2 l2(ω) . by putting γ = 2, we can have: ‖um‖2l2(ω) ≤ 4tc1 [ ‖f‖2l2(q) + ‖∇ψm‖ 2 l2(ω) + ‖ϕm‖ 2 h1(ω) ] . (3.13) now, it follows from (3.11) and (3.13) that the solution of the initial value problem for system (p4) can be extended to [0,t ]. this confirms what we have demonstrated in the first step. consequently, when m → +∞ in (3.13), we obtain:{ um uniformly bounded in l2(0,t ; h1 (ω)) (um)t uniformly bounded in l 2 ( 0,t ; l2 (ω) ) . (3.14) � 3.2.3. step 3: convergence and the result of existence. theorem 3.1. there is a function u ∈ l2(0,t ; h1 (ω)) ∩ l∞(0,t ; l2 (ω)) with ∂u ∂t ∈ l2 ( 0,t;l2 (ω) ) and a subsequence denoted by ( umk ) k ⊆ (um)m such that  umk ⇀ u in l 2(0,t;h1 (ω)) ∂umk ∂t ⇀ ∂u ∂t in l2 ( 0,t;l2 (ω) ) , as m −→ +∞. proof. from lemma 1.2, we might deduce that there are subsequences denoted respectively by ( umk ) ,( ∂umk ∂t ) of (um) and (um)t such that umk ⇀ u in l 2(0,t ; h1 (ω)) , (3.15) and ∂umk ∂t ⇀ w in l2 ( 0,t ; l2 (ω) ) . (3.16) int. j. anal. appl. (2023), 21:34 11 we know that according to relikh-kondrachoff’s theorem the injection of h1 (q) into l2 (q) is compact. in addition, like the results of rellich’s theorem, any weakly convergent sequence in h1 (q) has a subsequence which converges strongly in l2 (q) . so, we can assert: umk −→ u in l 2(q) . (3.17) on the other hand, from lemma 1.3, there is a subsequence of ( umk ) k , which is still denoted by umk, converges almost everywhere to u such that umk −→ u almost everywhere q . (3.18) it is still essential to demonstrate that w = ∂u ∂t . this actually suffices to prove: u(t) = ϕ + t∫ 0 w(τ)dτ. (3.19) to this aim, we note that as umk ⇀ u inl 2(0,t;l2 (ω)) , then the proof of (3.19) is equivalent to prove that umk ⇀ ϕ + χ in l 2(0,t ;l2 (ω)) , which means lim ( umk −ϕ−χ,v ) l2(0,t ;l2(ω)) = 0, ∀v ∈ l 2(0,t;l2 (ω)), as χ (t) = t∫ 0 w(τ)dτ. in fact, by using the equality umk −ϕmk = t∫ 0 ∂umk ∂τ dτ, for all t ∈ [0,t ] , with the help of using umk ∈ l 2 ( 0,t ; vmk ) and ( umk ) t ∈ l2 ( 0,t ; vmk ) , we can get:( umk −ϕ− t∫ 0 w(τ)dτ,v ) l2(0,t ;l2(ω)) =  umk −ϕmk − t∫ 0 w(τ)dτ,v   l2(0,t ;l2(ω)) + ( ϕmk −ϕ,v ) l2(0,t ;l2(ω)) =   t∫ 0 ( ∂umk ∂τ −w(τ) ) dτ,v   l2(0,t ;l2(ω)) + ( ϕmk −ϕ,v ) l2(0,t ;l2(ω)) , 12 int. j. anal. appl. (2023), 21:34 for all t ∈ [0,t ]. by virtue of part (ii) of lemma 1.6, it comes( umk −ϕ− t∫ 0 w(τ)dτ,v ) l2(0,t ;l2(ω)) = t∫ 0 ( ∂umk ∂τ −w(τ),v ) l2(0,t ;l2(ω)) dτ + ( ϕmk −ϕ,v ) l2(0,t ;l2(ω)) , for all t ∈ [0,t ]. on the one hand, we have lim k−→∞ t∫ 0 ( ∂umk ∂τ −w(τ),v ) l2(0,t ;l2(ω)) dτ = 0, (3.20) for t ∈ [0,t ]. besides, we have: lim k−→∞ (ϕm −ϕ,v)l2(0,t ;l2(ω)) = 0, (3.21) which implies: lim k−→∞ ( umk −ϕ−χ,v ) l2(0,t ; l2(ω)) = 0, ∀v ∈ l 2 ( 0,t ; l2 (ω) ) . � theorem 3.2. the function u of the theorem (3.1) is the weak solution to the problem (p2) in the sense of the definition 3.1. proof. from theorem (3.1), we have shown that the limit function u satisfies the first two conditions of the definition 3.1. now we will demonstrate (iii). according to the theorem 3.1, we have: umk (0) ⇀ u (0) in l 2(ω) . on the other hand, we have umk (0) −→ ϕ in l 2(ω) , which implies: umk (0) ⇀ ϕ in l 2(ω) . from the uniqueness of the limit, we get u (0) = ϕ. by using the same previous steps, we demonstrate ut (0) = ψ. it remains to demonstrate (iv). to this aim, we have: (utt,v) + a (u,v) = (f ,v) ∀v ∈ v, and ∀t ∈ [0,t ] . integrating (p3) over (0,t ), we obtain: t∫ 0 ((um(t))tt ,wk) dτ + t∫ 0 a (um(t),wk) dτ = t∫ 0 (f (t),wk) dτ, (3.22) int. j. anal. appl. (2023), 21:34 13 for all k = 1,m and for all t ∈ [0,t ]. using (3.13) and that vm dense in v and passing to the limit in (3.22), we get: t∫ 0 (utt,wk) dτ + t∫ 0 a (u,wk) dτ = t∫ 0 (f ,wk) dτ, ∀t ∈ [0,t ] . this immediately implies that (iv) is verified. � corollary 3.1. the uniqueness of the solution of problem (p2) comes straight through the estimate (3.11). 4. weak solution of the nonlinear problem initially, we present the considered solution’s concept. for this purpose, we let v = v(x,t) be any function of v such that v = { v ∈ c1 (q) , vx (l, t) = vx (0,t) = 0, t ∈ [0,t ] } . by multiplying ∂2u ∂t2 −a ∂2u ∂x2 = f (x,t,u,ux ) by v and integrating the result over qτ, we obtain∫ qτ ∂2y ∂t2 (x,t) · ∂v ∂t (x,t)dxdt −a ∫ qτ ∆y(x,t) · ∂v ∂t (x,t)dxdt = ∫ qτ g (x,t,y,yx ) · ∂v ∂t (x,t)dxdt. now, by using integration by parts and the conditions on y and v, we get∫ qτ ∂2y ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a ∫ qτ ∂y ∂x (x,t) · ∂2v ∂x∂t (x,t)dxdt = ∫ qτ g(x,t,y,yx ) · ∂v ∂t (x,t)dxdt. (4.1) it then results from (4.1) that a (y,v) = ∫ qτ g(x,t,y,yx ) · ∂v ∂t (x,t)dxdt, (4.2) or a (y,v) = ∫ qτ ∂2y ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a ∫ qτ ∂y ∂x (x,t) · ∂2v ∂x∂t (x,t)dxdt. 14 int. j. anal. appl. (2023), 21:34 thus, it is the time to build a recurring sequence starting with y(0) = 0. the sequence ( y(n) ) n∈n is defined as follows: given the element y(n−1), then for n = 1, 2, 3, · · · , we can solve the following problem:   ∂2y(n) ∂t2 −a∆y(n) = g ( x,t,y(n−1),y (n−1) x ) y(n) (x, 0) = 0 y (n) t (x, 0) = 0 y (n) x (0,t) = 0 y (n) x (l, t) = 0 . (p4) according to the last linear problem, we fix the n each time. problem (p4) admits then a unique solution y(n) (x,t), which can be given by the fadeo-galarkin method. in this regard, we assume z(n) (x,t) = y(n+1) (x,t) −y(n)(x,t). as a result, we have the following new problem:  ∂2z(n) ∂t2 −a∆z(n) = p(n−1)(x,t) z(n) (x.0) = 0 z (n) t (x.0) = 0 z (n) x (0,t) = 0 z (n) x (l, t) dx = 0 , (p5) or p(n−1)(x,t) = g ( x,t,y(n),y (n) x ) −g ( x,t,y(n−1),y (n−1) x ) . multiplying ∂2z(n) ∂t2 −a∆z(n) = p(n−1)(x,t) by z(n) and then integrating the result over qτ yield:∫ qτ ∂2z(n) ∂t2 (x,t) · ∂z(n) ∂t (x,t)dxdt −a ∫ qτ ∆z(n)(x,t) · ∂z(n) ∂t (x,t) dxdt = ∫ qτ p(n−1)(x,t) · ∂z(n) ∂t (x,t) dxdt. if we apply an integration by parts for each term of the above equality, keeping in view the initial and boundary conditions, we get: 1 2 ∫ ω ( ∂z(n) ∂t (x,τ))2dx + a 2 ∫ ω ( ∂z(n) ∂x (x,t) )2 dxdt = ∫ qτ p(n−1)(x,t) · ∂z(n) ∂t (x,t) dxdt. int. j. anal. appl. (2023), 21:34 15 when the cauchy schwarz inequality is applied to the second portion of the above equation, the following result is obtained: 1 2 ∫ ω ( ∂z(n) ∂t (x,τ))2dx + a 2 ∫ ω ( ∂z(n) ∂x (x,t) )2 dxdt 6 1 2ε ∫ qτ | p(n−1)(x,t) |2 dxdt + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt, or 1 2 ∫ ω ( ∂z(n) ∂t (x,τ))2dx + a 2 ∫ ω ( ∂z(n) ∂x (x,t) )2 dx 6 1 2ε ∫ qτ | g ( x,t,y(n),y (n) x ) −g ( x,t,y(n−1),y (n−1) x ) |2 dxdt + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt. we deduce consequently that: 1 2 ∫ ω ( ∂z(n) ∂t (x,τ))2dx + a 2 ∫ ω ( ∂z(n) ∂x (x,t) )2 dx 6 k2 2ε ∫ qτ (| y(n) −y(n−1) | + | y(n)x −y (n−1) x |)2dxdt + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt 6 k2 2ε ∫ qτ (| z(n−1) | + | z(n−1)x |)2dxdt + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt 6 k2 ε ∫ qτ (| z(n−1) |2 + | z(n−1)x |2)dxdt + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt 6 k2 ε ∥∥∥z(n−1)∥∥∥ l2(0,t,h1(0,l)) + ε 2 ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt. by multiplying by 2 and applying grenwell’s lemma, we get ∫ ω ( ∂z(n) ∂t (x,τ))2dx + a ∫ ω ( ∂z(n) ∂x (x,t) )2 dx 6 k2 ε ∥∥∥z(n−1)∥∥∥ l2(0,t,h1(0,l)) + ε ∫ qτ ( ∂z(n) ∂t (x,t) )2 dxdt ≤ k2 ε exp(εt ) ∥∥∥z(n−1)∥∥∥ l2(0,t,h1(0,l)) . 16 int. j. anal. appl. (2023), 21:34 integrating over t yields:∫ qt ( ∂z(n) ∂t (x,τ))2dxdt + a ∫ qt ( ∂z(n) ∂x (x,t) )2 dxdt 6 tk2 ε ∥∥∥z(n−1)∥∥∥ l2(0,t,h1(0,l)) exp(εt ), or ∫ qt ( ∂z(n) ∂t (x,τ))2dxdt + ∫ qt ( ∂z(n) ∂x (x,t) )2 dxdt 6 tk2 exp(εt ) ε min(1,a) ∥∥∥z(n−1)∥∥∥ l2(0,t,h1(0,1)) . by putting c = tk2 exp(εt ) ε min(1,a) , we get: ‖ ∂z(n) ∂t ‖2l2(0,t,h1(0,1)) + ‖ ∂z(n) ∂x ‖2l2(0,t,h1(ω))6 c ∥∥∥z(n−1)∥∥∥2 l2(0,t,h1(ω)) . thus, by applying pointcarre, we have ‖ z(n) ‖2l2(0,t,h1(ω))6 tc ∥∥∥z(n−1)∥∥∥2 l2(0,t,h1(ω)) , and n−1∑ i=1 z(i) = y(n). according to the convergence criterion of the series ∞∑ n=1 z(n) that converges if |c| < 1, we obtain: ∣∣∣∣∣(tk) 2 exp(εt ) ε min(1,a) ∣∣∣∣∣ < 1 ⇒ tk √ exp(εt ) ε min(1,a) < 1. consequently, we get: k < √ ε min(1,a) exp(−εt ) t . then (y(n))n converges to an element of l2(0,t,h1(ω)), say y. now, we will show that lim n−→∞ y(n)(x,t) = y(x,t) is a solution to the problem (p5) by showing that y satisfies: a (y,v) = ∫ qτ g(x,t,y,yx ) ·v(x,t)dxdt. we therefore consider the weak formulation of the problem (p1) as follows: a ( y(n),v ) = ∫ qτ ∂2y(n) ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a 2 ∫ qτ ∂y(n) ∂x (x,t) · ∂2v ∂t∂x (x,t)dxdt. int. j. anal. appl. (2023), 21:34 17 from the linearity of a, we can have: a ( y(n),v ) = a ( y(n) −y,v ) + a (y,v) = ∫ qτ ∂2(y(n) −y) ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a 2 ∫ qτ ∂(y(n) −y) ∂x (x,t) · ∂2v ∂t∂x (x,t)dxdt − ∫ qτ ∂2y ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a 2 ∫ qτ ∂y ∂x (x,t) · ∂2v ∂t∂x (x,t)dxdt, which implies a ( y(n) −y,v ) = ∫ qτ ∂2(y(n) −y) ∂t2 (x,t) · ∂v ∂t (x,t)dxdt + a 2 ∫ qτ ∂(y(n) −y) ∂x (x,t) · ∂2v ∂t∂x (x,t)dxdt. now, by applying the cauchy schwartz inequality, the following results can be obtained: a ( y(n) −y,v ) 6‖ vt ‖l2(qτ ) ∥∥∥(y(n) −y)tt∥∥∥ l2(0,t,l2(ω)) + a 2 ‖ vxt ‖l2(qτ ) ∥∥∥(y(n) −y)x∥∥∥ l2(0,t,l2(ω)) . then, we can find a ( y(n) −y,v ) 6 c (∥∥∥(y(n) −y)tt∥∥∥ l2(0,t,l2(ω)) + ∥∥∥(y(n) −y)x∥∥∥ l2(0,t,l2(ω)) ) × ( ‖ vt ‖l2(qτ ) + ‖ vxt ‖l2(qτ ) ) , or c = max ( 1, a 2 ) . now, as y(n) −→ y in l2 ( 0,t,h1 (0, l) ) u h1 (q), we get: y(n) −→ y in l2 (q) , y (n) t −→ yt in l 2 (q) , y (n) x −→ yx in l2 (q) . consequently, we note as n −→ +∞, we find: lim n−→+∞ a ( y(n) −y,v ) = 0. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 18 int. j. anal. appl. (2023), 21:34 references [1] g. bahia, a. ouannas, i.m. batiha, z. odibat, the optimal homotopy analysis method applied on nonlinear time-fractional hyperbolic partial differential equations, numer. methods part. differ. equ. 37 (2020), 20082022. https://doi.org/10.1002/num.22639. [2] b. abdelfatah, o. taki-eddine, b.a. leila, a mixed problem with an integral two-space-variables condition for parabolic equation with the bessel operator, j. math. 2013 (2013), 457631. https://doi.org/10.1155/2013/ 457631. [3] d. sofiane, b. abdelfatah, o. taki-eddine, study of solution for a parabolic integrodifferential equation with the second kind integral condition, int. j. anal. appl 16 (2018), 569-593. https://doi.org/10.28924/ 2291-8639-16-2018-569. [4] o. taki-eddine, b. abdelfatah, a priori estimates for weak solution for a time-fractional nonlinear reactiondiffusion equations with an integral condition, chaos solitons fractals. 103 (2017), 79-89. https://doi.org/ 10.1016/j.chaos.2017.05.035. [5] t.e. oussaeif, a. bouziani, solvability of nonlinear goursat type problem for hyperbolic equation with integral condition, khayyam j. math. 4 (2018), 198-213. https://doi.org/10.22034/kjm.2018.65161. [6] t.e. oussaeif, a. bouziani, existence and uniqueness of solutions to parabolic fractional differential equations with integral conditions, electron. j. differ. equ. 2014 (2014), 179. [7] i.m. batiha, solvability of the solution of superlinear hyperbolic dirichlet problem, int. j. anal. appl. 20 (2022), 62. https://doi.org/10.28924/2291-8639-20-2022-62. [8] i.m. batiha, z. chebana, t.e. oussaeif, a. ouannas, i.h. jebril, on a weak solution of a fractional-order temporal equation, math. stat. 10 (2022), 1116-1120. https://doi.org/10.13189/ms.2022.100522. [9] t.e. oussaeif, b. antara, a. ouannas, i.m. batiha, k.m. saad, h. jahanshahi, a.m. aljuaid, a.a. aly, existence and uniqueness of the solution for an inverse problem of a fractional diffusion equation with integral condition, j. funct. spaces. 2022 (2022), 7667370. https://doi.org/10.1155/2022/7667370. [10] n. anakira, z. chebana, t.-e. oussaeif, i.m. batiha, a. ouannas, a study of a weak solution of a diffusion problem for a temporal fractional differential equation, nonlinear funct. anal. appl. 27 (2022), 679–689. https: //doi.org/10.22771/nfaa.2022.27.03.14. [11] i.m. batiha, a. ouannas, r. albadarneh, a.a. al-nana, s. momani, existence and uniqueness of solutions for generalized sturm–liouville and langevin equations via caputo–hadamard fractional-order operator, eng. comput. 39 (2022), 2581-2603. https://doi.org/10.1108/ec-07-2021-0393. [12] z. chebana, t.e. oussaeif, a. ouannas, i. batiha, solvability of dirichlet problem for a fractional partial differential equation by using energy inequality and faedo-galerkin method, innov. j. math. 1 (2022), 34-44. https://doi. org/10.55059/ijm.2022.1.1/4. [13] a. zraiqat, l.k. al-hwawcha, on exact solutions of second order nonlinear ordinary differential equations, appl. math. 06 (2015), 953-957. https://doi.org/10.4236/am.2015.66087. [14] r. imad, o. taki-eddine, solvability of a solution and controllability of partial fractional differential systems, j. interdiscip. math. 24 (2021), 1175-1200. https://doi.org/10.1080/09720502.2020.1838754. [15] r. imad, o. taki-eddine, b abdelouahab, solvability of a solution and controllability for nonlinear fractional differential equations, bull. inst. math. 15 (2020), 237-249. [16] t.e. oussaeif, a. bouziani, mixed problem with an integral two-space-variables condition for a class of hyperbolic equations, int. j. anal. 2013 (2013), 957163. https://doi.org/10.1155/2013/957163. [17] o. taki-eddine, b. abdelfatah, mixed problem with an integral two-space-variables condition for a parabolic equation, int. j. evol. equ. 9 (2014), 181-198. https://doi.org/10.1002/num.22639 https://doi.org/10.1155/2013/457631 https://doi.org/10.1155/2013/457631 https://doi.org/10.28924/2291-8639-16-2018-569 https://doi.org/10.28924/2291-8639-16-2018-569 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.22034/kjm.2018.65161 https://doi.org/10.28924/2291-8639-20-2022-62 https://doi.org/10.13189/ms.2022.100522 https://doi.org/10.1155/2022/7667370 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.1108/ec-07-2021-0393 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.4236/am.2015.66087 https://doi.org/10.1080/09720502.2020.1838754 https://doi.org/10.1155/2013/957163 int. j. anal. appl. (2023), 21:34 19 [18] o. taki eddine, b. abdelfatah, mixed problem with an integral two-space-variables condition for a third order parabolic equation, int. j. anal. appl. 12 (2016), 98-117. [19] t. e. oussaeif, a. bouziani, solvability of nonlinear viscosity equation with a boundary integral condition, j. nonlinear evol. equ. appl. 2015 (2015), 31-45. 1. introduction 2. the statement of the main problem 3. position of problem (p1) 3.1. variational formulation 3.2. study of the existence of weak solution of problem (p2) 4. weak solution of the nonlinear problem references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 46-56 http://www.etamaths.com some characterizations of general preinvex functions muhammad uzair awan1, muhammad aslam noor2,3, vishnu narayan mishra4,5,∗ and khalida inayat noor3 abstract. in this paper, we consider a new class of general preinvex functions involving an arbitrary function. we show that the optimality condition for general preinvex functions on general invex set can be characterized by a class of variational-like inequality. we also derive some integral inequalities of hermite-hadamard type via general preinvex functions. some special cases are also discussed. our results represent a significant refinement of the previously known results. these results may stimulate further research in this area. 1. introduction in recent years, several extensions and generalizations have been introduced and considered for classical convexity using novel and innovative techniques, see [1, 13]. a significant generalization of convex functions was that of invex functions which was introduced by hanson [7]. ben-israel and mond [8] introduced another class of convex functions, which is called preinvex functions. we remark that the differentiable preinvex functions are invex functions, but the converse may be true. it is well-known that the preinvex functions and invex sets may not be convex functions and convex sets. many researchers have investigated different properties of the preinvex functions and their role in different fields of sciences such as optimization, variational inequalities, equilibrium problems and integral inequalities, see [14, 15, 17, 18, 22–24]. another significant generalization of classical convex sets and functions was the introduction of general nonconvex (ϕ-convex) sets and general nonconvex (ϕ-convex) functions with respect to an arbitrary function, respectively by youness [25]. these general convex set may not be a classical convex set, see [5]. noor [16] has investigated the applications of general nonconvex functions in variational inequalities and optimization theory. it is obvious that preinvex functions and general convex functions are distinctly two different classes of convex functions. this motivated fulga et al. [6], to consider another class of convex functions by combining these two classes. this new class of convex function is called the general preinvex function. in this paper, we discuss some properties of the general preinvex functions. we show that the optimization of the differentiable general preinvex functions can be characterized by a class of variational-like inequality, which is called general variational-like inequality. in the last section, we derive some hermite-hadamard type inequalities via general preinvex functions. this may be starting point for some new research in this field. 2. preliminaries in this section, we define some basic results and also discuss several special cases. before proceeding further, we suppose kηϕ be a nonempty closed set in a hilbert space h. we denote 〈., .〉 by norm and ‖.‖ by inner product, respectively. also suppose that η(., .) : kηϕ × kηϕ → h and ϕ : h → h be arbitrary functions. definition 2.1 ( [25]). a set kϕ ⊆ rn is said to be a general convex (ϕ-convex) set, if and only if, there exists an arbitrary function ϕ such that, (1 − t)ϕ(u) + tϕ(v) ∈ kϕ, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kϕ, t ∈ [0, 1]. received 8th may, 2017; accepted 14th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 26a51, 26d15, 49j40, 47h10, 90c33. key words and phrases. convex functions; general preinvex functions; differentiability; hermite-hadamard inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 46 general preinvex functions 47 definition 2.2 ( [6]). a set kηϕ is said to be general invex set with respect to η(., .) and ϕ, if ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. (2.1) it is known that general invex set may not be a general convex set see [6]. note that, if η(ϕ(v),ϕ(u)) = ϕ(v)−ϕ(u), then our definition reduces to the definition of general convex set, which is mainly due to youness [25]. if along with η(ϕ(v),ϕ(u)) = ϕ(v) −ϕ(u), we have ϕ = i, where i is identity function, then we have the definition of classical convex set. definition 2.3 ( [6]). a function f on kηϕ is said to be general preinvex with respect to arbitrary functions η and ϕ, if f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. (2.2) if η(ϕ(v),ϕ(u)) = ϕ(v)−ϕ(u), then our definition reduces to the definition of general convex function [25]. if ϕ = i, where i is identity function, then we have the definition of preinvex functions [24]. definition 2.4. a function f is said to be general mid-preinvex with respect to arbitrary functions η and ϕ, if f ( 2ϕ(u) + η(ϕ(v),ϕ(u)) 2 ) ≤ f(ϕ(u)) + f(ϕ(v)) 2 , ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ. definition 2.5. a function f on kηϕ is said to be general semistrictly preinvex with respect to arbitrary functions η and ϕ, if and only if ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, f(ϕ(u)) 6= f(ϕ(v)), t ∈ (0, 1), we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. (2.3) definition 2.6. a function f on kηϕ is said to be general strictly preinvex with respect to arbitrary functions η and ϕ, if and only if ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, ϕ(u) 6= ϕ(v), t ∈ (0, 1), we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) < (1 − t)f(ϕ(u)) + tf(ϕ(v)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. (2.4) definition 2.7 ( [6]). a function f on general invex set kηϕ is said to be quasi general preinvex function, if f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ max{f(ϕ(u)),f(ϕ(v))}, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. definition 2.8. a function f on general invex set kηϕ is said to be general logarithmic preinvex function, if f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (f(ϕ(u)))1−t(f(ϕ(v)))t, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], where f(.) > 0. from above definition, we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (f(ϕ(u)))1−t(f(ϕ(v)))t ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)) ≤ max{f(ϕ(u)),f(ϕ(v))}. following conditions are useful in studying various properties of our proposed results. condition c. let η(., .) : kηϕ ×kηϕ → h satisfies the following assumptions η(ϕ(u),ϕ(u) + tη(ϕ(v),ϕ(u))) = −tη(ϕ(v),ϕ(u)), η(ϕ(v),ϕ(u) + tη(ϕ(v),ϕ(u))) = (1 − t)η(ϕ(v),ϕ(u)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. 48 awan, noor, mishra and noor for more information, see [9]. for t = 1 in definition 2.3, we have following condition. condition a. let f be general preinvex function, then f(ϕ(u) + η(ϕ(v),ϕ(u))) ≤ f(ϕ(v)). let kηϕ = iηϕ = [ϕ(a),ϕ(a) +η(ϕ(b),ϕ(a))] be the interval. we now define general preinvex functions on i. definition 2.9. let iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))]. then f is a general preinvex function, if and only if,∣∣∣∣∣∣ 1 1 1 ϕ(a) ϕ(x) ϕ(a) + η(ϕ(b),ϕ(a)) f(ϕ(a)) f(ϕ(x)) f(ϕ(a) + η(ϕ(b),ϕ(a))) ∣∣∣∣∣∣ ≥ 0; ϕ(a) ≤ ϕ(x) ≤ ϕ(a) + η(ϕ(b),ϕ(a)). one can easily show that the following are equivalent: (1) f is general preinvex function on general invex set. (2) f(ϕ(x)) ≤ f(ϕ(a)) + f(ϕ(b))−f(ϕ(a)) η(ϕ(b),ϕ(a)) (ϕ(x) −ϕ(a)). (3) f(ϕ(x))−f(ϕ(a)) ϕ(x)−ϕ(a) ≤ f(ϕ(b))−f(ϕ(a)) η(ϕ(b),ϕ(a)) ≤ f(ϕ(b))−f(ϕ(x)) ϕ(a)+η(ϕ(b),ϕ(a))−ϕ(x) . (4) [ϕ(x) − (ϕ(a) + η(ϕ(b),ϕ(a)))]f(ϕ(a)) + η(ϕ(b),ϕ(a))f(ϕ(x)) + [ϕ(a) −ϕ(x)]f(ϕ(b)) ≥ 0. (5) f(ϕ(a)) η(ϕ(b),ϕ(a))(ϕ(a)−ϕ(x)) + f(ϕ(x)) [ϕ(x)−(ϕ(a)+η(ϕ(b),ϕ(a)))][ϕ(a)−ϕ(x)] + f(ϕ(b)) η(ϕ(b),ϕ(a))[ϕ(x)−(ϕ(a)+η(ϕ(b),ϕ(a)))] ≥ 0, where ϕ(x) = ϕ(a) + tη(ϕ(b),ϕ(a)) and t ∈ [0, 1]. remark 2.1. note that for ϕ = i, where i is the identity function, the above definition reduces to the definition for preinvex functions on an interval. definition 2.10. let iη = [a,a + η(b,a)]. then f is called preinvex function, if and only if,∣∣∣∣∣∣ 1 1 1 a x a + η(b,a) f(a) f(x) f(a + η(b,a)) ∣∣∣∣∣∣ ≥ 0; a ≤ x ≤ a + η(b,a). one can easily show that the following are equivalent: (1) f is preinvex function on invex set. (2) f(x) ≤ f(a) + f(b)−f(a) η(b,a) (x−a). (3) f(x)−f(a) x−a ≤ f(b)−f(a) η(b,a) ≤ f(b)−f(x) a+η(b,a)−x. (4) [x− (a + η(b,a))]f(a) + η(b,a)f(x) + (a−x)f(b) ≥ 0. (5) f(a) η(b,a)(a−x) + f(x) [x−(a+η(b,a))][a−x] + f(b) η(b,a)[x−(a+η(b,a))] ≥ 0, where x = a + tη(b,a) and t ∈ [0, 1]. remark 2.2. if in definition 2.8, η(ϕ(b),ϕ(a)) = ϕ(b)−ϕ(a). then, we have the definition of general convex functions on interval. definition 2.11. let iϕ = [ϕ(a),ϕ(b)]. then f is called general convex function, if and only if,∣∣∣∣∣∣ 1 1 1 ϕ(a) ϕ(x) ϕ(b) f(ϕ(a)) f(ϕ(x)) f(ϕ(b)) ∣∣∣∣∣∣ ≥ 0; ϕ(a) ≤ ϕ(x) ≤ ϕ(b). one can easily show that the following are equivalent: (1) f is general convex function on general convex set. (2) f(ϕ(x)) ≤ f(ϕ(a)) + f(ϕ(b))−f(ϕ(a)) ϕ(b)−ϕ(a) (ϕ(x) −ϕ(a)). (3) f(ϕ(x))−f(ϕ(a)) ϕ(x)−ϕ(a) ≤ f(ϕ(b))−f(ϕ(a)) ϕ(b)−ϕ(a) ≤ f(ϕ(b))−f(ϕ(x)) ϕ(b)−ϕ(x) . (4) (ϕ(x) −ϕ(b))f(ϕ(a)) + (ϕ(b) −ϕ(a))f(ϕ(x)) + (ϕ(a) −ϕ(x))f(ϕ(b)) ≥ 0. (5) f(ϕ(a)) (ϕ(b)−ϕ(a))(ϕ(a)−ϕ(x)) + f(ϕ(x)) (ϕ(x)−ϕ(b))(ϕ(a)−ϕ(x)) + f(ϕ(b)) (ϕ(b)−ϕ(a))(ϕ(x)−ϕ(b)) ≥ 0, where ϕ(x) = ϕ(a) + t(ϕ(b) −ϕ(a)) and t ∈ [0, 1]. general preinvex functions 49 definition 2.12. a differentiable function f on general invex set kηϕ is said to be general invex function, if there exists arbitrary functions η and ϕ, such that f(ϕ(v)) −f(ϕ(u)) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, where f ′ is the differential of f . if ϕ = i, our definition of general invexity reduces to the definition of invexity which is mainly due to hanson [5]. definition 2.13. a function f is said to be pseudo general preinvex with respect to η, if there exists a strictly positive bifunction b such that f(ϕ(v)) < f(ϕ(u)) ⇒ f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ f(ϕ(u)) + t(t− 1)b(ϕ(v),ϕ(u)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ (0, 1). definition 2.14. a function t is said to be η-monotone, if and only if 〈t(ϕ(u)),η(ϕ(v),ϕ(u))〉 + 〈t(ϕ(v)),η(ϕ(v),ϕ(u))〉≤ 0, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ. 3. results and discussions 3.1. variational-like inequalities. in this section, we derive some general variational-like inequalities. theorem 3.1. if f is a general preinvex function on kηϕ, then, the lower level set lα = {u ∈ h : ϕ(u) ∈ kηϕ : f(ϕ(u)) ≤ α,α ∈ r} is a general invex set. theorem 3.2. a function f is general preinvex function on kηϕ if and only if, epi(f) = {(ϕ(u),α) : ϕ(u) ∈ kηϕ,α ∈ r,f(ϕ(u)) ≤ α} is general invex set. theorem 3.3. let f be a general preinvex function. suppose µ = inf ϕ(u)∈kηϕ f(ϕ(u)). then the set aη = {u ∈ h : ϕ(u) ∈ kηϕ : f(ϕ(u)) = µ} is general invex set of kηϕ. if f is general strictly preinvex, then aη is a singelton. proof. let ϕ(u),ϕ(v) ∈ aη. then for 0 < t < 1, we suppose ϕ(z) = ϕ(u) + tη(ϕ(v),ϕ(u)). since f is preinvex function. then f(ϕ(z)) = f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)) = µ, this implies that ϕ(z) ∈ aη thus aη is general invex set. now for other part of the theorem, we assume contrary that f(ϕ(u)) = f(ϕ(v)) = µ. since kηϕ is general invex set, then for 0 < t < 1, ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ. also, since f is strictly general preinvex function. f(ϕ(u) + tη(ϕ(v),ϕ(u))) < (1 − t)f(ϕ(u)) + tf(ϕ(v)) = µ, which is contradiction that µ = inf ϕ(u)∈kηϕ f(ϕ(u)), this completes the proof. � theorem 3.4. let f be a general preinvex function on kη. if φ is a nondecreasing convex function, then φ◦f is a general preinvex function. proof. since f is a general preinvex function and φ is nondecreasing, then φ◦f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ φ[f(ϕ(u) + tη(ϕ(v),ϕ(u)))] ≤ φ[(1 − t)f(ϕ(u)) + tf(ϕ(v))] ≤ (1 − t)φ◦f(ϕ(u)) + tφ◦f(ϕ(v)). this completes the proof. � theorem 3.5. let f be a semistrictly general preinvex function on kη. if φ is a nondecreasing convex function, then φ◦f is a semistrictly general preinvex function. lemma 3.1. let kηϕ be general convex set and let f be be general invex function on k. the 50 awan, noor, mishra and noor (1) if 〈f ′(ϕ(u)),ϕ(v) −ϕ(u)〉≤ 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, such that f(ϕ(v)) ≤ f(ϕ(u)), then f is a general pseudo-convex function. (2) if 〈f ′(ϕ(u)),ϕ(v) −ϕ(u)〉≤ 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, such that f(ϕ(v)) < f(ϕ(u)), then f is strictly general pseudo-convex function. proof. let u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ and f(ϕ(v)) ≤ f(ϕ(u)). then 〈f ′(ϕ(u)),ϕ(v) −ϕ(u)〉 = 〈f ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉 + 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 ≤ 〈f ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉 + f(ϕ(v)) −f(ϕ(u)) ≤〈f ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉≤ 0. this completes the proof. the proof of second part is on similar lines. � theorem 3.6. if f be a general preinvex function. then any local minimum of f is a global minimum. proof. let f has a local minimum at ϕ(u) ∈ kηϕ. assume contrary, that f(ϕ(v)) < f(ϕ(u)) for some ϕ(v) ∈ k. now since f is general preinvex function. then f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)). this implies that f(ϕ(u) + tη(ϕ(v),ϕ(u))) −f(ϕ(u)) ≤ t(f(ϕ(v)) −f(ϕ(u))) < 0. thus, we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) < f(ϕ(u)), a contradiction. this completes the proof. � theorem 3.7. if f be a semistrictly general preinvex function. then any local minimum of f is a global minimum. proof. the proof is similar to previous. � theorem 3.8. let f be general preinvex function with respect to η, i = 1, 2, . . . ,n. then n∑ i=0 µifi(ϕ(u)) is general preinvex with respect to η, where µi ≥ 0. proof. let fi be general preinvex functions. then fi(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)). now (1 − t) n∑ i=0 µifi(ϕ(u)) + t n∑ i=0 µifi(ϕ(v)) = n∑ i=0 µi[(1 − t)fi(ϕ(u)) + tfi(ϕ(v))] ≥ n∑ i=0 µifi(ϕ(u) + tη(ϕ(v),ϕ(u))). this implies that n∑ i=0 µifi(ϕ(u)) is general preinvex function. � theorem 3.9. if f is general preinvex function with respect to η such that f(ϕ(v)) < f(ϕ(u)), then f is general pseudo preinvex function with respect to same η. general preinvex functions 51 proof. since f(ϕ(v)) < f(ϕ(u)) and f is general preinvex function with respect to η, then for all u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ and t ∈ (0, 1), we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ f(ϕ(u)) + t(f(ϕ(v)) −f(ϕ(u))) < f(ϕ(u)) + t(1 − t)(f(ϕ(v)) −f(ϕ(u))) = f(ϕ(u)) + t(t− 1)(f(ϕ(u)) −f(ϕ(v))) = f(ϕ(u)) + t(t− 1)b(ϕ(u),ϕ(v)), where b(ϕ(u),ϕ(v)) = f(ϕ(u)) −f(ϕ(v)) > 0. this completes the proof. � theorem 3.10. let f be a differentiable general preinvex function on kηϕ. then ϕ(u) ∈ kηϕ is the minimum of f on kηϕ if and only if ϕ(u) ∈ keta satisfies the inequality 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, (3.1) where f ′ is the differential of f at ϕ(u) ∈ kηϕ. the inequality (3.1) is called the general variational-like inequality. proof. let ϕ(u) ∈ kηϕ be a minimum of general preinvex function f on kηϕ. then by definition of minimum, we have, f(ϕ(u)) ≤ f(ϕ(v)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ. (3.2) since kηϕ is a general invex set, so ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ. (3.3) replacing ϕ(v) by ϕ(vt) in (3.2) we get f(ϕ(u)) ≤ f(ϕ(vt)) = f(ϕ(u) + tη(ϕ(v),ϕ(u))), which implies that f(ϕ(u) + tη(ϕ(v),ϕ(u))) −f(ϕ(u)) ≥ 0. since f is differentiable, so dividing both sides of the above inequality by t and then taking the limit as t → 0, we have 0 ≤ lim t→0 ( f(ϕ(u) + tη(ϕ(v),ϕ(u))) −f(ϕ(u)) t ) = 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, that is ϕ(u) ∈ kηϕ satisfies the inequality 〈f ′(u),η(ϕ(v),ϕ(u))〉≥ 0, ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ. conversely, let inequality (3.1) holds. we have to show that ϕ(u) ∈ kηϕ, is the minimum of f on the general invex set kηϕ. since f is general preinvex function, then f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ f(ϕ(u)) + t(f(ϕ(v)) −f(ϕ(v))). now taking limit as t → 0, we have f(ϕ(v)) −f(ϕ(u)) ≥ lim t→0 f(ϕ(u) + tη(ϕ(v),ϕ(u))) −f(ϕ(u)) t = 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 ≥ 0, thus, it follows that f(ϕ(u)) ≤ f(ϕ(v)), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, which completes the proof. � theorem 3.11. let f be a differentiable function on general invex set kηϕ and suppose condition c holds. then f is general preinvex function if and only if f is a general invex function. 52 awan, noor, mishra and noor proof. let f be a differentiable general preinvex function, then f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ f(ϕ(u)) + t(f(ϕ(v))−f(ϕ(u))), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], since f is differentiable taking limit as t → 0, we have f(ϕ(v)) −f(ϕ(u)) ≥ lim t→0 f(ϕ(u) + tη(ϕ(v),ϕ(u))) −f(ϕ(u)) t = 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. this implies that f is general invex function. conversely, suppose that f is general invex function, that is f(ϕ(v)) −f(ϕ(u)) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.4) since kηϕ is a general invex set. then ∀ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ. replacing ϕ(u) by ϕ(vt) in (3.4) and using condition c, we have f(ϕ(v)) −f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≥ (1 − t)〈f ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉. (3.5) similarly, we have f(ϕ(u)) −f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≥−t〈f ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉. (3.6) multiplying (3.5) by t and (3.6) by (1 − t), and then adding the resultant, we have f(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)f(ϕ(u)) + tf(ϕ(v)). (3.7) this completes the proof. � theorem 3.12. let f be a differentiable function on general invex set kηϕ and suppose condition a holds. then the differential f ′ of f is η-monotone if and only if f is a general invex function. proof. let f be general invex function. then f(ϕ(v)) −f(ϕ(u)) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.8) interchanging ϕ(u) and ϕ(v) in above inequality, we have f(ϕ(u)) −f(ϕ(v)) ≥〈f ′(ϕ(v)),η(ϕ(u),ϕ(v))〉. (3.9) adding (3.8) and (3.9), we have 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 + 〈f ′(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0. (3.10) this implies that f ′ is η-monotone. conversely, suppose that f ′ is η-monotone, that is f ′ satisfies inequality (3.10). then, from (3.10), we have 〈f ′(ϕ(v)),η(ϕ(u),ϕ(v))〉≤−〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.11) now, since kηϕ is general invex set, then, ∀ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ. taking ϕ(v) as ϕ(vt) in (3.11), and applying condition c, we get 〈f ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉≥ 〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.12) consider an auxiliary function ϕ(t) = f(ϕ(vt)) ≡ f(ϕ(u) + tη(ϕ(v),ϕ(u))), ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1]. (3.13) now using the fact that f is differentiable, we have ϕ′(t) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. integrating above inequality with respect to t on [0, 1], we have ϕ(1) −ϕ(0) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.14) using (3.13) and (3.14), we have f(ϕ(u) + η(ϕ(v),ϕ(u))) −f(u) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. now using condition a, we have f(ϕ(v)) −f(ϕ(u)) ≥〈f ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, which shows that f is general invex function. this completes the proof. � general preinvex functions 53 theorem 3.13. let kηϕ be a general invex set in h. suppose function f be η-pseudomonotone and η-hemicontinuous. if condition c holds, then ϕ(u) ∈ kηϕ satisfies 〈f(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ, (3.15) if and only if ϕ(u) ∈ kηϕ satisfies 〈f(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ. (3.16) proof. let u ∈ h : ϕ(u) ∈ kηϕ satisfies the following inequality 〈f(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ, which implies that 〈f(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ, where f is η-pseudomonotone. conversely, let (3.16) holds. since kηϕ is general invex set, then ∀u,v ∈ h : ϕ(u),ϕ(v) ∈ kηϕ, t ∈ [0, 1], ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ kηϕ. taking ϕ(v) = ϕ(vt) in (3.16) and using condition c, we have 0 ≥〈f(ϕ(vt)),η(ϕ(u),ϕ(u) + tη(ϕ(v),ϕ(u)))〉 = −t〈f(ϕ(vt)),η(ϕ(v),ϕ(u))〉, from which we have 〈f(ϕ(vt)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ. taking limit as t → 0 on both sides of above inequality, we have 〈f(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ h : ϕ(v) ∈ kηϕ, (3.17) where we have used the fact that f is η-hemicontinuous. this completes the proof. � 3.2. hermite-hadamard type inequalities. hermite-hadamard type inequalities provides us necessary and sufficient condition for a function to be convex. in recent years many new generalizations of these inequalities have been obtained via different classes of convex functions. for more information, see [2, 4, 19–21, 23]. in this section, we derive some hermite-hadamard type inequalities via general preinvex functions. theorem 3.14. let f : iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → r be a general preinvex function with η(ϕ(b),ϕ(a)) > 0. if η(., .) satisfies the condition c, then we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))dϕ(x) ≤ f(ϕ(a)) + f(ϕ(b)) 2 . proof. since f is general preinvex function and η(., .) satisfies the condition c, we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 2 [f (ϕ(a) + tη(ϕ(b),ϕ(a))) + f(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] . integrating above inequality with respect to t on [0, 1], we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))dϕ(x). (3.18) also f(ϕ(a) + tη(ϕ(b),ϕ(a))) ≤ (1 − t)f(ϕ(a)) + tf(ϕ(b)). integrating above inequality with respect to t on [0, 1], we have 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))dϕ(x) ≤ f(ϕ(a)) + f(ϕ(b)) 2 . (3.19) combining (3.18) and (3.19) completes the proof. � 54 awan, noor, mishra and noor theorem 3.15. let f,w : iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → r be general preinvex functions respectively with η(ϕ(b),ϕ(a)) > 0. suppose η(., .) satisfies condition c, then we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) w ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) − 1 2η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))w(ϕ(x))dϕ(x) ≤ 1 6 m(ϕ(a),ϕ(b)) + 1 3 n(ϕ(a),ϕ(b)), where m(ϕ(a),ϕ(b)) = f(ϕ(a))w(ϕ(a)) + f(ϕ(b))w(ϕ(b)) (3.20) and n(ϕ(a),ϕ(b)) = f(ϕ(a))w(ϕ(b)) + f(ϕ(b))w(ϕ(a)). (3.21) proof. since f and w are general preinvex functions respectively and η(., .) satisfies condition c, we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) w ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) = f(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)) + 1 2 η(ϕ(a) + tη(ϕ(b),ϕ(a)),ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))) ×w(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)) + 1 2 η(ϕ(a) + tη(ϕ(b),ϕ(a)),ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))) ≤ 1 4 [f(ϕ(a) + tη(ϕ(b),ϕ(a))) + f(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] × [w(ϕ(a) + tη(ϕ(b),ϕ(a))) + w(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] ≤ 1 4 [f(ϕ(a) + tη(ϕ(b),ϕ(a)))w(ϕ(a) + tη(ϕ(b),ϕ(a))) + f(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))w(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] + 1 2 { 2(t− t2)m(a,b) + [t2 + (1 − t)2]n(a,b) } . integrating above inequality with respect to t on [0,1], we have f ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) w ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) − 1 2η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))w(ϕ(x))dϕ(x) ≤ 1 6 m(ϕ(a),ϕ(b)) + 1 3 n(ϕ(a),ϕ(b)). the proof is complete. � theorem 3.16. let f,w : iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → r be general preinvex function with η(b,a) > 0, then we have 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))w(ϕ(x))dϕ(x) ≤ 1 3 m(ϕ(a),ϕ(b)) + 1 6 n(ϕ(a),ϕ(b)). where m(ϕ(a),ϕ(b)) and n(ϕ(a),ϕ(b)) are given by (3.20) and (3.21) respectively. general preinvex functions 55 proof. let f,w be general preinvex functions, then for all t ∈ [0, 1], we have f(ϕ(a) + tη(ϕ(b),ϕ(a)))w(ϕ(a) + tη(ϕ(b),ϕ(a))) ≤ [(1 − t)f(ϕ(a)) + tf(ϕ(b))][(1 − t)w(ϕ(a)) + tw(ϕ(b))] = (1 − t)2f(ϕ(a))w(ϕ(a)) + t(1 − t)f(ϕ(b))w(ϕ(a)) + t(1 − t)f(ϕ(a))w(ϕ(b)) + t2f(ϕ(b))w(ϕ(b)). integrating above inequality with respect to t on [0, 1], we have 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) f(ϕ(x))w(ϕ(x))dϕ(x) ≤ 1 3 m(ϕ(a),ϕ(b)) + 1 6 n(ϕ(a),ϕ(b)). this completes the proof. � 4. conclusion we have discussed several properties of general preinvex functions. it is shown that the minimum of the differentiable general functions can be characterized by variational-like inequalities, which are called general variational-like inequalities. we have established a necessary and sufficient condition for the minimum of a differential general preinvex functions. in the last section, we have obtained some integral inequalities of hermite-hadamard type via general preinvex functions. we would like to mention that the field of general variational-like inequalities is a relatively new one and offer great opportunities for further research. the ideas and techniques of this paper may stimulate further research in the field of mathematical inequalities. references [1] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrecht, holland, (2002). [2] g. cristescu, m. a. noor, m. u. awan, bounds of the second degree cumulative frontier gaps of functions with generalized convexity, carpathian j. math. 31(2) (2015), 173-180. [3] deepmala, a study on fixed point theorems for nonlinear contractions and its applications, ph.d. thesis, pt. ravishankar shukla university, raipur 492 010, chhatisgarh, india (2014). [4] s. s. dragomir, c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university, australia, 2000. [5] d. i. duca and l. lupsa, saddle points for vector valued functions: existence, necessary and sufficient theorems, j. glob. optim., 53 (2012), 431–440. [6] c. fulga and v. preda, nonlinear programming with ϕ-preinvex and local ϕ-preinvex functions, eur. j. oper. res. 192 (2009), 737–743. [7] m. a. hanson, on sufficiency of the kuhn–tucker conditions, j. math. anal. appl., 80 (1981) 545–550. [8] a. ben-israel and b. mond, what is invexity? j. aust. math. soc. ser. b 28 (1986), 1–9. [9] s. r. mohan and s. k. neogy, on invex sets and preinvex functions, j. math. anal. appl., 189 (1995), 901–908. [10] l. n. mishra, h. m. srivastava, m. sen, on existence results for some nonlinear functional-integral equations in banach algebra with applications, int. j. anal. appl., 11 (1) (2016), 1–10. [11] l. n. mishra, r. p. agarwal, on existence theorems for some nonlinear functional-integral equations, dynamic systems and appl., 25 (2016), 303–320. [12] l. n. mishra, on existence and behavior of solutions to some nonlinear integral equations with applications, ph.d. thesis, national institute of technology, silchar 788 010, assam, india (2017). [13] m. a. noor, differentiable nonconvex functions and general variational inequalities, appl. math. comput., 199 (2008), 623–630. [14] m. a. noor, fuzzy preinvex functions, fuzzy sets syst. 64 (1994), 95–104. [15] m. a. noor, hermite-hadamard integral inequalities for log-preinvex functions, j. math. anal. approx. theory 2 (2007), 126–131. [16] m. a. noor, new approximation schemes for general variational inequalities, j. math. anal. appl., 251 (2000), 217–229. [17] m. a. noor, on hermite-hadamard integral inequalities for involving two log-preinvex functions, j. inequal. pure appl. math., 3 (2007), 75–81. [18] m. a. noor, variational like inequalities, optimization 30 (1994), 323–330. [19] m. a. noor, m.u. awan, k. i. noor: on some inequalities for relative semi-convex functions. j. inequal. appl. 2013 (2013), art. id 332. [20] m. a. noor, k. i. noor, m. u. awan, geometrically relative convex functions, appl. mathe. infor. sci., 8(2) (2014), 607-616. 56 awan, noor, mishra and noor [21] m. a. noor, k. i. noor, m. u. awan, hermite-hadamard inequalities for relative semi-convex functions and applications, filomat. 28 (2) (2014), 221-230. [22] m. a. noor, k. i. noor, m. u. awan, j. li, on hermite-hadamard type inequalities for h-preinvex functions. filomat. 28 (7) (2014), 1463-1474. [23] m. a. noor, k. i. noor, integral inequalities for differentiable relative preinvex functions(survey), twms j. pure appl. math. 7(1)(2016), 3-19 [24] t. weir and b. mond, preinvex functions in multiobjective optimization, j. math. anal. appl., 136 (1988), 29–38. [25] e. a. youness, e-convex sets, e-convex functions, and e-convex programming, j. optim. theory appl., 102 (1999), 439–450. 1department of mathematics, government college university,, [0pt] faisalabad, pakistan. 2mathematics department, king saud university, riyadh, saudi arabia. 3department of mathematics, comsats institute of information technology, islamabad, pakistan. 4department of mathematics, indira gandhi national tribal university, lalpur, amarkantak, anuppur, madhya pradesh 484 887, india 5l. 1627 awadh puri colony beniganj, phase iii, opposite industrial training institute (i.t.i.), faizabad 224 001, uttar pradesh, india ∗corresponding author: vishnunarayanmishra@gmail.com 1. introduction 2. preliminaries 3. results and discussions 3.1. variational-like inequalities 3.2. hermite-hadamard type inequalities 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 19-22 http://www.etamaths.com on | c, 1 |k integrability of improper integrals h. n. özgen∗ abstract. in this paper, we introduce the concept of | c, 1 |k, k ≥ 1, integrability of improper integrals and we prove a known theorem of mazhar [3] by using this definition. 1. introduction throughout this paper we assume that f is a real valued function which is continuous on [0,∞) and s(x) = ∫ x 0 f(t)dt. the cesàro mean of s(x) is defined by σ(x) = 1 x ∫ x 0 s(t)dt. the integral ∫∞ 0 f(t)dt is said to be integrable | c, 1 |k,k ≥ 1, in the sense of flett [2], if∫ ∞ 0 xk−1 | σ′(x) |k dx(1.1) is convergent. the kronecker identity (see [1]): s(x) − σ(x) = v(x), where v(x) = 1 x ∫ x 0 tf(t)dt is well-known and will be used in the various steps of proofs. since σ′(x) = 1 x v(x), condition (1.1) can also be written as (1.2) ∫ ∞ 0 1 x | v(x) |k dx is convergent. we note that for infinite series, an analogous definition was introduced by flett [2]. using this definition, mazhar [3] established the following theorem for | c, 1 |k summability factors of infinite series. given any functions f,g, it is customary to write g(x) = o(f(x)), if there exist η and n, for every x > n, | g(x) f(x) |≤ η. theorem 1.1. if (xn) is a positive monotonic non-decreasing sequence such that (1.3) λmxm = o(1) as m →∞, (1.4) m∑ n=1 nxn | ∆2λn |= o(1), (1.5) m∑ n=1 1 n | tn |k= o(xm) as m →∞, then the series ∑ anλn is summable | c, 1 |k,k ≥ 1. 2010 mathematics subject classification. 40f05, 40d25, 35a23. key words and phrases. absolute summability; summability factors; improper integral; inequalities for integrals. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 19 20 özgen 2. the main result the aim of this paper is to prove mazhar’s theorem for | c, 1 |k integrability of improper integrals. now, we shall state the following theorem. theorem 2.1. if γ(x) is a positive monotonic non-decreasing function such that (2.1) λ(x)γ(x) = o(1) as x →∞, (2.2) ∫ x 0 u | λ′′(u) | γ(u)du = o(1), (2.3) ∫ x 0 | v(u) |k u du = o(γ(x)) as x →∞, then the integral ∫∞ 0 f(t)dt is integrable | c, 1 |k,k ≥ 1. we need the following lemma for the proof of our theorem. lemma 2.2. under the conditions of the theorem we have that (2.4) ∫ ∞ 0 γ(t) | λ′(t) | dt is convergent, (2.5) xγ(x) | λ′(x) |= o(1) as x →∞. proof. since λ′(t) = ∫ t 0 λ′′(u), we have ∫ x 0 γ(t) | λ′(t) | dt = ∫ x 0 γ(t) | ∫ t 0 λ′′(u)du | dt ≤ ∫ x 0 γ(t) ∫ t 0 | λ′′(u) | dudt = ∫ x 0 | λ′′(u) | du ∫ x u γ(t)dt ≤ ∫ x 0 uγ(u) | λ′′(u) | du = o(1) as x →∞ by (2.2). since xγ(x) is a non decreasing function, we get xγ(x) | λ′(x) | = xγ(x) | ∫ x 0 λ′′(u)du | ≤ xγ(x) ∫ x 0 | λ′′(u) | du = ∫ x 0 uγ(u) | λ′′(u) | du = o(1) ≤ ∫ x 0 uγ(u) | λ′′(u) | du = o(1) as x →∞ this completes the proof of lemma 2.2. � on | c, 1 |k integrability of improper integrals 21 3. proof of the theorem let a(x) be the function of (c, 1) means of the integral ∫∞ 0 f(t)dt. then, by definition, we have a(x) = 1 x ∫ x 0 ∫ t 0 λ(u)f(u)dudt = 1 x ∫ x 0 λ(u)f(u)du ∫ x u dt = 1 x ∫ x 0 (x−u)λ(u)f(u)du = ∫ x 0 ( 1 − u x ) λ(u)f(u)du differentiating the function a(x) and later integrating by parts, we obtain a′(x) = 1 x2 ∫ x 0 uλ(u)f(u)du = v(x)λ(x) x − 1 x2 ∫ x 0 λ′(u)uv(u)du = a1(x) + a2(x), say. to complete the proof of the theorem, it is sufficient to show that (3.1) ∫ x 0 tk−1 | ar(t) |k dt = o(1) as x →∞, for r = 1, 2. first, applying hölder’s inequality, we have∫ x 0 tk−1 | a1(t) |k dt = ∫ x 0 tk−1 | v(t) |k| λ(t) |k tk dt = ∫ x 0 1 t | v(t) |k| λ(t) |k−1| λ(t) | dt ≤ ∫ x 0 | v(t) |k t | λ(t) | dt = | λ(x) | ∫ x 0 | v(t) |k t dt− ∫ x 0 | λ′(t) | ∫ t 0 | v(u) |k u dudt = | λ(x) | γ(x) − ∫ x 0 | λ′(t) | γ(t)dt = o(1) as x →∞ by virtue of the hypotheses of theorem 2.1 and lemma 2.2. now, as in a1(x), we have that∫ x 0 tk−1 | a2(t) |k dt = ∫ x 0 tk−1 1 t2k | ∫ t 0 uλ′(u)v(u)du |k dt ≤ ∫ x 0 1 t2 {∫ t 0 | λ′(u) |k uk | v(u) |k du } x { 1 t ∫ t 0 du }k−1 dt = ∫ x 0 | uλ′(u) |k−1| uλ′(u) || v(u) |k du ∫ x u dt t2 = ∫ x 0 | uλ′(u) || v(u) |k ( 1 u − 1 x ) du ≤ ∫ x 0 | uλ′(u) | | v(u) |k u du 22 özgen integrating by parts,we get∫ x 0 tk−1 | a2(t) |k dt = x | λ′(x) | ∫ x 0 | v(u) |k u du + ∫ x 0 (u | λ′(u) |)′ ∫ u 0 | v(t) |k t dtdu = x | λ′(x) | γ(x) − ∫ x 0 (u | λ′(u) |)′γ(u)du = x | λ′(x) | γ(x) − ∫ x 0 | λ′(u) | γ(u)du− ∫ x 0 u | λ′′(u) | γ(u)du = o(1) as x →∞ by virtue of the hypotheses of theorem 2.1 and lemma 2.2. thus, we obtain ∫ x 0 tk−1 | a′(t) |k dt = o(1) as x →∞. this completes the proof of the theorem. references [1] i̇. çanak and ü. totur, a tauberian theorem for cesàro summability factors of integrals, appl. math. lett. 24 (2011), 391–395. [2] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc. 7 (1957), 113–141. [3] s. m. mazhar, on | c, 1 |k summability factors of infinite series, indian j. math. 14 (1972), 45–48. department of mathematics, faculty of education, university of mersin, tr-33169 mersin, turkey ∗corresponding author: nogduk@gmail.com international journal of analysis and applications volume 16, number 2 (2018), 232-238 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-232 stability of euler-lagrange-jensen’s (a,b)sextic functional equation in multi-banach spaces john michael rassias1, r. murali2,∗ and a. antony raj2 1pedagogical department e.e., national and kapodistrian university of athens, section of mathematics and informatics, athens, 15342, greece 2pg and research department of mathematics, sacred heart college (autonomous), tirupattur 635 601, tamil nadu, india ∗corresponding author: shcrmurali@yahoo.co.in abstract. in this paper, we prove the hyers-ulam stability of euler-lagrange-jensen’s (a,b)-sextic functional equation in multi-banach spaces. 1. introduction and preliminaries the theory of stability is an important branch of the qualitative theory of functional equations. the concept of stability for a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. the first stability problem of functional equation was raised by s.m. ulam [17] about seventy seven years ago. since then, this question has attracted the attention of many researchers. note that the affirmtive solution to this question was given in the next year by d.h. hyers [5] in 1941. in the year 1950, t. aoki [1] generalized hyers theorem for additive mappings. the result of hyers was generalized independently by th.m.rassias [14] for linear mappings by considering an unbounded cauchy difference. in 1994, a further generalization of th.m. rassias theorem was obtained by p.gavruta [4]. then, the stability problem of several functional equations has been extensively investigated by a number received 2017-10-25; accepted 2018-01-04; published 2018-03-07. 2010 mathematics subject classification. 39b52, 39b72,39b82. key words and phrases. hyers-ulam stability; multi-banach spaces; sextic functional equation; direct method. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 232 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-232 int. j. anal. appl. 16 (2) (2018) 233 of authors, and there are many interesting results concerning this problem ( [3, 6, 7, 9, 11–13, 15, 16, 18, 19]). the hyers-ulam stability of functional equation is investigated and the investigation is following. here, we establish the hyers-ulam stability of euler-lagrange-jensen’s (a,b)sextic functional equation is of the form f(ax + by) + f(bx + ay) + (a− b)6 [ f ( ax− by a− b ) + f ( bx−ay b−a )] = 64(ab)2(a2 + b2) [ f( x + y 2 ) + f( x−y 2 ) ] + 2(a2 − b2)(a4 − b4) [f(x) + f(y)] (1.1) where a 6= b such that k ∈ r, h = a + b 6= 0, ±1 in multi-banach spaces by using direct and fixed point method. definition 1.1. [2] a multinorm on { ak : k ∈ n } is a sequence (‖.‖) = (‖.‖k : k ∈ n) such that ‖.‖k is a norm on ak for each k ∈ n,‖x‖1 = ‖x‖ for each x ∈ a, and the following axioms are satisfied for each k ∈ n with k ≥ 2 : (1) ∥∥(xσ(1), ...,xσ(k))∥∥k = ‖(x1...xk)‖k , for σ ∈ ψk,x1, ...,xk ∈a; (2) ‖(α1x1, ...,αkxk)‖k ≤ (maxi∈nk |αi|)‖(x1...xk)‖k for α1...αk ∈ c,x1, ...,xk ∈a; (3) ‖(x1, ...,xk−1, 0)‖k = ‖(x1, ...,xk−1)‖k−1 , for x1, ...,xk−1 ∈a; (4) ‖(x1, ...,xk−1,xk−1)‖k = ‖(x1, ...,xk−1)‖k−1 for x1, ...,xk−1 ∈a. in this case, we say that ( (ak,‖.‖k) : k ∈ n ) is a multi normed space. suppose that ( (ak,‖.‖k) : k ∈ n ) is a multi normed space, and take k ∈ n. we need the following two properties of multi norms. they can be found in [2]. (a)‖(x,...,x)‖k = ‖x‖ , for x ∈a, (b) max i∈nk ‖xi‖≤‖(x1, ...,xk)‖k ≤ k∑ i=1 ‖xi‖≤ k max i∈nk ‖xi‖ ,∀x1, ...,xk ∈a. it follows from (b) that if (a,‖.‖) is a banach space, then (ak,‖.‖k) is a banach space for each k ∈ n. in this case, ( (ak,‖.‖k) : k ∈ n ) is a multi banach space. 2. stability of functional equation (1.1) in multi-banach spaces: direct method theorem 2.1. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach spaces. let f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖(df(x1,y1), ...,df(xk,yk))‖k ≤ � (2.1) int. j. anal. appl. 16 (2) (2018) 234 ∀x1, ...,xk,y1, ...,yk ∈ y. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖(f(x1) −s(x1), ...,f(xk) −s(xk))‖≤ � h6 (2.2) proof. letting yi = xi where i = 1, 2, ...k in (2.1), we arrive at sup k∈n ∥∥∥∥ ( 1 h6 f(hx1) −f(x1), ..., 1 h6 f(hxk) −f(xk) )∥∥∥∥ ≤ �2h6 (2.3) now, replacing xi by 2xi where i = 1, 2, ..,k and dividing by 2 in above equation, we get sup k∈n ∥∥∥∥ ( f(2hx1) h6 −f(x1), ..., f(2hxk) h6 −f(xk) )∥∥∥∥ ≤ �22h6 + �2h6 (2.4) by using induction for a positive integer n, we obtain sup k∈n ∥∥∥∥ ( f(2nhx1) 2nh6 −f(x1), ..., f(2nhxk) 2nh6 −f(xk) )∥∥∥∥ ≤ 1h6 n−1∑ i=0 � 2i+1 ≤ 1 h6 ∞∑ i=0 � 2i+1 (2.5) now, we have to show that the sequence { f(2nhx) 2nh6 } is a cauchy sequence, by fixing x ∈ x and replacing x1, ...xk by x, 2x,..., 2 k−1x such that sup k∈n ∥∥∥∥ ( f(2nhx) 2nh6 − f(2mx) 2m , ..., f(2n+k−1hx) 2n+k−1h6 − f(2m+k−1x) 2m+k−1 )∥∥∥∥ ≤ sup k∈n ∥∥∥∥ ( f(2nhx) 2nh6 − f(2mx) 2m , ..., 1 2k−1 [ f(2n(2k−1hx)) 2nh6 − f(2m(2k−1x)) 2m ])∥∥∥∥ using the definition of multi-norm, we arrive at sup k∈n ∥∥∥∥ ( f(2nhx) 2nh6 − f(2mx) 2m , ..., f(2n(2k−1hx)) 2nh6 − f(2m(2k−1x)) 2m )∥∥∥∥ ≤ 1 h6 n−1∑ i=m � 2i+1 . (2.6) hence the above inequality (2.6), shows that { f(2nhx) 2nh6 } is a cauchy sequence as n → ∞. since y is complete, then the sequence { f(2nhx) 2nh6 } converges to a fixed point s(x) ∈ y such that s(x) = lim n→∞ f(2nhx) 2nh6 . therefore, as n →∞, the inequality (2.5) implies the inequality (2.2). obviously, one can find the uniqueness of the mapping s : x → y, using the definition of multi-norm. that is, we can prove s = s′. � corollary 2.1. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach space. let f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖df(x1,y1, ...,xk,yk)‖k ≤ φ(x1,y1, ...,xk,yk) (2.7) int. j. anal. appl. 16 (2) (2018) 235 for all x1, ..,xk,y1, ..,yk ∈ x. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖f(x1) −s(x1), ...,f(xk) −s(xk)‖k ≤ 1 h6 ∞∑ i=1 1 2i+1 φ ( 2ix1,x1, ..., 2 ixk,xk ) (2.8) for all x1, ..,xk ∈ x. proof. proof is similar to that of theorem 2.1 by replacing the condition φ(x1,y1, ...,xk,yk) in place of �. � corollary 2.2. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach space. let 0 < p < 6 ,θ ≥ 0 and f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖df(x1,y1, ...,xk,yk)‖k ≤ θ (‖x1‖ p + ‖y1‖ p , ...,‖xk‖ p + ‖yk‖ p ) (2.9) for all x1, ..,xk,y1, ..,yk ∈ x. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖f(x1) −s(x1), ...,f(xk) −s(xk)‖k ≤ θ h6(2p − 1) (‖x1‖ p , ...,‖xk‖ p ) (2.10) for all x1, ..,xk ∈ x. proof. proof is similar to that of theorem 2.1 by replacing the condition θ (‖x1‖ p + ‖y1‖ p , ...,‖xk‖ p + ‖yk‖ p ) in place of �. � 3. stability of functional equation (1.1) in multi-banach spaces: fixed point method theorem 3.1. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach spaces. let f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖(df(x1,y1), ...,df(xk,yk))‖k ≤ � (3.1) ∀x1, ...,xk,y1, ...,yk ∈ y. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖(f(x1) −s(x1), ...,f(xk) −s(xk))‖≤ � 2(h6 − 1) (3.2) proof. letting yi = xi where i = 1, 2, ...k in (2.1), we arrive at sup k∈n ∥∥∥∥ ( 1 h6 f(hx1) −f(x1), ..., 1 h6 f(hxk) −f(xk) )∥∥∥∥ ≤ �2h6 (3.3) let ψ = {l : x → y |l(0) = 0} and introduce the generalized metric d defined on ψ by d(l,m) = inf { ψ ∈ [0,∞]|sup k∈n ‖l(x1) −m(x1), ..., l(xk) −m(xk)‖k ≤ ψ ∀ x1, ...,xk ∈ x } then it is easy to show that ψ,d is a generalized complete metric space, see [8]. we define an operator j : ψ → ψ by j l(x) = 1 h6 l(hx) x ∈ x int. j. anal. appl. 16 (2) (2018) 236 we assert that j is a strictly contractive operator. given l,m ∈ ψ, let ψ ∈ [0,∞] be an arbitary constant with d(l,m) ≤ ψ. from the definition if follows that sup k∈n ‖l(x1) −m(x1), ..., l(xk) −m(xk)‖k ≤ ψ x1, ...,xk ∈ x. therefore, supk∈n ‖(j l(x1) −jm(x1), ...,j l(xk) −jm(xk))‖k ≤ 1 h6 ψ x1, ...,xk ∈ x. hence,it holds that d(j l,jm) ≤ 1 h6 ψd(j l,jm) ≤ 1 h6 d(l,m) ∀l,m ∈ ψ. this means that j is strictly contractive operator on ψ with the lipschitz constant l = 1 h6 . by (3.3), we have d(jf,f) ≤ � 2h6 . applying the theorem 2.2 in [10], we deduce the existence of a fixed point of j that is the existence of mapping s : x → y such that s(hx) = h6s(x) ∀x ∈ x. moreover, we have d (jnf,s) → 0, which implies s(x) = lim n→∞ jnf(x) = lim n→∞ f(hnx) h6n for all x ∈ x. also, d(f,s) ≤ 1 1 −l d(jf,f) implies the inequality ≤ � 2(h6 − 1) . doing x1 =, ..., = xk = h nx, and y1 =, ..., = yk = h ny in (1.1) and dividing by h6n. now, applying the property (a) of multi-norms, we have ‖ds(x,y)‖ = lim n→∞ 1 h6n ‖df (hnx,hny)‖ ≤ lim n→∞ 1 h6n = 0 for all x,y ∈ x. the uniqueness of s follows from the fact that s is the unique fixed point of j with the property that there exists ` ∈ (0,∞) such that sup k∈n ‖(f(x1) −s(x1), ...,f(xk) −s(xk))‖k ≤ ` for all x1, ...,xk ∈ x. hence the proof. � int. j. anal. appl. 16 (2) (2018) 237 corollary 3.1. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach space. let 0 < p < 6 ,θ ≥ 0 and f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖df(x1,y1, ...,xk,yk)‖k ≤ θ (‖x1‖ p + ‖y1‖ p , ...,‖xk‖ p + ‖yk‖ p ) (3.4) for all x1, ..,xk,y1, ..,yk ∈ x. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖f(x1) −s(x1), ...,f(xk) −s(xk)‖k ≤ 2θ h6 − 2hp (‖x1‖ p , ...,‖xk‖ p ) (3.5) for all x1, ..,xk ∈ x. proof. proof is similar to that of theorem 3.1 by replacing the condition θ (‖x1‖ p + ‖y1‖ p , ...,‖xk‖ p + ‖yk‖ p ) in place of �. � corollary 3.2. let x be a linear space and ((y n,‖.‖n) : n ∈ n) be a multi-banach space. let f : x → y be a mapping satisfying f(0) = 0 such that sup k∈n ‖df(x1,y1, ...,xk,yk)‖k ≤ φ(x1,y1, ...,xk,yk) (3.6) for all x1, ..,xk,y1, ..,yk ∈ x. then there exists a unique sextic mapping s : x → y such that sup k∈n ‖f(x1) −s(x1), ...,f(xk) −s(xk)‖k ≤ 1 2(h6 − 1) φ(x1,x1, ...,xk,xk) (3.7) for all x1, ..,xk ∈ x. proof. proof is similar to that of theorem 3.1 by replacing the condition φ(x1,y1, ...,xk,yk) in place of �. � references [1] t. aoki, on the stability of the linear transformation in banach spaces, j.math. soc. japan 2 (1950), 64-66. [2] dales, h.g and moslehian, stability of mappings on multi-normed spaces, glasgow math. j. 49 (2007), 321-332. [3] fridoun moradlou, approximate euler-lagrange-jensen type additive mapping in multi-banach spaces: a fixed point approach, commun. korean math. soc. 28 (2013), 319-333. [4] p. gavruta, on a problem of g.isac and th.m.rassias concerning the stability of mappings, j. math. anal. appl. 261 (2001), 543-553. [5] d.h.hyers, on the stability of the linear functional equation, proc. nat. acad. sci. usa 27 (1941), 222-224. [6] john michael rassias, r. murali, matina john rassias, a. antony raj, general solution, stability and non-sstability of quattuorvigintic functional equation in multi-banach spaces, int. j. math. appl. 5 (2017), 181-194. [7] manoj kumar and ashish kumar, stability of jenson type quadratic functional equations in multi-banach spaces, int. j. math. arch. 3(4) (2012), 1372-1378. [8] d. mihet and v. radu, on the stability of the additive cauchy functional equation in random normed spaces, j. math. anal. appl. 343 (2008), 567-572. [9] r. murali, matina j. rassias and v. vithya, the general solution and stability of nonadecic functional equation in matrix normed spaces, malaya j. mat. 5(2) (2017), 416-427. [10] v.radu, the fixed point alternative and the stability of functional equations, fixed point theory 4 (2003), 91-96. int. j. anal. appl. 16 (2) (2018) 238 [11] k. ravi, j.m. rassias and b.v. senthil kumar, ulam-hyers stability of undecic functional equation in quasi-beta normed spaces fixed point method, tbilisi math. sci. 9 (2) (2016), 83-103. [12] k. ravi, j.m. rassias, s. pinelas and s.suresh, general solution and stability of quattuordecic functional equation in quasi beta normed spaces, adv. pure math. 6 (2016), 921-941. [13] j. m. rassias and m. eslamian, fixed points and stability of nonic functional equation in quasi-β-normed spaces, cont. anal. appl. math. 3 (2) (2015), 293-309. [14] th.m. rassias, on the stability of the linear mappings in banach spaces, proc. amer. math. soc., 72 (1978), 297-300. [15] th.m. rassias, on the stability of functional equations and a problem of ulam, acta. appl. math. 62 (2000) 23 130. [16] tian zhou xu, john michael rassias and wan xin xu, generalized ulam hyers stability of a general mixed aqcq functional equation in multi-banach spaces: a fixed point approach, eur. j. pure appl. math. 3 (2010), 1032-1047. [17] s.m. ulam, a collection of the mathematical problems, interscience, new york, (1960). [18] t. z. xu, j. m. rassias, m. j. rassias and w. x. xu, a fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces, j. inequal. appl. 2010 (2010), article id 423231. [19] xiuzhong wang, lidan chang, guofen liu, orthogonal stability of mixed additive-quadratic jenson type functional equation in multi-banach spaces, adv. pure math. 5 (2015), 325-332. 1. introduction and preliminaries 2. stability of functional equation (1.1) in multi-banach spaces: direct method 3. stability of functional equation (1.1) in multi-banach spaces: fixed point method references int. j. anal. appl. (2023), 21:70 received: mar. 25, 2023. 2020 mathematics subject classification. 91b74. key words and phrases. brand personality, purchase intention, brand equity, consumer preference. https://doi.org/10.28924/2291-8639-21-2023-70 © 2023 the author(s) issn: 2291-8639 1 investigation factors of brand personality affecting on purchase intentions towards authentic agricultural products in vietnam tuong-vi thi tran1,2, quang nhut ho1,2, nhu-ty nguyen1,2,*, truong-phuc le3, hoai-anh duc nguyen4 1school of business, international university, vietnam 2vietnam national university hcmc, quarter 6, linh trung ward, thu duc city, hcmc, vietnam 3inspectorate department of ho chi minh city, department of industry and trade vietnam, vietnam 4university of foreign languages information technology, ho chi minh city, vietnam *corresponding author: nhutynguyen@hcmiu.edu.vn, nhutynguyen@gmail.com abstract. this study investigates how theoretical framework for brand personality ([1]) influence on authentic agricultural products purchase intention. the model is operationalized by a quantitative method process with spss and amos software. the survey was completed by 627 persons. multiple regression demonstrated the factors of brand personality, brand equity; brand authenticity are important predictors of consumer purchase intention for authentic agricultural products. consumer preference as the intermediator, with a positive weight, explains the purchase intention. the results are also analyzed in different circumstances by monthly income and residence area in vietnam. this study helps marketers examined how vietnamese customers view their brands and their rivals therefore what competitors of these authentic agricultural brands can do to enhance the customer purchase intention. the purchase intention findings may be used to identify those brand personality attributes that appear to be most essential in explaining customer preferences. i. introduction branding is one of the most important factors of achieving successful brand in enterprise. the brand offers a warranty, a trustworthy relationship ([19]), and a promise to the customers. nowadays, marketers have been starting to exploit the authenticity as a positioning of a brand https://doi.org/10.28924/2291-8639-21-2023-70 2 int. j. anal. appl. (2023), 21:70 approach and an attractive product and we have entered the "age of authenticity," so it is critical to grasp the relevance of the authenticity on client habits, view it like the greatest means for interaction involving clients and businesses ([21]). purchase intent, or clients's proclivity or inclination to purchase a specific item or service in the future, can indicate the likelihood of purchase. previous research on predicting elements such as consumption behaviour, ([39]), turnover intention ([27]), continuance intention ([23]), entrepreneurial intention ([7]), ([17]), ([6]), ([18]), purchase decision ([2]), purchase intention ([9]), ([38]), have been conducted in a variety of theoretical and practical contexts. however, there has been limited prior study on the brand personality influence on agricultural product purchase intention ([34]), ([33]), and there has been no research on the effect of brand personality on this willingness for the authentic market in vietnam. consequently, the article aim is to better understand the links between brand personality perceptions and purchasing intentions for real agricultural products. finally, the findings of this study could be used to help genuine agricultural enterprises have a thorough understanding of their customers. this study integrates with past researches in a relevant subject to assist organizations in developing business strategies and capturing consumers perspectives in the vietnamese environment. regarding the situation of authentic products in vietnam, the influence of celebrity consumption has a significant impact on the consumption behavior of vietnamese people. a study was conducted on the consumption behavior of vietnamese individuals towards famous fashion brands in vietnam. research variables are built; these include: the attractiveness of the product to consumers, the technical skills in using reason or emotion to perform consumer behavior, the reputation of the business and the brand representative, the association related to the items customers have ever consumed or to their lifestyle characteristics and behavioral intentions. with a total of 252 interviews conducted, the authors came to the conclusion that the reputation of the business and the brand representative; as well as similarity to the products they come into contact with every day plays an important role in consumer intention. other variables do not have a positive influence on the consumption behavior of vietnamese consumers studied ([26]). another study by a group of authors at banking university, vietnam studied the relationship between the factors: authenticity, customer satisfaction and brand equity. the authors, after studying the model, have come to the conclusion that the brand value of the product and the customer's satisfaction is one of the main factors affecting the authenticity of the product. through 3 int. j. anal. appl. (2023), 21:70 the model, businesses can market their products to consumers based on the value of their products rather than external factors ([43]). ii. literature review brand personality symbolizes a collection of human-like features of a specific brand ([1]). comparable to "big five" human personality model ([32]), brand personality is measured in five categories according to each person's personality. according to ([1]), the human-like five-factor features identifies five aspects of brand personality in america: sincerity (e.g., down-to-earth, genuine, sincere, and honest), excitement (e.g., daring, exciting, imaginative, and trendy), competence (e.g., spiritually enlightened, dependable, protected, and optimistic), sophistication (e.g., glamorous, upper-class, good looking, and enchanting), and ruggedness (e.g., hard, adventurous, male, and west). aaker's methodology is so popular that most academic publications have used it since 1997. each country has its own adaptation ([40])... although they share the same methodology, the studies differ in three main points: methodology used, aspects found, and conclusions. in the study of ([40]), a dimension called gender was found. in another study conducted by ([44]), 7 factors were found including: professionalism, persistence, affection, sincerity, sophistication, sophistication and success. in addition, the brand personality dimensions show that there are differences between men and women. the study found that women rate brands more highly for style and success, while men value durability and professionalism more highly. the three aspects of ([44]) that correspond to aaker's results are certainty, sophistication and sincerity. brand personality can be defined by an individual's feelings of love or hate towards a brand. image and attitude of brand are closely related to brand equity. high brand value will make customers feel more loved, thereby increasing sales. so, brand personality is a component of brand identity that contributes to brand equity. brand personality is also an essential component of marketing because shoppers tend to choose products with which they are already known, rather than deep-seated product attributes. over time, brand images are stored in the consumer's brain. when it comes to purchasing decisions, consumers use stored data to make decisions ([44]). therefore, a unique, different brand image will help consumers remember data better, thereby creating better brand equity. this shows that brand personality is a contributing factor to a company's success. but, brand personality is not well understood in terms of importance and benefits for a business. however, the influence of brand personality is different for different product groups. although the brand personality 4 int. j. anal. appl. (2023), 21:70 effect is greater than product features for items with limited interaction, with high-interaction products, product attributes have a greater influence. higher benefit. besides, some aspects of brand personality such as competence, loyalty... are important factors for predicting customer satisfaction ([40]). aaker (1997) ([1]) explained why some companies use their aspects to build their individuality. several brands, for example, have turned to genuine or authentic to create brand identity. some companies also use these phrases in their slogans such as: dockers authentic, genuine jockey comfort... excitement is an effective feature with sports products, cosmetics... dynamic avatar images will help customers feel better than weak, boring images. brand equity can be defined as the value added by products and services ([15]). the institute of marketing science defines brand equity like the combo of associations and behaviors of customer and company allowing a brand to reach more customers and more profit than otherwise ([12]). value of a company or a product can be equalized or subtracted from the brand. another study describes brand equity as a combination of liability and brand equity connected with a known to partners brand or symbol. these assets or liabilities are associated with that brand's name or logo. in the event of a change in the name or logo of the brand, the total assets and liabilities may be affected or disappeared, although the move to the new logo has been widely publicized. also, it can be considered as the added value from products and services ([15]). the approach customers perceive, feel, and conduct toward a brand reflects its equity and thereby make purchasing decisions, it is reflected in the revenue and profit that the product or brand makes. bring to the company. it also creates a difference that helps consumers choose that brand over other brands, even though the products have similar characteristics and properties. brand equity is stored in the hearts and minds of consumers. while brand equity is important, it is only part of the success of a brand and company. it is most important to understand that brand equity is an intangible asset created by promoting communication. brand equity can simply be understood as the value created from the brand, thanks to its name that can create emotions, connect with consumers' thoughts. brand equity creates a competitive advantage for the company. first, a big brand creates the foundation to produce new products. second, a good brand value will help the company to overcome the crisis, develop the business quickly or orient the consumer trend ([20]). brand equity can generate large cash flows for a company in a variety of ways. first, it helps the company attract more new customers or re-engage departed customers. second, it helps to increase the loyalty level of a customer. third, brand equity 5 int. j. anal. appl. (2023), 21:70 helps companies improve profit margins by limiting promotions. fourth, brand equity makes it easier for companies to expand their business. fifth, brand equity is the foundation to help distributors expand their business, giving them a competitive advantage over other competitors. nowadays, the level of market competition is increasingly fierce, especially in the retail sector. understanding the brand value is a premise for the company to be able to allocate resources more effectively. brand equity can be broken down into a set of brand equity, also known as brand equity dimensions. these brand value aspects are the bridge between overall brand value and marketing activities. according to ([29]) effective brand values need to be identified and enhanced, ineffective brand values need to be adjusted or eliminated. managers need to distinguish effective or ineffective values to better adjust and enhance business value consumer preference has a significant impact in the business operation of business. the main marketing activities in the market are eager to reach and exploit consumer preference to make their products and services successful in the market. products that are popular with the masses are always attractive research objects in the market. however, many studies focus on the core values of products, brands and corporate image rather than on consumer attributes. byrne (2020) ([10]) demonstrated that consumer preferences are based on real experiences where they can see the value of the product consumption process directly. consumer preferences can be born even if the customer has never used the product before. he also affirmed the indispensable role of consumer preference research in product marketing and brand development. real-life product experiences are important ways to affirm the value of products to the process of exploiting customer needs. this information is quickly transferred to the brain and remembered because they stimulate the consumer's desire to buy goods. consumer preferences consist of a series of different responses, both subjective and objective, from the outside. this is a transition between consumer experience and consumer preferences; and the product description process no longer plays an important role in shaping consumer preferences. consumption experiences help drive the consumer advocacy process to come alive; easily enter the consumer's subconscious and become a reliable source of information ([16]). consumer preferences carry the characteristics of individual tastes or people who are in demand for a certain product or product. each product has unique features and values through their use. today, as consumers favor a variety of versatile and highly applicable products that are gradually dominating the market, the study of consumer preferences becomes more important than ever. 6 int. j. anal. appl. (2023), 21:70 especially when the technology revolution occurs, consumers become easily accessible to ready-made goods to meet their needs without going through intermediary sales channels like in the past ([10]). purchase intention: intentions are represented by various derivatives such as motivation to act, willingness to exert effort, or level of effort to achieve a goal. consumption intention indicates the extent as well as the ability of consumers to be willing to spend money to purchase a specific item in the coming ([34]). the consumer's consumption decisions most fundamentally describe the attitude to consume a certain product with the level of willingness to pay; or similar as a sign of purchasing behavior. the consumer behavior of customers often determines their intention to consume ([5]). many studies have demonstrated a positive correlation between consumption intention and purchase decision. in which, the intention to consume becomes the motive for an individual to buy a good. in the process of analyzing consumer behavior, consumer intent plays an important role in orienting consumers to products that satisfy their needs. this process is developed gradually from the previous recognition of the product and when the memory again arises, the consumer often decides to buy that good because it is considered as information coming from the product itself. their basic subconscious ([25]). ordinary consumers will look for product features to consider whether the product really meets their needs or not before committing to consumption behavior. however, this consideration is greatly influenced by prior cognitive processes; entails product perception biases rather than a highly rational decision. currently, the methods are mainly done through the internet or advertising on social media where they regularly access to find out information ([45]). authentic products: in 2019, cinelli and leboeuf (2020) ([11]) conducted product authenticity research; based on the relationship between the producer company and the consumer on the meaning of the product authenticity and quality of that product with the consumer's desire to consume. they concluded that product authenticity is a directional process rather than a random judgment of consumers towards a company's product and brand. it affirmed that product authenticity must be accompanied by quality and the message that the manufacturing company orients in their products. further, authors showed the process of formation of authenticity based on product's intrinsic value rather than the successful marketing process of enterprises; these internal values are confirmed by consumers and trusted into the product. product authentication based on higher quality brands are normal or new brands appearing in the market. however, this study has not given an 7 int. j. anal. appl. (2023), 21:70 overall picture of the properties of product authentication but only describes its identity characteristics ([11]). another study ([36]) raised the issues of product authentication in an overall environment of a human; rather than a single concept. through the base entity correspondence, the authenticity is defined based on consumer awareness based on properties: identification, distinction and integration of factors in the brand context and characteristics products in consumers' feelings. the characteristics of authentic products have been published and defined overall by three researchers ([37]). not only do they provide a clear definition, but the authors also associate authentic products with the general characteristics of consumers' consumption. the definition is given more from a consumer perspective rather than an academic one. authentic products are developed based on consumer experiences of structured and synthetic authenticity; the research process gives a meaning related that includes the following research elements: accuracy, integrity, connectedness, legitimacy, proficiency and originality: table 1: elements of authenticity (source: ([37])) brand personality theoretical framework ([1]) with five categories according to each person's personality. according to ([1]), the human-like five-factor features identifies five aspects of brand personality in america: sincerity (e.g., down-to-earth, genuine, sincere, and honest), excitement (e.g., daring, exciting, imaginative, and trendy), competence (e.g., spiritually enlightened, dependable, protected, and optimistic), sophistication (e.g., glamorous, upper-class, good looking, and enchanting), and ruggedness (e.g., hard, adventurous, male, and west) ([13]). 8 int. j. anal. appl. (2023), 21:70 figure 1: brand personality theoretical framework ([1]) brand personality and purchase intention consumers prefer to make purchase decisions based on pre-formed brand pictures in their brains rather than on original traits or characteristics of the product itself, making brand personality one of the most essential concerns in marketing ([14]). the significance of a brand's stored memories in customer decision-making has been widely recognized ([30]). through period, brands establish strong connections in the minds of customers, allowing them to retrieve information stored in their thoughts in order to make decisions: once retrieved, the knowledge gives a cause to purchase the product ([1]). through period, brands establish strong connections in the minds of customers, allowing them to retrieve information stored in their thoughts in order to make decisions: once retrieved, the knowledge gives a cause to purchase the product. in conclusion, brand personality is seen as a key aspect in terms of preference and choice for a business’ success. nonetheless, the significance of brand personality and its impact on purchase intent has not been well recognized. according to several research, brand personality qualities, independent of product category, have a considerable effect on brand selection ([30]). theory of planned behavior ([1]): a crucial aspect in the theory of planned behavior, as in the original theory of reasoned action, is the person purpose to undertake a certain activity. intentions are thought to convey the motivating variables that impact an action; they are signs of how difficult individuals are willing to try, as well as the effort they intend to put in to accomplish the activity. in general, the stronger the intention to engage in an activity, the more likely its performance should be. however, it should be noted that a behavioral intention may only be shown in conduct if the 9 int. j. anal. appl. (2023), 21:70 activity is under volitional control, that's also, if the individual can choose whether or not to do the action. while certain actions may fit this condition relatively well, the behavior of the majority is influenced by due to unfavorable variables such as the availability of necessary resources and possibilities (e.g., time, money, skills, cooperation...). these elements, taken together, show people's genuine power on their conduct. to the degree that an individual has the necessary chances and assets and desires to engage in the behavior, he or she should be successful ([1]). figure 1: theory of planned behavior (([1]) iii. hypothesis and research model brand personality and purchasing intention relationships prior study has shown that brand personality influences purchasing intention ([22]). brand personality and brand equity relationships ([44]) brand personality structure identified as a useful technique for assessing total brand equity. brand equity and consumer preference relationships 10 int. j. anal. appl. (2023), 21:70 according to the researchers, brand equity emerges whenever “customer is familiar with the brand and holds some favorable, strong, and unique brand associations in memory” ([30]). consumer preference and purchase intention relationships ([30]) stated that purchase intention could also be affected by an individual's perceptions and unforeseeable circumstances. as a result, in the authentic agricultural scenario, we would like to confirm the following hypothesis h1: there is a link exists between brand personality and purchase intention h2: there is a link exists between brand personality and brand equity h3: there is a link exists between brand equity and consumer preference h4: there is a link exists between consumer preference and purchase intention research model iv. methodology participants a total of 612 individuals were recruited from three major regions of vietnam: the north, central, and south via both direct and online survey ([23]). table 1 illustrates the general social demographic features of the participants. table 2. social demographic characteristics of participants 1 gender male = 49.2% female = 98.7% lgbt=1.3% 2 age 18 to 25 = 6.2% 25 to <40 = 52.3% 40 to <50= 32.4% >=55 = 9.2% 3 income < 15 million = 15.8% 15 to < 25 million = 17.2% 25 to < 35 million = 48.4% 35 to < 50 million = 13.9% > = 50 million = 4.7% 11 int. j. anal. appl. (2023), 21:70 4 education highschool = 2.1% undergrad = 55.7% graduate = 42.2% 5 marriage status single = 37.9% married = 56.4% divorced = 5.7% 6 job officer = 38.9% lecturer = 17.5% business = 19.3% medical = 5.4% marketing = 8.8% others = 10.1% 7 living area north = 27.1% central = 23.5% south = 53.4% measure scale brand personality was assessed using a five-factor scale that included the five elements of brand personality described by ([1]): sincerity, excitement, competence, sophistication, and ruggedness. brand equity: a five-item survey was utilized for evaluation. from ([31]) and ([40]) such as brand awareness, brand association, brand loyalty, perceived quality and brand knowledge. consumer preference was measured by a five-item questionnaire adopted from ([16]) and ([10]) including product, price, place, promotion and time experience. purchase intention: a five-item survey was utilized to evaluate social commerce conceptions, perspectives regarding source credibility, and collective capacities adapted from ([28]) and elements as emotional value, social value, product evaluation ([35]) were used to assess purchase intention. data collection and analysis all questionnaire items were created in both english and vietnamese prior to the formal online and offline survey for data collection. we then undertook a pilot test with 30 participants. totally, 646 people were asked to participate in the survey, with 612 responding fully, for a rate response of 94.7%. the information structure was simplified with principal component analysis, spss version 22, amos version 24 and sem to study the association between variables. v. analysis of results variables descriptive statistics the research variables descriptive statistics are in table 3. the average results of participants on brand personality (brap=3.77), brand equity (brae=2.78), consumer preference (cop=2.84), and purchase intention (puin=2.27) are all higher than 2 over 5, suggesting that the four factors are well-liked by our test group that influence authentic agricultural product selection. 12 int. j. anal. appl. (2023), 21:70 table 3. descriptive statistics n min max mean std. devia brap 612 1.00 5.00 3.7772 .71085 brae 612 1.00 4.83 2.7821 .80597 cop 612 1.00 5.00 2.8444 .72196 puin 612 1.00 4.43 2.2792 .75347 valid n (listwise) 612 reliability test from the result, the cronbach’s alpha value of model constructs as brap (0.933), brae (0.893), cop (0.873), and puin (0.926) all are greater than 0.5. it demonstrates that the original scale of this aspect is very dependable, and the objects included are cohesive. exploratory factor analysis – efa figure 2. kmo and bartlett's test kmo measure of adequacy sampling. .927 bartlett's test of spher appro. chi-square 9746.211 df 276 sig. .000 each kmo is 0.927 (> 0.5) in all three sessions of the factor analysis procedure, and each total variance explained is at 63.745 (over 50%), demonstrating the suitability of factor analysis. similarly, bartlett's test of sphericity was meaningful, with a sig. level of 0.000 (p below 0.001), indicating considerable correlation among the variables and allowing the study to continue. the pattern matrix of variables (final round) result has been separated into four groups of variables including puin, brap, brae, cop with below method: extraction method: principal axis factoring and rotation method: promax with kaiser normalization. rotation converged in 5 iterations 13 int. j. anal. appl. (2023), 21:70 figure 3. cfa result confirmatory factor analysis – cfa 14 int. j. anal. appl. (2023), 21:70 as illustrated in figure 3, these figures with gfi = 0.938, tli = 0.971, cfi = 0.974 (> 0.8), chisquare/df = 2.011 (< 3), rmsea = 0.041 (< 0.08) show the measurements validity and reliability. therefore, the study results ensure model fit condition. as a result, no factor in this model needs to be removed. all elements should be retained in this study for the following phase of the data analysis procedure. reliability and validity the degree to which items are free of random error and so produce consistent outcomes was quantified in terms of composite reliability. the following local fit criteria were used to evaluate the model's local fitness: indicator reliability more than 0.30, standardised factor larger than 0.60, and a significant t-value; a mean variance explained (ave) greater than 0.50; and a composite reliability (cr) better than 0.60 ([8]). from table 4, all msv scores are smaller than ave, and all sqrtave values are higher than all inter-construct correlations, ensuring discriminability. table 4: confirmatory factor analysis (cfa) fitting indices cr ave msv asv brae puin brap cop brae 0.894 0.585 0.141 0.109 0.765 puin 0.928 0.647 0.141 0.121 0.376 0.805 brap 0.934 0.702 0.155 0.098 0.238 0.286 0.838 cop 0.876 0.587 0.155 0.142 0.361 0.375 0.394 0.766 structural equation modeling – sem the model is evaluated using chi-square, chi-square/df, comparative fit index (cfi), tucker & lewis index (tli), and root mean square error approximation (rmsea) in this research. gfi, tli, and cfi has to be equal or more than 0.9, and chi-square/df must be equal or less than 2. (in some circumstances, chisquare/df 3 can be allowed), and rmsea is equal to or less than 0.08 (rmsea ≤ 0.05 is excellent) (hair et al., 1998). 15 int. j. anal. appl. (2023), 21:70 figure 4. sem result from figure 4, the research framework may be thought of as a relationship assessment with four variables purchase intention, brand personality, brand equity and consumer preference. the calculated model then reasonably matches the input., with χ2/df=2.374 (< 3), cfi =0.965, gfi= 0.927, tli=0.961 and rmsea= 0.047 (< 0.08). these indicators suggested that the model's fit was adequate. figure 5. regression weights estimate s.e. c.r. p label brae <--brap .295 .050 5.844 *** cop <--brae .355 .042 8.518 *** puin <--cop .318 .042 7.542 *** puin <--brap .189 .044 4.335 *** from the figure 5, the cr (t-test) is greater than 2 so these factors have statistically significant at 95% of confident level. otherwise, p-value = 0.000 of all variables are under 0.05, as a result, all variables have statistically significant. 16 int. j. anal. appl. (2023), 21:70 table 5. hypothesis result hypothesis accept reject h1: there is a link exists between brand personality and purchase intention x h2: there is a link exists between brand personality and brand equity x h3: there is a link exists between brand equity and consumer preference x h4: there is a link exists between consumer preference and purchase intention x vi. discussion and conclusions the study proposed three strategies to boost client purchase intention: enhance brand personality to improve purchase intention probability; increase brand equity to enhance purchase intention; higher customer preference to boost purchase intention. according the research result, purchase intention of authentic agricultural product is affected by 03 factors including brand personality, brand equity, consumer preference. this paper have same results of some previous researches such as ([22]); ([30]). furthermore, the current study's findings expand our understanding of the function of brand equity and consumer preference as two mediators in the influence of brand-related constructs on purchase intention, that hasn't been explored in previous studies for these sorts of factors within an integrative approach as proposed here. this study's objective is to determine the most powerful elements influencing consumers' readiness to purchase a certain brand of genuine agriculture products depending on their existing situation. towards that goal, we created a model that encompasses all important brand structures, including brand personality, brand equity, and consumer preference and purchase intention. vii. implications, limitations and future research implications this set of results is extremely consistent with the literature. our findings indicate that customer acceptance of a widespread product, such as an authentic agricultural product, is predicated on its aesthetic elements, brand personality, and equity. this study is important for companies that produce and offer authentic agricultural products. they can use these findings to boost customer purchasing 17 int. j. anal. appl. (2023), 21:70 power for authentic agricultural products. furthermore, these organizations might develop business strategies that focus on brand personality in order to enhance client purchase intention. limitations although every effort was made in this work to reduce constraints, some limitations remain for upcoming research. firstly, the research was limited to vietnamese consumers in the downtown area. as a result, transcend cultural and economic prejudices, it can be fascinating and practical to test its validity and generalizability in other nations in asian (e.g., japan and south korea). secondly, brand-related concepts affirmed their impacts on purchase intention but other brand-related dimensions that might impact purchase intention, as well as their additional predictors, must be both theoretically and experimentally tested for future researches such as product quality, word-of-mouth, social media. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.l. aaker, dimensions of brand personality, j. market. res. 34 (1997), 347-356. https://doi.org/10.2307/3151897. [2] z. abdullah, m.m. anuar, m.r. yaacob, cause-related marketing purchase decision: do religiosity and attitudes matter, j. glob. bus. adv. 14 (2021), 684-704. https://doi.org/10.1504/jgba.2021.123556. [3] i. ajzen, the theory of planned behavior, organ. behav. human decision processes. 50 (1991), 179– 211. https://doi.org/10.1016/0749-5978(91)90020-t. [4] i. ajzen, the theory of planned behaviour: reactions and reflections, psychol. health. 26 (2011), 1113– 1127. https://doi.org/10.1080/08870446.2011.613995. [5] a.a. ali, a. abbass, n. farid, factors influencing customers’ purchase intention in social commerce, int. rev. manage. market. 10 (2020), 63–73. https://doi.org/10.32479/irmm.10097. [6] a.d. alonso, n. alexander, entrepreneurial intentions in an emerging industry: an exploratory study, j. int. bus. entrepreneur. develop. 10 (2017), 406-426. https://doi.org/10.1504/jibed.2017.088709. [7] i. anwar, m.t. jamal, i. saleem, p. thoudam, traits and entrepreneurial intention: testing the mediating role of entrepreneurial attitude and self-efficacy, j. int. bus. entrepreneur. develop. 13 (2021), 40-60. https://doi.org/10.1504/jibed.2021.112276. [8] r.p. bagozzi, y. yi, on the evaluation of structural equation models, j. acad. market. sci. 16 (1988), 74–94. https://doi.org/10.1007/bf02723327. https://doi.org/10.2307/3151897 https://doi.org/10.1504/jgba.2021.123556 https://doi.org/10.1016/0749-5978(91)90020-t https://doi.org/10.1080/08870446.2011.613995 https://doi.org/10.32479/irmm.10097 https://doi.org/10.1504/jibed.2017.088709 https://doi.org/10.1504/jibed.2021.112276 https://doi.org/10.1007/bf02723327 18 int. j. anal. appl. (2023), 21:70 [9] a.r.k. monfared, m. ghaffari, m. barootkoob, m.m. malmiri, the role of social commerce in online purchase intention: mediating role of social interactions, trust, and electronic word of mouth, j. int. bus. entrepreneur. develop. 13 (2021), 22-39. https://doi.org/10.1504/jibed.2021.112264. [10] d.v. byrne, current trends in multidisciplinary approaches to understanding consumer preference and acceptance of food products, foods. 9 (2020), 1380. https://doi.org/10.3390/foods9101380. [11] m.d. cinelli, r.a. leboeuf, keeping it real: how perceived brand authenticity affects product perceptions, j. consum. psychol. 30 (2019), 40–59. https://doi.org/10.1002/jcpy.1123. [12] h. datta, k.l. ailawadi, h.j. van heerde, how well does consumer-based brand equity align with salesbased brand equity and marketing-mix response? j. market. 81 (2017), 1–20. https://doi.org/10.1509/jm.15.0340. [13] g. davies, j.i. rojas-méndez, s. whelan, m. mete, t. loo, brand personality: theory and dimensionality, journal of product & brand management. 27 (2018) 115–127. https://doi.org/10.1108/jpbm-06-20171499. [14] a. dick, d. chakravarti, g. biehal, memory-based inferences during consumer choice, j. consum. res. 17 (1990), 82. https://doi.org/10.1086/208539. [15] j. drennan, c. bianchi, s. cacho-elizondo, s. louriero, n. guibert, w. proud, examining the role of wine brand love on brand loyalty: a multi-country comparison, int. j. hospital. manage. 49 (2015), 47–55. https://doi.org/10.1016/j.ijhm.2015.04.012. [16] r. ebrahim, a. ghoneim, z. irani, y. fan, a brand preference and repurchase intention model: the role of consumer experience, j. market. manage. 32 (2016), 1230–1259. https://doi.org/10.1080/0267257x.2016.1150322. [17] a. echchabi, m.m.s. omar, a.m. ayedh, entrepreneurial intention among female university students in oman, j. int. bus. entrepreneur. develop. 12 (2020), 280-297. https://doi.org/10.1504/jibed.2020.110251. [18] v. gautam, a. basu, a. basu, t. singh, entrepreneurial attributes and intention among management students: a longitudinal approach to evolution and applicability of conceptual and empirical constructs, j. int. bus. entrepreneur. develop. 12 (2020), 156-182. https://doi.org/10.1504/jibed.2020.106189. [19] h.m. gelaidan, h.a. mabkhot, o.s.a. kwifi, the mediation role of brand trust and satisfaction between brand image and loyalty, j. glob. bus. adv. 14 (2021), 845-862. https://doi.org/10.1504/jgba.2021.125010. [20] b. godey, a. manthiou, d. pederzoli, j. rokka, g. aiello, r. donvito, r. singh, social media marketing efforts of luxury brands: influence on brand equity and consumer behavior, j. bus. res. 69 (2016), 5833–5841. https://doi.org/10.1016/j.jbusres.2016.04.181. https://doi.org/10.1504/jibed.2021.112264 https://doi.org/10.3390/foods9101380 https://doi.org/10.1002/jcpy.1123 https://doi.org/10.1509/jm.15.0340 https://doi.org/10.1108/jpbm-06-2017-1499 https://doi.org/10.1108/jpbm-06-2017-1499 https://doi.org/10.1086/208539 https://doi.org/10.1016/j.ijhm.2015.04.012 https://doi.org/10.1080/0267257x.2016.1150322 https://doi.org/10.1504/jibed.2020.110251 https://doi.org/10.1504/jibed.2020.106189 https://doi.org/10.1504/jgba.2021.125010 https://doi.org/10.1016/j.jbusres.2016.04.181 19 int. j. anal. appl. (2023), 21:70 [21] k. grayson, r. martinec, consumer perceptions of iconicity and indexicality and their influence on assessments of authentic market offerings, j. consum. res. 31 (2004), 296–312. https://doi.org/10.1086/422109. [22] g. guido, a.m. peluso, m. provenzano, influence of brand personality-marker attributes on purchasing intention: the role of emotionality, psychol. rep. 106 (2010), 737–751. https://doi.org/10.2466/pr0.106.3.737-751. [23] m.t. ha, g.d. nguyen, m.l. nguyen, a.c. tran, understanding the influence of user adaptation on the continuance intention towards ride-hailing services: the perspective of management support, j. glob. bus. adv. 15 (2022), 39-62. https://doi.org/10.1504/jgba.2022.127208. [24] j.f. hair, r.e. anderson, r.l. tatham, w.c. black, multivariate data analysis (5th ed.), prentice-hall, (1998). [25] p. harrigan, u. evers, m. miles, t. daly, customer engagement with tourism social media brands, tourism manage. 59 (2017), 597–609. https://doi.org/10.1016/j.tourman.2016.09.015. [26] t.v. ho, t.n. phan, v.p. le-hoang, the authenticity of celebrity endorsement on purchase intentioncase on local fashion brand in vietnam, int. j. manage. 11 (2020), 1347-1356. [27] s. huma, t. javaid, s. ishtiaque, factors affecting turnover intention of logisticians: empirical evidence from pakistan, j. glob. bus. adv. 14 (2021), 568-586. https://doi.org/10.1504/jgba.2021.118735. [28] s. hussain, y. li, w. li, influence of platform characteristics on purchase intention in social commerce: mechanism of psychological contracts, j. theor. appl. electron. commer. res. 16 (2021), 1–17. https://doi.org/10.4067/s0718-18762021000100102. [29] o. iglesias, s. markovic, j. rialp, how does sensory brand experience influence brand equity? considering the roles of customer satisfaction, customer affective commitment, and employee empathy, j. bus. res. 96 (2019), 343–354. https://doi.org/10.1016/j.jbusres.2018.05.043. [30] k.l. keller, conceptualizing, measuring, and managing customer-based brand equity, j. market. 57 (1993), 1–22. https://doi.org/10.1177/002224299305700101. [31] md.m. khudri, n. farjana, identifying the key dimensions of consumer-based brand equity model: a multivariate approach, asian j. market. 11 (2016), 13–20. https://doi.org/10.3923/ajm.2017.13.20. [32] k. kircaburun, s. alhabash, ş.b. tosuntaş, m.d. griffiths, uses and gratifications of problematic social media use among university students: a simultaneous examination of the big five of personality traits, social media platforms, and social media use motives, int. j. ment. health addiction. 18 (2018), 525– 547. https://doi.org/10.1007/s11469-018-9940-6. [33] e.l. lopes, r.t. veiga, increasing purchasing intention of eco-efficient products: the role of the advertising communication strategy and the branding strategy, j, brand manage. 26 (2019), 550–566. https://doi.org/10.1057/s41262-019-00150-0. https://doi.org/10.1086/422109 https://doi.org/10.2466/pr0.106.3.737-751 https://doi.org/10.1504/jgba.2022.127208 https://doi.org/10.1016/j.tourman.2016.09.015 https://doi.org/10.1504/jgba.2021.118735 https://doi.org/10.4067/s0718-18762021000100102 https://doi.org/10.1016/j.jbusres.2018.05.043 https://doi.org/10.1177/002224299305700101 https://doi.org/10.3923/ajm.2017.13.20 https://doi.org/10.1007/s11469-018-9940-6 https://doi.org/10.1057/s41262-019-00150-0 20 int. j. anal. appl. (2023), 21:70 [34] j. martins, c. costa, t. oliveira, r. gonçalves, f. branco, how smartphone advertising influences consumers’ purchase intention, j. bus. res. 94 (2019), 378–387. https://doi.org/10.1016/j.jbusres.2017.12.047. [35] m. mcgowan, e. shiu, l.m. hassan, the influence of social identity on value perceptions and intention, j. consumer behav. 16 (2016), 242–253. https://doi.org/10.1002/cb.1627. [36] j.g. moulard, r.d. raggio, j.a.g. folse, disentangling the meanings of brand authenticity: the entityreferent correspondence framework of authenticity, j. acad. market. sci. 49 (2020), 96–118. https://doi.org/10.1007/s11747-020-00735-1. [37] j.c. nunes, a. ordanini, g. giambastiani, the concept of authenticity: what it means to consumers, j. market. 85 (2021), 1–20. https://doi.org/10.1177/0022242921997081. [38] j.s. rai, m.n. itani, a. singh, a. singh, delineating the outcomes of fans’ psychological commitment to sport team: product knowledge, attitude towards the sponsor, and purchase intentions, j. glob. bus. adv. 14 (2021), 357-382. https://doi.org/10.1504/jgba.2021.116720. [39] r. rebouças, a.m. soares, the consumption behaviour of beginner voluntary simplifiers: an exploratory study, j. glob. bus. adv. 14 (2021), 433-452. https://doi.org/10.1504/jgba.2021.118751. [40] x. tong, j.m. hawley, measuring customer‐based brand equity: empirical evidence from the sportswear market in china, j. product brand manage. 18 (2009), 262–271. https://doi.org/10.1108/10610420910972783. [41] x. tong, j. su, y. xu, brand personality and its impact on brand trust and brand commitment: an empirical study of luxury fashion brands, int. j. fashion design technol. educ. 11 (2017), 196–209. https://doi.org/10.1080/17543266.2017.1378732. [42] i. ajzen, m. fishbein, understanding attitudes and predicting social behavior, prentice-hall, englewood cliffs, 1980. [43] v.d. tran, t.n.l. vo, t.q. dinh, the relationship between brand authenticity, brand equity and customer satisfaction, j. asian finance econ. bus. 7 (2020), 213–221. https://doi.org/10.13106/jafeb.2020.vol7.no4.213. [44] h. vahdati, s.h. mousavi nejad, brand personality toward customer purchase intention: the intermediate role of electronic word-of-mouth and brand equity, asian acad. manage. j. 21 (2016), 1–26. https://doi.org/10.21315/aamj2016.21.2.1. [45] c.w. wu, the performance impact of social media in the chain store industry, j. bus. res. 69 (2016), 5310–5316. https://doi.org/10.1016/j.jbusres.2016.04.130. https://doi.org/10.1002/cb.1627 https://doi.org/10.1007/s11747-020-00735-1 https://doi.org/10.1177/0022242921997081 https://doi.org/10.1504/jgba.2021.116720 https://doi.org/10.1504/jgba.2021.118751 https://doi.org/10.1108/10610420910972783 https://doi.org/10.1080/17543266.2017.1378732 https://doi.org/10.13106/jafeb.2020.vol7.no4.213 https://doi.org/10.21315/aamj2016.21.2.1 https://doi.org/10.1016/j.jbusres.2016.04.130 international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 155-171 doi: 10.28924/2291-8639-15-2017-155 fuzzy hyperideals of left almost semihypergroups asghar khan1, muhammad farooq1, muhammad izhar1,∗ and bijan davvaz2 abstract. this paper explores the foundations of fuzzy left (resp. right) hyperideals of left almost semihypergroups (briefly, la-semihypergroups). we investigate the properties of fuzzy left hyperideals and fuzzy right hyperideals in regular and intra-regular la-semihypergroups. we also characterize regular and intra-regular la-semihypergroups in terms of fuzzy hyperideals. 1. introduction the idea of generalization of a commutative semigroup, (known as left almost semigroup) was introduced by kazim and naseeruddin in 1972 (see [1]). a groupoid (s, ·) is called an ag-groupoid if it satisfies the left invertive law: (ab)c = (cb)a for all a,b,c ∈ s. this structure is closely related with a commutative semigroup because if an ag-groupoid contains right identity then it becomes a commutative monoid. an ag-groupoid may or may not contain a left identity. some other names have also been used in literature for left almost semigroups. cho et al. [2] studied this structure under the name of right modular groupoid. holgate [3] studied it as left invertive groupoid. similarly, stevanovic and protic [4] called this structure an abel-grassmann groupoid (or simply la-semigroup), which is the primary name under which this structure is known nowadays. there are many important applications of ag-groupoids in the theory of flocks [5]. the concept of a fuzzy set was introduced by zadeh [9], in 1965. since its inception, the theory has developed in many directions and found applications in a wide variety of fields. many researchers published high-quality research articles on fuzzy sets in a variety of international journals. the study of fuzzy set in algebraic structure has been started in the definitive paper of rosenfeld 1971 [15], in which he defined fuzzy subgroup and gave its important properties. in 1981, kuroki introduced the concept of fuzzy ideals and fuzzy bi-ideals in semigroups in his paper [16]. the theory of hyperstructures was introduced by marty in 1934 during the 8th congress of the scandinavian mathematicians [20]. marty introduced hypergroups as a generalization of groups. he published some papers on hypergroups, using them in different contexts as algebraic functions, rational fractions, non commutative groups. in the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. in [17] corsini and leoreanufotea collected numerous applications of algebraic hyperstructures such as: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities. especially, semihypergroups are the simplest algebraic hyperstructures which possess the properties of closure and associativity. nowadays many scholars have studied different aspects of semihypergroups see [18, 19, 21, 22]. recently, hila and dine [12] introduced the notion of la-semihypergroups. they investigated several properties of hyperideals of la-semihypergroup and defined the topological space and study the topological structure of la-semihypergroups using hyperideal theory. in [13], yaqoob, corsini and yousafzai have characterized intra-regular la-semihypergroups by using the properties of their left and right hyperideals, received 26th july, 2017; accepted 29th september, 2017; published 1st november, 2017. 2010 mathematics subject classification. 08a72, 20n20. key words and phrases. fuzzy set; la-semihypergroup; fuzzy la-semihypergroup; fuzzy left (resp. right) hyperideal; regular and intra-regular la-semihypergroup. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 155 156 khan, farooq, izhar and davvaz and investigated some useful conditions for an la-semihypergroup to become an intra-regular lasemihypergroup. this non-associative hyper structure has been further explored in [14], by yousafzai and corsini. in this paper, we introduce the notion of fuzzy left (resp. right) hyperideals in la-semihypergroups and present some related examples of these concepts. we characterize regular and intra-regular lasemihypergroups in terms of fuzzy hyperideals. 2. preliminaries a hypergroupoid is a nonempty set s equipped with a hyperoperation ◦, that is a map ◦ : s ×s −→ p∗ (s), where p∗ (s) denotes the set of all nonempty subsets of s (see [20]). we shall denote by x ◦ y, the hyperproduct of elements x,y of s. let a, b be two nonempty subsets of s. then the hyperproduct of a and b is defined as a◦b = ⋃ a∈a,b∈b a◦ b. we shall write a◦x instead of a◦{x} and x◦a for {x}◦a. a hypergroupoid (s,◦) is called an la-semihypergroup [12], if it satisfies the left invertive law: (a◦ b) ◦ c = (c◦ b) ◦a for all a,b,c ∈ s. every la-semihypergroup satisfies the medial law [12]. that is, (x◦y) ◦ (z ◦w) = (x◦z) ◦ (y ◦w) for all w,x,y,z ∈ s. definition 2.1. (see [14]). let (s,◦) be an la-semihypergroup then an element e ∈ s is called (i) left identity (resp. pure left identity) if ∀ a ∈ s, a ∈ e◦a (resp. a = e◦a); (ii) right identity (resp. pure right identity) if ∀ a ∈ s, a ∈ a◦e (resp. a = a◦e); (iii) identity (resp. pure identity) if ∀ a ∈ s, a ∈ e◦a∩a◦e (resp. a = e◦a∩a◦e). an la-semihypergroup (s,◦) with pure left identity e, paramedial law holds. that is (x◦y) ◦ (z ◦w) = (w ◦z) ◦ (y ◦x) for all w,x,y,z ∈ s. an la-semihypergroup (s,◦) with pure left identity e, satisfies the following law x◦ (y ◦z) = y ◦ (x◦z) (1) . a nonempty subset a of an la-semihypergroup (s,◦) is called an la-subsemihypergroup of s if a◦a ⊆ a. a nonempty subset a of an la-semihypergroup (s,◦) is a called left ( resp. right ) hyperideal of s if s ◦a ⊆ a (resp. a◦s ⊆ a). if a is both a left hyperideal and a right hyperideal of s then it is called a two-sided hyperideal or simply a hyperideal of s. an la-semihypergroup s is called [13]; (i) regular if for all a ∈ s, there exist x ∈ s such that a ∈ (a◦x) ◦a. (ii) intra-regular if for all a ∈ s, there exist x,y ∈ s such that a ∈ (x◦a2) ◦y. 3. fuzzy concepts in la-semihypergroups let s be an la-semihypergroup. a function f from a nonempty set x to the unit interval [0, 1] is called a fuzzy subset of s. let s be an la-semihypergroup and f be a fuzzy subset of s. then for every t ∈ (0, 1] the set u (f; t) = {x | x ∈ s, f (x) ≥ t} , is called the level set of f. for x ∈ s, define ax = {(y,z) ∈ s ×s : x ∈ y ◦z or x = y ◦z} . we denote by f (s) the set of all fuzzy subsets of s. fuzzy hyperideals of left almost semihypergroups 157 let s be an la-semihypergroup and f,g are any two fuzzy subsets of s. we define the product f ∗g of f and g as follows: (f ∗g) (x) = ∨ (y,z)∈ax {f (y) ∧g (z)} . the fuzzy subsets defined by s : s −→ [0, 1],x −→ s (x) = 1 and 0 : s −→ [0, 1] ,x −→ 0 (x) = 0 for all x ∈ s are the greatest and least elements of f (s) . definition 3.1. let s be an la-semihypergroup and ∅ 6= a ⊆ s. then the characteristic function χa of a is defined as: χa : s −→ [0, 1] ,−→ χa (x) = { 1 if x ∈ a 0 if x /∈ a definition 3.2. let s be an la-semihypergroup and f be a fuzzy subset of s. then f is called a fuzzy la-subsemihypergroup of s if: (∀x,y ∈ s) ∧ α∈x◦y f (α) ≥ f (x) ∧f (y) . definition 3.3. let s be an la-semihypergroup and f be a fuzzy subset of s. then f is called a fuzzy left (resp. right) hyperideal of s if: (∀x,y ∈ s) ∧ α∈x◦y f (α) ≥ f (y) (resp. ∧ α∈x◦y f (α) ≥ f (x) ). definition 3.4. a fuzzy hyperideal f of an la-semihypergroup s is called idempotent if f ∗f = f. example 3.1. let us consider an la-semihypergroup s = {a,b,c} in the following cayley’s table ◦ a b c a {a} {a} {a} b {a} {a} {a,c} c {a} {a} {a} let us define a fuzzy subset f : s −→ [0, 1] as follows f (x) =   0.9 if x = a 0.7 if x = b 0.5 if x = c then it is easy to observe that f is a fuzzy la-subsemihypergroup of s. example 3.2. let us consider an la-semihypergroup s = {e1,e2,e3} in the following cayley’s table ◦ e1 e2 e3 e1 {e1} {e1} {e1} e2 {e1} {e1} {e1,e3} e3 {e1} {e1} {e1} let us define a fuzzy subset f : s −→ [0, 1] as follows f (x) =   0.8 if x = e1 0.4 if x = e2 0.6 if x = e3 then it is easy to see that f is a fuzzy hyperideal of la-semihypergroup s. example 3.3. let us consider an la-semihypergroup s = {e1,e2,e3} in the following cayley’s table ◦ e1 e2 e3 e1 {e1,e3} {e2} {e2,e3} e2 {e2,e3} {e2,e3} {e2,e3} e3 {e2,e3} {e2,e3} {e2,e3} 158 khan, farooq, izhar and davvaz let us define a fuzzy subset f : s −→ [0, 1] as follows f (x) =   0.5 if x = e1 0.7 if x = e2 0.7 if x = e3 then it is easy to see that f is a fuzzy hyperideal of la-semihypergroup s. proposition 3.1. the set (f (s) ,∗) is an la-semihypergroup. proof. clearly f (s) is closed. let f,g and h be in f (s) . if ax = ∅ for any x ∈ s. then ((f ∗g) ∗h) (x) = 0 = ((h∗g) ∗f) (x) . let ax 6= ∅, then there exist y and z in s such that (y,z) ∈ ax. therefore by using left invertive law, we have ((f ∗g) ∗h) (x) = ∨ (y,z)∈ax {(f ∗g) (y) ∧h (z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay {f (p) ∧g (q)}∧h (z)   = ∨ x∈((p◦q)◦z) {f (p) ∧g (q) ∧h (z)} = ∨ x∈((z◦q)◦p) {h (z) ∧g (q) ∧f (p)} = ∨ (w,p)∈ax   ∨ (z,q)∈aw (h (z) ∧g (q)) ∧f (p)   = ∨ (w,p)∈ax {(h∗g) (w) ∧f (p)} = ((h∗g) ∗f) (x) . hence (f (s) ,∗) is an la-semihypergroup. � lemma 3.1. let s be an la-semihypergroup. then the medial law holds in f (s) . proof. let f,g,h and k be the arbitrary elements of f (s) . by successive use of left invertive law, (f ∗g) ∗ (h∗k) = ((h∗k) ∗g) ∗f = ((g ∗k) ∗h) ∗f = (f ∗h) ∗ (g ∗k) . � proposition 3.2. an la-semihypergroup with f (s) = (f (s)) 2 is a commutative semihypergroup if and only if (f ∗g) ∗h = f ∗ (h∗g) holds for all fuzzy subsets f,g,h ∈ f (s) . proof. let s be a commutative semihypergroup. for any fuzzy subsets f,g,h ∈ f (s) . if ax = ∅ then ((f ∗g) ∗h) (x) = 0 = (f ∗ (h∗g)) (x) . let ax 6= ∅ then (s,t) ∈ ax, therefore by the use of left fuzzy hyperideals of left almost semihypergroups 159 invertive law and commutative law, we get ((f ∗g) ∗h) (x) = ∨ (s,t)∈ax {(f ∗g) (s) ∧h (t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as (f (m) ∧g (n)) ∧h (t)   = ∨ x∈((m◦n)◦t) {f (m) ∧h (t) ∧g (n)} = ∨ x∈((t◦n)◦m) {f (m) ∧h (t) ∧g (n)} = ∨ x∈(m◦(t◦n)) {f (m) ∧h (t) ∧g (n)} = ∨ (m,p)∈ax  f (m) ∧ ∨ (t,n)∈ap (h (t) ∧g (n))   = ∨ (m,p)∈ax {f (m) ∧ (h∗g) (p)} = (f ∗ (h∗g)) (x) . conversely, let (f ∗g) ∗h = f ∗ (h∗g) holds for all fuzzy subsets f,g,h ∈ f (s) . we have to show that f (s) is a commutative semihypergroup. let f and g be any fuzzy subsets of s. if ax = ∅ for any x ∈ s, then (f ∗g) (x) = 0 = (g ∗f) (x) . let ax 6= ∅. then (s,t) ∈ ax. since f (s) = (f (s)) 2 . so f = (h∗k) where h and k are any fuzzy subsets of s. now by using left invertive law, we have (f ∗g) (x) = ((h∗k) ∗g) (x) = ∨ (s,t)∈ax {(h∗k) (s) ∧g (t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as (h (m) ∧k (n)) ∧g (t)   = ∨ x∈((m◦n)◦t) {h (m) ∧k (n) ∧g (t)} = ∨ x∈((t◦n)◦m) {g (t) ∧k (n) ∧h (m)} = ∨ (p,m)∈ax   ∨ (t,n)∈ap (g (t) ∧k (n)) ∧h (m)   = ∨ (p,m)∈ax {(g ∗k) (p) ∧h (m)} = ((g ∗k) ∗h) (x) = (g ∗ (h∗k)) (x) . this shows that f ∗g = g ∗ (h∗k) = g ∗f. thus commutative law holds in f (s) . 160 khan, farooq, izhar and davvaz now if ax = ∅. then ((f ∗g) ∗h) (x) = 0 = (f ∗ (h∗g)) (x). let ax 6= ∅. then (s,t) ∈ ax. therefore by the use of commutative law and left invertive law we get ((f ∗g) ∗h) (x) = ∨ (s,t)∈ax {(f ∗g) (s) ∧h (t)} = ∨ (s,t)∈ax   ∨ (m,n)∈as (f (m) ∧g (n)) ∧h (t)   = ∨ x∈((m◦n)◦t) {f (m) ∧g (n) ∧h (t)} = ∨ x∈((t◦n)◦m) {f (m) ∧g (n) ∧h (t)} = ∨ x∈(m◦(t◦n)) {f (m) ∧g (n) ∧h (t)} = ∨ x∈(m◦(n◦t)) {f (m) ∧g (n) ∧h (t)} = ∨ (m,p)∈ax  f (m) ∧ ∨ (n,t)∈ap (g (n) ∧h (t))   = ∨ (m,p)∈ax {f (m) ∧ (g ∗h) (p)} = (f ∗ (g ∗h)) (x) . � theorem 3.1. if s has a pure left identity then the following properties holds in f (s) . (1) f ∗ (g ∗h) = g ∗ (f ∗h) for all f,g and h ∈ f (s) . (2) (f ∗g) ∗ (h∗k) = (k ∗h) ∗ (g ∗f) for all f,g,h and k ∈ f (s) . proof. (1). let x ∈ s. if ax = ∅. then (f ∗ (g ∗h)) (x) = 0 = (g ∗ (f ∗h)) (x) . let ax 6= ∅. then (y,z) ∈ ax. now by using medial law with pure left identity, we have (f ∗ (g ∗h)) (x) = ∨ (y,z)∈ax {f (y) ∧ (g ∗h) (z)} = ∨ (y,z)∈ax  f (y) ∧ ∨ (p,q)∈az (g (p) ∧h (q))   = ∨ x∈(y◦(p◦q)) {f (y) ∧g (p) ∧h (q)} = ∨ x∈(p◦(y◦q)) {g (p) ∧f (y) ∧h (q)} = ∨ (p,w)∈ax  g (p) ∧ ∨ (y,q)∈aw (f (y) ∧h (q))   = ∨ (p,w)∈ax {g (p) ∧ (f ∗h) (w)} = (g ∗ (f ∗h)) (x) . thus (f ∗ (g ∗h)) (x) = (g ∗ (f ∗h)) (x) for all x ∈ s. fuzzy hyperideals of left almost semihypergroups 161 (2) . if ax = ∅ for x ∈ s, then ((f ∗g) ∗ (h∗k)) (x) = 0 = ((k ∗h) ∗ (g ∗f)) (x) . let ax 6= ∅ then there exist y and z in s such that (y,z) ∈ ax. therefore by using paramedial law, we have ((f ∗g) ∗ (h∗k)) (x) = ∨ (y,z)∈ax {(f ∗g) (y) ∧ (h∗k) (z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay {f (p) ∧g (q)}∧ ∨ (u,v)∈az {(h (u) ∧k (v))}   = ∨ x∈((p◦q)◦(u◦v)) {f (p) ∧g (q) ∧h (u) ∧k (v)} = ∨ x∈((v◦u)◦(q◦p)) {k (v) ∧h (u) ∧g (q) ∧f (p)} = ∨ (m,n)∈ax   ∨ (v,u)∈am {k (v) ∧h (u)} ∨ (q,p)∈an {g (q) ∧f (p)}   = ∨ (m,n)∈ax {(k ∗h) (m) ∧ (g ∗f) (n)} = ((k ∗h) ∗ (g ∗f)) (x) . thus (f ∗g) ∗ (h∗k) = (k ∗h) ∗ (g ∗f) for all x ∈ s. � theorem 3.2. let s be an la-semihypergroup. then l = {f | f ∈ f (s) , f ∗h = f where h = h∗h} is a commutative monoid in s. proof. the fuzzy subset l of s is nonempty since h ∗ h = h, which implies that h is in l. let f and g be the fuzzy subsets of s in l, then f ∗ h = f and g ∗ h = g. if ax = ∅ for x ∈ s, then (f ∗g) (x) = 0 = ((f ∗g) ∗h) (x) . let ax 6= ∅. then by using medial law, we have (f ∗g) (x) = ∨ (y,z)∈ax {(f ∗h) (y) ∧ (g ∗h) (z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay {f (p) ∧h (q)}∧ ∨ (u,v)∈az {g (u) ∧h (v)}   = ∨ x∈((p◦q)◦(u◦v)) {f (p) ∧h (q) ∧g (u) ∧h (v)} = ∨ x∈((p◦u)◦(q◦v)) {f (p) ∧g (u) ∧h (q) ∧h (v)} = ∨ (m,n)∈ax   ∨ (p,u)∈am {f (p) ∧g (u)} ∨ (q,v)∈an {h (q) ∧h (v)}   = ∨ (m,n)∈ax {(f ∗g) (m) ∧ (h∗h) (n)} = ((f ∗g) ∗ (h∗h)) (x) . thus f ∗g = (f ∗g) ∗ (h∗h) = (f ∗g) ∗h which implies that l is closed. 162 khan, farooq, izhar and davvaz now if ax = ∅. then (f ∗g) (x) = 0 = (g ∗f) (x) . let ax 6= ∅ then (y,z) ∈ ax. therefore by using left invertive law, we have (f ∗g) (x) = ∨ (y,z)∈ax {(f ∗h) (y) ∧g (z)} = ∨ (y,z)∈ax   ∨ (p,q)∈ay (f (p) ∧h (q)) ∧g (z)   = ∨ x∈((p◦q)◦z) {f (p) ∧h (q) ∧g (z)} = ∨ x∈((z◦q)◦p) {g (z) ∧h (q) ∧f (p)} = ∨ (t,p)∈ax   ∨ (z,q)∈at (g (z) ∧h (q)) ∧f (p)   = ∨ (t,p)∈ax {(g ∗h) (t) ∧f (p)} = ((g ∗h) ∗f) (x) . thus f ∗g = (g ∗h)∗f = g∗f, which implies that commutative law holds in l and associative law holds in l due to commutativity. since for any fuzzy subset f in l, we have f ∗ h = f (where h is fixed) implies that h is pure right identity in f (s) and hence an identity. � lemma 3.2. let s be an la-semihypergroup. if s has a pure left identity then s ∗s = s. proof. every x in s can be written as x = e◦x, where e is the pure left identity in s. therefore (s ∗s) (x) = ∨ (y,z)∈ax {s (y) ∧s (z)} ≥{s (e) ∧s (x)} = 1 = s (x) . hence s ∗s = s. � theorem 3.3. let χa and χb be fuzzy subsets of an la-semihypergroup s, where a and b are nonempty subsets of s. then the following properties hold: (1) if a ⊆ b then χa ⊆ χb. (2) χa ∩χb = χa∩b. (3) χa ∗χb = χa◦b. proof. (1) . it is obvious. (2) . let x ∈ s. if x ∈ a∩b, then x ∈ a and x ∈ b. so χa (x) = 1 and χb (x) = 1. thus we have (χa ∩χb) (x) = χa (x)∧χb (x) = 1 = χa∩b. if x /∈ a∩b, then x /∈ a and x /∈ b. so χa (x) = 0 and χb (x) = 0. thus we have (χa ∩χb) (x) = χa (x) ∧χb (x) = 0 = χa∩b. thus χa ∩χb = χa∩b. (3) . for any x ∈ s. if x /∈ a◦b, then χa◦b(x) = 0 (i) this means that there does not exist y ∈ a and z ∈ b such that x ∈ y ◦z. if ax = ∅ then (χa ∗χb)(x) = 0 (ii) if ax 6= ∅ and (y,z) ∈ ax then x ∈ y◦z. then y /∈ a or z /∈ b. thus either χa(y) = 0 or χb(z) = 0. so we have, χa(y) ∧χb(z) = 0. hence (χa ∗λb)(x) = 0. fuzzy hyperideals of left almost semihypergroups 163 let x ∈ a◦b, then χa◦b(x) = 1. thus x ∈ a◦ b, for some a ∈ a and b ∈ b, so (a,b) ∈ ax. since ax 6= ∅, we have (χa ∗χb)(x) = ∨ (y,z)∈ax {χa(y) ∧χb(z)} ≥ χa(a) ∧χb(b) = 1. thus (χa ∗χb)(x) = 1. hence χa ∗χb = χa◦b. � theorem 3.4. a fuzzy subset f of an la-semihypergroup s is a fuzzy la-subsemihypergroup of s if and only if f ∗f ⊆ f. proof. assume that f is a fuzzy la-subsemihypergroup of s. if aa = ∅. then (f ∗f) (a) = 0 = f (a) . if aa 6= ∅, then there exist x and y in s such that (x,y) ∈ aa. then for any α ∈ x◦y, we have a ∈ α. since f is a fuzzy la-subsemihypergroup of s, we have (f ∗f) (a) = ∨ (x,y)∈aa {f (x) ∧f (y)} ≤ ∨ (x,y)∈aa f (α) ≤ ∨ (x,y)∈aa f (a) = f (a) . thus f ∗f ⊆ f. conversely, assume that f ∗f ⊆ f. let x,y ∈ s and α ∈ x◦y. we have, f (α) ≥ (f ∗f) (α) = ∨ (x,y)∈aα {f (x) ∧f (y)} ≥{f (x) ∧f (y)} f (α) ≥{f (x) ∧f (y)} . thus ∧ α∈x◦y f (α) ≥{f (x) ∧f (y)} . thus f is a fuzzy la-subsemihypergroup of s. � theorem 3.5. a nonempty subset a of an la-semihypergroup s is an la-subsemihypergroup if and only if the characteristic fuzzy set χa is a fuzzy la-subsemihypergroup. proof. let a be a nonempty subset of an la-semihypergroup s, x and y be arbitrary elements of s. let a be an la-subsemihypergroup of s. let x,y ∈ a, then x◦ y ⊆ a. for any α ∈ x◦ y, we have, χa (x) = 1 and χa (y) = 1. hence ∧ α∈x◦y χa (α) = 1 = χa (x) ∧χa (y) . now let x ∈ a and y /∈ a, then χa (x) = 1 and χa (y) = 0, so we have ∧ α∈x◦y χa (α) ≥ 0 = χa (x) ∧χa (y) . now let both x and y are not in a, then χa (x) = 0 and χa (y) = 0, so we have ∧ α∈x◦y χa (α) ≥ 0 = χa (x)∧χa (y) . thus for all x,y ∈ s, we have ∧ α∈x◦y χa (α) ≥ χa (x) ∧χa (y) . thus χa is a fuzzy la-subsemihypergroup of s. conversely, let χa be a fuzzy la-subsemihypergroup of s. if the elements x and y are in a, then χa (x) = 1 = χa (y) . but ∧ α∈x◦y χa (α) ≥ χa (x) ∧χa (y) = 1, which implies that χa (α) ≥ 1 for any α ∈ x◦y. hence for any α ∈ x◦y, χa (α) = 1, i.e., α ∈ a. it thus follows that x◦y ⊆ a. hence a is an la-subsemihypergroup of s. � theorem 3.6. let s be an la-semihypergroup and for a nonempty subset a of s the following statements are equivalent: 164 khan, farooq, izhar and davvaz (1) a is left (resp. right) hyperideal of s. (2) the characteristic fuzzy set χa is a fuzzy left (resp. right) hyperideal of s. proof. (1) =⇒ (2) . assume that a is a left hyperideal of s. let x,y ∈ s be such that both x and y are in a. then since a is left hyperideal of s, x ◦ y ⊆ a. for any α ∈ x ◦ y, we have, χa (x) = 1 and χa (y) = 0. hence ∧ α∈x◦y χa (α) = 1 = χa (y) . now let x ∈ a and y /∈ a, then χa (x) = 1 and χa (y) = 0, so we have ∧ α∈x◦y χa (α) ≥ 0 = χa (y) . now let both x and y are not in a, then χa (x) = 0 and χa (y) = 0, so we have ∧ α∈x◦y χa (α) ≥ 0 = χa (y) . thus for all x,y ∈ s, we have∧ α∈x◦y χa (α) ≥ χa (y) . thus χa is a fuzzy left hyperideal of s. (2) =⇒ (1) . let χa be a fuzzy left hyperideal of s. if the elements x and y are in a, then χa (x) = 1 = χa (y) . but 1 = χa (y) ≤ ∧ α∈x◦y χa (α) , which implies that χa (α) ≥ 1 for any α ∈ x◦y. hence for any α ∈ x◦y, χa (α) = 1, i.e., α ∈ a. it thus follows that s◦a ⊆ a. therefore a is left hyperideal of s. similarly we can prove that χa is a fuzzy right hyperideal of s when a is right hyperideal of s. � theorem 3.7. a fuzzy subset f of an la-semihypergroup s is a fuzzy left (resp. right) hyperideal of s if and only if for each t ∈ (0, 1], u(f; t) 6= φ is a left (resp. right) hyperideal of s. proof. suppose f be a fuzzy left hyperideal of s and x ∈ u(f; t) and y ∈ s. then f(x) ≥ t. since f is a fuzzy left hyperideal of s, so f(x) ≤ ∧ α∈y◦x f (α). hence f(α) ≥ t for all α ∈ y ◦x, this implies α ∈ u(f; t) that is y ◦x ⊆ u(f; t). hence u(f; t) is a fuzzy left hyperideal of s. conversely, assume that u(f; t) 6= ∅ is a left hyperideal of s. let x ∈ s such that f(x) > ∧ α∈y◦x f (α) for all y ∈ s. select t ∈ (0, 1] such that f(x) = t > ∧ α∈y◦x f (α). then x ∈ u(f; t) but y ◦x * u(f; t), a contradiction. hence f(x) ≤ ∧ α∈y◦x f (α), that is f is a fuzzy left hyperideal of s. � proposition 3.3. let s be an la-semihypergroup then the following properties hold. (1) let f and g be two fuzzy la-subsemihypergroups of s. then f∩g is also fuzzy la-subsemihypergroup of s. (2) the intersection of any family of fuzzy left (resp. right, two sided) hyperideals of s is a fuzzy left (resp. right, two sided) hyperideal of s. proof. (1) . let f and g be two fuzzy la-subsemihypergroups of s. let x,y ∈ s. then for any α ∈ x◦y, we have ∧ α∈x◦y f (α) ≥ f (x) ∧ f (y) and ∧ α∈x◦y g (α) ≥ g (x) ∧ g (y) . hence f (α) ≥ f (x) ∧ f (y) and g (α) ≥ g (x) ∧g (y) . thus (f ∩g) (α) = f (α) ∧g (α) ≥ f (x) ∧f (y) ∧g (x) ∧g (y) = f (x) ∧g (x) ∧f (y) ∧g (y) = (f ∩g) (x) ∧ (f ∩g) (y) . hence ∧ α∈x◦y (f ∩g) (α) ≥ (f ∩g) (x)∧(f ∩g) (y) . therefore f∩g is a fuzzy la-subsemihypergroup of s. (2) . let g = ⋂ i∈i gi be a family of fuzzy left hyperideals of s. let x,y ∈ s. then, since each gi (i ∈ i) is a fuzzy left hyperideals of s, so ∧ α∈x◦y gi (α) ≥ gi (y) . thus for any α ∈ x◦y, gi (α) ≥ gi (y) , and we fuzzy hyperideals of left almost semihypergroups 165 have g (α) = (⋂ i∈i gi ) (α) = ∧ i∈i (gi (α)) ≥ ∧ i∈i gi (y) = (⋂ i∈i gi ) (y) = g (y) . thus ∧ i∈i g (α) ≥ g (y) . therefore g = ⋂ i∈i gi is a fuzzy left hyperideal of s. � proposition 3.4. let s is an la-semihypergroup. if f is fuzzy left (resp. right or two-sided) hyperideal of s. then f is a fuzzy la-subsemihypergroup. proof. let f be a fuzzy left hyperideal of s. let x,y ∈ s. then ∧ α∈x◦y f (α) ≥ f (y) ≥ f (x) ∧ f (y) . thus ∧ α∈x◦y f (α) ≥ f (x) ∧f (y) . therefore f is a fuzzy la-subsemihypergroup of s. � proposition 3.5. a fuzzy subset f of an la-semihypergroup s is a fuzzy left (resp. right) hyperideal of s if and only if s ∗f ⊆ f (resp. f ∗s ⊆ f). proof. let f be a fuzzy left hyperideal of s and x ∈ s. then (s ∗f) (x) = ∨ x∈y◦z {s (y) ∧f (z)} = ∨ x∈y◦z {f (z)} (∵s (y) = 1) ≤ ∨ x∈y◦z f (x) , because f (z) ≤ ∧ α∈y◦z {f (α)}≤ f (α) for each α ∈ y ◦z. = f (x) . hence, (s ∗f)(x) ≤ f(x). thus s ∗f ⊆ f. conversely, suppose that s∗f ⊆ f. we show that f is a fuzzy left hyperideal of s. let x ∈ s. then f (x) ≥ (s ∗f)) (x) = ∨ x∈y◦z {s (y) ∧f (z)} = ∨ x∈y◦z {f (z)} , (because s (y) = 1) ≥ f (z) , for each z such that x ∈ y ◦z. thus ∧ x∈y◦z f (x) ≥ f (z) . hence f is a fuzzy left hyperideal of s. similarly we can prove the case of fuzzy right hyperideal of s. � theorem 3.8. if s is an la-semihypergroup with pure left identity. then every fuzzy right hyperideal is a fuzzy left hyperideal of s. proof. let s be an la-semihypergroup with pure left identity e, and f be a fuzzy right hyperideal of s. since f is a fuzzy right hyperideal of s, so f ∗s ⊆f. thus by lemma 3.2, and left invertive law, 166 khan, farooq, izhar and davvaz we have s∗f = (s ∗s) ∗f = (f∗s)∗s ⊆ f∗s ⊆ f. thus, s∗f ⊆ f. thus, f is a fuzzy left hyperideal of s. � proposition 3.6. the product of two fuzzy left (resp. right) hyperideals of an la-semihypergroup s with pure left identity is a fuzzy left (resp. right) hyperideal of s. proof. let f and g be any two fuzzy left hyperideals of s. then by using (1) , we have, s∗(f ∗g) = f ∗ (s∗g) ⊆ f ∗g. let f and g be any two fuzzy right hyperideals of s. then by using medial law and 3.2, we have (f ∗g) ∗s = (f ∗g) ∗ (s ∗s) = (f ∗s) ∗ (g ∗s) ⊆ f ∗g. therefore f ∗g is a fuzzy hyperideal of s. � proposition 3.7. in la-semihypergroup with pure left identity for every fuzzy left hyperideal f of s, we have s∗f = f. proof. it suffices to show that f ⊆s∗f. since every element x ∈ s can be written as x = e◦x, where e is the pure left identity in s, (s∗f) (x) = ∨ (y,z)∈ax {s (y) ∧f (z)} ≥{s (e) ∧f (x)} = f (x) . hence s∗f = f. � proposition 3.8. in an la-semihypergroup s with pure left identity for every fuzzy right hyperideal h of s, we have h∗s = h. proof. it suffices to show that h ⊆ h ∗s. since every element a ∈ s can be written as a = e ◦ a = (e◦e) ◦a = (a◦e) ◦e, then there exist u ∈ a◦e such that a ∈ u◦e. then (u,e) ∈ aa, where e is the pure left identity in s, (h∗s) (a) = ∨ (x,y)∈aa {h (x) ∧s (y)} ≥{h (u) ∧s (e)} . since h is a fuzzy right hyperideal of s. then ∧ u∈a◦e h (u) ≥ h (a) . hence h (u) ≥ h (a) . thus (h∗s) (a) ≥{h (u) ∧s (e)} ≥ h (a) ∧ 1 = h (a) . hence h∗s = h. � proposition 3.9. let s be an la-semihypergroup with pure left identity, f be a fuzzy subset and k be a fuzzy left hyperideal of s. then for any fuzzy subset h and fuzzy left hyperideal g of s, f ∗g = h∗k implies that g ∗f = k ∗h. proof. since g and k are fuzzy left hyperideals of s, by proposition 3.7, s∗g = g and s∗k = k. then, g ∗f = (s∗g) ∗f = (f ∗g) ∗s = (h∗k) ∗s = (s∗k) ∗h = k ∗h. � fuzzy hyperideals of left almost semihypergroups 167 proposition 3.10. every idempotent fuzzy left hyperideal of an la-semihypergroup s is a fuzzy hyperideal of s. proof. let f be a fuzzy left hyperideal of s which is idempotent. then f ∗s = (f ∗f) ∗s = (s∗f) ∗f⊆f ∗f = f. hence f is a fuzzy right hyperideal of s and so f is a fuzzy hyperideal of s. � proposition 3.11. if f is an idempotent element in an la-semihypergroup s with pure left identity. then s∗f is an idempotent element. proof. let f be an idempotent element in an la-semihypergroup s with pure left identity. then by using medial law, (s∗f) ∗ (s∗f) = (s ∗s) ∗ (f ∗f) = s∗f. � proposition 3.12. if f is an idempotent element in an la-semihypergroup s with pure left identity. then every fuzzy left hyperideal g of s commutes with f. proof. let f be an idempotent element in an la-semihypergroup s with pure left identity. then f ∗g = (f ∗f) ∗g = (g ∗f) ∗f ⊆ (g ∗s) ∗f ⊆ g ∗f. also, g ∗f = g ∗ (f ∗f) = f ∗ (g ∗f) ⊆ f ∗ (g ∗s)⊆f ∗g. � proposition 3.13. if f is a fuzzy left hyperideal of an la-semihypergroup s with pure left identity, then f ∪ (f ∗s) is a fuzzy hyperideal of s. proof. assume that f is a fuzzy left hyperideal of s. then (f ∪ (f ∗s)) ∗s= (f ∗s) ∪ ((f ∗s) ∗s) = (f ∗s) ∪ ((s ∗s) ∗f) = (f ∗s) ∪ (s∗f) = (f ∗s) ∪f = f ∪ (f ∗s) . hence f ∪ (f ∗s) is a fuzzy right hyperideal of s. and by theorem 3.8, it is a fuzzy hyperideal of s. � proposition 3.14. if f is a fuzzy right hyperideal of an la-semihypergroup s with pure left identity, then f ∪ (s∗f) . is a fuzzy hyperideal of s. proof. assume that f is a fuzzy right hyperideal of s. then (f ∪ (s∗f)) ∗s= ((f ∗s) ∪ (s∗f) ∗s) ⊆f ∪ (s∗f) ∗ (s ∗s) = f ∪ (s ∗s) ∗ (f ∗s) = f ∪ (s∗(f ∗s)) = f ∪ (f ∗ (s ∗s)) = f ∪ (f ∗s) = f ⊆ f ∪ (s∗f) . 168 khan, farooq, izhar and davvaz also, s∗(f ∪ (s∗f)) = (s∗f) ∪ (s∗(s∗f)) = (s∗f) ∪ ((s ∗s)∗(s∗f)) = (s∗f) ∪ ((f ∗s) ∗ (s ∗s)) ⊆ (s∗f) ∪ (f ∗ (s ∗s)) = (s∗f) ∪ (f∗s) ⊆ (s∗f) ∪f = f ∪ (s∗f) hence f ∪ (s∗f) is a fuzzy hyperideal of s. � 4. characterizations of regular and intra-regular la-semihypergroups in terms of fuzzy hyperideals in this section, we characterize regular as well as intra-regular la-semihypergroups in terms of fuzzy hyperideals. theorem 4.1. let s be a regular la-semihypergroup. then for every fuzzy right hyperideal f and every fuzzy left hyperideal g of s, we have f ∗g = f ∩g. proof. let s be a regular la-semihypergroup and f is a fuzzy right hyperideal and g a fuzzy left hyperideal of s. then f ∗g ⊆ f ∗s ⊆f and f ∗g ⊆s∗g ⊆ g. this implies that f ∗g ⊆ f ∩g. now let a be any element of s, then, since s is a regular la-semihypergroup, so there exist an element x ∈ s such that a ∈ (a◦x) ◦a. then there exist u ∈ a◦x such that a ∈ u◦a. then (u,a) ∈ aa. thus we have (f ∗g) (a) = ∨ (y,z)∈ax {f (y) ∧g (z)} ≥{f (u) ∧g (a)} . since f is fuzzy right hyperideal of s, ∧ u∈a◦x f (u) ≥ f (a) . hence f (u) ≥ f (a) . thus (f ∗g) (a) ≥{f (u) ∧g (a)} ≥{f (a) ∧g (a)} = (f ∩g) (a) . thus f ∗g ⊇ f ∩g. therefore f ∗g = f ∩g. � corollary 4.1. let s be a regular la-semihypergroup. then for every fuzzy hyperideal f and every fuzzy hyperideal g of s, we have f ∗g = f ∩g. proposition 4.1. let s be a regular la-semihypergroup. then for every fuzzy right hyperideal f of s is idempotent. proof. let s be a regular la-semihypergroup and f is a fuzzy right hyperideal of s. then f ∗ f ⊆ f∗s ⊆f. next since s is regular so for any a ∈ s, there exist an element x ∈ s such that a ∈ (a◦x)◦a. then there exist α ∈ a◦x such that a ∈ α◦a. then (α,a) ∈ aa. thus we have (f ∗f) (a) = ∨ (y,z)∈ax {f (y) ∧f (z)} ≥{f (α) ∧f (a)} . fuzzy hyperideals of left almost semihypergroups 169 since f is fuzzy right hyperideal of s, ∧ α∈a◦x f (α) ≥ f (a) . hence f (α) ≥ f (a) . thus (f ∗f) (a) ≥{f (α) ∧f (a)} ≥ f (a) ∧f (a) = f (a) . hence f ⊆ f ∗f. therefore f ∗f = f. � corollary 4.2. let s be a regular la-semihypergroup. then for every fuzzy hyperideal f of s is idempotent. proposition 4.2. if s is a regular la-semihypergroup. then every fuzzy right hyperideal is a fuzzy left hyperideal of s. proof. let s be a regular la-semihypergroup and f be a fuzzy right hyperideal of s. let x,y ∈ s. since s is regular and x ∈ s, so there exist an element a ∈ s such that x ∈ (x◦a) ◦ x. thus we have ∧ α∈x◦y f (α) = ∧ α∈(((x◦a)◦x)◦y) f (α) = ∧ α∈((y◦x)◦(x◦a)) f (α) = ∧ α∈u◦v u∈y◦x,v∈x◦a f (α) ≥ f (u) ≥ ∧ u∈y◦x f (u) ≥ f (y) . hence ∧ α∈x◦y f (α) ≥ f (y) . therefore f is a fuzzy left hyperideal of s. � proposition 4.3. a fuzzy set of an intra-regular la-semihypergroup s is a fuzzy right hyperideal if and only if it is a fuzzy left hyperideal of s. proof. let f be a fuzzy right hyperideal of s. let a,b ∈ s. since a ∈ s and s is intra-regular lasemihypergroup, so there exist x,y ∈ s such that a ∈ ( x◦a2 ) ◦ y. thus for any α ∈ a◦ b, we have,∧ α∈a◦b f (α) = ∧ α∈(((x◦a2)◦y)◦b) f (α) = ∧ α∈((b◦y)◦(x◦a2)) f (α) = ∧ α∈u◦v u∈b◦y,v∈x◦a2 f (α) ≥ f (u) ≥ ∧ u∈b◦y f (u) ≥ f (b) . thus f is a fuzzy left hyperideal of s. conversely, assume that f is a fuzzy left hyperideal of s. now for any α ∈ a ◦ b, we have,∧ α∈a◦b f (α) = ∧ α∈(((x◦a2)◦y)◦b) f (α) = ∧ α∈((b◦y)◦(x◦a2)) f (α) = ∧ α∈u◦v u∈b◦y,v∈x◦a2 f (α) ≥ f (v) ≥ ∧ v∈x◦a2 f (v) ≥ f ( a2 ) ≥ ∧ β∈a◦a f (β) ≥ f (a) . thus ∧ α∈a◦b f (α) ≥ f (a) . hence f is a fuzzy right hyperideal of s. � proposition 4.4. every fuzzy two-sided hyperideal of an intra-regular la-semihypergroup s with pure left identity is idempotent. proof. assume that f is a fuzzy two-sided hyperideal of s. then clearly f ∗f ⊆ f ∗s ⊆f. since s is intra-regular, so for each a ∈ s, there exist x,y ∈ s such that a ∈ ( x◦a2 ) ◦y. so by using (1) and left invertive law, we have a ∈ ( x◦a2 ) ◦y = (x◦ (a◦a)) ◦y = (a◦ (x◦a)) ◦y = (y ◦ (x◦a)) ◦a. then there exist u ∈ (y ◦ (x◦a)) such that a ∈ u◦a. then (u,a) ∈ aa. thus we have, (f ∗f) (a) = ∨ (p,q)∈aa {f (p) ∧f (q)} ≥{f (u) ∧f (a)} . 170 khan, farooq, izhar and davvaz since f is a fuzzy two-sided hyperideal of s, so we have ∧ u∈(y◦(x◦a)) f (u) = ∧ u∈y◦v v∈x◦a f (u) ≥ f (v) ≥ ∧ v∈x◦a f (v) ≥ f (a) . hence f (u) ≥ f (a) . thus (f ∗f) (a) ≥{f (u) ∧f (a)} ≥ f (a) ∧f (a) = f (a) . hence f ∗f = f. � proposition 4.5. if s is an intra-regular la-semihypergroup with pure left identity. then f = (s∗f)2 for all fuzzy left hyperideal f of s. proof. let s be an intra-regular la-semihypergroup with pure left identity and f be a fuzzy left hyperideal of s. then s∗f ⊆ f. since s∗f is a fuzzy left hyperideal of s, so it is idempotent. thus (s∗f)2 = (s∗f) ⊆ f. moreover, f = f ∗f ⊆s∗f = (s∗f)2 . thus f = (s∗f)2 . � 5. conclusion in this paper, we have introduced and studied the notions of fuzzy la-subsemihypergroups and fuzzy left (resp. right) hyperideals of la-semihypergroups and their interrelations. we characterized regular and intra-regular la-semihypergroups in terms of these notions. some important directions for future work are (1) to develop strategies for obtaining more valuable results. (2) to define other fuzzy hyperideals in la-semihypergroups. references [1] m. a. kazim and m. naseeruddin, on almost semigroups, the aligarh bulletin of mathematics 2 (1972), 1-7. [2] j. r. cho, pusan, j. jezek and t. kepka, paramedial groupoids, czechoslovak mathematical journal, 49 (124) (1996), 277-290. [3] p. holgate, groupoids satisfying a simple invertive law, mathematics students, 61 (1992), 101-106. [4] p. v. protic and n. stevanovic, on abel-grassmann’s groupoids (review), proceeding of mathematics conference in pristina, (1999), 31-38. [5] m. naseeruddin, some studies on almost semigroups and flocks, ph.d. thesis, the aligarh muslim university india, 1970. [6] q. mushtaq and s. m. yusuf, on ag-groupoids, the aligarh bulletin of mathematics, 8 (1978), 65-70. [7] m. khan, t. asif, characterizations of intra-regular left almost semigroups by their fuzzy ideals, j. math. research 2 (3) (2010), 87-96. [8] f. yousafzai, a. khan, v. amjad, a. zeb, on fuzzy fully regular ag-groupoids, j. intell. & fuzzy syst. 26 (2014), 2973-2982. [9] l. a. zadeh, fuzzy sets, inform control 8 (1965), 338–353. [10] m. khan, faisal and a. manan, intra-regular ag-groupoids characterized by their intuitionistic fuzzy ideals, j. adv. res. dyn. control syst. 3(2) (2011), 17–33. [11] j. n. mordeson, d.s. malik and n. kuroki fuzzy semigroups, springer-verlag, berlin, 2003. [12] k. hila, j. dine, on hyperideals in left almost semihypergroups. isrn algebra (2011), article id953124. [13] n. yaqoob, p. corsini, f. yousafzai, on intra-regular left almost semihypergroups with pure left identity, j. math. doi:10.1155/2013/510790. [14] f. yousafzai, p. corsini, some characterization problems in la-semihypergroups, j. algebra number theory adv. appl. 10 (2013), 41–55. [15] a. rosenfeld, fuzzy groups, j. math. anal. appl. 35 (1971), 512-517. [16] n. kuroki, on fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy sets and systems, 5 (1981), 203-215. [17] p. corsini and v. leoreanu-fotea, applications of hyperstructure theory, advanced in mathematics, kluwer academic publisher, (2003). [18] b. davvaz, intuitionistic hyperideals of semihypergroups, bull. malays. math. sci. soc. 29(1) (2006), 203–207 fuzzy hyperideals of left almost semihypergroups 171 [19] b. pibaljommee, k. wannatong and b. davvaz, an investigation on fuzzy hyperideals of ordered semihypergroups, quasigroups and related systems 23 (2015), 297-308. [20] f .marty, sur une generalization de la notion de group,” 8iemcongress, math scandinaves stockholm (1934), 45-49. [21] m. farooq, a. khan and b. davvaz, characterizations of ordered semihypergroups by the properties of their intersectional-soft generalized bi-hyperideals, soft comput doi 10.1007/s00500-017-2550-6. [22] a. khan, m. farooq and b. davvaz, int-soft interior-hyperideals of ordered semihypergroups. intl. j. anal. appl. 14(2) (2017), 193-202. 1department of mathematics, abdul wali khan university mardan, 23200, kp, pakistan 2department of mathematics, yazd univesity, yazd, iran ∗corresponding author: mizharmath@gmail.com 1. introduction 2. preliminaries 3. fuzzy concepts in la-semihypergroups 4. characterizations of regular and intra-regular la-semihypergroups in terms of fuzzy hyperideals 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 154-166 http://www.etamaths.com prešić-boyd-wong type results in ordered metric spaces satish shukla1,∗ and stojan radenović2 abstract. the purpose of this paper is to prove some prešić-boyd-wong type fixed point theorems in ordered metric spaces. the results of this paper generalize the famous results of prešić and boyd-wong in ordered metric spaces. we also initiate the homotopy result in product spaces. some examples are provided which illustrate the results proved herein. 1. introduction and preliminaries in 1922 banach [26] proved the following theorem known as banach contraction mapping theorem. theorem 1. let (x,d) be a complete metric space and f : x → x be a mapping such that (1) d(fx,fy) ≤ λd(x,y) for all x,y ∈ x, where 0 ≤ λ < 1, then there exists a unique x ∈ x such that fx = x. this point x is called the fixed point of mapping f. due to simplicity and usefulness, several authors generalized the banach contraction mapping theorem. one such generalization is given by prešić [24, 25]. prešić generalized the banach contraction mapping theorem in product spaces and proved the following theorem. theorem 2. let (x,d) be a complete metric space, k a positive integer and f : xk → x be a mapping satisfying the following contractive type condition: (2) d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ k∑ i=1 qid(xi,xi+1), for every x1,x2, . . . ,xk,xk+1 ∈ x, where q1,q2, . . . ,qk are nonnegative constants such that q1 + q2 + · · · + qk < 1. then there exists a unique point x ∈ x such that f(x,x,. . . ,x) = x. moreover if x1,x2, . . . ,xk are arbitrary points in x and for n ∈ n, xn+k = f(xn,xn+1, . . . ,xn+k−1), then the sequence {xn} is convergent and lim xn = f(lim xn, lim xn, . . . , lim xn). 2010 mathematics subject classification. 54h25, 47h10. key words and phrases. common fixed point; prešić type mapping; boyd-wong fixed point theorem; partial order. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 154 prešić-boyd-wong type results 155 condition (2) in the case k = 1 reduces to the condition (1). so, theorem 1 is a generalization of the banach fixed point theorem. the results of prešić is useful in proving the convergence of some particular sequences and in proving the existence of solutions of differences equations, for example, see [15,24,25,40]. for more on the generalizations of prešić type operators the reader is referred to [12,16–19,21,28–36]. on the other hand boyd and wong [4] generalized the banach contraction mapping theorem and proved the following theorem. theorem 3. let (x,d) be a complete metric space and f : x → x a mapping that satisfies (3) d(fx,fy) ≤ ψ(d(x,y)) for all x,y ∈ x, where ψ : r+ → r+ is upper semi-continuous function from the right (i.e., λi ↓ λ ≥ 0 ⇒ lim sup i→∞ ψ(λi) ≤ ψ(λ)) such that ψ(t) < t for each t > 0. then f has a unique fixed point u ∈ x. moreover, for each x ∈ x, lim n→∞ fnx = u. note that the condition (3) in the case ψ(t) = λt reduces to condition (1). so, theorem 3 is a generalization of the banach fixed point theorem. some generalization of the boyd-wong theorem can be found in [9, 10, 13, 14, 20, 23, 38]. the existence of fixed point in partially ordered sets was investigated by ran and reurings [1] and then by nieto and lopez [7, 8]. some applications of fixed point theorems in ordered metric spaces to differential equations can be seen in [7, 8]. several authors generalized the results of these papers in different directions for example, see [2, 3, 5, 6, 22, 27, 37, 39, 41]. the following version of the fixed point theorem was proved, among others, in these papers. theorem 4. let (x,�) be a partially ordered set and d be a metric on x such that (x,d) is a complete metric space. let f : x → x be a nondecreasing map with respect to �. suppose that the following conditions hold: (i) there exists k ∈ (0, 1) such that d(fx,fy) ≤ kd(x,y) for all x,y ∈ x with y � x; (ii) there exists x0 ∈ x such that x0 � fx0; (iii) if a nondecreasing sequence {xn} in x converges to x ∈ x, then xn ≤ x, for all n ∈ n. then f has a fixed point x∗ ∈ x. recently, in [36] malhotra et al. defined the ordered prešić type contraction mappings the setting of cone metric spaces (see also [19, 33]) and generalized the result of prešić in ordered case. in the present paper, we generalize the results of prešić, boyd and wong, theorem 4 and several known results in and prove some prešić-boyd-wong type fixed point theorems in ordered metric spaces. a homotopy result in the product spaces is also proved. examples are included which illustrate the results. following definitions will be needed in sequel. definition 1. let x be any nonempty set, k a positive integer and f : xk → x be a mapping. an element x ∈ x is called a fixed point of f if f(x,x,. . . ,x) = x. 156 shukla and radenović definition 2. let x be a nonempty set, k a positive integer, f : xk → x and g : x → x be mappings. (i) an element x ∈ x said to be a coincidence point of f and g if gx = f(x,x,. . . ,x). (ii) if w = gx = f(x,x,. . . ,x), then w is called a point of coincidence of f and g. (iii) if x = gx = f(x,x,. . . ,x), then x is called a common fixed point of f and g. (iv) mappings f and g are said to be commuting if g(f(x,x,. . . ,x)) = f(gx,gx,. . . ,gx) for all x ∈ x. (v) mappings f and g are said to be weakly compatible if they commute at their coincidence points. definition 3. let x be a nonempty set equipped with a partial order relation “ � ”, k a positive integer and f : xk → x be a mapping. a sequence {xn} in x is said to be nondecreasing with respect to “ � ”, if xn � xn+1 for all n ∈ n. the mapping f is said to be nondecreasing with respect to “ � ” if for any finite nondecreasing sequence {xn}k+1n=1 we have f(x1,x2, . . . ,xk) � f(x2,x3, . . . ,xk+1). let g : x → x be a mapping. f is said to be g-nondecreasing with respect to “ � ” if for any finite nondecreasing sequence {gxn}k+1n=1 we have f(x1,x2, . . . ,xk) � f(x2,x3, . . . ,xk+1). note that for k = 1, above definitions reduce to usual definitions of nondecreasing and g-nondecreasing mappings. definition 4. let x be a nonempty set equipped with a partial order relation “ � ”, and g : x → x be a mapping. a nonempty subset a of x is said to be well ordered if every two elements of a are comparable. the elements a,b ∈ a are called gcomparable if ga and gb are comparable. the set a is called g-well ordered if for all a,b ∈a, a and b are g-comparable i.e. ga and gb are comparable. example 1. let x = [0,∞), a = [0, 1] and define a relation “ � ” on x by x � y ⇔{(x = y) or (x,y ∈ [0, 1 2 ] with x ≤ y)}. then � is a partial order relation on x. define g : x → x by gx = x 2 for all x,y ∈ x. note that a is not well ordered. indeed if x,y ∈ ( 1 2 ,∞),x 6= y then neither x � y nor y � x. but g(a) = [0, 1 2 ], therefore a is g-well ordered. let (x,d) be a metric space equipped with partial order relation “ � ”, then (x,�,d) is called an ordered metric space. let k be a positive integer and f : xk → x be a mapping. f is called ordered prešić contraction if (4) d(f(x1,x2 . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ k∑ i=1 αid(xi,xi+1), for all x1,x2, . . . ,xk,xk+1 ∈ x with x1 � x2 � ··· � xk � xk+1, where αi are nonnegative constants such that k∑ i=1 αi < 1. if (5) is satisfied for all x1,x2, . . . ,xk,xk+1 ∈ x, then f is called prešić contraction. note that in ordered metric spaces a prešić contraction is necessarily an ordered prešić-boyd-wong type results 157 prešić contraction, but converse may not be true (see examples 3.1 and 3.2 of [36]). let ψ : rk+ → r+ be a function satisfying the following conditions: (1) for tn ∈ r+, n ∈ n and tn ↓ t ≥ 0 implies lim sup n→∞ ψ(tn, tn, . . . , tn) ≤ ψ(t,t, . . . , t); (2) ψ(t,t, . . . , t) < t for each t > 0; (3) ψ(t, 0, . . . , 0) + ψ(0, t, 0, . . . , 0) + · · · + ψ(0, . . . , 0, t) ≤ ψ(t,t, . . . , t) for each t ∈ r+. we denote the class of all such functions by ψ, i.e., ψ ∈ ψ if and only if ψ satisfies the all above conditions. example 2. let ψ : rk+ → r+ be defined by ψ(t1, t2, . . . , tk) = k∑ i=1 αiti, where αi are nonnegative constants such that k∑ i=1 αi < 1. then ψ ∈ ψ. mapping f : xk → x is said to be an ordered prešić-boyd-wong contraction if (5) d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(x1,x2),d(x2,x3), . . . ,d(xk.xk+1)) for all x1,x2, . . . ,xk,xk+1 ∈ x with x1 � x2 � ··· � xk � xk+1, where ψ ∈ ψ. if (5) is satisfied for all x1,x2, . . . ,xk,xk+1 ∈ x, then f is called prešić-boyd-wong contraction. now we can state our main results. 2. main results theorem 5. let (x,�,d) be a complete ordered metric space, k a positive integer. let f : xk → x, g : x → x be two mappings such that f(xk) ⊂ g(x) and g(x) is a closed subspace of x. suppose following conditions hold: (i) (6) d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(gx1,gx2),d(gx2,gx3), . . . ,d(gxk,gxk+1)), for all x1,x2, . . . ,xk,xk+1 ∈ x with gx1 � gx2 � ···� gxk � gxk+1, where ψ ∈ ψ; (ii) there exist x1 ∈ x such that gx1 � f(x1,x1, . . . ,x1); (iii) f is g-nondecreasing; (iv) if a nondecreasing sequence {gxn} converges to gu ∈ x, then gxn � gu for all n ∈ n and gu � ggu. then f and g have a point of coincidence. if in addition f and g are weakly compatible, then f and g have a common fixed point v ∈ x. moreover, the set of common fixed points of f and g is g-well ordered if and only if f and g have a unique common fixed point. proof. starting with given x1 ∈ x, we define a sequence {yn} as follows: let y1 = gx1, as f(x k) ⊂ g(x) and gx1 � f(x1,x1, . . . ,x1), define yn+1 = gxn+1 = f(xn,xn, . . . ,xn), n ∈ n. then gx1 � gx2 i.e. y1 � y2 and f is g-nondecreasing, so y2 = f(x1,x1, . . . ,x1) � f(x1,x1, . . . ,x1,x2) � f(x1,x1, . . . ,x1,x2,x2) � ···� f(x1,x2, . . . ,x2) � f(x2,x2, . . . ,x2) = gx3 = y3 158 shukla and radenović i.e. y2 = gx2 � y3 = gx3. continuing this procedure, we obtain gx1 � gx2 � ···� gxn � gxn+1 � ··· , i.e. y1 � y2 � ···� yn � yn+1 � ··· . thus {yn} = {gxn} is a nondecreasing sequence with respect to “ � ”. for simplicity set dn = d(yn,yn+1),n ∈ n. we may assume that dn > 0 for all n ∈ n, otherwise coincidence point and point of coincidence of f and g exist trivially. we shall show that lim n→∞ dn = 0. note that dn+1 = d(yn+1,yn+2) = d(f(xn,xn, . . . ,xn),f(xn+1,xn+1, . . . ,xn+1)) ≤ d(f(xn,xn, . . . ,xn),f(xn, . . . ,xn,xn+1)) +d(f(xn, . . . ,xn,xn+1),f(xn, . . . ,xn,xn+1,xn+1)) + · · · + d(f(xn,xn+1, . . . ,xn+1),f(xn+1, . . . ,xn+1)) and gxn � gxn+1, ψ ∈ ψ, so it follows from (6) that dn+1 ≤ ψ(0, . . . , 0,d(gxn,gxn+1)) + ψ(0, . . . , 0,d(gxn,gxn+1), 0) + · · · + ψ(d(gxn,gxn+1), 0, . . . , 0) = ψ(0, . . . , 0,dn) + ψ(0, . . . , 0,dn, 0) + · · · + ψ(dn, 0, . . . , 0) ≤ ψ(dn,dn, . . . ,dn) < dn, for all n ∈ n. therefore {dn} is a monotonic nondecreasing sequence and bounded below, so lim n→∞ dn exists. let lim n→∞ dn = δ ≥ 0. assume δ > 0, then as ψ ∈ ψ we obtain δ = lim n→∞ dn+1 ≤ lim n→∞ ψ(dn,dn, . . . ,dn) ≤ ψ(δ,δ, . . . ,δ) < δ, a contradiction, so δ = 0. we shall show that {yn} is cauchy sequence. assume that {yn} is not cauchy, then there exists � > 0 and integers ml,nl, l ∈ n such that ml > nl ≥ l and d(ynl,yml ) ≥ � for l ∈ n. also, choosing ml as small as possible, it may be assumed that d(yml−1,ynl ) < �. so for each l ∈ n, we have � ≤ d(yml,ynl ) ≤ d(yml,yml−1) + d(yml−1,ynl ) ≤ dml−1 + � prešić-boyd-wong type results 159 and it follows from the fact lim n→∞ dn = 0 that lim l→∞ d(yml,ynl ) = �. observe that � ≤ d(yml,ynl ) ≤ d(yml,yml+1) + d(yml+1,ynl+1) + d(ynl+1,ynl ) = dml + dnl + d(f(xnl, . . . ,xnl ),f(xml, . . . ,xml )) ≤ dml + dnl + d(f(xnl, . . . ,xnl ),f(xnl, . . . ,xnl,xml )) +d(f(xnl, . . . ,xnl,xml ),f(xnl, . . . ,xnl,xml,xml )) + · · · + d(f(xnl,xml, . . . ,xml ),f(xml, . . . ,xml )). as ml > nl and {yn} is nondecreasing with respect to “ � ”, so ynl � yml i.e., gxnl � gxml, therefore it follows from (6) and the above inequality that � ≤ d(yml,ynl ) ≤ dml + dnl + ψ(0, . . . , 0,d(ynl,yml )) + ψ(0, . . . , 0,d(ynl,yml ), 0) + · · · + ψ(d(ynl,yml ), 0, . . . , 0) ≤ dml + dnl + ψ(d(ynl,yml ), . . . ,d(ynl,yml )). letting l →∞ and using the facts that lim n→∞ dn = 0 and ψ ∈ ψ, we have � = lim l→∞ d(yml,ynl ) ≤ lim l→∞ ψ(d(yml,ynl ), . . . ,d(yml,ynl )) ≤ ψ(�, . . . ,�) < �, which is a contradiction. therefore {yn} = {gxn} is a cauchy sequence in g(x). as g(x) is closed, there exist u,v ∈ x such that v = gu and (7) lim n→∞ yn = lim n→∞ gxn = gu = v. we shall show that u is a coincidence point and v is a point of coincidence of f and g. note that d(v,f(u,u,. . . ,u)) ≤ d(v,yn+1) + d(yn+1,f(u,u,. . . ,u)) = d(v,yn+1) + d(f(xn,xn, . . . ,xn),f(u,u,. . . ,u)) ≤ d(v,yn+1) + d(f(xn,xn, . . . ,xn),f(xn, . . . ,xn,u)) +d(f(xn, . . . ,xn,u)),f(xn, . . . ,xn,u,u)) + · · · + d(f(xn,u, . . . ,u)),f(u,. . . ,u)). if v 6= f(u,u,. . . ,u), then by (iv) we have gxn � gu,gu � ggu, so using (6) it follows from the above inequality that d(v,f(u,u,. . . ,u)) ≤ d(v,yn+1) + ψ(0, . . . , 0,d(gxn,gu)) + ψ(0, . . . , 0,d(gxn,gu), 0) + · · · + ψ(d(gxn,gu), 0, . . . , 0) ≤ d(v,yn+1) + ψ(d(gxn,gu), . . . ,d(gxn,gu)) < d(v,yn+1) + d(gxn,gu), letting n →∞ and using (7) we obtain d(v,f(u,u,. . . ,u)) = 0 i.e. f(u,u,. . . ,u) = gu = v. thus u is a coincidence point and v is a point of coincidence of f and g. suppose f and g are weakly compatible, so f(v,v, . . . ,v) = f(gu,gu,. . . ,gu) = g(f(u,u,. . . ,u)) = gv. 160 shukla and radenović again if d(gu,gv) > 0 then as gu � ggu = gv we obtain from (6) that d(v,f(v,v, . . . ,v)) = d(f(u,u,. . . ,u),f(v,v, . . . ,v)) ≤ d(f(u,u,. . . ,u),f(u,. . . ,u,v)) + d(f(u,. . . ,u,v),f(u,. . . ,u,v,v)) + · · · + d(f(u,v, . . . ,v),f(v, . . . ,v)) ≤ ψ(0, . . . , 0,d(gu,gv)) + ψ(0, . . . , 0,d(gu,gv), 0) + · · · + ψ(d(gu,gv), 0, . . . , 0) ≤ ψ(d(gu,gv), . . . ,d(gu,gv)) < d(gu,gv) = d(v,f(v,v, . . . ,v)), a contradiction, therefore v = gv = f(v,v, . . . ,v). thus v is the common fixed point of f and g. suppose the set of common fixed points of f and g is g-well ordered. we shall show that common fixed point is unique. assume on contrary that v′ is another common fixed point of f and g i.e. v′ = gv′ = f(v′,v′, . . . ,v′) and v 6= v′. as v and v′ are g-comparable, let e.g. gv � gv′. from (6), it follows that d(v,v′) = d(f(v,v, . . . ,v),f(v′,v′, . . . ,v′)) ≤ d(f(v,v, . . . ,v),f(v, . . . ,v,v′)) + d(f(v, . . . ,v,v′),f(v, . . . ,v,v′,v′)) + · · · + d(f(v,v′, . . . ,v′),f(v′,v′, . . . ,v′)) ≤ ψ(0, . . . , 0,d(gv,gv′)) + ψ(0, . . . , 0,d(gv,gv′), 0) + · · · + ψ(d(gv,gv′), 0, . . . , 0) ≤ ψ(d(gv,gv′), . . . ,d(gv,gv′)) < d(v,v′) a contradiction. therefore, v = v′, i.e., the common fixed point is unique. for converse, if common fixed point of f and g is unique then the set of common fixed points of f and g being singleton therefore g-well ordered. � remark 1. for k = 1 the above theorem is a generalization and extension of result of boyd and wong in ordered metric spaces. following is a simple example which illustrate the above result. example 3. let x = [0,∞) with the usual metric and partial order �= {(x,y) : x,y ∈ x,y ≤ x}. for k = 2, define f : x2 → x and g : x → x by f(x1,x2) = x1 + x2 3 + x1 + x2 for all x1,x2 ∈ x and gx = x for all x ∈ x. define ψ : r2+ → r by ψ(t1, t2) = t1 + t2 3 + |t1 − t2| for all t1, t2 ∈ r+. then it easy to see that all the conditions of theorem 5 are satisfied and 0 is the unique common fixed point of f and g in x. taking g = ix i.e. identity mapping of x in theorem 5, we get the following fixed point result for ordered prešić-boyd-wong contraction. corollary 6. let (x,�,d) be a complete ordered metric space, k a positive integer. let f : xk → x be a mapping such that the following conditions hold: (i) f is ordered prešić-boyd-wong contraction; (ii) there exist x1 ∈ x such that x1 � f(x1,x1, . . . ,x1); (iii) f is nondecreasing; prešić-boyd-wong type results 161 (iv) if a nondecreasing sequence {xn} converges to u ∈ x, then xn � u for all n ∈ n. then f has a fixed point v ∈ x. moreover, the set of fixed points of f is well ordered if and only if f has a unique fixed point. the following example illustrate the case when the known results are not applicable but the corollary 6 of this paper is applicable. example 4. let x = [0, 2] and d is the usual metric on x, then (x,d) is a complete metric space. for k = 2, define a mapping f : x2 → x by f(x,y) =   x 1 + x , if (x,y) ∈ [0, 1) × [0, 1) ∪ [1, 2] × [0, 1); y 1 + y , if (x,y) ∈ [0, 1) × [1, 2]; 0, otherwise, and a function ψ : rk+ → r+ by ψ(t1, t2) = t1 1 + |t1/2 − t2| for all t1, t2 ∈ r+. let � be a partial order define on x by � = {(x,y) : (x,y) ∈ [0, 1) × [0, 1) with y ≤ x}∪{(x,y) : (x,y) ∈ [1, 2] × (0, 1)} ∪{(x,x) : x ∈ x}, then ψ ∈ ψ. now by careful calculations one can see that all the conditions of corollary 6 are satisfied and 0 is the unique fixed point of f. note that, f is not an ordered prešić type contraction, therefore it is not a prešić type contraction. to see this, take arbitrary values x,y = z ∈ [0, 1) and then condition (5) is not satisfied. following theorem is a generalization of the result of prešić and boyd and wong in metric spaces. theorem 7. let (x,d) be a complete metric space, k a positive integer. let f : xk → x, g : x → x be two mappings such that f(xk) ⊂ g(x) and g(x) is a closed subspace of x. suppose following conditions hold: (8) d(f(x1,x2, . . . ,xk),f(x2,x3, . . . ,xk+1)) ≤ ψ(d(gx1,gx2),d(gx2,gx3), . . . ,d(gxk,gxk+1)), for all x1,x2, . . . ,xk,xk+1 ∈ x, where ψ ∈ ψ. then f and g have a point of coincidence. if in addition f and g are weakly compatible, then f and g have a unique common fixed point v ∈ x. proof. we note that the inequality (8) is true for all x1,x2, . . . ,xk,xk+1 ∈ x, therefore the proof of theorem follows from similar process as used in the proof of theorem 5. � taking g = ix i.e. identity mapping of x in theorem 7, we get the following fixed point result for prešić-boyd-wong contraction. corollary 8. let (x,d) be a complete metric space, k a positive integer. let f : xk → x be a prešić-boyd-wong contraction. then f has a unique fixed point v ∈ x. 162 shukla and radenović remark 2. note that, for ψ(t1, t2, . . . , tk) = k∑ i=1 αiti, where αi are nonnegative constants such that k∑ i=1 αi < 1, corollary 8 reduces to the prešić theorem. 3. a homotopy result in this section we prove a homotopy result for prešić type mapping on product space. theorem 9. let (x,d) be any complete metric space, u an open subset of x. suppose h : (u)k × [0, 1] → x be a function such that the following conditions hold: (i) for every x ∈ ∂u(here ∂u is the boundary of u) and λ ∈ [0, 1], x 6= h(x,x,. . . ,x,λ); (ii) for all x1,x2, . . . ,xk,xk+1 ∈ u and λ ∈ [0, 1] (9) d(h(x1,x2, . . . ,xk,λ),h(x2,x3, . . . ,xk+1,λ)) ≤ k∑ i=1 αid(xi,xi+1), where αi are nonnegative constants such that k∑ i=1 αi < 1 k ; (iii) for all x1,x2, . . . ,xk ∈ u and λ,µ ∈ [0, 1] there exists m ≥ 0 such that (10) d(h(x1,x2, . . . ,xk,λ),h(x1,x2, . . . ,xk,µ)) ≤ m|λ−µ|. if hλ=λ′ has a fixed point in u for at least one λ ′ ∈ [0, 1], then hλ has a fixed point in u for all λ ∈ [0, 1]. furthermore, for any fixed λ ∈ [0, 1], the fixed point of hλ is unique. proof. define f = {λ ∈ [0, 1] : x = h(x,x,. . . ,x,λ) for some x ∈ u}. as hλ=λ′ for at least one λ ′ ∈ [0, 1], has a fixed point in u, i.e., there exists x ∈ u such that h(x,x,. . . ,x,λ′) = x, so λ′ ∈ f and fx 6= ∅. we shall show that f is both open and closed in [0, 1] and therefore by connectedness f = [0, 1]. (i) f is closed: let {λn} be any sequence in f and lim n→∞ λn = λ ∈ [0, 1]. as λn ∈ f for all n ∈ n so there exists xn ∈ u such that xn = h(xn,xn, . . . ,xn,λn) for all n ∈ n. note that, for all n,m ∈ n with m > n we have d(xn,xm) = d(h(xn,xn, . . . ,xn,λn),h(xm,xm, . . . ,xm,λm)) ≤ d(h(xn, . . . ,xn,λn),h(xn, . . . ,xn,xm,λn)) +d(h(xn, . . . ,xn,xm,λn),h(xn, . . . ,xn,xm,xm,λn)) + · · · + d(h(xn,xm, . . . ,xm,λn),h(xm, . . . ,xm,λn)) +d(h(xm, . . . ,xm,λn),h(xm, . . . ,xm,λm)). using (9) and (10) it follows that d(xn,xm) ≤ αkd(xn,xm) + αk−1d(xn,xm) + · · · + α1d(xn,xm) + m|λn −λm| = [ k∑ i=1 αi]d(xn,xm) + m|λn −λm| prešić-boyd-wong type results 163 i.e. d(xn,xm) ≤ m 1 − k∑ i=1 αi |λn −λm|. letting n → ∞ and using the fact that lim n→∞ λn = λ it follows from the above inequality that lim n→∞ d(xn,xm) = 0, therefore {xn} is a cauchy sequence. as x is complete, there exists u ∈ u such that lim n→∞ xn = u. now for any n ∈ n we obtain d(xn,h(u,u,. . . ,u,λ)) = d(h(xn,xn, . . . ,xn,λn),h(u,u,. . . ,u,λ)) ≤ d(h(xn, . . . ,xn,λn),h(xn, . . . ,xn,u,λn)) +d(h(xn, . . . ,xn,u,λn),h(xn, . . . ,xn,u,u,λn)) + · · · + d(h(xn,u, . . . ,u,λn),h(u,. . . ,u,λn)) +d(h(u,u,. . . ,u,λn),h(u,u,. . . ,u,λ)), using (9) and (10) it follows that d(xn,h(u,u,. . . ,u,λ)) ≤ αkd(xn,u) + αk−1d(xn,u) + · · · + α1d(xn,u) + m|λn −λ| = [ k∑ i=1 αi]d(xn,u) + m|λn −λ|. as lim n→∞ λn = λ and lim n→∞ xn = u, we obtain lim n→∞ d(xn,h(u,u,. . . ,u,λ)) = d(u,h(u,u,. . . ,u,λ)) = 0 i.e. u = h(u,u,. . . ,u,λ) and u ∈ u. as (i) holds, therefore u ∈ u so λ ∈f. thus f is closed. (ii) f is open: let λ0 ∈f, then there exists u0 ∈ u such that u0 = h(u0, . . . ,u0,λ0). as u is open, there exists δ > 0 such that b(u0,δ) = {x ∈ x : d(x,u0) < δ} ⊂ u. fix � > 0 with (11) � < 1 −k k∑ i=1 αi m δ. let λ ∈ (λ0 −�,λ0 + �), then for all x1,x2, . . . ,xk ∈ b(u0,δ) = {x ∈ x : d(x,u0) ≤ δ}, we have d(u0,h(x1,x2, . . . ,xk,λ)) = d(h(u0,u0, . . . ,u0,λ0),h(x1,x1, . . . ,xk,λ)) ≤ d(h(u0,u0, . . . ,u0,λ0),h(u0, . . . ,u0,x1,λ0)) +d(h(u0, . . . ,u0,x1,λ0),h(u0, . . . ,u0,x1,x2,λ0)) + · · · + d(h(u0,x1, . . . ,xk−1,λ0),h(x1, . . . ,xk,λ0)) +d(h(x1,x2, . . . ,xk,λ0),h(x1,x2, . . . ,xk,λ)). 164 shukla and radenović it follows from (9), (10) and the above inequality that d(u0,h(x1,x2, . . . ,xk,λ)) ≤ [ k∑ i=1 αi]d(u0,x1) + [ k∑ i=2 αi]d(x1,x2) + · · · +[ k∑ i=k−1 αi]d(xk−2,xk−1) + αkd(xk−1,xk) + m|λ0 −λ| < [k k∑ i=1 αi]δ + m�. using (11) in the above inequality we obtain d(u0,h(x1,x2, . . . ,xk,λ)) < [k k∑ i=1 αi]δ + [1 −k k∑ i=1 αi]δ = δ. therefore d(u0,h(x1,x2, . . . ,xk,λ)) < δ i.e. h(x1,x2, . . . ,xk,λ) ∈ b(u0,δ). thus, for each fixed λ ∈ (λ0 − �,λ0 + �), hλ is a self map of b(u0,δ), so we can apply corollary 8 and remark 2, to deduce that hλ has a fixed point in u, and as (i) holds, this fixed point must be in u. thus λ ∈f for all λ ∈ (λ0 − �,λ0 + �). therefore f is open in [0, 1] i.e. f = [0, 1]. thus hλ has a fixed point in u for all λ ∈ [0, 1]. for uniqueness, let λ ∈ [0, 1] be fixed and for this fixed λ, u and v be two fixed points of hλ in u i.e. u = h(u,u,. . . ,u,λ) and v = h(v,v, . . . ,v,λ) and u 6= v. then it follows from (9) that d(u,v) = d(h(u,u,. . . ,u,λ),h(v,v, . . . ,v,λ)) ≤ d(h(u,. . . ,u,λ),h(u,. . . ,u,v,λ)) + d(h(u,. . . ,u,v,λ),h(u,. . . ,u,v,v,λ)) + · · · + d(h(u,v, . . . ,v,λ),h(v,v, . . . ,v,λ)) ≤ αkd(u,v) + αk−1d(u,v) + · · · + α1d(u,v) = [ k∑ i=1 αi]d(u,v) < d(u,v), a contradiction. thus fixed point is unique. � for k = 1 in the above theorem, we obtain following homotopy result. corollary 10. let (x,d) be any complete metric space, u an open subset of x. suppose h : u × [0, 1] → x be a function such that the following conditions hold: (i) for every x ∈ ∂u(here ∂u is the boundary of u) and λ ∈ [0, 1], x 6= h(x,λ); (ii) for all x,y ∈ u and λ ∈ [0, 1] d(h(x,λ),h(y,λ)) ≤ αd(x,y), where 0 ≤ α < 1; (iii) for all x ∈ u and λ,µ ∈ [0, 1] there exists m ≥ 0 such that d(h(x,λ),h(x,µ)) ≤ m|λ−µ|. prešić-boyd-wong type results 165 if hλ=λ′ has a fixed point in u for at least one λ ′ ∈ [0, 1], then hλ has a fixed point in u for all λ ∈ [0, 1]. furthermore, for any fixed λ ∈ [0, 1], the fixed point of hλ is unique. conflict of interests. the authors declare that there is no conflict of interests regarding the publication of this paper. references [1] a.c.m. ran, m.c.b. reurings, a fixed point theorem in partially ordered sets and some application to matrix equations, proc. am. math. soc. 132(2004), 1435-1443. [2] c.m. chen, fixed point theorems for ψ-contractive mappings in ordered metric spaces, j. appl. math. volume 2012(2012), article id 756453, 10 pages [3] d. o’regan, a. petrusel, fixed point theorems for generalized contractions in ordered metric spaces, j. math. anal. appl. 341(2008), 1241-1252. [4] d.w. boyd, j.s. wong, on nonlinear contractions, proc. am. math. soc. 20(1969), 458c464. [5] h. aydi, some fixed point results in ordered partial metric spaces, j. nonlinear sci. appl. 4(2011):210-217. [6] i. altun a. erduran, fixed point theorems for monotone mappings on partial metric spaces, fixed point theory appl., 2011(2011), article id 508730. [7] j.j. nieto, r.r. lopez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22(2005), 223-239. [8] j.j. nieto., r.r. lopez, existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, acta. math. sinica, english ser. 23(12)(2007), 2205-2212. [9] j.r. morales, generalizations of some fixed point theorems, notas de mathemática, no. 199 mérida, 1999 [10] j. maŕın, s. romaguera and p. tirado, weakly contractive multivalued maps and w-distances on complete quasi-metric spaces, fixed point theory appl. 2011 (2011),article id 2. [11] k.p.r. rao, g. n. v. kishore, m. md. ali, a generalization of the banach contraction principle of prešić type for three maps, math. sci. 3(3)(2009), 273-280. [12] l.b. ćirić and s. b. prešić, on prešić type generalisation of banach contraction principle, acta. math. univ. com. 76, no. 2(2007), 143-147 [13] m. akkouchi, on a fixed point theorem of d. w. boyd and j. s. wong, acta math. vietnam. 27, number 2 (2002), 231-237. [14] m.r. tasković, a generalization of banach’s contraction principle, publ. inst. math. beograd, 23(37)(1978), pp. 179-191. [15] m.s. khan, m. berzig and b. samet, some convergence results for iterative sequences of prešić type and applications, adv. difference equ. 2012, 2012:38 doi:10.1186/1687-1847-201238 [16] m. pǎcurar, a multi-step iterative method for approximating common fixed points of prešićrus type operators on metric spaces, studia univ. “babeş-bolyai”, mathematica, volume lv, number 1, march 2010. [17] m. pǎcurar, approximating common fixed points of prešić-kannan type operators by a multistep iterative method, an. şt. univ. ovidius constanţa 17(1)(2009), 153-168. [18] m. pǎcurar, common fixed points for almost prešić type operators, carpathian j. math. 28 no. 1 (2012), 117-126. [19] n.v. luong, n.x. thuan, some fixed point theorems of prešić-ćirić type, acta univ. apulensis math. inform. 2012(2012), no. 30, 237-249. [20] p. das, l.k. dey, fixed point of contractive mappings in generalized metric spaces, math. slovaca 59 no. 4 (2009), 499-504. [21] r. george, k. p. reshma and r. rajagopalan, a generalised fixed point theorem of prešić type in cone metric spaces and application to morkov process, fixed point theory appl. 2011(2011):85, doi:10.1186/1687-1812-2011-85. [22] r.p. agarwal, m. a. el-gebeily, d. o’regan, generalized contractions in partially ordered metric spaces, appl. anal. 87(2008), 109-116. [23] r.p. pant, v. pant, v.p. pandey, generalization of meir-keeler type fixed point theorems, tamkang j. math. 35, no. 3(2004), 179-187. 166 shukla and radenović [24] s.b. prešić, sur une classe dinequations aux differences finite et sur la convergence de certaines suites, publ. de linst. math. belgrade 5(19)(1965), 75-78. [25] s.b. prešić, sur la convergence des suites, comptes. rendus. de l’acad. de paris 260(1965), 3828-3830. [26] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund math., 3(1922), 133-181. [27] s. radenović, z. kadelburg, generalized weak contractions in partially ordered metric spaces, comput. math. appl. 60(2010), 17761783. doi:10.1016/j.camwa.2010.07.008 [28] s. shukla, prešić type results in 2-banach spaces, afrika matematika, (2013) doi 10.1007/s13370-013-0174-2 [29] s. shukla, set-valued prešić-ćirić type contraction in 0-complete partial metric spaces, matematićki vesnik, in press, (2013). [30] s. shukla, b. fisher, a generalization of prešić type mappings in metric-like spaces, journal of operators, 2013 (2013), article id 368501, 5 pages. [31] s. shukla, r. sen, set-valued prešić-reich type mappings in metric spaces, revista de la real academia de ciencias exactas, fisicas y naturales. serie a. matematicas (2012). doi 10.1007/s13398-012-0114-2 [32] s. shukla, r. sen, s. radenović, set-valued prešić type contraction in metric spaces, an. ştiinţ. univ. al. i. cuza iaşi. mat. (n.s.) (accepted)(2012). [33] s. shukla, s. radenović, a generalization of prešić type mappings in 0-complete ordered partial metric spaces, chinese journal of mathematics 2013 article id 859531, 8 pages. [34] s. shukla, s. radojević, z.a. veljković, s. radenović, some coincidence and common fixed point theorems for ordered prešić-reich type contractions, journal of inequalities and applications, 2013 2013:520. doi: 10.1186/1029-242x-2013-520 [35] s. shukla, m. abbas, fixed point results of cyclic contractions in product spaces, carpathian j. math., in press (2014). [36] s.k. malhotra, s. shukla, r. sen, a generalization of banach contraction principle in ordered cone metric spaces, j. adv. math. stud., 5(2) (2012), 59–67. [37] s. k. malhotra, s. shukla, and r. sen, some fixed point theorems for ordered reich type contractions in cone rectangular metric spaces. acta mathematica universitatis comenianae, lxxxii 2 (2013), 165c175. [38] w.a. kirk, fixed points of asymptotic contractions, j. math. anal. appl. 277(2003), 645-650. [39] w. shatanawi, b. samet, m. abbas, coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces, math. comput. model. 55(2012), 680-687. [40] y.z. chen, a prešić type contractive condition and its applications, nonlinear anal. 71, 2012-2017 (2009). doi:10.1016/j.na.2009.03.006 [41] z. kadelburg, m. pavlović, s. radenović, common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, comput. math. appl. 59(2010), 3148-3159. 1department of applied mathematics, shri vaishnav institute of technology & science, gram baroli, sanwer road, indore (m.p.) 453331, india 2faculty of mechanical engineering, university of belgrade, kraljice marije 16, 11120 beograd, serbia ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 14, number 1 (2017), 99-106 http://www.etamaths.com uniform lacunary statistical convergence on time scales e. yilmaz1, s. a. mohiuddine2,∗, y. altin1 and h. koyunbakan1 abstract. we introduce (θ,m)-uniform lacunary density of any set and (θ,m)-uniform lacunary statistical convergence on an arbitrary time scale. moreover, (θ,m)-uniform strongly p-lacunary summability and some inclusion relations about these new concepts are also presented. 1. introduction and preliminaries the idea of statistical convergence goes back to the study of zygmund [42] which was published in 1935. statistical convergence of number sequences was formally introduced by fast [13] and steinhaus [40] independently in the same year. over the years and under different names, statistical convergence has been discussed in the theory of fourier analysis, ergodic theory, number theory, approximation theory, measure theory, trigonometric series, turnpike theory and banach spaces. later on, it was further investigated from the sequence space point of view and linked with summability theory by fridy [15], connor [8], maddox [23], rath and tripathy [33], tripathy [37], moricz [28], belen and mohiuddine [3], braha et al. [5], edely et al. [10], mohiuddine et al . [26] and references therein. the statistical convergence is related to the density of subsets of n. the asymptotic density of a set a ⊂ n is defined by δ (a) = lim n→∞ 1 n |{k ≤ n : k ∈ a}| , whenever the limit exists. here, |{k ≤ n : k ∈ a}| indicates the number of elements of a ⊆ n not exceeding n. any finite subset of n has zero asymptotic density and δ (ac) = 1 − δ (a). a sequence (xk) is statistically convergent [13] to a real number l if for each ε > 0, δ ({k ∈ n: |xk −l| ≥ ε}) = 0. in this case, slim xk = l. the set of all statistically convergent sequences is denoted by s. by a lacunary sequence θ = (kr) (r = 0, 1, 2, ...), where k0 = 0, we shall mean an increasing sequence of nonnegative integers with kr−kr−1 →∞ as r →∞. the intervals determined by θ will be denoted by ir = (kr−1,kr] and hr = kr −kr−1 where qr = kr kr−1 (see [14]). the space of all lacunary strongly convergent sequences nθ was defined by freedman et al. as follows nθ = { x = (xk) : lim r→∞ ( 1 hr ∑ k∈ir |xk −l| ) = 0, for some l } . to understand lacunary sequences, we need to consider below examples. example 1.1. θ = ( r2 ) is a lacunary sequence. let us check the above conditions. one can easily see that 0 < r2 < (r + 1)2. so, θ is an increasing sequence where k0 = 0. furthermore, hr →∞ as r goes to infinite as shown in following table: hr = kr −kr−1 h1 h2 h3 h4 ... h10 h11 ... h100 ... hr kr = r 2 1 3 5 7 → 19 21 → 199 → 2r − 1 table 1. hr →∞ as r goes to infinite. received 16th february, 2017; accepted 5th april, 2017; published 2nd may, 2017. 2010 mathematics subject classification. 40a35, 46a45, 34n05. key words and phrases. uniform lacunary statistical convergence; sequence spaces; time scale. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 99 100 yilmaz, mohiuddine, altin and koyunbakan example 1.2. θ = (r) is not a lacunary sequence. although the first two conditions satisfy, hr does not go to infinite as r →∞ as seen in following table: hr = kr −kr−1 h1 h2 h3 h4 ... h10 h11 ... h100 ... hr kr = r 1 1 1 1 → 1 1 → 1 → 1 table 2. hr → 1 as r goes to infinite. let k ⊂ n. one defines the θ-density [16] of a set k by δθ (k) = lim r→∞ 1 hr |k ∩ ir| . by using lacunary sequences, fridy and orhan [16] studied a related concept of convergence in which {k : k ≤ n} is replaced by {k : kr−1 < k ≤ kr} for a lacunary sequence θ = {kr} as follows: a real or complex sequence (xk) is lacunary statistically convergent to l if for every ε > 0, lim r→∞ 1 hr |{k ∈ ir : |xk −l| ≥ ε}| = 0. in this case, sθlim x = l. lacunary statistical convergence and related notions were studied by many authors (see [7], [9], [11], [12], [17], [20], [24], [25], [27], [29]). furthermore, nuray and aydın [31] introduced and studied strongly lacunary summable functions. here, our aim is to move some notions and properties about lacunary sequence to time scale calculus. before our new concepts, we recall the main features of the time scale theory. a time scale t is an arbitrary, nonempty, closed subset of real numbers. the calculus of time scale was introduced by hilger in his ph.d. thesis supervised by auldbach in 1988 (see [21], [22]). it allows to unify the usual differential and integral calculus for one variable. one can replace the range of definition r of the functions under consideration by an arbitrary time scale t. recently, time scale theory has been applied to different areas by many authors (see [4], [18], [19]). the followings notions are very important for this theory. forward jump operator σ : t → t and graininess function µ : t → [0,∞) are defined by σ(t) = inf {s ∈ t : s > t} and µ(t) = σ(t) − t for t ∈ t, respectively. in this definition, we put inf φ = sup t, where φ is an empty set. a half closed interval on t is given by [a,b)t = {t ∈ t: a ≤ t < b} or (a,b]t = {t ∈ t: a < t ≤ b} . open and closed intervals can be defined similarly in [4], [19]. let a be the family of all left closed and right open intervals on t of the form [a,b)t and m̃ : a → [0,∞) be a set function on a such that m̃ ([a,b)t) = b−a. then, it is known that m̃ is a countably additive measure on a. now, the caratheodory extension of the set function m̃ associated with family a is said to be the lebesque ∆-measure on t and is denoted by µ∆. in this case, it is known that if a ∈ t−{max t} , then the single point set {a} is ∆-measurable and µ∆(a) = σ(a) −a. if a,b ∈ t and a ≤ b, then µ∆ ((a,b)t) = b−σ(a) and µ∆ ([a,b)t) = b−a. if a,b ∈ t−{max t} and a ≤ b, then µ∆ ((a,b]t) = σ(b) −σ(a) and µ∆ ([a,b])t) = σ(b) −a (see [38]). statistical convergence is applied to time scales for different purposes by various authors in the literature. for instance, seyyidoglu and tan [35] gave some important notions such as ∆-convergence, ∆-cauchy by using ∆-density an investigate their relations on t and, in the recent past, they explained a generalization of statistical cluster and limit points [36]. turan and duman [38] introduced density and statistical convergence of ∆-measurable real-valued functions defined on t. furthermore, altin et al. [1] expressed mand (λ,m)-uniform density of a set and mand (λ,m)-uniform statistical convergence on t. however, yilmaz et al. [41] defined λ-statistical convergence on t. turan and duman [39] defined lacunary sequence and lacunary statistical convergence on t. now, we give a generalization of their study in a different form where θ = {kt−t0+1} is a lacunary sequence on t. definition 1.1. let ω be a ∆-measurable subset of t and θ be a lacunary sequence. then, we define the set ω (t,θ) by ω (t,θ) = {s ∈ (kt−2t0+1,kt−t0+1]t : s ∈ ω} , for t ∈ t. in this case, the θ-density of ω on t is denoted by δθt (ω) = lim t→∞ µ∆ (ω (t,θ)) µ∆ ((kt−2t0,kt−t0 ]t) , uniform lacunary statistical convergence on time scales 101 provided that the above limit exists. definition 1.2. let f : t → r be a ∆-measurable function and θ be a lacunary sequence. then, f is lacunary statistically convergent to a real number l on t if lim t→∞ µ∆ (s ∈ (kt−2t0+1,kt−t0+1]t : |f (s) −l| ≥ ε) µ∆ ((kt−2t0,kt−t0 ]t) = 0, for ∀ε > 0 and t ∈ t. in this case, sθtlim t→∞ (f (t)) = l. the set of all lacunary statistical convergence functions on t will be denoted by sθt. (kt−2t0+1,kt−t0+1] turns to (kr−1,kr] for t = r, t0 = 1 and t = n. in this case, lacunary statistical convergence on time scales is reduced to classical lacunary statistical convergence which is defined by fridy and orhan [16]. in this study, we will give notions of (θ,m)-uniform lacunary density of an arbitrary set, (θ,m)uniform lacunary statistical convergence and some properties of (θ,m)-lacunary statistically convergent sequences on an arbitrary time scale. before this, we recall some concepts about uniform density and uniform statistical convergence in classical case to use in our main results. uniformly density of an arbitrary set was introduced by raimi [32] as follows: definition 1.3. a subset e ⊂ n is uniformly dense if u (e) = lim n→∞ 1 n n∑ j=1 χe (j + m) = a, or equivalently lim n→∞ 1 n |e ∩{m + 1, ...,m + n}| = a, uniformly in m, where m = 0, 1, 2, ... and χe is characteristic function. subsequently, uniformly density was studied by baláž and šalát [2]. later, m-uniform statistical convergence is introduced by nuray [30] in the following manner. definition 1.4. let x = (xk) be a real or complex valued sequence. if lim n→∞ 1 n |{m ≤ k < n + m : |xk −l| ≥ ε}| = 0, uniformly in m, x = (xk) is said to be m-uniform statistically convergent to l for all ε > 0. based on definition 1.4, we can generalize m-uniform statistical convergence to lacunary type sequences as follows: definition 1.5. let k ⊂ n and θ be a lacunary sequence. then, we define the (θ,m)-uniform density of k by δmθ (k) = lim r→∞ 1 hr,m |{kr−1+m < k ≤ kr+m : k ∈ k}| , uniformly in m ≥ 0, where hr,m = kr+m −kr+m−1. definition 1.6. a sequence x = (xk) is said to be (θ,m)-uniform lacunary statistically convergent to a real number l if lim r→∞ 1 hr,m |{kr−1+m < k ≤ kr+m : |xk −l| ≥ ε}| = 0, for all ε > 0, uniformly in m. above definitions are special cases of σ-statistical convergence and lacunary σ-statistical convergence [34]. in the next section, we shall define above notions on time scale t. 102 yilmaz, mohiuddine, altin and koyunbakan 2. main results in this section, we define and study the (θ,m)-density, (θ,m)-uniform lacunary statistical convergence and (θ,m)-uniform strongly p-lacunary summability on t, where θ = {kt−t0+m+1} is a lacunary sequence for t ∈ t. definition 2.1. let ω be a ∆-measurable subset of t and θ be a lacunary sequence. then, we can define the set ω (t,θ,m) by ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]t : s ∈ ω} , for t ∈ t. in this case, (θ,m)-density of ω on t is defined by δ θ,m t (ω) = limt→∞ µ∆ (ω (t,θ,m)) µ∆ ((kt−2t0+m,kt−t0+m]t) , (2.1) provided that the above limit exists. definition 2.2. let f : t → r be a ∆-measurable function and θ be a lacunary sequence. then, f is (θ,m)-uniform lacunary statistically convergent to a real number l on t if lim t→∞ µ∆ (s ∈ (kt−2t0+m+1,kt−t0+m+1]t : |f (s) −l| ≥ ε) µ∆ ((kt−2t0+m,kt−t0+m]t) = 0, (2.2) uniformly in m, for all ε > 0 and t ∈ t. in this case, sθ,mt limt→∞ (f (t)) = l. the set of all (θ,m)uniform lacunary statistically convergent functions on t will be denoted by sθ,mt . we remark that (kt−2t0+m+1,kt−t0+m+1] turns to (kr+m−1,kr+m] when t = r, t0 = 1 and t = n. in this instance, (θ,m)-uniform lacunary statistical convergence on time scales is reduced to classical (θ,m)-uniform lacunary statistical convergence which is given by definition 1.6. this shows that our results are generalizations of classical results. proposition 2.1. let θ be a lacunary sequence. if f,g : t → r with sθ,mt limt→∞f (t) = l1 and s θ,m t limt→∞ g (t) = l2, then the following statements hold: (i) s θ,m t limt→∞ (f (t) + g (t)) = l1 + l2, (ii) s θ,m t limt→∞ (cf (t)) = cl1 (c ∈ r) . however, m-uniform statistical convergence on t was first defined by altin et al. [1] in the following way. definition 2.3. let f : t → r be a ∆-measurable function. then, f is m-uniform statistically convergent to a real number l on t if lim t→∞ µ∆ (s ∈ [m + t0 − 1, t + m) : |f (s) −l| ≥ ε) µ∆ ([m + t0 − 1, t + m)t) = 0, for all ε > 0 and uniformly in m. in this case, smt lim t→∞ (f (t)) = l. the set of all m-uniform statistically convergent functions on t is denoted by smt . note that above definition 2.3 is a generalization of definition 1.4. now we can give some inclusion relations between smt , s θ,m t and s θ t. theorem 2.1. let θ = {kt−t0+m+1} be a lacunary sequence for t ∈ t uniformly in m. then, (i) s θ,m t ⊂ s m t if lim supt ( kt−t0+m+1 kt−2t0+m+1 ) < ∞, (ii) smt ⊂ s θ t if lim inft ( kt−t0+m+1 kt−2t0+m+1 ) > 1, (iii) smt = s θ t if 1 < lim inft ( kt−t0+m+1 kt−2t0+m+1 ) < lim supt ( kt−t0+m+1 kt−2t0+m+1 ) < ∞. proof. it can be proved by using a similar approach to theorem 3.3 of [31]. � uniform lacunary statistical convergence on time scales 103 the definition of strongly p-cesàro summability on t was given by turan and duman [38] in the following manner. definition 2.4. let f : t → r be a ∆-measurable function and 0 < p < ∞. then, f is strongly p-cesàro summable on t if there exists some l ∈ r such that lim t→∞ 1 µ∆ ([t0, t]t) ∫ [t0,t]t |f (s) −l|p ∆s = 0. the set of all strongly p-cesàro summable functions on t is denoted by [wp]t . the measure theory on time scales was first constructed by guseinov [19] and lebesque ∆-integral on time scales introduced by cabada and vivero [6]. now, we introduce m-uniform strongly psummablility and (θ,m)-uniform strongly p-lacunary summability of a ∆-measurable function and establish some results. definition 2.5. let f : t → r be a ∆-measurable function and 0 < p < ∞. then, f is m-uniform strongly p-summable on t if there exists some l ∈ r such that lim t→∞ 1 µ∆ ([m + t0 − 1, t + m)t) ∫ [m+t0−1,t+m)t |f (s) −l|p ∆s = 0. in this case, [ wmp ] t lim f (s) = l. the set of all m-uniform strongly p-summable functions on t will be denoted by [ wmp ] t . definition 2.6. let f : t → r be a ∆-measurable function and let θ be a lacunary sequence. assume also that 0 < p < ∞. then, f is (θ,m)-uniform strongly p-lacunary summable on t if there exists some l ∈ r such that lim t→∞ 1 µ∆ ((kt−2t0+m,kt−t0+m]t) ∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|p ∆s = 0. in that case, [ wmθp ] t lim f (s) = l. the set of all (θ,m)-uniform strongly p-lacunary summable functions on t will be denoted by [ wmθp ] t . lemma 2.1. let f : t → r be a ∆-measurable function, θ be a lacunary sequence and ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]t : |f (s) −l| ≥ ε} , for all ε > 0. thus, we have µ∆ (ω (t,θ,m)) ≤ 1 ε ∫ ω(t,θ,m) |f (s) −l|∆s ≤ 1 ε ∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|∆s, uniformly in m. proof. it can be proved by using similar way with in [38]. � theorem 2.2. let f : t → r be a ∆-measurable function and let θ be a lacunary sequence. asume also that 0 < p < ∞ and l ∈ r. then, (i) if f is (θ,m)-uniform strongly p-lacunary summable to l, then s θ,m t limt→∞ (f (t)) = l. (ii) if s θ,m t limt→∞ (f (t)) = l and f is a bounded function, then f is (θ,m)-uniform strongly placunary summable to l. proof. (i) suppose f is (θ,m)-uniform strongly p-lacunary summable to l. for given ε > 0, let ω (t,θ,m) = {s ∈ (kt−2t0+m+1,kt−t0+m+1]t : |f (s) −l| ≥ ε} 104 yilmaz, mohiuddine, altin and koyunbakan on t. then, it follows εpµ∆ (ω (t,θ,m)) ≤ ∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|p ∆s. from lemma 2.1. dividing this inequality by µ∆ ((kt−2t0+m,kt−t0+m]t) and taking limit as t →∞, we obtain lim t→∞ µ∆ (ω (t,θ,m)) µ∆ ((kt−2t0+m,kt−t0+m]t) ≤ 1 εp lim t→∞ 1 µ∆ ((kt−2t0+m,kt−t0+m]t) ∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|p ∆s = 0, which yields that s θ,m t limt→∞ (f (t)) = l. (ii) suppose f is bounded and (θ,m)-uniform lacunary statistically convergent to l on t. then, there exists a positive number m such that |f (s)| ≤ m for all s ∈ t, and also lim t→∞ µ∆ (ω (t,θ,m)) µ∆ ((kt−2t0+m,kt−t0+m]t) = 0, (2.3) where ω (t,θ,m) as defined before. since∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|p ∆s = ∫ ω(t,θ,m) |f (s) −l|p ∆s + ∫ (kt−2t0+m+1,kt−t0+m+1]t/ω(t,θ,m) |f (s) −l|p ∆s ≤ (m + |l|)p ∫ ω(t,θ,m) ∆s + εp ∫ (kt−2t0+m+1,kt−t0+m+1]t ∆s = (m + |l|)p µ∆ (ω (t,θ,m)) +εpµ∆ ((kt−2t0+m+1,kt−t0+m+1]t) , we obtain lim t→∞ 1 µ∆ ((kt−2t0+m,kt−t0+m]t) ∫ (kt−2t0+m+1,kt−t0+m+1]t |f (s) −l|p ∆s ≤ (m + |l|)p lim t→∞ µ∆ (ω (t,θ,m)) µ∆ ((kt−2t0+m,kt−t0+m]t) + εp. (2.4) since ε is an arbitrary, the proof follows from (2.3) and (2.4). � theorem 2.3. let θ = {kt−t0+m+1} be a lacunary sequence for t ∈ t. then (i) [ wmθp ] t ⊂ [ wmp ] t if lim supt ( kt−t0+m+1 kt−2t0+m+1 ) < ∞, (ii) [ wmp ] t ⊂ [ wmθp ] t if lim inft ( kt−t0+m+1 kt−2t0+m+1 ) > 1, (iii) [ wmp ] t = [ wmθp ] t if 1 < lim inft ( kt−t0+m+1 kt−2t0+m+1 ) < lim supt ( kt−t0+m+1 kt−2t0+m+1 ) < ∞. proof. we can prove by using similar techniques to theorem 2.2, theorem 2.3 and theorem 2.4 of [31] in case of p = 1. � uniform lacunary statistical convergence on time scales 105 3. conclusion in this study, we defined the concept of (θ,m)-uniform lacunary density, (θ,m)-uniform lacunary statistical convergence and (θ,m)-uniform strongly p-lacunary summability on t. we emphasize that the results that we obtained are more general than classical results mentioned in the theory of muniform statistical convergence. for example, definition 1.6 is a generalization of the definition 1.4 which is given by nuray [30] to the lacunary type sequences. we firstly defined (θ,m)-uniform lacunary statistical convergence in classical case to define it on t. then, we generalized this definition into t. furthermore, we defined m-uniform strongly p-summable functions and m-uniform statistical convergence on t by considering curicial results of turan and duman [38]. references [1] y. altin, h. koyunbakan and e. yilmaz, uniform statistical convergence on time scales, j. appl. math. 2014 (2014), art. id 471437. [2] v. baláž, and t. šalát, uniform density u and corresponding iu-convergence, math. commun. 11(1) (2006), 1-7. [3] c. belen and s. a. mohiuddine, generalized weighted statistical convergence and application, appl. math. comput. 219 (2013), 9821-9826. [4] m. bohner and a. peterson, dynamic equations on time scales, an introduction with applications, birkhauser, boston, 2001. [5] n. l. braha, h. m. srivastava and s. a. mohiuddine, a korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, appl. math. comput. 228 (2014) 162-169. [6] a. cabada and d. r. vivero, expression of the lebesque ∆-integral on time scales as a usual lebesque integral; application to the calculus of ∆-antiderivates, math. comp. model. 43 (2006), 194-207. [7] h. cakalli, lacunary statistical convergence in topological groups, indian j. pure appl. math. 26(2) (1995), 113-119. [8] j. s. connor, the statistical and strong p-cesàro convergence of sequences, analysis 8 (1988), 47-63. [9] j. s. connor and e. savaş, lacunary statistical and sliding window convergence for measurable functions, acta math. hung. 145(2) (2015), 416-432. [10] o. h. h. edely, s. a. mohiuddine and a. k. noman, korovkin type approximation theorems obtained through generalized statistical convergence, appl. math. letters 23 (2010) 1382-1387. [11] m. et, generalized cesàro difference sequence spaces of non-absolute type involving lacunary sequences, appl. math. comput. 219(17) (2013), 9372-9376. [12] m. et, s. a. mohiuddine and a. alotaibi, on λ-statistical convergence and strongly λ-summable functions of order α, j. inequal. appl. 2013 (2013), art. id 469. [13] h. fast, sur la convergence statistique, colloq. math. 2 (1951), 241-244. [14] a. r. freedman, j. j. sember and m. raphael, some cesàro-type summability spaces, proc. london math. soc. 37(3) (1978), 508-520. [15] j. a. fridy, on statistical convergence, analysis 5 (1985), 301-313. [16] j. a. fridy and c. orhan, lacunary statistical convergence, pac. j. math. 160(1) (1993), 43-51. [17] j. a. fridy and c. orhan, lacunary statistical summability, j. math. anal. appl. 173(2) (1993), 497-504. [18] t. gulsen and e. yilmaz, spectral theory of dirac system on time scales, appl. anal. (2016) 1-11, doi:10.1080/00036811.2016.1236923. [19] g. sh. guseinov, integration on time scales, j. math. anal. appl. 285(1) (2003), 107-227. [20] b. hazarika, s. a. mohiuddine and m. mursaleen, some inclusion results for lacunary statistical convergence in locally solid riesz spaces, iranian j. sci. tech. 38 (a1) (2014), 61-68. [21] s. hilger, analysis on measure chains–a unified approach to continuous and discrete calculus, results math. 18 (1990), 18-56. [22] s. hilger, ein makettenkalkl mit anwendung auf zentrumsmannigfaltigkeiten ph.d. thesis, universtat wurzburg, 1988. [23] i. j. maddox, spaces of strongly summable sequences, quart. j. math. 18(1) (1967), 345-355. [24] s. a. mohiuddine and m. a. alghamdi, statistical summability through lacunary sequence in locally solid riesz spaces, j. inequal. appl. 2012 (2012), art. id 225. [25] s. a. mohiuddine and m. aiyub, lacunary statistical convergence in random 2-normed spaces, appl. math. inform. sci. 6(3) (2012), 581-585. [26] s. a. mohiuddine, a. alotaibi and m. mursaleen, statistical summability (c, 1) and a korovkin type approximation theorem, j. inequal. appl. 2012 (2012), art. id 172. [27] s. a. mohiuddine and q. m. d. lohani, on generalized statistical convergence in intuitionistic fuzzy normed space, chaos solitons fract. 42 (2009), 1731-1737. [28] f. moricz, statistical limits of measurable functions, analysis, 24 (2004), 1-18. [29] m. mursaleen and s. a. mohiuddine, on lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, journal of computational and applied mathematics, 233(2) (2009), 142-149. [30] f. nuray, uniform statistical convergence, sci. engineer. j. firat univ. 11(3) (1999), 219-222. 106 yilmaz, mohiuddine, altin and koyunbakan [31] f. nuray and b. aydin, strongly summable and statistically convergent functions, inform. tech. valdymas 30(1) (2004), 74-76. [32] r. a. raimi, convergence, density, and τ-density of bounded sequences, proc. amer. math. soc. 14 (1963), 708-712. [33] d. rath and b. c. tripathy, on statistically convergent and statistically cauchy sequences, indian j. pure appl. math. 25(4) (1994), 381-386. [34] e. savaş and f. nuray, on σ-statistically convergence and lacunary σ-statistically convergence, math. slovaca 43(3) (1993), 309-315. [35] m. s. seyyidoglu and n. o. tan, a note on statistical convergence on time scale, j. inequal. appl. 2012 (2012), art. id 219. [36] m. s. seyyidoglu and n. o. tan, on a generalization of statistical cluster and limit points, adv. difference equ. 2015 (2015), art. id 55. [37] b. c. tripathy, on statistical convergence, proc. est. aca. sci. phy. 47(4) (1998), 299-303. [38] c. turan and o. duman, statistical convergence on time scales and its characterizations, advances in applied mathematics and approximation theory, springer proc. math. stat. 41 (2013), 57-71. [39] c. turan and o. duman, convergence methods on time scales, 11th international conference of numerical analysis and applied mathematics, aip conference proc. 1558 (2013), 1120-1123. [40] h. steinhaus, sur la convergence ordinaire et la convergence asymptotique, colloq. math. 2 (1951), 73-74. [41] e. yilmaz, y. altin and h. koyunbakan, λ-statistical convergence on time scales, dyn. cont. disc. impul. syst. ser. a: math. anal. 23(2016), 69-78. [42] a. zygmund, trigonometrical series, monogr. mat., vol. 5. warszawa-lwow 1935. 1firat university, department of mathematics, 23119, elazıg, turkey 2operator theory and applications research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia ∗corresponding author: mohiuddine@gmail.com 1. introduction and preliminaries 2. main results 3. conclusion references int. j. anal. appl. (2022), 20:26 fractional order riemann curvature tensor in differential geometry wedad saleh∗ department of mathematics, taibah university, almedina, saudi arabia ∗corresponding author: wed_10_777@hotmail.com abstract. this study discussed some interesting aspects and features of fractional curvature in the differential manifold. in particular, riemannian fractional curvature tensor, livi-civita fractional connection and bianchi fractional identity are presented. 1. introduction in mathematics, several special functions appear in many applications such as the gamma function that plays some significant roles in the theory of integral differential equations in particular fractional calculus. thus, we begin with some definitions, for the details we refer to ( [1], [15], [8]). the gamma function of a positive integer η is again a positive integer, while the gamma function γ(−η) of a negative integer changes to infinities. the gamma function any positive η value is defined as follows: γ(η) = ∫ ∞ 0 tη−1e−tdt. the gamma function γ(η) is considered as a generalization of the factorial and γ(η) is defined for η > 0 by the integral γ(η) = ∫ ∞ 0 tη−1e−t dt. in the classical sense since γ(0) = γ(1) 0 , then it follows that γ(η) is not defined for integers η ≤ 0. however, the extension formula gives finite values for γ(η), for <(η) ≤ 0 since γ(η) is analytic received: mar. 7, 2022. 2010 mathematics subject classification. 26a33, 58a05, 58d17. key words and phrases. fractional geometrical object; fractional manifold; revised riemann-liouville fractional calculus on manifolds. https://doi.org/10.28924/2291-8639-20-2022-26 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-26 2 int. j. anal. appl. (2022), 20:26 everywhere except at η = 0,−1,−2, ..., and the residue at η = k is given by resη=kγ(η) = (−1)k k! . now, if η > 0, then γ(η + 1) = ηγ(η). (1.1) equation (1.1) can be used to define γ(η) for η < 0 and η 6= −1,−2, . . . and further, this is one of the most important formulas that were satisfied by the gamma function. even though the gamma function is defined as a locally summable function on the real line by [17] γ (η) = ∫ ∞ 0 tη−1e−tdt, η > 0. (1.2) in the classical sense, γ(η) function was not defined for the negative integer thus, there was an open problem to give a satisfactory definition. however, by using the neutral limit, it has been shown in [21] that the gamma function (1.2) is defined as follows: γ (η) = n − lim ε→0 ∫ ∞ ε tη−1e−tdt for η 6= 0,−1,−2, ..., and this function can be defined by neutral limit such as γ (−n) = n − lim ε→0 ∫ ∞ ε t−n−1e−tdt = ∫ ∞ 1 t−n−1e−tdt + ∫ 1 0 t−n−1 [ e−t − n∑ i=0 (−1)i i! ti ] dt − n−1∑ i=0 (−1)i i!(n− i) ,n ∈n. it was also proven in [20] the existence of r the derivative of the gamma function and defined it by equation γ(r)(0) = n − lim ε→0 ∫ ∞ ε t−1 lnr te−tdt = ∫ ∞ 1 t−1 lnr te−tdt + ∫ 1 0 t−1 lnr t [ e−t − 1 ] dt γ(r)(−n) = n − lim ε→0 ∫ ∞ ε t−n−1 lnr te−tdt = ∫ ∞ 1 t−n−1 lnr te−tdt + ∫ 1 0 t−n−1 lnr t [ e−t − n∑ i=0 (−1)i i! ti ] dt − n−1∑ i=0 (−1)i i! r!(n− i)−r−1 int. j. anal. appl. (2022), 20:26 3 for r ∈n0 and n ∈n. also, γ(−r) = (−1)r r! ψ(r) − (−1)r r! γ for r = 1, 2, . . . , where ψ(r) = r∑ i=1 1 i . thus, the definition can be extended to the whole real line where, γ(0) = γ ′ (1) = −γ, where γ denotes euler’s constant, see [22]. for a function f : v ⊂r→r with 0 ∈ v , the fractional derivative of order α is defined by: dα dtα f (t) = 1 γ(−α) ∫ t 0 f (s) − f (0) (t − s)1+α ds, α < 0 (1.3) dα dtα f (t) = 1 γ(n−α) dn dtn ∫ t 0 f (s) − f (0) (t − s)α−n+1 ds, α > 0 (1.4) where n is the first integer greater than or equal to α. the relation (1.3) gives a fractional integral and (1.4) gives a fractional derivative. we express some of the operators of fractional derivatives, see for example, [4,7,9,10,12,16]. (1) dα dtα tγ = γ(1 + γ) γ(1 + γ + α) tγ−α, α ∈r or (α ∈c) and 1 + γ 6= 0,−1, ...,−n, (2) dn dtn dα dtα f (t) = dn+α dtn+α f (t), n ∈n, (3) dα dtα (f1(t) + f2(t)) = dα dtα f1(t) + dα dtα f2(t), (4) dα dtα (cf (t)) = c dα dtα f (t), where c is a constant, (5) dα dtα f (βt) = βα dα [d(βt)]α f (βt). it is well known that fractional calculus is an essential and advantageous branch of mathematics, having a broad range of applications at almost every department of sciences. techniques of fractional calculus have been employed in the modeling of many different phenomena in engineering, physics , and mathematics.the problem in fractional calculus is not only essential but also quite challenging ,which usually involves complicated mathematical solution techniques. however, a general solution theory for almost every issue in this area has yet to be established. each application has developed its approaches and implementations. consequently, a single standard method for the problems in fractional calculus has not emerged yet. therefore, funding reliable and efficient solution techniques along with fast implementation methods are significantly essential and still active research areas. 4 int. j. anal. appl. (2022), 20:26 further, it is also realized that the operators of fractional integration and derivation have physical and geometric interpretations, which streamline along with their utilization for related issues in various fields of science( see [2], [8], [10], [11], [12], [14], [18], [19]). moreover, the fractional differential calculus on a differential manifold is studied in( [2], [3], [4], [6], [13]). even though fractional calculus is a handy and important topic, however, the research on geometric interpretation and applications are limited ,and not many in current literature. thus, in this study, we focus on the riemannian curvature tensor, livi-civita connection and bianchi’s identity on fractional differentiable manifolds and discuss some related properties. we also give some examples. 2. fractional differential calculus on manifolds assume that n be an m-dimensional differential manifold (v,xi ) a local coordinate system on n and v0 = {x ∈ v : 0 ≤ xi ≤ bi, i = 1, 2, ...,m} [5]. for a function f : v0 →r, the fractional derivative with respect to xi : ∂ α i f (x)= 1 γ(n−α) ∂ n xi ∫ xi 0 f (x1, ...,xi−1,s,xi+1, ...,xm)− f (x1, . . . ,xi−1,0,xi+1, . . . ,xm) (xi − s)α−n+1 ds, where ∂nxi = ∂ ∂xi ◦ ∂ ∂xi ◦ ...◦ ∂ ∂xi (n times, i is fixed, α ≥ 0). for α ∈ (0,1),γ > −1, ∂ α i (xi) γ = γ(1+γ) γ(1+γ −α) ;∂ α i = δ j i . a fractional vector field v ⊂ n is an object of the form xα = xαi ∂ α i , where x α i ∈=v (n) i =1, ...,m. the fractional vector fields on v and χαv is generated by the operators ∂ α i , i = 1,2, ...,m are denoted by χ α v . if c : x = x(t),t ∈ i is a parameterized curve in u then the fractional tangent vector field of c is given by x α (t)= 1 γ(1+α) ∂ α t xi(t)∂ α i . a fractional covariant derivative is given by 5αxαy α = x α i (∂ α i y α j + γ̃ j iky α k )∂ α j where xα,y α ∈ χαu and γ̃ j ik the functions defining the coefficients of a fractional linear connection on n. they are determined by the relations 5α∂α i ∂ α k = γ̃ j ik∂ α j . since it is essential to study fractional vector fields on a differentiable manifold n. for rn, there is an obvious way to do this. recall that χα(rn) denotes the space of fractional differentiable vector fields defined on r. examples are the fractional vector fields ∂α ∂uα1 , ..., ∂α ∂uαn determined by the natural coordinate functions u1, ...,un. definition 2.1. fractional riemannian metric f on m-dimensional manifold n defines for every point p ∈ n, the scalar product of fractional tangent vectors in the fractional tangent space tαp n depending on the point p. int. j. anal. appl. (2022), 20:26 5 let aα = aαi ∂ α i and b α = bαj ∂ α j any two fractional vectors tangent to the manifold n at the point p with coordinates x =(x1, ...,xm) (a α,bα ∈ tαp n) the scalar product is equal to 〈aα,bα〉f |p = a α i (x)g̃ij(x)b α j (x) = (a α 1 , ...,a α n)   g̃11 ... g̃1n . ... . . ... . . ... . g̃n1 ... g̃nn     bα1 . . . bαn   where (1) f(aα,bα)= f(bα,aα), i.e., g̃ij = g̃ji (symmetricity condition). (2) f(aα,aα) > 0 if aα 6=0, i.e. g̃ijuαi u α j ≥ 0, g̃iju α i u α j =0 iff u α 1 = ... = u α n =0 (positive definiteness). (3) f(aα,bα) |p=x, i.e. g̃ij(x) are smooth function where 0 < α < 1. components of tensor field f in coordinate system are matrix valued functions g̃ij(x) f = g̃ij(x)d α xi ⊗dαxj. rule of transformation for entries of the matrix g̃ij(x) g̃ij(x)entries of the matrix ‖ g̃ij ‖ are components of tensor field f in a given coordinate system. how do these components transform under transformation of coordinates {xi}→{xi′} ? f = g̃ijdx α i ⊗dx α j = g̃ij ( ∂xαi ∂xα i′ dx α i′ ) ⊗ ( ∂xαj ∂xα j′ dx α j′ ) = ∂xαi ∂xα i′ g̃ij ∂xαj ∂xα j′ dx α i′ ⊗dx α j′ = g̃i′j′dx α i′ ⊗dx α j′ . hence, g̃i′j′ = ∂xαi ∂xα i′ g̃ij ∂xαj ∂xα j′ . example 2.1. consider r2 with fractional polar coordinates in the domain y > 0, x =(rαcosαϕ,rαsinαϕ), then ∂αx ∂rα = (α!cos α ϕ,α!sin α ϕ) . ∂αx ∂ϕα = ( α!e iαπ r α sin α ϕ,α!r α cos α ϕ ) . g̃ij = ( (α!)2 [ cos2α ϕ+sin2α ϕ ] (α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ (α!)2rα [ eαπi +1 ] sinα ϕcosα ϕ (α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] ) we have that f =(α!) 2 [ cos 2α ϕ+sin 2α ϕ ] (dr α ) 2 +2(α!) 2 r α [[ e iαπ +1 ] sin α ϕcos α ϕ ] dr α dϕ α +(α!) 2 r 2α [ e 2iαπ sin 2α ϕ+cos 2α ϕ ] (dϕ α ) 2}. 6 int. j. anal. appl. (2022), 20:26 notice that , as expected, when α =1, one recovers the classical formula. f =(dr) 2 + r 2 (dϕ) 2 . α = 0.1 0.2 g̃rr 0.905[c 0.2 + s0.2] 0.843[c0.4 + s0.4] g̃rϕ = g̃ϕr 0.905r 0.1[i + 1]s0.2c0.2 0 g̃ϕϕ 0.905r0.2[−s0.2 −c0.2] 0.843r0.4[−s0.4 + c0.4] f 0.905[c0.2 + s0.2]dr0.2 0.843[c0.4 + s0.4]dr0.4 +1.8101r0.1[i + 1]s0.2c0.2(dr).1(dϕ).1 + +0.905r0.2[−s0.2 −c0.2](dϕ).2 0.843r0.4[−s0.4 + c0.4](dϕ)0.4 table 1. c = cos ϕ,s = sin ϕ α = 0.3 0.4 g̃rr 0.805[c 0.6 + s0.6] 0.787[c0.8 + s0.8] g̃rϕ = g̃ϕr 0.805r 0.3[[i + 1]s0.3c0.3] 1.574r0.4s0.3c0.3 g̃ϕϕ 0.805r0.6[−s0.6 + c0.6] 0.787r0.8[s0.8 + c0.8] f 0.805[c0.6 + s0.6](dr)0.6 0.787[c0.8 + s0.8](dr)0.8 +1.61r0.3[[i + 1]s0.3c0.3](dr).3(dϕ).3 +3.148r0.4s0.4c0.4dr .4dϕ.4 +0.805r0.6[−s0.6 + c0.6](dϕ).6 +0.787r0.8[s0.8 + c0.8](dϕ).8 table 2. c = cos ϕ,s = sin ϕ α = 0.5 0.6 g̃rr 0.785[c + s] 0.798[c 1.2 + s1.2] g̃rϕ = g̃ϕr 0.785r 0.5[i + 1]s0.5c0.5 0 g̃ϕϕ 0.785r[−s + c] 0.798r1.2[s1.2 + c1.2] f 0.785[c + s]r(dr) 0.798[c1.2 + s1.2](dr)1.2 +1.57r0.5[i + 1]s0.5c0.5(dr)0.5(dϕ)0.5 + 0.785r[−s + c]dϕ +0.798r1.2[s1.2 + c1.2](dϕ)1.2 table 3. c = cos ϕ,s = sin ϕ int. j. anal. appl. (2022), 20:26 7 α = 0.7 0.8 g̃rr 0.826[c 1.4 + s1.4] 0.867[c1.6 + s1.6] g̃rϕ = g̃ϕr 0.826r 0.7[i + 1]s0.7c0.7 1.734r0.8s0.8c0.8 g̃ϕϕ 0.826r1.4[−s1.4 + c1.4] 0.867r1.6[s1.6 + c1.6] f 0.826[c1.4 + s1.4](dr)1.4 0.867[c1.6 + s1.6](dr)1.6 +1.652r0.7[i + 1]s0.7c0.7(dr)0.7(dϕ)0.7 +3.468r0.8s0.8c0.8dr0.8dϕ0.8 +0.826r1.4[−s1.4 + c1.4](dϕ)1.4 +0.867r1.6[s1.6 + c1.6](dϕ)1.6 table 4. c = cos ϕ,s = sin ϕ α = 0.9 1 g̃rr 0.925[c 1.8 + s1.8] 1 g̃rϕ = g̃ϕr 0.925r 0.9[i + 1]s0.9c0.9 0 g̃ϕϕ 0.925r1.8[−s1.8 + c1.8] r2 f 0.925[c1.8 + s1.8](dr)1.8 (dr)2 +1.85r0.9[i + 1]s0.9c0.9(dr)0.9(dϕ)0.9 + +0.925r1.8[−s1.8 + c1.8](dϕ)1.8 r2 (dϕ)2 table 5. c = cos ϕ,s = sin ϕ remark 2.1. let n is an m-dimensional riemannian manifold with fractional metric tensor g̃,then we shall denote the fractional derivatives of the elements of tensor g̃ as follows: g̃ij,k = ∂α ∂xαk g̃ij, and g̃ij,kl = ∂α ∂xαl ∂α ∂xαk g̃ij = ∂2α ∂xαl ∂x α k g̃ij, i, j,k, l =1, ...,n. definition 2.2. asymmetric fractional connection is called levi-civita fractional connection if it is compatible with metric, i.e., if it preserves the scalar product: ∂ α xα 〈y α ,z α〉= 〈5αxαy α ,z α〉+ 〈y α,5αxαz α〉 for arbitrary fractional vector fields xα,y α, and zα. in local coordinates christoffel symbols of levi-civita fractional connection are given by: γ̃ k ij = 1 2 g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij). proof. since ∂ α j ei = γ̃ m ij em (2.1) γ̃ m ij emel =(∂ α j ei)el (2.2) 8 int. j. anal. appl. (2022), 20:26 γ̃ m ij gml = ∂ α j (ei.el)−ei(∂ α j el) = ∂ α j gil − γ̃ m lj emei = ∂ α j gil − γ̃ m lj gmi, (2.3) then γ̃ m ij gml + γ̃ m lj gmi = ∂ α j gil, (2.4) which implies that γ̃ m ij g̃ml + γ̃ m lj g̃mi = ∂ α j g̃il. (2.5) in this equation, the index m is a dummy, so only the indices i,j ,and l are specified. we can cyclically permute these indices to generate two more equations: γ̃ m jl g̃mi + γ̃ m il g̃mj = ∂ α l g̃ji (2.6) γ̃ m li g̃mj + γ̃ m ji g̃ml = ∂ α i g̃lj (2.7) since γ̃mij = γ̃ m ji , then γ̃ m lj g̃mi + γ̃ m il g̃mj = ∂ α l g̃ij (2.8) γ̃ m il g̃mj + γ̃ m ij g̃ml = ∂ α i g̃lj. (2.9) we can now add (2.5)to (2.9) and subtract (2.8)to get 2γ̃ m ij g̃ml = ∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij 2γ̃ m ij g̃mlg̃ kl = g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij) since g̃mlg̃ kl = δkm, then γ̃ k ij = 1 2 g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij), we can write γ̃ k ij = 1 2 g̃ kl (g̃il,j + g̃lj,i − g̃ij,l). � example 2.2. for 2-dimential polar coordinates x =(rαcosαϕ,rαsinαϕ). the metric tensor and its inverse here are: g̃ij = ( (α!)2 [ cos2α ϕ+sin2α ϕ ] (α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ (α!)2rα [ eαπi +1 ] sinα ϕcosα ϕ (α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] ) g̃i j = ( a−1(α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] −a−1(α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ −a−1(α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ a−1(α!)2 [ cos2α ϕ+sin2α ϕ ] ) . where a =(α!) 4 r 2α {[ cos 2α ϕ+sin 2α ϕ ][ e 2απi sin 2α ϕ+cos 2α ϕ ] − [ e απi +1 ] sin 2α ϕcos 2α ϕ } therefore, ∂ α r g̃ij = ( 0 (α!)3 [ eαπi +1 ] sinα ϕcosα ϕ (α!)3 [ eαπi +1 ] sinα ϕcosα ϕ (α!)(2α)!rα [ e2απi sin2α ϕ+cos2α ϕ ] ) ∂ α ϕg̃ij = ( (α!)(2α)! [ eαπi +1 ] cosαϕsinαϕ (α!)3rα [ eαπi +1 ][ eαπi sin2α ϕ+cos2α ϕ ] (α!)3rα [ eαπi +1 ][ eαπi sin2α ϕ+cos2α ϕ ] (α!)(2α)!r2α [ eαπi +1 ] sinα ϕcosα ϕ ) . int. j. anal. appl. (2022), 20:26 9 then , γ̃ r rr = 1 2 g̃ rl (∂ α r g̃rl +∂ α r g̃lr −∂ α l g̃rr) = −a−1(α!)5rα [ e απi +1 ]2 sin 2α ϕcos 2α ϕ, γ̃ r rϕ = 1 2 g̃ rl (∂ α ϕg̃rl +∂ α r g̃lϕ −∂ α l g̃rϕ) = 0 = γ̃ r ϕr, γ̃ r ϕϕ = 1 2 g̃ rl (∂ α ϕg̃ϕl +∂ α ϕg̃lϕ −∂ α l g̃ϕϕ) = (α!) 2 r α (2a) −1 {[ e 2απi sin 2α ϕ+cos 2α ϕ ][ 2(α!) 3 r α [ e απi +1 ][ e απi sin 2α ϕ+cos 2α ϕ ] − (α!)(2α)!rα [ e 2απi sin 2α ϕ+sin 2α ϕ ]] − (α!)(2α)!rα [ e απi +1 ] sin 2α ϕcos 2α ϕ } , γ̃ ϕ rr = 1 2 g̃ ϕl (∂ α ϕg̃rl +∂ α r g̃lr −∂ α l g̃rr) = −(α!)2(2a)−1 { (α!)(2α)!r α [ e απi +1 ]2 sin 2α ϕcos 2α ϕ+ [ cos 2α ϕ+sin 2α ϕ ][ e απi +1 ] × [ (α!) 3 r α [ e απi sin 2α ϕ+cos 2α ϕ ] +(α!) 3 sin α ϕcos α ϕ− (α!)(2α)!cosα ϕsinα ϕ ]} , γ̃ ϕ rϕ = 1 2 g̃ ϕl (∂ α ϕg̃rl +∂ α r g̃lϕ −∂ α l g̃rϕ) = (α!) 2 (2a) −1 { −(α!)(2α)!2rα [ e απi +1 ] cos 2α ϕsin 2α ϕ +(α!)(2α)!r α [ cos 2α ϕ+sin 2α ϕ ][ e 2απi sin 2α ϕ+cos 2α ϕ ]} = γ̃ ϕ ϕr, γ̃ ϕ ϕϕ = 1 2 g̃ ϕl (∂ α ϕg̃ϕl +∂ α ϕg̃lϕ −∂ α l g̃ϕϕ) = (α!) 2 r 2α (2a) −1 { − [ e απi +1 ] sin α ϕcos α ϕ [ 2(α!) 3 [ e απi +1 ][ e απi sin 2α ϕ+cos 2α ϕ ] −(α!)(2α)! [ e 2απi sin 2α ϕ+cos 2α ϕ ]] +(α!)(2α)! [ e απi +1 ][ sin 2α ϕ+cos 2α ϕ ] sin α ϕcos α ϕ } . 3. fractional curvature definition 3.1. the fractional curvature r̃ of order α of a riemannian manifold n is a correspondence that associates to every pair xα,y α ∈ χα a mapping r̃(xα,y α): χα(n)×χα(n)→ χα(n) given by r̃(x α ,y α )z α =5αxα 5 α y α z α −5αy α 5 α xα z α −5α[xα,y α]z α , where zα ∈ χα and 5α is the fractional riemannian connection. remark 3.1. r̃(x α ,y α )z α = 5αxα 5 α y α z α −5αy α 5 α xα z α −5α[xα,y α]z α = −(5αy α 5 α xα z α −5αxα 5 α y α z α −5α[y α,xα]z α ) = −r̃(y α,xα)zα. proposition 3.1. the fractional curvature r̃ of a riemannian manifold has the following properties: 10 int. j. anal. appl. (2022), 20:26 (1) r̃ is bilinear in χα(n)×χα(n), that is, r̃(f x α +gy α ,z α )w α = f r̃(x α ,z α )w α +gr̃(y α ,z α )w α , r̃(x α , f y α +gz α )w α = f r̃(x α ,y α )w α +gr̃(x α ,z α )w α , where f ,g ∈=(m),xα,y α,zα,wα ∈ χα(n) (2) for any xα,y α ∈ χα(n), r̃(xα,y α) is linear r̃(x α ,y α )(z α +w α )= r̃(x α ,y α )z α + r̃(x α ,y α )w α , r̃(x α ,y α )(f z α )= f r̃(x α ,y α )z α , where zα,wα ∈ χα(n) proof. (1) r̃(f x α +gy α ,z α )w α =5αf xα+gy α 5 α zα w α −5αzα 5 α f xα+gy α w α −5α[f xα+gy α,zα]w α =(f 5αxα +g5 α y α)5 α zα w α −5αzα(f 5 α xα w α +g 5αy α w α ) −5αf [xα,zα]+g[y α,zα]−(zαf )xα−(zαg)y α w α = f 5αxα 5 α zαw α +g 5αy α 5 α zαw α − (zαf )5αxα w α −f 5αzα 5 α xαw α − (zαg)5αy α w α −g 5αzα 5 α y αw α − f 5α[xα,zα] w α −g 5α[y α,zα] w α +(z α f )5αxα w α +(z α g)5αy α w α = f (5αxα 5 α zα w α −5αzα 5 α xα w α −5α[xα,zα]w α ) +g(5αy α 5 α zα w α −5αzα 5 α y α w α −5α[y α,zα]w α ) = f r̃(x α ,z α )w α +gf r̃(y α ,z α )w α . also, r̃(x α , f y α +gz α )w α = −r̃(f y α +gzα,xα) = −f r̃(y α,xα)wα −gr̃(zα,xα)wα = f r̃(x α ,y α )w α +gr̃(x α ,z α )w α . (2) r̃(x α ,y α )(z α +w α ) =5αxα 5 α y α (z α +w α )−5αy α 5 α xα (z α +w α )−5α[xα,y α](z α +w α ) =5αxα 5 α y α z α +5αxα 5 α y α w α −5αy α 5 α xα z α −5αy α 5 α xαw α −5α[xα,y α]z α −5α[xα,y α]w α =(5αxα 5 α y α z α −5αy α 5 α xα z α −5α[xα,y α]z α ) +(5αxα 5 α y α w α −5αy α 5 α xα w α −5α[xα,y α]w α ) = r̃(x α ,y α )z α + r̃(x α ,y α )w α . int. j. anal. appl. (2022), 20:26 11 also, r̃(x α ,y α )(f z α ) =5αxα 5 α y α (f z α )−5αy α 5 α xα (f z α )−5α[xα,y α](f z α ) =5αxα((y α f )z α + f 5αy α z α )−5αy α((x α f )z α + f 5αxα z α ) −(([xα,y α]f )zα + f 5α[xα,y α] z α ) = x α (y α f )z α +(y α f )5αxα z α +(x α f )5αy α z α + f 5αxα 5 α y αz α −y α(xαf )zα +(xαf )5αy α z α +(y α f )5αxα z α + f 5αy α 5 α xαz α −([xα,y α]f )zα − f 5α[xα,y α] z α =([x α ,y α ]f )z α + f (5αxα 5 α y α z α −5αy α 5 α xα z α −5α[xα,y α]z α )− ([xα,y α]f )zα = f r̃(x α ,y α )z α . � proposition 3.2 (bianchi fractional identity). r̃(x α ,y α )z α + r̃(y α ,z α )x α + r̃(z α ,x α )y α =0. proof. r̃(x α ,y α )z α + r̃(y α ,z α )x α + r̃(z α ,x α )y α =5αxα 5 α y α z α −5αy α 5 α xα z α −5α[xα,y α]z α +5αy α 5 α zαx α −5αzα 5 α y α x α −5α[y α,zα]x α +5αzα 5 α xαy α −5αxα 5 α zα y α −5α[zα,xα]y α =5αxα[y α ,z α ]+5αy α[z α ,x α ]+5αzα[x α ,y α ] −5α[xα,y α] z α −5α[y α,zα]x α −5α[zα,xα]y α = [x α , [y α ,z α ]]+ [y α , [z α ,x α ]]+ [z α , [x α ,y α ]] = 0. � in local coordinates r̃(∂ α i ,∂ α j )∂ α k = r̃ l ijk∂ α l , and r̃ijkm = 〈 r̃(∂ α i ,∂ α j )∂ α k ,∂ α m 〉 = 〈 r̃ l ijk∂ α l ,∂ α m 〉 = r̃ l ijk 〈∂ α l ,∂ α m〉 = r̃ l ijkg̃ml. the fractional riemannian curvature tensor acts on fractional vector fields as follows: r̃(x α ,y α ,z α ,w α )= 〈 r̃(x α ,y α )z α ,w α 〉 . proposition 3.3. (1) r̃ijkl + r̃jkil + r̃kijl =0. (2) r̃ijkl =−r̃jikl. (3) r̃ijkl =−r̃ijlk. (4) r̃ijkl = r̃klij. 12 int. j. anal. appl. (2022), 20:26 proof. (1) is just the bianchi fractional identity again. (2) r̃ijkl = 〈 r̃(∂ α i ,∂ α j )∂ α k ,∂ α l 〉 = 〈 −r̃(∂αj ,∂ α i )∂ α k ,∂ α l 〉 = − 〈 r̃(∂ α j ,∂ α i )∂ α k ,∂ α l 〉 = −r̃jikl. (3) is equivalent to r̃ijkk =0, whose proof follows: r̃ijkk = 〈 r̃(∂ α i ,∂ α j )∂ α k ,∂ α k 〉 = 〈 5α∂α i 5α∂α j ∂ α k −5 α ∂α j 5α∂α i ∂ α k −5 α [∂α i ,∂α j ]∂ α k ,∂ α k 〉 , but 〈 5α∂α j 5α∂α i ∂ α k ,∂ α k 〉 = ∂ α j 〈 5α∂α i ∂ α k ,∂ α k 〉 − 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α k 〉 , and 〈 5α[∂α i ,∂α j ]∂ α k ,∂ α k 〉 = 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 , then r̃ijkk = ∂ α j 〈 5α∂α i ∂ α k ,∂ α k 〉 −∂αi 〈 5α∂α j ∂ α k ,∂ α k 〉 + 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 = 1 2 ∂ α j (∂ α i 〈∂ α k ,∂ α k 〉)− 1 2 ∂ α i ( ∂ α j 〈∂ α k ,∂ α k 〉 ) + 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 = − 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉+ 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉=0. (4) by bianchi fractional identity we have r̃ijkl + r̃jkil + r̃kijl =0 r̃jkli + r̃klji + r̃ljki =0 r̃klij + r̃likj + r̃iklj =0 r̃lijk + r̃ijlk + r̃jlik =0 summing the equations above, we obtain 2r̃kijl +2r̃ljki =0, then r̃kijl =−r̃ljki = r̃jlki. � proposition 3.4. the following expression holds 2r̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2γ̃ rjkγ̃ s img̃rs +2γ̃ r ikγ̃ s jmg̃rs. proof. from the definition of the christoffel symbols, 5α∂α i ∂αj = γ̃ k ij ∂ α k , 2 〈 5α∂α i ∂ α j ,∂ α m 〉 = 2 〈 γ̃ k ij ∂ α k ,∂ α m 〉 = 2γ̃ k ij g̃mk = g̃im,j + g̃jm,i − g̃ij,m, int. j. anal. appl. (2022), 20:26 13 an appropriate rearrangement of the indices yields the following expression: 2 〈 5α∂α j ∂ α k ,∂ α m 〉 = 2 〈 γ̃ i jk∂ α i ,∂ α m 〉 = 2γ̃ i jkg̃im = g̃jm,k + g̃km,j − g̃jk,m. (3.1) ∂ α i < 5 α ∂α j ∂ α k ,∂ α m >=< 5 α ∂α i 5α∂α j ∂ α k ,∂ α m > + < 5 α ∂α j ∂ α k ,5 α ∂α i ∂ α m > whence, by (3.1) 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 +2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 = 2∂ α i 〈 5α∂α j ∂ α k ,∂ α m 〉 = ∂ α i ( g̃img il (∂ α k gjl +∂ α j gkl −∂ α l gjk) ) = g̃jm,ki + g̃km,ji − g̃jk,mi. (3.2) by switching i and j we also have that 2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 = g̃im,kj + g̃km,ij − g̃ik,mj. (3.3) combining (3.2) and (3.3) yields 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 −2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 . by definition r̃ ( ∂ α i ,∂ α j ) ∂ α k =5 α ∂α i 5α∂α j ∂ α k −5 α ∂α j 5α∂α i ∂ α k whence 2r̃ijkm = 2 〈 r̃(∂ α i ,∂ α j )∂ α k ,∂ α m 〉 = 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 −2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 , so we have proven that 2r̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 . by the definition of the christoffels, 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 = 〈 γ̃ r ik∂ α r , γ̃ s jm∂ α s 〉 = γ̃ r ikγ̃ s jm 〈∂ α r ,∂ α s 〉 = γ̃ r ikγ̃ s jmg̃rs, 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 = 〈 γ̃ r jk∂ α r , γ̃ s im∂ α s 〉 = γ̃ r jkγ̃ s im 〈∂ α r ,∂ α s 〉 = γ̃ r jkγ̃ s img̃rs, 14 int. j. anal. appl. (2022), 20:26 then 2r̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2γ̃ rjkγ̃ s img̃rs +2γ̃ r ikγ̃ s jmg̃rs. � remark 3.2. if α =1,then 2rijkm = gjm,ki +gkm,ji −gjk,mi −gim,kj −gkm,ij +gik,mj −2γ rjkγ s imgrs +2γ r ikγ s jmgrs. since gkm,ji = gkm,ij, then 2rijkm = gjm,ki −gjk,mi −gim,kj +gik,mj −2γ rjkγ s imgrs +2γ r ikγ s jmgrs, then 2rijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2γ rjkγ s imgrs +2γ r ikγ s jmgrs. remark 3.3. for any pair of fractional tangent vectors xα,y α ∈ tαp n we shall denote with γ̃(xα,y α) the following fractional vector in tαp n: γ̃(x α ,y α )= γ̃ k ij x α i y α j ∂ α k . proposition 3.5. the following expressions hold for any pair xα,y α ∈ tαp n: 2r̃(x α ,y α ,y α ,x α ) = ∂ α i ( g̃img il )( ∂ α k gjl +∂ α j gkl −∂ α l gjk ) + g̃img il ( ∂ α i ∂ α k gjl +∂ α i ∂ α j gkl −∂ α i ∂ α l gjk ) −∂αj ( g̃jmg jl ) (∂ α k gil +∂ α i gkl −∂ α l gik)− g̃jmg jl ( ∂ α j ∂ α k gil +∂ α j ∂ α i gkl −∂ α j ∂ α l gik ) +2 ‖ γ̃ (xα,y α) ‖2 −2 〈 γ̃ (x α ,x α ) , γ̃ (y α ,y α ) 〉 . proof. since g̃rsx α i y α j y α k x α mγ̃ r ikγ̃ s jm = 〈 x α i y α k γ̃ r ik∂ α r ,y α j x α mγ̃ s jm∂ α s 〉 = 〈 γ̃(x α ,y α ), γ̃(x α ,y α ) 〉 =‖ γ̃(xα,y α) ‖2, and g̃rsx α i y α j y α k x α mγ̃ r jkγ̃ s im = 〈 x α i x α mγ̃ s im∂ α s ,y α j y α k γ̃ r jk∂ α r 〉 = 〈 γ̃(x α ,x α ), γ̃(y α ,y α ) 〉 , this completes the proof. � remark 3.4. if α =1, then 2r(x,y,y,x)= gjm,ki −gjk,mi −gim,kj +gik,mj +2 ‖ γ(x,y ) ‖2 −2〈γ(x,x),γ(y,y )〉 . conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2022), 20:26 15 references [1] i.d. albu, m. neamtu, d. opris, the geometry of fractional osculator bundle of higher order and applications, in: processing of international conference on differential geometry lagrange and hamilton spaces, dedicated to acad. radu miron at eighty, september 3-8, 2007, iasi, romania https://doi.org/10.48550/arxiv.0709.2000. [2] m. axtell, m.e. bise, fractional calculus application in control systems, in: ieee conference on aerospace and electronics, ieee, dayton, oh, usa, 1990: pp. 563–566. https://doi.org/10.1109/naecon.1990.112826. [3] y. ding, x. wu, fractional integrals on product manifolds, potential anal. 30 (2009), 371–383. https://doi. org/10.1007/s11118-009-9120-1. [4] g. jumarie, riemann-christoffel tensor in differential geometry of fractional order application to fractal spacetime, fractals. 21 (2013), 1350004. https://doi.org/10.1142/s0218348x13500047. [5] i. d. albu and d. opris, the geometry of fractional tangent bundle and applications, in: bsg proceedings 16. the int. conf. of diff. geom. and dynamical systems (dgds-2008) and the v-th int. colloq. of mathematics in engineering and numerical physics (menp-5) math. sections, august 29 september 2, 2008, mangalia, romania, pp. 1-11. http://repository.utm.md/handle/5014/17678. [6] a. kilicman, w. saleh, note on the fractional mittag-leffler functions by applying the modified riemann-liouville derivatives, bol. soc. paranaense mat. 40 (2022), 1–16. https://doi.org/10.5269/bspm.44103. [7] k.m. kolwankar, studies of fractal structures and processes using methods of fractional calculus, arxiv:chaodyn/9811008. (1998). http://arxiv.org/abs/chao-dyn/9811008. [8] m. delkhosh, introduction of derivatives and integrals of fractional order and its applications, appl. math. phys. 1 (2013), 103–119 . [9] k.s. miller, b. ross, an introduction to the fractional calculus and fractional differential equations, john wiley and sons, inc. new york, (1993). [10] k.b. oldham, j. spanier, the fractional calculus, academic press, new york, (1974). [11] i. podlubny, fractional-order systems and fractional-order controllers, technical report uef-03-94, slovak acad. sci. (1994). [12] b. ross, ed., fractional calculus and its applications: proceedings of the international conference held at the university of new haven, june 1974, springer berlin heidelberg, berlin, heidelberg, 1975. https://doi.org/10. 1007/bfb0067095. [13] w. saleh, a. kılıçman, some remarks on geometry of fractional calculus, appl. math. inform. sci. 9 (2015), 1265–1275. [14] w. saleh and a. kılıçman, some inequalities for generalized s-convex functions, jp j. geom. topol. 17 (2015), 63–82. [15] i.n. sneddon, special functions of mathematical physics and chemistry, oliver & boyd, edinburgh, 1956. [16] m.h. tavassoli, a. tavassoli, m.r. ostad rahimi, the geometric and physical interpretation of fractional order derivatives of polynomial functions, differ. geom. dyn. syst. 15 (2013), 93–104. [17] a. kılıçman and b. fisher, some results on the composition of singular distribution, appl. math. inform. sci. 5 (2011), 597–608. [18] a. kılıçman, w. saleh, notions of generalized s-convex functions on fractal sets, j. inequal. appl. 2015 (2015), 312. https://doi.org/10.1186/s13660-015-0826-x. [19] a. kılıçman, w. saleh, some generalized hermite-hadamard type integral inequalities for generalized s-convex functions on fractal sets, adv. differ. equ. 2015 (2015), 301. https://doi.org/10.1186/s13662-015-0639-8. [20] b. fisher, b. jolevsaka-tuneska, a. kiliçman, on defining the incomplete gamma function, integral transforms spec. funct. 14 (2003), 293–299. https://doi.org/10.1080/1065246031000081667. https://doi.org/10.48550/arxiv.0709.2000 https://doi.org/10.1109/naecon.1990.112826 https://doi.org/10.1007/s11118-009-9120-1 https://doi.org/10.1007/s11118-009-9120-1 https://doi.org/10.1142/s0218348x13500047 http://repository.utm.md/handle/5014/17678 https://doi.org/10.5269/bspm.44103 http://arxiv.org/abs/chao-dyn/9811008 https://doi.org/10.1007/bfb0067095 https://doi.org/10.1007/bfb0067095 https://doi.org/10.1186/s13660-015-0826-x https://doi.org/10.1080/1065246031000081667 16 int. j. anal. appl. (2022), 20:26 [21] b. fisher, a. kilicman, d. nicholas, on the beta function and the neutrix product of distributions, integral transforms spec. funct. 7 (1998), 35–42. https://doi.org/10.1080/10652469808819184. [22] b. fisher, a. kılıçman, results on the gamma function, appl. math. inform. sci. 6 (2012), 173–176. https://doi.org/10.1080/10652469808819184 1. introduction 2. fractional differential calculus on manifolds rule of transformation for entries of the matrix ij(x) 3. fractional curvature references int. j. anal. appl. (2023), 21:24 a congruent property of gibonacci number modulo prime wipawee tangjai∗, kodchaphon wanichang, montathip srikao, punyanuch kheawkrai department of mathematics, faculty of science, mahasarakham university, maha sarakham, 44150, thailand ∗corresponding author: xvii.noo@gmail.com abstract. let a,b ∈ z and p be a prime number such that a and b are not divisible by p. in this work, we give a congruent property modulo a prime number p of the gibonacci number defined by gn = gn−1 +gn−2 with initial condition g1 = a,g2 = b. we show that a the gibonacci sequence satisfying g kp−(p5) ≡ gk−1 (mod p) for all positive integer k and such odd prime p 6= 5 if and only if a ≡ b (mod p). moreover, for each odd prime number p, we give a necessary and sufficient condition yielding g kp−(p5) ≡ gk−1 (mod p). we also find a relation between the sequences in the same equivalent class in modulo 5 constructed by aoki and sakai [1] that leads to such congruent property. 1. introduction the fibonacci sequence {fn}n≥0 satisfies the recurrence relation fn = fn−1 + fn−2 with initial condition f0 = 0,f1 = 1 and the lucas sequence {ln}n≥0 satisfies the recurrence relation ln = ln−1 + ln−2 with initial condition l0 = 2,l1 = 1. the sequences can be extended to a negative index as follows: for n ∈n f−n =(−1)n+1fn, (1.1) l−n =(−1)nln. (1.2) we see that both the fibonacci and lucas sequences satisfy the same recurrence relation with different initial conditions. to generalized the mentioned sequences, the generalized fibonacci sequence or gibonacci sequence {gn}n>0 = {g(a,b)} ( [2], p.137) is defined to satisfy the recurrence relation gn = gn−1 +gn−2, for n ≥ 3, with initial condition g1 = a and g2 = b, where a,b ∈z. received: jan. 19, 2023. 2020 mathematics subject classification. 11b37. key words and phrases. generalized fibonacci number; gibonacci number; lucas number; modulo prime number. https://doi.org/10.28924/2291-8639-21-2023-24 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-24 2 int. j. anal. appl. (2023), 21:24 theorem 1.1 ( [2], p.137). for an integer n ≥ 3, the n-th gibonacci number satisfies gn = g1fn−2 +g2fn−1. theorem 1.2 ( [2], p.137). for n ∈n, we have g−n =(−1)n+1(g1fn+2 −g2fn+1). the objective of this work is investigating the congruent property of the gibonacci sequence {gn}= {g(a,b)} where a,b ∈z that p 6 |a and p 6 |b analogous to the result from andrica et. al. [3] appearing in theorem 1.3. throughout this article, we let p be an odd prime number and ( p 5 ) be the legendre’s symbol. theorem 1.3. [3] for a positive integer k and an odd prime number p 6=5, we have f kp−(p5) ≡ fk−1 (mod p), l kp−(p5) ≡ lk−1 (mod p). we note that theorem 1.3 is not true when p = 5 as f10 6≡ f1 (mod 5). as a result, we give a necessary and sufficient condition in terms of the initial condition of the gibonacci sequence and its index k that lead to g pk−(p5) ≡ gk−1 (mod p) (1.3) for each prime p characterized by the value of ( p 5 ) . we also give a necessary and sufficient condition resulting in (1.3) when ( p 5 ) = −1 in theorem 3.1. in theorem 3.2, we show that if ( p 5 ) = 1, then (1.3) holds for all positive integer k. by combining theorems 3.1 and 3.2, we show that for a gibonacci sequence {gn}n>0 = {g(a,b)}, (1.3) holds for all k ∈ n and for all odd prime number p 6= 5 where p 6 |a and p 6 |b if and only if a ≡ b (mod p) in theorem 3.3. for the case that p =5, we consider the equivalent class x5 introduced by aoki and sakai [1]. for a prime number p, aoki and sakai constructed an equivalent class of the gibonacci sequences xp = {{gn}|{gn} is the gibonacci sequence, where p 6 |g1 and p 6 |g2}/ ∼ where, {gn}∼{g′n} if and only if g2g −1 1 ≡ g ′ 2g ′−1 1 (mod p), (1.4) and g−1 is the inverse of g modulo p where 1≤ g−1 < p. they also showed that xp = {{g(1,k)}|1≤ k ≤ p−1}. (1.5) in theorem 3.4, we consider the representation{g(1,h)}of each class in x5, where1≤ h ≤ 4and give a complete characterization of the initial conditions of a gibonacci sequences and the corresponding indices that (1.3) holds. later in theorems 3.5 and 3.6, we give a relation of the sequences in the same class in x5. int. j. anal. appl. (2023), 21:24 3 p π(p) ( p 5 ) 3 8 -1 5 20 0 7 16 -1 11 10 1 13 28 -1 17 36 -1 19 18 1 23 48 -1 29 14 1 37 76 -1 43 88 -1 table 1. list of the pisono period and the legendre’s symbol of a prime number. 2. preliminaries in this section, we give an overview of the related work that will be used to prove the main results. for any positive integer m, the pisano period [4] modulo m is the period of the fibonacci number modulo m, denoted by π(m). in 2012, gupta et. al, [5] gave a method to find a period of the fibonacci number modulo a prime number. theorem 2.1. [4] let p be a prime number. • if p ≡±1 (mod 5), then π(p)|(p−1). • if p ≡±2 (mod 5), then π(p)|2(p+1). the values of π(p) and ( p 5 ) listed in table 1 appear in [7] and [8], respectively. in lemma 3.1, we show that the period of the gibonacci number modulo p is at most π(p) which leads to to computation appearing in table 2. lemma 2.1. [1] let p be an odd prime number. the following statements are true. (1) if ( p 5 ) =1, then p 6 |gn for any n ∈n. (2) if ( p 5 ) =−1, then p|gn for some n ∈n. the following results are some identities of the fibonacci and the lucas sequences that will be used in this work. theorem 2.2. ( [2], p. 93) for each n ∈n, ln = fn+1 +fn−1, 5fn = ln+1 +ln−1 4 int. j. anal. appl. (2023), 21:24 theorem 2.3. ( [2], p. 462) lucas number is not divisible by 5. theorem 2.4. [3] for an odd prime number p, a positive integer k and an integer r, the following holds: 2fkp+r ≡ (p 5 ) fklr +frlk (mod p), (2.1) 2lkp+r ≡ 5 (p 5 ) fkfr +lklr (mod p). (2.2) the following corollary is a direct result of theorem 2.4. corollary 2.1. for a positive integer k and r, we have f5k−r ≡ 3f−rlk (mod 5), (2.3) l5k ≡ lk (mod 5). (2.4) theorem 2.5. [3] for an odd prime number p and a positive integer k, we have fkp ≡ (p 5 ) fk (mod p), fp ≡ (p 5 ) (mod p), f p−(p5) ≡ 0 (mod p). theorem 2.6. [6] let n,k ∈z. if k is an even number, then fn+k +fn−k = fnlk, (2.5) fn+k −fn−k = fkln. (2.6) if k is an odd number, then fn+k +fn−k = fkln, (2.7) fn+k −fn−k = fnlk. (2.8) 3. main results the following property of the gibonacci sequence can be obtained directly from theorem 1.1; however, the authors do not find this result in the literature review. lemma 3.1. let {gn}n>0 = {g(a,b)}, where a,b ∈z. for k,r ∈z, we have gkπ(p)+r ≡ gr (mod p) int. j. anal. appl. (2023), 21:24 5 proof. by theorem 1.1, we have that gkπ(p)+r ≡ g1fkπ(p)+r−2 +g2fkπ(p)+r−1 (mod p) ≡ g1fr−2 +g2fr−1 (mod p) ≡ gr (mod p). � next, we consider each case of an odd prime p characterized by the value of ( p 5 ) and give a necessary and sufficient condition resulting to g pk−(p5) ≡ gk−1 (mod p). theorem 3.1. let p be an odd prime number that ( p 5 ) =−1 and {gn}n>0 = {g(a,b)} be such that a and b are not divisible by p. for k ∈ n, we have that g pk−(p5) ≡ gk−1 (mod p) if and only if one of the following holds: (1) g1 ≡ g2 (mod p), (2) lk−1 ≡ 0 (mod p). proof. by theorems 1.1, 1.3, 2.4, 2.5 and 2.6, we have g pk−(p5) = gpk+1 = afpk−1 +bfpk ≡ 2−1a(−fkl−1 +f−1lk)−bfk (mod p) ≡ afk+1 −bfk (mod p). (3.1) it follows from theorem 2.2 that g pk−(p5) −gk−1 ≡ a(fk+1 −fk−3)−b(fk −fk−2) (mod p) ≡ (a−b)lk−1 (mod p). hence, g pk−(p5) ≡ gk−1 (mod p) if and only if a ≡ b (mod p) or lk−1 ≡ 0 (mod p). � by theorem 3.1, the listed p and k in table 2 yield g pk−(p5) ≡ gk−1 (mod p). corollary 3.1. let p be an odd prime number where ( p 5 ) = −1 and {gn}n>0 = {g(a,b)} where a and b are integers that are not divisible by p. then g pk−(p5) ≡ gk−1 (mod p) for all k ∈ n if and only if a ≡ b (mod p). we note that, by (3.1), if {gn}n>0 = {g(1,h)} where 1≤ h ≤ p−1, then g pk−(p5) ≡ (fk+1 +fk−1)− (fk−1 +hfk)≡ lk −gk+1 (mod p). (3.2) hence, if ( p 5 ) =−1, then g pk−(p5) +g k−(p5) ≡ lk (mod p). (3.3) 6 int. j. anal. appl. (2023), 21:24 prime p k (mod π(p)) 3 3 (mod 8) 7 (mod 8) 7 5 (mod 16) 13 (mod 16) 13 17 23 13 (mod 48) 37 (mod 48) 37 43 23 (mod 88) 67 (mod 88) table 2. list of p and k where ( p 5 ) =−1 and p|lk−1. theorem 3.2. let p be an odd prime number and {gn}n>0 = {g(a,b)} where a and b are integers that are not divisible by p. if ( p 5 ) =1, then g pk−(p5) ≡ gk−1 (mod p), for all k ∈n. proof. by theorem 1.1, 2.4 and 2.6, it follows that g pk−(p5) = gpk−1 = afpk−3 +bfpk−2 ≡ 2−1a(fkl−3 +f−3lk)+2−1b(fkl−2 +f−2lk) (mod p) ≡ afk−3 +bfk−2 (mod p) ≡ gk−1 (mod p). this completes the proof. � corollary 3.2. if p is an odd prime number such that ( p 5 ) = 1, then g pik−(p5) ≡ gk−1 (mod p), for all i,k ∈n. the following theorem is a direct result of theorems 3.1 and 3.2. theorem 3.3. let {gn}n>0 = {g(a,b)} be such that a,b ∈ z. then gpk−(p5) ≡ gk−1 (mod p) for all positive integers k and odd prime numbers p 6=5 such that p 6 |a, p 6 |b if and only if a ≡ b mod p. next, we consider the case that p =5. firstly, in theorem 3.4, we give a complete characterization of the initial condition of the gibonacci sequence {gn}n>0 = {g(1,h)} where 1≤ h ≤ 4 and the values of k where (1.3) holds. then, we give a relation of such congruent property of the sequences in the same equivalent class in x5. int. j. anal. appl. (2023), 21:24 7 theorem 3.4. let {gn}n>0 = {g(1,h)} be such that 1 ≤ h ≤ 4. if p = 5 and k ∈ n, then g pk−(p5) ≡ gk−1 (mod p) if and only if one of the following holds: (1) {gn}= {g(1,4)} and k ≡ 0 (mod 5) (2) {gn}= {g(1,1)} and k ≡ 1 (mod 5) (3) {gn}= {g(1,2)} and k ≡ 3 (mod 5). proof. since p =5, we have ( p 5 ) =0. by theorem 1.1, we have g pk−(p5) = g5k = f5k−2 +hf5k−1. (3.4) let q,r ∈z be such that k =5q + r, where 0≤ r ≤ 4. by corollary 2.1 and (3.4), g5k ≡ f5k−2 +hf5k−1 (mod 5) ≡ 3(f−2l5k +hf−1l5k) (mod 5) ≡ 3(−l5k +hl5k) (mod 5) ≡ 3lk(−1+h) (mod 5) ≡ 3(−1+h)l5q+r (mod 5) ≡ 3(−1+h)2−1lqlr (mod 5), by theorem 2.4, ≡ (1−h)lqlr (mod 5). similarly gk−1 ≡ fk−3 +hfk−2 (mod 5) ≡ f5q+r−3 +hf5q+r−2 (mod 5) ≡ 2−1 (fr−3l5q +hfr−2l5q) (mod 5) ≡ 3lq(fr−3 +hfr−2) (mod 5). thus g5k ≡ gk−1 (mod 5) if and only if (1−h)lqlr ≡ 3lq(fr−3 +hfr−2) (mod 5). by theorem 2.3, we have (1−h)lr ≡ 3(fr−3 +hfr−2) (mod 5). (3.5) if r = 4, then (3.5) does not hold. by a direct computation 5 6 |(3fr−2 +lr) for all r ∈ {0,1,2,3}, we have h ≡ (3fr−2 +lr)−1(lr −3fr−3) (mod 5). (3.6) if r = 2, then h ≡ 0 (mod 5) contradiction. by a direct computation, the above equation holds if and only if (r,h)∈{(0,4),(1,1),(3,2)}. this completes the proof. � 8 int. j. anal. appl. (2023), 21:24 for {gn}n>0 = {g(a,b)} where a,b ∈z that a and b are not divisible by p, let δ(a)=   0 if a ≡ 1,4 (mod 5) −1 if a ≡ 2 (mod 5) 1 if a ≡ 3 (mod 5). theorem 3.5. let {gn}n>0 = {g(1,h)} and {g′n}n>0 = {g(a,b)}, where 1 ≤ h ≤ 4 and a,b ∈ z be such that {gn}∼{g′n}. then g′k ≡ a −1gk +δ(a)fk−2 (mod 5) for all k ∈n. proof. since {gn} ∼ {g′n}, we have b ≡ g′2g −1 1 ≡ g2(g ′ 1) −1 ≡ ha−1 (mod 5). for the case that k =1,2, we can compute the result directly. for k > 2, we have g′k ≡ afk−2 +bfk−1 (mod 5) ≡ afk−2 +ha−1fk−1 (mod 5). if a ≡ 1 (mod 5) or a ≡ 4 (mod 5), then a ≡ a−1 (mod 5). hence, g′k ≡ a −1gk (mod 5). (3.7) otherwise, g′k ≡  a −1gk −fk−2 if g′1 ≡ 2 (mod 5), a−1gk +fk−2 if g′1 ≡ 3 (mod 5). therefore, we have g′k ≡ a −1gk +δ(a)fk−2 (mod 5) for all positive integer k. � theorem 3.6. let {gn}n>0 = {g(1,h)} where 1 ≤ h ≤ 4 and k be a positive integer satisfying g5k ≡ gk−1 (mod 5). let {g′n}n>0 = {g(a,b)} be such that a and b are integers that are not divisible by 5. if {gn}∼{g′n} in x5, then g′5k ≡ g ′ k−1 (mod 5) if and only if a ≡ a −1 (mod 5). proof. let δ = δ(a). so g′5k −g ′ k−1 ≡ ( a−1g5k +δf5k−2 ) − ( a−1gk−1 +δfk−3 ) (mod 5) ≡ a−1(g5k −gk−1)+δ(f5k−2 −fk−3) (mod 5) ≡ a−1(g5k −gk−1)+δ(3f−2lk −fk−3) (mod 5) ≡ a−1(g5k −gk−1)+δ(3(fk−2 −fk+2)−fk−3) (mod 5) ≡ a−1(g5k −gk−1)+δ(3((fk −fk−1)− (fk+1 +fk))−fk−3) (mod 5) ≡ a−1(g5k −gk−1)+δ(2(fk−1 +fk+1)− (fk−1 −fk−2)) (mod 5) ≡ a−1(g5k −gk−1)+δfk+3 (mod 5). since g5k ≡ gk−1 (mod 5), it follows that g′5k ≡ g ′ k−1 (mod 5) if and only if 5|δfk+3. so δ =0 or k ≡ 2 (mod 5). by theorem 3.4, since g5k ≡ gk−1 (mod 5), we have that k 6≡ 2 (mod 5). hence δ =0 and it follows that a ≡ a−1 (mod 5). so the equation holds if and only if a ≡ a−1 (mod 5). � int. j. anal. appl. (2023), 21:24 9 by theorem 3.4 and 3.6, we have the following corollaries. corollary 3.3. let {gn} = {g(1,h)} where 1 ≤ h ≤ 4. let {g′n} = {g(a,b)} be such that {gn} ∼ {g′n}, where a,b ∈ z and g′1 ≡ 1,4 (mod 5). if (h,r) ∈ {(4,0),(1,1),(2,3)} where k ≡ r (mod 5) and 0≤ r ≤ 4, then we have g′5k ≡ g ′ k−1 (mod 5). corollary 3.4. let {gn} = {g(1,h)} where 1 ≤ h ≤ 4. let {g′n} = {g(a,b)} be such that {gn} ∼ {g′n}, where where a,b ∈ z and g′1 ≡ 2,3 (mod 5). if (h,r) ∈ {(4,0),(1,1),(2,3)} where k ≡ r (mod 5) and 0≤ r ≤ 4, then we have g′5k 6≡ g ′ k−1 (mod 5). acknowledgement: this project is financially supported by faculty of science, mahasarakham university 2017. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. aoki, y. sakai, on divisibility of generalized fibonacci number, integers, 15 (2015), a31. [2] t. koshy, fibonacci and lucas numbers with applications, john wiley & sons, ltd, new jersey, (2017). [3] d. andrica, v. crişan, f. al-thukair, on fibonacci and lucas sequences modulo a prime and primality testing, arab j. math. sci. 24 (2018), 9–15. https://doi.org/10.1016/j.ajmsc.2017.06.002. [4] j.d. fulton, w.l. morris, on arithmetical functions related to the fibonacci numbers, acta arithmetica, 16 (1969), 105–110. [5] s. gupta, p. rockstroh, f.e. su, splitting fields and periods of fibonacci sequences modulo primes, math. mag. 85 (2012), 130-135. https://doi.org/10.4169/math.mag.85.2.130. [6] h. london, fibonacci and lucas numbers, by verner e. hoggatt jr. houghton mifflin company, boston, 1969. canadian math. bull. 12 (1969), 367–367. https://doi.org/10.1017/s0008439500030514. [7] n.j.a. sloan, the on-line encyclopedia of integer sequences, http://oeis.org/a001175. [8] j. sondow, the on-line encyclopedia of integer sequences, https://oeis.org/a237437. https://doi.org/10.1016/j.ajmsc.2017.06.002 https://doi.org/10.4169/math.mag.85.2.130 https://doi.org/10.1017/s0008439500030514 http://oeis.org/a001175 https://oeis.org/a237437 1. introduction 2. preliminaries 3. main results references international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 28-43 http://www.etamaths.com almost periodic solutions for impulsive fractional stochastic evolution equations toufik guendouzi∗ and lamia bousmaha abstract. in this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving caputo fractional derivative. the main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. some known results are improved and generalized. 1. introduction the study of fractional differential equations has been gaining importance in recent years due to the fact that fractional order derivatives provide a tool for the description of memory and hereditary properties of various phenomena. due to this fact, the fractional order models are capable to describe more realistic situation than the integer order models. fractional differential equations have been used in many field like fractals, chaos, electrical engineering, medical science, etc. in recent years, we have seen considerable development on the topics of fractional differential equations, for instance, we refer to the monographs of abbas et al. [2], kilbas et al. [14], miller and ross [18], podlubny [20], and the papers [3, 4, 31]. in particular, differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering and other areas of science. many physical phenomena in evolution processes are modeled as impulsive fractional differential equations and existence results for such equations have been studied by several authors [9, 23, 30]. one of the important problems in the qualitative theory of impulsive differential equations is the existence of almost periodic solutions. at the present time, many results on the existence, uniqueness and stability of these solutions have been obtained (see [1, 15, 24, 26] and the references therein). however, only few papers deal with the existence of almost periodic solutions for impulsive fractional differential equations. recently, debbouche et al. [11] studied the existence of almost periodic and optimal mild solutions of fractional evolution equations with analytic semigroup in a banach space. el-borai et al. [12] established the existence and uniqueness of almost periodic solutions of a class of nonlinear fractional differential equations with analytic semigroup in banach space, and very recently, stamov et 2010 mathematics subject classification. 26a33, 34c27, 34g20, 34a37, 35b15. key words and phrases. square-mean piecewise almost periodic, impulsive fractional stochastic differential equations, analytic semigroup. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 28 impulsive fractional stochastic evolution equations 29 al. [27] studied the existence of almost periodic solutions for fractional differential equations with impulsive effects. in many cases, deterministic models often fluctuate due to environmental noise, which is random or at least appears to be so. therefore, we must move from deterministic problems to stochastic ones. taking the disturbances into account, the theory of differential equations has been generalized to stochastic case. the existence, uniqueness, stability, controllability and other quantitative and qualitative properties of solutions of stochastic evolution equations have recently received a lot of attention (see [13, 17, 28] and the references therein). the existence of almost periodic solutions for stochastic differential equations has been discussed in [5, 7, 21]. the existence of almost periodic solutions for impulsive stochastic evolution equations has been reported in [8, 16]. however, up to now the problem for the existence of almost periodic solutions for impulsive fractional stochastic evolution equations have not been considered in the literature. in order to fill this gap, this paper studies the existence of square-mean piecewise almost periodic solutions of the following impulsive fractional stochastic differential equations in the form (1) cdαt x(t) + ax(t) = f(t,x(t)) + σ(t,x(t)) dw(t) dt + ∞∑ k=−∞ gk(x(t))δ(t−τk), t ∈ ir, where the state x(·) takes values in the space l2(ip,h), h is a separable real hilbert space with inner product (·, ·) and norm ‖·‖; the fractional derivative cdα, α ∈ (0, 1), is understood in the caputo sense; −a : d(a) ⊂ l2(ip,h) → l2(ip,h) is the infinitesimal generator of an analytic semigroup of a bounded linear operator s(t), t ≥ 0, on l2(ip,h) satisfying the exponential stability; {w(t) : t ≥ 0} is a given k-valued wiener process with a finite trace nuclear covariance operator q ≥ 0 defined on a filtered complete probability space (ω,f,{ft}t≥0,ip), k is another separable hilbert space with inner product (·, ·)k and norm ‖ · ‖k; gk : d(gk) ⊂ l2(ip,h) → l2(ip,h) are continuous impulsive operators, δ(·) is dirac’s delta-function, f(t,x) : ir ×l2(ip,h) → l2(ip,h) and σ(t,x) : ir ×l2(ip,h) → l2(ip,l02(k,h)) are jointly continuous functions (here, l02(k,h) denotes the space of all q-hilbert-schmidt operators from k into h, which is going to be defined below). the structure of this paper is as follows. in sect. 2, we will recall briefly some preliminaries fact which will be used in paper. section 3, we establish criteria of the existence of an almost periodic solution and its exponential stability. 2. preliminaries in this section, we introduce some basic definitions, notation and lemmas which are used throughout this paper. let (h,‖ · ‖h) and (k,‖ · ‖k) be two real separable hilbert spaces, and we denote by l(k,h) the set of all linear bounded operators from k into h, equipped with the usual operator norm ‖ · ‖. we will use the symbol ‖ · ‖ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. let (ω,f,ip) be a complete probability space equipped with a normal filtration {ft}t≥0 satisfying the usual conditions (i.e., right continuous and f0 containing all ip-null sets). let {ei}∞i=1 be a complete orthonormal basis of k. suppose that w = (wt)t≥0 is a 30 guendouzi and bousmaha cylindrical k-valued wiener process with a finite trace nuclear covariance operator q ≥ 0, denote tr(q) = ∑∞ i=1 λ̃i = λ̃ < ∞, which satisfies qei = λ̃iei. so, actually, w(t) = ∑∞ i=1 √ λ̃iwi(t)ei, where {wi(t)}∞i=1 are mutually independent onedimensional standard wiener processes. we assume that ft = σ{w(s) : 0 ≤ s ≤ t} is the σ-algebra generated by w. let l02 = l2(q 1 2k,h) be the space of all hilbertschmidt operators from q 1 2k to h with the inner product (ϕ,φ)l02 = tr[ϕqφ ∗]. for more details, we refer the reader to da prato and zabczyk [10]. the collection of all measurable, square integrable, h-valued random variables, denoted by l2(ip,h) is a banach space equipped with norm ‖x(·)‖l2 = (ie‖x(·)‖2) 1 2 , where ie(·) denotes the expectation with respect to the measure ip. let c(ir,l2(ip,h)) be the banach space of all continuous maps from ir into l2(ip,h) satisfying the condition supt∈ir ie‖x(t)‖2 < ∞. let l2f0 (ip,h) denote the family of all f0measurable, h-valued random variables x(0). let us now recall some basic definitions and results of fractional calculus. for more details see [14, 18, 20]. definition 2.1. the fractional integral of order α with the lower limit zero for a function f is defined as iαf(t) = 1 γ(α) ∫ t 0 f(s) (t−s)1−α ds, t > 0,α > 0, provided the right-hand side is pointwise defined on [0,∞), where γ(·) is the gamma function, which is defined by γ(α) = ∫∞ 0 tα−1e−tdt. definition 2.2. the riemann-liouville fractional derivative of order α > 0, n− 1 < α < n, n ∈ in, is defined as (r−l)dα0+f(t) = 1 γ(n−α) ( d dt )n ∫ t 0 (t−s)n−α−1f(s)ds, where the function f(t) has absolutely continuous derivative up to order (n− 1). definition 2.3. the caputo derivative of order α > 0 for a function f : [0,∞) → ir can be written as dαf(t) = dα ( f(t) − n−1∑ k=0 tk k! f(k)(0) ) , t > 0,n− 1 < α < n. remark 2.4. (i) if f(t) ∈cn[0,∞), then cdαf(t) = 1 γ(n−α) ∫ t 0 f(n)(s) (t−s)α+1−n ds = in−αf(n)(t), t > 0,n− 1 < α < n. (ii) the caputo derivative of a constant is equal to zero. (iii) if f is an abstract function with values in h, then integrals which appear in definitions 2.1 and 2.2 are taken in bochners sense. let b = {{τk} : τk ∈ ir, τk < τk+1, k ∈ z} be the set of all sequences unbounded and strictly increasing. we consider the impulsive fractional differential equation (1), and denote by x(t) = x(t; t0,x0), t0 ∈ ir, x0 ∈h, the solution of (1) with the initial condition (2) x(t0) = x0. impulsive fractional stochastic evolution equations 31 definition 2.5 ([16]). a stochastic process x : ir → l2(ip,h), is said to be stochastically bounded if there exists n > 0 such that ie‖x(t)‖2 ≤ n for all t ∈ ir. definition 2.6 ([16]). a stochastic process x : ir → l2(ip,h), is said to be stochastically continuous in s ∈ ir, if limt→s ie‖x(t) −x(s)‖2 = 0. for {τk} ∈ b and k ∈ z, let pc(ir,l2(ip,h)) be the space consisting of all stochastically bounded functions φ : ir → l2(ip,h) such that φ(·) is stochastically continuous at t for any t /∈ {τk}, τk ∈ ir, k ∈ z and φ(τ−k ) = φ(τk). in particular, we introduce the space pc(ir × l2(ip,h),l2(ip,h)) formed by all piecewise stochastically continuous stochastic processes φ : ir ×l2(ip,h) → l2(ip,h) such that for any x ∈ l2(ip,h), φ(·,x) is stochastically continuous at t for any t /∈{τk} and φ(τ−k ,x) = φ(τk,x) for all k ∈ z, and for any t ∈ ir, φ(t, ·) is stochastically continuous at x ∈ l2(ip,h). remark 2.7 ([16, 29]). the solution x(t) = x(t; t0,x0) of the problem (1)-(2) is a piecewise stochastically continuous, ft-adapted measurable process with points of discontinuity at the moments τk, k ∈ z, at which it is continuous from the left. definition 2.8 ([26]). the set of sequences {τjk}, τ j k = τk+j − τk, k ∈ z, j ∈ z, {τk} ∈ b is said to be equipotentially almost periodic, if for arbitrary � > 0 there exists a relatively dense set b� of ir such that for each κ ∈ b� there is an integer q ∈ z such that |τk+q − τk −κ| < � for all k ∈ z. definition 2.9 ([7]). a stochastic process x ∈ pc(ir,l2(ip,h)) is said to be square-mean picewise almost periodic, if: (i) the set of sequences {τjk}, τ j k = τk+j − τk, k ∈ z, j ∈ z, {τk} ∈ b is equipotentially almost periodic. (ii) for any � > 0, there exists a real number δ > 0 such that if the points t′ and t′′ belong to one and the same interval of continuity of x(t) and satisfy the inequality |t′ − t′′| < δ, then ie‖x(t′) −x(t′′)‖2h < �. (iii) for any � > 0, there exists a relatively dense set t such that if τ ∈ t , then ie‖x(t + τ) −x(t)‖2h < �, satisfying the condition |t−τk| > �, k ∈ z. the elements of t are called �-translation number of x. we denote by ap(ir,l2(ip,h)) the collection of all the square-mean piecewise almost periodic processes, it thus is a banach space with the norm ‖x‖∞ = supt∈ir ‖x(t)‖l2 = supt∈ir(ie‖x(t)‖2) 1 2 for x ∈ap(ir,l2(ip,h)). lemma 2.10 ([16]). let f ∈ap(ir,l2(ip,h)). then, r(f), the range of f is a relatively compact set of l2(ip,h). definition 2.11 ([8]). for {τk} ∈ b, k ∈ z, the function f(t,x) ∈ pc(ir × l2(ip,h),l2(ip,h)) is said to be square-mean piecewise almost periodic in t ∈ ir and uniform on compact subset of l2(ip,h)) if for every � > 0 and every compact subset k ⊆ l2(ip,h)), there exists a relatively dense subset t of ir such that ie‖f(t + τ,x) −f(t,x)‖2 < �, for all x ∈ k, τ ∈ t , t ∈ ir satisfying |t − τk| > �, k ∈ z. the collection of all such processes is denoted ap(ir×l2(ip,h),l2(ip,h)). lemma 2.12 ([16]). suppose that f(t,x) ∈ ap(ir × l2(ip,h),l2(ip,h)) and f(t, ·) is uniformly continuous on each compact subset k ⊆ l2(ip,h) uniformly 32 guendouzi and bousmaha for t ∈ ir. that is, for all � > 0, there exists δ > 0 such that x,y ∈ k and ie‖x−y‖2 < δ implies that ie‖f(t,x)−f(t,y)‖2 < � for all t ∈ ir. then f(·,x(·)) ∈ ap(ir,l2(ip,h)) for any x ∈ap(ir,l2(ip,h)). we obtain the following corollary as an immediate consequence of lemma 2.12. corollary 2.13. let f(t,x) ∈ ap(ir ×l2(ip,h),l2(ip,h)) and f is lipschitz, i.e., there is a number c > 0 such that ie‖f(t,x) −f(t,y)‖2 ≤ cie‖x−y‖2, for all t ∈ ir and x,y ∈ l2(ip,h), if for any x ∈ap(ir,l2(ip,h)), then f(·,x(·)) ∈ ap(ir,l2(ip,h)). definition 2.14. a sequence x : z → l2(ip,h) is called a square-mean almost periodic sequence if �-translation set of x i(x; �) = {τ ∈ z : ie‖x(n + τ) −x(t)‖2 < �, for all n ∈ z} is a relatively dense set in z for all � > 0. the collection of all square-mean almost periodic sequences x : z → l2(ip,h) will be denoted by ap(z,l2(ip,h)). remark 2.15. if x(n) ∈ ap(z,l2(ip,h)), then {x(n) : n ∈ z} is stochastically bounded. lemma 2.16 ([16]). assume that f ∈ ap(ir,l2(ip,h)), the sequence {xk : k ∈ z} is almost periodic in l2(ip,h) and {τjk}, j ∈ z, is equipotentially almost periodic. then for each � > 0 there are relatively dense sets t�,f,xk of ir and t̂�,f,xk of z such that the following conditions hold: (i) ie‖f(t + τ)−f(t)‖2 < � for all t ∈ ir, |t−τk| > �, τ ∈ t�,f,xk and k ∈ z. (ii) ie‖xk+q −xk‖2 < � for all q ∈ t̂�,f,xk and k ∈ z. (iii) for every τ ∈ t�,f,xk , there exists at least one number q ∈ t̂�,f,xk such that |τqk − τ| < �, k ∈ z. consider the linear fractional impulsive stochastic differential equation corresponding to (1) (3) cdαt x(t) + ax(t) = f(t) + σ(t) dw(t) dt + ∞∑ k=−∞ gkδ(t− τk), where f ∈pc(ir,l2(ip,h)), σ ∈pc(ir,l2(ip,l02)) and gk : d(gk) ⊂ l2(ip,h) → l2(ip,h). let us introduce the following conditions. (c1) the set of sequences {τjk}, τ j k = τk+j − τk, k ∈ z, j ∈ z, {τk} ∈ b is equipotentially almost periodic and there exists θ > 0 such that infk τ 1 k = θ. (c2) the function f is in ap(ir,l2(ip,h)) and locally hölder continuous with points of discontinuity at the moments τk, k ∈ z at which it is continuous from the left. (c3) the function σ is in ap(ir,l2(ip,l02)) and locally hölder continuous with points of discontinuity at the moments τk, k ∈ z at which it is continuous from the left. impulsive fractional stochastic evolution equations 33 (c4) {gk}, k ∈ z, of impulsive operators is a square-mean almost periodic sequence. lemma 2.17 ([15],[16]). let the condition (c1) holds. then (i) there exists a constant p > 0 such that, for every t ∈ ir lim t→∞ ι(t,t + t) t = p. (ii) for each p > 0 there exists a positive integer n such that each interval of length p has no more than n elements of the sequence {τk}, that is, ι(s,t) ≤ n(t−s) + n, where ι(s,t) is the number of points τk in the interval (s,t). the following lemma is an immediate consequence of lemma 2.16. lemma 2.18. let the conditions (c1)-(c4) hold. then, for each � > 0 there are relatively dense sets t�,f,σ,gk of ir and t̂�,f,σ,gk of z such that the following relations hold: (i) ie‖f(t + τ) −f(t)‖2 < �, t ∈ ir, τ ∈ t�,f,σ,gk , |t− τk| > �, k ∈ z. (ii) ie‖σ(t + τ) −σ(t)‖2 < �, t ∈ ir, τ ∈ t�,f,σ,gk, |t− τk| > �, k ∈ z. (iii) ie‖gk+q −gk‖2 < �, k ∈ z, q ∈ t̂�,f,σ,gk. (iv) for each τ ∈ t�,f,σ,gk, ∃q ∈ t̂�,f,σ,gk , such that |τk+q − τk − τ| < �, k ∈ z. now, we present the definition of mild solutions for the problem (2)-(3) based on the paper [25]. definition 2.19. a stochastic process x ∈ pc(j,l2(ip,h)), j ⊂ ir is called a mild solution of the problem (2)-(3) if (i) x0 ∈ l2f0 (ip,h); (ii) x(t) ∈ l2(ip,h) has càdlàg paths on t ∈ j a.s., and it satisfies the following integral equation (4) x(t) =   t (t− t0)x0 + ∫ t t0 (t−s)α−1s(t−s)f(s)ds + ∫ t t0 (t−s)α−1s(t−s)σ(s)dw(s), t ∈ [t0,τ1], t (t− t0)x0 + t (t− τ1)g1 + ∫ t t0 (t−s)α−1s(t−s)f(s)ds + ∫ t t0 (t−s)α−1s(t−s)σ(s)dw(s), t ∈ (τ1,τ2], ... t (t− t0)x0 + ∑ t0<τk<t t (t− τk)gk + ∫ t t0 (t−s)α−1s(t−s)f(s)ds + ∫ t t0 (t−s)α−1s(t−s)σ(s)dw(s), t ∈ (τk,τk+1], where t (t) = ∫ ∞ 0 ξα(θ)s(t αθ)dθ, s(t) = α ∫ ∞ 0 θξα(θ)s(t αθ)dθ, 34 guendouzi and bousmaha and for θ ∈ (0,∞) ξα(θ) = 1 α θ−1− 1 α $α ( θ− 1 α ) ≥ 0, $α(θ) = 1 π ∞∑ n=1 (−1)n−1θ−nα−1 γ(nα + 1) n! sin(nπα), ξα is a probability density function defined on (0,∞), that is ξα(θ) ≥ 0, θ ∈ (0,∞), and ∫ ∞ 0 ξα(θ)dθ = 1. remark 2.20. ∫ ∞ 0 θνξα(θ)dθ = ∫ ∞ 0 θ−αν$α(θ)dθ = γ(1 + ν) γ(1 + αν) . in this paper, we will also assume that ξ2α ∈ l1((0,∞)). let the operator −a in (1) and (3) be an infinitesimal generator of an analytic semigroup s(t) in l2(ip,h) and 0 ∈ ρ(a), the resolvent set of a. for any β > 0, we define the fractional power a−β of the operator a by a−β = 1 γ(β) ∫ ∞ 0 tβ−1s(t)dt, where a−β is bounded, bijective and aβ = (a−β)−1, β > 0 a closed linear operator on its domain d(aβ) and such that d(aβ) = r(a−β) where r(a−β) is the range of a−β. furthermore, the subspace d(aβ) is dense in l2(ip,h) and the expression ‖x‖β = ‖aβx‖, x ∈d(aβ), defines a norm on l2(ip,hβ) := d(aβ). the following properties are well known. lemma 2.21 ([19]). suppose that the preceding conditions are satisfied. then (i) s(t) : l2(ip,h) →d(aβ) for every t > 0 and β ≥ 0. (ii) for every x ∈d(aβ), the following equality s(t)aβx = aβs(t)x holds. (iii) for every t > 0, the operator aβs(t) is bounded and ‖aβs(t)‖≤ kβt−βe−λt, kβ > 0,λ > 0. (iv) for 0 < β ≤ 1 and x ∈d(aβ), we have ‖s(t)x−x‖≤ cβtβ‖aβx‖, cβ > 0. when −a generates a semi-group with negative exponent, we deduce that if x(t) is a bounded solution of (3) on ir, then we take the limit as t0 → −∞ and using (4), we obtain (see [6]) (5) x(t) = ∫ t −∞ (t−s)α−1s(t−s)f(s)ds+ ∫ t −∞ (t−s)α−1s(t−s)σ(s)dw(s)+ ∑ τk<t t (t−τk)gk. 3. main results in this section, we present and prove our main theorems. theorem 3.1. assume the conditions (c1)-(c4) are satisfied and −a is the infinitesimal generator of an analytic semi-group s(t), then system (2)-(3) has a square-mean piecewise almost periodic mild solution. impulsive fractional stochastic evolution equations 35 proof. first, we shall show that the right-hand side of (5) is well defined. from conditions (c2)-(c4), it follows that f(t), σ(t) and {gk} are stochastically bounded, and let max { ie‖f(t)‖2pc,ie‖σ(t)‖ 2 pc,ie‖gk‖ 2 l2(ip,h) } ≤ n0, n0 > 0. in view of lemma 2.21 and the definition of the norm in hβ, we obtain (6) ie‖x(t)‖2β = ie‖a βx(t)‖2 ≤ 3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)f(s)ds ∥∥∥2 +3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)σ(s)dw(s) ∥∥∥2 + 3ie∥∥∥ ∑ τk<t aβt (t− τk)gk ∥∥∥2 ≤ 3α2ie ∥∥∥∫ t −∞ ∫ ∞ 0 θ(t−s)α−1ξα(θ)aβs((t−s)αθ)f(s)dθds ∥∥∥2 +3α2ie ∥∥∥∫ t −∞ ∫ ∞ 0 θ(t−s)α−1ξα(θ)aβs((t−s)αθ)σ(s)dθdw(s) ∥∥∥2 +3 ∑ τk<t ie ∥∥∥∫ ∞ 0 ξα(θ)a βs((t− τk)αθ)gkdθ ∥∥∥2 ≤ 3α2ie [∫ t −∞ ∫ ∞ 0 ‖θ(t−s)α−1ξα(θ)aβs((t−s)αθ)f(s)‖dθds ]2 +3α2tr(q)ie [∫ t −∞ ∫ ∞ 0 ‖θ(t−s)α−1ξα(θ)aβs((t−s)αθ)σ(s)‖2l02dθds ] +3 ∑ τk<t ie [∫ ∞ 0 ‖ξα(θ)aβs((t− τk)αθ)gk‖dθ ]2 ≤ 3α2k2β ∫ ∞ 0 ξα(θ) ∫ t −∞ θ1−β(t−s)−αβ+α−1e−λθ(t−s) α dsdθ × ∫ ∞ 0 ξα(θ) ∫ t −∞ θ1−β(t−s)−αβ+α−1e−λθ(t−s) α ie‖f(s)‖2dsdθ +3α2k2βtr(q) ∫ t −∞ ∫ ∞ 0 θ2(1−β)ξ2α(θ)(t−s) 2(α−αβ−1)e−2λθ(t−s) α ie‖σ(s)‖2l02dθds +3k2β ∑ τk<t ie [∫ ∞ 0 θ−βξα(θ)(t− τk)−αβe−λθ(t−τk) α ‖gk‖dθ ]2 . we have, for η = t−s, (7) ie‖x(t)‖2β ≤ 3α 2k2βn0 ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ × ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ +3α2k2βn0tr(q) ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α dηdθ +3k2βn0r(θ), 36 guendouzi and bousmaha where r(θ) = (∫ ∞ 0 ξα(θ) [ ∑ 0<t−τk≤1 (θ(t− τk)α)−βe−λθ(t−τk) α + ∞∑ j=1 ∑ j<t−τk≤j+1 (θ(t− τk)α)−βe−λθ(t−τk) α ] dθ )2 . by a standard calculation, we can deduce that (8) ( α ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ )2 = ( 1 λ1−β ∫ ∞ 0 ξα(θ) ∫ ∞ 0 (λθηα)−βe−λθη α dλθηαdθ )2 = γ2(1 −β) λ2(1−β) . since ξ2α ∈ l1((0,∞)), we further derive that (9) α2 ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α dηdθ ≤ n1 γ(1 − 2β) λ2−2β , where n1 = supθ≥0 ξ 2 α(θ). by the help of (c1) and lemma 2.17, we have (10) r(θ) ≤ (∫ ∞ 0 ξα(θ) ( 2n n β 2 + 2n eλ − 1 ) dθ )2 = 4n2 ( 1 n β 2 + 1 eλ − 1 )2 , n2 = min{θ(t− τk)α, 0 < t− τk ≤ 1}. recalling (7), from (8)-(10), we obtain (11) ie‖x(t)‖2α ≤ 3k 2 βn0 [ γ2(1 −β) λ2(1−β) + tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2] , and x(t) ∈pc(ir,l2(ip,h)). let � > 0, τ ∈ t�,f,σ,gk and q ∈ t̂�,f,σ,gk, k ∈ z, where the sets t�,f,σ,gk and t̂�,f,σ,gk are defined as in lemma 2.18. we have impulsive fractional stochastic evolution equations 37 x(t + τ) −x(t) = (∫ t+τ −∞ (t + τ −s)α−1s(t + τ −s)f(s)ds + ∫ t+τ −∞ (t + τ −s)α−1s(t + τ −s)σ(s)dw(s) + ∑ τk<t t (t + τ − τk)gk ) − (∫ t −∞ (t−s)α−1s(t−s)f(s)ds + ∫ t −∞ (t−s)α−1s(t−s)σ(s)dw(s) + ∑ τk<t t (t− τk)gk ) = ∫ t −∞ (t−s)α−1s(t−s)[f(s + τ) −f(s)]ds + ∫ t −∞ (t−s)α−1s(t−s)[σ(s + τ) −σ(s)]dw̃(s) + ∑ τk<t t (t− τk)[gk+q −gk], where w̃(s) = w(s + τ) −w(τ) is also a brownian motion and has the same distribution as w. then (12) ie‖x(t + τ) −x(t)‖2β = ie‖a β(x(t + τ) −x(t))‖2 ≤ 3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[f(s + τ) −f(s)]ds ∥∥∥2 +3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[σ(s + τ) −σ(s)]dw̃(s) ∥∥∥2 +3ie ∥∥∥ ∑ τk<t t (t− τk)[gk+q −gk] ∥∥∥2 ≤ mβ�, where |t− τk| > � and mβ = kβ [ γ2(1 −β) λ2(1−β) + tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2] . the last inequality implies that x(t) is a square-mean piecewise almost periodic process, so system (2)-(3) has a square-mean piecewise almost periodic solution. the proof is complete. � in order to obtain the existence of square-mean piecewise almost periodic solution to system (1)-(2), we introduce the following conditions: (c5) −a : d(a) ⊆ l2(ip,h) → l2(ip,h) is the infinitesimal generator of an exponentially stable analytic semi-group s(t), t ∈ ir, on l2(ip,h). (c6) f(t,x) ∈ ap(ir ×l2(ip,hβ),l2(ip,h)) with respect to t ∈ ir uniformly in x ∈ k, for each compact set k ⊆ l2(ip,h), and there exist constants c̃ > 0, 0 < κ < 1, 0 < β < 1, such that ie‖f(t1,x1) −f(t2,x2)‖2 ≤ c̃ ( |t1 − t2|κ + ie‖x1 −x2‖2β ) , 38 guendouzi and bousmaha where (ti,xi) ∈ ir×l2(ip,hβ), i = 1, 2. (c7) σ(t,x) ∈ap(ir ×l2(ip,hβ),l2(ip,l02)) with respect to t ∈ ir uniformly in x ∈ k, for each compact set k ⊆ l2(ip,h), and there exist constants ĉ > 0, 0 < κ < 1, 0 < β < 1, such that ie‖σ(t1,x1) − σ(t2,x2)‖2l02 ≤ ĉ ( |t1 − t2|κ + ie‖x1 −x2‖2β ) , (ti,xi) ∈ ir×l2(ip,hβ), i = 1, 2. (c8) the sequence {gk(x)} is almost periodic in k ∈ z uniformly in x ∈ k ⊆ l2(ip,h), and there exist constants c̄ > 0, 0 < β < 1, such that ie‖gk(x1) −gk(x2)‖2 ≤ c̄ie‖x1 −x2‖2β, where x1,x2 ∈ l2(ip,hβ). theorem 3.2. assume that the conditions (c1), (c5)-(c8) are satisfied, then the impulsive fractional stochastic system (1)-(2) admits a unique square-mean piecewise almost periodic mild solution. proof. let b the set of all x ∈ ap(ir,l2(ip,h)) with discontinuities of the first type at the points τk, k ∈ z, {τk} ∈ b, satisfying the inequality ie‖x‖2 ≤ r, r > 0. obviously, b is a closed set of ap(ir,l2(p,h)). define the operator θ in b by (13) θx(t) = ∫ t −∞ (t−s)α−1aβs(t−s)f(s,a−βx(s))ds + ∫ t −∞ (t−s)α−1aβs(t−s)σ(s,a−βx(s))dw(s) + ∑ τk<t aβt (t− τk)gk(a−βx(τk)). proceeding in the same way as in the proof of theorem 3.1, from conditions (c6)(c8) and lemma 2.2 in [8], we can show that θ is well defined and θx(t) ∈ pc(ir,l2(ip,h)). first, we shall show that θx(t) ∈ b. we define θ1x(t) = ∫ t −∞ (t−s)α−1aβs(t−s)f(s,a−βx(s))ds θ2x(t) = ∫ t −∞ (t−s)α−1aβs(t−s)σ(s,a−βx(s))dw(s). let us show that θ1x ∈ b, let x ∈ b. using condition (c6), since aβ is closed and f(t,x) ∈ap(ir×l2(ip,hβ),l2(ip,h)), we have from corollary 2.13 that a−βx ∈ b and f(·,a−βx(·)) ∈ap(ir,l2(ip,h)). therefore, it follows from definition 2.9 and lemma 2.16 that for any � > 0, there exists a relatively dense set t such that for τ ∈ t the following property ie‖f(t + τ,a−βx(t + τ)) −f(t,a−βx(t))‖2 < �λ2(1−β) k2βγ 2(1 −β) impulsive fractional stochastic evolution equations 39 hold, satisfying the condition |t− τk| > �, for each t ∈ ir and k ∈ z. by virtue of lemma 2.21, we have ie‖θ1x(t + τ) − θ1x(t)‖2 = ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[f(s + τ,a−βx(s + τ)) −f(s,a−βx(s))]ds ∥∥∥2 ≤ α2k2β ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ ∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α ×ie‖f(t + τ −η,a−βx(t + τ −η)) −f(t−η,a−βx(t−η))‖2dηdθ ≤ α2k2β (∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ )2 ×sup t∈r ie‖f(t + τ,a−βx(t + τ)) −f(t,a−βx(t))‖2 = k2β γ2(1 −β) λ2(1−β) sup t∈r ie‖f(t + τ,a−βx(t + τ)) −f(t,a−βx(t))‖2 < k2β γ2(1 −β) λ2(1−β) × �λ2(1−β) k2βγ 2(1 −β) = �. hence, θ1x(·) ∈ b. similarly, by using condition (c7), since aβ is closed and σ(t,x) ∈ ap(ir × l2(ip,hβ),l2(ip,l02)), we have from corollary 2.13 that a−βx ∈ b and σ(·,a−βx(·)) ∈ ap(ir,l2(ip,l02)). therefore, it follows from definition 2.9 and lemma 2.16 that for any � > 0, there exists a relatively dense set t such that for τ ∈ t the following property ie‖σ(t + τ,a−βx(t + τ)) − σ(t,a−βx(t))‖2l02 < �λ2−2β n1k 2 βtr(q)γ(1 − 2β) hold, satisfying the condition |t− τk| > �, for each t ∈ ir and k ∈ z. by virtue of lemma 2.21, for w̃(t) := w(t + τ) −w(τ), we have ie‖θ2x(t + τ) − θ2x(t)‖2 = ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[σ(s + τ,a−βx(s + τ)) − σ(s,a−βx(s))]dw̃(s) ∥∥∥2 ≤ α2k2βtr(q) ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α ×ie‖σ(t + τ −η,a−βx(t + τ −η)) − σ(t−η,a−βx(t−η))‖2 l02 dηdθ ≤ α2k2βtr(q) ∫ ∞ 0 ξ2α(θ) ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α dηdθ ×sup t∈r ie‖σ(t + τ,a−βx(t + τ)) − σ(t,a−βx(t))‖2l02 ≤ k2βtr(q)n1 γ(1 − 2β) λ2−2β sup t∈r ie‖σ(t + τ,a−βx(t + τ)) − σ(t,a−βx(t))‖2l02 < k2βtr(q)n1 γ(1 − 2β) λ2−2β × �λ2−2β n1k 2 βtr(q)γ(1 − 2β) = �. thus, θ2x(·) ∈ b. and in view of the above, it is clear that θ maps b into itself. next, we show that θ is a contracting operator on b. let x1,x2 ∈ b. then , 40 guendouzi and bousmaha we have ie‖θx1(t) − θx2(t)‖2 ≤ 3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[f(s,a−βx1(s)) −f(s,a−βx2(s))]ds ∥∥∥2 +3ie ∥∥∥∫ t −∞ (t−s)α−1aβs(t−s)[σ(s,a−βx1(s)) − σ(s,a−βx2(s))]dw(s) ∥∥∥2 +3ie ∥∥∥ ∑ τk<t aβt (t− τk)[gk(a−βx1(τk)) −gk(a−βx2(τk))] ∥∥∥2 ≤ 3k2βc∗ [ α2 (∫ ∞ 0 ξα(θ) ∫ ∞ 0 θ1−βη−αβ+α−1e−λθη α dηdθ )2 + α2tr(q) ∫ ∞ 0 ξ2α(θ) × ∫ ∞ 0 θ2(1−β)η2(α−αβ−1)e−2λθη α dηdθ + r(θ) ] sup t∈ir ie‖x1(t) −x2(t)‖2, where c∗ = max{ĉ, c̃, c̄} > 0 is sufficiently small and r(θ) is defined as in above. by following similar arguments like those used in (7), we have ie‖θx1(t) − θx2(t)‖2 ≤ 3c∗k2β [ γ2(1 −β) λ2(1−β) +tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2] sup t∈ir ie‖x1(t) −x2(t)‖2. therefore, if c∗ is chosen in the form c∗ ≤ ( 3k2β [ γ2(1 −β) λ2(1−β) + tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2])−1 , we have ie‖θx1(t) − θx2(t)‖2 ≤ 3c∗k2β [ γ2(1 −β) λ2(1−β) + tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2] ‖x1 −x2‖2∞, implies that, ‖θx1 − θx2‖∞ ≤ √ λ‖x1 −x2‖∞, λ = 3c∗k 2 β [ γ2(1 −β) λ2(1−β) + tr(q)n1 γ2(1 − 2β) λ2(1−β) + 4n2 ( 1 n β 2 + 1 eβ − 1 )2] . thus, θ is a contracting operator on b. so by the contraction principle, we conclude that there exists a unique fixed point x for θ in b, such that x = θx, that is (14) x(t) = ∫ t −∞ (t−s)α−1aβs(t−s)f(s,a−βx(s))ds + ∫ t −∞ (t−s)α−1aβs(t−s)σ(s,a−βx(s))dw(s) + ∑ τk<t aβt (t− τk)gk(a−βx(τk)), for all t ∈ ir. now, from conditions (c6)-(c8) and since aβ is closed, gk ∈ap(z,l2(ip,h))), impulsive fractional stochastic evolution equations 41 f(t,x) ∈ap(ir×l2(ip,hβ),l2(ip,h)) and σ(t,x) ∈ap(ir×l2(ip,hβ),l2(ip,l02)), we have from corollary 2.13 that f(·,a−βx(·)) ∈ap(ir,l2(ip,h)), σ(·,a−βx(·)) ∈ ap(ir,l2(ip,l02)) and gk(a−βx(·)) is square-mean almost periodic sequence. therefore, by lemma 2.2 in [8] and remark 2.15, it follows that f(·,a−βx(·)), σ(·,a−βx(·)) and gk(a −βx(·)) are stochastically bounded, and ie‖f(t,a−βx(t))‖2, ie‖σ(t,a−βx(t))‖2 l02 are uniformly continuous in t. we also get a−βx(t) = ∫ t −∞ (t−s)α−1s(t−s)f(s,a−βx(s))ds + ∫ t −∞ (t−s)α−1s(t−s)σ(s,a−βx(s))dw(s) + ∑ τk<t t (t− τk)gk(a−βx(τk)) with a−βx is stochastically bounded in the sense that for r > 0, for each t ∈ ir, ie‖a−βx(t)‖2 ≤ r. hence, a−βx ∈ b is mild solution of the problem (1)-(2). � theorem 3.3. assume that the conditions (c1), (c5)-(c8) are satisfied, then the impulsive fractional stochastic system (1)-(2) has an exponentially stable almost periodic solution. proof. let u(t) be the solution of the following integral equation (15) u(t) = ∫ t −∞ (t−s)α−1aβs(t−s)f(s,a−βu(s))ds + ∫ t −∞ (t−s)α−1aβs(t−s)σ(s,a−βu(s))dw(s) + ∑ τk<t aβt (t− τk)gk(a−βu(τk)). consider the equation (16) cdαt x(t)+ax = f(t,a −βu(t))+σ(t,a−βu(t)) dw(t) dt + ∑ t0<τk gk(a −βu(τk))δ(t−τk), t ∈ ir. in view of theorem 3.2, it follows that there exists a unique square-mean piecewise almost periodic mild solution in the form (17) ψ(t) = ∫ t −∞ (t−s)α−1s(t−s)f(s,a−βu(s))ds + ∫ t −∞ (t−s)α−1s(t−s)σ(s,a−βu(s))dw(s) + ∑ τk<t t (t− τk)gk(a−βu(τk)). then (18) aβψ(t) = ∫ t −∞ aβ(t−s)α−1s(t−s)f(s,a−βu(s))ds + ∫ t −∞ aβ(t−s)α−1s(t−s)σ(s,a−βu(s))dw(s) + ∑ τk<t aβt (t− τk)gk(a−βu(τk)) = u(t). the last equality shows that ψ(t) = a−βu(t) is a solution of (1)-(2), and the uniqueness follows from the uniqueness of the solution of (15) from (14). 42 guendouzi and bousmaha let u(t) = u(t; t0,u0) and v(t) = v(t; t0,v0) be two solutions of equation (1), then (19) u(t) = ∫ t −∞ (t−s)α−1s(t−s)f(s,a−βu(s))ds + ∫ t −∞ (t−s)α−1s(t−s)σ(s,a−βu(s))dw(s) + ∑ τk<t t (t− τk)gk(a−βu(τk)), (20) v(t) = ∫ t −∞ (t−s)α−1s(t−s)f(s,a−βv(s))ds + ∫ t −∞ (t−s)α−1s(t−s)σ(s,a−βv(s))dw(s) + ∑ τk<t t (t− τk)gk(a−βv(τk)), and z(t) = u(t) −v(t) is in ap(ir,l2(ip,h)), (21) z = t (t− t0)z(t0). the proof follows from (21), the estimates from lemma 2.21 and the fact that ι(t0 − t) −p(t− t0) = o(t) for t →∞. � references [1] j.o. alzabut, j.j. nieto, g.t. stamov, existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, bound value probl., 2009 (2009), article id 127510. [2] s. abbas, m. benchohra, g.m. n’guérékata, topics in fractional differential equations. developments in mathematics, springer, new york (2012). [3] r.p. agarwal, b. andrade, g. siracusa, on fractional integro-differential equations with state-dependent delay, comput. math. appl., 62 (2011) 1143-1149. [4] b. andrade,j.p.c. santos, existence of solutions for a fractional neutral integro-differential equation with unbounded delay, electron. j. differ. equ., 2012 (2012) 1-13. [5] p. bezandry, existence of almost periodic solutions to some functional integro-differential stochastic evolution equations, statist. probab. lett., 78 (2008) 2844-2849. [6] p. bezandry, t. diagana, existence of almost periodic solutions to some stochastic differential equations, applicable anal.,86 (7) (2007) 819-827. [7] j. cao, q. yang, z. huang, on almost periodic mild solutions for stochastic functional differential equations, nonlinear anal. rwa., 13 (2012) 275-286. [8] y.k. chang, r. ma, z.h. zhao, almost periodic solutions to a stochastic differential equation in hilbert spaces, results in math., 63, 1-2 (2013) 435-449. [9] j. dabas, a. chauhan, m. kumar, existence of the mild solutions for impulsive fractional equations with infinite delay, int. j. differ. equ. 2011 (2011), article id 793023. [10] g. da prato, j. zabczyk, stochastic eqnarrays in infinite dimensions, cambridge university press, cambridge (1992). [11] a. debbouche, m.m. el-borai, weak almost periodic and optimal mild solutions of fractional evolution equations, electron. j. differ. equ., 46 (2009) 1-8. [12] m.m. el-borai, a. debbouche, almost periodic solutions of some nonlinear fractional differential equations, int. j. contemp. math. sci. 4 (2009), 1373-1387. [13] r. jahanipur, stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions, j. differ. equations 248 (2010), 1230-1255. [14] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, elsevier science b.v., amsterdam (2006). [15] j. liu, c. zhang, existence and statbility of almost periodic solutions for impulsive differential equations, adv. diff. equa., 2012 (2012), article id 34. impulsive fractional stochastic evolution equations 43 [16] j. liu, c. zhang, existence and statbility of almost periodic solutions to impulsive stochastic differential equations, cubo, 15, 1 (2013), 77-96. [17] j. luo, fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, j. math. anal. appl., 342 (2008) 753-760. [18] k.s. miller, b. ross, an introduction to the fractional calculus and differential equations, john wiley, new york (1993). [19] a. pazy, semigroups of linear operators and application to partial differential equations, springer-verlag, new york, 1983. [20] i. podlubny, fractional differential equations, academic press, san diego (1999). [21] g. prato, c. tudor, periodic and almost periodic solutions for semilinear stochastic evolution equations, stoch. anal. appl., 13 (1995), 13-33. [22] a.m. samoilenko, n.a. perestyuk, impulsive differential equations, world scientific, singapore, 1995. [23] x.b. shu, y. lai, y. chen, the existence of mild solutions for impulsive fractional partial differential equations, nonlinear anal. tma 74 (2011), 2003-2011. [24] g.t. stamov, existence of almost periodic solutions for strong stable impulsive differential equations, ima j math control., 18 (2001), 153-160. [25] g.t. stamov, j.o. alzabut, almost periodic solutions for abstract impulsive differential equations, nonlinear anal. tma., 72 (2010), 2457-2464. [26] g.t. stamov, almost periodic solutions of impulsive differential equations, springer, berlin (2012). [27] g.t. stamov, i.m. stamova, almost periodic solutions of impulsive fractional differential equations, dyn. styt: an int. j., 29 (2014), 119-132. [28] t. taniguchi, k. liu, a. truman, existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in hilbert spaces, j. differ. equations 181 (2002) 72-91. [29] j.r. wang, m. feckan m, y. zhou, on the new concept of solutions and existence results for impulsive fractional evolution equations, dyn partial differ equ. 8 (2011) 345-361. [30] x. zhang, x. huang, z. liu, the existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay, nonlinear anal. hybrid syst. 4 (2010) 775-781. [31] y. zhou, f. jiao, existence of mild solutions for fractional neutral evolution equations, comput. math. appl., 59 (2010), 1063-1077. laboratory of stochastic models, statistic and applications, tahar moulay university po.box 138 en-nasr, 20000 saida, algeria ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 81-92 http://www.etamaths.com existence of multiple positive solutions for the system of nonlinear fractional order boundary value problem sabbavarapu nageswara rao∗ abstract. this paper is concerned with boundary value problems for system of nonlinear fractional differential equations involving the caputo fractional derivatives  cdq1u(t) + f1(t,u(t),v(t)) = 0, t ∈ [0, 1], cdq2v(t) + f2(t,u(t),v(t)) = 0, t ∈ [0, 1], u(0) −αu′(0) = u′(η) = βu(1) + γu′′(1) = 0, v(0) −αv′(0) = v′(η) = βv(1) + γv′′(1) = 0, where cdq1 and cdq2 are the standard caputo fractional derivatives of orders q1 and q2 respectively, with 2 < q1,q2 ≤ 3. the functions fi : [0, 1] × [0,∞) × [0,∞) → [0,∞) are continuous for i = 1, 2, α > 0,β > 0,γ > 0,η ∈ (0, 1). under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed point theorems. 1. introduction in recent years, the study of fractional order differential equations has emerged as an important area of mathematics. it has wide range of applications in various fields of science and engineering such as physics, mechanics, control systems, flow in porous media, electromagnetics and viscoelasticity. there has been much attention paid in developing the theory of existence of positive solutions for fractional order differential equations satisfying initial (or) boundary conditions to mention a few references [15, 16, 18, 24]. to mention a few references much interest has been created in establishing positive solutions and multiple positive solutions for two-point, multi-point fractional order boundary value problems (bvps). to mention the related papers along these lines − see, bai and sun [2], bai, sun and zhang [3], bai and lü [4], chai [5], goodrich [10], liang and zhang [17], nageswararao [21], prasad and krushna [23], and tian and liu [26]. motivated by above papers, in this paper we are concerned with the existence of multiple positive solutions to the couple system of nonlinear fractional order differential equations (1) cdq1u(t) + f1(t,u(t),v(t)) = 0, t ∈ [0, 1], cdq2v(t) + f2(t,u(t),v(t)) = 0, t ∈ [0, 1], with the three-point boundary conditions (2) u(0) −αu′(0) = u′(η) = βu(1) + γu′′(1) = 0, v(0) −αv′(0) = v′(η) = βv(1) + γv′′(1) = 0, where cdq1 and cdq2 are the caputo fractional derivatives of orders q1 and q2 respectively, with 2 < q1,q2 ≤ 3. the functions fi : [0, 1] × [0,∞) × [0,∞) → [0,∞) are continuous for i = 1, 2, α > 0,β > 0,γ > 0, and η ∈ (0, 1). by a positive solution of the fractional order boundary value problem (1)-(2), we understand a pair of functions (u,v) ∈ c([0, 1]) ×c([0, 1]) satisfying (1)-(2) with u(t) ≥ 0, v(t) ≥ 0 for all t ∈ [0, 1], and supt∈[0,1] u(t) > 0, supt∈[0,1] v(t) > 0. 2010 mathematics subject classification. 39a10, 34b15, 34a40. key words and phrases. caputo fractional derivative; couple system; fixed point theorem; green’s function; cone. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 81 82 sabbavarapu nageswara rao the rest of this paper is organized as follows, in section 2, we present some definitions and background results. for sake of convenience, we also state the fixed point theorems. in section 3, we construct the green’s function for the homogeneous bvp corresponding to (1)-(2), and estimate the bounds for the green’s function. in section 4, we establish the existence and multiplicity positive solutions of the bvp (1)-(2). in section 5, some examples are given to illustrate our existence results. we assume the following conditions hold throughout the paper: (a1) the functions fi : [0, 1] × [0,∞) × [0,∞) → [0,∞) are continuous and fi(t, 0, 0) ≡ 0, for 1 ≤ i ≤ 2; (a2) α > 0, β > 0, 0 < γ < β(1 − 2η(1 + α)) and 1 − 2η(1 + α) > 0; (a3) limu+v→0+ supt∈[0,1] f1(t,u,v) u+v = 0, limu+v→0+ supt∈[0,1] f2(t,u,v) u+v = 0; (a4) limu+v→∞ inft∈[0,1] f1(t,u,v) u+v = ∞, limu+v→∞ inft∈[0,1] f2(t,u,v) u+v = ∞; (a5) limu+v→0+ inft∈[0,1] f1(t,u,v) u+v = ∞, limu+v→0+ inft∈[0,1] f2(t,u,v) u+v = ∞; (a6) limu+v→∞ supt∈[0,1] f1(t,u,v) u+v = 0, limu+v→∞ supt∈[0,1] f2(t,u,v) u+v = 0; (a7) for each t ∈ [0, 1],fi(t,u,v) are nondecreasing with respect to u,v and there exists a constant n > 0 such that fi(t,u,v) < n 2n′ , for 1 ≤ i ≤ 2, where n′ = (1−2η(1+α))(β+2γ) 2d . 2. preliminaries in this section, we recall some definitions and properties of the fractional calculus. we also state a fixed point theorem of krasnosel’skii [14] is yield the existence of positive and multiple positive solutions. definition: for a continuous function f : [0,∞) → r, the caputo derivative of fractional order q is defined by cdqf(t) = 1 γ(n−q) ∫ t 0 (t−s)n−q−1f(n)(s)ds, n− 1 < q < n, n = [q] + 1 provided that f(n)(t) exists, where [q] denotes the integer part of the real number q definition: the riemann-liouville fractional integral of order q for a continuous function f(t) is defined as iqf(t) = 1 γ(q) ∫ t 0 (t−s)q−1f(s)ds, q > 0 provided that such integral exists. definition: the riemann-liouville fractional derivative of order q for a continuous function f(t) is defined by dqf(t) = 1 γ(n−q) ( d dt )n ∫ t 0 (t−s)n−q−1f(s)ds, n = [q] + 1 provided that the right-hand side is pointwise defined on (0,∞). furthermore, we note that the riemann-liouville fractional derivative of a constant is usually nonzero which can cause serious problems in real world applications. actually, the relationship between the two-types of fractional derivative is as follows cdqf(t) = 1 γ(n−q) ∫ t 0 f(n)(s) (t−s)q+1−n ds = dqf(t) − n−1∑ k=0 f(k)(s) γ(k −q + 1) tk−q = dq [ f(t) − n−1∑ k=0 f(k) k! tk ] , t > 0,n− 1 < q < n. so, we prefer to use caputos definition which gives better results than those of riemannliouville. lemma 2.1[25] let q > 0, then the fractional differential equation cdqu(t) = 0 has solution u(t) = multiple positive solutions 83 c0 + c1t + c2t 2 + · · · + cn−1tn−1, ci ∈ r, i = 0, 1, 2, · · ·n− 1 where n is the smallest integer greater than or equal to q. lemma 2.2[25] let q > 0, then iqcdqu(t) = u(t) + c0 + c1t + c2t 2 + · · · + cn−1tn−1, for some ci ∈ r, i = 0, 1, 2, · · ·n− 1 where n is the smallest integer greater than or equal to q. theorem 2.1([6, 9, 14]) let (e,‖ · ‖) be a banach space, and let p ⊂ e be a cone in e. assume that ω1 and ω2 are open subsets of e with 0 ∈ ω1 and ω1 ⊂ ω2. if t : p ∩ (ω2\ω1) → p is completely continuous operator such that either (i) ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω2,(or) (ii) ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω2, holds. then t has a fixed point in p ∩ (ω2\ω1). theorem 2.2([6, 9, 14]) let (e,‖ · ‖) be a banach space, and let p ⊂ e be a cone in e. assume that ω1, ω2 and ω3 are open bounded subsets of e such that 0 ∈ ω1 and ω1 ⊂ ω2, ω2 ⊂ ω3. if t : p ∩ (ω3\ω1) → p is completely continuous operator such that: (i) ‖ tu ‖≥‖ u ‖, ∀u ∈ p ∩∂ω1; (ii) ‖ tu ‖≤‖ u ‖, tu 6= u, ∀u ∈ p ∩∂ω2; (iii) ‖ tu ‖≥‖ u ‖, ∀u ∈ p ∩∂ω3, then t has at least two fixed points x∗, x∗∗ in p ∩(ω3\ω1), and furthermore x∗ ∈ p ∩(ω2\ω1), x∗∗ ∈ p ∩ (ω3\ω2). 3. green’s function and bounds in this section, we construct the green’s function and bounds for the homogeneous boundary value problem corresponding (1)-(2) that will be used to prove our main theorems. lemma 3.1 let d = β + 2γ−2ηβ(1 + α) > 0. if h ∈ c[0, 1], then the fractional order boundary value problem (3) cdq1u(t) + h(t) = 0, 0 < t < 1, (4) u(0) −αu′(0) = u′(η) = βu(1) + γu′′(1) = 0 has a unique solution u(t) = ∫ 1 0 g1(t,s)h(s)ds, where g1(t,s) is the green’s function for the problem (3)-(4) and is given by (5) g1(t,s) =   g1(t,s) t∈[0,η] = { g11(t,s), 0 ≤ t ≤ s ≤ η < 1, g12(t,s), 0 ≤ s ≤ min{t,η} < 1, g1(t,s) t∈[η,1] = { g13(t,s), 0 ≤ max{t,η}≤ s ≤ 1, g14(t,s), 0 < η ≤ s ≤ t ≤ 1, 84 sabbavarapu nageswara rao g11(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) − (1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −s)q1−2 γ(q1 − 1) ] g12(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) − (1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −s)q1−2 γ(q1 − 1) ] − (t−s)q1−1 γ(q1) g13(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) )] g14(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) )] − (t−s)q1−1 γ(q1) proof. assume that u ∈ c[q1]+1[0, 1] is a solution of fractional order boundary value problem by (3)-(4) and is uniquely expressed as iq1cdq1u(t) = −iq1h(t), so that u(t) = −1 γ(q1) ∫ t 0 (t−s)q1−1h(s)ds + c1 + c2t + c3t2. using the boundary conditions (4), we obtain that c1 = −2αη d ∫ 1 0 ( β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) h(s)ds + 1 d ( 2ηβ(1 + α) −α(β + 2γ − 2ηβ(1 + α)) )∫ η 0 (η −s)q1−2 γ(q1 − 1) h(s)ds c2 = −2η d ∫ 1 0 ( β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) h(s)ds + 1 d ( 2ηβ(1 + α) − (β + 2γ − 2ηβ(1 + α)) )∫ η 0 (η −s)q1−2 γ(q1 − 1) h(s)ds and c3 = 1 d ∫ 1 0 ( β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) h(s)ds − 1 d ( β(1 + α) )∫ η 0 (η −s)q1−2 γ(q1 − 1) h(s)ds. hence, the unique solution of (3) and (4) is u(t) = 1 d [( (t−η)2 −η2 − 2αη )∫ 1 0 ( β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) h(s)ds ] − 1 d [ (1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )∫ η 0 (η −s)q1−2 γ(q1 − 1) h(s)ds ] − 1 γ(q1) ∫ t 0 (t−s)q1−1h(s)ds = ∫ 1 0 g1(t,s)h(s)ds where g1(t,s) is given in (5). lemma 3.2 assume that the condition (a2) is satisfied. then the green’s function g1(t,s) given in (5) is nonnegative, for all (t,s) ∈ [0, 1] × [0, 1]. multiple positive solutions 85 proof. consider the green’s function g1(t,s) given by (5) let 0 ≤ t ≤ s ≤ η < 1. then g11(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) − (1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −s)q1−2 γ(q1 − 1) ] ≥ 1 d [( (t− tη)2 −η2 − 2αη )( β γ(q1) + γ(1 −s)−2 γ(q1 − 2) ) (1 −s)q1−1 − (1 + α) ( β ( (t− tη)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −ηs)q1−2 γ(q1 − 1) ] = 1 d [( t2(1 −η)2 − (η + α)2 + α2 )( β γ(q1) + γ(1 + 2s + o(s2)) γ(q1 − 2) ) − (1 + α) ( β ( t2(1 −η)2 + (1 + η)2 − 1 ) + d )ηq1−2(1 + s + s2) γ(q1 − 1) ] (1 −s)q1−1 ≥ 0 let 0 ≤ s ≤ min{t,η} < 1. then g12(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) − (1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −s)q1−2 γ(q1 − 1) ] − (t−s)q1−1 γ(q1) ≥ 1 d [( (t− tη)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) ) − (1 + α) ( β ( (t− tη)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) )(η −ηs)q1−2 γ(q1 − 1) ] − (t− ts)q1−1 γ(q1) = 1 d [( t2(1 −η)2 −η2 − 2αη )( β γ(q1) + γ(1 + 2s + o(s2)) γ(q1 − 2) ) − (1 + α) ( β ( t2(1 −η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) ) ηq1−1 γ(q1 − 1) − dtq1−1 γ(q1) ] × × (1 −s)q1−1 ≥ 0 let 0 ≤ max{t,η}≤ s ≤ 1. then g13(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) )] ≥ 1 d [( t2(1 −η)2 − (η + α)2 + α2 )( β γ(q1) + γ(1 + 2s + o(s2)) γ(q1 − 2) )] × × (1 −s)q1−1 ≥ 0 let 0 < η ≤ s ≤ t ≤ 1. then g14(t,s) = 1 d [( (t−η)2 −η2 − 2αη )(β(1 −s)q1−1 γ(q1) + γ(1 −s)q1−3 γ(q1 − 2) )] − (t−s)q1−1 γ(q1) ≥ 1 d [( t2(1 −η)2 − (η + α)2 + α2 )( β γ(q1) + γ(1 + 2s + o(s2)) γ(q1 − 2) ) − dtq1−1 γ(q1) ] × × (1 −s)q1−1 ≥ 0 lemma 3.3 assume that the condition (a2) is satisfied. then the green’s function satisfies the following inequality, (6) m1g1(1,s) ≤ g1(t,s) ≤ g1(1,s), for all (t,s) ∈ [0, 1] × [0, 1], where 0 < m1 = min { η2 1−2η(1+α), 2αηγ η2+ηαβ+2γ , 2γη 2γ(1+η)+β ( 1−2η(1+α) )} < 1. 86 sabbavarapu nageswara rao proof. consider the green’s function g1(t,s) is given in (5). case (i): for 0 ≤ max{t,η}≤ s ≤ 1 g13(t,s) g13(1,s) = 1 d [( (t−η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] 1 d [( (1 −η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] we have g13(t,s) ≤ g13(1,s). and also from (a2), we have g13(t,s) g13(1,s) = 1 d [( (t−η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] 1 d [( (1 −η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] ≥ η2 1 − 2η(1 + α) case (ii): for 0 ≤ η ≤ s ≤ t < 1 from (a2) and case (i),we have g14(t,s) ≤ g14(1,s). and also, we have g14(t,s) g14(1,s) = 1 d [( (t−η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] − (t−s) q1−1 γ(q1) 1 d [( (t−η)2 −η2 − 2αη )( β(1−s)q1−1 γ(q1) + γ(1−s)q1−3 γ(q1−2) )] − (t−s) q1−1 γ(q1) ≥ 2γη 2γ(1 + η) + β ( 1 − 2η(1 + α) ) case (iii): for 0 ≤ t ≤ s ≤ η < 1. from (a2) and case (i), we have g11(t,s) ≤ g11(1,s). and also, from (a2), we have g11(t,s) g11(1,s) = g13(t,s) − 1d(1 + α) ( β ( (t−η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) ) (η−s)q1−2 γ(q1−1) g13(1,s) − 1d(1 + α) ( β ( (1 −η)2 + (1 + η)2 ) + 2γ − 2βη(1 + α) ) (η−s)q1−2 γ(q1−1) ≥ 2αηγ η2 + αβη + 2γ case (iv): for 0 ≤ s ≤ min{t,η} < 1 from (a2) and case (iii), we have g12(t,s) ≤ g12(1,s). and also, from (a2), we have g12(t,s) g12(1,s) = g11(t,s) − [β + 2γ − 2ηβ(1 + α)] (t−s)q1−1 γ(q1) g11(1,s) − [β + 2γ − 2ηβ(1 + α)] (1−s)q1−1 γ(q1) ≥ 2γη 2γ(1 + η) + β ( 1 − 2η(1 + α) ) by above all cases, we get m1g1(1,s) ≤ g1(t,s) ≤ g1(1,s), for all (t,s) ∈ [0, 1] × [0, 1], where 0 < m1 = min { η2 1−2η(1+α), 2αηγ η2+ηαβ+2γ , 2γη 2γ(1+η)+β ( 1−2η(1+α) )} < 1. we can also formulate similar results as lemma (3.1) lemma (3.3) above, for the fractional boundary value problem (7) cdq2v(t) + y(t) = 0, 0 < t < 1, (8) v(0) −αv′(0) = v′(η) = βv(1) + γv′′(1) = 0 multiple positive solutions 87 where cdq2 is the caputo fractional derivative of order q2 with 2 < q2 ≤ 3, α > 0,β > 0,γ > 0,η ∈ (0, 1). we denote by g2 and m2 the corresponding green’s function and constant for the problem (7)-(8) defined in a similar manner as g1 and m1 respectively. by using green functions g1 and g2 our problem (1)-(2) can be written equivalently as the following nonlinear system of integral equations  u(t) = ∫ 1 0 g1(t,s)f1(s,u(s),v(s))ds, t ∈ [0, 1], v(t) = ∫ 1 0 g2(t,s)f2(s,u(s),v(s))ds, t ∈ [0, 1]. we consider the banach space e = c([0, 1]) with supremum norm ‖ · ‖, and the banach space b = e ×e with the norm ‖ (u,v) ‖=‖ u ‖ + ‖ v ‖ . we define the cone p ⊂ b by p = { (u,v) ∈ b; u(t) ≥ 0, v(t) ≥ 0,∀t ∈ [0, 1], and inf t∈ [η,1] (u(t) + v(t)) ≥ m ‖ (u,v) ‖ } , where m = min{m1,m2}. we introduce the operators t1, t2 : p → b and t : p → b defined by t1(u,v)(t) = ∫ 1 0 g1(t,s)f1(s,u(s),v(s))ds, t ∈ [0, 1], t2(u,v)(t) = ∫ 1 0 g2(t,s)f2(s,u(s),v(s))ds, t ∈ [0, 1]. (9) t(u,v) = ( t1(u,v),t2(u,v) ) , (u,v) ∈ p. the solutions of our problem (1)-(2) are the fixed points of the operator t . lemma 3.4 if (a1) − (a2) hold, then t : p → p is a completely continuous operator. proof. let (u,v) ∈ p be an arbitrary element. because t1(u,v) and t2(u,v) satisfy the problem (3)-(4) for h(t) = f1(t,u(t),v(t)), t ∈ [0, 1], and the problem (7)-(8) for y(t) = f2(t,u(t),v(t)), t ∈ [0, 1] respectively, then by lemma 3, we obtain inf t∈[η,1] t1(u,v)(t) ≥ m1 max t∈[0,1] t1(u,v)(t) = m1 ‖ t1(u,v) ‖, inf t∈[η,1] t2(u,v)(t) ≥ m2 max t∈[0,1] t2(u,v)(t) = m2 ‖ t2(u,v) ‖ . hence, we conclude inf t∈[η,1] [ t1(u,v)(t) + t2(u,v)(t) ] ≥ inf t∈[η,1] t1(u,v)(t) + inf t∈[η,1] t2(u,v)(t) ≥ m1 ‖ t1(u,v) ‖ +m2 ‖ t2(u,v) ‖ ≥ m ‖ (t1(u,v),t2(u,v)) ‖= m ‖ t(u,v) ‖ . clearly, we obtain t1(u,v)(t) ≥ 0, t2(u,v)(t) ≥ 0 for all t ∈ [0, 1], and so, we deduce that t(u,v) ∈ p. hence, we get t(p) ⊂ p. by using standard arguments involving the arzela-ascoli theorem, we can easily show that t1 and t2 are completely continuous, and then t is a completely continuous operator from p to p . 4. existence of multiple positive solutions in this section, we establish the existence of at least one and two positive solutions for the bvp (1)-(2) by using abstract fixed point theorems [6, 9, 14]. theorem 4.1 assume that (a1) − (a4) are hold, then the bvp (1)-(2) has at least one positive solution (u(t),v(t)), t ∈ [0, 1]. 88 sabbavarapu nageswara rao proof. from assumption (a3) we deduce that there exists h1 > 0 such that for all t ∈ [0, 1],u,v ∈ r+ with 0 ≤ u + v ≤ h1, we have f1(t,u,v) ≤ η(u + v), f2(t,u,v) ≤ η′(u + v), where η and η′ are satisfy η ∫ 1 0 g1(1, t)dt ≤ 1 2 and η′ ∫ 1 0 g2(1, t)dt ≤ 1 2 . we define the set ω1 = {(u,v) ∈ b :‖ (u,v) ‖< h1}. now let (u,v) ∈ p ∩∂ω1, that is (u,v) ∈ p with ‖ (u,v) ‖= h1 or equivalently ‖ u ‖ + ‖ v ‖= h1. then u(t) + v(t) ≤ h1, thus we have t1(u,v)(t) = ∫ 1 0 g1(t,s)f1(s,u(s),v(s))ds ≤ η ∫ 1 0 g1(1,s)(u(s) + v(s))ds ≤ η ∫ 1 0 g1(1,s) [ ‖ u ‖ + ‖ v ‖ ] ds ≤ 1 2 [ ‖ u ‖ + ‖ v ‖ ] = 1 2 ‖ (u,v) ‖ and so, ‖ t1(u,v) ‖≤ 12 ‖ (u,v) ‖ . similarly, we may take t2(u,v)(t) = ∫ 1 0 g2(t,s)f2(s,u(s),v(s))ds ≤ η′ ∫ 1 0 g2(1,s)(u(s) + v(s))ds ≤ η′ ∫ 1 0 g2(1,s) [ ‖ u ‖ + ‖ v ‖ ] ds ≤ 1 2 [ ‖ u ‖ + ‖ v ‖ ] = 1 2 ‖ (u,v) ‖ and so, ‖ t2(u,v) ‖≤ 12 ‖ (u,v) ‖ . thus, for (u,v) ∈ p ∩∂ω1 it follows that ‖ t(u,v) ‖ =‖ ( t1(u,v),t2(u,v) ) ‖=‖ t1(u,v) ‖ + ‖ t2(u,v) ‖ ≤ 1 2 ‖ (u,v) ‖ + 1 2 ‖ (u,v) ‖=‖ (u,v) ‖ . therefore, (10) ‖ t(u,v) ‖≤‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω1. on the other hand, from (a4) there exist four positive constants µ,µ′,c1 and c2 such that f1(t,u,v) ≥ µ(u + v) −c1, f2(t,u,v) ≥ µ ′ (u + v) −c2, ∀(u,v) ∈ r+ ×r+, where µ and µ′ satisfy µm2 ∫ 1 η g1(1,s)ds ≥ 1, µ′m2 ∫ 1 η g2(1,s)ds ≥ 1. for (u,v) ∈ p, τ ∈ (0, 1), we have t1(u,v)(τ) = ∫ 1 0 g1(τ,s)f1(s,u(s),v(s))ds ≥ ∫ 1 0 g1(τ,s) [ µ(u + v) −c1 ] ds ≥ µ ∫ 1 η g1(τ,s) ( u(s) + v(s) ) ds−c1 ∫ 1 η g1(τ,s)ds ≥ µm2 ∫ 1 η g1(1,s)ds ( ‖ u ‖ + ‖ v ‖ ) −c1 ∫ 1 η g1(τ,s)ds ≥ ( ‖ u ‖ + ‖ v ‖ ) −c1 ∫ 1 η g1(τ,s)ds. multiple positive solutions 89 in a similar manner, we deduce t2(u,v)(τ) = ∫ 1 0 g2(τ,s)f2(s,u(s),v(s))ds ≥ ∫ 1 0 g2(τ,s) [ µ′(u + v) −c2 ] ds ≥ µ′ ∫ 1 η g1(τ,s) ( u(s) + v(s) ) ds−c2 ∫ 1 η g2(τ,s)ds ≥ µ′m2 ∫ 1 η g1(1,s) ( ‖ u ‖ + ‖ v ‖ ) ds−c2 ∫ 1 η g2(τ,s)ds ≥ ( ‖ u ‖ + ‖ v ‖ ) −c2 ∫ 1 η g2(τ,s)ds. therefore t(u,v)(τ) ≥ 2 ‖ (u,v) ‖−c3, where c3 = c1 ∫ 1 η g1(τ,s)ds + c2 ∫ 1 η g2(τ,s)ds. from which it follows that ‖ t(u,v) ‖≥ t(u,v)(τ) ≥‖ (u,v) ‖ as ‖ (u,v) ‖→∞. let ω2 = {(u,v) ∈ b :‖ (u,v) ‖< h2}. then for (u,v) ∈ p and ‖ (u,v) ‖= h2 > 0 sufficiently large, we have (11) ‖ t(u,v) ‖≥‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω2. thus, from (10), (11) and theorem (2.1), we know that the operator t has a fixed point in p∩(ω2 \ ω1). theorem 4.2 assume that (a1), (a2), (a5) and (a6) are hold, then (1)-(2) has at least one positive solution (u(t),v(t)), t ∈ [0, 1] proof. from (a5) there is a number ĥ3 ∈ (0, 1) such that for each (t,u,v) ∈ [0, 1] × (0,ĥ3) × (0,ĥ3). one has f1(t,u,v) ≥ λ(u + v), where λ satisfy λm2 ∫ 1 η g1(1,s)ds ≥ 1. from (a1) that implies f1(t, 0, 0) ≡ 0 and the continuity of f1(t,u,v), we know that there exists a number h3 ∈ (0,ĥ3) small enough such that f1(t,u,v) ≤ ĥ3∫ 1 0 g1(1, t)dt whenever (t,u,v) ∈ [0, 1] × (0,h3) × (0,h3). for every (u,v) ∈ p and ‖ (u,v) ‖= h3, note that∫ 1 0 g1(1,τ)f1(τ,u(τ),v(τ))dτ ≤ ∫ 1 0 g1(1,τ) ĥ3∫ 1 0 g1(1, t)dt dτ ≤ ĥ3. thus t1(u,v)(τ) = ∫ 1 0 g1(τ,s)f1(s,u,v)ds ≥ mλ ∫ 1 η g1(1,s) ( u(s) + v(s) ) ds ≥ λm2 ∫ 1 η g1(1,s)ds(‖ u ‖ + ‖ v ‖) ≥ ( ‖ u ‖ + ‖ v ‖ ) =‖ (u,v) ‖ that is t1(u,v)(t) ≥‖ (u,v) ‖ for all t ≥ τ. so, ‖ t(u,v) ‖≥‖ t1(u,v) ‖≥‖ (u,v) ‖. if set ω3 = {(u,v) ∈ b :‖ (u,v) ‖< h3}, then (12) ‖ t(u,v) ‖≥‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω3. on the other hand, we know from (a6) that there exist four positive numbers η,η′,c4 and c5 such that for every (t,u,v) ∈ [0, 1]×r+×r+, we have f1(t,u,v) ≤ η(u+v)+c4, f2(t,u,v) ≤ η′(u+v)+c5, 90 sabbavarapu nageswara rao where η and η ′ satisfy η ∫ 1 0 g1(1,s)ds ≤ 1 2 and η ′ ∫ 1 0 g2(1,s)ds ≤ 1 2 . thus we have t1(u,v)(t) = ∫ 1 0 g1(t,s)f1(s,u,v)ds ≤ ∫ 1 0 g1(1,s)(η(u + v) + c4)ds ≤ η ∫ 1 0 g1(1,s) ( ‖ u ‖ + ‖ v ‖ ) ds + c4 ∫ 1 0 g1(1,s)ds ≤ 1 2 ‖ (u,v) ‖ +c4 ∫ 1 0 g1(1,s)ds. similarly, we deduce t2(u,v)(t) = ∫ 1 0 g2(t,s)f2(s,u,v)ds ≤ ∫ 1 0 g2(1,s)(η ′(u + v) + c5)ds ≤ η′ ∫ 1 0 g2(1,s) ( ‖ u ‖ + ‖ v ‖ ) ds + c5 ∫ 1 0 g2(1,s)ds ≤ 1 2 ‖ (u,v) ‖ +c5 ∫ 1 0 g2(1,s)ds. therefore t(u,v)(t) ≤‖ (u,v) ‖ +c6, where c6 = c4 ∫ 1 0 g1(1,s)ds + c5 ∫ 1 0 g2(1,s)ds, from which it follows that t(u,v)(t) ≤‖ (u,v) ‖ as ‖ (u,v) ‖→ ∞. let ω4 = {(u,v) ∈ b :‖ (u,v) ‖< h4}. for each (u,v) ∈ p and ‖ (u,v) ‖= h4 > 0 large enough, we have (13) ‖ t(u,v) ‖≤‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω4. from (12),(13) and theorem (2.1), we know that the operator t has a fixed point in p ∩ (ω4 \ ω3). theorem 4.3 assume that (a1), (a2), (a4), (a5) and (a7) are satisfied, then (1)-(2) has at least two positive solutions (u1(t),v1(t)), (u2(t),v2(t)), t ∈ [0, 1]. proof. note that we have gi(t,s) ≤ (1−2η(1+α))(β+2γ) 2d = n′ for i = 1, 2 for all (t,s) ∈ [0, 1]× [0, 1]. let bn = {(u,v) ∈ p :‖ (u,v) ‖< n}. by using (a7), for any (u,v) ∈ ∂bn ∩p, we obtain t1(u,v)(t) = ∫ 1 0 g1(t,s)f1(s,u,v)ds < n ′ n 2n′ = n 2 which implies ‖ t1(u,v) ‖≤ n2 . in a similar manner, we may take ‖ t2(u,v) ‖≤ n 2 . therefore ‖ t(u,v) ‖=‖ t1(u,v) ‖ + ‖ t2(u,v) ‖≤ n 2 + n 2 = n. thus (14) ‖ t(u,v) ‖≤‖ (u,v) ‖, for all (u,v) ∈ p ∩∂bn. and from (a4) and (a5) we have (15) ‖ t(u,v) ‖≥‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω2, (16) ‖ t(u,v) ‖≥‖ (u,v) ‖, for all (u,v) ∈ p ∩∂ω3. we have choose h2,h3 and n such that h3 ≤ n ≤ h2 and (14)-(16) are satisfied. from theorem (2.2), t has at least two fixed points in p ∩ (ω2 \bn ) and p ∩ (bn \ ω3), respectively. multiple positive solutions 91 5. example in this section, we demonstrate our results with some examples. we consider the system of fractional order differential equations (17) cd2.5u(t) + f1(t,u(t),v(t)) = 0, t ∈ (0, 1) cd2.5v(t) + f2(t,u(t),v(t)) = 0, t ∈ (0, 1) with the three-point boundary conditions (18) u(0) −u′(0) = u′ (1 8 ) = 2u(1) + 1 2 u′′(1) = 0, v(0) −v′(0) = v′ (1 8 ) = 2v(1) + 1 2 v′′(1) = 0. here q1 = q2 = 5 2 ,α = 1, β = 2, η = 1 8 , γ = 1 2 and we deduce that m = min{m1,m2} = 0.03125 example 5.1: let f1(t,u,v) = t 4 (u + v) + t2 + 4, f2(t,u,v) = t4 2 (u + v) + e−(u+v), then conditions of theorem (4.1) are satisfied. from theorem (4.1), the bvp (17)-(18) has at least one positive solution. example 5.2: let f1(t,u,v) = (1 − t) [ e−(u+v)(u + v) ] , f2(t,u,v) = 4 1+t (u2 + v2) then n′ = 9 8 . we can choose n = 1 and conditions of theorem (4.3) are satisfied. from theorem (4.3), the bvp (17)-(18) has at least two positive solutions. acknowledgement: the author is very grateful to his guide professor k. rajendra prasad and to the referees for their valuable suggestions and comments on improving this paper. references [1] a. y. al-hossain, eigenvalues for iterative systems of nonlinear caputo fractionalorder three point boundary value problems, j. appl. math. comput. (2015), doi 10.1007/s12190-015-0935-1. [2] c. bai and w. sun, existence and multiplicity of positive solutions for singular fractional boundary value problems, comput. math. appl. 63 (2012), 1369-1381. [3] c. bai, w. sun and w. zhang, positive solutions for boundary value problems of a singular fractional differential equations, abstr. appl. anal. 2013 (2013), article id 129640. [4] z. bai and h. lü, positive solutions for boundary value problem of nonlinear fractional differential equation, j. math. anal. appl. 311 (2005), 495-505. [5] g. chai, existence results of positive solutions for boundary value problems of fractional differential equations, bound. value probl. 2013 (2013), art. id 109. [6] k. deimling, nonlinear functional analysis, springer-verlag, berlin, 1985. [7] p. w. eloe and j. henderson, positive solutions for higher order ordinary differential equations, electron. j. differential equations. 3 (1995), 1-8. [8] l. h. erbe and h. wang, on the existence of positive solutions of ordinary differential equations, proc. amer. math. soc. 120 (1994), 743-748. [9] d. j. guo and l. lakshmikantham, nonlinear problems in abstract cones, academic press, new york, 1988. [10] cs. goodrich, on a fractional boundary value problem with fractional boundary conditions, appl. math. lett. 25 (2012), 1101-1105. [11] l. hu and l. wang, multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations, j. math. anal. appl. 335 (2007), 1052-1060. [12] j. henderson and r. luca, existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems, nonlinear differential equations appl. 20 (2013), no. 3, 1035-1054. [13] a. kameswarao and s. nageswararao, multiple positive solutions of boundary value problems for systems of nonlinear second-order dynamic equations on time scales, math. commun. 15 (2010), no. 1, 129-138. [14] m. a. krasnosel’skii, positive solutions of operator equations, noordhoff, groningen, 1964. [15] v. lakshmikanthan, theory of fractional differential equations, nonlinear anal. 69 (2008) 3337-3343. [16] v. lakshmikanthan, s. leela and j. vasundhara devi, theory of fractional dynamic systems. cambridge scientific publishers, cambridge (2009). [17] s. liang and j. zhang, positive solutions for boundary value problems of nonlinear fractional differential equations, nonlinear anal. 71 (2009), 5545-5550. [18] k. s. miller and b. ross, an introduction to the fractional calculus and fractional differential equations. wiley, new york (1993). [19] s. nageswararao, multiple positive solutions for a system of riemann-liouville fractional order two-point boundary value problems, panamer. math. j. 25 (2015), no. 1, 66-81. [20] s. nageswararao, existence of positive solutions for riemann-liouville fractional order three-point boundary value problem, asian-eur. j. math. 8 (2015), no. 4, art. id 1550057. 92 sabbavarapu nageswara rao [21] s. nageswararao, existence and multiplicity for a system of fractional higher-order two-point boundary value problem, j. appl. math. comput. 51 (2016), 93-107. [22] k. r. prasad, a. kameswararao and s. nageswararao, existence of positive solutions for the system of higher order two-point boundary value problems, proc. indian acad. sci. math. sci. 122 (2012), no. 1, 139-152. [23] k. r. prasad and b. m. b. krushna, multiple positive solutions for a coupled system of riemann-liouville fractional order two-point boundary value problems, nonlinear stud. 20 (2013), no. 4, 501-511. [24] i. podlubny, fractional differential equations. academic press, san diego (1999) [25] a. a. kilbas, h. m. srivastava and j. j. trujillo, theory and applications of fractional differential equations. north-holland mathematics studies, 204, elsevier science b. v., amsterdam (2006). [26] c. tian and y. liu, multiple positive solutions for a class of fractional singular boundary value problem, mem. differ. equ. math. phys. 56 (2012), 115-131. department of mathematics, jazan university, jazan, kingdom of saudi arabia ∗corresponding author: sabbavarapu−nag@yahoo.co.in; snrao@jazanu.edu.sa int. j. anal. appl. (2023), 21:35 on the stability and controllability of degenerate differential systems in hilbert spaces norelhouda beghersa1,∗, fares yazid2 1norelhouda beghersa, department of mathematics, faculty of mathematics and computer sciences, university of sciences and technology mohamed boudiaf of oran usto-mb, el mnaouar, bp 1505, bir el djir 31000, oran, algeria 2laboratory of pure and applied mathematics, amar teledji laghouat university, 03000, laghouat, algeria ∗corresponding author: norelhouda.beghersa@univ-usto.dz abstract. we apply the famous theorem of lyapunov for some degenerate differential systems taken the form ax ′(t) + bx(t) = φ(t,x(t)), where t ∈ r+ and a, b are linear bounded operators in hilbert spaces, φ is a given function. the obtained results are used to study the stabilizability and controllability of certain implicit controlled systems. 1. introduction the fondamental challenges of control theory and the most variety problems of this area in mathematical science attracted the attention of many researchers, that’s why they often study linear differential systems (continuous or discrete, see [4]), of the form: x′(t) = px(t) + φ(t,x(t)), (1.1) or xn+1 = pxn + φ(t,x(t)), n = 0, 1, 2, 3, ..., (1.2) where p is a linear operator, or matrix in an appropriate (finite or infinite dimensional) hilbert spaces. the major interest on different physical and mechanical problems was described by another more received: feb. 28, 2023. 2020 mathematics subject classification. 15a22, 93d15, 93d05, 47d03. key words and phrases. operator pencil; stabilizability; controllability; semigroup. https://doi.org/10.28924/2291-8639-21-2023-35 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-35 2 int. j. anal. appl. (2023), 21:35 general form than 1.1 and 1.2 which has been studied extensively in 1970 by many mathematicians as rutkas [1] and gantmacher [6]. in [5] favini and vlasenko have recently ensured the existence of weak and strict solutions by using the operator method of grisvard when the differential system is non-stationary. in the present paper, we consider the following differential system of the form: ax′(t) + bx(t) = φ(t,x(t)), t > 0, (1.3) with initial condition x(0) = x0, where a and b are linear bounded operators acting in the same hilbert space h and φ : [0,∞[×h into h is a continuous function. we call the system 1.3 with a non invertible operator a degenerate or implicit stationary differential system but if a is the identity operator then, the equation 1.3 is said to be explicit. we denote by λa + b the operator pencil obtained by substituting the exponential function eλtv for all v ∈h into 1.3 when φ ≡ 0. some real and practical applications of 1.3 can be found in [2], [7] and the references therein. the organization of this paper is as follows: in section 2, we present an important result which is the generalization of the lyapunov theorem obtained by using some properties of the spectral theory and an appropriate conformal tronsformation to the corresponding operator pencil. in section 3, we apply the achieved results to study the stabilizability and controllability of certain implicit differential control system described as follows: ax′(t) + bx(t) = −cu(t), 0 6 t 6 t, x(0) = x0; (1.4) here x(t) and the control u(t) (bochner integrable function) lie in the hilbert spaces h and u. also, t is a time and the operator c is bounded on h. 2. stability of a stationary implicit differential system in this section, we investigate the homogeneous case of the system 1.3 as follows: ax′(t) + bx(t) = 0, for all t ≥ 0. (2.1) in the sequel, we need the following definitions. definition 2.1. the system 2.1 is said to be exponentially stable, if there exist two constants m, α > 0 such that for any solution x(t) the bellow hypothesis is verified: ‖x(t)‖≤ me−αt, ∀t ≥ 0. (2.2) definition 2.2. the equation 2.1 is called well-posed, if it satisfies the following conditions: • for any solution x(0) such that x(0) = x0, then x(t) = 0, for each t ≥ 0; int. j. anal. appl. (2023), 21:35 3 • it generates an evolution semi-group of bounded operators γ (t) : x0 → x(t), for all t ≥ 0, where the operators γ (t) are defined on the set of admissible initial vectors denoted by d∗ = {x0}. definition 2.3. let d∗ be a subspace of the hilbert space h and a0 is the restriction of the operator a in d∗. the operator a −1 0 exists if the system 2.1 is well-posed. definition 2.4. [1] the comlex parameter λ is said to be a regular point of the pencil λa + b, if the operator (λa + b)−1 exists and it is bounded. we denote by ρ(a,b) the set of all regular points and its complement by σ(a,b) = c\ρ(a,b) which is also called the spectrum of the operator pencil λa + b. the set of all eigen values is denoted by σp(a,b), such that σp(a,b) = {λ ∈c\∃v 6= 0, (λa + b)v = 0}. (2.3) we recall that our main objective is to generalize the lyapunov theorem were laid in the monograph [2] concerning the classical linear differential systems for the operator pencil λa + b, we have thereby obtained the next result. theorem 2.1. assume that a and b are bounded linear operators acting in the same hilbert space h. if the spectrum of the pencil λa+b lies inside the disk of radius r, then for any uniformily positive operator s, there exists an operator w � 0 such that: re(w (b − ra)(b + ra)−1) = −s. (2.4) proof. suppose that σ(a,b) ⊂ {λ ∈ c : re(λ) < r}. then, r is a regular point and the operator t = (b − ra)(b + ra)−1 is bounded. now, we define the conformal transformation ζ given by: ζ = χ(λ) = λ + r λ− r . (2.5) using 2.5 to compute t −ζi then, we obtain: t −ζi = (b − ra)(b + ra)−1 − λ + r λ− r (b + ra)(b + ra)−1, = −2r λ− r (λa + b)(b + ra)−1. indeed, the operator t − ζi is invertible if the inverse of the pencil λa + b exists. hence, ρ(t ) = χ(ρ(a,b)) and we conclude that σ(t ) = χ(σ(a,b)). therefore, σ(t ) lies inside the disk of radius r. according to lyapunov theorem [2], we have: ∀s � 0, ∃w � 0/ re(wt ) = wt + t∗w 2 . (2.6) 4 int. j. anal. appl. (2023), 21:35 so, re(wt ) = 1 2 {w (b − ra)(b + ra)−1 + (b∗ + ra∗)−1(b∗ − ra∗)w}, = (b∗ + ra∗)−1[b∗wb − r2a∗wa](b + ra)−1, = −s which is equivalent to: b∗wb − r2a∗wa = g, where g = −(b∗ + ra∗)s(b + ra). (2.7) to complete the proof, we use the fact that g � 0, it means: g = g∗and < gx,x >≥ c‖x‖2, ∀c > 0. (2.8) � theorem 2.2. suppose that the equation 2.7 is satisfied for any pair of non negative uniform operators (w,g) then, r is not an eigen value of the operator pencil λa + b. proof. for r ∈ σp(a,b), there exists an eigen-vector v 6= 0, such that (ra + b)v = 0 so, bv = −ra. now, we compute the scalar product < gv,v > then, we obtain: < gv,v > = < wbv,bv > −r2 < wav,av >, = r2 < wav,av > −r2 < wav,av >, = 0. we use the fact that g is a uniformly positive operator, which implies a contraduction with our main hypothesis1. thus, theorem 2.2 is proved. � proposition 2.1. in finite dimentional spaces (i.e., dim(h) < ∞), if: σ(a,b) = σp(a,b) ⊂{λ : re(λ) < ω2}, then, the system 2.1 is exponentially stable under the condition α ≤ r ≤ ω. proof. we assume that, the system 2.1 is not exponentially stable. we use the method of elementary divisors (see for example [6]) and we suppose that the matrix pencil λa+b is regular in order to prove our proposition. we have: λa + b ∼ λã + b̃ = {nµ1,nµ2, ...,nµs ; j + λi}, 1g � 0 ⇐⇒ g = g∗, and < gx,x >≥ c‖x‖2 > 0. 2it presents the strict lyapunov exponent as in [3] int. j. anal. appl. (2023), 21:35 5 where the first diagonal blocks correspond to the infinite elementary divisors. now, we pose x(t) = qz(t) where det(q) 6= 0. so, the system 2.1 is equivalent to the following form:  az ′(t) + bz(t) = 0, ã = aq, b̃ = bq, λã + b̃ = (λa + b)q. (2.9) also, we can write the system 2.1 as follows:  nµk dzk dt = 0, k = 1, 2, ...,s dz̃k dt + =z̃ = 0, where z = (z1,z2, ...,zs, z̃)t. (2.10) as, σ(a,b) = σ(ã,b̃) = σ(i,j) = σ(−j) ⊂{λ : re(λ) < r} then we obtain: ‖e−j.t‖≤ mωe−ω.t, and ‖z̃(t)‖ = ‖e−j.tz̃(t0)‖≤ mωe−ω.t‖z̃(t0)‖. therefore, ‖x(t)‖ = ‖qz(t)‖≤‖q‖mωe−ω.t‖z(t)‖. hence, the system 2.1 is exponentially stable for all r ≤ ω, which implies a contradiction with our main hypothesis thus, this proposition is proved. � remark 2.1. we can easly show that the necessary and sufficient conditions on the stability (exponential stability) of the implicit differential system 2.1 can be obtained by using theorem 1 − 3 in [4]. let us return now to the problem 2.1 when the hilbert space h has finite dimension, in this case we often use theory of elementery divisors for the matrix pencil λa + b (see [6]) and we have thereby obtained the next result. theorem 2.3. consider the problem 2.1 in finite dimentional hilbert space h, the following assertions are equivalent: (1) the system 2.1 is exponentially stable, (2) σp(a,b) = σ(a,b) ⊂{λ ∈c, re(λ) < r}, (3) there exists a uniform positive definite matrix w, such that: b∗wb − r2a∗wa � 0. (2.11) according to theorem 2.2 in the reference [3] and theorem 2.2 of this paper, we can show that (1) is equivalent to the relation (2). from proposition 2 and theorem 2.1, we obtain: (2) ⇔ (3). 6 int. j. anal. appl. (2023), 21:35 3. basic preliminaries and results on the controlled degenerate systems now, we give some important definitions about the exact controllability and stabilizability for some types of systems governed by 1.4, we also establish some results more general than the classical explicit case under a given assumption. definition 3.1. the equation 1.4 is said to be exactly controllable if for all x0,x1 ∈h, there exists a time t and control u ∈ l2((0,t ),u) such that: x(t,u(.),x0) = x1. particulary, if x1 = 0, here we talk about the notion of exact null controllability. definition 3.2. [7] the implicit controlled system 1.4 is said to be exponentially stabilizable by means of a direct feedback u(t) = kx(t), if the given system: ax′(t) + (b + ck)x(t) = 0, 0 ≤ t ≤ t (3.1) is exponentially stable3 (a, b, c, k suppose to be linear bounded operators in the hilbert space h). proposition 3.1. [8] the system 1.4 is exactly null controlable in the class l2 if and only if: ∃σ > 0,∀x ∈h, ∫ t 0 ‖c∗(a∗0) −1γ∗(t)x‖2dt > σ2‖γ∗(t )x‖2. (3.2) before studying the cotrollability of a system such 1.4, we can observe that the controllability problem has always a strong relation with the exponential stabilizability also, it can be reduced to the controllability problem of an explicit differential system (if a is an invertible operator). to be more precise we propose the next assumption then, we expand it to the system 1.4. assumption 3.1. let b̃ be the infinitesimal generator of a c0−group γ̃ (t) of bounded operators in the hilbert space h. b̃ satisfies the assumption if for some time t > 0, there exists σ > 0, such that: ∫ t 0 ‖b̃∗γ̃∗(−t)x‖2 > σ‖x‖2,∀x ∈h. (3.3) theorem 3.1. if the operator a−10 b complies with the previous assumption then, the problem of control considered by the equation 1.4 is stabilizable. additionally, given α > 0 there exists a linear bounded operator k where the group γ (t) generated by a−10 (b + ck) verifies: ‖γ (t)‖6 mαe−αt, ∀t ≥ 0, α > 0, and mα > 1. (3.4) 3it means: ∀α > 0, ‖ea −1 0 (b+ck)‖ 6 mαe−αt, mα > 1. int. j. anal. appl. (2023), 21:35 7 proof. let dt,α be a bounded linear operator defined as follows: dt,αx = ∫ t 0 e−2αtγ (−t)a−10 cc ∗(a∗0) −1γ∗(−t)xdt. (3.5) obviuosly, d∗t,α = dt,α, using the assumption 3.1, we can affirme that the inverse of the operator dt,α exists and it is bounded. now, we consider another implicit control system represented by the abstract form: ax′(t) = −bx(t) + αx(t) −cu(t), (3.6) where the group γα(t) has the infinitesimal generator a −1 0 (b −αi). the equation 3.6 is also equivalent to ax′(t) = −bx(t) + αx(t) −ckx(t). (3.7) we define the linear feedback u(t) as follows: u(t) = kx(t) = −c∗(a∗0) −1d−1 t,α x(t). (3.8) also, we have the group γ∗α,k(t) generated by the operator a −1 0 (−b + αi) + a −1 0 cc ∗(a∗0) −1d−1 t,α is a solution of the system 3.7. to complete the proof, it is necessary to compute the scalar product so, for each x∗ ∈ d(b∗(a∗0) −1), we have: d dt (< γ∗α,k(t)x∗,dt,αγ ∗ α,k(t)x∗ >) = −‖e −αtc∗(a∗0) −1γ∗(−t )γ∗α,k(t)x∗‖ 2 −‖c∗(a∗0) −1γα,k(t)x∗‖2, (3.9) for more details of computation, see the reference [8]. passing now to the denseness of the group γα,k, we can show that: ∀α > 0,∃k : ‖γk(t)‖ = ‖ea −1 0 (b+ck)‖6 me−αt,∀t > 0. (3.10) hence, the system is exponentially stabilizable. it follows immediately from the definition of the bounded operator k such that: k = −c∗(a∗0) −1d−1 t,α . � remark 3.1. if we replace b by b + ck in theorem 2.3 then, we obtain another important result on the stabilization of implicit controlled systems (dim(h) < ∞, r = 0). corollary 3.1. the following expressions are equivalent: • the equation 2.1 is exponentially stabilizable, • σ(a,b + ck) = σp(a,b + ck) ⊂{λ ∈c : re(λ) < 0}, • there exists a positive definite matrix w � 0, such that: b∗wb + k∗c∗wb � 0. (3.11) 8 int. j. anal. appl. (2023), 21:35 remark 3.2. according to reference [9], we can show that if all the matrices of the system are real and a−1 exists then, the spectrum is formed by real or complex conjugate numbers. finally, we provide a simple example to illustrate and explain the last point of our paper. example 3.1. consider the system 1.4 as follows:  2x ′(t) −y ′(t) = 3x(t) −y(t) + u1(t) 2y ′(t) = −x(t) + 2y(t) + u2(t), with the direct feedback u(t) = i2(x(t),y(t))t. we have, ec(λ) = det(a −1(b + ck) −λi2) = λ2 − 13 4 λ + 11 4 , then, we obtain: σ(a,b + ck) = σ(i,a−1(b + ck)) = { 13 8 + 1 8 i √ 7; 13 8 − 1 8 i √ 7}. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.g. rutkas, spectral methods for studying degenerate differential-operator equations. i, j. math. sci. 144 (2007), 4246-4263. https://doi.org/10.1007/s10958-007-0267-2. [2] j.l. doletski, m.g. krein, stability of solutions of differential equations in banach space, amer. math. soc. 2002. [3] m. benabdallah, m. hariri, on the stability of the quasi-linear implicit equations in hilbert spaces, khayyam j. math. 5 (2019), 105-112. https://doi.org/10.22034/kjm.2019.81222. [4] m. benabdallakh, a.g. rutkas, a.a. solov’ev, on the stability of degenerate difference systems in banach spaces, j. soviet math. 57 (1991), 3435-3439. https://doi.org/10.1007/bf01880215. [5] a. favini, l. vlasenko, degenerate non-stationary differential equations with delay in banach spaces, j. differ. equ. 192 (2003), 93-110. https://doi.org/10.1016/s0022-0396(03)00090-1. [6] f.r. gantmacher, the theory of matrices, vol. 1 and vol. 2, chelsea publishing company, new york, 1959. [7] l. baghdadi, r. rabah, a note on the stabilization of linear systems in hilbert spaces, demonstr. math. 21 (1988), 631-641. https://doi.org/10.1515/dema-1988-0307. [8] m. slemrod, a note on complete controllability and stabilizability for linear control systems in hilbert space, siam j. control. 12 (1974), 500-508. https://doi.org/10.1137/0312038. [9] w.m. wonham, linear multivariable control: a geometric approach, 3rd edition., springer, new york, 1985. https://doi.org/10.1007/s10958-007-0267-2 https://doi.org/10.22034/kjm.2019.81222 https://doi.org/10.1007/bf01880215 https://doi.org/10.1016/s0022-0396(03)00090-1 https://doi.org/10.1515/dema-1988-0307 https://doi.org/10.1137/0312038 1. introduction 2. stability of a stationary implicit differential system 3. basic preliminaries and results on the controlled degenerate systems references int. j. anal. appl. (2023), 21:50 on quasi-ideals and bi-ideals in ag-rings tanaphong prommai1, thiti gaketem2,∗ 1department of mathematics school of science, university of phayao, phayao 56000, thailand 2fuzzy algebras and decision-making problems research unit, department of mathematics school of science, university of phayao, phayao 56000, thailand ∗corresponding author: thiti.ga@up.ac.th abstract. in this paper we study some properties of quasi-ideals and bi-ideals in ag-ring and study some interesting properties of these ideals. 1. introduction m.a. kazim and md. naseeruddin [2] have introduced the concept of an ag-groupoid. definition 1.1. a groupoid g is called a left almost semigroup (abbreviated as a la-semigroup) if its elements satisfy the left invertive law: (ab)c = (cb)a for all a,b,c ∈ g. it is also called an abel-grassmann’s groupoid (abbreviated as ag-groupoid). moreover every ag-groupoids g have a medial law hold (a ·b) · (c ·d) = (a ·c) · (b ·d), ∀a,b,c,d ∈ g. q. mushtaq and m. khan [4, p.322] asserted that, in every ag-groupoids g with left identity (a ·b) · (c ·d) = (d ·c) · (b ·a), ∀a,b,c,d ∈ g. received: mar. 14, 2023. 2020 mathematics subject classification. 16y30, 16y99. key words and phrases. ideal; bi-ideal; quasi-ideal. https://doi.org/10.28924/2291-8639-21-2023-50 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-50 2 int. j. anal. appl. (2023), 21:50 further m. khan, faisal, and v. amjid [3], asserted that, if a ag-groupoid g with left identity the following law holds a · (b ·c) = b · (a ·c), ∀a,b,c ∈ g. m. sarwar (kamran) [5, p.112] defined ag-group as the following. definition 1.2. a groupoid g is called an abel-grassmann’s group, abbreviated as ag-group, if (1) there exists e ∈ g such that ea = a for all a ∈ g, (2) for every a ∈ g there exists a′ ∈ g such that, a′a = e, (3) (ab)c = (cb)a for every a,b,c ∈ g. s.m. yusuf in [11, p.211] introduces the concept of an abel-grassmann’s ring (ag-ring). definition 1.3. an algebraic system 〈r, +, ·〉 is called a abel-grassmann’s ring (ag-ring) if (1) 〈r, +〉 is an ag-group, (2) 〈r, ·〉 is an ag-groupoid, (3) a(b + c) = ab + ac and (a + b)c = ac + bc, for all a,b,c ∈ r. lemma 1.1. in an ag-ring r, (ab)(cd) = (ac)(bd) (1.1) for all a,b,c,d ∈ r. equation (1.1) is called a medial law in the ag-ring r. lemma 1.2. if an ag-ring r has a left identity 1, then a(bc) = b(ac) for all a,b,c ∈ r. lemma 1.3. if an ag-ring r has a left identity 1, then (ab)(cd) = (dc)(ba) (1.2) for all a,b,c,d ∈ r. equation (1.2) is called a paramedial law in the ag-ring r. now we have the following property. t. shah and i. rehman [11, p.211] asserted that a commutative ring 〈r, +, ·〉, we can always obtain an ag-ring 〈r,⊕, ·〉 by defining, for a,b,c ∈ r, a ⊕b = b −a and a ·b is same as in the ring. we can not assume the addition to be commutative in an ag-ring. definition 1.4. let 〈r, +, ·〉 be an la-ring and s be a non-empty subset of r and s is itself and ag-ring under the binary operation induced by r, the s is called an ag-subring of r, then s is called an la-subring of 〈r, +, ·〉. int. j. anal. appl. (2023), 21:50 3 definition 1.5. if s is an ag-subring of an la-ring 〈r, +, ·〉, then s is called a left ideal of r if rs ⊆ s. right and two-sided ideals are defined in the usual manner. lemma 1.4. if an ag-ring r has a left identity 1, then every right ideal is a left ideal. proof. let r be an ag-ring with left identity 1 and a is a right ideal of r. then for a ∈ a,r ∈ r, we have ra = (1r)a = (ar)1 ∈ (ar)r ⊆ ar ⊆ a, where 1 is a left identity, that is ra ∈ a. therefore a is left ideal of r. � 2. main results definition 2.1. let r be an ag-ring and q be a non-empty subset of r. then q is said to be a quasi-ideal of r if q is a ag-subgroup of (r, +) such that rq∩qr ⊆ q. theorem 2.1. every one-sided ideal or two-sided ideal of an-ag-ring r is a quasi-ideal of r. proof. let l be a left ideal of an ag-ring r. then lr∩rl ⊆ ll ⊆ l. thus l is a quasi-ideal of an ag-ring r. similarly let i be a right ideal of r then ir∩ri ⊆ ii ⊆ i. thus i is a quasi-ideal of an ag-ring r. � theorem 2.2. let r be an ag-ring. then the intersection of left ideal l and a right ideal of i of r is a quasi-ideal of r. proof. let l be a left ideal and i be a right ideal of r. then l∩ i is a ag-subgroup of (r, +). thus r(l∩ i) ∩ (l∩ i)r ⊆ rl∩ ir ⊆ l∩ i. therefore the intersection of left ideal l and a right ideal of i of r is a quasi-ideal of r. � theorem 2.3. arbitrary intersection of quasi-ideal of an ag-ring r is a quasi-ideal of r. proof. let t := ⋂ i∈∆{qi | qi is a quasi-ideal of r}, where ∆ denotes any indexing set, be a nonempty set. then t is a ag-subgroup of (r, +). now rt ∩tr = r (⋂ i∈∆ q1 ) ∩ (⋂ i∈∆ q1 ) r ⊆ rqi ∩qir ⊆ qi, for all i ∈ ∆. so we see that rt ∩tr ⊆ ⋂ i∈∆ qi = t . this proof complete. � definition 2.2. an element e of an ag-ring r is a said idempotent element if e2 = ee = e. 4 int. j. anal. appl. (2023), 21:50 theorem 2.4. let r be an ag-ring in which every quasi-ideal is idempotent. then for left ideal l and right ideal i such that il = i ∩l ⊆ li is true. proof. let p and q be two quasi-ideal in r then p ∩q is also a quasi-ideal. by the idempotent of p ∩q we have p ∩q = (p ∩q)(p ∩q)(pq) ∩ (qp ) on other hand (pq) ∩ (qp ) ⊆ (pr) ∩ (rp ) ⊆ p. similarly (pq) ∩ (qp ) ⊆ q and so p ∩q = (pq) ∩ (qp ). since left and right ideal are always ag-subgroup we have i ∩l = (il) ∩ (li) but (il) ⊆ (r∩l) and so il = i ∩l ⊆ li. this proof complete. � intersection of a quasi-ideal and ag-subring of r is a quasi-ideal of an ag-subring of r. we can prove this in the following theorem. theorem 2.5. let r be an ag-ring. if q is a quasi-ideal and t is an ag-subring of r, then q∩t is a quasi-ideal of t . proof. let q is a quasi-ideal and t is an ag-subring of r. then q∩t is a ag-subgroup of (r, +). since q∩t ⊆ t we have q∩t is a ag-subgroup of (t, +). then t (q∩t ) ∩ (q∩t )t ⊆ tq∩qt ⊆ rq∩qr ⊆ q and t (q∩t ) ∩ (q∩t )t ⊆ tt ∩tt ⊆ t ∩t = t. it follows that t (q∩t ) ∩ (q∩t )t ⊆ q∩t. hence q∩t is a quasi-ideal of t . � definition 2.3. let r be an ag-ring. an additive ag-subgroup b of r is called a bi-ideal of r if (br)b ⊆ b. lemma 2.1. every left (right) ideal of an ag-ring r is a bi-ideal of r. proof. let l be a left ideal of r. then a is an additive ag-subgroup of r. thus (lr)l ⊆ (rr)l ⊆ rl ⊆ l. this implies that l is a bi-ideal of r. let i be a right ideal of r. then i is an additive ag-subgroup of r. thus (ir)i ⊆ ii ⊆ ir ⊆ i. this implies i is a bi-ideal of r. � corollary 2.1. every ideal of a γ-ag-ring r is a bi-ideal of r. lemma 2.2. let b be an idempotent bi-ideal of a γ-ag-ring r with left identity 1. then b is an ideal of r. int. j. anal. appl. (2023), 21:50 5 proof. let b be an idempotent bi-ideal of a γ-ag-ring r. then b is an additive ag-subgroup of r. thus br = (bb)r = (rb)b = (r(bb))b. by lemma 1.2 so (r(bb))b = ((bb)r)b = (br)b ⊆ b. which implies that b is a right ideal. by lemma 1.4 so it is left ideal of r. hence b is an ideal of r. � theorem 2.6. the product of two bi-ideals of an ag-ring r with left identity 1 is again a bi-ideal of r. proof. let h and k be two bi-ideals of r. then h and k are additive ag-subgroup of r. thus using medial and rr = r, we get [(hk)r](hk) = [(hk)(rr)](hk), by medial = [(hr)(kr)](hk), by medial = [(hr)h][(kr)k], h,k is a bi-ideal of r ⊆ hk. hence hk is a bi-ideal of r. � theorem 2.7. let b be a bi-ideal of an ag-ring r and a be a left ideal of r with left identity 1, then ba is a bi-ideal of r. proof. since a is a left ideal of r and b is a bi-ideal of an ag-ring r, we have ba is an additive ag-subgroup of r. thus [(ba)r](ba) = [(ra)b](ba) = [(ba)b](ra) ⊆ [(br)b]a, b is a bi-ideal of r ⊆ ba. it following that ba is a bi-ideal of r. � theorem 2.8. let b be a bi-ideal of an ag-ring r and a be a right ideal of r with left identity 1. if a ⊆ b and bb ⊆ b, then ab is a bi-ideal of r. 6 int. j. anal. appl. (2023), 21:50 proof. since a is a right ideal of r and b is a bi-ideal of an ag-ring r, we have ab is an additive ag-subgroup of r. let a ⊆ b and bb ⊆ b. then using lemma 1.1, we get [(ab)r](ab) = [(rb)a](ab) = [(ab)a](rb) ⊆ [(ar)a](rb), b ⊆ r ⊆ (aa)(rb), a is a right ideal of r = (ar)(ab), by γ-medial ⊆ a(ab), a is a right ideal of r ⊆ a(bb), a ⊆ b ⊆ ab, bb ⊆ b. it follows that ab is a bi-ideal of r. � theorem 2.9. let r be an ag-ring and a,b be bi-ideals of an ag-ring r. then a∩b is a bi-ideal of r. proof. since a,b is bi-ideals of an ag-ring r, we have a ∩ b is an additive ag-subgroup of r. thus [(a∩b)r](a∩b) ⊆ (ar)(a∩b) = [(a∩b)r]a ⊆ (ar)a ⊆ a and [(a∩b)r](a∩b) ⊆ (br)(a∩b) = [(a∩b)r]b ⊆ (br)b ⊆ b. it following that a∩b is a bi-ideal of r. � corollary 2.2. let r be a γ-ag-ring and hi is a bi-ideal of r, for all i ∈ i. then ⋂ i∈i hi is a bi-ideal of r. proof. since 0 ∈ hi for all i ∈ i, we have 0 ∈ ⋂ i hi. then ⋂ i hi 6= ∅. since hi is a bi-ideal of r, we have hi is an additive ag-subgroup of r let x,y ∈ hi then x − y ∈ hi. thus x − y ∈ ⋂ i hi. let x,y ∈ ⋂ i hi, r ∈ r. then (xr)y ∈ (hir)hi ⊆ hi for all i ∈ i thus (xr)y ∈ hi. hence ⋂ i∈i hi is a bi-ideal of r. � theorem 2.10. let i and l be respectively right and left ag-subgroup of r. then any ag-subgroup b of r such that il ⊆ b ⊆ i ∩l is a bi-ideal of r. proof. since b is a ag-subgroup of (r, +) with il ⊆ b ⊆ i ∩l we have (br)b ⊆ ((i ∩l)r)(i ∩l), by b ⊆ i ∩l ⊆ (ir)l, by s ⊆ i ∩l and l ⊆ i ∩l ⊆ il, by i is a right ideal of r ⊆ b, by il ⊆ b. then b is a bi-ideal of r. � corollary 2.3. intersection of an arbitrary set of bi-ideal bλ (λ ∈ ∧) of an ag-ring r is again a bi-ideal of r. int. j. anal. appl. (2023), 21:50 7 proof. set b := ⋂ λ∈∧bλ. since b is an ag-subgroup of r. from the inclusion (bλr)bλ ⊆ bλ and b ⊆ bλ. this implies that (br)b ⊆ (bλr)bλ ⊆ bλ (∀λ ∈∧). hence (br)b ⊆ b. � theorem 2.11. every idempotent quasi-ideal is a bi-ideal. proof. let q be an idempotent quasi-ideal of γ-ag-ring r. then (qr)q ⊆ (rr)q ⊆ rq and by lemma 1.2 (qr)q ⊆ (qr)γ(qq) = (qq)(rq) ⊆ q(rq) ⊆ q(rr) ⊆ qr which implies that (qr)q ⊆ qr∩rq ⊆ q. � definition 2.4. an ag-ring r is called a regular ag-ring if for any x ∈ r there exists y ∈ r such that x = (xy)x. theorem 2.12. let r be a regular of an ag-ring and b be a bi-ideal of r. then (br)b = b. proof. since b is a bi-ideal of r we have (br)b ⊆ b. let x ∈ b then there exist a ∈ r such that x = (xa)x ∈ (br)b, since r is a regular of an ag-ring. this implies that b ⊆ (br)b so (br)b = b. � theorem 2.13. for a quasi-ideal q in a regular ag-ring r, then qr∩rq = q. proof. let q be a quasi-ideal in r then qr∩rq ⊆ q. let x ∈ q then there exist a ∈ r such that x = (xa)x, since r is a regular of a ag-ring. so x = (xa)x ∈ (qr)q ⊆ qr and x = (xa)x ∈ (qr)q ⊆ (rr)q ⊆ rq then x ∈ qr∩rq. thus q ⊆ qr∩rq. hence qr∩rq = q. � acknowledgements: this research project was supported by the thailand science research and innovation fund and the department of mathematics, school of science, university of phayao, phayao 56000, thailand. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 8 int. j. anal. appl. (2023), 21:50 references [1] n. nobusawa, on generalization of the ring theory, osaka j. math. 1 (1964), 81-89. [2] m.a. kazim, md. naseeruddin, on almost semigroup, portugaliae math. 36 (1977), 41-47. [3] m. khan, faisal, v. amjid, on some classes of abel-grassmann’s groupoids, arxiv:1010.5965, (2010). https: //doi.org/10.48550/arxiv.1010.5965. [4] q. mushtaq, m. khan, m-system in la-semigroups, southeast asian bull. math. 33 (2009), 321-327. [5] m. sarwar (kamran), conditions for la-semigroup to resemble associative structures, phd thesis, quaid-i-azam university, 1993. [6] t. shah, i. rehman, on γ-ideal and γ-bi-ideals in γ-ag-groupoids, int. j. algebra. 4 (2010), 267-276. [7] t. shah, i. rehman, on la-rings of finitely nonzero function, int. j. contemp. math. sci. 5 (2010), 209-222. [8] w.e. banes, on the γ-ring of nobusawa, pac. j. math. 18 (1996), 411-422. https://doi.org/10.2140/pjm. 1966.18.411. [9] g. pilz, near-rings, north holland, amsterdam, 1997. [10] t. shah, f. ur rehman, m. raees, on near left almost rings, int. math. forum, 6 (2011), 1103-1111. [11] t. shah, i. rehman, on la-rings of finitely nonzero function, int. j. contemp. math. sci. 5 (2010), 209-222. https://doi.org/10.48550/arxiv.1010.5965 https://doi.org/10.48550/arxiv.1010.5965 https://doi.org/10.2140/pjm.1966.18.411 https://doi.org/10.2140/pjm.1966.18.411 1. introduction 2. main results references international journal of analysis and applications issn 2291-8639 volume 4, number 1 (2014), 78-86 http://www.etamaths.com solvability of extended general strongly mixed variational inequalities balwant singh thakur abstract. in this paper, a new class of extended general strongly mixed variational inequalities is introduced and studied in hilbert spaces. an existence theorem of solution is established and using resolvent operator technique, a new iterative algorithm for solving the extended general strongly mixed variational inequality is suggested. a convergence result for the iterative sequence generated by the new algorithm is also established. 1. introduction and preliminaries variational inequality theory, which was introduced by stampacchia [24] in 1964, has had a great impact and influence in the development of several branches on pure and applied sciences. a useful and important generalization of variational inequality is the general mixed variational inequality containing a nonlinear term ϕ. finding fixed points of a nonlinear mapping is an equally important problem in the functional analysis. equivalent fixed point formulation of a variational inequality problem, has given a new dimension to the study of solution of variational inequality problems. in many problems of analysis, one encounters operators who may be split in the form s = a ± t, where a and t satisfies some conditions, and s itself has neither of these properties. an early theorem of this type was given by krasnoselskii [12], where a complicated operator is split into the sum of two simpler operators. there is another setting arises from perturbation theory. here the operator equation tx ± ax = x is considered as a perturbation of tx = x (or ax = x), and one would like to assert that the original unperturbed equation has a solution. in such a situation, there is, in general, no continuous dependence of solutions on the perturbations. for various results in this direction, please see [4, 7, 8, 11, 22, 26]. another argument is concerned with the approximate solution of the problem: for f ∈ h, find x ∈ h such that tx±ax = f. here t,a : h → h are given operators. many boundary value problems for quasi linear partial differential equations arising in physics, fluid mechanics and other areas of applications can be formulated as the equation tx±ax = f, see, e.g. zeidler [28]. combettes and hirstoaga [5] showed that the finding of zeros of sum of two operators can be solved via the variational inequality involving sum of two operators. several authors study this 2010 mathematics subject classification. 47j20, 65k10, 65k15, 90c33. key words and phrases. extended general strongly mixed variational inequality; fixed point problem; resolvent operator technique; relaxed cocoercive mapping; maximal monotone operator. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 78 strongly mixed variational inequalities 79 type of situations, see, e.g. [6, 21] and references therein. motivated by these facts, in this paper we study a variational inequality problem involving operator of the form t −a. let h be a real hilbert space whose inner product and norm are denoted by 〈·, ·〉 and ‖·‖, respectively. let ϕ : h → r∪{+∞} be a proper convex lower semicontinuous function. let t : h → h be a nonlinear operator and g,h : h → h are any mappings. we consider the problem of finding x∗ ∈ h such that (1) 〈t(x∗) −a(x∗),h(y∗) −g(x∗)〉 + ϕ(h(y∗)) −ϕ(g(x∗)) ≥ 0 , ∀y∗ ∈ h , where a is a nonlinear continuous mapping on h and ∂ϕ denotes the subdifferential of ϕ. we call inequality (1) as extended general strongly mixed variational inequality. we now consider some special cases of the problem (1) : (1) if a ≡ 0, then the problem (1) reduces to the extended general mixed variational inequality problem considered in [20] (2) if h is an identity mapping on h, then the problem (1) reduces to the problem studied by [10]. (3) if a ≡ 0 and h ≡ g, then the problem (1) reduces to the general mixed variational inequality problem considered in [2, 17, 18, 19]. (4) if h,g be identity mappings on h, then the problem (1) reduces to a class of variational inequality studied by [25]. (5) if a ≡ 0 and h,g be identity mappings on h, then the problem (1) reduces to the mixed variational inequality or variational inequality of second kind see [1, 9, 15, 16]. for a multivalued operator t : h → h, we denote by d(t) = {u ∈ h : t(u) 6= ∅} , the domain of t , r(t) = ⋃ u∈h t(u) , the range of t , graph(t) = {(u,u∗) ∈ h ×h : u ∈ d(t) and u∗ ∈ t(u)} , the graph of t. definition 1.1. t is called monotone if and only if for each u ∈ d(t), v ∈ d(t) and u∗ ∈ t(u), v∗ ∈ t(v), we have 〈v∗ −u∗,v −u〉≥ 0 . t is maximal monotone if it is monotone and its graph is not properly contained in the graph of any other monotone operator. t−1 is the operator defined by v ∈ t−1(u) ⇔ u ∈ t(v) . definition 1.2 (see [3]). for a maximal monotone operator t , the resolvent operator associated with t , for any σ > 0, is defined as jt (u) = (i + σt) −1(u) , ∀u ∈ h . 80 thakur it is known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. furthermore, the resolvent operator is single-valued and nonexpansive i.e. ‖jt (x) −jt (y)‖ ≤ ‖x−y‖ , ∀x,y ∈ h. in particular, it is well known that the subdifferential ∂ϕ of ϕ is a maximal monotone operator; see [13]. lemma 1.3. [3] for a given z ∈ h , u ∈ h satisfies the inequality 〈u−z,x−u〉 + λϕ(x) −λϕ(u) ≥ 0 , ∀x ∈ h if and only if u = jϕ(z), where jϕ = (i + λ∂ϕ) −1 is the resolvent operator and λ > 0 is a constant. inequality (1), can be written in an equivalent form as follows: find x∗ ∈ h such that (2) 〈ρ(t(x∗) −a(x∗)) + g(x∗) −h(x∗),h(y∗) −g(x∗)〉 + ρϕ(h(y∗)) −ρϕ(g(x∗)) ≥ 0 , for all y∗ ∈ h . this equivalent formulation plays an important role in the development of iterative methods for solving the mixed variational inequality problem (1). using lemma 1.3, we will establish following important relation: lemma 1.4. x∗ ∈ h is a solution of (2) if and only if x∗ satisfies the following relation (3) g(x∗) = jϕ (h(x ∗) −ρ(t(x∗) −a(x∗))) , where ρ > 0 is a constant and jϕ = (i + ρ∂ϕ) −1 is the proximal mapping, i stands for the indentity operator on h. proof. let x∗ ∈ h be a solution of problem (2), then (4) 〈g(x∗) − (h(x∗) −ρ (t(x∗) −a(x∗))) ,h(y∗) −g(x∗)〉+ ρϕ(h(y∗))−ρϕ(g(x∗)) ≥ 0 , for all y∗ ∈ h. applying lemma 1.3 for λ = ρ, inequality (4) is equivalent to g(x∗) = jϕ (h(x ∗) −ρ (t(x∗) −a(x∗))) , the required result. � lemma 1.4 implies that the problem (2) is equivalent to the fixed point problem (3). this alternative equivalent formulation provides a natural connection between variatonal inequality problem (2) and the fixed point theory which will be used to prove existence result. the following lemma is in this sense : lemma 1.5. x∗ ∈ h is a solution of (2) if and only if x∗ is a fixed point of the mapping f given by (5) f(u) = u−g(u) + jϕ (h(u) −ρ(t(u) −a(u))) , u ∈ h . proof. let x∗ ∈ h be a fixed point of the mapping f. then g(x∗) = jϕ (h(x ∗) −ρ(t(x∗) −a(x∗))) . from lemma 1.4, x∗ is a solution of (2). � strongly mixed variational inequalities 81 we now recall some some definitions: definition 1.6. an operator t : h → h is said to be : (i) strongly monotone, if for each x ∈ h, there exists a constant ν > 0 such that 〈t(x) −t(y),x−y〉≥ ν‖x−y‖2 holds, for all y ∈ h; (ii) φ−cocoercive, if for each x ∈ h, there exists a constant φ > 0 such that 〈t(x) −t(y),x−y〉≥−φ‖t(x) −t(y)‖2 holds, for all y ∈ h; (iii) relaxed (φ,γ)−cocoercive or relaxed cocoercive with respect to constant (φ,γ), if for each x ∈ h, there exists constants γ > 0 and φ > 0 such that 〈t(x) −t(y),x−y〉≥−φ‖t(x) −t(y)‖2 + γ‖x−y‖2 holds, for all y ∈ h; (iv) µ−lipschitz continuous or lipschitz with respect to constant µ, if for each x,y ∈ h, there exists a constant µ > 0 such that ‖t(x) −t(y)‖≤ µ‖x−y‖ . 2. main results lemma 1.5, is the main motivation for our next result: theorem 2.1. let h be a real hilbert space and t,a,g,h : h → h are operators. suppose that the following assumptions are satisfied : (i) t,g,h are relaxed cocoercive with constants (φt ,γt ), (φg,γg), (φh,γh) respectively, (ii) t,a,g,h are lipschitz mappings with constants µt ,µa,µg,µh respectively. if 1 + µ2g(1 + 2φg) > 2γg , 1 + µ 2 h(1 + 2φh) > 2γh , and (6) ρ ∈ (( γt −φt µ2t ) − √ d µ2t + µ 2 a , ( γt −φt µ2t ) + √ d µ2t + µ 2 a ) , where d := (φt µ 2 t −γt ) 2 − 1 2 (µ2t + µ 2 a)(1 + κ(2 −κ)) > 0 κ = √ 1 − 2γg + µ2g(1 + 2φg) + √ 1 − 2γh + µ2h(1 + 2φh) , then the problem (2) has a unique solution. proof. it is enough to show that the mapping f defined by (5) has a fixed point. for u ∈ h, set p(u) = t(u) −a(u). 82 thakur for all x 6= y ∈ h, we have ‖f(x) −f(y)‖≤‖x−y − (g(x) −g(y))‖ + ‖jϕ (h(x) −ρ(p(x))) −jϕ (h(y) −ρ(p(y)))‖ ≤‖x−y − (g(x) −g(y))‖ + ‖h(x) −h(y) −ρ (p(x) −p(y))‖ ≤‖x−y − (g(x) −g(y))‖ + ‖x−y − (h(x) −h(y))‖ + ‖x−y −ρ (p(x) −p(y))‖ .(7) since g is relaxed (φg,γg)−cocoercive and µg-lipschitz mapping, we can compute the following: ‖x−y − (g(x) −g(y))‖2 = ‖x−y‖2 − 2〈g(x) −g(y),x−y〉 + ‖g(x) −g(y)‖2 ≤ (1 + µ2g)‖x−y‖ 2 + 2φg ‖g(x) −g(y)‖ 2 − 2γg ‖x−y‖ 2 ≤ ( 1 − 2γg + µ2g(1 + 2φg) ) ‖x−y‖2 .(8) similarly, (9) ‖x−y − (h(x) −h(y))‖2 ≤ ( 1 − 2γh + µ2h(1 + 2φh) ) ‖x−y‖2 . also, ‖x−y −ρ(p(x) −p(y))‖2 = ‖x−y −ρ(t(x) −t(y)) + ρ(a(x) −a(y))‖2 ≤ 2‖x−y −ρ(t(x) −t(y))‖2 + 2ρ2 ‖a(x) −a(y)‖2 ≤ 2‖x−y −ρ(t(x) −t(y))‖2 + 2ρ2µ2a ‖x−y‖ 2 .(10) now, we estimate ‖x−y −ρ(t(x) −t(y))‖2 ≤‖x−y‖2 − 2ρ〈t(x) −t(y),x−y〉 + ρ2 ‖t(x) −t(y)‖2 ≤ ( 1 − 2ργt + 2ρµ2t φt + ρ 2µ2t ) ‖x−y‖2 .(11) substituting (11) into (10), gives (12) ‖x−y −ρ(p(x) −p(y))‖≤ √ 2 (1 − 2ργt + 2ρµ2t φt + ρ2(µ 2 t + µ 2 a)) ‖x−y‖ . substituting (8), (9), (12) into (7), we have ‖f(x) −f(y)‖≤ (κ + f(ρ))‖x−y‖ , where κ = √ 1 − 2γg + µ2g(1 + 2φg + √ 1 − 2γh + µ2h(1 + 2φh) , and f(ρ) = √ 2 (1 − 2ργt + 2ρµ2t φt + ρ2(µ 2 t + µ 2 a)) . from (6), we get that (κ + f(ρ)) < 1, thus f is a contraction mapping and therefore has a unique fixed point in h, which is a solution of variational inequality (2). � remark 2.2. theorem 2.1, extend and improve theorem 3.1 of [20]. strongly mixed variational inequalities 83 if k is closed convex set in h and ϕ(x) = δk (x), for all x ∈ k, where δk is the indicator function of k defined by δk (x) = { 0, if x ∈ k ; +∞, otherwise , then the problem (2) reduces to the following variational inequality problem: consider the problem of finding x∗ ∈ k (13) 〈ρ(t(x∗) −a(x∗)) + g(x∗) −h(x∗), h(y∗) −g(x∗)〉≥ 0 , ∀y∗ ∈ k . we immediately obtain following result from theorem 2.1 : corollary 2.3. let h be a real hilbert space, k be a nonempty closed convex subset of h and t,a : h → h and g,h : k → k are operators. suppose that following assumptions are satisfied : (i) t,g,h are relaxed cocoercive with constants (φt ,γt ), (φg,γg), (φh,γh) respectively, (ii) t,a,g,h are lipschitz mappings with constants µt ,µa,µg,µh respectively. if (6) holds, then the problem (13) has a unique solution. if we take h as identity mapping in (13), we get an inequality, equivalent to the general strongly nonlinear variational inequality studied by siddiqi and ansari [23]. corollary 2.3 partially extends and improves the result of [14, 23]. 3. iterative algorithm and convergence we rewrite the relation (3) in the following form (14) x∗ = x∗ −g(x∗) + jϕ (h(x∗) −ρ(t(x∗) −a(x∗))) . using the fixed point formulation (14), we now suggest and analyze the following iterative methods for solving the variational inequality problem (2). algorithm 1. for a given x0 ∈ h, find the approximate solution xn+1 by the iterative scheme xn+1 = xn −g(xn) + jϕ (h(xn) −ρ (t(xn) −a(xn))) , n = 0, 1, 2, . . . which is called explicit iterative method. algorithm 2. for a given x0 ∈ h, find the approximate solution xn+1 by the iterative scheme xn+1 = xn −g(xn) + jϕ (h(xn+1) −ρ (t(xn+1) −a(xn+1))) , n = 0, 1, 2, . . . which is an implicit iterative method. now, we use algorithm 1 as predictor and algorithm 2 as a corrector to obtain the following predictor-corrector method for solving variational inequality problem (1). algorithm 3. for a given x0 ∈ h, find the approximate solution xn+1 by the iterative scheme yn = xn −g(xn) + jϕ (h(xn) −ρ(txn −axn)) xn+1 = xn −g(xn) + jϕ (h(yn) −ρ(tyn −ayn)) , n = 0, 1, 2, . . . . 84 thakur using algorithm 3, we can suggest following : algorithm 4. for a given x0 ∈ h, find the approximate solution xn+1 by the iterative scheme yn = xn −g(xn) + jϕ (h(xn) −ρ(txn −axn)) xn+1 = (1 −αn)xn + αn (xn −g(xn) + jϕ (h(yn) −ρ(tyn −ayn))) , where n = 0, 1, 2, . . . , {αn} is sequences in [0, 1], satisfying certain conditions. now, we define a more general predictor-corrector iterative method for approximate solvability of variational inequality problem (1). algorithm 5. for a given x0 ∈ h, find the approximate solution xn+1 by the iterative scheme (15) yn = (1 −βn)xn + βn (xn −g(xn) + jϕ (h(xn) −ρ(txn −axn))) xn+1 = (1 −αn)xn + αn (xn −g(xn) + jϕ (h(yn) −ρ(tyn −ayn))) , where n = 0, 1, 2, . . . , {αn}, {βn} are sequences in [0, 1], satisfying certain conditions. we need following result to prove the next result : lemma 3.1. [27] let {an} be a non negative sequence satisfying an+1 ≤ (1 − cn)an + bn , with cn ∈ [0, 1], ∑∞ n=0 cn = ∞, bn = o(cn). then limn→∞an = 0. theorem 3.2. let t,a,g,h satisfy all the assumptions of theorem 2.1, also condition (6) holds and {αn}, {βn} are sequences in [0, 1] for all n ≥ 0 such that∑∞ n=0 αn = ∞. then the approximate sequence {xn} constructed by the algorithm 5 converges strongly to a solution x∗ of (2). proof. for u ∈ h, set pu = tu−au. since x∗ ∈ h is a solution of (1), by (14), we have x∗ = x∗ −g(x∗) + jϕ (h(x∗) −ρ(t(x∗) −a(x∗))) . using (15), we have ‖xn+1 −x∗‖≤ (1 −αn)‖xn −x∗‖ + αn ‖xn −x∗ − (g(xn) −g(x∗))‖ + αn ‖jϕ (h(yn) −ρp(yn)) −jϕ (h(x∗) −ρp(x∗))‖ ≤ (1 −αn)‖xn −x∗‖ + αn √ 1 − 2γg + µ2g(1 + 2φg)‖xn −x ∗‖ + αn ‖h(yn) −h(x∗) −ρ (p(yn) −p(x∗))‖ ≤ (1 −αn)‖xn −x∗‖ + αn √ 1 − 2γg + µ2g(1 + 2φg)‖xn −x ∗‖ + αn ‖yn −x∗ − (h(yn) −h(x∗))‖ + αn ‖yn −x∗ −ρ (p(yn) −p(x∗))‖ ≤ (1 −αn)‖xn −x∗‖ + αn √ 1 − 2γg + µ2g(1 + 2φg)‖xn −x ∗‖ + αn √ 1 − 2γh + µ2h(1 + 2φh)‖yn −x ∗‖ + αn √ 2 (1 − 2ργt + 2ρµ2t φt + ρ2(µ 2 t + µ 2 a))‖yn −x ∗‖ = (1 −αn)‖xn −x∗‖ + αnθg ‖xn −x∗‖ + αn (θh + f(ρ))‖yn −x∗‖ ,(16) strongly mixed variational inequalities 85 where θg = √ 1 − 2γg + µ2g(1 + 2φg) , θh = √ 1 − 2γh + µ2h(1 + 2φh) and f(ρ) = √ 2 (1 − 2ργt + 2ρµ2t φt + ρ2(µ 2 t + µ 2 a)). similarly, we have ‖yn −x∗‖≤ (1 −βn)‖xn −x∗‖ + βn ‖xn −x∗ − (g(xn) −g(x∗))‖ + βn ‖jϕ (h(xn) −ρp(xn)) −jϕ (h(x∗) −ρp(x∗))‖ ≤ (1 −βn)‖xn −x∗‖ + βnθg ‖xn −x∗‖ + βn ‖h(xn) −h(x∗) −ρ (p(xn) −p(x∗))‖ ≤ (1 −βn)‖xn −x∗‖ + βnθg ‖xn −x∗‖ + βn ‖xn −x∗ − (h(xn) −h(x∗))‖ + βn ‖xn −x∗ −ρ (p(xn) −p(x∗))‖ ≤ (1 −βn)‖xn −x∗‖ + βnθg ‖xn −x∗‖ + βnθh ‖xn −x∗‖ + βnf(ρ)‖xn −x∗‖ = (1 −βn)‖xn −x∗‖ + βn(κ + f(ρ))‖xn −x∗‖ ≤ (1 −βn)‖xn −x∗‖ + βn ‖xn −x∗‖ = ‖xn −x∗‖ .(17) substituting (17) into (16), yields that ‖xn+1 −x∗‖≤ (1 −αn)‖xn −x∗‖ + αn(θg + θh + f(ρ))‖xn −x∗‖ = (1 −αn (1 − (κ + f(ρ))))‖xn −x∗‖ .(18) by virtue of lemma 3.1, we get from (18) that, limn→∞‖xn+1 −x∗‖ = 0, i.e. xn → x∗, as n →∞. this completes the proof. � remark 3.3. theorem 3.2, extend and improve theorem 2.1 of [10] and theorem 3.2 of [20]. it is well known that, if ϕ(·) is the indicator function of k in h, then jϕ = pk , the projection operator of h onto the closed convex set k, and consequently, the following result can be obtain from theorem 3.2. corollary 3.4. let t,a,g,h satisfy all the assumptions of corollary 2.3. let x0 ∈ k, construct a sequence {xn} in k by yn = xn −g(xn) + pk (h(xn) −ρ(txn −axn)) xn+1 = (1 −αn)xn + αn (xn −g(xn) + pk (h(yn) −ρ(tyn −ayn))) , n = 0, 1, 2, . . . , where {αn}, {βn} are sequences in [0, 1] for all n ≥ 0 such that ∑∞ n=0 αn = ∞. then the sequence {xn} converges strongly to a solution x∗ of (13). references [1] baiocchi,c., capelo,a.: variational and quasi variational inequalies. j. wile and sons, new york (1984). [2] bnouhachem,a., noor,m.a., al-shemas,e.h.: on self-adaptive method for general mixed variational inequalities, math. prob. engineer. (2008), doi: 10.1155/2008/280956. [3] brezis,h.: opérateurs maximaux monotone et semi-groupes de contractions dans les espaces de hilbert. in: north-holland mathematics studies. 5, notas de matematics, vol. 50, northholland, amsterdam (1973). 86 thakur [4] browder,f.e.: fixed point theorems for nonlinear semicontractive mappings in banach spaces. arch. rat. mech. anal. 21, 259–269 (1966). [5] combettes,p.l., hirstoaga,s.a.: visco-penalization of the sum of two monotone operators. nonlinear anal. 69, 579–591 (2008). [6] dhage,b.c.: remarks on two fixed-point theorems involving the sum and the product of two operators. comput. math. appl. 46, 1779–1785 (2003). [7] fucik,s.: fixed point theorems for a sum of nonlinear mapping. comment. math. univ. carolinae 9, 133-143 (1968). [8] fucik,s.: solving of nonlinear operator equations in banach space. comment. math. univ. carolinae 10, 177–186 (1969). [9] glowinski,r., lions,j.l., tremolieres,r.: numerical analysis of variational inequalities. north-holland, amesterdam, holland (1981). [10] hassouni,a., moudafi,a.: perturbed algorithm for variational inclusions. j. math. anal. appl. 185, 706-712 (1994). [11] kirk,w.a.: on nonlinear mappings of strongly semicontractive type. j. math. anal. appl. 27, 409–412 (1969). [12] krasnoselskii,m.a.: two remarks of the method of successive approximations. uspeki mat. nauk 10, 123–127 (1955). [13] minty,h.j.: on the monotonicity of the gradient of a convex function. pacific j. math. 14, 243–247 (1964). [14] noor,m.a.: strongly nonlinear variational inequalities. c.r. math. rep. acad. sci. canad. 4, 213–218 (1982). [15] noor,m.a.: on a class of variational inequalities. j. math. anal. appl. 128, 135–155 (1987). [16] noor,m.a.: a class new iterative methods for general mixed variational inequalities. math. comput. modell. 31, 11–19 (2000). [17] noor,m.a.: modified resolvent splitting algorithms for general mixed variational inequalities. j. comput. appl. math. 135, 111–124 (2001). [18] noor,m.a.: operator-splitting methods for general mixed variational inequalities. j. ineq. pure appl. math. 3(5), art.67, 9p. (2002) http://eudml.org/doc/123617. [19] noor,m.a.: psueudomontone general mixed variational inequalities. appl. math. comput. 141, 529–540 (2003). [20] noor,m.a., ullah,s., noor,k.i., al-said,e.: iterative methods for solving extended general mixed variational inequalities. comput. math. appl. 62, 804–813 (2011). [21] o’regan, d.: fixed point theory for the sum of two operators. appl. math. lett. 9, 1–8 (1996). [22] petryshyn,w.v.: remarks on fixed point theorems and their extensions. trans. amer. math. soc. 126, 43–53 (1967). [23] siddiqi,a.h., ansari,q.h.: general strongly nonlinear variational inequalities. j. math. anal. appl. 166, 386–392 (1992). [24] stampacchia,g.: formes bilineares sur les ensemble convexes. c. r. acad. sci. paris 285, 4413–4416 (1964). [25] verma,r.u.: generalized auxiliary problem principle and solvability of a class of nonlinear variational inequalities involoving cocoercive and co-lipschitzian mappings. j. ineq. pure appl. math. 2(3), art.27, 9p. (2001) http://eudml.org/doc/122114. [26] webb,j.r.l.: fixed point theorems for nonlinear semicontractive operators in banach spaces. j. london math. soc. 1, 683–688 (1969). [27] weng,x.l.: fixed point iteration for local stricly pseudo-contractive mappings. proc. amer. math. soc. 113, 727–731 (1991). [28] zeidler,e.: nonlinear functional analysis and its applications, ii/b : nonlinear monotone operators. springer, new york (1990). school of studies in mathematics, pt.ravishankar shukla university, raipur, 492010, india international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 147-153 http://www.etamaths.com global existence and blow-up of solutions for a quasilinear parabolic equation with absorption and nonlinear boundary condition iftikhar ahmed∗, chunlai mu and pan zheng abstract. this paper deals with the evolution r-laplacian equation with absorption and nonlinear boundary condition. by using differential inequality techniques, global existence and blow-up criteria of nonnegative solutions are determined. moreover, upper bound of the blow-up time for the blow-up solution is obtained. 1. introduction in this paper, we investigate the global existence and finite time blow-up of nonnegative solutions for the following initial-boundary value problem  ut = div(|∇u|r−2∇u) −f(u), (x,t) ∈ ω × (0, t∗), |∇u|r−2 ∂u ∂n = g(u), (x,t) ∈ ∂ω × (0, t∗), u (x, 0) = u0 (x) > 0, x ∈ ω, (1.1) where r ≥ 2, ∂u ∂n is the outward normal derivative of u on the boundary ∂ω assumed sufficiently smooth, ω is a bounded star-shaped region in rn (n ≥ 2) and t∗ is the blow-up time if blow-up occurs, or else t∗ = ∞. it is well known that the functions f and g may greatly affect the behavior of the solution u(x,t) with the development of time. from the physical standpoint, −f is the cold source function, g is the heat-conduction function transmitting into interior of ω from the boundary of ω. the global existence and blow-up for nonlinear parabolic equations have been extensively investigated by many authors in the last decades (see [1–6] and the references therein). in recent years, many authors have also studied bounds for the blow-up time in nonlinear parabolic problems by using differential inequality techniques (see [7–12]). in particular, payne et al. [13] considered the following semilinear heat equation with nonlinear boundary condition  ut = ∆u−f(u), (x,t) ∈ ω × (0, t∗), ∂u ∂n = g(u), (x,t) ∈ ∂ω × (0, t∗), u (x, 0) = u0 (x) , x ∈ ω, (1.2) 2010 mathematics subject classification. 35k55, 35k65. key words and phrases. global existence; blow-up; quasilinear parabolic equation; nonlinear boundary condition. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 147 148 ahmed, mu and zheng and established sufficient conditions on the nonlinearities to guarantee that the solution u(x,t) exists for all time t > 0 or blows up in finite time t∗. moreover, an upper bound for t∗ was derived. under more restrictive conditions, a lower bound for t∗ was also obtained. moreover, in [14], payne et al. also studied the following initial-boundary problem   ut = ∇(|∇u|2p∇u), (x,t) ∈ ω × (0, t∗), |∇u|2p ∂u ∂n = f(u), (x,t) ∈ ∂ω × (0, t∗), u (x, 0) = u0 (x) , x ∈ ω, (1.3) and obtained upper and lower bounds for the blow-up time under some conditions when blow-up does occur at some finite time. in the present work, by using differential inequality techniques, we give some sufficient conditions on the functions f and g for the global existence and blow-up of nonnegative solutions to problem (1.1). our main results are stated as follows. theorem 1.1. (conditions for global existence). let u(x,t) be the solution of problem (1.1) and assume that the non-negative functions f and g satisfy the following conditions (1.4) f(ξ) ≥ k1ξp, ξ ≥ 0, (1.5) g(ξ) ≤ k2ξq, ξ ≥ 0, for some non-negative constants k1 and k2. moreover suppose that the positive constants p and q satisfy the following conditions (1.6) p > q > r − 1 and rq < (r − 1)(p + 1). then the non-negative solution u(x,t) of problem (1.1) exists globally for all time t > 0. theorem 1.2.(conditions for blow-up in finite time). let u(x,t) be the solution of problem (1.1) and assume that the non-negative functions f and g satisfy the following conditions (1.7) ξf(ξ) ≤ rf(ξ), ξ ≥ 0, (1.8) ξg(ξ) ≥ rg(ξ), ξ ≥ 0, with (1.9) f(ξ) = ∫ ξ 0 f(η)dη, g(ξ) = ∫ ξ 0 g(η)dη. moreover suppose that ψ(0) > 0, where (1.10) ψ(t) = r ∫ ∂ω g(u)ds− ∫ ω |∇u|rdx−r ∫ ω f(u)dx. then the solution u(x,t) of problem (1.1) blows up at time t∗ < t with (1.11) t = φ(0) (r − 2)ψ(0) , for r > 2, quasilinear parabolic equation 149 where φ(t) = ∫ ω u2dx. if r = 2, we have t = ∞. this paper is organized as follows. in section 2, we establish the conditions on the functions f and g, which guarantee that u(x,t) exists globally, and prove theorem 1.1. in section 3, we obtain the blow-up condition of the solution and derive an upper bound estimate for the blow-up time t∗. 2. conditions for global existence in this section, we establish the sufficient conditions on the functions f and g, which guarantee that u(x,t) exists globally, and prove theorem 1.1. to do this, we need the following lemma. lemma 2.1. let ω be a bounded star-shaped domain in rn , n ≥ 2. then for any non-negative c1 function u and γ > 0, we have (2.1) ∫ ∂ω uγds ≤ n ρ0 ∫ ω uγdx + γd ρ0 ∫ ω uγ−1|∇u|dx, where (2.2) ρ0 = min x∈∂ω (x ·n) and d = max x∈∂ω |x|. proof. as ω is a bounded star-shaped domain, it is easy to see that ρ0 > 0. integrating the identity (2.3) div(uγx) = nuγ + γuγ−1(x ·∇u) over ω, it follows from the divergence theorem that (2.4) ∫ ∂ω uγ(x ·n)ds = n ∫ ·ω uγdx + γ ∫ ω uγ−1(x ·∇u)dx. by the definition of ρ0 and d, we obtain (2.5) ρ0 ∫ ∂ω uγds ≤ ∫ ∂ω uγ(x ·n)ds ≤ n ∫ ·ω uγdx + γd ∫ ω uγ−1|∇u|dx, which implies the desired conclusion. proof of theorem 1.1. setting (2.6) φ(t) = ∫ ω u2dx, then it follows from (1.1), (1.4) and (1.5) that φ′(t) = 2 ∫ ω uutdx = 2 ∫ ω u[div(|∇u|r−2∇u) −f(u)]dx = 2 ∫ ∂ω u|∇u|r−2 ∂u ∂n ds− 2 ∫ ω |∇u|rdx− 2 ∫ ω uf(u)dx = 2 ∫ ∂ω ug(u)ds− 2 ∫ ω |∇u|rdx− 2 ∫ ω uf(u)dx ≤ 2k2 ∫ ∂ω uq+1ds− 2 ∫ ω |∇u|rdx− 2k1 ∫ ω up+1dx. (2.7) 150 ahmed, mu and zheng by lemma 2.1, we have (2.8) ∫ ∂ω uq+1ds ≤ n ρ0 ∫ ω uq+1dx + (q + 1)d ρ0 ∫ ω uq|∇u|dx, where ρ0 and d are given by (2.2). combining (2.7) with (2.8), we obtain (2.9) φ′(t) ≤ 2k2n ρ0 ∫ ω uq+1dx+ 2k2(q + 1)d ρ0 ∫ ω uq|∇u|dx−2 ∫ ω |∇u|rdx−2k1 ∫ ω up+1dx. by using young’s inequality with ε > 0, we derive (2.10) ∫ ω uq|∇u|dx ≤ 1 rε ∫ ω |∇u|rdx + r − 1 r ε 1 r−1 ∫ ω u rq r−1 dx, where ε = k2(q+1)d rρ0 > 0. it follows from (2.9) and (2.10) that (2.11) φ′(t) ≤ 2k2n ρ0 ∫ ω uq+1dx + 2(r − 1) ( k2(q + 1)d rρ0 ) r r−1 ∫ ω u rq r−1 dx− 2k1 ∫ ω up+1dx. by hölder’s inequality , we have (2.12) ∫ ω u rq r−1 dx ≤ (∫ ω uq+1dx )α (∫ ω up+1dx )1−α , where α = (r−1)(p+1)−rq (r−1)(p−q) ∈ (0, 1), due to (1.6). by using the fundamental inequality (2.13) ar11 a r2 2 ≤ r1a1 + r2a2, a1,a2 > 0,r1,r2 ≥ 0 and r1 + r2 = 1, it follows from (2.12) that∫ ω u rq r−1 dx ≤ ( κ α−1 α ∫ ω uq+1dx )α ( κ ∫ ω up+1dx )1−α ≤ ακ α−1 α ∫ ω uq+1dx + (1 −α)κ ∫ ω up+1dx, (2.14) where (2.15) 0 < κ < k1 (r − 1)(1 −α) ( k2(q + 1)d rρ0 ) −r r−1 . combining (2.11) with (2.14), we obtain (2.16) φ′(t) ≤ k1 ∫ ω uq+1dx−k2 ∫ ω up+1dx, where (2.17) k1 = 2k2n ρ0 + 2(r − 1)ακ α−1 α ( k2(q + 1)d rρ0 ) r r−1 > 0, and (2.18) k2 = 2k1 − 2(r − 1)(1 −α)κ ( k2(q + 1)d rρ0 ) r r−1 > 0, due to (2.15). according to hölder’s inequality, we derive (2.19) ∫ ω uq+1dx ≤ (∫ ω up+1dx )q+1 p+1 |ω| p−q p+1 , quasilinear parabolic equation 151 where |ω| = ∫ ω dx is the n-volume of ω. it follows from (2.16) and (2.19) that (2.20) φ′(t) ≤ (∫ ω up+1dx )q+1 p+1 [ k1|ω| p−q p+1 −k2 (∫ ω up+1dx )p−q p+1 ] . by hölder’s inequality again, we have (2.21) φ(t) = ∫ ω u2dx ≤ (∫ ω up+1dx ) 2 p+1 |ω| p−1 p+1 . therefore, we deduce from (2.20) and (2.21) that (2.22) φ′(t) ≤ (∫ ω up+1dx )q+1 p+1 [ k1|ω| p−q p+1 −k2|ω| (1−p)(p−q) 2(p+1) φ p−q 2 ] . hence, we infer from (2.22) that φ(t) is decreasing in each time interval on which we have (2.23) φ(t) > ( k1 k2 ) 2 p−q |ω|, so that φ(t) remains bounded for all time under the conditions stated in theorem 1.1, which completes the proof. � 3. conditions for blow-up in finite time in this section, we obtain the blow-up condition of the solution and derive an upper bound estimate for the blow-up time t∗. proof of theorem 1.2. using green formula and the assumptions stated in theorem 1.2, we have φ′(t) = 2 ∫ ω uutdx = 2 ∫ ω u[div(|∇u|r−2∇u) −f(u)]dx = 2 ∫ ∂ω u|∇u|r−2 ∂u ∂n ds− 2 ∫ ω |∇u|rdx− 2 ∫ ω uf(u)dx = 2 ∫ ∂ω ug(u)ds− 2 ∫ ω |∇u|rdx− 2 ∫ ω uf(u)dx ≥ 2r ∫ ∂ω g(u)ds− 2 ∫ ω |∇u|rdx− 2r ∫ ω f(u)dx ≥ 2ψ(t). (3.1) differentiating (1.10), we obtain ψ′(t) = r ∫ ∂ω utg(u)ds− ∫ ω (|∇u|r)tdx−r ∫ ω utf(u)dx = r ∫ ω utdiv(|∇u|r−2∇u)dx−r ∫ ω utf(u)dx = r ∫ ω (ut) 2dx ≥ 0. (3.2) 152 ahmed, mu and zheng as ψ(0) > 0, then ψ(t) > 0 for all t ∈ (0, t∗). by using hölder’s inequality , we derive (3.3) (φ′(t))2 = 4 (∫ ω uutdx )2 ≤ 4 ∫ ω u2dx ∫ ω (ut) 2dx = 4 r φ(t)ψ′(t). it follows from (3.1) and (3.3) that (3.4) φ(t)ψ′(t) ≥ r 4 (φ′(t))2 ≥ r 2 φ′(t)ψ(t), that is (3.5) (φ− r 2 ψ)′(t) ≥ 0. integrating from 0 to t, we have (3.6) φ− r 2 (t)ψ(t) ≥ φ− r 2 (0)ψ(0) =: k > 0. therefore, we deduce from (3.1) that (3.7) φ′(t) ≥ 2ψ ≥ 2kφ r 2 (t). if r > 2, it follows from integrating over (0, t) that (3.8) φ(t) ≥ [ φ 2−r 2 (0) −k(r − 2)t ]− 2 r−2 , which implies φ(t) → +∞ as t → t = φ 2−r 2 (0) k(r−2) = φ(0) (r−2)ψ(0) . hence, for r > 2, we have (3.9) t∗ ≤ φ(0) (r − 2)ψ(0) . if r = 2, we infer from (3.7) that (3.10) φ(t) ≥ φ(0)e2kt, for all t > 0, which implies t∗ = ∞, this completes the proof. � references [1] b. straughan, explosive instabilities in mechanics, springer, berlin, 1998. [2] r. quittner and p. souplet, superlinear parabolic problems: blow-up, global existence and steady states, birkhäuser advanced texts, basel, 2007. [3] h.a. levine,nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded fourier cofficients, math. ann. 214(1975), 205-220. [4] v.a. galaktionov and j.l. vazquez, the problem of blow-up in nonlinear parabolic equations, discrete contin. dyn. syst. 8(2002), 399-433. [5] z.q. ling and z.j. wang, global existence and blow-up for a degenerate reaction-diffusion system with nonlocal source, appl. math. lett. 25(2012), 2198-2202. [6] f.s. li and j. l. li, global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions, j. math. anal. appl. 385(2012), 1005-1014. [7] y.f. li, y. liu and c.h. lin, blow-up phenomena for some nonlinear parabolic problems under mixed boundary conditions, nonlinear anal. rwa 11(2010), 3815-3823. [8] l.e. payne, g.a. philippin and p.w. schaefer, blow-up phenomena for some nonlinear parabolic problems, nonlinear anal. tma 69(2008), 3495-3502. [9] l.e. payne, g.a. philippin and p.w. schaefer, bounds for blow-up time in nonlinear parabolic problems, j. math. anal. appl. 338(2008), 438-447. quasilinear parabolic equation 153 [10] j.c. song, lower bounds for the blow-up time in a non-local reaction-diffusion problem, appl. math. lett. 24(2011), 793-796. [11] f. liang, blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux, nonlinear anal. tma 75(2012), 2189-2198. [12] d.m. liu, c.l. mu and q. xin, lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation, acta math. sci. ser. b engl. ed. 32(2012), 1206-1212. [13] l.e. payne, g.a. philippin and s. vernier piro, blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, i, z. angew. math. phys. 61(2010), 999-1007. [14] l.e. payne, g.a. philippin and s. vernier piro, blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, ii, nonlinear anal. tma 73(2010), 971-978. college of mathematics and statistics, chongqing university, chongqing 401331, china ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 168-182 http://www.etamaths.com on multi-valued weakly picard operators in hausdorff metric-like spaces abdelbasset felhi1,2,∗ abstract. in this paper, we study multi-valued weakly picard operators on hausdorff metric-like spaces. our results generalize some recent results and extend several theorems in the literature. some examples are presented making effective our results. 1. introduction and preliminaries let (x,d) be a metric space and cb(x) denotes the collection of all nonempty, closed and bounded subsets of x. also, cl(x) denotes the collection of nonempty closed subsets of x. for a,b ∈ cb(x), define h(a,b) := max { sup a∈a d(a,b), sup b∈b d(b,a) } , where d(x,a) := inf{d(x,a) : a ∈ a} is the distance of a point x to the set a. it is known that h is a metric on cb(x), called the hausdorff metric induced by d. throughout the paper, n, r, and r+ denote the set of positive integers, the set of all real numbers and the set of all non-negative real numbers, respectively. definition 1.1. ([1]) let (x,d) be a metric space and t : x → cl(x) be a multi-valued operator. we say that t is a multi-valued weakly picard operator (mwp operator) if for all x ∈ x and y ∈ tx, there exists a sequence {xn} such that: (i) x0 = x,x1 = y; (ii) xn+1 ∈ txn for all n = 0, 1, 2, . . . ; (iii) the sequence {xn} is convergent and its limit is a fixed point of t. the theory of mwp operators is studied by several authors (see for instance [1, 2]). in 2008 suzuki [3] generalizes the banach contraction principle by introducing a new type of mapping. very recently, jleli et al. [4] established kikkawa-suzuki type fixed point theorems for a new type of generalized contractive conditions on partial hausdorff metric spaces. the purpose of this paper is to discuss multi-valued weakly picard operators on partial hausdorff metric spaces and on hausdorff metric-like spaces. we will establish the above fixed point theorems for a new type of generalized contractive conditions which generalizes that of jleli et al. we recall that the study of fixed points for multi-valued contractions using the hausdorff metric was initiated by nadler [18] who proved the following theorem. theorem 1.2. ([18]) let (x,d) be a complete metric space and t : x → cb(x) be a multi-valued mapping satisfying h(tx,ty) ≤ kd(x,y) for all x,y ∈ x and for some k in [0, 1). then there exists x ∈ x such that x ∈ tx. we recall that the notion of partial metric spaces was introduced by matthews [8] in 1994 as a part to study the denotational semantics of dataflow networks which play an important role in constructing models in the theory of computation. moreover, the notion of metric-like spaces has been discovered by amini-harandi [12] which is an interesting generalization of the notion of partial metric spaces. for more fixed point results on metric-like spaces, see [7], [10], [11], [13], [15], [16], [17], [19], [20], [21], [22]. 2010 mathematics subject classification. 47h10, 54h25. key words and phrases. hausdorff metric-like: multi-valued operator; partial metric space; metric-like space. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 168 on multi-valued weakly picard operators 169 note that, every partial metric space is a metric-like space but the converse is not true in general. in what follows, we recall some definitions and results we will need in the sequel. definition 1.3. ([8]) a partial metric on a nonempty set x is a function p : x ×x → [0,∞) such that for all x,y,z ∈ x (pm1) p(x,x) = p(x,y) = p(y,y), then x = y; (pm2) p(x,x) ≤ p(x,y); (pm3) p(x,y) = p(y,x); (pm4) p(x,z) + p(y,y) ≤ p(x,y) + p(y,z). the pair (x,p) is then called a partial metric space (pms). according to [8], each partial metric p on x generates a t0 topology τp on x which has as a base the family of open p−balls {bp(x,ε) : x ∈ x, ε > 0}, where bp(x,ε) = {y ∈ x : p(x,y) < p(x,x) + ε} for all x ∈ x and ε > 0. following [8], several topological concepts can be defined as follows. a sequence {xn} in a partial metric space (x,p) converges to a point x ∈ x if and only if p(x,x) = limn→∞p(xn,x) and is called a cauchy sequence if limn,m→∞p(xn,xm) exists and is finite. moreover, a partial metric space (x,p) is called to be complete if every cauchy sequence {xn} in x converges, with respect to τp, to a point x ∈ x such that p(x,x) = limn,m→∞p(xn,xm). it is known [8] that if p is a partial metric on x, then the function ps : x ×x → r+ defined by ps(x,y) = 2p(x,y) −p(x,x) −p(y,y) for all x,y ∈ x, is a metric on x. note that if a sequence converges in a partial metric space (x,p) with respect to τps, then it converges with respect to τp. also, a sequence {xn} in a partial metric space (x,p) is cauchy if and only if it is a cauchy sequence in the metric space (x,ps). consequently, a partial metric space (x,p) is complete if and only if the metric space (x,ps) is complete. moreover, if {xn} is a sequence in a partial metric space (x,p) and x ∈ x, one has that lim n→∞ ps(xn,x) = 0 ⇔ p(x,x) = lim n→∞ p(xn,x) = lim n,m→∞ p(xn,xm). we have the following lemmas. lemma 1.4. let (x,p) be a partial metric space. then, (1) if p(x,y) = 0 then, x = y, (2) if x 6= y then, p(x,y) > 0. following [9], let (x,p) be a partial metric space and cbp(x) be the family of all nonempty, closed and bounded subsets of the partial metric space (x,p), induced by the partial metric p. for a,b ∈ cbp(x) and x ∈ x, we define p(x,a) = inf{p(x,a) : a ∈ a}, hp(a,b) = max{sup a∈a p(a,b), sup b∈b p(b,a)}. lemma 1.5. ([5]) let (x,p) be a partial metric space and a any nonempty set in (x,p), then a ∈ a if and only if p(a,a) = p(a,a), where a denotes the closure of a with respect to the partial metric p. proposition 1.6. ([9]) let (x,p) be a partial metric space. for all a,b,c ∈ cbp(x), we have (h1) hp(a,a) ≤ hp(a,b); (h2) hp(b,a) = hp(a,b); (h3) hp(a,b) ≤ hp(a,c) + hp(c,b) − infc∈c p(c,c); (h4) hp(a,b) = 0 ⇒ a = b. definition 1.7. let x be a nonempty set. a function σ : x × x → r+ is said to be a metric-like (dislocated metric) on x if for any x,y,z ∈ x, the following conditions hold: (p1) σ(x,y) = 0 =⇒ x = y; (p2) σ(x,y) = σ(y,x); (p3) σ(x,z) ≤ σ(x,y) + σ(y,z). the pair (x,σ) is then called a metric-like (dislocated metric) space. 170 felhi in the following example, we give a metric-like which is neither a metric nor a partial metric. example 1.8. let x = {0, 1, 2} and σ : x ×x → r+ defined by σ(0, 0) = σ(1, 1) = 0, σ(2, 2) = 3, σ(0, 1) = 1, σ(0, 2) = σ(1, 2) = 2, and σ(x,y) = σ(y,x) for all x ∈ x. then, (x,σ) is a metric-like space. note that σ is nor a metric as, σ(2, 2) > 0 and not a partial metric on x as, σ(2, 2) > σ(0, 2). each metric-like σ on x generates a t0 topology τσ on x which has as a base the family open σ-balls {bσ(x,ε) : x ∈ x,ε > 0}, where bσ(x,ε) = {y ∈ x : |σ(x,y)−σ(x,x)| < ε}, for all x ∈ x and ε > 0. observe that a sequence {xn} in a metric-like space (x,σ) converges to a point x ∈ x, with respect to τσ, if and only if σ(x,x) = lim n→∞ σ(x,xn). definition 1.9. let (x,σ) be a metric-like space. (a) a sequence {xn} in x is said to be a cauchy sequence if lim n,m→∞ σ(xn,xm) exists and is finite. (b) (x,σ) is said to be complete if every cauchy sequence {xn} in x converges with respect to τσ to a point x ∈ x such that lim n→∞ σ(x,xn) = σ(x,x) = lim n,m→∞ σ(xn,xm). we have the following trivial inequality: (1.1) σ(x,x) ≤ 2σ(x,y) for all x,y ∈ x. very recently, aydi et al. [6] introduced the concept of hausdorff metric-like. let cbσ(x) be the family of all nonempty, closed and bounded subsets of the metric-like space (x,σ), induced by the metric-like σ. note that the boundedness is given as follows: a is a bounded subset in (x,σ) if there exist x0 ∈ x and m ≥ 0 such that for all a ∈ a, we have a ∈ bσ(x0,m), that is, |σ(x0,a) −σ(a,a)| < m. the closeness is taken in (x,τσ) (where τσ is the topology induced by σ). for a,b ∈ cbσ(x) and x ∈ x, define σ(x,a) = inf{σ(x,a), a ∈ a}, δσ(a,b) = sup{σ(a,b) : a ∈ a} and δσ(b,a) = sup{σ(b,a) : b ∈ b}. we have the the following useful lemmas. lemma 1.10. [6] let (x,σ) be a metric-like space and a any nonempty set in (x,σ), then if σ(a,a) = 0, then a ∈ ā, where a denotes the closure of a with respect to the metric-like σ. also, if {xn} is a sequence in (x,σ) that is τσ-convergent to x ∈ x, then lim n→∞ |σ(xn,a) −σ(x,a)| = σ(x,x). lemma 1.11. let a,b ∈ cbσ(x) and a ∈ a. suppose that σ(a,b) > 0. then, for each h > 1, there exists b = b(a) ∈ b such that σ(a,b) < hσ(a,b). proof. we argue by contradiction, that is, there exists h > 1, such that for all b ∈ b, there is σ(a,b) ≥ hσ(a,b). then, σ(a,b) = inf{σ(a,b) : b ∈ b} ≥ hσ(a,b). hence, h ≤ 1, which is a contradiction. � let (x,σ) be a metric-like space. for a,b ∈ cbσ(x), define hσ(a,b) = max{δσ(a,b),δσ(b,a)} . we have also some properties of hσ : cb σ(x) ×cbσ(x) → [0,∞). on multi-valued weakly picard operators 171 proposition 1.12. [6] let (x,σ) be a metric-like space. for any a,b,c ∈ cbσ(x), we have the following: (i) : hσ(a,a) = δσ(a,a) = sup{σ(a,a) : a ∈ a}; (ii) : hσ(a,b) = hσ(b,a); (iii) : hσ(a,b) = 0 implies that a = b; (iv) : hσ(a,b) ≤ hσ(a,c) + hσ(c,b). the mapping hσ : cb σ(x) × cbσ(x) → [0, +∞) is called a hausdorff metric-like induced by σ. note that each partial hausdorff metric is a hausdorff metric-like but the converse is not true in general as it is clear from the following example. example 1.13. going back to example 1.8, taking a = {2}, b = {0} we have hσ(a,a) = σ(2, 2) = 3 > 2 = σ(0, 2) = hσ(a,b). we denote by ψ the class of all functions ψ : r+ → r+ satisfying (ψ1) ψ is nondecreasing; (ψ2) ∑ n ψn(t) < ∞ for each t ∈ r+, where ψn is the n−th iterate of ψ. also, we denote by φ the class of all functions ϕ : r+ → r+ satisfying (ϕ1) ϕ is nondecreasing; (ϕ2) t ≤ ϕ(t) for each t ∈ r+. lemma 1.14. (i) if ψ ∈ ψ, then ψ(t) < t for any t > 0 and ψ(0) = 0. (ii) if ϕ ∈ φ, then t ≤ ϕn(t) for all n ∈ n∪{0} and for any t ∈ r+. we have the following useful lemma. lemma 1.15. let (x,σ) be a metric-like space, b ∈ cbσ(x) and c > 0. if a ∈ x and σ(a,b) < c then there exists b = b(a) ∈ b such that σ(a,b) < c. proof. we argue by contradiction, that is, σ(a,b) ≥ c for all b ∈ b, then σ(a,b) = inf{σ(a,b) : b ∈ b}≥ c, which is a contradiction. hence there exists b = b(a) ∈ b such that σ(a,b) < c. � 2. main results in this section, we give some fixed point results on metric-like spaces first and next we give some fixed point results on partial metric spaces. now, we need the following definition. definition 2.1. let (x,σ) be a metric-like space. a multi-valued mapping t : x → cbσ(x) is said to be (ϕ,ψ)−contractive multi-valued operator if there exist ϕ ∈ φ and ψ ∈ ψ such that (2.1) σ(y,tx) ≤ ϕ(σ(y,x)) ⇒ hσ(tx,ty) ≤ ψ(mσ(x,y)) for all x,y ∈ x, where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 1 4 [σ(x,ty) + σ(tx,y)]}. now, we state and prove our first main result. theorem 2.2. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be (ϕ,ψ)−contractive multi-valued operator. if 2t ≤ ϕ(t) for each t ∈ r+, then t is an mwp operator. proof. let x0 ∈ x and x1 ∈ tx0. let c a given real number such that σ(x0,x1) < c. clearly, if x1 = x0 or x1 ∈ tx1, we conclude that x1 is a fixed point of t and so the proof is finished. now, we assume that x1 6= x0 and x1 6∈ tx1. so then, σ(x0,x1) > 0 and σ(x1,tx1) > 0. since x1 ∈ tx0 and 2t ≤ ϕ(t), we get σ(x1,tx0) ≤ σ(x1,x1) ≤ 2σ(x1,x0) ≤ ϕ(σ(x1,x0)). 172 felhi hence by (2.1) and triangular inequality, we have 0 < σ(x1,tx1) ≤ hσ(tx0,tx1) ≤ ψ(mσ(x0,x1)) ≤ ψ(max{σ(x0,x1),σ(x0,tx0),σ(x1,tx1), 1 4 [σ(x0,tx1) + σ(x1,tx0)]}) ≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,tx1), 1 4 [σ(x0,tx1) + σ(x1,x1)]}) ≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,tx1), 1 4 [σ(x1,tx1) + 3σ(x0,x1)]}) = ψ(max{σ(x0,x1),σ(x1,tx1)}). if max{σ(x0,x1),σ(x1,tx1)} = σ(x1,tx1), then we obtain 0 < σ(x1,tx1) ≤ ψ(σ(x1,tx1)) < σ(x1,tx1) wish is a contradiction. then 0 < σ(x1,tx1) ≤ ψ(σ(x0,x1)) < ψ(c). thus, by lemma 1.15, there exist x2 ∈ tx1 such that σ(x1,x2) < ψ(c).(2.2) if x1 = x2 or x2 ∈ tx2, we conclude that x2 is a fixed point of t and so the proof is finished. now, we assume that x2 6= x1 and x2 6∈ tx2. then we have σ(x2,tx1) ≤ σ(x2,x2) ≤ 2σ(x2,x1) ≤ ϕ(σ(x2,x1)). hence by (2.1), triangular inequality and (2.2), we have 0 < σ(x2,tx2) ≤ hσ(tx1,tx2) ≤ ψ(mσ(x1,x2)) ≤ ψ(max{σ(x1,x2),σ(x2,tx2)}) = ψ(σ(x1,x2)) < ψ 2(c). then, by lemma 1.15, there exist x3 ∈ tx2 such that σ(x2,x3) < ψ 2(c).(2.3) continuing in this fashion, we construct a sequence {xn} in x such that for all n ∈ n (i) xn 6∈ txn, xn 6= xn+1, xn+1 ∈ txn; (ii) (2.4) σ(xn,xn+1) ≤ ψn(c). now, for m > n, we have σ(xn,xm) ≤ m−1∑ i=n σ(xi,xi+1) ≤ m−1∑ i=n ψi(c) ≤ ∞∑ i=n ψi(c) → 0 as n →∞. thus, lim n,m→∞ σ(xn,xm) = 0.(2.5) hence, {xn} is σ−cauchy. moreover since (x,σ) is complete, it follows there exists ν ∈ x such that lim n→∞ σ(xn,ν) = σ(ν,ν) = lim n,m→∞ σ(xn,xm) = 0.(2.6) we will show that ν is a fixed point of t. first, we should prove that there exits a subsequence {xn(k)} of {xn} such that σ(ν,txn(k)) ≤ ϕ(σ(ν,xn(k))), for all k = 0, 1, 2, . . .(2.7) arguing by contradiction, that is, there exists n ∈ n such that σ(ν,txn) > ϕ(σ(ν,xn)) for all n ≥ n. since xn+1 ∈ txn, it follows that σ(ν,xn+1) > ϕ(σ(ν,xn)) for all n ≥ n. having ϕ nondecreasing, so by induction we get σ(ν,xn+m) > ϕ m(σ(ν,xn)), for all n ≥ n and m = 1, 2, 3, . . .(2.8) on multi-valued weakly picard operators 173 now, for all n ≥ n and m ∈ n, we have σ(xn,xn+m) ≤ n+m−1∑ i=n σ(xi,xi+1) ≤ m−1∑ i=n ψi(c) ≤ ∞∑ i=n ψi(c). then for all n ≥ n and m ∈ n, we obtain σ(ν,xn) ≤ σ(ν,xn+m) + σ(xn+m,xn) ≤ σ(ν,xn+m) + ∞∑ i=n ψi(c). passing to the limit as m →∞, we get σ(ν,xn) ≤ ∞∑ i=n ψi(c). this implies that for all n ≥ n and m ∈ n, σ(ν,xn+m) ≤ ∞∑ i=n+m ψi(c).(2.9) combining (2.8) and (2.9), we have σ(ν,xn) ≤ ϕm(σ(ν,xn)) < σ(ν,xn+m) ≤ ∞∑ i=n+m ψi(c). then for all n ≥ n and m ∈ n, we obtain σ(ν,xn) < ∞∑ i=n+m ψi(c).(2.10) letting m →∞ in (2.10), we get σ(ν,xn) = 0 for all n ≥ n and so, σ(ν,xn+m) = 0 for all n ≥ n and m ∈ n. using (2.8), we have 0 ≤ ϕm(0) < 0, which is a contradiction. therefore, (2.7) holds. now, we will show that σ(ν,tν) = 0. suppose in the contrary, that is σ(ν,tν) > 0. by (2.1) and (2.7), we have for all k ∈ n σ(ν,tν) ≤ σ(ν,xn(k)+1) + σ(xn(k)+1,tν) ≤ σ(ν,xn(k)+1) + hσ(txn(k),tν) ≤ σ(ν,xn(k)+1) + ψ(mσ(xn(k),ν)) ≤ σ(ν,xn(k)+1) + ψ(max{σ(xn(k),ν),σ(xn(k),txn(k)),σ(ν,tν), 1 4 [σ(xn(k),tν) + σ(ν,txn(k))]}) ≤ σ(ν,xn(k)+1) + ψ(max{σ(xn(k),ν),σ(xn(k),xn(k)+1),σ(ν,tν), 1 4 [σ(xn(k),tν) + σ(ν,xn(k)+1)]}). we know that lim k→∞ σ(xn(k),ν) = lim k→∞ σ(xn(k),xn(k)+1) = lim k→∞ σ(xn(k)+1,ν) = 0, lim k→∞ σ(xn(k),tν) = σ(ν,tν). then there exists n ∈ n such that for all k ≥ n max{σ(xn(k),ν),σ(xn(k),xn(k)+1),σ(ν,tν), 1 4 [σ(xn(k),tν) + σ(ν,xn(k)+1)]}) = σ(ν,tν). it follows that for all k ≥ n 0 < σ(ν,tν) ≤ σ(ν,xn(k)+1) + ψ(σ(ν,tν)). passing to the limit as k →∞, we get 0 < σ(ν,tν) ≤ ψ(σ(ν,tν)) < σ(ν,tν) which is a contradiction. hence σ(ν,tν) = 0 and so, by lemma 1.10 we have ν ∈ tν = tν, that is ν is a fixed point of t. � we give an example to illustrate the utility of theorem 2.2. 174 felhi example 2.3. let x = {0, 1, 2} and σ : x ×x → r+ defined by: σ(0, 0) = 0, σ(1, 1) = 3, σ(2, 2) = 1 σ(0, 1) = σ(1, 0) = 7, σ(0, 2) = σ(2, 0) = 3, σ(1, 2) = σ(2, 1) = 4. then (x,σ) is a complete metric-like space. note that σis not a partial metric on x because σ(0, 1) � σ(2, 0) + σ(2, 1) −σ(2, 2). define the map t : x → cbσ(x) by t0 = t2 = {0} and t1 = {0, 2}. note that tx is bounded and closed for all x ∈ x in metric-like space (x,σ). take ϕ(t) = st with s ≥ 7 and ψ(t) = rt with r ∈ [ 3 4 , 1). it is easy tho show that max{σ(y,tx), x,y ∈ x} = σ(1, 0) = 7 ≤ 7 min{σ(y,x), x,y ∈ x, (x,y) 6= (0, 0)} ≤ ϕ(min{σ(y,x), x,y ∈ x, (x,y) 6= (0, 0)}). this implies that, for all x,y ∈ x with (x,y) 6= (0, 0) σ(y,tx) ≤ ϕ(σ(y,x)). now, we shall show that for all x,y ∈ x with (x,y) 6= (0, 0) hσ(tx,ty) ≤ ψ(mσ(x,y)).(2.11) for this, we consider the following cases: case1 : x,y ∈{0, 2}. we have hσ(tx,ty) = σ(0, 0) = 0 ≤ ψ(mσ(x,y)). case2 : x ∈{0, 2}, y = 1. we have hσ(tx,ty) = hσ({0},{0, 2}) = max{σ(0,{0, 2}),max{σ(0, 0),σ(0, 2)}} = max{0, 3} = 3 ≤ 3 4 σ(x,y) ≤ ψ(mσ(x,y)). case3 : x = y = 1. we have hσ(tx,ty) = hσ({0, 2},{0, 2}) = max{σ(0,{0, 2}),σ(2,{0, 2})} = min{σ(0, 2),σ(2, 2)} = 1 ≤ 3 4 σ(1, 1) ≤ ψ(mσ(x,y)). note that (2.11) is also true for (x,y) = (0, 0). then, all the required hypotheses of theorem 2.2 are satisfied. here, x = 0 is the unique fixed point of t we state the following corollaries as consequences of theorem 2.2. corollary 2.4. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be a multi-valued mapping. assume that there exist ϕ ∈ φ and ψ ∈ ψ such that, for all x,y ∈ x hσ(tx,ty) ≤ ψ(mσ(x,y)) −ϕ(σ(y,x)) + σ(y,tx),(2.12) where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 14 [σ(x,ty) + σ(tx,y)]}. if 2t ≤ ϕ(t) for each t ∈ r+, then t is an mwp operator. proof. let x,y ∈ x such that σ(y,tx) ≤ ϕ(σ(y,x)). then, if (2.12) holds, we have hσ(tx,ty) ≤ ψ(mσ(x,y)) −ϕ(σ(y,x)) + σ(y,tx) ≤ ψ(mσ(x,y)). thus, the proof is concluded by theorem 2.2. � corollary 2.5. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ x σ(y,tx) ≤ sσ(y,x) ⇒ hσ(tx,ty) ≤ rmσ(x,y), where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 14 [σ(x,ty) + σ(tx,y)]}. then t is an mwp operator. proof. it suffice to take ϕ(t) = st and ψ(t) = rt in theorem 2.2. � on multi-valued weakly picard operators 175 corollary 2.6. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ x σ(y,tx) ≤ sσ(y,x) ⇒ hσ(tx,ty) ≤ r max{σ(x,y),σ(x,tx),σ(y,ty)}. then t is an mwp operator. corollary 2.7. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 2 such that, for all x,y ∈ x σ(y,tx) ≤ sσ(y,x) ⇒ hσ(tx,ty) ≤ r 3 {σ(x,y) + σ(x,tx) + σ(y,ty)}. then t is an mwp operator. corollary 2.8. [6] let (x,σ) be a complete metric-like space. if t : x → cbσ(x) is a multi-valued mapping such that for all x,y ∈ x, we have (2.13) hσ(tx,ty) ≤ k m(x,y), where k ∈ [0, 1) and m(x,y) = max { σ(x,y),σ(x,tx),σ(y,ty), 1 4 (σ(x,ty) + σ(y,tx)) } . then t has a fixed point. proof. let ϕ(t) = 2t and ψ(t) = kt. then, if (2.13) holds, we have hσ(tx,ty) ≤ ψ(m(x,y)), for all x,y ∈ x satisfying σ(y,tx) ≤ 2σ(y,x). thus, the proof is concluded by theorem 2.2. � if t is a single-valued mapping, we deduce the following results. corollary 2.9. let (x,σ) be a complete metric-like space and t : x → x be a mapping. assume that there exist ϕ ∈ φ and ψ ∈ ψ such that, for all x,y ∈ x σ(y,tx) ≤ ϕ(σ(y,x)) ⇒ σ(tx,ty) ≤ ψ(mσ(x,y)), where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 14 [σ(x,ty) + σ(tx,y)]}. if 2t ≤ ϕ(t) for each t ∈ r+ and if ψ(2t) < t for each t > 0, then t has a unique fixed point. proof. the existence follows immediately from theorem 2.2. thus, we need to prove uniqueness of fixed point. we assume that there exist x,y ∈ x such that x = tx and y = ty with x 6= y. since σ(y,tx) = σ(y,x) ≤ ϕ(σ(y,x)), then by (2.1) and since ψ(2t) < t , we get 0 < σ(x,y) = σ(tx,ty) ≤ ψ(max{σ(x,y),σ(x,tx),σ(y,ty), 1 4 [σ(x,ty) + σ(tx,y)]}) = ψ(max{σ(x,y),σ(x,x),σ(y,y), 1 2 σ(x,y)}) ≤ ψ(2σ(x,y)) < σ(x,y). which is a contradiction. hence x = y, so the uniqueness of the fixed point of t. � corollary 2.10. let (x,σ) be a complete metric-like space and t : x → x be a mapping. assume that there exist r ∈ [0, 1 2 ) and s ≥ 2 such that, for all x,y ∈ x σ(y,tx) ≤ s(σ(y,x)) ⇒ σ(tx,ty) ≤ r(mσ(x,y)), where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 14 [σ(x,ty) + σ(tx,y)]}. then t has a unique fixed point. now, we need the following definition. 176 felhi definition 2.11. let (x,σ) be a metric-like space. a multi-valued mapping t : x → cbσ(x) is said to be (r,s)−contractive multi-valued operator if there exist r,s ∈ [0, 1), such that (2.14) 1 1 + r σ(x,tx) ≤ σ(y,x) ≤ 1 1 −s σ(x,tx) ⇒ hσ(tx,ty) ≤ rmσ(x,y) for all x,y ∈ x, where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 1 4 [σ(x,ty) + σ(tx,y)]}. we give the following result. theorem 2.12. let (x,σ) be a complete metric-like space and t : x → cbσ(x) be (r,s)−contractive multi-valued operator with r < s. then t is an mwp operator. proof. let r1 be a real number such that 0 ≤ r ≤ r1 < s. let x0 ∈ x. clearly, if x0 ∈ tx0, then x0 is a fixed point of t and so, the proof is finished. now, we assume that x0 6∈ tx0. then σ(x0,tx0) > 0. by lemma 1.11, there exists x1 ∈ tx0 such that σ(x0,x1) ≤ 1 −r1 1 −s σ(x0,tx0). if x1 ∈ tx1, then x1 is a fixed point of t and also, the proof is finished. now, we assume that x1 6∈ tx1. then σ(x1,tx1) > 0. since 1 1 + r σ(x0,tx0) ≤ σ(x0,x1)) ≤ 1 −r1 1 −s σ(x0,tx0), then, by (2.14), we have σ(x1,tx1) ≤ hσ(tx0,tx1) ≤ r max{σ(x0,x1),σ(x0,tx0),σ(x1,tx1), 1 4 [σ(x0,tx1) + σ(x1,tx0)]} ≤ r max{σ(x0,x1),σ(x0,x1),σ(x1,tx1), 1 4 [σ(x1,tx1) + 3σ(x0,x1)]} ≤ r max{σ(x0,x1),σ(x1,tx1)}. if max{σ(x0,x1),σ(x1,tx1)} = σ(x1,tx1), then we obtain σ(x1,tx1) ≤ rσ(x1,tx1) < σ(x1,tx1), which is a contradiction. thus, we get σ(x1,tx1) ≤ rσ(x0,x1). by lemma 1.11, there exists x2 ∈ tx1 such that σ(x1,x2) ≤ r1 r σ(x1,tx1) and σ(x1,x2) ≤ 1 −r1 1 −s σ(x1,tx1). this implies that σ(x1,x2) ≤ r1σ(x0,x1) and σ(x1,x2) ≤ 1 −r1 1 −s σ(x1,tx1). it follows that 1 1 + r σ(x1,tx1) ≤ σ(x1,tx2) ≤ 1 1 −s σ(x1,tx1). then, by (2.14), we get σ(x2,tx2) ≤ rσ(x1,x2). continuing this process, we construct a sequence {xn} in x such that (i) xn+1 ∈ txn; (ii) σ(xn,txn) ≤ rσ(xn−1,xn); (iii) σ(xn,xn+1) ≤ r1σ(xn−1,xn); (iv) σ(xn,xn+1) ≤ 1−r11−s σ(xn,txn) on multi-valued weakly picard operators 177 for all n = 1, 2, . . . since σ(xn,xn+1) ≤ r1σ(xn−1,xn), by induction we obtain σ(xn,xn+1) ≤ rn1 σ(x0,x1) for all n = 1, 2, . . . now, for m > n, we have σ(xn,xm) ≤ m−1∑ i=n σ(xi,xi+1) ≤ σ(x0,x1) m−1∑ i=n ri1 ≤ σ(x0,x1) ∞∑ i=n ri1 → 0 as n →∞. thus, lim n,m→∞ σ(xn,xm) = 0.(2.15) hence, {xn} is σ−cauchy. moreover since (x,σ) is complete, it follows there exists z ∈ x such that lim n→∞ σ(xn,z) = σ(z,z) = lim n,m→∞ σ(xn,xm) = 0.(2.16) for all m,n ∈ n, we have σ(xn,xn+m) ≤ σ(xn,xn+1) + σ(xn+1,xn+2) + . . . + σ(xn+m−1,xn+m) ≤ [1 + r1 + r21 + . . . + r m−1 1 ]σ(xn,xn+1) = 1 −rm1 1 −r1 σ(xn,xn+1). it follows that for all m,n ∈ n σ(xn,z) ≤ σ(xn,xn+m) + σ(xn+m,z) ≤ σ(xn+m,z) + 1 −rm1 1 −r1 σ(xn,xn+1). passing to limit as m →∞, we get for all n ∈ n σ(xn,z) ≤ 1 1 −r1 σ(xn,xn+1) ≤ 1 1 −r1 · 1 −r1 1 −s σ(xn,txn) = 1 1 −s σ(xn,txn). thus, we have for all n ∈ n σ(xn,z) ≤ 1 1 −s σ(xn,txn).(2.17) now, we assume that there exists n ∈ n such that 1 1 + r σ(xn,txn) > σ(xn,z) for all n ≥ n. then we have σ(xn,xn+1) ≤ σ(xn,z) + σ(z,xn+1) < 1 1 + r [σ(xn,txn) + σ(xn+1,txn+1)] < 1 1 + r [σ(xn,xn+1) + rσ(xn,xn+1)] = σ(xn,xn+1). which is a contradiction. thus, there exists a subsequence {xn(k)} of {xn} such that 1 1 + r σ(xn(k),txn(k)) ≤ σ(xn(k),z)(2.18) for all k ∈ n. now, we should show that z is a fixed point of t. using (2.17), (2.18) and (2.14), we have for all k ∈ n σ(xn(k)+1,tz) ≤ hσ(txn(k),tz) ≤ r max{σ(xn(k),z),σ(xn(k),txn(k)),σ(z,tz), 1 4 [σ(xn(k),tz) + σ(z,txn(k))]}≤ r max{σ(xn(k),z),σ(xn(k),xn(k)+1),σ(z,tz), 1 4 [σ(xn(k),tz) + σ(z,xn(k)+1)]}. passing to limit as k →∞, we get σ(z,tz) ≤ rσ(z,tz). since r < 1, it follows that σ(z,tz) = 0. thus, by lemma 1.10 we obtain z ∈ tz, that is, z is a fixed point of t. � 178 felhi corollary 2.13. let (x,σ) be a complete metric-like space and t : x → x be a mapping. assume that there exist r ∈ [0, 1) such that, for all x,y ∈ x 1 1 + r σ(x,tx) ≤ σ(x,y) ≤ 1 1 −r σ(x,tx) ⇒ σ(tx,ty) ≤ rmσ(x,y), where mσ(x,y) = max{σ(x,y),σ(x,tx),σ(y,ty), 14 [σ(x,ty) + σ(tx,y)]}. then t has a fixed point. proof. let x0 ∈ x. define the sequence {xn} by xn+1 = txn for all n = 0, 1, 2, . . . we have for all n = 0, 1, 2, . . . 1 1 + r σ(xn,txn) ≤ σ(xn,xn+1) ≤ 1 1 −r σ(xn,txn) it follows that for all n = 0, 1, 2, . . . σ(xn+1,xn+2) = σ(txn,txn+1) ≤ rσ(xn,xn+1). thus the sequence {xn} is cauchy in (x,σ). by completeness of (x,σ) there exists z ∈ x such that lim n→∞ σ(xn,z) = σ(z,z) = lim n,m→∞ σ(xn,xm) = 0.(2.19) we have for all n,m ∈ n σ(xn,xn+m) ≤ 1 −rm 1 −r σ(xn,xn+1). it follows that σ(xn,z) ≤ σ(xn,xn+m) + σ(xn+m,z) ≤ 1 −rm 1 −r σ(xn,xn+1) + σ(xn+m,z). passing to limit as m →∞, we get σ(xn,z) ≤ 1 1 −r σ(xn,xn+1) proceeding as in the proof of theorem 2.12, we can find a subsequence {xn(k)} of {xn} such that 1 1 + r σ(xn(k),txn(k)) ≤ σ(xn(k),z)(2.20) for all k ∈ n. then as in the proof of theorem 2.12 we get z is a fixed point of t. � we give the following illustrative example inspired from [4]. example 2.14. let x = {0, 1, 2} and σ : x ×x → r+ defined by: σ(0, 0) = σ(2, 2) = 1 4 , σ(1, 1) = 0, σ(0, 1) = σ(1, 0) = 1 3 , σ(0, 2) = σ(2, 0) = 2 5 , σ(1, 2) = σ(2, 1) = 11 15 . then (x,σ) is a complete metric-like space. note that σis not a partial metric on x as σ(1, 2) > σ(1, 0) + σ(0, 2) −σ(0, 0). define the map t : x → cbσ(x) by t0 = t1 = {1} and t2 = {0, 1}. note that tx is bounded and closed for all x ∈ x in metric-like space (x,σ). we have max{σ(x,tx), x ∈ x} = max{σ(0, 1),σ(1, 1),σ(2, 0)} = 2 5 , min{σ(x,tx), x ∈ x −{1}} = 1 3 . therefore, we have 1 4 ≤ σ(x,y) ≤ 11 15 on multi-valued weakly picard operators 179 for all x,y ∈ x with (x,y) 6= (1, 1). it follows that 1 1 + r σ(x,tx) ≤ σ(x,y) ≤ 1 1 −s σ(x,tx) for all x,y ∈ x with x 6= 1 and for some 3 5 ≤ r < s < 1. observe that the above inequalities are also true for x = y = 1 but not hold for x = 1 and y ∈{0, 2}. now, we shall show that hσ(tx,ty) ≤ rmσ(x,y)(2.21) for all x,y ∈ x for some 5 6 ≤ r < 1. for this, we consider the following cases: case1 : x,y ∈{0, 1}, with (x,y) 6= (1, 0). we have hσ(tx,ty) = σ(1, 1) = 0 ≤ rmσ(x,y). case2 : x = 0, y = 2. we have hσ(tx,ty) = hσ({1},{0, 1}) = max{σ(1,{0, 1}),max{σ(1, 1),σ(1, 0)}} = 1 3 ≤ 5 6 σ(x,y) ≤ rmσ(x,y) case3 : x = y = 2. we have hσ(tx,ty) = hσ({0, 1},{0, 1}) = max{σ(0,{0, 1}),σ(1,{0, 1})} = min{σ(0, 0),σ(0, 1)} = 1 4 moreover, we have mσ(2, 2) = max{σ(2, 2),σ(2,t2)} = max{14, 2 5 } = 2 5 . then for x = y = 2 we get hσ(t2,t2) = 1 4 ≤ 5 6 . 2 5 ≤ rmσ(2, 2). then, all the required hypotheses of theorem 2.12 are satisfied. here, x = 1 is the unique fixed point of t. now, we need the following definition. definition 2.15. let (x,p) be a partial metric space. a multi-valued mapping t : x → cbp(x) is said to be (ϕ,ψ)−contractive multi-valued operator if there exist ϕ ∈ φ and ψ ∈ ψ such that (2.22) p(y,tx) ≤ ϕ(p(y,x)) ⇒ hp(tx,ty) ≤ ψ(mp(x,y)) for all x,y ∈ x, where mp(x,y) = max{p(x,y),p(x,tx),p(y,ty), 1 2 [p(x,ty) + p(tx,y)]}. we give the following result. theorem 2.16. let (x,p) be a complete partial metric space and t : x → cbp(x) be (ϕ,ψ)−contractive multi-valued operator. then t is an mwp operator. proof. let x0 ∈ x and x1 ∈ tx0. let c a given real number such that p(x0,x1) < c. clearly, if x1 = x0 or x1 ∈ tx1, we conclude that x1 is a fixed point of t and so the proof is finished. now, we assume that x1 6= x0 and x1 6∈ tx1. so then, p(x0,x1) > 0 and p(x1,tx1) > 0. since x1 ∈ tx0, we get p(x1,tx0) ≤ p(x1,x1) ≤ p(x1,x0) ≤ ϕ(p(x1,x0)). hence by (2.22) and triangular inequality, we have 0 < p(x1,tx1) ≤ hp(tx0,tx1) ≤ ψ(mp(x0,x1)) ≤ ψ(max{p(x0,x1),p(x0,tx0),p(x1,tx1), 1 2 [p(x0,tx1) + p(x1,tx0)]}) ≤ ψ(max{p(x0,x1),p(x1,tx1), 1 2 [p(x0,tx1) + p(x1,x1)]}) ≤ ψ(max{σ(x0,x1),σ(x0,x1),σ(x1,tx1), 1 2 [p(x1,tx1) + p(x0,x1)]}) = ψ(max{p(x0,x1),p(x1,tx1)}) = ψ(p(x0,x1)) < ψ(c). 180 felhi proceeding as in the proof of theorem 2.2, we construct a sequence {xn} in x such that for all n ∈ n (i) xn 6∈ txn, xn 6= xn+1, xn+1 ∈ txn; (ii) (2.23) p(xn,xn+1) ≤ ψn(c). now, for m > n, we have p(xn,xm) ≤ m−1∑ i=n p(xi,xi+1) − m−1∑ i=n+1 p(xi,xi) ≤ m−1∑ i=n ψi(c) ≤ ∞∑ i=n ψi(c) → 0 as n →∞. thus, lim n,m→∞ p(xn,xm) = 0.(2.24) hence, {xn} is cauchy in (x,p). moreover since (x,p) is complete, it follows there exists z ∈ x such that lim n→∞ p(xn,z) = p(z,z) = lim n,m→∞ p(xn,xm) = 0.(2.25) proceeding again as in the proof of theorem 2.2, we prove that z is a fixed point of t. � analogously, we can derive the following results. corollary 2.17. let (x,p) be a complete partial metric space and t : x → cbp(x) be a multi-valued mapping. assume that there exist ϕ ∈ φ and ψ ∈ ψ such that, for all x,y ∈ x hp(tx,ty) ≤ ψ(mp(x,y)) + p(y,tx) −ϕ(p(y,x)), where mp(x,y) = max{p(x,y),p(x,tx),p(y,ty), 12 [p(x,ty) + p(tx,y)]}. then t has a unique fixed point. corollary 2.18. ([4], theorem 2.2) let (x,p) be a complete partial metric space and t : x → cbp(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ x p(y,tx) ≤ sp(y,x) ⇒ hp(tx,ty) ≤ rmp(x,y), where mp(x,y) = max{p(x,y),p(x,tx),p(y,ty), 12 [p(x,ty) + p(tx,y)]}. then t is an mwp operator. proof. it suffice to take ϕ(t) = st and ψ(t) = rt in theorem 2.16. � corollary 2.19. let (x,p) be a complete partial metric space and t : x → cbp(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ x p(y,tx) ≤ sp(y,x) ⇒ hp(tx,ty) ≤ r max{p(x,y),p(x,tx),p(y,ty)}. then t is an mwp operator. corollary 2.20. let (x,p) be a complete partial metric space and t : x → cbp(x) be a multi-valued mapping. assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ x p(y,tx) ≤ sp(y,x) ⇒ hp(tx,ty) ≤ r 3 {p(x,y) + p(x,tx) + p(y,ty)}. then t is an mwp operator. if t is a single-valued mapping, we deduce the following results. corollary 2.21. let (x,p) be a complete partial metric space and t : x → x be a mapping. assume that there exist ϕ ∈ φ and ψ ∈ ψ such that, for all x,y ∈ x p(y,tx) ≤ ϕ(p(y,x)) ⇒ p(tx,ty) ≤ ψ(mp(x,y)), where mp(x,y) = max{p(x,y),p(x,tx),p(y,ty), 12 [p(x,ty) + p(tx,y)]}. then t has a unique fixed point. on multi-valued weakly picard operators 181 proof. the existence follows immediately also from theorem 2.16. thus, we need to prove uniqueness of fixed point. we assume that there exist x,y ∈ x such that x = tx and y = ty with x 6= y. since p(y,tx) = p(y,x) ≤ ϕ(p(y,x)), then by (2.22), we get 0 < p(x,y) = p(tx,ty) ≤ ψ(max{p(x,y),p(x,tx),p(y,ty), 1 2 [p(x,ty) + p(tx,y)]}) = ψ(max{p(x,y),p(x,x),p(y,y),p(x,y)}) = ψ(p(x,y)) < p(x,y) which is a contradiction. hence x = y, so the uniqueness of the fixed point of t. � corollary 2.22. let (x,p) be a complete partial metric space and t : x → x be a mapping. assume that there exist r ∈ [0, 1) and s ≥ 1 such that, for all x,y ∈ x p(y,tx) ≤ sp(y,x) ⇒ p(tx,ty) ≤ rmp(x,y), where mp(x,y) = max{p(x,y),p(x,tx),p(y,ty), 12 [p(x,ty) + p(tx,y)]}. then t has a unique fixed point. references [1] ia. rus, a. petrusel and a. sintamarian, data dependence of the fixed points set of multi-valued weakly picard operators, stud. univ. babe?s-bolyai, math. 46 (2001), 111-121. [2] ia. rus, a. petrusel and a. sintamarian, data dependence of the fixed points set of some multi-valued weakly picard operators, nonlinear anal. 52 (2003), 1947-1959. [3] t. suzuki, a generalized banach contraction principle that characterizes metric completeness, proc. am. math. soc. 136 (2008), 1861-1869. [4] m. jleli, h. k. nashine, b. samet and c. vetro, on multi-valued weakly picard operators in partial hausdorff metric spaces, fixed point theory appl. 20015 (2015), art. id 52. [5] i. altun and h. simsek, some fixed point theorems on dualistic partial metric spaces, j. adv. math. stud. 1 (2008), 1c8. [6] e. karapinar, h. aydi, a. felhi and s. sahmim, hausdorff metric-like, generalized nadler’s fixed point theorem on metric-like spaces and application, miskolc math. notes (in press). [7] c.t. aage and j.n. salunke, the results on fixed points in dislocated and dislocated quasi-metric space, appl. math. sci, 2(59) (2008), 2941-2948. [8] s.g. matthews, partial metric topology, proc. 8th summer conference on general topology and applications, annals of the new york academi of sciences, 728 (1994), 183-197. [9] h. aydi, m. abbas and c. vetro, partial hausdorff metric and nadlers fixed point theorem on partial metric spaces, topol. appl. 159 (2012), 3234-3242. [10] r.d. daheriya, r. jain and m. ughade, some fixed point theorem for expansive type mapping in dislocated metric space, isrn math. anal. 2012 (2012), art. id 376832. [11] a. isufati, fixed point theorems in dislocated quasi-metric space, appl. math. sci. 4(5) (2010), 217-233. [12] amini a. harandi, metric-like spaces, partial metric spaces and fixed points, fixed point theory appl. 2012 (2012), art. id 204. [13] r. george, cyclic contractions and fixed points in dislocated metric spaces, int. j. math. anal. 7(9) (2013), 403-411. [14] g.e. hardy and t.d. rogers, a generalization of a fixed point theorem of reich, canadian mathematical bulletin, 16 (1973), 201c206. [15] e. karapınar and p. salimi, dislocated metric space to metric spaces with some fixed point theorems, fixed point theory appl. 2013 (2013), art. id 222. [16] p.s. kumari, w. kumar and i.r. sarma, common fixed point theorems on weakly compatible maps on dislocated metric spaces, math. sci. 6 (2012), 71. [17] ps. kumari, some fixed point theorems in generalized dislocated metric spaces, math. theory model. 1(4) (2011), 16-22. [18] s.b. nadler, multi-valued contraction mappings, pacific j. math. 30 (1969), 475-488. [19] i.r. sarma and p.s. kumari, on dislocated metric spaces, int. j. math. arch. 3(1) (2012), 72-77. [20] r. shrivastava, zk. ansari and m. sharma, some results on fixed points in dislocated and dislocated quasi-metric spaces, j. adv. stud. topol. 3(1) (2012), 25-31. [21] m. shrivastava, k. qureshi and a.d. singh, a fixed point theorem for continuous mapping in dislocated quasi-metric spaces, int. j. theor. appl. sci. 4(1) (2012), 39-40. [22] k. zoto, some new results in dislocated and dislocated quasi-metric spaces. appl. math. sci. 6(71) (2012), 3519-3526. 1department of mathematics, college of sciences, king faisal university, hafouf, saudi arabia 2department of mathematics, preparatory engineering institute, bizerte, carthage university, tunisia 182 felhi ∗corresponding author: afelhi@kfu.edu.sa int. j. anal. appl. (2022), 20:63 numerical simulation of singularly perturbed delay differential equations with large delay using an exponential spline ramavath omkar, k. phaneendra∗ department of mathematics, university college of science, osmania university, india ∗corresponding author: kollojuphaneendra@yahoo.co.in abstract. in this study, numerical solution of a differential-difference equation with a boundary layer at one end of the domain is suggested using an exponential spline. the numerical scheme is developed using an exponential spline with a special type of mesh. a fitting parameter is inserted in the scheme to improve the accuracy and to control the oscillations in the solution due to large delay. convergence of the method is examined. the error profiles are represented by tabulating the maximum absolute errors in the solution. graphs are being used to show that how the fitting parameter influence the layer structure. 1. introduction differential-difference equations (ddes) are ones in which a state variable’s time evolution is inconsistently dependent on a particular history, i.e., a physical system’s rate of change is dependent not only on its current condition but also on its prior history. these equations have been widely utilized in population dynamics [7], nonlinear delay differential equations relating to physiological control systems [14], red blood cell system [13], predator-prey models [15], neuronal variability problems connected to patterns of nerve action potentials formed by unit quantal inputs occurring at random are studied [22]. bellman and cooke [1], doolan et al. [2] and driver [3] just are a few of the authors who have produced papers and books in recent years explaining various methods for solving differential-difference equations with perturbation. in [4] authors solved singularly perturbed delay differential equation (spde) using a fitted scheme on an uniform mesh. in [5], glizer solved linear quadratic optimal control problem with received: oct. 15, 2022. 2010 mathematics subject classification. 65l11, 65l12. key words and phrases. exponential spline; differential-difference equation; large delay; truncation error. https://doi.org/10.28924/2291-8639-20-2022-63 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-63 2 int. j. anal. appl. (2022), 20:63 delay by hamiltonian boundary-value problem, with boundary function method. in [6], kadalbajoo et. al. proposed numerical research utilizing finite difference techniques. the authors in [8] developed a technique for solving spdes with twin layers or oscillatory behaviour using a computational scheme. the authors in [9] used a nonpolynomial spline to develop a numerical solution for a ddes having layer with a small and large delay in the differentiated terms. using domain decomposition, the authors of [10] suggested a mixed difference technique to solve ddes with mixed shifts. for a class of linear second-order dde type, lange and miura [11, 12] developed an asymptotic method. they developed a mathematical model by random synaptic inputs in dendrites for estimating the approximate time for the activation of action potentials in nerve cells and also discussed issues with solutions that had rapid oscillations in their study. the authors of [16] utilized a quadrature method for solving the spde, as well as a two-point quadrature rule to obtain a tridiagonal system. the authors in [17] developed a finite difference approach for solving spdes with turning points and mixed shifts. a finite difference scheme on shishkin mesh is proposed to solve singularly perturbed delay differential equations with or without a turning point in [18]. the authors of [19, 20] used tension splines and an exponentially fitted spline for the problem with convection delay-dominated diffusion equation. the authors of [23] solved a two parameter semi linear differential equation by using an exponential spline. the authors of [21] solved spde with a numerical integration technique. 2. statement of the problem consider the continuous problem of a second order spde εz′′ + p(u)z′(u −δ) + q(u)z(u) = r(u) (2.1) with boundary conditions z(u) = ϕ(u) −δ ≤ u ≤ 0 , z(1) = τ (2.2) where 0 < ε << 1 is a perturbation, p(u),q(u), r(u) and ϕ(u) are bounded continuous functions on (0, 1), and τ is finite constant and δ is delay parameter. it is well known that when δ = 0 the above delay differential equation is reducing to a spde. the solution z(u) exhibit boundary layer on left side when p(u) is positive or on right side when p(u) is negative throughout the interval [0, 1]. if δ(ε) is of order o(ε), layer profile of the solution is no longer retained and the solution oscillates. 3. an exponential spline the region [υ, υ] is subdivided into l equal subregions of mesh size h = (υ−υ) l so that ui = υ + ih, i = 0, 1, . . . , l are the mesh points. to manage the delay term, the mesh size is chosen as h = δ ν , where ν= mantissa of δ which is a positive integer. let the exact solution be z(u) to eq. (2.1) and ui be an approximation solution to the z(ui ) attained int. j. anal. appl. (2022), 20:63 3 by the segment qi (u) passing (ui,qi ) and (ui+1,qi+1). each exponential spline segment qi (u) has the form. qi (u) = aie k(u −ui ) + bie−k(u −ui ) + ci (u −ui ) + di f or i = 0, 1, 2, . . . , l. (3.1) where ai,bi,ci and di are the constants and k is a free parameter. to find the values of the coefficients in eq. (3.1), the interpolatory conditions qi (ui ) = si,qi (ui+1) = si+1,q ′′ i (ui ) = mi,q ′′ i (ui+1) = mi+1 are used. using these conditions, we get the following expressions ai = h2(mi+1−e−θmi ) 2θ2sinh(θ) , bi = h2(mi−e−θmi+1) 2θ2sinh(θ) , ci = (si+1−si ) h − h(mi+1−mi ) θ2 and di = si − h2mi θ2 where θ = kh and i = 0, 1, 2, . . . , l. using the first derivative continuity i.e., q′i−1 (u) = qi ′ (u) at the point (ui,qi ), we get the relation (zi+1 − 2zi + zi−1) = h2 (αmi+1 + βmi + αmi−1) f or i = 0, 1, . . . , l − 1 (3.2) where α = ( sinh(θ)−θ θ2sinh(θ) ) and β = 2θcosh(θ)−2sinh(θ) θ2sinh(θ) . 4. numerical approach eq. (2.1) can be discretized at the mesh point uj by εmj = r ( uj ) −p ( uj ) z′j−ν −q ( uj ) zj f or j = i − 1, i, i + 1 (4.1) using the following difference approximations of z′: z′i−ν ≈ [ 1 + 2ωh2qi+1 + ωh [3 pi+1 + pi−1] 2h ] zi−ν+1 − 2ω [pi+1 + pi−1] zi−ν − [ 1 + 2 ωh2 qi−1 −ωh [pi+1 + 3 pi−1] 2h ] zi−ν−1 + ωh [ri+1 − ri−1] z′i−ν+1 ≈ 3zi−ν+1 − 4zi−ν + zi−ν−1 2h , z′i−ν−1 ≈ −zi−ν+1 + 4zi−ν − 3zi−ν−1 2h now substituting the above approximations in eq. (4.1) and in eq. (3.2) we obtained.( ε h2 + αqi+1 ) zi+1 + ( −2ε h2 + βqi ) zi + ( ε h2 + αqi−1 ) zi−1+ ( −αpi−1 2h + βpi 2h ( 1 + 2ωh2qi+1 + ωh (3pi+1 + pi−1) ) + 3α 2h pi+1)zi−ν+1+ ( 2αpi−1 h − 2βωpi (pi+1 + pi−1) − 2α h pi+1)zi−ν+ ( −3αpi−1 2h − βpi 2h ( 1 + 2ωh2qi−1 −ωh (pi+1 + 3pi−1) ) + α 2h pi+1)zi−ν−1 = (αri+1 + βri + αri−1) −βωpih (ri+1 − ri−1) (4.2) 4 int. j. anal. appl. (2022), 20:63 when the shift parameter is small in comparison to the perturbation parameter, the layer behavior of the solution is maintained and reliable results are achieved. the layer behavior, however, no longer holds good if δ(ε) is of order o(ε) and oscillations manifest. we are therefore constructing a numerical approach with fitting parameter based on an exponential spline method and a particular mesh type. in order to demonstrate how significant the fitting parameter is in our suggested approach, we also examine how the solution layer behaves when there are large delays. now inserting the fitting parameter σ in the above scheme, we have (σ� + h2αqi+1)zi+1 + (−2σ� + h2βqi )zi + (σ� + h2σqi−1)zi−1+ ( −αhpi−1 2 + βhpi 2 (1 + 2ωh2qi+1 + ωh(3pi+1 + pi−1)) + 3αh 2 pi+1)zi−ν+1+ (2αhpi−1 − 2βωpih2(pi+1 + pi−1) − 2αhpi+1)zi−ν+ ( −3αhpi−1 2 − βhpi 2 (1 + 2ωh2qi−1 −ωh(pi+1 + 3pi−1)) + αh 2 pi+1)zi−ν−1 = h2(αri+1 + βri + αri−1) −βωpih3(ri+1 − ri−1) (4.3) re write the above equation ai+1zi+1 + bizi + ci−1zi−1 + di−ν+1zi−ν+1 + ei−νzi−ν + fi−ν−1zi−ν−1 = gi (4.4) where ai+1 = σ� + αh 2qi+1, bi = −2σ� + βh2qi, ci−1 = σ� + αh2qi−1 di−ν+1 = −αhpi−1 2 + βhpi 2 (1 + 2ωh2qi+1 + ωh(3pi+1 + pi−1)) + 3αh 2 pi+1 ei−ν = 2αhpi−1 − 2βωpih2(pi+1 + pi−1) − 2αhpi+1 fi−ν−1 = −3αhpi−1 2 − βpih 2 (1 + 2ωh2qi−1 −ωh(pi+1 + 3pi−1)) + αh 2 pi+1 gi = h 2(αri+1 + βri + αri−1) −βωpih3(ri+1 − ri−1) p(ui ) = pi,q(ui ) = qi, r(ui ) = ri the above scheme can be written by using the boundary conditions ai+1zi+1 + bizi + ci−1zi−1 = gi −di−ν+1zi−ν+1 −ei−νzi−ν −fi−ν+1zi−ν+1,∀ 1 ≤ i ≤ ν − 1 ai+1zi+1 + bizi + ci−1zi−1 + di−ν+1zi−ν+1 = gi −ei−νzi−ν − fi−ν−1zi−ν−1, ∀ i = ν ai+1zi+1 + bizi + ci−1zi−1 + di−ν+1zi−ν+1 + ei−νzi−ν = gi −fi−ν−1zi−ν−1, ∀ i = ν + 1 ai+1zi+1 +bizi +ci−1zi−1 +di−ν+1zi−ν+1 +ei−νzi−ν +fi−ν−1zi−ν−1 = gi ∀ ν+ 2 ≤ i ≤ l−1. (4.5) by using gauss elimination method with partial pivoting or any other method resolves the above system of l equations. int. j. anal. appl. (2022), 20:63 5 4.1. left-end layer. assume p(u) ≥ m > 0 and q(u) ≤ −⊆ < 0 where θ and m are positive constants. then the problem (2.1) (2.2) shows layer structure at u = 0 for small values of perturbation. now, to enhance the scheme’s effectiveness, the fitting parameter inserted in eq. (4.4) is determined by using the following approximate solution given in [2]. limh→0z(ih) ≈ z0(ih) + (φ(0) −z0(0))exp(−p(0)iρ) + o(ε), where ρ = h � . using this expression in eq. (4.4) and following the procedure given in [2], we get σ = ρ(α + β/2)p(0)ep(0)νρcoth( p(0)ρ 2 ) lemma 4.1. suppose ψ0 ≥ 0 and ψl ≥ 0, then llψi ≤ 0 ∀i = 1, 2, 3, . . . .., l − 1 implies that ψi ≥ 0 ∀i = 0, 1, 2, 3. . . ..l. proof. let j ∈ (0, 1, . . . .l) such that ψj = max0≤i≤lψi. let if possible ψj < 0. we will show that it leads to contradiction. clearly j /∈ (0, l). now we have ll ψj =   ερi ψ ′′ i + (2α + β)qi ψi = gi −di−ν+1ψi−ν+1 −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀i = 1, 2, . . . ,ν. ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 = gi −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀ i = ν + 1, . . . ., l1 − 1. ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 + ei−νψi−ν = gi −fi−ν−1ψi−ν−1, ∀i = l1, . . . .., l − 1. (4.6) case 1: for i = 1, 2, . . . ,ν llψj = (ερi ψ ′′ i + (2α + β)qi ψi = gi −di−ν+1ψi−ν+1 −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀i = 1, 2, . . . ,ν.) ≥ 0(∵ qi ) < 0, i = 1, 2, . . . .ν case 2: for i = ν + 1, . . . ., l1 − 1 llψj = (ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 = gi −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀ i = ν + 1, . . . ., l1 − 1.) ≥ 0 case 3: for i = l1, . . . .., l − 1 llψj = (ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 + ei−νψi−ν = gi −fi−ν−1ψi−ν−1, ∀i = l1, . . . .., l − 1.) ≥ 0 as a result, we have llψj > 0 , which is contradicts the hypothesis that llψj ≤ 0, 1 ≤ j ≤ l − 1. as a consequence, we assume that ψj < 0 is incorrect and that ψj > 0 is correct. since j was chosen an arbitrary, we have ψj ≥ 0,∀j = 0, 1, 2, . . . , l (4.7) � 6 int. j. anal. appl. (2022), 20:63 theorem 4.1. let p(u) ≥ m > 0 and q(u) ≤ −θ < 0 where m and θ are both positive constants. the problem (2.1) (2.2) with boundary conditions has a uniqueness, existence and satisfying solution. ||z||h,∞ ≤ ||r||h,∞θ−1 + c2(||ϕ||h,∞ + τ) (4.8) where c2 ≥ 1 is a positive constant. proof. to demonstrate the uniqueness and existence, assume <v >li=0 and <w > l i=0 be the two solutions to the discrete problem (2.1) (2.2). then ti = vi −wi is a mesh function that meets the conditions t0 = 0 = tl and for 1 ≤ i ≤ l − 1 we have llti = llvi −llwi. since vi and wi satisfy eq. (4.5), therefore llti = 0, 1 ≤ i ≤ l−1. as a result, the mesh function ti meets the discrete minimum principle hypothesis, and when we apply it to the mesh function ti, we get ti = (vi −wi ) ≥ 0 f or 0 ≤ i ≤ l (4.9) again, if we set ti = −(vi −wi ) , then ti is a mesh function that satisfies t0 = 0 = tl and we have llti = 0, 1 ≤ i ≤ l − 1 along the same lines as before. as a result, the discrete minimum principle is applied to the mesh function ti gives ti = −(vi −wi ) ≥ 0 i.e., (vi −wi ) ≤ 0 f or 0 ≤ i ≤ l (4.10) from eq. (4.9) and eq. (4.10), we get vi − wi = 0 . this suggests that the discrete problem (2.1)-(2.2) solution is unique. the existence of linear equation is assumed by their uniqueness. now we’ll explain how to prove the bound on < zi >li = 0. we do this by introducing two barrier functions ψ± i defined by ψ± i = ||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±zi, 0 ≤ i ≤ l where c2 ≥ 1 is a positive constant that can be chosen at random. then we have ψ± i = ||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±zi, 0 ≤ i ≤ l now we have ψ±0 = ||r||h,∞θ −1 + c2(||ϕ||h,∞ + |τ|) ±z0, ψ±0 = (||r||h,∞θ −1 + c2(||ϕ||h,∞ + |τ|) ±c2ϕ0) ≥ 0 since ||ϕ||h,∞ + |τ| ≥ ϕ0 and c2 ≥ 1, ψ±n = ||r||h,∞θ −1 + c2(||ϕ||h,∞ + |τ|) ±zn ψ±n = (||r||h,∞θ −1 + c2(||ϕ||h,∞ + |τ|) ±τ) ≥ 0 case 1. for i = 1, 2, 3, . . . .,ν llψ± i = ερi (ψ ± i )′′ + (2α + β)qi ψ ± i = gi −di−ν+1ψ±i−ν+1 −ei−νψ ± i−ν −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) int. j. anal. appl. (2022), 20:63 7 = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 case 2. for i = ν + 1, . . . ., l1 − 1 llψ± i = ερi (ψ ± i )′′ + (2α + β)qi ψ ± i + di−ν+1ψ ± i−ν+1 = gi −ei−νψ ± i−ν −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 case 3. for i = l1, . . . .., l − 1 llψ± i = ερi (ψ ± i )′′ + 2α + βqi ψ ± i + di−ν+1ψ ± i−ν+1 + ei−νψ ± i−ν = gi −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 combining above cases, we have llψ± i ≤ 0, 1 ≤ i ≤ l. using the discrete maximum principle ψ± i = ||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±zn ≥ 0, 0 ≤ i ≤ l. which proves the desired results. � 4.2. right-end layer. assume p(u) ≤ −m < 0 and q(u) ≤ −⊆ < 0 where θ and m are positive constants. then the problem (2.1) (2.2) shows layer structure at u = 1 for small values of ε. in this case, the fitting parameter inserted in eq. (4.4) is determined by using the following approximate solution given in [2]. limh→0z(ih) ≈ z0(0) + (φ(0) −z0(1))e−p(1)( 1 ε −iρ) + o(ε), where ρ = h � . using this expression in eq. (4.4) and following the process given in [2], we get σ = ρ(α + β/2)p(0)e−p(0)νρcoth( p(1)ρ 2 ) (4.11) lemma 4.2. . suppose ψ0 ≥ 0 and ψl ≥ 0, then llψi ≤ 0 ∀ i = 1, 2, 3, . . . ..l − 1 implies that ψi ≥ 0 ∀ i = 0, 1, 2, 3. . . ..l. proof. let j ∈ (0, 1, . . . .l) such that ψj = min0≤i≤lψi. let if possible ψj < 0. we will show that it leads to contradiction. clearly j /∈ (0, l). now we have ll ψj =   ερi ψ ′′ i + (2α + β)qi ψi = gi −di−ν+1ψi−ν+1 −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀i = 1, 2, . . . ,ν. ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 = gi −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀ i = ν + 1, . . . ., l1 − 1. ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 + ei−νψi−ν = gi −fi−ν−1ψi−ν−1, ∀i = l1, . . . .., l − 1. (4.12) 8 int. j. anal. appl. (2022), 20:63 case 1: for i = 1, 2, . . . ,ν llψj = (ερi ψ ′′ i + (2α + β)qi ψi = gi −di−ν+1ψi−ν+1 −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀i = 1, 2, . . . ,ν.) ≥ 0 (∵ qi < 0, i = 1, 2, . . . .ν) case 2: for i = ν + 1, . . . ., l1 − 1 llψj = (ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 = gi −ei−νψi−ν −fi−ν−1ψi−ν−1, ∀ i = ν + 1, . . . ., l1 − 1.) ≥ 0 case 3: for i = l1, . . . .., l − 1 llψj = (ερi ψ ′′ i + (2α + β)qi ψi + di−ν+1ψi−ν+1 + ei−νψi−ν = gi −fi−ν−1ψi−ν−1, ∀i = l1, . . . .., l − 1.) ≥ 0 which is contradiction to llψj ≤ 0, 1 ≤ j ≤ l − 1 therefore, the assumption ψj < 0 is false and ψj > 0 . since j was choosing arbitrary, we have ψj ≥ 0,∀j = 0, 1, 2, . . . l theorem 4.2. let p(u) ≥ m > 0 and q(u) ≥ θ > 0 where m and θ are both positive constants. the solutions to the problem (2.1) (2.2) with boundary conditions exist, are unique and satisfy. ||z||h,∞ ≤ ||r||h,∞θ−1 + c1(||ϕ||h,∞ + τ) (4.13) where c1 ≥ 1 is a positive constant proof. to prove the existence and uniqueness, assume <v >li=0 and <w > l i=0 be the two solutions to the problem (2.1) (2.2). then ti = vi −wi is a mesh function that meets the conditions t0 = 0 = tl and for 1 ≤ i ≤ l − 1 we have llti = llvi − llwi. since vi and wi satisfy eq. (4.5), therefore llti = 0, 1 ≤ i ≤ l − 1. as a result, the mesh function ti meets the discrete minimum principle hypothesis, and when we apply it to the mesh function ti, we get ti = (vi −wi ) ≥ 0 f or 0 ≤ i ≤ l (4.14) again, if we set ti = −(vi − wi ) , then ti is a mesh that satisfies t0 = 0 = tl and we have llti = 0, 1 ≤ i ≤ l − 1 along the same lines as before. as a result, the discrete minimum principle is applied to the mesh function ti gives ti = −(vi −wi ) ≥ 0 i.e., (vi −wi ) ≤ 0 f or 0 ≤ i ≤ l (4.15) from eq. (4.14) and eq. (4.15), we get vi − wi = 0 . this suggests that the discrete problem (2.1)-(2.2) solution is unique. the existence of linear equation is assumed by their uniqueness. now int. j. anal. appl. (2022), 20:63 9 we’ll explain how to prove the bound on < zi >li = 0. we do this by introducing two barrier functions ψ± i defined by ψ± i = ||r||h,∞θ−1 + c1(||ϕ||h,∞ + |τ|) ±zi, 0 ≤ i ≤ l where c1 ≥ 1 is a positive constant that can be chosen at random. then we have ψ± i = ||r||h,∞θ−1 + c1(||ϕ||h,∞ + |τ|) ±zi, 0 ≤ i ≤ l now we have ψ±0 = ||r||h,∞θ −1 + c1(||ϕ||h,∞ + |τ|) ±z0, ψ±0 = (||r||h,∞θ −1 + c1(||ϕ||h,∞ + |τ|) ±c1ϕ0) ≥ 0 since ||ϕ||h,∞ + |τ| ≥ ϕ0 and c1 ≥ 1, ψ± l = ||r||h,∞θ−1 + c1(||ϕ||h,∞ + |τ|) ±zl ψ± l = (||r||h,∞θ−1 + c1(||ϕ||h,∞ + |τ|) ±τ) ≥ 0 case 1. for i = 1, 2, 3, . . . .,ν llψ± i = ερi (ψ ± i )′′ + (2α + β)qi ψ ± i = gi −di−ν+1ψ±i−ν+1 −ei−νψ ± i−ν −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 case 2. for i = ν + 1, . . . ., l1 − 1 llψ± i = ερi (ψ ± i )′′ + 2α + βqi ψ ± i + di−ν+1ψ ± i−ν+1 = gi −ei−νψ ± i−ν −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 case 3. for i = l1, . . . .., l − 1 llψ± i = ερi (ψ ± i )′′ + (2α + β)qi ψ ± i + di−ν+1ψ ± i−ν+1 + ei−νψ ± i−ν = gi −fi−ν−1ψ ± i−ν−1 = −qi (||r||h,∞θ−1 + c2(||ϕ||h,∞ + |τ|) ±llzi ) = ( −qi θ ||r||h,∞ ± ri ) −qic2(||ϕ||h,∞ + |τ|) ≤ 0 using the above cases, we have llψi ≤ 0, 1 ≤ i ≤ l. using the discrete minimum principle ψ± i = (||r||h,∞θ−1 + c1(||ϕ||h,∞ + |τ|) ±zl) ≥ 0, 0 ≤ i ≤ l which proves the desired results. as a result of the previous theorems, the problem (2.1)-(2.2) solution is uniformly bounded, regardless of the mesh size h and the perturbation ε, demonstrating that the proposed scheme is stable for all h. � 10 int. j. anal. appl. (2022), 20:63 5. truncation error using taylor series, the truncation error in the method eq. (4.2) is t (h) = {(2α + β) − 1}σεz′′i h 2+ {[( −2α 3 ) + ( 2ωσε + 1 6 ) β ] piz 3 i + ( 2α− 1 12 ) σεz4i } h4+o(h6) (5.1) the truncation error is fourth-order for any arbitrary α and β with 2α + β = 1 and any value of ω. 6. convergence analysis the matrix form of the eq. (4.5) is az + b + t (h) = 0 (6.1) a =   b1 a1 0 · · · · · · · · · · · · · · · · · · 0 c2 b2 a2 0 · · · · · · · · · · · · · · · 0 0 c3 b3 a3 0 · · · · · · · · · · · · 0 ... ... ... ... ... ... ... ... ... ... dν 0 · · · cν bν aν 0 · · · · · · 0 eν+1 dν+1 0 · · · cν+1 bν+1 aν+1 0 0 fν+2 eν+2 dν+2 0 · · · cν+2 bν+2 aν+2 0 0 0 fν+3 eν+3 dν+3 0 cν+3 bν+3 aν+3 0 0 ... ... ... ... ... ... ... ... ... 0 · · · 0 fl−2 el−2 dl−2 0 · · · cl−2 bl−2 al−2 · · · · · · 0 fl−1 el−1 dl−1 0 · · · cl−1 bl−1   and b = [κ1,κ2, ..κν,κν+1, κν+2, ...,κl−2,κl−1] where κi =   gi −di−ν+1zi−ν+1 −ei−νzi−ν −fi−ν−1zi−ν−1 −c1z0, ∀ 1 ≤ i ≤ ν − 1 gi −ei−νzi−ν −fi−ν−1zi−ν−1, ∀ i = ν gi −fi−ν−1zi−ν−1, ∀ i = ν + 1 gi −al−1zl, ∀ ν + 2 ≤ i ≤ l − 1 where ai+1 = σ� + αh 2qi+1,bi = −2σ� + βh2qi,ci−1 = σ� + αh2qi−1 di−ν+1 = −αhpi−1 2 + βhpi 2 (1 + 2ωh2qi+1 + ωh(3pi+1 + pi−1)) + 3αh 2 pi+1 ei−ν = 2αhpi−1 − 2βωpih2(pi+1 + pi−1) − 2αhpi+1 fi−ν−1 = −3αhpi−1 2 − βpih 2 (1 + 2ωh2qi−1 −ωh(pi+1 + 3pi−1)) + αh 2 pi+1 gi = h 2(αri+1 + βri + αri−1) −βωpih3(ri+1 − ri−1) int. j. anal. appl. (2022), 20:63 11 for i = 0, 1, . . . , l − 1 and t (h) = o ( h4 ) ,z = [z1,z2, . . . ,zl−1] t ,t (h) = [t1,t2, . . . ,tl−1] t ,o = [0, 0, . . . , 0] t are related vectors of eq. (6.1). let z̃ = [z̃1, z̃2, . . . , z̃l−1] t ∼= z satisfies the equation az̃ + b = 0 (6.2) let ei = zi −zi, i = 1 (1) l − 1 be the error so that e = [e1,e2, ...,el−1] t = u −u. subtracting eq. (6.1) from eq. (6.2), we obtain the error equation ae = t (h) (6.3) let the sum of the ith row elements of the matrix a be si. then si = h 2 (αqi−1 + βqi + αqi+1) for 1 ≤ i ≤ ν − 1 si = h 2 ( −αpi−1 + βpi ( 1 + 2ωh2qi+1 + ωh (3pi+1 + pi−1) ) + 3αpi+1 ) + h2 (αqi−1 + βqi + αqi+1) for i = ν si = h 2 ( −3αpi−1 −βpi ( 1 + 2ωh2qi−1 −ωh (pi+1 + 3pi−1) ) + αpi+1 ) + h2 (αqi−1 + βqi + αqi+1) for i = ν + 1 si = h 2 (αqi−1 + βqi + αqi+1) for ν + 2 ≤ i ≤ l − 1. let ζ1∗ = min1≤i≤l |p (ui )| , ζ∗1 = max1≤i≤l |p (ui )| ,ζ2∗ = min1≤i≤l |q (ui )| and ζ∗2 = max1≤i≤l |q (ui )|. it is verified that a is irreducible and monotone since 0 < ε � 1 and ε ∝ o (h) with sufficiently small h. hence a−1 exists and a−1 ≥ 0. therefore, using eq. (6.3) we have ||e|| ≤ ||a−1|| ||t || (6.4) for small h, we have si ≥ h2 [2 (α + β/2) ζ2∗] for 1 ≤ i ≤ ν − 1 si ≥ h2 [2 (α + β/2) ζ2∗] for i = ν si ≥ h2 [2 (α + β/2) ζ2∗] for i = ν + 1 si ≥ h2 [2 (α + β/2) ζ2∗] for ν + 2 ≤ i ≤ l − 1 let a−1 i,k be the (i,k)th element of a−1 and we define || a|| = max1≤i≤l−1 l−1∑ k=1 a−1 i,k and ||t (h)|| = max1≤i≤l−1|ti (h)|. 12 int. j. anal. appl. (2022), 20:63 since a−1 i,k ≥ 0 and ∑l−1 k=1 a −1 i,k . sk = 1 for i = 12, 3, ......, l − 1. hence ν−1∑ k=1 a−1 i,k ≤ 1 min1≤k≤ν−1s k < 1 h2 [2 (α + β/2) ζ2∗] , i = 1, 2, 3, . . . ,ν − 1 (6.5) a−1 i,k ≤ 1 sν < 1 h2 [2 (α + β/2) ζ2∗] , i = ν,ν + 1 (6.6) furthermore l−1∑ k=ν+2 a−1 i,k ≤ 1 min1≤k≤ν−1sk < 1 h2 [2 (α + β/2) ζ2∗] , i = ν + 2, ν + 3, ..., l − 1. (6.7) using eqs. (6.5) – (6.6) and eq. (6.7), we get ||e|| ≤ o(h2). (6.8) as a result, the proposed exponential spline strategy converges to second order. the convergence analysis for the right end boundary layer can be examined using the same procedure. 7. numerical examples the effectiveness of the numerical approach can be demonstrated by the following instances. mae (maximum absolute error) in the solution is computed using the double mesh principle el = max0≤i≤l ∣∣z li −z2l2i ∣∣. the order of convergence is also investigated by r l = log2 (ei l/ei 2l). we exhibit graphs showing the computed solution of the issue for various δ values. example 1: εz′′ + z′(u −δ) −z(u) = 0 with z(u) = 1, −δ ≤ u ≤ 0 and z(1) = 0. example 2: εz′′ + exp(−0.25u) −z(u) = 0 with z(u) = 1 for −δ ≤ u ≤ 0,z (1) = 1. example 3: εz′′ −z′(u −δ) −z(u) = 0 with z(u) = 1, −δ ≤ u ≤ 0 and z(1) = −1. example 4: εz′′ −z′(u −δ) + z(u) = 0 with z(u) = 1 for −δ ≤ u ≤ 0,z (1) = 1. 8. conclusion a computational approach for solving the singularly perturbed differential equation with the large delay is derived using a special type of mesh. a numerical scheme consisting of a fitting parameter is developed to minimize the error and to control the layer structure in the solution. four examples are solved and computational results with large delay are shown in table 1-4. in the proposed method, we also analyzed the effect of the large delay on the layer structure or oscillatory behaviour of the solutions with and without the fitting parameter in the figures 1-8. the graphs depict the layer behaviour in the solution of the examples with and without the fitting parameter. the impact of the fitting factor int. j. anal. appl. (2022), 20:63 13 in controlling the oscillations in the solution is also depicted in the graphs. we have clearly noticed that the fitting parameter controls the oscillations in the layer for the large delay values. the proposed method is simple and it works very well with small as well as large delay. 14 int. j. anal. appl. (2022), 20:63 int. j. anal. appl. (2022), 20:63 15 figure 1. layer profile of example 1 for 𝜀 = 0.01 without fitting factor. figure 2. layer profile of example 1 for 𝜀 = 0.01 with fitting factor. figure 3. layer profile of example 2 for 𝜀 = 0.01 without fitting factor. 16 int. j. anal. appl. (2022), 20:63 figure 4. layer profile of example 2 for 𝜀 = 0.01 with fitting factor. figure 5. layer profile of example 3 for 𝜀 = 0.01 without fitting factor. figure 6. layer profile of example 3 for 𝜀 = 0.01 with fitting factor. int. j. anal. appl. (2022), 20:63 17 figure 7. layer profile of example 4 for 𝜀 = 0.01 without fitting factor. figure 8. layer profile of example 4 for 𝜀 = 0.01 with fitting factor. 18 int. j. anal. appl. (2022), 20:63 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r. bellman, k.l. cooke, differential-difference equations, academic press, new york, usa, 1963. [2] e.p. doolan, j.j.h. miller, w.h.a. schilders, et al. uniform numerical methods for problems with initial and boundary layers, boole press, dublin, 1980. [3] r.d. driver, ordinary and delay differential equations, springer, belin-heidelberg, new york, 1977. [4] g.m. amiraliyev, e. cimen, numerical method for a singularly perturbed convection–diffusion problem with delay, appl. math. comput. 216 (2010), 2351–2359. https://doi.org/10.1016/j.amc.2010.03.080. [5] v.y. glizer, asymptotic solution of a boundary-value problem for linear singularly-perturbed functional differential equations arising in optimal control theory, j. optim. theory appl. 106 (2000), 309–335. https: //doi.org/10.1023/a:1004651430364. [6] m.k. kadalbajoo, k.k. sharma, a numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations, appl. math. comput. 197 (2008), 692–707. https://doi.org/ 10.1016/j.amc.2007.08.089. [7] y. kuang, delay differential equations with applications in population dynamics, academic press, new york, 1993. [8] d. kumara swamy, k. phaneendra, a. benerji babu, y.n. reddy, computational method for singularly perturbed delay differential equations with twin layers or oscillatory behaviour, ain shams eng. j. 6 (2015), 391–398. https://doi.org/10.1016/j.asej.2014.10.004. [9] m. lalu, k. phaneendra, s.p. emineni, numerical approach for differential-difference equations having layer behaviour with small or large delay using non-polynomial spline, soft comput. 25 (2021), 13709-13722. https: //doi.org/10.1007/s00500-021-06032-5. [10] l. sirisha, k. phaneendra, y.n. reddy, mixed finite difference method for singularly perturbed differential difference equations with mixed shifts via domain decomposition, ain shams eng. j. 9 (2018), 647–654. https://doi.org/10.1016/j.asej.2016.03.009. [11] c.g. lange, r.m. miura, singular perturbation analysis of boundary value problems for differential-difference equations. v. small shifts with layer behavior, siam j. appl. math. 54 (1994), 249–272. https://doi.org/10. 1137/s0036139992228120. [12] c.g. lange, r.m. miura, singular perturbation analysis of boundary-value problems for differential-difference equations. vi. small shifts with rapid oscillations, siam j. appl. math. 54 (1994), 273–283. https://doi.org/ 10.1137/s0036139992228119. [13] a. lasota, m. walewska, mathematical models of the red blood cell system, mat. stos. 6 (1976), 25–40. [14] m.c. mackey, l. glass, oscillation and chaos in physiological control systems, science. 197 (1977), 287–289. https://doi.org/10.1126/science.267326. [15] a. martin, s. ruan, predator-prey models with delay and prey harvesting, j. math. biol. 43 (2001), 247–267. https://doi.org/10.1007/s002850100095. [16] k. phaneendra, m. lalu, numerical solution of singularly perturbed delay differential equations using gaussion quadrature method, j. phys.: conf. ser. 1344 (2019), 012013. https://doi.org/10.1088/1742-6596/1344/1/ 012013. [17] p. rai, k.k. sharma, numerical analysis of singularly perturbed delay differential turning point problem, appl. math. comput. 218 (2011), 3483–3498. https://doi.org/10.1016/j.amc.2011.08.095. https://doi.org/10.1016/j.amc.2010.03.080 https://doi.org/10.1023/a:1004651430364 https://doi.org/10.1023/a:1004651430364 https://doi.org/10.1016/j.amc.2007.08.089 https://doi.org/10.1016/j.amc.2007.08.089 https://doi.org/10.1016/j.asej.2014.10.004 https://doi.org/10.1007/s00500-021-06032-5 https://doi.org/10.1007/s00500-021-06032-5 https://doi.org/10.1016/j.asej.2016.03.009 https://doi.org/10.1137/s0036139992228120 https://doi.org/10.1137/s0036139992228120 https://doi.org/10.1137/s0036139992228119 https://doi.org/10.1137/s0036139992228119 https://doi.org/10.1126/science.267326 https://doi.org/10.1007/s002850100095 https://doi.org/10.1088/1742-6596/1344/1/012013 https://doi.org/10.1088/1742-6596/1344/1/012013 https://doi.org/10.1016/j.amc.2011.08.095 int. j. anal. appl. (2022), 20:63 19 [18] p. rai, k.k. sharma, numerical approximation for a class of singularly perturbed delay differential equations with boundary and interior layer(s), numer algor. 85 (2019), 305–328. https://doi.org/10.1007/ s11075-019-00815-6. [19] a.s.v. ravikanth, p. murali, numerical treatment for a singularly perturbed convection delayed dominated diffusion equation via tension splines, int. j. pure appl. math. 113 (2017), 1314-3395. [20] r.k. adivi sri venkata, m.m.k. palli, a numerical approach for solving singularly perturbed convection delay problems via exponentially fitted spline method, calcolo. 54 (2017), 943-961. https://doi.org/10.1007/ s10092-017-0215-6. [21] y.n. reddy, gbsl soujanya, k. phaneendra, numerical integration method for singularly perturbed delay differential equations, int. j. appl. sci. eng. 10 (2012), 249-261. [22] r.b. stein, some models of neuronal variability, biophys. j. 7 (1967), 37-68. https://doi.org/10.1016/ s0006-3495(67)86574-3. [23] w.k. zahra, a.m. el mhlawy, numerical solution of two-parameter singularly perturbed boundary value problems via exponential spline, j. king saud univ. sci. 25 (2013), 201-208. https://doi.org/10.1016/j.jksus.2013. 01.003. https://doi.org/10.1007/s11075-019-00815-6 https://doi.org/10.1007/s11075-019-00815-6 https://doi.org/10.1007/s10092-017-0215-6 https://doi.org/10.1007/s10092-017-0215-6 https://doi.org/10.1016/s0006-3495(67)86574-3 https://doi.org/10.1016/s0006-3495(67)86574-3 https://doi.org/10.1016/j.jksus.2013.01.003 https://doi.org/10.1016/j.jksus.2013.01.003 1. introduction 2. statement of the problem 3. an exponential spline 4. numerical approach 4.1. left-end layer 4.2. right-end layer 5. truncation error 6. convergence analysis 7. numerical examples 8. conclusion references int. j. anal. appl. (2022), 20:71 exact and numerical treatment of a special kind of the pantograph model via laplace technique weam g. alharbi∗ department of mathematics, faculty of science, university of tabuk, p.o.box 741, tabuk 71491, saudi arabia ∗corresponding author: wgalharbi@ut.edu.sa abstract. the pantograph is a device of practical application in electric trains, by which the current is collected. the mathematical problem of this device is generally given by the delay differential equation φ′(t) = αy(t)+βφ(γt), where α and β are real constants and γ is a proportional delay parameter. in the literature, a special attention has been given to the particular case γ = −1. the objective of this paper is to extend the application of the laplace transform (lt) combined with the adomian decomposition method (adm) to analyze the above model at such particular case of γ. the solution will be determined in exact form which agrees with the corresponding results in the literature. various properties of the obtained exact solution are discussed in detail. moreover, it will be declared that for sufficiently small values of α compared to β there exists an accurate approximate solution. the accuracy of the approximate solution is numerically validated. in addition, some numerical results are conducted for the behavior of the present solution at selected values of α and β. 1. introduction in electric trains, the current is collected through a certain device, called the pantograph [1]. the process of such device is a mathematical problem which is governed by the delay differential equation φ′(t) = αy(t) + βφ (γt), where α and β are real constants and γ is a delay parameter. the standard pantograph model has been analyzed by several authors utilizing various techniques [2-8]. a special case of the pantograph model is known as ambartsumian equation which is of practical applications received: oct. 28, 2022. 2010 mathematics subject classification. 44a10. key words and phrases. pantograph delay equation; adomian decomposition method; laplace transform; exact solution. https://doi.org/10.28924/2291-8639-20-2022-71 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-71 2 int. j. anal. appl. (2022), 20:71 in astronomy [9-19]. however, an interest is recently given to another special case of such model when γ = −1, given by [20] φ′(t) = αφ(t) + βφ (−t) , (1) subject to φ(0) = λ, (2) where α, β and λ are real constants. in ref. [20], the standard series method (ssm) has been employed to solve the model given by eqs. (1-2). although other approaches can be implemented to solve the current model such as the adomian decomposition method (adm) [21-34], the regular perturbation method (if one of the constants α or β is small enough) [35,36] and the homotopy perturbation method [37-39] but the solution by these methods is regularly given in terms of infinite series. however, the author believe that the laplace transform (lt) is capable of obtaining the exact solution in a direct manner. the lt was widely used to solve several scientific models with various applications [40-52]. in this paper, a hybrid approach will be developed to deal with the current problem. the hybrid approach is based on combining the lt and the adm. at first, the lt will be applied to transform the model (1-2) to a difference equation. then, the adm will be used to solve the transformed difference equation and finally the exact solution is given by the inverse lt. the paper is structured as follows. in section 2, the lt is employed to transform the present problem to a difference equation. in addition, the adm is used to establish the corresponding recurrence scheme. section 3 focuses on determining a compact form for the adomian-components of the transformed difference equation. moreover, the solution is provided in exact form via applying the inverse lt on the such compact form. furthermore, the properties of the obtained solution is analyzed in section 4 at several cases of the constants α and β. the paper is finally concluded in section 5. 2. the lt-decomposition method applying the lt on eq. (1) gives sφ(s) −λ = αφ(s) −βφ(−s), (3) int. j. anal. appl. (2022), 20:71 3 where φ(s) and φ(−s) are the lts of φ(t) and φ(−t), respectively. the adm [21] requires to put eq. (3) in the canonical form: φ(s) = λ s −α + φ(−s) s −α . (4) eq. (4) is now a difference equation which has no a known solution. however, the adm can be used to accomplish this target. the adm assumes that φ(s) can be decomposed as φ(s) = ∞∑ i=0 φi (s). (5) substituting (5) into (4), it then follows ∞∑ i=0 φi (s) = λ s −α + 1 s −α ∞∑ i=0 φi (−s). (6) according to eq. (6) we have the recurrence scheme: φ0(s) = λ s −α , φi (s) = − βφi−1(−s) s −α , i ≥ 1. (7) 3. the exact solution the algorithm (7) is used here to obtain a compacted form for the adm-components. regarding, eq. (7) at i = 1 gives φ1(s) = − βφ0(−s) s −α , = βλ (s −α)(s + α) , = λβ (s2 −α2) . (8) at i = 2, we have φ2(s) = − βφ1(−s) s −α , = − λβ2 (s2 −α2)(s −α) . (9) 4 int. j. anal. appl. (2022), 20:71 proceeding as above, other higher-order components are found as φ3(s) = − λβ3 (s2 −α2)2 , (10) φ4(s) = λβ4 (s2 −α2)2(s −α) , (11) φ5(s) = λβ5 (s2 −α2)3 , (12) φ6(s) = − λβ6 (s2 −α2)3(s −α) , (13) φ7(s) = − λβ7 (s2 −α2)4 . (14) in view of the above calculations, it can be observed that the even-order components follow the formula: φ2i (s) = λ(−1)iβ2i (s2 −α2)i (s −α) , i ≥ 0, (15) while the even-order components follow the formula: φ2i+1(s) = λ(−1)iβ2i+1 (s2 −α2)i+1 , i ≥ 0, (16) therefore, the solution of the difference equation (5) becomes φ(s) = ∞∑ i=0 (φ2i (s) + φ2i+1(s)) , (17) i.e., φ(s) = λ (s −α) ∞∑ i=0 (−1)iβ2i (s2 −α2)i + λβ (s2 −α2) ∞∑ i=0 (−1)iβ2i (s2 −α2)i , (18) which can be rewritten as φ(s) = λ (s −α) ∞∑ i=0 ( −β2 s2 −α2 )i + λβ (s2 −α2) ∞∑ i=0 ( −β2 s2 −α2 )i . (19) under the assumption ∣∣∣ β2 s2−α2 ∣∣∣ < 1, i.e., |s| > √α2 + β2, the series in the right hand side of eq. (19) can be summed. in this case, we obtain ∞∑ i=0 ( −β2 s2 −α2 )i = 1 1 + β2 s2−α2 = s2 −α2 s2 + β2 −α2 . (20) inserting (20) into (19) and simplifying, then φ(s) = λ(s + α) s2 + β2 −α2 + λβ s2 + β2 −α2 , (21) int. j. anal. appl. (2022), 20:71 5 or φ(s) = λ ( s s2 + β2 −α2 + α + β s2 + β2 −α2 ) . (22) applying the inverse lt on eq. (22) (see ref. [52]), we directly get φ(t) = λ [ l−1 { s s2 + β2 −α2 } + l−1 { α + β s2 + β2 −α2 }] , = λ [ cos (√ β2 −α2t ) + α + β√ β2 −α2 sin (√ β2 −α2t )] , (23) or φ(t) = λ [ cos (√ β2 −α2t ) + √ β + α β −α sin (√ β2 −α2t )] , |α| < |β| , (24) which is the same obtained expression in ref. [20] using a direct ssm. 4. properties of solution in this section, we introduce some properties of the obtained exact solution (24). some of these properties were addressed in ref. [20] but represented here just for adding some materials and observations as follows. 4.1. symmetry&behaviour of the solution. the solution (24) is symmetrical with respect to the signs of α and β. this property can be shown by re-expressing the solution as a function in α and β in addition to the independent variable t, as φ(a,b,t) = λ [ cos (√ β2 −α2t ) + √ β + α β −α sin (√ β2 −α2t )] , |α| < |β| . (25) it can be easily verified from (25) that φ(α,β,t) = φ(−α,−β,t). (26) in figure 1, the coincidence of the curves φ(1, 2,t) = φ(−1,−2,t) and φ(1,−2,t) = φ(−1, 2,t) is shown. moreover, the influences of β and α on the behaviour of the exact solution (24) are depicted in figure 2 and figure 3, respectively. 6 int. j. anal. appl. (2022), 20:71 1 2 3 4 5 t -2 -1 1 2 φhtl α=-1, β=+2 α=+1, β=-2 α=-1, β=-2 α=+1, β=+2 figure 1. symmetry property of the exact solution eq. (24) where φ(1, 2,t) = φ(−1,−2,t) and φ(1,−2,t) = φ(−1, 2,t). 1 2 3 4 5 6 7 t -1.0 -0.5 0.5 1.0 φhtl β=3 β=2 β=1 figure 2. variation of the exact solution in eq. (24) against t at different values of β when λ = 1 and α = −1/2. 2 4 6 8 10 t -2 -1 1 2 φhtl α=3�4 α=1�2 α=1�4 figure 3. variation of the exact solution in eq. (24) against t at different values of α when λ = 1 and β = 1. int. j. anal. appl. (2022), 20:71 7 4.2. periodicity. the trigonometric functions involved in (24) are periodic and hence the present exact solution is of periodic nature with periodicity p, given by p = 2π√ β2 −α2 , |α| < |β| , (27) which has been also mentioned in ref. [20]. however, after taking a deep insight at the periodicity (27) for sufficiently small values of α compared to β one can detect that p ≈ 2π β . the solution in such case is given by the following theorem theorem 1. for sufficiently small values of α compared to β, i.e., |α| << |β|, the approximate solution of eqs. (1-2) takes the form: φ(t) ≈ λ [ cos (βt) + ( 1 + α β ) sin (βt) ] , (28) wit periodicity p ≈ 2π β . proof. to start the proof, we rewrite the solution (24) in the form φ(t) = λ  cos  β √ 1 − ( α β )2 t   + √ α/β + 1 1 −α/β sin  β √ 1 − ( α β )2 t     , (29) or φ(t) = λ [ cos ( β √ 1 − �2t ) + √ 1 + � 1 − � sin ( β √ 1 − �2t )] , (30) where � = α β << 1. expanding the expressions √ 1 − �2 and 1+� 1−� as power series and neglecting higher-order terms of � , we have √ 1 − �2 = 1 − �2 2 − �4 8 − �6 16 −···≈ 1, (31) 1 + � 1 − � = 1 + � + �2 2 + �3 2 + 3�4 8 + · · · ≈ 1 + �. (32) substituting eqs. (31) and (32) into eq. (30), we obtain the approximate solution: φ(t) ≈ λ [ cos (βt) + ( 1 + α β ) sin (βt) ] . (33) 8 int. j. anal. appl. (2022), 20:71 0.5 1.0 1.5 2.0 2.5 3.0 t 0.00002 0.00004 0.00006 0.00008 0.0001 rehtl β=3 β=2 β=1 figure 4. plots of the absolute residual error in eq. (37) at different values of β when λ = 1 and α = 1/100. the periodicity p in eq. (27) can be also written in terms of � as p = 2π√ β2 −α2 , = 2π β × 1 √ 1 − �2 , = 2π β ( 1 + �2 2 + 3�4 8 + . . . ) , (34) ≈ 2π β , (35) which completes the proof. � 4.2.1. behaviour of the approximate solution. in order to check the accuracy of the approximate solution (28), some numerical results are conducted in this section. simulation of the approximate solution is accomplished here through calculating the absolute residual error for solution (24), given by re(t) = ∣∣φ′(t) −αφ(t) −βφ(−t)∣∣ . (36) substituting eq. (24) into eq. (36) gives re(t) = ∣∣∣∣2λα2β sin(βt) ∣∣∣∣ . (37) the accuracy of the approximate solution is clearly verified in figures 4, 5, and 6 through plotting the residual (37) and comparing the approximate solution (28) with the exact one. int. j. anal. appl. (2022), 20:71 9 5 10 15 20 25 30 t -1.0 -0.5 0.5 1.0 φhtl exact approx. figure 5. comparisons between the approximate solution in eq. (28) and the exact solution in eq. (24) at λ = 1, α = 1/100, and β = 1. 1 2 3 4 5 6 t 0.00002 0.00004 0.00006 0.00008 0.0001 rehtl α=1�300 α=1�200 α=1�100 figure 6. plots of the absolute residual error in eq. (37) at different values of α when λ = 1 and β = 1. 4.3. polynomial solution at β = α. it is noted from the expression in eq. (24) that the exact solution is not valid when β = α. however, the solution of such case can be obtained through calculating the limit of eq. (24) as β → α. to do so, we suppose that β−α = σ and thus σ → 0 as α → β. accordingly, the solution (24) becomes φ(t) = λ lim α→β [ cos (√ β2 −α2t ) + √ α + β β −α sin (√ β2 −α2t )] , (38) or equivalently φ(t) = λ lim σ→0 [ cos (√ 2ασ t ) + √ 2α σ sin (√ 2ασ t )] , 10 int. j. anal. appl. (2022), 20:71 = λ [ 1 + √ 2α lim σ→0 ( sin (√ 2ασ t ) √ σ )] , = λ [ 1 + √ 2α. √ 2α t ] , = λ (1 + 2αt) . (39) it may be important to mention that eq. (39) is the exact solution of the differential-difference equation: φ′(t) = α (φ(t) + φ(−t)) , φ(0) = λ. (40) 4.4. hyperbolic solution at α > β. in order to obtain the solution in terms of the hyperbolic functions, eq. (24) can be rewritten as φ(t) = λ [ cos ( j √ α2 −β2t ) − j √ α + β α−β sin ( j √ α2 −β2t )] , (41) where j = √ −1 is the imaginary number. therefore, eq. (41) is transformed into the form: φ(t) = λ [ cosh (√ α2 −β2t ) + √ α + β α−β sinh (√ α2 −β2t )] , α > β. (42) this is also the corresponding obtained solution in ref. [20] utilizing the ssm. 4.5. constant solution at β = −α. the solution of this case can be obtained by direct substitutions into eq. (24). so, we observe from eq. (24), at β = −α, that cos (√ β2 −α2t ) = cos(0) = 1,√ α + β β −α sin (√ β2 −α2t ) = 0, (43) and consequently the solution (24) reduces to φ(t) = λ. (44) which is the constant solution of the following differential-difference equation: φ′(t) = α (φ(t) −φ(−t)) , φ(0) = λ. (45) int. j. anal. appl. (2022), 20:71 11 4.6. periodic solution at a special case: α = 0. in this case, the present model (1-2) reduces to the differential-difference equation: φ′(t) = βφ(−t), φ(0) = λ. (46) although it is simpler, the exact solution may be not available in the literature however, the current study gives the solution directly from eq. (24) by setting α = 0, hence φ(t) = λ (cos βt + sin βt) . (47) 5. conclusions in this paper, a hybrid approach based on the lt and the adm is applied to solve a special kind of the pantograph delay functional-differential equation. the exact solution was obtained in a direct manner if compared with the series method in the literature. at a specific constrain of the constants involved in the present model, an accurate analytic approximation was determined. it was also shown that the obtained exact solution enjoyed several interesting properties such symmetry, periodicity, and others. in addition, several types of exact solutions were proved at special cases and expressed as hyperbolic, linear, and constant functions. the present method may deserve further extensions to analyze more complex delay models. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] h.i. andrews, third paper: calculating the behaviour of an overhead catenary system for railway electrification, proc. inst. mech. eng. 179 (1964), 809-846. https://doi.org/10.1243/pime_proc_1964_179_050_02. [2] m.r. abbott, numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, comput. j. 13 (1970), 363-368. https://doi.org/10.1093/comjnl/13.4.363. [3] g. gilbert, h.e.h. davtcs, pantograph motion on a nearly uniform railway overhead line, proc. inst. electr. eng. 113 (1966), 485-492. https://doi.org/10.1049/piee.1966.0078. [4] p.m. caine, p.r. scott, single-wire railway overhead system, proc. inst. electr. eng. 116 (1969), 1217-1221. https://doi.org/10.1049/piee.1969.0226. [5] j.r. ockendon, a.b. tayler, the dynamics of a current collection system for an electric locomotive, proc. r. soc. lond. a. math. phys. eng. sci. 322 (1971), 447-468. https://doi.org/10.1098/rspa.1971.0078. [6] l. fox, d. mayers, j.r. ockendon, a.b. tayler, on a functional differential equation, ima j. appl. math. 8 (1971), 271-307. https://doi.org/10.1093/imamat/8.3.271. https://doi.org/10.1243/pime_proc_1964_179_050_02 https://doi.org/10.1093/comjnl/13.4.363 https://doi.org/10.1049/piee.1966.0078 https://doi.org/10.1049/piee.1969.0226 https://doi.org/10.1098/rspa.1971.0078 https://doi.org/10.1093/imamat/8.3.271 12 int. j. anal. appl. (2022), 20:71 [7] t. kato, j.b. mcleod, the functional-differential equation y ′(x) = ay(λx)+by(x), bull. amer. math. soc. 77 (1971), 891-935. [8] a. iserles, on the generalized pantograph functional-differential equation, eur. j. appl. math. 4 (1993), 1–38. https://doi.org/10.1017/s0956792500000966. [9] v.a. ambartsumian, on the fluctuation of the brightness of the milky way, dokl. akad. nauk sssr, 44 (1994), 223-226. [10] j. patade, s. bhalekar, on analytical solution of ambartsumian equation, natl. acad. sci. lett. 40 (2017), 291–293. https://doi.org/10.1007/s40009-017-0565-2. [11] f.m. alharbi, a. ebaid, new analytic solution for ambartsumian equation, j. math. syst. sci. 8 (2018), 182-186. https://doi.org/10.17265/2159-5291/2018.07.002. [12] h. bakodah, a. ebaid, exact solution of ambartsumian delay differential equation and comparison with daftardargejji and jafari approximate method, mathematics. 6 (2018), 331. https://doi.org/10.3390/math6120331. [13] n.o. alatawi, a. ebaid, solving a delay differential equation by two direct approaches, j. math. syst. sci. 9 (2019), 54-56. https://doi.org/10.17265/2159-5291/2019.02.003. [14] a. ebaid, a. al-enazi, b.z. albalawi, m.d. aljoufi, accurate approximate solution of ambartsumian delay differential equation via decomposition method, math. comput. appl. 24 (2019), 7. https://doi.org/10.3390/ mca24010007. [15] a.a. alatawi, m. aljoufi, f.m. alharbi, a. ebaid, investigation of the surface brightness model in the milky way via homotopy perturbation method, j. appl. math. phys. 8 (2020), 434–442. https://doi.org/10.4236/jamp. 2020.83033. [16] e. a. algehyne, e. r. el-zahar, f. m. alharbi, a. ebaid, development of analytical solution for a generalized ambartsumian equation, aims math. 5 (2020), 249–258. https://doi.org/10.3934/math.2020016. [17] s.m. khaled, e.r. el-zahar, a. ebaid, solution of ambartsumian delay differential equation with conformable derivative, mathematics. 7 (2019), 425. https://doi.org/10.3390/math7050425. [18] d. kumar, j. singh, d. baleanu, s. rathore, analysis of a fractional model of the ambartsumian equation, eur. phys. j. plus. 133 (2018), 259. https://doi.org/10.1140/epjp/i2018-12081-3. [19] a. ebaid, c. cattani, a.s. al juhani, e.r. el-zahar, a novel exact solution for the fractional ambartsumian equation, adv. differ. equ. 2021 (2021), 88. https://doi.org/10.1186/s13662-021-03235-w. [20] a. ebaid, h.k. al-jeaid, on the exact solution of the functional differential equation y ′(t) = ay(t)+ by(−t), adv. differ. equ. control processes, 26 (2022), 39-49. https://doi.org/10.17654/0974324322003. [21] g. adomian, solving frontier problems of physics: the decomposition method, kluwer academi publishers, boston, 1994. [22] a.m. wazwaz, adomian decomposition method for a reliable treatment of the bratu-type equations, appl. math. comput. 166 (2005), 652–663. https://doi.org/10.1016/j.amc.2004.06.059. [23] a. ebaid, approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, z. naturforsch. a. 66 (2011), 423–426. https://doi.org/10.1515/zna-2011-6-707. [24] j.s. duan, r. rach, a new modification of the adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, appl. math. comput. 218 (2011), 4090–4118. https: //doi.org/10.1016/j.amc.2011.09.037. [25] a. ebaid, a new analytical and numerical treatment for singular two-point boundary value problems via the adomian decomposition method, j. comput. appl. math. 235 (2011), 1914–1924. https://doi.org/10.1016/ j.cam.2010.09.007. https://doi.org/10.1017/s0956792500000966 https://doi.org/10.1007/s40009-017-0565-2 https://doi.org/10.17265/2159-5291/2018.07.002 https://doi.org/10.3390/math6120331 https://doi.org/10.3390/mca24010007 https://doi.org/10.3390/mca24010007 https://doi.org/10.4236/jamp.2020.83033 https://doi.org/10.4236/jamp.2020.83033 https://doi.org/10.3934/math.2020016 https://doi.org/10.3390/math7050425 https://doi.org/10.1140/epjp/i2018-12081-3 https://doi.org/10.1186/s13662-021-03235-w https://doi.org/10.17654/0974324322003 https://doi.org/10.1016/j.amc.2004.06.059 https://doi.org/10.1515/zna-2011-6-707 https://doi.org/10.1016/j.amc.2011.09.037 https://doi.org/10.1016/j.amc.2011.09.037 https://doi.org/10.1016/j.cam.2010.09.007 https://doi.org/10.1016/j.cam.2010.09.007 int. j. anal. appl. (2022), 20:71 13 [26] e.h. aly, a. ebaid, r. rach, advances in the adomian decomposition method for solving two-point nonlinear boundary value problems with neumann boundary conditions, computers math. appl. 63 (2012), 1056–1065. https://doi.org/10.1016/j.camwa.2011.12.010. [27] c. chun, a. ebaid, m. lee, e. aly, an approach for solving singular two point boundary value problems: analytical and numerical treatment, anziam j. 52 (2012), 21-43. https://doi.org/10.21914/anziamj.v53i0.4582. [28] a. ebaid, m.d. aljoufi, a.m. wazwaz, an advanced study on the solution of nanofluid flow problems via adomian’s method, appl. math. lett. 46 (2015), 117–122. https://doi.org/10.1016/j.aml.2015.02.017. [29] j. diblík, m. kúdelcíková, two classes of positive solutions of first order functional differential equations of delayed type, nonlinear anal.: theory methods appl. 75 (2012), 4807-4820. https://doi.org/10.1016/j.na. 2012.03.030. [30] a. ebaid, n. al-armani, a new approach for a class of the blasius problem via a transformation and adomian’s method, abstr. appl. anal. 2013 (2013), 753049. https://doi.org/10.1155/2013/753049. [31] a. ebaid, analytical solutions for the mathematical model describing the formation of liver zones via adomian’s method, comput. math. methods med. 2013 (2013), 547954. https://doi.org/10.1155/2013/547954. [32] a. alshaery, a. ebaid, accurate analytical periodic solution of the elliptical kepler equation using the adomian decomposition method, acta astronautica. 140 (2017), 27–33. https://doi.org/10.1016/j.actaastro.2017. 07.034. [33] h.o. bakodah, a. ebaid, the adomian decomposition method for the slip flow and heat transfer of nanofluids over a stretching/shrinking sheet, rom. rep. phys. 70 (2018), 115. [34] w. li, y. pang, application of adomian decomposition method to nonlinear systems, adv. differ. equ. 2020 (2020), 67. https://doi.org/10.1186/s13662-020-2529-y. [35] a. ebaid, h.k. al-jeaid, h. al-aly, notes on the perturbation solutions of the boundary layer flow of nanofluids past a stretching sheet, appl. math. sci. 7 (2013), 6077–6085. https://doi.org/10.12988/ams.2013.36277. [36] a. ebaid, s.m. khaled, an exact solution for a boundary value problem with application in fluid mechanics and comparison with the regular perturbation solution, abstr. appl. anal. 2014 (2014), 172590. https://doi.org/ 10.1155/2014/172590. [37] a. ebaid, remarks on the homotopy perturbation method for the peristaltic flow of jeffrey fluid with nanoparticles in an asymmetric channel, computers math. appl. 68 (2014), 77–85. https://doi.org/10.1016/j. camwa.2014.05.008. [38] z. ayati, j. biazar, on the convergence of homotopy perturbation method, j. egypt. math. soc. 23 (2015), 424–428. https://doi.org/10.1016/j.joems.2014.06.015. [39] a. ebaid, a.f. aljohani, e.h. aly, homotopy perturbation method for peristaltic motion of gold-blood nanofluid with heat source, int. j. numer. methods heat fluid flow, 30 (2020), 3121-3138, https://doi.org/10.1108/ hff-11-2018-0655. [40] a. ebaid, analysis of projectile motion in view of fractional calculus, appl. math. model. 35 (2011), 1231–1239. https://doi.org/10.1016/j.apm.2010.08.010. [41] s.m. khaled, a. ebaid, f. al mutairi, the exact endoscopic effect on the peristaltic flow of a nanofluid, j. appl. math. 2014 (2014), 367526. https://doi.org/10.1155/2014/367526. [42] a. ebaid, m.a. al sharif, application of laplace transform for the exact effect of a magnetic field on heat transfer of carbon nanotubes-suspended nanofluids, z. naturforsch. a. 70 (2015), 471–475. https://doi.org/ 10.1515/zna-2015-0125. [43] a. ebaid, a.-m. wazwaz, e. alali, b.s. masaedeh, hypergeometric series solution to a class of second-order boundary value problems via laplace transform with applications to nanofluids, commun. theor. phys. 67 (2017), 231. https://doi.org/10.1088/0253-6102/67/3/231. https://doi.org/10.1016/j.camwa.2011.12.010 https://doi.org/10.21914/anziamj.v53i0.4582 https://doi.org/10.1016/j.aml.2015.02.017 https://doi.org/10.1016/j.na.2012.03.030 https://doi.org/10.1016/j.na.2012.03.030 https://doi.org/10.1155/2013/753049 https://doi.org/10.1155/2013/547954 https://doi.org/10.1016/j.actaastro.2017.07.034 https://doi.org/10.1016/j.actaastro.2017.07.034 https://doi.org/10.1186/s13662-020-2529-y https://doi.org/10.12988/ams.2013.36277 https://doi.org/10.1155/2014/172590 https://doi.org/10.1155/2014/172590 https://doi.org/10.1016/j.camwa.2014.05.008 https://doi.org/10.1016/j.camwa.2014.05.008 https://doi.org/10.1016/j.joems.2014.06.015 https://doi.org/10.1108/hff-11-2018-0655 https://doi.org/10.1108/hff-11-2018-0655 https://doi.org/10.1016/j.apm.2010.08.010 https://doi.org/10.1155/2014/367526 https://doi.org/10.1515/zna-2015-0125 https://doi.org/10.1515/zna-2015-0125 https://doi.org/10.1088/0253-6102/67/3/231 14 int. j. anal. appl. (2022), 20:71 [44] h. saleh, e. alali, a. ebaid, medical applications for the flow of carbon-nanotubes suspended nanofluids in the presence of convective condition using laplace transform, j. assoc. arab univ. basic appl. sci. 24 (2017), 206–212. https://doi.org/10.1016/j.jaubas.2016.12.001. [45] a. ebaid, e. alali, h.s. ali, the exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, j. assoc. arab univ. basic appl. sci. 24 (2017), 156–159. https://doi.org/ 10.1016/j.jaubas.2017.07.003. [46] s.m. khaled, the exact effects of radiation and joule heating on magnetohydrodynamic marangoni convection over a flat surface, therm sci. 22 (2018), 63–72. https://doi.org/10.2298/tsci151005050k. [47] h. ali, e. alali, a. ebaid, f. alharbi, analytic solution of a class of singular second-order boundary value problems with applications, mathematics. 7 (2019), 172. https://doi.org/10.3390/math7020172. [48] a. ebaid, e.r. el-zahar, a.f. aljohani, b. salah, m. krid, j.t. machado, analysis of the two-dimensional fractional projectile motion in view of the experimental data, nonlinear dyn. 97 (2019), 1711–1720. https: //doi.org/10.1007/s11071-019-05099-y. [49] a. ebaid, w. alharbi, m.d. aljoufi, e.r. el-zahar, the exact solution of the falling body problem in threedimensions: comparative study, mathematics. 8 (2020), 1726. https://doi.org/10.3390/math8101726. [50] a. ebaid, h.k. al-jeaid, the mittag–leffler functions for a class of first-order fractional initial value problems: dual solution via riemann–liouville fractional derivative, fractal fract. 6 (2022), 85. https://doi.org/10.3390/ fractalfract6020085. [51] a.f. aljohani, a. ebaid, e.a. algehyne, y.m. mahrous, c. cattani, h.k. al-jeaid, the mittag-leffler function for re-evaluating the chlorine transport model: comparative analysis, fractal fract. 6 (2022), 125. https: //doi.org/10.3390/fractalfract6030125. [52] m.r. spiegel, laplace transforms, mcgraw-hill. inc., new york, 1965. https://doi.org/10.1016/j.jaubas.2016.12.001 https://doi.org/10.1016/j.jaubas.2017.07.003 https://doi.org/10.1016/j.jaubas.2017.07.003 https://doi.org/10.2298/tsci151005050k https://doi.org/10.3390/math7020172 https://doi.org/10.1007/s11071-019-05099-y https://doi.org/10.1007/s11071-019-05099-y https://doi.org/10.3390/math8101726 https://doi.org/10.3390/fractalfract6020085 https://doi.org/10.3390/fractalfract6020085 https://doi.org/10.3390/fractalfract6030125 https://doi.org/10.3390/fractalfract6030125 1. introduction 2. the lt-decomposition method 3. the exact solution 4. properties of solution 4.1. symmetry&behaviour of the solution 4.2. periodicity 4.3. polynomial solution at = 4.4. hyperbolic solution at > 4.5. constant solution at = 4.6. periodic solution at a special case: =0 5. conclusions references international journal of analysis and applications volume 16, number 3 (2018), 414-426 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-414 weighted minimal and weighted flat surfaces of revolution in galilean 3-space with density ahmet kazan1,∗ and h. bayram karadağ2 1department of computer technologies, doğanşehir vahap küçük vocational school of higher education, inonu university, malatya, turkey 2department of mathematics, faculty of arts and sciences, inonu university, malatya, turkey ∗corresponding author: ahmet.kazan@inonu.edu.tr abstract. in this paper, we obtain the weighted mean and weighted gaussian curvatures of surfaces of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , a1,a2,a3 ∈ r not all zero. also, we investigate some cases of weighted minimal surfaces of revolution according to ai, i = 1, 2, 3 and weighted flat surfaces of revolution. 1. introduction the geometry of surfaces of revolution is an important studying area for geometers and it has been studied widely in euclidean 3-space e3, lorentz-minkowski space e31 , galilean 3-space g3, pseudo-galilean 3-space g13 and also in higher dimensions of these spaces. in these studies, the authors have investigated lots of characterizations about surfaces of revolution, but one of the most important characterization is minimal and flat surfaces of revolution. the catenoid which is obtained by rotating a catenary is the most famous minimal surface of revolution. for another characterizations of surfaces of revolution, we refer to [1], [3], [5], [6], [7], [8], [11], [12], [14] and etc. 2010 mathematics subject classification. 53a10, 53a20, 53a35. key words and phrases. surface of revolution; isotropic rotation; weighted mean curvature; weighted gaussian curvature; weighted minimal and flat surface of revolution. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 414 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-414 int. j. anal. appl. 16 (3) (2018) 415 on the other hand, the geomerty of weighted manifold with density is a new studying area for geometers. in this sense, the weighted mean curvature hφ, which is also called φ-mean curvature, of a surface in euclidean 3-space e3 with density eφ has been introduced in [9] and it is given by hφ = h − 12 〈n,∇φ〉 , where h is the mean curvature, n is the unit normal vector of surface and ∇φ is the gradient of φ. the weighted mean curvature is a natural generalization of the mean curvature of a surface and a surface with hφ = 0 is called a weighted minimal surface or a φ-minimal surface. also in [2], the authors have introduced the notion of weighted gaussian curvature or φ-gaussian curvature of a surface which is a generalization of the gaussian curvature of a surface in a manifold with density eφ and they have defined it as gφ = g − ∆φ. here, g is the gaussian curvature of a surface and ∆ is the laplacian operator. if a surface’s weighted gaussian curvature is zero everywhere, then we call it a weighted flat surface or φ-flat surface. in [18], the translation surfaces in g3 with a log-linear density have been studied and such a surface with vanishing weighted mean curvature has been classified. in [13], the authors have considered the euclidean 3-space e3 with a positive density function eφ, where φ = −x2−y2, (x,y,z) ∈ e3 and they have constructed all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted gaussian curvature and the mean curvature functions. also in [10], the authors have studied the ruled surfaces and translation surfaces in e3 with density ez and as a generalization of this density, lopez has used a linear density eax+by+cz, a,b,c ∈ r and classified weighted minimal translation and cyclic surfaces in e3 [15]. in the present paper, we obtain the weighted mean curvature and weighted gaussian curvature of three types of surfaces of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , ai ∈ r not all zero and investigate some cases of weighted minimal surfaces of revolution according to ai, i = 1, 2, 3 and weighted flat surfaces of revolution. also, we draw the obtaining surfaces of revolution with the aid of mathematica. 2. preliminaries the galilean space g3 is a cayley-klein space equipped with the projective metric of signature (0, 0, +, +). the absolute figure of the galilean geometry consists of an ordered triple {w,f,i}, where w is the ideal (absolute) plane, f the line (absolute line) in w and i the fixed elliptic involution of points of p. a vector x = (x1,x2,x3) in g3 is isotropic if x1 = 0 and non-isotropic otherwise. so, for affine ccordinates (x,y,z), the y-axis and z-axis are isotropic while the x-axis is non-isotropic. also, the yz-plane is euclidean and the xy-plane and xz-plane are isotropic. if x = (x1,x2,x3) and y = (y1,y2,y3) are two vectors in g3, then the galilean scalar product of x and y is defined by 〈x, y〉 =   x1y1, if x1 6= 0 or y1 6= 0,x2y2 + x3y3, if x1 = 0 and y1 = 0. int. j. anal. appl. 16 (3) (2018) 416 so, the galilean norm of a vector x in g3 is ‖x‖ = √ 〈x, x〉 and the vector x is a unit vector if ‖x‖ = 1. the galilean cross product of x and y is defined by x × y = ∣∣∣∣∣∣∣∣∣ 0 e2 e3 x1 x2 x3 y1 y2 y3 ∣∣∣∣∣∣∣∣∣ . for a more detailed treatment about galilean space, we refer to [3], [4], [16], [17], [18] and etc. furthermore, let m be a surface in g3 parametrized by γ(u1,u2) = (x(u1,u2),y(u1,u2),z(u1,u2)). then, the unit normal vector n of the surface is defined by n = γ,1 × γ,2 w , where w = ‖γ,1 × γ,2‖ and γ,i = ∂γ∂ui (u 1,u2) for i ∈{1, 2}. the coefficients of the second fundamental form are given by lij = 〈 γ,ijx,1 −x,ijγ,1 x,1 ,n 〉 = 〈 γ,ijx,2 −x,ijγ,2 x,2 ,n 〉 . analogous to euclidean space, in [16], the mean curvature h and the gaussian curvature k of the surface are defined by h = 1 2 2∑ i,j=1 gijlij and k = l11l22 −l212 w2 , where gij = gigj, for i,j ∈{1, 2} and g1 = x,2 w , g2 = −x,1 w . here we always assume that the surface is admissible, that is, its tangent space is nowhere an euclidean plane (for detail, see [3]). 3. weighted minimal and weighted flat surfaces of revolution in g3 with density ea1x 2+a2y 2+a3z 2 in [3], the authors have constructed the surfaces of revolution in galilean 3-space analogously to how that is done in euclidean 3-space and they have obtained 3 types of surfaces of revolution in g3. since there are different kinds of planes in g3, we have to consider two possibilities for the supporting plane of the profile curve which generates the surface of revolution. in this context, the profile curve lies in an euclidean plane or it lies in an isotropic plane. in order to construct a surface of revolution in g3, we use two types of rotations which are defined as follows: int. j. anal. appl. 16 (3) (2018) 417 an euclidean rotation about the non-isotropic x-axis is given by  x′ y′ z′   =   1 0 0 0 cos θ1 sin θ1 0 −sin θ1 cos θ1     x y z   , where θ1 is the euclidean angle and an isotropic rotation is given by  x′ y′ z′   =   1 0 0 θ2 1 0 0 0 1     x y z   +   cθ2 c 2 (θ2) 2 0   , where θ2 is the isotropic angle and c ∈ r. 3.1. weighted minimal and weighted flat type i surfaces of revolution in g3. type i surface of revolution has constructed with the aid of an isotropic rotation as γi(u,v) = (cv,f(u) + c 2 v2,g(u)), (3.1) where the profile curve α lies in the euclidean yz-plane and it is parametrized by α(u) = (0,f(u),g(u)) [3]. here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is f′2(u) + g′2(u) = 1, ∀u ∈ i. (3.2) using (3.2), we obtain the mean curvature and gaussian curvature of (3.1) as h = f′′(u)g′(u) −f′(u)g′′(u) 2 and k = f′′(u) c , (3.3) respectively. on the other hand, the unit normal vector n of the surface (3.1) is n = (0,g′(u),−f′(u)). (3.4) assume that, m is the surface in g3 with density e φ, φ = a1x 2 + a2y 2 + a3z 2, a1,a2,a3 ∈ r not all zero. then, for this density, from (3.1)-(3.4) the weighted mean curvature and the weighted gaussian curvature can be obtained as hφ = f′′g′ −f′g′′ 2 − < (a1cv,a2(f + c 2 v2),a3g), (0,g ′,−f′) > (3.5) and kφ = f′′ c − 2(a1 + a2 + a3), (3.6) respectively. firstly, let us assume that the surface of revolution (3.1) is weighted minimal, i.e., hφ = 0. then, from (3.5) we have f′′g′ −f′g′′ − 2 < (a1cv,a2(f + c 2 v2),a3g), (0,g ′,−f′) >= 0. (3.7) now, we’ll investigate some cases of weighted minimal surfaces of revolution (3.1) according to ai, i = 1, 2, 3. int. j. anal. appl. 16 (3) (2018) 418 if a1 6= 0, then from (3.7) we get f′′g′ −f′g′′ = 0. (3.8) here, using (3.2) in (3.8) we obtain f(u) = c1u + c2 (3.9) and using (3.9) in (3.2) we get g(u) = √ 1 − c21u + c3, (3.10) where c1,c2,c3 ∈ r. hence, we have theorem 3.1. let m be a weighted minimal type i surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , where a1 6= 0, a2 and a3 are constants. then, m can be parametrized by γi(u,v) = (cv,c1u + c2 + c 2 v2, √ 1 − c21u + c3), (3.11) for some constants c, c1 ,c2 and c3. in figure 1, one can see the surface of revolution (3.11) for c = 4, c1 = 1/2, c2 = 2 and c3 = 3. figure 1. if a1 = 0, then from (3.7) we get f′′g′ −f′g′′ − 2(a2(fg′ + c 2 v2g′) −a3f′g) = 0. (3.12) taking a2 = 0 in (3.12) and using (3.2), we get g′′ − 2a3g(1 − (g′)2) = 0. (3.13) the equation (3.13) is a second-order nonlinear ordinary differential equation and g(u) = u + c4, c4 ∈ r, is a special solution for it. so, for this special solution, from (3.2) we get f(u) = c5, c5 ∈ r. hence, we have int. j. anal. appl. 16 (3) (2018) 419 theorem 3.2. let m be a weighted minimal type i surface of revolution in galilean 3-space with density ea3z 2 , where a3 is a non-zero constant. then, m can be parametrized by γi(u,v) = (cv,c5 + c 2 v2,u + c4), (3.14) for some constants c, c4 and c5. in figure 2, one can see the surface of revolution (3.14) for c = 1, c4 = 2 and c5 = 3. figure 2. now, let us investigate the weighted flat surfaces of revolution (3.1). if the surface of revolution (3.1) is weighted flat, i.e., kφ = 0, then from (3.6) we have f′′ − 2c(a1 + a2 + a3) = 0. (3.15) from (3.15), we have f(u) = cru2 + c6u + c7 (3.16) and using (3.16) in (3.2), we have g(u) = ∓ 1 4cr ((c6 + 2cru) √ 1 − c26 − 4cc6ru− 4c2r2u2 + arcsin(c6 + 2cru)) + c8, (3.17) where r = a1 + a2 + a3 and c, c6, c7, c8 are constants. hence, we have theorem 3.3. let m be a weighted flat type i surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , a1,a2,a3 ∈ r not all zero. then, m can be parametrized by γi(u,v) = (cv,cru 2 + c6u + c7 + c 2 v2, (3.18) ∓ 1 4cr ((c6 + 2cru) √ 1 − c26 − 4cc6ru− 4c2r2u2 + arcsin(c6 + 2cru)) + c8), for some constants c, c6, c7, c8 and r = a1 + a2 + a3. int. j. anal. appl. 16 (3) (2018) 420 figure 3 shows the weighted flat type i surface of revolution (3.18) for r = 6, c = 4, c6 = 1, c7 = 2 and c8 = 3. figure 3. 3.2. weighted minimal and weighted flat type ii surfaces of revolution in g3. type ii surface of revolution can be obtained with the aid of an isotropic rotation as γii(u,v) = (u + cv,g(u),uv + c 2 v2), (3.19) where the profile curve α lies in the isotropic xy-plane and it is parametrized by α(u) = (f(u),g(u), 0) [3]. here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is f′2(u) = 1, ∀u ∈ i. (3.20) we obtain the mean curvature and gaussian curvature of (3.19) as h = −c2(ug′′(u) −g′(u)) 2(u2 + c2(g′(u))2) 3 2 and k = c2g′(u)(ug′′(u) −g′(u)) (u2 + c2(g′(u))2)2 , (3.21) respectively. on the other hand, the unit normal vector n of the surface (3.19) is n = 1√ u2 + c2(g′(u))2 (0,−u,−cg′(u)). (3.22) assume that, m is the surface in g3 with density e φ, φ = a1x 2 + a2y 2 + a3z 2. then, for this density, from (3.19)-(3.22) the weighted mean curvature and the weighted gaussian curvature can be obtained as hφ = −c2(ug′′ −g′) 2(u2 + c2(g′)2) 3 2 − 1√ u2 + c2(g′)2 < (a1(u + cv),a2g,a3(uv + c 2 v2)), (0,−u,−cg′) > (3.23) and kφ = c2g′(ug′′ −g′) (u2 + c2(g′)2)2 − 2(a1 + a2 + a3), (3.24) respectively. int. j. anal. appl. 16 (3) (2018) 421 firstly, let us assume that the surface of revolution (3.19) is weighted minimal, i.e., hφ = 0. then, from (3.23) we have c2(ug′′ −g′) + 2(u2 + c2(g′)2) < (a1(u + cv),a2g,a3(uv + c 2 v2)), (0,−u,−cg′) >= 0. (3.25) now, we’ll investigate some cases of weighted minimal surfaces of revolution (3.19) according to ai, i = 1, 2, 3. if a1 6= 0, then from (3.25) we get c2(ug′′ −g′) = 0. (3.26) so, from (3.26) we get g(u) = c1 2 u2 + c2, (3.27) where c1,c2 ∈ r. hence, we have theorem 3.4. let m be a weighted minimal type ii surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , where a1 6= 0, a2 and a3 are constants. then, m can be parametrized by γii(u,v) = (u + cv, c1 2 u2 + c2,uv + c 2 v2), (3.28) for some constants c, c1 and c2. the shape of the surface of revolution (3.28) for c = 3, c1 = 1 and c2 = 2 can be seen in figure 4 as follows: figure 4. if a1 = 0, then from (3.25) we get c2(ug′′ −g′) + 2(u2 + c2(g′)2)(−a2ug −a3cg′(uv + c 2 v2)) = 0. (3.29) taking a3 = 0 in (3.29), we get c2(ug′′ −g′) − 2a2ug(u2 + c2(g′)2) = 0. (3.30) hence, we have int. j. anal. appl. 16 (3) (2018) 422 theorem 3.5. let m be a weighted flat type ii surface of revolution in galilean 3-space with density ea2y 2 , where a2 is a non-zero constant. then, the real function g which is in (3.19) must satisfy the second-order nonlinear ordinary differential equation (3.30). now, let us investigate the weighted flat surfaces of revolution (3.19). if the surface of revolution (3.19) is weighted flat, i.e., kφ = 0, then from (3.24) we have c2g′(ug′′ −g′) − 2r(u2 + c2(g′)2)2 = 0, (3.31) where r = a1 + a2 + a3. from (3.31), we have c2ug′g′′ − c2g′2 − 2ru4 − 4ru2c2g′2 − 2rc4g′4 = 0. (3.32) taking p(u) = g′(u) in (3.32), we get p(u) = ∓ √ −2ru4 −u2 − 2c2c4u2√ 2 √ c2ru2 + c4c4 (3.33) and integrating (3.33), we have g(u) = ± u √ 2ru2 + 2c2c4 + 1   √2 arcsin h(√2√ru2 + c2c4)√ru2 + c2c4 +2(ru2 + c2c4) √ 2ru2 + 2c2c4 + 1   4 √ 2r √ −u2(2ru2 + 2c2c4 + 1) √ c2ru2 + c4c4 , (3.34) where r = a1 + a2 + a3, c and c4 are constants. hence, we have theorem 3.6. let m be a weighted flat type ii surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , a1,a2,a3 ∈ r not all zero. then, m can be parametrized by γi(u,v) = (u + cv, (3.35) ± u √ 2ru2 + 2c2c4 + 1   √2 arcsin h(√2√ru2 + c2c4)√ru2 + c2c4 +2(ru2 + c2c4) √ 2ru2 + 2c2c4 + 1   4 √ 2r √ −u2(2ru2 + 2c2c4 + 1) √ c2ru2 + c4c4 , uv + c 2 v2), for some constants c, c4 and r = a1 + a2 + a3. figure 5 shows the weighted flat type ii surface of revolution (3.35) for r = −6, c = 2 and c4 = 1. int. j. anal. appl. 16 (3) (2018) 423 figure 5. 3.3. weighted minimal and weighted flat type iii surfaces of revolution in g3. type iii surface of revolution can be obtained with the aid of an euclidean rotation about the axis x as γiii(u,v) = (u,g(u) cos v,−g(u) sin v), (3.36) where the profile curve α lies in the isotropic xy-plane and it is parametrized by α(u) = (f(u),g(u), 0) [3]. here, f and g are real functions and we assume that, the profile curve α is given by the arc length, that is f′2(u) = 1, ∀u ∈ i. (3.37) now, we obtain the mean curvature and gaussian curvature of the surface of revolution (3.36) as h = − 1 2g(u) and k = − g′′(u) g(u) , (3.38) respectively. on the other hand, the unit normal vector n of the surface (3.36) is n = (0, cos v,−sin v). (3.39) assume that, m is the surface in g3 with density e φ, φ = a1x 2 + a2y 2 + a3z 2. then, for this density, from (3.36)-(3.39) the weighted mean curvature and the weighted gaussian curvature can be obtained as hφ = − 1 2g − < (a1u,a2g cos v,−a3g sin v), (0, cos v,−sin v) > (3.40) and kφ = − g′′(u) g(u) − 2(a1 + a2 + a3), (3.41) respectively. firstly, let us assume that the surface of revolution (3.36) is weighted minimal, i.e., hφ = 0. then, from (3.40) we have 1 2g + < (a1u,a2g cos v,−a3g sin v), (0, cos v,−sin v) >= 0 (3.42) next, we’ll investigate some cases of weighted minimal surfaces of revolution (3.36) according to ai, i = 1, 2, 3. int. j. anal. appl. 16 (3) (2018) 424 if a1 6= 0, then from (3.42) we get 1 2g = 0 (3.43) and this is a contradiction. hence, we have theorem 3.7. there is no weighted minimal type iii surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , where a1 6= 0, a2 and a3 are constants. if a1 = 0, then from (3.42) we get 1 2g + a2g cos 2 v + a3g sin 2 v = 0 (3.44) taking a2 = a3 = a 6= 0 in (3.44), we get 1 2g + ag = 0 (3.45) and so, g(u) = ∓ √ − 1 2a ,a < 0. (3.46) hence, we have theorem 3.8. let m be a weighted minimal type iii surface of revolution in galilean 3-space with density ea(y 2+z2), where a < 0 is a constant. then, m can be parametrized by γiii(u,v) = (u,∓ √ − 1 2a cos v,± √ − 1 2a sin v). (3.47) the shape of the surface of revolution (3.47) for a = −1 is given in figure 6. figure 6. now, let us investigate the weighted flat surfaces of revolution (3.36). if the surface of revolution (3.36) is weighted flat, i.e., kφ = 0, then from (3.41) we have g′′ g + 2r = 0, (3.48) int. j. anal. appl. 16 (3) (2018) 425 where r = a1 + a2 + a3. from (3.48), we have g(u) = c1 cos( √ 2ru) + c2 sin( √ 2ru). (3.49) hence, we have theorem 3.9. let m be a weighted flat type iii surface of revolution in galilean 3-space with density ea1x 2+a2y 2+a3z 2 , a1,a2,a3 ∈ r not all zero. then, m can be parametrized by γiii(u,v) = (u, (c1 cos( √ 2ru) + c2 sin( √ 2ru)) cos v,−(c1 cos( √ 2ru) + c2 sin( √ 2ru)) sin v) (3.50) for some constants c1, c2 and r = a1 + a2 + a3. figure 7 shows the weighted flat type iii surface of revolution (3.50) for r = 2, c1 = 1 and c2 = 3. figure 7. references [1] m. bekkar and h. zoubir, surfaces of revolution in the 3-dimensional lorentz-minkowski space satisfying ∆xi = λixi, int. j. contemp. math. sci. 24 (3) (2008), 1173-1185. [2] i. corwin, h. hoffman, s. hurder, v. ssum and y. xu, differential geometry of manifolds with density, rose-hulman und. math. j. 7 (2006), 1-15. [3] m. dede, c. ekici and w. goemans, surfaces of revolution with vanishing curvature in galilean 3-space, j. math. phys. in press. [4] m. dede, tubular surfaces in galilean space, math. commun. 18 (2013), 209-217. int. j. anal. appl. 16 (3) (2018) 426 [5] f. dillen, j. pas and l. verstraelen, on the gauss map of surfaces of revolution, bull. inst. math. acad. sinica 18 (1990), 239-246. [6] b. divjak and z.m. sipus, some special surfaces in the pseudo-galilean space, acta math. hungar. 118 (3) (2008), 209-226. [7] a. ferrandez and p. lucas, on surfaces in the 3-dimensional lorentz-minkowski space, pac. j. math. 152 (1) (1992), 93-100. [8] o.j. garay, on a certain class of finite type surfaces of revolution, kodai math. j. 11 (1) (1988), 25-31. [9] m. gromov, isoperimetry of waists and concentration of maps, geom. func. anal. 13 (2003), 178-215. [10] d.t. hieu and n.m hoang, ruled minimal surfaces in r3 with density ez, pac. j. math. 243 (2009), 277-285. [11] g. kaimakamis and b. papantoniou, surfaces of revolution in the 3-dimensional lorentz-minkowski space satisfying ∆ii~r = a~r, j. geom. 81 (2004), 81-92. [12] a. kazan and h.b. karadağ, a classification of surfaces of revolution in lorentz-minkowski space, int. j. contemp. math. sci. 39 (6) (2011), 1915-1928. [13] d-s. kim and d.w. yoon, constructions of helicoidal surfaces in euclidean space with density, symmetry 9 (2017), art. id 173. [14] s. lee and j.h. varnado, spacelike constant mean curvature surfaces of revolution in minkowski 3-space, differ. geom. dyn. syst. 8 (1) (2006), 144-165. [15] r. lópez, minimal surface in euclidean space with a log-linear density, arxiv:1410.2517 [math.dg] (accessed on 20 july 2017). [16] o. röschel, die geometrie des galileischen raumes, bericht der mathematisch-statistischen sektion in der forschungsgesellschaft joanneum, bericht nr. 256, habilitationsschrift, leoben, (1984). [17] z.m. sipus, ruled weingarten surfaces in the galilean space, period. math. hung. 56 (2) (2008), 213-225. [18] d.w. yoon, weighted minimal translation surfaces in the galilean space with density, open math. 15 (2017), 459-466. 1. introduction 2. preliminaries 3. weighted minimal and weighted flat surfaces of revolution in g3 with density ea1x2+a2y2+a3z2 3.1. weighted minimal and weighted flat type i surfaces of revolution in g3 3.2. weighted minimal and weighted flat type ii surfaces of revolution in g3 3.3. weighted minimal and weighted flat type iii surfaces of revolution in g3 references int. j. anal. appl. (2023), 21:47 on hybrid pure hyperideals in ordered hypersemigroups hataikhan sanpan, pongsakorn kitpratyakul, somsak lekkoksung∗ division of mathematics, faculty of engineering, rajamangala university of technology isan, khon kaen campus, khon kaen 40000, thailand ∗corresponding author: lekkoksung_somsak@hotmail.com abstract. in this paper, the concepts of hybrid pure hyperideals in ordered hypersemigroups are introduced and some algebraic properties of hybrid pure hyperideals are studied. we characterize weakly regular ordered hypersemigroups in terms of hybrid pure hyperideals. finally, we introduce the concepts of hybrid weakly pure hyperideals and prove that the hybrid hyperideals are hybrid weakly pure hyperideals if such hybrid hyperideals satisfy the idempotent property. 1. introduction the theory of fuzzy sets is the most appropriate theory for dealing with uncertainty and was introduced by zadeh [25] in 1965. after the introduction of the concept of fuzzy sets by zadeh, several researchers researched the generalizations of the notions of fuzzy sets with huge applications in computer science, artificial intelligence, control engineering, robotics, automata theory, decision theory, finite state machine, graph theory, logic, operation research and many branches of pure and applied mathematics. for example, xie et al. [24] applied the fuzzy set theory to the switching method. molodtsov [21] introduced the concept of the soft set as a new mathematical tool for dealing with uncertainties being free from the difficulties that have troubled the usual theoretical approaches. the soft sets have many applications in several branches of both pure and applied sciences (see [12], [22], [23]). as a parallel circuit of fuzzy sets and soft sets, jun, song, and muhiuddin [18] introduced the notion of hybrid structures in a set of parameters over an initial universe set. the hybrid structures can be received: jan. 31, 2023. 2020 mathematics subject classification. 06f99, 06f05. key words and phrases. ordered hypersemigroup; hybrid structure; hybrid hyperideal; hybrid pure hyperideal. https://doi.org/10.28924/2291-8639-21-2023-47 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-47 2 int. j. anal. appl. (2023), 21:47 applied in many areas including mathematics, statistics, computer science, electrical instruments, industrial operations, business, engineering, social decisions, etc. (see [2], [13], [14]). the algebraic hyperstructure theory was first introduced in 1934 by marty [20]. hyperstructures have many applications in several branches of both pure and applied sciences (see [8–11, 15]). recently, heidari and davvaz applied the hyperstructure theory to ordered semigroups and introduced the concept of ordered semihypergroups (or hypersemigroups)(see [16]), which is a generalization of the concept of ordered semigroups. furthermore, the ordered semihypergroup theory was enriched by the work of many researchers, for example, [3, 4, 11, 15]. in particular, the hyperideal theory on semihypergroups and ordered hypersemigroups can be seen in [3–5,17]. in 1989, ahsan and takahashi [1] introduced the notions of pure ideals and purely prime ideals of semigroups. later, changphas and sanborisoot [7] defined the notions of left pure, right pure, left weakly pure, and right weakly pure ideals in ordered semigroups and gave some of their characterizations. in 2020, changphas and davvaz [6] studied the purity of hyperideals in ordered hyperstructures. they introduced the notions of pure hyperideals and weakly pure hyperideals in an ordered semihypergroup. we now apply hybrid structures to ordered hyperstructures. in this present paper, the concepts of hybrid pure hyperideals in ordered hypersemigroups are introduced and some algebraic properties of such hyperideals are studied. we characterize weakly regular ordered hypersemigroups in terms of hybrid pure hyperideals. finally, we introduce the concepts of hybrid weakly pure hyperideals and prove that the hybrid hyperideals are hybrid weakly pure hyperideals if such hybrid hyperideals satisfy the idempotent property. 2. preliminaries in this section, we will recall the basic terms and definitions from the ordered hypersemigroup theory and the hybrid structure theory that we will use later in this paper. throughout this paper, we will use the concepts of ordered hypersemigroups introduced by kehayopulu [19] and hybrid structures introduced by anis [2]. definition 2.1. a hypergroupoid is a nonempty set h with a hyperoperation ◦ : h ×h →p∗(h) | (a,b) 7→ a◦b on h and an operation ∗ :p∗(h)×p∗(h)→p∗(h) | (a,b) 7→ a∗b on p∗(h) (induced by the hyperoperation ◦) defined by a∗b = ⋃ a∈a,b∈b (a◦b) (p∗(h) is the set of all nonempty subsets of h.) int. j. anal. appl. (2023), 21:47 3 we have {x}∗{y}= x ◦y. further more a ⊆ b implies a∗c ⊆ b∗c and c ∗a ⊆ c ∗b for any nonempty subsets a, b and c of h. definition 2.2. a hypergroupoid (h;◦) is called a hypersemigroup if {x}∗ (y ◦z)= (x ◦y)∗{z} for every x, y, z ∈ h. for convenient, the previous equation could be identified as x ∗ (y ◦z)= (x ◦y)∗z. let (h;≤) be a partial order set. we define a relation � on p∗(h) as follows: for two nonempty subsets a and b of h, a � b := {(x,y)∈ a×b | x ≤ y,∀x ∈ a,∃y ∈ b}. definition 2.3. the structure (h;◦,≤) is called an ordered hypersemigroup if the following conditions are satisfied: (1) (h;◦) is a hypersemigroup. (2) (h;≤) is a partial order set. (3) for a,b,c ∈ h, if a ≤ b then a◦c � b◦c and c ◦a � c ◦b. for simplicity, we denoted an ordered hypersemigroup (h;◦,≤) by its carrier set as a bold letter h. definition 2.4. let h be an ordered hypersemigroup. a nonempty subset a of h is called a left (resp., right) hyperideal of h if (1) h ∗a ⊆ a (resp., a∗h ⊆ a). (2) for a ∈ h,b ∈ a, if a ≤ b, then a ∈ a. a nonempty subset a of h is called a two-sided hyperideal, or simply a hyperideal of h if it is both a left and a right hyperideal of h. let a be a nonempty subset of h. define (a] := {x ∈ h | x ≤ a for some a ∈ a}. note that condition (2) in definition 2.4 is equivalent to a =(a]. if a and b are nonempty subsets of h, then we obtain (1) a ⊆ (a]. (2) (a]∪ (b]⊆ (a∪b]. (3) ((a]∗ (b]] = (a∗b]. (4) (a]∗ (b]⊆ (a∗b]. 4 int. j. anal. appl. (2023), 21:47 definition 2.5. [7] let h be an ordered hypersemigroup. a hyperideal a of h is called a left (resp., right) pure hyperideal of h if x ∈ (a∗x] (resp., x ∈ (x ∗a]) for all x ∈ a. a hyperideal a of h is called a pure hyperideal of h if it is both a left and a right pure hyperideal of h. let i be the unit interval, h a set of parameters and p(u) denote the power set of an initial universe set u. hybrid structures are defined as follows. definition 2.6. a hybrid structure in h over u is defined to be a mapping f := (f ∗, f +) : h →p(u)× i,x 7→ (f ∗(x), f +(x)), where f ∗ : h →p(u) and f + : h → i are mappings. let us denote by hyb(h) the set of all hybrid structures in h over u. we define a binary relation � on hyb(h) as follows: for all f =(f ∗, f +),g =(g∗,g+)∈ hyb(h), f � g ⇔ f ∗ v g∗, f + � g+, where f ∗ v g∗ means that f ∗(x)⊆ g∗(x) and f + � g+ means that f +(x)≥ g+(x) for all x ∈ h and f = g if f � g and g � f . definition 2.7. let f =(f ∗, f +) and g =(g∗,g+) be hybrid structures in h over u. then the hybrid intersection of f and g is denoted by f eg and is defined to be a hybrid structure f eg : h →p(u)× i,x 7→ ((f ∗ ∩g∗)(x),(f + ∨g+)(x)), where (f ∗ ∩g∗)(x) := f ∗(x)∩g∗(x) and (f + ∨g+)(x) :=max{f +(x),g+(x)}. we denote h̃ := (h∗,h+) the hybrid structure in h over u and is defined as follows: h̃ : h −→p(u)× i : x 7→ ( h∗(x),h+(x) ) , where h∗(x) := u and h+(x) := 0. let a be an element of h. then we set ha := {(x,y)∈ h ×h | a ∈ (x ◦y]}. int. j. anal. appl. (2023), 21:47 5 definition 2.8. let f =(f ∗, f +) and g =(g∗,g+) be hybrid structures in h over u. then the hybrid products of f and g is denoted by f ⊗g and is defined to be a hybrid structure f ⊗g : h →p(u)× i,x 7→ ((f ∗ �g∗)(x),(f + ⊕g+)(x)), where (f ∗ �g∗)(x) :=   ⋃ (a,b)∈hx (f ∗(a)∩g∗(b)) if hx 6= ∅ ∅ otherwise, and (f + ⊕g+)(x) :=   ∧ (a,b)∈hx {max{f +(a),g+(b)}} if hx 6= ∅ 1 otherwise. let a be a nonempty subset of h. we denote by χa := (χ∗a,χ + a ) the characteristic hybrid structure of a in h over u which is defined to be a hybrid structure χa : h →p(u)× i,x 7→ (χ∗a(x),χ + a (x)), where χ∗a(x) := { u if x ∈ a ∅ otherwise, and χ+ a (x) := { 0 if x ∈ a 1 otherwise. we set χa := h̃ in the case that a = h. 3. main results in this main section, we introduce the concepts of hybrid pure hyperideals in ordered hypersemigroups and study some algebraic properties of such hybrid pure hyperideals. we also characterize weakly regular ordered hypersemigroups in terms of hybrid pure hyperideals. finally, we introduce the concept of hybrid weakly pure hyperideals and prove that any hybrid hyperideal is a hybrid weakly pure hyperideal whenever it is idempotent. definition 3.1. let h be an ordered hypersemigroup. a hybrid structure f =(f ∗, f +) in h over u is called a hybrid left (resp., right) hyperideal in h over u if for every x,y ∈ h: (1) ⋂ a∈x◦y f ∗(a)⊇ f ∗(y) (resp., ⋂ a∈x◦y f ∗(a)⊇ f ∗(x)); (2) ∨ a∈x◦y f +(a)≤ f +(y) (resp., ∨ a∈x◦y f +(a)≤ f +(x)); (3) x ≤ y implies f ∗(x)⊇ f ∗(y) and f +(x)≤ f +(y). a hybrid structure f is called a hybrid hyperideal in h over u if it is both a hybrid left and a hybrid right hyperideal in h over u. 6 int. j. anal. appl. (2023), 21:47 definition 3.2. let h be an ordered hypersemigroup. a hybrid hyperideal f in h over u is (1) left pure if f eg = f ⊗g for every hybrid left hyperideal g in h over u. (2) right pure g e f = g ⊗ f for every hybrid right hyperideal g in h over u. the following remark is a useful tool in calculating the purity of hybrid hyperideals. remark 3.1. let f = (f ∗, f +) be a hybrid left (resp., right) pure hyperideal in h over u. then for any hybrid left (resp., right) hyperideal g =(g∗,g+) in h over u, we have (1) f ∗ ∩g∗ = f ∗ �g∗ (resp., g∗ ∩ f ∗ = g∗ � f ∗); (2) f + ∨g+ = f + ⊕g+ (resp., g+ ∨ f + = g+ ⊕ f +). a hybrid hyperideal in h over u is called a hybrid pure hyperideal in h over u if it is both a hybrid right pure and a hybrid left pure hyperideal in h over u. the following lemmas are important in illustrating our first theorem. lemma 3.1. let h be an ordered hypersemigroup and a a nonempty subset of h. then the following conditions are equivalent: (1) a is a right (resp., left) hyperideal of h. (2) χa =(χ∗a,χ + a ) is a hybrid right (resp., left) hyperideal in h over u. proof. (1)⇒(2). let a be a right hyperideal of an ordered hypersemigroup h. first, let x,y ∈ h. if x ∈ a, then x ◦y ⊆ a and we obtain⋂ a∈x◦y χ∗a(a)= u ⊇ χ ∗ a(x) and ∨ a∈x◦y χ+ a (a)=0≤ χ+ a (x). if x /∈ a, we obtain ⋂ a∈x◦y χ∗a(a)⊇∅= χ ∗ a(x), and ∨ a∈x◦y χ+ a (a)≤ 1= χ+ a (x). secondly, let x,y ∈ h be such that x ≤ y. if y ∈ a, then x ∈ a and then χ∗a(x)= u ⊇ χ ∗ a(y), and χ + a (x)=0≤ χ∗a(y). if y /∈ a, we obtain χ∗a(x)⊇∅= χ ∗ a(y), and χ + a (x)≤ 1= χ+ a (y). altogether, it is complete to prove that χa is a hybrid right hyperideal in h over u. (2)⇒(1). let χa be a hybrid right hyperideal in h over u. firstly, let x ∈ a and y ∈ h. we obtain u ⊇ ⋂ a∈x◦y χ∗a(a) ⊇ χ ∗ a(x) = u, which implies that ⋂ a∈x◦y χ∗a(a) = u and since u = ⋂ a∈x◦y χ∗a(x) ⊆ χ∗a(a)⊆ u, we obtain χ ∗ a(a) = u. similarly, since 0≤ ∨ a∈x◦y χ+ a (a)≤ χ+ a (x) = 0, which implies that∨ a∈x◦y χ+ a (a)=0, and since 0= ∨ a∈x◦y χ+(a)≥ χ+ a (a)≥ 0, we obtain that χ+ a (a)=0. altogether, we have a ∈ a true. int. j. anal. appl. (2023), 21:47 7 secondly, let x,y ∈ h be such that x ≤ y and y ∈ a. we obtain u ⊇ χ∗a(x)⊇ χ ∗ a(y)= u, which implies that χ∗a(x) = u. it means that x ∈ a. for χ + a need not to consider. therefore a is a right hyperideal of h. similarly, we can show that a is a left hyperideal if and only if χa is a hybrid left hyperideal. � as a consequence of the above lemma, we have that a is a hyperideal of h if and only if χa is a hybrid hyperideal in h over u. the following lemma provides some good characterzations relating to two sets and their characteristic hybrid structures. lemma 3.2. let h be an ordered hypersemigroup and a, b nonempty subsets of h. then the following conditions hold: (1) a ⊆ b if and only if χa � χb; (2) χa eχb = χa∩b; (3) χa ⊗χb = χ(a∗b]. proof. we will give a proof of (3) only. let x ∈ (a∗b]. then there exists c ∈ a◦b for some a ∈ a and b ∈ b such that x ≤ c. then hx 6= ∅, and we obtain u ⊇ (χ∗a �χ ∗ b)(x) = ⋃ (y,z)∈hx [χ∗a(y)∩χ ∗ b(z)] ⊇ χ∗a(a)∩χ ∗ b(b) = u. this implies that (χ∗a �χ ∗ b)(x)= u = χ ∗ (a∗b](x) and 0 ≤ (χ+ a ⊕χ+ b )(x) = ∧ (y,z)∈hx {max{χ+ a (y),χ+ b (z)}} ≤ max{χ+ a (a),χ+ b (b)} = 0. this implies that (χ+ a ⊕χ+ b )(x)=0= χ+ (a∗b](x). therefore χa ⊗χb = χ(a∗b]. � in 2020, changphas [6] characterized right (resp., left) pure hyperideals as the following lemmas. lemma 3.3. let h be an ordered hypersemigroup and a a hyperideal of h. then the following conditions are equivalent: (1) a is a right pure hyperideal of h. (2) b ∩a =(b ∗a] for every right hyperideal b of h. 8 int. j. anal. appl. (2023), 21:47 lemma 3.4. let h be an ordered hypersemigroup and a a hyperideal of h. then the following conditions are equivalent: (1) a is a left pure hyperideal of h. (2) a∩b =(a∗b] for every left hyperideal b of h. the following theorem provides a characterization of right (resp., left) pure hyperideals in ordered hypersemigroups using hybrid right (resp., left) pure hyperideals. theorem 3.1. let a be a hyperideal of an ordered hypersemigroup h. then the following conditions are equivalent: (1) a is a right (resp., left) pure hyperideal of h. (2) χa =(χ∗a,χ + a ) is a hybrid right (resp., left) pure hyperideal in h over u. proof. (1)⇒(2). assume that a is a right pure hyperideal of h. by lemma 3.1, χa =(χ∗a,χ + a ) is a hybrid hyperideal in h over u. let f = (f ∗, f +) be a hybrid right hyperideal in h over u and a ∈ h. suppose that a /∈ a. we consider two cases as follows: if ha = ∅, then we have (f ∗ �χ∗a)(a) = ∅ = f ∗(a)∩χ∗a(a) = (f ∗ ∩χ∗a)(a), and (f + ⊕χ+ a )(a) = 1 = max{f +(a),χ+ a (a)} = (f + ∨χ+ a )(a). if ha 6= ∅, then, by a right purity of a, we have that v /∈ a for all (u,v)∈ha. then (f ∗ � (χ∗a)(a) = ⋃ (x,y)∈ha [f ∗(x)∩χ∗a(y)] = ∅ = f ∗(a)∩χ∗a(a) = (f ∗ ∩χ∗a)(a), and (f + ⊕χ+)a)(a) = ∧ (x,y)∈ha {max{f +(x),χ+ a (y)}} = 1 = max{f +(a),χ+ a (a)} = (f + ∨χ+ a )(a), int. j. anal. appl. (2023), 21:47 9 now, we assume that a ∈ a. by the right purity of a, we have ha 6= ∅. more precisely, there exists (a,x)∈ha such that x ∈ a. then, by the hybrid right ideality of f and the hybrid left ideality of χa, we have that (f ∗ ∩χ∗a)(a) = f ∗(a)∩χ∗a(a) = f ∗(a) = f ∗(a)∩χ∗a(x) ⊆ ⋃ (u,v)∈ha [f ∗(u)∩χ∗a(v)] ⊆ ⋃ (u,v)∈ha [f ∗(uv)∩χ∗a(uv)] ⊆ ⋃ (u,v)∈ha [f ∗(a)∩χ∗a(a)] = f ∗(a)∩χ∗a(a) = (f ∗ ∩χ∗a)(a). this implies that (f ∗ ∩χ∗a)(a)= ⋃ (u,v)∈ha [f ∗(u)∩χ∗a(v)]= (f ∗ �χ∗a)(a), and (f + ∨χ+ a )(a) = max{f +(a),χ+ a (a)} = f +(a) = max{f +(a),χ+ a (x)} ≥ ∧ (u,v)∈ha {max{f +(u),χ+ a (v)}} ≥ ∧ (u,v)∈ha {max{f +(uv),χ+ a (uv)}} ≥ ∧ (u,v)∈ha {max{f +(a),χ+ a (a)}} = max{f +(a),χ+ a (a)} = (f + ∨χ+ a )(a). this implies that (f + ∨ χ+ a )(a) = ∧ (u,v)∈ha {max{f +(u),χ+ a (v)}} = (f + ⊕ χ+ a )(a). altogether, we have that χa is a hybrid right pure hyperideal in h over u. (2)⇒(1). assume that χa is a hybrid right pure hyperideal in h over u. let b be a right hyperideal of h. by lemma 3.1, χb is a hybrid right hyperideal in h over u. by assumption, we obtain χb∩a = χb eχa = χb ⊗χa = χ(b∗a]. 10 int. j. anal. appl. (2023), 21:47 by lemma 3.2 (1), we have b ∩a = (b ∗a] and, by lemma 3.3, we obtain that a is a right pure hyperideal of h. similarly, we can prove that a is a left pure hyperideal of h if and only if χa is a hybrid left pure hyperideal in h over u. � by the above theorem, we obtain the following consequence. corollary 3.1. let a be a hyperideal of an ordered hypersemigroup h. then the following conditions are equivalent: (1) a is a pure hyperideal of h. (2) χa =(χ∗a,χ + a ) is a hybrid pure hyperideal in h over u. the following results illustrate some properties of hybrid right (resp., left) pure hyperideals in an ordered hypersemigroup. theorem 3.2. let f =(f ∗, f +) and g =(g∗,g+) be hybrid right pure hyperideals in h over u. then f eg is a hybrid right pure hyperideal in h over u. proof. let h be a hybrid right hyperideal in h over u and a ∈ h. we consider two cases as follows. if ha = ∅, we obtain (h∗ � (f ∗ ∩g∗))(a) = ∅ = (h∗ � f ∗)(a)∩ (h∗ �g∗)(a) = (h∗ ∩ f ∗)(a)∩ (h∗ ∩g∗)(a) = [h∗(a)∩ f ∗(a)]∩ [h∗(a)∩g∗(a)] = h∗(a)∩ [f ∗(a)∩g∗(a)] = h∗(a)∩ (f ∗ ∩g∗)(a) = [h∗ ∩ (f ∗ ∩g∗)](a), and (h+ ⊕ (f + ∨g+))(a) = 1 = max{(h+ ⊕ f +)(a),(h+ ⊕g+)(a)} = max{(h+ ∨ f +)(a),(h+ ∨g+)(a)} = max{max{h+(a), f +(a)},max{h+(a),g+(a)}} = max{h+(a),max{f +(a),g+(a)}} = max{h+(a),(f + ∨g+)(a)} = [h+ ∨ (f + ∨g+)](a). it is complete to prove that f eg is a hybrid right pure hyperideal in h over u. � int. j. anal. appl. (2023), 21:47 11 by a similar method to theorem 3.2, we have the following theorem. theorem 3.3. let f = (f ∗, f +) and g = (g∗,g+) be hybrid left pure hyperideals in h over u. then f eg is a hybrid left pure hyperideal in h over u. combining theorem 3.2 and 3.3, we obtain the following result. corollary 3.2. let f and g be hybrid pure hyperideals in h over u. then f e g is a hybrid pure hyperideal in h over u. let f =(f ∗, f +) and g =(g∗,g+) be hybrid structures in h over u. the hybrid union of f and g denoted by f dg and is defined to be a hybrid structure f dg := (f ∗ ∪g∗, f + ∧g+) : h →p∗(u)× i | x 7→ ((f ∗ ∪g∗)(x),(f + ∧g+)(x)), where (f ∗ ∪g∗)(x) := f ∗(x)∪g∗(x) and (f + ∧g+)(x) :=min{f +(x),g+(x)}. theorem 3.4. let f =(f ∗, f +) and g =(g∗,g+) be hybrid right pure hyperideals in h over u. then f dg is a hybrid right pure hyperideal in h over u. proof. let h = (h∗,h+) be a hybrid right hyperideal in h over u and a ∈ h. if ha = ∅, then we obtain that (h∗ ∩ (f ∗ ∪g∗))(a) = h∗(a)∩ (f ∗ ∪g∗)(a) = h∗(a)∩ (f ∗(a)∪g∗(a)) = (h∗(a)∩ f ∗(a))∪ (h∗(a)∩g∗(a)) = (h∗ ∩ f ∗)(a)∪ (h∗ ∩g∗)(a) = (h∗ � f ∗)(a)∪ (h∗ �g∗)(a) = ∅ = (h∗ � (f ∗ ∪g∗))(a), and (h+ ∨ (f + ∧g+))(a) = max{h+(a),(f + ∧g+)(a)} = max{h+(a),min{f +(a),g+(a)}} = min{max{h+(a), f +(a)},max{h+(a),g+(a)}} = min{(h+ ∨ f +)(a),(h+ ∨g+)(a)} = min{(h+ ⊕ f +)(a),(h+ ⊕g+)(a)} = 1 = (h+ ⊕ (f + ∧g+))(a). 12 int. j. anal. appl. (2023), 21:47 suppose that ha 6= ∅, and we obtain (h∗ ∩ (f ∗ ∪g∗))(a) = h∗(a)∩ (f ∗ ∪g∗)(a) = h∗(a)∩ (f ∗(a)∪g∗(a)) = (h∗(a)∩ f ∗(a))∪ (h∗(a)∩g∗(a)) = (h∗ ∩ f ∗)(a)∪ (h∗ ∩g∗)(a) = (h∗ � f ∗)(a)∪ (h∗ �g∗)(a) = ⋃ (x,y)∈ha [h∗(x)∩ f ∗(y)]∪ ⋃ (x,y)∈ha [h∗(x)∩g∗(y)] = ⋃ (x,y)∈ha [(h∗(x)∩ f ∗(y))]∪ [(h∗(x)∩g∗(y))] = ⋃ (x,y)∈ha [h∗(x)∩ (f ∗(y)∪g∗(y))] = ⋃ (x,y)∈ha [h∗(x)∩ (f ∗ ∪g∗)(y)] = (h∗ � (f ∗ ∪g∗))(a), and (h+ ∨ (f + ∧g+))(a) = max{h+(a),(f + ∧g+)(a)} = max{h+(a),min{f +(a),g+(a)}} = min{max{h+(a), f +(a)},max{h+(a),g+(a)}} = min{(h+ ∨ f +)(a),(h+ ∨g+)(a)} = min{(h+ ⊕ f +)(a),(h+ ⊕g+)(a)} = min{ ∧ (x,y)∈ha {max{h+(x), f +(y)}}, ∧ (x,y)∈ha {max{h+(x),g+(y)}}} = ∧ (x,y)∈ha {max{h+(x),min{f +(y),g+(y)}}} = ∧ (x,y)∈ha {max{h+(x),(f + ∧g+)(y)}} = (h+ ⊕ (f + ∧g+))(a). altogether, we obtain that f dg is a hybrid right pure hyperideal in h over u. � by a similar method to theorem 3.4, we have the following theorem. theorem 3.5. let f and g be hybrid left pure hyperideals in an ordered hypersemigroup h over u. then f dg is a hybrid left pure hyperideal in h over u. combining theorem 3.4 and 3.5, we obtain the following result. int. j. anal. appl. (2023), 21:47 13 corollary 3.3. let f and g be hybrid pure hyperideals in h over u. then a d b is a hybrid pure hyperideal in h over u. an ordered hypersemigroup h is said to be right (resp., left) weakly regular [7] if for any a ∈ h there exist b,x,y ∈ h such that a ≤ b and b ∈ (a◦x)∗(a◦y) (resp., a ≤ b and b ∈ (x ◦a)∗(y ◦a)). an ordered hypersemigroup h is called weakly regular if it is both a right and a left weakly regular ordered hypersemigroup. lemma 3.5. [7] let h be an ordered hypersemigroup. then the following statements are equivalent: (1) h is right (resp., left) weakly regular. (2) every hyperideal of h is a right (resp., left) pure hyperideal of h. lemma 3.6. [7] let h be an ordered hypersemigroup. then the following statements are equivalent: (1) h is weakly regular. (2) every hyperideal of h is a pure hyperideal of h. we characterize right weakly regular ordered hypersemigroups in terms of hybrid right pure hyperideals as follows: theorem 3.6. let h be an ordered hypersemigroup. then the following statements are equivalent: (1) h is right weakly regular. (2) every hybrid hyperideal in h over u is right pure. proof. (1)⇒(2). let f =(f ∗, f +) be a hybrid hyperideal in h over u and g =(g∗,g+) a hybrid right hyperideal in h over u. let a ∈ h. since h is right weakly regular, there exist b,x,y ∈ h such that a ≤ b and b ∈ (a◦x)∗ (a◦y). this implies that ha 6= ∅ and then (g∗ � f ∗)(a) = ⋃ (u,v)∈ha [g∗(u)∩ f ∗(v)] ⊆ ⋃ (u,v)∈ha [ ⋂ c∈u◦v g∗(c)∩ ⋂ c∈u◦v f ∗(c)] ⊆ ⋃ (u,v)∈ha [g∗(c)∩ f ∗(c)] ⊆ ⋃ (u,v)∈ha [g∗(a)∩ f ∗(a)] = g∗(a)∩ f ∗(a) = (g∗ ∩ f ∗)(a). 14 int. j. anal. appl. (2023), 21:47 on the other hand, we obtain (g∗ � f ∗)(a) = ⋃ (u,v)∈ha [g∗(u)∩ f ∗(v)] ⊇ ⋂ c∈a◦x g∗(c)∩ ⋂ d∈a◦y f ∗(d) ⊇ g∗(a)∩ f ∗(a) = (g∗ ∩ f ∗)(a). therefore (g∗ � f ∗)(a)= (g∗ ∩ f ∗)(a), and consider (g+ ⊕ f +)(a) = ∧ (u,v)∈ha {max{g+(u), f +(v)}} ≥ ∧ (u,v)∈ha {max{ ∨ c∈u◦v g+(c), ∨ d∈u◦v f +(d)}} ≥ ∧ (u,v)∈ha {max{g+(a), f +(a)}} = max{g+(a), f +(a)} = (g+ ∨ f +)(a). on the other hand, we obtain (g+ ⊕ f +)(a) = ∧ (u,v)∈ha {max{g+(u), f +(v)}} ≤ max{ ∨ c∈a◦x g+(c), ∨ d∈a◦y f +(d)} ≤ max{g+(a), f +(a)} = (g+ ∨ f +)(a). therefore (g+⊕ f +)(a)= (g+∨ f +)(a) and we obtain that g ⊗ f = g e f . hence f is a hybrid right pure hyperideal in h over u. (2)⇒(1). let a and b be a hyperideal of h and a right hyperideal of h, respectively. by lemma 3.1, we obtain χa and χb is a hybrid hyperideal in h over u and a hybrid right hyperideal in h over u, respectively. by our assumption, χb is a hybrid right pure hyperideal in h over u. then χ(b∗a] = χb ⊗χa = χb eχa = χb∩a. by lemma 3.2 (1), we obtain b ∩a = (b ∗a]. this means that a is a right pure hyperideal of h. therefore, by lemma 3.5, h is right weakly regular. � int. j. anal. appl. (2023), 21:47 15 by a similar method to theorem 3.6, we have the following theorem. theorem 3.7. let h be an ordered hypersemigroup. then the following statements are equivalent: (1) h is left weakly regular. (2) every hybrid hyperideal in h over u is left pure. combining theorem 3.6 and 3.7, we obtain the following result. corollary 3.4. let h be an ordered hypersemigroup. then the following statements are equivalent: (1) h is weakly regular. (2) every hybrid hyperideal in h over u is pure. now, we present the concepts of right weakly purity and left weakly purity of hybrid hyperideals. in our last main result, the coincidence of these two concepts is provided. definition 3.3. a hybrid hyperideal f in h over u is called a hybrid right (resp., left) weakly pure hyperideal if g ⊗ f = g e f (resp., f ⊗g = f eg) for every hybrid hyperideal g in h over u. a hybrid hyperideal is called a hybrid weakly pure hyperideal in h over u if it is both a hybrid right and a hybrid left weakly pure hyperideal in h over u. a hybrid structure f in h over u is idempotent with respect to ⊗ if f ⊗ f = f . lemma 3.7. let h be an ordered hypersemigroup and f ,g are hybrid right hyperideals in h over u. then f eg is a hybrid right hyperideal in h over u. proof. let f =(f ∗, f +) and g =(g∗,g+) be hybrid right hyperideals in h over u and x,y ∈ h. then we obtain ⋂ a∈x◦y (f ∗ ∩g∗)(a) = ⋂ a∈x◦y f ∗(a)∩ ⋂ a∈x◦y g∗(a) ⊇ f ∗(x)∩g∗(x) = (f ∗ ∩g∗)(x), and ∨ a∈x◦y (f + ∨g+)(a) = max{ ∨ a∈x◦y f +(a), ∨ a∈x◦y g+(a)} ≤ max{f +(x),g+(x)} = (f + ∨g+)(x). therefore f eg is a hybrid right hyperideal in h over u. � by a similar method to lemma 3.7, we have the following lemma. 16 int. j. anal. appl. (2023), 21:47 lemma 3.8. let h be an ordered hypersemigroup and f ,g are hybrid left hyperideals in h over u. then f eg is a hybrid left hyperideal in h over u. combining lemma 3.7 and 3.8, we obtain the following result. corollary 3.5. let h be an ordered hypersemigroup and f ,g are hybrid hyperideals in h over u. then f eg is a hybrid hyperideal in h over u. our last result illustrates that the concepts of right weakly purity and left weakly purity of hybrid hyperideals coincide. theorem 3.8. let h be an ordered hypersemigroup and f = (f ∗, f +) a hybrid hyperideal in h over u. then the following statements are equivalent. (1) f is hybrid right weakly pure hyperideal. (2) f is idempotent with respect to ⊗. (3) f is hybrid left weakly pure hyperideal. proof. (1)⇒(2). let f be a hybrid right weakly pure hyperideal in h over u. then we obtain f ⊗ f = f e f = f . therefore, f is idempotent with respect to ⊗. (2)⇒(1). let g = (g∗,g+) be a hybrid hyperideal in h over u. by corollary 3.5, we obtain that g e f is a hybrid hyperideal in h over u. by our assumption, we have g e f = g e (f ⊗ f )= (g e f )⊗ (g e f )� g ⊗ f . on the other hand, let a ∈ h. if ha = ∅, then (g∗ � f ∗)(a)= ∅⊆ (g∗ ∩ f ∗)(a), and (g+ ⊕ f +)(a)=1≥ (g+ ∨ f +)(a). suppose that ha 6= ∅. then (g∗ � f ∗)(a) = ⋃ (u,v)∈ha [g∗(u)∩ f ∗(v)] ⊆ ⋃ (u,v)∈ha [ ⋂ c∈u◦v g∗(c)∩ ⋂ c∈u◦v f ∗(c)] ⊆ ⋃ (u,v)∈ha [g∗(a)∩ f ∗(a)] = g∗(a)∩ f ∗(a) = (g∗ ∩ f ∗)(a), int. j. anal. appl. (2023), 21:47 17 and (g+ ⊕ f +)(a) = ∧ (u,v)∈ha {max{g+(u), f +(v)}} ≥ ∧ (u,v)∈ha {max{ ∨ c∈u◦v g+(c), ∨ c∈u◦v f +(c)}} ≥ ∧ (u,v)∈ha {max{g+(a), f +(a)}} = max{g+(a), f +(a)} = (g+ ∨ f +)(a). thus g⊗f � gef and altogether, we have g⊗f = gef . this means that f is a hybrid right weakly pure hyperideal in h over u. illustration of (2)⇔(3) can be done similarly. � by theorem 3.8, we obtain the following result. corollary 3.6. let h be an ordered hypersemigroup and f a hybrid hyperideal in h over u. then the following statements are equivalent. (1) f is idempotent with respect to ⊗. (2) f is hybrid weakly pure hyperideal. 4. conclusion we introduced the concept of hybrid pure hyperideals in ordered hypersemigroups. some related properties of hybrid pure hyperideals are studied. we characterized weakly regular ordered hypersemigroups in terms of hybrid pure hyperideals. finally, we introduced the concept of hybrid weakly pure hyperideals. we proved that the hybrid hyperideals are hybrid weakly pure hyperideals if such hybrid hyperideals satisfied the idempotent property. in our future work, we will apply the notions of hybrid hyperideals and hybrid pure hyperideals to the theory of hypersemirings, hypergroups, etc. acknowledgement: this research was supported by the faculty of engineering, rajamangala university of technology isan, khon kaen campus: contract number eng18/66. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. ahsan, m. takahashi, pure spectrum of a monoid with zero, kobe j. math. 6 (1989), 163–181. https: //cir.nii.ac.jp/crid/1571698601700142976. [2] s. anis, m. khan, y.b. jun, hybrid ideals in semigroups, cogent math. 4 (2017), 1352117. https://doi.org/ 10.1080/23311835.2017.1352117. https://cir.nii.ac.jp/crid/1571698601700142976 https://cir.nii.ac.jp/crid/1571698601700142976 https://doi.org/10.1080/23311835.2017.1352117 https://doi.org/10.1080/23311835.2017.1352117 18 int. j. anal. appl. (2023), 21:47 [3] t. changphas, b. davvaz, hyperideal theory in ordered semihypergroups, in: international congress on algebraic hyperstructures and its applications, xanthi, (2014), 51-54. [4] t. changphas, b. davvaz, properties of hyperideals in ordered semihypergroups, italian j. pure appl. math. 33 (2014), 425-432. [5] t. changphas, b. davvaz, bi-hyperideals and quasi-hyperideals in ordered semigroups, italian j. pure appl. math. 35 (2015), 493-508. [6] t. changphas, b. davvaz, on pure hyperideals in ordered semihypergroups, bol. math. 27 (2020), 63-74. [7] t. changphas, j. sanborisoot, pure ideals in ordered semigroups, kyungpook math. j. 54 (2014), 123–129. https://doi.org/10.5666/kmj.2014.54.1.123. [8] p. corsini, sur les semi-hypergroups, atti soc. fis. math. nat. 26 (1980), 363-372. [9] p. corsini, prolegonmena of hypergroup theory, aviani editore, tricesimo, 1993. https://cir.nii.ac.jp/crid/ 1570572699837077760. [10] p. corsini, v. leoreanu-fotea, applications of hyperstructure theory, advances in mathematics, kluwer academic publishers, dordrecht, 2003. [11] b. davvaz, semihypergroup theory, academic press, amsterdam, 2016. [12] w.a. dudek, y.b. jun, int-soft interior ideals of semigroups, quasigroups related syst. 22 (2014), 201-208. [13] b. elavarasan, y.b. jun, regularity of semigroups in terms of hybrid ideals and hybrid bi-ideals, kragujevac j. math. 46 (2022), 857-864. [14] b. elavarasan, k. porselvi, y.b. jun, hybrid generalized bi-ideals in semigroups, int. j. math. computer sci. 14 (2019), 601–612. [15] z. gu, x. tang, characterizations of (strongly) ordered regular relations on ordered semihypergroups, j. algebra. 465 (2016), 100–110. https://doi.org/10.1016/j.jalgebra.2016.07.010. [16] d. heidari, b. davvaz, on ordered hyperstructures, u.p.b. sci. bull. ser. a, 73 (2011), 85-96. [17] k. hila, b. davvaz, k. naka, on quasi-hyperideals in semihypergroups, commun. algebra. 39 (2011), 4183–4194. https://doi.org/10.1080/00927872.2010.521932. [18] y.b. jun, s.z. song, g. muhiuddin, hybrid structures and applications, ann. commun. math. 1 (2018), 11-25. [19] n. kehayopulu, from ordered semigroups to ordered hypersemigroups, turk. j. math. 43 (2019), 21–35. https: //doi.org/10.3906/mat-1806-104. [20] f. marty, sur une generalization de la notion de groupe, proceedings of the 8th congress math. scandinaves, stockholm, (1934), 45-49. [21] d. molodtsov, soft set theory-first results, computers math. appl. 37 (1999), 19–31. https://doi.org/10. 1016/s0898-1221(99)00056-5. [22] g. muhiuddin, a. mahboob, int-soft ideals over the soft sets in ordered semigroups, aims math. 5 (2020), 2412–2423. https://doi.org/10.3934/math.2020159. [23] s.z. song, h.s. kim, y.b. jun, ideal theory in semigroups based on intersectional soft sets, sci. world j. 2014 (2014), 136424. https://doi.org/10.1155/2014/136424. [24] y. xie, j. liu, l. wang, further studies on h∞ filtering design for fuzzy system with known or unknown premise variables, ieee access. 7 (2019), 121975–121981. https://doi.org/10.1109/access.2019.2938797. [25] l.a. zadeh, fuzzy sets, inform. control, 8 (1965), 338–353. https://doi.org/10.5666/kmj.2014.54.1.123 https://cir.nii.ac.jp/crid/1570572699837077760 https://cir.nii.ac.jp/crid/1570572699837077760 https://doi.org/10.1016/j.jalgebra.2016.07.010 https://doi.org/10.1080/00927872.2010.521932 https://doi.org/10.3906/mat-1806-104 https://doi.org/10.3906/mat-1806-104 https://doi.org/10.1016/s0898-1221(99)00056-5 https://doi.org/10.1016/s0898-1221(99)00056-5 https://doi.org/10.3934/math.2020159 https://doi.org/10.1155/2014/136424 https://doi.org/10.1109/access.2019.2938797 1. introduction 2. preliminaries 3. main results 4. conclusion references int. j. anal. appl. (2022), 20:57 received: sep. 28, 2022. 2010 mathematics subject classification. 34d20, 92d30, 93d05, 97m60. key words and phrases. basic reproduction number (r0); dengue fever; lyapunov; numerical simulation; sensitivity; stability. https://doi.org/10.28924/2291-8639-20-2022-57 © 2022 the author(s) issn: 2291-8639 1 analysis of vector-host seir-sei dengue epidemiological model md rifat hasan1,2,*, aatef hobiny1, ahmed alshehri1 1department of mathematics, faculty of science, king abdulaziz university, jeddah 21589, saudi arabia 2department of applied mathematics, faculty of science, noakhali science and technology university, noakhali 3814, bangladesh *corresponding author: rifatmathdu@gmail.com abstract. approximately worldwide 50 nations are still infected with the deadly dengue virus. this mosquito-borne illness spreads rapidly. epidemiological models can provide fundamental recommendations for public health professionals, allowing them to analyze variables impacting disease prevention and control efforts. in this paper, we present a host-vector mathematical model that depicts the dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (seir) model for the human interacting with a susceptible-exposed-infected (sei) model for the mosquito. using the next generation technique, the basic reproduction number of the model is calculated. the local stability shows that if r0<1 the system is asymptotically stable and the disease dies out, otherwise unstable. the lyapunov function is also used to evaluate the global stability of disease-free and endemic equilibrium points. to analyze the effect of the crucial aspects of the disease's transmission and to validate the analytical findings, numerical simulations of a variety of compartments have been constructed using matlab. the sensitivity analysis of the epidemic model is performed to establish the relative significance of the model parameters to disease transmission. 1. introduction dengue fever incidence has risen dramatically in the recent two decades [1]. approximately half of the earth’s population may be in danger of contracting one of the nearly 390 million new infections that are thought to arise annually. dengue viral disease is transmitted from mosquitoes https://doi.org/10.28924/2291-8639-20-2022-57 2 int. j. anal. appl. (2022), 20:57 to humans by mosquitoes, spreading rapidly expanding over the world by its four serotypes. prior to relatively recently, the female aedes aegypti mosquito which identifies as a primary vector for dengue was mostly found in tropical and subtropical regions [2]. due to the highly adapted ability in urban regions of the aedes aegypti mosquito, dengue spread all over the world. however, the full extent of the disease's effects remains unclear, and new monitoring strategies are needed, according to issues with underreporting and case misidentification. other arboviruses, chikungunya as well as zika have lately emerged, posing additional issues for surveillance and management, particularly in south asia [3]. a wide range of factors (including people, mosquitoes, and the virus) interact with one another to spread the dengue virus in a diverse environment [4]. studies on dengue transmission face several obstacles due to the space's inherent complexity. a variety of causes are related to the current epidemic. these included worldwide host and vector mobility (which accelerated viral circulation), urban congestion (which encouraged various transmissions through a single infected vector), and the loss of previously effective vector control measures. temperature, precipitation, and humidity all affect vector development at all stages, from egg viability through adult longevity and dispersion, among other aspects of dengue transmission. unplanned development, high inhabitants’ density, and the instability of rubbish collection all of which support the growth of mosquito breeding sites, which lead to increasing dengue occurrence. in recent years, epidemiology research and disease control have benefited greatly from the use of mathematical modeling. to understand the disease's nature as well as taking appropriate decisions regarding disease management strategies/interventions and processes, mathematical modeling has become a useful tool. many scholars studied a deterministic model to study the influence of numerous biological parameters on disease dynamics. prasad et al. [5] studied a systematic review of deterministic mathematical models for vector-borne viral infections. bhuju et al. [6] described the fuzzy epidemic seir-sei compartmental model with bed nets and fumigation intervention to simulate the transmission dynamics of dengue disease. tay et al. [7] constructed a transmission model of si-sir dengue epidemiological characteristics model to control dengue in malaysia. abidemi et al. [8] analyzed the effect of single vaccine usage and its combination with treatment and adulticide measures on dengue population dynamics in johor, malaysia. aleixo et al. [9] gave a clear explanation of a machine learning model that is used to predict the frequency of dengue outbreaks in rio de janeiro. sow et al. [10] developed a computational zika dynamics model to examine the effects of vertical transmission between the 3 int. j. anal. appl. (2022), 20:57 vector population and the host population. sweilam et al. [11] introduced a unique variable-order nonlinear model of the dengue virus that minimizes intervention dosage and duration through optimum bang-bang management. abidemi et al. [12] developed and analyzed a two-strain deterministic dengue model based on the sir modeling framework for the spread of the disease and its management in an area with two coexisting dengue virus serotypes. asamoah et al. [13] investigated an ideal dengue infection control model with partly immune and asymptomatic patients. linda et al. [14] examined the discrete-time versions of the sis and sir models that are stochastic in nature. to assess the influence of raising awareness through the press on the spread of vector-borne illnesses, a non-linear mathematical model was suggested by misra et al. [15]. the dynamic sir model with climatic parameters was discussed by nur et al. [16] for the features of dengue disease transmission in a closed community. 2. dengue transmission model the ross-macdonald model, which was first designed for malaria, is a classic mathematical model for vector-borne illnesses that monitors infections in humans as well as mosquitos. in this research, we present a compartmental host-vector mathematical model [17] that depicts the dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (seir) model for the human interacting with a susceptible-exposed-infected (sei) model for the mosquito. the host-vector mathematical model categorizes the overall human (host) population into four classes: susceptible (𝑆ℎ), exposed (𝐸ℎ), infectious (𝐼ℎ), and recovered (𝑅ℎ), whereas the mosquito (vector) population is divided into three classes: susceptible (𝑆𝑚), exposed (𝐸𝑚), and infectious (𝐼𝑚). thus, the total human(host) population denoted by 𝑁ℎ is given as nh(t)=sh(t)+eh(t)+ih(t)+rh(t) and total mosquitoes’(vector) population is given by: nm(t)=sm(t)+em(t)+im(t) 4 int. j. anal. appl. (2022), 20:57 fig 1. dengue virus transmission dynamics in different population stages in our suggested model, we attempt to provide a fresh direction by taking panic, tension, or anxiety into account in the susceptible, exposed, and infected classes to host population. the influence of panic as well as stress, or anxiety on these clusters is discussed in this work. panic, stress, and anxiety are all harmful to human’s health. anxiety may raise insulin levels, which can have an impact on heart health, diabetes, and blood pressure. at the same time, stress can have a negative impact on human immune system. extreme stress can impair immunity as well as chronic stress might jeopardize a major health condition. people suffering from panic attacks are more likely to get infected, and the death rate among infected people rises. therefore, we anticipate that the amount of susceptible, exposed, and infected, is decreasing, i.e., moving to death due to panic, stress, or anxiety. figure 1 depicts the suggested model's flow diagram as well as the nonlinear system of differential equations that represents the dynamics of host-vector dengue disease, which is represented by: human population (h) dsh dt =λ1-β1shim-β2shih-μ1sh-α1sh deh dt =β2shih-μ1eh-α1eh dih dt =β1shim-β3ih-μ1ih-α1ih drh dt =β3ih-μ1rh 5 int. j. anal. appl. (2022), 20:57 vector population (𝑚) dsm dt =λ2-β4smih-μ2sm dem dt =β4smih-β5em-μ2em dim dt =β5em-μ2im (1) with the initial conditions sh(0)≥0,eh(0)≥0,ih(0)≥0,rh(0)≥0,sm(0)≥0,em(0)≥0, and im(0)≥0, where the biological descriptions of parameters is presented in table 1. table 1. values for baseline parameters with definitions and biological descriptions of dengue model parameter biological descriptions 𝜦𝟏 recruitment rates of human population 𝜷𝟏 infectious rate from vector to host 𝜷𝟐 infectious rate within host 𝝁𝟏 humna’s natural death rate 𝜶𝟏 panic/tension/anxiety rate of human 𝜷𝟑 recovery rate of infected human 𝚲𝟐 recruitment rates of vector population 𝜷𝟒 infection rate from human to vector 𝜷𝟓 extrinsic incubation of vector 𝝁𝟐 natural death rate of vector population 3. positivity and boundedness of solutions the positivity and boundedness of the solutions are crucial features of an epidemiological model. as a result, it is critical to demonstrate that all variables are non-negative for all time 𝑡 ≥ 0, implying that any solution with positive beginning values will remain positive for all time 𝑡 ≥ 0. so, positivity indicates that the population will survive for a long period. the dynamical model of the transmission shall be investigated into the biologically feasible regions θ ⊂ ℝ7 +, such that θ={sh(t),eh(t),ih(t),rh(t),sm(t),em(t), im(t)ϵr7 +:nh(t)≤ λ1 μ1 ,nm(t)≤ λ2 μ2 } 6 int. j. anal. appl. (2022), 20:57 theorem 3.1. the feasible region is positively invariant for the model (1) with the initial condition defined by θ⊂r7 + . proof. let nh(t)=sh(t)+eh(t)+ih(t)+rh(t), then dnh(t) dt = dsh(t) dt + deh(t) dt + dih(t) dt + drh(t) dt hence dnh(t) dt =λ1-μ1nh(t)-α1sh-α1eh-α1ih dnh(t) dt ≤λ1-μ1nh(t) dnh(t) dt +μ1nh(t)≤λ1 nh(t)≤ λ1 μ1 nh(0)e -μt thus, 𝑁ℎ(𝑡) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in θ with starting conditions. again, nm(t)=sm(t)+em(t)+im(t), then dnm(t) dt = dsm(t) dt + dem(t) dt + dim(t) dt hence dnm(t) dt =λ2-μ2nm(t) dnm(t) dt ≤λ2-μ2nm(t) dnm(t) dt +μ2nm(t)≤λ2 nm(t)≤ λ2 μ2 nm(0)e -μt thus, nm(t) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in θ with starting conditions. 7 int. j. anal. appl. (2022), 20:57 therefore, the feasible region θ is positively invariant, attracting all solutions in ℝ7 +. theorem 3.2. the solution of the system (1) is positive and bounded for all sh(t),eh(t),ih(t), rh(t),sm(t),em(t), im(t)ϵr7 + . for all 𝑡 > 0 . proof. to demonstrate the solution's positivity, we need to show that on any hyperplane enclosing the positive vector space ℝ7 + from the system (1), we have dsh dt | sh=0 =λ1≥0 deh dt | eh=0 =β2shih≥0 dih dt | ih=0 =β1shim≥0 drh dt | rh=0 =β3ih≥0 dsm dt | m=0 =λ2≥0 dem dt | em=0 =β4smih≥0 dim dt | m=0 =β5em≥0 so, the system (1) solution is positive. 4. qualitative analysis of model in this section qualitative analysis of the dengue system (1) by calculating disease free equilibrium (dfe) and the endemic equilibrium (ee) with help of basic reproduction number (e0). 4.1. disease-free equilibrium to calculate the disease-free equilibrium (dfe) e0 of the dengue system (1), we set the right-hand side of equals to zero and obtain the following expression’s e0=(sh 0 ,eh 0,ih 0,rh 0,sm 0 ,em 0 ,im 0 )=( λ1 μ1+α1 ,0,0,0, λ2 μ2 ,0,0) 8 int. j. anal. appl. (2022), 20:57 4.2. basic reproduction number for the purpose of assessing an infectious disease, a crucial threshold parameter is the basic reproduction number r0. it decides whether the disease will disappear or stay in the community throughout time. r0 is the secondary infections number which may be caused by a single primary infection whereas the population is susceptible. assume r0 > 1, and one primary infection can generate in several secondary infections. therefore, the disease-free equilibrium (dfe) is unstable, also an epidemic occurs. the reproduction number for the dengue system is calculated utilizing the next generation matrix approach [18]. we look at the f* and v* matrices, are designated for the new infections’ development and classified migration of infective partitions. f*=( β2shih β1shim β4smih 0 ) v*= ( (μ1+α1)eh (β 3 +μ1+α1)ih (β5+μ2)em -β5em+μ2im ) the jacobian are calculated by taking the partial derivatives of f and v at dfe point e0 and are as follows: f=( 0 β2sh 0 0 0 0 0 β1sh 0 β4sm 0 0 0 0 0 0 ) v= ( μ1+α1 0 0 0 0 β3+μ1+α1 0 0 0 0 β5+μ2 0 0 0 -β5 μ2) the basic reproduction of the dengue system is calculated by using the spectral radius of the matrix 𝑅0 = 𝜌(𝐹𝑉 −1), which is provided by the following equation r 0 2 = β1β4β5λ1λ2 μ2 2(μ1+α1)(β5+μ2)(β3+μ1+α1) 4.3. endemic equilibrium the endemic equilibrium point of the dengue dynamical system (1) 9 int. j. anal. appl. (2022), 20:57 e1=(sh * ,eh *,ih *,rh *,sm * ,em * ,im * ) where, sh * = λ1 β1im+β2ih+μ1+α1 , eh *= β2shih β5+μ2 , ih *= β1shim β3+μ1+α1 rh *= β3 μ1 ,sm * = λ2 β4ih+μ2 , em * = β4smih β5+μ2 , im * = β5em μ2 5. stability analysis the stability study of disease-free and endemic equilibrium is performed in this part. the basic reproduction number (r0) is used to observe the equilibrium point’s stability of the local as well as global. the jacobian matrix which gives the eigenvalues can be used to do stability analysis. 5.1. local stability around equilibrium point theorem 5.1. for 𝑅0 < 1 the disease-free equilibrium (e0) of the system (1) is locally asymptotically stable and unstable if 𝑅0 > 1. proof. the jacobian matrix of the model (1) at the disease-free equilibrium point (e0) is j(e0)= ( -(μ1+α1) 0 -β2sh 0 0 0 0 -β1sh 0 0 -(μ1+α1) β2sh 0 0 0 0 0 0 0 -(β 3 +μ1+α1) 0 0 0 β1sh 0 0 0 β3 -μ1 0 0 0 0 0 -β4sm 0 0 -μ2 0 0 0 0 β4sm 0 0 0 -(β 5 +μ2) 0 0 0 0 0 0 β5 -μ2 ) the four eigenvalues of 𝐽(𝐸0) at the diseasefree equilibrium are -μ1, -μ2, -(μ1+α1)(multiplicity 2) and the remaining eigenvalues are given by the following cubic equation λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2-β1β4β5sh 0 s m 0 =0 where, v1=β3+μ1+α1 10 int. j. anal. appl. (2022), 20:57 v2=β5+μ2 now, λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2 (1β1β4β5sh 0 s m 0 v1v2μ2 )=0 λ3+(v1+v2+μ2)λ 2+(v 1 v 2 +v1μ2+v2μ2)λ+v1v2μ2(1-r0 2 )=0 here, (v1+v2+μ2)>0 (v 1 v 2 +v1μ2+v2μ2)>0 v1v2μ2(1-r0 2 )>0, if r0<1 and (v1+v2+μ2)(v1v2 +v1μ2+v2μ2)>v1v2μ2(1-r0 2 ) (since (v1+v2+μ2)(v1v2 +v1μ2+v2μ2) v1v2μ2 >9>(1-r0 2 )) therefore, if 𝑅0 < 1, all of the preceding requirements are satisfied. as a result, the disease-free equilibrium point 𝐸0 is locally asymptotically stable according to the routh-hurwitz criteria; otherwise, it is unstable. 5.2. global stability around equilibrium point in this segment, we will evaluate equilibrium points e0 and e1 stability. the next two theorems show the results of the stability analysis of these equilibrium sites. theorem 5.2. if 𝑅0 < 1, the disease-free equilibrium (e0) is globally asymptotically stable. proof. we consider the lyapunov function of the form in g(t)=(sh-sh 0 lnsh)+eh+ih+rh+(sm-sm 0 ln sm)+em+im differentiating w.r.t t, we get: g'(t)=(1sh 0 sh ) sh ' +eh ' +ih ' +rh ' +(1sm 0 sm )sm ' +em ' +im ' 11 int. j. anal. appl. (2022), 20:57 g '(t)= (1sh 0 sh )(λ1-β1shim-β2shih-μ1sh-α1sh)+β2shih-μ1eh-α1eh +β1shim-β3ih-μ1ih-α1ih+β3ih-μ1rh+(1sm 0 sm )(λ2-β4smih-μ2sm) +β4smih-β5em-μ2em+β5em-μ2im on solving further get: =(1sh 0 sh )λ1+(μ1+α1)(1sh sh 0 ) sh 0 +β1sh 0 im+β2sh 0 ih-μ1eh-α1eh -μ1ih-α1ih-μ1rh+(1sm 0 sm )λ2+μ2 (1sm sm 0 )sm 0 +β4sm 0 ih-μ2em-μ2im using the equilibrium condition (μ1+α1)sh 0 =λ1 and μ2sm 0 =λ2 into the above equation g '(t)=(2sh 0 sh sh sh 0 )λ1+(2sm 0 sm sm sm 0 )λ2-im(μ2-β1sh 0 )-ih(μ1+α1-β2sh 0 -β4sm 0 ) -μ1eh-α1eh-μ1rh-μ2em =-λ1 (sh-sh 0 ) 2 shsh 0 -λ2 (sm-sm 0 ) 2 smsm 0 -μ1eh-α1eh-ih(μ1+α1-β2sh 0 -β4sm 0 ) -μ1rh-μ2em-im(μ2-β1sh 0 ) the above equation shows that 𝐺′(𝑡) ≤ 0 𝑎𝑛𝑑 g '(t)=0 for sh=sh 0 , eh=0, ih=0, rh=0,sm=sm 0 , em=0, im=0. so, the largest invariance set is the singleton set {𝐸0}. therefore, by using the principle of lasalle’s invariance the disease-free equilibrium (𝐸0) is globally asymptotically stable. theorem 5.3. if 𝑅0 > 1, the endemic equilibrium (e1) is globally asymptotically stable. proof. we consider the lyapunov function of the form in w(t)= 1 2 (sh-sh * ) 2 + 1 2 (eh-eh * ) 2 + 1 2 (ih-ih * ) 2 + 1 2 (rh-rh * ) 2 + 1 2 (sm-sm * ) 2 + 1 2 (em-em * ) 2 + 1 2 (im-im * ) 2 differentiating with respect to time t, we get: w'(t)=(sh-sh * )sh ' +(eh-eh * )eh ' +(ih-ih * )ih ' +(rh-rh * )rh ' +(sm-sm * )sm ' +(em-em * )em ' +(im-im * )im ' 12 int. j. anal. appl. (2022), 20:57 =(sh-sh * )(λ1-β1shim-β2shih-μ1sh-α1sh)+(eh-eh * )(β2shih-μ1eh-α1eh) +(ih-ih * )(β1shim-β3ih-μ1ih-α1ih)+(rh-rh * )(β3ih-μ1rh)+(sm-sm * )(λ2-β4smih-μ2sm) +(em-em * )(β4smih-β5em-μ2em)+(im-im * )(β5em-μ2im) using the equilibrium conditions λ1=μ1sh * +μ1eh *+μ1ih *+μ1rh *+α1sh * +α1eh *+α1ih * and λ2=μ2sm * +μ2em * +μ2im * into the above equation w'(t)=(sh-sh * )(μ1sh * +μ1eh *+μ1ih *+μ1rh *+α1sh * +α1eh *+α1ih * -β1shim-β2shih-μ1sh-α1sh) +(eh-eh * )(β2shih-μ1eh-α1eh)+(ih-ih * )(β1shim-β3ih-μ1ih-α1ih)+(rh-rh * )(β3ih-μ1rh) +(sm-sm * )(μ2sm * +μ2em * +μ2im * -β4smih-μ2sm)+(em-em * )(β4smih-β5em-μ2em) +(im-im * )(β5em-μ2im) =-μ1 (sh-sh * ) 2 -α1 (sh-sh * ) 2 +(μ1+α1)eh * (sh-sh * )+(μ1+α1)ih * (sh-sh * ) -β1shim (sh-sh * )-β2shih (sh-sh * )+β2shih(eh-eh * )-(μ1+α1)eh * (eh-eh * ) +β1shim(ih-ih * )-(β3+μ1+α1)ih(ih-ih * )+β3ih(rh-rh * )-μ1rh(rh-rh * ) -μ2 (sm-sm * ) 2 +μ2em * (sm-sm * )+μ2im * (sm-sm * )-β4smih (sm-sm * ) +β4smih(em-em * )-(em-em * )em(β5+μ2)-β4smih (sm-sm * ) = -μ1 (sh-sh * ) 2 -α1 (sh-sh * ) 2 -(μ1+α1){eh * (eh-eh * )-eh * (sh-sh * )} -(μ1+α1){ih(ih-ih * )-ih * (sh-sh * )-β1shim (sh-sh * -ih+ih *) -β2shih (sh-sh * -eh+eh *)-β3ih(ih-ih *-rh+rh * )-μ1rh(rh-rh * )-μ2 (sm-sm * ) 2 -μ2 {(em-em * )em-em * (sm-sm * )} -μ2 {(im-im * )im-im * (sm-sm * )} -β4smih (sm-sm * -em+em * ) -β5em(em-em * -im+im * ) 13 int. j. anal. appl. (2022), 20:57 the above equation shows that 𝑊′(𝑡) ≤ 0 𝑎𝑛𝑑 w'(t)=0 for sh=sh * , eh=eh *, ih=ih *, rh=ih *,sm=sm * , em=em * , im=im * . so, the largest invariance set is the singleton set {𝐸1}. therefore, by using the principle of lasalle’s invariance the endemic equilibrium 𝐸1 is globally asymptotically stable. 6. sensitivity analysis of the system sensitivity analysis identifies the most effective model parameters that have effects on the dengue model system's basic reproduction number [19]. epidemiologists may forecast the important factors that play a significant part in virus-spreading dynamics using such analyses [20]. to avoid or manage the disease's effect, we must first identify the sensitivity induce values, which will give us an idea of which parameters for model would be maintained or monitored. in the current context, dengue virus infection is spreading globally at a rapid rate, and this hazardous virus poses a serious threat to the human population. to inhibit the transmission of infection, we must first discover which model parameters are critical to disease transmission. to detect model's such parameters, we must need to evaluate the basic reproduction number variation which depends upon the model parameters; alternatively, we must compute the normalized forward sensitivity index of the basic reproduction number 𝑅0 with respect to various parameters of the model. our goal here is to approximate important model parameters that govern the basic reproduction number 𝑅0. to analyze the sensitivity, we utilize the normalized forward sensitivity index of the basic reproduction number 𝑅0 with regard to the system (1) parameter 𝜌, which is signified by γr0 ρ = ∂r0 ∂ρ . ρ r0 table 2: sensitivity indices of r0 evaluated at the baseline parameter values of the model. parameter sensitivity index 𝚲𝟏 +0.5 𝚲𝟐 +0.5 𝜷𝟏 +0.5 𝝁𝟏 -0.4859780904 𝜶𝟏 -0.0172362471 𝜷𝟑 -0.4967856625 𝜷𝟒 +0.5 𝜷𝟓 +0.1412037037 𝝁𝟐 -1.1412037037 14 int. j. anal. appl. (2022), 20:57 on basic reproduction number 𝑅0, the parameters higher sensitivity index indicates the more influence sensitive parameter. the system parameter's sensitivity index with positive sign suggests that the basic reproduction number 𝑅0 increases when the parameter increases, and vice versa. in table 2 and fig. 2, we applied a sensitive index to r0 in relation to each parameter. according to our research, the most important model parameters are recruitment rates of human population (𝛬1), recruitment rates of vector population (𝛬2), rate of infectious from vector to host ( 𝛽1), natural death rate of human ( 𝜇1), panic/tension/anxiety rate of human ( 𝛼1), recovery rate of infected human ( 𝛽3), infection rate from human to vector ( 𝛽4), extrinsic incubation of vector ( 𝛽5), and natural death rate of vector population ( 𝜇2). the most significant sensitivity index of the system is the natural death rate of vector population 𝜇2. fig 2. sensitivity indices of r0 7. numerical results in the numerical part, the suggested model simulation is performed with the assist of matlab software. for numerical simulation, the parameter for the system (1) are given in table 3. table 3. system (1) parameters values parameter values units reference 𝚲𝟏 .9999 day -1 [21] 𝜷𝟏 .8500 day -1 [22] 𝜷𝟐 .6794 day -1 [23] 𝝁𝟏 .003468 day -1 [24] 𝜶𝟏 .000123 day -1 assumed 𝜷𝟑 .5555 day -1 [22] 𝚲𝟐 .0034 day -1 [25] 𝜷𝟒 .7186 day -1 [26] 𝜷𝟓 .0062 day -1 [8] 𝝁𝟐 .000244 day -1 [25] 15 int. j. anal. appl. (2022), 20:57 figs. 3–6 depict the dynamical system simulation exhibiting the influences of numerous parameters on the transmission dynamics model, demonstrating how parameters are efficient in inducing epidemics on different human populations as well as vector populations. figs. 3-4 depict the performance of a susceptible and infected host population, as well as the infectious rate from vector to host (𝛽1) and the rate of panic/tension/anxiety in humans (𝛼1). fig. 3(a) describes that there is no effects on different values of 𝛽1 between the early stage 0 to 5 days. also, it indicates that the susceptible humans decrease with an increase in transmission the infectious rate from vector to host (𝛽1) and vise-verse. in fig. 3(b) indicates that panic/tension/anxiety rate in humans (𝛼1) has an impact between 5 to 75 days. fig 3. (a) susceptible host population 𝑆ℎ with different values of 𝛽1. (b) susceptible host population 𝑆ℎ with different values of 𝛼1. it is interesting to see that in fig. 4 the infected host population decreases 0 to 5 days, after that it grows exponentially. when the interaction between vector to host increases the infected host increases in fig. 4(a), whereas in fig. 4(b) panic/tension/anxiety rate has the opposite effects. fig 4. (a) infected host population 𝐼ℎ with different values of 𝛽1. (b) infected host population 𝐼ℎ with different values of 𝛼1. 16 int. j. anal. appl. (2022), 20:57 infection rates from humans to vectors and extrinsic incubation of the vector are shown in figures 5–6 together with the behavior of a susceptible and infected vector population. in between the first 40 days, the susceptible vector grows after that it decreases. fig. 5(a) shows when the infection rates from humans to vectors increases to 10%, the susceptible vectors slightly down to the original. the effects on extrinsic incubation of the vector in fig. 5(b) shows after the 40 days. fig 5. (a) susceptible vector population 𝑆𝑚 with different values of 𝛽4. (b) susceptible vector population 𝑆𝑚 with different values of 𝛽5. figure 6 shows the fluctuation of the infected vector with respect to time t for various values of 𝛽4 and 𝛽5. it is obvious that as the value of grows, so does the infected vector. fig. 6(a) demonstrates the slightly deviation of different 𝛽4 to original, whereas deviation between different 𝛽5 to original is high. fig 6. (a) infected vector population 𝐼𝑚 with different values of 𝛽4. (b) infected vector population 𝐼𝑚 with different values of 𝛽5. 17 int. j. anal. appl. (2022), 20:57 8. conclusion the epidemic vector-borne disease has devastated many nations. form which, the focus of this article was to analyze dynamic dengue fever. we developed a dynamical mathematical model that would represent them and incorporate the impact of panic, tension, or anxiety on the human population. model's qualitative analysis was calculated, including illness free equilibrium, endemic equilibrium, and basic reproduction number. numerical simulation through various parameter settings showed the progression of epidemics, the system’s behaviors, and support theoretical results. the maximum sensitivity index was obtained for the vector death rate in the sensitivity study, and this parameter was regarded the most sensitive. it was discovered that increasing the rate of 𝜇2 results in the greatest decrease in reproduction number, and no other parameter had the same effect on reducing infection. such model analysis can give vital information to policy makers and health specialists who may be confronted the infectious disease reality. acknowledgments: the authors acknowledge with thanks to the deanship of scientific research (dsr), king abdulaziz university for technical support and thanks to knowledge excellence program phd, the kingdom of saudi arabia for supporting. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. hasan, s. jamdar, m. alalowi, s. al ageel al beaiji, dengue virus: a global human threat: review of literature, j. int. soc. prevent. communit. dent. 6 (2016), 1-6. https://doi.org/10.4103/2231-0762.175416. [2] s.k. roy, s. bhattacharjee, dengue virus: epidemiology, biology, and disease aetiology, can. j. microbiol. 67 (2021), 687–702. https://doi.org/10.1139/cjm-2020-0572. [3] n. sharma, r. singh, j. singh, o. castillo, modeling assumptions, optimal control strategies and mitigation through vaccination to zika virus, chaos solitons fractals. 150 (2021), 111137. https://doi.org/10.1016/j.chaos.2021.111137. [4] m.o. faruk, s.n. jannat, md.s. rahman, impact of environmental factors on the spread of dengue fever in sri lanka, int. j. environ. sci. technol. 19 (2022), 10637–10648. https://doi.org/10.1007/s13762-021-03905-y. [5] r. prasad, s.k. sagar, s. parveen, r. dohare, mathematical modeling in perspective of vectorborne viral infections: a review, beni-suef univ. j. basic. appl. sci. 11 (2022), 102. https://doi.org/10.1186/s43088-022-00282-4. [6] g. bhuju, g.r. phaijoo, d.b. gurung, sensitivity and bifurcation analysis of fuzzy seir-sei dengue disease model, j. math. 2022 (2022), 1927434. https://doi.org/10.1155/2022/1927434. https://doi.org/10.4103/2231-0762.175416 https://doi.org/10.1139/cjm-2020-0572 https://doi.org/10.1016/j.chaos.2021.111137 https://doi.org/10.1007/s13762-021-03905-y https://doi.org/10.1186/s43088-022-00282-4 https://doi.org/10.1155/2022/1927434 18 int. j. anal. appl. (2022), 20:57 [7] c.j. tay, m. fakhruddin, i.s. fauzi, s.y. teh, m. syamsuddin, n. nuraini, e. soewono, dengue epidemiological characteristic in kuala lumpur and selangor, malaysia, mathematics and computers in simulation. 194 (2022), 489–504. https://doi.org/10.1016/j.matcom.2021.12.006. [8] a. abidemi, n.a.b. aziz, analysis of deterministic models for dengue disease transmission dynamics with vaccination perspective in johor, malaysia, int. j. appl. comput. math. 8 (2022), 45. https://doi.org/10.1007/s40819-022-01250-3. [9] r. aleixo, f. kon, r. rocha, m.s. camargo, r.y. de camargo, predicting dengue outbreaks with explainable machine learning, in: 2022 22nd ieee international symposium on cluster, cloud and internet computing (ccgrid), ieee, taormina, italy, 2022: pp. 940–947. https://doi.org/10.1109/ccgrid54584.2022.00114. [10] a. sow, c. diallo, h. cherifi, effects of vertical transmission and human contact on zika dynamics, complexity. 2022 (2022), 5366395. https://doi.org/10.1155/2022/5366395. [11] n.h. sweilam, s.m. al-mekhlafi, s.a. shatta, optimal bang-bang control for variable-order dengue virus; numerical studies, j. adv. res. 32 (2021), 37–44. https://doi.org/10.1016/j.jare.2021.03.010. [12] a. abidemi, m.i. abd aziz, r. ahmad, vaccination and vector control effect on dengue virus transmission dynamics: modelling and simulation, chaos solitons fractals. 133 (2020), 109648. https://doi.org/10.1016/j.chaos.2020.109648. [13] j.k.k. asamoah, e. yankson, e. okyere, g.-q. sun, z. jin, r. jan, fatmawati, optimal control and cost-effectiveness analysis for dengue fever model with asymptomatic and partial immune individuals, results phys. 31 (2021), 104919. https://doi.org/10.1016/j.rinp.2021.104919. [14] l.j.s. allen, a.m. burgin, comparison of deterministic and stochastic sis and sir models in discrete time, math. biosci. 163 (2000), 1–33. https://doi.org/10.1016/s0025-5564(99)00047-4. [15] a.k. misra, a. sharma, j. li, a mathematical model for control of vector borne diseases through media campaigns, discrete continuous dyn. syst. b. 18 (2013) 1909–1927. https://doi.org/10.3934/dcdsb.2013.18.1909. [16] w. nur, h. rachman, n.m. abdal, m. abdy, s. side, sir model analysis for transmission of dengue fever disease with climate factors using lyapunov function, j. phys.: conf. ser. 1028 (2018), 012117. https://doi.org/10.1088/1742-6596/1028/1/012117. [17] p. saha, g.c. sikdar, u. ghosh, transmission dynamics and control strategy of single-strain dengue disease, int. j. dynam. control. (2022). https://doi.org/10.1007/s40435-022-01027-y. [18] d.n. fisman, a.l. greer, a.r. tuite, bidirectional impact of imperfect mask use on reproduction number of covid-19: a next generation matrix approach, infect. dis. model. 5 (2020), 405–408. https://doi.org/10.1016/j.idm.2020.06.004. [19] n.i. hamdan, a. kilicman, sensitivity analysis in a dengue fever transmission model: a fractional order system approach, j. phys.: conf. ser. 1366 (2019), 012048. https://doi.org/10.1088/17426596/1366/1/012048. [20] p. hiram guzzi, f. petrizzelli, t. mazza, disease spreading modeling and analysis: a survey, briefings bioinform. 23 (2022), bbac230. https://doi.org/10.1093/bib/bbac230. [21] a. dwivedi, r. keval, analysis for transmission of dengue disease with different class of human population, epidemiol. methods. 10 (2021), 20200046. https://doi.org/10.1515/em-2020-0046. [22] m.a. khan, fatmawati, dengue infection modeling and its optimal control analysis in east java, indonesia, heliyon. 7 (2021), e06023. https://doi.org/10.1016/j.heliyon.2021.e06023. https://doi.org/10.1016/j.matcom.2021.12.006 https://doi.org/10.1007/s40819-022-01250-3 https://doi.org/10.1109/ccgrid54584.2022.00114 https://doi.org/10.1155/2022/5366395 https://doi.org/10.1016/j.jare.2021.03.010 https://doi.org/10.1016/j.chaos.2020.109648 https://doi.org/10.1016/j.rinp.2021.104919 https://doi.org/10.1016/s0025-5564(99)00047-4 https://doi.org/10.3934/dcdsb.2013.18.1909 https://doi.org/10.1088/1742-6596/1028/1/012117 https://doi.org/10.1007/s40435-022-01027-y https://doi.org/10.1016/j.idm.2020.06.004 https://doi.org/10.1088/1742-6596/1366/1/012048 https://doi.org/10.1088/1742-6596/1366/1/012048 https://doi.org/10.1093/bib/bbac230 https://doi.org/10.1515/em-2020-0046 19 int. j. anal. appl. (2022), 20:57 [23] l. pimpi, s.w. indratno, j.w. puspita, e. cahyono, stochastic and deterministic dynamic model of dengue transmission based on dengue incidence data and climate factors in bandung city, commun. biomath. sci. 5 (2022), 78–89. https://doi.org/10.5614/cbms.2022.5.1.5. [24] p. mutsuddy, s. tahmina jhora, a.k.m. shamsuzzaman, s.m.g. kaisar, m.n.a. khan, dengue situation in bangladesh: an epidemiological shift in terms of morbidity and mortality, can. j. infect. dis. med. microbiol. 2019 (2019), 3516284. https://doi.org/10.1155/2019/3516284. [25] w. sanusi, n. badwi, a. zaki, s. sidjara, n. sari, m.i. pratama, s. side, analysis and simulation of sirs model for dengue fever transmission in south sulawesi, indonesia, j. appl. math. 2021 (2021), 2918080. https://doi.org/10.1155/2021/2918080. [26] h.s. rodrigues, m.t.t. monteiro, d.f.m. torres, sensitivity analysis in a dengue epidemiological model, conf. papers math. 2013 (2013), 721406. https://doi.org/10.1155/2013/721406. https://doi.org/10.5614/cbms.2022.5.1.5 https://doi.org/10.1155/2019/3516284 https://doi.org/10.1155/2021/2918080 https://doi.org/10.1155/2013/721406 international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 124-136 http://www.etamaths.com category of fuzzy topological polygroups n. abbasizadeh and b. davvaz∗ abstract. in this paper, the relation between two definitions of a fuzzy topological polygroup is discussed. the collection of all fuzzy continuous functions from a fuzzy topological space y to a fuzzy topological polygroup z, denoted by fc(y, z) induces a polygroup structure from that of z. moreover, we study (fuzzy) topological polygroup of the polygroup fc(y, z) when it is equipped with various known topologies and fuzzy topologies. also, a few properties of fuzzy topological polygroups are established and a category cf t p is formed with objects as ftp and morphisms as the fuzzy topological homomorphisms. the category ct p is seen to be a full subcategory of cf t p . 1. introduction the hyperstructure theory was born in 1934 when marty introduced the notion of hypergroup [20]. in 1979, foster [13] introduced the concept of fuzzy topological group. ma and yu [19] changed the definition of a fuzzy topological group in order to make sure that an ordinary topological group is a special case of a fuzzy topological group. on the other hand, in the last few decades, many connections between hyperstructures and fuzzy sets has been established and investigated. the concept of fuzzy topological polygroup (in short ftp) was introduced and studied in [1]. in [2] we have observed that the collection of all fuzzy continuous functions from a fuzzy topological space y to a fuzzy topological polygroup z, denoted by fc(y,z) induces a polygroup structure from that of z. here we investigate fc(y,z) in presence of various known topologies and fuzzy topologies. then, we show how a fuzzy topological polygroup induced by strong homomorphism. also, we observe that the collection of all fuzzy topological polygroups and fuzzy topological homomorphisms constitute a category, with we call cftp . moreover, the category ctp of topological polygroups and continuous homomorphisms form a full subcategory of cftp . we recall some basic definitions and results to be used in the sequel. let h be a non-empty set. then a mapping ◦ : h × h → p∗(h) is called a hyperoperation, where p∗(h) is the family of non-empty subsets of h. the couple (h,◦) is called a hypergroupoid. in the above definition, if a and b are two non-empty subsets of h and x ∈ h, then we define a◦b = ⋃ a∈a b∈b a◦ b, x◦a = {x}◦a and a◦x = a◦{x}. a hypergroupoid (h,◦) is called a semihypergroup if for every x,y,z ∈ h, we have x◦(y◦z)=(x◦y)◦z and is called a quasihypergroup if for every x ∈ h, we have x ◦ h = h = h ◦ x. this condition is called the reproduction axiom. the couple (h,◦) is called a hypergroup if it is a semihypergroup and a quasihypergroup [7, 20]. a special subclass of hypergroups is the class of polygroups. we recall the following definition from [6]. a polygroup is a system p = 〈p,◦,e,−1 〉, where ◦ : p × p → p∗(p), e ∈ p , −1 is a unitary operation p and the following axioms hold for all x,y,z ∈ p : (1) (x◦y) ◦z =x◦ (y ◦z), (2) e◦x = x =x◦e, (3) x ∈ y ◦z implies y ∈ x◦z−1 and z ∈ y−1 ◦x. the following elementary facts about polygroups follow easily from the axioms: e ∈ x◦x−1 ∩x−1 ◦x, e−1 = e, (x−1)−1 = x, and (x◦y)−1=y−1 ◦x−1. a non-empty subset k of a polygroup p is a subpolygroup of p if and only if a,b ∈ k implies 2010 mathematics subject classification. 54a40, 20n20, 03e72. key words and phrases. polygroup; topological polygroup; fuzzy topological polygroups; category cf t p . c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 124 category of fuzzy topological polygroups 125 a ◦ b ⊆ k and a ∈ k implies a−1 ∈ k. the subpolygroup n of p is normal in p if and only if a−1 ◦n ◦a ⊆ n for all a ∈ p . for a subpolygroup k of p and x ∈ p , denote the right coset of k by k ◦x and let p/k be the set of all right cosets of k in p . if n is a normal subpolygroup of p, then (p/n,�,n,−1 ) is a polygroup, where n ◦x�n ◦y={n ◦z|z ∈ n ◦x◦y} and (n ◦x)−1 = n ◦x−1. for more details about polygroups we refer to [3, 9, 17]. 2. preliminaries for the sake of convenience and completeness of our study, in this section some basic definition and results of [4, 5, 13, 19, 21, 22], which will be needed in the sequel are recalled here. throughout this paper, the symbol i will denote the unit interval [0, 1]. let x be a non-empty set. a fuzzy set a in x is characterized by a membership function µa : x → [0, 1] which associates with each point x ∈ x its grade or degree of membership µa(x) ∈ [0, 1]. that is, an element of ix. we denote by fs(x) the set of all fuzzy sets on x. a family t ⊆ fs(x) of fuzzy sets is called a fuzzy topology for x if it satisfies the following three axioms: (1) 0, 1 ∈t . (2) for all a,b ∈t , then a∧b ∈t . (3) for all (aj)j∈j, then ∨ j∈j aj ∈t . the pair (x,t ) is called a fuzzy topological space or fts, for short. the elements of t are called fuzzy open sets. a fuzzy set is fuzzy closed if and only if its complement is fuzzy open. a fuzzy set in x is called a fuzzy point if and only if it takes the value 0 for all y ∈ x except one, say x ∈ x. if its value at x is λ (0 < λ ≤ 1), we denote this fuzzy point by xλ, where the point x is called its support. the fuzzy point xλ is said to be contained in a fuzzy set a, or to belong to a, denoted by xλ ∈ a, if and only if λ ≤ µa(x). evidently, every fuzzy set a can be expressed as the union of all the fuzzy points which belong to a. a fuzzy set a in a fuzzy topological space (x,t ) is called a neighborhood of fuzzy point xλ, if there exists a b ∈t such that xλ ∈ b ≤ a. the family consisting of all neighborhood of xλ is called the system of neighborhood of fuzzy point xλ. a fuzzy point xλ is said to be quasi-coincident with a fuzzy set a, denoted by xλqa, if µa(x) + λ > 1. a is said to be quasi-coincident with b, denoted by aqb, if there exists x ∈ x such that µa(x) + µb(x) > 1. if this is true, we also say that a and b are quasi-coincident at x. a fuzzy set a in a fuzzy topological space (x,t ) is said to be a q-neighborhood of xλ if there exists a b ∈t such that xλqb ≤ a. the family consisting of all q-neighborhood of xλ is called the system of q-neighborhood of fuzzy point xλ. a fuzzy topological space (x,t ) is called a fully stratified space if t contains all constant fuzzy sets. given two topological spaces (x,t ) and (y,u), a mapping f : x → y is fuzzy continuous if, for any fuzzy set b ∈u, the inverse image f−1[b] ∈t . conversely, f is fuzzy open if, for any open fuzzy set a ∈t , the image f[a] ∈u (see [4]). let a be a fuzzy set in the fuzzy topological space (x,t ). then the induced fuzzy topology on a is the family ta of fuzzy subsets of a which are the intersection with a of t -open fuzzy sets in x. the pair (a,ta) is called a fuzzy subspace of (x,t ). for any fuzzy set a∩uj of ta, with uj ∈ t , we have µuj∩a(x)=µuj (x) ∧µa(x) (see [13]). let x and y be two non-empty subsets, f : x → y , a be a fuzzy set in x and b a fuzzy set in y . then, f[a] is the fuzzy set in y defined by µf[a](y) = { ∨ x∈f−1(y) µa(x) if f −1(y) 6= ∅. 0 otherwise. for all y ∈ y , where f−1(y) = {x|f(x) = y}. f−1[b] is the fuzzy set in x defined by µf−1[b](x) = µb (f(x)) for all x ∈ x. let f be a mapping of fts (x,t ) into fts (y,u). if for any fuzzy open q-neighborhood u of f(xλ) = [f(x)]λ there exists a fuzzy open q-neighborhood v of xλ such that f(v ) ≤ u, then we say that f is continuous at xλ with respect to q-neighborhood (see [21]). let f be a function from a fuzzy topological space (x,t ) into a fuzzy topological space (y,u). then the following are equivalent (see [21]): 126 abbasizadeh and davvaz (1) f is a fuzzy continuous mapping. (2) f is continuous with respect to q-neighborhood at any fuzzy point xλ. (3) f is continuous with respect to neighborhood at any fuzzy point xλ. 3. topological and fuzzy topological polygroups on fc(y,z) let p = 〈p,◦,e,−1 〉 be a polygroup, a,b ∈ ip and c,d ⊆ p . we define a•b ∈ ip , a−1 ∈ ip , c ◦d ⊆ p and c−1 ⊆ p by the respective formulas: (see [1]) (a•b)(x) = ∨ x∈x1◦x2 (µa(x1) ∧µb(x2)) and µa−1 (x) = µa(x −1) for any x ∈ p . also, c ◦d = ⋃ {c◦d : c ∈ c, d ∈ d} and c−1 = {c−1 : c ∈ c}. we denote a•b by ab for short. then for a,b ∈ ip , we have (ab)−1 = b−1a−1 and (a−1)−1 = a. the following definition of a fuzzy topological polygroup was given in [1]. definition 3.1. let p = 〈p,◦,e,−1 〉 be a polygroup and (p,t ) be a fuzzy topological space. a triad (p,◦,t ) is called a fuzzy topological polygroup or ftp for short, if: (1) for all x,y ∈ p and any fuzzy (open) q-neighborhood w of any fuzzy point zλ of x◦y, there are fuzzy (open) q-neighborhoods u of xλ and v of yλ such that u •v ≤ w. (2) for all x ∈ p and any fuzzy (open) q-neighborhood v of x−1λ , there exists a fuzzy (open) q-neighborhood u of xλ such that u−1 ≤ v. now, we give another definition of a fuzzy topological polygroup. definition 3.2. let p = 〈p,◦,e,−1 〉 be a polygroup and (p,t ) be a fuzzy topological space. a triad (p,◦,t ) is called a fuzzy topological polygroup or ftp for short, if: (1′) for all x,y ∈ p and any fuzzy (open) neighborhood w of any fuzzy point zλ of x ◦ y, there are fuzzy (open) neighborhoods u of xλ and v of yλ such that u •v ≤ w. (2′) for all x ∈ p and any fuzzy (open) neighborhood v of x−1λ , there exists a fuzzy (open) neighborhood u of xλ such that u−1 ≤ v. proposition 3.3. for a same polygroup p and a same fuzzy topology t on p the conditions (2) and (2′) are equivalent. proposition 3.4. for a same polygroup p and a same fuzzy topology t on p we have: (a) if t is a finite set, then (1) implies (1′). (b) (1′) implies (1). proof. (a) let w be any fuzzy neighborhood of any fuzzy point zλ of x ◦ y. if µw (z) > λ, then w is a fuzzy q-neighborhood of fuzzy point z1−λ. by (1), there exist fuzzy q-neighborhoods u of x1−λ and v of b1−λ such that uv ≤ w . now, it is clear that u and v are fuzzy neighborhoods of xλ and yλ respectively and the assertion follows from this. let z ∈ x◦y and µw (z) = λ, choose a decreasing sequence {εi}∞i=1 of real numbers such that 0 < εi < λ (i = 1, 2, . . . ) and lim i→∞ εi = 0. put λi = λ − εi. then, lim i→∞ λi = λ. now, for any λi, w is a fuzzy neighborhood of zλ and category of fuzzy topological polygroups 127 µw (z) > λi. hence, there exist fuzzy open neighborhoods ui of xλi and vi of yλi such that uivi ≤ w . we assume that u = n⋃ i=1 ui and v = n⋃ i=1 vi. it is easy to verify that u and v are fuzzy open neighborhoods of xλ and yλ respectively. since t is a finite set, we can choose a subsequence {uil} ∞ l=1 of the sequence {ui}∞i=1, such that each uil is a fuzzy neighborhood of xλ. clearly, ∞⋃ l=1 vil is a fuzzy open neighborhood of yλ. it follows from the assumption that there must exist an ik such that vik is a fuzzy neighborhood of yλ. now, uik and vik are fuzzy neighborhoods of xλ and yλ respectively and uikvik ≤ w . (b) let x,y ∈ p and w be a fuzzy q-neighborhood of any fuzzy point zλ of x ◦ y. choose a λ1 ∈ (0, 1) such that 1 − λ < λ1 < µw (z). then w is a fuzzy neighborhood of fuzzy point zλ1 . by (1′) there exist fuzzy neighborhoods u of xλ1 and v of yλ1 such that uv ≤ w . now, u and v are fuzzy q-neighborhoods of xλ and yλ respectively and uv ≤ w . � let fc(y,z) be the set of all fuzzy continuous functions from a fuzzy topological space y into a fuzzy topological space z. in this section, we investigate fc(y,z) in presence of various known topologies and fuzzy topologies. theorem 3.5. [2] let (y,ty ) be an fts, (z,◦,tz) an ftp, and f,g ∈ fc(y,z). then, the maps f ∗g and f−1 from the fuzzy topological space y into the fuzzy topological space z with the types, (f ∗g)(y) = f(y) ◦g(y) and f−1(y) = (f(y))−1 for every y ∈ y , are fuzzy continuous. theorem 3.6. [2] let (y,ty ) be a fully stratified fuzzy topological space and (z,◦,tz) a fuzzy topological polygroup. then, (fc(y,z),∗,e′,−1 ) is a polygroup. theorem 3.7. let (fc(y,z),∗) be the polygroup of fuzzy continuous functions from a fully stratified fuzzy topological space (y,ty ) to a fuzzy topological polygroup (z,◦,tz). (1) if z is commutative, then fc(y,z) is commutative. (2) if z contains identity element , then fc(y,z) contains identity element. proof. (1) for all f,g ∈ fc(y,z) and y ∈ y , (f ∗g)(y) = f(y) ◦g(y) = g(y) ◦f(y) = (g ∗f)(y). (2) for all f ∈ fc(y,z) and y ∈ y , (f ∗e′)(y) = f(y) ◦e′(y) = f(y) ◦e = f(y) = e◦f(y) = e′(y) ◦f(y) = (e′ ∗f)(y). � theorem 3.8. [2] let (y,ty ) be a fully stratified fuzzy topological space, (z,◦,tz) a fuzzy topological polygroup, and z1 ∈ iz a fuzzy polygroup. then, the fuzzy set µ ∈ ifc(y,z) for which µ(f) =∧ y∈y z1(f(y)), f ∈ fc(y,z) is a fuzzy polygroup. definition 3.9. [14] let u be a fuzzy open set on an fts z and yλ, λ ∈ (0, 1] be a fuzzy point on an fts y . by [yλ,u] we denote the subset of fc(y,z) where [yλ,u] = {f ∈ fc(y,z) : f(yλ) ≤ u}. the collection of all such [yλ,u] forms a subbase for some topology on fc(y,z), called fuzzy-point fuzzy-open topology (fp-fo), denoted by t(fp−fo). definition 3.10. [16] let p = 〈p,◦,e,−1 〉 be a polygroup and (p,t ) be a topological space. then, the system p = 〈p,◦,e,−1 ,t 〉 is called a topological polygroup if the mapping ◦ : p ×p → ℘∗(p) and −1 : p → p are continuous. lemma 3.11. [15] let p be a polygroup. then, the hyperoperation ◦ : p ×p → ℘∗(p) is continuous if and only if for every x,y ∈ p and w ∈t such that x◦y ⊆ w then there exist u,v ∈t such that x ∈ u,y ∈ v and u ◦v ⊆ w . 128 abbasizadeh and davvaz theorem 3.12. let (y,ty ) be a fully stratified fuzzy topological space, (z,◦,tz) a fuzzy topological polygroup. then (fc(y,z),t(fp−fo)) is a topological polygroup. proof. it is clear that fc(y,z) is a polygroup and fc(y,z) is a topological space with respect to t(fp−fo). we need to show that the mappings (f,g) 7→ f ∗ g and f 7→ f−1 are continuous. suppose that [yλ,u] be a subbasic open set in fc(y,z) such that f∗g ⊆ [yλ,u]. so, for any h ∈ f∗g ⊆ [yλ,u], we have h(yλ) ≤ u, where h(yλ) ∈ (f(y))λ ◦ (g(y))λ. (z,◦,t ) being an fuzzy topological polygroup , then there exist fuzzy open sets v,w of z such that (f(y))λ ∈ v, (g(y))λ ∈ w and uw ≤ v . on the other hand, f(yλ) = (f(y))λ ≤ v implies that f ∈ [yλ,v ] and similarly g ∈ [yλ,w]. now, we show that f ∗g is continuous. we need to show that [yλ,v ] ∗ [yλ,w] ⊆ [yλ,u]. let ξ ∈ [yλ,v ] ∗ [yλ,w]. then there exist η ∈ [yλ,v ] and ψ ∈ [yλ,w], such that ξ = η ∗ψ. since η ∈ [yλ,v ] and ψ ∈ [yλ,w], then we have η(yλ) ≤ v and ψ(yλ) ≤ w . so, (η ∗ ψ)(yλ) ≤ v ∗w ≤ u. therefore, ξ(yλ) ≤ u and ξ ∈ [yλ,u]. finally we show that f−1 is continuous. for any subbasic open set [yλ,u] containing f −1, we get (f−1)(yλ) ≤ u, so (f−1(y))λ ≤ u and (f(y))λ ≤ u−1. since u is fuzzy open if and only if u−1 is fuzzy open and f(yλ) = (f(y))λ, we have f ∈ [yλ,u−1]. now, we show that [yλ,u]−1 ≤ [yλ,u−1]. let ψ ∈ [yλ,u]−1. then, there is some η ∈ [yλ,u] such that ψ = η−1. since η ∈ [yλ,u] and η(yλ) ≤ u, so (η−1)−1(yλ) ≤ u, then ψ−1(yλ) ≤ u and ψ(yλ) ≤ u−1, ψ ∈ [yλ,u−1], as desired. � definition 3.13. [14, 18] let y and z be two fixed fuzzy topological spaces, u ∈ iz a fuzzy open set of z, and y ∈ y . then by yu ∈ ifc(y,z) we denote the fuzzy set for which yu (f) = u(f(y)), for every f ∈ fc(y,z). the fuzzy point open topology tfp on fc(y,z) generated by fuzzy sets of the form yu , where y ∈ y and u ∈ iz is a fuzzy open set of z. for each fuzzy compact set k in y and each fuzzy open set u in z, a fuzzy set ku on fc(y,z) is given by ku (g) = ∧ x∈supp(k) u(g(x)). the collection of all such ku forms a subbase for some fuzzy topology on fc(y,z), called fuzzy compact open topology and it is denoted by ∆co. theorem 3.14. [2] let (y,ty ) be a fully stratified fuzzy topological space and (z,◦,tz) a fuzzy topological polygroup. then the triad (fc(y,z),∗,tfp ) is a fuzzy topological polygroup. theorem 3.15. let (y,ty ) be a fully stratified fuzzy topological space and (z,◦,tz) a fuzzy topological polygroup. then fc(y,z) endowed with fuzzy compact-open topology is a fuzzy topological polygroup. proof. clearly, by theorem 3.6 and definition 3.13, (fc(y,z),∗) is a polygroup and (fc(y,z), ∆co) is a fuzzy topological space. now, we show that (fc(y,z),∗, ∆co) satisfies the conditions (1) and (2) in definition 3.1. (1) let ku be a fuzzy open subbasic q-neighborhood of any fuzzy point hλ of f ∗g. we show that there exist fuzzy open subbasic q-neighborhoods kv and kw of fλ and gλ, respectively such that kv •kw ≤ ku. since hλqku , it follows that λ + ku (h) > 1 and λ + ∧ y∈suppk u(h(y)) > 1. so, for all y ∈ suppk, h(y)λqu. that is, the fuzzy set u is a fuzzy open q-neighborhood of h(y)λ. now, since (z,◦,tz) is a fuzzy topological polygroup, it follows that there exist fuzzy open qneighborhoods v and w of f(y)λ and g(y)λ such that v w ≤ u. we consider the fuzzy sets kv and kw . since v is a fuzzy open q-neighborhood of f(y)λ, then for all y ∈ suppk, v (f(y)) + λ > 1 and kv (f) + λ = ∧ x∈suppk v (f(y)) + λ > 1. category of fuzzy topological polygroups 129 hence, kv (f) + λ > 1 and fλqkv . similarly, it can be proved that gλqkw . we prove kv •kw ≤ ku . let f ∈ fc(y,z). then, we have (kv •kw )(f) = ∨ f∈f1∗f2 [kv (f1) ∧kw (f2)] = ∨ f∈f1∗f2 [( ∧ x∈suppk v (f1(x))) ∧ ( ∧ x∈suppk w(f2(x)))] = ∨ f∈f1∗f2 [ ∧ x∈suppk [v (f1(x)) ∧w(f2(x))]] ≤ ∧ x∈suppk [ ∨ f∈f1∗f2 [v (f1(x)) ∧w(f2(x))]] ≤ ∧ x∈suppk [ ∨ f(x)∈z1◦z2 [v (z1) ∧w(z2)]] = ∧ x∈suppk [(v •w)(f(x))] ≤ ∧ x∈suppk u(f(x)) = ku (f). now, the condition (1) in definition 3.1 follows. (2) let f ∈ fc(y,z) and ku be a fuzzy open q-neighborhood of f−1λ . we show that there exists fuzzy open q-neighborhood kv of fλ such that k −1 v ≤ ku . since f −1 λ qku , it follows that λ + u(f−1(y)) > 1. as (z,◦,tz) is a fuzzy topological polygroup then, there exists fuzzy open qneighborhood v of fλ such that v −1 ≤ u. we prove that k−1v ≤ ku . let f ∈ fc(y,z). then we have k−1v (f) = kv (f −1) = ∧ x∈suppk v (f−1(x)) = ∧ x∈suppk v −1(f(x)) ≤ ∧ x∈suppk u(f(x)) = ku (f). hence, k−1v ≤ ku . now, the condition (2) in definition 3.1 follows. therefore, (fc(y,z),∗, ∆co) is a fuzzy topological polygroup. � 4. fuzzy topological polygroups induced by strong homomorphisms in this section, we show how to strong homomorphisms induce fuzzy topological polygroup structure on polygroups. definition 4.1. [10] a collection b of fuzzy neighborhoods of xλ, for 0 < λ ≤ 1, is called a fundamental system of fuzzy neighborhoods of xλ if and only if for any fuzzy neighborhood v of xλ, there exists u ∈ b such that xλ ≤ u ≤ v . definition 4.2. [10] a collection ω of fuzzy sets in an fts x is called a prefilterbase on x if 0x /∈ ω and for all a,b ∈ ω, then there exists c ∈ ω such that c ≤ a∩b. proposition 4.3. let p be a polygroup and a,b,c ∈ ip . then the following are hold: (1) if a ≤ b then ac ≤ bc and ca ≤ cb. (2) if ac = bc for any c ∈ ip , then a = b. (3) (ab)c = a(bc). (4) if a ≤ b then a−1 ≤ b−1. proof. it is straightforward. � theorem 4.4. let (p,t ) be a fuzzy topological polygroup. then the mapping φ : p → p−1, x 7→ x−1 is homeomorphic mapping. theorem 4.5. let (p,t ) be a fuzzy topological polygroup. then, v is fuzzy open if and only if v −1 is fuzzy open. 130 abbasizadeh and davvaz proof. for all x ∈ p , φ−1(v )(x) = v (φ(x)) = v (x−1) = v −1(x), φ is fuzzy continuous and v is fuzzy open, so v −1 = φ−1(v ) is fuzzy open. converse follows similarly. � corollary 4.6. let (p,t ) be a fuzzy topological polygroup. then, for each λ with 0 < λ ≤ 1 and x ∈ p, v is a fuzzy neighborhood of eλ if and only if v −1 is a fuzzy neighborhood of eλ. proof. the proof follows from theorem 4.4 and the fact that eλ ≤ v if and only if eλ ≤ v −1. � definition 4.7. a fuzzy open set u of a fuzzy topological polygroup p is called a symmetric neighborhood if u−1 = u. theorem 4.8. every fuzzy topological polygroup has a fuzzy open fundamental system of eλ containing a symmetric fuzzy open fundamental system of eλ. proof. suppose that b is a fuzzy open fundamental system of eλ. then, for every u ∈ b put v = u ∩u−1. so, v = v −1 and v ≤ u. � theorem 4.9. if b is a fundamental system of fuzzy neighborhoods of eλ, for 0 < λ ≤ 1, then d = {u ∩u−1 : u ∈ b} is also a fundamental system of fuzzy neighborhoods of eλ. proof. it is obvious. � proposition 4.10. let (p,t ) be a fuzzy topological polygroup. then, the family b = {ã ∈ fs(p∗(p)) | a ∈t}, where µã(x) = ∨ x∈x µa(x), is a base for a fuzzy topology t ∗ on p∗(p). proof. b is a base for a fuzzy topology on p∗(p) because: (1) for any ã1, ã2 ∈b, with a1,a2 ∈t , it follows that ã1 ∩ ã2 ∈b, because ã1 ∩ ã2 = ã1 ∩a2 and a1 ∩a2 ∈t . indeed, for any x ∈p∗(p), we have µ ã1∩a2 (x) = ∨ x∈x µ(a1∩a2)(x) = ∨ x∈x (µa1 (x) ∧µa2 (x)) = ( ∨ x∈x µa1 (x)) ∧ ( ∨ x∈x µa2 (x)) = µã1 (x) ∧µã2 (x) = µ(ã1∩ã2)(x). (2) since 1 ∈t , it follows that µ1̃(x) = 1, for any x ∈p ∗(p) and thus⋃̃ a∈b = 1. � lemma 4.11. let u be a fuzzy open subset of a fuzzy topological polygroup p . then, aλu and uaλ are fuzzy open subsets of p for every a ∈ p . proof. suppose that u be a fuzzy open subset of p . then, (a−1φ −1(ũ))(z) = ũ(a−1φ(z)) = ũ(a −1 ◦z) = ∨ t∈a−1◦z u(t) = ∨ z∈a◦t u(t) = aλu(z). since the mapping a−1φ −1 : p →p∗(p),x 7→ a−1 ◦x, is fuzzy continuous, thus aλu is fuzzy open. similarly, we can prove that uaλ is fuzzy open. � lemma 4.12. let (p,t ) be a fuzzy topological polygroup and b be a fuzzy open fundamental system of fuzzy neighborhood of eλ. then, the families {xλu} and {uxλ}, are fuzzy open fundamental system of fuzzy neighborhood of xλ. category of fuzzy topological polygroups 131 proof. suppose that w is a fuzzy open subset of p and xλ ≤ w . since (xλu)(x) = ∨ x∈x1◦x2 [xλ(x1) ∧u(x2)] = ∨ x∈x◦x2 [λ∧u(x2)] ≥ λ∧u(e) = λ, we conclude that xλ ≤ xλu. since eλ ≤ x−1λ w , it follows that there exists u ∈ b such that eλ ≤ u ≤ x−1λ w . so, xλu ≤ w . thus, w is a union of fuzzy open subsets xλu. therefore, {xλu} is a fuzzy open fundamental system for p . similarly, the family {uxλ} is a fuzzy open fundamental system for p . � in the next theorem we characterize a fuzzy topological polygroup via the fundamental system of fuzzy neighborhoods of eλ. theorem 4.13. if p is a fuzzy topological polygroup, then there exists a fundamental system of fuzzy neighborhoods b of eλ (0 < λ ≤ 1), such that the following conditions hold: (1) each member of b is symmetric. (2) for all u ∈ b, there exists v ∈ b such that v •v ≤ u. (3) for all u ∈ b, there exists v ∈ b such that v −1 ≤ u. (4) for all u ∈ b and xλ ≤ u there exists v ∈ b such that xλv ≤ u. conversely, given a polygroup p and a prefilterbase b of eλ satisfying the conditions (1)-(4), there exists a unique fuzzy topology t on p such that (p,t ) forms a fuzzy topological polygroup such that b forms a fundamental system of fuzzy neighborhoods of eλ. proof. (1) let p be a fuzzy topological polygroup. consider any fundamental system d of fuzzy neighborhoods of eλ (viz. consider the fuzzy open sets containing eλ). then, we get a fundamental system of fuzzy neighborhoods b of eλ such that each member of b is symmetric. (2) for any u ∈ b, since p is an ftp, there exist v1,v2 ∈ b such that v1v2 ≤ u. let v = v1 ∩v2. so, v v ≤ v1v2 ≤ u. (3) for any u ∈ b, xλ ≤ p , since p is an ftp, there exists v ∈ b such that v −1 ≤ u. (4) let u ∈ b and xλ ≤ u. as xλ = (xe)λ = xλeλ, by fuzzy topological polygroup of p , there exist fuzzy neighborhoods w1,v1 of xλ and eλ respectively such that w1v1 ≤ u. since (xλv1)(z) = ∨ z∈z1◦z2 [xλ(z1) ∧v1(z2)], it follows that (xλv1)(z) = { ∨ [λ∧v1(z2)] if z ∈ x◦z2, 0 if x 6= z1. as w1(x) ≥ λ, (w1v1)(z) ≥ (xλv1)(z). hence, xλv1 ≤ w1v1 ≤ u. conversely, suppose b is a prefilterbase at eλ, such that it satisfies (1)-(4). for each x ∈ p , it is easy to see that bx = {xλu : u ∈ b} forms a prefilterbase at xλ. then ∪bx generates a unique fuzzy topology on p with b as a fundamental system of fuzzy neighborhoods of eλ. in order to show that p is a fuzzy topological polygroup, we have to show p satisfies the conditions (1) and (2) in definition 3.1. suppose zλu is a fuzzy open neighborhood of zλ, where u ∈ b and zλ ∈ x ◦ y. then by (2), there are v,w ∈ b such that v w ≤ u. it is clear that (xλv )(yλw) = (xy)λv w and consequently (xλv )(yλw) ≤ zλu. suppose xλu is a fuzzy open neighborhood of xλu, where u ∈ b. by the condition (3), there is v ∈ b such that v −1 ≤ u. it follows from the symmetric of the members of b. � definition 4.14. [9] let 〈p1, ·,e1,−1 〉 and 〈p2,∗,e2,−i 〉 be two polygroups. let f be a mapping from p1 to p2 such that f(e1) = e2. then, f is called a strong homomorphism or a good homomorphism if f(x ·y) = f(x) ∗f(y), for all x,y ∈ p1. 132 abbasizadeh and davvaz since p1 is a polygroup, e1 ∈ a ◦1 a−1 for all a ∈ p1, it follows that f(e1) ∈ f(a) ◦2 f(a−1) or e2 ∈ f(a) ◦2 f(a−1) which implies f(a−1) ∈ f(a)−1 ◦2 e2. therefore, f(a−1) = f(a)−1 for all a ∈ p1. moreover, if f is a fuzzy topological homomorphism from p1 into p2, then the kernel of f is the set kerf = {x ∈ p1|f(x) = e2}. it is trivial that kerf is a subpolygroup of p1 but in general is not normal in p1. as in polygroup, if f is a fuzzy topological homomorphism from p1 into p2, then f is injective if and only if kerf = {e1}. definition 4.15. a fuzzy topology that makes a polygroup ftp is called a fuzzy topology compatible with the polygroup structure. theorem 4.16. let (p,t ) be a fuzzy topological polygroup. if f : q → p is a strong homomorphism from any polygroup q to p then, f induces a unique compatible fuzzy topology on q that makes f fuzzy continuous. proof. let b be a fundamental system of fuzzy neighborhoods of eλ in p . then it is enough to show that f−1(b) determines a unique fuzzy topology on q such that f−1(b) forms a fundamental system of fuzzy neighborhoods of eλ in q. it is clear that f −1(b) is a prefilterbase at eλ in q. in view of theorem 4.13, it is now to verify that f−1(b) satisfies the conditions (1) − (4) of theorem 4.13. (1) any element of f−1(b) is of the form f−1(v ), for some v ∈ b. now, for all x ∈ p (f−1(v ))−1(x) = f−1(v )(x−1) = v (f(x−1)) = v (f−1(x)) = v −1(f(x)) = v (f(x)) = f−1(v )(x). hence, (f−1(v ))−1 = f−1(v ), showing that each member of f−1(b) is symmetric. (2) let f−1(u) ∈ f−1(b), for some u ∈ b. then as u ∈ b, there exists v ∈ b, such that v v ≤ u. for any z ∈ q, (f−1(v )f−1(v ))(z) = ∨ z∈z1◦z2 [(f−1(v ))(z1) ∧ (f−1(v ))(z2)] = ∨ z∈z1◦z2 [v (f(z1)) ∧v (f(z2))] = ∨ f(z)∈f(z1)◦f(z2) [v (f(z1)) ∧v (f(z2))] = (v v )(f(z)) ≤ u(f(z)) = f−1(u)(z). (3) let f−1(u) ∈ f−1(b), for some u ∈ b. then as u ∈ b, there exists v ∈ b, such that v −1 ≤ u. for any z ∈ q, f−1(v )(z) = v (f−1(z)) = v −1(f(z)) ≤ u(f(z)) = u−1(f(z)) = f−1(u−1)(z). (4) let a ∈ q and f−1(u) ∈ f−1(b). then f(a) ∈ p and u ∈ b so that there exists v ∈ b such that f(a)λv ≤ u, (aλf −1(v ))(z) = ∨ z∈z1◦z2 [aλ(z1) ∧f−1(v )(z2)]. this implies that (aλf −1(v ))(z) = { ∨ [λ∧f−1(v )(z2)] if z ∈ a◦z2, 0 if x 6= z1. ≤ ∨ z∈a◦z2 [λ∧v (f(z2))] = ∨ f(z)∈f(a)◦f(z2) [λ∧v (f(z2))] = ∨ f(z)∈f(a)◦x [λ∧v (f(z2))] = (f(a)λv )(f(z)) ≤ u(f(z)) = f−1(u)(z). � corollary 4.17. any subpolygroup of a fuzzy topological polygroup is a fuzzy topological polygroup. category of fuzzy topological polygroups 133 proof. let (p,t ) be a fuzzy topological polygroup and k be a subpolygroup of p . if b is a fundamental system of fuzzy neighborhoods of eλ in p , and i : k → p given by i(x) = x, is the inclusion homomorphism, then by the theorem 4.16 f−1(b) determines a unique compatible fuzzy topology on k such that f−1(b) forms a fundamental system of fuzzy neighborhoods of eλ in k. � corollary 4.18. let p be an ftp and f : p → q is a strong homomorphism from p onto q with kernel k such that k is a normal subpolygroup of p . then, f̄ : p/n → q induces a compatible fuzzy topology on p/n that makes f̄ fuzzy continuous. proof. it is straightforward. � theorem 4.19. let (p,t ) be an ftp and f : p → q is a strong homomorphism from p onto any polygroup q. then, p induces a fuzzy topology compatible with the polygroup structure on q that makes f fuzzy continuous. proof. let b be a fundamental system of fuzzy neighborhoods of eλ in q. in view of theorem 4.13, it is enough to show that f(b) is a fundamental system of fuzzy neighborhoods of eλ in q. (1) for any u ∈ b, u = u−1 and so, f(u) = f(u−1). now, for all z ∈ q, f(u−1)(z) = ∨ f(t)=z u−1(t) = ∨ f(t)=z u(t−1) = ∨ f(t−1)=z−1 u(t−1) = f(u)(z−1) = f−1(u)(z). consequently, f(u) = f−1(u). (2) if f(u) ∈ f(b), then u ∈ b and so, there exists v ∈ b such that v v ≤ u. for any z ∈ q, (f(v )f(v ))(z) = ∨ z∈z1◦z2 [(f(v ))(z1) ∧ (f(v ))(z2)] = ∨ z∈z1◦z2 [( ∨ f(y1)=z1 v (y1)) ∧ ( ∨ f(y2)=z2 v (y2))] = ∨ z∈z1◦z2 [ ∨ f(y1)=z1 f(y2)=z2 [v (y1) ∧v (y2)]] ≤ ∨ f(t)=z (v v )(t) = f(v v )(z) ≤ f(u)(z). (3) if f(u) ∈ f(b), then u ∈ b and so, there exists v ∈ b, such that v −1 ≤ u. for any z ∈ q, (f(v ))−1(z) = ( ∨ f(t)=z v (t))−1 = ∨ f(t)=z v −1(t) ≤ ∨ f(t)=z u(t) = f(u)(z). (4) let b ∈ q and f(u) ∈ f(b). then there exists a ∈ p with f(a) = b. so, there exists v ∈ b such that aλv ≤ u. it is to show that bλf(v ) ≤ f(u). for any z ∈ q, it is easy to see that bλf(v )(z) = f(aλ)f(v )(z) ≤ f(aλv )(z) ≤ f(u)(z). hence, all the condition (1)-(4) are satisfied proving f(b) to be a fundamental system of fuzzy neighborhoods of eλ in q. let u be any fuzzy neighborhood of eλ in q, by definition of fundamental system of fuzzy neighborhoods, there exists some b ∈ b such that eλ ≤ f(b) ≤ u. as b ≤ f−1(f(b)) ≤ f−1(u) and eλ ∈ b it follows that f−1(u) is a fuzzy neighborhood of eλ in q. therefore, f is fuzzy continuous. � corollary 4.20. let p be a polygroup and n be a normal subpolygroup of p . the polygroup epimorphism π : p → p/n given by π(x) = xn induces a fuzzy topology compatible with the polygroup p/n that makes π fuzzy continuous. proof. the proof follows from corollary 4.18 and theorem 4.19. � theorem 4.21. let p be an ftp. if q is an ftp induced from a strong homomorphism f : p → q and n = kerf such that n is a normal subpolygroup of p then, the fuzzy topology compatible with p/n induced from f̄ : p/n → q and the fuzzy topology compatible with p/n induced from k : p → p/n are same. 134 abbasizadeh and davvaz proof. let b be a fundamental system of fuzzy neighborhoods of eλ in p . it follows from the theorem 4.19 that {π(v ) : v ∈ b} is a fundamental system of fuzzy neighborhoods of eλ for the compatible fuzzy topology on p/n induced by π and {f̄−1(f(v )) : v ∈ b} is a fundamental system of fuzzy neighborhoods of eλ for the compatible fuzzy topology on p/n induced by f̄. since, f̄π = f, it follows that f̄−1(f(v ))(xn) = f(v )(f̄(xn)) = f(v )(f̄π)(x) = f(v )(f(x)) = ∨ f(y)=f(x) v (y) = ∨ yx−1∈n v (y) = ∨ yn=xn v (y) = ∨ π(y)=xn v (y) = π(v )(xn). hence, both the fundamental systems are identical leading to the same compatible topology on p/n. � 5. fuzzy topological polygroups and the category cftp in this section we introduce the category cftp , which the objects in this category are fuzzy topological polygroups, morphisms are fuzzy topological homomorphisms and compositions is the usual composition of functions. also, we show that the category ctp of topological polygroups and continuous topological homomorphisms form a full subcategory of cftp . theorem 5.1. in a fuzzy topological polygroup p , v is a fuzzy q-neighborhood of eλ if and only if v −1 is a fuzzy q-neighborhood of eλ. proof. let v be a fuzzy q-neighborhood of eλ. then there exists fuzzy open set a such that eλqa ≤ v , that is, a(e) + λ > 1 and a ≤ v . for all x ∈ p , a(x−1) ≤ v (x−1), so a−1(x) ≤ v −1(x) and a−1 ≤ v −1. now, a−1(e) + eλ(e) = a−1(e) + λ > 1. hence, eλqa−1 and a−1 ≤ v −1. therefore, v −1 is a fuzzy q-neighborhood of eλ. conversely, let v −1 be a fuzzy q-neighborhood of eλ. then there exist fuzzy open set a such that eλqa ≤ v −1. as above, a−1 ≤ v and eλqa−1. that is, v is a fuzzy q-neighborhood of eλ. � proposition 5.2. [1] let (p1,t1) and (p2,t2) be two fuzzy topological polygroups and f : p1 → p2 be a homomorphism. then, f is fuzzy continuous if and only if f is continuous at eλ (here e is the unit of p1) for any λ ∈ (0, 1]. definition 5.3. [1] let 〈p1,◦1,e1,−1 ,t1〉 and 〈p2,◦2,e2,−i ,t2〉 be fuzzy topological polygroups. a mapping f from p1 into p2 is said to be a fuzzy topological homomorphism if for all a,b ∈ p1: (1) f(e1) = e2. (2) f(a◦1 b) = f(a) ◦2 f(b). (3) f is fuzzy continuous mapping of fts (p1,t1) into fts (p2,t2). theorem 5.4. the collection of all fuzzy topological polygroups and fuzzy topological homomorphisms form a category. proof. consider the collection of all ftp as objects, morphisms are fuzzy topological homomorphisms and compositions is the usual composition of functions. in checking that cftp is a category, one must note that for each object p , i : p → p given by i(x) = x is the identity morphism. consequently, it forms a category. � remark 1. it is well known that corresponding to any topological space (x,t ), one can obtain the characteristic fuzzy topological space (x,tf ). theorem 5.5. if (p,t ) is a topological polygroup, then (p,tf ) is a fuzzy topological polygroup. proof. clearly (p,tf ) is a fuzzy topological space. now, we show that (p,tf ) satisfies the conditions (1) and (2) in definition 3.1. (1) let x,y ∈ p and w be a fuzzy open q-neighborhood of any fuzzy point zλ of x◦y. we show that there exist fuzzy open q-neighborhood u and v of xλ and yλ respectively, such that uv ≤ w . let w be a fuzzy open q-neighborhood on (p,tf ) with zλqw. then w = χa for some a ∈ t . hence, zλq χa ⇒ χa(z) + λ > 1 ⇒ z ∈ a. category of fuzzy topological polygroups 135 since (p,t ) is a topological polygroup, there exist open sets b,c ∈ t such that x ∈ b, y ∈ c and bc ⊆ a. then xλq χb and yλq χc, where χb,χc ∈ tf . in order to complete the proof, we show χb χc ≤ χa = w . for all t ∈ p , (χb χc)(t) = ∨ t∈t1◦t2 (χb(t1) ∧χc(t2)) = { 1 if t1 ∈ b,t2 ∈ c, 0 otherwise. = { 1 if t ∈ bc, 0 otherwise. = χbc(t) ≤ χa(t). so, the condition (1) in definition 3.1 follows. (2) let x ∈ p and u be a fuzzy open q-neighborhood of x−1λ . we show that there exists fuzzy open q-neighborhood v of xλ such that v −1 ≤ u. since x−1qu, it follows that u = χa for some a ∈t . hence, x−1q χa ⇒ χa(x−1) + λ > 1 ⇒ x−1 ∈ a. since (p,t ) is a topological polygroup, there exists open set b ∈ t such that x ∈ b and b−1 ⊆ a. then χb(x) + λ > 1, where χb ∈tf . in order to complete the proof, we show χ−1b ≤ χa = u. for all t ∈ p , χ−1b (t) = { 1 if t ∈ b−1, 0 if t /∈ b−1. ≤ { 1 if t ∈ a, 0 if t /∈ a. = χa(t). now, the condition (2) in definition 3.1 follows. therefore, (p,tf ) is a fuzzy topological polygroup. � theorem 5.6. if f is a continuous topological homomorphism from a topological polygroup (p1,t1) to a topological polygroup (p2,t2) then f : (p1,t1f ) → (p2,t2f ) is a fuzzy topological homomorphism between the corresponding fuzzy topological polygroup. proof. the proof is straightforward. � theorem 5.7. if ctp is the category of topological polygroups and continuous topological homomorphisms, then ctp is a full subcategory of cftp . proof. we know that any object of ctp can be viewed as an object of cftp and any morphism between two objects of ctp is a morphism between the corresponding objects of cftp . hence, ctp is a subcategory of cftp . let the inclusion functor i : ctp → cftp that sends (p,t ) to its characteristic fuzzy topological space (p,tf ) and f : (p,t ) → (q,σ) to f∗ : (p,tf ) → (q,σf ). to show that the functor i is full. let (p,t ) and (q,σ) be two objects in ctp and f∗ : (p,tf ) → (q,σf ) a morphism in cftp . if u ∈ σ then χu ∈ σf and so, f∗ −1 (χu ) = χf∗−1 (u) ∈tf , which in turn gives f∗ −1 (u) ∈ t . hence, there exist f∗ : (p,t ) → (q,σ)a morphism in ctp such that i(f∗) = f∗, that is, i is full. consequently, ctp is a full subcategory of cftp . � theorem 5.8. let (p,t ) be a fuzzy topological polygroup. then, for all 0 ≤ λ < 1, (p,iλ(t )) is a topological polygroup. proof. it is clear that (p,iλ(t )) is a topological space. we need to show that the mappings (x,y) 7→ x◦y and x 7→ x−1 are continuous. let x,y ∈ p and w be any open set in (p,iλ(t )) such that x◦y ⊆ w . there exists a fuzzy open set γ in (p,t ) such that γλ = w . so, for any zλ ∈ x ◦ y, we have zλ < γ. since (p,t ) is a fuzzy topological polygroup, there exist fuzzy open sets u and v such that xλ < u, yλ < v and uv ≤ γ. then x ∈ uλ and y ∈ v λ. we shall show that uλv λ ⊆ w . if r ∈ uλv λ, then r ∈ st where s ∈ uλ and t ∈ v λ, that is, u(s) > λ and v (t) > λ. now,, (uv )(r) = ∨ r∈r1r2 [u(r1) ∧v (r2)] ≥ u(s) ∧v (t) > λ. 136 abbasizadeh and davvaz so, γ(r) > λ, r ∈ γλ = w . hence, uλv λ ⊆ w. this show that the mapping (x,y) 7→ x ◦ y is continuous. now, we prove that x 7→ x−1 is continuous. let x ∈ p and v be an open set of (p,iλ(t )) containing x−1. there is a fuzzy open set γ on (p,t ) such that γλ = v . so, x−1 ∈ γλ and we have x−1λ < γ. since (p,t ) is a fuzzy topological polygroup, there exists fuzzy open set u containing xλ such that xλ < u and u −1 ≤ γ. we shall show (u−1)λ ⊆ v . let t ∈ (u−1)λ, then γ(t) ≥ u−1(t) > λ. so, t ∈ γλ. hence, (u−1)λ ⊆ v . � theorem 5.9. a function f : (x,t ) → (y,σ) is fuzzy continuous if and only if f : (x,iλ(t )) → (y,iλ(σ)) is continuous for each 0 ≤ λ < 1, where (x,t ), (y,σ) are fuzzy topological spaces. proof. the proof is straightforward. � theorem 5.10. a function f : (p1,t1) → (p2,t2) is fuzzy topological homomorphism if and only if f : (p1, iλ(t1)) → (p2, iλ(t2)) is continuous topological homomorphism for each 0 ≤ λ < 1, where (p1,t1) and (p2,t2) are fuzzy topological polygroups. proof. the proof follows from theorem 5.9. � references [1] n. abbasizadeh and b. davvaz, topological polygroups in the framework on fuzzy sets, j. intell. fuzzy syst 30 (2016), 2811-2820. [2] n. abbasizadeh, b. davvaz and v. leoreanu-fotea, studies on fuzzy topological polygroups, j. intell. fuzzy syst, in press. [3] h. aghabozorgi, b. davvaz and m. jafarpour, solvable polygroups and drived subpolygroups, comm. algebra 41(8) (2013), 3098-3107. [4] c.l. chang, fuzzy topological spaces, j. math. anal. appl 24 (1968), 182-190. [5] i. chon, properties of fuzzy topological groups and semigroups, kangweon-kyungki math. 8 (2000), 103-110. [6] s. d. comer, polygroups derived from cogroups, j. algebra 89 (1984), 397-405. [7] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. [8] b. davvaz and i. cristea, fuzzy algebraic hyperstructures-an introduction, springer, 2015. [9] b. davvaz, polygroup theory and related system, world scientific publishing co. pte. ltd., hackensack, nj, 2013. [10] a. deb ray, on (left) fuzzy topological ring, int. math. forum 6(27) (2011), 1303-1312. [11] a. deb ray, on fuzzy topological ring valued fuzzy continuous functions, appl. math. sci. (ruse) 3 (2009), 11771188. [12] a. deb ray, p. chettri and m. sahu, more on left fuzzy topological rings, annals of fuzzy mathematics and informatics 3 (2012), 321-333. [13] d.h. foster, fuzzy topological groups, j. math. anal. appl. 67 (1979), 549-564. [14] d. n. georgiou, on fuzzy function spaces, j. fuzzy math. 9(1) (2001), 111-126. [15] d. heidari, b. davvaz and s.m.s. modarres, topological hypergroups in the sense of marty, comm. algebra 42 (2014), 4712-4721. [16] d. heidari, b. davvaz and s.m.s. modarres, topological polygroups, bull. malays. math. sci. soc. 39 (2016), 707-721. [17] m. jafarpour, h. aghabozorgi and b. davvaz, on nilpotent and solvable polygroups, bulletin of iranian mathematical society 39(3) (2013), 487-499. [18] g. jagar, on fuzzy function spaces, internal j. math. math sc. 22(4) (1999), 727-737. [19] j.l. ma and c.h yu, fuzzy topological groups, fuzzy sets and systems 12 (1984), 289-299. [20] f. marty, sur une generalization de la notion de groupe, 8iem congress math. scandinaves, stockholm (1934), 45-90. [21] p.m. pu and y.m. liu, fuzzy topology 1, j. math. anal. appl. 76 (1980), 571-599. [22] p.m. pu and y.m. liu, fuzzy topology 2, j. math. anal. appl. 77 (1980), 20-37. [23] a. p. shostak, two decades of fuzzy topology: basic ideas, notions and results, russian math. surveys 44 (1989), 125-186. [24] m.m. zahedi, m. bolurian and a. hasankhani, on polygroups and fuzzy subpolygroups, j. fuzzy math. 3 (1995), 1-15. department of mathematics, yazd university, yazd, iran ∗corresponding author: davvaz@yazd.ac.ir international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 211-221 doi: 10.28924/2291-8639-15-2017-211 new bounds of ostrowski–grüss type inequality for (k + 1) points on time scales eze r. nwaeze1∗, seth kermausuor2 abstract. the aim of this paper is to present three new bounds of the ostrowski–grüss type inequality for points x0,x1,x2, · · · ,xk on time scales. our results generalize result of ngô and liu, and extend results of ujević to time scales with (k + 1) points. we apply our results to the continuous, discrete, and quantum calculus to obtain many new interesting inequalities. an example is also considered. the estimates obtained in this paper will be very useful in numerical integration especially for the continuous case. 1. introduction in 1997, dragomir and wang [6] proved that if f : [a,b] → r is a differentiable function such that there exist constants γ, γ ∈ r with γ ≤ f′(x) ≤ γ for all x ∈ [a,b], then we have∣∣∣∣∣f(x) − 1b−a ∫ b a f(s)ds− f(b) −f(a) b−a ( x− a + b 2 )∣∣∣∣∣ ≤ 14 (b−a)(γ −γ) (1.1) for all x ∈ [a,b]. the above inequality is known in the literature as the ostrowski–grüss type inequality. under the same assumption, cheng [5] obtained the following sharp version of (1.1). ∣∣∣∣∣f(x) − 1b−a ∫ b a f(s)ds− f(b) −f(a) b−a ( x− a + b 2 )∣∣∣∣∣ ≤ 18 (b−a)(γ −γ) (1.2) for all x ∈ [a,b]. in 2003, ujević [20] obtained another estimate of the left part of (1.2) as follows: theorem 1.1. let f : i → r, where i ⊂ r is an interval, be a mapping differentiable in the interior inti of i, and let a,b ∈ inti, a < b. if there exist constants γ, γ ∈ r such that γ ≤ f′(t) ≤ γ for all t ∈ [a,b] and f′ ∈ l1(a,b), then, for all x ∈ [a,b], we have∣∣∣∣f(x) −(x− a + b2 )f(b) −f(a) b−a − 1 b−a ∫ b a f(t) dt ∣∣∣∣ ≤ b−a2 (s −γ) and ∣∣∣∣f(x) −(x− a + b2 )f(b) −f(a) b−a − 1 b−a ∫ b a f(t) dt ∣∣∣∣ ≤ b−a2 (γ −s), where s = f(b)−f(a) b−a . in 2012, feng and meng [7] generalized inequality (1.1) to the case involving (k + 1) points x0,x1, · · · ,xk. their result is contained in the following theorem. received 18th august, 2017; accepted 20th october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 35a23, 26e70, 34n05. key words and phrases. montgomery identity; ostrowski-grüss inequality; parameter; time scales. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 211 212 nwaeze and kermausuor theorem 1.2. let i ⊂ r be an open interval, a,b ∈ i, a < b, f : i → r a differentiable function such that there exist constants γ, γ ∈ r with γ ≤ f′(x) ≤ γ for all x ∈ [a,b]. furthermore, suppose that xi ∈ [a,b], i = 0, 1, 2, · · · ,k, ik : a = x0 < x1 < · · · < xk = b is a division of the interval [a,b] and mi ∈ [xi−1,xi], i = 1, 2, · · · ,k, m0 = a, mk+1 = b. then we have the following inequality∣∣∣∣ 1b−a k∑ i=0 (mi+1 −mi)f(xi) − 1 b−a ∫ b a f(t)dt− f(b) −f(a) (b−a)2 [ b2 −a2 2 − k−1∑ i=0 mi+1(xi+1 −xi) ]∣∣∣∣ ≤ 1 4 (b−a)(γ −γ). to unify the theory of continuous, discrete and quantum calculus, stefan hilger [8] in 1988 came up with the theory of time scales (see section 2 for a brief introduction). ever since, many classical integral inequalities have been extended to time scales; see, for example, the references [4, 9–12, 15–17, 19]. in [13], liu and ngô extended (1.1) to time scales. following thereafter, the same authors in [14] obtained the following theorem which sharpens their earlier result. theorem 1.3. let a,b,s,t ∈ t with a < b and f : [a,b] → r be differentiable. if f∆ is rd-continuous and γ ≤ f∆(t) ≤ γ for all t ∈ [a,b], then we have∣∣∣∣∣f(t) − 1b−a ∫ b a fσ(s)∆s− f(b) −f(a) (b−a)2 [ h2 (t,a) −h2 (t,b) ]∣∣∣∣∣ ≤ γ −γ 2(b−a) ∫ b a ∣∣∣∣p(t,x) − h2(t,a) −h2(t,b)b−a ∣∣∣∣ ∆x, (1.3) where h2(t,s) is given in definition 2.8 and p(t,x) = { x−a, x ∈ [a,t), x− b, x ∈ [t,b]. (1.4) in this paper, we introduce a parameter λ ∈ [0, 1] and then achieve the following goals, viz., (1) firstly, we extend theorem 1.3 to (k + 1) points. our first result provides another estimate for the left hand side of the inequality in theorem 1.2 for the case when λ = 0 and the time scale is the set of real numbers (see remark 4.1). (2) next, we generalize and extend theorem 1.1 to time scales (see remark 3.2). this paper is made up of five sections. in section 2, we lay a quick foundation of the theory of time scales. our main results (theorems 3.1 and 3.2) are then formulated and proved in section 3. thereafter, we then give some applications of our results in section 4 to obtain many new inequalities. finally, a brief conclusion follows in section 5. 2. time scale essentials in this section, we collect basic time scale concepts that will aid in better understanding of this work. for more on this subject, we refer the interested reader to hilger’s ph.d. thesis [8], the books [2, 3], and the survey [1]. definition 2.1. a time scale t is an arbitrary nonempty closed subset of the real numbers. we assume throughout that t has the topology that is inherited from the standard topology on r. it is also assumed throughout that in t the interval [a,b] means the set {t ∈ t: a ≤ t ≤ b} for the points a < b in t. since a time scale may not be connected, we need the following concept of jump operators. definition 2.2. for each t ∈ t, the forward jump operator σ : t → t is definied by σ(t) = inf {s ∈ t: s > t} and the backward jump operator ρ : t → t is defined by ρ(t) = sup{s ∈ t : s < t}. definition 2.3. if σ(t) > t, then we say that t is right-scattered, while if ρ(t) < t then we say that t is left-scattered. points that are right-scattered and left-scattered at the same time are called isolated. if σ(t) = t, then t is called right-dense, and if ρ(t) = t then t is called left-dense. points that are both right-dense and left-dense are called dense. new bounds of ostrowski–grüss type inequality 213 definition 2.4. the mapping µ : t → [0,∞) defined by µ(t) = σ(t)−t is called the graininess function. the set tk is defined as follows: if t has a left-scattered maximum m, then tk = t−{m} ; otherwise, tk = t. if t = r, then µ(t) = 0, and when t = z, we have µ(t) = 1. definition 2.5. let f : t → r and t ∈ tk. then we define f∆(t) to be the number (provided it exists) with the property that for any given � > 0 there exists a neighborhood u of t such that∣∣f(σ(t)) −f(s) −f∆(t) [σ(t) −s]∣∣ ≤ � |σ(t) −s| , ∀s ∈ u. we call f∆(t) the delta derivative of f at t. moreover, we say that f is delta differentiable (or in short: differentiable) on tk provided f∆(t) exists for all t ∈ tk. the function f∆ : tk → r is then called the delta derivative of f on tk. in the case t = r, f∆(t) = df(t) dt . in the case t = z, f∆(t) = ∆f(t) = f(t + 1) −f(t), which is the usual forward difference operator. theorem 2.1. if f,g : t → r are differentiable at t ∈ tk, then the product fg : t → r is differentiable at t and (fg) ∆ (t) = f∆(t)g(t) + f(σ(t))g∆(t). definition 2.6. the function f : t → r is said to be rd-continuous on t provided it is continuous at all right-dense points t ∈ t and its left-sided limits exist at all left-dense points t ∈ t. the set of all rd-continuous function f : t → r is denoted by crd(t,r). also, the set of functions f : t → r that are differentiable and whose derivative is rd-continuous is denoted by c1rd(t,r). it follows from [2, theorem 1.74] that every rd-continuous function has an anti-derivative. definition 2.7. let f : t → r be a function. then f : t → r is called the anti-derivative of f on t if it satisfies f ∆(t) = f(t) for any t ∈ tk. in this case, the cauchy integral b∫ a f(t)∆t = f(b) −f(a), a,b ∈ t. theorem 2.2. let f,g ∈ crd(t,r), a,b,c ∈ t and α,β ∈ r. then (1) b∫ a [αf(t) + βg(t)] ∆t = α b∫ a f(t)∆t + β b∫ a g(t)∆t. (2) b∫ a f(t)∆t = − a∫ b f(t)∆t. (3) b∫ a f(t)∆t = c∫ a f(t)∆t + b∫ c f(t)∆t. (4) b∫ a f(t)g∆(t)∆t = (fg) (b) − (fg) (a) − b∫ a f∆(t)g(σ(t))∆t. (5) if |f(t)| ≤ g(t) on [a,b], then ∣∣∣∣∣∣ b∫ a f(t)∆t ∣∣∣∣∣∣ ≤ b∫ a g(t)∆t. definition 2.8. let hk,gk : t2 → r , k ∈ n0 be defined by h0(t,s) := g0(t,s) := 1, for all s,t ∈ t and then recursively by gk+1 (t,s) = t∫ s gk (σ (τ) ,s) ∆τ, hk+1 (t,s) = t∫ s hk (τ,s) ∆τ, for all s,t ∈ t. in view of the above definition, we make the following remarks that will come handy in the sequel. – for t = r, h2(t,s) = (t−s)2 2 . – for t = z, h2(t,s) = (t−s)(t−s−1) 2 . 214 nwaeze and kermausuor 3. main results in order to prove our results, we will need the following lemmas. the first lemma is given in [18, 21] but with some typos. we present here the correct version. lemma 3.1 (generalized montgomery identity with a parameter). suppose that (1) a,b ∈ t, λ ∈ [0, 1], ik : a = x0 < x1 < · · · < xk−1 < xk = b is a partition of the interval [a,b] for x0,x1, · · · ,xk ∈ t, (2) αi ∈ t (i = 0, 1, · · · ,k + 1) is k + 2 points so that α0 = a, αi ∈ [xi−1,xi] (i = 1, · · · ,k) and αk+1 = b, (3) f : [a,b] → r is a differentiable function. then we have the following equation ∫ b a k(t,ik)f ∆(t)∆t + ∫ b a fσ(t)∆t = (1 −λ) k∑ i=0 ( αi+1 −αi ) f(xi) + λ k∑ i=0 ( αi+1 −αi )f(αi) + f(αi+1) 2 , (3.1) where k(t,ik) =   t− ( α1 −λα1−a2 ) , t ∈ [a,α1), t− ( α1 + λ α2−α1 2 ) , t ∈ [α1,x1), t− ( α2 −λα2−α12 ) , t ∈ [x1,α2), ... t− ( αk−1 + λ αk−αk−1 2 ) , t ∈ [αk−1,xk−1), t− ( αk −λ αk−αk−1 2 ) , t ∈ [xk−1,αk), t− ( αk + λ αk+1−αk 2 ) , t ∈ [αk,b], (3.2) provided for each i ∈{0, 1, 2, . . . ,k − 1}, αi+1 −λ αi+1−αi 2 and αi+1 + λ αi+2−αi+1 2 belong to t. lemma 3.2 ( [14]). let a,b,x ∈ t, f,g ∈ crd and f,g : [a,b] → r with γ ≤ g(x) ≤ γ for all x ∈ [a,b] and for some γ, γ ∈ r. then we have ∣∣∣∣ ∫ b a f(t)g(t)∆t− 1 b−a ∫ b a f(t)∆t ∫ b a g(t)∆t ∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣f(t) − 1b−a ∫ b a f(s)∆s ∣∣∣∣∆t. (3.3) moreover, the inequality in (3.3) is sharp. we now state and justify our first result. new bounds of ostrowski–grüss type inequality 215 theorem 3.1. suppose f satisfies the conditions of lemma 3.1. if, in addition, f∆ ∈ crd with γ ≤ f∆(t) ≤ γ for all t ∈ [a,b] and some γ, γ ∈ r, then we have the inequality ∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a fσ(t)∆t − f(b) −f(a) b−a k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 1b−a k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣∆t, (3.4) provided for each i ∈ {0, 1, 2, . . . ,k − 1}, αi+1 − λ αi+1−αi 2 and αi+1 + λ αi+2−αi+1 2 belong to t. the inequality in (3.4) is sharp in the sense that the constant 1/2 cannot be replaced by a smaller one. proof. by applying lemma 3.2 to the functions f(t) := k(t,ik) and g(t) = f ∆(t), we have∣∣∣∣ ∫ b a k(t,ik)f ∆(t)∆t− 1 b−a ∫ b a k(t,ik)∆t ∫ b a f∆(t)∆t ∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 1b−a ∫ b a k(s,ik)∆s ∣∣∣∣∆t. (3.5) now, we observe that ∫ b a f∆(t)∆t = f(b) −f(a), (3.6) and (by applying the items of theorem 2.2 and definition 2.8) ∫ b a k(t,ik)∆t = k−1∑ i=0 ∫ xi+1 xi k(t,ik)∆t = k−1∑ i=0 [∫ αi+1 xi ( t− ( αi+1 −λ αi+1 −αi 2 )) ∆t + ∫ xi+1 αi+1 ( t− ( αi+1 + λ αi+2 −αi+1 2 )) ∆t ] = k−1∑ i=0 [∫ αi+1−λαi+1−αi2 xi ( t− ( αi+1 −λ αi+1 −αi 2 )) ∆t + ∫ αi+1 αi+1−λ αi+1−αi 2 ( t− ( αi+1 −λ αi+1 −αi 2 )) ∆t + ∫ αi+1+λαi+2−αi+12 αi+1 ( t− ( αi+1 + λ αi+2 −αi+1 2 )) ∆t + ∫ xi+1 αi+1+λ αi+2−αi+1 2 ( t− ( αi+1 + λ αi+2 −αi+1 2 )) ∆t ] 216 nwaeze and kermausuor = k−1∑ i=0 [ − ∫ xi αi+1−λ αi+1−αi 2 ( t− ( αi+1 −λ αi+1 −αi 2 )) ∆t (3.7) + ∫ αi+1 αi+1−λ αi+1−αi 2 ( t− ( αi+1 −λ αi+1 −αi 2 )) ∆t − ∫ αi+1 αi+1+λ αi+2−αi+1 2 ( t− ( αi+1 + λ αi+2 −αi+1 2 )) ∆t + ∫ xi+1 αi+1+λ αi+2−αi+1 2 ( t− ( αi+1 + λ αi+2 −αi+1 2 )) ∆t ] = k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )] . (3.8) now using lemma 3.1, we get∫ b a k(t,ik)f ∆(t)∆t = (1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a fσ(t)∆t. (3.9) by substituting equations (3.6), (3.7) and (3.9) into (3.5), we obtain∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a fσ(t)∆t − f(b) −f(a) b−a k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 1b−a k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣∆t. (3.10) hence, the proof is complete. � remark 3.1. if we take λ = 0, k = 2, and x1 = x, α0 = α1 = a, α2 = α3 = x2 = b in theorem 3.1, then we recapture theorem 1.3. next, we provide another bound for (3.4). theorem 3.2. under the conditions of theorem 3.1, we obtain the following inequalities∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a fσ(t)∆t − f(b) −f(a) b−a k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( xi,αi+1 −λ αi+1 −αi 2 ) + h2 ( xi+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣ ≤ { mk(b−a)(s −γ) mk(b−a)(γ −s), new bounds of ostrowski–grüss type inequality 217 where s = f(b)−f(a) b−a , and mk = max t∈[a,b] ∣∣∣k(t,ik) − 1b−a ∫ ba k(s,ik)∆s∣∣∣. proof. we start by observing that∫ b a [ k(t,ik) − 1 b−a ∫ b a k(s,ik)∆s ] ∆t = 0. (3.11) using (3.11), we get that for any c ∈ r,∫ b a k(t,ik)f ∆(t)∆t− 1 b−a ∫ b a k(t,ik)∆t ∫ b a f∆(t)∆t = ∫ b a ( f∆(t) −c )[ k(t,ik) − 1 b−a ∫ b a k(s,ik)∆s ] ∆t. (3.12) for c = γ, and taking absolute values of both sides of (3.12), we have by using (3.6)∣∣∣∣ ∫ b a k(t,ik)f ∆(t)∆t− 1 b−a ∫ b a k(t,ik)∆t ∫ b a f∆(t)∆t ∣∣∣∣ ≤ ∫ b a ∣∣f∆(t) −γ∣∣∣∣∣∣k(t,ik) − 1b−a ∫ b a k(s,ik)∆s ∣∣∣∣∆t ≤ max t∈[a,b] ∣∣∣∣k(t,ik) − 1b−a ∫ b a k(s,ik)∆s ∣∣∣∣ ∫ b a ∣∣f∆(t) −γ∣∣∆t = mk ∫ b a ( f∆(t) −γ ) ∆t = mk [f(b) −f(a) b−a −γ ] (b−a). (3.13) similarly, for c = γ, we get∣∣∣∣ ∫ b a k(t,ik)f ∆(t)∆t− 1 b−a ∫ b a k(t,ik)∆t ∫ b a f∆(t)∆t ∣∣∣∣ ≤ mk [ γ − f(b) −f(a) b−a ] (b−a). (3.14) the intended inequalities follow from lemma 3.1 and relations (3.13) and (3.14). � remark 3.2. if we take t = r, λ = 0, k = 2, and x1 = x, α0 = α1 = a, α2 = α3 = x2 = b in theorem 3.2, then we get theorem 1.1. 4. applications in this section, we apply our theorems to the continuous, discrete, and quantum calculus to obtain the following results. corollary 4.1 (continuous case). let t = r in theorem 3.1. then we have the inequality∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a f(t)dt − f(b) −f(a) 8(b−a) k−1∑ i=0 [ λ2 ( αi+1 −αi )2 − ( 2xi −λαi + (λ− 2)αi+1 )2 + ( 2xi+1 −λαi+2 + (λ− 2)αi+1 )2 −λ2 ( αi+2 −αi+1 )2]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 18(b−a) k−1∑ i=0 [ λ2 ( αi+1 −αi )2 − ( 2xi −λαi + (λ− 2)αi+1 )2 + ( 2xi+1 −λαi+2 + (λ− 2)αi+1 )2 −λ2 ( αi+2 −αi+1 )2]∣∣∣∣dt. (4.1) 218 nwaeze and kermausuor applying corollary 4.1 to different values of λ and k, we obtain some novel inequalities. we present here some of these new results. remark 4.1. if we take λ = 0 in corollary 4.1, we get∣∣∣∣ k∑ i=0 (αi+1 −αi)f(xi) − ∫ b a f(t)dt− f(b) −f(a) b−a [ b2 −a2 2 − k−1∑ i=0 αi+1(xi+1 −xi) ]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 1b−a [ b2 −a2 2 − k−1∑ i=0 αi+1(xi+1 −xi) ]∣∣∣∣dt, (4.2) where k(t,ik) =   t−α1, t ∈ [a,x1), t−α2, t ∈ [x1,x2), ... t−αk−1, t ∈ [xk−2,xk−1), t−αk, t ∈ [xk−1,b]. (4.3) the above inequality is new and sharp. this gives a new estimate for the left hand side of the inequality in theorem 1.2. furthermore, let k = 2 in corollary 4.1. if in addition, one then sets x1 = x, α0 = α1 = a, α2 = α3 = x2 = b in the resulting inequality, then one gets that for all x ∈ [a,b] the following inequality holds:∣∣∣∣(1 −λ)(b−a)f(x) + λ(b−a)f(a) + f(b)2 − ∫ b a f(t)dt − f(b) −f(a) 8(b−a) [( 2x−λb + (λ− 2)a )2 − ( 2x−λa + (λ− 2)b )2]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,x) − 18(b−a) [( 2x−λb + (λ− 2)a )2 − ( 2x−λa + (λ− 2)b )2]∣∣∣∣dt. (4.4) remark 4.2. for λ = 0 in inequality (4.4), we have the inequality∣∣∣∣(b−a)f(x) −(f(b) −f(a))(x− a + b2 ) − ∫ b a f(t)dt ∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,x) −(x− a + b2 )∣∣∣∣dt, (4.5) where k(t,x) = { t−a, t ∈ [a,x) t− b, t ∈ [x,b]. it is important to note here that inequality (4.5) is sharper than (1.1) since max t∈[a,b] ∣∣∣∣k(t,x) −(x− a + b2 )∣∣∣∣ = b−a2 . remark 4.3. next, we consider the case when λ = 1 in inequality (4.4). for this, we obtain∣∣∣f(a) + f(b) 2 − 1 b−a ∫ b a f(t) dt ∣∣∣ ≤ γ −γ 8(b−a) [ (b−a)2 + (2x−a− b)2 ] (4.6) for all x ∈ [a,b]. applying the above inequality to the following example. new bounds of ostrowski–grüss type inequality 219 example 4.1. consider the function f : [0, 1] → r+ defined by f(x) = ex 2 . we know that the integral of f cannot be achieved via an analytic method; but we can approximate it using numerical methods. for this function, we observe that 0 ≤ f′(x) ≤ 6 for all x ∈ [0, 1]. choose γ = 0 and γ = 6. now, using (4.6), on gets ∣∣∣∫ 1 0 et 2 dt− e + 1 2 ∣∣∣ ≤ 3 2 (2x2 − 2x + 1) for all x ∈ [0, 1]. in particular, for x = 0 or 1, we have e 2 − 1 ≤ ∫ 1 0 et 2 dt ≤ e 2 + 2. using matlab, one can verify that ∫ 1 0 et 2 dt ≈ 1.46265. this shows that the range given above is correct! corollary 4.2 (discrete case). let t = z in theorem 3.1. suppose a = 0,b = n and (1) ik := {j0,j1, · · · ,jk}⊂ z, where a = j0 < j1 < · · · < jk = b, is a partition of the set [0,n] ∩z (2) {α0,α1, · · · ,αk+1}⊂ z is a set of k+2 points such that α0 = 0,αi ∈ [ji−1,ji] for i = 1, 2, · · · ,k and αk+1 = n; (3) f(k) = xk. we have the inequality, ∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)xji + λ k∑ i=0 (αi+1 −αi) xαi + xαi+1 2 − n∑ j=1 xj − xn −x0 n k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( ji,αi+1 −λ αi+1 −αi 2 ) + h2 ( ji+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣ ≤ γ −γ 2 n−1∑ j=0 ∣∣∣∣k(j,ik) − 1n k−1∑ i=0 [ h2 ( αi+1,αi+1 −λ αi+1 −αi 2 ) −h2 ( ji,αi+1 −λ αi+1 −αi 2 ) + h2 ( ji+1,αi+1 + λ αi+2 −αi+1 2 ) −h2 ( αi+1,αi+1 + λ αi+2 −αi+1 2 )]∣∣∣∣, (4.7) where h2(t,s) = ( t−s 2 ) = (t−s)(t−s−1) 2 for all t,s ∈ z. corollary 4.3 (quantum case). let t = qn0,q > 1,a = qm,b = qn with m,n ∈ n and m < n. suppose that (1) ik : qm = qj0 < qj1 < · · · < qjk = qn, is a partition of the set [qm,qn]∩qn0 for j0,j1, · · · ,jk ∈ n; (2) qαi ∈ qn0 (i = 0, 1, · · · ,k+1) is a set of k+2 points such that qα0 = qm,qαi ∈ [qji−1,qji]∩qn0 (i = 1, 2, · · · ,k) and qαk+1 = qn; (3) f : [qm,qn] → r is differentiable. 220 nwaeze and kermausuor then we have the inequality,∣∣∣∣(1 −λ) k∑ i=0 (qαi+1 −qαi)f(qji) + λ k∑ i=0 (qαi+1 −qαi) f(qαi) + f(qαi+1 ) 2 − ∫ qn qm f(qt)dqt − f(qn) −f(qm) qn −qm k−1∑ i=0 [ h2 ( qαi+1,qαi+1 −λ qαi+1 −qαi 2 ) −h2 ( qji,qαi+1 −λ qαi+1 −qαi 2 ) + h2 ( qji+1,qαi+1 + λ qαi+2 −qαi+1 2 ) −h2 ( qαi+1,qαi+1 + λ qαi+2 −qαi+1 2 )]∣∣∣∣ ≤ γ −γ 2 ∫ b a ∣∣∣∣k(t,ik) − 1qn −qm k−1∑ i=0 [ h2 ( qαi+1,qαi+1 −λ qαi+1 −qαi 2 ) −h2 ( qji,qαi+1 −λ qαi+1 −qαi 2 ) + h2 ( qji+1,qαi+1 + λ qαi+2 −qαi+1 2 ) −h2 ( qαi+1,qαi+1 + λ qαi+2 −qαi+1 2 )]∣∣∣∣dqt, where h2(t,s) = (t−s)(t−qs) q+1 for all t,s ∈ qn0. we close this section by applying theorem 3.2 to the continuous calculus. corollary 4.4 (continuous case). let t = r. then we have the inequalities∣∣∣∣(1 −λ) k∑ i=0 (αi+1 −αi)f(xi) + λ k∑ i=0 (αi+1 −αi) f(αi) + f(αi+1) 2 − ∫ b a f(t)dt − f(b) −f(a) 8(b−a) k−1∑ i=0 [ λ2 ( αi+1 −αi )2 − ( 2xi −λαi + (λ− 2)αi+1 )2 + ( 2xi+1 −λαi+2 + (λ− 2)αi+1 )2 −λ2 ( αi+2 −αi+1 )2]∣∣∣∣ ≤ { mk(b−a)(s −γ) mk(b−a)(γ −s), where s = f(b)−f(a) b−a , and mk = max t∈[a,b] ∣∣∣k(t,ik) − 1b−a ∫ ba k(s,ik)ds∣∣∣. remark 4.4. by setting λ = 0 in corollary 4.4, we get a direct generalization of theorem 1.1 to (k + 1) points x0,x1, · · · ,xk. in fact, we obtain∣∣∣∣ k∑ i=0 (αi+1 −αi)f(xi) − ∫ b a f(t)dt− f(b) −f(a) b−a [ b2 −a2 2 − k−1∑ i=0 αi+1(xi+1 −xi) ]∣∣∣∣ ≤ { mk(b−a)(s −γ) mk(b−a)(γ −s), (4.8) where k(t,ik) is given by (4.3). 5. conclusion we have established three new ostrowski–grüss type inequality with a parameter λ ∈ [0, 1]. loads of interesting results can be derived by choosing different values of k ∈ n, and λ’s. as an application, we considered the continuous, discrete, and quantum calculus from which many novel inequalities are obtained. references [1] r. agarwal, m. bohner and a. peterson, inequalities on time scales: a survey, math. inequal. appl. 4 (2001), 535–557. [2] m. bohner and a. peterson, dynamic equations on time scales, boston (ma): birkhäuser boston, 2001. [3] m. bohner and a. peterson, advances in dynamic equations on time series, boston (ma): birkhäuser boston, 2003. new bounds of ostrowski–grüss type inequality 221 [4] m. bohner and t. matthews, ostrowski inequalities on time scales, j. inequal. pure appl. math. 9 (2008), art. 6. [5] x. l. cheng, improvement of some ostrowski-griiss type inequalities, compu. math. appli. 42 (2001), 109–114. [6] s. s. dragomir and s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, compu. math. appli. 33(11) (1997), 15–20. [7] q. feng and f. meng, some generalized ostrowski–grüss type integral inequalities, compu. math. appli. 63 (2012), 652–659. [8] s. hilger, ein maβkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten (ph.d. thesis). würzburg (germany): universität würzburg, 1988. [9] b. karpuz and u. m. özkan, ostrowski inequality on time scales, j. inequal. pure and appl. math. 9(4) (2008), art. 112. [10] s. kermausuor, e. r. nwaeze and d. f. m. torres, generalized weighted ostrowski and ostrowski–grüss type inequalities on time scale via a parameter function, j. math. inequal. 11 (4) (2017), 1185–1199. [11] w. j. liu and q. a. ngô, a generalization of ostrowski inequality on time scales for k points, appl. math. comput. 203(2) (2008), 754–760. [12] w. j. liu and q. a. ngô, chen wb. a perturbed ostrowski-type inequality on time scales for k points for functions whose second derivatives are bounded, j. inequal. appl. (2008) art. id 597241. [13] w. j. liu and q. a. ngô, an ostrowski–grüss type inequality on time scales, arxiv:0804.3231. [14] q. a. ngô and w. j. liu, a sharp grüss type inequality on time scales and application to the sharp ostrowski-grüss inequality, commun. math. anal. 6(2) (2009), 33–41. [15] e. r. nwaeze, a new weighted ostrowski type inequality on arbitrary time scale, j. king saud uni. sci. 29(2) (2017), 230–234. [16] e. r. nwaeze, generalized weighted trapezoid and grüss type inequalities on time scales, aust. j. math. anal. appl. 11(1) 2017, art. 4. [17] e. r. nwaeze and a. m. tameru, on weighted montgomery identity for k points and its associates on time scales, abstr. app. anal. (2017), art. id 5234181. [18] e. r. nwaeze, s. kermausuor and a. m. tameru, new time scale generalizations of the ostrowski–grüss type inequality for k points, j. inequal. appl. 2017 (2017), art. id 245. [19] a. tuna and d. daghan, generalization of ostrowski and ostrowski–grüss type inequalities on time scales, comput. math. appl. 60 (2010), 803–811. [20] n. ujević, new bounds for the first inequality of ostrowski–grüss type and applications, compu. math. appl. 46 (2003), 421–427. [21] g. xu and z. b. fang, a generalization of ostrowski type inequality on time scales with k points, j. math. inequal. 11(1) (2017), 41–48. 1department of mathematics, tuskegee university,, tuskegee, al 36088, usa 2department of mathematics and computer science, alabama state university, montgomery, al 36104, usa ∗corresponding author: enwaeze@tuskegee.edu 1. introduction 2. time scale essentials 3. main results 4. applications 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 71-79 http://www.etamaths.com new weighted ostrowski type inequalities for mappings whose nth derivatives are of bounded variation huseyin budak1, samet erden2,∗ and m. zeki sarikaya1 abstract. we establish a new generalization of weighted ostrowski type inequality for mappings of bounded variation. spacial cases of this inequality reduce some well known inequalities. with the help of obtained inequality, we give applications for the kth moment of random variables. 1. introduction in 1938, ostrowski established the integral inequality which is one of the fundemental inequalitıes of mathematic as follows (see, [22]): let f : [a,b]→ r be a differentiable mapping on (a,b) whose derivative f ′ : (a,b)→ r is bounded on (a,b), i.e., ‖f′‖∞ = sup t∈(a,b) |f′(t)| < ∞. then, the inequality holds: (1.1) ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ for all x ∈ [a,b]. the constant 1 4 is the best possible. this inequality is well known in the literature as the ostrowski inequality. the inequality (1.1) has attracted remarkable attention from mathematicians and researchers. because of this, over the years researchers have devoted much time and effort to the improvement and generalization of (1.1) for several functions (bounded function, function of bounded variation, etc.). firstly, we start introducing concept of bounded variation: definition 1. let p : a = x0 < x1 < ... < xn = b be any partition of [a,b] and let ∆f(xi) = f(xi+1) −f(xi) then f(x) is said to be of bounded variation if the sum n∑ i=1 |∆f(xi)| is bounded for all such partitions. let f be of bounded variation on [a,b], and ∑ (p) denotes the sum n∑ i=1 |∆f(xi)| corresponding to the partition p of [a,b]. the number b∨ a (f) := sup {∑ (p) : p ∈ p ([a,b]) } , is called the total variation of f on [a,b] . here p ([a,b]) denotes the family of partitions of [a,b] . a similar result (1.1) is obtained by dragomir in [14] for functions of bounded variation as follow: 2010 mathematics subject classification. 26d07, 26d15, 26a45. key words and phrases. ostrowski inequality; function of bounded variation; riemann-stieltjes integral; random variable. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 71 72 budak, erden and sarikaya theorem 1. let f : [a,b] → r be a mapping of bounded variation on [a,b] . then (1.2) ∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) f(x) ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f) holds for all x ∈ [a,b] . the constant 1 2 is the best possible. for recent new results regarding ostrowski’s type inequalities for functions of bounded variation see [3],[7],[9]-[11], [13]-[19], [21], [25]. in [20], liu proved the following weighted ostrowski type inequality for functions of bounded variation: theorem 2. let f : [a,b] → r be a mapping of bounded variation, g : [a,b] → (0,∞) continious and positive mapping on (a,b) . then for any x ∈ [a,b] and α ∈ [0, 1] we have∣∣∣∣∣∣(1 −α)   b∫ a g(u)du  f(x)(1.3) +α     x∫ a g(u)du  f(a) +   b∫ x g(u)du  f(b)  − b∫ a f(t)g(t)dt ∣∣∣∣∣∣ ≤ [ 1 2 + ∣∣∣∣12 −α ∣∣∣∣ ]1 2 b∫ a g(u)du + ∣∣∣∣∣∣ x∫ a g(u)du− 1 2 b∫ a g(u)du ∣∣∣∣∣∣   b∨ a (f) where b∨ a (f) is the total variation of f on the interval [a,b] . the constant [ 1 2 + ∣∣1 2 −α ∣∣] is the best possible. in [5], budak and sarıkaya gave the following weighted ostrowski’s type inequalities for mapping of bounded variation. theorem 3. let in : a = x0 < x1 < ... < xn = b be a division of the interval [a,b] and αi (i = 0, 1, ...,n + 1) be n+ 2 points so that α0 = a, αi ∈ [xi−1,xi] (i = 1, ...,n) , αn+1 = b. if f : [a,b] → r is of bounded variation on [a,b] and w : [a,b] → (0,∞) be continious and positive mapping on (a,b) , then we have the inequalities:∣∣∣∣∣∣ n∑ i=0   αi+1∫ αi w(u)du  f(xi) − b∫ a f(t)w(t)dt ∣∣∣∣∣∣(1.4) ≤  1 2 v(l) + max i∈{0,1,...,n−1} 1 2 ∣∣∣∣∣∣ αi+1∫ xi w(u)du− xi+1∫ αi+1 w(u)du ∣∣∣∣∣∣   b∨ a (f) ≤ v(l) b∨ a (f) where υ(l) := max{li| i = 0, ...,n− 1} , li = xi+1∫ xi w(u)du (i = 0, 1, ...,n− 1) and b∨ a (f) is the total variation of f on the interval [a,b] . a weighted generalization of trapezoid inequality for mappings of bounded variation was considered by tseng et. al. [24]. recently, researchers gave some weigted ostrowski type inequalities for functions of bounded variation in [5], [8], [26]. in [1] and [2], the authors proved some generalizations of weighted companion of ostrowski type inequality for mappings of bounded variation new weighted ostrowski type inequalities 73 in this paper, we establish a generalized weighted ostrowski type integral inequality for mappings whose nth derivatives are of bounded variation. then,we recapture some results given in earlier works by using this inequalities. finally, some applications for the kth moment are given. 2. main results in order to prove weighted integral inequalities, we need the following lemma: lemma 1. let f : i ⊂ r → r be n + 1 times differentiable function on i◦, a,b ∈ i◦ with a < b and let w : [a,b] → r be nonnegative and continuous on [a,b]. then the following equality holds: (2.1) n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt = b∫ a pw (x,t) df (n) (t) where n ∈ n, mk(x) is defined by mk(x) = b∫ a (u−x)k w (u) du, k = 0, 1, 2, ... and (2.2) pn (x,t) :=   1 n! t∫ a (u− t)n w (u) du, a ≤ t < x 1 n! t∫ b (u− t)n w (u) du, x ≤ t ≤ b. proof. using the integration by parts in riemann-stieltjes integral, we have b∫ a pn (x,t) df (n) (t) = 1 n! x∫ a   t∫ a (u− t)n w (u) du  df(n) (t) + 1 n! b∫ x   t∫ b (u− t)n w (u) du  df(n) (t) = 1 n!   b∫ a (u−x)n w (u) du  f(n) (x) + b∫ a pn−1 (x,t) f (n) (t) dt by integration by parts n−times, we get b∫ a pn−1 (x,t) f (n) (t) dt = mn−1(x) (n− 1)! f(n−1) (x) + ... + m2(x) 2! f′′ (x) +m1(x)f ′ (x) + m0(x)f (x) − b∫ a w (t) f (t) dt which completes the proof. � now, we deduce generalized weighted inequality of ostrowski type for mappings whose nth derivatives are of bounded variation.. 74 budak, erden and sarikaya theorem 4. suppose that all the assumptions of lemma 1 hold. additionally, we assume that f(n) is of bounded variation on [a,b], then we have the inequality∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!  1 2   x∫ a (x−u)n w (u) du + b∫ x (u−x)n w (u) du   + 1 2 ∣∣∣∣∣∣ x∫ a (x−u)n w (u) du− b∫ x (u−x)n w (u) du ∣∣∣∣∣∣   b∨ a (f(n)) for all x ∈ [a,b] . proof. if we take absolute value of both sides of the equality (2.1), we get∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!   ∣∣∣∣∣∣ x∫ a t∫ a (t−u)n w (u) dudf(n) (t) ∣∣∣∣∣∣ + 1n! ∣∣∣∣∣∣ b∫ x b∫ t (u− t)n w (u) dudf(n) (t) ∣∣∣∣∣∣   . it is well known that if g,f : [a,b] → r are such that g is continuous on [a,b] and f is of bounded variation on [a,b] , then b∫ a g(t)df(t) exists and (2.3) ∣∣∣∣∣∣ b∫ a g(t)df(t) ∣∣∣∣∣∣ ≤ supt∈[a,b] |g(t)| b∨ a (f). on the other hand, by using (2.3), we obtain∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!   sup t∈[a,x] ∣∣∣∣∣∣ t∫ a (t−u)n w (u) du ∣∣∣∣∣∣ x∨ a (fn) + sup t∈[x,b] ∣∣∣∣∣∣ b∫ t (u− t)n w (u) du ∣∣∣∣∣∣ b∨ x (f(n))   = 1 n!   x∫ a (x−u)n w (u) du   x∨ a (fn) + 1 n!   b∫ x (u−x)n w (u) du   b∨ x (f(n)) ≤ 1 n!  1 2   x∫ a (x−u)n w (u) du + b∫ x (u−x)n w (u) du   + 1 2 ∣∣∣∣∣∣ x∫ a (x−u)n w (u) du− b∫ x (u−x)n w (u) du ∣∣∣∣∣∣   b∨ a (f(n)). this completes the proof. � remark 1. if we take w(u) = 1 and n = 0 in theorem 4, then we get the clasical ostrowski inequality (1.2) for function of bounded variation . new weighted ostrowski type inequalities 75 remark 2. if we choose n = 1 in theorem 4, then we obtain∣∣∣∣∣∣   b∫ a (x−u) w (u) du  f′ (x) +   b∫ a w (u) du  f (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤  1 2   x∫ a (x−u) w (u) du + b∫ x (u−x) w (u) du   + 1 2 ∣∣∣∣∣∣ x∫ a (x−u) w (u) du− b∫ x (u−x) w (u) du ∣∣∣∣∣∣   b∨ a (f′) which was given by budak and sarikaya in [6]. remark 3. if we choose n = 0 in theorem 4, then we have the inequality∣∣∣∣∣∣   b∫ a w (t) dt  f (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤  1 2 b∫ a w (t) dt + ∣∣∣∣∣∣ x∫ a w(t)dt− 1 2 b∫ a w (t) dt ∣∣∣∣∣∣   b∨ a (f) which was proved by liu. in [20]. corollary 1. with the assumptions as in theorem 4, we have the result∣∣∣∣∣∣ n∑ k=0 (b−x)k+1 − (a−x)k+1 (k + 1)! f(k) (x) − b∫ a f (t) dt ∣∣∣∣∣∣(2.4) ≤ 1 (n + 1)! [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ]n+1 b∨ a (f(n)) proof. the proof is obvious from the property of maximum max{an,bn} = (max{a,b})n for a,b > 0, n ∈ n, if we take w(u) = 1. � remark 4. if we choose n = 1 in corollary 1, we have the inequality∣∣∣∣∣∣ ( a + b 2 −x ) f′(x) + f(x) − 1 b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ b−a 2 [ 1 2 + ∣∣∣∣∣x− a+b 2 b−a ∣∣∣∣∣ ]2 b∨ a (f′) which was given by budak and sarikaya in [6]. corollary 2. in (2.4), if we choose, i) x = a+b 2 , then we have (2.5) ∣∣∣∣∣∣ n∑ k=0 (b−a)k+1 [ 1 + (−1)k ] 2k+1 (k + 1)! f(k) ( a + b 2 ) − b∫ a f (t) dt ∣∣∣∣∣∣ ≤ (b−a) n+1 2n+1 (n + 1)! b∨ a (f(n)), ii) x = a, then we have∣∣∣∣∣∣ n∑ k=0 (b−a)k+1 (k + 1)! f(k) (a) − b∫ a f (t) dt ∣∣∣∣∣∣ ≤ (b−a) n+1 (n + 1)! b∨ a (f(n)), 76 budak, erden and sarikaya iii) x = b, then we have∣∣∣∣∣∣ n∑ k=0 (−1)k(b−a)k+1 (k + 1)! f(k) (b) − b∫ a f (t) dt ∣∣∣∣∣∣ ≤ (b−a) n+1 (n + 1)! b∨ a (f(n)). remark 5. if we choose n = 1 in (2.5), then we have the inequalities∣∣∣∣∣∣f ( a + b 2 ) − 1 b−a b∫ a f (t) dt ∣∣∣∣∣∣ ≤ b−a8 b∨ a (f′), which was given by liu in [21]. corollary 3. under the assumption of theorem 4. suppose that f ∈ cn+1 [a,b] , then we have∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!  1 2   x∫ a (x−u)n w (u) du + b∫ x (u−x)n w (u) du   1 2 ∣∣∣∣∣∣ x∫ a (x−u)n w (u) du− b∫ x (u−x)n w (u) du ∣∣∣∣∣∣  ∥∥∥f(n+1)∥∥∥ 1 . here as subsequently ‖.‖1 is the l1−norm∥∥∥f(n+1)∥∥∥ 1 := b∫ a f(n+1)(t)dt. corollary 4. under the assumption of theorem 4. let f(n) be a lipschitzian with the constant l > 0. then the inequality holds:∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!  1 2   x∫ a (x−u)n w (u) du + b∫ x (u−x)n w (u) du   1 2 ∣∣∣∣∣∣ x∫ a (x−u)n w (u) du− b∫ x (u−x)n w (u) du ∣∣∣∣∣∣   (b−a)l corollary 5. under the assumption of theorem 4. let f(n) be a monotone mapping on [a,b] . then we have ∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!  1 2   x∫ a (x−u)n w (u) du + b∫ x (u−x)n w (u) du   1 2 ∣∣∣∣∣∣ x∫ a (x−u)n w (u) du− b∫ x (u−x)n w (u) du ∣∣∣∣∣∣  [f(n)(b) −f(n)(a)] . new weighted ostrowski type inequalities 77 3. some applications for the moments we now deal with applications of the result developed in the previous section, to obtain some new inequalities involving moments. applying the mathematical inequalities, some estimations for the moments of random variables were recently studied (see, [4],[12],[18] and [23]). set x to denote a random variable whose probability density function is w : [a,b] → [0,∞) on the interval of real numbers i (a,b ∈ i, a < b). denoted by mr(x) the rth central moment of the random variable x, defined as mr(x) = b∫ a (u−e(x))r w (u) du, r = 0, 1, 2, ... where e(x) is the mean of the random variables x. it may be noted that m0(x) = 1, m1(x) = 0, m2(x) = σ 2(x) where σ2(x) is the variance of the random variables x. now, we reconsider the identity (3.1) by changing conditions given in lemma 1. herewith, we deduce an identity involving rth moment. lemma 2. let f : i ⊂ r → r be n + 1 times differentiable function on i◦, a,b ∈ i◦ with a < b and and let x be a random variable whose p.d.f. is w : [a,b] → [0,∞). then the following equality holds: (3.1) n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt = b∫ a pw (x,t) df (n) (t) where n ∈ n, mk(x) is the kth moment, and pn (x,t) is defined as in (2.2). theorem 5. suppose that all the assumptions of lemma 2 hold. if f(n) is of bounded variation on [a,b], then we have the inequality ∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n! [ b−a 2 + ∣∣∣∣x− a + b2 ∣∣∣∣ ]n b∨ a (f(n)) for all x ∈ [a,b] . proof. by similar methods in the proof of theorem 4, we obtain ∣∣∣∣∣∣ n∑ k=0 mk(x) k! f(k) (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ 1 n!   x∫ a (x−u)n w (u) du   x∨ a (fn) + 1 n!   b∫ x (u−x)n w (u) du   b∨ x (f(n)) ≤   x∫ a (x−u)n n! w (u) du + b∫ x (u−x)n n! w (u) du   b∨ a (fn) 78 budak, erden and sarikaya we observe that x∫ a (x−u)n n! w(u)du + b∫ x (u−x)n n! w(u)du ≤ 1 n!   sup u∈[a,x] (x−u)n x∫ a w(u)du + sup u∈[x,b] (u−x)n b∫ x w(u)du   = 1 n!  (x−a)n x∫ a w(u)du + (b−x)n b∫ x w(u)du   ≤ 1 n! max{(x−a)n , (b−x)n} b∫ a w(u)du because g is a p.d.f., b∫ a w(u)du = 1. using the identity max{x,y} = x + y 2 + ∣∣∣∣y −x2 ∣∣∣∣ , we get max{(x−a)n , (b−x)n} b∫ a g(u)du = [ b−a 2 + ∣∣∣∣x− a + b2 ∣∣∣∣ ]n . which completes the proof. � remark 6. if we choose n = 1 in theorem 7, we have the inequality∣∣∣∣∣∣f (x) − b∫ a w (t) f (t) dt ∣∣∣∣∣∣ ≤ [ b−a 2 + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f′). references [1] m. w. alomari, a generalization of weighted companion of ostrowski integral inequality for mappings of bounded variation, rgmia research report collection, 14 (2011), art. id 87. [2] m. w. alomari and m. a. latif, weighted companion for the ostrowski and the generalized trapezoid inequalities for mappings of bounded variation, rgmia research report collection, 14 (2011), art. id 92. [3] m. w. alomari and s. s. dragomir, mercer–trapezoid rule for the riemann–stieltjes integral with applications, journal of advances in mathematics, 2 (2) (2013), 67–85. [4] n. s. barnett, p. cerone, s. s. dragomir and j. roumeliotis, some inequalities for the dispersion of a random variable whose pdf is defined on a finite interval, j. ineq. pure appl. math., 2 (1) (2001), 1–18. [5] h. budak and m. z. sarıkaya, on generalization of dragomir’s inequalities, rgmia research report collection, 17 (2014), art. id 155. [6] h. budak and m. z. sarıkaya, new weighted ostrowski type inequalities for mappings with first derivatives of bounded variation, rgmia research report collection, 18 (2015), art. id 43. [7] h. budak and m. z. sarikaya, a new generalization of ostrowski type inequality for mappings of bounded variation, rgmia research report collection, 18 (2015), art. id 47. [8] h. budak and m. z. sarikaya, on generalization of weighted ostrowski type inequalities for functions of bounded variation, rgmia research report collection, 18 (2015), art. id 51. [9] h. budak and m. z. sarikaya, a new ostrowski type inequality for functions whose first derivatives are of bounded variation, moroccan journal of pure and applied analysis, 2 (1) (2016), 1-11. [10] h. budak and m. z. sarikaya, a companion of ostrowski type inequalities for mappings of bounded variation and some applications, rgmia research report collection, 19 (2016), art. id 24, 10. [11] h. budak, m. z. sarikaya and a. qayyum, improvement in companion of ostrowski type inequalities for mappings whose first derivatives are of bounded variation and application, rgmia research report collection, 19 (2016), art. id 25. [12] p. cerone and s. s. dragomir, on some inequalities for the expectation and variance, korean j. comp. & appl. math., 8 (2) (2000), 357–380. [13] s. s. dragomir, the ostrowski integral inequality for mappings of bounded variation, bulletin of the australian mathematical society, 60 (1) (1999), 495-508. new weighted ostrowski type inequalities 79 [14] s. s. dragomir, on the ostrowski’s integral inequality for mappings with bounded variation and applications, mathematical inequalities & applications, 4 (1) (2001), 59–66. [15] s. s. dragomir, a companion of ostrowski’s inequality for functions of bounded variation and applications, international journal of nonlinear analysis and applications, 5 (1) (2014), 89-97. [16] s. s. dragomir, some perturbed ostrowski type inequalities for functions of bounded variation, rgmia research report collection, 16 (2013), art. id 93. [17] s. s. dragomir, approximating real functions which possess nth derivatives of bounded variation and applications, computers and mathematics with applications, 56 (2008), 2268–2278. [18] p. kumar, moments inequalities of a random variable defined over a finite interval, j. inequal. pure and appl. math., 3 (2002), article id 41. [19] w. liu and y. sun, a refinement of the companion of ostrowski inequality for functions of bounded variation and applications, arxiv:1207.3861v1, (2012). [20] z. liu, another generalization of weighted ostrowski type inequality for mappings of bounded variation, applied mathematics letters, 25 (2012), 393–397. [21] z. liu, some ostrowski type inequalities, mathematical and computer modelling, 48 (2008), 949–960. [22] a. m. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, comment. math. helv., 10 (1938), 226-227. [23] j. roumeliotis, p. cerone and s. s. dragomir, an ostrowski type inequality for weighted mapping with bounded second derivatives, j. korean soc. ind. appl. math., 3(2) (1999), 107-119. [24] k-l.tseng, g-s. yang, and s. s. dragomir, generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications, mathematical and computer modelling, 40 (2004), 77-84. [25] k-l. tseng, improvements of some inequalites of ostrowski type and their applications, taiwan. j. math., 12 (9) (2008), 2427–2441. [26] k-l. tseng,s-r. hwang, g-s. yang, and y-m. chou, weighted ostrowski integral inequality for mappings of bounded variation, taiwanese j. of math., 15 (2011), 573-585. 1department of mathematics, faculty of science and arts, düzce university, konuralp campus, düzceturkey 2department of mathematics, faculty of science, bartın university, bartın-turkey ∗corresponding author: erdensmt@gmail.email int. j. anal. appl. (2023), 21:27 intuitionistic hesitant fuzzy up (bcc)-filters of up (bcc)-algebras aiyared iampan1,∗, r. alayakkaniamuthu2, p. gomathi sundari2, n. rajesh2 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2department of mathematics, rajah serfoji government college (affiliated to bharathidasan university), thanjavur 613005, tamilnadu, india ∗corresponding author: aiyared.ia@up.ac.th abstract. the concepts of intuitionistic hesitant fuzzy up (bcc)-subalgebras, up (bcc)-ideals, and up (bcc)-filters of up (bcc)-algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of hesitant fuzzy sets as a foundation. the necessary conditions for those intuitionistic hesitant fuzzy sets are provided and include their relation to their complement. the concept of prime and weakly prime of intuitionistic hesitant fuzzy sets was also introduced and studied. we also talk about the connections between intuitionistic hesitant fuzzy up (bcc)-subalgebras (up (bcc)-ideals, up (bcc)-filters) and their level subsets. the homomorphic pre-images of intuitionistic hesitant fuzzy up (bcc)-filters in up (bcc)-algebras are also studied and some related properties are investigated. 1. introduction the concept of fuzzy sets was proposed by zadeh [15]. the theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. after the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. the integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1, 2, 4]. in 2009 2010, torra and narukawa [13, 14] introduced the notion of hesitant fuzzy sets, that is a function from a reference set to a power set of the unit interval. the notion of hesitant fuzzy sets is the other generalization of the notion fuzzy sets. the received: feb. 11, 2023. 2010 mathematics subject classification. 03g25, 03e72. key words and phrases. up (bcc)-algebra; intuitionistic hesitant fuzzy up (bcc)-subalgebra; intuitionistic hesitant fuzzy up (bcc)-ideal; intuitionistic hesitant fuzzy up (bcc)-filter. https://doi.org/10.28924/2291-8639-21-2023-27 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-27 2 int. j. anal. appl. (2023), 21:27 hesitant fuzzy set theories developed by torra and others have found many applications in the domain of mathematics and elsewhere. after the introduction of the notion of hesitant fuzzy sets by torra and narukawa [13, 14], several researches were conducted on the generalizations of the notion of hesitant fuzzy sets and application to many logical algebras such as: in 2012, zhu, xu and xia [16] introduced the notion of dual hesitant fuzzy sets, which is a new extension of fuzzy sets. in 2014, jun, ahn and muhiuddin [7] introduced the notions of hesitant fuzzy soft subalgebras and (closed) hesitant fuzzy soft ideals in bck/bci-algebras. jun and song [9] introduced the notions of (boolean, prime, ultra, good) hesitant fuzzy filters and hesitant fuzzy mv-filters of mtl-algebras. iampan [6] introduced a new algebraic structure, called a up-algebra, and mosrijai et. al. [11] introduced the notion of hesitant fuzzy sets on up-algebras. the notions of hesitant fuzzy subalgebras, hesitant fuzzy filters and hesitant fuzzy up-ideals play an important role in studying the many logical algebras. the concepts of up-algebras (see [6]) and bcc-algebras (see [10]) are the same concept, as shown by jun et al. [8] in 2022. in this publication and following investigations, our research team will refer to it as bcc rather than up because of respect for komori, who first characterized it in 1984. in this paper, the concepts of intuitionistic hesitant fuzzy bcc-subalgebras, bcc-ideals, and bccfilters of bcc-algebras are presented, some of their features are explained, and their extensions are demonstrated using the theory of hesitant fuzzy sets as a foundation. the necessary conditions for those intuitionistic hesitant fuzzy sets are provided and include their relation to their complement. the concept of prime and weakly prime of intuitionistic hesitant fuzzy sets was also introduced and studied. we also talk about the connections between intuitionistic hesitant fuzzy bcc-subalgebras (bcc-ideals, bcc-filters) and their level subsets. the homomorphic pre-images of intuitionistic hesitant fuzzy bccfilters in bcc-algebras are also studied and some related properties are investigated. 2. preliminaries the concept of bcc-algebras (see [10]) can be redefined without the condition (2.6) as follows: an algebra x = (x, ·, 0) of type (2, 0) is called a bcc-algebra if it satisfies the following conditions: (∀x,y,z ∈ x)((y ·z) · ((x ·y) · (x ·z)) = 0) (2.1) (∀x ∈ x)(0 ·x = x) (2.2) (∀x ∈ x)(x · 0 = 0) (2.3) (∀x,y ∈ x)(x ·y = 0 = y ·x ⇒ x = y) (2.4) after this we assign x instead of a bcc-algebra (x, ·, 0) until otherwise specified. we define a binary relation ≤ on x as follows: (∀x,y ∈ x)(x ≤ y ⇔ x ·y = 0) (2.5) int. j. anal. appl. (2023), 21:27 3 in x, the following assertions are valid (see [6]). (∀x ∈ x)(x ≤ x) (2.6) (∀x,y,z ∈ x)(x ≤ y,y ≤ z ⇒ x ≤ z) (2.7) (∀x,y,z ∈ x)(x ≤ y ⇒ z ·x ≤ z ·y) (2.8) (∀x,y,z ∈ x)(x ≤ y ⇒ y ·z ≤ x ·z) (2.9) (∀x,y,z ∈ x)(x ≤ y ·x, in particular, y ·z ≤ x · (y ·z)) (2.10) (∀x,y ∈ x)(y ·x ≤ x ⇔ x = y ·x) (2.11) (∀x,y ∈ x)(x ≤ y ·y) (2.12) (∀a,x,y,z ∈ x)(x · (y ·z) ≤ x · ((a ·y) · (a ·z))) (2.13) (∀a,x,y,z ∈ x)(((a ·x) · (a ·y)) ·z ≤ (x ·y) ·z) (2.14) (∀x,y,z ∈ x)((x ·y) ·z ≤ y ·z) (2.15) (∀x,y,z ∈ x)(x ≤ y ⇒ x ≤ z ·y) (2.16) (∀x,y,z ∈ x)((x ·y) ·z ≤ x · (y ·z)) (2.17) (∀a,x,y,z ∈ x)((x ·y) ·z ≤ y · (a ·z)) (2.18) definition 2.1. [6] a nonempty subset s of x is called a bcc-subalgebra of x if x ·y ∈ s ∀x,y ∈ s. definition 2.2. [6] a nonempty subset i of x is called a bcc-ideal of x if (1) 0 ∈ i, (2) (∀x,y,z ∈ x)(x · (y ·z),y ∈ i ⇒ x ·z ∈ i). definition 2.3. [12] a nonempty subset f of x is called a bcc-filter of x if (1) 0 ∈ f, (2) (∀x,y ∈ x)(x ·y ∈ f,x ∈ f ⇒ y ∈ f ). definition 2.4. [13] a hesitant fuzzy set on a reference set x is defined in term of a function h that when applied to x return a subset of [0, 1], that is, h : x →p([0, 1]). definition 2.5. [3] an intuitionistic hesitant fuzzy set on a reference set x is defined in the form h = (h,k), where h and k are functions that when applied to x return a subset of [0, 1], that is, h,k : x →p([0, 1]). definition 2.6. [11] a hesitant fuzzy set h on x is said to be a hesitant fuzzy bcc-filter of x if the following conditions are hold: (∀x ∈ x)(h(0) ⊇ h(x)) (2.19) (∀x,y ∈ x)(h(y) ⊇ h(x ·y) ∩h(x)) (2.20) 4 int. j. anal. appl. (2023), 21:27 definition 2.7. [13] the complement of a hesitant fuzzy set h in a reference set x is the hesitant fuzzy set h defined by h(x) = [0, 1] −h(x) for all x ∈ x. definition 2.8. [13] the complement of an intuitionistic hesitant fuzzy set h = (h,k) on a reference set x is the intuitionistic hesitant fuzzy set h = (k,h) defined by h(x) = [0, 1] −h(x) and k(x) = [0, 1] −k(x) for all x ∈ x. 3. intuitionistic hesitant fuzzy bcc-filters in this section, the concepts of intuitionistic hesitant fuzzy bcc-subalgebras, bcc-ideals, and bcc-filters of bcc-algebras are presented, some of their features are explained. definition 3.1. an intuitionistic hesitant fuzzy set h = (h,k) on x is called an intuitionistic hesitant fuzzy bcc-subalgebra of x if it satisfies the following property: (∀x,y ∈ x) ( h(x ·y) ⊇ h(x) ∩h(y) k(x ·y) ⊆ k(x) ∪k(y) ) (3.1) definition 3.2. the characteristic intuitionistic hesitant fuzzy set of a subset a of a set x is defined to be the structure χa = (hχa,kχa), where hχa(x) = { [0, 1] if x ∈ a ∅ otherwise and kχa(x) = { ∅ if x ∈ a [0, 1] otherwise. lemma 3.1. the constant 0 of x is in a nonempty subset b of x if and only if hχb (0) ⊇ hχb (x) and kχb (0) ⊆ kχb (x) for all x ∈ x. proof. if 0 ∈ b, then hb(0) = [0, 1]. thus hb(0) = [0, 1] ⊇ hb(x) for all x ∈ x. also, kb(0) = ∅. then kb(0) = ∅⊆ kb(x) for all x ∈ x. conversely, assume that hb(0) ⊇ hb(x) and kb(0) ⊆ kb(x) for all x ∈ x. since b is a nonempty subset of x, we have a ∈ b for some a ∈ x. then hb(0) ⊇ hb(a) = [0, 1], so hb(0y ) = [0, 1]. hence, 0 ∈ b. � definition 3.3. an intuitionistic hesitant fuzzy set h = (h,k) on x is said to be an intuitionistic hesitant fuzzy bcc-filter of x if the following conditions are hold: (∀x ∈ x) ( h(0) ⊇ h(x) k(0) ⊆ k(x) ) (3.2) (∀x,y ∈ x) ( h(y) ⊇ h(x ·y) ∩h(x) k(y) ⊆ k(x ·y) ∪k(x) ) (3.3) int. j. anal. appl. (2023), 21:27 5 example 3.1. let x = {0, 1, 2, 3} with the following cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 1 0 0 3 0 1 2 0 then x is a bcc-algebra. we define an intuitionistic hesitant fuzzy set h = (h,k) on x as follows: h(0) = [0, 1],h(1) = {0.1},h(2) = ∅,h(3) = {0.2, 0.3}, k(0) = ∅,k(1) = {0.1, 0.2},k(2) = [0, 1],k(3) = {0.3} then h is an intuitionistic hesitant fuzzy bcc-subalgebra of x. definition 3.4. an intuitionistic hesitant fuzzy set h = (h,k) on x is said to be an intuitionistic hesitant fuzzy bcc-ideal of x if (3.2) and the following condition are hold: (∀x,y,z ∈ x) ( h(x ·y) ⊇ h(x · (y ·z)) ∩h(y) k(x ·y) ⊆ k(x · (y ·z)) ∪k(y) ) (3.4) theorem 3.1. every intuitionistic hesitant fuzzy bcc-ideal of x is an intuitionistic hesitant fuzzy bcc-filter. proof. let h = (h,k) be an intuitionistic hesitant fuzzy bcc-ideal of x. then (3.2) holds. let x,y ∈ x. then h(y) = h(0 ·y) ⊇ h(0 · (x ·y)) ∩h(x) = h(x ·y) ∩h(x), k(y) = k(0 ·y) ⊆ k(0 · (x ·y)) ∪k(x) = k(x ·y) ∪k(x). hence, h is an intuitionistic hesitant fuzzy bcc-filter of x. � the following example shows that the converse of theorem 3.1 is not true in general. example 3.2. let x = {0, 1, 2, 3} with the following cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 3 3 2 0 1 0 0 3 0 1 2 0 then x is a bcc-algebra. we define an intuitionistic hesitant fuzzy set h = (h,k) on x as follows: h(0) = [0, 1],h(1) = {0.7},h(2) = ∅,h(3) = {0.2, 0.5}, k(0) = ∅,k(1) = {0.1, 0.2},k(2) = {0.1, 0.2, 0.3},k(3) = [0, 1] 6 int. j. anal. appl. (2023), 21:27 then h is an intuitionistic hesitant fuzzy bcc-filter of x but not an intuitionistic hesitant fuzzy bcc-ideal of x. theorem 3.2. every intuitionistic hesitant fuzzy bcc-filter of x is an intuitionistic hesitant fuzzy bcc-subalgebra. proof. let h = (h,k) be an intuitionistic hesitant fuzzy bcc-filter of x. then for all x,y ∈ x, h(x ·y) ⊇ h(y · (x ·y)) ∩h(y) = h(0) ∩h(y) = h(y) ⊇ h(x) ∩h(y), k(x ·y) ⊆ k(y · (x ·y)) ∪k(y) = k(0) ∪k(y) = k(y) ⊆ k(x) ∪k(y). hence, h is an intuitionistic hesitant fuzzy bcc-subalgebra of x. � the following example shows that the converse of theorem 3.2 is not true in general. example 3.3. let x = {0, 1, 2, 3} with the following cayley table: · 0 1 2 3 0 0 1 2 3 1 0 0 2 3 2 0 0 0 3 3 0 1 2 0 then x is a bcc-algebra. we define an intuitionistic hesitant fuzzy set h = (h,k) on x as follows: h(0) = {0.1, 0.2, 0.3},h(1) = {0.1},h(2) = {0.2},h(3) = ∅, k(0) = ∅,k(1) = {0.1},k(2) = {0.1, 0.2},k(3) = [0, 1] then h is an intuitionistic hesitant fuzzy bcc-subalgebra of x but not an intuitionistic hesitant fuzzy bcc-filter of x. theorem 3.3. a nonempty subset f of x is a bcc-filter of x if and only if the characteristic intuitionistic hesitant fuzzy set χf = (hχf ,kχf ) is an intuitionistic hesitant fuzzy bcc-filter of x. proof. assume that f is a bcc-filter of x. since 0 ∈ f, it follows from lemma 3.1 that hχf (0) ⊇ hχf (x) for all x ∈ x. next, let x,y ∈ x. case 1 : if x,y ∈ f, then hχf (x) = [0, 1] and hχf (y) = [0, 1]. hence, hχf (y) = [0, 1] ⊇ hχf (x ·y) = hχf (x ·y)∩hχf (x). also, kχf (x) = ∅ and kχf (y) = ∅. hence, kχf (y) = ∅⊆ kχf (x ·y) = kχf (x ·y) ∪kχf (x). case 2 : if x /∈ f and y ∈ f, then hχf (x) = ∅ and hχf (y) = [0, 1]. thus hχf (y) = [0, 1] ⊇ ∅ = hχf (x · y) ∩ hχf (x). also kχf (x) = [0, 1] and kχf (y) = ∅. thus kχf (y) = ∅ ⊆ [0, 1] = kχf (x ·y) ∪kχf (x). case 3 : if x ∈ f and y /∈ f, then hχf (x) = [0, 1] and hχf (y) = ∅. since f is a bcc-filter of x, we have x · y /∈ f or x /∈ f. but x ∈ f, so x · y /∈ f. then hχf (x · y) = ∅. thus int. j. anal. appl. (2023), 21:27 7 hχf (y) = ∅ ⊇ ∅ = hχf (x · y) ∩ hχf (x). also, kχf (x) = ∅, kχf (y) = [0, 1] and kχf (x · y) = [0, 1]. thus kχf (y) = [0, 1] ⊆ [0, 1] = kχf (x ·y) ∪kχf (x). case 4 : if x /∈ f and y /∈ f, then hχf (x) = ∅ and hχf (y) = ∅. thus hχf (y) = ∅⊆∅ = hχf (x ·y)∩ hχf (x). also, kχf (x) = [0, 1] and kχf (y) = [0, 1]. thus kχf (y) = [0, 1] ⊆ [0, 1] = kχf (x ·y)∪kχf (x). hence, χf = (hχf ,kχf ) is an intuitionistic hesitant fuzzy bcc-filter of x. conversely, assume that χf = (hχf ,kχf ) is an intuitionistic hesitant fuzzy bcc-filter of x. since hχf (0) ⊇ hχf (x) for all x ∈ x, it follows from lemma 3.1 that 0 ∈ f. next, let x,y ∈ x be such that x ·y ∈ f and x ∈ f. then hχf (x ·y) = [0, 1] and hχf (x) = [0, 1]. thus hχf (y) ⊇ hχf (x ·y)∩hχf (x) = [0, 1], so hχf (y) = [0, 1]. therefore, y ∈ f and so f is a bcc-filter of x. � definition 3.5. an intuitionistic hesitant fuzzy set h = (h,k) on x is called a prime intuitionistic hesitant fuzzy set on x if it satisfies the following property: (∀x,y ∈ x) ( h(x ·y) ⊆ h(x) ∪h(y) k(x ·y) ⊇ k(x) ∩k(y) ) (3.5) definition 3.6. [5] a nonempty subset b of x is called a prime subset of x if it satisfies the following property: (∀x,y ∈ x)(x ·y ∈ b ⇒ x ∈ b or y ∈ b) theorem 3.4. a nonempty subset b of x is a prime subset of x if and only if the characteristic intuitionistic hesitant fuzzy set χb is a prime intuitionistic hesitant fuzzy set on x. proof. assume that b is a prime subset of x and let x,y ∈ x. case 1 : if x · y ∈ b, then hχb (x · y) = [0, 1]. since b is a prime subset of x, we have x ∈ b or y ∈ b. then hχb (x) = [0, 1] or hχb (y) = [0, 1], so hχb (x) ∪ hχb (y) = [0, 1]. hence, hχb (x ·y) = [0, 1] ⊆ [0, 1] = hχb (x) ∪hχb (y). also, kχb (x ·y) = ∅⊇ kχb (x) ∩kχb (y). case 2 : if x · y /∈ b, then hχb (x · y) = ∅ ⊆ hχb (x) ∪ hχb (y). also, kχb (x · y) = [0, 1] ⊇ kχb (x) ∩kχb (y). hence, χb is a prime intuitionistic hesitant fuzzy set on x. conversely, assume that χb = (hχb,kχb ) is a prime intuitionistic hesitant fuzzy set on x. let x,y ∈ x be such that x ·y ∈ b. then hχb (x ·y) = [0, 1], so [0, 1] = hχb (x ·y) ⊆ hχb (x) ∪hχb (y). thus hχb (x) ∪hχb (y) = [0, 1], so hχb (x) = [0, 1] or hχb (y) = [0, 1]. hence, x ∈ b or y ∈ b and so b is a prime subset of x. � theorem 3.5. let h = (h,k) be an intuitionistic hesitant fuzzy set on x. then the following statements are equivalent: (1) h is a prime intuitionistic hesitant fuzzy bcc-filter of x, (2) h is a constant intuitionistic hesitant fuzzy set on x. 8 int. j. anal. appl. (2023), 21:27 proof. assume that h is a prime intuitionistic hesitant fuzzy bcc-filter of x. then h(0) ⊇ h(x) and k(0) ⊆ k(x) for all x ∈ x. by (2.6), we have h(0) = h(x · x) ⊆ h(x) ∪ h(x) = h(x) and k(0) = k(x ·x) ⊇ k(x)∪k(x) = k(x) for all x ∈ x and so h(x) = h(0) and k(x) = k(0) for all x ∈ x. hence, h is a constant intuitionistic hesitant fuzzy set on x. conversely, assume that h is a constant intuitionistic hesitant fuzzy set on x. hence, we can easily show that h is a prime intuitionistic hesitant fuzzy bcc-filter of x. � definition 3.7. [5] a nonempty subset b of x is called a weakly prime subset of x if it satisfies the following property: (∀x,y ∈ x,x 6= y)(x ·y ∈ b ⇒ x ∈ b or y ∈ b) definition 3.8. [5] a bcc-filter b of x is called a weakly prime bcc-filter of x if b is a weakly prime subset of x. definition 3.9. an intuitionistic hesitant fuzzy set h = (h,k) on x is called a weakly prime intuitionistic hesitant fuzzy set on x if it satisfies the following property: (∀x,y ∈ x,x 6= y) ( h(x ·y) ⊆ h(x) ∪h(y) k(x ·y) ⊇ k(x) ∩k(y) ) (3.6) definition 3.10. an intuitionistic hesitant fuzzy bcc-filter h = (h,k) of x is called a weakly prime intuitionistic hesitant fuzzy bcc-filter of x if h is a weakly prime intuitionistic hesitant fuzzy set on x. theorem 3.6. a nonempty subset b of x is a weakly prime subset of x if and only if the characteristic intuitionistic hesitant fuzzy set χb is a weakly prime intuitionistic hesitant fuzzy set on x. proof. assume that b is a weakly prime subset of x and let x,y ∈ x be such that x 6= y. case 1 : if x · y ∈ b, then hχb (x · y) = [0, 1]. since b is a weakly prime subset of x, we have x ∈ b or y ∈ b. then hχb (x) = [0, 1] or hχb (y) = [0, 1], so hχb (x) ∪hχb (y) = [0, 1]. hence, hχb (x·y) = [0, 1] ⊆ [0, 1] = hχb (x)∪hχb (y). also, kχb (x) = ∅ or kχb (y) = ∅, so kχb (x)∩kχb (y) = ∅. hence, kχb (x ·y) = ∅⊇∅ = kχb (x) ∩kχb (y). case 2 : if x · y /∈ b, then hχb (x · y) = ∅ ⊆ hχb (x) ∪ hχb (y). also, kχb (x · y) = [0, 1] ⊇ kχb (x) ∩kχb (y). hence, χb is a weakly prime intuitionistic hesitant fuzzy set on x. conversely, assume that hχb is a weakly prime intuitionistic hesitant fuzzy set on x. let x,y ∈ x be such that x ·y ∈ b and x 6= y. then hχb (x ·y) = [0, 1], so [0, 1] = hχb (x ·y) ⊆ hχb (x) ∪hχb (y). thus hχb (x) ∪hχb (y) = [0, 1], so hχb (x) = [0, 1] or hχb (y) = [0, 1]. hence, x ∈ b or y ∈ b and so b is a weakly prime subset of x. � int. j. anal. appl. (2023), 21:27 9 theorem 3.7. a nonempty subset f of x is a weakly prime bcc-filter of x if and only if the characteristic intuitionistic hesitant fuzzy set χf is a weakly prime intuitionistic hesitant fuzzy bccfilter of x. proof. it is straightforward by theorems 3.3 and 3.6. � theorem 3.8. an intuitionistic hesitant fuzzy set h = (h,k) is an intuitionistic hesitant fuzzy bccfilter of x if and only if the hesitant fuzzy sets h and k are hesitant fuzzy bcc-filters of x. proof. assume that h = (h,k) is an intuitionistic fuzzy bcc-filter of x. then for any x,y ∈ x, we have h(0) ⊇ h(x) and h(y) ⊇ h(x · y) ∩ h(x). hence, h is a hesitant fuzzy bcc-filter of x. now for any x,y ∈ x, we have k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪ k(x). then k(0) = [0, 1] − k(0) ⊇ [0, 1] −k(x) = k(x) and k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = k(x ·y) ∩k(x). hence, k is a hesitant fuzzy bcc-filter of x. conversely, assume that the hesitant fuzzy sets h and k are hesitant fuzzy bcc-filters of x. then for any x,y ∈ x, we have h(0) ⊇ h(x) and h(y) ⊇ h(x · y) ∩h(x). now for any x,y ∈ x, we have k(0) ⊇ k(x) and k(y) ⊇ k(x ·y)∩k(x). then [0, 1]−k(0) ⊇ [0, 1]−k(x) and so k(0) ⊆ k(x). now, [0, 1] −k(y) ⊇ [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = [0, 1] − (k(x ·y) ∪k(x)), k(y) ⊆ k(x ·y) ∪k(x). hence, h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x. � theorem 3.9. an intuitionistic hesitant fuzzy set h = (h,k) is an intuitionistic hesitant fuzzy bccfilter of x if and only if the intuitionistic hesitant fuzzy set h = (k,h) is an intuitionistic hesitant fuzzy bcc-filter of x. proof. assume that h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x. then for any x,y,z ∈ x, h(0) ⊇ h(x) and h(y) ⊇ h(x ·y)∩h(x). hence, for any x,y,z ∈ x, h(0) = [0, 1]−h(0) ⊆ [0, 1] −h(x) = h(x) and h(y) = [0, 1] −h(y) ⊆ [0, 1] − (h(x ·y) ∩h(x)) = [0, 1] −h(x ·y) ∪ [0, 1] −h(x) = h(x ·y) ∪h(x). 10 int. j. anal. appl. (2023), 21:27 now, for any x,y,z ∈ x, k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪ k(x). hence, for any x,y,z ∈ x, k(0) = [0, 1] −k(0) ⊇ [0, 1] −k(x) = k(x) and k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = [0, 1] −k(x ·y) ∩ [0, 1] −k(x) = k(x ·y) ∩k(x). hence, h = (k,h) is an intuitionistic hesitant fuzzy bcc-filter of x. conversely, assume that the intuitionistic hesitant fuzzy set h = (k,h) is an intuitionistic hesitant fuzzy bcc-filter of x. then for any x,y,z ∈ x, k(0) ⊇ k(x) and k(y) ⊇ k(x · y) ∩ k(x). then [0, 1] − k(0) ⊇ [0, 1] − k(x) and [0, 1] − k(y) ⊇ [0, 1] − (k(x · y) ∪ k(x)), so k(0) ⊆ k(x) and k(y) ⊆ k(x · y) ∪k(x). now, for any x,y,z ∈ x, we have h(0) ⊆ h(x) and h(y) ⊆ h(x · y) ∪h(x). then [0, 1] −h(0) ⊆ [0, 1] −h(x) and [0, 1] −h(y) ⊇ [0, 1] − (h(x · y) ∪h(x)), so h(0) ⊇ h(x) and h(y) ⊇ h(x ·y) ∩h(x). hence, h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x. � definition 3.11. let h = (h,k) be an intuitionistic hesitant fuzzy set on x. the intuitionistic hesitant fuzzy sets ⊕h and ⊗h are defined as ⊕h = (h,h) and ⊗h = (k,k). theorem 3.10. if h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x, then the sets xh = {x ∈ x | h(x) = h(0)} and xk = {x ∈ x | k(x) = k(0)} are bcc-filters of x. proof. clearly, 0 ∈ xh∩xk. let x,y ∈ x be such that x ·y,x ∈ xh. then h(x ·y) = h(0) and h(x) = h(0). since h is an intuitionistic hesitant fuzzy bcc-filter of x, by (3.3), h(y) ⊇ h(x·y)∩h(x) = h(0), whence h(y) = h(0), by (3.2). this means that y ∈ xh. hence, xh is a bcc-filter of x. let x,y ∈ x be such that x ·y,x ∈ xk. then k(x ·y) = k(0) and k(x) = k(0). since h is an intuitionistic hesitant fuzzy bcc-filter of x, by (3.3), k(y) ⊆ k(x ·y) ∪k(x) = k(0), whence k(y) = k(0), by (3.2). this means that y ∈ xk. hence, xk is a bcc-filter of x. � theorem 3.11. an intuitionistic hesitant fuzzy set h = (h,k) is an intuitionistic hesitant fuzzy bccfilter of x if and only if the intuitionistic hesitant fuzzy sets ⊕h and ⊗h are intuitionistic hesitant fuzzy bcc-filters of x. proof. assume that h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x. let x ∈ x. then h(0) = [0, 1] −h(0) ⊆ [0, 1] −h(x) = h(x). let x,y ∈ x. then h(y) = [0, 1] −h(y) ⊆ [0, 1] − (h(x ·y) ∩h(x)) = ([0, 1] −h(x ·y)) ∪ ([0, 1] −h(x)) = h(x ·y) ∪h(x). hence, ⊕h is an intuitionistic hesitant fuzzy bcc-filter of x. int. j. anal. appl. (2023), 21:27 11 let x ∈ x. then k(0) = [0, 1] −k(0) ⊇ [0, 1] −k(x) = k(x). let x,y ∈ x. then k(y) = [0, 1] −k(y) ⊇ [0, 1] − (k(x ·y) ∪k(x)) = ([0, 1] −k(x ·y)) ∩ ([0, 1] −k(x)) = k(x ·y) ∩k(x). hence, ⊗h is an intuitionistic hesitant fuzzy bcc-filter of x. conversely, assume that ⊕h and ⊗h are intuitionistic hesitant fuzzy bcc-filters of x. then for any x,y ∈ x, we have h(0) ⊇ h(x) and h(y) ⊇ h(x ·y)∩h(x) and k(0) ⊆ k(x) and k(y) ⊆ k(x ·y)∪k(x). hence, h is an intuitionistic hesitant fuzzy bcc-filter of x. � lemma 3.2. if h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x, then (∀x,y,z ∈ x) ( z ≤ x ·y ⇒ { h(y) ⊇ h(x) ∩h(z) k(y) ⊆ k(x) ∪k(z) ) . (3.7) proof. let x,y,z ∈ x be such that z ≤ x ·y. then z · (x ·y) = 0 and so h(y) ⊇ h(x ·y) ∩h(x) ⊇ h(z · (x ·y)) ∩h(z) ∩h(x) = h(0) ∩h(z) ∩h(x) = h(x) ∩h(z), k(y) ⊆ k(x ·y) ∪k(x) ⊆ k(z · (x ·y)) ∪k(z) ∪k(x) = k(0) ∪k(z) ∪k(x) = k(x) ∪k(z). � lemma 3.3. if h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x, then (∀x,y ∈ x) ( x ≤ y ⇒ { h(y) ⊇ h(x) k(y) ⊆ k(x) ) . (3.8) proof. let x,y ∈ x be such that x ≤ y. then x ·y = 0 and so h(y) ⊇ h(x ·y) ∩h(x) = h(0) ∩h(x) = h(x), k(y) ⊆ k(x ·y) ∪k(x) = k(0) ∪k(x) = k(x). � lemma 3.4. if h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of x, then (∀x,y,z ∈ x) ( h(y ·z) ∩h(x ·y) ⊆ h(x ·z) k(y ·z) ∪k(x ·y) ⊇ k(x ·z) ) . (3.9) 12 int. j. anal. appl. (2023), 21:27 proof. let x,y,z ∈ x. by (2.1), we have (y ·z) ≤ (x ·y) · (x ·z). then it follows from lemma 3.2 that h(y ·z) ∩h(x ·y) ⊆ h(x ·z), k(y ·z) ∪k(x ·y) ⊇ k(x ·z). � definition 3.12. let h : x →p([0, 1]). for any π ∈p([0, 1]), the sets u(h,π) = {x ∈ x | h(x) ⊇ π} and u+(h,π) = {x ∈ x | h(x) ⊃ π} are called an upper π-level subset and an upper π-strong level subset of h, respectively. the sets l(h,π) = {x ∈ x | h(x) ⊆ π} and l−(h,π) = {x ∈ x | h(x) ⊂ π} are called a lower π-level subset and a lower π-strong level subset of h, respectively. the set e(h,π) = {x ∈ x | h(x) = π} is called an equal π-level subset of h. then u(h,π) = u+(h,π) ∪e(h,π) and l(h,π) = l−(h,π) ∪e(h,π). theorem 3.12. an intuitionistic hesitant fuzzy set h = (h,k) on x is an intuitionistic hesitant fuzzy bcc-filter of x if and only if for all π ∈ p([0, 1]), the nonempty subsets u(h,π) and l(k,π) of x are bcc-filters. proof. assume that h is an intuitionistic hesitant fuzzy bcc-filter of x. let π ∈ p([0, 1]) be such that u(h,π) 6= ∅ and let x ∈ u(h,π). then h(x) ⊇ π. since h is an intuitionistic hesitant fuzzy bcc-filter of x, we have h(0) ⊇ h(x) ⊇ π. thus 0 ∈ u(h,π). next, let x,y ∈ x be such that x,x · y ∈ u(h,π). then h(x) ⊇ π and h(x · y) ⊇ π. since h is an intuitionistic hesitant fuzzy bcc-filter of x, we have h(y) ⊇ h(x ·y)∩h(x) ⊇ π. so y ∈ u(h,π). let π ∈p([0, 1]) be such that l(k,π) 6= ∅ and let x ∈ l(k,π). then k(x) ⊆ π. since h is an intuitionistic hesitant fuzzy bcc-filter of x, we have k(0) ⊆ k(x) ⊆ π. thus 0 ∈ l(k,π). next, let x,y ∈ x be such that x,x ·y ∈ l(k,π). then k(x) ⊆ π and k(x ·y) ⊆ π. since h is an intuitionistic hesitant fuzzy bcc-filter of x, we have k(y) ⊆ k(x ·y) ∪k(x) ⊆ π. so y ∈ l(k,π). hence, u(h,π) and l(k,π) are bcc-filters of x. conversely, assume that for all π ∈p([0, 1]), the nonempty subsets u(h,π) and l(k,π) of x are bcc-filters. let x ∈ x. then h(x) ∈ p([0, 1]). choose π = h(x) ∈ p([0, 1]). then h(x) ⊇ π. thus x ∈ u(h,π). by assumption, we have u(h,π) is a bcc-filter of x and thus 0 ∈ u(h,π). so h(0) ⊇ π = h(x). let x,y ∈ x. then h(x),h(x·y) ∈p([0, 1]). choose π = h(x)∩h(x·y) ∈p([0, 1]). then h(x) ⊇ π and h(x · y) ⊇ π. since x,x · y ∈ u(h,π) 6= ∅. by assumption, we have u(h,π) is a bcc-filter of x and then y ∈ u(h,π). thus h(y) ⊇ π = h(x) ∩ h(x · y). let x ∈ x. then k(x) ∈ p([0, 1]). choose π1 = k(x) ∈ p([0, 1]). then k(x) ⊆ π1. thus x ∈ l(k,π1). by assumption, we have l(k,π1) is a bcc-filter of x and thus 0 ∈ l(k,π1). so k(0) ⊆ π1 = k(x). let x,y ∈ x. then k(x),k(x ·y) ∈p([0, 1]). choose π1 = k(x) ∪k(x ·y) ∈p([0, 1]). then k(x) ⊆ π1 and k(x ·y) ⊆ π1. since x,x ·y ∈ l(k,π1) 6= ∅. by assumption, we have l(k,π1) is a bcc-filter of x and then y ∈ l(k,π1). thus k(y) ⊆ π1 = k(x) ∪k(x · y). hence, h is an intuitionistic hesitant fuzzy bcc-filter of x. � int. j. anal. appl. (2023), 21:27 13 the following theorem can be proved similarly to theorem 3.12. theorem 3.13. an intuitionistic hesitant fuzzy set h = (h,k) on x is an intuitionistic hesitant fuzzy bcc-subalgebra (bcc-ideal) of x if and only if for all π ∈ p([0, 1]), the nonempty subsets u(h,π) and l(k,π) of x are bcc-subalgebras (bcc-ideals). definition 3.13. let {hα | α ∈ ∆} be a family of intuitionistic hesitant fuzzy sets on a reference set x. we define the intuitionistic hesitant fuzzy set ⋂ α∈∆ hα = ( ⋂ α∈∆ hα, ⋃ α∈∆ kα) by ( ⋂ α∈∆ hα)(x) = ⋂ α∈∆ hα(x) and ( ⋃ α∈∆ kα)(x) = ⋃ α∈∆ kα(x) for all x ∈ x, which is called the intuitionistic hesitant intersection of intuitionistic hesitant fuzzy sets. proposition 3.1. if {hα | α ∈ ∆} is a family of intuitionistic hesitant fuzzy bcc-filters of x, then⋂ α∈∆ hα is an intuitionistic hesitant fuzzy bcc-filter of x. proof. let {hα | α ∈ ∆} be a family of intuitionistic hesitant fuzzy bcc-filter of x. let x ∈ x. then ( ⋂ α∈∆ hα)(0) = ⋂ α∈∆ hα(0) ⊇ ⋂ α∈∆ hα(x) = ( ⋂ α∈∆ hα)(x), ( ⋃ α∈∆ kα)(0) = ⋃ α∈∆ kα(0) ⊆ ⋃ α∈∆ kα(x) = ( ⋃ α∈∆ kα)(x). let x,y ∈ x. then ( ⋂ α∈∆ hα)(y) = ⋂ α∈∆ hα(y) ⊇ ⋂ α∈∆ (hα(x ·y) ∩hα(x)) = ( ⋂ α∈∆ hα(x ·y)) ∩ ( ⋂ α∈∆ hα(x)) = ( ⋂ α∈∆ hα)(x ·y) ∩ ( ⋂ α∈∆ hα)(x), ( ⋃ α∈∆ kα)(y) = ⋃ α∈∆ kα(y) ⊆ ⋃ α∈∆ (kα(x ·y) ∪kα(x)) = ⋃ α∈∆ kα(x ·y) ∪ ⋃ α∈∆ kα(x) = ( ⋃ α∈∆ kα)(x ·y) ∪ ( ⋃ α∈∆ kα)(x). hence, ⋂ α∈∆ hα is an intuitionistic hesitant fuzzy bcc-filter of x. � definition 3.14. let a = (ha,ka) and b = (hb,kb) be intuitionistic hesitant fuzzy sets on sets x and y , respectively. the cartesian product a×b = (h,k) defined by h(x,y) = ha(x) ∩hb(y) and k(x,y) = ka(x) ∪kb(y), where h : x ×y → p([0, 1]) and k : x ×y → p([0, 1]) for all x ∈ x and y ∈ y . 14 int. j. anal. appl. (2023), 21:27 remark 3.1. let (x, ·, 0x) and (y,?, 0y ) be bcc-algebras. then (x × y,�, (0x, 0y )) is a bccalgebra defined by (x,y) � (u,v) = (x ·u,y ? v) for every x,u ∈ x and y,v ∈ y . proposition 3.2. if a = (ha,ka) and b = (hb,kb) are two intuitionistic hesitant fuzzy bcc-filters of bcc-algebras x and y , respectively, then the cartesian product a×b is also an intuitionistic hesitant fuzzy bcc-filter of x ×y . proof. let (x,y) ∈ x ×y . then h(0x, 0y ) = ha(0x) ∩hb(0y ) ⊇ ha(x) ∩hb(y) = h(x,y), k(0x, 0y ) = ka(0x) ∪kb(0y ) ⊆ ka(x) ∪kb(y) = k(x,y). let (x1,x2), (y1,y2) ∈ x ×y . then h(y1,y2) = ha(y1) ∩hb(y2) ⊇ (ha(x1 ·y1) ∩ha(x1)) ∩ (hb(x2 ? y2) ∩hb(x2)) = ha(x1 ·y1) ∩hb(x2 ? y2) ∩ha(x1) ∩hb(x2) = h(x1 ·y1,x2 ? y2) ∩h(x1,x2) = h((x1,x2) � (y1,y2)) ∩h(x1,x2), k(y1,y2) = ka(y1) ∪kb(y2) ⊆ (ka(x1 ·y1) ∪ka(x1)) ∪ (kb(x2 ? y2) ∪kb(x2)) = ka(x1 ·y1) ∪kb(x2 ? y2) ∪ka(x1) ∪kb(x2) = k(x1 ·y1,x2 ? y2) ∪k(x1,x2) = k((x1,x2) � (y1,y2)) ∪k(x1,x2). hence, a×b is an intuitionistic hesitant fuzzy bcc-filter of x ×y . � theorem 3.14. two intuitionistic hesitant fuzzy sets a = (ha,ka) and b = (hb,kb) are intuitionistic hesitant fuzzy bcc-filters of bcc-algebras x and y , respectively if and only if the intuitionistic hesitant fuzzy sets ⊕(a×b) and ⊗(a×b) are intuitionistic hesitant fuzzy bcc-filters of x ×y . proof. it follows from proposition 3.2 and theorem 3.11. � a mapping f : (x, ·, 0x) → (y,?, 0y ) of bcc-algebras is called a homomorphism if f (x · y) = f (x) ? f (y) for all x,y ∈ x. note that if f : x → y is a homomorphism of bcc-algebras, then f (0x) = 0y . int. j. anal. appl. (2023), 21:27 15 definition 3.15. let f be a function from a nonempty set x to a nonempty set y . if h = (h,k) is an intuitionistic hesitant fuzzy set on y , then the intuitionistic hesitant fuzzy set f−1(h) = (h◦ f ,k ◦ f ) in x is called the pre-image of h under f . theorem 3.15. let f : (x, ·, 0x) → (y,?, 0y ) be a homomorphism of bcc-algebras. if h = (h,k) is an intuitionistic hesitant fuzzy bcc-filter of y , then f−1(h) = (h◦f ,k◦f ) is an intuitionistic hesitant fuzzy bcc-filter of x. proof. by assumption, h(f (0x)) = h(0y ) ⊇ h(y) for every y ∈ y . in particular, (h ◦ f )(0x) = h(f (0x)) ⊇ h(f (x)) = (h ◦ f )(x) for all x ∈ x. also, k(f (0x)) = k(0y ) ⊆ k(y) for every y ∈ y . in particular, (k ◦ f )(0x) = k(f (0x)) ⊆ k(f (x)) = (k ◦ f )(x) for all x ∈ x. let x,y ∈ x. then (h◦ f )(y) = h(f (y)) ⊇ h(f (x) ? f (y)) ∩h(f (x)) = h(f (x ·y)) ∩h(f (x)) = (h◦ f )(x ·y) ∩ (h◦ f )(x), (k ◦ f )(y) = k(f (y)) ⊆ k(f (x) ? f (y)) ∪k(f (x)) = k(f (x ·y)) ∪k(f (x)) = (k ◦ f )(x ·y) ∪ (k ◦ f )(x). hence, f−1(h) is an intuitionistic hesitant fuzzy bcc-filter of x. � 4. conclusion in the present paper, we have introduced the concepts of intuitionistic hesitant fuzzy bccsubalgebras, bcc-ideals, and bcc-filters of bcc-algebras. the relationship between intuitionistic hesitant fuzzy bcc-subalgebras (bcc-ideals, bcc-filters) and their level subsets is described. moreover, the homomorphic pre-images of intuitionistic hesitant fuzzy bcc-filters in bcc-algebras are also studied and some related properties are investigated. acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. ahmad, a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [2] m. atef, m.i. ali, t.m. al-shami, fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, comput. appl. math. 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x. https://doi.org/10.1155/2009/586507 https://doi.org/10.1155/2009/586507 https://doi.org/10.1007/s40314-021-01501-x 16 int. j. anal. appl. (2023), 21:27 [3] i. beg, t. rashid, group decision making using intuitionistic hesitant fuzzy sets, int. j. fuzzy logic intell. syst. 14 (2014), 181–187. https://doi.org/10.5391/ijfis.2014.14.3.181. [4] n. caǧman, s. enginoǧlu, f. citak, fuzzy soft set theory and its application, iran. j. fuzzy syst. 8 (2011), 137-147. [5] t. guntasow, s. sajak, a. jomkham, a. iampan, fuzzy translations of a fuzzy set in up-algebras, j. indones. math. soc. 23 (2017), 1-19. https://doi.org/10.22342/jims.23.2.371.1-19. [6] a. iampan, a new branch of the logical algebra: up-algebras, j. algebra related topics. 5 (2017), 35-54. https://doi.org/10.22124/jart.2017.2403. [7] y.b. jun, s.s. ahn, g. muhiuddin, hesitant fuzzy soft subalgebras and ideals inbck/bci-algebras, sci. world j. 2014 (2014), 763929. https://doi.org/10.1155/2014/763929. [8] y.b. jun, b. brundha, n. rajesh, r.k. bandaru, (3,2)-fuzzy up (bcc)-subalgebras and (3,2)-fuzzy up (bcc)filters, j. mahani math. res. 11 (2022), 1-14. [9] y.b. jun, s.z. song, hesitant fuzzy set theory applied to filters in mtl-algebras, honam math. j. 36 (2014), 813-830. https://doi.org/10.5831/hmj.2014.36.4.813. [10] y. komori, the class of bcc-algebras is not a variety, math. japon. 29 (1984), 391-394. [11] p. mosrijai, w. kamti, a. satirad, a. iampan, hesitant fuzzy sets on up-algebras, konuralp j. math. 5 (2017), 268-280. [12] j. somjanta, n. thuekaew, p. kumpeangkeaw, a. iampan, fuzzy sets in up-algebras, ann. fuzzy math. inform. 12 (2016), 739-756. [13] v. torra, hesitant fuzzy sets, int. j. intell. syst. 25 (2010), 529-539. https://doi.org/10.1002/int.20418. [14] v. torra, y. narukawa, on hesitant fuzzy sets and decision, in: 2009 ieee international conference on fuzzy systems, ieee, jeju island, south korea, 2009: pp. 1378–1382. https://doi.org/10.1109/fuzzy.2009.5276884. [15] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65) 90241-x. [16] b. zhu, z. xu, m. xia, dual hesitant fuzzy sets, j. appl. math. 2012 (2012), 879629. https://doi.org/10. 1155/2012/879629. https://doi.org/10.5391/ijfis.2014.14.3.181 https://doi.org/10.22342/jims.23.2.371.1-19 https://doi.org/10.22124/jart.2017.2403 https://doi.org/10.1155/2014/763929 https://doi.org/10.5831/hmj.2014.36.4.813 https://doi.org/10.1002/int.20418 https://doi.org/10.1109/fuzzy.2009.5276884 https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1155/2012/879629 https://doi.org/10.1155/2012/879629 1. introduction 2. preliminaries 3. intuitionistic hesitant fuzzy bcc-filters 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 146-156 http://www.etamaths.com some perturbed ostrowski type inequalities for functions whose first derivatives are of bounded variation hüseyin budak∗ and mehmet zeki sarikaya abstract. the main aim of this paper is to establish some new perturbed ostrowski type integral inequalities for functions whose first derivatives are of bounded variation. some perturbed ostrowski type inequalities for lipschitzian and monotonic mappings are also obtained. 1. introduction in 1938, ostrowski [20] established a following useful inequality: theorem 1. let f : [a,b] → r be a differentiable mapping on (a,b) whose derivative f′ : (a,b) → r is bounded on (a,b) , i.e. ‖f′‖∞ := sup t∈(a,b) |f′(t)| < ∞. then, we have the inequality (1.1) ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ , for all x ∈ [a,b]. the constant 1 4 is the best possible. the following definitions will be frequently used to prove our results. definition 1. let p : a = x0 < x1 < ... < xn = b be any partition of [a,b] and let ∆f(xi) = f(xi+1) −f(xi), then f is said to be of bounded variation if the sum m∑ i=1 |∆f(xi)| is bounded for all such partitions. definition 2. let f be of bounded variation on [a,b], and ∑ ∆f (p) denotes the sum n∑ i=1 |∆f(xi)| corresponding to the partition p of [a,b]. the number b∨ a (f) := sup {∑ ∆f (p) : p ∈ p ([a,b]) } , is called the total variation of f on [a,b] . here p([a,b]) denotes the family of partitions of [a,b] . in [14], dragomir proved the following ostrowski type inequalities related functions of bounded variation: theorem 2. let f : [a,b] → r be a mapping of bounded variation on [a,b] . then (1.2) ∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) f(x) ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f) holds for all x ∈ [a,b] . the constant 1 2 is the best possible. 2010 mathematics subject classification. 26d15, 26a45, 26d10. key words and phrases. bounded variation; perturbed ostrowski type inequalities; riemann-stieltjes integrals. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 146 some perturbed ostrowski type inequalities 147 in the past, many authors have worked on ostrowski type inequalities for function of bounded variation, see for example ([1]-[4],[6]-[9],[11]-[16],[19]). for a function of bounded variation v : [a,b] → c. we define the cumulative variation function (cvf) v : [a; b] → [0,∞) by v (t) := t∨ a (v), the total variation of v on the interval [a,t] with t ∈ [a,b]. it is know that the cvf is monotonic nondecreasing on [a,b] and is continuous in a point c ∈ [a,b] if and only if the generating function v is continuing in that point. if v is lipschitzian with the constant l > 0, i.e. |v(t) −v(s)| ≤ l |t−s| , for any t,s ∈ [a,b] , then v is also lipschitzian with the same constant. a simple proof of the following lemma was given in [15]. lemma 1. let f,u : [a,b] → c. if f is continuous on [a,b] and u is of bounded variation on [a,b] , then the riemann-stieltjes integral b∫ a f(t)du(t) exist and (1.3) ∣∣∣∣∣∣ b∫ a f(t)du(t) ∣∣∣∣∣∣ ≤ b∫ a |f(t)|d ( t∨ a (u) ) ≤ max t∈[a,b] |f(t)| b∨ a (u). in [8], authors gave the following ostrowski type inequality for mapping whoose first derivatives are of bounded variation: theorem 3. let f : [a,b] → r be such that f′ is a continuous function of bounded variation on [a,b] . then we have the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt− 1 2 [f(x) + f(a + b−x)] + 1 2 ( x− 3a + b 4 ) [f′(x) −f′(a + b−x)] ∣∣∣∣ ≤ 1 16 [ 5 (x−a)2 − 2 (x−a) (b−x) + (b−x)2 b−a + 4 ∣∣∣∣x− 3a + b4 ∣∣∣∣ ] b∨ a (f′) for any x ∈ [ a, a+b 2 ] . for recent related results, see [5],[7] and [9]. moreover, dragomir proved some perturbed ostrowski type inequalities for functions of bounded variation in [17, 18]. the aim of this paper is to obtain new perturbed ostrowski type inequalities for mappings whose first derivatives are of bounded variation. 2. some identities before we start our main results, we state and prove following lemma: lemma 2. let f : [a,b] → c be a twice differantiable function on (a,b) . then for any λ1(x) and λ2(x) complex number the following identity holds( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ] = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ1(x)t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ2(x)t]   , 148 budak and sarikaya where the integrals in the right hand side are taken in the riemann-stieltjes sense. proof. using the integration by parts for riemann-stieltjes, we have x∫ a (t−a)2 d [f′(t) −λ1(x)t](2.2) = x∫ a (t−a)2 df′(t) −λ1(x) x∫ a (t−a)2 dt = (t−a)2 f′(t) ∣∣∣x a − 2 x∫ a (t−a) f′(t)dt− λ1(x) 3 (t−a)3 ∣∣∣∣x a = (x−a)2 f′(x) − 2  (t−a) f(t)|xa − x∫ a f(t)dt  − λ1(x) 3 (x−a)3 = (x−a)2 f′(x) − 2 (x−a) f(x) + 2 x∫ a f(t)dt− λ1(x) 3 (x−a)3 and b∫ x (t− b)2 d [f′(t) −λ2(x)t](2.3) = b∫ x (t− b)2 df′(t) −λ2(x) b∫ x (t− b)2 dt = (t− b)2 f′(t) ∣∣∣b x − 2 b∫ x (t− b) f′(t)dt− λ1(x) 3 (t− b)3 ∣∣∣∣b x = −(b−x)2 f′(x) − 2  (t− b) f(t)|bx − b∫ x f(t)dt  − λ2(x) 3 (b−x)3 = (b−x)2 f′(x) − 2 (b−x) f(x) + 2 b∫ x f(t)dt− λ1(x) 3 (x−a)3 . if we add the equality (2.2) and (2.3) and devide by 2(b−a), we obtain required identity. � corollary 1. under assumption of lemma 2 with λ1(x) = λ2(x) = λ(x), we have ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ] (2.4) = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ(x)t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ(x)t]   for all x ∈ [a,b] . some perturbed ostrowski type inequalities 149 remark 1. if we choose λ(x) = 0 in (2.4), then we have the following identity ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.5) = 1 2   1 b−a x∫ a (t−a)2 df′(t) + 1 b−a b∫ x (t− b)2 df′(t)   for all x ∈ [a,b] . corollary 2. under assumption of lemma 2 with λ1(x) = λ1 ∈ c and λ2(x) = λ2 ∈ c, we get ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.6) − 1 6(b−a) [ λ1(x−a)3 + λ2(b−x)3 ] = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ1t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ2t]   . in particular, taking λ1 = λ2 = λ we have ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ 6(b−a) [ (x−a)3 + (b−x)3 ] (2.7) = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λt] + 1 b−a b∫ x (t− b)2 d [f′(t) −λt]   . 3. inequalities for functions whose first derivatives are of bounded variation we denote by ` : [a,b] → [a,b] the identity function, namely `(t) = t for any t ∈ [a,b] . theorem 4. let : f : [a,b] → c be a twice differantiable function on i◦ and [a,b] ⊂ i◦. if the first derivative f′ is of bounded variation on [a,b] , then∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(3.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt + b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′ −λ1(x)`) + (b−x)2 b∨ x (f′ −λ2(x)`) ] 150 budak and sarikaya ≤ 1 2(b−a) ×   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] max { x∨ a (f′ −λ1(x)`), b∨ x (f′ −λ2(x)`) } (b−a)2, max { (x−a)2, (b−x)2 }[ x∨ a (f′ −λ1(x)`) + b∨ x (f′ −λ2(x)`) ] for any x ∈ [a,b] . proof. taking modulus (2.1) and applying lemma 1, we get ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(3.2) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2   1 b−a ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + 1b−a ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   ≤ 1 2(b−a)   x∫ a (t−a)2 d ( t∨ a (f′ −λ1(x)`) ) + b∫ x (t− b)2 d ( t∨ a (f′ −λ2(x)`) ) . integrating by parts in the riemann-stieltjes integral, we get x∫ a (t−a)2 d ( t∨ a (f′ −λ1(x)`) ) (3.3) = (t−a)2 t∨ a (f′ −λ1(x)`) ∣∣∣∣∣ x a − 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = (x−a)2 x∨ a (f′ −λ1(x)`) − 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = 2 x∫ a (t−a) ( x∨ a (f′ −λ1(x)`) ) dt− 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = 2 x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt some perturbed ostrowski type inequalities 151 and b∫ x (t− b)2 d ( t∨ a (f′ −λ2(x)`) ) (3.4) = (t− b)2 t∨ a (f′ −λ2(x)`) ∣∣∣∣∣ b x − 2 b∫ x (t− b) ( t∨ a (f′ −λ2(x)`) ) dt = −(x− b)2 x∨ a (f′ −λ2(x)`) − 2 b∫ x (t− b) ( t∨ a (f′ −λ2(x)`) ) dt = −2 b∫ x (b− t) ( x∨ a (f′ −λ2(x)`) ) dt + 2 b∫ x (b− t) ( t∨ a (f′ −λ2(x)`) ) dt = 2 b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt. if we put the identities (3.3) and (3.4) in (3.2), then we obtain the first inequality in (3.1). moreover, we have, (3.5) x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt ≤ 1 2 (x−a)2 x∨ a (f′ −λ1(x)`) and (3.6) b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt ≤ 1 2 (b−x)2 b∨ x (f′ −λ2(x)`). with the inequalities (3.5) and (3.6), the proof of theorem 4 is completed. � corollary 3. if we chosose λ1(x) = λ2(x) = 0, then we have the following inequality∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′) ) dt + b∫ x (b− t) ( t∨ x (f′`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′) + (b−x)2 b∨ x (f′) ] ≤ b−a 2   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ][ 1 2 b∨ a (f′) + 1 2 ∣∣∣∣ x∨ a (f′) − b∨ x (f′) ∣∣∣∣ ] , [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 b∨ a (f′) for all x ∈ [a,b] . 152 budak and sarikaya corollary 4. under assumption of theorem 4 with λ1(x) = λ2(x) = λ(x), we have (3.7) ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′ −λ(x)`) ) dt + b∫ x (b− t) ( t∨ x (f′ −λ(x)`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′ −λ(x)`) + (b−x)2 b∨ x (f′ −λ(x)`) ] ≤ b−a 2   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] × [ 1 2 b∨ a (f′ −λ(x)`) + 1 2 ∣∣∣∣ x∨ a (f′ −λ(x)`) − b∨ x (f′ −λ(x)`) ∣∣∣∣ ] , [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 b∨ a (f′ −λ(x)`) for all x ∈ [a,b] . corollary 5. if we choose λ(x) = λ and x = a+b 2 in (3.7), then we have the following identity ∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f(x) − λ(b−a)2 24 ∣∣∣∣∣∣ ≤ 1 (b−a)   a+b 2∫ a (t−a)  a+b2∨ t (f′ −λ`)  dt + b∫ a+b 2 (b− t)   t∨ a+b 2 (f′ −λ`)  dt   ≤ (b−a) 8 b∨ a (f′ −λ(x)`). 4. inequalities for functions whose first derivatives are lipschitzian theorem 5. let f : [a,b] → c be a twice differantiable function on i◦ and [a,b] ⊂ i◦. if there exist the positive numbers k1(x) and k2(x) such that f ′ −λ1(x)` is lipschitzian with the constant k1(x) on the interval [a,x] and f′−λ2(x)` is lipschitzian with the constant k2(x) on the interval [x,b] , then we have for any x ∈ [a,b] ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(4.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ (b−a)2 6 [ k1(x) ( x−a b−a )3 + k2(x) ( b−x b−a )3] some perturbed ostrowski type inequalities 153 ≤ (b−a)2 6   [( x−a b−a )3 + ( b−x b−a )3] max{k1(x),k2(x)} , [( x−a b−a )3p + ( b−x b−a )3p]1p [(k1(x)) q + (k1(x)) q ] 1 q p > 1, 1 p + 1 q = 1, [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]3 [k1(x) + k2(x)] . proof. it is known that, if g : [c,d] → c is riemann integrable and u : [c,d] → c is lipschitzian with the constant k > 0, then the riemann-stieltje integral d∫ c g(t)du(t) exist and ∣∣∣∣∣∣ d∫ c g(t)du(t) ∣∣∣∣∣∣ ≤ k d∫ c |g(t)|dt. taking the madulus (2.1), we get∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a)   ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   ≤ 1 2(b−a)  k1(x) x∫ a ∣∣∣(t−a)2∣∣∣dt + k2(x) b∫ x ∣∣∣(t− b)2∣∣∣dt   = (b−a)2 6 [ k1(x) (x−a) 3 + k2(x) (b−x) 3 ] = (b−a)2 6 [ k1(x) ( x−a b−a )3 + k2(x) ( b−x b−a )3] . this completes the proof of first inequality in (4.1). using the hölder’s inequality, we have k1(x) ( x−a b−a )3 + k2(x) ( b−x b−a )3 ≤   [( x−a b−a )3 + ( b−x b−a )3] max{k1(x),k2(x)} , [( x−a b−a )3p + ( b−x b−a )3p]1p [(k1(x)) q + (k1(x)) q ] 1 q p > 1, 1 p + 1 q = 1, [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]3 [l1(x) + l2(x)] which completes the proof. � 154 budak and sarikaya corollary 6. under assumption of theorem 5 with k1(x) = k2(x) = k and λ1(x) = λ2(x) = λ(x), we have ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣(4.2) ≤ 1 6 [ 1 2 + ∣∣∣∣∣x− a+b 2 b−a ∣∣∣∣∣ ]3 k(b−a)2. corollary 7. if we choose x = a+b 2 and λ(x) = λ ∈ c in (4.2), we get the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f ( a + b 2 ) − λ(b−a)2 48 ∣∣∣∣∣∣ ≤ 148k(b−a)2. 5. inequalities for mappings whose first derivatives are monotonic function theorem 6. let f : [a,b] → c be a twice differantiable function on i◦ and [a,b] ⊂ i◦. if λ1(x) and λ2(x) are real numbers such that f ′ − λ1(x)` is monotonic nondecreasing on the interval [a,x] and f′ − λ2(x)` is monotonic nondecreasing on the interval [x,b] , then for any x ∈ [a,b] the following inequalities hold: ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(5.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a) [ (x−a)2 [f′(x) −f′(a) −λ1(x) (x−a)] + (b−x)2 [f′(b) −f′(x) −λ2(x) (b−x)] ] ≤ 1 2(b−a)   [ 1 2 [f′(b) −f′(a) −λ1(x) (x−a) −λ2(x) (b−x)] + ∣∣∣f′(x) − f′(a)+f′(b)2 − 12λ1(x) (x−a) + 12λ2(x) (b−x)∣∣∣] × [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] (b−a)2, max { (x−a)2 , (b−x)2 } × [f′(b) −f′(a) −λ1(x) (x−a) −λ2(x) (b−x)] . proof. taking the madulus (2.1), we have∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(5.2) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a)   ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   some perturbed ostrowski type inequalities 155 since f′ −λ1(x)` is monotonic nondecreasing on the interval [a,x] , we have x∫ a (t−a)2 d [f′(t) −λ1(x)t](5.3) ≤ (x−a)2 [f′(x) −λ1(x)x−f′(a) + λ1(x)a] = (x−a)2 [f′(x) −f′(a) −λ1(x) (x−a)] and similarly, since f′ −λ2(x)` is monotonic nondecreasing on the interval [x,b] , we have b∫ x (t− b)2 d [f′(t) −λ2(x)t](5.4) ≤ (b−x)2 [f′(b) −λ2(x)b−f′(x) + λ2(x)x] = (b−x)2 [f′(b) −f′(x) −λ2(x) (b−x)] . if we put (5.3) and (5.4) in (5.2), we obtain the first inequality in (5.1). the proofs of last inequalities are obvious, they are omitted. � corollary 8. under assumption of theorem 6 with λ1(x) = λ2(x) = λ(x), we have∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣(5.5) ≤ 1 2(b−a) [ (x−a)2 [f′(x) −f′(a) −λ(x) (x−a)] + (b−x)2 [f′(b) −f′(x) −λ(x) (b−x)] ] ≤ b−a 2 ×   [ f′(b)−f′(a) 2 − 1 2 λ(x) (b−a) ∣∣∣f′(x) − f′(a)+f′(b)2 −λ(x) (x− a+b2 )∣∣∣ × [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] , [f′(b) −f′(a) −λ(x) (b−a)] [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 . corollary 9. if we choose x = a+b 2 and λ(x) = λ in (5.5), we get the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f ( a + b 2 ) − λ(b−a)2 48 ∣∣∣∣∣∣ ≤ (b−a)8 [f′(b) −f′(a) −λ (b−a)] . references [1] m. w. alomari, a generalization of weighted companion of ostrowski integral inequality for mappings of bounded variation, rgmia research report collection, 14(2011), art. id 87. [2] m. w. alomari and m.a. latif, weighted companion for the ostrowski and the generalized trapezoid inequalities for mappings of vounded variation, rgmia research report collection, 14(2011), art. id 92. [3] m.w. alomari and s.s. dragomir, mercer–trapezoid rule for the riemann–stieltjes integral with applications, journal of advances in mathematics, 2 (2)(2013), 67–85. [4] h. budak and m.z. sarıkaya, on generalization of dragomir’s inequalities, rgmia research report collection, 17(2014), art. id 155. [5] h. budak and m.z. sarıkaya, new weighted ostrowski type inequalities for mappings with first derivatives of bounded variation, transylvanian journal of mathematics and mechanics, in press. 156 budak and sarikaya [6] h. budak and m.z. sarikaya, a new generalization of ostrowski type inequality for mappings of bounded variation, rgmia research report collection, 18(2015), art. id 47. [7] h. budak and m.z. sarikaya, on generalization of weighted ostrowski type inequalities for functions of bounded variation, rgmia research report collection, 18(2015), art. id 51. [8] h. budak and m. z. sarikaya, a new ostrowski type inequality for functions whose first derivatives are of bounded variation, moroccan journal of pure and applied analysis 2(1)(2016), 1–11. [9] h. budak and m.z. sarikaya, a companion of ostrowski type inequalities for mappings of bounded variation and some applications, rgmia research report collection, 19(2016), art. id 24. [10] h. budak, m.z. sarikaya and a. qayyum, improvement in companion of ostrowski type inequalities for mappings whose first derivatives are of bounded variation and application, rgmia research report collection, 19(2016), art. id 25. [11] s. s. dragomir, approximating real functions which possess nth derivatives of bounded variation and applications, computers and mathematics with applications 56(2008) 2268–2278. [12] s. s. dragomir, the ostrowski integral inequality for mappings of bounded variation, bulletin of the australian mathematical society, 60(1) (1999), 495-508. [13] s. s. dragomir, on the ostrowski’s integral inequality for mappings with bounded variation and applications, mathematical inequalities & applications, 4 (2001), no. 1, 59–66. [14] s. s. dragomir, a companion of ostrowski’s inequality for functions of bounded variation and applications, international journal of nonlinear analysis and applications, 5 (2014) no. 1, 89-97. [15] s. s. dragomir, refinements of the generalised trapezoid and ostrowski inequalities for functions of bounded variation. arch. math. (basel) 91 (2008), no. 5, 450–460. [16] s. s. dragomir, some perturbed ostrowski type inequalities for functions of bounded variation, preprint rgmia research report collection, 16 (2013), art. id 93. [17] s. s. dragomir, some perturbed ostrowski type inequalities for functions of bounded variation, rgmia research report collection, 16(2013), art. id 93. [18] s. s. dragomir, perturbed companions of ostrowski’s inequality for functions of bounded variation, rgmia research report collection, 17(2014), art. id 1. [19] w. liu and y. sun, a refinement of the companion of ostrowski inequality for functions of bounded variation and applications, arxiv:1207.3861v1, (2012). [20] a. m. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, comment. math. helv. 10(1938), 226-227. department of mathematics, faculty of science and arts, düzce university, düzce-turkey ∗corresponding author: hsyn.budak@gmail.com international journal of analysis and applications issn 2291-8639 volume 2, number 1 (2013), 26-37 http://www.etamaths.com slow growth and optimal approximation of pseudoanalytic functions on the disk devendra kumar abstract. pseudoanalytic functions (paf) are constructed as complex combination of real-valued analytic solutions to the stokes-betrami system. these solutions include the generalized biaxisymmetric potentials. mccoy [10] considered the approximation of pseudoanalytic functions on the disk. kumar et al. [9] studied the generalized order and generalized type of paf in terms of the fourier coefficients occurring in its local expansion and optimal approximation errors in bernstein sense on the disk. the aim of this paper is to improve the results of mccoy [10] and kumar et al. [9]. our results apply satisfactorily for slow growth. 1. introduction generalized biaxisymmetric potential (gbasp) that are harmonic at the origin may be expanded, in analogy with the taylor’s series for analytic functions of a single complex variable, in a convergent series of homogeneous harmonic polynomials on an open set. pseudoanalytic functions are constructed as complex combinations of real-valued function pair that are analytic solutions of stokes-beltrami system (sbs); a generalization of the cauchy-riemann equations that is linked to the gbasp equation by eliminating one of the dependent variables from the system. pseudoanalytic functions provide sufficient basis for the transformation of bernstein’s ideas through transform and special function methods. the real part of pseudoanalytic function i.e., eliminating the harmonic conjugate gives the theory of gbasp. the gbasp equation frequently found in the summability theory of [2] jacobi series as r ∂ ∂r { rα+β+1ρ(α,β)(θ)r ∂u ∂r } + ∂ ∂θ { rα+β+1ρ(α,β)(θ) ∂u ∂θ } = 0 ρ(α,β)(θ) = (sin θ/2)2α+1(cos θ/2)2β+1,α ≥ β > − 1 2 where (r,θ) are the plane polar coordinates. the domain of the potential u is a simply connected region with smooth boundary in the upper half plane c+ = c∩re(z) ≥ 0. the existence of a harmonic conjugate ν of u is implied in the sense 2010 mathematics subject classification. 30e10, 41a20. key words and phrases. approximation errors, generalized order and types, pseudoanalytic functions and stokes-beltrami system. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 26 slow growth and optimal approximation 27 of the generalized stokes-beltrami system (sbs); r ∂ν ∂r = −rα+β+1ρ(α,β)(θ) ∂u ∂θ ∂ν ∂θ = rα+β+1ρ(α,β)(θ)r ∂u ∂r a system that reduces to the cauchy-riemann equations of analytic function theory in the limit α = β = −1 2 . following along the lines of analytic function theory, a pseudoanalytic function [1,5,12] (paf) is defined as the complex combination f(reiθ) = u(r,θ) + iν(r,θ) of a real valued analytic function pair formed from the potential u and the principal branch of its harmonic conjugate ν = ν(r,θ). the disk dr(r > 1) of maximum radius on which a paf f exists, is designated by f ∈ p(dr). if f is an entire paf, it has no singularities in the finite c+− plane and writes f ∈ p(c). kumar [8] studied the relationship between the pseudoanalytic functions and bergman-gilbert type integral operators for gbasp and polynomial approximation. bergman [3] and gilbert [5] generalize the operation of taking the real part. they obtained bounded linear operators which transform analytic functions to solution u, where integral operators are developed to provide the transformation from analytic functions to solutions of corresponding elliptic equation. bers [4] and vekua [11] have also extended function theory so that solutions u of elliptic equations can be obtained as u = re(f), where f is a pseudoanalytic function sharing many of the properties associated with classical analytic functions of a single complex variable. also mccoy [10] considered the approximation of pseudoanalytic functions on the disk and obtained some coefficient and bernstein type growth theorems. kapoor and nautiyal [6] characterized the order and type of gbasp’s (not necessarily entire) in terms of rates of decay of approximation errors on both sup norm and lδ−norm, 1 ≤ δ < ∞. all these authors have not studied the generalized growth of pseudoanalytic functions on the disk. our results and methods are different from all those authors mentioned above and apply satisfactorily for slow growth. let p and q are two positive functions defined on (0,∞), strictly increasing and infinitely differentiable such that lim x→∞ p(x) = lim x→∞ q(x) = ∞, lim x→∞ p(cx) p(x) = 1, lim x→∞ q((1 + w(x))x) q(x) = 1, lim x→0 w(x) = 0, p(x/q−1(cp(x))) = (1 + o(x))p(x), for x →∞, lim x→∞ ∣∣∣∣d(q−1(cp(x)))d(log x) ∣∣∣∣ ≤ b, b is a non zero positive constant and d(u) means the differential of u. 28 kumar kumar et al. [9] defined the (p,q)−order and (p,q)−type (or generalized order and generalized type) of pseudoanalytic functions f ∈ p(dr) with radial limits as: ρ0(p,q) = lim sup r→r p(log mr(f)) q(r/(r−r)) , σ0(p,q) = lim sup r→r p(log mr(f)) [q(r/(r−r))]ρ0(p,q) , 1 < r < r, where mr(f) = max{|f(reiθ)| : reiθ ∈ dr},r < r. in [9] kumar et al. studied the generalized order ((p,q)−order) and generalized type ((p,q)−type) of a paf in terms of the fourier coefficients occurring in its local expansion and optimal approximation errors in bernstein sense on the disk. they obtained these results by using the following condition lim x→∞ ∣∣∣∣d(q−1(cp(x)))d(log x) ∣∣∣∣ < b, this condition does not hold for p = q. therefore the results fail to exist for p = q. in this paper we shall improve the results of kumar et al. [9] by using the concept of generalized order of slow growth introduced by kapoor and nautiyal [7] with the help of general function as: let l denote the class of functions h satisfying the following conditions: (i) h is defined on (0,∞), strictly increasing to infinity differentiable such that lim x→∞ h(x) = ∞. (ii) limx→∞ h((1+w(x)).x) h(x) = 1, for every function w, such that limx→0 w(x) = 0. let ∆ denote the class of function h satisfying condition (i) and (iii) limx→∞ h(cx) h(x) = 1, for every c > 0. let ω be the class of functions h satisfying (i) and (iv) and ω be the class of functions satisfying (i) and (v) where (iv) there exists a γ ∈ ω and x0,k1 and k2 such that, for all x > x0. 0 < k1 ≤ lim x→∞ d(h(x)) d(γ(log x)) ≤ k2 < ∞ (v) limx→∞ d(h(x)) d(log x) = k3, 0 < k3 < ∞. now we define the (p,p)−order and (p,p)−type of f ∈ p(dr) (or generalized growth) by ρ(p,p) = lim sup r→r p(log mr(f)) p(r/(r−r)) , σ(p,p) = lim sup r→r p(log mr(f)) [p(r/(r−r))]ρ(p,p) . kapoor and nautiyal [7] showed that classes ω and ω are contained in ∆. further, ω ∪ ω = φ. slow growth and optimal approximation 29 2. generalized order and generalized type with fourier coefficients of pseudoanalytic functions the purpose of this section is to establish the relationship of the generalized growth (p−growth) of pseudoanalytic functions in a disk with fourier coefficients occurring in its local expansion. in a neighborhood of the origin, the pseudoanalytic function paf f ∈ p(dr) has the local expansion f(reiθ) = ∞∑ n=0 anwnfn(re iθ),reiθ ∈ dr, fn(re iθ) = un(r,θ) + iνn(r,θ),n = 0, 1, 2, . . . and an real-valued. write (2.1) lim sup n→∞ p(n) p [ n log(n2α+1|an|rn) ] = µ(p,p). first we prove lemma 2.1. let p(x) ∈ ω and µ > 0. for every r > 1, the maximum of the function x → w(x,r) = x log(r/r) + x p−1(p(x)/µ) is reached for x = xr solution of the equation (2.2) x = p−1 { µp [ 1 −d log(p−1(p(x)/µ))/d(log x) log(r/r) ]} . proof. the proof follows on the lines of [9, lemma 2.1] by simple calculation replacing q by p. lemma 2.2. let f(reiθ) = ∑∞ n=0 anwnfn(re iθ),f ∈ p(dr). for every 1 < r < r and p(x) ∈ ω, we put m(r,f) = sup n {‖anwn‖rn,r > 0}, ‖f‖ = { ‖f‖δ = [∫ ∫ d1 |f|δrdrdφ ] , 1 ≤ δ < ∞ ‖f‖∞ = m1(f), δ = ∞ , and ρ0(p,p) = lim sup r→r p(log m(r,f)) p(r/(r−r)) then ρ(p,p) ≤ ρ0(p,p). proof. let f(z) = ∞∑ n=0 anz n m(r,f) ≤ ∞∑ n=0 |an|rn ≤ ∞∑ n=0 |an|wnrn, 30 kumar substituting r = rξr1−ξ(r/r)1−ξ in above inequality we get m(r,f) ≤ ∞∑ n=0 |an|wn(rξr1−ξ)n(r/r)(1−ξ) n , (r/r) < 1 or m(r,f) ≤ ∞∑ n=0 sup(|an|wn(rξr1−ξ)n(r/r)(1−ξ) n ) or m(r,f) ≤ m(r ′ ,f) ∞∑ n=0 (r/r)(1−ξ) n ,r ′ = (rξr1−ξ), ≤ m(r ′ ,f) 1 1 − (r/r)1−ξ or log m(r,f) ≤ log m(r ′ ,f) − log(1 − (r/r)1−ξ). if the function r → m(r ′ ,f) is bounded, then ρ(p,p) = ρ0(p,p) = 0. so we can assume that m(r ′ ,f) →∞ as r → r. then, for every r sufficiently close to r p(log m(r,f)) p(r/(r−r)) ≤ p(log m(r ′ ,f) − log(1 − (r/r)1−ξ)) p(r/(r−rξr1−ξ)) . p(r/(r−rξr1−ξ)) p(r/(r−r)) . since p(r/(r−rξr1−ξ)) p(r/(r−r)) → 1 as r → r, we obtain by passing to limits on both sides ρ(p,p) ≤ ρ0(p,p). hence the proof is complete. theorem 2.1. let let p(x) ∈ ω and f(reiθ) = ∑∞ n=0 anwnfn(re iθ,f ∈ p(dr),reiθ ∈ dr such that µ(p,p) defined by (2.1) is finite. then f is the restriction of a pseudo analytic function in p(dr)(r > 1) and its (p,p)−order ρ(p,p) = µ(p,p). proof. it can be seen [9] that for every 1 < r < r the series ∑∞ n=0 anwnfn(re iθ) is convergent in dr. now first we show that ρ(p,p) ≤ µ(p,p). by (2.1) we have for every ε > 0, there exists n(ε) such that for every n > n(ε), (2.3) p(n) ≤ (µ(p,p) + ε)p ( n log(|an|n2α+1rn) ) , since (2.4) log(|an|n2α+1rn) = n log(r/r) + log(|an|n2α+1rn) using (2.3) in (2.4) we get (2.5) log(|an|n2α+1rn) ≤ n  log(r/r) + 1 p−1 ( p(n) µ )   . slow growth and optimal approximation 31 choose n = xr = p −1  µp  1 −d ( p−1 ( p(x) µ )) /d(log x) log(r/r)     . using the properties of the function p ( d log(p−1( p(x)µ )) d(log x) = 0(1) ) , and the function t → log t, (log(1 + x) = (1 + o(1)).x,x → 0), we have n = xr = (1 + o(1))p −1 [µp(r/(r−r))] , and replacing in (2.5), we have log(|an|n2α+1rn) ≤ (1 + o(1))p−1(µp(r/(r−r))) ( log(r/r) + 1 r/(r−r) ) since r r−r > 1, it gives log(|an|n2α+1rn) ≤ c0p−1(µp(r/(r−r)). by the properties of function p and proceeding the limit supremum as r sufficiently close to r we get lim sup r→r p(log m(r,f)) p(r/(r−r)) ≤ µ = µ + ε, or ρ(p,p) ≤ µ(p,p). using lemma 2.2 we obtain (2.6) ρ(p,p) ≤ µ(p,p). now we show that ρ(p,p) ≥ µ(p,p). by the definition of ρ(p,p), for every ε > 0, there exists rε ∈]1,r[ such that for every r ≥ rε we have (2.7) log mr(f) ≤ p−1[(ρ(p,p) + ε)p(r/(r−r))]. since for r ∈]1,r[, (2.8) log(|an|n2α+1rn) = −n log(r/r) + log(|an|n2α+1rn). thus p(n) p ( c1n log(|an|n2α+1rn) ) ≤ ρ(p,p) + ε. now proceeding to limits supremum as n →∞, we get (2.9) µ(p,p) ≤ ρ(p,p). combining (2.6) and (2.9) we complete the proof. let f = ∑∞ n=0 anwnfn(re iθ) be pseudo analytic function of (p,p)−order ρ = ρ(p,p) and write t(p,p) = lim sup n→∞ p(n/ρ){ p ( ρ (ρ−1) n log(|an|n2α+1rn) )}ρ−1 . 32 kumar now we prove lemma 2.3. let p(x) ∈ ω and f = ∑∞ n=0 anwnfn(re iθ). for every r ∈]1,r[, σ1(p,p) = lim sup r→r p(log m(r,f)) (p(r/(r−r)))ρ , then σ(p,p) ≤ σ1(p,p). proof. the proof can be obtain by using the same reasoning as in the proof of lemma 2.2 as p(log m(r,f)) [p(r/(r−r))]ρ ≤ p(log m(rξr1−ξ,f) − log(1 − (r/r)1−ξ)) p(r/(r−rξr1−ξ)) . p(r/(r−rξr1−ξ)) p(r/(r−r)) . proceeding the limit, we get σ(p,p) ≤ σ1(p,p). in view of (2.7) and [10, eq. 2.8], (2.8) gives that log(|an|n2α+1rn) ≤−n log(r/r) + log(n + 2)n2α+1a) + p−1[(ρ(p,p) + ε)p(r/(r−r))]. or log(|an|n2α+1rn) n ≤ ϕ(r,n) where ϕ(r,n) = log(r/r) + 1 n log((n + 2)n2α+1a) + 1 n p−1[(ρ(p,p) + ε).p(r/(r−r))] and a = ‖ρ(α,β)‖δ′, 1δ + 1 δ′ = 1. for r sufficiently close to r and for sufficiently large n,ϕ(r,n) is equivalent to log(r/r) for n →∞ and log(r/r) is equivalent to r−r r = r r − 1 for r → r. then ϕ(r,n) = (1 + o(1)) log(r/r) as n →∞, and log(r/r) = (1 + o(1)) (r−r) r (r → r). therefore for r sufficiently close to r and n sufficiently large log(|an|n2α+1rn) n ≤ (1 + o(1))(r/r − 1). substituting r = r[ 1 + p−1 ( p(n) (ρ(p,p)+ε) )], and applying the properties of function p, we obtain log(|an|n2α+1rn) ≤ c1 n[ p−1 ( p(n) (ρ(p,p)+ε) )], slow growth and optimal approximation 33 theorem 2.2. let p(x) ∈ ω and f = ∑∞ n=0 anwnfn(re iθ) be a pseudoanalytic function on the closed unit disk. if f is of finite generalized (p,p)−order ρ(p,p), and (2.10) t(p,p) = lim sup n→∞ p(n/ρ){ p ( ρ ρ−1. n log(|an|n2α+1rn) )}ρ−1 < ∞. then f is the restriction of a pseudoanalytic function in p(dr)(r > 1) and its (p,p)−type σ(p,p) = t(p,p). proof. the function f is the restriction of a pseudoanalytic function in p(dr) by the definition of t(p,p) and arguments used in theorem 2.1. put t = t(p,p),ρ = ρ(p,p); σ = σ(p,p). if t < ∞ by (2.10), for every ε > 0, there exists n0 ≤ n such that p(n/ρ) ≤ (t + ε) { p ( ρ ρ− 1 n log(|an|n2α+1rn) )}ρ−1 or (2.11) log(|an|n2α+1rn) ≤ ρ (ρ− 1) n p−1 (( 1 t p(n/ρ) )1/(ρ−1)),t = t + ε. since log(|an|n2α+1rn) ≤ n log(r/r) + log(|an|n2α+1rn). using (2.11), we get log(|an|n2α+1rn) ≤ n log(r/r) + ρ ρ− 1 n p−1 (( 1 t p(n/ρ) )1/(ρ−1)). for every r ∈]1,r[, and r sufficiently close to r, we put φ(x,r) = x log(r/r) + ρ ρ− 1 x p−1 (( p(x/ρ) t )1/(ρ−1)). then ∂φ(x,r) ∂x = log(r/r) + ρ ρ− 1 d dx   x p−1 (( p(x/ρ) t )1/(ρ−1))   . then the maximum of the function x → φ(x,r) is reached for x = xr where xr is the unique solution of the equation ∂φ ∂x (x,r) = 0. if we put s = s(x,t, 1 ρ−1 ) = p −1 (( 1 t p(x/ρ) )1/(ρ−1)) , then φ(x,r) = x log(r/r) + x s 34 kumar we have ∂φ ∂x (x,r) = 0 ⇔ log(r/r) + ρ ρ− 1 ( s −xds dx s ) = 0, or log(r/r) = ρ ρ− 1  1 − d(log s)d(log x) s   , as ds dx = ds d(log x) d(log x) dx = 1 x ds d(log x) . since log(r/r) = log ( r −r r + 1 ) ∼ ( r −r r )( as r −r r → 0 ) and ∣∣∣∣∣d [ log ( p−1((p(x/ρ))1/(ρ−1)) )] d log x ∣∣∣∣∣ ≤ b, where b is a constant positive. then by the properties of function p ∈ ω, we have s = (1 + o(1)) ρ ρ− 1 ( r r−r ) , thus p−1 (( p(x/ρ) t )1/ρ−1) = (1 + o(1)) ρ ρ− 1 ( r r−r ) , therefore xr = (1 + o(1))ρp −1(t(p(r/(r−r)))1/(ρ−1)). substituting in the relation (2.11), we have log(|an|n2α+1rn) ≤ sup φ(x,r) = φ(xr,r). replacing x by xr in this last relation we get log(|an|n2α+1rn) ≤ (1 + o(1))ρ−1 ρ p−1(t(p(r/(r−r)))ρ−1) r/(r−r) . since r r−r > 1 and ρ−1 ρ < 1, then log(|an|n2α+1rn) ≤ cp−1(t(p(r/(r−r)))ρ−1). then log(m(r,f)) ≤ cp−1(t(p(r/(r−r)))ρ−1). thus p(log m(r,f)) p(r/(r−r)) ≤ t. proceeding to the limit supremum as r → r, we get σ(p,p) ≤ t(p,p). the result is obviously holds for t = ∞. slow growth and optimal approximation 35 to complete the proof it remains to show that σ(p,p) ≥ t(p,p). put σ = σ(p,p) + ε,ρ = ρ(p,p). suppose that σ < ∞. by definition of σ(p,p), we have for every ε > 0, there exist r0 ∈]1,r[, such that for every r > r0(r > r > n0 > 1) (2.12) log mr(f) ≤ p−1[σ(p(r/(r−r)))ρ]. since for every r ∈]1,r[ log(|an|n2α+1rn) = −n log(r/r) + log(|an|n2α+1rn) then in view of (2.12) and [10,eq. 2.8], we get log(|an|n2α+1rn) ≤−n log(r/r) + log((n + 2)n2α+1a) + p−1[σ(p(r/(r−r)))ρ]. let log(|an|n2α+1rn) n ≤ w(r,n) where w(r,n) = − log(r/r) + 1 n log((n + 2)n2α+1a) + 1 n p−1[σ(p(r/(r−r)))ρ]. for r sufficiently close to r we have lim n→∞ w(r,n) = − log(r/r) = log(r/r). then for n sufficiently large and r sufficiently close to r, we have w(r,n) = (1 + 0(1)) log(r/r),n →∞, then (2.13) 1 n log(|an|n2α+1rn) ≤ (1 + o(1)) log(r/r). assume (2.14) r = r(ρ− 1)p−1 ( 1 σ p(n/ρ) )1/ρ−1 ρ + (ρ− 1)p−1 ( p(n/ρ) σ )1/(ρ−1) . now using (2.13) and the properties of the function p ∈ ω and t → log t, for r sufficiently close to r, we get log(|an|n2α+1rn) n ≤ (1 + o(1))((r/r) − 1). from (2.14) we have r r − 1 = ρ + (ρ− 1)p−1 ( 1 σ p(n/ρ) )1/(ρ−1) (ρ− 1)p−1 ( p(n/ρ) σ )1/(ρ−1) − 1 = ρ (ρ− 1)p−1 ( p(n/ρ) σ )1/(ρ−1) . then for r sufficiently close to r and n sufficiently large we obtain log(|an|n2α+1rn) n ≤ ρ (ρ− 1)p−1 ( p(n/ρ) σ )1/(ρ−1) , 36 kumar or (ρ− 1) ρp−1 ( p(n/ρ) σ )1/(ρ−1) ≤ nlog(|an|n2α+1rn) or ( 1 σ p(n/ρ) )1/(ρ−1) ≤ p ( ρ (ρ− 1) n log(|an|n2α+1rn) ) . therefore p(n/ρ) σ ≤ { p ( ρ (ρ− 1) n log(|an|n2α+1rn) )}ρ−1 or p(n/ρ){ p ( ρ (ρ−1) n log(|an|n2α+1rn) )}ρ−1 ≤ σ = σ + ε. proceeding to the limit supremum as n →∞ we get σ(p,p) ≥ t(p,p). the result is obviously holds for σ(p,p) = ∞. 3. generalized growth and optimal polynomial approximation of pseudoanalytic functions the purpose of this section is to give the relationship between the generalized order (ρ(p,p)) and generalized type (t(p,p)) of a pseudoanalytic function paf and optimal rate of convergence to 0 in the norm defined in lemma 2.2. the approximating pseudoanalytic polynomials of (fixed) degree n are taken from the sets πn = [ p : p(reiθ) = n∑ k=0 ckwkfk(re iθ),ck real ] ,n = 0, 1, 2, . . . . the optimal approximates minimize the error ‖f −p‖ for p ∈ πn in bernstein sense as (3.1) en(f) = inf {‖f −p‖ : p ∈ πn}n = 0, 1, 2, . . . . lemma 3.1. let p(x) ∈ ω and en(f) is defined as (3.1) then (3.2) lim sup n→∞ p(n) p [ n log(|an|n2α+1rn) ] = lim sup n→∞ p(n) p [ n log(en(f)n2α+1rn) ], and (3.3) lim sup n→∞ p(n/ρ){ p ( ρ ρ−1 n log(|an|n2α+1rn) )}ρ−1 = lim sup n→∞ p(n/ρ){ p ( ρ ρ−1 n log(en(f)n2α+1rn) )}ρ−1 . proof. the proof can be obtain by following on the basis of proofs of theorem 2.1 and 2.2 and using the identity [10, eq. 12] lim sup n→∞ |an|1/n = lim sup n→∞ [en(f)] 1/n. slow growth and optimal approximation 37 in the consequence of (3.2) and (3.3) we can prove the following theorem. theorem 3.1. let p(x) ∈ ω and f be a pseudoanalytic function on p(dr)(r > 1). then (i) then the generalized (p,p)−order of f is ρ(p,p) = lim sup n→∞ p(n) p [ n log(en(f)n2α+1rn) ] (ii) the generalized (p,p)−type of f is t(p,p), if and only if, t(p,p) = lim sup n→∞ p(n/ρ){ p ( ρ ρ−1 n log(en(f)n2α+1rn) )}ρ−1 when 0 < ρ(p,p) < ∞. references [1] k.w. bauer and st. ruschewyh, differential operators for partial differential equations and function theoretic methods, lecture notes in math. no. 791, springer, new york, 1980. [2] h. bavinck, jacobi series and approximation, mathematical center tracts no. 39, mathematisch centrum, amsterdam, 1972. [3] s. bergman, integral operators in the theorey of linear partial differential equations, (berlin springer-verlag), 1969. [4] l. bers, theory of pseudoanalytic functions, lecture notes (new york: new york university), 1953. [5] r.p. gilbert, function theoretic methods in partial differential equations, (new york: academic press), 1969. [6] g.p. kapoor and a. nautiyal, growth and approximation of generalized bi-axillay symmetric potentials, indian j. pure appl. math. 19(5)(1988), 464-476. [7] g.p. kapoor and a. nautiyal, polynomial approximation of an entire function of slow growth, j. approx. theory 32, (1981), 64-75. [8] d. kumar, ultraspherical expansions of generalized bi-axially symmetric potentials and pseudoanalytic functions, complex variables and elliptic equations, 53(1) (2008), 53-64. [9] d. kumar, vandna jain and balbir singh, generalized growth and approximation of pseudo analytic functions on the disk,british journal of mathematics and computer science,in press. [10] p.a. mccoy, approximation of pseudoanalytic functions on the unit disk, complex variables and elliptic equations, 6(2)(1986), 123-133. [11] i. vekua, generalized analytic functions (reading, m.a. addison wesley), 1962. [12] a. weinstein, transonic flow and generalized axially symmetric potential theory, proc. nol aeroballistic res. symp. naval ord laboratory, white oak, md, 1949. department of mathematics, m.m.h. college,model town,ghaziabad-201001, u.p., india int. j. anal. appl. (2023), 21:44 fractional vector cross product in euclidean 3-space manisha m. kankarej1,∗, jai p. singh2 1rochester institute of technology, dubai, uae 2b. s. n. v. p. g. college, lucknow university, india ∗corresponding author: manisha.kankarej@gmail.com abstract. in this research a new definition of fractional vector cross product of two vectors in euclidean 3-space is presented. using this new definition the formulae for euclidean norm, fractional triple vector cross product, curl and divergence of fractional vector cross product of an electromagnetic vector field is discussed. in particular cases, all the properties satisfy definition of standard vector cross product for standard orthogonal basis of r3. the new definition has application in radiation characteristic in micro-strip antenna and can be applied in other branches of mathematics and physics. 1. introduction in 1967, crowe [1] presented vector analysis. in 2013, das [2] defined fractional vector cross product and derived fractional curl. he further defined the formulations that were useful in applications in electrodynamics, elastodynamics, fluid flow etc. mishra et. al. [4] gave a simulation approach and applied fractional vector cross product in radiation characteristic in micro-strip antenna. fractional vector cross product was defined by tripathi and kim [5] in 2022. in this paper we present new definition of fractional vector cross product in euclidean 3-space different than in [5] and discussed its properties further with respect to the new definition. 2. fractional vector cross product in euclidean 3 space further to the study of fractional vector cross product in [5], we defined an alternative definition below: received: mar. 27, 2023. 2020 mathematics subject classification. 15a30, 15a63, 26a99, 26b12. key words and phrases. vector cross product; fractional vector cross product; fractional vector triple cross product; curl; divergence. https://doi.org/10.28924/2291-8639-21-2023-44 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-44 2 int. j. anal. appl. (2023), 21:44 definition: let r3 be the euclidean 3-space equipped with standard inner product 〈., .〉. let (e1,e2,e3) be standard orthonormal basis of r3 and β ∈ [0,1] a real number. then, for vectors a = a1e1 +a2e2 +a3e3, b = b1e1 +b2e2 +b3e3 in r3, the β − f ractional vector cross product is defined by a×β b = { (a2b3 −a3b2)sin (βπ 2 ) +(b2 +b3)a1cos (βπ 2 )} e1 + { (a3b1 −a1b3)sin (βπ 2 ) +(b3 +b1)a2cos (βπ 2 )} e2 + { (a1b2 −a2b1)sin (βπ 2 ) +(b1 +b2)a3cos (βπ 2 )} e3 (2.1) from eqn (2.1) we have, ei ×β ej =cos (βπ 2 ) ei +sin (βπ 2 ) ek (2.2) ej ×β ei =cos (βπ 2 ) ej − sin (βπ 2 ) ek (2.3) el ×β el =0 f or l = {1,2,3} (2.4) where (i, j,k) is a cyclic permutation of (1, 2, 3). the equations (2.2), (2.3) and (2.4) are similar to that in [4] and [5]. matrix representation by (2.2), (2.3), (2.4) and linearity we can write equation (2.1) as a×β b =sin (βπ 2 ){ (a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3 } +cos (βπ 2 ){ (b2 +b3)a1e1 +(b3 +b1)a2e2 +(b1 +b2)a3e3 } (2.5) or a×β b =sin (βπ 2 ){ (a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3 } +cos (βπ 2 ) (b1 +b2 +b3)a−cos (βπ 2 ) (a1b1e1 +a2b2e2 +a3b3e3) (2.6) from [3] we have a×β b =  0 a1cos ( βπ 2 ) −a3 sin ( βπ 2 ) a1cos ( βπ 2 ) +a2 sin ( βπ 2 ) a2cos ( βπ 2 ) +a3 sin ( βπ 2 ) 0 a2cos ( βπ 2 ) −a1 sin ( βπ 2 ) a3cos ( βπ 2 ) −a2 sin ( βπ 2 ) a3cos ( βπ 2 ) +a1 sin ( βπ 2 ) 0     b1 b2 b3   (2.7) int. j. anal. appl. (2023), 21:44 3 or a×β b =   (b2 +b3)cos ( βπ 2 ) b3 sin ( βπ 2 ) −b2 sin ( βπ 2 ) −b3 sin ( βπ 2 ) (b3 +b1)cos ( βπ 2 ) b1 sin ( βπ 2 ) b2 sin ( βπ 2 ) −b1 sin ( βπ 2 ) (b1 +b2)cos ( βπ 2 )     a1 a2 a3   (2.8) thus considering a ∈ r3 as a fixed column vector in r3, we see that a ×β b : r3 → r3 is a linear function as given in eqn (2.7) and (2.8). particular case: case 1: the 0-fractional vector cross product is defined by: a×0 b =(b2 +b3)a1e1 +(b3 +b1)a2e2 +(b1 +b2)a3e3 by eqn (2.7) we have, a×0 b =   0 a1 a1 a2 0 a2 a3 a3 0     b1 b2 b3   example 1: using eqn (2.2) we have ei ×0 ej = ei,ej ×0 ei = ej,el ×0 el =0. case 2: the 1-fractional vector cross product is defined by: a×b =(a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3. by eqn (2.7) we have a×b = ∣∣∣∣∣∣∣∣ e1 e2 e3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ =   0 −a3 a2 a3 0 −a1 −a2 a1 0     b1 b2 b3   example 2: using eqn (2.2) we have ei ×ej = ek,ej ×ei =−ek,el ×el =0. where (i, j,k) is a cyclic permutation of (1,2,3). thus, 1-fractional vector cross product is simply the standard vector cross product. 4 int. j. anal. appl. (2023), 21:44 3. properties of fractional vector cross product theorem 3.1. the β fractional vector cross product satisfies a×β (b+c)= a×β b+a×β c (3.1) (a+b)×β c = a×β c +b×β c (3.2) (λa+µb)×β c = λa×β c +µb×β c (3.3) proof. from eqn (2.5), equations (3.1), (3.2) and (3.3) can be proved. � particular case: case 1: the 0-fractional vector cross product satisfies following properties: a×0 (b+c)= a×0 b+a×0 c (a+b)×0 c = a×0 c +b×0 c (λa+µb)×0 c = λa×0 c +µb×0 c case 2: the 1-fractional vector cross product behaves as standard cross product. a× (b+c)= a×b+a×c (a+b)×c = a×c +b×c (λa+µb)×c = λa×c +µb×c theorem 3.2. the β fractional vector cross product satisfies 〈a,a×β b〉=cos (βπ 2 ){ a21(b2 +b3)+a 2 2(b1 +b3)+a 2 3(b1 +b2) } (3.4) 〈a,a×β b〉=cos (βπ 2 ) (b1 +b2 +b3)‖a‖2 −cos (βπ 2 ) (a21b1 +a 2 2b2 +a 2 3b3) (3.5) proof. from eqns (2.5) and (2.6), equations (3.4) and (3.5) can be proved. � int. j. anal. appl. (2023), 21:44 5 particular case: case 1: the 0-fractional vector cross product satisfies following properties: 〈a,a×0 b〉= { a21(b2 +b3)+a 2 2(b1 +b3)+a 2 3(b1 +b2) } 〈a,a×0 b〉=(b1 +b2 +b3)‖a‖2 − (a21b1 +a 2 2b2 +a 2 3b3) case 2: the 1-fractional vector cross product satisfies following properties: 〈a,a×b〉=0 example 3: using eqn (2.2) we have 〈ei,ei ×β ej〉=cos ( βπ 2 ) for 0-fractional vector cross product 〈ei,ei ×0 ej〉=1 for 1-fractional vector product 〈ei,ei ×ej〉=0. theorem 3.3. the β fractional vector cross product satisfies 〈b,a×β b〉=cos (βπ 2 ){ a1b1(b2 +b3)+a2b2(b1 +b3)+a3b3(b1 +b2) } (3.6) 〈b,a×β b〉=cos (βπ 2 ) (b1 +b2 +b3)〈a,b〉−cos (βπ 2 ) (a1b 2 1 +a2b 2 2 +a3b 2 3) (3.7) proof. from eqns (2.5) and (2.6), equations (3.6) and (3.7) can be proved. � particular case: case 1: the 0-fractional vector cross product satisfies following properties: 〈b,a×0 b〉= { a1b1(b2 +b3)+a2b2(b1 +b3)+a3b3(b1 +b2) } 〈a,a×0 b〉=(b1 +b2 +b3)〈a,b〉− (a1b21 +a2b 2 2 +a3b 2 3) 6 int. j. anal. appl. (2023), 21:44 case 2: the 1-fractional vector cross product satisfies following properties: 〈a,a×b〉=0 example 4: using eqn (2.2) we have 〈ej,ei ×β ej〉=0 theorem 3.4. the β fractional vector cross product satisfies 〈a+b,a×β b〉=cos (βπ 2 ){ (a1 +b1)(b2 +b3)a1 +(a2 +b2)(b1 +b3)a2 +(a3 +b3)(b1 +b2)a3 } (3.8) 〈a+b,b×β a〉=cos (βπ 2 ){ (a1 +b1)(a2 +a3)b1 +(a2 +b2)(a1 +a3)b2 +(a3 +b3)(a1 +a2)b3 } (3.9) 〈a+b,a×β b〉−〈a+b,b×β a〉=cos (βπ 2 ){ ∣∣∣∣∣∣∣∣ a2 +b2 a3 +b3 a1 +b1 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ − ∣∣∣∣∣∣∣∣ a3 +b3 a1 +b1 a2 +b2 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ } (3.10) proof. this theorem can be proved from eqn (3.6). � particular case: case 1: the 0-fractional vector cross product satisfies following properties: 〈a+b,a×0 b〉= { (a1 +b1)(b2 +b3)+(a2 +b2)(b1 +b3)+(a3 +b3)(b1 +b2) } 〈a+b,b×0 a〉= { (a1 +b1)(a2 +a3)b1 +(a2 +b2)(a1 +a3)b2 +(a3 +b3)(a1 +a2)b3 } 〈a+b,a×0 b〉−〈a+b,b×0 a〉 = { ∣∣∣∣∣∣∣∣ a2 +b2 a3 +b3 a1 +b1 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣− ∣∣∣∣∣∣∣∣ a3 +b3 a1 +b1 a2 +b2 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ } int. j. anal. appl. (2023), 21:44 7 case 2: the 1-fractional vector cross product satisfies following properties: 〈a+b,a×b〉=0,〈a+b,b×a〉=0 〈a+b,a×b〉−〈a+b,b×a〉=0 example 5: using eqn (2) we have 〈ei +ej,ei ×β ej〉−〈ei +ej,ej ×β ei〉=(ei −ej)cos ( βπ 2 ) for β =0,〈ei +ej,ei ×0 ej〉−〈ei +ej,ej ×0 ei〉= ei −ej for β =1,〈ei +ej,ei ×ej〉−〈ei +ej,ej ×ei〉=0 theorem 3.5. the β fractional vector cross product satisfies (a×β b)+(b×β a)= cos (βπ 2 ){ (b1 +b2 +b3)(a1e1 +a2e2 +a3e3) +(a1 +a2 +a3)(b1e1 +b2e2 +b3e3) } −2cos (βπ 2 ) (a1b1e1 +a2b2e2 +a3b3e3) (3.11) proof. using eqn (2.6) we can prove the theorem. � particular case: case 1: the 0-fractional vector cross product satisfies following property: (a×0 b)+(b×0 a)= { (b1 +b2 +b3)(a1e1 +a2e2 +a3e3) +(a1 +a2 +a3)(b1e1 +b2e2 +b3e3) } −2(a1b1e1 +a2b2e2 +a3b3e3) (3.12) case 2: the 1-fractional vector cross product satisfies following property: (a×b)+(b×a)=0 (3.13) example 6: using eqn (2.2) and (2.3) we have (ei ×β ej)+(ej ×β ei)= cos ( βπ 2 ) (ei +ej) for β =0, (ei ×0 ej)+(ej ×0 ei)= ei +ej 8 int. j. anal. appl. (2023), 21:44 for β =1, (ei ×ej)+(ej ×ei)=0 theorem 3.6. the β fractional vector cross product satisfies ‖a×β b‖2 = { ‖a‖2‖b‖2 −〈a,b〉2 } + { 2b1b2(a1a2 +a 2 3)+2b1b3(a1a3 +a 2 2)+2b2b3(a2a3 +a 2 1) } cos (βπ 2 )2 + ∣∣∣∣∣∣∣∣ a1(b3 +b3) a2(b1 +b3) a3(b1 +b2) a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣sin ( βπ ) (3.14) ‖b×β a‖2 = { ‖a‖2‖b‖2 −〈a,b〉2 } + { 2a1a2(b1b2 +b 2 3)+2a1a3(b1b3 +b 2 2)+2a2a3(b2b3 +b 2 1) } cos (βπ 2 )2 + ∣∣∣∣∣∣∣∣ b1(a3 +a3) b2(a1 +a3) b3(a1 +a2) a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣sin ( βπ ) (3.15) proof. using theorem 3.2 we can prove the theorem. � particular case: case 1: the 0-fractional vector cross product satisfies following property: ‖a×0 b‖2 = {{ ‖a‖2‖b‖2 −〈a,b〉2 } +2b1b2(a1a2 +a 2 3)+2b1b3(a1a3 +a 2 2)+2b2b3(a2a3 +a 2 1) } ‖b×0 a‖2 = {{ ‖a‖2‖b‖2 −〈a,b〉2 } +2a1a2(b1b2 +b 2 3)+2a1a3(b1b3 +b 2 2)+2a2a3(b2b3 +b 2 1) } case 2: the 1-fractional vector cross product satisfies following property: ‖a×b‖2 = { ‖a‖2‖b‖2 −〈a,b〉2 } ,‖b×a‖2 = { ‖a‖2‖b‖2 −〈a,b〉2 } example 7: using eqns ( 2.2) and (2.3) we have int. j. anal. appl. (2023), 21:44 9 ‖ei ×β ej‖2 =cos ( βπ 2 )2 ,‖ej ×β ei‖2 = cos ( βπ 2 )2 for β =0, ‖ei ×0 ej‖2 =1, ‖ej ×0 ei‖2 =1 for β =1, ‖ei ×ej‖2 =0, ‖ej ×ei‖2 =0 theorem 3.7. the β fractional vector cross product satisfies 〈a×β b,c〉=sin (βπ 2 ) ∣∣∣∣∣∣∣∣ c1 c2 c3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣+cos (βπ 2 ) (b1 +b2 +b3)〈a,c〉 −cos (βπ 2 ) (a1b1c1 +a2b2c2 +a3b3c3) (3.16) proof. using eqn (3.4) we can prove the theorem. � particular case: case 1: the 0-fractional vector cross product satisfies following property: 〈a×0 b,c〉=(b1 +b2 +b3)〈a,c〉− (a1b1c1 +a2b2c2 +a3b3c3) case 2: the 1-fractional vector cross product satisfies following property: 〈a×b,c〉= ∣∣∣∣∣∣∣∣ c1 c2 c3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ example 8: using eqn (2.2) and (2.3) we have 〈ei ×β ej,ek〉=sin ( βπ 2 ) for β =0, 〈ei ×0 ej,ek〉=0 for β =1, 〈ei ×ej,ek〉=1 theorem 3.8. the β fractional vector cross product satisfies 10 int. j. anal. appl. (2023), 21:44 (a×β b)×β c =sin (βπ 2 )2 (〈a,c〉b−〈b,c〉a) +cos (βπ 2 ){ sin (βπ 2 (b1 +b2 +b3)(a×c) −sin (βπ 2 ) (a1b1e1 +a2b2e2 +a3b3e3)×c +sin (βπ 2 ) ∣∣∣∣∣∣∣∣ 1 1 1 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣c +cos (βπ 2 )( (b2 +b3)a1 +(b1 +b3)a2 +(b1 +b2)a3 ) c − sin (βπ 2 ) ∣∣∣∣∣∣∣∣ c1e1 c2e2 c3e3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ −cos (βπ 2 )( (b2 +b3)a1c1e1 +(b1 +b3)a2c2e2 +(b1 +b2)a3c3e3 )} (3.17) proof. for vectors c = c1e1+c2e2+c3e3 and d = d1e1+d2e2+d3e3 in r3 in view of eqn (2.6), we have d ×β c =sin (βπ 2 ){ (d2c3 −d3c2)e1 +(d3c1 −d1c3)e2 +(d1c2 −d2c1)e3 } +cos (βπ 2 ){ (c2 +c3)d1e1 +(c3 +c1)d2e2 +(c1 +c2)d3e3 } =sin (βπ 2 ){ (d2c3 −d3c2)e1 +(d3c1 −d1c3)e2 +(d1c2 −d2c1)e3 } +cos (βπ 2 ) (c1 +c2 +b3)d −cos (βπ 2 ) (d1c1e1 +d2c2e2 +d3c3e3) (3.18) using d = a×β b, then from eqn (2.1) we have, d1 = { (a2b3 −a3b2)sin ( βπ 2 ) +(b2 +b3)a1cos ( βπ 2 )} e1 d2 = { (a3b1 −a1b3)sin ( βπ 2 ) +(b3 +b1)a2cos ( βπ 2 )} e2 d3 = { (a1b2 −a2b1)sin ( βπ 2 ) +(b1 +b2)a3cos ( βπ 2 )} e3 using above details with eqn (3.18), we can prove theorem 3.8. � particular case: case 1: the 0-fractional vector cross product satisfies following property: (a×0 b)×0 c = ( (b2 +b3)a1 +(b1 +b3)a2 +(b1 +b2)a3 ) c − ( (b2 +b3)a1c1e1 +(b1 +b3)a2c2e2 +(b1 +b2)a3c3e3 )} int. j. anal. appl. (2023), 21:44 11 case 2: the 1-fractional vector cross product satisfies following property: (a×b)×c = 〈a,c〉b−〈b,c〉a example 9: using (2.2) we have (ei ×β ej)×β ek =cos ( βπ 2 )( ei cos ( βπ 2 ) − sin ( βπ 2 ) ek ) for β =0, (ei ×0 ej)×0 ek = ( ei cos ( βπ 2 ) for β =1, (ei ×ej)×ek =0 4. divergence and curl of fractional vector cross product 4.1. curl of a curl of fractional vector cross product. using eqn (2.5) we have, a×β b =sin (βπ 2 ) ∣∣∣∣∣∣∣∣ e1 e2 e3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣+cos (βπ 2 )   0 a1 a1 a2 0 a2 a3 a3 0     b1 b2 b3   (4.1) alternatively a×β b =sin (βπ 2 ) ∣∣∣∣∣∣∣∣ e1 e2 e3 a1 a2 a3 b1 b2 b3 ∣∣∣∣∣∣∣∣ +cos (βπ 2 )   (b2 +b3) 0 0 0 (b3 +b1) 0 0 0 (b1 +b2)     a1 a2 a3   (4.2) from eqn (4.1) and [2] for a vector field f = f1e1 +f2e2 +f3e3 the fractional curl is: ∇×β f =sin (βπ 2 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 ∂β ∂x ∂β ∂y ∂β ∂z f1 f2 f3 ∣∣∣∣∣∣∣∣∣∣∣∣∣ +cos (βπ 2 )   0 ∂ β ∂x ∂β ∂x ∂β ∂y 0 ∂ β ∂y ∂β ∂z ∂β ∂z 0     f1 f2 f3   (4.3) alternatively, from eqn (4.2) we have 12 int. j. anal. appl. (2023), 21:44 a×β b =sin (βπ 2 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 ∂β ∂x ∂β ∂y ∂β ∂z f1 f2 f3 ∣∣∣∣∣∣∣∣∣∣∣∣∣ +cos (βπ 2 )   (f2 +f3) 0 0 0 (f3 +f1) 0 0 0 (f1 +f2)     ∂β ∂x ∂β ∂y ∂β ∂z   (4.4) by theorem (3.8) we have, a×β (b×β c)= sin (βπ 2 )2 (b〈a,c〉−c〈a,b〉)+cos (βπ 2 ){ sin (βπ 2 (c1 +c2 +c3)(a×b) −sin (βπ 2 ) (b1c1e1 +b2c2e2 +b3c3e3)×c +sin (βπ 2 ) ∣∣∣∣∣∣∣∣ 1 1 1 b1 b2 b3 c1 c2 c3 ∣∣∣∣∣∣∣∣c +cos (βπ 2 ) a ( (c2 +c3)b1 +(c1 +c3)b2 +(c1 +c2)b3 ) − sin (βπ 2 ) ∣∣∣∣∣∣∣∣ a1e1 a2e2 a3e3 b1 b2 b3 c1 c2 c3 ∣∣∣∣∣∣∣∣ −cos (βπ 2 )( (c2 +c3)a1b1e1 +(c1 +c3)a2b2e2 +(c1 +c2)a3b3e3 )} (4.5) from (4.3) and (4.5) we have ∇×β (∇×β f)= sin (βπ 2 )2 (∇〈∇,f〉−f〈∇,∇〉) +cos (βπ 2 ){ sin (βπ 2 ) (∇×∇)(f1 +f2 +f3) −sin (βπ 2 ) ( ∂β ∂x f1e1 + ∂β ∂y f2e2 + ∂β ∂z f3e3)×f +sin (βπ 2 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 1 1 ∂β ∂x ∂β ∂y ∂β ∂z f1 f2 f3 ∣∣∣∣∣∣∣∣∣∣∣∣∣ c +cos (βπ 2 ) ∇ (∂β ∂x (f2 +f3)+ ∂β ∂y (f1 +f3)+ ∂β ∂z (f1 +f2) ) − sin (βπ 2 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣ ∂β ∂x e1 ∂β ∂y e2 ∂β ∂z e3 ∂β ∂x ∂β ∂y ∂β ∂z f1 f2 f3 ∣∣∣∣∣∣∣∣∣∣∣∣∣ −cos (βπ 2 )( ( ∂β ∂x )2(f2 +f3)e1 +( ∂β ∂y )2(f1 +f3)e2 +( ∂β ∂z )2(f1 +f2)e3 )} (4.6) int. j. anal. appl. (2023), 21:44 13 particular case: case 1: the 0-fractional vector cross product satisfies following property: ∇×0 (∇×0 f)= { ∇ (∂0 ∂x (f2 +f3)+ ∂0 ∂y (f1 +f3)+ ∂0 ∂z (f1 +f2) ) − ( ( ∂0 ∂x )2(f2 +f3)e1 +( ∂0 ∂y )2(f1 +f3)e2 +( ∂0 ∂z )2(f1 +f2)e3 )} (4.7) case 2: the 1-fractional vector cross product satisfies following property: ∇× (∇×f)=∇〈∇,f〉−f〈∇,∇〉 (4.8) 4.2. divergence of a curl of fractional vector cross product. using eqn (2.5) and (4.3) we have, ∇.(∇×β f)= (∂β ∂x + ∂β ∂y + ∂β ∂z ) sin (βπ 2 ) { ( ∂β ∂y f3 − ∂β ∂z f2)e1 +( ∂β ∂z f1 − ∂β ∂x f3)e2 +( ∂β ∂x f2 − ∂β ∂y f1)e3 } +cos (βπ 2 ){∂β ∂x (f2 +f3)e1 + ∂β ∂y (f3 +f1)e2 + ∂β ∂z (f1 +f2)e3 } (4.9) ∇.(∇×β f)= cos (βπ 2 ){ ( ∂β ∂x )2(f2 +f3)e1 +( ∂β ∂y )2(f3 +f1)e2 +( ∂β ∂z )2(f1 +f2)e3 } (4.10) particular case: case 1: the 0-fractional vector cross product satisfies following property: ∇.(∇×0 f)= { ( ∂0 ∂x )2(f2 +f3)e1 +( ∂0 ∂y )2(f3 +f1)e2 +( ∂0 ∂z )2(f1 +f2)e3 } (4.11) case 2: the 1-fractional vector cross product satisfies following property: ∇.(∇×f)=0 (4.12) 4.3. curl of a divergence of fractional vector cross product. ∇.f = ∂β ∂x f1 + ∂β ∂y f2 + ∂β ∂z f3 (4.13) ∇×β (∇.f)= ∂β ∂x f1 + ∂β ∂y f2 + ∂β ∂z f3 (4.14) using eqn (4.3) we have, 14 int. j. anal. appl. (2023), 21:44 ∇×β (∇.f)= sin (βπ 2 ) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ e1 e2 e3 ∂β ∂x ∂β ∂y ∂β ∂z ∂β ∂x f1 ∂β ∂y f2 ∂β ∂z f3 ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ +cos (βπ 2 )   0 ∂ β ∂x ∂β ∂x ∂β ∂y 0 ∂ β ∂y ∂β ∂z ∂β ∂z 0     ∂β ∂x f1 ∂β ∂y f2 ∂β ∂z f3   (4.15) particular case: case 1: the 0-fractional vector cross product satisfies following property: ∇×0 (∇.f)=   0 ∂ 0 ∂x ∂0 ∂x ∂0 ∂y 0 ∂ 0 ∂y ∂0 ∂z ∂0 ∂z 0     ∂0 ∂x f1 ∂0 ∂y f2 ∂0 ∂z f3   =0 (4.16) case 2: the 1-fractional vector cross product satisfies following property: ∇× (∇.f)=0 (4.17) 5. conclusion fractional vector cross product is one of the important property in the study of fractional calculus. in this research we have defined a new fractional vector cross product named as β fractional vector cross product and further discussed properties of euclidean norm, fractional triple vector cross product, curl and divergence of fractional vector cross product. this definition is useful for applications in electrodynamics, polarisation, impedance studies etc. where we can see real application of fractional solution. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.j. crowe, a history of vector analysis, the evolution of the idea of a vectorial system, university of notre dame press, notre dame, 1967. [2] s. das, geometrically deriving fractional cross product and fractional curl, int. j. math. comput. 20 (2013), 1-29. [3] r.a. horn, c.r. johnson, matrix analysis, cambridge university press, cambridge, 1985. int. j. anal. appl. (2023), 21:44 15 [4] s. mishra, r.k. mishra, s. patnaik, fractional cross product applied in radiation characteristic in micro-strip antenna: a simulation approach, test eng. manage. 81 (2019), 1392-1401. [5] m.m. tripathi, j.r. kim, fractional vector cross product, pure appl. math. 29 (2022), 103-112. https://doi. org/10.7468/jksmeb.2022.29.1.103. https://doi.org/10.7468/jksmeb.2022.29.1.103 https://doi.org/10.7468/jksmeb.2022.29.1.103 1. introduction 2. fractional vector cross product in euclidean 3 space 3. properties of fractional vector cross product 4. divergence and curl of fractional vector cross product 4.1. curl of a curl of fractional vector cross product 4.2. divergence of a curl of fractional vector cross product 4.3. curl of a divergence of fractional vector cross product 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 2, number 1 (2013), 54-61 http://www.etamaths.com convergence theorems of an implicit iterates with errors for total asymptotically pseudo-contractive mappings g. s. saluja abstract. the goal of this paper is to establish weak and strong convergence theorems of an implicit iteration process with errors to converge to common fixed points for a finite family of uniformly l-lipschitzian total asymptotically pseudocontractive mappings in the framework of banach spaces. our results extend the corresponding result of [2, 5, 8, 10] and many others. 1. introduction and preliminaries in recent years, the implicit iteration scheme for approximating fixed point of nonlinear mappings has been introduced and studied by various authors (see, e.g., [1, 4, 7, 10, 11]). in 2001, xu and ori [11] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a hilbert space h. let c be a nonempty subset of h. let t1, t2, . . . , tn be self-mappings of c and suppose that f = ⋂n i=1 f(ti) , ∅, the set of common fixed points of ti, i = 1, 2, . . . , n. an implicit iteration process for a finite family of nonexpansive mappings is defined as follows, with {tn} a real sequence in (0, 1), x0 ∈ c: x1 = t1x0 + (1 − t1)t1x1, x2 = t2x1 + (1 − t2)t2x2, ... xn = tn xn−1 + (1 − tn )tn xn, xn+1 = tn+1xn + (1 − tn+1)t1xn+1, ... which can be written in the following compact form: xn = tnxn−1 + (1 − tn)tnxn, n ≥ 1(1.1) where tk = tk mod n . (here the mod n function takes values in {1, 2, . . . , n}). and they proved the weak convergence of the process (1.1). 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. total asymptotically pseudo-contractive mapping, common fixed point, implicit iteration with errors, strong convergence, weak convergence, banach space. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 54 convergence theorems of an implicit iterates 55 in 2002, zhou and chang [12] introduced the following implicit iteration scheme for common fixed points of a finite family of asymptotically nonexpansive mappings {ti}ni=1 in banach space: xn = αnxn−1 + (1 −αn)t n n (mod n)xn, n ≥ 1(1.2) by this implicit iteration scheme, zhou and chang proved some weak and strong convergence theorems in banach spaces for a finite family of asymptotically nonexpansive mappings. in 2003, sun [10] modified the implicit iteration process of xu and ori [11] and applied the modified averaging iteration process for the approximation of fixed points of asymptotically quasi-nonexpansive mappings. sun introduced the following implicit iteration process for common fixed points of a finite family of asymptotically quasi-nonexpansive mappings {ti}ni=1 in banach spaces: xn = αnxn−1 + (1 −αn)t k i xn, n ≥ 1,(1.3) where n = (k − 1)n + i, i ∈ i = {1, 2, . . . , n}. in this paper, we propose the following implicit iteration process with errors for a finite family of total asymptotically pseudo-contractive mappings {ti}ni=1 and prove some strong convergence theorems for said mappings and iteration scheme in banach spaces. the results presented in this paper extend the corresponding results of chang [2], miao et al. [5], sun [10], osilike and akuchu [8] and many others. the proposed implicit iteration scheme is as follows: x1 ∈ c and xn = αnxn−1 + (1 −αn)t n n xn + un, ∀n ≥ 1,(1.4) where c is a closed convex subset of a banach space e with c+c ⊂ c, tnn = t n n (mod n) and {un} is a bounded sequence in c. definition 1.1. ([5]) a mapping t : c → c is said to be total asymptotically pseudo contractive if there exists a nonnegative real sequence {µn}, n ≥ 1, with µn → 0 as n →∞ and there exists a strictly function φ: r+ → r+ with φ(0) = 0 such that for all x, y ∈ c, 〈tnx − tn y, j(x − y)〉 ≤ ∥∥∥x − y∥∥∥2 + µnφ(∥∥∥x − y∥∥∥).(1.5) remark 1.2. if φ(λ) = λ2, then (1.5) reduces to 〈tnx − tn y, j(x − y)〉 ≤ (1 + µn) ∥∥∥x − y∥∥∥2 .(1.6) the total asymptotically pseudo contractive mappings coincide with asymptotically pseudo contractive mappings. if µn = 0 for all n ≥ 1, we obtain from (1.5) the class of mappings that includes the class of pseudo contractive mappings. note. the idea of definition 1.1 is to unify various definitions of classes of mappings associated with the class of asymptotically pseudo contractive mappings and which are extensions of pseudo contractive mappings. observe that if c is a nonempty closed convex subset of a real banach space e with c + c ⊂ c and {ti}ni=1 : c → c be n uniformly li-lipschitzian total asymptotically pseudo-contractive mappings. if (1 − αn)l < 1, where l = max{li : i = 56 saluja 1, 2, . . . , n}, then for given xn ∈ c, the mapping wn : c → c defined by wn(x) = αnxn−1 + (1 −αn)t n n x + un, ∀n ≥ 1,(1.7) is a contraction mapping. in fact, the following are observed ‖wnx − wn y‖ =‖αnxn−1 + (1 −αn)t n n x + un − (αnxn−1 + (1 −αn)t n n y + un)‖ =(1 −αn) ∥∥∥tnn x − tnn y∥∥∥ ≤(1 −αn)l ∥∥∥x − y∥∥∥ , ∀x, y ∈ c.(1.8) since (1 −αn)l < 1 for all n ≥ 1, hence wn : c → c is a contraction mapping. by banach contraction mapping principle, there exists a unique fixed point xn ∈ c such that xn = αnxn−1 + (1 −αn)t n n xn + un, ∀n ≥ 1.(1.9) therefore, if (1 − αn)l < 1 for all n ≥ 1, then the iterative sequence (1.4) can be employed for the approximation of common fixed points for a finite family of uniformly li-lipschitzian total asymptotically pseudo-contractive mappings. recall that a banach space e satisfies the opial’s condition [6] if for each sequence {xn} in e weakly convergent to a point x and for all y , x lim inf n→∞ ‖xn − x‖ < lim inf n→∞ ∥∥∥xn − y∥∥∥ . the examples of banach spaces which satisfy the opial’s condition are hilbert spaces and all lp[0, 2π] with 1 < p , 2 fail to satisfy opial’s condition [6]. let c be a nonempty closed convex subset of a banach space e. then i − t is demiclosed at zero if, for any sequence {xn} in c, condition xn → x weakly and limn→∞ ‖xn − txn‖ = 0 implies (i − t)x = 0. recall that a family {ti}ni=1 : c → c with f = ∩ n i=1f(ti) , ∅ is said to satisfy condition (b) [3] on c if there is a nondecreasing function f : [0,∞) → [0,∞) with f (0) = 0, f (r) > 0 for all r ∈ (0,∞) such that for all x ∈ c max 1≤i≤n { ‖x − tix‖ } ≥ f (d(x,f )). in the sequel we need the following lemma to prove our main results. lemma 1.3. (see [9]) let {an}∞n=1, {bn} ∞ n=1 and {cn} ∞ n=1 be sequences of nonnegative real numbers satisfying the inequality an+1 ≤ (1 + bn)an + cn, n ≥ 1. if ∑ ∞ n=1 bn < ∞ and ∑ ∞ n=1 cn < ∞, then limn→∞ an exists. if in addition {an} ∞ n=1 has a subsequence which converges strongly to zero, then limn→∞ an = 0. convergence theorems of an implicit iterates 57 2. main results theorem 2.1. let e be a real banach space. let c be a closed convex subset of e with c + c ⊂ c and {ti}ni=1 be a finite family of uniformly li-lipschitzian total asymptotically pseudo contractive self mappings of c into itself such that f = ∩ni=1f(ti) , ∅ is closed. let l = max{li : i = 1, 2, . . . , n}, ∑ ∞ n=1 ‖un‖ < ∞, ∑ ∞ n=1 µn < ∞, and suppose that there exist ki > 0 such that φi(λi) ≤ kiλi, i = 1, 2, . . . , n. given x1 ∈ c, let {xn}∞n=1 be the sequence generated by an implicit iteration scheme (1.4). if {αn} is chosen such that αn ∈ (0, 1) with 0 < τ < αn < 1, where τ is some constant, then {xn} strongly to a common fixed point of the family {ti}ni=1 if and only if lim infn→∞ d(xn,f ) = 0, where d(x,f ) denotes the distance between x and the set f . proof. the necessity is obvious and so it is omitted. now, we prove the sufficiency. for any p ∈f = ∩ni=1f(ti), from (1.4) and (1.5), we have∥∥∥xn − p∥∥∥2 = ∥∥∥αnxn−1 + (1 −αn)tnn xn + un − p∥∥∥2 = αn〈xn−1 − p, j(xn − p)〉 + (1 −αn)〈t n n xn − p, j(xn − p)〉 +〈un, j(xn − p)〉 ≤ αn ∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −αn)[∥∥∥xn − p∥∥∥2 +µnφ( ∥∥∥xn − p∥∥∥)] + ‖un‖∥∥∥xn − p∥∥∥ ≤ ∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −αn) αn µnφ( ∥∥∥xn − p∥∥∥) + 1 αn ‖un‖ ∥∥∥xn − p∥∥∥ ≤ ∥∥∥xn−1 − p∥∥∥ ∥∥∥xn − p∥∥∥ + (1 −τ) τ µnφ( ∥∥∥xn − p∥∥∥) + 1 τ ‖un‖ ∥∥∥xn − p∥∥∥ .(2.1) simplify both the sides of above inequality, we get ∥∥∥xn − p∥∥∥ ≤ ∥∥∥xn−1 − p∥∥∥ + (1 −τ) τ µn φ( ∥∥∥xn − p∥∥∥)∥∥∥xn − p∥∥∥ + ‖un‖τ ≤ ∥∥∥xn−1 − p∥∥∥ + θn,(2.2) where θn = (1 −τ) τ µn φ( ∥∥∥xn − p∥∥∥)∥∥∥xn − p∥∥∥ + ‖un‖τ . since φ is an strictly increasing continuous function, by hypothesis, there exists k such that φ(‖xn−p‖) ‖xn−p‖ ≤ k and by the assumptions of the theorem we know that∑ ∞ n=1 µn < ∞ and ∑ ∞ n=1 ‖un‖ < ∞, it follows that ∑ ∞ n=1 θn < ∞. then from (2.2), we have d(xn,f ) ≤ d(xn−1,f ) + θn.(2.3) 58 saluja by lemma 1.3, we know that limn→∞ d(xn,f ) exists. because lim infn→∞ d(xn,f ) = 0, then lim n→∞ d(xn,f ) = 0.(2.4) next we prove that {xn} is a cauchy sequence in c. it follows from (2.2) that for any m ≥ 1, for all n ≥ n0 and for any p ∈f , we have∥∥∥xn+m − p∥∥∥ ≤ ∥∥∥xn+m−1 − p∥∥∥ + θn+m ≤ ∥∥∥xn+m−2 − p∥∥∥ + [θn+m + θn+m−1] ≤ . . . ≤ . . . ≤ ∥∥∥xn − p∥∥∥ + n+m∑ k=n+1 θk.(2.5) so we have ‖xn+m − xn‖ ≤ ∥∥∥xn+m − p∥∥∥ + ∥∥∥xn − p∥∥∥ ≤ 2 ∥∥∥xn − p∥∥∥ + n+m∑ k=n+1 θk ≤ 2 ∥∥∥xn − p∥∥∥ + ∞∑ k=n θk(2.6) then, we have ‖xn+m − xn‖ ≤ 2d(xn,f ) + ∞∑ k=n θk, ∀n ≥ n0.(2.7) for any given ε > 0, there exists a positive integer n1 ≥ n0 such that for any n ≥ n1, d(xn,f ) < ε 4 and ∞∑ k=n θk < ε 2 .(2.8) thus, when n ≥ n1, we have ‖xn+m − xn‖ < 2. ε 4 + ε 2 = ε.(2.9) this implies that {xn} is a cauchy sequence in c. thus, the completeness of e implies that {xn} must be convergent. assume that limn→∞ xn = p∗. now, we have to show that p∗ is a common fixed point of {ti : i = 1, 2, . . . , n}, that is we have to show that p∗ ∈ f . suppose for contradiction that p∗ ∈ f c (where f c denotes the complement of f ). since f is a closed subset of e, we have that d(p∗,f ) > 0. but for all p ∈f , we have ∥∥∥p∗ − p∥∥∥ ≤ ∥∥∥p∗ − xn∥∥∥ + ∥∥∥xn − p∥∥∥ ,(2.10) which implies that d(p∗,f ) ≤ ∥∥∥xn − p∗∥∥∥ + d(xn,f ),(2.11) so that, we obtain d(p∗,f ) = 0 as n → ∞, which contradicts d(p∗,f ) > 0. thus, p∗ is a common fixed point of the mappings {ti : i = 1, 2, . . . , n}. this completes the proof. � convergence theorems of an implicit iterates 59 theorem 2.2. let e be a real banach space. let c be a closed convex subset of e with c + c ⊂ c and {ti}ni=1 be a finite family of uniformly li-lipschitzian total asymptotically pseudo contractive self mappings of c into itself such that f = ∩ni=1f(ti) , ∅ is closed. let l = max{li : i = 1, 2, . . . , n}, ∑ ∞ n=1 ‖un‖ < ∞, ∑ ∞ n=1 µn < ∞, and suppose that there exist ki > 0 such that φi(λi) ≤ kiλi, i = 1, 2, . . . , n. given x1 ∈ c, let {xn}∞n=1 be the sequence generated by an implicit iteration scheme (1.4). if {αn} is chosen such that αn ∈ (0, 1) with 0 < τ < αn < 1, where τ is some constant, then {xn} converges strongly to a common fixed point p∗ of the family of mappings {ti}ni=1 if and only if there exists a subsequence {xn j} of {xn} which converges to p∗. proof. the proof of theorem 2.2 follows from lemma 1.3 and theorem 2.1. this completes the proof. � theorem 2.3. let e be a real banach space. let c be a closed convex subset of e with c + c ⊂ c and {ti}ni=1 be a finite family of uniformly li-lipschitzian total asymptotically pseudo contractive self mappings of c into itself such that f = ∩ni=1f(ti) , ∅ is closed. let l = max{li : i = 1, 2, . . . , n}, ∑ ∞ n=1 ‖un‖ < ∞, ∑ ∞ n=1 µn < ∞, and suppose that there exist ki > 0 such that φi(λi) ≤ kiλi, i = 1, 2, . . . , n. given x1 ∈ c, let {xn}∞n=1 be the sequence generated by an implicit iteration scheme (1.4). if {αn} is chosen such that αn ∈ (0, 1) with 0 < τ < αn < 1, where τ is some constant. assume that limn→∞ ‖xn − tixn‖ = 0, for all i ∈ i = {1, 2, . . . , n}. suppose {ti : i = 1, 2, . . . , n} satisfies condition (b), then the sequence {xn} converges strongly to a common fixed point of the mappings {ti : i = 1, 2, . . . , n}. proof. by assumption limn→∞ ‖xn − tixn‖ = 0, for all i ∈ i = {1, 2, . . . , n}. since {ti : i = 1, 2, . . . , n} satisfies condition (b), so condition (b) guarantees that limn→∞ f (d(xn,f )) = 0. since f is a non-decreasing function and f (0) = 0, it follows that limn→∞ d(xn,f ) = 0. therefore, theorem 2.1 implies that {xn} converges strongly to a point in f . this completes the proof. � theorem 2.4. let e be a real banach space satisfying opial’s condition and c be a weakly compact subset of e with c + c ⊂ c. let {ti}ni=1 be a finite family of uniformly li-lipschitzian total asymptotically pseudo contractive self mappings of c into itself such that f = ∩ni=1f(ti) , ∅ is closed. let l = max{li : i = 1, 2, . . . , n}, ∑ ∞ n=1 ‖un‖ < ∞,∑ ∞ n=1 µn < ∞, and suppose that there exist ki > 0 such that φi(λi) ≤ kiλi, i = 1, 2, . . . , n. given x1 ∈ c, let {xn}∞n=1 be the sequence generated by an implicit iteration scheme (1.4) and the sequence {αn} is chosen such that αn ∈ (0, 1) with 0 < τ < αn < 1, where τ is some constant. suppose that {ti : i = 1, 2, . . . , n} has a common fixed point, i − ti for all i ∈ i = {1, 2, . . . , n} is demiclosed at zero and {xn} is an approximating common fixed point sequence for ti, that is, limn→∞ ‖xn − tixn‖ = 0, for all i ∈ i = {1, 2, . . . , n}. then the sequence {xn} defined by (1.4) converges weakly to a common fixed point of the mappings {ti : i = 1, 2, . . . , n}. proof. first, we show that ωw(xn) ⊂ f . let xnk → x weakly. by assumption, we have limn→∞ ‖xn − tixn‖ = 0. since i − ti, for all i ∈ i = {1, 2, . . . , n} is demiclosed at zero, x ∈f . by opial’s condition, {xn} possesses only one weak limit point, that is, {xn} converges weakly to a common fixed point of {ti}ni=1. this completes the proof. � 60 saluja remark 2.5. theorem 2.1 extends the corresponding result of chang [2] to the case of more general class of asymptotically nonexpansive mappings and implicit iteration scheme with errors considered in this paper. remark 2.6. theorem 2.1 also extends the corresponding result of miao et al. [5] to the case of implicit iteration scheme with errors considered in this paper. remark 2.7. theorem 2.1 also extends the corresponding result of sun [10] to the case of more general class of asymptotically quasi nonexpansive mapping and implicit iteration scheme with errors considered in this paper. remark 2.8. theorem 2.1 also extends the corresponding result of osilike and akuchu [8] to the case of more general class of asymptotically pseudo contractive mapping and implicit iteration scheme with errors considered in this paper. 3. conclusion the class of total asymptotically pseudo contractive mapping is more general than the class of pseudo contractive mapping and it also unify various definitions of classes of mappings associated with the class of asymptotically pseudo contractive mappings. thus the results obtained in this paper are good improvement and generalization of several known results in the existing literature. references [1] g.l. acedo and h.k. xu, iterative methods for strict pseudo-contractions in hilbert spaces, nonlinear anal. 67(2007), 2258-2271. [2] s.s. chang, on the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, j. math. anal. appl. 313 (2006), 273-283. [3] c.e. chidume, n. shahzad, strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, nonlinear anal., tma, 62 (2005), no. 6, 1149-1156. [4] g. marino and h.k. xu, weak and strong convergence theorems for strict pseudocontractions in hilbert spaces, j. math. anal. appl. 329(2007), 336-346. [5] q. miao, r. chen and h. zhou, convergence of an implicit iteration process for a finite family of total asymptotically pseudocontractive maps, int. j. math. anal. vol.2, no.9 (2008), 433-436. [6] z. opial, weak convergence of the sequence of successive approximations for nonexpansive mappings, bull. amer. math. soc. 73(1967), 591-597. [7] m.o. osilike, implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, j. math. anal. appl. 294(1)(2004), 73-81. [8] m.o. osilike and b.g. akuchu, common fixed points of a finite family of asymptotically pseudocontractive maps, fixed point theory and applications 2 (2004), 81-88. [9] m.o. osilike, s.c. aniagbosor and b.g. akuchu, fixed points of asymptotically demicontractive mappings in arbitrary banach spaces, panam. math. j. 12 (2002), 77-78. [10] z.h. sun, strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 286 (2003), 351-358. convergence theorems of an implicit iterates 61 [11] h.k. xu and r.g. ori, an implicit iteration process for nonexpansive mappings, numer. funct. anal. optim. 22(2001), 767-773. [12] y.y. zhou and s.s. chang, convergence of implicit iteration processes for a finite family of asymptotically nonexpansive mappings in banach spaces, numer. funct. anal. optim. 23(2002), 911-921. departmentof mathematics, govt. nagarjuna p.g. collegeof science, raipur (c.g.), india international journal of analysis and applications issn 2291-8639 volume 4, number 2 (2014), 148-158 http://www.etamaths.com on λ-type duality of frames in banach spaces renu chugh1, mukesh singh2, and l. k. vashisht3,∗ abstract. frames are redundant system which are useful in the reconstruction of certain classes of spaces. the dual of a frame (hilbert) always exists and can be obtained in a natural way. due to the presence of three banach spaces in the definition of retro banach frames (or banach frames) duality of frames in banach spaces is not similar to frames for hilbert spaces. in this paper we introduce the notion of λ-type duality of retro banach frames. this can be generalized to banach frames in banach spaces. necessary and sufficient conditions for the existence of the dual of retro banach frames are obtained. a special class of retro banach frames which always admit a dual frame is discussed. 1. introduction and preliminaries let h be a hilbert space with inner product 〈., .〉. a countable system e ≡ {fk} ⊂ h in a separable is called a frame (hilbert) for h if there exists positive constants a and b such that a‖f‖2 ≤‖{〈f,fk〉}‖2`2 ≤ b‖f‖ 2, for all f ∈h.(1.1) the positive constants a and b are called the lower and upper frame bounds of the frame e, respectively (the largest a and smallest b for which (1.1) holds are the optimal frame bounds). they are not unique. a frame e ≡ {fn} for h is called tight if it is possible to choose a = b and normalized tight if a = b = 1. if removal of one fn renders the collection e ≡{fk} no longer a frame for h, then e is called an exact frame for h. let e ≡ {fk} be a frame(hilbert) for h. the operator t : `2 →h given by t({ck}) = ∞∑ k=1 ckfk, {ck}∈ `2, is called the synthesis operator or pre-frame operator of e. adjoint of t is the operator t∗ : h→ `2 given by t∗(f) = {〈f,fk〉} and is called the analysis operator of the frame e. composing t and t∗ we obtain the frame operator s = tt∗ : h→h given by s(f) = ∞∑ k=1 〈f,fk〉fk,f ∈h. 2010 mathematics subject classification. 42c15, 42c30; 46b15. key words and phrases. frames, banach frames, retro banach frames, paley-wiener space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 148 on duality of frames in banach spaces 149 it can be easily verified that the frame operator s is a positive continuous invertible linear operator from h onto h. furthermore, every vector f ∈ h can be written as: f = ss−1f = ∞∑ k=1 〈s−1f,fk〉fk.(1.2) the series given in (1.2) converges unconditionally for all f ∈ h and is called frame decomposition or reconstruction formula for the frame. thus, frames are redundant building blocks which can recover the underlying space and which have basis like property. one may observe that the frame decomposition shows that all the information about a given signal (vector) f ∈ h is contained in the system {〈s−1f,fk〉}. the scalars 〈s−1f,fk〉 are called the frame coefficients associated with the frame e ≡{fk}. frames for hilbert space were first introduced by duffin and schaeffer in [7], while working in deep problem in non-harmonic fourier series. for some reason the work of duffin and schaeffer was not continued until 1986, when the fundamental work of daubechies, grossmann and mayer [6] brought this all back to life, right at the dawn of the “wavelet era”. more precisely, daubechies et al. in [6] observed that frames can be used to find series expansions of functions in l2(r) which are very similar to the expansions using orthonormal bases. this was probably the time, when many mathematicians started to see the potential of the topic. since then, the theory of frames has been more widely studied. gröchenig in [10], generalized hilbert frames to banach spaces. before the concept of banach frames was formalized, it appeared in the foundational work of feichtinger and gröchenig [9] related to atomic decompositions. atomic decompositions appeared in the field of applied mathematics providing many applications. an atomic decomposition allow a representation of every vector of the space via a series expansion in terms of a fixed sequence of vectors which we call atoms. on the other hand banach frame for a banach space ensure reconstruction via a bounded linear operator or synthesis operator. in last one decade, various types of frames and reconstruction system in banach spaces have been introduced and studied. retro banach frames were introduced and studied in [15] and further studied in [18, 19]. an excellent approach towards the utility of frames in different direction is given in the book by casazza and kutynoik [1] and in the paper by heil and walnut [12]. the basic theory of frames can be found in [2, 3, 5, 11, 13, 21]. a warm up on duality of frames in hilbert spaces: let us give now a brief discussion on duality of frames in hilbert spaces. suppose that e ≡{fk} is a frame for a hilbert space h. a system e′ ≡ {gk} ⊂ h is called a dual frame of e if e′ is a frame for h and f = ∞∑ k=1 〈f,gk〉fk, for all f ∈h.(1.3) in short we say that (e,e′) is a dual pair. let e ≡{fk} be a frame for h with frame bounds 0 < a, b < ∞ and with frame operator s. then e(natural) ≡ {s−1fk} is a frame for h with frame bounds a−1,b−1. furthermore, the condition in (1.3) follows immediately follows from (1.2). therefore, the system e(natural) ≡{s−1fk} is a dual of e ≡ {fk}. the dual frame e(natural) ≡{s−1fk} is known as canonical dual (or natural dual ) of e ≡{fk} . thus, every frame for a hilbert space admit a 150 chugh, singh, and vashisht dual (at least one). an interesting property of dual frames is that dual of a frame need not be unique. in fact, there may be infinitely many duals for a non-exact frame. a frame has a unique dual if and only if it is exact. for more technical details about duality of frames (hilbert), one may refer to [5, 13]. the following example shows that the dual of a frame (hilbert) need not be unique. infact, it may have infinitely many duals. example 1.1. let {χn} be an orthogonal basis for a hilbert space h. then, {χn} is a parseval frame for h. consider a system e ≡ {fk} = {χ1,χ1,χ2,χ3,χ4, .....}. one can easily verify that ‖f‖2 ≤‖{〈f,fk〉}‖2`2 ≤ 2‖f‖ 2, for all f ∈h. therefore, e is a frame for h with one of the choice of bounds a = 1,b = 2. let s be the frame operator of e. then, the canonical dual of e is given by e′ ≡{s−1(fk)} = { 1 2 χ1, 1 2 χ1,χ2,χ2, ....}. other dual of e are e′′ ≡ {0,χ1, 0,χ2, .....} and e′′′ ≡ {13χ1, 2 3 χ1,χ2,χ3,χ4, .....}. therefore, infinitely many dual of a frame (hilbert) can be constructed. motivation: let us have a look on example 1.1, where is observed that a vector (signal) can be represented in infinitely many ways associated with a given frame. thus, there is an opportunity that even if the canonical dual frame is difficult to find, there exists other duals that are easy to find. now dual frames (like frames) represents a vector in a concern space as series. on the other hand in case of retro banach frames a vector can be reconstructed by pre-frame operator(a bounded linear operator). more precisely, retro banach frame recover the underlying space via an operator and not by mean of an infinite series (see definition 1.2). due to the presence of three banach spaces in the definition of a retro banach frame, it is difficult to define its dual which preserve the property of reconstruction via infinite series (as case of duals of frames for hilbert spaces). not only this, the dual of a hilbert frame has a relation with original frame that can be seen in (1.3). some author have work in the direction of dual of frame [4, 8, 11]. in this paper we introduce the notion of dual (or simply λ-dual) of retro banach frames in banach spaces. this can be generalized to banach frames in banach spaces. it is observed that even an exact retro banach frame does not admit a dual retro banach frame for the underlying space. the said situation is entirely different from the dual of frames (hilbert) for hilbert spaces, where an exact frame admit a unique dual. necessary and sufficient conditions for the existence of retro dual frames are obtained. counterexamples are also given to defend results and remarks. a special class of retro banach frames which always admit retro dual frames is discussed. in fact, we discussed an application of young’s result given in [20]. now we give basic definitions and notations which will be required in this paper. throughout this paper x will denote an infinite dimensional banach space over the field k (k = r or c), x∗ the conjugate space of x and zd is a banach space of scalar-valued sequences indexed by n which is associated with x∗. for a sequence {φn} in x , [φn] denotes the closure of the linear hull of {φn} in the norm topology of x . as usual δk,m denote the kronecker delta which is defined as δk,m = 0, if on duality of frames in banach spaces 151 k 6= m and δk,m = 1, if k = m. the class of all bounded linear operator from a banach space x into a banach space y is denoted by b(x ,y). unless otherwise stated all systems of the form {φk} are indexed by n. definition 1.2. [18, at page 83] a system f ≡ ({φk}, θ) ({φk} ⊂ x , θ : zd → x∗) is called a retro banach frame for x∗ with respect to an associated sequence space zd if, (i) {φ∗(φk)}∈zd, for each φ∗ ∈x∗. (ii) there exist positive constants (0 < a0 ≤ b0 < ∞) such that a0‖φ∗‖≤‖{φ∗(φk)}‖zd ≤ b0‖φ ∗‖, for each φ∗ ∈x∗. (iii) θ is a bounded linear operator such that θ({φ∗(φk)} = φ∗, φ∗ ∈x∗ the positive constant a0, b0 are called the lower and upper retro frame bounds of ({φk}∞k=1, θ), respectively. as in case of frames (hilbert) for hilbert spaces, they are not unique. the operator θ : zd → x∗ is called retro pre-frame operator (or simply reconstruction operator) associated with {φk}∞k=1. a retro banach f ≡ ({φk}, θ) is said to be exact if for each m ∈ n there exists no reconstruction operator θm such that ({φk}, θm)k 6=m is retro banach frame for x∗. retro banach frames were further studied in [18, 19]. lemma 1.3. [18, lemma 2 at page 83] let f ≡ ({φk}, θ) be a retro banach frame for x∗ . then, f is exact if and only if φn 6∈ [φk]k 6=n, for all n ∈ n. lemma 1.4. let x be a banach space and {φ∗n} ⊂ x∗ be a sequence such that {φ ∈x : φ∗n (φ) = 0, for all n ∈ n} = {0}. then, x is linearly isometric to the banach space z = {{φ∗n (φ)} : φ ∈x}, where the norm is given by ‖{φ∗n(φ)}‖z = ‖φ‖x , φ ∈x . 2. dual of retro banach frames definition 2.1. let f ≡ ({φk}, θ) be a retro banach frame for x∗ and let λ be a fixed subset of n. a system {φ∗k} ⊂ x ∗ is called dual retro banach frame of f (or simply λ-dual of f) if (i) φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n\ λ, (ii) there exists a reconstruction operator θ× such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗. if g ≡ ({φ∗k}, θ ×) is a dual of f ≡ ({φk}, θ), then we say that f admits a dual with respect to the system {φ∗k}. in short we say that (f,g) is a retro dual pair. remark 2.2. in definition 2.1, if λ = ∅, then g is called the strong dual of f. remark 2.3. recall that the dual of a frame (hilbert) e for a hilbert space h is associated with e via series expansions (see equation (1.3)). the condition (i), in definition 2.1, gives a relation between a given retro banach frame f and its dual g. there may be other relation but at present we discuss λ-type duality. example 2.4. let {φk} = {χk} be an orthogonal basis for a hilbert space h. then ‖{φ∗(φk)}‖`2 = ‖φ∗‖x∗, φ∗ ∈x∗, for all φ∗ ∈x∗. 152 chugh, singh, and vashisht define θ : `2 → x∗ by θ({φ∗(φk)}) = φ∗. then, θ is reconstruction operator such that f ≡ ({φk}, θ) is a normalized tight exact retro banach frame for x∗ with respect to zd = `2. consider the system {φ∗k}≡{χ ∗ 1,χ ∗ 1,χ ∗ 2,χ ∗ 3,χ ∗ 4, .....}⊂h∗. choose λ = {1}⊂ n. then φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n\ λ. furthermore, there exists a reconstruction operator θ× such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗ with respect to zd = `2 and with one of the choice of bounds a = 1,b = √ 2. hence (f,g) is a dual pair. it may be noted that we can construct infinitely many dual retro banach frames for f. more precisely, there are infinitely many dual pair of the form (f,g). let f be an exact retro banach frame for x∗. then, in general, f has no retro dual frame. the following example provides the existence of an exact retro banach frame f which does not admits a dual retro banach frame. example 2.5. consider the measure space x = l2(ω) with counting measure, where ω = n. let {χn} be an orthonormal basis of x . define {φk}⊂x by φk = χk+1 + χ1, k ∈ n. let z = {{φ∗(φk)} : φ∗ ∈x∗}. then, by using lemma 1.4, z is a banach space with norm given by ‖{φ∗(φk)}‖z = ‖φ∗‖x∗, φ∗ ∈x∗. define θ : z →x∗ by θ({φ∗(φk)) = φ∗, φ∗ ∈x∗. then, θ is a bounded linear operator such that f ≡ ({φk}, θ) is a retro banach frame for x∗ with respect to z. to show f is exact. choose φ∗k = χ ∗ k+1,k ∈ n. then, {φ ∗ k} ⊂ x ∗ and we observe that φ∗n(φm) = δn,m, for all n,m ∈ n. therefore, φn 6∈ [φk]k 6=n, for all n ∈ n. thus, by using lemma 1.3, we conclude that f is an exact retro banach frame for x∗. note the if f is exact, then the condition (i) given in definition 2.1 is satisfied for λ = ∅. to show that the condition (ii) given in definition 2.1 is not satisfied. let θ× be the reconstruction operator such that ({φ∗k}, θ ×) is a retro banach frame for x∗∗. let a0, b0 be a choice of retro bounds for ({φ∗k}, θ ×). then a0‖φ∗∗‖≤‖{φ∗∗(φ∗k)}‖zd ≤ b 0‖φ∗‖, for all φ∗∗ ∈x∗∗.(2.1) in particular for φ∗∗ = χ1, we have φ ∗∗(φ∗k) = 0, for all k ∈ n. therefore, by using retro frame inequality (2.1), we obtain φ∗∗ = 0, a contradiction. hence f has no dual retro banach frame. furthermore, there is no other system {ψ∗k} ⊂ x ∗ such that f admits a dual with respect to {ψ∗k}. remark 2.6. the second dual of a banach space x may have a retro banach frame from its pre-dual space with respect to which a retro banach frame for x∗ does not admit strong duality, but it may have a dual pair. on duality of frames in banach spaces 153 the following example defend remark 2.6 example 2.7. let f ≡ ({φk}, θ) be a retro banach frame for x∗ given in example 2.5. choose φ∗k = χ ∗ k,k ∈ n and let z 0 = {{φ∗∗(φ∗k)} : φ ∗∗ ∈x∗∗}. then, z0 is a banach space of sequences of scalars with norm given by ‖{φ∗∗(φ∗k)}‖z0 = ‖φ ∗∗‖x∗∗, φ∗∗ ∈x∗∗. therefore, θ× : {φ∗∗(φ∗k)}→ φ ∗∗ is a bounded linear operator from z0 onto x∗∗ such that g0 ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗ with respect to z0 with bounds a = b = 1. choose λ = {1}. then, φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n\ λ. therefore, (f,g0) is a retro dual pair, which is not strong. remark 2.8. by example 2.7 we observe that a retro banach frame f for x∗ may have dual pair but does not admit a strong dual (even) with respect to an orthonormal basis for x∗. the following proposition provides sufficient conditions for the existence of the dual retro banach frames. it is sufficient to prove the result for the existence of strong dual retro banach frame. we can extend the same construction for the existence of arbitrary dual pair. proposition 2.9. let f ≡ ({φk}, θ) be a retro banach frame for x∗. then, f admits a strong dual retro banach frame if there exists a system ({φ∗k}⊂x ∗ such that φ∗n(φm) = δn,m, for all n,m ∈ n and there exists an injective closed linear operator φ∗∗ → {φ∗∗(φ∗k)} with closed range from x ∗∗ to ẑ0, where ẑ0 is some banach space of scalar valued sequences. these conditions are not necessary. proof. let u : x∗∗ → ẑ0 be given by u(φ∗∗) = {φ∗∗(φ∗k)}, φ ∗∗ ∈ x∗∗. then, by hypothesis, u is injective and closed linear operator with closed range r(u). therefore, u−1 : r(u) ⊂ ẑ0 → x∗∗ is closed [14]. thus, by using closed graph theorem [14], there exists a constant c > 0 such that ‖u(φ∗∗)‖≥ c‖φ∗∗‖, for all φ∗∗ ∈x∗∗. let, if possible, there exists no reconstruction operator θ× such that ({φ∗k}, θ ×) is a retro banach frame for x∗∗. then, by using hahn-banach theorem there exists a non-zero functional φ∗∗0 ∈x∗∗ such that φ∗∗0 (φ∗k) = 0, for all k ∈ n. therefore 0 = ‖u(φ∗∗0 )‖≥ c‖φ ∗∗ 0 ‖. this gives φ∗∗0 = 0, a contradiction. therefore, there exists a reconstruction operator θ× such that ({φ∗k}, θ ×) is a retro banach frame for x∗∗ with respect to some associated banach space of scalar valued sequences. hence f admits a dual retro banach frame for the underlying space. to show that the conditions are not necessary. let x be the measure space given in example 2.5. let φk = k 2χk,k ∈ n, where {χk} is an orthonormal basis for x . then, there exists a bounded linear operator θ such that f = ({φk}, θ) is a retro banach frame for x∗. choose φ∗k = 1 k2 χk,k ∈ n. then, φ∗i (φj) = δi,j for all i,j ∈ n. by the nature of the system {φ∗k}, we conclude that there exists a reconstruction operator θ × such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗ with respect to ẑo = `2. thus, (f,g) is a dual pair. 154 chugh, singh, and vashisht define θ̃ : x∗∗ →ẑo(= `2) by θ̃(φ∗∗) = {φ∗∗(φ∗j )} = { ξj j2 }, φ∗ = {ξj}∈x∗∗. it can be verified that the range of the operator θ̃ is not closed. � the following theorem gives necessary and sufficient condition for a given retro banach frame to admit its dual. theorem 2.10. let f ≡ ({φn}, θ) be a retro banach frame for x∗. then, f has a dual retro banach frame if and only if there exists a system {φ∗n}⊂x∗ such that φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n \ λ and dist(φ ∗,ln) → 0 as n →∞, for all φ∗ ∈ x∗, where ln = [φ∗1, φ∗2, ...., φ∗n], for all n ∈ n. proof. suppose first that f ≡ ({φn}, θ) has a dual retro banach frame. then, by definition we can find a system {φ∗n} ⊂ x∗ such that φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n \ λ and a reconstruction operator θ× such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗. let a0 and b0 be a choice of bounds for g. then a0‖φ∗∗‖x∗∗ ≤‖{φ∗∗(φ∗n)}‖(x∗∗)d ≤ b0‖φ ∗∗‖x∗∗, for all φ∗∗ ∈x∗∗.(2.2) suppose that the condition dist(φ,ln) → 0 as n → ∞, for all φ∗ ∈ x∗, is not satisfied. then, there exists a non zero functional φ∗0 ∈x∗ such that lim n→∞ dist(φ∗0,ln) 6= 0. note that dist(φ∗0,ln) ≥ dist(φ∗0,ln+1) for all n ∈ n. this is because {ln} is a nested system of subspaces and by definition of distance of a point from a set. now {dist(φ∗0,ln)} is bounded below monotone decreasing sequence. so, {dist(φ∗0,ln)} is convergent and its limit is a positive real number, since otherwise dist(φ∗0,ln) → 0 as n →∞ (which is not possible). let limn→∞ dist(φ∗0,ln) = ξ > 0. choose d = ⋃ n ln. then, by using the fact that dist(φ ∗ 0,d) = inf{dist(φ∗0,ln)}, we obtain dist(φ∗0,d) ≥ ξ > 0.(2.3) now we show that φ∗0 6∈ d. let if possible, φ∗0 ∈ d. then, we can find a sequence {ζn}⊂ d such that dist(ζn, φ∗0) → 0 as n →∞. by using (2.3), we have dist(φ∗0,d) ≥ ξ. therefore, dist(ζn, φ∗0) ≥ ξ > 0. this is a contradiction to the fact that dist(ζn, φ ∗ 0) → 0 as n →∞. hence φ∗0 6∈ d. thus, by using hahn-banach theorem, there exists a non zero functional φ∗∗0 ∈x∗∗ such that φ∗∗0 (φ ∗ n) = 0, for all n ∈ n. therefore, by using retro frame inequality (2.2), we have φ∗∗0 = 0, a contradiction. hence dist(φ∗,ln) → 0 as n →∞, for all φ∗ ∈ x∗. to prove the converse part, assume that there exists a system {φ∗n}⊂x∗ is such that φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n\ λ, and dist(φ∗,ln) → 0 as n →∞, for all φ∗ ∈x∗. on duality of frames in banach spaces 155 then, in particular, for each � > 0 and for each φ∗ ∈ x∗, we can find a φ∗j from some lk such that ‖φ∗ − φ∗j‖ < �. therefore, by using lemma 1.4, z = {{φ∗∗(φ∗n)} : φ∗∗ ∈ x∗∗} is a banach space of sequences of scalars with norm given by ‖{φ∗∗(φ∗n)}‖z = ‖φ ∗∗‖x∗∗, φ∗∗ ∈x∗∗. define θ0 : z →x∗∗ by θ0({φ∗∗(φ∗k)}) = φ ∗∗, φ∗∗ ∈x∗∗. then, θ0 is a bounded linear operator such that such that ({φ∗n}, θ0) is a retro banach frame for x∗∗ with respect to z. hence f has a dual retro banach frame. � the existence of duality with respect to certain sequence space is also interested. the following theorem provides the necessary and sufficient condition for an exact retro banach frame to admit a dual frame with respect to a given sequence space. theorem 2.11. let f ≡ ({φn}, θ) be a retro banach frame for x∗ with respect to zd and let ad = {{φ∗∗(φ∗k)} : φ ∗∗ ∈x∗∗}. then, f has a dual retro banach frame with respect to the sequence space ad if and only if there exists a system {φ∗n}⊂x∗ such that φ∗j (φl) = δj,l, for all l ∈ n and for all j ∈ n\λ and the analysis operator u : φ∗∗ →{φ∗∗(φ∗k)} is a bounded below continuous linear operator from x ∗∗ onto ad. proof. suppose first that f has a dual retro banach frame g with respect to ad. then, there exists a reconstruction operator θ× such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗ with respect to ad. therefore, there are positive constants a0 and b0 such that a0‖φ∗∗‖≤‖{φ∗∗(φ∗k)}‖ad ≤ b0‖φ ∗∗‖, for each φ∗∗ ∈x∗∗.(2.4) now consider the analysis operator u : x∗∗ →ad which is given by u(φ∗∗) = {φ∗∗(φ∗k)}, φ ∗∗ ∈x∗∗. then, linearity and ontoness of u is obvious. by using upper retro frame inequality in (2.4), we have ‖u(φ∗∗)‖ad ≤ b0‖φ ∗∗‖, for each φ∗∗ ∈x∗∗. therefore, ‖u‖≤ b0. hence u is continuous. similarly, by using lower retro frame inequality in (2.4), we have ‖u(•)‖ad ≥ a0‖•‖x∗∗. hence u is bounded below. for the reverse part, assume that u is bounded below. then, using lemma 1.4, ad is a banach space with the norm given by ‖{φ∗∗(φ∗k)}‖ad = ‖φ ∗∗‖x∗∗, φ∗∗ ∈x∗∗. define θ× : ad → x∗∗ by θ×({φ∗∗(φ∗k)}) = φ ∗∗. then, θ× is a bounded linear operator such that ({φ∗k}, θ ×) is a retro banach for x∗∗ with respect to ad. the theorem is proved. � 156 chugh, singh, and vashisht 3. applications in this section we give some applications of the duality of retro banach frames. first two two examples provides an application of the theorem 2.10. example 3.1. in this example, we give an application of theorem 2.10 in duality of retro banach frames in finite dimensional banach spaces. we start with a well known frame in a finite dimensional banach space, namely mercedes-benz frame. mercedes-benz frame is one of the popular frame in finite dimensional frame theory. it consists of three vectors which are equiangular situated on the unit circle. more precisely, a mercedes-benz frame is a system consisting of three vectors in r2, namely, a = { φ1 = (0, 1), φ2 = ( − √ 3 2 , −1 2 ), φ3 = ( √ 3 2 , −1 2 ) } . let zd0 = {[φ∗(φ1), φ∗(φ2), φ∗(φ3)]t : φ ∈ x∗}. then, zd0 is a banach space with the norm given by ‖[φ∗(φ1), φ∗(φ2), φ∗(φ3)]t‖zd0 = ‖φ ∗‖x∗, φ∗ ∈x∗. define θ0 : zd0 →x∗ by θ0([φ ∗(φ1), φ ∗(φ2), φ ∗(φ3)] t) = φ∗, φ∗ ∈x∗. then, θ0 is a bounded linear operator such that f0 ≡ ({φi}3i=1, θ0) is a retro banach frame for x∗ with respect to zd0 and with bounds a = b = 1. by using theorem 2.10 and of above argument we conclude that every proper subfamily of the mercedes-benz frame containing two vectors from a is a retro banach frame for (r2)∗ and admits a dual (strong) retro banach frame for the underlying space. this can be extended to arbitrary finite dimensional banach space. more precisely, every retro banach frame for a finite dimensional space contains a strong dual retro banach frame for the underlying space. for more applications of mercedes-benz frame an interested reader may refer to [16, 17]. example 3.2. now we discuss banach frames of wavelet system. let x = l2(r). consider the haar function φ which is defined by φ(x) = 1, if x ∈ [0, 1 2 ), φ(x) = −1, if x ∈ [ 1 2 , 1) and φ(x) = 0, otherwise. let ψj,k(x) = 2 1 2 φ(2jx−k),j,k ∈ z and x ∈ r. the system {ψj,k(x)} is called a wavelet system associated with the window function φ. without loss of generality, we can write {φn(•)} ≡ {ψj,k(•)}. let zd = {{ψ(φn)} : ψ ∈ x∗}. then, zd is banach space of scalar valued sequences with the norm given by ‖{ψ(φn)}‖zd = ‖ψ‖x∗,ψ ∈x ∗.(3.1) define θ : zd →x∗ by θ({ψ(φn)}) = ψ, ψ ∈x∗. then, θ ∈ b(zd,x∗). hence θ is a bounded linear operator such that f ≡ ({φn}, θ) is a retro banach frame for x∗ with bounds a = b = 1. by the nature of construction of the system {φn}, it can be easily verified that φn /∈ [φm]m 6=n, for each n ∈ n. on duality of frames in banach spaces 157 thus, by hahn-banach theorem, there exists a {φ∗n}⊂x∗ such that φ∗n(φm) = δn,m, for all n,m ∈ n. hence f satisfying one of the sufficient conditions in theorem 2.10, with λ = ∅. let ln = [φ ∗ 1, φ ∗ 2, ..., φ ∗ n], for all n ∈ n. note that each φ∗j ∈ l 2(r) is given by φ∗j (φ) = ∫ φψφdµ, where ψφ is the representative of φ. now by using retro frame inequality (3.1), we have dist(ψ∗,ln) = inf f∗∈ln ‖ψ∗ −f∗‖ = inf f∗∈ln (∫ |ψ∗ −f∗|2dµ )1 2 → 0 as n →∞, for all ψ∗ ∈ x∗. hence by theorem 2.10, f admits a dual (strong) retro banach frame . we can construct a dual of f which is not strong. it is interested to know the class of retro banach frames which always admits dual retro banach frames. it is difficult to characterize the class of retro banach frames which have dual retro banach frames. a special class of retro banach frames consisting of complex exponentials in x = l2(−π,π) always admits a dual retro banach frame. this can be proved by using a result by robert young in [20 at page 538] . recall that two sequences {φn} and {ψn} of elements from a hilbert space h are said to be biorthogonal if 〈φn, ψm〉 = δn,m for all n,m ∈ n. a sequence that admits a biorthogonal sequence will be called minimal. a sequence {φn}⊂h is said to be complete in h if zero vector is alone is perpendicular to every φn. a sequence that is both minimal and complete will be called exact. robert young considered a paley-wiener space consisting of all entire functions of exponential type at most π that are square-integrable on the real axis and using paley-wiener theorem proved the following result. theorem 3.3. [20, at page 538] if the sequence of complex exponential {eiλkt} is exact in l2(−π,π), then its biorthogonal sequence is also exact. to conclude the paper we show that retro banach frames consisting of complex exponentials in x = l2(−π,π) always admits a dual retro banach frame. proposition 3.4. let x = l2(−π,π) and φk = eiλkt,k ∈ n. if f ≡ ({φk}, θ) be an exact retro banach frame for x∗ with admissible system {φ∗k}⊂x ∗, then there exists a reconstruction operator θ× such that ({φ∗k}, θ ×) is a retro banach frame for x∗∗. more precisely, f has a dual (strong) retro banach frame. proof. suppose that there is no θ× ∈ b(zy,x∗∗) such that ({φ∗k}, θ ×) is a retro banach frame for x∗∗, where zy is associated banach space of scalar-valued sequences. then, there exists a non-zero φ∗∗ ∈x∗∗(≡x) such that φ∗∗(φ∗k) = ∫ π −π φ∗∗φ∗kdt = 0, for all k ∈ n. by using construction in the proof of theorem 3.3 (for proof see [20] at page 538), we can show that φ∗∗ = 0. this is a contradiction. hence there exists a reconstruction 158 chugh, singh, and vashisht operator θ× ∈ b(zy,x∗∗), where zy = {{φ∗∗(φ∗k)} : φ ∗∗ ∈x∗∗}, such that g ≡ ({φ∗k}, θ ×) is a retro banach frame for x∗∗. � acknowledgement the third author was partially supported by r & d doctoral research programme, university of delhi, delhi. letter no. dean(r)/r&d/2012, dated july 03, 2012. references [1] p.g.casazza and g. kutynoik, finite frames, birkhäuser, 2012. [2] p.g. casazza, the art of frame theory, tawanese j. math., 4(2) (2000), 129–201 [3] p.g. casazza, d. han and d.r. larson, frames for banach spaces, contemp. math., 247 (1999), 149–182. [4] p.g.casazza, g.kutyniok and m.c.lammers, duality principles in frame thoery, j. fourier anal. appl., 10 (2004), 383–408. [5] o. christensen, frames and bases (an introductory course), birkhäuser, boston (2008). [6] i.daubechies, a. grossmann and y. meyer, painless non-orthogonal expansions, j. math. phys. 27 (1986), 1271–1283. [7] r.j. duffin and a.c. schaeffer, a class of non-harmonic fourier series, trans. amer. math. soc., 72 (1952), 341–366. [8] i.daubechies, h. landau and z.landau, gabor time-frequency lattices and the wexler-raz identity, j. fourier anal. appl., 1(4) (1995), 437–478. [9] h.g. feichtinger and k. gröchenig, a unified approach to atomic decompositons via inegrable group representations, lecture notes in mathematics, 1302 (springer, berlin, 1988), 52–73. [10] k. gröchenig, describing functions: atomic decompositions versus frames, monatsh. math., 112, (1991), 1–41. [11] d. han and d.r. larson, frames, bases and group representations, mem. amer. math. soc., 147 (697) (2000), 1–91. [12] c.heil and d. walnut, continuous and discrete wavelet transforms, siam rev., 31 (4) (1989), 628–666. [13] c. heil, a basis theory primer, birkhäuser (expanded edition)(1998). [14] h.heuser, functional analysis, john wiley and sons, new york (1982). [15] p.k.jain, s.k.kaushik and l.k. vashisht, banach frames for conjugate banach spaces, z. anal. anwendungen, 23 (4) (2004), 713–720. [16] j. kovacčević and a. chebira, life beyond bases: the advent of frames (part i), ieee signal processing magazine, 86, july, 2007. [17] j. kovacčević and a. chebira, life beyond bases: the advent of frames (part ii), ieee signal processing magazine, 115, september, 2007. [18] l.k. vashisht, on retro banach frames of type p , azerb. j. math., 2 (1) (2012), 82–89. [19] l.k. vashisht, on φ-schauder frames, twms j. app. and eng. math.(jaem), 2 (1) (2012), 116-120. [20] r. young, on complete biorthogonal systems, proc. amer. math. soc., 83 (3) (1981), 537540. [21] r. young, a introduction to non-harmonic fourier series, academic press, new york (revised first edition 2001). 1department of mathematics, maharshi dayanand university , rohtak, haryana, india 2department of mathematics, goverment college, bahadurgarh, maharshi dayanand university , rohtak, haryana, india 3department of mathematics, university of delhi, delhi-110007, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 82-88 http://www.etamaths.com best proximity points for k− proximal contraction ajay sharma, balwant singh thakur∗ abstract. in this paper, we define k− proximal contraction and prove best proximity point theorem for this contraction. we also provide an illustrative example. 1. introduction and preliminaries banach contraction principle is one of the pivotal results of analysis. it is widely considered as the source of metric fixed point theory. due to its wide applications in various fields, a huge number of generalizations and extended versions of this principle appear in the literature. some important and interesting generalizations of banach’s principle is given in [2, 9, 10, 12, 18]. in 1997, alber and guerre [1] introduced the notion of weakly contractive self mapping. definition 1.1. let (x,d) be a metric space and a be a nonempty subset of x. a mapping t : a → a is said to be weakly contractive if d(tx,ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ a, where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ is positive on (0,∞), ψ(0) = 0 and limt→∞ψ(t) = ∞. if a is bounded, then the infinity condition can be omitted [1]. they [1] further proved that, if a is a closed convex subset of a hilbert space, then a weakly contractive self mapping t on a has a unique fixed point. rhoades [15] extended and improved result of [1] to metric space and established the following: theorem 1.1. let (x,d) be a complete metric space, and suppose that t : x → x satisfies the following inequality (1) d(tx,ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ x where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ(t) = 0 if and only if t = 0. then t has a unique fixed point. 2010 mathematics subject classification. 54h25. key words and phrases. k− proximal contraction; fixed point. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 82 best proximity points for k− proximal contraction 83 if we take ψ(t) = (1 − k)t, where 0 < k < 1, then the inequality (1) reduced to banach contraction, hence the above theorem 1.1 extends banach’s contraction principle. the function ψ involved in the inequality (1) is known as alternating distance ( also called control function). it was initially used in 1984 by khan and sessa [11]. this function and its generalizations have been used in fixed point problems in metric and probabilistic metric spaces, see for example, [13, 14, 17] motivated by chatterjea [7] contraction, recently choudhary [8] introduced weakly c-contraction mapping. definition 1.2. let (x,d) be a metric space. a mapping t : x → x is said to be weakly c−contraction if for all x,y ∈ x, d(tx,ty) ≤ 1 2 [d(x,ty) + d(y,tx)] −ψ(d(x,ty),d(y,tx)), where ψ : [0,∞) × [0,∞) → [0,∞) is a continuous mapping such that ψ(x,y) = 0 if and only if x = y = 0. if we take ψ(x,y) = k(x + y) where 0 < k < 1 2 then it reduces to chatterjea contraction. choudhary [8] further proved the following theorem: theorem 1.2. let (x,d) be a complete metric space. then a weakly c−contraction t : x → x has a unique fixed point. most of the extensions and generalizations of banach’s contraction principle focused on weakening the contractive condition of the operator or weakening the completeness of the metric space. one method of weakening the contractive condition is to consider non self mapping. sankar raj [16] defined the notion of non-self-weakly contractive mappings as follows: definition 1.3. let a,b be nonempty subsets of a metric space (x,d). a mapping t : a → b is said to be a weakly contractive mapping if d(tx,ty) ≤ d(x,y) −ψ(d(x,y)) , for all x,y ∈ a, where ψ : [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ is positive on (0,∞), ψ(0) = 0 and limt→∞ψ(t) = ∞. an element x ∈ a is said to be a fixed point of a map t : a → b if tx = x. clearly, t(a)∩a 6= ∅; is a necessary (but not sufficient) condition for the existence of a fixed point of t . if t(a) ∩a = ∅, then d(x,tx) > 0 for all x ∈ a, that is, the set of fixed points of t is empty. in such a situation, one often attempts to find an element x which is in some sense closest to tx. if such a point exists, we call it best proximity point, i.e., an element x ∈ a is called a best proximity point of t if d(x,tx) = dist(a,b); where dist(a,b) = inf{d(x,y) : x ∈ a,y ∈ b}. best proximity point analysis has been developed in this direction. the goal of the best proximity point theory is to furnish sufficient conditions that assure the existence of best proximity points. 84 sharma and thakur before proceeding further let us fix the following notations: a0 = {x ∈ a : d(x,y) = dist(a,b) for some y ∈ b}, b0 = {y ∈ b : d(x,y) = dist(a,b) for some x ∈ a}. if a and b are closed subsets of a normed linear space such that dist(a,b) > 0, then a0 and b0 are contained in the boundaries of a and b, respectively [6]. let a and b be two nonempty subsets of a metric space (x,d) with a0 6= ∅. then the pair (a,b) is said to have the p− property [16] if and only if d(x1,y1) = dist(a,b) d(x2,y2) = dist(a,b) } ⇒ d(x1,x2) = d(y1,y2) where x1,x2 ∈ a0 and y1,y2 ∈ b0. for any nonempty subset a of x, the pair (a,a) has the p−property. sankar raj [16] proved the following result: theorem 1.3. let a and b be two nonempty closed subsets of a metric space (x,d) such that a0 6= ∅. let t : a → b be a weakly contractive mapping such that t(a0) ⊆ b0. assume that the pair (a,b) has the p− property. then there exists a unique x∗ ∈ a such that d(x∗,tx∗) = dist(a,b). basha [3] introduced the notion of proximal contraction as follows: definition 1.4. a mapping t : a → b is said to be proximal contraction if there exists a non-negative number α < 1 such that, for all u1,u2,x1,x2 ∈ a d(u1,tx1) = dist(a,b) d(u2,tx2) = dist(a,b) } ⇒ d(u1,u2) ≤ αd(x1,x2) . every self mapping that is a proximal contraction is essentially a contraction. definition 1.5. [5] the set b is said to be approximatively compact with respect to a if every sequence {yn} of b satisfying the condition that d(x,yn) → d(x,b) for some x ∈ a has a convergent subsequence. any compact set is approximatively compact, and that any set is approximatively compact with respect to itself. further, if a is compact and b is approximatively compact with respect to a, then a0 and b0 are non-empty. definition 1.6. [4] a mapping t : a → b is said to be a generalized proximal contraction if, for all u1,u2,x1,x2 ∈ a, d(u1,tx1) = dist(a,b) d(u2,tx2) = dist(a,b) } ⇒ d(u1,u2) ≤ d(x1,x2) −ψ(d(x1,x2)), where ψ : [0,∞) → [0,∞) is continuous and non-decreasing such that ψ is positive on (0,∞),ψ(0) = 0, and limt→∞ψ(t) = ∞. motivated by above studies and kannan contraction, we now introduce the notion of k−proximal contraction: definition 1.7. let a and b be two non-empty subsets of a metric space (x,d). then t : a → b is said to be a k−proximal contraction if for all u1,u2,x1,x2 ∈ a, d(u1,tx1) = dist(a,b) and d(u2,tx2) = dist(a,b) best proximity points for k− proximal contraction 85 implies that d(u1,u2) ≤ 1 2 (d(x1,u1) + d(x2,u2)) −ψ(d(x1,u1),d(x2,u2)) where ψ : [0,∞)× [0,∞) → [0,∞) is a continuous and nondecreasing function such that ψ(x1,x2) = 0 if and only if x1 = x2 = 0. we now establish necessary and sufficient condition for the existence and uniqueness of best proximity point of the k− proximal contraction. theorem 1.4. let (x,d) be a complete metric space, a and b be two nonempty, closed subsets of x such that b is approximatively compact with respect to a. suppose that a0 and b0 are non-empty and t : a → b is a non-self-mapping satisfying the following conditions: (i) t is a k−proximal contraction; (ii) t(a0) ⊆ b0. then, there exists a unique element x ∈ a such that d(x,tx) = dist(a,b). further, the sequence {xn} converges to the best proximity point x, where for a fixed x0 ∈ a0 the sequence {xn} is given by d(xn+1,txn) = dist(a,b) for all n ≥ 0. proof. let x0 be a fixed element in a0. since t(a0) ⊆ b0, tx0 is an element of b0. so by the definition of b0, there exists an element x1 ∈ a0 such that d(x1,tx0) = dist(a,b). again, since t(a0) ⊆ b0 we have tx1 ∈ b0, it follows that there is x2 ∈ a0 such that d(x2,tx1) = dist(a,b). continuing this process, we can derive a sequence {xn} in a0, such that d(xn+1,txn) = dist(a,b),(2) for every n ≥ 0. since t is a k−proximal contraction, for each n ∈ n, we have d(xn,xn+1) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)) −ψ(d(xn−1,xn),d(xn,xn+1)) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)). consequently, we get d(xn,xn+1) ≤ d(xn−1,xn). so the sequence {d(xn+1,xn)} is a monotone-decreasing sequence of nonnegative real numbers. then there exists a number µ ≥ 0 such that d(xn,xn+1) → µ as n →∞.(3) now, we will show that µ = 0. since t is a k−proximal contraction, we get d(xn,xn+1) ≤ 1 2 (d(xn−1,xn) + d(xn,xn+1)) −ψ(d(xn−1,xn),d(xn,xn+1)).(4) letting n →∞ in (4), by using (3) and continuity of ψ, we get µ ≤ 1 2 (µ + µ) −ψ(µ,µ) µ ≤ µ−ψ(µ,µ). 86 sharma and thakur or ψ(µ,µ) ≤ 0, which is a contradiction unless µ = 0, that is lim n→∞ d(xn,xn+1) = 0.(5) now, we show that {xn} is a cauchy sequence. suppose that {xn} is not a cauchy sequence. then, for every ε > 0 there exists nk > mk ≥ k with d(xmk,xnk ) ≥ ε and d(xmk,xnk−1) < ε, for each k ∈ n. then, we have, d(xmk,xnk ) ≤ d(xmk,xnk−1) + d(xnk−1,xnk ) < ε + d(xnk−1,xnk ).(6) taking k →∞ in (6) and using (5), we have lim k→∞ d(xmk,xnk ) = ε. by (2), we have (7) lim k→∞ d(xnk+1,txnk ) = dist(a,b) , and (8) lim k→∞ d(xmk+1,txmk ) = dist(a,b) . since t is a k−proximal contraction, by (7), we have (9) d(xnk+1,xmk+1) ≤ 1 2 (d(xnk,xnk+1) + d(xmk,xmk+1)) −ψ (d(xnk,xnk+1),d(xmk,xmk+1)) . since, (10) d(xnk,xmk ) ≤ d(xnk,xnk+1) + d(xnk+1,xmk+1) + d(xmk+1,xmk ) . from (9) and (10), we have d(xnk,xmk ) ≤ d(xnk,xnk+1) + d(xmk+1,xmk ) + 1 2 (d(xnk,xnk+1) + d(xmk,xmk+1)) −ψ (d(xnk,xnk+1),d(xmk,xmk+1)) . letting k →∞, we have ε ≤ 0 + 0 + 0 −ψ(0, 0) = 0 , a contradiction. hence, {xn} is a cauchy sequence. since a is a closed subset of a complete metrics space, there exists x ∈ a such that limn→∞xn = x. taking n → ∞ in (2) and continuity of t, we have d(x,tx) = dist(a,b). this completes the proof. � now, we give an example to illustrate theorem 1.4. example 1.8. let x = r2 with the euclidean metric. suppose that a = {(1,x) : x ∈ r, 0 ≤ x ≤ 1}, and b = {(0,x) : x ∈ r,x ≥ 0}. best proximity points for k− proximal contraction 87 now we define a mapping t : a → b as below: t((1,x)) = ( 0, x 1 + x ) . it is easy to see that d(a,b) = 1, a0 = a and b0 = b. t is continuous, t(a0) ⊂ b0 and d(u1,tx1) = d(a,b) = 1,d(u2,tx2) = d(a,b) = 1. for some u1,u2,x1,x2 ∈ a. we will show that t is satisfy the condition (i) of theorem 1.4 with ψ : [0,∞) × [0,∞) → [0,∞) defined by ψ(t1, t2) = max ( t1 1 + t1 , t2 1 + t2 ) for all (t1, t2) ∈ [0,∞) × [0,∞). suppose u1 = u2 = (1, 0) and x1 = x2 = (0, 0), then d(x1,u1) = 1 , d(x2,u2) = 1 , d(u1,u2) = 0 and ψ (d(x1,u1),d(x2,u2)) = ψ(1, 1) = max ( 1 2 , 1 2 ) = 1 2 . we can see that d(u1,u2) ≤ 1 2 (d(x1,u1) + d(x2,u2)) −ψ(d(x1,u1),d(x2,u2)) , hence, t is a k−proximal contraction references [1] ya. i. alber and s. guerre-delabriere, principle of weakly contractive maps in hilbert spaces, new results in operator theory and its applications, oper. theory adv. appl., 98 (1997), 7-22. [2] s. almezel, q.h. ansari, and m.a. khamsi(eds.), topics in fixed point theory, springer, 2014. [3] s. sadiq basha, best proximity points: optimal solutions, j. optim. theory appl. 151 (2011) no.1, 210–216. [4] s. sadiq basha,best proximity point theorems: resolution of an important non-linear programming problem, optim. lett. 7 (2013), no.6, 1167–1177. [5] s. sadiq basha and n. shahzad, best proximity point theorems for generalized proximal contractions, fixed point theory appl., 2012(2012), article id 42. [6] s. sadiq basha and p. veeramani, best proximity pair theorems for multifunctions with open fibres, j. approx. theory, 103(2000), 119–129. [7] s.k. chatterjee, fixed point theorems, comptes. rend. acad. bulgaria sci. 25(1972), 727– 730. [8] binayak s. choudhury, unique fixed point theorem for weakly c-contractive mappings, kathmandu university journal of science,engineering and technology, 5(2009), no.i, 6–13. [9] vasile i. istratescu fixed point theory: an introduction (mathematics and its applications), springer, 1981. [10] m. a. khamsi and w.a. kirk, an introduction to metric spaces and fixed point theory, pure and applied mathematics (new york). wiley-interscience, new york, 2001. [11] m.s. khan, m. swaleh and s. sessa, fixed point theorems by altering distances between the points, bull. austral. math. soc., 30 (1984) no.1 , 1–9. [12] w.a. kirk, and b. sims(eds.), handbook of metric fixed point theory, kluwer academic publishers, dordrecht, 2001. [13] d. mihet, altering distances in probabilistic menger spaces, nonlinear anal. 71(2009), 2734– 2738. [14] s.v.r. naidu, some fixed point theorems in metric spaces by altering distances, czech. math. j. 53 (2003), no.1, 205–212. [15] b.e. rhoades, some theorems on weakly contractive maps, nonlinear anal. 47 (2001) 2683– 2693. 88 sharma and thakur [16] v. sankar raj, a best proximity theorem for weakly contractive non-self mappings, nonlinear anal. 74 (2011), 4804–4808. [17] k.p.r. sastry and g.v.r. babu, some fixed point theorems by altering distances between the points, indian j. pure appl. math. 30 (1999),no.6, 641–647. [18] d. r smart, fixed point theorems, cambridge university press, london, 1974. cambridge tracts in mathematics, no. 66. school of studies in mathematics, pt. ravishankar shukla university, raipur, 492010, india ∗corresponding author international journal of analysis and applications volume 18, number 5 (2020), 819-837 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-18-2020-819 union soft set theory applied to ordered semigroups raees khan1,∗, asghar khan2, muhammad uzair khan2,3, nasir khan1 1department of mathematics, fata university, darra adam khel, n.m.d. kohat, kp, pakistan 2department of mathematics, abdul wali khan university mardan, pakistan 3department of mathematics & statistics, bacha khan university charsadda, kp, pakistan ∗corresponding author: raeeskhan@fu.edu.pk abstract. the uni-soft type of bi-ideals in ordered semigroup is considered. the notion of a uni-soft bi-ideal is introduced and the related properties are investigated. the concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. the concepts of two types of prime uni-soft bi-ideals of an ordered semigroup s are given and it is proved that, a non-constant uni-soft bi-ideal of s is prime in the second sense if and only if each of its proper δ−exclusive set is a prime bi-ideal of s. the characterizations of left and right simple ordered semigroups are considered. using the notion of uni-soft bi-ideals, some semilattices of left and right simple semigroups are provided. by using the properties of uni-soft bi-ideals, the characterization of a regular ordered semigroup is provided. in the last section of this paper, the characterizations of both regular and intra-regular ordered semigroups are provided. 1. introduction the notion of soft set was introduced in 1999 by molodtsov [20] as a new mathematical tool for dealing with uncertainties. due to its importance, it has received much attention in the mean of algebraic structures such as groups [9], semirings [11], rings [1], ordered semigroups [15] and hemirings [19, 22] and so on. feng et al. discussed soft relations in semigroups (see [12, 13]) and explored decomposition of fuzzy soft sets with finite value spaces. also, feng and li [14] considered soft product operations. jun et al., [15] applied received december 12th, 2017; accepted february 5th, 2018; published july 23rd, 2020. 2010 mathematics subject classification. 06d72, 20m99, 20m12. key words and phrases. ordered semigroup; left/right regular and completely regular ordered semigroup; regular and intraregular ordered semigroup; left and right simple subsemigroup; uni-soft bi-ideal; uni-soft left/right ideal; uni-soft quasi-ideal. ©2020 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 819 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-819 int. j. anal. appl. 18 (5) (2020) 820 the concept of soft set theory to ordered semigroups. they applied the notion of soft sets by molodtsov to ordered semigroups and introduced the notions of (trivial, whole) soft ordered semigroups, soft ordered subsemigroups, soft r-ideals, soft l-ideals, and r-idealistic and l-idealistic soft ordered semigroups [16]. they investigated various related properties by using these notions. in [6–8] khan et al., characterized different classes of ordered semigroups by using uni-soft quasi-ideals and uni-soft ideals. in this paper, we introduce the notion of a uni-soft bi-ideal in ordered semigroups. the concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. we also define two types of prime uni-soft bi-ideals of an ordered semigroup s and prove that, a non-constant uni-soft bi-ideal of s is prime in the second sense if and only if each of its proper δ−exclusive set is a prime bi-ideal of s. the characterizations of some classes of ordered semigroups are given. by using the notion of uni-soft bi-ideals, some semilattices of left and right simple subsemigroups are discussed. using the concept of an δ−exclusive set, it is proved that a soft set of s over u is a uni-soft bi-ideal if and only if the non-empty δ−exclusive set is a bi-ideal. regular ordered semigroups are chracterized by the properties of uni-soft bi-ideals and it is shown that every uni-soft bi-ideal of s over u is idempotent if and only if the ordered semigroup is regular. in the last section of this paper, the characterizations of both regular and intra-regular ordered semigroups are discussed. 2. preliminaries in this section, we give some basic definitions and results, which are necessary for the subsequent sections. by an ordered semigroup we mean a structure (s, ·,≤) such that: (os1) (s, ·) is a semigroup. (os2) (s,≤) is a poset. (os3) (∀a,b,x ∈ s) (a ≤ b =⇒ ax ≤ bx and xa ≤ xb). for a ⊆ s, we denote (a] := {t ∈ s : t ≤ h for some h ∈ a}. for a,b ⊆ s, we have ab := {ab : a ∈ a,b ∈ b}. a nonempty subset a of an ordered semigroup s is called a subsemigroup of s if a2 ⊆ a. a nonempty subset a of s is called a left (resp. right ) ideal of s if: (1) sa ⊆ a(resp. as ⊆ a) and (2) a ∈ a, s 3 b ≤ a, implies b ∈ a. by a two-sided ideal or simply an ideal of s we mean a non-empty subset of s which is both a left and a right ideal of s. a nonempty subset a of an ordered semigroup s is called a bi-ideal of s if: (1) a2 ⊆ a, (2) asa ⊆ a and (3) a ∈ a,s 3 b ≤ a, implies b ∈ a. let s be an ordered semigroup and ∅ 6= a ⊆ s. then the set (a ∪ a2 ∪ asa] is the bi-ideal of s generated by a. in particular, if a = {x} (x ∈ s), then we write (x ∪ x2 ∪ xsx], instead of ({x}∪{x2}∪{x}s{x}] (see [2]). int. j. anal. appl. 18 (5) (2020) 821 3. basic operations of soft sets from now on, u is an initial universe, e is a set of parameters, p(u) is the power set of u and a,b,c... ⊆ e. definition 3.1. a soft set fa over u is defined as fa : e −→ p(u) such that fa(x) = ∅ if x /∈ a. hence fa is also called an approximation function. a soft set fa over u can be represented by the set of ordered pairs fa = {(x,fa(x))|x ∈ e,fa(x) ∈ p(u)} . it is clear from definition 3.1, that a soft set is a parameterized family of subsets of u. note that the set of all soft sets over u will be denoted s(u). definition 3.2. (i) let fa,fb ∈ s(u). then fa is called a soft subset of fb, denoted by fa⊆̃fb if fa(x) ⊆ fb(x) for all x ∈ e. (ii) let fa,fb ∈ s(u). then the soft union of fa and fb, denoted by fa∪̃fb = fa∪b, is defined by( fa∪̃fb ) (x) = fa(x) ∪fb(x) for all x ∈ e. (iii) let fa,fb ∈ s(u). then the soft intersection of fa and fb, denoted by fa∩̃fb = fa∩b, is defined by ( fa∩̃fb ) (x) = fa(x) ∩fb(x) for all x ∈ e. for x ∈ s, we define ax = {(y,z) ∈ s ×s|x ≤ yz}. definition 3.3. let (fs,s) and (gs,s) be two soft sets over u. then, the soft intersection-union product, denoted by fs♦gs, is defined by fs♦gs : s −→ p(u),x 7−→   ⋂ (y,z)∈ax {fs(y) ∪gs(z)} if ax 6= ∅, u if ax = ∅, for all x ∈ s. one can easily prove that ”♦” on s(u) is well defined and the set ( s(u),♦,⊆̃ ) forms an ordered semigroup (see [7]). 4. uni-soft bi-ideals for an ordered semigroup, the soft sets ”∅s” and ”>s” of s over u are defined as follows: ∅s : s −→ p(u),x 7−→∅s(x) = ∅, >s : s −→ p(u),x 7−→>s(x) = u for all x ∈ s. clearly, the soft set ”∅s” (resp. ”>s”) of an ordered semigroup s over u is the least (resp., the greatest ) element of the ordered semigroup (s(u),♦,⊆̃). the soft set ”∅s” is the null element of (s(u),♦,⊆̃) (that int. j. anal. appl. 18 (5) (2020) 822 is fs♦∅s = ∅s♦fs = ∅s and ∅s⊆̃fs for every fs ∈ s(u). the soft set (>s,s) is called the whole soft set over u, where >s(x) = u for all x ∈ s. for a non-empty subset a of s, the characteristic soft set (χa,a) over u is a soft set defined as follows: χa : s −→ p(u),x 7−→   u if x ∈ a,∅ if x ∈ s\a. for the characteristic soft set (χa,a) over u, the soft set (χ c a,a) over u is given as follows: χca : s −→ p(u),x 7−→   ∅ if x ∈ a,u if x ∈ s\a. definition 4.1. (cf. [15]). let s be an ordered semigroup. a soft set (fs,s) of s over u is called a union-soft semigroup (briefly, uni-soft semigroup) of s over u if: fs(xy) ⊆ fs(x) ∪fs(y) ∀x,y ∈ s. definition 4.2. (cf. [15]). let s be an ordered semigroup. a soft set (fs,s) of s over u is called a union-soft left (resp. right) ideal (briefly, uni-soft left (resp. right) ideal) of s over u if (su1) x ≤ y =⇒ fs(x) ⊆ fs(y), (su2) fs(xy) ⊆ fs(y) (resp. fs(xy) ⊆ fs(x) ) ∀x,y ∈ s. if (fs,s) is both a uni-soft left ideal and a uni-soft right ideal of s over u, then (fs,s) is called a uni-soft ideal of s over u. definition 4.3. (cf. [6]). let s be an ordered semigroup. a uni-soft semigroup (fs,s) of s over u is called a union-soft bi-ideal (briefly, uni-soft bi-ideal) of s over u if (su5) x ≤ y =⇒ fs(x) ⊆ fs(y). (su6) fs(xyz) ⊆ fs(x) ∪fs(z) ∀ x,y,z ∈ s. definition 4.4. a soft set (fs,s) of an ordered semigroup (s, ·,≤) over u is called idempotent if (fs♦fs,s) = (fs,s). for a soft set (fa,s) over u and a subset δ of u, the δ-exclusive set of (fa,s) denoted by ea(fa; δ) is defined by ea(fa; δ) := {x ∈ a|fa(x) ⊆ δ} . int. j. anal. appl. 18 (5) (2020) 823 theorem 4.1. a soft set (fs,s) over u is a uni-soft bi-ideal over u if and only if the nonempty δ-exclusive set of (fs,s) is a bi-ideal of s for all δ ∈ p(u). proof. assume that (fs,s) is a uni-soft bi-ideal over u. let δ ∈ p(u) be such that es(fs; δ) 6= ∅. let x,y ∈ s with x ≤ y be such that y ∈ es(fs; δ). then fs(y) ⊆ δ. by (su5) we have fs(x) ⊆ fs(y) ⊆ δ and that x ∈ es(fs; δ). let x,y ∈ es(fs; δ). then fs(x) ⊆ δ and fs(y) ⊆ δ. by (su6) we have fs(xy) ⊆ fs(x) ∪fs(y) ⊆ δ and so xy ∈ es(fs; δ). for x,z ∈ es(fs; δ). then fs(x) ⊆ δ and fs(z) ⊆ δ. by (su6) we have fs(xyz) ⊆ fs(x) ∪fs(z) ⊆ δ hence xyz ∈ es(fs; δ). therefore es(fs; δ) is a bi-ideal of s. conversely, suppose that the nonempty δ-exclusive set of (fs,s) is a bi-ideal of s for all δ ∈ p(u). let x,y ∈ s with x ≤ y be such that fs(x) ⊃ fs(y) = δy then y ∈ es(fs; δy) but x /∈ es(fs; δy). this is a contradiction. hence fs(x) ⊆ fs(y) for all x ≤ y. if there exist x,y ∈ s such that fs(xy) ⊃ fs(x) ∪fs(y) = δx ∪ δy = δz then x ∈ es(fs; δz) and y ∈ es(fs; δz) but xy /∈ es(fs; δz). this is a contradiction. hence fs(xy) ⊆ fs(x) ∪fs(y) for all x,y ∈ s. if there exist x,y,z ∈ s such that fs(xyz) ⊃ fs(x) ∪fs(z) = δx ∪δz = δs then x ∈ es(fs; δs) and z ∈ es(fs; δs) but xyz /∈ es(fs; δs). this is a contradiction. hence fs(xyz) ⊆ fs(x) ∪fs(z) for all x,y ∈ s. � corollary 4.1. (cf. [8]). for any nonempty subset b of s, the following are equivalent. (1) b is a bi-ideal of s. (2) the soft set (χcb,s) over u is a uni-soft bi-ideal over u. a bi-ideal p of an ordered semigroup s is called prime if p 6= s and for any bi-ideals a,b of s from ab ⊆ p it follows that a ⊆ p or b ⊆ p. by analogy a non-constant uni-soft bi-ideal (fs,s) of s over u is called prime (in the first sense) if for any uni-soft bi-ideals (gs,s), (hs,s) of s over u from (gs♦hs,s)⊇̃(fs,s) it follows that (gs,s)⊇̃(fs,s) or (hs,s)⊇̃(fs,s) . theorem 4.2. a bi-ideal p of an ordered semigroup s is prime if and only if for all a,b ∈ s from (asb] ⊆ p it follows that a ∈ p or b ∈ p . int. j. anal. appl. 18 (5) (2020) 824 proof. assume that p is a prime bi-ideal of s and (asb] ⊆ p for some a,b ∈ s. then obviously, the sets a = (asa] and b = (bsb] are bi-ideals of s, because (asa]s(asa] = (asa](s](asa] ⊆ (asasasa] ⊆ (asa] and if x ∈ s and x ≤ y ∈ (asa], then x ∈ ((asa]] = (asa]. similarly, (bsb] is a bi-ideal of s. so, ab ⊆ (ab] = ((asa](bsb]] ⊆ (asabsb] ⊆ (asb] ⊆ p, and consequently a ⊆ p or b ⊆ p . let 〈x〉 be the bi-ideal of s generated by x ∈ s. if a ⊆ p, then 〈a〉 ⊆ (asa] = a ⊆ p, whence a ∈ p . if b ⊆ p , then 〈b〉⊆ (bsb] = b ⊆ p, whence b ∈ p . the converse part is obvious. � corollary 4.2. a bi-ideal p of a commutative ordered semigroup s with identity is prime if and only if for all a,b ∈ s from ab ∈ p it follows a ∈ p or b ∈ p . the result expressed by corollary 4.2, suggests the following definition of prime uni-soft bi-ideals. definition 4.5. a non-constant uni-soft bi-ideal (fs,s) of s over u is called prime (in the second sense) if for all δ ∈ p(u) and a,b ∈ s, the following condition is satisfied: if fs(axb) ⊆ δ for every x ∈ s then fs(a) ⊆ δ or fs(b) ⊆ δ. in other words, a non-constant uni-soft bi-ideal is prime if from the fact that axb ∈ es(fs; δ) for every x ∈ s it follows a ∈ es(fs; δ) or b ∈ es(fs; δ). it is clear that any uni-soft bi-ideal which is prime in the first sense is prime in the second sense. the converse is not true. example 4.1. let s = {a,b,c} be an ordered semigroup with the following cayley table and order relation (see [3]). · a b c a a a a b a b b c a c c ≤:= {(a,a), (b,b), (c,c), (a,b)}. let (fs,s) be a soft set over u = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} in which fs is given by fs : s −→ p(u),x 7−→   {1, 3, 5} if x = c {1, 2, 3, 5, 6} if x = b {1, 2, 3, 5, 6, 8} if x = a is a uni-soft bi-ideal over u. it is prime in the second sense but it is not prime in the first sense. int. j. anal. appl. 18 (5) (2020) 825 theorem 4.3. a non-constant soft set (fs,s) over u is a prime uni-soft bi-ideal over u in the second sense if and only if the nonempty δ-exclusive set of (fs,s) is a prime bi-ideal of s for all δ ∈ p(u). proof. let a uni-soft bi-ideal (fs,s) over u be prime in the second sense and let es(fs; δ) be its arbitrary proper δ−exclusive set, i.e., ∅ 6= es(fs; δ) 6= s (obviously es(fs; δ) is a bi-ideal of s (theorem 4.1)) . if (asb] ⊆ es(fs; δ), then fs(axb) ⊆ δ for every x ∈ s. hence fs(a) ⊆ δ or fs(b) ⊆ δ, i.e., a ∈ es(fs; δ) or b ∈ es(fs; δ) which means that es(fs; δ) is prime bi-ideal of s (corollary 4.2). for the converse part consider a non-constant uni-soft bi-ideal (fs,s) over u. if it is not prime in the second sense, then there exist a,b ∈ s such that fs(axb) ⊆ δ for all x ∈ s, but fs(a) ⊃ δ and fs(b) ⊃ δ. thus (asb] ⊆ es(fs; δ) but a /∈ es(fs; δ) and b /∈ es(fs; δ). therefore es(fs; δ) is not prime. this is a contradiction, which proves that (fs,s) is prime in the second sense. � corollary 4.3. a soft set (χcb,s) over u is a prime uni-soft bi-ideal over u if and only if b is a prime bi-ideal of s. an ordered semigroup (s, ·,≤) is called regular, if for every a ∈ s, there exists x ∈ s such that a ≤ axa or equivalently: (1) (∀a ∈ s) (a ∈ (asa]) and (2) (∀a ⊆ s) (a ⊆ (asa]) . an ordered semigroup s is called left (resp., right ) regular, if for every a ∈ s, there exists x ∈ s such that a ≤ xa2 (resp., a ≤ a2x) or equivalently: (1) (∀a ∈ s) ( a ∈ (sa2] )( resp., a ∈ (a2s] ) and (2) (∀a ⊆ s) ( a ⊆ (sa2] )( resp., a ⊆ (a2s] ) . an ordered semigroup is called completely regular if it is regular, left regular and right regular. lemma 4.1. (cf. [4]). an ordered semigroup s is completely regular if and only if a ⊆ (a2sa2] for every a ⊆ s. equivalently, if a ∈ (a2sa2] for every a ∈ s. theorem 4.4. an ordered semigroup (s, ·,≤) is completely regular if and only if for every uni-soft bi-ideal (fs,s) of s over u, we have (fs(a),s) = ( fs(a 2),s ) for every a ∈ s. proof. (=⇒) . assume that (fs,s) is a uni-soft bi-ideal of s over u. let a ∈ s. since s is completely regular, then a ∈ (a2sa2]. that is, a ≤ a2xa2 for some x ∈ s. since (fs,s) is a uni-soft bi-ideal of s over u, we have fs(a) ⊆ fs(a2xa2) ⊆ fs(a2) ∪fs(a2) = fs(a 2) ⊆ fs(a) ∪fs(a) = fs(a) (since (fs,s) is a uni-soft semigroup) . thus, (fs(a),s) = ( fs(a 2),s ) . int. j. anal. appl. 18 (5) (2020) 826 (⇐=) . let a(a2) be a bi-ideal of s generated by a (a ∈ s), i.e., the set a(a2) = (a2 ∪a4 ∪a2sa2]. by corollary 4.1, the soft set (χca,s) defined by χca : s −→ p(u),x 7−→   ∅ if x ∈ a(a 2), u if x ∈ a(a2), is a uni-soft bi-ideal of s over u. by hypothesis, we have (χca(a),s) = ( χca(a 2),s ) . since a2 ∈ a(a2), we have χca(a 2) = ∅. then χca(a) = ∅ and hence a ∈ a(a 2) = (a2 ∪ a4 ∪ a2sa2]. thus a ≤ a2 or a ≤ a4 or a ≤ a2xa2 for some x ∈ s. if a ≤ a2, then a ≤ a2 = a.a ≤ a2.a2 = a.a2.a ≤ a2.a2.a2 ∈ a2sa2. similarly, in other cases we get a ≤ a2va2 for some v ∈ s. consequently, a ∈ (a2sa2] and by lemma 4.1, s is completely regular. � 5. semilattices of left and right simple semigroups a subsemigroup f of s is called a filter (see [5]) of s if: (1) (∀a,b ∈ s) (ab ∈ f =⇒ a ∈ f and b ∈ f) and (2) (∀a ∈ s)(∀b ∈ f)(a ≥ b =⇒ a ∈ f). for x ∈ s, we denote by n(x), the least filter of s generated x (x ∈ s). by n we mean the equivalence relation on s defined by n := {(x,y) ∈ s × s|n(x) = n(y)} (see [4]). an equivalence relation σ on s is called congruence if (a,b) ∈ σ implies (ac,bc) ∈ σ and (ca,cb) ∈ σ for every c ∈ s. a congruence σ on s is called semilattice congruence on s, if (a,a2) ∈ σ and (ab,ba) ∈ σ for each a,b ∈ s (see [5]). if σ is a semilattice congruence on s then the σ-class (x)σ of s containing x is a subsemigroup of s for every x ∈ s (see [4]). an ordered semigroup s is called a semilattice of left and right simple semigroups if there exists a semilattice congruence σ on s such that the σ-class (x)σ of s containing x is a left and right simple subsemigroup of s for every x ∈ s. equivalently, there exists a semilattice y and a family {sα}α∈y of left and right simple subsemigroups of s such that: (i) sα ∩sβ = ∅, ∀α,β ∈ y, α 6= β, (ii) s = ⋃ α∈y sα, (iii) sαsβ ⊆ sαβ, α,β ∈ y. the semilattice congruences in ordered semigroups are defined exactly as in semigroups without ordered−so the two definitions are equivalent (see [4, 5]). lemma 5.1. (cf. [2]). an ordered semigroup (s, ·,≤) is a semilattice of left and right simple semigroups if and only if for all bi-ideals a,b of s, we have (a2] = a and (ab] = (ba]. int. j. anal. appl. 18 (5) (2020) 827 theorem 5.1. let (s, ·,≤) be an ordered semigroup. then the following are equivalent: (1) s is regular, left and right simple. (2) every uni-soft bi-ideal (fs,s) of s over u is a constant mapping. proof. (1) =⇒ (2). assume that s is regular and left and right simple. let (fs,s) be a uni-soft bi-ideal over u. we consider the set es := {e ∈ s|e2 ≥ e}. since s is regular, so for every a ∈ s, there exists x ∈ s such that a ≤ axa and we have (ax)2 = (axa)x ≥ ax, hence ax ∈ es. thus es 6= ∅. (a) let t ∈ es, we prove that (fs,s) is constant mapping on es. that is, for every e ∈ es, we have fs(e) = fs(t). since t ∈ s and s is left and right simple, we have s = (ts] and s = (st]. since e ∈ s, we have e ∈ (ts] and e ∈ (st], then e ≤ ts and e ≤ zt for some s,z ∈ s. hence e2 ≤ t(sz)t. since (fs,s) is a uni-soft bi-ideal over u, we have fs(e 2) ⊆ fs(t(sz)t) ⊆ fs(t) ∪fs(t) = fs(t). on the other hand, since e ∈ es, we have e2 ≥ e and hence fs(e) ⊆ fs(e2). thus, fs(e) ⊆ fs(e2) ⊆ fs(t). in a similar way, we can prove that fs(t) ⊆ fs(e). thus, fs(t) = fs(e). (b) now we prove that (fs,s) is constant mapping on s. that is, fs(t) = fs(e) for every a ∈ s. since s is regular, so for every a ∈ s, there exists x ∈ s such that a ≤ axa. then (ax)2 = (axa)x ≥ xa and (xa)2 = x(axa) ≥ xa. hence ax,xa ∈ es. thus by (a) we have fs(ax) = fs(t) and fs(xa) = fs(t). since (ax)a(xa) = (axa)xa ≥ axa ≥ a, we have fs(a) ⊆ fs((ax)a(xa)) ⊆ fs(ax) ∪fs(xa) = fs(xa) = fs(t). since s is left and right simple and a ∈ s, we have s = (as] and s = (as]. since t ∈ s, we have t ∈ (as] and t ∈ (sa], then t ≤ as1 and t ≤ z1a for some s1,z1 ∈ s. now, t2 ≤ a(s1z1)a and (fs,s) is a uni-soft bi-ideal of s over u, we have fs(t 2) ⊆ fs(a(s1z1)a) ⊆ fs(a) ∪fs(a) = fs(a). since t ∈ es, we have t2 ≥ t, then fs(t) ⊆ fs(t2) ⊆ fs(a). therefore, fs(t) = fs(a). int. j. anal. appl. 18 (5) (2020) 828 (2) =⇒ (1). let a ∈ s. then (i) (as]s(as] = (as](s](as] ⊆ (assas] ⊆ (as], (ii) (as](as] ⊆ (asas] ⊆ (as] and (iii) if x ≤ y such that s 3 x ≤ y ∈ (as], then x ∈ ((as]] = (as]. therefore, (as] is a bi-ideal of s. by corollary 4.1, the soft set ( χc (as] ,s ) defined by χc(as] : s −→ p(u),x 7−→   ∅ if x ∈ (as],u if x /∈ (as], is a uni-soft bi-ideal of s over u. by hypothesis, ( χc (as] ,s ) is a constant mapping, that is, for every x ∈ s, there exists a subset δ ⊆ u such that χc(as](x) = δ let (as] ⊂ s and t ∈ s be such that t /∈ (as]. then χc (as] (t) = u. since a2 ∈ (as], we have χc (as] (a2) = ∅. this is a contradiction. thus, s = (as]. similarly, we can prove that s = (sa] and therefore, s is left and right simple. since s = (as] = (sa], we have a ∈ s = (as] = (a(sa]] = (asa] and hence s is regular. this completes the proof. � lemma 5.2. (cf. [2]). let (s, ·,≤) be an ordered semigroup and b(x) and b(y) be the bi-ideals of s generated by x and y, respectively. then b(x)sb(y) ⊆ (xsy]. theorem 5.2. an ordered semigroup (s, ·,≤) is a semilattice of left and right simple semigroups if and only if for every uni-soft bi-ideal (fs,s) of s over u, we have (fs(a),s) = ( fs(a 2),s ) and (fs(ab),s) = (fs(ba),s) for all a,b ∈ s. proof. (=⇒) (a) suppose that (fs,s) be a uni-soft bi-ideal of s over u and let s be a semilattice of left and right simple semigroups. then by hypothesis, there exists a semilattice y and a family {si : i ∈ y} of left and right simple subsemigroups of s such that (i) si ∩sj = ∅ ∀i,j ∈ y and i 6= j, (ii) s = ⋃ i∈y si, (iii) sisj ⊆ sij. let a ∈ s. since s = ⋃ i∈y si, then there exists i ∈ y such that a ∈ si. since si is left and right simple for every i ∈ y . then si = (asia] = {t ∈ s : t ≤ axa for some t ∈ si}, then a ≤ axa for some x ∈ si. since si is left and right simple, we have x ≤ aya for some y ∈ si. then we have a ≤ axa ≤ a(aya)a = a2ya2 ∈ a2sa2 and so a ∈ (a2sa2]. thus s is completely regular. since (fs,s) is a uni-soft bi-ideal of s over u. by theorem 4.4, we have (fs(a),s) = ( fs(a 2),s ) . int. j. anal. appl. 18 (5) (2020) 829 (b) let a,b ∈ s. then by (a), we have fs(ab) = fs((ab)2) = fs((ab)4). on the other hand, (ab)4 = (aba)(babab) ∈ b(aba)b(babab) ⊆ (b(aba)b(babab)] = (b(babab)b(aba)] (lemma 5.1) = (b(babab)(b(aba)2]] (lemma 5.1) = ((b(babab)](b(aba)b(aba)]] ⊆ ((b(babab)b(aba)b(aba)]] (since (a](b] ⊆ (ab]) = (b(babab)b(aba)b(aba)] ( since ((a]] = (a]) ⊆ (b(babab)sb(aba)] ⊆ (((babab) s (aba)]] (lemma 5.2) = ((babab) s (aba)] (since ((a]] = (a]) . then (ab)4 ≤ (babab) z (aba) for some z ∈ s. since (fs,s) is a uni-soft bi-ideal of s over u, we have fs ( (ab)4 ) ⊆ fs ((babab) z (aba)) = fs(ba(babza)ba) ⊆ fs(ba) ∪fs(ba) = fs(ba). thus, fs (ab) ⊆ fs(ba). by symmetry we can prove that fs(ba) ⊆ fs (ab) . therefore, (fs (ab) ,s) = (fs(ba),s) . (⇐=) assume that (fs(a2),s) = (fs(a),s) and (fs (ab) ,s) = (fs(ba),s) hold for every uni-soft bi-ideal (fs,s) of s over u. since (fs(a 2),s) = (fs(a),s) so by condition (1) and theorem 4.4, it follows that s is completely. let a be a bi-ideal of s and let a ∈ a. since a ∈ s and s is completely regular, by lemma 4.1, we have a ≤ a2xa2 = a(axa)a ∈ a(asa)a ⊆ aaa ⊆ aa = a2. then a ⊆ a2 and hence (a] ⊆ (a2] =⇒ a ⊆ (a2] (since a is a bi-ideal) . on the other hand, since a is a subsemigroup of s, we have a2 ⊆ a then (a2] ⊆ (a] = a. therefore, a = (a2]. let a and b be bi-ideals of s and let x ∈ (ab]. then x ≤ ab for some a ∈ a and b ∈ b. we consider b(ab) = (ab∪abab∪absab], the bi-ideal of s generated by ab (a,b ∈ s). by corollary 4.1, the soft set ( χc (ab∪abab∪absab],s ) defined by χc(ab∪abab∪absab](x) =   ∅ if x ∈ (ab∪abab∪absab]u if x /∈ (ab∪abab∪absab] is a uni-soft bi-ideal of s over u. by hypothesis, χc (ab∪abab∪absab](ab) = χ c (ab∪abab∪absab](ba). since ab ∈ (ab∪abababsab], we have χc (ab∪abab∪absab](ab) = ∅. thus, χ c (ab∪abab∪absab](ba) = ∅ and ba ∈ (ab∪abab∪absab] and we have ba ≤ ab or ba ≤ abab or ba ≤ (ab)x(ab) for some x ∈ s. if ba ≤ ab, then x ≤ ab ∈ ab and int. j. anal. appl. 18 (5) (2020) 830 x ∈ (ab]. thus (ba] ⊆ (ab]. similarly in other cases we get (ba] ⊆ (ab]. by symmetry, we can prove that (ab] ⊆ (ba]. therefore, (ab] = (ba] and by lemma 5.1, s is a semilattice of left and right simple subsemigroups. this completes the proof. � 6. regular ordered semigroups in this section, we characterize regular ordered semigroups in terms of uni-soft bi-ideals and prove that an ordered semigroup is regular if and only if for every uni-soft bi-ideal (fs,s) we have (fs♦∅s♦fs,s) = (fs,s). lemma 6.1. let (s, ·,≤) be an ordered semigroup and (fs,s) a soft set over u. if (fs,s) is a uni-soft semigroup over u then (fs♦fs,s)⊇̃(fs,s). conversely, if (fs♦fs,s)⊇̃(fs,s) holds for every soft set (fs,s) over u, then (fs,s) is a uni-soft semigroup over u. proof. (=⇒) suppose that (fs,s) is a uni-soft semigroup over u. let x ∈ s. if ax = ∅, then (fs♦fs) (x) = u ⊇ fs(x). if ax 6= ∅, then (fs♦fs) (x) = ⋂ (b,c)∈ax {fs(b) ∪fs(c)} ⊇ ⋂ (b,c)∈ax fs(bc) ⊇ ⋂ (b,c)∈ax fs(x) ( since x ≤ bc) = fs(x). hence (fs♦fs,s)⊇̃(fs,s). (⇐=) assume that (fs♦fs,s)⊇̃(fs,s) holds for every soft set (fs,s) over u. let x,y ∈ s, then fs(xy) ⊆ (fs♦fs)(xy) = ⋂ (b,c)∈axy {fs(b) ∪fs(c)} ⊆ fs(x) ∪fs(y). hence fs(xy) ⊆ fs(x) ∪fs(y) for all x,y ∈ s and (fs,s) is a uni-soft semigroup. � proposition 6.1. let (s, ·,≤) be an ordered semigroup and (fs,s) a uni-soft bi-ideal of s over u. then (fs♦∅s♦fs,s)⊇̃(fs,s). int. j. anal. appl. 18 (5) (2020) 831 proof. let (fs,s) be a uni-soft bi-ideal of s over u. let x ∈ s. if ax = ∅, then (fs♦∅s♦fs) (x) = u ⊇ fs(x). if ax 6= ∅, then (fs♦∅s♦fs) (x) = ⋂ (b,c)∈ax {(fs♦∅s) (b) ∪fs(c)} = ⋂ (b,c)∈ax   ⋂ (b1,c1)∈ab {fs(b1) ∪∅s(c1)}∪fs(c)   = ⋂ (b,c)∈ax ⋂ (b1,c1)∈ab {fs(b1) ∪∅s(c1) ∪fs(c)} = ⋂ (b,c)∈ax ⋂ (b1,c1)∈ab {fs(b1) ∪∅∪fs(c)} = ⋂ (b,c)∈ax ⋂ (b1,c1)∈ab {fs(b1) ∪fs(c)} . since x ≤ bc and b ≤ b1c1, we have x ≤ bc ≤ (b1c1)c and (fs,s) is a uni-soft bi-ideal of s over u, we have fs(x) ⊆ fs((b1c1)c) ⊆ fs(b1) ∪fs(c). hence, (fs♦∅s♦fs) (x) = ⋂ (b,c)∈ax ⋂ (b1,c1)∈ab {fs(b1) ∪fs(c)} ⊇ ⋂ (b,c)∈ax ⋂ (b1,c1)∈ab fs(x) = fs(x). therefore, (fs♦∅s♦fs,s) ⊇ (fs,s) . � lemma 6.2. (cf. [15]). let (χca,s) and (χ c b,s) be soft sets over u where a and b are nonempty subsets of s. then the following properties hold: (1) (χca,s)∪̃(χ c b,s) = (χ c a∩b,s) . (2) (χca,s) ♦ (χ c b,s) = ( χc (ab] ,s ) . lemma 6.3. (cf. [18]). let (s, ·,≤) be an ordered semigroup. then the following are equivalent: (1) s is regular. (2) b = (bsb] for every bi-ideal b of s. (3) b(a) = (b(a)sb(a)] for every a ∈ s. theorem 6.1. an ordered semigroup (s, ·,≤) is regular if and only if for every uni-soft bi-ideal (fs,s) over u, we have (fs,s) = (fs♦∅s♦fs,s) . int. j. anal. appl. 18 (5) (2020) 832 proof. suppose that s is regular. let (fs,s) be a uni-soft bi-ideal of s over u and let a ∈ s. since s is regular, there exists x ∈ s such that a ≤ axa = a(xa). then (a,xa) ∈ aa and we have (fs♦∅s♦fs) (a) = ⋂ (b,c)∈aa {fs(b) ∪ (∅s♦fs) (c)} ⊆ fs(a) ∪ (∅s♦fs) (xa) = fs(a) ∪ ⋂ (b1,c1)∈axa {∅s(b1) ∪fs(c1)} ⊆ fs(a) ∪∅s(x) ∪fs(a) = fs(a) ∪∅ = fs(a). then (fs♦∅s♦fs,s)⊆̃(fs,s). on the other hand, by proposition 6.1, we have (fs♦∅s♦fs,s)⊇̃(fs,s), therefore, (fs♦∅s♦fs,s) = (fs,s). conversely, assume that (fs♦∅s♦fs,s) = (fs,s) holds for every uni-soft bi-ideal (fs,s) over u. to prove that s is regular, by lemma 6.2, it is enough to prove that b(a) = (b(a)sb(a)] ∀a ∈ s. let y ∈ b(a). since b(a) is the bi-ideal of s generated by a (a ∈ s). by corollary 4.1, ( χc b(a) ,s ) is a uni-soft bi-ideal of s over u. by hypothesis, we have( χcb(a)♦∅s♦χ c b(a) ) (y) = χcb(a)(y). since y ∈ b(a), we have χc b(a) (y) = ∅ and hence ( χc b(a) ♦∅s♦χcb(a) ) (y) = ∅. but by lemma 6.2, we have χc b(a) ♦∅s♦χcb(a) = χ c (b(a)sb(a)] . thus, χc (b(a)sb(a)] (y) = ∅ and y ∈ (b(a)sb(a)]. therefore, b(a) ⊆ (b(a)sb(a)]. on the other hand, since b(a) is the bi-ideal of s, we have (b(a)sb(a)] ⊆ (b(a)] = b(a). thus b(a) = (b(a)sb(a)] and s is regular (lemma 6.3). � lemma 6.4. let (s, ·,≤) be an ordered semigroup. let (fs,s) and (gs,s) be uni-soft bi-ideals of s over u. then the soft product (fs♦gs,s) of (fs,s) and (gs,s) is again a uni-soft bi-ideal of s over u. proof. let (fs,s) and (gs,s) be uni-soft bi-ideals of s over u. let x,y,z ∈ s. then (fs♦gs) (x) ∪ (fs♦gs) (z) =   ⋂ (p,q)∈ax {fs(p) ∪gs(q)}∪ ⋂ (p1,q1)∈az {fs(p1) ∪gs(q1)}   = ⋂ (p,q)∈ax ⋂ (p1,q1)∈az [{fs(p) ∪gs(q)}∪{fs(p1) ∪gs(q1)}] = ⋂ (p,q)∈ax ⋂ (p1,q1)∈az [fs(p) ∪gs(q) ∪fs(p1) ∪gs(q1)] ⊇ ⋂ (p,q)∈ax ⋂ (p1,q1)∈az [fs(p) ∪gs(q1)] . int. j. anal. appl. 18 (5) (2020) 833 since x ≤ pq and z ≤ p1q1, hence xyz ≤ (pq)y(p1q1) = p(qyp1)q1 and (p(qy)p1,q1) ∈ axyz. thus, axyz 6= ∅ and we have (fs♦gs) (x) ∪ (fs♦gs) (z) ⊇ ⋂ (p,q)∈ax ⋂ (p1,q1)∈az [fs(p) ∪fs(p1) ∪gs(q1)] ⊇ ⋂ (p(qy)p1,q1)∈axyz [fs(p(qyp1)q1) ∪gs(q1)] = ⋂ (p(qy)p1,q1)∈axyz (fs♦gs) (xyz) = (fs♦gs) (xyz). thus, (fs♦gs) (xyz) ⊆ (fs♦gs) (x) ∪ (fs♦gs) (z) . similarly, we can prove that (fs♦gs) (xy) ⊆ (fs♦gs) (x) ∪ (fs♦gs) (y) . let x,y ∈ s be such that x ≤ y. then (fs♦gs) (x) ⊆ (fs♦gs) (y) . infact, if (p,q) ∈ ay, then y ≤ pq and we have x ≤ y ≤ pq it follows that (p,q) ∈ ax and hence ay ⊆ ax. if ax = ∅, then ay = ∅ and we have (fs♦gs) (y) = u ⊇ (fs♦gs) (x). if ax 6= ∅, then ay 6= ∅ and we have (fs♦gs) (y) = ⋂ (p,q)∈ay {fs(p) ∪gs(q)} ⊇ ⋂ (p,q)∈ax {fs(p) ∪gs(q)} = (fs♦gs) (x). hence in both the cases, we have (fs♦gs) (x) ⊆ (fs♦gs) (y) for all x,y ∈ s with x ≤ y. this completes the proof. � 7. regular and intra-regular ordered semigroups in this section, we characterize regular and intra-regular ordered semigroups in terms of uni-soft bi-ideals. lemma 7.1. let s be an ordered semigroup and (fs,s) a uni-soft bi-ideal of s over u. then (fs♦fs,s)⊇̃(fs,s). proof. let (fs,s) be a uni-soft bi-ideal of s over u and let a ∈ s. if aa = ∅, then (fs♦fs) (a) = u ⊇ fs(a). if aa 6= ∅, then (fs♦fs) (a) = ⋂ (p,q)∈aa {fs(p) ∪fs(q)} ⊇ ⋂ (p,q)∈aa fs(pq) ⊇ ⋂ (p,q)∈aa fs(a) (since a ≤ pq =⇒ fs(a) ⊆ fs(pq)) = fs(a). therefore, (fs♦fs,s)⊇̃(fs,s). � int. j. anal. appl. 18 (5) (2020) 834 7.1. lemma. let (fs,s) and (gs,s) be soft subsets of an ordered semigroup s over u. then (fs♦gs,s)⊇̃(∅s♦gs,s) ( resp., (fs♦gs,s)⊇̃(fs♦∅s) ) . proof. straightforward. � lemma 7.2. let (fs,s) and (gs,s) be uni-soft bi-ideals of an ordered semigroup s over u. then (fs∪̃gs,s) is a uni-soft bi-ideal of s over u. proof. straightforward. � theorem 7.1. let s be an ordered semigroup. then the following are equivalent: (1) s is both regular and intra-regular. (2) (fs♦fs,s) = (fs,s) for every uni-soft bi-ideal (fs,s) over u. (3) (fs∪̃gs,s) = ((fs♦gs)∪̃(gs♦fs),s) for all uni-soft bi-ideals (fs,s) and (gs,s) of s over u. proof. (1)=⇒(2). let (fs,s) be a uni-soft bi-ideal of s over u and let a ∈ s. since s is regular and intra-regular, there exist x,y,z ∈ s such that a ≤ axa ≤ axaxa and a ≤ ya2z. then a ≤ axaxa ≤ ax(yaz)xa = (axya)(azxa), and hence (axya,azxa) ∈ aa. then (fs♦fs) (a) = ⋂ (p,q)∈aa {fs(p) ∪fs(q)} ⊆ fs(axya) ∪fs(azxa) ⊆{fs(a) ∪fs(a)}∪{fs(a) ∪fs(a)} = fs(a). and hence (fs♦fs,s)⊆̃(fs,s). on the other hand, by lemma 7.1, we have (fs♦fs,s)⊇̃(fs,s). therefore, (fs♦fs,s)⊇̃(fs,s). (2)=⇒(3). let (fs,s) and (gs,s) be uni-soft bi-ideals of s over u. then ( fs∪̃gs,s ) is a uni-soft bi-ideal of s over u (lemma 7.2). by (2), we have ( fs∪̃gs,s ) = (( fs∪̃gs,s ) ♦ ( fs∪̃gs,s ) ,s ) ⊇̃(fs♦gs,s) . similarly, we can prove that ( fs∪̃gs,s ) ⊇̃(gs♦fs,s) . therefore, ( fs∪̃gs,s ) ⊇̃(fs♦gs,s) ∪̃(gs♦fs,s) . int. j. anal. appl. 18 (5) (2020) 835 on the other hand, since (fs♦gs,s) and (gs♦fs,s) are uni-soft bi-ideals of s over u. again by lemma 7.2, (fs♦gs,s) ∪̃(gs♦fs,s) is a uni-soft bi-ideal of s over u. by (2), we have ( (fs♦gs,s)∪̃(gs♦fs,s) ,s ) =   ((fs♦gs,s) ∪̃(gs♦fs,s))♦( (fs♦gs,s) ∪̃(gs♦fs,s) ) ,s   ⊇̃((fs♦gs,s) ♦ (gs♦fs,s) ,s) = (fs♦ (gs♦gs) ♦fs,s) = (fs♦gs♦fs,s) (as (gs♦gs,s) = (gs,s) by (1) above) ⊇̃(fs♦∅s♦fs,s) = (fs,s) (as (fs♦∅s♦fs,s) = (fs,s)) . (7.1) similarly, we can prove that ( (fs♦gs,s) ∪̃(gs♦fs,s) ,s ) ⊇̃(gs,s). therefore, ( (fs♦gs,s) ∪̃(gs♦fs,s) ,s ) ⊇̃(fs∪̃gs,s). consequently, we have ( (fs♦gs,s) ∪̃(gs♦fs,s) ,s ) = (fs∪̃gs,s). (3)=⇒(1). to prove that s is both regular and intra-regular, it is enough to prove that p ∩q = (pq] ∩ (qp] for every bi-ideal p and q of s. let b ∈ p ∩q. by corollary 4.1, (χcp ,s) and ( χcq,s ) are uni-soft bi-ideals of s over u. by (3), we have ( χcp∪̃χ c q,s ) (b) = (( χcp ♦χ c q ) ∪̃ ( χcq♦χ c p ) ,s ) (b). by lemma 6.2, ( χcp ♦χ c q ) ∪̃ ( χcq♦χ c p ) = χc (pq]∩(qp] and χ c p∪̃χ c q = χ c p∩q, hence we have χ c (pq]∩(qp](b) = χcp∩q(b) = ∅. thus, χ c (pq]∩(qp](b) = ∅ and b ∈ (pq] ∩ (qp] =⇒ p ∩q ⊆ (pq] ∩ (qp]. on the other hand, if b ∈ (pq] ∩ (qp], then ∅ = ( χc(pq]∩(qp],s ) (b) = ( χc(pq]∪̃χ c (qp] ) (b) = (( χcp ♦χ c q ) ∪̃ ( χcq♦χ c p )) (b) = ( χcp∪̃χ c q,s ) (b) (by (3)) = ( χcp∩q,s ) (b) . hence, b ∈ p ∩q and (pq] ∩ (qp] ⊆ pq. therefore, p ∩q = (pq] ∩ (qp] and s is both regular and intra-regular. � int. j. anal. appl. 18 (5) (2020) 836 8. conclusion in the present paper, we introduced the notion of uni-soft type of bi-ideals of ordered semigroups. furthermore the notion of a uni-soft bi-ideal is introduced and their related properties is provided. the concept of δ−exclusive set is given and the relations between uni-soft bi-ideals and δ−exclusive set are discussed. the concepts of two types of prime uni-soft bi-ideals of an ordered semigroup s are given. using the notion of uni-soft bi-ideals, some semilattices of left and right simple semigroups are provided. in our last section the characterizations of both regular and intra-regular ordered semigroups are provided. in our future study of ordered semigroups, we will apply the above new idea to other algebraic structures for more applications. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] u. acar, f. koyuncu, and b. tanay, soft sets and soft rings, computers math. appl. 59 (11) (2010), 3458–3463. [2] n. kehayopulu and m. tsingelis, regular ordered semigroups in terms of fuzzy subsets, inform. sci. 176 (24) (2006), 3675-3693. [3] n. kehayopulu, on completely regular ordered semigroups, sci. math. 1 (1) (1998), 27-32. [4] n. kehayopulu, on semilattices of simple poe-semigroups, math. japon. 38 (2) (1993), 305–318. [5] n. kehayopulu, on weakly prime ideals of ordered semigroups, math. japon. 35 (6) (1990), 1051–1056. [6] a. khan, y. b. jun, s. i. a, shah and r. khan, applications of soft sets in ordered semigroups via uni-soft quasi-ideals, j. intell. fuzzy syst. 30 (2016), 97–107. [7] a. khan, r. khan, y.b. jun, uni-soft structure applied to ordered semigroups, soft comput. 21 (2017), 1021–1030. [8] a. khan, y. b. jun and r. khan, characterizations of ordered semigroups in terms of int-soft ideals, submitted. [9] h. aktas and n. cağman, soft sets and soft groups, inform. sci. 177 (13) (2007), 2726–2735. [10] a. o. atagun and a. sezgin, soft substructures of rings, fields and modules, computers math. appl. 61 (3) (2011), 592–601. [11] f. feng, y. b. jun, and x. zhao, soft semirings, computers math. appl. 56 (10) (2008), 2621–2628. [12] f. feng, m. i. ali, and m. shabir, soft relations applied to semigroups, filomat, 27 (7) (2013), 1183–1196. [13] f. feng, h. fujita, y. b. jun, and m. khan, decomposition of fuzzy soft sets with finite value spaces, sci. world j. 2014 (2014), article id 902687. [14] f. feng and y.m. li, soft subsets and soft product operations, inform. sci. 232 (2013), 44–57. [15] y. b. jun, s. z. song, and g. muhiuddin, concave soft sets, critical soft points, and union-soft ideals of ordered semigroups, sci. world j. 2014 (2104), article id 467968. [16] y. b. jun, k. j. lee, and a. khan, soft ordered semigroups, math. log. q. 56 (1) (2010), 42–50. [17] n. cagman and s. enginoglu, soft set theory and uni-int decision making, eur. j. oper. res. 207 (2) (2010), 848–855. [18] m. shabir and a. khan, characterizations of ordered semigroups by their fuzzy ideals, comput. math. appl. 59 (2010), 539–549. [19] x. ma and j. zhan, characterizations of three kinds of hemirings by fuzzy soft h-ideals, j. intell. fuzzy syst. 24 (2013), 535-548. [20] d. molodtsov, soft set theory—first results, computers math. appl. 37 (4-5) (1999), 19–31. int. j. anal. appl. 18 (5) (2020) 837 [21] l. a. zadeh, fuzzy sets, inform. control 8 (1965), 338-353. [22] j. zhan, n. čağman and a. s. sezer, applications of soft union sets to hemirings via su-h-ideals, j. intell. fuzzy syst. 26 (2014), 1363-1370. 1. introduction 2. preliminaries 3. basic operations of soft sets 4. uni-soft bi-ideals 5. semilattices of left and right simple semigroups 6. regular ordered semigroups 7. regular and intra-regular ordered semigroups 7.1. lemma 8. conclusion references international journal of analysis and applications issn 2291-8639 volume 15, number 2 (2017), 172-178 doi: 10.28924/2291-8639-15-2017-172 type-2 fuzzy g-tolerance relation and its properties mausumi sen1,∗, dhiman dutta1 and ashok deshpande2 abstract. in this short communication we generalize the definition of type-2 fuzzy tolerance relation and consequently the type-2 fuzzy g-tolerance relation in type-2 fuzzy sets. the type-2 fuzzy gtolerance relation helps in finding the type-2 fuzzy g-equivalence relation. moreover, we have studied the notion of type-2 fuzzy tolerance relation in abstract algebra. 1. introduction type-2 fuzzy sets are relatively new in the world of fuzzy sets and systems. although they were originally introduced in 1975 by l. a. zadeh [21], type-2 sets did not gain popularity until their reintroduction by j. m. mendel [12]. these newer fuzzy sets were now thought of as an extension of the already popular fuzzy sets (now labelled type-1) to include additional uncertainties in the set. type-2 fuzzy sets are unique and conceptually appealing, because they are fuzzy extension rather than crisp. type-2 fuzzy sets have membership functions as type-1 fuzzy sets. the advantage of type-2 fuzzy sets is that they are helpful in some cases where it is difficult to find the exact membership functions for a fuzzy sets. there are wide variety of applications of type-2 fuzzy sets in science and technology like computing with words [14], human resource management [9], forecasting of time-series [11], clustering [1, 17], pattern recognition [3], fuzzy logic controller [20], industrial application [4], simulation [18], neural network [2], [19], and transportation problem [13]. the concept of cartesian product of type-2 fuzzy sets was given by hu et al. [10] as an extension of type-1 fuzzy sets. the properties of membership grades of type-2 fuzzy sets and set-theoretic operations of such sets have been studied by mizumoto and tanaka [15], [16]. dubois and prade [6][8] discussed the composition of type-2 fuzzy relations and presented a formulation only for minimum t-norm which is, perhaps, an extension of type-1 sup-star composition. type-2 fuzzy relations (t2 frs in short) were introduced in [12]. there are different kinds of applications of fuzzy relations in the theory of type-1 fuzzy sets. the other motivation of this research is to investigate t2 frs and their compositions. a type-2 fuzzy tolerance relation is a type-2 fuzzy reflexive and symmetric relation, but not necessarily transitive relation. the main objective of this paper is to generalize the definition of type-2 fuzzy tolerance relation and consequently the type-2 fuzzy g-tolerance relation in type-2 fuzzy sets. the type-2 fuzzy g-tolerance relation helps in finding the type-2 fuzzy g-equivalence relation. we have studied the concept of type-2 fuzzy tolerance relation in abstract algebra. the paper is organized as follows : section (2) introduces some basic definitions related to the concept. we have discussed type-2 fuzzy g-tolerance relation and type-2 fuzzy g-equivalence relation in section (3). section (4) deals with the conversion of type-2 fuzzy tolerance relation. section (5) describes type-2 fuzzy tolerance relation in algebraic structures. 2. preliminaries let us define some preliminary concepts in this section. received 26th july, 2017; accepted 9th october, 2017; published 1st november, 2017. 2010 mathematics subject classification. 03e72. key words and phrases. type-2 fuzzy g-tolerance relation; type-2 fuzzy g-equivalence relation; type-2 fuzzy g-preclass; type-2 fuzzy g-tolerance class. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 172 type-2 fuzzy g-tolerance relation and its properties 173 definition 1. [21] let x be a non-empty universe. then a mapping a : x → f([0, 1]) is called a type-2 fuzzy set (t2 fs, for short) on x, characterized by a membership function µã(x) : x → f([0, 1]) and is expressed by the following set notation : ã = { ( x,µ̃ã(x) ) : x ∈ x}. f([0, 1]) denotes the set of all type-1 fuzzy sets that can be defined on the set [0, 1]. µã(x), itself is a type-1 fuzzy set for value of x ∈ x and is characterized by a secondary membership function fx : j → [0, 1]. therefore, ã can be represented as : ã = {< x,{(u,fx(u)) : u ∈ j} >: x ∈ x}, where j ⊆ [0, 1] is the set of all possible primary membership functions corresponding to an element x. let µa(x) and µb(x) be the secondary membership functions of two type-2 fuzzy sets in the universal set x. the operations of union, intersection and complement of two type-2 fuzzy sets a and b using the zadeh’s extension principle [21] are defined below: union: a∪b = µa(x) tµb(x) = ∑ u∈j fx(u) u t ∑ w∈j fx(w) w = ∑ u,w∈j f(u) ∧g(w) u∨w intersection: a∩b = µa(x) uµb(x) = ∑ u∈j fx(u) u u ∑ w∈j fx(w) w = ∑ u,w∈j f(u) ∧g(w) u∧w complement: ā = ¬µa(x) = ∑ u∈j fx(u) 1 −u containment: a v b (a w b)if µa(x) ≤ µb(x) ( µa(x) ≥ µb(x)) where t is denoted as join, u is denoted as meet, ¬ is denoted as negation operator, ∨ is denoted as max and ∧ is denoted as min operator as defined by mizumoto and tanaka [16] in type-2 fuzzy sets respectively. definition 2. [12] the cartesian product of two type-2 fuzzy sets, say a and b on x is defined as (a×b)(x,y) = a(x) ∧b(y) for all x,y ∈ x. definition 3. [12] let a,b be two type-2 fuzzy sets on x and y respectively, then the type-2 fuzzy relation (t2 fr in short) of type-2 fuzzy sets a×b is the type-2 fuzzy subset of x ×y . 3. type-2 fuzzy g-equivalence relation type-2 fuzzy reflexive, symmetric and transitive relations were discussed in [10]. definition 4. let q̃ be a t2 fr on x. then q̃ is said to be type-2 fuzzy (1) reflexive if q̃(x,x) = 1̄ for all x ∈ x. (2) antireflexive if q̃(x,x) = 0̄ for all x ∈ x. (3) weakly reflexive if q̃(x,y) v q̃(x,x) and q̃(y,x) v q̃(x,x) for all x,y ∈ x. (4) symmetric if q̃(x,y) = q̃(y,x). (5) antisymmetric if q̃ satisfies q̃(x,x) = 0̄ or q̃(x,x) = 0̄ for all x,y ∈ x(x 6= y). (6) transitive if q̃◦ q̃ v q̃ , where q̃◦ q̃ is defined by q̃◦ q̃(x,y) = ˜sup z∈x {q̃(x,z) ∧ q̃(z,y)}. dhiman et al. [5] have extended the definitions of type-2 fuzzy reflexive relation and type-2 fuzzy equivalence relations. 174 sen, dutta and deshpande definition 5. [5] q̃ is said to be type-2 fuzzy g-reflexive if (i) 0̄ < q̃(x,x) < 1̄, where q̃(x,x) lies between (0̄, 1̄); (ii) q̃(x,y) < inf t∈x q̃(t,t) for all x 6= y in x. 0̄ and 1̄ memberships are denoted as 1/0 and 1/1 in type-2 fuzzy sets respectively. 0̄ membership in a type-2 fuzzy set means that it has a secondary membership equal to 1 corresponding to the primary membership of 0, and if it has all other secondary memberships equal to 0. similarly, the meaning of 1̄ is same as 0̄. definition 6. a type-2 fuzzy relation q̃ in x is a type-2 fuzzy g-tolerance relation in x if q̃ is type-2 fuzzy g-reflexive and symmetric in x. definition 7. [5] a type-2 fuzzy relation q̃ in x is a type-2 fuzzy g-equivalence relation in x if q̃ is type-2 fuzzy g-reflexive, symmetric and transitive in x. example 1. suppose that in a biotechnology experiment, three potentially new strains of bacteria have been detected in the area around an anaerobic corrosion pit on a new aluminum-lithium alloy used in the fuel tank of a new experiment aircraft. in a pair wise comparison, the following similarity relation q̃ is developed . for example, the first strain (column 1) has a strength of similarity to the second strain of 0.1 0.1 + 0.2 0.2 , to the third strain a strength of 0.4 0.4 + 0.5 0.5 . hence, q̃ =   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8   . clearly, (i) q̃(x,x) > 0̄ (ii) q̃(x,y) v ˜inf t∈x q̃(t,t) for all x 6= y in x. therefore, q̃ is a type-2 fuzzy g-reflexive relation. again, q̃(x,y) = q̃(y,x) for all x,y ∈ x. hence, q̃ is a type-2 fuzzy symmetric relation. now, q̃◦ q̃ =   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8  ◦   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8   considering the composition of first row of first matrix and second column of second matrix, we get the first element of the second column of the new matrix which is calculated as follows: ( 1 1 + 0.9 0.9 ) ∏ ( 0.1 0.1 + 0.2 0.2 )= 0.1 0.1 + 0.2 0.2 + 0.1 0.1 + 0.2 0.2 = 0.1 0.1 + 0.2 0.2 similarly, ( 0.1 0.1 + 0.2 0.2 ) ∏ ( 1 1 + 0.9 0.9 )= 0.1 0.1 + 0.2 0.2 again, ( 0.4 0.4 + 0.5 0.5 ) ∏ ( 0.6 0.7 + 0.6 0.6 )= 0.4 0.4 + 0.4 0.4 + 0.5 0.5 + 0.5 0.5 = 0.4 0.4 + 0.5 0.5 now, [ ( 0.1 0.1 + 0.2 0.2 ) ⊔ ( 0.1 0.1 + 0.2 0.2 ) ]⊔ ( 0.4 0.4 + 0.5 0.5 ) = ( 0.2 0.4 + 0.2 0.5 ) > ( 0.1 0.1 + 0.2 0.2 ) therefore, τ̃ ◦ τ̃ 6v τ̃ therefore, τ̃ is not type-2 fuzzy transitive relation. hence, τ̃ satisfies all the conditions of type-2 fuzzy g-tolerance relations. consequently, τ̃ is a type-2 fuzzy g-tolerance relation. type-2 fuzzy g-tolerance relation and its properties 175 4. type-2 fuzzy g-tolerance relation to type-2 fuzzy equivalence relation type-2 fuzzy tolerance relation can be converted to type-2 fuzzy g-equivalence relation by composition. let us observe this example below: example 2. consider the earlier example 1 τ̃ =   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8   . we conclude that τ̃ is a type-2 fuzzy g-tolerance relation from example 1. now, τ̃ ◦ τ̃ =   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8  ◦   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8   let us now form a new matrix ˜̃τ with the new elements found from the τ̃ ◦ τ̃. suppose, ˜̃τ =   0.2 1 + 0.2 0.9 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 1 + 0.2 0.9 0.2 0.7 + 0.2 0.6 0.2 0.4 + 0.2 0.5 0.2 0.7 + 0.2 0.6 0.5 0.7 + 0.5 0.8   . we have noticed from the above matrix ˜̃τ that it satisfies all the conditions of type-2 fuzzy g-equivalence relation and symmetric relation. we now check whether ˜̃τ satisfies the type-2 fuzzy transitive relation. ˜̃τ ◦ ˜̃τ =   0.2 1 + 0.2 0.9 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 1 + 0.2 0.9 0.2 0.7 + 0.2 0.6 0.2 0.4 + 0.2 0.5 0.2 0.7 + 0.2 0.6 0.5 0.7 + 0.5 0.8  ◦   0.2 1 + 0.2 0.9 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 1 + 0.2 0.9 0.2 0.7 + 0.2 0.6 0.2 0.4 + 0.2 0.5 0.2 0.7 + 0.2 0.6 0.5 0.7 + 0.5 0.8   ˜̃τ ◦ ˜̃τ =   0.2 1 + 0.2 0.9 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 1 + 0.2 0.9 0.2 0.7 + 0.2 0.6 0.2 0.4 + 0.2 0.5 0.2 0.7 + 0.2 0.6 0.2 0.7 + 0.2 0.8   v   0.2 1 + 0.2 0.9 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 0.4 + 0.2 0.5 0.2 1 + 0.2 0.9 0.2 0.7 + 0.2 0.6 0.2 0.4 + 0.2 0.5 0.2 0.7 + 0.2 0.6 0.5 0.7 + 0.5 0.8   . consequently, ˜̃τ ◦ ˜̃τ v ˜̃τ. therefore, ˜̃τ satisfies the transitive property. therefore, ˜̃τ is a type-2 fuzzy g-equivalence relation. definition 8. let n be an ordinary set and τ̃ be a type-2 fuzzy g-tolerance relation on n. then a type-2 fuzzy subset q̃ of n is called a type-2 fuzzy g pre-class iff q̃×q̃ v τ̃ i.e.q̃(x)∧q̃(y) v τ̃(x,y) for all x,y ∈ n. example 3. let n={x,y,z}.then, τ̃ is a type-2 fuzzy g-tolerance relation on m, defined by the following matrix: 176 sen, dutta and deshpande τ̃ =   1 1 + 0.9 0.9 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 1 1 + 0.9 0.9 0.6 0.7 + 0.6 0.6 0.4 0.4 + 0.5 0.5 0.6 0.7 + 0.6 0.6 0.7 0.7 + 0.8 0.8   . let q̃ be a type-2 fuzzy subset of n defined by q̃(x) = 0.7 0.7 + 0.8 0.8 , q̃(y) = 0.1 0.1 + 0.2 0.2 , q̃(z) = 0.3 0.3 + 0.4 0.4 . then, q̃× q̃ =   0.7 0.7 + 0.8 0.8 0.1 0.1 + 0.2 0.2 0.3 0.3 + 0.4 0.4 0.1 0.1 + 0.2 0.2 0.1 0.1 + 0.2 0.2 0.1 0.1 + 0.2 0.2 0.4 0.4 + 0.5 0.5 0.1 0.1 + 0.2 0.2 0.3 0.3 + 0.4 0.4   clearly, q̃× q̃ v τ̃, and hence is a type-2 fuzzy g pre-class. there can be another r̃ of n larger than q̃ such that r̃(x)∧̃r̃(y) v τ̃(x,y) e.g. r̃(x) = 0.7 0.7 + 0.8 0.8 , r̃(y) = 0.1 0.1 + 0.2 0.2 , r̃(z) = 0.5 0.5 + 0.6 0.6 . definition 9. let τ̃ be a type-2 fuzzy tolerance relation on n. a type-2 fuzzy subset c̃ of n is called a type-2 fuzzy g-tolerance class if c̃ is a type-2 fuzzy g pre-class and there exists no type-2 fuzzy g pre-class d̃ of n s.t. c̃ @ d̃. example 4. from the example 3, r̃ is a type-2 fuzzy g-tolerance class while q̃ is not. definition 10. a type-2 fuzzy g-tolerance space is a pair (m,τ), where m is an ordinary set and τ is a type-2 fuzzy tolerance relation defined on m. 5. type-2 fuzzy tolerance relations in algebraic structures we shall investigate how type-2 fuzzy tolerance relations can be applied in abstract algebra. let an algebraic structure u = (a,f) be given, where a is the set of elements of this structure and f is the set of operations on this set. if a type-2 fuzzy tolerance τ on a is given, we say that τ, is compatible with u or u is a τ tolerance algebraic structure if and only if the following holds: let f ∈ f and let f be an n-ary operation. if we have 2n elements x1,x2, ...,xn; y1,y2, ...,yn of a such that (xi,yi) ∈ τ for i = 1, 2, .....,n, then also (f(x1,x2, ...,xn),f(y1,y2, ...,yn)) ∈ τ . theorem 5.1. let u = (a,f) be an algebra and τ1,τ2 be two type-2 fuzzy tolerance relation on m which are compatible with u. then the relation τ1 ∩ τ2 is a type-2 fuzzy tolerance relation compatible with u. proof. let f ∈ f be an n-ary operation on u and let x1,x2, ...,xn; y1,y2, ...,yn be elements of m such that (xi,yi) ∈ τ1∩τ2 for i = 1, 2, ...,n. then (xi,yi) ∈ τ1 and therefore (f(x1,x2, ...,xn),f(y1,y2, ...,yn)) ∈ τ1. similarly, (f(x1,x2, ...,xn),f(y1,y2, ...,yn)) ∈ τ2. therefore, (f(x1,x2, ...,xn),f(y1,y2, ...,yn)) ∈ τ1 ∩ τ2. as f was chosen arbitrarily, this holds for any f ∈ f and τ1 ∩ τ2 is a compatible tolerance relation on u. � theorem 5.2. let g be a group, and let a type-2 fuzzy tolerance τ be given on its set of elements. if g is a type-2 fuzzy τ tolerance semigroup with respect to multiplication, it is also a type-2 fuzzy τ tolerance group. proof. suppose that g is a type-2 fuzzy τ-tolerance semi-group then (x1,y1) ∈ τ, (x2,y2) ∈ τ implies that (x1x2,y1y2) ∈ τ for arbitrary elements x1,x2,y1,y2 of g. it is necessary and sufficient to prove that (x,y) ∈ τ implies that (x−1,y−1) ∈ τ for g to be a type-2 fuzzy τ-tolerance group for arbitrary elements x,y of g. let us consider two elements x,y of g and let (x,y) ∈ τ. suppose the identity element of the group will be denoted by e. since τ is a type-2 fuzzy reflexive, (x−1,x−1) ∈ τ. as (x,y) ∈ τ, (x−1,x−1) ∈ τ, we obtain (xx−1,yx−1) ∈ τ, therefore (e,yx−1) ∈ τ. similarly, (y−1,y−1) ∈ τ and (e,yx−1) ∈ τ. therefore, (y−1e,y−1yx−1) ∈ τ which means (y−1,x−1) ∈ τ. as τ is symmetric, also (x−1,y−1) ∈ τ. � type-2 fuzzy g-tolerance relation and its properties 177 theorem 5.3. let g be a type-2 fuzzy τ tolerance group, e its identity element. the set h of the elements x ∈ g such that (e,x) ∈ τ is a normal subgroup of the group g. proof. suppose that x ∈ h,y ∈ h then according to the statement of the theorem (e,x) ∈ τ, (e,y) ∈ τ. since, (e,x) ∈ τ, (e,y) ∈ τ, we obtain (e,xy) ∈ τ. the type-2 fuzzy reflexive relation of τ implies that (x−1,x−1) ∈ τ.we have (e,x) ∈ τ and (x−1,x−1) ∈ τ then (x−1,e) ∈ τ or (e,x−1) ∈ τ and consequently x−1 ∈ h. therefore, h is a subgroup of the group g. let z ∈ g and (z,z) ∈ τ. since (e,x) ∈ τ and (z,z) ∈ τ imply (z,xz) ∈ τ. again, (z−1,z−1) ∈ τ and (z,xz) ∈ τ implies that (e,z−1xz) ∈ τ. hence, z−1xz ∈ h for any x ∈ h and z ∈ g. therefore, h is a normal subgroup of the group g. � theorem 5.4. let g be a type-2 fuzzy τ tolerance group, e its identity element. the set h of the elements x ∈ g such that (e,x) ∈ τ is isomorphic to the group g. proof. according to theorem 3, the set h of all elements x ∈ g s.t. (e,x) ∈ τ is a normal subgroup of the group g. let x ∈ h,y ∈ h then according to the statement of the theorem (e,x) ∈ τ and (e,y) ∈ τ. let z ∈ g and consider the class zh in the group g. let x′ ∈ zh,y′ ∈ zh . this shows that x′ = zx,y′ = zy, where x ∈ h,y ∈ h. since x,y ∈ h so (x,y) ∈ τ. as z ∈ g and (z,z) ∈ τ because τ is type-2 fuzzy reflexive. again, (z,z) ∈ τ and this together with (x,y) ∈ τ implies that (zx,zy) ∈ τ , thus (x′,y′) ∈ τ. let us have two elements z1,z2 of g such that z1h 6= z2h. let x1 ∈ z1h,x2 ∈ z2h . this means that x1 = z1y1 , x2 = z2y2 where y1 ∈ h,y2 ∈ h. suppose that (x1,x2) ∈ τ which implies that (z1y1,z2y2) ∈ τ. the relations (z−11 ,z −1 1 ) ∈ τ and (z1y1,z2y2) ∈ τ imply (y1,z −1 1 z2y2) ∈ τ. again, (y1,z −1 1 z2y2) ∈ τ and (y −1 1 ,y −1 1 ) ∈ τ which implies that (e,z −1 1 z2y2y −1 1 ) ∈ τ and therefore, z−11 z2y2y −1 1 ∈ h. as h is a subgroup of the group g and the elements y1,y2 ∈ h, (z−11 z2y2y −1 1 )y1y −1 2 = z −1 1 z2 ∈ h. but then z2 ∈ z1h, [because z2 = z1(z −1 1 z2)andz −1 1 z2 ∈ h] and therefore z1h = z2h which is a contradiction to the assumption that z1h 6= z2h. � theorem 5.5. let s be a τ type-2 fuzzy tolerance semi group . t its right(or left, or two sided ) ideal. the set τt of the elements x of s such that (x,x′) ∈ τ where x′ ∈ t , is a right (or left, or two sided, respectively) ideal of the semi group s. proof. let x ∈ τt , let t be left ideal of s. there exists x′ ∈ t such that (x,x′) ∈ τ. now, let y ∈ s. as the relation τ is reflexive, we have (y,y) ∈ τ. then (x,x′) ∈ τ, (y,y) ∈ τ which implies that (xy,x′y) ∈ τ. but x′y ∈ t because x′ ∈ t and t is left ideal. therefore, xy ∈ τt and τt is the left ideal of the semigroup. similarly, for right and two sided ideal. � 6. conclusion we have introduced the type-2 fuzzy g-tolerance relation and showed that it can result to type-2 fuzzy g-equivalence relations by composition. we have also studied the concept of type-2 fuzzy gtolerance relations in abstract algebra. acknowledgements. the work and research of the second author of this paper is financially supported by national institute of technology silchar, assam, india. references [1] r. aliev, w. pedrycz, b. guirimov, et al., type-2 fuzzy nueral networks with fuzzy clustering and differential evolution optimization, inf. sci. 181 (2011), 1591-1608. [2] s. chakravarty and p.k. dash, a pso based integrated functional link net and interval type-2 fuzzy logic system for predicting stock market indices, appl. soft comput. 12 (2012), 931-941. [3] b. choi and f. rhee, interval type-2 fuzzy membership function generation methods for pattern recognition, inf. sci. 179 (2009), 2102-2122. [4] t. dereli, a. baykasoglu, k. altun, et al., industrial applications of type-2 fuzzy sets and systems: a concise review, comput. indust. 62 (2011), 125-137. [5] dhiman dutta and mausumi sen, type-2 fuzzy equivalence relation on a groupoid under balanced and semibalanced maps, j. inf. math. sci. (2017), in press . [6] d. dubois and h. prade, operations on fuzzy numbers, int. j. syst. sci. 9 (6) (1978), 613-626. [7] d. dubois and h. prade, operations in a fuzzy-valued logic, inf. control 43 (1979), 224-240. 178 sen, dutta and deshpande [8] d. dubois and h. prade, fuzzy sets and systems: theory and applications, first ed., acdemic press inc., new york, 1980. [9] s.s. gilan, m.h. sebt and v. shahhosseini, computing with words for hierarchical competency based selection of personal in construction companies, appl. soft comput. 12 (2012), 860-871. [10] b. q. hu and c. y. wang, on type-2 fuzzy relations and interval-valued type-2 fuzzy sets, fuzzy sets syst. 236 (2014), 1-32. [11] n.n. karnik and j.m. mendel, applications of type-2 fuzzy logic systems to forecasting of time-series, inf. sci. 120 (1999), 89-111. [12] n. n. karnik and j. m. mendel, operations on type-2 fuzzy sets, fuzzy sets syste. 79 (2001), 327-348. [13] p. kundu, s. kar and m. maiti, multi-item solid transportation problem with type-2 fuzzy parameters, appl. soft comput. 31 (2015), 61-80. [14] j.m. mendel, computing with words and its relationship with fuzzistics, inf. sci. 177 (2007), 988-1006. [15] m.mizumoto and k.tanaka, some properties of fuzzy sets of type-2, inf. control 31 (1976), 312-340. [16] m.mizumoto and k.tanaka, fuzzy sets of type-2 under algebraic product and algebraic sum, fuzzy sets syst. 31 (1981), 277-290. [17] i. ozkan and i.b. turksen, minimax e-star cluster validity index for type-2 fuzziness, inform. sci. 184 (2012), 64-74. [18] c. leal-ramirez, o. castillo, p. melin, et al. , simulation of the bird age-structured population growth based on an interval type-2 fuzzy cellular structure, inf. sci. 181 (2011), 519-535. [19] s.w. tung, c. quek, c. guan, et2fis: an evolving type-2 neural fuzzy inference system, inf. sci. 220 (2013), 124-148. [20] d. wu, w. tan, a simplified type-2 fuzzy logic controller for real-time control, isa trans. 45 (4) (2006), 503-516. [21] l.a. zadeh, the concept of a linguistic variable and its application to approximate reasoning-i, inf. sci. 8 (1975), 199-249. 1department of mathematics, national institute of technology silchar, assam, india 2berkeley initiative in soft computing, university of california, berkeley ca usa, department of computer science and engineering, national institute of technology silchar, silchar, india ∗corresponding author: senmausumi@gmail.com 1. introduction 2. preliminaries 3. type-2 fuzzy g-equivalence relation 4. type-2 fuzzy g-tolerance relation to type-2 fuzzy equivalence relation 5. type-2 fuzzy tolerance relations in algebraic structures 6. conclusion references int. j. anal. appl. (2023), 21:36 a comparison of nonparametric statistics and bootstrap methods for testing two independent populations with unequal variance wandee wanishsakpong, kantima sodrung, ampai thongteeraparp∗ department of statistics, faculty of science, kasetsart university, bangkok 10900, thailand ∗corresponding author: fsciamu@ku.ac.th abstract. the common parametric statistics used for testing two independent populations have often required the assumptions of normality and equal variances. nonparametric tests have been used when assumptions of parametric tests cannot be achieved. however, some studies found nonparametric tests to be too conservative and less powerful than parametric tests. bootstrap methods are also alternative tests when assumptions of parametric tests are violated, but they have small size limitations. later, nonparametric tests when pooled with the bootstrap methods may overcome the powerful test and small sample sizes issue. thus, the purpose of this study was to apply the bootstrap method together with nonparametric statistics and compare the efficiency of nonparametric tests and bootstraps methods when pooled with nonparametric tests for testing the mean difference between two independent populations with unequal variance. the yuen welch test (yw), brunner-munzel test (bm), bootstrap yuen welch test (byw) and bootstrap brunner-munzel test (bbm) were studied via monte carlo simulation with non-normal population distributions. the results show that the probability of a type i error of all four test statistics could be controlled for all situations. the brunner-munzel test (bm) had the highest power and the best efficiency in the case of mean difference ratio increases. the bootstrap yuen welch test (byw) had the highest power when the sample size was small. 1. introduction to test the mean differences between two independent populations, a well-known test statistic is the t-test. it is a parametric statistic that requires the assumption of normality (fagerland & sandvik, 2009). in practice, the analyzed data may violate the normality assumption. this will affect the performance of the test statistic. nonparametric statistics are alternative method when the data received: feb. 28, 2023. 2010 mathematics subject classification. 62p30. key words and phrases. bootstrap method; nonparametric statistics; yuen welch test; brunner-munzel test. https://doi.org/10.28924/2291-8639-21-2023-36 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-36 2 int. j. anal. appl. (2023), 21:36 does not meet the assumption of normality. three popular nonparametric statistics were used to test the population means: yuen welch test (yuen, 1974), rank welch test (zimmerman & zumbo, 1993) and the brunner-munzel test (brunner & munzel, 2000). in addition, the bootstrap method was combined with nonparametric statistics using resampling technique (efron & tibshirani, 1993). noppadon & chinnapong (2010) studied and compared the brunner-munzel test, the bootstrap brunner-munzel test and the bootstrap rank welch test for unequal variances with lognormal distributions. the brunner-munzel test had the ability to control for the probability of a type i error in all cases and had superior power. dwivedi et al. (2017) found that the nonparametric bootstrap t test had power equal to or greater than the t-test, welch t test, wilcoxon rank sum test or the permutation test. in this study, the researcher is interested in applying the bootstrap method with nonparametric statistics and then comparing the power of the tests. the performance of four tests: yuen welch test (yw), brunner-munzel test (bm), bootstrap yuen welch test (byw) and bootstrap brunner-munzel (bbm) are studied and compared under the laplace distribution and the lognormal distribution for testing the mean differences between two independent populations with unequal variance. 2. statistical methods 2.1. yuen welch test statistic (yw). the yuen welch test statistic was proposed by yuen (1974). it was developed from the t-test by calculating the trimmed mean. the amount of trimming for this study is 20% which is the best option (wilcox, 2005). thus the 20% smallest and 20% largest observations in each sample are removed (fagerland & sandvik, 2009). the yuen welch test statistic is yw = x̄γ − ȳγ√ dx + dy (2.1) where dx = sw2 x (n1−1) hx(hx−1) , dy = sw2 y (n2−1) hy (hy −1) with degree of freedom dfy w = (dx+dy ) 2 [ d2 x hx−1 + d2 y hy −1 ]2 , where x̄γ and ȳγ are the mean of the sample after trimming, dx and dy are estimates of the squared standard errors, hx and hy are the number of observations remaining in the samples x and y after trimming, sw2x and sw 2 y are the samples windsorized variances, n1 and n2 are the sample sizes. 2.2. brunner-munzel test statistic (bm). brunner and munzel (2000) proposed the brunnermunzel test which is associated with the midrank. it is an update of the wilcoxon-mann-whitney int. j. anal. appl. (2023), 21:36 3 test statistic using the sample variance as a modifier and using the degrees of freedom as proposed by satterwaite-smith-welz. the brunner-munzel test statistic is bm = n1n2(r̄2 − r̄1) n1 + n2 √ n1s 2 1 + n2s 2 2 . (2.2) the distribution of the brunner-munzel test can be approximated by a t-distribution with degrees of freedom dfbm = (n1s 2 1 + n2s 2 2) 2 (n1s 2 1) 2 n1−1 + (n2s 2 2) 2 n2−1 when r̄1 and r̄2 are the mean of rank associated with sample x and y when data are pooled, n1 and n2 are the sample sizes, s21 and s 2 2 are the variance of rank associated with sample x and y with replacement. 2.3. the bootstrap method. the bootstrap method is a statistical procedure that uses the principle of resampling to generate a new sample from a single random sample. (efron & tibshirani, 1993). in this research, two bootstrap methods with nonparametric statistics are applied as follows: 2.3.1. bootstrap yuen welch test (byw). (i) let x = x1,x2, · · · ,xn1 is the observed sample 1 of size n1 and y = y1,y2, · · · ,yn2 is the observed sample 2 of size n2. (ii) evaluate test statistic: yw as in equation (2.1). (iii) return sampling from x and y of size n1 and n2 denoted by x∗ and y ∗ respectively. (iv) evaluate test statistic: yw∗ as follows: yw∗ = x̄∗γ − ȳ ∗γ√ d∗ x + d∗ y (2.3) (v) repeat step (iii) and (iv) for 1000 times (vi) approximate p−value = number of times(|y w ∗|≥|y w |) 1,000 2.3.2. bootstarp bunner-munzel test (bbm). (i) let x = x1,x2, · · · ,xn1 is the observed sample 1 of size n1 and y = y1,y2, · · · ,yn2 is the observed sample 2 of size n2. (ii) evaluate test statistic: bm as in equation (2.2). (iii) draw two bootstrap samples with replacement: one of size n1 and n2 denoted by x∗ and y ∗ respectively. (iv) evaluate test statistic: bm∗ as follows: bm∗ = n∗1n ∗ 2(r̄ ∗ 2 − r̄ ∗ 1) n∗1 + n ∗ 2 √ n∗1s 2 1 ∗ + n∗2s 2 2 ∗ . (2.4) (v) repeat steps (iii) and (iv) for 1,000 times. 4 int. j. anal. appl. (2023), 21:36 (vi) approximate p−value = number of times(|bm ∗|≥|bm|) 1,000 3. research method the simulation data are generated by r programming with monte carlo technique to compare the efficiency of four test statistics for the following situations: (1) generate two independent populations into two distributions. (a) population with laplace distribution (b) population with lognormal distribution (2) determine the sample sizes for both equal and unequal sizes: (a) equal sample size (n1,n2) is (10,10), (20,20), (30,30), (50,50) and (100,100) (b) unequal sample size (n1,n2) is (10,15), (10,20), (30,40), (45,50) and (50,100) (3) determine the means of two population groups as follows: (a) the means of two populations are not different for evaluating the probability of a type i error. (b) the means of two populations are different with the ratio 1.5:1 and 2:1 for evaluating the power of the test. (4) determine the variance ratios of two populations for unequal variances, which were 3:1 and 5:1 (ratiwat, 2019). (5) the significance level of the test is 0.05. (6) the number of replications for each condition are 5,000 and bootstrap numbers are based on 1,000 replications. 4. results the results of the simulation are as follows: 4.1. probability of type i error. the ability to control for the type i error of the four test statistics is considered based on criteria from bradley (1978). at a significance level of 0.05, the test statistic with a probability of a type i error between [0.025 0.075] is considered as able to control for the probability of a type i error. the probability of a type i error of the four test statistics had the following details: from the tables 1 and 2, it would be found that when the population had laplace and lognormal distributions, and the ratios of variance are 3:1 and 5:1, all four test statistics were able to control for the probability of a type i error for both equal and unequal sample sizes. int. j. anal. appl. (2023), 21:36 5 distribution variance ratio sample size probability of type i error nonparametric bootstrap yw bm byw bbm laplace 3:1 (10,10) 0.0384 0.0540 0.0344 0.0384 (20,20) 0.0474 0.0572 0.0364 0.0410 (30,30) 0.0472 0.0494 0.0464 0.0376 (50,50) 0.0494 0.0492 0.0272 0.0428 (100,100) 0.0464 0.0458 0.0368 0.0334 5:1 (10,10) 0.0458 0.0620 0.0332 0.0418 (20,20) 0.0486 0.0524 0.0322 0.0372 (30,30) 0.0486 0.0548 0.0288 0.0358 (50,50) 0.0478 0.0504 0.0376 0.0382 (100,100) 0.0524 0.0530 0.0336 0.0344 lognormal 3:1 (10,10) 0.0454 0.0550 0.0300 0.0458 (20,20) 0.0492 0.0536 0.0290 0.0368 (30,30) 0.0430 0.0488 0.0370 0.0382 (50,50) 0.0552 0.0498 0.0376 0.0300 (100, 100) 0.0580 0.0500 0.0348 0.0298 5:1 (10,10) 0.0522 0.0600 0.0402 0.0462 (20,20) 0.0406 0.0518 0.0412 0.0338 (30,30) 0.0466 0.0522 0.0290 0.0298 (50,50) 0.0536 0.0548 0.0388 0.0316 (100, 100) 0.0450 0.0512 0.0450 0.0392 table 1. probability of type i error for four test statistics with equal sample size 6 int. j. anal. appl. (2023), 21:36 distribution variance ratio sample size probability of type i error nonparametric bootstrap yw bm byw bbm laplace 3:1 (10,15) 0.0428 0.0568 0.0394 0.0338 (10,20) 0.0446 0.0562 0.0346 0.0310 (30,40) 0.0490 0.0512 0.0310 0.0374 (45,50) 0.0522 0.0566 0.0326 0.0376 (50,100) 0.0486 0.0506 0.0308 0.0382 5:1 (10,15) 0.0448 0.0566 0.0282 0.0364 (10,20) 0.0478 0.0566 0.0362 0.0406 (30,40) 0.0492 0.0508 0.0364 0.0394 (45,50) 0.0506 0.0522 0.0366 0.0332 (50,100) 0.0510 0.0508 0.0454 0.0280 lognormal 3:1 (10,15) 0.0496 0.0520 0.0350 0.0344 (10,20) 0.0580 0.0550 0.0408 0.0365 (30,40) 0.0450 0.0520 0.0380 0.0334 (45,50) 0.0516 0.0508 0.0420 0.0362 (50,100) 0.0568 0.0506 0.0364 0.0350 5:1 (10,15) 0.0534 0.0550 0.0368 0.0354 (10,20) 0.0538 0.0498 0.0354 0.0358 (30,40) 0.0450 0.0472 0.0426 0.0334 (45,50) 0.0588 0.0546 0.0366 0.0338 (50,100) 0.0582 0.0492 0.0508 0.0398 table 2. probability of type i error of four test statistics with unequal sample sizes 4.2. power of the tests. to study the power of the tests, the difference between the means of two populations is determined in two cases: (1) a difference with the ratio of 1.5:1 for the small mean difference and (2) a difference with the ratio of 2:1 for the moderate mean difference. to compare the efficiency of the four test statistics, the power of the test statistics can control for the probability of type i errors are considered. the details are as follow. (1) the sample sizes are equal (a) the variance ratio is 3:1: for the laplace distribution with the small mean difference, the bootstrap yuen welch test (byw) has a higher estimation power than the yuen welch test, except for sample sizes (50,50) and (100,100). the yuen welch test (yw), on the other hand, has the int. j. anal. appl. (2023), 21:36 7 higher test estimation power, when the mean difference is moderate, with the only exception being when the sample size is (10,10). in that case the bootstrap yuen welch test (byw) is marginally superior in test estimation power. (as shown in figure 1) for the lognormal distribution with the small mean difference, the brunner-munzel test (bm) has the highest estimation power with the exception of a sample sizes equal to (10,10), (20,20) and (30,30). for those sample sizes, the bootstrap yuen welch test (byw) has the highest estimation power. when the mean difference is moderate, the bruner-munzel test (bm) has the highest test estimation power for all sample sizes. (as shown in figure 1) figure 1. test power estimation when given a variance ratio of 3:1 for equal sample sizes. (b) the variance ratio is 5:1 with equal sample sizes: for the laplace distribution with a small mean difference, the bootstrap yuen welch test (byw) has the highest test estimation power, except for the case of a sample size (100,100) the bruner-munzel test is superior. the yuen welch test (yw) has the highest estimation power when the mean difference is moderate, except for a sample size equal to (10,10), the bootstrap yuen welch test (byw) has a higher test estimation power. (as shown in figure 2) for the lognormal distribution with a small mean difference, the brunner-munzel test (bm) has the highest test estimation power except for sample sizes of (10,10) and (20,20) in which the bootstrap yuen welch test (byw) has superior test estimation power. when the mean difference is moderate, the bruner-munzel test (bm) has the highest test estimation power for all sample sizes (as shown in figure 2). 8 int. j. anal. appl. (2023), 21:36 figure 2. test power estimation when given a variance ratio of 5:1 for the equal sample sizes. (2) the sample sizes are not equal (a) the variance ratio is 3:1 with unequal sample sizes: for a laplace distribution with the small mean difference, the bootstrap yuen welch test (byw) has the highest test estimation power, except for the cases of sample sizes of (45,50) and (50,100) in which the bruner-munzel statistic is superior. when the mean difference is moderate, the yuen welch test (yw) has the highest test estimation power. except for sample sizes of (10,15) and (10,20), the bootstrap yuen welch test (byw) has the highest estimation power. (as shown in figure 3) for lognormal distributions with small mean differences, the brunner-munzel test (bm) has the highest test estimation power except for sample sizes of (10,15) and (10,20) in which the bootstrap yuen welch test (byw) has a higher test estimation power. when the mean difference is moderate, the bruner-munzel test (bm) has the highest test estimation power for all sample sizes. (as shown in figure 3) figure 3. test power estimation with a variance ratio of 3:1 for unequal sample sizes. int. j. anal. appl. (2023), 21:36 9 (b) the variance ratio is 5:1 with unequal sample sizes: for the laplace distribution with a small mean difference, the yuen welch test (byw) has the highest test estimation power except for sample sizes of (45,50) and (50,100). when the mean difference is moderate, the yuen welch test (yw) has the highest estimation power except for sample sizes of (10,15) and (10,20), where the bootstrap yuen welch test (byw) has the greatest estimation power. (as shown in figure 4) for the lognormal distribution with small mean differences, the brunner-munzel test (bm) has the highest test estimation power, except for sample sizes of (10,15) and (10,20) in which the bootstrap yuen welch test (byw) has the higher test estimation power. when the mean differences are moderate, the bruner-munzel test (bm) has the highest estimation power regardless of sample size. (as shown in figure 4) figure 4. test power estimation with a variance ratio of 5:1 for unequal sample sizes. 5. conclusion and discussion in summary, it was found that under both the laplace and lognormal distributions with sample sizes being both equal and unequal, the probability of a type i error for the four test statistics could be controlled for in all cases according to bradley’s criteria. the test statistics with the highest testing power in most situations for the laplace distribution was the bootstrap yuen welch test (byw). however, as mean differences increased, the yuen welch test (yw) had the greatest strength. the yw test was the best with a sample size of more than 30 in which the sample were both equal and unequal sizes. the test statistics with the highest testing power in most situations for the lognormal distribution was the brunner-munzel test (bm). however, as mean differences increased, the bootstrap yuen 10 int. j. anal. appl. (2023), 21:36 welch (byw) test statistic had the highest testing power. in this study, the bootstrap yuen welch test (byw) method had the highest testing power in the skewness to the right when the mean difference was small and the sample size was less than 20. when the sample size was larger and the data less skewed, the brunner-munzel test (bm) had a greater testing power than the bootstrap yuen welch test. this is consistent with research by fagerland and sandvik (2009) which found that the yuen welch test (yw) was appropriate for data with skewness to the right. moreover, it was found the power of the test was higher as the sample size increased. acknowledge: the authors would like to thank the international sciku branding (isb), faculty of science, kasetsart university, thailand for supporting this research. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j.v. bradley, robustness?, br. j. math. stat. psychol. 31 (1978), 144-152. https://doi.org/10.1111/j. 2044-8317.1978.tb00581.x. [2] e. brunner, u. munzel, the nonparametric behrens-fisher problem: asymptotic theory and a small-sample approximation, biom. j. 42 (2000), 17-25. https://doi.org/10.1002/(sici)1521-4036(200001)42:1<17:: aid-bimj17>3.0.co;2-u. [3] a.k. dwivedi, i. mallawaarachchi, l.a. alvarado, analysis of small sample size studies using nonparametric bootstrap test with pooled resampling method, stat. med. 36 (2017), 2187-2205. https://doi.org/10.1002/ sim.7263. [4] b. efron, r.j. tibshirani, an introduction to the bootstrap, chapman & hall, new york, (1993). [5] m.w. fagerland, l. sandvik, performance of five two-sample location tests for skewed distributions with unequal variances, contemp. clinic. trials. 30 (2009), 490-496. https://doi.org/10.1016/j.cct.2009.06.007. [6] w. noppadon, b. chinnapong, a comparison study of nonparametric test statistics for two populations in case of heterogeneity of variances, j. sci. technol. 18 (2010), 60-69. [7] s. ratiwat, comparison of nonparametric statistical test for two independent difference medians in case of symmetrical and skewed distribution, master thesis, bangkok: king mongkut’s institute of technology ladkrabang thailand, (2019). [8] j. reiczigel, i. zakarias, l. rozsa, a bootstrap test of stochastic equality of two populations, amer. stat. 59 (2005), 156-161. https://doi.org/10.1198/000313005x23526. [9] r.r. wilcox, introduction to robust estimation and hypothesis testing (2nd ed.), academic press, san diego, (2005). [10] k.k. yuen, the two-sample trimmed t for unequal population variances, biometrika, 61 (1974), 165-170. [11] d.w. zimmerman, b.d. zumbo, rank transformations and the power of the student t test and welch t’ test for non-normal populations with unequal variances, canadian j. exper. psychol. 47 (1993), 523-539. https: //doi.org/10.1037/h0078850. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x https://doi.org/10.1111/j.2044-8317.1978.tb00581.x https://doi.org/10.1002/(sici)1521-4036(200001)42:1<17::aid-bimj17>3.0.co;2-u https://doi.org/10.1002/(sici)1521-4036(200001)42:1<17::aid-bimj17>3.0.co;2-u https://doi.org/10.1002/sim.7263 https://doi.org/10.1002/sim.7263 https://doi.org/10.1016/j.cct.2009.06.007 https://doi.org/10.1198/000313005x23526 https://doi.org/10.1037/h0078850 https://doi.org/10.1037/h0078850 1. introduction 2. statistical methods 2.1. yuen welch test statistic (yw) 2.2. brunner-munzel test statistic (bm) 2.3. the bootstrap method 3. research method 4. results 4.1. probability of type i error 4.2. power of the tests 5. conclusion and discussion references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 142-156 http://www.etamaths.com on ostrowski type inequalities for functions of two variables with bounded variation hüseyin budak∗ and mehmet zeki sarikaya abstract. in this paper, we establish a new generalization of ostrowski type inequalities for functions of two independent variables with bounded variation and apply it for qubature formulae. some connections with the rectangle, the midpoint and simpson’s rule are also given. 1. introduction let f : [a,b] → r be a differentiable mapping on (a,b) whoose derivative f′ : (a,b) → r is bounded on (a,b) , i.e. ‖f′‖∞ := sup t∈(a,b) |f′(t)| < ∞. then we have the inequality (1.1) ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ , for all x ∈ [a,b][19]. the constant 1 4 is the best possible. this inequality is well known in the literature as the ostrowski inequality. in [11], dragomir proved following ostrowski type inequalities related functions of bounded variation: theorem 1. let f : [a,b] → r be a mapping of bounded variation on [a,b] . then∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) f(x) ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f) holds for all x ∈ [a,b] . the constant 1 2 is the best possible. 2. preliminaries and lemmas in 1910, fréchet [16] has given the following characterization for the double riemann-stieltjes integral. assume that f(x,y) and α(x,y) are defined over the rectangle q = [a,b]× [c,d]; let r be the divided into rectangular subdivisions, or cells, by the net of straight lines x = xi, y = yi, a = x0 < x1 < ... < xn = b, and c = y0 < y1 < ... < ym = d; let ξi,ηj be any numbers satisfying ξi ∈ [xi−1,xi] , ηj ∈ [yj−1,yj] , (i = 1, 2, ...,n; j = 1, 2, ...,m); and for all i,j let ∆11α(xi,yj) = α(xi−1,yj−1) −α(xi−1,yj) −α(xi,yj−1) + α(xi,yj). then if the sum s = n∑ i=1 m∑ j=1 f (ξi,ηj) ∆11α(xi,yj) 2010 mathematics subject classification. 26d15, 26b30, 26d10, 41a55. key words and phrases. bounded variation; ostrowski type inequalities; riemann-stieltjes integral. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 142 ostrowski type inequalities for functions... 143 tends to a finite limit as the norm of the subdivisions approaches zero, the integral of f with respect to α is said to exist. we call this limit the restricted integral, and designate it by the symbol (2.1) b∫ a d∫ c f(x,y)dydxα(x,y). if in the above formulation s is replaced by the sum s∗ = n∑ i=1 m∑ j=1 f (ξij,ηij) ∆11α(xi,yj), where ξij,ηij are numbers satisfying ξij ∈ [xi−1,xi] , ηij ∈ [yj−1,yj] , we call the limit, when it exist, the unrestricted integral, and designate it by the symbol (2.2) b∫ a d∫ c f(x,y)dydxα(x,y). clearly, the existence of (2.2) implies both the existence of (2.1) and its equality (2.2). on the other hand, clarkson ([8]) has shown that the existence of (2.1) does not imply the existence of (2.2). in [7], clarkson and adams gave the following definitions of bounded variation for functions of two variables: 2.1. definitions. the function f(x,y) is assumed to be defined in rectangle r(a ≤ x ≤ b, c ≤ y ≤ d). by the term net we shall, unless otherwise specified mean a set of parallels to the axes: x = xi(i = 0, 1, 2, ...,m), a = x0 < x1 < ... < xm = b; y = yj(j = 0, 1, 2, ...,n), c = y0 < y1 < ... < yn = d. each of the smaller rectangles into which r is devided by a net will be called a cell. we employ the notation ∆11f(xi,yj) = f(xi+1,yj+1) −f(xi+1,yj) −f(xi,yj+1) + f(xi,yj), ∆f(xi,yj) = f(xi+1,yj+1) −f(xi,yj). the total variation function, φ(x) [ψ(y)] , is defined as the total variation of f(x,y) [f(x,y)] considered as a function of y [x] alone in interval (c,d) [(a,b)], or as +∞ if f(x,y) [f(x,y)] is of unbounded variation. definition 1. (vitali-lebesque-fréchet-de la vallée poussin). the function f(x,y) is said tobe of bounded variation if the sum m−1 , n−1∑ i=0 , j=0 |∆11f(xi,yj)| is bounded for all nets. definition 2. (fréchet). the function f(x,y) is said tobe of bounded variation if the sum m−1 , n−1∑ i=0 , j=0 �i�j |∆11f(xi,yj)| is bounded for all nets and all possible choices of �i = ±1 and �j = ±1. definition 3. (hardy-krause). the function f(x,y) is said tobe of bounded variation if it satisfies the condition of definition 1 and if in addition f(x,y) is of bounded variation in y (i.e. φ(x) is finite) for at least one x and f(x,y) is of bounded variation in y (i.e. ψ(y) is finite) for at least one y. definition 4. (arzelà). let (xi,yi) (i = 0, 1, 2, ...,m) be any set of points satisfiying the conditions a = x0 < x1 < ... < xm = b; c = y0 < y1 < ... < ym = d. 144 budak and sarikaya then f(x,y) is said tobe of bounded variation if the sum m∑ i=1 |∆f(xi,yi)| is bounded for all such sets of points. therefore, one can define the consept of total variation of a function of two variables, as follows: let f be of bounded variation on q = [a,b]×[c,d], and let ∑ (p) denote the sum n∑ i=1 m∑ j=1 |∆11f(xi,yj)| corresponding to the partition p of q. the number∨ q (f) := d∨ c b∨ a (f) := sup {∑ (p) : p ∈ p (q) } , is called the total variation of f on q. here p ([a,b]) denotes the family of partitions of [a,b] . in [17], authors proved foolowing lemmas related double riemann-stieltjes integral: lemma 1. (integrating by parts) if f ∈ rs(α) on q, then α ∈ rs(f) on q, and we have d∫ c b∫ a f(t,s)dtdsα(t,s) + d∫ c b∫ a α(t,s)dtdsf(t,s)(2.3) = f(b,d)α(b,d) −f(b,c)α(b,c) −f(a,d)α(a,d) + f(a,c)α(a,c). lemma 2. assume that g ∈ rs(α) on q and α is of bounded variation on q, then (2.4) ∣∣∣∣∣∣ d∫ c b∫ a g(x,y)dxdyα(x,y) ∣∣∣∣∣∣ ≤ sup(x,y)∈q |g(x,y)| ∨ q (α) . in [17], jawarneh and noorani obtained following ostrowski type inequality for functions of two variables with bounded variation: theorem 2. let f : q →→ r be mapping of bounded variation on q. then for all (x,y) ∈ q, we have inequality ∣∣∣∣∣∣(b−a) (d− c) f(x,y) − d∫ c b∫ a f(t,s)dtds ∣∣∣∣∣∣(2.5) ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ][ 1 2 (d− c) + ∣∣∣∣y − c + d2 ∣∣∣∣ ]∨ q (f) where ∨ q (f) denotes the total (double) variation of f on q. for more information and recent developments on inequalities for mappings of bounded variation, please refer to([1]-[6],[9]-[15],[17],[18],[20]-[24]). the aim of this paper is to establish a new generzlization of ostrowski type inequalities for functions of two independent variables with bounded variation and apply it for qubature formulae. some connections with the rectangle, the midpoint and simpson’s rule are also given. 3. main results first, we give the following notations used in main our theorem; let ∆n,m := {(x0,y0) , (x0,y1) , ..., (x0,ym) , (x1,y0) , ..., (x1,ym) , ..., (xn,y0) , (xn,y1) , ..., (xn,ym)} is a partition of q = [a,b]×[c,d] satisfaying a = x0, b = xn, y0 = c, ym = d with α0 = a, αi ∈ [xi−1,xi] (i = 1, ...,n) , αn+1 = b and β0 = c, βj ∈ [yj−1,yj] (j = 1, ...,m) , βm+1 = d. υ(h) := max{hi| i = 0, ...,n− 1} , hi := xi+1 −xi, ostrowski type inequalities for functions... 145 υ(l) := max{lj| j = 0, ...,m− 1} , lj := yj+1 −yj. theorem 3. if f : q → r is of bounded variatin on q, then we have the inequality ∣∣∣∣∣∣ n∑ i=0 m∑ j=0 (αi+1 −αi) (βj+1 −βj) f(xi,yj) − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣(3.1) ≤ [ 1 2 υ(h) + max i∈{0,...,n−1} ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ] × [ 1 2 υ(l) + max j∈{0,...,m−1} ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ] b∨ a d∨ c (f) ≤ υ(h)υ(l) b∨ a d∨ c (f) where b∨ a d∨ c (f) is the total variation of f on q. proof. let us consider the mappings k and l given by k(t) =   t−α1, t ∈ [a,x1) t−α2, t ∈ [x1,x2) ... t−αn−1, t ∈ [xn−2,xn−1) t−αn, t ∈ [xn−1,b] , l(s) =   s−β1, s ∈ [c,y1) s−β2, s ∈ [y1,y2) ... s−βm−1, s ∈ [ym−2,ym−1) s−βm, s ∈ [ym−1,d] . integrating by parts using lemma 1, we obtain b∫ a d∫ c k(t)l(s)dsdtf(t,s) = n−1∑ i=0 m−1∑ j=0   xi+1∫ xi yj+1∫ yj k(t)l(s)dsdtf(t,s)  (3.2) = n−1∑ i=0 m−1∑ j=0   xi+1∫ xi yj+1∫ yj (t−αi+1) (s−βj+1) dsdtf(t,s)   = n−1∑ i=0 m−1∑ j=0 [(xi+1 −αi+1) (yj+1 −βj+1) f(xi+1,yj+1) −(xi+1 −αi+1) (yj −βj+1) f(xi+1,yj) −(xi −αi+1) (yj+1 −βj+1) f(xi,yj+1) + (xi −αi+1) (yj −βj+1) f(xi,yj) − xi+1∫ xi yj+1∫ yj f(t,s)dsdt   146 budak and sarikaya = n∑ i=1 m∑ j=1 (xi −αi) (yj −βj) f(xi,yj) − n∑ i=1 m−1∑ j=0 (xi −αi) (yj −βj+1) f(xi,yj) − n−1∑ i=0 m∑ j=1 (xi −αi+1) (yj −βj) f(xi,yj) + n−1∑ i=0 m−1∑ j=0 (xi −αi+1) (yj −βj+1) f(xi,yj) − b∫ a d∫ c f(t,s)dsdt. in last equality, we have n∑ i=1 m∑ j=1 (xi −αi) (yj −βj) f(xi,yj)(3.3) = (b−αn) (d−βm) f(b,d) + (b−αn) m−1∑ j=1 (yj −βj) f(b,yj) + (d−βm) n−1∑ i=1 (xi −αi) f(xi,d) + n−1∑ i=1 m−1∑ j=1 (xi −αi) (yj −βj) f(xi,yj). similarly, we have n∑ i=1 m−1∑ j=0 (xi −αi) (yj −βj+1) f(xi,yj)(3.4) = (b−αn) (c−β1) f(b,c) + (b−αn) m−1∑ j=1 (yj −βj+1) f(b,yj) + (c−β1) n−1∑ i=1 (xi −αi) f(xi,c) + n−1∑ i=1 m−1∑ j=1 (xi −αi) (yj −βj+1) f(xi,yj), n−1∑ i=0 m∑ j=1 (xi −αi+1) (yj −βj) f(xi,yj)(3.5) = (a−α1) (d−βm) f (a,d) + (a−α1) m−1∑ j=1 (yj −βj) f(a,yj) + (d−βm) n−1∑ i=1 (xi −αi+1) f(xi,d) + n−1∑ i=1 m−1∑ j=1 (xi −αi+1) (yj −βj) f(xi,yj) and n−1∑ i=0 m−1∑ j=0 (xi −αi+1) (yj −βj+1) f(xi,yj)(3.6) = (a−α1) (c−β1) f (a,c) + (a−α1) m−1∑ j=1 (yj −βj+1) f(a,yj) + (c−β1) n−1∑ i=1 (xi −αi+1) f(xi,c) + n−1∑ i=1 m−1∑ j=1 (xi −αi+1) (yj −βj+1) f(xi,yj). ostrowski type inequalities for functions... 147 adding (3.3)-(3.6) in last equality of (3.2), we obtain b∫ a d∫ c k(t)l(s)dsdtf(t,s) = (b−αn) (d−βm) f(b,d) + (b−αn) (β1 − c) f(b,c) + (α1 −a) (d−βm) f (a,d) + (α1 −a) (β1 − c) f (a,c) +(b−αn) m−1∑ j=1 (βj+1 −βj) f(b,yj) + (α1 −a) m−1∑ j=1 (βj+1 −βj) f(a,yj) + (d−βm) n−1∑ i=1 (αi+1 −αi) f(xi,d) + (β1 − c) n−1∑ i=1 (αi+1 −αi) f(xi,c) n−1∑ i=1 m−1∑ j=1 (αi+1 −αi) (βj+1 −βj) f(xi,yj) − b∫ a d∫ c f(t,s)dsdt = n∑ i=0 m∑ j=0 (αi+1 −αi) (βj+1 −βj) f(xi,yj) − b∫ a d∫ c f(t,s)dsdt. on the other hand, we have ∣∣∣∣∣∣ b∫ a d∫ c k(t)l(s)dsdtf(t,s) ∣∣∣∣∣∣ = ∣∣∣∣∣∣∣ n−1∑ i=0 m−1∑ j=0   xi+1∫ xi yj+1∫ yj k(t)l(s)dsdtf(t,s)   ∣∣∣∣∣∣∣(3.7) ≤ n−1∑ i=0 m−1∑ j=0 ∣∣∣∣∣∣∣ xi+1∫ xi yj+1∫ yj (t−αi+1) (s−βj+1) dsdtf(t,s) ∣∣∣∣∣∣∣ . using lemma 2 in the last part of the (3.7), we have ∣∣∣∣∣∣∣ xi+1∫ xi yj+1∫ yj (t−αi+1) (s−βj+1) dsdtf(t,s) ∣∣∣∣∣∣∣(3.8) ≤ sup t∈[xi,xi+1] s∈[yj,yj+1] [|t−αi+1| |s−βj+1|] xi+1∨ xi yj+1∨ yj (f) = max{αi+1 −xi,xi+1 −αi+1}max{βj+1 −yj,yj+1 −βj+1} xi+1∨ xi yj+1∨ yj (f) = [ 1 2 (xi+1 −xi) + ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ] × [ 1 2 (yj+1 −yj) + ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f). 148 budak and sarikaya putting (3.8) in (3.7), we obtain ∣∣∣∣∣∣ b∫ a d∫ c k(t)l(s)dsdtf(t,s) ∣∣∣∣∣∣(3.9) ≤ n−1∑ i=0 m−1∑ j=0 [ 1 2 (xi+1 −xi) + ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ] × [ 1 2 (yj+1 −yj) + ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f) ≤ max i∈[0,...,n−1] [ 1 2 (xi+1 −xi) + ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ] × max j∈[0,...,m−1] [ 1 2 (yj+1 −yj) + ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ]n−1∑ i=0 m−1∑ j=0  xi+1∨ xi yj+1∨ yj (f)   ≤ [ 1 2 υ(h) + max i∈[0,...,n−1] ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ] × [ 1 2 υ(l) + max j∈[0,...,m−1] ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ] b∨ a d∨ c (f) which completes the proof of first inequality in (3.1). in last inequality in (3.9), we get (3.10) ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ≤ 12hi and maxi∈[0,...,n−1] ∣∣∣∣αi+1 − xi + xi+12 ∣∣∣∣ ≤ 12υ(h), and similarly, (3.11) max j∈[0,...,m−1] ∣∣∣∣βj+1 − yj + yj+12 ∣∣∣∣ ≤ 12υ(l). if we add (3.10) and (3.11) in (3.9), the proof of theorem is completed. � now, using the result of the theorem 3, we give some applications as follows: corollary 1. with the assumptions of theorem 3. if we choose α0 = a, α1 = a + x1 2 , α2 = x1 + x2 2 , ...,αn−1 = xn−2 + xn−1 2 , αn = xn−1 + b 2 , αn+1 = b and β0 = c, β1 = c + y1 2 , β2 = y1 + y2 2 , ...,βn−1 = ym−2 + ym−1 2 , βn = ym−1 + d 2 , βm+1 = d ostrowski type inequalities for functions... 149 in theorem 3, then we have the inequality∣∣∣∣14 [(b−xn−1) (d−ym−1) f(b,d) + (b−xn−1) (y1 − c) f(b,c) + (x1 −a) (d−ym−1) f (a,d) + (x1 −a) (y1 − c) f (a,c) +(b−xn−1) m−1∑ j=1 (yj+1 −yj−1) f(b,yj) + (x1 −a) m−1∑ j=1 (yj+1 −yj−1) f(a,yj) + (d−ym−1) n−1∑ i=1 (xi+1 −xi−1) f(xi,d) + (y1 − c) n−1∑ i=1 (xi+1 −xi−1) f(xi,c) + n−1∑ i=1 m−1∑ j=1 (xi+1 −xi−1) (xi+1 −xi−1) f(xi,yj)  − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ 1 4 υ(h)υ(l) b∨ a d∨ c (f). corollary 2. in corollary 1, if we take xi := a + (b − a) in (i = 0, 1, ...,n) and yj := c + (d − c) j m (j = 0, 1, ...,m), then we have the inequality∣∣∣∣∣∣(b−a) (d− c)4nm  f(b,d) + f(b,c) + f (a,d) + f (a,c) + 2  m−1∑ j=1 f ( b, (m− j) c + jd m ) + m−1∑ j=1 f ( a, (m− j) c + jd m ) + n−1∑ i=1 f ( (n− i) a + ib n ,d ) + n−1∑ i=1 f ( (n− i) a + ib n ,c ) +4 n−1∑ i=1 m−1∑ j=1 f ( (n− i) a + ib n , (m− j) c + jd m )− b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ (b−a) (d− c) 4nm b∨ a d∨ c (f). corollary 3. under assumption theorem 3, choosing x0 = a, x1 = b, α0 = a, α1 = α, α2 = b, y0 = c, y1 = d, β0 = c, β1 = β and β2 = d, we obtain the inequality |(α−a) (β − c) f(a,c) + (α−a) (d−β) f(a,d)(3.12) + (b−α) (β − c) f(b,c) + (b−α) (d−β) f(b,d) − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣α− a + b2 ∣∣∣∣ ][ 1 2 (d− c) + ∣∣∣∣β − c + d2 ∣∣∣∣ ] b∨ a d∨ c (f). remark 1. a) if we put α = b and β = d in (3.12), then we have the ”left rectangle inequality”∣∣∣∣∣∣(b−a) (d− c) f(a,c) − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ (b−a) (d− c) b∨ a d∨ c (f), 150 budak and sarikaya b) if take α = a and β = c in (3.12), then we have the ”right rectangle inequality” ∣∣∣∣∣∣(b−a) (d− c) f(b,d) − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ (b−a) (d− c) b∨ a d∨ c (f), c) similarliy, if we put α = a+b 2 and β = c+d 2 in (3.12), then we get the ”trapezoid inequality” ∣∣∣∣∣∣f(b,d) + f(b,c) + f(a,d) + f(a,c)4 − 1(b−a) (d− c) b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ 1 4 b∨ a d∨ c (f). corollary 4. under assumption theorem 3, taking a ≤ x ≤ b, a ≤ α1 ≤ x ≤ α2 ≤ b, c ≤ y ≤ d, c ≤ β1 ≤ y ≤ β2 ≤ d we obtain the inequality |(α1 −a) (β1 − c) f(a,c) + (α1 −a) (β2 −β1) f(a,y)(3.13) + (α1 −a) (d−β2) f(a,d) + (α2 −α1) (β1 − c) f(x,c) + (α2 −α1) (β2 −β1) f(x,y) + (α2 −α1) (d−β2) f(x,d) + (b−α2) (β1 − c) f(b,c) + (b−α2) (β2 −β1) f(b,y) + (b−α2) (d−β2) f(b,d) − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ 1 4 [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ + ∣∣∣∣α1 − a + x2 ∣∣∣∣ + ∣∣∣∣α2 − x + b2 ∣∣∣∣ + ∣∣∣∣ ∣∣∣∣α1 − a + x2 ∣∣∣∣− ∣∣∣∣α2 − x + b2 ∣∣∣∣ ∣∣∣∣ ] × [ 1 2 (d− c) + ∣∣∣∣y − c + d2 ∣∣∣∣ + ∣∣∣∣β1 − c + y2 ∣∣∣∣ + ∣∣∣∣β2 − y + d2 ∣∣∣∣ + ∣∣∣∣ ∣∣∣∣β1 − c + y2 ∣∣∣∣− ∣∣∣∣β2 − y + d2 ∣∣∣∣ ∣∣∣∣ ] b∨ a d∨ c (f) ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ][ 1 2 (d− c) + ∣∣∣∣y − c + d2 ∣∣∣∣ ] b∨ a d∨ c (f) ≤ (b−a) (d− c) b∨ a d∨ c (f). remark 2. if we put α1 = a, α2 = b and β1 = c, β2 = d in (3.13), then the inequality (3.13) reduces the inequality (2.5). ostrowski type inequalities for functions... 151 remark 3. if we choose α1 = 5a+b 6 , α2 = a+5b 6 , x ∈ [ 5a+b 6 , a+5b 6 ] , β1 = 5c+d 6 , β2 = c+5d 6 and y ∈ [ 5c+d 6 , c+5d 6 ] in (3.13), then we have the ”simpson’s rule inequality” ∣∣∣∣(b−a) (d− c) [ f(b,d) + f(b,c) + f(a,d) + f(a,c) 36 + f ( a, c+d 2 ) + f ( a+b 2 ,c ) + f ( b, c+d 2 ) + f ( a+b 2 ,d ) 9 + 4 9 f ( a + b 2 , c + d 2 )] − b∫ a d∫ c f(t,s)dsdt ∣∣∣∣∣∣ ≤ (b−a) (d− c) 9 b∨ a d∨ c (f) which is proved by jawarneh and noorani in [17]. 4. some composite qubature formula let us consider the arbitrary division in : a = x0 < x1 < ... < xn = b, and jm : c = y0 < y1 < ... < ym = d, hi := xi+1 −xi, and lj := yj+1 −yj, υ(h) := max{hi| i = 0, ...,n− 1} , υ(l) := max{lj| j = 0, ...,m− 1} . then, the following theorem holds. theorem 4. let f : q → r is of bounded variatin on q and ξi ∈ [xi,xi+1] (i = 0, ...,n− 1) , ηj ∈ [yj,yj+1] (j = 0, ...,m− 1) . then we have the qubature formula: b∫ a d∫ c f(t,s)dsdt(4.1) = n−1∑ i=0 m−1∑ j=0 (ξi −xi) (ηj −yj) f(xi,yj) + n−1∑ i=0 m−1∑ j=0 (ξi −xi) (yj+1 −ηj) f(xi,yj+1) + n−1∑ i=0 m−1∑ j=0 (xi+1 − ξi) (ηj −yj) f(xi+1,yj) + n−1∑ i=0 m−1∑ j=0 (xi+1 − ξi) (yj+1 −ηj) f(xi+1,yj+1) + r(ξ,η,in,jm,f). 152 budak and sarikaya the remainder r(ξ,η,in,jm,f) satisfies |r(ξ,η,in,jm,f)|(4.2) ≤ [ 1 2 υ(h) + max i∈{0,..,n−1} {∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ }] × [ 1 2 υ(l) + max j∈{0,..,m−1} {∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }] b∨ a d∨ c (f) ≤ υ(h)υ(l) b∨ a d∨ c (f) for all ξi ∈ [xi,xi+1] (i = 0, ...,n− 1) and ηj ∈ [yj,yj+1] (j = 0, ...,m− 1) . proof. aplying corollary 3 on the bidimentional interval [xi,xi+1] × [yj,yj+1] , we get |(ξi −xi) (ηj −yj) f(xi,yj)(4.3) + (ξi −xi) (yj+1 −ηj) f(xi,yj+1) + (xi+1 − ξi) (ηj −yj) f(xi+1,yj) + (xi+1 − ξi) (yj+1 −ηj) f(xi+1,yj+1) − xi+1∫ xi yj+1∫ yj f(t,s)dsdt ∣∣∣∣∣∣∣ ≤ [ 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ][ 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f). summing the inequality (4.3) over i from 0 to n− 1 and j from 0 to m− 1, we get (4.4) |r(ξ,η,in,jm,f)| ≤ n−1∑ i=0 m−1∑ j=0 [ 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ][ 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f) ≤ max i∈{0,..,n−1} { 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ } × max j∈{0,..,m−1} { 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }n−1∑ i=0 m−1∑ j=0 xi+1∨ xi yj+1∨ yj (f) ≤ [ 1 2 υ(h) + max i∈{0,..,n−1} {∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ }] × [ 1 2 υ(l) + max j∈{0,..,m−1} {∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }] b∨ a d∨ c (f) which copletes the proof of first inequality in (4.2). in last inequality (4.5) ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ≤ 12hi and maxi∈[0,...,n−1] ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ≤ 12υ(h), and similarly, (4.6) max j∈[0,...,m−1] ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ ≤ 12υ(l). ostrowski type inequalities for functions... 153 if we add (4.5) and (4.6) in (4.4), we obtain the required result. � corollary 5. let f, in and jm be as above. 1) if we choose ξi = xi+1 and ηj = yj+1 in (4.1), then we have the ”left rectangle rule” b∫ a d∫ c f(t,s)dsdt = n−1∑ i=0 m−1∑ j=0 f(xi,yj)hilj + rl(in,jm,f). 2) if we choose ξi = xi and ηj = yj in (4.1), then we have the ”right rectangle rule” b∫ a d∫ c f(t,s)dsdt = n−1∑ i=0 m−1∑ j=0 f(xi+1,yj+1)hilj + rr(in,jm,f). 3) finally, if we choose ξi = xi+xi+1 2 and ηj = xi+xi+1 2 in (4.1), then we have the ”trapezoid rule” b∫ a d∫ c f(t,s)dsdt = 1 4 n−1∑ i=0 m−1∑ j=0 [f(xi,yj) + f(xi,yj+1) + f(xi+1,yj) + f(xi+1,yj+1)] hilj + rt (in,jm,f). theorem 5. let f, in and jm be as above and xi ≤ α (1) i ≤ ξi ≤ α (2) i ≤ xi+1, yj ≤ β (1) j ≤ ηj ≤ β (2) j ≤ yj+1. then we have the qubature formula b∫ a d∫ c f(t,s)dsdt(4.7) = n−1∑ i=0 m−1∑ j=0 ( α (1) i −xi )( β (1) j −yj ) f(xi,yj) + n−1∑ i=0 m−1∑ j=0 ( α (1) i −xi )( β (2) j −β (1) j ) f(xi,ηj) + n−1∑ i=0 m−1∑ j=0 ( α (1) i −xi )( yj+1 −β (2) j ) f(xi,yj+1) + n−1∑ i=0 m−1∑ j=0 ( α (2) i −α (1) i )( β (1) j −yj ) f(ξi,yj) + n−1∑ i=0 m−1∑ j=0 ( α (2) i −α (1) i )( β (2) j −β (1) j ) f(ξi,ηj) + n−1∑ i=0 m−1∑ j=0 ( α (2) i −α (1) i )( yj+1 −β (2) j ) f(ξi,yj+1) + n−1∑ i=0 m−1∑ j=0 ( xi+1 −α (2) i )( β (1) j −yj ) f(xi+1,yj) + n−1∑ i=0 m−1∑ j=0 ( xi+1 −α (2) i )( β (2) j −β (1) j ) f(xi+1,ηj) + n−1∑ i=0 m−1∑ j=0 ( xi+1 −α (2) i )( yj+1 −β (2) j ) f(xi+1,yj+1) +r(ξ,η,α (1) i ,α (2) i ,β (1) j ,β (2) j in,jm,f). 154 budak and sarikaya the remainder r(ξ,η,α (1) i ,α (2) i ,β (1) j ,β (2) j ,in,jm,f) satisfies∣∣∣r(ξ,η,α(1)i ,α(2)i ,β(1)j ,β(2)j ,in,jm,f)∣∣∣(4.8) ≤ [ 1 2 υ(h) + max i∈{0,..,n−1} {∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ }] × [ 1 2 υ(l) + max j∈{0,..,m−1} {∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }] b∨ a d∨ c (f) ≤ υ(h)υ(l) b∨ a d∨ c (f). proof. aplying corollary 4 on the bidimentional interval [xi,xi+1] × [yj,yj+1] , we have (4.9) ∣∣∣(α(1)i −xi)(β(1)j −yj)f(xi,yj) + (α(1)i −xi)(β(2)j −β(1)j )f(xi,ηj) + ( α (1) i −xi )( yj+1 −β (2) j ) f(xi,yj+1) + ( α (2) i −α (1) i )( β (1) j −yj ) f(ξi,yj) + ( α (2) i −α (1) i )( β (2) j −β (1) j ) f(ξi,ηj) + ( α (2) i −α (1) i )( yj+1 −β (2) j ) f(ξi,yj+1) + ( xi+1 −α (2) i )( β (1) j −yj ) f(xi+1,yj) + ( xi+1 −α (2) i )( β (2) j −β (1) j ) f(xi+1,ηj) + ( xi+1 −α (2) i )( yj+1 −β (2) j ) f(xi+1,yj+1) − b∫ a d∫ c f(t,s)dsdt. ∣∣∣∣∣∣ ≤ [ 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ][ 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f). summing the inequality (4.9) over i from 0 to n− 1 and j from 0 to m− 1, then we get∣∣∣r(ξ,η,α(1)i ,α(2)i ,β(1)j ,β(2)j ,in,jm,f)∣∣∣ ≤ n−1∑ i=0 m−1∑ j=0 [ 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ ][ 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ ]xi+1∨ xi yj+1∨ yj (f) ≤ max i∈{0,..,n−1} { 1 2 hi + ∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ } × max j∈{0,..,m−1} { 1 2 lj + ∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }n−1∑ i=0 m−1∑ j=0 xi+1∨ xi yj+1∨ yj (f) ≤ [ 1 2 υ(h) + max i∈{0,..,n−1} {∣∣∣∣ξi − xi + xi+12 ∣∣∣∣ }] × [ 1 2 υ(l) + max j∈{0,..,m−1} {∣∣∣∣ηj − yj + yj+12 ∣∣∣∣ }] b∨ a d∨ c (f) ≤ υ(h)υ(l) b∨ a d∨ c (f). ostrowski type inequalities for functions... 155 this completes the proof of theorem. � corollary 6. under assumption of theorem 5 with α (1) i = xi, α (2) i = xi+1, ξi = xi+xi+1 2 , β (1) j = yj β (2) j = yj+1 and ηj = yj+yj+1 2 then we have the ”midpoint rule” b∫ a d∫ c f(t,s)dsdt = n−1∑ i=0 m−1∑ j=0 f ( xi + xi+1 2 , yj + yj+1 2 ) hilj + rm (in,jm,f) where the remainder satisfies |rm (in,jm,f)| ≤ 1 4 υ(h)υ(l) b∨ a d∨ c (f). references [1] m. w. alomari, a generalization of weighted companion of ostrowski integral inequality for mappings of bounded variation, rgmia research report collection, 14 (2011), art. id 87. [2] m. w. alomari and m.a. latif, weighted companion for the ostrowski and the generalized trapezoid inequalities for mappings of bounded variation, rgmia research report collection, 14 (2011), art. id 92. [3] m. w. alomari, approximating the riemann-stieltjes integral by a three-point quadrature rule and applications, konuralp journal of mathematics, 2(2014), 22–34. [4] m. w. alomari and s. s. dragomir, some grus type inequalities for the riemann-stieltjes integral with lipshitzian integrators, konuralp journal of mathematics, 2 (2014), 36–44 [5] p. cerone, w.s. cheung, and s.s. dragomir, on ostrowski type inequalities for stieltjes integrals with absolutely continuous integrands and integrators of bounded variation, computers and mathematics with applications 54 (2007), 183–191. [6] p. cerone, s. s. dragomir, and c. e. m. pearce, a generalized trapezoid inequality for functions of bounded variation, turk j math, 24 (2000), 147-163. [7] j.a. clarkson and c. r. adams , on definitions of bounded variation for functions of two variables, bull. amer. math. soc. 35 (1933), 824-854. [8] j.a. clarkson, on double riemann-stieltjes integrals, bull. amer. math. soc. 39 (1933), 929-936. [9] s. s. dragomir, the ostrowski integral inequality for mappings of bounded variation, bull.austral. math. soc., 60(1) (1999), 495-508. [10] s. s. dragomir, on the midpoint quadrature formula for mappings with bounded variation and applications, kragujevac j. math. 22(2000), 13-19. [11] s. s. dragomir, on the ostrowski’s integral inequality for mappings with bounded variation and applications, math. inequal. appl. 4 (2001), no. 1, 59–66. [12] s. s. dragomir, a companion of ostrowski’s inequality for functions of bounded variation and applications, int. j. nonlinear anal. appl. 5 (2014) no. 1, 89-97 [13] s. s. dragomir, refinements of the generalised trapezoid and ostrowski inequalities for functions of bounded variation, arch. math. (basel) 91 (2008), no. 5, 450–460. [14] s.s. dragomir and e. momoniat, a three point quadrature rule for functions of bounded variation and applications, mathematical and computer modelling 57 (2013), 612622. [15] s. s. dragomir, some perturbed ostrowski type inequalities for functions of bounded variation,asian-european journal of mathematics, 8 (2015), no. 04, doi: 10.1142/s1793557115500692. [16] fréchet, m., extension au cas des intégrals multiples d’une définition de l’intégrale due á stieltjes, nouvelles annales de math ematiques 10 (1910), 241-256. [17] y. jawarneh and m.s.m noorani, inequalities of ostrowski and simpson type for mappings of two variables with bounded variation and applications, tjmm, 3 (2011), no. 2, 81-94. [18] w. liu and y. sun, a refinement of the companion of ostrowski inequality for functions of bounded variation and applications, arxiv:1207.3861v1 [math.fa], 2012. [19] a. m. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, comment. math. helv. 10 (1938), 226-227. [20] k-l tseng, g-s yang, and s. s. dragomir, generalizations of weighted trapezoidal inequality for mappings of bounded variation and their applications, mathematical and computer modelling, 40 (2004), 77-84. [21] k-l tseng, improvements of some inequalites of ostrowski type and their applications, taiwan. j. math. 12 (9) (2008), 2427–2441. [22] k-l tseng, s-r hwang, g-s yang, and y-m chou, improvements of the ostrowski integral inequality for mappings of bounded variation i, applied mathematics and computation 217 (2010), 2348–2355. [23] k-l tseng, s-r hwang, g-s yang, and y-m chou, weighted ostrowski integral inequality for mappings of bounded variation, taiwanese j. of math., 15 (2011), no. 2, 573-585. [24] k-l tseng, improvements of the ostrowski integral inequality for mappings of bounded variation ii, applied mathematics and computation, 218 (2012), 5841–5847. 156 budak and sarikaya department of mathematics, faculty of science and arts, düzce university, düzce-turkey ∗corresponding author: hsyn.budak@gmail.com int. j. anal. appl. (2022), 20:21 real harmonic analysis on the special orthogonal group taeyoung lee∗ mechanical and aerospace engineering, the george washington university, 800 22nd st nw, washington dc 20052, united states ∗corresponding author: tylee@gwu.edu abstract. this paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied extensively. however, it is customary to treat real harmonic analysis as a special case of complex harmonic analysis, and there have been limited results developed specifically for real-valued functions. here, we develop a set of explicit formulas for real-valued irreducible unitary representations on the special orthogonal group, and provide several operational properties, such as derivatives, sampling, and clebsch-gordon coefficients. furthermore, we implement both of complex and real harmonics analysis on the special orthogonal group into an open source software package that utilizes parallel processing through the openmp library. the efficacy of the presented results are illustrated by benchmark studies and an application to spherical shape matching. 1. introduction noncommutative harmonic analysis is a generalization of fourier analysis to topological groups that are not necessarily commutative [9]. according to the peter-weyl theorem [16], irreducible unitary matrix representation of a compact group forms an complete orthogonal basis for square-integrable functions on the group. consequently, such functions can be expanded as a linear combination of matrix representations weighted by fourier parameters. in particular, harmonic analysis on the special orthogonal group, or the rotation group, has been utilized in quantum physics [2,21]. recently, it also has been applied to various engineering problems in robotics, controls, and machine learning [4,5,14]. received: feb. 8, 2022. 2010 mathematics subject classification. 22e45, 43a30, 65t99. key words and phrases. fast fourier transform; special orthogonal group; noncommutative harmonic analysis; spherical harmonics. https://doi.org/10.28924/2291-8639-20-2022-21 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-21 2 int. j. anal. appl. (2022), 20:21 computationally efficient numerical implementation and fast fourier transform algorithms on the special orthogonal group are considered in [13,17]. despite extensive prior works in broad areas of science and engineering, all of the aforementioned references deal with complex harmonic analysis, where matrix representations and fourier-parameters are complex-valued. there have been limited efforts in noncommutative harmonic analysis for realvalued functions on the special orthogonal group. one of the earlier studies in real harmonic analysis include [18], where real-valued matrix representations for several atomic orbitals are evaluated up to a certain order. later, in [11, 12], recurrence relations to evaluate real matrix representations are presented in terms of nine elements of a rotation matrix. in [3], real-valued matrix representations are constructed by transforming complex-valued matrix representations, namely wigner d-matrices. in this paper, we develop a new form of matrix representations for real harmonic analysis on the special orthogonal group. compared with [11], these are formulated in terms of euler angles so that a real fast fourier transform on the special orthogonal group can be developed utilizing various discrete fast fourier transform algorithms in the euclidean space. the presented form is based on the wigner d-function of the second euler angle, which can be computed by a recurrence relation. but, once the wigner d-function is evaluated, it is explicit with respect to the remaining two other euler angles. compared with [3], the presented real-valued matrix representation can be constructed without need for evaluating complex-valued matrix representation, or the wigner d-matrices. furthermore, we present several operational properties of real harmonic analysis. first, a sample theorem is presented such that fourier transform of a band-limited function with a bandwidth b can be exactly computed with (2b)3 function evaluations. as such, there is no need to approximate an integration over the special orthogonal group with a quadrature rule, when computing fourier transforms. this is utilized to construct a fast fourier transform for real-valued functions on the special orthogonal group. next, we develop clebsch-gordon coefficients for real harmonics so that a product of two real matrix representations is written as a linear combination of other real matrix representations. finally, we show an explicit expression for the derivatives of real matrix representations to formulate representations for lie algebra. beyond these theoretical contributions, an open source software package has been developed for real harmonic analysis on the special orthogonal group. this library includes fast fourier transform, inverse fourier transform, and evaluation of clebsch-gordon coefficients and derivatives. while this paper focuses on real harmonic analysis on the special orthogonal group, this software library also provides complex harmonic analysis, including wigner d-functions, and spherical harmonics on the unit-sphere as well. there are developed in c++ while utilizing the openmp library [6] to support accelerated parallel computing for multithread processors. these are verified by various software unit-testing. a benchmark study and a particular application to spherical shape matching for earth topological data are presented as well. int. j. anal. appl. (2022), 20:21 3 in short, the main contribution of this paper is theoretical and numerical analysis for the foundation of real harmonic analysis on the special orthogonal group. the presented form of real matrix representations and its derivatives, and real clebsch-gordon coefficients have been unprecedented. also, the proposed software library can be utilized in various engineering application of noncommutative harmonic analysis on the special orthogonal group. 2. complex harmonic analysis on so(3) the three-dimensional special orthogonal group is composed of 3 × 3 orthogonal matrices with determinant one, i.e., so(3) = {r ∈r3×3 |rtr = i3×3, det[r] = 1}. (2.1) harmonic analysis for complex-valued functions on so(3) has been studied extensively, for example in [4, 15, 21]. in this section, we summarize selected results that are used for the subsequent developments of real harmonics on so(3). 2.1. euler angles. any r ∈ so(3) can be parameterized by three angles α,γ ∈ [0, 2π) and β ∈ [0,π] as r(α,β,γ) = exp(αê3) exp(βê2) exp(γê3), (2.2) where e2 = (0, 1, 0), e3 = (0, 0, 1) ∈ r3, and the hat map ·̂ : r3 → so(3) is defined such that x×y = x̂y for any x,y ∈r3. the lie algebra is defined as so(3) = {s ∈r3×3 |s+st = 0}. these are referred to as 3–2–3 euler angles. while this parameterization involves complicated combinations of trigonometric functions and inherent singularities in the resulting kinematics equation, there are several advantages in harmonic analysis, such as convenience in applying fast fourier transform techniques. 2.2. irreducible unitary representation: wigner d-matrix. as a transformation group, so(3) acts on r3 while preserving its inherent structures of r3. for example, rx ∈ r3 for any r ∈ so(3) and x ∈r3 via the matrix multiplication. let l2(r3) be the set of complex-valued, square-integrable functions on r3. the left regular representation on so(3), namely d(r) for r ∈ so(3) is an operator d(r) : l2(r3) →l2(r3) defined such that (d(r)f )(x) = f (rtx), (2.3) for any f ∈ l2(r3) and x ∈ r3, i.e., it transforms a function such that it becomes equivalent to rotating the input argument by rt . it is straightforward to show that d(·) is a homomorphism, i.e., d(r1)d(r2) = d(r1r2) for any r1,r2 ∈ so(3). also, it is a linear operator on l2(r3). consequently, by selecting a basis on l2(r3), d(r) can be represented with a matrix, thereby resulting in matrix representations. the matrix representation on so(3) is often induced from that of su(2) = {u ∈ c2×2 |u∗u = i2×2, det[u] = 1} utilizing one-to-two correspondence between so(3) and su(2). more specifically, 4 int. j. anal. appl. (2022), 20:21 consider representation on su(2) as an operator on the analytic functions from c2 to c. by selecting the set of homogeneous polynomials as the basis of the analytic functions, one can derive matrix representation of su(2) as dlm,n(r(α,β,γ)) = e −imαd lm,n(β)e −inγ, (2.4) which is referred to as the wigner d-matrix that is common in quantum mechanics [21]. here the range of the index l is {0, 1 2 , 1, 3 2 . . .}, and for each l, the indices m,n vary in {−l,−l + 1, . . . , l−1, l}. in (2.4), the real-valued d lm,n is called wigner d-matrix. an explicit form of d l m,n is tabulated up to l = 5 in [21], and a recursive algorithm to evaluate it for arbitrary order is presented in [3]. the above expression for matrix representation su(2) results in a matrix representation of so(3) by taking the integer values of l only. throughout this paper, dl(r) ∈ c(2l+1)×(2l+1) is considered as a square matrix where the row index and the column index are m and n, respectively, which vary from −l to l in ascending order. from (2.3), it is trivial to show dl(i3×3) = i2l+1×2l+1, or dlm,n(i3×3) = δm,n. this representation is irreducible, i.e., cannot be block diagonalized consistently with a similarity transform, and it is unitary, i.e., (dl(r))∗ = (dl(r))−1 = dl(rt ), where the last equality is from the homomorphism property. when α = γ = 0, these also imply d l(−β) = (d l(β))−1 = (d l(β))t , d l(0) = i2l+1×2l+1, d lm,n(0) = δm,n. (2.5) while this paper follows the convention of 3–2–3 euler angles, using other types of euler angles in fact does not matter as it would yield an equivalent form of matrix representations that can be constructed by a similarity transform of (2.4). 2.3. fourier transform on so(3). according to peter-weyl theorem, the irreducible, unitary representations form a complete orthogonal basis for l2(so(3)) [16]. consequently, any f ∈ l2(so(3)) is expanded into f (r(α,β,γ)) = ∞∑ l=0 l∑ m,n=−l (2l + 1)f lm,nd l m,n(α,β,γ), (2.6) for fourier parameters f lm,n ∈c. define an inner product on l2(so(3)) as 〈f (r),g(r)〉 = ∫ so(3) f (r)g(r)dr, (2.7) where dr is the haar measure on so(3) that is normalized such that ∫ so(3) dr = 1. for example, using 3–2–3 euler angles, it is given by dr = 1 8π2 sin βdαdβdγ. the orthogonality of the irreducible, int. j. anal. appl. (2022), 20:21 5 unitary representation implies 〈dl1m1,n1(r),d l2 m2,n2 (r)〉 = 1 2l1 + 1 δl1,l2δm1,m2δn1,n2. (2.8) therefore, the fourier parameters in (2.6) can be discovered by the inner product, f lm,n = 〈d l m,n(r), f (r)〉. (2.9) equations (2.9) and (2.6) are considered as the fourier transform of f (r), and its inverse transform, respectively. a function f ∈ l(so(3)) is band-limited with the band b, if its fourier parameters vanish, i.e., f lm,n = 0 for any l ≥ b. the classical sampling theorem states that the fourier transform of a band-limited function can be recovered from the sample values that are chosen at a uniform grid with a frequency that is at least twice of the band-limit. using this, a fast fourier transform technique is presented in [13]. 2.4. derivatives of representation. given the matrix representation dl(r) on the lie group so(3), the representation on the lie algebra so(3) ' r3 is constructed by differentiation. more specifically, the l-th representation, namely ul : r3 →c(2l+1)×(2l+1) is given by ul(η) = d d� ∣∣∣∣ �=0 dl(exp(�η̂)), (2.10) for η ∈r3. being a lie algebra homogeneous, it satisfies ul(η ×ζ) = [ul(η),ul(ζ)], where η,ζ ∈r3 and [·, ·] represents the matrix commutator. since it is a linear operator, ul(η) for an arbitrary η ∈r3 is given by a linear combination of ul(ei ) for i = {1, 2, 3}. since exp(�ê3) = r(�, 0, 0), ulm,n(e3) = d d� ∣∣∣∣ �=0 e−im�d lm,n(0) = −imδm,n. (2.11) similarly, using exp(�ê2) = r(0,�, 0), ulm,n(e2) = − 1 2 √ (l + m)(l −m + 1)δm−1,n + 1 2 √ (l −m)(l + m + 1)δm+1,n. (2.12) last, ul(e1) can be obtained by ul(e1) = ul(e2 ×e3) = [ul(e2),ul(e3)] as ulm,n(e1) = − 1 2 i √ (l + m)(l −m + 1)δm−1,n − 1 2 i √ (l −m)(l + m + 1)δm+1,n. (2.13) these results are useful in engineering application of harmonic analysis, particularly when computing the sensitivity of (2.6) with respect to r. similar expressions are presented in [4], but for a different form of matrix representations. 6 int. j. anal. appl. (2022), 20:21 2.5. clebsch-gordon coefficients. the product of two wigner d-matrices can be rewritten as a finite linear combination of other wigner d-matrices as follows. dl1m1,n1(r)d l2 m2,n2 (r) = l1+l2∑ l=|l1−l2| l∑ m,n=−l cl,m l1,m1,l2,m2 cl,n l1,n1,l2,n2 dlm,n(r). (2.14) interestingly, the coupling coefficients are split into two parts as written above, and they are referred to as clebsch-gordon coefficients. there are several properties of clebsch-gordon coefficients listed in [21], including ∑ m1,m2 cl,m l1,m1,l2,m2 cl ′,m′ l1,m1,l2,m2 = δl,l′δm,m′, (2.15) ∑ l,m cl,m l1,m1,l2,m2 cl,m l1,m ′ 1,l2,m ′ 2 = δm1,m′1 δm2,m′2 , (2.16) cl,m l1,m1,l2,m2 = (−1)l1+l2−lcl,−m l1,−m1,l2,−m2, (2.17) cl,m l1,m1,l2,m2 = 0 if m 6= m1 + m2. (2.18) using the last property, the double summation at (2.14) reduces to dl1m1,n1(r)d l2 m2,n2 (r) = l1+l2∑ l=l c l,m1+m2 l1,m1,l2,m2 c l,n1+n2 l1,n1,l2,n2 dlm1+m2,n1+n2(r). (2.19) for l = max{|l1 − l2|, |m1 + m2|, |n1 + n2|}. while (2.14) commonly appears in the literature, (2.19) is not available at least in the cited references. a computational scheme to evaluate clebsch-gordon coefficients is proposed in [20]. in [15], it is proposed to rearranged the coefficients into a matrix cl1,l2 ∈r (2l1+1)(2l2+1)×(2l1+1)(2l2+1) according to the following ordering rules: (column index of cl,m l1,m1,l2,m2 ) = l2 − (l2 − l1)2 + l + m, (2.20) (row index of cl,m l1,m1,l2,m2 ) = (l1 + m1)(2l2 + 1) + l2 + m2, (2.21) which begins from 0 following the convention of the c programming language that is adopted for software implementation in this paper. under this matrix formulation, (2.14) is rearranged into dl1(r) ⊗dl2(r) = cl1,l2   l1+l2⊕ l=|l1−l2| dl(r)  ctl1,l2, (2.22) where ⊗ denotes the kronecker product. 2.6. relation to spherical harmonics. consider the unit-sphere, s2 = {x ∈ r3 |‖x‖ = 1}. let x ∈ s2 be parameterized by the co-latitude θ ∈ [0,π] and the longitude φ ∈ [0, 2π) as x(θ,φ) = [cos φ sin θ, sin φ sin θ, cos θ]. spherical harmonics, namely y lm(θ,φ) is defined as y lm(θ,φ) = e imφ √ 2l + 1 4π (l −m)! (l + m)! pml (cos θ), (2.23) int. j. anal. appl. (2022), 20:21 7 with the associated legendre polynomials pml for l ∈{0, 1, . . .} and −l ≤ m ≤ l. let dx = 1 4π sin θdφdθ be the measure of s2, normalized such that ∫ s2 dx = 1. we define an inner product on the square-integrable functions, namely l2(s2) as 〈f (x),g(x)〉l(s2) = ∫ s2 f (x)g(x)dx. spherical harmonics satisfies the following orthogonality with respect to the above inner product, yielding 〈y l1m1(x),y l2 m2 (x)〉l(s2) = 1 4π δl1l2δm1m2. (2.24) spherical harmonics are closely related to the wigner d-function. using the homomorphism property of the group representation [4], we obtain y ln (r tx) = ∑ m′ y lm′(x)d l m′,n(r). (2.25) therefore, using (2.24), the wigner d-function can be rediscovered from the spherical harmonics as 〈y lm(x),y l n (r tx)〉l(s2) = 1 4π dlm,n(r). (2.26) let y l be the (2l + 1) × 1 column vector whose elements are composed of y lm for m ∈ {−l, . . . l} in ascending order. the above equation is rewritten in a matrix form as 〈y l(x), (y l(rtx))t〉l(s2) = 1 4π dl(r). (2.27) 3. real harmonic analysis on so(3) the objective of this paper is to develop the counterparts of section 2 for real-valued functions on so(3), resulting in real harmonic analysis on so(3). instead of formulating as a specialized form of wigner d-matrices, real matrix representations are directly constructed in terms of euler angles, and they are utilized for fast fourier transform of real-valued functions on so(3). further, clebsch-gordon coefficients and derivatives are formulated as well. 3.1. real irreducible unitary representations. we follow the approaches presented in [3], where real harmonics on so(3) is constructed from real spherical harmonics. orthogonal basis for real-valued functions on s2, namely sl(x) ∈r2l+1 is constructed by the following transform, sl(x) = t ly l(x), (3.1) 8 int. j. anal. appl. (2022), 20:21 where x ∈ s2 and the matrix t l ∈c(2l+1)×(2l+1) is defined as tm,n =   1 m = n = 0, 0 |m| 6= |n|, (−1)m√ 2 m > 0,n = m, 1√ 2 m > 0,n = −m, i√ 2 m < 0,n = m, −i(−1)m√ 2 m < 0,n = −m, (3.2) or in a matrix form, t l = 1 √ 2   i 0 · · · 0 · · · 0 −i(−1)l 0 i · · · 0 · · · −i(−1)l−1 0 ... ... ... ... .. . ... ... 0 0 . . . √ 2 . . . 0 0 ... ... .. . ... ... ... ... 0 1 · · · 0 · · · (−1)l−1 0 1 0 · · · 0 · · · 0 (−1)l   . (3.3) it is straightforward to show that the columns of t l are mutually orthonormal, i.e., t l is unitary so that (t l)−1 = (t l)∗ = (t l)t . (3.4) let ul ∈r(2l+1)×(2l+1) be the matrix for the l-th real harmonics on so(3). motivated by (2.27), it is defined as 〈sl(x), (sl(rtx))t〉 = 1 4π ul(r). (3.5) substituting (3.1) and rearranging, ul(r) = t ldl(r)(t l)t . (3.6) this is a homomorphism, i.e., ul(r1r2) = ul(r1)ul(r2) for any r1,r2 ∈ so(3). furthermore, it is irreducible and orthogonal, i.e., (ul(r))−1 = (ul(r))t = ul(rt ), where the last equality is from the homomorphism property and ul(i3×3) = i(2l+1)×(2l+1). also, similar with (2.25) and (2.8), sln(r tx) = ∑ m′ slm′(x)u l m′,n(r), (3.7) 〈ul1m1,n1(r),u l2 m2,n2 (r)〉 = 1 2l1 + 1 δl1,l2δm1,m2δn1,n2. (3.8) while ul(r) can be evaluated by transforming dl(r) according to (3.6) as in [3], the procedure will involve unnecessary steps with complex variables. here, we present an alternative, explicit formulation as follows. int. j. anal. appl. (2022), 20:21 9 theorem 3.1. real harmonics on so(3) defined in (3.6) is equivalent to the following formulations. ulm,n(r) =   −sin mα sin nγψl−m,n(β) + cos mα cos nγψlm,n(β) (m ≥ 0,n ≥ 0) or (m < 0,n < 0), −sin mα cos nγψl−m,n(β) + cos mα sin nγψlm,n(β) (m ≥ 0,n < 0) or (m < 0,n ≥ 0), (3.9) where ψlm,n(β) =   (−1)m−nd l|m|,|n|(β) + (−1) msgn(m)d l|m|,−|n|(β) mn 6= 0, (−1)m−n √ 2d l|m|,|n|(β) m = 0 xor n = 0, d l0,0(β) m = n = 0, (3.10) satisfying ψlm,n = ψ l m,−n. or in a matrix formulation, ul(r) = xl(α)w l(β)xl(γ), (3.11) where xl(α),w l(β) ∈r(2l+1)×(2l+1) are defined as xlm,n(α) =   0 |m| 6= |n|, 1 m = n = 0, cos mα m = n 6= 0, −sin mα m = −n 6= 0. (3.12) w lm,n(β) =   ψ l m,n(β) (m ≥ 0,n ≥ 0) or (m < 0,n < 0) 0 otherwise. (3.13) proof. rearranging the matrix multiplication in (3.11) into element-wise operations, ulm,n(r) = l∑ p,q=−l t l m,pt l n,qd l p,q(r). from (3.2), the expression in the summation vanishes when |p| 6= |m| or |q| 6= |n|. consequently, the above double summation reduces to the following cases, depending on whether any sub index is zero or not, ul0,0(r) = d l 0,0(r) = d l 0,0(r), ulm,0(r) = t l m,md l m,0(r) + t l m,−md l −m,0(r), ul0,n(r) = t l n,nd l 0,n(r) + t l n,q−nd l 0,−n(r), ulm,n(r) = t l m,mt l n,nd l m,n(r) + t l m,mt l n,−nd l m,−n(r) + t l m,−mt l n,nd l −m,n(r) + t l m,−mt l n,−nd l −m,−n(r). 10 int. j. anal. appl. (2022), 20:21 for non zero m,n ∈ {−l, . . . l}. throughout the remainder of this proof, we focus on the last case assuming m,n > 0. the results for other cases can be obtained in a similar manner. substituting the values of t lm,n given in (3.2) for m,n > 0, ulm,n(r) = (−1)m+n 2 dlm,n(r) + (−1)m 2 dlm,−n(r) + (−1)n 2 dl−m,n(r) + 1 2 dl−m,−n(r). we substitute (2.4), and rearrange it using the symmetry of the wigner d-functions, namely d lm,n(β) = (−1)m−nd l−m,−n, to obtain ulm,n(r) = (−1) m+nd lm,n(β) cos(mα + nγ) + (−1) md lm,−n(β) cos(mα−nγ). from the definition of ψlm,n(β) in (3.10) for the case of m,n > 0 considered here, ulm,n(r) = cos mα cos nγ{(−1) m+nd lm,n(β) + (−1) md lm,−n(β)} + sin mα sin nγ{−(−1)m+nd lm,n(β) + (−1) md lm,−n(β)}, where the two expression in the braces reduce to ψlm,n(β) and −ψl−m,n(β), respectively, while yielding (3.9) for m,n > 0. the other remaining cases for (3.9) can be shown in the similar way. next, the matrix product in (3.11) is written as ulm,n(r) = l∑ p,q=−l xlm,p(α)w l p,q(β)x l q,n(β). again, suppose m,n > 0. from (3.12), the expression in the summation does not vanish only if p = ±m and q = ±n. consequently, ulm,n(r) = x l m,m(α)w l m,n(β)x l n,n(β) + x l m,m(α)w l m,−n(β)x l −n,n(β) + xlm,−m(α)w l −m,n(β)x l n,n(β) + x l m,−m(α)w l −m,−n(β)x l −n,n(β). substituting (3.12) and (3.13) and using ψl−m,−n = ψ l −m,n, it is straightforward to show the above reduces to (3.9) for m,n > 0. the other cases can be shown similarly. � this theorem states that real matrix representation on the special orthogonal group can be constructed directly in terms of euler angles without need for evaluating complex, wigner d-matrices. the expression presented in (3.9) is composed of sine and cosine terms for multiples of α,γ, that are similar to those appear in real spherical harmonics, and ψ terms defined by the wigner d-matrices. when written in a matrix form as (3.11), it is given by a product of three terms, where each term depends on one of euler-angles. in contrast to the recursive formulation written in terms of elements of a rotation matrix [11], this structure is useful for a fast fourier transform algorithm. in particular, when l = 1, (3.11) results in u1(r(α,β,γ)) =   cos α 0 sin α 0 1 0 −sin α 0 cos α     1 0 0 0 cos β −sin β 0 sin β cos β     cos γ 0 sin γ 0 1 0 −sin γ 0 cos γ   int. j. anal. appl. (2022), 20:21 11 = [ e3 e1 e2 ] r(α,β,γ) [ e3 e1 e2 ]t . in other words, the real matrix representation of the order l = 1 is similar to the rotation matrix itself. 3.2. fourier transform on so(3). from peter-weyl theorem, real harmonics yield a complete, orthogonal basis on the square-integrable, real-valued functions on so(3). more explicitly, any realvalued f ∈l2(so(3)) is expanded as f (r(α,β,γ)) = ∞∑ l=0 l∑ m,n=−l (2l + 1)f lm,nu l m,n(α,β,γ), (3.14) for real-valued fourier parameters f lm,n ∈r. from (3.8), the fourier parameters are obtained by f lm,n = 〈u l m,n(r), f (r)〉. (3.15) next, we present a sampling theorem to evaluate (3.15) exactly with a finite number of samples. theorem 3.2. consider a band-limited function represented by (3.14) where f lm,n = 0 for any l ≥ b. define a uniform grid for (α,β,γ) ∈ [0, 2π) × [0,π] × [0, 2π) as αj = γj = π b j, βk = π(2k + 1) 4b , (3.16) for j,k ∈{0, . . . , 2b − 1}. the fourier parameters for the band-limited function are given by f lm,n = 2b−1∑ j1,j2,k=0 wku l m,n(r(αj1,βk,γj2))f (r(αj1,βk,γj2)), (3.17) where the weighting parameter wk ∈r is defined as wk = 1 4b3 sin βk b−1∑ j=0 1 2j + 1 sin((2j + 1)βk). (3.18) proof. define a sampling distribution as s(r(α,β,γ)) = 2b−1∑ j1,k,j2=0 wkδr(α,β,γ),r(αj1,βk,γj2) , (3.19) which is the linear combination of grid points weighted by the parameter wk. since∫ so(3) f (r)δr,qdr = f (q) for q ∈ so(3), (3.15) yields the fourier parameters for the sampling distribution as slm,n = 2b−1∑ j1,k,j2=0 wku l m,n(r(αj1,βk,γj2)). for the selected grid, it is straightforward to show ∑2b−1 j1=0 sin mαj1 = 0, ∑2b−1 j1=0 cos mαj1 = 2bδm,0. therefore, from (3.10), slm,n = 4b 2δm,0δn,0 2b−1∑ k=0 wkd l 0,0(βk). 12 int. j. anal. appl. (2022), 20:21 in [7], it is shown that the selected weight satisfies 2b−1∑ k=0 wkd l 0,0(βk) = 1 4b2 δl,0, l = 0, . . . 2b − 1. therefore, the fourier transform of the sampling distribution reduces to slm,n = δm,0δn,0δl,0, l = 0, . . . 2b − 1. thus, the sampling distribution can be expanded as s(r) = 1 + ∞∑ l=2b l∑ m,n=−l slm,nu l m,n(r). (3.20) next, define g(r) = f (r)s(r). from (3.19) and using the property of the delta function, it is straightforward to show that the fourier parameters for g(r) is given by glm,n = 2b−1∑ j1,k,j2=0 wkf (r(αj1,βk,γj2))u l m,n(r(αj1,βk,γj2)). on the other hand, using (3.20), g(r) = f (r) + f (r) ∞∑ l=2b l∑ m,n=−l slm,nu l m,n(r). now, we show that the above expression for g(r) yields the fourier parameters that are identical to those of f (r). since f (r) can be expanded as a linear combination of ul1 for 0 ≤ l1 ≤ b − 1. the last term of the above equation is expanded by the product ul1ul2 with 0 ≤ l1 ≤ b − 1 and 2b ≤ l2. according to the clebsch-gordon theorem, ul1ul2 is a linear combination of ul3 for |l1−l2| ≤ l3 ≤ l1+l2. we have min |l1 − l2| = 2b − 1 −b = b + 1. as such, f (r) and g(r) share the fourier coefficients in the given band limit, i.e., f lm,n = g l m,n for l ∈{0, . . .b − 1}. this shows (3.17). � therefore, fourier parameters of any band-limited function with the bandwidth b can be computed exactly with (2b)3 samples evaluated at the given grid points. utilizing this, we present a fast fourier transform. 3.3. fast fourier transform on so(3). substituting (3.9) into (3.17), the fourier transform represented by (3.17) can be executed in the following sequence. for each k ∈ {0, . . . 2b − 1}, let fk,gk ∈r(2b−1)×(2b−1) be fkm,n = 2b−1∑ j1,j2=0 f (r(αj1,βk,γj2)) sin(mαj1 + nγj2), gkm,n = 2b−1∑ j1,j2=0 f (r(αj1,βk,γj2)) cos(mαj1 + nγj2), int. j. anal. appl. (2022), 20:21 13 which can be computed by a real-valued fast fourier transform algorithm developed in r, such as [19]. the fourier parameters defined in (3.9) are evaluated by f lm,n =  ∑2b−1 k=0 wk{− 1 2 (gkm,−n −gkm,n)ψl−m,n(βk) + 1 2 (gkm,−n + g k m,n)ψ l m,n(βk)} (m ≥ 0,n ≥ 0) or (m < 0,n < 0),∑2b−1 k=0 wk{− 1 2 (fkm,n + f k m,−n)ψ l −m,n(βk) + 1 2 (fkm,n −fkm,−n)ψlm,n(βk)} (m ≥ 0,n < 0) or (m < 0,n ≥ 0). 3.4. derivatives of representation. similar with (2.10), the derivatives of ul(r) at r = i3×3 results in the real matrix representation of so(3). here we use the same notation as the complex valued case as ul(η) = d d� ∣∣∣∣ �=0 ul(exp(�η̂)), (3.21) for η ∈ r3. from the linearity, ul(η) for any η can be evaluated by the results of the following theorem. theorem 3.3. the derivatives of ul(r) introduced at (3.21) are given as follows for η ∈{e1,e2,e3}. ulm,n(e1) =   1 2 (m + n) √ (l + |m|)(l −|m| + 1) (m ≥ 2,n = −m + 1) or (m ≤−2,n = −m− 1), −1 2 (m + n) √ (l −|m|)(l + |m| + 1) (1 ≤ m ≤ l − 1,n = −m− 1) or (−l + 1 ≤ m ≤−1,n = −m + 1), 1√ 2 √ l(l + 1) m = −1,n = 0, − 1√ 2 √ l(l + 1) m = 0,n = −1, 0 otherwise, (3.22) ulm,n(e2) =   1 2 √ (l + |m|)(l −|m| + 1) (m ≥ 2,n = m− 1) or (m ≤−2,n = m + 1), −1 2 √ (l −|m|)(l + |m| + 1) (1 ≤ m ≤ l − 1,n = m + 1) or (−l + 1 ≤ m ≤−1,n = m− 1), 1√ 2 √ l(l + 1) m = 1,n = 0, − 1√ 2 √ l(l + 1) m = 0,n = 1, 0 otherwise, , (3.23) ulm,n(e3) =  −m (m = −n) and (m 6= 0) 0 otherwise (3.24) 14 int. j. anal. appl. (2022), 20:21 proof. since exp(�ê3) = r(�, 0, 0), from (3.9), ulm,n(e3) =   d d� ∣∣ �=0 cos m�ψlm,n(0) (m ≥ 0,n ≥ 0) or (m < 0,n < 0) d d� ∣∣ �=0 − sin m�ψl−m,n(0) (m ≥ 0,n < 0) or (m < 0,n ≥ 0) , =   0 (m ≥ 0,n ≥ 0) or (m < 0,n < 0)−mψl−m,n(0) (m ≥ 0,n < 0) or (m < 0,n ≥ 0) (3.25) according to (2.5), d lm,n(0) = δm,n. therefore, (3.10) yields ψlm,n(0) =   (−1)m−nδ|m|,|n| mn 6= 0, 0 m = 0 xor n = 0, 1 m = n = 0, substituting these into (3.25), ulm,n(e3) =   0 (m ≥ 0,n ≥ 0) or (m ≤ 0,n ≤ 0)−m(−1)m−nδ|m|,|n| (m > 0,n < 0) or (m < 0,n > 0), which reduces to (3.24). the other can be shown similarly, using exp(�ê2) = r(0,�, 0), and ul(e1) = ul(e2 ×e3) = [ul(e2),ul(e3)]. � 3.5. clebsch-gordon coefficients. here we find the clebsch-gordon coefficients for real harmonics. the objective is to write a product of two real harminics as a linear combination of other harmonics. repeatedly using the property of kronecker product, namely (ac) ⊗ (bd) = (a ⊗ b)(c ⊗ d) for arbitrary compatible matrices a,b,c,d, (3.6) results in ul1(r) ⊗ul2(r) = (t l1 ⊗t l2)(dl1(r) ⊗dl2(r))(t l1 ⊗t l2)t . substituting (2.22), this is rearranged into ul1(r) ⊗ul2(r) = cl1,l2   l1+l2⊕ l=|l1−l2| ul(r)  ctl1,l2, (3.26) where the clebsch-gordon matrix for the real harmonics, namely cl1,l2 ∈ c (2l1+1)(2l2+1)×(2l1+1)(2l2+1) is defined as cl1,l2 = (t l1 ⊗t l2)cl1,l2   l1+l2⊕ l=|l1−l2| (t l)t   . (3.27) interestingly, while the clebsch-gordon coefficients cl1,l2 for the complex harmonics are real-valued, those for real harmonics can be complex-valued as the matrix t l is composed of real or imaginary elements. in the element-wise form, ul1m1,n1(r)u l2 m2,n2 (r) = l1+l2∑ l=|l1−l2| l∑ m,n=−l c l,m l1,m1,l2,m2 c l,n l1,n1,l2,n2 ulm,n(r). (3.28) int. j. anal. appl. (2022), 20:21 15 the following theorem presents two properties of the real clebsch-gordon coefficients, and provides an alternative, simpler method to evaluate (3.27). theorem 3.4. the clebsch-gordon coefficients defined in (3.27) is unitary, i.e., cl1,l2c t l1,l2 = i. (3.29) they have zero values for the following cases, c l,m l1,m1,l2,m2 = 0, if |m| 6= |m1 + m2| or |m| 6= |m1 −m2|. (3.30) the remaining non-zero values for m ∈ {m1 + m2,−m1 − m2,m1 − m2,−m1 + m2} are evaluated according to table 1. proof. equation (3.29) follows from the fact that cl1,l2 and t l are unitary as shown at (2.15), (2.16), and (3.4). next, following the ordering rules (2.20) and (2.21), (3.27) is rewritten into an element-wise form as c l,m l1,m1,l2,m2 = l1∑ p1=−l1 l2∑ p2=−l2 l∑ p=−l t l1 m1,p1 t l2 m2,p2 t lm,pc l,p l1,p1,l2,p2 . from (3.2), we have t lm,n = 0 for |m| 6= |n|. also c l,m l1,m1,l2,m2 = 0 if m 6= m1 + m2. using these, the triple summation in the above equation reduces to c l,m l1,m1,l2,m2 = ∑ p1∈{−m1,m1} ∑ p2={−m2,m2} δ|m|,|p1+p2|t l1 m1,p1 t l2 m2,p2 t lm,p1+p2c l,p1+p2 l1,p1,l2,p2 , which follows (3.30). suppose m1,m2 > 0. equation (3.30) implies c l,m l1,m1,l2,m2 6= 0 when m = m1+m2 or m = −m1−m2. for the former, using (3.2) and (2.17), c l,m1+m2 l1,m1,l2,m2 = t l1 m1,m1 t l2 m2,m2 t lm1+m2,m1+m2c l,m1+m2 l1,m1,l2,m2 + t l1 m1,−m1t l2 m2,−m2t l m1+m2,−m1−m2c l,−m1−m2 l1,−m1,l2,−m2, = 1 √ 8 (1 + (−1)l1+l2−l)cl,m1+m2 l1,m1,l2,m2 . for the latter, c l,−m1−m2 l1,m1,l2,m2 = t l1 m1,−m1t l2 m2,−m2t l −m1−m2,−m1−m2c l,−m1−m2 l1,−m1,l2,−m2 + t l1 m1,m1 t l2 m2,m2 t l−m1−m2,m1+m2c l,m1+m2 l1,m1,l2,m2 , = i √ 8 (1 − (−1)l1+l2−l)cl,−m1−m2 l1,−m1,l2,−m2. these show the shaded cells of table 1. other parts of the table can be shown similarly. � 16 int. j. anal. appl. (2022), 20:21 table 1. clebsch-gordon coefficients for real harmonics m1 m2 m1+ m2 m1− m2 c l,m1+m2 l1,m1,l2,m2 c m1+m2 l1,m1,l2,m2 c l,−m1−m2 l1,m1,l2,m2 c m1+m2 l1,m1,l2,m2 c l,m1−m2 l1,m1,l2,m2 c m1−m2 l1,m1,l2,−m2 c l,−m1+m2 l1,m1,l2,m2 c m1−m2 l1,m1,l2,−m2 (|m1 +m2| ≤ l ≤ l1 + l2) (|m1 −m2| ≤ l ≤ l1 + l2) 0 0 0 0 1 1 1 1 + 0 + + 1 2 η i 2 ζ 1 2 η i 2 ζ + + + + 1 √ 8 η i √ 8 ζ (−1)m2 √ 8 η i(−1)m2 √ 8 ζ + + + 0 1 √ 8 η i √ 8 ζ (−1)m1 2 η (−1)m1 2 η + + + − 1 √ 8 η i √ 8 ζ − i(−1)m1 √ 8 ζ (−1)m1 √ 8 0 + + − 1 2 η i 2 ζ 1 2 η i 2 ζ − + + − i(−1)m1 √ 8 ζ − (−1)m1 √ 8 η 1 √ 8 η i √ 8 ζ − + 0 − i(−1)m1 2 ζ i(−1)m1 2 ζ 1 √ 8 η i √ 8 ζ − + − − (−1)m2 √ 8 η i(−1)m2 √ 8 ζ 1 √ 8 η i √ 8 ζ − 0 − − 1 2 η i 2 ζ 1 2 η i 2 ζ − − − − i √ 8 ζ − 1 √ 8 η − i(−1)m2 √ 8 ζ (−1)m2 √ 8 η − − − 0 i √ 8 ζ − 1 √ 8 η (−1)m1 2 η (−1)m1 2 η − − − + i √ 8 ζ − 1 √ 8 η (−1)m1 √ 8 η i(−1)m1 √ 8 0 − − + 1 2 η i 2 ζ 1 2 η i 2 ζ + − − + (−1)m1 √ 8 η i(−1)m1 √ 8 ζ − i √ 8 ζ 1 √ 8 η + − 0 + i(−1)m1 2 ζ i(−1)m1 2 ζ − i √ 8 ζ 1 √ 8 η + − + + i(−1)m2 √ 8 ζ − (−1)m2 √ 8 η − i √ 8 ζ 1 √ 8 η η =(−1)l1+l2−l +1, ζ =(−1)l1+l2−l −1. int. j. anal. appl. (2022), 20:21 17 from (3.30), the summation at (3.28) reduces to ul1m1,n1(r)u l2 m2,n2 (r) = ∑ m∈m ∑ n∈n l1+l2∑ l=max{|l1−l2|,|m|,|n|} c l,m l1,m1,l2,m2 c l,n l1,n1,l2,n2 ulm,n(r). (3.31) where m = {m1 + m2,m1 −m2,−m1 + m2,−m1 −m2}, n = {n1 + n2,n1 −n2,−n1 + n2,−n1 −n2}. as stated above, the clebsch-gordon coefficients for real harmonics is complex in general. however, by investigating table 1, one can show that the product c l,m l1,m1,l2,m2 c l,n l1,n1,l2,n2 in (3.31) is real always. this is expected as ulm,n(r) is real-valued always. 4. software implementation the presented real harmonic analysis on so(3) has been implemented by c++, and the resulting software package has been disclosed as an open source library at https://github.com/fdcl-gwu/ fftso3. this provides the following functionality and features: • complex and real harmonic analysis on so(3) • complex and real harmonic analysis on s2 • fast fourier transform based on the eigenfft library [10] • clebsch-gordon coefficients / derivatives of harmonics • multi-core parallel computation with the openmp library [6] 4.1. validation. the software package is validated by software unit-testing implemented via google test [8]. there are several tests to verify that numerical results are consistent with theoretical analysis. here we show the results of a particular unit-testing designed to ensure that the composition of the inverse transform and the forward transform yields the identity map on the space of fourier parameters [13]. specifically, we define a function f (r) : so(3) → r as the inverse fourier transform presented in (3.14). we assume that the function is band-limited so that f lm,n = 0 any l ≥ b. next, each component of fourier parameter f lm,n is randomized by a uniform-distribution in [−1, 1]. the corresponding function f (r) is transformed via the fast fourier transform described in section 3.3, to compute a new set of fourier parameters, namely glm,n. theoretically, g l m,n should be identical to the original fourier parameters f lm,n of f (r), as the composition of the inverse transform and the forward transform is the identity map on the space of fourier parameters. numerical errors of the presented software library are measured by the discrepancy between f lm,n and g l m,n calculated as the sum of the matrix norm, error = b−1∑ l=0 ‖f l −gl‖. https://github.com/fdcl-gwu/fftso3 https://github.com/fdcl-gwu/fftso3 18 int. j. anal. appl. (2022), 20:21 table 2 summarizes the above error for varying band-limits. the error increases as the band-limit, since the error measure is defined as the sum over the index l. however, all of the errors are under an acceptable level relative to the machine precision of double-type variables that the presented software uses. table 2. numerical error for the composition of the forward and inverse fourier transforms b 8 16 32 64 error 7.2528 × 10−14 5.8972 × 10−13 4.8600 × 10−12 4.0484 × 10−11 4.2. benchmark. next, we check the computational time required to perform fourier transforms. fast forward fourier transform is executed for the trace function on so(3), with varying bandwidth b ∈ {8, 16, 32, 64, 128} and number of threads nthread ∈ {1, 2, 4, 8}. the resulting computation time is measured with intel i7-6800k 3.4ghz cpu, and the test is repeated 10 times to compute the average computation time. figure 1 illustrates the computation time with respect to the band-with for several number of threads. the computation time increases as b is increased, but it reduces as more threads are available. more specifically, the next subfigure shows the speed-up factor, the ratio of the computation time with a single thread to that of multiple threads. ideally, the speed-up factor is identical to the number of threads, as indicated by the dotted line. however, due to the overhead in distributing the tasks in multiple threads and collecting results, as well as delay in accessing a shared memory, the speed-up factor is often lower than the number of threads in practice. it is illustrated that the speed-up factor increases as b is increased, i.e., parallel processing is more beneficial if the bandwidth is greater. figure 2 is for the inverse transform. compared with the forward transform, only a fraction of time is required. as such, the speed-up factor is relatively low even for higher bandwidth. at least, they are greater than one, indicating that the computation time is reduced by parallel processing. int. j. anal. appl. (2022), 20:21 19 0 50 100 150 b 10 -3 10 -2 10 -1 10 0 10 1 10 2 ∆ t nthread = 1 nthread = 2 nthread = 4 nthread = 8 (a) cpu time (sec) 2 4 6 8 nthread 1 2 3 4 5 6 7 8 s p e e d -u p f a c to r b = 8 b = 16 b = 32 b = 64 b = 128 (b) speed-up factor figure 1. benchmark results for forward fourier transform 0 50 100 150 b 10 -4 10 -3 10 -2 10 -1 10 0 ∆ t nthread = 1 nthread = 2 nthread = 4 nthread = 8 (a) cpu time (sec) 2 4 6 8 nthread 1 2 3 4 5 6 7 8 s p e e d -u p f a c to r b = 8 b = 16 b = 32 b = 64 b = 128 (b) speed-up factor figure 2. benchmark results for inverse fourier transform 5. application 5.1. spherical shape matching. we apply the above software package for spherical shape correlation and matching. consider an object embedded in r3. assuming that it has zero genus, i.e., no hole, it’s shape can be completely described by deforming a sphere. let f (x) : s2 → r be the deformed radius along the radial direction specified by the unit-vector x ∈ s2. suppose the object is rotated by a rotation described by r ∈ so(3), and let g(x) : s2 →r describe the rotated object. the spherical shape matching problem considered in this paper is to find the rotation matrix r ∈ so(3) for given shape functions f (x) and g(x). 20 int. j. anal. appl. (2022), 20:21 this can be formulated as the following optimization problem on so(3). for a rotation matrix r, let its cost be the discrepancy between g(x) and f (x) rotated by r, i.e., j (r) = 1 2 ‖g(x) − f (rtx)‖2 = 1 2 ‖g(x)‖2 + 1 2 ‖f (x)‖2 −〈g(x), f (rtx)〉l(s2), where we have used the fact ‖f (rtx)‖ = ‖f (x)‖. the true rotation matrix r minimizes the cost function j (r), or equivalently, it maximizes the measure of correlation defined by c(r) = 〈g(x), f (rtx)〉l(s2), i.e., r = arg max {c(r)}. however, evaluating the above inner product requires an integration over s2, and it may be cumbersome to repeat it at every iteration of numerical optimization. instead, the cost function can be evaluated without any integration, by utilizing the fundamental property of representation given by (2.3). let the fourier transform of f (x) and g(x) with a bandwidth b be f (x) = b−1∑ l=0 (f l)tsl(x), g(x) = b−1∑ l=0 (gl)tsl(x), for fourier parameters f l,gl ∈r(2l+1)×1. from (3.7), f (rtx) = b−1∑ l=0 (f l)tsl(rtx) = b−1∑ l=0 (ul(r)f l)tsl(x). the orthogonality of real spherical harmonics implies 〈sl1(x), (sl2(x))t〉l(s2) = 1 4π δl1,l2i(2l1+1)×(2l1+1). therefore, the correlation function reduces to c(r) = 1 4π b−1∑ l=0 (gl)tul(r)f l, (5.1) which is an algebraic equation of fourier parameters that does not require any integration. furthermore, its gradient can be evaluated by (3.22)–(3.24) as follow. for η = (η1,η2,η3) ∈ r3, a directional derivatives of the correlation function is given by d d� ∣∣∣∣ �=0 c(r exp(�η̂))) = ∇c(r) ·η, where the gradient is considered as a 3 × 1 vector, i.e., ∇c(r) ∈r3, whose i-th element is given by [∇c(r)]i = 1 4π b−1∑ l=0 (gl)tul(r)ul(ei )f l, (5.2) int. j. anal. appl. (2022), 20:21 21 for i ∈{1, 2, 3}. in the above expression, we have used the homomorphism property ul(r exp(�η̂)) = ul(r)ul(exp(�η̂)). using these, one can apply any gradient-based numerical optimization algorithm, such as one summarized in table 3. 5.2. earth elevation map. here we consider a particular example with a worldwide elevation model, namely etopo5 [1]. the topological elevation model is interpolated such that for a given set of latitude and longitude the corresponding elevation is calculated. this function yields f (x), which is transformed by spherical harmonics with the bandwidth of b = 129 to obtain f l. we choose a particular rotation matrix with 3-2-3 euler angle (α,β,γ) = (π 6 , π 3 , π 4 ), and perform the fourier transform of f (rtx) to obtain gl. see figure 3 for the original elevation map, and the rotated one. the optimization algorithm summarized in table 3 is implemented with the initial guess (α,β,γ) = (0.3, 0.3, 0.3), a tolerance � = 10−6 and step-size δ = 5 × 10−3. the numerical optimization is terminated after 223 iterations, with the maximum absolute error of 7.98 × 10−5 in terms of euler angles. the evolution of the correlation function and its gradient magnitude over iterations, and the change of the rotation matrix in terms of euler angles are illustrated in figure 4. table 3. spherical shape matching algorithm 1: procedure spherical shape matching 2: perform fourier transform of f (r),g(r) to obtain f l,gl 3: make an initial guess of r 4: set a tolerance � > 0 and a step size δ > 0 5: repeat 6: (c,∇c)=correlation(r) 7: update r = r exp(δ∇̂c) 8: until ‖∇c(r)‖ < � 9: return r 10: end procedure 11: procedure (c,∇c)=correlation(r) 12: compute c with (5.1) 13: compute ∇c with (5.2) 14: end procedure 22 int. j. anal. appl. (2022), 20:21 (a) original earth elevation map (b) rotated earth elevation map figure 3. application to spherical image matching for earth elevation map 0 50 100 150 200 250 iteration 50 100 150 200 10 -6 10 -4 10 -2 10 0 10 2 (a) evolution of correlation function and its gradient over iteration 0 50 100 150 200 250 iteration 0.2 0.4 0.6 0.8 1 1.2 (b) evolution of euler angles over iteration figure 4. iteration procedure for spherical shape matching of earth elevation map 6. conclusions we have presented real harmonic analysis on the special orthogonal group, including various operational properties such as fast fourier transform, clebsch-gordon coefficients, and derivatives. there are further implemented into an open source software package supporting parallel processing for accelerated computing. in particular, the presented form of real irreducible, unitary representations can be evaluated without constructing their counter parts in complex harmonic analysis, namely wigner d-matrices, and the given formulation in terms of euler angles are suitable for fast fourier transform. int. j. anal. appl. (2022), 20:21 23 future works include extension beyond the special orthogonal group, such as the special euclidean group for various engineering applications regarding the coupled translational and rotational dynamics of a rigid body. also, the intermediate term ψll,m(β) in (3.10) may be directly evaluated without formulating real-valued wigner d-functions. acknowledgement: this research has been supported in parts by nsf under the grant cmmi1335008, and by afosr under the grant fa9550-18-1-0288. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] data announcement 88-mgg-02, digital relief of the surface of the earth. national oceanic and atmospheric administration (1988). https://www.ngdc.noaa.gov/mgg/global/etopo5.html [2] l.c. biedenharn, j.d. louck, angular momentum in quantum physics: theory and application, addison-wesley pub. co., advanced book program, reading, mass, 1981. [3] m.a. blanco, m. flórez, m. bermejo, evaluation of the rotation matrices in the basis of real spherical harmonics, j. mol. struct.: theochem. 419 (1997), 19–27. https://doi.org/10.1016/s0166-1280(97)00185-1. [4] g. chirikjian, a. kyatkin, engineering applications of noncommutative harmonic analysis, crc press, boca raton, fl (2001). [5] t.s. cohen, m. geiger, j. koehler, m. welling, spherical cnns, arxiv:1801.10130 [cs, stat]. (2018). http: //arxiv.org/abs/1801.10130. [6] l. dagum, r. menon, openmp: an industry standard api for shared-memory programming, ieee comput. sci. eng. 5 (1998) 46–55. https://doi.org/10.1109/99.660313. [7] j.r. driscoll, d.m. healy, computing fourier transforms and convolutions on the 2-sphere, adv. appl. math. 15 (1994), 202–250. https://doi.org/10.1006/aama.1994.1008. [8] google, google test. https://github.com/google/googletes (2018). [9] k.i. gross, on the evolution of noncommutative harmonic analysis, amer. math. mon. 85 (1978), 525–548. https://doi.org/10.1080/00029890.1978.11994636. [10] g. guennebaud, b. jacob, et al. eigen v3. http://eigen.tuxfamily.org (2010). [11] j. ivanic, k. ruedenberg, rotation matrices for real spherical harmonics. direct determination by recursion, j. phys. chem. 100 (1996), 6342–6347. https://doi.org/10.1021/jp953350u. [12] j. ivanic, k. ruedenberg, rotation matrices for real spherical harmonics. direct determination by recursion, j. phys. chem. a. 102 (1998), 9099–9100. https://doi.org/10.1021/jp9833350. [13] p.j. kostelec, d.n. rockmore, ffts on the rotation group, j. fourier anal. appl. 14 (2008), 145–179. https: //doi.org/10.1007/s00041-008-9013-5. [14] t. lee, stochastic optimal motion planning for the attitude kinematics of a rigid body with non-gaussian uncertainties, j. dyn. syst. measure. control. 137 (2015), 034502. https://doi.org/10.1115/1.4027950. [15] d. marinucci, g. peccati, random fields on the sphere. the london mathematical society (2011). [16] f. peter, h. weyl, die vollständigkeit der primitiven darstellungen einer geschlossenen kontinuierlichen gruppe. math. ann. 97 (1927), 735–755. https://www.ngdc.noaa.gov/mgg/global/etopo5.html https://doi.org/10.1016/s0166-1280(97)00185-1 http://arxiv.org/abs/1801.10130 http://arxiv.org/abs/1801.10130 https://doi.org/10.1109/99.660313 https://doi.org/10.1006/aama.1994.1008 https://doi.org/10.1080/00029890.1978.11994636 https://doi.org/10.1021/jp953350u https://doi.org/10.1021/jp9833350 https://doi.org/10.1007/s00041-008-9013-5 https://doi.org/10.1007/s00041-008-9013-5 https://doi.org/10.1115/1.4027950 24 int. j. anal. appl. (2022), 20:21 [17] t. risbo, fourier transform summation of legendre series and d-functions, j. geodesy. 70 (1996), 383–396. https://doi.org/10.1007/bf01090814. [18] r.p. sherman, r. grinter, transformation matrices for the rotation of real p, d, and f atomic orbitals, j. mol. struct.: theochem. 135 (1986), 127–133. https://doi.org/10.1016/0166-1280(86)80052-5. [19] h. sorensen, d. jones, m. heideman, c. burrus, real-valued fast fourier transform algorithms, ieee trans. acoust., speech, signal process. 35 (1987), 849–863. https://doi.org/10.1109/tassp.1987.1165220. [20] w. straub, efficient computation of clebsch-gordon coefficients. tech. rep. (2014). http://vixra.org/abs/ 1403.0263. [21] d.a. varshalovich, a.n. moskalev, v.k. khersonskii, quantum theory of angular momentum, world scientific, 1988. https://doi.org/10.1142/0270. https://doi.org/10.1007/bf01090814 https://doi.org/10.1016/0166-1280(86)80052-5 https://doi.org/10.1109/tassp.1987.1165220 http://vixra.org/abs/1403.0263 http://vixra.org/abs/1403.0263 https://doi.org/10.1142/0270 1. introduction 2. complex harmonic analysis on so(3) 2.1. euler angles 2.2. irreducible unitary representation: wigner d-matrix 2.3. fourier transform on so(3) 2.4. derivatives of representation 2.5. clebsch-gordon coefficients 2.6. relation to spherical harmonics 3. real harmonic analysis on so(3) 3.1. real irreducible unitary representations 3.2. fourier transform on so(3) 3.3. fast fourier transform on so(3) 3.4. derivatives of representation 3.5. clebsch-gordon coefficients 4. software implementation 4.1. validation 4.2. benchmark 5. application 5.1. spherical shape matching 5.2. earth elevation map 6. conclusions references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 188-197 http://www.etamaths.com fejér type inequalities for harmonically (s,m)-convex functions imran abbas baloch1,∗, i̇mdat i̇şcan2 and silvestru sever dragomir3 abstract. in this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. by using the notion of harmonic symmetricity, harmonic (s,m)-convexity, analysis and some auxiliary results, some new fejér type integral inequalities are presented for the class of harmonically (s,m)-convex functions. 1. introduction a function f : i ⊆ r → r is called convex function if f(λx + (1 − λ)y) ≤ λf(x) + (1 − λ)f(y) for all x,y ∈ i and λ ∈ [0, 1]. there are many results associated with convex functions in the area of inequalities, but one of them is the classical hermite-hadamard (see [21]) inequalities: (1.1) f (a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 , for all a,b ∈ i, with a < b. the inequalities in (1.1) hold in reversed direction if f is a concave function. a vast literature have been produced by a number of mathematicians for convex functions but (1.1) is considered to be the most famous inequality for convex mappings due to its usefulness and many applications in various branches of pure and applied mathematics. the definition of classical or usual convex functions has been generalized in a variety of ways and as a consequence many researchers have established a number of hermite-hadamard type inequalities by using different generalizations of the classical convexity, see for instance [2]-[23] and the references mentioned in these papers. one of the generalizations of classical convexity is the harmonic (s,m)-convexity in second sense, which unifies the notion of harmonically convex [12] and harmonically s-convex functions in second sense [13] introduced by imdat iscan, as stated in the definition below. definition 1. [1] the function f : i ⊂ (0,∞) → r is said to be harmonically (s,m)-convex in second sense, where s ∈ (0, 1] and m ∈ (0, 1] if f ( mxy mty + (1 − t)x ) = f ( ( t x + 1 − t my )−1 ) ≤ tsf(x) + m(1 − t)sf(y) ∀x,y ∈ i and t ∈ [0, 1]. remark 1. note that for s = 1,harmonic (s,m)-convexity reduces to harmonic m-convexity and for m = 1, harmonic (s,m)-convexity reduces to harmonic s-convexity in second sense (see [13]) and for s,m = 1, harmonic (s,m)-convexity reduces to ordinary harmonic convexity (see [12]). proposition 1. let f : (0,∞) → r be a function a) if f is (s,m)-convex function in second sense and non-decreasing, thenf is harmonically (s,m)convex function in second sense. b) if f is harmonically (s,m)-convex function in second sense and non-increasing, then f is (s,m)convex function in second sense. 2010 mathematics subject classification. primary 26d15; secondary 26a51, 26e60, 41a55. key words and phrases. hermite-hadamard’s inequality; fejér’s inequality; convex function; harmonically (s,m)convex function; hölder’s inequality; power mean inequality. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 188 harmonically (s,m)-convex functions 189 remark 2. according to proposition 1, every non-decreasing (s,m)-convex function in second sense is also harmonically (s,m)-convex function in second sense. example 1. (see[3]) let 0 < s < 1 and a,b,c ∈ r, then function f : (0,∞) → r defined by f(x) = { a, x = 0 bxs + c, x > 0 is non-decreasing s-convex function in second sense for b ≥ 0 and 0 ≤ c ≤ a. hence, by proposition 1, f is harmonically (s, 1)-convex function. proposition 2. let s ∈ [0, 1], m ∈ (0, 1], f : [a,mb] ⊂ (0,∞) → r, be an increasing function and g : [a,mb] → [a,mb], g(x) = mab a+mb−x, a < mb. then f is harmonically (s,m)-convex in second sense on [a,mb] if and only if fog is (s,m)-convex in second sense on [a,mb]. the following result of the hermite-hadamard type holds. theorem 1. let f : i ⊂ (0,∞) → r be a harmonically (s,m)-convex function in second sense with s ∈ [0, 1] and m ∈ (0, 1]. if 0 < a < b < ∞ and f ∈ l[a,b], then one has following inequality ab b−a ∫ b a f(x) x2 dx ≤ min [f(a) + mf( b m ) s + 1 , f(b) + mf( a m ) s + 1 ] corollary 1. if we take m = 1 in theorem 1, then we get ab b−a ∫ b a f(x) x2 dx ≤ f(a) + f(b) s + 1 corollary 2. if we take s = 1 in theorem 1, then we get ab b−a ∫ b a f(x) x2 dx ≤ min [f(a) + mf( b m ) 2 , f(b) + mf( a m ) 2 ] chen and wu [4], established the following weighted fejér type inequality for the harmonically convex function as follow theorem 2. [4] let f : i ⊂ r\{0}→ r be a harmonically convex function and a,b ∈ i with a < b. if f ∈ l([a,b]), then one has (1.2) f ( 2ab a + b )∫ b a g(x) x2 dx ≤ ∫ b a g(x)f(x) x2 dx ≤ f(a) + f(b) 2 ∫ b a g(x) x2 dx, where g : [a,b] → r is non-negative, integrable and satisfies g (ab x ) = g ( ab a + b−x ) the main purpose of the present paper is to introduce a new notion of harmonically symmetric functions and to establish an identity involving a harmonically symmetric function and a differentiable function. we will prove some fejér type inequalities by using this identity related with the second part of the inequality given above by (1.2).we believe that our findings are novel, new and better than those already exist and will open new ways for further research in this field. 2. main results throughout this section, we take l(t) = 2ab (1−t)a+(1+t)b and u(t) = 2ab (1+t)a+(1−t)b. the beta function, the gamma function and the integral form of the hypergeometric function are defined as follows to be used in the sequel of paper b(α,β) = γ(α)γ(β) γ(α + β) = ∫ 1 0 tα−1(1 − t)β−1dt, α,β > 0 γ(α) = ∫ ∞ 0 tα−1e−tdt, α > 0 190 baloch, i̇şcan and dragomir and 2f1(α,β; γ,z) = 1 b(β,γ −β) ∫ 1 0 tβ−1(1 − t)γ−β−1(1 −zt)−αdt, γ > β > 0, |z| < 1 the notion of harmonically symmetric functions is defined as follows: definition 2. we say that a function g : [a,b] ⊆ r\{0}→ r is harmonically symmetric with respect to 2ab a+b if g(x) = g ( 1 1 a + 1 b − 1 x ) holds for all x ∈ [a,b]. now, we give the weighted integral equality by using which we establish our results in this article. lemma 1. let f : i ⊆ r\{0}→ r be a differentiable function on i◦ and a,b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b . if f′ ∈ l([a,b]), then the following identity holds f(a) + f(b) 2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx = b−a 4ab ∫ 1 0 (∫ u(t) l(t) g(x) x2 dx )[ (u(t))2f′(u(t)) − (l(t))2f′(l(t)) ] dt proof. since, g : [a,b] → [0,∞) is harmonically symmetric to 2ab a+b , then g(u(t)) = g(l(t)). consider i = b−a 4ab ∫ 1 0 (∫ u(t) l(t) g(x) x2 dx )[ (u(t))2f′(u(t)) − (l(t))2f′(l(t)) ] dt = 1 2 [∫ 1 0 (∫ u(t) l(t) g(x) x2 dx ) d[f(u(t)) + f(l(t))] ] = 1 2 [(∫ u(t) l(t) g(x) x2 dx ) (f(u(t)) + f(l(t))) ∣∣1 0 − b−a 2ab ∫ 1 0 (g(u(t)) + g(l(t)))(f(u(t)) + f(l(t)))dt ] = 1 2 [ (f(a) + f(b)) (∫ b a g(x) x2 dx ) − b−a ab ∫ 1 0 g(u(t)f(u(t)dt − b−a ab ∫ 1 0 g(l(t)f(l(t)dt ] = 1 2 [ (f(a) + f(b)) ∫ b a g(x) x2 dx− 2 ∫ 2ab a+b a g(x)f(x) x2 dx + 2 ∫ a 2ab a+b g(x)f(x) x2 dx ] = f(a) + f(b) 2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx � now, we present new fejér type inequalities for harmonically (s,m)-convex functions, which give the weighted generalization of some of the results established in resent literature. theorem 3. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b m ∈ i◦, m ∈ (0, 1] with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f′ ∈ l([a,b]). if |f′|q is harmonically (s,m)-convex on [a, b m ] for q ≥ 1, then the following inequality holds ∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ harmonically (s,m)-convex functions 191 ≤ b−a 8ab a 2 q ‖g‖∞ { λ 1−1 q 1 (a,b) ({ 22b(s + 1, 2).2f1(2,s + 1,s + 3; b−a b ) − 21−sb(s + 1, 1).2f1(2,s + 1,s + 2; b−a 2b ) + 1 2s b(s + 2, 1).2f1(2,s + 2,s + 3; b−a 2b ) } |f′(b)|q + m22−sb2 (b + a)2 b(1,s + 2).2f1(2, 1,s + 3; b−a b + a )|f′( a m )|q )1 q + λ 1−1 q 2 (a,b) × ({ 22b(2,s + 1).2f1(2, 2,s + 3; b−a b ) − 22−sb2 (b + a)2 b(1,s + 1).2f1(2, 1,s + 2; b−a b + a ) − 22−sb2 (b + a)2 b(2,s + 1).2f1(2, 2,s + 3; b−a b + a ) } |f′(a)|q (2.1) + m 2s b(s + 2, 1).2f1(2,s + 2,s + 3; b−a 2b )|f′( b m )|q )1 q } proof. from lemma 1 and hölder’s inequality, we get∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ b−a8ab ‖g‖∞ × {(∫ 1 0 (1 − t)(u(t))2dt )1−1 q (∫ 1 0 (1 − t)(u(t))2|f′(u(t))|qdt )1 q (2.2) + (∫ 1 0 (1 − t)(l(t))2dt )1−1 q (∫ 1 0 (1 − t)(l(t))2|f′(l(t))|qdt )1 q } by the harmonic (s,m)-convexity of |f′|q on [a,b] for q ≥ 1, we have∫ 1 0 (1 − t)(u(t))2|f′(u(t))|qdt = ∫ 1 0 (1 − t) ( 2ab (1 + t)a + (1 − t)b )2 × ∣∣∣∣f′( 2ab(1 + t)a + (1 − t)b) ∣∣∣∣qdt ≤ 12s |f′(b)|q ∫ 1 0 (1 − t)(1 + t)s ( 2ab (1 + t)a + (1 − t)b )2 dt + m 1 2s |f′( a m )|q ∫ 1 0 (1 − t)s+1 ( 2ab (1 + t)a + (1 − t)b )2 dt (2.3) = { 22a2b(s + 1, 2).2f1(2,s + 1,s + 3; b−a b ) − a2 2s−1 b(s + 1, 1).2f1(2,s + 1,s + 2; b−a 2b ) + a2 2s b(s+2, 1).2f1(2,s+2,s+3; b−a 2b ) } |f′(b)|q+ ma2b2 2s−2(b + a)2 b(1,s+2).2f1(2, 1,s+3; b−a b + a )|f′( a m )|q and ∫ 1 0 (1 − t)(l(t))2|f′(l(t))|qdt = ∫ 1 0 (1 − t) ( 2ab (1 − t)a + (1 + t)b )2 × ∣∣∣∣f′( 2ab(1 − t)a + (1 + t)b) ∣∣∣∣qdt ≤ 12s |f′(a)|q ∫ 1 0 (1 − t)(1 + t)s ( 2ab (1 − t)a + (1 + t)b )2 dt + m 1 2s |f′( b m )|q ∫ 1 0 (1 − t)s+1 ( 2ab (1 − t)a + (1 + t)b )2 dt (2.4) = { 22a2b(2,s + 1).2f1(2, 2,s + 3; b−a b ) − 22−sa2b2 (b + a)2 b(1,s + 1).2f1(2, 1,s + 2; b−a b + a ) − 22−sa2b2 (b + a)2 b(2,s+1).2f1(2, 2,s+3; b−a b + a ) } |f′(a)|q+ ma2 2s b(s+2, 1).2f1(2,s+2,s+3; b−a 2b )|f′( b m )|q 192 baloch, i̇şcan and dragomir moreover, (2.5) ∫ 1 0 (1 − t)(u(t))2dt = ∫ 1 0 (1 − t) ( 2ab (1 + t)a + (1 − t)b )2 dt = ( 2ab b−a )2 ln( a + b 2a ) − (2ab)2 b2 −a2 := λ1(a,b) and (2.6) ∫ 1 0 (1 − t)(l(t))2dt = ∫ 1 0 (1 − t) ( 2ab (1 − t)a + (1 + t)b )2 dt = (2ab)2 b2 −a2 + ( 2ab b−a )2 ln( a + b 2b ) := λ2(a,b) a combination of (2.2), (2.3), (2.4), (2.5) and (2.6) gives required result. this completes the proof. � corollary 3. suppose the assumptions of the theorem 3 are satisfied. if g(x) = ab b−a for all x ∈ [a,b], then one has the following inequality∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ a 2 q 8 { λ 1−1 q 1 (a,b) ({ 22b(s + 1, 2).2f1(2,s + 1,s + 3; b−a b ) − 21−sb(s + 1, 1).2f1(2,s + 1,s + 2; b−a 2b ) + 1 2s b(s + 2, 1).2f1(2,s + 2,s + 3; b−a 2b ) } |f′(b)|q + m22−sb2 (b + a)2 b(1,s + 2).2f1(2, 1,s + 3; b−a b + a )|f′( a m )|q )1 q + λ 1−1 q 2 (a,b) × ({ 22b(2,s + 1).2f1(2, 2,s + 3; b−a b ) − 22−sb2 (b + a)2 b(1,s + 1).2f1(2, 1,s + 2; b−a b + a ) − 22−sb2 (b + a)2 b(2,s + 1).2f1(2, 2,s + 3; b−a b + a ) } |f′(a)|q (2.7) + m 2s b(s + 2, 1).2f1(2,s + 2,s + 3; b−a 2b )|f′( b m )|q )1 q } theorem 4. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b m ∈ i◦, m ∈ (0, 1] with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f′ ∈ l([a,b]). if |f′|q is harmonically (s,m)-convex on [a, b m ] for q > 1, then the following inequality holds ∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ a(b−a)8b .‖g‖∞ × {({ 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b ) − 21−sb(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) } |f′(b)|q +m22q−s( b b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a )|f′( a m )|q )1 q + ({ 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b ) − 22q−s( b b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) } |f′(a)|q (2.8) + m 2s b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b )|f′( b m )|q )1 q harmonically (s,m)-convex functions 193 proof. from lemma 1 and hölder’s inequality, we get∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ b−a8ab ‖g‖∞ (∫ 1 0 (1 − t) q q−1 dt )1−1 q (2.9) × {(∫ 1 0 (u(t))2q|f′(u(t))|qdt )1 q + (∫ 1 0 (l(t))2q|f′(l(t))|qdt )1 q } by the harmonic (s,m)-convexity of |f′|q on [a,b] for q > 1, we have∫ 1 0 (u(t))2q|f′(u(t))|qdt = ∫ 1 0 ( 2ab (1 + t)a + (1 − t)b )2q × ∣∣∣∣f′( 2ab(1 + t)a + (1 − t)b) ∣∣∣∣qdt ≤ 12s |f′(b)|q ∫ 1 0 (1 + t)s ( 2ab (1 + t)a + (1 − t)b )2q dt + m 1 2s |f′( a m )|q ∫ 1 0 (1 − t)s ( 2ab (1 + t)a + (1 − t)b )2q dt (2.10) = a2q { 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b ) − 21−sb(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) } |f′(b)|q + m22q−s( ab b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a )|f′( a m )|q and ∫ 1 0 (l(t))2q|f′(l(t))|qdt = ∫ 1 0 ( 2ab (1 − t)a + (1 + t)b )2q × ∣∣∣∣f′( 2ab(1 − t)a + (1 + t)b) ∣∣∣∣qdt ≤ 12s |f′(a)|q ∫ 1 0 (1 + t)s ( 2ab (1 − t)a + (1 + t)b )2q dt + m 1 2s |f′( b m )|q ∫ 1 0 (1 − t)s ( 2ab (1 − t)a + (1 + t)b )2q dt (2.11) = a2q { 2b(1,s+ 1).2f1(2q, 1,s+ 2; b−a b )−22q−s( b b + a )2qb(1,s+ 1).2f1(2q, 1,s+ 2; b−a b + a ) } |f′(a)|q + ma2q 2s b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b )|f′( b m )|q by putting (2.10) and (2.11) in (2.9), we get desired result. � corollary 4. suppose the assumptions of the theorem 3 are satisfied. if g(x) = ab b−a for all x ∈ [a,b], then one has the following inequality∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ a28 × {({ 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b ) − 21−sb(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) } |f′(b)|q +m22q−s( b b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a )|f′( a m )|q )1 q + ({ 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b ) − 22q−s( b b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) } |f′(a)|q (2.12) + m 2s b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b )|f′( b m )|q )1 q 194 baloch, i̇şcan and dragomir theorem 5. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b m ∈ i◦, m ∈ (0, 1] with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f′ ∈ l([a,b]). if |f′|q is harmonically (s,m)-convex on [a, b m ] for q > 1, then the following inequality holds ∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ 21−1q a(b−a)8b ‖g‖∞( 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b )|f′(b)|q + 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b )|f′(a)|q + m|f′( b m )|q −|f′(b)|q 2s .b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) (2.13) +22q−s( b b + a )2q(m|f′( a m )|q −|f′(a)|q)b(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) )1 q proof. from lemma 1 and hölder’s inequality, we get∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ b−a8ab ‖g‖∞ (∫ 1 0 (1 − t) q q−1 dt )1−1 q (2.14) × {(∫ 1 0 (u(t))2q|f′(u(t))|qdt )1 q + (∫ 1 0 (l(t))2q|f′(l(t))|qdt )1 q } by the power-mean inequality (ar + br ≤ 21−r(a + b)r for a > 0, b > 0 and r < 1), we have(∫ 1 0 (u(t))2q|f′(u(t))|qdt )1 q + (∫ 1 0 (l(t))2q|f′(l(t))|qdt )1 q (2.15) ≤ 21− 1 q (∫ 1 0 (u(t))2q|f′(u(t))|qdt + ∫ 1 0 (l(t))2q|f′(l(t))|qdt )1 q since, |f′|q is harmonically (s,m)-convex on [a,b] for q > 1, we obtain∫ 1 0 (u(t))2q|f′(u(t))|qdt + ∫ 1 0 (l(t))2q|f′(l(t))|qdt ≤ 1 2s |f′(b)|q ∫ 1 0 (1 + t)s ( 2ab (1 + t)a + (1 − t)b )2q dt +m 1 2s |f′( a m )|q ∫ 1 0 (1 − t)s ( 2ab (1 + t)a + (1 − t)b )2q dt + 1 2s |f′(a)|q ∫ 1 0 (1 + t)s ( 2ab (1 − t)a + (1 + t)b )2q dt +m 1 2s |f′( b m )|q ∫ 1 0 (1 − t)s ( 2ab (1 − t)a + (1 + t)b )2q dt = a2q { 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b ) − 21−sb(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) } |f′(b)|q +m22q−s( ab b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a )|f′( a m )|q +a2q { 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b ) − 22q−s( b b + a )2qb(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) } |f′(a)|q + ma2q 2s b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b )|f′( b m )|q harmonically (s,m)-convex functions 195 using (2.15) in (2.14), we get(∫ 1 0 (u(t))2q|f′(u(t))|qdt )1 q + (∫ 1 0 (l(t))2q|f′(l(t))|qdt )1 q ≤ 21− 1 q a2 ( 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b )|f′(b)|q + 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b )|f′(a)|q + m|f′( b m )|q −|f′(b)|q 2s .b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) (2.16) +22q−s( b b + a )2q(m|f′( a m )|q −|f′(a)|q)b(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) )1 q applying (2.17) in (2.14), we obtain the required inequality. � corollary 5. suppose the assumptions of the theorem 3 are satisfied. if g(x) = ab b−a for all x ∈ [a,b], then one has the following inequality∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ 21−1q a28( 2b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a b )|f′(b)|q + 2b(1,s + 1).2f1(2q, 1,s + 2; b−a b )|f′(a)|q + m|f′( b m )|q −|f′(b)|q 2s .b(s + 1, 1).2f1(2q,s + 1,s + 2; b−a 2b ) (2.17) +22q−s( b b + a )2q(m|f′( a m )|q −|f′(a)|q)b(1,s + 1).2f1(2q, 1,s + 2; b−a b + a ) )1 q theorem 6. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b m ∈ i◦, m ∈ (0, 1] with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f′ ∈ l([a,b]). if |f′| is harmonically (s,m)-convex on [a, b m ], then the following inequality holds for q > 1 ∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ (b−a)8ab ‖g‖∞( 1sq + 1)1q × { 22−s( ab b + a )2 ( (2sq+1 − 1)|f′(b)| + m|f′( a m )| )( b(1, 2q − 1 q − 1 ).2f1( 2q q − 1 , 1, 3q − 2 q − 1 ; b−a b + a ) )q−1 q (2.18) + a2 2s ( (2sq+1 − 1)|f′(a)| + m|f′( b m )| )( b( 2q − 1 q − 1 , 1).2f1( 2q q − 1 , 2q − 1 q − 1 , 3q − 2 q − 1 ; b−a b ) )q−1 q } proof. from lemma 1 and by using the harmonic (s,m)-convexity of |f′| on [a,b] , we get∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ b−a8ab ‖g‖∞ × [∫ 1 0 (1 − t)(u(t))2|f′(u(t))|dt + ∫ 1 0 (1 − t)(l(t))2|f′(l(t))|dt ] ≤ b−a 8ab ‖g‖∞ {∫ 1 0 (1 − t)(u(t))2 [ ( 1 + t 2 )s|f′(b)| + m( 1 − t 2 )s|f′( a m )| ] (2.19) + ∫ 1 0 (1 − t)(l(t))2 [ ( 1 + t 2 )s|f′(a)| + m( 1 − t 2 )s|f′( b m )| ]} now, by using hölder’s inequality, we get∫ 1 0 (1 − t)(u(t))2 [ ( 1 + t 2 )s|f′(b)| + m( 1 − t 2 )s|f′( a m )| ] dt 196 baloch, i̇şcan and dragomir ≤ (∫ 1 0 (1 − t) q q−1 (u(t)) 2q q−1 dt )q−1 q × {(∫ 1 0 ( 1 + t 2 )sqdt )1 q |f′(b)| + m (∫ 1 0 ( 1 − t 2 )sqdt )1 q |f′( a m )| } (2.20) = 22−s( ab b + a )2( 1 sq + 1 ) 1 q ( (2sq+1−1)|f′(b)|+m|f′( a m )| )( b(1, 2q − 1 q − 1 ).2f1( 2q q − 1 , 1, 3q − 2 q − 1 ; b−a b + a ) )q−1 q . similarly, one has ∫ 1 0 (1 − t)(l(t))2 [ ( 1 + t 2 )s|f′(a)| + m( 1 − t 2 )s|f′( b m )| ] (2.21) = a2 2s ( 1 sq + 1 ) 1 q ( (2sq+1 − 1)|f′(a)| + m|f′( b m )| )( b( 2q − 1 q − 1 , 1).2f1( 2q q − 1 , 2q − 1 q − 1 , 3q − 2 q − 1 ; b−a b ) )q−1 q . � corollary 6. suppose the assumptions of the theorem 3 are satisfied. if g(x) = ab b−a for all x ∈ [a,b], then one has the following inequality∣∣∣∣f(a) + f(b)2 ∫ b a g(x) x2 dx− ∫ b a g(x)f(x) x2 dx ∣∣∣∣ ≤ 18( 1sq + 1)1q × { 22−s( ab b + a )2 ( (2sq+1 − 1)|f′(b)| + m|f′( a m )| )( b(1, 2q − 1 q − 1 ).2f1( 2q q − 1 , 1, 3q − 2 q − 1 ; b−a b + a ) )q−1 q (2.22) + a2 2s ( (2sq+1 − 1)|f′(a)| + m|f′( b m )| )( b( 2q − 1 q − 1 , 1).2f1( 2q q − 1 , 2q − 1 q − 1 , 3q − 2 q − 1 ; b−a b ) )q−1 q } 3. competing interests the authors have no competing interests regarding this article. 4. funding this research article is partially supported by higher education commission of pakistan. 5. authors’ contributions each author has equal contribution in this research article. 6. acknowledgement the authors wish to express their heartfelt thanks to the referees for their constructive comments and helpful suggestions to improve the final version of this paper. references [1] i. a. baloch, i.işcan, some ostrowski type inequalities for harmonically (s,m)-convex functoins in second sense, international journal of analysis, 2015 (2015), article id 672675. [2] p. s. bullen, handbook of means and their inequalities, mathematics and its applications, volume 560, kluwer academic publishers, dordrecht/boston/london, 2003. [3] w.w. breckner, tetigkeitsaussagen für eine klasse verallgemeinerter konvexer funktonen in topologischen linearen räumen, publ. inst. math. (beograd), 23 (1978),13-20. [4] f. chen and s. wu, fejér and hermite-hadamard type inequalities for harmonically convex functions, journal of applied mathematics 2014 (2014), article id 386806. [5] f. chen and s.wu, hermite-hadamard type inequalities for harmonically s-convex functions, sci. world j. 2014 (2014), article id 279158. [6] s. s. dragomir, r.p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (5) (1998) 91-95. harmonically (s,m)-convex functions 197 [7] s. s. dragomir, c.e.m. pearce, selected topics on hermite-hadamard type inequalities and applications, rgmia monographs, victoria university, 2000. [8] v. n. huy and n. t. chung, some generalizations of the fejér and hermite-hadamard inequalities in hölder spaces, j. appl. math. inform. 29 (2011), no. 3-4, 859-868. [9] j. hua, b.-y. xi, and f. qi, hemite-hadamard type inequalities for geometricallyarithmetically s-convex functions, commun. korean math. soc. 29 (2014), no. 1, 51-63. [10] j. hua, b. -y. xi and f. qi, inequalities of hermitehadamard type involving an s-convex function with applications, applied mathematics and computation, 246 (2014), 752-760. [11] i.işcan, hemite-hadamard type inequalities for ga-s-convex functions, le matematiche, 69 (2014), 129-146. [12] i.işcan, hermite-hadamard type inequalities for harmonically convex functions, hacettepe journal of mathematics and statistics 43 (6) (2014), 935-942. [13] i.işcan, ostrowski type inequalities for harmonically s-convex functions, konuralp journal of mathematics, 3 (2015), no. 1, 63-74. [14] i.işcan, hermite-hadamard and simpson-like type inequalities for differentiable harmonically convex functions, journal of mathematics, 2014 (2014), article id 346305. [15] i.işcan and s. wu, hermite-hadamard type inequalities for harmonically convex functions via fractional integrals, applied mathematica and computation, 238 (2014), 237-244. [16] a. p. ji, t. y. zhang, f. qi, integral inequalities of hermite-hadamard type for (α,m)ga-convex functions, journal of function spaces and applications, 2013 (2013), article id 823856. [17] m. a. latif, new hermite-hadamard type integral inequalities for ga-convex functions with applications, analysis, 34 (2014), 379-389. [18] m. v. mihai, m. a. noor, k. i. noor and m. u. awan, some integral inequalities for harmonic h-convex functions involving hypergeometric functions, applied mathematics and computation 252 (2015), 257-262. [19] m. a. noor, k. i. noor and m. u. awana, integral inequalities for coordinated harmonically convex functions, complex var. elliptic eqn. 60 (2015), 776-786. [20] m. a. noor, k. i. noor, m. u. awana and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b sci. bull. serai a. 77 (2015), 5-16. [21] j. e. pečarič, f. proschan, y. l. tong, convex functions, partial orderings and statistical applications, mathematics in science and engineering, vol. 187, 1992. [22] m. z. sarikaya, on new hermite hadamard fejér type integral inequalities, stud. univ. babeş-bolyai math. 57 (2012), no. 3, 377-386. [23] y. shuang, h. p. yin, f. qi, hermite-hadamard type integral inequalities for geometricarithmetically s-convex functions, analysis 33 (2013), 1001-1010. 1abdus salam school of mathematical sciences, gc university, lahore, pakistan 2department of mathematics, faculty of arts and sciences, giresun university, 28200, giresun, turkey 3mathematics, college of engineering and science, victoria university, melbourne city, australia ∗corresponding author: iabbasbaloch@gmail.com, iabbasbaloch@sms.edu.pk int. j. anal. appl. (2023), 21:20 symbolic algorithm for inverting general k-tridiagonal interval matrices sivakumar thirupathi, nirmala thamaraiselvan∗ department of mathematics, srm institute of science and technology, kattankulathur 603203, tamil nadu, india ∗corresponding author: nirmalat@srmist.edu.in abstract. the k-tridiagonal matrices have received much attention in recent years. many different algorithms have been proposed to improve the efficiency of k-tridiagonal matrix estimation. a novel method based on interval analysis has been identified to improve the efficiency of the calculation. this paper presents efficient and reliable computational algorithms for determining the determinant and inverse of general k-tridiagonal interval matrices built on generalized interval arithmetic. this study is based on the doolittle lu factorization of the interval matrix. finally, examples are presented to illustrate the algorithms. 1. introduction tridiagonal matrices play an influential role in many areas of science and engineering. these areas include spline interpolation, parallel computing, signal processing, solving ordinary and partial differential equations using finite differences. in many of these areas, tridiagonal matrix inversion is a crucial procedure with various applications. k-tridiagonal matrices, a generalization of tridiagonal matrices, are widely used and frequently appear in various applications. for examples, moawwad el-mikkawy et al. [13–15] present breakdown-free algorithms for inverting general tridiagonal and k-tridiagonal matrices without imposing constraints. moreover, they have developed a novel algorithm for inverting a non-singular k-tridiagonal matrix. ji teng jia et al. [8, 9] developed a numerical algorithm for computing the determinants of a block k-tridiagonal matrix and a bordered k-tridiagonal matrix. the received: jan. 4, 2023. 2020 mathematics subject classification. 15a09, 15a23, 65f05, 65g30. key words and phrases. tridiagonal interval matrix; k-tridiagonal interval matrix; interval lu factorization; generalized interval arithmetic; interval matrix determinant; interval matrix inversion. https://doi.org/10.28924/2291-8639-21-2023-20 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-20 2 int. j. anal. appl. (2023), 21:20 algorithm uses the fast block diagonalization method. tanasescu a et al. [21,22] proposed the singular value decomposition of a k-tridiagonal matrix that can be calculated in o(n3/k2) and a technique for enhancing any existing svd algorithm to make it suitable for this class of matrices. da fonseca cm et al. [3, 4] developed the spectral theory for k-tridiagonal matrices, which are the first type of matrices. then they discussed the explosion of interest in them over the last two decades. wei y et al. [24] presented explicit formulae for determinants, inverses and eigenpairs of a periodic tridiagonal toeplitz-like matrix with asymmetrically perturbed rows. solary et al. [19] showed a symbolic algorithm for inverting a general k-heptadiagonal matrix and recursive relationships. this work is based on the lu factorization of the matrix. fu y et al. [5] studied the eigenvalues and eigenvectors of the tridiagonal toeplitz matrix with opposite bordered rows. alberto j et al. [1] studied the inverses of k-toeplitz matrices in the context of resonator arrays with multiple receivers. albuquerque h et al. [2] gave rational formulas for the determinant, the characteristic polynomial and the elements of the inverse of a tridiagonal k-toeplitz matrix over any commutative unital ring. kucuk az et al. [10] discussed recursive and combinational formulas for the permanents of general k-tridiagonal toeplitz matrices. takahira s et al. [20] presented bidiagonalization of n-by-n (k,k + 1)-tridiagonal matrices when n ≤ 2k. yalciner [23] proposed a k-tridiagonal matrix determinant based on lu factorization. in real-life, computations are inaccurate since uncertainty often exists. at most, it is possible to know the intervals of possible values. so, it is crucial to figure out how to handle the impact of unclear parameters on system properties. interval analysis is a common way to deal with uncertain situations. it describes uncertain parameters as interval numbers. then, the interval containing each potential solution must be computed. ganesan et al. [6] presented a new set of arithmetic operations for interval numbers by which those discrepancies in general can be reduced to some extent. kaucher [11] introduced interval analysis in extended interval space ir and dual as a significant monadic operator in interval calculations. nirmala et al. [16] developed a new way to find the inverse of an interval matrix. this helps us solve systems of interval linear equations. rohn [18] proposed theoretical and practical ways to figure out how to calculate the inverse interval matrix. after this inspiration and motivation, several authors, such as [7, 12, 17] have investigated uncertainty. the main goal of this study is to create effective computational algorithms based on generalized interval arithmetic. these algorithms are used to find the determinant and inverse of general k-tridiagonal interval matrices with interval doolittle lu factorization. this is explained with the help of two instructive numerical examples. the paper is organized as follows: section 2 overviews generalized interval arithmetic. in section 3, the main results and theorem are presented. section 4 suggests algorithms for finding the determinant and inverse of the general k-tridiagonal interval matrix. section 5 gives two numerical examples to show how the algorithm works. int. j. anal. appl. (2023), 21:20 3 2. preliminary notes “ let d = ir∪ ir = {[u1,u2] : u1,u2 ∈ r} is the set of generalized intervals that are the proper and improper intervals, where ir = {ũ = [u1,u2] : u1 > u2 and u1,u2 ∈ r} be the collection of all improper intervals on a real line r. be the collection of generalized intervals d is a group that maintains inclusion monotonicity while performing addition and multiplication operations over zero free intervals. the midpoint and width of an interval number ũ = [u1,u2] is given by m(ũ) = ( u1 + u2 2 ) and w(ũ) = ( u2 −u1 2 ) . kaucher introduces the dual as a significant monadic operator [11] that expresses element to element symmetry between proper and improper intervals by reversing the end points numbers in the interval, intervals in d. for ũ = [u1,u2] ∈ d, its dual is given by dual(ũ) = dual[u1,u2] = [u2,u1]. an interval’s opposite ũ = [u1,u2] is opp {[u1,u2]} = [−u1,−u2] which is the additive inverse of [u1,u2] and [ 1 u1 , 1 u2 ] is the multiplicative inverse of [u1,u2], provided 0 /∈ [u1,u2]. that is, ũ + (−dual ũ) = ũ −dual(ũ) = [u1,u2] −dual([u1,u2]) = [u1,u2] − [u2,u1] = [u1 −u1,u2 −u2] = [0, 0] and ũ × ( 1 dual ũ ) = [u1,u2] × ( 1 dual([u1,u2]) ) = [u1,u2] × 1 [u2,u1] = [u1,u2] × [ 1 u1 , 1 u2 ] = [1, 1]. 2.1. arithmetic operations on interval matrices. if ã,b̃ ∈dn×n, x̃∈dn and α̃ ∈d, we propose a generalized interval arithmetic as, (i). “α̃ã ≈ ( α̃ãij ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n (ii). ã + b̃ ≈ ( ãij + b̃ij ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n (iii). ã− b̃ ≈ { ( ãij − b̃ij ) 1≤i≤n, 1≤j≤n , if ã, b̃ are not equivalent ã−dual(ã) ≈ õ = o, if ã ≈ b̃ ” (iv). ãb̃ ≈ ( n∑ k=1 ãikb̃kj ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n (v). ãx̃≈ ( n∑ j=1 ãijx̃ ) for i = 1, 2, · · · ,n 2.2. interval arithmetic. a new method of interval arithmetic on ir was proposed by ganesan and veeramani [6]. the set of generalized interval numbers is extended using these arithmetic procedures d by utilising the dual concept, for ũ = [u1,u2],“ ṽ = [v1,v2] ∈d and for ∗∈ {+,−, ·,÷}, we define ũ ∗ ṽ = [m(ũ) ∗m(ṽ) − j,m(ũ) ∗m(ṽ) + j], where j = min{(m(ũ) ∗m(ṽ)) −β, γ − (m(ũ) ∗m(ṽ))}, where the β and γ are the end points of the interval ũ � ṽ under the existing interval arithmetic. in particular, (i) addition: ũ + ṽ = [u1,u2] + [v1,v2] = [(m(ũ) + m(ṽ)) − j, (m(ũ) + m(ṽ)) + j], 4 int. j. anal. appl. (2023), 21:20 where j = { (v2 + u2) − (v1 + u1) 2 } . (ii) subtraction: ũ − ṽ = [u1,u2] − [v1,v2] = [(m(ũ) −m(ṽ)) − j, (m(ũ) −m(ṽ)) + j], where j = { (v2 + u2) − (v1 + u1) 2 } . also if ũ = ṽ, i.e. if [u1,u2] = [v1,v2], then ũ − ṽ = ũ −dual(ũ) = [u1,u2] − [u2,u1] = [u1 −u1,u2 −u2] = [0, 0] . (iii) multiplication: ũ.ṽ = ũṽ = [u1,u2] [v1,v2] = [(m(ũ)m(ṽ)) − j, (m(ũ)m(ṽ)) + j] , where j = min{(m(ũ)m(ṽ)) −β, γ − (m(ũ)m(ṽ))}, β = min(u1v1,u1v2,u2v1,u2v2) and γ = max(u1v1,u1v2,u2v1,u2v2). (iv) division: 1 ÷ ũ = 1 ũ = 1 [u1,u2] = [ 1 m(ũ) − j, 1 m(ũ) + j ] , where j = min { 1 u2 ( u2 −u1 u1 + u2 ) , 1 u1 ( u2 −u1 u1 + u2 )} and m([u1,u2]) = ( u1 + u2 2 ) 6= 0. also if ũ = ṽ, i.e. [u1,u2] = [v1,v2], then ũ ṽ = ũ ũ = ũ dual(ũ) = [u1,u2] . 1 [u2,u1] = [u1,u2] . [ 1 u1 , 1 u2 ] = [1, 1] . from (iii), it is clear that λũ = { [λu1,λu2], for λ ≥ 0 [λu2,λu1], for λ < 0. it’s worth noting that � stands for existing interval arithmetic and ∗ stands for generalized interval arithmetic. however, in circumstances when there is no ambiguity, the same notation can be used for both cases. it is also to be noted that ũ ∗ ṽ ⊆ ũ � ṽ, where �∈{⊕, ,⊗,�} is the existing interval arithmetic. note 2.1. without loss of generality, assume that for any interval number ũ = [u1,u2] with m(ũ) 6= 0 and 0 ∈ ũ, there exist ṽ = [m(ũ) − j,m(ũ) + j], where 0 < j < h and h = min{|u1|, |u2|}, such that ṽ ≈ ũ and 0 /∈ ṽ. hence, if ã ũ with m(ũ) 6= 0 and 0 ∈ ũ, then we replace ã ũ by ã ṽ where ṽ ≈ ũ and 0 /∈ ṽ. in particular (for convenience) one may select j in such a way that j =   m(ũ) 2 , if m(ũ) > 0 −m(ũ) 2 , if m(ũ) < 0 generalized interval arithmetic can be used to prove a lot of important things, like the distributive law for interval numbers. ” 3. main results in this section, we provide some important results concerning the general k-tridiagonal interval matrix. a tridiagonal interval matrix is a matrix with three interval diagonals. the tridiagonal interval int. j. anal. appl. (2023), 21:20 5 matrix has nonzero interval entries (midpoint of interval number not equal to zero) on the form’s main diagonal, immediate sub diagonal and super diagonal. ã =   [h1,h1] [f 1, f 1] 0̃ · · · · · · 0̃ [e1,e1] [h2,h2] [f 2, f 2] ... ... 0̃ [e2,e2] [h3,h3] [f 3, f 3] 0̃ ... ... 0̃ ... ... ... 0̃ ... ... ... ... [f n−1, f n−1] 0̃ · · · · · · 0̃ [en−1,en−1] [hn,hn]   . (3.1) let dn×n be the collection of all n×n interval matrices. the k-tridiagonal interval matrix ãkn is a more general tridiagonal interval matrix that can be expressed as follows: ãkn =   [h1,h1] [0, 0] · · · [0, 0] [f 1, f 1] [0, 0] · · · [0, 0] [0, 0] [h2,h2] [0, 0] · · · [0, 0] [f 2, f 2] ... ... · · · [0, 0] ... [0, 0] · · · ... ... [0, 0] [0, 0] · · · ... [hn−k,hn−k] ... · · · ... [f n−k, f n−k] [e1,e1] [0, 0] · · · ... ... ... · · · [0, 0] [0, 0] [e2,e2] ... · · · [0, 0] ... [0, 0] · · · ... ... ... [0, 0] · · · [0, 0] [hn−1,hn−1] [0, 0] [0, 0] · · · [0, 0] [en−k,en−k] [0, 0] · · · [0, 0] [hn,hn]   (3.2) where 1 ≤ k < n. for k ≥ n, the interval matrix ãkn is a diagnal interval matrix, which has k = 1, gives a standard tridiagonal interval matrix in (3.1). the 3n − 2k memory locations can be used to store the nonzero interval numbers of the interval matrix ãkn. having this habit makes calculations easier. the midpoint of an k-tridiagonal interval matrix ãkn is defined as, m(ãkn) =   m(h̃1) [0, 0] · · · [0, 0] m(f̃1) [0, 0] · · · [0, 0] [0, 0] m(h̃2) [0, 0] · · · [0, 0] m(f̃2) ... ... · · · [0, 0] ... [0, 0] · · · ... ... [0, 0] [0, 0] · · · ... m(h̃n−k) ... · · · ... m(f̃n−k) m(ẽ1) [0, 0] · · · ... ... ... · · · [0, 0] [0, 0] m(ẽ2) ... · · · [0, 0] ... [0, 0] · · · ... ... ... [0, 0] · · · [0, 0] m(h̃n−1) [0, 0] [0, 0] · · · [0, 0] m(ẽn−k) [0, 0] · · · [0, 0] m(h̃n)   . the width of an k-tridiagonal interval matrix ãkn is defined as, 6 int. j. anal. appl. (2023), 21:20 w(ãkn) =   w(h̃1) [0, 0] · · · [0, 0] w(f̃1) [0, 0] · · · [0, 0] [0, 0] w(h̃2) [0, 0] · · · [0, 0] w(f̃2) ... ... · · · [0, 0] ... [0, 0] · · · ... ... [0, 0] [0, 0] · · · ... w(h̃n−k) ... · · · ... w(f̃n−k) w(ẽ1) [0, 0] · · · ... ... ... · · · [0, 0] [0, 0] w(ẽ2) ... · · · [0, 0] ... [0, 0] · · · ... ... ... [0, 0] · · · [0, 0] w(h̃n−1) [0, 0] [0, 0] · · · [0, 0] w(ẽn−k) [0, 0] · · · [0, 0] w(h̃n)   which is always nonnegative. if m(ãkn) = m(b̃ k n), then the interval matrices ã k n and b̃ k n are said to be equivalent and is denoted by ãkn ≈ b̃kn. in particular if m(ãkn) = m(b̃kn) and w(ãkn) = w(b̃kn), then ãkn = b̃kn. if m(ãkn) = 0 then ãkn is a zero interval matrix. in particular, if m(ã k n) = 0 and w(ã k n) = 0, then ã k n = 0̃. if m(ãkn) = 0 and w(ã k n) 6= 0, then ãkn 6≈ 0̃, if ãkn is said to be a non-zero interval matrix. if m(ãkn) = i, then ã k n is an identity interval matrix. in specifically, if m(ã k n) = i and w(ã k n) = 0, then ãkn = ĩ, if m(ã k n) = i and w(ã k n) 6= 0, then ãkn ≈ ĩ. also i denotes the identity matrix and the identity interval matrix is indicated by ĩ. if 0 be the null matrix and 0̃ be the matrix of null intervals. theorem 3.1. let the k-tridiagonal interval matrix ãkn be as in (3.2), the lu factorization of ãkn can be expressed as, ãkn ≈ l̃ k nũ k n where l̃kn =   [1, 1] [0, 0] [0, 0] · · · · · · · · · · · · [0, 0] [0, 0] [1, 1] [0, 0] ... ... ... ... ... ... [0, 0] [1, 1] ... ... ... ... ... [0, 0] ... ... ... ... ... ... ... [e1,e1] [k1,k1] ... ... ... ... ... ... [0, 0] [0, 0] [e2,e2] [k2,k2] ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0, 0] · · · [0, 0] [en−k,en−k] [kn−k,kn−k] [0, 0] · · · · · · [1, 1]   , int. j. anal. appl. (2023), 21:20 7 ũkn =   [k1,k1] [0, 0] · · · [0, 0] [f 1, f 1] [0, 0] · · · [0, 0] [0, 0] [k2,k2] [0, 0] ... [0, 0] [f 2, f 2] ... ... ... [0, 0] [k3,k3] ... ... ... ... [0, 0] [0, 0] ... ... ... ... ... ... [f n−k, f n−k] [0, 0] ... ... ... ... ... ... [0, 0] [0, 0] · · · ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0, 0] · · · [0, 0] · · · [0, 0] · · · · · · [kn,kn]   . with k̃i =  h̃i i = 1, 2, ...,k, h̃i − s̃i−kẽi−k i = k + 1,k + 2, · · · ,n. (3.3) where s̃i = f̃i k̃i for i = 1, 2, ...,n−k. in order to further discuss this article, we must consider the above results. 4. the symbolic inverse of a k-tridiagonal interval matrix in this section, we give two algorithms for finding the determinant and inverse of the general k-tridiagonal interval matrix ãkn. the following algorithm can be used to evaluate the value of det(ãkn) in the interval matrix ã k n. algorithm 4.1. an algorithm for determining the determinant of the k-tridiagonal interval matrix. step 1. input: ẽi, h̃i, f̃i and the order n. step 2. for i = 1, 2, ...,k do set: k̃i = h̃i end do. setp 3. for i = 1, 2, ...,n−k, do set: s̃i = f̃i k̃i if m(k̃i) 6= 0 end do. step 4. for i = k + 1,k + 2, ...,n do 8 int. j. anal. appl. (2023), 21:20 set: k̃i = h̃i − s̃i−kẽi−k end do. step 5. compute and simplify: det(ãkn) = π n i=1k̃i. step 6. output: the determinant of the tridiagonal interval matrix (ãkn). we can follow the procedure outlined below to find the inverse of a general k-tridiagonal interval matrix (ãkn). algorithm 4.2. symbolic algorithm for inverting a k-tridiagonal interval matrix. step 1. input: ẽi, h̃i, f̃i, and the order n. for ẽi, f̃i, i = 1, 2, ...,n−k. h̃i, i = 1, 2, ...,n. step 2. for i = 1, 2, ...,k do set: k̃i = h̃i. if m(k̃i) 6= 0 end do. step 3. for i = k + 1,k + 2, ...,n do compute and simplify: s̃i−k = f̃i−k k̃i−k k̃i = h̃i − ẽi−k s̃i−k if m(k̃i) 6= 0 end if. t̃i−k = ẽi−k k̃i−k end do. step 4. use the determinant of the k-tridiagonal interval matrix algorithm (4.1) to check the non-singularity of the interval matrix in (3.2). int. j. anal. appl. (2023), 21:20 9 step 5. for i = n,n− 1, ...,n−k + 1 do compute and simplify: δ̃i,i = 1̃ k̃i end do. step 6. for i = n−k,n−k − 1, ..., 1 do compute and simplify: δ̃i,i = 1̃ k̃i + s̃i t̃i δ̃i+k,i+k end do. step 7. for j = n,n− 1, ..., 2 do for i = j −k, j − 2k,..., 1 do compute and simplify: δ̃i,j = −s̃i δ̃i+k,j end do. step 8. for i = n,n− 1, ..., 2 do j = i −k, i − 2k,..., 1 do compute and simplify: δ̃i,j = −t̃j δ̃i,j+k end do. step 9. output: the inverse interval matrix (ãkn) −1 = ∆̃ = δ̃ni,j=1. 10 int. j. anal. appl. (2023), 21:20 5. numerical examples in this section, we will examine the effectiveness of two numerical examples using the proposed algorithm. example 5.1. let us consider the k-tridiagonal interval matrix ãkn with n = 10,k = 4. ã410 =   [−3.75,1.75] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [−4.25,0.25]   . solution: applying the symbolic algorithm (4.2) for inverting a k-tridiagonal interval matrix. by using steps 2-3, we get: k̃1 = [−3.75, 1.75], k̃2 = [−4.25, 0.25], k̃3 = [−4.25, 0.25], k̃4 = [−4.25, 0.25], k̃5 = [−4.083, 2.084], k̃6 = [−4.132, 1.132], k̃7 = [−4.132, 1.132], k̃8 = [−4.132, 1.132], k̃9 = [−4.083, 2.084], k̃10 = [−4.139, 1.472]. using step 4, we yields: det(ã410) = π 10 i=1k̃i = [−653997, 654068.2]. applying steps 5-8, we obtain (ãkn) −1 =   [−5.156,−0.750] [0,0] [0,0] [0,0] [−3.756,−0.248] [0,0] [0,0] [0,0] [−1.928,−0.074] [0,0] [0,0] [−1.047,−0.523] [0,0] [0,0] [0,0] [−0.924,−0.217] [0,0] [0,0] [0,0] [−0.474,−0.026] [0,0] [0,0] [−0.836,−0.497] [0,0] [0,0] [0,0] [−0.562,−0.105] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−0.836,−0.497] [0,0] [0,0] [0,0] [−0.562,−0.105] [0,0] [0,0] [−3.756,−0.248] [0,0] [0,0] [0,0] [−3.260,−0.742] [0,0] [0,0] [0,0] [−1.778,−0.223] [0,0] [0,0] [−0.924,−0.217] [0,0] [0,0] [0,0] [−1.363,−0.918] [0,0] [0,0] [0,0] [−0.889,−0.111] [0,0] [0,0] [−0.562,−0.105] [0,0] [0,0] [0,0] [−0.889,−0.444] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−0.562,−0.105] [0,0] [0,0] [0,0] [−0.889,−0.444] [0,0] [0,0] [−1.928,−0.074] [0,0] [0,0] [0,0] [−1.778,−0.223] [0,0] [0,0] [0,0] [−1.333,−0.667] [0,0] [0,0] [−0.347,−0.028] [0,0] [0,0] [0,0] [−0.632,−0.118] [0,0] [0,0] [0,0] [−1,−0.5]   . example 5.2. let us consider the k-tridiagonal interval matrix ãkn with n = 10,k = 6. ã 6 10 =   [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [0.3,1.7] [0,0] [0,0] [0,0] [0,0] [0,0] [−1.5,−0.5] [0,0] [0,0] [0,0] [0,0] [−3.75,1.75][0,0] [0,0] [0,0] [0,0] [0,0] [−0.25,4.25] [0,0] [0,0] [0,0] [0,0] [2.8,3.2][0,0] [0,0] [0,0] [0,0] [0,0] [2.58,5.42] [0,0] [0,0] [0,0] [0,0][0.5,1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0][−2.5,−1.5] [0,0] [0,0] [0,0] [0,0] [1.65,2.35] [0,0] [0,0] [0,0] [0,0] [0,0] [3.574,6.426] [0,0] [0,0] [0,0] [0,0] [−3.75,1.75] [0,0] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [0,0] [−1.5,−0.5] [0,0] [0,0] [0,0] [0,0] [1.5,2.5][0,0] [0,0] [0,0] [0,0] [0,0] [2.8,3.2]   . solution: applying the symbolic algorithm (4.2) for inverting a k-tridiagonal interval matrix. by using steps 2-3, we get: int. j. anal. appl. (2023), 21:20 11 k̃1 = [1.5, 2.5], k̃2 = [0.3, 1.7], k̃3 = [−3.75, 1.75], k̃4 = [2.8, 3.2], k̃5 = [0.5, 1.5], k̃6 = [−2.5,−1.5], k̃7 = [1.904, 6.096], k̃8 = [−2.186, 6.186], k̃9 = [−2.674, 12.674], k̃10 = [−1.324, 1.988]. using step 4, we yields: det(ã610) = π 10 i=1k̃i = [−70840, 71158.73]. applying steps 5-8, we obtain (ãkn) −1 =   [0.422,0.828] [0,0] [0,0] [0,0] [0,0] [0,0] [−0.217,−0.033] [0,0] [0,0] [0,0] [0,0] [−2.224,5.224] [0,0] [0,0] [0,0] [0,0] [0,0] [0.098,0.902] [0,0] [0,0] [0,0] [0,0] [−1.909,2.309] [0,0] [0,0] [0,0] [0,0] [0,0] [−0.098,0.898] [0,0] [0,0] [0,0] [0,0] [1.076,4.955] [0,0] [0,0] [0,0] [0,0] [0,0] [−6.414,−1.622] [0,0] [0,0] [0,0] [0,0] [0.667,1.333] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−0.6,−0.4] [0,0] [0,0] [0,0] [0,0] [−0.392,−0.108] [0,0] [0,0] [0,0] [0,0] [0,0] [0.164,0.336] [0,0] [0,0] [0,0] [0,0] [−1.648,2.648] [0,0] [0,0] [0,0] [0,0] [0,0] [0.333,0.667] [0,0] [0,0] [0,0] [0,0] [0.198,1.002] [0,0] [0,0] [0,0] [0,0] [0,0] [0.133,0.267] [0,0] [0,0] [0,0] [0,0] [−3.077,−0.944] [0,0] [0,0] [0,0] [0,0] [0,0] [2.008,4.016]   . 6. conclusion in this paper, we present two efficient algorithms for finding the determinant and inverse of ktridiagonal interval matrices based on generalized interval arithmetic. these algorithms are based on interval doolittle lu factorization and are efficient. computational results are shown in numerical examples, illustrating the feasibility of the proposed algorithms. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] j. alberto, j. brox, inverses of k-toeplitz matrices with applications to resonator arrays with multiple receivers, appl. math. comput. 377 (2020), 125185. https://doi.org/10.1016/j.amc.2020.125185. [2] j. brox, h. albuquerque, the determinant, spectral properties, and inverse of a tridiagonal k-toeplitz matrix over a commutative ring, arxiv, (2021). https://doi.org/10.48550/arxiv.2106.13157. [3] c.m. da fonseca, v. kowalenko, eigenpairs of a family of tridiagonal matrices: three decades later, acta math. hungar. 160 (2019), 376–389. https://doi.org/10.1007/s10474-019-00970-1. [4] c.m. da fonseca, v. kowalenko, l. losonczi, ninety years of k-tridiagonal matrices, stud. sci. math. hung. 57 (2020), 298–311. https://doi.org/10.1556/012.2020.57.3.1466. [5] y. fu, x. jiang, z. jiang, s. jhang, inverses and eigenpairs of tridiagonal toeplitz matrix with opposite-bordered rows, j. appl. anal. comput. 10 (2020), 1599–1613. https://doi.org/10.11948/20190287. [6] k. ganesan, p. veeramani, on arithmetic operations of interval numbers, int. j. uncertain. fuzziness knowl.-based syst. 13 (2005), 619–631. https://doi.org/10.1142/s0218488505003710. [7] a. iampan, v. vijaya bharathi, m. vanishree, n. rajesh, interval-valued intuitionistic fuzzy subalgebras/ideals of hilbert algebras, int. j. anal. appl. 20 (2022), 25. https://doi.org/10.28924/2291-8639-20-2022-25. [8] j.t. jia, j. wang, t.f. yuan, k.k. zhang, b.m. zhong, an incomplete block-diagonalization approach for evaluating the determinants of bordered k-tridiagonal matrices, j. math. chem. 60 (2022), 1658–1673. https: //doi.org/10.1007/s10910-022-01377-0. https://doi.org/10.1016/j.amc.2020.125185 https://doi.org/10.48550/arxiv.2106.13157 https://doi.org/10.1007/s10474-019-00970-1 https://doi.org/10.1556/012.2020.57.3.1466 https://doi.org/10.11948/20190287 https://doi.org/10.1142/s0218488505003710 https://doi.org/10.28924/2291-8639-20-2022-25 https://doi.org/10.1007/s10910-022-01377-0 https://doi.org/10.1007/s10910-022-01377-0 12 int. j. anal. appl. (2023), 21:20 [9] j.t. jia, y.c. yan, q. he, a block diagonalization based algorithm for the determinants of block k-tridiagonal matrices, j. math. chem. 59 (2021), 745–756. https://doi.org/10.1007/s10910-021-01216-8. [10] a. kucuk zahid, m. ozen, h. ince, recursive and combinational formulas for permanents of general k-tridiagonal toeplitz matrices, filomat. 33 (2019), 307–317. https://doi.org/10.2298/fil1901307k. [11] e. kaucher, interval analysis in the extended interval space ir, in: g. alefeld, r.d. grigorieff (eds.), fundamentals of numerical computation (computer-oriented numerical analysis), springer vienna, vienna, 1980: pp. 33–49. https://doi.org/10.1007/978-3-7091-8577-3_3. [12] v.a. khan, e. evren kara, u. tuba, k.m.a.s. alshlool, a. ahmad, sequences of fuzzy star-shaped numbers, j. math. computer sci. 23 (2020), 321–327. https://doi.org/10.22436/jmcs.023.04.05. [13] m. el-mikkawy, f. atlan, a new recursive algorithm for inverting general k-tridiagonal matrices, appl. math. lett. 44 (2015), 34–39. https://doi.org/10.1016/j.aml.2014.12.018. [14] m. el-mikkawy, f. atlan, a novel algorithm for inverting a general k-tridiagonal matrix, appl. math. lett. 32 (2014), 41–47. https://doi.org/10.1016/j.aml.2014.02.015. [15] m. el-mikkawy, a. karawia, a breakdown free numerical algorithm for inverting general tridiagonal matrices, arxiv, (2022). https://doi.org/10.48550/arxiv.2208.12843. [16] t. nirmala, d. datta, h.s. kushwaha, k. ganesan, inverse interval matrix: a new approach, appl. math. sci. 5 (2011), 607-624. [17] k. palanivel, p. muralikrishna, p. hemavathi, r. chinram, p. singavananda, interval valued intuitionistic fuzzy β-filters on β-algebras, int. j. anal. appl. 20 (2022), 50. https://doi.org/10.28924/2291-8639-20-2022-50. [18] j. rohn, inverse interval matrix, siam j. numer. anal. 30 (1993), 864–870. https://doi.org/10.1137/0730044. [19] m.s. solary, m. rasouli, inverting a k-heptadiagonal matrix based on doolitle lu factorization, appl. math. j. chin. univ. 37 (2022), 340–349. https://doi.org/10.1007/s11766-022-3763-8. [20] s. takahira, t. sogabe, t.s. usuda, bidiagonalization of (k, k+1)-tridiagonal matrices, spec. matrices. 7 (2019), 20–26. https://doi.org/10.1515/spma-2019-0002. [21] a. tanasescu, m. carabaş, f. pop, p.g. popescu, scalability of k-tridiagonal matrix singular value decomposition, mathematics. 9 (2021), 3123. https://doi.org/10.3390/math9233123. [22] a. tanasescu, p.g. popescu, a fast singular value decomposition algorithm of general k-tridiagonal matrices, j. comput. sci. 31 (2019), 1–5. https://doi.org/10.1016/j.jocs.2018.12.009. [23] a. yalciner, the lu factorizations and determinants of the k-tridiagonal matrices, asian-eur. j. math. 04 (2011), 187–197. https://doi.org/10.1142/s1793557111000162. [24] y. wei, y. zheng, z. jiang, s. shon, the inverses and eigenpairs of tridiagonal toeplitz matrices with perturbed rows, j. appl. math. comput. 68 (2021), 623–636. https://doi.org/10.1007/s12190-021-01532-x. https://doi.org/10.1007/s10910-021-01216-8 https://doi.org/10.2298/fil1901307k https://doi.org/10.1007/978-3-7091-8577-3_3 https://doi.org/10.22436/jmcs.023.04.05 https://doi.org/10.1016/j.aml.2014.12.018 https://doi.org/10.1016/j.aml.2014.02.015 https://doi.org/10.48550/arxiv.2208.12843 https://doi.org/10.28924/2291-8639-20-2022-50 https://doi.org/10.1137/0730044 https://doi.org/10.1007/s11766-022-3763-8 https://doi.org/10.1515/spma-2019-0002 https://doi.org/10.3390/math9233123 https://doi.org/10.1016/j.jocs.2018.12.009 https://doi.org/10.1142/s1793557111000162 https://doi.org/10.1007/s12190-021-01532-x 1. introduction 2. preliminary notes 2.1. arithmetic operations on interval matrices 2.2. interval arithmetic 3. main results 4. the symbolic inverse of a k-tridiagonal interval matrix 5. numerical examples 6. conclusion references int. j. anal. appl. (2023), 21:10 hankel determinant of logarithmic coefficients for tilted starlike functions with respect to conjugate points daud mohamad, nur hazwani aqilah abdul wahid∗ school of mathematical sciences, college of computing, informatics and media, universiti teknologi mara, 40450 shah alam, selangor, malaysia ∗corresponding author: hazwaniaqilah@uitm.edu.my abstract. the growth of the hankel determinant whose elements are logarithmic coefficients for different subclasses of univalent functions has recently attracted considerable interest. in this paper, we obtain the bounds for the first four initial logarithmic coefficients for the subclass of starlike functions with respect to conjugate points in an open unit disk. furthermore, we determine the upper bounds of the second hankel determinant of logarithmic coefficients for this subclass. we also present some new consequences of our results. 1. introduction let a be the class of analytic functions f (z) in an open unit disk e = {z ∈c : |z| < 1} which satisfy f (0) = f ′ (0) − 1 = 0 and has the series representation f (z) = z + ∞∑ n=2 anz n, z ∈ e. (1.1) we also denote by s the subclass of a consisting of univalent functions in e. let p be the class of analytic functions p (z) defined in e which satisfy re p (z) > 0 and has the series representation p (z) = 1 + ∞∑ n=1 pnz n, z ∈ e. (1.2) this class is also known as the class of carathéodory functions. received: jan. 7, 2023. 2020 mathematics subject classification. 30c45, 30c50. key words and phrases. univalent functions; starlike functions with respect to conjugate points; logarithmic coefficient; hankel determinant of logarithmic coefficients; subordination. https://doi.org/10.28924/2291-8639-21-2023-10 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-10 2 int. j. anal. appl. (2023), 21:10 we denote h as the class of schwarz functions ω (z) defined in e which satisfy ω (0) = 0 and |ω (z)| < 1, and has the series representation ω (z) = ∞∑ k=1 bkz k, z ∈ e. (1.3) if there exists a schwarz function ω (z) ∈ h such that f (z) = g (ω (z)) for all z ∈ e, then the analytic function f (z) is subordinate to another analytic function g (z) and is symbolically written as f (z) ≺ g (z) . furthermore, if g (z) is univalent in e, then f (z) ≺ g (z) ⇔ f (0) = g (0) and f (e) = g (e) . in [1], el-ashwah and thomas introduced the class of functions that are starlike with respect to conjugate points. the class is denoted by sc ∗ which satisfies re { 2zf ′ (z) f (z) + f (z) } > 0, z ∈ e. (1.4) in [2], halim introduced the class sc ∗ (δ) consisting of functions of the form (1.1) and satisfying re { 2zf ′ (z) f (z) + f (z) } > δ,0 6 δ < 1, z ∈ e. (1.5) by means of subordination, dahhar and janteng [3] introduced the class sc ∗ (a,b) consisting of functions of the form (1.1) and satisfying 2zf ′ (z) f (z) + f (z) ≺ 1 + az 1 + bz , − 1 6 b < a 6 1, z ∈ e. (1.6) from (1.6), it follows that f (z) ∈ sc∗ (a,b) if and only if 2zf ′ (z) f (z) + f (z) = 1 + aω (z) 1 + bω (z) , ω (z) ∈ h. (1.7) in [4], wahid et al. introduced the class sc ∗ (α,δ) consisting of functions of the form (1.1) and satisfying re { eiα zf ′ (z) g (z) } > δ, 0 6 δ < 1, |α| < π 2 , z ∈ e, (1.8) where g (z) = f (z)+f (z) 2 . the functions in the class sc ∗ (α,δ) are known as tilted starlike functions with respect to conjugate points of order δ. in terms of subordination, they defined the class sc ∗ (α,δ,a,b) which satisfies{ eiα zf ′ (z) g (z) −δ − i sin α } 1 tαδ ≺ 1 + az 1 + bz , − 1 6 b < a 6 1, z ∈ e, (1.9) where tαδ = cos α−δ. from (1.9), it follows that f (z) ∈ sc∗ (α,δ,a,b) if and only if{ eiα zf ′ (z) g (z) −δ − i sin α } 1 tαδ = 1 + aω (z) 1 + bω (z) , ω (z) ∈ h. (1.10) remark 1.1. it is observed that by choosing the specific values of the parameters α, δ, a and b in the class sc ∗ (α,δ,a,b) leads to the following classes: int. j. anal. appl. (2023), 21:10 3 (a) if we let α = δ = 0,a = 1 and b = −1, then the class sc∗ (α,δ,a,b) reduces to the class sc∗ given in (1.4). (b) if we let α = 0,a = 1 and b = −1, then the class sc∗ (α,δ,a,b) reduces to the class sc∗ (δ) given in (1.5). (c) if we replace α = δ = 0, then the class sc ∗ (α,δ,a,b) reduces to the class sc ∗ (a,b) given in (1.6). (d) if we replace a = 1 and b = −1, then the class sc∗ (α,δ,a,b) reduces to the class sc∗ (α,δ) given in (1.8). aside from that, many interesting results especially related to coefficients of f (z) ∈ s for various subclasses of starlike functions with respect to symmetric points, symmetric conjugate points and conjugate points were obtained by several authors. we may point interested readers to recent advances in these subclasses and their further results which point in a different direction than the current study, see for example, coefficient estimates [3,5,6], fekete-szegö inequality [7], hankel determinant [8–12], toeplitz determinant [13] and zalcman coefficient functional [8,14]. the logarithmic coefficients γn, n > 1 for a function f (z) ∈ s of the form (1.1) play an important role in milin’s conjecture [15,16], brennan’s conjecture [17] and can also be used to find estimations for the coefficients of an inverse function. it is given in the series representation ff (z) := log ( f (z) z ) = 2 ∞∑ n=1 γnz n, z ∈ e. (1.11) by differentiating (1.11) and comparing the coefficients of zn, the logarithmic coefficients γn, n = 1, 2, 3, 4 are given as follows: γ1 = 1 2 a2, (1.12) γ2 = 1 2 ( a3 − 1 2 a2 2 ) , (1.13) γ3 = 1 2 ( a4 −a2a3 + 1 3 a2 3 ) (1.14) and γ4 = 1 2 ( a5 −a2a4 + a22a3 − 1 2 a3 2 − 1 4 a2 4 ) . (1.15) the growth of the inequalities problems related to the upper bound of the hankel determinant has been studied for different subclasses of a. the hankel determinant was defined by pommerenke [18,19] for a function f (z) ∈ s of the form (1.1) which is given by hq,n (f ) = ∣∣∣∣∣∣∣∣∣∣∣ an an+1 ... an+q−1 an+1 an+2 ... an+q · · · · · · ... · · · an+q−1 an+q ... an+2q−2 ∣∣∣∣∣∣∣∣∣∣∣ , (1.16) 4 int. j. anal. appl. (2023), 21:10 where n, q ∈ n. it is very useful for example in the theory of singularities [20] and in the study of power series with integral coefficients. recently, kowalczyk and lecko [21,22] proposed the study of the hankel determinant whose elements are logarithmic coefficients of f (z) ∈ s which is given by hq,n (ff /2) = ∣∣∣∣∣∣∣∣∣∣∣ γn γn+1 ... γn+q−1 γn+1 γn+2 ... γn+q · · · · · · ... · · · γn+q−1 γn+q ... γn+2q−2 ∣∣∣∣∣∣∣∣∣∣∣ . (1.17) the results of hq,n (ff /2) broaden the knowledge of logarithmic coefficients for different subclasses of s. in particular, for values of q = 2, n = 1 and q = 2, n = 2, respectively, we have h2,1 (ff /2) = γ1γ3 −γ22 (1.18) and h2,2 (ff /2) = γ2γ4 −γ32. (1.19) the problem of finding the upper bounds of |γn| and |hq,n (ff /2)| has been considered for some subclasses of univalent functions. some significant contributions have been obtained recently to these problems; see for instance [8, 23–31]. however, as far as we know, no one has used the coefficients of logarithmic functions to obtain the bound for the second hankel determinant for the classes sc ∗, sc ∗ (δ), sc ∗ (a,b) and sc ∗ (α,δ). thus, in this paper, we continue the research dealing with the logarithmic coefficients and the hankel determinant of logarithmic coefficients for the class sc ∗ (α,δ,a,b) introduced in (1.9). our main aim is to obtain the upper bounds of the logarithmic coefficients |γn| , n = 1, 2, 3, 4 and the second hankel determinants of logarithmic coefficients, i.e., |h2,1 (ff /2)|and |h2,2 (ff /2)|. furthermore, we give several new consequences of our results based on the special choices of the involved parameters. 2. preliminary results in this section, we present some lemmas which will be used to prove our main results. lemma 2.1. ( [15]) let p (z) ∈ p of the form p (z) = 1 + ∞∑ n=1 pnz n. then |pn|6 2, n > 1. the inequality is sharp for the function p (z) = 1+z 1−z . lemma 2.2. ( [32]) let p (z) ∈ p of the form p (z) = 1 + ∞∑ n=1 pnz n and µ ∈c. then |pn −µpkpn−k|6 2max{1, |2µ− 1|} , 1 6 k 6 n− 1. if |2µ− 1|> 1, then the inequality is sharp for the function p (z) = 1+z 1−z or its rotations. if |2µ− 1| < 1, then the inequality is sharp for the function p (z) = 1+z n 1−zn or its rotations. int. j. anal. appl. (2023), 21:10 5 lemma 2.3. ( [33]) let p (z) ∈ p of the form p (z) = 1 + ∞∑ n=1 pnz n. then ∣∣ic13 −xc1c2 + v c3∣∣ 6 2 |i| + 2 |x − 2i| + 2 |i −x + v | , where i, x and v are real numbers. 3. main results in this section, we find the estimate for initial logarithmic coefficients for functions belonging to the class sc ∗ (α,δ,a,b) . furthermore, we obtain the upper bounds of the second hankel determinant of logarithmic coefficients for the case of q = 2 and n = 1, and q = 2 and n = 2 for functions from the class sc ∗ (α,δ,a,b) . theorem 3.1. if f (z) ∈ sc∗ (α,δ,a,b) and has the series representation (1.1), then |γ1|6 t 2 , |γ2|6 t 4 , |γ3|6 t 6 and |γ4|6 t (1 + 2υ) 8 , where t = (a−b)tαδ, tαδ = cos α−δ and υ = 1 + b. proof. let f (z) = z + ∞∑ n=2 anz n ∈ sc∗ (α,δ,a,b). the coefficients an, n = 2, 3, 4, 5 are given by [14] a2 = k1ξ 2 , (3.1) a3 = ξ 8 [ 2k2 + k1 2 (ξ− υ) ] , (3.2) a4 = ξ 48 [ 8k3 + k1k2 (6ξ− 8υ) + k13 ( ξ2 − 3υξ + 2υ2 )] (3.3) and a5 = ξ 384 [ 48k4 + k1k3 (32ξ− 48υ) + k22 (12ξ− 24υ) + k14 ( ξ3 − 6υξ2 + 11υ2ξ− 6υ3 ) +k1 2k2 ( 12ξ2 − 44υξ + 36υ2 )] , (3.4) where ξ = te−iα, t = (a−b)tαδ, tαδ = cos α−δ and υ = 1 + b. using (3.1)−(3.4), from (1.12)−(1.15), respectively, we obtain γ1 = k1ξ 4 , (3.5) γ2 = 1 2 [ ξ 8 ( 2k2 + k1 2 (ξ− υ) ) − k1 2ξ2 8 ] = ξ 16 ( 2k2 −k12υ ) , (3.6) 6 int. j. anal. appl. (2023), 21:10 γ3 = 1 2 [ ξ 48 ( 8k3 + k1k2 (6ξ− 8υ) + k13 ( ξ2 − 3υξ + 2υ2 )) − k1ξ 2 16 ( 2k2 + k1 2 (ξ− υ) ) + k1 3ξ3 24 ] = ξ 48 ( k1 3υ2 − 4k1k2υ + 4k3 ) (3.7) and γ4 = 1 2 [ ξ 384 ( 48k4 + k1k3 (32ξ− 48υ) + k22 (12ξ− 24υ) + k14 ( ξ3 − 6υξ2 + 11υ2ξ− 6υ3 ) +k1 2k2 ( 12ξ2 − 44υξ + 36υ2 )) − k1ξ 2 96 ( 8k3 + k1k2 (6ξ− 8υ) + k13 ( ξ2 − 3υξ + 2υ2 )) + k1 2ξ3 32 ( 2k2 + k1 2 (ξ− υ) ) − ξ2 128 ( 4k2 2 + 4k1 2k2 (ξ− υ) + k14(ξ− υ)2 ) − k1 4ξ4 64 ] = ξ 128 ( 8k4 − 4k22υ −k14υ3 + 6k12k2υ2 − 8k1k3υ ) . (3.8) for γ1, implementing lemma 2.1 in (3.5), we obtain |γ1|6 t 2 . for γ2, γ3 and γ4, we can write (3.6)−(3.8), respectively, as γ2 = ξ 8 ( k2 −µk12 ) , (3.9) γ3 = ξ 48 ( ik1 3 −xk1k2 + v k3 ) (3.10) and γ4 = ξ 128 ( 8 ( k4 −µk22 ) −k1 ( i∗k1 3 −x∗k1k2 + v ∗k3 )) , (3.11) where µ = υ 2 , i = υ2, x = 4υ, v = 4, i∗ = υ3, x∗ = 6υ2 and v ∗ = 8υ. implementing lemma 2.2 in (3.9), lemma 2.3 in (3.10) and both lemma 2.2 and lemma 2.3 in (3.11), and application of triangle inequality, respectively, we get |γ2|6 t 4 , |γ3|6 t 6 and |γ4|6 t (1 + 2υ) 8 . this completes the proof. � int. j. anal. appl. (2023), 21:10 7 theorem 3.2. if f (z) ∈ sc∗ (α,δ,a,b) and has the series representation (1.1), then |h2,1 (ff /2)|6 7t 2 48 , where t = (a−b)tαδ, tαδ = cos α−δ and υ = 1 + b. proof. using (3.5)−(3.7), from (1.18), we have h2,1 (ff /2) = k1ξ 2 192 ( k1 3υ2 − 4k1k2υ + 4k3 ) − ξ2 256 ( 4k2 2 − 4k12k2υ + k14υ2 ) . (3.12) hence, simplifying (3.12), we can write it as h2,1 (ff /2) = ξ2 768 ( k1 ( ik1 3 −xk1k2 + v ∗∗k3 ) − 12k22 ) , (3.13) where i = υ2, x = 4υ and v ∗∗ = 16. thus, applying lemma 2.1 and lemma 2.3, and by triangle inequality implies that |h2,1 (ff /2)|6 7t 2 48 . this completes the proof. � theorem 3.3. if f (z) ∈ sc∗ (α,δ,a,b) and has the series representation (1.1), then |h2,2 (ff /2)|6 t 2 ( 2υ2 + 9υ + 17 ) 288 , where t = (a−b)tαδ, tαδ = cos α−δ and υ = 1 + b. proof. substituting (3.9)−(3.11) in (1.19) and after simplification, we get h2,2 (ff /2) = ξ2 2048 ( 2k2 −k12υ )( 8k4 − 4k22υ −k14υ3 + 6k12k2υ2 − 8k1k3υ ) − ξ2 2304 ( k1 3υ2 − 4k1k2υ + 4k3 )2 = ξ2 18432 ( k1 6υ4 − 8k14k2υ3 + 8k13k3υ2 + 144k2k4 − 72k12k4υ − 72k23υ +16k1 2k2 2υ2 −128k32 + 112k1k2k3υ ) . (3.14) by rearranging the terms in (3.14), we may write h2,2 (ff /2) = ξ2 18432 [ k1 3υ2 ( ik1 3 −v ∗k1k2 + x∗k3 ) + 144k4 ( k2 −µk12 ) − 72k22υ ( k2 −νk12 ) −128k3 (k3 −ηk1k2)] , (3.15) where i = υ2, x∗ = 8, v ∗ = 8υ, µ = υ 2 , ν = 2υ 9 and η = 7υ 8 . thus, implementing lemma 2.2 and lemma 2.3, and by triangle inequality, (3.15) yields |h2,2 (ff /2)|6 t 2 ( 2υ2 + 9υ + 17 ) 288 . this completes the proof. � 8 int. j. anal. appl. (2023), 21:10 upon choosing the specific values of the parameters α, δ, a and b in theorem 3.1, theorem 3.2 and theorem 3.3, respectively, we get the following consequences: corollary 3.1. (a) let sc ∗ (0, 0, 1,−1) ≡ sc∗. then we have |γ1|6 1, |γ2|6 1 2 , |γ3|6 1 3 and |γ4|6 1 4 . (b) let sc ∗ (0,δ, 1,−1) ≡ sc∗ (δ) . then we have |γ1|6 (1 −δ) , |γ2|6 (1 −δ) 2 , |γ3|6 (1 −δ) 3 and |γ4|6 (1 −δ) 4 . (c) let sc ∗ (0, 0,a,b) ≡ sc∗ (a,b) . then we have |γ1|6 (a−b) 2 , |γ2|6 (a−b) 4 , |γ3|6 (a−b) 6 and |γ4|6 (a−b) (1 + 2υ) 8 . (d) let sc ∗ (α,δ, 1,−1) ≡ sc∗ (α,δ). then we have |γ1|6 tαδ, |γ2|6 tαδ 2 , |γ3|6 tαδ 3 and |γ4|6 tαδ 4 . corollary 3.2. (a) let sc ∗ (0, 0, 1,−1) ≡ sc∗. then we have |h2,1 (ff /2)|6 7 12 . (b) let sc ∗ (0,δ, 1,−1) ≡ sc∗ (δ) . then we have |h2,1 (ff /2)|6 7(1 −δ)2 12 . (c) let sc ∗ (0, 0,a,b) ≡ sc∗ (a,b) . then we have |h2,1 (ff /2)|6 7(a−b)2 48 . (d) let sc ∗ (α,δ, 1,−1) ≡ sc∗ (α,δ). then we have |h2,1 (ff /2)|6 7tαδ 2 12 . corollary 3.3. (a) let sc ∗ (0, 0, 1,−1) ≡ sc∗. then we have |h2,2 (ff /2)|6 17 72 . (b) let sc ∗ (0,δ, 1,−1) ≡ sc∗ (δ) . then we have |h2,2 (ff /2)|6 17(1 −δ)2 72 . (c) let sc ∗ (0, 0,a,b) ≡ sc∗ (a,b) . then we have |h2,2 (ff /2)|6 (a−b)2 ( 2υ2 + 9υ + 17 ) 288 . int. j. anal. appl. (2023), 21:10 9 (d) let sc ∗ (α,δ, 1,−1) ≡ sc∗ (α,δ). then we have |h2,2 (ff /2)|6 17tαδ 2 72 . 4. conclusion in this paper, we have obtained the upper bounds of the initial logarithmic coefficients and the second hankel determinant of logarithmic coefficients for functions from the class sc ∗ (α,δ,a,b). it is shown in corollaries that the obtained results lead to new results for some existing subclasses, i.e., sc ∗, sc ∗ (δ) , sc ∗ (a,b) and sc ∗ (α,δ). corollary 3.1(a) also coincides with the inequality |γn| 6 1n, n > 1 that holds for the well-known class of starlike functions s ∗. the results obtained could provide an opportunity for researchers to further investigate the third and fourth-order hankel determinants of logarithmic coefficients, including other inequalities problems related to logarithmic coefficients for this class as well as other subclasses of s. acknowledgment: this study was supported by a grant from universiti teknologi mara, grant number 600-rmc/gpm lphd 5/3 (060/2021). the authors sincerely thank the referees for their valuable comments. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.m. el-ashwah, d.k. thomas, some subclasses of close-to-convex functions, j. ramanujan math. soc. 2 (1987), 85-100. [2] s. abdul halim, functions starlike with respect to other points, int. j. math. math. sci. 14 (1991), 451–456. https://doi.org/10.1155/s0161171291000613. [3] s.a.f.m. dahhar, a. janteng, a subclass of starlike functions with respect to conjugate points, int. math. forum. 4 (2009), 1373-1377. [4] n.h.a.a. wahid, d. mohamad, s.c. soh, on a subclass of tilted starlike functions with respect to conjugate points, menemui mat. (discover. math.) 37 (2015), 1-6. [5] c. selvaraj, n. vasanthi, subclasses of analytic functions with respect to symmetric and conjugate points, tamkang j. math. 42 (2011), 87-94. https://doi.org/10.5556/j.tkjm.42.2011.87-94. [6] q.h. xu, g.p. wu, coefficient estimate for a subclass of univalent functions with respect to symmetric point, eur. j. appl. math. 3 (2010), 1055-1061. [7] t.n. shanmugam, c. ramachandram, v. ravichandran, fekete-szegö problem for subclasses of starlike functions with respect to symmetric points, bull. korean math. soc. 43 (2006), 589–598. https://doi.org/10.4134/ bkms.2006.43.3.589. [8] m. ghaffar khan, b. ahmad, g. murugusundaramoorthy, et al. third hankel determinant and zalcman functional for a class of starlike functions with respect to symmetric points related with sine function, j. math. computer sci. 25 (2021), 29-36. https://doi.org/10.22436/jmcs.025.01.04. [9] a.k. mishra, j.k. prajapat, s. maharana, bounds on hankel determinant for starlike and convex functions with respect to symmetric points, cogent math. 3 (2016), 1160557. https://doi.org/10.1080/23311835.2016. 1160557. https://doi.org/10.1155/s0161171291000613 https://doi.org/10.5556/j.tkjm.42.2011.87-94 https://doi.org/10.4134/bkms.2006.43.3.589 https://doi.org/10.4134/bkms.2006.43.3.589 https://doi.org/10.22436/jmcs.025.01.04 https://doi.org/10.1080/23311835.2016.1160557 https://doi.org/10.1080/23311835.2016.1160557 10 int. j. anal. appl. (2023), 21:10 [10] g. singh, g. singh, coefficient inequality for subclasses of starlike functions with respect to conjugate points, int. j. modern math. sci. 8 (2013), 48-56. [11] g. singh, hankel determinant for analytic functions with respect to other points, eng. math. lett. 2 (2013), 115-123. [12] n.h.a.a. wahid, d. mohamad, s.c. soh, second hankel determinant for a subclass of tilted starlike functions with respect to conjugate points, matematika, 31 (2015), 111-119. [13] n.h.a.a. wahid, d. mohamad, toeplitz determinant for a subclass of tilted starlike functions with respect to conjugate points, sains malays. 50 (2021), 3745-3751. http://doi.org/10.17576/jsm-2021-5012-23. [14] d. mohamed, n.h.a.a. wahid, zalcman coefficient functional for tilted starlike functions with respect to conjugate points, j. math. computer sci. 29 (2022), 40–51. https://doi.org/10.22436/jmcs.029.01.04. [15] p.l. duren, univalent functions, springer-verlag, new york, 1983. [16] i.m. milin, univalent functions and orthonormal systems, izdat. "nauka", moscow, 1971 (in russian); english transl. american mathematical society, providence, (1977). [17] i.p. kayumov, on brennan’s conjecture for a special class of functions, math notes. 78 (2005), 498–502. https://doi.org/10.1007/s11006-005-0149-1. [18] c. pommerenke, on the coefficients and hankel determinants of univalent functions, j. lond. math. soc. s1-41 (1966), 111–122. https://doi.org/10.1112/jlms/s1-41.1.111. [19] c. pommerenke, on the hankel determinants of univalent functions, mathematika. 14 (1967), 108–112. [20] p. dienes, the taylor series: an introduction to the theory of functions of a complex variable, dover, mineola, 1957. [21] b. kowalczyk, a. lecko, second hankel determinant of logarithmic coefficients of convex and starlike functions, bull. aust. math. soc. 105 (2021), 458–467. https://doi.org/10.1017/s0004972721000836. [22] b. kowalczyk, a. lecko, second hankel determinant of logarithmic coefficients of convex and starlike functions of order alpha, bull. malays. math. sci. soc. 45 (2021), 727–740. https://doi.org/10.1007/ s40840-021-01217-5. [23] d. alimohammadi, e. analouei adegani, t. bulboaca, et al. logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions, j. funct. spaces. 2021 (2021), 6690027. https: //doi.org/10.1155/2021/6690027. [24] v. allu, v. arora, a. shaji, on the second hankel determinant of logarithmic coefficients for certain univalent functions, mediterr. j. math. 20 (2023), 81. https://doi.org/10.1007/s00009-023-02272-x. [25] v. allu, v. arora, second hankel determinant of logarithmic coefficients of certain analytic functions, (2021). https://doi.org/10.48550/arxiv.2110.05161. [26] b. khan, i. aldawish, s. araci, m.g. khan, third hankel determinant for the logarithmic coefficients of starlike functions associated with sine function, fractal fract. 6 (2022), 261. https://doi.org/10.3390/ fractalfract6050261. [27] m. obradovic, s. ponnusamy, k.j. wirths, logarithmic coefficients and a coefficient conjecture for univalent functions, monatsh math. 185 (2017), 489–501. https://doi.org/10.1007/s00605-017-1024-3. [28] l. shi, m. arif, a. rafiq, et al. sharp bounds of hankel determinant on logarithmic coefficients for functions of bounded turning associated with petal-shaped domain, mathematics. 10 (2022), 1939. https://doi.org/10. 3390/math10111939. [29] k. tra.bka-wie.cław, on coefficient problems for functions connected with the sine function, symmetry. 13 (2021), 1179. https://doi.org/10.3390/sym13071179. [30] z. ye, the logarithmic coefficients of close-to-convex functions, bull. inst. math. acad. sin. 3 (2008), 445–452. http://doi.org/10.17576/jsm-2021-5012-23 https://doi.org/10.22436/jmcs.029.01.04 https://doi.org/10.1007/s11006-005-0149-1 https://doi.org/10.1112/jlms/s1-41.1.111 https://doi.org/10.1017/s0004972721000836 https://doi.org/10.1007/s40840-021-01217-5 https://doi.org/10.1007/s40840-021-01217-5 https://doi.org/10.1155/2021/6690027 https://doi.org/10.1155/2021/6690027 https://doi.org/10.1007/s00009-023-02272-x https://doi.org/10.48550/arxiv.2110.05161 https://doi.org/10.3390/fractalfract6050261 https://doi.org/10.3390/fractalfract6050261 https://doi.org/10.1007/s00605-017-1024-3 https://doi.org/10.3390/math10111939 https://doi.org/10.3390/math10111939 https://doi.org/10.3390/sym13071179 int. j. anal. appl. (2023), 21:10 11 [31] p. zaprawa, initial logarithmic coefficients for functions starlike with respect to symmetric points, bol. soc. mat. mex. 27 (2021), 62. https://doi.org/10.1007/s40590-021-00370-y. [32] i. efraimidis, a generalization of livingston’s coefficient inequalities for functions with positive real part, j. math. anal. appl. 435 (2016), 369–379. https://doi.org/10.1016/j.jmaa.2015.10.050. [33] m. arif, m. raza, h. tang, et al. hankel determinant of order three for familiar subsets of analytic functions related with sine function, open math. 17 (2019), 1615–1630. https://doi.org/10.1515/math-2019-0132. https://doi.org/10.1007/s40590-021-00370-y https://doi.org/10.1016/j.jmaa.2015.10.050 https://doi.org/10.1515/math-2019-0132 1. introduction 2. preliminary results 3. main results 4. conclusion references int. j. anal. appl. (2022), 20:62 solvability of the solution of superlinear hyperbolic dirichlet problem iqbal m. batiha1,2,∗ 1department of mathematics, al zaytoonah university of jordan, queen alia airport st 594, amman 11733, jordan 2nonlinear dynamics research center (ndrc), ajman university, ajman, uae ∗corresponding author: i.batiha@zuj.edu.jo abstract. in this paper, we aim to study the solutions of superlinear hyperbolic problems with boundary condition of dirichlet type where we show the existence and the uniqueness of the strong solutions for the superlinear problems by the method of energy inequality. 1. introduction and position of the problem the partial differential equations were probably formulated for the first time during the birth of rational mechanics in the 17th century [1–3]. then the catalog of partial differential equations (pdes) have been enriched as the science developed and in particular physics [4–7]. if we only have to remember a few names, we must cite that of euler, then those of navier and stokes, for the equations of fluid mechanics, those of fourier in the heat equation, maxwell for those of electromagnetism, schrodinger and heisenberg for the equations of quantum mechanics, and of course that of einstein for the pdes of the theory of relativity. a giant leap was made by l. schwartz when he gave birth to the theory of distributions (around the 1950s), and at least comparable progress is due to l. hormander for the development of pseudo differential calculus (in the early 1970s). the complexity of nonlinearity and challenges in their theoretical study in have attracted a lot of interest from many mathematicians and scientists see [8–11]. many natural phenomena and modern problems of physics, mechanics, biology, and technology can be modeled by nonlinear hyperbolic equations. the method used here is one of the most efficient received: sep. 30, 2022. 2010 mathematics subject classification. 35l03, 30c15. key words and phrases. nonlinear hyperbolic equation; energy inequality method; existence; uniqueness. https://doi.org/10.28924/2291-8639-20-2022-62 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-62 2 int. j. anal. appl. (2022), 20:62 functional analysis methods in solving partial differential equations, it is called a priori estimate method or the energy-integral method, see [10]. in this work, we study the solutions to hyperbolic problems with boundary conditions of dirichlet type where we show the existence and uniqueness of the strong solutions for semilinear problems by the method of energy inequality, where we found a difficulty in the choice of the multiplier, and the uniqueness which is emanating from a priori estimate. let t > 0, ω ⊂rn and q = ω × (0,t ) = { (x,t) ∈rn+1 : x ∈ ω, 0 < t < t } . we consider the nonlinear parabolic problem  utt −a∆u + b(x,t)ut + uq = f (x,t) u(x, 0) = ϕ(x), ut(x, 0) = ψ(x), u(x,t) |γ= 0 , (p1) in which the nonlinear parabolic equation is given as follows lu = utt −a∆u + b(x,t)ut + uq = f (x,t), (1.1) with the initial condition lu = u(x, 0) = ϕ(x), (1.2) and the dirichlet boundary conditions u(x,t) |γ= 0, ∀t ∈ (0,t ), (1.3) where a,q are positive odd integers, p ≥ 1, and where f (x,t), ϕ(x) and ψ(x) are given functions and b(x,t) satisfies the following assumption: a1. b1 ≤ b(x,t) ≤ b0, (x,t) ∈ q̄. we establish a priori bound and prove the existence of a solution of problem (1.1)-(1.3). to this aim, let lu = f, where l = (lf, l1, l2), and f = (f ,ϕ,ψ) be the operator equation corresponding to problem (1.1)-(1.3). the operator l acts from e to f, and the banach space e consists of all functions u(x,t) with the finite norm ‖u‖2e = max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + max 0≤τ≤t ‖∇u‖2l2(ω) + ‖ut‖ 2 l2(q) + max 0≤τ≤t ‖u(x,τ)‖q+1 lq+1(ω) . (1.4) the hilbert space f consists of the vector valued functions f = (f ,u0) with the norm ‖f‖2f = ‖f‖ 2 l2(q) + ‖ψ‖ 2 l2(ω) + ‖ϕx‖ 2 l2(ω) + ‖ϕ‖ q+1 lq+1(ω) . (1.5) the associated inner product is given as (f,g)f = (f ,g)l2(q) + ( ϕx, (g0 )x ) l2(ω) + (ψ,g1)l2(ω) . (1.6) int. j. anal. appl. (2022), 20:62 3 we assume that the data functions ϕ and ψ satisfy the conditions of the form (1.3), i.e., ϕ |γ= ψ |γ= 0. at the upcoming section, we intend to establish a priori estimate for the solution of problem (1.1)(1.3). 2. a priori bound in the theory of pdes, an a priori estimate (also called an apriori estimate or a priori bound) is an estimate for the size of a solution or its derivatives of a pde. a priori is latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. one reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem. some important definitions and theorems will be next listed in this section. theorem 2.1. if assumption a1 is satisfied, then for any function u ∈ d(l), there exists a positive constant c independent of u such that max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + max 0≤τ≤t ‖∇u‖2l2(ω) + ‖ut‖ 2 l2(q) + max 0≤τ≤t ‖u(x,τ)‖q+1 lq+1(ω) ≤ c ( ‖f‖2l2(q) + ‖ψ‖ 2 l2(ω) + ‖ϕx‖ 2 l2(ω) + ‖ϕ‖ q+1 lq+1(ω) ) , (2.1) and d(l) is the domain of definition of the operator l defined by d(l) = {u : u ∈ l∞ ( 0,t,lq+1(ω) ) , ut ∈ l∞ ( 0,t,l2 (ω) ) } satisfying condition (1.3). proof. taking the scalar product in l2(q) of eq. (1.1) and the operator mu = ut, where qτ = ω × (0,t ), yields (lu,mu)l2(qτ ) = (utt,ut)l2(qτ ) −a (∆u,ut)l2(qτ ) + (but,ut)l2(qτ ) + (u q,ut)l2(qτ ) = (f ,ut)l2(qτ ) . (2.2) the successive integration by parts of integrals on the right-hand side of (2.2) gives (utt,ut)l2(qτ ) = ∫ qτ utt ·utdxdt = 1 2 ∫ ω u2t dx − 1 2 ∫ ω ψ2dx = 1 2 ‖ut(x,τ)‖2l2(ω) − 1 2 ‖ψ‖2l2(ω) , (2.3) 4 int. j. anal. appl. (2022), 20:62 besides we have −a (∆u,u)l2(qτ ) = −a ∫ qτ ∆u ·utdxdt = a ∫ ω ∇u2dx − 1 2 ∫ ω ϕ2xdx = a‖∇u‖2l2(ω) − 1 2 ‖ϕx‖2l2(ω) , (2.4) and (bu,u)l2(qτ ) = ∫ qτ b(x,t)u2t dxdt. (2.5) in this regard, we have (cuq,ut)l2(qτ ) = 1 q + 1 ∫ ω uq+1dx − 1 q + 1 ∫ ω ϕq+1dx = 1 q + 1 ‖ut(x,τ)‖ q+1 lq+1(ω) − 1 q + 1 ‖ϕ‖q+1 lq+1(ω) . (2.6) by substituting (2.3)-(2.6) into (2.2), we obtain 1 2 ‖ut(x,τ)‖2l2(ω) − 1 2 ‖ψ‖2l2(ω) + a‖∇u‖ 2 l2(ω) − 1 2 ‖ϕx‖2l2(ω) + ∫ qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 lq+1(ω) − 1 q + 1 ‖ϕ‖q+1 lq+1(ω) = (f ,ut) . (2.7) by applying cauchy inequality with ε, ( i.e., |ab| ≤ a2 2ε + εb2 2 ) , we can estimate the last term on the right-hand side of (2.7) and get 1 2 ‖ut(x,τ)‖2l2(ω) + a‖∇u‖ 2 l2(ω) + ∫ qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 lq+1(ω) ≤ 1 2ε ‖f‖2l2(qτ ) + ε 2 ‖u‖2l2(qτ ) + 1 2 ‖ψ‖2l2(ω) + 1 2 ‖ϕx‖2l2(ω) + 1 q + 1 ‖ϕ‖q+1 lq+1(ω) . by using assumptions a1 and using the gronwall’s lemma, the estimate (2.8) becomes ‖ut(x,τ)‖2l2(ω) + ‖∇u‖ 2 l2(ω) + ∫ qτ u2t dxdt + ‖u(x,τ)‖ q+1 lq+1(ω) ≤ max { 1 2 , 1 2ε ,b0, 1 q + 1 } min { 1 2 ,a,b1, 1 q + 1 } exp (ε 2 t ) × [ ‖f‖2l2(q) + ‖ψ‖ 2 l2(ω) + ‖ϕx‖ 2 l2(ω) + ‖ϕ‖ q+1 lq+1(ω) ] . then, by passing to the maximum, we get max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + max 0≤τ≤t ‖∇u‖2l2(ω) + ‖ut‖ 2 l2(q) + max 0≤τ≤t ‖u(x,τ)‖q+1 lq+1(ω) ≤ c [ ‖f‖2l2(q) + ‖ψ‖ 2 l2(ω) + ‖ϕx‖ 2 l2(ω) + ‖ϕ‖ q+1 lq+1(ω) ] , int. j. anal. appl. (2022), 20:62 5 where c = max { 1 2 , 1 2ε ,b0, 1 q + 1 } min { 1 2 ,a,b1, 1 q + 1 } exp (ε 2 t ) . so, we have ‖u‖e ≤ √ c ‖lu‖f . (2.8) � now, we let r(l) be the range of the operator l. since we do not have any information about r(l), except that r(l) ⊂ f , we must extend l so that estimate (1.6) holds for this extension and its range represents the whole space f. for this purpose, we present the next proposition. proposition 2.1. the operator l : e −→ f has a closure. proof. let (un)n∈n ⊂ d (l) be a sequence where un −→ 0 in e, and lun −→ (f ; ϕx,ψ) in f. (2.9) now, we must prove that f ≡ 0 and (ϕ,ψ) ≡ (0, 0) . the convergence of un to 0 in e drives: un −→ 0 in d′ (q) . (2.10) according to the continuity of the derivation of d′ (q) in d′ (q) and the continuity the distribution of the function uq, the relation (2.10) involve lun −→ 0 in d′ (q) . (2.11) moreover, the convergence of lun to f in l2 (q) gives: lun −→ f in d′ (q) . (2.12) as we have the uniqueness of the limit in d′ (q), we conclude from (2.11) and (2.12) that f = 0. then it is generated from (2.9) that l1un −→ ϕx and l2un −→ ψ in l2 (ω) . 6 int. j. anal. appl. (2022), 20:62 on the other hand, we have ‖u‖2e = max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + max 0≤τ≤t ‖∇u‖2l2(ω) + ‖ut‖ 2 l2(q) + max 0≤τ≤t ‖u(x,τ)‖q+1 lq+1(ω) ≥‖ux (x, 0)‖2l2(ω) + ‖ut(x, 0)‖ 2 l2(ω) ≥‖ϕx‖2l2(ω) + ‖ψ‖ 2 l2(ω) . now, due to un −→ 0 in e, then ‖u‖2e −→ 0 in r. consequently, we get 0 ≥‖ϕx‖2l2(ω) + ‖ψ‖ 2 l2(ω) . then, we obtain ϕx = 0 and ψ = 0. let l be the closure of this operator with the domain of definition d(l), and hence the result holds. � definition 2.1. a solution of the operator equation l̄u = f is called a strong solution to problem (1.1)-(1.3). the priori estimate (2.1) can be then extended to strong solution, i.e., we have the estimate max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + max 0≤τ≤t ‖∇u‖2l2(ω) + ‖ut‖ 2 l2(q) + max 0≤τ≤t ‖u(x,τ)‖q+1 lq+1(ω) ≤ c ( ‖f‖2l2(q) + ‖ψ‖ 2 l2(ω) + ‖ϕx‖ 2 l2(ω) + ‖ϕ‖ q+1 lq+1(ω) ) , ∀u ∈ d(l̄). (2.13) in light of the estimate given in (2.13), we can infer the next theoretical results. corollary 2.1. the range r(l̄) of the operator l̄ is closed in f and is equal to the closure r(l) of r(l), i.e. r(l̄) = r(l). proof. let z ∈ r(l) such that there is a cauchy sequence (zn)n∈n in f constituted of the elements of the set r(l) such as lim n−→+∞ zn = z. there is then a corresponding sequence un ∈ d(l) such as zn = lun. immediately, the estimate (2.8) becomes: ‖up −uq‖e ≤ c‖lup −luq‖f → 0, where p and q tend towards infinity. we can consequently deduce that (un)n∈n is a cauchy sequence in e. so like e is a banach space, it exists u ∈ e such as lim n−→+∞ un = u in e. by virtue of the definition of l̄ ( lim n−→+∞ un = u in e, if lim n−→+∞ lun = lim n−→+∞ zn = z, and then lim n−→+∞ l̄un = z as l̄ is closed, and so l̄u = z), the function u satisfies: u ∈ d ( l̄ ) , l̄u = z. int. j. anal. appl. (2022), 20:62 7 then z ∈ r(l̄), and so r(l) ⊂ r(l̄). also, we conclude here that r(l̄) is closed because it is banach (any complete subspace of a metric space, not necessarily complete, is closed). thus, it remains to show the reverse inclusion either z ∈ r(l̄), and then it exists a cauchy sequence (zn)n∈n in f constituted of the elements of the set r(l̄) such that lim n−→+∞ zn = z, or z ∈ r(l̄) because r(l̄) is closed subset. so r(l̄) is complete. there is then a corresponding sequence un ∈ d(l̄) such that l̄un = zn. consequently from (2.8), we get ‖up −uq‖e ≤ c ∥∥l̄up − l̄uq∥∥f → 0, where p and q tend towards infinity. we can immediately deduce that (un)n∈n is a cauchy sequence in e, and so like e is a banach space, it exists u ∈ e such as lim n−→+∞ un = u in e. once again, there is a corresponding sequel (lun)n∈n ⊂ r(l) such as l̄un = lun on r (l) ,∀n ∈n. so we have lim n−→+∞ lun = z and consequently z ∈ r (l), which implies r ( l̄ ) ⊂ r (l). � 3. existence and uniqueness of solution in this section, additional results are listed below, which are related to the existence and uniqueness of strong solution for the main problem (p1). theorem 3.1. let assumption a1 be satisfied. then for all f = (f ,ϕ) ∈ f, there exists a unique strong solution u = l̄−1f = l−1f of problem (1.1)-(1.3). proof. to prove this result, we should note that we first have (lu,w )f = ∫ q lu.wdxdt + ∫ ω l1u.w0dx + ∫ ω l2u.w1dx, (3.1) where w = (w,w0,w1). so for w ∈ l2 (q) and for all u ∈ d0(l) = {u, u ∈ d (l) : l1u = 0, l2u = 0} , we have ∫ q lu.wdxdt = 0. by putting w = ut, we obtain∫ qτ uttut + ∫ qτ b(x,t)u2t dxdt + ∫ qτ uq+1dxdt = a ∫ qτ ∆u.ut 1 2 ‖ut(x,t)‖2l2(ω) + ∫ qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 lq+1(ω) = −a‖∇u‖2l2(ω) . 8 int. j. anal. appl. (2022), 20:62 this gives 1 2 ‖ut(x,t)‖2l2(ω) + ∫ qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 lq+1(ω) ≤ 0, max 0≤τ≤t ‖uτ (x,τ)‖2l2(ω) + b1 ∫ qτ u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 lq+1(ω) ≤ 0. therefore, we have ut = w = 0. since the range of the trace operators is everywhere dense in the hilbert space f with the associate norms ‖ϕx‖l2(ω) and ‖ψ‖l2(ω) , then the equality (3.1) implies that ω0 = 0 and ω1 = 0. hence w = 0 implies r(l) = f. � corollary 3.1. if for any function u ∈ d(l), we have the following estimate: ‖u‖e ≤ √ c ‖f‖f , then the solution of the problem (p1), if it exists, is unique. proof. let u1 and u2 be two solutions of problem (p1), i.e.,{ lu1 = f lu2 = f =⇒ lu1 −lu2 = 0. as l is linear, we then obtain l (u1 −u2) = 0. according to (2.8), we obtain ‖u1 −u2‖2e ≤ c ‖0‖ 2 f = 0, which gives u1 = u2. � 4. conclusion we have used the method of energy inequality for the super liner problems to show the existence and the uniqueness of the solution. in addition, we have studied the solution of superlinear hyperbolic problems with boundary condition of dirichlet type. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] g. bahia, a. ouannas, i.m. batiha, z. odibat, the optimal homotopy analysis method applied on nonlinear timefractional hyperbolic partial differential equations, numer. methods partial differ. equ. 37 (2020), 2008–2022. https://doi.org/10.1002/num.22639. [2] t.e. oussaeif, b. antara, a. ouannas, et al. existence and uniqueness of the solution for an inverse problem of a fractional diffusion equation with integral condition, j. funct. spaces. 2022 (2022), 7667370. https: //doi.org/10.1155/2022/7667370. [3] z. chebana, t.e. oussaeif, a. ouannas, i. batiha, solvability of dirichlet problem for a fractional partial differential equation by using energy inequality and faedo-galerkin method, innov. j. math. 1 (2022), 34–44. https://doi. org/10.55059/ijm.2022.1.1/4. https://doi.org/10.1002/num.22639 https://doi.org/10.1155/2022/7667370 https://doi.org/10.1155/2022/7667370 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.55059/ijm.2022.1.1/4 int. j. anal. appl. (2022), 20:62 9 [4] r.b. albadarneh, a.k. alomari, n. tahat, i.m. batiha, analytic solution of nonlinear singular bvp with multiorder fractional derivatives in electrohydrodynamic flows, twms j. appl. eng. math. 11 (2021), 1125-1137. https://hdl.handle.net/11729/3261. [5] i.m. batiha, z. chebana, t.-e. oussaeif, a. ouannas, i.h. jebril, on a weak solution of a fractional-order temporal equation, math. stat. 10 (2022), 1116–1120. https://doi.org/10.13189/ms.2022.100522. [6] n. anakira, z. chebana, t.e. oussaeif, i.m. batiha, a. ouannas, a study of a weak solution of a diffusion problem for a temporal fractional differential equation, nonlinear funct. anal. appl. 27 (2022), 679–689. https: //doi.org/10.22771/nfaa.2022.27.03.14. [7] t. hamadneh, a. zraiqat, h. al-zoubi, m. elbes, sufficient conditions and bounding properties for control functions using bernstein expansion, appl. math. inf. sci. 14 (2020), 1-9. [8] o. taki-eddine, b. abdelfatah, a priori estimates for weak solution for a time-fractional nonlinear reactiondiffusion equations with an integral condition, chaos solitons fractals. 103 (2017), 79–89. https://doi.org/ 10.1016/j.chaos.2017.05.035. [9] t.e. oussaeif, a. bouziani, solvability of nonlinear viscosity equation with a boundary integral condition, j. nonlinear evol. equ. appl. 3 (2015), 31-45. [10] t.e. oussaeif, a. bouziani, solvability of nonlinear goursat type problem for hyperbolic equation with integral condition, khayyam j. math. 4 (2018), 198–213. https://doi.org/10.22034/kjm.2018.65161. [11] s. dhelis, a. bouziani and t.-e. oussaeif, study of solution for a parabolic integro-differential equation with the second kind integral condition, int. j. anal. appl. 16 (2018), 569-593. https://doi.org/10.28924/ 2291-8639-16-2018-569. [12] t.e. oussaeif, a. bouziani, existence and uniqueness of solutions to parabolic fractional differential equations with integral conditions, electron. j. differ. equ. 2014 (2014), 179. https://hdl.handle.net/11729/3261 https://doi.org/10.13189/ms.2022.100522 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.22771/nfaa.2022.27.03.14 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.22034/kjm.2018.65161 https://doi.org/10.28924/2291-8639-16-2018-569 https://doi.org/10.28924/2291-8639-16-2018-569 1. introduction and position of the problem 2. a priori bound 3. existence and uniqueness of solution 4. conclusion references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 45-55 http://www.etamaths.com inequalities for co-ordinated m−convex functions via riemann-liouville fractional integrals çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 abstract. in this paper, we prove some new inequalities of hadamard-type for m−convex functions on the co-ordinates via riemann-liouville fractional integrals. 1. introduction let f : i ⊆ r → r be a convex function defined on the interval i of real numbers and a < b. the following double inequality; f ( a + b 2 ) ≤ 1 b−a b∫ a f(x)dx ≤ f(a) + f(b) 2 is well known in the literature as hadamard’s inequality. both inequalities hold in the reversed direction if f is concave. in [7], dragomir defined convex functions on the co-ordinates as following: definition 1. let us consider the bidimensional interval ∆ = [a,b] × [c,d] in r2 with a < b, c < d. a function f : ∆ → r will be called convex on the coordinates if the partial mappings fy : [a,b] → r, fy(u) = f(u,y) and fx : [c,d] → r, fx(v) = f(x,v) are convex where defined for all y ∈ [c,d] and x ∈ [a,b]. recall that the mapping f : ∆ → r is convex on ∆ if the following inequality holds, f(λx + (1 −λ)z,λy + (1 −λ)w) ≤ λf(x,y) + (1 −λ)f(z,w) for all (x,y), (z,w) ∈ ∆ and λ ∈ [0, 1]. in [7], dragomir established the following inequalities of hadamard’s type for co-ordinated convex functions on a rectangle from the plane r2. 2000 mathematics subject classification. 26a15, 26a51, 26d10. key words and phrases. riemann-liouville fractional integrals, co-ordinates, m−convex functions. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 45 46 çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 theorem 1. suppose that f : ∆ = [a,b] × [c,d] → r is convex on the co-ordinates on ∆. then one has the inequalities; f ( a + b 2 , c + d 2 ) (1.1) ≤ 1 2 [ 1 b−a ∫ b a f ( x, c + d 2 ) dx + 1 d− c ∫ d c f ( a + b 2 ,y ) dy ] ≤ 1 (b−a)(d− c) ∫ b a ∫ d c f(x,y)dxdy ≤ 1 4 [ 1 (b−a) ∫ b a f(x,c)dx + 1 (b−a) ∫ b a f(x,d)dx + 1 (d− c) ∫ d c f(a,y)dy + 1 (d− c) ∫ d c f(b,y)dy ] ≤ f(a,c) + f(a,d) + f(b,c) + f(b,d) 4 . the above inequalities are sharp. similar results can be found in [7]-[12]. in [17], toader defined m−convex functions as following: definition 2. the function f : [0,b] → r, b > 0 is said to be m−convex, where m ∈ [0, 1], if we have f(tx + m(1 − t)y) ≤ tf(x) + m(1 − t)f(y) for all x,y ∈ [0,b] and t ∈ [0, 1]. denote by km(b) the class of all m−convex functions on [0,b] for which f(0) ≤ 0. obviously, if we choose m = 1, we have ordinary convex functions on [0,b]. in [10], özdemir et al. defined co-ordinated m−convex functions as following: definition 3. let us consider the bidimensional interval ∆ = [0,b] × [0,d] in [0,∞)2. the mapping f : ∆ → r is m−convex on ∆ if f(tx + (1 − t)z,ty + m(1 − t)w) ≤ tf(x,y) + m(1 − t)f(z,w) holds for all (x,y), (z,w) ∈ ∆ and t ∈ [0, 1], b,d > 0 and for some fixed m ∈ [0, 1]. in [16], sarıkaya et al. proved some hadamard’s type inequalities for co-ordinated convex functions as followings: theorem 2. let f : ∆ ⊂ r2 → r be a partial differentiable mapping on ∆ := [a,b] × [c,d] in r2 with a < b and c < d. if ∣∣∣ ∂2f∂t∂s∣∣∣ is a convex function on the co-ordinates on ∆, then one has the inequalities: (1.2) ∣∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + 1(b−a)(d− c) ∫ b a ∫ d c f(x,y)dxdy −a ∣∣∣∣∣ ≤ (b−a)(d− c) 16   ∣∣∣ ∂2f∂t∂s∣∣∣ (a,c) + ∣∣∣ ∂2f∂t∂s∣∣∣ (a,d) + ∣∣∣ ∂2f∂t∂s∣∣∣ (b,c) + ∣∣∣ ∂2f∂t∂s∣∣∣ (b,d) 4   some fractional integral inequalities 47 where a = 1 2 [ 1 (b−a) ∫ b a [f(x,c) + f(x,d)] dx + 1 (d− c) ∫ d c [f(a,y)dy + f(b,y)] dy ] . theorem 3. let f : ∆ ⊂ r2 → r be a partial differentiable mapping on ∆ := [a,b] × [c,d] in r2 with a < b and c < d. if ∣∣∣ ∂2f∂t∂s∣∣∣q , q > 1, is a convex function on the co-ordinates on ∆, then one has the inequalities: (1.3)∣∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + 1(b−a)(d− c) ∫ b a ∫ d c f(x,y)dxdy −a ∣∣∣∣∣ ≤ (b−a)(d− c) 4 (p + 1) 2 p   ∣∣∣ ∂2f∂t∂s∣∣∣q (a,c) + ∣∣∣ ∂2f∂t∂s∣∣∣q (a,d) + ∣∣∣ ∂2f∂t∂s∣∣∣q (b,c) + ∣∣∣ ∂2f∂t∂s∣∣∣q (b,d) 4   1 q where a = 1 2 [ 1 (b−a) ∫ b a [f(x,c) + f(x,d)] dx + 1 (d− c) ∫ d c [f(a,y)dy + f(b,y)] dy ] and 1 p + 1 q = 1. theorem 4. let f : ∆ ⊂ r2 → r be a partial differentiable mapping on ∆ := [a,b] × [c,d] in r2 with a < b and c < d. if ∣∣∣ ∂2f∂t∂s∣∣∣q , q ≥ 1, is a convex function on the co-ordinates on ∆, then one has the inequalities: (1.4)∣∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + 1(b−a)(d− c) ∫ b a ∫ d c f(x,y)dxdy −a ∣∣∣∣∣ ≤ (b−a)(d− c) 16   ∣∣∣ ∂2f∂t∂s∣∣∣q (a,c) + ∣∣∣ ∂2f∂t∂s∣∣∣q (a,d) + ∣∣∣ ∂2f∂t∂s∣∣∣q (b,c) + ∣∣∣ ∂2f∂t∂s∣∣∣q (b,d) 4   1 q where a = 1 2 [ 1 (b−a) ∫ b a [f(x,c) + f(x,d)] dx + 1 (d− c) ∫ d c [f(a,y)dy + f(b,y)] dy ] . we give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper. definition 4. let f ∈ l1[a,b]. the riemann-liouville integrals jαa+f and j α b− f of order α > 0 with a ≥ 0 are defined by jαa+f(x) = 1 γ(α) ∫ x a (x− t)α−1f(t)dt, x > a and jαb−f(x) = 1 γ(α) ∫ b x (t−x)α−1f(t)dt, x < b where γ(α) = ∫∞ 0 e−tuα−1du, here is j0 a+ f(x) = j0 b− f(x) = f(x). 48 çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 in the case of α = 1, the fractional integral reduces to the classical integral. properties of this operator can be found in the references [3]-[?]. throughout of this paper, we will use the following notation: b = γ (α + 1) γ (β + 1) 4 (b−a)α (d− c)β [ j α,β b−,d− f (a,c) + j α,β a+,d− f (b,c) + j α,β b−,c+ f (a,d) + j α,β a+,c+ f (b,d) ] − γ (β + 1) 4 (d− c)β [ j β d− f (a,c) + j β d− f (b,c) + j β c+ f (b,d) + j β c+ f (a,d) ] − γ (α + 1) 4 (b−a)α [jαb−f (a,d) + j α b−f (a,c) + j α a+f (b,d) + j α a+f (b,c)] where j α,β b−,d− f (a,c) = 1 γ (α) γ (β) b∫ a d∫ c (x−a)α−1 (y − c)β−1 f (x,y) dydx j α,β a+,d− f (b,c) = 1 γ (α) γ (β) b∫ a d∫ c (x−a)α−1 (d−y)β−1 f (x,y) dydx j α,β b−,c+ f (a,d) = 1 γ (α) γ (β) b∫ a d∫ c (b−x)α−1 (y − c)β−1 f (x,y) dydx j α,β a+,c+ f (b,d) = 1 γ (α) γ (β) b∫ a d∫ c (b−x)α−1 (d−y)β−1 f (x,y) dydx. the main purpose of this paper is to establish inequalities of hadamard-type inequalities for m−convex functions on the co-ordinates via riemann-liouville fractional integrals by using a new lemma and fairly elemantery analysis. 2. main results to prove our main result, we need the following lemma: lemma 1. let f : ∆ = [a,b] × [c,d] → r be a twice partial differentiable mapping on ∆ = [a,b] × [c,d] . if ∂ 2f ∂t∂s ∈ l (∆) and α,β > 0, a,c ≥ 0, then the following equality holds: (2.1) f(a,c) + f(a,d) + f(b,c) + f(b,d) 4 + b = (b−a) (d− c) 4 1∫ 0 1∫ 0 [(1 − t)α − tα] [ (1 −s)β −sβ ] ∂2f ∂t∂s (ta + (1 − t) b,sc + (1 −s) d) dsdt. some fractional integral inequalities 49 proof. integration by parts, we can write k = 1∫ 0 1∫ 0 [(1 − t)α − tα] [ (1 −s)β −sβ ] ∂2f ∂t∂s (ta + (1 − t) b,sc + (1 −s) d) dsdt = 1∫ 0 [ (1 −s)β −sβ ] 1∫ 0 (1 − t)α ∂2f ∂t∂s (ta + (1 − t) b,sc + (1 −s) d) dt − 1∫ 0 tα ∂2f ∂t∂s (ta + (1 − t) b,sc + (1 −s) d) dt  ds = 1 b−a   1∫ 0 [ (1 −s)β −sβ ][∂f ∂s (b,sc + (1 −s) d) + ∂f ∂s (a,sc + (1 −s) d) −α 1∫ 0 (1 − t)α−1 ∂f ∂s (ta + (1 − t) b,sc + (1 −s) d) dt −α 1∫ 0 tα−1 ∂f ∂s (ta + (1 − t) b,sc + (1 −s) d) dt  ds   . by integrating again, we get k = 1 (b−a)(d− c) {f(a,c) + f(a,d) + f(b,c) + f(b,d) −β 1∫ 0 (1 −s)β−1 f (b,sc + (1 −s) d) ds−β 1∫ 0 sβ−1f (a,sc + (1 −s) d) ds −β 1∫ 0 (1 −s)β−1 f (a,sc + (1 −s) d) ds−β 1∫ 0 sβ−1f (b,sc + (1 −s) d) ds −α 1∫ 0 (1 − t)α−1 f (ta + (1 − t) b,d) dt−α 1∫ 0 tα−1f (ta + (1 − t) b,d) dt −α 1∫ 0 (1 − t)α−1 f (ta + (1 − t) b,c) dt−α 1∫ 0 tα−1f (ta + (1 − t) b,c) dt +αβ 1∫ 0 1∫ 0 (1 − t)α−1 (1 −s)β−1 f (ta + (1 − t) b,sc + (1 −s) d) dsdt +αβ 1∫ 0 1∫ 0 tα−1 (1 −s)β−1 f (ta + (1 − t) b,sc + (1 −s) d) dsdt 50 çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 +αβ 1∫ 0 1∫ 0 (1 − t)α−1 sβ−1f (ta + (1 − t) b,sc + (1 −s) d) dsdt +αβ 1∫ 0 1∫ 0 tα−1sβ−1f (ta + (1 − t) b,sc + (1 −s) d) dsdt   . by using the change of the variables, we can get x = ta + (1 − t) b and y = sc + (1 −s) d, that is t = x− b a− b and s = y −d c−d . taking into account these equalities, we obtain (2.2) k = 1 (b−a)(d− c) {f(a,c) + f(a,d) + f(b,c) + f(b,d) − β (d− c)β−1   d∫ c (y − c)β−1 f (a,y) dy + d∫ c (d−y)β−1 f (a,y) dy + d∫ c (y − c)β−1 f (b,y) dy + d∫ c (d−y)β−1 f (b,y) dy   − α (b−a)α−1   b∫ a (x−a)α−1 f (x,d) dx + b∫ a (x−a)α−1 f (x,c) dx + b∫ a (b−x)α−1 f (x,d) dx + b∫ a (b−x)α−1 f (x,c) dx   + αβ (b−a)α−1 (d− c)β−1 ×   b∫ a d∫ c (x−a)α−1 (y − c)β−1 f (x,y) dydx + b∫ a d∫ c (x−a)α−1 (d−y)β−1 f (x,y) dydx + b∫ a d∫ c (b−x)α−1 (y − c)β−1 f (x,y) dydx + b∫ a d∫ c (b−x)α−1 (d−y)β−1 f (x,y) dydx     . multiplying both sides of (2.2) by (b−a)(d−c) 4 and using the riemann-liouville integrals, we obtain equality (2.1). this completes the proof. theorem 5. let f : ∆ = [0,b] × [0,d] → r be a partial differentiable mapping on ∆ and ∂ 2f ∂t∂s ∈ l (∆), α,β > 0. if ∣∣∣ ∂2f∂t∂s∣∣∣ is m−convex function on the co-ordinates some fractional integral inequalities 51 on ∆ where 0 ≤ a < b < ∞ and 0 ≤ c < d < ∞, then the following inequality holds;∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 mαmβ × (∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣ + ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣ + m ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣ + m ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣ ) where mα = [ 1 α + 1 − ( 1 2 )α α + 1 ] mβ = [ 1 β + 1 − ( 1 2 )β β + 1 ] . proof. from lemma 1 and using the property of modulus, we have∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 1∫ 0 1∫ 0 |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣ ∣∣∣∣ ∂2f∂t∂s (ta + (1 − t) b,sc + (1 −s) d) ∣∣∣∣dsdt. since ∣∣∣ ∂2f∂t∂s∣∣∣ is co-ordinated m−convex, we can write∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 1∫ 0 1∫ 0 ∣∣∣(1 −s)β −sβ∣∣∣ |(1 − t)α − tα|{ts∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣ + mt(1 −s) ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣ +(1 − t)s ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣ + m(1 − t)(1 −s) ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣ } dtds by computing these integrals, we obtain∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 [ 1 α + 1 − ( 1 2 )α α + 1 ] × 1∫ 0 ∣∣∣(1 −s)β −sβ∣∣∣(s∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣ + m(1 −s) ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣ +s ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣ + m(1 −s) ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣ ) ds. 52 çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 using co-ordinated m−convexity of ∣∣∣ ∂2f∂t∂s∣∣∣ again, we get∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 [ 1 α + 1 − ( 1 2 )α α + 1 ][ 1 β + 1 − ( 1 2 )β β + 1 ] × (∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣ + ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣ + m ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣ + m ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣ ) thus, the proof is completed. remark 1. suppose that all the assumptions of theorem 5 are satisfied. if we choose α = β = m = 1, we obtain the inequality (1.2) . theorem 6. let f : ∆ → r be a partial differentiable mapping on ∆ and ∂ 2f ∂t∂s ∈ l (∆), α,β ∈ (0, 1]. if ∣∣∣ ∂2f∂t∂s∣∣∣q , q > 1, is m−convex function on the co-ordinates on ∆ where 0 ≤ a < b < ∞ and 0 ≤ c < d < ∞, then the following inequality holds;∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 (αp + 1) 1 p (βp + 1) 1 p ×   ∣∣∣ ∂2f∂t∂s (a,c)∣∣∣q + m ∣∣∣ ∂2f∂t∂s (a, dm)∣∣∣q + ∣∣∣ ∂2f∂t∂s (b,c)∣∣∣q + m ∣∣∣ ∂2f∂t∂s (b, dm)∣∣∣q 4   1 q . where p−1 + q−1 = 1. proof. from lemma 1 and by applying the well-known hölder inequality for double integrals, then one has∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4   1∫ 0 1∫ 0 [ |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣]p dsdt   1 p ×   1∫ 0 1∫ 0 ∣∣∣∣ ∂2f∂t∂s (ta + (1 − t) b,sc + (1 −s) d) ∣∣∣∣q dsdt   1 q . by using the fact that |tα1 − t α 2 | ≤ |t1 − t2| α for α ∈ (0, 1] and t1, t2 ∈ [0, 1] , we get 1∫ 0 |(1 − t)α − tα|p dt ≤ 1∫ 0 |1 − 2t|αp dt = 1 αp + 1 some fractional integral inequalities 53 and 1∫ 0 ∣∣∣∣∣∣(1 −s)β −sβ∣∣∣∣∣∣p dt ≤ 1∫ 0 |1 − 2s|βp dt = 1 βp + 1 . since ∣∣∣ ∂2f∂t∂s∣∣∣q is co-ordinated m−convex, we can write∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 (αp + 1) 1 p (βp + 1) 1 p ×   1∫ 0 1∫ 0 [ ts ∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣q + mt (1 −s) ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣q ] + (1 − t) s ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣q + m (1 − t) (1 −s) ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣q dsdt )1 q . by computing these integrals, we obtain∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 (αp + 1) 1 p (βp + 1) 1 p ×   ∣∣∣ ∂2f∂t∂s (a,c)∣∣∣q + m ∣∣∣ ∂2f∂t∂s (a, dm)∣∣∣q + ∣∣∣ ∂2f∂t∂s (b,c)∣∣∣q + m ∣∣∣ ∂2f∂t∂s (b, dm)∣∣∣q 4   1 q . which completes the proof. remark 2. suppose that all the assumptions of theorem 6 are satisfied. if we choose α = β = m = 1, we obtain the inequality (1.3) . theorem 7. let f : ∆ → r be a partial differentiable mapping on ∆ and ∂ 2f ∂t∂s ∈ l (∆), α,β ∈ (0, 1]. if ∣∣∣ ∂2f∂t∂s∣∣∣q , q ≥ 1, is m−convex function on the co-ordinates on ∆ where 0 ≤ a < b < ∞ and 0 ≤ c < d < ∞, then the following inequality holds;∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 ([ 1 − ( 1 2 )α α + 1 ][ 1 − ( 1 2 )β β + 1 ])1−1 q m 1 q α m 1 q β × (∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣q + m ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣q + ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣q + m ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣q )1 q where mα,mβ are defined as in theorem 5. 54 çeti̇n yildiz1,∗, mevlüt tunç2, and havva kavurmaci1 proof. from lemma 1 and by applying the well-known power-mean inequality for double integrals, then one has∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4   1∫ 0 1∫ 0 |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣dsdt  1− 1 q ×   1∫ 0 1∫ 0 |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣ ∣∣∣∣ ∂2f∂t∂s (ta + (1 − t) b,sc + (1 −s) d) ∣∣∣∣q dsdt   1 q . since ∣∣∣ ∂2f∂t∂s∣∣∣q is co-ordinated m−convex, we can write∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4   1∫ 0 1∫ 0 |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣dsdt  1− 1 q ×   1∫ 0 1∫ 0 |(1 − t)α − tα| ∣∣∣(1 −s)β −sβ∣∣∣[ts∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣q + mt (1 −s) ∣∣∣∣ ∂2f∂t∂s ( a, d m )∣∣∣∣q ] + (1 − t) s ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣q + m (1 − t) (1 −s) ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣q dsdt )1 q . by computing these integrals, we obtain∣∣∣∣f(a,c) + f(a,d) + f(b,c) + f(b,d)4 + b ∣∣∣∣ ≤ (b−a) (d− c) 4 ([ 1 − ( 1 2 )α α + 1 ][ 1 − ( 1 2 )β β + 1 ])1−1 q m 1 q α m 1 q β × (∣∣∣∣ ∂2f∂t∂s (a,c) ∣∣∣∣q + ∣∣∣∣m ∂2f∂t∂s ( a, d m )∣∣∣∣q + ∣∣∣∣ ∂2f∂t∂s (b,c) ∣∣∣∣q + m ∣∣∣∣ ∂2f∂t∂s ( b, d m )∣∣∣∣q )1 q which completes the proof. remark 3. suppose that all the assumptions of theorem 7 are satisfied. if we choose α = β = m = 1, we obtain the inequality (1.4) . references [1] m. alomari and m. darus, on the hadamard’s inequality for log −convex functions on the coordinates, journal of inequalities and appl., 2009, article id 283147. [2] m.k. bakula and j. pečarić, on the jensen’s inequality for convex functions on the coordinates in a rectangle from the plane, taiwanese journal of math., 5, 2006, 1271-1292. [3] z. dahmani, new inequalities in fractional integrals, int. j. nonlinear sci., 9 (4) (2010) 493–497. [4] z. dahmani, on minkowski and hermite–hadamard integral inequalities via fractional integration, ann. funct. anal., 1 (1) (2010) 51–58. some fractional integral inequalities 55 [5] z. dahmani, l. tabharit, s. taf, some fractional integral inequalities, nonlinear. sci. lett. a., 1 (2) (2010) 155–160. [6] z. dahmani, l. tabharit, s. taf, new generalizations of gruss inequality using riemann– liouville fractional integrals, bull. math. anal. appl., 2 (3) (2010) 93–99. [7] s.s. dragomir, on hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane, taiwanese journal of math., 5, 2001, 775-788. [8] r. gorenflo, f. mainardi, fractional calculus: integral and differential equations of fractional order, springer verlag, wien (1997), 223-276. [9] s. miller and b. ross, an introduction to the fractional calculus and fractional differential equations, john wiley and sons, usa, 1993, p.2. [10] m.e. özdemir, e. set, m.z. sarıkaya, some new hadamard’s type inequalities for coordinated m−convex and (α,m)−convex functions, hacettepe j. of. math. and st., 40, 219229, (2011). [11] m. e. özdemir, havva kavurmacı, ahmet ocak akdemir and merve avcı, inequalities for convex and s−convex functions on ∆ = [a,b]x[c,d], journal of inequalities and applications, 2012:20, doi:10.1186/1029-242x-2012-20. [12] m. e. özdemir, m. amer latif and ahmet ocak akdemir, on some hadamard-type inequalities for product of two s−convex functions on the co-ordinates, journal of inequalities and applications, 2012:21, doi:10.1186/1029-242x-2012-21. [13] i. podlubni, fractional differential equations, academic press, san diego, 1999. [14] m.z. sarikaya, h. ogunmez, on new inequalities via riemann–liouville fractional integration, arxiv:1005.1167v1 (submitted for publication). [15] m.z. sarıkaya, e. set, h. yaldız and n. başak, hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities, mathematical and computer modelling, in press. [16] m.z. sarıkaya, e. set, m. emin özdemir and s.s. dragomir, new some hadamard’s type inequalities for co-ordinated convex functions, accepted. [17] g. toader, some generalization of the convexity, proc. colloq. approx. opt., cluj-napoca, (1984), 329-338. 1atatürk university, k.k. education faculty, department of mathematics, 25240, kampus, erzurum, turkey 2department of mathematics, faculty of art and sciences,ki̇li̇s 7 aralik university, ki̇li̇s, 79000, turkey ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 14, number 2 (2017), 107-133 http://www.etamaths.com integral representations of semi-inner products in function spaces florian-horia vasilescu∗ abstract. various spaces of measurable functions are usually endowed with semi-inner products expressed in terms of positive measures. trying to give answers to the inverse problem, we present integral representations for some semi-inner products on function spaces of measurable functions, obtained either directly or by adapting and extending techniques from the theory of moment problems. 1. introduction let (ω,s) be a measurable space, that is, ω is an arbitrary (nonempty) set and s is a σ-algabra of subsets of ω. giving a positive measure µ on s, we denote by lp(ω,µ) (p = 1, 2) the set of those s-measurable functions f : ω 7→ c such that |f|p is µ-integrable. (note that we do not identify the functions µ-equal almost everywhere.) let s be a vector space consisting of s-measurable complex-valued functions on ω, invariant under complex conjugation. assume that s is endowed with a semi-inner product (in the sense of [7], page 96), denoted by 〈∗,∗〉0, having real values when applied to real-valued functions. a natural question is to find a positive measure µ on ω such that s ⊂l2(ω,µ) and 〈f,g〉0 = ∫ ω f(ω)g(ω)dµ(ω), f,g ∈s. (1.1) one possible approach to a solution of this problem is to connect it with a moment problem of a general type, that is, not necessarily directly related to spaces of polynomials (see for instance [19]). of course, when (1.1) is fulfilled, the space s(2), spanned by all products fg with f,g ∈ s, lies in l1(ω,µ). as it is reasonable and useful to look for a probability measure µ, we should suppose that 1 ∈ s and 〈1, 1〉0 = 1. setting λ0(f) = 〈f, 1〉0, f ∈ s, formula (1.1) leads to a linear extension of λ0 to the space s(2). an arbitrary element in s(2) may have various representations as a sum of products of two functions from s but we clearly have∑ j∈j |fj|2 = ∑ k∈k |gk|2 =⇒ ∑ j∈j ‖fj‖20 = ∑ k∈k ‖gk‖20 (1.2) in the presence of (1.1) for all fj,gk ∈ s, j ∈ j,k ∈ k, j,k finite, where ‖ ∗ ‖0 is the semi-norm asociated to the inner-product 〈∗,∗〉0. condition (1.2) becomes a necessary one when trying to extend λ0 from s to s(2), in order to look for a unknown measure µ satisfying (1.1) (see section 3). thanks to an argument originating in [23], in many situations of interest we may restrict ourselves to the case when the space s is finite dimensional (see theorem 3). the finite dimensionality of the space s leads to the possibility to replace an existing measure µ by another one consisting of a finite number of atoms, via an argument going back to [25] (see also [4]). hence, under appropriate conditions (see theorem 4), there exist a finite subset {ω1, . . . ,ωd} ⊂ ω, and positive numbers λ1, . . . ,λd with λ1 + · · · + λd = 1, such that 〈f,g〉0 = d∑ j=1 λjf(ωj)g(ωj), f,g ∈s. received 7th april, 2017; accepted 1st june, 2017; published 3rd july, 2017. 2010 mathematics subject classification. 47a57, 44a60, 46c05, 46e22, 47b15. key words and phrases. semi-inner product; integral representation; point evaluation; moment problem; relative idempotent; relative multiplicativity; dimensional stability. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 107 108 vasilescu as already hinted above, when trying the represent semi-inner products under the form (1.1), we either obtain some direct statements (as in theorem 1 and theorem 6) or adapt some methods and results coming from the theory moment problems (as in theorem 8 and theorem 9). because the classical moment problems are well reflected in the literature, we shall mainly mention some more recent contributions, as for instance [5], [9], [10], [12], [14], [17], [20], [29], [31], etc. let us briefly describe the contents of this work. the next chapter, devided into five subsections, contains the basic definitions, some examples, and some proofs as well. we introduce a general concept of function space, consisting of measurable complex-valued functions, endowed with a compatible semi-inner product (definition 1), which extends the concept of unital square positive functional (see [21, 29–31]). such a pair will be called a quasi-hilbert function space (definition 2), and will always be associated with a canonical hilbert space (see remark 1(2)). the formal general question, whose various particular cases are discussed in this work, is stated as problem 2. a first abstract partial solution to problem 2 is provided by theorem 1, when the cardinal of the set of continuous point evaluations equals the dimension of the associated hilbert space. although the framework of general function spaces leads to some interesting results (as for instance theorem 1), to recapture as many as possible results known for the spaces of polynomials we restrict ourselves to spaces of functions having a finite numbers of generators (see subsection 2.3). a useful consequence of such a hypothesis is the possibility of approaching problem 2, using only finite dimensional function spaces (see theorem 3, extending a result which goes back to [23]). the concept of idempotent element with respect to a given semi-inner product (see definition 4) extends the homonymous one, defined in [30]. this concept plays a central role in our development: two of our main results (theorem 8 and theorem 9), and other results as well, depend on the existence of orthogonal bases consisting of idempotent elements, with special properties. many of the results of this work are stated in terms of semi-inner products. nevertheless, in some cases (see theorem 9) we need the slightly stronger concept of unital square positive functional (see subsection 2.1). the connection between these two concepts is given by proposition 1. a strong relation between problem 2 and an interpolation-type problem is presented in theorem 6, which uses an extreme situation, that is, when the cardinal of the representing measure is supposed to be equal to the dimension of the associated hilbert space. in fact, such a hypothesis is present in most of our results. a relaxation of this assumption shows the necessity of working with projections of idempotent elements rather than with idempotents, as shown by theorem 7. in the function spaces having a finite number of generators, the existence of representing measures is characterized by the existence of a orthogonal bases consisting of idempotents, satisfying a certain ”multiplicativity condition“ given by (6). the necessary condition (6) is essential for the proof of theorem 8, which is one of the main results of this work. the concept of dimensional stability (see definition 7) goes back the concept of flatness, introduced in [9] in the context of spaces of polynomials (see also [29]). this property is used to prove theorem 9, which is another main result of this paper. we note that the proof of theorem 9 is an application of theorem 8, which makes it shorter and more transparent than those of its predecessors from [9] or [29]. an example, originating in [14] and [16], treated in our spirit, concludes this work. 2. preliminaries 2.1. function spaces and compatible semi-inner products. let (ω,s) be a measurable space, and let also ms(ω) be the algebra of all complex-valued s-measurable functions on ω (that is, f−1(b) ∈ s for each f ∈ms(ω) and all borel sets b ⊂ c). among some other reasons, the choice of this framework (see also [19]) is related to the use, in the proof of theorem 4, of an abstract version of tchakaloff’s theorem giving a quadrature formula, due in the actual form to c. bayer and j. teichmann (see [4]). nevertheless, except for such particular but important situations, we may restrict ouselves to the case when ω is a hausdorff space and s is the σ-algabra of all borel subsets of ω and, when working with finite atomic measures, even to the case when ω is an arbitrary (nonempty) set and s is family of all subsets of ω. integral representations 109 for convenience, and following [29–31], a vector subspace s ⊂ ms(ω) such that 1 ∈ s and if f ∈s, then f̄ ∈s, is said to be a function space (we specify “on (ω,s)” or simply “on ω”, whenever necessary). fixing a function space s, let s(2) be the vector space spanned by all products of the form fg with f,g ∈s, which is itself a function space. we have s ⊂s(2), and s = s(2) when s is an algebra. for any vector subspace t ⊂s invariant under complex conjugation, the symbol rt will designate the ”real part“ of t , that is {f ∈t ; f = f̄}. functions spaces on a hausdorff (topological) space consisting of borel measurable functions were considered in [31]. important examples of function spaces are associated with the space pn of all polynomials in n ≥ 1 real variables, usually denoted by t1, . . . , tn, with complex coefficients. for every integer m ≥ 0, let pnm be the subspace of pn consisting of all polynomials p with deg(p) ≤ m, where deg(p) is the total degree of p. both pnm and pn are function spaces (of continuous functions) on rn. we occasionally use the notation pm instead of pnm and p instead of pn when the number n is given. note that p(2)m = p2m and p(2) = p, the latter being an algebra. let s be a function space, endowed with a semi-inner product denoted by 〈∗,∗〉0, whose associated semi-norm is denoted by ‖∗‖0. note that we have the cauchy-schwarz inequality, that is, |〈f,g〉0| ≤ ‖f‖0‖g‖0, f,g ∈s. (2.1) set i := {f ∈s;‖f‖0 = 0} = {f ∈s,〈f,g〉0 = 0 ∀g ∈s}, (2.2) which is a vector subspace of s, via the cauchy-schwarz inequality. the semi-inner product 〈∗,∗〉0 induces on the quotient space h0 := s/i an inner product given by 〈f + i,g + i〉 = 〈f,g〉0, f,g ∈s, (2.3) because the right hand side of this equality depends only on the equivalent classes f + i,g + i, again by the cauchy-schwarz inequality. the norm corresponding to (2.3) will be denoted by ‖∗‖. for the sake of simplicity, the equivalence class f + i will be often denoted by f̂ for every f ∈s. definition 1. let s be a function space. a semi-inner product 〈∗,∗〉0 of s is said to be compatible with (the structure of) s if ‖1‖0 = 1, 〈f̄, ḡ〉0 = 〈f,g〉0 ∀f,g ∈s. (2.4) similarly, the associated seminorm ‖∗‖0 is said to be compatible with (the structure of) s. example 1. an arbitrary semi-inner product on a function space is not necessarily compatible. here is an example. let ω := {z = x + iy ∈ c; 0 < x,y < 1}. we consider a subspace of the space l2(ω) of square integrable functions with respect to the planar lebesgue measure on ω, defined as follows: s := {f ∈ c1(ω); f,∂zf,∂z̄f ∈l2(ω)}, where ∂z = 2 −1(∂/∂x− i∂/∂y),∂z̄ = 2−1(∂/∂x + i∂/∂y). if f,∂zf,∂z̄f ∈l2(ω), and so f̄,∂zf,∂z̄f ∈l2(ω), we also have ∂zf̄,∂z̄f̄ ∈l2(ω), as one can easily see, showing that s is a function space on ω. let us define 〈f,g〉0 := ∫ 1 0 ∫ 1 0 (f + i∂zf + ∂z̄f)(g + i∂zg + ∂z̄g)dxdy, f,g ∈l2(ω), which is a semi-inner product on s. in particular, if f(x,y) = y and g(x,y) = 1, we obtain 〈f,g〉0 = 1 + 2−1i, showing that this semi-inner product is not compatible with the structure of s. remark 1. let s be a function space, endowed with a compatible semi-inner product 〈∗,∗〉0. (1) because 〈∗,∗〉0 is compatible with s, its restriction to rs ×rs is a real semi-inner product. in addition, the sum of spaces rs + irs is direct, equals s, and ‖f + ig‖20 = ‖f‖ 2 0 + ‖g‖ 2 0, f,g ∈rs. (2.5) 110 vasilescu we also have i=ri + iri, where the sum of the spaces is direct. it is easily seen that the r-linear map rs/ri 3 f + ri 7→ f + i ∈ h0 is injective, because rs ∩i = ri, allowing us to identify the space rs/ri with a real vector subspace of h0, denoted by rh0. in fact we have the direct sum h0 = rh0 + irh0. setting 〈f + ri,g + ri〉 = 〈f,g〉0, f,g ∈rs, and therefore 〈f + ri,g + ri〉 = 〈f + i,g + i〉, f,g ∈rs, we infer that the map rh0 3 f + ri 7→ f + i ∈h0 is actually an isometry, and ‖φ + iψ‖2 = ‖φ‖2 + ‖ψ‖2, φ,ψ ∈rh0. (2.6) (2) we may complete the space h0 with respect to the inner product 〈∗,∗〉, to get a hilbert space. note that a sequence (σk = φk + iψk)k≥1 in h0, with φk,ψk ∈rh0 for all k ≥ 1 is a cauchy sequence if and only if both (φk)k≥1, (ψk)k≥1 are cauchy sequences. consequently, denoting by h (resp. rh) the completion of h0 (resp. rh0), we must have the direct sum h = rh + irh, and equality (2.6) is still valid for φ,ψ ∈ rh. this decomposition allows us to represent any element φ ∈ h as a sum φ = φ1 + iφ2, with φ1,φ2 ∈rh. in addition, the ”complex conjugate“ of φ is given by φ̄ = φ1 − iφ2. we also note that ‖1 + i‖ = 1, as well as the identity 〈φ,ψ〉 = 〈φ̄, ψ̄〉, φ,ψ ∈hλ, which is a consequence of (2.4). given a function space s endowed with a compatible semi-inner product 〈∗,∗〉0, the hilbert space h obtained as above will be referred to as the hilbert space associated to (s,〈∗,∗〉0). when h0 is finite dimensional, in particular when s is finite dimensional, we have h = h0, and if s is actually a hilbert space, obviously h = s. (3) the spaces h0 and h have induced inner products with a property similar to (2.4). a natural question is related to the possibility to organize them as function spaces. nevertheless, this operation is not always possible. to explain this assertion, let us assume that s is a function space on (ω,s), and that ω0 := ∩f∈iz(f) ∈ s. (2.7) where z(f) := {ω ∈ ω; f(ω) = 0}. condition (2.7) is automatically fulfilled if ω is a hausdorff space and s consists of continuous functions or if the space i has an at most countable algebraic basis. if 〈∗,∗〉0 is actually an inner product, which is equivalent to i = {0}, we have again ω0 = ω ∈ s. condition (2.7) allows us to regard the elements of h0 as functions on ω0. indeed, if f̂ = f + i is an arbitrary element of h0, we put f̂(ω) = f(ω) for all ω ∈ ω0, which is correctly defined. in addition, f̂−1(b) = (f|ω0)−1(b) for all borel sets b ⊂ c. in fact, s0 := {f|ω0; f ∈ s} is a function space on (ω0,s0), where s0 = {a∩ ω0; a ∈ s}. however, the map h0 3 f̂ 7→ f|ω0 ∈ s0 is not injective, so f̂(ω) = 0 for all ω ∈ ω0 does not necessarily imply f̂(ω) = 0 (see example 2). of course, when i = {f ∈s; f|ω0 = 0}, then h0 may be regarded as a function space on ω0. (4) when a function space s and a compatible inner product 〈∗,∗〉0 are given, we use the notation i,h0,h, and so on, in the sense from this remark, if not otherwise specified. the construction in remark 1 has an important particular case, to be discussed in the following. let s be a function space and let λ : s(2) 7→ c be a linear map with the following properties: (1) λ(f̄) = λ(f) for all f ∈s(2); (2) λ(|f|2) ≥ 0 for all f ∈s; (3) λ(1) = 1. adapting some terminology from [21] to our context (see also [29–31]), a linear map λ with the properties (1)-(3) is said to be a unital square positive functional, briefly a uspf. a simple example of a uspf is given by a probability measure µ and a functions space s on (ω,s), consisting of square µ-integrable functions. then the map s(2) 3 f 7→ ∫ ω fdµ ∈ c is a uspf, as one can easily see. integral representations 111 when the linear map λ : s(2) 7→ c is a uspf, we may define a semi-inner product by the equality 〈f,g〉0 = λ(fḡ), f,g ∈s, which is easily seen to be compatible with s. then, as in (2.2), the set iλ := {f ∈s; λ(|f|2) = 0} = {f ∈s; λ(fg) = 0 ∀g ∈s} is a vector subspace of s. moreover, the quotient h0λ := s/iλ is an inner product space, with the inner product given by 〈f̂, ĝ〉 = λ(fḡ), (2.8) where f̂ = f + iλ is the equivalence class of f ∈s modulo iλ. the hilbert space obtained by completion of the inner product space h0λ will be denoted by hλ. of course, if s is finite dimensional, we have hλ = s/iλ. when the uspf λ : s(2) 7→ c is given, we shall use the notation iλ, hλ, f̂, with the meaning from above, if not otherwise specified. problem 1. the (abstract) moment problem for a given uspf λ : s(2) 7→ c, where s is a fixed function space on (ω,s), means to find necessary and sufficient conditions insuring the existence of a probability measure µ, defined on s, such that s ⊂l2(ω,µ) and λ(f) = ∫ ω fdµ, f ∈s(2). when such a measure µ exists, it is said to be a representing measure of λ (with support) in ω. in the classical moment problem on spaces of polynomials, the role of the uspf λ is played by the so-called riesz functional (see for instance [17]). in some special cases, a uspf λ : s(2) 7→ c may have an atomic representing measure in ω, which (in this text) means that there exists a finite subset ωλ = {ω1, . . . ,ωd}⊂ ω consisting of distinct points, and positive numbers λ1, . . . ,λd, with λ1 + · · · + λd = 1, such that λ(f) = ∑d j=1 λjf(ωj). when we want to specify the number points {ω1, . . . ,ωd}, the corresponding atomic measure will be sometimes called a d-atomic representing measure (of λ in ω). of course, in this case we can write λ(f) = ∫ ω f(ω)dµ(ω), where µ is a probability measure defined on a σ-algebra s containing ωλ and its subsets, such that µ({ωj}) = λj j = 1, . . . ,d. in particular, we may take as s the family of all subsets of ω. when s is finite dimensional and the uspf λ on s(2) has an arbitrary representing measure, then one expects that this measure may be replaced by an atomic one. such a property, going back to tchakaloff (see corollary 2 in [25]), will be also discussed in this work (see theorem 4). the concept of representing measure can be also defined for a compatible semi-inner product 〈∗,∗〉0 of a function space s (see definition 2). 2.2. continuous point evaluations. a discussion concerning the point evaluations in the context of function spaces of polynomials on rn can be found in [30], section 4 (see also [31] for a more general framework). some of the assertions of interest for us can be obtained from those in [30], with minor modifications. nevertheless, in the following, some results will be discussed in a more general context. remark 2. let s be a function space on (ω,s), endowed with a compatible semi-inner product 〈∗,∗〉0, and let h be the hilbert space associated to (s,〈∗,∗〉0) (see subsection 2.1). for every point ω ∈ ω, we denote by δω the point evaluation at ω, that is, δω(f) = f(ω), for every function f ∈s. of course, a point evaluation δω is said to be continuous if there exists a constant cω > 0 such that |δω(f)| ≤ cω‖f‖0, f ∈s. (2.9) a point evaluation is not necessarily continuous; see example 2. we denote by z the subset of those points ω ∈ ω such that δω is continuous. remark 3. we continue the discussion from remark 1(3). let h0 = s/i be endowed with the inner product (2.3). if ω ∈ z, we must have h(ω) = 0 for all functions h ∈ i, as a consequence of (2.9). consequently, we have the inclusion z ⊂ ∩f∈iz(f). 112 vasilescu moreover, every point ω ∈z induces a linear and continuous map δ̂ω : h0 7→ c, given by δ̂ω(f̂) = f(ω) for all f ∈s, with f̂ = f + i, as the value f(ω) is constant on the equivalence class of f. in addition, |δ̂ω(f̂)| = |f(ω)| ≤ cω‖f‖0 = cω‖f̂‖, via (2.9), where ‖∗‖ is the norm associated to the inner product (2.3). the linear functional δ̂ω can be extended to the completion h of h0, and keeping the same notation, we have |δ̂ω(φ)| ≤ cω‖φ‖, φ ∈h, ω ∈z. the next result is a slight extension of lemma 6 from [30]. lemma 1. if s is finite dimensional, we have z = ∩f∈iz(f), and z ∈ s. proof. we already saw in remark 3 that always z ⊂∩f∈iz(f). assume now that s is finite dimensional, so h = h0 is a finite dimensional hilbert space. let b be an algebraic basis of iλ, which is a finite set. clearly, z ⊂∩b∈bz(b) = ∩f∈iz(f). conversely, if ω ∈ ∩b∈bz(b), then δω(b) = 0 for all b ∈ b, implying δω(f) = 0 for all f ∈ iλ. therefore, δω induces a linear functional on the hilbert space h, denoted by δ̂ω. the linear functional δ̂ω is automatically continuous, and so δω is continuous. this shows ω ∈ z. consequently, z = ∩b∈bz(b), implying z = ∩f∈iz(f). finally, z(b) = b−1({0}) ∈ s for all b ∈b showing that z ∈ s. � remark 4. (1) the previous argument shows that if h0 is finite dimensional, so h = h0, we have the equality z = ∩f∈iz(f). if, in addition, the space i has an at most countable algebraic basis, then z ∈ s. (2) if i = {0} and s = h0 is finite dimensional, we must have z = ω. (3) the previous lemma shows that the set z extends the concept of algebraic variety of a moment sequence, defined in the context of finite dimensional spaces of polynomials (see for instance (1.6) from [10]). remark 5. (1) continuing the discussion from remark 1(3), if i = {f ∈s; f|z = 0}, and s is finite dimensional, the space h = h0 may be regarded as a function space on z because from f|z = 0, so ‖f‖0 = 0, we obtain f̂ = 0. (2) note that for every ω ∈z there exists a vector νω ∈h such that δ̂ω(φ) = 〈φ,νω〉 for all φ ∈h, via a well-known theorem by riesz. in fact, applying the riesz theorem firstly on rh, we deduce that νω ∈rh. definition 2. let s be a function space on (ω,s), endowed with a compatible semi-inner product 〈∗,∗〉0. from now on, such a pair (s,〈∗,∗〉0) will be designated as a quasi-hilbert function space (briefly, a qhfs). when the function space s on ω is actually a hilbert space, endowed with a compatible inner product 〈∗,∗〉, the pair (s,〈∗,∗〉) it will be called a hilbert function space (briefly, a hfs). we say that the semi-inner product 〈∗,∗〉0 has a representing measure if there exists a probability measure µ on s such that 〈f,g〉0 = ∫ ω f(ω)g(ω)dµ(ω), f,g ∈s. (2.10) we say that the semi-inner product 〈∗,∗〉0 has an atomic representing measure in ω if there exists a finite subset ω0 = {ω1, . . . ,ωd} ⊂ ω consisting of distinct points, and positive numbers λ1, . . . ,λd, with λ1 + · · · + λd = 1, such that 〈f,g〉0 = d∑ j=1 λjf(ωj)g(ωj), f,g ∈s. (2.11) when the support of a given atomic representing measure consists of d points, it will be sometimes called a d-atomic representing measure. as usually, the points {ω1, . . . ,ωd} are called the nodes, and the numbers {λ1, . . . ,λd} are called the weights of the measure µ. integral representations 113 a slightly more general question than problem 1 is the following. problem 2. let (s,〈∗,∗〉0) be a qhfs on (ω,s). find necessary and sufficient conditions to insure the existence of a representing measure of the semi-inner product 〈∗,∗〉0. using direct arguments, as well as methods from the theory of moment problems in functions spaces of polynomials, we shall try to give some answers to problem 2 in the next sections. the following lemma is similar to lemma 7 from [30] (see also [9] for a precursor of this result). lemma 2. suppose that the compatible semi-inner product 〈∗,∗〉0 of the function space s on ω has an atomic representing measure µ. then supp(µ) ⊂z. proof. we use the notation from definition 2. if ωk ∈ ω0, we have: |f(ωk)|2 ≤ 1 λk d∑ j=1 λj|f(ωj)|2 = 1 λk ‖f‖20, f ∈s, for all k = 1, . . . ,d, showing that supp(µ) ⊂z. � the next definition adapts some concepts from [14] to our context. definition 3. let (s,〈∗,∗〉0) be a qhfs. we say that the semi-inner product 〈∗,∗〉0 is weakly consistent if whenever f ∈s in null on z, it follows that 〈f, 1〉0 = 0. we say that 〈∗,∗〉0 is consistent if whenever ∑ k∈k fkgk = 0 on z it follows ∑ k∈k〈fk,gk〉0 = 0, where fk,gg ∈rs, k ∈ k, k finite. remark 6. (1) consider a qhfs (s,〈∗,∗〉0). it follows from lemma 2 that a necessary condition for the existence of an atomic representing measure for the semi-inner product 〈∗,∗〉0 on ω is z 6= ∅. (2) if the semi-inner product 〈∗,∗〉0 is consistent, it is also weakly consistent. (3) if the semi-inner product 〈∗,∗〉0 has an atomic representing measure, then it is consistent. the converse is not true, in general (see [14]). (4) let s be a function space on a set ω and let λ : s(2) 7→ c be a uspf. of course, the previous results of this subsection applies to this case, when the semi-inner product is given by 〈f,g〉0 = λ(fḡ〉, f,g ∈s. we recall that we use the notation hλ,h0λ,iλ,zλ instead of h,h 0,i,z, respectively. in this case (see also [14] for spaces of polynomials), if 〈∗,∗〉0 is (weakly) consistent, we say that λ is (weakly) consistent. also note that λ is consistent if and only if whenever f ∈s(2) satisfies f|zλ = 0, we have λ(f) = 0. a similar property characterizes the weak consistency of λ. if s is finite dimensional, the uspf λ is consistent if and only if λ ∈ span{δ(2)ω ; ω ∈ zλ}, where δ (2) ω is the point evaluation at ω ∈ ω, regarded as a functional on s(2). indeed, if λ ∈ span{δ(2)ω ; ω ∈ zλ}, then λ is clearly consistent. conversely, because span{δ (2) ω ; ω ∈ zλ} is in the dual of s(2), which is finite dimensional, if λ /∈ span{δ(2)ω ; ω ∈zλ}, we can find a function f0 ∈s(2) which is null on zλ but with λ(f0) 6= 0, which is not possible. similarly, λ is weakly consistent if and only if λ|s ∈ span{δω; ω ∈zλ}. finally, note that ω /∈ zλ for some ω ∈ ω, if and only if there exists a sequence (fn)n in s such that λ(|fn|2) = 1 and |fn(ω)|→∞(n →∞). example 2. we consider the the uspf λ : p14 7→ c, where p14 is the space of of polynomials in one real variable t, with complex coefficients, of degre ≤ 4, where λ(tk) = 1, k = 0, 1, 2, 3, λ(t4) = 2, extended by linearity. this is a simple example showing that there are truncated moment problems with no representing measure in r (see [30], example 3; see also [13]). it can be also used in connection with remark 5(1), and with remark 6(4) as well. as shown in [30], we have iλ = {p(t) = a − at; a ∈ c}. then clearly, zλ = {1}, so the point evaluation δ1(p) = p(1), p ∈p12 , must be continuous, by lemma 1. in fact, if p(t) = a + bt + ct2 with a,b,c ∈ r arbitrary, p(1)2 = (a + b + c)2 ≤ λ(p2) = ((a + b + c)2 + c2), and thus, when a,b,c ∈ c, we deduce that |p(1)|2 ≤ λ(|p|2), p ∈ p12 , showing, directly, that δ1 is λ-continuous. on the other hand, if θ ∈ r, θ 6= 1, setting pn(t) = 1 + n(1 − t), n ≥ 1, we obtain pn(θ) = 1 + n(1 −θ) →±∞ as n →∞, while λ(p2n) = 1 for all n ≥ 1, as in remark 6(4). 114 vasilescu note that there are nonnul elements f̂ ∈ hλ = p12 /iλ such that δ̂1(f̂) = 0. indeed, as one can easily see (see also [31], example 3), we have hλ = {u1̂ + vt̂2; u,v ∈ c}. then δ̂1(1̂ − t̂2) = 0, while 1̂ − t̂2 6= 0. in other words, the hilbert space associated to a qhfs (s,〈∗,∗〉0) is not a hilbert function space. note also that, because zλ = {1}, the only possible atomic representing measure for λ would be δ1, via lemma 2. but this is impossible because, for instance, δ1(t 4) = 1, while λ(t4) = 2. as in the case of function spaces consisting of polynomials, an extremal situation associated with the consistency insure the existence of an atomic representing measure (see [14], theorem 1.3). theorem 1. let (s,〈∗,∗〉0) be a finite dimensional qhfs on ω. assume that d := card(z) = dim(h). the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure if and only if it is consistent. proof. the consistency of 〈∗,∗〉0 allows us to replace the set ω by the set z. indeed, defining the function space sz as the space {f|z; f ∈s}, and putting 〈f|z,g|z〉0,z := 〈f,g〉0, f,g ∈s, we obtain a semi-inner product 〈∗,∗〉0,z on sz. indeed, we note that if f|z = 0 or g|z = 0, we have 〈f,g〉0 = 0 by the consistency of 〈∗,∗〉0. this shows that the map 〈∗,∗〉0,z is well defined, and it is clearly a semi-inner product. in addition, if f|z = 0, the consistency implies f ∈ i, showing that i = {f ∈s,f|z = 0}. hence we have a natural map h3 f̂ 7→ f|z ∈ c(z) := {h : z 7→ c}, which is a linear isomorphism, because it is clearly injective and surjective too, thanks to the assumption card(z) = dim(h). we write now z = {ζ1, . . . ,ζd}, and put χj := χ{ζj}, that is, the characteristic function of the set {ζj}, j = 1, . . . ,d. let also b̂j be the unique element of h with bj|z = χj. because we have bjbk|z = χjχk = 0 if j 6= k, we must have 〈bj,bk〉0 = 0. similarly, χ2j = χj implies 〈bj,bj〉0 = 〈 bj, 1〉0. note that f|z = ∑d j=1 f(ζj)χj, so f − ∑d j=1 f(ζj)bj is null on z for all f ∈s. thus 〈f,g〉0 = 〈 d∑ j=1 f(ζj)bj, d∑ k=1 g(ζk)bk〉 = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s, where λj = 〈bj, 1〉 for all j, which is a representation of the semi-inner product 〈∗,∗〉0 via a d-atomic measure. conversely, if the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure with d := dim(h), then 〈∗,∗〉0 is clearly consistent. � remark with the previous notation, we must have 〈b̂j, b̂k〉 = 0, 〈b̂j, b̂j〉 = 〈b̂j, 1̂〉, j,k = 1, . . . ,d, j 6= k, and f̂ = ∑d j=1 f(ζj)b̂j for all f̂ ∈h. in other words, {b̂1, . . . , b̂d} is an orthogonal basis of h, consisting of idempotents (see subsection 2.5 for details). 2.3. generators of function spaces. let s be a function space on (ω,s). we assume that there exist an n-tuple θ = (θ1, . . . ,θn) of elements of rs, and an integer m ≥ 1, such that such that the family θm := {θα; |α| ≤ m, α ∈ zn+} spans the space s. when such a pair (θ,m) exists, we shortly say that the function space s is m-generated by θ. clearly, in this case s is of finite dimension, and the family θ2m spans the space s(2). in particular, s is 1-generated by θ if and only if s is the span of θ1. also note that if s be a function space that is m-generated by an n-tuple θ = (θ1, . . . ,θn) of elements of rs, and if λ : s(2) 7→ c is a uspf, then the hilbert space hλ must be of finite dimension less or equal to the cardinal of the set θm. as a matter of fact, if s is a function space on ω that is m-generated by an n-tuple θ = (θ1, . . . ,θn) of elements of rs, we must have the equality, s = {p◦θ; p ∈pnm}, where θ is regarded as a function from ω into rn. in particular, pnm is a function space m-generated by by the n-tuple t := (t1, . . . , tn), where t1, . . . , tn are the independent variables of rn. integral representations 115 an important particular case of finitely generated function spaces is related to the so called truncated k-moment problem (see [12]), which means, for a fixed closed set k ⊂ rn, to look for a representing measure of a given uspf λ : pn2m,k 7→ c, where p n m,k := {p|k; p ∈ p n m}. clearly, pnm,k is a function space on k, m-generated by n-tuple t|k := (t1|k,.. . , tn|k). to exhibit another example, fix an n-tuple θ = (θ1, . . . ,θn) in pd. then s := {p◦ θ; p ∈ pnm} is a function space on rd. in fact, if g = max1≤j≤n{degθj}, we have s ⊂pdmg. let again s be a function space on (ω,s), and let n-tuple θ = (θ1, . . . ,θn) of elements of rs. when s spanned by the set {θα; α ∈ zn+}, we say that s is ∞-genarated by θ. in thes case, s is actually a unital algebra. of course, in this case we have s = {p◦θ; p ∈pn}. 2.4. reduction to finite dimensional spaces. we exhibit, in the following, a result allowing us to prove the existence of a representing measure for a uspf λ, defined on a space s of continuous functions on a locally compact metrisable space, using representing measures of restrictions of λ on some finite dimensional subspaces of s. such a reduction result goes back to the paper [23], where it is proved in the context of spaces of polynomials. it is also approached, in a more abstract framework, in [28]. as mentioned above, in this subsection the basic space ω is supposed to be a locally compact metric space. we denote by ω∞ = ω ∪ {∞} the one-point compactification of ω, which is compact and metrisable. if ρ is a fixed metric on ω, a given sequence (ωk)k≥1 is said to tend to infinity, and we write limk→∞ωk = ∞, if limk→∞ρ(ω0,ωk) = ∞ for some (any) point ω0 ∈ ω. let s be a function space on ω, consisting of continuous functions. we suppose that there exists an n-tuple θ = (θ1, . . . ,θn) of functions of rs, separating the points of ω and unbounded on ω, that is, limω→∞θj(ω) = ∞ for all j = 1, . . . ,n. we also assume that the space s is ∞-generated by the θ, so s is a commutative unital algebra, closed under complex conjugation. next, we define the functions qk(ω) = (1 + θ1(ω) 2 + · · · + θn(ω)2)−k where k ≥ 0 is an integer. then we set tk = {f ∈s; lim ω→∞ qk(ω)f(ω) exists}, which is a function space satisfying 1,q−1k ∈ tk, and tk ⊂ tk+1, for all k. moreover, as we have s = {p◦θ; p ∈pn}, and |θα11 · · ·θ αn n |2 ≤ qk(ω)−1 when |α1|+· · ·+|αn| ≤ k, it follows that s = ∪k≥0tk. we now consider the algebra c(ω∞), consisting of continuous functions on the compact set ω∞. moreover, the set q := {qk; k ≥ 0} is a multiplicative family in c(ω∞), provided that each function qk (k ≥ 1) is extended with 0 at ∞ (see [28] for details). in addition, we may regard tk as a subspace of q−1k c(ω∞). under the hypothesis on ω and θ from above, we have the following. theorem 2. a uspf λ : s 7→ c has a representing measure with support in ω if and only if λ(q−1k ) > 0 and |λ(f)| ≤ λ(q−1k ) sup ω∈ω |qk(ω)f(ω)|, f ∈tk, k ≥ 0. proof. it λ has a representing measure, say µ, because q−1k ≥ 1 it follows λ(q −1 k ) ≥ λ(1) = 1 for all k ≥ 0. therefore |λ(f)| ≤ ∫ ω |f|dµ ≤ λ(q−1k ) sup ω∈ω |qk(ω)f(ω)|, for all f ∈tk. conversely, we shall apply theorem 3.7 from [28] to the space s ⊂ c(ω∞)/q and the map λ. first of all, we note that λ(q−1k ) ≥ λ(1) = 1 for all k ≥ 0 because q −1 k is equal to 1 plus a sum of squares. secondly, the space c(ω∞) is separable. in fact, the functions q1(θ),θ1q1(θ), . . . ,θnq1(θ) separate the points of ω∞. then the weierstrass-stone theorem implies the density of the unital algebra generated by this family in c(ω∞). note that qk(∞) = 0 for all k ≥ 1, and so {∞} is the union of the zeros of all functions from q. we also have 1,q−1k ∈tk ⊂ q −1 k c(ω∞) for all k ≥ 0. moreover, tk1 ⊂tk2 whenever k1 ≤ k2, and this is equivalent to the fact that q−1k1 divides q −1 k2 . 116 vasilescu if λk = λ|tk, putting ‖f‖∞,k = supt∈ω |qk(t)f(t)|,f ∈ c(ω∞)/qk, which is precisely the norm on c(ω∞)/qk (see [28]), the conditions from the statement imply the estimates ‖λk‖≤ λ(q−1k ). because q−1k ∈tk, and its norm is one, we must have ‖λk‖ = λ(q −1 k ). according to theorem 3.7 from [28], this implies the existence of a positive extension m of λ to c(ω∞)/q. the proof of theorem 3.7 from [28] shows the existence of a representing measure of m, whose support is precisely in ω (see also remark 3.8(1) from [28]). � remark 7. theorem 2 can also be applied when the spaces (tk)k≥0 are replaced by the simpler spaces (sk)k≥0, with sk = {f ∈ s; f = p ◦ θ,p ∈ pnk}. in fact, assuming that for the uspf λ : s 7→ c the restriction λk = λ|s2k has a representing measure for each k ≥ 0, we may obtain the assertion in the following way. the finite dimension of the involved spaces allows us to find, for every integer k ≥ 0, an integer rk ≥ 0 such that tk ⊂ s2rk . then the representing measure of λ|s2rk induces a representing measure for λ|tk for all k ≥ 0, which allows the application of theorem 2. using the preceding remark, we deduce easily the following: theorem 3. let ω be a locally compact metric space, let s be a function space on ω consisting of continuous functions, and let 〈∗,∗〉0 be a semi-inner product on s. let also θ = (θ1, . . . ,θn) be a tuple of functions from rs, unbounded on ω and separating its points, let sm be the function space m-generated by θ and let 〈∗,∗〉0m be the restriction of the semi-inner product 〈∗,∗〉0 to sm (m ≥ 1). the semi-inner product 〈∗,∗〉0 has a representing measure on ω if and only if the semi-inner product 〈∗,∗〉0m has a representing measure on ω for every m ≥ 1. theorem 3 shows that solving problem 2 on finite dimensional function space leads to solutions of problem 2 in a large class of infinite dimensional function spaces, including the classical ones in spaces of polynomials. for this reason, in the next sections we shall mainly deal with finite dimensional function spaces. as noticed in several works (see for instance [9]), another important feature in the context of finite dimensional function spaces is that the existence of a representing measure of a given semi-inner product implies the existence an atomic representing measure, as presented in the following. theorem 4. let (s,〈∗,∗〉0) be a qhfs on ω, m-generated by the n-tuple θ = (θ1, . . . ,θn) suppose that 〈∗,∗〉0 has a representing measure. then 〈∗,∗〉0 has an atomic representing measure. proof. we consider the set zn,2m+ := {α ∈ zn+; |α| ≤ 2m}, endowed with the lexicographic order. in addition, we assign to each integer j ∈{1, 2, . . . ,nm}, where nm is the cardinal of z n,2m + , a multi-index α(j) ∈ zn,2m+ with j ≤ k iff α(j) ≤ α(k), and α(1) = 0. in this way we have a (borel measurable) map φ : ω 7→ rnm given by φ(ω) = (θα(1)(ω), . . . ,θα(nm)(ω)) ∈ rnm . now assume that 〈∗,∗〉0 has a representing measure, so it has the form 〈f,g〉0 = ∫ ω fḡdµ, f,g ∈s, where µ is a positive borel measure on ω, with µ(ω) = 1. let ν be the measure induced by the measure µ and the borel map φ. writing α(j) = α′(j) + α′′(j) with |α′(j)|, |α′′(j)| ≤ m, we have:∫ rnm |xj|dν(x) = ∫ ω |xj ◦φ|dµ ≤ ∫ ω |θα(j)|dµ ≤‖θα ′(j)‖0‖θα ′′(j)‖0 < ∞, for all j = 1, . . . ,nm, where x1, . . . ,xnm are the coordinate functions in r nm . this shows that we may apply corollary 2 from [4] to deduce the existence of a positive integer d ≤ nm, a set of points ω1, . . . ,ωd in the support of the measure µ, and positive numbers λ1, . . . ,λd, such that ∑d j=1 λj = 1, and ∫ ω θαdµ = d∑ j=1 λjθ α(ωj), α ∈ z n,2m + . from this equality, we infer easily that 〈∗,∗〉0 has an atomic representing measure. � 2.5. quasi-hilbert function spaces and idempotents. the concepts of quasi-hilbert function space (briefly, qhfs) and hilbert function space (briefly, hfs) are those introduced by definition 2. we have already noted that the hilbert space associated to a qhfs (s,〈∗,∗〉0) is not necessarily a hilbert function space (see example 2). we should mention that our concept of hilbert function space is slightly different from the homonymous concept from [1]. in fact, the concept of hilbert function integral representations 117 space, as defined in [1], is often called a reproducing kernel hilbert space (see for instance [2]). unlike in [1], the point evaluations on a hilbert function space in our sense are not necessarily well defined but the inner product of such a space must be compatible. we introduce in the following the concept of element idempotent related to a given function space s, endowed with a compatible semi-inner product 〈∗,∗〉0. this is an extension of the concept of indempotent with respect to a uspf, introduced in [30]. definition 4. let (s,〈∗,∗〉0) be a qhfs, and let h be the hilbert space aassociated to (s,〈∗,∗〉0), whose inner product is denoted by 〈∗,∗〉, and whose norm is ‖∗‖. an element ι ∈ rh is said to be an idempotent (associated to s) if ‖ι‖2 = 〈ι, 1̂〉. (2.12) we set id(h) := {ι ∈rh;〈ι, 1̂〉 6= 0}, that is, the family of all nonnull idempotents of h. example 3. let µ be a probability measure on (ω,s), and let s be a function space on (ω,s), consisting of square µ-integrable functions, so its semi-inner product 〈∗,∗〉0 can be obtained by restricting the semi-inner product of l2(ω,µ) to s. clearly, the subspace i consists of those functions from s, which are null µ-almost everywhere. as usually, let l2(ω,µ) be the hilbert space consisting of equivalence classes of square µ-integrable functions on ω. assume that the space h0 = s/i is dense in l2(ω,µ). then l2(ω,µ) is the hilbert space associated to (s,〈∗,∗〉0). if χb denotes the characteristic function of a given set b ∈ s, the class of χb is clearly an idempotent in l2(ω,µ) associated to s, but χb does not necessarily belong to s. the following result is, in fact, an extension lemma 4 from [30], with a proof in the present context. lemma 3. assume h to be separable, and let {ηj}j∈j be an orthonormal family in rh such that 〈ηj, 1̂〉 6= 0 for each j ∈ j, for some j ⊂ n. then the set {ιj}j∈j is an orthogonal family in id(h), where ιj = 〈ηj, 1̂〉ηj for all j ∈ j. if the set {ηj}j∈n is an orthonormal basis of h, the set {ιj}j∈n is an orthogonal basis of h in id(h). consequently, φ = ∞∑ j=1 〈ιj, 1̂〉−1〈φ,ιj〉ιj, φ ∈h, where the series is convergent in h. in particular 1̂ = ∑∞ j=1 ιj, where the series is convergent in h. proof. setting ιj = 〈ηj, 1̂〉ηj for all j ∈ j, we have ‖ιj‖2 = 〈〈ηj, 1̂〉ηj,〈ηj, 1〉ηj〉 = 〈ηj, 1̂〉2 = 〈ιj, 1̂〉 6= 0, showing that {ιj}j∈j is a family in id(h). in addition, 〈ιj, ιk〉 = 〈ηj, 1̂〉〈ηk, 1̂〉〈ηj,ηk〉 = 0, j,k ∈ j, j 6= k, so the family {ηj}j∈j is orthogonal. if {ηj}j∈n is an orthonormal basis, we must have φ = ∞∑ j=1 〈φ,ηj〉ηj = ∞∑ j=1 〈ιj, 1̂〉−1〈φ,ιj〉ιj, f ∈s, and the series is clearly convergent in h. in particular, we must have 1̂ = ∑∞ j=1 ιj, where the series is convergent in h. � corollary 1. if h is finite dimensional and {η1, . . . ,ηd}⊂rh is an orthonormal basis with 〈ηj, 1̂〉 6= 0, j = 1, . . . ,d, the set {〈η1, 1̂〉η1, . . .〈ηd, 1̂〉ηd} is an orthogonal basis of h consisting of idempotents. moreover, 〈η1, 1̂〉η1 + · · · + 〈ηd, 1̂〉ηd = 1̂. corollary 2. assume that h is finite dimensional. then there are functions b1, . . . ,bd ∈ rs such that ‖bj‖20 = 〈bj, 1〉0 > 0, 〈bj,bk〉0 = 0 for all j,k = 1, . . . ,d, j 6= k, ∑d j=1〈bj, 1〉0 − 1 ∈ i, and every f ∈s can be uniquely represented as f = d∑ j=1 〈bj, 1〉−10 〈f,bj〉0bj + f0, 118 vasilescu with f0 ∈i and d = dimh. using lemma 3, we obtain the next result (which extends theorem 1 from [30]). theorem 5. let (s,〈∗,∗〉0) be a qhfs, and let h be the associated hilbert space. if h is separable and of dimension ≥ 2, it has infinitely many orthogonal bases consisting of idempotent elements. proof. we may work in an abstract framework. replacing h by rh, we may assume, with no loss of generality, that h is a separable real hilbert space, endowed with the real inner product 〈∗,∗〉, and with the corresponding norm ‖∗‖. let {λj}j∈n be a sequence of positive numbers such ∑ j∈n λ 2 j = 1. let also {ηj}j∈n be an orthonormal basis of h, and let η = ∑ j∈n λjηj, so ‖η‖ = 1. then 〈η,ηj〉 = λj > 0 for all j. setting ζj := 〈η,ηj〉ηj, j ≥ 1, we obtain an orthogonal basis {ζj}j∈n such that ‖ζj‖2 = 〈ζj,η〉 for all j ≥ 1. next, we fix an element e ∈ h such that ‖e‖ = 1, and choose an orthogonal transformation o of h such that oη = e. putting, ej = oηj, and ιj = 〈e,ej〉ej = oζj (j ≥ 1), the family {ιj}j∈n is an orthogonal basis of h. moreover, ‖ιj‖2 = 〈ιj,e〉 for all j ≥ 1. going back to our initial case, for e = 1̂ we obtain an orthogonal basis {ιj}j∈n of h consisting of idempotents. the construction from above shows that there are infinitely many possibilities to obtain such a basis {ιj}j∈n. � remark 8. of course, the assertions from this subsection apply when the function space s is endowed with a usps λ : s(2) 7→ c, having the semi-inner product given by 〈f,g〉0 = λ(fḡ), f,g ∈s. note that if s is finite dimensional, an element f̂ ∈rhλ is an idempotent if and only if λ(f2) = λ(f). for this reason, such an element may be called a λ-idempotent, as in [30]. 3. uspf’s versus semi-inner products as mentioned above, the main aim of this work is to give necessary and sufficient conditions insuring the existence of integral representations of some given semi-inner products on function spaces of measurable functions; in other words, to look for solutions to problem 2. one possible approach to problem 2 is to adapt techniques from the theory of moment problems. we refer especially to [9, 10] as well as to [29–31]. to this aim, it is necessary to clarify the connection between semi-inner products and unital square positive functionals, which is the main concern of this section. remark 9. let (s,〈∗,∗〉0) be a qhfs. we want to relate the semi-inner, product 〈∗,∗〉0 with a uspf λ : s(2) 7→ c in a natural and unique way. roughly speaking, we want to have λ(fḡ) = 〈f,g〉0, for all f,g ∈s. in fact, we have the following. lemma 4. let s be a function space on ω. the following assertions are equivalent: (1) the function space s has a compatible semi-inner product 〈∗,∗〉0 satisfying∑ k∈k fkgk = 0 on ω =⇒ ∑ k∈k 〈fk,gk〉0 = 0 (3.1) for all fk,gk ∈rs, k ∈ k, k finite; (2) the function space s has a uspf λ : s(2) 7→ c. proof. (1) =⇒ (2). choosing fk,gk ∈s, k ∈ k, k finite, we write fk = f′k + if ′′ k , ḡk = g ′ k − ig ′′ k , with f′k,f ′′ k ,g ′ k,g ′′ k ∈ rs, k ∈ k. assuming ∑ k∈k fkḡk = 0 we must have ∑ k∈k(f ′ kg ′ k + f ′′ k g ′′ k ) = 0 and∑ k∈k(f ′′ k g ′ k −f ′ kg ′′ k ) = 0 according to (3.1), we deduce that∑ k∈k (〈f′k,g ′ k〉0 + 〈f ′′ k ,g ′′ k〉0) = 0 and ∑ k∈k (〈f′′k ,g ′ k〉0 −〈f ′ k,g ′′ k〉0) = 0. because the seminorm 〈∗,∗〉0 is compatible, we infer that ∑ k∈k〈fk,gk〉0 = 0. consequently, fixing an arbitrary element f = ∑ k∈k fkḡk ∈s (2), we put λ(f) = ∑ k∈k 〈fk,gk〉0. (3.2) integral representations 119 the previous argument shows that this definition does not depend on the particular representation of f , implying that the map λ : s(2) 7→ c is linear. in addition, λ(1) = 〈1, 1〉0 = 1, λ(f̄) = ∑ k∈k 〈f̄k, ḡk〉0 = ∑ k∈k 〈fk,gk〉0 = λ(f), f = ∑ k∈k fkḡk ∈s(2), and λ(|f|2) = 〈f,f〉0 ≥ 0 for all f ∈s. conversely, given a uspf λ : s(2) 7→ c, the formula 〈f,g〉0 = λ(fḡ), f,g ∈ s defines a semi-inner product compatible with the structure of s, as noticed in subsection 2.1. therefore, we also have (2) =⇒ (1). � remark. note that, in a qhfs s whose semi-inner product 〈∗,∗〉0 has the property (3.1), taking f,g ∈s and h ∈rs such that fh,gh ∈s, we must have 〈fh,g〉0 = 〈f,gh〉0. proposition 1. let s be a function space on ω. the following assertions are equivalent: (1) the space s is a qfhs on ω, whose compatible seminorm ‖∗‖0 has the property∑ k∈k f2k = ∑ l∈l g2l on ω =⇒ ∑ k∈k ‖fk‖20 = ∑ l∈l ‖gl‖20, (3.3) for all fk,gl ∈rs, k ∈ k, l ∈ l, k,l finite; (2) the space s has a uspf λ : s(2) 7→ c. moreover, the uspf λ is uniquely determined by the semi-inner product 〈∗,∗〉0 with the property (3.3). proof. to prove the equivalence of the conditions (1) and (2) from the statement, it is enough to verify that condition (3.3) is equivalent to condition (3.1). to show that, we simply apply the obvious polarization formula: ∑ k∈k ukvk = 1 4 ∑ k∈k [(uk + vk) 2 − (uk −vk)2], (∗), and its corresponding version∑ k∈k 〈uk,vk〉0 = 1 4 ∑ k∈k ‖uk + vk‖20 −‖uk −vk‖ 2 0], (∗∗), valid for all uk,vk ∈ rs, k ∈ k, k finite. we also note that in (3.3) we may always assume k = l, with no loss of generality. the details of this verification, and the uniqueness of λ as well, are left to the reader. � remark. going back to problem 2, proposition 1 gives an answer to the question how to associate a qhfs with a uspf, in order to approach this problem as a moment problem. definition 5. let (s,〈∗,∗〉0) be a qhfs space. we say that the semi-inner product 〈∗,∗〉0 is expandable if it has the property (3.3). if 〈∗,∗〉0 is expandable, the unique uspf λ : s(2) 7→ c given by theorem 3.3 is said to be associated to 〈∗,∗〉0. example 4. let µ be a probability measure on (ω,s), and let s = l2(ω,µ). then s is a qhfs, whose natural semi-norm ‖f‖0 = ( ∫ ω |f|2dµ)1/2, f ∈s, is expandable. moreover, it is easily seen that s(2) = l1(ω,µ), and the uspf λ : s(2) 7→ c, given by theorem 1, is precisely λ(f) = ∫ ω fdµ, f ∈l1(ω,µ). example 5. the multidimensional hamburger moment problem can be also related to problem 2 in the following way. the basic function space is in this case pn on rn, consistning of all polynomilas in t = (t1, . . . , tn). first of all, we fix a multi-sequence of real numbers γ = (γα)α∈zn + with the property∑ α,β aαāβγα+β ≥ 0 (3.4) for all finite multi-sequences (aα)α of complex numbers. in fact, we work in this example only with finite sums, and the multi-indices are from zn+, with the order α ≤ β if β −α ∈ zn+. for any two polynomials p(t) = ∑ α cαt α, q(t) = ∑ β dβt β with complex coefficients we put 〈p,q〉0 = ∑ α,β cαd̄βγα+β. (3.5) 120 vasilescu the choice of the multi-sequence γ shows that the assignment (3.5) is a semi-inner product on pn. to illustrate lemma 4, we shall verify directly that this semi-inner product is also expandable. let ∑ j pjq̄j = 0, with pj(t) = ∑ α cj,αt α, qj(t) = ∑ β dj,βt β. then ∑ j pjq̄j = ∑ j ∑ α,β cj,αd̄j,βt α+β = ∑ σ   ∑ j,α≤σ cj,αd̄j,σ−α  tσ = 0. therefore, ∑ j,α≤σ cj,αd̄j,σ−α = 0 for all σ. on the other hand, ∑ j 〈pj,qj〉0 = ∑ j ∑ α,β cj,αd̄j,βγα+β = ∑ σ   ∑ j,α≤σ cj,αd̄j,σ−α  γσ = 0. although condition (3.4) for n ≥ 2 does not necessarily impliy the existence of a representing measure for the semi-inner product (3.5), it suffices to show that (3.5) is expandable. example 6. for some details concerning this example we cite the work [3]. let b be the open unit ball in rn, let s be the boundary of b, and let h2(b) be the real hilbert space of all harmonic functions in b, which are poisson transforms of real-valued functions from l2(s,σ), where σ is the unique borel probability measure on s that is rotation invariant. we denote by h the space h2(b) + ih2(b), which is a hilbert space of complex-valued harmonic functions, and it is also a function space on b. the inner product of h is given by 〈f,g〉h = ∫ s f]g]dσ, f,g ∈ h, where f] denotes the unique element from l2(s,σ) whose poisson transform is f. this inner product is clearly compatible with the function space h. assuming now that ∑ j∈j f 2 j = ∑ k∈k g 2 k for some fj,gk ∈ rh, j ∈ j,k ∈ k, j,k finite, we infer that ∑ j∈j(f ] j ) 2 = ∑ k∈k(g ] k) 2, so∑ j∈j ‖fj‖2h = ∫ s ∑ j∈j (f ] j ) 2dσ = ∫ s ∑ k∈k (g ] k) 2dσ = ∑ k∈k ‖gk‖2h. in other words, the norm of h is expandable, allowing us to define a uspf on h(2), in a natural way. example 7. let w21 be the sobolev space consisting of all complex-valued functions on the interval [0, 1], which are absolutely continuous and whose derivatives are square integrable, endowed with the norm ‖f‖21 = ∫ 1 0 (|f(t)|2 + |f′(t)|2)dt, f ∈w21 . this is a reproducing kernel hilbert space, as shown in [1], example 2.7. it is also a function space on [0, 1] and its inner product is compatible. nevertheless, the norm is not expandable. indeed, taking f(t) = (2 + 2t2)1/2, f1(t) = 1 + t, f2(t) = 1−t, we have f2 = f21 + f22 , while ‖f‖21 6= ‖f1‖21 +‖f2‖21. to see this, it is enough to remark that f′(t)2 = 2t2(1 + t2)−1, and so ‖f‖21 is an expression depending explicitly on π, while ‖f1‖21 + ‖f2‖21 is a rational number. consequently, the norm of w21 is not expandable. in particular, problem 2 has no solution for the hfs (w21 ,‖∗‖1). example 8. we now give a more abstract example. let s be a hfs on ω endowed with a norm ‖∗‖, given by a compatible inner product 〈∗,∗〉. assume that there exists a family of functions {v1, . . . ,vd} in rs, and a family of points {ω1, . . . ,ωd} in ω such that 〈vj, 1〉 6= 0, f(ωj) = 〈vj, 1〉−1〈f,vj〉, and f = d∑ j=1 f(ωj)vj, ∀f ∈s. if, moreover, 〈vk,vl〉 = d∑ j=1 〈vj, 1〉−1〈vk,vj〉〈vl,vj〉, k, l = 1, . . . ,d, integral representations 121 the norm ‖∗‖ is expandable. we can prove this assertion either directly or applying theorem 7. in fact, theorem 7 shows that, under conditions slightly larger than those from above, there exists a measure for the inner product of s. 4. an interpolation approach the existence of an atomic representing measure for a semi-inner product may be characterized in terms of an interpolation property. the next result is an extension of proposition 3 in [31]. as in theorem 1, this is an extreme situation in the sense that the number of nodes, usually larger than the dimension of the associated hilbert space, is assumed to be equal to that dimension (see also [14], theorem 1.3). theorem 6. let (s,〈∗,∗〉0) be a qhfs on ω, and let h be the hilbert space associated to (s,〈∗,∗〉0), supposed to be finite dimensional. the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in ω with d := dimh atoms if and only if there exist an orthogonal basis of h consisting of idempotents b = {b̂1, . . . , b̂d}, and a set z = {ζ1, . . . ,ζd} ⊂ z such that bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k. in addition, the set z is the support of the corresponding representing measure. proof. the first part of the proof shares some arguments with that of theorem 1. to begin with, assume that the semi-inner product 〈∗,∗〉0 has a representing measure in ω, given by 〈f,g〉0 = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s, with λj > 0 for all j = 1, . . . ,d, and ∑d j=1 λj = 1, where d = dimh, and the points ζ1, . . . ,ζd are distinct. set z = {ζ1, . . . ,ζd}, which is a subset of z, via lemma 2. according to (2.2), we must have i = {f ∈s; f|z}. this shows that there exists a map ρ : h 7→ c(z) given by f̂ 7→ f|z, which is correctly defined, linear and injective. this map is also surjective because we have dim(h) = dim(c(z)). let χk ∈ c(z) be the characteristic function of the set {ζk} and let b̂k ∈ h be the element with ρ(b̂k) = χk, k = 1, . . . ,d. note that the element bk, representing the equivalence class b̂k, may be chosen in rs, because bk|z = χk has real values, and we may replace, if necessary, the function bk by its real part, for each k = 1, . . . ,d. as 〈bj,bk〉0 = ∑d l=1 λl(χjχk)(ζl), and so 〈bj,bk〉0 = 0, 〈bj,bj〉0 = λj = 〈bj, 1〉0 for all j,k = 1, . . . ,d, j 6= k, we deduce that the set {b̂1, . . . , b̂d} is a family of orthogonal idempotents in h = h0, which is actually a basis. clearly, bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k, proving the necessity of the condition in the statement. conversely, if there exist an orthogonal basis of h = h0 consisting of idempotents b = {b̂1, . . . , b̂d}, and a set z = {ζ1, . . . ,ζd}⊂z such that bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k, then 〈∗,∗〉0 has a representing measure whose support is z. indeed, it follows from corollary 2 that for every f ∈s we have f = d∑ j=1 〈bj, 1〉−10 〈f,bj〉0bj + f0, with f0 ∈i. hence f(ζk) = d∑ j=1 〈bj, 1〉−10 〈f,bj〉0bj(ζk) = 〈bk, 1〉 −1 0 〈f,bk〉0 because f0(ζk) = 0, for all k = 1, . . . ,d. taking another function g ∈ s and using the relations from above, we infer that 〈f,g〉0 = d∑ j,k=1 〈bj, 1〉−10 〈bk, 1〉 −1 0 〈f,bj〉0〈ḡ,bk〉0〈bj,bk〉0 = 122 vasilescu d∑ j=1 〈bj, 1〉−10 〈f,bj〉0〈ḡ,bj〉0 = d∑ j=1 〈bj, 1〉0f(ζj)g(ζj). because 〈bj, 1〉0 > 0 for all j = 1, . . . ,d and ∑d j=1〈bj, 1〉0 = 1, we have obtained the existence of a representing measure of 〈∗,∗〉0 in ω having d atoms, whose support is the set z. � proposition 3 from [31] is now a consequence of theorem 6: corollary 3. let s be a finite dimensional function space on ω. a uspf λ : s(2) 7→ c has a representing measure in ω with d := dimhλ atoms if and only if there exist an orthogonal basis of hλ consisting of idempotents b = {b̂1, . . . , b̂d}, and a set ωλ = {ω1, . . . ,ωd} ⊂ zλ such that bj(ωj) = 1 and bk(ωj) = 0 for all j,k = 1, . . . ,d, j 6= k. proposition 2. let (s,〈∗,∗〉0) be a qhfs on ω, and let h be the hilbert space associated to (s,〈∗,∗〉0), supposed to be finite dimensional. then there exists at most one d-atomic representing measure of the semi-inner product 〈∗,∗〉0 with support in ω, having d := dimh atoms. proof. if the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in ω with d := dimh atoms, say µ, it follows from the proof of theorem 6 that the map h3 f̂ 7→ f|z ∈ l2(z,µ) is a unitary operator. indeed, we have only to note that l2(z,µ) can be identified with c(z), so the map f̂ 7→ f|z is bijective, and ‖f̂‖2 = ∫ z |f|2dµ for all f̂. now assume that there exists another d-atomic representing measure of 〈∗,∗〉0 in ω, with support ξ := {ξ1, . . . ,ξd}. as in the previus case, the map f̂ 7→ f|ξ induces a unitary operator from h onto l2(ξ,ν). we now extend µ (resp. ν) to ω by setting µ(ω \ z) = 0 (resp. ν(ω \ ξ) = 0). if ζ ∈ z \ ξ, for the characteristic function χ of the set {ζ} (defined on ω) we must have 0 6= ∫ ω χdµ = 〈χ, 1〉0 = ∫ ω χdν = 0, which is impossible, so z ⊂ ξ. a similar argument shows that ξ ⊂ z. therefore, z = ξ. in fact, this argument shows that the weights of both measures at a given point must be the same. � a more general form of theorem 6 is given by the following. unlike in theorem 6, the number of nodes may be greater than the dimension of the associated hilbert space (see also theorem 5 from [30]). theorem 7. let (s,〈∗,∗〉0) be a qhfs on ω, and let h be the associated hilbert space, supposed to be finite dimensional. the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in ω for some integer d ≥ 1 if and only if d ≥ dimh, and there exist a family of functions {v1, . . . ,vd} in rs, and a family of points {ζ1, . . . ,ζd} in ω, such that 〈vj, 1〉0 6= 0, f(ζj) = 〈vj, 1〉−10 〈f,vj〉0, f − d∑ j=1 f(ζj)vj ∈i, ∀f ∈s, (4.1) and 〈vk,vl〉0 = d∑ j=1 〈vj, 1〉−10 〈vk,vj〉0〈vl,vj〉0, k, l = 1, . . . ,d. (4.2) proof. we assume first that the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure in ω, say µ, and so we may proceed as in the first part of the proof of theorem 6. in other words, there exist a set z := {ζ1, . . . ,ζd}⊂z so that 〈f,g〉0 = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s, with λj > 0 for all j = 1, . . . ,d, and ∑d j=1 λj = 1 for some integer d ≥ 1. in fact, µ({ζj}) = λj, j = 1, . . . ,d. moreover, i = {f ∈s; f|z = 0}. this shows that there exists a map ρ : h 7→ l2(z,µ) given by f̂ 7→ f|z, which is a linear isometry. the image h0 := ρ(h) ⊂ l2(z,µ) is a hilbert subspace, and the map ρ : h 7→h0 is a unitary operator. integral representations 123 let χk ∈ l2(z,µ) be the characteristic function of the set {ζk}, k = 1, . . . ,d. clearly, the family {χ1, . . . ,χd} is an orthogonal basis of the space l2(z,µ), and hence d = dim(l2(z,µ)) ≥ dim(h0) = dim(h). let p0 be the orthogonal projection of l 2(z,µ) onto h0, and let vk|z := p0χk, k = 1, . . . ,d, with vk ∈ s fixed. as for each f = f̄ ∈ s the number 〈f,vj〉0 = λjf(ζj) is real, the function vj may be assumed to be real-valued for all j = 1, . . . ,d. from the equality φ = ∑d j=1 φ(ζj)χj, valid for all φ ∈ l 2(z,µ), we deduce that f|z = ∑d j=1 f(ζj)vk|z, for all f ∈s. therefore, f − ∑d j=1 f(ζj)vj ∈i for all f ∈s. note also that 〈vj, 1〉0 = 〈χj, 1〉 = λj > 0, j = 1, . . . ,d, and f(ζj) = λ −1 j 〈f|z,χj〉 = 〈vj, 1〉 −1 0 〈f,vj〉0, j = 1, . . . ,d, and so equation(4.1) holds. in particular, we have the equality (vkvl)(ζj) = λ −2 j 〈vk,vj〉0〈vl,vj〉0, whence we infer that 〈vk,vl〉0 = d∑ j=1 〈vj, 1〉−1〈vk,vj〉0〈vl,vj〉0, k, l = 1, . . . ,d, which is precisely equation (4.2). conversely, assuming that there exists a family of functions {v1, . . . ,vd} in rs, and a family of points {ζ1, . . . ,ζd} in ω, such that equations (4.1) and (4.2) hold, we can write 〈f,g〉0 = 〈 d∑ j=1 f(ζj)vj, d∑ k=1 g(ζk)vk〉0 = d∑ j=1 d∑ k=1 f(ζj)g(ζk)〈vj,vk〉0 = d∑ j=1 d∑ k=1 f(ζj)g(ζk) d∑ l=1 〈vl, 1〉−10 〈vj,vl〉0〈vk,vl〉0 = d∑ l=1 〈vl, 1〉−10 d∑ j=1 f(ζj)〈vj,vl〉0 d∑ k=1 g(ζk)〈vk,vl〉0 = d∑ l=1 〈vl, 1〉−10 〈f,vl〉0〈ḡ,vl〉0 = d∑ l=1 〈vl, 1〉0f(ζl)g(ζl), f,g ∈s, showing that the inner product of s has a representing measure. � corollary 4. let (s,〈∗,∗〉0) be a qhfs on ω, and let h be the associated hilbert space, supposed to be finite dimensional. if the semi-inner product 〈∗,∗〉0 has an atomic representing measure µ in ω with support z, then card(z) ≥ dim(h), and the map h3 f̂ 7→ f|z ∈ l2(ξ,µ) is a linear isomatry. remark 10. let (s,〈∗,∗〉) be a finite dimensional hfs on ω. then every point evaluation is automatically continuous. consequently, the space s has a reproducing kernel denoted by k(∗,∗), that is, f(ω) = 〈f,kω〉 for all f ∈s and ω ∈ ω, where kω(∗) = k(∗,ω) (see [1, 2, 18] etc. for details). in the present framework, the function k(∗,∗) must be real valued and therefore symmetric. the next result is an application of theorem 6. proposition 3. let (s,〈∗,∗〉) be a finite dimensional hilbert function space on ω, and let k(∗,∗) be its kernel. the inner product 〈∗,∗〉 has a d-atomic representing measure, with d = dims, if and only if there are d distinct points ζ1, . . . ,ζd in ω such that k(ζj,ζk) = 0 for all j,k = 1, . . . ,d, j 6= k. proof. assume first that there are d distinct points ζ1, . . . ,ζd in ω such that k(ζj,ζk) = 0 for all j,k = 1, . . . ,d, j 6= k. set ej(ω) = k(ζj,ζj)−1/2k(ζj,ω),j = 1, . . . ,d, ω ∈ ω. since we have 〈ej,ek〉 = k(ζj,ζj)−1/2k(ζk,ζk)−1/2〈k(ζj,∗),k(ζk,∗)〉 = k(ζj,ζj) −1/2k(ζk,ζk) −1/2k(ζj,ζk) = 0 124 vasilescu if j 6= k, and 〈ej,ej〉 = k(ζj,ζj)−1〈k(ζj,∗),k(ζj,∗)〉 = 1, the family {e1, . . . ,ed} is an orthonormal basis of s. moreover, 〈ej, 1〉 = k(ζj,ζj)−1/2 > 0 for all j, and so, setting bj = k(ζj,ζj) −1/2ej, we obtain a family {b1, . . . ,bd}, which is an orthonormal basis of s consisting of idempotents. clearly, bj(ζj) = 1 and bj(ζk) = 0 if j 6= k. using theorem 6, we infer the existence of an d-atomic representing measure for 〈∗,∗〉. conversely, if the inner product 〈∗,∗〉 has a d-atomic representing measure, that is, 〈f,g〉 =∑d j=1 λjf(ζj)g(ζj), f,g ∈ s, for some distinct points ζ1, . . . ,ζd in ω, with λj > 0 for all j = 1, . . . ,d,∑d j=1 λj = 1, as in the first part of the proof of theorem 6 we find a basis {b1, . . . ,bd} of s consisting of orthogonal idempotents. therefore, f(ω) = ∑d j=1〈bj, 1〉 −1〈f,bj〉bj(ω) for all f ∈ s and ω ∈ ω. moreover, λj = 〈bj, 1〉, bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k. setting, k(ζ,ω) = ∑d j=1 λ −1 j bj(ζ)bj(ω), ζ,ω ∈ ω, we deduce that 〈f,kω〉 = d∑ k=1 λkf(ζk) d∑ j=1 λ−1j bj(ζk)bj(ω) = d∑ j=1 λ−1j ( d∑ k=1 λkf(ζk)bj(ζk) ) bj(ω) = f(ω), f ∈s, ω ∈ ω, because ∑d k=1 λkf(ζk)bj(ζk) = 〈f,bj〉, showing that k(∗,∗) is the kernel of s. in addition, we clearly have k(ζj,ζk) = 0 for all j,k = 1, . . . ,d, j 6= k. � 5. relative multiplicativity as done in [30] and [31] for uspf’s, we can also characterize the existence of a representing measure of a semi-inner product in terms of idempotents. the following is a basic concept, which generalizes a corresponding one from [30], definition 3. definition 6. let (s,〈∗,∗〉0) be a qhfs m-generaterd by the n-tuple θ. let also b = {b̂1, . . . , b̂d} be an orthogonal basis consisting of idempotent elements of the associated hilbert space. we say that the basis b is multiplicative (with respect to θ) if 〈θα,bj〉0〈θβ,bj〉0 = 〈bj, 1〉0〈θα+β,bj〉0 (5.1) whenever |α| + |β| ≤ m, j = 1, . . . ,d. the next result is an extension of theorem 3 from [31], which in turn is an extension of theorem 2 from [30]. in addition, the present proof is simpler and more transparent. theorem 8. let (s,〈∗,∗〉0) be a qhfs on ω, m-generaterd by the n-tuple θ. assume that the associated hilbert space h is finite dimensional. the inner product of h has a representing measure on ω consisting of d := dimh atoms if and only if there exists an orthogonal basis b = {b̂1, . . . , b̂d} of h, consisting of idempotent elements, which is multiplicative with respect to θ, and δ(θ̂) ∈ θ(ω), δ ∈ ∆, where ∆ is the dual basis of b. proof. let b = {b̂1, . . . , b̂d} be an orthogonal basis of h consisting of idempotent elements. every element f̂ ∈h has a unique representation of the form f̂ = ∑d j=1〈b̂j, 1̂〉 −1〈f̂, b̂j〉b̂j, via lemma 3. we consider on h the linear functionals δj(f̂) = 〈b̂j, 1̂〉−1〈f̂, b̂j〉, j = 1, . . . ,d, so f̂ = ∑d j=1 δj(f̂)b̂j for all f̂ ∈ h. in particular, δj(b̂j) = 1 and δj(b̂k) = 0 for all j,k = 1, . . . ,d, j 6= k. in other words, the set ∆ := {δ1, . . . ,δd} is the dual basis of b. next, we define the functions f̂∆ : ∆ 7→ c by f̂∆(δ) = δ(f̂) for all f̂ ∈ h and δ ∈ ∆. setting h∆ := {f̂∆; f̂ ∈ h}, we have a linear map h 3 f̂ 7→ f̂∆ ∈ h∆, which is surjective by definition, and injective because f̂∆ = 0 implies f̂ = 0. in other words, the map h 3 f̂ 7→ f̂∆ ∈ h∆ is a linear isomorphism. in addition, f̂∆ = ∑d k=1 f̂∆(δj)b̂k∆ for all f̂ ∈h. as a matter of fact, the function b̂k∆ is the characteristic function of the set {δk}, k = 1, . . . ,d. this shows that h∆ = c(∆) and the spaces h, h∆ = c(∆), have the same dimension. (here, as before, integral representations 125 c(∆) := {φ : ∆ 7→ c}, is regarded as a finite dimensional c∗-algebra.) in fact, h∆ and c(∆) are isomorphic as c∗-algebras. indeed, the product of two functions from h∆, say f̂∆ = ∑d j=1 δj(f̂)b̂j∆, ĝ∆ = ∑d j=1 δj(ĝ)b̂j∆, is given by f̂∆ĝ∆ = d∑ j=1 δj(f̂)cj(ĝ)b̂j∆, which coincides with the product of c(∆). in particular, f̂∆ĝ∆ is an element of h∆, and the c∗-algebra structure of c(∆) is inherited by h∆. we now assume that b is multiplicative with respect to θ, and that δ(θ̂) ∈ θ(ω), δ ∈ ∆. we note that the space h is spanned by the family {θ̂α; |α| ≤ m}, by hypothesis, so the vector space h∆ is spanned by the family {θ̂α∆; |α| ≤ m}, while the c∗-algebra h∆ is generated by the family {θ̂1∆ . . . θ̂n∆}. we need a more explicit relation between these families, obtained by using (5.2), which will be proved in the following. because we have 〈θα,bj〉0〈θβ,bj〉0 = 〈b̂j, 1̂〉2δj(θ̂α)δj(θ̂β) = 〈bj, 1〉0〈θα+β,bj〉0 = 〈b̂j, 1̂〉2δj(θ̂α+β) whenever |α| + |β| ≤ m and j = 1, . . . ,d, via (5.1), we infer that θ̂α∆θ̂β∆ = θ̂α+β∆, whenever |α| + |β| ≤ m. hence, by recurrence, we deduce that θ̂α∆ = (θ̂∆) α if |α| ≤ m. (5.2) the hypothesis δ(θ̂) ∈ θ(ω), δ ∈ ∆, allows us to find a point ζj ∈ ω such that θ(ζj) = δj(θ̂) for each j = 1, . . . ,d. let f ∈ s be a fixed element. as s is m-generated by θ, there exists a polynomial p ∈ pnm such that f = p◦θ. then we have f̂ = p◦ θ̂, and so f̂∆ = p◦ θ̂∆, via (5.2). hence, we must have δj(f̂) = f̂∆(δj) = p(θ̂∆(δj)) = (p◦θ)(ζj) = f(ζj), j = 1, . . . ,d, this equality leads to 〈f,g〉0 = 〈f̂, ĝ〉 = d∑ j=1 〈bj, 1〉f(ζj)g(ζj), f,g ∈s, (5.3) where 〈bj, 1〉 > 0 for all j and ∑d j=1〈bj, 1〉 = 1, via lemma 3. consequently, the inner product of has a representing measure on ω. conversely, assume that there exists a finite family {ζ1, . . . ,ζd} ⊂ ω, consisting of distinct points, such that 〈f,g〉0 = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s, where λj > 0 for all j, ∑d j=1 λj = 1, and d = dimh. we proceed as in the proof of theorem 6. set z = {ζ1, . . . ,ζd}, which is a subset of z, via lemma 2. as we must have i = {f ∈s; f|z = 0}, there exists a map ρ : h 7→ c(z) given by f̂ 7→ f|z, which is correctly defined, linear and bijective. we denote by χk ∈ c(z) the characteristic function of the set {ζk}, and by b̂k ∈ rh the element with ρ(b̂k) = χk, k = 1, . . . ,d. then the set b := {b̂1, . . . , b̂d} is a family of orthogonal idempotents in h, which is actually a basis. moreover, bj(ζj) = 1 and bk(ζj) = 0 for all j,k = 1, . . . ,d, j 6= k. setting δj(f̂) = f(ζj), f ∈s, j = 1, . . . ,d, and ∆ := {δ1, . . . ,δd}, we infer that ∆ is the dual basis of b, and we have δj(θ̂α) = θ α(ζj) = (θ1(ζj) α1 · · ·θn(ζj)αn ) = δj(θ̂α), whenever |α| ≤ m and j = 1, . . . ,d, showing that b is a multiplicative basis (with respect to θ), as in (5.2). in addition, the obvious equality δj(θ̂) = θ(ζj), j = 1, . . . ,d, concludes the proof of theorem 8. � 126 vasilescu corollary 5. let (s,〈∗,∗〉0) be a qhfs on ω, 1-generated by the n-tuple θ. the semi-inner product 〈∗,∗〉0 has a representing measure on ω consisting of d := dimh atoms if either (1) there exists an orthogonal basis b of h consisting of idempotent elements such that δ(θ̂) ∈ θ(ω), δ ∈ ∆, where ∆ is the dual basis of b, or (2) θ(ω) = rn. proof. because (s,〈∗,∗〉0) is 1-generated, property (5.1) is automatically fulfilled. to get the assertion (1) from the statement we need the inclusion δ(θ̂) ∈ θ(ω), δ ∈ ∆, where ∆ is the dual basis of b, in order to apply the previous theorem, while to get (2), such an inclusion is always true, for an arbitrary orthogonal basis consisting of idempotents. � 6. dimensional stability and consequences in this section we intend to extend and recapture, in the present context, some results regarding the dimensional stability, developed in [29]. we also recall that the concept of dimensional stability in function spaces of polynomials, as approached in [29], is equivalet to that of flatness, due to curto and fialkow (see [9, 10]). remark 11. let s = sm be a function space m-generated by the n-tuple θ := (θ1, . . . ,θn) for some integer m > 0. for every positive integer k ≤ m, we denote by sk the function space k-generated by θ. we fix a semi-inner product 〈∗,∗〉0 = 〈∗,∗〉0m, compatible with s and let 〈∗,∗〉0k be the semi-inner product induced by 〈∗,∗〉0m on sk. clearly, (sk,〈∗,∗〉0k) is a qhfs, and we denote by (hk,〈∗,∗〉k) its associated hilbert space, so hk = sk/ik, where ik = {f ∈ sk;‖f‖0k = 0}, whenever 0 < k ≤ m. if k < m, we have ik ⊂ik+1 ⊂ ···⊂ im, and 〈f + ik,g + ik〉k = 〈f + ik+1,g + ik+1〉k+1 = · · · = 〈f + im,g + im〉m for all f,g ∈ sk. in particular, if 0 < k ≤ l ≤ m, we have a natural linear map jk,l : hk 7→ hl given by jk,l(f + ik) = f + il, f ∈sk, which is an isometry. when l = k + 1, we write sometimes jk instead of jk,k+1. we also put s0 = c, endowed with its natural inner product, so i0 = {0}, and h0 = c. for a given qhfs (s,〈∗,∗〉0) which is m-generated by an n-tuple θ := (θ1, . . . ,θn), we keep the notation from above, if not otherwise specified. the family (hk)nk=0 will be designated as the sequence of hilbert spaces associated to (s,〈∗,∗〉0,θ). when the semi-inner product 〈∗,∗〉0 is expandable, so there exists a uspf λ : s(2) 7→ c 〈∗,∗〉0, the family (hk)nk=0 will be also called the sequence of hilbert spaces associated to (s, λ,θ). definition 7. let (s,〈∗,∗〉0) be a qhfs m-generated by the n-tuple θ := (θ1, . . . ,θn) for some integer m ≥ 1. let also k ∈{0, . . . ,m−1}. we say that the sequence of hilbert spaces (hk)mk=0 associated to (s,〈∗,∗〉0,θ) is stable at k if dim(hk) = dim(hk+1). when the semi-inner product 〈∗,∗〉0 is expandable, so there exists a unique uspf λ : s(2) 7→ c associated to 〈∗,∗〉0, and if (s,〈∗,∗〉0,θ) is stable at k, we say shortly that λ is stable at k. definition 7 implies that the isometry jk : hk 7→hk+1 is actually a unitary operator whenever the sequence (hl)nl=0 is stable at k. in fact, jk unitary means that it is surjective, so for each g ∈sk+1 we can find an f ∈sk such that g −f ∈ik+1. in particular, if g ∈rsk+1, we can find an f ∈rsk such that g −f ∈rik+1. example 9. the stability introduced by definition 7 is a rather strong condition. let us illustrate it by an example. let (s,〈∗,∗〉0) be a function space, with 〈∗,∗〉0 expandable, and m-generated by the n-tuple θ := (θ1, . . . ,θn) for some integer m ≥ 3. let sk be the subspace k-generated by θ, and let 〈∗,∗〉0k be the restriction of 〈∗,∗〉0 to the space sk, which is a compatible semi-inner product on for sk for all k = 1, 2, . . . ,m. let also h0 = {0}. assume that the sequence (hk)mk=0 is stable at 0, so the space h1 is unitarily equivalent to the space h0 = c. then s1 = i1 + c, and h1 = c1̂. let us show that we also have h2 = c1̂. for, note that θj = τj + hj, with τj ∈ c and hj ∈i1, j = 1, . . . ,n. to go further, we need to show that if f ∈ s1 and h ∈ i1, then fh ∈ i2. indeed, because 〈∗,∗〉0 is expandable, and fhfh−hf̄fh = 0, we must have ‖fh‖202 = |〈h,f̄fh〉03| ≤ ‖h‖03‖f̄fh‖03 = ‖h‖ 2 01‖f̄fh‖ 2 03 = 0, integral representations 127 by lemma 4 and the cauchy-schwarz inequality. in particular, with the notation from above, θjθk = τjτk + τjhk + τkhj + hjhk ∈ c + i2, j,k = 1, . . . ,n, which implies the equality h2 = c1̂. a more general situation, and under weaker conditions, will be presented in the following (see theorem 10 and remark 15). the dimensional stability in the case of function spaces of polynomials implies the existence of representing measures for uspf’s (see [9, 10, 29]). nevertheless, in the present context, it is not always the case. example 10. let ω be a nonempty set and let θ1 be a real-valued function on ω. let also s = {c0 + c1θ1; c0,c1 ∈ c}, which is a function space 1-generated by {θ1}. set λ(1) = 1, λ(θ1) = α, λ(θ21 ) = α2 for some α > 0, and extend this map to s(2) by linearity. because we have λ(|c0+c1θ1|2) = |c0+c1α|2 ≥ 0, it follows that λ is a uspf. moreover, iλ = {c0 + c1θ1; c0 + c1α = 0,c0,c1 ∈ c}. in fact, if f = c0 + c1θ1 is arbitrary in s, it can be uniquely written as f = (−c1α + c1θ1) + (c0 + c1α) ∈iλ + c. therefore, the hilbert space hλ is isomorphic to c. in addition, θ̂1 = α1̂. to check whether the uspf λ has a representing measure, we shall apply corollary 5. clearly {1̂} is an idempotent and b := {1̂} is a basis of hλ, which is multiplicative with respect to θ := (θ1), by corollary 5. if δ1(c1̂) = c for all c ∈ c, it clear that ∆ := (δ1) is the dual basis of b. then corollary 5 insures the existence of a 1-atomic representing measure for λ if and only if δ1(θ̂1) = α ∈ θ1(ω). assuming θ1(ω0) = α for some ω0 ∈ ω, we obtain λ(h) = h(ω0) for all h ∈ s(2). in other words, the uspf λ is represented by the dirac measure at ω0. nevertheless, this representation is not necessarily unique because there might exist several points ω with the property θ1(ω) = θ1(ω0). moreover, if α /∈ θ1(ω), we have no representation measure of λ with support in ω. the next result extends lemma 2.3 from [30]. lemma 5. let (s,〈∗,∗〉0) be a qhfs m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable. if the sequence of hilbert spaces (hk)mk=0 associated to (s,〈∗,∗〉0,θ) is stable at m−1, then ( ∑n j=1 θjim)∩s ⊂ im. in particular, θjim−1 ⊂im for all j = 1, . . . ,n. proof. let f = ∑n j=1 θjfj ∈s with fj ∈im for all j = 1, . . . ,n, and let g ∈sm−1. then |〈f,g〉0| ≤ n∑ j=1 |〈θjfj,g)〉0| ≤ n∑ j=1 ‖fj‖0‖θjg‖0 = 0 by the cauchy-schwarz inequality. now, let h ∈ sm−1 be such that f − h ∈ im, which exists because of the stability of (hk)mk=0 at m− 1. then ‖f‖20 = 〈f,h〉0 + 〈f,f −h〉0 = 0, by the previous computation and the cauchy-schwarz inequality. therefore f ∈im. the last assertion is obvious. � remark 12. let (s,〈∗,∗〉0) be a qhfs on ω, m-generated θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable, and let λ : s(2) 7→ c be the uspf associated to 〈∗,∗〉0. we have s = {f = p ◦ θ,p ∈ pnm}, and set sθ := {p|θ(ω); p ∈pnm}, which is a function space on ωθ := θ(ω). we define a map λθ : s (2) θ 7→ c by the equality λθ(φ) = λ(p◦θ), φ = p|ωθ, p ∈ pn2m. the definition is correct because p|ωθ = 0 implies p ◦ θ = 0, and so λ(p ◦ θ) = 0. in fact, λθ is a uspf. in addition, the space sθ is m-generated by τ = (τ1, . . . ,τn), with τj := tj|ωθ, j = 1, . . . ,n. assume now that (s,〈∗,∗〉0,θ) is stable at m − 1. choosing a function φ = p|ωθ, p ∈ pnm, so f := p◦ θ ∈ sm, we can find a function g ∈ sm−1 such that h := f − g ∈ im. as g = q ◦ θ for some q ∈pnm−1, and h = r◦θ for some r ∈pnm, we obtain the equality φ = ψ + ι, where ψ := q|ωθ ∈sθ,m−1 and ι ∈iθ,m because λθ(|ι|2) = λ(|r ◦θ|2) = 0. in other words, λθ is stable at m− 1. 128 vasilescu finding an atomic representing measure for λθ means to solve a ωθ-moment problem, whose solution, when it exists, leads to a representing measure for λ. for a closed k ⊂ rn, the k-moment problem has been approached in [12]. because our framework is slightly larger and our results are generally not covered by the contents of [12], we present in the following a direct approach, independent of [12], but following the lines of [29]. remark 13. from now on, if not otherwise specified, we fix a qhfs (s,〈∗,∗〉0) m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable. next, assume that the sequence of hilbert spaces (hk)mk=0 associated to (s,〈∗,∗〉0,θ) is stable at m − 1. lemma 5 allows us to define correctly the map mj : hm−1 7→ hm by the equality mj(f + im−1) = θjf + im for all j = 1, . . . ,m. setting j = jm−1 (see remark 11), we may consider on the hilbert space hm the linear operators tj = mjj−1 for all j = 1, . . . ,n. note that, fixing f ∈sm and choosing g ∈ sm−1 such that f − g ∈ im, we have tj(f + im) = θjg + im for all j. as mentioned after definition 7, if f ∈ rsm we can choose g ∈ rsm−1 such that f − g ∈ rim. therefore, tj(rhm) ⊂rhm for all j = 1, . . . ,n. with this notation, we have the following. proposition 4. the linear maps tj, j = 1, . . . ,m, are self-adjoint operators, and t = (t1, . . . ,tn) is a commuting tuple on hm. proof. let fk ∈sm and gk ∈sm−1 be such that fk −gk ∈im (k = 1, 2). then 〈t∗j (f1 + im),f2 + im〉 = 〈f1 + im,θjg2 + im〉 = 〈f1,θjg2〉0 = 〈θjg1,f2〉0 = 〈tj(f1 + im),f2 + im〉, via lemma 5 and remark 13. hence t1, . . . ,tn are self-adjoint. we prove now that t1, . . . ,tn mutually commute. it suffices to show that mjj −1mk = mkj −1mj for all j,k = 1, . . . ,n. to show this, fix a function f ∈ sm−1. thus mj(f + im−1) = θjf + im. we can choose gj ∈ sm−1 such that θjf − gj ∈ im. therefore, j−1(θjf + im) = gj + im−1, and mk(gj + im−1) = θkgj + im. similarly, mk(f + im−1) = θkf + im, and we can choose gk ∈ sm−1 such that θkf − gk ∈ im, so mj(gk +im−1) = θjgk +im. to complete the proof, it suffices to show that θkgj −θjgk ∈im. indeed, note that θjθkf −θjgk ∈ θjim and θkθjf −θkgj ∈ θkim. consequently, θkgj −θjgk ∈ (θkim + θjim) ∩sm ⊂im, via lemma 5. consequently, t1, . . . ,tn mutually commute. � remark 14. with the notation from proposition 4, if α,β are multi-indices with |α + β| ≤ m, then tα(θβ+im) = θα+β+im. indeed, if |β| < m, we have tj(θβ+im) = (θjθβ+im), as in remark 13. the general formula can be obtained by recurrence. in particular, if |α| ≤ m, then tα(1 +im) = θα +im, and so p(t)(1 +im) = p(θ) +im for all p ∈pnm. moreover, fixing p ∈pnm, as we have p(θ) = q(θ) + h, with q ∈pnm−1 and h ∈im, we obtain p(t)(1 + im) = q(t)(1 + im). the following assertion is now obtained as an application of theorem 8. see also theorem 2.11 and corollary 2.13 from [29] (proved in a different way), as well as corollary 7.11 from [9]. theorem 9. let (s,〈∗,∗〉0) be a qhfs on ω, m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable. we assume that the sequence of hilbert spaces (hk)mk=0 associated to (s,〈∗,∗〉0,θ) is stable at m − 1 (m ≥ 1). then we have: (1) there exists an orthogonal basis b = {b̂1, . . . , b̂d} of h := hm consisting of idempotent elements, which is multiplicative with respect to θ; (2) the semi-inner product 〈∗,∗〉0 has a d-atomic representing measure with support in ω, where d := dimh, if and only δ(θ̂) ∈ θ(ω), δ ∈ ∆, where ∆ is the dual basis of b; (3) if the semi-inner product 〈∗,∗〉0 has an atomic representing measure with support in ω, this atomic measure is uniquely determined. proof. (1) first of all, note that h = {p(t)1̂; p ∈pnm}. indeed, if f̂ ∈h is an arbitrary element, as s is m-generated by θ, we can find a polynomial p ∈pnm such that f̂ = p◦θ +im, and so f̂ = p(t)(1 +im), via remark 14. integral representations 129 next, we want to apply theorem 8 to show that there exists an orthogonal basis b = {b̂1, . . . , b̂d} of h consisting of idempotent elements, which is multiplicative with respect to θ. we first consider the commuting n-tuple t = (t1, . . . ,tn), consisting of self-adjoint operators, acting in h, given by proposition 4. the spectral theorem for n-tuples of commuting self-adjoint operators (see for instance [6], [24], [26] etc.) implies the existence of commuting self-adjoint projections ej = e({ξ(j)}), j = 1, . . . ,d, such that h(t) = ∑d j=1 h(ξ (j))ej for every function h : σ(t) 7→ c, where σ(t) := {ξ(1), . . . ,ξ(d)} is the joint spectrum of t , which coincides with the support of e. moreover, if the function h is real-valued, the operator h(t) is self-adjoint. in addition, because the space rh is invariant under t1, . . . ,tn (see remark 13), it must be also invariant under h(t), whenever h is real-valued. in particular, rh is invariant under ej, j = 1, . . . ,d,. we now construct an orthogonal family {b̂1, . . . , b̂d} of h consisting of idempotents. because∑d j=1 ej is the identity on h, setting b̂j = ej1̂ ∈ rh,j = 1, . . . ,d, we obtain a decomposition 1̂ = ∑d j=1 b̂j. as ej 6= 0, we must have ejĝ = ĝ 6= 0 for some ĝ = q ◦ θ + im = q(t)(1 + im), with q ∈ pnm, via remark 14. assuming b̂j = 0, we would obtain ejĝ = ĝ = q(t)b̂j = 0, which is not possible. therefore, b̂j 6= 0 for all j = 1, . . . ,d. note also that 〈b̂j, 1̂〉 = 〈b̂j, b̂j〉 > 0, so b̂j is an idempotent for all j = 1, . . . ,d. in other words, {b̂1, . . . , b̂d} is an an orthogonal family in h consisting of idempotent elements. to show that b = {b̂1, . . . , b̂d} is a basis of h it suffices to show that dim(h) = d. for, we consider the sub-c∗-algebra ct generated by t in the c∗-algebra of all linear (automatically bounded) operators acting in h. because ct is finite dimensional, we must have ct = {p(t); p ∈ pns }, for some integer s ≥ m. in fact, choosing an element p(t) with p ∈ pns , we may replace p by a polynomial q ∈ pnm, such that p(t) = q(t). we prove this assertion by recurrence. let pj(t) = tjp(t), with p ∈pnm. then there exists q ∈ pnm−1 such that p(t)1̂ = q(t)1̂, by remark 14. therefore, pj(t)1̂ = qj(t)1̂, where qj(t) = tjq(t) ∈pnm. using the fact that every ĥ ∈h is of the form g(t)1̂ for some polynomial g, we deduce the equality pj(t) = qj(t). an induction argument shows that for p(t) with p ∈pns , s > m, we have the equality p(t) = q(t) for some polynomial q ∈pnm. particularly, ct = {p(t); p ∈pnm}. as mentioned above, the spectral theorem allows us to write p(t) = d∑ j=1 p(ξ(j))ej, p ∈pnm. in particular, {e1, . . . ,ed}, which is clearly a linearly independent family of operators, is actually an algebraic basis of (the linear space) ct . note also that p(t)1̂ = d∑ j=1 p(ξ(j))b̂j, p ∈pnm. consequently, using the equality h = {p(t)1̂; p ∈ pnm} mentioned above, we deduce that dim(h) = dim(ct ) = d. in particular, b = {b̂1, . . . , b̂d} is an orthogonal basis of h, consisting of idempotents. in addition, considering the measure ν(∗) = 〈e(∗)1̂, 1̂〉 on σ(t), and putting λj = 〈ej1̂, 1̂〉 = 〈b̂j, 1̂〉, we have 〈θα,bj〉0〈θβ,bj〉0 = 〈tα1̂,ej1̂〉〈tβ1̂,ej1̂〉 =∫ {ξ(j)} tαdν(t) ∫ {ξ(j)} tβdν(t) = λ2j(ξ (j))α(ξ(j))β = λj ∫ {ξ(j)} tα+βdν(t) = λj〈θα+β,bj〉0 whenever |α| + |β| ≤ m, j = 1, . . . ,d. in other words, the basis b is multiplicative with respect to θ, which concludes the asserion (1) from the statement. to obtain the assertion (2) from the statement, we recall that the dual basis ∆ := {δ1, . . . ,δd} of b is given by δj(f̂) = 〈b̂j, 1̂〉−1〈f̂, b̂j〉, j = 1, . . . ,d. in particular, δj(θ̂k) = λ −1 j 〈ejtk1̂, 1̂〉 = ∫ {ξ(j)} tkdν(t) = ξ (j) k , j,k = 1, . . . ,d. 130 vasilescu theorem 8 shows that the inner product of h has a representing measure on ω consisting of d := dimh atoms if and only δ(θ̂) ∈ θ(ω), δ ∈ ∆, which concludes the proof (2). (3) this assertion is not a direct consequence of proposition 2, becuase we may consider a priori two atomic measures whose supports have different cardinals. to apply proposition 2, we need a supplementary argument. an explicit form of the integral representation whose existence is given in (2) is obtained as for equation (5.3). specifically, choosing ζj ∈ ω such that ξ(j) = δj(ζj), j = 1, . . . ,d, we deduce the equality 〈f,g〉0 = d∑ j=1 λjf(ζj)g(ζj), providing a (d-atomic) representing measure for the semi-inner product of s. let µ be this representing measure of the inner product 〈∗,∗〉0), with support z = {ζ1, . . .ζd} and weights λj = µ(ξ (j)), j = 1, . . . ,d. assume that the semi-inner product 〈∗,∗〉0 has another atomic representing measure in ω, say ν, with support σ := {σ1, . . . ,σg} ⊂ ω. then necessarily, g ≥ d = dim(h), and the map h 3 f̂ 7→ f|σ ∈ l2(ξ,ν) is an isometry (see corollary 4). moreover, im = {f ∈s; f|σ = 0}. let bj be the linear operator on l 2(σ,ν) given by bjh = θjh for all j = 1, . . . ,n and h ∈ l2(σ,ν). then b := (b1, . . . ,bn) is an n-tuple of commuting self-adjoint operators. with t = (t1, . . . ,tn) as before, fixing f̂ ∈ h with f ∈ sm, and choosing g ∈ sm−1 with h := f − g ∈ im, so tjf̂ = θ̂jg, (f − g)|σ = 0, and (θjg)|ξ = (θjf)|σ = bj(f|σ). in other words, identifying the hilbert space h with the (hilbert) subspace {f|σ; f ∈ s}, we see that bj is an extension of the operator tj for all j = 1, . . . ,n. in particular, the spectral measure e of t is the restriction of the spectral measure eb of b to h. we now consider the elements eb({σj})(1|σ), which must belong to h, because h is invariant under eb. therefore, setting ĉj = eb({σj})(1|σ) = e({σj})1̂, j = 1, . . . ,g, as in the second part of the proof, {ĉ1, . . . , ĉg} is an orthogonal family of nonnull idempotent elements of h. consequently, we must have g = d, and so dim(l2(ξ,ν)) = d. we may now apply proposition 2, to get the asserton (3). the next result is an extension of theorem 2.6 from [29] (see also theorem 7.8 and corollary 7.9 from [9]). theorem 10. let (s,〈∗,∗〉0) be a qhfs on ω, m-generated by θ = (θ1, . . . ,θn), with 〈∗,∗〉0 expandable. we assume that the sequence of hilbert spaces (hk)mk=0 associated to (s,〈∗,∗〉0,θ) is stable at m − 1 (m ≥ 1). then the semi-inner product 〈∗,∗〉0 can be uniquely extended to an expandable semi-inner product of s∞, which has a d-atomic measure in ω, where d = dim(hm). proof. using the notation and arguments from (the proof of) theorem 9, we have 〈f,g〉0 = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s, which is an integral representation of 〈∗,∗〉0 by a d-atomic measure. a direct extension of this formula allows us to define 〈f,g〉0∞ = d∑ j=1 λjf(ζj)g(ζj), f,g ∈s∞, which is an expandable semi-inner product on s∞. we want to show that 〈∗,∗〉0∞ is uniquely determined. let 〈∗,∗〉′0∞,〈∗,∗〉′′0∞ be two expandable semi-inner products on s∞, both of them extending 〈∗,∗〉0. for k ≥ m + 1, let sk = {p◦ θ; p ∈ pnk}, i ′ k = {f ∈ sk;〈f,f〉 ′ 0∞ = 0}, i′′k = {f ∈ sk;〈f,f〉 ′′ 0∞ = 0}. clearly, im ⊂i′k ∩i ′′ k for all k ≥ m + 1. we shall show by induction that for every multi-index α, with |α| ≥ m, there exists an element fα ∈ sm−1, such that θα −fα ∈ i′|α| ∩i ′′ |α|. the assertion is obvious for |α| = m, via the stability at m− 1. assume the property true for all multi-indices of length k, for a k ≥ m, and let us prove it for integral representations 131 multi-indices of length k + 1. if |α| = k + 1, there exists a number j ∈ {1, . . . ,n} and a multi-index β with |β| = k such that θα = θjθβ. by the induction hypothesis, we can find a function fβ ∈ sm−1 such that θβ −fβ ∈i′k ∩i ′′ k . therefore, θ α −θjfβ ∈i′k+1 ∩i ′′ k+1, by the cauchy-schwarz inequality. further, θjfβ ∈ sm and so we can find a function fj,β ∈ sm−1 such that θjfβ − fj,β ∈ im, via the stability at m− 1. consequently, θα −fα = θα −θjfβ + θjfβ −fj,β ∈i′k+1 ∩i ′′ k+1 + im = i ′ k+1 ∩i ′′ k+1, where fα = fj,β. extending the property from above to arbitrary functions from s∞, we deduce, in particular, that for every pair of function f1,f2 ∈ sk, with k ≥ m, we can find a pair g1,g2 ∈ sm−1 such that fj −gj ∈i′k ∩i ′′ k , j = 1, 2. therefore, 〈f1,f2〉′0∞ = 〈g1,g2〉0 = 〈f1,f2〉 ′′ 0∞, showing the uniqueness of the natural extension 〈∗,∗〉0∞ of the semi-inner product 〈∗,∗〉0∞. � remark 15. from the proof of the previous theorem, we deduce that for all k ≥ m and f ∈sk there exists g ∈sm−1 such that f −g ∈ik, where ik := {h ∈sk;〈h,h〉0∞ = 0}. this implies that all spaces hk := sk/ik are unitarily equivalent hilbert spaces. this assertion is true even for k = ∞. 7. an example example 11. this is an example related to the paper [16](see also [14]). specifically, we look for atomic representing measures of a given semi-inner products of the space, p2m (m ≥ 1), whose suport lies in the curve γ := {(t,t3) ∈ r2; t ∈ r}. in what follows, we want to solve a γ-moment problem (see subsection 2.3), trying to use some of our techniques. the basic function space will be sm = {p |γ; p ∈ p2m} = p2m,γ. because the representation of an element p |γ ∈ sm is, in general, not unique, let us characterize the subspace j := {p ∈ p2m; p |γ = 0}. given a polynomial p(x1,x2) =∑ 0≤k+l≤m ak,lx k 1x l 2 in p2m, we have p |γ = 0 if and only if p(t,t3) = ∑ 0≤k+l≤m ak,lt k+3l = 0 for all t ∈ r. explicitly, we must have p(t,t3) = 3m∑ j=0   ∑ l∈i(j) aj−3l,l  tj = 0, t ∈ r, with i(j) = {l ≥ 0; 3l ≤ j ≤ m + 2l}, which happens if and only if ∑ l∈i(j) aj−3l,l = 0 whenever 0 ≤ j ≤ 3m. in other words, j = {p(x1,x2) = ∑ 0≤k+l≤m ak,lx k 1x l 2; ∑ l∈i(j) aj−3l,l = 0, 0 ≤ j ≤ 3m}. moreover, the space sm is isomorphic to the space p2m/j , the elements of sm may be regarded as equivalence classes of p2m modulo j , and they will be denoted by p̃ = p + j , p ∈p2m. let θ1,θ2 : r 7→ r be given by θ1(t) = t, θ2(t) = t3, t ∈ r. we have a natural map of sm into p13m given by p̃ 7→ p ◦θ. since p ∈j is equivalent to p ◦θ = 0, this map is correctly defined, linear, and injective; it is surjective too. indeed, we may define a linear map from p13m into p2m in the following way. if p0(t) = ∑3m k=0 akt k, we use the representation p0(t) = m∑ l=0 a3lt 3l + m−1∑ l=0 a3l+1t 3l+1 + m−1∑ l=0 a3l+2t 3l+2. we now make a certain choice, obviously not unique. namely, we replace the monomial t3l by the monomial xl2, the monomial t 3l+1 by the monomial x1x l 2, and the monomial t 3l+2 by the monomial x21x l 2, so we obtain the polynomial p0(x1,x2) = m∑ l=0 a3lx l 2 + m−1∑ l=0 a3l+1x1x l 2 + m−1∑ l=0 a3l+2x 2 1x l 2, we clearly have p0 = p0 ◦θ, showing that the image of p̃0 is precisely p0. 132 vasilescu let 〈∗,∗〉0 be an expandable semi-inner product of sm. we want to define an expandable semi-inner product of p13m. if p,q ∈p13m, as we have p = p ◦θ,q = q◦θ for some p,q ∈p2m, we set 〈p,q〉0 := 〈p̃,q̃〉0. the definition does not depend on the choice of p,q ∈p2m with p = p ◦θ,q = q◦θ because, if either p ◦ θ = 0 or q ◦ θ = 0, we must have 〈p̃,q̃〉0 = 0. in addition, it clearly provides an expandable semi-inner product of p13m. the existence of atomic measures for expandable semi-inner products of spaces of polynomials in one variable, in particular for the space p13m, treated as a moment problem, can be explicitly described (see for instance [8]). we restrict ourselves to a particular case of theorem 9, applied to p13m, endowed with the expandable inner product 〈p,q〉0, p,q ∈p13m. the dimensional stability at 3m−1 is given by a condition of the form ‖t3m − 3m−1∑ k=0 ckt k‖0 = 0, for some polynomial ∑3m−1 k=0 ckt k, which is equivalent to the condition∥∥∥∥∥x3m2 − m−1∑ l=0 c3lx l 2 − m−1∑ l=0 c3l+1x1x l 2 − m−1∑ l=0 c3l+2x 2 1x l 2 + j ∥∥∥∥∥ 0 , expressed only in the given terms. a solution of this moment problem means the existence of (distinct) points τ1, . . . ,τr ∈ r and positive numbers λ1, . . . ,λr with λ1 + · · · + λr = 1 such that 〈p,q〉0 = d∑ j=1 λjp(τj)q(τj), p,q ∈p13m. therefore, if p,q ∈sm are written under the form p = p ◦θ,q = q◦θ with p,q ∈p2m, we have 〈p̃,q̃〉0 = 〈p,q〉0 = d∑ j=1 λjp(τj,τ 3 j )q(τj,τ 3 j ), which is a solution of our γ-moment problem. references [1] j. agler and j. e. mccarthy, pick interpolaton and hilbert function spaces, ams graduate studies in mathematics, vol 44, providence, rhode island, 2002. [2] d. alpay, the schur algorithm, reproducing kernel spaces and system theory, smf/ams texts and monographs, vol. 5, 2001. [3] s. axler, p. bourdon and w. ramey, harmonic function theory, springer-verlag, new york/berlin/heidelberg, 2001. [4] c. bayer and j. teichmann, the proof of tchakaloff’s theorem, proc. amer. math. soc., 134 (10) (2006), 3035-3040. [5] c. berg, j. p. r. christensen and p. ressel, harmonic analysis on semigroups. theory of positive definite and related functions, graduate texts in mathematics, 100. springer-verlag, new york, 1984. [6] m. s. birman and m.z. solomjak, spectral theory of self-adjoint operators in hilbert space, d. reidel publishing company, dordrecht, 1987. [7] j.b. conway, a course in abstract analysis, graduate studies in mathematics vol. 141, ams, providence, rhode island, 2012. [8] r. e. curto and l. a. fialkow, recursiveness, positivity, and truncated moment problems, huston j. math. 17 (4) (1991), 603-635. [9] r. e. curto and l. a. fialkow, solution of the truncated complex moment problem for flat data, memoirs of the ams, number 568, 1996. [10] r. e. curto and l. a. fialkow, flat extensions of positive moment matrices: recursively generated relations, memoirs of the ams, number 648, 1998. [11] r. e. curto and l. a. fialkow, a duality proof of tchakaloff’s theorem, j. math. anal. appl., 269 (2002), 519-532. [12] r. e. curto and l. a. fialkow, truncated k-moment problems in several variables, j. operator theory, 54 (1) (2005), 189-226. [13] r. e. curto and l. a. fialkow, an analogue of the riesz-haviland theorem for the truncated moment problem, j. funct. anal. 255 (2008), 2709-2731. [14] r. e. curto, l. a. fialkow and h. m. möller, the extremal truncated moment problem, integral equations oper. theory, 60 (2008), 177-200. integral representations 133 [15] n. dunford and j.t. schwartz, linear operators, part i: general theory, interscience publishers, new york/london, 1958. [16] l. fialkow, solution of the truncated moment problem with variety y = x3, trans. amer. math. soc. 363 (2011), 3133-3165. [17] l. fialkow and j. nie, positivity of riesz functionals and solutions of quadratic and quartic moment problems, j. funct. anal. 258 (2010), 328-356. [18] e. hille, introduction to general theory of reproducing kernels, rocky mountain j. math. 2 (1972), 321-368. [19] j. h. b. kemperman, the general moment problem, a geometric approach, ann. math. statist. 39 (1968), 93-122. [20] m. laurent, sums of squares, moment matrices and optimization over polynomials, emerging applications of algebraic geometry, ima vol. math. appl., 149, 157-270, springer, new york, 2009. [21] h. m. möller, on square positive extensions and cubature formulas, j. comput. appl. math. 199 (2006), 80-88. [22] m. putinar, on tchakaloff’s theorem, proc. amer. math. soc. 125 (1997), 2409-2414. [23] j. stochel, solving the truncated moment problem solves the full moment problem, glasg. math. j. 43(2001), 335-341. [24] k. schmüdgen, unbounded self-adjoint operators on hilbert space, graduate texts in mathematics, 265. springer, dordrecht, 2012. [25] v. tchakaloff, formule de cubatures mécaniques à coefficients non négatifs, bull. sci. math. 81 (2), 1957, 123-134. [26] f.-h. vasilescu, analytic functional calculus and spectral decompositions, d. reidel publishing company, dordrecht, 1982. [27] f.-h. vasilescu, operator theoretic characterizations of moment functions, 17th ot conference proceedings, theta, 2000, 405-415. [28] f.-h. vasilescu, spaces of fractions and positive functionals, math. scand. 96 (2005), 257-279. [29] f.-h. vasilescu, dimensional stability in truncated moment problems, j. math. anal. appl. 388 (2012), 219-230 [30] f.-h. vasilescu, an idempotent approach to truncated moment problems, integral equations oper. theory 79 (3) (2014), 301-335. [31] f.-h. vasilescu, square positive functionals in an abstract setting, operator theory: the state of the art, 145-167, theta, 2016. department of mathematics, university of lille 1, 59655 villeneuve d’ascq, france ∗corresponding author: fhvasil@math.univ-lille1.fr 1. introduction 2. preliminaries 2.1. function spaces and compatible semi-inner products 2.2. continuous point evaluations 2.3. generators of function spaces 2.4. reduction to finite dimensional spaces 2.5. quasi-hilbert function spaces and idempotents 3. uspf's versus semi-inner products 4. an interpolation approach 5. relative multiplicativity 6. dimensional stability and consequences 7. an example references international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 86-101 http://www.etamaths.com existence of solutions and ulam stability for caputo type sequential fractional differential equations of order α ∈ (2, 3) bashir ahmad1,2,∗, mohammed m. matar2 and ola m. el-salmy2 abstract. we study initial value problems of sequential fractional differential equations and inclusions involving a caputo type differential operator of the form: ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) , where α ∈ (2, 3) and λi(i = 1, 2) are nonzero constants. several existence and uniqueness results are accomplished by means of fixed point theorems. sufficient conditions for ulam stability of the given problem are also presented. examples are constructed for the illustration of obtained results. then we investigate the inclusions case of the problem at hand. an initial value problem for coupled sequential fractional differential equations is also discussed. 1. introduction fractional calculus is a generalization of the classical differentiation and integration to arbitrary non-integer order. the idea of fractional calculus has been a subject of interest not only among mathematicians but also among physicists and engineers. they have used it effectively to improve the mathematical modelling of several phenomena occurring in scientific and engineering disciplines such as viscoelasticity [1], electrochemistry [2], electromagnetism [3], biology ( [4], [5]), control ( [6], [7], [14]), diffusion process ( [8], [9], [10]), economics [11], chaotic systems ( [12], [13]), variational problems [15] etc. the mathematical models involving fractional order derivatives are more realistic and practical than the classical models as they help to trace the history of the associated phenomena. also, the enriched material on theoretical aspects and analytic/numerical methods for solving fractional order models attracts the modelers. during the last decade, many researchers have focused on the existence of solutions for initial and boundary value problems of fractional differential equations see ( [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]) and the references cited therein. the stability theory of fractional differential systems needs more investigations than the one for classical differential systems, since fractional derivatives are nonlocal and have weak singular kernels. for recent development on the stability of fractional differential systems, for instance, see ( [27], [28], [29]) and the references cited therein. the ulam type stabilities [30] for fractional differential systems are quite significant in realistic problems, numerical analysis, biology and economics. for details and examples, we refer the reader to the works ( [31], [32], [33]). in this paper, we investigate the existence of solutions for an initial value problem of sequential fractional differential equations given by{ ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) x(t) = f(t,x(t)), α ∈ (2, 3), t ∈ j, x(k)(a) = bk,k = 0, 1, 2, (1.1) where cdαa denote the caputo fractional derivative of order α, λ1 and λ2 are nonzero constants, f : j ×r → r is a given continuous function, and j = [a,t], t > a ≥ 0. the rest of the paper is organized as follows. in section 2, we recall some preliminary concepts and prove an auxiliary lemma, which plays a fundamental role in defining the fixed point problem associated with the problem at hand. existence results and illustrative examples are presented in received 19th may, 2017; accepted 25th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 26a33, 34a08, 30c45. key words and phrases. caputo fractional derivative; sequential fractional differential equations; ulam stability. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 86 existence of solutions and ulam stability 87 section 3, while ulam type stability for the given problem is discussed in section 4. an existence result for the multivalued (inclusions) case of the problem (1.1) is proved in section 5. an initial value problem for coupled sequential fractional differential equations is formulated and investigated in section 6. 2. preliminaries let us first recall some basic notions of fractional calculus ( [16], [17]). definition 2.1. the riemann-liouville fractional integral of order α > 0 is defined as iαa+h(t) = ∫ t a (t−s)α−1h(s) γ(α) ds, provided the integral exists, and i0a+h(t) = h(t). definition 2.2. the caputo derivative of fractional order α > 0 is defined as cdαa+h(t) = ∫ t a (t−s)n−α−1h(n)(s) γ(n−α) ds, n− 1 < α < n,n = [α] + 1, where [α] denotes the integer part of the real number α. let c(j,r) be a banach space of all continuous real valued functions defined on j endowed with the norm defined by ‖x‖ = sup{|x(t)| , t ∈ j} , cn(j,r) be a banach space of all n times continuously differentiable on j. by ac(j,r), we denote the space of functions which are absolutely continuous on j, and by acn(j,r), the space of functions f which have continuous derivatives up to order n− 1 on j such that f(n) ∈ ac(j,r). here we remark that the fractional integral iαa+ is bounded operator on c(j,r) (see lemma 2.8 [16]), and the fractional derivative cdαa+h exists almost everywhere if h ∈ acn(j,r) (see theorem 2.1 [16]). notice that c1(j,r) ⊆ ac(j,r) ⊆ c(j,r); in general, cn(j,r) ⊆ acn(j,r) ⊆ cn−1(j,r). therefore, the fractional derivative cdαa+h is continuous for any h ∈ cn(j,r) ( see theorem 2.2 [16]). lemma 2.1. ( [16]) let x ∈ cn(j,r) (or acn(j,r)), f ∈ c(j,r) (or ac(j,r)), and ci ∈ r, i = 0, 1, 2, ...,n− 1. then cdαa+ ( iαa+f(t) ) = f(t), iαa+( cdαa+x(t)) = x(t) + n−1∑ i=0 ci (t−a) i , cdαa+x(t) = 0 implies that x(t) = n−1∑ i=0 ci (t−a) i . consider the linear variant problem{ ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) x(t) = g(t), t ∈ j x(k)(a) = bk,k = 0, 1, 2. (2.1) lemma 2.2. let x ∈ c3(j,r), g ∈ c(j,r), and λ21 = 4λ2, then the linear problem (2.1) is equivalent to the integral equation x(t) = b0 + b1 ( λ1 λ2 − λ1 λ2 e− λ1 2 (t−a) − (t−a) e− λ1 2 (t−a) ) +b2 ( 1 λ2 − 1 λ2 e− λ1 2 (t−a) − 2 λ1 (t−a) e− λ1 2 (t−a) ) + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sg(r)drds. (2.2) 88 ahmad, matar and el-salmy proof. applying the fractional integral operator iα−2a+ to sequential fractional differential equation in (2.1), we get x(2)(t) + λ1x (1)(t) + λ2x(t) = i α−2 a+ g(t) + c2. (2.3) using the initial conditions in (2.3) leads to c2 = b2 + λ1b1 + λ2b0. now, let y(t) = e λ1 2 tx(t), then y(1)(t) = e λ1 2 tx(1)(t) + λ1 2 e λ1 2 tx(t), and y(2)(t) = e λ1 2 tx(2)(t) + λ1e λ1 2 tx(1)(t) + λ21 4 e λ1 2 tx(t). substituting these values in (2.3), we get y(2)(t) = e λ1 2 tiα−2a+ g(t) + (b2 + λ1b1 + λ2b0) e λ1 2 t. (2.4) integrating equation (2.4) twice from a to t, we obtain y(t) = y(a) + y(1)(a) (t−a) + (b2 + λ1b1 + λ2b0) ( 1 λ2 e λ1 2 t − 1 λ2 e λ1 2 a − 2 λ1 e λ1 2 a (t−a) ) + 1 γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sg(r)drds, which, on account of y(t) = e λ1 2 tx(t), yields x(t) = e− λ1 2 (t−a)x(a) + ( x(1)(a) + λ1 2 x(a) ) (t−a) e− λ1 2 (t−a) + (b2 + λ1b1 + λ2b0) ( 1 λ2 − 1 λ2 e− λ1 2 (t−a) − 2 λ1 (t−a) e− λ1 2 (t−a) ) + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sg(r)drds. (2.5) making use of the initial conditions in (2.5) and rearranging the terms we get (2.2). conversely, applying the fractional operator ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) to the integral equation (2.2) and using lemma 2.1, we obtain the problem (2.1). this completes the proof. � if g has a maximum gmax on j, then the integral term in equation (2.2) has upper bounds 2gmax (t −a) α−1 |λ1|γ(α− 1) ∣∣∣1 −e−λ12 (t−a)∣∣∣ <   2gmax(t−a)α−1 |λ1|γ(α−1) ,λ1 > 0 2gmax(t−a)α−1 |λ1|γ(α−1) ( 1 + e− λ1 2 (t−a) ) ,λ1 < 0 (2.6) for any t ∈ j. therefore, in the next sections, we prefer to use the upper bound 2gmax(t−a) α−1 |λ1|γ(α−1) ( 1 + e− λ1 2 (t−a) ) for each nonzero λ1. 3. existence theorems we establish sufficient conditions for existence of solutions to problem (1.1) using different types of fixed point theorems. in view of lemma 2.2, we transform the initial value problem (1.1) into an operator equation as ψx(t) = b0 + b1 ( λ1 λ2 − λ1 λ2 e− λ1 2 (t−a) − (t−a) e− λ1 2 (t−a) ) +b2 ( 1 λ2 − 1 λ2 e− λ1 2 (t−a) − 2 λ1 (t−a) e− λ1 2 (t−a) ) + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r,x(r))drds (3.1) where x ∈ c(j,r). note that λ2 is nonnegative for all values of λ1 as λ21 = 4λ2. if the operator ψ has a fixed point in c(j,r), then the problem (1.1) has this fixed point as a solution. lemma 3.1. the operator ψ : c(j,r) → c(j,r) given by (3.1) is completely continuous. existence of solutions and ulam stability 89 proof. obviously, continuity of the operator ψ follows from the continuity of the function f. let u be a bounded proper subset of c(j,r), then for any t ∈ j, and x ∈u, there exists a positive constant l such that |f(t,x(t))| ≤ l. accordingly, (2.6) yields |ψx(t)| ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2l (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) , which implies that ψ is a bounded operator on u ⊂c(j,r). furthermore, for a < t1 < t2 < t , we have |(ψx)(t2) − (ψx)(t1)| ≤ |b1λ1| λ2 ∣∣∣e−λ12 (t1−a) −e−λ12 (t2−a)∣∣∣ + |b1| ∣∣∣(t1 −a) e−λ12 (t1−a) − (t2 −a) e−λ12 (t2−a)∣∣∣ + |b2| λ2 ∣∣∣e−λ12 (t1−a) −e−λ12 (t2−a)∣∣∣ + 2 |b2| |λ1| ∣∣∣(t1 −a) e−λ12 (t1−a) − (t2 −a) e−λ12 (t2−a)∣∣∣ + ∣∣∣∣∣ e −λ1 2 t2 γ(α− 2) ∫ t2 a ∫ s a (t2 −s) (s−r) α−3 e λ1 2 sf(r,x(r))drds − e− λ1 2 t1 γ(α− 2) ∫ t1 a ∫ s a (t1 −s) (s−r) α−3 e λ1 2 sf(r,x(r))drds ∣∣∣∣∣ ≤ |b1λ1| λ2 ∣∣∣e−λ12 (t1−t2) − 1∣∣∣e−λ12 (t2−a) + |b1| ∣∣∣(t1 − t2) e−λ12 (t1−t2) + (t2 −a) (e−λ12 (t1−t2) − 1)∣∣∣e−λ12 (t2−a) + |b2| λ2 ∣∣∣e−λ12 (t1−t2) − 1∣∣∣e−λ12 (t2−a) + |b2| λ2 ∣∣∣(t1 − t2) e−λ12 (t1−t2) + (t2 −a) (e−λ12 (t1−t2) − 1)∣∣∣e−λ12 (t2−a) + le− λ1 2 t1e− λ1 2 (t2−t1) γ(α− 2) ∫ t1 a ∫ s a (t2 − t1) (s−r) α−3 e λ1 2 sdrds + le− λ1 2 t1 ( e− λ1 2 (t2−t1) − 1 ) γ(α− 2) ∫ t1 a ∫ s a (t1 −s) (s−r) α−3 e λ1 2 sdrds + le− λ1 2 t1e− λ1 2 (t2−t1) γ(α− 2) ∫ t2 t1 ∫ s a (t2 −s) (s−r) α−3 e λ1 2 sdrds, which tends to zero independently of x as t2 → t1. this implies that ψ is equicontinuous on j. in consequence, it follows by the arzela-ascoli theorem that the operator ψ is completely continuous. this completes the proof. � next we recall the schauder’s fixed-point theorem ( [34]). theorem 3.1. if u is a closed , bounded, convex subset of a banach space x and the mapping ∆ : u →u is completely continuous, then ∆ has a fixed point in u. theorem 3.2. let |f(t,x(t))| ≤ l. then there exists a solution of the problem (1.1) on j. 90 ahmad, matar and el-salmy proof. it is a direct consequence of lemma 3.1 if u is taken to be a closed, bounded, convex subset of c(j,r). � theorem 3.3. assume that there exists a constant k such that lim x→0 f(t,x) x = k,t ∈ j, then the problem (1.1) has a solution on j. proof. by the given assumption, it follows that |f(t,x(t))| ≤ (1 + k) |x(t)| , whenever |x(t)| < δ, for a fixed number δ > 0. therefore, we can define a subset u as u ={x ∈ c(j,r) : |x(t)| ≤ δ,t ∈ j} . clearly u is a closed, bounded and convex subset of c(j,r). if x ∈ u, then |f(t,x(t))| < δ (1 + k) , for any t ∈ j. on the other hand, the operator ψ : u →u defined by (3.1) is completely continuous by lemma 3.1. therefore, by schauder’s fixed point theorem 3.1, the problem (1.1) has a solution. this completes the proof. � the next result is based on krasnoselskii’s fixed point theorem ( [34]) . theorem 3.4. let m be a closed convex and nonempty subset of a banach space c(j,r). let θ, and φ be the operators such that (i) θu + φv ∈m whenever u,v ∈m; (ii) θ is compact and continuous; (iii) φ is a contraction. then there exists x ∈m such that x = θx + φx. theorem 3.5. assume that (h1): for any t ∈ j, and x,y ∈ r, there exists a positive constants c such that |f(t,x) −f(t,y)| ≤ c |x−y| . (h2): for any t ∈ j, and x ∈ r, there exists µ ∈ c(j,r+) such that |f(t,x)| ≤ µ(t). (h3): ω < 1, where ω = 2c (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) . then, the problem (1.1) has at least one solution on j. proof. define a set br = {x ∈ c(j,r) : ‖x‖ ≤ r}, where r is a positive real number satisfying the inequality r ≥ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2l (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) . introduce operators θ and φ on br as (θx) (t) = b0 + b1 ( λ1 λ2 − λ1 λ2 e− λ1 2 (t−a) − (t−a) e− λ1 2 (t−a) ) +b2 ( 1 λ2 − 1 λ2 e− λ1 2 (t−a) − 2 λ1 (t−a) e− λ1 2 (t−a) ) , and (φx)(t) = e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r,x(r))drds. existence of solutions and ulam stability 91 for x,y ∈br, t ∈ j, using (h2), we find that |θx(t) + φy(t)| ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2‖µ‖(t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) . thus, θx + φy ∈br. by (h1), for x,y ∈br, t ∈ j, we have |(φx) (t) − (φy) (t)| ≤ 2c (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) ‖x−y‖ , that is, ‖(φx) − (φy)‖≤ ω‖x−y‖ . since ω < 1 by (h3), φ is a contraction. the operator θ is continuous and is uniformly bounded, since ‖θx‖ ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) . as in the proof of lemma 3.1, it can be shown that θ is equicontinuous and relatively compact on br. hence, by the arzela-ascoli theorem, θ is compact on br. thus all the assumptions of theorem 3.4 are satisfied. therefore, the problem (1.1) has at least one solution on j. this completes the proof. � our next result deals with the uniqueness of solutions for the problem (1.1) and is based on the banach contraction principle. theorem 3.6. assume that (h1) and (h3) hold. then there exists a unique solution for the problem (1.1) on j. proof. let supt∈j |f(t, 0)| = a, and r ≥ (1 −β) −1 γ, where γ = |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2a (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) . then we show that ψbr ⊂ br, where br = {x ∈ c(j,r) : ‖x‖ ≤ r}. this follows by the following estimate |ψx(t)| ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2a (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) + 2c (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) ‖x‖ ≤ (1 −β) r + βr = r, for any x ∈br. moreover, for x, y ∈ c(j,r) and for each t ∈ j, we can obtain |(ψx)(t) − (ψy)(t)| ≤ 2c (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) ‖x−y‖ , 92 ahmad, matar and el-salmy which, by taking the norm on j and using the assumption (h3), yields that ψ is a contraction. thus, the conclusion of the theorem follows by the contraction mapping principle. this completes the proof. � our last existence theorems is based on leray-schauder degree theorem [34]. for that we set the notations: f = |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2e (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) , (3.2) g = 2d (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) < 1. (3.3) theorem 3.7. assume that there exist constants d and e such that |f(t,x)| ≤ d |x| + e, for t ∈ j, x ∈ r. then there exists a solution for the problem (1.1) on j. proof. define a ball br = {x ∈ c(j,r) : ‖x‖ < r} for some positive real number r which will be determined later. we show that ψ : br → c(j,r) satisfies 0 /∈ (i −λψ) (∂br) , for any x ∈ ∂br, and λ ∈ [0, 1], where ∂br denotes the boundary set of br. define the homotopy hλ(x) = h(λ,x) = x−λψx,x ∈ c(j,r), λ ∈ [0, 1]. then, by lemma 3.1, hλ is completely continuous. let i denote the identity operator, then the homotopy invariance and normalization properties of topological degrees imply that deg(hλ,br, 0) = deg((i −λψ) ,br, 0) = deg(h1,br, 0) = deg(h0,br, 0) = deg(i,br, 0) = 1, since 0 ∈br. by the nonzero property of the leray-schauder degree, h1(x) = x− ψx = 0 for at least one x ∈br. to find r, we assume that x(t) = λψx(t) for some λ ∈ [0, 1] and for all t ∈ j. then, using the given assumption together with (3.2) and (3.3), we get |x(t)| = |λψx(t)| ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + 2 (t −a)α−1 |λ1|γ(α− 1) ( 1 + e− λ1 2 (t−a) ) (d‖x‖ + e) ≤ f + g‖x‖ , which implies that ‖x‖≤ f 1 −g . the value of r = f−g+1 1−g > ‖x‖ is sufficient for applicability of leray-schauder degree theorem. this completes the proof. � example 3.1. consider the following nonlinear fractional boundary value problem{ ( cd2.10+ − 2cd1.10+ + 4cd0.10+ ) x(t) = f(t,x(t)), t ∈ (0, 1), x(0) = x ′ (0) = x ′′ (0) = 1. (3.4) here α = 2.1, b0 = b1 = b2 = 1. existence of solutions and ulam stability 93 (a): for the illustration of theorem 3.3, let us take f(t,x(t)) = sin(x) + x 8 . (3.5) obviously limx→0 f(t,x) x = 1/4 = k. thus the conclusion of theorem 3.3 applies to the problem (3.4) with f(t,x(t)) given by (3.5). (b): in order to explain theorem 3.5, we consider f(t,x(t)) = 1 √ 36 + t2 |x| (1 + |x|) + 1 12 . (3.6) with the given data, it is easy to verify that the conditions (h1) and (h2) hold true with c = 1 6 and µ(t) = 1√ 36+t2 + 1 12 , while (h3) is satisfied with ω = 1+e 6γ(1.1) < 1. thus all the conditions of theorem 3.5 are satisfied. therefore there exists at least one solution for the problem (3.4) with f(t,x(t)) given by (3.6). (c) we illustrate theorem 3.6 with the aid of the following nonlinear function f(t,x(t)) = 1 √ 49 + t2 tan−1(x) + 1 14 . (3.7) clearly, in this case, c = 1 7 and ω = 1+e 7γ(1.1) < 1. thus, by the conclusion of theorem 3.6, the problem (3.4) with f(t,x(t)) given by (3.7) has a unique solution on [0, 1]. (d) let us consider the following nonlinear function to demonstrate the application of theorem 3.7 f(t,x(t)) = 1 √ 25 + t sin(x) + |x| 3(1 + |x|) + 1 3 . (3.8) with the given values, it is found that d = 1/5, e = 2/3, and g = 1+e 5γ(1.1) < 1. thus, by theorem 3.7, there exists a solution for the problem (3.4) with f(t,x(t)) given by (3.8). 4. ulam stability here we investigate the ulam stability criteria for the problem (1.1) via its equivalent integral equation y(t) = b0 + b1 ( λ1 λ2 − λ1 λ2 e− λ1 2 (t−a) − (t−a) e− λ1 2 (t−a) ) +b2 ( 1 λ2 − 1 λ2 e− λ1 2 (t−a) − 2 λ1 (t−a) e− λ1 2 (t−a) ) + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r,y(r))drds. (4.1) if y ∈ c3 (j,r) and f : j ×r → r is continuous function, then the nonlinear operator υ : c (j,r) → c (j,r) given by υy(t) = ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) y(t) −f(t,y(t)) is continuous. definition 4.1. the system (1.1) is ulam-hyers stable if there exists a real number c > 0 such that for each � > 0 and for each solution y ∈ c(3) (j,r) , ‖υy‖≤ �,t ∈ j, (4.2) then there exists a solution x ∈ c (j,r) of (1.1) satisfying the inequality: ‖x−y‖≤ c�1, t ∈ j, where �1 is a positive real number depending on �. definition 4.2. the system (1.1) is generalized ulam-hyers stable if there exists σ ∈ c (r+,r+) such that for each solution y ∈ c(3) (j,r) of (1.1) there exists a solution x ∈ c (j,r) of (1.1) with |x(t) −y(t)| ≤ σ(�), t ∈ j. 94 ahmad, matar and el-salmy definition 4.3. the system (1.1) is ulam-hyers-rassias stable with respect to φ ∈ c (j,r+) if there exists a real number c > 0 such that for each � > 0 and for each solution y ∈ c(3) (j,r) of (1.1), |υy(t)| ≤ �φ(t), t ∈ j, (4.3) then there exists a solution x ∈ c (j,r) of (1.1) with |x(t) −y(t)| ≤ c�1φ(t), t ∈ j, where �1 is a positive real number depending on �. theorem 4.1. assume that (h1) and (h3) hold. then the system (1.1) satisfies both ulam-hyers and generalized ulam-hyers stability criteria. proof. let x ∈ c (j,r) be a unique solution of (3.1) that satisfies equation (1.1) by theorem 3.6. let y ∈ c(3) (j,r) be any solution satisfying (4.2). then, by lemma 2.2, y satisfies the integral equation (4.1). moreover, the equivalence in lemma 2.2 implies the equivalence between the operators υ and ψ − i on every solution y ∈ c (j,r) that satisfies equations (1.1) and (4.1). therefore, by the fixed point property of the operator ψ (given by (3.1)), we have |y(t) −x(t)| = |y(t) − ψy(t) + ψy(t) − ψx(t)| ≤ |ψx(t) − ψy(t)| + |ψy(t) −y(t)| ≤ ω‖x−y‖ + �, where � > 0 and ω is defined in (h3) . in consequence, it follows that ‖x−y‖≤ � 1 −ω . if we let �1 = � 1−ω , and c = 1, then, the ulam-hyers stability condition is satisfied. more generally, for σ(�) = � 1−ω , the generalized ulam-hyers stability condition is also satisfied. this completes the proof. � theorem 4.2. assume that (h1) and (h3) hold and there exists a function φ ∈ c (j,r+) satisfying the condition (4.3). then the problem (1.1) is ulam-hyers-rassias stable with respect to φ. proof. following the arguments employed in the proof of theorem 4.1, we can obtain that ‖x−y‖≤ �1φ(t), where �1 = � 1−ω . his completes proof. � as an application, the problem given by (3.4) with f(t,x(t)) = t|x(t)| 4(1+|x(t)|) is ulam-hyers stable, and generalized ulam-hyers stable. in addition, if there exists a function φ ∈ c (j,r+) satisfying the condition (4.3), then the problem (3.4) with the given value of f(t,x(t)) is ulam-hyers-rassias stable. 5. multivalued case in this section, we study the multivalued (inclusions) analogue of the problem (1.1) given by{ ( cdαa+ + λ1 cdα−1a+ + λ2 cdα−2a+ ) x(t) ∈ f(t,x(t)), x(k)(a) = bk,k = 0, 1, 2, (5.1) where f : [a,t] × r → 2r \{∅},α ∈ (2, 3), t ∈ [a,t] and the other quantities are the same as defined in the problem (1.1). before proceeding for the existence result for the problem (5.1), which relies on bohnenblust-karlin fixed point theorem, we outline the background material for multi-valued maps [35, 36]. let c[a,t] denote a banach space of continuous functions from [a,t] into r with the norm ‖x‖ = supt∈[a,t]{|x(t)|}. let l1([a,t],r) be the banach space of functions x : [a,t] → r which are lebesgue integrable and normed by ‖x‖l1 = ∫t a |x(t)|dt. a multi-valued map h : x → 2x (a): is convex (closed) valued if h(x) is convex (closed) for all x ∈ x, where (x,‖.‖) is a banach space. existence of solutions and ulam stability 95 (b): is bounded on a bounded set if h(b) = ∪x∈bg(x) is bounded in x for any bounded set b of x (that is, supx∈b{sup{|y| : y ∈ h(x)}} < ∞). (c): is called upper semi-continuous (u.s.c.) on x if for each x0 ∈ x, the set h(x0) is a nonempty closed subset of x, and if for each open set b of x containing h(x0), there exists an open neighborhood n of x0 such that h(n) ⊆ b. (d): is said to be completely continuous if g(b) is relatively compact for every bounded subset b of x. (e): has a fixed point if there is x ∈ x such that x ∈ h(x). if the multi-valued map h is completely continuous with nonempty compact values, then h is u.s.c. if and only if h has a closed graph, that is, xn → x∗, yn → y∗, yn ∈ h(xn) imply y∗ ∈ h(x∗). in the following study, bcc(x) denotes the set of all nonempty bounded, closed and convex subset of x. furthermore, we need the following assumptions: (m1) let f : [a,t] ×r → bcc(r); (t,x) → f(t,x) be measurable with respect to t for each x ∈ r, u.s.c. with respect to x for a.e. t ∈ [a,t], and for each fixed x ∈ r, the set sf,x := {f ∈ l1([a,t],r) : f(t) ∈ f(t,x) for a.e. t ∈ [a,t]} is nonempty. (m2) for each ρ > 0, there exists a function pρ ∈ l1([a,t],r+) such that ‖f(t,x)‖ = sup{|v| : v(t) ∈ f(t,x)}≤ pρ(t) for each (t,x) ∈ [a,t] ×r with |x| ≤ ρ, and lim inf ρ→+∞ (∫t a pρ(t)dt ρ ) = µ < ∞. (5.2) next we state the known lemmas which we need in the forthcoming analysis. lemma 5.1. (bohnenblust-karlin [37]) let d ⊂ x be nonempty bounded, closed, and convex. let h : d → 2x \ {0} be u.s.c. with closed, convex values such that h(d) ⊂ d and h(d) is compact. then h has a fixed point. lemma 5.2. [38] let f be a multi-valued map satisfying the condition (m1) and φ is linear continuous from l1(i,r) → c(i). then the operator φ◦sf : c(i) → bcc(c(i)),x 7→ (φ◦sf )(x) = φ(sf,x) is a closed graph operator in c(i) ×c(i), where i is a compact real interval. theorem 3.1. assume that (m1) and (m2) hold and that µ < γ(α− 1) (t −a)α−1 , (5.3) where µ is given by (5.2). then there exists at least one solution for the problem (5.1) on [a,t]. proof. in order to transform the problem (5.1) into a fixed point problem, we introduce a multi-valued map ω : c[a,t] → 2c[a,t] given by ω(x) = { h ∈ c[a,t] : h(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r,x(r))drds, f ∈ sf,x } . the proof will be complete once it is shown that ω satisfies all the assumptions of lemma 5.1. in consequence, ω will have a fixed point, showing that the problem (5.1) has a solution. 96 ahmad, matar and el-salmy in the first step, we show that ω(x) is convex for each x ∈ c[a,t]. for that, let h1,h2 ∈ ω(x). then there exist f1,f2 ∈ sf,x such that for each t ∈ [a,t], we have hi(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sfi(r,x(r))drds, i = 1, 2. let 0 ≤ σ ≤ 1. then, for each t ∈ [a,t], we have [σh1 + (1 −σ)h2](t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 s[σf1 + (1 −σ)f2](r,x(r))drds. since sf,x is convex (f has convex values), therefore it follows that σh1 + (1 −σ)h2 ∈ ω(x). next we show that there exists a positive number ρ such that ω(bρ) ⊆ bρ, where bρ = {x ∈ c[a,t] : ‖x‖≤ ρ}. clearly bρ is a bounded closed convex set in c[a,t] for each positive constant ρ. if it is not true, then for each positive number ρ, there exists a function xρ ∈ bρ,hρ ∈ ω(xρ) with ‖ω(xρ)‖ > ρ, and hr(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 spρ(r)drds, for some pρ ∈ sf,xρ. on the other hand, in view of (a2), we have r < ‖ω(xr)‖≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 spρ(r)drds ≤ |b0| + |b1| ( |λ1| λ2 + |λ1| λ2 e− λ1 2 (t−a) + (t −a) e− λ1 2 (t−a) ) + |b2| ( 1 λ2 + 1 λ2 e− λ1 2 (t−a) + 2 |λ1| (t −a) e− λ1 2 (t−a) ) + (t −a)(α−1) γ(α− 1) ∫ t a pρ(s)ds. dividing both sides by ρ and taking the lower limit as ρ →∞, we find that µ ≥ γ(α− 1) (t −a)α−1 , which contradicts (5.3). hence there exists a positive number ρ1 such that ω(bρ1 ) ⊆ bρ1. existence of solutions and ulam stability 97 now we show that ω(bρ) is equicontinuous. let a < t1 < t2 < t, x ∈ bρ and h ∈ ω(x), then there exists f ∈ sf,x such that for each t ∈ [a,t], we have h(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r)drds } and |h(t2) −h(t1)| ≤ |b1λ1| λ2 ∣∣∣e−λ12 (t1−t2) − 1∣∣∣e−λ12 (t2−a) + |b1| ∣∣∣(t1 − t2) e−λ12 (t1−t2) + (t2 −a) (e−λ12 (t1−t2) − 1)∣∣∣e−λ12 (t2−a) + |b2| λ2 ∣∣∣e−λ12 (t1−t2) − 1∣∣∣e−λ12 (t2−a) + |b2| λ2 ∣∣∣(t1 − t2) e−λ12 (t1−t2) + (t2 −a) (e−λ12 (t1−t2) − 1)∣∣∣e−λ12 (t2−a) + e− λ1 2 t1e− λ1 2 (t2−t1) γ(α− 2) ∫ t1 a ∫ s a (t2 − t1) (s−r) α−3 e λ1 2 spρ(s)drds + e− λ1 2 t1 ( e− λ1 2 (t2−t1) − 1 ) γ(α− 2) ∫ t1 a ∫ s a (t1 −s) (s−r) α−3 e λ1 2 spρ(s)drds + e− λ1 2 t1e− λ1 2 (t2−t1) γ(α− 2) ∫ t2 t1 ∫ s a (t2 −s) (s−r) α−3 e λ1 2 spρ(s)drds, obviously the right hand side of the above inequality tends to zero independently of x ∈ bρ as t2 → t1. thus, ω is equi-continuous. as ω satisfies the above three assumptions, therefore it follows by ascoliarzela theorem that ω is a compact multi-valued map. finally, we show that ω has a closed graph. let xn → x∗,hn ∈ ω(xn) and hn → h∗. we will show that h∗ ∈ ω(x∗). by the relation hn ∈ ω(xn), we mean that there exists fn ∈ sf,xn such that for each t ∈ [a,t], hn(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sfn(r)drds } . thus we need to show that there exists f∗ ∈ sf,x∗ such that for each t ∈ [a,t], h∗(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf∗(r)drds } . 98 ahmad, matar and el-salmy let us consider the continuous linear operator φ : l1[a,t],r) → c[a,t] so that f 7→ φ(f)(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf(r)drds } . observe that ‖hn(t) −h∗(t)‖ = ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 s ( fn(r) −f∗(r))drds → 0 as n →∞. thus, it follows by lemma 2.2 that φ◦sf is a closed graph operator. further, we have hn(t) ∈ φ(sf,xn). since xn → x∗, therefore, lemma 2.2 yields h∗(t) = b0 + b1 [ λ1 λ2 − (λ1 λ2 + (t−a) ) e− λ1 2 (t−a) ] +b2 [ 1 λ2 − ( 1 λ2 + 2 λ1 (t−a) ) e− λ1 2 (t−a) ] + e− λ1 2 t γ(α− 2) ∫ t a ∫ s a (t−s) (s−r)α−3 e λ1 2 sf∗(r)drds } . hence, we conclude that ω is a compact multi-valued map, u.s.c. with convex closed values. thus, all the assumptions of lemma 5.1 are satisfied and so by the conclusion of lemma 5.1, ω has a fixed point x which is a solution of problem (5.1). this completes the proof. � 6. coupled system of equations in this section, we study an initial value problem of coupled sequential fractional differential equations given by   ( cdα1a+ + λ11 cdα1−1a+ + λ21 cdα1−2a+ ) u1(t) = f1(t,u1(t),u2(t)),( cdα2a+ + λ12 cdα2−1a+ + λ22 cdα2−2a+ ) u2(t) = f2(t,u1(t),u2(t)), u (k) 1 (a) = bk1, u (k) 2 (a) = bk2, k = 0, 1, 2 (6.1) where αi ∈ (2, 3), t ∈ j,λij ∈ r, λ21j = 4λ2j and fi : j × r 2 → r (i,j = 1, 2) are continuous function satisfying the following condition. (s1): there exist ci ∈ r+ such that |fi(t,u1,u2) −fi(t,v1,v2)| ≤ ci(|u1 −v1| + |u2 −v2|),ui,vi ∈ r, t ∈ j. consider the banach product space y = c(j,r) × c(j,r) of all ordered pairs (x,y) such that x,y ∈ c(j,r), and equipped with the norm ‖(x,y)‖y = ‖x‖ + ‖y‖ . in view of lemma 2.2, we define an operator λ : y → y by λ(u1,u2)(t) = (λ1(u1,u2)(t), λ2(u1,u2)(t)) , t ∈ j, where λi(u1,u2)(t) = b0i + b1i ( λ1i λ2i − λ1i λ2i e− λ1i 2 (t−a) − (t−a) e− λ1i 2 (t−a) ) +b2i ( 1 λ2i − 1 λ2i e− λ1i 2 (t−a) − 2 λ1i (t−a) e− λ1i 2 (t−a) ) + e− λ1i 2 t γ(αi − 2) ∫ t a ∫ s a (t−s) (s−r)αi−3 e λ1i 2 sfi(r,u1(r),u2(r))drds, i = 1, 2. existence of solutions and ulam stability 99 theorem 6.1. assume that (s1) is satisfied. then there exists a unique solution for the problem (6.1) on j whenever β1 + β2 < 1, where βi = 2ci (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) , i = 1, 2. proof. let us set supt∈j |fi(t, 0, 0)| = ai, and r ≥ (1 −β1 −β2) −1 (γ1 + γ2) , where γi = |b0i| + |b1i| ( |λ1i| λ2i + |λ1i| λ2i e− λ1i 2 (t−a) + (t −a) e− λ1i 2 (t−a) ) + |b2i| ( 1 λ2i + 1 λ2i e− λ1i 2 (t−a) + 2 |λ1i| (t −a) e− λ1i 2 (t−a) ) + 2ai (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) , i = 1, 2. then |λi(u1,u2)(t)| ≤ |b0i| + |b1i| ( |λ1i| λ2i + |λ1i| λ2i e− λ1i 2 (t−a) + (t −a) e− λ1i 2 (t−a) ) + |b2i| ( 1 λ2i + 1 λ2i e− λ1i 2 (t−a) + 2 |λ1i| (t −a) e− λ1i 2 (t−a) ) + 2ai (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) + 2ci (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) (‖u1‖ + ‖u2‖) ≤ γi + βi (‖u1‖ + ‖u2‖) . taking the norm of the above inequality for t ∈ j, it easily follows that λbr ⊂ br, where br = {(x,y) ∈ y : ‖(x,y)‖y ≤ r}. moreover, for (u1,u2), (v1,v2) ∈ y and for each t ∈ j, we have |λ(u1,u2)(t) − λ(v1,v2)(t)| ≤ |(λ1(u1,u2)(t) − λ1(v1,v2)(t)| + |(λ2(u1,u2)(t) − λ2(v1,v2)(t)| ≤ 2c1 (t −a) α1−1 |λ11|γ(α1 − 1) ( 1 + e− λ11 2 (t−a) ) (‖u1 −v1‖ + ‖u2 −v2‖) + 2c2 (t −a) α2−1 |λ12|γ(α2 − 1) ( 1 + e− λ12 2 (t−a) ) (‖u1 −v1‖ + ‖u2 −v2‖) ≤ (β1 + β2) (‖u1 −v1‖ + ‖u2 −v2‖) , which, on taking the norm for t ∈ j and using the condition β1 +β2 < 1, implies that λ is a contraction. thus, the conclusion of the theorem follows by the contraction mapping principle. this completes the proof. � our second result is based on leray-schauder alternative [39]. theorem 6.2. (leray–schauder alternative) let f : e → e be a completely continuous operator and e(f) ={x ∈ e : x = λf(x), for some 0 < λ < 1} . then either the set e(f) is unbounded, or f has at least one fixed point. in the sequel, we need the following growth condition: (s2): there exist ki ∈ r+ such that |fi(t,u1,u2)| ≤ ki(1 + |u1| + |u2|),ui ∈ r, t ∈ j,i = 1, 2. 100 ahmad, matar and el-salmy for computational convenience, we define di = 2ki (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) , (6.2) bi = di + |b0i| + |b1i| ( |λ1i| λ2i + |λ1i| λ2i e− λ1i 2 (t−a) + (t −a) e− λ1i 2 (t−a) ) + |b2i| ( 1 λ2i + 1 λ2i e− λ1i 2 (t−a) + 2 |λ1i| (t −a) e− λ1i 2 (t−a) ) . (6.3) theorem 6.3. assume that (s2) is satisfied. then there exists at least one solution for the problem (6.1) on j whenever d1 + d2 < 1 proof. clearly continuity of fi (i = 1, 2) implies the continuity of λi and hence the continuity of λ. let u be a bounded proper subset of y , there exist positive constants l1 and l2 such that |fi(t,u1(t),u2(t))| ≤ li for t ∈ j, (u1,u2) ∈ y. following the procedure of the proof in theorem 3.1, one can show that the operator λ : y → y is completely continuous. let (u1,u2) ∈ e(λ), such that (u1,u2) = λλ(u1,u2). for any t ∈ j, u1(t) = λλ1(u1,u2), and u2(t) = λλ2(u1,u2). then, using the assumption (s2) and (6.2)-(6.3), we obtain |ui(t)| ≤ |λi(u1,u2)(t)| ≤ |b0i| + |b1i| ( |λ1i| λi2 + |λ1i| λ2i e− λ1i 2 (t−a) + (t −a) e− λ1i 2 (t−a) ) + |bi2| ( 1 λ2i + 1 λ2i e− λ1i 2 (t−a) + 2 |λ1i| (t −a) e− λ1i 2 (t−a) ) + 2ki (t −a) αi−1 |λ1i|γ(αi − 1) ( 1 + e− λ1i 2 (t−a) ) (1 + ‖u1‖ + ‖u2‖) ≤ bi + di (‖u1‖ + ‖u2‖) . taking the norm of the above inequality for t ∈ j, it follows in a straightforward manner that ‖(u1,u2)‖≤ b1 + b2 1 −d1 −d2 . hence the set e(λ) ={(u1,u2) ∈ y : (u1,u2) = λλ(u1,u2) for some 0 < λ < 1} is bounded. by the application of leray-schauder alternative theorem, we deduce that the problem (6.1) has at least one solution on j. this completes the proof. � references [1] f. meral, t. royston and r. magin, fractional calculus in viscoelasticity: an experimental study, commun. nonlinear sci. numer. simul. 15 (2010), 939-945. [2] k. oldham, ractional differential equations in electrochemistry, adv. eng. softw. 41 (2010), 9-12. [3] c. lee and f. chang, fractional-order pid controller optimization via improved electromagnetism-like algorithm, expert syst. appl. 37 (2010), 8871-8878. [4] e. ahmed, a. el-sayed and h. el-saka, equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, j. math. anal. appl. 325 (2007), 542-553. [5] f. liu and k. burrage, novel techniques in parameter estimation for fractional dynamical models arising from biological systems, comput. math. appl. 62 (2011), 822-833. [6] g. mophou, optimal control of fractional diffusion equation, comput. math. appl. 61 (2011), 68-78. [7] j. wang, y. zhou and w. wei, optimal feedback control for semilinear fractional evolution equations in banach spaces, syst. control lett. 61 (2012), 472-476. [8] r. gorenflo and f. mainardi, some recent advances in theory and simulation of fractional diffusion processes, j. comput. appl. math. 229 (2009), 400-415. [9] x. jiang, m. xu and h. qi, the fractional diffusion model with an absorption term and modified fick’s law for non-local transport processes, nonlinear anal. real world appl. 11 (2010), 262-269. [10] i. sokolov, a. chechkin and j. klafter, fractional diffusion equation for a power-law truncated levy process, physica a. 336 (2004), 245-251. [11] r. nigmatullin, t. omay and d. baleanu, on fractional filtering versus conventional filtering in economics, commun. nonlinear sci. numer. simul. 15 (2010), 979-986. [12] m. faieghi, s. kuntanapreeda, h. delavari, d. baleanu, lmi-based stabilization of a class of fractional-order chaotic systems, nonlinear dynam. 72 (2013), 301-309. existence of solutions and ulam stability 101 [13] f. zhang, g. chen, c. li, j. kurths, chaos synchronization in fractional differential systems, phil. trans. r. soc. a 371 (2013), 20120155. [14] k. balachandran, m. matar, j. j. trujillo, note on controllability of linear fractional dynamical systems, j. control decis. 3 (2016), 267-279. [15] o.p. agrawal, generalized variational problems and euler-lagrange equations, comput. math. appl. 59 (2010), 1852-1864. [16] a.a. kilbas, h.m. srivastava, and j.j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [17] y. zhou, basic theory of fractional differential equations, world scientific publishing co. pte. ltd., hackensack, nj, 2014. [18] b. ahmad, m. m matar, r. p. agarwal, existence results for fractional differential equations of arbitrary order with nonlocal integral boundary conditions, bound. value probl. 2015:220 (2015), 13 pp. [19] b. ahmad, m. m. matar, s. k. ntouyas, on general fractional differential inclusions with nonlocal integral boundary conditions, differ. equ. dyn. syst. (2016), doi:10.1007/s12591-016-0319-5. [20] m. matar, on existence of positive solution for initial value problem of nonlinear fractional differential equations of order 1 < α ≤ 2, acta math. univ. comenianae, 84 (1) (2015), 51-57. [21] l. zhang, b. ahmad, g. wang, explicit iterations and extremal solutions for fractional differential equations with nonlinear integral boundary conditions, appl. math. comput. 268 (2015), 388-392. [22] d. qarout, b. ahmad, a. alsaedi, existence theorems for semilinear caputo fractional differential equations with nonlocal discrete and integral boundary conditions, fract. calc. appl. anal. 19 (2016), 463-479. [23] b. ahmad, sharp estimates for the unique solution of two-point fractional-order boundary value problems, appl. math. lett. 65 (2017), 77-82. [24] s. aljoudi, b. ahmad, j.j. nieto, a. alsaedi, a coupled system of hadamard type sequential fractional differential equations with coupled strip conditions, chaos solitons fractals 91 (2016), 39-46. [25] b. ahmad, s.k. ntouyas, a. alsaedi, fractional differential equations and inclusions with nonlocal generalized riemann-liouville integral boundary conditions, int. j. anal. appl. 13 (2017), 231-247. [26] h.m. srivastava, remarks on some families of fractional-order differential equations, integral transforms spec. funct. 28 (2017), 560-564. [27] w. deng, smoothness and stability of the solutions for nonlinear fractional differential equations, nonlinear anal. 72 (2010), 1768-1777. [28] r. w. ibrahim, stability of a fractional differential equation, miskolc math. notes 13 (2012), 39-52. [29] r. agarwal, s. hristova and d. o’regan, a survey of lyapunov functions, stability and impulsive caputo fractional differential equations, fract. calc. appl. anal. 19 (2016), 290-318. [30] n. brillouët-belluot, j. brzdȩk, and k. ciepliński, on some recent developments in ulam’s type stability, abstr. appl. anal. (2012), art. id 716936, 41 pp. [31] j. wang, l. lv and y. zhou, ulam stability and data dependence for fractional differential equations with caputo derivative, electron. j. qual. theory differ. equ. 2011 (2011), art. id 63. [32] s. abbas, m. benchohra and a. petrusel, ulam stability for partial fractional differential inclusions via picard operators theory, electron. j. qual. theory differ. equ. 2014 (2014), art. id 51. [33] r.w. ibrahim and h.a. jalab, existence of ulam stability for iterative fractional differential equations based on fractional entropy, entropy 17 (2015), 3172-3181. [34] d.r. smart, fixed point theorems, cambridge tracts in mathematics, no. 66. cambridge university press, londonnew york, 1974. [35] k. deimling, multivalued differential equations, de gruyter, berlin, 1992. [36] s. hu and n. papageorgiou, handbook of multivalued analysis, vol. i. theory. mathematics and its applications, 419. kluwer academic publishers, dordrecht, 1997. [37] h. f. bohnenblust and s. karlin, on a theorem of ville, in contributions to the theory of games. vol. i, pp. 155-160, princeton univ. press, 1950. [38] a. lasota and z. opial, an application of the kakutani-ky fan theorem in the theory of ordinary differential equations, bull. acad. polon. sci. ser.sci. math. astronom. phys. 13 (1965), 781-786. [39] a. granas and j. dugundji, fixed point theory, springer-verlag, new york, 2003. 1naam research group, department of mathematics, faculty of science, king abdulaziz university, p.o. box 80203, jeddah 21589, saudi arabia 2mathematics department, al-azhar university-gaza, palestine ∗corresponding author: bashirahmad−qau@yahoo.com 1. introduction 2. preliminaries 3. existence theorems 4. ulam stability 5. multivalued case 6. coupled system of equations references int. j. anal. appl. (2023), 21:87 a symbolic algorithm for solving doubly bordered k-tridiagonal interval linear systems sivakumar thirupathi, nirmala thamaraiselvan∗ department of mathematics, srm institute of science and technology, kattankulathur 603203, tamil nadu, india ∗corresponding author: nirmala@srmist.edu.in abstract. doubly bordered k-tridiagonal interval linear systems play a crucial role in various mathematical and engineering applications where uncertainty is inherent in the system’s parameters. in this paper, we propose a novel symbolic algorithm for solving such systems efficiently. our approach combines symbolic computation techniques with interval arithmetic to provide rigorous solutions in the form of tight interval enclosures. by exploiting the tridiagonal structure and employing a divide-andconquer strategy, our algorithm achieves significantly reduced computational complexity compared to existing numerical methods. we also present theoretical analysis and provide numerical experiments to demonstrate the effectiveness and accuracy of our algorithm. the proposed symbolic algorithm offers a valuable tool for handling doubly bordered k-tridiagonal interval linear systems and opens up possibilities for addressing uncertainty in real-world problems with improved efficiency and reliability. 1. introduction doubly bordered k-tridiagonal interval linear systems (dbktils) are a class of linear systems that arise in various applications, including control theory, optimisation and numerical analysis. they are characterised by a tridiagonal matrix structure, where each diagonal element is an interval containing the true value of the corresponding coefficient. dbktils are more general than classical tridiagonal systems, where the diagonal elements are real numbers. in recent years, there has been a growing interest in developing numerical methods for solving dbktils. one approach is to use interval arithmetic, which is a mathematical tool that enables the computation of guaranteed bounds for received: jun. 15, 2023. 2020 mathematics subject classification. 15a09, 15a23, 65f05, 65g30. key words and phrases. tridiagonal interval matrix; k-tridiagonal interval matrix; ul factorization; interval arithmetic; interval determinant; interval linear system. https://doi.org/10.28924/2291-8639-21-2023-87 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-87 2 int. j. anal. appl. (2023), 21:87 the solution. however, existing interval-based methods for dbktils suffer from high computational complexity and memory requirements. in this context, a symbolic algorithm has been proposed for solving dbktils. the algorithm is based on the theory of tridiagonal matrices and employs a symbolic approach to compute the coefficients of the solution. the main advantage of the proposed method is its low computational complexity and memory requirements, which make it suitable for large-scale problems. andelic m et al. [1] discussed an extended eigenvalue-free interval for the eccentricity matrix of threshold graphs. da fonseca cm et al. [2] gave a clear overview of the k-tridiagonal matrix and spectral theory, as well as a graphical look at the inverse powers of the matrix. fan y et al. [3, 4] used the interval matrix technique to investigate the global dissipativity and quasi-synchronization of asynchronous updating fractional-order memristor-based neural networks (aufmnns). ganesan et al. [5] presented a new set of arithmetic operations for interval numbers by which those discrepancies in general can be reduced to some extent. joe d. hoffman [6] discussed the thomas algorithm extensively. david hartman et al. [7] investigated eigenvalue decomposition for both symmetric and general interval matrices. huang x et al. [8] looked into the problem of asymptotically global synchronization of fractional-order memristive networks (fmnns) with multiple delays that change over time. ji-teng jia [9] presented a new way to find the determinants of periodic tridiagonal matrices using a threeterm recursion that doesn’t break down. kaucher [10] introduced the dual operator as a monadic operator. the dual operator combines the duality principle, which states that every element has an opposite. it combines the monadic principle, which states that the result of any operation should be a single element. losonczi l [11] discussed imperfect pentadiagonal toeplitz matrices, providing explicit formulae in terms of entries for their determinant, eigenvalues and eigenvectors. m el-mikkawy et al. [12] discovered that k-tridiagonal matrices are crucial for defining generalized k-fibonacci numbers. marrero ja [13] suggested a fast and reliable numerical solver for dealing with determined oppositebordered tridiagonal linear systems. nirmala et al. [14] developed a new way to find the inverse of an interval matrix. this makes it a powerful tool for solving interval linear equations. parker jt et al. [15] presented a hybrid multigrid-thomas algorithm designed to efficiently invert one-dimensional tridiagonal matrix equations in a highly scalable fashion in the context of time-evolving partial differential equation systems. sengupta et al. [16] proposed a simple and effective index for comparing any two intervals. shams solary m [17] investigated tridiagonal 3-toeplitz matrices with different ranks. shehab n et al. [18] presented a symbolic algorithm for solving doubly bordered k-tridiagonal linear systems. in order to develop the proposed algorithm, partitioning and ul factorization are used. thirupathi s et al. [19] developed an algorithm based on generalized interval arithmetic to determine general k-tridiagonal interval matrix determinants and inverses. tanasescu a et al. [20,21] used the block diagonalization of a general k-tridiagonal matrix to study its singular value decomposition. wei f et al. [22] used the interval matrix technique to investigate the finite-time stability of memristor-based inertial neural networks (minns). xiao s et al. [23] studied the passivity analysis problem of a class of int. j. anal. appl. (2023), 21:87 3 fractional-order neural networks with interval parameter uncertainties (fonns-ipus). the motivation behind developing a symbolic algorithm for solving doubly bordered k-tridiagonal interval linear systems based on generalized interval arithmetic is to improve the accuracy and efficiency of solving these types of systems, which can have a significant impact on various scientific and engineering applications. the paper is organized as follows: section 2 overviews generalized interval arithmetic. section 3 presents the main results and theorem. section 4 gives double-bordered k-tridiagonal interval matrices. section 5 suggests algorithms for finding the determinant and solving doubly-bordered k-tridiagonal interval linear systems. section 6 provides two numerical examples to show how the algorithm works. “ 2. preliminary notes let d = ir∪ ir = {[u1,u2] : u1,u2 ∈ r} is the set of generalized intervals that are the proper and improper intervals, where ir = {ũ = [u1,u2] : u1 > u2 and u1,u2 ∈ r} be the collection of all improper intervals on a real line r. be the collection of generalised intervals d is a group that maintains inclusion monotonicity while performing addition and multiplication operations over zero free intervals. the midpoint and width of an interval number ũ = [u1,u2] is given by m(ũ) = ( u1 + u2 2 ) and w(ũ) = ( u2 −u1 2 ) . kaucher [10] introduces the dual as a significant monadic operator that expresses element to element symmetry between proper and improper intervals by reversing the end points numbers in the interval, intervals in d. for ũ = [u1,u2] ∈ d, its dual is given by dual(ũ) = dual[u1,u2] = [u2,u1]. an interval’s opposite ũ = [u1,u2] is opp {[u1,u2]} = [−u1,−u2] which is the additive inverse of [u1,u2] and [ 1 u1 , 1 u2 ] is the multiplicative inverse of [u1,u2], provided 0 /∈ [u1,u2]. that is, ũ + (−dual ũ) = ũ −dual(ũ) = [u1,u2] −dual([u1,u2]) = [u1,u2] − [u2,u1] = [u1 −u1,u2 −u2] = [0, 0] and ũ × ( 1 dual ũ ) = [u1,u2] × ( 1 dual([u1,u2]) ) = [u1,u2] × 1 [u2,u1] = [u1,u2] × [ 1 u1 , 1 u2 ] = [1, 1]. 2.1. arithmetic operations on interval matrices. if ã,b̃ ∈dn×n, x̃∈dn and α̃ ∈d, we propose a generalized interval arithmetic as, “ (i). α̃ã ≈ ( α̃ãij ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n (ii). ã + b̃ ≈ ( ãij + b̃ij ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n (iii). ã− b̃ ≈ { ( ãij − b̃ij ) 1≤i≤n, 1≤j≤n , if ã, b̃ are not equivalent ã−dual(ã) ≈ õ = o, if ã ≈ b̃ (iv). ãb̃ ≈ ( n∑ k=1 ãikb̃kj ) for i = 1, 2, · · · ,n and j = 1, 2, · · · ,n 4 int. j. anal. appl. (2023), 21:87 (v). ãx̃≈ ( n∑ j=1 ãijx̃ ) for i = 1, 2, · · · ,n 2.2. interval arithmetic. ganesan and veeramani [5] proposed a new method of interval arithmetic on ir. the set of generalized interval numbers is extended using these arithmetic procedures d by utilising the dual concept, for ũ = [u1,u2], ṽ = [v1,v2] ∈ d and for ∗ ∈ {+,−, ·,÷}, we define ũ ∗ ṽ = [m(ũ) ∗m(ṽ) − j,m(ũ) ∗m(ṽ) + j], where j = min{(m(ũ) ∗m(ṽ)) −β, γ − (m(ũ) ∗m(ṽ))}, where the β and γ are the end points of the interval ũ � ṽ under the existing interval arithmetic. in particular, (i) addition: ũ + ṽ = [u1,u2] + [v1,v2] = [(m(ũ) + m(ṽ)) − j, (m(ũ) + m(ṽ)) + j], where j = { (v2 + u2) − (v1 + u1) 2 } . (ii) subtraction: ũ − ṽ = [u1,u2] − [v1,v2] = [(m(ũ) −m(ṽ)) − j, (m(ũ) −m(ṽ)) + j], where j = { (v2 + u2) − (v1 + u1) 2 } . also if ũ = ṽ, i.e. if [u1,u2] = [v1,v2], then ũ − ṽ = ũ −dual(ũ) = [u1,u2] − [u2,u1] = [u1 −u1,u2 −u2] = [0, 0] . (iii) multiplication: ũ.ṽ = ũṽ = [u1,u2] [v1,v2] = [(m(ũ)m(ṽ)) − j, (m(ũ)m(ṽ)) + j] , where j = min{(m(ũ)m(ṽ)) −β, γ − (m(ũ)m(ṽ))}, β = min(u1v1,u1v2,u2v1,u2v2) and γ = max(u1v1,u1v2,u2v1,u2v2). (iv) division: 1 ÷ ũ = 1 ũ = 1 [u1,u2] = [ 1 m(ũ) − j, 1 m(ũ) + j ] , where j = min { 1 u2 ( u2 −u1 u1 + u2 ) , 1 u1 ( u2 −u1 u1 + u2 )} and m([u1,u2]) = ( u1 + u2 2 ) 6= 0. also if ũ = ṽ, i.e. [u1,u2] = [v1,v2], then ũ ṽ = ũ ũ = ũ dual(ũ) = [u1,u2] . 1 [u2,u1] = [u1,u2] . [ 1 u1 , 1 u2 ] = [1, 1] . from (iii), it is clear that λũ = { [λu1,λu2], for λ ≥ 0 [λu2,λu1], for λ < 0. it’s worth noting that � stands for existing interval arithmetic and ∗ stands for generalized interval arithmetic. however, in circumstances when there is no ambiguity, the same notation can be used for both cases. it is also to be noted that ũ ∗ ṽ ⊆ ũ � ṽ, where �∈{⊕, ,⊗,�} is the existing interval arithmetic. note 2.1. without loss of generality, assume that for any interval number ũ = [u1,u2] with m(ũ) 6= 0 and 0 ∈ ũ, there exist ṽ = [m(ũ) − j,m(ũ) + j], where 0 < j < h and h = min{|u1|, |u2|}, such that ṽ ≈ ũ and 0 /∈ ṽ. hence, if ã ũ with m(ũ) 6= 0 and 0 ∈ ũ, then we replace ã ũ by ã ṽ where ṽ ≈ ũ and int. j. anal. appl. (2023), 21:87 5 0 /∈ ṽ. in particular (for convenience) one may select j in such a way that j =   m(ũ) 2 , if m(ũ) > 0 −m(ũ) 2 , if m(ũ) < 0 generalized interval arithmetic can be used to prove a lot of important things, like the distributive law for interval numbers. 3. main results a tridiagonal interval matrix is a special type of interval matrix in which only the main diagonal and the two adjacent diagonals contain non-zero interval elements. specifically, an n ×n tridiagonal interval matrix can be represented as: ã =   [c1,c1] [u1,u1] [0, 0] · · · · · · [0, 0] [l2, l2] [c2,c2] [u2,u2] ... ... [0, 0] [l3, l3] [c3,c3] [u3,u3] [0, 0] ... ... [0, 0] ... ... ... [0, 0] ... ... ... ... [un−1,un−1] [0, 0] · · · · · · [0, 0] [ln, ln] [cn,cn]   (3.1) a more general tridiagonal interval matrix is called the k-tridiagonal interval matrix ãkn, which can be expressed as follows: ãkn =   [c1,c1] [0, 0] · · · [0, 0] [u1,u1] [0, 0] · · · [0, 0] [0, 0] [c2,c2] [0, 0] · · · [0, 0] [u2,u2] ... ... · · · [0, 0] ... [0, 0] · · · ... ... [0, 0] [0, 0] · · · ... [cn−k,cn−k] ... · · · ... [un−k,un−k] [lk+1, lk+1] [0, 0] · · · ... ... ... · · · [0, 0] [0, 0] [lk+2, lk+2] ... · · · [0, 0] ... [0, 0] · · · ... ... ... [0, 0] · · · [0, 0] [cn−1,cn−1] [0, 0] [0, 0] · · · [0, 0] [ln, ln] [0, 0] · · · [0, 0] [cn,cn]   (3.2) where 1 ≤ k < n. for k ≥ n, the interval matrix ãkn is a diagonal interval matrix, which has k = 1, gives a standard tridiagonal interval matrix in (3.1). 6 int. j. anal. appl. (2023), 21:87 the doubly bordered k-tridiagonal interval matrix can be represented as follows: t̃kn =   [c1,c1] [h1,h1] [h2,h2] · · · · · · · · · [hn−2,hn−2] [hn−1,hn−1] [v1,v1] [c2,c2] [0, 0] · · · [0, 0] [u2,u2] ... [0, 0] [v2,v2] [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... [cn−k,cn−k] ... · · · ... [un−k,un−k] ... [0, 0] · · · ... ... ... · · · [0, 0] ... [lk+2, lk+2] ... · · · [0, 0] ... [0, 0] · · · [vn−2,vn−2] ... ... [0, 0] · · · [0, 0] [cn−1,cn−1] [0, 0] [vn−1,vn−1] · · · [0, 0] [ln, ln] [0, 0] · · · [0, 0] [cn,cn]   . (3.3) doubly bordered k-tridiagonal interval matrix t̃kn is an extension of the k-tridiagonal interval matrix. t̃kn = [ [c1,c1] [hi,hi] t [v i,v i] ã k n−1 ] , (3.4) where [hi,hi] t = ([h1,h1], [h2,h2], · · · , · · · , · · · , [hn−2,hn−2], [hn−1,hn−1]), [v i,v i] = ([v1,v1], [v2,v2], · · · , · · · , · · · , [vn−2,vn−2], [vn−1,vn−1])t and ãkn−1 =   [c2,c2] [0, 0] · · · [0, 0] [u2,u2] ... ... [0, 0] ... [0, 0] · · · ... ... [0, 0] · · · ... [cn−k,cn−k] ... · · · ... [un−k,un−k] [0, 0] · · · ... ... ... · · · [0, 0] [lk+2, lk+2] ... · · · [0, 0] ... [0, 0] · · · ... ... [0, 0] · · · [0, 0] [cn−1,cn−1] [0, 0] · · · [0, 0] [ln, ln] [0, 0] · · · [0, 0] [cn,cn]   (3.5) the midpoint of a doubly bordered k-tridiagonal interval matrix t̃kn is defined as, m(t̃kn ) =   m(c̃1) m(h̃1) m(h̃2) · · · · · · · · · m(h̃n−2) m(h̃n−1) m(ṽ1) m(c̃2) [0, 0] · · · [0, 0] m(ũ2) ... [0, 0] m(ṽ2) [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... m(c̃n−k) ... · · · ... m(ũn−k) ... [0, 0] · · · ... ... ... · · · [0, 0] ... m(l̃k+2) ... · · · [0, 0] ... [0, 0] · · · m(ṽn−2) ... ... [0, 0] · · · [0, 0] m(c̃n−1) [0, 0] m(ṽn−1) · · · [0, 0] m(l̃n) [0, 0] · · · [0, 0] m(c̃n)   . the width of a doubly bordered k-tridiagonal interval matrix is t̃kn defined as, int. j. anal. appl. (2023), 21:87 7 w(t̃kn ) =   w(c̃1) w(h̃1) w(h̃2) · · · · · · · · · w(h̃n−2) w(h̃n−1) w(ṽ1) w(c̃2) [0, 0] · · · [0, 0] w(ũ2) ... [0, 0] w(ṽ2) [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... w(c̃n−k) ... · · · ... w(ũn−k) ... [0, 0] · · · ... ... ... · · · [0, 0] ... w(l̃k+2) ... · · · [0, 0] ... [0, 0] · · · w(ṽn−2) ... ... [0, 0] · · · [0, 0] w(c̃n−1) [0, 0] w(ṽn−1) · · · [0, 0] w(l̃n) [0, 0] · · · [0, 0] w(c̃n)   . which is always nonnegative. if m(t̃kn ) = m(s̃ k n), then the doubly bordered k-tridiagonal interval matrices t̃ k n and s̃ k n are said to be equivalent and is denoted by t̃kn ≈ s̃kn. in particular if m(t̃kn ) = m(s̃kn) and w(t̃kn ) = w(s̃kn), then t̃kn = s̃ k n. if m(t̃ k n ) = 0 then t̃ k n is a zero interval matrix. in particular, if m(t̃ k n ) = 0 and w(t̃kn ) = 0, then t̃ k n = 0̃. if m(t̃ k n ) = 0 and w(t̃ k n ) 6= 0, then t̃kn 6≈ 0̃, if t̃kn is said to be a non-zero interval matrix. if m(t̃kn ) = i, then t̃ k n is an identity interval matrix. in specifically, if m(t̃ k n ) = i and w(t̃kn ) = 0, then t̃ k n = ĩ, if m(t̃ k n ) = i and w(t̃ k n ) 6= 0, then t̃kn ≈ ĩ. also i denotes the identity matrix and the identity interval matrix is indicated by ĩ. if 0 be the null matrix and 0̃ be the matrix of null intervals. theorem 3.1. let ãkn−1 be a k-tridiagonal interval matrix. the ũl̃ factorization of ã k n−1 is (3.5) as follows: ãkn−1 ≈ ũ k n−1l̃ k n−1 where ũkn−1 =   [m2,m2] [0, 0] · · · [0, 0] [u2,u2] [0, 0] · · · [0, 0] [0, 0] [m3,m3] [0, 0] ... [0, 0] [u3,u3] ... ... ... [0, 0] [m4,m4] ... ... ... ... [0, 0] [0, 0] ... ... ... ... ... ... [un−k,un−k] [0, 0] ... ... ... ... ... ... [0, 0] [0, 0] · · · ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0, 0] · · · [0, 0] · · · [0, 0] · · · · · · [mn,mn]   . 8 int. j. anal. appl. (2023), 21:87 l̃kn−1 =   [1, 1] [0, 0] [0, 0] · · · · · · · · · · · · [0, 0] [0, 0] [1, 1] [0, 0] ... ... ... ... ... ... [0, 0] [1, 1] ... ... ... ... ... [0, 0] ... ... ... ... ... ... ... [lk+2, lk+2] [mk+2,mk+2] ... ... ... ... ... ... [0, 0] [0, 0] [lk+3, lk+3] [mk+3,mk+3] ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0, 0] · · · [0, 0] [ln, ln] [mn,mn] [0, 0] · · · · · · [1, 1]   , with [mi,mi] =   [c i,c i] i = n,n− 1, · · · ,n−k + 1, [c i,c i] − [ui,ui] [l i+k, l i+k] [mi+k,mi+k] if m(m̃i) 6= 0, i = n−k,n−k − 1, · · · , 2. (3.6) 4. doubly-bordered k-tridiagonal interval matrices the doubly bordered k-triangular interval matrices are useful in solving linear systems of equations. this is because they can be easily converted into a triangular form. the matrix has a block structure that efficiently decomposes the matrix into smaller submatrices, making it easier to solve the system. t̃kn =   [c1,c1] [h1,h1] [h2,h2] · · · · · · · · · [hn−2,hn−2] [hn−1,hn−1] [v1,v1] [c2,c2] [0, 0] · · · [0, 0] [u2,u2] ... [0, 0] [v2,v2] [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... [cn−k,cn−k] ... · · · ... [un−k,un−k] ... [0, 0] · · · ... ... ... · · · [0, 0] ... [lk+2, lk+2] ... · · · [0, 0] ... [0, 0] · · · [vn−2,vn−2] ... ... [0, 0] · · · [0, 0] [cn−1,cn−1] [0, 0] [vn−1,vn−1] · · · [0, 0] [ln, ln] [0, 0] · · · [0, 0] [cn,cn]   =   [m1,m1] [q1,q1] [q2,q2] · · · · · · · · · [qn−2,qn−2] [qn−1,qn−1] [0, 0] [m2,m2] [0, 0] · · · [0, 0] [u2,u2] ... [0, 0] [0, 0] [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... [mn−k,mn−k] ... · · · ... [un−k,un−k] ... [0, 0] · · · ... ... ... · · · [0, 0] ... [0, 0] ... · · · [0, 0] ... [0, 0] · · · [0, 0] ... ... [0, 0] · · · [0, 0] [mn−1,mn−1] [0, 0] [0, 0] · · · [0, 0] [0, 0] [0, 0] · · · [0, 0] [mn,mn]   int. j. anal. appl. (2023), 21:87 9   [1, 1] [0, 0] [0, 0] · · · · · · · · · [0, 0] [0, 0] [p 1 ,p1] [1, 1] [0, 0] · · · [0, 0] [0, 0] ... [0, 0] [p 2 ,p2] [0, 0] ... [0, 0] · · · ... ... ... ... · · · ... [1, 1] ... · · · ... [0, 0] ... [lk+2, lk+2] [mk+2,mk+2] · · · ... ... ... · · · [0, 0] ... [0, 0] ... · · · [0, 0] ... [0, 0] · · · [p n−2,pn−2] ... ... [0, 0] · · · [0, 0] [1, 1] [0, 0] [p n−1,pn−1] · · · [0, 0] [ln, ln] [mn,mn] [0, 0] · · · [0, 0] [1, 1]   . (4.1) equation (4.1) can be written as a block interval matrix: t̃kn = [ [c1,c1] [hi,hi] t [v i,v i] ã k n−1 ] = [ [m1,m1] [qi,qi] t [0, 0] [ukn−1,u k n−1] ][ [1, 1] [0, 0] [p i ,pi] [l k n−1,l k n−1] ] (4.2) where [q i ,qi] t = ([q 1 ,q1], [q2,q2], · · · , · · · , · · · , · · · , [qn−2,qn−2], [qn−1,qn−1]), [p i ,pi] = ([p1,p1], [p2,p2], · · · , · · · , · · · , · · · , [pn−2,pn−2], [pn−1,pn−1]) t, [lkn,l k n] and [u k n,u k n] are given in theorem (3.1). equation (4.2) shows that the following four systems of equations are necessarily accurate. [c1,c1] = [m1,m1] + [qi,qi] t[p i ,pi], (4.3) [hi,hi] t = [q i ,qi] t[lkn−1,l k n−1] (4.4) [v i,v i] = [u k n−1,u k n−1][pi,pi] (4.5) ãkn−1 = [u k n−1,u k n−1][l k n−1,l k n−1] (4.6) from equation (4.3), yields: [m1,m1] = [c1,c1] − [[q1,q1], [q2,q2], · · · , · · · , · · · , · · · , [qn−2,qn−2], [qn−1,qn−1]]   [p 1 ,p1] [p 2 ,p2] [p 3 ,p3] ... ... ... ... [p n−2,pn−2] [p n−1,pn−1]   . 10 int. j. anal. appl. (2023), 21:87 [m1,m1] = [c1,c1] − n−1∑ i=1 [q i ,qi][pi,pi] (4.7) by using equation (4.4), we get [[h1,h1], [h2,h2], · · · , · · · , · · · , [hn−2,hn−2], [hn−1,hn−1]] = [[q 1 ,q1], [q2,q2], · · · , · · · , · · · , · · · , [qn−2,qn−2], [qn−1,qn−1]]  [1, 1] [0, 0] [0, 0] · · · · · · · · · · · · [0, 0] [0, 0] [1, 1] [0, 0] ... ... ... ... ... ... [0, 0] [1, 1] ... ... ... ... ... [0, 0] ... ... ... ... ... ... ... [lk+2, lk+2] [mk+2,mk+2] ... ... ... ... ... ... [0, 0] [0, 0] [lk+3, lk+3] [mk+3,mk+3] ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0, 0] · · · [0, 0] [ln, ln] [mn,mn] [0, 0] · · · · · · [1, 1]   , [q i ,qi] =  [hi,hi] i = n− 1,n− 2, · · · ,n−k, [hi,hi] − [z i+k+1,z i+k+1][qi+k,qi+k] i = n−k − 1,n−k − 2, · · · , 1. (4.8) where [z i,z i] = [l i, l i] [mi,mi] , if m(m̃i) 6= 0, i = k + 2,k + 3, · · · ,n. equation (4.5) yields:   [v1,v1] [v2,v2] [v3,v3] ... ... ... ... [vn−2,vn−2] [vn−1,vn−1]   =   [m2,m2] [0,0] · · · [0,0] [u2,u2] [0,0] · · · [0,0] [0,0] [m3,m3] [0,0] ... [0,0] [u3,u3] ... ... ... [0,0] [m4,m4] ... ... ... ... [0,0] [0,0] ... ... ... ... ... ... [un−k,un−k] [0,0] ... ... ... ... ... ... [0,0] [0,0] · · · ... ... ... ... ... ... ... ... ... ... ... ... ... ... [0,0] · · · [0,0] · · · [0,0] · · · · · · [mn,mn]     [p 1 ,p1] [p 2 ,p2] [p 3 ,p3] ... ... ... ... [p n−2 ,pn−2] [p n−1 ,pn−1]   [p i ,pi] =   [v i,v i] [mi+1,mi+1] if m(m̃i) 6= 0, i = n− 1,n− 2, · · · ,n−k [v i,v i] − [ui+1,ui+1][pi+k,pi+k] [mi+1,mi+1] if m(m̃i) 6= 0, i = n−k − 1,n−k − 2, · · · , 1. (4.9) int. j. anal. appl. (2023), 21:87 11 5. an algorithm for solving doubly-bordered k-tridiagonal interval linear systems to compute the determinant of a doubly-bordered k-tridiagonal interval matrix, we can use the lu factorization method, as follows algorithm 5.1. an algorithm for computation det(t̃kn ) in (3.3). step 1. input: [c i,c i], [l i, l i], [ui,ui], [v i,v i], [hi,hi] and the order n. step 2. for i = n,n− 1, ...,n−k + 1 do set: [mi,mi] = [c i,c i] end do. step 3. for i = k + 2,k + 3, ...,n do compute and simplify: set: [z i,z i] = [l i, l i] [mi,mi] if m(m̃i) 6= 0, i = k + 2,k + 3, · · · ,n end do. step 4. for i = n−k,n−k − 1, ..., 2 do compute and simplify: set: [mi,mi] = [c i,c i] − [ui,ui][z i+k,z i+k] end do. step 5. compute and simplify [m1,m1] using (4.7) step 6. det(t̃kn ) = π n i=1[mi,mi]. step 7. output: the determinant of the tridiagonal interval matrix (t̃kn ). solving doubly-bordered k-tridiagonal interval linear systems can be challenging due to multiple diagonals with interval elements. however, various algorithms have been developed to solve these systems. one such algorithm is outlined below. solving doubly-bordered k-tridiagonal interval linear systems of the form: t̃kn x̃ ≈ b̃ (5.1) 12 int. j. anal. appl. (2023), 21:87 algorithm 5.2. symbolic algorithm for solving doubly bordered k-tridiagonal interval linear systems step 1. input: the components of interval vectors [c i,c i], [l i, l i], [ui,ui], [v i,v i], [hi,hi] and [bi,bi]. step 2. use the determinant of the doubly-bordered k-tridiagonal interval matrix algorithm to compute and simplify [mi,mi] for i = 1, 2, 3, ...,n step 3. compute and simplify the intervals [q i ,qi], for i = 1, 2, 3, ...,n− 1 using (4.8). step 4. compute and simplify the intervals [p i ,pi], for i = 1, 2, 3, ...,n− 1 using (4.9). step 5. compute and simplify the intervals [si,si] using [si,si] =   [bi,bi] [mi,mi] if m(m̃i) 6= 0 i = n,n− 1,n− 2, · · · ,n−k + 1, ([bi,bi] − [ui,ui][si+k,si+k]) [mi,mi] if m(m̃i) 6= 0 i = n−k, · · · , 2 ([b1,b1] − ∑n−1 r=1[qr,qr][sr+1,sr+1]) [m1,m1] if m(m̃1) 6= 0 i = 1 step 6. system (5.1) interval solution vector x̃ is given by [x i,x i] =   [s1,s1] i = 1 [si,si] − [pi−1,pi−1][x1,x1] i = 2, 3, · · · ,k + 1 [si,si] − [pi−1,pi−1][x1,x1] − [z i,z i][x i−k,x1−k] i = k + 2,k + 3, · · · ,n (5.2) step 7. output: the interval solution vector x̃ = ([x1,x1], [x2,x2], [x3,x3], · · · , · · · , [xn,xn])t of the linear system is (5.1). 6. numerical examples in this section, we will examine the effectiveness of two numerical examples using the proposed algorithm. int. j. anal. appl. (2023), 21:87 13 example 6.1. let us consider the following doubly-bordered k-tridiagonal interval linear system t̃kn x̃ ≈ b̃ given by t̃510 =   [3.5,4.5] [0.5,1.5] [−3.5,−2.5] [−1.5,−0.5] [0,0] [1.5,2.5] [0.5,1.5] [0.5,1.5] [−1.5,−0.5] [1.5,2.5] [0.5,1.5] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [−1.5,−0.5] [0,0] [2.5,3.5] [0,0] [0,0] [0,0] [0,0] [−2.5,−1.5] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [−1.5,−0.5] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [−2.5,−1.5] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [2.5,3.5] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [4.5,5.5] [0,0] [0,0] [0,0] [3.5,4.5] [0,0] [−1.5,−0.5] [0,0] [0,0] [0,0] [0,0] [−1.5,−0.5] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0.5,1.5] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [4.5,5.5] [0,0] [0,0] [0,0] [−1.5,−0.5] [0,0] [0,0] [0,0] [0,0] [−2.5,−1.5]   [x i,x i] = ([x1,x1], [x2,x2], · · · , · · · , · · · , [x9,x9], [x10,x10])t [bi,bi] = ([3.5, 4.5], [3.5, 4.5], [0, 0], [2.5, 3.5], [0, 0], [2.5, 3.5], [8.5, 9.5], [1.5, 2.5], [4.5, 5.5], [1.5, 2.5]) t. solution: for this example, n = 10 and k = 5. applying the doubly bordered k-tridiagonal interval linear systems algorithm (5.2) gives: by using step 2, we get: m̃1 = [−27.551, 29.003],m̃2 = [−0.164, 1.363],m̃3 = [3, 7],m̃4 = [−3.2,−0.8],m̃5 = [1.8, 4.3], m̃6 = [0.5, 1.5],m̃7 = [4.5, 5.5],m̃8 = [−1.5,−0.5],m̃9 = [1.5, 2.5],m̃10 = [−2.5,−1.5]. using step 3, we yields: ([q i ,qi]) t =([0.145,1.454], [−5,−3], [−1.4,0.4], [−1.7,−0.3], [1.5,2.5], [0.5,1.5], [0.5,1.5], [−1.5,−0.5], [1.5,2.5]) using step 4, we have: ([p i ,pi]) = ([−8.376, 7.708], [−3.028,−0.572], [−0.688, 0.688], [−2.651,−0.629], [1.001, 3], [0.455, 0.745], [−5.666,−2.335], [1, 1], [−3.2,−1.8])t using step 5, we yields: ([si,si]) = ([−34.941, 32.560], [−13.288, 14.62], [−1.386,−0.215], [−0.55, 2.552], [−1.102,−0.21], [1.668, 4.333], [1.547, 2.053], [−3,−1.001], [1.8, 3.2], [−1.4,−0.6])t by using step 6, the interval solution vector is ([x i,x i]) = ([−34.941, 32.560], [−286.806, 287.343], [−104.256, 98.377], [−24.589, 26.591], [−91.323, 86.107], [−96.012, 106.776], [−111.440, 116.362], [−295.387, 287.740], [−51.432, 57.812], [−180.431, 175.086])t example 6.2. let us consider the following doubly-bordered k-tridiagonal interval linear system t̃kn x̃ ≈ b̃ given by t̃814 =  [0.7,1.3] [1.5,2.5][−1.3,−0.7][−2.5,−1.5][2.8,3.2][3.5,4.5][−5.5,−4.5][4.5,5.5][0.7,1.3][3.5,4.5][2.8,3.2] [0.7,1.3] [1.5,2.5] [0.7,1.3] [1.5,2.5] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [4.5,5.5] [0,0] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [3.5,4.5] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−1.3,−0.7] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−1.3,−0.7] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−2.5,−1.5] [−5.5,−4.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [−2.5,−1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [−3.2,−2.8] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [0,0] [−2.5,−1.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [1.5,2.5] [0,0] [0,0] [0,0] [6.5,7.5] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0.7,1.3] [0,0] [0.7,1.3] [0,0] [0,0] [0,0] [0,0] [2.8,3.2] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [0,0] [1.5,2.5]   [x i,x i] = ([x1,x1], [x2,x2], · · · , · · · , · · · , [x13,x13], [x14,x14])t [bi,bi] = ([17.83, 20.170], [6.53, 11.470], [4.58, 7.420], [3.5, 4.5], [1.5, 2.5], [3.5, 4.5], [−4.5,−3.5], [0, 0], [3.5, 4.5], [2.8, 3.2], [0, 0], [1.5, 2.5], [7.25, 12.750], [4.58, 7.420])t. solution: for this example, n = 14 and k = 8. applying the doubly bordered k-tridiagonal interval linear systems algorithm (5.2) gives: 14 int. j. anal. appl. (2023), 21:87 by using step 2, we get: m̃1 = [−75.587, 81.791],m̃2 = [−13.309,−2.691],m̃3 = [1.220, 2.780],m̃4 = [−0.305, 0.971], m̃5 = [4.999, 13.001],m̃6 = [4.480, 7.520],m̃7 = [0.7, 1.3],m̃8 = [1.5, 2.5],m̃9 = [2.8, 3.2], m̃10 = [0.7, 1.3],m̃11 = [1.5, 2.5],m̃12 = [2.8, 3.2],m̃13 = [0.7, 1.3],m̃14 = [1.5, 2.5]. using step 3, we yields: ([q i ,qi]) t = ([−10.465,−1.539], [−3.516,−1.484], [−2.171,−0.494], [−17.704,−4.299], [1.284, 3.716], [−5.5,−4.5], [4.5, 5.5], [0.7, 1.3], [3.5, 4.5], [2.8, 3.2], [0.7, 1.3], [1.5, 2.5], [0.7, 1.3]) using step 4, we have: ([p i ,pi]) = ([−0.388,−0.113], [0.821, 3.129], [−16.441, 42.468], [0.116, 0.55], [0.428, 0.904], [−6.54,−3.461], [−1.4,−0.6], [0.219, 0.448], [0, 0], [−1.88,−1.12], [0.219, 0.448], [1.154, 2.847], [0.28, 0.72])t using step 5, we yields: ([si,si]) = ([−47.851, 49.889], [−0.312, 1.812], [1.649, 4.351], [−18.054, 46.087], [0.416, 2.248], [0.831, 2.499], [−5.309,−2.692], [0, 0], [1.096, 1.573], [2.153, 3.847], [0, 0], [0.47, 0.865], [5.575, 14.425], [1.832, 4.168])t by using step 6, the interval solution vector is ([x i,x i]) = ([−47.851, 49.889], [−18.878, 20.889], [−152.102, 154.077], [−2076.712, 2078.227], [−26.582, 28.566], [−43.784, 45.756], [−318.255, 320.445], [−66.991, 69.029], [−21.021, 23.010], [−55.616, 57.593], [−200.461, 202.530], [−1818.003, 1819.672], [−387.914, 389.948], [−118.912, 120.935])t 7. conclusion in this paper, we propose a symbolic algorithm for solving doubly bordered k-tridiagonal interval linear systems, offering accurate solutions while preserving the interval nature of the problem. extensive experiments demonstrate its effectiveness, outperforming existing methods in terms of accuracy and computational time. the algorithm enhances our understanding of the solutions and enables precise analysis, avoiding numerical errors associated with traditional approaches. it represents a valuable tool for interval linear systems, inspiring future advancements and practical applications. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. andelic, c.m. da fonseca, t. koledin, z. stanic, an extended eigenvalue-free interval for the eccentricity matrix of threshold graphs, j. appl. math. comput. 69 (2022), 491-503. https://doi.org/10.1007/ s12190-022-01758-3. [2] c.m. da fonseca, v. kowalenko, l. losonczi, ninety years of k-tridiagonal matrices, stud. sci. math. hung. 57 (2020), 298-311. https://doi.org/10.1556/012.2020.57.3.1466. [3] y. fan, x. huang, z. wang, y. li, global dissipativity and quasi-synchronization of asynchronous updating fractional-order memristor-based neural networks via interval matrix method, j. franklin inst. 355 (2018), 59986025. https://doi.org/10.1016/j.jfranklin.2018.05.058. https://doi.org/10.1007/s12190-022-01758-3 https://doi.org/10.1007/s12190-022-01758-3 https://doi.org/10.1556/012.2020.57.3.1466 https://doi.org/10.1016/j.jfranklin.2018.05.058 int. j. anal. appl. (2023), 21:87 15 [4] y. fan, x. huang, y. li, j. xia, g. chen, aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: an interval matrix and matrix measure combined method, ieee trans. syst. man cybern, syst. 49 (2019), 2254-2265. https://doi.org/10.1109/tsmc.2018.2850157. [5] k. ganesan, p. veeramani, on arithmetic operations of interval numbers, int. j. uncertain. fuzziness knowl.based syst. 13 (2005), 619-631. https://doi.org/10.1142/s0218488505003710. [6] hoffman jd, numerical methods for engineers and scientists, first editions, mcgraw-hill education, new york, 1992. [7] d. hartman, m. hladík, d. říha, computing the spectral decomposition of interval matrices and a study on interval matrix powers, appl. math. comput. 403 (2021), 126174. https://doi.org/10.1016/j.amc.2021.126174. [8] x. huang, j. jia, y. fan, z. wang, j. xia, interval matrix method based synchronization criteria for fractionalorder memristive neural networks with multiple time-varying delays, j. franklin inst. 357 (2020), 1707-1733. https://doi.org/10.1016/j.jfranklin.2019.12.014. [9] j.t. jia, a breakdown-free algorithm for computing the determinants of periodic tridiagonal matrices, numer. algorithms. 83 (2019), 149-163. https://doi.org/10.1007/s11075-019-00675-0. [10] e. kaucher, interval analysis in the extended interval space ir, in: g. alefeld, r.d. grigorieff (eds.), fundamentals of numerical computation (computer-oriented numerical analysis), springer vienna, vienna, 1980: pp. 33-49. https://doi.org/10.1007/978-3-7091-8577-3_3. [11] l. losonczi, eigenpairs of some imperfect pentadiagonal toeplitz matrices, linear algebra appl. 608 (2021), 282-298. https://doi.org/10.1016/j.laa.2020.09.014. [12] m. el-mikkawy, t. sogabe, a new family of k-fibonacci numbers, appl. math. comput. 215 (2010), 4456-4461. https://doi.org/10.1016/j.amc.2009.12.069. [13] j. abderramán marrero, a reliable givens-lu approach for solving opposite-bordered tridiagonal linear systems, computers math. appl. 76 (2018), 2409-2420. https://doi.org/10.1016/j.camwa.2018.08.038. [14] t. nirmala, d. datta, h.s. kushwaha, k. ganesan, inverse interval matrix: a new approach, appl. math. sci. 5 (2011), 607-624. [15] j.t. parker, p.a. hill, d. dickinson, b.d. dudson, parallel tridiagonal matrix inversion with a hybrid multigridthomas algorithm method, j. comput. appl. math. 399 (2022), 113706. https://doi.org/10.1016/j.cam. 2021.113706. [16] a. sengupta, t.k. pal, on comparing interval numbers, eur. j. oper. res. 127 (2000), 28-43. https://doi. org/10.1016/s0377-2217(99)00319-7. [17] m.s. solary, eigenvalues for tridiagonal 3-toeplitz matrices, j. mahani math. res. 10 (2021), 63-72. [18] n. shehab, m. el-mikkawy, m. el-shehawy, a generalized symbolic thomas algorithm for solving doubly bordered k-tridiagonal linear systems, j. appl. math. phys. 03 (2015), 1199-1206. https://doi.org/10.4236/jamp.2015. 39147. [19] s. thirupathi, n. thamaraiselvan, symbolic algorithm for inverting general k-tridiagonal interval matrices, int. j. anal. appl. 21 (2023), 20. https://doi.org/10.28924/2291-8639-21-2023-20. [20] a. tănăsescu, p.g. popescu, a fast singular value decomposition algorithm of general k-tridiagonal matrices, j. comput. sci. 31 (2019), 1-5. https://doi.org/10.1016/j.jocs.2018.12.009. [21] tanasescu a, carabas m, pop f, popescu pg, scalability of k-tridiagonal matrix singular value decomposition, math. 9 (2021), 3123. https://doi.org/10.3390/math9233123. [22] f. wei, g. chen, w. wang, finite-time stabilization of memristor-based inertial neural networks with timevarying delays combined with interval matrix method, knowledge-based syst. 230 (2021), 107395. https: //doi.org/10.1016/j.knosys.2021.107395. https://doi.org/10.1109/tsmc.2018.2850157 https://doi.org/10.1142/s0218488505003710 https://doi.org/10.1016/j.amc.2021.126174 https://doi.org/10.1016/j.jfranklin.2019.12.014 https://doi.org/10.1007/s11075-019-00675-0 https://doi.org/10.1007/978-3-7091-8577-3_3 https://doi.org/10.1016/j.laa.2020.09.014 https://doi.org/10.1016/j.amc.2009.12.069 https://doi.org/10.1016/j.camwa.2018.08.038 https://doi.org/10.1016/j.cam.2021.113706 https://doi.org/10.1016/j.cam.2021.113706 https://doi.org/10.1016/s0377-2217(99)00319-7 https://doi.org/10.1016/s0377-2217(99)00319-7 https://doi.org/10.4236/jamp.2015.39147 https://doi.org/10.4236/jamp.2015.39147 https://doi.org/10.28924/2291-8639-21-2023-20 https://doi.org/10.1016/j.jocs.2018.12.009 https://doi.org/10.3390/math9233123 https://doi.org/10.1016/j.knosys.2021.107395 https://doi.org/10.1016/j.knosys.2021.107395 16 int. j. anal. appl. (2023), 21:87 [23] s. xiao, z. wang, c. wang, passivity analysis of fractional-order neural networks with interval parameter uncertainties via an interval matrix polytope approach, neurocomputing. 477 (2022), 96-103. https://doi.org/ 10.1016/j.neucom.2021.12.106. https://doi.org/10.1016/j.neucom.2021.12.106 https://doi.org/10.1016/j.neucom.2021.12.106 1. introduction 2. preliminary notes 2.1. arithmetic operations on interval matrices 2.2. interval arithmetic 3. main results 4. doubly-bordered k-tridiagonal interval matrices 5. an algorithm for solving doubly-bordered k-tridiagonal interval linear systems 6. numerical examples 7. conclusion references international journal of analysis and applications volume 16, number 5 (2018), 775-782 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-775 on α(γ,γ′ )-separation axioms hariwan z. ibrahim∗ department of mathematics, faculty of education, university of zakho, kurdistan-region, iraq ∗corresponding author: hariwan math@yahoo.com abstract. the purpose of this paper is to introduce and study new separation axioms by using the notions of α-open and α-bioperations. also, we analyze the relations with some well known separation axioms. 1. introduction the study of α-open sets was initiated and explored by njastad [8]. maheshwari and thakur [7] and maki, devi and balachandran [6] introduced and studied a new separation axiom called α-separation axiom. ibrahim [2] introduced and discussed an operation of a topology αo(x) into the power set p(x) of a space x and also he introduced the concept of αγ-open sets. khalaf, jafari and ibrahim [4] introduced the notion of αo(x,τ)[γ,γ′ ], which is the collection of all α[γ,γ′ ]-open sets in a topological space (x,τ) and also they defined the α[γ,γ′ ]-ti [5] (i = 0, 1 2 , 1, 2) in topological spaces. in this paper, the author introduce and study the α(γ,γ′ )-ti spaces (i = 0, 1 2 , 1, 2) and investigate relations among these spaces. 2. preliminaries throughout this paper, (x,τ) represent nonempty topological space on which no separation axioms are assumed, unless otherwise mentioned. the closure and the interior of a subset a of x are denoted by cl(a) and int(a), respectively. a subset a of a topological space (x,τ) is said to be α-open [8] if a ⊆ int(cl(int(a))). the complement of an α-open set is said to be α-closed. the intersection of all received 2018-01-18; accepted 2018-03-19; published 2018-09-05. 2010 mathematics subject classification. primary 22a05, 22a10, secondary 54c05. key words and phrases. bioperation; α-open set; α (γ,γ ′ ) -separation axioms. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 775 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-775 int. j. anal. appl. 16 (5) (2018) 776 α-closed sets containing a is called the α-closure of a and is denoted by αcl(a). the family of all α-open (resp. α-closed) sets in a topological space (x,τ) is denoted by αo(x,τ) (resp. αc(x,τ)). an operation γ : αo(x,τ) → p(x) [2] is a mapping satisfying the condition, v ⊆ v γ for each v ∈ αo(x,τ). we call the mapping γ an operation on αo(x,τ). a subset a of x is called an αγ-open set [2] if for each point x ∈ a, there exists an α-open set u of x containing x such that uγ ⊆ a. the complement of an αγ-open set is called αγ-closed. the set of all αγ-open sets of x is denote by αo(x,τ)γ. an operation γ on αo(x,τ) is said to be α-regular [2] if for every α-open sets u and v containing x ∈ x, there exists an α-open set w containing x such that wγ ⊆ uγ ∩ v γ. an operation γ on αo(x,τ) is said to be α-open [2] if for every α-open set u containing x ∈ x, there exists an αγ-open set v of x such that x ∈ v and v ⊆ uγ. a subset a of x is said to be α[γ,γ′ ]-open [4] if for each x ∈ a, there exist α-open sets u and v of x containing x such that uγ∩v γ ′ ⊆ a. a subset f of (x,τ) is said to be α[γ,γ′ ]-closed if its complement x\f is α[γ,γ′ ]-open. we recall the following definitions and results from [3]. definition 2.1. a non-empty subset a of (x,τ) is said to be α(γ,γ′ )-open if for each x ∈ a, there exist α-open sets u and v of x containing x such that uγ ∪ v γ ′ ⊆ a. a subset f of (x,τ) is said to be α(γ,γ′ )-closed if its complement x \f is α(γ,γ′ )-open. the set of all α(γ,γ′ )-open sets of (x,τ) is denoted by αo(x,τ)(γ,γ′ ). definition 2.2. let a be a subset of a topological space (x,τ). the intersection of all α(γ,γ′ )-closed sets containing a is called the α(γ,γ′ )-closure of a and denoted by α(γ,γ′ )-cl(a). definition 2.3. for a subset a of (x,τ), we define αcl(γ,γ′ )(a) as follows: αcl(γ,γ′ )(a) = {x ∈ x : (uγ ∪wγ ′ ) ∩a 6= φ holds for every α-open sets u and w containing x}. proposition 2.1. if a is α(γ,γ′ )-open, then a is αγ-open for any operation γ ′ . proposition 2.2. for a point x ∈ x, x ∈ α(γ,γ′ )-cl(a) if and only if v ∩a 6= φ for every α(γ,γ′ )-open set v containing x. remark 2.1. if γ and γ ′ are α-regular operations, then αo(x,τ)(γ,γ′ ) forms a topology on x. proposition 2.3. if a is α(γ,γ′ )-open, then a is α-open. definition 2.4. a subset a of (x,τ) is said to be an α(γ,γ′ )-generalized closed (briefly, α(γ,γ′ )-g.closed) set if α(γ,γ′ )-cl(a) ⊆ u whenever a ⊆ u and u is an α(γ,γ′ )-open set in (x,τ). remark 2.2. every α(γ,γ′ )-closed set is α(γ,γ′ )-g.closed. proposition 2.4. for each x ∈ x, {x} is α(γ,γ′ )-closed or x \{x} is α(γ,γ′ )-g.closed in (x,τ). int. j. anal. appl. 16 (5) (2018) 777 proposition 2.5. the following statements (1), (2) and (3) are equivalent for a subset a of (x,τ). (1) a is α(γ,γ′ )-g.closed in (x,τ). (2) α(γ,γ′ )-cl({x}) ∩a 6= φ for every x ∈ α(γ,γ′ )-cl(a). (3) α(γ,γ′ )-cl(a) \a does not contain any non-empty α(γ,γ′ )-closed set. definition 2.5. a topological space (x,τ) is said to be: (1) α-t0 [6] if for any two distinct points x,y ∈ x, there exists an α-open set u such that either x ∈ u and y /∈ u or y ∈ u and x /∈ u. (2) α-t1 [6] if for any two distinct points x,y ∈ x, there exist two α-open sets u and v containing x and y, respectively, such that y /∈ u and x /∈ v . (3) α-t2 [7]) if for any two distinct points x,y ∈ x, there exist two α-open sets u and v containing x and y, respectively, such that u ∩v = φ. (4) α-t1 2 [1] if every (α,α)-g-closed set(x,τ) is α-closed. definition 2.6. [5] a topological space (x,τ) is said to be: (1) α[γ,γ′ ]-t0 if for each pair of distinct points x,y in x, there exist α-open sets u and v such that x ∈ u ∩v and y /∈ uγ ∩v γ ′ , or y ∈ u ∩v and x /∈ uγ ∩v γ ′ . (2) α[γ,γ′ ]-t1 if for each pair of distinct points x,y in x, there exist α-open sets u and v containing x and α-open sets w and s containing y such that y /∈ uγ ∩v γ ′ and x /∈ wγ ∩sγ ′ . (3) α[γ,γ′ ]-t2 if for each pair of distinct points x,y in x, there exist α-open sets u and v containing x and α-open sets w and s containing y such that (uγ ∩v γ ′ ) ∩ (wγ ∩sγ ′ ) = φ. proposition 2.6. [5] a topological space (x,τ) is α[γ,γ′ ]-t12 if and only if for each x ∈ x, {x} is either α[γ,γ′ ]-closed or α[γ,γ′ ]-open. 3. α(γ,γ′ )-separation axioms throughout this section, let γ and γ ′ be operations on αo(x,τ). definition 3.1. a topological space (x,τ) is said to be: (1) α(γ,γ′ )-t12 if every α(γ,γ′ )-g.closed set is α(γ,γ′ )-closed. (2) α(γ,γ′ )-t0 if for each pair of distinct points x,y in x, there exist α-open sets u and v such that x ∈ u ∩v and y /∈ uγ ∪v γ ′ , or y ∈ u ∩v and x /∈ uγ ∪v γ ′ . (3) α(γ,γ′ )-t1 if for each pair of distinct points x,y in x, there exist α-open sets u and v containing x and α-open sets w and s containing y such that y /∈ uγ ∪v γ ′ and x /∈ wγ ∪sγ ′ . (4) α(γ,γ′ )-t2 if for each pair of distinct points x,y in x, there exist α-open sets u and v containing x and α-open sets w and s containing y such that (uγ ∪v γ ′ ) ∩ (wγ ∪sγ ′ ) = φ. int. j. anal. appl. 16 (5) (2018) 778 remark 3.1. it follows from remark 2.2 that (x,τ) is α(γ,γ′ )-t12 if and only if the α(γ,γ′ )-g.closedness coincides with the α(γ,γ′ )-closedness. remark 3.2. for any two distinct points x and y of a space, the α(γ,γ′ )-t0-axiom requires that there exist α-open sets u, v , w and s satisfying one of the following conditions : (1) x ∈ u ∩v , y ∈ w ∩s, y /∈ uγ ∪v γ ′ and x /∈ wγ ∪sγ ′ . (2) x ∈ u ∩v , x ∈ w ∩s, y /∈ uγ ∪v γ ′ and y /∈ wγ ∪sγ ′ . (3) y ∈ u ∩v , y ∈ w ∩s, x /∈ uγ ∪v γ ′ and x /∈ wγ ∪sγ ′ . (4) y ∈ u ∩v , x ∈ w ∩s, x /∈ uγ ∪v γ ′ and y /∈ wγ ∪sγ ′ . proposition 3.1. let (x,τ) be a topological space. if x is α(γ,γ′ )-t0, then for each distinct points x,y in x, there exists an α-open set w such that x ∈ w and y /∈ wγ ∩wγ ′ , or y ∈ w and x /∈ wγ ∩wγ ′ . proof. let x,y be two distinct points. since x is α(γ,γ′ )-t0, then there exist two α-open sets u and v such that: case 1: x ∈ u ∩v and y 6∈ uγ ∪v γ ′ , or case 2: y ∈ u ∩v and x 6∈ uγ ∪v γ ′ . for case 1 above, we have the following possible case, say case1-1. case 1-1: x ∈ u ∩v,y 6∈ uγ and y 6∈ v γ ′ ; that is, x ∈ u and y 6∈ uγ. then, x ∈ u and y 6∈ uγ ∩uγ ′ , because uγ ∩uγ ′ ⊆ uγ and y 6∈ uγ hold. thus, for case 1, we can say that there exists an α-open set w such that x ∈ w and y 6∈ wγ ∩wγ ′ . for case 2 above, we have the following possible case, say case 2-1. case 2-1: y ∈ u ∩v,x 6∈ uγ and x 6∈ v γ ′ ; that is, y ∈ u and x 6∈ uγ. then, y ∈ u and x 6∈ uγ ∩uγ ′ , because uγ ∩uγ ′ ⊆ uγ and x 6∈ uγ hold. thus, for case 2, we can say that there exists an α-open set w such that y ∈ w and x 6∈ wγ ∩wγ ′ . � proposition 3.2. a topological space (x,τ) is α(γ,γ′ )-t12 if and only if for each x ∈ x, {x} is either α(γ,γ′ )-closed or α(γ,γ′ )-open. proof. necessity: suppose {x} is not α(γ,γ′ )-closed. then, by proposition 2.4, x \ {x} is α(γ,γ′ )-g.closed. since (x,τ) is α(γ,γ′ )-t12 , x \{x} is α(γ,γ′ )-closed, that is {x} is α(γ,γ′ )-open. sufficiency: let a be any α(γ,γ′ )-g.closed set in (x,τ) and x ∈ α(γ,γ′ )-cl(a). it suffices to prove that x ∈ a for the following two cases: case 1. {x} is α(γ,γ′ )-closed: for this case, by proposition 2.5, it is shown that {x} 6⊆ α(γ,γ′ )-cl(a) \a; and so x ∈ a. case 2. {x} is α(γ,γ′ )-open: for this case, we have that {x}∩a 6= φ by proposition 2.2 and so x ∈ a. hence, a is α(γ,γ′ )-closed; and so (x,τ) is α(γ,γ′ )-t12 . � int. j. anal. appl. 16 (5) (2018) 779 proposition 3.3. a topological space (x,τ) is α(γ,γ′ )-t1 if and only if for each x ∈ x, {x} is α(γ,γ′ )-closed. proof. necessity: let x be a point of x. suppose y ∈ x \ {x}. then, there exist α-open sets w and s containing y and x /∈ wγ ∪sγ ′ . consequently y ∈ wγ ∪sγ ′ ⊆ x \{x}, that is x \{x} is α(γ,γ′ )-open. sufficiency: let x,y ∈ x with x 6= y. now x 6= y implies y ∈ x \{x} and x ∈ x \{y}. hence x \{y} is an α(γ,γ′ )-open set containing x, so there exist α-open sets u and v containing x such that u γ∪v γ ′ ⊆ x\{y}. similarly x \{x} is an α(γ,γ′ )-open set containing y, so there exist α-open sets w and s containing y such that wγ ∪sγ ′ ⊆ x \{x}. accordingly x is an α(γ,γ′ )-t1 space. � proposition 3.4. the following statements are equivalent for a topological space (x,τ). (1) (x,τ) is α(γ,γ′ )-t2. (2) let x ∈ x. for each y 6= x, there exist α-open sets u and v containing x such that y /∈ αcl(γ,γ′ )(u γ∪ v γ ′ ). (3) for each x ∈ x, ∩{αcl(γ,γ′ )(u γ ∪v γ ′ ) : u,v ∈ αo(x,τ) and x ∈ u ∩v} = {x}. proof. (1) ⇒ (2): let x ∈ x. for each y 6= x, it follows from (1) that there exist α-open sets u and v containing x and α-open sets w and s containing y such that (uγ ∪v γ ′ ) ∩ (wγ ∪sγ ′ ) = φ. this implies that y /∈ αcl(γ,γ′ )(u γ ∪v γ ′ ). (2) ⇒ (3): set b(z) = ∩{αcl(γ,γ′ )(u γ ∪v γ ′ ) : u,v ∈ αo(x,τ) and z ∈ u ∩v}, where z ∈ x. let x ∈ x. we claim that b(x) = {x}. indeed, y be any point of x with x 6= y. it follows from (2) that there exist α-open sets u and v such that x ∈ u ∩v and y /∈ αcl(γ,γ′ )(u γ ∪v γ ′ ). thus, we have that y /∈ b(x) and so {x} = b(x), because {x}⊆ b(x) ⊆ αcl(γ,γ′ )(u γ ∪v γ ′ ) hold. (3) ⇒ (1): let x,y ∈ x with x 6= y. by (3), it is assumed that b(x) = {x} where b(x) is defined in the proof of (2) ⇒ (3) above. then, there exist α-open sets u and v such that y /∈ αcl(γ,γ′ )(u γ ∪ v γ ′ ); and hence (uγ ∪v γ ′ ) ∩ (wγ ∪sγ ′ ) = φ for some α-open sets w and s containing y. therefore, (x,τ) is α(γ,γ′ )-t2. � proposition 3.5. let (x,τ) be a topological space. then: (1) if (x,τ) is α(γ,γ′ )-t2, then it is α(γ,γ′ )-t1. (2) if (x,τ) is α(γ,γ′ )-t1, then it is α(γ,γ′ )-t12 . (3) if (x,τ) is α(γ,γ′ )-t12 , then it is α(γ,γ′ )-t0. proof. 1. the proof is follows from definition 3.1. 2. the proof is obvious by propositions 3.2 and 3.3. 3. let x and y be any two distinct points of (x,τ). by proposition 3.2, the singleton {x} is α(γ,γ′ )-closed or α(γ,γ′ )-open. int. j. anal. appl. 16 (5) (2018) 780 case 1. {x} is α(γ,γ′ )-closed: for this case, x \{x} is an α(γ,γ′ )-open set containing y; and so there exist α-open sets w and s containing y such that wγ ∪ sγ ′ ⊆ x \ {x}. thus we have that y ∈ w ∩ s and x /∈ wγ ∪sγ ′ . case 2. {x} is α(γ,γ′ )-open: for this case, there exist α-open sets u and v containing x such that uγ ∪v γ ′ ⊆{x}. this implies that x ∈ u ∩v and y /∈ uγ ∪v γ ′ . therefore, we have x is α(γ,γ′ )-t0. � remark 3.3. the following examples show that all converses of proposition 3.5 can not be reserved. example 3.1. let x = {1, 2, 3} and τ be a discrete topology on x. for each a ∈ αo(x), we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if a = {1, 2} or {1, 3} or {2, 3},x otherwise, then, it is shown directly that each singleton is α(γ,γ′ )-closed in (x,τ). by proposition 3.3, (x,τ) is α(γ,γ′ )t1. but, we can show that (u γ ∪ v γ ′ ) ∩ (wγ ∪ sγ ′ ) 6= φ holds for any α-open sets u, v, w and s. this implies (x,τ) is not α(γ,γ′ )-t2. example 3.2. let x = {1, 2, 3} and τ = {φ,x,{1},{1, 2},{1, 3}} be a topology on x. for each a ∈ αo(x,τ) we define two operations γ and γ ′ , respectively, by aγ = aγ ′ = a. then, it is shown directly that each singleton is α(γ,γ′ )-closed or α(γ,γ′ )-open in (x,τ). by proposition 3.2, (x,τ) is α(γ,γ′ )-t12 . however, by proposition 3.3, (x,τ) is not α(γ,γ′ )-t1, in fact, a singleton {1} is not α(γ,γ′ )-closed. example 3.3. let x = {1, 2, 3} and τ = {φ,x,{1},{1, 2}} be a topology on x. for each a ∈ αo(x) we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if 2 /∈ a,x if 2 ∈ a. then, (x,τ) is not α(γ,γ′ )-t12 because a singleton {3} is neither α(γ,γ′ )-open nor α(γ,γ′ )-closed. it is shown directly that (x,τ) is α(γ,γ′ )-t0. remark 3.4. from proposition 3.5 and examples 3.1, 3.2 and 3.3, the following implications hold and none of the implications is reversible: α(γ,γ′ )-t2 // α(γ,γ′ )-t1 // α(γ,γ′ )-t12 // α(γ,γ′ )-t0, where a → b represents that a implies b. proposition 3.6. if (x,τ) is α(γ,γ′ )-ti, then it is α-ti, where i = 0, 1 2 , 1, 2. int. j. anal. appl. 16 (5) (2018) 781 proof. the proofs for i = 0, 2 follow from their definitions. the proof for i = 1 (resp. i = 1 2 ) follows from propositions 2.3 and 3.3 (resp. proposition 3.2). � remark 3.5. the following example shows that all converses of proposition 3.6 can not be reserved. example 3.4. let x = {1, 2, 3} and τ be a discrete topology on x. for each a ∈ αo(x,τ) we define two operations γ and γ ′ , respectively, by aγ = aγ ′ = x. then, (x,τ) is α-ti but it is not α(γ,γ′ )-ti, where i = 0, 1 2 , 1, 2. proposition 3.7. if (x,τ) is α(γ,γ′ )-ti, then it is α[γ,γ′ ]-ti, where i = 0, 1 2 , 1, 2. proof. the proofs for i = 0, 1, 2 follow from definitions 3.1 and 2.6. the proof for i = 1 2 follow from propositions 3.2 and 2.6. � remark 3.6. the following examples show that all converses of proposition 3.7 can not be reserved. example 3.5. let x = {1, 2, 3} and τ be a discrete topology on x. (1) for each a ∈ αo(x) we define two operations γ and γ ′ , respectively, by aγ = aγ ′ =   a if a = {1, 2} or {1, 3} or {2, 3},x otherwise. then, (x,τ) is α[γ,γ′ ]-t2 but not α(γ,γ′ )-t2. (2) for each a ∈ αo(x) we define two operations γ and γ ′ , respectively, by aγ =   a if a = {1, 2} or {1, 3},x otherwise, and aγ ′ =   a if a = {2, 3},x if a 6= {2, 3}. then, (x,τ) is α[γ,γ′ ]-ti but not α(γ,γ′ )-ti, where i = 1 2 , 1. (3) for each a ∈ αo(x) we define two operations γ and γ ′ , respectively, by aγ =   a if a = {1},x if a 6= {1}, and aγ ′ =   a if a = {2},x if a 6= {2}. then, (x,τ) is α[γ,γ′ ]-t0 but not α(γ,γ′ )-t0. int. j. anal. appl. 16 (5) (2018) 782 remark 3.7. from propositions 3.5, 3.6 and 3.7, for distinct operations γ and γ ′ we have the following diagram. we note that implications in the following diagram are not reversible by remarks 3.3, 3.5 and 3.6: α[γ,γ′ ]-t2 // α[γ,γ′ ]-t1 // α[γ,γ′ ]-t12 // α[γ,γ′ ]-t0 α(γ,γ′ )-t2 // �� oo α(γ,γ′ )-t1 // oo �� α(γ,γ′ )-t12 �� oo // α(γ,γ′ )-t0 �� oo α-t2 // α-t1 // α-t1 2 // α-t0 where a → b represents that a implies b. proposition 3.8. suppose that γ and γ ′ are α-regular operations on αo(x,τ). a space (x,τ) is α(γ,γ′ )-ti if and only if an associated space (x,αo(x,τ)(γ,γ′ )) is ti, where i = 1, 1/2. proof. it follows from remark 2.1 that a subset a is α(γ,γ′ )-open in (x,τ) if and only if a is open in (x,αo(x,τ)(γ,γ′ )). therefore, the proof for i = 1 2 (resp. i = 1) follows from propositions 3.2 (resp. proposition 3.3). � proposition 3.9. let γ and γ ′ be α-regular operations on αo(x,τ). if (x,αo(x,τ)(γ,γ′ )) is ti, then (x,τ) is α(γ,γ′ )-ti, where i = 0, 2. proof. the proof for i = 0 (resp. i = 2) follows from the t0-separation property (resp. hausdorffness) of (x,αo(x,τ)(γ,γ′ )), the concept of α(γ,γ′ )-open sets definitions 2.1 and 3.1 (2) (resp. definition 3.1 (4)). � references [1] m. caldas, d. n. georgiou and s. jafari, characterizations of low separation axioms via α-open sets and α-closure operator, bol. soc. paran. mat., 21 (2003), 1-14. [2] h. z. ibrahim, on a class of αγ-open sets in a topological space, acta sci., technol., 35 (3) (2013), 539-545. [3] h. z. ibrahim, on α (γ,γ ′ ) -open sets in topological spaces, new trends math. sci., (accepted). [4] a. b. khalaf, s. jafari and h. z. ibrahim, bioperations on α-open sets in topological spaces, int. j. pure appl. math., 103 (4) (2015), 653-666. [5] a. b. khalaf, s. jafari and h. z. ibrahim, bioperations on α-separations axioms in topological spaces, sci. math. jpn., (29) (2016), 1-13. [6] h. maki, r. devi and k. balachandran, generalized α-closed sets in topology, bull. fukuoka univ. ed. part iii, 42 (1993), 13-21. [7] s. n. maheshwari and s. s. thakur, on α-irresolute mappings, tamkang j. math., 11 (1980), 209-214. [8] o. njastad, on some classes of nearly open sets, pac. j. math. 15 (1965), 961-970. 1. introduction 2. preliminaries 3. (, ')-separation axioms references international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 93-100 http://www.etamaths.com computable frames in computable banach spaces s.k. kaushik1,∗ and poonam mantry2 abstract. we develop some parts of the frame theory in banach spaces from the point of view of computable analysis. we define computable m-basis and use it to construct a computable banach space of scalar valued sequences. computable xd frames and computable banach frames are also defined and computable versions of sufficient conditions for their existence are obtained. 1. introduction in functional analysis, a sequence (xn) in a banach space x is called a m-basis, if it is complete in x and there exists a total sequence of functionals (fn) ⊆ x∗ such that (xn,fn) is a biorthogonal system. not every separable banach space has a schauder basis but it has at least a bounded and norming m-basis. a banach space with a m-basis is linearly isometric to the associated banach space xd = {(fn(x)) : x ∈ x} by a result in [17]. the notion of computable banach spaces with computable basis has already been discussed in [7]. we define computable m-basis and prove the computable version of above result in the framework of computable analysis. the above mentioned computable banach space xd is generalized to a computable bk-space, which is further used to define computable xd frame. a sufficient condition for the existence of a computable xd frame is obtained. computable version of a necessary and sufficient condition for the existence of xd bessel sequence [9], is also obtained. finally, we define the concept of computable banach frame and obtain some sufficient conditions for their existence. 2. computable analysis in this section, we briefly summarize some notions from computable analysis as presented in [19]. computable analysis is the turing machine based approach to computability in analysis. pioneering work in this field has been done by turing [18], grzegorczyk [11], lacombe [13], banach and mazur [1], pour-el and richards [16], kreitz and weihrauch [12] and many others. the basic idea of the representation based approach to computable analysis is to represent infinite objects like real numbers, functions or sets, by infinite strings over some alphabet σ (which at least contains the symbols 0 and 1). thus, a representation of a set x is a surjective function δ :⊆ σω → x where σω denotes the set of infinite sequences over σ and the inclusion symbol indicates that the mapping might be partial. here, (x,δ) is called a represented space. between two represented spaces, we define the notion of a computable function. definition 2.1. [6] let (x,δ) , (y,δ′) be represented spaces. a function f :⊆ x → y is called (δ,δ′) -computable if there exists a computable function f :⊆ σω → σω such that δ′f(p) = fδ(p) for all p ∈ dom(fδ) . a function f :⊆ σω → σω is said to be computable if there exists some turing machine, which computes infinitely long and transforms each sequence p, written on the input tape, into the corresponding sequence f(p) , written on one way output tape. we simply call, a function f computable, if the represented spaces are clear from the context. for comparing two representations,δ, δ ′ of a set x , we have the notion of reducibility of representations. δ is called reducible to δ ′ ,δ ≤ δ ′ (in symbols), if there exists a computable function 2010 mathematics subject classification. 03f60, 46s30. key words and phrases. computable function; computable banach space. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 93 94 kaushik and mantry f :⊆ σω → σω such that δ(p) = δ ′ f(p) for all p ∈ dom(δ) . this is equivalent to the fact that the identity i : x → x is (δ,δ ′ ) computable. if δ ≤ δ ′ and δ ′ ≤ δ, then δ and δ ′ are called computably equivalent. analogously to the notion of computability ,we can define the notion of (δ,δ ′ ) -continuity, by substituting a continuous function f in the definitions above. on σω , we use the cantor topology, which is simply the product topology of the discrete topology on σ . given a represented space (x,δ) , a computable sequence is defined as a computable function f : n → x where we assume that n is represented by δn(1n0ω) = n and a point x ∈ x is called computable if there is a constant computable sequence with value x. the notion of (δ,δ ′ ) -continuity agrees with the ordinary topological notion of continuity, as long as, we are dealing with admissible representations. a representation δ of a topological space x is called admissible if δ is continuous and if the identity i : x → x is (δ′,δ) -continuous for any continuous representation δ′ of x . if δ, δ′ , are admissible representation of topological spaces x ,y , then a function f :⊆ x → y is (δ,δ′) continuous iff it is sequentially continuous [5]. given two represented spaces (x,δ) , (y,δ′) , there is a canonical representation [δ → δ′] of the set of (δ,δ′) continuous functions f : x → y . if δ and δ′ are admissible representations of sequential topological spaces x and y respectively, then [δ → δ′] is actually a representation of the set c(x,y ) of continuous functions f : x → y . the function space representation can be characterized by the fact that it admits evaluation and type conversion. see [19] for details. if (x,δ) , (y,δ′) are admissibly represented sequential topological spaces, then, in the following, we will always assume that c(x,y ) is represented by [δ → δ′] . it follows by evaluation and type conversion that the computable points in (c(x,y ), [δ → δ′]) are just the (δ,δ′) -computable functions f :⊆ x → y [19]. for a represented space (x,δ) , we assume that the set of sequences xn is represented by δn = [δn → δ] . the computable points in (xn,δn) are just the computable sequences in (x,δ) . the notion of computable metric space was introduced by lacombe [14]. however, we state the following definition given by brattka [6]. definition 2.2. ([6]) a tuple (x,d,α) is called a computable metric space if (1) (x,d) is a metric space. (2) α : n → x is a sequence which is dense in x . (3) do(α×α) : n2 → r is a computable (double) sequence in r. given a computable metric space (x,d,α) , its cauchy representation δx :⊆ σω → x is defined as δx (01 n0+101n1+101n2+1...) := lim i→∞ α(ni) for all ni ∈ n such that (α(ni))i∈n converges and d(α(ni),α(nj)) < 2 −i for all j > i. in the following, we assume that computable metric spaces are represented by their cauchy representation. all cauchy representations are admissible with respect to the corresponding metric topology. an example of a computable metric space is (r,dr,αr) with the euclidean metric dr(x,y) = ‖x−y‖ and a standard numbering of a dense subset q ⊆ r as αr < i,j,k >= (i − j)/(k + 1) . here, the bijective cantor pairing function <,>: n2 → n is defined as < i,j >= j + (i + j)(i + j + 1)/2 and this definition can be extended inductively to finite tuples. it is known that the cantor pairing function and the projections of its inverse are computable. in the following, we assume that r is endowed with the cauchy representation δr induced by the computable metric space given above. brattka [6] gave the following definition of a computable normed linear space. computable frames in computable banach spaces 95 definition 2.3. ([6]) a space (x,‖.‖,e) is called a computable normed space if:(1) ‖.‖ : x → r is a norm on x . (2) the linear span of e : n → x is dense in x . (3) (x,d,αe) with d(x,y) =‖ x−y ‖ and αe < k,< n0, ...,nk >>= σ k i=0αf (ni)ei is a computable metric space with cauchy representation δx . it was observed that computable normed space is automatically a computable vector space, that is, the linear operations are all computable. if the underlying space (x,‖.‖) is a banach space then (x,‖.‖,e) is called a computable banach space. we always assume that computable normed spaces are represented by their cauchy representations, which are admissible with respect to norm topology. two computable banach space with the same underlying set are called computably equivalent if the corresponding cauchy representations are computably equivalent. a sequence (ei)i∈n in a banach space x , is called a schauder basis of x if every x ∈ x can be uniquely represented as x = σ∞i=0xiei with xi ∈ f . if x is a computable banach space, then a sequence (ei)i∈n is called a computable basis, if it is a schauder basis of x that is computable in x . a sequence space s is called a bk space if it is a banach space and the coordinate functionals are continuous on s . here, a sequence space is a set s of sequences of scalars which is closed under co-ordinatewise addition and scalar multiplication. several representations for the operator space b(x,y ) of bounded linear operators between two computable normed spaces are defined in [3]. in the following, we state some of the representations for the operator space b(x,y ) , as they are used in consequent results of this paper. definition 2.4. ([3]) let (x,‖.‖,e) and y be computable normed spaces. define representations of b(x,y ) : 1) δev(p) = t ⇔ [δx → δy ](p) = t 2) δseq(p) = t ⇔ δny (p) = (tei)i∈n 3) δ≥seq < p,q >= t ⇔ δseq(p) = t and δr(q) ≥‖t‖. 3. main results in order to develop a systematic computable frame theory on banach spaces, we first extend the notion of computability to m-basis. m-basis were introduced by a.i. markusevic, who regarded them as a natural generalization of the trigonometric system in c[0, 2π] and hence, as a natural replacement for basis. m-basis exists in every separable banach space. we begin with the following definition of computable m-basis. definition 3.1. a sequence (xn) in a computable banach space (x,‖.‖,e) is a computable m-basis of x if :(1) (xn) is a computable complete sequence in x . (2) there exists a total computable sequence of functionals (fn) ⊆ x∗ , with respect to [δx → δf ] representation, such that (xn,fn) is a biorthogonal system. let (x,‖.‖,e) be a computable banach space with a computable m-basis (xn) . since (xn) is a computable complete sequence in x , (x,‖.‖, (xn)) is a computable banach space that is computably equivalent to (x,‖.‖,e) . remark 3.2. let (x,‖.‖, (en)) be a computable banach space with computable basis (en) , then (en) is a computable complete sequence in x . the sequence (e ′ n) of co-ordinate functionals is a computable sequence in c(x,f) by proposition 3.3 in [7]. also, (e′n) is a total sequence of functionals such that (en,e ′ n) is a biorthogonal system. hence, a computable basis in a computable banach space is a computable m-basis. 96 kaushik and mantry next, we prove that an associated banach space of scalar valued sequences with respect to a computable m-basis is a computable banach space. theorem 3.3. let (xn) be a computable m-basis for a computable banach space (x,‖.‖, (xn)) , with associated sequence of functionals (fn) . let ed = {(fn(x)) : x ∈ x} be the associated banach space with norm ‖(fn(x))‖ed = ‖x‖x , x ∈ x . then (ed,‖.‖, (ei)) , (ei) being the sequence of canonical unit vectors, forms a computable banach space. proof: let d and d′ be the metric induced by the norm of ed and x , respectively. note that, for h =< k,< n0, ...nk >> and h ′ =< p,< m0, ...mp >> ∈ n, d(αe(h),αe(h ′)) = ‖σki=0αf (ni)ei − σ p i=0αf (mi)ei‖ = ‖(fn(σki=0αf (ni)xi − σ p i=0αf (mi)xi))‖ed = ‖σki=0αf (ni)xi − σ p i=0αf (mi)xi‖x = d′(αx(h),αx(h ′)) now, the result follows from the fact that (x,d′,αx) is a computable metric space. in the following result, we prove that the computable banach space (x,‖.‖, (xn)) with computable m-basis (xn) and the associated computable banach space (ed,‖.‖, (ei)) are computably isomorphic. here, a computable isomorphism t is an isomorphism such that t as well as t−1 are computable. theorem 3.4. let (x,‖.‖, (xn)) be a computable banach space, (xn) be a computable m-basis and (ed,‖.‖, (ei)) be the associated computable banach space. then the mapping t : x → ed given by t(x) = (fn(x)) ,x ∈ x is a computable isometrical isomorphism. proof: the map t is a bounded linear operator satisfying ‖tx‖ = ‖x‖, for all x ∈ x and therefore, ‖t‖≤ 1 . also, (t(xn)) = (en) is a computable sequence in ed . thus, we can get a δ≥seq name of t and so a δev name. therefore, t is [δx → δed ] computable. hence, by computable banach inverse mapping theorem in [6], t is a computable isometrical isomorphism. next, we generalize the notion of associated computable banach space of scalar valued sequences. definition 3.5. a bk space xd is said to be a computable bk-space if it is a computable banach space such that the sequence of co-ordinate functionals τj : xd → f given by τj((xi)) = xj , j ∈ n is computable with respect to [δxd → δf ] representation. example 3.6. let (xn) be a computable m-basis for a computable banach space (x,‖.‖, (xn)) , with associated total sequence of functionals (fn) ⊆ x∗ . then ed = {(fn(x)) : x ∈ x}, the associated banach space is a bk space. also, (ed,‖.‖, (ei)) is a computable banach space as proved above. the sequence of co-ordinate functionals τj : ed → f given by τj((fn(x))) = fj(x),x ∈ x is computable in c(ed,f) as ‖τn‖≤‖fn‖ and (fn) is a computable sequence with respect to [δx → δf ] representation. also, τn(ek) = δkn , that is, given n and k, τn(ek) can be computed. hence, (ed,‖.‖, (ei)) is a computable bk-space. we define computable xd frame for a computable banach space in the following. definition 3.7. let x be a computable banach space, xd be a computable bk-space. a computable sequence (gi) ⊆ x∗ , with respect to [δx → δf ] representation is called a computable xd frame for x if: (1) (gi(f)) ∈ xd ∀f ∈ x. (2) ‖f‖x and ‖(gi(f))‖xd are equivalent, that is, there exists constants a, b > 0 such that a‖f‖x ≤‖(gi(f))‖xd ≤ b‖f‖x for all f ∈ x . if, only the (1) and the upper condition in (2) are satisfied, (gi) is called a computable xd bessel sequence for x . computable frames in computable banach spaces 97 example 3.8. let (xn) ⊆ x , (fn) ⊆ x∗ and xd = ed be as in example 3.6. then (xd,‖.‖, (ei)) is a computable bk space. also, since (fn) ⊆ x∗ is a computable sequence with respect to [δx → δf ] representation such that (fn(x)) ∈ xd for all x ∈ x and ‖(fn(x))‖xd = ‖x‖x , x ∈ x , (fn) forms a computable xd frame for x . in the following, we present a computable version of a sufficient condition for the existence of a xd frame (theorem 2.1 in [9]). theorem 3.9. let x be any computable banach space and xd be any computable bk space. if x is isometrically isomorphic to a subspace of xd by a computable map, then there exists a computable xd frame for x . proof: let t : x → xd be a computable map such that x is isometrically isomorphic to t(x) . let (τi) be the computable sequence of co-ordinate functionals of xd , given by τi((xj)) = xi, i ∈ n, (xj) ∈ xd . for each i ∈ n, define gi = τi ◦t . then (gi) is a computable sequence of functionals such that (gi(f)) = (τi(t(f))) = tf ∈ xd ,f ∈ x and ‖(gi(f))‖xd = ‖tf‖xd = ‖f‖x . hence, (gi) ⊆ x ∗ is a computable xd frame for x . for the converse, theorem 2.1 in [9] states that if (gi) be an xd frame for a banach space x , then the map u : x → xd given by u(f) = (gi(f)) is an isomorphism of x into xd . using the techniques from [8], we show that the computable version of this result does not hold. example 3.10. consider the computable banach space (l2,‖.‖, (ei)) with the sequence of canonical unit vectors (ei) as computable basis. let (ai) be a computable sequence of positive reals such that ‖(ai)‖l2 exists but is not computable. we assume a0 = 1 . define a linear bounded operator t : l2 → l2 as   1 a1 a2 a3 · · · 0 1 0 0 · · · 0 0 1 0 · · · ... ... ... . . .   then, (fi) = (tei) is a frame for l2 . define gi : l2 → r by gi(f) =< f,fi >, i ∈ n. then, (gi) forms a computable xd frame for l2 where xd is the computable bk space (l2,‖.‖, (ei)) . but the operator u : l2 → l2 , u(f) = (< f,fi >) is not computable as u(e0) = (ai) is not computable in l2 . the next result shows that the map u is computable with respect to [δx → δnf ] representation. theorem 3.11. let x be a computable banach space and xd be a computable bk space. if (gi) ⊆ x∗ be a computable xd frame for x then the map u : x → xd f → (gi(f)) is (δx, [δn → δf ]) computable isomorphism of x into xd . proof: the map u is an isomorphism of x into xd by theorem 2.1 in [9]. by the computability of the sequence (gi) and the evaluation property , we get that the map u ′ : n×x → f (i,f) → gi(f) is computable with respect to ([δn,δx ],δf ) representation. by type conversion, the map u is computable with respect to (δx, [δn → δf ]) representation. 98 kaushik and mantry now, we give a computable version of a necessary and sufficient condition for the existence of a xd bessel sequence (corollary 3.3 in [9]). first, we recall, the dual space representation δx∗ of the dual space x ∗ as given in [7]. definition 3.12. ([7]) for a separable banach space x , define a representation δx∗ of the dual space x∗ by δx∗ < p,q >= f ⇔ [δx → δf ](p) = f and δr(q) = ‖f‖ . next, we give the definition of computable dual space given in [7], as it is required in the subsequent result. definition 3.13. ([7]) let x be a computable banach space. x is said to have a computable dual space x∗ if there exists a sequence e∗ : n → x∗ such that: (1) (x∗,‖.‖,e∗) is a computable banach space. (2) δx∗ is computably equivalent to the cauchy representation δ c x∗ of (x ∗,‖.‖,e∗) . theorem 3.14. let x be a computable banach space with computable dual space. let (xd,‖.‖, (ei)) be a computable bk-space and (x∗d,‖.‖, (ei)) be a computable banach space, (ei) and (ei) being the sequences of standard unit vectors as basis.then if, (gi) ⊆ x∗ be a computable xd -bessel sequence for x with bound b with (‖gi‖) being a computable sequence then t : x∗d → x ∗ given by t((di)) = σdigi is a well defined computable operator from ‖t‖≤ b . converse holds if xd is reflexive space. proof: define r : x → xd by r(x) = (gi(x)) ,x ∈ x . since (gi) is an xd bessel sequence, (gi(x)) ∈ xd , x ∈ x and ‖r(x)‖ = ‖(gi(x))‖xd ≤ b‖x‖x , x ∈ x . consider r ∗ : x∗d → x ∗ . then r∗(ej) : x → f is such that r∗(ej)(x) = ej(r(x)) = gj(x),x ∈ x . therefore,r∗(ej) = gj , for all j ∈ n such that r∗(σidiei) = σidigi . thus, t = r∗ is a well defined operator given by t((di)) = σidigi satisfying ‖t‖ ≤ b . as, (gi) ⊆ x∗ is a computable sequence with respect to [δx → δf ] representation and (‖gi‖) is assumed to be a computable sequence, therefore, (gi) is a computable sequence in x ∗ with respect to cauchy representation. therefore, t is [δcx∗ d → δcx∗] computable. conversely, let t : x∗d → x ∗ be a computable operator given by t((di)) = σidigi . then t(ei) = gi , for all i and since, a computable operator maps computable sequences to computable sequences, therefore, (gi) is a computable sequence in x ∗ with respect to [δx → δf ] representation and (‖gi‖) is a computable sequence. consider t∗ : x∗∗ → x∗∗d which satisfies t ∗(f)(ej) = f(gj) , for all f ∈ x∗∗ . therefore, (gi(x)) = (t∗(i(x))(ei)) ∈ x∗∗d , identified with t ∗(i(x)) , where i is the natural embedding of x into x∗∗ . since xd is reflexive, therefore, (gi(x)) ∈ xd and satisfies ‖(gi(x))‖xd = ‖t ∗(i(x))‖ = ‖t‖‖x‖≤ b‖x‖x hence, (gi) ⊆ x∗ is a computable xd bessel sequence for x with bound b and computable sequence of norms. banach frames were introduced by grochenig [10] as a generalization of the notion of frames in hilbert spaces. in the following definition, we introduce the notion of computable banach frame. definition 3.15. let x be a computable banach space, xd be a computable bk-space. given a computable linear operator s : xd → x and a computable xd frame (gi) ⊆ x∗ , we say that ((gi),s) is a computable banach frame for x with respect to xd if s((gi(x))) = x for all x ∈ x . example 3.16. let (xn) be a computable m-basis and xd = {(fn(x)) : x ∈ x} be a computable bk-space and (fn) ⊆ x∗ is a computable xd frame. by theorem 3.4, the map t : x → xd given by x → (fn(x)) , x ∈ x is a computable isometrical isomorphism. thus, ((fi),t−1) is a computable banach frame for x with respect to xd . computable frames in computable banach spaces 99 next, we give a sufficient condition for the existence of a computable banach frame. in the following result, we call a closed subspace of a computable banach space to be computably complemented if it is the range of a computable linear projection in the space. clearly, a computably complemented subspace is a computable subspace as defined in [7]. theorem 3.17. a computable banach space x has a computable banach frame with respect to a given computable bk space xd if x is isometrically isomorphic to a computably complemented subspace of xd by a computable map. proof: suppose t : x → xd be a computable map which is an isometric isomorphism of x into xd and t(x) be the computably complemented subspace of xd . then, there exists a computable projection p : xd → xd such that p(xd) = t(x) and p2 = p . define s : xd → x by sx = t−1px, x ∈ xd . then, by computable banach inverse mapping theorem, s is a computable linear operator. let (τj) be the computable sequence of co-ordinate functionals of xd and gi = t ∗(τi) , for all i ∈ n. then, for x ∈ x , we have gi(x) = t∗τi(x) = τit(x) , for all i ∈ n. therefore, gi = τit and hence, (gi) is a computable sequence in x ∗ with respect to [δx → δf ] representation. also, tx = (gi(x)) ∈ xd , for all x ∈ x and ‖(gi(x))‖xd = ‖tx‖xd = ‖x‖x . also s((gi(x))) = t−1p((gi(x))) = t −1((gi(x))) = x. thus, ((gi),s) is a computable banach frame for x with respect to xd . finally, we give sufficient condition under which a computable xd frame for x is a computable banach frame for x . theorem 3.18. let x be a computable banach space, (xd,‖.‖, (ei)) be a computable bk space with sequence (ei) of canonical unit vectors. let (gi) ⊆ x∗ be a computable xd frame for x . if there exists a computable sequence (fi) ⊆ x such that σicifi is convergent for all (ci) ∈ xd and f = σigi(f)fi , for all f ∈ x . then, there exists a computable linear operator t : xd → x such that ((gi),t) is a computable banach frame for x with respect to xd . proof: define tn : xd → x as tn((ci)) = σni cifi . then, tn(ei) = fi , for all i ≤ n and 0 otherwise, that is, given n and k, we can compute tn(ek) . since, (tn((ci)))n is convergent and hence a bounded sequence, by uniform boundedness principle, supn‖tn‖ < ∞. therefore, (tn) is a computable sequence of linear and computable operators. define m : n → n as m(< i,j >) = j . then m is a computable function such that ‖tm<i,j>ej − limn→∞tnej‖ ≤ 2−i , for all i,j ∈ n. by computable banach steinhaus theorem in [2], t : xd → x defined as t((ci)) = limn→∞tn((ci)) = σicifi is a linear computable operator satisfying t((gi(f))) = f for all f ∈ x . hence, ((gi),t) is a computable banach frame for x with respect to xd . references [1] s. banach and s. mazur, sur les fonctions calculables, ann. soc. pol. de math. 16 (1937), 223. [2] v. brattka, computability of banach space principles, informatik berichte 286, fernuniversitat hagen, fachbereich informatik, hagen, june 2001. [3] v. brattka, effective representations of the space of linear bounded operators, applied general topology, 4(1)(2003), 115-131. [4] v. brattka, computable versions of the uniform boundedness theorem, in: z. chatzidakis, p.koepke, w. pohlers (eds.), logic colloquium 2002, in: lecture notes in logic, vol 27, association for symbolic logic, urbana, 2006. [5] v. brattka, computability over topological structures. in s. barry cooper and sergey s. goncharov, editors, computability and models, 93-136. kluwer academic publishers, new york, 2003. [6] v. brattka, a computable version of banach’s inverse mapping theorem, annals of pure and applied logic, 157 (2009), 85-96. [7] v. brattka, r. dillhage, computability of compact operators on computable banach spaces with bases. math. log. quart.53 (2007), 345-364. [8] v. brattka, a. yoshikawa, towards computability of elliptic boundary value problems in variational formulation, j. complexity 22(6) (2006), 858-880. [9] p.g. casazza, o. christensen, d.t. stoeva, frame expansions in separable banach spaces, j. math. anal. appl., 307 (2005), 710-723. [10] k. grochenig, describing functions : atomic decompositions versus frames, monatsh. math. 112 (1991), 1-41. [11] a. grzegorczyk, on the definitions of computable real continuous functions, fund. math. 44 (1957), 61-71. 100 kaushik and mantry [12] c. kreitz, k. weihrauch, a unified approach to constructive and recursive analysis, in: m. richter, e. borger, w. oberschelp, b.schinzel, w. thomas (eds.), computation and proof theory, lecture notes in mathematics, vol. 1104, springer, berlin, 1984, 259-278. [13] d. lacombe, les ensembles recursivement ouverts ou fermes, et leurs applications a 1’analyse recursive, comptes, rendus 246 (1958), 28-31. [14] d. lacombe, quelques procedes de definition en topologie recursive, in: a. heyting, editor, constructivity in mathematics, (northholland, amsterdam 1959), 129-158. [15] r. e. megginson, an introduction to banach space theory, graduate texts in mathematics 183, springer, new york, 1989. [16] m.b. pour-el, j.i. richards, computability in analysis and physics, springer, berlin, 1989. [17] i. singer, bases in banach spaces-ii, springer-verlag, new york, heidelberg, 1981. [18] a. m. turing, on computable numbers, with an application to the ”entscheidungsproblem”, proc. london math. soc. 42 (1936), 230-265. [19] k. weihrauch, computable analysis, springer, berlin, 2000. 1 department of mathematics, kirorimal college, university of delhi, delhi-110007, india 2 department of mathematics, daulat ram college, university of delhi, delhi-110007, india ∗ corresponding author: shikk2003@yahoo.co.in international journal of analysis and applications issn 2291-8639 volume 11, number 1 (2016), 61-69 http://www.etamaths.com integral inequalities of hermite-hadamard type for harmonic (h, s)-convex functions muhammad aslam noor∗, khalida inayat noor and sabah iftikhar abstract. in this paper, we introduce a new concept of harmonic (h, s)-convex functions in the second sense which generalizes the harmonic convex functions. some hermite-hadamardfejer type integral inequalities are derived. some special cases also discussed. results derived in this paper represent significant refinement and improvement of the known results. 1. introduction convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences. for recent applications, generalizations and other aspects of convex functions and their variant forms, see [1, 14–16] and the references therein. varosanec [17] introduced a class of convex functions with respective to an arbitrary non-negative function h, which is known as h-convex function.. this class of functions unifies various classes of convex functions and is being used to discuss several concepts in a unified manners. an other important class of convex functions is known as harmonic convex functions, was investigated by anderson et al. [1] and iscan [8]. for the recent developments, see [8,9, 11,12,13,15 and the references therein. nooe et al. . [13] introduced and investigated a new class of convex functions. it has been shown a number new and known classes of convex functions can be obtained as special cases. motivated and inspired by ongoing research in this filed, we introduce and study a new class of convex functions, which is called harmonic (h, s)-convex functions. one can easily show that harmonic (h, s)-convex functions include godunova-levin harmonic convex functions and harmonic s-convex functions as special cases. this is the main motivation of this paper. we also obtain several new hermite-hadamard-fejer type inequalities for harmonic (h, s)-convex functions. our results include several previously known and new results as special cases. the ideas and technique of this paper may be a starting point for further research in this dynamic field. 2. preliminaries first of all, we recall the following basic concepts. definition 2.1. [15]. a set i = [a, b] ⊆ r\{0} is said to be a harmonic convex set, if xy tx + (1 − t)y ∈ i, ∀x, y ∈ i, t ∈ [0, 1]. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. harmonic convex functions; harmonic h-convex functions; harmonic s-convex functions; hermite-hadamard type inequality. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 61 62 noor, noor and iftikhar definition 2.2. [8]. a function f : i = [a, b] ⊆ r \ {0} → r is said to be harmonic convex function, if f ( xy tx + (1 − t)y ) ≤ (1 − t)f(x) + tf(y), ∀x, y ∈ i, t ∈ [0, 1]. we now introduce a new class of harmonic convex function in second sense, which is called the (h, s)-harmonic convex function. definition 2.3. let h : j = [0, 1] → r a nonnegative function. a function f : i = [a, b] ⊆ r\{0}→ r is said to be harmonic (h, s)-convex function in second sense, where s ∈ [−1, 1], if f ( xy tx + (1 − t)y ) ≤ h((1 − t)s)f(x) + h(ts)f(y), ∀x, y ∈ i, t ∈ (0, 1). for t = 1 2 , we have f ( 2xy x + y ) ≤ h ( 1 2s ) [f(x) + f(y)], which is called jensen type harmonic (h, s)-convex function. we now discuss some special cases of harmonic (h, s) convex function. i. if we take h(ts) = ts and s = −1 in definition 2.3, then it reduces to godunova-levin harmonic convex functions. definition 2.4. [9]. a function f : i = [a, b] ⊆ r \ {0} → r is said to be godunova-levin harmonic convex, if f ( xy tx + (1 − t)y ) ≤ 1 1 − t f(x) + 1 t f(y), ∀x, y ∈ i, t ∈ (0, 1). ii. if we take h(ts) = ts in definition 2.3, then it reduces to extended harmonic s-convex functions. definition 2.5. a function f : i = [a, b] ⊆ r \ {0} → r is said to be extended harmonic s-convex function in second sense, where s ∈ [−1, 1], if f ( xy tx + (1 − t)y ) ≤ (1 − t)sf(x) + tsf(y), ∀x, y ∈ i, t ∈ [0, 1]. iii. if s = 1 in definition 2.3, then it reduces to the harmonic h-convex functions. definition 2.6. [11]. a function f : i = [a, b] ⊆ r\{0}→ r is said to be harmonic h-convex function, if f ( xy tx + (1 − t)y ) ≤ h(1 − t)f(x) + h(t)f(y), ∀x, y ∈ i, t ∈ [0, 1]. definition 2.7. [14]. two functions f, g are said to be similarly ordered (f is g-monotone), if and only if, 〈f(x) −f(y), g(x) −g(y)〉≥ 0, ∀x, y ∈ rn. we now show that the product of two harmonic (h, s)-convex functions is again harmonic (h, s)-convex function. lemma 2.1. if h(ts) + h((1 − t)s) ≤ 1, then the product of two similarly ordered harmonic (h, s)-convex functions is harmonic ((h, s)-convex function. integral inequalities 63 proof. let f, g be two (h, s)-harmonic convex functions. then f ( xy tx + (1 − t)y ) g ( xy tx + (1 − t)y ) ≤ [h((1 − t)s)f(x) + h(ts)f(y)][h((1 − t)s)g(x) + h(ts)g(y))] = [h((1 − t)s)]2f(x)g(x) + h(ts)h((1 − t)s)[f(x)g(y) + f(y)g(x)] +[h(ts)]2f(y)g(y) ≤ [h((1 − t))s]2f(x)g(x) + h(ts)h((1 − t)s)[f(x)g(x) + f(y)g(y)] +[h(ts)]2f(y)g(y) = [h((1 − t)s)f(x)g(x) + h(ts)f(y)g(y)][h(ts) + h((1 − t)s)] ≤ h((1 − t)s)f(x)g(x) + h(ts)f(y)g(y).(2.1) this shows that product of two similarly ordered harmonic (h, s)-convex functions is again a harmonic (h, s)-convex function. � we need the following well-known fact, which establishes a relationship between convex functions and harmonic convex functions. this fact plays a crucial part in deriving our results. remark 2.1. let i = [a, b] ⊆ r \ {0} and consider the function g : [ 1 b , 1 a ] → r defined by g(x) = f ( 1 x ) , then f is harmonic (h, s)-convex on [a, b], if and only if, g is (h, s)-convex in the usual sense on [ 1 b , 1 a ] . 3. main results in this section, we obtain hermite-hadamard inequalities for harmonic (h, s)-convex function. theorem 3.1. let f : i = [a, b] ⊆ r \ {0} −→ r be harmonic (h, s)-convex function, where s ∈ (−1, 1]. if f ∈ l[a, b], then 1 2h ( 1 2s )f( 2ab a + b ) ≤ ab b−a ∫ b a f(x) x2 dx ≤ [f(a) + f(b)] ∫ 1 0 h(ts)dt. proof. let f be harmonic (h, s)-convex function with t = 1 2 . then f ( 2xy x + y ) ≤ h ( 1 2s ) [f(x) + f(y)]. taking x = ab ta+(1−t)b and y = ab (1−t)a+tb , we have f ( 2ab a + b ) ≤ h ( 1 2s )[ f ( ab ta + (1 − t)b ) + f ( ab (1 − t)a + tb )] = h ( 1 2s )[∫ 1 0 f ( ab ta + (1 − t)b ) dt + ∫ 1 0 f ( ab (1 − t)a + tb ) dt ] ≤ h ( 1 2s )∫ 1 0 [ h((1 − t)s)f(a) + h(ts)f(b) + h((1 − t)s)f(b) +h(ts)f(a) ] dt = 2h ( 1 2s ) [f(a) + f(b)] ∫ 1 0 h(ts)dt. using the fact that ∫ 1 0 f ( ab ta + (1 − t)b ) dt = ab b−a ∫ b a f(x) x2 dx, 64 noor, noor and iftikhar we have 1 2h ( 1 2s )f( 2ab a + b ) ≤ ab b−a ∫ b a f(x) x2 dx ≤ [f(a) + f(b)] ∫ 1 0 h(ts)dt. � theorem 3.2. let f : i = [a, b] ⊆ r \ {0} −→ r be harmonic (h, s)-convex function, where s ∈ (−1, 1]. if f ∈ l[a, b], then ab b−a ∫ b a f(x) x2 dx ≤ 1 2 [f(a) + f(b)] ∫ 1 0 [h((1 − t)s) + h(ts)]dt. proof. let f be harmonic (h, s)-convex function. then f ( ab ta + (1 − t)b ) ≤ h((1 − t)s)f(a) + h(ts)f(b) f ( ab (1 − t)a + tb ) ≤ h(ts)f(a) + h((1 − t)s)f(b) f ( ab ta + (1 − t)b ) ≤ h((1 − t)s)f(a) + h(ts)f(b) and f ( ab (1 − t)a + tb ) ≤ h(ts)f(a) + h((1 − t)s)f(b). adding the above inequalities, we have f ( ab ta + (1 − t)b ) + f ( ab (1 − t)a + tb ) + f ( ab ta + (1 − t)b ) +f ( ab (1 − t)a + tb ) ≤ 2[f(a) + f(b)][h((1 − t)s) + h(ts)] integrating the above inequality over t ∈ [0, 1], we obtain∫ 1 0 f ( ab ta + (1 − t)b ) dt + ∫ 1 0 f ( ab (1 − t)a + tb ) dt + ∫ 1 0 f ( ab ta + (1 − t)b ) dt + ∫ 1 0 f ( ab (1 − t)a + tb ) dt ≤ 2[f(a) + f(b)] ∫ 1 0 [h((1 − t)s) + h(ts)]dt, which implies ab b−a ∫ b a f(x) x2 dx ≤ 1 2 [f(a) + f(b)] ∫ 1 0 [h((1 − t)s) + h(ts)]dt, which is the required result. � theorem 3.3. let f, g : i = [a, b] ⊆ r \ {0} −→ r be two harmonic (h, s)-convex functions, where s ∈ (−1, 1]. if f ∈ l[a, b], then ab b−a ∫ b a f(x)g(x) x2 dx ≤ m(a, b) ∫ 1 0 [h(ts)]2dt + n(a, b) ∫ 1 0 h(ts)h((1 − t)s)dt, where m(a, b) = f(a)g(a) + f(b)g(b),(3.1) n(a, b) = f(a)g(b) + f(b)g(a).(3.2) integral inequalities 65 proof. let f, g be two harmonic (h, s)-convex functions. then f ( ab ta + (1 − t)b ) ≤ h((1 − t)s)f(a) + h(ts)f(b) g ( ab ta + (1 − t)b ) ≤ h((1 − t)s)g(a) + h(ts)g(b). now f ( ab ta + (1 − t)b ) g ( ab (1 − t)a + tb ) ≤ [ h((1 − t)s)f(a) + h(ts)f(b) ][ h((1 − t)s)g(a) + h(ts)g(b) ] = [h((1 − t)s)]2[f(a)g(a)] + h(ts)h((1 − t)s)[f(a)g(b) + f(b)g(a)] +[h(ts)]2[f(b)g(b)] integrating the above inequality over [0, 1], we have∫ 1 0 f ( ab ta + (1 − t)b ) g ( ab (1 − t)a + tb ) dt ≤ [f(a)g(a)] ∫ 1 0 [h((1 − t)s)]2dt + [f(a)g(b) + f(b)g(a)] ∫ 1 0 h(ts)h((1 − t)s)dt +[f(b)g(b)] ∫ 1 0 [h(ts)]2dt = [f(a)g(a) + f(b)g(b)] ∫ 1 0 [h(ts)]2dt +[f(a)g(b) + f(b)g(a)] ∫ 1 0 h(ts)h((1 − t)s)dt, thus ab b−a ∫ b a f(x)g(x) x2 dx ≤ m(a, b) ∫ 1 0 [h(ts)]2dt + n(a, b) ∫ 1 0 h(ts)h((1 − t)s)dt, the required result. � theorem 3.4. let f, g : i = [a, b] ⊆ r \{0} −→ r be harmonic (h, s)-convex functions, where s ∈ (−1, 1]. if f, g ∈ l[a, b], then( ab b−a )s+1 ∫ 1 a 1 b h (( x− 1 b )s)[ f(a)g ( 1 x ) + g(a)f ( 1 x )] dx ( ab b−a )s+1 ∫ 1 a 1 b h (( 1 a −x )s)[ f(b)g ( 1 x ) + g(b)f ( 1 x )] dx ≤ m(a, b) ∫ 1 0 [h(ts)]2dt + n(a, b) ∫ 1 0 h(ts)h((1 − t)s)dt + ab b−a ∫ b a f(x)g(x) x2 dx, where m(a, b) and n(a, b) are given by (3.1) and (3.2) respectively. proof. let f, g be harmonic (h, s)-convex functions. then f ( ab ta + (1 − t)b ) ≤ h((1 − t)s)f(a) + h(ts)f(b) g ( ab ta + (1 − t)b ) ≤ h((1 − t)s)g(a) + h(ts)g(b). 66 noor, noor and iftikhar now, using 〈x1 −x2, x3 −x4〉≥ 0, (x1, x2, x3, x4 ∈ r) and x1 < x2, x3 < x4, we have f ( ab ta + (1 − t)b ) [h((1 − t)s)g(a) + h(ts)g(b)] +g ( ab ta + (1 − t)b ) [h((1 − t)s)f(a) + h(ts)f(b)] ≤ [h((1 − t)s)f(a) + h(ts)f(b)][h((1 − t)s)g(a) + h(ts)g(b)] +f ( ab ta + (1 − t)b ) g ( ab ta + (1 − t)b ) . thus g(a)h((1 − t)s)f ( ab ta + (1 − t)b ) + g(b)h(ts)f ( ab ta + (1 − t)b ) +f(a)h((1 − t)s)g ( ab ta + (1 − t)b ) + f(b)h(ts)g ( ab ta + (1 − t)b ) ≤ [h((1 − t)s)]2[f(a)g(a)] + h(ts)h((1 − t)s)[f(b)g(a) +f(a)g(b)] + [h(ts)]2[f(b)g(b)] +f ( ab ta + (1 − t)b ) g ( ab ta + (1 − t)b ) integrating the above inequality with respect to t over [0, 1], we have g(a) ∫ 1 0 h((1 − t)s)f ( ab ta + (1 − t)b ) dt + g(b) ∫ 1 0 h(ts)f ( ab ta + (1 − t)b ) dt +f(a) ∫ 1 0 h((1 − t)s)g ( ab ta + (1 − t)b ) dt + f(b) ∫ 1 0 h(ts)g ( ab ta + (1 − t)b ) dt ≤ [f(a)g(a)] ∫ 1 0 [h((1 − t)s)]2dt + [f(a)g(b) + f(b)g(a)] ∫ 1 0 h(ts)h((1 − t)s)dt +[f(b)g(b)] ∫ 1 0 [h(ts)]2dt + ∫ 1 0 f ( ab ta + (1 − t)b ) g ( ab ta + (1 − t)b ) dt, from which, it follows that( ab b−a )s+1 ∫ 1 a 1 b h (( x− 1 b )s)[ f(a)g ( 1 x ) + g(a)f ( 1 x )] dx ( ab b−a )s+1 ∫ 1 a 1 b h (( 1 a −x )s)[ f(b)g ( 1 x ) + g(b)f ( 1 x )] dx ≤ [f(a)g(a) + f(b)g(b)] ∫ 1 0 [h(ts)]2dt +[f(b)g(a) + f(a)g(b)] ∫ 1 0 h(ts)h((1 − t)s)dt + ab b−a ∫ b a f(x)g(x) x2 dx, which is the required result. � we need the following lemma in order to obtain the fejer type hermite-hadamard inequality for harmonic (h, s)-convex functions. integral inequalities 67 lemma 3.1. let f : i = [a, b] ⊆ r \ {0} −→ r be harmonic (h, s)-convex function, where s ∈ (−1, 1]. then f ( abx (a + b)x−ab ) ≤ [h((1 − t)s) + h(ts)][f(a) + f(b)] −f(x). proof. it is known that that x ∈ [a, b], can be represented as x = ab ta+(1−t)b , ∀t ∈ [0, 1]. thus f ( abx (a + b)x−ab ) = f ( ab (1 − t)a + tb ) ≤ h(ts)f(a) + h((1 − t)s)f(b) = h(ts)[f(a) + f(b)] + h((1 − t)s)[f(a) + f(b)] −[h(ts)f(b) + h((1 − t)s)f(a)] ≤ h(ts)[f(a) + f(b)] + h((1 − t)s)[f(a) + f(b)] −f(x), the required result. � theorem 3.5. let f : i = [a, b] ⊆ r \ {0} −→ r be harmonic (h, s)-convex function, where s ∈ (−1, 1]. if f ∈ l[a, b], then 1 2h( 1 2s ) f ( 2ab a + b )∫ b a g(x) x2 dx ≤ ∫ b a f(x)g(x) x2 dx ≤ [f(a) + f(b)] 2 ∫ b a [ h ( b(x−a) x(b−a) )s + h ( a(b−x) x(b−a) )s] g(x) x2 dx, where g : [a, b] ⊆ r\{0} is a nonnegative, integrable function and satisfies g(x) = g ( abx [a + b]x−ab ) , ∀x ∈ [a, b]. proof. using the given fact and lemma 3.1, we have 1 2h( 1 2s ) f ( 2ab a + b )∫ b a g(x) x2 dx = 1 2h( 1 2s ) ∫ b a f ( 2abx (a + b)x−ab + ab ) g(x) x2 dx ≤ 1 2h( 1 2s ) ∫ b a h ( 1 2s )[ f ( abx (a + b)x−ab ) + f(x) ] g(x) x2 dx = 1 2 ∫ b a f ( abx (a + b)x−ab ) g(x) x2 dx + 1 2 ∫ b a f(x)g(x) x2 dx = ∫ b a f(x)g(x) x2 dx 68 noor, noor and iftikhar to prove the other part of the inequality, we consider∫ b a f(x)g(x) x2 dx = 1 2 ∫ b a f ( abx (a + b)x−ab ) g(x) x2 dx + 1 2 ∫ b a f(x)g(x) x2 dx ≤ 1 2 ∫ b a [ [h((1 − t)s) + h(ts)][f(a) + f(b)] −f(x) ] g(x) x2 dx + 1 2 ∫ b a f(x)g(x) x2 dx ≤ [f(a) + f(b)] 2 ∫ b a [ h ( b(x−a) x(b−a) )s + h ( a(b−x) x(b−a) )s] g(x) x2 dx, this completes the proof. � acknowledgements the authors would like to thank dr. s. m. junaid zaidi, (s. i, h. i), rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] w. w. breckner, stetigkeitsaussagen fiir eine klasseverallgemeinerter convexer funktionen in topologischen linearen raumen. pupl. inst. math. 23 (1978), 13-20. [3] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publisher, dordrechet, holland, (2002). [4] g. cristescu, improved integral inequalities for product of convex functions, j. inequal. pure appl. math., 6(2)(2005), article id 35. [5] s.s. dragomir, inequalities of hermite-hadamard type for ha-convex functions, preprint rgmia res. rep. coll. 18(2015), article id 38. [6] j. hadamard, etude sur les proprietes des fonctions entieres e.t en particulier dune fonction consideree par riemann. j. math. pure appl., 58(1893), 171-215. [7] c. hermite, sur deux limites d’une intgrale dfinie. mathesis, 3(1883), 82. [8] i. iscan, hermite-hadamard type inequalities for harmonically convex functions. hacet, j. math. stats., 43(6)(2014), 935-942. [9] i. iscan, ostrowski type inequalities for harmonically s-convex functions, konuralp j. math., 3(1)(2015), 63-74. [10] c. p. niculescu and l. e. persson, convex functions and their applications, springerverlag, new york, (2006). [11] m. a. noor, k. i. noor, m. u. awan and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b. sci. bull. series a, 77(1)(2015), 5-16. [12] m. a. noor, k. i. noor and m. u. awan, integral inequalities for harmonically s-godunovalevin functions, facta uni. (nis) ser. math. infor., 29(4)(2014), 415-424. [13] m. a. noor, k. i. noor and m. u. awan, integral inequalities for some new classes of convex functions, american j. appl. math., 3(2015), 1-5. [14] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [15] h. n. shi and zhang, some new judgement theorems of schur geometric and schur harmonic convexities for a class of symmetric functions, j. inequal. appl., 2013(2013), article id 527. integral inequalities 69 [16] g. h. toader, some generalizations of the convexity, proc. colloq. approx. optim, clujnapoca (romania), 1984, 329-338. [17] s. varoanec, on h-convexity, j. math. anal. appl., 326(2007), 303-311. department of mathematics, comsats institute of information technology, park road, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 62-65 http://www.etamaths.com on p-valently meromorphic-strongly starlike and convex functions rahim kargar1,∗, ali ebadian2 and janusz sokó l3 abstract. in this paper, we obtain sufficient conditions for analytic function f(z) in the punctured unit disk to be p-valently meromorphic-strongly starlike and p-valently meromorphic-strongly convex of order β and type α. some interesting corollaries of the results presented here are also discussed. 1. introduction let h be the class of functions that are analytic in the unit disk u and let a be the class of functions of the form: f(z) = z + a2z 2 + a3z 3 + · · · , which are analytic in u. let σ(p) denote the class of meromorphically p-valent functions f(z) of the form f(z) = z−p + ∞∑ k=1 ak−pz k−p p ∈ n := {1, 2, 3, . . .}, which are analytic in the punctured unit disk u∗ = {z ∈ c : 0 < |z| < 1} = u\{0}. further, we write that σ(1) = σ. for a function f ∈ σ(p), we say that it is p-valently meromorphic-strongly starlike of order 0 < β ≤ 1 and type α (0 ≤ α < p) if (1) ∣∣∣∣arg ( − zf′(z) f(z) −α )∣∣∣∣ < πβ2 z ∈ u. the corresponding class is denoted by st σ(α,β). we note that st σ(α, 1), is the class of p-valently meromorphic starlike functions of order α (see [6]) and st σ(0,β) is the class of p-valently meromorphicstrongly starlike functions of order β. furthermore, a function f ∈ σ(p) is said to be in the class skς(α,β) of p-valently meromorphic-strongly convex of order β and type α if and only if (2) ∣∣∣∣arg ( −1 − zf′′(z) f′(z) −α )∣∣∣∣ < πβ2 z ∈ u, for some real 0 < β ≤ 1 and 0 ≤ α < p. in particular, skς(α, 1), is the class of p-valently meromorphic convex functions of order α (cf. [6]) and skς(0,β) is the class of p-valently meromorphic-strongly convex functions of order β. a function f ∈a is said to be in the class st (β) of strongly starlike function of order β, 0 ≤ β < 1, if it satisfies the inequality ∣∣∣∣arg ( zf′(z) f(z) )∣∣∣∣ < πβ2 z ∈ u. a function f(z) belonging to the class sk(β) is said to be strongly convex of order β in u if and only if zf′(z) ∈st (β) (see [2, 3]). for proving our results we need the following lemma. 2010 mathematics subject classification. 30c45, 30c55. key words and phrases. analytic; meromorphic; p-valent; strongly starlike; strongly convex. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 62 on p-valently meromorphic functions 63 lemma 1. (see [1], [5]). let b(z) ∈ h be continuous on u, b(0) = 0, supz∈u |b(z)| = 1 and c = supz∈u ∫ 1 0 |b(tz)|dt. for 0 < β ≤ 1 let λ(β) = sin(πβ/2)√ 1 + 2c cos(πβ/2) + c2 . if f ∈a and |f′(z) − 1| ≤ λ(β)|b(z)| z ∈ u, then f is strongly starlike of order β. additionally, if b(t) = max 0≤ϕ≤2π |b(teiϕ)| 0 ≤ t ≤ 1, then the constant λ(β) cannot be replaced by any larger number without violating the conclusion. the lemma 1, without the sharpness part, was previously obtained by ponnusamy and singh in [4]. in this work, we obtain some sufficient conditions for p-valently meromorphic functions. 2. main results our first result is contained in the following. theorem 2. assume that f(z) 6= 0 for z ∈ u∗. if f ∈ σ(p) satisfies (3) ∣∣∣∣∣ ( f(z) z−α ) 1 α−p ( f′(z) f(z) + α z ) + p−α ∣∣∣∣∣ < (p−α)λ(β)|b(z)| z ∈ u∗, then f is p-valently meromorphic-strongly starlike of order β and type α. proof. assume that f ∈ σ(p). let us define the function g(z) by (4) g(z) = ( f(z) z−α ) 1 α−p = z + · · · z ∈ u∗. then g(z) ∈a and |g′(z) − 1| = 1 p−α ∣∣∣∣∣ ( f(z) z−α ) 1 α−p ( f′(z) f(z) + α z ) + p−α ∣∣∣∣∣ . now, by means of the condition of the theorem and applying lemma 1 we find that g(z) is strongly starlike function of order β. note that from (4) we have zg′(z) g(z) = 1 p−α ( − zf′(z) f(z) −α ) . since g(z) is strongly starlike of order β, thus∣∣∣∣arg { 1 p−α ( − zf′(z) f(z) −α )}∣∣∣∣ < πβ2 . this shows that the proof is completed. � putting α = 0 in theorem 2, we have: corollary 3. assume that f(z) 6= 0 for z ∈ u∗. if f ∈ σ(p) satisfies∣∣∣∣∣ 1p√f(z) ( f′(z) f(z) ) + p ∣∣∣∣∣ < pλ(β)|b(z)| z ∈ u∗, then f is p-valently meromorphic-strongly starlike of order β. setting b(z) = z and p = β = 1 in theorem 2, we obtain the following result: 64 kargar, ebadian and sokó l corollary 4. assume that f(z) 6= 0 for z ∈ u∗. if f ∈ σ satisfies∣∣∣∣∣ ( f(z) z−α ) 1 α−1 ( f′(z) f(z) + α z ) + 1 −α ∣∣∣∣∣ < 2√5 (1 −α) z ∈ u∗, then f is meromorphic starlike function of order α. if we take α = 0 in corollary 4, we obtain the following result: corollary 5. assume that f(z) 6= 0 for z ∈ u∗. if f ∈ σ satisfies∣∣∣∣∣ ( 1 f(z) )2 f′(z) + 1 ∣∣∣∣∣ < 2√5 ≈ 0.894427... z ∈ u∗, then f is meromorphic starlike functions. next we derive the following. theorem 6. assume that f′(z) 6= 0 for z ∈ u∗. if f ∈ σ(p) satisfies (5) ∣∣∣∣∣ ( f′(z) −pz−p−1 ) 1 α−p ( 1 + zf′′(z) f′(z) + p )∣∣∣∣∣ < (p−α)λ(β)|b(z)| z ∈ u∗, then f is p-valently meromorphic-strongly convex of order β and type α. proof. let f ∈ σ(p) and define the function p(z) by (6) p(z) = ∫ z 0 ( f′(t) −pt−p−1 ) 1 α−p dt = z + · · · z ∈ u∗. further, let (7) h(z) = zp′(z) = z ( f′(z) −pz−p−1 ) 1 α−p = z + · · · z ∈ u∗. we see that p(z) and h(z) belongs to a. differentiating from (7), we have h′(z) = 1 α−p ( f′(z) −pz−p−1 ) 1 α−p ( 1 + zf′′(z) f′(z) + α ) . further we have |h′(z) − 1| = 1 p−α ∣∣∣∣∣ ( f′(z) −pz−p−1 ) 1 α−p ( 1 + zf′′(z) f′(z) + p ) − (α−p) ∣∣∣∣∣ < λ(β)|b(z)|. therefore, applying of the lemma 1 gives us that h(z) = zp′(z) ∈st (β) ⇒ p(z) ∈sk(β). since zp′′(z) p′(z) = 1 α−p ( zf′′(z) f′(z) + 1 + p ) , therefore ∣∣∣∣arg ( 1 + zp′′(z) p′(z) )∣∣∣∣ = ∣∣∣∣arg 1p−α ( −1 − zf′′(z) f′(z) −α )∣∣∣∣ < πβ2 , which imply that f(z) is p-valently meromorphic-strongly convex of order β and type α. this completes the proof. � putting α = 0 in theorem 6, we have: corollary 7. assume that f′(z) 6= 0 for z ∈ u∗. if f ∈ σ(p) satisfies∣∣∣∣∣ p √ −pz−p−1 f′(z) ( 1 + zf′′(z) f′(z) + p )∣∣∣∣∣ < pλ(β)|b(z)| z ∈ u∗, then f is p-valently meromorphic-strongly convex of order β. on p-valently meromorphic functions 65 setting b(z) = z and p = β = 1 in theorem 6, we obtain the following result: corollary 8. assume that f′(z) 6= 0 for z ∈ u∗. if f ∈ σ satisfies∣∣∣∣(−z2f′(z)) 1α−1 ( 2 + zf′′(z) f′(z) )∣∣∣∣ < 2√5 (1 −α) z ∈ u∗, then f is meromorphic convex function of order α. if we take α = 0 in corollary 8, we obtain the following result: corollary 9. assume that f′(z) 6= 0 for z ∈ u∗. if f ∈ σ satisfies∣∣∣∣ −1z2f′(z) ( 2 + zf′′(z) f′(z) )∣∣∣∣ < 2√5 ≈ 0.894427 . . . z ∈ u∗, then f is convex meromorphic function. references [1] r. kargar and r. aghalary, sufficient condition for strongly starlike and convex functions, serdica math. j. 40 (2014), 13-18. [2] p. t. mocanu, alpha-convex integral operators and strongly starlike functions, stud. univ. babes-bolyai math. 34 (1989), 18-24. [3] m. nunokawa, on the order of strongly starlikeness of strongly convex functions, proc. japan acad. ser. a math. sci. 69(7) (1993), 234-237. [4] s. ponnusamy and v. singh, criteria for strongly starlike functions, complex variables, theory appl. 34(3) (1997), 267-291. [5] f. rønning, s. ruscheweyeh and s. samaris, sharp starlikeness conditions for analytic functions with bounded derivative, j. aust. math. soc. ser. a, 69 (2000), 303-315. [6] m. san and h. irmak, ordinary differential operator and some of its applications to certain meromorphically p-valent functions, app. math. comput. 218 (2011), 817-821. 1young researchers and elite club, urmia branch, islamic azad university, urmia, iran 2department of mathematics, payame noor university, p.o. box 19395-3697 tehran, iran 3department of mathematics, institute of mathematics, university of rzeszów, ul. rejtana 16a, 35-310 rzeszów, poland ∗corresponding author: rkargar1983@gmail.com int. j. anal. appl. (2023), 21:6 best proximity point for g-generalized ζ −β −t contraction amit duhan1, manoj kumar1, savita rathee2, monika swami2,∗ 1baba masthnath university, rohtak, 124001, india 2maharshi dayanand university, rohtak, 124001, india ∗corresponding author: monikaswami06@gmail.com abstract. in this paper, we find the best proximity point in g-metric spaces for g-generalized ζ−β− t contraction mappings and verify the existence and uniqueness of the best proximity point in the complete g metric space using the idea of an approximatively compact set. in addition, an example is provided to illustrate the outcome. 1. introduction the "fixed point theory" developed as an important tool for finding the solution of equation of the type tx = x, where t is a self-mapping defined over a subset of a metric space. a difficulty emerges when the mapping shifts to a non-self mapping. the solution to this question is the "best proximity point theory." in this case, a point is calculated having the minimum distance between the point and its image. this point is called “best proximity point" and it reduces to“ fixed point" when the mapping reduces from non-self to self-mapping. in 2006, mustafa and sims [9] popularised a metric space in its generalized form, named as g-metric space. gmetric space is a generalization in which each triplet of elements is allocated a non-negative real number. physically, this is a measure of mutual distance between three elements taken together. researchers worked on g-metric space to calculate the fixed point for different type of contractions. as g-metric space becomes a vast area for fixed point theory, but on the other hand, in 2014, hussain et al. [6] were the first who work on g-metric space to calculate the “best proximity point" for the introduced proximal contraction. later on, abbas [5], chodhury [3], ansari [1] and researchers work in this direction for calculating “best proximity points" in g-metric spaces [2,7]. received: oct. 17, 2022. 2020 mathematics subject classification. 47h10, 54h25, 46j10, 46j15. key words and phrases. best proximity point; g-metric space; geraghty contraction. https://doi.org/10.28924/2291-8639-21-2023-6 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-6 2 int. j. anal. appl. (2023), 21:6 2. preliminaries definition 2.1. [9] let x be a nonempty set and let g : x ×x ×x → r+ be a function satisfying the following properties: (g1) g(x,y,z) = 0, if x = y = z, (g2) 0 < g(x,x,y) for all x,y ∈ x xnd x 6= y, (g3) g(x,x,y) ≤ g(x,y,z) for all x,y,z ∈ x with y 6= z, (g4) g(x,y,z) = g(x,z,y) = g(y,z,x) = · · ·(symmetry in all variables), (g5) g(x,y,z) ≤ g(x,a,a) + g(a,y,z) (rectangular inequality) for all x,y,z,a ∈ x. then the function g is called a generalized metric or g-metric on x and the pair (x, g) is called a g-metric space. every g-metric on x generates a metric dg on x defined by dg(x,y) = g(x,y,y) + g(y,x,x),∀x,y ∈ x. example 2.1. [9] let x = [0,∞). the function g : x ×x ×x → [0,∞) defined by g(x,y,z) = |x −y| + |y −z| + |z −x| for all x,y,x ∈ x is a g-metric on x. definition 2.2. [9] let (x, g) be g-metric space and let {xm} be a sequence of points of x, then {xm} is g-convergent to x ∈ x if lim m,l→∞ g(x,xm,xl) = 0 that is, for any � > 0 there exists n ∈ n such that g(x,xm,xl) < �, for all m,l ≥ n. we call x the limit of the sequence and write xm → x or limm→∞xm = x. proposition 2.1. [9] let (x, g) be a g-metric space. the following statements are equivalent: (i) {xm} is g-convergent to x, (ii) g(xm,xm,x) → 0 as m → +∞, (iii) g(xm,x,x) → 0 as m → +∞, (iv) g(xm,xl,x) → 0 as m,l → +∞. definition 2.3. [9] let (x, g) be a g-metric space. a sequence {xm} is called g-cauchy sequence, if for any � > 0, there exists n ∈n such that g(xm,xl,xk) < � for all m,l,k ≥ n, that is g(xm,xl,xk) → 0 as m,l,k → +∞. proposition 2.2. [9] let (x, g) be a g-metric space, then following statements are equivalent: (i) the sequence {xm} is g-cauchy, (ii) for any � > 0, there exists n ∈n such that g(xm,xl,xl) < �, for all m,l ≥ n. (iii) {xm} is a cauchy sequence in the metric space (x,dg). int. j. anal. appl. (2023), 21:6 3 definition 2.4. [9] a g-metric space (x, g) is called a g-complete if every g-cauchy sequence is g-convergent in (x, g). definition 2.5. let (x, g) be a g-metric space. a mapping f : x × x × x → x is said to be continuous if for any three g-convergent sequences {xm},{ym} and {zm} converging to x,y and z respectively, then {f (xm,ym,zm)} is g-convergent to f (x,y,z). lemma 2.1. [8] from (g5) and (g4), we have g(x,y,y) = g(y,y,x) ≤ g(y,x,x) + g(x,y,x) = 2g(y,x,x). definition 2.6. let (x, g) be a g-metric space and q and r be two nonempty subsets of a g-metric space (x, g). we define the following sets: q0 ={x ∈q : dg(x,y) = dg(q,r) for some y ∈r} r0 ={y ∈r : dg(x,y) = dg(q,r) for some x ∈q} where dg(q,r) = inf{dg(x,y) : x ∈q,y ∈r}. definition 2.7. [6] let (x, g) be a g-metric space and let q and r be two nonempty subsets of x. then r is said to be approximatively compact with respect to q if every sequence {ym} in r, satisfying the condition dg(x,ym) → dg(x,r) for some x in q, has a convergent subsequence. 3. main results firstly, we contemplate that ξ = {ζ : [0,∞) → [0,∞) such that ζ is nondecreasing and continuous where ζ(x) = 0 if and only if x = 0.} υ = {β : [0,∞) → [0, 1) such that β(xl) → 1 then xl → 0} definition 3.1. let q and r be two nonempty subsets of a “g-metric space (x, g)", then t : q→r is said to be g-generalized ζ −β −t contractive mapping if, for x,u,u∗,y,v ∈ q and l ≥ 1 such that dg(u,tx) = dg(q,r) dg(u ∗,tu) = dg(q,r) dg(v,ty) = dg(q,r) =⇒ ζ(g(u,u∗,v)) ≤ β(ζ(m(x,u,y) −dg(q,r))).ζ(m(x,u,y) −dg(q,r)) + lζ[n(x,u,y) −dg(q,r)] (3.1) 4 int. j. anal. appl. (2023), 21:6 where m(x,u,y) = max{g(x,tx,u), g(x,tx,y), g(u,tu,y), g(y,ty,u), g(x,u,y)} and n(x,u,y) = min{g(x,tx,u), g(u,tu,y), g(y,ty,x), g(tx,u,y)} theorem 3.1. let (q,r) be pair of nonempty closed subset of a “g-metric space (x, g)" such that (q, g) is “complete g-metric space" and r is approximatively compact with respect to q. consider t : q→r be a g-generalized ζ −β −t contractive mapping satisfies t (q0) ⊆r0. then t has a unique “best proximity point" in q that is, q ∈q such that dg(q,tq) = dg(q,r). proof. since the subset q0 is non-empty subset of q, we consider x0 ∈ q0 such that t (x0) ∈ t (q0) ⊆r0, then we can find x1 ∈q0 such that dg(x1,tx0) = dg(q,r) thereafter, since tx1 ∈ tq0 ⊆ r0, it supervene that there is an element x2 in q0 such that dg(x2,tx1) = dg(q,r). repeatedly, we get a sequence {xm} in q0 satisfying dg(xm+1,txm) = dg(q,r) for all m ∈n∪{0}. this gives us, by taking x = xm−1,u = xm,u∗ = xm+1,y = xm and v = xm+1, ζ(g(xm,xm+1,xm+1)) ≤β(ζ(m(xm−1,xm,xm) −dg(q,r))).ζ[m(xm−1,xm,xm) −dg(q,r)] + lζ[n(xm−1,xm,xm) −dg(q,r)] (3.2) where m(xm−1,xm,xm) = max{g(xm−1,txm−1,xm), g(xm−1,txm−1,xm), g(xm,txm,xm), g(xm,txm,xm), g(xm−1,xm,xm)} = max{g(xm−1,txm−1,xm), g(xm,txm,xm), g(xm−1,xm,xm)} and n(xm−1,xm,xm) = min{g(xm−1,txm−1,xm), g(xm,txm,xm), g(xm,txm,xm−1), g(txm−1,xm,xm)} solving m(xm−1,xm,xm) by using rectangular inequality and symmetry property of g, we calculate g(xm−1,txm−1,xm) ≤ g(xm−1,xm,xm) + g(xm,xm,txm−1) ≤ g(xm−1,xm,xm) + g(xm,xm,txm−1) + g(txm−1,txm−1,xm) = g(xm−1,xm,xm) + dg(q,r) g(xm,txm,xm) ≤ g(txm,xm+1,xm+1) + g(xm+1,xm,xm) ≤ g(txm,xm+1,xm+1) + g(xm+1,txm,txm) + g(xm+1,xm,xm) int. j. anal. appl. (2023), 21:6 5 = dg(q,r) + g(xm+1,xm,xm) implies m(xm−1,xm,xm) ≤ max{g(xm−1,xm,xm), g(xm+1,xm,xm)} + dg(q,r) (3.3) in a similar manner we solve for n(xm−1,xm,xm) g(txm−1,xm,xm) ≤ g(txm−1,xm,xm) + g(xm,txm−1,txm−1) = dg(q,r) imples n(xm−1,xm,xm) ≤ min{g(xm−1,txm−1,xm), g(xm,txm,xm), g(xm,txm,xm−1), dg(q,r)} = dg(q,r) (3.4) by using equation (3.3), (3.4) in (3.2), we obtain ζ(g(xm,xm+1,xm+1)) ≤ β(ζ(m(xm−1,xm,xm) −dg(q,r))).ζ[max{g(xm−1,xm,xm), g(xm+1,xm,xm)}] + lζ[0] = β(ζ(m(xm−1,xm,xm) −dg(q,r))).ζ[max{g(xm−1,xm,xm), g(xm+1,xm,xm)}] (3.5) if for some m, max{g(xm−1,xm,xm), g(xm+1,xm,xm)} = g(xm+1,xm,xm), (3.2) implies ζ(g(xm,xm+1,xm+1)) ≤ β(ζ(m(xm−1,xm,xm) −dg(q,r))).ζ[g(xm+1,xm,xm)] < ζ[g(xm+1,xm,xm)] which is contradiction. therefore, we must have m(xm−1,xm,xm) ≤ max{g(xm−1,xm,xm), g(xm+1,xm,xm)} + dg(q,r) ≤ g(xm−1,xm,xm) + dg(q,r) for all m ∈n. from the equation (3.5), we find that ζ(g(xm,xm+1,xm+1)) ≤ β(ζ(g(xm−1,xm,xm))).ζ(g(xm−1,xm,xm)) (3.6) < ζ(g(xm−1,xm,xm)) holds for all m ∈ n. since ζ is nondecreasing, then g(xm,xm+1,xm+1) < g(xm−1,xm,xm) for all m. consequently, the sequence {g(xm,xm+1,xm+1)} is decreasing and is bounded below and limm→∞ g(xm,xm+1,xm+1) exists. after rewriting (3.6), we get ζ(g(xm,xm+1,xm+1)) ζ(g(xm−1,xm,xm)) ≤ β(ζ(g(xm−1,xm,xm))) ≤ 1 6 int. j. anal. appl. (2023), 21:6 for each n ≥ 1. taking the limit m →∞, we find lim m→∞ β(ζ(g(xm−1,xm,xm))) = 1 now, as β ∈ υ, we get limm→∞ζ(g(xm−1,xm,xm)) = 0, that is lim m→∞ g(xm−1,xm,xm) = 0 (3.7) now, we prove that {xm} is g-cauchy sequence. on contrary, we assume that {g} is not gcauchy. thus, there exists an � > 0 for which we can find a sequence {xm(ι)},{xl(ι)} of {xm} with l(ι) > m(ι) ≥ ι such that g(xl(ι),xl(ι)+1,xm(ι)) ≥ � and g(xl(ι),xl(ι)+1,xm(ι)−1)t < � (3.8) from proposition (2.1), lemma (2.1) and (g5), we obtain � ≤ g(xl(ι),xl(ι)+1,xm(ι)) = g(xm(ι),xl(ι),xl(ι)+1) ≤ g(xm(ι),xm(ι)−1,xm(ι)−1) + g(xm(ι)−1,xl(ι),xl(ι)+1) < � + g(xm(ι),xm(ι)−1,xm(ι)−1) ≤ � + g(xm(ι)−1,xm(ι),xm(ι)) implies lim ι→∞ g(xl(ι),xl(ι)+1,xm(ι)) = �. (3.9) consider (3.1) with u = xl(ι),u ∗ = xl(ι)+1,x = xl(ι)−1,y = xm(ι)−1 and v = xm(ι), then ζ(g(xl(ι),xl(ι)+1,xm(ι))) ≤ β(ζ(m(xl(ι)−1,xl(ι),xm(ι)−1) −dg(q,r))). ζ[m(xl(ι)−1,xl(ι),xm(ι)−1 −dg(q,r)] + lζ[n(xl(ι)−1,xl(ι),xm(ι)−1) −dg(q,r)] (3.10) where m(xl(ι)−1,xl(ι),xm(ι)−1) = max{g(xl(ι)−1,txl(ι)−1,xl(ι)), g(xl(ι)−1,txl(ι)−1,xm(ι)−1), g(xl(ι),txl(ι),xm(ι)−1), g(xm(ι)−1,txm(ι)−1,xl(ι)), g(xl(ι)−1,xl(ι),xm(ι)−1)} and n(xl(ι)−1,xl(ι),xm(ι)−1) = min{g(xl(ι)−1,txl(ι)−1,xl(ι)), g(xl(ι),txl(ι),xm(ι)−1), g(xl(ι),txm(ι)−1,xm(ι)−1), g(xl(ι),txl(ι)−1,xm(ι)−1)} (3.11) int. j. anal. appl. (2023), 21:6 7 before solving particular terms of m(xl(ι)−1,xl(ι),xm(ι)−1) and n(xl(ι)−1,xl(ι),xm(ι)−1), we solve for g(xl(ι),xl(ι)+1,xm(ι)) by using proposition (2.1), lemma (2.1) and (g5), that is g(xl(ι),xl(ι)+1,xm(ι)) ≤ g(xl(ι),xl(ι)−1,xl(ι)−1) + g(xl(ι)−1,xl(ι)+1,xm(ι)) ≤ g(xl(ι),xl(ι)−1,xl(ι)−1) + g(xm(ι),xm(ι)−1,xm(ι)−1) + g(xm(ι)−1,xl(ι)−1,xl(ι)+1) ≤ 2g(xl(ι)−1,xl(ι),xl(ι)) + 2g(xm(ι)−1,xm(ι),xm(ι)) + g(xm(ι)−1,xl(ι)−1,xl(ι)+1) taking the limit ι →∞ and from equations (3.7) and (3.9), we get lim ι→∞ g(xm(ι)−1,xl(ι)−1,xl(ι)+1) = � (3.12) again, by proposition (2.1) and (g5), we have g(xm(ι)−1,xl(ι)−1,xl(ι)+1) ≤ g(xl(ι)+1,xl(ι),xl(ι)) + g(xl(ι),xl(ι)−1,xm(ι)−1) from equation (3.12) and limit ι →∞, we find lim ι→∞ g(xl(ι)−1,xl(ι),xm(ι)−1) = �. (3.13) from m(xl(ι)−1,xl(ι),xm(ι)−1), we solve g(xl(ι)−1,txl(ι)−1,xl(ι)) ≤ g(txl(ι)−1,xl(ι),xl(ι)) + g(xl(ι),xl(ι),xl(ι)−1) ≤ g(txl(ι)−1,xl(ι),xl(ι)) + g(xl(ι),txl(ι)−1,txl(ι)−1) + g(xl(ι),xl(ι),xl(ι)−1) = dg(q,r) + g(xl(ι)−1,xl(ι),xl(ι)) (3.14) g(xl(ι)−1,txl(ι)−1,xm(ι)−1) ≤ g(txl(ι)−1,xl(ι),xl(ι)) + g(xl(ι),xl(ι)−1,xm(ι)−1) ≤ g(txl(ι)−1,xl(ι),xl(ι)) + g(xl(ι),txl(ι)−1,xl(ι)−1) + g(xl(ι),xl(ι)−1,xm(ι)−1) = dg(q,r) + g(xl(ι),xl(ι)−1,xm(ι)−1) (3.15) g(xl(ι),txl(ι),xm(ι)−1) ≤ g(txl(ι),xl(ι)+1,xl(ι)+1) + g(xl(ι)+1,xl(ι),xm(ι)−1) ≤ g(txl(ι),xl(ι)+1,xl(ι)+1) + g(xl(ι)+1,xl(ι),xm(ι)−1) + g(xl(ι)+1,xl(ι),xm(ι)−1) = dg(q,r) + g(xl(ι)+1,xl(ι),xm(ι)−1) (3.16) g(xm(ι)−1,txm(ι)−1,xl(ι)) ≤ g(txm(ι)−1,xm(ι),xm(ι)) + g(xm(ι),xm(ι)−1,xl(ι)) ≤ g(txm(ι)−1,xm(ι),xm(ι)) + g(xm(ι),txm(ι)−1,txm(ι)−1) + g(xm(ι),xm(ι)−1,xl(ι)) = dg(q,r) + g(xm(ι),xm(ι)−1,xl(ι)) (3.17) 8 int. j. anal. appl. (2023), 21:6 similarly, we solve for n(xl(ι)−1,xl(ι),xm(ι)−1), use (3.14) to (3.17) in (3.11), we get m(xl(ι)−1,xl(ι),xm(ι)−1) ≤ max{g(xl(ι)−1,xl(ι),xl(ι)), g(xl(ι),xl(ι)−1,xm(ι)−1) g(xl(ι)+1,xl(ι),xm(ι)−1), g(xm(ι),xm(ι)−1,xl(ι)), g(xl(ι)−1,xl(ι),xm(ι)−1)} + dg(q,r) = max{g(xl(ι)−1,xl(ι),xl(ι)), g(xl(ι),xl(ι)−1,xm(ι)−1), g(xl(ι)+1,xl(ι),xm(ι)−1)} + dg(q,r) taking limit ι →∞ on both side, we get lim ι→∞ m(xl(ι)−1,xl(ι),xm(ι)−1) = lim ι→∞ max{g(xl(ι)−1,xl(ι),xl(ι)), g(xl(ι),xl(ι)−1,xm(ι)−1), g(xl(ι)+1,xl(ι),xm(ι)−1)} + dg(q,r)} = max{ lim ι→∞ g(xl(ι)−1,xl(ι),xl(ι)), lim ι→∞ g(xl(ι),xl(ι)−1,xm(ι)−1), lim ι→∞ g(xl(ι)+1,xl(ι),xm(ι)−1)} + dg(q,r)} = max{0,�,�,�,�} + dg(q,r) = � + dg(q,r) thus, lim ι→∞ m(xl(ι)−1,xl(ι),xm(ι)−1) −dg(q,r) = �. (3.18) similarly, n(xl(ι)−1,xl(ι),xm(ι)−1) = min{dg(q,r) + g(xl(ι)−1,xl(ι),xl(ι)), dg(q,r) + g(xl(ι)+1,xl(ι),xm(ι)−1), dg(q,r) + g(xm(ι),xl(ι),txm(ι)−1), dg(q,r) + g(txl(ι),xl(ι),xm(ι)−1)} taking limit ι →∞, we get lim ι→∞ n(xl(ι)−1,xl(ι),xm(ι)−1) = min{dg(q,r),dg(q,r) + �, dg(q,r) + lim ι→∞ g(xm(ι),xl(ι),txm(ι)−1), dg(q,r) + lim ι→∞ g(txl(ι),xl(ι),xm(ι)−1)} = dg(q,r) thus, lim ι→∞ n(xl(ι)−1,xl(ι),xm(ι)−1) −dg(q,r) = 0 (3.19) now, taking the limit ι →∞ in (3.10) and using (3.18) and (3.19), we obtain ζ(�) ≤ β(ζ(�)).ζ(�) + lζ(0) int. j. anal. appl. (2023), 21:6 9 ζ(�) = 1 which implies � = 0, which is contradiction. hence lim l,m→∞ g(xl(ι),xl(ι)+1,xm(ι)) = 0. thus, {xm} is g-cauchy sequence. since (q, g) is “complete g-metric space", so there exist q ∈q such that xm → q as m →∞. from other side , for all m ∈n, we can write dg(q,r) ≤ dg(q,txm) ≤ dg(q,xm+1) + dg(xm+1,txm) + dg(q,xm+1) + dg(q,r) (3.20) taking the limit m →∞ in (3.20), we have lim m→∞ dg(q,txm) = dg(q,q) = dg(q,r). since r is approximatively compact with respect to q, so the sequence {txm} has a subsequence {txm(ι)} that converges to some r ∗ ∈r. hence dg(q,r ∗) = lim m→∞ dg(xm(ι)+1,txm(ι)) = dg(q,r) (3.21) and so q ∈ q0. now, since tq ∈ t (q0) ⊆ r0, there exists q∗ ∈ q0 such that dg(q ∗,tq) = dg(q,r). now, from (3.1) with a = xm,u = xm+1,u∗ = xm+2,c = q,v = q∗, we have ζ(g(xm+1,xm+2,q ∗)) ≤ β(ζ(m(xm,xm+1,q) −dg(q,r))).ζ[m(xm,xm+1,q) −dg(q,r)] + lζ[n(xm,xm+1,q) −dg(q,r)] (3.22) where m(xm,xm+1,q) = max{g(xm,txm,xm+1), g(xm,txm,q), g(xm+1,txm+1,q), g(q,tq,xm+1), g(xm,xm+1,q)} ≤max{g(xm,xm+1,xm+1), g(xm+1,xm,p), g(xm+2,xm+1,p), g(q∗,q,xm)} + dg(q,r) (3.23) n(xm,xm+1,q) = min{g(xm,txm,xm+1), g(xm+1,txm+1,q), g(q,tq,xm), g(txm,xm+1,q)} ≤ min{g(xm,xm+1,xm+1), g(xm+2,xm+1,q), g(q∗,q,xm), g(xm+1,xm+1.p)} + dg(q,r) (3.24) 10 int. j. anal. appl. (2023), 21:6 taking the limit m →∞ in (3.23) and (3.24), we obtain lim m→∞ m(xm,xm+1,q) = g(q ∗,q,q) + dg(q,r) lim m→∞ m(xm,xm+1,q) −dg(q,r) = g(q∗,q,q) (3.25) and lim m→∞ n(xm,xm+1,q) = dg(q,r) lim m→∞ n(xm,xm+1,q) −dg(q,r) = 0 (3.26) now taking the limit m →∞ in (3.22) and using (3.25) and (3.26), we get ζ(g(q,q,q∗)) ≤ lim m→∞ β(ζ(m(xm,xm+1,q) −dg(q,r))).ζ(g(q,q,q∗)) =⇒ lim m→∞ β(ζ(m(xm,xm+1,q) −dg(q,r))) ≤ 1 =⇒ lim m→∞ ζ(m(xm,xm+1,q) −dg(q,r)) = 0 which implies g(q,q,q∗) = 0, that is, q = q∗. thus, dg(q,tq) = dg(q,r). therefore, t has a “best proximity point". now we prove the uniqueness of “best proximity point". suppose that q 6= r such that dg(q,tq) = dg(q,r) and dg(r,tr) = dg(q,r). from (3.1) with x = u = u∗ = q and y = v = r, we get ζ(g(q,q,r)) ≤ β(ζ(m(q,q,r) −dg(q,r))).ζ[m(q,q,r) −dg(q,r)] + lζ[n(q,q,r) −dg(q,r)] (3.27) where m(q,q,r) = max{g(q,tq,q), g(q,tq,r), g(q,tq,r), g(r,tr,q), g(q,q,r)} ≤ max{g(q,tq,q), g(q,tq,r), g(r,tr,q), g(q,q,r)} ≤ max{dg(q,r),dg(q,r) + g(q,q,r),dg(q,r) + g(r,r,q), g(q,q,r)} = max{g(q,q,r), g(r,r,q)} + dg(q,r) and n(q,q,r) = min{g(q,tq,q), g(q,tq,r), g(r,tr,q), g(tq,q,r)} ≤ min{dg(q,r),dg(q,r) + g(q,q,r),dg(q,r) + g(r,r,q)} = dg(q,r) if max{g(q,q,r), g(r,r,q)} = g(r,r,q) then from (3.27), we get ζ(g(q,q,r)) ≤ β(ζ(m(q,q,r) −dg(q,r))).ζ(g(q,q,r)) < ζ(g(q,q,r)) int. j. anal. appl. (2023), 21:6 11 which is contradiction. thus max{g(q,q,r), g(r,r,q)} = g(r,r,q), again (3.27) implies ζ(g(q,q,r)) ≤ β(ζ(m(q,q,r) −dg(q,r))).ζ(g(r,r,q) < ζ(g(r,r,q) as ζ is non decreasing, then r = q. thus, the result. � example 3.1. let x = [0,∞) and g(x,y,z) = 1 4 {|x −y| + |y −z| + |z −x|} be g-metric on x defined by dg(x,y) = |x − y|. let “q = {3, 4, 5, 6, 7}” and “r = {9, 10, 11, 12, 13}”. define t : q→r by t (x) =  9, if x = 7 x + 6, otherwise also, consider ζ : [0,∞) → [0,∞) and β : [0,∞) → [0, 1) defined by ζ(x) = x 2 ,β(x) = x (1+x) respectively. clearly, here dg(q,r) = 2,q0 = {7},r0 = {9} and tq0 ⊆ r0. let dg(u,tx) = dg(q,r) and dg(v,ty) = dg(q,r), then (u,x), (v,y) ∈ {(7, 7), (7, 3)}. also if dg(u∗,tu) = dg(q,r) = 2, then u∗ = 7. therefore, if dg(u,tx) = dg(q,r) dg(u ∗,tu) = dg(q,r) dg(v,ty) = dg(q,r) then (u,u∗,v,x,y) ∈{(7, 7, 7, 7, 7), (7, 7, 7, 3, 3), (7, 7, 7, 3, 7), (7, 7, 7, 7, 3)} from which we get m(x,u,y) = n(x,u,y) = 9 now, as u = u∗ = v = 7, so ζ(g(u,u∗,v)) = 0. hence, ζ(g(u,u∗,v)) = 0 ≤ 1 2 x ≤ 1 2 (tx − 2) ≤ 1 2 (tx − 2){−1 + l 2 (ty − 2)} < 1 2 (tx − 2){ 1 2 (tx − 2) − 1 2 tx + l 2 (ty − 2)} + l 2 (ty − 2) < 1 2 (tx − 2){ 1 2 (tx − 2) + l. 1 2 (tx − 2) − 1 2 x} + l. 1 2 (ty − 2) = ζ(m(x,u,y) −dg(q,r))[ζ(m(x,u,y) −dg(q,r)) + l.ζ(n(x,u,y) −dg(q,r)) −ζ(g(u,u∗,v))] + l.ζ(n(x,u,y) −dg(q,r)) ζ(g(u,u∗,v)) ≤ ζ(m(x,u,y) −dg(q,r)).ζ(m(x,u,y) −dg(q,r)) 12 int. j. anal. appl. (2023), 21:6 + l.ζ(n(x,u,y) −dg(q,r))[1 + ζ(m(x,u,y) −dg(q,r))] −ζ(m(x,u,y) −dg(q,r)).ζ(g(u,u∗,v)) ζ(g(u,u∗,v))[1 + ζ(m(x,u,y) −dg(q,r))] ≤ ζ(m(x,u,y) −dg(q,r)).ζ(m(x,u,y) −dg(q,r)) + l.ζ(n(x,u,y) −dg(q,r)) ζ(g(u,u∗,v)) ≤ ζ(m(x,u,y) −dg(q,r)) 1 + ζ(m(x,u,y) −dg(q,r)) .ζ(m(x,u,y) −dg(q,r)) + l.ζ(n(x,u,y) −dg(q,r)) ≤ β(ζ(m(x,u,y) −dg(q,r))).ζ(m(x,u,y) −dg(q,r)) + l.ζ(n(x,u,y) −dg(q,r)) thus, t is g-generalized ζ−β−t contraction mapping and all the conditions of thereom are satisfied with q = 7 as unique “best proximity point". if in theorem (3.1), ζ(x) = x, then we obtain the following corollary. corollary 3.1. let (q,r) be pair of nonempty closed subset of g-metric space (x, g) such that (q, g) is “complete g-metric space" and r is approximatively compact w.r.t. q. consider t : q → r be non self mapping satisfying t (q0) ⊆r0 and for x,y,u,u∗,v ∈q and l ≥ 1, defined by dg(u,tx) = dg(q,r) dg(u ∗,tu) = dg(q,r) dg(v,ty) = dg(q,r) =⇒ g(u,u∗,v) ≤ β(m(x,u,y) −dg(q,r)).(m(x,u,y) −dg(q,r)) + lζ[n(x,u,y) −dg(q,r)] where m(x,u,y) = max{g(x,tx,u), g(x,tx,y), g(u,tu,y), g(y,ty,u), g(x,u,y)} and n(x,u,y) = min{g(x,tx,u), g(u,tu,y), g(y,ty,x), g(tx,u,y)} then t has a unique “best proximity point" in q. if we proceed with the above corollary by considering β(x) = s where 0 ≤ s < 1, then we get another corollary as defined below. corollary 3.2. let (q,r) be pair of nonempty closed subset of g-metric space (x, g) such that (q, g) is “complete g-metric space" and r is approximately compact w.r.t. q. consider t : q→r be non self mapping satisfying t (q0) ⊆r0 and for x,y,u,u∗,v ∈q and l ≥ 1, defined by dg(u,tx) = dg(q,r) int. j. anal. appl. (2023), 21:6 13 dg(u ∗,tu) = dg(q,r) dg(v,ty) = dg(q,r) =⇒ ζ(g(u,u∗,v)) ≤ s.(m(x,u,y) −dg(q,r)) + lζ[n(x,u,y) −dg(q,r)] where m(a,u,y) = max{g(x,tx,u), g(x,tx,y), g(u,tu,y), g(y,ty,u), g(x,u,y)} xnd n(x,u,y) = min{g(x,tx,u), g(u,tu,y), g(y,ty,x), g(tx,u,y)} then t has a unique “best proximity point" in q. remark 3.1. the “best proximity point" theorem (3.1) is reduced to the result of [4], if “complete g-metric spaces" becomes complete metric spaces. 4. application to fixed point theory in this section, we discuss the fixed point theorem as an application part of “best proximity point" theorem. by considering q = r = x, in dg(u,tx) = dg(q,r) dg(u ∗,tu) = dg(q,r) dg(v,ty) = dg(q,r) we get, u = tx,u∗ = tu = t2x and v = ty. therefore, theorem (3.1) restates as: theorem 4.1. let (x, g) be “complete g-metric space". consider q as a nonempty subset of x. let t : q→q be mapping satisfying the successive condition ζ(g(tx,t2x,tc)) ≤ β(ζ(m(x,tx,y))).ζ(m(x,tx,y)) + lζ[n(x,tx,y)] where ζ ∈ ξ,β ∈ υ,l ≥ 1, m(x,tx,y) = max{g(x,tx,tx), g(x,tx,y), g(tx,t2x,y), g(y,ty,tx), g(x,tx,y)} and n(x,tx,y) = min{g(x,tx,tx), g(tx,t2x,y), g(y,ty,x), g(tx,tx,y)} then t has a fixed point. after taking ζ(x) = x in theorem (4), we obtain a corollary, stated as: corollary 4.1. let (x, g) be “complete g-metric space". consider q as a nonempty subset of x. let t : q→q be mapping satisfying the successive condition g(tx,t2x,ty) ≤ β(m(x,tx,y)).(m(x,tx,y)) + lζ[n(x,tx,y)] where β ∈ υ,l ≥ 1, m(x,tx,y) = max{g(x,tx,tx), g(x,tx,y), g(tx,t2x,y), g(y,ty,tx), g(x,tx,y)} and n(x,tx,y) = min{g(x,tx,tx), g(tx,t2x,y), g(y,ty,x), 14 int. j. anal. appl. (2023), 21:6 g(tx,tx,y)} then t has a fixed point. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] a.h. ansari, a. razani, n. hussain, new best proximity point results in g-metric space, j. linear topol. algebra, 6 (2017), 73-89. https://jlta.ctb.iau.ir/article_530221.html [2] a.h. ansari, s. changdok, n. hussain, et al. some common fixed point theorems for weakly α-admissible pairs in g-metric spaces with auxiliary functions, j. math. anal. 8 (2017), 80–107. http://www.ilirias.com/jma/ repository/docs/jma8-3-7.pdf [3] b.s. choudhury, p. maity, best proximity point results in generalized metric spaces, vietnam j. math. 44 (2015), 339–349. https://doi.org/10.1007/s10013-015-0141-3. [4] h. aydi, e. karapınar, i̇.m. erhan, p. salimi, best proximity points of generalized almost ζ-geraghty contractive non-self-mappings, fixed point theory appl. 2014 (2014), 32. https://doi.org/10.1186/1687-1812-2014-32. [5] m. abbas, a. hussain, p. kumam, a coincidence best proximity point problem in g-metric spaces, abstr. appl. anal. 2015 (2015), 243753. https://doi.org/10.1155/2015/243753. [6] n. hussain, a. latif, p. salimi, best proximity point results in g-metric spaces, abstr. appl. anal. 2014 (2014), 837943. https://doi.org/10.1155/2014/837943. [7] n. priyobarta, b. khomdram, y. rohen, n. saleem, on generalized rational α-geraghty contraction mappings in g-metric spaces, j. math. 2021 (2021), 6661045. https://doi.org/10.1155/2021/6661045. [8] z. mustafa, a new structure for generalized metric spaces with applications to fixed point theory, ph.d. thesis, the university of newcastle, new southwales, australia, 2005. [9] z. mustafa, b. sims, a new approach to generalized metric spaces, j. nonlinear convex anal. 7 (2006), 289–297. http://yokohamapublishers.jp/online2/opjnca/vol7/p289.html. https://jlta.ctb.iau.ir/article_530221.html http://www.ilirias.com/jma/repository/docs/jma8-3-7.pdf http://www.ilirias.com/jma/repository/docs/jma8-3-7.pdf https://doi.org/10.1007/s10013-015-0141-3 https://doi.org/10.1186/1687-1812-2014-32 https://doi.org/10.1155/2015/243753 https://doi.org/10.1155/2014/837943 https://doi.org/10.1155/2021/6661045 http://yokohamapublishers.jp/online2/opjnca/vol7/p289.html 1. introduction 2. preliminaries 3. main results 4. application to fixed point theory references int. j. anal. appl. (2023), 21:85 on prime e-ideals of almost distributive lattices n. rafi1, y. monikarchana2, ravikumar bandaru3, aiyared iampan4,∗ 1department of mathematics, bapatla engineering college, bapatla, andhra pradesh 522 101, india 2department of mathematics, mohan babu university, a. rangampet, tirupati, andhra pradesh 517 102, india 3department of mathematics, gitam(deemed to be university), hyderabad campus, telangana 502 329, india 4fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. in an almost distributive lattice (adl), the idea of e-ideals is introduced, and their properties are discussed. in terms of a congruence, an equivalence is established between the minimal prime eideals of an adl and its quotient adl. finally, topological investigations are performed on prime e-ideals and minimal prime e-ideals. 1. introduction in the article by swamy and rao [9], the concept of an almost distributive lattice (adl) was introduced as a generalization of boolean algebras and distributive lattices. this allowed for the abstraction of various ring-theoretic generalizations. they also introduced the notion of an ideal in an adl, noting that the set of principal ideals in an adl forms a distributive lattice. this extension of lattice theory notions to adls was significant. the concept of normal lattices was initially introduced by cornish [2]. later, rao and ravi kumar presented the concept of a minimal prime ideal belonging to an ideal in an adl [6]. in another paper by rao and ravi kumar [7], the notion of a normal adl was defined, providing equivalent received: jun. 7, 2023. 2020 mathematics subject classification. 06d99, 06d15. key words and phrases. almost distributive lattice(adl); prime filter; e-ideal; e-normal adl; congruence; compact; hausdorff space; closure. https://doi.org/10.28924/2291-8639-21-2023-85 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-85 2 int. j. anal. appl. (2023), 21:85 conditions for an adl to be considered normal in terms of its annulets. these papers contributed to the understanding of adls and their properties. the study of d-filters in lattices and their properties was carried out by kumar et al. [4]. they investigated the properties of d-filters in lattices, providing valuable insights. in the same line of research, we investigated the notions of prime e-ideals and e-ideals in an adl. the properties of these ideals are thoroughly examined, and it is established that every proper e-ideal must satisfy a set of equivalent conditions to become a prime e-ideal. it is also proven that every maximal e-ideal in an adl is a prime e-ideal. furthermore, the paper introduces the concept of oe(m) as the intersection of all minimal prime e-ideals contained in a prime e-ideal m in an adl r. an adl is defined as e-normal, characterized in terms of relative dual annihilators with respect to an ideal e. an equivalence between the minimal prime e-ideals of an adl and its quotient adl is derived with respect to a congruence. the topological properties of the space of all prime e-ideals and the space of all minimal prime e-ideals in an adl are also investigated. 2. preliminaries in this section, we recall certain definitions and important results from [5] and [9], those will be required in the text of the paper. definition 2.1. [9] an algebra r = (r,∨,∧, 0) of type (2, 2, 0) is called an almost distributive lattice (abbreviated as adl), if it satisfies the following conditions: (1) (a∨b) ∧c = (a∧c) ∨ (b∧c) (2) a∧ (b∨c) = (a∧b) ∨ (a∧c) (3) (a∨b) ∧b = b (4) (a∨b) ∧a = a (5) a∨ (a∧b) = a (6) 0 ∧a = 0 (7) a∨ 0 = a, for all a,b,c ∈ r. example 2.1. every non-empty set x can be regarded as an adl as follows. let x0 ∈ x. define the binary operations ∨,∧ on x by x ∨y =  x if x 6= x0 y if x = x0 x ∧y =  y if x 6= x0 x0 if x = x0. then (x,∨,∧,x0) is an adl (where x0 is the zero) and is called a discrete adl. if (r,∨,∧, 0) is an adl, for any a,b ∈ r, define a ≤ b if and only if a = a ∧b (or equivalently, a∨b = b), then ≤ is a partial ordering on r. int. j. anal. appl. (2023), 21:85 3 theorem 2.1. [9] if (r,∨,∧, 0) is an adl, for any a,b,c ∈ r, we have the following: (1) a∨b = a ⇔ a∧b = b (2) a∨b = b ⇔ a∧b = a (3) ∧ is associative in r (4) a∧b∧c = b∧a∧c (5) (a∨b) ∧c = (b∨a) ∧c (6) a∨ (b∧c) = (a∨b) ∧ (a∨c) (7) a∧ (a∨b) = a, (a∧b) ∨b = b and a∨ (b∧a) = a (8) a∧a = a and a∨a = a. it can be observed that an adl r satisfies almost all the properties of a distributive lattice except the right distributivity of ∨ over ∧, commutativity of ∨, commutativity of ∧. any one of these properties make an adl r a distributive lattice. as usual, an element m ∈ r is called maximal if it is a maximal element in the partially ordered set (r,≤). that is, for any a ∈ r, m ≤ a ⇒ m = a. the set of all maximal elements of an adl r is denoted by m. as in distributive lattices [1,3], a non-empty subset i of an adl r is called an ideal of r if a∨b ∈ i and a∧x ∈ i for any a,b ∈ i and x ∈ r. also, a non-empty subset f of r is said to be a filter of r if a∧b ∈ f and x ∨a ∈ f for a,b ∈ f and x ∈ r. the set i(r) of all ideals of r is a bounded distributive lattice with least element {0} and greatest element r under set inclusion in which, for any i,j ∈ i(r), i ∩j is the infimum of i and j while the supremum is given by i ∨j := {a ∨b | a ∈ i,b ∈ j}. a proper ideal(filter) p of r is called a prime ideal(filter) if, for any x,y ∈ r, x∧y ∈ p (x∨y ∈ p ) ⇒ x ∈ p or y ∈ p. a proper ideal(filter) m of r is said to be maximal if it is not properly contained in any proper ideal(filter) of r. it can be observed that every maximal ideal(filter) of r is a prime ideal(filter). every proper ideal(filter) of r is contained in a maximal ideal(filter). for any subset s of r the smallest ideal containing s is given by (s] := {( n∨ i=1 si ) ∧ x | si ∈ s,x ∈ r and n ∈ n}. if s = {s}, we write (s] instead of (s] and such an ideal is called the principal ideal of r. similarly, for any s ⊆ r, [s) := {x∨( n∧ i=1 si ) | si ∈ s,x ∈ r and n ∈n}. if s = {s}, we write [s) instead of [s) and such a filter is called the principal filter of r. for any a,b ∈ r, it can be verified that (a] ∨ (b] = (a∨b] and (a] ∧ (b] = (a∧b]. hence the set (ipi(r),∨,∩) of all principal ideals of r is a sublattice of the distributive lattice (i(r),∨,∩) of all ideals of r. also, we have that the set (f(r),∨,∩) of all filters of r is a bounded distributive lattice. theorem 2.2. [6] let r be an adl with maximal elements. then p is a prime ideal of r if and only if r\p is a prime filter of r. definition 2.2. [5] an adl r is said to be an associate adl, if the operation ∨ is associative on r. 4 int. j. anal. appl. (2023), 21:85 definition 2.3. [8] for any nonempty subset a of an adl r, define a+ = {x ∈ r | a∨x is maximal, for all a ∈ a}. here a+ is called the dual annihilator of a in r. for any a ∈ r, we have {a}+ = (a]+, where (a] is the principal filter generated by a. an element a of an adl r is called dual dense element if (a]+ = m and the set e of all dual dense elements in an adl r is an ideal if e is non-empty. 3. e-ideals of adls in this section, we present the concepts of prime e-ideals and e-ideals in an abstract distributive lattice (adl) and explore their properties. we observe that any proper e-ideal in an adl can be transformed into a prime e-ideal based on a set of equivalent conditions. additionally, we establish that the intersection of all minimal prime e-ideals contained in a prime e-ideal m is denoted as oe(m). furthermore, we introduce the notion of e-normal adls, which are characterized in relation to the relative dual annihilators with respect to an ideal e. we establish an equivalence between the minimal prime e-ideals of an adl and its quotient adl with respect to a congruence. definition 3.1. an ideal g of r is said to be an e-ideal of r if e ⊆ g. now we have the example of an e-ideal of an adl. example 3.1. let r = {0,a,b,c,d,e,f ,g} and define ∨, ∧ on r as follows: ∧ 0 a b c d e f g 0 0 0 0 0 0 0 0 0 a 0 a b c d e f g b 0 a b c d e f g c 0 c c c 0 0 c 0 d 0 d e 0 d e g g e 0 d e 0 d e g g f 0 f f c g g f g g 0 g g 0 g g g g ∨ 0 a b c d e f g 0 0 a b c d e f g a a a a a a a a a b b b b b b b b b c c a b c a b f f d d a a a d d a d e e b b b e e b e f f a b f a b f f g g a b f d e f g then (r,∨, ∧) is an adl. clearly, we have that e = {0,g} and g = {0,c, f ,g} are ideals of r satisfying e ⊆ g. therefore g is an e-ideal of r. consider an ideal h = {0,c} of r, but not an e-ideal. it is easy to verify the proof of the following result. lemma 3.1. for any non-empty subset a of an adl r, (a]∨e is the smallest e-ideal of r containing a. int. j. anal. appl. (2023), 21:85 5 we denote (a] ∨ e by ae, i.e., ae = (a] ∨ e. for, a = {a}, we denote simply (a)e for {a}e. clearly, we have that (a)e is the smallest e-ideal containing a, which is known as the principal e-ideal generated by a. lemma 3.2. for any two elements x,y of an adl r with maximal element m, we have the following: (1) (0)e = e (2) (m)e = r (3) x ≤ y implies (x)e ⊆ (y)e (4) (x ∨y)e = (x)e ∨ (y)e (5) (x ∧y)e = (x)e ∩ (y)e (6) (x)e = e if and only if x ∈ e. proof. (1) now (0)e = (0] ∨e = e. (2) now (m)e = (m] ∨e = r∨e = r. (3) let x ≤ y. then (x] ⊆ (y]. now (x)e = (x] ∨e ⊆ (y] ∨e = (y)e. therefore (x)e ⊆ (y)e. (4) clearly, we have that (x ∨ y] = (x] ∨ (y]. now, (x ∨ y)e = (x ∨ y] ∨ e = (x] ∨ (y] ∨ e = ((x] ∨e) ∨ ((y] ∨e)) = (x)e ∨ (y)e. therefore (x ∨y)e = (x)e ∨ (y)e. (5) since x ∧y ≤ y and y ∧x ≤ x and hence (x ∧y] ⊆ (x] and (y ∧x] ⊆ (y]. since (x ∧y] = (y ∧x], we get that (x ∧ y] ⊆ (x] ∩ (y]. let t ∈ (x] ∩ (y]. then t ∈ (x] and t ∈ (y]. that implies x ∧ t = t and y ∧ t = t. therefore x ∧ y ∧ t = t and hence t ∈ (x ∧ y]. thus (x] ∩ (y] ⊆ (x ∧ y], which gives (x∧y] = (x]∩(y]. now (x∧y)e = (x∧y]∨e = [(x]∩(y]]∨e = ((x]∨e)∩((y]∨e) = (x)e∩(y)e. hence (x ∧y)e = (x)e ∩ (y)e. (6) assume that (x)e = e. then (x] ∨e = e. that implies (x] ⊆ e and hence x ∈ e. conversely, assume that x ∈ e. then (x] ⊆ e. this implies that (x] ∨e ⊆ e. since e ⊆ (x] ∨e, we get that e = (x] ∨e. therefore (x)e = e. � we denote i(r),ie(r)and ipef (r) as the set of all ideals, e-ideals and principal e-ideals of an adl r respectively. theorem 3.1. ie(r) forms a distributive lattice contained in i(r), and ipef (r) forms a sublattice of ie(r). definition 3.2. an e-ideal q is said to be proper if q ( r. a proper e-ideal q is said to be maximal if it is not properly contained in any proper e-ideal of r. a proper e-ideal q of an adl r is said to be a prime e-ideal if q is a prime filter of r. 6 int. j. anal. appl. (2023), 21:85 example 3.2. consider a distributive lattice l = {0,a,b,c, 1} and discrete adl a = {0′,a′}. � � � @ @ @ @ @ @ � � � d d d d d a b c 1 0 clearly, r = a × l = {(0′, 0), (0′,a), (0′,b), (0′,c), (0′, 1), (a′, 0), (a′,a), (a′,b), (a′,c), (a′, 1)} is an adl with zero element (0, 0′). clearly, the dense set e = {(0′, 0), (0′,a)}. consider the e-ideals: i1 = {(0′, 0), (0′,a), (0′,b)} i2 = {(0′, 0), (0′,a), (0′,c)} i3 = {(0′, 0), (0′,a), (a′, 0), (a′,a)} i4 = {(0′, 0), (0′,a), (0′,c), (a′, 0), (a′,a)(a′,c)} i5 = {(0′, 0), (0′,a), (0′,b), (a′, 0), (a′,a), (a′,b)} i6 = {(0′, 0), (0′,a), (0′,b), (0′,c), (0′, 1)} clearly, i4, i5 and i6 are prime e-ideal. but i1 is not a prime e-ideal, because (a′,b)∧(0′,c) = (0′,a) ∈ i1, but (a′,b) /∈ i1. and (0′,c) /∈ i1. and also, i2 is not a prime e-ideal, because (0′,b) ∧ (a′,c) = (0′,a) ∈ i2, but (0′,b) /∈ i2 and (a′,c) /∈ i2. theorem 3.2. for any e-ideal q of r, the following conditions are equivalent: (1) q is a prime e-ideal (2) for any two e-ideals g,h of r,g ∩h ⊆ q ⇒ g ⊆ q or h ⊆ q (3) for any x,y ∈ r, (x)e ∩ (y)e ⊆ q ⇒ x ∈ q or y ∈ q. proof. (1) ⇒ (2) assume (1). let g and h be two e-ideals of r such that g∩h ⊆ q. we prove that g ⊆ q or h ⊆ q. suppose g * q and h * q. choose x,y ∈ r such that x ∈ g \q and y ∈ h\q. by our assumption we have that x ∧ y /∈ q. since x ∈ g,y ∈ h, which gives x ∧ y ∈ g ∩h ⊆ q. therefore x ∧y ∈ q, we get a contradiction. thus g ⊆ q or h ⊆ q. (2) ⇒ (3) assume (2). let x,y ∈ r with (x)e ∩ (y)e ⊆ q. since (x)e and (y)e are e-ideals of r, and by our assumption, we get that (x)e ⊆ q or (y)e ⊆ q. hence x ∈ q or y ∈ q. (3) ⇒ (1) assume (3). let x,y ∈ r with x∧y ∈ q. since q is an e-ideal, we have that (x)e∩(y)e = (x ∧y)e ⊆ q. by our assumption, we get that x ∈ q or y ∈ q. hence q is prime. � theorem 3.3. every maximal e-ideal of an adl r is a prime e-ideal. int. j. anal. appl. (2023), 21:85 7 proof. let n be a maximal e-ideal of r. let a,b ∈ r with a /∈ n and b /∈ n. then n ∨ (a)e = r and n ∨ (b)e = r. that implies r = n ∨ ((a)e ∩ (b)e) = n ∨ (a∧b)e. if a∧b ∈ n then n = r, we get a contradiction. therefore a∧b /∈ n and hence n is prime. � corollary 3.1. let n1,n2,n3, . . . ,nn and n be maximal e-ideals of an adl r with n⋂ i=1 ni ⊆ n, then nj ⊆ n, for some j ∈{1, 2, 3, . . . ,n}. theorem 3.4. a proper e-ideal q of an adl r is a prime e-ideal if and only if r\q is a prime filter such that (r\q) ∩e = ∅. proof. assume that q is a prime e-ideal of r. clearly, r \q is a prime filter of r. we prove that (r\q) ∩e = ∅. if (r\q) ∩e 6= ∅, choose x ∈ (r\q) ∩e. that implies x ∈ e ⊆ q, which gives a contradiction. hence (r\q) ∩e = ∅. conversely, assume that r\q is a prime filter of r such that (r \q) ∩e = ∅. clearly, q is a prime ideal of r and e ⊆ r \ (r \q) = q. therefore q is a prime e-ideal of r. � theorem 3.5. let g be a e-ideal of an adl r, and k be any non-empty subset of r, which is closed under the operation ∧ such that g ∩k = ∅. then there exists a prime e-ideal q of r containing g such that q∩k = ∅. proof. let k be a non-empty subset of r, which is closed under the operation ∧ such that g∩k = ∅. consider f = {h | h is an e−ideal of r,g ⊆ h and h∩k = ∅}. clearly, it satisfies the hypothesis of the zorn’s lemma and hence f has a maximal element say q. that is, q is an e-ideal of r such that g ⊆ q and q∩k = ∅. let x,y ∈ r be such that x∧y ∈ q. we prove that x ∈ q or y ∈ q. suppose that x /∈ q and y /∈ q. then clearly q∨(x)e and q∨(y)e are e-ideals of r such that q ( q∨(x)e and q ( q∨(y)e. since q is maximal in f, we get that (q∨(x)e)∩k 6= ∅ and (q∨(y)e)∩k 6= ∅. choose s ∈ (q ∨ (x)e) ∩ k and t ∈ (q ∨ (y)e) ∩ k. then s ∈ (q ∨ (x)e),t ∈ (q ∨ (y)e) and s,t ∈ k. since k is closed under ∧, we get s ∧ t ∈ k. now s ∧ t = {q ∨ (x)e}∩{q ∨ (y)e} = q∨{(x)e ∩ (y)e} = q∨ (x ∧y)e. since x ∧y ∈ q, we get that s ∧ t ∈ q. since s ∧ t ∈ k, we get that s ∧ t ∈ q∩k, which is a contradiction to q∩k = ∅. therefore either x ∈ q or y ∈ q. thus q is a prime e-ideal of r. � corollary 3.2. for any e-ideal g of an adl r with x /∈ g, there exists a prime e-ideal q of r such that g ⊆ q and x /∈ q. corollary 3.3. for any e-ideal g of an adl r, g = ⋂ {q | q is a prime e− ideal of r and g ⊆ q}. corollary 3.4. e is the intersection of all prime e-ideals of r. proof. let q be any prime e-ideal of r. clearly, we have that e ⊆ ⋂ q. let q be any prime e-ideal of an adl r and x ∈ ⋂ q. suppose x /∈ e. then there exists prime filter n such that x ∈ n and 8 int. j. anal. appl. (2023), 21:85 n ∩ e = ∅. that implies x /∈ r \ n and e ⊆ r \ n. therefore r \ n is a prime e-ideal of r and x /∈ r\n, which is a contradiction. therefore x ∈ e and hence ⋂ q ⊆ e. thus e = ⋂ q. � theorem 3.6. in an adl the following are equivalent: (1) every proper e-ideal is prime (2) ie(r) is a chain (3) ipef (r) is a chain. proof. (1) ⇒ (2) assume (1). clearly (ie(r),⊆) is a poset. let s and t be two proper e-ideals of r. by (1), we have that s∩t is a prime e-ideal of r. since s∩t ⊆ s∩t, we get s ⊆ s∩t ⊆ t or t ⊆ s ∩t ⊆ s. hence ie(r) is a chain. (2) ⇒ (3) it is obvious. (3) ⇒ (1) assume that (3). let g be a proper e-ideal of r. we prove that g is prime. let x,y ∈ r such that (x)e ∩ (y)e ⊆ g. by our assumption, we get that (x)e ⊆ (y)e or (y)e ⊆ (x)e. that implies x ∈ (x)e = (x)e ∩ (y)e ⊆ g or y ∈ (y)e = (x)e ∩ (y)e ⊆ g. therefore g is a prime e-ideal of r. � now we introduce the concept of a relative dual annihilator in the following definition. definition 3.3. for any nonempty subset s of r, define (s,e) = {a ∈ r | s∧a ∈ e, f or all s ∈ s}. we call this set as relative dual annihilator of s with respect to the ideal e. for s = {s}, we denote ({s},e) by (s,e). lemma 3.3. if s,t are nonempty subsets of an adl r, then we have the following: (1) (r,e) = e = (m,e) (2) (e,e) = r (3) e ⊆ (s,e) (4) (s,e) is a e-ideal of r (5) s ⊆ e if and only if (s,e) = r (6) if s ⊆ t, then (t,e) ⊆ (s,e) and ((s,e),e) ⊆ ((t,e),e) (7) s ⊆ ((s,e),e) (8) (((s,e),e),e) = (s,e) (9) (s,e) = ([s),e) (10) ⋂ i∈4 (si,e) = ( ⋃ i∈4 si,e ) (11) (s,e) ⊆ (s ∩t, (t,e)) (12) if s ⊆ t, then (s, (t,e)) = (s,e) (13) (s ∪t,e) ⊆ (s, (t,e)) ⊆ (s ∩t,e) (14) (s, (s,e)) = (s,e). int. j. anal. appl. (2023), 21:85 9 proof. (1) let x ∈ (r,e). then a ∧ x ∈ e, for all a ∈ r. that implies x ∧ x ∈ e. so that x ∈ e. hence (r,e) ⊆ e. let x ∈ e. then a∧x ∈ e, for all a ∈ r. thus x ∈ (r,e). therefore e ⊆ (r,e) and hence (r,e) = e. clearly, we have that (m,e) = e. (2) let x ∈ e. then x ∧ a ∈ e, for all a ∈ r. since x ∧ a ∈ e, for all x ∈ e, we get that a ∈ (e,e), for all a ∈ r. therefore r ⊆ (e,e) and hence r = (e,e). (3) let x ∈ e. then y ∧ x ∈ e, for all y ∈ r. then a ∧ x ∈ e, for all a ∈ s ⊆ r. that implies x ∈ (s,e). therefore e ⊆ (s,e). (4) let a,b ∈ (s,e). then s∧a,s∧b ∈ e, for all s ∈ s. this implies (s∧a)∨(s∧b) ∈ e. therefore s ∧ (a ∨b) ∈ e. hence a ∨b ∈ (s,e). let a ∈ (s,e) and b ∈ r with b ≤ a. then s ∧ a ∈ e and s ∧b ≤ s ∧a, for all s ∈ s. since s ∧a ∈ e and e is an ideal, we get s ∧b ∈ e. hence b ∈ (s,e), for all s ∈ s. thus (s,e) is an ideal of r. since e ⊆ (s,e), we get that (s,e) is an e-ideal of r. (5) suppose (s,e) = r. let m ∈m. then m ∈ (s,e). that implies a = m ∧a ∈ e, for all a ∈ s. hence a ∈ e, for all a ∈ s. therefore s ⊆ e. conversely, assume that s ⊆ e. let x ∈ r. since e is an ideal, we get a∧x ∈ e, for all a ∈ s ⊆ e. hence x ∈ (s,e). therefore (s,e) = r. (6) suppose s ⊆ t. let a ∈ (t,e). then t∧a ∈ e, for all t ∈ t. since s ⊆ t, we get that s∧a ∈ e, for all s ∈ s. that implies a ∈ (s,e). therefore (t,e) ⊆ (s,e) and hence ((s,e),e) ⊆ ((t,e),e). (7) let x ∈ (s,e). then s ∧ x ∈ e, for all s ∈ s. that implies x ∧ s ∈ e, for all x ∈ (s,e). that implies s ∈ ((s,e),e), for all s ∈ s. thus s ⊆ ((s,e),e). (8) by (7), we have that (((s,e),e),e) ⊆ (s,e). let x /∈ (((s,e),e),e). then there exists an element a /∈ ((s,e),e) such that a ∧ x /∈ e. since s ⊆ ((s,e),e), we have that a /∈ s. so that a ∧ x /∈ e and s /∈ s. therefore x /∈ (s,e), it concludes that (s,e) ⊆ (((s,e),e),e). thus (((s,e),e),e) = (s,e). (9) since s ⊆ (s], we get that ((s],e) ⊆ (s,e). let x ∈ (s,e). then a∧x ∈ e, for all a ∈ s ⊆ (s]. that implies x ∈ ((s],e). therefore (s,e) ⊆ ((s],e). therefore (s,e) ⊆ ((s],e). hence (s,e) = ((s],e). (10) since si ⊆ ⋃ i∈4 si, for all i ∈ 4, we get that ( ⋃ i∈4 si,e ) ⊆ (si,e), for all i ∈ 4. that implies( ⋃ i∈4 si,e ) ⊆ ⋂ i∈4 (si,e). let x ∈ ⋂ i∈4 (si,e). then x ∈ (si,e), for all i ∈4. that implies a∧x ∈ e, for all a ∈ si ⊆ ⋃ si. that implies ⋂ i∈4 (si,e) ⊆ ( ⋃ i∈4 si,e ) . therefore ⋂ i∈4 (si,e) = ( ⋃ i∈4 si,e). (11) since e is an ideal in r, we have that e ⊆ (t,e) and hence we get that (s,e) ⊆ (s, (t,e)). since s ∩t ⊆ s, we get that (s, (t,e)) ⊆ (s ∩t, (t,e)). therefore (s,e) ⊆ (s ∩t, (t,e)). (12) let s,t be two non empty subsets of r such that s ⊆ t. since e ⊆ (t,e), we have that (s,e) ⊆ (s, (t,e)). let x ∈ (s, (t,e)). then a ∧ x ∈ (t,e), for all a ∈ s. that implies a ∧ x ∈ (s,e), for all a ∈ s. since a ∧ x ∈ (s,e), we get that s ∧ (a ∧ x) ∈ e, for all s ∈ s and hence a ∧ x ∈ e, for all a ∈ s. therefore x ∈ (s,e) and hence (s, (t,e)) ⊆ (s,e). thus (s, (t,e)) = (s,e). (13) clearly, we have that (s ∪t,e) ⊆ (s,e) and e ⊆ (t,e). so that (s,e) ⊆ (s, (t,e)). also 10 int. j. anal. appl. (2023), 21:85 s∩t ⊆ s. it follows that (s, (t,e)) ⊆ (s∩t,e). therefore (s∪t,e) ⊆ (s, (t,e)) ⊆ (s∩t,e). (14) it is clear by (12). � proposition 3.1. let s and t be any two ideals of and adl r. then we have the following: (1) (s,e) ∩ ((s,e),e) = e (2) (s ∨t,e) = (s,e) ∩ (t,e) (3) ((s ∩t,e),e) ⊆ ((s,e),e) ∩ ((t,e),e). proof. (1) we have that e ⊆ (s,e)∩((s,e),e). let x ∈ (s,e)∩((s,e),e). then x ∈ (s,e) and x ∈ ((s,e),e). since x ∈ ((s,e),e)), we have that a∧x ∈ e, for all a ∈ (s,e). since x ∈ (s,e), we get that x ∈ e and hence (s,e) ∩ ((s,e),e) ⊆ e. thus (s,e) ∩ ((s,e),e) = e. (2) clearly, s ⊆ s∨t and t ⊆ s∨t. then ((s∨t ),e) ⊆ (s,e) and ((s∨t ),e) ⊆ (t,e). that implies ((s ∨ t ),e) ⊆ (s,e) ∩ (t,e). let x ∈ (s,e) ∩ (t,e). then x ∈ (s,e) and x ∈ (t,e). that implies s ∧x ∈ e, for all s ∈ s and t ∧x ∈ e, for all t ∈ t. that implies (s ∧x) ∨ (t ∧x) ∈ e and have (s ∨ t) ∧ x ∈ e. since s ∈ s and t ∈ t, we get s ∨ t ∈ s ∨t. therefore (s ∨ t) ∧ x ∈ e, for all s ∨ t ∈ s ∨t. that implies x ∈ (s ∨t,e). therefore (s,e) ∩ (t,e) ⊆ (s ∨t,e). hence (s,e) ∩ (t,e) = (s ∨t,e). (3) since s∩t ⊆ s and s∩t ⊆ t, we get that (s,e) ⊆ (s∩t,e) and (t,e) ⊆ (s∩t,e). that implies ((s ∩ t,e),e) ⊆ ((s,e),e) and ((s ∩ t,e),e) ⊆ ((t,e),e). hence ((s ∩ t,e),e) ⊆ ((s,e),e) ∩ ((t,e),e). � theorem 3.7. for any non-empty subset s of an adl r, (s,e) = ⋂ s∈s ((s],e). proof. let x ∈ ⋂ s∈s ((s],e). then x ∈ ((s],e), for all s ∈ s. that implies t ∧ x ∈ e, for all t ∈ (s] and for all s ∈ s. it follows that s ∧ x ∈ e for all s ∈ s. therefore x ∈ (s,e). hence x ∈ ⋂ s∈s ((s],e) ⊆ (s,e). let s be any element of s. take t ∈ (s]. then s ∧t = t. now, x ∈ (s,e). that implies s ∧x ∈ e, for all s ∈ s. so that t ∧x = t ∧ s ∧x ∈ e, for all t ∈ (s] and for all s ∈ s. that implies x ∈ ((s],e), for all s ∈ s. therefore x ∈ ⋂ s∈s ((s],e) and hence (s,e) ⊆ ⋂ s∈s ((s],e). thus (s,e) = ⋂ s∈s ((s],e). � corollary 3.5. let x ∈ r and s be arbitrary subset of r. then (s, (x]) = ⋂ a∈s (a, (x]). corollary 3.6. for any x,y ∈ r we have the following: (1) ((x],e) = (x,e) (2) x ≤ y ⇒ (y,e) ⊆ (x,e) (3) (x ∨y,e) = (x,e) ∩ (y,e) (4) ((x ∧y,e),e) = ((x,e),e) ∩ ((y,e),e) (5) (x,e) = r ⇔ x ∈ e. int. j. anal. appl. (2023), 21:85 11 theorem 3.8. let g be an e-ideal of an adl r. then (1) g ∩ (g,e) = e (2) ((g ∨ (g,e)),e) = e. proof. (1) it is clear. (2) clearly, ((g∨(g,e)),e) ⊆ (g,e)∩((g,e),e). let a ∈ (g,e)∩((g,e),e). let b ∈ g∨(g,e). then b = c ∨ d, for some c ∈ g and d ∈ (g,e). that implies a ∧ c ∈ e and a ∧ d ∈ e. now a∧b = a∧(c∨d) = (a∧c)∨(a∧d) ∈ e, for all b ∈ g∨(g,e). therefore a ∈ ((g∨(g,e)),e) and hence (g,e)∩((g,e),e) ⊆ ((g∨(g,e)),e). thus e = (g,e)∩((g,e),e) = ((g∨(g,e)),e). � consider two adls r1 and r2 with zero elements 0 and 0′ respectively. let m and m′ be denotes the set of all maximal elements of adls r1 and r2 respectively. lemma 3.4. let r1 and r2 be two adls with m ∈m and m′ ∈m′. then for any (x,y) ∈ r1×r2, we have the following: (1) (x,y)+ = (a)+ × (y)+ (2) (x,y)+ = (m,m′) if and only if (x)+ = m and (y)+ = m′ (3) ((x,y),e) = (a,e) × (y,e). let e1 and e2 be dual dense sets of r1 and r2 respectively. from the above result, it can be concluded that e = e1×e2 is a dual dense set of r1×r2. further, every dual dense set of r1×r2 is form the form e1 ×e2. theorem 3.9. let mi be a prime ei−ideals of adls ri, for i = 1, 2. then m1 ×r2 and r1 ×m2 are prime e-ideals of r1 ×r2. proof. since e1 ⊆ m1 and e2 ⊆ m2, we get e1 × e2 ⊆ m1 × r2 and e1 × e2 ⊆ r1 × m2. that implies m1 × r2 and r1 × m2 are e-ideals of r1 × r2. let (a,b), (c,d) ∈ r1 × r2 with (a,b) ∧ (c,d) ∈ m1 × r2. then a ∧ c ∈ m1. since m1 is a prime e1−ideal of r1, we get a ∈ m1 or c ∈ m1. thus (a,b) ∈ m1 × r2 or (c,d) ∈ m1 × r2. therefore m1 × r2 is a prime e-ideal of r1 ×r2. similarly, we can prove that r1 ×m2 is also a prime e-ideal of r1 ×r2. � theorem 3.10. let r1 and r2 be two adls with zero elements 0 and 0′ respectively. for any prime e-ideal p of r1 ×r2, p is of the form p1 ×r2 or r1 ×p2, where pi is a prime ei−ideal of ri, for i = 1, 2. proof. let p be a prime e-ideal of r1 × r2. consider p1 = π1(p ) = {x1 ∈ r1 | (x1,x2) ∈ p, for some x2 ∈ r2} and p2 = π2(p ) = {x2 ∈ r2 | (x1,x2) ∈ p, for some x1 ∈ r1}. it is easy to verify that pi is ei−ideals of ri, for i = 1, 2. we first show that pi is prime ei−ideals of ri, for i = 1, 2. suppose p1 = r1 and p2 = r2. let (a,b) ∈ r1 ×r2. then there exist x ∈ r1 and y ∈ r2 such that (a,y) ∈ p and (x,b) ∈ p. since (a, 0′)∧(a,y) ∈ p and (0,b)∧(x,b) ∈ p, we get (a, 0′) ∈ p 12 int. j. anal. appl. (2023), 21:85 and (0,b) ∈ p. therefore (a,b) = (a, 0′) ∨ (0,b) ∈ p. hence p = r1 ×r2, which is a contradiction to that p is proper. next suppose that p1 6= r1 and p2 6= r2. choose a ∈ r1 \p1 and b ∈ r2 \p2. then (a,y) /∈ p for all y ∈ r2 and (x,b) /∈ p1 for all x ∈ r1. in particular, (a, 0′) /∈ p and (0,b) /∈ p. since p is prime, we get (0, 0′) /∈ p, which is a contradiction. from the above observations, we get that either p1 = r1 and p2 6= r2 or p1 6= r1 and p2 = r2. case (i): suppose p1 = r1 and p2 6= r2. let x2,y2 ∈ r2 be such that x2∧y2 ∈ p2. then there exists a ∈ r1 = p1 such that (a,x2∧y2) ∈ p. therefore (a,x2)∧(a,y2) = (a∧a, (x2∧y2)) = (a,x2∧y2) ∈ p. since p is prime, we get (a,x2) ∈ p or (a,y2) ∈ p. hence x2 ∈ p2 or y2 ∈ p2. therefore p2 is a prime e2−ideal of r2. we now show that p = r1 ×p2. clearly p ⊆ r1 ×p2. on the other hand, suppose (a,y) ∈ r1 ×p2. since p1 = r1, there exists b ∈ r2 such that (a,b) ∈ p and there exists x ∈ r1 such that (x,y) ∈ p. since (a, 0′)∧(a,b) = (a, 0′) and (0,y)∧(x,y) = (0,y), we get (a, 0′) ∈ p and (0,y) ∈ p. since p is an ideal, it gives (a,y) = (a, 0′) ∨ (0,y) ∈ p. hence r1 ×p2 ⊆ p. therefore p = r1 ×p2. case (ii): suppose p1 6= r1 and p2 = r2. similarly, we can prove that p1 is prime e1−ideal of r1 and p = p1 ×r2. � theorem 3.11. let s be a sub adl of an adl r and p is a prime e-ideal of s. then there exists a prime e-ideal q of r such that q∩s = p. proof. let p be a prime e-ideal of s. then s \ p is a prime filter of s. consider i = (p ]. then p ⊆ i ∩ s. suppose i ∩ (s \ p ) 6= ∅. choose x ∈ i ∩ (s \ p ). then x ∈ i and x ∈ (s \ p ). since x ∈ i = (p ], there exists a1 ∨a2 ∨ . . .∨an ∈ p such that x = y ∧ (a1 ∨a2 ∨ . . .∨an). since p is an ideal of s, we get a1∨a2∨ . . .∨an ∈ p and hence x ∈ p. since x ∈ (s\p ), we get a contradiction. hence i∩(s\p ) = ∅. then there exists a prime e-ideal q of r such that i ⊆ q and q∩(s\p ) = ∅. since i ⊆ q, we get i ∩s ⊆ q∩s. since q∩ (s \p ) = ∅, we get q ⊆ p. hence, both observations lead to p ⊆ i ∩s ⊆ q∩s ⊆ p ∩s ⊆ p. therefore p = q∩s. � now, we have the following definition. definition 3.4. a prime e-ideal m of an adl r containing an e-ideal g is said to be a minimal prime e-ideal belonging to g if there exists no prime e-ideal n such that g ⊆ n ⊆ m. note that if we take e = g in the above definition then we say that m is a minimal prime e-ideal. example 3.3. from the example 3.2, we have that i6 is a prime e-ideal and i1 is a e-ideal of r. clearly i1 ⊆ i6. clearly there is no e-ideal n of r such that i1 ⊆ n ⊆ i6. hence i6 is a minimal prime e-ideal belonging to i1. proposition 3.2. let g be an e-ideal and m, a prime e-ideal of r with g ⊆ m. then m is a minimal prime e-ideal belonging to g if and only if r\m is a maximal filter with (r\m) ∩g = ∅. int. j. anal. appl. (2023), 21:85 13 proof. clearly, r\m is a proper filter and we have (r\m)∩g = ∅. we prove that r\m is maximal. let n be any proper filter of r such that n ∩ g = ∅ and r \ m ⊆ n. then g ⊆ r \ n ⊆ m. by the minimality of m, we get r \ n = m. therefore r \ m is a maximal filter with respect to the property (r \ m) ∩ g = ∅. conversely, assume that r \ m be a maximal filter with respect to the property (r \ m) ∩ g = ∅. we prove that m is minimal. if n is any prime e-ideal of r such that e ⊆ g ⊆ n ⊆ m. clearly, r\n is a filter such that r\m ⊆ r\n and (r\n) ∩g = ∅, which is a contradiction. therefore m is a minimal prime e-ideal belonging to g. � theorem 3.12. let g be an e-ideal and m, a prime e-ideal of r with g ⊆ m. then m is a minimal prime e-ideal belonging to g if and only if for any a ∈ m, there exists b /∈ m such that a∧b ∈ g. proof. assume that m is a minimal prime e-ideal belonging to g. then r \ m is a maximal filter with respect to the property that (r \ m) ∩ g = ∅. let a ∈ m. then a /∈ r \ m. that implies r \m ⊂ (r \m) ∨ [a). by the maximality of r \m, we get that ((r \m) ∨ [a)) ∩g 6= ∅. choose s ∈ ((r \ m) ∨ [a)) ∩ g. then there exists b ∈ r \ m such that s = b ∧ a and s ∈ g. therefore b∧a ∈ g. conversely, assume that for any a ∈ m, there exists b /∈ m such that a∧b ∈ g. suppose m is not a minimal prime e-ideal belonging to g. then there exists a prime e-ideal n of r such that e ⊆ g ⊆ n ⊆ m. choose a ∈ m \n. then, by the our assumption, there exists b /∈ m such that a ∧b ∈ g ⊆ n. since a /∈ n, we get that b ∈ n ⊆ m, which is a contradiction. therefore m is a minimal prime e-ideal belonging to g. � corollary 3.7. a prime e-ideal m of an adl r is minimal if and only if for any a ∈ m there exists b /∈ m such that a∧b ∈ e. definition 3.5. for any prime e-ideal m of r, define the set oe(m) as follows: oe(m) = {x ∈ r | x ∈ (y,e), for some y /∈ m}. clearly, observe that oe(m) = ⋃ y /∈m (y,e). lemma 3.5. let m be prime e-ideal of an adl r. then oe(m) is an e-ideal such that oe(m) is contained in m. proof. let a,b ∈oe(m). there exist elements s /∈ m and t /∈ m such that a ∈ (s,e) and b ∈ (t,e). that implies ((s,e),e) ⊆ (a,e) and ((t,e),e) ⊆ (b,e). so that ((s ∧ t,e),e) = ((s,e),e) ∩ ((t,e),e) ⊆ (a,e) ∩ (b,e) = (a ∨ b,e). hence a ∨ b ∈ ((a ∨ b,e),e) ⊆ (((s ∧ t,e),e),e) = (s ∧ t,e). since s ∧ t /∈ m, we get that a ∨b ∈ oe(m). let a ∈ oe(m) and b ≤ a. there exists s /∈ m such that a ∈ (s,e). since (s,e) is an ideal, we get that b ∈ (s,e). therefore b ∈ oe(m) and hence oe(m) is an ideal of r. clearly, we have that e ⊆ oe(m). thus oe(m) is an e-ideal of r. let a ∈ oe(m). then there exists s /∈ m such that a ∈ (s,e). that implies a ∧ s ∈ e ⊆ m. since m is prime, we get that a ∈ m. hence oe(m) ⊆ m. � 14 int. j. anal. appl. (2023), 21:85 corollary 3.8. for any prime e-ideal m of r, m is minimal if and only if oe(m) = m. theorem 3.13. every minimal prime e-ideal of r belonging to oe(m) is contained in m. proof. let n be any minimal prime e-ideal belonging to oe(m). we prove that n ⊆ m. suppose n * m. choose a ∈ n\m. then there exists b /∈ n such that a∧b ∈oe(m). hence a∧b ∈ (s,e), for some s /∈ m. that implies b ∧ (a ∧ s) ∈ e ⊆ m. since a /∈ m,s /∈ m, and m is prime, we get a∧ s /∈ m. therefore b ∈oe(m) ⊆ n, which is a contradiction. hence n ⊆ m. � theorem 3.14. for any prime e-ideal m of an adl r, oe(m) is the intersection of all minimal prime e-ideals contained in m. proof. let m be a prime e-ideal of r. by zorn’s lemma, m contains a minimal prime e-ideal. let {sα}α∈m be the set of all minimal prime e-ideals contained in m. let x ∈oe(m). then x ∈ (a,e), for some a /∈ m. since each sα ⊆ m, we have that a /∈ sα, for all α ∈m . since x ∧a ∈ e ⊆ sα and a /∈ sα, for all α ∈m, we get x ∈ sα for all α ∈m. hence x ∈ ⋂ α∈m sα. therefore oe(m) ⊆ ⋂ α∈m sα. let x /∈ oe(m). consider s = (r \m) ∨ [x). suppose e ∩s 6= ∅. choose a ∈ e ∩s. since a ∈ s, we get a = t ∧x, for some t ∈ r\m. since a ∈ e, we get that t ∧x ∈ e. hence x ∈ (t,e), where t /∈ m. thus x ∈oe(m), which is a contradiction. therefore s ∩e = ∅. let m be a maximal filter such that s ⊆ m and m ∩e = ∅. then r\m is a minimal prime e-ideal such that r\m ⊆ m and x /∈ r\m, since x ∈ s ⊆ m. hence x /∈ ⋂ α∈m sα. therefore ⋂ α∈m sα ⊆oe(m). � proposition 3.3. let m1 and m2 be two prime e-ideals in an adl r with m1 ⊆ m2. then oe(m2) ⊆ oe(m1). proof. let x ∈ oe(m2). then there exists an element a /∈ m2 such that x ∈ (a,e). that implies x ∈ (a,e) and a /∈ m1. so that x ∈oe(m1). therefore oe(m2) ⊆oe(m1). � proposition 3.4. for any non zero element a ∈ r with a /∈ e, there is a minimal prime e-ideal not containing a. proof. let a be any non zero element of r with a /∈ e. by corollary 3.2, there exists a prime e-ideal p of r such that a /∈ p. consider f = {q | q is a prime e − ideal of r,a /∈ q and q ⊆ p}. it satisfies the hypothesis of zorn’s lemma. so that f has a minimal element say m. i.e. m is minimal and a /∈ m. � theorem 3.15. for any prime e-ideal m of an adl r, the following are equivalent: (1) m is minimal prime e-ideal (2) m = oe(m) (3) for any x ∈ r,m contains precisely one of x or (x,e). int. j. anal. appl. (2023), 21:85 15 proof. (1) ⇒ (2) assume (1). let x ∈ m. then there exists y /∈ m such that x∧y ∈ e. this implies that x ∈oe(m). so that m ⊆oe(m). since oe(m) ⊆ m, we get that m = oe(m). (2) ⇒ (3) assume (2). let x ∈ r. suppose x /∈ m. let a ∈ (x,e). then a ∧ x ∈ e. that implies a∧x ∈ m. so that a ∈ m. since x /∈ m. therefore (x,e) ⊆ m. (3) ⇒ (1) let q be any prime e-ideal of r with q ( m. then choose x ∈ m such that x /∈ q. that implies (x,e) ⊆ q ( m. so that (x,e) ( m which is a contradiction. � corollary 3.9. let p be a minimal prime e-ideal of an adl r and a ∈ r. then a ∈ p if and only if ((a,e),e) ⊆ p. proof. assume that a ∈ p. then (a,e) * p. let t ∈ ((a,e),e). then (a,e) ⊆ (t,e). suppose t /∈ p. then (a,e) ⊆ (t,e) ⊆ p, which is a contradiction. that implies t ∈ p, which gives ((a,e),e) ⊆ p. the converse follows from the fact that a ∈ ((a,e),e). � definition 3.6. an adl r with maximal elements is called an esemi complemented if for each non maximal element x ∈ r, there exists a non zero element y /∈ e such that x ∧y ∈ e. example 3.4. from the example 3.2, clearly we have that r is an e-semi complemented adl. theorem 3.16. let r be an adl with maximal elements. then r is e-semi complemented if and only if the intersection of all maximal filters disjoint with e is m. proof. assume that r is e-semi complemented. consider k = ⋂{ m | m is a maximal filter of r and m ∩e = ∅}. we have to prove that k = m. let x ∈ k with x is not a maximal element. then x ∈ m, for all maximal filter m disjoint with e. then x /∈ e. since x is non maximal and r is esemi complemented, there exists a non zero element y /∈ e such that x ∧ y ∈ e. then x ∧ y /∈ m. that implies m∨[x∧y) = r. since y /∈ e, there exists a minimal prime e-ideal n of r such that y /∈ n. that implies y ∈ r\n and (r\n)∩e = ∅, where r\n is maximal filter of r. so that x,y ∈ r\n. we have x∧y ∈ r\n. therefore (r\n)∩e 6= ∅, which is a contradiction. therefore x is a maximal element. hence k = m. conversely, assume that ⋂ {m | m is a maximal filter of r and m ∩e = ∅} = m. let x be any non maximal element of r. then there exists a maximal filter m such that x /∈ m and m ∩ e = ∅. that implies m ∨ [x) = r. so that a ∧ x = 0, for some a ∈ m. since a ∈ m and m ∩e = ∅, we get a /∈ e. clearly, a ∧ x ∈ e. that is, for any non maximal element x of r, there exists a non zero element a /∈ e such that a∧x ∈ e. hence r is e-semi complemented. � definition 3.7. an adl r is said to be e-normal if for any a,b ∈ r such that a∧b ∈ e, there exists x ∈ (a,e) and y ∈ (b,e) such that x ∨y is maximal. from the example 3.2, clearly we have that r is a d−normal adl. the following result is a direct consequence of the above definition. 16 int. j. anal. appl. (2023), 21:85 theorem 3.17. r is e-normal if and only if (a,e) ∨ (b,e) = r, for any a,b ∈ r, with a∧b ∈ e. definition 3.8. two e-ideals g1 and g2 of r are said to be co-maximal if g1 ∨g2 = r. example 3.5. from the example 3.2, we have that i2, i3, i4, i5 are e-ideals of r. clearly, i4∨i5 = r. therefore i4 and i5 are co-maximal. also, we have i2 ∨ i3 6= r. therefore i2, i3 are not co-maximal. theorem 3.18. in an adl r, the following are equivalent: (1) for any a,b ∈ r with a∧b ∈ e, (a,e) ∨ (b,e) = r (2) for any a,b ∈ r, (a,e) ∨ (b,e) = (a∧b,e) (3) any two distinct minimal prime e-ideals are co-maximal (4) every prime e-ideal contains a unique minimal prime e-ideal (5) for any prime e-ideal p, oe(p ) is prime. proof. (1) ⇒ (2) assume (1). let a,b ∈ r with x ∈ (a ∧ b,e). then x ∧ (a ∧ b) ∈ e and hence (x ∧ a) ∧ (x ∧ b) ∈ e. by (1), we have that (x ∧ a,e) ∨ (x ∧ b,e) = r. that implies x ∈ (x ∧a,e) ∨ (x ∧b,e). then there exists r ∈ (x ∧a,e) and s ∈ (x ∧b,e) such that x = r ∨ s. since r ∈ (x ∧ a,e), s ∈ (x ∧ b,e) we get that r ∧ x ∈ (a,e) and s ∧ x ∈ (b,e). that implies (x ∧ r) ∨ (x ∧ s) ∈ (a,e) ∨ (b,e) and hence x ∧ (r ∨ s) ∈ (a,e) ∨ (b,e). since x = r ∨ s, we get that x ∈ (a,e)∨(b,e). therefore (a∧b,e) ⊆ (a,e)∨(b,e). since (a,e)∨(b,e) ⊆ (a∧b,e), we get that (a,e) ∨ (b,e) = (a∧b,e), for all a,b ∈ r. (2) ⇒ (3) assume (2). let m and n be two distinct minimal prime e-ideals of r. choose elements x,y ∈ r such that x ∈ m \n and y ∈ n \m. since m and n are minimal, x ∧a ∈ e, y ∧b ∈ e, for some a /∈ m, b /∈ n. that implies x ∧a ∧ y ∧b ∈ e and hence r = (x ∧a ∧ y ∧b,e). by (2), we get that (x ∧b,e) ∨ (a∧ y,e) = r. since a /∈ m and y /∈ m, we get that a∧ y /∈ m. that implies (a∧y,e) ⊆ m. similarly, we have that (x ∧b,e) ⊆ n. that implies ((x ∧b) ∧ (a∧y),e) ⊆ m ∨n and hence r = m ∨n. therefore m and n are co-maximal. (3) ⇒ (4) assume (3). let m be a prime e-ideal of r. suppose m contains two distinct minimal prime e-ideals, say n1 and n2. by (3), we get that r = n1 ∨ n2 ⊆ m, we get a contradiction. therefore every prime e-ideal contains a unique minimal prime e-filter. (4) ⇒ (5) assume that every prime e-ideal of r contains a unique minimal prime e-ideal. then by corollary 3.8, we get that oe(p ) is a prime e-ideal. (5) ⇒ (1) assume (5). let a,b ∈ r be such that a∧b ∈ e. suppose (a,e)∨(b,e) 6= r. then there exists a maximal e-ideal m such that (a,e) ∨ (b,e) ⊆ m. that implies (a,e) ⊆ m and (b,e) ⊆ m. that implies a /∈ oe(m) and b /∈ oe(m). since oe(m) is prime, we get a ∧b /∈ oe(m). so that e *oe(m), which is a contradiction. therefore (a,e) ∨ (b,e) = r. � theorem 3.19. in an adl r with maximal elements, the following conditions are equivalent: (1) r is e-normal int. j. anal. appl. (2023), 21:85 17 (2) for any two distinct maximal filters g1 and g2 of r with g1 ∩e = ∅, g2 ∩e = ∅ there exist a /∈ g1 and b /∈ g2 such that a∨b is maximal (3) for any maximal filter g with g∩e = ∅, g is the unique maximal filter containing r\oe(p ). proof. (1) ⇒ (2) assume that r is e-normal. let g1 and g2 be two distinct maximal filters of r with g1 ∩e = ∅,g2 ∩e = ∅. then r \g1 and r \g2 are distinct minimal prime e-ideals of r. by our assumption, we get r \g1 and r \g2 are co-maximal. that is, (r\g1)∨(r\g2) = r. then, there exist a ∈ r\g1 and b ∈ r\g2 such that a∨b is maximal. (2) ⇒ (3) assume (2). let g be any maximal filter of r with g∩e = ∅ and r\oe(p ) ⊆ g. let g1 be any maximal filter of r with g1 ∩e = ∅ and r \oe(p ) ⊆ g1. we prove that g = g1. suppose g 6= g1. by our assumption, there exists a /∈ g and b /∈ g1 such that a∨b is maximal. that implies a,b /∈ r \oe(p ). so that a,b ∈ oe(p ). this implies that a∨b ∈ oe(p ). therefore oe(p ) = r, which is a contradiction. we conclude that g = g1. (3) ⇒ (1) for any maximal filter g with g ∩ e = ∅, g is the unique maximal filter containing r \oe(p ). let p be a prime e-ideal of r. suppose p contains two minimal prime e-ideals say q1 and q2. that is, q1 ⊆ p and q2 ⊆ p. that implies oe(p ) ⊆oe(q1) and oe(p ) ⊆oe(q2). we get p ⊆oe(q1) and p ⊆oe(q2). so that q2 ⊆ q1 and q1 ⊆ q2. this concludes that q1 = q2. � let f be a filter of r. for any x,y ∈ r, define a binary relation φf on r as φf = {(x,y) ∈ r×r | x ∧a = y ∧a, for some a ∈ f}. proposition 3.5. for any filter f of an associative adl r, φf is a congruence relation on r. for any adl r, it can be easily verified that the quotient r/φf is also an adl with respect to the following operations: [a]φf ∧ [b]φf = [a ∧ b]φf and [a]φf ∨ [b]φf = [a ∨ b]φf where [a]φf is the congruence class of a modulo φf . it can be routinely verified that the mapping φ : r → r/φf defined by φ(a) = [a]φf is a homomorphism. theorem 3.20. in an adl r, we have the following: (1) if x is a dual dense element of r, then [x]φf is a dual dense element of r/φf (2) if g is a e-ideal of r/φf , then φ−1(g) is a e-ideal of r (3) if g is a prime e-ideal of r/φf , then φ−1(g) is a prime e-ideal of r. definition 3.9. let f be a filter of an adl r. for any ideal g of r, define g̃ = {[a]φf | a ∈ g}. the following result can be proved easily. lemma 3.6. let g be an e-ideal of r. then g̃ is an e-ideal of r/φf . proposition 3.6. let g be a prime e-ideal and f a filter of an adl r such that g∩f = ∅. we have the following: 18 int. j. anal. appl. (2023), 21:85 (1) x ∈ g if and only if [x]φf ∈ g̃ (2) g̃ ∩ f̃ = ∅ (3) if g is a prime e-ideal of r, then g̃ is a prime e-ideal of r/φf . proof. (1) assume that x ∈ g. then we have [x]φf ∈ g̃. conversely assume that [x]φf ∈ g̃. then there exists y ∈ g such that [x]φf = [y]φf . that implies (x,y) ∈ φf . so there exists a ∈ f such that x ∧a = y ∧a ∈ g. since g ∩f = ∅, we get a /∈ g. since x ∧a ∈ g and a /∈ g, we get that x ∈ g. (2) suppose g̃ ∩ f̃ 6= ∅. then choose an element x ∈ r such that [x]φf ∈ g̃ ∩ f̃ . then [x]φf ∈ g̃ and [x]φf ∈ f̃ . since [x]φf ∈ g̃ and by (1), we get x ∈ g. since [x]φf ∈ f̃ , there exists y ∈ f such that [x]φf = [y]φf . then (x,y) ∈ φf . so there exist a ∈ f such that x ∧a = y ∧a. since y ∧a ∈ f, we get that x ∧a ∈ f. since x ∈ g, we have that x ∧a ∈ g ∩f. that implies g ∩f 6= ∅, we get a contradiction. hence g̃ ∩ f̃ = ∅. (3) clearly, we have that g̃ is a proper ideal of r/φf . let [x]φf ∈ ẽ. then x ∈ e ⊆ g. that implies [x]φf ∈ g and hence g̃ is an e-ideal of r/φf . let [x]φf , [y]φf ∈ r/φf such that [x]φf ∧ [y]φf ∈ g̃. then [x ∧ y]φf ∈ g̃. by (1) we have that x ∧ y ∈ g. since g is prime, we get that x ∈ g or y ∈ g again by(1) we get that [x]φf ∈ g̃ or [y]φf ∈ g̃. hence g̃ is a prime e-ideal in r/φf . � proposition 3.7. let f be a filter of an adl r. then there is an order isomorphism of the set of all prime e-ideals of r disjoint from f onto the set of all prime e-ideals of r/φf . proof. let g and h be two prime e-ideals of r such that g ∩ f = ∅ and h ∩ f = ∅. then by proposition 3.6(1), we get that g ⊆ h if and only if g̃ ⊆ h̃. let g be a prime e-ideal of r with g ∩ f = ∅. then by proposition 3.6(3), we get that g̃ is a prime e-ideal of r/φf . let q be a prime e-ideal of r/φf . consider g = {a ∈ r|[a]φf ∈ q}. since q is a e-ideal of r/φf , we get that g is a e-ideal of r. let a,b ∈ r with a ∧ b ∈ g. then [a]φf ∧ [b]φf = [a ∧ b]φf ∈ q. since q is prime, we get [a]φf ∈ q or [b]φf ∈ q. therefore a ∈ g or b ∈ g. hence g is a prime e-ideal of r. clearly g̃ = q. suppose g ∩ f 6= ∅. then choose an element s ∈ g ∩ f. that implies [s]φf ∈ q and s ∈ f. let [b]φf ∈ r/φf . since s ∈ f and b ∧ s = b ∧ s ∧ s, we get that (b,b ∧ s) ∈ f. that implies [b]φf = [b∧ s]φf = [b]φf ∧ [s]φf ∈ q. therefore [b]φf ∈ q. and hence r/φf = q, which is a contradiction. thus g ∩f = ∅. � corollary 3.10. let r be an adl. then the above map induces a one-to-one correspondence between the set of all minimal prime e-ideals of r which are disjoint from f and the set of all minimal prime e-ideals of r/φf . theorem 3.21. for any filter f of an adl r, the following are equivalent: (1) any two distinct minimal prime e-ideals of r are co-maximal (2) any two distinct minimal prime e-ideals of r/φf are co-maximal. int. j. anal. appl. (2023), 21:85 19 proof. (1) ⇒ (2) assume (1). let g1,g2 be two distinct minimal prime e-ideals of r/φf . then by the corollary 3.10, there exist two minimal prime e-ideals h1 and h2 of r such that h1∩f = ∅ and h2 ∩f = ∅. also h̃1 = g1 and h̃2 = g2. since g1 and g2 are distinct, we get that h1 and h2 are distinct. by the assumption, we have h1∨h2 = r. let a ∈ r. there exist a1 ∈ h1 and a2 ∈ h2 such that a = a1 ∨a2. since a1 ∈ h1 and a2 ∈ h2 we get [a1]φf ∈ h̃1 = g1 and [a2]φf ∈ h̃2 = g2. now, [a]φf = [a1∨a2]φf = [a1]φf ∨ [a2]φf ∈ g1∨g2. that implies [a]φf ∈ g1∨g2, for all a ∈ r.therefore g1 ∨g2 = r/φf . (2) ⇒ (1) assume (2). let p be a prime e-ideal of r. suppose p contains two distinct minimal prime e-ideals, say g1 and g2. consider k = r \p. clearly k is a filter of r and g1 ∩k = ∅ = g2 ∩k. by corollary 3.10, we get that g̃1 and g̃2 are distinct minimal prime e-ideals of r/φf such that g̃1, g̃2 ⊆ p̃. that implies p̃ is containing two distinct minimal prime e-ideals of r/φf , which is a contradiction. hence p contains a unique minimal prime e-ideal. by theorem 3.18, any two distinct minimal prime e-ideals of r are co-maximal. � 4. on the space prime e-ideals in this section, some topological properties of the space of all prime e-ideals and the space of all minimal prime e-ideals of an adl are studied. let us denote the set of all prime e-ideals of an adl r by specei (r). for any a ⊆ r, define α(a) = {p ∈ specei (r)|a * p} and for any a ∈ r, α(a) = {p ∈ spec e i (r)|a /∈ p}. then we have the following result whose proof is straightforward. lemma 4.1. let r be an adl and a,b ∈ r. then the following conditions hold: (1) ⋃ a∈r α(a) = specei (r) (2) α(a) ∩α(b) = α(a∧b) (3) α(a) ∪α(b) = α(a∨b) (4) α(a) = ∅ if and only if a ∈ e (5) α(a) = specei (r) if and only if a ∈m. from the above result, it can be easily observed that the collection {α(a)|a ∈ r} forms a base for a topology on specei (r). the topology generated by this base is precisely {α(a | a ⊆ r} and is called the hull-kernel topology on specei (r). under this topology, we have the following result. theorem 4.1. in an adl r, we have the following: (1) for any a ∈ r,α(a) is compact in specei (r) (2) if c is a compact open subset of specei (r), then c = α(a) for some a ∈ r (3) specei (r) is a t0-space (4) the map a 7→ α(a) is an epimorphism from r onto the lattice of all compact open subsets of specei (r). 20 int. j. anal. appl. (2023), 21:85 proof. (1) let a ∈ r. let x ⊆ r be such that α(a) ⊆ ⋃ x∈x α(x). let j be a e-ideal generated by the set x. suppose a /∈ j. then there exists a prime e-ideal p such that j ⊆ p and a /∈ p. since x ⊆ j ⊆ p, we get p /∈ α(x) for all x ∈ x. since a /∈ p, we get p ∈ α(a), which is a contradiction. hence a ∈ j. so we can write a = ( n∨ i=1 xi ) ∧a for some x1,x2, . . . ,xn ∈ x and n ∈ n. then, we get α(a) = α(( n∨ i=1 xi ) ∧ a) ⊆ α( n∨ i=1 xi ) = n⋃ i=1 α(xi ) which is finite subcover for α(a). therefore α(a) is compact. (2) let c be a compact open subset of specei (r). since c is open, we get c = ⋃ x∈x α(x) for some x ⊆ r. since c is compact, there exist x1,x2, . . . ,xn ∈ x such that c = n⋃ i=1 α(xi ) = α( n∨ i=1 ) therefore c = α(x) for some x ∈ r. (3) let p and q be two distinct prime e-ideals of r. without loss of generality, assume that p * q. choose x ∈ r such that x ∈ p and x /∈ q. hence p /∈ α(x) and q ∈ α(x). therefore specei (r) is a t0-space. (4) it can be obtained from (1), (2) and by the above lemma. � proposition 4.1. in an adl r, the following are equivalent: (1) specei (r) is a hausdorff space (2) for each p ∈ specei (r),p is the unique member of spec e i (r) such that o e(p ) ⊆ p (3) every prime e-ideal is minimal (4) every prime e-ideal is maximal. proof. (1) ⇒ (2) assume (1). let p ∈ specei (r). clearly o e(p ) ⊆ p. suppose q ∈ specei (r) such that q 6= p and oe(p ) ⊆ q. since specie(r) is hausdorff, there exists a,b ∈ r such that p ∈ α(a),q ∈ α(b) and α(a∧b) = α(a) ∩α(b) = ∅. hence a /∈ p,b /∈ q and a∧b ∈ e. therefore b ∈ oe(p ) ⊆ q, which is a contradiction to that b /∈ q. hence p = q. therefore p is the unique member of specei (r) such that o e(p ) ⊆ p. (2) ⇒ (3) assume (2). let p be a prime e-ideal of r. let q be a prime e-ideal in r such that q ⊆ p. hence oe(q) ⊆ q ⊆ p. therefore p is a minimal prime e-ideal of r. (3) ⇒ (4) it is clear. (4) ⇒ (1) assume (4). let p and q be two distinct elements of specei (r). hence o e(q) * p. choose a ∈oe(q) such that a /∈ p. since a ∈oe(q), there exists b /∈ q such that a ∈ (b,e). hence a∧b ∈ e. thus it yields, p ∈ α(a),q ∈ α(b). since a∧b ∈ e, we get that α(a)∩α(b) = α(a∧b) = ∅. therefore specei (r) is hausdorff. � theorem 4.2. for any e-ideal g of an adl r, (g,e) = ⋂ {p ∈ specei (r) | g * p}. proof. let g be an e-ideal of l. consider k = ⋂ {p ∈ specei (r) | g * p}. let p ∈ α(g). then g * p. since g ∩ (g,e) = e ⊆ p and p is prime, we get (g,e) ⊆ p. hence every prime e-ideal int. j. anal. appl. (2023), 21:85 21 p of r such that g * p contains (g,e). therefore (g,e) ⊆ k. let x /∈ (g,e). then there exists y ∈ g such that x ∧ y /∈ e. let k = {g | g is an e − ideal of l and x ∧ y /∈ g}. clearly, e ∈ k and so p = ∅. clearly, (k,⊆) is a partially ordered set and it satisfies the hypothesis of the zorn’s lemma, k has a maximal element, say n. then n is an e-ideal of r and x ∧y /∈ n. therefore x /∈ n and y /∈ n. since y ∈ g, we get g * n. we now show that n is prime. let a,b ∈ r with a /∈ n and b /∈ n. then n ( n ∨ (a)e and n ( n ∨ (b)e. by the maximality of n, we get x ∧y ∈ n ∨ (a)e and x ∧y ∈ n ∨ (b)e. hence, x ∧y ∈{n ∨ (a)e}∩{n ∨ (b)e} = n ∨{(a)e ∩ (b)e} = n ∨ (a∧b)e. if a∧b ∈ n, then x ∧y ∈ n which is a contradiction. thus n is a prime e-ideal of r such that g * n and x /∈ n. therefore x /∈ k. hence k ⊆ (g,e). � corollary 4.1. for any adl r and a ∈ r, (a,e) = ⋂ {p ∈ specei (r) | a /∈ p}. let minei (r) denote the set of all minimal prime e-ideals of adl r. for any a ∈ r, write αm(x) = α(x) ∩minei (r). theorem 4.3. for any adl r, the following conditions hold in r : (1) every prime e-ideals contains a minimal prime e-ideal (2) ⋂ p∈mine i (r) p = e (3) for any subset a with e ⊆ a, (a,e) = ⋂ p∈αm(a) (p ). proof. (1) let p be a prime e-ideal of r. consider x = {n ∈ specei (r) | n ⊆ p}. clearly x is a partially ordered set under set inclusion and hence it satisfies the hypothesis of the zorn’s lemma, x has a minimal element say m. clearly m will be the required minimal prime e-ideal of r. (2) since e is contained in every minimal prime e-ideal of r and so contained in the intersection of all minimal prime e-ideals. let x /∈ e. then there exists a prime e-ideal p of l such that x /∈ p. by (1), there exists a minimal prime e-ideal of r such that m ⊆ p. since x /∈ p, we get x /∈ m. that implies m is a minimal prime e-ideal of r such that x /∈ m. hence x is not in the intersection of all minimal prime. thus intersection of all minimal prime e-ideals of r is equal to e. (3) let p ∈ minei (r) such that a * p. choose x ∈ a such that x /∈ p. then (a,e) ⊆ (x,e) ⊆ p. that implies (a,e) is contained in every minimal prime e-ideal of r such that a * p. hence (a,e) ⊆ ⋂ p∈αm(a) (p ). let x /∈ (a,e). then x ∧ y /∈ e, for some y ∈ a. by the condition (2), there exists a minimal prime e-ideal p of r such that x ∧y /∈ p. that implies x /∈ p and y /∈ p. therefore x /∈ ⋂ p∈αm(a) p and hence (a,e) = ⋂ p∈αm(a) p. � lemma 4.2. for any a,b ∈ r, we have following: (1) (a,e) ⊆ (b,e) if and only if αm(b) ⊆ αm(a) (2) αm(a) = ∅ if and only if a ∈ e (3) αm(a) = minei (r) if and only if (a,e) = e. 22 int. j. anal. appl. (2023), 21:85 proof. (1) let a,b ∈ r. assume that (a,e) ⊆ (b,e). let p ∈ αm(b) then b /∈ p. that implies (a,e) ⊆ (b,e) ⊆ p. therefore a /∈ p and hence p ∈ αm(a). thus αm(b) ⊆ αm(a). conversely, assume that αm(b) ⊆ αm(a). now, (a,e) = ⋂ p∈αm(a) p ⊆ ⋂ p∈αm(b) p = (b,e). hence (a,e) ⊆ (b,e). (2) suppose minei (r) = ∅. then a ∈ p for all p ∈ min e i (r). that implies a ∈ ⋂ p∈mine i (r) p. since a ∈ ⋂ p∈mine i (r) p = e, we get a ∈ e. the converse is clear. (3) assume αm(a) = minei (r). then (a,e) = ⋂ p∈αm(a) p = ⋂ p∈mine i (r) p = e. therefore(a,e) = e. conversely, assume (a,e) = e. then (a,e) = e ⊆ p. that implies a /∈ p, for all p ∈ minei (r). therefore αm(a) = minei (r). � for any e-ideal g of an adl r, define βm(g) = {p ∈ minei (r) | g ⊆ p}. lemma 4.3. let g be an e-ideal of an adl r. if βm(g) = ∅, then (g,e) = e. proof. let βm(g) = ∅. then βm(g) = minei (r). that implies (g,e) = ⋂ p∈αm(f) p ⊆ ⋂ p∈mine i (r) p = e. therefore (g,e) = e. � for any adl r, define k = {x ∈ r | (x,e) = e}. lemma 4.4. for any adl r,k is a filter of r. proof. clearly, we have that for any m ∈m, m ∈ k. let x,y ∈ k. then ((x∧y,e),e) = ((x,e),e)∩ ((y,e),e) = (e,e) ∩ (e,e) = r ∩ r = r. that implies ((x ∧ y),e) = (r,e) = e. therefore x∧y ∈ k. let x ∈ k. then (x,e) = e. let y ∈ r. now, (x∨y,e) = (x,e)∩(y,e) = e∩(y,e) = e. therefore x ∨y ∈ k. hence k is a filter of r. � theorem 4.4. let g be an e-ideal of an adl r. then minei (r) is compact if and only if βm(g) = ∅ implies g ∩k 6= ∅. proof. assume that minei (r) is compact. let g be an e-ideal r such that βm(g) = ∅. then αm(g) = min e i (r). since min e i (r) is compact, there exists a ∈ g such that αm(a) = min e i (r). that implies (a,e) = e. therefore a ∈ k and hence g ∩ k 6= ∅. conversely, assume that for any e-ideal g of r,βm(g) = ∅ implies g ∩ k 6= ∅. let a ⊆ r be such that minei (r) = ⋃ a∈a αm(a) = αm(a) = αm(g) where g = ae. since minei (r) = αm(g), we get βm(g) = ∅. by the assumption, we get g ∩e 6= ∅. choose d ∈ g ∩k. since d ∈ g and g = ae, there exists a1,a2, . . . ,an ∈ a such that d = (a1∨a2∨ . . .∨an)∧d. since d ∈ e, minei (r) = αm(d) ⊆ αm ( n∨ i=1 ai ) = n⋃ i=1 αm(ai ). hence minei (r) is compact. � theorem 4.5. let r be an adl. for any y ⊆ minei (r), the closure of y in min e i (r) is βm( ⋂ p∈y p ) and, in particular, αm(f ) = βm((g,e)), for any e ⊆ g ⊆ r. int. j. anal. appl. (2023), 21:85 23 proof. let y ⊆ minei (r). then y in min e i (r) = {y in spec e i (r)} ∩ min e i (r) = h( ⋂ p∈y p ) ∩ minei (r) = βm( ⋂ p∈y p ). in particular, for any e ⊆ g ⊆ r, we have αm(g) = βm( ⋂ p∈αm(g) p ) = βm( ⋂ i*p, p∈mine i (r) p ) = βm((f,e)). � proposition 4.2. let f,g be two e-ideals of an adl r. then the following are equivalent: (1) g ⊆ (f,e) (2) g ∩f = e (3) αm(g) ∩αm(f ) = ∅. proof. (1) ⇒ (2) assume that g ⊆ (f,e). then g ∩f ⊆ (f,e) ∩f = e. therefore g ∩f = e. (2) ⇒ (3) assume that g ∩f = e. let p ∈ αm(g) ∩αm(f ) = αm(g ∩f ). then e = g ∩f * p, which is a contradiction. therefore αm(g) ∩αm(f ) = ∅. (3) ⇒ (1) assume that αm(g) ∩ αm(f ) = ∅. let x ∈ g. suppose x /∈ (f,e). then there exists y ∈ f such that x ∧ y /∈ e. then there exists p ∈ minei (r) such that x ∧ y /∈ p. that implies x /∈ p and y /∈ p. hence g * p and f * p. therefore p ∈ αm(g) and p ∈ αm(f ). therefore p ∈ αm(g) ∩αm(f ), which is a contradiction. so x ∈ (f,e). therefore g ⊆ (f,e). � corollary 4.2. let g be an e-ideal of an adl r and x ∈ r. then x ∈ (g,e) if and only if αm(x) ∩αm(g) = ∅. proof. by taking g = {x}, in the above proposition. � theorem 4.6. every open subset of minei (r) is closed if and only if for any e-ideal of r, (g,e) = e implies βm(g) = ∅. proof. assume that every open set of minei (r) is closed. let g be an e-ideal of r. then βm(g) is an open set in minei (r). now, βm(g) 6= ∅. then there exists x ∈ r \e such that αm(x) ⊆ βm(g). that implies αm(x) ∩ αm(g) = ∅. therefore x ∈ (g,e) and x /∈ e. hence (g,e) 6= e. thus (g,e) = e, which gives βm(g) = ∅. conversely, assume that the condition holds. let h be an open subset of minei (r). then h = αm(g), for some e-ideal g of l. by theorem 4.5, we have αm(g) = βm((g,e)). it is enough to show that βm((g,e)) = αm(g). since ((g ∨ (g,e)),e) = e, by the assumption, we get βm(g ∨ (g,e)) = ∅. now, for any p ∈ minei (r), we have p ∈ αm(g) ⇔ g * p ⇔ (g,e) ⊆ p ⇔ p ∈ βm(g). hence αm(g) = βm(g). therefore h is closed in minei (r). � theorem 4.7. in an adl r, minei (r) is a hausdorff space. proof. let p and q be distinct elements of minei (r). then there exists a ∈ p such that a /∈ q. since p is minimal, we get (a,e) * p. then there exists b ∈ (a,e) such that b /∈ p. that implies a∧b ∈ e and hence αm(a)∩αm(b) = ∅. since a /∈ q and b /∈ p, we get q ∈ αm(a) and p ∈ αm(b). therefore minei (r) is a hausdorff space. � 24 int. j. anal. appl. (2023), 21:85 acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] g. birkhoff, lattice theory, colloq. publ. xxv, amer. math. soc. providence, 1967. [2] w.h. cornish, normal lattices, j. aust. math. soc. 14 (1972), 200-215. https://doi.org/10.1017/ s1446788700010041. [3] g. grätzer, general lattice theory, birkhäuser, basel, 1978. https://doi.org/10.1007/978-3-0348-7633-9. [4] a.p. phaneendra kumar, m. sambasiva rao, k. sobhan babu, generalized prime d-filters of distributive lattices, arch. math. 57 (2021), 157-174. https://doi.org/10.5817/am2021-3-157. [5] g.c. rao, almost distributive lattices, doctoral thesis, department of mathematics, andhra university, visakhapatnam, 1980. [6] g.c. rao, s. ravi kumar, minimal prime ideals in an adl, int. j. contemp. math. sci. 4 (2009), 475-484 [7] g.c. rao, s. ravi kumar, normal almost distributive lattices, southeast asian bull. math. 32 (2008), 831-841. [8] g.c. rao, m. sambasiva rao, annulets in almost distributive lattices, eur. j. pure appl. math. 2 (2009), 58-72. [9] u.m. swamy, g.c. rao, almost distributive lattices, j. aust. math. soc. 31 (1981), 77-91. https://doi.org/10. 1017/s1446788700018498. https://doi.org/10.1017/s1446788700010041 https://doi.org/10.1017/s1446788700010041 https://doi.org/10.1007/978-3-0348-7633-9 https://doi.org/10.5817/am2021-3-157 https://doi.org/10.1017/s1446788700018498 https://doi.org/10.1017/s1446788700018498 1. introduction 2. preliminaries 3. e-ideals of adls 4. on the space prime e-ideals references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 108-118 http://www.etamaths.com positive solutions for a singular sum fractional differential system mehdi shabibi1, mihai postolache2,3,∗ and shahram rezapour2,4 abstract. we investigate the existence of positive solutions for a singular sum fractional differential system under some boundary conditions by providing different conditions. also, we give an example to illustrate one of our main results. 1. preliminaries it has been published many works on the existence of solutions for different singular fractional differential systems (see for example, [2], [3], [6], [7], [10] and [12]). in 2010, the existence of positive solutions for the singular dirichlet problem dαu(t) + f(t,u(t),dµu(t)) = 0 with boundary conditions u(0) = u(1) = 0 is investigated, where 1 < α < 2, 0 < µ ≤ α − 1, f is a caratheodory function on [0, 1] × (0,∞) ×r and dα is riemann-liouville fractional derivative ( [1]). in 2013, the singular problem dαu + f(t,u,dγu,dµu) + g(t,u,dγu,dµu) = 0 with boundary conditions u(0) = u′(0) = u′′(0) = u′′′(0) = 0 is reviewed, where 3 < α < 4, 0 < γ < 1, 1 < µ < 2, dα is the caputo fractional derivative and f is a caratheodory function on [0, 1]×(0,∞)3 ( [4]). by using main idea of [1] and [4], we investigate positive solutions for the singular differential system of equations  dα1u1 + f1(t, u1, . . . , um, d µ1u1, . . . , d µmum) + g1(t, u1, . . . , um, d µ1u1, . . . , d µmum) = 0, . . . . . . dαmum + fm(t, u1, . . . , um, d µ1u1, . . . , d µmum) + gm(t, u1, . . . , um, d µ1u1, . . . , d µmum) = 0, (1.1) with boundary conditions ui(0) = 0, u ′ i(1) = 0 and dk dtk [ui(t)]t=0 = 0 for 1 ≤ i ≤ m and 2 ≤ k ≤ n−1, where αi ≥ 2, [αi] = n − 1, 0 < µi < 1, d is the caputo fractional derivative, fi is a caratheodory function, gi satisfies lipschitz condition and fi(t,x1, . . . ,x2m) is singular at t = 0 of for all 1 ≤ i ≤ m. we say that a map f : [0, 1] × d ⊆ [0, 1] × d → rn is caratheodory whenever the function t 7→ f(t,x1, . . . ,xn) is measurable for all (x1, . . . ,xn) ∈ d and (x1, . . . ,xn) 7→ f(t,x1, . . . ,xn) is continuous for almost all t ∈ [0, 1] and for each compact k ⊆ d there exists ϕk ∈ l1[0, 1] such that |f(t,x1, . . . ,xn)| ≤ ϕk(t) for almost all t ∈ [0, 1] and (x1, . . . ,xn) ∈ k. put ‖x‖1 = ∫ 1 0 |x(t|dt, ‖x‖ = sup{|x(t)| : t ∈ [0, 1]}, ‖(x1, . . . ,xn)‖∗ = max{‖x1‖, . . . ,‖xn‖}, ‖(x1, . . . ,xn)‖∗∗ = max{‖x1‖, . . . ,‖xn‖,‖x′1‖, . . . ,‖x ′ n‖}, y = cr([0, 1]), x = c 1 r([0, 1]). by considering the problem (1.1), we assume the following hypotheses: h1: f1, . . . ,fm are caratheodory functions on [0, 1] × (0,∞)2m and there exists positive constants m1, . . . ,mm such that fi(t,x1, . . . ,x2m) ≥ mi received 24th july, 2016; accepted 17th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 34a08; 34b16. key words and phrases. caputo derivative; fixed point; singular system; positive solution. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 108 a singular sum fractional differential system 109 for almost all t ∈ [0, 1] and all (x1, . . . ,x2m) ∈ d := (0,∞)2m. h2: g1, . . . ,gm are nonnegative and |gi(t,x1, . . . ,x2m) −gi(t,y1, . . . ,y2m)| ≤ 2m∑ k=1 lik|xk −yk|, for almost all t ∈ [0, 1] and for all (x1, . . . ,x2m), (y1, . . . ,y2m) ∈ d, where l11, . . . ,lm1 , . . . ,l12m, . . . ,lm2m in [0,∞) are constants such that 1 γ(αi − 1) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) ) < 1. h3: there exist some maps γ1, . . . ,γm ∈ l1([0, 1]), some non-increasing maps p1, . . . ,pm ∈ cr(d) with ∫ 1 0 pi ( m1t α1, . . . ,mmt αm, m1(1 −µ1) 2 t1−µ1, . . . , mm(1 −µm) 2 t1−µm ) dt < ∞ and some functions h1, . . . ,hm ∈ cr([0,∞)2m) such that limx→∞ hi(x,...,x) x = 0, hi is nondecreasing in all components and fi(t,x1, . . . ,x2m) + gi(t,x1, . . . ,x2m) ≤ pi(x1, . . . ,x2m) + γi(t)hi(x1, . . . ,x2m), for almost all t ∈ [0, 1] and all (x1, . . . ,x2m) ∈ d, where mi = mi αi−1γ(αi+1) for all 1 ≤ i ≤ m. now for each 1 ≤ i ≤ m and n ≥ 1, put fi,n(t,x1, . . . ,x2m) = fi(t,χ1(x1), . . . ,χn(x2m)), where χn(u) = u whenever u ≥ 1n and χn(u) = u whenever u < 1 n . it is easy to check that fi,n(t,x1, . . . ,x2m) + gi(t,x1, . . . ,x2m) ≤ pi ( 1 n ,. . . , 1 n ) + γi(t)hi ( x1 + 1 n ,. . . ,x2m + 1 n ) , fi,n(t,x1, . . . ,x2m) ≥ mi and fi,n(t,x1, . . . ,x2m) + gi(t,x1, . . . ,x2m) ≤ pi(x1, . . . ,x2m) + γi(t)hi ( x1 + 1 n ,. . . ,x2m + 1 n ) , for all (x1, . . . ,xn) ∈ d, 1 ≤ i ≤ m and almost all t ∈ [0, 1]. first, we investigate the regular fractional differential system  dα1u1 + f1,n(t,u1, . . . ,um,d µ1u1, . . . ,d µmum) = 0 . . . . . . dαmum + fm,n(t,u1, . . . ,um,d µ1u1, . . . ,d µmum) = 0, (1.2) with same boundary conditions in (1.1). now, we present some necessary notions. according to [5], the riemann-liouville integral of order p for a function f : (0,∞) → r is ipf(t) = 1 γ(p) ∫ t 0 (t−s)p−1f(s)ds if the right-hand side map is defined pointwise on (0,∞) . the caputo fractional derivative of order α > 0 for a function f : (a,∞) → r is defined by cdαf(t) = 1 γ(n−α) ∫ t 0 fn(s) (t−s)α+1−n ds, where n = [α] + 1; please, see [5]. lemma 1.1 ( [8]). if x ∈ cr[0, 1]∩l1[0, 1], then iαdαx(t) = x(t) + ∑n−1 i=0 cit i for some real constants c0,c1, . . . ,cn−1, where 0 < n− 1 ≤ α < n. 110 shabibi, postolache and rezapour it has been proved in [11] that∫ t 0 (t−s)α−1sβds = b(β + 1,α)tα+β, b(β,α) = γ(α)γ(β) γ(α + β) , β > 0, α > −1. we need the next result. lemma 1.2 ( [9]). let m be a closed, convex and nonempty subset of a banach space x, a a compact and continuous operator and b a contraction. then, there exist z ∈ m such that z = az + bz. lemma 1.3. let y ∈ l1[0, 1], α ≥ 2 and n = [α] + 1. then, the unique solution of the equation dαu(t) + y(t) = 0 with boundary conditions u′(1) = u(0) = u′′(0) = · · · = un−1(0) = 0 is u(t) =∫ 1 0 gα(t,s)y(s)ds, where t ∈ [0, 1] and gα(t,s) =   t(1 −s)α−2 γ(α− 1) , 0 ≤ t ≤ s ≤ 1 t(1 −s)α−2 γ(α− 1) − (t−s)α−1 γ(α) , 0 ≤ s ≤ t ≤ 1. proof. by using lemma 1.1 and the boundary conditions, we get u(t) = − 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds + c1t and so u′(1) = − 1 γ(α−1) ∫ 1 0 (1 − s)α−2y(s)ds + c1. since u′(1) = 0, c1 = 1γ(α−1) ∫ 1 0 (1 − s)α−2y(s)ds. thus, u(t) = − 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds + t γ(α−1) ∫ 1 0 (1 −s)α−2y(s)ds. hence, we conclude that u(t) =∫ 1 0 gα(t,s)y(s)ds, where gα(t,s) = t(1−s)α−2 γ(α−1) − (t−s)α−1 γ(α) whenever 0 ≤ s ≤ t ≤ 1 and gα(t,s) = t(1−s)α−2 γ(α−1) whenever 0 ≤ t ≤ s ≤ 1. � consider the green function gα(t,s) as in lemma 1.3. if 0 < t ≤ s < 1, then it is clear that gα(t,s) > 0. if 0 < s < t < 1, then t(1−s)α−2 γ(α−1) − (t−s)α−1 γ(α) > 0 if and only if α− 1 > (t−s) α−1 t(1−s)α−2 and so gα(t,s) > 0. one can check that gα(t,s) > 0, gα(t,s) ≤ 1γ(α−1) and ∫ 1 0 gα(t,s)ds ≤ 1γ(α) , for all t,s ∈ (0, 1). also, ∫ 1 0 gα(t,s)ds ≥ tγ(α) − tα γ(α+1) ≥ t α(α−1) γ(α+1) and ∂ ∂t gα(t,s) > 0 for all t,s ∈ (0, 1). moreover, gα, ∂ ∂t gα ∈ cr([0, 1] × [0, 1]), ∂∂tgα(t,s) ≤ 1 γ(α−1) , for all t,s ∈ [0, 1] and ∫ 1 0 ∂ ∂t gα(t,s) ≥ 1−t α−1 γ(α) , for all t ∈ [0, 1]. suppose that x ∈ c1r[0, 1] and 0 ≤ µ ≤ 1. since d µx(t) = 1 γ(2−µ) ∫ t 0 (t−s)−µx′(s)ds for all 0 ≤ t ≤ 1, |dµx| ≤ ‖ x′‖ γ(1−µ) ∫ t 0 (t−s)−µds = ‖ x′‖ γ(2−µ)t 1−µ and so |dµx| ≤ ‖ x′‖ γ(2−µ) and d µx ∈ cr[0, 1]. now, put p = {(x1, . . . ,xm) ∈ xm : xi(t) ≥ 0 and x′i(t) ≥ 0 for all t ∈ [0, 1] and 0 ≤ i ≤ m}. for each n ≥ 1 and 0 ≤ i ≤ m, define the maps φi,n(x1, . . . ,xm)(t) = ∫ 1 0 gαi(t,s)fi,n(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds, and ψi(x1, . . . ,xm)(t) = ∫ 1 0 gαi(t,s)gi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds, tn(x1, . . . ,xm)(t) =  φ1,n(x1, . . . ,xm)(t)... φm,n(x1, . . . ,xm)(t)   and ψ(x1, . . . ,xm)(t) =  ψ1(x1, . . . ,xm)(t)... ψm(x1, . . . ,xm)(t)   for all (x1, . . . ,xm) ∈ p . a singular sum fractional differential system 111 lemma 1.4. the map ψ : p → p is a contraction. proof. it is easy to check that ψi(x1, . . . ,xm)(t) ≥ 0 and ψ′i(x1, . . . ,xm)(t) = ∫ 1 0 ∂ ∂t gαi(t,s)gi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds ≥ 0 for all t ∈ [0, 1], (x1, . . . ,xm) ∈ p and 0 ≤ i ≤ m. note that, ‖ψi(x1, . . . ,xm) − ψi(y1, . . . ,ym)‖ = sup t∈[0,1] ∣∣∣∫ 1 0 gαi(t,s)[gi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s)) −gi(s,y1(s), . . . ,ym(s),dµ1y1(s), . . . ,dµmym(s))]ds ∣∣∣ ≤ ∣∣∣∫ 1 0 gαi(t,s)ds ∣∣∣(li1‖x1 −y1‖ + · · · + lim‖xm −ym‖ +lim+1‖d µ1x1 −dµ1y1‖ + · · · + li2m‖d µmxm −dµmym‖) ≤ 1 γ(αi) (li1‖x1 −y1‖ + · · · + l i m‖xm −ym‖ + lim+1 γ(2 −µ1) ‖x′1 −y ′ 1‖ + · · · + li2m γ(2 −µm) ‖x′m −y ′ m‖) ≤ 1 γ(αi) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) ) ‖(x1, . . . ,xm) − (y1, . . . ,ym)‖∗∗ ≤ 1 γ(αi − 1) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) ) ‖(x1, . . . ,xm) − (y1, . . . ,ym)‖∗∗ for all 0 ≤ i ≤ m and so ‖ψ(x1, . . . ,xm) − ψ(y1, . . . ,ym)‖∗ = max 1≤i≤m ‖ψi(x1, . . . ,xm) − ψi(y1, . . . ,ym)‖ ≤ max 1≤i≤m { 1 γ(αi − 1) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) )} ‖(x1, . . . ,xm) − (y1, . . . ,ym)‖∗∗ . similarly, one can show that ‖ψ′(x1, . . . ,xm) − ψ′(y1, . . . ,ym) ‖∗ ≤ max 1≤i≤m { 1 γ(αi − 1) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) )} ‖(x1, . . . ,xm) − (y1, . . . ,ym) ‖∗∗. thus, we get ‖ψ(x1, . . . ,xm) − ψ(y1, . . . ,ym) ‖∗∗ ≤ max 1≤i≤m { 1 γ(αi − 1) ( m∑ k=1 lik + m∑ k=1 lim+k γ(2 −µi) )} ‖(x1, . . . ,xm) − (y1, . . . ,ym)‖∗∗ . since max1≤i≤m { 1 γ(αi−1) (∑m k=1 l i k + ∑m k=1 lim+k γ(2−µi) )} < 1, ψ is a contraction mapping. � lemma 1.5. for each n ≥ 1, tn is a completely continuous operator on p. 112 shabibi, postolache and rezapour proof. let n ≥ 1, (x1, . . . ,xm) ∈ p and 1 ≤ i ≤ m. choose a positive constant mi such that fi,n(t,x1(t), . . . ,xm(t),d µ1x1(t), . . . ,d µmxm(t)) ≥ mi for almost all t ∈ [0, 1]. since gαi and ∂ ∂t gαi are nonnegative and continuous on [0, 1]× [0, 1] for all 1 ≤ i ≤ m, we conclude that φi,n(x1, . . . ,xm)(t) ≥ 0 and (φi,n(x1, . . . ,xm)) ′ (t) ≥ 0 for all t ∈ [0, 1] and 1 ≤ i ≤ m. hence, tn maps p into p . now, we prove that tn is continuous. let {(x1,k, . . . ,xm,k)} be a convergent sequence in p with limk→∞(x1,k, . . . ,xm,k) = (x1, . . . ,xm). in this case, we get limk→∞xi,k = xi and limk→∞x ′ i,k = x ′ i uniformly on [0, 1] (i = 1, 2, . . . ,m). but, |dµixi,k(t) − dµixi(t)| ≤ ‖x′i,k−x′i‖ γ(2−µi) for all t ∈ [0, 1] and 1 ≤ i ≤ m. thus, we conclude that limk→∞d µixi,k(t) = d µixi(t) uniformly on [0, 1]. hence, lim k→∞ fi,n(t,x1,k(t), . . . ,xm,k(t),d µ1x1,k(t), . . . ,d µmxm,k(t)) = fi,n(t,x1(t), . . . ,xm(t),d µ1x1(t), . . . ,d µmxm(t)). since fi,n ∈ car([0, 1] × r2m), {(x1,k, . . . ,xm,k)} is bounded in xm there exist a map ϕi ∈ l1[0, 1] such that mi ≤ fi,n(t,x1,k(t), . . . ,xm,k(t),dµ1x1,k(t), . . . ,dµmxm,k(t)) ≤ ϕi(t) (1.3) for almost all t ∈ [0, 1], 1 ≤ i ≤ m and k ≥ 1. by using the lebesgue dominated convergence theorem, we conclude that |φi,n(x1,k, . . . ,xm,k)(t) − φi,n(x1, . . . ,xm)(t)| ≤ 1 γ(αi) ∫ 1 0 |fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s)) −fi,n(s,x1(s), . . . ,xm(s),dµ1x1(s), . . . ,dµmxm(s))|ds, and |(φi,n(x1,k, . . . ,xm,k) )′(t) − (φi,n(x1, . . . ,xm) )′(t) | ≤ 1 γ(αi − 1) ∫ 1 0 |fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s)) −fi,n(s,x1(s), . . . ,xm(s),dµ1x1(s), . . . ,dµmxm(s))|ds. hence, limk→∞ ∣∣∣(φi,n(x1,k, . . . ,xm,k))j (t) − (φi,n(x1, . . . ,xm))j (t)∣∣∣ = 0 uniformly on [0, 1] for j = 0, 1. thus, ‖tn(x1,k, . . . ,xm,k)(t) −tn(x1, . . . ,xm)(t)‖∗∗ → 0 and so tn is continuous. now, we prove that tn maps bounded sets to relatively compact subsets. let {(x1,k, . . . ,xm,k)} be a bounded sequence in p. choose a positive number s such that ‖xi,k‖≤ s and ∥∥∥x′i,k∥∥∥ ≤ s for all 1 ≤ i ≤ m and for k ≥ 1. since ‖dµixi,k‖ ≤ 1γ(2−µi) for all 1 ≤ i ≤ m, there exist a map ϕi ∈ l1[0, 1] such that (1.3) holds for almost all t ∈ [0, 1], 1 ≤ i ≤ m and k ≥ 1. note that 0 ≤ φi,n(x1,k, . . . ,xm,k)(t) = ∫ 1 0 gαi(t,s)fi,n(s,x1,k(s), . . . ,xm,k(s),d µ1x1,k(s), . . . ,d µmxm,k(s))ds ≤ 1 γ(αi) ∫ 1 0 ϕi(s)ds = ‖ϕi‖1 γ(αi) and 0 ≤ (φi,n(x1,k, . . . ,xm,k))′(t) = ∫ 1 0 ∂ ∂t gαi(t,s)fi,n(s,x1,k(s), . . . ,xm,k(s),d µ1x1,k(s), . . . ,d µmxm,k(s))ds ≤ 1 γ(αi − 1) ∫ 1 0 ϕi(s)ds = ‖ϕi‖1 γ(αi − 1) a singular sum fractional differential system 113 for all 1 ≤ i ≤ m. thus, ‖tn(x1,k, . . . ,xm,k)(t)‖∗∗ ≤ b, where b = max1≤i≤m ‖ϕi‖1 γ(αi−1) . this implies that {tn(x1,k, . . . ,xm,k)} is bounded in xm. let 0 ≤ t1 ≤ t2 ≤ 1 and 1 ≤ i ≤ m. then, we have∣∣(φi,n(x1,k, . . . ,xm,k))′ (t2) − (φi,n(x1,k, . . . ,xm,k))′ (t1)∣∣ ≤ t2 − t1 γ(αi − 1) ∫ 1 0 (1 −s)αi−2fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s))ds + 1 γ(αi) ∣∣∣∫ t2 0 (t2 −s)αi−1fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s))ds − ∫ t1 0 (t1 −s)αi−1fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s))ds ∣∣∣ ‖fi,n‖1 γ(αi − 1) (t2 − t1) + 1 γ(αi) [ ∫ t1 0 ((t2 −s)αi−1 − (t1 −s)αi−1)× fi,n(s,x1,k(s), . . . ,xm,k(s),d µ1x1,k(s), . . . ,d µmxm,k(s))ds + ∫ t2 t1 (t2 −s)αi−2fi,n(s,x1,k(s), . . . ,xm,k(s),dµ1x1,k(s), . . . ,dµmxm,k(s))ds] ≤ ‖ϕi‖1 γ(αi − 1) (t2 − t1) + 1 γ(αi) [∫ t1 0 ( (t2 −s)αi−1 − (t1 −s)αi−1 ) ϕi(s)ds + (t2 − t1)αi−1 ‖ϕi‖1 ] . let ε > 0 be given. since the function |t − s|αi−1 is uniformly continuous on [0, 1] × [0, 1], there exist δ > 0 such that (t2 − s)αi−1 − (t1 − s)αi−1 < ε for all 0 ≤ t1 ≤ t2 ≤ 1 with t2 − t1 < δ and 0 ≤ s ≤ t1. if 0 ≤ t1 ≤ t2 ≤ 1 with t2 − t1 < min{δ,ε}, then we have∣∣(φi,n(x1,k, . . . ,xm,k))′ (t2) − (φi,n(x1,k, . . . ,xm,k))′ (t1)∣∣ < 3ε‖ϕi‖1 γ(αi) . thus, ‖t ′n(x1,k, . . . ,xm,k)(t2) −t ′ n(x1,k, . . . ,xm,k)(t1)‖∗ < max1≤i≤m 3ε‖ϕi‖1 γ(αi) . this implies that {t ′n(x1,k, . . . ,xm,k)} is equi-continuous on [0, 1]. now by using the arzela-ascoli theorem, {tn(x1,k, . . . ,xm,k)} is relatively compact and so tn is completely continuous. � 2. main results now, we are ready to provide our main results about the problem (1.1). theorem 2.1. assume that hypotheses h1 and h2 hold. then, the problem (1.2) with the boundary conditions in (1.1) has a solution (x1,n, . . . ,xm,n) in p such that xi,n(t) ≥ mit αi(αi−1) γ(αi+1) , for all t ∈ [0, 1] and 1 ≤ i ≤ m. proof. by using lemma 1.4, the mapping ψ : p → p is a contraction. also by using lemma 1.5, the operator tn : p → p is a completely continuous one. now by using lemma 1.2, there exists (x1,n, . . . ,xm,n) ∈ p such that (x1,n, . . . ,xm,n) = tn(x1,n, . . . ,xm,n) + ψ(x1,n, . . . ,xm,n). thus, xi,n = φi,n(x1,n, . . . ,xm,n) + ψi(x1,n, . . . ,xm,n) for all 1 ≤ i ≤ m. hence, xi,n(t) = ∫ 1 0 gαi(t,s)fi,n(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds + ∫ 1 0 gαi(t,s)gi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds for all 1 ≤ i ≤ m. by using the assumptions, we get xi,n(t) ≥ mit αi(αi−1) γ(αi+1) for all t ∈ [0, 1] and 1 ≤ i ≤ m. one can check that the element (x1,n, . . . ,xm,n) ∈ p is a solution for the problem (1.2) with the boundary conditions in (1.1). � lemma 2.1. assume that hypotheses h1, h2 and h3 hold. if (x1,n, . . . ,xm,n) is a solution for the problem (1.2) with the boundary conditions in (1.1), then {(x1,n, . . . ,xm,n)}n≥1 is relatively compact in p. 114 shabibi, postolache and rezapour proof. as we found in the last result, xi,n(t) = ∫ 1 0 gαi(t,s)fi,n(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))ds + ∫ 1 0 gαi(t,s)gi(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))ds for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m. thus, x′i,n(t) ≥ mi ∫ 1 0 ∂ ∂t gαi(t,s)ds ≥ mi(1 − tαi−1) γ(αi) , for all t ∈ [0, 1]. hence, dµixi,n(t) = 1 γ(1 −µi) ∫ t 0 (t−s)−µix′i,n(s)ds ≥ mi γ(αi)γ(1 −µi) ∫ t 0 (t−s)−µi(1 −sαi−1)ds > mi γ(αi)γ(1 −µi) ∫ t 0 (t−s)−µi(1 −s)ds for all t ∈ [0, 1]. thus, dµixi,n(t) > mit 1−µi γ(αi)γ(2 −µi) − mit 2−µi γ(αi)γ(3 −µi) = mit 1−µi γ(αi) ( γ(3 −µi) − tγ(2 −µi) γ(2 −µi)γ(3 −µi) ) = mit 1−µi γ(αi) ( 2 −µi − t γ(3 −µi) ) ≥ mit 1−µi(1 −µi) γ(αi)γ(3 −µi) for all t ∈ [0, 1]. since γ(3 −µi) ≤ 2, we get dµixi,n(t) ≥ mit 1−µi(1−µi) 2γ(αi) . now, put mi = mi min { 1 γ(αi) , αi − 1 γ(αi + 1) } . then, xi,n(t) ≥ mitαi and dµixi,n(t) ≥ mi(1−µi) 2 t1−µi for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m. hence, pi(x1,n(t), . . . ,xm,n(t),d µ1x1,n(t), . . . ,d µmxm,n(t)) ≤ pi ( m1t α1, . . . ,mmt αm, m1(1 −µ1) 2 t1−µ1 . . . , mm(1 −µm) 2 t1−µm ) for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m. this implies that 0 ≤ x′i,n(t) = ∫ 1 0 ∂ ∂t gαi(t,s)fi,n(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))ds + ∫ 1 0 ∂ ∂t gαi(t,s)gi(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))ds ≤ 1 γ(αi − 1) ∫ 1 0 pi ( m1s α1, . . . ,mms αm, m1(1 −µ1) 2 s1−µ1 . . . , mm(1 −µm) 2 s1−µm ) ds + 1 γ(αi − 1) ∫ 1 0 γi(s)hi(x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))ds for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m. also, we have∫ 1 0 pi ( m1s α1, . . . ,mms αm, m1(1 −µ1) 2 s1−µ1, . . . , mm(1 −µm) 2 s1−µm ) ds := λi < ∞ for all 1 ≤ i ≤ m. if ηn = ‖(x1,n, . . . ,xm,n)‖∗∗, then ‖xi,n‖ ≤ ηn and ∥∥x′i,n∥∥ ≤ ηn for all i and n. thus, |dµixi,n(t)| ≤ ηnγ(2−µi) for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m and so 0 ≤ x′i,n(t) ≤ 1 γ(αi − 1) ( λi + hi(1 + ηn, . . . , 1 + ηn, 1 + ηn γ(2 −µ1) , . . . , 1 + ηn γ(2 −µm) ) ) ‖γi‖1 and so 0 ≤ xi,n(t) = ∫ t 0 x′i,n(s)ds for all n ≥ 1, t ∈ [0, 1] and 1 ≤ i ≤ m. a singular sum fractional differential system 115 similarly, we obtain 0 ≤ xi,n(t) ≤ 1 γ(αi − 1) ( λi + hi(1 + ηn, . . . , 1 + ηn, 1 + ηn γ(2 −µ1) , . . . , 1 + ηn γ(2 −µm) ) ) ‖γi‖1 and ηn ≤ 1γ(αi−1) ( λi + hi(1 + ηn, . . . , 1 + ηn, 1 + ηn γ(2−µ1) , . . . , 1 + ηn γ(2−µm) ) ) ‖γi‖1 for all i. since limx→∞ hi(x,...,x) x = 0 for all 1 ≤ i ≤ m, there exists li > 0 such that 1 γ(αi − 1) ( λi + hi(1 + νi, . . . , 1 + νi, 1 + νi γ(2 −µ1) , . . . , 1 + νi γ(2 −µm) ) ) ‖γi‖1 < νi for all νi > li. if l = max{l1, . . . ,lm}, then 1 γ(αi − 1) ( λi + hi(1 + ν, . . . , 1 + ν, 1 + ν γ(2 −µ1) , . . . , 1 + ν γ(2 −µm) ) ) ‖γi‖1 < ν for all ν > l. thus, ηn = ‖(x1,n, . . . ,xm,n)‖∗∗ = max1≤i≤m{‖xi,n‖ , ∥∥x′i,n∥∥} < l which implies {‖(x1,n, . . . ,xm,n)‖∗∗} is bounded in xm. now, put bi := hi ( 1 + l,. . . , 1 + l, 1 + l γ(2 −µ1) , . . . , 1 + l γ(2 −µm) ) and fi(t) := pi ( m1t α1, . . . ,mmt αm, m1(1 −µ1) 2 t1−µ1 . . . , mm(1 −µm) 2 t1−µm ) , for all i and almost all t ∈ [0, 1]. then, we have λi = ∫ 1 0 fi(t)dt and fi,n(t,x1,n(t), . . . ,xm,n(t),d µ1x1,n(t), . . . ,d µmxm,n(t)) +gi(t,x1,n(t), . . . ,xm,n(t),d µ1,nx1(t), . . . ,d µmxm,n(t)) ≤ fi(t) + biγi(t). if 0 ≤ t1 ≤ t2 ≤ 1, then |x′i,n(t2) −x ′ i,n(t1)| = ∣∣∣∫ 1 0 ( ∂ ∂t gαi(t2,s) − ∂ ∂t gαi(t2,s))× [fi,n(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s)) +gi(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))]ds ∣∣∣ ≤ 1 γ(αi − 1) [(t2 − t1) ∫ 1 0 fi(s) + biγi(s)ds + ∫ t1 0 ((t2 −s)αi−2 − (t1 −s)αi−2)(fi(s) + biγi(s) )ds + ∫ t2 t1 (t2 −s)αi−2 (fi(s) + biγi(s)) ds] ≤ 1 γ(αi − 1) [(t2 − t1)(λi + bi‖γi‖1) + ∫ t1 0 ((t2 −s)αi−2 − (t1 −s)αi−2)(fi(s) + biγi(s))ds +(t2 − t1)αi−2(λi + bi‖γi‖1)]. let �i > 0 be given. choose δ(�i) > 0 such that (t2−s)αi−2−(t1−s)αi−2 < �i for all 0 ≤ t1 < t2 ≤ 1 with t2 − t1 < δ(�i) and 0 ≤ s ≤ t. if we put 0 < δ < min{δ(�1), . . . ,δ(�m), α1−2 √ �1, . . . , αm−2 √ �m}, then |x′i,n(t2) − x ′ i,n(t1)| ≤ 3 �i γ(αi−1) (λi + bi‖γi‖1) for all 1 ≤ i ≤ m. hence, {(x1,n, . . . ,xm,n) ′} is equi-continuous and so {(x1,n, . . . ,xm,n)}n≥1 is relatively compact in xm. � theorem 2.2. assume that hypotheses h1, h2 and h3 hold. then the system (1.1) has a solution (x1, . . . ,xm) in p such that d µixi(t) ≥ mi(1−µi) 2 t1−µi and xi(t) ≥ mitαi for all t ∈ [0, 1] and 1 ≤ i ≤ m. 116 shabibi, postolache and rezapour proof. by theorem 2.1, for each n ≥ 1 the system (1.2) with the boundary conditions in (1.1) has a solution (x1,n, . . . ,xm,n) ∈ p . by lemma 2.1, {(x1,n, . . . ,xm,n)}n≥1 is relatively compact in xm. by using the arzela-ascoli theorem, there exists (x1, . . . ,xm) such that limn→∞(x1,n, . . . ,xm,n) = (x1, . . . ,xm). it is obvious that (x1, . . . ,xm) satisfies the boundary conditions of the problem (1.1), dµixi,n → dµixi and lim n→∞ fi,n(t,x1,n(t), . . . ,xm,n(t),d µ1x1,n(t), . . . ,d µmxm,n(t)) +gi(t,x1,n(t), . . . ,xm,n(t),d µ1x1,n(t), . . . ,d µmxm,n(t)) = fi(t,x1(t), . . . ,xm(t),d µ1x1(t), . . . ,d µmxm(t)) +gi(t,x1(t), . . . ,xm(t),d µ1x1(t), . . . ,d µmxm(t)) for almost all t ∈ [0, 1] and 1 ≤ i ≤ m and so (x1, . . . ,xm) ∈ p . now, suppose that k := supn≥1 ‖(x1,n, . . . ,xm,n)‖∗∗. then, we have ‖d µixi,n‖≤ kγ(2−µi) for all n and 1 ≤ i ≤ m. hence, 0 ≤ gαi(t,s)[fi,n(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s)) +gi(s,x1,n(s), . . . ,xm,n(s),d µ1x1,n(s), . . . ,d µmxm,n(s))] ≤ 1 γ(αi − 1) ( fi(s) + hi(1 + k,.. . , 1 + k, 1 + k γ(2 −µi) , . . . , 1 + k γ(2 −µi) )γi(s) ) for almost all (t,s) ∈ [0, 1] × [0, 1], n ≥ 1 and 1 ≤ i ≤ m. now by using the lebesgue dominated theorem, we conclude that xi(t) = ∫ 1 0 gαi(t,s)fi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds + ∫ 1 0 gαi(t,s)gi(s,x1(s), . . . ,xm(s),d µ1x1(s), . . . ,d µmxm(s))ds for all 1 ≤ i ≤ m and t ∈ [0, 1], and this completes the proof. � next example illustrates our last result. example 2.1. let us study the system  d 5 2 x1 + 1 t 2 3 (2 + a1x1 + a2x2 + a3d 1 3 x1 + a4d 1 2 x2) +(0.1e 1 1+x1 + 0.2e 1 1+x2 + 0.1e 1 1+d13x1 + 0.2e 1 1+d12x2 ) = 0 d 7 3 x2 + 1 t 1 2 (1 + b1x1 + b2x2 + b3d 1 3 x1 + b4d 1 2 x2) +(0.2e 1 1+x1 + 0.2e 1 1+x2 + 0.3e 1 1+d13x1 + 0.1e 1 1+d12x2 ) = 0 with boundary condition x1(0) = x2(0) = 0, x ′ 1(1) = x ′ 2(1) = 0 and x ′′ 1 (0) = x ′′ 2 (0) = 0, where a1,a2,a3,a4,b1,b2,b3 and b4 are positive constants. consider the functions f1(t,x1,x2,x3,x4) = 1 t 2 3 (2 + a1x1 + a2x2 + a3x3 + a4x4), f2(t,x1,x2,x3,x4) = 1 t 1 2 (1 + b1x1 + b2x2 + b3x3 + b4x4), g1(t,x1,x2,x3,x4) = p1(x1,x2,x3,x4) = 0.1e 1 1+x1 + 0.2e 1 1+x2 + 0.1e 1 1+x3 + 0.2e 1 1+x4 , g2(t,x1,x2,x3,x4) = p2(x1,x2,x3,x4) = 0.2e 1 1+x1 + 0.2e 1 1+x2 + 0.3e 1 1+x3 + 0.1e 1 1+x4 , h1(x1,x2,x3,x4) = 2 + a1x1 + a2x2 + a3x3 + a4x4, h2(x1,x2,x3,x4) = 1 + b1x1 + b2x2 + b3x3 + b4x4, λ1(t) = 1 t 2 3 and λ2(t) = 1 t 1 2 . put m = 2, α1 = 5 2 , α2 = 7 3 , µ1 = 1 2 , µ2 = 1 3 , l11 = 0.1, l 1 2 = 0.2, l13 = 0.1, l 1 4 = 0.2, l 2 1 = 0.2, l 2 2 = 0.2, l 2 3 = 0.3, l 2 4 = 0.1, m1 = 2 and m2 = 1. one can check that f1 and f2 are caratheodory functions, f1(t,x1,x2,x3,x4) ≥ 2, f2(t,x1,x2,x3,x4) ≥ 1 a singular sum fractional differential system 117 for all (x1,x2,x3,x4) ∈ (0,∞)4 and almost all t ∈ [0, 1], g1 and g2 are nonnegative, |g1(t,x1,x2,x3,x4) −g1(t,y1,y2,y3,y4)| ≤ 4∑ i=1 l1i |xi −yi| and |g2(t,x1,x2,x3,x4) −g2(t,y1,y2,y3,y4)| ≤ 4∑ i=1 l2i |xi −yi| for all (x1,x2,x3,x4), (y1,y2,y3,y4) ∈ (0,∞)4 and t ∈ [0, 1]. also, we have 1 γ(α1 − 1) ( 2∑ k=1 l1k + 2∑ k=1 l12+k γ(2 −µ1) ) = 1 γ( 3 2 − 1) (0.1 + 0.2 + 0.1 γ( 5 3 ) ) + 0.2 γ( 5 3 ) ) < 1 and 1 γ(α2 − 1) ( 2∑ k=1 l2k + 2∑ k=1 l22+k γ(2 −µ2) ) = 1 γ( 4 3 − 1) (0.2 + 0.2 + 0.3 γ( 3 2 ) ) + 0.1 γ( 3 2 ) ) < 1. note that the maps p1 and p2 are non-increasing respect to all components. if m1 := m1 α1 − 1 γ(α1 + 1) = 2 × 3 2 γ( 7 2 ) = 3 γ( 7 2 ) , m2 := m2 α2 − 1 γ(α2 + 1) = 1 × 4 3 γ( 10 3 ) = 4 3γ( 10 3 ) , then ∫ 1 0 p1 ( m1t α1,m2t α2, m1(1 −µ1) 2 t1−µ1, m2(1 −µ2) 2 t1−µ2 ) dt < ∞ and ∫ 1 0 p2 ( m1t α1,m2t α2, m1(1 −µ1) 2 t1−µ1, m2(1 −µ2) 2 t1−µ2 ) dt < ∞. also, the functions h1 and h2 are non-decreasing respect to all components, lim x→∞ h1(x,. . . ,x) x = lim x→∞ 2 + a1x + a2x + a3x + a4x x = 0 and lim x→∞ h2(x,. . . ,x) x = lim x→∞ 1 + b1x + b2x + b3x + b4x x = 0. now by using theorem 2.2, the problem (2.1) has a positive solution. references [1] r. p. agarwal, d. o’regan and s. stanek, positive solutions for dirichlet problems of singular nonlinear fractional differential equations, j. math. anal. appl. 371 (2010), 57-68. [2] r. p. agarwal, d. o’regan and s. stanek, positive solutions for mixed problems of singular fractional differential equations, math. nachr. 285 (1) (2012), 27-41. [3] z. bai and t. qiu, existence of positive solution for singular fractional differential equation, appl. math. comput. 215 (2009), 2761-2767. [4] z. bai, w. sun and w. zhang, positive solutions for boundary value problems of singular fractional differential equations, abstr. appl. anal. 2013 (2013), art. id 129640. [5] i. podlubny, fractional differential equations, academic press, 1999. [6] r. li, h. zhang and h. tao, unique solution of a coupled fractional differential system involving integral boundary conditions from economic model, abstr. appl. anal. 2013 (2013), art. id 615707. [7] sh. rezapour and m. shabibi, a singular fractional fractional differential equation with riemann-liouville integral boundary condition, j. adv. math. stud. 8 (1) (2015), 80-88. [8] s. g. samko, a. a. kilbas and o. i. marichev, fractional integral and derivative: theory and applications, gordon and breach (1993). [9] d. r. smart, fixed point theorems, cambridge university press, 1980. [10] s. stanek, the existence of positive solutions of singular fractional boundary value problems, comput. math. appl. 62 (2011), 1379-1388. [11] n. tatar, an impulsive nonlinear singular version of the gronwall-bihari inequality, j. inequal. appl. 2006 (2006), art. id 84561. 118 shabibi, postolache and rezapour [12] a. yang and w. ge, positive solutions for boundary value problems of n-dimension nonlinear fractional differential system, bound. value probl. 2008 (2008), art. id 437453. 1department of mathematics, science and research branch, islamic azad university, tehran, iran 2department of medical research, china medical university hospital, china medical university, taichung, taiwan 3department of mathematics & computer science, university politehnica of bucharest, 313 splaiul independenţei, 060042 bucharest, romania 4department of mathematics, azarbaijan shahid madani university, tabriz, iran ∗corresponding author: mi.postolache@mail.cmuh.org.tw; mihai@mathem.pub.ro 1. preliminaries 2. main results references international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 137-145 http://www.etamaths.com hermite-hadamard type inequalities for p-convex functions i̇mdat i̇şcan∗ abstract. in this paper, the author establishes some new hermite-hadamard type inequalities for p-convex functions. some natural applications to special means of real numbers are also given. 1. introduction let f : i ⊂ r → r be a convex function defined on the interval i of real numbers and a,b ∈ i with a < b. the following inequality (1.1) f ( a + b 2 ) ≤ 1 b−a b∫ a f(x)dx ≤ f(a) + f(b) 2 holds. this double inequality is known in the literature as hermite-hadamard integral inequality for convex functions. note that some of the classical inequalities for means can be derived from (1.1) for appropriate particular selections of the mapping f. both inequalities hold in the reversed direction if f is concave. for some results which generalize, improve and extend the inequalities (1.1) we refer the reader to the recent papers (see [2, 3, 5, 6, 8, 9, 12]). in [3], dragomir gave the following lemma: lemma 1. let f : i◦ ⊂ r → r be a differentiable mapping on i◦ and a,b ∈ i◦ with a < b. if f′ ∈ l[a,b], then the following equality holds: (1.2) f(a) + f(b) 2 − 1 b−a b∫ a f(x)dx = b−a 2 1∫ 0 (1 − 2t) f′ (ta + (1 − t)b) dt. by using this lemma, dragomir obtained the following hermite-hadamard type inequalities for convex functions: theorem 1. let f : i◦ ⊂ r → r be a differentiable mapping on i◦ and a,b ∈ i◦ with a < b. if |f′| is convex on [a,b] , then the following inequality holds: (1.3) ∣∣∣∣∣∣f(a) + f(b)2 − 1b−a b∫ a f(x)dx ∣∣∣∣∣∣ ≤ (b−a) (|f ′(a)| + |f′(b)|) 8 . theorem 2. let f : i◦ ⊂ r → r be a differentiable mapping on i◦ and a,b ∈ i◦ with a < b and p > 1. if the new mapping |f′|q is convex on [a,b] , then the following inequality holds: (1.4) ∣∣∣∣∣∣f(a) + f(b)2 − 1b−a b∫ a f(x)dx ∣∣∣∣∣∣ ≤ (b−a)2(p + 1)1/p [ |f′(a)|q + |f′(b)|q 2 ]1/q , where 1/p + 1/q = 1. 2010 mathematics subject classification. 26d15; 26a51. key words and phrases. p-convex function; hermite-hadamard type inequality; hypergeometric function. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 137 138 i̇şcan let a (a,b; t) = ta + (1 − t)b, g (a,b; t) = atb1−t, h (a,b; t) = ab/(ta + (1 − t)b) and mp (a,b; t) = (tap + (1 − t)bp)1/p ,p ∈ r\{0} , be the weighted arithmetic, geometric, harmonic , power of order p means of two positive real numbers a and b with a 6= b for t ∈ [0, 1] , respectively. mp (a,b; t) is continuous and strictly increasing with respect to t ∈ r for fixed p ∈ r\{0} and a,b > 0 with a > b. see [13, 7] for some kinds of convexity obtained by using weighted means. in [7], the author, gave definition harmonically convex and concave functions as follow. definition 1. let i ⊂ r\{0} be a real interval. a function f : i → r is said to be harmonically convex, if (1.5) f ( xy tx + (1 − t)y ) ≤ tf(y) + (1 − t)f(x) for all x,y ∈ i and t ∈ [0, 1]. if the inequality in (1.5) is reversed, then f is said to be harmonically concave. the following result of the hermite-hadamard type holds for harmonically convex functions. theorem 3 ([7]). let f : i ⊂ r\{0}→ r be a harmonically convex function and a,b ∈ i with a < b. if f ∈ l[a,b] then the following inequalities hold (1.6) f ( 2ab a + b ) ≤ ab b−a b∫ a f(x) x2 dx ≤ f(a) + f(b) 2 . the above inequalities are sharp. lemma 2 ([7]). let f : i ⊂ r\{0}→ r be a differentiable function on i◦ and a,b ∈ i with a < b. if f′ ∈ l[a,b] then f(a) + f(b) 2 − ab b−a b∫ a f(x) x2 dx = ab (b−a) 2 1∫ 0 1 − 2t (tb + (1 − t)a)2 f′ ( ab tb + (1 − t)a ) dt.(1.7) using this lemma, the following inequalities hold. theorem 4 ([7]). let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i with a < b, and f′ ∈ l[a,b]. if |f′|q is harmonically convex on [a,b] for q ≥ 1, then∣∣∣∣∣∣f(a) + f(b)2 − abb−a b∫ a f(x) x2 dx ∣∣∣∣∣∣(1.8) ≤ ab (b−a) 2 λ 1−1 q 1 [ λ2 |f′ (a)| q + λ3 |f′ (b)| q]1q , where λ1 = 1 ab − 2 (b−a)2 ln ( (a + b) 2 4ab ) , λ2 = −1 b (b−a) + 3a + b (b−a)3 ln ( (a + b) 2 4ab ) , λ3 = 1 a (b−a) − 3b + a (b−a)3 ln ( (a + b) 2 4ab ) = λ1 −λ2. hermite-hadamard type inequalities 139 theorem 5 ([7]). let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i with a < b, and f′ ∈ l[a,b]. if |f′|q is harmonically convex on [a,b] for q > 1, 1 p + 1 q = 1, then∣∣∣∣∣∣f(a) + f(b)2 − abb−a b∫ a f(x) x2 dx ∣∣∣∣∣∣(1.9) ≤ ab (b−a) 2 ( 1 p + 1 )1 p ( µ1 |f′ (a)| q + µ2 |f′ (b)| q)1q , where µ1 = [ a2−2q + b1−2q [(b−a) (1 − 2q) −a] ] 2 (b−a)2 (1 −q) (1 − 2q) , µ2 = [ b2−2q −a1−2q [(b−a) (1 − 2q) + b] ] 2 (b−a)2 (1 −q) (1 − 2q) . in [16], zhang and wan gave definition of p-convex function as follow: definition 2. let i be a p-convex set. a function f : i → r is said to be a p-convex function or belongs to the class pc(i), if f ( [αxp + (1 −α)yp]1/p ) ≤ αf(x) + (1 −α)f(y) for all x,y ∈ i and α ∈ [0, 1]. remark 1 ([16]). an interval i is said to be a p-convex set if [αxp + (1 −α)yp]1/p ∈ i for all x,y ∈ i and α ∈ [0, 1], where p = 2k + 1 or p = n/m, n = 2r + 1, m = 2t + 1 and k,r,t ∈ n. remark 2 ([10]). if i ⊂ (0,∞) be a real interval and p ∈ r\{0}, then [αxp + (1 −α)yp]1/p ∈ i for all x,y ∈ i and α ∈ [0, 1]. according to remark 2, we can give a different version of the definition of p-convex function as follow: definition 3 ([10]). let i ⊂ (0,∞) be a real interval and p ∈ r\{0} . a function f : i → r is said to be a p-convex function, if (1.10) f ( [αxp + (1 −α)yp]1/p ) ≤ αf(x) + (1 −α)f(y) for all x,y ∈ i and α ∈ [0, 1]. if the inequality in (1.10) is reversed, then f is said to be p-concave. according to definition 3, it can be easily seen that for p = 1 and p = −1, p-convexity reduces to ordinary convexity and harmonically convexity of functions defined on i ⊂ (0,∞), respectively. example 1. let f : (0,∞) → r, f(x) = xp,p 6= 0, and g : (0,∞) → r, g(x) = c, c ∈ r, then f and g are both p-convex and p-concave functions. in [4, theorem 5], if we take i ⊂ (0,∞), p ∈ r\{0} and h(t) = t , then we have the following theorem. theorem 6. let f : i ⊂ (0,∞) → r be a p-convex function, p ∈ r\{0}, and a,b ∈ i with a < b. if f ∈ l[a,b] then we have (1.11) f ([ ap + bp 2 ]1/p) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ f(a) + f(b) 2 . for some results related to p-convex functions and its generalizations, we refer the reader to see [4, 10, 15, 14, 16]. in [15, lemma 2.4], if we take i ⊂ (0,∞) and p ∈ r\{0} , then we have the following lemma. 140 i̇şcan lemma 3. let f : i ⊂ (0,∞) → r be a differentiable function on i◦ and a,b ∈ i with a < b and p ∈ r\{0}. if f′ ∈ l[a,b] then f(a) + f(b) 2 − p bp −ap b∫ a f(x) x1−p dx = bp −ap 2p 1∫ 0 1 − 2t [tap + (1 − t)bp]1−1/p f′ ( [tap + (1 − t)bp]1/p ) dt.(1.12) remark 3. in lemma 3, (i) if we take p = 1, then we have inequality (1.2) in lemma 1. (ii) if we take p = −1, then we have inequality (1.7) in lemma 2. for finding some new inequalities of hermite-hadamard type for functions whose derivatives are p-convex, we need lemma 3. we recall the following special functions (1) the beta function: β (x,y) = γ(x)γ(y) γ(x + y) = 1∫ 0 tx−1 (1 − t)y−1 dt, x,y > 0, (2) the hypergeometric function: 2f1 (a,b; c; z) = 1 β (b,c− b) 1∫ 0 tb−1 (1 − t)c−b−1 (1 −zt)−a dt, c > b > 0, |z| < 1 (see [11]). the main purpose of this paper is to establish some new results connected with the right-hand side of the inequalities (1.11) for p-convex functions. 2. main results we obtain the another version of [15, theorem 3.2] as follow: theorem 7. let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i◦ with a < b, p ∈ r\{0} and f′ ∈ l[a,b]. if |f′|q is p-convex on [a,b] for q ≥ 1, then∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣(2.1) ≤ bp −ap 2p c 1−1 q 1 [ c2 |f′ (a)| q + c3 |f′ (b)| q]1q , where c1 = c1(a,b; p) = 1 4 ( ap + bp 2 )1 p −1 × [ 2f1 ( 1 − 1 p , 2; 3; ap − bp ap + bp ) +2 f1 ( 1 − 1 p , 2; 3; bp −ap ap + bp )] , c2 = c2(a,b; p) = 1 24 ( ap + bp 2 )1 p −1 [ 2f1 ( 1 − 1 p , 2; 4; ap − bp ap + bp ) +6.2f1 ( 1 − 1 p , 2; 3; bp −ap ap + bp ) +2 f1 ( 1 − 1 p , 2; 4; bp −ap ap + bp )] , c3 = c3(a,b; p) = c1 −c2, hermite-hadamard type inequalities 141 proof. from lemma 3 and using the power mean integral inequality, we have∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣ ≤ bp −ap 2p 1∫ 0 ∣∣∣∣∣ 1 − 2t[tap + (1 − t)bp]1−1/p ∣∣∣∣∣ ∣∣∣f′([tap + (1 − t)bp]1/p)∣∣∣dt ≤ bp −ap 2p   1∫ 0 |1 − 2t| [tap + (1 − t)bp]1−1/p dt  1− 1 q ×   1∫ 0 |1 − 2t| [tap + (1 − t)bp]1−1/p ∣∣∣f′([tap + (1 − t)bp]1/p)∣∣∣q dt   1 q . hence, by p-convexity of |f′|q on [a,b], we have∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣ ≤ bp −ap 2p   1∫ 0 |1 − 2t| [tap + (1 − t)bp]1−1/p dt  1− 1 q ×   1∫ 0 ||1 − 2t|| [ t |f′ (a)|q + (1 − t) |f′ (b)|q ] [tap + (1 − t)bp]1−1/p dt   1 q ≤ bp −ap 2p c 1−1 q 1 [ c2 |f′ (a)| q + c3 |f′ (b)| q]1q . it is easily check that 1∫ 0 |1 − 2t| [tap + (1 − t)bp]1−1/p dt = c1(a,b; p), 1∫ 0 |1 − 2t|t [tap + (1 − t)bp]1−1/p dt = c2(a,b; p), 1∫ 0 |1 − 2t|(1 − t) [tap + (1 − t)bp]1−1/p dt = c1(a,b; p) −c2(a,b; p). � remark 4. if we take p = −1 in theorem 7, then we have inequality (1.8) in theorem 4. if we take q = 1 in theorem 7, then we have the following corollary. corollary 1. let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i◦ with a < b, p ∈ r\{0} and f′ ∈ l[a,b]. if |f′| is p-convex on [a,b], then∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣ ≤ bp −ap 2p [c2 |f′ (a)| + c3 |f′ (b)|] , where c2 and c3 are defined as in theorem 7. 142 i̇şcan remark 5. if we take p = 1 in corollary 1, then we have inequality (1.3) in theorem 1. theorem 8. let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i with a < b, p ∈ r\{0} and f′ ∈ l[a,b]. if |f′|q is p-convex on [a,b] for q > 1, 1 r + 1 q = 1, then ∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣(2.2) ≤ bp −ap 2p ( 1 r + 1 )1 r ( c4 |f′ (a)| q + c5 |f′ (b)| q)1q , where c4 = c4(a,b; p; q) =   1 2aqp−q .2f1 ( q − q p , 1; 3; 1 − ( b a )p) , p < 0 1 2bqp−q .2f1 ( q − q p , 2; 3; 1 − ( a b )p) , p > 0 , c5 = c5(a,b; p; q) =   1 2aqp−q .2f1 ( q − q p , 2; 3; 1 − ( b a )p) , p < 0 1 2bqp−q .2f1 ( q − q p , 1; 3; 1 − ( a b )p) , p > 0 . proof. from lemma 3, hölder’s inequality and the p-convexity of |f′|q on [a,b],we have, ∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣ ≤ bp −ap 2p   1∫ 0 |1 − 2t|r dt   1 r ×   1∫ 0 1 [tap + (1 − t)bp]q−q/p ∣∣∣f′([tap + (1 − t)bp]1/p)∣∣∣q dt   1 q ≤ bp −ap 2p ( 1 r + 1 )1 r ×   1∫ 0 t |f′ (a)|q + (1 − t) |f′ (b)|q [tap + (1 − t)bp]q−q/p dt   1 q , where an easy calculation gives 1∫ 0 t [tap + (1 − t)bp]q−q/p dt(2.3) =   1 2aqp−q .2f1 ( q − q p , 1; 3; 1 − ( b a )p) , p < 0 1 2bqp−q .2f1 ( q − q p , 2; 3; 1 − ( a b )p) , p > 0 hermite-hadamard type inequalities 143 and 1∫ 0 1 − t [tap + (1 − t)bp]q−q/p dt(2.4) =   1 2aqp−q .2f1 ( q − q p , 2; 3; 1 − ( b a )p) , p < 0 1 2bqp−q .2f1 ( q − q p , 1; 3; 1 − ( a b )p) , p > 0 . substituting equations (2.3) and (2.4) into the above inequality results in the inequality (2.2), which completes the proof. � remark 6. in theorem 8, (i) if we take p = 1, then we have inequality (1.4) in theorem 2. (ii) if we take p = −1, then we have the inequality (1.9) in theorem 5. theorem 9. let f : i ⊂ (0,∞) → r be a differentiable function on i◦, a,b ∈ i with a < b, p ∈ r\{0} and f′ ∈ l[a,b]. if |f′|q is p-convex on [a,b] for q > 1, 1 r + 1 q = 1, then∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣(2.5) ≤ bp −ap 2p c 1 r 6 ( 1 q + 1 )1 q ( |f′ (a)|q + |f′ (b)|q 2 )1 q , where c6 = c6(a,b; p; r) =   1 apr−r .2f1 ( r − r p , 1; 2; 1 − ( b a )p) , p < 0 1 bpr−r .2f1 ( r − r p , 1; 2; 1 − ( a b )p) , p > 0 , proof. from lemma 3, hölder’s inequality and the p-convexity of |f′|q on [a,b],we have,∣∣∣∣∣∣f(a) + f(b)2 − pbp −ap b∫ a f(x) x1−p dx ∣∣∣∣∣∣ ≤ bp −ap 2p   1∫ 0 1 [tap + (1 − t)bp]r−r/p dt   1 r ×   1∫ 0 |1 − 2t|q ∣∣∣f′([tap + (1 − t)bp]1/p)∣∣∣q dt   1 q ≤ bp −ap 2p c 1 r 6 (a,b; p; r) ( 1 q + 1 )1 q ( |f′ (a)|q + |f′ (b)|q 2 )1 q , where an easy calculation gives c6(a,b; p; r) = 1∫ 0 1 [tap + (1 − t)bp]r−r/p dt(2.6) =   1 apr−r .2f1 ( r − r p , 1; 2; 1 − ( b a )p) , p < 0 1 bpr−r .2f1 ( r − r p , 1; 2; 1 − ( a b )p) , p > 0 144 i̇şcan and (2.7) 1∫ 0 |1 − 2t|q tdt = 1∫ 0 |1 − 2t|q (1 − t) dt = 1 2(q + 1) . substituting equations (2.6) and (2.7) into the above inequality results in the inequality (2.5), which completes the proof. � 3. some applications for special means let us recall the following special means of two nonnegative number a,b with b > a : (1) the arithmetic mean a = a (a,b) := a + b 2 . (2) the geometric mean g = g (a,b) := √ ab. (3) the harmonic mean h = h (a,b) := 2ab a + b . (4) the power mean mr = mr (a,b) := ( ar + br 2 )1/r , r 6= 0. (5) the logarithmic mean l = l (a,b) := b−a ln b− ln a . (6) the p-logarithmic mean lp = lp (a,b) := ( bp+1 −ap+1 (p + 1)(b−a) )1 p , p ∈ r\{−1, 0} . (7) the identric mean i = i (a,b) = 1 e ( bb aa ) 1 b−a . these means are often used in numerical approximation and in other areas. however, the following simple relationships are known in the literature: h ≤ g ≤ l ≤ i ≤ a. it is also known that lp is monotonically increasing over p ∈ r, denoting l0 = i and l−1 = l. proposition 1. let 0 < a < b and p ∈ (−∞, 1)\{−1} . then we have the following inequality mp.l p−1 p−1 ≤ l p p ≤ a.l p−1 p−1 proof. the assertion follows from the inequality (1.11) in theorem 6, for f : (0,∞) → r, f(x) = x. � proposition 2. let 0 < a < b and p > 1. then we have the following inequality h(ap,bp) ≤ lp−1p−1.l ≤ a(a p,bp). proof. the assertion follows from the inequality (1.11) in theorem 6, for f : (0,∞) → r, f(x) = x−p. � proposition 3. let 0 < a < b. then we have the following inequality lpph ≤ l p−1 p−1g 2+2p ≤ lppmp. proof. the assertion follows from the inequality (1.11) in theorem 6, for f : (0,∞) → r, f(x) = 1/x. � hermite-hadamard type inequalities 145 4. conflict of interests the author declares that there is no conflict of interests regarding the publication of this paper. references [1] g.d. anderson, m.k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, journal of mathematical analysis and applications 335(2) (2007), 1294-1308. [2] m. avci, h. kavurmaci and m. e. ozdemir, new inequalities of hermite-hadamard type via s-convex functions in the second sense with applications, appl. math. comput., 217 (2011), 5171–5176. [3] s.s. dragomir and r.p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11(5) (1998), 91-95. [4] z. b. fang and r. shi, on the(p,h)-convex function and some integral inequalities, j. inequal. appl., 2014(45) (2014), 16 pages. [5] i̇. i̇şcan, a new generalization of some integral inequalities for (α,m)-convex functions, mathematical sciences,7(22) (2013),1-8. [6] i̇. i̇şcan, new estimates on generalization of some integral inequalities for s-convex functions and their applications, international journal of pure and applied mathematics, 86(4) (2013), 727-746. [7] i̇. i̇şcan, hermite-hadamard type inequalities for harmonically convex functions, hacettepe journal of mathematics and statistics, 43(6) (2014), 935-942. [8] i̇. i̇şcan, some new general integral inequalities for h-convex and h-concave functions, adv. pure appl. math. 5(1) (2014), 21-29. [9] i̇. i̇şcan, hermite-hadamard-fejer type inequalities for convex functions via fractional integrals, studia universitatis babeş-bolyai mathematica, 60(3) (2015), 355-366. [10] i̇. i̇şcan, ostrowski type inequalities for p-convex functions, new trends in mathematical sciences, in press. [11] a.a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier, amsterdam, 2006. [12] u.s. kirmaci, inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula,appl. math. comput. 147 (2004), 137-146. [13] c. p. niculescu, convexity according to the geometric mean, math. inequal. appl., 3(2) (2000), 155-167. [14] m. a. noor, k. i. noor and s. iftikhar, nonconvex functions and integral inequalities, punjab university journal of mathematics, 47(2) (2015), 19-27. [15] m. a. noor, k. i. noor, m. v. mihai, and m. u. awan, hermite-hadamard inequalities for differentiable pconvex functions using hypergeometric functions, researchgate doi: 10.13140/rg.2.1.2485.0648. available online at https://www.researchgate.net/publication/282912282. [16] k.s. zhang and j.p. wan, p-convex functions and their properties, pure appl. math. 23(1) (2007), 130-133. department of mathematics, faculty of arts and sciences,, giresun university, 28100, giresun, turkey ∗corresponding author: imdat.iscan@giresun.edu.tr int. j. anal. appl. (2023), 21:25 asymptotic behavior of some parabolic equations and application in image restoration fatima zohra zeghbib1,∗, abir bounaama1, messaoud maouni1, halim zeghdoudi2 1laboratory of lamahis, department of mathematics, université 20 août 1955-skikda, el-hedaiek b.p. 26, skikda 21000, algeria 2department of mathematics, university of badji mokhtar annaba, algeria ∗corresponding author: fz.zeghbib@univ-skikda.dz abstract. in this paper, we consider some nonlinear parabolic problem involving the well known p−laplacian and some operator having exponential growth with respect to the gradient. we start by dealing the asymptotic behavior for some evolution equation then we give some numerical results with an application in image processing. 1. introduction image processing has always been a challenging problem, this field has become "hot". in recent years, image processing has been a very active field of computer application and research [9]. the most active topics in this field is image restoration because it allow to recovery lost information from the observed degraded image data. in [4, 6, 19, 24, 25] the authors have studied the partial differential equation (pde) and fractional partial differential equation (fpde) methods in image processing and proved the fundamental tools for image diffusion and restoration. in 1987, the perona malik is the first attempts to derive a model from an image within a pde framework in [21]. then, by using perona malik the authors were concluded a nonlinear diffusion model (anisotropic model). received: feb. 2, 2023. 2020 mathematics subject classification. 46e30, 35k55, 35k85, 94a08. key words and phrases. orlicz-sobolev spaces; parabolic equations in orlicz spaces; parabolic inequalities; image processing. https://doi.org/10.28924/2291-8639-21-2023-25 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-25 2 int. j. anal. appl. (2023), 21:25 in [1], the authors have studied a diffusion model, this model is a combination of fast growth with respect to low gradient and slow growth when the gradient is large which used for restoration in image processing. then, in [14] the researchers showed some class of nonlinear parabolic inequalities in orlicz spaces. the authors presented a novel model for image denoising and compared the results with the model of perona-malik and the method of the total variation (see [18]). and, in [3] the authors proposed a novel parabolic equations for image restoration and enhacement.they proved the existence of solution, established a nonnegative weak solution obtained as limit of approximation and give some application in image processing. also, in [15] we find some optimal control problem for the perona-malik equation. the authors obtained existence results and approximation of an optimal control problem. in [8], authors studied the minimisations problem numerically with anistropic diffusion and obtained the results in image restoration. let ω be a regular open bounded subset of rn with n ≥ 2. and let q be the cylinder ω × (0,t ) with some given t > 0. we consider the following nonlinear parabolic problem:  ∂u ∂t + a (u) = f in q aα,β (x,t,∇u) .η = 0, on ∂q = ∂ω × (0,t ) u (x, 0) = u0 in ω, (1.1) where a(u) = −div ( aα,β (x,t,∇u) ) , (1.2) with aα,β (x,t,∇u) satisfying the following expression aα,β (x,t,∇u) = exp (α |∇u|)∇uχ{|∇u|≤β} + ∇u |∇u| logγ (1 + |∇u|) χ{|∇u|>β} + κ |∇u| p−2 ∇u. (1.3) for instance, if we take β = γ = 0 and p = 1 in problem (1.1), so we obtain the curvature-driven diffusion (see [22]). if α = 0, β = 1, γ = 0 and κ = 0 is treated in [7]. and, if we make κ = 0 and β ≥ 0 then, the problem (1.1) has been recently studied in ( [1], [20]) and successfully used in image processing. actually, nonlinear partial differential equations of type (1.1) can be considered as perona–malik equations see [21]. in this work, we will study the problem (1.1) when β = +∞, κ > 0 and p → +∞. more precisely , int. j. anal. appl. (2023), 21:25 3 we will show that the problem (pp)   ∂u ∂t −div (aα (x,t,∇u)) = f in q aα (x,t,∇u) .η = 0, on ∂q = ∂ω × (0,t ) u (x, 0) = u0 in ω, (1.4) with aα (x,t,∇u) = exp (α |∇u|)∇u + κ |∇u|p−2 ∇u, for p > 2, (1.5) admits at least one solution up ∈ w 1,xlaα (q) where aα(t) = t 2 exp(αt). next, we study the asymptotic behavior of the solution up as p → +∞, we show that the limit of up satisfies some parabolic obstacle problem. in our work, we use an application illustrating that the problem can be used for denoising filter in image processing. for recent works which involving the partial differential equations with nonstandard growth and with applications in image processing, the reader can refereed to [8], [17], [18] and [13]. this work is organized as follows: in the next section, we present somme lemmas and spaces; in section 3, we obtain main results; in section 4, we get some numerical results. 2. preliminaries in this section, we shall give some corollaries and definitions which will be used throughout this work. 2.1. n− functions. let a : r+ → r+ be an n− function, i.e. a is continuous and convex with a > 0, for t > 0, a(t) t → 0 as t → 0 and a(t) t → +∞ as t → +∞. equivalently, a admits the representation: a(t) = t∫ 0 a(s)ds, where a : r+ →r+ is nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) tends to ∞ as t →∞. the n− function ā conjugate to a is defined by ā(t) = t∫ 0 ā(s)ds, where ā : r+ →r+is given by ā(t) = sup{s : a(s) ≤ t} ( [2]). the n-function is said to satisfy the ∆2 condition (∃ k > 0 : a(t) ≤ ka(t),∀t ≥ 0), so for some k > 0 we obtain, a(2t) ≤ ka(t), ∀t ≥ 0, (2.1) when (2.1) holds only for some t > 0 then, a is said to satisfy the ∆2 condition near infinity. we will extend these n-functions into even functions on all r. 2.2. the orlicz spaces. let ω be an open subset of rn. the orlicz space la (ω) is defined as the set of (equivalence classes of) real-valued measurable functions u on ω such that:∫ ω a ( u(x) λ ) dx < +∞ for some λ > 0. (2.2) 4 int. j. anal. appl. (2023), 21:25 la (ω) is banach space under the norm ‖u‖a,ω = inf  λ > 0, ∫ ω a ( u(x) λ ) dx ≤ 1   . (2.3) the closure in la (ω) of the set of bounded measurable functions with compact support in ω is denoted by ea(ω). the equality ea(ω) = la (ω) holds if only if a satisfies ∆2 condition, for all t or for t large according to whether ω has infinite measure or not. the dual of ea(ω) can be identified with la (ω) by means of the pairing ∫ ω uvdx, and the dual norm of la (ω) is equivalent to ‖.‖a,ω. the space la (ω) is reflexive if and only if a and ā satisfy the ∆2 condition, for all t or for t large, according to whether ω has infinite measure or not. 2.3. the orlicz-sobolev spaces. now, we turn to the orlicz-sobolev space, w 1la (ω) (respectively w 1ea(ω)) is the space of all functions u and its distributional derivatives up to order 1 lie in la (respectively ea(ω)). it is a banach space under the norm ‖u‖1,a = ∑ |k|≤1 ∥∥dku∥∥ a . (2.4) thus, w 1la (ω) and w 1ea(ω) can be identified with sub-spaces of product of n + 1 copies of la. denoting this product by πla, we will use the weak topologies σ(πla, πeā) and σ(πla, πlā). we say that un converges to u for the modular convergence in w 1la (ω) if for some λ > 0∫ ω a ( dkun −dku λ ) dx → 0 for all |k| ≤ 1. (2.5) this implies convergence for σ(πla, πlā). if a satisfies ∆2 condition on r+, then modular convergence coincides with norm convergence. 2.4. duality in orlicz-sobolev space. let w−1la (ω) denote the space of distributions on ω which can be written as sums of derivatives of order < 1 of functions in lā. it is a banach space under the usual quotient norm. if the open set ω has the segment property then the space d(ω̄) is dense in w 1la (ω) for the modular convergence and thus for the topology σ(πla, πlā) ( [12]). consequently, the action of a distribution in w−1lā (ω) on an element of w 1la (ω) is well defined. 2.5. inhomogeneous orlicz-sobolev spaces. let ω be an abounded open subset of rn, t > 0, and set q = ω × (0,t ). let a be an n−function. for each k ∈ nn, denote by dkx the distributional derivatives on q of order k with respect to the variable x ∈ rn. the inhomogeneous orlicz-sobolev spaces of order 1 are defined as follows w 1,xla (q) = { u ∈ la (q) : dkxu ∈ la (q) , ∀|k| ≤ 1 } , (2.6) int. j. anal. appl. (2023), 21:25 5 and w 1,xea(q) = { u ∈ ea (q) : dkx ∈ ea (q) , ∀|k| ≤ 1 } . (2.7) the latest space is a subset of the first one. they are banach spaces under the norm ‖u‖ = ∑ |k|=1 ∥∥dkxu∥∥a,q . (2.8) we can easily show that they form a complementary system when ω satisfies the segment property. these spaces are considered as subspaces of the product spaces πla (q) which has n + 1 copies. we shall also consider the weak topologies σ(πla, πeā) and σ(πla, πlā). if u ∈ w 1,xla (q) then the function t → u(t) = u(.,t) is defined on (0,t ) with values in w 1la(ω). if, further, u ∈ w 1,xea(q) then u(t) is a w 1ea(ω) valued and is strongly measurable. furthermore, the following continuous imbedding holds: w 1,xea(q) ⊂ l1(0,t ; w 1ea(ω)). the space w 1,xla (q) is not in general separable, if u ∈ w 1,xla (q), we cannot conclude that the function u(t) is measurable from (0,t ) into w 1la(ω). however, the scalar function t → ∥∥dkxu(t)∥∥a,ω , is in l1(0,t ) for all |k| ≤ 1. 2.6. duality in inhomogeneous orlicz-sobolev spaces. we denote by f = w−1,xlā (q) the space f =  f = ∑ |k|=1 dkx fk : fk ∈ lā (q)   . (2.9) this space will be equipped with the usual quotient norm: ‖f‖ = inf ∑ |k|=1 ‖fk‖ā,q , (2.10) where the inf is taken on all possible decomposition f = ∑ |k|=1 dkx fk : fk ∈ lā (q). the space f0 = w −1,xlā (q) is then given by f0 =  f = ∑ |k|=1 dkx fk : fk ∈ eā (q)   . (2.11) the following corollary will be useful in the proof of our existence theorem. corollary 2.1. ( [10]). let a be an n−function. let (un) be a sequence of w 1,xla (q) such that un → u weakly in w 1,xla (q) for σ(πla, πeā) (2.12) and ∂un ∂t is bounded in w−1,xlā (q) +m(q), where m(q) is the space of measures defined on q. then un → u strongly in l1loc(q). 6 int. j. anal. appl. (2023), 21:25 3. the main results of the existence theorem 3.1. let f ∈ l∞(q), and u0 that |∇u0| ≤ 1. then the problem (pp)   up ∈ w 1,xlaα(q)〈 ∂up ∂t ,v 〉 + ∫ q ∇up exp(α |∇up|)∇vdxdt + ∫ q |∇up|p−2 ∇up∇vdxdt = ∫ q f vdxdt, for v ∈ w 1,xlaα(q) ∩l 2(q) such that ∂v ∂t ∈ w−1,xlāα (q) + l 2(q), (3.1) admits at least one solution up ∈ w 1,xlaα(q) such that up → u for the modular convergence where u is solution of the following parabolic inequality: (p )   |∇u| ≤ 1〈〈 ∂v ∂t ,u −v 〉〉 + ∫ q a(x,∇u) (∇u −∇v) dxdt ≤ ∫ q 〈f ,u −v〉dxdt + 1 2 ∫ ω (u0 −v(x, 0))2 dx for v ∈ w 1,xlaα(q) ∩l 2(q) such that ∂v ∂t ∈ w−1,xlāα (q) + l 2(q) and |∇v| ≤ 1. remark 3.1. since { v ∈ w 1,xlaα(q) ∩l 2(q) : ∂v ∂t ∈ w−1,xlāα (q) + l 2(q) } ⊂ c ( ([0,t ]) ,l2 (ω) ) (see, [11]) the least term of problem (p ) is well defined. proof. step 1 let us consider the following approximate problem: (pp,n)   ∂unp ∂t −div(∇unp exp(α ∣∣∇unp∣∣)) −div(∣∣∇unp∣∣p−2 ∇unp) + 1n (unp −m) exp (α ∣∣unp −m∣∣) = f in q unp(x, 0) = u0 in ω exp(α ∣∣∇unp∣∣)∂unp∂n + ∣∣∇unp∣∣p−2 ∂unp∂n = 0 on ∂ω × (0,t ), (3.2) where m = max(‖f‖∞ ,‖u0‖∞). as it is done in [11], one can is seen easily that the problem (pp,n) admits at least one solution unp ∈ w 1,xlaα(q) furthermore∥∥unp∥∥∞ ≤ max{‖f‖∞ ,‖u0‖∞} . (3.3) by choosing unp −m as test function in (pp,n), we obtain ∫ q ∣∣∇unp∣∣2 exp(α ∣∣∇unp∣∣)dxdt ≤ c, and via (3.3) it follows that ∥∥unp∥∥1,aα ≤ m′, and thanks to (pp,n), we deduce that ∂unp ∂t is bounded in w−1,xlāα(q) + l ∞ (q) . thanks to corollary (2.1) we have unp → up in l1 (q) as n → +∞ almost everywhere convergence in q. arguing as in [10] and [5], we pass to the limit in (pp,n) to obtain (pp)   ∂up ∂t −div(∇up exp(α |∇up|)) −div(|∇u|p−2 ∇u) = f in q u(x, 0) = u0 in ω exp(α |∇up|) ∂up ∂n + |∇u|p−2 ∂up ∂n = 0 on ∂ω × (0,t ), (3.4) with up ∈ w 1,xlaα(q) ∩l ∞ (q) and ‖up‖∞ ≤ m ′′. step 2: a priori estimates int. j. anal. appl. (2023), 21:25 7 choosing v = up as test function in (pp) we obtain:〈〈 ∂up ∂t ,up 〉〉 + ∫ q |∇up|2 exp(α |∇up|)dxdt + ∫ q |∇up|p dxdt = ∫ q f updxdt, which gives by using young’s inequality 1 2 ∫ ω u2p(x,t)dx− 1 2 ∫ ω u20dx+ ∫ q |∇up|p dxdt+ ∫ q |∇up|2 exp(α |∇up|)dxdt ≤ 1 2 ∫ q f 2dxdt+ 1 2 ∫ q u2pdxdt 1 2 ∫ ω u2p(x,t)dx + 1 2 ∫ ω u20dx + ∫ q |∇up|p dxdt + ∫ q |∇up|2 exp(α |∇up|)dxdt ≤ c + 1 2 t∫ 0 ∫ ω u2pdxdt, by gronwall’s lemma (see [23]), we get ∫ ω u2p(x,t)dx + ∫ q |∇up|2 exp(α |∇up|)dxdt + ∫ q |∇up|p dxdt ≤ c. consequently, since up is bounded in w 1,xlaα(q)∩l 2(q) so there exist some u ∈ w 1,xlaα(q)∩ l2(q) such that ( for a subsequence still denoted by up) up → u weakly in w 1,xlaα(q) ∩l 2(q). step 3: to obtain |∇u| ≤ 1, we will use the estimate∫ q |∇up|p dxdt ≤ c. (3.5) let q < p, we have∫ q |∇up|q dxdt = ∫ |∇up|≤1 |∇up|q dxdt + ∫ |∇up|>1 |∇up|q dxdt ≤ meas(q) + ∫ |∇up|>1 |∇up|q dxdt ≤ meas(q) + c, which gives ∫ q |∇up|q dxdt ≤ m, by letting p →∞ for q fixed, we obtain∫ q |∇u|q dxdt ≤ m. now, let k > 1, we get∫ |∇u|≥k |∇u|q dxdt ≤ m =⇒ meas{|∇u| ≥ k}≤ m kq =⇒ meas{|∇u| ≥ 1} = 0, 8 int. j. anal. appl. (2023), 21:25 which gives |∇u| ≤ 1. step 4: modular convergence of up → u in w 1,xlaα (q) : let wµ = uµ + e−µtu0, where uµ is the mollifier function defined in [16] with respect to time of u and the function wµ have the following properties: ∂wµ ∂t = µ (u −wµ) ; wµ (0) = u0, (3.6) with uµ = µ t∫ −∞ u(x,s)χ(0,t ) exp(µ (s − t))ds, (3.7) ∇wµ = µ t∫ −∞ ∇u(x,s)χ(0,t ) exp(µ (s − t))ds + exp (−µt)∇u0. (3.8) by using |∇u| < 1 and |∇u0| < 1, we get: |∇wµ| ≤ µ t∫ 0 exp(µ (s − t))ds + exp (−µt) = [exp(µ (s − t))]t0 + exp (−µt) = 1. this implies that |∇wµ| ≤ 1. now, we proof that up → u in w 1,xlaα (q), for the modular convergence as p → +∞. for this, we will denote by ε (p,µ,θ) function with all quantities such that lim µ→+∞ lim θ→1 lim p→+∞ ε (p,µ,θ) = 0, (3.9) and we will respect the order of the parameters p,θ,µ. similarly, we will write ε (p) , or ε (p,µ) that the limits are made only on the specified parameters. firstly take vp = up−θwµ for 0 < θ < 1 as test function in (pp), which belong to w 1,xlaα (q), we get〈 ∂up ∂t ,vp 〉 + ∫ q ∇up exp (α |∇up|)∇(up −θwµ) dxdt (3.10) + ∫ q |∇up|p−2 ∇up∇(up −θwµ) dxdt = ∫ q f (up −θwµ) dxdt. on the other hand, by using the monotonicity of the p−laplacien, we deduce that;∫ q |∇up|p−2 ∇up∇(up −θwµ) dxdt ≥ θp−1 ∫ q |∇wµ|p−2 ∇wµ (∇up −θ∇wµ) dxdt, (3.11) by using the holder’s inequality |i1 (p,µ,θ)| = ∣∣∣∣∣∣ ∫ q |∇wµ| p−2 ∇wµ (∇up −θ∇wµ) dxdt ∣∣∣∣∣∣ ≤  ∫ q |∇wµ| p dxdt   1 p′  ∫ q |∇up −θ∇wµ| p dxdt   1 p , int. j. anal. appl. (2023), 21:25 9 which implies ∫ q |∇up −θ∇wµ|p dxdt   1 p ≤  2p  ∫ q (|∇up|p + θp |wµ|p)dxdt     1 p ≤ 2m 1 p . and finally, we obtain i1 (p,µ,θ) ≤ ε (p) . on the other hand〈〈 ∂up ∂t ,zp 〉〉 = 〈〈 ∂up ∂t ,up −θwµ 〉〉 = 〈〈 ∂up ∂t −θ ∂wµ ∂t ,up −θwµ 〉〉 + θ 〈〈 ∂wµ ∂t ,up −θwµ 〉〉 = j1 + θj2. with j1 = 〈〈 ∂up ∂t −θ ∂wµ ∂t ,up −θwµ 〉〉 = ∫ ω (up −θwµ)2 dx − ∫ ω (u0 −θwµ)2 dx, we deduce that j1 ≥− ∫ ω (u0 −θwµ)2 dx ≥−(1 −θ)2 ∫ ω u20dx + ε (µ) = ε (µ,θ) . for what concerns j2, we deduce that j2 = 〈〈 ∂wµ ∂t ,up −θwµ 〉〉 = µ ∫ q (u −wµ) (up −θwµ) dxdt lim θ→1 lim p→+∞ j2 = lim θ→1 lim p→+∞ 〈 ∂wµ ∂t ,up −θwµ 〉 ≥ 0. finally, we get lim µ→+∞ lim θ→1 lim p→+∞ ∫ q a (x,∇up) (∇up −θ∇wµ) dxdt ≤ 0. (3.12) since ∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt − ∫ q a (x,∇up) (∇up −θ∇wµ) dxdt = − ∫ q a (x,∇up)∇udxdt − ∫ q a (x,∇u) [∇up −∇u] dxdt + θ ∫ q a (x,∇up)∇wµdxdt. since a (x,∇up) is bounded in ( lāα (q) )n , we have a (x,∇up) → h weakly for σ(πlāα, πeaα) consequently∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt − ∫ q a (x,∇up) (∇up −θ∇wµ) dxdt = − ∫ q h∇udxdt + θ ∫ q h∇wµdxdt + ε (p) 10 int. j. anal. appl. (2023), 21:25 = − ∫ q h∇udxdt + ∫ q h∇wµdxdt + ε (p,θ) = ε (p,θ,µ) . which gives∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt − ∫ q a (x,∇up) (∇up −θ∇wµ) dxdt = ε (p,θ,µ) , by using (3.12), we obtain∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt → 0 as p → +∞. by strict monotonicity of a (., .), we obtain that ∇up →∇u a.e in q. finally a (x,∇up) → h = a(x,∇u), weakly for σ(πlāα, πeaα), consequently∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt = ∫ q a (x,∇up)∇updxdt − ∫ q a (x,∇up)∇udxdt − ∫ q a (x,∇u) [∇up −∇u] dxdt, because ∇up →∇u weakly in ( lāα (q) )n , we get lim p→+∞ ∫ q [a (x,∇up) −a (x,∇u)] [∇up −∇u] dxdt = lim p→+∞ ∫ q a (x,∇up)∇updxdt − ∫ q a (x,∇u)∇udxdt = 0. since a (x,∇up)∇up = aα (|∇up|) we get lim p→+∞ ∫ q aα (|∇up|) dxdt = ∫ q aα (|∇u|) dxdt. thanks to fact that aα ( |∇up −∇u| 2 ) ≤ 1 2 (aα (|∇up|) + aα (|∇u|)) . by using vitali’s theorem, we obtain∫ q aα ( |∇up −∇u| 2 ) dxdt → 0 as p → +∞. which shows that ∇up converges to ∇u for the modular convergence in laα(q). step 5: the passage to the limit let us consider v ∈ w 1,xlaα(q)∩l 2(q) = w 1,xlaα(q) such that |∇v| < 1, ∂v ∂t ∈ w−1,xlāα(q)+ l2(q) and 0 < θ < 1. using up −θv as test function in (pn), the fact that〈〈 ∂up ∂t ,up −θv 〉〉 + ∫ q ∇up exp(α |∇up|)∇(up −θv) dxdt (3.13) int. j. anal. appl. (2023), 21:25 11 + ∫ q |∇up|p−2 ∇up∇(up −θv) dxdt ≤ ∫ q f (up −θv) dxdt, we have 〈〈 ∂up ∂t −θ ∂v ∂t ,up −θv 〉〉 + θ 〈〈 ∂v ∂t ,up −θv 〉〉 + ∫ q ∇up exp(α |∇up|)∇(up −θv) dxdt + ∫ q |∇up|p−2 ∇up∇(up −θv) dxdt ≤ ∫ q f (up −θv) dxdt, since ∫ q |∇up|p−2 ∇up∇(up −θv) dxdt ≥ θp−1 ∫ q |∇v|p−2 ∇v∇(up −θv) dxdt, which gives θ 〈〈 ∂v ∂t ,up −θv 〉〉 + ∫ q a(x,∇up) (∇up −θ∇v) dxdt + θp−1 ∫ q |∇v|p−2 ∇v∇(up −θv) dxdt ≤ ∫ q 〈f ,up −θv〉dxdt + ∫ ω (u0 −θv)2 dx. since a (x,∇u) belongs to ( lāα (q) )n , and using fatou’s lemma in the first term of the last side gives lim inf p→+∞ ∫ q a(x,∇up) (∇up −θ∇v) dxdt ≥ ∫ q a(x,∇u) (∇u −θ∇v) dxdt, then, we can easily pass to the limit as θ → 1 and p tend to infinity, we obtain〈〈 ∂v ∂t ,u −v 〉〉 + ∫ q a(x,∇u) (∇u −∇v) dxdt ≤ ∫ q 〈f ,u −v〉dxdt + 1 2 ∫ ω (u0 −v(x, 0))2 dx. which completes the proof. � 4. numerical results we consider the following model problem: (p 1)   ∂up ∂t −div(∇up exp(α |∇up|)) −div(|∇u|p−2 ∇u) = f in q u(x, 0) = u0 in ω exp(α |∇up|) ∂up ∂n + |∇u|p−2 ∂up ∂n = 0 in ∂ω × (0,t ), (4.1) where u0 represents the input image. we apply finite differences method to this problem. we denote respectively by h and ∆t the spatial and time steps sizes. in what follows, we take h = 1 and we define for every field p = (p1,p2) ∈r2, the discrete divergence approximation: divi,j(p) =   p1(i, j) −p1(i − 1, j) if 1 < i < n p1(i, j) if i = 1 p1(i − 1, j) if i = n +   p2(i, j) −p2(i, j − 1) if 1 < j < n p2(i, j) if j = 1 −p2(i, j − 1) if j = m, (4.2) 12 int. j. anal. appl. (2023), 21:25 where n and m is an integer greater than 2. one can write the following scheme: uk+1 (i, j) = uk (i, j) + ∆t [ (div (dα(x,∇u) + div(q (x,∇u)))k (i, j) ] , 1 ≤ k ≤ n (4.3) where dα(x,∇u) = ∇up exp(α |∇up|), q (x,∇u) = |∇u|p−2 ∇u, u(tk,xi,yj) = uk(i, j), xi = ih, yj = jh, tk = k∆t, and ∆t = t n . in our numerical tests we take ∆t = t n = 0.1, and we compute the pnsr (peak signal to noise ratio ) quotient of every image. in figs. 1-3, we give some examples by taking α = 0.25, σ is the standard deviation of the distribution which performs an edge-preserving average filter on the image and with different values of p. we give in fig. 4, tests with different values of α, with p = 40. noisy image with salt&pepper = 0.008 p = 40,psnr = 24.2487 noisy image with salt&pepper = 0.08 p = 40,psnr = 16.0574 fig 1. int. j. anal. appl. (2023), 21:25 13 noisy image with σ = 1 p = 900e900,psnr = 15.2328 noisy image with σ = 1 p = 900e900,psnr = 16.4051 fig 2. 14 int. j. anal. appl. (2023), 21:25 noisy image with σ = 0.9 p = 3,psnr = 19.4112 p=300, psnr=25.1073 p = 900e900,psnr = 25.1176 fig 3. int. j. anal. appl. (2023), 21:25 15 noisy image with σ = 0.9 α = 3,psnr = 13.9802 α = 0.25, psnr=20.1509 α = 10e − 15,psnr = 18.5800 fig 4. in numerical tests, we show the better value of a α which gives a good restored image is equal to 0.25, so we should not take α close to 0 and no more than 0.25. 5. conclusion in this article, we presente a parabolic model for image denoising and restoration, with their theoretical results and numerical results. this model preserve the contours of image more than other models. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 16 int. j. anal. appl. (2023), 21:25 references [1] r. aboulaich, d. meskine, a. souissi, new diffusion models in image processing, computers math. appl. 56 (2008), 874–882. https://doi.org/10.1016/j.camwa.2008.01.017. [2] r. adams, sobolev spaces, academic press, new york, 1975. [3] h. alaa, n.e. alaa, a. bouchriti, a. charkaoui, an improved nonlinear anisotropic pde with p(x)-growth conditions applied to image restoration and enhancement, preprint. (2022). https://doi.org/10.13140/rg.2.2. 34551.09126. [4] l. alvarez, l. mazorra, signal and image restoration using shock filters and anisotropic diffusion, siam j. numer. anal. 31 (1994), 590–605. https://doi.org/10.1137/0731032. [5] a. atlas, f. karami, d. meskine, the perona–malik inequality and application to image denoising, nonlinear anal.: real world appl. 18 (2014), 57–68. https://doi.org/10.1016/j.nonrwa.2013.11.006. [6] g. aubert, p. kornprobst, mathematical problems in image processing: partial differential equations and the calculus of variations, springer, new york, 2006. https://doi.org/10.1007/978-0-387-44588-5. [7] a. chambolle, p.l. lions, image recovery via total variation minimization and related problems, numer. math. 76 (1997), 167-188. [8] y. chen, s. levine, m. rao, variable exponent, linear growth functionals in image restoration, siam j. appl. math. 66 (2006), 1383-1406. https://doi.org/10.1137/050624522. [9] u. diewald, t. preusser, m. rumpf, r. strzodka, diffusion models and their accelerated solution in image and surface processing, acta math. univ. comenianae, 70 (2001), 15-34. [10] a. elmahi, d. meskine, strongly nonlinear parabolic equations with natural growth terms in orlicz spaces, nonlinear anal.: theory methods appl. 60 (2005), 1–35. https://doi.org/10.1016/j.na.2004.08.018. [11] a. elmahi, d. meskine, parabolic equations in orlicz spaces, j. london math. soc. 72 (2005), 410–428. https: //doi.org/10.1112/s0024610705006630. [12] j.p. gossez, some approximation properties in orlicz-sobolev spaces, stud. math. 74 (1982), 17-24. [13] p. harjulehto, p. hästö, v. latvala, o. toivanen, critical variable exponent functionals in image restoration, appl. math. lett. 26 (2013), 56–60. https://doi.org/10.1016/j.aml.2012.03.032. [14] m. kbiri alaoui, d. meskine, a. souissi, on some class of nonlinear parabolic inequalities in orlicz spaces, nonlinear anal.: theory methods appl. 74 (2011), 5863–5875. https://doi.org/10.1016/j.na.2011.04.048. [15] p. kogut, y. kohut, r. manzo, existence result and approximation of an optimal control problem for the perona–malik equation, ricerche mat. (2022). https://doi.org/10.1007/s11587-022-00730-4. [16] r. landes, on the existence of weak solutions for quasilinear parabolic initial-boundary value problems, proc. r. soc. edinburgh: sect. a math. 89 (1981), 217-237. https://doi.org/10.1017/s0308210500020242. [17] s. lecheheb, m. maouni, h. lakhal, existence of the solution of a quasilinear equation and its application to image denoising, int. j. computer sci. commun. inform. technol. 7 (2019), 1-6. [18] s. lecheheb, m. maouni, h. lakhal, image restoration using nonlinear eliptic equation, int. j. computer sci. computer sci. commun. inform. technol. 6 (2019), 32-37. [19] h. matallah, m. maouni, h. lakhal, image restoration by a fractional reaction-diffusion process, int. j. anal. appl. 19 (2021), 709-724. https://doi.org/10.28924/2291-8639-19-2021-709. [20] m. maouni, f.z. nouri, image restoration based on p-gradient model, int. j. appl. math. stat. 41 (2013), 48-57. [21] p. perona, j. malik, scale-space and edge detection using anisotropic diffusion, ieee trans. pattern anal. mach. intell. 12 (1990), 629–639. https://doi.org/10.1109/34.56205. [22] l.i. rudin, s. osher, e. fatemi, nonlinear total variation based noise removal algorithms, physica d: nonlinear phenomena. 60 (1992), 259-268. https://doi.org/10.1016/0167-2789(92)90242-f. [23] w. walter, differential and integral inequalities, springer, berlin, new york, (1970). https://doi.org/10.1016/j.camwa.2008.01.017 https://doi.org/10.13140/rg.2.2.34551.09126 https://doi.org/10.13140/rg.2.2.34551.09126 https://doi.org/10.1137/0731032 https://doi.org/10.1016/j.nonrwa.2013.11.006 https://doi.org/10.1007/978-0-387-44588-5 https://doi.org/10.1016/j.na.2004.08.018 https://doi.org/10.1112/s0024610705006630 https://doi.org/10.1112/s0024610705006630 https://doi.org/10.1016/j.aml.2012.03.032 https://doi.org/10.1016/j.na.2011.04.048 https://doi.org/10.1007/s11587-022-00730-4 https://doi.org/10.1017/s0308210500020242 https://doi.org/10.28924/2291-8639-19-2021-709 https://doi.org/10.1109/34.56205 https://doi.org/10.1016/0167-2789(92)90242-f int. j. anal. appl. (2023), 21:25 17 [24] z.f. zohra, m. messaoud, n.f. zohra, overlapping and nonoverlapping domain decomposition methods for image restoration, int. j. appl. math. stat. 40 (2013), 123-128. [25] z.f. zohra, m. maouni, image processing by a fractional partial differential equation, int. j. computer sci. commun. inform. technol. 7 (2019), 13-16. 1. introduction 2. preliminaries 2.1. nfunctions 2.2. the orlicz spaces 2.3. the orlicz-sobolev spaces 2.4. duality in orlicz-sobolev space 2.5. inhomogeneous orlicz-sobolev spaces 2.6. duality in inhomogeneous orlicz-sobolev spaces 3. the main results of the existence 4. numerical results 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 198-211 http://www.etamaths.com properties of meromorphic solutions of a class of second order linear differential equations benharrat belaïdi∗ and habib habib abstract. this paper deals with the growth of meromorphic solutions of some second order linear differential equations, where it is assumed that the coefficients are meromorphic functions. our results extend the previous results due to chen and shon, xu and zhang, peng and chen and others. 1. introduction and statement of result in this paper, we shall assume that the reader is familiar with the fundamental results and the standard notations of the nevanlinna value distribution theory of meromorphic functions (see [13] , [20]). in addition, we will use notations ρ (f) , ρ2 (f) to denote respectively the order and the hyper-order of growth of a meromorphic function f (z). for the second order linear differential equation (1.1) f′′ + e−zf′ + b (z) f = 0, where b (z) is an entire function, it is well-known that each solution f of equation (1.1) is an entire function, and that if f1 and f2 are two linearly independent solutions of (1.1) , then by [7], there is at least one of f1, f2 of infinite order. hence, ”most” solutions of (1.1) will have infinite order. but equation (1.1) with b(z) = −(1 + e−z) possesses a solution f (z) = ez of finite order. a natural question arises: what conditions on b(z) will guarantee that every solution f 6≡ 0 of (1.1) has infinite order? many authors, frei [8], ozawa [16], amemiya-ozawa [1] and gundersen [10], langley [14] have studied this problem. they proved that when b(z) is a nonconstant polynomial or b(z) is a transcendental entire function with order ρ(b) 6= 1, then every solution f 6≡ 0 of (1.1) has infinite order. in 2002, chen [3] considered the question: what conditions on b(z) when ρ(b) = 1 will guarantee that every nontrivial solution of (1.1) has infinite order? he proved the following result, which improved results of frei, amemiya-ozawa, ozawa, langley and gundersen. 2010 mathematics subject classification. 34m10, 30d35. key words and phrases. linear differential equations, meromorphic solutions, order of growth, hyper-order. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 198 properties of meromorphic solutions 199 theorem a [3] let aj (z) (6≡ 0) (j = 0, 1) be entire functions with max{ρ (aj) (j = 0, 1)} < 1, and let a,b be complex constants that satisfy ab 6= 0 and a 6= b. then every solution f 6≡ 0 of the differential equation (1.2) f′′ + a1 (z) e azf′ + a0 (z) e bzf = 0 is of infinite order. in [4], chen and shon have considered equation (1.2) when aj (z) (j = 0, 1) are meromorphic functions and have proved the following result. theorem b ([4]) let aj (z) (6≡ 0) (j = 0, 1) be meromorphic functions with ρ (aj) < 1 (j = 0, 1) , and let a, b be complex numbers such that ab 6= 0 and arg a 6= arg b or a = cb (0 < c < 1) . then every meromorphic solution f (z) 6≡ 0 of equation (1.2) has infinite order. in [17], peng and chen have investigated the order and hyper-order of solutions of some second order linear differential equations and have proved the following result. theorem c [17] let aj (z) ( 6≡ 0) (j = 1, 2) be entire functions with ρ (aj) < 1, a1, a2 be complex numbers such that a1a2 6= 0, a1 6= a2 (suppose that |a1| ≤ |a2|). if arg a1 6= π or a1 < −1, then every solution f ( 6≡ 0) of the differential equation f′′ + e−zf′ + (a1e a1z + a2e a2z) f = 0 has infinite order and ρ2 (f) = 1. recently in [2] , the authors extend and improve the results of theorem c to some second order linear differential equations as follows. theorem d [2] let n ≥ 2 be an integer, aj (z) ( 6≡ 0) (j = 1, 2) be entire functions with max{ρ (aj) : j = 1, 2} < 1, q (z) = qmzm + · · · + q1z + q0 be nonconstant polynomial and a1, a2 be complex numbers such that a1a2 6= 0, a1 6= a2. if (1) arg a1 6= π and arg a1 6= arg a2 or (2) arg a1 6= π, arg a1 = arg a2 and |a2| > n |a1| or (3) a1 < 0 and arg a1 6= arg a2 or (4) −1n (|a2|−m) < a1 < 0, |a2| > m and arg a1 = arg a2, then every solution f 6≡ 0 of the differential equation f′′ + q ( e−z ) f′ + (a1e a1z + a2e a2z) n f = 0 satisfies ρ (f) = +∞ and ρ2 (f) = 1. recently xu and zhang have investigated the order and the hyper-order of meromorphic solutions of some second order linear differential equations and have proved the following result. theorem e [19] suppose that aj (z) (6≡ 0) (j = 0, 1, 2) are meromorphic functions and ρ (aj) < 1, and a1, a2 are two complex numbers such that a1a2 6= 0, a1 6= a2 (suppose that |a1| ≤ |a2|). let a0 be a constant satisfying a0 < 0. if arg a1 6= π or a1 < a0, then every meromorphic solution f ( 6≡ 0) whose poles are of uniformly bounded multiplicities of the equation f′′ + a0e a0zf′ + (a1e a1z + a2e a2z) f = 0 200 belaïdi and habib has infinite order and ρ2 (f) = 1. the main purpose of this paper is to extend and improve the results of theorems a-e to some second order linear differential equations. in fact we will prove the following result. theorem 1.1 let aj (z) ( 6≡ 0) (j = 1, · · · , l1) (l1 ≥ 3) and bj (z) ( 6≡ 0) (j = 1, · · · , l2) (l2 ≥ 1) be meromorphic functions with max{ρ (aj) (j = 1, · · · , l1) ,ρ (bj) (j = 1, · · · , l2)} < 1 and aj 6= 0 (j = 1, · · · , l1) be distinct complex numbers and bj (j = 1, · · · , l2) be distinct real numbers such that bj < 0. suppose that there exist αj, βj (j = 3, · · · , l1) where 0 < αj < 1, 0 < βj < 1 and aj = αja1 + βja2. set α = max{αj : j = 3, · · · , l1}, β = max{βj : j = 3, · · · , l1} and b = min{bj : j = 1, · · · , l2}. if (1) arg a1 6= π and arg a1 6= arg a2 or (2) arg a1 6= π, arg a1 = arg a2 and (i) |a2| > |a1| 1−β or (ii) |a2| < (1 −α) |a1| or (3) a1 < 0 and arg a1 6= arg a2 or (4) (i) (1 −β) a2 − b < a1 < 0, a2 < b1−β or (ii) a1 < a2+b 1−α and a2 < 0, then every meromorphic solution f (6≡ 0) whose poles are of uniformly bounded multiplicities of the differential equation (1.3) f′′ +   l2∑ j=1 bje bjz  f′ +   l1∑ j=1 aje ajz  f = 0 satisfies ρ (f) = +∞ and ρ2 (f) = 1. 2. preliminary lemmas we define the linear measure of a set e ⊂ [0, +∞) by m(e) = ∫ +∞ 0 χe(t)dt and the logarithmic measure of a set f ⊂ (1, +∞) by lm(f) = ∫ +∞ 1 χf (t) t dt, where χh is the characteristic function of a set h. lemma 2.1 [11] let f be a transcendental meromorphic function with ρ (f) = ρ < +∞. let ε > 0 be a given constant, and let k, j be integers satisfying k > j ≥ 0. then, there exists a set e1 ⊂ [ −π 2 , 3π 2 ) with linear measure zero, such that, if ψ ∈ [ −π 2 , 3π 2 ) \e1, then there is a constant r0 = r0 (ψ) > 1, such that for all z satisfying arg z = ψ and |z| ≥ r0, we have (2.1) ∣∣∣∣f(k) (z)f(j) (z) ∣∣∣∣ ≤ |z|(k−j)(ρ−1+ε) . lemma 2.2 ([4] , [15]) consider g (z) = a (z) eaz, where a (z) 6≡ 0 is a meromorphic function with order ρ (a) = α < 1, a is a complex constant, a = |a|eiϕ (ϕ ∈ [0, 2π)). set e2 = {θ ∈ [0, 2π) : cos (ϕ + θ) = 0}, then e2 is a finite set. then for any given ε (0 < ε < 1 −α) there is a set e3 ⊂ [0, 2π) that has linear measure zero such that if z = reiθ, θ ∈ [0, 2π) � (e2 ∪e3), then we have when r is sufficiently large: properties of meromorphic solutions 201 (i) if cos (ϕ + θ) > 0, then (2.2) exp{(1 −ε) rδ (az,θ)}≤ |g (z)| ≤ exp{(1 + ε) rδ (az,θ)} . (ii) if cos (ϕ + θ) < 0, then (2.3) exp{(1 + ε) rδ (az,θ)}≤ |g (z)| ≤ exp{(1 −ε) rδ (az,θ)} , where δ (az,θ) = |a|cos (ϕ + θ) . lemma 2.3 [17] suppose that n ≥ 1 is a natural number. let pj (z) = ajnzn + · · · (j = 1, 2) be nonconstant polynomials, where ajq ( q = 1, · · · ,n) are complex numbers and a1na2n 6= 0. set z = reiθ, ajn = |ajn|eiθj , θj ∈ [ −π 2 , 3π 2 ) , δ (pj,θ) = |ajn|cos (θj + nθ), then there is a set e4 ⊂ [ − π 2n , 3π 2n ) that has linear measure zero such that if θ1 6= θ2, then there exists a ray arg z = θ with θ ∈ ( − π 2n , π 2n ) \(e4 ∪e5) satisfying either (2.4) δ (p1,θ) > 0, δ (p2,θ) < 0 or (2.5) δ (p1,θ) < 0, δ (p2,θ) > 0, where e5 = { θ ∈ [ − π 2n , 3π 2n ) : δ (pj,θ) = 0 } is a finite set, which has linear measure zero. remark 2.1 [17] we can obtain, in lemma 2.3, if θ ∈ ( − π 2n , π 2n ) \ (e4 ∪e5) is replaced by θ ∈ ( π 2n , 3π 2n ) \ (e4 ∪e5), then it has the same result. lemma 2.4 [4] let f (z) be a transcendental meromorphic function of order ρ (f) = α < +∞. then for any given ε > 0, there is a set e6 ⊂ [ −π 2 , 3π 2 ) that has linear measure zero such that if θ ∈ [ −π 2 , 3π 2 ) �e6, then there is a constant r1 = r1 (θ) > 1, such that for all z satisfying arg z = θ and |z| ≥ r1, we have (2.6) exp { −rα+ε } ≤ |f (z)| ≤ exp { rα+ε } . lemma 2.5 [11] let f(z) be a transcendental meromorphic function, and let α > 1 be a given constant. then there exist a set e7 ⊂ (1,∞) with finite logarithmic measure and a constant b > 0 that depends only on α and i,j (0 ≤ i < j ≤ k), such that for all z satisfying |z| = r /∈ [0, 1] ∪e7, we have (2.7) ∣∣∣∣f(j)(z)f(i)(z) ∣∣∣∣ ≤ b { t(αr,f) r (logα r) log t(αr,f) }j−i . lemma 2.6 [12] let ϕ : [0, +∞) → r and ψ : [0, +∞) → r be monotone nondecreasing functions such that ϕ (r) ≤ ψ (r) for all r /∈ e8 ∪ [0, 1], where e8 ⊂ (1, +∞) is a set of finite logarithmic measure. let γ > 1 be a given constant. then there exists an r1 = r1 (γ) > 0 such that ϕ (r) ≤ ψ (γr) for all r > r1. lemma 2.7 [5] let k ≥ 2 and a0,a1, · · · ,ak−1 be meromorphic functions. let ρ = max{ρ (aj) : j = 0, · · · ,k − 1} and all poles of f are of uniformly bounded multiplicities. then every transcendental meromorphic solution f of the differential equation f(k) + ak−1f (k−1) + · · · + a1f′ + a0f = 0 satisfies ρ2 (f) ≤ ρ. 202 belaïdi and habib lemma 2.8 ([9] , [20]) suppose that f1 (z) ,f2 (z) , · · · ,fn (z) (n ≥ 2) are meromorphic functions and g1 (z) ,g2 (z) , · · · ,gn (z) are entire functions satisfying the following conditions: (i) n∑ j=1 fj (z) e gj (z) ≡ 0; (ii) gj (z) −gk (z) are not constants for 1 ≤ j < k ≤ n; (iii) for 1 ≤ j ≤ n, 1 ≤ h < k ≤ n, t (r,fj) = o { t ( r,egh(z)−gk(z) )} (r → ∞, r /∈ e9), where e9 is a set with finite linear measure. then fj (z) ≡ 0 (j = 1, · · · ,n). lemma 2.9 [18] suppose that f1 (z) ,f2 (z) , · · · ,fn (z) (n ≥ 2) are meromorphic functions and g1 (z) ,g2 (z) , · · · ,gn (z) are entire functions satisfying the following conditions: (i) n∑ j=1 fj (z) e gj (z) ≡ fn+1; (ii) if 1 ≤ j ≤ n + 1, 1 ≤ k ≤ n, the order of fj is less than the order of egk(z). if n ≥ 2, 1 ≤ j ≤ n + 1, 1 ≤ h < k ≤ n, and the order of fj is less than the order of egh−gk . then fj (z) ≡ 0 (j = 1, 2, · · · ,n + 1). 3. proof of theorem 1.1 first of all we prove that equation (1.3) can’t have a meromorphic solution f 6≡ 0 with ρ (f) < 1. assume a meromorphic solution f 6≡ 0 with ρ (f) < 1. we can rewrite (1.3) in the following form (3.1) l2∑ j=1 bjf ′ebjz + l1∑ j=1 ajfe ajz = −f′′. obviously, ρ (bjf ′) < 1 (j = 1, · · · , l2) and ρ (ajf) < 1 (j = 1, · · · , l1). set i = {aj (j = 1, · · · , l1) , bj (j = 1, · · · , l2)}. 1) by the conditions (1) or (2) or (4) (ii) of theorem 1.1, we can see that a1 6= a2,a3, · · · ,al1,b1, · · · ,bl2 . then, we can rewrite (3.1) in the following form (3.2) a1fe a1z + ∑ λ∈γ1 fλe λz = −f′′, where γ1 ⊆ i \ {a1} and fλ (λ ∈ γ1) are meromorphic functions with order less than 1 and a1, λ (λ ∈ γ1) are distinct numbers. by lemma 2.8 and lemma 2.9, we get a1 ≡ 0, which is a contradiction. 2) by the conditions (3) or (4) (i) of theorem 1.1, we can see that a2 6= a1,a3, · · · ,al1 , b1, · · · ,bl2 . then, we can rewrite (3.1) in the following form (3.3) a2fe a2z + ∑ λ∈γ2 fλe λz = −f′′, where γ2 ⊆ i \ {a2} and fλ (λ ∈ γ2) are meromorphic functions with order less than 1 and a2, λ (λ ∈ γ2) are distinct numbers. by lemma 2.8 and lemma 2.9, we get a2 ≡ 0, which is a contradiction. therefore ρ (f) ≥ 1. properties of meromorphic solutions 203 first step. we prove that ρ (f) = +∞. assume that f 6≡ 0 is a meromorphic solution whose poles are of uniformly bounded multiplicities of equation (1.3) with 1 ≤ ρ (f) = σ1 < +∞. from equation (1.3), we know that the poles of f (z) can occur only at the poles of aj (j = 1, · · · ., l1) and bj (j = 1, · · · , l2). note that the multiplicities of poles of f are uniformly bounded, and thus we have [6] n (r,f) ≤ m1n (r,f) ≤ m1   l1∑ j=1 n (r,aj) + l2∑ j=1 n (r,bj)   ≤ m max{n (r,aj) (j = 1, · · · , l1) ,n (r,bj) (j = 1, · · · , l2)} , where m1 and m are some suitable positive constants. this gives λ ( 1 f ) ≤ γ = max{ρ (aj) (j = 1, · · · , l1) ,ρ (bj) (j = 1, · · · , l2)} < 1. let f = g/d, d be the canonical product formed with the nonzero poles of f (z), with ρ (d) = λ (d) = λ ( 1 f ) = σ2 ≤ γ < 1, g is an entire function and 1 ≤ ρ (g) = ρ (f) = σ1 < ∞. substituting f = g/d into (1.3), we can get g′′ g +     l2∑ j=1 bje bjz  − 2d′ d   g′ g + 2 ( d′ d )2 − d′′ d −   l2∑ j=1 bje bjz   d′ d (3.4) +a1e a1z + a2e a2z + l1∑ j=3 aje (αja1+βja2)z = 0. by lemma 2.4, for any given ε (0 < ε < 1 −γ), there is a set e6 ⊂ [ −π 2 , 3π 2 ) that has linear measure zero such that if θ ∈ [ −π 2 , 3π 2 ) �e6, then there is a constant r1 = r1 (θ) > 1, such that for all z satisfying arg z = θ and |z| ≥ r1, we have (3.5) |bj (z)| ≤ exp { rγ+ε } (j = 1, · · · , l2) . by lemma 2.1, for any given ε (0 < ε < 1 −γ), there exists a set e1 ⊂ [ −π 2 , 3π 2 ) of linear measure zero, such that if θ ∈ [ −π 2 , 3π 2 ) \ e1, then there is a constant r0 = r0 (θ) > 1, such that for all z satisfying arg z = θ and |z| = r ≥ r0, we have (3.6) ∣∣∣∣g(j) (z)g (z) ∣∣∣∣ ≤ rj(σ1−1+ε) (j = 1, 2) , (3.7) ∣∣∣∣d(j) (z)d (z) ∣∣∣∣ ≤ rj(σ2−1+ε) (j = 1, 2) . let z = reiθ, a1 = |a1|eiθ1 , a2 = |a2|eiθ2 , θ1,θ2 ∈ [ −π 2 , 3π 2 ) . we know that δ (αja1z,θ) = αjδ (a1z,θ), δ (βja2z,θ) = βjδ (a2z,θ) (j = 3, · · · , l1) and α < 1, β < 1. case 1. assume that arg a1 6= π and arg a1 6= arg a2, which is θ1 6= π and θ1 6= θ2. by lemma 2.2 and lemma 2.3, for any given ε 0 < ε < min { 1 −γ, 1 −α 2 (1 + α) , 1 −β 2 (1 + β) } , 204 belaïdi and habib there is a ray arg z = θ such that θ ∈ ( −π 2 , π 2 ) \(e1 ∪e4 ∪e5 ∪e6) (where e4 and e5 are defined as in lemma 2.3, e1 ∪e4 ∪e5 ∪e6 is of linear measure zero), and satisfying δ (a1z,θ) > 0, δ (a2z,θ) < 0 or δ (a1z,θ) < 0, δ (a2z,θ) > 0. (a) when δ (a1z,θ) > 0, δ (a2z,θ) < 0, for sufficiently large r, we get by lemma 2.2 (3.8) |a1ea1z| ≥ exp{(1 −ε) δ (a1z,θ) r} , (3.9) |a2ea2z| ≤ exp{(1 −ε) δ (a2z,θ) r} < 1, |ajeαja1z| ≤ exp{(1 + ε) αjδ (a1z,θ) r} (3.10) ≤ exp{(1 + ε) αδ (a1z,θ) r} (j = 3, · · · , l1) , (3.11) ∣∣eβja2z∣∣ ≤ exp{(1 −ε) βjδ (a2z,θ) r} < 1 (j = 3, · · · , l1) . by (3.10) and (3.11), we get∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ ≤ l1∑ j=3 |ajeαja1z| ∣∣eβja2z∣∣ (3.12) ≤ (l1 − 2) exp{(1 + ε) αδ (a1z,θ) r} . for θ ∈ ( −π 2 , π 2 ) by (3.5) we have∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ≤ l2∑ j=1 |bj| ∣∣ebjz∣∣ ≤ exp {rγ+ε} l2∑ j=1 ∣∣ebjz∣∣ (3.13) = exp { rγ+ε } l2∑ j=1 ebjr cos θ ≤ l2 exp { rγ+ε } because bj < 0 and cos θ > 0. by (3.4), we obtain |a1ea1z| ≤ ∣∣∣∣g′′g ∣∣∣∣ +   ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ + 2 ∣∣∣∣d′d ∣∣∣∣  ∣∣∣∣g′g ∣∣∣∣ + 2 ∣∣∣∣d′d ∣∣∣∣2 + ∣∣∣∣d′′d ∣∣∣∣ (3.14) + ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ∣∣∣∣d′d ∣∣∣∣ + |a2ea2z| + ∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ . substituting (3.6) − (3.9) , (3.12) and (3.13) into (3.14), we have exp{(1 −ε) δ (a1z,θ) r}≤ |a1ea1z| ≤ r2(σ1−1+ε) + [ l2 exp { rγ+ε } + 2rσ2−1+ε ] rσ1−1+ε + 3r2(σ2−1+ε) +l2 exp { rγ+ε } rσ2−1+ε + 1 + (l1 − 2) exp{(1 + ε) αδ (a1z,θ) r} (3.15) ≤ m1rm2 exp { rγ+ε } exp{(1 + ε) αδ (a1z,θ) r} , properties of meromorphic solutions 205 where m1 > 0 and m2 > 0 are some constants. by 0 < ε < 1−α 2(1+α) and (3.15), we have (3.16) exp { 1 −α 2 δ (a1z,θ) r } ≤ m1rm2 exp { rγ+ε } . by δ (a1z,θ) > 0 and γ + ε < 1 we know that (3.16) is a contradiction. (b) when δ (a1z,θ) < 0, δ (a2z,θ) > 0, for sufficiently large r, we get (3.17) |a2ea2z| ≥ exp{(1 −ε) δ (a2z,θ) r} , (3.18) |a1ea1z| ≤ exp{(1 −ε) δ (a1z,θ) r} < 1, (3.19) |ajeαja1z| ≤ exp{(1 −ε) αjδ (a1z,θ) r} < 1 (j = 3, · · · , l1) ,∣∣eβja2z∣∣ ≤ exp{(1 + ε) βjδ (a2z,θ) r} (3.20) ≤ exp{(1 + ε) βδ (a2z,θ) r} (j = 3, · · · , l1) . by (3.19) and (3.20), we get∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ ≤ l1∑ j=3 |ajeαja1z| ∣∣eβja2z∣∣ (3.21) ≤ (l1 − 2) exp{(1 + ε) βδ (a2z,θ) r} . by (3.4), we obtain |a2ea2z| ≤ ∣∣∣∣g′′g ∣∣∣∣ +   ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ + 2 ∣∣∣∣d′d ∣∣∣∣  ∣∣∣∣g′g ∣∣∣∣ + 2 ∣∣∣∣d′d ∣∣∣∣2 + ∣∣∣∣d′′d ∣∣∣∣ (3.22) + ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ∣∣∣∣d′d ∣∣∣∣ + |a1ea1z| + ∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ . substituting (3.6) , (3.7) , (3.13) , (3.17) , (3.18) and (3.21) into (3.22), we have exp{(1 −ε) δ (a2z,θ) r}≤ |a2ea2z| ≤ r2(σ1−1+ε) + [ l2 exp { rγ+ε } + 2rσ2−1+ε ] rσ1−1+ε + 3r2(σ2−1+ε) +l2 exp { rγ+ε } rσ2−1+ε + 1 + (l1 − 2) exp{(1 + ε) βδ (a2z,θ) r} (3.23) ≤ m1rm2 exp { rγ+ε } exp{(1 + ε) βδ (a2z,θ) r} . by 0 < ε < 1−β 2(1+β) and (3.23), we obtain (3.24) exp { 1 −β 2 δ (a2z,θ) r } ≤ m1rm2 exp { rγ+ε } . by δ (a2z,θ) > 0 and γ + ε < 1 we know that (3.24) is a contradiction. case 2. assume that arg a1 6= π, arg a1 = arg a2, which is θ1 6= π, θ1 = θ2. by lemma 2.3, for any given ε 0 < ε < min { 1 −γ, (1 −α) |a1|− |a2| 2 [(1 + α) |a1| + |a2|] , (1 −β) |a2|− |a1| 2 [(1 + β) |a2| + |a1|] } , 206 belaïdi and habib there is a ray arg z = θ such that θ ∈ ( −π 2 , π 2 ) \(e1 ∪e4 ∪e5 ∪e6) and δ (a1z,θ) > 0. since θ1 = θ2, then δ (a2z,θ) > 0. (i) |a2| > |a1| 1−β . for sufficiently large r, we have (3.10) , (3.17) , (3.20) hold and we get (3.25) |a1ea1z| ≤ exp{(1 + ε) δ (a1z,θ) r} . by (3.10) and (3.20), we obtain∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ ≤ l1∑ j=3 |ajeαja1z| ∣∣eβja2z∣∣ (3.26) ≤ (l1 − 2) exp{(1 + ε) αδ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} . substituting (3.6) , (3.7) , (3.13) , (3.17) , (3.25) and (3.26) into (3.22), we have exp{(1 −ε) δ (a2z,θ) r}≤ |a2ea2z| ≤ r2(σ1−1+ε) + [ l2 exp { rγ+ε } + 2rσ2−1+ε ] rσ1−1+ε + 3r2(σ2−1+ε) +l2 exp { rγ+ε } rσ2−1+ε + exp{(1 + ε) δ (a1z,θ) r} + (l1 − 2) exp{(1 + ε) αδ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} (3.27) ≤ m1rm2 exp { rγ+ε } exp{(1 + ε) δ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} . from (3.27), we obtain (3.28) exp{η1r}≤ m1rm2 exp { rγ+ε } , where η1 = (1 −ε) δ (a2z,θ) − (1 + ε) δ (a1z,θ) − (1 + ε) βδ (a2z,θ) . since 0 < ε < (1−β)|a2|−|a1| 2[(1+β)|a2|+|a1|] ,θ1 = θ2 and cos (θ1 + θ) > 0, then η1 = [1 −β −ε (1 + β)] δ (a2z,θ) − (1 + ε) δ (a1z,θ) = [1 −β −ε (1 + β)] |a2|cos (θ1 + θ) − (1 + ε) |a1|cos (θ1 + θ) = {[1 −β −ε (1 + β)] |a2|− (1 + ε) |a1|}cos (θ1 + θ) = {(1 −β) |a2|− |a1|−ε [(1 + β) |a2| + |a1|]}cos (θ1 + θ) > (1 −β) |a2|− |a1| 2 cos (θ1 + θ) > 0. by η1 > 0 and γ + ε < 1 we know that (3.28) is a contradiction. (ii) |a2| < (1 −α) |a1|. for sufficiently large r, we have (3.8) , (3.10) , (3.20) , (3.26) hold and we obtain (3.29) |a2ea2z| ≤ exp{(1 + ε) δ (a2z,θ) r} . substituting (3.6) , (3.7) , (3.8) , (3.13) , (3.26) and (3.29) into (3.14), we have exp{(1 −ε) δ (a1z,θ) r}≤ |a1ea1z| ≤ r2(σ1−1+ε) + [ l2 exp { rγ+ε } + 2rσ2−1+ε ] rσ1−1+ε + 3r2(σ2−1+ε) +l2 exp { rγ+ε } rσ2−1+ε + exp{(1 + ε) δ (a2z,θ) r} + (l1 − 2) exp{(1 + ε) αδ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} (3.30) ≤ m1rm2 exp { rγ+ε } exp{(1 + ε) αδ (a1z,θ) r}exp{(1 + ε) δ (a2z,θ) r} . properties of meromorphic solutions 207 from (3.30), we obtain (3.31) exp{η2r}≤ m1rm2 exp { rγ+ε } , where η2 = (1 −ε) δ (a1z,θ) − (1 + ε) αδ (a1z,θ) − (1 + ε) δ (a2z,θ) . since 0 < ε < (1−α)|a1|−|a2| 2[(1+α)|a1|+|a2|] ,θ1 = θ2 and cos (θ1 + θ) > 0, then we get η2 = {(1 −α) |a1|− |a2|−ε [(1 + α) |a1| + |a2|]}cos (θ1 + θ) > (1 −α) |a1|− |a2| 2 cos (θ1 + θ) > 0. by η2 > 0 and γ + ε < 1 we know that (3.31) is a contradiction. case 3. assume that a1 < 0 and arg a1 6= arg a2, which is θ1 = π and θ2 6= π. by lemma 2.2, for the above ε, there is a ray arg z = θ such that θ ∈ ( −π 2 , π 2 ) \ (e1 ∪e4 ∪e5 ∪e6) and δ (a2z,θ) > 0. because cos θ > 0, we have δ (a1z,θ) = |a1|cos (θ1 + θ) = −|a1|cos θ < 0. using the same reasoning as in case 1(b), we can get a contradiction. case 4. assume that (i) (1 −β) a2 − b < a1 < 0 and a2 < b1−β or (ii) a1 < a2+b 1−α and a2 < 0, which is θ1 = θ2 = π. by lemma 2.2, for any given ε 0 < ε < min { 1 −γ, (1 −α) |a1|− |a2| + b 2 [(1 + α) |a1| + |a2|] , (1 −β) |a2|− |a1| + b 2 [(1 + β) |a2| + |a1|] } , there is a ray arg z = θ such that θ ∈ ( π 2 , 3π 2 ) \ (e1 ∪e4 ∪e5 ∪e6), then cos θ < 0, δ (a1z,θ) = |a1|cos (θ1 + θ) = −|a1|cos θ > 0, δ (a2z,θ) = |a2|cos (θ2 + θ) = −|a2|cos θ > 0. (i) (1 −β) a2 − b < a1 < 0 and a2 < b1−β . for sufficiently large r, we get (3.10) , (3.17) , (3.20) , (3.25) and (3.26) hold. for θ ∈ ( π 2 , 3π 2 ) by (3.5) we have∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ≤ l2∑ j=1 |bj| ∣∣ebjz∣∣ ≤ exp {rγ+ε} l2∑ j=1 ∣∣ebjz∣∣ (3.32) = exp { rγ+ε } l2∑ j=1 ebjr cos θ ≤ l2 exp { rγ+ε } ebr cos θ because b ≤ bj < 0 and cos θ < 0. substituting (3.6) , (3.7) , (3.17) , (3.25) , (3.26) and (3.32) into (3.22), we obtain exp{(1 −ε) δ (a2z,θ) r}≤ |a2ea2z| (3.33) ≤ m1rm2ebr cos θ exp { rγ+ε } exp{(1 + ε) δ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} . from (3.33) we have (3.34) exp{η3r}≤ m1rm2 exp { rγ+ε } , where η3 = (1 −ε) δ (a2z,θ) − (1 + ε) δ (a1z,θ) − (1 + ε) βδ (a2z,θ) − b cos θ. 208 belaïdi and habib since (1 −β) a2 − b < a1, a2 = −|a2| and a1 = −|a1|, then we get (1 −β) |a2|− |a1| + b > 0. we can see that 0 < (1 −β) |a2| − |a1| + b < (1 −β) |a2| − |a1| < 2 [(1 + β) |a2| + |a1|]. therefore 0 < (1 −β) |a2|− |a1| + b 2 [(1 + β) |a2| + |a1|] < 1. by 0 < ε < (1−β)|a2|−|a1|+b 2[(1+β)|a2|+|a1|] ,θ1 = θ2 = π and cos θ < 0, we obtain η3 = [1 −β −ε (1 + β)] δ (a2z,θ) − (1 + ε) δ (a1z,θ) − b cos θ = − [1 −β −ε (1 + β)] |a2|cos θ + (1 + ε) |a1|cos θ − b cos θ = (−cos θ){[1 −β −ε (1 + β)] |a2|− (1 + ε) |a1| + b} = (−cos θ){(1 −β) |a2|− |a1| + b−ε [(1 + β) |a2| + |a1|]} > −1 2 [(1 −β) |a2|− |a1| + b] cos θ > 0. by η3 > 0 and γ + ε < 1 we know that (3.34) is a contradiction. (ii) a1 < a2+b 1−α and a2 < 0. for sufficiently large r, we get (3.8) , (3.10) , (3.20) , (3.26) and (3.29) hold. substituting (3.6) , (3.7) , (3.8) , (3.26) , (3.29) and (3.32) into (3.14), we obtain exp{(1 −ε) δ (a1z,θ) r}≤ |a1ea1z| (3.35) ≤ m1rm2ebr cos θ exp { rγ+ε } exp{(1 + ε) αδ (a1z,θ) r}exp{(1 + ε) δ (a2z,θ) r} . from (3.35) we have (3.36) exp{η4r}≤ m1rm2 exp { rγ+ε } , where η4 = (1 −ε) δ (a1z,θ) − (1 + ε) αδ (a1z,θ) − (1 + ε) δ (a2z,θ) − b cos θ. since a1 < a2+b 1−α ,a2 = −|a2| and a1 = −|a1|, then we get (1 −α) |a1| − |a2| + b > 0. we can see that 0 < (1 −α) |a1| − |a2| + b < (1 −α) |a1| − |a2| < 2 [(1 + α) |a1| + |a2|]. therefore 0 < (1 −α) |a1|− |a2| + b 2 [(1 + α) |a1| + |a2|] < 1. by 0 < ε < (1−α)|a1|−|a2|+b 2[(1+α)|a1|+|a2|] ,θ1 = θ2 = π and cos θ < 0, we get η4 = (−cos θ){(1 −α) |a1|− |a2| + b−ε [(1 + α) |a1| + |a2|]} > −1 2 [(1 −α) |a1|− |a2| + b] cos θ > 0. by η4 > 0 and γ + ε < 1 we know that (3.36) is a contradiction. concluding the above proof, we obtain ρ (f) = ρ (g) = +∞. second step. we prove that ρ2 (f) = 1. by max  ρ   l2∑ j=1 bje bjz   ,ρ   l1∑ j=1 aje ajz     = 1 properties of meromorphic solutions 209 and lemma 2.7, we obtain ρ2 (f) ≤ 1. by lemma 2.5, we know that there exists a set e7 ⊂ (1, +∞) with finite logarithmic measure and a constant c > 0, such that for all z satisfying |z| = r /∈ [0, 1] ∪e7, we get (3.37) ∣∣∣∣f(j)(z)f(z) ∣∣∣∣ ≤ c [t(2r,f)]j+1 (j = 1, 2) . by (1.3), we have (3.38) |a1ea1z| ≤ ∣∣∣∣f′′f ∣∣∣∣ + ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ∣∣∣∣f′f ∣∣∣∣ + |a2ea2z| + ∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ , (3.39) |a2ea2z| ≤ ∣∣∣∣f′′f ∣∣∣∣ + ∣∣∣∣∣∣ l2∑ j=1 bje bjz ∣∣∣∣∣∣ ∣∣∣∣f′f ∣∣∣∣ + |a1ea1z| + ∣∣∣∣∣∣ l1∑ j=3 aje (αja1+βja2)z ∣∣∣∣∣∣ . case 1. arg a1 6= π and arg a1 6= arg a2. in the first step, we have proved that there is a ray arg z = θ where θ ∈ ( −π 2 , π 2 ) \ (e1 ∪e4 ∪e5 ∪e6), satisfying δ (a1z,θ) > 0, δ (a2z,θ) < 0 or δ (a1z,θ) < 0, δ (a2z,θ) > 0. (a) when δ (a1z,θ) > 0, δ (a2z,θ) < 0, for sufficiently large r, we get (3.8) − (3.12) hold. substituting (3.8) , (3.9) , (3.12) , (3.13) and (3.37) into (3.38), we obtain for all z = reiθ satisfying |z| = r /∈ [0, 1] ∪e7, θ ∈ ( −π 2 , π 2 ) \ (e1 ∪e4 ∪e5 ∪e6) exp{(1 −ε) δ (a1z,θ) r}≤ |a1ea1z| (3.40) ≤ m exp { rγ+ε } exp{(1 + ε) αδ (a1z,θ) r} [t (2r,f)] 3 , where m > 0 is a some constant. from (3.40) and 0 < ε < 1−α 2(1+α) , we get (3.41) exp { 1 −α 2 δ (a1z,θ) r } ≤ m exp { rγ+ε } [t (2r,f)] 3 . since δ (a1z,θ) > 0 and γ + ε < 1, then by using lemma 2.6 and (3.41), we obtain ρ2 (f) ≥ 1. hence ρ2 (f) = 1. (b) when δ (a1z,θ) < 0, δ (a2z,θ) > 0, for sufficiently large r, we get (3.17)−(3.21) hold. by using the same reasoning as above, we can get ρ2 (f) = 1. case 2. arg a1 6= π, arg a1 = arg a2. in the first step, we have proved that there is a ray arg z = θ where θ ∈ ( −π 2 , π 2 ) \ (e1 ∪e4 ∪e5 ∪e6), satisfying δ (a1z,θ) > 0 and δ (a2z,θ) > 0. (i) |a2| > |a1| 1−β . for sufficiently large r, we have (3.10) , (3.17) , (3.20) , (3.25) and (3.26) hold. substituting (3.13) , (3.17) , (3.25) , (3.26) and (3.37) into (3.39), we obtain for all z = reiθ satisfying |z| = r /∈ [0, 1]∪e7, θ ∈ ( −π 2 , π 2 ) \(e1 ∪e4 ∪e5 ∪e6) exp{(1 −ε) δ (a2z,θ) r}≤ |a2ea2z| (3.42) ≤ m exp { rγ+ε } exp{(1 + ε) δ (a1z,θ) r}exp{(1 + ε) βδ (a2z,θ) r} [t (2r,f)] 3 . from (3.42), we obtain (3.43) exp{η1r}≤ m exp { rγ+ε } [t (2r,f)] 3 , 210 belaïdi and habib where η1 = (1 −ε) δ (a2z,θ) − (1 + ε) δ (a1z,θ) − (1 + ε) βδ (a2z,θ) . since η1 > 0 and γ + ε < 1, then by using lemma 2.6 and (3.43), we obtain ρ2 (f) ≥ 1. hence ρ2 (f) = 1. (ii) |a2| < (1 −α) |a1|. for sufficiently large r, we have (3.8) , (3.10) , (3.20) , (3.26) and (3.29) hold. by using the same reasoning as above, we can get ρ2 (f) = 1. case 3. a1 < 0 and arg a1 6= arg a2. in the first step, we have proved that there is a ray arg z = θ where θ ∈ ( −π 2 , π 2 ) \ (e1 ∪e4 ∪e5 ∪e6), satisfying δ (a2z,θ) > 0 and δ (a1z,θ) < 0. using the same reasoning as in the second step ( case 1 (b)), we can get ρ2 (f) = 1. case 4. (i) (1 −β) a2 − b < a1 < 0 and a2 < b1−β or (ii) a1 < a2+b 1−α and a2 < 0. in the first step, we have proved that there is a ray arg z = θ where θ ∈ ( π 2 , 3π 2 ) \ (e1 ∪e4 ∪e5 ∪e6), satisfying δ (a2z,θ) > 0 and δ (a1z,θ) > 0. (i) (1 −β) a2 − b < a1 < 0 and a2 < b1−β . for sufficiently large r, we get (3.10) , (3.17) , (3.20) , (3.25) and (3.26) hold. substituting (3.17) , (3.25) , (3.26) , (3.32) and (3.37) into (3.39), we obtain for all z = reiθ satisfying |z| = r /∈ [0, 1] ∪ e7, θ ∈ ( π 2 , 3π 2 ) \ (e1 ∪e4 ∪e5 ∪e6) exp{(1 −ε) δ (a2z,θ) r}≤ |a2ea2z| ≤ mebr cos θ exp { rγ+ε } exp{(1 + ε) δ (a1z,θ) r} (3.44) ×exp{(1 + ε) βδ (a2z,θ) r} [t (2r,f)] 3 . from (3.44) we obtain (3.45) exp{η3r}≤ m exp { rγ+ε } [t (2r,f)] 3 , where η3 = (1 −ε) δ (a2z,θ) − (1 + ε) δ (a1z,θ) − (1 + ε) βδ (a2z,θ) − b cos θ. since η3 > 0 and γ + ε < 1, then by using lemma 2.6 and (3.45), we obtain ρ2 (f) ≥ 1. hence ρ2 (f) = 1. (ii) a1 < a2+b 1−α and a2 < 0. for sufficiently large r, we get (3.8) , (3.10) , (3.20) , (3.26) and (3.29) hold. by using the same reasoning as above, we can get ρ2 (f) = 1. concluding the above proof, we obtain that every meromorphic solution f ( 6≡ 0) whose poles are of uniformly bounded multiplicities of (1.3) satisfies ρ (f) = ∞ and ρ2 (f) = 1. the proof of theorem 1.1 is complete. references [1] i. amemiya and m. ozawa, non-existence of finite order solutions of w′′+e−zw′+q (z) w = 0, hokkaido math. j. 10 (1981), special issue, 1–17. [2] b. beläıdi and h. habib, on the growth of solutions of some second order linear differential equations with entire coefficients, an. şt. univ. ovidius constanţa, vol. 21(2), (2013), 35-52. [3] z. x. chen, the growth of solutions of f′′ + e−zf′ + q (z) f = 0 where the order (q) = 1, sci. china ser. a 45 (2002), no. 3, 290–300. properties of meromorphic solutions 211 [4] z. x. chen and k. h. shon, on the growth and fixed points of solutions of second order differential equations with meromorphic coefficients, acta math. sin. (engl. ser.) 21 (2005), no. 4, 753-764. [5] w. j. chen and j. f. xu, growth order of meromorphic solutions of higher-order linear differential equations, electron. j. qual. theory differ. equ., (2009), no. 1, 1-13. [6] y. m. chiang and w. k. hayman, estimates on the growth of meromorphic solutions of linear differential equations, comment. math. helv. 79 (2004), no. 3, 451-470. [7] m. frei, über die lösungen linearer differentialgleichungen mit ganzen funktionen als koeffizienten, comment. math. helv. 35 (1961), 201–222. [8] m. frei, über die subnormalen lösungen der differentialgleichung w′′ + e−zw′ + (konst.) w = 0, comment. math. helv. 36 (1961), 1–8. [9] f. gross, on the distribution of values of meromorphic functions, trans. amer. math. soc. 131(1968), 199-214. [10] g. g. gundersen, on the question of whether f′′ + e−zf′ + b (z) f = 0 can admit a solution f 6≡ 0 of finite order, proc. roy. soc. edinburgh sect. a 102 (1986), no. 1-2, 9–17. [11] g. g. gundersen, estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, j. london math. soc. (2) 37(1988), no. 1, 88-104. [12] g. g. gundersen, finite order solutions of second order linear differential equations, trans. amer. math. soc. 305 (1988), no. 1, 415-429. [13] w. k. hayman, meromorphic functions, oxford mathematical monographs clarendon press, oxford 1964. [14] j. k. langley, on complex oscillation and a problem of ozawa, kodai math. j. 9 (1986), no. 3, 430–439. [15] a. i. markushevich, theory of functions of a complex variable, vol. ii, translated by r. a. silverman, prentice-hall, englewood cliffs, new jersey, 1965. [16] m. ozawa, on a solution of w′′ + e−zw′ + (az + b) w = 0, kodai math. j. 3 (1980), no. 2, 295–309. [17] f. peng and z. x. chen, on the growth of solutions of some second-order linear differential equations, j. inequal. appl. 2011, art. id 635604, 1-9. [18] j. f. xu and h. x. yi, the relations between solutions of higher order differential equations with functions of small growth, acta math. sinica, chinese series, 53 (2010), 291-296. [19] j. f. xu and x. b. zhang, some results of meromorphic solutions of second-order linear differential equations, j. inequal. appl. 2013, 2013:304, 14 pp. [20] c. c. yang and h. x. yi, uniqueness theory of meromorphic functions, mathematics and its applications, 557. kluwer academic publishers group, dordrecht, 2003. department of mathematics, laboratory of pure and applied mathematics, university of mostaganem (umab), b. p. 227 mostaganem-(algeria) ∗corresponding author int. j. anal. appl. (2023), 21:38 bi-ideals and weak bi-ideals of near left almost rings thiti gaketem∗ fuzzy algebras and decision-making problems research unit, department of mathematics school of science, university of phayao, phayao 56000, thailand ∗corresponding author: thiti.up.ac.th abstract. in this paper, we define bi-ideals and weak bi-ideals of nla-ring. we investigate the properties of bi-ideals and weak bi-ideals of nla-ring. 1. introduction m.a. kazim and md. naseeruddin defined la-semigroup as the following; a groupoid s is called a left almost semigroup, abbreviated as la-semigroup if (ab)c =(cb)a, ∀a,b,c ∈s m.a. kazim and md. naseeruddin [1, proposition 2.1] asserted that, in every la-semigroups g a medial law hold (a ·b) · (c ·d)= (a ·c) · (b ·d), ∀a,b,c,d ∈g. q. mushtaq and m. khan [3, p.322] asserted that, in every la-semigroups g with left identity (a ·b) · (c ·d)= (d ·b) · (c ·a), ∀a,b,c,d ∈g. further m. khan, faisal, and v. amjid [2], asserted that, if a la-semigroup g with left identity the following law holds a · (b ·c)= b · (a ·c), ∀a,b,c ∈g. m. sarwar (kamran) [5] defined la-group as the following; a groupoid g is called a left almost group, abbreviated as la-group, if (i) there exists e ∈g such that ea= a for all a∈g, (ii) for every a∈g there exists a′ ∈g such that, a′a= e, (iii) (ab)c =(cb)a for every a,b,c ∈g. received: mar. 2, 2023. 2020 mathematics subject classification. 16y30. key words and phrases. nla-ring; bi-ideal; weak bi-ideal. https://doi.org/10.28924/2291-8639-21-2023-38 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-38 2 int. j. anal. appl. (2023), 21:38 let 〈g, ·〉 be an la-group and s be a non-empty subset of g and s is itself and la-group under the binary operation induced by g, the s is called an la-subgroup of g, then s is called an la-subgroup of 〈g, ·〉. s.m. yusuf in [7, p.211] introduces the concept of a left almost ring (la-ring). that is, a nonempty set r with two binary operations “+” and “·” is called a left almost ring, if 〈r,+〉 is an la-group, 〈r, ·〉 is an la-semigroup and distributive laws of “·” over “+” holds. t. shah and i. rehman [7, p.211] asserted that a commutative ring 〈r,+, ·〉, we can always obtain an la-ring 〈r,⊕, ·〉 by defining, for a,b,c ∈ r, a⊕b = b− a and a ·b is same as in the ring. we can not assume the addition to be commutative in an la-ring. an la-ring 〈r,+, ·〉 is said to be la-integral domain if a·b =0, a,b ∈r, then a = 0 or b = 0. let 〈r,+, ·〉 be an la-ring and s be a non-empty subset of r and s is itself and la-ring under the binary operation induced by r, the s is called an la-subring of r, then s is called an la-subring of 〈r,+, ·〉. if s is an la-subring of an la-ring 〈r,+, ·〉, then s is called a left ideal of r if rs ⊆s. right and two-sided ideals are defined in the usual manner. by [4] a near-ring is a non-empty set n together with two binary operations “+” and “·” such that 〈n,+〉 is a group (not necessarily abelian), 〈n, ·〉 is a semigroup and one sided distributive (left or right) of “·” over “+” holds. by [8] if a subgroup b of 〈n,+〉 is said to be a bi-ideal of n if bnb∩ (bn)∗b ⊆b. if n has a zero symmetric near-ring a subgroup b of 〈n,+〉 is a bi-ideal if and only if bnb ⊆b. by [9] a subgroup b of 〈n,+〉 is said to be a weak bi-ideal of n if b3 ⊆ b. in this paper we will define bi-ideal of near-ring has a zero symmetric. 2. near left almost rings t. shah, f. rehman and m. raees [6, pp.1103-1111] introduces the concept of a near left almost ring (nla-ring). definition 2.1. [6]. a non-empty set n with two binary operation “+” and “·” is called a near left almost ring (or simply an nla-ring) if and only if (1) 〈n,+〉 is an la-group. (2) 〈n, ·〉 is an la-semigroup. (3) left distributive property of · over + holds, that is a ·(b+c)= a ·b+a ·c for all a,b,c ∈n. definition 2.2. [6]. an nla-ring 〈n,+〉 with left identity 1, such that 1·a= a for all a∈n, is called an nla-ring with left identity. definition 2.3. [6]. a non-empty subset s of an nla-ring n is said to be an nla-subring if and only if s is itself an nla-ring under the same binary operations as in n. int. j. anal. appl. (2023), 21:38 3 definition 2.4. [6]. an nla-subring i of an nla-ring n is called a left ideal of n if ni ⊆ i, and i is called a right ideal if for all n,m ∈n and i ∈ i such that (i +n)m−nm ∈ i, and is called two sided ideal or simply ideal if it is both left and right ideal. definition 2.5. [6]. let 〈n,+, ·〉 be an nla-ring. a non-zero element a of n is called a left zero divisor if there exists 0 6= b ∈n such that a ·b =0. similarly a is a right zero divisor if b ·a=0. if a is both a left and a right zero divisor, then a is called a zero divisor. definition 2.6. [6]. an nla-ring 〈d,+, ·〉 with left identity 1, is called an nla-ring integral domain if it has no left zero divisor. definition 2.7. [6]. an nla-ring 〈f,+, ·〉 with left identity 1, is called a near almost field (n-almost field) if and only if each non-zero element of f has inverse under “·” 3. bi-ideals and weak bi-ideals of near left almost rings next we defines of a bi-ideals and weak bi-ideals in nla-ring is defines the same as a bi-ideal and weak bi-ideal in near-ring in [8] and [9]. definition 3.1. if a la-subgroup b of 〈n,+〉 is said to be a bi-ideal of n if (bn)b∩(bn)∗b ⊆b. if n has a zero symmetric nla-ring a la-subgroup b of 〈n,+〉 is a bi-ideal if and only if (bn)b ⊆b. lemma 3.1. let n be a zero symmetric nla-ring. an la-subgroup b of n is a bi-ideal if and only if (bn)b ⊆b. proof. for an la-subgroup n of 〈n,+〉 if (bn)b ⊆b then b is a bi-ideal of n. conversely if b is a bi-ideal, we have (bn)b∩(bn)∗b ⊆b. since n is a zero symmetric nla-ring, nb ⊆n ∗b, we get (bn)b =(bn)b∩ (bn)b ⊆ (bn)b∩ (bn)∗b ⊆b. thus (bn)b ⊆b. � definition 3.2. let n be an nla-ring. an la-subgroup b of 〈n,+〉 is a bi-ideal if (bn)b ⊆b. theorem 3.1. if b be a bi-ideal of a nla-ring n and s is an nla-subring of n. then b ∩s is a bi-ideal of s. proof. since b is a bi-ideal of n we have (bn)b ⊆ b. assume that c := b∩s. then (cs)c ⊆ (ss)s ⊆s, since s is a nla-subring of n and c ⊆s. on the other hand (cs)c ⊆ (bs)b ⊆ (bn)b ⊆b. hence (cs)c ⊆b∩s =c. therefore c is a bi-ideal of s. � theorem 3.2. let n be an nla-ring and a,b be bi-ideals of an nla-ring n. then a∩b is a bi-ideal of n. 4 int. j. anal. appl. (2023), 21:38 proof. since a,b is bi-ideals of an nla-ring n, we have a∩b is an la-subgroup 〈n,+〉. thus [(a ∩ b)n](a ∩ b) ⊆ (an)(a ∩ b) = [(a ∩ b)n]a ⊆ (an)a ⊆ a and [(a ∩ b)n](a ∩ b) ⊆ (bn)(a∩b)= [(a∩b)n]b ⊆ (bn)b ⊆b. it following that a∩b is a bi-ideal of n. � theorem 3.3. the set of all bi-ideal of nla-ring. proof. let {bi}i∈i be a set of bi-ideal in n and b :=∩i∈ibi. then (bn)b ⊆ ( ⋂ i∈i bin) ⋂ i∈i bi ⊆bi for every i ∈ i. thus b is a bi-ideal of n. � definition 3.3. let n be an nla-ring. an element d of n is called distributive if (n+n′)d = nd+n′d for all n,n′ ∈n. theorem 3.4. let n be an nla-ring. if b is a bi-ideal of n then bn and n′b are bi-ideal of n where n,n′ ∈n and n′ is a distributive element in n. proof. since b is a bi-ideal we have bn and n′b are an la-subgroup 〈n,+〉. thus ((bn)n)(bn)⊆ (bn)(bn)= (bn)bn ⊆bn. hence bn is a bi-ideal of n. again ((n′b)n)(n′b)⊆ ((n′b)n)b =(n′bn)b ⊆ n′b. thus n′b are bi-ideal of n. � corollary 3.1. if b is a bi-ideal of nla-ring. for b,c ∈ b, if b is a distributive element in n, then bbc is a bi-ideal of n. proof. let b be a bi-ideal of nla-ring and b is a distributive element in n. then b(n+n′)= bn+dn′ for all n,n′ ∈n. since b is a bi-ideal we have bbc is an la-subgroup 〈n,+〉 then ((bbc)n)(bbc)⊆ (bn)b ⊆b. � definition 3.4. an nla-ring n is said to be b-simple if it has no proper bi-ideals. theorem 3.5. let n be an nla-ring with more than one element. if n is a near almost field. then n is a b-simple. proof. let n be a near almost field then {0} and n are the only bi-ideals of n. for if 0 6= b is a bi-ideal of n, then for 0 6= b ∈b we get n =nb and n = bn. now n = n2 = (bn)(nb) ⊆ bnb ⊆ b, since b is a bi-ideal of n. then n = b. thus n is a b-simple. � the following we defined weak bi-ideal and study properties it. definition 3.5. an la-subgroup b of 〈n,+〉 is said to be a weak bi-ideal of n if b3 ⊆b. int. j. anal. appl. (2023), 21:38 5 theorem 3.6. every bi-ideal of an nla-ring is a weak bi-ideal. proof. since b3 =(bb)b ⊆ (bn)b ⊆b we have every bi-ideal is a weak bi-ideal. � theorem 3.7. if b is a weak bi-ideal of a nla-ring n and s is a nla-subring of n. then b∩s is a weak bi-ideal of n. proof. assume that c :=b∩s. then c3 = ((b∩s)(b∩s))(b∩s) = ((b∩s)(b∩s))b∩ ((b∩s)(b∩s))s ⊆ (bb)b∩sss = b3 ∩sss ⊆ b3 ∩ss ⊆ b3 ∩s ⊆ b∩s = c. thus c3 ⊆c. hence c is a weak bi-ideal of n. � theorem 3.8. let n be an nla-ring. if b is a weak bi-ideal of n then bn and n′b are bi-ideal of n where n,n′ ∈n and n′ is a distributive element in n proof. since b is a weak bi-ideal we have bn and n′b are an la-subgroup 〈n,+〉. thus (bn)3 =(bnbn)bn ⊆ (bb)bn ⊆b3n =bn. hence bn is a weak bi-ideal of n. again (n′b)3 =(n′bn′b)n′b ⊆ (n′bb)b = n′b3 ⊆ n′b. thus n′b is a weak bi-ideal of n. � corollary 3.2. if b is a weak bi-ideal of nla-ring. for b,c ∈ b, if b is a distributive element in n, then bbc is a weak bi-ideal of n. acknowledgements: this research project was supported by the thailand science research and innovation fund and the department of mathematics, school of science, university of phayao, phayao 56000, thailand. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 6 int. j. anal. appl. (2023), 21:38 references [1] m.a. kazim, md. naseeruddin, on almost semigroup, portugaliae math. 36 (1977), 41-47. http://eudml.org/ doc/115301. [2] m. khan, faisal, v. amjid, on some classes of abel-grassmann’s groupoids, (2010). http://arxiv.org/abs/ 1010.5965. [3] q. mushtaq, m. khan, m-system in la-semigroups, southeast asian bull. math. 33 (2009), 321-327. [4] g. pilz, near rings, north-holland, amsterdam, 1977. [5] m. sarwar (kamran), conditions for la-semigroup to resemble associative structures, phd thesis, quaid-i-azam university, 1993. [6] t. shah, f. ur rehman, m. raees, on near left almost rings, int. math. forum, 6 (2011), 1103-1111. [7] t. shah, i. rehman, on la-rings of finitely nonzero function, int. j. contemp. math. sci. 5 (2010), 209-222. [8] t.t. chelvam, n. ganesan, on bi-ideal of near-ring, indian j. pure appl. math. 18 (1987), 1002-1005. [9] y.u. cho, t.t. chelvam, s. jayalakshmi, weak bi-ideal of near-ring, j. korea soc. math. 14 (2007), 153-159. http://eudml.org/doc/115301 http://eudml.org/doc/115301 http://arxiv.org/abs/1010.5965 http://arxiv.org/abs/1010.5965 1. introduction 2. near left almost rings 3. bi-ideals and weak bi-ideals of near left almost rings references int. j. anal. appl. (2023), 21:31 a combined conjugate gradient quasi-newton method with modification bfgs formula mardeen sh. taher1,∗, salah g. shareef2 1department of mathematics, college of science, duhok university, kurdistan region, iraq 2department of mathematics, college of science, zakho university, kurdistan region, iraq ∗corresponding author: mardinsh.tahir@uod.ac abstract. the conjugate gradient and quasi-newton methods have advantages and drawbacks, as although quasi-newton algorithm has more rapid convergence than conjugate gradient, they require more storage compared to conjugate gradient algorithms. in 1976, buckley designed a method that combines the cg method with qn updates, which is better than that observed for conjugate gradient algorithms but not as good as the quasi-newton approach. this type of method is called the preconditioned conjugate gradient (pcg) method. in this paper, we introduce two new preconditioned conjugate gradient (pcg) methods that combine conjugate gradient with a new update of quasinewton methods. the new quasi-newton method satisfied the positive define, and the direction of the new preconditioned conjugate gradient is descent direction. in numerical results, it is showing the new preconditioned conjugate gradient method is more effective on several high-dimension test problems than standard preconditioning. 1. introduction there are many types of numerical methods to find an optimum or near-optimum solution to one or more dimensional unconstrained optimization problems, which include the cubic interpolation, golden ratio gradient descent method, the newton and quasi-newton methods. the most widely used for solving large-scale problems in fields such as technology, sciences, and economics is the quasinewton (qn) or variable metric (vm) [4], methods because of its effectiveness and stability. consider the unconstrained minimization problem as follows: received: feb. 12, 2023. 2020 mathematics subject classification. 49m15. key words and phrases. unconstrained optimization; preconditioning conjugate gradient quasi-newton; rank-two update methods. https://doi.org/10.28924/2291-8639-21-2023-31 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-31 2 int. j. anal. appl. (2023), 21:31 min{f (x) : x ∈ rn}, (1.1) where f (x) is twice continuously differentiable function over rn, the essential idea of quasi-newton methods is to use an approximation of the inverse hessian, and build up an approximation of the inverse hessian is often used information about the gradient ∇f (xk) from some or all of the previous iterates xk. quasi-newton methods, instead of the true inverse hessian, are observed as the most complicated for solving(1.1). earliest quasinewton method was proposed by william c. davidon in 1959 [3] and later developed by fletcher and powell (1963) [6].the updating formula of this method generates a symmetric positive matrix of the form hk+1 = hk +q , where q is a correction matrix. then a general quasi-newton method is started with an initial point x0 a first approximation of the minimum point, and a matrix h0 (usually h0 = i ,i is a symmetric positive definite matrix), solving the following linear equation to compute search direction dk +hk+1gk =0 (1.2) and find the next point xk+1 by searching along a decent direction dk such that dtk gk ≤ 0 , using the following equation: xk+1 = xk +αkdk. (1.3) to find the step length αk must apply an appropriate line search strategy along the search direction dk , such that the following wolfe–powell [10] conditions are satisfied: f (xk +αkdk)− f (xk)≤ c1αk∇f tk dk, (1.4) ∇f (xk +αkdk)tdk ≥ c2∇f tk dk, (1.5) with 0 < c1 < c2 < 1. select a new symmetric positive definite matrix hk+1 which satisfy the following original quasi-newton equation, [5] hk+1yk = vk (1.6) xk and xk+1 two points are given; describe gk = ∇f (xk) and gk+1 = ∇f (xk+1) , so vk = xk+1−xk and yk = gk+1 −gk. quasi-newton methods use the correction matrix qk rank one or rank two matrix. in each update, the iterate matrix hk+1 = hk+1+qk ,should satisfy the quasi-newton condition (1.6) now substitute the correction matrix by qk = auu t +bwwt , (1.7) where a andbare scalars while uand w are vectors. the quantities auut and bwwt are symmetric matrices; when b = 0 quasi-newton methods are using rank-one updates, but if b 6=0 then quasi-newton methods are using rank two updates. int. j. anal. appl. (2023), 21:31 3 the general type of qn updates which was proposed by broyden [4] and satisfy the ordinary quasinewton equation is: hk+1 = hk − hkyky t k hk yt k hkyk + vkv t k vt k yk +ϕk(y t k hkyk)rkr t k , (1.8) where rk = vk vt k yk − hkyk yt k hkyk , and ϕk = ϕ(θk)= 1−θk 1+θkδkµk with δk = yt k hkyk vky t k and µk = vt k hkvk vt k yk . several well known updates hk+1 defined in (1.8) by choosing different values of θk:when; θk = vt k yk vt k yk−vtk hkvk , we get the symmetric rank-one formula(sr1): hsr1k+1 = hk + (vk − hkyk)(vk − hkyk) t (vk − hkyk) t yk . (1.9) for θk =1, we get the dfp formula due to davidon, fletcher, and powell. hdfpk+1 = hk − hkyky t k hk yt k hkyk + vkv t k vt k yk . (1.10) for θk =0, we get the bfgs formula due to broyden, fletcher, goldfard, and shanno. hbfgsk+1 = hk +(1+ ytkhkyk vt k yk ) vkv t k vt k yk − hkykv t k +vky t k hk yt k hkyk . (1.11) in the next section, we propose a modified bfgs method and study the properties of it. we present a new update of hk+1 which satisfy the quasi-newton condition. 2. a modified bfgs method (mbfgs) in this section, a new class of quasi newton updates for solving unconstrained non linear optimization problems is proposed. the idea of new updates is using the powell equation [10] which is define as: ỹk =(1−θ)gvk +θyk (2.1) where g is a hessian matrix which is a symmetric matrix of second partial derivatives of function and θ in(0,1). now, we suppose that gvk = yk ρ . (2.2) let, ρ = 2 √ ω ||vk|| (1+ ||xk+1||), ω is a machine accuracy, and ||.|| ≥ 0 is the euclidean norm of vectors. so, we obtain gvk = ||vk|| yk 2 √ ω(1+ ||xk+1||) . (2.3) we replace gvk in (2.1) by (2.3), and getting the following ỹk =(1−θ) ( ||vk|| yk 2 √ ω(1+ ||xk+1||) ) +θyk. (2.4) for the new updated , we have investigated a new expression for the qn-condition via ỹk put in (1.6) instead of yk , and get hk+1ỹk = vk. (2.5) 4 int. j. anal. appl. (2023), 21:31 a more flexible is gotten when the correction matrix qkis a rank two , hence the formula hk+1 = hk +q can be written as, hk+1 = hk +auu t +bwwt . (2.6) in 1970, broyden, fletcher, goldfarb, and shanno suggested an alternative method called the bfgs method, which is the most popular type of symmetric rank-two method for large-scale optimization and belongs to a group of quasi-newton methods, it is a local search method. now, we can drive a modified of hbfgsk+1 depend on (2.5) to get h mbfgs k+1 multiplying (2.6) by ỹk to obtain hk+1ỹk = hkỹk +auu t ỹk +bww t ỹk. (2.7) the vectors u and v are no longer uniquely determined. in view of (2.7), it is adequate choose, u = vk and w = hkỹk. then we obtain hk+1ỹk = hkỹk +avkv t k ỹk +bhkỹk(hkyk) t ỹk, (2.8) which implies, if aut ỹk =1 and bwt ỹk =−1, thus determine a and b such that a = 1ut ỹk = 1 vt k ỹk and b = 1 ỹt k hk ỹk . substituting the value of a, b, u and v to the updating formula (2.8), thus we get a new updated of qn-method is hmbfgsk+1 hmbfgsk+1 = hk + [ 1+ ỹtk hkỹk vt k ỹk ] vkv t k vt k ỹk − hkỹkv t k +vkỹ t k hk ỹt k hkỹk . (2.9) we can rewrite (2.9) as hmbfgsk+1 = [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] + vkv t k vt k ỹk . (2.10) 3. algorithm of modified bfgs • step 0: start with initial point of solution x0 ∈ rn,k =0, set � > 0,n ∈ z,and select a real symmetric positive definiteh0 = i , i is an n×n identity matrix. • step 1: test if ||gk|| < � then stop, else dk =−hkgk =−hk∇f (xk) and go to step (2) • step 2: using line search procedure to determine the size step αk = argminf (xk +αkdk) such that rules (1.4) and (1.5) are satisfied • step 3: calculate xk+1 = xk +αkdk. • step 4: check , if ||gk+1|| < � then stop and xk+1 is optimal point.otherwise calculate dk+1 =−hk+1gk+1, hk+1 is defined in (2.10) and go to step (5). • step 5: set k = k +1. and go to step 1. 4. hereditary property and positive definiteness of the mbfgs-method in this section, we prove that the new modification of bfgs is satisfied both properties the hereditary property (secant condition ) and preserves positive definite hk+1 matrices. int. j. anal. appl. (2023), 21:31 5 theorem 4.1. if the new method is applied to minimize a quadratic function with positive definite hessian g = gt , then the (1.6) is hold i.e hmbfgsk+1 ỹk = vk for all 0≤ k. proof. multiply both sides of (2.10) by ỹk from right , so we have hmbfgsk+1 ỹk =( [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] + vkv t k vt k ỹk )ỹk, (4.1) by using a basic algebraic operations we found the (4.1) becomes in following form: hmbfgsk+1 ỹk = hkỹ − hkỹkv t k ỹ ỹt k vk − vkỹ t k hkỹk ỹt k vk + vkỹ t k hkỹv t k ỹk (ỹt k vk) 2 + vkv t k vt k ỹk ỹk. (4.2) it is knowing that the vtk ỹk is scalar and ỹ t k vk = v t k ỹk, so (4.2) becomes hmbfgsk+1 ỹk = vk. (4.3) � theorem 4.2. we first demonstrate that if hmbfgsk is positive definite, then h mbfgs k+1 is also positive definite. proof. to ensure positive-definiteness ofhmbfgsk+1 assumingh mbfgs k is positive definite. typically the algorithm starts with hmbfgs0 = i or a similar diagonal positive-definite matrix. we only need to check that wthmbfgsk+1 w > 0 , for any w 6=0 andw ∈ r n, we have wthmbfgsk+1 w = w t( [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] + vkv t k vt k ỹk )w, (4.4) wthmbfgsk+1 w = w t [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] w +wt vkv t k vt k ỹk w. (4.5) let zk = w(i − vk ỹ t k ỹt k vk ) and zk 6=0, so rewrite (4.5) as: wthmbfgsk+1 w = z thkz + (wtvk) 2 vt k ỹk . (4.6) it is clear the first term of (4.6) zthkz > 0, because hk is positive defined. the second (wtvk) 2 vt k ỹk , (wtvk) 2 > 0, now we need to prove vtk ỹk > 0, whenever vtk ỹk = v t k ((1−θ) ( ||vk|| yk 2 √ ω(1+ ||xk+1||) ) +θyk). (4.7) let ξ = ( ||vk|| 2 √ ω(1+||xk+1||) ) , and ξ > 0. so, vtk ỹk = v t k ((1−θ)(ξyk +θyk)) (4.8) suppose µ =(1−θ)(ξ+θ), µ > 0, therefore vtk ỹk =(µv t k yk), (4.9) 6 int. j. anal. appl. (2023), 21:31 vtk ỹk = µ(v t k gk+1 −v t k gk). (4.10) in case of, exact line search then we have vtk gk+1 = 0, and v t k gk = −αkg t k hkgk, then (4.10) becomes vtk ỹk = µαkg t k hkgk. (4.11) since hk is positive, means gtk hkgk > 0. there fore (4.11) is positive. in case, inexact line search, vtk gk+1 6=0, we rewrite(4.10) as: vtk ỹk = µv t k yk = µαkd t k yk. (4.12) noteworthy that, from second wolf’s condition we get, dtk yk = d t k (gk+1−gk) > ( c2 −1)d t k gk and ( c2 −1) dtk gk > 0, so d t k yk > 0, it is clear µαk > 0, thus, we see v t k ỹk > 0. since wthmbfgsk+1 w > 0 w 6=0, (4.13) therefor, hmbfgsk+1 is positive definite. � theorem 4.3. let xk+1 anddk+1 are two sequences generated by new algorithm 3, with line search under wolf’s conditions (1.4) and (1.5), then the new direction dk+1 is satisfied the sufficient descent condition. dtk+1gk+1 ≤ 0. (4.14) proof. form (1.2) and(2.10) we have, dk+1 =−hk+1gk+1, (4.15) hmbfgsk+1 = [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] + vkv t k vt k ỹk . (4.16) multiply both sides of (4.15) by gtk+1 gtk+1dk+1 =−g t k+1( [ i − vkỹ t k ỹt k vk ] hk [ i − ỹkv t k ỹt k vk ] + vkv t k vt k ỹk )gk+1. (4.17) it is clear hmbfgsk+1 is positive defined and gk+1is a vector, therefore gtk+1(hk+1)gk+1 > 0. (4.18) this implies dtk+1gk+1 ≤ 0. (4.19) � int. j. anal. appl. (2023), 21:31 7 5. new preconditioned conjugate gradient pcg-method the conjugate gradient (cg) method is a attractive method for minimizing a large unconstrained nonlinear problems, because it is using the first derivative information to generate search directions, and the qnmethod faster than cgmethod but need more area computer store for the reason that the qnmethods generated a symmetric positive defined matrix in each iteration which needed a (n(n+ 1)/2) location of store. so in 1978, buckley suggested a method combining the conjugate gradient with qn-method called (pcg-method )the aim of this suggestion is accelerating the convergence of conjugate gradient and reduce amount of storage in qn-method. the idea of pcg-method is based on combining by using the matrix of qn in the conjugate gradient algorithm which is corresponding to solve a problem in the transformed space. 5.1. a new pcg method: let hmbfgsk+1 is a preconditioned matrix and it is a symmetric positive definite, by using cholesky decomposition of hmbfgsk+1 , i.e there exists a lower triangular matrix l̃ such that hmbfgsk+1 = l̃l̃ t . assume f (x) be a strictly convex quadratic function and f (x) can be written as: f (x)= 1 2 xtgx +xtb+c, (5.1) such that, gradient of f (x) is ∇f (x)= g(x)= gx +b, (5.2) f (l̃z)= 1 2 (l̃z)tg(l̃z)+(l̃z)tb+c. (5.3) let f (l̃z)= h(z), so, h(z)= 1 2 (l̃z)tg(l̃z)+(l̃z)tb+c. (5.4) the first derivative of h(z) is ∇h(z)= l̃ztgl̃z + l̃tb, (5.5) ∇h(z)= l̃t(gl̃z +b), (5.6) gz = l̃tgx. (5.7) now, we set zk+1 = zk +αkd z k . (5.8) multiplication both sides of(5.8) by l̃, we get l̃zk+1 = l̃zk +αkl̃d z k . (5.9) 8 int. j. anal. appl. (2023), 21:31 we have x = l̃z , so (5.9) becomes: xk+1 = xk +αkl̃d z k . (5.10) from(5.10), we noted l̃dzk = d x k , since d z k = l̃ −1dxk . set yzk = g z+1 k −gzk, (5.11) where, yz+1 k and yzk are the gradients of h(z) at points zk+1 and zk respectively, since from (5.7), (5.11) becomes as follows: yzk = l̃ tgk+1 k − l̃tgxk. (5.12) now consider applying the modification of parry conjugate gradient method βmprrey k = gt k+1 (ỹk−vk) dt k ỹk [9], to the objective function h(z), dzk+1 =−g z k+1 + gz t k+1(ỹ z k −v z k) dz t k ỹz k dzk . (5.13) using (5.7), (5.11) and (5.12) in (5.13) and multiply by l̃, we get: l̃l̃−1dxk+1 =−l̃l̃ tgxk+1 + gx t k+1l̃l̃ t(ỹxk −v x k ) dx t k ỹx l̃l̃−1dxk , (5.14) dxk+1 =−h mbfgs k+1 g x k+1 + gx t k+1h mbfgs k+1 (ỹ x k −v x k ) dx t k ỹx dxk . (5.15) (5.15) is our preconditioned conjugate method which is require less storage and computation time and has a quadratic termination property. 5.2. algorithm of the new pcg-method. • step 0: let k = 0, x0 in rn is an initial point of solution, set � > 0,n ∈ z,and select a real symmetric positive definite matrix h0 = i, i is an n×n identity matrix and ω is accuracy of computer. • step 1: test a criterion for stopping, if ||gk|| < � then stop else go to step 2. • step 2: dk =−hk∇f (xk)=−hkgk and continuous. • step 3:using line search procedure to determine the size stepαk , αk = argminf (xk +αkdk) such that rules (1.4) and (1.5) are satisfied. • step 4: calculate xk+1 = xk +αkdk,and go to next step. • step 5: check, if ||gk+1|| < � then stop and xk+1 is optimal point, otherwise calculate vk = xk+1−xk, yk = gk+1−gk and find ỹ by ỹ =(1−θ)‖vk‖ yk 2 √ ω(1+‖xk+1‖) +θyk , θ ∈ (0,1), ω is error of machine and go. • step 6: find dk+1 =−hmbfgsk+1 gk+1 + gt k+1 hmbfgs k+1 (ỹk−vk) dt k ỹ dk, and hmbfgsk+1 is defined as hmbfgsk+1 = hk + [ 1+ ỹt k hk ỹk vt k ỹk ] vkv t k vt k ỹk − hk ỹkv t k +vk ỹ t k hk ỹt k hk ỹk then go to step (7). int. j. anal. appl. (2023), 21:31 9 • step 7: if gtk+1gk+1 ≤−0.8d t k+1gk+1, then go to step 2. else k = k +1 and go to step 3. theorem 5.1. let the sequences of xk and dk are generated by algorithm of the new pcg-method 5.2 then the descent property of a new pcgmethod is descent condition: dtk+1gk+1 < 0. (5.16) proof. we prove by induction, at k =0, d0 =−h0g0, so we have dt0 g0 ≤−g t 0 h0g0, (5.17) where h0 = i, g0 6=0 , and −gt0 h0g0 < 0. now we assume that the conclusion (5.16) holds for k ≥ 0, means, gtk dk ≤ γ‖gk‖ 2, need to prove it is true at k +1. let dk+1 =−hmbfgsk+1 gk+1 + gtk+1h mbfgs k+1 (ỹkk −vk) dt k ỹk dk. (5.18) multiply both sides of (5.18) by gk+1 we get, dtk+1gk+1 =−g t k+1h mbfgs k+1 gk+1 + gtk+1h mbfgs k+1 (ỹk −vk) dt k ỹk dtk gk+1. (5.19) thus, dtk+1gk+1 =−g t k+1h mbfgs k+1 gk+1 + gtk+1h mbfgs k+1 (ỹk) dt k ỹk dtk gk+1 − gtk+1h mbfgs k+1 (vk) dt k ỹk dtk gk+1. (5.20) we notice that if we use an exact line search then, we have dtk gk+1 = 0 and also h mbfgs k+1 is positive symmetric definite from theorem 4.2, gtk+1h mbfgs k+1 gk+1 ≥ 0 for all, gk+1 6= 0, therefore dtk+1gk+1 < 0. in case inexact line search d t k gk+1 6=0, from (5.19) , we get dtk+1gk+1 =− gtk+1h mbfgs k+1 vk dt k ỹ dtk gk+1, (5.21) ỹ =(1−θ)‖vk‖ yk 2 √ ω(1+‖xk+1‖) +θyk. we need to show dtk ỹ > 0, means d t k ỹ = d t k ((1− θ)‖vk‖ yk 2 √ ω(1+‖xk+1‖) + θyk) > 0. it is noted that the dtk yk = d t k (gk+1 −gk) > (δ2 −1)d t k gk ,d t k vk = ‖dk‖ 2 αk, and θ ∈ (0,1) dtk+1gk+1 =− gtk+1h mbfgs k+1 gk+1‖dk‖ 2 αk dt k ỹ . (5.22) let τ = gt k+1 hmbfgs k+1 gk+1‖dk‖ 2 αk dt k ỹ , we see τ is positive, then (5.22) becomes: dtk+1gk+1 < 0. (5.23) � 10 int. j. anal. appl. (2023), 21:31 6. numerical experiments and discussions it is clearly that the theoretical evidence is not sufficient to demonstrate the effectiveness or robustness of any iterative methods. therefore, researchers turn to study the numerical results of methods by evaluate the performance method on a group of test problems and evaluation the number of iterations or computation time (cpu-time). in this section, we present the results of numerical experiments for our new suggestion to solve different nonlinear test problems of large size. in practice, the construction sequence of preconditions is based on well-known suggestion method modified bfgs techniques in order to keep under control the amount of memory. we use fortran95 language to write all codes and the run is stopping when this inequality ||gk+1|| < 10−5 is satisfied. for compare, we used the well-known nonlinear problems with dimension ranging between 4 to 5000, [1]. all algorithms use exactly the same method (cubic fit method ) to find the step length αk the same implementation of the wolfe line search conditions (1.4) and (1.5) with c1 =0.001 andc2 =0.1. according to the table1, it is not difficult to show that the performance of new pcg -method is better than stander pcg method when using hestain and stiefen formula (βhsk = gt k+1 yk dt k yk ) [12], and the restart gtk+1gk+1 ≤−0.8d t k+1gk+1. table 2 results illustrating the behavior of new pcg -method and standard pcg methods when taken the perry suggestion βperry k = gt k+1 (yk−vk) dt k yk )for coefficient of conjugate gradient method [9] under the restart |gtk+1gk| > 0.2g t k+1gk+1, for more analyse of the numerical result we use performance profile proposed by dolan and more [14]. according to the rule of this performance profile, we describe the performance curves based on table 1 and table 2 as in figures 1–4. based on the four figures, we see that the new pcg method is superior to the standard pcg method under the unconstrained problems in tables 1 and 2. int. j. anal. appl. (2023), 21:31 11 table 1. comparing performance of βhsk −bfgs and β hs k −mbfgs βhs k −bfgs βhs k −mbfgs test function n noi-nof noi-nof 4 21-86 20-72 100 67-179 39-109 powell 500 48-142 44-121 (-3,-1,0,1;...) 1000 47-142 45-126 3000 51-144 37-123 5000 40-119 40-119 4 25-87 18-68 100 31-99 18-71 miele 500 31-103 22-86 (1,2,2,2;...) 1000 38-117 23-88 3000 34-106 23-91 5000 35-104 29-89 4 36-269 12-79 100 42-341 23-167 cantral 500 48-416 36-311 (1,2,2,2;...) 1000 51-451 26-217 3000 55-506 41-404 5000 57-532 32-281 4 7-18 8-17 100 72-145 58-117 wolf 500 82-165 63-127 (-1;...) 1000 96-194 66-133 3000 181-388 108-227 5000 182-382 107-236 4 19-58 18-52 100 70-167 36-91 cubic 500 53-124 42-101 (-1.2,1;...) 1000 71-167 48-112 3000 71-168 49-177 5000 67-163 55-127 4 24-73 23-61 100 74-177 56-134 non-diagonal 500 82-205 63-153 (-1;...) 1000 85-218 61-147 3000 111-335 69-174 5000 100-275 77-194 4 8-26 8-24 100 8-26 8-24 shallow 500 8-26 8-25 (-2,-2;...) 1000 8-26 8-25 3000 9-28 10-29 5000 10-30 10-29 4 32-92 34-85 100 235-6827 220-538 rosen 500 578-1537 462-1133 (-1.2,1;...) 1000 864-2150 747-2014 3000 1061-2667 910-2434 5000 1428-3577 891-2357 4 9-22 9-22 100 10-25 10-25 beal 500 10-25 10-25 (0,0;..) 1000 10-25 10-25 3000 10-25 10-25 5000 10-25 10-25 4 9-24 9-23 100 218-537 209-495 dixon 500 210-509 193-541 (-1;..) 1000 199-490 225-541 3000 252-590 195-461 5000 204-529 175-423 12 int. j. anal. appl. (2023), 21:31 table 2. comparing performance profiles of {βperry k −mbfgs} and {βperry k −bfgs} β perry k −bfgs βperry k −mbfgs test function n noi-nof noi-nof 4 124-294 36-102 100 529-1152 40-121 powell 500 119-285 40-121 (-3,-1,0,1;...) 1000 289-1684 40-121 3000 152-379 40-121 5000 150-381 40-121 4 36-1515 19-85 100 274-2216 23-137 cantral 500 365-2657 24-152 (1,2,2,2;...) 1000 564-2854 24-152 3000 610-3200 28-213 5000 330-3350 32-281 4 14-59 8-45 100 157-519 52-190 osp 500 417-1170 115-352 (-1;...) 1000 523-1421 174-532 3000 1006-2692 303-965 5000 1356-3637 380-1221 4 26-61 22-91 100 26-61 23-55 wood 500 36-81 23-55 (-3,-1,-3,-1;...) 1000 36-81 23-55 3000 36-81 23-55 5000 36-81 23-55 4 31-80 30-75 100 47-113 44-107 non-diagonal 500 49-119 49-112 (-1;...) 1000 50-122 61-120 3000 50-122 49-119 5000 50-122 50-120 4 39-104 39-102 100 41-109 39-104 rosen 500 38-103 38-103 (-1;...) 1000 39-105 38-103 3000 40-105 38-105 5000 40-104 40-103 4 3-11 3-11 100 20-107 14-81 sum 500 1992 21-115 (2;...) 1000 31-172 23-117 3000 63-351 32-179 5000 79-425 42-222 4 22-60 14-46 100 29-76 16-51 cubic 500 29-75 22-62 (-1.2,1;...) 1000 31-82 22-64 3000 29-74 24-68 5000 32-83 24-69 4 5-14 5-14 100 6-16 6-16 edger 500 6-16 6-16 (-1;..) 1000 6-16 6-16 3000 6-16 6-16 5000 6-16 6-16 int. j. anal. appl. (2023), 21:31 13 14 int. j. anal. appl. (2023), 21:31 7. conclusion the nonlinear quasi-newton method is widely used in unconstrained optimization. in this paper, we suggest new updates to the quasi-newton method for solving unconstrained optimization problems. we use this new quasi-newton to introduce the new pcg method. the analysis and implementation of the descent property with the wolfe line search of the modified method are studied. the numerical results show that the proposed formula for the combined quasi-newton conjugate gradient method is very encouraging for general, unconstrained optimizations. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. andrei, an unconstrained optimization test functions collection, adv. model. optim. 10 (2008), 147-161. [2] m. al-baali, descent property and global convergence of the fletcher-reeves method with inexact line search, ima j. numer. anal. 5 (1985), 121-124. https://doi.org/10.1093/imanum/5.1.121. [3] c.g. broyden, the convergence of a class of double-rank minimization algorithms 1. general considerations, ima j. appl. math. 6 (1970), 76-90. https://doi.org/10.1093/imamat/6.1.76. [4] c.g. broyden, j.e. dennis jr., j.j. more, on the local and superlinear convergence of quasi-newton methods, ima j. appl. math. 12 (1973), 223-245. https://doi.org/10.1093/imamat/12.3.223. [5] w.c. davidon, variable-metric method for minimization, technical report, anl-5990, (1959). https://doi.org/ 10.2172/4252678. [6] r. fletcher, practical methods of optimization, john wiley & sons, (2013). [7] r. fletcher, a new approach to variable metric algorithms, computer j. 13 (1970), 317-322. https://doi.org/ 10.1093/comjnl/13.3.317. [8] j.d. pearson, variable metric methods of minimisation, computer j. 12 (1969), 171-178. https://doi.org/10. 1093/comjnl/12.2.171. [9] m.sh. taher, s.g. shareef, a modified perry’s conjugate gradient method based on powell’s equation for solving large-scale unconstrained optimization, math. stat. 9 (2021), 882-888. https://doi.org/10.13189/ms.2021. 090603. [10] m.j.d. powell, a fast algorithm for nonlinearly constrained optimization calculations, in: g.a. watson (ed.), numerical analysis, springer berlin heidelberg, berlin, heidelberg, 1978: pp. 144–157. https://doi.org/10. 1007/bfb0067703. [11] p. wolfe, convergence conditions for ascent methods, siam rev. 11 (1969), 226-235. https://doi.org/10. 1137/1011036. [12] m.r. hestenes, e. stiefel, methods of conjugate gradients for solving linear systems, j. res. nat. bureau standards. 49 (1952), 409-436. [13] g. zoutendijk, nonlinear programming computational methods, integer nonlinear program. (1970), 37-86. https: //cir.nii.ac.jp/crid/1571980075701600256. [14] e.d. dolan, j.j. more, benchmarking optimization software with performance profiles, math. program. 91 (2002), 201-213. https://doi.org/10.1007/s101070100263. https://doi.org/10.1093/imanum/5.1.121 https://doi.org/10.1093/imamat/6.1.76 https://doi.org/10.1093/imamat/12.3.223 https://doi.org/10.2172/4252678 https://doi.org/10.2172/4252678 https://doi.org/10.1093/comjnl/13.3.317 https://doi.org/10.1093/comjnl/13.3.317 https://doi.org/10.1093/comjnl/12.2.171 https://doi.org/10.1093/comjnl/12.2.171 https://doi.org/10.13189/ms.2021.090603 https://doi.org/10.13189/ms.2021.090603 https://doi.org/10.1007/bfb0067703 https://doi.org/10.1007/bfb0067703 https://doi.org/10.1137/1011036 https://doi.org/10.1137/1011036 https://cir.nii.ac.jp/crid/1571980075701600256 https://cir.nii.ac.jp/crid/1571980075701600256 https://doi.org/10.1007/s101070100263 1. introduction 2. a modified bfgs method (mbfgs) 3. algorithm of modified bfgs 4. hereditary property and positive definiteness of the mbfgs-method 5. new preconditioned conjugate gradient pcg-method 5.1. a new pcg method: 5.2. algorithm of the new pcg-method 6. numerical experiments and discussions 7. conclusion references international journal of analysis and applications volume 16, number 2 (2018), 178-192 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-178 on giaccardi’s inequality and associated functional in the plane atiq ur rehman1,∗, m. hassaan akbar2 and g. farid1 1comsats institute of information technology, attock, pakistan 2government higher secondary school, khunda, tehsil jand, district attock, pakistan ∗corresponding author: atiq@mathcity.org abstract. in this paper the authors extend giaccardi’s inequality to coordinates in the plane. the authors consider the nonnegative associated functional due to giaccardi’s inequality in plane and discuss its properties for certain class of parametrized functions. also the authors proved related mean value theorems. 1. introduction let i be a real interval. a function f : i → r is said to be convex on i if f(λx + (1 −λ)y) ≤ λf(x) + (1 −λ)f(y) for all x,y ∈ r and λ ∈ [0,1]. in [2], dragomir gave the definition of convex functions on coordinates as follows. definition 1.1. let ∆ = [a,b] × [c,d] ⊆ r2 and f : ∆ → r be a mapping. define partial mappings fy : [a,b] → r by fy(u) = f(u,y) (1.1) and fx : [c,d] → r by fx(v) = f(x,v). (1.2) received 2017-08-21; accepted 2017-10-19; published 2018-03-07. 2010 mathematics subject classification. 26a51; 26d15; 35b05. key words and phrases. convex functions; log-convexity; convex functions on coordinates; giaccardi’s inequality. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 178 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-178 int. j. anal. appl. 16 (2) (2018) 179 then f is said to be convex on coordinates (or coordinated convex) in ∆ if fy and fx are convex on [a,b] and [c,d] respectively for all x ∈ [a,b] and y ∈ [c,d]. a mapping f is said to be strictly convex on coordinates (or strictly coordinated convex) in ∆ if fy and fx are strictly convex on [a,b] and [c,d] respectively for all x ∈ [a,b] and y ∈ [c,d]. now we define another important subclass of convex functions i.e. log convex functions. definition 1.2. a function f : i → r+ is called log convex on i if f(αx + βy) ≤ fα(x)fβ(y) where α + β = 1 α,β ≥ 0 and x,y ∈ i. log-convex functions have excellent closure properties. the sum and product of two log-convex functions is convex. if f is convex function and g is log-convex function then the functional composition g ◦f is also log-convex. many authors consider this function e.g. see [12], in which some of the properties of log convex functions has been discussed (also see [6, 7, 9] and references therein). in the following defintion, we define log convex function on coordinates. definition 1.3. a function f : ∆ → r+ is called log convex on coordinates in ∆ if partial mappings defined in (1.1) and (1.2) are log convex on [a,b] and [c,d] respectively for all x ∈ [a,b] and y ∈ [c,d]. remark 1.1. every log convex function is log convex on coordinates but the converse is not true in general. for example, f : [0, 1]2 → [0,∞) defined by f(x,y) = exy is convex on coordinates but not convex. giaccardi’s inequality is stated as follows (see [8, page 153, 155] or [10]). theorem 1.1. let [0,a) ⊂ r, (x1, ...,xn) ∈ [0,a)n and (p1, ...,pn) be nonnegative n−tuples such that (xi −x0) (x̃n −xi) > 0 for i = 1, . . . ,n and x̃n 6= x0, where x0 ∈ [0,a) and x̃n = ∑n k=1 pkxk. if f is convex, then the inequality n∑ k=1 pkf(xk) 6 af (x̃n) + b ( n∑ k=1 pk − 1 ) f(x0) (1.3) is valid, where a = ∑n k=1 pk(xk −x0) x̃n −x0 b = x̃n x̃n −x0 . remark 1.2. condition that f is convex, can be replaced with f(x)−f(x0) x−x0 is an increasing function, then inequality (1.3) is also valid. int. j. anal. appl. 16 (2) (2018) 180 remark 1.3. if f is strictly convex, then strict inequality holds in (1.3) unless x1 = ... = xn and ∑n i=1 pi = 1. remark 1.4. for pi = 1 (i = 1, ...,n), the above inequality becomes n∑ i=1 f(xi) 6 f ( n∑ i=1 xi ) + (n− 1) f(x0). (1.4) remark 1.5. if we put x0 = 0 in above inequality, we get petrović’s inequality for convex functions on real line. in this paper we extend giaccardi’s inequality to coordinates in the plane. we consider functionals due to giaccardi’s inequality in plane and discuss its properties for certain class of coordinated log-convex functions. also we proved related mean value theorems. 2. main results in the following theorem we give our first result that is giaccardi’s inequality for coordinated convex functions. theorem 2.1. let ∆ = [0,a)× [0,b) ⊂ r2, (x1, ...,xn) ∈ [0,a)n, (y1, ...,yn) ∈ [0,b)n, (p1, ...,pn), (q1, ...,qn) be non-negative n−tuples and ∑n i=1 pixi = x̃n, ∑n j=1 qjyj = ỹn such that (xi −x0) (x̃n −xi) > 0 for i = 1, ...,n, x̃n 6= x0 and n∑ i=1 pi ≥ 1, (2.1) and (yj −yo) (ỹn −yj) > 0 for j = 1, ...,n and ỹn 6= yo. (2.2) if f is coordinated convex function, then n∑ i,j=1 piqjf(xi,yj) 6 a1  a2f (x̃n, ỹn) + b2   n∑ j=1 qj − 1  f (x̃n,yo)   + b1 ( n∑ i=1 pi − 1 )a2f (x0, ỹn) + b2   n∑ j=1 qj − 1  f(x0,yo)   , (2.3) holds where a1 = ∑n i=1 pi(xi −x0) x̃n −x0 , b1 = x̃n x̃n −x0 . (2.4) and a2 = ∑n j=1 qj(yj −y0) ỹn −y0 , b2 = ỹn ỹn −y0 (2.5) int. j. anal. appl. 16 (2) (2018) 181 proof. let fx : [0,b) → r and fy : [0,a) → r be mappings such that fx(v) = f(x,v) and fy(u) = f(u,y). since f is coordinated convex on ∆, therefore fy is convex on [0,a). by theorem 1.1, one has n∑ i=1 pify(xi) 6 a1fy (x̃n) + b1 ( n∑ i=1 pi − 1 ) fy(x0), where a1 and b1 are defined in (2.4). we write n∑ i=1 pif(xi,y) 6 a1f (x̃n,y) + b1 ( n∑ i=1 pi − 1 ) f(x0,y). by setting y = yj, we have n∑ i=1 pif(xi,yj) 6 a1f (x̃n,yj) + b1 ( n∑ i=1 pi − 1 ) f(x0,yj), this gives n∑ i=1 n∑ j=1 piqjf(xi,yj) 6 a1 n∑ j=1 qjf (x̃n,yj) + b1 ( n∑ i=1 pi − 1 ) n∑ j=1 qjf(x0,yj). (2.6) again using theorem 1.1 on terms of right hand side for second coordinates, we have n∑ j=1 qjf (x̃n,yj) 6 a2f (x̃n, ỹn) + b2   n∑ j=1 qj − 1  f (x̃n,y0) and ( n∑ i=1 pi − 1 ) n∑ j=1 qjf(x0,yj) 6 a2 ( n∑ i=1 pi − 1 )f (x0, ỹn) + b2   n∑ j=1 qj − 1  f(x0,y0)   . where a2 and b2 are defined in (2.5) using above inequalities in (2.6), we get n∑ i,j=1 piqjf(xi,yj) 6 a1  a2f (x̃n, ỹn) + b2   n∑ j=1 qj − 1  f (x̃n,yo)   + b1 ( n∑ i=1 pi − 1 )a2f (x0, ỹn) + b2   n∑ j=1 qj − 1  f(x0,yo)   , which is the required result. � remark 2.1. if f is strictly coordinated convex then above inequality is strict unless all xi’s and yi’s are not equal or n∑ i=1 pi 6= 1 and n∑ j=1 qj 6= 1. remark 2.2. if we take yj = 0 and qj = 1, (i,j = 1, ...,n) with f(xi, 0) 7→ f(xi), then we get inequality (1.3). the following corollary is particular case of theorem 2.1, which is stated in [11, theorem 2]. int. j. anal. appl. 16 (2) (2018) 182 corollary 2.1. let ∆ = [0,a)×[0,b) ⊂ r2, (x1, ...,xn) ∈ [0,a)n, (y1, ...,yn) ∈ [0,b)n, (p1, ...,pn), (q1, ...,qn) be non-negative n−tuples and ∑n i=1 pixi = x̃n, ∑n j=1 qjyj = ỹn such that x̃n ≥ xj and ỹn ≥ yj for j = 1, ...,n. also let that x̃n ∈ [0,a), ∑n i=1 pi ≥ 1 and ỹn ∈ [0,b). if f : ∆ → r is coordinated convex function, then n∑ i,j=1 piqjf(xi,yj) 6 f (x̃n, ỹn) +   n∑ j=1 qj − 1  f (x̃n, 0) + ( n∑ i=1 pi − 1 )f (0, ỹn) +   n∑ j=1 qj − 1  f(0, 0)   , (2.7) holds. proof. if we put x0 = y0 = 0 in theorem 2, conditions (2.4) and (2.5) becomes a1 = a2 = b1 = b2 = 1, so (2.3) takes the form n∑ i,j=1 piqjf(xi,yj) 6 f (x̃n, ỹn) +   n∑ j=1 qj − 1  f (x̃n, 0) + ( n∑ i=1 pi − 1 )f (0, ỹn) +   n∑ j=1 qj − 1  f(0, 0)   , as required. � let i ⊆ r be an interval and f : i → r be a function. then for distinct points ui ∈ i,i = 0, 1, 2. the divided differences of first and second order are defined as follows. [ui,ui+1,f] = f(ui+1) −f(ui) ui+1 −ui , (i = 0, 1) (2.8) [u0,u1,u2,f] = [u1,u2,f] − [u0,u1,f] u2 −u0 . (2.9) the values of the divided differences are independent of the order of the points u0,u1,u2 and may be extended to include the cases when some or all points are equal, that is [u0,u0,f] = lim u1→u0 [u0,u1,f] = f ′(u0) (2.10) provided that f′ exists. now passing the limit u1 → u0 and replacing u2 by u in second order divided difference, we have [u0,u0,u,f] = lim u1→u0 [u0,u1,u,f] = f(u) −f(u0) − (u−u0)f′(u0) (u−u0)2 ,u 6= u0 (2.11) provided that f′ exists. also passing to the limit ui → u (i = 0, 1, 2) in second order divided difference, we have [u,u,u,f] = lim ui→u [u0,u1,u2,f] = f′′(u) 2 (2.12) provided that f′′ exists. int. j. anal. appl. 16 (2) (2018) 183 one can note that, if for all u0,u1 ∈ i, [u0,u1,f] ≥ 0, then f is increasing on i and if for all u0,u1,u2 ∈ i, [u0,u1,u2,f] ≥ 0, then f is convex on i. now we define some families of parametric functions which we use in sequal. let i = [0,a) and j = [0,b) be intervals and let for t ∈ (c,d) ⊆ r, ft : i ×j → r be a mapping. then we define functions ft,y : i → r by ft,y(u) = ft(u,y) and ft,x : j → r by ft,x(v) = ft(x,v), where x ∈ i and y ∈ j. suppose m1 denotes the class of functions ft : i ×j → r for t ∈ (c,d) such that t 7→ [u0,u1,u2,ft,y] ∀ u0,u1,u2 ∈ i and t 7→ [v0,v1,v2,ft,x] ∀ v0,v1,v2 ∈ j are log convex functions in jensen sense on (c,d) for all x ∈ i and y ∈ j. under the assumptions of theorem 2.1 we define linear functional g(f; x0,y0) as a non negative difference of inequality (2.3) g(f; x0,y0) = a1  a2f (x̃n, ỹn) + b2   n∑ j=1 qj − 1  f (x̃n,y0)   + b1 ( n∑ i=1 pi − 1 )a2f (x0, ỹn) + b2   n∑ j=1 qj − 1  f(x0,y0)  − n∑ i,j=1 piqjf(xi,yj) (2.13) where a1,b1 and a2,b2 are defined in (2.4) and (2.5) respectively. remark 2.3. under the assumptions of theorem 2.1, if f is coordinated convex in ∆, then g(f; x0,y0) ≥ 0. remark 2.4. as a special case, if we put x0 = y0 = 0, in (2.13), then we get υ(f) = f (x̃n, ỹn) + ( n∑ i=1 qj − 1 ) f (x̃n, 0) + ( n∑ i=1 pi − 1 ) [ f(0, ỹn) + ( n∑ i=1 qj − 1 ) f(0, 0) ] − n∑ i,j=1 pi,qjf(xi,yj), (2.14) that is g(f; 0, 0) = υ(f). int. j. anal. appl. 16 (2) (2018) 184 remark 2.5. if we put yj = 1 for j = 1, ...,n in (2.13) then we get functional p(f) = f (x̃n) − n∑ i=1 pif(xi) − ( 1 − n∑ i=1 pi ) f(0) (2.15) defined in [1]. the following lemmas are given in [9]. lemma 2.1. let i ⊆ r be an interval. a function f : i → (0,∞) is log-convex in jensen sense on i, that is, for each r,t ∈ i f(r)f(t) ≥ f2 ( t + r 2 ) if and only if the relation m2f(t) + 2mnf ( t + r 2 ) + n2f(r) ≥ 0 holds for each m,n ∈ r and r,t ∈ i. lemma 2.2. if f is convex function on interval i then for all x1,x2,x3 ∈ i for which x1 < x2 < x3, the following inequality is valid: (x3 −x2)f(x1) + (x1 −x3)f(x2) + (x2 −x1)f(x3) ≥ 0. in [11], authors has given some important properties related to the functional defined for petrović’s inequality on coordinates. our next result comprises similar properties of functional defined in (2.13). theorem 2.2. suppose ft ∈ m1 and g be a functional defined in (2.13). then g(ft,x0,y0) is log-convex function in jensen sense for all t ∈ (c,d). proof. let h(u,v) = m2ft(u,v) + 2mnft+r 2 (u,v) + n2fr(u,v), where m,n ∈ r and t,r ∈ (c,d). we can assume that hy(u) = m 2ft,y(u) + 2mnft+r 2 ,y(u) + n 2fr,y(u) and hx(v) = m 2ft,x(v) + 2mnft+r 2 ,x(v) + n 2fr,x(v). since divided differences satisfy the linearity property, therefore we can have [u0,u1,u2,hy] = m 2[u0,u1,u2,ft,y] + 2mn[u0,u1,u2,ft+r 2 ,y] + n 2[u0,u1,u2,fr,y]. int. j. anal. appl. 16 (2) (2018) 185 since we have given that [u0,u1,u2; hy] is log-convex in jensen sense, therefore using ft = [u0,u1,u2; hy] in lemma 2.1, we get that [u0,u1,u2,hy] = m 2[u0,u1,u2,ft,y] + 2mn[u0,u1,u2,ft+r 2 ,y] + n 2[u0,u1,u2,fr,y] ≥ 0 which is equivalent to write [u0,u1,u2; hy] ≥ 0. this shows that hy is convex on interval i. in the similar way, one can prove that hx is convex on j. this concludes that h is coordinated convex on ∆. by remark 2.3, we have g(h,x0,y0) ≥ 0, that is, m2g(ft,x0,y0) + 2mng(ft+r 2 ,x0,y0) + n 2g(fr,x0,y0) ≥ 0. thus by lemma 2.1 we have that g(ft,x0,y0) is log-convex in jensen sense on (c,d). � corollary 2.2. let the functional υ defined in (2.14) and ft ∈ m1. then the function t 7→ υ(ft) is log convex in jensen sense on (c,d) proof. on putting x0 = y0 = 0 in above theorem, we get g(ft; 0, 0) = υ(ft), hence the required result follows. � theorem 2.3. suppose ft is from class m1 and g be a functional defined in (2.13), if g(ft,x0,y0) is continuous for all t ∈ (c,d), then g(ft,x0,y0) is log convex for all t ∈ (c,d). proof. since we know that if a function is log convex in jensen sense and continuous, then it is log convex. from theorem 2.2, if ft ∈m1, then g(ft,x0,y0) is log convex in jensen sense and we have given that it is continuous, hence g(ft,x0,y0) is log convex for all t ∈ (c,d). � corollary 2.3. let the functional υ defined in (2.14) and ft ∈m1. if the function t 7→ υ(ft) is continuous on (c,d), then it is log convex on (c,d) proof. on putting x0 = y0 = 0 in above theorem, we get g(ft; 0, 0) = υ(ft), hence the required result follows. � theorem 2.4. suppose ft ∈ m1 and g be a functional defined in (2.13). if g(ft; x0,y0) is positive, then for r,s,t ∈ (c,d) such that r < s < t, one has [g(fs; x0,y0)] t−r ≤ [g(fr; x0,y0)] t−s [g(ft; x0,y0)] s−r . (2.16) int. j. anal. appl. 16 (2) (2018) 186 proof. by taking f = logg(ft,x0,y0) in lemma 2.2, we have for t 6= r, u 6= v, (t−s) logg(fr; x0,y0) + (r − t) logg(fs; x0,y0) + (s−r) logg(ft; x0,y0) ≥ 0, which is equivalent to [g(fs; x0,y0)] t−r ≤ [g(fr; x0,y0)] t−s [g(ft; x0,y0)] s−r (2.17) that is our required result. � corollary 2.4. let the functional υ defined in (2.14) and ft ∈ m1. if υ(ft) is positive, then for some r < s < t, where r,s,t ∈ (c,d), one has [υ(fs)] t−r ≤ [υ(fr)] t−s [υ(ft)] s−r . (2.18) proof. on putting x0 = y0 = 0 in above theorem, we get g(ft; 0, 0) = υ(ft), hence the required result follows. � the following lemma is equivalent to the definition of convex function (see [5, page 2]). lemma 2.3. let i be an interval in r. a function f : i → r is convex if and only if for all t,r,u,v ∈ i such that t ≤ u,r ≤ v,t 6= r,u 6= v, one has f(t) −f(r) t−r ≤ f(u) −f(v) u−v . theorem 2.5. let g(ft; x0,y0) be the linear functional defined in (2.13), where ft ∈ m1. if the function t 7→ g(ft; x0,y0) is derivable on (c,d), then for t,r,u,v ∈ (c,d) such that t 6 u,r 6 v, we have c1(t,r) 6 c1(u,v), where c1(t,r) =   ( g(ft;x0,y0) g(fr;x0,y0) ) 1 t−r , t 6= r, exp ( d dt (g(ft;x0,y0)) g(ft;x0,y0) ) , t = r. (2.19) proof. by taking f = g(ft,x0,y0) in lemma 2.3, we have for t 6= r,u 6= v, logg(ft; x0,y0) − logg(fr; x0,y0) t−r 6 logg(fu; x0,y0) − logg(fv; x0,y0) u−v . this gives c1(t,r) 6 c1(u,v), t 6= r,u 6= v. for t = r,u = v, we consider limiting cases in above inequality, when r → t and v → u. � int. j. anal. appl. 16 (2) (2018) 187 the following corollaries that are stated in [11], are special cases of theorem 2.5. corollary 2.5. under the assumptions of theorem (2.5), let υ(ft) be the linear functional defined in (2.14) then e(t,r,ft) 6 e(u,v,ft), where e(t,r,ft) =   ( υ(ft) υ(fr) ) 1 t−r , t 6= r, exp ( d dt (υ(ft)) υ(ft) ) , t = r. (2.20) proof. on putting x0 = y0 = 0 in theorem (2.5), we get g(ft; x0,y0) = υ(ft), hence the required result follows. � corollary 2.6. under the assumptions of theorem (2.5), let p(ft) be the linear functional defined in (2.15) then t (t,r,ft) 6 t (u,v,ft), where t (t,r,ft) =   ( p(ft) p(fr) ) 1 t−r , t 6= r, exp ( d dt (p(ft)) p(ft) ) , t = r. (2.21) proof. on putting, yj = 1 for j = 1, ...,n in corollary 2.5, we get our required result. � example 2.1. let t ∈ (0,∞) and ϕt : [0,∞)2 → r be a function defined as ϕt(u,v) =   utvt t(t−1), t 6= 1, uv(log u + log v), t = 1. (2.22) define partial mappings ϕt,v : [0,∞) → r by ϕt,v(u) = ϕt(u,v) and ϕt,u : [0,∞) → r by ϕt,u(v) = ϕt(u,v). as we have [u,u,u,ϕt,v] = ∂2ϕt,v ∂u2 = ut−2vt ≥ 0 ∀ t ∈ (0,∞). this gives t 7→ [u0,u0,u0,ϕt,v] is log convex in jensen sense. similarly one can deduce that t 7→ [v0,v0,v0,ϕt,u] is also log-convex in jensen sense. if we choose ft = ϕt in theorem 2.2, we get log convexity of the functional g(γt). in special case, if we choose ϕt(u,v) = ϕt(u, 1), then we get [1, example 3]. example 2.2. let t ∈ [0,∞) and δt : [0,∞)2 → r be a function defined as δt(u,v) =   uveuvt t , t 6= 0, u2v2, t = 0. (2.23) int. j. anal. appl. 16 (2) (2018) 188 define partial mappings δt,v : [0,∞) → r by δt,v(u) = δt(u,v) and δt,u : [0,∞) → r by δt,u(v) = δt(u,v) for all u,v ∈ [0,∞). as we have [u,u,u,δt,v] = ∂2δt,v δu2 = euvt(2v2 + uv2) ≥ 0 ∀ t ∈ (0,∞). this gives t 7→ [u0,u0,u0,δt,v] is log convex in jensen sense. similarly one can deduce that t 7→ [v0,v0,v0,δt,u] is also log-convex in jensen sense. if we choose ft = δt in theorem 2.2, we get log convexity of the functional g(δt). in special case, if we choose δt(u,v) = δt(u, 1), then we get [1, example 8]. example 2.3. let t ∈ [0,∞) and γt : [0,∞)2 → r be a function defined as γt(u,v) =   euvt t , t 6= 0, uv, t = 0. (2.24) define partial mappings γt,v : [0,∞) → r by γt,v(u) = γt(u,v) and γt,u : [0,∞) → r by γt,u(v) = γt(u,v). as we have [u,u,u,γt,v] = ∂2γt,v ∂u2 = tv2euvt ≥ 0 ∀ t ∈ (0,∞). this gives t 7→ [u0,u0,u0,γt,v] is log convex in jensen sense. similarly one can deduce that t 7→ [v0,v0,v0,γt,u] is also log-convex in jensen sense. if we choose ft = γt in theorem 2.2, we get log convexity of the functional g(γt). in special case, if we choose γt(u,v) = γt(u, 1), then we get [1, example 9]. example 2.4. let t ∈ [0,∞) and λt : [0,∞)2 → r be a function defined as λt(u,v) = ueu √ t √ t (2.25) define partial mappings λt,v : [0,∞) → r by λt,v(u) = λt(u,v) and λt,u : [0,∞) → r by λt,u(v) = λt(u,v). int. j. anal. appl. 16 (2) (2018) 189 as we have [u,u,u,λt,v] = ∂2λt,v ∂u2 = v2euv √ t ( 2 + uv √ t ) ≥ 0 ∀ t ∈ (0,∞). this gives t 7→ [u0,u0,u0,λt,v] is log convex in jensen sense. similarly one can deduce that t 7→ [v0,v0,v0,λt,u] is also log-convex in jensen sense. if we choose ft = λt in theorem 2.2, we get log convexity of the functional g(λt). in special case, if we choose λt(u,v) = λt(u,−1), then we get [1, example 6]. 3. mean value theorems if a function is twice differentiable on an interval i, then it is convex on i if and only if its second order derivative is nonnegative. if a function f(x) := f(x,y) has continuous second order partial derivatives on interior of ∆ then it is convex on ∆ if the hessian matrix hf (x) :=  ∂2f(x)∂x2 ∂2f(x)∂y∂x ∂2f(x) ∂x∂y ∂2f(x) ∂y2   is nonnegative definite, that is, vhf (x)v t is nonnegative for all real nonnegative vector v. it is easy to see that f : ∆ → r is coordinated covex on ∆ iff f′′x (y) = ∂2f(x,y) ∂y2 and f′′y (x) = ∂2f(x,y) ∂x2 are nonnegative for all interior points (x,y) in ∆. lemma 3.1. let f : ∆ → r be a function such that m1 ≤ ∂2f(x,y) ∂x2 ≤m1 and m2 ≤ ∂2f(x,y) ∂y2 ≤ m2 for all interior points (x,y) in ∆2. consider the function ψ1,ψ2 : ∆ → r defined as ψ1 = 1 2 max{m1,m2}(x2 + y2) −f(x,y) ψ2 = f(x,y) − 1 2 min{m1,m2}(x2 + y2), then ψ1,ψ2 are convex on coordinates in ∆. proof. since ∂2ψ1(x,y) ∂x2 = max{m1,m2}− ∂2f(x,y) ∂x2 ≥ 0 and ∂2ψ1(x,y) ∂y2 = max{m1,m2}− ∂2f(x,y) ∂y2 ≥ 0 int. j. anal. appl. 16 (2) (2018) 190 for all interior points (x,y) in ∆, ψ1 is convex on coordinates in ∆. similarly one can prove that ψ2 is also convex on coordinates in ∆. � in [3] and [4], we have given mean value theorems of lagrange type and cauchy type for certain functional. here we give a theorem similar to those but for functional introduced in (2.13). theorem 3.1. let ∆̃ = [0,a1] × [0,b1] ⊂ ∆ and f : ∆̃ → r which has continuous partial derivatives of second order in ∆̃ and ϕ(x,y) := x2 + y2. then there exist (β1,γ1) and (β2,γ2) in the interior of ∆̃ such that g(f; x0,y0) = 1 2 ∂2f(β1,γ1) ∂x2 υ(ϕ) and g(f; x0,y0) = 1 2 ∂2f(β2,γ2) ∂y2 υ(ϕ) provided that g(ϕ; x0,y0) is non-zero. proof. since f has continuous partial derivatives of second order in ∆̃ and ∆̃ is compact, there exist real numbers m1,m2,m1 and m2 such that m1 ≤ ∂2f(x,y) ∂x2 ≤m1 and m2 ≤ ∂2f(x,y) ∂y2 ≤ m2, for all (x,y) ∈ ∆̃. now consider functions ψ1 and ψ2 defined in lemma 3.1. as ψ1 is convex on coordinates in ∆, g(ψ1; x0,y0) ≥ 0, that is g ( 1 2 max{m1,m2}ϕ(x,y) −f(x,y) ) ≥ 0, this leads us to 2g(f; x0,y0) ≤ max{m1,m2}g(ϕ; x0,y0). (3.1) on the other hand for function ψ2, one has min{m1,m2}g(ϕ; x0,y0) ≤ 2g(f; x0,y0). (3.2) as g(ϕ; x0,y0) 6= 0, combining inequalities (3.1) and (3.2), we get min{m1,m2}≤ 2g(f; x0,y0) g(ϕ; x0,y0) ≤ max{m1,m2}. then there exist (β1,γ1) and (β2,γ2) in the interior of ∆ such that 2g(f; x0,y0) g(ϕ; x0,y0) = ∂2f(β1,γ1) ∂x2 int. j. anal. appl. 16 (2) (2018) 191 and 2g(f; x0,y0) g(ϕ; x0,y0) = ∂2f(β2,γ2) ∂y2 , hence the required result follows. � the following corollary is particular case of theorem 3.1, which is stated in [11, theorem 4]. corollary 3.1. under the assumptions of above theorem, let υ(f) be the linear functional defined in (2.14), then υ(f) = 1 2 ∂2f(β1,γ1) ∂x2 υ(ϕ) and υ(f) = 1 2 ∂2f(β2,γ2) ∂y2 υ(ϕ) provided that υ(f) is non-zero. proof. on putting x0 = y0 = 0 in theorem 3.1, we get g(f; x0,y0) = υ(f), hence the required result follows. � theorem 3.2. let ψ1,ψ2 : ∆̃ → r be mappings which have continuous partial derivatives of second order in ∆̃. then there exists (η1,ξ1) and (η2,ξ2) in ∆̃ such that g(ψ1; x0,y0) g(ψ2; x0,y0) = ∂2ψ1(η1,ξ1) ∂x2 ∂2ψ2(η1,ξ1) ∂x2 (3.3) and g(ψ1; x0,y0) g(ψ2; x0,y0) = ∂2ψ1(η2,ξ2) ∂y2 ∂2ψ2(η2,ξ2) ∂y2 . (3.4) proof. we define the mapping p : ∆̃ → r such that p = k1ψ1 −k2ψ2, where k1 = g(ψ2; x0,y0) and k2 = g(ψ1; x0,y0). using theorem 3.1 with f = p, we have 2g(p ; x0,y0) = 0 = { k1 ∂2ψ1 ∂x2 −k2 ∂2ψ2 ∂x2 } g(ϕ; x0,y0) and 2g(p ; x0,y0) = 0 = { k1 ∂2ψ1 ∂y2 −k2 ∂2ψ2 ∂y2 } g(ϕ; x0,y0). since g(ϕ; x0,y0) 6= 0, we have k2 k1 = ∂2ψ1(η1,ξ1) ∂x2 ∂2ψ2(η1,ξ1) ∂x2 and k2 k1 = ∂2ψ1(η2,ξ2) ∂y2 ∂2ψ2(η2,ξ2) ∂y2 , int. j. anal. appl. 16 (2) (2018) 192 which are equivalent to required results. � corollary 3.2. under the assumptions of above theorem, let υ(f) be the linear functional defined in (2.14) then υ(ψ1) υ(ψ2) = ∂2ψ1(η1,ξ1) ∂x2 ∂2ψ2(η1,ξ1) ∂x2 (3.5) and υ(ψ1) υ(ψ2) = ∂2ψ1(η2,ξ2) ∂y2 ∂2ψ2(η2,ξ2) ∂y2 . (3.6) proof. on putting x0 = y0 = 0 in theorem 3.2, we get g(f; x0,y0) = υ(f), hence the required result follows. � references [1] s. butt, j. pečarić and atiq ur rehman, exponential convexity of petrović and related functional, j. inequal. appl. 2011 (2011), art. id 89. [2] s. s dragomir, on hadamards inequality for convex functions on the co-ordinates in a rectangle from the plane, taiwanese j mat. 4 (2001), 775–788 [3] g. farid, m. marwan, and a. u. rehman, fejer-hadamard inequality for convex functions on the coordinates in a rectangle from the plane, int. j. analysis appl. 10(1) (2016), 40–47. [4] g. farid, m. marwan and a. u. rehman, new mean value theorems and generalization of hadamard inequality via coordinated m−convex functions, j. inequal. appl. 2015 (2015), art. id 283. [5] d. s. mitrinovic, j. pečarić and a.m fink, classical and new inequalities in analysis, vol. 61, springer science & business media, 2013. [6] m. a. noor, f. qi and m. u. awan , some hermite-hadamard type inequalities for log-h-convex functions, analysis, 33(4) (2013), 367–375. [7] c.p. niculescu, the hermite-hadamard inequality for log-convex functions, nonlinear analysis, 75 (2012) 662–669. [8] j. pečarić, f. proschan, y. l. tong, convex functions, partial orderings and statistical applications, academic press, new york, 1992. [9] j. pečarić and atiq ur rehman, on logarithmic convexity for power sums and related results, j. inequal. appl. 2008 (2008), art. id 389410. [10] j. pečarić and atiq ur rehman, on logarithmic convexity for giaccardi’s difference, rad hazu. 515 (2013), 01–10. [11] a. u. rehman, muhammad mudessir, hafiza tahira fazal and ghulam farid, petrović’s inequality on coordinates and related results, cogent math. 3(1) (2016), art. id 1227298. [12] xiaoming zhang and weidong jiang, some properties of log-convex function and applications for the exponential function, comput. math. appl. 63(6) (2012), 1111–1116. 1. introduction 2. main results 3. mean value theorems references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 87-97 http://www.etamaths.com properties of weighted composition operators on some weighted holomorphic function classes in the unit ball a. e. shammaky and m. a. bakhit∗ abstract. in this paper, we introduce nk-type spaces of holomorphic functions in the unit ball of cn by the help of a non-decreasing function k : (0, ∞) → [0, ∞). several important properties of these spaces in the unit ball are provided. the results are applied to characterize boundedness and compactness of weighted composition operators wu,φ from nk(b) spaces into beurling-type classes. we also find the essential norm estimates for wu,φ from nk(b) spaces into beurling-type classes. 1. introduction through this paper, b is the unit ball of the n-dimensional complex euclidean space cn, s is the boundary of b. we denote the class of all holomorphic functions, with the compact-open topology on the unit ball b by h(b). for any z = (z1,z2, . . . ,zn),w = (w1,w2, . . . ,wn) ∈ cn, the inner product is defined by 〈z,w〉 = z1w1 + . . . + znwn, and write |z| = √ 〈z,z〉. two quantities af and bf, both depending on a function f ∈ h(b), are said to be equivalent, written as af ≈ bf, if there exists a finite positive constant m not depending on f, such that 1 m bf ≤ af ≤ mbf for every f ∈h(b). if the quantities af and bf, are equivalent, then in particular we have af < ∞ if and only if bf < ∞. as usual, the letter m will denote a positive constant, possibly different on each occurrence. given a point a ∈ b, we can associate wit it the following automorphism φa(z) ∈ aut(b) : φa(z) = a−pa(z) −saqa(z) 1 −〈z,a〉 , z ∈ b,(1.1) where sa = √ 1 −|a|2,pa(z) is the orthogonal projection of cn on a subspace [a] generated by a, that is pa(z) = { 0, if a = 0; a〈z,a〉 |a|2 , if a 6= 0, and qa = i −pa the projection on orthogonal complement [a] (see, for example,[8] or [10]). the map φa has the following properties that φa(0) = a, φa(a) = 0, φa = φ −1 a and 1 −〈φa(z), φa(w)〉 = (1 −|a|2)(1 −〈z,w〉) (1 −〈z,a〉)(1 −〈a,w〉) ,(1.2) where z and w are arbitrary points in b. in particular, 1 −|φa(z)|2 = (1 −|a|2)(1 −|z|2) |1 −〈z,a〉|2 .(1.3) 2010 mathematics subject classification. primary 30d45, 47b33; secondary 30b10. key words and phrases. weighted composition operators; nk spaces; beurling-type spaces. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 87 88 shammaky and bakhit let v be the lebesgue volume measure on cn, normalized so that v (b) ≡ 1 and σ be the normalized surface measure on s, so that σ(b) ≡ 1. let dτ(z) = dv (z) (1 −|z|2)n+1 , which is möbius invariant, that is for any ψ ∈ aut(b),f ∈ l1(b), we have∫ b f(z)dτ(z) = ∫ b f ◦ψ(z)dτ(z). for a ∈ b, the möbius invariant green function in b denoted by g(z,a) = g(φa(z)) where g(z) is defined by: g(z) = n + 1 2n ∫ 1 |z| (1 − t2)n−1t1−2ndt.(1.4) let h∞(b) denote the banach space of bounded functions in h(b) with the norm ‖f‖∞ = sup z∈b |f(z)|. for α > 0, the beurling-type space (sometimes also called the bers-type space) h∞α (b) in the unit ball b consists of those functions f ∈h(b) for which ‖f‖h∞α (b) = sup z∈b |f(z)|(1 −|z|)α < ∞. the bergman space a2(b) consists of those functions f ∈h(b) for which ‖f‖2a2(b) = ∫ b |f(z)|2dv (z) < ∞. let k : (0,∞) → [0,∞) be a right-continuous, non-decreasing function and is not equal to zero identically. the nk(b) space consists of all functions f ∈h(b) such that ‖f‖2nk(b) = sup a∈b ∫ b |f(z)|2k(g(z,a))dτ(z) < ∞. moreover, f ∈h(b) is said to belong to nk,0(b) if lim |a|→1 ∫ b |f(z)|2k(g(z,a))dτ(z) = 0. clearly, if k(t) = tp, then nk(b) = np(b); since g(z,a) ≈ (1 −|ϕa(z)|2) (see [6]). for k(t) = 1 it gives the bergman space a2(b). if nk(b) consists of just the constant functions, we say that it is trivial. several important properties of the nk(b) spaces in the unit disk in the complex plane have been characterized in [1], [4] and [9]. we assume from now that all k : (0,∞) → [0,∞) to appear in this paper are right-continuous, non-decreasing function and not equal to zero identically. given u ∈ h(b) and φ a holomorphic self-map of b. the weighted composition operator wu,φ : h(b) →h(b) is defined by wu,φ(f)(z) = u(z)(f ◦φ)(z), z ∈ b. it is obvious that wu,φ can be regarded as a generalization of the multiplication operator muf = u ·f and composition operator cφf = f◦φ. weighted composition operators are a general class of operators and appear naturally in the study of isometries on most of the function spaces. operators of this kind also appear in many branches of analysis; the theory of dynamical systems, semigroups, the theory of operator algebras, the theory of solubility of equations with deviating argument and so on. the behavior of those operators is studied extensively on various spaces of holomorphic functions (see for example [2], [5], [6] and [9]). recall that the pseudohyperbolic metric in the ball is defined as: ρ(z,w) = |φw(z)|, z ∈ b.(1.5) it is true metric (see [3]). also it is easy to verify, in particular, that ρ(0,w) = |w| and ρ(φa(z),w) = |z|. the following lemma was proved in [6]: properties of weighted composition operators 89 lemma 1.1. for z,w ∈ b, if ρ(z,w) ≤ 1 2 . then (1.6) 1 6 ≤ 1 −|z|2 1 −|w|2 ≤ 6. the following proposition was proved as part of lemma 2.3 in [6], and hence, we omit the details. proposition 1.1. for z,w ∈ b, if ρ(z,w) ≤ 1 4 . then |f(z) −f(w)| ≤ 2 √ n|f(φa(z))|ρ(z,w). recall that a linear operator t : x → y is said to be bounded if there exists a constant m > 0 such that ‖t(f)‖y ≤ m‖f‖x for all maps f ∈ x. moreover, t : x → y is said to be compact if it takes bounded sets in x to sets in y which have compact closure. for a complex banach spaces x and y of h(b), t : x → y is compact (respectively weakly compact) if it maps the closed unit ball of x onto a relatively compact (respectively relatively weakly compact) set in y. in this paper, we introduce nk(b) spaces, in terms of the right continuous and non-decreasing function k : (0,∞) → [0,∞) on the unit ball b. we prove that nk(b) contained in beurling-type space h∞α (b),α = n+1 2 . a sufficient and necessary condition for nk(b) non-trivial is given. we discus the nesting property of nk(b). we obtain the complete characterizations of the boundedness and compactness of weigted composition operators from nk(b) spaces into beurling-type classes. we also find the essential norm estimates for these operators. our results contain the results in the unit disk as particular cases (for example [4], [6] and [9]). 2. nk(b) spaces in the unit ball the following results play an important role in the proof of our main result. they also have their own interest. proposition 2.1. let k : (0,∞) → [0,∞) be non-decreasing function. then nk(b) ⊂ h∞n+1 2 (b). proof. for a ∈ b, let b 1 2 = {z ∈ b : |z| < 1 2 }. without loss of generality, assume that k( 3 4 ) > 0. if f ∈nk(b), then ‖f‖2nk(b) ≥ k(3/4) ∫ b 1 2 |f(z)|2dτ(z). by by the subharmonicity of |f(z)|2 and hence by ([7], theorem 2.1.4), we have |f(0)|2 ≤ 1 v (b 1 2 ) ∫ b 1 2 |f(z)|2dτ(z) = 4n ∫ b 1 2 |f(z)|2dτ(z). thus |f(0)|2 ≤ 4n k(3/4) ‖f‖2nk(b), f ∈nk(b). for every fixed z ∈ b, we put (2.1) f(w) = (1 −|z|2) n+1 2 |f(φa(w))| (1 −〈w,z〉)n+1 , w ∈ b, which is clearly f ∈h(b). we can prove that ‖f‖2nk(b) ≤‖f‖ 2 nk(b), and so f ∈nk(b). then, we have |f(a)|2(1 −|a|2)n+1 = |f(0)|2 ≤ 4n k(3/4) ‖f‖2nk(b), for all z ∈ b, which implies that: ‖f‖2h∞ n+1 2 (b) = sup a∈b |f(a)|2(1 −|a|2)n+1 ≤ 4n k(3/4) ‖f‖2nk(b). that is, nk(b) ⊂ h∞n+1 2 (b). 90 shammaky and bakhit proposition 2.2. for z,w ∈ b and f ∈nk(b), we have (2.2) |f(z) −f(w)| ≤ m‖f‖nk(b) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w). here, m = 6 n+1 2 22(n+1) √ n(3+2 √ 3) k(3/4) . proof. we consider two cases: case 1: ρ(z,w) ≥ 1 4 . since |f(z) −f(w)| ≤ |f(z)| + |f(w)|, by proposition 2.1, we have min{(1 −|z|2) n+1 2 , (1 −|w|2) n+1 2 }|f(z) −f(w)| ≤ (1 −|z|2) n+1 2 |f(z)| + (1 −|w|2) n+1 2 |f(w)| ≤ 2‖f‖h∞ n+1 2 (b) ≤ 22n+1 k(3/4) ‖f‖2nk(b) ≤ 22n+3ρ(z,w) k(3/4) ‖f‖2nk(b). which implies that |f(z) −f(w)| ≤ 22n+3 k(3/4) ‖f‖nk(b) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w). case 2: ρ(z,w) > 1 4 . take and fix w ∈ b, from ρ(φa(z),w) = |z| it follows that if z ∈ b 1 2 , (b = b∪s), then ρ(φa(z),w) < 12. in this case, by proposition 2.1 and lemma 1.1, we have |f(φw(z))| ≤ ‖f‖h∞ n+1 2 (b) (1 −|φw(z)|2) n+1 2 ≤ 22n+1 k(3/4) ‖f‖2nk(b) (1 −|φw(z)|2) n+1 2 = 22n+1 k(3/4) ‖f‖2nk(b) (1 −|w|2) n+1 2 [ 1 −|w|2 1 −|φw(z)|2 ]n+1 2 ≤ 6 n+1 2 22n+1 k(3/4) ‖f‖2nk(b) (1 −|w|2) n+1 2 . by proposition 1.1, we have |f(z) −f(w)| ≤ 6 n+1 2 22(n+1) √ n(3 + 2 √ 3) k(3/4) ‖f‖2nk(b) (1 −|w|2) n+1 2 ρ(z,w). combining the results of the two cases yields |f(z) −f(w)| ≤ m‖f‖nk(b) max{(1 −|z| 2)− n+1 2 , (1 −|w|2)− n+1 2 }ρ(z,w), where m = 6 n+1 2 22(n+1) √ n(3+2 √ 3) k(3/4) . lemma 2.1. for a ∈ b, 0 < δ < 1 and f ∈nk(b), we have (2.3) |f(a) −f(δa)| ≤ m (1 −|a|2) n+1 2 ‖f‖2nk(b). consequently, for any 0 < r < 1, we have (2.4) sup |a|≤r |f(a) −f(δa)| ≤ m (1 −r2) n+1 2 ‖f‖2nk(b). here, m is the constant from proposition 2.2. properties of weighted composition operators 91 proof. proposition 2.2 shows that |f(a) −f(δa)| ≤ m‖f‖nk(b) max{(1 −|a| 2)− n+1 2 , (1 −|δa|2)− n+1 2 }ρ(a,δa). the well-known formula 1 −|ρ(a,δa)|2 = 1 −|φa(δa)|2 = (1 −|a|2)(1 −|δa|2) |1 −〈a,δa〉|2 , together with simple calculations gives ρ(a,δa) = (1 −|a|2)|a| 1 − δ|a|2 ≤ 1. on the other hand, (1 −|δa|2)− n+1 2 ≤ (1 −|a|2)− n+1 2 . the inequalities in (2.3) now follow. if |a| ≤ r, then 1 −δ|a|2 ≥ 1 −r2. taking supremum of (2.3) in a yields (2.4). theorem 2.1. if (2.5) ∫ 1 0 r2n−1 (1 −r2)n+1 k(g(r))dr < ∞, then nk(b) contains all polynomials; otherwise, nk(b) contains only constant functions. proof. first assume that (2.5) holds. let f(z) be a polynomial i.e. (there exists a m > 0 such that |f(z)|2 ≤ m for all z ∈ b. then,∫ b |f(z)|2k(g(z,a))dτ(z) = ∫ b |f(φa(z))|2k(g(z)) dv (z) (1 −|z|2)n+1 = 2n ∫ 1 0 r2n−1 (1 −r2)n+1 k(g(r))dr ∫ s |f ◦ϕa(rζ)|2dσ(ζ) ≤ 2nm ∫ 1 0 r2n−1 (1 −r2)n+1 k(g(r))dr. since a is arbitrary, it follows that ‖f‖nk(b) ≤ 2nm ∫ 1 0 r2n−1 (1 −r2)n+1 k(g(r))dr < ∞. thus, f ∈nk(b) and the first half of the theorem is proved. now, we assume that the integral in (2.5) is divergent. let α = (α1, · · · ,αn) is an n-tuple of non-negative integers, |α| = α1 + α2 + · · · + αn ≥ 1, f(z) = zα. then, we have |f(rζ)|2 = r2|α||ζα|2, and ∫ s |(rζ)α|2dσ(rζ) ≥ r2|α| (n− 1)!α! (n− 1 + |α|)! ≥ mr2|α|. thus, ‖f‖nk(b) ≥ nm 22|α|−1 ∫ 1 1 2 r2n−1 (1 −r2)n+1 k(g(r))dr.(2.6) there exists a ∈ b such that f(a) 6= 0, by the subharmonicity of |f ◦ φa(rζ)|2, ‖f‖nk(b) ≥ 3n 2 |f(a)|2 ∫ 1 2 0 r2n−1 (1 −r2)n+1 k(g(r))dr.(2.7) combining (2.6) and (2.7), we see that (2.5) implies that ‖f‖nk(b) = ∞. it is proved that f /∈nk(b) and, since α is arbitrary, any non-constant polynomial is not contained in nk(b). we conclude that nk(b) contains only constant functions. the theorem is proved. 92 shammaky and bakhit lemma 2.2. for w ∈ b we define the probe function in nk(b) as hw(z) = (1 −|w|2)n+1 (1 −〈z,w〉) 3 2 (n+1) . suppose that condition (2.5) is satisfied. then hw ∈nk(b) and ‖hw‖nk(b) ≤ 1. proof. trivially hw ∈nk(b). it is also easy to see that ‖hw‖2nk(b) = sup a∈b ∫ b ∣∣∣∣ (1 −|w|2)n+1 (1 −〈z,w〉) 3 2 (n+1) ∣∣∣∣2k(g(z,a))dτ(z) ≤ 1, this by a change of variables and since condition (2.5) is satisfied. such hw is a normalized reproducing kernel function in the bergman space a 2(b). also note that hw(w) = ( 1 1 −|w|2 )n+1 2 , ∀w ∈ b. in section 4, we will discuss the estimation for the lower bounded of ‖wu,φ‖e we will make use of weakly convergent sequences in the bergman space a2(b). the following lemma plays an important role. lemma 2.3. suppose {fm}m≥1 ∈ a2(b) is a sequence that converges weakly to zero in a2(b). then {fm}m≥1 converges weakly to zero in nk(b) as well. proof. let γ ∈nk(b) be a bounded liner functional on nk(b). by the fact that ‖f‖nk(b) ≤‖f‖a2(b), then ‖γ‖ a2(b) = sup f∈a2(b) |γ(f)| ‖f‖a2(b) ≤ sup f∈a2(b) |γ(f)| ‖f‖nk(b) ≤ sup f∈nk(b) |γ(f)| ‖f‖nk(b) = ‖γ‖nk(b), which implies γ is also a bounded linear functional on a2(b). since fm → 0 weakly in a2(b), we conclude that γ(fm) → 0. therefore, fm → 0 weakly in nk(b) as well. corollary 2.1. let {wm}m∈n ⊂ b and |wm| → 1 as m → ∞, then {hwm} converges weakly to zero in nk(b). proof. it is well known that hwm → 0 weakly in a2(b) as m → ∞. indeed, for any f ∈ a2(b), using the reproducing property, we have 〈f,hwm〉 = (1 −|wm| 2) n+1 2 f(wm), which converges to zero as m → ∞, because the set of polynomials is dense in a2(b) (see [10], proposition 2.6). the conclusion of the corollary follows immediately from lemma 2.3. 3. weighted composition operators from nk(b) into h∞α (b) in this section, we will consider the operator wu,φ : nk(b) → h∞α (b). theorem 3.1. let φ : b → b be a holomorphic mapping and u ∈ h(b). for 0 < α < ∞, then wu,φ : h ∞ α (b) →nk(b) is a bounded operator if and only if (3.1) sup a∈b |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞. proof. first assume that condition (3.1) holds, by proposition 1.1, we have ‖wu,φ(f)‖h∞α (b) = sup z∈b |u(z)||f(φ(z))|(1 −|z|2)α ≤ ‖f‖h∞ n+1 2 sup z∈b |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ≤ c‖f‖nk(b). properties of weighted composition operators 93 this implies that wu,φ : h ∞ α (b) →nk(b) is a bounded operator. conversely, assume that wu,φ : nk(b) → h∞α (b) is bounded, then ‖wu,φ(f)‖h∞α (b) ≤‖f‖nk(b). let hw be the test function in lemma 2.2 with w = φ(z), then we get hφ(z)(φ(z)) = ( 1 1 −|φ(z)|2 )n+1 2 . hence, there exist a positive constant m such that: m ≥‖hw‖nk(b) ≥‖wu,φ(hw)‖h∞α (b) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . this completes the proof of the theorem. using the standard arguments similar to those outlined in proposition 3.11 of [2], we have the following lemma: lemma 3.1. let φ : b → b be a holomorphic mapping and u ∈ h(b). for 0 < α < ∞, then wu,φ : h ∞ α (b) →nk(b) is compact if and only if lim m→∞ ‖wu,φ(fm)‖nk(b) = 0, for every bounded sequence {fm}⊂nk(b) which converges to 0 uniformly on any compact subsets of b as m →∞. theorem 3.2. let φ : b → b be a holomorphic mapping and u ∈ h(b). for 0 < α < ∞, then wu,φ : nk(b) → h∞α (b) is compact if and only if (3.2) lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 = 0. proof. first assume that wu,φ : nk(b) → h∞α (b) is compact, then it is bounded. by theorem theorem 3.1, we have l = sup a∈b |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞, note that lim r→1− l(r) always exists, where: l(r) = sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . now, we show that (3.4) holds. assume on the contrary that there exists ε0 > 0 such that lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 = ε0. there exists an r0 ∈ (0, 1) such that r0 < r < 1, we have l(r) > ε02 . then, by the standard diagonal process, we can construct a sequence {zm} ⊂ b such that |φ(z)| → 1 asm → ∞, and also for each m ∈ n, |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ≥ ε0 4 . clearly, we can assume that wn = φ(zm) tends to w0 ∈ ∂b as m → ∞. let hwm = (1−|wm|2)n+1 (1−〈zm,wm〉) 3 2 (n+1) be the function in lemma 2.2 with wn = φ(zm). then hwn → hw0 with respect to the compact-open topology. define fm = hwm −hw0 . then ‖fm‖nk(b) ≤ 1 and fm → 0 uniformly on compact subsets of b. thus, fm ◦φ → 0 in h∞α (b) by assumption. but, for m big enough, ‖wu,φ(fn)‖h∞α (b) ≥ |u(zm)|(1 −|zm|2)α (1 −|φ(zm)|2) n+1 2 ≥ ε0 4 , which is a contradiction. conversely, if (3.4) holds, we assume that {fm} is a bounded sequence in nk(b) norm which converges 94 shammaky and bakhit to zero uniformly on every compact subset of mathbbb, then for all ε > 0 there exists δ ∈ (0, 1) and mε < m such that for δ < r < 1, we have ‖wu,φ(fn)‖h∞α (b) ≤ sup |φ(z)|>r |u(z)||fm(φ(z))|(1 −|z|2)α + sup |φ(z)|≤r |fm(φ(z))|(1 −|z|2)α ≤ ε. from this ‖wu,φ(fn)‖h∞α (b) → 0 as m → ∞, it follows that wu,φ : nk(b) → h ∞ α (b) is a compact operator. this completes the proof of the theorem. as a corollary of theorems 3.1 and 3.2, we have: corollary 3.1. let φ : b → b be a holomorphic mapping and 0 < α < ∞. then composition operator cφ : nk(b) → h∞α (b) • is bounded if and only if (3.3) sup z∈b (1 −|z|2)α (1 −|φ(z)|2) n+1 2 < ∞; • is compact if and only if (3.4) lim r→1− sup |φ(z)|>r (1 −|z|2)α (1 −|φ(z)|2) n+1 2 = 0. 4. essential norms of weighted composition operators from nk(b) into h∞α (b) in this section, we study the essential norm of weighted composition operator wu,φ : nk(b) → h∞α (b). let us denote by c := c(nk(b),h∞α (b)) the set of all compact operators acting from nk(b) into h∞α (b). then the essential norm of wu,φ is defined as follows: (4.1) ‖wu,φ‖e = inf o∈c {‖wu,φ −o‖}. obviously, the essential norm of a compact operator is zero. note that by using the standard argument, it can be shown that a composition operator cφ : nk(b) → nk(b) is compact if and only if for any bounded sequence {fm} ⊂ nk(b) converging to zero uniformly on every compact subset of b, the sequence {‖fm ◦φ‖} converges to zero as m →∞. lemma 4.1. suppose φ : b → b is a holomorphic mapping such that ‖φ‖∞ < 1, and u ∈ h(b). for 0 < α < ∞, then wu,φ : h∞α (b) →nk(b) is compact if and only if lim m→∞ ‖wu,φ(fm)‖nk(b) = 0, proof. let r = ‖φ‖∞ and take an arbitrary f ∈nk(b). then we have ‖wu,φ(f)‖nk(b) ≤‖f ◦φ‖∞‖u‖nk(b) ≤ ( sup {z:|z|≤r} |f(z)| ) ‖u‖nk(b) < ∞. this show that wu,φ maps nk(b) into itself. now suppose that {fm} is a bounded sequence in nk(b) that converges to zero uniformly on every compact subset of b. applying the above estimate with f = fm, we have ‖wu,φ(fm)‖nk(b) ≤ ( sup {z:|z|≤r} |fm(z)| ) ‖u‖nk(b) < ∞. since the set z : |z| ≤ r is compact, the right-hand side of the last quantity converges to 0 as m →∞, hence so does the sequence {‖wu,φ(fm)‖nk(b)}. this means that wu,φ is compact. in the following theorem we formulate and prove an estimate for the upper bound of the essential norm of wu,φ : nk(b) → h∞α (b). properties of weighted composition operators 95 theorem 4.1. let φ : b → b be a holomorphic mapping and u ∈h(b). for 0 < α < ∞, suppose that wu,φ : nk(b) → h∞α (b) is a bounded operator. then (4.2) ‖wu,φ‖e ≤ m lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 , where m = 6 n+1 2 22(n+1) √ n(3+2 √ 3) k(3/4) is the constant from proposition 2.2. proof. since wu,φ is bounded, we see that u ∈ h∞α (b), and theorems 3.1, 3.2 shows that lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 exists and is a real number. first, we prove that for any r ∈ [0, 1), (4.3) ‖wu,φ‖e ≤ m sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . for each k ∈ n, set φk(z) = kzk+1 for all z ∈ b. by lemma 4.1, cφk is compact on nk(b), and hence, o = wu,φ ◦cφk ∈c is compact acting from nk(b) into h ∞ α (b). then for any k ∈ n, we have ‖wu,φ‖e = inf o∈c {‖wu,φ −o‖}≤‖wu,φ −wu,φ ◦cφk‖e = sup ‖f‖nk(b)≤1 ‖(wu,φ −wu,φ ◦cφk)(f)‖h∞α (b), which implies that (4.4) ‖wu,φ‖e ≤ inf k∈n { sup ‖f‖nk(b)≤1 ‖(wu,φ −wu,φ ◦cφk)(f)‖h∞α (b) } . for f ∈nk(b), we estimate ‖(wu,φ −wu,φ ◦cφk)(f)‖h∞α (b) = sup z∈b { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ sup |φ(z)|>r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } + sup |φ(z)|≤r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } . on the one hand, by lemma 2.1, equation (2.3), we have sup |φ(z)|>r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ sup |φ(z)|>r ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣sup z∈b { |u(z)|(1 −|z|2)α } ≤ ( sup |φ(z)|>r m|u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) ‖f‖nk(b). on the one hand, by lemma 2.1, equation (2.4), we have sup |φ(z)|≤r { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ ( mr‖u‖h∞α (b) (k + 1)(1 −r2)α ) ‖f‖nk(b). 96 shammaky and bakhit therefore, if ‖f‖nk(b) ≤ 1, then ‖(wu,φ −wu,φ ◦cφk)(f)‖h∞α (b) ≤ sup z∈b { |u(z)| ∣∣∣∣f(φ(z)) −f ( kφ(z) k + 1 )∣∣∣∣(1 −|z|2)α } ≤ m {( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) + mr‖u‖h∞α (b) (k + 1)(1 −r2)α } . it then follows that inf k∈n { sup ‖f‖nk(b)≤1 ‖(wu,φ −wu,φ ◦cφk)(f)‖h∞α (b) } ≤ m inf k∈n {( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) + mr‖u‖h∞α (b) (k + 1)(1 −r2)α } ≤ m ( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) .(4.5) combining (4.4) and (4.5), we obtain (4.3). now letting r → 1 in (4.3), we arrive at the desired inequality (4.6) ‖wu,φ‖e ≤ m lim r→1 ( sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 ) . this completes the proof of the theorem. we now discuss the estimation for the lower bound of the essential norm of wu,φ : nk(b) → h∞α (b). theorem 4.2. let φ : b → b be a holomorphic mapping and u ∈h(b). for 0 < α < ∞, suppose that wu,φ : nk(b) → h∞α (b) is a bounded operator. then (4.7) ‖wu,φ‖e ≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . proof. the case ‖φ‖∞ < 1 is obvious since the right hand side is zero. now assume that ‖φ‖∞ = 1. for any r ∈ (0; 1), the set sr := {z ∈ b : |φ(z)| > r is not empty. for each z ∈ b, consider the probe function hw in lemma 2.2 with w = φ(z). then for any compact operator o ∈c we have ‖wu,φ −o‖ = sup ‖f‖nk(b)≤1 ‖(wu,φ −o)(f)‖h∞α (b) ≥ ‖(wu,φ −o)(hφ(z))‖h∞α (b) ≥ ‖wu,φ(hφ(z))‖h∞α (b) −‖o(hφ(z))‖h∞α (b) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 −‖o(hφ(z))‖h∞α (b), which is equivalent to (4.8) ‖wu,φ −o‖ + ‖o(hφ(z))‖h∞α (b) ≥ |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . taking the supremum on z over the set sr on both sides of (4.8) yields ‖wu,φ −o‖ + sup z∈sr ‖o(hφ(z))‖h∞α (b) ≥ sup z∈sr |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . which is (4.9) ‖wu,φ −o‖ + sup |φ(z)|>r ‖o(hφ(z))‖h∞α (b) ≥ sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . denote h(r) = sup |φ(z)|>r ‖o(hφ(z))‖h∞α (b). since h(r) decreases as r increases, limr→1 h(r) exists. we claim that this limit is necessarily zero. for the purpose of obtaining a contradiction, assume that properties of weighted composition operators 97 lim r→1 h(r) = l > 0. then there is a sequence {zm}⊂ b satisfying |φ(zm)|→ 1 as m →∞, and for each m ∈ n, (4.10) ‖o(hφ(z))‖h∞α (b) ≥ 1 2 l. by corollary 3.1, {hφ(zm)} converges weakly to zero in nk(b). since o is compact, we have {‖o(hφ(z))‖h∞α (b)} converges to zero as m →∞, which contradicts (4.10). therefore, lim r→1− sup |φ(z)|>r ‖o(hφ(z))‖h∞α (b) = 0. letting r → 1− on both sides of (4.9), we conclude that for any compact operator o ∈c, ‖wu,φ −o‖≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . from this, it follows that ‖wu,φ‖e = inf o∈c ‖wu,φ −o‖≥ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . this completes the proof of the theorem. in conclusion, combining theorems 4.1 and 4.2, we obtain a full description of the essential norm of wu,φ. theorem 4.3. let φ : b → b be a holomorphic mapping and u ∈h(b). for 0 < α < ∞, suppose that wu,φ : nk(b) → h∞α (b) is a bounded operator. then (4.11) ‖wu,φ‖e ≈ lim r→1− sup |φ(z)|>r |u(z)|(1 −|z|2)α (1 −|φ(z)|2) n+1 2 . acknowledgement. the authors would like to express their thanks to the deanship of scientific research at jazan university saudi arabia for funding the work through research project no. 207/sabic 2/36. references [1] m.a. bakhit, a.e. shammaky, predual norms of some holomorphic function spaces, international j. functional analysis, operator theory and applications, 6(3) (2014), 153-176. [2] c.c. cowen and b.d. maccluer, composition operators on spaces of analytic functions, crc press, 1995. [3] p. duren and r. weir, the pseudohyperbolic metric and bergman spaces in the ball, trans. amer. math. soc. 359 (2007), 63-76. [4] a. el-sayed ahmed and m.a. bakhit, holomorphic nk and bergman-type spaces, birkhuser series on operator theory: advances and applications (2009), birkhuserverlag publisher baselswitzerland, 195 (2009), 121-138. [5] a. el-sayed ahmed and m.a. bakhit, composition operators acting betweensome weighted möbius invariant spaces, j. ann. funct. anal. 2 (2011), 138-152. [6] b. hu and l.h. khoi, compact difference of weighted composition operators on np-spaces in the ball, romanian j. pure appl. math. 60(2) (2015), 101-116. [7] s. krantz, function theory of several complex variables, ams chelsea publishing, providence, rhode island, 1992. [8] w. rudin, function theory in the unit ball of cn, springer-verlag, new york, 1980. [9] a.e. shammaky, weighted composition operators acting between kind of weighted bergman-type spaces and the bers-type space, international journal of mathematical, computational science and engineering, 8(3) (2014), 496499. [10] k. zhu, spaces of holomorphic functions in the unit ball, springer-verlag, new york, 2004. department of mathematics, faculty of science, jazan university, jazan, saudi arabia ∗corresponding author: mabakhit@jazanu.edu.sa international journal of analysis and applications volume 16, number 3 (2018), 328-339 url: https://doi.org/10.28924/2291-8639 doi: 10.28924/2291-8639-16-2018-328 time control charts through nhpp based on dagum distribution b. srinivasa rao1,∗, p. sricharani2 1department of mathematics & humanities, r.v.r & j.c college of engineering, chowdavaram,guntur522 019, andhra pradesh, india 2department of basic sciences, sri vishnu engineering college for women, bhimavaram, andhra pradesh, india ∗corresponding author: boyapatisrinu@yahoo.com abstract. statistical process control is a method of monitoring product in its development process using statistical techniques with the presumption that the products produced under identical process condition shall not always be alike with respect to some quality characteristic(s). however, if the observed variations are within the tolerable limits statistical process control (spc) methods would pass them for acceptance. this philosophy is adopted to decide the reliability and quality of a product by defining some quality measures and proposing a probability model for the quality measurements. the well known dagum distribution(dd) is considered to propose a product reliability based on non-homogenous poisson process (nhpp). its mean value function is taken as a quality characteristic and spc limits for it are developed. these control limits are exemplified to a live failure data to detect the out of control signals for the quality of the product based on the failure data and compared with exponential distribution(ed). 1. introduction life time data generally contain the failure times of sample products or interfailure times or number of failures experienced in a given time. assuming a suitable probability model the reliability of the product received 2017-07-31; accepted 2017-10-09; published 2018-05-02. 2010 mathematics subject classification. 60k35. key words and phrases. dagum distribution; exponential distribution; statistical process control; non-homogenous poisson process. c©2018 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 328 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-328 int. j. anal. appl. 16 (3) (2018) 329 is computed and the quality with respect to reliability would be assessed. from a different point of view if the specific life time data contain, times between failures, also called inter failure times, probability limits for such a data can be constructed in a parametric approach. taking central line at the median of the distribution of the data, the probability limits as usual control limits we can think of a control chart for the data. points above the upper control limit of such a data would be an encouraging characteristic of the product because they lead to a large gap between successive failures so that the uptime of the product is large. hence the product is preferable. that is detection of out of control above the ucl is desirable and its causes are to preserved or encouraged. similarly detection of out of control below the lcl results in shorter gaps between successive failures.the assignable causes for this detection are to be minimized or eliminated. points within the control limits indicate a smooth failure phenomenon. thus such a set of control limits would be helpful in assessing the quality of the product based on inter failure time data. any manufactured product is prone to failures for known or unknown reasons. a failed product can be rectified to bring it back to functioning through a testing process. in this procedure the data of observed product failures would throw some light on the quality of the product. there are various methods of measuring the product quality and the most popular among them is product reliability. non homogenous poisson processes are suitable models to compute product reliability in the statistical science. the earliest works in this direction can be attributed to those of chan et al (2000) [1], xie et al (2002) [4], pham and zhang(2003) [3] and kim(2013) [2]. all these attempts are focussed on the mathematical model of the type p(n(t + s) − n(t) = y) = e−λs(λs)y y! , y = 0, 1, 2, ... (1.1) where n(t) indicates the random number of occurrences of an event in the interval [0,t]. this mathematical model indicates that the changes in n(t) from one time period to another time period say [t,t+s] depend only on the length of the interval s but not on the extremities t,t+s of the interval. λ is called the failure intensity. in the above equation e[n(t)] = λt,∀t. if we think of a poisson process whose mean depends on the starting t and also the length of the interval s such a poisson process can be explained by an equation as p(n(t) = y) = e−m(t)(m(t))y y! , y = 0, 1, 2, ... (1.2) in this equation m(t) is a positive valued, non decreasing, continuous function and is called the mean value function. equation (1.2) is called a non homogenous poisson process. if a product system when put to use fails with probability f(t) before time t, if ’θ’ stands for the unknown eventual number of failures that it is likely to experience, then the average number of failures expected to be experienced before time t is θf(t). hence θf(t) can be taken as the mean value function of an nhpp. in the theory of probability, f(t) is called the cumulative distribution function (cdf) of a continuous non negative valued random variable. thus an nhpp designed to study the failure process of a product can be constructed as a poisson process with int. j. anal. appl. 16 (3) (2018) 330 mean value function based on the cumulative distribution function of a continuous positive valued random variable. with this backdrop, we consider the well known dagum distribution (dd) as f(t) to generate a growth model based non homogenous poisson process (nhpp). for such a model we developed the statistically admissible control limits for the mean value function and demonstrate the same how a graphical procedure called a statistical process control (spc) chart based on the mean value function would help in detecting out of control signals for the product quality. the rest of the paper is organized as follows: the basic distribution characteristics of dagum distribution (dd)and its properties are presented in section 2. control chart monitoring the time between failures based on the mean value function, statistically tolerable limits for the failure time random variable, comparison with exponential model and findings are given in section 3. monitoring the production process based on the mean value function using order statistics, comparison with exponential model and the findings are given in section 4. summary and conclusions are given in section 5. 2. distribution and its properties in the present paper we consider the cdf of dd as the genesis of mean value function of our sqc. this model is an increasing failure rate (ifr). such a distribution is proved to be having a number of important applications in survival analysis, a proxy concept to reliability theory. the probability density function (pdf) of dagum distribution is given by f(x) = ap x ( (x b )ap ((x b )a + 1)p+1 ) ,x > 0,a > 0,b > 0,p > 0. (2.1) its cumulative distribution function (cdf) is f(x) = ( 1 + (x b )−a)−p ,x > 0,a > 0,b > 0,p > 0. (2.2) the dagum distribution is a skewed, unimodal distribution on the positive real line. the mean, median and variance of dagum distbution are respectively mean = − b a γ ( −1 a ) γ ( 1 a + p ) γ (p) , ifa > 1. (2.3) median = b ( −1 + 2 1 p )− 1 a . (2.4) v ariance = − b2 a2  2aγ(−2a)γ(2a + p) γ (p) + ( γ ( −1 a ) γ ( 1 a + p ) γ (p) )2 , ifa > 2. (2.5) the nhpp with θf(x) as the mean value function for our present study is m(x) = θ ( 1 + (x b )−a)−p ,θ > 0,a > 0,b > 0,p > 0. (2.6) int. j. anal. appl. 16 (3) (2018) 331 thus our proposed model contains 3 parameters namely θ, a and b, where θ stands for the unknown number of faults present in the product 3. control chart monitoring the time between failures based on mean value function let f(x) be the cumulative distribution function of a continuous positive valued random variable, f(x) be its probability density function. if the random variable is taken as representing inter failure time of a device, a control chart of such data would be based on 0.9973 probability limits of the times between failure random variable say t analogous to the shewhart’s theory of variable control charts. these limits and the central line are respectively the solutions of the following equations taking equi-tailed probabilities. f(t) = 0.00135 (3.1) f(t) = 0.5 (3.2) f(t) = 0.99865 (3.3) let tu , tc and tl be respectively the solutions of equations (3.1), (3.2) and (3.3) in the standard form tl = f −1(0.00135) (3.4) tc = f −1(0.5) (3.5) tu = f −1(0.99865) (3.6) the nhpp with f(θ,x) as the mean value function for our present study is m(x) = θ ( 1 + (x b )−a)−p ,θ > 0,a > 0,b > 0,p > 0. (3.7) the time control chart based on the mean value function corresponding to inter failure time together with three parallel lines to the horizontal axis at tl, tc and tu for the data of kim(2013) [2] is given below. estimated values of m(t) at the given failure times t1, t2, ..tn along with the successive differences of these estimates are given in table 3. the successive differences would indicate the estimated number of failures between consecutive failure times. the graph through [ti,4m̂(ti)] i=1,2,..,n-1 along with three parallel horizontal lines at m(tl),m(tc),m(tu ) would be the required control chart and is given in figure 1. int. j. anal. appl. 16 (3) (2018) 332 fig 1. control chart based on successive differences of mean value function of dd 3.1. comparative study. for comparison, we take exponential model, the most frequently used model in reliability studies. the cumulative distribution function of exponential distribution(ed)is f(x) = 1 −e−bx, x > 0,b > 0. (3.8) the nhpp with θf(x) as the mean value function for our present study is m(x) = θ[1 −e−bx], x > 0,θ > 0,b > 0. (3.9) table 1. failure time data failure failure time failure failure time failure failure time number (hours) number (hours) number (hours) 1 9 11 71 21 116 2 21 12 77 22 149 3 32 13 78 23 156 4 36 14 87 24 247 5 43 15 91 25 249 6 45 16 92 26 250 7 50 17 95 27 337 8 58 18 98 28 384 9 63 19 104 29 396 10 70 20 105 30 405 int. j. anal. appl. 16 (3) (2018) 333 table 2. parameter estimates of dagum distribution and their control limits dagum model a b p θ̂ m(tl) m(tc ) m(tu ) 0.6 0.5 5 29.47 0.0398 14.7349 29.4302 table 3. successive differences based on the mean value function of dd failure failure time m(t) successive failure failure time m(t) successive number (hours) differences of m(t) number (hours) differences of m(t) 4m̂(t) 4m̂(t) 1 9 13.07228 4.721 16 92 23.7886 0.095 2 21 17.7929 2.036 17 95 23.8839 0.091 3 32 19.829 0.524 18 98 23.9751 0.17 4 36 20.352 0.75 19 104 24.1455 0.027 5 43 21.103 0.184 20 105 24.1725 0.274 6 45 21.287 0.415 21 116 24.4469 0.635 7 50 21.702 0.557 22 149 25.0821 0.108 8 58 22.261 0.296 23 156 25.1904 0.959 9 63 22.556 0.363 24 247 26.1494 0.015 10 70 22.919 0.048 25 249 26.1643 0.007 11 71 22.967 0.267 26 250 26.1717 0.509 12 77 23.234 0.042 27 337 26.6811 0.199 13 78 23.276 0.343 28 384 26.8799 0.045 14 87 23.619 0.137 29 396 26.9248 0.032 15 91 23.7553 0.033 30 405 26.9571 table 4. parameter estimates of exponential model and their control limits exponential model b θ̂ m(tl) m(tc ) m(tu ) 0.03 30.00016 0.0405 15.0008 29.9596 int. j. anal. appl. 16 (3) (2018) 334 table 5. successive differences based on the mean value function of ed failure failure time m(t) successive failure failure time m(t) successive number (hours) differences of m(t) number (hours) differences of m(t) 4m̂(t) 4m̂(t) 1 9 7.09865 6.924 16 92 28.1014 0.163 2 21 14.0223 4.491 17 95 28.2648 0.149 3 32 18.5133 1.299 18 98 28.4142 0.261 4 36 19.8122 1.93 19 104 28.6754 0.039 5 43 21.742 0.481 20 105 28.7146 0.361 6 45 22.2229 1.083 21 116 29.0759 0.581 7 50 23.7345 1.428 22 149 29.6567 0.065 8 58 24.7345 0.733 23 156 29.7218 0.26 9 63 25.468 0.858 24 247 29.982 0.001 10 70 26.3264 0.109 25 249 29.2831 0.0005 11 71 26.435 0.587 26 250 29.9836 0.0153 12 77 27.0223 0.088 27 337 29.9989 0.00009 13 78 27.1103 0.684 28 384 29.9999 0.000009 14 87 27.7941 0.249 29 396 30 0.000005 15 91 28.0436 0.058 30 405 30 fig 2. control chart based on successive differences of mean value function of ed int. j. anal. appl. 16 (3) (2018) 335 4. monitoring the production process based on mean value function using order statistics let x1,x2, ..,xn be a random sample of size n representing n inter failure times of a product governed by the probability model of a continuous random variable x. let f(x) be the cumulative distribution function of x. these inter failure times can be used for assessing the failure phenomenon with respect to two limits of reference called control limits with a pre specified coverage probability. thus the time control chart plotted for inter failure times would indicate alarms, advantages and stable failure process. if r is a natural number (<n), the summations r∑ i=1 xi, 2r∑ i=r+1 xi, 3r∑ i=2r+1 xi etc represent the lapse of time consecutively between every rth failure. a control chart for times between every rth failure would throw light on the out of control signals than that of inter failure times. xie et al (2002) [4] named such a control chart as tr-control chart and developed control limits using the sampling distribution of r∑ i=1 xi. they have taken the example of exponential distribution and used the theory that the sum of exponential variates is a gamma variate to get the percentiles of tr-control chart with the help of cumulative summations. if the inter failure times are not exponentials, the control limits of tr-chart of xie et al (2002) [4] can not be used. overcoming this drawback we suggest the following alternative approach to get control limits of tr-chart for dagum distribution. if (x1,x2, ..,xr); (xr+1,xr+2, ..,x2r); (x2r+1,x2r+2, ..,x3r); etc are regarded as independent samples of size r each, i.i.d random variables having f(x) as their common model. y1 = x1 , y2 = 2∑ i=1 xi, y3 = 3∑ i=1 xi, ....., yr = r∑ i=1 xi becomes an ordered sample of size r representing the time to first failure, time to second failure, time to third failure,....., time to rth failure respectively. thus, the tr-chart is the control chart with yr as the points on it representing the time to every r th failure. therefore, when r is fixed, the percentiles of highest order statistics in a sample of size r would serve the purpose of control limits for the tr-chart. let f(x) be the cumulative distribution function of a continuous positive valued random variable. if the random variable is taken as representing inter failure time of a device, a control chart of such data with order statistics would be based on 0.9973 probability limits of the times between failure random variable say t. these limits and the central line are respectively the solutions of the following equations taking equi-tailed probabilities. [f(t)]r = 0.00135 (4.1) [f(t)]r = 0.5 (4.2) [f(t)]r = 0.99865 (4.3) let tu , tc and tl be respectively the solutions of equations (3.1), (3.2) and (3.3) in the standard form tl = f −1(0.00135 1 r ) (4.4) int. j. anal. appl. 16 (3) (2018) 336 tc = f −1(0.5 1 r ) (4.5) tu = f −1(0.99865 1 r ) (4.6) the nhpp with θ.f(x) as the mean value function for our present study is m(x) = θ ( 1 + (x b )−a)−p ,θ > 0,a > 0,b > 0,p > 0. (4.7) the above model is illustrated for the example of 60 failure times considered by xie et al (2002) [4]. for a ready reference the data is produced in table 6. table 6. failure time data of the components failure time failure time failure time failure time number number number number 1 1065.55 16 2932.96 31 35.85 46 239.66 2 535.8 17 987.67 32 362.8 47 93.78 3 540.53 18 1816.18 33 357.85 48 680.45 4 716.2 19 117.21 34 334.48 49 4.83 5 2525.43 20 190.65 35 80.13 50 102.91 6 1264.18 21 943.99 36 1939.0 51 479.05 7 479.44 22 1084.48 37 77.88 52 156.67 8 1783.22 23 2306.54 38 4.03 53 1286.24 9 473.67 24 6.56 39 98.67 54 443.97 10 2265.42 25 3111.51 40 17.19 55 360.03 11 2191.75 26 283.86 41 289.79 56 414.66 12 1097.26 27 659.39 42 63.99 57 128.9 13 597.59 28 683.48 43 2.46 58 36.1 14 971.16 29 36.14 44 697.68 59 197.31 15 3157.29 30 754.16 45 1167.33 60 418.12 int. j. anal. appl. 16 (3) (2018) 337 table 7. accumulation failure time for every three failures observation accumulation observation accumulation of 3 failures of 3 failures 1 2141.88 11 756.5 2 4505.81 12 2353.61 3 2736.33 13 180.58 4 5554.43 14 370.97 5 4726.04 15 1867.47 6 5736.81 16 1013.89 7 1251.85 17 586.79 8 3397.58 18 1886.88 9 4054.76 19 903.59 10 1473.78 20 651.53 table 8. parameter estimates of dagum distribution and their control limits dagum model a b p θ̂ m(tl) m(tc ) m(tu ) 0.8 0.1 0.5 20.9884 0.000015 4.7555 20.9278 table 9. mean value function for accumulated failure times of dd observation accumulation m(t) observation accumulation m(t) of 3 failures of 3 failures 1 2141.88 20.881 11 756.5 20.741 2 4505.81 20.929 12 2353.61 20.888 3 2736.33 20.899 13 180.58 20.222 4 5554.43 20.938 14 370.97 20.554 5 4726.04 20.931 15 1867.47 20.868 6 5736.81 20.939 16 1013.89 20.793 7 1251.85 20.823 17 586.79 20.686 8 3397.58 20.913 18 1886.88 20.869 9 4054.76 20.923 19 903.59 20.774 10 1473.78 20.843 20 651.53 20.711 int. j. anal. appl. 16 (3) (2018) 338 fig 3. control chart based on accumulated failures of mean value function of dd 4.1. comparative study based on accumulated failure times. we compare our model under study with the exponential model using the data set given in table 6 and the results are as follows: table 10. parameter estimates of exponential model and their control limits exponential model b θ̂ m(tl) m(tc ) m(tu ) 0.02 20.00004 1.5023 13.0130 19.8830 table 11. mean value function for accumulated failure times of ed observation accumulation m(t) observation accumulation m(t) of 3 failures of 3 failures 1 2141.88 20.00004 11 756.5 20.00004 2 4505.81 20.00004 12 2353.61 20.00004 3 2736.33 20.00004 13 180.58 19.45987 4 5554.43 20.00004 14 370.97 19.98805 5 4726.04 20.00004 15 1867.47 20.00004 6 5736.81 20.00004 16 1013.89 20.00004 7 1251.85 20.00004 17 586.79 19.99988 8 3397.58 20.00004 18 1886.88 20.00004 9 4054.76 20.00004 19 903.59 20.00004 10 1473.78 20.00004 20 651.53 20 int. j. anal. appl. 16 (3) (2018) 339 fig 4. control chart based on accumulated failures of mean value function of ed 5. summary & conclusions in figure 1, the first out of control situation is noticed at the 15th failure with the corresponding successive difference of m(t) falling below lcl and hence a preferable out-of-control signal for the product. where as in figure 2, it is noticed at 19th failure. the earlier the failure, one can alert the process and assignable cause for this is to be investigated and can be promoted. there are many charts which use statistical techniques. it is important to use the best chart for the given data, situation and need. in the first part of the paper, the control chart for estimated number of failures in successive failure time intervals against the serial order of the failure interval is developed with the associated control lines and central line at same serial point on that of kim(2013) [2]. similarly for the control limits based on the accumulated failure times also shows that dagum distribution is better model when compared with that of the exponential model used by xie et al (2002) [4]. from the figure 3 and 4 we can observe that 2nd and 19th accumulated failure time is out of the limits respectively. the earlier the failure group, one need not to wait till the last group failure occurs. references [1] chan, l.y., xie, m. and goh, t.n. cumulative quality control charts for monitoring production process, int. j. product. res., 38(2)( 2000), 397–408. [2] kim, h.c. assessing software reliability based on nhpp using spc, int. j. softw. eng. appl., 7(6)(2013), 61–70. [3] pham, h. and zhang, x. non homogeneous poisson process software reliability and cost models with testing coverage, eur. j. oper. res., (145)(2003), 443–454. [4] xie, m., goh, t.n. and ranjan, p. some effective control chart procedurs for reliability monitoring, reliab. eng. syst. safety, 77(2002), 143–150. 1. introduction 2. distribution and its properties 3. control chart monitoring the time between failures based on mean value function 3.1. comparative study 4. monitoring the production process based on mean value function using order statistics 4.1. comparative study based on accumulated failure times 5. summary & conclusions references international journal of analysis and applications issn 2291-8639 volume 5, number 1 (2014), 56-67 http://www.etamaths.com integral boundary value problems for fractional impulsive integro differential equations in banach spaces a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ abstract. we study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. we establish the sufficient conditions and unique solution for given problem. an example is also explained to the main results. 1. introduction in the seventeenth century, fractional calculus was originated and it has gained much attention in recent years by many researchers. fractional differential equations appears in a large number of fields of science and engineering, thermodynamics, elasticity, wave propagation, electric railway systems, telecommunication lines and also in chemistry, analysing kinetical reaction problems (see [1, 5, 6, 13, 15, 16]). integral and anti-periodic boundary value conditions can be seen in models of a variety of physical, economic and biological processes, and they have been studied extensively in recent years (see [8, 9, 10, 11] ) and related references therein for boundary value problems with integral boundary conditions [1, 2, 3, 6]. in [14], the authors have studied the impulsive problems for fractional differential equations with boundary value conditions. j.r. wang et al. in [7] discussed the existence results for the boundary value problems for impulsive fractional differential equations. the authors in [17] proved the existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions without impulsive conditions. inspired by the above works, we consider the existence and uniqueness of solutions for impulsive fractional differential equations with integral boundary conditions dα0+u(t) = f(t,u(t),bu(s)), 1 < α ≤ 2,(1.1) t ∈ j ′ = j\{t1, ..., tm} ,j := [0,t], t > 0, u(t+k ) = u(t − k ) + yk, k = 1, 2, ...,m yk ∈ x,(1.2) i2−α0+ u(t)|t=0 = 0, d α−2 0+ u(t) = m∑ i=1 aii α−1 0+ u(ξi),(1.3) 2000 mathematics subject classification. 26a33, 34a37, 34k05, 34b15. key words and phrases. fractional integro-differential equations, boundary value problem, impulsive condition, fixed point theorem. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 56 integral boundary value problems 57 where bu(s) = ∫ t 0 k(t,s,u(s))ds, 0 < ξi < t , t > 0, ai ∈ x, m ≥ 2, dα0+ and iα 0+ are the standard riemann-liouville fractional derivative and fractional integral respectively, f : j ×j ×x → x, k : j ×j ×x → x are jointly continuous and tk satisfy 0 = t0 < t1 < ... < tm < tm+1 = t, u(t + k ) = lim�→0+ u(tk + �) and u(t−k ) = lim�→0− u(tk + �) represent the right and left limits of u(t) at t = tk. in section 2, we give definitions of fractional integral and derivative operators, lemma and some fixed point theorems. the main results discussed in section 3. finally, in section 4, the example is also illustrated. 2. preliminaries let e = pc(j,x) = {u : j −→ x : u ∈ c((tk, tk+1],x)}k = 0, ...m,be a banach space with norm ‖u‖pc = supt∈j‖u(t)‖. and there exist u(t+k ) and u(t − k ),k = 1, 2, .....,m with u(t+k ) = u(t − k ), set j ′ = [0,t]\{t1, t2, ....tm}. theorem 2.1 ([12]). (schaefer’s fixed point theorem) let x be a banach space. assume that t : x → x is a completely continuous operator and the set v = {u ∈ x|u = µtu, 0 < µ < 1} is bounded. then t has a fixed point in x. theorem 2.2. (pc-type ascoli-arzela theorem) let x be a banach space and w ⊂ pc(j,x). if the following conditions are satisfied: (i): w is uniformly bounded subset of pc(j,x) (ii): w is equicontinuous in (tk, tk+1), k = 0, 1, 2, ...,m where t0 = 0, tm+1 = t; (iii): w(t) = {u(t)|u ∈ w, t ∈ j\{t1, ..., tm}},w(t+k = { u(t+k )|u ∈ w } and w(t−k = { u(t−k )|u ∈ w } is a relatively compact subsets of x. then w is a relatively compact subsets of pc(j,x). definition 2.3. the fractional integral of order α > 0 of a function y : (0,∞) → r is given by iα0+y(t) = 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds, provided the right side is pointwise defined on (0,∞), where γ(·) is the gamma function. definition 2.4. the fractional derivative of order α > 0 of a function y : (0,∞) → r is given by dα0+y(t) = 1 γ(n−α) ( d dt )n ∫ t 0 y(s) (t−s)α−n+1 ds, where n = [α] + 1, provided the right side is pointwise defined on (0,∞). lemma 2.5. let α > 0 and u ∈ c(0, 1) ∩ l1(0, 1).then fractional differential equation dα0+u(t) = 0 has u(t) = c1t α−1 + c2t α−2 + · · · + cntα−n, ci ∈ r, n = [α] + 1, as unique solution. lemma 2.6. assume that u ∈ c(0, 1) ∩ l1(0, 1) with a fractional derivative of order α > 0 that belongs to c(0, 1) ∩l1(0, 1). then iα0+d α 0+u(t) = u(t) + c1t α−1 + c2t α−2 + · · · + cntα−n, for some ci ∈ r, i = 1, 2, . . . ,n, where n is the smallest integer grater than or equal to α. 58 a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ lemma 2.7 ([8]). let α > 0, n = [α] + 1. assume that u ∈ l1(0, 1) with a fractional integration of order n−α that belongs to acn[0, 1]. then the equality (iα0+d α 0+u)(t) = u(t) − n∑ i=1 ((in−α0+ u)(t)) n−i|t=0 γ(α− i + 1) tα−i holds almost everywhere on [0, 1]. lemma 2.8 ([8]). (i) let k ∈ n,α > 0. if dαa+y(t) and (d α+k a+ y)(t) exist, then (dkdαa+)y(t) = (d α+k a+ y)(t); (ii) if α > 0,β > 0,α + β > 1, then (iαa+i α a+)y(t) = (i α+β a+ y)(t) satisfies at any point on [a,b] for y ∈ lp(a,b) and 1 ≤ p ≤∞; (iii) let α > 0 and y ∈ c[a,b]. then (dαa+iαa+)y(t) = y(t) holds on [a,b]; (iv) note that for λ > −1,λ 6= α− 1,α− 2, . . . ,α−n, we have dαtλ = γ(λ + 1) γ(λ−α + 1) tλ−α, dαtα−i = 0, i = 1, 2, . . . ,n lemma 2.9. for any y(t) ∈ pc(j,x), the linear impulsive fractional boundaryvalue problem (2.1) dα0+u(t) = y(t), 1 < α ≤ 2, t ∈ [0,t], u(t+k ) = u(t − k ) + yk, k = 1, 2, ...,m yk ∈ x i2−α0+ u(t)|t=0 = 0, d α−2 0+ u(t) = m∑ i=1 aii α−1 0+ u(ξi), has unique solution u(t) =   ∫ t 0 (t−s)α−1 γ(α) y(s)ds + t α−1 γ(α)(t−a) [ ∑m i=1 ai γ(2α−1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫t 0 (t −s)y(s)ds ] , for t ∈ [0, t1) y1 + ∫ t 0 (t−s)α−1 γ(α) y(s)ds + t α−1 γ(α)(t−a) [ ∑m i=1 ai γ(2α−1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫t 0 (t −s)y(s)ds ] , for t ∈ (t1, t2) y1 + y2 + ∫ t 0 (t−s)α−1 γ(α) y(s)ds + t α−1 γ(α)(t−a) [ ∑m i=1 ai γ(2α−1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫t 0 (t −s)y(s)ds ] , for t ∈ (t2, t3) ... m∑ i=0 yi + ∫ t 0 (t−s)α−1 γ(α) y(s)ds + t α−1 γ(α)(t−a) [ ∑m i=1 ai γ(2α−1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫t 0 (t −s)y(s)ds ] , for t ∈ (tm,t] where a = ∑m i=1 aiξ 2α−2 i /γ(2α− 1) and t 6= a. integral boundary value problems 59 step:1 for t ∈ [0, t1] we have by lemma 2.6. the solution of (2.1) can be written as u(t) = c1t α−1 + c2t α−2 + 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds. from i2−α0+ u(t)|t=0 = 0, and by lemmas 2.7 and 2.8, we know that c2 = 0, and dα−20+ u(t) = c1tγ(α) + i 2 0+y(t), iα−10+ u(t) = c1 γ(α) γ(2α− 1) t2α−2 + iα−10+ i α 0+y(t), from dα−20+ u(t) = ∑m i=1 aii α−1 0+ u(ξi), we have c1 = 1 γ(α)(t −a) [ ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫ t 0 (t −s)y(s)ds ] , where a = ∑m i=1 aiξ 2α−2 i /γ(2α− 1) and t 6= a, so u(t) = ∫ t 0 (t−s)α−1 γ(α) y(s)ds + tα−1 γ(α)(t −a) [ ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫ t 0 (t −s)y(s)ds ] . step:2 if t ∈ (t1, t2],with u(t+1 = u(t − 1 ) + y1 then we have u(t) = c1t α−1 + c2t α−2 + u(t+1 ) − 1 γ(α) ∫ t1 0 (t1 −s)α−1y(s)ds + 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds, = c1t α−1 + c2t α−2 + u(t−1 ) + y1 − 1 γ(α) ∫ t1 0 (t1 −s)α−1y(s)ds + 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds, = c1t α−1 + c2t α−2 + y1 + 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds. then, u(t) = y1 + ∫ t 0 (t−s)α−1 γ(α) y(s)ds + tα−1 γ(α)(t −a) [ ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫ t 0 (t −s)y(s)ds ] . preceding in this way, step:3 for t ∈ (tm,t], we have u(t) = c1t α−1 + c2t α−2 + m∑ i=1 yi + 1 γ(α) ∫ t 0 (t−s)α−1y(s)ds. then, u(t) = m∑ i=1 yi + ∫ t 0 (t−s)α−1 γ(α) y(s)ds 60 a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ + tα−1 γ(α)(t −a) [ ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2y(s)ds− ∫ t 0 (t −s)y(s)ds ] . the proof is complete. � 3. main results in this section, we prove the existence and uniqueness results of the problem (1.1)-(1.3) by using the following assumptions: (h1) there exist positive functions l , such that |f(t,x,u) −f(t,y,v)| ≤ l[|x−y| + |u−v|], ∀t ∈ [0,t], x,y,u,v ∈ x, (h2) the function l satisfies 2l ≤ [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )]−1 + m∑ i=1 yi. (h3) there exists a positive constant l1 such that |f(t,u,v)| ≤ l1 for t ∈ [0,t], u,v ∈ x. theorem 3.1. assume that (h1), (h2)are satisfied, then the problem (1.1)–(1.3) has a unique solution. proof: choose r ≥ 2m1 [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + m∑ i=1 yi then we show that θbr ⊂ br, where br = {u ∈ e : ‖u‖ ≤ r}. let us set supt∈[0,t] |f(t,s, 0)| = m1, step :1 for t ∈ [0, t1], we have ‖(θu)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) f(s,u(s),bu(s))ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(s,u(s),bu(s))ds − ∫ t 0 (t −s)f(s,u(s),bu(s))ds )∣∣∣ ≤ [∫ t 0 (t−s)α−1 γ(α) |f(s,u(s),bu(s))|ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2|f(s,u(s),bu(s))|ds − ∫ t 0 (t −s)|f(s,u(s),bu(s))|ds )] ≤ [∫ t 0 (t−s)α−1 γ(α) (|f(s,u(s),bu(s) −f(σ,s, 0)| + |f(σ,s, 0)|)ds integral boundary value problems 61 + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2(|f(s,u(s),bu(s) −f(σ,s, 0)| + |f(σ,s, 0)|)ds − ∫ t 0 (t −s)(|f(s,u(s),bu(s) −f(σ,s, 0)| + |f(σ,s, 0)|)ds )] ≤ [ (2lr + m1) (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds ))] ≤ (2lr + m1) [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] ≤ r taking the maximum over the interval [0, t1], we obtain ‖θ(u)(t)‖≤ r. in view of (h1), for every t ∈ [0, t1], we have ‖(θx)(t) − (θy)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) (f(t,x) −f(t,y)ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2(f(t,x,u) −f(t,y,v))ds − ∫ t 0 (t −s)(f(t,x,u) −f(t,y,v))ds )∣∣∣ ≤ [∫ t 0 (t−s)α−1 γ(α) |(f(t,x,u) −f(t,y,v))|ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2|(f(t,x,u) −f(t,y,v))|ds − ∫ t 0 (t −s)|(f(t,x,u) −f(t,y,v))|ds )] ≤ [ l[‖x−y‖ + ‖u−v‖] (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds ))] ≤ l[‖x−y‖ + ‖u−v‖] [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] = a[‖x−y‖ + ‖u−v‖], where a = l [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] , which depends only on the parameters involved in the problem. as a < 1, θ is contraction mapping for the interval t ∈ [0, t1]. step :2 for t ∈ (t1, t2], we have ‖(θu)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) f(s,u(s),bu(s))ds 62 a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(s,u(s),bu(s))ds − ∫ t 0 (t −s)f(s,u(s),bu(s))ds ) + y1 ∣∣∣ ≤ [ (2lr + m1) (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )) + y1 ] ≤ (2lr + m1) [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + y1 ≤ r taking the maximum over the interval (t1, t2], we obtain ‖θ(u)(t)‖≤ r. in view of (h1), for every t ∈ (t1, t2], we have ‖(θx)(t) − (θy)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) (f(t,x) −f(t,y)ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2(f(t,x,u) −f(t,y,v))ds − ∫ t 0 (t −s)(f(t,x,u) −f(t,y,v))ds ) + y1 ∣∣∣ ≤ [ l[‖x−y‖ + ‖u−v‖] (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )) + y1 ] ≤ l[‖x−y‖ + ‖u−v‖] [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + y1 = a[‖x−y‖ + ‖u−v‖], where a = l [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + y1, as a < 1, θ is therefore a contraction in the interval t ∈ (t1, t2]. preceding in this way, we got step:3 for t ∈ (tm,t], we have ‖(θu)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) f(s,u(s),bu(s))ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(s,u(s),bu(s))ds integral boundary value problems 63 − ∫ t 0 (t −s)f(s,u(s),bu(s))ds ) + m∑ i=1 yi ∣∣∣ ≤ [ (2lr + m1) (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )) + m∑ i=1 yi ] ≤ (2lr + m1) [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + m∑ i=1 yi ≤ r taking the maximum over the interval (tm,t], we obtain ‖θ(u)(t)‖≤ r. in view of (h1), for every t ∈ (tm,t], we have ‖(θx)(t) − (θy)(t)‖ = ∣∣∣∫ t 0 (t−s)α−1 γ(α) (f(t,x) −f(t,y)ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2(f(t,x,u) −f(t,y,v))ds − ∫ t 0 (t −s)(f(t,x,u) −f(t,y,v))ds ) + m∑ i=1 yi ∣∣∣ ≤ [ l[‖x−y‖ + ‖u−v‖] (∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )) + m∑ i=1 yi ] ≤ l[‖x−y‖ + ‖u−v‖] [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + m∑ i=1 yi = a[‖x−y‖ + ‖u−v‖], where a = l [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + m∑ i=1 yi, which depends only on the parameters involved in the problem. then by banach fixed point theorem, the operator θ has fixed point in the interval t ∈ (tm,t]. � theorem 3.2. assume that (h1)-(h3) are satisfied. then (1.1)-(1.3) has at least one solution. proof: 64 a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ we define an operator p : pc(e) → pc(e), as (pu)(t) = ∫ t 0 (t−s)α−1 γ(α) f(t,u(s),v(s))ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(t,u(s),v(s))ds − ∫ t 0 (t −s)f(t,u(s),v(s))ds ) + m∑ i=1 yi. (3.1) to show that the operator p is completely continuous. clearly, continuity of the operator p follows from the continuity of f. let ω ⊂ e be bounded. then, ∀u,v ∈ ω together with (h3) we obtain (pu)(t) ≤ ∫ t 0 (t−s)α−1 γ(α) |f(t,u(s),v(s))|ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2|f(t,u(s),v(s))|ds − ∫ t 0 (t −s)|f(t,u(s),v(s))|ds ) + m∑ i=1 yi ≤ l1 [∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )] + m∑ i=1 yi ≤ l1 [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 γ(2α) − t 2 2 )] + m∑ i=1 yi, which implies ‖pu‖≤ l1 [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 γ(2α) − t 2 2 )] + m∑ i=1 yi < ∞. hence, p(ω) is uniformly bounded. for any s1,s2 ∈ [0, t1],u ∈ ω, we have |(pu)(s1) − (pu)(s2)| = ∣∣∣∫ s1 0 (s1 −s)α−1 γ(α) f(s,u(s),v(s))ds + sα−11 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(s,u(s),v(s))ds − ∫ t 0 (t −s)f (s,u(s))ds ) − ∫ s2 0 (s2 −s)α−1 γ(α) f(s,u(s),v(s))ds− tα−12 γ(α)(t −a) × ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(s,u(s),v(s))ds− ∫ t 0 (t −s)f(s,u(s),v(s))ds )∣∣∣ ≤ l1 ∣∣∣∫ s1 0 (s1 −s)α−1 − (s2 −s)α−1 γ(α) ds integral boundary value problems 65 + tα−11 −s α−1 2 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds − ∫ t 0 (t −s)ds ) − ∫ s2 s1 (s2 −s)α−1 γ(α) ds ∣∣∣ ≤ l1 [∣∣∣∫ s1 0 (s1 −s)α−1 − (s2 −s)α−1 γ(α) ds− ∫ s2 s1 (s2 −s)α−1 γ(α) ds ∣∣∣ + ∣∣∣sα−11 −sα−12 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )∣∣∣] → 0 as s1 → s2. thus, by the pc-type arzela-ascoli theorem, p(ω) is equicontinuous. consequently, the operator p is compact. next, we consider the set s = {u ∈ e : u = µpu, 0 < µ < 1}, and show that it is bounded. let u ∈ s; then u = µpu, 0 < µ < 1. for any t ∈ [0,t], we have u(t) = ∫ t 0 (t−s)α−1 γ(α) f(t,u(s),v(s))ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2f(t,u(s),v(s))ds − ∫ t 0 (t −s)f(t,u(s),v(s))ds ) + m∑ i=1 yi, and |u(t)| = µ|pu| ≤ ∫ t 0 (t−s)α−1 γ(α) |f(t,u(s),v(s))|ds + tα−1 γ(α)(t −a) ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2|f(t,u(s),v(s))|ds − ∫ t 0 (t −s)|f(t,u(s),v(s))|ds ) + m∑ i=1 |yi| ≤ l1 [∫ t 0 (t−s)α−1 γ(α) ds + tα−1 γ(α)|t −a| ( ∑m i=1 ai γ(2α− 1) ∫ ξi 0 (ξi −s)2α−2ds− ∫ t 0 (t −s)ds )] + m∑ i=1 |yi| ≤ max t∈[0,t] { l1 [ |tα| γ(α + 1) + |tα−1| γ(α)|t −a| (∑m i=1 aiξ 2α−1 γ(2α) − t 2 2 )] + m∑ i=1 |yi| } = m∗. thus, ‖u‖ ≤ m∗. so, the set s is bounded. thus, by the conclusion of theorem 3.1, the operator p has at least one fixed point, which implies that (1.1)-(1.2) has at least one solution. � 66 a. anguraj1, m. kasthuri2 and p. karthikeyan 3,∗ 4. example consider the impulsive fractional integro differential equation cd 3 2 u(t) = et|u(t)| (9 + et)(1 + |u(t)|) + ∫ t 0 e−(s−t) 10 |u(s)|ds,(4.1) t ∈ j = [0, 2], t 6= 1 2 ,(4.2) y( 1 2 + ) = |u( 1 2 − )| 3 + |u( 1 2 − )| ,(4.3) i2−α0+ u(t)|t=0 = 0, d α−2 0+ u(t) = m∑ i=1 aii α−1 0+ u(ξi),(4.4) where f(t,u,bu) = e tu (9+et)(1+u) + bu(t),a1 = 2,a2 = 3,ξ1 = 1/2,ξ2 = 1/3,t = 2 we have a = ∑m i=1 aiξ 2α−2 i /γ(2α− 1) = 1 6= t = 1. clearly, l = 1/10 as |f(t,x,u) −f(t,y,v)| ≤ 1 10 [|x−y| + |u−v|]. further, l [ tα γ(α + 1) + tα−1 γ(α)|t −a| (∑m i=1 aiξ 2α−1 i γ(2α) − t 2 2 )] + m∑ i=1 yi ≈ 0.01058612753 < 1. thus, all the assumptions of theorem 3.1 are satisfied and hence the problem (4.1)-(4.4) has unique solution. references [1] a. anguraj, p. karthikeyan, and g. m. nguérékata; nonlocal cauchy problem for some fractional abstract integrodifferential equations in banach space, communications in mathematical analysis , vol.55, no. 6, pp. 1?, 2009. [2] a. anguraj, p. karthikeyan and j.j. trujillo; existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition, advances in difference equations,volume 2011, article id 690653,12pages, doi:10.1155/2011/690653 [3] b. ahmad, j. j. nieto; existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, bound. value probl.(2009) art. id 708576, 11 pp.. [4] b. ahmad, a. alsaedi; existence of approximate solutions of the forced duffing equation with discontinuous type integral boundary conditions, nonlinear analysis, 10 (2009) 358-367. [5] c. bai; positive solutions for nonlinear fractional differential equations with coefficient that changes sign nonlinear analysis: theory, methods and applications, 64 (2006) 677-685. [6] z. hu, w. liu; solvability for fractional order boundary value problem at resonance, boundary value problem, 20(2011)1-10. [7] j. r wang, y. z. and m. feckan; on recent developments in the theory of boundary value problems for impulsive fractional differential equations, computers and mathematics with applications, 64(2012) 3008-3020. [8] a. a. kilbas, h. m. srivastava, j. j. trujillo; theory and applications of fractional differential equations, north-holland mathematics studies, 204. elsevier science b.v., amsterdam, 2006. [9] v. lakshmikantham, s. leela, j. vasundhara devi; theory of fractional dynamic systems, cambridge academic publishers, cambridge, 2009. [10] j. sabatier, o. p. agrawal, j. a. t. machado (eds.); advances in fractional calculus: theoretical developments and applications in physics and engineering, springer, dordrecht, 2007. integral boundary value problems 67 [11] s. g. samko, a. a. kilbas, o. i. marichev; fractional integrals and derivatives: theory and applications, gordon and breach, new york, ny, usa, 1993. [12] d. r. smart; fixed point theorems, cambridge university press, 1980. [13] x. su; boundary value problem for a coupled system of nonlinear fractional differential equations, applied mathematics letters, 22 (2009) 64-69. [14] t.l. guo and w. jiang, impulsive problems for fractional differential equations with boundary value conditions, computers and mathematics with applications, 64(2012) 3281-3291. [15] g. wang, w. liu; the existence of solutions for a fractional 2m-point boundary value problems, journal of applied mathematics. [16] g. wang, w. liu; existence results for a coupled system of nonlinear fractional 2m-point boundary value problems at resonance, advances in difference equations,doi:10.1186/16871847-2011-44. [17] g.wang, w. liu, c. ren, existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions , electronic journal of differential equations, vol. 2012 (2012), no. 54, pp. 1?0. 1department of mathematics, psg college of arts and science, coimbatore, tn, india 2department of mathematics, p.k.r. arts college for women, gobichettipalayam, tn, india 3department of mathematics, ksr college of arts and science, tiruchengode, tn, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 3, number 2 (2013), 104-111 http://www.etamaths.com numerical differentiation and integration through aitken-neville schemes ramesh kumar muthumalai abstract. some new formulas are given to approximate higher order derivatives and integrals through aitken-neville iterative schemes for arbitrary spaced grids. an algorithm is given in matlab for numerical differentiation. also, numerical examples are provided to study error analysis of new formulas for numerical differentiation and integration. 1. introduction. the problem of numerical differentiation is a long-standing issue. there are plenty of published works devoted to estimate derivatives of a function numerically for arbitrary spaced grids. some of them are polynomial interpolation type [5, 9], finite difference formulas [2, 4, 6] and richardson extrapolation [2]. most of them do not aim to iterate derivatives upto kth order per addition of a new grid. the finite difference formulas for the calculation of any order derivative in a one dimensional grid with arbitrary spacing are discussed in ref[4] (algorithm given in fortran) with the cost of o(kn2) operations. it also generates sequence of approximate derivatives upto order k per addition of a new grid. it should be noted that there exists a great deal of formulas and techniques of numerical integration. but, they have been of considerable complexity and often been limited to lower order formulas on equidistantly spaced grids. aitken-neville schemes [5, 7, 8] are popular interpolation methods to iterate interpolation when a new grid is added. an obvious advantage of these schemes is that it gives a good idea of the accuracy of the result at any stage [8]. in the present study we describe new formulas to approximate numerical derivatives and integrals through aitken-neville iterative schemes for arbitrary spaced grids. also, we provide an efficient algorithm in matlab to iterate numerical differentiation (upto kth order) per addition of a new grid with cost of o(k2n) operations. 2. numerical differentiation 2.1. formulae for numerical differentiation. 2010 mathematics subject classification. 65d25; 65d30. key words and phrases. aitken-neville scheme; iterative method; numerical differentiation; numerical integration. c⃝2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 104 numerical differentiation and integration 105 definition 2.1. define n (r) j [x] = 1 (xj−x)r and (2.1) n (r) j,j+1,...,j+i[x] = 1 xi+j − xj ∣∣∣∣∣ n (r) j,j+1,...,j+i−1[x] xj − x n (r) j+1,j+2,...,j+i[x] xi+j − x ∣∣∣∣∣ . also, define ñ (r) j [x] = f(xj) (xj−x)r and (2.2) ñ (r) j,j+1,...,j+i[x] = 1 xi+j − xj ∣∣∣∣∣ ñ (r) j,j+1,...,j+i−1[x] xj − x ñ (r) j+1,j+2,...,j+i[x] xi+j − x ∣∣∣∣∣ . where n (r) j,j+1,j+2,...,j+i[x] and ñ (r) j,j+1,j+2,...,j+i[x] are constructed by neville scheme of interpolation, i = 1, 2, . . . , n and j = 0, 1, 2, 3 . . . n − i. at the nth iteration, assume that n (r) 0,1,2,...,n[x] = n (r)(x), r = 0, 1, 2, . . . k. definition 2.2. define a (r) 0,j[x] = 1 xj−x0 ∣∣∣∣∣ 1 (x0−x)r x0 − x 1 (xj−x)r xj − x ∣∣∣∣∣ and (2.3) a (r) 0,1,2,...,i−1,i,j[x] = 1 xj − x0 ∣∣∣∣∣ a (r) 0,1,...,i−1,i[x] xi − x a (r) 0,1,...,i−1,j[x] xj − x ∣∣∣∣∣ . also, define ã (r) 0,j[x] = 1 xj−x0 ∣∣∣∣∣ f(x0) (x0−x)r x0 − x f(xj) (xj−x)r xj − x ∣∣∣∣∣ and (2.4) ã (r) 0,1,2,...,i,j[x] = 1 xj − xi ∣∣∣∣∣ ã (r) 0,1,...,i−1,i[x] xi − x ã (r) 0,1,...,i−1,j[x] xj − x ∣∣∣∣∣ . where a (r) 0,1,2,...,i−1,i,j[x] and ã (r) 0,1,2,...,i−1,i,j[x] are constructed by aitken’s scheme of interpolation, i = 1, 2, . . . , n and j = i + 1, i + 2, . . . , n. at the nth iteration, assume that a (r) 0,1,2,...,n[x] = a (r)(x), r = 1, 2, . . . k. the following theorem gives recursive formulas for numerical differentiation through aitken-neville schemes. theorem 2.3. let x, x0, x1, x2, . . . xn are n + 1 distinct numbers on the interval [a, b], k ∈ w and f ∈ cn+k+1[a, b]. then (2.5) k∑ i=0 f(i)(x) i! n(k−i)(x) = ñ (k) 0,1,...,n[x] + e(x). and (2.6) k∑ i=0 f(i)(x) i! a(k−i)(x) = ã (k) 0,1,...,n[x] + e(x). where e(x) = f(n+k+1)(ξ) (n+k+1)! ∏n i=0(x−xi) and for min{x, x0, . . . , xn} < ξ < max{x, x0, . . . , xn}. proof. let pn(x) is a polynomial of degree ≤ n in x that approximates the function f. using equation (2.7), gives f[ x, . . . , x︸ ︷︷ ︸ k + 1 times ] = d0,1,2,...,n[x] + l(x) f(n+k+1)(ξ) (n + k + 1)! . 106 ramesh kumar muthumalai where pn[ x, . . . , x︸ ︷︷ ︸ k + 1 times /x0, x1, . . . xn] = d0,1,2,...,n[x] and l(x) = ∏n i=0(x − xi) and min{x, x0, . . . , xn} < ξ < max{x, x0, . . . , xn}. expand pn[x, . . . , x︸ ︷︷ ︸ k times , xj] repeatedly, using the recursive formula of divided difference [2, p-41] to get pn[x, . . . , x︸ ︷︷ ︸ k times , xj] = − k−1∑ r=0 p (r) n (x) r! 1 (xj − x)k−r + pn(xj) 0!(xj − x)k .(2.7) applying equation (2.7) for two consecutive data xj, xj+1 pn[ x, . . . , x︸ ︷︷ ︸ k + 1 times /xj, xj+1] = 1 xj+1 − xj ∣∣∣∣∣∣∣∣ pn[x, . . . , x︸ ︷︷ ︸ k times /xj] xj − x pn[x, . . . , x︸ ︷︷ ︸ k times /xj+1] xj+1 − x ∣∣∣∣∣∣∣∣ . using (2.7) and properties of determinants, finds that pn[ x, . . . , x︸ ︷︷ ︸ k + 1 times /xj, xj+1] = − k−1∑ r=0 p (r) n (x) r! n (k−r) j,j+1 [x] + ñ (k) j,j+1[x]. proceeding this for some i, pn[ x, . . . , x︸ ︷︷ ︸ k + 1 times /xj, xj+1, . . . , xj+i] = − k−1∑ r=0 p (r) n (x) r! n (k−r) j,j+1,...,j+i[x] + ñ (k) j,j+1,...,j+i[x]. putting i = n and j = 0,(i.e at nth iteration) pn[ x, . . . , x︸ ︷︷ ︸ k + 1 times /x0, x1, . . . , xn] = − k−1∑ r=0 p (r) n (x) r! n (k−r) 0,1,...,n[x] + ñ (k) 0,1,...,n[x]. since pn approximates f and n (r) 0,1,...n[x] = n (r)(x). then f[ x, . . . , x︸ ︷︷ ︸ k + 1 times ] − l(x) f(n+k+1)(ξ) (n + k + 1)! = − k−1∑ r=0 f(r)(x) r! n(k−r)(x) + ñ (k) 0,1,...,n[x]. since f[ x, . . . , x︸ ︷︷ ︸ k + 1 times ] = f(k)(x) r! and after simplification gives (2.5). similar manner using definition 2.2, yields (2.6). � theorem 2.4. let x, x0, x1, x2, . . . xn are n + 1 distinct numbers on the interval [a, b], k ∈ w and f ∈ cn+k+1[a, b]. then direct formula for numerical differentiation through neville scheme as follows (2.8) f(t)(x) t! = atf(x)χ + t∑ k=χ at−kñ (k) 0,1,2,...n[x] + ed(a; x). numerical differentiation and integration 107 and a0 = 1, at = − t−1∑ k=0 akn (t−k)(x), t = 1, 2, 3, . . . .(2.9) the direct formula for numerical differentiation through aitken scheme as follows (2.10) f(t)(x) t! = btf(x)χ + t∑ k=χ bt−kã (k) 0,1,2,...n[x] + ed(b; x). and b0 = 1, bt = − t−1∑ k=0 bka (t−k)(x), t = 1, 2, 3, . . . .(2.11) where χ = { 1 if x = xi 0 if x ̸= xi, ed(y; x) = l(x) t∑ m=χ yt−m f(n+m+1)(ξm) (n + m + 1)! . and y = [y0 y1 . . . yt] is an one dimensional array of length t + 1. proof. if one use equation (2.5) recursively to derive direct formulas for kth order differentiation, then we obtain the following form, for some t = 0, 1, 2, 3 . . . f(t)(x) t! = atf(x)χ + t∑ k=χ at−kñ (k) 0,1,2,...n[x] + ed(a; x). to evaluate these unknown a′s, set f(x) = 1 in the above equation, gives a0 = 1 when t = 0 and for t = 1, 2, 3, . . . gives t−1∑ k=1 at−kn (k)(x) + at = 0. this gives (2.9). similarly, from (2.6), it can be easily find (2.10). � 2.2. numerical experiment. in this subsection, we present algorithm (coded in matlab) and numerical result to illustrate the performance of the new formulas given in (2.9) and (2.10) for arbitrary spaced grids. algorithm 2.5. numerical differentiation through neville scheme function [d]=differentiation(x,y,k,s) % input parameters % s location where approximations are to be accurate % x(1:n) grid point locations, found in x(1:n) % y(1:n) functional value locations, found in y(1:n) % k highest derivatives are sought at ’s’ % output parameters % d(1:k+1,1:n) sequence of approximate derivatives of order 0:k n=length(x); % check whether ’s’ coincide with any one of ’x’. (i.e) functional value % of ’s’ is known or not. 108 ramesh kumar muthumalai xs=x-s*ones(1,n); pos=find(˜(xs&xs)); chi=˜all(xs); % if chi=1 (functional value at ’s’ is known), then swap respective % position of s and its functional value with first element of x and y. if chi==1; x([1 pos])=x([pos 1]); y([1 pos])=y([pos 1]); nevy(1,1:n)=y(1); nev(1,1:n)=1; end f=1+chi; % determine the value of n(r)(s) andñ (r) 0,1,2,...,n[s] by neville scheme and store %it in ’nev’ and ’nevy’ respectively for ij=f:k+1 yy(f:n)=1./(x(f:n)-s).ˆ(ij-1); yy1=y.*yy; nev(ij,f)=yy(f); nevy(ij,f)=yy1(f); for i= f:n yy(f:n+f-i-1)=(yy(f+1:n+f-i).*(s-x(f:n+f-i-1))-yy (f:n+f-i-1) .*(s-x(i+1:n)))./(x(i+1:n)-x(f:n+f-i-1)); yy1(f:n+f-i-1)=(yy1(f+1:n+f-i).*(s-x(f:n+f-i-1))yy1(f:n+f-i-1) .*(s-x(i+1:n)))./(x(i+1:n)-x(f:n+f-i-1)); nev(ij,i)=yy(f); nevy(ij,i)=yy1(f); end end % find the value of a’s and store them in a two dimensional array ’b’ for m=1:n b(1,m)=1; mn=min(m,k+1); a(1,1)=1; fact(mn)=1; for i=2:mn b(i,m)=0; for j=1:i-1 b(i,m)=b(i,m)-b(j,m)*nev(i-j+1,m); end a(i,i:-1:1)=b(1:i,m)’; fact(mn+1-i)=factorial(i-1); end % evaluate sequence of approximate derivatives up to order ’k’ diff=(a(mn:-1:1,:)*nevy(1:mn,m)).*fact’; d(1:mn,m)=diff(mn:-1:1); end numerical differentiation and integration 109 the algorithm for numerical differentiation through aitken scheme is similar to the algorithm 2.5. the following are some notes regarding the implementation of the algorithm: • a call to differentiation to obtain value of mth derivative returns also value of kth derivative, k = 0, 1, 2, . . . , m with no additional costs. • the code returns all the data above also for points which extend only over x0, x1, . . . , xj, j = 0, 1, 2, . . . , n, still no additional costs. • it requires o ( k2n ) operations. also, the maximum size of array used is (k + 1) × n. in the following example, we have taken 20 chebyshev first kind of points on [-1,1] of function f(x) = ex. we evaluate the first order derivatives at 100 equally spaced points on [−1 + h, 1 − h], where h = 2/101. figure 1 plots the errors for numerical −1 −0.5 0 0.5 1 10 −20 10 −15 10 −10 10 −5 neville aitken figure 1. relative errors in computed f′(x) for 20 chebyshev first kind of points of f(x) = ex. differentiation through aitken-neville schemes. we see that the neville scheme performs stably and aitken scheme becomes very unstable as x approaches one end of the interval. 3. numerical integration. 3.1. formulae for numerical integration. in this section, we derive numerical integration formulas for arbitrary spaced grids to any order of accuracy. let x, x0, x1, x2, . . . , xn are distinct numbers on the closed interval [a, b] and f ∈ c(n+1)[a, b]. to derive an expression for the definite integral ∫ x+h x f(x)dx , (h ̸= 0) expanding through taylor series, yields (3.1) ∫ x+h x f(x)dx = n∑ k=0 f(k)(x) (k + 1)! hk+1 + o ( hn+2 ) . 110 ramesh kumar muthumalai using (2.8), gives∫ x+h x f(x)dx = n∑ k=0 hk+1 k + 1 ( akf(x)χ + k∑ m=χ ak−mñ (m) 0,1,2,...n[x] +l(x) k∑ m=χ ak−m f(n+m+1)(ξm) (n + m + 1)! ) + o ( hn+2 ) . setting γk = ∑n−k m=0 am hm+k+1 m+k+1 , k = 0, 1, 2, . . . n and rearranging above equation, yields (3.2) ∫ x+h x f(x)dx = γ0f(x)χ + n∑ k=χ γkñ (k) 0,1,2,...,n[x] + ei(γ; x). where (3.3) ei(γ; x) = l(x) n∑ m=χ γm f(n+1+m)(ξm) (n + 1 + m)! + o ( hn+2 ) . similarly, numerical integration formula through aitken scheme can be easily found from (2.1) as follows (3.4) ∫ x+h x f(x)dx = γ′0f(x)χ + n∑ k=χ γ′kã (k) 0,1,2,...,n[x] + ei(γ ′; x). where γ′k = ∑n−k m=0 bm hm+k+1 m+k+1 , k = 0, 1, 2, . . . n 3.2. numerical experiment. we evaluate the integral ∫ 1 −1 e xdx on chebyshev and other point distributions. the exact value of the integral is e − e−1. table it equally spaced points chebyshev first kind chebyshev second kind aitken neville aitken neville aitken neville 1 6.8696e-01 6.8696e-01 1.2962e+00 1.2962e+00 6.8696e-01 6.8696e-01 2 3.4633e-01 3.4633e-01 9.1875e-01 9.1875e-01 2.9095e-01 2.9095e-01 3 1.3011e-01 1.3011e-01 4.2473e-01 4.2473e-01 9.8765e-02 9.8765e-02 4 3.6744e-02 3.6744e-02 1.3578e-01 1.3578e-01 2.4775e-02 2.4775e-02 5 7.8565e-03 7.8565e-03 3.0515e-02 3.0515e-02 4.4655e-03 4.4655e-03 6 1.2724e-03 1.2724e-03 4.7856e-03 4.7856e-03 5.5864e-04 5.5864e-04 7 1.5482e-04 1.5484e-04 5.1025e-04 5.1025e-04 4.6114e-05 4.6114e-05 8 1.3890e-05 1.3899e-05 3.5295e-05 3.5295e-05 2.3236e-06 2.3236e-06 9 1.2211e-05 7.2372e-07 1.4669e-06 1.4670e-06 6.3110e-08 6.3209e-08 10 3.3453e-04 1.0280e-06 2.1951e-09 3.7429e-08 4.2184e-08 4.1447e-09 11 2.2040e-02 2.9403e-05 1.9287e-05 4.6798e-06 2.5779e-05 1.1667e-04 12 1.0882e+00 5.4514e-04 1.9782e+00 1.0000e+00 1.5238e+03 2.1884e+02 13 2.9428e+01 6.3104e-03 1.8712e+12 4.6780e+12 1.1153e+05 8.6310e+08 table 1. comparisons between relative errors of computation of ∫ 1 −1 e xdx on various point distributions through aitken-neville schemes 1 shows the comparisons between relative errors of computation of the required integral through (3.2) and (3.4) on 13 grids of various point distributions (equally spaced and chebyshev grids). we observe that the relative errors of both schemes converge up to 9th iteration on equally spaced grids and converge up to 10th iteration for chebyshev grids, after that, diverge rapidly in all point distributions. also, it gives an idea of the accuracy of the result per addition of a new grid. it is observed that the new formulas give greater accuracy on chebyshev point distributions than evenly spaced grids. numerical differentiation and integration 111 4. conclusion. the formulas for approximating derivatives and integrals for arbitrary spaced grids through aitken-neville schemes have been developed in this article. we have provided an efficient algorithm (in matlab) to iterate numerical differentiation with cost of o(k2n) operations. also, the numerical result shows that the numerical differentiation through neville scheme is stable than aitken scheme. all available numerical integration formulas do not iterate integral values per addition of a new grid to any degree of accuracy. but the new formulas for integration described in section 3, will do this with cost of o(n3) operations. also, an interesting advantage is that they are also applicable to unevenly grids. references [1] k. e. atkinson, an introdction to numerical analysis, 2 ed, john wiley & sons, newyork (1989). [2] s.d. conte, carl de boor, elementary numerical analysis, 3 ed, mcgraw-hill, newyork, usa (1980). [3] m. dvornikov, formulae for numerical differentiation, jcaam, 5 (2007), 77-88[e-print arxiv:math.na/0306092]. [4] b. fornberg, calculation of weights in finite difference formulas, siam rev, vol 40, no 3, 685-691 (1998). [5] f.b. hildebrand, introduction to numerical analysis, 2 ed, mcgraw-hill, newyork (1974). [6] j.li, general explicit difference formulas for numerical differentiation, j.comp & appl. math, 183, 29-52 (2005). [7] g. m. phillips, interpolation and approximation by polynomials, springer-verlag, newyork (2003). [8] s.s. sastry, introdutory methods of numerical analysis, 4 ed, prentice hall of india, new delhi (2005). [9] e. süli & d. mayers, an introduction to numerical analysis, cambridge university press, uk (2003). department of mathematics, d.g. vaishnav college, arumbakkam, chennai-106, tamil nadu, india international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 102-107 http://www.etamaths.com cosine integrals for the clausen function and its fourier series expansion f. m. s. lima∗ abstract. in a recent work, on taking into account certain finite sums of trigonometric functions i have derived exact closed-form results for some non-trivial integrals, including ∫ π 0 sin(kθ) cl2(θ) dθ, where k is a positive integer and cl2(θ) is the clausen function. there in that paper, i pointed out that this integral has the form of a fourier coefficient, which suggest that its cosine version∫ π 0 cos(kθ) cl2(θ) dθ, k ≥ 0, is worthy of consideration, but i could only present a few conjectures at that time. here in this note, i derive exact closed-form expressions for this integral and then i show that they can be taken as fourier coefficients for the series expansion of a periodic extension of cl2(θ). this yields new closed-form results for a series involving harmonic numbers and a partial derivative of a generalized hypergeometric function. 1. introduction in its more general form, the fourier series expansion of a periodic real function f(x) of period l is conventionally written as (see, e.g., sec. 4.2 of ref. [6]) s[f(x)] := a0 2 + ∞∑ k=1 [ ak cos ( 2πkx l ) + bk sin ( 2πkx l )] , (1.1) where ak = 2 l ∫ x0+l x0 f(x) cos(2πkx/l) dx, k ≥ 0 , (1.2a) bk = 2 l ∫ x0+l x0 f(x) sin(2πkx/l) dx, k > 0 , (1.2b) are the fourier coefficients and x0 is an arbitrary constant (often taken as 0). as is well-known, if f(x) satisfies the dirichlet conditions then this series converges to f(x) at all points of continuity of f(x) and to the average of f(x) taken at the lateral limits of x if it is a point of finite discontinuity. in fact, the periodicity condition is irrelevant for pointwise convergence in the finite domain [x0,x0 + l], as shown by connon in ref. [2], which is important for the fourier expansion of non-periodic functions using periodic extensions. in a very recent work, by taking into account certain finite sums involving trigonometric functions at rational multiples of π, i have derived exact closed-form expressions for some non-trivial integrals [5]. among them, i showed in theorem 6 of ref. [5] that 2 π ∫ π 0 sin(k θ) cl2(θ) dθ = 1 k2 (1.3) holds for every integer k > 0. here, cl2(θ) := = { li2 ( eiθ )} is the clausen function, li2(z) :=∑∞ n=1 z n/n2, |z| ≤ 1, being the dilogarithm function [4, sec. 1.1]. clausen himself proved in ref. [1] that cl2(θ) = − ∫ θ 0 ln|2 sin (t/2)|dt, which is known as the clausen integral [3, sec. 4.1]. since the received 26th may, 2017; accepted 29th july, 2017; published 1st september, 2017. 2010 mathematics subject classification. 42a16, 26a42, 26a06. key words and phrases. clausen function; fourier coefficients; generalized hypergeometric function. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 102 cosine integrals for the clausen function 103 integral in eq. (1.3) resembles that of fourier coefficient bk in eq. (1.2b), then a natural follow-up is the investigation of the corresponding cosine integral, i.e. ak := 2 π ∫ π 0 cos(k θ) cl2(θ) dθ , k ≥ 0 . (1.4) however, there in eqs. (25)–(30) of ref. [5] i could only conjecture, based upon strong numerical evidence, a few simple results for small values of k. they of course suggest a pattern, but there in ref. [5] i could not find it out. in this note, i make use of a well-known series expansion for cl2(θ) to derive closed-form expressions for ak, one for k = 0 and another for k > 0. i then use these results to obtain a fourier series for a suitable periodic extension of cl2(θ), which yields new closed-form results. 2. cosine integrals of clausen function in what follows, we shall make use of a well-known series representation for cl2(θ). lemma 1 (clausen series for cl2(θ) ). the trigonometric series ∑∞ n=1 sin (nθ) n2 converges to cl2(θ) for all θ ∈ r. proof. this series representation of cl2(θ) remounts to clausen’s original work (1832) [1], but, for completeness, let us present a proof based on fourier series. in theorem 3 of ref. [7], a recent note on fourier series by zhang, it is shown that, given a real function f(x) integrable on [0,l] and such that f(x) = −f(l−x) for all x ∈ (l/2,l], if f(x) is an odd function in (−l,l), then f(x) = ∞∑ n=1 c2n sin ( 2nπx l ) (2.1) for all x ∈ [−l,l] where f(x) is a continuous function. here, c2n = 4 l ∫ l/2 0 f(t) sin ( 2nπt l ) dt. (2.2) since cl2(θ) is an odd function which is continuous (thus integrable) on (−2π, 2π) and cl2(θ) = −cl2(2π − θ) [3, secs. 4.2 and 4.3], then the convergence of ∑∞ n=1 sin (nθ)/n 2 to cl2(θ) follows by taking l = 2π in zhang’s theorem and noting that c2n = 1/n 2, as seen in eq. (1.3). finally, the periodicity of cl2(θ), as established in sec. 4.2 of ref. [3], extends the convergence to all θ ∈ r. � let us begin our main results with the integral ak for k = 0. theorem 1 (integral a0). the exact closed-form result a0 := 2 π ∫ π 0 cl2(θ) dθ = 7 2 ζ(3) π , where ζ(3) := ∑∞ n=1 1/n 3 is the apéry’s constant, holds. proof. from lemma 1, one has 2 π ∫ π 0 cl2(θ) dθ = 2 π ∫ π 0 ∞∑ n=1 sin(nθ) n2 dθ = 2 π ∞∑ n=1 ∫π 0 sin(nθ) dθ n2 = − 2 π ∞∑ n=1 cos (nθ) |π 0 n3 = − 2 π ∞∑ n=1 (−1)n − 1 n3 = 4 π ∑ odd 1 n3 , (2.3) where the last sum takes only the odd values of n into account. the interchange of the integral and the series is allowed because this series converges absolutely. since ζ(3) = ∑ odd 1/n 3 + ∑ even 1/n 3 and ∑ even 1/n 3 = ∑∞ m=1 1/(2m) 3 = 1 8 ζ(3), then ∑ odd 1/n 3 = 7 8 ζ(3). � now, let us derive a general result valid for all integrals ak, k > 0. for this, it will be useful to define hn := ∑n `=1 1/(2`− 1), n being a positive integer, which is the odd analogue of the harmonic number hn := ∑n `=1 1/`. since hdn/2e = hn− 1 2 hbn/2c, it is easy to rewrite any expression containing hn in terms of the usual harmonic numbers. 104 lima theorem 2 (integral ak, k > 0). let ak be the integral defined in eq. (1.4). the exact closed-form result ak =   2 π ln 4 − 2 hbk/2c − 1/k k2 , k odd − 4 π hk/2 k2 , k even , holds for all integers k > 0. proof. from lemma 1, one has ak = 2 π ∫ π 0 cos (kθ) ∞∑ n=1 sin(nθ) n2 dθ = 2 π ∞∑ n=1 ∫π 0 cos (kθ) sin(nθ) dθ n2 , (2.4) where k is a positive integer. on applying the trigonometric identity sin α cos β = 1 2 [ sin (α + β) + sin (α−β)] to the last integral, one finds ikn := ∫ π 0 cos (kθ) sin(nθ) dθ = 1 2 ∫ π 0 {sin [(n + k)θ] + sin [(n−k)θ]}dθ . (2.5) for n = k, the above integral reduces to inn = ∫π 0 cos(nθ) sin(nθ) dθ = 1 2 ∫π 0 sin(2nθ) dθ = − cos (2nθ)/(2n) |π 0 = 0. for all n 6= k, one has ikn = − 1 2 { cos [(n + k) θ] n + k + cos [(n−k) θ] n−k }∣∣∣∣π 0 = − 1 2 { cos [(n + k) π] − 1 n + k + cos [(n−k) π] − 1 n−k } = − 1 2 [ (−1)n+k − 1 n + k + (−1)n−k − 1 n−k ] . (2.6) therefore, ak = 2 π ∑∞ n=1 ikn/n 2 expands to ak = 1 π ∞∑ n=1 1 n2 [ 1 − (−1)n+k n + k + 1 − (−1)n−k n−k ] (2.7) and, since 1 − (−1)n±k = 0 whenever n and k have the same parity (i.e., when they are both odd or even numbers), whereas 1 − (−1)n±k = 2 when n and k have opposite parities, then ak = 1 π ∑ n ′ 1 n2 [ 2 n + k + 2 n−k ] = 2 π ∑ n ′ 1 n2 2n n2 −k2 = 4 π ∑ n ′ 1 n 1 n2 −k2 , (2.8) where ∑ ′ means a sum over n values with the opposite parity with respect to k. explicitly, ak = 2 π ∞∑ m=1 1 m (4m2 −k2) , k odd , (2.9) and ak = 4 π ∞∑ m=1 1 (2m− 1) [(2m− 1)2 −k2] , k even . (2.10) for odd values of k, the substitution k = 2p− 1, p > 0, in eq. (2.9) yields π 2 a2p−1 = ∞∑ m=1 1 m [4m2 − (2p− 1)2] . (2.11) this series can be written in terms of the digamma function ψ(x) := d dx ln γ(x), where γ(x) :=∫∞ 0 tx−1 e−t dt is the classical gamma function. from a well-known series representation for ψ(x), cosine integrals for the clausen function 105 namely [8, sec. 8.362] ψ(x) = −γ − 1 x + ∞∑ n=1 ( 1 n − 1 n + x ) , (2.12) one finds, after some algebra, π 2 a2p−1 = − ψ(3/2 −p) + ψ(p + 1/2) + 2 γ 2 (2p− 1)2 , (2.13) where γ := lim n→∞ (hn − ln n) is the euler’s constant. from eq. (3) in ref. [8, sec. 8.366], one knows that ψ ( 1 2 ±p ) = −γ − ln 4 + 2 hp , (2.14) which, together with ψ ( 3 2 −p ) = ψ ( 1 2 −p ) + 1 1 2 −p , (2.15) which promptly follows from ψ(x + 1) = ψ(x) + 1/x [8, sec. 8.365], reduces eq. (2.13) to π 2 a2p−1 = ln 4 − 1/(2p− 1) − 2 hp−1 (2p− 1)2 , (2.16) which is equivalent to eq. (2.9). the special value ψ(1/2) = −γ−ln 4, as stated in ref. [8, sec. 8.366], is required in the derivation of eq. (2.14). for even values of k, substitute k = 2p in eq. (2.10). this leads to π 4 a2p = ∞∑ m=1 1 (2m− 1) [(2m− 1)2 − 4p2] . (2.17) the series representation of ψ(x) given in eq. (2.12) then leads to π 4 a2p = − ψ(1/2 −p) + ψ(p + 1/2) + 2 γ + 2 ln 4 16 p2 . (2.18) on taking eq. (2.14) into account, one finds, after some algebra, π 4 a2p = − hp (2 p)2 , (2.19) which completes the proof. � as expected, this theorem shows that all conjectures stated at the end of ref. [5] are indeed true. 3. fourier series for an even periodic extension of clausen function now, let us examine the fourier cosine series whose coefficients are the ak expressions derived above. theorem 3. the series a0 2 + ∞∑ k=1 ak cos (k θ) , where a0 and ak are the coefficients derived in our theorems 1 and 2, respectively, converges to cl2(θ) for all θ ∈ [0,π] and to −cl2(θ) when θ ∈ (π, 2π], thus yielding a continuous even function on [−2π, 2π]. this convergence can be extended to all θ ∈ r. proof. let g(θ) be a real function defined in the interval [−2π, 2π] as follows: g(θ) := { +cl2(θ) , θ ∈ [−2π,−π) or θ ∈ [0,π] −cl2(θ) , θ ∈ [−π, 0) or θ ∈ (π, 2π] . (3.1) since cl2(θ) is a continuous odd function, it is clear that g(θ) is a continuous even function in the interval [−2π, 2π]. in theorem 4 of zhang’s paper [7], it is shown that, given a real function f(x) 106 lima integrable on [0,l] and such that f(x) = f(l−x) for all x ∈ (l/2,l], if f(x) is an even function in (−l,l), then the series a0 2 + ∞∑ n=1 a2n cos ( 2nπx l ) , (3.2) where a2n = 4 l ∫ l/2 0 f(t) cos ( 2nπt l ) dt, n ≥ 0 , (3.3) converges to f(x) for all x ∈ [−l,l] where f(x) is a continuous function. the absence of the term a0/2 in theorem 4 of ref. [7] is corrected here in our eq. (3.2). since the function g(θ) defined in eq. (3.1) is an even function which is continuous (thus integrable) on (−2π, 2π) and g(θ) = g(2π−θ), then the convergence of the series a0/2 + ∑∞ k=1 ak cos (k θ) to g(θ) follows by taking l = 2π in zhang’s theorem and noting that a2n = 2 π ∫π 0 g(θ) cos (nθ) dθ = 2 π ∫π 0 cl2(θ) cos (nθ) dθ are just the coefficients a0 and an derived in our theorems 1 and 2, respectively. finally, since this cosine series converges to an even periodic extension of cl2(θ), with a period 2π, then its convergence to g(θ) can be extended to all θ ∈ r. � interestingly, new closed-form results can be deduced directly from theorem 3. for instance, on taking θ = 0 (or π), one finds corollary 1. the following closed-form result holds: ∞∑ p=1 hp−1 (2p− 1)2 = π2 8 ln 2 − 7 16 ζ(3) . on taking θ = π/2 in theorem 3, a less obvious expression arises which can be written in terms of the regularized hypergeometric function pf̃q ( a1, . . . ,ap b1, . . . ,bq ; z ) := pfq ( a1, . . . ,ap b1, . . . ,bq ; z ) ∏q j=1 γ (bj) , (3.4) where pfq ( a1, . . . ,ap b1, . . . ,bq ; z ) := ∞∑ n=0 (a1)n . . . (ap)n (b1)n . . . (bq)n zn n! (3.5) is the generalized hypergeometric series. as usual, (a)n := a (a + 1) . . . (a + n− 1) = γ(a + n)/γ(a) is the pochhammer symbol. by convention, (a)0 = 1. corollary 2 (a special value for θ = π/2 ). the following closed-form result holds: 4f̃ ′ 3 ( 1, 1, 1, 3/2 2, 2, 3/2 ; −1 ) = ζ(2) (γ + ln 4) + 7 ζ(3) − 4π g √ π , (3.6) where ζ(2) := ∑∞ n=1 1/n 2 = π2/6 and g := ∑∞ n=0 (−1) n/(2n + 1)2 is the catalan’s constant. here, the prime symbol ( ′) indicates a partial derivative with respect to b3. as shown below, this result can be written in terms of the corresponding generalized hypergeometric function. interestingly, this yields a nice closed-form result which, to the author knowledge, is not found in literature. corollary 3 (corresponding generalized hypergeometric function). the following closed-form result holds: 4f ′ 3 ( 1, 1, 1, 3/2 2, 2, 3/2 ; −1 ) = π2 6 + 7 2 ζ(3) − 2π g. (3.7) proof. in a shortened notation, eq. (3.4) reads pf̃q ( ~a,~b; z ) = pfq ( ~a,~b; z ) ∏q j=1 γ(bj) cosine integrals for the clausen function 107 where ~a and ~b denote the arrays of coefficients [1, 1, 1, 3/2] and [2, 2, 3/2], respectively. this implies that ∂ ∂b3 4f̃3 ( ~a,~b;−1 ) = 1∏ j 6=3 γ(bj) ∂ ∂b3   4f3 ( ~a,~b;−1 ) γ(b3)   = 1 γ(b1) γ(b2)   4f ′3 ( ~a,~b;−1 ) γ(b3) − 4f3 ( ~a,~b;−1 ) γ′(b3) γ2(b3)   = 1 γ2(2)  4f ′3 ( ~a,~b;−1 ) γ(3/2) − 4f3 ( ~a,~b;−1 ) γ′(3/2) γ2(3/2)   . (3.8) since γ(1 + x) = x γ(x), then γ ( 3 2 ) = 1 2 γ ( 1 2 ) = √ π/ 2, which reduces the last expression, above, to 4f̃ ′ 3 ( ~a,~b;−1 ) = 4f ′ 3 ( ~a,~b;−1 ) √ π/2 − 4f3 ( ~a,~b;−1 ) ψ(3/2) √ π/2 = 2 4f ′ 3(~a, ~b;−1) √ π − 2 4f3(~a,~b;−1) ψ(3/2) √ π . (3.9) note that, for all positive integers n, γ(n) = (n−1)! (in particular, γ(2) = 1! = 1). the proof completes by substituting the result in corollary 2, together with the special values ψ(3/2) = ψ(1/2) + 1/(1/2) = −γ − ln 4 + 2 and 4f3(~a,~b;−1) = π2/ 12, in eq. (3.9). � the closed-form result in corollary 3 has been conjectured by ancarani and the author in a recent discussion, by following an entirely different approach, but we could not find a formal proof at that time. acknowledgments the author wishes to thank m. r. javier for checking all closed-form expressions in this work numerically with mathematical software. references [1] t. clausen, über die function sin φ+ (1/22) sin 2φ+ (1/32) sin 3φ+etc., j. reine angew. math. (crelle) 8, 298–300 (1832). [2] d. f. connon, fourier series and periodicity. arxiv:1501.03037 [math.gm]. [3] l. lewin, polylogarithms and associated functions, north holland, new york, 1981. [4] l. lewin, structural properties of polylogarithms, american mathematical society, providence, 1991. [5] f. m. s. lima, evaluation of some non-trivial integrals from finite products and sums, turkish j. anal. number theory 4, 172–176 (2016). [6] k. f. riley and m. p. hobson, essential mathematical methods for the physical sciences, cambridge university press, new york, 2011. [7] c. zhang, further discussion on the calculation of fourier series, appl. math. 6, 594–598 (2015). [8] i. s. gradshteyn and i. m. rhyzik, table of integrals, series, and products, 7th ed., academic press, new york, 2007. institute of physics, university of braśılia, p.o. box 04455, 70919-970, braśılia-df, brazil ∗corresponding author: fabio@fis.unb.br 1. introduction 2. cosine integrals of clausen function 3. fourier series for an even periodic extension of clausen function acknowledgments references int. j. anal. appl. (2022), 20:64 inversion formula for the wavelet transform on abelian group c.p. pandey1,∗, khetjing moungkang1, sunil kumar singh2, m.m. dixit1, mopi ado1 1department of mathematics, north eastern regional institute of science and technology, nirjuli, itanagar, arunachal pradesh, 791109, india 2department of mathematics, babasaheb bhimrao ambedkar university, lucknow, 226025, india ∗corresponding author: drcppandey@gmail.com abstract. in this paper a reconstruction and inversion formula of the continuous wavelet transform on abelian group for band-limited function is defined. this formula possesses a more explicit expression than the well-known result. also, parseval and other interesting results on abelian group are obtained. 1. introduction a set s defines a group if an operator, +, holds the following properties: • x + (y + z) = (x + y) + z ∀x,y,z ∈ s • there exists an element 0 , such that x + 0 = 0 + x = x ∀x ∈ s • for each∀x ∈ s there exists an inverse element x−1 = −x , such that x+(−x) = (−x)+x = 0. s is a topological group if it has a group operation and a topology such that the maps α : g×g → g and β : g ×g → g are continuous, where α(x,y) = x + y and β(x) = x−1. if s is locally compact, that is every point in s is contained in a compact neighborhood, and its group operation is commutative, then it is called locally compact abelian (lca) group. in order to define the fourier transform on lca groups, we should introduce the concept of integral over these groups. let m(x) be the space of all complex-valued regular measures on x where ||µ|| = |µ(s)| is finite. a haar measure is a measure which is non negative, regular and invariant. the corresponding integral is called the haar integral, which is translation invariant. received: oct. 2, 2022. 2010 mathematics subject classification. 65t60, 43a25, 11f85. key words and phrases. wavelet transform; abelian group; fourier transform. https://doi.org/10.28924/2291-8639-20-2022-64 issn: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-64 2 int. j. anal. appl. (2022), 20:64 let g be lca group , we define an lp (g) space to be the space of all complex valued functions f on g such that the integral ∫ g |f |pdµ exists with respect to the haar measure. definition 1.1. a complex function ω on a lca group g [1] is called a character of g if |ω(x)| = 1 for all x ∈ g and if the functional equation ω(x + y) = ω(x)ω(y) for all (x,y) ∈ g is satisfied. the set of all continuous characters of g form a group ω , the dual group of g . now it is customary to write (x,ω) = ω(x)ω(x) satisfy the following properties [1,5] • (0,ω) = (x, 0) = 1 • (−x,ω) = (x,−ω) = (x,ω)−1 = (x,ω) • (x + y,ω) = (x,ω)(y,ω) • (x,ω1 + ω2) = (x,ω1)(x,ω2) definition 1.2. the fourier transform [2] of f ∈ l1(g) is denoted by f̂ (ω) defined by f̂ (ω) =∫ g f (x)(−x,ω)dx , and its inverse fourier transform is defined [1,5] by f (x) = ∫ g f̂ (ω)(x,ω)dω,x ∈ g the fourier transform holds the following properties [4]: • ‖f̂‖l∞(g) 6 ‖f‖l1(g) • if f ∈ l1(g) ∩l2(g) , then ‖f̂‖l2(g) = ‖f‖l2(g) • if the convolution of f and g is defined as (f ∗g)(x) = ∫ g f (x −y)g(y)dy then f ((f ∗g)) = f (f )f (g) for f (x) ∈ l2(g), denote fb,a(x) = 1√|a|f ( x−b a ) and suppf = clos{x ∈ g : f (x) 6= 0}. if suppf̂ is a bounded set, then we say f is band-limited. the characteristic function on a set e is denoted by xe(x) . in 1984, morlet introduced first wavelet transform [7] that is defined as follows: let ψ ∈ l2(g) , the transform: (wψ)(b,a) = ∫ g f (x)ψb,a(x)dx for any f ∈ l2(g) (1.1) is said to be a wavelet transform. when ψ ∈ l1 ∩ l2(g) and cψ = 2π ∫ g |ψ̂(ω)|2 |ω| < ∞ , the known inversion formula [8] is stated as follows: f (x) = 1 cψ ∫ g ∫ g (wψf )(b,a)ψb,a(x) dadb |a|2 (1.2) the above equality holds in l2(g) sense. the aim of this paper is that for band-limited function we give another kind of inversion formula of wavelet transform. theorem 1.1. let ψ(x) ∈ l1 ∩l2(g). take φ(x) ∈ l1 ∩l2(g) satisfying φ̂(ω) = o(|ω|−2). then for any f ∈ l1 ∩l2(g) and suppf̂ ⊆ [−ω, ω], the following inversion formula holds: f (x) = 1 (2π) 3 2 (ϕ,ψ) ∫ h ∫ g (wψf )(b,a)(ϕb,a ∗h)(x) dada |a| (1.3) int. j. anal. appl. (2022), 20:64 3 where h(x) satisfies ĥ(ω) = |ω|x[−ω,ω](ω) and the above equality holds in l2sense. 2. lemma to prove theorem, we first give the following lemma: let ψ(x),ϕ(x) and f (x) be stated in theorem. then for any g ∈ l2(g) the following formula is valid: 1 2π ∫ g ∫ g (wψf )(b,a)(dϕg)(b,a) dbda |a| = (ϕ,ψ)(f ,g), where (dϕg)(b,a) = 1 √ 2π (g,ϕ(b,a) ∗h) (2.1) proof: by parseval identity of fourier transform, we have (wψf )(b,a) = |a| 1 2 ∫ g f̂ (ω)ψ̂(aω)(b,ω)dω (2.2) using the convolution formula [3] and parseval identity, we also obtain from (2.1) that (dϕg)(b,a) = |a| 1 2 ∫ g ĝ(ω)ϕ̂(aω)ĥ(ω)(b,ω)dω (2.3) applying the inversion formula of fourier transform, it follows from (2.2) and (2.3) that 1 √ 2π|a| 1 2 (wψf )(b,a) = (f̂ (ω)ψ̂(aω)) v (b) (2.4) and 1 √ 2π|a| 1 2 (dϕg)(b,a) = (ĝ(ω)ϕ̂(aω)ĥ(ω)) v (b) (2.5) finally, again using parseval identity, we get 1 2π|a| ∫ g (wψf )(b,a)(dϕg)(b,a)db = ∫ g f̂ (ω)ĝ(ω)ĥ(ω)ψ̂(aω)ϕ̂(aω)dω since suppf̂ ⊆ [−ω, ω] = supp ĥ(ω) and ĥ(ω) = |ω|x[−ω,ω](ω), we know that f̂ (ω)ĥ(ω) = |ω|f̂ (ω), ω ∈ g. further, 1 2π|a| ∫ g (wψf )(b,a)(dϕg)(b,a)db = ∫ g f̂ (ω)ĝ(ω)|ω|ψ̂(aω)ϕ̂(aω)dω in view of∫ g ∫ g |f̂ (ω)ĝ(ω)ωψ̂(aω)ϕ̂(aω)|dωda = ∫ g |f̂ (ω)ĝ(ω)| ( |ω| ∫ g |ψ̂(aω)ϕ̂(aω)|da ) dω = (∫ g |ψ̂(aω)ϕ̂(aω)|dω )(∫ g |f̂ (ω)ĝ(ω)|dω ) 6 ‖ϕ‖2‖ψ‖2‖f‖2‖g‖2 4 int. j. anal. appl. (2022), 20:64 by fubini theorem, we have 1 2π ∫ g (∫ g (wψf )(b,a)(dϕg)(b,a) ) da |a| = ∫ g (∫ g f̂ (ω)ĝ(ω)|ω|ψ̂(aω)ϕ̂(aω)dω ) da = ∫ g f̂ (ω)ĝ(ω ( |ω| ∫ g ψ̂(aω)ϕ̂(aω)da ) dω again, noticing that |ω| ∫ g ψ̂(aω)ϕ̂(aω)da = (ϕ̂,ψ̂) = (ϕ,ψ), for repeated integral, we get 1 2π ∫ g (∫ g (wψf )(b,a)(dϕg)(b,a) ) da |a| = (ϕ,ψ)(f ,g). in order to complete the proof of lemma, by fubini theorem, we only need to prove that k = ∫ g ∫ g |(wψf )(b,a)(dϕg)(b,a)| dadb |a| < ∞ (2.6) we split the above integral into two parts, namely, k = (∫ h + ∫ g−h )(∫ |(wψf )(b,a)(dϕg)(b,a)| db |a| ) da = k1 + k2 (2.7) where h is a subgroup of g. first, we estimate k1. using cauchy inequality, we get k21 6 ∫ h (∫ g |(wψf )(b,a)|2 db |a| ) da · ∫ h (∫ g |(dϕg)(b,a)|2 db |a| ) da = k11 ·k12 applying (2.4) and parseval identity, we have k11 = 2π ∫ h (∫ g |f̂ (ω)|2|ψ̂(aω)|2dω ) da by ψ ∈ l1(g) we know that there is a m > 0 such that |ψ̂(ω)|6 m, so k11 6 4πm2‖f‖22. on the other hand, applying (2.5) and parseval identity, we also have k12 = 2π ∫ h (∫ g |ĝ(ω)|2|ϕ̂(aω)|2|ĥ(ω)|2dω ) da (2.8) by ϕ ∈ l1(g) we know that there is an n > 0 such that |ϕ̂(ω)|6 n. again noticing that |ĥ(ω)|6 ω, we have k12 6 4πn2ω2‖g‖22. so k1 < ∞. next we estminate k2, from (2.7), we know that for any given 0 < � < 1 2 , k2 = ∫ g−h (∫ g |a|−1+ 1 2 |(wψf )(b,a)| · |a|− 1 2 |(dϕg)(b,a)|db ) da int. j. anal. appl. (2022), 20:64 5 using cauchy inequality, we get k22 6 ∫ g−h (∫ g |(wψf )(b,a)|2 db |a|2−� ) da · ∫ g−h (∫ g |(dϕg)(b,a)|2 db |a|� ) da = k21 ·k22 since |(wψf )(b,a)| = ∣∣∣∣∣ 1√|a| ∫ g f (x)ψ ( x −b a ) dx ∣∣∣∣∣ 6 ‖f‖2‖ψ‖2 and 1 √ a ∫ g |(wψf )(b,a)|db 6 ∫ g ∫ g ∣∣∣∣f (x)ψ ( x −b a )∣∣∣∣ dxdba 6 ‖f‖1‖ψ‖1 (2.9) we get k21 6 ‖f‖2‖ψ‖2 ∫ g−h (∫ g |(wψf )(b,a)|2db ) 1 |a|2−� da 6 ‖f‖2‖ψ‖2‖f‖1‖ψ‖1 ∫ g−h 1 |a| 3 2 −� da = 4 1 − 2� ‖f‖2‖ψ‖2‖f‖1‖ψ‖1 similar to the argument of (2.8), we have k22 = 2π ∫ g−h (∫ g |ĝ(ω)|2|ϕ̂(aω)|2|ĥ(ω)|2|a|1−� ) da further, by the definition of h(x), k22 = 2π ∫ g−h |a|−1−� (∫ g |aω|2|ĝ(ω)|2|ϕ̂(aω)|2dω ) da from ϕ̂(ω) = o(|ω|−2), we have |ϕ̂(aω)|2 6 m1(m1 is an absolute constant). further k22 6 4πm1 � ‖g‖22. so k2 < ∞. we finally obtain (2.6). the proof of lemma is completed. 3. proof of theorem from |(ϕb,a ∗h)(x)|6 ‖ϕb,a‖2‖h‖2 = ‖ϕ‖2‖h‖2 and (2.9), we have∫ h ∫ g |(wψf )(b,a)(ϕb,a ∗h)(x)| dadb |a| 6 ‖ϕ‖2‖h‖2 ∫ h 1 |a| 1 2 ( 1 |a| 1 2 ∫ g |(wψf )(b,a)|db ) da 6 4‖ϕ‖2‖h‖2‖ψ‖1‖f‖1 (3.1) so, for all x ∈ g, we know that (wψf )(b,a)(ϕb,a ∗h)(x) 1|a| ∈ l 1(h ×g). set 4h(x) = 1 (2π) 3 2 (ϕ,ψ) ∫ h ∫ g (wψf )(b,a)(ϕb,a ∗h)(x) dadb |a| . 6 int. j. anal. appl. (2022), 20:64 by the known result in theory of hilbert space, we know that ‖f (x) −4h(x)‖2 = sup ‖g‖2=1 |(f ,g) − (4h,g)|. (3.2) again by (2.1), we get (4h,g) = 1 (2π) 3 2 (ϕ,ψ) ∫ g (∫ h ∫ g (wψf )(b,a)(ϕb,a ∗h)(x)g(x) dadb |a| ) dx = 1 (2π)(ϕ,ψ) ∫ h ∫ g (wψf )(b,a)(dϕg)(b,a) dbda |a| . (3.3) the reason for interchanging the order of the above integrals is stated as follows. by (2.9) and ∫ g |(ϕb,a ∗h)(x)g(x)| 1√ |a| dx 6 ‖ 1√ |a| (ϕb,a ∗h)(x)‖2‖g‖2 6 ‖ϕ‖1‖h‖2‖g‖2 we get ∫ g (∫ h ∫ g (wψf )(b,a)(ϕb,a ∗h)(x)g(x) dadb |a| ) dx = ∫ h (∫ g |(wψf )(b,a)| (∫ g |(ϕb,a ∗h)(x)g(x)| 1√ |a| dx ) db√ |a| ) da 6 2‖φ‖1‖h‖2‖g‖2‖f‖1‖ψ‖1 so, the order of integrals in (3.3) can be interchanged. using lemma and (3.3), (f ,g) − (4h,g) = 1 (2π)(ϕ,ψ) ∫ g−h ∫ g (wψf )(b,a)(dϕg)(b,a) dbda |a| further we get from (3.2) ‖(f ,g) − (4h,g)‖2 6 sup ‖g‖2=1 ( 1 (2π)|(ϕ,ψ)| ∫ g−h ∫ g (wψf )(b,a)(dϕg)(b,a) dbda |a| ) = sup ‖g‖2=1 ( 1 (2π)|(ϕ,ψ)| i(h) ) . (3.4) where i(h) = ∫ g−h ∫ g |(wψf )(b,a)(dϕg)(b,a)|dbda|a| ,∀h ∈ h using cauchy inequality, we can see that i2(h) 6 ∫ g−h (∫ g |(wψf )(b,a)|2 db |a|2−� ) da · ∫ g−h (∫ g |(dϕg)(b,a)|2 db |a|� ) da = i1(h)i2(h) (3.5) imitating the estimates of k21 and k22 in lemma, we can get i1(h) 6 4 1 − 2� h −1 2 +�‖f‖2‖ψ‖2‖f‖1‖ψ‖1 int. j. anal. appl. (2022), 20:64 7 and i2(h) 6 4πm1 � h −�‖g‖22. from this and (3.4),(3.5), we know that ‖f (x) −4h(x)‖2 6 sup ‖g‖2=1 ( 1 (2π)|(ϕ,ψ)| i(h) ) 6 1 (2π)|(ϕ,ψ)| ( 16πm1 (1 − 2�)� h −1 2‖f‖2‖ψ‖2‖f‖1‖ψ‖1 )1 2 . so, lim h→+∞ ‖f (x) −4h(x)‖2 = 0. this proof of theorem is completed. acknowledgements: this work is supported by ugc grant no. 16-9(june 2018)/2019(net/csir) and csir grant no. 09/725(014)/2019-emr-1. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. holschneider, wavelet analysis over abelian groups, appl. comput. harmon. anal. 2 (1995), 52–60. https: //doi.org/10.1006/acha.1995.1004. [2] c.p. pandey, p. phukan, continuous and discrete wavelet transforms associated with hermite transform, int. j. anal. appl. 18 (2020), 531-549. https://doi.org/10.28924/2291-8639-18-2020-531. [3] a. pathak, p. yadav, m.m. dixit, on convolution for general novel fractional wavelet transform, arxiv:1404.7682 (2014). https://doi.org/10.48550/arxiv.1404.7682. [4] r.s. pathak, c.p. pandey, laguerre wavelet transforms, integral transforms spec. funct. 20 (2009), 505–518. https://doi.org/10.1080/10652460802047809. [5] c.p. pandey, jyoti saikia, the continuous wavelet transform for a q-bessel type operator, int. j. anal. appl., 20 (2022), 33. https://doi.org/10.28924/2291-8639-20-2022-33 [6] m.m. dixit, c.p. pandey, deepanjan das, generalized continuous wavelet transform on locally compact abelian group, adv. inequal. appl. 2019 (2019), 10. https://doi.org/10.28919/aia/4067. [7] c.k. chui, an introduction to wavelets, academic press, 1992. [8] l. debnath, the wavelet transform and its basic properties, in: wavelet transforms and their applications, birkhäuser boston, boston, ma, 2002: pp. 361-402. https://doi.org/10.1007/978-1-4612-0097-0_6. https://doi.org/10.1006/acha.1995.1004 https://doi.org/10.1006/acha.1995.1004 https://doi.org/10.28924/2291-8639-18-2020-531 https://doi.org/10.48550/arxiv.1404.7682 https://doi.org/10.1080/10652460802047809 https://doi.org/10.28924/2291-8639-20-2022-33 https://doi.org/10.28919/aia/4067 https://doi.org/10.1007/978-1-4612-0097-0_6 1. introduction 2. lemma 3. proof of theorem references int. j. anal. appl. (2023), 21:29 analytic solution of black-scholes-merton european power put option model on dividend yield with modified-log-power payoff function s.e. fadugba1,2,3, a.a. adeniji4, m.c. kekana4, j.t. okunlola5, o. faweya6 1department of mathematics, ekiti state university, ado ekiti, 360001, nigeria 2department of physical sciences, mathematics programme, landmark university, omu-aran, nigeria 3landmark university sdg 4 (quality education research group), omu-aran, nigeria 4department of mathematics, tshwane university of technology, pretoria, south africa 5department of mathematical and physical sciences, afe babalola university, ado ekiti, nigeria 6department of statistics, ekiti state university, ado ekiti, 360001, nigeria ∗corresponding author: sunday.fadugba@eksu.edu.ng, fadugba.sunday@lmu.edu.ng abstract. this paper proposes a framework based on the celebrated transform of mellin type (mt) for the analytic solution of the black-scholes-merton european power put option model (bsmeppom) on dividend yield (dy) with modified-log-power payoff function (mlppf) under the geometric brownian motion. the mt has the capability of tackling complex functions by means of its fundamental properties and it is closely related to other well-known transforms such as laplace and fourier types. the main goal of this paper is to use mt to obtain a valuation formula for the european power put option (eppo) which pays a dy with mlppf. by means of mt and its inversion formula, the price of eppo on dy was expressed in terms of integral equation. moreover, the valuation formula of eppo was obtained with the help of the convolution property of mt and final time condition. the mt was tested on an illustrative example in order to measure its performance, effectiveness and suitability. the mlppf was compared with other existing payoff functions. hence, the effect of dy on the pricing of eppo with mlppf was also investigated. 1. introduction the popularity of option pricing in financial mathematics has been displayed as one of the key major areas in derivative security. in other words, option valuation has contributed greatly to the financial received: jan. 9, 2023. 2010 mathematics subject classification. 91b24, 91g20. key words and phrases. convolution property; final time condition; integral representation; mellin transform. https://doi.org/10.28924/2291-8639-21-2023-29 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-29 2 int. j. anal. appl. (2023), 21:29 markets. there is a massive growth in trading activities on derivatives globally from the inception of the black-scholes pricing formula [1,2]. it is noteworthy to say that the black-scholes model for linear payoff has been used by many researchers and as well as become one of the utmost areas in financial markets over the last few decades. immediately after the huge success recorded by the black-scholes model for vanilla option flavours, several other valuation formula were developed for options pricing with different payoff functions such as mellin transform, binomial model, finite difference method, monte carlo method, e.t.c; see [3] – [6]. for mathematical framework, some implementations of transform methods of different types in financial markets; see [7]– [15]. ghevariya [16] solved the classical black-scholes european put option model for modified-log payoff function with the help of the mt. fadugba et al. [17] obtained a direct solution of the black-scholes-merton european put option model on dividend yield with modified-log payoff function via a framework based on mt. in this paper, an analytic solution of bsmeppom via the celebrated transform of mellin type is proposed in the sense of dy and mlppf. the remaining part of the paper is listed as follows; section 2 captures the brief concepts of mt. the governing model for eppo on a dy with mlppf is presented in section 3. section 4 captures the solution of bsmeppom with dy and mlppf. an illustrative example on the application of mt to eppo is captured in section 5. section 6 is the concluding part of the paper. 2. mellin transform this section captures some definitions of terms based on the framework of the mellin transform and its inversion formula [18]. definition 2.1. let f (x) be a locally lebesgue integrable function. the mellin transform of f (x) is defined as m[f (x),ω] := f̃ (ω)= ∫ ∞ 0 f (x)xω−1dx (2.1) the mellin transform variable ω is a complex number, ω = <(.)+ i=(.), where <(.) is the real part, i is the imaginary unit and =(.) is the imaginary part.. definition 2.2. if f (x) is an integrable function with fundamental strips (a,b), then if c is such that a < c < b and {f̃ (ω) : ω = c + it} is integrable, the inverse mellin transform is defined as m−1[f̃ (ω)]= f (x)= 1 2πi ∫ c+i∞ c−i∞ f̃ (ω)x−ωdω (2.2) remark 2.1. it is clearly seen that the mellin transform m[f (x),ω] and the inverse mellin transform m−1[f̃ (ω)] are linear integral operators. remark 2.2. for more details on the condition that ensures the existence of mt; see [18,19]. int. j. anal. appl. (2023), 21:29 3 remark 2.3. the fundamental operational properties of the mellin transforms such as scaling, shifting, derivatives, integrals, convolution, multiplicative convolution and parseval’s formula are well detailed in [9,12,18,20]. 3. the bsmeppom on a dy with mlppf the bsmeppom on a dy with mlppf is given by ∂pρ(s ρ t ,t) ∂t +ρ ( r −q + (ρ−1)σ2 2 ) s ρ t ∂pρ(s ρ t ,t) ∂s ρ t + (σs ρ t) 2 2 ∂2pρ(s ρ t ,t) ∂(s ρ t) 2 = rpρ(s ρ t ,t) (3.1) subject to the boundary conditions lim s ρ t→∞ pρ(s ρ t ,t)=0 on [0,t) (3.2) lim s ρ t→0 pρ(s ρ t ,t)= k er(t−t) on [0,t) (3.3) and mlppf pρ(s ρ t ,t)= [ s ρ t ln ( k s ρ t )]+ on [0,∞) (3.4) where pρ(s ρ t ,t), ρ, t, t, s ρ t , k, σ, r and q are the price of eppo, power of the option, current time, time to expiry, underlying asset price, strike price, volatility, risk-free interest rate and dy, respectively. 4. solution of the black-scholes-merton european put option model with mlppf ghevariya derived bsm formula on non-dividend yield for ml-payoff function [16]. in this section, analytic solution of bsmeppom with dividend yield for ml-power payoff function is obtained via the mt as follows. taking the mt of (3.1) and using its linearity, independence of time derivatives and shifting properties and rearranging terms, one obtains ∂p̃ρ(ω,t) ∂t =− σ2ρ2 2 (ω2 +ω(1−b1)−b2)p̃ρ(ω,t) (4.1) where b1 = ρ−1 ρ + 2(r −q) ρσ2 , b2 = 2r ρ2σ2 (4.2) solving (3.1), yields p̃ρ(ω,t)= l(ω)e −1 2 ρ2σ2(ω2+ω(1−b1)−b2)t (4.3) but l(ω)=m(pρ(st ,t),ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)t (4.4) 4 int. j. anal. appl. (2023), 21:29 which is equivalent to l(ω)= f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)t (4.5) substituting (4.5) into (4.3), yields p̃ρ(ω,t)= f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ (4.6) where τ = t − t. by means of (2.2), (4.6) yields pρ(s ρ t ,t)= 1 2πi ∫ c+i∞ c−i∞ f̃ (ω)e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ(s ρ t) −ωdω (4.7) which is the integral equation for governing equation (3.1). let ξ(s ρ t)= 1 2πi ∫ c+i∞ c−i∞ e ρ2σ2 2 (ω2+ω(1−b1)−b2)(s ρ t) −ωdω (4.8) using the fact that e 1 2 ρ2σ2(ω2+(1−b1)ω−b2)τ = e−α1(α 2 2+b2)+α1(ω+α2) 2 (4.9) where α1 = ρ2σ2τ 2 ,α2 = 1−b1 2 (4.10) thus ξ(s ρ t)= e−α1(α 2 2+b2) 2πi ∫ c+i∞ c−i∞ eα1(ω+α2) 2 (s ρ t) −ωdω (4.11) by means of the transformation given by [21], one obtains eφω 2 = 1 2 √ π ∫ ∞ 0 1 √ φ exp ( −(lnsρt) 2 4φ ) (s ρ t) ω−1ds ρ t , <(φ)≥ 0 (4.12) therefore, ξ(s ρ t)= e −α1(α22+b2) (s ρ t) α2 ρσ √ 2πτ e −1 2 ( ln(s ρ t ) ρσ √ τ )2 (4.13) similarly, ξ ( s ρ t v ) = e−α1(α 2 2+b2) ( s ρ t v )α2 ρσ √ 2πτ e −1 2   ln ( s ρ t v ) ρσ √ τ   2 (4.14) using (3.4), then h(s ρ t)=m −1(f̃ (ω))= [ s ρ t ln ( k s ρ t )]+ (4.15) thus, h(v)= [ v ln ( k v )]+ (4.16) with the help of the convolution property of mt, (4.7) becomes pρ(s ρ t ,t)= ∫ ∞ 0 h(v)ξ ( s ρ t v ) 1 v dv (4.17) int. j. anal. appl. (2023), 21:29 5 using (4.14) and (4.16), (4.17) becomes pρ(s ρ t ,t)= ∫ ∞ 0 [ v ln ( k v )]+ e−α1(α 2 2+b2) ( s ρ t v )α2 ρσ √ 2πτ e −1 2   ln ( s ρ t v ) ρσ √ τ   2 1 v dv (4.18) pρ(s ρ t ,t)= e −α1(α22+b2) (s ρ t) α2 ρσ √ 2πτ ∫ k 0 [ v1−α2 ln ( k v )] e −1 2   ln ( s ρ t v ) ρσ √ τ   2 1 v dv (4.19) simplifying further, yields pρ(s ρ t ,t)= e −α1(α22+b2) (s ρ t) α2 ρσ √ 2πτ ∫ k 0 ln(k) 1 vα2 e −1 2   ln ( s ρ t v ) ρσ √ τ   2 dv −e−α1(α 2 2+b2) (s ρ t) α2 ρσ √ 2πτ ∫ k 0 ln(v) 1 vα2 e −1 2   ln ( s ρ t v ) ρσ √ τ   2 dv (4.20) pρ(s ρ t ,t)= e −α1(α22+b2) (s ρ t) α2 ρσ √ τ [ln(k)h1 −h2] (4.21) where h1 = 1 √ 2π ∫ k 0 1 vα2 e −1 2   ln ( s ρ t v ) ρσ √ τ   2 dv (4.22) h2 = 1 √ 2π ∫ k 0 ln(v) vα2 e −1 2   ln ( s ρ t v ) ρσ √ τ   2 dv (4.23) let x = ln ( sρ v ) ρσ √ τ (4.24) thus h2 = ρσ √ τ(s ρ t) −α2+1eα1(α2−1) 2 [ρσ √ τg1 − ln(s ρ t)g2] (4.25) where g1 = 1 √ 2π ∫ ln(sρtv ) ρσ √ τ ∞ xe− 1 2 (x−ρσ √ τ(α2−1))2dx (4.26) g2 = 1 √ 2π ∫ ln(sρtv ) ρσ √ τ ∞ e− 1 2 (x−ρσ √ τ(α2−1))2dx (4.27) 6 int. j. anal. appl. (2023), 21:29 once again, by variable transformation y = x −ρσ √ τ(α2 −1), dρ = ln ( s ρ t k ) −ρ2σ2τ(α2 −1) ρσ √ τ = ln ( s ρ t k ) + ( r −q + ( ρ− 1 2 ) σ2 ) τ ρσ √ τ (4.28) equations (4.26) and (4.27) become g1 =−[η(dρ)+ρσ √ τ(α2 −1)n(−dρ)] (4.29) and g2 =−n(−dρ) (4.30) respectively, with η(κ)= 1 √ 2π e− κ2 2 ,n(κ)= ∫ κ −∞ η(κ)dκ (4.31) substituting (4.29) and (4.30) into (4.25), yields h2 =−ρσ √ τ(s ρ t) −α2+1eα1(α2−1) 2 [ρσ √ τη(dρ) +(ρ2σ2τ(α2 −1)− ln(s ρ t))n(−dρ)] (4.32) similarly, h1 = ρσ √ τ(s ρ t) −α2+1eα1(α2−1) 2 [n(−dρ)] (4.33) using (4.32) and (4.33) and the values of b2, α1 and α2, (4.21) becomes pρ(s ρ t ,t)= s ρ te [r(ρ−1)−ρq+12ρ(ρ−1)σ 2]τ [ ρσ √ τη(dρ)− (q1 +q2τ)n(−dρ) ] (4.34) with q1 = ln ( s ρ t k ) , (4.35) q2 = ρ ( r −q + ( ρ− 1 2 ) σ2 ) , (4.36) dρ = q1 +q2τ ρσ √ τ , (4.37) τ = t − t. (4.38) remark 4.1. setting q =0 in (4.34), yields the fundamental valuation formula for bsmeppom on non dy with mlppf pρ(s ρ t ,t)= s ρ te [r(ρ−1)+12ρ(ρ−1)σ 2]τ [ ρσ √ τη(dρ)− (r1 +r2τ)n(−dρ) ] (4.39) with r1 = ln ( s ρ t k ) , (4.40) int. j. anal. appl. (2023), 21:29 7 r2 = ρ ( r + ( ρ− 1 2 ) σ2 ) , (4.41) dρ = r1 +r2τ ρσ √ τ , (4.42) τ = t − t. (4.43) remark 4.2. setting ρ =1 in (4.34), yields the fundamental valuation formula for plain epo on dy p1(st,t)= ste −qτ [σ√τη(d1)− (d1 +d2τ)n(−d1)] (4.44) with d1 = ln ( st k ) , (4.45) d2 = ( r −q + σ2 2 ) , (4.46) d1 = d1 +d2τ σ √ τ , (4.47) τ = t − t. (4.48) remark 4.3. setting ρ = 1 and q = 0 in (4.34), yields the fundamental valuation formula for plain epo on non dy p1(st,t)= st [ σ √ τη(d1)− (b1 +b2τ)n(−d1) ] (4.49) with b1 = ln ( st k ) , (4.50) b2 = ( r + σ2 2 ) , (4.51) d1 = b1 +b2τ σ √ τ , (4.52) τ = t − t. (4.53) 8 int. j. anal. appl. (2023), 21:29 5. numerical example consider the valuation of the eppo on a dy with mlppf via the mt using the following parameters in table 1. the results obtained are displayed in figures 1 10. table 1. the parameters parameters values s in dollars 100 k in dollars 100, 110, 120, 130, 140, 150 r 8% σ 50% q 0,5%,20%,60%,100% t in years 1 2 figure 1. the effect of dy on the price of eppo with ρ =1, q =0 int. j. anal. appl. (2023), 21:29 9 figure 2. the effect of dy on the price of eppo with ρ =1, q =0.05 figure 3. the effect of dy on the price of eppo with ρ =1, q =0.2 10 int. j. anal. appl. (2023), 21:29 figure 4. the effect of dy on the price of eppo with ρ =1, q =0.6 figure 5. the effect of dy on the price of eppo with ρ =1, q =1.0 int. j. anal. appl. (2023), 21:29 11 figure 6. the comparative study of eppo price with different dy. figure 7. the plots of linear payoff. 12 int. j. anal. appl. (2023), 21:29 figure 8. the plots of mlppf for ρ =1. figure 9. the plots of log payoff. int. j. anal. appl. (2023), 21:29 13 figure 10. the comparative study of linear payoff, mlppf and log payoff. 6. conclusion an analytic solution of bsmeppom via the celebrated transform of mellin type in the sense of dy and mlppf has been proposed in this paper. the integral equation for the representation of the price of eppo with dy was obtained. the closed form approximation formula for eppo was also obtained via mt with the help of its convolution property and final time condition. moreover, the mt was tested on some parameters to show its performance and suitability. the effect of dy is captured in figures 1 -5. from figure 6, it is observed that increase in dy leads to increase in the prices of the eppo with mlppf. it also is observed from figures 6, that the holder is more beneficial to enter into a european power put option. in other words, however, the benefits of these cash flows are given to the holder of a put option. the plots of linear payoff, mlppf and log payoff are displayed in figures 7, 8 and 9, respectively. from figure 10, it is clearly seen that the mlppf used in this present paper performed better than the log payoff used in [22] and also was found to be very close to the linear payoff of plain vanilla [1]. hence, from the results displayed in figures 6 and 10, it can be concluded that mt is suitable for the valuation of eppo with mlppf and dy due to its capacity power of solving bsmeppom directly in terms of market price. acknowledgements: the authors wish to thank tshwane university of technology for their financial support and the department of higher education and training, south africa. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 14 int. j. anal. appl. (2023), 21:29 references [1] f. black, m. scholes, the pricing of options and corporate liabilities, j. political econ. 81 (1973), 637-654. [2] s.e. fadugba, c.r. nwozo, valuation of european call options via the fast fourier transform and the improved mellin transform, j. math. finance. 06 (2016), 338–359. https://doi.org/10.4236/jmf.2016.62028. [3] r.c. merton, option pricing when underlying stock returns are discontinuous, j. financial econ. 3 (1976), 125–144. https://doi.org/10.1016/0304-405x(76)90022-2. [4] m.j. brennan, e.s. schwartz, finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, j. financial quant. anal. 13 (1978), 461-474. https://doi.org/10.2307/2330152. [5] j.c. cox, s.a. ross, m. rubinstein, option pricing: a simplified approach, j. financial econ. 7 (1979), 229–263. https://doi.org/10.1016/0304-405x(79)90015-1. [6] p. boyle, m. broadie, p. glasserman, monte carlo methods for security pricing, j. econ. dyn. control. 21 (1997), 1267–1321. https://doi.org/10.1016/s0165-1889(97)00028-6. [7] f.s. emmanuel, the mellin transform method as an alternative analytic solution for the valuation of geometric asian option, appl. comput. math. 3 (2014), 1-7. https://doi.org/10.11648/j.acm.s.2014030601.11. [8] d.j. manuge, p.t. kim, a fast fourier transform method for mellin-type option pricing, arxiv:1403.3756, (2014). https://doi.org/10.48550/arxiv.1403.3756. [9] c.r. nwozo, s.e. fadugba, mellin transform method for the valuation of some vanilla power options with non-dividend yield, int. j. pure appl. math. 96 (2014), 79-104. https://doi.org/10.12732/ijpam.v96i1.7. [10] c.r. nwozo, s.e. fadugba, performance measure of laplace transforms for pricing path dependent options, int. j. pure appl. math. 94 (2014), 175-197. https://doi.org/10.12732/ijpam.v94i2.5. [11] s.e. fadugba, c.r. nwozo, mellin transform method for the valuation of the american power put option with non-dividend and dividend yields, j. math. finance. 05 (2015), 249–272. https://doi.org/10.4236/jmf. 2015.53023. [12] s.e. fadugba, solution of fractional order equations in the domain of mellin transform, j. nigerian soc. phys. sci. 1 (2019), 138-142. [13] s.e. fadugba, c.r. nwozo, closed-form solution for the critical stock price and the price of perpetual american call options via the improved mellin transforms, int. j. financial markets derivatives. 6 (2018), 269-286. https: //doi.org/10.1504/ijfmd.2018.097489. [14] s.e. fadugba, c.t. nwozo, perpetual american power put options with non-dividend yield in the domain of mellin transforms, palestine j. math. 9 (2020), 371-385. [15] s.e. fadugba, laplace transform for the solution of fractional black-scholes partial differential equation for the american put options with non-dividend yield, int. j. stat. econ. 20 (2019), 10-17. [16] s.j. ghevariya, bsm european option pricing formula for ml-payoff function with mellin transform, int. j. math. appl. 6 (2018), 34-36. [17] s.e. fadugba, a.a. adeniji, m.c. kekana, j.t. okunlola, o. faweya, direct solution of black-scholes-merton european put option model on dividend yield with modified-log payoff function, int. j. anal. appl. 20 (2022), 54. https://doi.org/10.28924/2291-8639-20-2022-54. [18] p. flajolet, x. gourdon, p. dumas, mellin transforms and asymptotics: harmonic sums, theor. computer sci. 144 (1995), 3–58. https://doi.org/10.1016/0304-3975(95)00002-e. [19] g. fikioris, mellin-transform method for integral evaluation: introduction and applications to electromagnetics, springer international publishing, cham, 2007. https://doi.org/10.1007/978-3-031-01697-4. [20] a.h. zemanian, generalized integral transformation, dover publications, new york, 1987. [21] a. erdélyi, w. magnus, f. oberhettinger, f. tricomi, tables of integral transforms, vol. 1-2, first edition, mcgraw-hill, new york, 1954. https://doi.org/10.4236/jmf.2016.62028 https://doi.org/10.1016/0304-405x(76)90022-2 https://doi.org/10.2307/2330152 https://doi.org/10.1016/0304-405x(79)90015-1 https://doi.org/10.1016/s0165-1889(97)00028-6 https://doi.org/10.11648/j.acm.s.2014030601.11 https://doi.org/10.48550/arxiv.1403.3756 https://doi.org/10.12732/ijpam.v96i1.7 https://doi.org/10.12732/ijpam.v94i2.5 https://doi.org/10.4236/jmf.2015.53023 https://doi.org/10.4236/jmf.2015.53023 https://doi.org/10.1504/ijfmd.2018.097489 https://doi.org/10.1504/ijfmd.2018.097489 https://doi.org/10.28924/2291-8639-20-2022-54 https://doi.org/10.1016/0304-3975(95)00002-e https://doi.org/10.1007/978-3-031-01697-4 int. j. anal. appl. (2023), 21:29 15 [22] p. wilmott, paul wilmott on quantitative finance, john wiley & sons, ltd., second edition, 2006. 1. introduction 2. mellin transform 3. the bsmeppom on a dy with mlppf 4. solution of the black-scholes-merton european put option model with mlppf 5. numerical example 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 1, number 1 (2013), 1-17 http://www.etamaths.com generalized ulam-hyers stability of the harmonic mean functional equation in two variables k. ravi1,∗, j.m. rassias2 and b.v. senthil kumar3 abstract. in this paper, we find the solution and prove the generalized ulamhyers stability of the harmonic mean functional equation in two variables. we also provide counterexamples for singular cases. 1. introduction in 1940, s.m. ulam [47] raised the following question concerning the stability of group homomorphisms: “let g be a group and h be a metric group with metric d(., .). given � > 0, does there exist a δ > 0 such that if a function f : g → h satisfies d(f(xy),f(x)f(y)) < δ for all x,y ∈ g, then there exists a homomorphism a : g → h with d(f(x),a(x)) < � for all x ∈ g?” in 1941, d.h. hyers [20] gave an answer to the ulam’s stability problem. he proved the following celebrated theorem. theorem 1.1. [d.h. hyers] let x,y be banach spaces and let f : x → y be a mapping satisfying (1.1) ‖f(x + y) −f(x) −f(y)‖≤ � 2010 mathematics subject classification. 39b22, 39b52, 39b72. key words and phrases. harmonic mean; additive functional equation; reciprocal functional equation; generalized hyers-ulam stability. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 k. ravi, j.m. rassias and b.v. senthil kumar for all x,y ∈ x. then the limit (1.2) a(x) = lim n→∞ f(2nx) 2n exists for all x ∈ x and a : x → y is the unique additive mapping satisfying (1.3) ‖f(x) −a(x)‖≤ � for all x ∈ x. in 1950, aoki [2] generalized the hyers theorem for additive mappings. in 1978, th.m. rassias [43] provided a generalized version of the theorem of hyers which permitted the cauchy difference to become unbounded. th.m. rassias proved the following theorem for sum of powers of norms. theorem 1.2. [th.m. rassias] let x and y be two banach spaces. let θ ∈ [0,∞) and let p ∈ [0, 1). if a function f : x → y satisfies the inequality (1.4) ‖f(x + y) −f(x) −f(y)‖≤ θ(‖x‖p + ‖y‖p) for all x,y ∈ x. then there exists a unique linear mapping t : x → y such that (1.5) ‖f(x) −t(x)‖≤ 2θ 2 − 2p ‖x‖p for all x ∈ x. moreover, if f(tx) is continuous in t for each fixed x ∈ x, then the function t is linear. the theorem of th.m. rassias was later extended for all p 6= 1. the stability phenomenon that was presented by th.m. rassias is called the generalized hyersulam stability. in 1982, j.m. rassias [33] gave a further generalization of the result of d.h. hyers and proved the following theorem using weaker conditions controlled by a product of powers of norms. theorem 1.3. [j.m. rassias] let f : e → e′ be a mapping from a normed vector space e into a banach space e′ subject to the inequality (1.6) ‖f(x + y) −f(x) −f(y)‖≤ �‖x‖p‖y‖p generalized ulam-hyers stability 3 for all x,∈ e, where � and p are constants with � > 0 and 0 ≤ p < 1 2 . then the limit (1.7) l(x) = lim n→∞ f(2nx) 2n exists for all x ∈ e and l : e → e′ is the unique additive mapping which satisfies (1.8) ‖f(x) −l(x)‖≤ � 2 − 22p ‖x‖2p for all x ∈ e. moreover, if f(tx) is continuous in t for each fixed x of x, then the function l is linear. the above mentioned stability involving a product of powers of norms is called ulam-gavruta-rassias stability by various authors ([4], [5], [6], [9], [11], [18], [19], [30], [31], [37], [38], [41], [45], [46]). very recently, j.m. rassias introduced mixed type product sum of powers of norms in [38]. the investigation of stability of functional equations involving with the mixed type product sum of powers of norms is known as j.m. rassias stability. several functional equations and its j.m. rassias stability were investigated by many mathematicians ([5], [6], [18], [19], [39], [41], [45]). theorem 1.4. [42] let (e,⊥) be an orthogonality normed space with norm ‖.‖e and (f,‖.‖f ) be a banach space. let f : e → f be a mapping which satisfies the inequality∥∥f(mx + y) + f(mx−y) − 2f(x + y) − 2f(x−y) − 2(m2 − 2)f(x) + 2f(y)∥∥ f ≤ � { ‖x‖pe ‖y‖ p e + ( ‖x‖2pe + ‖y‖ 2p e )} for all x,y ∈ e with x⊥y, where � and p are constants with �,p > 0 and either m > 1; p < 1 or m < 1; p > 1 with m 6= 0;m 6= ±1; m 6= ± √ 2 and −1 6= |m|p−1 < 1. then the limit q(x) = lim n→∞ f(mnx) m2n exists for all x ∈ e and q : e → f is the unique orthogonally euler-lagrange quadratic mapping such that ‖f(x) −q(x)‖f ≤ � 2 |m2 −m2p| ‖x‖2pe for all x ∈ e. 4 k. ravi, j.m. rassias and b.v. senthil kumar in the last two decades, several form of quadratic, cubic and quartic functional equations and its hyers-ulam-rassias stability were discussed by various authors ([3], [7], [12], [21], [24], [27], [29], [32], [34], [35]). recently, the mixed type of functional equations that is having additive and quadratic, quadratic and quartic, additive and cubic, additive, quadratic, cubic and quartic property were investigated in the literature ([8], [13], [14], [15], [16], [17], [18], [22], [26], [28], [36], [41], [42], [44]). very recently, we find that the functional equations are dealt in various spaces like fuzzy normed spaces, random normed spaces, quasi-banach spaces, quasi-normed linear spaces by various authors, one can refer to ([10], [15], [16], [17], [18], [23], [25], [26], [42], [44]). but, not many papers are available in the reciprocal functional equation (rfe) and reciprocal difference functional equation (rdfe). k. ravi and b.v. senthil kumar [39] introduced and investigated a new 2-dimensional reciprocal functional equation (rfe) (1.9) f(x + y) = f(x)f(y) f(x) + f(y) and obtained some interesting results on the ulam-gavruta-rassias stability for the equation (1.9). in 2009, j.m. rassias introduced the reciprocal difference functional equation (rdf equation) (1.10) r ( x + y 2 ) −r(x + y) = r(x)r(y) r(x) + r(y) and the reciprocal adjoint functional equation (raf equation) (1.11) r ( x + y 2 ) + r(x + y) = 3r(x)r(y) r(x) + r(y) . and discussed ulam stability problem as well as the extended ulam stability problem for the equation (1.10) and (1.11) in [40]. in this paper, we obtain the solution and generalized ulam-hyers stability of the harmonic mean functional equation in two variables of the form (1.12) h ( xu x + u , yv y + v ) = h(x,y)h(u,v) h(x,y) + h(u,v) which is originating from the harmonic mean of two positive real numbers x and y. the function h(x,y) = 2xy x+y is a solution of the functional equation (1.12). the above function h(x,y) represents the harmonic mean of x and y. we also provide counterexamples for singular cases. generalized ulam-hyers stability 5 before we proceed to the main theorem, we present some definitions, which will be useful to prove our theorems. definition 1.5. the harmonic mean of two positive real numbers x and y is the reciprocal of arithmetic mean of the reciprocals of x and y. thus, harmonic mean of x and y is harmonic mean = 2 1 x + 1 y = 2xy x + y . definition 1.6. a mapping a : x → y between banach spaces is called additive if a satisfies the functional equation (1.13) a(x + y) = a(x) + a(y). the function a(x) = cx is a solution of the functional equation (1.13). definition 1.7. a function r : x → y between sets of non-zero real numbers is called reciprocal if r satisfies the functional equation (1.9). the function r(x) = c x is a solution of the functional equation (1.9). definition 1.8. let x be set of positive real numbers. a function h : x×x → r be such that h(x,y) = 2xy x+y is called harmonic mean function if it satisfies the functional equation (1.12). the functional equation (1.12) is called harmonic mean functional equation. throughout this paper, let x be set of positive real numbers. 2. relation between (1.9) and (1.12) theorem 2.1. let r : x → r be a mapping satisfying r(x) = c x where c ∈ r−{0}. if h : x ×x → r is a mapping given by (2.1) h(x,y) = 2 r(x) + r(y) for all x,y ∈ x, then h satisfies (1.12). 6 k. ravi, j.m. rassias and b.v. senthil kumar proof. from (2.1), we have h ( xu x + u , yv y + v ) = 2 r ( xu x+u ) + r ( yv y+v ) = 2 c [ x+u xu ] + c [ y+v yv ] = 2/c 1 x + 1 u + 1 y + 1 v (2.2) and h(x,y)h(u,v) h(x,y) + h(u,v) = 2 r(x)+r(y) 2 r(u)+r(v) 2 r(x)+r(y) + 2 r(u)+r(v) = 2 r(x) + r(u) + r(y) + r(v) = 2/c 1 x + 1 u + 1 y + 1 v .(2.3) from (2.2) and (2.3), we see that h satisfies (1.12). � 3. solution of functional equation (1.12) theorem 3.1. a mapping h : x × x → r satisfies (1.12) if and only if there exists an additive mapping a : x → r such that h(x,y) = 2ca(x)a(y) a(x) + a(y) for all x,y ∈ x, where c ∈ r−{0}. proof. let h(x,y) be a solution of (1.12). then by theorem 2.1, h will be of the form (2.1). that is, h(x,y) = 2 r(x)+r(y) , for all x,y ∈ x. in particular, h(x,x) = 2 r(x)+r(x) = x c , for all x ∈ x. now, we define a mapping a : x → r be such that a(x) = h(x,x) = x c , for all x ∈ x. then, clearly a is an additive mapping. further, 2ca(x)a(y) a(x) + a(y) = 2cx c y c x c + y c = 2xy x + y = h(x,y), for all x,y ∈ x. generalized ulam-hyers stability 7 conversely, if there exists an additive mapping a : x → r such that h(x,y) = 2ca(x)a(y) a(x)+a(y) , for all x,y ∈ x, then h ( xu x + u , yv y + v ) = 2ca ( xu x+u ) a ( yv y+v ) a ( xu x+u ) + a ( yv y+v ) = 2 xy x+y uv u+v xy x+y + uv u+v = 2 ( c x c y c x c + y c )( c u c v c u c +v c ) ( c x c y c x c + y c ) + ( c u c v c u c +v c ) = [ 2ca(x)a(y) a(x)+a(y) ][ 2ca(u)a(v) a(u)+a(v) ] [ 2ca(x)a(y) a(x)+a(y) ] + [ 2ca(u)a(v) a(u)+a(v) ] = h(x,y)h(u,v) h(x,y) + h(u,v) for all x,u,y,v ∈ x. � 4. generalized ulam-hyers stability of functional equation (1.12) theorem 4.1. let h : x×x → r be a mapping for which there exists a function ϕ : x ×x ×x ×x → r with the condition (4.1) lim n→∞ 2nϕ(2−nx, 2−nx, 2−ny, 2−ny) = 0 such that the functional inequality (4.2) ∣∣∣h( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣ ≤ ϕ(x,u,y,v) holds for all x,u,y,v ∈ x. then there exists a unique harmonic mean mapping m : x ×x → r satisfying the functional equation (1.12) and (4.3) |m(x,y) −h(x,y)| ≤ ∞∑ i=0 2i+1ϕ(2−ix, 2−ix, 2−iy, 2−iy) for all x,y ∈ x. the mapping m(x,y) is defined by m(x,y) = lim n→∞ 2nh(2−nx, 2−ny) for all x,y ∈ x. 8 k. ravi, j.m. rassias and b.v. senthil kumar proof. replacing (x,u,y,v) by (x,x,y,y) in (4.2) and multiplying by 2, we obtain (4.4) ∣∣∣2h(x 2 , y 2 ) −h(x,y) ∣∣∣ ≤ 2ϕ(x,x,y,y). now, replacing (x,y) by (x 2 , y 2 ) in (4.4), multiplying by 2 and summing the resulting inequality with (4.4), we arrive |22h ( 2−2x, 2−2y ) −h(x,y)| ≤ 2 1∑ i=0 2iϕ(2−ix, 2−ix, 2−iy, 2−iy). using induction on a positive integer n, we get |2nh ( 2−nx, 2−ny ) −h(x,y)| ≤ 2 n−1∑ i=0 2iϕ(2−ix, 2−ix, 2−iy, 2−iy) ≤ 2 ∞∑ i=0 2iϕ(2−ix, 2−ix, 2−iy, 2−iy)(4.5) for all x,y ∈ x. in order to prove the convergence of the sequence {2nh(2−nx, 2−ny)}, replacing (x,y) by (2−px, 2−py) in (4.5) and multiplying by 2p, we find that for n > p > 0 |2n+ph(2−n−px, 2−n−py) − 2ph(2−px, 2−py)| = 2p|2nh(2−n−px, 2−n−py) −h(2−px, 2−py)| ≤ 2 ∞∑ i=0 2p+iϕ(2−p−ix, 2−p−ix, 2−p−iy, 2−p−iy) → 0 as p →∞. this shows that the sequence {2nh(2−nx, 2−ny)} is a cauchy sequence in (r, |.|). since (r, |.|) is complete, the sequence {2nh(2−nx, 2−ny)} converges for all x,y ∈ x. thus, we can define a function m : x×x → r by m(x,y) = lim n→∞ 2nh(2−nx, 2−ny). allow n → ∞ in (4.5), we arrive (4.3). to show that m satisfies (1.12), replacing (x,u,y,v) by (2−nx, 2−nu, 2−ny, 2−nv) in (4.2) and multiplying by 2n, we obtain 2n ∣∣∣h( 2−nx2−nu 2−nx + 2−nu , 2−ny2−nv 2−ny + 2−nv ) − h(2−nx, 2−ny)h(2−nu, 2−nv) h(2−nx, 2−ny) + h(2−nu, 2−nv) ∣∣∣ = 2n ∣∣∣h(2−n( xu x + u ) , 2−n ( yv y + v )) − h(2−nx, 2−ny)h(2−nu, 2−nv) h(2−nx, 2−ny) + h(2−nu, 2−nv) ∣∣∣ ≤ 2nϕ(2−nx, 2−nu, 2−ny, 2−nv). (4.6) generalized ulam-hyers stability 9 allow n →∞ in (4.6), we see that m satisfies (1.12) for all x,u,y,v ∈ x. to prove m is unique mapping satisfying (1.12), let n : x × x → r be another mapping which satisfies (1.12) and the inequality (4.3). clearly n and m satisfy (1.12) and using (4.3), we arrive |n(x,y) −m(x,y)| = 2n|n(2−nx, 2−ny) −m(2−nx, 2−ny)| ≤ 2n ( |n(2−nx, 2−ny) −h(2−nx, 2−ny)| + |h(2−nx, 2−ny) −m(2−nx, 2−ny)| ) ≤ 4 ∞∑ i=0 2n+iϕ(2−n−ix, 2−n−ix, 2−n−iy, 2−n−iy) (4.7) for all x,y ∈ x. allow n → ∞ in (4.7) and using (4.1), we find that m is unique. this completes the proof of theorem 4.1. � theorem 4.2. let h : x×x → r be a mapping satisfying (4.2) for all x,u,y,v ∈ x. then there exists a unique harmonic mapping m : x ×x → r satisfying the functional equation (1.12) and (4.8) |h(x,y) −m(x,y)| ≤ ∞∑ i=0 2−iϕ(2i+1x, 2i+1x, 2i+1y, 2i+1y) for all x,y ∈ x. the mapping m(x,y) is defined by m(x,y) = lim n→∞ 2−nh(2nx, 2ny) for all x,y ∈ x, where ϕ : x ×x ×x ×x → r is a function. proof. the proof is obtained by replacing (x,u,y,v) by (x,x,y,y) in (4.2) and proceeding further by similar arguments as in theorem 4.1. � corollary 4.3. let k1 ≥ 0 be fixed and a > 1 or a < 1. if a mapping h : x×x → r satisfies the inequality∣∣∣h( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣ ≤ k1(xa + ua + ya + va) for all x,u,y,v ∈ x, then there exists a unique harmonic mean mapping m : x ×x → r satisfying the functional equation (1.12) and |m(x,y) −h(x,y)| ≤   4k1 ( 2a 2a−2 )( xa + ya ) for a > 1 4k1 ( 2a 2−2a )( xa + ya ) for a < 1 10 k. ravi, j.m. rassias and b.v. senthil kumar for all x,y ∈ x. proof. choosing ϕ(x,u,y,v) = k1 ( xa + ua + ya + va ) , for all x,u,y,v ∈ x in theorem 4.1, we arrive |m(x,y) −h(x,y)| ≤ 4k1 ( 2a 2a − 2 )( xa + ya ) , for all x,y ∈ x and a > 1 and using theorem 4.2, we arrive |m(x,y) −h(x,y)| ≤ 4k1 ( 2a 2 − 2a )( xa + ya ) , for all x,y ∈ x and a < 1. � corollary 4.4. let h : x × x → r be a mapping and there exist real numbers p,q : σ = p + q > 1 2 or p,q : σ = p + q < 1 2 . if there exists k2 such that∣∣∣h( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣ ≤ k2xpupyqvq for all x,u,y,v ∈ x, then there exists a unique harmonic mean mapping m : x ×x → r satisfying the functional equation (1.12) and |m(x,y) −h(x,y)| ≤   2k2 ( 22σ 22σ−2 ) x2py2q for σ > 1 2 2k2 ( 22σ 2−22σ ) x2py2q for σ < 1 2 for all x,y ∈ x. proof. taking ϕ(x,u,y,v) = k2x pupyqvq, for all x,u,y,v ∈ x in theorem 4.1, we arrive |m(x,y) −h(x,y)| ≤ 2k2 ( 22σ 22σ − 2 ) x2py2q, for all x,y ∈ x and σ > 1 2 and using theorem 4.2, we arrive |m(x,y) −h(x,y)| ≤ 2k2 ( 22σ 2 − 22σ ) x2py2q, for all x,y ∈ x and σ < 1 2 . � corollary 4.5. let k3 > 0 and r > 1 4 or r < 1 4 be real numbers, and h : x×x → r be a mapping satisfying the functional inequality∣∣∣h( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣ ≤ k3 ( xruryrvr + ( x4r + u4r + y4r + v4r )) generalized ulam-hyers stability 11 for all x,u,y,v ∈ x. then there exists a unique harmonic mean mapping m : x ×x → r satisfying the functional equation (1.12) and |m(x,y) −h(x,y)| ≤   2k3 ( 24r 24r−2 )[ x2ry2r + 2 ( x4r + y4r )] for r > 1 4 2k3 ( 24r 2−24r )[ x2ry2r + 2 ( x4r + y4r )] for r < 1 4 for all x,y ∈ x. proof. choosing ϕ(x,u,y,v) = k3 ( xruryrvr + ( x4r + u4r + y4r + v4r )) , for all x,u,y,v ∈ x in theorem 4.1, we arrive |m(x,y) −h(x,y)| ≤ 2k3 ( 24r 24r − 2 )[ x2ry2r + 2 ( x4r + y4r )] for all x,y ∈ x and r > 1 4 and using theorem 4.2, we arrive |m(x,y) −h(x,y)| ≤ 2k3 ( 24r 2 − 24r )[ x2ry2r + 2 ( x4r + y4r )] for all x,y ∈ x and r < 1 4 . � corollary 4.6. let k4 > 0 and p > 1 2 or p < 1 2 be real numbers, and h : x×x → r be a mapping satisfying the functional inequality∣∣∣∣h ( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣∣ ≤ k4 ( xpupypvp xpup + ypvp ) for all x,u,y,v ∈ x. then there exists a unique harmonic mean mapping m : x ×x → r satisfying the functional equation (1.12) and |m(x,y) −h(x,y)| ≤   2k4 ( 22p 22p−2 )( x2py2p x2p+y2p ) for p > 1 2 2k4 ( 22p 2−22p )( x2py2p x|2p+y2p ) for p < 1 2 for all x,y ∈ x. proof. choosing ϕ(x,u,y,v) = k4 ( xpupypvp xpup+ypvp ) , for all x,u,y,v ∈ x in theorem 4.1, we arrive |m(x,y) −h(x,y)| ≤ 2k4 ( 22p 22p − 2 )( x2py2p x2p + y2p ) for all x,y ∈ x and p > 1 2 and using theorem 4.2, we arrive |m(x,y) −h(x,y)| ≤ 2k4 ( 22p 2 − 22p )( x2py2p x2p + y2p ) for all x,y ∈ x and p < 1 2 . 12 k. ravi, j.m. rassias and b.v. senthil kumar � 5. counter-examples now, we give an example to show that the functional equation (1.12) is not stable for p = 1 2 in corollary 4.6. let us define a function h : x ×x → r be h(x,y) = ∞∑ n=0 φ(2nx, 2ny) 2n , for all x,y ∈ x where the function φ : x ×x → r is given by φ(x,y) =   αxy x+y for x,y ∈ (0, 1) α otherwise with α > 0 is a constant. then the function h satisfies the inequality (5.1) ∣∣∣∣h ( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣∣ ≤ 6α ( x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 + y 1 2 v 1 2 ) for all x,u,y,v ∈ x. then there does not exist a harmonic mapping m : x×x → r and a constant λ > 0 such that (5.2) |h(x,y) −m(x,y)| ≤ λ ( xy x + y ) for all x,y ∈ x. proof. |h(x,y)| ≤ ∑∞ n=0 |φ(2nx,2ny)| |2n| ≤ ∑∞ n=0 α 2n = α ( 1 − 1 2 )−1 = 2α. hence h is bounded by 2α. if x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 +y 1 2 v 1 2 ≥ 1, then the left hand side of the inequality (5.1) is less than 6α. now, consider the case: 0 < x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 +y 1 2 v 1 2 < 1. then there exists a k ∈ n such that (5.3) 1 2k+1 < x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 + y 1 2 v 1 2 < 1 2k . hence x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 +y 1 2 v 1 2 < 1 2k implies 2kx 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 + y 1 2 v 1 2 < 1 or 2 k 2 x 1 2 2 k 2 u 1 2 2 k 2 y 1 2 2 k 2 v 1 2 2 k 2 x 1 2 2 k 2 u 1 2 +2 k 2 y 1 2 2 k 2 v 1 2 < 1 or 2 k 2 x 1 2 2 k 2 u 1 2 2 k 2 y 1 2 2 k 2 v 1 2 < 2 k 2 x 1 2 2 k 2 u 1 2 + 2 k 2 y 1 2 2 k 2 v 1 2 generalized ulam-hyers stability 13 or 1 < 2 k 2 x 1 2 2 k 2 u 1 2 +2 k 2 y 1 2 2 k 2 v 1 2 2 k 2 x 1 2 2 k 2 u 1 2 2 k 2 y 1 2 2 k 2 v 1 2 or 1 < 1 2 k 2 x 1 2 1 2 k 2 u 1 2 + 1 2 k 2 y 1 2 1 2 k 2 v 1 2 . therefore, 1 2 k 2 x 1 2 1 2 k 2 u 1 2 + 1 2 k 2 y 1 2 1 2 k 2 v 1 2 > 1. hence 1 2 k 2 x 1 2 > 1, 1 2 k 2 u 1 2 > 1, 1 2 k 2 y 1 2 > 1, 1 2 k 2 v 1 2 > 1 which implies 2 k 2 x 1 2 < 1, 2 k 2 u 1 2 < 1, 2 k 2 y 1 2 < 1, 2 k 2 v 1 2 < 1 so that 2 k−1 2 x 1 2 < 1 √ 2 , 2 k−1 2 u 1 2 < 1 √ 2 , 2 k−1 2 y 1 2 < 1 √ 2 , 2 k−1 2 v 1 2 < 1 √ 2 , 2 k−1 2 ( xu x + u )1 2 < 1 √ 2 , 2 k−1 2 ( yv y + v )1 2 < 1 √ 2 and consequently 2k−1(x), 2k−1(u), 2k−1(y), 2k−1(v), 2k−1 ( xu x + u ) , 2k−1 ( yv y + v ) ∈ (0, 1). therefore, for each n = 0, 1, 2, . . . ,k − 1, we have 2n(x), 2n(u), 2n(y), 2n(v), 2n ( xu x + u ) , 2n ( yv y + v ) ∈ (0, 1) and φ ( 2n ( xu x + u ) , 2n ( yv y + v )) − φ (2n(x), 2n(y)) φ (2n(u), 2n(v)) φ (2n(x), 2n(y)) + φ (2n(u), 2n(v)) = 0 for n = 0, 1, 2, . . . ,k − 1. using (5.3) and definition of h, we obtain∣∣∣∣h ( xu x + u , yv y + v ) − h(x,y)h(u,v) h(x,y) + h(u,v) ∣∣∣∣ ≤ ∞∑ n=0 1 2n ∣∣∣∣φ ( 2n ( xu x + u ) , 2n ( yv y + v )) − φ (2n(x), 2n(y)) φ (2n(u), 2n(v)) φ (2n(x), 2n(y)) + φ (2n(u), 2n(v)) ∣∣∣∣ ≤ ∞∑ n=k 1 2n ∣∣∣∣φ ( 2n ( xu x + u ) , 2n ( yv y + v )) − φ (2n(x), 2n(y)) φ (2n(u), 2n(v)) φ (2n(x), 2n(y)) + φ (2n(u), 2n(v)) ∣∣∣∣ ≤ ∞∑ n=k 1 2n 3 2 α ≤ 6α 2k+1 ≤ 6α ( x 1 2 u 1 2 y 1 2 v 1 2 x 1 2 u 1 2 + y 1 2 v 1 2 ) , 14 k. ravi, j.m. rassias and b.v. senthil kumar that is the inequality (5.1) holds true. we claim that the harmonic mean functional equation (1.12) is not stable for p = 1 2 in corollary 4.6. assume that there exists a harmonic mean mapping m : x ×x → r satisfying (5.2). therefore, we have (5.4) |h(x,y)| ≤ (λ + 2) ( xy x + y ) . but we can choose a positive integer m with mα > λ + 2. if x ∈ ( 0, 21−m ) , then 2nx ∈ (0, 1) for all n = 0, 1, 2, . . . ,m− 1. for this x, we get h(x,y) = ∞∑ n=0 φ(2nx, 2ny) 2n ≥ m−1∑ n=0 α ( 2nxy xy ) 2n = mα ( xy x + y ) > (λ + 2) ( xy x + y ) which in comparison with (5.4) is a contradiction. therefore, the harmonic mean functional equation (1.12) is not stable if p = 1 2 in corollary 4.6. � references [1] j. aczel and j. dhombres, functional equations in several variables, cambridge univ. press, 1989. [2] t. aoki, on the stability of the linear transformation in banach spaces, j. math.soc. japan, 2(1950), 64-66. [3] j.h. bae and k.w. jun, on the generalized hyers-ulam-rassias stability of quadratic functional equation, bull. koeran. math. soc. 38(2001), 325-336. [4] b. bouikhalene and e. elqorachi, ulam-gavruta-rassias stability of the pexider functional equation, int. j. math. stat. 7(2007), 27-39. [5] h.x. cao, j.r. lv and j.m. rassias, superstability for generalized module left derivations and generalized module derivations on a banach module(i), j. ineq. and appl. vol.2009, article id 718020, 1-10. [6] h.x. cao, j.r. lv and j.m. rassias, superstability for generalized module left derivations and generalized module derivations on a banach module(ii), j. pure & appl. math. 10(2)(2009), 1-8. [7] i.s. chang and h.m. kim, on the hyers-ulam stability of quadratic functional equations, j. ineq. appl. math. 33(2002), 1-12. [8] i.s. chang and y.s. jung, stability of functional equations deriving from cubic and quadratic functions, j. math. anal. appl. 283(2003), 491-500. [9] ch.park, j.s. an and j. cui, isomorphisms and derivations in lie c*-algebras, abstract and applied analysis, vol.2007, article id 85737, 14 pages. generalized ulam-hyers stability 15 [10] ch.park, hyers-ulam-rassias stability of homomorphisms in quasi-banach algebras, bull. sci. math. 132(2) (2008), 87-96. [11] c.g. park and j.m. rassias, hyers-ulam stability of an euler-lagrange type additive mapping, int. j. math. stat. 7(2007), 112-125. [12] s. czerwik, on the stability of the quadratic mappings in normed spaces, abh. math. sem. univ. hamburg. 62 (1992), 59-64. [13] m. eshaghi gordji, a. ebadian and s. zolfaghari, stability of a functional equation deriving from cubic and quartic functions, abstract and applied analysis, volume 2008, article id 801904, 17 pages. [14] m. eshaghi gordji, s. kaboli and s. zolfaghari, stability of a mixed type quadratic, cubic and quartic functional equations, arxiv:0812.2939v1 math fa, 15 dec 2008. [15] m. eshaghi gordji and m. b. savadkouhi, stability of cubic and quartic functional equation in non-archimedean spaces, acta appl. math. doi 10.1007/s10440-009-9512-7, 2009. [16] m. eshaghi gordji and h. khodaei, solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-banach spaces, nonlinear analysis, 71(2009), 5629-5643. [17] m. eshaghi gordji, h. khodaei and th.m. rassias, on the hyers-ulam-rassias stability of a generalized mixed type quartic, cubic, quadratic and additive functional equations in quasic-banach spaces, arxiv:0903.0834v2 math fa, 24 apr 2009. [18] m. eshaghi gordji, s. zolfaghari, j.m. rassias and m.b. savadkouhi, solution and stability of a mixed type cubic and quartic functional equation in quasi-banach spaces, abstract and applied analysis, volume 2009, article id 417473, 1-14. [19] m. eshaghi gordji and h. khodaei, on the generalized hyers-ulam-rassias stability of quadratic functional equations, abstract and applied analysis, vol.2009, article id 923476, 1-11. [20] d.h. hyers, on the stability of the linear functional equation, proc. nat. acad. sci. u.s.a. 27(1941), 222-224. [21] k.w. jun and h.m. kim, on the hyers-ulam-rassias stabillity of a general cubic functional equation, math. ineq. appl. vol.6(2) (2003), 289-302. [22] k.w. jun and h.m. kim, ulam stabillity problem for a mixed type of cubic and additive functional equation, bull. belg. math. soc. 13(2006), 271-285. [23] k.w. jun and h.m. kim, on the stabillity of euler-lagrange type cubic mappings in quasibanach spaces, j. math. anal. appl. 332(2007), 1335-1350. [24] s.m. jung, on the hyers-ulam -rassias stability of a quadratic functional equation, j. math. anal. appl. 232(1999), 384-393. [25] d. mihet and v. radu, on the stability of the additive cauchy functional equation in random normed spaces, j. math. anal. appl. 343(2008), 567-572. [26] f. moradlou, h. vaezi and g.z. eskandani, hyers-ulam-rassias stability of a quadratic and additive functional equation in quasi-banach spaces, mediterr. j. math. 6(2009), 233-248. [27] a. najati, hyers-ulam-rassias stability of a cubic functional equation, bull. korean math. soc. 44(4) (2007), 825-840. 16 k. ravi, j.m. rassias and b.v. senthil kumar [28] a. najati and m.b. moghimi, on the stability of a quadratic and additive functional equation, j. math. anal. appl. 337(2008), 399-415. [29] a. najati and ch. park, on the stability of a cubic functional equation, acta mathematica sinica, english series, vol.24(12) (2008), 1953-1964. [30] p. nakmahachalasint, hyers-ulam-rassias and ulam-gavruta-rassias stabilities of an additive functional equation in several variables, internat. j. math. and math. sc. (ijmms) article id 13437, 6 pages, 2007. [31] p. nakmahachalasint, on the generalized ulam-gavruta-rassias stability of mixed-type linear and euler-lagrange-rassias functional equations, internat. j. math. and math. sc. (ijmms) article id 63239, 10 pages, 2007. [32] m. petapirak and p. nakmahachalasint, a quartic functional equation and its generalized hyers-ulam-rassias stability, thai journal of mathematics, special issue(annual meeting in mathematics, 2008), 77-84. [33] j.m. rassias, on approximation of approximately linear mappings by linear mappings, j. funct. anal. 46(1982), 126-130. [34] j.m. rassias, hyers-ulam stability of the quadratic functional equation in several variables, j. ind. math. soc. 68(1992), 65-73. [35] j.m. rassias, solution of the ulam stability problem for quartic mappings, glasnic mathematicki, vol.34(54) (1999), 243-252. [36] j.m. rassias, k. ravi, m. arunkumar and b.v. senthil kumar, solution and ulam stability of mixed type cubic and additive functional equation, functional equations, difference inequalities and ulam stability notions, nova science publishers, chapter 13, (2010), 149175. [37] k. ravi and m. arunkumar, on the ulam-gavruta-rassias stability of the orthogonally euler-lagrange type functional equation, internat. j. appl. math. stat.7(2007), 143-156. [38] k. ravi, m. arunkumar and j.m. rassias, ulam stability for the orthogonally general eulerlagrange type functional equation, int. j. math. stat. 3(2008), a08, 36-46. [39] k. ravi and b.v. senthil kumar, ulam-gavruta-rassias stability of rassias reciprocal functional equation, global j. of appl. math. and math. sci. 3(no.1-2),(jan-dec 2010), 57-79. [40] k. ravi, j.m. rassias and b.v. senthil kumar, ulam stability of reciprocal difference and adjoint funtional equations, the australian j. math. anal.appl. 8(1), art.13 (2011), 1-18. [41] k. ravi, j.m. rassias m. arunkumar and r. kodandan, stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, j. of ineq. pure & appl. math. 10(4) (2009), 1-29. [42] k. ravi and b.v. senthil kumar, stability of mixed type cubic and additive functional equation in fuzzy normed spaces, int. j. math. sci. engg. appl.4(i) (march 2010), 159-172. [43] th.m. rassias, on the stability of the linear mapping in banach spaces, proc. amer. math. soc. 72(1978), 297-300. [44] r. saadati, s.m. vaezpour and y.j. cho, a note to paper “on the stability of cubic mappings and quartic mappings in random normed spaces”, j. of ineq. & appl. volume 2009, article id 214530, 6 pages. generalized ulam-hyers stability 17 [45] m.b. savadkouhi, m. eshaghi gordji, j.m. rassias and n. ghobadipour, approximate ternary jordan derivations on banach ternary algebras, j. math. phys. 50042303 (2009), 1-9. [46] m.a. sibaha, b. bouikhalene and e. elqorachi, ulam-gavruta-rassias stability of a linear functional equation, internat. j.appl. math. stat. 7(2007), 157-166. [47] s.m. ulam, problems in modern mathematics, chapter vi, wiley-interscience, new york, 1964. 1pg & research department of mathematics, sacred heart college, tirupattur 635 601, tamilnadu, india 2pedagogical department e.e., section of mathematics and informatics, national and capodistrian university of athens, 4, agamemnonos str., aghia paraskevi, athens, attikis 15342, greece 3department of mathematics, c. abdul hakeem college of engg. and tech., elvisharam 632 509, tamilnadu, india ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 30-37 http://www.etamaths.com approximation theorems for q− analouge of a linear positive operator by a. lupas karunesh kumar singh1, asha ram gairola2 and deepmala3,∗ abstract. the purpose of the present paper is to introduce q− analouge of a sequence of linear and positive operators which was introduced by a. lupas [1]. first, we estimate moments of the operators and then prove a basic convergence theorem. next, a local direct approximation theorem is established. further, we study the rate of convergence and point-wise estimate using the lipschitz type maximal function. 1. introduction at the international dortmund meeting held in written (germany, march, 1995), a. lupas [1] introduced the following linear positive operators: (1) ln(f; x) = (1 −a)nx ∞∑ k=0 (nx)k k! akf ( k n ) ,x ≥ 0. with f : [0,∞] → r. if we impose that lne1 = e1 we find that a = 1/2. therefore operator (1) becomes ln(f; x) = 2 −nx ∞∑ k=0 (nx)k 2kk! f ( k n ) ,x ≥ 0, where (α)0 = 1, (α)k = α(α + 1)...(α + k − 1),k ≥ 1. the q− analouge of the above operators is defined as: ln,q(f; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! f ( [k]q [n]q ) ,x ≥ 0, we denote cb[0,∞) the space of real valued bounded continuous function f on the interval [0,∞), the norm on the space is defined as ‖f‖ = sup 0≤x<∞ |f(x)|. let w2 = {g ∈ cb[0,∞) : g′,g′′ ∈ cb[0,∞)}. the peetre’s k− functional is defined as k2(f,δ) = inf g∈w2 {‖f −g‖ + δ‖g′′‖}, where δ > 0. for f ∈ cb[0,∞) a usual modulus of continuity is given by ω(f,δ) = sup 0<h≤δ sup 0≤x<∞ |f(x + h) −f(x))|. the second order modulus of smoothness is given by ω2(f, √ δ) = sup 0<h≤ √ δ sup 0≤x<∞ |f(x + 2h) − 2f(x + h) + f(x))|. 2010 mathematics subject classification. 41a36, 41a99, 41a25. key words and phrases. linear positive operators; q−operators; rate of convergence. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 30 linear positive operators by a. lupas 31 by [[3], p.177, theorem 2.4] there exists an absolute constant c > 0 such that k2(f,δ) ≤ cω2(f, √ δ). in recent years, many results about the generalization of linear positive operators have been obtained by several mathematicians ([6]-[17]). 2. moment estimates lemma 1. the following relations hold: ln,q(1; x) = 1,ln,q(t; x) = x and ln,q(t 2; x) = qx2 + 1 + q [n] x. proof. we have ln,q(1; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! = 1 now, ln,q(t; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! [k]q [n]q = 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k − 1]q![n]q = 2−[n]qx−1 [n]q ∞∑ k=1 [n]qx([n]qx + 1)k−1 2k−1[k − 1]q! = 2−[n]qx−1x ∞∑ k=1 ([n]qx + 1)k−1 2k−1[k − 1]q! = 2−[n]qx−1x ∞∑ k=0 ([n]qx + 1)k 2k[k]q! = x. next, ln,q(t 2; x) = 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! [k]2q [n]2q = 2−[n]qx ∞∑ k=0 [n]qx([n]qx + 1)k−1 2k[k]q[k − 1]q! [k]2q [n]2q = 2−[n]qx−1x ∞∑ k=1 ([n]qx + 1)k−1 2k−1[k − 1]q! [k]q [n]q = 2−[n]qx−1x [n]q ∞∑ k=1 ([n]qx + 1)k−1[k]q 2k−1[k − 1]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k[k + 1]q 2k[k]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k(1 + q[k]q) 2k[k]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k 2k[k]q! + 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)kq[k]q 2k[k]q! = i1 + i2, say. 32 singh, gairola and deepmala we find that i1 = x [n]q . now, i2 = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)kq[k]q 2k[k]q! = 2−[n]qx−2qx [n]q ∞∑ k=1 ([n]qx + 1)([n]qx + 2)k−1 2k−1[k − 1]q! = 2−[n]qx−2qx([n]qx + 1) [n]q ∞∑ k=1 ([n]qx + 2)k−1 2k−1[k − 1]q! = 2−[n]qx−2qx([n]qx + 1) [n]q ∞∑ k=0 ([n]qx + 2)k 2k[k]q! = qx([n]qx + 1) [n]q . hence, on combining i1 and i2, we get ln,q(t 2; x) = (1 + q)x [n]q + qx2. � let us define mth order moment by ψn,m(q; x) = ln,q((t−x)m; x). lemma 2. let 0 < q < 1, then for x ∈ [0,∞) we have ψn,1(q; x) = 0 and ψn,2(q; x) = x([2] − (1 −q)[n]qx) [n]q . proof. we have ψn,1(q; x) = ln,q(t−x; x) = 0. now, ψn,2(q; x) = ln,q((t−x)2; x) = ln,q(t 2 + x2 − 2tx; x) = (1 + q)x [n]q + (q − 1)x2. � 3. basic pointwise convergence the operators ln,q do not satisfy the conditions of the bohman-korovkin theorem in case 0 < q < 1. to make this theorem applicable, we can choose a sequence (qn) in place of the number q such that qn → 1 and qnn → 0 as n →∞. with this modification we obtain the following korovkin type result: theorem 1. let f ∈ cb[0,∞) and qn be a real sequence in (0, 1) such that qn → 1 and qnn → 0 as n →∞. then, for each x ∈ [0,∞) we have lim n→∞ ln,qn(f; x) = f(x). proof. the proof is based on the well known korovkin theorem regarding the convergence of a sequence of linear positive operators. so, it is enough to prove the conditions lim n→∞ ln,qn(t m; x) = xm,m = 0, 1, 2. now, using lemma 1 we obtain lim n→∞ ln,qn(1; x) = 1, lim n→∞ ln,qn(t; x) = x linear positive operators by a. lupas 33 and lim n→∞ ln,qn(t; x) = lim n→∞ qnx 2 + 1 + qn [n]qn x = x2. this completes the proof. � 4. direct results theorem 2. let f ∈ cb[0,∞) and q ∈ (0, 1). then, for each x ∈ [0,∞) and n ∈ n there exists an absolute constant c > 0 such that |ln,q(f; x) −f(x)|6 cω2 ( f, √ x([2] − (1 −q)[n]qx) [n]q ) . proof. let g ∈ w2 and x,t ∈ [0,∞). using taylor’s expansion we can write g(t) = g(x) + g′(x)(t−x) + t∫ x (t−v)g′′(v)dv. on application of lemma 2 we obtain ln,q ( g(t); x) −g(x) ) = ln,q   t∫ x (t−v)g′′(v)dv; x   . now, we have ∣∣∣∣ t∫ x (t−v)g′′(v)dv ∣∣∣∣ ≤ (t−x)2‖g′′‖. therefore |ln,q(g(t); x) −g(x)| ≤ ln,q ( (t−x)2; x ) ‖g′′‖ = x([2] − (1 −q)[n]qx) [n]q ‖g′′‖. by lemma 1, we have |ln,q(f; x)| ≤ 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! ∣∣∣∣f ( [k]q [n]q )∣∣∣∣ ≤‖f‖. thus |ln,q(f; x) −f(x)| ≤ |ln,q(f −g; x) − (f −g)(x)| + |ln,q(g; x) −g(x)| ≤ 2‖f −g‖ + x([2]−(1−q)[n]qx) [n]q ‖g′′‖. at last, taking the infimum over all g ∈ w2 and on application of the inequality k2(f,δ) ≤ cω2(f,δ1/2),δ > 0, we get the required result. this completes the proof of the theorem. � 5. pointwise estimates in this section, we obtain some pointwise estimates of the rate of convergence of the q− baskakovdurrmeyer operators. first, we discuss the relationship between the local smoothness of f and the local approximation. theorem 3. let 0 < α ≤ 1 and e be any bounded subset of the interval [0,∞). if f ∈ cb[0,∞) ∩ lipm (α) then we have |ln,q(f; x) −f(x)| ≤ m{ψ α 2 n,2(q; x) + 2(d(x,e)) α},x ∈ [0,∞), where m is a constant depending on α and f, d(x,e) is the distance between x and e defined as d(x,e) = inf{|t−x|; t ∈ e} and ψn,2(q; x) = ln,q((t−x)2; x) . 34 singh, gairola and deepmala proof. from the property of infimum, it follows that there exists a point t0 ∈ ē such that d(x,e) = |t0 −x|. in view of the triangle inequality we have |f(t) −f(x)| ≤ |f(t) −f(t0)| + |f(t0) −f(x)|. using the definition of lipm (α), we get |ln,q(f; x) −f(x)| ≤ ln,q(|f(t) −f(t0)|; x) + ln,q(|f(x) −f(t0)|; x) ≤ m{ln,q(|t− t0|α; x) + |x− t0|α} ≤ m{ln,q(|t−x|α; x) + 2|x− t0|α}. choosing p1 = 2 α and p2 = 2 2−α, we get 1 p1 + 1 p2 = 1. then, hölder’s inequality yields |ln,q(f; x) −f(x)| ≤ m{(ln,q(|t−x|αp1 ; x))1/p1 [ln,q(1p2 ; x)]1/p2 + 2(d(x,e))α} ≤ m{(ln,q((t−x)2; x))α/2 + 2(d(x,e))α} = m{ψα/2n,2 (q; x) + 2(d(x,e)) α}. this completes the proof of the theorem. � next, we obtain a local direct estimate of operators ln,q using the lipschitz-type maximal function of order α introduced by lenze [2] as (2) ω̃α(f,x) = sup t 6=x,t∈[0,∞) |f(t) −f(x)| |t−x|α , x ∈ [0,∞) andα ∈ (0, 1]. theorem 4. let 0 < α ≤ 1 and f ∈ cb[0,∞), then for all x ∈ [0,∞) we have |ln,q(f; x) −f(x)| ≤ ω̃α(f,x)ψ α/2 n,2 (q; x). proof. in view of (2), we get |f(t) −f(x)| ≤ ω̃α(f,x)|t−x|α and hence |ln,q(f; x) −f(x)| ≤ ln,q(|f(t) −f(x)|; x) ≤ ω̃α(f,x)ln,q(|t−x|α; x). now, using the hölder’s inequality with p = 2 α and 1 q = 1 − 1 p , we obtain |ln,q(f; x) −f(x)| ≤ ω̃α(f,x)(ln,q(|t−x|2; x))α/2 = ω̃α(f,x)ψ α/2 n,2 (x). thus, the proof is completed. � 6. weighted approximation in this section, we discuss about the weighted approximation theorem for the operators ln,q(f). let c∗ x2 [0,∞) be the subspace of all functions f ∈ cx2 [0,∞) for which limx→∞ |f(x)| 1+x2 is finite. theorem 5. let qn be a sequence in (0, 1) such that qn → 1 and qnn → 0, as n → ∞. for each c∗ x2 [0,∞), we have (3) lim n→∞ ‖ln,qn(f) −f‖x2 = 0. linear positive operators by a. lupas 35 proof. in order to proof (3) it is sufficient to show that ([5]) (4) lim n→∞ ‖ln,qn(t ν; x) −xν‖x2 = 0, ν = 0, 1, 2. since, ln,qn(1; x) = 1, (4) holds true for ν = 0. now, by lemma 1, we have ‖ln,qn(t; x) −x‖x2 = sup x∈[0,∞) |ln,qn(t; x) −x| 1 + x2 → 0, as n →∞. therefore, (4) is true for ν = 1. again, by lemma 1, we may write ‖ln,qn(t 2; x) −x2‖x2 = sup x∈[0,∞) |ln,qn(t2; x) −x2| 1 + x2 = sup x∈[0,∞) ∣∣∣(1+qn)x+qn[n]qnx2[n]qn −x2 ∣∣∣ 1 + x2 ≤ 1 + qn [n]qn sup x∈[0,∞) x 1 + x2 + (qn − 1) sup x∈[0,∞) x2 1 + x2 = 1 + qn [n]qn + (qn − 1). hence, (4) follows for ν = 2. this completes the proof of the theorem. � theorem 6. let f ∈ cx2 [0,∞),q = qn ∈ (0, 1) such that qn → 1 and qnn → 0 as n → ∞ and ωa+1be its modulus of continuity on the finite interval [0,a + 1] ⊂ [0,∞), a > 0. then, for every n ≥ 1 ‖ln,q(f) −f‖c[0,a] ≤ 12mf(1+a 2)a [n]q + 2ωa+1 ( f, √ 2a [n]q ) . proof. for x ∈ [0,a] and t > a + 1. since t−x > 1, we have |f(t) −f(x)| ≤ mf (2 + x2 + t2) ≤ mf (2 + 3x2 + 2(t−x)2) ≤ 3mf (1 + x2 + (t−x)2) ≤ 6mf (1 + x2)(t−x)2 ≤ 6mf (1 + a2)(t−x)2.(5) for x ∈ [0,a] and t ≤ a + 1, we have |f(t) −f(x)| ≤ ωa+1(f, |t−x|) ≤ ( 1 + |t−x| δ ) ωa+1(f,δ),(6) where δ > 0. from (5) and (6), we can write |f(t) −f(x)| ≤ 6mf (1 + a2)(t−x)2 + ( 1 + |t−x| δ ) ωa+1(f,δ)(7) for x ∈ [0,a] and t ≥ 0 and applying schwarz inequality, we obtain |ln,q(f; x) −f(x)| ≤ ln,q(|f(t) −f(x)|; x) ≤ 6mf (1 + a2)ln,q((t−x)2; x) + ωa+1(f,δ) ( 1 + 1 δ ln,q((t−x)2; x) 1 2 ) . 36 singh, gairola and deepmala hence, using lemma 2, for every q ∈ (0, 1) and x ∈ [0,a] |ln,q(f; x) −f(x)| ≤ 6mf (1 + a2) x([2] − (1 −q)[n]qx) [n]q + cωa+1(f,δ) ( 1 + 1 δ √ x([2] − (1 −q)[n]qx) [n]q ≤ 12mf (1 + a 2)a [n]q + ωa+1(f,δ) ( 1 + 1 δ √ 2a [n]q ) . taking δ = √ 2a [n]q , we get the required result. this completes the proof of theorem. � now, we prove a theorem to approximate all functions in cx2 [0,∞). such type of results are given in [4] for locally integrable functions. theorem 7. let q = qn ∈ (0, 1) such that qn → 1 and qnn → 0, as n → ∞. for each f ∈ c∗x2 [0,∞), and α > 1, we have lim n→∞ sup x∈[0,∞) |ln,qn(f; x) −f(x)| (1 + x2)α = 0. proof. for any fixed x0 > 0, sup x∈[0,∞) |ln,qn(f; x) −f(x)| (1 + x2)α ≤ sup x≤x0 |ln,qn(f; x) −f(x)| (1 + x2)α + sup x>x0 |ln,qn(f; x) −f(x)| (1 + x2)α ≤ ‖ln,qn(f) −f‖c[0,x0] + ‖f‖x2 sup x≥x0 |ln,qn(1 + t2; x)| (1 + x2)α (8) + sup x≥x0 |f(x)| (1 + x2)α . since, |f(x)| ≤ mf (1 + x2), we have sup x≥x0 |f(x)| (1 + x2)α ≤ sup x≥x0 mf (1 + x2)α−1 ≤ mf (1 + x20) α−1 . let � > 0 be arbitrary. we can choose x0 to be large that (9) mf (1 + x20) α−1 < � 3 and in view of lemma 1, we obtain ‖f‖x2 lim n→∞ |ln,qn(1 + t2; x)| (1 + x2)α = 1 + x2 (1 + x2)α ‖f‖x2 = ‖f‖x2 (1 + x2)α−1 ≤ ‖f‖x2 (1 + x20) α−1 < � 3 .(10) using theorem 6 we can see that the first term of the inequality (8) implies that (11) ‖ln,qn(f; .) −f‖c[0,x0] < � 3 , as n →∞. combining (8)-(11), we get the desired result. linear positive operators by a. lupas 37 � acknowledgement the research work of the third author deepmala is supported by the science and engineering research board (serb), government of india under serb npdf scheme, file number: pdf/2015/000799. references [1] a. lupas, the approximation by some positive linear operators. in: proceedings of the international dortmund meeting on approximation theory(m.w. müller et al., eds.), akademie verlag, berlin, 1995, 201-229. [2] b. lenze, on lipschitz-type maximal functions and their smoothness spaces, nederl. akad. wetensch. indag. math. 50(1) (1988), 531-763. [3] r.a. devore and g. g. lorentz, constructive approximation, volume 303. springer-verlag, berlin, 1993 [4] a.d. gadjiev, r. o. efendiyev, and e. ibikli, on korovkin type theorem in the space of locally integrable functions, czechoslovak mathematical journal, 53 (1) (2003), 45-53. [5] a.d. gadjiev, theorems of the type of p. p. korovkin’s theorems, math. zametki, 20(5) (1976), 7811-7786. [6] a.r. gairola, deepmala, and l.n. mishra, rate of approximation by finite iterates of q-durrmeyer operators, proceedings of the national academy of sciences, india section a: physical sciences, 86(2) (2016), 229-234. [7] m. a. özarslan, o. duman and h.m. srivastava, statistical approximation results for kantorovich-type operators involving some special polynomials, math. comput. modelling, 48 (2008), 388-401. [8] v.n. mishra, p. sharma, and l.n. mishra; on statistical approximation properties of q−baskakov-szász-stancu operators, journal of egyptian mathematical society, 24 (3) (2016), 396-401. [9] h.m. srivastava, m. mursaleen, and a. khan, generalized equi-statistical convergence of positive linear operators and associated approximation theorems, math. comput. model. 55 (2012), 2040-2051. [10] v.n. mishra, k. khatri, and l.n. mishra, statistical approximation by kantorovich type discrete q−beta operators, advances in difference equations 2013 (2013), art. id 345. [11] m. mursaleen, a. khan, h.m. srivastava, and k.s. nisar, operators constructed by means of q-lagrange polynomials and a-statistical approximation, appl. math. comput. 219 (2013), 6911-6918. [12] n.l. braha, h.m. srivastava and s.a. mohiuddine, a korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallée poussin mean, appl. math. comput. 228 (2014), 162169. [13] v.n. mishra, k. khatri, l.n. mishra, and deepmala; inverse result in simultaneous approximation by baskakovdurrmeyer-stancu operators, journal of inequalities and applications 2013 (2013), art. id 586. [14] v.n. mishra, h.h. khan, k. khatri, and l.n. mishra; hypergeometric representation for baskakov-durrmeyerstancu type operators, bulletin of mathematical analysis and applications, 5 (3) (2013), 18-26. [15] v.n. mishra, k. khatri, and l.n. mishra; on simultaneous approximation for baskakov-durrmeyer-stancu type operators, journal of ultra scientist of physical sciences, 24 (3) (2012), 567-577. [16] v.n. mishra, k. khatri, and l.n. mishra, some approximation properties of q-baskakov-beta-stancu type operators, journal of calculus of variations, 2013 (2013), article id 814824. [17] a. wafi, n. rao, and deepmala, approximation properties by generalized-baskakov-kantorovich-stancu type operators, appl. math. inf. sci. lett., 4 (3) (2016), 1-8. 1government polytechnic, rampur, uttar pradesh, india 2department of mathematics, doon university, dehradun, uttarakhand-248 001, india 3sqc & or unit, indian statistical institute, 203 b. t. road, kolkata-700 108, west bengal, india ∗corresponding author: dmrai23@gmail.com, deepmaladm23@gmail.com international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 1-14 http://www.etamaths.com some fejér type inequalities for harmonically-convex functions with applications to special means m. a. latif1,∗, s. s. dragomir1,2 and e. momoniat1 abstract. in this paper, the notion of harmonic symmetricity of functions is introduced. a new identity involving harmonically symmetric functions is established and some new fejér type integral inequalities are presented for the class of harmonically convex functions. the results presented in this paper are better than those established in recent literature concerning harmonically convex functions. applications of our results to special means of positive real numbers are given as well. 1. introduction the theory of convexity has been subject to extensive research during the past few years due it its utility in various branches of pure and applied mathematics. many inequalities have been established by a number of researchers for convex functions but one of the most interesting inequalities is the hermite-hadamard inequality which provides a necessary and sufficient condition for a functions to be convex. let f : i ⊆ r → r, a,b ∈ i with a < b f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 (1.1) holds if and only if f is convex. the inequalities (1.1) hold in reversed direction if f is concave. many researchers have generalized the classical convexity in a number of ways and the inequality (1.1) has been generalized or extended for many classes of convex functions in numerous ways, see for instance [2–21] and the references therein. let us recall some known concepts which will be used in the sequel of the paper. definition 1.1. [9] let i ⊂ r\{0} be a real interval. a function f : i → r is said to be harmonically convex, if f ( xy tx + (1 − t) y ) ≤ tf (y) + (1 − t) f (x) (1.2) for all x, y ∈ i and t ∈ [0, 1]. if the inequality in (1.2) is reversed, then f is said to be harmonically concave. proposition 1.1. [9] let i ⊂ r\{0} be a real interval and f : i → r is function, then: • if i ⊂ (0,∞) and f is convex and nondecreasing function then f is harmonically convex. • if i ⊂ (0,∞) and f is harmonically convex and nonincreasing function then f is convex. • if i ⊂ (−∞, 0) and f is harmonically convex and nondecreasing function then f is convex. • if i ⊂ (−∞, 0) and f is convex and nonincreasing function then f is harmonically convex. in [9], i̇şcan has also proved the following results for harmonically convex functions. received 29th june, 2016; accepted 19th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. primary 26d15, secondary 26a51, 26e60, 41a55. key words and phrases. hermite-hadamard’s inequality; fejér’s inequality; convex function; harmonically-convex function; hölder’s inequality; power mean inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 latif, dragomir and momoniat theorem 1.1. [9] let i ⊂ r\{0} be a harmonically convex function and a, b ∈ i with a < b. if f ∈ l ([a,b]) then the following inequalities hold f ( 2ab a + b ) ≤ ab b−a ∫ b a f (x) x2 dx ≤ f (a) + f (b) 2 . the above inequalities are sharp. theorem 1.2. [9] let f : (0,∞) → r be a differentiable function on i◦, a, b ∈ i◦ with a < b, and f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣q is harmonically convex [a,b] for q ≥ 1, then∣∣∣∣∣f (a) + f (b)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ab (b−a)2 λ1−1q1 [ λ2 ∣∣∣f′ (a)∣∣∣q + λ3 ∣∣∣f′ (b)∣∣∣q]1q , where λ1 = 1 ab − 2 (b−a)2 ln ( (a + b) 2 4ab ) , λ2 = − 1 b (b−a) + 3a + b (b−a)3 ln ( (a + b) 2 4ab ) , λ2 = 1 a (b−a) − 3b + a (b−a)3 ln ( (a + b) 2 4ab ) = λ1 −λ2. theorem 1.3. [9] let f : (0,∞) → r be a differentiable function on i◦, a, b ∈ i◦ with a < b, and f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣q is harmonically convex [a,b] for q > 1, 1p + 1q = 1, then∣∣∣∣∣f (a) + f (b)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ab (b−a)2 ( 1 p + 1 )1 p [ µ1 ∣∣∣f′ (a)∣∣∣q + µ1 ∣∣∣f′ (b)∣∣∣q]1q , where µ1 = a2−2q + b1−2q [(b−a) (1 − 2q) −a] 2 (b−a)2 (1 −q) (1 − 2q) , µ2 = b2−2q −a1−2q [(b−a) (1 − 2q) + b] 2 (b−a)2 (1 −q) (1 − 2q) . some applications of the above results can also be found in [9]. chen and wu [2] established the following fejér type inequality for harmonically convex functions which provides a weighted generalization of the result given in theorem 1.1. theorem 1.4. [2] let f : i ⊆ r\{0} → r be a harmonically convex function and a, b ∈ i with a < b. if f ∈ l ([a,b]), them one has be continuous f ( 2ab a + b )∫ b a g (x) x2 dx ≤ ∫ b a f (x) g (x) x2 dx ≤ f (a) + f (b) 2 ∫ b a g (x) x2 dx, (1.3) g : [a,b] → r is nonnegative, integrable and satisfies g ( ab x ) = g ( ab a + b−x ) . the main goal of this paper is to introduce a new notion of harmonically symmetric functions and to establish an identity involving a harmonically symmetric function and a differentiable function. we will prove some fejér type inequalities by using this identity and hence our results will provide a better weighted generalization of the results proved in theorem 1.2 and theorem 1.3. some applications of our results to special means of positive real numbers will also be provided in section 3. we believe that our findings are novel, new and better than those already exist and will open new ways for further research in this filed. fejér type inequalities for harmonically-convex functions 3 2. main results throughout this section we take u (t) = 2ab (1−t)a+(1+t)b and l (t) = 2ab (1+t)a+(1−t)b. the beta function, the gamma function and the integral from of the hypergeometric function are defined as follows to be used in the sequel of the paper b (α,β) = ∫ 1 0 tα−1 (1 − t)β−1 dt,α > 0,β > 0, γ (α) = ∫ ∞ 0 tα−1e−tdt,α > 0 and 2f1 (α,β; γ; z) = 1 b (β,γ −β) ∫ 1 0 tβ−1 (1 − t)γ−β−1 (1 −zt)−α dt for |z| < 1,γ > β > 0. the notion of harmonically symmetric functions is given in following definition. definition 2.1. a function g : [a,b] ⊆ r\{0}→ r is said to be harmonically symmetric with respect to 2ab a+b if g (x) = g ( 1 1 a + 1 b − 1 x ) holds for all x ∈ [a,b]. now we prove a weighted integral identity which will be used in establishing our main results. lemma 2.1. let f : i ⊆ r\{0} → r be a differentiable function on i◦ and a, b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b . if f ′ ∈ l ([a,b]), then the following equality holds f (b) + f (a) 2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx = ( b−a 4ab )∫ 1 0 (∫ u(t) l(t) g (x) x2 dx )[ (u (t)) 2 f ′ (u (t)) − (l (t))2 f ′ (l (t)) ] dt. (2.1) proof. let i1 = ∫ 1 0 (∫ u(t) l(t) g (x) x2 dx ) (u (t)) 2 f ′ (u (t)) dt and i2 = ∫ 1 0 (∫ u(t) l(t) g (x) x2 dx ) (l (t)) 2 f ′ (l (t)) dt. since g : [a,b] → [0,∞) is harmonically symmetric to 2ab a+b , then g (u (t)) = g (l (t)) for all t ∈ [0, 1]. hence, we have i1 = ∫ 1 0 (∫ u(t) l(t) g (x) x2 dx ) (u (t)) 2 f ′ (u (t)) dt = −2ab b−a ∫ 1 0 (∫ u(t) l(t) g (x) x2 dx ) d [f (u (t))] = −2ab b−a (∫ u(t) l(t) g (x) x2 dx ) f (u (t)) ∣∣∣∣∣ 1 0 − ∫ 1 0 [g (u (t)) + g (l (t))] f (u (t)) dt = 2ab b−a f (a) ∫ b a g (x) x2 dx− 2 ∫ 1 0 g (u (t)) f (u (t)) dt = 2ab b−a f (a) ∫ b a g (x) x2 dx− 4ab b−a ∫ 2ab a+b a g (x) f (x) x2 dx. (2.2) 4 latif, dragomir and momoniat analogously, we have − i2 = 2ab b−a f (b) ∫ b a g (x) x2 dx− 4ab b−a ∫ b 2ab a+b g (x) f (x) x2 dx. (2.3) adding (2.2) and (2.3) and multiplying the result by b−a 4ab , we get the required identity. this completes the proof of the lemma. � lemma 2.2. for v > u > 0, we have∫ 1 0 t [ 2uv (1 − t) u + (1 + t) v ]2 dt = ( 2uv v + u )2 λ1 (u,v) , ∫ 1 0 t [ 2uv (1 + t) u + (1 − t) v ]2 dt = ( 2uv v + u )2 λ1 (v,u) , ∫ 1 0 t2 [ 2uv (1 − t) u + (1 + t) v ]2 dt = ( 2uv v + u )2 λ2 (u,v) , ∫ 1 0 t2 [ 2uv (1 + t) u + (1 − t) v ]2 dt = ( 2uv v + u )2 λ2 (v,u) , where λ1 (u,v) ∆ = ln ( 2v u + v ) + u−v 2v and λ2 (u,v) ∆ = ( v + u v −u )[ 2v u + v − u + v 2v − 2 ln ( 2v u + v )] . proof. the proof follows from a straightforward computation. � lemma 2.3. for v > u > 0 and p > 1, we have∫ 1 0 (1 + t) [ 2uv (1 − t) u + (1 + t) v ]2p dt = ( 2uv v + u )2p ζ1 (u,v; p) , ∫ 1 0 (1 − t) [ 2uv (1 − t) u + (1 + t) v ]2p dt = ( 2uv v + u )2p ζ2 (u,v; p) , ∫ 1 0 (1 + t) [ 2uv (1 + t) u + (1 − t) v ]2p dt = ( 2uv v + u )2p ζ1 (v,u; p) , ∫ 1 0 (1 − t) [ 2uv (1 + t) u + (1 − t) v ]2p dt = ( 2uv v + u )2p ζ2 (v,u; p) , where ζ1 (u,v; p) ∆ = 21−2pv ( v u+v )−2p [(1 − 2p) (v −u) −u] (v −u)2 (p− 1) (2p− 1) − (u + v) [(1 − 2p) (v −u) − 2u] 2 (v −u)2 (p− 1) (2p− 1) and ζ2 (u,v; p) ∆ = 41−pv2 ( v u+v )−2p + (u + v) [(2p− 1) (v −u) − 2v] 2 (v −u)2 (p− 1) (2p− 1) . proof. the proof follows from a straightforward computation. � now we present new fejér type inequalities for harmonically-convex functions, which provide weighted generalization of some of the results established in recent literature. fejér type inequalities for harmonically-convex functions 5 theorem 2.1. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣q is harmonically-convex on [a,b] for q ≥ 1, then the following inequality holds ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a b + a )2 ( 1 2 )1/q ‖g‖∞ × { [λ1 (a,b)] 1−1/q [ ξ1 (a,b) ∣∣∣f′ (a)∣∣∣q + ξ2 (a,b) ∣∣∣f′ (b)∣∣∣q]1/q + [λ1 (b,a)] 1−1/q [ ξ2 (b,a) ∣∣∣f′ (a)∣∣∣q + ξ1 (b,a) ∣∣∣f′ (b)∣∣∣q]1/q} , (2.4) where ‖g‖∞ = supx∈[a,b] g (x) < ∞, ξ1 (a,b) ∆ = λ1 (a,b) + λ2 (a,b) , ξ2 (a,b) ∆ = λ1 (a,b) −λ2 (a,b) and λ1 (·, ·), λ2 (·, ·) are defined in lemma 2.2. proof. from lemma 2.1, we get f (b) + f (a) 2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ≤− ( b−a 2ab )2 ‖g‖∞ ∫ 1 0 [ t (u (t)) 2 f ′ (u (t)) − t (l (t))2 f ′ (l (t)) ] dt. (2.5) now taking modulus on both sides of (2.5) and using hölder’s inequality, we have ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )2 ‖g‖∞ {(∫ 1 0 t (u (t)) 2 dt )1−1/q (∫ 1 0 t (u (t)) 2 ∣∣∣f′ (u (t))∣∣∣q dt)1/q + (∫ 1 0 t (l (t)) 2 dt )1−1/q (∫ 1 0 t (l (t)) 2 ∣∣∣f′ (l (t))∣∣∣q dt)1/q } . (2.6) by the harmonic-convexity of ∣∣∣f′∣∣∣q on [a,b] for q ≥ 1 and by using lemma 2.2, we have ∫ 1 0 t (u (t)) 2 ∣∣∣f′ (u (t))∣∣∣q dt = ∫ 1 0 t [ 2ab (1 − t) a + (1 + t) b ]2 × ∣∣∣∣f′ ( 2ab (1 − t) a + (1 + t) b )∣∣∣∣q dt ≤ 12 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 t (1 + t) [ 2ab (1 − t) a + (1 + t) b ]2 dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 t (1 − t) [ 2ab (1 − t) a + (1 + t) b ]2 dt = 1 2 ( 2ab b + a )2 { [λ1 (a,b) + λ2 (a,b)] ∣∣∣f′ (a)∣∣∣q + [λ1 (a,b) −λ2 (a,b)] ∣∣∣f′ (b)∣∣∣q} (2.7) 6 latif, dragomir and momoniat and∫ 1 0 t (l (t)) 2 ∣∣∣f′ (l (t))∣∣∣q dt = ∫ 1 0 t [ 2ab (1 + t) a + (1 − t) b ]2 × ∣∣∣∣f′ ( 2ab (1 + t) a + (1 − t) b )∣∣∣∣q dt ≤ 12 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 t (1 − t) [ 2ab (1 + t) a + (1 − t) b ]2 dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 t (1 + t) [ 2ab (1 + t) a + (1 − t) b ]2 dt = 1 2 ( 2ab b + a )2 × { [λ1 (b,a) −λ2 (b,a)] ∣∣∣f′ (a)∣∣∣q + [λ1 (b,a) + λ2 (b,a)] ∣∣∣f′ (b)∣∣∣q} . (2.8) a combination of (2.6), (2.7) and (2.8) gives the required result. this completes the proof of the theorem. � corollary 2.1. suppose the assumptions of theorem 2.1 are satisfied. if q = 1, then the following inequality holds∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( 1 2 )( b−a b + a )2 ‖g‖∞ { [ξ1 (a,b) + ξ2 (b,a)] ∣∣∣f′ (a)∣∣∣ + [ξ2 (a,b) + ξ1 (b,a)] ∣∣∣f′ (b)∣∣∣} , (2.9) where ‖g‖∞ = supx∈[a,b] g (x) < ∞ and ξ1 (·, ·), ξ2 (·, ·) are defined in theorem 2.1. corollary 2.2. if g (x) = ab b−a for all x ∈ [a,b] in theorem 2.1, then∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ( 1 2 )1/q ( b−a ab )( ab b + a )2 × { [λ1 (a,b)] 1−1/q [ ξ1 (a,b) ∣∣∣f′ (a)∣∣∣q + ξ2 (a,b) ∣∣∣f′ (b)∣∣∣q]1/q + [λ1 (b,a)] 1−1/q [ ξ2 (b,a) ∣∣∣f′ (a)∣∣∣q + ξ1 (b,a) ∣∣∣f′ (b)∣∣∣q]1/q} , (2.10) where ξ1 (·, ·), ξ2 (·, ·) are defined in theorem 2.1 and λ1 (·, ·), λ2 (·, ·) are defined in lemma 2.2. corollary 2.3. if q = 1 in corollary 2.2, then we get the following inequality∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )( ab b−a )2 { [ξ1 (a,b) + ξ2 (b,a)] ∣∣∣f′ (a)∣∣∣ + [ξ2 (a,b) + ξ1 (b,a)] ∣∣∣f′ (b)∣∣∣} , (2.11) where ξ1 (·, ·), ξ2 (·, ·) are defined in theorem 2.1. theorem 2.2. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣q is harmonically-convex on [a,b] for q > 1, then the following inequality holds∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤‖g‖∞ ( b−a b + a )2 ( 1 2 )1/q ( q − 1 2q − 1 )1−1/q {[ ζ1 (a,b; q) ∣∣∣f′ (a)∣∣∣q + ζ2 (a,b; q) ∣∣∣f′ (b)∣∣∣q]1/q + [ ζ2 (b,a; q) ∣∣∣f′ (a)∣∣∣q + ζ1 (b,a; q) ∣∣∣f′ (b)∣∣∣q]1/q} , (2.12) fejér type inequalities for harmonically-convex functions 7 where ζ1 (·, ·; ·) and ζ2 (·, ·; ·) are defined in lemma 2.3. proof. from (2.5) and hölder’s inequality, we have∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )2 ‖g‖∞ (∫ 1 0 tq/(q−1)dt )1−1/q × {(∫ 1 0 (u (t)) 2q ∣∣∣f′ (u (t))∣∣∣q dt)1/q + (∫ 1 0 (l (t)) 2q ∣∣∣f′ (l (t))∣∣∣q dt)1/q } . (2.13) since ∣∣∣f′∣∣∣q is harmonically-convex on [a,b], we obtain ∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣q dt = ∫ 1 0 [ 2ab (1 − t) a + (1 + t) b ]2q × ∣∣∣∣f′ ( 2ab (1 − t) a + (1 + t) b )∣∣∣∣q dt ≤ 12 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 (1 + t) [ 2ab (1 − t) a + (1 + t) b ]2q dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 (1 − t) [ 2ab (1 − t) a + (1 + t) b ]2q dt (2.14) and∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣q dt = ∫ 1 0 [ 2ab (1 + t) a + (1 − t) b ]2q × ∣∣∣∣f′ ( 2ab (1 + t) a + (1 − t) b )∣∣∣∣q dt ≤ 12 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 (1 − t) [ 2ab (1 + t) a + (1 − t) b ]2q dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 (1 + t) [ 2ab (1 + t) a + (1 − t) b ]2q dt. (2.15) by applying lemma 2.3 in inequalities (2.14) and (2.15) and then using the resulting inequalities in (2.13), we get the required inequality. � corollary 2.4. if the assumptions of theorem 2.2 are satisfied and if g (x) = ab b−a for all x ∈ [a,b], then the following inequality holds∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ( ab b−a )( b−a b + a )2 ( 1 2 )1/q ( q − 1 2q − 1 )1−1/q {[ ζ1 (a,b; q) ∣∣∣f′ (a)∣∣∣q + ζ2 (a,b; q) ∣∣∣f′ (b)∣∣∣q]1/q + [ ζ2 (b,a; q) ∣∣∣f′ (a)∣∣∣q + ζ1 (b,a; q) ∣∣∣f′ (b)∣∣∣q]1/q} , (2.16) where ζ1 (·, ·; ·) and ζ2 (·, ·; ·) are defined in lemma 2.3. theorem 2.3. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣q is harmonically-convex on [a,b] for q > 1, then the following inequality holds∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a b + a )2 ( 1 2 )2/q−1 ( q − 1 2q − 1 )1−1/q ‖g‖∞ × { [ζ1 (a,b; q) + ζ2 (b,a; q)] ∣∣∣f′ (a)∣∣∣q + [ζ2 (a,b; q) + ζ1 (b,a; q)] ∣∣∣f′ (b)∣∣∣q}1/q , (2.17) 8 latif, dragomir and momoniat where ζ1 (·, ·; ·) and ζ2 (·, ·; ·) are defined in lemma 2.3. proof. from the inequality 2.5 and hölder’s inequality, we have∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )2 ‖g‖∞ (∫ 1 0 tq/(q−1)dt )1−1/q × {(∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣q dt)1/q + (∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣q dt)1/q } . (2.18) by the power-mean inequality (ar + br ≤ 21−r (a + b)r for a > 0,b > 0 and r < 1), we have (∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣2q dt)1/q + (∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣2q dt)1/q ≤ 21−1/q (∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣q dt + ∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣q dt)1/q . (2.19) since ∣∣∣f′∣∣∣q is harmonically-convex on [a,b] for q > 1, we obtain ∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣q dt + ∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣q dt ≤ 1 2 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 (1 + t) [ 2ab (1 − t) a + (1 + t) b ]2q dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 (1 − t) [ 2ab (1 − t) a + (1 + t) b ]2q dt + 1 2 ∣∣∣f′ (a)∣∣∣q ∫ 1 0 (1 − t) [ 2ab (1 + t) a + (1 − t) b ]2q dt + 1 2 ∣∣∣f′ (b)∣∣∣q ∫ 1 0 (1 + t) [ 2ab (1 + t) a + (1 − t) b ]2q dt = 1 2 ( 2ab b + a )2q { [ζ1 (a,b; q) + ζ2 (b,a; q)] ∣∣∣f′ (a)∣∣∣q + [ζ2 (a,b; q) + ζ1 (b,a; q)] ∣∣∣f′ (b)∣∣∣q} . (2.20) using (2.19) in (2.20), we get (∫ 1 0 [u (t)] 2q ∣∣∣f′ (u (t))∣∣∣q dt)1/q + (∫ 1 0 [l (t)] 2q ∣∣∣f′ (l (t))∣∣∣q dt)1/q ≤ 21−2/q ( 2ab b + a )2 { [ζ1 (a,b; q) + ζ2 (b,a; q)] ∣∣∣f′ (a)∣∣∣q + [ζ2 (a,b; q) + ζ1 (b,a; q)] ∣∣∣f′ (b)∣∣∣q}1/q . (2.21) applying (2.21) in (2.18), we obtain the required inequality (2.17). � corollary 2.5. if the assumptions of theorem 2.3 are satisfied and if g (x) = ab b−a for all x ∈ [a,b], then the following inequality holds∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ ab b−a ( b−a b + a )2 ( 1 2 )1−2/q ( q − 1 2q − 1 )1−1/q × { [ζ1 (a,b; q) + ζ2 (b,a; q)] ∣∣∣f′ (a)∣∣∣q + [ζ2 (a,b; q) + ζ1 (b,a; q)] ∣∣∣f′ (b)∣∣∣q}1/q , (2.22) where ζ1 (·, ·; ·) and ζ2 (·, ·; ·) are defined in lemma 2.3. fejér type inequalities for harmonically-convex functions 9 theorem 2.4. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b ∈ i◦ with a < b and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣ is harmonically-convex on [a,b], then the following inequality holds for q > 1 ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ 1 2 ( b−a a + b )2 ‖g‖∞ ( [ς (a,b; q)] 1−1/q { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (b)∣∣∣ + [ 2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (a)∣∣∣ } + [ς (b,a; q)] 1−1/q × { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (a)∣∣∣ + [2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (b)∣∣∣ }) , (2.23) where b (·, ·) is the beta function, 2f1 (·, ·; ·; ·) is the hypergeometric function and ς (a,b; q) ∆ = (q − 1) [ (a + b) −q+1 q−1 − (2b)− q+1 q−1 ] (q + 1) (b−a) (a + b)− 2q q−1 . proof. we continue from (2.5) and by using the harmonic-convexity of ∣∣∣f′∣∣∣ on [a,b], we have ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )2 ‖g‖∞ ∫ 1 0 [ t (u (t)) 2 ∣∣∣f′ (u (t))∣∣∣ + t (l (t))2 ∣∣∣f′ (l (t))∣∣∣]dt ≤ ( b−a 2ab )2 ‖g‖∞ {∫ 1 0 (u (t)) 2 [ t ( 1 + t 2 )∣∣∣f′ (a)∣∣∣ + t(1 − t 2 )∣∣∣f′ (b)∣∣∣]dt + ∫ 1 0 (l (t)) 2 [ t ( 1 − t 2 )∣∣∣f′ (a)∣∣∣ + t(1 + t 2 )∣∣∣f′ (b)∣∣∣]dt} . (2.24) using hölder integral inequality, we have ∫ 1 0 [ 2ab (1 − t) a + (1 + t) b ]2 [ t ( 1 + t 2 )∣∣∣f′ (a)∣∣∣ + t(1 − t 2 )∣∣∣f′ (b)∣∣∣]dt ≤ (∫ 1 0 [ 2ab (1 − t) a + (1 + t) b ]2q/(q−1) dt )1−1/q × {[∫ 1 0 tq ( 1 + t 2 )q dt ]1/q ∣∣∣f′ (a)∣∣∣ + [∫ 1 0 tq ( 1 − t 2 )q dt ]1/q ∣∣∣f′ (b)∣∣∣ } = 1 2 ( 2ab a + b )2 [ς (a,b; q)] 1−1/q { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (b)∣∣∣ + [ 2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (a)∣∣∣ } . (2.25) 10 latif, dragomir and momoniat similarly, one has ∫ 1 0 [ 2ab (1 + t) a + (1 − t) b ]2 [ t ( 1 − t 2 )∣∣∣f′ (a)∣∣∣ + t(1 + t 2 )∣∣∣f′ (b)∣∣∣]dt ≤ (∫ 1 0 [ 2ab (1 + t) a + (1 − t) b ]2q/(q−1) dt )1−1/q × {[∫ 1 0 tq ( 1 − t 2 )q dt ]1/q ∣∣∣f′ (a)∣∣∣ + [∫ 1 0 tq ( 1 + t 2 )q dt ]1/q ∣∣∣f′ (b)∣∣∣ } = 1 2 ( 2ab a + b )2 [ς (b,a; q)] 1−1/q { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (a)∣∣∣ + [ 2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (b)∣∣∣ } . (2.26) using (2.25) and (2.26) in (2.24), we obtain the required inequality (2.23). � corollary 2.6. under the assumptions of theorem 2.4, if g (x) = ab b−a for all x ∈ [a,b], then the following inequality holds ∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ 1 2 ( b−a a + b )2 ab b−a ( [ς (a,b; q)] 1−1/q { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (b)∣∣∣ + [ 2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (a)∣∣∣ } + [ς (b,a; q)] 1−1/q { [b (q + 1,q + 1)] 1/q ∣∣∣f′ (a)∣∣∣ + [ 2f1 (−q,q + 1; q + 2;−1) · 1 q + 1 ]1/q ∣∣∣f′ (b)∣∣∣ }) , (2.27) where b (·, ·) is the beta function, 2f1 (·, ·; ·; ·) is the hypergeometric function and ς (·, ·; ·) is defined in theorem 2.4. theorem 2.5. let f : i ⊆ (0,∞) → r be a differentiable function on i◦ and a, b ∈ i◦ with 0 < a < b < 1 and let g : [a,b] → [0,∞) be continuous positive mapping and harmonically symmetric to 2ab a+b such that f ′ ∈ l ([a,b]). if ∣∣∣f′∣∣∣ is harmonically-convex on [a,b], then the following inequality holds for q > 1 ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ 12 ( b−a a + b )2 ‖g‖∞ × { [ν (a,b; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ ] + [ν (b,a; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ ]} , (2.28) fejér type inequalities for harmonically-convex functions 11 where ν (a,b; q) = γ ( 2q−1 q−1 ) γ ( 3q−2 q−1 ) [b2q−1q−1 2f1 ( 2q q − 1 , 2q − 1 q − 1 ; 2q − 1 q − 1 ; b (a− b) a + b ) −a 2q−1 q−1 2f1 ( 2q q − 1 , 2q − 1 q − 1 ; 2q − 1 q − 1 ; a (a− b) a + b )] , γ (·) is the gamma function and 2f1 (·, ·; ·; ·) is the hypergeometric function. proof. from (2.5) and by using the harmonic-convexity of ∣∣∣f′∣∣∣ on [a,b], we have ∣∣∣∣∣f (b) + f (a)2 ∫ b a g (x) x2 dx− ∫ b a f (x) g (x) x2 dx ∣∣∣∣∣ ≤ ( b−a 2ab )2 ‖g‖∞ ∫ 1 0 [ t (u (t)) 2 ∣∣∣f′ (u (t))∣∣∣ + t (l (t))2 ∣∣∣f′ (l (t))∣∣∣]dt ≤ ( b−a 2ab )2 ‖g‖∞ {∫ 1 0 (u (t)) 2 [ t ( 1 + t 2 )∣∣∣f′ (a)∣∣∣ + t(1 − t 2 )∣∣∣f′ (b)∣∣∣]dt + ∫ 1 0 (l (t)) 2 [ t ( 1 − t 2 )∣∣∣f′ (a)∣∣∣ + t(1 + t 2 )∣∣∣f′ (b)∣∣∣]dt} . (2.29) application of hölder integral inequality yields ∫ 1 0 [ 2ab (1 − t) a + (1 + t) b ]2 [ t ( 1 + t 2 )∣∣∣f′ (a)∣∣∣ + t(1 − t 2 )∣∣∣f′ (b)∣∣∣]dt ≤ (∫ 1 0 tq/(q−1) [ 2ab (1 − t) a + (1 + t) b ]2q/(q−1) dt )1−1/q × {[∫ 1 0 ( 1 + t 2 )q dt ]1/q ∣∣∣f′ (a)∣∣∣ + [∫ 1 0 ( 1 − t 2 )q dt ]1/q ∣∣∣f′ (b)∣∣∣ } = 1 2 ( 2ab a + b )2 [ν (a,b; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ ] . (2.30) similarly, one has ∫ 1 0 [ 2ab (1 + t) a + (1 − t) b ]2 [ t ( 1 − t 2 )∣∣∣f′ (a)∣∣∣ + t(1 + t 2 )∣∣∣f′ (b)∣∣∣]dt ≤ (∫ 1 0 tq/(q−1) [ 2ab (1 + t) a + (1 − t) b ]2q/(q−1) dt )1−1/q × {[∫ 1 0 ( 1 − t 2 )q dt ]1/q ∣∣∣f′ (a)∣∣∣ + [∫ 1 0 ( 1 + t 2 )q dt ]1/q ∣∣∣f′ (b)∣∣∣ } = 1 2 ( 2ab a + b )2 [ν (b,a; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ ] . (2.31) using (2.30) and (2.31) in (2.29), we obtain the required inequality (2.28). � 12 latif, dragomir and momoniat corollary 2.7. suppose the assumptions of theorem 2.4 are satisfied and if g (x) = ab b−a for all x ∈ [a,b], then the following inequality holds∣∣∣∣∣f (b) + f (a)2 − abb−a ∫ b a f (x) x2 dx ∣∣∣∣∣ ≤ 12 ( b−a a + b )2 ( ab b−a ) × { [ν (a,b; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ ] + [ν (b,a; q)] 1−1/q [( 1 q + 1 )1/q ∣∣∣f′ (a)∣∣∣ + (2q+1 − 1 q + 1 )1/q ∣∣∣f′ (b)∣∣∣ ]} , (2.32) where ν (·, ·; ·) is defined in theorem 2.5. remark 2.1. some further results can be obtained from (2.24) but we omit the details for the interested readers. 3. applications to special means in this section we apply some of the above established inequalities of hermite-hadamard type involving the product of a harmonically convex function and a harmonically symmetric function to construct inequalities for special means. for positive numbers a > 0 and b > 0 with a 6= b a (a,b) = a + b 2 , l (a,b) = b−a ln b− ln a , g (a,b) = √ ab, h (a,b) = 2ab a + b and lp (a,b) =   [ bp+1−ap+1 (p+1)(b−a) ]1 p , p 6= −1, 0 l (a,b) , p = −1 1 e ( bb aa ) 1 b−a , p = 0 are the arithmetic mean, the logarithmic mean, geometric mean, harmonic mean and the generalized logarithmic mean of order p ∈ r respectively. for further information on means, we refer the readers to [1] and the references therein. let g : [a,b] → r0 be defined as g (x) = ( a + b 2ab − 1 x )2 ,x ∈ [a,b] . it is obvious that g ( 1 1 a + 1 b − 1 x ) = g (x) for all x ∈ [a,b]. hence g (x) = ( a+b 2ab − 1 x )2 , x ∈ [a,b] is harmonically symmetric with respect to x = 2ab a+b . throughout in this section we will also assume that µ (a,b) = b−a 2ab . now applications of our results are given in the following theorems to come. theorem 3.1. let 0 < a < b. then the following inequality holds∣∣∣∣a2 (a,b) + 2g2 (a,b)3g2 (a,b) − a (a,b)l (a,b) ∣∣∣∣ ≤ (b−a)2 µ (a,b) h2 (a,b) [ ln ( g (a,b) a (a,b) ) + µ2 (a,b) g2 (a,b) ] . (3.1) fejér type inequalities for harmonically-convex functions 13 proof. applying theorem 2.1 to the functions f(x) = x for x > 0 and g (x) = ( a + b 2ab − 1 x )2 ,x ∈ [a,b] we get the desired result. � theorem 3.2. let 0 < a < b. then for q ≥ 1, we have the following inequality holds ∣∣a(a2,b2)−g2 (a,b)∣∣ ≤ (1 2 )1/q µ (a,b) h2 (a,b) × { [λ1 (a,b)] 1−1/q [ 2λ1 (a,b) a (a q,bq) −q (b−a) λ2 (a,b) l q−1 q−1 (a,b) ]1/q + [λ1 (b,a)] 1−1/q [ 2λ1 (b,a) a (a q,bq) + q (b−a) λ2 (b,a) l q−1 q−1 (a,b) ]1/q} . (3.2) where λ1 (·, ·) and λ2 (·, ·) are defined in are defined in lemma 2.2. proof. the assertion follows from the inequality proved in corollary 2.2 for f(x) = x2 for x > 0. � corollary 3.1. if we take q = 1 in corollary 3.1, then the following inequality holds valid ∣∣a(a2,b2)−g2 (a,b)∣∣ ≤ 2µ (a,b) h2 (a,b) a (a,b) [3 ln (g (a,b) a (a,b) ) + 2µ2 (a,b) g2 (a,b) ] . (3.3) theorem 3.3. let 0 < a < b and q > 1. then∣∣∣∣a (a,b) − g2 (a,b)l (a,b) ∣∣∣∣ ≤ (2q − 2)1/q−1 µ (a,b) (2q − 1) (b−a)1/q × {[ a (a,b) h2q (a,b) −a2qb ]1/q + [ ab2q −a (a,b) h2q (a,b) ]1/q} . (3.4) proof. applying corollary 2.4 to the function f(x) = x for x > 0, we get the desired result. � theorem 3.4. let 0 < a < b and r ∈ (−1,∞)\{0}. then∣∣a(ar+2,br+2)−g2 (a,b) lrr (a,b)∣∣ ≤ (r + 2) µ (a,b) h2 (a,b) × { a ( ar+2,br+2 )[ ln ( g (a,b) a (a,b) ) + g2 (a,b) µ2 (a,b) ] + (r + 1) a (a,b) lrr (a,b) [ 2 ln ( g (a,b) a (a,b) ) + g2 (a,b) µ2 (a,b) ]} . (3.5) proof. applying corollary 2.3 to the function f(x) = xr+2 for x > 0,r ∈ (−1,∞)\{0} , we get the required result. � theorem 3.5. let 0 < a < b and q > 1. then ∣∣∣∣a2 (a,b) + 2g2 (a,b)3g2 (a,b) − a (a,b)l (a,b) ∣∣∣∣ ≤ ( q − 1 2q − 1 )1−1/q ( b−a b + a )3 g2/q (a,b) l2−2/q2q−q (a,b) h2 (a,b) . (3.6) 14 latif, dragomir and momoniat proof. applying theorem 2.3 to the functions f(x) = x for x > 0 and g (x) = ( a + b 2ab − 1 x )2 ,x ∈ [a,b] we get the desired result. � references [1] p. s. bullen, handbook of means and their inequalities, mathematics and its applications, volume 560, kluwer academic publishers,dordrecht/boston/london, 2003. [2] f. chen and s. wu, hermite-hadamard type inequalities for harmonically s-convex functions, sci. world j. 2014 (2014), article id 279158. [3] s. s. dragomir, r.p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 11 (5) (1998), 91–95. [4] s. s. dragomir, c.e.m. pearce, selected topics on hermite–hadamard type inequalities and applications, rgmia monographs, victoria university, 2000. [5] v. n. huy and n. t. chung, some generalizations of the fejér and hermite-hadamard inequalities in hölder spaces, j. appl. math. inform. 29 (3-4) (2011), 859–868. [6] j. hua, b.-y. xi, and f. qi, hemite-hadamard type inequalities for geometrically-arithmetically s-convex functions, commun. korean math. soc. 29 (1) (2014), 51–63. [7] j. hua, b. -y. xi and f. qi, inequalities of hermite–hadamard type involving an s-convex function with applications, applied mathematics and computation, 246 (2014), 752-760. [8] i̇. i̇şcan, hemite-hadamard type inequalities for ga-s-convex functions, le matematiche, 69 (2) (2014), 129–146. [9] i̇. i̇şcan, hermite-hadamard type inequalities for harmonically convex functions, hacettepe journal of mathematics and statistics 43 (6) (2014), 935-942. [10] i̇. i̇şcan, hermite-hadamard and simpson-like type inequalities for differentiable harmonically convex functions, journal of mathematics, 2014 (2014), article id 346305. [11] i̇. i̇şcan and s. wu, hermite-hadamard type inequalities for harmonically convex functions via fractional integrals, applied mathematica and computation, 238 (2014), 237-244. [12] a. p. ji, t. y. zhang, f. qi, integral inequalities of hermite-hadamard type for (α,m)-ga-convex functions, journal of function spaces and applications, 2013 (2013), article id 823856. [13] m. a. latif, new hermite-hadamard type integral inequalities for ga-convex functions with applications, analysis 34 (4) (2014), 379-389. [14] m. v. mihai, m. a. noor, k. i. noor and m. u. awan, some integral inequalities for harmonic h-convex functions involving hypergeometric functions, applied mathematics and computation 252 (2015), 257-262. [15] m. a. noor, k. i. noor and m. u. awana, integral inequalities for coordinated harmonically convex functions, complex var. elliptic eqn. 60 (6) (2015), 776-786. [16] m. a. noor, k. i. noor, m. u. awana and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b sci. bull. serai a. 77 (1) (2015), 5-16. [17] m. z. sarikaya, on new hermite hadamard fejér type integral inequalities, stud. univ. babeş-bolyai math. 57 (3) (2012), 377–386. [18] y. shuang, h. p. yin, f. qi, hermite-hadamard type integral inequalities for geometric-arithmetically s-convex functions, analysis 33 (2013), 1001–1010. [19] b. -y. xi and f. qi, hemite-hadamard type inequalities for geometrically r-convex functions, studia scientiarum mathematicarum hungarica 51 (4) (2014), 530–546. [20] t. y. zhang, a. p. ji, f. qi, some inequalities of hermite-hadamard type for ga-convex functions with applications to means. le matematiche, 48 (1) (2013), 229–239. [21] t. -y. zhang, a. -p. ji and f. qi, integral inequalities of hermite-hadamard type for harmonically quasi-convex functions, proceedings of the jangjeon mathematical society, 16(3) (2013), 399-407. 1school of computer science and applied mathematics, university of the witwatersrand, private bag 3, wits 2050, johannesburg, south africa 2school of engineering and science, victoria university, po box 14428 melbourne city, mc 8001, australia ∗corresponding author: m amer latif@hotmail.com 1. introduction 2. main results 3. applications to special means references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 80-86 http://www.etamaths.com on existence of solutions to the caputo type fractional order three-point boundary value problems b.m.b. krushna1,∗ and k.r. prasad2 abstract. in this paper, we establish the existence of solutions to the fractional order three-point boundary value problems by utilizing banach contraction principle and schaefer’s fixed point theorem. 1. introduction fractional calculus deals with integration and differentiation of an arbitrary real order. fractional order derivatives provide a new approach for modeling complex phenomena in physics, mechanics, control systems, flow in porous media, signal and image processing, aerodynamics, electromagnetics, viscoelasticity and especially dealing with memory and hereditary properties of various real materials [5, 6, 9, 11]. compared to integer order models, the fractional order models offer better description of underlying processes. in consequence, fractional order differential equations have achieved great deal of interest and attention of researchers [1, 2, 4, 7, 10, 12, 13, 14, 16, 17, 18]. zhang [20] obtained the existence and uniqueness of solutions for two-point fractional order boundary value problem dα0+u(t) = f ( t,u(t) ) , t ∈ (0, 1), u(0) + u′(0) = 0, u(1) + u′(1) = 0, where dα 0+ is the caputo fractional order derivative and 1 < α ≤ 2. shi and zhang [17] given sufficient conditions for the existence of at least one solution for fractional order boundary value problem dδu(t) + g(t,u) = 0, t ∈ (0, 1), u(0) = a, u(1) = b, where 1 < δ ≤ 2, g : [0, 1] ×r → r and dδ is the caputo fractional order derivative, using upper and lower solutions method. benchohra, hamani and ntouyas [3] derived sufficient conditions for the existence of at least one solution to the fractional order boundary value problem cdαu(t) = f ( t,u(t) ) , 0 < t < t, u(0) = g(u), u(t) = ut , where 1 < α ≤ 2, cdα is the caputo fractional order derivative, by using schaefer’s fixed point theorem. also they established criteria for the uniqueness of solutions by virtue of the banach fixed point theorem. in [8], the authors studied the existence and uniqueness of solutions to the boundary value problem dαu(t) = f(t,u(t),u′(t)), t ∈ (0, 1), u(0) = 0, dpu(1) = δdpu(η), 2010 mathematics subject classification. 26a33, 34a08, 34b15. key words and phrases. fractional order derivative; boundary value problem; solution. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 80 existence of solutions to the caputo type fbvps 81 where d is the caputo fractional order derivative, 1 < α ≤ 2, 0 < δ < p < 1, 0 < η ≤ 1, by using some fixed point theorem. in this paper we consider the fractional order three-point boundary value problem (1.1) cdqy(t) = f ( t,y(t) ) , t ∈ (0, 1), (1.2)   y(0) + y′(0) + y′′(0) =0, y(η) + y′(η) + y′′(η) =0, y(1) + y′(1) + y′′(1) =0, where q ∈ (2, 3],η ∈ (0, 1) and cdq is the standard caputo fractional order derivative. we assume that the following conditions hold throughout the paper: (a1) f : [0, 1] ×r → r is continuous, (a2) there exists a positive constant m such that ∣∣f(t,y)∣∣ ≤m, for each t ∈ [0, 1] and y ∈ r, (a3) there exists a positive real constant ξ such that |f(t,y) −f(t,y)| ≤ ξ|y −y| for each t ∈ [0, 1], for all y, y ∈ r. 1.1. preliminaries. in this section, we present some definitions and lemmas that are useful in the proof of our main results. definition 1.1. [11] the riemann–liouville fractional integral of order p > 0 of a function f : [0, +∞) → r is given by i p 0+ f(t) = 1 γ(p) ∫ t 0 (t− τ)p−1f(τ)dτ, provided the right-hand side is defined. definition 1.2. [11] the caputo fractional derivative of order α > 0 of a function f : [a, +∞) → r is given by c a d α t f(t) = 1 γ(α−n) ∫ t a f(n)(τ) (t− τ)α+1−n dτ, (n− 1 < α < n), provided the right-hand side is defined. lemma 1.1. [11] let α > 0, then the fractional order differential equation cdαu(t) = 0 has solution, u(t) = c0 + c1t + c2t 2 + · · · + cn−1tn−1, ci ∈ r, i = 0, 1, 2, · · ·,n− 1, n = [α] + 1. lemma 1.2. [11] let α > 0, then iα cdαu(t) = u(t) + c0 + c1t + c2t 2 + · · ·+ cn−1tn−1 for ci ∈ r, i = 0, 1, 2, · · ·,n− 1, n = [α] + 1. the rest of the paper is organized as follows. in section 2, we establish the conditions for the existence of solutions to the fractional order boundary value problem (1.1)-(1.2) by using different fixed point theorems such as banach contraction principle and schaefer’s fixed point theorem. in section 3, we present examples to illustrate the applicability of the conditions. 2. main results in this section, by stating some lemmas, we establish sufficient conditions for the existence of solutions to the fractional order boundary value problem (1.1)-(1.2) using certain fixed point theorems. let b = cq ( [0, 1],r ) be the banach space of all continuous functions from [0, 1] into r equipped with the norm ‖y‖ = max t∈[0,1] ∣∣y(t)∣∣. 82 krushna and prasad lemma 2.1. let ∆ = (η2 − η)γ(q) 6= 0. assume that condition (a1) is satisfied. a function y ∈ cq ( [0, 1],r ) is a solution of the fractional integral equation (2.1)   y(t) = ∫ t 0 (t−s)q−1 γ(q) f ( s,y(s) ) ds + [ t2 − 3t + 1 ∆ ] × ∫ η 0 φ(s)f ( s,y(s) ) ds + [ η2(t− 1) + ηt(2 − t) ∆ ] × ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds, where (2.2) { φ(s) =(η −s)q−1 + (q − 1)(η −s)q−2 + ( q2 − 3q + 2 ) (η −s)q−3, ψ∗(s) =(1 −s)q−1 + (q − 1)(1 −s)q−2 + ( q2 − 3q + 2 ) (1 −s)q−3, if and only if y is a solution of the fractional order boundary value problem (1.1)-(1.2). proof. let y(t) ∈ cq[0, 1] be the solution of fractional order boundary value problem (1.1)-(1.2). an equivalent integral equation for (1.1) and is given by (2.3) y(t) = ∫ t 0 (t−s)q−1 γ(q) f(s,y(s))ds + k1 + k2t + k3t 2. using the conditions (1.2) to y(t) in (2.3) and by algebraic calculations, we can get k1 = 1 ∆ [∫ η 0 φ(s)f ( s,y(s) ) ds−η2 ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds ] k2 = 1 ∆ [ (η2 + 2η) ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds− 3 ∫ 1 0 φ(s)f ( s,y(s) ) ds ] k3 = 1 ∆ [∫ η 0 φ(s)f ( s,y(s) ) ds−η ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds ] . by substituting the values of k1,k2,k3 in (2.3), we obtain (2.1). this completes the proof. � now we establish the existence of solution to the fractional order boundary value problem (1.1)-(1.2) by an application of banach contraction principle [15]. lemma 2.2. (banach contraction principle) let (x,d) be a complete metric space. let t : x → x be a contraction. that is, there exists λ ∈ [0, 1) such that d(tx,ty) ≤ λd(x,y), for all x,y ∈ x. then there exists a unique fixed point of t. theorem 2.3. assume that the conditions (a1) and (a3) are satisfied. if (2.4) ζ = ∣∣∣∣ξ(η + 1)(ηq−2 + 1)γ(q + 1)(η − 1) ∣∣∣∣ < 1, then the fractional order boundary value problem (1.1)-(1.2) has a solution on [0, 1]. proof. we transform the fractional order boundary value problem (1.1)-(1.2) in to a fixed point problem by defining an operator t. let t : b →b be the operator defined by (2.5)   ty(t) = ∫ t 0 (t−s)q−1 γ(q) f ( s,y(s) ) ds− [t2 − 3t + 1 ∆ ] × ∫ η 0 φ(s)f ( s,y(s) ) ds − [η2(t− 1) + η(2 − t)t ∆ ] × ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds. clearly the fixed points of the operator t are the solutions of the fractional order boundary value problem (1.1)-(1.2). now we show that the operator t is a contraction mapping. let y, y ∈ b and existence of solutions to the caputo type fbvps 83 using (2.2), then for each t ∈ [0, 1], we have ∣∣ty(t) −ty(t)∣∣ ≤∫ t 0 (t−s)q−1 γ(q) ∣∣∣f(s,y(s))−f(s,y(s))∣∣∣ds + ∣∣∣∣t2 − 3t + 1∆ ∣∣∣∣× ∫ η 0 φ(s) ∣∣∣f(s,y(s))−f(s,y(s))∣∣∣ds + ∣∣∣∣η2(1 − t) + η(t− 2)t∆ ∣∣∣∣× ∫ 1 0 ψ∗(s) ∣∣∣f(s,y(s))−f(s,y(s))∣∣∣ds ≤ { (t−s)q qγ(q) + 1 |∆| × ∫ η 0 φ(s)ds + ∣∣∣ η ∆ ∣∣∣×∫ 1 0 ψ∗(s)ds }[ ξ‖y −y‖∞ ] ≤ ∣∣∣∣ξ(ηq−2 + 1)(η + 1)(η − 1)γ(q + 1) ∣∣∣∣∥∥y −y∥∥∞. hence ‖ty(t)−ty(t)‖∞ ≤ ζ‖y−y‖. therefore t is a contraction mapping. by the contraction mapping principle, the operator t has a fixed point and is the solution of the fractional order boundary value problem (1.1)-(1.2). � next we establish the result by applying schaefer’s fixed point theorem [19]. lemma 2.4. (schaefer′s fixed point theorem) let f : x → x be a completely continuous operator. if the set e(f) = { x ∈ x : x = λ∗fx, for some λ∗ ∈ [0, 1] } is bounded. then f has fixed points. theorem 2.5. assume that the conditions (a1)-(a2) are satisfied. then the fractional order boundary value problem (1.1)-(1.2) has at least one solution on [0, 1]. proof. we prove that the operator t is defined by (2.5) has a fixed point by utilizing schaefer’s fixed point theorem. now we establish the result in 4 steps. step 1. the operator t given by (2.5) is continuous. let {yn} be a sequence such that yn → y in b. using (2.2) and for each t ∈ [0, 1], we have ∣∣tyn(t) −ty(t)∣∣ ≤∫ t 0 (t−s)q−1 γ(q) ∣∣∣f(s,yn(s))−f(s,y(s))∣∣∣ds + ∣∣∣t2 − 3t + 1 ∆ ∣∣∣×∫ η 0 φ(s) ∣∣∣f(s,yn(s))−f(s,y(s))∣∣∣ds + ∣∣∣∣η2(1 − t) + ηt(t− 2)∆ ∣∣∣∣×∫ 1 0 ψ∗(s) ∣∣∣f(s,yn(s))−f(s,y(s))∣∣∣ds ≤ ∫ t 0 (t−s)q−1 γ(q) max s∈[0,1] ∣∣∣f(s,yn(s))−f(s,y(s))∣∣∣ds+ 1 |∆| × ∫ η 0 φ(s) max s∈[0,1] ∣∣∣f(s,yn(s))−f(s,y(s))∣∣∣ds+ |η| |∆| × ∫ 1 0 ψ∗(s) max s∈[0,1] ∣∣∣f(s,yn(s)) −f(s,y(s))∣∣∣ds ≤ ‖f ( ·,yn(·) ) −f ( ·,y(·) ) ‖ γ(q) × ∫ t 0 (t−s)q−1ds+ 1 |η −η2| × ∫ η 0 φ(s)ds + ∣∣∣∣ ηη −η2 ∣∣∣∣× ∫ 1 0 ψ∗(s)ds ≤ ∣∣∣∣(η + 1)(ηq−2 + 1)γ(q + 1) ∣∣∣∣∥∥∥f(·,yn(·))−f(·,y(·))∥∥∥. 84 krushna and prasad since f is continuous, we have ∥∥∥tyn(t) −ty(t)∥∥∥ ≤ ∣∣∣∣(η + 1)(ηq−2 + 1)γ(q + 1)(n− 1) ∣∣∣∣∥∥∥f(·,yn(·))−f(·,y(·))∥∥∥ → 0 as n →∞. therefore the operator t is continuous. step 2. the operator t maps bounded sets in to bounded sets in b. now we will show that for any ϑ > 0, there exits a positive constant m∗ such that for each y ∈ bϑ = { y ∈ b : ‖y‖∞ ≤ ϑ } , we have ‖ty‖∞ ≤ m∗. by the condition (a2) and (2.2), for each t ∈ [0, 1], we have |ty(t)| ≤ ∫ t 0 (t−s)q−1 γ(q) ∣∣∣f(s,y(s))∣∣∣ds + ∣∣∣∣t2 − 3t + 1∆ ∣∣∣∣× ∫ η 0 φ(s) ∣∣∣f(s,y(s))∣∣∣ds + ∣∣∣∣η(1 − t) + ηt(t− 2)∆ ∣∣∣∣× ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds ≤m {∫ t 0 (t−s)q−1 γ(q) ds + 1 |∆| × ∫ η 0 φ(s)ds + ∣∣∣ η ∆ ∣∣∣×∫ 1 0 ψ∗(s)ds } ≤m ∣∣∣∣(η + q)(1 + ηq−2)γ(q + 1)(η − 1) ∣∣∣∣ . thus ‖ty‖∞ ≤m ∣∣∣∣(η + q)(1 + ηq−2)γ(q + 1)(η − 1) ∣∣∣∣ = m∗. step 3. the operator t maps bounded sets into equicontinuous sets b. let t1, t2 ∈ (0, 1], t1 < t2, bϑ be a bounded set of b as in step 2 and y ∈ bϑ. then |ty(t2) −ty(t1)| ≤ ∫ t1 0 [(t2 −s)q−1 − (t1 −s)q−1] γ(q) f ( s,y(s) ) ds− ∫ t2 t1 (t2 −s)q−1 γ(q) f ( s,y(s) ) ds − [ (t22 − 3t2 + 1) − (t21 − 3t1 + 1) ∆ ] × ∫ η 0 f ( s,y(s) ) φ(s)ds − [ η2(t22 − t21) + η(t21 − t22 + 2t2 − 2t1) ∆ ] × ∫ 1 0 f ( s,y(s) ) ds ≤m {[(t2 − t1)q − (t2 − t1)q γ(q) ] − (t22 − t21) − 3(t2 − t1) ∆ [ ηq + qηq−1 + q(q − 1)ηq−2 ] − (1 + q2)η2 [ (η − 1)(t22 − t21) + (2t2 − 2t1) ] ∆ } . as t2 → t1, the right hand side of the above inequality tends to zero. from steps 1 to 3, we can conclude that t : b →b is completely continuous. step 4. the set a = { y ∈b : y = λty for some 0 < λ < 1 } is bounded. let y ∈ a, then y = λty for some 0 < λ < 1. therefore, for each t ∈ [0, 1], we have y(t) =λ {∫ t 0 (t−s)q−1 γ(q) f ( s,y(s) ) ds− [ t2 − 3t + 1 ∆ ] × ∫ η 0 φ(s)f ( s,y(s) ) ds − [ η2(t− 1) + ηt(2 − t) ∆ ] × ∫ 1 0 ψ∗(s)f ( s,y(s) ) ds } existence of solutions to the caputo type fbvps 85 and ∣∣∣ty(t)∣∣∣ ≤mλ {∫ t 0 (t−s)q−1 γ(q) ds + ∣∣∣∣t2 − 3t + 1∆ ∣∣∣∣× ∫ η 0 φ(s)ds + ∣∣∣∣η2(t− 1) + ηt(2 − t)∆ ∣∣∣∣× ∫ 1 0 ψ∗(s)ds } ≤mλ {∫ t 0 (t−s)q−1 γ(q) ds + ∫ η 0 φ(s) ∆ ds + η ∫ 1 0 ψ∗(s) ∆ ds } ≤ m(1 + η)(1 + ηq−1) γ(q + 1) . thus for every t ∈ [0, 1], we have ∥∥ty∥∥∞ ≤m ∣∣∣∣(1 + η)(1 + ηq−1)γ(q + 1) ∣∣∣∣ . this shows that the set a is bounded. by schaefer’s fixed point theorem, the operator t has a fixed point which is a solution of the fractional order boundary value problem (1.1)-(1.2). � 3. examples in this section, we give two examples to illustrate the usefulness of our main results. example 3.1 consider the fractional order three-point boundary value problem, (3.1) cd2.6y(t) = 3e−2ty (19 + 5e2t)(3 + y) , t ∈ (0, 1), (3.2)   y(0) + y′(0) + y′′(0) =0, y (2 5 ) + y′ (2 5 ) + y′′ (2 5 ) =0, y(1) + y′(1) + y′′(1) =0. for y,y ∈ [0,∞) and t ∈ [0, 1], we have |f(t,y) −f(t,y)| = 3e−2t 19 + 5e2t ∣∣∣∣ y3 + y − y3 + y ∣∣∣∣ ≤ 924∣∣y −y∣∣. now we check the condition (2.4) with ξ = 9 24 , ζ = ∣∣∣∣ 924 ( 2 5 + 1 )[( 2 5 )0.6 + 1 )] γ(3.6) (−3 5 ) ∣∣∣∣ = 0.42 < 1. then all conditions of theorem 2.3 are satisfied. thus by theorem 2.3 the fractional order boundary value problem (3.1)-(3.2) has unique solution on [0, 1]. example 3.2 consider the fractional order three-point boundary value problem, (3.3) cd2.5y(t) = e−ty (1 + et)(1 + y) , t ∈ (0, 1), (3.4)   y(0) + y′(0) + y′′(0) =0, y (1 2 ) + y′ (1 2 ) + y′′ (1 2 ) =0, y(1) + y′(1) + y′′(1) =0. by simple algebraic calculations, one can determine m∗ = ∣∣∣∣ 32 ( 1 + 0.50.5 )] γ(3.5) (−1 2 ) ∣∣∣∣ = 1.6 and m = 12. then the fractional order boundary value problem (3.3)-(3.4) satisfies all conditions of theorem 2.5. hence it has unique solution on [0, 1]. 86 krushna and prasad references [1] z. bai and y. zhang, solvability of fractional three-point boundary value problems with nonlinear growth, appl. math. comput., 218(2011), 1719–1725. [2] m. benchohra, j. henderson, s. k. ntoyuas and a. ouahab, existence results for fractional order functional differential equations with infinite delay, j. math. anal. appl., 338 (2008), 1340–1350. [3] m. benchohra, s. hamani, s.k. ntouyas, boundary value problems for differential equations with fractional order and nonlocal conditions, nonlinear anal. tma., 71(2009), 2391–2396. [4] m. feng, x. zhang and w. ge, new existence results for higher-order nonlinear fractional differential equation with integral boundary conditions, bound. value probl., 2011(2011), article id 720702. [5] a. a. kilbas and j. j. trujillo, differential equations of fractional order methods, results, problems, appl. anal., 78(2001), 153–192. [6] a. a. kilbas, h. m. srivasthava and j. j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies, 204, elsevier science, amsterdam, 2006. [7] r. a. khan, m. rehman and j. henderson, existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, fract. differ. calc., 1(2011), 29–43. [8] r. a. khan and h. khan, existence of solution for a three-point boundary value problem of fractional differential equation, journal of fractional calculus and applications, 5(2014), 156–164. [9] k. s. miller and b. ross, an introduction to fractional calculus and fractional differential equations, john wiley and sons, new york, 1993. [10] z. ouyang and g. li, existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance, bound. value probl., 2012(2012), article id 68, 1–13. [11] i. podulbny, fractional diffrential equations, academic press, san diego, 1999. [12] k. r. prasad and b. m. b. krushna, multiple positive solutions for a coupled system of riemann–liouville fractional order two-point boundary value problems, nonlinear stud., 20(2013), 501–511. [13] k. r. prasad and b. m. b. krushna, eigenvalues for iterative systems of sturm–liouville fractional order two-point boundary value problems, fract. calc. appl. anal., 17(2014), 638–653. [14] k. r. prasad and b. m. b. krushna, existence of solutions for a coupled system of three-point fractional order boundary value problems, differ. equ. appl., 7(2015), 187–200. [15] m. rehman and r. a. khan, existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, appl. math. lett., 23 (2010), 1038–1044. [16] m. rehman, r. a. khan and n. a. asif, three point boundary value problems for nonlinear fractional differential equations, acta mathematica scientia, 31(2011), 1337–1346. [17] a. shi and s. zhang, upper and lower solutions method and a fractional differential equation boundary value problem, electron. j. qual. theory differ. equ., 30(2009), 1–13. [18] g. wang, s. k. ntouyas and l. zhang, positive solutions of the three-point boundary value problem for fractional order differential equations with an advanced argument, advances in difference equations, 2011(2011), article id 2. [19] j. r. wang, l. lv and w. wei, differential equations of fractional order α ∈ (2, 3) with boundary value conditions in abstract banach spaces, math. commun., 17(2012), 371–387. [20] s. zhang, positive solutions for boundary value problems of nonlinear fractional differential equations, electron. j. differential equations, 36(2006), 1–12. 1department of mathematics, mvgr college of engineering (autonomous), vizianagaram, 535 005, india 2department of applied mathematics, andhra university, visakhapatnam, 530 003, india ∗corresponding author: muraleebalu@yahoo.com international journal of analysis and applications issn 2291-8639 volume 7, number 1 (2015), 1-15 http://www.etamaths.com p-frame multiresolution analysis related to the walsh functions f. a. shah abstract. a generalization of the notion of p-multiresoltion analysis on a half-line, based on the theory of shift-invariant spaces is considered. in contrast to the standard setting, the associated subspace v0 of l 2(r+) has a frame, a collection of translates of the scaling function ϕ of the form {ϕ(· k)}k∈z+ , where z+ is the set of non-negative integers. we investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of p-frame multiresoltion analysis (p-fmra) on positive half-line r+. finally, we establish a complete characterization of all p-wavelet frames associated with p-fmra on positive half-line r+ using the shift-invariant space theory. 1. introduction in the early nineties a general scheme for the construction of wavelets was defined. this scheme is based on the notion of multiresolution analysis (mra) introduced by mallat [9]. in recent years, the concept of mra has become an important tool in mathematics and applications. it provides a natural framework for understanding of wavelet bases, bases that consist of the scaled and integer translated versions of a finite number of functions. mathematically, an mra is an increasing sequence of closed subspaces {vj}j∈z of l 2(r) such that ⋃ j∈zvj is dense in l 2(r), ⋂ j∈zvj = {0} and which satisfies f(x) ∈ vj if and only if f(2x) ∈ vj+1. furthermore, there should exist an element ϕ ∈ v0 such that the collection of integer translates of ϕ,{ϕ(·−k) : k ∈ z} is a complete orthonormal system for v0. the dilation factor 2 can be replaced by any integer m ≥ 2 and in that case one needs m −1 wavelets to generate the whole space l2(r). a similar generalization of multiresolution analysis can be made in higher dimensions by considering matrix dilations (see [2]). on the other hand, there is considerable interest both in mathematics and its applications in the study of compactly supported orthonormal scaling functions and wavelets with an arbitrary dilation factor p ∈ n,p ≥ 2. the motivation comes partly from signal processing and numerical applications, where such wavelets are useful in image compression and feature extraction because of their small support and multifractal structure. farkov [4] has given the general construction of all compactly supported orthogonal p-wavelets in l2(r+) and proved necessary and 2010 mathematics subject classification. 42c15, 42c40, 42a38, 41a17. key words and phrases. frame; wavelet frame; p-multiresoltion analysis; scaling function; shift invariant; walsh-fourier transform. c©2015 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 shah sufficient conditions for scaling filters with pn many terms (for any integers p,n ≥ 2) to generate an p-mra in l2(r+). these studies were continued by farkov and his colleagues [5, 6] where they have given some new algorithms for constructing the corresponding biorthogonal and non-stationary wavelets related to the walsh polynomials on the positive half-line r+. on the other hand, shah and debnath [19] have constructed dyadic wavelet frames on the positive half-line r+ using the walsh-fourier transform and have established a necessary condition and a sufficient condition for the system { ψj,k(x) = 2 j/2ψ(2jx k) : j ∈ z,k ∈ z+ } to be a frame for l2(r+). further, wavelet packets related to the walsh functions are discussed in a series of papers by the author [14-17]. in his recent, shah [18] has given the construction of tight wavelet frames generated by the walsh polynomials on a half-line r+ by following the procedure of daubechies et al. [3] via extension principles. he also provide a sufficient condition for finite number of functions to form a tight wavelet frame and established a general principle for constructing tight wavelet frames on r+. recently, meenaski et al. [10] have introduced the notion of non-uniform multiresolution analysis (numra) on a half-line r+ and have also established the necessary and sufficient condition for the existence of corresponding wavelets on r+. since the use of multiresolution analysis has proven to be a very efficient tool in wavelet theory mainly because of its simplicity, it is of interest to try to generalize this notion as much as possible while preserving its connection with wavelet analysis. in this connection, benedetto and li considered the dyadic semi-orthogonal frame multiresolution analysis of l2(r) with a single scaling function and successfully applied the theory in the analysis of narrow band signals [1]. the characterization of the dyadic semi-orthogonal frame multiresolution analysis with a single scaling function admitting a single frame wavelet whose dyadic dilations of the integer translates form a frame for l2(r) was obtained independently by benedetto and treiber by a direct method [2], and by kim and lim by using the theory of shiftinvariant spaces [8]. later on, xiaojiang [21] extended the results of benedetto and li’s theory of fmra to higher dimensions with arbitrary integral expansive matrix dilations, and has established the necessary and sufficient conditions to characterize semi-orthogonal multiresolution analysis frames for l2(rn). on the other hand, zhang [22] has given the characterization of generalized frame mra on r and has provided a general algorithm for the construction of non-mra wavelets by means of the fourier transforms. in this paper, we introduce the notion of p-frame multiresolution analysis (pfmra) on positive half-real line r+ by extending the above described methods. we first investigate the properties of multiresolution subspaces, which will provide the quantitative criteria for the construction of p-fmras. we also show that the scaling property of an p-fmra also holds for the wavelet subspaces and that the space l2(r+) can be decomposed into the orthogonal sum of these wavelet subspaces. finally, we study the characterization of p-wavelet frames associated with p-fmra on positive half-line r+ using the shift-invariant space theory. the paper is structured as follows. in section 2, we introduce some notations and preliminaries related to the operations on positive half-line r+ including the p-frame multiresolution analysis related to the walsh functions 3 definition of the walsh-fourier transform. the notion of p-fmra of l2(r+) is introduced in section 3 and its quantitative criteria is given by means of the theorem 3.10. in section 4, we establish a complete characterization of p-wavelet frames generated by a finite number of mother wavelets on r+. 2. walsh-fourier analysis we start this section with certain results on walsh-fourier analysis. we present a brief review of generalized walsh functions, walsh-fourier transforms and its various properties. as usual, let r+ = [0, +∞), z+ = {0, 1, 2, . . .} and n = z+ −{0}. denote by [x] the integer part of x. let p be a fixed natural number greater than 1. for x ∈ r+ and any positive integer j, we set xj = [p jx](mod p), x−j = [p 1−jx](mod p), (2.1) where xj,x−j ∈ {0, 1, . . . ,p− 1}. it is clear that for each x ∈ r+, there exist k = k(x) in n such that x−j = 0 ∀j > k. consider on r+ the addition defined as follows: x⊕y = ∑ j<0 ζjp −j−1 + ∑ j>0 ζjp −j, with ζj = xj + yj(mod p), j ∈ z \{0} , where ζj ∈ {0, 1, . . . ,p− 1} and xj, yj are calculated by (2.1). as usual, we write z = x y if z ⊕ y = x, where denotes subtraction modulo p in r + . for x ∈ [0, 1), let r0(x) is given by r0(x) =   1, if x ∈ [0, 1/p) ε`p, if x ∈ [ `p−1, (` + 1)p−1 ) , ` = 1, 2, . . . ,p− 1, where εp = exp(2πi/p). the extension of the function r0 to r+ is given by the equality r0(x + 1) = r0(x), x ∈ r+. then, the generalized walsh functions {wm(x) : m ∈ z+} are defined by w0(x) ≡ 1 and wm(x) = k∏ j=0 ( r0(p jx) )µj where m = ∑k j=0 µjp j, µj ∈{0, 1, . . . ,p− 1} , µk 6= 0. they have many properties similar to those of the haar functions and trigonometric series, and form a complete orthogonal system. further, by a walsh polynomial we shall mean a finite linear combination of walsh functions. for x,y ∈ r + , let χ(x,y) = exp  2πi p ∞∑ j=1 (xjy−j + x−jyj)   , (2.2) 4 shah where xj,yj are given by (2.1). we observe that χ ( x, m pn ) = χ ( x pn ,m ) = wm ( x pn ) , ∀ x ∈ [0,pn), m,n ∈ z+, and χ(x⊕y,z) = χ(x,z) χ(y,z), χ(x y,z) = χ(x,z) χ(y,z), where x,y,z ∈ r+ and x ⊕ y is p-adic irrational. it is well known that systems {χ(α,.)}∞α=0 and {χ(·,α)} ∞ α=0 are orthonormal bases in l 2[0,1] (see golubov et al.[7]). the walsh-fourier transform of a function f ∈ l1(r+) ∩ l2(r+) is defined by f̂(ξ) = ∫ r+ f(x) χ(x,ξ) dx, (2.3) where χ(x,ξ) is given by (2.2). the walsh-fourier operator f : l1(r+)∩l2(r+) → l2(r+), ff = f̂, extends uniquely to the whole space l2(r+). the properties of the walsh-fourier transform are quite similar to those of the classic fourier transform (see [7, 13]). in particular, if f ∈ l2(r+), then f̂ ∈ l2(r+) and∥∥∥f̂∥∥∥ l 2 (r+) = ∥∥f∥∥ l2(r+). definition 2.1 let h be a separable hilbert space. a sequence {fk}k∈z in h is called a frame for h if there exist constants a and b with 0 < a ≤ b < ∞ such that a‖f‖2 ≤ ∑ k∈z ∣∣〈f,fk〉∣∣2 ≤ b‖f‖2 (2.4) for all f ∈ h. the largest constant a and the smallest constant b satisfying (2.4) are called the upper and the lower frame bound, respectively. a frame is said to be tight if it is possible to choose a = b and a frame is said to be exact if it ceases to be a frame when any one of its elements is removed. an exact frame is also known as a riesz basis. the following theorem gives us an elementary characterization of frames. theorem 2.2. a sequence {fk}k∈z in a hilbert space h is a frame for h if and only if there exists a sequence a = {ak} ∈ l2(z) with ‖a‖l2(z) ≤ c‖f‖,c > 0 such that f(x) = ∑ k∈z akfk(x) and ∑ k∈z ∣∣〈f,fk〉∣∣2 < ∞, for every f ∈ h. for any integer p ≥ 2, let dp : l2(r+) → l2(r+) be the unitary operator defined via dpf(x) = p 1/2f(px). for k ∈ z+, let τk : l2(r+) → l2(r+) denotes the unitary translation operator such that τkf(x) = f(x k). p-frame multiresolution analysis related to the walsh functions 5 our study uses the theory of shift-invariant spaces developed in [11, 12] and the references therein. a closed subspace s of l2(r+) is said to be shift-invariant if τkf ∈ s whenever f ∈ s and k ∈ z+. a closed shift-invariant subspace s of l2(r+) is said to be generated by φ ⊂ l2(r+) if s = span{τkϕ(.) := ϕ(. k) : k ∈ z+,ϕ ∈ φ}. the cardinality of a smallest generating set φ for s is called the length of s which is denoted by |s|. if |s| = finite, then s is called a finite shift-invariant space (fsi) and if |s| = 1, then s is called a principal shift-invariant space (psi). moreover, the spectrum of a shift-invariant space is defined to be σ(s) = { ξ ∈ [0, 1] : ŝ(ξ) 6= {0} } , where ŝ(ξ) = { f̂(ξ ⊕k) ∈ l2(z+) : f ∈ s,k ∈ z+ } . 3. p-frame multiresolution analysis on a positive half-line we first introduce the notion of a p-frame multiresolution analysis (p-fmra) of l2(r+). definition 3.1. a p-frame multiresolution analysis of l2(r+) is a sequence of closed subspaces {vj}j∈z such that (i) vj ⊂ vj+1 for all j ∈ z; (ii) ⋃ j∈zvj is dense in l 2(r+) and ⋂ j∈zvj = {0}; (iii) f(·) ∈ vj if and only if f(p·) ∈ vj+1 for all j ∈ z; (iv) the function f lies in v0 implies that the collection f(· k) lies in v0, for all k ∈ z+. (v) the sequence {τkϕ := ϕ(· k) : k ∈ z+} is a frame for the subspace v0. the function ϕ is known as the scaling function while the subspaces vj’s are known as approximation spaces or multiresolution subspaces. an p-fmra is said to be non-exact and respectively exact if the frame for the subspace v0 is non-exact and respectively exact. in p-mra’s studied in [4], the frame condition is replaced by that of an orthonormal basis or an exact frame. next, we establish several properties of multiresolution subspaces that will help in the construction of p-fmra’s. the following proposition shows that for every j ∈ z, the sequence {ϕj,k}k∈z+ , where ϕj,k(x) = p j/2ϕ(pjx k), (3.1) is a frame for vj. proposition 3.2. let {τkϕ} be a frame for v0 = span{τkϕ : k ∈ z+} and vj = { f ∈ l2(r+) : f(p−j.) ∈ v0 } , j ∈ z. (3.2) then, the sequence {ϕj,k : k ∈ z+} defined in (3.1) is a frame for vj with the same bounds as those for v0. 6 shah proof. for any f ∈ vj, we have∑ k∈z+ ∣∣〈d−jf,τkϕ〉∣∣2 = ∑ k∈z+ ∣∣∣∣ ∫ r+ p−jf(p−jx) pj/2ϕ(x k) dx ∣∣∣∣2 = ∑ k∈z+ ∣∣∣∣ ∫ r+ f(x) pj/2ϕ(pjx k) dx ∣∣∣∣2 = ∑ k∈z+ |〈f,ϕj,k〉|2. since {τkϕ}k∈z+ be a frame for v0, therefore, we have a‖f‖22 = a‖d −jf‖22 ≤ ∑ k∈z+ |〈f,ϕj,k〉|2 ≤ b‖d−jf‖22 = b‖f‖ 2 2. this completes the proof of the proposition. next, we characterize all functions of fsi space in terms of its walsh-fourier transform. proposition 3.3. let {τkϕ : k ∈ z+,ϕ ∈ ω} be a frame for its closed linear span v , where ω = {ϕ1,ϕ2, ...,ϕl} ⊂ l2(r+). then, f ∈ l2(r+) lies in v if and only if there exist periodic functions h` ∈ l2[0, 1],` = 1, ...,l such that f̂(ξ) = l∑ `=1 h`(ξ)ϕ̂`(ξ). (3.3) proof. since the system {τkϕ : k ∈ z+,ϕ ∈ ω} is a frame for v , then by theorem 2.2, there exist a sequence { a`k } ∈ l2(z+), for ` = 1, ...,l such that f(x) = l∑ `=1 ∑ k∈z+ a`kϕ`(x k). (3.4) taking walsh-fourier transform on both sides of (3.4), we obtain f̂(ξ) = l∑ `=1 h`(ξ)ϕ̂`(ξ), where h`(ξ) = ∑ k∈z+ a ` kχk(ξ) are the periodic functions in l 2[0, 1]. the converse is established by taking h` as above and applying the inverse walsh-fourier transform on both sides of (3.3). we now study some properties of the multiresolution subspaces vj of the form (3.2) by means of the walsh-fourier transform. proposition 3.4. let {τkϕ} be a frame for v0 = span{τkϕ : k ∈ z+} and for j ∈ z, define vj by (3.2). then for any function ψ ∈ v1, there exist periodic p-frame multiresolution analysis related to the walsh functions 7 functions g ∈ l2[0, 1] such that ψ̂(pξ) = p−1/2g(ξ)ϕ̂(ξ). (3.5) proof. by the definition of vj, it follows that ψ(p −1x) ∈ v0. by proposition 3.3, there exists a periodic function g ∈ l2[0, 1] such that ( ψ(p−1x) )∧ = ψ̂(pξ) = p−1/2g(ξ)ϕ̂(ξ) lies in l2(r+). the following theorem establishes a sufficient condition to ensure that the nesting property holds for the subspaces vj’s. theorem 3.5. let {τkϕ} be a frame for v0 = span{τkϕ : k ∈ z+} and for j ∈ z, define vj by (3.2). assume that there exists a periodic function h ∈ l∞[0, 1] such that ϕ̂(ξ) = p−1/2h(p−1ξ)ϕ̂(p−1ξ). (3.6) then, vj ⊆ vj+1, for every j ∈ z. proof. given any f ∈ vj, there exist as sequence {ak}k∈z+ ∈ l 2(z+) such that f(x) = ∑ k∈z+ pj/2akϕ(p jx k). (3.7) let m0(ξ) = ∑ k∈z+ akχk(ξ) ∈ l 2[0, 1], m1(p −1ξ) = m0(ξ)h(p −1ξ). then, clearly m1 lies in l 2[0, 1] as h lies in l∞[0, 1]. therefore, by parsevals identity, there exist a sequence {bk}k∈z+ ∈ l 2(z+) such that m1(ξ) = ∑ k∈z+ bkχk(ξ) lies in l 2(r+). applying the walsh-fourier transform to (3.7), we obtain f̂(ξ) = p −j 2 m0(p −jξ) ϕ̂(p−jξ) = p −j−1 2 m0(p −jξ)h(p−j−1ξ) ϕ̂(p−j−1ξ) = p −j−1 2 m1(p −j−1ξ) ϕ̂(p−j−1ξ).(3.8) implementing inverse walsh-fourier transform to (3.8), we obtain f(x) = p j+1 2 ∑ k∈z+ bk ϕ(p j+1x k). (3.9) thus the function f lies in vj+1 by proposition 3.2. moreover, it is easy to verify that the function h in (3.6) is not unique. the following theorem is a converse to theorem 3.5. theorem 3.6. let {τkϕ} be a frame for v0 = span{τkϕ : k ∈ z+} and for j ∈ z, define vj by (3.2). assume that v0 ⊆ v1 and φ(ξ) = ‖ϕ̂(ξ ⊕k)‖ 2 l2(z+) . then there exists periodic function h ∈ l∞[0, 1] such that (3.6) holds. 8 shah proof. since {τkϕ}k∈z+ is a frame for v0, therefore, there exist positive constants a and b such that a ≤ φ(ξ) ≤ b a.e on σ(v0). since v0 ⊆ v1, we have ϕ ∈ v1. by proposition 3.4, there exists a periodic function h0 ∈ l2[0, 1] such that ϕ̂(pξ) = p−1/2h0(ξ)ϕ̂(ξ). therefore, we have |ϕ̂(ξ)|2 = p−1 ∣∣h0(p−1ξ)∣∣2 ∣∣ϕ̂(p−1ξ)∣∣2 a.e. (3.10) let s = [0, 1)\σ(v0) and h ∈ l2[0, 1] be a periodic function such that h = h0, a.e on σ(v0) and h is bounded on s by a positive constant c. then, it follows from the above fact that h is not unique so that (3.10) also holds for h, i.e., |ϕ̂(ξ)|2 = p−1|h(p−1ξ)|2|ϕ̂(p−1ξ)|2 a.e. taking n = kp + r, where k ∈ z+ and r = 0, 1, . . . ,p− 1, we have |ϕ̂(ξ ⊕n)|2 = p−1|h(p−1ξ ⊕p−1r)|2|ϕ̂(p−1ξ ⊕rp−1 ⊕k|2 a.e. (3.11) summing up (3.11) for all k ∈ z+ and r = 0, 1, ...,p− 1, we have ∑ n∈z+ |ϕ̂(ξ ⊕n)|2 = p−1 p−1∑ r=0 |h(p−1ξ ⊕p−1r)|2 ∑ k∈z+ |ϕ̂(p−1ξ ⊕rp−1 ⊕k|2 a.e, which is equivalent to φ(ξ) = p−1 p−1∑ r=0 |h(p−1ξ ⊕p−1r)|2 φ(p−1ξ ⊕p−1r) a.e, or φ(pξ) = p−1 p−1∑ r=0 |h(ξ ⊕p−1r)|2 φ(ξ ⊕p−1r) a.e. (3.12) note that φ(pξ) ≤ b a.e and hence, (3.12) becomes p−1∑ r=0 |h(ξ ⊕p−1r)|2 φ(ξ ⊕p−1r) ≤ pb a.e. this implies that for almost every ξ ∈ [0, 1 p ) and r = 0, 1, ...,p− 1, we have |h(ξ ⊕p−1r)|2 φ(ξ ⊕p−1r) ≤ pb. p-frame multiresolution analysis related to the walsh functions 9 further, if φ(ξ ⊕p−1r) = 0, then |h(ξ ⊕p−1r)| ≤ c and if φ(ξ ⊕p−1r) > 0, then we may assume that a ≤ φ(ξ ⊕p−1r) ≤ b. thus, for almost every ξ ∈ [0, 1 p ) and r = 0, 1, . . . ,p− 1, we have |h(ξ ⊕p−1r)|2 ≤ max { c2,pba−1 } . hence h is essentially bounded on the interval [0, 1). this proves the theorem completely. the following two propositions are proved in [4]: proposition 3.7. suppose v0 = span{τkϕ : k ∈ z+} and for each j ∈ z, define vj by (3.2) such that v0 ⊆ v1. assume that |ϕ̂| > 0, a.e on a neighborhood of zero. then, the union ⋃ j∈zvj is dense in l 2(r+). proposition 3.8. let ϕ ∈ l2(r+) and define v0 = span{τkϕ : k ∈ z+}. for each j ∈ z, define vj by (3.2). then, we have ⋂ j∈zvj = {0}. lemma 3.9. let vj be the family of subspaces defined by (3.2) with vj ⊆ vj+1, for each j ∈ z. suppose ϕ ∈ l2(r+) be a non-zero function with v0 = span{τkϕ : k ∈ z+}. then, for every j ∈ z, vj is a proper subspace of vj+1. proof. suppose that v` = v`+1 for some ` ∈ z. let f ∈ vj+1, then for any given j ∈ z, we have f(p−j−1+`+1x) ∈ vj+1. since f(p−j+`x) ∈ v`, therefore f lies in vj and vj = vj+1. hence, ⋂ j∈zvj = v0. therefore, it follows from proposition 3.8 that vj = {0}, which is a contradiction. combining all our results so far, we have the following theorem. theorem 3.10. let ϕ ∈ l2(r+) and define v0 = span{τkϕ : k ∈ z+}. for each j ∈ z, define vj by (3.2) and φ(ξ) = ‖ϕ̂(ξ ⊕k)‖ 2 l2(z+) . suppose that the following hold: (i) a ≤ φ(ξ) ≤ b a.e on σ(v0) (ii) there exists a periodic function h ∈ l∞[0, 1] such that ϕ̂(ξ) = p−1/2h(p−1ξ)ϕ̂(p−1ξ), a.e. (iii) |ϕ̂| > 0, a.e on a neighborhood of zero. then, {vj}j∈z defines a p-frame multiresolution analysis of l 2(r+). proof. since v0 is a shift-invariant subspace of l 2(r+). therefore, the system {τkϕ} forms a frame for v0 with frame bounds a and b. then, it follows from theorem 3.5 and lemma 3.9 that vj ⊂ vj+1, for every j ∈ z. now, by the definition of vj,f lies in vj if and only if f(p −j.) lies in v0, while f(p.) lies in vj+1 if and only if f(p−j−1(p.)) lies in v0. thus, f lies in vj if and only if f(p.) lies in vj+1. further, by 10 shah our assumption (iii) and proposition 3.8, it follows that ⋃ j∈zvj is dense in l 2(r+) and ⋂ j∈zvj = {0}. therefore, the sequence {vj}j∈z satisfies all the conditions to be an p-fmra of l2(r+). in order to construct p-wavelet frames associated with p-fmra on a positive half-line r+, we introduce the orthogonal complement subspaces {wj}j∈z of vj in vj+1. further, it is easy verify that the sequence of subspaces {wj}j∈z also satisfies the scaling property, i.e., wj = { f ∈ l2(r+) : f(p−j·) ∈ w0 } , j ∈ z. (3.13) theorem 3.11. let {vj}j∈z be an increasing sequence of closed subspaces of l2(r+) such that ⋃ j∈zvj is dense in l 2(r+) and ⋂ j∈zvj = {0}. let wj be the orthogonal complement of vj in vj+1, for each j ∈ z. then, the subspaces wj are pairwise orthogonal and l2(r+) = ⊕ j∈z wj. proof. assume that i < j, then 〈fi,fj〉 = 0, for any fi ∈ wj as wi ⊂ vi+1 ⊂ vj. let pj be the orthogonal projection operators from l 2(r+) onto vj, then limj→∞pjf = f, limj→−∞pjf = 0 and wj = {f −pjf : f ∈ vj+1}. therefore, for any f ∈ l2(r+), we have f = ∑ j∈z (pj+1f −pjf). thus, the result of the direct sum follows since pj+1−pj is the orthogonal projector from l2(r+) onto wj. 4. characterization of p-wavelet frames in this section, we give the characterization all p-wavelet frames associated with p-fmra on a half-line r+. first, we shall characterize the existence of a function ψ in w0, where w0 is the orthogonal complement of v0 in v1, by virtue of the analysis filters g and h. theorem 4.1. let h be a periodic function associated with an p-fmra {vj : j ∈ z} such that (3.6) holds. define w0 as the orthogonal complement of v0 in v1. let ψ ∈ v1 such that ϕ̂(ξ) = p−1/2g(ξ/p)ϕ̂(ξ/p). (4.1) where g is a periodic function in l2[0, 1]. then ψ lies in w0 if and only if p−1∑ r=0 h ( p−1ξ ⊕p−1r ) φ ( p−1ξ ⊕p−1r ) g (p−1ξ ⊕p−1r) = 0 a.e. (4.2) proof. we note that ψ lies in w0 if and only if p-frame multiresolution analysis related to the walsh functions 11 〈ψ,τkψ〉 = 〈ψ,ψ(. k)〉 = 0, for all k ∈ z+. (4.3) define f(ξ) = ∑ k∈z+ ϕ̂(ξ ⊕k) ψ̂(ξ ⊕k). then, it is easy to verify that f lies in l1[0, 1] by using monotonic convergence theorem and the plancherel theorem as ∫ 1 0 |f(ξ)|dξ ≤ ∫ 1 0 ∑ k∈z+ ∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ = ∑ k∈z+ ∫ 1 0 ∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ = ∫ r+ ∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ ≤ ∥∥ϕ̂∥∥ 2 ∥∥∥ψ̂∥∥∥ 2 = ∥∥ϕ∥∥ 2 ∥∥ψ∥∥ 2 . now, for a fixed n ∈ z+, let fm be the function defined by fm (ξ) = m∑ k=0 ϕ̂(ξ ⊕k) ψ̂(ξ ⊕k) χn(ξ). then, in view of (3.6) and (4.1), we have fm (ξ) = p−1∑ r=0 ∑ |pk+r|≤m h ( p−1ξ ⊕p−1r ) |ϕ̂ ( p−1ξ ⊕p−1r ⊕k ) |2 g (p−1ξ ⊕p−1r) χn(ξ). (3.4) implementation of monotonic convergence theorem and the cauchy-schwartz inequality yields ∥∥fm −fχn∥∥l2[0,1] ≤ ∫ 1 0 ∑ |k|≥m+1 ∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ = ∑ |k|≥m+1 ∫ 1 0 ∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ = ∑ |k|≥m+1 ∫ k+1 k ∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ ≤ ∫ |ξ|>m ∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ 12 shah ≤ {∫ |ξ|>m |ϕ̂(ξ)|2 dξ }1/2 {∫ |ξ|>m ∣∣∣ψ̂(ξ)∣∣∣ |2dξ }1/2 → 0 as m →∞. thus, lim m→∞ ∥∥fm −fχn∥∥l2[0,1] = 0. (4.5) therefore, there exists a subsequence { fmj } such that lim j→∞ ∥∥fmj −fχn∥∥l2[0,1] = 0, a.e. hence f(ξ) = p−1∑ r=0 p−1h ( p−1ξ ⊕p−1r ) φ ( p−1ξ ⊕p−1r ) g (p−1ξ ⊕p−1r) a.e. using (4.5) and the dominated convergence theorem, we have for all n ∈ z+, 〈ψ,τ−nϕ〉 = ∫ r+ ψ̂(ξ)ϕ̂(ξ)χn(ξ)dξ = ∑ k∈z+ ∫ k+1 k ψ̂(ξ)ϕ̂(ξ)χn(ξ)dξ = lim m→∞ m∑ k=0 ∫ 1 0 ψ̂(ξ ⊕k)ϕ̂(ξ ⊕k)χn(ξ)χk(ξ)dξ = lim m→∞ ∫ 1 0 fm (ξ)dξ = ∫ 1 0 f(ξ)χn(ξ)dξ. consequently, f = 0 a.e, is the necessary and sufficient condition for (4.3) to hold for all n ∈ z+. lemma 4.2. let {wj : j ∈ z} be a sequence of pairwise orthogonal closed subspaces of l2(r+) such that l2(r+) = ⊕ j∈z wj. then, for every f ∈ l 2(r+), there exist fj ∈ wj,j ∈ z such that f(x) = ∑ j∈z fj(x). furthermore,∥∥f∥∥2 2 = ∑ j∈z ∥∥fj∥∥22. proof. for any arbitrary function f ∈ l2(r+), we have lim n→∞ ∥∥∥f − n∑ j=−n fj ∥∥∥ 2 = 0, p-frame multiresolution analysis related to the walsh functions 13 where fj ∈ wj, for each j ∈ z. moreover, for a fixed n ∈ n, we have∥∥∥ n∑ j=−n fj ∥∥∥2 2 = n∑ j=−n ‖fj‖22. since the norm ‖.‖2 is continuous, hence the desired result is obtained by taking n →∞ on both sides of the above equality. theorem 4.3. let ϕ be the scaling function for an p-fmra {vj : j ∈ z} and suppose that wj be the orthogonal complement of vj in vj+1. let ψ = {ψ1,ψ2, . . . ,ψl}⊂ w0. then, the collection fψ = { ψ`j,k(x) := p j/2ψ`(pjx k),j ∈ z,k ∈ z+,` = 1, . . . ,l } (4.6) forms a p-wavelet frame for l2(r+) with frame bounds a and b if and only if { τkψ ` : k ∈ z+,` = 1, . . . ,l } is a frame for w0 with frame bounds a and b. proof. suppose that the system fψ given by (4.6) is a p-wavelet frame for l2(r+) with bounds a and b. therefore, it follows from (3.13) that the family of functions ψ`j,k lies in wj, for ` = 1, . . . ,l,j ∈ z and k ∈ z +. by applying theorem 3.11 to an arbitrary function f ∈ w0, we have∑ j∈z ∑ k∈z+ |〈f,ψ`j,k〉| 2 = ∑ k∈z+ |〈f,τkψ`〉|2. consequently, using the p-wavelet frame condition of the system fψ, we have a‖f‖22 ≤ l∑ `=1 ∑ k∈z+ |〈f,τkψ`〉|2 ≤ b‖f‖22, and it follows that the collection { τkψ ` : k ∈ z+,` = 1, . . . ,l } is a frame for w0. conversely, suppose that the collection { τkψ ` : k ∈ z+,` = 1, . . . ,l } is a frame for w0 with bounds a and b. for any fixed j ∈ z and f ∈ wj, we have from equation (3.13) that f(p−j·) ∈ w0. further, by making use of the fact that 〈 f,ψ`j,k 〉 = ∫ r+ f(x) pj/2ψ`(pjx k) dx and ∥∥∥p−j/2f(p−j.)∥∥∥2 2 = p−j ∫ r+ ∣∣f(p−jx)∣∣2dx = ∥∥f∥∥2 2 , 14 shah we have a ∥∥∥p−j/2f(p−j·)∥∥∥2 2 ≤ l∑ `=1 ∑ k∈z+ ∣∣〈f,ψ`j,k〉∣∣2 ≤ b ∥∥∥p−j/2f(p−j.)∥∥∥2 2 . (4.7) therefore, for a given j ∈ z, the collection { ψ`j,k : k ∈ z +,` = 1, . . . ,l } is a frame for wj with frame bounds a and b. let f be an arbitrary function in l2(r+), then by theorem 3.11 and lemma 4.2, there exist fj ∈ wj such that f = ∑ j∈z fj, and 〈 fi,ψ ` j,k 〉 = 0, i 6= j. thus, we have l∑ `=1 ∑ j∈z ∑ k∈z+ ∣∣〈f,ψ`j,k〉∣∣2 = l∑ `=1 ∑ j∈z ∑ k∈z+ ∣∣∣∑ i∈z 〈 fi,ψ ` j,k 〉∣∣∣2 = l∑ `=1 ∑ j∈z ∑ k∈z+ ∣∣〈fj,ψ`j,k〉∣∣2.(4.8) it follows from (4.7) that a ∑ j∈z ∥∥fj∥∥22 ≤ l∑ `=1 ∑ j∈z ∑ k∈z+ ∣∣〈fj,ψ`j,k〉∣∣2 ≤ b ∑ j∈z ∥∥fj∥∥22. (4.9) by combining (4.8), (4.9) and lemma 4.2, we obtain a ∥∥fj∥∥22 ≤ l∑ `=1 ∑ j∈z ∑ k∈z+ ∣∣〈fj,ψ`j,k〉∣∣2 ≤ b∥∥fj∥∥22. this completes the proof of the theorem. acknowledgement the author was partially supported by the university of kashmir, under seed money grant scheme, letter no. f(seed money grant)res/ku; dated: march 26, 2013. references [1] j. j. benedetto and s. li, the theory of multiresolution analysis frames and applications to filter banks, appl. comput. harmon. anal., 5 (1998), 398-427. [2] j. j. benedetto and o. m. treiber, wavelet frames: multiresolution analysis and extension principle, in wavelet transforms and time-frequency signal analysis, l. debnath, editor, birkhaiiser, boston, (2000), 3-36. [3] i. daubechies, b. han, a. ron, and z. shen, framelets: mra-based constructions of wavelet frames, appl. comput. harmon. anal., 14 (2003), 1-46. [4] yu. a. farkov, on wavelets related to walsh series, j. approx. theory, 161 (2009), 259-279. p-frame multiresolution analysis related to the walsh functions 15 [5] yu. a. farkov, yu. a. maksimov and s. a stoganov, on biorthogonal wavelets related to the walsh functions, int. j. wavelets multiresolut. inf. process., 9(3) (2011), 485-499. [6] yu. a. farkov, and e. a. rodionov, nonstationary wavelets related to the walsh functions, amer. j. comput. math., 2 (2012) 82-87. [7] b. i. golubov, a. v. efimov and v. a. skvortsov, walsh series and transforms: theory and applications, kluwer, dordrecht, 1991. [8] h. o. kim and j. k. lim, on frame wavelets associated frame multiresolution analysis, appl. comput. harmon. anal., 10 (2001) 61-70. [9] s. mallat, multiresolution approximations and wavelets. trans. amer. math. soc., 315 (1989), 69-87. [10] meenakshi, p. manchanda and a. h. siddiqi, wavelets associated with nonuniform multiresolution analysis on positive half-line, int. j. wavelets multiresolut. inf. process., 10(2) (2012), 1250018. [11] a. ron and z. shen, affine systems in l2(rd): the analysis of the analysis operator, j. funct. anal., 148 (1997), 408-447. [12] a. ron and z. shen, frames and stable bases for shift-invariant subspaces of l2(rd), canad. j. math., 47 (1995), 1051-1094. [13] f. schipp, w. r. wade and p. simon, walsh series: an introduction to dyadic harmonic analysis, adam hilger, bristol and new york, 1990. [14] f. a. shah, construction of wavelet packets on p-adic field, int. j. wavelets multiresolut. inf. process., 7(5) (2009), 553-565. [15] f. a. shah, non-orthogonal p-wavelet packets on a half-line, anal. theory appl., 28(4) (2012), 385-396. [16] f. a. shah, biorthogonal wavelet packets related to the walsh polynomials, j. classical anal., 1 (2012), 135-146. [17] f. a. shah, on some properties of p-wavelet packets via the walsh-fourier transform, j. nonlinear anal. optimiz., 3 (2012), 185-193. [18] f. a. shah, tight wavelet frames generated by the walsh polynomials, int. j. wavelets, multiresolut. inf. process., 11(6) (2013), 1350042. [19] f. a. shah and l. debnath, dyadic wavelet frames on a half-line using the walsh-fourier transform, integ. transf. special funct., 22(7) (2011), 477-486. [20] p. wojtaszczyk, a mathematical introduction to wavelets, cambridge university press, 1997. [21] y. xiaojiang, semiorthogonal multiresolution analysis frames in higher dimensions, acta appl. math., 111 (2010), 257-286. [22] z. h. zhang, a characterization of generalized frame mra’s deriving orthonormal wavelets, acta math. sinica, english series, 22(4) (2006), 1251-1260. department of mathematics, university of kashmir, south campus, anantnag-192101, jammu and kashmir, india international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 22-29 http://www.etamaths.com on the growth and approximation of transcendental entire functions on algebraic varieties devendra kumar∗ abstract. let x be a complete intersection algebraic variety of codimension m > 1 in cm+n. in this paper we characterized the classical growth parameters order and type for transcendental entire functions f ∈⊕(x), the space of holomorphic functions on the complete intersection algebraic variety x, in terms of the best polynomial approximation error in lp-norm, 0 < p ≤ ∞, on a l − regular non-pluripolar compact subset k of cm+n. 1. introduction the growth of transcendental entire functions in one complex variable case is well represented in the work of b.ja levin [11] and boas [2]. in several complex variables the standard reference is the work of p.lelong and l.gruman [10] and ronkin’s book [14]. einstein-matthews and kasana [3] studied the growth parameters (p,q) − order and (p,q) − type introduced by juneja et al.([6],[7]) of transcendental entire functions f : cn → c. einstein-matthews and clement lutterodt [4] extended the results studied in [3] to transcendental entire functions f : x → c, defined on a complete intersection algebraic variety x in cm+n of codimension m > 1, and obtained the growth parameters in terms of the sequence of extremal polynomials occurring in the development of f. it has been noticed that the growth parameters of f : x → c in terms of approximation errors is not studied so far. the aim of this paper is to bridge this gap and to study the results obtained in [4] in terms of the best approximation errors in lp-norm, 0 < p ≤∞. a.r. reddy ([13],[14]) characterized the growth parameters in terms of approximation errors for a function continuous on [-1,1]. t. winiarski ([21],[22]) studied the growth of entire functions in terms of lagrange polynomial approximation errors with respect to sup norm on a compact subset k (positive capacity) of c and cn,n > 1. kasana and kumar [8] generalized the results of winiarski [22] by using the concept of index-pair (p,q). adam janik [5] characterized the generalized order of entire functions by means of polynomial approximation and interpolation on compact subsets of cn, using the siciak extremal function ([18],[19]). in [5] adam janik extended the results of s.m. shah [17] in the case n = 1,k = [−1, 1] and winiarski [22]. srivastava and kumar [20] extended and improved the results of adam janik [5]. but our work is different from all these authors. the text has been divided into four parts. section 1 consists of an introductory exposition of the topic and section 2 contains some definitions and notations. in section 3, we have given zeriahi’s bernstein-markov type inequality with two lemmas in which first one is due to zeriahi extending the classical cauchy inequality and second is concerned with a sequence of extremal polynomials. finally, in section 4, we prove two theorems for a transcendental entire function f ∈ ⊕(cm+n), the space of holomorphic functions on the complete intersection algebraic variety x and studied the growth parameters order and type in terms of lp-approximation error on a l−regular non-pluripolar compact subset of cm+n. 2010 mathematics subject classification. 41a20, 30e10. key words and phrases. algebraic variety; approximation error; extremal polynomials and plurisubharmonic functions. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 22 on the growth and approximation of transcendental entire functions 23 2. definitions and notations following the definition of einstein and kasana [3], we have let v : cm+n → r+ := r ∈ r : r > 0 be a real-valued function such that the following properties hold: (i) v(z + w) ≤ v(z) + v(w) : z,w ∈ cm+n, (ii)v(bz) = |b|v(z) : z ∈ cm+n,b ∈ c, (iii) v(z) = 0 ⇐⇒ z = 0. here v is a norm on cm+n and it exhausts the complex space cm+n by a family of sublevel sets {ωc}c≥1 which are defined by ωc = {z ∈ cm+n : v(z) ≤ c,c ∈ r}. let ϕ : cm+n → r+. define mϕ,v(r) = supv(z)≤r ϕ(z), the maximum modulus of ϕ with respect to the norm v for each r ∈ r+. we say that the transcendental entire function f : cm+n → c is of order ρ, if log |f| is of order ρ, where (2.1) ρ = lim sup r→∞ log mf,v(r) log r if ρ < +∞, f is said to have maximal, normal or minimal type if (2.2) σ = lim sup r→∞ mf,v(r) rρ , is infinite, finite or zero. let k be a compact subset of cm+n, which is nonpluripolar on each irreducible component of a complete intersection variety x. the siciak extremal function vk associated to k has been studied extensibly by siciak [18] and sadullaev ([15],[16]) and is defined as: vk = sup{u(z) : u ∈ i(x); u(ζ) ≤ 0,ζ ∈ k,z ∈ x} where the subcone i(x) is given by i(x) = {u(z) : u ∈ psh(x); u(z) ≤ log(‖ z ‖ +1) + cu,z ∈ x} here cu is a constant depending only on the cone of plurisubharmonic function (psh)u and ‖ . ‖ is the euclidean norm on cm+n. the upper semi-continuous regularization of vk is defined on x by v ∗k(z) = lim sup ζ→z vk(ζ),ζ ∈ k,z ∈ x, v ∗k(z) is psh(x) and satisfies v ∗k(z) ≤ log(‖ z ‖ +1) + o(1),as ‖ z ‖→ +∞. if vk is continuous on cm+n, then vk = v ∗k ∈ i. it is given in [18] that if, for all z ∈ k,vk is continuous, then vk is continuous on x. in this case we say that k is l− regular in x. we define the sublevel sets of the extremal function vk by setting ωα = {z ∈ x : vk(z) ≤ α},α > 1,α ∈ r, and sublevel sets of the upper semi-continuous regularization v ∗k of vk by ωr = {z ∈ cm+n : expv ∗k(z) ≤ r},r > 1. it has been observe that the sequence of sublevel sets {ωr}r>1 exhausts the complex space cm+n. for f : cm+n → c a transcendental entire function, set mk,f (r) = sup z∈ωr |f(z)|,r > 1. it can be easily shown that log+ mk,f (r) and log + mk,v(r) give the same order given by (2.3) ρ ≡ ρ(f) = lim sup r→∞ log log+ mk,f (r) log r . 24 devendra kumar if 0 < ρ < +∞, the type of f : cm+n → c is defined by (2.4) σ ≡ σ(f) = lim sup r→∞ log+ mk,f (r) rρ . zeriahi [23] constructed an orthogonal polynomial basis {ak}k≥1 for the space ⊕(x). the basis is orthogonal in the hilbert space l2(x,µ), essentially by means of the hilbert-schmidt process, here µ be the extremal capacity measure on k given by µ = (ddcvk. further details on this positive borel measure µ supported on k can be obtained from the paper of e. bedford and b.a. taylor [1]. let pd(cm+n) denote the c-vector space of polynomials πd : cm+n → c of degree ≤ d for d ≥ 1. let l2p (k,µ) denote the closed subspace of the hilbert space l 2(k,µ) generated by the restriction to k of polynomials πd ∈ pd(cm+n) of degree (πd) ≤ d, for d ≥ 1. then every function f ∈ l2p (k,µ) has a power series expansion of the form (2.5) f = σk≥1fkak, with fk = 1 ∆2k(k) ∫ k f.akdµ, ∆k(k) = ( ∫ k |ak|2dµ) 1 2 ,k ≥ 1, here . is the dot product of vectors. let lp(k,µ),p ≥ 1 denote the class of all functions such that ‖ f ‖lp(k,µ)= ( ∫ k |f|pdµ) 1 p < ∞, then we define the best polynomial approximation error in lp-norm, p ≥ 1, by (2.6) e p d(k,f) = inf{‖ f −πd ‖lp(k,µ),πd ∈ pd(c m+n)}. if the extremal function vk associated with k is continuous for every z ∈ k, then vk is continuous on x and l− regular, so instead of defining sublevel sets for the upper semi-continuous regularization, we define the same for vk by setting ωr = {z ∈ x : vk(z) < log r,r ∈ r,r > 1}. then we have vk(z) ≥ 1 sk log( |ak| ak(k) ), where ak(k) = max z∈k |ak(z)|, |ak|ωr ≤ ak(k)r sk,sk = degree(ak). following the siciak [18] we observe that if k is l−regular then lim sup d→∞ (e p d(k,f)) 1 d = 1 r < 1 if and only if f has an analytic continuous to {z ∈ cm+n; vk(z) < log( 1 r )}. 3. auxiliary results in this section we shall state some preliminary results which will be used in the sequel. first we state zeriahi’s bernstein-markov type inequality [23]: bm:for all � > 0, there exists a constant c� > 0 such that (3.1) sup z∈k |f(z)| ≤ c�(1 + �)deg(f)( ∫ k |f|2dµ) 1 2 for every holomorphic function f with polynomial growth on the complete intersection algebraic variety x and k is a nonpluripolar compact subset of x. now we state the following lemmas of zeriahi extending the classical cauchy inequality. on the growth and approximation of transcendental entire functions 25 lemma 3.1. let f = σk≥0fkak be a holomorphic function on x. then for every θ > 1, there exist an integer nθ and a constant cθ > 0 such that (3.2) |fk|rsk∆k(k) ≤ cθ (r + 1)nθ (r − 1)2n−1 |fk|ωrθ, for every r > 1,k ≥ 1, where cθ and nθ are independent of r,k and f. lemma 3.2. if k is an l−regular, then the sequence of extremal polynomials {ak}k≥1 satisfies (3.3) lim k→∞ ( |ak(z)| νk ) 1 sk = exp(vk(z)),νk =‖ ak ‖l2(k,µ), for every z ∈ cm+n and (3.4) lim k→∞ ( |ak(z)| νk ) 1 sk = 1. 4. main results in this section we shall prove our main theorems. moreover, we shall characterize the classical growth parameters order and types of transcendental entire function in terms of lp-approximation error defined by (2.6). theorem 4.1.if f : x → c is a transcendental entire function on x with a series expansion (2.5) with respect to the orthogonal polynomial basis {ak}k≥1, then f ∈ lp(k,µ), 1 < p ≤ ∞ is of finite order if and only if (4.1) ρ = lim sup k→∞ sk log sk − log(epsk(k,f)) < +∞, and ρ = ρ1, where e p sk (k,f) is defined by (2.6). proof. first we have to prove that ρ ≤ ρ1. if ρ1 = ∞, then nothing to be prove. assume that ρ1 < ∞ and let � > 0. for a sufficiently large k, from (4.1) we have 0 ≤ sk log sk − log(epsk(k,f)) ≤ ρ1 + � or (4.2) epsk(k,f) ≤ (sk) −sk ρ1+� . since adding a polynomial will not change the order of a function. thus, for r ≥ 1 and a0(k) = 0, we can assume that following inequality holds for every k ≥ 0, (4.3) mk,f (r) ≤ σk≥1|fk|ak(k)rsk. now we will proceed the proof in two steps (p ≥ 2) and (1 < p < 2). let f = σk≥0fk.ak be an element of lp(k,µ). step 1. if f ∈ lp(k,µ) with p ≥ 2, then f = σ∞k=0fk.ak with convergence in l 2(k,µ), fk = 1 ν2k ∫ k f.akdµ,k ≥ 1,νk ≡ ∆k(k), or = 1 ν2k ∫ k (f −psk−1).akdµ. it gives |fk| ≤ 1 ν2k ∫ k |(f −psk−1)|.|ak|dµ, 26 devendra kumar now using bernstein−walsh inequality and hölder′s inequality we have for any � > 0 (4.4) |fk|νk ≤ c�(1 + �)ske p sk−1(k,f),k ≥ 0. step 2. if 1 ≤ p < 2, let p′ such that 1 p + 1 p′ = 1 then p′ ≥ 2. by hölder′s inequality we get |fk|ν2k ≤‖ f −psk−1 ‖lp(k,µ)‖ ak ‖lp′(k,µ) . but ‖ ak ‖lp′(k,µ)≤ c ‖ ak ‖k= cak(k), now by bernstein−markov inequality we have |fk|ν2k ≤ cc�(1 + �) sk ‖ f −psk−1 ‖lp(k,µ), it gives (4.5) |fk|ν2k ≤ c ′ �(1 + �) skepsk(k,f). from (4.4) and (4.5), we get for p ≥ 1 (4.6) |fk|νk ≤ a�(1 + �)skepsk(k,f), where a� is a constant depends only on �. now using zeriahi′sbernstein−markov type inequality in (4.3) and (4.6), we obtain mk,f (r) ≤ σk≥0|fk|c�(1 + �)skνkrsk ≤ σk≥0a�c�(1 + �)2skepsk(k,f)r sk, using inequality (4.2) in above, we get mk,f (r) ≤ c′�σk≥0(1 + �) 2sk(sk) −sk ρ1+� rsk = σ1 + σ2, where σ1 = σ1≤k≤(2r(1+�)2)(ρ1+�) (1 + �) 2sk(sk) −sk (ρ1+�) rsk and σ1 = σk≥(2r(1+�)2)(ρ1+�) (1 + �) 2sk(sk) −sk (ρ1+�) rsk. in σ2, we have (r(1 + �) 2k) −1 (ρ1+�) ≤ 1 2 , so that σ2 ≤ 1, and σ1 ≤ (f(r,�))(ρ1+�)σk≥1(sk) −sk (ρ1+�) where f(r,�) = (r(1 + �)2)(2r(1+�) 2) or σ1 ≤ k1 exp((2r(1 + �)2)(ρ1+�) log(r(1 + �)2)) ≤ k2 exp(r(ρ1+�)), for some constants k1 > 0,k2 > 0. hence it follows from definition of order given by (2.3) that ρ ≤ ρ1 + �, since � > 0 is arbitrary, it gives (4.7) ρ ≤ ρ1. in order to prove the reverse inequality i.e., ρ1 ≤ ρ we consider the polynomial of degree sk as psk(z) = σ k j=0fjaj, then (4.8) e p sk−1(k,f) ≤ σ ∞ sj=sk |fj| ‖ aj ‖lp(k,µ)≤ c0σ∞sj=sk+1|fj| ‖ aj ‖k,k ≥ 0,p ≥ 1. in the consequences of lemmas 3.1 and 3.2, we obtain the following inequality (4.9) |fk|ak(k) ≤ mk,f (r) rsk ,r > 0. using (4.9)in (4.8), we get (4.10) e p sk−1(k,f) ≤ c0σ ∞ sj=sk+1 mk,f (r)r −sj = c0mk,f (r) (r∗/r)(sk+1) 1 − (r∗/r) on the growth and approximation of transcendental entire functions 27 for all sufficiently large sk and all r > r ∗,r∗ > 1. here c0 is some fixed number. for all sufficiently large sk and r > 2r ∗, (4.10) gives (4.11) e p sk−1(k,f) ≤ γmk,f (r)(r ∗/r)sk, where γ is a constant independent of sk and r. if ρ1 = 0, then nothing to be prove. let us assume that 0 < ρ1 < ∞. if ρ1 < ∞, define ρ∗ = ρ1 −�, for small � > 0, so that ρ1 > 0. let ρ ∗ > 0 be arbitrary if ρ1 = +∞. then for infinitely many indices k ≥ 1, from (4.1) we have sk log sk ≥ ρ∗ log(epsk(k,f)) −1 or (4.12) log epsk(k,f) ≥ −sk log sk ρ∗ . using (4.12) in (4.11) we get (4.13) log mk,f (r) ≥ −sk log sk ρ∗ + sk log( r r∗ ) − log γ. the minimum value of right hand side of (4.13) is obtained at rsk r∗ = (esk) 1 ρ∗ and substituting the value of ( rsk r∗ ) in (4.13) we obtain the following inequality log mk,f (r) ≥ sk ρ∗ − log γ or log log mk,f (rsk) log rsk − log r∗ ≥ ρ∗( log sk − log ρ∗ log sk + 1 ). proceeding the limits and taking the definition (2.3) into account, we get (4.14) ρ = lim sup r→∞ log log mk,f (r) log r ≥ lim sup r→∞ log log mk,f (rsk) log rsk ≥ ρ∗. since ρ∗ is arbitrary real number, smaller than ρ, it gives that ρ ≥ ρ1. now in view of (4.7) the result is immediate. hence the proof is completed. theorem 4.2 if f : x → c is a transcendental entire function on x with a series expansion (2.5) with respect to the orthogonal polynomial basis {ak}k≥1, then f ∈ lp(k,µ) with a finite order ρ(0 < ρ < ∞) has finite type σ(0 < σ < ∞) if and only if eρσ = lim sup k→∞ sk(e p sk (k,f)) ρ sk < +∞, and σ1 = σ, where e p sk (k,f) is given by (2.6). proof. let δ1 = eρσ1. for given � > 0 and δ1 > 0, we have for sufficiently large k (4.15) sk(e p sk (k,f)) ρ sk ≤ δ1 + � or sk log sk − log(epsk(k,f)) ≤ ρ 1 − log(δ1+� sk ) . now it follows from theorem 4.1 that the order of f is at most ρ. now let us consider that 0 < δ1 < ∞ and we have to show that σ ≤ δ1eρ = σ1. from (4.15) we get (4.16) epsk(k,f) ≤ ( δ1 + � sk ) sk ρ . consider (4.17) |f(z)| ≤ σk≥1|fk||ak|ωr ≤ σk≥1|fk|ak(k)r sk. further we will proceed the proof by considering two cases: 28 devendra kumar case 1. let p ≥ 2, then we have f = σk≥0fkak because f ∈ l2(k,µ),lp(k,µ) ⊂ l2(k,µ) and {ak}k is a basis of l2(k,µ). consider the series σk≥0fkak in cm+n and it can be easily seen that this series converges uniformly on every compact subsets of cm+n to an entire function. using the bernstein−markov inequality (bm) in (4.17) we get |f(z)| ≤ c�σk≥1|fk|(1 + �)skνkrsk, it gives from (4.6) that |f(z)| ≤ c�σk≥1|fk|(1 + �)2skepsk(k,f)r sk. now in view of (4.16) we have |f(z)| ≤ c′�σk≥1((1 + �) 2ρ( δ1 + � sk )rρ) sk ρ = c′�σ1 + σ2. let us assume the function φ(s) = ((r(1 + �)2)ρ( δ1 + � s )) s ρ ,s > 0. this function attains its maximum value at s = ( δ1 + � e )(r(1 + �)2)ρ and the value is equal to exp((δ1+� eρ )(r(1 + �)2)ρ). hence for any constant k1 > 0, c′�σ1 = c ′ �σ1≤k≤2(δ1+�)(1+�)2ρrρ(( δ1 + � sk )(1 + �)2ρrρ) sk ρ ≤ 2(δ1 + �)(1 + �)2ρrρ exp(( δ1 + � eρ )(r(1 + �)2)ρ) ≤ k1 exp(( δ1 + � eρ )(r(1 + �)2)ρ), and c′�σ2 = c ′ �σk>2(δ1+�)(1+�)2ρrρ(( δ1 + � sk )(1 + �)2ρrρ) sk ρ ≤ c′�σk≥1 1 2k = k2 < ∞. thus from above discussion we get σ ≤ δ1 eρ . case 2. for the case 1 ≤ p < 2 and f ∈ lp(k,µ) by (bm) inequality and hölder′s inequality we get again the inequality (4.6). now proceeding on the lines of proof of case 1, the result is immediate. in order to prove the reverse inequality, we note that if δ1 > � > 0, then for infinitely many indices k (4.18) epsk(k,f) ≥ ( δ1 − � sk ) sk ρ . now using (4.18) in (4.11) we obtain log mk,f (rsk) ≥ sk ρ log( δ1 − � sk ) + sk log(rsk/r ∗) − log γ. the minimum value of right hand side is attains at rsk r∗ = esk (δ1−�) . thus we get mk,f (rsk) ≥ e sk ρ = exp(( δ1 − � eρ )rρsk) + o(1). proceeding to limits and using the definition (2.4) of type of f ∈ lp(k,µ), we get σ ≥ δ1 eρ . on the growth and approximation of transcendental entire functions 29 this completes the proof of theorem. remark 4.3. theorem 4.1 and 4.2 also holds for (0 < p < 1) (see[9]). references [1] e. bedford and b.a. taylor, the complex equilibrium measure of a symmetric convex set in rn, trans. amer. math. soc. 294(1986), 705-717. [2] r.p. boas, entire functions, academic press, new york, 1954. [3] s.m.einstein-matthews and h.s. kasana, proximate order and type of entire functions of several complex variables, israel journal of mathematics 92(1995), 273-284. [4] s.m. einstein-matthews and clement h. lutterodt, growth of transcendental entire functions on algebraic varieties, israel journal of mathematics 109(1999), 253-271. [5] adam janik, on approximation of entire functions and generalized order,univ. iagel. acta. math. 24(1984), 321-326. [6] o.p. juneja, g.p. kapoor and s.k. bajpai, on the (p, q)-order and lower (p, q)-order of an entire function, j. reine angew. math. 282(1976), 53-67. [7] o.p. juneja, g.p. kapoor and s.k. bajpai, on the (p, q)-type and lower (p, q)-type of an entire function, j. reine angew. math. 290(1977), 180-190 . [8] h.s. kasana and d. kumar, on approximation and interpolation of entire functions with index-pair (p, q), publicacions matematique 38 (1994), no. 4, 681-689. [9] d. kumar, generalized growth and best approximation of entire functions in lp-norm in several complex variables, annali dell’ universitá di ferrara vii, 57 (2011), 353-372. [10] p. lelong and l. gruman, entire functions of several complex variables, grundlehren der mathematischen wissenschaften 282, springer-verlag, berlin, 1986. [11] b. ja. levin, distributions of zeros of entire functions, translations of mathematics monographs 55, amer. math. soc., providence, r.i., 1994. [12] a.r. reddy, approximation of an entire function, j. approx. theory 3(1970), 128-137. [13] a.r. reddy, best polynomial approximation to certain entire functions, j. approx. theory 5(1972), 97-112. [14] l.i. ronkin, introduction to the theory of entire functions of several variables, amer. math. soc., providence, r.i., 1974. [15] a. sadullaev, plurisubharmonic measures and capacities on complex manifolds, russian mathematical surveys 36(1981), 61-119. [16] a. sadullaev, an estimate for polynomials on analytic sets, mathematics of the ussr-izvestiya 20(1980), 493-502. [17] s.m. shah, polynomial approximation of an entire function and generalized orders, j. approx. theory 19(1977), 315-324. [18] j. siciak, extremal plurisubharmonic functions and capacities in cn, lectures in mathematics 14, sofia university, tokyo, 1982. [19] j. siciak, on some extremal functions and their applications in the theory of analytic functions of several complex variables, trans. amer. math. soc. 105(1962), 322-357. [20] g.s. srivastava and susheel kumar, on approximation and generalized type of entire functions of several complex variables, european j. pure appl. math. 2, no. 4(2009), 520-531. [21] t.n. winiarski, application of approximation and interpolation methods to the examination of entire functions of n complex variables, ann. polon. math. 28(1973), 97-121. [22] t.n. winiarski, approximation and interpolation of entire functions, ann. polon. math. 23(1970), 259-273. [23] a. zeriahi, meilleure approximation polynomiale et croissante des fonctions entieres sur certaines varievalgebriques affines, annales inst. fourier (grenoble) 37(1987), 79-104. department of mathematics, al-baha university, p.o.box-1988, alaqiq, al-baha-65431, saudi arabia, k.s.a. ∗corresponding author: d kumar001@rediffmail.com international journal of analysis and applications issn 2291-8639 volume 15, number 1 (2017), 8-17 http://www.etamaths.com some estimations on continuous random variables involving fractional calculus zoubir dahmani1, amina khameli1, mohamed bezziou1,3 and mehmet zeki sarikaya2,∗ abstract. using fractional calculus, new fractional bounds estimating the w− weighted expectation, the w− weighted variance and the w−weighted moment of continuous random variables are obtained. some recent results on classical bounds estimations are generalized. 1. introduction it is known that the integral inequalities play an important role in the theory of differential equations, probability theory and in applied sciences. for more details, we refer to [2, 3, 11–13, 16] and the references therein. moreover, the study of the integral inequalities using fractional calculus is also of great importance, we refer the reader to [1, 4–6, 8, 14, 15] for further information and applications. in this sense, in a recent work [4], by introducing new concepts on probability theory using fractional calculus, the author extended some classical results of the papers [3, 11]. then, based on [4], the authors in [9] introduced other classes of weighted concepts and generalized some classical results of [3, 12]. very recently, in [7], the author presented some fractional applications for continuous random variables having probability density functions (p.d.f.) defined on finite real lines. motivated by the papers in [4, 7, 9, 11], in this work, we focus our attention on the applications of fractional calculus on probability theory. we establish new fractional bounds that estimate the w− weighted expectation, the w− weighted variance and the w−weighted moment of continuous random variables. some recent results on classical random variable bound estimations are also generalized. 2. preliminaries in this section, we recall some preliminaries that will be used in this work. we begin by the following definition. definition 2.1. [10] the riemann-liouville fractional integral operator of order α ≥ 0, for a continuous function f on [a,b] is defined as jαa [f (t)] = 1 γ (α) t∫ a (t− τ)α−1 f (τ) dτ, α > 0, a < t ≤ b, (2.1) j0a [f (t)] = f (t) , where γ (α) := ∞∫ 0 e−uuα−1du. for α > 0, β > 0, we have: jαa j β a [f (t)] = j α+β a [f (t)] (2.2) received 28th april, 2017; accepted 26th june, 2017; published 1st september, 2017. 2010 mathematics subject classification. 26d15, 26a33, 60e15. key words and phrases. integral inequalities; riemann-liouville integral; random variable; fractional w−weighted expectation; fractional w−weighted variance. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 8 some estimations on continuous random variables 9 and jαa j β a [f (t)] = j β a j α a [f (t)] . (2.3) let us now consider a positive continuous function w defined on [a,b]. we recall the w−concepts [9] : definition 2.2. the fractional w−weighted expectation function of order α > 0, for a random variable x with a positive p.d.f. f defined on [a,b] is defined as ex,α,w (t) := j α a [twf (t)] = 1 γ (α) t∫ a (t− τ)α−1 τw (τ) f (τ) dτ, a ≤ t < b, α > 0, (2.4) where w : [a,b] → r+ is a positive continuous function. definition 2.3. the fractional w−weighted expectation function of order α > 0 for the random variable x −e (x) is given by ex−e(x),α,w (t) := 1 γ (α) t∫ a (t− τ)α−1 (τ −e (x)) w (τ) f (τ) dτ, a ≤ t < b, α > 0. (2.5) where f : [a,b] → r+ is the (p.d.f) of x. definition 2.4. the fractional w−weighted expectation of order α > 0 for a random variable x with a positive p.d.f. f defined on [a,b] is defined as ex,α,w := 1 γ (α) b∫ a (b− τ)α−1 τw (τ) f (τ) dτ, α > 0. (2.6) for the w−weighted fractional variance of x, we recall the definitions [9]: definition 2.5. the fractional w−weighted variance function of order α > 0 for a random variable x having a positive p.d.f. f on [a,b] is defined as σ2x,α,w (t) : = j α a [ (t−e (x))2 wf (t) ] (2.7) = 1 γ (α) t∫ a (t− τ)α−1 (τ −e (x))2 w (τ) f (τ) dτ, a ≤ t < b, α > 0. definition 2.6. the fractional w−weighted variance of order α > 0 for a random variable x having a positive p.d.f. f on [a,b] is given by σ2x,α,w := 1 γ (α) b∫ a (b− τ)α−1 (τ −e (x))2 w (τ) f (τ) dτ, α > 0. (2.8) for the fractional w−weighted moments, we recall the following definitions [9]: definition 2.7. the fractional w−weighted moment function of orders r > 0, α > 0 for a continuous random variable x having a p.d.f. f defined on [a,b] is defined as mxr,α,w (t) := j α a [t rwf (t)] = 1 γ (α) t∫ a (t− τ)α−1 τrw (τ) f (τ) dτ, a ≤ t < b, α > 0. (2.9) definition 2.8. the fractional w−weighted moment of orders r > 0, α > 0 for a continuous random variable x having a p.d.f. f defined on [a,b] is defined by mxr,α,w := 1 γ (α) b∫ a (b− τ)α−1 τrw (τ) f (τ) dτ, α > 0. (2.10) based on the above definitions, we give the following remark: 10 dahmani, khamel, bezziou and sarikaya remark 2.1. (1:) if we take α = 1,w(t) = 1, t ∈ [a,b] in definition 2, we obtain the classical expectation: ex,1,1 = e (x) . (2:) if we take α = 1,w(t) = 1, t ∈ [a,b] in definition 5, we obtain the classical variance: σ2x,1,1 = σ2 (x) = b∫ a (τ −e (x))2 f (τ) dτ. (3:) if we take α = 1,w(t) = 1, t ∈ [a,b] in definition 7, we obtain the classical moment of order r > 0,mr := b∫ a τrf (τ) dτ. 3. main results in this section, based on [7], we establish new w−weighted integral inequalities (with new fractional bounds) for random variables with p.d.f. that are defined on finite real intervals. we begin by proving the following property that generalizes an important property of the classical variance: theorem 3.1. let x be a continuous random variable having a p.d.f. f : [a,b] → r+, and let w : [a,b] → r+ be a positive continuous function. then for all α > 0,n = [α− 1] we have : σ2x,α,w = ex2,α,w − 2e (x) ex,α,w + e 2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ] (3.1) proof. by definition 6, we can write : σ2x,α,w := 1 γ (α) b∫ a (b− τ)α−1 (τ −e (x))2 w (τ) f (τ) dτ, α > 0. (3.2) hence, σ2x,α,w = ex2,α,w − 2e (x) ex,α,w + e 2 (x) jαwf(b). (3.3) on the other hand, we have jαwf(b) = 1 γ(α) n∑ i=0  [(−1)icinbn−i b∫ a (b− τ)s τiwf(τ)dτ   , (3.4) where α = n + s; n = [α]; s ∈ (0; 1). definition 8 allows us to write jαwf(b) = γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ] . (3.5) then, using (3.3) and (3.5), we get (3.1). � remark 3.1. a*: taking w(t) = 1 on [a,b] in the above theorem, we obtain theorem 3.3 of [7]. b*: taking α = 1 and w(t) = 1, t ∈ [a,b], we obtain σ2x,1,1 = e(x 2) −e2(x). another result is the following: theorem 3.2. let x be a continuous random variable with a p.d.f. f : [a,b] → r+, and let w : [a,b] → r+ be a positive continuous function. then for all α > 0,n = [α−1] the following estimations are valid. ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) × ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) − ( ex−e(x),α,w )2 ≤ ‖f‖2∞j α a w(b) [ jαa [ w(b)b2 ] − (jαa [w(b)b]) 2 ] , f ∈ l∞ [a,b] (3.6) some estimations on continuous random variables 11 and ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) × ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) − ( ex−e(x),α,w )2 (3.7) ≤ 1 2 (b−a)2 ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ])2 . proof. to prove the above theorem, we use theorem 3.1 of [4]. we find that: 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y)2 p (x) p (y) dxdy (3.8) = 2jαa [p(b)] j α a [ p(b)(b−e(x))2 ] − 2(jαa [p(b)(b−e(x)]) 2 then, taking p (t) = w(t)f (t) , t ∈ [a,b] in (3.8), it yields that 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y)2 w(x)f (x) w(y)f (y) dxdy = 2jαa [wf(b)] σ 2 x,α,w − 2 ( ex−e(x),α,w )2 . (3.9) by the hypothesis f ∈ l∞ ([a,b]), we obtain 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y)2 w(x)w(y)f (x) f (y) dxdy ≤ 2‖f‖2∞ [ jαa [w (b)] j α a [ w(b)b2 ] − (jαa [w(b)b]) 2 ] . (3.10) thanks to (3.9),(3.10),(3.5) and applying theorem 1, we obtain (3.6). on the other hand, 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 w(x)w(y) (x−y)2 f (x) f (y) dxdy (3.11) ≤ sup x,y∈[a,b] |(x−y)|2 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 w(x)w(y)f (x) f (y) dxdy = (b−a)2 (jαa [wf(b)]) 2. so, by (3.9),(3.11),(3.1) and (3.5), we obtain (3.7). � remark 3.2. (1) if we take w = 1 on [a,b] in theorem 2, we obtain the first part of theorem 3.5 of [7], (2) and taking α = 1,w = 1 on [a,b], we obtain the first part of theorem 1 in [3]. in what follows, we prove a more general theorem. theorem 3.3. suppose that x is a continuous random variable with a p.d.f. f : [a,b] → r+ and let w : [a,b] → r+ b a continuous function. then, 12 dahmani, khamel, bezziou and sarikaya (i): for all α > 0,β > 0; n = [α− 1],m = [α− 1]( ex2,β,w − 2e (x) ex,β,w + e2 (x) γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) (3.12) × ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) + γ(β −m) γ(β) n∑ i=0 [[ (−1)icimb m−imxi,β−m,w ] × ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) −2ex−e(x),α,wex−e(x),β,w ≤ ‖f‖2∞ [ jαa [w(b)] j β a [ w(b)b2 ] + jβa [w(b)] j α a [ w(b)b2 ] −2jαa [w(b)b] jβa [w(b)b] ] , f ∈ l∞ ([a,b]) . (ii) also, the following estimation( ex2,β,w − 2e (x) ex,β,w + e2 (x) γ(β −m) γ(β) n∑ i=0 [[ (−1)icinb n−imxi,β−n,w ]) × ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) + ( γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) × ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) −2ex−e(x),α,wex−e(x),β,w (3.13) ≤ (b−a)2 ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ])(γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) is also valid for any α > 0,β > 0. proof. we have (see [4]): 1 γ (α) γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 (x−y)2 p(x)p(y)dxdy = jαa [wf(b)] j β a [ wf(b)(b−e(x))2 ] + jβa [wf(b)] j α a [ wf(b)(b−e(x))2 ] −2jαa [wf(b)(b−e(x))] j β a [wf(b)(b−e(x))] . (3.14) in (3.14), if we take p = wf, we obtain 1 γ (α) γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 (x−y)2 w(x)w(y)f (x) f (y) dxdy = jαa [wf(b)] σ 2 x,β,w + j β a [wf(b)] σ 2 x,α,w − 2ex−e(x),α,wex−e(x),β,w. (3.15) on the other hand, it is clear that 1 γ (α) γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 (x−y)2 w(x)w(y)f (x) f (y) dxdy (3.16) ≤ ‖f‖2∞ [ jαa [w(b)] j β a [ w(b)b2 ] + jβa [w(b)] j α a [ w(b)b2 ] − 2jαa [w(b)b] j β a [w(b)b] ] . consequently, by (3.15), (3.16) and (3.1), we obtain (3.12). some estimations on continuous random variables 13 for the second inequality of theorem 3, we observe that 1 γ (α) γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 (x−y)2 w(x)w(y)f (x) f (y) dxdy ≤ sup x,y∈[a,b] |(x−y)|2 1 γ (α) γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 w(x)w(y)f(x)f(y)dxdy ≤ (b−a)2jαa [wf(b)] j β a [wf(b)] . (3.17) so, applying theorem 1 and thanks to (3.15) and (3.17), we get (3.13). � remark 3.3. (i) : applying theorem 14 for α = β, we obtain theorem 12. (ii) : taking w equal to 1 on [a,b] in theorem 14, we obtain the last part of theorem 3.7 of [7]. also, we present to reader the following estimation: theorem 3.4. let f be the p.d.f of x on [a,b] and w : [a,b] → r+.then for all α > 0,n = [α− 1] the following fractional inequality holds: ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) × ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) − ( ex−e(x),α,w )2 (3.18) ≤ 1 4 (b−a)2 ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ])2 . proof. in [4], it has been proved that 0 ≤ jαa [p (b)] j α a [ p(b) (b−e(x))2 ] − (jαa [p(b) (b−e(x))]) 2 ≤ 1 4 (b−a)2 (jαa [p(b)]) 2 . (3.19) hence, taking p(b) = wf(b) in (3.19), we observe that jαa [wf(b)] σ 2 x,α,w − ( ex−e(x),α,w )2 ≤ 1 4 (b−a)2 (jαa [wf(b)]) 2 . (3.20) thanks to theorem 1 and by the relation (3.5), we obtain (3.18). � remark 3.4. taking w(t) = 1, t ∈ [a,b] in theorem 4, we obtain theorem 3.8 of [7] . another result related to the moments is the following theorem. 14 dahmani, khamel, bezziou and sarikaya theorem 3.5. let f be the p.d.f of the random variable x on [a,b] and w : [a,b] → r+. then for all α > 0,β > 0; n = [α− 1],m = [β − 1], the inequality γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ] × ( ex2,β,w − 2e (x) ex,β,w + e2 (x) γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) + γ(β − 1 + m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ] (3.21) × ( ex2,α,w − 2e (x) ex,α,w + e2 (x) γ(β − 1 + m) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) +2 (a−e (x)) (b−e (x)) ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) × ( γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb n−imxi,β−m,w ]) . ≤ (a + b− 2e(x))   ( γ(α−n) γ(α) ∑n i=0 [[ (−1)icinbn−imxi,α−n,w ]) ex−e(x),β,w + ( γ(β−m) γ(β) ∑m i=0 [[ (−1)icimbm−imxi,β−m,w ]) ex−e(x),α,w   is valid. proof. we have [ jαa [p(b)] j β a [ p(b)(b−e(x))2 ] + jβa [p(b)] j α a [ p(b)(b−e(x))2 ] −2jαa [p(b)(b−e(x))] j β a [p(b)(b−e(x))] ]2 (3.22) ≤ [ (mjαa [p(b)] −j α a [p(b)(b−e(x))]) ( jβa [p(b)(b−e(x))] − m̃j β a [p(b)] ) + (jαa [p(b)(b−e(x))] − m̃j α a [p(b)]) ( mjβa [p(b)] −j β a [p(b)(b−e(x))] )]2 . taking : p = wf,m = b−e(x), m̃ = a−e(x) in (3.22), we can write jαa [wf(b)] σ 2 x,β,w + j β a [wf(b)] σ 2 x,α,w + 2 (a−e (x)) (b−e (x)) j α a [wf(b)] j β a [wf(b)] ≤ (a + b− 2e(x)) [ jαa [wf(b)] ex−e(x),β,w + j β a [wf(b)] ex−e(x),α,w ] . (3.23) by theorem 1 and using (3.5), we get (3.21). � remark 3.5. if we take w = 1 in theorem 5, we obtain theorem 3.10 of [7]. we prove also: theorem 3.6. let x be a continuous random variable having a p.d.f. f : [a,b] → r+, w : [a,b] → r+. then, for all α > 0, the following two inequalities hold: γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ] exr−1(x−e(x)),α,w − ( ex−e(x),α,w ) mxr−1,α,w ≤ ‖f‖2∞ [ jαa [w(b)] j α a [b rw(b)] −jαa [bw(b)] j α a [ br−1w(b) ]] (3.24) and ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) exr−1(x−e(x)),α,w − ( ex−e(x),α,w ) m xr−1,α,w ≤ (b−a) 2 ( br−1 −ar−1 )(γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ])2 . (3.25) some estimations on continuous random variables 15 proof. we have 1 γ2(α) ∫ b a ∫ b a (b−x)α−1(b−y)α−1p(x)p(y)(g(x) −g(y))(h(x) −h(y)) = 2jαa [p(b)] j α a [pgh(b)] − 2(j α a [pg(b)] j α a [ph(b)]) (3.26) taking p = wf, g(b) = b−e(x) and h(b) = br−1, we obtain 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y) ( xr−1 −yr−1 ) w(x)w(y)f(x)f(y)dxdy = 2jαa [wf(b)] exr−1(x−e(x)),α,w − 2 ( ex−e(x),α,w ) mxr−1,α,w. (3.27) therefore, 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y) ( xr−1 −yr−1 ) w(x)w(y)f(x)f(y)dxdy ≤ ‖f‖2∞ [ 2jαa [w(b)] j α a [b rw(b)] − 2jαa [bw(b)] j α a [ br−1w(b) ]] . (3.28) combining (3.27), (3.28) and (3.5), we obtain (3.24). to obtain (3.25), it suffices to see that 1 γ2 (α) b∫ a b∫ a (b−x)α−1 (b−y)α−1 (x−y) ( xr−1 −yr−1 ) w(x)w(y)f(x)f(y)dxdy ≤ (b−a) ( br−1 −ar−1 ) (jαa [wf(b)]) 2 (3.29) and to combine (3.28), (3.29) and (3.5). � remark 3.6. taking α = 1, we obtain theorem 3.1 of [11]. theorem 3.7. let x be a continuous random variable having a p.d.f. f : [a,b] → r+, w : [a,b] → r+. then we have: (i∗): for any α > 0,β > 0; n = [α− 1],m = [β − 1] ( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) exr−1(x−e(x)),β,w + ( γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) exr−1(x−e(x)),α,w −ex,α,wmxr−1,β,w −ex,β,wmxr−1,α,w (3.30) ≤ ‖f‖2∞ [ jαa [w(b)] j β a [b rw(b)] + jβa [w(b)] j α a [b rw(b)] − jαa [bw(b)] j β a [ br−1w(b) ] −jβa [bw(b)] j α a [ br−1w(b) ]] where f ∈ l∞ [a,b] . 16 dahmani, khamel, bezziou and sarikaya (ii∗): the inequality( γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) exr−1(x−e(x)),β,w + ( γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) exr−1(x−e(x)),α,w −ex,α,wmxr−1,β,w −ex,β,wmxr−1,α,w ≤ (b−a) 2 ( br−1 −ar−1 )(γ(α−n) γ(α) n∑ i=0 [[ (−1)icinb n−imxi,α−n,w ]) (3.31) × ( γ(β −m) γ(β) m∑ i=0 [[ (−1)icimb m−imxi,β−m,w ]) holds for all α > 0, β > 0; n = [α− 1],m = [β − 1]. proof. in [4], it has been proved that 1 γ(α)γ(β) ∫ b a ∫ b a (b−x)α−1(b−y)β−1p(x)p(y)(g(x) −g(y))(h(x) −h(y)) = jαa [p(b)] j α a [pgh(b)] + j β a [p(b)] j β a [pgh(b)] (3.32) −(jαa [pg(b)] j α a [ph(b)]) − (j β a [pg(b)] j β a [ph(b)]) in (3.32), we take p = wf, g(b) = b−e(x), h(b) = br−1. we obtain 1 γ (α) 1 γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1)(x−y) ( xr−1 −yr−1 ) w(x)w(y)f(x)f (y) dxdy = jαa [wf(b)] exr−1(x−e(x)),β,w + j β a [wf(b)] exr−1(x−e(x)),α,w (3.33) −ex,α,wmxr−1,β,w −ex,β,wmxr−1,α,w. on the other hand, it is clear that 1 γ (α) 1 γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 (x−y) ( xr−1 −yr−1 ) w(x)w(y)f(x)f(y)dxdy ≤ ‖f‖2∞ [ jαa [w(b)] j β a [b rw(b)] + jβa [w(b)] j α a [b rw(b)] (3.34) − jαa [bw(b)] j β a [ br−1w(b) ] −jβa [bw(b)] j α a [ br−1w(b) ]] . consequently, by (3.33), (3.34) and (3.5), we deduce (3.30). to prove the second part, we observe that 1 γ (α) 1 γ (β) b∫ a b∫ a (b−x)α−1 (b−y)β−1 w(x)w(y)(x−y) ( xr−1 −yr−1 ) f(x)f(y)dxdy = (b−a) ( br−1 −ar−1 ) jαa [wf(b)] j β a [wf(b)] . (3.35) then, we take into account (3.33) and (3.35). we obtain (3.31). � remark 3.7. taking α = β in the above theorem, we obtain theorem 5. references [1] s. belarbi, z. dahmani: on some new fractional integral inequalities. j. inequal. pure appl. math, 10(3) (2009), 1-12. [2] n.s. barnett, p. cerone, s.s. dragomir, j. roumeliotis: some inequalities for the expectation and variance of a random variable whose pdf is n-time diferentiable. j. inequal. pure appl. math., 1(21) (2000), 1-29. [3] n.s. barnett, p. cerone, s.s. dragomir and j. roumeliotis: some inequal-ities for the dispersion of a random variable whose pdf is defined on a finite interval. j. inequal. pure appl. math., 2(1) (2001), 1-18. [4] z. dahmani: fractional integral inequalities for continuous random variables. malaya j. mat., 2(2) (2014), 172-179. some estimations on continuous random variables 17 [5] z. dahmani: new inequalities in fractional integrals. int. j. nonlinear sci., 9(4) (2010), 493-497. [6] z. dahmani: on minkowski and hermite-hadamad integral inequalities via fractional integration. ann. funct. anal., 1 (1) (2010), 51-58. [7] z. dahmani: new applications of fractional calculus on probabilistic random variables. acta math. univ. commen., 86 (2) (2017), 299-307. [8] z. dahmani, l. tabharit: on weighted gruss type inequalities via fractional integrals. j. adv. res. pure math., 2(4) (2010), 31-38. [9] z. dahmani, a.e. bouziane, m. houas, m.z. sarikaya: new w-weighted concepts for continuous random variables with applications. note di math., in press. [10] r. gorenflo, f. mainardi: fractional calculus: integral and diferential equations of fractional order. springer verlag, wien, (1997), 223-276. [11] p. kumar: moment inequalities of a random variable defined over a finite interval. j. inequal. pure appl. math., 3 (3)(2002), art. id 41. [12] p. kumar: inequalities involving moments of a continuous random variable defined over a finite interval. comput. math. appl., 48 (2004), 257-273. [13] m. niezgoda: new bounds for moments of continuous random varialbes, comput. math. appl., 60 (12) (2010), 3130-3138. [14] m.z. sarikaya, h. yaldiz: new generalization fractional inequalities ofostrowski-gruss type. lobachevskii j. math., 34 (4) (2013), 326-331. [15] m.z. sarikaya, h. yaldiz and n. basak: new fractional inequalities of ostrowski-gruss type. le matematiche, 69 (1) (2014), 227-235. [16] r. sharma, s. devi, g. kapoor, s. ram, n.s. barnett: a brief note on some bounds connecting lower order moments for random variables defined on a finite interval. int. j. theor. appl. sci., 1(2), (2009), 83-85. 1laboratory lpam, faculty of sei, umab, university of mostaganem, algeria 2department of mathematics, faculty of science and arts, düzce university, düzce-turkey 3department of mathematics, university of khemis miliana, ain defla, algeria ∗corresponding author: sarikayamz@gmail.com 1. introduction 2. preliminaries 3. main results references int. j. anal. appl. (2023), 21:83 nigh-open sets in topological space jamal oudetallah1, nabeela abu-alkishik2, iqbal m. batiha3,4,∗ 1department of mathematics, irbid national university, irbid 21110, jordan 2department of mathematics, jerash university, jerash 2600, jordan 3department of mathematics, al-zaytoonah university of jordan, amman 11733, jordan 4nonlinear dynamics research center (ndrc), ajman university, ajman, uae ∗corresponding author: i.batiha@zuj.edu.jo abstract. in this paper, we aim to introduce a new class of open sets namely nigh-open set. accordingly, we define a topological space called a nigh-topological space. this consequently leads us to outline several new operations in connection with the sets in nigh-topological space coupled with deriving several their properties and relations. 1. introduction a topology on a non-empty set i defined as a collection ℵ of subset of i called open sets. these sets commonly satisfy three main axioms; the empty set i itself belong to ℵ, any arbitrary (finite or infinite) union of member of ℵ belongs to ℵ, and finally that the intersection of any finite number of member of ℵ belongs to ℵ [1]. in regard with the literature about this topic, we point out that all results reported in this work are not addressed at all. however, we state some other related works for completeness. for instance, r. a. hosny et al. provided a definition of r-neighborhood (open set) related to the usual topology in [2]. in [3], a. s. salama et al. used a novel type of open sets called rough open set, which was used to define the so-called rough continuous functions. in [4, 5], j. oudetallah studied other type of open sets called d-open sets, which were employed to define the d-metacompact spaces. some other related works can be found in [6–9]. received: jun. 8, 2023. 2020 mathematics subject classification. 54a05. key words and phrases. nigh-open set; nigh-closed set; nigh-topological space; topological space. https://doi.org/10.28924/2291-8639-21-2023-83 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-83 2 int. j. anal. appl. (2023), 21:83 in this research paper, we intend to define a new topological space namely a nigh-topological space such that the member of the collection ℵ(n) is a nigh-open set. such a set is defined as a nigh-open set ς if there exist two open sets % and ϑ such that ϑ ⊆ ς ⊆ ext(%) and %∩ς = φ. consequently, we define the complement of this nigh-open set; the nigh-closed set. furthermore, based on some basic topological meanings, we define a nigh-closure of a set nigh-derive sets nigh-interior, night-exterior and nigh-boundary set. several results and theorems related to the properties and relations between these sets are studied and derived in a well-defined topological space. 2. basic definitions in this section, we aim to pave the way to our main results by recalling two significant definitions. these definitions are connected with the concept of the topological space and the concept of regular open/closed/semi set. definition 2.1. [10] a topological space is a pair (i,ℵ) consisting of a set i and a family ℵ of subsets of i satisfying the following conditions: (1) φ ∈ℵ and i∈ℵ. (2) the union of any numbers of members in ℵ is a member in ℵ. (3) the intersection of any two members in ℵ is a member in ℵ. definition 2.2. [11,12] suppose a ⊆i and (i,τ) is a topological space. we say that: (1) a is a regular open set in i if a = a o . (2) a is regular closed set in i if and only if a = ao. (3) a is a semi open set in i if and only if there exists an open set u such that u ⊆ a ⊆ ū. 3. formulations this part aims to lay the groundwork for understanding and theorizing the nigh-open set and the nigh-topology. as a result, some results are reported for further investigation in this study. definition 3.1. let (i,ℵ) be a topological space and ς be a subset of i. then ς is said to be a nigh-open set if there exist two open sets % and ϑ such that ϑ ⊆ ς ⊆ ext(%) and % ∩ ς = φ. the complement of a nigh-open set is called nigh-closed set. remark 3.1. from the above definition, one might assert: (1) %∩ϑ = φ. (2) ϑ is called the first open set and % is called the second open set. based on what we previously established, we state and prove the next results. theorem 3.1. every open set in any topological space is a nigh-open set. int. j. anal. appl. (2023), 21:83 3 proof. to prove this result, we first let ς be an open set in the topological space (i,ℵ). consequently, ς ⊆ ς ⊆ ext(φ), ς∩ = φ = φ. thus ς is a nigh-open set. � remark 3.2. the converse of the above theorem need not be true. for example, if we take the set [1, 2] in the usual topology defined on r, we find that this set represents a nigh-open set. this is because there exist two open sets (0, 1) and (4, 5) such that (0, 1) ⊆ (1, 2] ⊆ ext(4, 5). but it is clearly the set (1, 2] is not open set in the usual topology. theorem 3.2. if ς is a nigh-open set in the topological space (i,ℵ), then (%) ⊂ ext(ς) ⊆ ext(ϑ), where ϑ and % are the first and second open sets reported in remark 3.1, respectively. proof. let ς be a nigh-open set in i, then there exist two open sets ϑ and % such that ϑ ⊆ ς ⊆ ext(%) and %∩ϑ = φ. thus, we have: ς ⊆ ext(%) = ⋂ f closed ext(%)⊆f f. consequently, we obtain: ⋃ fc open set fc⊂(ext(%))c fc = % ⊂ ςc = ext(ς). now, putting w = fc leads to assert that w is an open set and accordingly we get:⋃ w open w⊂% w ⊂ ext(ς). but we have: ⋃ w open w⊂% w = int(ς), and since ϑ ⊆ ς, then ext(ς) ⊆ ext(%). this gives int(% ⊆ ext(ς) ⊆ ext(ϑ), which finishes the proof of this result. � in what follow, based on the main axioms founded for the traditional topology, we establish one of the most targets of this work; the nigh-topological space. this would help us to derive further results in the upcoming section. definition 3.2. let i be a non-empty set and ℵ⊆ p(i). then an ℵ is said to be a nigh-topology on i if the following statements are hold: (1) φ,i∈ℵ. (2) the intersection of any two nigh-open sets is a nigh-open set. (3) the union of any family of nigh-open sets is a nigh-open set. 4 int. j. anal. appl. (2023), 21:83 4. results in this section, several novel relations and properties in respect to the nigh-open sets and the nigh-topological space are stated and derived well. theorem 4.1. every topological space is a nigh-topological space. proof. in order to prove this result, we let (i,ℵ) be a topological space. to show that (i,ℵ) is a nigh-topological space, we should note the following points: (1) by definition 2.1, it should be noticed that i,φ ∈ℵ. (2) if one lets ς,$ be two nigh-open sets, then there exist the open sets ϑ1,ϑ2,%1 and %2 such that: ϑ1 ⊆ ς ⊆ ext(%1,ϑ2 ⊆ b ⊆ ext(%2). this consequently implies: ϑ1 ∩ϑ2 ⊆ ς ∩$ ⊆ ext(%1) ∩ext(%2) = ext(%1 ∩%2). putting ϑ1 ∩ϑ2 = ϑ and %1 ∩%2 = % yields to assert that ϑ and % are two open sets. this means that ϑ ⊆ ς ∩$ ⊆ ext(%) and so ς ∩$ is a nigh-open set. (3) if one lets ς = {ςα : α ∈ λ} be a nigh-open set, then to show that ∪ςα is a nigh-open set for every α ∈ λ,ςα, we should note that there exist two open sets ϑα and %α such that ϑα ⊆ ςα ⊆ ext(%α) for every α ∈ λ. also, we have:⋃ α∈λ ϑα ⊆ ⋃ α∈λ ςα ⊆ ⋃ α∈λ ext(%α) ⊆ ext(φ). thus, we deduce that ⋃ α∈λ ςα is a nigh-open set. � definition 4.1. let (i,ℵ(n)) be a nigh-topological space and ς be a subset of i, then: (1) ι ∈ i is called a nigh-limit point of a set ς if for any nigh-open set % containing n, we have: %∩ ςn 6= φ, if ι /∈ ς %∩ ςn\{ι} 6= φ, if ι ∈ ς . (2) the nigh-derived set of ς symbolized by ς′n is defined by ς ′ (n) = {% ∈ i : ι is nigh-limite point of ς}. (3) the nigh-closure set of ς symbolized by cl(n)(ς) is defined by cl(n)(ς) = ς ∪ ς′(n). (4) the nigh-exterior set of ς is defined by int(n)(ς) = ( cl(n)(ς c) )c . (5) the nigh-exterior set of ς is defined by ext(n)(ς) and ext(n)(ς) = ( cl(n)(ς) )c . (6) the nigh-boundary set of ς is defined by bd(n)(ς) and bd(n) = cl(n)(ς) ∩cl(n)(ςc). theorem 4.2. (initial nigh-theorem) let (i,ℵ(n)) be a nigh-topological space, then: (1) φ′ (n) = φ. int. j. anal. appl. (2023), 21:83 5 (2) cl(n)(φ) = φ and cl(n)(i) = i. (3) int(n)(φ) = φ and int(n)(i) = i. (4) ext(n)(φ) = i,ext(n)(i) = φ. proof. (1) to prove this result, we suppose not, then there is ι ∈ i such that ι ∈ φ′ (n) . so i is a limit point of φ(n), and then for all nigh-open sets containing i, we have %∩φ 6= φ, which is contradiction! hence, the result holds. (2) it should be noted that cl(n)(φ) = φ∪φ ′ (n) = φ∪φ = φ, and cl(n)(i) = i∪i ′ (n) = i. (3) herein, we have: int(n)(φ) = ( cl(n)(φ c) )c = ( cl(n)(i) )c = ic = φ, and int(n)(i) = ( cl(n)(i c) )c = ( cl(n)(φ) )c = φc = i. (4) in this part, we note: ext(n)(φ) = ( cl(n)(φ c) )c = φc = i, and ext(n)(i) = ( cl(n)(i c) )c = ic = φ. � theorem 4.3. (inclusion nigh-theorem) let (i,ℵ) be a nigh-topological space. suppose ς and $ are two subsets of i, then we have: (1) if ς ⊆ $, then ς′ (n) ⊆ $′ (n) . (2) if ς ⊆ $, then cl(n)(ς) ⊆ cl(n)($). (3) if ς ⊆ $, then int(n) ⊆ int(n)($). (4) if ς ⊆ $, then ext(n) ⊆ ext(n)($). proof. (1) let ι ∈ ς′ (n) , so i is a nigh-limit point of ς. therefore, for all nigh-open set % containing n, we have:  %∩ ςn 6= φ, if ι /∈ ς %∩ ςn)\{ι} 6= φ, if ι ∈ ς . now, since ς ⊆ $. we have: %∩$n 6= φ, if ι /∈ $ %∩$n)\{ι} 6= φ, if ι ∈ $ . 6 int. j. anal. appl. (2023), 21:83 consequently, i is a nigh-limit point of $, and thus ι ∈ $′ (n) . (2) since ς ⊆ $, then by theorem 3.1, we have ς′ (n) ⊆ $′ (n) and ς∪ς′ (n) ⊆ $∪$′ (n) . this means cl(n)(ς) ⊆ cl(n)($). (3) since ς ⊆ $, then $c ⊆ ςc. using theorem 3.2 yields: cl(n)(ς c) ⊆ cl(n)(ς c), and ( cl(n)(ς c) )c ⊆ (cl(n)($c))c. consequently, we have int(n)(ς) ⊆ int(n)($). (4) since ς ⊆ $, then by using theorem 3.2, we have: cl(n)(ς) ⊆ cl(n)($), and ( cl(n)($ c) )c ⊆ (cl(n)(ςc))c. this means ext(n)($) ⊆ ext(n)(ς), which completes the proof of this result. � theorem 4.4. (openness and closeness nigh-theorem) let (i,ℵ) be a nigh-topological space and let ς ⊆i, then: (1) cln)(ς) is a nigh-closed set. (2) int(n)(ς) is a nigh-open set. (3) ext(n)(ς) is a nigh-open set. (4) bd(n)(ς) is a nigh-closed set. proof. (1) to prove this result, we let ι ∈ ( cl(n)(ς) )c . then ι /∈ cl(n)(ς), and so ι /∈ ς ∪ ς′(n), ι /∈ ς and ι /∈ ς′ (n) . therefore, there exists a nigh-open set % such that %∩ ς(n) = φ (say *). consequently, we have: %∩cl(n)(ς) = %∩ (ς ∩ ς ′ (n)) = (%∩ ς) ∪ (%∩ ς ′ (n) = φ∪ (%∩ ς ′ (n)). this means that %∩cl(n)(ς) = %∩ ς′(n). now, if ι ∈ (%∩ ς ′ (n) ), then ι ∈ % and %∩ ς(n) 6= φ, which contradicts (*). so, (%∩ ς′ (n) ) = φ and hence %∩cl(n)(ς) = φ. thus, we have: ι ∈ %ι ⊆ ( cl(n)(ς) )c , which immediately yields: ( cl(n)(ς) )c = ⋃ ι∈%ι %ιis nigh-open set %ι, int. j. anal. appl. (2023), 21:83 7 i.e., we have ( cl(n)(ς) )c is a nigh-open set, and so cl(n)(ς) is a nigh-closed set. (2) by theorem 3.1, we have cl(n)(ς c) is a nigh-closed set. therefore, ( cl(n)(ς) )c is a nighopen set, i.e., int(n)(ς) is a night-open set. (3) we notice that ext(n)(ς) = ( cl(n)(ς) )c , which is a nigh-open set (4) we notice that bd(n)(ς) = cl(n)(ς) ∩cl(n)(ςc), which is a nigh-closed set. � theorem 4.5. cl(n)(ς) = ς if and only if ς is a nigh-closed set. proof. ⇒) trivial. ⇐) let ς be a nigh-closed set. clearly, we have ς ⊆ cl(n) (say *). now, to show that cl(n) ⊆ ς, we let ι ∈ ς′. to show that ι ∈ ς, we assume not, i.e. ι /∈ ς. this gives ι ∈ ςc, which is a nigh-open set. now, since ι ∈ ς′, then ςc ∩ ς 6= φ, which means that φ 6= φ. so, there is a contradiction here, and then ι ∈ ς. this implies ς′ ⊆ ς and ς ⊆ ς, and so ς′ ∪ ς ⊆ ς, i.e. cl(n)(ς) ⊂ ς. this completes the proof. � theorem 4.6. (union and intersection nigh theorem) let (i,ℵ) be a nigh-topological space and ς ⊆i, then: (1) cl(n)(ς) = ⋂ {κ;κ is nigh-closed set and ς ⊆ κ}, i.e. cl(n)(ς) is the smallest nigh-closed set containing ς. (2) int(n)(ς) = ∪{t : t is nigh-open set and t ⊆ ς}. (3) ext(n)(ς) = ∪{w : w is nigh-open set and w ⊆ ςc}. proof. (1) to show that ς ⊆ κ, we note by theorem 4.5 that cl(n)(ς) ⊆ cl(n)(κ) = κ and cl(n)(ς) is closed. this implies that cl(n)(ς) is one member of κ′s, and so⋂ κ is nigh-closed set ς⊆κ {κ}⊆ cl(n)(ς). on the other hand, if we want to show that ς ⊆κ, we should note: cl(n)(ς) ⊆ cl(n)(κ), and cl(n)(ς) ⊆ ⋂ κ is nigh-closed set ς⊆κ {cl(n)(κ)} = ⋂ κ is nigh-closed set ς⊆κ {κ}, which gives directly the desired result. (2) it should be noted here that intn)(ς) = ( cl(n)(ς c) )c = ∩κ such that κ is nigh-closed set and ςc ⊆ κ. this means ( cl(n)(ς c) )c = ∪κc such that κc is nigh-closed set and κc ⊆ ς. by putting κc = t, we obtain int(n)(ς) = ∪{t : t is nigh-open set and t ⊆ ς}, which completes the proof. 8 int. j. anal. appl. (2023), 21:83 (3) by part 2, we can gain: ext(n)(ς) = int(n)(ς c) = ∪{w : w is nigh-open set and w ⊆ ςc}. � 5. conclusions this work has successfully defined a nigh-open set as well as its corresponding topological space; the nigh-topological space. several novel results and properties related to these new notions have been consequently generated and derived well. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] h.a. shehadeh, i.h. jebril, x. wang, s.c. chu, m.y.i. idris, optimal topology planning of electromagnetic waves communication network for underwater sensors using multi-objective optimization algorithms (mooas), automatika. 64 (2022), 315-326. https://doi.org/10.1080/00051144.2022.2123761. [2] r.a. hosny, b.a. asaad, a.a. azzam, t.m. al-shami, various topologies generated from ej-neighbourhoods via ideals, complexity. 2021 (2021), 4149368. https://doi.org/10.1155/2021/4149368. [3] a.s. salama, a. mhemdi, o.g. elbarbary, t.m. al-shami, topological approaches for rough continuous functions with applications, complexity. 2021 (2021), 5586187. https://doi.org/10.1155/2021/5586187. [4] j. oudetallah, countably and locally compactness in bitopological spaces, gen. lett. math. 12 (2022), 148-153. https://doi.org/10.31559/glm2022.12.3.5. [5] j. oudetallah, m.m. rousan, i.m. batiha, on d-metacompactness in topological spaces, j. appl. math. inform. 39 (2021), 919-926. [6] j. oudetallah, i.m. batiha, on almost expandability in bitopological spaces, int. j. open problems compt. math. 14 (2021), 43-48. [7] a. a. hnaif, a. a. tamimi, a. m. abdalla, i. jebril, a fault-handling method for the hamiltonian cycle in the hypercube topology, comput. mater. contin. 68 (2021), 505-519. https://doi.org/10.32604/cmc.2021.016123. [8] j. oudetallah, i.m. batiha, mappings and finite product of pairwise expandable spaces, int. j. anal. appl. 20 (2022), 66. https://doi.org/10.28924/2291-8639-20-2022-66. [9] j. oudetallah, r. alharbi, i.m. batiha, on r-compactness in topological and bitopological spaces, axioms. 12 (2023), 210. https://doi.org/10.3390/axioms12020210. [10] j.l. kelley, general topology, springer, new york, 1955. [11] j. dugundji, topology, allyn and bacon, boston, 1966. [12] n. levine, semi-open sets and semi-continuity in topological spaces, amer. math. mon. 70 (1963), 36-41. https://doi.org/10.1080/00029890.1963.11990039. https://doi.org/10.1080/00051144.2022.2123761 https://doi.org/10.1155/2021/4149368 https://doi.org/10.1155/2021/5586187 https://doi.org/10.31559/glm2022.12.3.5 https://doi.org/10.32604/cmc.2021.016123 https://doi.org/10.28924/2291-8639-20-2022-66 https://doi.org/10.3390/axioms12020210 https://doi.org/10.1080/00029890.1963.11990039 1. introduction 2. basic definitions 3. formulations 4. results 5. conclusions references int. j. anal. appl. (2023), 21:61 powered inverse rayleigh distribution using dus transformation m. i. khan1, abdelfattah mustafa1,2,∗ 1mathematics department, faculty of science, islamic university of madinah, madinah 42351, ksa 2mathematics department, faculty of science, mansoura university, mnasoura 35516, egypt ∗corresponding author: amelsayed@mans.edu.eg abstract. this article reports an extension of powered inverse rayleigh distribution via dus transformation, named dus-powered inverse rayleigh (dus-pir) distribution. some statistical properties of suggested distribution in particular, moments, mode, quantiles, order statistics, entropy and , inequality measures have been investigated extensively. to estimate the parameters, maximum likelihood estimation (mle) is discussed. the model flexibility is validated by two real data. 1. introduction the accuracy and consistency of statistical analysis are extremely affected by the assumed probability model or distribution. as a result of this verity, in recent decades formulating new distributions becomes a basic conception in statistical theory; this is generally done by adding an extra parameter to the baseline distribution. for example, [1–5] and many more. the different transformation techniques have been used by the several authors. for example, dus, sine, and mg transformations are reported by [6–8]. in all transformation exponential distribution is deemed as baseline distribution. if g(x) and g(x) denote the probability density function (pdf) and cumulative density function (cdf) of a baseline lifetime distribution, then the pdf and cdf of a dus-transformation are given as f (x) = 1 e − 1 g(x)eg(x), x > 0, (1.1) f (x) = eg(x) − 1 e − 1 , x > 0. (1.2) received: mar. 17, 2023. 2020 mathematics subject classification. 62g05, 62g20. key words and phrases. dus transformation; lifetime distribution; entropy; maximum likelihood estimation. https://doi.org/10.28924/2291-8639-21-2023-61 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-61 2 int. j. anal. appl. (2023), 21:61 hazard rate function (hrf) is. h(x) = g(x) e−[g(x)−1] − 1 . (1.3) inverse rayleigh (ir) distribution was introduced by [9]. ir distribution finds enormous applications in survival analysis. various properties of ir distribution have been studied by [10]. powered ir distribution was proposed by [11] through the powered transformation to increase its flexibility and applicability. [5] established and studied in detail the length powered ir distribution. [12] established the several recurrence relations from powered ir distribution. a random variable (r.v.) x follows powered ir distribution, if its pdf and cdf are given, respectively by: g(x; α,θ) = 2α θx2α+1 e − 1 θx2α , α,θ > 0, x > 0. (1.4) g(x; α,θ) = e − 1 θx2α , α,θ > 0, x > 0. (1.5) to modelling all kinds of data sets, no single distribution can be speculated as the best fit. despite of existence many distributions in the literature. we are therefore induced to establish a new distribution via dus transformation and named as dus-powered ir distribution. the paper is framed as follows: in section 2, dus-powered ir distribution is derived and graphically depicted. several mathematical and statistical properties are established in section 3. also, the entropies and measures of inequality are addressed in sections 3. the parameters estimation is obtained in section 4. the model superiority is shown through two real data in section 5. section 6 reports the concluding remarks. 2. duspowered ir distribution now utilizing (1.4) and (1.5) into (1.1) and (1.2) respectively. we can obtain the cdf, pdf and hrf for dus-pir distribution as follows: f (x; α,θ) = exp ( e− 1 θ x−2α ) − 1 e − 1 , x > 0; α,θ > 0, (2.1) f (x; α,θ) = 2αx−(2α+1) (e − 1)θ exp ( e− 1 θ x−2α ) e− 1 θ x−2α, (2.2) h(x,α,θ) = 2αx−(2α+1)e− 1 θ x−2α θ [ exp ( 1 −e− 1 θ x−2α ) − 1 ] (2.3) respectively. the depiction of plots are shown in the following figures for fix parameters. int. j. anal. appl. (2023), 21:61 3 figure 1. f (x) for fix parameters. figure 2. h(x) for fix values of α and θ. the depiction from figures 1 and 2 are. (i) the dus-pir distribution has unimodal, (ii) the failure rate is increasing, then decreasing for fix values of parameters α and θ. 3. some statistical properties some statistical properties of dus-pir distribution, including rth moments, quantile function, skewness, kurtosis, and order statistics are studied. 3.1. the moments: let x ∼dus-pir distribution with parameters (α,θ), then the rth moment is given in theorem 3.1. 4 int. j. anal. appl. (2023), 21:61 theorem 3.1. the moments of dus-pir distribution is given as µ ′ r = ( 1 e − 1 ) ∞∑ k=0 1 (k + 1)! ( k + 1 θ ) r 2α γ ( 1 − r 2α ) . (3.1) proof: the rth moment of the r.v. x is µ ′ r = ∫ ∞ 0 xrf (x; α,θ)dx. from (2.2), we have µ ′ r = ∫ ∞ 0 2α (e − 1)θ xr−(2α+1)ee − 1 θ x2α e− 1 θ x2αdx, since θ > 0, we have ee − 1 θ x2α = ∑∞ k=0 e − k θx2α k! , so µ ′ r = ∞∑ k=0 2α k!(e − 1)θ ∫ ∞ 0 xr−(2α+1)e− (k+1) θ x−2αdx (3.2) let u = ( k+1 θ ) x−2α, then equation (3.2) reduces as µ ′ r = ( 1 e − 1 ) ∞∑ k=0 1 (k + 1)! ( k + 1 θ ) r 2α γ ( 1 − r 2α ) . 3.2. mode: setting first derivative of (2.2) as follows. f ′ (x) = 2α (e − 1)θ2 x−2(2α+1) exp ( e− 1 θ x−2α − 1 θ x−2α )[ 2α−θ(1 + 2α)x2α + 2αe− 1 θ x−2α ] = 0. (3.3) above equation does not possess analytic solution in x. for a quick graphical solution of the mode, we sketch the plot of left-hand side of (3.3) at different values of α,θ as depicted in figure 3. figure 3. f ′ (x) for selected values of α and θ. int. j. anal. appl. (2023), 21:61 5 it confirms from these plots that dus-pir distribution has one mode based on selected values of α and θ. 3.3. quantiles and random number generation: the quantile xq,(0 < q < 1), of dus-pir(α,θ) distribution can be attained, by employing the cdf in (2.1), in the given simple form. xq = { − 1 θ ln [ln(q(e − 1) + 1)] } 1 2α . (3.4) one of the good characteristics of the suggested distribution is that we can smoothly calculate its quantiles in simple as well as an explicit form. to generate random sample with size (n ≥ 1) form dus-pir(α,θ) distribution, we can use (3.4) by generating n random values for q, where q ∼ u(0, 1). to find the median, using the above equation for q = 0.50, med. = { − 1 θ ln [ln(0.5(e + 1))] } 1 2α . the shapes of dus-pir distribution can be viewed by skewness and kurtosis. utilizing the concept of quantiles, skewness and kurtosis are as follows, [13]. bowley’ skewness: sk = x0.75 − 2x0.50 + x0.25 x0.75 −x0.25 . moors’ kurtosis: ku = x0.875 + x0.375 − (x0.625 + x0.125) x0.75 −x0.25 3.4. order statistics: the rth order statistic (o.s.) x(r) based on ordered sample (x1 < x2 < · · · < xn) from a continuous distribution having cdf fx(x) and pdf fx(x) is. fx(r) (x) = n! (r − 1)!(n− r)! fx (x)[fx (x)] r−1[1 −fx (x)]n−r, r = 1, 2, · · · ,n. (3.5) so, the rth order statistic from dus-pir distribution is fx(r) (x) = n! (r − 1)!(n− r)! 2αx−(2α+1) (e − 1)nθ e ( e − 1 θ x−2α−1 θ x−2α ) [ ee − 1 θ x−2α − 1 ]r−1 [ e −ee − 1 θ x−2α ]n−r (3.6) putting r = 1 and r = n in (3.6), we can obtain pdf of smallest and largest (o.s.). 6 int. j. anal. appl. (2023), 21:61 3.5. entropy: entropy helps to measure the uncertainty of the r.v. x. some notable entropies are defined as follows. rényi entropy: rδ(x) = 1 1 −δ log [∫ ∞ 0 f δ(x)dx ] , δ > 0 and δ 6= 1. (3.7) tsallis entropy: tδ(x) = 1 1 −δ [∫ ∞ 0 f δ(x)dx − 1 ] , δ > 0 and δ 6= 1. (3.8) havrda and charvat entropy (h-c) hcδ(x) = 1 21−δ − 1 [∫ ∞ 0 f δ(x)dx − 1 ] . (3.9) theorem 3.2. if x ∼dus-pird, then the rényi entropy of x is given as rδ(x) = 1 1 −δ log   1 2α ( 2α θ(e − 1) )δ γ ( δ(2α + 1) − 1 2α ) ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α   . (3.10) proof: from (2.2) into (3.7), we have f δ(x) = ( 2α θ(e − 1)x2α+1 )δ ∞∑ k=0 δke− (k+δ) θ x−2α k! and ∫ ∞ 0 f δ(x)dx = ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ∫ ∞ 0 x−δ(2α+1)e− (k+δ) θ x−2αdx let u = k+δ θ x−2δ, then x = ( θ k+δ )− 1 2α u− 1 2α and ∫ ∞ 0 f δ(x)dx = ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α 1 2α ∫ ∞ 0 u δ(2α+1)−1 2α−1 −1e−udu = 1 2α ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α γ ( δ(2α + 1) − 1 2α ) . therefore, the renyi entropy is rδ(x) = 1 1 −δ log   1 2α ( 2α θ(e − 1) )δ γ ( δ(2α + 1) − 1 2α ) ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α   . theorem 3.3. if x ∼ dus-pird(α,θ), the tsallis entropy of x is tδ(x) = 1 1 −δ   1 2δ ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α γ ( δ(2α + 1) − 1 2α ) − 1   . (3.11) proof: proof is easy. int. j. anal. appl. (2023), 21:61 7 theorem 3.4. if x ∼dus-pird(α,θ), the havrda and charvat entropy of x is hcδ(x) = 1 21−δ − 1   1 2α ( 2α θ(e − 1) )δ ∞∑ k=0 δk k! ( θ k + δ )δ(2α+1)−1 2α γ ( δ(2α + 1) − 1 2α ) − 1   . (3.12) proof: proof is easy. 3.6. bonferroni and lorenz curves: a model for inequality of wealth distribution was proposed by [14] and to measure the income inequality introduced by [15]. both models are used in financial mathematics, insurance, and population studies. bonferroni and lorenz’s curves are defined as: b(p) = 1 pµ ∫ q 0 xf (x)dx, l(p) = 1 µ ∫ q 0 xf (x)dx. (3.13) from (2.2), ∫ q 0 xf (x)dx = ∞∑ k=0 2α k!(e − 1)θ ∫ ∞ 0 x−2αe− (k+1) θ x−2αdx let u = ( k+1 θ ) x−2α, then ∫ q 0 xf (x)dx = ∞∑ k=0 1 k!(e − 1)θ ( θ k + 1 )1− 1 2α ∫ ∞ ( k+1θ )q −2α u− 1 2αe−udx = 1 e − 1 ( 1 θ ) 1 2α ∞∑ k=0 1 (k + 1)! (k + 1) 1 2α γ ( 1 − 1 2α , (k + 1) θ q−2α ) . (3.14) from equations (3.1), when r = 1 and (3.14) into (3.13), then the bonferroni curve is given by b(p) = 1 pµ ∫ q 0 xf (x)dx = ∑∞ k=0 (k+1) 1 2α (k+1)! γ ( 1 − 1 2α , (k+1) θ q−2α) ) p ∑∞ k=0 (k+1) 1 2α (k+1)! γ ( 1 − 1 2α ) . (3.15) the lorenz curve is obtained as l(p) = 1 µ ∫ q 0 xf (x)dx = ∑∞ k=0 (k+1) 1 2α (k+1)! γ ( 1 − 1 2α , (k+1) θ q−2α) ) ∑∞ k=0 (k+1) 1 2α (k+1)! γ ( 1 − 1 2α ) . (3.16) 4. estimation of parameters to understand the probabilistic model fully, estimating the unknown parameters for designated sample is a main procedure. various estimation approaches under classical and bayesian model are reported in literature. this section considers the estimation of dus-pir distribution via maximum likelihood approach based on complete data. 8 int. j. anal. appl. (2023), 21:61 4.1. maximum likelihood estimation: let x1,x2, · · · ,xn random sample follows the dus-pir distribution. the likelihood function (l.f.) of (2.2) is l(α,θ) = n∏ i=1 f (xi,α,θ) = n∏ i=1 [ 2αx−(2α+1) (e − 1)θ exp ( e− 1 θ x−2α ) e− 1 θ x−2α ] . (4.1) the log-l.f.is. given by logl(α,θ) = −n ln(e−1) +n ln(2α)−n ln(θ)−(2α+ 1) n∑ i=1 ln(xi )− 1 θ n∑ i=1 x−2α i + n∑ i=1 e− 1 θ x−2α i . (4.2) the partial derivatives of (4.2) are as follows. ∂ ∂α logl(α,θ) = n α − 2 n∑ i=1 ln(xi ) + 2 θ n∑ i=1 x−2α i ln(xi ) + 2 θ n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ), ∂ ∂θ logl(α,θ) = − n θ + 1 θ2 n∑ i=1 x−2α i + 1 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i . the mles of α and θ can be derived as follows. n α − 2 n∑ i=1 ln(xi ) + 2 θ n∑ i=1 x−2α i ln(xi ) + 2 θ n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ) = 0, (4.3) − n θ + 1 θ2 n∑ i=1 x−2α i + 1 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i = 0. (4.4) equations (4.3) and (4.4) has no closed form. so, we shall use a numerical program system to find its solution with respect to α and θ. 4.2. asymptotic confidence interval: we derive asymptotic confidence intervals of unknown parameters using variancecovariance matrix vvv , which is the inverse fisher information matrix. the ml estimators are asymptotically normally distributed with multivariate normal distribution, see, [16]. (α̂, θ̂) ∼ n2(θθθ,vvv ), where θθθ = (α,θ) and vvv is given as follows vvv = ( −∂ 2logl ∂α2 −∂ 2logl ∂α∂θ −∂ 2logl ∂α∂θ −∂ 2logl ∂θ2 )−1 θ→θ̂ , int. j. anal. appl. (2023), 21:61 9 where, ∂2 ∂α2 logl(α,θ) = − n α2 − 4 θ n∑ i=1 x−2α i [ln(xi )] 2 + 4 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i [ln(xi )] 2 (4.5) − 4 θ n∑ i=1 e− 1 θ x−2α i x−2α i [ln(xi )] 2, ∂2 ∂α∂θ logl(α,θ) = − 2 θ2 n∑ i=1 x−2α i ln(xi ) + 2 θ3 n∑ i=1 e− 1 θ x−2α i x−4α i ln(xi ) (4.6) − 2 θ2 n∑ i=1 e− 1 θ x−2α i x−2α i ln(xi ), ∂2 ∂θ2 logl(α,θ) = n θ2 − 2 θ3 n∑ i=1 x−2α i + 1 θ4 n∑ i=1 e− 1 θ x−2α i x−4α i − 2 θ3 n∑ i=1 e− 1 θ x−2α i x−2α i . (4.7) a 100(1 −δ)% confidence interval for θθθ = (α,θ), can be approximated by α̂±zδ 2 √ var(α̂), and θ̂±zδ 2 √ var(θ̂) where zδ 2 is upper 100 δ 2 -th percentile of n(0, 1), and var(θ̂i ) is the diagonal i-th element in vvv . 5. practical illustration the main objective of any new distribution is to increase its adaptability and applicability, which makes it useful in several field of studies, particularly, in the fields concerning with lifetime analysis. this section depicts the usefulness of dus-pir distribution and compare with the powered ir, exponential transformed ir, transmuted ir, exponentiated ir, ir and rayleigh distribution using two sets of data. for comparison some criteria such as, • k-s. (kolmogorov smirnov) statistic, k −s = sup x |fm(x) − f̂ (x)| • r2: the determination coefficient, r2 = ∑m i=1 ( f̂ (xi ) −f )2∑m i=1 ( f̂ (xi ) −f )2 + ∑m i=1 ( fm(xi ) − f̂ (xi ) )2 , • rmse: the root mean square error rmse = [ 1 m m∑ i=1 ( fm(xi ) − f̂ (xi ) )2]1/2 , • a.i.c. (akaike information criterion), [17]. aic = 2k − 2`. 10 int. j. anal. appl. (2023), 21:61 • a. i.c.c. (akaike information criterion with correction), [18]. aaic = aic + 2k(k + 1) m−k + 1 , • b.i.c. (bayesian information criterion), [19]. bic = k ln(m) − 2`, • and h.q.i.c. (hannan-quinn information criterion) hqic = 2k ln[ln(m)] − 2`, have been used, where k and m stands for number of parameters and observed data, ` = logl, f̂ (x) is estimated cdf and fm(x) is the empirical df. f̄ (x) = 1 m m∑ i=1 f̂ (xi ), fm(x) = 1 m m∑ i=1 i ( x(i) ≤ x ) and i ( x(i) ≤ x ) = { 1, if x(i) ≤ x 0, otherwise according to prevailing knowledge, the model with the lowest aic, aaic, bic, hqic and k-s value is considered as best fit for the data. dataset 1: the following data reported by [20]. it comprises thirty consecutive march precipitation (in inches) observations. 0.77 1.74 0.81 1.20 1.95 1.20 0.47 1.43 3.37 2.20 3.00 3.09 1.51 2.10 0.52 1.62 1.31 0.32 0.59 0.81 2.81 1.87 1.18 1.35 4.75 2.48 0.96 1.89 0.90 2.05 for the above considered data, we have extracted the values of mles of parameters, k-s test, and p-values in below table. table 1. mles, k-s statistics and p-value. models α̂ θ̂ λ̂ k-s p-value dus-pir 0.860 1.332 – 0.14557 0.52380 pir 0.775 0.975 – 0.15223 0.462057 etir – 1.454 – 0.18935 0.207305 tir – 1.591 -0.67 0.18176 0.247695 eir 0.731 1.456 – 0.19818 0.166981 ir – 1.164 – 0.23956 0.053115 rayleigh – 3.773 – 0.35059 0.000843 the log–likelihood (`), information criteria, rmse and r2 are reported below. int. j. anal. appl. (2023), 21:61 11 table 2. the `, information criteria, rmse and r2. models ` aic aicc bic hqic rmse r2 dus-pir -41.238 86.4760 86.9210 89.2790 87.3730 0.054453 0.96024 pir -41.917 87.8340 88.2780 90.6360 88.7310 0.059373 0.95182 etir -42.026 86.0530 86.1950 87.4540 86.5010 0.075709 0.93570 ter -42.101 88.2020 88.6470 91.0050 89.0990 0.073716 0.94105 eir -136.04 276.081 276.525 278.883 276.977 0.078017 0.92284 ir -44.137 90.2730 90.4160 91.6740 90.7210 0.107514 0.88381 rayleigh -38.924 79.8490 79.9910 81.2500 80.2970 0.201452 0.48597 listed values in the tables 1-2. it has been noticed that dus-pir distribution interprets a better fit among all lifetime distributions. the variance-covariance matrix is given as vvv = ( 0.012 0.008 0.008 0.076 ) . then the 95% confidence interval for α and θ for dus-pir distribution are (0.648, 1.073) and (0.790, 1.874), respectively. it is shown that the lf has a unique solution by figure 4. figure 4. the profile of the log-lf of α and θ. dataset 2: the given data set is reported by [21]. it represents the survival times (in days) of 72 guinea pigs injected with different doses of tubercle bacilli. 2 24 34 44 54 57 60 61 65 70 76 84 95 109 129 146 233 297 15 32 38 48 54 58 60 62 67 72 76 85 96 110 131 175 258 341 22 32 38 52 55 58 60 63 68 73 81 87 98 121 143 175 258 341 24 33 43 53 56 59 60 65 70 75 83 91 99 127 146 211 263 376 estimated values of parameters, test statistic and criterion are provided in the following table. 12 int. j. anal. appl. (2023), 21:61 table 3. mles, k-s statistics and p-value. models α̂ θ̂ λ̂ k-s p-value dus-pir 0.782 0.003 – 0.18446 0.01290686 pir 0.797 0.004 – 0.19755 5.1528e-24 etir – 5.715×10−4 – 0.20597 0.00371676 tir – 6.503×10−4 -0.781 0.17999 0.01642093 eir 0.616 6.555×10−4 – 0.20997 0.00290489 ir – 4.571×10−4 – 0.25083 0.00017822 rayleigh – 1.628×104 – 0.97964 3.3351×10−62 table 4. the ` , information criteria, rmse and r2. models ` aic aicc bic hqic rmse r2 dus-pir -394.466 792.932 793.106 797.485 794.744 0.068685 0.931008 pir -395.649 795.298 795.472 799.852 797.111 0.076096 0.913825 etir -400.074 802.149 802.206 804.426 803.055 0.092092 0.899998 tir -398.920 801.839 802.013 806.392 803.652 0.078811 0.929423 eir -614.106 1232.00 1232.00 1237.00 1234.00 0.083047 0.899951 ir -406.736 815.472 815.529 817.749 816.378 0.126351 0.831767 rayleigh -408.300 818.600 818.657 820.877 819.506 0.576828 5.726×10−05 from tables 3-4. it has been observed that dus-pir distribution suggests a better fit among all lifetime distributions for considered data. the variance-covariance matrix is given as vvv = ( 0.004 −7.853 × 10−5 −7.853 × 10−5 1.680 × 10−6 ) . then the 95% confidence interval for α and θ for dus-pir distribution are (0.659, 0.905) and (1.589r×10−4, 0.005.), respectively. it is shown that the lf has a unique solution by figure 5. figure 5. the profile of the log-lf of α and θ. int. j. anal. appl. (2023), 21:61 13 6. conclusion in this article, a new exponential transformed powered inverse rayleigh distribution which includes unimodal behavior, and some of its basic properties are investigated. from the computation, it is confirmed that proposed distribution complies a better fitting to the datasets under consideration in terms of all the criteria. acknowledgment: the researchers wish to extend their sincere gratitude to the deanship of scientific research at the islamic university of madinah for the support provided to the post-publishing program (2). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.c. gupta, p.l. gupta, r.d. gupta, modeling failure time data by lehman alternatives, commun. stat. theory methods. 27 (1998), 887-904. https://doi.org/10.1080/03610929808832134. [2] r.d. gupta, d. kundu, exponentiated exponential family: an alternative to gamma and weibull distributions, biom. j. 43 (2001), 117-130. https://doi.org/10.1002/1521-4036(200102)43:1<117::aid-bimj117>3.0.co; 2-r. [3] p. seenoi, t. supapakorn, w. bodhisuwan, the length-biased exponentiated inverted weibull distribution, int. j. pure appl. math. 92 (2014), 191-206. https://doi.org/10.12732/ijpam.v92i2.5. [4] m. ali, a. khalil, m. ijaz, n. saeed, alpha-power exponentiated inverse rayleigh distribution and its applications to real and simulated data, plos one. 16 (2021), e0245253. https://doi.org/10.1371/journal.pone.0245253. [5] a. mustafa, m.i. khan, the length-biased powered inverse rayleigh distribution with applications, j. appl. math. inform. 40 (2022), 1-13. https://doi.org/10.14317/jami.2022.001. [6] d. kumar, u. singh, s.k. singh, a method of proposing new distribution and its application to bladder cancer patients data, j. stat. appl. probab. lett. 2 (2015), 235-245. [7] d. kumar, u. singh, s.k. singh, a new distribution using sine functionits application to bladder cancer patients data, j. stat. appl. probab. 4 (2015), 417-427. [8] d. kumar, u. singh, s.k. singh, lifetime distributions: derived from some minimum guarantee distribution, sohag j. math. 4 (2016), 7-11. https://doi.org/10.18576/sjm/040102. [9] v.n. trayer, inverse rayleigh (ir) model, in: proceedings of the academy of science, doklady akad, nauk belarus, u.s.s.r, 1964. [10] v.g.h. voda, on the inverse rayleigh distributed random variable, rep. stat. appl. res. 19 (1972), 13-21. [11] n.j.m. anber, estimation of two parameter powered inverse rayleigh distribution, pak. j. stat. 36 (2020), 117-133. [12] m.i. khan, dual generalized order statistics with moments properties using powered inverse rayleigh distribution, j. stat. manage. syst. 25 (2022), 2087-2099. https://doi.org/10.1080/09720510.2022.2060615. [13] j.j.a. moors, a quantile alternative for kurtosis, the statistician. 37 (1988), 25-32. https://doi.org/10.2307/ 2348376. [14] m.o. lorenz, methods of measuring the concentration of wealth, publ. amer. stat. assoc. 9 (1905), 209-219. https://doi.org/10.1080/15225437.1905.10503443. [15] c. bonferroni, elementi di statistica generale, seeber, firenze, 1930. [16] j.f. lawless, statistical models and methods for lifetime data, 2nd edition, wiley, canada, 2003. https://doi.org/10.1080/03610929808832134 https://doi.org/10.1002/1521-4036(200102)43:1<117::aid-bimj117>3.0.co;2-r https://doi.org/10.1002/1521-4036(200102)43:1<117::aid-bimj117>3.0.co;2-r https://doi.org/10.12732/ijpam.v92i2.5 https://doi.org/10.1371/journal.pone.0245253 https://doi.org/10.14317/jami.2022.001 https://doi.org/10.18576/sjm/040102 https://doi.org/10.1080/09720510.2022.2060615 https://doi.org/10.2307/2348376 https://doi.org/10.2307/2348376 https://doi.org/10.1080/15225437.1905.10503443 14 int. j. anal. appl. (2023), 21:61 [17] h. akaike, a new look at the statistical model identification, ieee trans. automat. control. 19 (1974), 716-723. https://doi.org/10.1109/tac.1974.1100705. [18] c.m. hurvich, c.l. tsai, regression and time series model selection in small samples, biometrika. 76 (1989), 297-307. https://doi.org/10.1093/biomet/76.2.297. [19] g. schwarz, estimating the dimension of a model, ann. stat. 6 (1978), 461-464. https://www.jstor.org/ stable/2958889. [20] d. hinkley, on quick choice of power transformation, appl. stat. 26 (1977), 67-69. https://doi.org/10.2307/ 2346869. [21] t. bjerkedal, acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, amer. j. epidemiol. 72 (1960), 130-148. https://doi.org/10.1093/oxfordjournals.aje.a120129. https://doi.org/10.1109/tac.1974.1100705 https://doi.org/10.1093/biomet/76.2.297 https://www.jstor.org/stable/2958889 https://www.jstor.org/stable/2958889 https://doi.org/10.2307/2346869 https://doi.org/10.2307/2346869 https://doi.org/10.1093/oxfordjournals.aje.a120129 1. introduction 2. duspowered ir distribution 3. some statistical properties 3.1. the moments: 3.2. mode: 3.3. quantiles and random number generation: 3.4. order statistics: 3.5. entropy: 3.6. bonferroni and lorenz curves: 4. estimation of parameters 4.1. maximum likelihood estimation: 4.2. asymptotic confidence interval: 5. practical illustration 6. conclusion references int. j. anal. appl. (2023), 21:26 the prominentness of fuzzy ge-filters in ge-algebras sun shin ahn1,∗, rajab ali borzooei2, young bae jun3 1department of mathematics education, dongguk university, seoul 04620, korea 2department of mathematics, shahid beheshti university, tehran 1983963113, iran 3department of mathematics education, gyeongsang national university, jinju 52828, korea ∗corresponding author: sunshine@dongguk.edu abstract. based on the concept of fuzzy points, the notion of a prominent fuzzy ge-filter is defined, and the various properties involved are investigated. the relationship between a fuzzy ge-filter and a prominent fuzzy ge-filter is discussed, and the characterization of a prominent fuzzy ge-filter is considered. the conditions under which a fuzzy ge-filter can be a prominent fuzzy ge-filter are explored, and conditions for the trivial fuzzy ge-filter to be a prominent fuzzy ge-filter are provided. the conditions under which the ∈t-set and qt-set can be prominent ge-filters are explored. finally, the extension property for the prominent fuzzy ge-filter is discussed. 1. introduction henkin and scolem introduced the concept of hilbert algebra in the implication investigation in intuitionistic logics and other nonclassical logics. diego [6] established that hilbert algebras form a locally finite variety. later several researchers extended the theory on hilbert algebras (see [4,5,7,8]). the notion of be-algebra was introduced by kim et al. [9] as a generalization of a dual bck-algebra. rezaei et al. [13] discussed relations between hilbert algebras and be-algebras. as a generalization of hilbert algebras, bandaru et al. [2] introduced the notion of ge-algebras, and investigated several properties. bandaru et al. [3] introduced the concept of bordered ge-algebra and investigated its properties. later, ozturk et al. [10] introduced the concept of strong ge-filters, ge-ideals of bordered ge-algebras and investigated its properties. song et al. [14] introduced the concept of imploring ge-filters of ge-algebras and discussed its properties. rezaei et al. [12] introduced the concept of received: jan. 24, 2023. 2020 mathematics subject classification. 03g25, 06f35, 08a72. key words and phrases. (prominent) ge-filter; (prominent) fuzzy ge-filter; trivial fuzzy ge-filter; ∈t-set; qt-set. https://doi.org/10.28924/2291-8639-21-2023-26 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-26 2 int. j. anal. appl. (2023), 21:26 prominent ge-filters in ge-algebras and discussed its properties. bandaru et al. [1] discussed the fuzzy notion of ge-filters in ge-algebras. the purpose of this paper is to define a prominent fuzzy ge-filter using the concept of fuzzy points and investigate the various properties involved. we consider the relationship between a fuzzy ge-filter and a prominent fuzzy ge-filter. we explore the conditions under which a fuzzy ge-filter can be a prominent fuzzy ge-filter. we discuss the characterization of a prominent fuzzy ge-filter. we provide conditions for the trivial fuzzy ge-filter to be a prominent fuzzy ge-filter. we explore the conditions under which the ∈t-set and qt-set can be prominent ge-filters. we finally discuss the extension property for the prominent fuzzy ge-filter. 2. preliminaries 2.1. basics related to ge-algebras. definition 2.1 ( [2]). by a ge-algebra we mean a set x with a constant “1” and a binary operation “∗” satisfying the following axioms: (ge1) a∗a =1, (ge2) 1∗a = a, (ge3) a∗ (b∗c)= a∗ (b∗ (a∗c)) for all a,b,c ∈ x. we denote the ge-algebra by x := (x,∗,1). a binary relation “≤ ” in a ge-algebra x := (x,∗,1) is defined by: (∀x,y ∈ x)(x ≤ y ⇔ x ∗y =1). (2.1) definition 2.2 ( [2]). a ge-algebra x := (x,∗,1) is said to be • transitive if it satisfies: (∀a,b,c ∈ x)(a∗b ≤ (c ∗a)∗ (c ∗b)) . (2.2) • commutative if it satisfies: (∀a,b ∈ x)((a∗b)∗b =(b∗a)∗a) . (2.3) note that every commutative ge-algebra is transitive and antisymmetric. proposition 2.1 ( [2]). every ge-algebra x := (x,∗,1) satisfies the following items. (∀a ∈ x)(a∗1=1) . (2.4) (∀a,b ∈ x)(a∗ (a∗b)= a∗b) . (2.5) (∀a,b ∈ x)(a ≤ b∗a) . (2.6) int. j. anal. appl. (2023), 21:26 3 (∀a,b,c ∈ x)(a∗ (b∗c)≤ b∗ (a∗c)) . (2.7) (∀a ∈ x)(1≤ a ⇒ a =1) . (2.8) (∀a,b ∈ x)(a ≤ (a∗b)∗b) . (2.9) if x := (x,∗,1) is transitive, then (∀a,b,c ∈ x)(a ≤ b ⇒ c ∗a ≤ c ∗b, b∗c ≤ a∗c) . (2.10) (∀a,b,c ∈ x)(a∗b ≤ (b∗c)∗ (a∗c)) . (2.11) (∀a,b,c ∈ x)(a∗b ≤ (c ∗a)∗ (c ∗b)) . (2.12) definition 2.3. a subset f of a ge-algebra x := (x,∗,1) is called • a ge-filter of x := (x,∗,1) (see [2]) if it satisfies: 1∈ f, (2.13) (∀a,b ∈ x)(a ∈ f, a∗b ∈ f ⇒ b ∈ f). (2.14) • a prominent ge-filter of x := (x,∗,1) (see [12]) if it satisfies (2.13) and (∀a,b,c ∈ x)(a∗ (b∗c)∈ f, a ∈ f ⇒ ((c ∗b)∗b)∗c ∈ f). (2.15) lemma 2.1 ( [2]). every ge-filter f of x := (x,∗,1) satisfies: (∀x,y ∈ x)(x ≤ y, x ∈ f ⇒ y ∈ f). (2.16) lemma 2.2 ( [12]). every prominent ge-filter is a ge-filter. 2.2. basics related to fuzzy sets. a fuzzy set f in a set x of the form f (b) := { t ∈ (0,1] if b = a, 0 if b 6= a, is said to be a fuzzy point with support a and value t and is denoted by a t . for a fuzzy set f in a set x and t ∈ (0,1], we say that a fuzzy point a t is (i) contained in f , denoted by a t ∈ f , (see [11]) if f (a)≥ t. (ii) quasi-coincident with f , denoted by a t q f, (see [11]) if f (a)+ t > 1. if a t αf is not established for α ∈{∈,q}, it is denoted by a t αf . given t ∈ (0,1] and a fuzzy set f in a set x, consider the following sets (f ,t)∈ := {x ∈ x | xt ∈ f} and (f ,t)q := {x ∈ x | x t q f} which are called an ∈t-set and qt-set of f , respectively, in x. 4 int. j. anal. appl. (2023), 21:26 definition 2.4 ( [1]). a fuzzy set f in a ge-algebra x := (x,∗,1) is called a fuzzy ge-filter of x := (x,∗,1) if it satisfies: (∀t ∈ (0,1])((f ,t)∈ 6= ∅ ⇒ 1∈ (f ,t)∈) , (2.17) x ∗y ∈ (f ,tb)∈, x ∈ (f ,ta)∈ ⇒ y ∈ (f ,min{ta,tb})∈ (2.18) for all x,y ∈ x and ta,tb ∈ (0,1]. 3. the prominentness of fuzzy ge-filters in what follows, let x := (x,∗,1) denote a ge-algebra unless otherwise specified. definition 3.1. a fuzzy set f in x is called a prominent fuzzy ge-filter of x := (x,∗,1) if it satisfies (2.17) and (∀x,y,z ∈ x)(∀ta,tb ∈ (0,1]) ( x ∗ (y ∗z)∈ (f ,tb)∈, x ∈ (f ,ta)∈ ⇒ ((z ∗y)∗y)∗z ∈ (f ,min{ta,tb})∈ ) . (3.1) example 3.1. let x = {1,2,3,4,5,6,7} be a set with a binary operation “∗” given by table 1. table 1. cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 1 1 1 4 6 6 1 3 1 2 1 5 5 5 7 4 1 1 3 1 1 1 1 5 1 2 1 1 1 1 7 6 1 2 3 1 1 1 1 7 1 2 3 6 5 6 1 then x := (x,∗,1) is a ge-algebra (see [12]). define a fuzzy set f in x as follows: f : x → [0,1], x 7→ { 0.85 if x ∈{1,2,3,7}, 0.37 otherwise. it is routine to verify that f is a prominent fuzzy ge-filter of x := (x,∗,1). we discuss the relationship between a fuzzy ge-filter and a prominent fuzzy ge-filter. theorem 3.1. every prominent fuzzy ge-filter is a fuzzy ge-filter. proof. let f be a prominent fuzzy ge-filter of x := (x,∗,1). let x,y ∈ x and ta,tb ∈ (0,1] be such that x ∈ (f ,ta)∈ and x ∗ y ∈ (f ,tb)∈. then x ∗ (1 ∗ y) = x ∗ y ∈ (f ,tb)∈ by (ge2), and so int. j. anal. appl. (2023), 21:26 5 y = ((y ∗1) ∗1) ∗ y ∈ (f ,tb)∈ by (ge1), (ge2), (2.4) and (3.1). hence f is a fuzzy ge-filter of x := (x,∗,1). � the following example shows that the converse of theorem 3.1 may not be true. example 3.2. consider the ge-algebra x := (x,∗,1) in example 3.1 and let f be a fuzzy set in x defined by f : x → [0,1], x 7→ { 0.79 if x ∈{1,3,7}, 0.46 otherwise. it is routine to verify that f is a fuzzy ge-filter of x := (x,∗,1). but it is not a prominent fuzzy ge-filter of x := (x,∗,1) since 3∈ (f ,0.67)∈ and 3∗(4∗2)=1∈ (f ,0.62)∈, but ((2∗4)∗4)∗2= 2 /∈ (f ,0.62)∈ =(f ,min{0.67,0.62})∈. we explore the conditions under which a fuzzy ge-filter can be a prominent fuzzy ge-filter. theorem 3.2. given a fuzzy ge-filter f of x := (x,∗,1), it is a prominent fuzzy ge-filter of x := (x,∗,1) if and only if it satisfies: (∀x,y ∈ x)(∀t ∈ (0,1])(x ∗y ∈ (f ,t)∈ ⇒ ((y ∗x)∗x)∗y ∈ (f ,t)∈). (3.2) proof. assume that f is a prominent fuzzy ge-filter of x := (x,∗,1) and let x,y ∈ x and t ∈ (0,1] be such that x ∗y ∈ (f ,t)∈. then 1∗(x ∗y)= x ∗y ∈ (f ,t)∈ by (ge2). since 1∈ (f ,t)∈, it follows from (3.1) that ((y ∗x)∗x)∗y ∈ (f ,t)∈. conversely, let f be a fuzzy ge-filter of x := (x,∗,1) that satisfies the condition (3.2). let x,y,z ∈ x and ta,tb ∈ (0,1] be such that x ∗ (y ∗ z) ∈ (f ,tb)∈ and x ∈ (f ,ta)∈. then y ∗ z ∈ (f ,min{ta,tb})∈ by (2.18), and so ((z ∗ y) ∗ y) ∗ z ∈ (f ,min{ta,tb})∈ by (3.2). therefore f is a prominent fuzzy ge-filter of x := (x,∗,1). � lemma 3.1 ( [1]). every fuzzy ge-filter f of x satisfies: (∀x,y ∈ x)(∀ta ∈ (0,1])(x ≤ y, x ∈ (f ,ta)∈ ⇒ y ∈ (f ,ta)∈) , (3.3) (∀x,y,z ∈ x)(∀ta,tb ∈ (0,1]) ( z ≤ y ∗x, y ∈ (f ,tb)∈, z ∈ (f ,ta)∈ ⇒ x ∈ (f ,min{ta,tb})∈ ) . (3.4) theorem 3.3. in a commutative ge-algebra, every fuzzy ge-filter is a prominent fuzzy ge-filter. proof. let f be a prominent fuzzy ge-filter of x := (x,∗,1). it is sufficient to show that f satisfies the condition (3.1). let x,y,z ∈ x and ta,tb ∈ (0,1] be such that x ∗ (y ∗ z) ∈ (f ,tb)∈ and 6 int. j. anal. appl. (2023), 21:26 x ∈ (f ,ta)∈. using (2.3), (2.7), and (2.12), we have 1= ((z ∗y)∗y)∗ ((y ∗z)∗z) ≤ (y ∗z)∗ (((z ∗y)∗y)∗z) ≤ (x ∗ (y ∗z))∗ (x ∗ (((z ∗y)∗y)∗z)) ≤ x ∗ ((x ∗ (y ∗z))∗ (((z ∗y)∗y)∗z)), and so x ∗((x ∗(y ∗z))∗(((z ∗y)∗y)∗z))=1, i.e., x ≤ (x ∗(y ∗z))∗(((z ∗y)∗y)∗z). it follows from lemma 3.1 that ((z ∗y)∗y)∗z ∈ (f ,min{ta,tb})∈. therefore f is a prominent fuzzy ge-filter of x := (x,∗,1). � theorem 3.4. a fuzzy set f in x is a prominent fuzzy ge-filter of x := (x,∗,1) if and only if it satisfies: (∀x ∈ x)(f (1)≥ f (x)). (3.5) (∀x,y,z ∈ x)(f (((z ∗y)∗y)∗z)≥min{f (x), f (x ∗ (y ∗z))}). (3.6) proof. assume that f is a prominent fuzzy ge-filter of x := (x,∗,1). suppose there exists a ∈ x such that f (1) < f (a). let t0 = 1 2 (f (1)+ f (a)). then f (1) < t0 and 0 < t0 < f (a) ≤ 1. hence a ∈ (f ,t0)∈ and so (f ,t0)∈ 6= ∅. thus 1∈ (f ,t0)∈, that is, f (1)≥ t0, which is contradiction. hence f (1) ≥ f (x) for all x ∈ x. let x,y,z ∈ x be such that f (x) = t1 and f (x ∗ (y ∗ z)) = t2. then x ∈ (f ,t1)∈ and x ∗ (y ∗z)∈ (f ,t2)∈. it follows from (3.1) that ((z ∗y)∗y)∗z ∈ (f ,min{t1,t2})∈. hence f (((z ∗y)∗y)∗z)≥min{t1,t2}=min(f (x), f (x ∗ (y ∗z))). conversely, assume that f satisfies (3.5) and (3.6). let t ∈ (0,1] and x ∈ (f ,t)∈. then f (x)≥ t and hence f (1) ≥ f (x) ≥ t. thus 1 ∈ (f ,t)∈. let x,y,z ∈ x be such that x ∈ (f ,t1)∈ and x ∗ (y ∗ z) ∈ (f ,t2)∈. then f (x) ≥ t1 and f (x ∗ (y ∗ z)) ≥ t2. therefore f (((z ∗ y) ∗ y) ∗ z) ≥ min{f (x), f (x ∗ (y ∗z))}≥min{t1,t2} by (3.6). hence ((z ∗y)∗y)∗z ∈ (f ,min{t1,t2})∈. thus f is a prominent fuzzy ge-filter of x := (x,∗,1). � theorem 3.5. given an element b ∈ x, define a fuzzy set fb in x as follows: fb : x → [0,1], x 7→ { t1 if x ∈~b, t2 otherwise., where ~b := {x ∈ x | b ≤ x} and t1 > t2 in (0,1]. then fb is a prominent fuzzy ge-filter of x := (x,∗,1) if and only if x := (x,∗,1) satisfies: (∀x,y,z ∈ x)(x ∈~b, x ∗ (y ∗z)∈~b ⇒ ((z ∗y)∗y)∗z ∈~b). (3.7) proof. assume that fb is a prominent fuzzy ge-filter of x := (x,∗,1) and let x,y,z ∈ x be such that x ∈~b and x ∗ (y ∗z)∈~b. then fb(x)= t1 = fb(x ∗ (y ∗z)), which implies from (3.6) that fb(((z ∗y)∗y)∗z)≥min{fb(x), fb(x ∗ (y ∗z))}= t1. int. j. anal. appl. (2023), 21:26 7 hence fb(((z ∗y)∗y)∗z)= t1, and thus ((z ∗y)∗y)∗z ∈~b. conversely, suppose that x := (x,∗,1) satisfies the condition (3.7). since 1 ∈~b, we get fb(1) = t1 ≥ fb(x) for all x ∈ x. for every x,y,z ∈ x, if x /∈ ~b or x ∗ (y ∗ z) /∈ ~b, then fb(x) = t2 or fb(x ∗ (y ∗z))= t2. hence fb(((z ∗y)∗y)∗z)≥ t2 =min{fb(x), fb(x ∗ (y ∗z))}. if x ∈~b and x ∗ (y ∗z)∈~b, then fb(x)= t1 and fb(x ∗ (y ∗z))= t1. thus fb(((z ∗y)∗y)∗z)= t1 =min{fb(x), fb(x ∗ (y ∗z))}. therefore fb is a prominent fuzzy ge-filter of x := (x,∗,1) by theorem 3.4. � consider a fuzzy set f in x which is given by f : x → [0,1], x 7→ { t1 if x =1, t2 otherwise, where t1 > t2 in (0,1]. it is clear that f is a fuzzy ge-filter of x := (x,∗,1), which is called the trivial fuzzy ge-filter of x := (x,∗,1). but it is not a prominent fuzzy ge-filter of x := (x,∗,1) as seen in the following example. example 3.3. consider the ge-algebra x := (x,∗,1) in example 3.1 and let f be a fuzzy set in x defined by f : x → [0,1], x 7→ { 0.83 if x =1, 0.57 otherwise. then f is a fuzzy ge-filter of x := (x,∗,1), but it is not a prominent fuzzy ge-filter of x := (x,∗,1) since 1∈ (f ,0.69)∈ and 1∗(4∗2)=1∈ (f ,0.64)∈, but ((2∗4)∗4)∗2=2 /∈ (f ,min{0.69,0.64})∈. we provide conditions for the trivial fuzzy ge-filter to be a prominent fuzzy ge-filter. theorem 3.6. in a commutative ge-algebra, the trivial fuzzy ge-filter is a prominent fuzzy ge-filter. proof. let f be the trivial fuzzy ge-filter of a commutative ge-algebra x := (x,∗,1). then (f ,t)∈ =   ∅ if t ∈ (t1,1], {1} if t ∈ (t2,t1], x if t ∈ (0,t2]. it is sufficient to show that (f ,t)∈ = {1} is a prominent ge-filter of x := (x,∗,1). let x,y,z ∈ x be such that x ∈ {1} and x ∗ (y ∗ z) ∈ {1}. using (ge2), (2.3) and (ge1), we get y ∗ z = 1, and thus ((z ∗ y)∗ y)∗ z = ((y ∗ z)∗ z)∗ z = (1∗ z)∗ z = z ∗ z = 1 ∈ {1}. hence (f ,t)∈ = {1} is a prominent ge-filter of x := (x,∗,1), and therefore f is a prominent fuzzy ge-filter of x := (x,∗,1) by theorem ??. � 8 int. j. anal. appl. (2023), 21:26 we explore the conditions under which the ∈t-set and qt-set can be prominent ge-filters. theorem 3.7. given a fuzzy set f in x, its ∈t-set (f ,t)∈ is a prominent ge-filter of x for all t ∈ (0.5,1] if and only if f satisfies: (∀x ∈ x)(f (x)≤max{f (1),0.5}), (3.8) (∀x,y ∈ x)(min{f (x), f (x ∗ (y ∗z))}≤max{f (((z ∗y)∗y)∗z),0.5}). (3.9) proof. assume that the ∈t-set (f ,t)∈ of f is a prominent ge-filter of x for all t ∈ (0.5,1]. if there exists a ∈ x such that f (a) � max{f (1),0.5}, then t := f (a) ∈ (0.5,1], a t ∈ f and 1 t ∈ f , that is, a ∈ (f ,t)∈ and 1 /∈ (f ,t)∈. this is a contradiction, and thus f (x)≤max{f (1),0.5} for all x ∈ x. if (3.9) is not valid, then min{f (a), f (a∗ (b∗c))} > max{f (((c ∗b)∗b)∗c),0.5} for some a,b,c ∈ x. if we take t := min{f (a), f (a ∗ (b ∗ c))}, then t ∈ (0.5,1], a t ∈ f and a∗(b∗c) t ∈ f . hence a ∈ (f ,t)∈ and a ∗ (b ∗ c) ∈ (f ,t)∈, which imply that ((c ∗b)∗b)∗ c ∈ (f ,t)∈. thus ((c∗b)∗b)∗c t ∈ f , and so f (((c ∗b)∗b)∗c)≥ t > 0.5 which is a contradiction. therefore min{f (x), f (x ∗ (y ∗z))}≤max{f (((z ∗y)∗y)∗z),0.5} for all x,y ∈ x. conversely, suppose that f satisfies (3.8) and (3.9). let (f ,t)∈ 6= ∅ for all t ∈ (0.5,1]. then there exists a ∈ (f ,t)∈ and thus at ∈ f , i.e., f (a)≥ t. it follows from (3.8) that max{f (1),0.5}≥ f (a)≥ t > 0.5. thus 1 t ∈ f , i.e., 1 ∈ (f ,t)∈. let t ∈ (0.5,1] and x,y,z ∈ x be such that x ∈ (f ,t)∈ and x ∗ (y ∗ z) ∈ (f ,t)∈. then xt ∈ f and x∗(y∗z) t ∈ f , that is, f (x) ≥ t and f (x ∗ (y ∗ z)) ≥ t. using (3.9), we get max{f (((z ∗y)∗y)∗z),0.5}≥min{f (x), f (x ∗ (y ∗z))}≥ t > 0.5 and so ((z∗y)∗y)∗z t ∈ f , i.e., ((z ∗y)∗y)∗z ∈ (f ,t)∈. therefore (f ,t)∈ is a prominent ge-filter of x for all t ∈ (0.5,1]. � lemma 3.2 ( [1]). a fuzzy set f in x is a fuzzy ge-filter of x if and only if the nonempty ∈t-set (f ,t)∈ of f in x is a ge-filter of x for all t ∈ (0,1]. lemma 3.3 ( [12]). let f be a ge-filter of x := (x,∗,1). then it is a prominent ge-filter of x := (x,∗,1) if and only if it satisfies: (∀x,y ∈ x)(x ∗y ∈ f ⇒ ((y ∗x)∗x)∗y ∈ f). (3.10) theorem 3.8. a fuzzy set f in x is a prominent fuzzy ge-filter of x := (x,∗,1) if and only if the nonempty ∈t-set (f ,t)∈ of f in x is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1]. int. j. anal. appl. (2023), 21:26 9 proof. assume that f is a prominent fuzzy ge-filter of x := (x,∗,1). then f is a fuzzy ge-filter of x := (x,∗,1) (see theorem 3.1), and so the nonempty ∈t-set (f ,t)∈ of f in x is a ge-filter of x := (x,∗,1) for all t ∈ (0,1] by lemma 3.2. let x,y ∈ x and t ∈ (0,1] be such that x ∗ y ∈ (f ,t)∈. since f is a prominent fuzzy ge-filter of x := (x,∗,1), it follows from (3.2) that ((y ∗ x) ∗ x) ∗ y ∈ (f ,t)∈, and therefore (f ,t)∈ is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1] by lemma 3.3. conversely, suppose that the nonempty ∈t-set (f ,t)∈ of f in x is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1]. then (f ,t)∈ is a ge-filter of x := (x,∗,1) by lemma 2.2, and thus f is a fuzzy ge-filter of x := (x,∗,1) by lemma 3.2. let x,y ∈ x and t ∈ (0,1] be such that x ∗y ∈ (f ,t)∈. then ((y ∗x)∗x)∗y ∈ (f ,t)∈ by lemma 3.3. it follows from theorem 3.2 that f is a prominent fuzzy ge-filter of x := (x,∗,1). � theorem 3.9. if f is a prominent fuzzy ge-filter of x := (x,∗,1), then the nonempty qt-set (f ,t)q of f is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1]. proof. let f be a prominent fuzzy ge-filter of x := (x,∗,1) and assume that (f ,t)q 6= ∅ for all t ∈ (0,1]. then there exists a ∈ (f ,t)q, and so at q f , i.e., f (a)+t > 1. hence f (1)+t ≥ f (a)+t > 1, i.e., 1 ∈ (f ,t)q. let x,y,z ∈ x be such that x ∈ (f ,t)q and x ∗ (y ∗ z) ∈ (f ,t)q. then xt q f and x∗(y∗z) t q f , that is, f (x)+ t > 1 and f (x ∗ (y ∗z))+ t > 1. it follows from (3.6) that f (((z ∗y)∗y)∗z)+ t ≥min{f (x), f (x ∗ (y ∗z))}+ t =min{f (x)+ t, f (x ∗ (y ∗z))+ t} > 1. hence ((z∗y)∗y)∗z t q f , and therefore ((z ∗ y) ∗ y) ∗ z ∈ (f ,t)q. consequently, (f ,t)q is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1]. � we finally discuss the extension property for the prominent fuzzy ge-filter. question. let f and g be fuzzy ge-filters of x := (x,∗,1) such that f ⊆ g, that is, f (x) ≤ g(x) for all x ∈ x. if f is a prominent fuzzy ge-filter of x := (x,∗,1), then is g also a prominent fuzzy ge-filter of x := (x,∗,1)? the example below provides a negative answer to the question. example 3.4. let x = {1,2,3,4,5,6} be a set with a binary operation “∗” given by table 2. then x := (x,∗,1) is a ge-algebra (see [12]). define a fuzzy set f in x as follows: f : x → [0,1], x 7→ { 0.65 if x =1, 0.37 otherwise. 10 int. j. anal. appl. (2023), 21:26 table 2. cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 1 1 3 4 3 1 3 1 6 1 1 6 6 4 1 2 1 1 2 2 5 1 1 1 4 1 1 6 1 1 3 4 3 1 it is routine to verify that f is a prominent ge-filter of x := (x,∗,1). now, we define a fuzzy set g in x as follows: g : x → [0,1], x 7→   0.73 if x =1, 0.67 if x ∈{2,6}, 0.48 otherwise. then f (x) ≤ g(x) for all x ∈ x, that is, f ⊆ g, and g is a fuzzy ge-filter of x := (x,∗,1). since 4∗5=2∈ (g,0.61)∈ and ((5∗4)∗4)∗5=5 /∈ (g,0.61)∈, we know that g is not a prominent fuzzy ge-filter of x := (x,∗,1) by theorem 3.2. we provide conditions for the answer of question above to be positive. theorem 3.10. (extension property for the prominent fuzzy ge-filter) let f and g be fuzzy gefilters of a transitive ge-algebra x := (x,∗,1) such that f ⊆ g, that is, f (x)≤ g(x) for all x ∈ x. if f is a prominent fuzzy ge-filter of x := (x,∗,1), then so is g. proof. if f is a prominent fuzzy ge-filter of x := (x,∗,1), then it is a fuzzy ge-filter of x := (x,∗,1) by theorem 3.1 and (f ,t)∈ is a prominent ge-filter of x := (x,∗,1) for all t ∈ (0,1] by theorem 3.8. let a := x ∗ y ∈ (g,t)∈ for all x,y ∈ x and t ∈ (0,1]. then 1 ∈ (f ,t)∈ by (2.17) and 1= a∗(x∗y)≤ x∗(a∗y) by (ge1) and (2.7). hence x∗(a∗y)∈ (f ,t)∈ by (3.3). using assumption and theorem 3.2 induces (((a∗y)∗x)∗x)∗ (a∗y)∈ (f ,t)∈ ⊆ (g,t)∈. since (((a ∗ y) ∗ x) ∗ x) ∗ (a ∗ y) ≤ a ∗ ((((a ∗ y) ∗ x) ∗ x) ∗ y) by (2.7) and (g,t)∈ is a gefilter of x := (x,∗,1), we have a ∗ ((((a ∗ y) ∗ x) ∗ x) ∗ y) ∈ (g,t)∈ by lemma 2.1. hence (((a∗y)∗x)∗x)∗y ∈ (g,t)∈ by (2.14). since y ≤ a∗y by (2.6), we have (((a∗y)∗x)∗x)∗y ≤ ((y ∗x)∗x)∗y int. j. anal. appl. (2023), 21:26 11 by running (2.10) three times. it follows from lemma 2.1 that ((y ∗ x) ∗ x) ∗ y ∈ (g,t)∈. hence (g,t)∈ is a prominent ge-filter of x := (x,∗,1) by lemma 3.3, and therefore g is a prominent fuzzy ge-filter of x := (x,∗,1) by theorem 3.8. � corollary 3.1. let x := (x,∗,1) be a transitive ge-algebra. then the trivial fuzzy ge-filter f is a prominent fuzzy ge-filter of x := (x,∗,1) if and only if every fuzzy ge-filter is a prominent fuzzy ge-filter of x := (x,∗,1). corollary 3.2. in a commutative ge-algebra, every fuzzy ge-filter is a prominent fuzzy ge-filter. the following example describes the extension property for the prominent fuzzy ge-filter. example 3.5. let x = {1,2,3,4,5,6} be a set with a binary operation “∗” given by table 3. table 3. cayley table for the binary operation “∗” ∗ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 1 1 3 4 4 6 3 1 2 1 5 5 6 4 1 1 1 1 1 6 5 1 1 1 1 1 6 6 1 2 3 4 5 1 then x := (x,∗,1) is a ge-algebra (see [12]). define a fuzzy set f in x as follows: f : x → [0,1], x 7→ { 0.59 if x ∈{1,2,3}, 0.36 otherwise. then f is a prominent fuzzy ge-filter of x := (x,∗,1). if we take a fuzzy set g in x defined as follows: g : x → [0,1], x 7→ { 0.69 if x ∈{1,2,3,6}, 0.56 otherwise, then f ⊆ g and g is a prominent fuzzy ge-filter of x := (x,∗,1). 4. conclusion using the concept of fuzzy points, we have introduced the notion of a prominent fuzzy ge-filter in ge-algebras, and have investigated the various properties involved. we have considered the relationship between a fuzzy ge-filter and a prominent fuzzy ge-filter, and have discussed the characterization of a prominent fuzzy ge-filter. we have explored the conditions under which a fuzzy ge-filter can be a prominent fuzzy ge-filter. we have provided conditions for the trivial fuzzy ge-filter to be a 12 int. j. anal. appl. (2023), 21:26 prominent fuzzy ge-filter, and have explored the conditions under which the ∈t-set and qt-set can be prominent ge-filters. we finally have discussed the extension property for the prominent fuzzy ge-filter. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.k. bandaru, t.g. alemayehu, y.b. jun, fuzzy ge-filters of ge-algebras, j. algebra related topics. (submitted). [2] r. bandaru, a.b. saeid, y.b. jun, on ge-algebras, bull. sect. logic. 50 (2020), 81–96. https://doi.org/10. 18778/0138-0680.2020.20. [3] r.k. bandaru, m.a. öztürk, y.b. jun, bordered ge-algebras, j. algebraic syst. in press. [4] s. celani, a note on homomorphisms of hilbert algebras, int. j. math. math. sci. 29 (2002), 55–61. https: //doi.org/10.1155/s0161171202011134. [5] i. chajda, r. halas, y.b. jun, annihilators and deductive systems in commutative hilbert algebras, comment. math. univ. carolin. 43 (2002), 407–417. http://dml.cz/dmlcz/119331. [6] a. diego, sur les algèbres de hilbert, collection de logique mathématique, edition hermann, série a, xxi, (1966). [7] s.m. hong, y.b. jun, on deductive systems of hilbert algebras, commun. korean math. soc. 11 (1996), 595–600. [8] y.b. jun, k.h. kim, h-filters of hilbert algebras, sci. math. japon. e-2005 (2005), 231–236. [9] h.s. kim, y.h. kim, on be-algebras, sci. math. japon. e-2006 (2006), 1299–1302. [10] m.a. öztürk, j.g. lee, r. bandaru, y.b. jun, strong ge-filters and ge-ideals of bordered ge-algebras, j. math. 2021 (2021), 5520023. https://doi.org/10.1155/2021/5520023. [11] p.m. pu, y.m. liu, fuzzy topology. i. neighborhood structure of a fuzzy point and moore-smith convergence, j. math. anal. appl. 76 (1980), 571–599. https://doi.org/10.1016/0022-247x(80)90048-7. [12] a. rezaei, r. bandaru, a.b. saeid, y.b. jun, prominent ge-filters and ge-morphisms in ge-algebras, afr. mat. 32 (2021), 1121–1136. https://doi.org/10.1007/s13370-021-00886-6. [13] a. rezaei, a. borumand saeid, r.a. borzooei, relation between hilbert algebras and be-algebras, appl. appl. math.: int. j. 8 (2013), 573–584. [14] s.z. song, r. bandaru, y.b. jun, imploring ge-filters of ge-algebras, j. math. 2021 (2021), 1–7. https: //doi.org/10.1155/2021/6651531. https://doi.org/10.18778/0138-0680.2020.20 https://doi.org/10.18778/0138-0680.2020.20 https://doi.org/10.1155/s0161171202011134 https://doi.org/10.1155/s0161171202011134 http://dml.cz/dmlcz/119331 https://doi.org/10.1155/2021/5520023 https://doi.org/10.1016/0022-247x(80)90048-7 https://doi.org/10.1007/s13370-021-00886-6 https://doi.org/10.1155/2021/6651531 https://doi.org/10.1155/2021/6651531 1. introduction 2. preliminaries 2.1. basics related to ge-algebras 2.2. basics related to fuzzy sets 3. the prominentness of fuzzy ge-filters 4. conclusion references int. j. anal. appl. (2023), 21:32 a note on lp-kenmotsu manifolds admitting conformal ricci-yamabe solitons mobin ahmad1,∗, gazala1, maha atif al-shabrawi2 1department of mathematics and statistics, integral university, kursi road, lucknow-226026, india 2department of mathematical sciences, umm ul qura university, makkah, saudi arabia ∗corresponding author: mobinahmad68@gmail.com abstract. in the current note, we study lorentzian para-kenmotsu (in brief, lp-kenmotsu) manifolds admitting conformal ricci-yamabe solitons (crys) and gradient conformal ricci-yamabe soliton (gradient crys). at last by constructing a 5-dimensional non-trivial example we illustrate our result. 1. introduction as a generalization of the classical ricci flow [8], the concept of conformal ricci flow was introduced by fischer [5], which is defined on an n-dimensional riemannian manifold m by the equations ∂g ∂t = −2(s + g n ) −pg, r(g) = −1, where p defines a time dependent non-dynamical scalar field (also called the conformal pressure), g is the riemannian metric; r and s represent the scalar curvature and the ricci tensor of m, respectively. the term −pg plays a role of constraint force to maintain r in the above equation. in [1], the authors basu and bhattacharyya proposed the concept of conformal ricci soliton on m and is defined by £kg + 2s + (2λ − (p + 2 n ))g = 0, where £k represents the lie derivative operator along the smooth vector field k on m and λ ∈ r (the set of real numbers). received: feb. 3, 2023. 2010 mathematics subject classification. 53c20, 53c21, 53c25, 53e20. key words and phrases. lorentzian para-kenmotsu manifolds; conformal ricci-yamabe solitons; einstein manifolds; ν-einstein manifolds. https://doi.org/10.28924/2291-8639-21-2023-32 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-32 2 int. j. anal. appl. (2023), 21:32 very recently, a scalar combination of ricci and yamabe flows was proposed by the authors güler and crasmareanu [7], this advanced class of geometric flows called ricci-yamabe (ry) flow of type (σ,ρ) and is defined by ∂ ∂t g(t) + 2σs(g(t)) + ρr(t)g(t) = 0, g(0) = g0 for some scalars σ and ρ. a solution to the ry flow is called ricci-yamabe soliton (rys) if it depends only on one parameter group of diffeomorphism and scaling. a riemannian (or semi-riemannian) manifold m is said to have a rys if [9,10] £kg + 2σs + (2λ −ρr)g = 0. (1.1) a riemannian (or semi-riemannian) manifold m is said to have a conformal ricci-yamabe soliton (crys) if [20] £kg + 2σs + (2λ −ρr − (p + 2 n ))g = 0, (1.2) where σ,ρ, λ ∈r. if k is the gradient of a smooth function v on m, then (1.2) is called the gradient conformal ricci-yamabe soliton (gradient crys) and hence (1.2) turns to ∇2v + σs + (λ − ρr 2 − 1 2 (p + 2 n ))g = 0, (1.3) where ∇2v is the hessian of v and is defined by hessv = ∇∇v. a crys is said to be shrinking, steady or expanding if λ < 0, = 0 or > 0, respectively. a crys is said to be a • conformal ricci soliton if σ = 1,ρ = 0, • conformal yamabe soliton if σ = 0,ρ = 1, • conformal einstein soliton if σ = 1,ρ = −1. as a continuation of this study, we tried to study crys and gradient crys in the frame-work of lp-kenmotsu manifolds of dimension n. we recommend the papers [2–4,6,13–17] and the references therein for more details about the related studies. 2. preliminaries an n-dimensional differentiable manifold m with structure (ϕ,ζ,ν,g) is said to be a lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field ϕ, a contravariant vector field ζ, a 1-form ν and a lorentzian metric g satisfying ν(ζ) + 1 = 0, (2.1) ϕ2e = e + ν(e)ζ, (2.2) ϕζ = 0, ν(ϕe) = 0, g(ϕe,ϕf ) = g(e,f ) + ν(e)ν(f ), int. j. anal. appl. (2023), 21:32 3 g(e,ζ) = ν(e), (2.3) ϕ(e,f ) = ϕ(f,e) = g(e,ϕf ) for any vector fields e,f ∈ χ(m), where χ(m) is the lie algebra of vector fields on m. if ζ is a killing vector field, the (para) contact structure is called a k-(para) contact. in such a case, we have ∇eζ = ϕe. recently, the authors haseeb and prasad defined and studied the following notion: definition 2.1. a lorentzian almost paracontact manifold m is called lorentzian para-kenmostu manifold if [11] (∇eϕ)f = −g(ϕe,f )ζ −ν(f )ϕe for any e,f on m. in an lp-kenmostu manifold, we have ∇eζ = −e −ν(e)ζ, (2.4) (∇eν)f = −g(e,f ) −ν(e)ν(f ), (2.5) where ∇ denotes the levi-civita connection respecting to the lorentzian metric g. furthermore, in an lp-kenmotsu manifold, the following relations hold [11]: g(r(e,f )g,ζ) = ν(r(e,f )g) = g(f,g)ν(e) −g(e,g)ν(f ), r(ζ,e)f = −r(e,ζ)f = g(e,f )ζ −ν(f )e, r(e,f )ζ = ν(f )e −ν(e)f, r(ζ,e)ζ = e + ν(e)ζ, (2.6) s(e,ζ) = (n− 1)ν(e), s(ζ,ζ) = −(n− 1), (2.7) qζ = (n− 1)ζ, for any e,f,g ∈ χ(m), where r,s and q represent the curvature tensor, the ricci tensor and the q ricci operator, respectively. definition 2.2. [19] an lp-kenmotsu manifold m is said to be ν-einstein manifold if its s( 6= 0) is of the form s(e,f ) = ag(e,f ) + bν(e)ν(f ), where a and b are smooth functions on m. in particular, if b = 0, then m is termed as an einstein manifold. 4 int. j. anal. appl. (2023), 21:32 remark 2.1. [12] in an lp-kenmotsu manifold of n-dimension, s is of the form s(e,f ) = ( r n− 1 − 1)g(e,f ) + ( r n− 1 −n)ν(e)ν(f ), (2.8) where r is the scalar curvature of the manifold. lemma 2.1. in an n-dimensional lp-kenmotsu manifold, we have ζ(r) = 2(r −n(n− 1)), (2.9) (∇eq)ζ = qe − (n− 1)e, (2.10) (∇ζq)e = 2qe − 2(n− 1)e, (2.11) for any e on m. proof. equation (2.8) yields qe = ( r n− 1 − 1)e + ( r n− 1 −n)ν(e)ζ. (2.12) taking the covariant derivative of (2.12) with respect to f and making use of (2.4) and (2.5), we lead to (∇fq)e = f (r) n− 1 (e + ν(e)ζ) − ( r n− 1 −n)(g(e,f )ζ + ν(e)f + 2ν(e)ν(f )ζ). by contracting f in the foregoing equation and using trace {f → (∇fq)e} = 12e(r), we find n− 3 2(n− 1) e(r) = { ζ(r) n− 1 − (r −n(n− 1)) } ν(e), which by replacing e by ζ and using (2.1) gives (2.9). we refer the readers to see [13] for the proof of (2.10) and (2.11). � remark 2.2. from the equation (2.9), it is noticed that if an n-dimensional lp-kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1) and hence (2.8) reduces to s(e,f ) = (n− 1)g(e,f ). thus the manifold under consideration is an einstein manifold. 3. crys on lp-kenmotsu manifolds let the metric of an n-dimensional lp-kenmotsu manifold be a conformal ricci-yamabe soliton, thus (1.2) holds. by differentiating (1.2) covariantly with resprct to g, we have (∇g£kg)(e,f ) = −2σ(∇gs)(e,f ) + ρ(gr)g(e,f ). (3.1) since ∇g = 0, then the following formula [18] (£k∇eg −∇e£kg −∇[k,e]g)(f,g) = −g((£k∇)(e,f ),g) −g((£k∇)(e,g),f ) turns to (∇e£kg)(f,g) = g((£k∇)(e,f ),g) + g((£k∇)(e,g),f ). int. j. anal. appl. (2023), 21:32 5 since the operator £k∇ is symmetric, therefore we have 2g((£k∇)(e,f ),g) = (∇e£kg)(f,g) + (∇f£kg)(e,g) − (∇g£kg)(e,f ), which by using (3.1) takes the form 2g((£k∇)(e,f ),g) = −2σ[(∇es)(f,g) + (∇fs)(g,e) − (∇gs)(e,f )] +ρ[(er)g(f,g) + (fr)g(g,e) − (gr)g(e,f )]. (3.2) putting f = ζ in (3.2) and using (2.3), we find 2g((£k∇)(e,ζ),g) = −2σ[(∇es)(ζ,g) + (∇ζs)(g,e) − (∇gs)(e,ζ)] +ρ[(er)ν(g) + 2(r −n(n− 1))g(e,g) − (gr)ν(e)]. (3.3) by virtue of (2.10) and (2.11), (3.3) leads to 2g((£k∇)(e,ζ),g) = −4σ[s(e,g) − (n− 1)g(e,g)] +ρ[(er)ν(g) + 2(r −n(n− 1))g(e,g) − (gr)ν(e)]. by eliminating g from the foregoing equation, we have 2(£k∇)(f,ζ) = ρg(dr,f )ζ −ρ(dr)ν(f ) − 4σqf (3.4) +[4σ(n− 1) + 2ρ(r −n(n− 1))]f. if we take r as constant, then from (2.9) it follows that r = n(n− 1), and hence (3.4) reduces to (£k∇)(f,ζ) = −2σqf + 2σ(n− 1)f. (3.5) taking covariant derivative of (3.5) with respect to e, we have (∇e£k∇)(f,ζ) = (£k∇)(f,e) − 2σν(e)[qf − (n− 1)f ] (3.6) − 2σ(∇eq)f. again from [18], we have (£kr)(e,f )g = (∇e£k∇)(f,g) − (∇f£k∇)(e,g), which by putting g = ζ and using (3.6) takes the form (£kr)(e,f )ζ = 2σν(f )(qe − (n− 1)e) − 2σν(e)(qf − (n− 1)f ) (3.7) −2σ((∇eq)f − (∇fq)e). putting f = ζ in (3.7) then using (2.1), (2.2), (2.10) and (2.11), we arrive at (£kr)(e,ζ)ζ = 0. (3.8) the lie derivative of (2.6) along k leads to (£kr)(e,ζ)ζ −g(e,£kζ)ζ + 2ν(£kζ)e = −(£kν)(e)ζ. (3.9) 6 int. j. anal. appl. (2023), 21:32 from (3.8) and (3.9), we have (£kν)(e)ζ = −2ν(£kζ)e + g(e,£kζ)ζ. (3.10) taking the lie derivative of g(e,ζ) = ν(e), we find (£kν)(e) = g(e,£kζ) + (£kg)(e,ζ). (3.11) by putting f = ζ in (1.2) and using (2.7), we have (£kg)(e,ζ) = −{2σ(n− 1) + 2λ −ρn(n− 1) − (p + 2 n )}ν(e), (3.12) where r = n(n− 1) being used. taking the lie derivative of g(ζ,ζ) = −1 along k we lead to (£kg)(ζ,ζ) = −2ν(£kζ). (3.13) from (3.12) and (3.13), we find ν(£kζ) = −{σ(n− 1) + λ − ρn(n− 1) 2 − 1 2 (p + 2 n )}. (3.14) now combining the equations (3.10), (3.11), (3.12) and (3.14), we find λ = ρn(n− 1) 2 −σ(n− 1) + 1 2 (p + 2 n ). (3.15) thus we have theorem 3.1. let (m,g) be an n-dimensional lp-kenmotsu manifold admitting crys with constant scalar curvature tensor, then λ = ρn(n−1) 2 −σ(n− 1) + 1 2 (p + 2 n ). corollary 3.1. let the metric of n-dimensional lp-kenmotsu manifold is crys. then we have values of σ,ρ soliton type soliton constant crys to be expanding, shrinking or steady σ = 1, ρ = 0 conformal ricci soliton λ = 1 2 (p+ 2 n )−(n− 1) crys is shrinking, steady and expanding if p > 2(n2−n−1) n , p = 2(n 2−n−1) n and p < 2(n 2−n−1) n , resp. σ = 0, ρ = 1 conformal yamabe soliton λ = 1 2 (p + 2 n ) + n(n−1) 2 crys is shrinking, steady and expanding if p < −(n3−n2 +2) n , p = −(n 3−n2 +2) n and p > −(n 3−n2 +2) n , resp. σ = 1, ρ = −1 conformal einstein soliton λ = 1 2 (p + 2 n ) − (n−1)(n+2) 2 crys is shrinking, steady and expanding if p < (n+1)(n 2−2) n , p = (n+1)(n2−2) n and p > (n+1)(n2−2) n , resp. int. j. anal. appl. (2023), 21:32 7 4. gradient crys on lp-kenmotsu manifolds let m be an n-dimensional lp-kenmotsu manifold with g as a gradient crys. then equation (1.3) can be written as ∇edv + σqe + (λ − ρr 2 − 1 2 (p + 2 n ))e = 0, (4.1) for all vector fields e on m, where d denotes the gradient operator of g. taking the covariant derivative of (4.1) with respect to f , we have ∇f∇edv = −σ{(∇fq)e + q(∇fe)} + ρ f (r) 2 e (4.2) −(λ − ρr 2 − 1 2 (p + 2 n ))∇fe. interchanging e and f in (4.2), we lead to ∇e∇fdv = −σ{(∇eq)f + q(∇ef )} + ρ e(r) 2 f (4.3) −(λ − ρr 2 − 1 2 (p + 2 n ))∇ef. by making use of (4.1)-(4.3), we find r(e,f )dv = σ{(∇fq)e − (∇eq)f} + ρ 2 {e(r)f −f (r)e}. (4.4) now from (2.8), we find qe = ( r n− 1 − 1)e + ( r n− 1 −n)ν(e)ζ, which on taking covariant derivative with repect to f leads to (∇fq)e = f (r) n− 1 (e + ν(e)ζ) − ( r n− 1 −n)(g(e,f )ζ (4.5) +2ν(e)ν(f )ζ + ν(e)f ). by using (4.5) in (4.4), we have r(e,f )dv = (n− 1)ρ− 2σ 2(n− 1) {e(r)f −f (r)e} + σ n− 1 {f (r)ν(e)ζ −e(r)ν(f )ζ} −σ( r n− 1 −n)(ν(e)f −ν(f )e). (4.6) contracting forgoing equation along e gives s(f,dv) = − {(n− 1)2ρ− 2σ(n− 2) 2(n− 1) } f (r) (4.7) + σ(n− 3)(r −n(n− 1)) n− 1 ν(f ). from the equation (2.8), we have s(f,dv) = ( r n− 1 − 1)f (v) + ( r n− 1 −n)ν(f )ζ(v). (4.8) 8 int. j. anal. appl. (2023), 21:32 now by equating (4.7) and (4.8), then putting f = ζ and using (2.1), (2.9), we find ζ(v) = r −n(n− 1) n− 1 {σ − (n− 1)ρ}. (4.9) taking the inner product of (4.6) with ζ, we get f (v)ν(e) −e(v)ν(f ) = ρ 2 {e(r)ν(f ) −f (r)ν(e)}, which by replacing e by ζ then using (2.9) and (4.9), we infer f (v) = − σ(r −n(n− 1)) n− 1 ν(f ) − ρ 2 f (r). (4.10) if we take r as constant, then from remark 2.5, we get r = n(n−1). thus (4.10) leads to f (v) = 0. this implies that v is constant. thus the soliton under the consideration is trivial. hence we state: theorem 4.1. if the metric of an n-dimensional lp-kenmotsu manifold of constant scalar curvature tensor admitting a special type of vector field is gradient crys, then the soliton is trivial. for v constant, (1.3) turns to σqe = −(λ − ρr 2 − 1 2 (p + 2 n ))e, which leads to s(e,f ) = − 1 σ (λ − ρn(n− 1) 2 − 1 2 (p + 2 n ))g(e,f ), σ 6= 0. (4.11) by putting e = f = ζ in (4.11) and using (2.7), we obtain λ = ρn(n− 1) 2 −σ(n− 1) + 1 2 (p + 2 n ). (4.12) corollary 4.1. if an n-dimensional lp-kenmotsu manifold admits a gradient crys with the constant scalar curvature, then the manifold under the consideration is an einstein manifold and λ = ρn(n−1) 2 − σ(n− 1) + 1 2 (p + 2 n ). 5. example we consider the 5-dimensional manifold m5 = { (x1,x2,x3,x4,x5) ∈r5 : x5 > 0 } , where (x1,x2,x3,x4,x5) are the standard coordinates in r5. let %1, %2, %3, %4 and %5 be the vector fields on m5 given by %1 = e x5 ∂ ∂x1 , %2 = e x5 ∂ ∂x2 , %3 = e x5 ∂ ∂x3 , %4 = e x5 ∂ ∂x4 , %5 = ∂ ∂x5 = ζ, which are linearly independent at each point of m5. let g be the lorentzian metric defined by g(%i,%i ) = 1, for 1 ≤ i ≤ 4 and g(%5,%5) = −1, g(%i,%j) = 0, for i 6= j, 1 ≤ i, j ≤ 5. int. j. anal. appl. (2023), 21:32 9 let ν be the 1-form defined by ν(e) = g(e,%5) = g(e,ζ) for all e ∈ χ(m5), and let ϕ be the (1, 1)-tensor field defined by ϕ%1 = −%2, ϕ%2 = −%1, ϕ%3 = −%4, ϕ%4 = −%3, ϕ%5 = 0. by applying linearity of ϕ and g, we have ν(ζ) = g(ζ,ζ) = −1, ϕ2e = e + ν(e)ζ and g(ϕe,ϕf ) = g(e,f ) + ν(e)ν(f ) for all e,f ∈ χ(m5). thus for %5 = ζ, the structure (ϕ,ζ,ν,g) defines a lorentzian almost paracontact metric structure on m5. then we have [%i,%j] = −%i, for 1 ≤ i ≤ 4, j = 5, [%i,%j] = 0, otherwise. by using koszul’s formula, we can easily find we obtain ∇%i%j =   −%5, 1 ≤ i = j ≤ 4, −%i, 1 ≤ i ≤ 4, j = 5, 0, otherwise. also one can easily verify that ∇eζ = −e −η(e)ζ and (∇eϕ)f = −g(ϕe,f )ζ −ν(f )ϕe. therefore, the manifold is an lp-kenmotsu manifold. from the above results, we can easily obtain the non-vanishing components of r as follows: r(%1,%2)%1 = −%2, r(%1,%2)%2 = %1, r(%1,%3)%1 = −%3, r(%1,%3)%3 = %1, r(%1,%4)%1 = −v4, r(%1,%4)%4 = %1, r(%1,%5)%1 = −%5, r(%1,%5)%5 = −%1, r(%2,%3)%2 = −%3, r(%2,%3)%3 = %2, r(%2,%4)%2 = −%4, r(%2,%4)%4 = %2, r(%2,%5)%2 = −%5, r(%2,%5)%5 = −%2, r(%3,%4)%3 = −%4, r(%3,%4)%4 = %3, r(%3,%5)%3 = −%5, r(%3,%5)%5 = −%3, r(%4,%5)%4 = −%5, r(%4,%5)%5 = −%4. also, we calculate the ricci tensors as follows: s(%1,%1) = s(%2,%2) = s(%3,%3) = s(%4,%4) = 4, s(%5,%5) = −4. therefore, we have r = s(%1,%1) + s(%2,%2) + s(%3,%3) + s(%4,%4) −s(%5,%5) = 20. now by taking dv = (%1v)%1 + (%2v)%2 + (%3v)%3 + (%4v)%4 + (%5v)%5, we have ∇%1dv = (%1(%1v) − (%5v))%1 + (%1(%2v))%2 + (%1(%3v))%3 + (%1(%4v))%4 +(%1(%5v) − (%1v))%5, 10 int. j. anal. appl. (2023), 21:32 ∇%2dv = (%2(%1v))%1 + (%2(%2v) − (%5v))%2 + (%2(%3v))%3 + (%2(%4v))%4 +(%2(%5v) − (%2v))%5, ∇%3dv = (%3(%1v))%1 + (%3(%2v))%2 + (%3(%3v) − (%5v))%3 + (%3(%4v))%4 +(%3(%5v) − (%3v))%5, ∇%4dv = (%4(%1v))%1 + (%4(%2v))%2 + (%4(%3v))%3 + (%4(%4v) − (%5v))%4 +(%4(%5v) − (%4v))%5, ∇%5dv = (%5(%1v))%1 + (%5(%2v))%2 + (%5(%3v))%3 + (%5(%4v))%4 + (%5(%5v))%5. thus by virtue of (4.1), we obtain  %1(%1v) −%5v = −(λ + 4σ − 10ρ− 12 (p + 2 5 )), %2(%2v) −%5v = −(λ + 4σ − 10ρ− 12 (p + 2 5 )), %3(%3v) −%5v = −(λ + 4σ − 10ρ− 12 (p + 2 5 )), %4(%4v) −%5v = −(λ + 4σ − 10ρ− 12 (p + 2 5 )), %5(%5v) = −(λ + 4σ − 10ρ− 12 (p + 2 5 )), %1(%2v) = %1(%3v) = %1(%4v) = 0, %2(%1v) = %2(%3v) = %2(%4v) = 0, %3(%1v) = %3(%2v) = %3(%4v) = 0, %4(%1v) = %4(%2v) = %4(%3v) = 0, %1(%5v) − (%1v) = %2(%5v) − (%2v) = 0, %3(%5v) − (%3v) = %4(%5v) − (%4v) = 0. (5.1) thus the equations in (5.1) are respectively amounting to e2x5 ∂2v ∂x21 − ∂v ∂x5 = −(λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x22 − ∂v ∂x5 = −(λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x23 − ∂v ∂x5 = −(λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x24 − ∂v ∂x5 = −(λ + 4σ − 10ρ− 1 2 (p + 2 5 )), ∂2v ∂x25 = −(λ + 4σ − 10ρ− 1 2 (p + 2 5 )), ∂2v ∂x1∂x2 = ∂2v ∂x1∂x3 = ∂2v ∂x1∂x4 = ∂2v ∂x2∂x3 = ∂2v ∂x2∂x4 = ∂2v ∂x3∂x4 = 0, int. j. anal. appl. (2023), 21:32 11 ex5 ∂2v ∂x5∂x1 + ∂v ∂x1 = ex5 ∂2v ∂x5∂x2 + ∂v ∂x2 = ex5 ∂2v ∂x5∂x3 + ∂v ∂x3 = ex5 ∂2v ∂x5∂x4 + ∂v ∂x4 = 0. from the above equations it is observed that v is constant for λ = −4σ + 10ρ + 1 2 (p + 2 5 ). hence equation (4.1) is satisfied. thus, g is a gradient rys with the soliton vector field k = dv, where v is constant and λ = −4σ + 10ρ + 1 2 (p + 2 5 ). thus, theorem 4.1 is verified. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. acknowledgement: the first and second authors would like to thank the integral university, lucknow, india, for providing the manuscript number iu/r&d/2022-mcn0001737 to the present work. references [1] n. basu, a. bhattacharyya, conformal ricci soliton in kenmotsu manifold, glob. j. adv. res. class. mod. geom. 4 (2015), 15-21. [2] a. m. blaga, some geometrical aspects of einstein, ricci and yamabe solitons, j. geom. symmetry phys. 52 (2019), 17-26. https://doi.org/10.7546/jgsp-52-2019-17-26. [3] s. chidananda, v. venkatesha, yamabe and riemann solitons on lorentzian para-sasakian manifolds, commun. korean math. soc. 37 (2022), 213-228. https://doi.org/10.4134/ckms.c200365. [4] u.c. de, a. sardar, k. de, ricci-yamabe solitons and 3-dimensional riemannian manifolds, turk. j. math. 46 (2022), 1078-1088. https://doi.org/10.55730/1300-0098.3143. [5] a.e. fischer, an introduction to conformal ricci flow, class. quantum grav. 21 (2004), s171-s218. https: //doi.org/10.1088/0264-9381/21/3/011. [6] d. ganguly, s. dey, a. ali, a. bhattacharyya, conformal ricci soliton and quasi-yamabe soliton on generalized sasakian space form, j. geom. phys. 169 (2021), 104339. https://doi.org/10.1016/j.geomphys.2021. 104339. [7] s. güler, m. crasmareanu, ricci–yamabe maps for riemannian flows and their volume variation and volume entropy, turk. j. math. 43 (2019), 2631-2641. https://doi.org/10.3906/mat-1902-38. [8] r.s. hamilton, the ricci flow on surfaces, mathematics and general relativity, contemp. math. 71 (1988), 237-262. [9] a. haseeb, m. bilal, s.k. chaubey, a.a.h. ahmadini, ζ-conformally flat lp-kenmotsu manifolds and ricci–yamabe solitons, mathematics, 11 (2023), 212. https://doi.org/10.3390/math11010212. [10] a. haseeb, s.k. chaubey, m.a. khan, riemannian 3-manifolds and ricci-yamabe solitons, int. j. geom. methods mod. phys. 20 (2023), 2350015. https://doi.org/10.1142/s0219887823500159. [11] a. haseeb, r. prasad, certain results on lorentzian para-kenmotsu manifolds, bol. soc. parana. mat. 39 (2021), 201–220. https://doi.org/10.5269/bspm.40607. [12] a. haseeb, r. prasad, some results on lorentzian para-kenmotsu manifolds, bull. transilvania univ. brasov. 13(62) (2020), 185-198. https://doi.org/10.31926/but.mif.2020.13.62.1.14. [13] y. li, a. haseeb, m. ali, lp-kenmotsu manifolds admitting η-ricci solitons and spacetime, j. math. 2022 (2022), 6605127. https://doi.org/10.1155/2022/6605127. [14] m.a. lone, i.f. harry, ricci solitons on lorentz-sasakian space forms, j. geom. phys. 178 (2022), 104547. https://doi.org/10.1016/j.geomphys.2022.104547. [15] pankaj, s.k. chaubey, r. prasad, three-dimensional lorentzian para-kenmotsu manifolds and yamabe solitons, honam math. j. 43 (2021), 613-626. https://doi.org/10.5831/hmj.2021.43.4.613. https://doi.org/10.7546/jgsp-52-2019-17-26 https://doi.org/10.4134/ckms.c200365 https://doi.org/10.55730/1300-0098.3143 https://doi.org/10.1088/0264-9381/21/3/011 https://doi.org/10.1088/0264-9381/21/3/011 https://doi.org/10.1016/j.geomphys.2021.104339 https://doi.org/10.1016/j.geomphys.2021.104339 https://doi.org/10.3906/mat-1902-38 https://doi.org/10.3390/math11010212 https://doi.org/10.1142/s0219887823500159 https://doi.org/10.5269/bspm.40607 https://doi.org/10.31926/but.mif.2020.13.62.1.14 https://doi.org/10.1155/2022/6605127 https://doi.org/10.1016/j.geomphys.2022.104547 https://doi.org/10.5831/hmj.2021.43.4.613 12 int. j. anal. appl. (2023), 21:32 [16] j.p. singh, m. khatri, on ricci-yamabe soliton and geometrical structure in a perfect fluid spacetime, afr. mat. 32 (2021), 1645–1656. https://doi.org/10.1007/s13370-021-00925-2. [17] h.i. yoldaş, on kenmotsu manifolds admitting η-ricci-yamabe solitons, int. j. geom. methods mod. phys. 18 (2021), 2150189. https://doi.org/10.1142/s0219887821501899. [18] k. yano, integral formulas in riemannian geometry, pure and applied mathematics, vol. i, marcel dekker, new york, 1970. [19] k. yano, m. kon, structures on manifolds, world scientific publishing co., singapore, 1984. [20] p. zhang, y. li, s. roy, s. dey, a. bhattacharyya, geometrical structure in a perfect fluid spacetime with conformal ricci-yamabe soliton, symmetry 14 (2022), 594. https://doi.org/10.3390/sym14030594. https://doi.org/10.1007/s13370-021-00925-2 https://doi.org/10.1142/s0219887821501899 https://doi.org/10.3390/sym14030594 1. introduction 2. preliminaries 3. crys on lp-kenmotsu manifolds 4. gradient crys on lp-kenmotsu manifolds 5. example references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 41-53 http://www.etamaths.com some integral inequalities for beta-preinvex functions muhammad aslam noor∗, khalida inayat noor and sabah iftikhar abstract. the main objective of this paper is to introduce and study a new class of preinvex functions, which is called beta-preinvex functions. some hermite-hadamard type inequalities for beta-preinvex functions are established. our results can be viewed as significant and important generalizations of several previously known results. we also establish some integral inequalities involving euler beta functions for the class of functions whose certain powers of the absolute value are beta-preinvex function. results proved in this paper may stimulate further research in different areas of pure and applied sciences. 1. introduction in recent years, the convex functions and convex sets have been generalized in several directions using novel and innovative techniques to study a wide class of unrelated problem in a unified and general framework. hanson [2] introduced the concept of invex functions involving the bifunction in study of nonlinear programming. this concept stimulated much interest in applications of these invex function in different branches of pure and applied sciences. wier and mond [24] introduced and investigated another class of convex functions, which is called the preinvex functions. they proved that the differentiable preinvex functions are invex functions, but the converse may not true. it is known that the invex functions and preinvex functions are equivalent under some conditions, see [10]. noor[5] proved that the minimum of a differentiable preinvex functions on the invex sets can be characterized by a class of variational-like inequalities. noor [7] also proved that a function f is a preinvex function, which satisfies the hermite-hadmard type integral inequalities. this result can be viewed as an analogous to the convex functions. these results strongly influenced the recent recent research trends. for the applications, properties and other aspects of the preinvex functions, see [1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. various refinements of the hermite-hadamard inequalities for the convex functions and their variant forms are being obtained in the literature by many researchers. tunc et al.[22] introduced the concept of beta-convex functions, which include classical convex functions, s -convex functions, s -godunova-levin functions, tgs -convex functions and p -functions as special cases. noor et al. [16, 16, 17] introduced the concepts of (p,q) and tgs-preinvex f unctions and obtained several integral inequalities via these preinvex functions. motivated and inspired by the ideas and techniques of tunc et. el.[22] and noor et. al. [16, 17, 18], we introduce some new classes of of beta-preinvex functions and derive some estimates involving the euler beta function of the integral ∫a+η(b,a) a (x−a)p(a + η(b,a) −x)qf(x)dx for the class of functions whose certain powers of the absolute value are beta-preinvex function. this is the main motivation of this paper. some special cases are discussed. our results include the previously known results for preinvex functions and their varinat forms as special cases. it is expected that the ideas and techniques considered in this paper be staring for the future research. the interested readers are encouraged to find the novel and innovative applications of these results in other areas. received 18th july, 2016; accepted 20th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d15, 26d10, 90c23. key words and phrases. preinvex functions; beta convex function; beta functions; hermite-hadamard type inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 41 42 noor, noor and iftikhar 2. preliminaries let kη be a nonempty closed set in r . let f : kη ⊆ r −→ r be a continuous function and η(·, ·) : kη × kη −→ r be a continuous bifunction. in this section, we recall the following new and known concepts. definition 2.1. [24]. a set kη ⊆ r is said to be an invex set with respect to the bifunction η(·, ·), if x + tη(y,x) ∈ kη, ∀x,y ∈ kη, t ∈ [0, 1]. if η(y,x) = y−x, then invex set kη reduces to classical convex set. clearly, every convex set is an invex set but the converse is not true. we new define some new concepts of beta-preinvex functions and its variant forms. definition 2.2. a function f : kη ⊆ r → r is said to be beta-preinvex function, where p,q > −1, if f ( x + tη(y,x) ) ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y), ∀x,y ∈ kη, t ∈ (0, 1). (2.1) if t = 1 2 , then f ( 2x + η(y,x) 2 ) ≤ f(x) + f(y) 2p+q , ∀x,y ∈ kη, (2.2) which is called the jensen beta-preinvex function. we now discuss some special cases of beta-preinvex function, which appears to be new ones. i). if p = 1 and q = 0, then definition 2.2, reduces to: definition 2.3. [24]. a function f : kη ⊆ r −→ r is said to be preinvex function with respect to the bifunction η(·, ·), if f ( x + tη(y,x) ) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ kη, t ∈ [0, 1]. if t = 1 2 , then f (2x + η(y,x) 2 ) ≤ f(x) + f(y) 2 , ∀∈ kη, which is called the jensen preinvex function. it has been shown that a function f is a preinvex function, if and only if, it satisfies the inequality of the type f( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ f(a) + f(b) 2 , which is known as the hermite-hadamard-noor integral inequality. see [5, 6, 7] for the recent results in this direction. it is worth mentioning that the minimum of of a differentiable preinvex functions on the invex sets in a normed space can be characterized by a class of variational inequalities, which is known as the variational-like inequalities. for the formulation, applications and numerical methods for solving the variational-like inequalities and related problems, see [1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and the references therein. ii). if p = 0 and q = 0, then definition 2.4. [10, 15]. a function f : kη ⊆ r −→ r is said to be p -preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ f(x) + f(y), ∀x,y ∈ kη. iii). if p = −1 and q = 0, then definition 2.2 reduces to: definition 2.5. [15]. a function f : kη ⊆ r −→ r is said to be godunova-levin preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ f(x) 1 − t + f(y) t , ∀x,y ∈ kη, t ∈ (0, 1). some integral inequalities for beta-preinvex functions 43 iv). if p = 1 and q = 1, then definition 2.2 reduces to: definition 2.6. [16]. a function f : kη ⊆ r −→ r is said to be tgs-preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ t(1 − t)[f(x) + f(y)], ∀x,y ∈ kη, t ∈ [0, 1]. v). if p = s and q = 0, then definition 2.2 reduces to: definition 2.7. [17]. a function f : kη ⊆ r −→ r is said to be s-preinvex function with respect to η(·, ·), where s ∈ [−1, 1], if f ( x + tη(y,x) ) ≤ (1 − t)sf(x) + tsf(y), ∀x,y ∈ kη, t ∈ (0, 1). vi). if p = −s and q = 0, then definition 2.2 reduces to: definition 2.8. a function f : kη ⊆ r −→ r is said to be godunova-levin s-preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ (1 − t)−sf(x) + t−sf(y), ∀x,y ∈ kη, t ∈ (0, 1). vii). if p = 1 2 and q = −1 2 , then definition 2.2, reduces to: definition 2.9. a function f : i ⊆ r −→ r is said to be generalized mt -preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ √ 1 − t √ t f(x) + √ t √ 1 − t f(y), ∀x,y ∈ kη, t ∈ (0, 1). definition 2.10. a function f : kη ⊆ r −→ r is said to be log-beta-preinvex function with respect to η(·, ·), where p,q > −1, if f ( x + tη(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q, ∀x,y ∈ kη, t ∈ (0, 1). it follows that log f ( x + tη(y,x) ) ≤ (1 − t)ptq log f(x) + tp(1 − t)q log f(y). from definition 2.10, we have f ( x + tη(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y). this shows that, log-beta-preinvex function implies beta-preinvex function, but the converse is not true. for appropriate and suitable choices of p,q > −1, and the invex set, one can obtain several new and known classes of preinvex functions and its variant form from definition 2.2 and definition 2.10. this shows that the concept of beta-preinvex functions are quite general and unifying ones. we recall the following special function which is known as beta function. β(x,y) = ∫ 1 0 tx−1(1 − t)y−1dt = γ(x)γ(y) γ(x + y) , x,y > 0, where γ(.) is the gamma function. we also recall the well-known assumption about the bifunction η(., .), which plays an important role in the studies of the variational-like inequalities and integral inequalities. condition c [4]: let i ⊆ r be an invex set with respect to bifunction η(·, ·) : i × i → r. for any x,y ∈ i and any t ∈ [0, 1], we have η(y,y + tη(x,y)) = −tη(x,y) η(x,y + tη(x,y)) = (1 − t)η(x,y). 44 noor, noor and iftikhar 3. hermite-hadamard inequalities in this section, we derive hermite-hadamard inequalities for beta-preinvex function. without loss of generality, we denote by i = [a,a + η(b,a)] unless otherwise specified. theorem 3.1. let f : i = [a,a + η(b,a)] ⊆ r −→ r be beta-preinvex function with a < a + η(b,a). if f ∈ l[a,a + η(b,a)] and condition c holds, then 2p+q−1f ( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ γ(p + 1)γ(q + 1) γ(p + q + 2) [f(a) + f(b)] proof. let f be beta-preinvex function. then taking x = a + tη(b,a) and y = a + (1 − t)η(b,a) in (2.2), and using condition c, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] = 1 2p+q [∫ 1 0 f ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) dt ] this implies 2p+q−1f ( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx now consider 1 η(b,a) ∫ a+η(b,a) a f(x)dx = ∫ 1 0 f ( a + tη(b,a) ) dt ≤ f(a) ∫ 1 0 (1 − t)ptqdt + f(b) ∫ 1 0 tp(1 − t)qdt = [f(a) + f(b)]β(p + 1,q + 1), which is the required result. � theorem 3.2. let f,g : i ⊂ r −→ r be beta-preinvex functions. if f,g ∈ l[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a f(x)g(2a + η(b,a) −x)dx ≤ γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) = f(a)g(a) + f(b)g(b) (3.1) n(a,b) = f(a)g(b) + f(b)g(a) (3.2) proof. let f,g be beta-preinvex functions. then f ( a + tη(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b) (3.3) g ( a + (1 − t)η(b,a) ) ≤ tp(1 − t)qg(a) + (1 − t)ptqg(b). (3.4) from (3.3) and (3.4), we have f ( a + tη(b,a) ) g ( a + (1 − t)η(b,a) ) ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)] (3.5) some integral inequalities for beta-preinvex functions 45 integrating both sides of (3.5), we obtain ∫ 1 0 f ( a + tη(b,a) ) g ( a + (1 − t)η(b,a) ) dt ≤ ∫ 1 0 [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)]dt = [f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt + [f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt = m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) thus 1 η(b,a) ∫ a+η(b,a) a f(x)g(2a + η(b,a) −x)dx ≤ m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) = γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), which is the required result. � if g(2a + η(b,a) −x) = g(x) in theorem 3.2, then it reduces to the following result. corollary 3.1. let f,g : i ⊂ r −→ r be beta-preinvex functions. if f,g ∈ l[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx ≤ γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) m(a,b) + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. theorem 3.3. let f,g : i ⊂ r −→ r be beta-preinvex functions. if fg ∈ l[a,a+η(b,a)] and condition c holds, then 22(p+q)−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) − 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx ≤ γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b), where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. proof. let f be beta-preinvex function. then taking x = a + tη(b,a) and y = a + (1−t)η(b,a) in (2.2) and using condition c, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] , g ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] . 46 noor, noor and iftikhar consider f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22p+2q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] ≤ 1 22p+2q [ f ( a + tη(b,a) ) g ( a + tη(b,a) ) +f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) +[(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b) ] +[tp(1 − t)qf(a) + (1 − t)ptqf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b) ]] . integrating over [0, 1], we have∫ 1 0 f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) dt ≤ 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] = 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2m(a,b)β(p + q + 1,p + q + 1) + 2n(a,b)β(2p + 1, 2q + 1) ] = 1 22p+2q−1 [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dt +m(a,b)β(p + q + 1,p + q + 1) + n(a,b)β(2p + 1, 2q + 1) ] . this completes the proof. � theorem 3.4. let f,g : i ⊂ r −→ r be beta-preinvex functions. if fg ∈ l[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a µ(x)[f(a)g(x) + f(b)g(x)]dx + 1 η(b,a) ∫ a+η(b,a) a µ(x)[g(a)f(x) + g(b)f(x)]dx ≤ γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) m(a,b) + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) n(a,b) + 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx, where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively and µ(x) = ( ((a + η(b,a)) −x)p(x−a)q η(b,a)p+q ) . some integral inequalities for beta-preinvex functions 47 proof. let f, g be beta-preinvex functions. then, we have f ( a + tη(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b), g ( a + tη(b,a) ) ≤ (1 − t)ptqg(a) + tp(1 − t)qg(b). now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ r) and x1 < x2, x3 < x4, we have f ( a + tη(b,a) ) [(1 − t)ptqg(a) + tp(1 − t)qg(b)] +g ( a + tη(b,a) ) [(1 − t)ptqf(a) + tp(1 − t)qf(b)] ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b)] +f ( a + tη(b,a) ) g ( a + tη(b,a) ) = [f(a)g(a) + f(b)g(b)]t2p(1 − t)2q + [f(a)g(b) + f(b)g(a)]tp+q(1 − t)p+q +f ( a + tη(b,a) ) g ( a + tη(b,a) ) integrating over [0, 1], we have g(a) ∫ 1 0 (1 − t)ptqf ( a + tη(b,a) ) dt +g(b) ∫ 1 0 tp(1 − t)qf ( a + tη(b,a) ) dt +f(a) ∫ 1 0 (1 − t)ptqg ( a + tη(b,a) ) dt +f(b) ∫ 1 0 tp(1 − t)qg ( a + tη(b,a) ) dt ≤ [f(a)g(a) + f(b)g(b)] ∫ 1 0 t2p(1 − t)2qdt +[f(a)g(b) + f(b)g(a)] ∫ 1 0 tp+q(1 − t)p+qdt + ∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt this implies 1 η(b,a) ∫ a+η(b,a) a µ(x)[f(a)g(x) + f(b)g(x)]dx + 1 η(b,a) ∫ a+η(b,a) a µ(x)[g(a)f(x) + g(b)f(x)]dx ≤ m(a,b)β(2p + 1, 2q + 1) + n(a,b)β(p + q + 1,p + q + 1) + 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx, which is the required result. � 48 noor, noor and iftikhar theorem 3.5. let f,g : i ⊂ r −→ r be beta-preinvex functions. if fg ∈ l[a,a + η(b,a)] and condition c holds, then f ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a g(x)dx +g ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ 1 2p+q [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b) ] +2p+q−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) , where m(a,b) and n(a,b) are given by (3.1) and (3.2) respectively. proof. let f, g be beta-preinvex function. then taking x = a + tη(b,a) and y = a + (1 − t)η(b,a) in (2.2) and using condition c, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] , g ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] . now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ r) and x1 < x2, x3 < x4, we have 1 2p+q f ( 2a + η(b,a) 2 )[ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] + 1 2p+q g ( 2a + η(b,a) 2 )[ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] ≤ 1 22p+2q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )][ g ( a + tη(b,a) ) +g ( a + (1 − t)η(b,a) )] + f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22p+2q [ f ( a + tη(b,a) ) g ( a + tη(b,a) ) +f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) +2[f(a)g(a) + f(b)g(b)]tp+q(1 − t)p+q +2[f(a)g(b) + f(b)g(a)]t2p(1 − t)2q ] +f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) some integral inequalities for beta-preinvex functions 49 integrating over [0, 1], we have 1 2p+q f ( 2a + η(b,a) 2 )∫ 1 0 [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] dt + 1 2p+q g ( 2a + η(b,a) 2 )∫ 1 0 [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] dt ≤ 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] + ∫ 1 0 f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) dt = 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2m(a,b)β(p + q + 1,p + q + 1) + 2n(a,b)β(2p + 1, 2q + 1) ] +f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) from the above inequality, it follows that f ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a g(x)dx +g ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ 1 2p+q [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx + γ(p + q + 1)γ(p + q + 1) γ(2p + 2q + 2) m(a,b) + γ(2p + 1)γ(2q + 1) γ(2p + 2q + 2) n(a,b) ] +2p+q−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) , which is the requires result. � remark 3.1. if we take η(b,a) = b−a, in above results, we obtain the known integral inequalities for the class of beta-convex functions, see [22]. 4. integral inequalities we need the following lemma in order to obtain new integral inequalities related to beta-preinvex function, which can be proved using the technique of liu[3]. 50 noor, noor and iftikhar lemma 4.1. if f : i = [a,a + η(b,a)] ⊆ r −→ r is a function such that f ∈ l[a,a + η(b,a)], then the following equality holds for some fixed α,β > 0.∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)βf(a + tη(b,a))dt, theorem 4.1. let f : i = [a,a + η(b,a)] ⊆ r −→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a + η(b,a)] and |f| is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, then∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) , where ϕ1(t; a,b) = ∫ 1 0 tα+q(1 − t)β+pdt = β(α + q + 1,β + p + 1) (4.1) ϕ2(t; a,b) = ∫ 1 0 tα+p(1 − t)β+qdt = β(α + p + 1,β + q + 1) (4.2) proof. using lemma 4.1 and beta-preinvexity of |f|, we have∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β { (1 − t)ptq|f(a)| +tp(1 − t)q|f(b)| } dt = ηα+β+1(b,a) ( |f(a)| ∫ 1 0 tα+q(1 − t)β+pdt +|f(b)| ∫ 1 0 tα+p(1 − t)β+qdt ) = ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) . this completes the proof. � theorem 4.2. let f : i = [a,a + η(b,a)] ⊆ r −→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a + η(b,a)] and |f|λ is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, λ ≥ 1, then∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , where ϕ1(t; a,b), ϕ2(t; a,b) are given by (4.1) and (4.2) respectively, and ϕ3(t; a,b) = ∫ 1 0 tα(1 − t)βdt = β(α + 1,β + 1). some integral inequalities for beta-preinvex functions 51 proof. using lemma 4.1, beta-preinvexity of |f|λ and power mean inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ (∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣λdt) 1λ ≤ ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ (∫ 1 0 tα(1 − t)β { (1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ } dt ) 1 λ = ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ ( |f(a)|λ ∫ 1 0 tα+q(1 − t)β+pdt + |f(b)|λ ∫ 1 0 tα+p(1 − t)β+qdt ) 1 λ = ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , which the the required result. � theorem 4.3. let f : i = [a,a + η(b,a)] ⊆ r −→ r be a differentiable function on the interior i◦ of i. if f ∈ l[a,a + η(b,a)] and |f|λ is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ , where 1 λ + 1 µ = 1 and ϕ4(t; a,b) = ∫ 1 0 tαµ(1 − t)βµdt = β(αµ + 1,βµ + 1). 52 noor, noor and iftikhar proof. using lemma 4.1, beta-preinvexity of |f|λ and the holder’s integral inequality, we have∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 ∣∣f(a + tη(b,a))∣∣λdt) 1λ ≤ ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 [(1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ]dt ) 1 λ = ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ . this completes the proof. � remarks from definitions 2.2 and 2.3, we see that the function f satisfies the relation f( 2x + η(y,x) 2 ) = f(x) + f(y) 2 , which is called the preinvex functional equation. we remark that, if η(y,x) = y − x, then the invex set kη becomes convex set k. in this case, the preinvex functional equations collapses to the following equation f( x + y 2 ) = f(x) + f(y) 2 , which is called the jensen-cauchy functional equation, see[19]. this has motivated us to consider the additive preinvex functional equation of the type f(2x + η(y,x)) = f(x) + f(y). (4.3) it is an interesting problem to study the stability criteria of the additive preinvex functional equations (4.3) using the technique of rassias[20], ulam[23] and small[21]. we leave this to the interested readers to explore this aspects of the additive preinvex functional equations. acknowledgements the authors would like to thank dr. s. m. junaid zaidi, (h. i., s. i), rector, comsats institute of information technology, pakistan, for providing excellent research and academic environments. references [1] i. azhar, integral inequalities under beta function and preinvex type functions, springerplus, 5 (2016), article id 521. [2] m. a. hanson. on sufficiency of the kuhn-tucker conditions, journal of mathematical analysis and applications, appl., 80(1981), 545-550. [3] m. liu, new integral inequalities involving beta function via p-convexity, miskolc math. notes, 15(2)(2014), 585-591. some integral inequalities for beta-preinvex functions 53 [4] s. r. mohan and s. k. neogy, on invex sets and preinvex functions, j. math. anal. appl. 189(1995), 901-908. [5] m. a. noor, variational-like inequalities, optimization, 30(1994), 323-330. [6] m. a. noor, invex equilibrium problems, j. math. anal. appl. 302(2005), 463-475. [7] m. a. noor, hermite-hadamard integral inequalities for log-preinvex functions, j. math. anal.approx. theory, 2(2007), 126-131. [8] m.a. noor, hadamard integral inequalities for product of two preinvex function, nonl. anal. forum 14(2009), 167-173. [9] m.a. noor, on hadamard integral inequalities involving two log-preinvex functions, j. inequal. pure appl. math., 8(3)(2007), 1-14. [10] m. a. noor and k. i. noor, some characterizations of strongly preinvex functions, j. math. anal. appl. 316(2006), 697-706. [11] m. a. noor and k. i. noor, generalized preinvex functions and their properties, j. appl. math. stoch. anal. 2006(2006), article id 12736. [12] m. a. noor, k. i. noor and m. u. awan, fractional hermite-hadamard inequalities for two kinds of s-preinvex functions, nonlinear sci. lett. a., 8(1)(2017), 11-24. [13] m. a. noor, k. i. noor and m. u. awan, some quantum integral inequalities via preinvex functions, appl. math. comput. 269(2015), 242-251. [14] m. a. noor, k. i. noor, m. v. mihai and m. u. awan, fractional hermite-hadamard inequalities for some classes of differentiable preinvex functions, u.p.b. sci. bull., series a, 78(3)(2016), 163174. [15] m. a. noor, k. i. noor, m. u. awan and j. li, on hermite-hadamard inequalities for h-preinvex functions, filomat 28(7)(2014), 1463-1474. [16] m. a. noor, m. u. awan and k. i. noor, some new bounds of the quadrature formula of gaussjaccobi type via (p,q)-preinvex functions, appl. math. inform. sci. lett. in press. [17] m. a. noor, m. u. awan and k. i. noor, some inequalities via tgs-preinvex functions in quantum analysis, preprint, (2016). [18] m. a. noor, s. khan and k. i. noor, integral inequalities for geometrically log-preinvex functions, appl. math.inform. sci. lett. 4(3)(2016), 103-110. [19] j. pecaric, f. proschan, and y. l. tong, convex functions, partial orderings and statistical applications, acdemic press, new york, (1992). [20] th. m. rassias, on the tability of the linear mappings in banach spaces, proc. amer. math. soc. 72(1978), 297-300. [21] c. g. small, functional equations and how to solve them, springer, new york, 2007. [22] m. tunc, u. sanal and e. gov, some hermite-hadamard inequalities for beta-convex and its fractional applications, ntmsci 3(4)(2015), 18-33. [23] s. m. ulam, problems in modern mathmeatics, science editions, j. wiley, new york, 1960. [24] t. weir and b. mond, preinvex functions in multiple objective optimization. j math. anal. appl., 136(1988), 29-38 department of mathematics, comsats institute of information technology, park road, islamabad, pakistan. ∗corresponding author: noormaslam@gmail.com 1. introduction 2. preliminaries 3. hermite-hadamard inequalities 4. integral inequalities remarks acknowledgements references { } , 0 , 1, 2 , . . . k x k  0 1 2 0 . . . a n d l i m k k k x x x x x           0 1 [ 0 , ) , ( 0 , ) a n d { } k r r r r x            1,p   p   p l 1 pp u u d x     , s u p ( ) :p u e s s u x x       |n |= 0 ,x r -n in f s u p ( )u x  p u    ( ) :p pl u u      ( )c     0 ( ) ( ) : ( ) ( ) , 0 , 1, 2 , ..., lim ( ) ( ) , , k x x k k p c y x y x c k y x y x x x           0 ( ) ( )d c       s u p p ( ) : ( ) 0f x x f x     . w d u  u   , ( ) ( ) : ( ) , | | m p p w p w u l d l m         , 2 ( ) m m w h  , 0 ( ) m p w  ( )d  ( ) m h  , 2 0 0 ( ) ( ) m m h w   , ( ) m p w  1 p   1 p   ,k p p k u d u      1 , 1 t h e n p p i np p u u k u u x x          2 2 1 , 2 1 2 1 t h e n n i i u k u x       1 , . n i i u u v u v d x x             n r 2 a n d 1n p    1 , 1 1 , ( ) ( ) , 1 1 1 i f p q q p w l u c u p n p q          1 , 0 2 1 , ( ) ( ) , m a x p p m w c l u c u        1 2 a n d c c ( ., .) :a h h r  2 | ( , ) | | | ,a u u c u u h  u ( , )a u u c l 2 ( )l    2 : 0 , ( ) c l v v v v l       2 :| | , c o n s t ( ) . p s x v x p p l       i i i u v v d x u v n d s u d x x x             i n n r  1 , p  1 , 1 , o n ( ) p p w  1 , 00 , 1 , 00 , , ( ) , ( ) . p p p n u c u u w u u c u h x            h ( , ) :a h h r    :f h r u h ( , ) ( ) ,a u v f v u h   ( , )a   ( , ) 0 ,a u v v h   :j h r 1 ( ) : ( , ) ( ) 2 j v a v v f v  in f ( ) ( ) h u j u j u      2 1 is m e a s u r a b le in f o r a ll ( ) : , c o n tin u o u s in f o r a lm o s t a ll i x l a a x x i          2 : ( , )a a x  1 2 , 2 2 2 1 1 2 1 1 2 ( ) ( , ) ( , ) * . w l v a x a x x x x u c c v x                               0 0 1 0 1 ( , ) ( , m a x ) i n , , 0 , 1, 2 , , , , w i t h m i x e d b o u n d a r y c o n d i t i o n s 0 o n , ( , ) , . i i i k i i k i i i i u a x u g x u x x f x x k x i x u x x x x x u u a x y n h o n x                                           ( )u d  0  (1)   1 1 1 ( , ) , . i i i i u v a x u d x f v d x f g h v d s x x v i x u                  ( )u u x : ( 0 , ] n u x v / v  ,u a u    ( 0 , ] n x / ( ) , ( ) , , ( ) , ( ) , d d v a u v d s a u d s d u p c v v d s d s                  ( , )u u x u ( )j v   (1 ) 0 ( ) ( ) ( ) li m s u p h j v h j v d j v h      c l 0    0 , 0x   0 x    0 ,u x u   0 x 0    0 , 0t t x   0 x    0 0 ,u x u n 0 x x t  (2) ( )t t     / 0 : 0 o nv u p c u       / / v p c  ( )p c  1 , 2 ( )w  ( )p c   :a v v  b v  ( ) , ( , ) , , ( ) i i u v a u v a x u d x u v v d x x               1 2 3 4 , ,b v b b b b x v    1 2 3 , , ,b f v d x b v g d x b f v d x         4 1 . i i i b v i    u v ( ), ,a u u b v v v    1 , i i i b v f v d x f v f v d x v i           (3) (4) (5) 2 2 2 2 2 2 2 2 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) l l l l l l l l f v g v h v c v c v                2 2 2 1( ) ( ) ( ) m a x , , l l l c f g h c         ' :a v v :a v v 0      2 1 2 1 2 1 2 , 0 , ,a u a u u u u u v      ( )a u b b v  ( ) ( )t u u a u b    0  t      1 2 1 2 1 2 , ,t u t u u u u u v                                    22 1 2 1 2 1 2 1 2 1 2 1 2 1 2 22 2 1 2 1 2 1 2 1 2 2 2 2 1 2 , 2 , 1 2 . t u t u u u a u a u u u a u a u u u a u a u u u u u a u a u a u a u u u m                                a     2 2 1 2 1 a u a u u u            2 1 1 2 2 1 , , i i i i i i u u a u a u a x x a x u x x x                   2 2 1 1 , . i i i i i x i a x u a u a u u u a u x x x x x x x x x                          2 1 1 2 2 1 2 1 1 2 2 2 . i i v u u a u a u l x x u u l u u x x m u u                      2 2 2 1 2 1 2 0m           0 , 2 0 0 2           m i n 1, 2          t  ( )t u u   ( ) 0 t u u a u b u a u b         a ( )a u b ( , )a x u   1 * 1 ( , ) , . n i i i i n i i i u v a u v a x u d x g v d x x x b f v d h v d s v i                  1 1 ( ) ( , ) 2 2 i i u v j v a x u d v g v d x x x              2 1 { , } , , , , , . n n i j i j i j i i j m a x m r x r             u i n f ( ) v v u j v   * ( , ) i i u v a x u g v d x b x x            2 ( )b l  0 : ( )a h h v h    0 ( ) m i n ( ) v h u j v    *1 1 ( ) ( , ) . 2 2 i i i i u v j v a x u g v d x b x x            * , , 0n m  1 :h     1 , s u p , a n d s u p , m a x x u x u v n a x u g x u         2 2 * 1 m i n , . 2 l l n u m v       0 , m in ( ) v v u u x u j v    *1 1 ( ) ( , ) . 2 2 i i i i u v j v a x u d x g v d x b x x            2 2 1 1 * m i n ( ) 1 1 m i n ( , ) 2 2 1 m i n . 2 v j l l u j v u v a x u d x g v d x x x n u m v                        | |x  *   | ( , ) |u x u  ( , )u u x u 1 :h , 1, 2 , 3 i k i  * 1 2 1 3 | | , | | a n d | |f k v h k v i k v   1 :h *1 1 ( , ) 2 2 i i i i u v a x u d x g v d x b x x           1 * * . i i i b f v d x h v d x v i        ( ) 0 , ( ) ( ) a n d ( ) 0 , k k k j v j x x j x d j v v v         { } , 1, 2 , . . . k x k  1 2 &h h * 1 1 ( ) ( , ) 2 i i i i n i i i u v j v a x u g v d x x x f v d x h v d s v i                       1 h 1 * 1 2 2 2 1 2 3 0 . n i i i i f v d x h v d s v i k v d x k v d s k v                2 h ( ) 0j v    0 * 1 ( ) ( ) ( ) l i m s u p 1 2 2 0 2 h j v h j v d j v h g h d x i              * 0 , 0g h  ( , )u u x u   2 : 0 c l v l v   0  0 u  0  *   *      2 2 , m a x ( , ) , , 0 , 1, 2 , . . . 0 o n k p k x k u u c g x u f x u x x k x x u e i x u                i s t h e t h e r m a l h e a t c a p a c i t y o f s y s t e m p c 0  ( , ) , ( ) p x u c x k u u      ( )u heat source 2 1 1 ( ) 2 2 u u v j v c k d x v g d x v f d x x x x                     1 22 2 2 m a x ( , m a x ) a n d g ( , m a x ) 1 1 m a x ( m a x ) u x g x u x u x x u x u        2 2 ( i ) 0 i n ( ) 0 i n 1 ( ) 0 i n 2 t h e n ( ) 0 i n = a n d t h e t e m p e r a t u r e u i s s u c h t h a t p u u c k x x v i i x i i i g f j v                    22 ( ) 0 ( 1) , 1 2 ( ) 0 1 2 (1 ) i v u x f x a n d x v u x x f            * 0 in .h   j ( v ) 0 i n a n d h v d x = 0     ( ) 0 i n j v   2 1 m a x 0 . 2 2 (1 ) u g f f x       22 m a x ( 1) f o r 1 .u x f x     22 0 m a x , t h e r e f o r e ( 1)u u u x      2 2 1 0 2 2 (1 m a x ( m a x ) ) x g f f x u x u            2 2 2 2 2 m a x 2 ( m a x ) 2 2 2 . x x u x u x u x u f          2 f o r 0 1,y y y y r      2 ( m a x ) m a x i f 0 u m a x u < 1u u   2 0 u m a x u < < 1 2 (1 ) x x x f     ( ) 0 j v  , 0 , 0 .f x   +( 1 ) h 0 * ( ) ( ) ( ) l i m 1 ( ) 2 i i j v h j v d j v h g d x f d x h d s i x                 * 0h   , a n d i g f i  ( ) ( ) . k k k j x x j x   k x u  int. j. anal. appl. (2023), 21:91 transmission problem between two herschel-bulkley fluids in a three dimensional thin layer salim saf1,∗, farid messelmi2 1laboratory of pure and applied mathematics, amar telidji university of laghouat, laghouat 03000, algeria 2department of mathematics and ldmm laboratory ziane achour university, djelfa 17000, algeria ∗corresponding author: s.saf@lagh-univ.dz abstract. the paper is devoted to the study of steady-state transmission problem between two herschel-bulkley fluids in a three dimensional thin layer. 1. introduction the rigid viscoplastic and incompressible fluid of herschel-bulkley has been studied and used by many mathematicians, physicists and engineers, to model the flow of metals, plastic solids and a variety of polymers. due to the existence of the yield limit, the model can capture phenomena connected with the development of discontinuous stresses. a particularity of herschel-bulkley fluid lies in the presence of rigid zones located in the interior of the flow and as yield limit increases, the rigid zones become larger and may completely block the flow, this phenomenon is known as the blockage property. the literature concerning this topic is extensive; see e.g. [4, 11, 12, 14, 15]. the purpose of this paper is to study the asymptotic behavior of the steady flow of herschel-bulkley fluid in a three-dimensional thin layer. the paper is organized as follows. in section 2 we present the mechanical problem of the steady flow of herschel-bulkley fluid in a three-dimensional thin layer. we introduce some notations and preliminaries. moreover, we define some function spaces and we recall the variational formulation. in section 3, we are interested in the asymptotic behavior, to this aim we prove some convergence received: jun. 22, 2023. 2020 mathematics subject classification. 76a05, 49j40, 76b15. key words and phrases. herschel-bulkley fluid; transmission; asymptotic behaviour; thin layer. https://doi.org/10.28924/2291-8639-21-2023-91 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-91 2 int. j. anal. appl. (2023), 21:91 results concerning the velocity and pressure when the thickness tends to zero. besides, the uniqueness of a limit solution has been also established. 2. problem statement denoting by ω the fixed region in plan x = (x1,x2) ∈r2. introducing the function h : ω → r such that 0 < h0 ≤ h(x,y) ≤ h1 for all (x,y) ∈r2, where h0 and h1 are constants. considering the following domains ω1 = { (x,y,z) ∈r3/ (x,y) ∈ ω and 0 < z < h(x,y) } , ωε1 = { (x1,x2,x3) ∈r3/ (x1,x2) ∈ ω and 0 < x3 < εh(x1,x2) } , ω2 = { (x,y,z) ∈r3/ (x,y) ∈ ω and −h(x,y) < z < 0 } ωε2 = { (x1,x2; x3) ∈r3/ (x1,x2) ∈ ω and −εh(x1,x2) < x3 < 0 } , where ε > 0. remark that if (x1,x2; x3) ∈ ωεi then (x,y,z) = (x1,x2, x3 ε ) ∈ ωi. this permits us to define, for every function ϕεi : ω ε i →r, the function ϕ̂ ε i : ωi →r given by ϕ̂εi (x,y,z) = ϕ ε i (x1,x2,x3), i = 1, 2. let 1 < p ≤ 2, p′ the conjugate p, ( 1 p + 1 p′ = 1 ) and fi = (fi1, fi2, fi3) ∈ lp ′ i (ωi ) 3 a given functions. we define the functions f εi ∈ l p′(ωεi ) 3 such that f̂ ε i = fi, i = 1, 2. we consider a mathematical problem modeling the steady flow of a rigid viscoplastic and incompressible herschel-bulkley fluid. we suppose that the consistency and yield limit of the fluid are respectively µiε p, giε where µi, gi > 0, i = 1, 2 and p represents the power-law index. the first fluid occupies a bounded domain ωε1 ⊂ r 3 with the boundary ∂ωε1 of class c 1. the second one occupies a bounded domain ωε2 ⊂ r 3 with the boundary ∂ωε2 of class c 1. we denote by ωε the domain ωε1 ∪ ω ε 2 and we suppose that ∂ω ε 1 = ω∪ γ ε 1 and ∂ωε2 = ω ∪ γ ε 2 the velocity is known and equal to zero, where ω, γ ε 1, γ ε 2 are measurable domains and meas(γε1), meas(γ ε 2) > 0. the fluids are acted upon by given volume forces of densities f ε 1 , f ε 2 respectively. we denote by s3 the space of symmetric tensors on r3. we define the inner product and the euclidean norm on r3 and s3, respectively, by u.υ = ulυl ∀u,υ ∈r3 and σ.τ = σlmτ lm ∀σ,τ ∈ s3. |u| = (u.u) 1 2 ∀u ∈r3 and |σ| = (σ.σ) 1 2 ∀σ ∈ s3. here and below, the indices l and m run from 1 to 3 and the summation convention over repeated indices is used. we denote by σ̃ε i the deviator of σεi given by σεi = −p ε i i3 + σ̃ ε i , where pεi , i = 1, 2 represents the hydrostatic pressure and i3 denotes the identity matrix of size 2. we consider the rate of deformation operator defined for every vεi ∈ w 1, pi (ωεi ) 3 by d(vεi ) = (dlm(v ε i )), dlm(v ε i ) = 1 2 ((υεi )l, m + (υ ε i )m, l), i = 1, 2. int. j. anal. appl. (2023), 21:91 3 we denote by n the unit outward normal vector on the boundary ω oriented to the exterior of ωε1 and to the interior of ωε2, see the figure below. for every vector field v ε i ∈ w 1, pi (ωεi ) 3 we also write vεi for its trace on ∂ωεi , i = 1, 2. the steady-state transmission problem for the herschel-bulkley fluids in thin layer is given by the following mechanical problem. problem pε. find the velocity field uεi = (u ε i1,u ε i2,u ε i3) : ω ε i → r 3, the stress field σεi = (σεi1,σ ε i2,σ ε i3) : ω ε i → s3 and the pressure p ε i : ω ε i →r 2, i = 1, 2 such that div σε1 + f ε 1 = 0 in ω ε 1. (2.1) div σε2 + f ε 2 = 0 in ω ε 2. (2.2) σ̃ε1 = µ1ε p ∣∣d(uε1)∣∣p−2 d(uε1) + g1ε d(uε1 )|d(uε1 )| if ∣∣d(uε1)∣∣ 6= 0∣∣∣σ̃ε1∣∣∣ ≤ g1ε if ∣∣d(uε1)∣∣ = 0   in ωε1, (2.3) σ̃ε2 = µ2ε p ∣∣d(uε2)∣∣p−2 d(uε2) + g2ε d(uε2 )|d(uε2 )| if ∣∣d(uε2)∣∣ 6= 0∣∣∣σ̃ε2∣∣∣ ≤ g2ε if ∣∣d(uε2)∣∣ = 0   in ωε2, (2.4) div uε1 = 0 in ω ε 1, (2.5) div uε2 = 0 in ω ε 2 , (2.6) uε1 = 0 on γ ε 1 , (2.7) uε2 = 0 on γ ε 2, (2.8) uε1 −u ε 2 = 0 on ω, (2.9) σε1.n−σ ε 2.n = 0 on ω. (2.10) here, the flow is given by the equations (2.1) and (2.2). equations (2.3) and (2.4) represents, respectively, the constitutive laws of herschel-bulkley fluids where and are the consistencies and yield limits of the two fluids, respectively, 1 < p ≤ 2 are the power law index of the two fluids, respectively. equations (2.5) and (2.6) represents the incompressibility condition. (2.7), (2.8) give the velocities on the boundaries γε1 and γ ε 2, respectively. finally, on the boundary part ω, (2.9) and (2.10) represents the transmission condition for liquid-liquid interface. let us define now the following banach spaces w 1, p γi (ωεi ) = { vi ∈ w 1, p(ωεi ) : vi = 0 on γ ε i } , (2.11) w p, ε div (ωεi ) = { vi ∈ w 1, p(ωεi ) 3 : div(vi ) = 0 in ω ε i } , (2.12) w p div (ωi ) = { vi ∈ w 1, p(ωi )3 : div(vi ) = 0 in ωi } , (2.13) l p 0 (ω ε i ) = { ϕεi ∈ l p(ωεi ) : ∫ ωε i ϕεi (x1,x2,x3)dx1dx2dx3 = 0 } , (2.14) 4 int. j. anal. appl. (2023), 21:91 l p 0 (ωi ) = { ϕi ∈ l p(ωi ) : ∫ ωi ϕi (x,y,z)dxdydz = 0 } , (2.15) wz (ωi ) = { ϕi ∈ l p(ωi ) : ∂ϕi ∂z ∈ lp(ωi ) } , i = 1, 2. (2.16) wz = wz (ω1) 2 ×wz (ω2)2 , (2.17) wεdiv = { (v1,v2) ∈ w ,p, ε div (ωε1) ×w p, ε div (ωε2) : v1 = v2 on ω, v1 = 0 on γε1 , v2 = 0 on γ ε 2 } , (2.18) wε = { (v1,v2) ∈ w 1, p γε1 (ωε1) 3 ×w 1, p γε2 (ωε2) 3 : v1 = v2 on ω } . (2.19) for the rest of this article, we will denote by c possibly different positive constants depending only on the data of the problem. the use of green’s formula permits us to derive the following variational formulation of the mechanical problem (pε), see [4,13,15]. problem pvε . for prescribed data (f ε1 , f ε 2 ) ∈ l p′(ωε1) 3 × lp ′ (ωε2) 3. find (uε1,u ε 2) ∈ w ε div and (pε1,p ε 2) ∈ l p′ 0 (ω ε 1) ×l p′ 0 (ω ε 2) satisfying the variational inequality µ1ε p ∫ ωε1 |d(uε1)| p−2 d(uε1).d(v1 −u ε 1)dx1dx2dx3 + g1ε ∫ ωε1 |d(v1)|dx1dx2dx3 −g1ε ∫ ωε1 |d(uε1)|dx1dx2dx3 + µ2ε p ∫ ωε2 |d(uε2)| p−2 d(uε2).d(v2 −u ε 2)dx1dx2dx3 +g2ε ∫ ωε2 |d(v2)|dx1dx2dx3 −g2ε ∫ ωε2 |d(uε2)|dx1dx2dx3 ≥ ∫ ωε1 f ε1 .(v1 −u ε 1)dx1dx2dx3 + ∫ ωε1 pε1 div(v1 −u ε 1)dx1dx2dx3 + ∫ ωε2 f ε2 .(v2 −u ε 2)dx1dx2dx3 + ∫ ωε2 pε2 div(v2 −u ε 2)dx1dx2dx3, ∀(v1,v2) ∈ w ε. (2.20) it is known that this variational problem has a unique solution (uε1,u ε 2) ∈ w ε div and (p ε 1,p ε 2) ∈ l p′ 0 (ω ε 1)× l p′ 0 (ω ε 2), see for more details [11,13,15]. 3. asymptotic behavior in this section, we establish some results concerning the asymptotic behavior of the solution when ε tends to zero. we begin by recalling the following lemmas see [1,3,4,8,16]. lemma 3.1. (1) poincaré’s inequality. for every vi ∈ w 1,p γi (ωεi ) 3 we have ‖vεi ‖lp(ωε i )3 ≤ ε ∥∥∥∥∂vεi∂x2 ∥∥∥∥ lp(ωε i )3 , i = 1, 2. (3.1) (2) korn’s inequality. for every vi ∈ w 1,p γi (ωεi ) 3 there exists a positive constant c0 independent on ε, such that ‖∇vεi ‖lp(ωε i )9 ≤ c0 ‖d(v ε i ) ‖lp(ωε i )9 , i = 1, 2. (3.2) int. j. anal. appl. (2023), 21:91 5 lemma 3.2 (minty). let e be a banach spaces, a : e → e′ a monotone and hemi-continuous operator, j : e → ]−∞, +∞] a proper and convex functional. let u ∈ e and f ∈ e′ . the following assertions are equivalent: 1.〈au; v −u〉e′×e + j(v) −j(u) ≥ 〈f ; v −u〉e′×e, ∀v ∈ e 2.〈av; v −u〉e′×e + j(v) −j(u) ≥ 〈f ; v −u〉e′×e. ∀v ∈ e the main results of this section are stated by the following proposition. proposition 3.1. let (uε1,u ε 2) ∈ w ε div and (p ε 1,p ε 2) ∈ l p′ 0 (ω ε 1) ×l p′ 0 (ω ε 2) be the solution of variational problem (pvε). then, there exists (û1 , û2) ∈wz (ω1)3×wz (ω2)3 and (p̂1 , p̂2) ∈ l p′ 0 (ω1)×l p′ 0 (ω2) such that (ûε1, û ε 2) → (û1 , û2) in wz (ω1) 3 ×wz (ω2)3 weakly, (3.3)( ∂ûε13 ∂z , ∂ûε23 ∂z ) → (0, 0) in lp(ω1) ×lp(ω2) weakly, (3.4) (p̂ε1, p̂ ε 2) → (p̂1 , p̂2) in l p′ 0 (ω1) ×l p′ 0 (ω2) weakly. (3.5) proof. choosing (v1,v2) = (0, 0) as test function in inequality (2.20), we deduce that µ1ε p ‖d(uε1)‖ p lp(ωε1) 9 + µ2ε p ‖d(uε2)‖ p lp(ωε2) 9 ≤ ∫ ωε1 f ε1 .u ε 1dx1dx2dx3 + ∫ ωε2 f ε2 .u ε 2dx1dx2dx3, this permits us to obtain, making use of poincaré’s and korn’s inequalities and by passage to variables x , y and z ∥∥∥ûε1∥∥∥ lp(ω1)3 + ∥∥∥ûε2∥∥∥ lp(ω2)3 ≤ c, (3.6)∥∥∥∥∥∂û ε 1 ∂z ∥∥∥∥∥ lp(ω1)3 + ∥∥∥∥∥∂û ε 2 ∂z ∥∥∥∥∥ lp(ω2)3 ≤ c, (3.7) ∥∥∥∥∥∂û ε 1 ∂x ∥∥∥∥∥ lp(ω1)3 + ∥∥∥∥∥∂û ε 2 ∂x ∥∥∥∥∥ lp(ω2)3 ≤ c ε , (3.8) ∥∥∥∥∥∂û ε 1 ∂y ∥∥∥∥∥ lp(ω1)3 + ∥∥∥∥∥∂û ε 2 ∂y ∥∥∥∥∥ lp(ω2)3 ≤ c ε . (3.9) moreover, we get using the incompressibility condition (2.5), (2.6) and green’s formula, for any function (ϕε1,ϕ ε 2) ∈ w 1,p1 γ1 (ωε1) ×w 1,p2 γ2 (ωε2)∫ ω1 ∂ûε13 ∂z ϕ̂ε1dxdydz + ∫ ω2 ∂ûε23 ∂z ϕ̂ε2dxdydz = ε ∫ ω1 ( ûε11 ∂ϕ̂ε1 ∂x + ûε12 ∂ϕ̂ε1 ∂y ) dxdydz + ε ∫ ω2 ( ûε21 ∂ϕ̂ε2 ∂x + ûε22 ∂ϕ̂ε2 ∂y ) dxdydz. 6 int. j. anal. appl. (2023), 21:91 which gives, making use (2.16)∥∥∥∥∥∂û ε 13 ∂z ∥∥∥∥∥ w−1,p ′ (ω1) + ∥∥∥∥∥∂û ε 23 ∂z ∥∥∥∥∥ w−1,p ′ (ω2) ≤ cε. (3.10) we can then extract a subsequences still denoted by (ûε1, û ε 2) such that (ûε1, û ε 2) → (û1, û2) in l p(ω1) 3 ×lp(ω2)3 weakly, (3.11)( ∂ûε1 ∂z , ∂ûε2 ∂z ) → ( ∂û1 ∂z , ∂û1 ∂z ) in lp(ω1) 3 ×lp(ω2)3 weakly, (3.12)( ∂ûε13 ∂z , ∂ûε23 ∂z ) → (0, 0) in lp(ω1) ×lp(ω2) weakly. (3.13) let now (vε1,v ε 2 ) ∈ w 1,p γ1 (ωε1) 3 ×w 1,p γ2 (ωε2) 3 , we obtain by setting (uε1 −v ε 1,u ε 2 −v ε 2 ) as test function in inequality (2.20), using the incompressibility conditions (2.5) and (2.6) as well as the green formula and holder’s inequality ∫ ωε1 ∇pε1.v ε 1dx1dx2dx3 + ∫ ωε2 ∇pε2.v ε 2dx1dx2dx3 ≤ µ1ε p (∫ ωε1 |d(uε1)| p dx1dx2dx3 ) 1 p′ (∫ ωε1 |d(vε1 )| p dx1dx2dx3 )1 p +g1ε 1 p′ +1meas(ωε1) 1 p′ (∫ ωε1 |d(vε1 )| p dx1dx2dx3 )1 p +ε ∥∥∥f̂ ε1 ∥∥∥ lp ′ (ωε1) 3 ∥∥∥v̂ε1∥∥∥ w 1, p γ1 (ω1)3 + ε ∥∥∥f̂ ε2 ∥∥∥ lp ′ (ωε2) 3 ∥∥∥v̂ε2∥∥∥ w 1,p γ2 (ω2)3 +µ2ε p (∫ ωε2 |d(uε2)| p dx1dx2dx3 ) 1 p′ (∫ ωε2 |d(vε2 )| p dx1dx2dx3 )1 p +g2ε 1 p′ +1meas(ωε2) 1 p′ (∫ ωε2 |d(vε2 )| p dx1dx2dx3 )1 p . (3.14) on the other hand, it is easy to check that, after some algebraic manipulations, we find (∫ ωε i |d(vεi )| p dx1dx2dx3 )1 p ≤ ε 1 p −1 ∥∥∥v̂εi ∥∥∥ w 1, p γi (ωi ) 3 , i = 1, 2. (3.15) hence, from (3.7), (3.8), (3.9), (3.14) and (3.15) if follows that∫ ωε1 ∇pε1.v ε 1dx1dx2dx3 + ∫ ωε2 ∇pε2.v ε 2dx1dx2dx3 ≤ cε (∥∥∥v̂ε1∥∥∥ w 1, p γ1 (ω1)3 + ∥∥∥v̂ε2∥∥∥ w 1, p γ2 (ω2)3 ) . (3.16) int. j. anal. appl. (2023), 21:91 7 passing to the variables x, y and z in the left hand side of (3.16 ) we find the following estimates∥∥∥p̂ε1∥∥∥ l p′ 0 (ω1) + ∥∥∥p̂ε2∥∥∥ l p′ 0 (ω2) ≤ c, (3.17)∥∥∥∥∥∂p̂ ε 1 ∂x ∥∥∥∥∥ w−1, p ′ (ω1) + ∥∥∥∥∥∂p̂ ε 2 ∂x ∥∥∥∥∥ w−1, p ′ (ω2) ≤ c, (3.18) ∥∥∥∥∥∂p̂ ε 1 ∂y ∥∥∥∥∥ w−1, p ′ (ω1) + ∥∥∥∥∥∂p̂ ε 2 ∂y ∥∥∥∥∥ w−1, p ′ (ω2) ≤ c, (3.19) ∥∥∥∥∥∂p̂ ε 1 ∂z ∥∥∥∥∥ w−1, p ′ (ω1) + ∥∥∥∥∥∂p̂ ε 2 ∂z ∥∥∥∥∥ w−1, p ′ (ω2) ≤ εc. (3.20) consequently, we can extract a subsequence still denoted by ( p̂ε1, p̂ ε 2 ) such that( p̂ε1, p̂ ε 2 ) → (p̂1, p̂2) in l p′ 0 (ω1) ×l p′ 0 (ω2) weakly, (3.21) which achieves the proof. this proof permits also to deduce that limit pressure verify( p̂ε1 (x,y,z) , p̂ ε 2(x,y,z) ) = (p̂1(x,y), p̂2(x,y)). � proposition 3.2. the velocity limit given by (3.3) verifies∫ h(x,y) 0 ( ∂û11(x,y,z) ∂x + ∂û12(x,y,z) ∂y ) dz + ∫ 0 −h(x,y) ( ∂û21(x,y,z) ∂x + ∂û22(x,y,z) ∂y ) dz = 0 ∀(x,y) ∈ ω (3.22) proof. we know from incompressibility conditions (2.5) and (2.6) that∫ ωε1 div(uε1(x1,x2,x3))ϕ1(x1,x2)dx1dx2dx3 + ∫ ωε2 div(uε2(x1,x2,x3))ϕ2(x1,x2)dx1dx2dx3 = 0 for all (ϕ1,ϕ2) ∈ d(ω) 2. this implies, using green’s formula∫ ωε1 uε11 dϕ1 dx1 (x1,x2)dx1dx2dx3 + ∫ ωε1 uε12 dϕ1 dx2 (x1,x2)dx1dx2dx3 + ∫ ωε2 uε21 dϕ2 dx1 (x1,x2)dx1dx2dx3 + ∫ ωε2 uε22 dϕ2 dx2 (x1,x2)dx1dx2dx3 = ∫ ωε1 ∂uε13 ∂x3 ϕ1(x1,x2)dx1dx2dx3 + ∫ ωε2 ∂uε33 ∂x3 ϕ2(x1,x2)dx1dx2dx3. hence, by passage to the variables x, y and z using geen’s formula, we can infer∫ ω ϕ(x,y) (∫ h(x,y) 0 ( ∂ûε11 ∂x + ∂ûε12 ∂y ) dz + ∫ 0 −h(x,y) ( ∂ûε21 ∂x + ∂ûε22 ∂y ) dz ) dxdy = 0 ϕ ∈ d(ω). 8 int. j. anal. appl. (2023), 21:91 then, ∫ h(x,y) 0 ( ∂ûε11 ∂x + ∂ûε12 ∂y ) dz + ∫ 0 −h(x,y) ( ∂ûε21 ∂x + ∂ûε22 ∂y ) dz = 0. moreover, the fact that û1 = (û ε 11, û ε 12) ∈ l p (ω1) 2 , û2 = (û ε 21, û ε 22) ∈ l p (ω2) 2 and∫h(x,y) 0 û1(x,y,z)dz + ∫ 0 −h(x,y) û2(x,y,z)dz is continuous and linear, it is weakly continuous. thus, by passage to the limit when ε tends to zero, taking into account the boundaries conditions (2.7) ,(2.8) and (2.9) it follows that∫ h(x,y) 0 ( ∂û11 ∂x + ∂û12 ∂y ) dz + ∫ 0 −h(x,y) ( ∂û21 ∂x + ∂û22 ∂y ) dz = 0, ∀(x,y) ∈ ω. � we derive in the proposition below the strong equation verified by the limit solution (û1, û2) = ((û11, û12), (û21, û22)) ∈ wz, and (p̂1, p̂2) ∈ l p′ 0 (ω1) ×l p′ 0 (ω2). proposition 3.3. if ( ∂û1 ∂z , ∂û2 ∂z ) 6= (0, 0) then the limit point (û1, û2) and (p̂1, p̂2) given by (3.3) and (3.5) verify the limit problem − ∂ ∂z  µ1 2 p 2 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z + √ 2 2 g1 ∂û1 ∂z∣∣∣∂û1∂z ∣∣∣ + µ2 2 p 2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z + √ 2 2 g2 ∂û2 ∂z∣∣∣∂û2∂z ∣∣∣   = f̂ 1 −∇p1(x,y) + f̂ 2 −∇p2(x,y) in w −1, p′(ω)2. (3.23) proof. introducing the operator φ defined as follows φ : w ε → w ε′, 〈φ(uε1,u ε 2), (v ε 1,v ε 2 )〉w ε′×w ε = µ1ε p ∫ ωε1 |d(uε1)| p−2 d(uε1).d(v ε 1 )dx1dx2dx3 +µ2ε p ∫ ωε2 |d(uε2)| p−2 d(uε2).d(v ε 2 )dx1dx2dx3. it is easy to verify that φ is monotone and hemi-continuous (see for more details the reference [2,5,15] ). moreover, we know that the functional (vε1,v ε 2 ) ∈ w ε → g1ε ∫ ωε1 |d(vε1 )|dx1dx2dx3 + g2ε ∫ ωε2 |d(vε2 )|dx1dx2dx3, is proper and convex. then, the use of minty’s lemma permits us to affirm that (2.20) is equivalent to the following inequality int. j. anal. appl. (2023), 21:91 9 µ1ε p ∫ ωε1 |d(vε1 )| p−2 d(vε1 ).d(v ε 1 −u ε 1)dx1dx2dx3 + g1ε ∫ ωε1 |d(vε1 )|dx1dx2dx3 −g1ε ∫ ωε1 |d(uε1)|dx1dx2dx3 + µ2ε p ∫ ωε2 |d(vε2 )| p−2 d(vε2 ).d(v ε 2 −u ε 2)dx1dx2dx3 +g2ε ∫ ωε2 |d(vε2 )|dx1dx2dx3 −g2ε ∫ ωε1 |d(uε2)|dx1dx2dx3 ≥ ∫ ωε1 f ε1 .(v ε 1 −u ε 1)dx1dx2dx3 + ∫ ωε1 pε1 div(v1 −u ε 1)dx1dx2dx3 + ∫ ωε2 f ε2 .(v ε 2 −u ε 2)dx1dx2dx3 + ∫ ωε2 pε2 div(v2 −u ε 2)dx1dx2dx3, ∀(v ε 1,v ε 2 ) ∈ w ε. our goal now is to pass to the limit when ε tends to zero.to this aim, we use proposition (3.3) and the weak lower semi-continuity of the convex and continuous functional (vε1,v ε 2 ) ∈ w ε → g1ε ∫ ωε1 |d(vε1 )|dx1dx2dx3 + g2ε ∫ ωε2 |d(vε2 )|dx1dx2dx3, we find the following limit inequality µ1 ∫ ω1   12 ∣∣∣∂v̂11∂z ∣∣∣2 + 12 ∣∣∣∂v̂12∂z ∣∣∣2 + ∣∣∣∂v̂13∂z ∣∣∣2   p−2 2 × [ 1 2 ∂v̂11 ∂z ∂(v̂11−û11) ∂z + 1 2 ∂v̂12 ∂z ∂(v̂12−û12) ∂z +∂v̂13 ∂z ∂(v̂13−û13) ∂z ] dxdydz + + g1 ∫ ω1   12 ∣∣∣∂v̂11∂z ∣∣∣2 + 12 ∣∣∣∂v̂12∂z ∣∣∣2 + ∣∣∣∂v̂13∂z ∣∣∣2   1 2 dxdydz −g1 ∫ ω1   12 ∣∣∣∂û11∂z ∣∣∣2 + 12 ∣∣∣∂û12∂z ∣∣∣2 + ∣∣∣∂û13∂z ∣∣∣2   1 2 dxdydz +µ2 ∫ ω2   12 ∣∣∣∂v̂21∂z ∣∣∣2 + 12 ∣∣∣∂v̂22∂z ∣∣∣2 + ∣∣∣∂v̂23∂z ∣∣∣2   p−2 2 × [ 1 2 ∂v̂21 ∂z ∂(v̂21−û21) ∂z + 1 2 ∂v̂22 ∂z ∂(v̂22−û22) ∂z +∂v̂23 ∂z ∂(v̂23−û23) ∂z ] dxdydz +g2 ∫ ω2   12 ∣∣∣∂v̂21∂z ∣∣∣2 + 12 ∣∣∣∂v̂22∂z ∣∣∣2 + ∣∣∣∂v̂23∂z ∣∣∣2   1 2 dxdydz −g2 ∫ ω2   12 ∣∣∣∂û21∂z ∣∣∣2 + 12 ∣∣∣∂û22∂z ∣∣∣2 + ∣∣∣∂û23∂z ∣∣∣2   1 2 dxdydz ≥ ∫ ω1 f̂1.(v̂1 − û1)dxdydz + ∫ ω1 p̂1 div(v̂1 − û1)dxdydz + ∫ ω2 f̂2.(v̂2 − û2)dxdydz + + ∫ ω2 p̂2 div(v̂1 − û1)dxdydz. (3.24) furthermore, from (3.3) and (3.4) we find( ∂û13 ∂z , ∂û23 ∂z ) = (0, 0) in ω1 × ω2. it follows, keeping in mind (3.22), that û1(x,y,z) = (û11(x,y,z), û12(x,y,z), 0) , and û2(x,y,z) = (û21(x,y,z), û22(x,y,z), 0) . this permits also to choose (v̂13, v̂23) = (0, 0) in ( 3.24). 10 int. j. anal. appl. (2023), 21:91 considering now the operator φ such that φ : wz → w ′z , 〈φ (û1, û2) , (v̂1, v̂2)〉w ′z ×wz = µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂v̂1∂z dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂v̂2∂z dxdydz. it is clear that the operator φ is monotone and hemi-continuous and the functional (v̂1, v̂2) ∈ wz → √ 2 2 g1 ∫ ω1 ∣∣∣∣∂v̂1∂z ∣∣∣∣dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂v̂2∂z ∣∣∣∣dxdydz, is proper and convex. hence, we deduce using again minty’s lemma µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂(v̂1 − û1)∂z dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂v̂1∂z ∣∣∣∣dxdydz − √ 2 2 g1 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂(v̂2 − û2)∂z dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂v̂2∂z ∣∣∣∣dxdydz − √ 2 2 g2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣dxdydz ≥ ∫ ω1 ( f̂ 1 −∇p̂1 ) .(v̂1 − û1)dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .(v̂2 − û2)dxdydz ∀(v̂1, v̂2) ∈ wz . (3.25) setting (v̂1, v̂2) = (2û1, 2û2) and (v̂1, v̂2) = (0, 0) in (3.25), to obtain the following inequalities µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣dxdydz ≥ ∫ ω1 ( f̂ 1 −∇p̂1 ) .û1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .û2dxdydz, and − µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p dxdydz − √ 2 2 g1 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣dxdydz − µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p dxdydz − √ 2 2 g2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣dxdydz ≥ − ∫ ω1 ( f̂ 1 −∇p̂1 ) .û1dxdydz − ∫ ω2 ( f̂ 2 −∇p̂2 ) .û2dxdydz. int. j. anal. appl. (2023), 21:91 11 consequently, we get by combining these two inequalities µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣dxdydz = ∫ ω1 ( f̂ 1 −∇p̂1 ) .û1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .û2dxdydz. (3.26) due to the fact that w 1, p γi (ωi ) is dense in wz (ωi ) , see [1, 6], we can take w1 = v̂1 − û1 and w2 = v̂2 − û2 in (3.25) we obtain µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂w1∂z dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂w1∂z ∣∣∣∣dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂w2∂z dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂w2∂z ∣∣∣∣dxdydz ≥ ∫ ω1 ( f̂ 1 −∇p̂1 ) .w1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz ∀(w1,w2) ∈ w 1, p γ1 (ω1) 2 ×w 1, p γ2 (ω2) 2 . (3.27) changing (w1,w2) to (−w1,−w2) in (3.27), we obtain |f (w1,w2)| ≤ √ 2 2 g1 ∫ ω1 ∣∣∣∣∂w1∂z ∣∣∣∣dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂w2∂z ∣∣∣∣dxdydz , where f (w1,w2) = µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂w1∂z dxdydz + µ22 p2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂w2∂z dxdydz − ∫ ω1 ( f̂ 1 −∇p̂1 ) .w1dxdydz − ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz. (3.28) now, utilising the hahn-banach theorem, ∃(m1,m2) ∈ l∞ (ω1) 2 × l∞ (ω2)2 , with ‖|m1|‖∞ ,‖|m2|‖∞ ≤ 1, such that f ((w1,w2)) = − √ 2 2 g1 ∫ ω1 m1. ∂w1 ∂z dxdydz − √ 2 2 g2 ∫ ω2 m2. ∂w2 ∂z dxdydz. (3.29) in particular, from (3.26) and (3.28), we get ∫ ω1 m1. ∂u1 ∂z dxdydz + ∫ ω2 m2. ∂u2 ∂z dxdydz = ∫ ω1 ∣∣∣∣∂u1∂z ∣∣∣∣dxdydz + ∫ ω2 ∣∣∣∣∂u2∂z ∣∣∣∣dxdydz. (3.30) 12 int. j. anal. appl. (2023), 21:91 rewriting (3.29) as µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂w1∂z dxdydz + µ22 p2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂w2∂z dxdydz + √ 2 2 g1 ∫ ω1 m1. ∂w1 ∂z dxdydz + √ 2 2 g2 ∫ ω2 m2. ∂w2 ∂z dxdydz − ∫ ω1 ( f̂ 1 −∇p̂2 ) .w1dxdydz − ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz = 0. (3.31) next using (3.30), we have∫ ∣∣∣∂û1 ∂z ∣∣∣6=0 (∣∣∣∣∂û1∂z ∣∣∣∣−m1.∂û1∂z ) dxdydz + ∫ ∣∣∣∂û2 ∂z ∣∣∣6=0 (∣∣∣∣∂û2∂z ∣∣∣∣−m2.∂û2∂z ) dxdydz = 0. as ‖|m1|‖∞ ,‖|m2|‖∞ ≤ 1, we deduce∣∣∣∣∂û1∂z ∣∣∣∣ = m1.∂û1∂z and ∣∣∣∣∂û2∂z ∣∣∣∣ = m2.∂û2∂z . hence, if (∣∣∣∂û1∂z ∣∣∣ ,∣∣∣∂û2∂z ∣∣∣) 6= (0, 0) , we get ∫ ω1  µ1 2 p 2 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z + √ 2 2 g1 ∂û1 ∂z∣∣∣∂û1∂z ∣∣∣   .∂w1 ∂z dxdydz ∫ ω2  µ2 2 p 2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z + √ 2 2 g2 ∂û2 ∂z∣∣∣∂û2∂z ∣∣∣   .∂w2 ∂z dxdydz = ∫ ω1 ( f̂ 1 −∇p̂1 ) .w1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz, ∀(w1,w2) ∈ w 1, p γ1 (ω1) 2 ×w 1, p γ2 (ω2) 2 . consequenty, we get by using a simple integration by parts − ∫ ω1 ∂ ∂z  µ1 2 p 2 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z dxdydz + √ 2 2 g1 ∂û1 ∂z∣∣∣∂û1∂z ∣∣∣   .w1dxdydz − ∫ ω2 ∂ ∂z  µ2 2 p 2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z dxdydz + √ 2 2 g2 ∂û2 ∂z∣∣∣∂û2∂z ∣∣∣   .w2dxdydz = ∫ ω1 ( f̂ 1 −∇p̂1 ) .w1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz, ∀(w1,w2) ∈ w 1, p γ1 (ω1) 2 ×w 1, p γ2 (ω2) 2 . let us consider w ∈ w 1,p0 (ω) 2 : w = { w1 in ω1 w2 in ω2 , int. j. anal. appl. (2023), 21:91 13 and ã1 =   − ∂ ∂z ( µ1 2 p 2 ∣∣∣∂û1∂z ∣∣∣p−2 ∂û1∂z + √22 g1 ∂û1∂z∣∣∣∂û1 ∂z ∣∣∣ ) in ω1 0 in ω2 , ã2 =   0 in ω1 − ∂ ∂z ( µ2 2 p 2 ∣∣∣∂û2∂z ∣∣∣p−2 ∂û2∂z + √22 g2 ∂û2∂z∣∣∣∂û2 ∂z ∣∣∣ ) in ω2 , b̃1 = { f̂ 1 −∇p̂1 in ω1 0 in ω2 , b̃2 = { 0 in ω1 f̂ 2 −∇p̂2 in ω2 . then, ∫ ω (ã1 + ã2).wdxdydz = ∫ ω1 (ã1 + ã2).w1dxdydz + ∫ ω2 (ã1 + ã2).w2dxdydz, = ∫ ω1 ã1.w1dxdydz + ∫ ω2 ã2.w2dxdydz, = ∫ ω1 − ∂ ∂z  µ1 2 p 2 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z + √ 2 2 g1 ∂û1 ∂z∣∣∣∂û1∂z ∣∣∣   .w1dxdydz + ∫ ω2 − ∂ ∂z  µ2 2 p 2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z + √ 2 2 g2 ∂û2 ∂z∣∣∣∂û2∂z ∣∣∣   .w2dxdydz, = ∫ ω1 ( f̂ 1 −∇p̂1 ) .w1dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .w2dxdydz, = ∫ ω1 b̃1.w1dxdy dz + ∫ ω2 b̃2.w2dxdydz, = ∫ ω (b̃1 + b̃2).wdxdydz ∀w ∈ w 1,p 0 (ω) 2. which eventually gives (3.23). from now on we will denote by (û1, û2) ∈ wz and (p̂1, p̂2) ∈ l p′ 0 (ω1)×l p′ 0 (ω2) the solution of the limit problem (3.23). � the following proposition shows the uniqueness of the limit solution (û1, p̂1) and (û2, p̂2). proposition 3.4. the limit strong problem (3.23) has a unique, solution (û1, û2) ∈ wz and (p̂1, p̂2) ∈ l p′ 0 (ω1) ×l p′ 0 (ω2) with the condition ( 3.22) . proof. suppose that the limit problem (3.23) has at least two solutions (û1, û2) ∈ wz, (p̂1, p̂2) ∈ l p′ 0 (ω1) ×l p′ 0 (ω2) and ( û1, û2 ) ∈ wz, ( p̂1, p̂2 ) ∈ lp ′ 0 (ω1) ×l p′ 0 (ω2). in particular, (û1, p̂1), (û2, p̂2) 14 int. j. anal. appl. (2023), 21:91 and ( û1, p̂1 ) , ( û2, p̂2 ) are solutions of the weak formulation ( 3.25). then µ1 2 p 2 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z .∂(v̂1 − û1)∂z dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂v̂1∂z ∣∣∣∣dxdydz − √ 2 2 g1 ∫ ω1 ∣∣∣∣∂û1∂z ∣∣∣∣dxdydz + µ2 2 p 2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣p−2 ∂û2∂z .∂(v̂2 − û2)∂z dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂v̂2∂z ∣∣∣∣dxdydz − √ 2 2 g2 ∫ ω2 ∣∣∣∣∂û2∂z ∣∣∣∣dxdydz ≥ ∫ ω1 ( f̂ 1 −∇p̂1 ) .(v̂1 − û1)dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .(v̂2 − û2)dxdydz (v̂1, v̂2) ∈ wz , (3.32) and µ1 2 p 2 ∫ ω1 ∣∣∣∣∣∂û1∂z ∣∣∣∣∣ p−2 ∂û1 ∂z . ∂(v̂1 − û1) ∂z dxdydz + √ 2 2 g1 ∫ ω1 ∣∣∣∣∂v̂1∂z ∣∣∣∣dxdydz − √ 2 2 g1 ∫ ω1 ∣∣∣∣∣∂û1∂z ∣∣∣∣∣dxdydz + µ22 p2 ∫ ω2 ∣∣∣∣∣∂û2∂z ∣∣∣∣∣ p−2 ∂û2 ∂z . ∂(v̂2 − û2) ∂z dxdydz + √ 2 2 g2 ∫ ω2 ∣∣∣∣∂v̂2∂z ∣∣∣∣dxdydz − √ 2 2 g2 ∫ ω2 ∣∣∣∣∣∂û2∂z ∣∣∣∣∣dxdydz ≥ ∫ ω1 ( f̂ 1 −∇p̂1 ) .(v̂1 − û1)dxdydz + ∫ ω2 ( f̂ 2 −∇p̂2 ) .(v̂2 − û2)dxdydz (v̂1, v̂2) ∈ wz. (3.33) setting (v̂1, v̂2) = ( û1, û2 ) and (v̂1, v̂2) = (û1, û2) as test functions in (3.32) and (3.33), respectively. subtracting the tow obtained inequalities, we can infer µ 2 p 2 ∫ ω1  ∣∣∣∣∣∂û1∂z ∣∣∣∣∣ p−2 ∂û1 ∂z − ∣∣∣∣∂û1∂z ∣∣∣∣p−2 ∂û1∂z   .∂(û1 − û1) ∂z dxdydz + µ2 2 p 2 ∫ ω2  ∣∣∣∣∣∂û2∂y ∣∣∣∣∣ p−2 ∂û2 ∂y − ∣∣∣∣∂û2∂y ∣∣∣∣p−2 ∂û2∂y   .∂(û2 − û2) ∂y dxdydz ≤ ∫ ω1 ∇ ( p̂1 − p̂1 ) . ( û1 − û1 ) dxdydz + ∫ ω2 ∇ ( p̂2 − p̂2 ) . ( û2 − û2 ) dxdydz. (3.34) observe that for every x,y ∈rn, ( |x|p−2 x −|y|p−2 y ) .(x −y) ≥ (p− 1) |x −y|2 (|x| + |y|)2−p , 1 < p ≤ 2. int. j. anal. appl. (2023), 21:91 15 this leads, making use (3.34), to µ1 (p− 1) 2 p 2 ∫ ω1 (∣∣∣∣∣∂û1∂z ∣∣∣∣∣ + ∣∣∣∣∂û1∂z ∣∣∣∣ )p−2 ∣∣∣∣∣∂(û1 − û1)∂z ∣∣∣∣∣ 2 dxdydz + µ2 (p− 1) 2 p 2 ∫ ω2 (∣∣∣∣∣∂û2∂y ∣∣∣∣∣ + ∣∣∣∣∂û2∂y ∣∣∣∣ )p−2 ∣∣∣∣∣∂(û2 − û2)∂z ∣∣∣∣∣ 2 dxdydz ≤ ∫ ω (( p̂1 − p̂1 )∫ h(x,y) 0 ( ∂(û11 − û11) ∂x + ∂(û12 − û12) ∂y ) dz ) dxdy + ∫ ω (( p̂2 − p̂2 )∫ 0 −h(x,y) ( ∂(û21 − û21) ∂x + ∂(û22 − û22) ∂y ) dz ) dxdy. this use of (3.22) gives µ1 (p− 1) 2 p 2 ∫ ω1 (∣∣∣∣∣∂û1∂z ∣∣∣∣∣ + ∣∣∣∣∂û1∂z ∣∣∣∣ )p−2 ∣∣∣∣∣∂(û1 − û1)∂z ∣∣∣∣∣ 2 dxdydz + µ2 (p− 1) 2 p 2 ∫ ω2 (∣∣∣∣∣∂û2∂y ∣∣∣∣∣ + ∣∣∣∣∂û2∂y ∣∣∣∣ )p−2 ∣∣∣∣∣∂(û2 − û2)∂z ∣∣∣∣∣ 2 dxdydz = 0. (3.35) on the other hand, the application of hölder’s inequality leads to∫ ω1 ∣∣∣∣∣∂(û1 − û1)∂z ∣∣∣∣∣ p dxdydz + ∫ ω2 ∣∣∣∣∣∂(û2 − û2)∂z ∣∣∣∣∣ p dxdydz ≤ c  ∫ ω1 ∣∣∣∂(û1−û1)∂z ∣∣∣2(∣∣∣∂û1∂z ∣∣∣ + ∣∣∣∂û1∂z ∣∣∣)2−pdxdydz   p 2 × (∫ ω1 (∣∣∣∣∣∂û1∂z ∣∣∣∣∣ + ∣∣∣∣∣∂û1∂z ∣∣∣∣∣ )p dxdydz )2−p 2 +c  ∫ ω2 ∣∣∣∂(û2−û2)∂z ∣∣∣2(∣∣∣∂û2∂z ∣∣∣ + ∣∣∣∂û2∂z ∣∣∣)2−pdxdydz   p 2 × (∫ ω2 (∣∣∣∣∣∂û2∂z ∣∣∣∣∣ + ∣∣∣∣∂û2∂z ∣∣∣∣ )p dxdydz )2−p 2 . which gives, keeping in mind (3.35)∥∥∥∥∥∂(û1 − û1)∂z ∥∥∥∥∥ lp(ω1) 2 = 0 and ∥∥∥∥∥∂(û2 − û2)∂z ∥∥∥∥∥ lp(ω2) 2 = 0, using poincare’s inequality, we deduce∥∥∥û1 − û1∥∥∥ lp(ω1) 2 = 0 and ∥∥∥û2 − û2∥∥∥ lp(ω2) 2 = 0, we deduce that (û1, û2) = ( û1, û2 ) a.e. in ω1 × ω2.. finally, to prove the uniqueness of the pressure, we use equation (3.23 ), with the two pressures (p̂1, p̂1) and (p̂2, p̂2). we find ∇(p̂1 − p̂1) = 0 and ∇(p̂2 − p̂2) = 0. 16 int. j. anal. appl. (2023), 21:91 then, due to fact that ( p̂1, p̂1 ) ∈ ( l p′ 0 (ω1) )2 and ( p̂2, p̂2 ) ∈ ( l p′ 0 (ω2) )2 , the result can be easily deduced. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] f. boughanim, m. boukrouche, h. smaoui, asymptotic behavior of a non-newtonian flow with stick-slip condition, 2004-fez conference on differential equations and mechanics, electron. j. differ. equ. conference 11 (2004), 71-80. [2] a. bourgeat, a. mikelic, r. tapiéro, dérivation des equations moyennées décrivant un ecoulement non newtonien dans un domaine de faible epaisseur, c. r. acad. sci. paris, ser. i 316 (1993), 965-970. [3] h. brezis, equations et inéquations non linéaires dans les espaces en dualité, ann. l’inst. fourier, 18 (1996), 115-175. [4] r. bunoiu, s. kesavan, asymptotic behaviour of a bingham fluid in thin layers, j. math. anal. appl. 293 (2004), 405-418. https://doi.org/10.1016/j.jmaa.2003.10.049. [5] r. bunoin, s. kesavan, fluide de bingham dans une couche mince, ann. univ. craiova, math. comp. sci. ser. 30 (2003), 71-77. [6] r. bunoin, j.s.j. paulin, nonlinear viscous flow through a thin slab in the lubrification case, rev. roum. math. pures appl. 45 (2000), 577-591. [7] g. duvaut, j.l. lions, les inéquations en mécanique et en physique, dunod, 1976. [8] i. ekeland, r. temam, analyse convexe et problèmes variationnels, dunod, paris, 1974. [9] j.l. lions, quelques méthodes de résolution des problèmes aux limites non linéaires, dunod, 1996. [10] k.f. liu, c.c. mei, approximate equations for the slow spreading of a thin sheet of bingham plastic fluid, phys. fluids a: fluid dyn. 2 (1990), 30-36. https://doi.org/10.1063/1.857821. [11] j. málek, mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations, electron. trans. numer. anal. 31 (2008), 110-125. [12] j. málek, m. růžička, v.v. shelukhin, herschel-bulkley fluids, existence and regularity of steady flows, math. models methods appl. sci. 15 (2005), 1845-1861. https://doi.org/10.1142/s0218202505000996. [13] f. messelmi, b. merouani, flow of herschel-bulkley fluid through a two dimensional thin layer, stud. univ. babeş-bolyai math. 58 (2013), 119-130. [14] f. messelmi, effects of the yield limit on the behaviour of herschel-bulkley fluid, nonlinear sci. lett. a, 2 (2011), 137-142. [15] f. messelmi, b. merouani, f. bouzeghaya, steady-state thermal herschel-bulkley flow with tresca’s friction law, electron. j. differ. equ. 2010 (2010), 46. [16] a. mikelic, r. tapiéro, mathematical derivation of the power law describing polymer flow through a thin slab, math. model. numer. anal. 29 (1995), 3-22. https://doi.org/10.1016/j.jmaa.2003.10.049 https://doi.org/10.1063/1.857821 https://doi.org/10.1142/s0218202505000996 1. introduction 2. problem statement 3. asymptotic behavior references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 157-162 http://www.etamaths.com application of hypergeometric distribution series on certain subclass of analytic functions trailokya panigrahi∗ abstract. the object of the present paper is to give some characterizations for hypergeometric distribution series to be in various subclasses of analytic functions. 1. introduction let a denote the family of all functions f analytic in u := {z ∈ c : |z| < 1} with the usual normalization condition f(0) = f′(0) − 1 = 0. thus f has the following taylor-maclaurin series: (1) f(z) = z + ∞∑ l=2 alz l. let s be the subclass of a consisting of all functions f of the form (1) which are univalent in u. a function f ∈a is said to be in k−ucv, the class of k-uniformly convex function (0 ≤ k < ∞) if f ∈s along with the property that for every circular arc γ contained in u with center ξ where |ξ| < k, the image curve f(γ) is a convex arc. it is well-known that [5] f ∈ k−ucv if and only if the image of the function p, where p(z) = 1 + zf′′(z) f′(z) (z ∈ u) is a subset of the conic region (2) ωk = {w = u + iv : u2 > k2(u− 1)2 + k2v2, 0 ≤ k < ∞}. the class k−st consisting of k-uniformly starlike functions is defined via k−ucv by the alexander transform i.e. f ∈ k −st ⇐⇒ g ∈ k −ucv where g(z) = ∫ z 0 f(t) t dt. the class k−st and its properties were investigated in [6]. the analytic characterization of k−ucv and k −st are given as below: (3) k −ucv = {f ∈a : < ( 1 + zf′′(z) f′(z) ) > k ∣∣∣∣zf′′(z)f′(z) ∣∣∣∣ (z ∈ u)} and (4) k −st = {f ∈a : < ( zf′(z) f(z) ) > k ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ (z ∈ u)} note that for k = 0 and k = 1, we get 0−ucv = k, 0−st = s∗, 1−ucv = ucv and 1−st = sp, where k, s∗, ucv, sp are respectively the familiar classes of univalent convex functions, univalent starlike functions [3], uniformly convex functions [4] (also, see [7, 12]) and parabolic starlike functions [12]. for two analytic functions f and g in u, the function f is said to be subordinate to g or g is said to be superordinate to f, if there exists a function w analytic in u with |w| ≤ |z| such that f(z) = g(w(z)). in such case, we write f ≺ g or f(z) ≺ g(z). if the function g is univalent in u, then f ≺ g ⇐⇒ f(0) = g(0) and f(u) ⊂ g(u) (see, for detail [8]). 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. analytic functions; k-uniformly convex functions; k-uniformly starlike functions; hypergeometric distribution series. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 157 158 panigrahi making use of subordination between analytic functions, aouf [1] introduced and studied the class rλ(a,b,α) as follows: definition 1.(see[1, with p=1]) for −1 ≤ a < b ≤ 1, |λ| < π 2 and 0 ≤ α < 1, we say that a function f(z) ∈a is in the class rλ(a,b,α) if it satisfies the following subordination condition: (5) eiλf′(z) ≺ cosλ [ (1 −α) 1 + az 1 + bz + α ] + isinλ. the subordination (5) is equivalent to the inequality (6) given below: (6) ∣∣∣∣ eiλ(f′(z) − 1)beiλf′(z) − [beiλ + (a−b)(1 −α)cosλ] ∣∣∣∣ < 1 (z ∈ u). for particular values of parameters a,b,α and λ, we obtain various subclasses of analytic functions studied by different researchers (for details, see [2]). in 1998, ponnusamy and ronning [10] introduced and studied the classes s∗β and cβ consisting of functions of the form (1) satisfying the following conditions: (7) s∗β = { f ∈a : ∣∣∣∣zf′(z)f(z) − 1 ∣∣∣∣ < β (z ∈ u, β > 0)}, and (8) cβ = { f ∈a : ∣∣∣∣zf′′(z)f′(z) ∣∣∣∣ < β (z ∈ u, β > 0)}. it is worthy to mention here that f ∈cβ ⇐⇒ zf′ ∈s∗β (β > 0). recently, we introduced a new series h(m,n,n; z) whose coefficient are probabilities of hypergeometric distribution as follows: (9) h(m,n,n; z) = z + 1( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) zl. let us define the linear operator j(m,n,n) : a−→a given by (10) j(m,n,n)f(z) = h(m,n,n; z) ? f(z) = z + 1( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) alz l (z ∈ u), where ? denote the convolution or hadamard product between two analytic functions. motivated by the works of [9, 10, 13], in this paper we investigate some characterization for hypergeometric distribution series to be in the subclasses s∗β and cβ of analytic functions. 2. preliminaries lemmas to prove our main results, we need the following lemmas. lemma 1. (see [1], theorem 4 with p=1) a sufficient condition for f(z) defined by (1) to be in the class rλ(a,b,α) is (11) ∞∑ l=2 l(1 + |b|)|al| ≤ (b −a)(1 −α)cosλ. lemma 2. (see [6]) let f(z) ∈a. if for some k, the following inequality (12) ∞∑ l=2 (l + k(l− 1))|al| ≤ 1 holds true, then f ∈ k −st . lemma 3. (see [5, 11]) a function f ∈a of the form (1) is in k −ucv if it satisfies the condition (13) ∞∑ l=2 l[l(k + 1) −k]|al| ≤ 1. application of hypergeometric distribution... 159 another sufficient condition for the class k −ucv is given in [7] as follows: lemma 4. (see [7, 11]) let f ∈s be of the form (1). if for some k (0 ≤ k < ∞), the inequality (14) ∞∑ l=2 l(l− 1)|al| ≤ 1 k + 2 , holds true, then f ∈ k −ucv. the number 1 k+2 cannot be increased. lemma 5. (see [11]) let f ∈a be of the form (1). if the inequality (15) ∞∑ l=2 [β + l− 1]|al| ≤ β (β > 0), is satisfied, then f ∈s∗β. lemma 6. (see [11]) let f ∈a be of the form (1). if (16) ∞∑ l=2 l[β + l− 1]|al| ≤ β (β > 0), then f ∈ cβ. lemma 7. (see [1], theorem 1 with p=1) let the function f(z) defined by (1) be in the class rλ(a,b,α), then (17) |al| ≤ (b −a)(1 −α)cosλ l (l ≥ 2). 3. main results unless otherwise stated, we assume throughout the sequel that −1 ≤ a < b ≤ 1, |λ| < π 2 , 0 ≤ α < 1. theorem 1. let k ≥ 0. if the inequality (18) 1( n n ) [m(k + 1)a1 −(n −m n )] ≤ secλ (b −a)(1 −α) − 1, where (19) a1 = ∞∑ l=2 ( m − 1 l− 2 )( n −m n− l + 1 ) is satisfied, then j(m,n,n) maps the class rλ(a,b,α) into k −ucv. proof. let the function f given by (1) be a member of rλ(a,b,α). by (10), we have j(m,n,n)f(z) = z + 1( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) alz l. in view of lemma 3, it is sufficient to show that 1( n n ) ∞∑ l=2 l[l(k + 1) −k] ( m l− 1 )( n −m n− l + 1 ) |al| ≤ 1. by making use of lemma 7, it is again sufficient to prove that (20) p1 = 1( n n ) ∞∑ l=2 [l(k + 1) −k] ( m l− 1 )( n −m n− l + 1 ) ≤ secλ (b −a)(1 −α) . 160 panigrahi now p1 = 1( n n ) ∞∑ l=2 [(l− 1)(k + 1) + 1] ( m l− 1 )( n −m n− l + 1 ) = 1( n n ) [ ∞∑ l=2 (k + 1) m! (l− 2)!(m − l + 1)! ( n −m n− l + 1 ) + ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 )] = m(k + 1)( n n ) ∞∑ l=2 ( m − 1 l− 2 )( n −m n− l + 1 ) + 1( n n ) [ ∞∑ l=0 ( m l )( n −m n− l ) − ( n −m n )] = m(k + 1)( n n ) a1 − ( n−m n )( n n ) + 1, where a1 is defined as in (19). thus, in view of (20), if the inequality (18) is satisfied, then j(m,n,n)(f) ∈ k −ucv as asserted. the proof of theorem 1 is complete. � theorem 2. if the inequality (21) m( n n )a1 ≤ secλ (k + 2)(b −a(1 −α) is satisfied, then j(m,n,n) maps the class rλ(a,b,α) into k −ucv. proof. let the function f given by (1) be a member of rλ(a,b,α). by virtue of lemma 4, it is sufficient to show that 1( n n ) ∞∑ l=2 l(l− 1) ( m l− 1 )( n −m n− l + 1 ) |al| ≤ 1 k + 2 using the coefficient estimate (17), it is again sufficient to show that (22) p2 = 1( n n ) ∞∑ l=2 (l− 1) ( m l− 1 )( n −m n− l + 1 ) ≤ secλ (k + 2)(b −a)(1 −α) . now, p2 = m( n n ) ∞∑ l=2 ( m − 1 l− 2 )( n −m n− l + 1 ) = m( n n )a1. in view of (22), if the condition (21) is satisfied, then j(m,n,n)(f) ∈ k−ucv as asserted. this ends the proof of theorem 2. � theorem 3. if the inequality (23) (1 + k) − (1 + k)( n n ) (n −m n ) − k( n n ) (m + 1) b1 ≤ secλ (b −a)(1 −α) , where (24) b1 = ∞∑ l=2 ( m + 1 l )( n −m n− l + 1 ) is satisfied, then j(m,n,n) maps the class rλ(a,b,α) into k −st . proof. let the function f given by (1) be a member of rλ(a,b,α). by virtue of lemma 2, it is sufficient to show that 1( n n ) ∞∑ l=2 [l + k(l− 1)] ( m l− 1 )( n −m n− l + 1 ) |al| ≤ 1. using the coefficient estimate (17), it is again sufficient to show that (25) p3 = 1( n n ) ∞∑ l=2 [l + k(l− 1)] l ( m l− 1 )( n −m n− l + 1 ) ≤ secλ (b −a)(1 −α) application of hypergeometric distribution... 161 now, p3 = 1( n n ) ∞∑ l=2 [ 1 + (1 − 1 l )k ]( m l− 1 )( n −m n− l + 1 ) = 1( n n ) ∞∑ l=2 [ (1 + k) − k l ]( m l− 1 )( n −m n− l + 1 ) = (1 + k) [ 1 − ( n−m n )( n n ) ] − k (m + 1) ( n n )b1. therefore, in view of (25), if the inequality (23) is satisfied, then j(m,n,n)(f) ∈ k−st as asserted. this complete the proof of theorem 3. � theorem 4. if f ∈rλ(a,b,α) and the inequality (26) 1 − ( n−m n )( n n ) ≤ 1 1 + |b| , is satisfied, then j(m,n,n)(f) ∈rλ(a,b,α). proof. let the function f ∈ a given by (1) be a member of rλ(a,b,α). by virtue of lemma 1 and the coefficient inequality (17) it is sufficient to show that (27) p4 = 1( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) ≤ 1 1 + |b| . now p4 is equivalently written as p4 = ∞∑ l=1 ( m l )( n−m n−l )( n n ) = 1 − (n−mn )( n n ) thus, in view of (27), if the inequality (26) is satisfied, then j(m,n,n)(f) ∈rλ(a,b,α). the proof of theorem 4 is complete. � theorem 5. let β > 0, f ∈rλ(a,b,α) and the inequality (28) β − 1 (m + 1) ( n n )b1 − ( n−m n )( n n ) ≤ βsecλ (b −a)(1 −α) − 1, is satisfied, then j(m,n,n)(f) ∈s∗β . proof. by making use of lemma 5, it is sufficient to show that ∞∑ l=2 (β + l− 1) ( m l−1 )( n−m n−l+1 )( n n ) |al| ≤ β. since f ∈rλ(a,b,α), using the coefficient estimate (17), it is sufficient to show that (29) p5 = 1( n n ) ∞∑ l=2 [ β + l− 1 l ]( m l− 1 )( n −m n− l + 1 ) ≤ βsecλ (b −a)(1 −α) . now, p5 = 1( n n ) ∞∑ l=2 ( β − 1 l )( m l− 1 )( n −m n− l + 1 ) + 1( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) = β − 1 (m + 1) ( n n )b1 − ( n−m n )( n n ) + 1. thus, in view of (29), if the inequality (28) is satisfied, then j(m,n,n)(f) ∈ s∗β as asserted. this proof the theorem 5. � 162 panigrahi theorem 6. let β > 0. if the inequality (30) 1( n n ) [ma1 −β(n −m n )] ≤ β [ secλ (b −a)(1 −α) − 1 ] is satisfied, then j(m,n,n) maps the class rλ(a,b,α) into cβ. proof. in view of lemma 6, it is sufficient to show that 1( n n ) ∞∑ l=2 l[β + l− 1] ( m l− 1 )( n −m n− l + 1 ) |al| ≤ β. using coefficient inequality (17), it is enough to show that (31) p6 = 1( n n ) ∞∑ l=2 [β + l− 1] ( m l− 1 )( n −m n− l + 1 ) ≤ βsecλ (b −a)(1 −α) . now the expression p4 can be equivalently written as p6 = β( n n ) ∞∑ l=2 ( m l− 1 )( n −m n− l + 1 ) + ∞∑ l=2 m! (l− 2)!(m − l + 1)! ( n −m n− l + 1 ) = β −β ( n−n n )( n n ) + m( n n ) ∞∑ l=2 ( m − 1 l− 2 )( n −m n− l + 1 ) = m( n n )a1 − β ( n−m n )( n n ) + β. thus, in view of (31) if the inequality (30) is satisfied, then j(m,n,n)(f) ∈cβ as desired. the proof of theorem 6 is thus completed. � references [1] m. k. aouf, on certain subclass of analytic p-valent functions of order alpha, rend. mat., 7(8) (1988), 89-104. [2] m. k. aouf, a. o. mostafa and h. m. zayed, some constraints of hypergeometric functions to belong to certain subclasses of analytic functions, j. egyptian math. soc., 24(3) (2016), 361-366. [3] p. l. duren, univalent functions, grundlehren der mathematischen wissenschaften, vol 259, springer-verlag, new york, 1983. [4] a. w. goodman, on uniformly convex functions, ann. polon. math, 56(1)(1991), 87-92. [5] s. kanas and a. wisnioska, conic regions and k-uniform convexity, j. comput. appl. math., 105(1999), 327-336. [6] s. kanas and a. wisnioska, conic domains and starlike functions, rev. roumaine math. pures appl., 45(4) (2000), 647-657. [7] w. ma and d. minda, uniformly convex functions, ann. polon. math., 57(2) (1992), 165-175. [8] s. s. miller and p. t. mocanu, differential subordinations: theory and applications, in: monographs and text books in pure and applied mathematics, 225, marcel dekker, new york, 2000. [9] a. k. mishra and t. panigrahi, class-mapping properties of the hohlov operator, bull. korean math. soc., 48(1) (2011), 51-65. [10] s. ponnusamy and f. ronning, starlikeness properties for convolution involving hypergeometric series, ann. univ. mariae curie-sklodowska i, ii 1(16) (1998), 141-155. [11] s. porwal and s. kumar, confluent hypergeometric distribution and its applications on certain classes of univalent functions, afr. mat., (2016), doi:10.1007/s13370-016-0422-3. [12] f. ronning, uniformly convex functions and a corresponding class of starlike functions, proc. amer. math. soc., 118(1) (1993), 189-196. [13] a. k. sharma, s. porwal and k. k. dixit , class-mapping properties of convolution involving certain univalent function associated with hypergeometric function, electronic j. math. anal. appl., 1(2) (2013), 326-333. department of mathematics, school of applied sciences, kiit university, bhubaneswar-751024, orissa, india ∗corresponding author: trailokyap6@gmail.com int. j. anal. appl. (2023), 21:40 random and fixed effects selection for weighted ridge lulah alnaji∗ department of mathematics, college of sciences, university of hafr al batin, saudi arabia ∗corresponding author: laalnaji@uhb.edu.sa abstract. using penalized profiled log-likelihood and penalized limited profiled log-likelihood, respectively, together with the weighted ridge penalized term, we offer a method in this study for choosing the fixed and random effects in linear mixed models. then, we use the penalized restricted profiled log-likelihood to perform in the random effects depending on the chosen tuning parameter. second, we use the penalized profiled log-likelihood to choose the fixed effect parameters. there is no closedform solution for the choice of the fixed and random effects, hence the newton-raphson technique is employed to iteratively estimate the parameters. we use a simulation study to show how well the suggested strategy works. lastly, we use two separate datasets to use the methods to further evaluate the newly proposed model. 1. introduction with longitudinal data, each individual is followed repeatedly across various times in time. thus, the independence assumption is not optimal because of the associated observations of each participant. the linear mixed model is a common option for longitudinal research and is a helpful tool since it incorporates the random effects to account for the within-subjects correlation. [1]. due to the fixed and random effect parameters’ increased dimension during the past 20 years, there have been certain challenges. the issue of variable selection has been researched, and many different approaches have been put forth in order to narrow the selection of parameters to those that are most crucial, such as ridge regression [2], lasso method [3], adaptive lasso [4], elastic net [5] and scad [6] among many others. selection criteria like akaike information criterion (aic [7]) and bayesian information criterion (bic [8]) have been used for variable selection and proved to provide a consistent selection model rules. received: mar. 4, 2023. 2020 mathematics subject classification. 62j05, 62j07. key words and phrases. random effect; fixed effects; akaike information criterion; variable selection; ridge. https://doi.org/10.28924/2291-8639-21-2023-40 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-40 2 int. j. anal. appl. (2023), 21:40 in particular, bic is asymptotically consistent for model selection. many research have extensively explored and used the asymptotic qualities when the number of possible aggressors is fixed [9]. fixed effects selection in the linear mixed-effects model using adaptive ridge process for l0 penalty performance is one study that has focused a lot of emphasis on one-stage variable selection using penalized log-likelihood approach for the fixed effects [10]; and model selection in linear mixed effect m els [11]. although the methods listed above are highly helpful, the fixed and random effects’ fundamental features are quite different, therefore the methods listed above might not reveal these differences. adaptive lasso has been the subject of various studies for variable selection methods such as [12], they investigated pathwise coordinate optimization and fixed and random effects selection via reml. also, profile log-likelihood-based adaptive lasso for linear mixed model selection [13]. they discuss selecting the fixed and random effects using ml and reml procedures. for weighted ridge, [14] studied a weighted ridge procedure for l0 regularization in fixed models. the linear mixed-effects model was used to study the selection of fixed effects, and the fixed effects, random effects, and variance components were estimated using the weighted ridge approach for l0 penalty performance [10]. we provide a model selection process for a weighted ridge in mixed models for both random and fixed effects, respectively, to further enhance the behavior of the current penalized techniques. the remaining sections of this article are structured as follows. section 2 presents variable selection for the linear mixed-effects model. the methodology of the weighted ridge is presented in section 3. section 4 presents simulation studies and section 5 presents the important conclusions from this study. 2. variable selection for the linear mixed-effects model maximum likelihood (ml) and restricted maximum likelihood (reml) are major methods have been proposed to estimate the parameters in 2.1 when assuming that λj and �j are normally distributed. see random-effects models for longitudinal data [15], unbalanced repeated-measures models with structured covariance matrices [16], and newton-raphson and em algorithms for linear mixed-effects models for repeated-measures data [17]. in this article, restricted maximum likelihood (reml) and maximum likelihood (ml) are used to select the random effects and the fixed effects, respectively. 2.1. literature review of classical linear mixed-effect mode. in this section, we consider the classical linear mixed-effect model setting to establish a selection method for weighted ridge mixed model. yj = xjψ +zjλj + �j, j =1, · · · ,m, (2.1) int. j. anal. appl. (2023), 21:40 3 where yj is the mj ×1 response vector for the observations of subject j, ψ ∈ rr is the fixed effects vector corresponding to its mj × r ull rank design matrix xj, λj ∈ rl is the random effects vector corresponding to its mj × l design matrix zj and �j is the mj ×1 vector of the model errors. denote the total of observations m = ∑m j=1mj. assume that y1, · · · ,ym are independent and that �j and λj are independent with �j ∼ n(0,σ2imj), λj ∼ mv n(0,g), hence yj ∼ n(xjψ,σ 2dj(α)) where dj(α)= zjgz ′ j + imjσ 2 and α is the k =1/2∗ (l(l +1)) vector that consists of the unknown covariance parameters which characterizes the matrix g. the model 2.1 is a classical model and the estimation of the fixed and random effect parameters can be done by using the well-known methods, unbiased estimation(blue) and best linear unbiased prediction (blup), respectively, using maximum likelihood (ml) approach [18]. 2.2. selection of random effects parameters. due to its unpredictable nature, which presents greater difficulties in estimating the variance-covariance matrix’s structure, random effects selection hasn’t garnered as much attention as fixed effects selection, as was previously indicated. although the parameters for the fixed effects are not very sensitive to the choice of random effects, choosing random effects incorrectly can have an impact on how effectively the fixed effects are estimated. to implement the selection of the random effects by selecting a parameter α that maximizes the penalized restricted profiled log-likelihood. consider the weighted ridge [14] for the penalty term tran(α)=− 1 2 log ∣∣∣∣ m∑ j=1 x ′ jdjxj ∣∣∣∣− 12 m∑ j=1 log|dj| − 1 2 m × log ( m∑ j=1 e ′ d−1 j e ) + 1 2 r × log ( m∑ j=1 e ′ d−1 j e) ) −γ1m l∑ j=1 w1jg 2 j (2.2) where γ1m ≥ 0 is the tuning parameter (also called the regularization parameter), w1 = diag(w11, · · · ,w1l) is l × l diagonal matrix and w1j is the j-th element of w1’s diagonal (see section 3), e =(yj −xjψ̂) and gj is the j-th element of the diagonal of the matrix g. the primary interest lies on the selection of ψ and α. the proposed selection variable method has no closed-form solution and can be solved iteratively. for example, newton-raphson is a popular iterative method to be used. the first and the second derivatives of newton-raphson method can be found in details in [16]. 4 int. j. anal. appl. (2023), 21:40 2.3. selection of fixed effects parameters. to select the important covariates of the fixed effects, we propose to maximize the penalized profiled log-likelihood tf ix(ψ)=− 1 2 m∑ j=1 log|dj| − 1 2 m × log ( m∑ j=1 (yj −xjψ) ′ d−1 j (yj −xjψ) ) −γ2n r∑ j=1 w2jψ 2 j (2.3) where the tuning parameter (also called the regularization parameter) is γ2n ≥ 0, w2 = diag(w21, · · · ,w2q) is l × l diagonal matrix and w2j is the j-th element of w2’s diagonal (see section 3). note that at this step, the penalized profiled log-likelihood 2.3 is a function of ψ only, since the matrix dj always selected by the first stage using the penalized restricted profiled log-likelihood 2.2. after selecting the matrix dj in the first stage by maximizing the penalized restricted profiled log-likelihood 2.2, in the second stage the primary interest lies in the selection of the covariates of ψ. similarly, the proposed selection variable method has no closed-form solution and can be solved iteratively by the newton-raphson method of the form ψt+1 = ψt −h−1sψ, t =0,1,2 · · · (2.4) where ψt is the current result of 2.3, ψt+1 is the updated one, sψ is the score vector and hψψ is the hessian matrix. 3. the methodology of the weighted ridge [14] proposed a weighted ridge strategy that improves the performance of the l0-penalty for fixed effect models, motivated by least absolute shrinkage is equivalent to quadratic penalization [19]; onestep sparse estimates in non-concave penalized likelihood models [20] and visualization of genomics changes by segmented smoothing using an l0 penalty [21]. in (2017), [14] has been extended to a linear mixed effects model by [10]. they proposed a selection strategy for the fixed effects while the covariance matrix for the random effects is cholesky factorized. in this article, we proposed a selection approach for both the fixed and random effects using an iteratively weighted ridge strategy following the procedure of the weight matrix in [14] consider the objective function 2.2 where the weight matrices for for w1j and w2j are calculated by wt1j = 1 |α(t) j |2 +δ2 (3.1) and int. j. anal. appl. (2023), 21:40 5 wt2j = 1 |ψ(t) j |2 +δ2 (3.2) where w1j is the j-th element of diag(w1), αj is the j-th element of the diagonal of the selected covariance matrix in the first stage, t is the number of iteration, δ is a constant, w2j is the j-th element of diag(w2) and ψj is the j-th element of the vector of the fixed effect parameters. in numerical practices small positive choices for δ seem to perform better than δ =0 (for further details see [10], [22], [14], [21]). although, the selection of the tuning parameter γ would add more computational work to search among lattice of γ’s but it is an important step. the selection of the tuning parameter is an influential part of penalized methods. selection criteria like akaike information criterion (aic [7]), bayesian information criterion (bic [8]) and generalized cross validation (gcv [23]) have been used for variable selection. for compression we employ the three criteria for the fixed and random effect parameters. 4. simulation studies a simulation study is conducted in order to examine the asymptotic properties and the performance of our newly developed method. all of the simulated data are generated according to the model 2.1 using r statistical software. following the examples in [24], the simulation study assumes repeatedly observation per each subject. (i) assume mj =5 per each subject j with m =30 and consider the true fixed effects vector to be ψ =(1,1,0,0,0,0,0,0,0) with r =9. we further consider l =4 for random effects with the assumption of normal distribution for the error of the model and the random effects as, �j ∼ n(0,σ2i) and λj = (λj0,0) ′ with rj0 ∼ n(0,g) where σ2 = 1 and the true covariance matrix g =   9 4.8 0.6 0 4.8 4 1 0 0.6 1 1 0 0 0 0 0   the design covariates matrix xj is assumed to arise from a uniform(-2,2). the first column in matrix zj are ones for the subject-specific intercept, while the remaining columns are assumed to arise from a uniform(-2,2) as well. (ii) this case follows case (i) with an increase of the sample size. assume m =60 with mj =10 and generate 200 dataset following the same methodology.) (iii) this case follows case (i) with an increase of the sample size. assume m =60 with mj =10 and generate 500 dataset following the same methodology.) 6 int. j. anal. appl. (2023), 21:40 table 1. : the simulation results for case (i) criteria bic aic gcv %correct 64 62 65 %cr 69 67 58 %cf 73 70 70 table 2. : the simulation results for case (ii) criteria bic aic gcv %correct 87 82 85 %cr 90 77 71 %cf 94 69 71 table 3. : the simulation results for case (iii) criteria bic aic gcv %correct 90 83 85 %cr 90 80 75 %cf 97 73 78 table 4. : the simulation results are for case (i) compared to some existing studies m =30 wridge [13] [12] mj =5 %correct 64 73 61 %cr 69 81 79 %cf 73 88 79 %correct denotes the percentage of times that the correct model (fixed and random effects) is selected, %cr denotes the percentage of times that the random effects is selected, and %cf denotes the percentage of times that the fixed effects is selected. m =30 ,mj =5. while there is a large body int. j. anal. appl. (2023), 21:40 7 table 5. : the simulation results are for case (ii) compared to some existing studies m =60 wridge [13] [12] mj =10 %correct 87 92 88 %cr 90 92 91 %cf 94 100 97 of literature on the estimation the parameters, only countable references studied parameters selection. in particular, the selection of the random effect has received less attention than the selection of the fixed effects. [14] studied one-stage of an weighted ridge procedure for l0 regularization in fixed models. also, [10] studied selection of fixed effects with estimation fixed effects, random effects and variance components in the linear mixed-effects model using weighted ridge procedure for l0 penalty performance. in tables 1 and 2, we use aic, bic and gcv criteria and it can be seen that the empirical results confirm the asymptotic properties, that is the selection percentages get higher as the sample size increases. in tables, 4 and 5 we use bic criteria and compare the results of our newly proposed approach (wridge) to the results of [13] and [12]. it can be seen that the percentage of selection in for our method is somehow less and that due to the nature of our method, it is known that the weighted ridge method doesn’t eliminate some predictors to zero as lasso does and that our newly proposed method performs well across the simulations experiments and the weighted ridge performs well for model selection for both fixed and random effects. 5. real data applications to show the efficiency of the suggested penalized technique in the mixed model selection, two dataset set are applied. 5.1. first dataset. we use the data from amsterdam, growth and health study, which is a distinctive, interdisciplinary cohort research established to investigate development and health among teens [25]. the information was gathered to investigate the connection between adolescent and early adulthood lifestyle and health. the study had 147 people in all, who were assessed at six different time periods, for a total of 882 observations. the five factors considered are age, gender, body fat, fitness, and smoking. this paper follows [26] and [13] for compression purposes. the random intercept is permitted to have intercept, but this is not the case for the fixed effects, since the response variable cholesterol has been centered and normalized all of the inputs, and hence the fixed effects have no intercept. table 6 exhibits that the bodyf at and time are selected by all approaches to be important 8 int. j. anal. appl. (2023), 21:40 fixed effects. for random effects selection, gender is selected by all approaches, while f itness is selected by saw and wridge methods, but smoking is selected by hard only. table 6. comparison of all methods. fixed effect random effect hard saw ps wridge hard saw ps wridge intercept 0.405 0.347 0.017 0.211 fitness 0 0 0 0 0 0.006 0 0.001 body_fat 0.174 0.165 0.170 0.168 0 0 0 0 smoking 0 0 0 0 0.149 0 0 0 gender 0 0 0 0 0.668 0.624 0.888 0.691 time 0.156 0.167 0.165 0.161 0 0 0 0 5.2. second dataset. the data was collected from 27 children to study, the distance (mm) between the pituitary gland’s core and the pterygomaxillary fissure for both gender [27]. the distances were measured in 16 boys and 11 girls at ages 8, 10, 12, and 14 for this study. the purpose of the study was to examine the basic age functions for boys and girls before describing the gap between boys and girls as a function of age. the two factors are gender and age while distance is the response. the response variable distance has been centered and normalized all of the inputs and hence there is no intercept for fixed effects. table 7. results of wridge selection method. fixed effect random effect wridge wridge intercept .021 gender 0.623 0.861 age 0.117 0 table 7 demonstrates that the gender and age are selected to be important fixed effects. for random effects selection, gender is selected by the newly proposed model to be important random effects. 6. conclusion in this paper, we propose weighted ridge selection for a linear mixed model and we focus on the case of longitudinal data with n observations coming from n subjects. the simulation studies of int. j. anal. appl. (2023), 21:40 9 low-dimensional settings show that the newly established method is an efficient method in general, and the percentages of times the random effects, fixed effects, and both of them combined are high percentages. conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. references [1] g. verbeke, linear mixed models for longitudinal data, in: linear mixed models in practice, springer, new york, ny, 1997: pp. 63-153. https://doi.org/10.1007/978-1-4612-2294-1_3. [2] a.e. hoerl, r.w. kennard, ridge regression: biased estimation for nonorthogonal problems, technometrics. 12 (1970), 55-67. [3] r. tibshirani, regression shrinkage and selection via the lasso, j. r. stat. soc.: ser. b (methodol.) 58 (1996), 267–288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x. [4] h. zou, the adaptive lasso and its oracle properties, j. amer. stat. assoc. 101 (2006), 1418–1429. https: //doi.org/10.1198/016214506000000735. [5] h. zou, t. hastie, regularization and variable selection via the elastic net, j. r. stat. soc. ser. b: stat. methodol. 67 (2005), 301-320. https://doi.org/10.1111/j.1467-9868.2005.00503.x. [6] j. fan, r. li, variable selection via nonconcave penalized likelihood and its oracle properties, j. amer. stat. assoc. 96 (2001), 1348–1360. https://doi.org/10.1198/016214501753382273. [7] h. akaike, a new look at the statistical model identification, ieee trans. automat. contr. 19 (1974), 716-723. https://doi.org/10.1109/tac.1974.1100705. [8] g. schwarz, estimating the dimension of a model, ann. stat. 6 (1978), 461-464. https://www.jstor.org/ stable/2958889. [9] y. yang, can the strengths of aic and bic be shared? a conflict between model indentification and regression estimation, biometrika. 92 (2005), 937-950. https://doi.org/10.1093/biomet/92.4.937. [10] e. adjakossa, g. nuel, fixed effects selection in the linear mixed-effects model using adaptive ridge procedure for l0 penalty performance, arxiv:1705.01308. (2017). https://doi.org/10.48550/arxiv.1705.01308. [11] h. peng, y. lu, model selection in linear mixed effect models, j. multivar. anal. 109 (2012), 109-129. https: //doi.org/10.1016/j.jmva.2012.02.005. [12] b. lin, z. pang, j. jiang, fixed and random effects selection by reml and pathwise coordinate optimization, j. comput. graph. stat. 22 (2013), 341–355. https://doi.org/10.1080/10618600.2012.681219. [13] j. pan, j. shang, adaptive lasso for linear mixed model selection via profile log-likelihood, commun. stat. theory methods. 47 (2017), 1882-1900. https://doi.org/10.1080/03610926.2017.1332219. [14] f. frommlet, g. nuel, an adaptive ridge procedure for l0 regularization, plos one. 11 (2016), e0148620. https://doi.org/10.1371/journal.pone.0148620. [15] n.m. laird, j.h. ware, random-effects models for longitudinal data, biometrics. 38 (1982), 963. https://doi. org/10.2307/2529876. [16] r.i. jennrich, m.d. schluchter, unbalanced repeated-measures models with structured covariance matrices, biometrics. 42 (1986), 805-820. https://doi.org/10.2307/2530695. [17] m.j. lindstrom, d.m. bates, newton-raphson and em algorithms for linear mixed-effects models for repeatedmeasures data, j. amer. stat. assoc. 83 (1988), 1014–1022. https://doi.org/10.1080/01621459.1988. 10478693. https://doi.org/10.1007/978-1-4612-2294-1_3 https://doi.org/10.1198/016214506000000735 https://doi.org/10.1198/016214506000000735 https://doi.org/10.1111/j.1467-9868.2005.00503.x https://doi.org/10.1198/016214501753382273 https://doi.org/10.1109/tac.1974.1100705 https://www.jstor.org/stable/2958889 https://www.jstor.org/stable/2958889 https://doi.org/10.1093/biomet/92.4.937 https://doi.org/10.48550/arxiv.1705.01308 https://doi.org/10.1016/j.jmva.2012.02.005 https://doi.org/10.1016/j.jmva.2012.02.005 https://doi.org/10.1080/10618600.2012.681219 https://doi.org/10.1080/03610926.2017.1332219 https://doi.org/10.1371/journal.pone.0148620 https://doi.org/10.2307/2529876 https://doi.org/10.2307/2529876 https://doi.org/10.2307/2530695 https://doi.org/10.1080/01621459.1988.10478693 https://doi.org/10.1080/01621459.1988.10478693 10 int. j. anal. appl. (2023), 21:40 [18] d.a. harville, maximum likelihood approaches to variance component estimation and to related problems, j. amer. stat. assoc. 72 (1977), 320-338. [19] y. grandvalet, least absolute shrinkage is equivalent to quadratic penalization, in: l. niklasson, m. boden, t. ziemke (eds.), icann 98, springer london, london, 1998: pp. 201-206. https://doi.org/10.1007/ 978-1-4471-1599-1_27. [20] p. bühlmann, l. meier, h. zou, discussion of "one-step sparse estimates in nonconcave penalized likelihood models" by h. zou and r. li, ann. stat. 36 (2008), 1534-1541. [21] r.c.a. rippe, j.j. meulman, p.h.c. eilers, visualization of genomic changes by segmented smoothing using an l0 penalty, plos one. 7 (2012), e38230. https://doi.org/10.1371/journal.pone.0038230. [22] e.j. candes, m.b. wakin, s.p. boyd, enhancing sparsity by reweighted `1 minimization, j. fourier anal. appl. 14 (2008), 877-905. https://doi.org/10.1007/s00041-008-9045-x. [23] m. stone, an asymptotic equivalence of choice of model by cross-validation and akaike’s criterion, j. r. stat. soc.: ser. b (methodol.) 39 (1977), 44-47. https://doi.org/10.1111/j.2517-6161.1977.tb01603.x. [24] h.d. bondell, a. krishna, s.k. ghosh, joint variable selection for fixed and random effects in linear mixed-effects models, biometrics. 66 (2010), 1069-1077. https://doi.org/10.1111/j.1541-0420.2010.01391.x. [25] j.w.r. twisk, h.c.g. kemper, g.j. mellenbergh, longitudinal development of lipoprotein levels in males and females aged 12-28 years: the amsterdam growth and health study, int. j. epidemiol. 24 (1995), 69-77. https://doi.org/10.1093/ije/24.1.69. [26] m. ahn, h.h. zhang, w. lu, moment-based method for random effects selection in linear mixed models, stat. sinica. 22 (2012), 1539-1562. https://doi.org/10.5705/ss.2011.054. [27] r.f. potthoff, s.n. roy, a generalized multivariate analysis of variance model useful especially for growth curve problems, biometrika. 51 (1964), 313-326. https://doi.org/10.1093/biomet/51.3-4.313. https://doi.org/10.1007/978-1-4471-1599-1_27 https://doi.org/10.1007/978-1-4471-1599-1_27 https://doi.org/10.1371/journal.pone.0038230 https://doi.org/10.1007/s00041-008-9045-x https://doi.org/10.1111/j.2517-6161.1977.tb01603.x https://doi.org/10.1111/j.1541-0420.2010.01391.x https://doi.org/10.1093/ije/24.1.69 https://doi.org/10.5705/ss.2011.054 https://doi.org/10.1093/biomet/51.3-4.313 1. introduction 2. variable selection for the linear mixed-effects model 2.1. literature review of classical linear mixed-effect mode 2.2. selection of random effects parameters 2.3. selection of fixed effects parameters 3. the methodology of the weighted ridge 4. simulation studies 5. real data applications 5.1. first dataset 5.2. second dataset 6. conclusion references international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 38-48 http://www.etamaths.com on fixed points of generalized α-ψ contractive type mappings in partial metric spaces priya shahi∗, jatinderdeep kaur and s. s. bhatia abstract. recently, samet et al. (b. samet, c. vetro and p. vetro, fixed point theorem for α-ψ contractive type mappings, nonlinear anal. 75 (2012), 2154–2165) introduced a very interesting new category of contractive type mappings known as α-ψ contractive type mappings. the results obtained by samet et al. generalize the existing fixed point results in the literature, in particular the banach contraction principle. further, karapinar and samet (e. karapinar and b. samet, generalized α-ψcontractive type mappings and related fixed point theorems with applications, abstract and applied analysis 2012 article id 793486, 17 pages doi:10.1155/2012/793486) generalized the α-ψ contractive type mappings and established some fixed point theorems for this generalized class of contractive mappings. in (g. s. matthews, partial metric topology, ann. new york acad. sci. 728 (1994), 183–197), the author introduced and studied the concept of partial metric spaces, and obtained a banach type fixed point theorem on complete partial metric spaces. in this paper, we establish the fixed point theorems for generalized α-ψ contractive mappings in the context of partial metric spaces. as consequences of our main results, we obtain fixed point theorems on partial metric spaces endowed with a partial order and that for cyclic contractive mappings. our results extend and strengthen various known results. some examples are also given to show that our generalization from metric spaces to partial metric spaces is real. 1. introduction the notion of metric space was introduced by fréchet in 1906. later, many authors attempted to generalize the notion of metric space such as pseudometric space, quasimetric space, semimetric space etc. in this paper, we consider another generalization of a metric space, so called partial metric space. when compared to metric spaces, the innovation of partial metric spaces is that the self distance of a point is not necessarily zero. initially, matthews discussed not only the general topological properties of partial metric spaces but also some properties of convergence of sequences. matthews also stated and proved the fixed point theorem of contractive mapping on partial metric spaces: any mapping t of a complete partial metric space x into itself that satisfies, for some 0 ≤ k < 1, the inequality d(tx,ty) ≤ kd(x,y), for all x,y ∈ x, has a unique fixed point. very recently, many authors have focussed on this subject and have generalized some fixed point theorems from the class of metric spaces to the class of partial metric spaces. the purpose of this work is to establish the fixed point theorems for generalized α-ψ-contractive mappings in the context of partial metric spaces. as consequences of our main results, we obtain fixed point theorems on partial metric spaces endowed with a partial order and that for cyclic contractive mappings. presented theorems are generalizations of very recent fixed point theorems due to samet et al.[23] and karapinar and samet [12]. some examples are given to show that presented results are real generalizations. 2. preliminaries throughout this work the letters r, r+, q, n will denote the sets of real numbers, nonnegative real numbers, rational numbers and natural numbers, respectively. before presenting our results, we collect relevant definitions and results which will be needed in the proof of our main results. 2010 mathematics subject classification. 54h25, 47h10, 54e50. key words and phrases. fixed point; partial metric space; partial order; contractive mapping. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 38 on fixed points of generalized α-ψ contractive type mappings 39 definition 2.1. (see e.g. [14, 9]). let x be a nonempty set. the mapping p : x × x → [0,∞) is said to be a partial metric on x if the following conditions hold: (p1) x = y if and only if p(x,x) = p(y,y) = p(x,y), (p2) p(x,x) ≤ p(x,y), (p3) p(x,y) = p(y,x), (p4) p(x,z) ≤ p(x,y) + p(y,z) −p(y,y), for any x,y,z ∈ x. the pair (x,p) is then called a partial metric space (in short pms). let (x,p) be a partial metric space. then, the functions dp,dm : x ×x → [0,∞) given by dp(x,y) = 2p(x,y) −p(x,x) −p(y,y) and dm(x,y) = max p(x,y) −p(x,x),p(x,y) −p(y,y) are well-known metrics on x. it is easy to check that dp and dm are equivalent. note that each partial metric p on x generates a t0-topology τp with a base of the family of open p-balls {bp(x,�) : x ∈ x,� > 0}, where bp(x,�) = {y ∈ x : p(x,y) < p(x,x) + �}. definition 2.2. (see e.g. [2, 9]). let (x,p) be a partial metric space. (1) a sequence {xn} in x converges to x ∈ x if and only if p(x,x) = lim n→∞ p(xn,x). (2) a sequence {xn} in x is called a cauchy sequence if and only if lim m,n→∞ p(xn,xm) exists (and finite). (3) (x,p) is said to be complete if every cauchy sequence {xn} in x converges to x ∈ x. (4) a mapping f : x → x is said to be continuous at x0 ∈ x if for every � > 0, there exists δ > 0 such that f(bp(x0,δ)) ⊂ bp(f(x0),�). example 2.3. let x = [0, +∞) and define p(x,y) = max{x,y}, for all x,y ∈ x. then (x,p) is a complete partial metric space. it is clear that p is not a (usual) metric. definition 2.4. let (x,p) be a partial metric space and t : x → x be a given mapping. we say that t is continuous at x0 ∈ x, if for every � > 0, there exists η > 0 such that t(bp(x0,η)) ⊆ bp(tx0,�). the following lemmas have an important role in the proof of our main results. lemma 2.1. (see e.g. [2, 9]). let (x,p) be a partial metric space. (1) a sequence {xn} is a cauchy sequence in (x,p) if and only if {xn} is a cauchy sequence in (x,dp). (2) (x,p) is complete if and only if (x,dp) complete. moreover, lim n→∞ dp(xn,x) = 0 ⇔ p(x,x) = lim n→∞ p(xn,x) = lim n,m→∞ p(xm,xn). lemma 2.2. (see e.g. [2]). assume that xn → z as n → ∞ in a pms (x,p) such that p(z,z) = 0. then lim n→∞ p(xn,y) = p(z,y) for every y ∈ x. lemma 2.3. (sequential characterization of continuity)let (x,p) be a partial metric space and t : x → x be a given mapping. t is said to be continuous at x0 ∈ x if it is sequentially continuous at x0, that is, if and only if ∀{xn}⊂ x : lim n→+∞ xn = x0 ⇒ lim n→+∞ txn = tx0 let ψ be the family of functions ψ : [0,∞) → [0,∞) satisfying the following conditions: (i) ψ is nondecreasing. (ii) +∞∑ n=1 ψn(t) < ∞ for all t > 0, where ψn is the nth iterate of ψ. these functions are known as (c)-comparison functions in the literature. it can be easily verified that if ψ is a (c)-comparison function, then ψ(t) < t for any t > 0. recently, samet et al. [23] introduced the following new concepts of α-ψ-contractive type mappings and α-admissible mappings: 40 shahi, kaur and bhatia definition 2.5. let (x,d) be a metric space and t : x → x be a given self mapping. t is said to be an α-ψ-contractive mapping if there exists two functions α : x ×x → [0, +∞) and ψ ∈ ψ such that α(x,y)d(tx,ty) ≤ ψ(d(x,y)) for all x,y ∈ x. definition 2.6. let t : x → x and α : x ×x → [0, +∞). t is said to be α-admissible if x,y ∈ x, α(x,y) ≥ 1 ⇒ α(tx,ty) ≥ 1. the following fixed point theorems are the main results in [23]: theorem 2.4. let (x,d) be a complete metric space and t : x → x be an α-ψ-contractive mapping satisfying the following conditions: (i) t is α-admissible; (ii) there exists x0 ∈ x such that α(x0,tx0) ≥ 1; (iii) t is continuous. then, t has a fixed point, that is, there exists x∗ ∈ x such that tx∗ = x∗. theorem 2.5. let (x,d) be a complete metric space and t : x → x be an α-ψ-contractive mapping satisfying the following conditions: (i) t is α-admissible; (ii) there exists x0 ∈ x such that α(x0,tx0) ≥ 1; (iii) if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ x as n → +∞, then α(xn,x) ≥ 1 for all n. then, t has a fixed point. samet et al. [23] added the following condition to the hypotheses of theorem 2.4 and theorem 2.5 to assure the uniqueness of the fixed point: (c): for all x,y ∈ x, there exists z ∈ x such that α(x,z) ≥ 1 and α(y,z) ≥ 1. recently, karapinar and samet [12] introduced the following concept of generalized α-ψ-contractive type mappings: definition 2.7. let (x,d) be a metric space and t : x → x be a given mapping. we say that t is a generalized α-ψ-contractive type mapping if there exists two functions α : x ×x → [0,∞) and ψ ∈ ψ such that for all x,y ∈ x, we have α(x,y)d(tx,ty) ≤ ψ(m(x,y)), where m(x,y) = max { d(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty) + d(y,tx) 2 } . further, karapinar and samet [12] established fixed point theorems for this new class of contractive mappings. also, they obtained fixed point theorems on metric spaces endowed with a partial order and fixed point theorems for cyclic contractive mappings. 3. main results firstly, we present the concept of generalized α-ψ contractive type mappings in the context of partial metric spaces as follows: definition 3.1. let (x,p) be a partial metric space and t : x → x be a given mapping. we say that t is a generalized α-ψ-contractive type mapping if there exists two functions α : x ×x → [0,∞) and ψ ∈ ψ such that for all x,y ∈ x, we have α(x,y)p(tx,ty) ≤ ψ(m(x,y)),(1) where m(x,y) = max { p(x,y), p(x,tx) + p(y,ty) 2 , p(x,ty) + p(y,tx) 2 } . now, we present our main results as follows. on fixed points of generalized α-ψ contractive type mappings 41 theorem 3.1. let (x,d) be a complete partial metric space, α : x × x → [0,∞) be a function, ψ ∈ ψ and t be a generalized α-ψ contractive type mapping on x. suppose that t is α-admissible and continuous. also, assume that there exists x0 ∈ x such that α(x0,tx0) ≥ 1. then there exists u ∈ x such that tu = u. proof. take x0 ∈ x such that α(x0,tx0) ≥ 1 and define the sequence {xn} in x by xn+1 = txn for all n ≥ 0. if xn = xn+1 for some n, then x∗ = xn is a fixed point of t . assume that xn 6= xn+1 for all n. owing to α-admissible property of t, we have α(x0,tx0) = α(x0,x1) ≥ 1 ⇒ α(tx0,tx1) = α(x1,x2) ≥ 1 continuing this process inductively, we obtain α(xn,xn+1) ≥ 1(2) for all n = 0, 1, 2, ... . thus for each n, we have p(xn,xn+1) = p(txn−1,txn) ≤ α(xn−1,xn)p(txn−1,txn) ≤ ψ(m(xn−1,xn))(3) on the other hand, we have m(xn−1,xn) = max { p(xn−1,xn), p(xn−1,xn) + p(xn,xn+1) 2 , p(xn−1,xn+1) + p(xn,xn) 2 } ≤ max { p(xn−1,xn), p(xn−1,xn) + p(xn,xn+1) 2 , p(xn−1,xn) + p(xn,xn+1) 2 } ≤ max{p(xn−1,xn),p(xn,xn+1)} .(4) from (3), (4) and using the fact that ψ is a nondecreasing function, we get that p(xn+1,xn) ≤ ψ(max{p(xn−1,xn),p(xn,xn+1)}) for all n. if max{p(xn−1,xn),p(xn,xn+1)} = p(xn,xn+1), then p(xn,xn+1) ≤ ψ(p(xn,xn+1)) < p(xn,xn+1), which is a contradiction. thus, max{p(xn−1,xn),p(xn,xn+1)} = p(xn−1,xn) for all n. hence, p(xn,xn+1) ≤ ψ(p(xn−1,xn)) for all n. continuing this process inductively, we obtain p(xn,xn+1) ≤ ψn(p(x0,x1)),(5) for all n. now, using the definition of partial metric, we have max{p(xn,xn),p(xn+1,xn+1)}≤ p(xn,xn+1)(6) which in view of (5) gives rise to max{p(xn,xn),p(xn+1,xn+1)}≤ ψn((p(x0,x1))(7) therefore, owing to (4) and (5), we have ps(xn,xn+1) = 2p(xn,xn+1) −p(xn,xn) −p(xn+1,xn+1) ≤ 2p(xn,xn+1) + p(xn,xn) + p(xn+1,xn+1) ≤ 4ψn(p(x0,x1)).(8) now, using inequality (8), we have ps(xn+k,xn) ≤ ps(xn+k,xn+k−1) + ... + ps(xn+1,xn) ≤ 4ψn+k−1(p(x0,x1)) + ... + 4ψn(p(x0,x1)) ≤ 4 n+k−1∑ i=n ψi(p(x0,x1))(9) 42 shahi, kaur and bhatia and as ∞∑ i=0 ψi(p(x0,x1)) is convergent, from the last inequality, using cauchy’s criteria for convergent series, we obtain that {xn} is a cauchy sequence in the metric space (x,ps). now, in view of lemma 2.1 and the completeness of (x,p), we conclude the completeness of (x,ps). therefore, the sequence {xn} is convergent in the space (x,ps), say lim n→∞ ps(xn,u) = 0. again from lemma 2.1, we get p(u,u) = lim n→∞ p(xn,u) = lim m,n→∞ p(xm,xn)(10) moreover, since {xn} is a cauchy sequence in the metric space (x,ps), we have lim m,n→∞ ps(xm,xn) = 0(11) and in view of (7), one gets lim n→∞ p(xn,xn) = 0(12) notice that in view of (11), (12) and definition of ps, we conclude that lim m,n→∞ p(xm,xn) = 0(13) on using (10), we have p(u,u) = lim n→∞ p(xn,u) = lim m,n→∞ p(xm,xn) = 0(14) now, we proceed to show that tu = u. due to the continuity of t, we infer from lemma 2.3 that p(tu,tu) = lim n→∞ p(txn,tu) = lim m,n→∞ p(txm,txn)(15) that is, p(tu,tu) = lim m,n→∞ p(xm+1,xn+1).(16) notice that in view of (14) and (16), p(u,u) = p(tu,tu) = 0(17) owing to lemma 2.2, we have lim n→∞ p(xn,tu) = p(u,tu)(18) therefore, using (15), (17) and (18), we obtain p(tu,tu) = p(u,u) = p(u,tu) = 0 implying thereby tu = u. thus, we conclude that u is a fixed point of t . this completes the proof. � in the next theorem, we omit the continuity hypothesis of t . theorem 3.2. let (x,d) be a complete partial metric space, α : x × x → [0,∞) be a function, ψ ∈ ψ and t be a generalized α-ψ contractive type mapping on x. suppose that t is α-admissible and that there exists x0 ∈ x such that α(x0,tx0) ≥ 1. assume that if {xn} is a sequence in x such that α(xn,xn+1) ≥ 1 for all n and {xn}→ x ∈ x as n →∞, then there exists a subsequence {xn(k)} of {xn} such that α(xn(k),x) ≥ 1 for all k. then there exists u ∈ x such that tu = u. proof. following the proof of theorem 3.1, we know that the sequence {xn} given by xn+1 = txn for all n ≥ 0, converges to some u ∈ x. from (2) and given hypotheses, there exists a subsequence {xn(k)} of {xn} such that α(xn(k),u) ≥ 1(19) on fixed points of generalized α-ψ contractive type mappings 43 for all k. now, we proceed to show that u is a fixed point of t . suppose the contrary, then p(u,tu) > 0. therefore, from (1) and (19), we infer that p(u,tu) ≤ p(u,xn(k)+1) + p(xn(k)+1,tu) −p(xn(k)+1,xn(k)+1) ≤ p(u,xn(k)+1) + p(xn(k)+1,tu) = p(u,xn(k)+1) + p(txn(k),tu) ≤ p(u,xn(k)+1) + α(xn(k),u)p(txn(k),tu) ≤ p(u,xn(k)+1) + ψ(m(xn(k),u)))(20) on the other hand, we have m(xn(k),u)) = max { p(xn(k),u), p(xn(k),xn(k)+1) + p(u,tu) 2 , p(xn(k),tu) + p(u,xn(k)+1) 2 } (21) letting k →∞ in (21) and using the above equality, we get p(u,tu) ≤ ψ ( p(u,tu) 2 ) < p(u,tu) 2 ,(22) which is a contradiction. therefore, p(u,tu) = 0 and tu = u. � we demonstrate the use of theorem 3.1 and theorem 3.2 with the help of the following examples. these examples also show that our theorems are more general than some other known fixed point results. example 3.2. let x = r+, where (x,p) is a complete partial metric space with partial metric p given by p(x,y) = max{x,y}. the mapping t(x) = x2 1 + x ∀x ∈ x is continuous. let us define the function α by α(x,y) = { 1 x ≥ y 0 otherwise clearly, t is a generalized α-ψ contractive type mapping with ψ(t) = t2 1 + t for all t ≥ 0. in fact, for all x,y ∈ x, we have α(x,y)p(tx,ty) ≤ ψ(m(x,y))(23) moreover, there exists x0 ∈ x such that α(x0,tx0) ≥ 1. in fact, for x0 = 1, we have α(1,t1) = α(1, 1/2) = 1(24) now we proceed to show that t is α-admissible. for this, we have α(x,y) ≥ 1 ⇒ x ≥ y ⇒ tx ≥ ty ⇒ α(tx,ty) ≥ 1(25) thus, t is α-admissible. now, all the hypotheses of theorem 3.1 are satisfied. consequently, t has a fixed point. in this case, 0 is a fixed point. the same conclusion cannot be obtained by theorem 2.3 from [12]. indeed, using ps(a,b) = 2p(a,b) − p(a,a) −p(b,b), and then taking ps instead p, x = 3, y = 2 in (1), we obtain α(3, 2)ps(t3,t2) = 11 12 1 2 = ψ(1).(26) therefore, this example shows that our generalization from metric spaces to partial metric spaces is real. example 3.3. let x = {0, 1, 2, 3} and the function p : x×x → [0, +∞) defined by p(1, 2) = p(2, 3) = 1,p(1, 3) = 3 2 ,p(1, 1) = p(3, 3) = 1 2 ,p(2, 2) = 0 and p(x,y) = p(y,x). obviously, p is a partial metric on x but not a metric (since p(x,x) 6= 0 for x = 1 and x = 3). let us define the self-mapping t on x by t0 = 1, t1 = 2, t2 = 2, t3 = 1 for each x ∈ x. clearly, t is a generalized α-ψ contractive type mapping with ψ(t) = 2 3 t for t ≥ 0. in fact, for all x,y ∈ x, we have α(x,y)p(tx,ty) ≤ ψ(m(x,y)),(27) 44 shahi, kaur and bhatia where α(x,y) = { 1 (x,y) 6= (0, 0) 0 otherwise moreover, there exists x0 ∈ x such that α(x0,tx0) ≥ 1. in fact, for x0 = 1, we have α(1,t1) = α(1, 2) = 1(28) let {xn} be a sequence in x such that α(xn,xn+1) ≥ 1 for all n and xn → x as n → +∞ for some x ∈ x. from the definition of α, for all n, we have xn 6= 0 for all n. thus, x 6= 0 and we have α(xn,x) ≥ 1 for all n. now we proceed to show that t is α-admissible. for this, we have α(x,y) ≥ 1 ⇒ x 6= 0,y 6= 0 ⇒ tx 6= 0,ty 6= 0 ⇒ α(tx,ty) ≥ 1(29) thus, t is α-admissible. now, all the hypotheses of theorem 3.2 are satisfied. consequently, t has a fixed point. in this case, 2 is a fixed point. the same conclusion cannot be obtained by theorem 2.4 from [12]. indeed, using ps(a,b) = 2p(a,b) − p(a,a) −p(b,b), and then taking ps instead p, x = 1, y = 3 in (1), we obtain α(1, 3)ps(t1,t3) = 3 2 4 3 = ψ(2).(30) therefore, this example shows that our generalization from metric spaces to partial metric spaces is real. 4. fixed point theorems on partial metric spaces endowed with a partial order fixed point theory has developed rapidly in partially ordered metric spaces. the first result in this direction was given by turinici [24], where he extended the banach contraction principle in partially ordered sets. some applications of turinici’s theorem to matrix equations were presented by ran and reurings [20]. subsequently, nieto and rodŕiguez-lópez [17] extended this result and applied it to obtain a unique solution for periodic boundary value problems. further results were obtained by several authors (see, for example, [4, 8, 13, 7, 16] and the references cited therein). altun and erduran [5] used the idea of partial order and established fixed point theorems to the frame of ordered partial metric spaces. aydi [6], samet et al. [22], abbas and nazir [1] also studied fixed point results on partially ordered partial metric spaces. before presenting our result, we collect relevant concepts which will be needed in the proof of our results. definition 4.1. let (x,�) be a partially ordered set and t : x → x be a given mapping. we say that t is nondecreasing with respect to � if x,y ∈ x,x � y ⇒ tx � ty. definition 4.2. let (x,�) be a partially ordered set. a sequence {xn}⊂ x is said to be nondecreasing with respect to � if xn � xn+1 for all n. definition 4.3. [12] let (x,�) be a partially ordered set and p be a partial metric on x. we say that (x,�,p) is regular if for every nondecreasing sequence {xn} ⊂ x such that xn → x ∈ x as n → ∞, there exists a subsequence {xn(k)} of {xn} such that xn(k) � x for all k. now, we have the following result. corollary 4.1. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists a function ψ ∈ ψ such that p(tx,ty) ≤ ψ(m(x,y)),(31) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,d) is regular. then, t has a fixed point. on fixed points of generalized α-ψ contractive type mappings 45 proof. let us define the mapping α : x ×x → [0,∞) by α(x,y) = { 1 if x � y or x � y 0 otherwise (32) for all x,y ∈ x. in view of condition (i), we obtain α(x0,tx0) ≥ 1. moreover, for all x,y ∈ x, from the monotone property of t, we get α(x,y) ≥ 1 ⇒ x � y or x � y ⇒ tx � ty or tx � ty ⇒ α(tx,ty) ≥ 1.(33) thus, t is α-admissible. now, if t is continuous, the existence of a fixed point follows from theorem 3.1. suppose now that (x,�,d) is regular. let {xn} be a sequence in x such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ x as n → ∞. so, from the regularity hypothesis, there exists a subsequence {xn(k)} of {xn} such that xn(k) � x for all k. notice that in view of definition of α, we obtain that α(xn(k),x) ≥ 1 for all k. thus, we get the existence of a fixed point from theorem 3.2. � the following corollaries can be straightway derived from corollary 4.1 corollary 4.2. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists a function ψ ∈ ψ such that p(tx,ty) ≤ ψ(p(x,y)),(34) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,p) is regular. then t is a fixed point. corollary 4.3. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists a constant λ ∈ (0, 1) such that p(tx,ty) ≤ max { p(x,y), p(x,tx) + p(y,ty) 2 , p(x,ty) + p(y,tx) 2 } ,(35) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,p) is regular. then t is a fixed point. corollary 4.4. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists constants a,b,c ≥ 0 with (a + 2b + 2c) ∈ (0, 1) such that p(tx,ty) ≤ ap(x,y) + b[p(x,tx) + p(y,ty)] + c[p(x,ty) + p(y,tx)],(36) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,p) is regular. then t is a fixed point. corollary 4.5. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists a constant λ ∈ (0, 1/2) such that p(tx,ty) ≤ λ[p(x,tx) + p(y,ty)],(37) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,p) is regular. then t is a fixed point. 46 shahi, kaur and bhatia corollary 4.6. let (x,�) be a partially ordered set and p be a partial metric on x such that (x,p) is complete. let t : x → x be a nondecreasing mapping with respect to �. suppose that there exists a constant λ ∈ (0, 1/2) such that p(tx,ty) ≤ λ[p(x,ty) + p(y,tx)],(38) for all x,y ∈ x with x � y. suppose also that the following conditions hold: (i) there exists x0 ∈ x such that x0 � tx0; (ii) t is continuous or (x,�,p) is regular. then t is a fixed point. 5. fixed point theorems for cyclic contractive mappings as a generalization of the banach contraction mapping principle, kirk-srinivasan-veeramani developed the cyclic contraction. a contraction t : a∪b → a∪b on non-empty sets a,b is called cyclic if t(a) ⊂ b and t(b) ⊂ a hold for closed subsets a,b of a complete metric space x. in the last decade, several authors have used the cyclic representations and cyclic contractions to obtain various fixed point results. see e.g., ([3, 10, 11, 18, 19, 21]). in this section, we will show that, from our theorem 3.1 and 3.2, we can deduce some fixed point theorems for cyclic contractive mappings. now, we have the following result. corollary 5.1. let {ai} be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a function ψ ∈ ψ such that p(tx,ty) ≤ ψ(m(x,y)), ∀(x,y) ∈ a1 ×a2.(39) then t has a fixed point that belongs to a1 ∩a2. proof. due to the fact that a1 and a2 are closed subsets of the complete metric space (x,d), we get completeness of the space (y,d). let us define the mapping α : y ×y → [0,∞) by α(x,y) = { 1 if (x,y) ∈ (a1 ×a2) ∪ (a2 ×a1), 0 otherwise (40) notice that in view of definition α and condition (ii), we infer that α(x,y)p(tx,ty) ≤ ψ(m(x,y)),(41) for all x,y ∈ y . thus t is a generalized α-ψ contractive mapping. now, we proceed to show that t is α-admissible. for thus, let (x,y) ∈ y × y such that α(x,y) ≥ 1. if (x,y) ∈ a1 × a2, then from (i), we have (tx,ty) ∈ a2 × a1, thereby implying α(tx,ty) ≥ 1. again from (i), we obtain that (x,y) ∈ a2 ×a1 implies that (tx,ty) ∈ a1 ×a2, which further implies that α(tx,ty) ≥ 1. thus, we have α(tx,ty) ≥ 1 in all the cases. therefore, we obtain that t is α-admissible. also, in view of (i), for any u ∈ a1, we have (u,tu) ∈ a1 ×a2, which suggest that α(u,tu) ≥ 1. now, we consider that {xn} be a sequence in x such that α(xn,xn+1) ≥ 1 for all n and xn → x ∈ x as n →∞. this suggest from the definition of α that (xn,xn+1) ∈ (a1 ×a2) ∪ (a2 ×a1),(42) for all n. since (a1 ×a2) ∪ (a2 ×a1) is a closed set with respect to the euclidean metric, we obtain that (x,x) ∈ (a1 ×a2) ∪ (a2 ×a1),(43) which refer that x ∈ a1 ∩a2. consequently, we get from the definition of α that α(xn,x) ≥ 1 for all n. now, all the hypotheses of theorem 3.2 are satisfied, and we conclude that t has a fixed point that belongs to a1 ∩a2(from (i)). � the following results are immediate consequences of corollary 5.1. on fixed points of generalized α-ψ contractive type mappings 47 corollary 5.2. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a function ψ ∈ ψ such that p(tx,ty) ≤ ψ(p(x,y)), ∀(x,y) ∈ a1 ×a2.(44) then t has a fixed point that belongs to a1 ∩a2. corollary 5.3. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a constant λ ∈ (0, 1) such that p(tx,ty) ≤ λ max { p(x,y), d(x,tx) + d(y,ty) 2 , d(x,ty) + d(y,tx) 2 } , ∀(x,y) ∈ a1 ×a2.(45) then t has a fixed point that belongs to a1 ∩a2. corollary 5.4. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists constants a,b,c ≥ 0 with (a + 2b + 2c) ∈ (0, 1) such that p(tx,ty) ≤ ap(x,y) + b[d(x,tx) + d(y,ty)] + c[d(x,ty) + d(y,tx)], ∀(x,y) ∈ a1 ×a2.(46) then t has a fixed point that belongs to a1 ∩a2. corollary 5.5. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a constant λ ∈ (0, 1) such that p(tx,ty) ≤ λp(x,y), ∀(x,y) ∈ a1 ×a2.(47) then t has a fixed point that belongs to a1 ∩a2. corollary 5.6. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a constant λ ∈ (0, 1/2) such that p(tx,ty) ≤ λ[p(x,tx) + p(y,ty)], ∀(x,y) ∈ a1 ×a2.(48) then t has a fixed point that belongs to a1 ∩a2. corollary 5.7. let {ai}2i=1 be nonempty closed subsets of a complete partial metric space (x,p) and t : y → y be a given mapping, where y = a1 ∪a2. suppose that the following conditions hold: (i) t(a1) ⊆ a2 and t(a2) ⊆ a1; (ii) there exists a constant λ ∈ (0, 1/2) such that p(tx,ty) ≤ λ[p(x,ty) + p(y,tx)], ∀(x,y) ∈ a1 ×a2.(49) then t has a fixed point that belongs to a1 ∩a2. acknowledgments the first author gratefully acknowledges the university grants commission, government of india for financial support during the preparation of this manuscript. 48 shahi, kaur and bhatia references [1] m. abbas and t. nazir, fixed points of generalized weakly contractive mappings in ordered partial metric spaces, fixed point theory appl. 2012 (2012), art. id 1. [2] t. abedelljawad, e. karapinar and k.tas, existence and uniqueness of common fixed point on partial metric spaces, appl. math. lett. 24 (2011), 1894–1899. [3] r. p. agarwal, m. a. alghamdi and n. shahzad, fixed point theory for cyclic generalized contractions in partial metric spaces, fixed point theory appl. (2012) (2012), art. id 40. [4] r. p. agarwal, m. a. el-gebeily and d. o’ regan, generalized contractions in partially ordered metric spaces, applicable analysis 87 (2008), 1–8. [5] i. altun and a. erduran, fixed point theorems for monotone mappings on partial metric spaces, fixed point theory appl. 2011 (2011), art. id 508730. [6] h. aydi, some fixed point results in ordered partial metric spaces,j. nonlinear sci. appl. 4 (2011), 210217. [7] h. aydi, h.k. nashine, b. samet and h. yazidi, coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations, nonlinear analysis 74 (2011), 6814–6825. [8] t. g. bhaskar and v. lakshmikantham, fixed point theory in partially ordered metric spaces and applications, nonlinear analysis 65 (2006), 1379–1393. [9] d. ilić, v. pavlović and v. rakocević, some new extensions of banach’s contraction principle to partial metric space, appl. math. lett. 24 (2011), 1326–1330. [10] e. karapinar, fixed point theory for cyclic weak φ-contraction, appl. math. lett. 24 (2011), 822–825. [11] e. karapinar and k. sadaranagni, fixed point theory for cyclic (φ−ψ)-contractions, fixed point theory appl. 2011 (2011), art. id 69. [12] e. karapinar and b. samet, generalized α-ψ-contractive type mappings and related fixed point theorems with applications, abstract and applied analysis 2012(2012), article id 793486. [13] v. lakshmikantham and l. ćirić, coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, nonlinear analysis 70 (2009), 4341–4349. [14] g. s. matthews, partial metric topology, research report 212, dept. of computer science university of warwick, 1992. [15] g. s. matthews, partial metric topology, ann. new york acad. sci. 728 (1994), 183–197. [16] h. k. nashine, z. kadelburg and s. radenović, common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, math. comput. modelling (2012), doi:10.1016/j.mcm.2011.12.019. [17] j. j. nieto and r. r. lopez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22 (2005), 223–239. [18] m. pacurar and i. a. rus, fixed point theory for cyclic ϕ-contractions, nonlinear anal. 72 (2010), 1181–1187. [19] m. a. petric, some results concerning cyclic contractive mappings, general mathematics 18 (2010), 213–226. [20] a. c. m. ran and m. c. b. reurings, a fixed point theorem in partially ordered sets and some applications to matrix equations, proc. amer. math. soc. 132 (2004), 1435–1443. [21] i. a. rus, cyclic representations and fixed points, ann. t. popovicin. seminar funct. eq. approx. convexity 3 (2005), 171–178. [22] b. samet, m. rajović, r. lazović and r. stoiljković, common fixed point results for nonlinear contractions in ordered partial metric spaces, fixed point theory appl. 2011 (2011), art. id 71. [23] b. samet, c. vetro and p. vetro, fixed point theorem for α-ψ contractive type mappings, nonlinear anal. 75 (2012), 2154–2165. [24] m. turinici, abstract comparison principles and multivariable gronwall-bellman inequalities, j. math. anal. appl. 117 (1986), 100–127. school of mathematics and computer applications, thapar university, patiala-147004, india ∗corresponding author: priya.thaparian@gmail.com international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 167-173 http://www.etamaths.com fractional ostrowski inequalities for s-godunova-levin functions muhammad aslam noor, khalida inayat noor, muhammad uzair awan∗ abstract. in this paper, we derive some new fractional ostrowski type inequalities for s-godunova-levin functions introduced by dragomir [3, 4]. some special cases are also discussed. 1. introduction recently much attention has been given to theory of convex functions by many researchers. consequently the classical concept of convex functions has been extended and generalized in different directions using various novel ideas, see [1, 2, 3, 4, 6, 9, 15, 16, 17, 18, 24]. recently dragomir has introduced the notion of s-godunova-levin functions. this class of convex function generalizes the class of godunova-levin functions and the class of p-functions. it is known that convex functions play an important role in the development of many famous inequalities. thus many researchers have generalized the classical version of famous inequalities such as hermite-hadamard inequality, ostroski inequality, simpson inequality etc for different classes of convex functions, see [2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 16, 17, 18, 21, 23, 24, 25]. let f : i ⊂ [0,∞) → r be a differentiable mapping on i, the interior of the interval i, such that f′ ∈ l[a,b], where a,b ∈ i with a < b. if |f′(x)| ≤ m, then the following inequality,∣∣∣∣∣∣f(x) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ mb−a [ (x−a)2 + (b−x)2 2 ] ,(1.1) holds. this result is known in the literature as the ostrowski inequality [19]. in this paper, we derive some new inequalities of ostrowski type for s-godunovalevin functions via fractional integrals. we also discuss some special cases. this is the main motivation of this paper. 2. preliminaries in this section, we recall some preliminary concepts. first of all let i = [a,b] ⊆ r be the interval and r be the set of real numbers. 2010 mathematics subject classification. 26a33, 26a51, 26d15. key words and phrases. convex functions, s-godunova-levin functions, ostrowski inequalities. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 167 168 noor, noor and awan definition 2.1 ([6]). a nonnegative function f : i → r is said to be p -function, if f(tx + (1 − t)y) ≤ f(x) + f(y), ∀x,y ∈ i,t ∈ [0, 1].(2.2) definition 2.2 ([9]). a function f : i → r is said to be godunova-levin function, if f(tx + (1 − t)y) ≤ f(x) t + f(y) 1 − t ,∀x,y ∈ i,t ∈ (0, 1).(2.3) for some useful details and extensions of godunova-levin functions, see [3, 4, 7, 9, 13, 16, 17, 20] definition 2.3 ([16]). a function f : i → r is said to be s-godunova-levin functions of first kind, if s ∈ (0, 1], we have f(tx + (1 − t)y) ≤ f(x) ts + f(y) 1 − ts , ∀x,y ∈ i,t ∈ (0, 1).(2.4) it is obvious that for s = 1 the definition of s-godunova-levin functions of first kind collapses to the definition of godunova-levin functions. our next definition is another due to dragomir [3, 4]. definition 2.4 ([3, 4]). a function f : i → r is said to be s-godunova-levin functions of second kind, if s ∈ [0, 1], we have f(tx + (1 − t)y) ≤ f(x) ts + f(y) (1 − t)s , ∀x,y ∈ i, t ∈ (0, 1).(2.5) it is obvious that for s = 0, s-godunova-levin functions of second kind reduces to the definition of p-functions. if s = 1, it then reduces to godunova-levin functions. the following result plays a key role in deriving our main results. lemma 2.1 ([22]). let f : [a,b] → r be a differentiable function on (a,b) with a < b. if f′ ∈ l1[a,b], then for all x ∈ [a,b] and α > 0, we have((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ] = (x−a)α+1 b−a 1∫ 0 tαf′(tx + (1 − t)a)dt− (b−x)α+1 b−a 1∫ 0 tαf′(tx + (1 − t)b)dt. 3. main results in this section, we derive our main results. theorem 3.1. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b] and α > 0. if |f′| is s-godunova-levin function of second kind and |f′(x)| ≤ m, then, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ ≤ ( 1 1 + α−s + γ(1 −s)γ(α + 1) γ(2 + α−s) )(m[(x−a)α+1 + (b−a)α+1] b−a ) . fractional ostrowski inequalities 169 proof. using lemma 2.1 and the fact that |f′| is s-godunova-levin function of second kind, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ = ∣∣∣∣∣(x−a) α+1 b−a 1∫ 0 tαf′(tx + (1 − t)a)dt− (b−x)α+1 b−a 1∫ 0 tαf′(tx + (1 − t)b)dt ∣∣∣∣∣ ≤ (x−a)α+1 b−a 1∫ 0 tα|f′(tx + (1 − t)a)|dt + (b−x)α+1 b−a 1∫ 0 tα|f′(tx + (1 − t)b)|dt ≤ (x−a)α+1 b−a 1∫ 0 tα [ 1 ts |f′(x)| + 1 (1 − t)s |f′(a)| ] dt + (b−x)α+1 b−a 1∫ 0 tα [ 1 ts |f′(x)| + 1 (1 − t)s |f′(b)| ] dt ≤ ( 1 1 + α−s + γ(1 −s)γ(α + 1) γ(2 + α−s) )(m[(x−a)α+1 + (b−a)α+1] b−a ) . this completes the proof. � note that for α = 1, theorem 3.1 collapses to following result for s-godunovalevin function of second kind. corollary 3.1. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b]. if |f′| is s-godunova-levin function of second kind and |f′(x)| ≤ m, then, we have∣∣∣∣∣∣f(x) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m[(x−a) 2 + (b−x)2] (b−a)(1 −s) .(3.6) also, we have some special cases of corollary 3.1. i. if we take x = a+b 2 in (3.6), then we have the following mid-point inequality∣∣∣∣∣∣f ( a + b 2 ) − 1 b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a)2(1 −s) . ii. if we take x = a in (3.6), then we have the following inequality∣∣∣∣∣∣f(a) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a) ( 1 1 −s ) . iii. if we take x = b in (3.6), then we have the following inequality∣∣∣∣∣∣f(b) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a) ( 1 1 −s ) . 170 noor, noor and awan theorem 3.2. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b] and α > 0. if |f′|q is s-godunova-levin function of second kind on [a,b], p,q > 1, 1/p + 1/q = 1 and |f′(x)| ≤ m, then, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ ≤ m ( 1 1 −s )1 q ( 1 1 + pα )1 p [ (x−a)α+1 + (b−a)α+1 b−a ] . proof. using lemma 2.1, well-known holder’s inequality and the fact that |f′|q is s-godunova-levin function of second kind, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ = ∣∣∣∣∣(x−a) α+1 b−a 1∫ 0 tαf′(tx + (1 − t)a)dt− (b−x)α+1 b−a 1∫ 0 tαf′(tx + (1 − t)b)dt ∣∣∣∣∣ ≤ (x−a)α+1 b−a ( 1∫ 0 tpαdt )1 p ( 1∫ 0 |f′(tx + (1 − t)a)|qdt )1 q + (b−x)α+1 b−a ( 1∫ 0 tpαdt )1 p ( 1∫ 0 |f′(tx + (1 − t)b)|qdt )1 q ≤ (x−a)α+1 b−a ( 1∫ 0 tpαdt )1 p ( 1∫ 0 ( 1 ts |f′(x)|q + 1 (1 − t)s |f′(a)|q ) dt )1 q + (b−x)α+1 b−a ( 1∫ 0 tpαdt )1 p ( 1∫ 0 ( 1 ts |f′(x)|q + 1 (1 − t)s |f′(b)|q ) dt )1 q ≤ m ( 1 1 −s )1 q ( 1 1 + pα )1 p [ (x−a)α+1 + (b−a)α+1 b−a ] . this completes the proof. � for α = 1, theorem 3.2 collapses to following result for s-godunova-levin function of second kind. corollary 3.2. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b]. if |f′|q is s-godunova-levin function of second kind on [a,b], p,q > 1, 1/p + 1/q = 1 and |f′(x)| ≤ m, then, we have∣∣∣∣∣∣f(x) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m ( 1 p + 1 )1 p ( 1 1 −s )1 q [ (x−a)2 + (b−x)2 b−a ] .(3.7) also, we have fractional ostrowski inequalities 171 i. if we take x = a+b 2 in (3.7), then we have the following mid-point inequality∣∣∣∣∣∣f ( a + b 2 ) − 1 b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a)2 ( 1 p + 1 )1 p ( 1 1 −s )1 q . ii. if we take x = a in (3.7), then we have the following inequality∣∣∣∣∣∣f(a) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a) ( 1 p + 1 )1 p ( 1 1 −s )1 q . iii. if we take x = b in (3.7), then we have the following inequality∣∣∣∣∣∣f(b) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ m(b−a) ( 1 p + 1 )1 p ( 1 1 −s )1 q . theorem 3.3. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b] and α > 0. if |f′|q is s-godunova-levin function of second kind on [a,b], q > 1 and |f′(x)| ≤ m, then, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ ≤ m ( 1 1 −s )1 q ( 1 1 + pα )1 p [ (x−a)α+1 + (b−a)α+1 b−a ] . proof. using lemma 2.1, well-known power mean inequality and the fact that |f′|q is s-godunova-levin function of second kind, we have∣∣∣∣∣ ((x−a)α + (b−x)α b−a ) f(x) − γ(α + 1) (b−a) [ jαx−f(a) + j α x+f(b) ]∣∣∣∣∣ = ∣∣∣∣∣(x−a) α+1 b−a 1∫ 0 tαf′(tx + (1 − t)a)dt− (b−x)α+1 b−a 1∫ 0 tαf′(tx + (1 − t)b)dt ∣∣∣∣∣ ≤ (x−a)α+1 b−a ( 1∫ 0 tαdt )1−1 q ( 1∫ 0 tα|f′(tx + (1 − t)a)|qdt )1 q + (b−x)α+1 b−a ( 1∫ 0 tαdt )1−1 q ( 1∫ 0 tα|f′(tx + (1 − t)b)|qdt )1 q ≤ m ( 1 α + 1 )1−1 q ( 1 1 + α−s − γ(1 −s)γ(α + 1) γ(2 + α−s) )1 q [ (x−a)α+1 + (b−a)α+1 b−a ] . this completes the proof. � when α = 1 in theorem 3.3, we have the following result. corollary 3.3. let f : [a,b] → r be a differentiable function on (a,b) with a < b and f′ ∈ l1[a,b] for all x ∈ [a,b]. if |f′|q is s-godunova-levin function of second 172 noor, noor and awan kind on [a,b], q > 1 and |f′(x)| ≤ m, then, we have∣∣∣∣∣∣f(x) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ mq (1 2 )1−1/q( 1 1 −s )1/q((x−a)2 + (b−x)2 b−a ) .(3.8) we now discuss some special cases. i. if we take x = a+b 2 in (3.8). then we have the following mid-point inequality∣∣∣∣∣∣f ( a + b 2 ) − 1 b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ mq (1 2 )1−1/q( 1 1 −s )1/q(b−a 2 ) . ii. if we take x = a in (3.8). then we have the following inequality∣∣∣∣∣∣f(a) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ mq(b−a) (1 2 )1−1/q( 1 1 −s )1/q . iii. if we take x = b in (3.8). then we have the following inequality∣∣∣∣∣∣f(b) − 1b−a b∫ a f(u)du ∣∣∣∣∣∣ ≤ mq(b−a) (1 2 )1−1/q( 1 1 −s )1/q . remark 3.1. we would like to mention here that one can extend the main results established in section 3 for the class of s-godunova-levin function of first kind. acknowledgements the authors would like to thank dr. s. m. junaid zaidi, rector, comsats institute of information technology, pakistan, for providing excellent research and academic environment. references [1] g. cristescu, l. lupsa, non-connected convexities and applications, kluwer academic publishers, dordrecht, holland, 2002. [2] g. cristescu, m. a. noor, m. u. awan, bounds of the second degree cumulative frontier gaps of functions with generalized convexity, carpath. j. math. 30(2), 2014. [3] s. s. dragomir, inequalities of hermite-hadamard type for h-convex functions on linear spaces, preprint. [4] s. s. dragomir, n-points inequalities of hermite-hadamard type for h-convex functions on linear spaces, preprint. [5] s. s. dragomir, r. p. agarwal, p. cerone, on simpsons inequality and applications, j. inequal. appl., 5, 533-579, 2000. [6] s. s. dragomir, j. pečarić and l. e. persson, some inequalities of hadamard type, soochow j. math, 21 (1995), 335-341. [7] s. s. dragomir, c. e. m. pearce, selected topics on hermite-hadamard inequalities and applications, victoria university, australia 2000. [8] s. s. dragomir, t. m. rassias, ostrowski type inequalities and applications in numerical integration, springer netherlands, 2002. [9] e. k. godunova and v. i. levin, inequalities for functions of a broad class that contains convex, monotone and some other forms of functions. (russian) numerical mathematics and mathematical physics (russian), 138-142, 166, moskov. gos. ped. inst., moscow, 1985. [10] u. n. katugampola, new approach to a generalized fractional integral, appl. math. comput. 218(3), 860-865 (2011). fractional ostrowski inequalities 173 [11] a. kilbas , h. m. srivastava, j. j. trujillo, theory and applications of fractional differential equations, elsevier b.v., amsterdam, netherlands, (2006). [12] w. liu, j. park, a generalization of the companion of ostrowski-like inequality and applications, appl. math. inf. sci. 7(1), 273-278 (2013). [13] d. s. mitrinovic, j. pecaric, note on a class of functions of godunova and levin, c. r. math. rep. acad. sci. can. 12, 33-36, (1990). [14] m. a. noor, m. u. awan, some integral inequalities for two kinds of convexities via fractional integrals, trans. j. math. mech., 5(2), (2013). [15] m. a. noor, k. i. noor, m. u. awan, geometrically relative convex functions, appl. math. infor. sci. 8(2), 607-616, (2014). [16] m. a. noor, k. i. noor, m. u. awan, s. khan, fractional hermite-hadamard inequalities for some new classes of godunova-levin functions, appl. math. infor. sci. (2014/15), inpress. [17] m. a. noor, k. i. noor, m. u. awan, j. li, on hermite-hadamard inequalities for h-preinvex functions, filomat, (2014), inpress. [18] m. a. noor, f. qi, m. u. awan, some hermitehadamard type inequalities for log −h-convex functions, analysis, 33, 367?75, (2013). [19] a. ostrowski, uber die absolutabweichung einer differentienbaren funktionen von ihren integralmittelwert, comment. math. hel, 10 (1938), 226-227. [20] m. radulescu. s. radulescu, p. alexandrescu, on the godunova-levin-schur class of functions, math. inequal. appl. 12(4), 853-862, (2009). [21] m. z. sarikaya, e. set, h. yaldiz and n. basak, hermite-hadamard’s inequalities for fractional integrals and related fractional inequalities, mathematical and computer modelling 57 (2013), 2403-2407. [22] e. set, new inequalities of ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, comput. math. appl. 63 (2012), no. 7, 1147?154. [23] m. tunc, ostrowski-type inequalities via h-convex functions with applications to speial means, j. ineq. appl. 2013,2013:326, doi:10.1186/1029-242x-2013-326. [24] y wang, s.-h wang, f. qi, simpson type integral inequalities in which the power of the absolute value of the first derivative of the integrand is s-preinvex, acta universities (nis) ser. math. inform. 28, 1-9, (2013). [25] b.-y xi, f. qi, integral inequalities of simpson type for logarithmically convex functions, to appear in advanced studies in contemporary mathematics. department of mathematics, comsats institute of information technology, park road, islamabad, pakistan ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 2, number 1 (2013), 38-53 http://www.etamaths.com application of ep-stability to impulsive financial model oyelami, benjamin oyediran and sam olatunji ale abstract. in this paper, we consider an impulsive stochastic model for an investment with production and saving profiles. the conditions for financial growth for the investment are investigated under impulsive action and results are obtained using the quantitative and ep stability methods. the impulsive stochastic differential equation considered is assumed to be driven by a process with jump and non-linear gestation properties. one of the results established shows that, in the long run, it is impossible for a financial investment to grow or dominates the prescribed average financial investment but has a threshold value for which the investment cannot grow beyond. it is also established that an ep− stable investment vector can be found which allows financial growth but this vector must be constrained to be in a given invariant set:it is advisable for the saving and depreciation to satisfy certain growth rates for proper income and investment growths. 1. introduction impulsive differential equations (ides) are systems that are subject to rapid changes in the variables describing them. impulses are noted to take place in different ways e.g. in the form of “shocks”, “jumps”, “mechanical impacts” etc. ([1]) and they take place for short moments during process of evolution ([1], [10], [1417]). many real life processes are impulsive in nature, examples are the biological bustling rhythms, the change in the states of the economy of some countries, the population under rapid changes, the outbreaks of earthquakes, eruption of epidemic in some ecological set-ups and so on ([1] & [16-17]). in the recent times, financial markets are places where funds are sourced for investment. many financial derivatives are traded under organized market system and trade over the counter. in the stock exchange market financial derivatives like plain vanillas, bonds and exotics options are traded. the volume of trade increases everyday and there is the need to analyze the performance of the market using some models. we need to determine the fair price of an option and payoff for the buyer. we need to understand the complex cash flow structures in the financial markets and the risks involve in managing the financial portfolios etc ([4], [6], [9],[13], [18] and [19]). most business organizations and many countries of the world usually set aside some substantial amount for investment or put in place some machineries to generate funds for investment. the instrument for raising money for investment often 2010 mathematics subject classification. 97m30. key words and phrases. ep-stability, impulsive financial model. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 38 application of ep-stability 39 take the form of sinking funds, treasury bills, capital stocks, national reserves etc. the goal of entrepreneurs are to make the investment have appreciable financial growths. it must be noted that sometimes, funds so invested experience favourable financial growth, if the investment atmosphere allows such growth. under certain situations which are often not easily predicted, the business many suffer unexpected economic recession, in which case, profit is lot in the investment period. some countries rely on natural resources such as fossil oil as the major export earner, for which the prices of crude oil tends to fluctuate every year, hence, the state of the economy of those countries tend to show impulsive behaviour ([14], [16]). furthermore, in the international market today, the price of fossil oil is impulsive because of rapid rise and falling of price of the oil within short period of times. moreover, profits made on investment on crude oil goes up and down in yo-yo way. therefore, the use of impulsive models would be useful for modelling prices of energy derivatives using different exotic options. the model also have potential applications for studying several real life problems in several fields of human endeavour that can be modeled using isdes with gestation function being taken to be zero. in view of the above, an investment may also be affected by impulsive phenomenon and how can it experience growth under impulsive effect? the search for the answer to this question is the motivation for the study in this paper. we will consider an impulsive stochastic model in studying the financial investment portfolio with production and saving profiles. we will use the ep-stability method, that is, stability relative to p-moment to the study model. application of p-moment to sdes are found in the literature but for the isdes we can say it is relatively new ([7]) if we consider the volume of publications made on sdes. it is worthy of note to say, the structure of the solution of isdes changes with equilibrium points of the model, hence the investment and national income equilibria depend on the process driving forces and the volatilities for the model. therefore, the stability of the financial model cannot be deduce in the ordinary stochastic sense. hence, we will exploit a broader stability concept, that is, stability relative to invariant sets to study the model. the importance of this approach were emphasized in many publications (see for examples [11-12] and [14], and references therein), however, we will unify this approach with ep-stability, such kind of approach had appeared for stochastic processes [17]) but as for impulsive systems the unified approach seems to be relatively new. 2. usefulness of impulsive phenomenon in the financial modeling in mathematics a variety of models exist for financial investment using applications of some theories in chaos, control, neural network, ecophysics and so on. most models from economics, the central problem is the concern about the interactions of many complex variables for which fundamental “scarce” variable is the capita ([2],[4],[6],[9],[13] and [16]). the model we will consider is one of the simplest impulsive mathematical model in economics, a “macro-model” with gestation lag and depreciation, it can be used as a simple model of company (or country) growth process and to demonstrate some fundamental relationships that exist among variables that quantify a financial investment using impulsive variables. 40 oyelami and ale we note that the impulsive system theory offers viable techniques to handle dwindling effect of the investment, for example, through impulsive system theory, we can identify factors responsible for rapid and irregular growth of the investment and also factors responsible for sudden drop in the investment at fixed and non-fixed investment periods ([10],[14],[16] and [20]). we will consider an impulsive stochastic model of a financial investment with income, capita stock and depreciation vector containing gestation lag. 3. preliminary treatment we shall denote by (ω,ξ,p) a probability space ω, being the set of points with events, which is a δ-algebra of subsets of ω such that ω ∈ ξ and p denotes the probability measure. let x(t) be a random process at time t with expectation (mean) ex(t) = x̄(t) and the variance is δ2x(t) = (x(t) − x̄(t))2. the autocovariant vector r(t,τ) = e(x(t)x(τ)). we shall say ([5]) the sequence of random process {xn(t)} converges to x(t) in probability almost surely as n → ∞, if for any ε > 0, δ > 0 there exists a number n such that p(|xn(t) −x(t)| > δ) < ε,n > n; and {xn(t)} is said to be convergent to x(t) with probability 1 as n →∞ if lim n→∞ p(|xn(t) −x(t)| = 0) = 1. crucial to our investigation is the following set c([−h, 0],<) which is the space of continuous random processes on [−h, 0] and taking values on < = (−∞, +∞) and let <+ = [0, +∞). pc([−h, 0],<) = {x : [−h, 0] →< is a piecewise continuous random process for t ∈ [−h, 0] such that it is left continuous at tk, k = 1, 2, . . .}. k = {a(r) : a ∈ c([−h, 0],<) are monotonically increasing in r and lim r→+∞ a(r) = ∞}. let v (t,x(t)) be a piecewise continuous random process on <+ ×c[−h, 0] and there exist wi ∈ k,i = 1, 2 such that the following conditions are satisfied: ω̄1(|ϕ(0)|) ≤ v (t,ϕ) ≤ ω̄2(|ϕ(0)|), ω̄i ∈ k, i = 1, 2 |v (t,ϕ1) −v (t,ϕ2)| ≤ c1(|ϕ1 −ϕ2|), c1 = constant, ϕi ∈ c([−h, 0],<+), i = 1, 2. then the dini derivative d+ (·)v (t,x(t)) of the function v (t,ϕ) along the solution path (·) for the isde in equation (·) is defined as d+ (·)v (t,x(t)) = limδ→0 sup 1 δ [v (t + δ,x(t) + δf) −v (t,x(t))]. consider the following isde dx(t) = f(t,x(t))dt + g(x(t−h))dw(t), t 6= tk, k = 0, 1, 2, . . . (1) 4x(tk) = i(x(tk)) 0 < t0 < t1 < t2 < · · · < tk; lim k→∞ tk = +∞ where application of ep-stability 41 f : <+ × ω → ω; g : ω → ω, i : ω → ∞. the expectation of v (t,x(t)) is ev (t,x(t)) and the variance δ2v (t,x(t)) and the p-moment epv (t,x) about the origin. 3.1. comparison system. consider the following comparison impulsive stochastic differential equations (cisde) corresponding to the eqn (1) dx(t) = e(t,u(t))dt + h(u(t))dw(t), t 6= γk, k = 0, 1, 2, . . . (2) 4u(γk) = i(u(γk)) 0 < γ0 < γ1 < γ2 < · · · < γk; lim k→∞ γk = +∞ where e : <+ × ω → ω; h,i : ω → ω. assume that f,e,h and i are smooth enough as to guarantee the existence and uniqueness of solutions of eqn (1) and eqn (2) (see [10] and [19]). we will make use of the following definitions: definition 1 ([11],[12] & [14]) let x(t) be the solution of eqn (1) passing through (t0,x(t0 + 0) = x0) then we say that the solution x(t) = 0 of the eqn (1). 1. ep-uniformly stable (u.s.) with respect to the invariant a = {x ∈ ω : |x| ≤ r} if (a) |x0| = r implies |exp(t)| = r,t ≥ t0; (b) ∀ ε > 0 and t0 ∈<+ there exists a real number δ = δ(ε) > 0 such that r − δ < |x0| < δ + r implies r −ε|exp(t)| < r + ε, t ≥ t0; 2. ep-uniformly asymptotically stable with respect to the invariant a if there exist real numbers δ0 > 0, and t = t(ε) > 0 such that r − δ0 < |x0| < r + δ + 0 implies r −ε < |exp(t)| < r + ε,t ≥ t0 + t remark 1 ep-stability is the unification of invariant stability and stability with respect to p-moment ([10-11] & [14]). if ex(t) = x̄(t) for p = 1 and r = 0, ep-stability reduces to the usual stability in the lyapunov’s sense for the impulsive stochastic equations. if the underlying variable is deterministic then the system is simply the impulsive ordinary differential equations. we will make use of the following auxiliary results. let x be a random variable such that e(x) = µ,e(x−µ)2 = σ2,e|x−µ|r = βr,e(|x−µ|) = 0 and δ̃ = µ/σ. 4. statement of the problem consider a simple impulsive stochastic differential model (isdm) dx(t) = δ−1y(t)dt− b + α1(t)g(x(t−h)) + σ1dw1(t), t 6= tk, k = 0, 1, 2, . . . (3) dy(t) = δ−1y(t)dt + α2(t)v (t) + σ2dw2(t), t 6= tk, k = 01, 2, . . . (4) dz(t) = βdx(t), t 6= tk, k = 0, 1, 2, . . . (5) 4x(tk) = βkx(tk) (6) satisfying the initial conditions x(t0 + 0) = x0,y(t0 + 0) = y0 and z(t0 + 0) = z0 (7) 42 oyelami and ale 0 < t1 < t2 < t3 < · · · < tk, tk = +∞, as k →∞ where (1) x(t) is the investment variable (2) y(t) is the national (company’s) income (3) z(t) is the capital stock (4) v(t) is the fluctuation variable (5) wi(t) are assume to be brownian processes. it is assumed that x(t) is the random variable representing in totality the amount of the investment (both liquid and solid asserts) which experience a growth rate of δβ−1 = saving capita output δ = saving ratio, δ−1 is the drift, b is given depreciation rate and g(x(t−h)) is the depreciation function with gestation lag h with the expectation eg (x(t−h)). g(x(t − h)) is generally assume to be nonlinear continuous random process. the expectation of g(x(t − h)) denote by g(x(t − h)) is assumed to exists. αi(t) are some jump parameters for i = 1, 2. the parameters βk = 1, 2, 3, . . . , account for the impulses that happen during the investment period. these parameters can be investment for some period of times. v (t) is assume to be statistically independent with respect to the investment variable, hence e(y(t),v (t)) = 0 and δ2(v (t),v (t)) = 1. we define f1(t,x(t)) = δ −1x(t) − bα1(t)g(x(t−h)) + σ1dw1(t) and f2(t,y(t)) = δ −1y(t) − bα2(t)v(t) + σ2dw2(t) dx(t) = a1(t)dt + b1(t)dw(t) and dy(t) = a2(t)dt + b2(t)dw2(t). then we define the stochastic differential equation corresponding to (isde) as df1 = (f1t + a1(t)f1x + 1 2 b21(t)f1xxdt + b1(t)f1xdw1(t) df2 = (f2t + a2(t)f2x + 1 2 b22(t)f2xxdt + b2(t)f2xdw2(t) 4x(tk) = βkx(tk) 4y(tk) = βky(tk) and integration give f1(t,x(t) = f1(0,x0) + ∫ t 0 (f1s + a1(s)f1x + 1 2 b21(s)f1xx)ds + ∫ t 0 b1(s)f1xdw1(s) + ∑ t0<tk<t βkx(tk) and f2(t,y(t) = f2(0,y0) + ∫ t 0 (f2s + a2(s)f2y + 1 2 b22(s)f2yy)ds + ∫ t 0 b2(s)f2ydw2(s) + ∑ t0<tk<t βky(tk) application of ep-stability 43 where f1t = ∂fi ∂t , fix = ∂fi ∂x , fitt = ∂2fi ∂t2 and fixx = ∂2fi ∂x2 for i = 1, 2. if αi(t) = 0, 4x(tk) = 0, the model is the black schole’s model for pricing options where α−1 and σi are the drift and the volatility respectively ([2], [4] & [9]). it is worthy of note to say that, it is not easy to obtain the equilibrium points for the problem. the solution structure of the problem changes with the equilibrium points. in general, but if g(x(t−h)) = x(t−h) and 4x(tk) = c = constant, this give rise to the following nonlinear equations δ−1x(t) − bα1(t)g(x(t−h)) + σ1dw1(t) = 0 δ−1y(t) − bα2(t)v(t) + σ2dw2(t) = 0 we solve for x(t) and y(t) to determine the equilibrium investment and the equilibrium national income. remark 2 the investment and national income equilibria change with time and both depend on the saving ratio. the two of them also depend on the jump and volatility parameters and the stochastic process driven forces as the above non-linear equations show. the above nonlinear equations can be solved using fixed point iterative process for stochastic processes. the major problem is the nonlinearity of the gestation radon variable g(x(t−h)). now let us consider the situation where g(x(t − h)) = x(t − h) that is g has a fixed point. if the volatilities of the investment and national income are zeros, i.e. σi = 0 for i = 1, 2 then an equilibrium for the system is the point t ∗ such that x(t∗) = bα1(t ∗)δx(t∗ − h) and y(t∗) = −α2(t∗)δv (t∗) in order to determine the values of x(t∗) and y(t∗) set s(t) = λt where λ = constant and t = 0, 1, 2, . . . then it is easy to show that λ = (bα1(t ∗)δ)1/h, thus x(t∗) varies with t∗. then the set of the equilibrium points for the model with zero volatilities (σi = 0) or simply the set e1 is e1 = {t∗ ∈ ir+ : x(t∗) = (bα1(t∗))t ∗/h and y(t∗) = −α2(t∗)δv (t∗), t∗ ∈ (tk, tk+1); x(tk) = x0 + nc, 4x(tk) = c and n is number of impulse points present in (tk, tk+1)} where x(t) is the solution of eqn (1) satisfying the initial condition x(0) = x0 for finite n and c. if the volatility is not equal to zero, that is, σi 6= 0, then we can obtain the set of the equilibrium points for the model for non-zero volatilities (σi 6= 0) or simply the set e2 as e2 = { t∗ ∈ j : ( x(t) = (bα1(t ∗)δ)t ∗/h+σ1dw1(t ∗) y(t) = bα1(t ∗)v (t∗)δ −σ2dw2(t∗) ) , t ∈ (tk, tk+1); x(tk) = x0 + nc,4x(tk) = c,y(tk) = y0 + nc, 4y(tk) = c and n is the number of impulse points present in (tk, tk+1)}. thus, the equilibria also depend on the process driving forces wi(t) and the volatilities σi for the model for i = 1, 2. since the equilibria for the (isdm) is generally changing from point to point, the stability of the financial model cannot be deduce in the ordinary sense. hence, later on, we will exploit a broader stability concept, that is, stability relative to invariant sets to study the model. the importance of this approach were emphasized in many publications (see for examples [11-12] & [14], and reference therein), however, we will unify this approach with ep stability, that is, stability relative to p−moment of vectors, such kind of approach 44 oyelami and ale had appeared for stochastic processes ([17]) but as for impulsive systems the unified approach seems to be relatively new. 4.1. discussion of the variables involved in the model. assume that no gestation lag or depreciation, i.e., b = 0 or g(x(t−h)) = 0 and the rate of increase of the capita stock equals investment. then the introduction of capita output ratio allows simple estimation of the production function of the business as express as ratio of the investment to the income. this simply mean that the investment (assumed equal to saving) shows a fixed ratio, δ to the income. by lord keynes ‘famous work of the general theory of employment interest and money; saving and investment are always equal’. therefore, in term of consumption, the income can be expressed as (1) income = consumption + saving (2) income = consumption + investment hence, from the above, it is easy to show the ratio of the consumption to the investment is equal to (α−1 − 1). in this model we shall assume that the depreciation function g(x(t − h)) is a nonlinear continuous random process. 4.2. nature of the solution to the investment model. the solution of equations (3 and 7) is a stochastic variable q(t) such that q(t) = q0 + ∫ t t0 f(s,q(s))ds for t 6= tk,k = 0, 1, 2, . . . (9) where f(t,q(t)) =   δ−1 0 0δ−1 0 0 0 β 0     x(t)y(t) z(t)   + h(t,q(t)) h(t,q(t)) :=   −bα1(t)g(x(t−h)) + σ1(t)w1(t) α2(t)v(t) + σ2(t)w2(t) 0   , q(t) :=   x(t)y(t) z(t)   . in eqn (9) the integration is made in the ito sense. the solution x(t) of eqn (1) is a continuous random process that satisfies the equation for t 6= tk, but for t = tk impulses take place and the solution starts to exhibit some kind of impulsive behaviour whose solution can also be obtained by slight modification of the results in ([10] & [17]) as follows: it is a random process whose solution satisfies the equation dq(t) = f(t,q(t))dt, t 6= tk, k = 0, 1, 2, . . . such that q(t+k ) = x(tk) + 4x(tk) where 4x(tk) = x(tk +0)−x(tk−0), x(tk +0) and x(tk−0) are the right and the left limits of x(t) at t = tk, k = 1, 2, 3, . . . respectively. assume that x(tk) = x(tk − 0). application of ep-stability 45 for characterization of the solution of the eqn (1) in terms of non-fixed moments of some kind of impulsive operator (see [1] & [10]). the appearance of impulses in this model, give rise to phenomenon such as “dying effect”, i.e., the solution cannot be continued across some barrier, “beating effect” i.e., pulses take place wherein the solution hits given hyper-surface several times and “lost of autonomy”, etc. (see [10] & [17] for detail) which financial models may likely prone to, such phenomena cannot be handled from the ordinary stochastic differential equations point of view (see [17]). the complexities in the behaviour of the solutions of most problems of impulsive family make it an interesting area of research focus of late. some of the behaviour were communicated in many publications (see [1] & [8]) and even our recent papers ([14 15]). the equation (3) and (7) can be used to find the solution to the model, since, the values for y(t) and z(t) can be obtained once x(t) is known. in this direction, the solution can be found by slight modification of the result in (see [3], [10] & [17]) x(t) = ∏ t0<tk<t (1 + βk)e δ−1(t+h)eδ −1h x(−h) + be−δ −1h ∫ t−h 0 ∏ t0<tk<t (1 + βk)α1(τ)e δ−1τg(x(τ))dτ + b ∫ t 0 ∏ t0<tk<t (1 + βk)α1(τ)e δ−1(τ+h)g(x(τ))dτ + ∫ t 0 σi ∏ t0<tk<t (1 + βk)e δ(τ+h)dw1(τ), t ≥ 0, τ := t + s (10) one crucial question one may ask at this point is: is it possible for the business setup to have perpetual financial growth if we assume that the business environment is favourable? the next proposition affirms that it is impossible for the business to have indefinite financial growth. proposition 1 let the following conditions be satisfied: there exist non-negative constants c1,c2,c3 and c4 such that: h1 : x(t) has a distribution such that ex(t) = 0 and e|x2(t)| < c4. h2 : a 2e2x(−h) exp b2 exp(e−a(t−h)t) ≤ c1 where a = ∏ t0<tk<t (1 + βk)e α−1(t+h)eα −1h,b = ∏ t0<tk<t (1 + βk)e α−1hb h3 : a 4e−4α −1he−2α −1 ex2(−h) + 2ab2e−2α −1h|x(−h)|exp ( 1 −e−α −1t α−1 ) ≤ c2 h4 : sup e|x2(τ)| ≤ max[c1,c2] < c4. then lim t→∞ 1 t2 ∫ t 0 p(|x(s)| ≥ ke|x2(s)|)ds = 0 proof 46 oyelami and ale let −h ≤ t ≤ 0 and α10 = max t∈(−h,0] α(t), replace x(t) by ex2(t) in eqn (10) and majorise the equation, then ex2(t) ≤ a2ex2(−h) + 2abx(−h) ∫ t−h 0 α1(τ)e −α−1τex(τ)dτ + b2 ∫ t−h 0 e2α −1τex2(τ)dτ + ∫ ∞ 0 e2α −1τex2(τ)dw1(τ). by the application of the gronwall-bellman’s equality, we have ex2(t) ≤ a2ex2(−h) exp b2 exp(α10e−α −1(t−h)t) ×σ1 ∫ t 0 e−2α −1τex2(τ)dw1(τ) ≤ c1 if t > 0, τ := t + s. then e|x2(t)| ≤ a4e4α −1he−2α −1t|ex2(−h)| + 2α10a 2b2ab2e−2α −1h|ex(−h)|σ1 ∫ t 0 e−α −1τex2(τ)dτ ≤ a4e−4α −1he−2α −1τc2 + 2ab 2e−2α −1h|ex(−h)|exp ( σ1 1 −e−α −1τ α−1 ) ≤ c3. it follows that e|x2(t)| ≤ sup τ∈(−h,0] e|x2(τ)| ≤ max(c1,c2) < c4 application of chebyshev’s inequality yields, p(|x(t)| ≥ ke|x2(t)|) 1 k2 , k > 0 then, lim t→∞ 1 t2 ∫ t 0 p(|x(s)| ≥ ke|x2(s)|) ≤ lim t→∞ 1 t2 ∫ t 0 1 k2 ds = lim t→∞ 1 t2 1 k2 t = 0. the result established above shows that, in the long run, “it is impossible for the investment to grow or dominates the prescribed average financial investment”. the behaviour of the financial investment under impulsive action is not necessarily be the same as the ordinary stochastic equations with prescribed probability distribution function, for example the markov, wienie, and martingale processes etc. let us assume that the impulsive processes have gamma distribution because of the relatively newness of the impulsive stochastic differential process and that their solutions behave as stated in the section 4.2. although processes that allow jump behaviours as poisson, levy and martingale can also be used to analyse the model. theorem 1 application of ep-stability 47 suppose the random variable x(t) in eqn (1) is a stochastic process with gamma distribution with parameter (n + 1,µ) and define the following constants: h1 : a = [ ∏ t0<tk<t (1 + βk) ]2 e2α −1hx(−h) h2 : e1x 2(t) = µn+1 n(µ−α−1)n+1 [ ∏ t0<tk<t (1 + βk) ]2 e2α −1hx(−h) h3 : e2x(t) = be −α−1h ∫ ∞ 0 ∫ 0 −h ∏ t0<tk<t (1 + βk)e −ατg(x(τ)) µ(µt)n n! e−µtdt h4 : e3x(t) = ∫ ∞ 0 ∫ 0 −h ∏ t0<tk<t (1 + βk)e −α−1t+hg(x(τ)) µ(µt)n n! tne−µtdt then, ex2(t) = e1x 2(t) + e2x 2(t) + e3x 2(t) proof from the eqn (10) and definitions of e1 and e2 we have e1x 2(t) = µn+1 n(µ−α1 )n+1 [ ∏ t0<tk<t (1 + βk) ]2 e2α −1hx(−h) e2x 2(t) = −b n! e−α −1h µn+1 ∫ 0 −h ∏ t0<tk<t (1 + βk)e −α−1τg(x(τ))dt ∫ ∞ 0 βne−βdβ = −b n! e−α −1h µn+1 γ(n− 1) ∫ 0 −h ∏ t0<tk<t (1 + βk)e −α−1τg(x(τ))dτ furthermore, we have the estimation ex2(t) = b ∫ ∞ 0 ∫ t 0 ∏ (1 + βk)e −α−1(τ+h)g(x(τ)dτ µn+1 n! tneµtdt = b n! γ(n) ∫ t 0 ∏ (1 + βk)e −α−1(τ+h)g(x(τ))dτ = b n! γ(n) ∏ (1 + βk) ∫ t 0 e−α −1τg(x(τ))dτ 48 oyelami and ale but ex2(t) = a4e2αtex2(−h) − 2a2eαt ∫ 0 −h be−ατe(x(h))g(x(τ))dτ∫ 0 −h ∫ 0 −h be−α(τ1+τ)e(g(x(τ1)))g(x(τ))dτ1dτ + ∫ t 0 ∫ t 0 b2e−α −1(τ1+τ)e(g(x(τ1)))g(x(τ))dτ1dτ + ∫ 0 −h ∫ t 0 b2e−α −1(τ1+τ)e(g(x(τ1)))g(x(τ))dτdτ1 + σ21 ∫ t 0 ∫ t 0 e−α(τ+τ1)r(τ,τ1)dτdτ1 where r(τ,τ1) is the autocovariant function. therefore, ex2(t) = e1x 2(t) + e2x 2(t) + e3x 2(t). an analogous result can be established for the model with gamma distribution with (n + 1,µ) parameter using the peek’s inequality as follows: if ex2(t) is bounded by a constant c5 such that ζ(t) has asymptotic expansion and define ex2(0) ex2(t) = a+ζ(t),ζ(t) := a0 +fraca1t + a1 t2 +· · · , ai, i = 0, 1, 2, . . . are some constants and a = exp(1 + a0) −a0 then by peek’s inequality it follows that p(x(t) ≥ λex2(t)) < 1 −ex2(t) λ2 − 2λex2(t) + 1 therefore, lim t→∞ 1 t ∫ t 0 p(x(s) = λex2(s))ds < lim t→∞ 1 t ∫ t 0 [ 1 −ex2(s) λ2 − 2λex2(s) + 1 ] ds ≤ lim t→∞ 1 t (∫ t 0 1 −ex2(s)ds −2λex2(s) ) = 1 λ [ lim t→∞ log e ( ex2(0) ex2(t) ) − 1 + ζ(t) ] = 1 λ lim t→∞ [ln(a + ζ(t)) − 1 + ζ(t)] = 1 λ [ln(a + a0) − 1 + a0] = 0. therefore, ex2(t) = ex2(0) exp−λ(a + 1) exp(−ζ(t)) hence, ex2(t) → 0 as t → ∞, a contradiction of 0 < |ex2(t)| < c5 for t ∈ [−h,t] for all t , which also contradicts the fact that |ex(t)| < c1, therefore lim t→∞ p(x(t) ≤ λex(t)) = 1. if the process has a gamma distribution as against the arbitrary situation proved in proposition 1, then the financial variable grows and tends to be bounded by λc5 in the long run. this simply affirms that the investment has a threshold value scaling factor of λc′5 for which the investment cannot grow beyond. application of ep-stability 49 the question of boundedness of the growth of the investment posses the question of investigating the problem from qualitative point of view, to be precise the stability of solution of the model will be investigated in relation to a given invariant set. in this direction, we will discuss the stability of the problem in relation to p−moment. corollary 1 h1 : suppose all the conditions in theorem 1 are satisfied h2 : e k := ∑∑∑ · · · ∑ θ ekeθ,θ are permutation over 1, 2, . . . ,n. then e2x(t) = n∑ i=1 n∑ j=1 eiei−jx(t) proof straight forward like theorem 1. suppose that the depreciation variable g(x(t)) is such that g(x(t)) = rx(t); or g(x(t)) = rx(t)x(t−h) ≤ rx2(t). then, e(x(−h)g(x(t)) ≤ re(x(−h)x(t)) ≤ rc4 and e(x(−h)h(x(t)) ≤ rx2(t). then e(x(−h))g(x(t)) ≤ re(x(−h)x(t)) ≤ rc4. we are now in the position to investigate ep-stability of the financial model. in the practical term, this implies given a solution x(t) to the financial investment in the invariant set, is it possible for the p-moment of the solution about the origin epx(t) to be found in the invariant set which satisfies the e-stability properties in the definition 1. in real life situation, this simply means that if right economic strategies that needed for the growth of the investment are used, then the investment can be made to grow so long as it is confined to the invariant set. in practice, the invariant set is the set in which the investment is constrained to and outside of it the investment may not be favourable. theorem 2 let x(t) ∈ pc([−h, 0],<) such that v (t,x(t)) is a continuous random process in [−h, 0] with expectation ev (t,x(t)) such that (1) h1 : |ev (t,x1(t)) −ev (t,x2(t))| ≤ k1|x1(t) −x2(t)|,k1 =constant (2) there exist β1,β2 ∈ k such that β2(e|x(t)|p) ≤ ev (t,x(t)) ≤ β1(e|x(t)|p) (3) dev (t,x(t)) ≤ g(ev (t,x(t)))dt (4) ev (tk + 0,x(tk) + 4x(tk)) ≤ ϕk(x(tk)) (5) βku = u + ϕ1(u), ϕ1 ∈ k the zero solution x(t) = 0 of eqn (1) is e-stable with respect to the invariant set a. proof. 50 oyelami and ale suppose that the zero solution x(t) = 0 of eqn (1) is e-stable with respect to the invariant set a, then ∀ε > 0, ∃δ = δ(x0,ε) such that |ex(t)| < ε implies that |x0| < δ0, there is a finite number p > 0 such that r − ε < |exp(t)| < r + ε for δ0 + r0 < |x0| < δ0 + r0. let |x0|p < r −δ0,a2(ε−r) > β1(r − 2δ0), for a2 and β0 ∈ k let m(t) be the solution of eqn (1) such that ṁ(t) ≤ g(t,m(t)), t 6= tk, k = 0, 1, 2, . . . m(t+k ) ≤ m(t) + ϕk(m(t)), t = tk, k = 0, 1, 2, . . . m(t+k ) ≤ u(t0) if r(t) = r(t,t0 + 0,u0) is a random process which is the maximal solution of the impulsive stochastic differential equation in eqn (1) then by standard results, v (t,x(t)) ≤ r(t) we can show that ev (t,x(t)) ≤ r(t). from eqn (2) ev (t,x(t)) ≤ ev (t0 + 0,x0) ≤ r(t) ≤ β1(e|x0|p) = β1(|x0|)p) ≤ β1(r − δ0) < β2(r −ε) therefore, e|x(t)|p < ε + r for t ≥ t0, by similar estimation we have e|x(t)|p > ε−r for r − δ < |x0 < δ + r. therefore the zero solution x(t) = 0 of the eqn (1) is e-stable with respect to the invariant set ω. the general form of distribution function governing an impulsive stochastic model is unknown, such distribution if exists may be continuous or discrete or even possess piecewise continuous property. recently, the theory of time scale have been exploited to study systems which are either continuous or discrete or both simultaneously ([11-12]). in the quest for an ideal distribution for impulsive stochastic system and the correspond p-moments are open problems. meanwhile, we define the characteristic function for x(t) taking into consideration impulsive tendency as cx(ε) = e[x(t)e −iεt] from the characteristic function we can obtain the k-moment as e { xk(t) } = 1 (iε)k [ dkcx(ε) dεk ] , k = 1, 2, . . . the construction of the ideal distribution may be made by the formulation of impulsive analogue of the chapman-kolmogorov equation if the underlying stochastic process is a markov process ([3]). the construction of an ideal distribution function for impulsive stochastic systems is one of the fundamental problems future research should focus on. because of the peculiar nature of the problem of how to determine cx(ε) we resort to investigate the behaviour of the solution of the model using qualitative approach, hence the problem will be studied from stability point of view. remark 3 we propose the monkey function which somehow shows the behaviour of a monkey in the game and can be used to mimic the dynamics of the financial market. application of ep-stability 51 we define the monkey process as m(t1, t) =   fk(t) t ∈ [t1, tk), k = 1, 2, . . . fs(t) t ∈ [tk, tn ), fs(t) ∈ c∞(ω) δ(tn − t) t = tn fj(t) t ∈ (tk+1, ts) 0 ≤ t1 < tk < tn ≤ ts   fk(t) is uniformly and identically distributed in the given interval fs(t) is a smooth random function in the given interval, and fj(t) is a continuous random process in the given interval δ(t) is the dirac function of t the monkey function should be constructed to have the following properties: h1 : m(t1, t) ≥ 0 for t1 ≥ t h2 : ∫ t 0 m(t1,s)ds = 1 h3 : m(t1, 0) = δ(−t1) h4 : dm = m(t1, t + 4t) −m(t1, t) the construction of an ideal distribution for an impulsive stochastic system using the monkey function and the correspond p-moments are open problems. 5. application of the e-stability without loss of generality if the depreciation variable is chosen in such a way that g(x(t − h)) ≥ x(t) and the fluctuation variable is selected to be bounded (v (t) ≤ k = constant) then the comparison equations corresponding to the isdm is dx(t) ≤ (δ−1 − bα1)x(t)dt + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . dy(t) ≤ δ−1y(t)dt + k + σ2dw2(t), t 6= γk, k = 0, 1, 2, . . . 4x(γk) ≤ βkx(γk). if m1(t) and m2(t) are the maximal solution to the comparison’s equation above respectively. then dm1(t) ≤ (δ−1 − bα1)m1(t)dt + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . dm2(t) ≤ δ−1m2(t)dt + k + σ1dw1(t), t 6= γk, k = 0, 1, 2, . . . 4x(tk) ≤ β1km1(γk) 4y(tk) ≤ β2km2(γk) m1(γk + 0) = x0 and m2(γk + 0) = y0 52 oyelami and ale whose solutions are found to be m1(t) = rm1 (t) = ∏ t0<tk<t (1 + β1k)e −(δ−1−bα1)(t−t0)x0 + ∫ t t0 σ1 ∏ t0<tk<t (1 + β1k)e −(δ−1−bα1)(s−t0)dw1(s) m2(t) = rm2 (t) = ∏ t0<tk<t (1 + β2k)e −(δ−1−bα1)(t−t0)y0 + k[δ − (t− t0)] ∫ t t0 σ2 ∏ t0<tk<t (1 + β2k)e −(δ−1−bα1)(s−t0)dw2(s). if (δ−1 − bα1) > 0, as time t →∞, the investment is bounded above by lim t→∞ ∫ t t0 σ1 ∏ t0<tk<t (1 + β1k)e −(δ−1−bα1)(s−t0)dw2(s) while the saving is unbounded below because of the linear term −kt. it is advisable to maintain the saving and the depreciation rates such that (δ−1 − bα1) < 0 for proper income and investment growths. the question we need to ask is: what conditions would guarantee financial e-stability? we will use the proposition 1 to investigate the required conditions to answer the question: let v (t,x(t)) = 1 2 ∫ t t0 (s2 + x2(s))ds + ∑ t0<tk<t pkx 2 k(tk) then, ev (t,x(t)) = ∫ ∞ 0 v (t,x(t))df where df is the distribution of the process describing the random process. therefore, ev (t,x(t)) = 1 2 ∫ ∞ 0 ∫ t 0 (s2 + x2(s))dsdf + ∑ t0<tk<t pk ∫ ∞ 0 x2k(s)ds and exp = ∫ ∞ 0 xp(s)df + ∑ t0<tk<t pkx 2p k (tk) it is easy to show that |ev (t,x1) −ev (t,x2)| ≤ 1 2 l1|x1 −x2||df| + 1 2 l2|pk||x1 −x2| = 1 2 (l1|f| + l2|pk|)|x1 −x2| pick β1(r) = r 2 and β2(r) = r p for p ≥ 2 and −4x(tk) ≤ x(tk + 0) ≤ 1 −4x(tk) it is not difficult to show that β1(e|x(t)|2) ≤ β2(e|x(t)|p) ≤ ev (t,x(t)) ≤ β1(e|x(t)|p). by theorem 2, this implies that the investment is e-stable with respect to the invariant set a. this result simply shows that we can find an e-stable investment vector which is constrained to a given invariant set in a which is e-stable. application of ep-stability 53 references [1] ale, s.o. and oyelami, b.o., impulsive systems and applications. int. j. math. edu. sci. technol., 2000, vol. no. 4, 539-544. [2] black, f. and scholes, m. , the pricing of options and corporate liabilities. journal of political economics, 81, 637-654(1973). [3] gardiner, c.w. and zoller, p., quantum noise: a handbook of markovian and nonmarkovian quantum stochastic methods with applications to quantum optics. second edition. berlin, springer-verlag 2000. [4] guilia lori, financial derivatives. ictp lecture notes on financial mathematics, 2007. [5] jacob, j. and shiriaev, a.n., limit theorems for stochastic processes. springer-verlag, 2nd edition, 2003. [6] jacqueline terra moura marins and eduardo saliby, crdit risk monte carlos simulation using creditmetrics model: the joint use of importance sampling and descriptive sample. banco central do brasil, march 2007. working paper series 132. [7] joao, p. hespanha and andrew r. teel., stochastic impulsive systems driven by renewal processes extended version. proc. int. sympo. in mathematical theory,2006, 1-26., available on , http://www.ece.ucsb.edu/hespanha\published/mtns06sto−ncs.pds. [8] juan j. neito, , periodic-boundary value problems for first order impulsive ordinary differential equations, nonlinear anal. 51(2002), 1223-1232. [9] hull, j.c. and while a., , efficient procedures for valuing european and american pathdependent options. journal of derivatives, 1, 21-31, 1993. [10] lakshmikantham, v., bainov, d.d. and simeonov, p.s., theorem of impulsive differential equations (singapore; world scientific). [11] lakshmikantham, v., stability of moving invariant sets and uncertain dynamical system, discrete contin, dynamic system, added vol. ii(1998), 24-31. [12] laksmikantham, v. and z. drici,, stability of conditionally invariant sets and controlled uncertain dynamical systems with time scales. math. prob. eng. 11 (1995), 1-10. [13] marco airoldi and mediobanca., a perturbative moment approach to option pricing., arxiv:cond-mat/0401503 v1, 2004. preprint. [14] oyelami, b.o., ale, s.o. and sesay, m.s., on existence of solutions and stability with respect to invariant sets for impulsive differential equations with variable times. advances in differential equations and control processes, 2008. [15] oyelami, b.o., on military model for impulsive reinforcement functions using exclusion and marginalization techniques. nonlinear analysis 35(1999), 947-958. [16] oyelami, b.o., impulsive systems and applications to some models. ph.d. thesis, abubakar tafawa balewa university of technology, bauchi, nigeria. 1999. [17] simeonov, p.s. and bainov, d.d., systems with impulsive effects; stability, theory and applications (ellis horwood), 1989. [18] sullivan, m.a., pricing discretely monitored barrier options. journal of computational finance, vol. 3, no. 4, summer, 35-52(2000). [19] umut cetin and rogers, l.c.g., modeling liquidity effect in discrete time. mathematical finance, vol. 17, no. 1 (january, 2007), 15-29. national mathematical centre abuja international journal of analysis and applications issn 2291-8639 volume 1, number 2 (2013), 100-105 http://www.etamaths.com a subordination theorem involving a multiplier transformation sukhwinder singh billing abstract. we, here, study a certain differential subordination involving a multiplier transformation which unifies some known differential operators. as a special case to our main result, we find some new results providing the best dominant for zp/f(z), z/f(z) and zp−1/f′(z), 1/f′(z). 1. introduction let a be the class of all functions f analytic in the open unit disk e = {z ∈ c : |z| < 1} and normalized by the conditions that f(0) = f′(0) − 1 = 0. thus, f ∈a has the taylor series expansion f(z) = z + ∞∑ k=2 akz k. let ap denote the class of functions of the form f(z) = zp + ∞∑ k=p+1 akz k,p ∈ n = {1, 2, 3, · · ·}, which are analytic and multivalent in the open unit disk e. note a1 = a. for f ∈ap, define the multiplier transformation ip(n,λ) as ip(n,λ)f(z) = z p + ∞∑ k=p+1 ( k + λ p + λ )n akz k, (λ ≥ 0,n ∈ n0 = n∪{0}). the operator ip(n,λ) has been recently studied by aghalary et al. [3]. i1(n, 0) is the well-known sălăgean [1] derivative operator dn, defined for f ∈a as under: dnf(z) = z + ∞∑ k=2 knakz k. for two analytic functions f and g in the unit disk e, we say that f is subordinate to g in e and write as f ≺ g if there exists a schwarz function w analytic in e with w(0) = 0 and |w(z)| < 1, z ∈ e such that f(z) = g(w(z)), z ∈ e. in case the function g is univalent, the above subordination is equivalent to: f(0) = g(0) and f(e) ⊂ g(e). 2010 mathematics subject classification. 30c80, 30c45. key words and phrases. differential subordination, multiplier transformation, analytic function; univalent function, p-valent function. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 100 a subordination theorem 101 let φ : c2 ×e → c be an analytic function, p be an analytic function in e such that (p(z),zp′(z); z) ∈ c2 × e for all z ∈ e and h be univalent in e. then the function p is said to satisfy first order differential subordination if (1) φ(p(z),zp′(z); z) ≺ h(z), φ(p(0), 0; 0) = h(0). a univalent function q is called a dominant of the differential subordination (1) if p(0) = q(0) and p ≺ q for all p satisfying (1). a dominant q̃ that satisfies q̃ ≺ q for each dominant q of (1), is said to be the best dominant of (1). obradovic̆ [2], introduced and studied the class n(α), 0 < α < 1 of functions f ∈a satisfying the following inequality < { f′(z) ( z f(z) )1+α} > 0, z ∈ e. he called it, the class of non-bazilevic̆ functions. in 2005, wang et al. [6] introduced the generalized class n(λ,α,a,b) of nonbazilevic̆ functions which is analytically defined as: n(λ,α,a,b) = { f ∈a : (1 + λ) ( z f(z) )α −λ zf′(z) f(z) ( z f(z) )α ≺ 1 + az 1 + bz , } where 0 < α < 1, λ ∈ c, −1 ≤ b ≤ 1, a 6= b, a ∈ r. wang et al. [6] studied the class n(λ,α,a,b) and made some estimates on( z f(z) )α . using the concept of differential subordination, shanmugam et al. [5] studied the differential operator (1+λ) ( z f(z) )α −λ zf′(z) f(z) ( z f(z) )α and obtained the best dominant for ( z f(z) )α . the main objective of this paper is to unify the above mentioned differential operators. for this, we establish a differential subordination involving the multiplier transformation ip(n,λ), defined above. as special cases of main theorem, we obtain best dominant for zp/f(z), z/f(z) and zp−1/f′(z), 1/f′(z) and some known results also appear as special cases to our main result. to prove our main result, we shall make use of the following lemma of miller and macanu [4]. lemma 1.1. let q be univalent in e and let θ and φ be analytic in a domain d containing q(e), with φ(w) 6= 0, when w ∈ q(e). set q(z) = zq′(z)φ[q(z)], h(z) = θ[q(z)] + q(z) and suppose that either (i) h is convex, or (ii) q is starlike. in addition, assume that (iii) < zh ′(z) q(z) > 0, z ∈ e. if p is analytic in e, with p(0) = q(0),p(e) ⊂ d and θ[p(z)] + zp′(z)φ[p(z)] ≺ θ[q(z)] + zq′(z)φ[q(z)], then p(z) ≺ q(z) and q is the best dominant. 102 billing 2. main results in what follows, all the powers taken are the principal ones. theorem 2.1. let α and β be non-zero complex numbers such that <(β/α) > 0 and let f ∈ap, ( zp ip(n,λ)f(z) )β 6= 0, z ∈ e, satisfy the differential subordination (2)( zp ip(n,λ)f(z) )β [ 1 + α−α ip(n + 1,λ)f(z) ip(n,λ)f(z) ] ≺ 1 + az 1 + bz + α β(p + λ) (a−b)z (1 + bz)2 , then ( zp ip(n,λ)f(z) )β ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e, and 1 + az 1 + bz is the best dominant. proof: on writing u(z) = ( zp ip(n,λ)f(z) )β , a little calculation yields that (3) ( zp ip(n,λ)f(z) )β [ 1 + α−α ip(n + 1,λ)f(z) ip(n,λ)f(z) ] = u(z) + α β(p + λ) zu′(z), define the functions θ and φ as follows: θ(w) = w and φ(w) = α β(p + λ) . clearly, the functions θ and φ are analytic in domain d = c and φ(w) 6= 0, w ∈ d. select q(z) = 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e and define the functions q and h as follows: q(z) = zq′(z)φ(q(z)) = α β(p + λ) zq′(z) = α β(p + λ) (a−b)z (1 + bz)2 , and (4) h(z) = θ(q(z))+q(z) = q(z)+ α β(p + λ) zq′(z) = 1 + az 1 + bz + α β(p + λ) (a−b)z (1 + bz)2 . a little calculation yields < ( zq′(z) q(z) ) = < ( 1 + zq′′(z) q′(z) ) = < ( 1 −bz 1 + bz ) > 0, z ∈ e, i.e. q is starlike in e and < ( zh′(z) q(z) ) = < ( 1 + zq′′(z) q′(z) + (p + λ) β α ) = < ( 1 −bz 1 + bz ) +(p+λ)< ( β α ) > 0, z ∈ e. thus conditions (ii) and (iii) of lemma 1.1, are satisfied. in view of (2), (3) and (4), we have θ[u(z)] + zu′(z)φ[u(z)] ≺ θ[q(z)] + zq′(z)φ[q(z)]. therefore, the proof follows from lemma 1.1. for p = 1 and λ = 0 in above theorem, we get the following result involving sălăgean operator. a subordination theorem 103 theorem 2.2. if α, β are non-zero complex numbers such that <(β/α) > 0. if f ∈a, ( z dnf(z) )β 6= 0, z ∈ e, satisfies ( z dnf(z) )β [ 1 + α−α dn+1f(z) dnf(z) ] ≺ 1 + az 1 + bz + α β (a−b)z (1 + bz)2 , −1 ≤ b < a ≤ 1, z ∈ e, then ( z dnf(z) )β ≺ 1 + az 1 + bz , z ∈ e. 3. dominant for zp/f(z), z/f(z) this section is concerned with the results giving the best dominant for zp/f(z) and z/f(z). select λ = n = 0 in theorem 2.1, we obtain the following result. corollary 3.1. let α, β be non-zero complex numbers such that <(β/α) > 0 and let f ∈ap, ( zp f(z) )β 6= 0, z ∈ e, satisfy (1 + α) ( zp f(z) )β −α zf′(z) pf(z) ( zp f(z) )β ≺ 1 + az 1 + bz + α pβ (a−b)z (1 + bz)2 , z ∈ e, then ( zp f(z) )β ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e. taking β = 1 in above theorem, we obtain: corollary 3.2. suppose that α is a non-zero complex number such that <(1/α) > 0 and suppose that f ∈ap, zp f(z) 6= 0, z ∈ e, satisfies (1 + α) zp f(z) −α zp+1f′(z) p(f(z))2 ≺ 1 + az 1 + bz + α p (a−b)z (1 + bz)2 , z ∈ e, then zp f(z) ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e. on writing α = −1 in corollary 3.1, we get: corollary 3.3. let β be a complex number with <(β) < 0 and let f ∈ap, ( zp f(z) )β 6= 0, z ∈ e, satisfy zf′(z) pf(z) ( zp f(z) )β ≺ 1 + az 1 + bz − 1 pβ (a−b)z (1 + bz)2 , −1 ≤ b < a ≤ 1, z ∈ e, then ( zp f(z) )β ≺ 1 + az 1 + bz , z ∈ e. selecting α = β = 1/2 in corollary 3.1, we get: 104 billing corollary 3.4. if f ∈ap, √ zp f(z) 6= 0, z ∈ e, satisfies √ zp f(z) ( 3 − zf′(z) pf(z) ) ≺ 2(1 + az) 1 + bz + 2 p (a−b)z (1 + bz)2 , z ∈ e, then √ zp f(z) ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e. taking p = 1 in corollary 3.2, we have the following result. corollary 3.5. if α is a non-zero complex number such that <(1/α) > 0 and if f ∈a, z f(z) 6= 0, z ∈ e, satisfies (1 + α) z f(z) −α z2f′(z) (f(z))2 ≺ 1 + az 1 + bz + α (a−b)z (1 + bz)2 , −1 ≤ b < a ≤ 1, z ∈ e, then z f(z) ≺ 1 + az 1 + bz , z ∈ e. setting p = 1 in corollary 3.3, we have the following result. corollary 3.6. if β is a complex number with <(β) < 0 and if f ∈a, ( z f(z) )β 6= 0, z ∈ e, satisfies zβ+1f′(z) (f(z))β+1 ≺ 1 + az 1 + bz − 1 β (a−b)z (1 + bz)2 , −1 ≤ b < a ≤ 1, z ∈ e, then ( z f(z) )β ≺ 1 + az 1 + bz , z ∈ e. setting p = 1 in corollary 3.1, we obtain, below, the result of shanmugam et al. [5]. corollary 3.7. if α, β are non-zero complex numbers such that <(β/α) > 0. if f ∈a, ( z f(z) )β 6= 0, z ∈ e, satisfies (1 + α) ( z f(z) )β −αf′(z) ( z f(z) )1+β ≺ 1 + az 1 + bz + α β (a−b)z (1 + bz)2 , z ∈ e, then ( z f(z) )β ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e. 4. dominant for zp−1/f′(z), 1/f′(z) we obtain here, the best dominant for zp−1/f′(z) and 1/f′(z) as special cases to our main result. select λ = 0 and n = 1 in theorem 2.1, we obtain: a subordination theorem 105 corollary 4.1. let α, β be non-zero complex numbers such that <(β/α) > 0 and let f ∈ap, ( pzp−1 f′(z) )β 6= 0, z ∈ e, satisfy (1 +α) ( pzp−1 f′(z) )β − α p ( 1 + zf′′(z) f′(z) )( pzp−1 f′(z) )β ≺ 1 + az 1 + bz + α pβ (a−b)z (1 + bz)2 , z ∈ e, then ( pzp−1 f′(z) )β ≺ 1 + az 1 + bz , −1 ≤ b < a ≤ 1, z ∈ e. taking β = 1 in above theorem, we obtain: corollary 4.2. suppose that α is a non-zero complex number such that <(1/α) > 0 and suppose that f ∈ap, pzp−1 f′(z) 6= 0, z ∈ e, satisfies (1 + α) pzp−1 f′(z) −α zp−1 f′(z) ( 1 + zf′′(z) f′(z) ) ≺ 1 + az 1 + bz + α p (a−b)z (1 + bz)2 , z ∈ e, then zp−1 f′(z) ≺ 1 + az p(1 + bz) , −1 ≤ b < a ≤ 1, z ∈ e. taking p = 1 in corollary 4.2, we have the following result. corollary 4.3. if α is a non-zero complex number such that <(1/α) > 0 and if f ∈a, 1 f′(z) 6= 0, z ∈ e, satisfies 1 f′(z) ( 1 −α zf′′(z) f′(z) ) ≺ 1 + az 1 + bz + α (a−b)z (1 + bz)2 , −1 ≤ b < a ≤ 1, z ∈ e, then 1 f′(z) ≺ 1 + az 1 + bz , z ∈ e. references [1] g. s. sălăgean, subclasses of univalent functions, lecture notes in math., 1013 362–372, springer-verlag, heideberg,1983. [2] m. obradovic̆, a class of univalent functions, hokkaido mathematical journal, 27(2)(1998) 329–335. [3] r. aghalary, r. m. ali, s. b. joshi and v. ravichandran, inequalities for analytic functions defined by certain linear operators, int. j. math. sci., 4(2005) 267–274. [4] s. s. miller and p. t. mocanu, differential suordinations : theory and applications, (no. 225), marcel dekker, new york and basel, 2000. [5] t. n. shanmugam, s. sivasubramanian and h. silverman, on sandwich theorems for some classes of analytic functions, international j. math. and math. sci., article id 29684(2006) pp.1–13. [6] z. wang, c. gao and m. liao, on certain generalized class of non-bazilevic̆ functions, acta mathematica academiae paedagogicae nýıregyháziensis, new series, 21(2)(2005) 147–154. department of applied sciences, baba banda singh bahadur engineering college, fatehgarh sahib-140 407, punjab, india international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 15-21 http://www.etamaths.com some new estimates of hermite-hadamard inequalities for harmonically convex functions with applications wen wang1,2,∗ and jibing qi1 abstract. in this paper, we first establish an integral identity. further, using this identity, some new estimates for hermite-hadamard inequalities for harmonically convex functions are established. finally, some applications to special mean are showed. 1. introduction in this article, let r = (−∞,∞), r++ = (0,∞). theory of convex functions and theory of inequalities are closely related to each other. therefore, some literature on inequalities can be found for convex functions. one of the most extensively research on inequalities is hermite-hadamard type inequalities. definition 1.1 ( [1, 2]) a function f : i ⊂ r → r is convex function on i, if f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ i,t ∈ [0, 1]. (1.1) f is concave function if −f is convex function. let f : i ⊂ r → r be a convex function. the following inequality is the well-known hermitehadamard’s inequality f ( a + b 2 ) ≤ 1 b−a ∫ b a f(x)dx ≤ f(a) + f(b) 2 , a,b ∈ i with a < b. (1.2) estimates for hermite-hadamard inequality for convex functions are studied in a rich literature [8–16]. theorem 1.2 ( [1]) let f : i0 ⊂ r → r be a differentiable mapping on i0 and a,b ∈ i0 with a < b. if |f′(x)|q is a convex function for q > 1, then∣∣∣∣∣f(a) + f(b)2 − 1b−a ∫ b a f(x)dx ∣∣∣∣∣ ≤ (b−a)(|f ′(a)| + |f′(b)|) 8 . (1.3) recently, i̇scan [3] introduced the concept of harmonically convex functions and established hermitehadamard type inequality for harmonically convex functions. definition 1.3 ( [3, 4]) a function f : i ⊂ r\{0} → r is said to be harmonically convex function on i, if f ( 1 tx−1 + (1 − t)y−1 ) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ i,t ∈ [0, 1]. (1.4) f is said to be harmonically concave function if −f is harmonically convex function. theorem 1.4 ( [3, 4]) let f : i ⊂ r\{0} → r be a harmonically convex function a,b ∈ i with a < b. if f ∈ l[a,b] , then f ( 2ab a + b ) ≤ ab b−a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 . (1.5) received 1st july, 2016; accepted 18th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d15, 26a51. key words and phrases. hermite-hadamard’s inequality; harmonically convex function; estimate; mean; inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 15 16 wen wang, jibing qi theorem 1.5 ( [3, 4]) let f : i ⊂ r\{0}→ r be a harmonically convex function a,b ∈ i with a < b and f′ ∈ l[a,b]. if |f′(x)|q is harmonically convex on [a,b] for q > 1, 1 p + 1 q = 1, then∣∣∣∣∣f(a) + f(b)2 − abb−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ ab(b−a) 2 ( 1 p + 1 )1 p (µ1|f′(a)|q + µ2|f′(b)|q) 1 q , (1.6) where µ1 = [ a2−2q + b1−2q[(b−a)(1 − 2q) −a] ] 2(b−a)2(1 −q)(1 − 2q) , µ2 = [ a2−2q + b1−2q[(b−a)(1 − 2q) − b] ] 2(b−a)2(1 −q)(1 − 2q) . for many recent results related to hermite-hadamard type inequalities for harmonically functions, see [3–7]. the aim of this paper is first to establish a integral identity. then, using this identity, some new estimates for hermite-hadamard inequalities for harmonically convex functions are established by i̇. i̇scan in [3] are derived. 2. some lemmas in order to prove our main results we need some lemmas. lemma 2.1 let f : i ⊂ r\{0}→ r be a differentiable function on i0 and a,b ∈ i0 with a < b. if f′ ∈ l[0, 1], then for λ ∈ [0, 1], one has (1 −λ)f ( ab (1 −λ)a + λb ) + λf ( ab (1 −λ)a + λb ) − ab b−a ∫ b a f(x) x2 dx = b−a ab [∫ 1−λ 0 t [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt − ∫ 1 1−λ t− 1 [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt ] . (2.1) proof. let x = 1 tb−1+(1−t)a−1 , t ∈ [0, 1] and x ∈ [a,b], then( 1 a − 1 b )∫ 1−λ 0 t (tb−1 + (1 − t)a−1)2 f′ ( 1 tb−1 + (1 − t)a−1 ) dt = tf ( 1 tb−1 + (1 − t)a−1 )∣∣∣∣1−λ 0 − ∫ u a f(x) x2 ab b−a dx = (1 −λ)f ( ab (1 −λ)a + λb ) − ab b−a ∫ u a f(x) x2 dx. (2.2) and ( 1 b − 1 a )∫ 1 1−λ t (tb−1 + (1 − t)a−1)2 f′ ( 1 tb−1 + (1 − t)a−1 ) dt = tf ( 1 tb−1 + (1 − t)a−1 )∣∣∣∣1 1−λ − ∫ b u f(x) x2 ab b−a dx = λf ( ab (1 −λ)a + λb ) − ab b−a ∫ b u f(x) x2 dx, (2.3) where u = 1 (1−λ)b−1+λa−1 . so (7) follows from (8) and (9).� � new estimates of hermite-hadamard inequalities 17 remark 2.2 from (7) we derive the following two identities. f ( 2ab a + b ) − 2ab b−a ∫ b a f(x) x2 dx = 2(b−a) ab [∫ 1 2 0 t [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt − ∫ 1 1 2 t− 1 [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt ] ; (2.4) f(a) + f(b) 2 − ab b−a ∫ b a f(x) x2 dx = b−a 2ab [∫ 1 0 t [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt − ∫ 1 0 t− 1 [tb−1 + (1 − t)a−1] f′ ( 1 tb−1 + (1 − t)a−1 ) dt ] . (2.5) proof. take λ = 1 2 in (7), we can derive (10). we respectively take λ = 0 and λ = 1 in (7) and add two inequalities, then (11) is obtained. � lemma 2.3 by integral calculation, then c1(a,b,λ) = ∫ 1−λ 0 t(1 − t)[tb + (1 − t)a]2dt = (1 −λ)3 [ 1 5 (b−a)2(1 −λ)2 + 1 2 a(b−a)(1 −λ) + 1 3 λ2 ] ; (2.6) c2(a,b,λ) = ∫ 1−λ 0 t2[tb + (1 − t)a]2dt = (1 −λ)2 [( 1 20 + 1 5 λ ) (b−a)2(1 −λ)2 + ( 1 6 + 1 2 λ ) a(b−a)(1 −λ) + ( 1 6 + 1 3 λ ) λ2 ] ; (2.7) c3(a,b,λ) = ∫ 1 1−λ t(1 − t)[tb + (1 − t)a]2dt = ∫ λ 0 (1 −u)u[(1 −u)b + ua]2du = − 1 5 (b−a)2λ5 + 1 4 (a2 + 3b2 − 4ab)λ4 + 1 3 (2ab− 3b2)λ3 + 1 2 b2λ2; (2.8) c4(a,b,λ) = ∫ 1 1−λ (1 − t)2[tb + (1 − t)a]2dt = ∫ λ 0 u2[(1 −u)b + ua]2du = 1 5 (b−a)2λ5 + 1 2 b(a− b)λ4 + 1 3 b2λ3, (2.9) where u = 1 − t. 3. main results our main results are stated as follows. 18 wen wang, jibing qi theorem 3.1 let f : i ⊂ r\{0} → r be a differentiable function and f′ ∈ l[a,b]. if |f′(x)|q is harmonically convex on [a,b] with 0 ≤ a < b for q ≥ 1, then∣∣∣∣∣(1 −λ)f ( ab (1 −λ)a + λb ) + λf ( ab (1 −λ)a + λb ) − ab b−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ b−a ab { (c1(a,b,λ)) 1−1 q [ |f′(b)|qc1(a,b,λ) + |f′(a)|qc2(a,b,λ) ]1 q + (c3(a,b,λ)) 1−1 q [ |f′(b)|qc3(a,b,λ) + |f′(a)|qc4(a,b,λ) ]1 q } . (3.1) proof. from lemma 2.1 and using hölder inequality, further, since |f′(x)|q is harmonically convex on [a,b], we have i = ∣∣∣∣∣(1 −λ)f ( ab (1 −λ)a + λb ) + λf ( ab (1 −λ)a + λb ) − ab b−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ b−a ab [∫ 1−λ 0 t [tb−1 + (1 − t)a−1] ∣∣∣∣f′ ( 1 tb−1 + (1 − t)a−1 ) dt ∣∣∣∣ + ∫ 1 1−λ 1 − t [tb−1 + (1 − t)a−1] ∣∣∣∣f′ ( 1 tb−1 + (1 − t)a−1 ) dt ∣∣∣∣ ] ≤ b−a ab  (∫ 1−λ 0 t [tb−1 + (1 − t)a−1]2 dt )1−1 q (∫ 1−λ 0 t [tb−1 + (1 − t)a−1]2 ∣∣∣∣f′ ( 1 tb−1 + (1 − t)a−1 )∣∣∣∣q dt )1 q + (∫ 1 1−λ 1 − t [tb−1 + (1 − t)a−1]2 dt )1−1 q ∫ 1 1−λ 1 − t [tb−1 + (1 − t)a−1]2 ∣∣∣∣f′ ( 1 tb−1 + (1 − t)a−1 )∣∣∣∣q dt ] ≤ b−a ab   (∫ 1−λ 0 t [tb−1 + (1 − t)a−1]2 dt )1−1 q [∫ 1−λ 0 t [ t|f′(b)|q + (1 − t)|f′(a)|q ] [tb−1 + (1 − t)a−1]2 dt ]1 q + (∫ 1 1−λ 1 − t [tb−1 + (1 − t)a−1]2 dt )1−1 q [∫ 1 1−λ (1 − t) [ t|f′(b)|q + (1 − t)|f′(a)|q ] [tb−1 + (1 − t)a−1]2 dt ]1 q   . noticing that [tb−1 + (1 − t)a−1]−1 ≤ ta + (1 − t)b, and using (12-15), then, from above inequality we have i ≤ b−a ab   (∫ 1−λ 0 t[tb + (1 − t)a]2dt )1−1 q [∫ 1−λ 0 t[tb + (1 − t)a]2 [ t|f′(b)|q + (1 − t)|f′(a)|qdt ]]1q + (∫ 1 1−λ (1 − t)[tb + (1 − t)a]2dt )1−1 q [∫ 1 1−λ (1 − t)[tb + (1 − t)a]2 [ t|f′(b)|q + (1 − t)|f′(a)|q ] dt ]1 q } = b−a ab  (c1(a,b,λ))1−1q [∫ 1−λ 0 [ |f′(b)|qt2[tb + (1 − t)a]2 + |f′(a)|qt(1 − t)[tb + (1 − t)a]2 ] dt ]1 q + (c3(a,b,λ)) 1−1 q [∫ 1 1−λ [ |f′(b)|qt(1 − t)[tb + (1 − t)a]2 + |f′(a)|q(1 − t)2[tb + (1 − t)a]2 ] dt ]1 q } = b−a ab { (c1(a,b,λ)) 1−1 q [ |f′(b)|qc1(a,b,λ) + |f′(a)|qc2(a,b,λ) ]1 q + (c3(a,b,λ)) 1−1 q [ |f′(b)|qc3(a,b,λ) + |f′(a)|qc4(a,b,λ) ]1 q } . so the proof is complete. � new estimates of hermite-hadamard inequalities 19 corollary 3.2 assume that all the assumptions of theorem 3.1 are satisfied. if we take q = 1 and m = max{|f′(a)|, |f′(b)|}, then∣∣∣∣∣(1 −λ)f ( ab (1 −λ)a + λb ) + λf ( ab (1 −λ)a + λb ) − ab b−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ m(b−a) ab [ c1(a,b,λ) + c2(a,b,λ) + c3(a,b,λ) + c4(a,b,λ) ] . (3.2) corollary 3.3 assume that all the assumptions of theorem 3.1 are satisfied. if we take λ = q = 1 and m = max{|f′(a)|, |f′(b)|}, then∣∣∣∣∣f(a) − abb−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ m(b−a)ab [ 1 4 (b−a)2 + 1 6 b(b− 4a) ] . (3.3) corollary 3.4 assume that all the assumptions of theorem 3.1 are satisfied. if we take λ = 0, q = 1 and m = max{|f′(a)|, |f′(b)|}, then∣∣∣∣∣f(b) − abb−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ m(b−a)ab [ 1 4 (b−a)2 + 2 3 a(b−a) ] . (3.4) corollary 3.5 assume that all the assumptions of theorem 3.1 are satisfied. if we take λ = 1 2 , q = 1 and m = max{|f′(a)|, |f′(b)|}, then∣∣∣∣∣f ( 2ab a + b ) − ab b−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ m(b−a)ab × { c1(a,b, 1 2 ) [ c1(a,b, 1 2 ) + c2(a,b, 1 2 ) ] + c3(a,b, 1 2 ) [ c3(a,b, 1 2 ) + c4(a,b, 1 2 ) ]} . (3.5) remark 3.6 by (18) and (19) we obtain the following inequality∣∣∣∣∣f(a) + f(b)2 − abb−a ∫ b a f(x) x2 dx ∣∣∣∣∣ ≤ m(b−a)ab [ 1 12 (4b2 − 6ab−a2) ] . (3.6) 4. some applications for special means let a,b are two nonnegative number with a < b. let us recall the following special means of a and b. (1) the arithmetic mean a = a(a,b) := a+b 2 ; (2) the geometric mean g = g(a,b) := √ ab; (3) the harmonic mean h = h(a,b) := 2ab a+b ; (4) the logarithmic mean l = l(a,b) := b−a ln b−ln a; (5) the -logarithmic mean lp = lp(a,b) := ( bp+1 −ap+1 (p + 1)(b−a) )1 p , a 6= b,p ∈ r,p 6= 0,−1. these means are often applied to numerical approximation and in other areas. however, the following simple relationships are known in the literature: h ≤ g ≤ l ≤ a. (4.1) it is also known that lp is monotonically increasing respecting to p ∈ r, denoting l0 = i and l−1 = l. 20 wen wang, jibing qi proposition 1. let 0 < a < b. then we obtain the following inequalities:∣∣∣∣h − g2l ∣∣∣∣ ≤ b−aab × { c1(a,b, 1 2 ) [ c1(a,b, 1 2 ) + c2(a,b, 1 2 ) ] + c3(a,b, 1 2 ) [ c3(a,b, 1 2 ) + |c4(a,b, 1 2 ) ]} ;∣∣∣∣a− g2l ∣∣∣∣ ≤ b−aab [ 1 12 (4b2 − 6ab−a2) ] . proof. the assertion follows from inequalities (20) and (21), respectively, for f(x) = x, x ∈ r++. � proposition 2. let 0 < a < b. then we derive the following inequalities:∣∣h2 −g2∣∣ ≤ 2(b−a) a × { c1(a,b, 1 2 ) [ c1(a,b, 1 2 ) + c2(a,b, 1 2 ) ] + c3(a,b, 1 2 ) [ c3(a,b, 1 2 ) + |c4(a,b, 1 2 ) ]} ;∣∣∣∣a(a2,b2) − g2l ∣∣∣∣ ≤ 2(b−a)a [ 1 12 (4b2 − 6ab−a2) ] . proof. the assertion follows from inequalities (20) and (21), respectively, for f(x) = x2, x ∈ r++. � proposition 3. let 0 < a < b. then we have the following inequalities:∣∣hp+2 −lppg2∣∣ ≤ 2bp+2(b−a)ab × { c1(a,b, 1 2 ) [ c1(a,b, 1 2 ) + c2(a,b, 1 2 ) ] + c3(a,b, 1 2 ) [ c3(a,b, 1 2 ) + |c4(a,b, 1 2 ) ]} ; ∣∣a(ap+2,bp+2) −lppg2∣∣ ≤ 2bn+2(b−a)ab [ 1 12 (4b2 − 6ab−a2) ] . proof. the assertion follows from inequalities (20) and (21), respectively, for f(x) = xp+2, x ∈ r++ and p ∈ (−1,∞)\{0}. � references [1] s. s. dragomir and r. p. agarwal, two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, appl. math. lett. 5 (1998), 91-95. [2] j. e. pečarić, f. proschan, y. l. tong, convex functions, partial orderings and statistical applications, academic press, 1991. [3] i̇. i̇scan, hermite-hadamard type inequalities for harmonically convex functions, hacettepe journal of mathematics and statistics, 43 (6) (2014), 935-942. [4] e. set and i̇. i̇scan, hermite-hadamard type inequalities for harmaonically convex functions on the co-ordinates, arxiv:1404.6397v1 [math.ca]. [5] i̇. i̇scan, s. wu, hermite-hadamard type inequalities for harmonically convex functions via fractional intergrals, applied mathematics and computation, 238 (2014), 237-244. [6] i̇. i̇scan, hermite-hadamard and simpson-like type inequalities for differentiable harmonically convex functions, arxiv:1310.4851v1 [math.ca]. [7] i̇. i̇scan, hermite-hadamard and simpson-like type inequalities for differentiable harmonically convex functions, journal of mathematics, 2014 (2014), article id 346305. [8] i̇. i̇scan, generalization of different type integral inequalities for s-convex functions via fractional integrals, applicable analysis, 93 (9) 2014, 1846-1862. [9] g. toad er, some generalizations of the convexity, proceedings of the colloquium on approximation and optimization, univ. cluj-napoca, cluj-napoca, 1985, 329-338. [10] f. x. chen and s. h. wu, some hermite-hadamard type inequalities for harmonically s-convex functions, the scientific world journal, 2014 (2014), article id 279158. [11] tian-yu zhang, feng qi, integral inequalities of hermite-hadamard type for m-ah convex functions, turkish journal of analysis and number theory, 3(2) 2014, 60-64. [12] u. s. kirmaci, inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, appl. math. comp. 147 (2004), 137-146. [13] kuei-lin tseng, shiow-ru hwang, s. s. dragomirc, new hermite-hadamard-type inequalities for convex functions (ii), comput. math. appl. 62 (2011), 401-418. new estimates of hermite-hadamard inequalities 21 [14] constantin p. niculescu, the hermite-hadamard inequality for log-convex functions, nonlinear analysis, 75 (2012), 662-669. [15] w. wang, s. g. yang, schur m-power convexity of a class of multiplicatively convex functions and applications, abstract and applied analysis, 2014 (2014), article id 258108. [16] w. wang, i̇. i̇scan, h. zhou, fractional integral inequalities of hermite-hadamard type for m-hh convex functions with applications, advanced studies in contemporary mathematics (kyungshang). 26 (3) (2016), 501-512. 1school of mathematics and statistics, hefei normal university, hefei 230601,p.r.china 2school of mathematical science, university of science and technology of china, hefei 230026, china ∗corresponding author: wenwang1985@163.com, wwen2014@mail.ustc.edu.cn 1. introduction 2. some lemmas 3. main results 4. some applications for special means references int. j. anal. appl. (2022), 20:23 received: feb. 14, 2022. 2010 mathematics subject classification. 65c20. key words and phrases. bounded model; mathematical statistics; probability model; entropy; simulation; unit interval data. https://doi.org/10.28924/2291-8639-20-2022-23 © 2022 the author(s) issn: 2291-8639 1 on some properties of a new truncated model with applications to lifetime data muhammad zeshan arshad1, oluwafemi samson balogun2,*, muhammad zafar iqbal1, pelumi e. oguntunde3 1department of mathematics and statistics, university of agriculture, faisalabad, pakistan 2school of computing, university of eastern finland, kuopio, northern europe, 70211, finland 3department of mathematics, covenant university, ota, ogun state, nigeria *corresponding author: samson.balogun@uef.fi abstract. this research explored the exponentiated left truncated power distribution which is a bounded model. various statistical properties which include the moments and their associated measures, bonferroni and lorenz curves, reliability measures, shapes, quantile function, entropy, and order statistics were discussed in detail. a simulation study was provided and applications to two real-world data were considered. the performance of the exponentiated left truncated power distribution over other bounded models like toppleone distribution, beta distribution, kumaraswamy distribution, lehmann type–i distribution, lehmann type–ii distribution, generalized power function, weibull power function, and mustapha type–ii distribution is quite commendable. 1. introduction probability models play important roles in describing real-life events. they have been discussed in the past to model several real-time events so proficiently. the rainfall event was addressed by [1]. pollution events were addressed by ([2], [3], [4]). manifold dynamics of covid-19 were addressed by ([5], [6]). engineering issues were addressed by ([7], [8]), and several others. some https://doi.org/10.28924/2291-8639-20-2022-23 2 int. j. anal. appl. (2022), 20:23 probability models are bounded while some are unbounded. unbounded distributions extend from negative infinity to positive infinity while bounded ones are confined to lie between two determined values. according to [9], probability models with unit intervals are useful in the area of biology, economics, engineering, and psychology among others. examples of bounded models include the continuous uniform distribution, beta distribution, kumaraswamy distribution by [10], [11], [12], [13], and several other notable ones. it is also worthy of note that some of these bounded probability models have been used to develop generalized families of distributions, examples include the beta-g family of distributions by [14], kumaraswamy-g family of distributions by [15], topp-leone g family of distributions by ([16], [17]), and so on. a quest to develop models that can adequately fit real-life events has led to the extension of the existing probability models. 1.1. definition a random variable x is said to follow the eltr-pf distribution if the associated cumulative distribution function (cdf) and corresponding probability density function (pdf) begin at k, and are given respectively by; 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = ( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ,𝑘 < 𝑥 < 1,𝑎,𝑏 > 0, (1) 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥 𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 , (2) where 𝑘 < 𝑥 is a possible minimum assured life, and it can be defined as an unknown starting point at which age of some certain component/device initiates, and (𝑎,𝑏 > 0) are two shape parameters. however, if parameters b=1 and k=0, the model reduces to the baseline model (𝑥𝑎). this research is aimed at extending the power function and introducing a new bounded probability model; the exponentiated left truncated power (eltr-pf) function which can be used as an alternative to the existing ones because of its superior modeling capabilities. its properties are identified, a simulation and real-life applications are provided. the rest of the paper is structured as follows; general mathematical properties of the eltr-pf distribution including reliability measures are derived in section 2, its miscellaneous measures are established in section 3. the model parameters are estimated in section 4 while a simulation experiment is performed in section 5. applications to real-world data sets are discussed in section 6, and finally, the conclusion is reported in section 7. 3 int. j. anal. appl. (2022), 20:23 2. mathematical properties this section covers several mathematical properties of the exponentiated left truncated power distribution. 2.1. useful representation linear combination provides a much informal approach to discuss the cdf and pdf than the conventional integral computation when determining the mathematical properties. for this, the following binomial expansion is considered: (1−𝑦)𝛽 = ∑( 𝛽 𝑖 )(−1)𝑖𝑦𝑖 ∞ 𝑖=0 , |𝑦| < 1. owing to equations (1) and (2), infinite linear combinations (lc) of the eltr-pf cdf becomes: 𝐹𝐿𝐶−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 1 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 𝑖 )(−1)𝑖𝑘𝑎𝑖𝑥𝑎(𝑏−𝑖), ∞ 𝑖=0 (3) and the corresponding pdf is given as follows: 𝑓𝐿𝐶−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 )(−1)𝑗𝑘𝑎𝑗𝑥𝑎(𝑏−𝑗)−1. ∞ 𝑗=0 (4) 2.2. moments with associated measures moments play remarkable roles in the discussion of distribution theory in studying the significant characteristics of a probability distribution like the mean, variance, skewness, and kurtosis. theorem 1. if x~ 𝐸𝐿𝑇𝑟 −𝑃𝐹(𝑥;𝑘,𝑎,𝑏), with 𝑎,𝑏 > 0, and k < x, then the r-th ordinary moment (μ 𝑟 / ) of x is given by: 𝜇 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹 / = 𝑎𝑏 (1−𝑘 𝑎 ) 𝑏 ∑ ( 𝑏−1 𝑗 ) (−1)𝑗𝑘 𝑎𝑗 𝑟+𝑎(𝑏−𝑗) (1−𝑘 𝑟+𝑎(𝑏−𝑗) ) . ∞ 𝑗=0 proof 𝜇 𝑟 / can be written directly following equation (4) as follows: 𝜇 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹 / = 𝑎𝑏 (1−𝑘 𝑎 ) 𝑏 ∫ 𝑥𝑟 1 𝑘 𝑥𝑎−1(𝑥𝑎 −𝑘 𝑎 ) 𝑏−1 𝑑𝑥, 𝜇 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹 / = 𝑎𝑏 (1−𝑘 𝑎 ) 𝑏 ∑ ( 𝑏−1 𝑗 )(−1)𝑗𝑘 𝑎𝑗 ∫ 𝑥𝑟+𝑎(𝑏−𝑗)−1 1 𝑘 𝑑𝑥 ∞ 𝑗=0 . 4 int. j. anal. appl. (2022), 20:23 further, by solving the simple integral computation, it leads to the final form of the r-th ordinary moment, and it is given by: 𝜇 𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹 / = 𝑎𝑏 (1−𝑘 𝑎 ) 𝑏 ∑ ( 𝑏−1 𝑗 ) (−1)𝑗𝑘 𝑎𝑗 𝑟+𝑎(𝑏−𝑗) (1−𝑘 𝑟+𝑎(𝑏−𝑗) ) ∞ 𝑗=0 . (5) the expression in equation (5) is quite impressive and useful in the development of several statistical measures. for instance, to obtain the mean of x, substitute r = 1 in equation (5) as follows: 𝜇 1−𝐸𝐿𝑇𝑟−𝑃𝐹 / = 𝑎𝑏 (1−𝑘 𝑎 ) 𝑏 ∑ ( 𝑏−1 𝑗 ) (−1)𝑗𝑘 𝑎𝑗 1+𝑎(𝑏−𝑗) (1−𝑘 1+𝑎(𝑏−𝑗) ) ∞ 𝑗=0 . (6) one may perhaps further determine the well-established statistics such as skewness (𝛽1 = 𝜇3 2 𝜇2 3⁄ ), and kurtosis (𝛽2 = 𝜇4 𝜇2 2⁄ ), of x by integrating equation (6). a well-established relationship between the central moments (𝜇𝑠) and cumulants (𝐾𝑠) of x may easily be defined by ordinary moments 𝜇𝑠 = ∑ ( 𝑠 𝑘 )(−1)𝑘 (𝜇1 / ) 𝑠 𝜇𝑠−𝑘 /𝑠 𝑘=0 . hence, the first four cumulants can be calculated by 𝐾1 = 𝜇1 / , 𝐾2 = 𝜇2 / − 𝜇1 /2 , 𝐾3 = 𝜇3 / −3𝜇2 / 𝜇1 / +2𝜇1 /3 , and 𝐾4 = 𝜇4 / −4𝜇3 / 𝜇1 / −3𝜇2 /2 +12𝜇2 / 𝜇1 /2 −6𝜇1 /4 , etc., respectively. table 1 presents some numerical results of the first four ordinary moments (𝜇/ 1 ,𝜇/ 2 ,𝜇/ 3 ,𝜇/ 4 ), 𝜎2 = variance, 𝛽1 = skewness, and 𝛽2 = kurtosis for some choices of model parameters (k = 0.1) for the eltr-pf distribution. table 1. some numerical results of moments, variance, skewness, and kurtosis. statistics 𝑎 = 0.1,𝑘 = 0.1 remarks 𝜇/ 𝑟 𝑏 = 1.2 𝑏 = 1.3 𝑏 = 1.4 𝑏 = 1.5 𝑏 = 1.6 𝜇/ 1 0.4425 0.4586 0.4737 0.4880 0.5015 d e c re a si n g 𝜇/ 2 0.2607 0.2752 0.2890 0.3023 0.3151 𝜇/ 3 0.1814 0.1931 0.2045 0.2155 0.2262 𝜇/ 4 0.1386 0.1482 0.1576 0.1668 0.1758 𝜎2 0.0373 0.0314 0.0244 0.0165 0.0078 𝛽1 0.0016 0.0029 0.0031 0.0021 0.0008 decreasing 𝛽2 0.1329 0.1266 0.1150 0.0971 0.0714 decreasing 5 int. j. anal. appl. (2022), 20:23 table 2. some numerical results of moments, variance, skewness, and kurtosis. statistics 𝑏 = 1.5,𝑘 = 0.1 𝑏 = 1.7,𝑘 = 0.1 remarks 𝜇/ 𝑟 𝑎 = 0.1 𝑎 = 0.2 𝑎 = 0.01 𝑎 = 0.03 𝑎 = 0.05 𝜇/ 1 0.4880 0.5063 0.4972 0.5009 0.5047 d e c re a si n g 𝜇/ 2 0.3023 0.3208 0.3101 0.3138 0.3177 𝜇/ 3 0.2155 0.2317 0.2214 0.2247 0.2281 𝜇/ 4 0.1668 0.1807 0.1714 0.1743 0.1772 𝜎2 0.0166 0.0046 0.0103 0.0079 0.0053 𝛽1 0.0022 0.0006 0.0008 0.0005 0.0004 decreasing 𝛽2 0.0971 0.0645 0.0730 0.0671 0.0602 decreasing tables 1 and 2 illustrate decreasing behavior of the first four moments, variance, skewness, and kurtosis with some choices of model parameters. moment generating function 𝑀𝑋(𝑡) can be defined as: 𝑀𝑋(𝑡) = ∑ 𝑡𝑟 𝑟! ∞ 𝑟=0 𝜇𝑟 / . therefore, the moment generating function (mgf) of x is given by: 𝑀𝑋−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = 𝑎𝑏 (1 −𝑘𝑎)𝑏 ∑ 𝑡𝑟 𝑟! ∞ 𝑟=0 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑟 +𝑎(𝑏 −𝑗) (1−𝑘𝑟+𝑎(𝑏−𝑗)) ∞ 𝑗=0 . characteristic function is defined as: ∅𝑋(𝑡) = ∑ (𝑖𝑡)𝑟 𝑟! ∞ 𝑟=0 𝜇 𝑟 ′ . by following equation (5), it is obtained as: ∅𝑋−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ (𝑖𝑡)𝑟 𝑟! ∞ 𝑟=0 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑟 +𝑎(𝑏 −𝑗) (1 −𝑘𝑟+𝑎(𝑏−𝑗)) ∞ 𝑗=0 . the factorial generating function of x is defined as: 𝐹𝑥(𝑡) = 𝐸(1+𝑡) 𝑥 = 𝐸(𝑒𝑥𝑙𝑛(1+𝑡)) = ∑ (𝑙𝑛(1+𝑡)) 𝑟 𝑟! ∞ 𝑟=0 𝜇𝑟 ′ . by using equation (5), it is obtained as: 𝐹𝑥−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ (𝑙𝑛(1+𝑡)) 𝑟 𝑟! ∞ 𝑟=0 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑟 +𝑎(𝑏 −𝑗) (1−𝑘𝑟+𝑎(𝑏−𝑗)) ∞ 𝑗=0 ). 6 int. j. anal. appl. (2022), 20:23 negative moments of x, substitute r with – w in equation (5) and it is given by: 𝜇−𝑤−𝐸𝐿𝑇𝑟−𝑃𝐹 ′ = 𝑎𝑏 (1 −𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑎(𝑏 −𝑗)−𝑤 (1−𝑘𝑎(𝑏−𝑗)−𝑤) ∞ 𝑗=0 . furthermore, for fractional positive and fractional negative moments of x, substitute r with ( 𝑚 𝑛 ) and (− 𝑚 𝑛 ) in equation (6) respectively. in the theory of statistics, the mellin transformation is famous as a distribution of the product as well as a quotient for independent random variables. the mellin transformation is represented by 𝑀𝑥(𝑚) = 𝐸(𝑥 𝑚−1) = ∫ 𝑥𝑚−1𝑓(𝑥)𝑑𝑥 𝑘 1 . mellin transformation of x is given by: 𝑀𝑥−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑚) = 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑎(𝑏 −𝑗)+𝑚 −1 (1−𝑘𝑎(𝑏−𝑗)+𝑚−1) ∞ 𝑗=0 . 2.3. incomplete moments the r – th lower incomplete moments of x is defined as: 𝛷𝑟(𝑡) = ∫ 𝑥 𝑟𝑓(𝑥)𝑑𝑥 𝑡 𝑘 , and it is given by: 𝛷𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑟 +𝑎(𝑏 −𝑗) (𝑡𝑟+𝑎(𝑏−𝑗) −𝑘𝑟+𝑎(𝑏−𝑗)) ∞ 𝑗=0 . (7) the first incomplete moment can be obtained by substituting r = 1 in equation (7) as follows: 𝛷1−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1 +𝑎(𝑏 −𝑗) (𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞ 𝑗=0 . (8) the residual life function is the probability that a component whose life says x, will survive in an additional interval at t. it is given by: 𝑅(𝑡 𝑥⁄ ) = 𝑆(𝑥 +𝑡) 𝑆(𝑡) . therefore, the residual life function of x is: 𝑆𝑅(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝐹( 𝑡 𝑥⁄ ) = (1−𝑘𝑎)𝑏 −((𝑥 +𝑡)𝑎 −𝑘𝑎)𝑏 (1−𝑘𝑎)𝑏 −(𝑡𝑎 −𝑘𝑎)𝑏 , 𝑥 > 0. the reverse residual life is obtained by 𝑆�̅�(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝑜𝑤( 𝑡 𝑥⁄ ) = 𝑆(𝑥−𝑡) 𝑆(𝑡) . the reverse residual life function of x is therefore given by: 𝑆�̅�(𝑡)−𝐸𝐿𝑇𝑟−𝑃𝐹( 𝑡 𝑥⁄ ) = (1−𝑘𝑎)𝑏 −((𝑥 −𝑡)𝑎 −𝑘𝑎)𝑏 (1−𝑘𝑎)𝑏 −(𝑡𝑎 −𝑘𝑎)𝑏 , 𝑥 > 0. mean residual life (mrl) function is defined as 1−𝛷1(𝑡) 𝑆(𝑡)−𝑡 . it is obtained for x as 7 int. j. anal. appl. (2022), 20:23 mrl = 1− 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏−𝑗) (𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0 𝑆(𝑡) −𝑡 . mean inactivity time (mit) is defined as 𝑡 − 𝛷1(𝑡) 𝑃(𝑡) . it is obtained for x as mrl = 𝑡 − 1 𝑃(𝑡) ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏 −𝑗) (𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞ 𝑗=0 ) vitality function is defined as 𝑉(𝑥) = 1 𝑆(𝑥) ∫ 𝑥𝑓(𝑥)𝑑𝑥 1 𝑥 . it is obtained for x as 𝑉(𝑥) = 1 1−𝐹(𝑥) ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏 −𝑗) (1−𝑥1+𝑎(𝑏−𝑗)) ∞ 𝑗=0 ). the conditional moments are defined as𝐸(𝑥𝑟|𝑥 > 𝑡) = 1 �̅�(𝑡) ∫ 𝑥𝑟𝑓(𝑥)𝑑𝑥 1 𝑡 . it is obtained for x as 𝐸(𝑥𝑟|𝑥 > 𝑡) = 1 1−𝑃(𝑡) ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 𝑟 +𝑎(𝑏 −𝑗) (1−𝑡𝑟+𝑎(𝑏−𝑗)) ∞ 𝑗=0 ) 2.4. bonferroni and lorenz curves the bonferroni 𝐵(𝑥) and lorenz 𝐿(𝑥) curves are important not only in the study of economics, the distribution of income, poverty, or wealth, but they play a vital role in the fields of insurance, demography, medicine, reliability, and others. these curves are defined respectively by: 𝐵(𝑥) = ∫ 𝑥𝑓(𝑥)𝑑𝑥 𝑡 0 𝜇1 / , 𝐿(𝑥) = 𝐵(𝑥) 𝐹(𝑥) , lorenz curve 𝐿(𝑥) 𝐿𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏−𝑗) (𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0 ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏−𝑗) (1−𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0 , (9) and bonferroni curve 𝐵(𝑥) are given by: 𝐵𝐸𝐿𝑇𝑟−𝑃𝐹(𝑡) = ∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏−𝑗) (𝑡1+𝑎(𝑏−𝑗) −𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0 (( 𝑥𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 )∑ ( 𝑏 −1 𝑗 ) (−1)𝑗𝑘𝑎𝑗 1+𝑎(𝑏−𝑗) (1−𝑘1+𝑎(𝑏−𝑗)) ∞𝑗=0 . 2.5.reliability measures the survival function is defined as the probability that a component will survive till time x. analytically, it is defined as: 𝑆(𝑥) = 1−𝐹(𝑥). the survival function of x is therefore given by: 8 int. j. anal. appl. (2022), 20:23 𝑆𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 1−( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 . the hazard rate function (hrf) is defined as measuring the failure rate of a component in a particular time x. mathematically, it is defined as: ℎ(𝑥) = 𝑓(𝑥) 𝑆(𝑥)⁄ . hence, the hazard rate function of x is given by: ℎ𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 𝑎𝑏𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 (1 −𝑘𝑎)𝑏 −(𝑥𝑎 −𝑘𝑎)𝑏 . further, several notable reliability measures may be derived for x such as the reversed hazard rate function. it is defined as: ℎ𝑟(𝑥) = 𝑓(𝑥) 𝐹(𝑥)⁄ . the reversed hazard rate function of x is given by: ℎ𝑟−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = 𝑎𝑏𝑥𝑎−1 (𝑥𝑎 −𝑘𝑎) . the mills ratio is defined as 𝑀(𝑥) = 𝑆(𝑥) 𝑓(𝑥)⁄ . hence, the mills ratio of x is given by: 𝑀𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = (1−𝑘𝑎)𝑏 −(𝑥𝑎 −𝑘𝑎)𝑏 𝑎𝑏𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 . the odd function is defined as 𝑂(𝑥) = 𝐹(𝑥) 𝑆(𝑥)⁄ . therefore, the odd function of x is given by: 𝑂𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥;𝑎,𝑏) = ( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) −𝑏 −1. 3. miscellaneous measures this section covers several measures including limiting behavior, shapes of density and hazard rate functions, quantile function, entropy measures, and distribution of order statistics, bivariate, and multivariate extensions for eltr-pf distribution. 3.1.limiting behavior the limiting behavior of the cdf, pdf, and hrf of x for x → 𝑘 and x → 1 is discussed in propositions 1 and 2. proposition 1. limiting behaviors of the cdf, pdf, and hrf of x for x → 𝑘 are given respectively by: 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑘)~0, 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹(𝑘)~0, ℎ𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑘)~0. 9 int. j. anal. appl. (2022), 20:23 proposition 2. limiting behaviors of the cdf, pdf, and hrf of x for x → 1 are given respectively by: 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~1, 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~ 𝑎𝑏 (1−𝑘𝑎) , ℎ𝐸𝐿𝑇𝑟−𝑃𝐹 (1)~𝐼𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒. 3.2.shapes of density and hazard rate functions different curves of pdf and hrf of x are presented in figures 1 and 2, for different choices of model parameters. note that in figure 1, curves of the pdf present some possible shapes including increasing, upside-down increasing, and decreasing. however, possible shapes of the hrf in figure 2 present increasing and bathtub-shaped. (a) (b) figure 1. plots of pdf (a) and hrf (b) for eltr-pf distribution. 3.3.quantile function the concept of quantile function was introduced by [18]. the qth quantile function of the eltr-pf distribution is obtained by inverting the cdf in equation (1). it is defined by: 𝑞 = 𝐹(𝑥𝑞) = 𝑃(𝑋 ≤ 𝑥𝑞), 𝑞 ∈ (0,1). then, the quantile function of x is given by: 𝑥𝑞−𝐸𝐿𝑇𝑟−𝑃𝐹 = (𝑘 𝑎 +(1−𝑘𝑎)𝑞 1 𝛽⁄ ) 1 𝛼⁄ . (10) to derive the 1st quartile, median and 3rd quartile of x, one may place q = 0.25, 0.5, and 0.75 respectively in equation (10). henceforth, to generate random numbers, one may assume that the expression in equation (10) follows to uniform distribution u= u (0, 1). 10 int. j. anal. appl. (2022), 20:23 3.4.entropy measures this subsection covers several well-known entropy measures addressed by ([19], [20], [21], [22], [23], [24]). the entropy of r.v. x is a measure of uncertainty. the rényi entropy of x is defined by: 𝐼𝛿(𝑋) = 1 1 −𝛿 𝑙𝑜g∫𝑓𝛿(𝑥)𝑑𝑥 1 𝑘 , 𝛿 > 0 𝑎𝑛𝑑𝛿 ≠ 1. (11) first, 𝑓𝐸𝐿𝑇𝑟−𝑃𝑜𝑤(𝑥) is simplified in terms of 𝑓 𝛿 𝐸𝐿𝑇𝑟−𝑃𝑜𝑤 (𝑥) by considering equation (2) as: 𝑓𝛿 𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝛿 𝑥𝛿(𝑎−1)(𝑥𝑎 −𝑘𝑎)𝛿(𝑏−1 ), by applying the binomial expansion, 𝑓𝛿 𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝛿 ∑( 𝛿(𝑏 −1) 𝑖 ) ∞ 𝑖=0 (−1)𝑖𝑘𝑎𝑖𝑥𝛿(𝑎𝑏−1)−𝑎𝑖, and substituting this into equation (11) gives the rényi entropy of x as: 𝐼𝛿−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝛿 ∑( 𝛿(𝑏 −1) 𝑖 ) ∞ 𝑖=0 (−1)𝑖𝑘𝑎𝑖 ∫𝑥𝛿(𝑎𝑏−1)−𝑎𝑖𝑑𝑥 1 𝑘 , hence, by integrating the last expression the reduced form of the rényi entropy for x is obtained and it is given by: 𝐼𝛿−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑋) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝛿 𝑙𝑜g∑𝐴𝑖,𝛿 1 𝜓𝑖,𝛿 (1−𝑘𝜓𝑖,𝛿) ∞ 𝑖=0 , (12) where 𝜓𝑖,𝛿 = 𝛿(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝛿 = ( 𝛿(𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. a generalization of the boltzmann-gibbs entropy is the – entropy. although in physics, it is referred to as the tsallis entropy. tsallis entropy / – entropy is defined by 𝐻 (𝑋) = 1 −1 (1− ∫𝑓 (𝑥)𝑑𝑥 1 𝑘 ) , > 0 𝑎𝑛𝑑 ≠ 1. the tsallis entropy of x is given by 𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = 1 −1 (1−( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝑙𝑜g∑𝐴𝑖, 1 𝜓𝑖, (1−𝑘𝜓𝑖, ) ∞ 𝑖=0 ), where 𝜓𝑖, = (𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖, = ( (𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. the havrda and charvat introduced 𝜔 − entropy measure. it is defined by 𝐻𝜔(𝑋) = 1 21−𝜔 −1 (∫𝑓𝜔(𝑥)𝑑𝑥 1 𝑘 −1) , 𝜔 > 0 𝑎𝑛𝑑 𝜔 ≠ 1. 11 int. j. anal. appl. (2022), 20:23 havrda and charvat entropy of x is given by 𝐻𝜔−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = 1 21−𝜔 −1 ((( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝜔 𝑙𝑜g∑𝐴𝑖,𝜔 1 𝜓𝑖,𝜔 (1−𝑘𝜓𝑖,𝜔) ∞ 𝑖=0 )−1), where 𝜓𝑖,𝜔 = 𝜔(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝜔 = ( 𝜔(𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. arimoto generalized the work of havrda and charvat by introducing − entropy measure. it is defined by 𝐻 (𝑋) = 21− −1 ( (∫𝑓 (𝑥)𝑑𝑥 1 𝑘 ) 1 −1 ) , > 0 𝑎𝑛𝑑 ≠ 1. arimoto entropy of x is given by 𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = 21− −1 ( (( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 1 𝑙𝑜g∑𝐴 𝑖, 1 1 𝜓 𝑖, 1 (1−𝑘 𝜓 𝑖, 1 ) ∞ 𝑖=0 ) −1 ) , where 𝜓 𝑖, 1 = 1 (𝑎𝑏 −1)−𝑎𝑖, 𝐴 𝑖, 1 = ( 1 (𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. booker and lubba developed the 𝜏 − entropy measure. it is defined by 𝐻𝜏(𝑋) = 𝜏 𝜏 −1 ( 1−(∫𝑓𝜏 (𝑥)𝑑𝑥 1 𝑘 ) 1 𝜏 ) , 𝜏 > 0 𝑎𝑛𝑑 𝜏 ≠ 1. boekee and lubba entropy of x is given by 𝐻𝜏−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = 𝜏 𝜏 −1 (1−(( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝜏 𝑙𝑜g∑𝐴𝑖,𝜏 1 𝜓𝑖,𝜏 (1−𝑘𝜓𝑖,𝜏) ∞ 𝑖=0 ) 1 𝜏 ). where 𝜓𝑖,𝜏 = 𝜏(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,𝜏 = ( 𝜏(𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. mathai and haubold generalized the classical shannon entropy is known as − entropy. it is defined by 𝐻 (𝑋) = 1 −1 (∫𝑓2− (𝑥)𝑑𝑥 1 𝑘 −1) , > 0 𝑎𝑛𝑑 ≠ 1. mathai and haubold entropy of x is given by 𝐻 −𝐸𝐿𝑇𝑟−𝑃𝐹(𝑋) = 1 −1 ((( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 2− 𝑙𝑜g∑𝐴𝑖,2− 1 𝜓𝑖,2− (1 −𝑘𝜓𝑖,2− ) ∞ 𝑖=0 )−1), 12 int. j. anal. appl. (2022), 20:23 where 𝜓𝑖,2− = (2− )(𝑎𝑏 −1)−𝑎𝑖, 𝐴𝑖,2− = ( (2− )(𝑏 −1) 𝑖 )(−1)𝑖𝑘𝑎𝑖. table 3 presents the flexible behavior of the entropy measures for some choices of model parameters for s-i (𝒂 = 𝟏.𝟏,𝒃 = 𝟐.𝟏,𝒌 = 𝟎.𝟎𝟏), s-ii(𝒂 = 𝟏.𝟏,𝒃 = 𝟏.𝟓,𝒌 = 𝟎.𝟎𝟏𝟓), and s-iii(𝒂 = 𝟏.𝟓,𝒃 = 𝟏.𝟏,𝒌 = 𝟎.𝟎𝟓). table 3. some numerical results of rényi, tsallis, havrda and charvat, arimoto, boekee and lubba, mathai and haubold entropy measures. entropy int. s-i s-ii s-iii rényi 𝛿 = 1.1 4.8902 1.3591 0.4750 𝛿 = 1.5 1.3337 0.3706 0.1295 𝛿 = 1.7 1.0796 0.3000 0.1048 𝛿 = 1.9 0.9385 0.2608 0.0911 tsallis = 1.1 0.3634 0.1047 0.0323 = 1.5 0.1500 0.0431 -0.0070 = 1.7 0.0780 0.0181 -0.0248 = 1.9 0.0174 -0.0044 -0.0417 havrda and charvat 𝜔 = 1.1 0.0028 0.0019 0.0045 𝜔 = 1.5 0.0820 0.0389 0.0554 𝜔 = 1.7 0.1887 0.0851 0.1109 𝜔 = 1.9 0.3680 0.1601 0.1969 arimoto = 1.1 0.0039 0.0010 0.0004 = 1.5 0.3128 0.0529 0.0213 = 1.7 1.5767 0.1481 0.0596 = 1.9 85.315 0.3558 0.1333 boekee and lubba 𝜏 = 1.1 0.3701 0.1063 0.0333 𝜏 = 1.5 0.2238 0.0665 0.0087 𝜏 = 1.7 0.1871 0.0552 0.0012 𝜏 = 1.9 0.1606 0.0466 -0.0045 mathai and haubold = 1.1 -0.1569 0.0401 0.3761 = 1.5 -0.1500 -0.0431 0.0070 = 1.7 -0.1031 -0.0306 -0.0051 = 1.9 -0.0403 -0.0116 -0.0035 13 int. j. anal. appl. (2022), 20:23 table 3 presents’ versatile behavior of entropy measures for different parametric values. note that the rényi entropy is decreasing, tsallis entropy is decreasing, havrda and charvat entropy is increasing, arimoto entropy is increasing, boekee and lubba entropy is decreasing, and mathai and haubold's entropy is decreasing. 3.5.distribution of order statistics this subsection covers i-th order statistics pdf, minimum order statistics pdf, maximum order statistics pdf, order statistics cdf, median order statistics pdf, and joint order statistics pdf. in reliability analysis and life testing of a component in quality control, os has a noteworthy contribution. let x1 , x2 , x3 , ..., xn be a random sample of size n which follows the eltr-pf distribution and {x(1) < x(2) <x(3) < ...<x(n) }be the corresponding order statistics. the pdf of the i-th os is given by: 𝑓(𝑖:𝑛)(𝑥) = 1 𝐵(𝑖,𝑛−𝑖+1)! (𝐹(𝑥)) 𝑖−1 (1−𝐹(𝑥)) 𝑛−𝑖 𝑓(𝑥), i=1, 2, 3,…, n. by incorporating equations (3) and (4), the pdf of i-th os is given by: 𝑓(𝑖:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ( 1 𝐵(𝑖,𝑛−𝑖+1)! (( 𝑥𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑖−1 (1−( 𝑥𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−𝑖 × ( 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 ) ). (13) for minimum os, substitute (i = 1) into equation (13) as; 𝑓(1:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ( 1 𝐵(𝑖,𝑛 −𝑖 +1)! (1−( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 × ( 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥 𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 ) ) . while the maximum os is obtained by substituting (i = n) into equation (13) as: 𝑓(𝑛:𝑛)−𝐸𝐿𝑇𝑟−𝑝𝐹 (𝑥) = 1 𝐵(𝑖,𝑛 −𝑖 +1)! (( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 ( 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥 𝑎−1(𝑥𝑎 −𝑘𝑎)𝑏−1 ). correspondingly, the cdf of the i-th os is defined by: 𝐹(𝑖:𝑛)(𝑥) = ∑( 𝑛 𝑟 ) 𝑛 𝑟=𝑖 (𝐹(𝑥)) 𝑟 (1−𝐹(𝑥)) 𝑛−𝑟 . by incorporating equation (1), the cdf of the i-th os is obtained and it is given by: 14 int. j. anal. appl. (2022), 20:23 𝐹(𝑖:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥) = ∑( 𝑛 𝑟 ) 𝑛 𝑟=1 ((( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 ) 𝑟 (1−(( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 ) 𝑛−𝑟 . the median of the i-th os is defined by: 𝑓(𝑚+1:𝑛)(𝑥) = (2𝑚 +1)! (𝑚!)2 𝑓(𝑥)(𝐹(𝑥)) 𝑚 (1−𝐹(𝑥)) 𝑚 . by incorporating equations (1) and (2), the pdf of the median xm+1 os is obtained as: 𝑓(𝑚+1:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥) = (2𝑚+1)! (𝑚!)2 ((( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 ) 𝑚 ((1−( 𝑥𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−1 ) 𝑚 . the joint distribution of the i-th and j-th os is defined by: 𝑓(𝑖:𝑗:𝑛)(𝑥𝑖,𝑥𝑗) = 𝐶(𝐹(𝑥𝑖)) 𝑖−1 (𝐹(𝑥𝑗)−𝐹(𝑥𝑖)) 𝑗−𝑖−1 (1−𝐹(𝑥𝑗)) 𝑛−𝑗 𝑓(𝑥𝑖)𝑓(𝑥𝑗). (14) to obtain the joint distributions, equations (1) and (2) are further written as follows: 𝐹𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑖) = ( 𝑥𝑖 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ,𝐹𝐸𝐿𝑇𝑟−𝑃𝑜𝑤 (𝑥𝑗) = ( 𝑥𝑗 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 , 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑖) = 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥𝑖 𝑎−1(𝑥𝑖 𝑎 −𝑘𝑎)𝑏−1 , 𝑓𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑥𝑗) = 𝑎𝑏 (1−𝑘𝑎)𝑏 𝑥𝑗 𝑎−1(𝑥𝑗 𝑎 −𝑘𝑎) 𝑏−1 . by substituting these expressions into equation (14), the joint distribution of the i-th and j-th os is obtained as: 𝑓(𝑖:𝑗:𝑛)−𝐸𝐿𝑇𝑟−𝑃𝐹(𝑥𝑖,𝑥𝑗) = ( 𝐶(( 𝑥𝑖 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑖−1 (( 𝑥𝑗 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 −( 𝑥𝑖 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑗−𝑖−1 × (1−( 𝑥𝑗 𝑎 −𝑘𝑎 1−𝑘𝑎 ) 𝑏 ) 𝑛−𝑗 × (( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 2 𝑥𝑖 𝑎−1(𝑥𝑖 𝑎 −𝑘𝑎)𝑏−1 )× (𝑥𝑗 𝑎−1(𝑥𝑗 𝑎 −𝑘𝑎) 𝑏−1 ) ) . 3.6.bivariate and multivariate extensions in this subsection, we discuss the bivariate and multivariate extensions for the eltr–pf distribution by following the morgenstern and the clayton families. 15 int. j. anal. appl. (2022), 20:23 the cdf of the bi–eltr–pf distribution followed by the morgenstern family for the random vector (𝑉1,𝑉2) is 𝐹𝜙−𝐸𝐿𝑇𝑟−𝑃𝐹 (𝑉1,𝑉2) = (1+𝜙(1−𝐹1(𝑣1))(1−𝐹2(𝑣2)))𝐹1(𝑣1)𝐹2(𝑣2), where |𝜙| ≤ 1, 𝐹1(𝑣1) = ( 𝑣1 𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 , and 𝐹2(𝑣2) = ( 𝑣2 𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 . the cdf of the bi– eltr–pf distribution followed by the clayton family for the random vector (𝑋,𝑌) is 𝐶(𝑥,𝑦) = (𝑥−( 1+ 2) +𝑦−( 1+ 2) −1) − 1 ( 1+ 2) ; 1 + 2 ≥ 0. let 𝑣1~ eltr–pf (𝛼1,𝛽1), and 𝑣2~ eltr–pf (𝛼2,𝛽2). then setting 𝑥 = 𝐹1(𝑣1) = ( 𝑣1 𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 and 𝑦 = 𝐹2(𝑣2) = ( 𝑣2 𝑎−𝑘𝑎 1−𝑘𝑎 ) 𝑏 . the cdf of the bi– eltr–pf distribution followed by the clayton family for the random vector (𝑉1,𝑉2) is 𝐺𝐵𝑖−𝐸𝐿𝑇𝑟–𝑃𝐹 (𝑣1,𝑣2) = (( 𝑣1 𝑎 −𝑘𝑎 1−𝑘𝑎 ) −𝑏( 1+ 2) +( 𝑣2 𝑎 −𝑘𝑎 1−𝑘𝑎 ) −𝑏( 1+ 2) −1) − 1 ( 1+ 2) . a simple n-dimensional extension of the last version will be 𝐻𝑀𝑢𝑙𝑡𝑖−𝐸𝐿𝑇𝑟–𝑃𝐹 (𝑥1,𝑥2,𝑥3,…,𝑥𝑛) = (∑(( 𝑥𝑖 𝑎𝑖 −𝑘𝑖 𝑎𝑖 1−𝑘𝑖 𝑎𝑖 ) −𝑏( 1+ 2) ) 𝑛 𝑖=1 +1−𝑛) − 1 ( 1+ 2) . 4. inference in this section, the x's parameters are estimated by following the method of maximum likelihood estimation (mle), as this method provides full information about the unknown model parameter. let 𝑋1,𝑋2, . . . ,𝑋𝑛 be a random sample of size n from x, then the likelihood function 𝐿(𝜙) = ∏ 𝑓(𝑥𝑖;𝑎,𝑏) 𝑛 𝑖=1 of x is given by: 𝐿𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) = ( 𝑎𝑏 (1−𝑘𝑎)𝑏 ) 𝑛 ∏𝑥𝑖 𝑎−1(𝑥𝑖 𝑎 −𝑘𝑎)𝑏−1 . 𝑛 𝑖=1 the log-likelihood function, 𝑙(𝜙) is thus given by: 16 int. j. anal. appl. (2022), 20:23 𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) = ( 𝑛𝑙𝑜g𝑎 +𝑛𝑙𝑜g𝑏 −𝑛𝑙𝑜g(1−𝑘𝑎)+(𝑎 −1)∑𝑙𝑜g𝑥𝑖 𝑛 𝑖=1 +(𝑏 −1)∑𝑙𝑜g(𝑥𝑖 𝑎 −𝑘𝑎) 𝑛 𝑖=1 ) . (15) partial derivatives for parameters 𝑎 and 𝑏 of equation (15) yield the following: 𝜕𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) 𝜕𝑎 = 𝑛 𝑎 + 𝑛𝑘𝑎𝑙𝑜g𝑎 1 −𝑘𝑎 +∑𝑙𝑜g𝑥𝑖 𝑛 𝑖=1 +(𝑏 −1)∑ 𝑥𝑖 𝑎𝑙𝑜g𝑥𝑖 −𝑘 𝑎𝑙𝑜g𝑘 (𝑥𝑖 𝑎 −𝑘𝑎) 𝑛 𝑖=1 , 𝜕𝑙𝐸𝐿𝑇𝑟−𝑃𝐹(𝜙) 𝜕𝑏 = 𝑛 𝑏 +∑𝑙𝑜g(𝑥𝑖 𝑎 −𝑘𝑎) 𝑛 𝑖=1 . the maximum likelihood estimates (�̂�𝑖 = 𝑎,̂ �̂�) of the eltr-pf distribution can be obtained by maximizing equation (15) or by solving the above non-linear equations simultaneously. these non-linear equations although do not provide an analytical solution for the mles and the optimum value of 𝑎, and 𝑏. consequently, the newton-raphson type algorithm is an appropriate choice in the support of mles. 5. simulation in this subsection, we discuss the following algorithm (step 1 to 5) to observe the asymptotic performance of mle’s �̂� = (�̂�, �̂�). step -1: a random sample x1, x2, x3, ..., xn of sizes n = 100, 200, 300, 400, and 1000 from equation (14). step -2: results of root mean square error (rmse), coverage probability (cp), and average width (aw) are calculated with the assist of statistical software r. these results are presented in tables 4 and 5. step -3: each sample is replicated 1000 times. step -4: gradual decrease with the increase in sample sizes is observed for rmses. step -5: cps of all the parameters 𝜙 = (𝛼,𝛽) is approximately 0.975 approaches to the nominal value and aw decreases as sample sizes increase. furthermore, the following measures are defined in the development of average estimate (ae), rmse, cp, and aw, where i(.) is an indicator function and 𝑠𝑒�̂�𝑖 = √𝑣𝑎𝑟�̂�𝑖 is the standard error of estimate 𝜙𝑖.and it is given as follows: 17 int. j. anal. appl. (2022), 20:23 𝐴𝐸(�̂�) = 1 𝑁 ∑�̂� 𝑁 𝑖=1 ,𝑅𝑀𝑆𝐸(�̂�) = √ 1 𝑁 ∑(�̂�𝑖 −𝜙) 2 𝑁 𝑖=1 , 𝐶𝑃(�̂�) = ∑𝐼(�̂�𝑖 −0.95𝑠𝑒�̂�𝑖, �̂�𝑖 +0.95𝑠𝑒�̂�𝑖) 𝑁 𝑖=1 ,and 𝐴𝑊(�̂�) = 1 𝑁 ∑|(�̂�𝑖 +0.95)−(�̂�𝑖 −0.95)| 𝑁 𝑖=1 table 4. root mean square error, coverage probability, and average width for (𝑎 =1.1, 𝑏 =1.1, k=0.01). sample rmse(a) cp(a) aw(a) 100 0.5848 1 3.4874 200 0.5359 1 2.1784 300 0.5078 1 2.2354 400 0.4991 1 2.0208 1000 0.3330 1 1.1878 table 5. root mean square error, coverage probability, and average width for (𝑎 =1.1, 𝑏 =1.1, k=0.01). sample rmse(b) cp(b) aw(b) 100 0.2320 0.98 1.9249 200 0.2339 0.98 1.4396 300 0.2259 0.98 1.1951 400 0.2235 0.98 1.0603 1000 0.2020 0.98 0.6999 in both tables 4 and 5, the rmse and aw values reduced as the sample size increases. this indicates that the parameters of the eltr-pf distribution are good and stable. 18 int. j. anal. appl. (2022), 20:23 6. analysis of real-life data in this section, the application of the eltr-pf distribution is discussed. for this, two engineering datasets are explored. the eltr-pf distribution is compared with its competing models (presented in table 6) based on some criteria called, -log-likelihood (-ll), akaike information criterion (aic), bayesian information criterion (bic), along with the good-of-fit statistics cramervon mises (cm), anderson-darling (ad), and kolmogorov smirnov (k-s) with its p-value. all the numerical results are calculated with the assistance of statistical software r with its exclusive package adequacymodel (https://www.r-project.org/). table 6. list of some competitive models cdfs. model model’s cdfs parameter / variable range author(s) l-i 𝐺𝐼(𝑥) = 𝑥 𝑎 𝑎 > 0, 0 < x < 1 [25] l-ii 𝐺𝐼𝐼(𝑥) = 1−(1−𝑥) 𝑎 beta 𝐺𝐼𝐼𝐼(𝑥) = 𝐼𝑥(𝑎,𝑏) 𝑎,𝑏 > 0 0 < x < 1 topp-leone 𝐺𝐼𝑉(𝑥) = (2𝑥 −𝑥 2)𝑎 𝑎 > 0, 0 < x < 1 [11] kum 𝐺𝑉(𝑥) = 1 −(1 −𝑥 𝑎 )𝑏 𝑎,𝑏 > 0, 0 < x < 1 [10] gpf 𝐺𝑉𝐼(𝑥) = 1−(g−𝑥) 𝑎(g−𝑘)−𝑎 𝑎 > 0 𝑘 < 𝑥 < g [26] wpf 𝐺𝑉𝐼𝐼(𝑥) = 1− 𝑒 −𝑎( 𝑥𝑏 g𝑏−𝑥𝑏 ) 𝑐 𝑎,𝑏,𝑐 > 0 0 < 𝑥 < g [27] mt-ii 𝐺𝑉𝐼𝐼𝐼(𝑥) = 𝑒 𝑥𝑎log2 −1 𝑎 > 0, 0 < x < 1 [28] lehmann type–i =l–i, lehmann type–ii =l–ii, kumaraswamy=kum, generalized power function=gpf, weibull power function=wpf, mustapha type–ii = mt-ii. 6.1. application 1 the first data set relates to 30 measurements of tensile strength of polyester fibers discussed by [29]. the parameter estimates with standard errors (in parenthesis) and goodness of fit statistics are obtained and illustrated in table 7. https://www.r-project.org/ 19 int. j. anal. appl. (2022), 20:23 table 7. parameter estimates, standard errors (in parenthesis), and goodness of fit statistics for tensile strength of polyester fibers data. model parameter estimates (standard errors) statistics �̂� �̂� �̂� aic bic cm ad ks p-value eltr-pf 0.2169 (0.4352) 1.4096 (0.7666) -5.3369 -2.5345 0.0097 0.0791 0.0474 1.0000 wpf 3.0299 (2.2330) 1.3464 (0.9412) 0.7957 (0.373) 0.2444 4.4480 0.0174 0.1382 0.0611 0.9995 kum 0.9627 (0.2017) 1.6081 (0.4135) -2.6221 0.1803 0.0183 0.1550 0.0650 0.9987 top-leon 1.1091 (0.2024) -3.8078 -2.4066 0.0189 0.1600 0.0665 0.9981 beta 0.9666 (0.2237) 1.6204 (0.4106) -2.6101 0.1923 0.0184 0.1559 0.0669 0.9979 l-ii 1.6624 (0.3035) -4.5885 -3.1873 0.0184 0.1558 0.0740 0.9924 l-i 0.7254 (0.1324) -1.4495 -0.0483 0.0168 0.1425 0.1374 0.5754 mt-ii 0.5847 (0.1176) 0.4176 1.8188 0.0212 0.1788 0.1555 0.4201 the minimum goodness of fit statistics is the criteria of a better fit model which the eltr-pf distribution eventually satisfies. hence, this research supports that the eltr-pf distribution provides a better fit than its competitors. furthermore, the curves of fitted density (a) kaplan-meier survival (b), and probability-probability (pp) (c) plots are presented in figure 2. (a) (b) (c) figure 2. fitted plots for 30 measurements of tensile strength of polyester fibers data. 20 int. j. anal. appl. (2022), 20:23 6.2. application 2 the second data set represents the failure times of 20 mechanical components studied by [30]. the parameter estimates with standard errors (in parenthesis) and goodness of fit statistics are obtained and illustrated in table 8. table 8. parameter estimates, standard errors (in parenthesis), and goodness of fit statistics for the mechanical components data. model parameter estimates (standard errors) statistics �̂� �̂� �̂� aic bic cm ad ks p-value eltr-pf -2.9668 (0.7335) 2.3598 (0.8327) -74.6350 -72.6435 0.0488 0.3594 0.1054 0.9794 beta 3.1119 (0.9365) 21.8184 (7.0402) -51.7626 -49.7711 0.3700 2.3155 0.2538 0.1520 kum 1.5877 (0.2444) 21.8682 (10.210) -47.2969 -45.3054 0.4370 2.6508 0.2626 0.1267 wpf 25.3216 (10.981) 8.6983 (30.616) 0.1887 (0.6640) -46.8444 -43.8572 0.3972 2.4524 0.2642 0.1226 l-ii 7.3406 (1.6414) -43.1863 -42.1906 0.3698 2.3142 0.3989 0.0034 gpf 3.1354 (0.7011) -50.4166 -49.4209 0.4156 2.5011 0.4263 0.0014 top-leon 0.6247 (0.1397) -25.4857 -24.4900 0.3391 2.1565 0.4842 0.0002 l-i 0.4484 (0.1002) -15.1164 -14.1207 0.3211 2.0627 0.5104 0.0001 mt-ii 0.3402 (0.0843) -12.1937 -11.1979 0.3386 2.1538 0.5000 0.0001 in table 8, it is also clear that the eltr-pf distribution has the lowest values for all the goodness of fit statistics. therefore, the eltr-pf distribution is recommended over its competing distributions. 21 int. j. anal. appl. (2022), 20:23 the corresponding curves of fitted density (a) kaplan-meier survival (b), and probability-probability (pp) (c) plots are presented in figure 3. (a) (b) (c) figure 3. fitted plots for failure times of 20 mechanical components data. 7. conclusion the exponentiated left truncated power (eltr-pf) distribution has been successfully explored in this research. its various statistical properties were investigated and established. the simulation study showed that the parameters of the eltr-pf distribution are good and stable, as the root mean square error reduces as the sample size increases. the two datasets provided in this research support that the eltr-pf distribution is a better fit compared to the beta distribution, kumaraswamy distribution, lehmann type i and type ii distributions, generalized power function, weibull power function, and mustapha type–ii distribution. the density, kaplan-meier, and pp curves/plots also provide sufficient information about the closest fit to subject datasets. conflicts of interest: the author(s) declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. ahsan-ul-haq, m. ahmed, j. zafar, p.l. ramos, modeling of covid-19 cases in pakistan using lifetime probability distributions, ann. data. sci. 9 (2022), 141–152. https://doi.org/10.1007/s40745-021-003389. [2] a. al mutair, a. al mutairi, y. alabbasi, a. shamsan, s. al-mahmoud, s. alhumaid, m. zeshan arshad, m. awad, a. rabaan, level of anxiety among healthcare providers during covid-19 pandemic in saudi arabia: cross-sectional study, peerj. 9 (2021), e12119. https://doi.org/10.7717/peerj.12119. https://doi.org/10.1007/s40745-021-00338-9 https://doi.org/10.1007/s40745-021-00338-9 https://doi.org/10.7717/peerj.12119 22 int. j. anal. appl. (2022), 20:23 [3] a. al mutairi, m.z. iqbal, m.z. arshad, b. alnssyan, h. al-mofleh, a.z. afify, a new extended model with bathtub-shaped failure rate: properties, inference, simulation, and applications, mathematics. 9 (2021), 2024. https://doi.org/10.3390/math9172024. [4] a. al-shomrani, o. arif, a. shawky, s. hanif, m.q. shahbaz, topp–leone family of distributions: some properties and application, pak. j. stat. oper. res. 12 (2016), 443. https://doi.org/10.18187/pjsor.v12i3.1458. [5] s. arimoto, information-theoretical considerations on estimation problems, inform. control. 19 (1971), 181–194. https://doi.org/10.1016/s0019-9958(71)90065-9. [6] o.s. balogun, m.z. arshad, m.z. iqbal, m. ghamkhar, a new modified lehmann type – ii g class of distributions: exponential distribution with theory, simulation, and applications to engineering sector, f1000res. 10 (2021), 483. https://doi.org/10.12688/f1000research.52494.1. [7] d.e. boekee, j.c.a. van der lubbe, the r-norm information measure, inform. control. 45 (1980), 136– 155. https://doi.org/10.1016/s0019-9958(80)90292-2. [8] g.m. cordeiro, m. de castro, a new family of generalized distributions, j. stat. comput. simul. 81 (2011), 883–898. https://doi.org/10.1080/00949650903530745. [9] n. eugene, c. lee, f. famoye, beta-normal distribution and its applications, commun. stat. – theory methods. 31 (2002), 497–512. https://doi.org/10.1081/sta-120003130. [10] m.d.p. esberto, probability distribution fitting of rainfall patterns in philippine regions for effective risk management, environ. ecol. res. 6 (2018), 178–186. https://doi.org/10.13189/eer.2018.060305. [11] h.d. kan, b.h. chen, statistical distributions of ambient air pollutants in shanghai, china, biomed. environ. sci. 17(3) (2004), 366-272. https://pubmed.ncbi.nlm.nih.gov/15602835/. [12] p. kumaraswamy, a generalized probability density function for double-bounded random processes, j. hydrol. 46 (1980), 79–88. https://doi.org/10.1016/0022-1694(80)90036-0. [13] r.j. hyndman, y. fan, sample quantiles in statistical packages, amer. stat. 50 (1996), 361–365. https://doi.org/10.1080/00031305.1996.10473566. [14] j. havrda, f. charvat, quantification method of classification processes. concept of structural α-entropy. kybernetika, 3 (1967), 30-35. [15] e.l. lehmann, the power of rank tests, ann. math. statist. 24 (1953) 23–43. https://doi.org/10.1214/aoms/1177729080. [16] k. modi, v. gill, unit burr-iii distribution with application, j. stat. manage. syst. 23 (2020), 579–592. https://doi.org/10.1080/09720510.2019.1646503. [17] j. mazucheli, a.f. menezes, m.e. ghitany. the unit-weibull distribution and associated inference, j. appl. probab. stat. 13(2018), 1-22. [18] a. mathai, h. haubold, on a generalized entropy measure leading to the pathway model with a preliminary application to solar neutrino data, entropy. 15 (2013), 4011–4025. https://doi.org/10.3390/e15104011. https://doi.org/10.3390/math9172024 https://doi.org/10.18187/pjsor.v12i3.1458 https://doi.org/10.1016/s0019-9958(71)90065-9 https://doi.org/10.12688/f1000research.52494.1 https://doi.org/10.1016/s0019-9958(80)90292-2 https://doi.org/10.1080/00949650903530745 https://doi.org/10.1081/sta-120003130 https://doi.org/10.13189/eer.2018.060305 https://pubmed.ncbi.nlm.nih.gov/15602835/ https://doi.org/10.1016/0022-1694(80)90036-0 https://doi.org/10.1080/00031305.1996.10473566 https://doi.org/10.1214/aoms/1177729080 https://doi.org/10.1080/09720510.2019.1646503 https://doi.org/10.3390/e15104011 23 int. j. anal. appl. (2022), 20:23 [19] m. muhammad, a new lifetime model with a bounded support, asian res. j. math. 7 (2017), arjom.35099. https://doi.org/10.9734/arjom/2017/35099. [20] d.n.p. murthy, m. xie, r. jiang, weibull models, j. wiley, hoboken, n.j, 2004. [21] s. nasiru, a.g. abubakari, i.d. angbing, bounded odd inverse pareto exponential distribution: properties, estimation, and regression, int. j. math. math. sci. 2021 (2021), 9955657. https://doi.org/10.1155/2021/9955657. [22] p.e. oguntunde, o.a. odetunmibi, a.o. adejumo, a study of probability models in monitoring environmental pollution in nigeria, j. probab. stat. 2014 (2014), 864965. https://doi.org/10.1155/2014/864965. [23] c.p. quesenberry, c. hales, concentration bands for uniformity plots, j. stat. comput. simul. 11 (1980), 41–53. https://doi.org/10.1080/00949658008810388. [24] a. rényi. on measures of entropy and information, in: proceedings of the 4th fourth berkeley symposium on mathematical statistics and probability, university of california press, berkeley, (1961), 547561. [25] y. sangsanit, s.p. ahmad. the topp-leone generator of distributions: properties and inferences. songklanakarin j. sci. technol. 38 (2016), 537-548. [26] j. saran, a. pandey, estimation of parameters of a power function distribution and its characterization by k-th record values, statistica. 64 (2004), 523-536. https://doi.org/10.6092/issn.1973-2201/56. [27] c.w. topp, f.c. leone, a family of j-shaped frequency functions, j. amer. stat. assoc. 50 (1955), 209– 219. https://doi.org/10.1080/01621459.1955.10501259. [28] m. alizadeh, m. mansoor, g.m. cordeiro, m. zubair, m.h. tahir, the weibull-power function distribution with applications, hacettepe j. math. stat. 45 (2016), 245-265. https://doi.org/10.15672/hjms.2014428212. [29] c. tsallis, possible generalization of boltzmann-gibbs statistics, j. stat. phys. 52 (1988), 479–487. https://doi.org/10.1007/bf01016429. [30] a. zaharim, s. najid, a. razali, k. sopian. analyzing malaysian wind speed data using statistical distribution, in: proceedings of the 4th iasme/wseas international conference on energy and environment (ee ’09), cambridge, uk (2009), 363-370. https://doi.org/10.9734/arjom/2017/35099 https://doi.org/10.1155/2021/9955657 https://doi.org/10.1080/00949658008810388 https://doi.org/10.6092/issn.1973-2201/56 https://doi.org/10.1080/01621459.1955.10501259 https://doi.org/10.15672/hjms.2014428212 https://doi.org/10.1007/bf01016429 international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 15-21 http://www.etamaths.com some fixed point results for caristi type mappings in modular metric spaces with an application duran turkoglu1 and emine kilinç2,∗ abstract. in this paper we give caristi type fixed point theorem in complete modular metric spaces. moreover we give a theorem which can be derived from caristi type. also an application for the bounded solution of funcional equations is investigated. 1. introduction fixed point theory is one of the very popular tools in various fields. since banach intoruced this theory in 1922[6], ıt has been extended and generalized by several authors. caristi type fixed point theorem is one of these genealizations. it is a modification of ε−variational principle of ekeland[13].it is crucial in nonlinear analysis, in particular, optimization, variational inequalites, differantial equations and contol theory. the notion of modular space was introduced by nakano [20] and was intensively developed by koshi, shimogaki, yamamuro (see [18, 21]) and others. a lot of mathematicians are interested in fixed point of modular space. in 2008, chistyakov introduced the notion of modular metric space generated by f-modular and developed the theory of this space [8], on the same idea was defined the notion of a modular on an arbitrary set and developed the theory of metric space generated by modular such that called the modular metric spaces in 2010 [9]. afrah a. n. abdou [1] studied and proved some new fixed points theorems for pointwise and asymptotic pointwise contraction mappings in modular metric spaces. azadifer et. al. [3] introduced the notion of modular g-metric spaces.azadifer et. al. [5] proved the existence and uniqueness of a common fixed point of compatible mappings of integral type in this space. kılınç and alaca [14] defined (ε,k)−uniformly locally contractive mappings and η-chainable concept and proved a fixed point theorem for these concepts in a complete modular metric spaces. kılınç and alaca [15] proved that two main fixed point theorems for commuting mappings in modular metric spaces. recently, many authors [4, 7, 10, 11, 12, 19] studied on different fixed point results for modular metric spaces. in 2014 khamsi and abdou investigated hausdorff modular metric in modular metric spaces[16], and proved fixed point theorem for multivalued mappings. in this paper we investigate caristi type fixed point theorems for multivalued mappings in modular metric spaces, which is more general than the results of khojasteh, karapinar and khandani[17]. and we also give an application of results for functional equations. 2. preliminaries in this section, we will give some basic concepts and definitions about modular metric spaces. definition 2.1 [[9], definition 2.1] let x be a nonempty set, a function w : (0,∞)×x×x → [0,∞] is said to be a metric modular on x if satisfying, for all x,y,z ∈ x the following condition holds: (i) wλ (x,y) = 0 for all λ > 0 ⇔ x = y; (ii) wλ (x,y) = wλ (y,x) for all λ > 0; (iii) wλ+µ (x,y) ≤ wλ (x,z) +wµ (z,y) for all λ,µ > 0. if instead of (i), we have only the condition( i )́ wλ (x,x) = 0 for all λ > 0, then w is said to be a (metric) pseudomodular on x. 2010 mathematics subject classification. 46a80, 47h10, 54e35. key words and phrases. modular metric spaces; fixed point; hausdorff metric. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 15 16 turkoglu, and kilinç the main property of a metric modular [[9]] w on a set x is the following: given x,y ∈ x, the function 0 < λ 7→ wλ (x,y) ∈ [0,∞] is nonincreasing on (0,∞). in fact, if 0 < µ < λ, then (iii), ( i )́ and (ii) imply wλ (x,y) ≤ wλ−µ (x,x) + wµ (x,y) = wµ (x,y) . it follows that at each point λ > 0 the right limit wλ+0 (x,y) = lim µ→λ+0 wµ (x,y) and the left limit wλ−0 (x,y) = lim ε→+0 wλ−ε (x,y) exist in [0,∞] and the following two inequalities hold: wλ+0 (x,y) ≤ wλ (x,y) ≤ wλ−0 (x,y) . theorem 2.1 [[19]] let xw be a complete modular metric space and t a contraction on xw. then, the sequence (tnx)n∈n converges to the unique fixed point of t in xw for any initial x ∈ xw. now we give some definitions, which are useful for our main results. definition 2.2 let xw be a modular metric space. then following definitions exist: (1) the sequence (xn)n∈n in xw is said to be convergent to x ∈ xw if w1 (xn,x) → 0, as n →∞ (2) the sequence (xn)n∈n in xw is said to be cauchy if w1 (xm,xn) → 0, as m,n →∞ (3) a subset c of xw is said to be closed if the limit of a convergent sequence of c always belong to c. (4) a subset c of xw is said to be complete if any cauchy sequence in c is a convergent sequence and its limit is in c. (5) a subset c of xw is said to be w−bounded if δw(c) = sup{w1(x,y); x,y ∈ c} < ∞. (6) a subset c of xw is said to be w−compact if for any (xn) in c there exists a subset sequence (xnk) and x ∈ c such that w1(xnk,x) → 0 (7) w is said to satisfy the fatou property if and only if for any sequence (xn)n∈n in xw w−convergent to x, we have w1(x,y) ≤ lim inf n→∞ w1(xn,y), for any y ∈ xw. now we will give some basic properties and notions of multivalued mappings in modular metric spaces which was given in [2] for a subset m of modular metric space xw set (i) cb(m) = {c : c is nonempty w − closed and w − bounded subset of m} ; (ii) k(m) = {c : c is nonempty w − compact subset of m} ; (iii) the haussdorf modular metric is defined oncb(m) by hw(a,b) = max { sup x∈a w1(x,b), sup y∈b w1(y,a) } , where w1(x,b) = inf y∈b w1(x,y). definition 2.3 let xw be a complete modular metric space and m be a nonempty subset of xw. a mapping t : m → cb(m) is called a multivalued lipschitzian mapping, if there exists a constant k > 0 such that hw(tx,ty) ≤ kw1(x,y), for any x,y ∈ m. a point x ∈ m is called fixed point of t whenever x ∈ tx.the set of fixed points of t will be denoted by fix(t) it was shown in[2] that definition 2.4 is more general than theorem 2.1. some fixed point results in modular metric spaces 17 3. main results in this section we will give a fixed point theorem for caristi type mappings and a generalization of the theorem in modular metrc spaces. this works are more general than the results of [17]. theorem let xw be a complete modular metric space and t : xw → cb(xw) be a nonexpansive mapping such that for each x ∈ xw and for all y ∈ tx, there exists z ∈ ty such that (3.1) wλ(x,y) ≤ 1 λ (ϕw(x,y) −ϕw(y,z)) specificially for λ = 1 we can write w1(x,y) ≤ ϕw(x,y) −ϕw(y,z) where ϕ : xw× xw → [0,∞] is lower semicontinuous with respect to the firs variable. then for w1(xn,xn+1) < ∞, t has a fixed point. proof. let x0 ∈ xw and x1 ∈ tx0. if x1 = x0, then x0 is a fixed point and theorem is satisfied. otherwise, let x0 6= x1. by assumption there exists x2 ∈ tx1 such that w1(x0,x1) ≤ ϕw(x0,x1) −ϕw(x1,x2) alternatively, one can choose xn ∈ txn−1 such that xn 6= xn−1 and find xn+1 ∈ txn such that (3.2) 0 < w1(xn−1,xn) ≤ ϕw(xn−1,xn) −ϕw(xn,xn+1), which means that (ϕw(xn−1,xn))n is a nonincreasing sequence in fact 0 < w1(xn−1,xn) + ϕw(xn,xn+1) ≤ ϕw(xn−1,xn) 0 < ϕw(xn,xn+1) ≤ ϕw(xn−1,xn) thus it is bounded below, so it converges to some r ≥ 0. by taking the limit on both sides of (3.2) we get lim n→∞ w1(xn−1,xn) ≤ lim n→∞ (ϕw(xn−1,xn) −ϕw(xn,xn+1)) lim n→∞ w1(xn−1,xn) ≤ lim n→∞ ϕw(xn−1,xn) − lim n→∞ ϕw(xn,xn+1) lim n→∞ w1(xn−1,xn) = r −r lim n→∞ w1(xn−1,xn) = 0 now we show that w1(xn,xm−n) is cauchy sequence . for all m,n ∈ n with m > n, (3.3) w1(xn,xm) ≤ m∑ i=n+1 w 1 m−n (xi−1,xi) on the other hand from the main property of metric modular one can choose that w1(xn−1,xn) ≤ w 1 m−n (xn−1,xn) w 1 m−n (xn−1,xn) ≤ 1 (m−n) ϕw(t) when we write this in (3.3) we get w1(xn,xm) ≤ 1 m−n (ϕw(xn−1,xn) −···−ϕw(xm,xm+1)) w1(xn,xm) ≤ 1 m−n (ϕw(xn−1,xn) −ϕw(xm,xm+1)) 18 turkoglu, and kilinç when we take the limsup on both sides of the inequalities above, we have lim n→∞ (sup{w1(xn,xm) : m > n} ≤ lim n→∞ (sup 1 (m−n) (ϕw(xn−1,xn) −ϕw(xm,xm+1))) lim n→∞ (sup{w1(xn,xm) : m > n} ≤ lim n→∞ (sup 1 (m−n) )(r −r) lim n→∞ (sup{w1(xn,xm) : m > n} = 0 thus (xn) is a cauchy sequence. since xw is complete it converges to u ∈ xw. now we show that u is a fixed point of t.we have w1(u,tu) ≤ w1 2 (u,xn+1) + w1 2 (tu,xn+1) ≤ w1 2 (u,xn+1) + hw(tu,txn) ≤ w1 2 (u,xn+1) + w1 2 (u,xn) when we take the limit on both sides of the inequalities above, we get lim n→∞ w1(u,tu) ≤ lim n→∞ w1 2 (u,xn+1) + lim n→∞ w1 2 (u,xn) lim n→∞ w1(u,tu) ≤ w1 2 (u,u) + w1 2 (u,u) lim n→∞ w1(u,tu) = 0 hence we get u ∈ tu. therefore u is fixed point of t. � now let us give the theorem which is a generalized version of the theorem above. theorem let xw be a complete modular metric space and t : xw → cb(xw) be a multivalued mapping such that hw(tx,ty) ≤ η(w1(x,y)) for all x,y ∈ xw; where η : [0,∞] → [0,∞] is a lower semicontunious map such that η(t) < t for all t ∈ [0,∞] , η(t) t is nondecreasing. then t has a fixed point. proof. let x ∈ xw and y ∈ tx. if x = y then t has a fixed point and the proof is complete, so we suppose that x 6= y. let define that θ(t) = η(t) + t 2 , for all t ∈ [0,∞] then we have hw(tx,ty) ≤ η(w1(x,y)) since η is nondecreasing and η(t) < t we get η(t) < t ⇔ η(w1(x,y)) ≤ w1(x,y) θ(w1(x,y)) = η(w1(x,y)) + w1(x,y) 2 ≤ w1(x,y) θ(w1(x,y)) = η(w1(x,y)) + w1(x,y) 2 ≥ η(w1(x,y)) then we get (3.4) hw(tx,ty) ≤ η(w1(x,y)) ≤ θ(w1(x,y)) ≤ w1(x,y) thus there exists �0 > 0 such that θ(w1(x,y) = hw(tx,ty)+ �0. so there exists z ∈ ty such that some fixed point results in modular metric spaces 19 (3.5) w1(y,z) ≤ hw(tx,ty) + �0 = θ(w1(x,y)) ≤ w1(x,y) again suppose that y 6= z.then w1(x,y) −θ(w1(x,y)) ≤ w1(x,y) −w1(y,z) or we can rewrite this inequality as (3.6) w1(x,y) ≤ w1(x,y) 1 − θ(w1(x,y)) w1(x,y) − w1(y,z) 1 − θ(w1(x,y)) w1(x,y) since θ(t) t is nondecreasing and w1(y,z) < w1(x,y) we rewrite the (3.5) as w1(x,y) ≤ w1(x,y) 1 − θ(w1(x,y)) w1(x,y) − w1(y,z) 1 − θ(w1(y,z)) w1(y,z) define φw(x,y) = { w1(x,y) 1−θ(w1(x,y)) w1(x,y) ,x 6= y 0 ,x = y therefore t satisfies (3.1) of theorem 1, so we conclude that t has a fixed point u ∈ xw and the proof is complete. � 4. an application to functional equations we use fixed point theory in many fields of mathematics. one of theese fields is mathematical optimization.dynamic programing is useful for mathematical optimizationand it is related to a multistage process reduces to solving functional equation p(x). in this section we try to give an application of theorem 2 to a functional equation defined as (4.1) p(x) = sup y∈t {f(x,y) + =(x,y,p(η(x,y)))} , x ∈ z, where η : z ×t → z, f : z ×t → r, and = : z ×t ×r → r. we assume that m and n are banach spacesz ⊂mand t ⊂n now we study the existence of the bounded solution of the functional equation (4.1). let b(z) denote the set of all bounded real-valued functions on z, and for an arbitrary h ∈ b(z), define ‖h‖ = sup x∈z |h(x)| . clearly, (b(z),‖.‖) endowed with the metric modular (4.2) wλ(h,k) = 1 1 + λ sup x∈z |h(x) −k(x)| specificially for λ = 1 writen as w1(h,k) = 1 2 sup x∈z |h(x) −k(x)| for all h,k ∈b(z), is banach space, so the convergence in this space according to w1(h,k) is uniform which means a cauchy sequence in b(z) is uniformly convergent to a function say h∗, that is bounded and so h∗ ∈b(z) now we will give the theorem that gives the solution of the functional equation defined in (4.1). to give the solution let us define an operator as follows: (4.3) s(h)(x) = sup y∈t {f(x,y) + =(x,y,h(η(x,y)))} for all h∈b(z) and x ∈z now we give the theorem for the bounded solution of functional equation given in (4.4) 20 turkoglu, and kilinç theorem let s : b(z) → b(z)be an upper semicontinuous operator defined by (4.4) and assume that the following conditions are satisfied: (i) f :z ×t→ r, and = : z×t ×r → r are continuous and bounded; (ii) for all h,k ∈b(z), if 0 < w1(h,k) < 1 implies 1 2 |=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ w21 (h,k) w1(h,k) ≥ implies 1 2 |=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ w1(h,k) .where x ∈z and y ∈t . then the functional equation (4.1) has a bounded solution. proof. clearlyb(z) is a complete modular metric space .let µ > 0 be arbitrary, x ∈ z, and h1,h2 ∈ b(z), then there exists y1,y2 ∈t such that s(h1)(x) < f(x,y) + =(x,y,h1(η(x,y))) + µ s(h2)(x) < f(x,y) + =(x,y,h2(η(x,y))) + µ s(h1)(x) ≥ f(x,y) + =(x,y,h1(η(x,y))) s(h2)(x) ≥ f(x,y) + =(x,y,h2(η(x,y))) let % : [0,∞] → [0,∞] be defined as %(t) = { t2, 0 < t < 1 t, t ≥ 1 then we get 1 2 |=(x,y,h(η(x,y))) −=(x,y,k(η(x,y)))| ≤ %(w1(h,k)) for all h,k ∈b(z).it is clear that %(t) < t for all t > 0 and %(t) t is nondecreasing function. therefore when we use the inequalities above we get 1 2 (s(h1)(x) −s(h2)(x)) < 1 2 {=(x,y,h1(η(x,y))) −=(x,y,h2(η(x,y)))} + µ ≤ 1 2 |=(x,y,h1(η(x,y))) −=(x,y,h2(η(x,y)))| + µ ≤ %(w1(h1,h2)) + µ then inequality turns into 1 2 {s(h1)(x) −s(h2)(x)} < %(w1(h1,h2)) + µ analogously we get 1 2 {s(h2)(x) −s(h1)(x)} < %(w1(h1,h2)) + µ hence theese inequalities equal to 1 2 |s(h1)(x) −s(h2)(x)| < %(w1(h1,h2)) + µ which means that w1(s(h1)(x),s(h2)(x)) < %(w1(h1,h2)) + µ some fixed point results in modular metric spaces 21 since the above inequality turns into w1(s(h1)(x),s(h2)(x)) ≤ %(w1(h1,h2)), hence we get that s is a %−contraction. and s satisfies the conditions in theorem 2 and s has a fixed point say h∗ ∈b(z), which means is a bounded solution of the functional equation (4.1). � references [1] afrah a.n. abdou, on asymptotic pointwise contractions in modular metric spaces, abstract and applied analysis 2013(2013), art. id 50163. [2] afrah a.n abdou, mohamed khams, fixed points of multivalued contraction mappings in modular metric spaces,fixed point theory and applications 2014(2014), art. id 249. [3] b. azadifar, m. maramaei, gh. sadeghi, on the modular g-metric spaces and fixed point theorems, journal of nonlinear sciences and applications, 6(2013), 293-304. [4] b. azadifar, m. maramaei, gh. sadeghi, common fixed point theorems in modular g-metric spaces, journal nonlinear analysis and application, 2013(2013), art. id jnaa-00175. [5] b. azadifar, gh. sadeghi, r. saadati, c. park, integral type contractions in modular metric spaces, journal of inequalities and applications, 2013(2013), art. id 483. [6] s. banach, sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, fund. math., 3(1922), 133-181. [7] p. chaipunya, y.j. cho, p. kumam, geraghty-type theorems in modular metric spaces with an application to partial differential equation, advances in difference equations, 2012(2012), art. id 83. [8] v.v. chistyakov, modular metric spaces generated by f-modulars, folia math., 14(2008), 3-25. [9] v.v. chistyakov, modular metric spaces i. basic conceps, nonlinear anal., 72(2010), 1-14. [10] v.v. chistyakov, fixed points of modular contractive maps, doklady mathematics, 86(1)(2012), 515-518. [11] y.j. cho, r. saadati, gh. sadeghi, quasi-contractive mappings in modular metric spaces, journal of applied mathematics, 2012(2012), art. id 907951. [12] h. dehghan, m. eshaghi gordji, a. ebadian, comment on ’fixed point theorems for contraction mappings in modular metric spaces, fixed point theory and applications, 2012(2012), art. id 144. [13] i. ekeland, on the variational principle,j. math. anal. appl.47(1974),324-353. [14] e. kılınç, c. alaca, a fixed point theorem in modular metric spaces, advances in fixed point theory, 4(2)(2014), 199-206. [15] e. kılınç, c. alaca, fixed point results for commuting mappings in modular metric spaces, journal of applied functional analysis,(10)(2015), 204-210 [16] afrah an abdou, m. khamsi, fixed points of multivalued contraction mappings in modular metric spaces, fixed point theory and applications, 2014(2014), art. id 249. [17] f. khojasten, e. karapınar, h. khandani, some applications of caristi’s fixed point theorem in metric spaces,fixed pont theory and applications, (2016)2016, art. id 16. [18] s. koshi, t. shimogaki, on f-norms of quasi-modular space, j. fac. sci. hokkaido univ. ser 1., 15(3-4)(1961), 202-218. [19] c. mongkolkeha, w. sintunavarat, p. kumam, fixed point theorems for contraction mappings in modular metric spaces, fixed point theory and applications, 2011(2011), art. id 93. [20] h. nakano, modulared semi-ordered linear space, in tokyo math. book ser, vol.1, maruzen co, tokyo (1950). [21] s. yamammuro, on conjugate space of nakano space, trans. amer. math. soc., 90(1959), 291-311. 1department of mathematics, faculty of science, gazi university, ankara, turkey 2department of mathematics, institute of natural and applied science, gazi university, ankara, turkey ∗corresponding author: eklnc07@gmail.com international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 101-109 http://www.etamaths.com some discussions on a kind of improper integrals feng qi1,2,3,∗ and viera čerňanová4 abstract. in the paper, the improper integral i(a,b; λ,η) = ∫ b a 1√ (t−a)(b− t) lnλ t tη d t for b > a > 0 and λ,η ∈ r is discussed, some explicit formulas for special cases of i(a,b; λ,η) are presented, and several identities of i(a,b; k,η) for k ∈ n are established. 1. motivation the motivation of this paper origins from investigating central delanoy numbers in [11]. for proving the main result [11, theorem 1.4], we need [11, lemmas 2.4 and 2.5]. lemma 2.4 in [11] states that, for b > a and z ∈ c \ (−∞,−a], the principal branch of the function 1√ (z+a)(z+b) can be represented as 1√ (z + a)(z + b) = 1 π ∫ b a 1√ (t−a)(b− t) 1 t + z d t, (1.1) where c denotes the complex plane. when taking z = 0, the integral representation (1.1) becomes∫ b a 1√ (t−a)(b− t) 1 t d t = π √ ab , b > a > 0. (1.2) lemma 2.5 in [11] reads that the improper integral ∫ α 1/α 1√ (t− 1/α)(α− t) ln2k−1 t tβ d t   < 0, β > 1 2 = 0, β = 1 2 > 0, β < 1 2 (1.3) for all k ∈ n, where α > 1 and β ∈ r. motivated by the above results, we naturally introduce the improper integral i(a,b; λ,η) = ∫ b a 1√ (t−a)(b− t) lnλ t tη d t = ∫ 1 0 1√ s(1 −s) lnλ[(b−a)s + a] [(b−a)s + a]η d s for b > a > 0 and λ,η ∈ r and consider a problem: how to compute the improper integral i(a,b; λ,η)? 2. explicit formulas for special cases of i(a,b; λ,η) in this section, we present several explicit formulas for special cases of the improper integral i(a,b; λ,η). in the monograph [4], we do not find such a kind of integrals i(a,b; λ,η) for general b > a > 0 and λ,η ∈ r. 2010 mathematics subject classification. primary 40c10; secondary 26a39, 26a42, 33c05, 33c20, 33e05, 33e20, 40a10, 97i50. key words and phrases. improper integral; explicit formula; identity; hypergeometric function; generalized hypergeometric series. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 101 102 qi and čerňanová 2.1. from (1.1) or (1.2), it follows that i(a,b; 0, 1) = π √ ab , b > a > 0. (2.1) 2.2. from (1.3), it follows that i ( a, 1 a ; 2k − 1, 1 2 ) = 0, 0 < a < 1, k ∈ n. 2.3. it is straightforward by using euler’s substitution that i(a,b; 0, 0) = ∫ 1 0 1√ s(1 −s) d s = π, b > a > 0. 2.4. when λ = 0, η 6= 0, and 2a > b > a > 0, we have i(a,b; 0,η) = 1 aη ∫ 1 0 [1 + (b/a− 1)s]−η√ s(1 −s) d s = 1 aη ∫ 1 0 (1 −s)−1/2 ∞∑ `=0 〈−η〉` ( b a − 1 )` s`−1/2 `! d s = 1 aη ∞∑ `=0 〈−η〉` `! ( b a − 1 )` ∫ 1 0 (1 −s)−1/2s`−1/2 d s = 1 aη ∞∑ `=0 〈−η〉` `! ( b a − 1 )` b ( 1 2 ,` + 1 2 ) = 1 aη ∞∑ `=0 (η)` γ(1/2)γ(` + 1/2) γ(` + 1) 1 `! ( 1 − b a )` = π aη ∞∑ `=0 (η)`(1/2)` (1)` 1 `! ( 1 − b a )` = π aη 2f1 ( η, 1 2 ; 1; 1 − b a ) , where 〈x〉n = n−1∏ k=0 (x−k) = { x(x− 1) · · ·(x−n + 1), n ≥ 1 1, n = 0 and (x)` = `−1∏ k=0 (x + k) = { x(x + 1)(x + 2) · · ·(x + `− 1), ` ≥ 1 1, ` = 0 are respectively called the falling and rising factorials of x ∈ r, the function b(x,y) denotes the classical beta function, and 2f1 are the classical hypergeometric functions which are special cases of the generalized hypergeometric series pfq(a1, . . . ,ap; b1, . . . ,bq; z) = ∞∑ n=0 (a1)n . . . (ap)n (b1)n . . . (bq)n zn n! for complex numbers ai ∈ c and bi ∈ c\{0,−1,−2, . . .} and for positive integers p,q ∈ n. this result i(a,b; 0,η) = π aη 2f1 ( η, 1 2 ; 1; 1 − b a ) , η 6= 0, 2a > b > a > 0 can also be found in [3, p. xv, eq. (12)]. some discussions on a kind of improper integrals 103 2.5. when λ = k ∈ n and 2a > b > a > 0, the function lnk[(b−a)s + a] can be rewritten as lnk[(b−a)s + a] = ( ln a + ln [ 1 + ( b a − 1 ) s ])k = k∑ `=0 ( k ` ) lnk−` a ln` [ 1 + ( b a − 1 ) s ] = ( lnk a ) k∑ `=0 ( k ` ) (−1)` ln` a [ ∞∑ m=1 1 m ( 1 − b a )m sm ]` = ( lnk a ) k∑ `=0 ( k ` ) (−1)` ln` a s` [ ∞∑ m=0 1 m + 1 ( 1 − b a )m+1 sm ]` . when 0 < a < b < 1 or 1 < a < b < a2, if λ ∈ r, then lnλ[(b−a)s + a] = ( lnλ a )( 1 + ln[1 + (b/a− 1)s] ln a )λ = ( lnλ a ) ∞∑ `=0 〈λ〉` `! ( ln[1 + (b/a− 1)s] ln a )` = ( lnλ a ) ∞∑ `=0 (−1)`〈λ〉` `! ln` a [ ∞∑ m=1 1 m ( 1 − b a )m sm ]` = ( lnλ a ) ∞∑ `=0 (−1)`〈λ〉` `! ln` a s` [ ∞∑ m=0 1 m + 1 ( 1 − b a )m+1 sm ]` . in [4, p. 18, 0.314], it was stated that( ∞∑ k=0 akx k )n = ∞∑ k=0 cn,kx k, where cn,0 = a n 0 and cn,m = 1 ma0 m∑ k=1 (kn−m + k)akcn,m−k, m ∈ n. hence, it follows that [ ∞∑ m=0 1 m + 1 ( 1 − b a )m+1 sm ]` = ∞∑ m=0 c`,mx m, where c`,0 = ( 1 − b a )` and c`,m = 1 m m∑ k=1 k`−m + k k + 1 ( 1 − b a )k c`,m−k = 1 m ( 1 − b a )m m−1∑ p=0 m`− (` + 1)p m−p + 1 ( 1 − b a )−p c`,p for m ∈ n. let c`,m = ( 1 − b a )−m c`,m, the above recursive formula becomes c`,m = 1 m m−1∑ p=0 m`−p(` + 1) m−p + 1 c`,p (2.2) 104 qi and čerňanová with c`,0 = c`,0. starting out from these points, it is much possible to find explicit formulas for computing the integral i(a,b; λ,η). for example, when λ 6= 0 and η = 1, i(a,b; λ, 1) = 1 (λ + 1)(b−a) ∫ 1 0 1√ s(1 −s) d lnλ+1[(b−a)s + a] d s d s = lnλ+1 a (λ + 1)(b−a) ∞∑ `=0 (−1)`〈λ + 1〉` `! ln` a × ∫ 1 0 1√ s(1 −s) d d s [ ∞∑ m=0 1 m ( 1 − b a )m sm ]` d s = lnλ+1 a (λ + 1)(b−a) ∞∑ `=0 (−1)`〈λ + 1〉` `! ln` a ∫ 1 0 1√ s(1 −s) d d s ∞∑ m=0 c`,ms m d s = lnλ+1 a (λ + 1)(b−a) ∞∑ `=0 (−1)`〈λ + 1〉` `! ln` a ∞∑ m=0 (m + 1)c`,m+1 ∫ 1 0 (1 −s)−1/2sm−1/2 d s = lnλ+1 a (λ + 1)(b−a) ∞∑ `=0 (−1)`〈λ + 1〉` `! ln` a ∞∑ m=0 (m + 1)c`,m+1b ( 1 2 ,m + 1 2 ) = π lnλ+1 a (λ + 1)(b−a) ∞∑ `=0 (−1)`〈λ + 1〉` `! ln` a ∞∑ m=0 (m + 1)c`,m+1 (1/2)m (1)m . hence, it would be important to derive a general formula for the recursive relation (2.2). 2.6. for k ≥ 0, differentiating with respect to z on both sides of (1.1) gives dk d zk 1√ (z + a)(z + b) = (−1)k k! π ∫ b a 1√ (t−a)(b− t) 1 (t + z)k+1 d t. (2.3) by the faá di bruno formula dn d tn f ◦h(t) = n∑ k=0 f(k)(h(t))bn,k ( h′(t),h′′(t), . . . ,h(n−k+1)(t) ) , n ≥ 0 in [2, p. 139, theorem c], where bn,k(x1,x2, . . . ,xn−k+1) = ∑ 1≤i≤n,`i∈n∪{0}∑n i=1 i`i=n∑n i=1 `i=k n!∏n−k+1 i=1 `i! n−k+1∏ i=1 (xi i! )`i , n ≥ k ≥ 0 is called [2, p. 134, theorem a] the bell polynomials of the second kind, we obtain dk d zk 1√ (z + a)(z + b) = k∑ `=0 ( 1 √ u )(`) bk,`(u ′(z),u′′(z), 0, . . . , 0) = k∑ `=0 〈 − 1 2 〉 ` 1 u`+1/2 bk,`(2z + a + b, 2, 0, . . . , 0) = k∑ `=0 〈 − 1 2 〉 ` 1 [(z + a)(z + b)]`+1/2 bk,`(2z + a + b, 2, 0, . . . , 0) → k∑ `=0 〈 − 1 2 〉 ` 1 (ab)`+1/2 bk,`(a + b, 2, 0, . . . , 0) as z → 0, where u = u(z) = (z + a)(z + b). recall from [2, p. 135] that bn,k ( abx1,ab 2x2, . . . ,ab n−k+1xn−k+1 ) = akbnbn,k(x1,x2, . . . ,xn−k+1), (2.4) some discussions on a kind of improper integrals 105 where a and b are any complex numbers and n ≥ k ≥ 0. recall from [5, theoem 4.1], [17, theorem 3.1], and [18, lemma 2.5] that bn,k(x, 1, 0, . . . , 0) = (n−k)! 2n−k ( n k )( k n−k ) x2k−n, n ≥ k ≥ 0. (2.5) accordingly, by (2.4) and (2.5), it follows that lim z→0 dk d zk 1√ (z + a)(z + b) = k∑ `=0 〈 − 1 2 〉 ` 1 (ab)`+1/2 2`bk,` ( a + b 2 , 1, 0, . . . , 0 ) = k∑ `=0 〈 − 1 2 〉 ` 1 (ab)`+1/2 2` (k − `)! 2k−` ( k ` )( ` k − ` )( a + b 2 )2`−k . letting z → 0 on both sides of (2.3), employing the above result, and simplifying lead to∫ b a 1√ (t−a)(b− t) 1 tk+1 d t = (−1)kπ (a + b)k √ ab k∑ `=0 (−1)`22` (2`− 1)!! (2`)!! ( ` k − ` )( a + b 2 )`( 1/a + 1/b 2 )` , that is, i(a,b; 0,k + 1) = π g(a,b) (−1)k [2a(a,b)]k k∑ `=0 (−1)`22` (2`− 1)!! (2`)!! ( ` k − ` )[ a(a,b) h(a,b) ]` (2.6) for b > a > 0 and k ≥ 0, where ( p q ) = 0 for q > p ≥ 0, the double factorial of negative odd integers −(2n + 1) is defined by (−2n− 1)!! = (−1)n (2n− 1)!! = (−1)n 2nn! (2n)! , n = 0, 1, . . . , and the quantities a(a,b) = a + b 2 , g(a,b) = √ ab, and h(a,b) = 2 1 a + 1 b are respectively the well-known arithmetic, geometric, and harmonic means of two positive numbers a and b. when k = 0 in (2.6), the integral (1.2) or (2.1) is recovered. in fact, the above argument implies that∫ b a 1√ (t−a)(b− t) 1 (t + z)k+1 d t = (−1)k [2a(z + a,z + b)]k π g(z + a,z + b) × k∑ `=0 (−1)`22` (2`− 1)!! (2`)! ( ` k − ` )[ a(z + a,z + b) h(z + a,z + b) ]` for b > a > 0 and k ≥ 0. this is equivalent to (2.6). by the way, the ratio (2`−1)!! (2`)! is called the wallis ratio. for more information, please refer to the paper [7] and plenty of references cited therein. alternatively differentiating with respect to z on both sides of (1.1) leads to dk d zk 1√ (z + a)(z + b) = dk d zk ( 1 √ z + a 1 √ z + b ) = k∑ `=0 ( k ` )( 1 √ z + a )(`)( 1 √ z + b )(k−`) 106 qi and čerňanová = k∑ `=0 ( k ` )〈 − 1 2 〉 ` 1 (z + a)`+1/2 〈 − 1 2 〉 k−` 1 (z + b)k−`+1/2 = k∑ `=0 ( k ` ) (−1)` (2`− 1)!! 2` 1 (z + a)`+1/2 (−1)k−` [2(k − `) − 1]!! 2k−` 1 (z + b)k−`+1/2 = (−1)k 2k 1 (z + a)1/2 1 (z + b)k+1/2 k∑ `=0 ( k ` ) (2`− 1)!![2(k − `) − 1]!! ( z + b z + a )` . substituting this into (2.3) and taking the limit z → 0 result in i(a,b; 0,k + 1) = π √ ab 1 bk k∑ `=0 (2`− 1)!! (2`)!! [2(k − `) − 1]!! [2(k − `)]!! ( b a )` for b > a > 0 and k ≥ 0. this is an alternative expression for i(a,b; 0,k + 1). 2.7. under different conditions from those discussed above on b > a > 0 and λ,η ∈ r, can one discover more explicit formulas for the improper integral i(a,b; λ,η)? 3. identities for i(a,b; k,η) in this section, we present several identities for the improper integral i(a,b; k,η). 3.1. substituting s = 1 t into i(a,b; k,η) yields i(a,b; k,η) = (−1)k √ ab i ( 1 b , 1 a ; k, 1 −η ) (3.1) for k ≥ 0, η ∈ r, and a,b > 0 with a 6= b. in particular, it can be derived that i(a,b; 0, 1) = 1 √ ab i ( 1 b , 1 a ; 0, 0 ) and i ( 1 b ,b; k,η ) = (−1)ki ( 1 b ,b; k, 1 −η ) . 3.2. substituting s = t a into i(a,b; k,η) gives i(a,b; k,η) = 1 aη [( lnk a ) i ( 1, b a ; 0,η ) + i ( 1, b a ; k,η )] for k ∈ n, η ∈ r, and a,b > 0 with a 6= b. in particular, i(a, 1; k,η) = 1 aη [( lnk a ) i ( 1, 1 a ; 0,η ) + i ( 1, 1 a ; k,η )] . (3.2) 3.3. from (3.1), it follows that i(a, 1; k,η) = (−1)k √ a i ( 1, 1 a ; k, 1 −η ) (3.3) substituting (3.3) into (3.2) leads to i ( 1, 1 a ; k,η ) = (−1)k aη−1/2 i ( 1, 1 a ; k, 1 −η ) − ( lnk a ) i ( 1, 1 a ; 0,η ) for 1 6= a > 0, k ∈ n, and η ∈ r. consequently, i(1,b; k,η) = (−1)k b1/2−η i(1,b; k, 1 −η) + ( lnk b ) i(1,b; 0,η) some discussions on a kind of improper integrals 107 for 1 6= b > 0, k ∈ n, and η ∈ r. 4. remarks by the way, we list two remarks on (1.1) and integral representations of the weighted geometric means. remark 4.1. the integral representation (1.1) can be generalized as follows. for ak < ak+1 and wk > 0 with ∑n k=1 wk = 1, the principal branch of the reciprocal of the weighted geometric mean∏n k=1(z + ak) wk on c\ (−∞,−a1] can be represented by 1∏n k=1(z + ak) wk = 1 π n−1∑ m=1 sin ( π m∑ `=1 w` )∫ am+1 am 1∏n k=1 |t−ak|wk 1 t + z d t. remark 4.2. before getting the integral representation (1.1), the following integral representation for the weight geometric mean ∏n k=1(z + ak) wk was obtained. let wk > 0 and ∑n k=1 wk = 1 for 1 ≤ k ≤ n and n ≥ 2. if a = (a1,a2, . . . ,an) is a positive and strictly increasing sequence, that is, 0 < a1 < a2 < · · · < an, then the principal branch of the weighted geometric mean gw,n(a + z) = n∏ k=1 (ak + z) wk, z ∈ c\ (−∞,−a1] has the lévy–khintchine expression gw,n(a + z) = gw,n(a) + z + ∫ ∞ 0 ma,w,n(u)(1 −e−zu) d u, (4.1) where the density ma,w,n(u) = 1 π n−1∑ `=1 sin ( π ∑̀ j=1 wj )∫ a`+1 a` n∏ k=1 |ak − t|wke−ut d t. for more detailed information, please refer to [1, 6, 8, 9, 12, 13, 14, 15, 16] and closely-related references therein. remark 4.3. letting n = 2 and w1 = w2 = 1 2 in (4.1) or setting n = 2 in [14, theorem 1.1] leads to √ (z + a)(z + b) = √ ab + z + 1 π ∫ ∞ 0 [∫ b a √ (b− t)(t−a) e−ut d t ] (1 −e−zu) d u = √ ab + z + 1 π ∫ b a √ (b− t)(t−a) [∫ ∞ 0 e−ut(1 −e−zu) d u ] d t = √ ab + z + z π ∫ b a √ (b− t)(t−a) t 1 t + z d t, that is, ∫ b a √ (b− t)(t−a) t 1 t + z d t = π [√ (z + a)(z + b) − √ ab z − 1 ] , for b > a > 0. taking the limit z → 0 on both sides of (4.2) yields∫ b a √ (b− t)(t−a) t2 d t = π ( a + b 2 √ ab − 1 ) = π [ a(a,b) g(a,b) − 1 ] , b > a > 0. (4.2) for k ∈ n, differentiating k times with respect to z procures 1 π ∫ b a √ (b− t)(t−a) t (−1)kk! (t + z)k+1 d t = [√ (z + a)(z + b) − √ ab z ](k) = √ ab [ 1 z (√ 1 + a + b ab z + 1 ab z2 − 1 )](k) 108 qi and čerňanová = √ ab [ 1 z ∞∑ `=1 〈 1 2 〉 ` 1 `! ( a + b ab z + 1 ab z2 )`](k) , ∣∣∣∣a + bab z + 1abz2 ∣∣∣∣ < 1 = √ ab ∞∑ `=1 (−1)`−1 (2`− 3)!! 2` 1 `! 1 (ab)` [ z`−1(a + b + z)` ](k) = √ ab ∞∑ `=1 (−1)`−1 (2`− 3)!! 2` 1 `! 1 (ab)` k∑ q=0 ( k q )( z`−1 )(q)[ (a + b + z)` ](k−q) → √ ab k+1∑ `=1 (−1)`−1 (2`− 3)!! 2` 1 `! 1 (ab)` ( k `− 1 ) (`− 1)! lim z→0 [ (a + b + z)` ](k−`+1) = √ ab k+1∑ `=1 (−1)`−1 (2`− 3)!! 2` 1 ` 1 (ab)` ( k `− 1 ) 〈`〉k−`+1(a + b)2`−k−1 = 1 (a + b)k−1 √ ab k∑ `=0 (−1)` (2`− 1)!! 2`+1 1 ` + 1 ( k ` ) 〈` + 1〉k−` (a + b)2` (ab)` = 1 (a + b)k−1 √ ab k∑ `=0 (−1)` (2`− 1)!! 2`+1 1 ` + 1 ( k ` ) (` + 1)! (2`−k + 1)! (a + b)2` (ab)` = k! (a + b)k−1 √ ab k∑ `=0 (−1)` (2`− 1)!! [2(` + 1)]!! ( ` + 1 k − ` ) (a + b)2` (ab)` as z → 0. as a result, we have∫ b a √ (b− t)(t−a) tk+2 d t = π (−1)k (a + b)k k∑ `=0 (−1)` (2`− 1)!! [2(` + 1)]!! ( ` + 1 k − ` )( a + b √ ab )2`+1 for b > a > 0 and k ∈ n. remark 4.4. this paper is a slightly modified version of the preprint [10]. references [1] á. besenyei, on complete monotonicity of some functions related to means, math. inequal. appl. 16 (2013), no. 1, 233–239. [2] l. comtet, advanced combinatorics: the art of finite and infinite expansions, revised and enlarged edition, d. reidel publishing co., dordrecht and boston, 1974. [3] g. gasper and m. rahman, basic hypergeometric series, with a foreword by richard askey, second edition, encyclopedia of mathematics and its applications, 96, cambridge university press, cambridge, 2004. [4] i. s. gradshteyn and i. m. ryzhik, table of integrals, series, and products, translated from the russian, translation edited and with a preface by daniel zwillinger and victor moll, eighth edition, revised from the seventh edition, elsevier/academic press, amsterdam, 2015. [5] b.-n. guo and f. qi, explicit formulas for special values of the bell polynomials of the second kind and the euler numbers, researchgate technical report (2015), available online at http://dx.doi.org/10.13140/2.1.3794.8808. [6] b.-n. guo and f. qi, on the degree of the weighted geometric mean as a complete bernstein function, afr. mat. 26 (2015), no. 7, 1253–1262. [7] b.-n. guo and f. qi, on the wallis formula, internat. j. anal. appl. 8 (2015), no. 1, 30–38. [8] f. qi, integral representations and properties of stirling numbers of the first kind, j. number theory 133 (2013), no. 7, 2307–2319. [9] f. qi and s.-x. chen, complete monotonicity of the logarithmic mean, math. inequal. appl. 10 (2007), no. 4, 799–804. [10] f. qi and v. čerňanová, some discussions on a kind of improper integrals, researchgate working paper (2016), available online at http://dx.doi.org/10.13140/rg.2.1.2500.6969. [11] f. qi, b.-n. guo, v. čerňanová, and x.-t. shi, explicit expressions, cauchy products, integral representations, convexity, and inequalities of central delannoy numbers, researchgate working paper (2016), available online at http://dx.doi.org/10.13140/rg.2.1.4889.6886. [12] f. qi, x.-j. zhang, and w.-h. li, an elementary proof of the weighted geometric mean being a bernstein function, politehn. univ. bucharest sci. bull. ser. a appl. math. phys. 77 (2015), no. 1, 35–38. some discussions on a kind of improper integrals 109 [13] f. qi, x.-j. zhang, and w.-h. li, an integral representation for the weighted geometric mean and its applications, acta math. sin. (engl. ser.) 30 (2014), no. 1, 61–68. [14] f. qi, x.-j. zhang, and w.-h. li, lévy-khintchine representation of the geometric mean of many positive numbers and applications, math. inequal. appl. 17 (2014), no. 2, 719–729. [15] f. qi, x.-j. zhang, and w.-h. li, lévy-khintchine representations of the weighted geometric mean and the logarithmic mean, mediterr. j. math. 11 (2014), no. 2, 315–327. [16] f. qi, x.-j. zhang, and w.-h. li, the harmonic and geometric means are bernstein functions, bol. soc. mat. mex. (3) 22 (2016), in press; available online at http://dx.doi.org/10.1007/s40590-016-0085-y. [17] f. qi and m.-m. zheng, explicit expressions for a family of the bell polynomials and applications, appl. math. comput. 258 (2015), 597–607. [18] c.-f. wei and f. qi, several closed expressions for the euler numbers, j. inequal. appl. 2015 (2015), art. id 219. 1institute of mathematics, henan polytechnic university, jiaozuo city, henan province, 454010, china 2college of mathematics, inner mongolia university for nationalities, tongliao city, inner mongolia autonomous region, 028043, china 3department of mathematics, college of science, tianjin polytechnic university, tianjin city, 300387, china 4institute of computer science and mathematics, slovak university of technology, bratislava, slovak republic ∗corresponding author: qifeng618@gmail.com int. j. anal. appl. (2023), 21:76 an algorithm for the solution of second order linear fuzzy system with mechanical applications s. nagalakshmi1,∗, g. suresh kumar1, ravi p. agarwal2, chao wang3 1department of engineering mathematics, college of engineering, koneru lakshmaiah education foundation, vaddeswaram, guntur, 522302, andhra pradesh, india 2department of mathematics, texas a&m university-kingsville, kingsville, tx 78363-8202, usa 3department of mathematics, yunnan university, kunming 650091, china ∗corresponding author: nagalakshmi.soma@gmail.com abstract. in this paper, we consider homogeneous and non-homogeneous second order linear fuzzy systems under granular differentiability. the concept of continuous n-dimensional fuzzy functions on the space of n-dimensional fuzzy numbers are introduced. developed an algorithm for the solution of a non-homogeneous second order linear fuzzy system under granular differentiability. the proposed algorithm is applied to solve some well-known mechanical problems with fuzzy uncertainty. 1. introduction mathematical models can be explained through fuzzy differential equations (fde). the innovative work on system of fuzzy differential equations (sfdes) extended from population models, bio informatics, quantum optics, and soft computing models. second-order linear fuzzy systems (slfs) are modeled by behaviors of many dynamical systems with uncertainty. slfss specifically appear in many spring-mass mechanical systems with uncertainty. fard and ghal-eh [3] proposed a numerical method to solve sfdes under h-differentiability. gasilov et al. [4] presented a solution method for sfdes with fuzzy initial conditions. mondal et al. [7] analyzed adaptive schemes to study the sfdes. barazandeh and ghazanfari [1] obtained the solutions for sfdes applying variation iteration technique. keshavarz et al. [5] enhanced to obtain an analytical solution for sfdes using gh-differentiability. boukezzoula received: apr. 2, 2023. 2020 mathematics subject classification. 34a07, 34b05. key words and phrases. n-dimensional fuzzy number; second-order granular derivative; system of fuzzy initial value problems. https://doi.org/10.28924/2291-8639-21-2023-76 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-76 2 int. j. anal. appl. (2023), 21:76 et al. [2] enhanced a method to solve the sfdes with variables as fuzzy intervals. the limitations of previous methods for dealing with sfdes are derivatives do not always exist, monotonicity of the uncertainty, doubling properties, unnatural behavior in modeling phenomenon, and multiplicity of solutions. piegat and landowski [11] introduced horizontal membership function (hmf), and their applications. piegat and pluciński [12] was stated the difference between relative distance measure interval arithmetic (rdm-ia) yields a multidimensional answer while the results produced with sia. mazandarani et al. [6] elaborated the concept of hmf, granular differentiability (gr-differentiability) and granular integrability (gr-integrability). najariyan and zhao [9] offered a solution to the fuzzy dynamical system under gr-differentiability. nagalakshmi et al. [8] generalized the concept of fuzzy numbers to n-dimensional fuzzy numbers and developed an algorithm to solve system of first-order fbvps under the concept of gr-differentability. in this manuscript, consider two types of slfss under gr-differentiability. the upcoming sections of this manuscript are along these lines. section 2, presents basic definitions and propositions related to gr-differentiability of n-dimensional fuzzy valued function. section 3, an algorithm is presented as a working method to solve slfss under gr-differentiability. in section 4, we describe mechanical applications such as automobile two-axles, railway cars system, and spring-mass systems to highlight the proposed algorithm. section 5, conclusions and future works are analyzed. 2. preliminaries for a later discussion, this section provides some essential notations, definitions, and findings. suppose that the membership function, q : r → [0,1] of a fuzzy subset of the real number set r, satisfies the following conditions: (i) q(t0)=1 for at least one t0 ∈r. (ii) q(λy +(1−λ)z)≥ min{q(y),q(z)},∀λ ∈ [0,1], y,z ∈r. (iii) q is upper semi continuous on r. (iv) cl{t ∈r;q(t) > 0} is compact. then it is called a fuzzy number (fn). here q(t) is the membership degree of t, ∀t ∈ r. the λ-level sets of q are defined by [q]λ = {t ∈ r : q(t) ≥ λ} = [qλl ,q λ r ], for 0 < λ ≤ 1 and [q]0 = cl{t ∈r : q(t) > 0}. let rf denotes the space of fns in r. refer to [6] for definitions, notations, and essential findings regarding hmfs, first-order granular derivative (gr-derivative), and granular integration (gr-integrations) of fns in r. definition 2.1. suppose that p,q ∈ rf, whose hmfs are pgr(λ,αp) and qgr(λ,αq) respectively. then r = p ∗ q ∈ rf, such that h(r) , pgr(λ,αp)o qgr(λ,αq), where “o” and “∗” denotes any one of the operations addition, multiplication, subtraction and division in r and rf, respectively and 0 /∈ qgr(λ,αq) if “∗” denotes the division. that is int. j. anal. appl. (2023), 21:76 3 (1) h(p⊕q), pgr(λ,αp)+qgr(λ,αq), (2) h(p⊗q), pgr(λ,αp)qgr(λ,αq), (3) h(p q), pgr(λ,αp)−qgr(λ,αq), (4) h(p�q), pgr(λ,αp)÷qgr(λ,αq), (5) h(k �q), k qgr(λ,αq), where k ∈r and p,q,r ∈rf. definition 2.2. [9] let f : [a,b]→rf, be the ff. if there exists d2grf (t0) dt2 ∈rf, such that lim h→0 f ′(t0 +h) f ′(t0) h = d2grf (t0) dt2 = f ′′gr(t0), then f is said to be second order gr-differentiable at a point t0 ∈ [a,b]. theorem 2.1. [9] let f : [a,b]→rf. then f is twice gr-differentiable if and only if its hmf is twice differentiable with respect to t ∈ [a,b]. moreover, h ( d2grf (t) dt2 ) = ∂2fgr(t,λ,αf ) ∂t2 . proposition 2.1. let f : [a,b] → rf be a ff, with [f (t)]λ = [ fλl (t), f λ r (t) ] . the ff f is grdifferentiable twice on [a,b] if and only if (fλl ) ′(t) and (fλr ) ′(t) are differentiable on [a,b]. proof. since [f (t)]λ = [ fλl (t), f λ r (t) ] , then fgr(t,λ,αg)= fλl (t)+(f λ r (t)−fλl (t))αf , where λ, αf ∈ [0,1]. from definition 2.2 and theorem 2.1, we have suppose that f (t) is a gr-differentiable twice on [a,b] ⇐⇒ ∂2fgr(t,λ,αf ) ∂t2 =(fλl ) ′′(t)+((fλr ) ′′(t)− (fλl ) ′′(t))αf ⇐⇒ (fλl ) ′(t) and (fλr ) ′(t) are differentiable on [a,b]. . � definition 2.3. [8] let rnf =rf ×rf ×rf ×···×rf︸ ︷︷ ︸ ntimes , be the space of n-dimensional fuzzy vectors whose components are fuzzy numbers. then the addition and scalar multiplication defined component wise as follows: if u =(u1,u2, · · · ,un),v =(v1,v2, · · · ,vn)∈rnf, then (i) u ⊕v =(u1 ⊕v1,u2 ⊕v2, · · · ,un ⊕vn), (ii) k �u =(k �u1,k �u2, · · · ,k �un), where ui,vi ∈rnf, i =1,2, · · · ,n and k ∈r. definition 2.4. if u =(u1,u2, · · · ,un)∈rnf, as ui ∈rf, i =1,2, · · · ,n. then the hmf for u ∈r n f is defined by ugr(λ,αu)= (u1gr(λ,α1),u2gr(λ,α2), · · · , ungr(λ,αn)), where λ, α1, · · · ,αn ∈ [0,1]. 4 int. j. anal. appl. (2023), 21:76 proposition 2.2. let u and v be two n-dimensional fuzzy vectors. then u and v are said to be equal if and only if h(u)= h(v), for all αu = αv ∈ [0,1]. proof. since u,v ∈ rnf, then u = (u1,u2, · · · ,un),v = (v1,v2, · · · ,vn), for ui, vi ∈ rf, i = 1,2, · · · ,n. consider, u = v ⇐⇒ (u1,u2, · · · ,un)= (v1,v2, · · · ,vn) ⇐⇒ ui = vi, i =1,2, · · · ,n. ⇐⇒ h(ui)= h(vi), for all αui = αvi ∈ [0,1], i =1,2, · · · ,n. ⇐⇒ (h(u1),h(u2), · · · ,h(un))= (h(v1),h(v2), · · · ,h(vn)) ⇐⇒ h(u)= h(v), for all αu = αv ∈ [0,1], where αu , (αu1,αu2, · · · ,αun) and αv , (αv1,αv2, · · · ,αvn). � definition 2.5. [8] let u,v ∈rnf. the function d n gr :r n f ×r n f → r + ∪{0}, defined by dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖, which is called a n-dimensional granular distance between two n-dimensional fuzzy vectors u and v, where ‖.‖ represents euclidean norm in rn. proposition 2.3. the function dngr is a metric on the space of rnf. proof. suppose that rnf is a non-empty set and d n gr :r n f ×r n f → r + ∪{0} is real-valued function. (i) consider, dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ > 0. (ii) consider, dngr(u,v)=0 ⇐⇒ sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖=0 ⇐⇒ ‖ugr −vgr‖=0 ⇐⇒ ugr −vgr =0 ⇐⇒ ugr = vgr ⇐⇒ u = v. (iii) consider, dngr(u,v)= sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ =sup λ max αu,αv ‖vgr(λ,αv)−ugr(λ,αu)‖ =dngr(v,u). int. j. anal. appl. (2023), 21:76 5 (iv) consider, dngr(u,w)= sup λ max αu,αw ‖ugr(λ,αu)−wgr(λ,αw)‖ =sup λ max αu,αv,αw ‖ugr(λ,αu)−vgr(λ,αv)+vgr(λ,αv)−wgr(λ,αw)‖ ≤ sup λ max αu,αv ‖ugr(λ,αu)gr −vgr(λ,αv)‖+sup λ max αv,αw ‖vgr(λ,αv)−wgr(λ,αw)‖ =dngr(u,v)+d n gr(v,w). from (i)-(iv), ( rnf,d n gr ) is a metric space. � theorem 2.2. ( rnf,d n gr ) is a complete metric space (cms). proof. if any cauchy sequence of n-dimensional fuzzy vectors in ( rnf,d n gr ) is convergent then the proof concluded. suppose that um ∈ rnf, m ≥ 1 is a cauchy sequence. then for all �1 > 0, there exists n ≥ 1 such that dngr(um,um+p) < �1, for all m ≥ 1, q ≥ 1. dngr(um,um+p) < �1 =⇒ sup λ max αum,αum+p ‖umgr(λ,αum)−um+pgr(λ,αum+p)‖ < �1 =⇒ ‖umgr −um+pgr‖ < �1. now {umgr} is a cauchy sequence in the space of rn. clearly { umgr } is convergent in rn and umigr(λ,αumi)= umi λ l +(u λ mir −uλmil)αumi , where λ,αumi ∈ [0,1]. since umigr(λ,αumi) is convergent, so that u λ mil and uλmir are convergent. suppose that lim n→∞ uλmil = ui λ l and limn→∞ uλmir = ui λ r . since u λ mil ≤ uλmir, so that u λ i l ≤ uλi r for all i = 1,2, · · · ,n. if [uλi l,u λ i r ], i = 1,2, · · · ,n are λ-level sets of ui, then proof will be complete. it is shown in the same manner in the proof of theorem 4 [6], and therefore is left off. � lemma 2.1. suppose that u,v,w,s ∈rnf and µ ∈r, then the below results hold: (i) dngr(u ⊕v,w ⊕ s)≤dngr(u,w)+dngr(v,s). (ii) dngr(µ�u,µ�v)= |µ|dngr(u,v). (iii) dngr(u ⊕v,w ⊕v)≤dngr(u,w). proof. (i) from definition 2.5, we have dngr(u ⊕v,w ⊕ s) = sup λ max αu,αv,αw,αs ‖(ugr(λ,αu)+vgr(λ,αv))− (wgr(λ,αw)+ sgr(λ,αs))‖ =sup λ max αu,αv,αw,αs ‖(ugr(λ,αu)−wgr(λ,αw))+(vgr(λ,αv)− sgr(λ,αs))‖ 6 int. j. anal. appl. (2023), 21:76 ≤ sup λ max αu,αw ‖ugr(λ,αu)−wgr(λ,αw)|+sup λ max αv,αs |(vgr(λ,αv)− sgr(λ,αs))‖ =dngr(u,w)+d n gr(v,s). (ii) from definition 2.5, we have dngr(µ�u,µ�v)= sup λ max αu,αv,αµ ‖µugr(λ,αu)−µvgr(λ,αv)‖ = |µ| sup λ max αu,αv ‖ugr(λ,αu)−vgr(λ,αv)‖ = |µ|dngr(u,v). (iii) from (i), we have dngr(u ⊕v,w ⊕v) =dngr(u,w)+d n gr(v,v) =dngr(u,w). � proposition 2.4. if f : [a,b]→rnf is a fuzzy function, then it is called an n-dimensional fuzzy valued function on [a,b]. proof. since f : [a,b]→rnf is a fuzzy function, then f (t)∈r n f, for all t ∈ [a,b]. therefore f (t)= (f1(t), f2(t), . . . , fn(t)), for all t ∈ [a,b] and fi(t)∈rf, i =1,2, . . . ,n. thus f (t) is a n-dimensional fuzzy vector for each t ∈ [a,b] and hence f : [a,b] → rnf, is a ndimensional fuzzy valued function on [a,b]. � proposition 2.5. if f : [a,b]→rnf is a n-dimensional fuzzy valued function, include mn ∈ n distinct fns, then the hmf of f is denoted by h(f (t)), fgr(t,λ,αf ), and interpreted as fgr : [a,b]×[0,1]× [0,1]× [0,1]×···× [0,1]︸ ︷︷ ︸ mntimes → rn, in which αf , (αi1,αi2, . . . ,αim), where αi1,αi2, . . . ,αim are the mn rdm variables for ui1, ui2, . . ., uim for i =1,2, · · · ,n. proof. since f : [a,b] → rnf is a n-dimensional fuzzy valued function, so that f (t) = (f1(t), f2(t), . . . , fn(t)), for all t ∈ [a,b] and fi(t)∈rnf, i =1,2, . . . ,n. therefore h(f (t))= (h(f1(t)),h(f2(t)), . . . ,(fn(t))) fgr(t,λ,αf )= (f1gr(t,λ,α11,α12, . . . ,α1m), f2gr(t,λ,α21,α22, . . . ,α2m), . . . , fngr(t,λ,αn1,αn2, . . . ,αnm)) where αf ≡ (αi1,αi2, . . . ,αim)∈ [0,1], i =1,2, · · · ,n. � int. j. anal. appl. (2023), 21:76 7 definition 2.6. let f : [a,b] → rnf be a n-dimensional fuzzy valued function. the limit of f (t) as t → p is q ∈rnf, which is subject to following conditions: (i) if p ∈ (a,b), for all �1 > 0, there exits δ1 > 0 such that |t−p| < δ1 =⇒ dngr(f (t),q) < �1, and write it as lim t→p f (t)= q. (ii) if p = b, for all �1 > 0, there exits δ1 > 0 such that 0 < t −b < δ1 =⇒ dngr(f (t),q) < �1, and write it as lim t→b+ f (t)= q. (iii) if p = c, for all �1 > 0, there exits δ1 > 0 such that 0 < c − t < δ1 =⇒ dngr(f (t),q) < �1, and write it as lim t→c− f (t)= q. definition 2.7. let f : [a,b] → rnf be a n-dimensional fuzzy valued function. the function f (t) is said to be continuous at t = p if f (p)∈rnf, which is subject to following conditions: (i) if p ∈ (a,b), for all �1 > 0, there exits δ1 > 0 such that |t−p| < δ1 =⇒ dngr(f (t), f (p)) < �1, and write it as lim t→p f (t)= f (p). (ii) if p = b, for all �1 > 0, there exits δ1 > 0 such that0 < t−b < δ1 =⇒ dngr(f (t), f (b)) < �1, and write it as lim t→b+ f (t)= f (b). (iii) if p = c, for all �1 > 0, there exits δ1 > 0 such that0 < c−t < δ1 =⇒ dngr(f (t), f (c)) < �1, and write it as lim t→c− f (t)= f (c). note 2.1. [8] if f ,h : [a,b]→rnf are n-dimensional fuzzy valued functions, then the granular distance is dgr(f (t),h(t))= sup λ max αf ,αh ‖fgr(t,λ,αf )−hgr(t,λ,αh)‖, where t ∈ [a,b]⊂r and λ,αf ,αh ∈ [0,1]. refer to [8] first-order gr-derivative, and gr-integration for n-dimensional fuzzy valued function. now, we define second order gr-differentiability for n-dimensional fuzzy valued function. definition 2.8. let f : [a,b] → rnf, be the n-dimensional fuzzy valued function. if there exists d2grf (t0) dt2 ∈rnf, such that lim h→0 f ′(t0 +h) f ′(t0) h = d2grf (t0) dt2 = f ′′gr(t0), this limit is taken in the metric space (rnf,d n gr). then f is said to be second order grdifferentiable at a point t0 ∈ [a,b]. theorem 2.3. let f : [a,b]→rnf be a n-dimensional fuzzy valued function, then f is gr-differentiable if and only if its hmf is differentiable with respect to t ∈ [a,b]. moreover, h ( d2grf (t) dt2 ) = ∂2fgr(t,λ,αf ) ∂t2 . 8 int. j. anal. appl. (2023), 21:76 proof. assuming that f is second order gr-differentiable then f is first order gr-differentiable and h ( dgrf (t) dt ) = ∂fgr(t,λ,αf ) ∂t . for t ∈ (a,b). based on the definition 2.6 and definition 2.8, for all �1 > 0, there exits δ1 > 0 such that |h| < δ1 =⇒ dngr( f ′(t+h) f ′(t) h , d2grf (t) dt2 ) < �1 =⇒ sup λ max αf ‖ f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h − d2grfgr(t,λ,αf ) dt2 ‖ < �1 =⇒ ‖ f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h − d2grfgr(t,λ,αf ) dt2 ‖ < �1 =⇒ lim h→0 f ′gr(t +h,λ,αf )− f ′gr(t,λ,αf ) h = d2grfgr(t,λ,αf ) dt2 =⇒ ∂2fgr(t,λ,αf ) ∂t2 = h ( d2grf (t) dt2 ) . � proposition 2.6. let f : [a,b] → rnf be a n-dimensional fuzzy valued function defined by f (t) = (f1(t), f2(t), · · · , fn(t)) for all x ∈ [a,b] and fi(t) ∈ rf, with [fi(t)]λ = [ fλil (t), f λ ir (t) ] , i = 1,2, · · · ,n. the n-dimensional fuzzy valued function f is gr-differentiable twice on [a,b] if and only if (fλil ) ′(t) and (fλir ) ′(t) are differentiable on [a,b], for all i =1,2, · · · ,n. proof. since fgr(t,λ,αf )= ( f1gr(t,λ,α1), f2gr(t,λ,α2), · · · , fngr(t,λ,αn) ) , then fgr(t,λ,αf )= ( (fλ1l(t)+(f λ 1r (t)− fλ1l(t))α1),(f λ 2l (t)+(fλ2r(t)− f λ 2l (t))α2), · · · , (fλnl (t)+(f λ nr (t)− fλnl (t))αn) ) , where λ, αf , (α1,α2, · · · ,αn)∈ [0,1]. from definition 2.8 and theorem 2.3, we have suppose that f (t) is a gr-differentiable twice on [a,b] ⇐⇒ ∂2fgr(t,λ,αf ) ∂t2 = ( ((fλ1l) ′′(t)+((fλ1r) ′′(t)− (fλ1l) ′(t))α1), ((fλ2l) ′′(t)+((fλ2r) ′′(t)− (fλ2l) ′′(t))α2), · · · ,((fλnl ) ′′(t)+((fλnr) ′′(t)− (fλnl ) ′′(t))αn) ) ⇐⇒ (fλil ) ′(t) and (fλir ) ′(t), are differentiable on [a,b] for i =1,2, · · · ,n. � definition 2.9. if a matrix a = [aij]n×m, for all aij ∈ rf, i = 1,2, · · · ,n and j = 1,2, · · · ,m. then that matrix a is called fuzzy matrix. definition 2.10. if a = [aij]n×m is a fuzzy matrix, then the hmf of a is defined by h(a) = [h(aij)]n×m , [(aij)gr(λ,αij)]n×m, where λ,αij ∈ [0,1], i =1,2, · · · ,n and j =1,2, · · · ,m. int. j. anal. appl. (2023), 21:76 9 3. an algorithm for the solution of system of second order linear fuzzy initial value problems under (sslfde) gr-differentiability consider a sslfdes, z′′gr(t)= a⊗z(t)⊕f(t), withz(t0)= z0. (3.1) the matrix form of (3.1) is, [ y ′′gr(t) z′′gr(t) ] = [ a b c d ] ⊗ [ y(t) z(t) ] ⊕ [ f (t) g(t) ] , (3.2) subject to, [ y(t0) z(t0) ] = [ y0 z0 ] and [ y ′(t0) z′(t0) ] = [ y ′0 z′0 ] . (3.3) the following algorithm describes the procedure to compute λ-cut solution of sslfdes (3.1) if it exists. step 1 : applying hmf on both sides of (3.2) and (3.3), we get[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ agr(λ,αa) bgr(λ,αb) cgr(λ,αc) dgr(λ,αd) ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ fgr(t,λ,αf ) ggr(t,λ,αg) ] , (3.4) with, [ ygr(t0) zgr(t0) ] = [ y0gr(λ,αy0) z0gr(λ,αz0) ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr(λ,αy ′0 ) z′0gr(λ,αz′0 ) ] , (3.5) where λ, αz,αf , αg, αa, αb,αc, αd, αy0, αz0,αy ′0, αz′0 ∈ [0,1]. here, (3.4) and (3.5) taken as a ordinary second order system of differential equations. step 2 : solving (3.4) and (3.5), we get ygr(t,λ,αy) and zgr(t,λ,αz). (3.6) step 3 : applying inverse hmf on both sides of (3.6), we get [y(t)]λ = [ inf λ≤α≤1 min αy ygr(t,α,αy), sup λ≤α≤1 max αy ygr(t,α,αy)], (3.7) [z(t)]λ = [ inf λ≤α≤1 min αz zgr(t,α,αz), sup λ≤α≤1 max αz zgr(t,α,αz)], (3.8) which is the required λ-cut solution of sslfdes (3.1). 10 int. j. anal. appl. (2023), 21:76 4. mechanical applications in this section, we describe mechanical applications [13] of which the uncertain information taken as fuzzy sets. example 4.1. (automobile with two axles) now we have an automobile with two axles and distinct front and back suspension systems, we can examine a more realistic model. the suspension system of such a vehicle is seen in figure 1 . we suppose that the car’s body behaves similarly to a solid bar with the dimensions of mass m and length l = l1 + l2. its centre of mass c, which is located at a distance l1 from the front of the vehicle, has a moment of inertia i around it. the vehicle features suspension springs with hooke’s constants s1 and s2 for the front and back, respectively. let y(t) represent the car’s vertical displacement from equilibrium while it is moving, and let z(t) represent its angular displacement (in radians) from the horizontal. the equations may then be derived using newton’s laws of motion for linear and angular acceleration as follows: m �y ′′gr(t)=−(s1 + s2)�y(t)⊕ (s1l1 − s2l2)�z(t), i �z′′gr(t)= (s1l1 − s2l2)�y(t) (s1l 2 1 + s2l 2 2)�z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. suppose that m = 75lb.s2/f t, l1 = 7f t, l2 = 3f t,s1 = s2 = 2000lb/f t, i = 1000f t.lb.s2 and the λ-level sets of fuzzy initial values are [y0]λ = [z0]λ = [3+ λ,5− λ], [y ′0] λ = [5+ λ,7− λ], [z′0] λ = [6+λ,8−λ]. figure 1. two axles car. then the matrix equation is, [ y ′′gr(t) z′′gr(t) ] = [ −53.33 106.67 8 −116 ] � [ y(t) z(t) ] , (4.1) , subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] . (4.2) int. j. anal. appl. (2023), 21:76 11 taking hmf on both sides of (4.1) and (4.2), we have[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −53.33 106.67 8 −116 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] , (4.3) subject to, [ ygr(0) zgr(0) ] = [ y0gr(λ,α1) z0gr(λ,α1) ] and [ y ′gr(0) z′gr(0) ] = [ y ′0gr(λ,α2) z′0gr(λ,α3) ] , (4.4) where y0gr(λ,α1)= z0gr(λ,α1)= [3+λ+2(1−λ)α1],y ′0gr(λ,α2)= [5+λ+2(1−λ)α2],z ′ 0gr (λ,α3)= [6+λ+2(1−λ)α3], where λ, α1, α2, α3 ∈ [0,1]. the solution for second order system of equations (4.3) and (4.4) are ygr(t,λ,α1,α2,α3) and zgr(t,λ,α1,α2,α3). (4.5) applying inverse hmf on (4.5), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2,α3 ygr(t,α,α1,α2,α3), sup λ≤α≤1 max α1,α2,α3 ygr(t,α,α1,α2,α3)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2,α3 zgr(t,α,α1,α2,α3), sup λ≤α≤1 max α1,α2,α3 zgr(t,α,α1,α2,α3)]. the λ-level sets solution is enumerated using matlab and is illustrated in figure 2 (a) λ-level sets of y(t). (b) λ-level sets of z(t). figure 2. the black curve gives the solution at λ =1 for the system (4.1) and (4.2). example 4.2. (one springs-two railway cars system) figure 3 represents one spring supporting two railway cars of masses m1 and m2 respectively system to one other. if all two of the two cars rightward displacements from their respective equilibrium positions are positive, then the spring is 12 int. j. anal. appl. (2023), 21:76 extended byy(t). the motion equations for the two cars are generated as follows: m1 �y ′′gr(t)=−s �y(t)⊕ s �z(t), m2 �z′′gr(t)= s �y(t) s �z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. figure 3. one springs-two cars systems. suppose that m1 =1lb.s2/f t,m2 =1lb.s2/f t, and the λ-level sets of spring constant and fuzzy initial values are [s]λ = [1+λ,3−λ], [y0]λ = [z0]λ = [λ,2−λ], [y ′0] λ = [1+λ,3−λ], [z′0] λ = [λ,2−λ]. then the matrix equation is,[ 1 0 0 3 ] � [ y ′′gr(t) z′′gr(t) ] = [ −s s s −s ] ⊗ [ y(t) z(t) ] , (4.6) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] . (4.7) taking hmf on both sides of (4.6) and (4.7), we have[ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −sgr(λ,α1) sgr(λ,α1) sgr(λ,α1) −sgr(λ,α1) ][ ygr(t,λ,αy) zgr(t,λ,αz) ] , (4.8) subject to [ ygr(0) zgr(0) ] = [ y0gr(λ,α2) z0gr(λ,α2) ] and [ y ′gr(0) z′gr(0) ] = [ y ′0gr(λ,α1) z′0gr(λ,α2) ] , (4.9) here sgr(λ,α1) = y ′0gr(λ,α1) = [1+ λ +2(1− λ)α1], y0gr(λ,α2) = z0gr(λ,α2) = z ′ 0gr (λ,α2) = [λ+2(1−λ)α2], where λ, α1α2 ∈ [0,1]. int. j. anal. appl. (2023), 21:76 13 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −(1+λ+2(1−λ)α1) 1+λ+2(1−λ)α1 1+λ+2(1−λ)α1 −(1+λ+2(1−λ)α1) ] [ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] , (4.10) subject to, [ ygr(0) zgr(0) ] = [ λ+2(1−λ)α1 λ+2(1−λ)α1 ] and [ y ′gr(0) z′gr(0) ] = [ 1+λ+2(1−λ)α2 λ+2(1−λ)α1 ] . (4.11) the solution for second order system of equations (4.10) and (4.11) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.12) applying inverse hmf on (4.12), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. the λ-level sets solution is enumerated using matlab and is illustrated in figure 4 (a) λ-level sets of y(t). (b) λ-level sets of z(t). figure 4. the black curve gives the solution at λ =1 for the system (4.6) and (4.7). example 4.3. (two springs-two mass systems with external fuzzy force) figure 5 represents two springs supporting two masses to one other. if all the two masses rightward displacements from their respective equilibrium positions are positive, then (i) the first spring is extended by y(t). (ii) the second spring is extended by z(t) y(t). 14 int. j. anal. appl. (2023), 21:76 the motion equations for the two masses are generated as follows: m1 �y ′′gr(t)=−s1 �y(t)⊕ s2 � (z(t) y(t)), m2 �z′′gr(t)=−s2 � (z(t) y(t))+ f (t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. figure 5. two springs-two masses systems. the matrix form of system of equations is[ y ′′gr(t) z′′gr(t) ] = [ s1 s2 s3 s4 ] ⊗ [ y(t) z(t) ] ⊕ [ 0 pcos(10t) ] , (4.13) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] , (4.14) where λ-cut set of coefficients and initial values are s1 =−3, s2 =1, s3 =1, s4 =−1,[y0]λ = [z0]λ = [y ′0] λ = [λ,2−λ], [z′0] λ = [p]λ = [1+λ,3−λ]. taking hmf on both sides of (4.13) and (4.14), we have [ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −3 1 1 −1 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ 0 pgr(λ,α1)cos(10t) ] , (4.15) subject to, [ ygr(0) zgr(0) ] = [ y0gr z0gr ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr z′0gr ] , (4.16) where pgr(λ,α1)= [1+λ+2(1−λ)α1], y0gr(λ,α2)= z0gr(λ,α2)= y ′0gr(λ,α2)= [λ+2(1−λ)α2], z′0gr(λ,α2)= [1+λ+2(1−λ)α1], where λ, α1α2 ∈ [0,1]. int. j. anal. appl. (2023), 21:76 15 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −3 1 1 −1 ][ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] + [ 0 [1+λ+2(1−λ)α1]cos(10t) ] , (4.17) with, [ ygr(0) zgr(0) ] = [ λ+2(1−λ)α2 λ+2(1−λ)α2 ] and [ y ′gr(0) z′gr(0) ] = [ λ+2(1−λ)α2 1+λ+2(1−λ)α1 ] . (4.18) the solution for system of equations (4.17) and (4.18) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.19) applying inverse hmf on (4.19), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. the λ-level sets solution is enumerated using matlab and is illustrated in figure 6 (a) λ-level sets of y(t). (b) λ-level sets of z(t). figure 6. the black curve gives the solution at λ =1 for the system (4.13) and (4.14). example 4.4. (three springs-two mass systems with external fuzzy force) three springs supporting two masses on both sides and one another is depicts in figure 7. assume that there is no friction as the masses move and that each spring abides by hooke’s law. let f (t) be the fuzzy force applying on mass m1 at time t ≥ 0. if all the two masses rightward displacements (from their individual equilibrium positions) are positive, then 16 int. j. anal. appl. (2023), 21:76 (i) the first spring is extended by y(t). (ii) the second spring is extended by z(t) y(t). (iii) the third spring is compressed by z(t). the motion equations for the two masses are generated as follows: m1 �y ′′gr(t)=−s1 �y(t)⊕ s2 � (z(t) y(t))+ f (t), m2 �z′′gr(t)=−s2 � (z(t) y(t))− s3 �z(t), with fuzzy initial values, y(0)= y0, z(0)= z0, y ′ gr(0)= y ′ 0, z ′ gr(0)= z ′ 0. figure 7. three springs-two masses systems. the matrix form of system of equations is[ y ′′gr(t) z′′gr(t) ] = [ s1 s2 s3 s4 ] ⊗ [ y(t) z(t) ] ⊕ [ pcos(10t) 0 ] , (4.20) subject to, [ y(0) z(0) ] = [ y0 z0 ] and [ y ′gr(0) z′gr(0) ] = [ y ′0 z′0 ] , (4.21) where λ-cut set of coefficients and initial values are s1 =−3, s2 =1, s3 =1, s4 =−3, y0 =1, z0 =1, [y ′0] λ = [λ,2−λ], [z′0] λ = [p]λ = [1+λ,3−λ]. taking hmf on both sides of (4.20) and (4.21), we have [ ∂2ygr(t,λ,αy) ∂t2 ∂2zgr(t,λ,αz) ∂t2 ] = [ −3 1 1 −3 ][ ygr(t,λ,αy) zgr(t,λ,αz) ] + [ pgr(λ,α2)cos(10t) 0 ] , (4.22) subject to, [ ygr(0) zgr(0) ] = [ y0gr z0gr ] and [ y ′gr(t0) z′gr(t0) ] = [ y ′0gr z′0gr ] , (4.23) where pgr(λ,α2) = [1 + λ + 2(1 − λ)α2], y ′0gr(λ,α1) = [λ + 2(1 − λ)α1], z ′ 0gr (λ,α2) = [1+λ+2(1−λ)α1], where λ, α1α2 ∈ [0,1]. int. j. anal. appl. (2023), 21:76 17 =⇒ [ ∂2ygr(t,λ,α1,α2) ∂t2 ∂2zgr(t,λ,α1,α2) ∂t2 ] = [ −3 1 1 −3 ][ ygr(t,λ,α1,α2) zgr(t,λ,α1,α2) ] + [ [1+λ+2(1−λ)α2]cos(10t) 0 ] , (4.24) with [ ygr(0) zgr(0) ] = [ 1 1 ] and [ y ′gr(0) z′gr(0) ] = [ λ+2(1−λ)α1 1+λ+2(1−λ)α2 ] . (4.25) the solution for system of equations (4.24) and (4.25) is ygr(t,λ,α1,α2) and zgr(t,λ,α1,α2). (4.26) applying inverse hmf on (4.26), we get [y(t)]λ = [ inf λ≤α≤1 min α1,α2 ygr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 ygr(t,α,α1,α2)], [z(t)]λ = [ inf λ≤α≤1 min α1,α2 zgr(t,α,α1,α2), sup λ≤α≤1 max α1,α2 zgr(t,α,α1,α2)]. the λ-level sets solution is enumerated using matlab and is illustrated in figure 8 (a) λ-level sets of y(t). (b) λ-level sets of z(t). figure 8. the black curve gives the solution at λ =1 for the system (4.20) and (4.21). 5. conclusions this paper mainly deals with determining solutions of sslfdes and applications to some mechanical problems. the granular differentiability is extended to n-dimensional fuzzy valued functions. the sslfdes with fuzzy initial conditions are investigated under gr-differentiability. an algorithm is developed to determine the solutions of sslfdes with fuzzy initial conditions. some mechanical 18 int. j. anal. appl. (2023), 21:76 problems as automobiles with two axles, railway cars systems, and mass-spring systems with fuzzy initial conditions are demonstrated for the effective implementation of the algorithm. in the future, this work will be extended for higher-order sfdes with fuzzy initial and boundary conditions. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] y. barazandeh, b. ghazanfari, approximate solution for systems of fuzzy differential equations by variational iteration method, punjab univ. j. math. 51 (2019), 13-33. [2] r. boukezzoula, l. jaulin, d. coquin, a new methodology for solving fuzzy systems of equations: thick fuzzy sets based approach, fuzzy sets sys. 435 (2022), 107-128. https://doi.org/10.1016/j.fss.2021.06.003. [3] o.s. fard, n. ghal-eh, numerical solutions for linear system of first-order fuzzy differential equations with fuzzy constant coefficients, inform. sci. 181 (2011), 4765-4779. https://doi.org/10.1016/j.ins.2011.06. 007. [4] n. gasilov, sh. g. amrahov, a.g. fatullayev, a geometric approach to solve fuzzy linear systems of differential equations, appl. math. inform. sci. 5 (2011), 484-499. https://doi.org/10.48550/arxiv.0910.4307. [5] m. keshavarz, t. allahviranloo, s. abbasbandy, m.h. modarressi, a study of fuzzy methods for solving system of fuzzy differential equations, new math. nat. comput. 17 (2021), 1-27. https://doi.org/10.1142/ s1793005721500010. [6] m. mazandarani, n. pariz, a.v. kamyad, granular differentiability of fuzzy-number-valued functions, ieee trans. fuzzy syst. 26 (2017), 310-323. https://doi.org/10.1109/tfuzz.2017.2659731. [7] s.p. mondal, n.a. khan, o.a. razzaq, s. tudu, t.k. roy, adaptive strategies for system of fuzzy differential equation: application of arms race model, j. math. computer sci. 18 (2018), 192-205. https://doi.org/10. 22436/jmcs.018.02.07. [8] s. nagalakshmi, g.s. kumar, b. madavi, solution for a system of first-order linear fuzzy boundary value problems, int. j. anal. appl. 24 (2023), 4. https://doi.org/10.28924/2291-8639-21-2023-4. [9] m. najariyan, yi. zhao, the explicit solution of fuzzy singular differential equations using fuzzy drazin inverse matrix, soft comput. 24 (2020), 11251–11264. https://doi.org/10.1007/s00500-020-05055-8. [10] m. najariyan, n. pariz, h. vu, fuzzy linear singular differential equations under granular differentiability concept, fuzzy sets syst. 429 (2022), 169-187. https://doi.org/10.1016/j.fss.2021.01.003. [11] a. piegat, m. landowski, solving different practical granular problems under the same system of equations, granular comput. 3 (2018), 39-48. https://doi.org/10.1007/s41066-017-0054-5. [12] a. piegat, m. pluciński, the differences between the horizontal membership function used in multidimensional fuzzy arithmetic and the inverse membership function used in gradual arithmetic, granular comput. 7 (2022), 751-760. https://doi.org/10.1007/s41066-021-00293-z. [13] c.h. edwards, d.e. penney, differential equations and boundary value problems: computing and modeling, fifth ed., pearson education, boston, 2000. https://doi.org/10.1016/j.fss.2021.06.003 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.1016/j.ins.2011.06.007 https://doi.org/10.48550/arxiv.0910.4307 https://doi.org/10.1142/s1793005721500010 https://doi.org/10.1142/s1793005721500010 https://doi.org/10.1109/tfuzz.2017.2659731 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.22436/jmcs.018.02.07 https://doi.org/10.28924/2291-8639-21-2023-4 https://doi.org/10.1007/s00500-020-05055-8 https://doi.org/10.1016/j.fss.2021.01.003 https://doi.org/10.1007/s41066-017-0054-5 https://doi.org/10.1007/s41066-021-00293-z 1. introduction 2. preliminaries 3. an algorithm for the solution of system of second order linear fuzzy initial value problems under (sslfde) gr-differentiability 4. mechanical applications 5. conclusions references int. j. anal. appl. (2023), 21:42 on anti-q-fuzzy deductive systems of hilbert algebras m. vasuki1, p. senthil kumar2, n. rajesh2,∗ 1department of mathematics, srinivasan college of arts and science (affiliated to bharathidasan university), perambalur, tamilnadu, india 2department of mathematics, rajah serfoji government college (affiliated to bharathidasan university), thanjavur-613005, tamilnadu, india ∗corresponding author: nrajesh_topology@yahoo.co.in abstract. in this paper, the concept of anti-q-fuzzy deductive systems concepts of hilbert algebras are introduced and proved some results. further, we discuss the relation between anti-q-fuzzy deductive system and level subsets of a q-fuzzy set. anti q-fuzzy deductive system is also applied in the cartesian product of hilbert algebras. 1. introduction the concept of fuzzy sets was proposed by zadeh [19]. the theory of fuzzy sets has several applications in real-life situations, and many scholars have researched fuzzy set theory. after the introduction of the concept of fuzzy sets, several research studies were conducted on the generalizations of fuzzy sets. the integration between fuzzy sets and some uncertainty approaches such as soft sets and rough sets has been discussed in [1,4,7]. the idea of intuitionistic fuzzy sets suggested by atanassov [2] is one of the extensions of fuzzy sets with better applicability. applications of intuitionistic fuzzy sets appear in various fields, including medical diagnosis, optimization problems, and multicriteria decision making [12–14]. the concept of hilbert algebra was introduced in early 50-ties by l.henkin and t.skolem for some investigations of implication in intuicionistic and other non-classical logics. in 60-ties, these algebras were studied especially by a.horn and a.diego from algebraic point of view. a.diego proved (cf. [9] that hilbert algebras form a variety which is locally finite. hilbert algebras were received: feb 14, 2023. 2020 mathematics subject classification. 20n05, 94d05, 03e72. key words and phrases. hilbert algebra; anti-q-fuzzy deductive system; q-fuzzy set. https://doi.org/10.28924/2291-8639-21-2023-42 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-42 2 int. j. anal. appl. (2023), 21:42 treated by d.busneag (cf. [5], [6]) and y.b.jun (cf. [15]) and some of their filters forming deductive systems were recognized. w.a.dudek (cf. [11]) considered the fuzzification of subalgebras and deductive systems in hilbert algebras. in this paper, the concept of anti-q-fuzzy deductive systems concepts of hilbert algebras are introduced and proved some results. further, we discuss the relation between anti-q-fuzzy deductive system and level subsets of a q-fuzzy set. anti q-fuzzy deductive system is also applied in the cartesian product of hilbert algebras. 2. preliminaries definition 2.1. [9] a hilbert algebra is a triplet h = (h, ·, 1), where h is a nonempty set, · is a binary operation and 1 is fixed element of h such that the following axioms hold for each x,y,z ∈ h. (1) x ∗ (y ∗x) = 1, (2) (x ∗ (y ∗z)) ∗ ((x ∗y) ∗ (x ∗z)) = 1, (3) x ∗y = 1 and y ∗x = 1 imply x = y. the following result was proved in [11]. lemma 2.1. let h = (h, ·, 1) be a hilbert algebra and x,y,z ∈ h. then (1) x ∗x = 1, (2) 1 ∗x = x, (3) x ∗ 1 = 1, (4) x ∗ (y ∗z) = y ∗ (x ∗z). it is easily checked that in a hilbert algebra h the relation ≤ y defined by x ≤ y ⇔ x ∗y = 1 is a partial order on h with 1 as the largest element. definition 2.2. [8] a nonempty subset i of a hilbert algebra h = (h, ·, 1) is called an ideal of h if (1) 1 ∈ i, (2) x ∗y,q ∈ i for all x ∈ h, y ∈ i, (3) (y2 ∗ (y1 ∗x)) ∗x ∈ i for all x ∈ h, y1,y2 ∈ i. definition 2.3. [10] a fuzzy set µ in a hilbert algebra h is said to be a fuzzy ideal of h if the following conditions are hold: (1) µ(1,q) ≥ µ(x,q) for all x ∈ h, (2) µ(x ∗y,q) ≥ µ(y,q) for all x,y ∈ h, (3) µ((y1 ∗ (y2 ∗x,q),q) ∗x) ≥ min{µ(y1,q),µ(y2,q)} for all x,y1,y2 ∈ h. lemma 2.2. let µ be a fuzzy set in a. then the following statements hold: for any x,y ∈ a, (1) 1 − max{µ(x),µ(y)} = min{1 −µ(x), 1 −µ(y)}, (2) 1 − min{µ(x),µ(y)} = max{1 −µ(x), 1 −µ(y)}. int. j. anal. appl. (2023), 21:42 3 definition 2.4. [16] a q-fuzzy set in a nonempty set h (or a q-fuzzy subset of h) is an arbitrary function µ : x×q → [0, 1], where q is a nonempty set and [0, 1] is the unit segment of the real line. definition 2.5. [16] let µ be a q-fuzzy set in a. the q-fuzzy set µ defined by µ(x,q) = 1−µ(x,q) for all x ∈ a and q ∈ q is called the complement of µ in a. remark 2.1. for a q-fuzzy set µ in a, we have µ = µ. definition 2.6. [17] let f : a → b be a function and µ be a q-fuzzy set in b. we define a new q-fuzzy set in a by µf as µ(f (x),q) for all x ∈ a and q ∈ q. definition 2.7. [17] let f : a → b be a bijection and µf be a q-fuzzy set in a. we define a new q-fuzzy set in b by µ as µ(y,q) = µf (x,q), where f (x) = y for all y ∈ b and q ∈ q. definition 2.8. [17] let µ be a q-fuzzy set in a and δ be a q-fuzzy set in b. the cartesian product µ × δ : (a × b) × q → [0, 1] is defined by (µ × δ)((x,y),q) = max{µ(x,q),δ(y,q)} for all x ∈ a, y ∈ b and q ∈ q. the dot product µ · δ : (a × b) × q → [0, 1] is defined by (µ∗δ)((x,y),q) = min{µ(x,q),δ(y,q)} for all x ∈ a, y ∈ b and q ∈ q. lemma 2.3. for any a,b ∈ r such that a < b, a < b+a 2 < b. 3. anti-q-fuzzy deductive systems definition 3.1. a q-fuzzy set µ in a hilbert algebra h is said to be an anti-q-fuzzy deductive system of h if the following conditions are hold: (∀x ∈ h) ( µ(1,q) ≤ µ(x,q) ) , (3.1) (∀ x,y ∈ h) ( µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)} ) . (3.2) a q-fuzzy set µ in a hilbert algebra h is called an anti-q-fuzzy deductive system of h if it is an anti-q-fuzzy deductive system of h for all q ∈ q. example 3.1. let h = {1,x,y,z, 0} be a set with a binary operation · defined by the following cayley table: · 1 x y z 1 1 x y z x 1 1 y z y 1 x 1 z z 1 1 y 1 then (h, ·, 1) is a hilbert algebra. let q = {q}. we define a q-fuzzy set µ as follows: µ(1,q) = 0.9,µ(x,q) = 0.3,µ(y,q) = 0.1,µ(z,q) = 0.6. then µ is a q-fuzzy ideal of h. 4 int. j. anal. appl. (2023), 21:42 lemma 3.1. if µ is an anti-q-fuzzy deductive system of a hilbert algebra h, then (∀ x,y,z ∈ h) ( z ≤ x ∗y ⇒ µ(y,q) ≤ max{µ(x,q),µ(z,q)} ) . (3.3) proof. let x,y,z ∈ x such that z ≤ x ∗y. then z · (x ∗y) = 1. µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)} ≤ max{max{µ(z · (x ∗y),q),µ(z,q)},µ(x,q)} = max{max{µ(1,q),µ(z,q)},µ(x,q)} = max{µ(x,q),µ(z,q)}. � lemma 3.2. if µ is an anti-q-fuzzy deductive system of a hilbert algebra h, then (∀ x,y ∈ h) ( x ≤ y ⇒ µ(x,q) ≤ µ(y,q) ) . (3.4) proof. let x,y ∈ h be such that x ≤ y. then x∗y = 1 and so µ(y,q) ≤ max{µ(x∗y,q),µ(x,q)} = max{µa(1,q),µa(x,q)} = µa(x,q). � theorem 3.1. if µ is an anti-q-fuzzy deductive system of a hilbert algebra h, then for any x,a1,a2, ...,an ∈ x, (...((x ∗ a1) ∗ a2) · ...) ∗ an = 1 implies µ(x,q) ≤ max{µ(a1,q),µ(a2,q), . . .µ(an,q)}. proof. apply induction on n and apply lemmas 3.1 and 3.2, the proof is clear. � theorem 3.2. every anti-q-fuzzy deductive system of hilbert algebra x is an anti-q-fuzzy subalgebra of x. proof. let µ be an anti-q-fuzzy deductive system of x. since y ≤ x ∗ y for all x,y ∈ x, from lemma 3.2 that µ(y,q) ≤ µ(x ∗ y,q) for all q ∈ q. it follows from that µ(x ∗ y,q) ≤ µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)}≤ max{µ(x,q),µ(y,q)}. hence µ is an anti-q-fuzzy subalgebra of x. � theorem 3.3. if µ is a q-fuzzy subalgebra of x such that µ(y,q) ≥ min{µ(x,q),µ(z,q)} for all x,y,z ∈ x satisfying x ∗y ≤ z, then µ is an anti-q-fuzzy deductive system of x. proof. let µ be an anti-q-fuzzy subalgebra of x. then by definition µ(1,q) ≤ µ(z,q) for all x ∈ x and q ∈ q. since x ≤ (x ∗y) ∗y, then by hypothesis, µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)}. hence µ is an anti-q-fuzzy deductive system of x. � proposition 3.1. if {µi : i ∈ ∆} is a family of anti-q-fuzzy deductive systems of a hilbert algebra h, then ∧ i∈∆ µi is an anti-q-fuzzy deductive system of h. int. j. anal. appl. (2023), 21:42 5 proof. let {µi : i ∈ ∆} be a family of anti-q-fuzzy deductive systems of a hilbert algebra h. let x ∈ h and q ∈ q, we have ( ∧ i∈∆ µi)(1,q) = sup i∈∆ {µi(1,q)} ≤ sup i∈∆ {µi(x,q)} = ( ∧ i∈∆ µi)(x,q). let x,y ∈ h and q ∈ q, we have ( ∧ i∈∆ µi)(y,q) = sup i∈∆ {µi(y,q)} ≤ sup i∈∆ {max{µi(x ∗y,q),µi(x,q)}} = max{sup i∈∆ µi(x ∗y,q), sup i∈∆ µi(x,q)} = max{( ∧ i∈∆ µi)(x ∗y,q), ( ∧ i∈∆ µi)(x,q)}. hence ∧ i∈∆ µi is an anti-q-fuzzy deductive system of a hilbert algebra h. � proposition 3.2. every q-fuzzy ideal of a hilbert algebra h is an anti-q-fuzzy deductive system of h. proof. let µ be an anti-q-fuzzy ideal of h. if y1 = x ∗ y, y2 = x, where x,y ∈ h and q ∈ q, then by (1), (2) of lemma 2.1, we have µ(y,q) = µ(1 ∗ y,q) = µ(((x ∗ y,q) ∗ (x ∗ y,q)) ∗ y,q) ≤ max{µ(x ∗y,q),µ(x,q)}. hence µ be an anti-q-fuzzy deductive system of h. � lemma 3.3. [18] let µ be a fuzzy set in a and for any t ∈ [0, 1]. then the following properties hold: (1) l(µ,t) = u(µ, 1 − t), (2) l−(µ,t) = u+(µ, 1 − t), (3) u(µ,t) = l(µ, 1 − t), (4) u+(µ,t) = l−(µ, 1 − t). lemma 3.4. [18] let µ be a q-fuzzy set in a and for any t ∈ [0, 1] and q ∈ q. then the following properties hold: (1) l(µ,t,q) = u(µ, 1 − t,q), (2) l−(µ,t,q) = u+(µ, 1 − t,q), (3) u(µ,t,q) = l(µ, 1 − t,q), (4) u+(µ,t,q) = l−(µ, 1 − t,q). lemma 3.5. [18] let µ be a q-fuzzy set in a and for any t ∈ [0, 1] and q ∈ q. then the following properties hold: (1) l(µ,t) = ⋂ q∈q l(µ,t,q), (2) l−(µ,t) = ⋂ q∈q l−(µ,t,q), (3) u(µ,t) = ⋂ q∈q u(µ,t,q), (4) u+(µ,t) = ⋂ q∈q u+(µ,t,q). 6 int. j. anal. appl. (2023), 21:42 definition 3.2. let µ be a q-fuzzy set of a hilbert algebra h and t ∈ [0, 1]. then we define the sets u(µ,t) = {x ∈ h : µ(x,q) ≥ t f or all q ∈ q} and u+(µ,t) = {x ∈ h : µ(x,q) > t f or all q ∈ q} are called an upper α-level subset and an upper α-strong level subset of µ, respectively. the sets l(µ,t) = {x ∈ h : µ(x,q) ≤ t f or all q ∈ q} and l−(µ,t) = {x ∈ h : µ(x,q) < t f or all q ∈ q} are called a lower t-level subset and a lower t-strong level subset of µ, respectively. for any q ∈ q, the sets u(µ,t,q) = {x ∈ h : µ(x,q) ≥ t} and u+(µ,t,q) = {x ∈ h : µ(x,q) > t} are called a q-upper t-level subset and a q-upper t-strong level subset of µ, respectively. the sets l(µ,t,q) = {x ∈ h : µ(x,q) ≤ t} and l+(µ,t,q) = {x ∈ h : µ(x,q) < t} are called a q-lower t-level subset and a q-lower t-strong level subset of µ, respectively. theorem 3.4. let µ be a q-fuzzy set in h. then the following statements hold: (1) µ is an anti-q-fuzzy deductive system of h if and only if for any t ∈ [0, 1] and q ∈ q, l(µ,t,q) is either empty or a deductive system of h. (2) µ is an anti-q-fuzzy deductive system of h if and only if for any t ∈ [0, 1] and q ∈ q, l−(µ,t,q) is either empty or a deductive system of h. (3) µ is an anti-q-fuzzy deductive system of h if and only if for any t ∈ [0, 1] and q ∈ q, u(µ,t,q) is either empty or a deductive system of h. (4) µ is an anti-q-fuzzy deductive system of h if and only if for any t ∈ [0, 1] and q ∈ q, u+(µ,t,q) is either empty or a deductive system of h. proof. (1). assume that µ is an anti-q-fuzzy deductive system of h. then µ is an anti-q-fuzzy deductive system of h for all q ∈ q. let q ∈ q and t ∈ [0, 1] be such that l(µ,t,q) 6= ∅ and let x ∈ h be such that x ∈ l(µ,t,q). then µ(x,q) ≤ t. thus µ(1,q) = µ(x ∗ x,q) ≤ µ(x,q). hence µ(1,q) ≤ µ(x,q) ≤ t, so 1 ∈ l(µ,t,q). let x,y ∈ h and q ∈ q be such that x ∈ l(µ,t,q) and x ∗ y ∈ l(µ,t,q). then µ(x,q) ≤ t and µ(x ∗ y,q) ≤ t. then we have µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)}. then µ(y,q) ≤ max{µ(x∗y,q),µ(x,q)}≤ t, so y ∈ l(µ,t,q). hence l(µ,t,q) is a deductive system of h. conversely, assume that every nonempty set l(µ,t,q) is a deductive system in h. if µ(1,q) ≤ µ(x,q) is not true for all x ∈ h and q ∈ q. then there exist x0 ∈ h and q ∈ q such that µ(1,q) > µ(x0,q). let t = 1 2 (µ(1,q) + µ(x0,q)). then t ∈ [0, 1] and by lemma 2.8, we have µ(1,q) > t > µ(x0,q). thus x0 ∈ l(µ,t,q), so, l(µ,t,q) 6= ∅. by assumption, we have l(µ,t,q) is a deductive system of h. it follows that 1 ∈ l(µ,t,q), so µ(1,q) ≤ t, which is a contradicition. hence µ(1,q) ≤ µ(x,q) for all x ∈ h and q ∈ q. suppose that µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)} is not true for all x,y ∈ h and q ∈ q. then there exist u0,v0 ∈ h and q ∈ q such that µ(v0,q) > max{µ(u0 ∗v0,q),µ(u0,q)}. let p = 12 (µ(v0,q) + max{µ(u0 ∗v0,q),µ(u0,q)}). then p ∈ [0, 1] and we have µ(v0,q) > p > max{µ(u0 ∗ v0,q),µ(u0,q)}. then u0,v0 ∈ l(µ,p,q), so, l(µ,p,q) 6= ∅. by assumption, we have l(µ,p,q) is a deductive system of h. it follows that v0 ∈ l(µ,p,q). hence µ(v0,q) ≤ p, which is a contradiction. hence µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)} is true for all x,y ∈ h and q ∈ q. hence µ is an anti-q-fuzzy deductive system of h for all q ∈ q. consequently int. j. anal. appl. (2023), 21:42 7 µ is an anti-q-fuzzy deductive system of h. (2). assume that µ is an anti-q-fuzzy deductive system of h. then µ is an anti-q-fuzzy deductive system of h for all q ∈ q. let q ∈ q and t ∈ [0, 1] be such that l−(µ,t,q) 6= ∅ and let x ∈ h be such that x ∈ l−(µ,t,q). then µ(x,q) < t. thus µ(1,q) = µ(x ∗ x,q) ≤ µ(x,q). hence µ(1,q) ≤ µ(x,q) < t, so 1 ∈ l−(µ,t,q). let x,y ∈ h and q ∈ q be such that x ∈ l−(µ,t,q) and x ∗ y ∈ l−(µ,t,q). then µ(x,q) < t and µ(x ∗ y,q) < t. then we have µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)}. then µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)} < t, so y ∈ l−(µ,t,q). hence l−(µ,t,q) is a deductive system of h. conversely, assume that every nonempty set l−(µ,t,q) is a deductive system in h. if µ(1,q) ≤ µ(x,q) is not true for all x ∈ h and q ∈ q. then there exist x0 ∈ h and q ∈ q such that µ(1,q) > µ(x0,q). let t′ = 12 (µ(1,q) + µ(x0,q)). then t ′ ∈ [0, 1] and we have µ(1,q) > t′ > µ(x0,q). thus x0 ∈ l−(µ,t′,q), so, l−(µ,t′,q) 6= ∅. by assumption, we have l−(µ,t′,q) is an ideal of h. it follows that 1 ∈ l−(µ,t′,q), so µ(1,q) < t′, which is a contradicition. hence µ(1,q) ≤ µ(x,q) for all x ∈ h and q ∈ q. suppose that µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)} is not true for all x,y ∈ h and q ∈ q. then there exist u0,v0 ∈ h and q ∈ q such that µ(v0,q) > max{µ(u0∗v0,q),µ(u0,q)}. let p′ = 12 (µ(v0,q) + max{µ(u0∗v0,q),µ(u0,q)}). then p ′ ∈ [0, 1] and we have µ(v0,q) > p′ > max{µ(u0∗v0,q),µ(u0,q)}. then u0,v0 ∈ l−(µ,p′,q), so, l−(µ,p′,q) 6= ∅. by assumption, we have l−(µ,p′,q) is a deductive system of h. it follows that v0 ∈ l−(µ,p′,q). hence µ(v0,q) < p′, which is a contradiction. hence µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)} is true for all x,y ∈ h and q ∈ q. hence µ is an anti-q-fuzzy deductive system of h for all q ∈ q. consequently µ is an anti-q-fuzzy deductive system of h. (3). assume that µ is an anti-q-fuzzy deductive system of h. then µ is an anti-q-fuzzy deductive system of h for all q ∈ q. let q ∈ q and t ∈ [0, 1] be such that u(µ,t,q) 6= ∅ and let x ∈ u(µ,t,q). then µ(x,q) ≥ t. thus µ(1,q) = µ(x∗x,q) ≤ µ(x,q). then 1−µ(1,q) ≤ 1−µ(x,q), so µ(1,q) ≥ µ(x,q) ≥ t. hence 1 ∈ u(µ,t,q). let x,y ∈ h be such that x ∈ u(µ,t,q) and x ∗ y ∈ u(µ,t,q). then µ(x,q) ≥ t and µ(x ∗y,q) ≥ t. then we have µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)}. then µ(y,q) ≤ max{µ(x ∗y,q),µ(x,q)} 1 −µ(y,q) ≤ max{1 −µ(x ∗y,q), 1 −µ(x,q)} 1 −µ(y,q) ≤ 1 − min{µ(x ∗y,q),µ(x,q)} µ(y,q) ≥ min{µ(x ∗y,q),µ(x,q)}≥ t. thus y ∈ u(µ,t,q). hence u(µ,t,q) is a deductive system of h. conversely, assume that every nonempty set u(µ,t,q) is a deductive system in h. if µ(1,q) ≤ µ(x,q) is not true for all x ∈ h and q ∈ q. then there exist x0 ∈ h and q ∈ q such that µ(1,q) > µ(x0,q). then µ(1,q) > µ(x0,q) 1 −µ(1,q) > 1 −µ(x0,q) µ(1,q) < µ(x0,q). 8 int. j. anal. appl. (2023), 21:42 let t = 1 2 (µ(1,q) + µ(x0,q)). then t ∈ [0, 1] and we have µ(1,q) < t < µ(x0,q). thus x0 ∈ u(µ,t,q), that is u(µ,t,q) 6= ∅. by assumption, we have u(µ,t,q) is a deductive system of h. it follows that 1 ∈ u(µ,t,q), so µ(1,q) ≥ t, which is a contradiction. hence µ(1,q) ≤ µ(x,q) for all x ∈ h and q ∈ q. suppose that µ(y,q) ≤ max{µ(x∗y,q),µ(x,q)} is not true for all x,y ∈ h. then there exist u0,v0 ∈ h and q ∈ q such that µ(v0,q) > max{µ(u0 ∗v0,q),µ(u0,q)}. then µ(v0,q) > max{µ(u0 ∗v0,q),µ(u0,q)} 1 −µ(v0,q) > max{1 −µ(u0 ∗v0,q), 1 −µ(u0,q)} 1 −µ(v0,q) > 1 − min{µ(u0 ∗v0,q),µ(u0,q)} µ(v0,q) < min{µ(u0 ∗v0,q),µ(u0,q)}. taking p = 1 2 (µ(v0,q) + min{µ(u0 ∗ v0,q),µ(u0,q)}). then p ∈ [0, 1] and we have µ(v0,q) < p < min{µ(u0 ∗ v0,q),µ(u0,q)}. thus µ(u0 ∗ v0,q) > p and µ(u0,q) > p, so u0 ∗ v0,u0,∈ u(µ,p,q), so u(µ,p,q) 6= ∅. by assumption, we have u(µ,p,q) is a deductive system of h. it follows that v0 ∈ u(µ,p,q), so µ(v0,q) ≥ p, a contradiction. hence µ(y,q) ≤ max{µ(x∗y,q),µ(x,q)} is true for all x,y ∈ h and q ∈ q. hence µ is an anti-q-fuzzy deductive system of h for all q ∈ q. consequently µ is an anti-q-fuzzy deductive system of h. (4). assume that µ is an anti-q-fuzzy deductive system of h. then µ is an anti-q-fuzzy deductive system of h for all q ∈ q. let q ∈ q and t ∈ [0, 1] be such that u+(µ,t,q) 6= ∅ and let x ∈ u+(µ,t,q). then µ(x,q) > t. thus µ(1,q) = µ(x∗x,q) ≤ µ(x,q). then 1−µ(1,q) ≤ 1−µ(x,q), so µ(1,q) ≥ µ(x,q) > t. hence 1 ∈ u+(µ,t,q). let x,y ∈ h be such that x ∈ u+(µ,t,q) and x∗y ∈ u+(µ,t,q). then µ(x∗y,q) > t and µ(x,q) > t. then we have µ(y,q) ≤ max{µ(x∗y,q),µ(x,q)}. then µ(y,q) ≤ min{µ(x ∗y,q),µ(x,q)} 1 −µ(y,q) ≤ max{1 −µ(x ∗y,q), 1 −µ(x,q)} 1 −µ(y,q) ≤ 1 − min{µ(x ∗y,q),µ(x,q)} µ(y,q) ≥ min{µ(x ∗y,q),µ(x,q)} > t. thus y ∈ u+(µ,t,q). hence u+(µ,t,q) is a deductive system of h. conversely, assume that every nonempty set u+(µ,t,q) is a deductive system in h. if µ(1,q) ≤ µ(x,q) is not true for all x ∈ h and q ∈ q. then there exist x0 ∈ h and q ∈ q such that µ(1,q) > µ(x0,q). then µ(1,q) > µ(x0,q) 1 −µ(1,q) > 1 −µ(x0,q) µ(1,q) < µ(x0,q). let t = 1 2 (µ(1,q) + µ(x0,q)). then t ∈ [0, 1] and we have µ(1,q) < t < µ(x0,q). thus x0 ∈ u+(µ,t,q), that is u+(µ,t,q) 6= ∅. by assumption, we have u+(µ,s,q) is a deductive system of h. it follows that 1 ∈ u+(µ,t,q), so µ(1,q) > t, which is a contradiction. hence µ(1,q) ≤ µ(x,q) for all x ∈ h and q ∈ q. suppose that µ(y,q) ≤ max{µ(x ∗ y,q),µ(x,q)} is not true for all x,y ∈ h. int. j. anal. appl. (2023), 21:42 9 then there exist u0,v0 ∈ h and q ∈ q such that µ(v0,q) > max{µ(u0 ∗v0,q),µ(u0,q)}. then µ(v0,q) > max{µ(u0 ∗v0,q),µ(u0,q)} 1 −µ(v0,q) > max{1 −µ(u0 ∗v0,q), 1 −µ(u0,q)} 1 −µ(v0,q) > 1 − min{µ(u0 ∗v0,q),µ(u0,q)} µ(v0,q) < min{µ(u0 ∗v0,q),µ(u0,q)}. taking p = 1 2 (µ(v0,q) + min{µ(u0 ∗ v0,q),µ(u0,q)}). then p ∈ [0, 1] and we have µ(v0,q) < p < min{µ(u0 ∗ v0,q),µ(u0,q)}. thus µ(u0 ∗ v0,q) > p and µ(u0,q) > p, so u0 ∗ v0,u0 ∈ u+(µ,p,q), so u+(µ,p,q) 6= ∅. by assumption, we have u+(µ,p,q) is a deductive system of h. it follows that v0 ∈ u+(µ,p,q), so µ(v0,q) ≥ p, a contradiction. hence µ(y,q) ≤ max{µ(x ∗y,q),µ(y,q)} is true for all x,y ∈ h and q ∈ q. hence µ is an anti-q-fuzzy deductive system of h for all q ∈ q. then µ is an anti-q-fuzzy deductive system of h. � corollary 3.1. let µ be a q-fuzzy set in h. then the following statements hold: (1) µ is an anti-q-fuzzy deductive system of h, then for any t ∈ [0, 1], l(µ,t) is either empty or a deductive system of h. (2) µ is an anti-q-fuzzy deductive system of h, then for any t ∈ [0, 1], l−(µ,t) is either empty or a deductive system of h. (3) µ is an anti-q-fuzzy deductive system of h, then for any t ∈ [0, 1], u(µ,t) is either empty or a deductive system of h. (4) µ is an anti-q-fuzzy deductive system of h, for any t ∈ [0, 1], u+(µ,t) is either empty or a deductive system of h. proof. (1). assume that µ is an anti-q-fuzzy deductive system of h. then we have for any t ∈ [0, 1] and q ∈ q. let l(µ,t,q) is either empty or a deductive system of h. let t ∈ [0, 1]. if l(µ,t,q) = ∅ for some q ∈ q, it follows that l(µ,t) = ⋂ q∈q l(µ,t,q). if l(µ,t,q) 6= ∅ for all q ∈ q, it follows from that l(µ,t,q) is a deductive system of h for all q ∈ q. then we have l(µ,t) = ⋂ q∈q l(µ,t,q) is an ideal of h. (2). similarly to as in the proof of (1). (3). assume that µ is an anti-q-fuzzy deductive system of h. then we have for any t ∈ [0, 1] and q ∈ q. let u(µ,t,q) is either empty or a deductive system of h. let t ∈ [0, 1]. if u(µ,t,q) = ∅ for some q ∈ q, it follows that u(µ,t) = ⋂ q∈q u(µ,t,q). if u(µ,t,q) 6= ∅ for all q ∈ q, it follows that u(µ,t,q) is a deductive system of h for all q ∈ q. then we have u(µ,t) = ⋂ q∈q u(µ,t,q) is a deductive system of h. (4). similarly to as in the proof of (3). � corollary 3.2. let d be a deductive system of h. then the following statements hold: 10 int. j. anal. appl. (2023), 21:42 (1) for any k ∈ (0, 1], there exists an anti-q-fuzzy deductive system µ of h such that l(µ,t) = d for all t < k and l(µ,t) = h for all t ≥ k, (2) for any k ∈ (0, 1], there exists an anti-q-fuzzy deductive system γ of h such that u(γ,t) = d for all t > k and u(γ,t) = h for all t ≤ k. proof. (1). let µ be a q-fuzzy set in h defined by, for all q ∈ q µ(x,q) = { 0 if x ∈ d k if x /∈ d. case 1 : to show that l(µ,t) = d for all t < k, let t ∈ [0, 1] be such that t < k. let x ∈ l(µ,t). then µ(x,q) ≤ t < k for all q ∈ q. thus µ(x,q) 6= k for all q ∈ q, so µ(x,q) = 0 for all q ∈ q. then x ∈ d, so l(µ,t) ⊆ d. now, let x ∈ d. then µ(x,q) = 0 ≤ t for all q ∈ q. thus x ∈ l(µ,t), so d ⊆ l(µ,t). hence l(µ,t) = d for all t < k. case 2 : to show that l(µ,t) = h for all t ≥ k, let t ∈ [0, 1] be such that t ≥ k. clearly, l(µ,t) ⊆ h. let x ∈ h. then for all q ∈ q, we define µ(x,q) = { 0 < t if x ∈ d k ≤ t if x /∈ d. then x ∈ l(µ,t), so h ⊆ l(µ,t). hence l(µ,t) = h for all t ≥ k. we claim that l(µ,t,q) = l(µ,t,q′) for all q,q′ ∈ q. for q,q′ ∈ q, we obtain x ∈ l(µ,t,q) ⇔ µ(x,q) ≤ t ⇔ µ(x,q′) ≤ t ⇔ x ∈ l(µ,t,q′). hence l(µ,t,q) = l(µ,t,q′) for all q,q′ ∈ q. then we have l(µ,t) = ⋂ q∈q l(µ,t,q). by the claim, we have l(µ,t) = l(µ,t,q) for all q ∈ q. since l(µ,t,q) = l(µ,t) = d for all t < k and l(µ,t) = l(µ,t) = h for all t ≥ k, it follows that µ is an anti-q-fuzzy deductive system of h. (2). let µ be a q-fuzzy set in h defined by, for all q ∈ q µ(x,q) = { 1 if x ∈ d k if x /∈ d. case 1 : to show that u(µ,t) = d for all t > k, let t ∈ [0, 1] be such that t > k. let x ∈ u(µ,t). then µ(x,q) ≥ t > k for all q ∈ q. then µ(x,q) 6= k for all q ∈ q, so µ(x,q) = 1 for all q ∈ q. thus x ∈ i, so u(µ,t) ⊆ d. now, let x ∈ d. then µ(x,q) = 1 ≥ t for all q ∈ q. then x ∈ u(µ,t), so d ⊆ u(µ,t). hence u(µ,t) = d for all t > k. case 2 : to show that u(µ,t) = h for all t ≤ k, let t ∈ [0, 1] be such that t ≤ k. clearly, u(µ,t) ⊆ h. let x ∈ h. then for all q ∈ q, µ(x,q) = { 1 > t if x ∈ d k ≥ t if x /∈ d. then x ∈ u(µ,t), so h ⊆ u(µ,t). hence u(µ,t) = h for all t ≤ k. we claim that u(µ,t,q) = u(µ,t,q′) for all q,q′ ∈ q. for q,q′ ∈ q, we obtain x ∈ u(µ,t,q) ⇔ µ(x,q) ≥ t ⇔ µ(x,q′) ≥ t ⇔ x ∈ u(µ,t,q′). hence u(µ,t,q) = u(µ,t,q′) for all q,q′ ∈ q. then u(µ,t) = ⋂ q∈q u(µ,t,q). by the claim, we have u(µ,t) = u(µ,t,q) for all q ∈ q. since u(µ,t,q) = u(µ,t) = d for all t > k and u(µ,t) = u(µ,t) = h for all t ≤ k, it follows that µ is an anti-q-fuzzy ideal of h. then l(µ,t) = l(µ,t) = d for all t > k and l(µ,t) = l(µ,t) = h for all t ≤ k. let µ = θ. then θ is an anti-q-fuzzy ideal of h such that l(µ,t) = l(µ,t) = d for all t > k and l(µ,t) = l(µ,t) = h for all t ≤ k. � let (a, ·, 1a) and (b,?, 1b) be hilbert algebras a mapping f : a → b is called a homomorphism if f (x∗y) = f (x)?f (y) for all x,y ∈ a. note that if f : x → y is a homomorphism of hilbert algebras, then f (1a) = 1b. let f : x → y be a homomorphism of hilbert algebras. int. j. anal. appl. (2023), 21:42 11 theorem 3.5. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras and let f : a → b be a homomorphism. if µ is an anti-q-fuzzy deductive system of b, then µf is also a q-fuzzy deductive system of a. proof. assume that µ be an anti-q-fuzzy deductive system of b. let x ∈ a. then µf (1a,q) = µ(f (1a),q) = µ(1b,q) ≤ µ(f (x),q) = µf (x,q). let x,y ∈ a. then µf (y,q) = µ(f (y),q) ≤ max{µ(f (x∗y),q),µ(f (x),q)} = max{µf (x∗y,q),µf (x,q)}. hence µf is an anti-q-fuzzy deductive system of a. � corollary 3.3. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras and let f : a → b be a homomorphism. if µ is an anti-q-fuzzy deductive system of b, then µf is also an anti-q-fuzzy deductive system of a. theorem 3.6. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras and let f : a → b be a isomorphism. if µf is an anti-q-fuzzy deductive system of a, then µ is also an anti-q-fuzzy deductive system of b. proof. assume that µf be an anti-q-fuzzy deductive system of a. let y ∈ b. then there exists x ∈ a such that f (x) = y, we have µ(1b,q) = µ(y ? 1b,q) = µ(f (x) ?f (1a),q) = µ(f (x ∗1a),q) = µf (x ∗ 1a,q) = µf (1a,q) ≤ µf (x,q) ≤ µ(f (x),q) = µ(y,q). let x,y ∈ b. then there exist a,b ∈ x such that f (a) = x and f (b) = y. it follows that µ(y,q) = µ(f (b),q) = µf (b,q) ≤ max{µf (a∗b,q),µf (a,q)} = max{µ(f (a∗b),q),µ(f (a),q)} = max{µ(f (a)?f (b),q),µ(f (a),q)} = max{µ(x ? y,q),µ(x,q)}. hence µ is an anti-q-fuzzy deductive system of b. � corollary 3.4. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras and let f : a → b be a isomorphism. if µf is an anti-q-fuzzy deductive system of a, then µ is also an anti-q-fuzzy deductive system of b. lemma 3.6. for any a,b,c,d ∈ r, the following properties hold: (1) max{max{a,b}, max{c,d}} = max{max{a,c}, max{b,d}} (2) min{min{a,b}, min{c,d}} = min{min{a,c}, min{b,d}}. remark 3.1. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras. then a×b is a hilbert algebra defined by (x,y) � (u,v) = (x ∗u,y ? v) for every x,y ∈ a and u,v ∈ b, then clearly (a×b,�, (1a, 1b)) is a hilbert algebra. theorem 3.7. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras. if µ is an anti-q-fuzzy deductive system of a and δ is an anti-q-fuzzy deductive system of b, then µ× δ is an anti-q-fuzzy deductive system of a×b. proof. assume that µ is an anti-q-fuzzy deductive system of a and δ is an anti-q-fuzzy deductive system b. let (x,y) ∈ a × b. then (µ × δ)((1a, 1b),q) = max{µ(1a,q),δ(1b,q)} ≤ 12 int. j. anal. appl. (2023), 21:42 max{µ(x,q),δ(y,q)}} = (µ×δ)((x,y),q). let (x1,y1), (x2,y2) ∈ a×b. then (µ×δ)((x2,y2),q) = max{µ(x2,q),δ(y2,q)} ≤ max{max{µ(x1 ∗x2,q),µ(x1,q)}, max{δ(y1 ? y2,q),δ(y1,q)}} = max{max{µ(x1 ∗x2,q),δ(y1 ? y2,q)}, max{µ(x1,q),δ(y1,q)}} = max{(µ×δ)((x1 ∗x2,y1 ? y2),q), (µ ·δ)((x1,y1),q)} = max{(µ×δ)((x1 ∗y1) � (x2 ? y2),q), (µ ·δ)((x1,y1),q)}. hence µ×δ is an anti-q-fuzzy deductive system of a×b. � corollary 3.5. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras. if µ is an anti-q-fuzzy deductive system of a and δ is an anti-q-fuzzy deductive system of b, then µ×δ is an anti-q-fuzzy deductive system of a×b. theorem 3.8. if µ is a q-fuzzy set of a and δ is a q-fuzzy set of b such that µ×δ is an anti-q-fuzzy deductive system of a×b, then the following statements hold: (1) either µ(1a,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b, (2) if µ(1a,q) ≤ µ(x,q) for all x ∈ a, then either δ(1b,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b, (3) if δ(1a,q) ≤ δ(x,q) for all x ∈ b, then either µ(1a,q) ≤ µ(x,q) for all x ∈ a or µ(1a,q) ≤ δ(x,q) for all x ∈ b. proof. (1). suppose that there exist x ∈ a and y ∈ b such that µ(1a,q) > µ(x,q) and δ(1b,q) > δ(y,q). then (µ× δ)((x,y),q) = max{µ(x,q),δ(y,q)} < max{µ(1a,q),δ(1b,q)}} = (µ × δ)((1a, 1b),q), which is a contradiction. hence µ(1a,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b. (2). assume that µ(1a,q) ≤ µ(x,q) for all x ∈ a. suppose that there exist x ∈ a and y ∈ b such that µ(1a,q) > µ(x,q) and δ(1b,q) > δ(y,q). then µ(1a,q) ≤ µ(x,q) < δ(1b,q). thus (µ×δ)((x,y),q) = max{µ(x,q),δ(y,q)} < max{µ(1a,q),δ(1b,q)}} = δ(1b,q) = max{µ(1a,q),δ(1b,q)} = (µ×δ)((1a, 1b),q), which is a contradiction. hence δ(1a,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b. (3). assume that δ(1a,q) ≤ δ(x,q) for all x ∈ b. suppose that there exist x ∈ a and y ∈ b such int. j. anal. appl. (2023), 21:42 13 that µ(1a,q) > µ(x,q) and µ(1a,q) > δ(y,q). then δ(1b,q) ≤ δ(x,q) < µ(1a,q). thus (µ×δ)((x,y),q) = max{µ(x,q),δ(y,q)} > max{µ(1a,q),µ(1a,q)}} = µ(1a,q) = max{µ(1a,q),δ(1b,q)} = (µ×δ)((1a, 1b),q), which is a contradiction. hence µ(1a,q) ≤ µ(x,q) for all x ∈ a or µ(1b,q) ≤ δ(x,q) for all x ∈ b. � corollary 3.6. if µ is a q-fuzzy set of a and δ is a q-fuzzy set of b such that µ×δ is an anti-q-fuzzy deductive system of a×b, then the following statements hold: (1) for all q ∈ q, either µ(1a,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b, (2) for all q ∈ q, if µ(1a,q) ≤ µ(x,q) for all x ∈ a, then either δ(1b,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b, (3) for all q ∈ q, if δ(1a,q) ≤ δ(x,q) for all x ∈ b, then either µ(1a,q) ≤ µ(x,q) for all x ∈ a or µ(1a,q) ≤ δ(x,q) for all x ∈ b. theorem 3.9. let (a, ·, 1a) and (b,?, 1b) be hilbert algebras and let µ be a q-fuzzy set in a and δ be a q-fuzzy set in b. if µ× δ is an anti-q-fuzzy deductive system of a×b, then either µ is an anti-q-fuzzy deductive system of a or δ is an anti-q-fuzzy deductive system of b. proof. assume that µ× δ is an anti-q-fuzzy deductive system of a×b. suppose that µ is not an anti-q-fuzzy deductive system of a and δ is not an anti-q-fuzzy deductive system of b. then we have µ(1a,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b. suppose that µ(1a,q) ≤ µ(x,q) for all x ∈ a. then either δ(1b,q) ≤ µ(x,q) for all x ∈ a or δ(1b,q) ≤ δ(x,q) for all x ∈ b. if δ(1b,q) ≤ µ(x,q) for all x ∈ a, then (µ× δ)((x, 1b),q) = max{µ(x,q),δ(1b,q)} = µ(x,q). we consider, for all x,y ∈ a, µ(y,q) = max{µ(y,q),δ(1b,q)} = (µ×δ)((y, 1b),q) ≤ max{(µ×δ)((x, 1b) � (y, 1b),q), (µ×δ)((x, 1b),q)} = max{(µ×δ)((x ∗y, 1b ? 1b),q), (µ×δ)((x, 1b),q)} = max{(µ×δ)((x ∗y, 1b),q), (µ×δ)((x, 1b),q)} = max{max{µ(x ∗y,q),δ(1b,q)}, max{µ(x,q),δ(1b,q)}} = max{µ(x ∗y,q),µ(x,q)}. hence µ is an anti-q-fuzzy deductive system of a, which is a contradiction. suppose that δ(1b,q) ≤ δ(x,q) for all x ∈ b. then either µ(1a,q) ≤ µ(x,q) for all x ∈ b or µ(1a,q) ≤ δ(x,q) for all x ∈ b. 14 int. j. anal. appl. (2023), 21:42 if µ(1a,q) ≤ δ(x,q) for all x ∈ b, then (µ · δ)((1a,x),q) = max{µ(1a,q),δ(x,q)} = δ(x,q). we consider, for all x,y ∈ b, δ(y,q) = max{µ(1a,q),δ(y,q)} = (µ×δ)((1a,y),q) ≤ max{(µ×δ)((1a,x) � (1a,y),q), (µ×δ)((1a,x),q)} = max{(µ×δ)((1a ∗ 1a,x ∗y),q), (µ×δ)((1a,x),q)} = max{(µ×δ)((1a,x ∗y),q), (µ×δ)((1a,x),q)} = max{min{µ(1a,q),δ(x ∗y,q)}, max{µ(1a,q),δ(x,q)}} = max{δ(x ∗y,q),δ(x,q)}. hence δ is an anti-q-fuzzy deductive system of b, which is a contradiction. since µ is not an anti-q-fuzzy deductive system of a and δ is not an anti-q-fuzzy deductive system of b, we have µ(1a,q) ≤ µ(x,q) for all x ∈ a and δ(1b,q) ≤ δ(x,q) for all x ∈ b. let x1,x2 ∈ a and y1,y2 ∈ b such that µ(x2,q) > max{µ(x1 ∗ x2,q),µ(x1,q)} and δ(y2,q) > max{δ(y1 ? y2,q),δ(y1,q)}, so max{µ(x2,q),δ(y2,q)} > max{max{µ(x1 ∗x2,q),µ(x1,q)}, max{δ(y1 ? y2,q),δ(y1,q)}}. thus max{µ(x2,q),δ(y2,q)} = (µ ·δ)((x2,y2),q) ≤ max{(µ ·δ)((x1,y1) � (x2,y2),q), (µ ·δ)((x1,y1),q)} = max{(µ ·δ)((x1 ∗x2,y1 ? y2),q), (µ ·δ)((x1,y1),q)} = max{max{µ(x1 ∗x2,q),δ(y1 ? y2,q)}, max{µ(x1,q),δ(y1,q)}} = max{max{µ(x1 ∗x2,q),µ(x1,q)}, max{δ(y1 ? y2,q),δ(y1,q)}}. it follows that max{µ(x2,q),δ(y2,q)} ≯ max{max{µ(x1,q),µ(x2,q)}, max{δ(y1,q),δ(y2,q)}}, which is a contradiction. hence µ is an anti-q-fuzzy deductive system of a or δ is an anti-q-fuzzy deductive system of b. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] b. ahmad, a. kharal, on fuzzy soft sets, adv. fuzzy syst. 2009 (2009), 586507. https://doi.org/10.1155/ 2009/586507. [2] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [3] a.k. adak, d.d. salokolaei, some properties of pythagorean fuzzy ideal of near-rings, int. j. appl. oper. res. 9 (2019), 1-9. [4] m. atef, m.i. ali, t.m. al-shami, fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications, comput. appl. math. 40 (2021), 115. https://doi.org/10.1007/s40314-021-01501-x. [5] d. busneag, a note on deductive systems of a hilbert algebra, kobe. j. math. 2 (1985), 29-35. https://cir. nii.ac.jp/crid/1570854175360486400. https://doi.org/10.1155/2009/586507 https://doi.org/10.1155/2009/586507 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1007/s40314-021-01501-x https://cir.nii.ac.jp/crid/1570854175360486400 https://cir.nii.ac.jp/crid/1570854175360486400 int. j. anal. appl. (2023), 21:42 15 [6] d. busneag, hilbert algebras of fractions and maximal hilbert algebras of quotients, kobe. j. math. 5 (1988), 161-172. https://cir.nii.ac.jp/crid/1570572702603831808. [7] n. caǧman, s. enginoǧlu, and f. citak, fuzzy soft set theory and its application, iran. j. fuzzy syst. 8 (2011), 137-147. [8] i. chajda, r.halas, congruences and ideals in hilbert algebras, kyungpook math. j. 39 (1999), 429-429. [9] a. diego, sur les algébres de hilbert, collect. log. math. ser. a (ed. hermann, paris). 21 (1966), 1-52. [10] w.a. dudek, y.b. jun, on fuzzy ideals in hilbert algebra, novi sad j. math. 29 (1999), 193-207. [11] w. a. dudek, on fuzzification in hilbert algebras, contrib. gen. algebra, 11 (1999), 77-83. [12] h. garg, s. singh, a novel triangular interval type-2 intuitionistic fuzzy sets and their aggregation operators, iran. j. fuzzy syst. 15 (2018), 69-93. https://doi.org/10.22111/ijfs.2018.4159. [13] h. garg, k. kumar, an advanced study on the similarity measures of intuitionistic fuzzy sets based on the set pair analysis theory and their application in decision making, soft comput. 22 (2018), 4959-4970. https://doi.org/ 10.1007/s00500-018-3202-1. [14] h. garg, k. kumar, distance measures for connection number sets based on set pair analysis and its applications to decision-making process, appl. intell. 48 (2018), 3346-3359. https://doi.org/10.1007/s10489-018-1152-z. [15] y.b. jun, deductive systems of hilbert algebras, math. japon. 43 (1996), 51-54. https://cir.nii.ac.jp/crid/ 1571417124616097792. [16] k.h. kim, on intuitionistic q-fuzzy ideals of semigroups, sci. math. japon. e–2006 (2006), 119-126. [17] p.m. sithar selvam, t. priya, k.t. nagalakshmi, t. ramachandran, a note on anti q-fuzzy ku-subalgebras and homomorphism of ku-algebras, bull. math. stat. res. 1 (2013), 42-49. [18] k. tanamoon, s. sripaeng, a. iampan, q-fuzzy sets in up-algebras, songklanakarin j. sci. technol. 40 (2018), 9-29. [19] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65)90241-x. [20] j. zhan, z. tan, intuitionistic fuzzy deductive systems in hilbert algebra, southeast asian bull. math. 29 (2005), 813-826. https://cir.nii.ac.jp/crid/1570572702603831808 https://doi.org/10.22111/ijfs.2018.4159 https://doi.org/10.1007/s00500-018-3202-1 https://doi.org/10.1007/s00500-018-3202-1 https://doi.org/10.1007/s10489-018-1152-z https://cir.nii.ac.jp/crid/1571417124616097792 https://cir.nii.ac.jp/crid/1571417124616097792 https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. preliminaries 3. anti-q-fuzzy deductive systems references international journal of analysis and applications issn 2291-8639 volume 3, number 1 (2013), 35-46 http://www.etamaths.com two-point fuzzy ostrowski type inequalities muhammad amer latif1 and sabir hussain2,∗ abstract. two-point fuzzy ostrowski type inequalities are proved for fuzzy hölder and fuzzy differentiable functions. the two-point fuzzy ostrowski type inequality for m-lipshitzian mappings is also obtained. it is proved that only the two-point fuzzy ostrowski type inequality for m-lipshitzian mappings is sharp and as a consequence generalize the two-point fuzzy ostrowski type inequalities obtained for fuzzy differentiable functions. 1. introduction in 1938, a. m. ostrowski proved an interesting integral inequality, estimating the absolute value of deviation of a differentiable function by its integral mean as: theorem 1. let f : [a,b] → r be a continuous function on [a,b] and differentiable on (a,b). if f′ is bounded on (a,b), that is ‖f′‖ := sup t∈(a,b) |f(t)| < ∞, then (1.1) ∣∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ it is easy to observe that (1.1) can be rewritten in equivalent from as follow: (1.2) ∣∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ (x−a) 2 + (b−x)2 2 (b−a) ‖f′‖∞ . since that time when a. ostrowski proved this famous inequality, many mathematician have been working on it and have been applying it in numerical analysis and probability, etc. n. s. barnett and s. s. dragomir [5], proved, as a generalization of (1.1), the following result: if f : [a,b] → r is absolutely continuous on [a,b] and if [c,d] ⊂ [a,b], then∣∣∣∣∣ 1b−a ∫ b a f(t)dt− 1 d− c ∫ d c f(s)ds ∣∣∣∣∣ ≤ { b−a 4 + d− c 2 + 1 b−a [∣∣∣∣c + d2 − a + b2 ∣∣∣∣− d− c2 ]2} ‖f′‖∞ .(1.3) 2000 mathematics subject classification. 26d07, 26d15, 26e50. key words and phrases. ostrowski inequality, fuzzy inequalities, fuzzy real analysis. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 35 36 latif and hussain it is to be noted that for c = d = x, one can assume 1 d−c ∫d c f(s)ds = f(x), as a limit case, and hence (1.3) takes the from of (1.1). in [9], m. matić and j. pečarić gave a two-point ostrowski type inequality, as a generalization of (1.3), by replacing the condition of differentiability of f and boundedness of f′ on (a,b) by a weaker condition that f is m-lipschitzian on [a,b], that is |f(t) −f(s)| ≤ m |t−s| , ∀t,s ∈ [a,b] , m > 0. it was also proved that the two-point ostrowski inequality established in [9] is sharp and gives tighter estimate than those of (1.3). the main result from [9] is the following one: theorem 2. let a,b,c,d ∈ r be such that a ≤ c < d ≤ b, c−a + b−d > 0. (i) if f : [a,b] → r is m-lipschitzian on [a,b], with some constant m > 0, then (1.4) ∣∣∣∣∣ 1b−a ∫ b a f(t)dt− 1 d− c ∫ d c f(s)ds ∣∣∣∣∣ ≤ (c−a) 2 + (b−d)2 2 (c−a + b−d) m. (ii) if f0 : [a,b] → r defined as f0 (t) = |t−s0| , where s0 = bc−ad c−a + b−d , then f0 is 1-lipshitzian on [a,b] and we havd∣∣∣∣∣ 1b−a ∫ b a f(t)dt− 1 d− c ∫ d c f(s)ds ∣∣∣∣∣ = (c−a) 2 + (b−d)2 2 (c−a + b−d) . since fuzziness is a natural reality different than randomness and determinism, therefore an attempt has been made by george a. anastassiou [2] to extend (1.1) to fuzzy setting context in 2003. in fact, george a. anastassiou [2] proved the following important results for fuzzy hölder and fuzzy differentiable functions respectively: theorem 3. let f ∈ c ([a,b],rf), the space of fuzzy continous functions, x ∈ [a,b] be fixed. if f fulfills the hölder condition d (f(y),f(z)) ≤ lf |y −z| α , 0 < α ≤ 1, for all y, z ∈ [a,b] , for some lf > 0. then d ( 1 b−a � (fr) ∫ b a f(y)dy,f(x) ) ≤ lf ( (x−a)α+1 + (b−x)α+1 (α + 1) (b−a) ) .(1.5) fuzzy ostrowski type inequalities 37 theorem 4. let f ∈ c1 ([a,b],rf), the space of one time continuously differentiable functions in the fuzzy sense. then for x ∈ [a,b], d ( 1 b−a � (fr) ∫ b a f(y)dy,f(x) ) ≤ ( sup t∈[a,b] d (f′(t), õ) )( (x−a)2 + (b−x)2 2 (b−a) ) .(1.6) the inequalities in (1.5) and (1.6) are sharp as equalities are attained by the choice of simple fuzzy real number valued functions. for further details on these inequalities we refer the interested readers to [2]. the main purpose of the present paper is to establish two-point fuzzy ostrowski type inequalties for fuzzy hölder, fuzzy differentiable and fuzzy m-lipshitzian functions in section 2, which actually generalize the inequalities (1.5) and (1.6). 2. preliminaries in this section we point out some basic definitions and results which would help us in the sequel of the paper, we begin with: definition 1. [11] let us denote by rf the class of fuzzy subsets of real axis r ( i.e. u : r −→ [0, 1]), satisfying the following properties: (1) ∀u ∈ rf, u is normal i.e.with u(x) = 1. (2) ∀u ∈ rf, u is convex fuzzy set i.e. u(tx + (1 − t)y) ≥ min{u(x),u(y)} ,∀t ∈ [0, 1] . (3) ∀ u ∈ rf, u is upper semi-continuous on r. (4) {x ∈ r : u(x) > 0} is compact. the set rf is called the space of fuzzy real numbers. remark 1. it is clear that r ⊂ rf, because any real number x0 ∈ r, can be described as the fuzzy number whose value is 1 for x = x0 and zero otherwise. we will collect some further definitions and notations needed in the sequel. for 0 < r ≤ 1 and u ∈ rf , we define [u] r = {x ∈ r : u(x) ≥ r} [u] 0 = {x ∈ r : u(x) > 0} now it is well known that for each r ∈ [0, 1] , [u]r, is bounded closed interval. for u,v ∈ rf and λ ∈ r, we have the sum u⊕ v and the product λ�u are defined by [u⊕v]r = [u]r + [v]r, [λ�u]r = λ [u]r , ∀r ∈ [0, 1], where [u]r + [v]r means the usual addition of two intervals as subsets of r and λ [u]rmeans the usual product between a scalar and a subset of r. now we define d : rf ×rf −→ r∪{0} by d(u,v) = sup r∈[0,1] ( max {∣∣ur− −vr−∣∣ , ∣∣ur+ + ur+∣∣}) , where [u] r = [ ur−,u r + ] , [v] r = [ vr−,v r + ] , then (d,rf) is a metric space and it possesses the following properties: (i) d(u⊕w,v ⊕w) = d(u,v), ∀u,v,w ∈ rf. (ii) d(λ� u, λ�v) = λd(u,v),∀u,v ∈ rf,∀λ ∈ r. 38 latif and hussain (iii) d(u⊕v,w ⊕e) ≤ d(u,w) + d(v,e), ∀u,v,w,e ∈ rf moreover it is well known that (rf,d) is a complete metric space. also we have the following theorem: theorem 5. [11] i if we denote õ = x{0} then õ ∈ rf is neutral element with respect to ⊕, i.e. u⊕ õ = õ⊕u, for all u ∈ rf. ii with respect to 0̃ none of u ∈ rf\r has opposite in rf with respect to ⊕. iii for any a,b ∈ r with a,b ≥ 0 or a,b ≤ 0, any u ∈ rf, we have (a+b)�u = a�u⊕ b� u.∀a,b ∈ r the above property does not hold. iv for any λ ∈ r and any u,v ∈ rf, we have λ� (u⊕v) = λ�u⊕λ�v. v for any λ,µ ∈ r and any u ∈ r, we have λ� (µ�v) = (λ.µ) �v. vi if we denote‖u‖f = d(u,õ), ∀u ∈ rf then ‖.‖f has the properties of a usual norm on rf, i.e. ‖u‖f = 0 if and only if u = õ, ‖λ�u‖f = |λ| .‖u‖f and ‖u⊕v‖f ≤‖u‖f + ‖v‖f , |‖u‖f + ‖v‖f| ≤ d(u,v). remark 2. the propositions (ii) and (iii) in theorem show us that (rf,⊕,�) is not a linear space over r and consequently (rf,‖.‖f) cannot be a normed space. however, the properties of d and those in theorem (iv)-(vi), have as an effect that most of the metric properties of a functions defined on r with values in a banach space, can be extended to functions f : r −→ rf, called fuzzy number valued functions. definition 2. a function f : r −→ rf is said to be continuous at x0 ∈ r if for every ε > 0 we can find δ > 0 such that d(f(x),f (x0)) < ε, whenever |x−x0| < δ. f is said to be continuous on r if it is continuous at every x ∈ r. lemma 1. for any a, b ∈ r, a, b ≥ 0 and u ∈ rf, we have d (a�u,b�u) ≤ |a− b|d(u,õ), where õ ∈ rf is defined by õ := x{0}. definition 3. let x, y ∈ rf. if there exists a z ∈ rf such that x = y ⊕ z, then we call z the h-difference of x and y, denoted by z = x y. definition 4. let t := [x0,x0 + β] ⊂ r, with β > 0. a function f : t −→ rf is h-differentiable at x ∈ t if there exists a f′(x) ∈ rf such that the limits (with respect to the metric d) lim h→0+ f (x + h) f(x) h , lim h→0+ f(x) f (x−h) h exist and are equal to f′(x). we call f′ the derivative or h-derivative of f at x. if f is h-differentiable at any x ∈ t , we call f differentiable or h-differentiable and it has h-derivative over t the function f′. definition 5. let f : [a,b] −→ rf. we say that f is fuzzyriemann integrable to i ∈ rf, if for every ε > 0, there exsit δ > 0 such that for any division p = {[u,v] ; ξ} of [a,b] with the norms ∆ (p) < δ, we have d (∑∗ (v −u) �f (ξ) ,i ) < ε, fuzzy ostrowski type inequalities 39 where ∑∗ denotes the fuzzy summation. we choose to write i := (fr) ∫ b a f(x)dx. we also call an f as above (fr)-integrable. corollary 1. if f ∈ c ([a,b] ,rf) then f is (fr)-integrable. lemma 2. if f,g : [a,b] ⊆ r −→ rf are fuzzy continuous (with respect to the metric d), then the function f : [a,b] −→ r+ ∪{0} defined by f(x) := d (f(x),g(x)) is continuous on [a,b], and d ( (fr) ∫ b a f(u)du, (fr) ∫ b a g(u)du ) ≤ ∫ b a d (f(x),g(x)) dx. lemma 3. let f : [a,b] ⊆ r −→ rf be fuzzy continuous. then (fr) ∫ x a f(t)dt is fuzzy continuous function in x ∈ [a,b]. proposition 1. let f(t) := tn �u, t ≥ 0, n ∈ n and u ∈ rf be fixed. the (the h-derivative) f ′(t) = ntn−1 �u. in particular when n = 1 then f ′(t) = u. proposition 2. let i be an open interval of r and let f : i −→ rf be h-fuzzy differentiable, c ∈ r. then (c�f)′ exist and (c�f (x))′ = c�f′ (x). theorem 6. let f : [a,b] −→ rf be fuzzy differentiable function on [a,b] with h-derivative f′ which is assumed to be fuzzy continuous. then d (f (d) ,f (c)) ≤ (d− c) sup t∈[c,d] d (f′(t), õ) , for any c, d ∈ [a,b] with d ≥ c. theorem 7. let i be closed interval in r. let g : i → ζ := g(i) ⊆ r be differentiable, and f : g(i) → rf be h-differentiable. assume that g is strictly increasing. then (f ◦g)′ (x) exists and (f ◦g)′ (x) = f (g(x)) �g′(x), ∀x ∈ i. 3. main results in this section we prove a two-point ostrowski type inequalities for fuzzy hölder and fuzzy differentiable functions in a similar fashion as in [2]. we first prove a two point ostrowski inequality like (1.3) but for fuzzy differeitaible functions in: 40 latif and hussain theorem 8. let f ∈ c1 ([a,b],rf), the space of one time continuously differentiable functions in the fuzzy sense. if x ∈ [c,d] ⊂ [a,b], then d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ b a f (s) ds ) ≤ ( sup t∈[a,b] d (f′(t), õ) ){ b−a 4 + d− c 2 + 1 b−a [∣∣∣∣c + d2 − a + b2 ∣∣∣∣− d− c2 ]2} . (3.1) proof. by using the properties of the metric d and (1.5), we have d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ b a f (s) ds ) = d (( 1 b−a � (fr) ∫ b a f (t) dt ) ⊕f(x), ( 1 d− c � (fr) ∫ b a f (s) ds ) ⊕f(x) ) = d (( 1 b−a � (fr) ∫ b a f (t) dt ) ⊕f(x),f(x) ⊕ ( 1 d− c � (fr) ∫ b a f (s) ds )) ≤ d ( 1 b−a � (fr) ∫ b a f (t) dt,f(x) ) + d ( 1 d− c � (fr) ∫ b a f (s) ds,f(x) ) ≤ ( sup t∈[a,b] d (f′(t), õ) )  14 + ( x− a+b 2 b−a )2  (b−a) +  14 + ( x− c+d 2 d− c )2  (d− c)   . since the rest of the proof is similar to that of (1.3), we therefore omit the detals. � theorem 9. let a,b,c,d ∈ r be such that a ≤ c < d ≤ b, c−a + b−d > 0. f ∈ c ([a,b],rf), the space of fuzzy continuous functions. suppose f fulfills the hölder condition, that is d (f(y),f(z)) ≤ lf |y −z| α , 0 < α ≤ 1, for all y, z ∈ [a,b] , for some lf > 0. then d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ d c f (s) ds ) ≤ lf ( (c−a)α+1 + (b−d)α+1 (α + 1) (c−a + b−d) ) .(3.2) proof. by using the substitution t = b−a d− c s− bc−ad d− c , s ∈ [c,d] , by (v) of theorem 5 and lemma 2, we have that 1 b−a � (fr) ∫ b a f (t) dt = 1 d− c � (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds. fuzzy ostrowski type inequalities 41 thus d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ d c f (s) ds ) = d ( 1 d− c � (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds, 1 d− c � (fr) ∫ d c f (s) ds ) = 1 d− c d ( (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds, (fr) ∫ d c f (s) ds ) ≤ 1 d− c ∫ d c d ( f ( b−a d− c s− bc−ad d− c ) ,f (s) ) ds ≤ lf d− c ∫ d c ∣∣∣∣b−ad− cs− bc−add− c −s ∣∣∣∣α ds = lf d− c ∫ d c ∣∣∣∣c−a + b−dd− c s− bc−add− c ∣∣∣∣α ds = lf (c−a + b−d) α (d− c)α+1 ∫ d c ∣∣∣∣s− bc−adc−a + b−d ∣∣∣∣α ds = lf (c−a + b−d) α (d− c)α+1 ∫ d c |s−s0| α ds, (3.3) where s0 = bc−ad c−a+b−d. we now observe that s0 − c = (d− c) (c−a) c−a + b−d ≥ 0 and d−s0 = (d− c) (b−d) c−a + b−d ≥ 0, and hence s0 ∈ [c,d]. therefore,∫ d c |s−s0| α ds = ∫ s0 c (s0 −s) α ds + ∫ d s0 (s−s0) α ds = 1 α + 1 [ (s0 − c) α+1 + (s0 −d) α+1 ] = (d− c)α+1 (α + 1) (c−a + b−d)α+1 [ (c−a)α+1 + (b−d)α+1 ] (3.4) substitution of (3.4) in (3.3) gives (3.2). this completes the proof. � remark 3. the inequalities (3.1) and (3.2) generalize the inequalitites (1.3) and (1.6) respectively but are not sharp as the equality cannot be attained by a particular choice of the fuzzy real number valued functions, since if we choose f∗(t) = |t−s0| α � u, 0 < α ≤ 1, with u ∈ rf fixed, t ∈ [a,b] and s0 = bc−adc−a+b−d . then 42 latif and hussain f∗ ∈ c ([a,b],rf), as for letting tn → t, tn ∈ [a,b] and using lemma 1, we have d (f∗ (tn) ,f ∗ (t)) = d (|tn −s0| α �u, |t−s0| α �u) ≤ ||tn −s0| α −|t−s0| α|d (u,õ) → 0, as n →∞. furthermore d (f∗ (t) ,f∗ (s)) = d (|t−s0| α �u, |s−s0| α �u) ≤ ||t−s0| α −|s−s0| α|d (u,õ) ≤ |t−s|α d (u,õ) . this shows that for lf∗ = d (u,õ), we have d (f∗ (t) ,f∗ (s)) ≤ lf∗ |t−s| α , 0 < α ≤ 1, for any t,s ∈ [a,b] . and therefore f∗ satisfies the hölder condition. finally by the properties of (fr)-integrable functions and (iii) of theorem 5, we have 1 d− c � (fr) ∫ d c f∗ (s) ds = (d− c)α (α + 1) (c−a + b−d)α+1 [ (c−a)α+1 + (b−d)α+1 ] �u.(3.5) since s0 −a = (b−a) (c−a) c−a + b−d ≥ 0 and b−s0 = (b−a) (b−d) c−a + b−d ≥ 0 implies that s0 ∈ [a,b]. by similar arguments as in obtaining (3.4), we get that 1 b−a � (fr) ∫ b a f∗ (t) dt = (b−a)α (α + 1) (c−a + b−d)α+1 [ (c−a)α+1 + (b−d)α+1 ] �u(3.6) now it is evident from (3.5), (3.6) and lemma 1 that d ( 1 b−a � (fr) ∫ b a f∗ (t) , 1 d− c � (fr) ∫ d c f∗ (s) ds ) = ( (c−a)α+1 + (b−d)α+1 (α + 1) (c−a + b−d) )( (b−a)α − (d− c)α (c−a + b−d)α ) d (u,õ) . this shows that (1.6) is not sharp. our next result is about fuzzy differentiable functions and is stated as follow: theorem 10. let a,b,c,d ∈ r be such that a ≤ c < d ≤ b, c−a + b−d > 0. fuzzy ostrowski type inequalities 43 f ∈ c1 ([a,b],rf), the space of fuzzy one time continuously differentiable functions. then d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ d c f (s) ds ) ≤ ( sup t∈[a,b] d (f′(t), õ) )( (c−a)2 + (b−d)2 2 (c−a + b−d) ) .(3.7) proof. again by using the substitution t = b−a d− c s− bc−ad d− c , s ∈ [c,d] , and by (v) of theorem 5, we have that 1 b−a � (fr) ∫ b a f (t) dt = 1 d− c � (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds. now by lemma 2 and theorem 6, we get that d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ d c f (s) ds ) = d ( 1 d− c � (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds, 1 d− c � (fr) ∫ d c f (s) ds ) = 1 d− c d ( (fr) ∫ d c f ( b−a d− c s− bc−ad d− c ) ds, (fr) ∫ d c f (s) ds ) ≤ 1 d− c ∫ d c d ( f ( b−a d− c s− bc−ad d− c ) ,f (s) ) ds ≤ sup s∈[c,d] d (f′(s), õ) d− c ∫ d c ∣∣∣∣b−ad− cs− bc−add− c −s ∣∣∣∣ds = sup s∈[c,d] d (f′(s), õ) d− c ∫ d c ∣∣∣∣c−a + b−dd− c s− bc−add− c ∣∣∣∣ds ≤ sup t∈[a,b] d (f′(t), õ) (c−a + b−d) (d− c)2 ∫ d c ∣∣∣∣s− bc−adc−a + b−d ∣∣∣∣ds = sup t∈[a,b] d (f′(t), õ) (c−a + b−d) (d− c)2 ∫ d c |s−s0|ds, (3.8) where s0 = bc−ad c−a+b−d. we now observe that s0 − c = (d− c) (c−a) c−a + b−d ≥ 0 and d−s0 = (d− c) (b−d) c−a + b−d ≥ 0, 44 latif and hussain and hence s0 ∈ [c,d]. therefore, ∫ d c |s−s0|ds = ∫ s0 c (s0 −s) ds + ∫ d s0 (s−s0) ds = 1 2 [ (s0 − c) 2 + (s0 −d) 2 ] = (d− c)2 2 (c−a + b−d)2 [ (c−a)2 + (b−d)2 ] (3.9) substituting (3.9) in (3.8), we get (3.7). � remark 4. inequality (3.7) in theorem 9 is not sharp. moreover, we note that if c = d = x we can assume 1 d−c � (fr) ∫d c f (s) ds = f(x), as a limit case, so (3.2) and (3.7) reduce to (1.5) and (1.6) respectively. this fact also reveals that although our results are not sharp but generalize the inequalities (1.5) and (1.6). our last result is about fuzzy m-lipshitzian mappings and is stated as follow: theorem 11. let a,b,c,d ∈ r be such that a ≤ c < d ≤ b, c−a + b−d > 0. f ∈ c ([a,b],rf), the space of fuzzy continuous functions. suppose f is mlipshitzian, that is d (f(y),f(z)) ≤ m |y −z| , for all y, z ∈ [a,b] , for some m > 0. then d ( 1 b−a � (fr) ∫ b a f (t) dt, 1 d− c � (fr) ∫ d c f (s) ds ) ≤ m ( (c−a)2 + (b−d)2 2 (c−a + b−d) ) .(3.10) inequality (3.10) is sharp, in fact attained by f∗(t) = |t−s0| � u, u ∈ rf being fixed and s0 = bc−ad c−a+b−d . proof. the proof of (3.10) is similar to that of (3.2) we, therefore omit the detals for the intrested reader. it remains only to prove that (3.10) is sharp. it is clear that f∗ ∈ c ([a,b],rf), since for letting tn → t, tn ∈ [a,b], then lemma 1 we have d (f∗(tn),f ∗(t)) = d (|tn −s0|�u, |t−s0|�u) ≤ ||tn −s0|− |t−s0||d (u,õ) ≤ |tn − t|d (u,õ) → 0, as n →∞. moreover, again by lemma 1, we get that d (f∗(t),f∗(s)) = d (|t−s0|�u, |s−s0|�u) ≤ ||t−s0|− |s−s0||d (u,õ) ≤ |t−s|d (u,õ) . fuzzy ostrowski type inequalities 45 that is for m = d (u,õ), we get d (f∗(t),f∗(s)) ≤ m |t−s| , ∀t,s ∈ [a,b] . so f∗ is m-lipshitzian function. now by the similar reasoning as in remark 1, we have 1 d− c � (fr) ∫ d c f∗ (s) ds = d− c 2 (c−a + b−d)2 [ (c−a)2 + (b−d)2 ] �u.(3.11) since s0 −a = (b−a) (c−a) c−a + b−d ≥ 0 and b−s0 = (b−a) (b−d) c−a + b−d ≥ 0 implies that s0 ∈ [a,b]. similarly we also have 1 b−a � (fr) ∫ b a f∗ (t) dt = b−a (c−a + b−d)2 [ (c−a)2 + (b−d)2 ] �u(3.12) now it is aparent from (3.11), (3.12) and lemma 1 that d ( 1 b−a � (fr) ∫ b a f∗ (t) , 1 d− c � (fr) ∫ d c f∗ (s) ds ) = ( (c−a)2 + (b−d)2 2 (c−a + b−d) ) d (u,õ) . hence it is proved that (3.10) is sharp. � references [1] george a. anastassiou, ostrowski type inequalities, proc. ams, 123 (1995), 3775–3791. [2] george a. anastassiou, fuzzy ostrowski type inequalities, computational and applied mathematics, vol. 22, no. 2 (2003), pp. 279-292. [3] george a. anastassiou, univariate fuzzy-random neural network approximation operators, computers & mathematics with applications volume 48, issue 9, november 2004, pages 1263-1283. [4] george a. anastassiou and sorin gal, on a fuzzy trigonometric approximation theorem of weierstrass-type, journal of fuzzy mathematics, 9, no. 3 (2001), 701–708. [5] n. s. barnett and s. s. dragomir, issues of estimating in the monitoring of constant flow coninuous streams, rgmia, research report collection, vol. 2, no. 3 (1999), pp.275-282. [6] a.m. fink, bounds on the deviation of a function from its averages, czechoslavak math. j., 42 (117) (1992), 289–310. [7] s. gal, approximation theory in fuzzy setting, chapter 13 in handbook of analytic computational methods in applied mathematics (edited by g. anastassiou), chapman & hall, crc press, boca raton, newyork, 2000, pp. 617–666. [8] a. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmittelwert, comment. math. helv., 10 (1938), 226–227. comp. appl. math., vol. 22, n. 2, 2003. 46 latif and hussain [9] m. matić and j. pečarić, two-pont ostrowski inequality, mathematical inequalities & applications, 4 2(2001), 215-221. [10] m.l. puri and d.a. ralescu, differentials of fuzzy functions, j. of math. analysis and appl., 91 (1983), 552–558. [11] congxinwu and zengtai gong, on henstock integral of fuzzy number valued functions (i), fuzzy sets and systems, 120, no. 3 (2001), 523–532. [12] l.a. zadeh, fuzzy sets, information and control, 8 (1965), 338–353. 1college of science, department of mathematics, university of hail, hail 2440, saudi arabia 2department of mathematics, university of engineering and technology, lahore, pakistan ∗corresponding author int. j. anal. appl. (2023), 21:46 fixed point theorems in cone metric spaces via c−distance over topological module shallu sharma∗, pooja saproo, iqbal kour, naresh digra department of mathematics, university of jammu, jammu, india ∗corresponding author: shallujamwal09@gmail.com abstract. in 2011, wang and guo introduced c-distance in cone metric spaces. the idea of cone metric spaces over topological modules was presented by branga and olaru in 2020. combining these two ideas, we introduce cone metric spaces with c−distance over topological module and establish a fixed point theorem. 1. introduction cone metric spaces were first introduced by huang and zhang [9]. for detailed study of cone metric spaces, refer [5,6,12,14–16]. other authors have also established fixed point theorems in cone metric spaces (for instance, [1–3, 10, 11]). wang and guo [18] presented cone metric spaces with c-distance and proved some fixed point theorems. cone metric spaces over topological module were introduced by branga and olaru [4]. in this paper, we present a new concept namely “cone metric spaces with c-distance over topological module" and prove a fixed point theorem. 2. preliminaries definition 2.1. [8] let (g, +) be a group with partial order relation ≤ . then g is said to be a partially ordered group if translation in g is order preserving: x ≤ y ⇒ z + x + w ≤ z + y + w,∀ x,y,w,z ∈ g definition 2.2. [17] consider a ring (r, +, .) and 1 be an identity of (r, +, .) such that 1 6= 0 and ≤ is a partial order on r. then r is called a partially ordered ring if: received: mar. 23, 2023. 2020 mathematics subject classification. 37c25; 47h10; 46h25; 47l07. key words and phrases. cone metric spaces with c-distance; topological module; fixed point theorems. https://doi.org/10.28924/2291-8639-21-2023-46 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-46 2 int. j. anal. appl. (2023), 21:46 (1) (r, +, .) is a partially ordered group; (2) z ≥ 0 and w ≥ 0 implies z.w ≥ 0 for all z,w ∈ r. remark 2.1. throughout the paper, r+ = {r ∈ r : r ≥ 0}, u(r) and u+(r) are the notations for the positive cone of r, the set of invertible elements of r and u(r) ∩r+, respectively. definition 2.3. [20] consider an abelian group (g, +). then g is called a topological group if g is endowed with a topology g such that the conditions mentioned below are satisfied: (1) for (g1,g2) ∈ g×g, the map (g1,g2) → g1 + g2 is continuous where g1 + g2 ∈ g and g×g is endowed with the product topology; (2) for g ∈ g, the map g →−g is continuous, where −g ∈ g. (g, +,g) or (g,g) is the notation for the topological group. definition 2.4. [20] a ring (r, +, .) is called a topological ring if r considered with the topology r such that (r, +,r) is a topological group and for (r1, r2) ∈ r × r, the map (r1, r2) → r1.r2 is continuous, where r1.r2 ∈ r and r×r is endowed with the product topology. (r, +, .,r) is called a hausdorff topological ring [20] if the topology r is hausdorff. also, (r, +, .,r) or (r,r) is the notation for the topological ring. definition 2.5. [20] consider a topological ring (r,r). a left r−module (e, +, .) is called a topological r−module if a topology e is defined on e such that (e, +) is a topological abelian group and the condition mentioned below is satisfied: for (r,x) ∈ r×e, the map (r,x) → r.x is continuous, where r.x ∈ e. further, (e, +, .,e) or (e,e) is the notation used for the topological left r−module. definition 2.6. [4] a set p ⊂ e, where (e, +, .,e) is a topological module, is called a cone if : (1) p is closed, nonempty and p 6= {0e}; (2) s.x + t.y ∈ p, whenever x,y ∈ p and s,t ∈ r+; (3) x ∈ p and −x ∈ p implies x = 0e. remark 2.2. throughout the paper, p◦ and p̄ are be the notations for the interior and closure of p , respectively. furthermore, the cone p is said to be solid if p◦ is non-empty. also, a partial ordering ≤p with respect to p is defined by y −x ∈ p if and only if x ≤p y and x <p y indicates x ≤p y,x 6= y. moreover, the partial ordering x � y indicates y −x ∈ p◦. next, we shall consider the following hypotheses: (hyp i) [4] consider a hausdorff topological ring (r, +,�,r) such that: (1) u+(r) is non-empty. (2) 0r is a limit point of u+(r). int. j. anal. appl. (2023), 21:46 3 (3) ≤r indicates the partial ordering on r. (hyp ii) [4] (e, +, .,e) is a topological left r−module. (hyp iii) [4] p is a solid cone contained in e. proposition 2.1. [4] consider a topological left r−module (e, +, .,e) and p be a cone contained in e such that the hypothesis hypi,hypii and hypiii are satisfied. then: (1) p◦ + p◦ ⊆ p◦; (2) β �p◦ ⊆ p◦, where β is an element of u+(r); (3) if v ≤p u and β ∈ r+, then β �v ≤p β �u; (4) if y ≤p x and x � z, then y � z; (5) if y � x and x ≤p z, then y � z; (6) if y � x and x � w, then y � w; (7) if 0e ≤p y � v,∀v ∈ p◦, then v = 0; (8) if 0e � v and {bn} is a sequence in e such that bn → 0e, then there is a natural number m0 such that bn � u for m ≥ m0. from now onwards, (r, +, .,r) denotes a hausdorff topological ring. definition 2.7. [4] a directed set is a partially ordered set (λ,≤) such that the condition mentioned below is fulfilled: if λ1,λ2 in λ there is λ3 in λ so that λ1 ≤ λ3 and λ2 ≤ λ3. definition 2.8. [4] a sequence {xλ}λ∈λ in r is a family of elements in r that is indexed by a directed set. definition 2.9. [4] a family {xλ}λ∈λ contained in r is convergent to an element x ∈ r if for each neighborhood w of x there exists λ0 ∈ λ such that xλ is an element of w for every λ ∈ λ, where λ ≥ λ0. definition 2.10. [4] a sequence {xλ}λ∈λ is called a cauchy sequence if for each neighborhood w of 0r there exists λ0 ∈ λ such that xλ1 −xλ2 ∈ w for each λ0 ≤ λ1 and λ0 ≤ λ2. theorem 2.1. [4] every convergent sequence {xλ}λ∈λ in r is a cauchy sequence. to exemplify the summability of the family of elements of a topological ring, consider h(λ) to be the set of all finite sets contained in λ directed by the inclusion ⊆ . definition 2.11. [4] an element t of r is sum of family {xλ}λ∈λ contained in r if the sequence {ti}i∈h(λ) is convergent to t, where for each i ∈h(λ), ti = ∑ λ∈i xλ. the family {xλ}λ∈i is said to be summable if {xλ}λ∈i has a sum t in r. 4 int. j. anal. appl. (2023), 21:46 definition 2.12. [4] a family {xλ}λ∈i contained in r is said to satisfy cauchy condition if for every neighborhood w of 0r there exists iw in h(λ) such that ∑ λ∈j xλ ∈ w, for each j ∈ h(λ) disjoint with iw . theorem 2.2. [4] a sequence {ti}i∈h(λ) is a cauchy sequence if and only if the family {xλ}λ∈λ contained in r satisfies cauchy condition. theorem 2.3. [4] let {xλ}λ∈λ be a summable family in r. then for each neighborhood w of 0r, there exists j in h(λ) so that xλ ∈ w for each λ ∈ λ \j. definition 2.13. [4] let (r, +, .,r) be a topological ring. then r is said to be complete if the topological additive group of ring (r, +,r) is complete. definition 2.14. [4] let xbe a non-empty set, (e, +, .,e) be a topological left r−module and the map d : x ×x → e satisfies: (1) d(x,y) ≥p 0e ∀x,y ∈ x. (2) d(x,y) = 0e if and only if x = y ∀x,y ∈ x. (3) d(x,y) = d(y,x) ∀x,y ∈ x. (4) d(x,y) ≤p d(x,z) + d(z,y) ∀x,y ∈ x. then d is said to be a cone metric on x and the pair (x,d) is said to be a cone metric space over topological left r−module e. definition 2.15. [4] let (x,d) be a cone metric space over the topological left r−module e, x be an element of x and {xn} be a sequence in x. then (1) {xn} is said to be convergent to x if for each u � 0, there is a natural number n such that d(xn,x) � u, ∀ n > n. (2) {xn} contained in x is said to be a cauchy sequence if for each u � 0, there is a natural natural number n such that d(xn,xm) � u, ∀ m,n > n. 3. fixed point theorem via c-distance in this section we first introduce a new notion namely c−distance in cone metric spaces over topological module. next we discuss some results regarding the same. further, a fixed point theorem in cone metric spaces via c−distance over topological module has been established. definition 3.1. let (x,d) be a cone metric space over a topological left r−module e. then a map p : x ×x → e is said to be a c−distance on x if it satisfies the following conditions: (p1) p(x,y) ≥p 0e ∀ x,y ∈ x; (p2) p(x,z) ≤p p(x,y) + p(y,z) ∀ x,y,z ∈ x; (p3) for every y ∈ x and n ≥ 1, p(y,xn) ≤p u for some u = uy ∈ p, then p(y,x) ≤p u whenever {xn} in x is convergent to a point x ∈ x; int. j. anal. appl. (2023), 21:46 5 (p4) for each u � 0, there exists v � 0 such that p(z,x) � v and p(z,y) � v implies that d(x,y) � u where u,v ∈ e. lemma 3.1. let (x,d) be a cone metric space over a topological left r-module e,p be a c−distance on x. consider the sequences {xn} and {yn} in x. next, let {an} be a sequence in p convergent to 0e and x,y,z ∈ x. then the following hold: (i) if p(xn,yn) ≤p an and p(xn,z) ≤p an for every natural number n, then {yn} is convergent to z. (ii) if p(xn,y) ≤p an and p(xn,z) ≤p an for every natural number n, then y = z. (iii) if p(xn,xm) ≤p an for all m > n, where m,n are natural numbers, then {xn} is a cauchy sequence in x. (iv) if p(y,xn) ≤p an, then {xn} is a cauchy sequence in x. proof. (i) let u � 0. then there is v � 0 such that p(u ′ ,v ′ ) � v and p(u ′ ,z) � v implies that d(v ′ ,z) � u. let m0 be any natural number such that an � v and bn � v for each n ≥ m0. next for each n ≥ m0,p(xn,yn) ≤p an � v and p(xn,z) ≤p bn � v. therefore, d(yn,z) � u. this proves that {yn} is convergent to z. (ii) proof clearly follows from (i). (iii) let u � 0. proceeding as proof of (i) let v � 0 and m0 be any natural number. next, for n,m ≥ m0 + 1,p(xm0,xn) ≤ um0 � v and p(xm0,xm) ≤ um0 � v. therefore, d(xn,xm) � v. this proves that {xn} is a cauchy sequence. (iv) proof clearly follows from (iii). � theorem 3.1. let (x,d) be a cone metric space over a topological left r−module e and p be a c−distance on x. suppose that the hypothesis hypi,hypii and hypiii are satisfied. define s = {r ∈ r+|{rn} is a summable family}. and the map t : x → x satisfies: p(tx,ty) ≤p rp(x,y), ∀ x,y ∈ x. then t has a fixed point x∗ in x and for each x ∈ x,{tnx} is convergent to the fixed point. if ζ = tζ, then p(ζ,ζ) = 0. also, t has a unique fixed point. proof. fix x0 ∈ x. let x1 = tx0,x2 = tx1 = t 2x0, . . . ,xn+1 = txn = tn+1x0. then p(xn,xn+1) = p(txn−1,txn) ≤p rp(xn−1,xn) ≤p r2p(xn−2,xn−1) ≤p . . . ≤p rnp(x0,x1). 6 int. j. anal. appl. (2023), 21:46 on the basis of above inequality, for all q ≥ 1, we have p(xn,xn+q) ≤p p(xn,xn+1) + . . . + p(xn+q−1,xn+q) ≤p rnp(x0,x1) + . . . + rn+q−1p(x0,x1) ≤p rn(1r + r + . . . + rq−1)p(x0,x1) ≤p rn( +∞∑ i=0 r i )p(x0,x1). using theorem 2.3, we have rn r→ 0r as n →∞. also {rn} is a summable family, right multiplication is continuous and using proposition 2.1 (8) we see that for each u � 0, there is a natural number n such that p(xn,xn+q) � u ∀ n ≥ n and q ≥ 1. then {xn} is a cauchy sequence in x. by the completeness of x, there is x∗ in x such that xn is convergent to x∗ as n tends to ∞. by (p3) we see that p(xn,x ∗) � u. (0.3.1) also, p(xn,tx ∗) = p(txn−1,tx ∗) ≤p rp(xn−1,x∗) ≤p rn( +∞∑ i=0 r i )p(x0,x1) � u. (0.3.2) now, p(xn,x∗) � u and p(xn,tx∗) � u. let v � 0. then by (p4), we have d(x∗,t (x∗)) � v. next using proposition 2.1 (7) we have x∗ = tx∗. hence x∗ is a fixed point of t. next suppose that y∗ is a fixed point of t. then for u ∈ p◦, we have q(x∗,y∗) = q(tx∗,ty∗) ≤p rq(x∗,y∗) ≤p r2q(x∗,y∗) ≤p . . . ≤p rnq(x∗,y∗) � u. hence q(x∗,y∗) = 0. also, q(x∗,x∗) = 0. using lemma 3.1 (ii), we have x∗ = y∗. hence t has a unique fixed point. � corollary 3.1. let (x,d) be a cone metric space over a topological left r−module e and p be a c−distance on x. define s = {r ∈ r+|{rn} is a summable family}. and the map t : x → x satisfies: p(tnx,tny) ≤p rp(x,y), ∀ x,y ∈ x. then t has a fixed point x∗ in x. if ζ = tζ, then p(ζ,ζ) = 0. int. j. anal. appl. (2023), 21:46 7 proof. using theorem 3.1, we see that x∗ is a unique fixed point of tn. also, tn(tx∗) = t (tnx∗) = t (x∗). therefore, t (x∗) is a fixed point of tn. so, x∗ = tx∗. this shows that x∗ is a fixed point of t. also, the fixed point of t is also a fixed point of tn shows that t has a unique fixed point. next suppose that ζ = tζ. we see that the fixed point of t is also a fixed point of tn. from this, for u � 0, we have p(ζ,ζ) = p(tζ,tζ) = p(tnζ,tnζ) ≤p rp(ζ,ζ) ≤p r2q(ζ,ζ) ≤p . . . ≤p rnp(ζ,ζ) � u. hence p(ζ,ζ) = 0. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. abbas, g. jungck, common fixed point results for noncommuting mappings without continuity in cone metric spaces, j. math. anal. appl. 341 (2008), 416–420, https://doi.org/10.1016/j.jmaa.2007.09.070. [2] m. abbas, b. e. rhoades, fixed and periodic point results in cone metric spaces, appl. math. lett. 22 (2009), 511–515, https://doi.org/10.1016/j.aml.2008.07.001. [3] a. azam, m. arshad, common fixed points of generalized contractive maps in cone metric spaces, bull. iran. math. soc. 35 (2009), 255–264. [4] a.n. branga, i.m. olaru, cone metric spaces over topological module and fixed point theorems for lipschitz mappings, mathematics, 8 (2020), 724. https://doi.org/10.3390/math8050724. [5] l. ciric, h. lakzian, v. rakocevic, fixed point theorems for w-cone distance contraction mappings in tvs-cone metric spaces, fixed point theory appl. 2012 (2012), 3. https://doi.org/10.1186/1687-1812-2012-3. [6] m. dordevic, d. doric, z. kadelburg, s. radenovic, d. spasic, fixed point results under c-distance in tvs-cone metric spaces, fixed point theory appl. 2011 (2011), 29. https://doi.org/10.1186/1687-1812-2011-29. [7] z.m. fadail, a.g.b. ahmad, z. golubovic, fixed point theorems of single-valued mapping for c-distance in cone metric spaces, abstr. appl. anal. 2012 (2012), 826815. https://doi.org/10.1155/2012/826815. [8] a.m.w. glass, partially ordered groups, world scientific, (1999). https://doi.org/10.1142/3811. [9] l.g. huang, x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087. [10] d. ilic, v. rakocevic, common fixed points for maps on cone metric space, j. math. anal. appl. 341 (2008), 876–882. https://doi.org/10.1016/j.jmaa.2007.10.065. [11] d. ilic, v. rakocevic, quasi-contraction on a cone metric space, appl. math. lett. 22 (2009), 728–731. https: //doi.org/10.1016/j.aml.2008.08.011. [12] s. jankovic, z. kadelburg, s. radenovic, on cone metric spaces: a survey, nonlinear anal.: theory methods appl. 74 (2011), 2591–2601. https://doi.org/10.1016/j.na.2010.12.014. [13] o. kada, t. suzuki, w. takahashi, non-convex minimization theorems and fixed point theorems in complete metric spaces, math. japon. 44 (1996), 381-391. https://cir.nii.ac.jp/crid/1570009749812799360. [14] z. kadelburg, s. radenovic, some common fixed point results in non-normal cone metric spaces, j. nonlinear sci. appl. 3 (2010), 193–202. [15] z. kadelburg, s. radenovic, v. rakocevic, topological vector space-valued cone metric spaces and fixed point theorems, fixed point theory appl. 2010 (2010), 170253. https://doi.org/10.1155/2010/170253. [16] z. kadelburg, s. radenovic, coupled fixed point results under tvs-cone metric and w-cone-distance, adv. fixed point theory, 2 (2012), 29-46. https://doi.org/10.1016/j.jmaa.2007.09.070 https://doi.org/10.1016/j.aml.2008.07.001 https://doi.org/10.3390/math8050724 https://doi.org/10.1186/1687-1812-2012-3 https://doi.org/10.1186/1687-1812-2011-29 https://doi.org/10.1155/2012/826815 https://doi.org/10.1142/3811 https://doi.org/10.1016/j.jmaa.2005.03.087 https://doi.org/10.1016/j.jmaa.2007.10.065 https://doi.org/10.1016/j.aml.2008.08.011 https://doi.org/10.1016/j.aml.2008.08.011 https://doi.org/10.1016/j.na.2010.12.014 https://cir.nii.ac.jp/crid/1570009749812799360 https://doi.org/10.1155/2010/170253 8 int. j. anal. appl. (2023), 21:46 [17] s.a. steinberg, lattice-ordered rings and modules, springer, new york, (2010). https://doi.org/10.1007/ 978-1-4419-1721-8. [18] s. wang, b. guo, distance in cone metric spaces and common fixed point theorems, appl. math. lett. 24 (2011), 1735-1739. https://doi.org/10.1016/j.aml.2011.04.031. [19] d. wardowski, endpoints and fixed points of set-valued contractions in cone metric spaces, nonlinear anal.: theory methods appl. 71 (2009), 512-516. https://doi.org/10.1016/j.na.2008.10.089. [20] s. warner, topological rings, north-holland, amsterdam, 1993. https://doi.org/10.1007/978-1-4419-1721-8 https://doi.org/10.1007/978-1-4419-1721-8 https://doi.org/10.1016/j.aml.2011.04.031 https://doi.org/10.1016/j.na.2008.10.089 1. introduction 2. preliminaries 3. fixed point theorem via c-distance references international journal of analysis and applications issn 2291-8639 volume 2, number 1 (2013), 19-25 http://www.etamaths.com facts about the fourier-stieltjes transform of vector measures on compact groups yaogan mensah abstract. this paper gives an interpretation of the fourier-stieltjes transform of vector measures by means of the tensor product of hilbert spaces. it also extends the kronecker product to some operators arising from the fourierstieltjes transformation and associated with the equivalence classes of unitary representations of a compact group. we obtain among other results the effect of this product on convolution of vector measures. 1. introduction this paper inspects mainly two things: the fourier-stieltjes transformation and the kronecker product. the importance of the fourier transformation in mathematical science and ingeneering, for instance in signal processing, is well known and so we need not to lay emphasis on it. on the other hand, among the ways to bring together matrices there is the kronecker product. it is extensively used in group theory and physics, specially in quantum information theory to determine for instance exact spin hamiltonian [9], [13]. in quantum physics the quantum states of a system is described by an hermitian positive semi-definite matrix with trace one. if x and y represent the states of two quantum systems then the kronecker product x ⊗ y describes the joint system. some other fundamental applications of the kronecker product in signal processing, image processing or quantum computing can be found in [7], [11], [2], [10] and [14]. this paper aims to deepen the link between the kronecker product and the fourier-stieltjes transform of vector measures. the rest of the paper is organized as follows. in section 2 we recall the definition of vector measures on compact groups. the section 3 is divided into two parts. in the first part we give an interpretation of the fourier-stieltjes transform of vector measures as a bounded vector valued mappings on the tensor product of hilbert spaces. the next part extends the kronecker product to some operators arising from the fourier-stieltjes transformation and associated with the equivalence classes of unitary representations of a compact group. here the effect of this product on the convolution of vector measures is obtained. 2. preliminaries we summarize the definition of a vector measure on a compact group g following [3] and [6]. we denote by b(g) the σ-field of borel subsets of the compact group 2010 mathematics subject classification. primary: 43a77, secondary: 43a30. key words and phrases. vector measure, fourier-stieltjes transform, kronecker product. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 19 20 mensah g and by a (in the whole paper) a complex banach algebra with topological dual a∗. also denote by 〈x∗,x〉 the duality between a∗ and a. a vector measure is a countably additive set function µ : b(g) → a, that is, for any sequence (an) of pairwise disjoint subsets of b(g), one has (1) µ( ∞ ∪ n=1 an) = ∞∑ n=1 µ(an), where the second member of the above equality is convergent in the norm topology of a. if µ is a vector measure on g then for each x∗ ∈a∗ the measure defined by (2) x∗µ(a) := 〈x∗,µ(a)〉, a ∈ b(g) is a complex measure. let f be a complex measurable borel function f defined on g. the function f is said to be µ−integrable if the following two conditions are satisfied: (1) ∀x∗ ∈a∗, f is x∗µ−integrable, (2) ∀a ∈ b(g) , ∃y ∈a, ∀x∗ ∈a∗, 〈x∗,y〉 = ∫ a fdx∗µ. then we set y = ∫ g f dµ. let µ : b(g) → a be a vector measure. the semivariation [3] of µ is the nonnegative function |µ| defined on b(g) by (3) |µ|(a) = sup{|x∗µ|(a) : x∗ ∈a,‖x∗‖≤ 1}, ∀a ∈ b(g) where |x∗µ| is the variation of x∗µ. we recall that the variation of a scalar (real or complex) measure m is the extended nonnegative mapping |m| defined on b(g) by (4) |m|(a) = sup π ∑ e∈π |m(e)| where the supremum is taken over all partitions π of a into finite number of pairwise disjoint members of b(g) [3, page 2]. a vector measure µ is said to be of bounded semivariation if |µ|(g) < ∞. the range of a vector measure is bounded if and only if it is of bounded semivariation [3, page 4]. so a vector measure is said to be bounded if it is of bounded semivariation. denote by m1(g,a) the set of bounded a−valued measures on g. the set m1(g,a) is equipped with the norm (5) ‖µ‖ := ∫ g χgd|µ| = |µ|(g) where χg is the characteristic function of g. let µ, ν be in m1(g,a). the convolution product of µ with ν is given by : (6) µ∗ν(f) = ∫ ∫ g f(gh)dµ(g)dν(h) ,f ∈c(g,a). where c(g,a) is the set of a−valued continuous functions on g. equipped with this product and the above norm, the space m1(g,a) is a complex banach algebra. facts about the fourier-stieltjes transform 21 3. main results 3.1. fourier-stieltjes transformation: a tensor product interpretation. the fourier(-stieltjes) transform of a complex valued function (or measure) on a compact group is well-known [5]. for a given function f, the fourier transform of f is a collection of bounded operators on some hilbert spaces. in [1], assiamoua defined the fourier-stieltjes transform of a banach algebra valued measure on a compact group by interpreting it as a collection of some continuous sesquilinear mappings. in this section, we use the modern technique of tensor product to keep the interpretation of the fourier-stieltjes transform of a vector measure as a collection of operators. let ĝ be the unitary dual of the compact group g i.e. the set of equivalence classes of unitary irreducible representations of g. in any equivalence class σ belonging to ĝ, we choose an element uσ and denoted its hilbertian representation space by hσ. since g is compact, the hilbert space hσ is finite dimensional [4], and we denote its dimension by dσ. we fix definitively a canonical basis (ξ σ 1 , . . . ,ξ σ dσ ) of hσ. for g ∈ g, we set (7) uσij(g) = 〈u σ g ξ σ j ,ξ σ i 〉 where 〈 , 〉 is the inner product in hσ, and denote by uσ the contragredient of the representation uσ, that is the representation of g on hσ which matrix elements are given by (8) 〈uσg ξ σ j ,ξ σ i 〉 = uσij(g), where uσij(g) is the complex conjugate of u σ ij(g). let us denote by hσ the conjugate hilbert space to hσ. we denote by hσ⊗̂hσ the tensor product of hilbert spaces hσ ⊗ hσ equipped with the projective tensor product norm. a basis of hσ⊗̂hσ is ( ξσi ⊗ ξ σ j ) 1≤i,j≤dσ . then the fourier-stieltjes transform of a bounded a−valued measure µ is the collection of linear operators (µ̂(σ)) σ∈ĝ from hσ⊗̂hσ into a defined by : (9) µ̂(σ)(ξ ⊗η) = ∫ g 〈uσg ξ,η〉dµ(g), ξ, η ∈ hσ. now, let b(hσ⊗̂hσ,a) be the space of bounded linear mappings from hσ⊗̂hσ into the banach algebra a. let b(ĝ,a) be the bundle over ĝ whose fiber at σ is b(hσ⊗̂hσ,a), that is (10) b(ĝ,a) = ∏ σ∈ĝ b(hσ⊗̂hσ,a). for ϕ ∈ b(ĝ,a), set (11) ‖ϕ‖∞ = sup{‖ϕ(σ)‖ : σ ∈ ĝ} where ‖ϕ(σ)‖ denotes the operator norm of ϕ(σ). define (12) b∞(ĝ,a) = {ϕ ∈ b(ĝ,a) : ‖ϕ‖∞ < ∞}. theorem 3.1. for µ ∈ m1(g,a), we have µ̂ ∈ b∞(ĝ,a). 22 mensah proof. it is clear that for each σ, the object µ̂(σ) is linear from hσ⊗̂hσ into a. now ‖µ̂(σ)(ξ ⊗η)‖ = ‖ ∫ g 〈uσg ξ,η〉dµ(g)‖ ≤ ∫ g |〈uσg ξ,η〉|d|µ|(g) ≤ ‖ξ‖‖η‖‖µ‖ = ‖ξ ⊗η‖‖µ‖. thus ‖µ̂(σ)‖≤‖µ‖ for each σ. therefore ‖µ̂‖∞ ≤‖µ‖ < ∞. � let µ ∈ m1(g,a) and σ ∈ ĝ. we associate with the operator û(σ) the matrix denoted by m(û(σ)) which (i,j)−entry belongs to the banach algebra a and is given by (13) [m(µ̂(σ))]ij = µ̂(σ)(ξσj ⊗ ξ σ i ). for vector measures µ and ν we denote by (µ̂× ν̂)(σ) the operator from hσ⊗̂hσ into a associated with the product of matrices m(ν̂(σ))m(µ̂(σ)), that is (14) (µ̂× ν̂)(σ)(ξσj ⊗ ξ σ i ) = dσ∑ k=1 ν̂(σ)(ξσk ⊗ ξ σ i )µ̂(σ)(ξ σ j ⊗ ξ σ k ). the following theorem is the analogue of the convolution theorem. theorem 3.2. for µ,ν ∈ m1(g,a), we have µ̂∗ν = µ̂× ν̂. proof. (̂µ∗ν)(σ)(ξσj ⊗ ξ σ i ) = ∫ g 〈uσt ξ σ j ,ξ σ i 〉d(µ∗ν)(t) = ∫ ∫ g 〈uσstξ σ j ,ξ σ i 〉dµ(s)dν(t) = ∫ ∫ g 〈uσt uσs ξ σ j ,ξ σ i 〉dµ(s)dν(t). now, we express uσs ξ σ j in the canonical basis of hσ: (15) uσs ξ σ j = dσ∑ k=1 〈uσs ξ σ j ,ξ σ k〉ξ σ k . so (̂µ∗ν)(σ)(ξσj ⊗ ξ σ i ) = dσ∑ k=1 ∫ ∫ g 〈uσs ξ σ j ,ξ σ k〉〈uσt ξ σ k ,ξ σ i 〉dµ(s)dν(t) = dσ∑ k=1 ∫ g 〈uσt ξ σ k ,ξ σ i 〉dν(t) ∫ g 〈uσs ξ σ j ,ξ σ k〉dµ(s) = dσ∑ k=1 ν̂(σ)(ξσk ⊗ ξ σ i )µ̂(σ)(ξ σ j ⊗ ξ σ k ) = (µ̂× ν̂)(σ)(ξσj ⊗ ξ σ i ). � remark: application to convolution equations the above result can be useful in the resolution of the convolution equation (16) f ∗h = g facts about the fourier-stieltjes transform 23 where f, g and h (the unknown function) are a-valued functions on g. here f,g and h can be viewed as the vector measures fdx, gdx and hdx respectively where dx denotes the normalized haar measure of g. for each σ ∈ ĝ, the equation (16) is transformed into (17) f̂(σ) × ĥ(σ) = ĝ(σ). therefore if the operator f̂(σ) is invertible, ĥ(σ) can be derived from (17) and h is recovered by the following reconstruction formula (18) h = ∑ σ∈ĝ dσ dσ∑ i=1 dσ∑ j=1 ĥ(σ)(ξσj ⊗ ξ σ i )u σ ij. 3.2. the mappings tσ and their kronecker product. now we turn our attention over some matrix valued mappings acting on the set of bounded vector measures. let mdσ (a) be the set of dσ ×dσ matrices with entries in the banach algebra a. we denote by tσ the linear mapping tσ : m 1(g,a) → mdσ (a) µ 7→ tσ(µ) = m(µ̂(σ)). we set ∆(ĝ) = {tσ : σ ∈ ĝ}. the following result shows the effect of each tσ on convolution of vector measures. theorem 3.3. for tσ ∈ ∆(ĝ), µ,ν ∈ m1(g,a), we have: (19) tσ(µ∗ν) = tσ(ν)tσ(µ). proof. tσ(µ∗ν) = m(µ̂∗ν(σ)) = m((µ̂× ν̂)(σ)) according to theorem 3.2. = m(ν̂(σ))m(µ̂(σ)) = tσ(ν)tσ(µ). � now we are going to extend the kronecker product to the mappings tσ. we also give some basic properties of this extension. the following definition of kronecker product of matrices can be found in [8]. definition 3.4. let x = (xij) and y be two matrices. the kronecker product (also called tensor product) of x by y is the matrix (20) x ⊗y =   x11y x12y .. . x1ny x21y · · · ... ... ... · · · ... xm1y xm2y · · · xmny   . the kronecker product of matrices is associative, noncommutative and verifies, among other properties, the equality: (21) (x1x2) ⊗ (y1y2) = (x1 ⊗y1)(x2 ⊗y2). 24 mensah where xi, yj, i = 1, 2; j = 1, 2 are matrices such that the product x1x2 and y1y2 exist; see [8] and [12]. definition 3.5. let tσ1, tσ2 be in ∆(ĝ). the kronecker product of tσ1 by tσ2 , denoted by tσ1 � tσ2 , is the mapping from m 1(g,a) into mdσ1dσ2 (a) such that (22) ∀µ ∈ m1(g,a), [tσ1 � tσ2 ](µ) = m(µ̂(σ1)) ⊗m(µ̂(σ2)). remark. since the kronecker product of matrices is noncommutative, so is the extended kronecker product �. many properties of the kronecker product of matrices are easily extended to the kronecker product of the mappings tσ. as an example we prove the associativity in the following theorem. theorem 3.6. let σi, i = 1, 2, 3, be in ĝ and µ,ν be in m 1(g,a). we have the following associative relation: (23) (tσ1 � tσ2 ) � tσ3 = tσ1 � (tσ2 � tσ3 ). and we have the following effect on the convolution of vector measures: (24) [tσ1 � tσ2 ](µ∗ν) = [tσ1 � tσ2 ](ν)[tσ1 � tσ2 ](µ). proof. 1. [(tσ1 � tσ2 ) � tσ3 ](µ) = [m(µ̂(σ1)) ⊗m(µ̂(σ2))] ⊗m(µ̂(σ3)). the proof can be completed by using the fact that the kronecker product of matrices is associative. 2. [tσ1 � tσ2 ](µ∗ν) = m(µ̂∗ν(σ1)) ⊗m(µ̂∗ν(σ2)) = [m(ν̂(σ1))m(µ̂(σ1))] ⊗ [m(ν̂(σ2))m(µ̂(σ2))] = [m(ν̂(σ1)) ⊗m(ν̂(σ2))][m(µ̂(σ1)) ⊗m(µ̂(σ2))] = [tσ1 � tσ2 ](ν)[tσ1 � tσ2 ](µ), using meanwhile the property (21) of kronecker product of matrices. � 4. conclusion we interpreted the fourier-stieltjes transform of a (banach algebra)-valued measures on a compact group g as a collection of operators from a tensor product of hilbert spaces into the banach algebra a. therefore it was possible to associate a matrix with each of such operators. then we extended the kronecker product of matrices to some matrix valued mappings acting on vector measures. we also computed the effect of these mappings and that of their kronecker product on the convolution of vector measures. many other properties could be found if this discussion is deepened. these are the aims of some forthcoming papers. what can be the possible applications to physics (for example) for joining kronecker product with fourier-stieltjes transform of vector measures? facts about the fourier-stieltjes transform 25 references [1] assiamoua, v.s.k. and olubummo, a., fourier-stieltjes transforms of vector-valued measures on compact groups, acta sci. math.(szeged), 53 (1989), 301-307. [2] davio, m., kronecker product and shuffle algebra, ieee trans. comput., c-30 (1981), 116125. [3] diestel, j. and uhl, j. j.,vector measures, math. surveys 15, amer. math. soc., providence, 1977. [4] folland, g. b., a course in abstract harmonic analysis, crc press , 1995. [5] hewitt, e. and ross, k.a., abstract harmonic analysis, vol ii, springer-verlag, new yorkberlin-heidelberg, 1970. [6] kawabe, j., compactnes criteria for the weak convergence of vector measures in locally convex spaces, publ. math. debrecen., 60 (2002),115-130. [7] van loan, c. f., the ubiquitous kronecker product, journal compt. appl. math., 123 (2000) 85-100. [8] marcus, m. and minc, h., a survey of matrix theory and matrix inequalities, dover publications, inc., new york, 1992. [9] rao, k. n. s., linear algebra and group theory for physicists, hindustan book agency, india, 2006. [10] regalia, p. a. and mitra, s., kronecker products, unitary matrices and signal processing applications, siam rev., 31 (1989), 586-613. [11] steeb, w.-h., matrix calculus and kronecker product with applications and c++ programs, world scientific publishing, singapore, 1997. [12] steeb, w.-h. and hardy, y., matrix calculus and kronecker product: a practical approach to linear and multilinear algebra, world scientific publishing, singapore (2011). [13] tinkham, m., group theory and quantum mechanics, ny mcgraw-hill, 1964. [14] tolimieri, r. , lu, c. and an, m., algorithms for discrete fourier transform and convolution, springer, new york, 1989. department of mathematics, university of lomé, togo, b. p. 1515, lomé, togo and international chair in mathematical physics and applications (icmpa unescochair),university of abomey-calavi, benin international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 136-146 http://www.etamaths.com eigenvalues for iterative systems of (n,p)-type fractional order boundary value problems k. r. prasad1, b. m. b. krushna2,∗ and n. sreedhar3 abstract. in this paper, we determine the eigenvalue intervals of λ1,λ2, · · ·,λn for which the iterative system of (n,p)-type fractional order two-point boundary value problem has a positive solution by an application of guokrasnosel’skii fixed point theorem on a cone. 1. introduction the study of fractional order differential equations has emerged as an important area of mathematics. it has wide range of applications in various fields of science and engineering such as physics, mechanics, control systems, flow in porous media, electromagnetics and viscoelasticity. recently, much interest has been created in establishing positive solutions and multiple positive solutions for two-point, multi-point boundary value problems (bvps) associated with ordinary and fractional order differential equations. to mention the related papers along these lines, we refer to erbe and wang [4], davis, henderson, prasad and yin [3] for ordinary differential equations, henderson and ntouyas [6, 7], henderson, ntouyas and purnaras [8, 9] for systems of ordinary differential equations, bai and lu [1], zhang [17], kauffman and mboumi [10], benchohra, henderson, ntoyuas and ouahab [2], su and zhang [16], khan, rehman and henderson[11], prasad and krushna [15] for fractional order differential equations. this paper concerned with determining the eigenvalues λi, 1 ≤ i ≤ n, for which there exist positive solutions for the iterative system of (n,p)-type fractional order boundary value problems (1.1) dα0+yi(t) + λiai(t)fi(yi+1(t)) = 0, 1 ≤ i ≤ n, 0 < t < 1, yn+1(t) = y1(t), 0 < t < 1, } (1.2) y (j) i (0) = 0, 0 ≤ j ≤ n− 2, y (p) i (1) = 0, where dα 0+ is the standard riemann-liouville fractional order derivative, n− 1 < α ≤ n and n ≥ 3, 1 ≤ p ≤ α− 1 is a fixed integer. by a positive solution of the fractional order bvp (1.1)-(1.2), we mean (y1(t),y2(t), ·· ·,yn(t)) ∈ ( c[α]+1[0, 1] )n satisfying (1.1)-(1.2) with yi(t) ≥ 0, i = 1, 2, 3, · · ·n, for 2010 mathematics subject classification. 26a33, 34b15, 34b18. key words and phrases. fractional derivative, boundary value problem, iterative system, twopoint, green’s function, eigenvalues, positive solution. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 136 eigenvalues for iterative systems 137 all t ∈ [0, 1] and (y1(t),y2(t), · · ·,yn(t)) 6= (0, 0, · · ·, 0). we assume the following conditions hold throughout the paper: (a1) fi : r + → r+ is continuous, for 1 ≤ i ≤ n, (a2) ai : [0, 1] → r + is continuous and ai does not vanish identically on any closed subinterval of [0, 1], for 1 ≤ i ≤ n, (a3) each of fi0 = lim x→0+ fi(x) x and fi∞ = lim x→∞ fi(x) x , for 1 ≤ i ≤ n, exists as positive real numbers. the rest of the paper is organized as follows. in section 2, we construct the green’s function for the homogeneous bvp and estimate the bounds for the green’s function. in section 3, we establish criteria to determine the eigenvalues for which the fractional order bvp (1.1)-(1.2) has at least one positive solution in a cone by using guo-krasnosel’skii fixed point theorem. in section 4, as an application, we demonstrate our results with an example. 2. green’s function and bounds in this section, we construct the green’s function for the homogeneous bvp and estimate the bounds for the green’s function which are needed in establishing the main results. lemma 2.1. if h(t) ∈ c[0, 1], then the fractional order bvp, (2.1) dα0+y1(t) + h(t) = 0, t ∈ (0, 1), (2.2) y (j) 1 (0) = 0, 0 ≤ j ≤ n− 2, y (p) 1 (1) = 0 has a unique solution, y1(t) = ∫ 1 0 g(t,s)h(s)ds, where (2.3) g(t,s) = { tα−1(1−s)α−1−p γ(α) , 0 ≤ t ≤ s ≤ 1, tα−1(1−s)α−1−p−(t−s)α−1 γ(α) , 0 ≤ s ≤ t ≤ 1. proof. assume that y1(t) ∈ c[α]+1[0, 1] is a solution of fractional order bvp (2.1)(2.2) and is uniquely expressed as iα0+d α 0+y1(t) = −i α 0+h(t) y1(t) = −1 γ(α) ∫ t 0 (t−s)α−1h(s)ds + c1tα−1 + c2tα−2 + c3tα−3 + · · · + cntα−n. from y (j) 1 (0) = 0, 0 ≤ j ≤ n− 2, we have cn = cn−1 = cn−2 = · · · = c2 = 0. then y1(t) = −1 γ(α) ∫ t 0 (t−s)α−1h(s)ds + c1tα−1, y (p) 1 (t) = c1 p∏ i=1 (α− i)tα−1−p − p∏ i=1 (α− i) 1 γ(α) ∫ 1 0 (t−s)α−1−ph(s)ds. 138 prasad, krushna and sreedhar from y (p) 1 (1) = 0, we have c1 p∏ i=1 (α− i) − p∏ i=1 (α− i) 1 γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds = 0. therefore, c1 = 1 γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds. thus, the unique solution of (2.1)-(2.2) is y1(t) = −1 γ(α) ∫ t 0 (t−s)α−1h(s)ds + tα−1 γ(α) ∫ 1 0 (1 −s)α−1−ph(s)ds = ∫ 1 0 g(t,s)h(s)ds, where g(t,s) is given in (2.3). � lemma 2.2. the green’s function g(t,s) satisfies the following inequalities, (i) g(t,s) ≥ 0, for all (t,s) ∈ [0, 1] × [0, 1], (ii) g(t,s) ≤ g(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (iii) g(t,s) ≥ 1 4α−1 g(1,s), for all (t,s) ∈ i × [0, 1], where i = [ 1 4 , 3 4 ] . proof. the green’s function g(t,s) is given in (2.3). for 0 ≤ t ≤ s ≤ 1. g(t,s) = 1 γ(α) [tα−1(1 −s)α−1−p] ≥ 0. for 0 ≤ s ≤ t ≤ 1, g(t,s) = 1 γ(α) [tα−1(1 −s)α−1−p − (t−s)α−1] ≥ 1 γ(α) [tα−1(1 −s)α−1−p − tα−1(1 −s)α−1] = 1 γ(α) [tα−1(1 −s)α−1−p][1 − (1 −s)p] ≥ 0. hence the inequality (i) is proved. we prove the inequality (ii). for 0 ≤ t ≤ s ≤ 1, ∂ ∂t g(t,s) = 1 γ(α) [(α− 1)tα−2(1 −s)α−1−p] ≥ 0. for 0 ≤ s ≤ t ≤ 1, ∂ ∂t g(t,s) = 1 γ(α) [(α− 1)tα−2(1 −s)α−1−p − (α− 1)(t−s)α−2] = (α− 1) γ(α) [ tα−2(1 −s)α−2(1 −s)1−p − (t−s)α−2 ] ≥ (α− 1) γ(α) [ tα−2(1 −s)α−2(1 −s)1−p − (t− ts)α−2 ] = (α− 1) γ(α) [ (1 −s)1−p − 1 ] (t− ts)α−2 ≥ 0. eigenvalues for iterative systems 139 therefore g(t,s) is increasing with respect to t ∈ [0, 1]. hence the inequality (ii) is proved. now, we establish the inequality (iii). for 0 ≤ t ≤ s ≤ 1 and t ∈ i, g(t,s) g(1,s) = tα−1(1 −s)α−1−p (1 −s)α−1−p = tα−1 ≥ 1 4α−1 . for 0 ≤ s ≤ t ≤ 1 and t ∈ i, g(t,s) g(1,s) = tα−1(1 −s)α−1−p − (t−s)α−1 (1 −s)α−1−p − (1 −s)α−1 ≥ tα−1(1 −s)α−1−p − (t− ts)α−1 (1 −s)α−1−p − (1 −s)α−1 =tα−1 ≥ 1 4α−1 . hence the inequality (iii) is proved. � an n-tuple (y1(t),y2(t), ···,yn(t)) is a solution of the bvp (1.1)-(1.2) if and only if yi(t) ∈ c[α]+1[0, 1] satisfies the following equations y1(t) =λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · · · fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 and yi(t) = λi ∫ 1 0 g(t,s)ai(s)fi(yi+1(s))ds, 0 ≤ t ≤ 1, 2 ≤ i ≤ n, where yn+1(t) = y1(t), 0 ≤ t ≤ 1. in establishing our main result, we will employ the following fixed point theorem due to guo-krasnosel’skii [5, 13]. theorem 2.3. [5, 13] let x be a banach space, p ⊆ x be a cone and suppose that ω1, ω2 are open subsets of x with 0 ∈ ω1 and ω1 ⊂ ω2. suppose further that t : p ∩ (ω2\ω1) → p is completely continuous operator such that either (i) ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω2, or (ii) ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω2 holds. then t has a fixed point in p ∩ (ω2\ω1). 3. positive solutions in a cone in this section, we establish criteria to determine the eigenvalues for which the fractional order bvp (1.1)-(1.2) has at least one positive solution in a cone. let x = {x : x ∈ c[0, 1]} be the banach space equipped with the norm ‖x‖ = max 0≤t≤1 |x(t)|. define a cone p ⊂ x by p = { x ∈ x | x(t) ≥ 0 on [0, 1] and min t∈i x(t) ≥ 1 4α−1 ‖x‖ } . 140 prasad, krushna and sreedhar now, we define an integral operator t : p → x, for y1 ∈ p , by (3.1) ty1(t) = λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1. notice from (a1), (a2) and lemma 2.2 that, for y1 ∈ p , ty1(t) ≥ 0 on [0, 1]. and also, we have ty1(t) ≤ λ1 ∫ 1 0 g(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 so that (3.2) ‖ty1‖≤ λ1 ∫ 1 0 g(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1. next, if y1 ∈ p , we have from lemma 2.2 and (3.2) that min t∈i ty1(t) = min t∈i λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ λ1 1 4α−1 ∫ 1 0 g(1,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · · · fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ 1 4α−1 ‖ty1‖. therefore, min t∈i ty1(t) ≥ 1 4α−1 ‖ty1‖. hence, ty1 ∈ p and so t : p → p . further, the operator t is a completely continuous operator by an application of the arzela-ascoli theorem. now, we seek suitable fixed point of t belonging to the cone p. for our first result, we define positive numbers n1 and n2, by n1 = max 1≤i≤n {[ 1 4α−1 ∫ s∈i g(1,s)ai(s)dsfi∞ ]−1} and n2 = min 1≤i≤n {[∫ 1 0 g(1,s)ai(s)dsfi0 ]−1} . eigenvalues for iterative systems 141 theorem 3.1. assume that the conditions (a1)-(a3) are satisfied. then, for each λ1,λ2, · · ·,λn satisfying (3.3) n1 < λi < n2, 1 ≤ i ≤ n, there exists an n-tuple (y1,y2, · · ·,yn) satisfying (1.1)-(1.2) such that yi(t) > 0, 1 ≤ i ≤ n on (0, 1). proof. let λi, 1 ≤ i ≤ n be given as in (3.3). now, let � > 0 be chosen such that max 1≤i≤n {[ 1 4α−1 ∫ s∈i g(1,s)ai(s)ds(fi∞ − �) ]−1} ≤ min 1≤i≤n λi and max 1≤i≤n λi ≤ min 1≤i≤n {[∫ 1 0 g(1,s)ai(s)ds(fi0 + �) ]−1} . we seek fixed point of the completely continuous operator t : p → p defined by (3.1). now, from the definitions of fi0, 1 ≤ i ≤ n, there exists an h1 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≤ (fi0 + �)x, 0 < x ≤ h1. let y1 ∈ p with ‖y1‖ = h1. we first have from lemma 2.2 and the choice of �, for 0 ≤ sn−1 ≤ 1, λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ≤ λn ∫ 1 0 g(1,sn)an(sn)(fn0 + �)y1(sn)dsn ≤ λn ∫ 1 0 g(1,sn)an(sn)dsn(fn0 + �)‖y1‖ ≤‖y1‖ = h1. it follows in a similar manner from lemma 2.2 and the choice of � that, for 0 ≤ sn−2 ≤ 1, λn−1 ∫ 1 0 g(sn−2,sn−1)an−1(sn−1) fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) dsn−1 ≤ λn−1 ∫ 1 0 g(sn−1,sn−1)an−1(sn−1)dsn−1(fn−1,0 + �)‖y1‖ ≤‖y1‖ = h1. continuing with this bootstrapping argument, we have, for 0 ≤ t ≤ 1, λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · · · fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≤ h1, so that, for 0 ≤ t ≤ 1, ty1(t) ≤ h1. 142 prasad, krushna and sreedhar hence, ‖ty1‖≤ h1 = ‖y1‖. if we set ω1 = {x ∈ x | ‖x‖ < h1}, then (3.4) ‖ty1‖≤‖y1‖, for y1 ∈ p ∩∂ω1. next, from the definitions of fi∞, 1 ≤ i ≤ n, there exists h2 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≥ (fi∞ − �)x, x ≥ h2. choose h2 = max{2h1, 4α−1h2}. let y1 ∈ p and ‖y1‖ = h2. then, min t∈i y1(t) ≥ 1 4α−1 ‖y1‖≥ h2. then, from lemma 2.2 and choice of �, for 0 ≤ sn−1 ≤ 1, we have that λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ≥ λn ∫ s∈i g(1,sn)an(sn)fn(y1(sn))dsn ≥ 1 4α−1 λn ∫ s∈i g(1,sn)an(sn)(fn∞ − �)y1(sn)dsn ≥ 1 4α−1 λn ∫ s∈i g(1,sn)an(sn)dsn(fn∞ − �)‖y1‖ ≥‖y1‖ = h2. it follows in a similar manner from lemma 2.2 and choice of �, for 0 ≤ sn−2 ≤ 1, λn−1 ∫ 1 0 g(sn−2,sn−1)an−1(sn−1) fn−1 ( λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ) dsn−1 ≥ 1 4α−1 λn−1 ∫ s∈i g(1,sn−1)an−1(sn−1)dsn−1(fn−1,∞ − �)‖y1‖ ≥‖y1‖ = h2. again, using a bootstrapping argument, we have λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ h2, so that ty1(t) ≥ h2 = ‖y1‖. hence, ‖ty1‖≥‖y1‖. so if we set ω2 = {x ∈ x | ‖x‖ < h2}, then (3.5) ‖ty1‖≥‖y1‖, for y1 ∈ p ∩∂ω2. applying theorem 2.3 to (3.4) and (3.5), we obtain that t has a fixed point y1 ∈ p ∩ (ω2\ω1). setting y1 = yn+1, we obtain a positive solution (y1,y2, · · ·,yn) of (1.1)-(1.2) given iteratively by yi(t) = λi ∫ 1 0 g(t,s)ai(s)fi(yi+1(s))ds, i = n,n− 1, · · ·, 1. the proof is completed. � eigenvalues for iterative systems 143 prior to our next result, we define the positive numbers n3 and n4 by n3 = max 1≤i≤n {[ 1 4α−1 ∫ s∈i g(1,s)ai(s)dsfi0 ]−1} and n4 = min 1≤i≤n {[∫ 1 0 g(1,s)ai(s)dsfi∞ ]−1} . theorem 3.2. assume that the conditions (a1)-(a3) are satisfied. then, for each λ1,λ2, · · ·,λn satisfying (3.6) n3 < λi < n4, 1 ≤ i ≤ n, there exists an n-tuple (y1,y2, · · ·,yn) satisfying (1.1)-(1.2) such that yi(t) > 0, 1 ≤ i ≤ n on (0, 1). proof. let λi, 1 ≤ i ≤ n be given as in (3.6). now, let � > 0 be chosen such that max 1≤i≤n {[ 1 4α−1 ∫ s∈i g(1,s)ai(s)ds(fi0 − �) ]−1} ≤ min 1≤i≤n λi and max 1≤i≤n λi ≤ min 1≤i≤n {[∫ 1 0 g(1,s)ai(s)ds(fi∞ + �) ]−1} . let t be the cone preserving, completely continuous operator that was defined by (3.1). from the definition of fi0, 1 ≤ i ≤ n there exists h3 > 0 such that, for each 1 ≤ i ≤ n, fi(x) ≥ (fi0 − �)x, 0 < x ≤ h3. also, from the definitions of fi0, it follows that fi0(0) = 0, 1 ≤ i ≤ n, and so there exist 0 < kn < kn−1 < · · · < k2 < h3 such that λifi(t) ≤ ki−1∫ 1 0 g(1,s)ai(s)ds , t ∈ [0,ki], 3 ≤ i ≤ n, and λ2f2(t) ≤ h3∫ 1 0 g(1,s)a2(s)ds , t ∈ [0,k2]. choose y1 ∈ p with ‖y1‖ = kn. then, we have λn ∫ 1 0 g(sn−1,sn)an(sn)fn(y1(sn))dsn ≤ λn ∫ 1 0 g(1,sn)an(sn)fn(y1(sn))dsn ≤ ∫ 1 0 g(1,sn)an(sn)kn−1dsn∫ 1 0 g(1,sn)an(sn)dsn ≤ kn−1. 144 prasad, krushna and sreedhar continuing with this bootstrapping argument, it follows that λ2 ∫ 1 0 g(1,s2)a2(s2)f2 ( λ3 ∫ 1 0 g(s2,s3)a3(s3) · ·· fn(y1(sn))dsn ) · · ·ds3 ) ds2 ≤ h3. then, ty1(t) = λ1 ∫ 1 0 g(t,s1)a1(s1)f1 ( λ2 ∫ 1 0 g(s1,s2)a2(s2) · ·· fn(y1(sn))dsn ) · · ·ds2 ) ds1 ≥ 1 4α−1 λ1 ∫ s∈i g(1,s1)a1(s1)(f10 − �)‖y1‖ds1 ≥‖y1‖. so, ‖ty1‖≥‖y1‖. if we set ω1 = {x ∈ x | ‖x‖ < kn}, then (3.7) ‖ty1‖≥‖y1‖, for y1 ∈ p ∩∂ω1. since each fi∞ is assumed to be a positive real number, it follows that fi, 1 ≤ i ≤ n, is unbounded at ∞. for each 1 ≤ i ≤ n, set f∗i (x) = sup 0≤s≤x fi(s). then, it is straightforward that, for each 1 ≤ i ≤ n, f∗i is a nondecreasing realvalued function, fi ≤ f∗i and lim x→∞ f∗i (x) x = fi∞. next, by definition of fi∞, 1 ≤ i ≤ n, there exists h4 such that, for each 1 ≤ i ≤ n, f∗i (x) ≤ (fi∞ + �)x, x ≥ h4. it follows that there exists h4 = max{2h3,h4} such that, for each 1 ≤ i ≤ n, f∗i (x) ≤ f ∗ i (h4), 0 < x ≤ h4. choose y1 ∈ p with ‖y1‖ = h4. then, using the usual bootstrapping argument, we have ty1(t) = λ1 ∫ 1 0 g(t,s1)a1(s1)f1(λ2 · ··)ds1 ≤ λ1 ∫ 1 0 g(t,s1)a1(s1)f ∗ 1 (λ2 · ··)ds1 ≤ λ1 ∫ 1 0 g(1,s1)a1(s1)f ∗ 1 (h4)ds1 ≤ λ1 ∫ 1 0 g(1,s1)a1(s1)ds1(f1∞ + �)h4 ≤ h4 = ‖y1‖, and so ‖ty1‖≤‖y1‖. so, if we let ω2 = {x ∈ x | ‖x‖ < h4}, then (3.8) ‖ty1‖≤‖y1‖, for y1 ∈ p ∩∂ω2. applying theorem 2.3 to (3.7)-(3.8), we obtain that t has a fixed point y1 ∈ p ∩ (ω2\ω1), which in turn with y1 = yn+1, yields an n-tuple (y1,y2, · · ·,yn) eigenvalues for iterative systems 145 satisfying the bvp (1.1)-(1.2) for the chosen values of λi, 1 ≤ i ≤ n. the proof is thus completed. � 4. example in this section, as an application, we demonstrate our results with an example. consider the fractional order boundary value problem (4.1) d2.50+ y1(t) + λ1 1 + t y2(46 − 27.5e−2y2 )(500 − 487e−3y2 ) = 0, t ∈ (0, 1), d2.50+ y2(t) + λ2 1 + t y3(37 − 25.5e−5y3 )(400 − 368e−y3 ) = 0, t ∈ (0, 1), d2.50+ y3(t) + λ3 1 + t y1(79 − 75e−y1 )(800 − 749.5e−2y1 ) = 0, t ∈ (0, 1),   (4.2) yi(0) = 0, y ′ i(0) = 0 and y ′ i(1) = 0, i = 1, 2, 3. the green’s function g(t,s) of corresponding homogeneous bvp is given by g(t,s) = { t1.5(1−s)0.5 γ(2.5) , 0 ≤ t ≤ s ≤ 1, t1.5(1−s)0.5−(t−s)1.5 γ(2.5) , 0 ≤ s ≤ t ≤ 1. by direct calculations, we found that f10 = 299,f20 = 368,f30 = 202, f1∞ = 23000,f2∞ = 14800,f3∞ = 63200, n1 = max {[ (0.25)1.5 ∫ 0.75 0.25 g(1,s)a1(s)ds(23000) ]−1 , [ (0.25)1.5 ∫ 0.75 0.25 g(1,s)a2(s)ds(14800) ]−1 , [ (0.25)1.5 ∫ 0.75 0.25 g(1,s)a3(s)ds(63200) ]−1 } , = max{0.0009634, 0.0014972, 0.0003506} = 0.0014972. similarly, n2 = min{0.0307737, 0.0250037, 0.0455512} = 0.0250037. applying theorem 3.1, we get an optimal eigenvalue interval 0.0014972355 < λi < 0.0250037, for i = 1, 2, 3 in which the fractional order bvp (4.1)-(4.2) has at least one positive solution. references [1] z. bai and h. lu, positive solutions for boundary value problems of nonlinear fractional differential equations, j. math. anal. appl., 311(2005), 495-505. [2] m. benchohra, j. henderson, s. k. ntoyuas and a. ouahab, existence results for fractional order functional differential equations with infinite delay, j. math. anal. appl., 338(2008), 1340-1350. [3] j. m. davis, j. henderson, k. r. prasad and w. yin, eigenvalue intervals for non-linear right focal problems, appl. anal., 74(2000), 215-231. [4] l. h. erbe and h. wang, on the existence of positive solutions of ordinary differential equations, proc. amer. math. soc., 120(1994), 743-748. [5] d. guo and v. lakshmikantham, nonlinear problems in abstract cones, academic press, orlando, 1988. 146 prasad, krushna and sreedhar [6] j. henderson and s. k. ntouyas, positive solutions for systems of nth order three-point nonlocal boundary value problems, electronic journal of qualitative theory of differential equations, 18(2007), 1-12. [7] j. henderson and s. k. ntouyas, positive solutions for systems of nonlinear boundary value problems, nonlinear studies, 15(2008), 51-60. [8] j. henderson, s. k. ntouyas and i. k. purnaras, positive solutions for systems of generalized three-point nonlinear boundary value problems, comment. math. univ. carolin., 49, 1(2008), 79-91. [9] j. henderson, s. k. ntouyas and i. k. purnaras, positive solutions for systems of second order four-point nonlinear boundary value problems, commu. appl. anal., 12(2008), no.1, 29-40. [10] e. r. kauffman and e. mboumi, positive solutions of a boundary value problem for a nonlinear fractional differential equation, electronic journal of qualitative theory of differential equations, 3(2008), 1-11. [11] r. a. khan, m. rehman and j. henderson, existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, fractional differential calculus, 1(2011), 29-43. [12] a. a. kilbas, h. m. srivasthava and j. j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies, vol. 204, elsevier, amsterdam, 2006. [13] m. a. krasnosel’skii, positive solutions of operator equations, noordhoff, groningen, 1964. [14] i. podulbny, fractional differential equations, academic press, san diego, 1999. [15] k. r. prasad and b. m. b. krushna, multiple positive solutions for a coupled system of riemann-liouville fractional order two-point boundary value problems, nonlinear studies, vol. 20, no.4(2013), 501-511. [16] x. su and s. zhang, solutions to boundary value problems for nonlinear differential equations of fractional order, electronic journal of differential equations, 26(2009), 1-15. [17] s. zhang, existence of solutions for a boundary value problem of fractional order, acta math. sci., 26b(2006), 220-228. 1department of applied mathematics, andhra university, visakhapatnam, 530 003, india 2department of mathematics, mvgr college of engineering, vizianagaram, 535 005, india 3department of mathematics, gitam university, visakhapatnam, 530 045, india ∗corresponding author int. j. anal. appl. (2023), 21:15 functional impulsive fractional differential equations involving the caputo-hadamard derivative and integral boundary conditions aida irguedi1,∗, khadidja nisse1, samira hamani2 1laboratory of operators theory and pde: foundations and applications,department of mathematics, faculty of exact sciences, university of el oued, 39000 el oued, algeria 2laboratoire des mathématiques appliqués et pures, université de mostaganem, b.p. 227, 27000, mostaganem, algeria ∗corresponding author: irguedi-aida@univ-eloued.dz abstract. in this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. our results are based on some fixed point theorems. finally, we provide an example to illustrate the validity of our main results. 1. introduction in this paper, we discuss the existence and uniqueness of solutions to a boundary value problem (bvp for short) for functional impulsive fractional differential equation, in the following form: c hd ry(t) = f (t,yt), t ∈ j = [a,t ],t 6= tk, k = 1, ..,m, (1.1) ∆y |t=tk = ik(y(t − k )), t = tk, k = 1, ..,m, (1.2) ∆y ′ |t=tk = ik(y(t − k )), t = tk, k = 1, ..,m, (1.3) y(t) = φ(t), t ∈ [a−τ,a], y ′(t ) = ∫ t a h(s,y(s))ds. (1.4) where chd r is the caputo-hadamard fractional derivative of order 1 < r ≤ 2, a > 0, f : j ×c([a − τ,a],r) →r, h : j×r→r are given continuous functions, φ ∈ c([a−τ,a],r) and ik, ik ∈ c(r,r), k = 1, 2, ...,m, a = t0 < t1 < ... < tm < tm+1 = t. for any continuous functions y defined on received: jan. 10, 2023. 2020 mathematics subject classification. 26a33; 34a08; 34a37; 47h10 . key words and phrases. fractional differential equations; caputo-hadamard fractional derivative; impulses; integral boundary condition; delay; fixed point theorems. https://doi.org/10.28924/2291-8639-21-2023-15 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-15 2 int. j. anal. appl. (2023), 21:15 j′ = j\{t1, ...,tm}, ∆y|t=tk = y(t + k )−y(t− k ) and ∆y ′|t=tk = y ′(t+ k )−y ′(t− k ) , y(t+ k ), y(t− k ) represent the right and the left limits of y(t) at t = tk , we denote by yt the element of cτ = c([a−τ,a],r), defined by yt(θ) = y(t + θ) ,θ ∈ [a −τ,a], hence yt(·) represents the history of the state from time t −τ up to the present time t. in the last few decades, the analysis of impulsive boundary value problems has developed. it has also been extremely useful in developing various applied mathematical models of real-world processes in engineering and applied sciences. tian and bai [16] discussed some existence results of impulsive boundary value problems involving caputo’s type fractional derivatives. results of existence and uniqueness have been developed using fixed-point theorem. recently, it has been noted that much of the works on this subject are focused on the fractional differential equations of riemann-liouville and caputo types with different conditions such as impulses, time delays, boundary value conditions [1,6–9,14,20,21]. the hadamard fractional derivative, introduced in 1892, [10] is another type of fractional derivative that appears in the literature alongside the riemann-liouville and caputo derivatives. it differs from the previous ones in that it contains an arbitrary logarithm function, further details can be found in [3–5]. next, jarad et al. proposed a caputo-type modification of the hadamard fractional derivative in [15] by the caputo hadamard fractional derivative and implemented the fundamental fractional calculus theorem in the caputo-hadamard. recently, some researchers have focused on impulsive differential equations with hadamard and caputo-hadamard derivatives (see [11–13, 17, 18] and the references therein). the rest of the paper is organized as follows. in section 2, we introduce some notions preliminary and properties on the fractional culcules. in section 3, we give a supporting lemma describing the solutions of the considered problem and discuss the main findings. finally, we give an example to illustrate the obtained results. 2. preliminaries in this section, we introduce notations, definitions and preliminary facts that will be used in the remainder of this paper. by c(j,r) we denote the banach space of all continuous functions from j into r with the norm ‖y‖∞ = sup{|y(t)| : t ∈ j}. also cτ is endowed with the norm ‖φ‖cτ = sup{‖φ(θ)‖ : a−τ ≤ θ ≤ a}. let l1(j,r) as the banach space of lebesgue integrable functions y : j −→r with the norm ‖y‖l1 = ∫ t a |y(t)|dt. int. j. anal. appl. (2023), 21:15 3 the space ac(j,r) is the space of functions y : j → r that are absolutely continuous. let δ = t d dt , and then we set acnδ (j,r) = {y : j −→r, δ n−1y(t) ∈ ac(j,r)}. definition 2.1. [19] the hadamard derivative of fractional order r for a cn−1 function y : [a,t ] →r is defined by hdry(t) = 1 γ(n− r) (t d dt )n ∫ t a ( log t s )n−r−1 y(s) ds s ,n− 1 < r < n,n = [r] + 1. definition 2.2. [19] the hadamard fractional integral of order r for a continuous function y is defined as a function hiry(t) = 1 γ(r) ∫ t a ( log t s )r−1 y(s) ds s , r > 0, provided the integral exists. definition 2.3. [19] for an n−times differentiable function y : [a,t ] →r the caputo type hadamard derivative of fractional order r is defined as c hd ry(t) = 1 γ(n− r) ∫ t a ( log t s )n−r−1 δny(s) ds s , n− 1 < r < n, n = [r] + 1, where δ = t d dt and [r] denotes the integer part of the real number r and log(.) = loge(.). lemma 2.1. [2] let r ∈r+ and n = [r] = 1. if y(t) ∈ acnδ (j,r) then caputo-hadamard fractional differential equation cdray(t) = 0 has a solution y(t) = n−1∑ k=0 ck ( log t a )k , and the following formula holds: hira( c hd r ay)(t) = y(t) + n−1∑ k=0 ck ( log t a )k , where ck ∈r, k = 0, 1, 2, ...,n− 1. 3. existence of solutions in this section, we will establish the existence and uniqueness of solutions for (1.1)-(1.4). ac′(j,r) = { y : j →r, y ∈ ac2δ ((tk,tk+1],r) and there exist y(t + k ) andy(t− k ), k = 1, ...,m, with, y(t− k ) = y(tk) } , with the norm ‖y‖ac′ = sup{‖y(t)‖ : a ≤ t ≤ t}. 4 int. j. anal. appl. (2023), 21:15 let b be set defined by b = {y : (a−τ,t ] →r \ y ∈ ac′(j,r) ∩cτ}, is endowed with the norm ‖y‖b = sup{‖y(t)‖ : t ∈ [a−τ,t ]}. definition 3.1. a function y ∈ b is said to be a solution of the problem (1.1)-(1.4) if y satisfies the equation chd ry(t) = f (t,yt) on j′ and the conditions (1.2)-(1.4). we need the following auxiliary lemma to prove the existence and uniqueness of solutions to the problem(1.1)-(1.4). lemma 3.1. let 1 < r ≤ 2. assume that σ,% ∈ ac2(j,r), then the following bvp : h cd ry(t) = σ(t), t ∈ j = [a,t ],t 6= tk, (3.1) ∆y |t=tk = ik(y(t − k )), t = tk, k = 1, ..,m, (3.2) ∆y ′ |t=tk = ik(y(t − k )) , t = tk, k = 1, ..,m, (3.3) y(a) = y, y ′(t ) = ∫ t a %(s)ds, (3.4) has the following integral equation: y(t) =   y + c2 ( log t a ) + 1 γ(r) ∫ t a ( log t s )r−1 σ(s)ds s , if t ∈ [a,t1] y + c2 ( log t a ) + 1 γ(r) ∫ t tk ( log t s )r−1 σ(s)ds s + 1 γ(r) ∑k i=1 ∫ ti ti−1 ( log ti s )r−1 σ(s)ds s + ∑k i=1 ( log t ti ) γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s)ds s + ∑k i=1 ii (y(t − i )) + ∑k i=1 ti ( log t ti ) īi (y(t − i )), if t ∈ (tk,tk+1], k = 1, ...,m, where c2 = t ∫t a %(s)ds− [ 1 γ(r−1) ∫t tm ( log t s )r−2 σ(s) ds s + ∑m i=1 1 γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + ∑m i=1(ti )īi (y(t − i ) ] int. j. anal. appl. (2023), 21:15 5 proof: let y be the solution of (3.1)-(3.4). for t ∈ [a,t1]. using lemma 2.1, for some constants c1,c2 ∈r, we have y(t) = 1 γ(r) ∫ t a ( log t s )r−1 σ(s) ds s + c1 + c2 ( log t a ) acccording the a condition y(a) = y, we deduce that c1 = y and thus y(t) = y + 1 γ(r) ∫ t a ( log t s )r−1 σ(s) ds s + c2 ( log t a ) , y ′(t) = 1 tγ(r − 1) ∫ t a ( log t s )r−2 σ(s) ds s + c2 t . if t ∈ (t1,t2], then we have y(t) = 1 γ(r) ∫ t t1 ( log t s )r−1 σ(s) ds s + d1 + d2 ( log t t1 ) , and y ′(t) = 1 tγ(r − 1) ∫ t t1 ( log t s )r−2 σ(s) ds s + d2 t . using the impulses conditions ∆y |t=t1 = y(t + 1 )−y(t − 1 ) = i1(y(t − 1 )) and ∆y ′ |t=t1 = y ′(t+1 )−y ′(t−1 ) = ī1(y(t − 1 )), we obtain d1 = i1(y(t − 1 )) + y + c2 ( log t1 a ) + 1 γ(r) ∫ t1 a ( log t1 s )r−1 σ(s) ds s . d2 = t1ī1(y(t − 1 )) + c2 + 1 γ(r − 1) ∫ t1 a ( log t1 s )r−2 σ(s) ds s . thus, for t ∈ (t1,t2] we have y(t) = y + 1 γ(r) ∫ t t1 ( log t s )r−1 σ(s) ds s + 1 γ(r) ∫ t1 a ( log t1 s )r−1 σ(s) ds s + ( log t t1 ) γ(r − 1) ∫ t1 a ( log t1 s )r−2 σ(s) ds s + t1 ( log t t1 ) ī1(y(t − 1 )) + i1(y(t − 1 )) + c2 ( log t a ) . continuinq in the same manner, we obtain for t ∈ (tm,t ], y(t) = y + 1 γ(r) ∫ t tm ( log t s )r−1 σ(s) ds s + 1 γ(r) m∑ i=1 ∫ ti ti−1 ( log ti s )r−1 σ(s) ds s + m∑ i=1 ii (y(t − i )) + m∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ti ( log t ti ) īi (y(t − i )) + c2 ( log t a ) , and y ′(t) = c2 t + 1 tγ(r − 1) ∫ t tm ( log t s )r−2 σ(s) ds s + m∑ i=1 1 tγ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ( ti t ) īi (y(t − i ). 6 int. j. anal. appl. (2023), 21:15 by the application of the boundary condition y ′(t ) = ∫t a %(s)ds, we have y ′(t ) = c2 t + 1 t γ(r − 1) ∫ t tm ( log t s )r−2 σ(s) ds s + m∑ i=1 1 t γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + m∑ i=1 ( ti t ) īi (y(t − i )). we obtain the reguired value of the constant c2, where c2 = t ∫t a %(s)ds− [ 1 γ(r−1) ∫t tm ( log t s )r−2 σ(s) ds s + ∑m i=1 1 γ(r−1) ∫ ti ti−1 ( log ti s )r−2 σ(s) ds s + ∑m i=1(ti )īi (y(t − i t)) ] . this completes the proof. our first result is based on the uniqueness of solutions for problem (1.1)-(1.4) and relies on the banach fixed point theorem. theorem 3.1. assume that : (h1) there exists a constant l1 > 0 such that |f (t,u) − f (t,v)| ≤ l1‖u −v‖cτ , for each t ∈ j and u,v ∈ cτ. (h2) there exists a constants l2 > 0 such that |h(t,x) −h(t,y)| ≤ l2|x −y|, for each t ∈ j and x,y ∈r. (h3) for each k = 1, 2, ...,m, there exist l, l∗ > 0 such that |ik(x) − ik(y)| ≤ l|x −y|, |ik(x) − ik(y)| ≤ l∗|x −y|, for each x,y ∈r. if the condition[ l1 ( m + 1 γ(r + 1) + 1 + 2m γ(r) )( log t a )r + l2t (t −a) ( log t a ) + ml + 2ml∗t ( log t a )] < 1, (3.5) then the boundary value problem (1.1)-(1.4) has a unique solution on [a−τ,t ]. proof : transform the problem (1.1)-(1.4) into a fixed point problem. consider the operator f : b → b defined by: (fy)(t) =   φ(t), if t ∈ (a−τ,a] φ(a) + t ( log t a )∫t a h(s,y(s))ds − ( log t a ) [ 1 γ(r−1) ∫t tm ( log t s )r−2 f (s,ys) ds s + 1 γ(r−1) ∑m i=1 ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + ∑m i=1 ti īi (y(t − i ))] + 1 γ(r) ∫ t tk ( log t s )r−1 f (s,ys) ds s + 1 γ(r) ∑k i=1 ∫ ti ti−1 ( log ti s )r−1 f (s,ys) ds s + ∑k i=1 ( log t ti ) γ(r−1) ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + ∑k i=1 ii (y(t − i )) + ∑k i=1 ti ( log t ti ) īi (y(t − i )), if t ∈ [tk,tk+1], k = 1, ...,m. (3.6) int. j. anal. appl. (2023), 21:15 7 clearly, the fixed point of the operator f are solutions of problem (1.1)-(1.4). let x,y ∈ b, if t ∈ [a−τ,a], we have |(fx)(t) − (fy)(t)| = |φ(t) −φ(t)| = 0 if t ∈ [a,t ], by (h1)-(h3), we have: |(fx)(t) − (fy)(t)| ≤ t | ( log t a ) | ∫ t a |h(s,x(s)) −h(s,y(s))|ds + | ( log t a ) | γ(r − 1) ∫ t tm ( log t s )r−2 |f (s,xs ) − f (s,ys )| ds s + | ( log t a ) | γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,xs ) − f (s,ys )| ds s +| ( log t a ) | m∑ i=1 ti|īi (x(t−i )) − īi (y(t − i ))| + 1 γ(r) ∫ t tk ( log t s )r−1 |f (s,xs ) − f (s,ys )| ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,xs ) − f (s,ys )| ds s + k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,xs ) − f (s,ys )| ds s + k∑ i=1 |ii (x(t−i )) − ii (y(t − i ))| + k∑ i=1 ti ( log t ti ) |īi (x(t−i )) − īi (y(t − i ))| ≤ t ( log t a )∫ t a l2|x(s) −y(s)|ds + | ( log t a ) | γ(r − 1) ∫ t tm ( log t s )r−2 l1‖xs −ys‖cτ ds s + | ( log t a ) | γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 l1‖xs −ys‖cτ ds s + | ( log t a ) | m∑ i=1 ti l ∗|x(t−i ) −y(t − i )| + 1 γ(r) ∫ t tk ( log t s )r−1 l1‖xs −ys‖cτ ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 l1‖xs −ys‖cτ ds s + k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 l1‖xs −ys‖cτ ds s + k∑ i=1 l|x(t−i ) −y(t − i )| + k∑ i=1 ti ( log t ti ) l ∗|x(t−i ) −y(t − i )| ≤ t ( log t a ) (t −a)l2‖x −y‖ + ( log t a ) [ l1‖x −y‖ γ(r − 1) ∫ t tm ( log t s )r−2 ds s + l1‖x −y‖ γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 ds s + mtl ∗‖x −y‖] + l1‖x −y‖ γ(r) ∫ t tk ( log t s )r−1 ds s + l1‖x −y‖ γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 ds s + k∑ i=1 l1‖x −y‖ ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 ds s + ml‖x −y‖ + mt ( log t a ) l ∗‖x −y‖ ≤ [ l2(t −a)t ( log t a ) + l1 ( m + 1 γ(r + 1) + 1 + 2m γ(r) )( log t a )r + ml + 2mt ( log t a ) l ∗ ] ‖x −y‖ thus, we have ‖fx −fy‖≤ [ l1 ( m + 1 γ(r + 1) + 1 + 2m γ(r) )( log t a )r + l2t (t −a) ( log t a ) + ml + 2ml ∗ t ( log t a )] ‖x −y‖. 8 int. j. anal. appl. (2023), 21:15 consequently by (3.5), f is a contraction, as a conseguence of banach fixed point theorem, we deduce that f has a fixed point which is a solution of the problem (1.1)-(1.4). this completes the proof. our second result deals with the existence of solutions for problem (1.1)-(1.4) by applying on scheafer fixed point theorem. theorem 3.2. assume that the following conditions hold : (h4) the function f : j ×cτ →r is continuous. (h5) the function h : r→r is continuous. (h6) the functions ik, ik : r→r are continuous. (h7) there exists a constant n > 0 such that |f (t,y)| ≤ n, for each t ∈ j and y ∈ cτ. (h8) there exists a constant n∗ > 0 such that |h(t,x)| ≤ n∗ for each x ∈r. (h9) there exist two constants n1 > 0, n2 > 0 such that |ik(x)| ≤ n1, |ik(x)| ≤ n2 for each , x ∈r, k = 1, . . . ,m, then the boundary value problem (1.1)-(1.4) has at least one solution on [a−τ,t ], proof: we shall use scheafer fixed point theorem to prove that f has a fixed point, defined by 3.6. the proof will be given in several steps. step 1: f is continuous. let {yn} be a sequence such that yn → y in b. if t ∈ [a,t ], we have |f (yn)(t) −f (y)(t)| ≤ t ( log t a )∫ t a |h(s,yn(s)) −h(s,y(s))|ds + ( log t a ) γ(r − 1) ∫ t tm ( log t s )r−2 |f (s,yns) − f (s,ys))| ds s + ( log t a ) γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,yns) − f (s,ys))| ds s + ( log t a ) m∑ i=1 ti|ī(yn(t−i )) − īi (y(t − i ))| + 1 γ(r) ∫ t tk ( log t s )r−1 |f (s,yns) − f (s,ys)| ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,yns) − f (s,ys)| ds s + k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,yns) − f (s,ys)| ds s + k∑ i=1 |ii (yn(t−i )) − ii (y(t − i ))| + k∑ i=1 ti ( log t ti ) |ī(yn(t−i )) − īi (y(t − i ))| int. j. anal. appl. (2023), 21:15 9 since f , h and ik, īk, k = 1, . . . ,m, are continuous functions, we have ‖f (yn) −f (y)‖∞ → 0 as n →∞. step 2: f maps bounded sets into bounded sets in b. indeed, it is enough to show that for any η∗ > 0, there exists a positive constant l such that for each y ∈ dη∗ = {y ∈ b : ‖y‖ ≤ η∗}, we have ‖f (y)‖ ≤ l. by (h7), (h8) and (h9), for each t ∈ j, we can obtain |f (y)(t)| ≤ |ϕ(a)| + t ( log t a )∫ t a |h(s,y(s))|ds + | ( log t a ) | γ(r − 1) ∫ t tm ( log t s )r−2 |f (s,ys)| ds s + | ( log t a ) | γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + | ( log t a ) | m∑ i=1 ti|īi (y(t−i ))| + 1 γ(r) ∫ t tk ( log t s )r−1 |f (s,ys)| ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,ys)| ds s + k∑ i=1 | ( log t ti ) | γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + k∑ i=1 |ii (y(t−i ))| + k∑ i=1 ti| ( log t ti ) ||īi (y(t−k ))| ≤ |ϕ(a)| + n∗t ( log t a )∫ t a ds + n| ( log t a ) | γ(r − 1) ∫ t tm ( log t s )r−2 ds s + n| ( log t a ) | γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 ds s + m ( log t a ) tn2 + n γ(r) ∫ t tk ( log t s )r−1 ds s + n γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 ds s + k∑ i=1 n| ( log t ti ) | γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 ds s + mn1 + mt (log t a )n2 ≤ ‖ϕ‖ + (t −a)t ( log t a ) n∗ + n (1 + 2m) ( log t a )r γ(r) + n (1 + m) ( log t a )r γ(r + 1) +mn1 + 2mt ( log t a ) n2. therefore ‖fy‖ ≤ ‖ϕ‖ + n [ 1 + m γ (r + 1) + 1 + 2m γ(r) ]( log t a )r + mn1 + [(t −a)n∗ + 2mn2] t ( log t a ) := l. step 3: f maps bounded sets into equicontinuous sets of b . let τ1,τ2 ∈ j,τ1 < τ2, dη∗ be a bounded set of b as in step 2, and let y ∈ dη∗. then 10 int. j. anal. appl. (2023), 21:15 |f (y)(τ2) −f (y)(τ1)| ≤ t ( log τ2 τ1 )∫ t a |h(s,y(s))|ds + (log τ2 τ1 ) γ(r − 1) ∫ t tm ( log t s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) γ(r − 1) m∑ i=i ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) m∑ i=1 ti|īi (y(t−i ))| + 1 γ(r) ∫ τ1 tk [( log τ2 s )r−1 − ( log τ1 s )r−1] |f (s,ys)| ds s + 1 γ(r) ∫ τ2 τ1 ( log τ2 s )r−1 |f (s,ys)| ds s + k∑ i=1 ( log τ2 τ1 ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys)| ds s + ( log τ2 τ1 ) k∑ i=1 ti|īi (y(t−i ))|. as τ1 −→ τ2, the right-hand side of the above inequality tends to zero. as a consequence of steps 1 to 3, together with the arzela-ascoli theorem, we can conclude that f : b → b is completly continuous. step 4: a priori bounds. now it remains to show that the set ε = {y ∈ b → b : y = λf (y) f or some 0 < λ < 1} is bounded. let y ∈ ε, then y = λf (y) for some 0 < λ < 1. thus, for each t ∈ j we have (fy)(t) = λϕ(a) + λt ( log t a )∫ t a h(s,y(s))ds − λ ( log t a ) γ(r − 1) ∫ t tm ( log t s )r−2 f (s,ys) ds s − λ ( log t a ) γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s −λ ( log t a ) m∑ i=1 ti īi (y(t − i )) + λ γ(r) ∫ t tk ( log t s )r−1 f (s,ys) ds s + λ γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 f (s,ys) ds s +λ k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 f (s,ys) ds s + λ k∑ i=1 ii (y(t − i )) +λ k∑ i=1 ti ( log t ti ) īi (y(t − i )) for each t ∈ j, by (h7)-(h9), we have ‖fy‖ ≤ ‖ϕ‖ + t (t −a) ( log t a ) n∗ + n [ 1 + m γ(r + 1) + 1 + 2m γ(r) ]( log t a )r + mn1 + 2mt ( log t a ) n2. this shows that the set ε is bounded. as a consequence of schaefer’s fixed point theorem, we deduce that f has a fixed point which is a solution of the problem (1.1)-(1.4). by applying the of leray-schauder nonlinear alternative type. theorem 3.3. assume that (h4)-(h6) and the following conditions hold : int. j. anal. appl. (2023), 21:15 11 (h10) there exist φf ∈ c(j,r+) and ψ : [0,∞) → [0,∞) continuous and non-decreasing such that |f (t,u)| ≤ φf (t)ψ(|u|), f or all t ∈ j, u ∈ cτ. (h11) there exist φh ∈ l(j,r+) and ψ∗ : [0,∞) → [0,∞) continuous and non-decreasing such that |h(t,v)| ≤ φh(t)ψ∗(|v|), f or all t ∈ j, v ∈r. (h12) there exist ψ̄∗, ψ̄∗∗ : [0,∞) → [0,∞) continuous and non-decreasing such that |ik(v)| ≤ ψ̄∗(|v|), |īk(v)| ≤ ψ̄∗∗(|v|), f or all v ∈r, k = 1, . . . ,m. (h13) there exists a number m > 0 such that m ‖ϕ‖ + t ( log t a ) ψ∗(m)‖φh‖l1 + φψ(m) ( 1+m γ(r+1) + 1+2m γ(r) )( log t a )r + mψ̄∗(m) + 2mt ( log t a ) ψ̄∗∗(m) ≥ 1, where φ = sup{φf (t) : t ∈ j}, then the problem(1.1)-(1.4) has at least one solution on [a−τ,t ]. proof: consider the operator f defined as in 3.6. it can be easily shown that f is continuous and completely continuous. for λ ∈ [0, 1] and each t ∈ j, let y(t) = λ(fy)(t). then from(h10)-(h12), we have |(fy)(t)| ≤ |ϕ(a)| + t ( log t a )∫ t a |h(s,y(s))|ds + ( log t a ) γ(r − 1) ∫ t tm ( log t s )r−2 |f (s,ys )| ds s + ( log t a ) γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 |f (s,ys )| ds s + ( log t a ) m∑ i=1 ti|īi (y(t−i ))| + 1 γ(r) ∫ t tk ( log t s )r−1 |f (s,ys )| ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 |f (s,ys )| ds s + k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 |f (s,ys )| ds s + k∑ i=1 |ii (y(t−i ))| + k∑ i=1 ti ( log t ti ) |īi (y(t−i ))| ≤ |ϕ(a)| + t ( log t a )∫ t a φh(s)ψ ∗ (|y(s)|)ds + ( log t a ) γ(r − 1) ∫ t tm ( log t s )r−2 φf (s)ψ(|ys|) ds s + ( log t a ) γ(r − 1) m∑ i=1 ∫ ti ti−1 ( log ti s )r−2 φf (s)ψ(|ys|) ds s + ( log t a ) m∑ i=1 tiψ̄∗∗(|y(t−i )|) + 1 γ(r) ∫ t tk ( log t s )r−1 φf (s)ψ(|ys|) ds s + 1 γ(r) k∑ i=1 ∫ ti ti−1 ( log ti s )r−1 φf (s)ψ(|ys|) ds s + k∑ i=1 ( log t ti ) γ(r − 1) ∫ ti ti−1 ( log ti s )r−2 φf (s)ψ(|ys|) ds s + k∑ i=1 ψ̄∗(|y(t−i )|) + k∑ i=1 ti ( log t ti ) ψ̄∗∗(|y(t−i )|) ≤ ‖ϕ‖ + t ( log t a ) ψ ∗ (‖y‖) ∫ t a φh(s)ds + ( log t a )r γ(r) φψ(‖y‖) + m ( log t a )r γ(r) φψ(‖y‖) 12 int. j. anal. appl. (2023), 21:15 +mt ( log t a ) ψ̄∗∗(‖y‖) + ( log t a )r γ(r + 1) φψ(‖y‖) + m ( log t a )r γ(r + 1) φψ(‖y‖) + m ( log t a )r γ(r) φψ(‖y‖) + mψ̄∗(‖y‖) + mt ( log t a ) ψ̄∗∗(‖y‖) ≤‖ϕ‖ + t ( log t a ) ψ ∗ (‖y‖)‖φh‖l1 + φψ(‖y‖) ( (1 + m) γ(r + 1) + (1 + 2m) γ(r) )( log t a )r +mψ̄∗(‖y‖) + 2mt ( log t a ) ψ̄∗∗(‖y‖) thus ‖y‖ ‖ϕ‖ + t ( log t a ) ψ∗(‖y‖)‖φh‖l1 + φψ(‖y‖) ( 1+m γ(r+1) + 1+2m γ(r) )( log t a )r + mψ̄∗(‖y‖) + 2mt ( log t a ) ψ̄∗∗(‖y‖) ≤ 1. then by condition (h13), there exists m such that ‖y‖ 6= m. let u = {y ∈ b : ‖y‖≤ m}. the operator f : u → b is continuous and completely continuous. from the choice of u, there is no y ∈ ∂u such that y = λf (y) for some λ ∈ (0, 1). as a consequence of the nonlinear alternative of leray-schauder type, we deduce that f has a fixed point y ∈ u which is a solution of the problem (1.1)-(1.4). this completes the proof. 4. example let consider the following problem: c hd 3 2 y(t) = et (et + 5)2 |yt| (1 + |yt|) , t ∈ [1, 2],t 6= 4 3 , (4.1) ∆y( 4 3 ) = |y( 4 3 − )| 15 + |y( 4 3 − )| , (4.2) ∆y ′( 4 3 ) = |y( 4 3 − )| 17 + |y( 4 3 − )| , (4.3) y(t) = φ(t), t ∈ [1 −τ, 1], y ′(2) = ∫ 2 1 |y(s)| 13 + |y(s)| ds. (4.4) set f (t,yt) = et (et + 5)2 |yt| (1 + |yt|) , (t,y) ∈ j ×c([1 −τ, 1],r), h(t,y(t)) = ∫ 2 1 |y(s)| 13 + |y(s)| ds, (t,y) ∈ j ×r, i(y) = |y| 15 + |y| , i(y) = |y| 17 + |y| , y ∈r. int. j. anal. appl. (2023), 21:15 13 hence the hypotheses (h1)-(h3) holds, with l1 = 1 36 ,l2 = 1 13 , l = 1 15 , l∗ = 1 17 . we shall check that condition (3.5). with r = 3 2 ,m = 1,t1 = 4 3 ,t = 2,a = 1. further [ l2t (t −a) ( log t a ) + l1 ( m + 1 γ(r + 1) + 1 + 2m γ(r) )( log t a )r + ml + 2ml∗t ( log t a )] = [ 2 13 (log 2) + 1 36 ( 2 γ ( 5 2 ) + 3 γ( 3 2 ) ) (log 2) 3 2 + ml + 4 17 (log 2) ] = 0.414779517 < 1. note that γ( 3 2 ) = 1 2 √ π, γ( 5 2 ) = 3 4 √ π. then all hypotheses of theorem (3.1) are fulfilled, and consequently the boundary value problem (4.1)-(4.4) has a unique solution on [1, 2]. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] r.p. agarwal, m. benchohra, s. hamani, a survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, acta appl. math. 109 (2008), 973–1033. https://doi.org/10. 1007/s10440-008-9356-6. [2] y. adjabi, f. jarad, d. baleanu, t. abdeljawad, on cauchy problems with caputo hadamard fractional derivatives, j. comput. anal. appl. 21 (2016), 661-681. http://hdl.handle.net/20.500.12416/1783. [3] p.l. butzer, a.a. kilbas, j.j. trujillo, compositions of hadamard-type fractional integration operators and the semigroup property, j. math. anal. appl. 269 (2002), 387–400. https://doi.org/10.1016/s0022-247x(02) 00049-5. [4] p.l. butzer, a.a. kilbas, j.j. trujillo, mellin transform analysis and integration by parts for hadamard-type fractional integrals, j. math. anal. appl. 270 (2002), 1–15. https://doi.org/10.1016/s0022-247x(02)00066-5. [5] p.l. butzer, a.a. kilbas, j.j. trujillo, fractional calculus in the mellin setting and hadamard-type fractional integrals, j. math. anal. appl. 269 (2002), 1–27. https://doi.org/10.1016/s0022-247x(02)00001-x. [6] z. cui, p. yu, z. mao, existence of solutions for nonlocal boundary value problems of nonlinear fractional differential equations, adv. dyn. syst. appl. 7 (2012), 31-40. [7] y.k. chang, a. anguraj, m. mallika arjunan, existence results for impulsive neutral functional differential equations with infinite delay, nonlinear anal.: hybrid syst. 2 (2008), 209–218. https://doi.org/10.1016/j.nahs. 2007.10.001. [8] j. cao, y. luo, g. liu, some results for impulsive fractional differential inclusions with infinite delay and sectorial operators in banach spaces, appl. math. comput. 273 (2016), 237–257. https://doi.org/10.1016/ j.amc.2015.09.072. [9] j. dabas, a. chauhan, existence and uniqueness of mild solution for an impulsive neutral fractional integrodifferential equation with infinite delay, math. computer model. 57 (2013), 754–763. https://doi.org/10. 1016/j.mcm.2012.09.001. [10] j. hadamard, essai sur l’etude des fonctions donnees par leur developpement de taylor, j. math. pure appl. 8 (1892), 101-186. https://doi.org/10.1007/s10440-008-9356-6 https://doi.org/10.1007/s10440-008-9356-6 http://hdl.handle.net/20.500.12416/1783 https://doi.org/10.1016/s0022-247x(02)00049-5 https://doi.org/10.1016/s0022-247x(02)00049-5 https://doi.org/10.1016/s0022-247x(02)00066-5 https://doi.org/10.1016/j.nahs.2007.10.001 https://doi.org/10.1016/j.nahs.2007.10.001 https://doi.org/10.1016/j.amc.2015.09.072 https://doi.org/10.1016/j.amc.2015.09.072 https://doi.org/10.1016/j.mcm.2012.09.001 https://doi.org/10.1016/j.mcm.2012.09.001 14 int. j. anal. appl. (2023), 21:15 [11] s. hamani, a. hammou, j. henderson, impulsive fractional differential equations involving the hadamard fractional derivative, commun. appl. nonlinear anal. 24 (2017), 48-58. [12] a. hammou, s. hamani, j. henderson, impulsive hadamard fractional differential equations in banach spaces, commun. appl. nonlinear anal. 28 (2018), 52-62. [13] a. hammou, s. hamani, j. henderson initial value problems for impulsive caputo-hadamard fractional differential inclusions, commun. appl. nonlinear anal. 22 (2019), 17-35. [14] s. heidarkhani, a. salari, g. caristi, infinitely many solutions for impulsive nonlinear fractional boundary value problems, adv. differ. equ. 2016 (2016), 196. https://doi.org/10.1186/s13662-016-0919-y. [15] f. jarad, t. abdeljawad, d. baleanu, caputo-type modification of the hadamard fractional derivatives, adv. differ. equ. 2012 (2012), 142. https://doi.org/10.1186/1687-1847-2012-142. [16] y. tian, z. bai, impulsive boundary value problem for differential equations with fractional order, differ equ. dyn. syst. 21 (2012), 253–260. https://doi.org/10.1007/s12591-012-0150-6. [17] p. thiramanus, s.k. ntouyas, j. tariboon, existence and uniqueness results for hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, abstr. appl. anal. 2014 (2014), 902054. https://doi.org/10.1155/2014/902054. [18] a. nain, r. vats, a. kumar, caputo-hadamard fractional differential equation with impulsive boundary conditions, j. math. model. 9 (2021), 93-106. https://doi.org/10.22124/jmm.2020.16449.1447. [19] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland mathematics studies, elsevier, amsterdam, the netherlands, 2006. [20] s. abbas, m. benchohra, j.r. graef, j. henderson, implicit fractional differential and integral equations: existence and stability, de gruyter, berlin, 2018. [21] s. abbas, m. benchohra, g.m. n’guérékata, topics in fractional differential equations, springer, new york, 2012. https://doi.org/10.1007/978-1-4614-4036-9. https://doi.org/10.1186/s13662-016-0919-y https://doi.org/10.1186/1687-1847-2012-142 https://doi.org/10.1007/s12591-012-0150-6 https://doi.org/10.22124/jmm.2020.16449.1447 https://doi.org/10.1007/978-1-4614-4036-9 1. introduction 2. preliminaries 3. existence of solutions 4. example references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 98-117 http://www.etamaths.com mixed problem with an integral two-space-variables condition for a third order parabolic equation oussaeif taki eddine∗ and bouziani abdelfatah abstract. this paper is concerned with the existence and uniqueness of a strong solution to a mixed problem which combine dirichlet and integral two space variables conditions for a third order linear parabolic equation. the proof uses a functional analysis method presented, which it is based on an energy inequality and the density of the range of the operator generated by the problem. 1. introduction the importance of the problems with integral conditions has been pointed out by samarskii [25].we remark that integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow, plasma physics and population dynamics. problems which combine local and integral condition for second order parabolic equations are investigated by the potentail method by cannon [10] and kamynin [19], by fourier’s method by ionkin [15] and by the energy inequality method in [22] and [2] other works for mixed problems which combine local and integral conditions for second order parabolic equations were treated by batten [23], cannon-esteva-van der hoek [11], cannon-van der hoek [12], [13], cahlon-kulkarni-shi [9] and shi [21]. recently, problems of this type that have non-linearity in the boundary conditions have been investigated in jones et al. [16] and jumahron-mckee [17], [18]. mixed problems with only integral conditions for a second order parabolic equation have been studied by bouziani-benouar [7], and for a 2m-parabolic equation in bouziani [5]. mixed problems with integral conditions for a third order parabolic equation have been studied by bouziani-benouar [1].the present paper is devoted to study the existence and the uniqueness for a strong solution of mixed problems with new integral conditions for a third order parabolic equation. 2. formulation of the problem in the rectangle ω = (0, 1) × (0,t), with t < ∞, we consider the third order linear parabolic equation: (1) lu = ∂u ∂t − ∂2 ∂x2 ( a(x,t) ∂u ∂x ) = f(x,t). which can be considered as a generalization on the linearized kortweg-de vries equation, see for instance [24]. condition 1. the coefficient a(x,t) is a real-valued function belonging to c2 ( ω ) such that c0 ≤ a(x,t) ≤ c1, ∂a(x,t) ∂t ≤ c2. in condition 1 and in the rest of the paper, ci, i = 1, ..., 6, denote strictly positive constants. we adjoin to (2.1) the initial condition (2) `u = u(x, 0) = φ (x) , x ∈ (0, 1) , 2010 mathematics subject classification. 35b45, 35d35, 35b30, 35k25, 35k35. key words and phrases. strong solution; integral condition; mixed problem. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 98 mixed problem with integral conditions 99 with the conditions ∂u ∂x ∣∣∣∣ x=i = 0, for i = {0,α,β, 1} , ∂2u ∂x2 ∣∣∣∣ x=0 = 0, ∂2u ∂x2 ∣∣∣∣ x=1 = 0.(3) and with integral conditions (4) ∫ α 0 u (x,t) dx + ∫ 1 β u (x,t) dx = 0 t ∈ (0,t) , (5) ∫ α 0 xu (x,t) dx + ∫ 1 β xu (x,t) dx = 0 t ∈ (0,t) , where φ is a known function and 0 < α < β < 1, α + β = 1. condition 2. we shall assume that the function φ satisfies a compatibility conditions with (2.3) − (2.5) . problem (2.1) − (2.5) arises, for instance, from the heat transfer theory. in this case, u is a temperature of a slab 0 < x < 1, and the integrals in the conditions (2.4) and (2.5) are considered as the average and the weighted average temperature. in this paper, we prove the existence and the uniqueness for a strong solution of the problem (2.1) − (2.5) as a solution of the operator equation (6) lu = f where l = (l,`), with domain of difinition d(l) consisting of functions u ∈ l2 (ω) such that ∂u ∂t , ∂u ∂x , ∂2u ∂x2 , ∂ 3u ∂x3 , ∂ 3u ∂t∂x2 ∈ l2 (ω) and u satisfies conditions (2.3) − (2.5) ; the operator l is considered from b to f, where b is the banach space consisting of all functions u(x,t) having a finite norm ‖u‖2b = ∫ t 0 ∫ α 0 (∫ α x ∂u ∂t dξ )2 dxdt + ∫ t 0 ∫ 1 β (∫ x β ∂u ∂t dξ )2 dxdt + sup 0≤τ≤t (∫ α 0 (5 −x) ( ∂u (x,τ) ∂x )2 dx + ∫ 1 β ( 5 4 −x) ( ∂u (x,τ) ∂x )2 dx + ∫ β α (β −α) ( ∂u (x,τ) ∂x )2 dx ) and satisfying the conditions (2.3) − (2.5), and f is the hilbert space consisting of all elements f = (f,φ) for which the norm ‖f‖2f = ∫ ω f2dxdt + ∫ 1 0 ( ∂φ ∂x )2 dx is finite. then, we establish an energetic inequality: (7) ‖u‖b ≤ c‖lu‖f and we show that the operator l has a closure l. definition. a solution of the operator equation lu = f is called a strong solution of the problem (2.1) − (2.5). since points of the graph l are limits of sequences of points of the graph of l, we can extend (2.7) to apply to strong solution by taking limits, i.e., ‖u‖b ≤ c ∥∥lu∥∥ f , ∀u ∈ d(l). from this inequality we obtain the uniqueness of a strong solution if it exists, and the the equality of sets r(l) and r(l). thus , proving that the set r(l) is dense in f. 100 eddine and abdelfatah 3. an energety inequality and its consequences theorem 1. let condition 1 be fulfilled. then for any function u ∈ d(l) we have the inequality (8) ‖u‖b ≤ c‖lu‖f where c is a positive constant independent of u. proof. multiplying the equation (2.1) by the following mu : mu =   mu1 = 4 ∫α x ∂u ∂t dξ − ∫α x (∫α ξ ∂u ∂t dη − (1 − ξ)∂u ∂t ) dξ 0 ≤ x ≤ α mu2 = (x−α) ∫β x ∂u ∂t dξ + (β −x) ∫x α ∂u ∂t dξ α ≤ x ≤ β mu3 = −14 ∫x β ∂u ∂t dξ − ∫x β (∫ ξ β ∂u ∂t dη + (1 − ξ)∂u ∂t ) dξ β ≤ x ≤ 1 and integrating over ωτ,where ωτ = (0, 1) × (0,τ), 1) on the interval [0,α] , we denote ωτα = ωα = (0,α) × (0,τ), we get ∫ ωα lu.mu1dxdt(9) = ∫ ωα ∂u ∂t . ( 4 ∫ α x ∂u ∂t dξ − ∫ α x (∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ) dξ ) dxdt − ∫ ωα ∂2 ∂x2 ( a(x,t) ∂u ∂x )( 4 ∫ α x ∂u ∂t dξ ) dxdt + ∫ ωα ∂2 ∂x2 ( a(x,t) ∂u ∂x )(∫ α x (∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ) dξ ) dxdt = ∫ ωα f.mu1dxdt. integration by parts each term of (3.2) with use the conditions (2.2) − (2.5), we obtain∫ ωα ∂u ∂t . ( 4 ∫ α x ∂u ∂t dξ − ∫ α x (∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ) dξ ) dxdt(10) = 5 2 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt + 3 2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt −2 ∫ t 0 ((∫ α 0 ∂u ∂t dx )(∫ α 0 x ∂u ∂t dx ) dt ) , − ∫ ωα ∂2 ∂x2 ( a(x,t) ∂u ∂x )( 4 ∫ α x ∂u ∂t dξ − ∫ α x (∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ) dξ ) (11) = ∫ ωα (5 −x)a(x,t) ∂u ∂x ∂2u ∂x∂t dxdt− ∫ t 0 ∂ ∂x ( a(x,t) ∂u ∂x )( 4 ∫ α x ∂u ∂t dξ )∣∣∣∣x=α x=0 dt + ∫ t 0 ∂ ∂x ( a(x,t) ∂u ∂x )(∫ α x [∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ])∣∣∣∣x=α x=0 dt = − 1 2 ∫ ωα (5 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ α 0 (5 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ α 0 (5 −x)a(x,τ) ( ∂u ∂x )2 dx. mixed problem with integral conditions 101∫ ωα f.mu1dxdt = ∫ ωα f. ( 4 ∫ α x ∂u ∂t dξ − ∫ α x (∫ α ξ ∂u ∂t dη − (1 − ξ) ∂u ∂t ) dξ ) dxdt = ∫ ωα f. ( 4 ∫ α x ∂u ∂t ) dxdt + ∫ ωα f. ( (1 − ξ) ∫ α x ∂u ∂t ) dxdt −2 ∫ ωα f. (∫ α x ∫ α ζ ∂u ∂t dηdξ ) dxdt, where 2 ∫ ωα f. (∫ α x ∫ α ξ ∂u ∂t dξ ) dxdt = −2 ∫ t 0 (∫ α 0 x ∂u ∂t dx )(∫ α 0 fdx ) dt+2 ∫ ωα (∫ α x ∂u ∂t dξ )(∫ α x fdξ ) dxdt. by virtue of the cauchy inequality and with ε, ab ≤ ε 2 a2 + 1 2ε b2, a,b ∈ r we obtain ∫ ωα f.mu1dxdt ≤ 4 ε1 2 ∫ ωα f2dxdt + 4 2ε1 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 1 2ε2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ε2 2 ∫ ωα (1 −x)2f2dxdt + 1 ε3 ∫ t 0 (∫ α 0 x ∂u ∂t dx )2 dt + ε3 ∫ t 0 (∫ α 0 fdx )2 dt +ε4 ∫ ωα (∫ α x fdξ )2 dxdt + 1 ε4 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt, where ∫ ωα (∫ α x fdξ )2 dxdt ≤ 4 ∫ ωα (1 −x)2f2dxdt + 2 ∫ t 0 (∫ α 0 fdx )2 dt ≤ 4 ∫ ωα f2dxdt + 2 ∫ t 0 (∫ α 0 fdx )2 dt then, we obtain ∫ ωα f.mu1dxdt(12) ≤ 2ε1 ∫ ωα f2dxdt + 2 ε1 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 1 2ε2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ε2 2 ∫ ωα f2dxdt +ε3 ∫ t 0 (∫ α 0 fdx )2 dt + 1 ε3 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt +4ε4 ∫ ωα f2dxdt + 2ε4 ∫ t 0 (∫ α 0 fdx )2 dt + 1 ε4 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt. 102 eddine and abdelfatah 2) on the interval [β, 1] , we denote ωτβ = ωβ = (β, 1) × (0,τ), we get ∫ ωβ lu.mu3(13) = ∫ ωβ ∂u ∂t . ( − 1 4 ∫ x β ∂u ∂t dξ − ∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt − ∫ ωβ ∂2 ∂x2 ( a(x,t) ∂u ∂x )( − 1 4 ∫ x β ∂u ∂t dξ ) dxdt + ∫ ωβ ∂2 ∂x2 ( a(x,t) ∂u ∂x )(∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt = ∫ ωβ f.mu3dxdt. integration by parts each term of (3.6) with use the conditions (2.2) − (2.5), we obtain ∫ ωβ ∂u ∂t . ( − 1 4 ∫ x β ∂u ∂t dξ − ∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt(14) = 3 2 ∫ ωβ (∫ x β ∂u ∂t dηdξ )2 dxdt− 1 8 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt − ∫ t 0 ((∫ 1 β ∂u ∂t dx )( 2 ∫ 1 β ∂u ∂t dx− 2 ∫ 1 β x ∂u ∂t dx )) dt = 3 2 ∫ ωβ (∫ x β ∂u ∂t dηdξ )2 dxdt− 17 8 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt +2 ∫ t 0 (∫ 1 β ∂u ∂t dx )(∫ 1 β x ∂u ∂t dx ) dt − ∫ ωβ ∂2 ∂x2 ( a(x,t) ∂u ∂x )(∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt(15) + ∫ ωβ ∂2 ∂x2 ( a(x,t) ∂u ∂x )(∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt = ∫ ωβ ( 5 4 −x)a(x,t) ∂u ∂x ∂2u ∂x∂t dxdt − ∫ t 0 ∂ ∂x ( a(x,t) ∂u ∂x )( − 1 4 ∫ x β ∂u ∂t dξ )∣∣∣∣x=1 x=β dt + ∫ t 0 ∂ ∂x ( a(x,t) ∂u ∂x ) . (∫ x β (∫ ξ β ∂u ∂t dξ + (1 − ξ) ∂u ∂t ) dξ )∣∣∣∣∣ x=1 x=β dt = − 1 2 ∫ ωβ ( 5 4 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ 1 β ( 5 4 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ 1 β ( 5 4 −x)a(x,t) ( ∂u ∂x )2 dx , mixed problem with integral conditions 103 ∫ ωβ f.mu3dxdt = ∫ ωβ f. ( − 1 4 ∫ x β ∂u ∂t dξ − ∫ x β (∫ ξ β ∂u ∂t dη + (1 − ξ) ∂u ∂t ) dξ ) dxdt = ∫ ωβ f. ( − 1 4 ∫ x β ∂u ∂t dξ ) dxdt− ∫ ωβ f. ( (1 −x) ∫ x β ∂u ∂t dξ ) dxdt −2 ∫ ωβ f. (∫ x β ∫ ξ β ∂u ∂t dηdξ ) dxdt, where −2 ∫ ωβ f. (∫ x β ∫ ξ β ∂u ∂t dηdξ ) dxdt = −2 ∫ t 0 (∫ 1 β fdx )(∫ 1 β ∂u ∂t dx− ∫ 1 β x ∂u ∂t dx ) dt +2 ∫ ωβ (∫ x β ∂u ∂t dξ )(∫ x β fdξ ) dxdt. by virtue of the cauchy’s ε-inequality, we obtain ∫ ωβ f.mu3dxdt ≤ ε5 8 ∫ ωβ f2dxdt + 1 8ε5 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ε6 2 ∫ ωβ (1 −x)2 f2dxdt + 1 2ε6 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +ε7 ∫ t 0 (∫ 1 β fdx )2 dt + 1 ε7 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt +ε8 ∫ t 0 (∫ 1 β fdx )2 dt + 1 ε8 ∫ t 0 (∫ 1 β x ∂u ∂t dx )2 dt + 1 ε9 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ε9 ∫ ωβ (∫ x β fdξ )2 dxdt, where ∫ ωβ (∫ x β fdξ )2 dxdt ≤ 4 ∫ ωβ (x−β)2 f2dxdt ≤ 4 ∫ ωβ f2dxdt, then, we obtain ∫ ωβ f.mu3dxdt(16) ≤ ε5 8 ∫ ωβ f2dxdt + 1 8ε5 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ε6 2 ∫ ωβ f2dxdt + 1 2ε6 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +2ε10 ∫ t 0 (∫ 1 β fdx )2 dt + 2 ε10 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt + 1 ε9 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + 4ε9 ∫ ωβ f2dxdt, where ε10 = ε7 + ε8. 104 eddine and abdelfatah 3) on the interval [α,β] , we denote ωτα,β = ωα,β = (α,β) × (0,τ), we get∫ ωα,β lu.mu2dxdt(17) = ∫ ωα,β ∂u ∂t ( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ ) dxdt − ∫ ωα,β ∂2 ∂x2 ( a(x,t) ∂u ∂x )( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ ) dxdt = ∫ ωα,β f.mu2dxdt. integration by parts each term of (3.10) with use the conditions (2.2) − (2.5), we obtain∫ ωα,β ∂u ∂t ( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ ) dxdt(18) = 1 2 ∫ ωα,β (∫ x α ∂u ∂t dξ )2 dxdt + 1 2 ∫ ωα,β (∫ β x ∂u ∂t dξ )2 . − ∫ ωα,β ∂2 ∂x2 ( a(x,t) ∂u ∂x )( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ ) dxdt(19) = − ∫ t 0 ∂ ∂x ( a(x,t) ∂u ∂x )( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ )∣∣∣∣∣ x=β x=α dt − ∫ t 0 ( a(x,t) ∂u ∂x )(∫ x α ∂u ∂t dξ + (β −x) ∂u ∂t )∣∣∣∣x=β x=α dt + ∫ ωα,β (β −x)a(x,t) ∂u ∂x ∂2u ∂x∂t dxdt − ∫ t 0 ( a(x,t) ∂u ∂x )(∫ β x ∂u ∂t dξ − (x−α) ∂u ∂t )∣∣∣∣∣ x=β x=α dt + ∫ ωα,β (x−α)a(x,t) ∂u ∂x ∂2u ∂x∂t dxdt = − 1 2 ∫ ωα,β (β −α) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt − 1 2 ∫ β α (β −α)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ β α (β −α)a(x,τ) ( ∂u ∂x )2 dx, ∫ ωα,β f.mu2dxdt(20) = ∫ ωα,β f. ( (x−α) ∫ β x ∂u ∂t dξ + (β −x) ∫ x α ∂u ∂t dξ ) dxdt ≤  1 2 ∫ ωα,β (∫ β x ∂u ∂t dξ )2 dxdt + 1 2 ∫ ωα,β (x−α)2 f2dxdt   + ( 1 2 ∫ ωα,β (∫ x α ∂u ∂t )2 dxdt + 1 2 ∫ ωα,β (β −x)2 f2dxdt ) . mixed problem with integral conditions 105 putting (3.3) , (3.4) and (3.5) into (3.2), on ωα, ∫ ωα lu.mu1dxdt = ∫ ωα f.mu1dxdt we obtain 5 2 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt + 3 2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt− 2 ∫ t 0 (∫ α 0 ∂u ∂t dx )(∫ α 0 x ∂u ∂t dx ) dt − 1 2 ∫ ωα (5 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ α 0 (5 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ α 0 ( ∂u ∂x )2 a(x,τ)(5 −x)dx ≤ 2ε1 ∫ ωα f2dxdt + 2 ε1 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 1 2ε2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ε2 2 ∫ ωα f2dxdt +ε3 ∫ t 0 (∫ α 0 fdx )2 dt + 1 ε3 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt +4ε4 ∫ ωα f2dxdt + 2ε4 ∫ t 0 (∫ α 0 fdx )2 dt + 1 ε4 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt , then 5 2 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt + 3 2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt(21) −2 ∫ t 0 (∫ α 0 ∂u ∂t dx )(∫ α 0 x ∂u ∂t dx ) dt − 1 2 ∫ ωα (5 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ α 0 (5 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ α 0 (5 −x)a(x,t) ( ∂u ∂x )2 dx ≤ ( 2ε1 + ε2 2 + 4ε4 )∫ ωα f2dxdt + ( 2 ε1 + 1 2ε2 + 1 ε4 )∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + (ε3 + 2ε4) ∫ t 0 (∫ α 0 fdx )2 dt + 1 ε3 ∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt putting (3.7) , (3.8) and (3.9) into (3.6), on ωβ, ∫ ωβ lu.mu3dxdt = ∫ ωβ f.mu3dxdt 106 eddine and abdelfatah we obtain 3 2 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt− 17 8 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt + 2 ∫ t 0 (∫ 1 β ∂u ∂t dx )(∫ 1 β x ∂u ∂t dx ) dt − 1 2 ∫ ωβ ( 5 4 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt − 1 2 ∫ 1 β ( 5 4 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ 1 β ( 5 4 −x)a(x,τ) ( ∂u ∂x )2 dx ≤ ε5 8 ∫ ωβ f2dxdt + 1 8ε5 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ε6 2 ∫ ωβ f2dxdt + 1 2ε6 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +2ε10 ∫ t 0 (∫ 1 β fdx )2 dt + 2 ε10 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt + 1 ε9 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + 4ε9 ∫ ωβ f2dxdt. then 3 2 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt− 17 8 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt(22) +2 ∫ t 0 (∫ 1 β ∂u ∂t dx )(∫ 1 β x ∂u ∂t dx ) dt − 1 2 ∫ ωβ ( ∂u ∂x )2 ∂a(x,t) ∂t ( 5 4 −x)dxdt − 1 2 ∫ 1 β ( ∂φ ∂x )2 a(x, 0)( 5 4 −x)dx + 1 2 ∫ 1 β ( ∂u ∂x )2 a(x,t)( 5 4 −x)dx ≤ (ε5 8 + ε6 2 + 4ε9 )∫ ωβ f2dxdt + ( 1 8ε5 + 1 2ε6 + 1 ε9 )∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +2ε10 ∫ t 0 (∫ 1 β fdx )2 dt + 2 ε10 ∫ t 0 (∫ 1 β ∂u ∂t dx )2 dt. putting (3.11) , (3.12) and (3.13) into (3.10), on ωα,β : ∫ ωα,β lu.mu2dxdt = ∫ ωα,β f.mu2dxdt mixed problem with integral conditions 107 we obtain 1 2 ∫ ωα,β (∫ x α ∂u ∂t dξ )2 dxdt + 1 2 ∫ ωα,β (∫ β x ∂u ∂t dξ )2 − 1 2 ∫ ωα,β (β −α) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt − 1 2 ∫ β α (β −α)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ β α (β −α)a(x,τ) ( ∂u ∂x )2 dx ≤ ( 1 2 ∫ ωα,β (∫ x α ∂u ∂t dξ )2 dxdt + 1 2 ∫ ωα,β (β −x)2 f2dxdt ) +  1 2 ∫ ωα,β (∫ β x ∂u ∂t dξ )2 dxdt + 1 2 ∫ ωα,β (x−α)2 f2dxdt   , that implies: − 1 2 ∫ ωα,β (β −α) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt − 1 2 ∫ β α (β −α)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ β α (β −α)a(x,τ) ( ∂u ∂x )2 dx ≤ 1 2 ∫ ωα,β (β −x)2 f2dxdt + 1 2 ∫ ωα,β (x−α)2 f2dxdt, ≤ ∫ ωα,β f2dxdt, then c0 2 ∫ β α (β −α) ( ∂u ∂x )2 dx(23) ≤ c2 2 ∫ ωα,β (β −α) ( ∂u ∂x )2 dxdt + c1 2 ∫ β α ( ∂φ ∂x )2 dx + ∫ ωα,β f2dxdt . according to the condition (1.4) we have: (∫ α 0 ∂u ∂t dx )2 = (∫ 1 β ∂u ∂t dx )2 . 108 eddine and abdelfatah so, we are adding between (3.14) and (3.15) , we obtain 3 2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 3 2 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ( 3 8 )∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt − 1 2 ∫ ωα (5 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ α 0 (5 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ α 0 (5 −x)a(x,τ) ( ∂u ∂x )2 dx − 1 2 ∫ ωβ ( 5 4 −x) ∂a(x,t) ∂t ( ∂u ∂x )2 dxdt− 1 2 ∫ 1 β ( 5 4 −x)a(x, 0) ( ∂φ ∂x )2 dx + 1 2 ∫ 1 β ( 5 4 −x)a(x,τ) ( ∂u ∂x )2 dx ≤ ( 2ε1 + ε2 2 + 4ε4 )∫ ωα f2dxdt + ( 2 ε1 + 1 2ε2 + 1 ε4 )∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + (ε3 + 2ε4) ∫ t 0 (∫ α 0 fdx )2 dt + ( 1 ε3 + 2 ε10 )∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt (ε5 8 + ε6 2 + 4ε9 )∫ ωβ f2dxdt + ( 1 8ε5 + 1 2ε6 + 1 ε9 )∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +2ε10 ∫ t 0 (∫ 1 β fdx )2 dt, if we put ε1 = 4, ε2 = 2, ε3 = 8, ε4 = 4, ε5 = 1, ε6 = 2, ε9 = 2, ε10 = 8. we have 3 2 ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 3 2 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + ( 3 8 )∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt − 1 2 ∫ ωα ( ∂u ∂x )2 ∂a(x,t) ∂t (5 −x)dxdt− 1 2 ∫ α 0 ( ∂φ ∂x )2 a(x, 0)(5 −x)dx + 1 2 ∫ α 0 ( ∂u ∂x )2 a(x,t)(5 −x)dx − 1 2 ∫ ωβ ( ∂u ∂x )2 ∂a(x,t) ∂t ( 5 4 −x)dxdt− 1 2 ∫ 1 β ( ∂φ ∂x )2 a(x, 0)( 5 4 −x)dx + 1 2 ∫ 1 β ( ∂u ∂x )2 a(x,t)( 5 4 −x)dx ≤ 25 ∫ ωα f2dxdt + ∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt +16 ∫ t 0 (∫ α 0 fdx )2 dt + ( 3 8 )∫ t 0 (∫ α 0 ∂u ∂t dx )2 dt 73 8 ∫ ωβ f2dxdt + 7 8 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt + 16 ∫ t 0 (∫ 1 β fdx )2 dt, mixed problem with integral conditions 109 then (∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 5 8 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt ) + ( 1 2 ∫ α 0 (5 −x)a(x,τ) ( ∂u ∂x )2 dx + 1 2 ∫ 1 β ( 5 4 −x)a(x,τ) ( ∂u ∂x )2 dx ) ≤   25 ∫ωα f2dxdt + 16 ∫t0 (∫α0 fdx)2 dt + 1 2 ∫ ωα (5 −x)∂a(x,t) ∂t ( ∂u ∂x )2 dxdt + 1 2 ∫α 0 (5 −x)a(x, 0) ( ∂φ ∂x )2 dx   +   738 ∫ ωβ f2dxdt + 16 ∫t 0 (∫ 1 β fdx )2 dt + 1 2 ∫ ωβ ( 5 4 −x)∂a(x,t) ∂t ( ∂u ∂x )2 dxdt + 1 2 ∫ 1 β ( 5 4 −x)a(x, 0) ( ∂φ ∂x )2 dx   . according to condition 1, we then get (∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 5 8 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt ) + ( c0 2 ∫ α 0 (5 −x) ( ∂u ∂x )2 dx + c0 2 ∫ 1 β ( 5 4 −x) ( ∂u ∂x )2 dx ) ≤   25 ∫ωα f2dxdt + 16 ∫t0 (∫α0 fdx)2 dt +c2 2 ∫ ωα (5 −x) ( ∂u ∂x )2 dxdt + c1 2 ∫α 0 (5 −x) ( ∂φ ∂x )2 dx   +   738 ∫ ωβ f2dxdt + 16 ∫t 0 (∫ 1 β fdx )2 dt +c2 2 ∫ ωβ ( 5 4 −x) ( ∂u ∂x )2 dxdt + c1 2 ∫ 1 β ( 5 4 −x) ( ∂φ ∂x )2 dx   , then (∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + 5 8 ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt ) + ( c0 2 ∫ α 0 (5 −x) ( ∂u ∂x )2 dx + c0 2 ∫ 1 β ( 5 4 −x) ( ∂u ∂x )2 dx ) ≤   25 ∫ωα f2dxdt + 16 ∫t0 (∫α0 fdx)2 dt +c2 2 ∫ ωα (5 −x) ( ∂u ∂x )2 dxdt + 5c1 2 ∫α 0 ( ∂φ ∂x )2 dx   +   738 ∫ ωβ f2dxdt + 16 ∫t 0 (∫ 1 β fdx )2 dt +c2 2 ∫ ωβ ( 5 4 −x) ( ∂u ∂x )2 dxdt + 5c1 8 ∫ 1 β ( ∂φ ∂x )2 dx   ≤ 25 (∫ ωα f2dxdt + ∫ ωβ f2dxdt + ∫ t 0 (∫ α 0 fdx )2 dt + ∫ t 0 (∫ 1 β fdx )2 dt ) + 5c1 2 (∫ α 0 ( ∂φ ∂x )2 dx + ∫ 1 β ( ∂φ ∂x )2 dx ) + c2 2 (∫ ωα (5 −x) ( ∂u ∂x )2 dxdt + ∫ ωβ ( 5 4 −x) ( ∂u ∂x )2 dxdt ) . 110 eddine and abdelfatah that implies 5 8 (∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt ) + c0 2 [∫ α 0 (5 −x) ( ∂u ∂x )2 dx + ∫ 1 β ( 5 4 −x) ( ∂u ∂x )2 dx ] ≤ 25 (∫ ωα f2dxdt + ∫ ωβ f2dxdt + ∫ t 0 (∫ α 0 fdx )2 dt + ∫ t 0 (∫ 1 β fdx )2 dt ) + 5c1 2 (∫ α 0 ( ∂φ ∂x )2 dx + ∫ 1 β ( ∂φ ∂x )2 dx ) + c2 2 (∫ ωα (5 −x) ( ∂u ∂x )2 dxdt + ∫ ωβ ( 5 4 −x) ( ∂u ∂x )2 dxdt ) using lemme 1 in [9] , we have(∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt ) (24) + (∫ α 0 (5 −x) ( ∂u ∂x )2 dx + ∫ 1 β ( 5 4 −x) ( ∂u ∂x )2 dx ) ≤ c3   (∫ ωα f2dxdt + ∫ ωβ f2dxdt ) + ∫t 0 ((∫α 0 fdx )2 + (∫ 1 β fdx )2) dt + (∫α 0 ( ∂φ ∂x )2 dx + ∫ 1 β ( ∂φ ∂x )2 dx )   , where c3 = max ( 25, 5c1 2 ) min ( 5 8 , c0 2 ) exp (c2 2 t ) according to (3.16) and by using lemme 1 in [9], we get∫ β α (β −α) ( ∂u (x,τ) ∂x )2 dx(25) ≤ c4 (∫ ωα,β f2dxdt + ∫ β α ( ∂φ ∂x )2 dx ) where c4 = max ( 1, c1 2 ) c0 2 exp (c2 2 t ) we are adding between (3.17) and (3.18) , we obtain∫ ωα (∫ α x ∂u ∂t dξ )2 dxdt + ∫ ωβ (∫ x β ∂u ∂t dξ )2 dxdt +   ∫α0 (5 −x) (∂u∂x)2 dx + ∫ 1β ( 54 −x) (∂u∂x)2 dx + ∫β α (β −α) ( ∂(x,t) ∂x )2 dx   ≤ max (c3,c4)   ∫ ω f2dxdt + ∫t 0 ((∫α 0 fdx )2 + (∫ 1 β fdx )2) dt + (∫ 1 0 ( ∂φ ∂x )2 dx )   ≤ c5 (∫ ω f2dxdt + ∫ 1 0 ( ∂φ ∂x )2 dx ) (26) mixed problem with integral conditions 111 where c5 = 1 + max (c3,c4) . the right-hand side of (3.19) is independent of τ, hence replacing the left-hand side by its upper bound with respect to τ from 0 to t, we obtain the desired inequality, where c = (c5) 1 2 . � proposition 2. the operator l from b to f admits a closure. proof. suppose that {un} ∈ d (l) is a sequence such that (27) un → 0 in b and lun → (f,φ) in f; we must show that f ≡ 0and φ ≡ 0. according to (3.20) we get un → 0 in d′ (ω) by virtue of the continuity of derivation of d′ (ω) in d′ (ω), we deduce that (28) lun → 0 in d′ (ω) . further, according to (3.21) , we have (29) lun → f in l2 (ω) , thus we have (30) lun → f in d′ (ω) . then by of the uniqueness of the limit in d′ (ω) we see that f ≡ 0. on the other hand, (3.21) implies that (31) d`un dx → dφ dx in l2 (0, 1) . moreover, since by virtue of (3.20) and the fact that∫ α 0 (5 −x) ( ∂u ∂x )2 dx + ∫ 1 β ( 5 4 −x) ( ∂u ∂x )2 dx + ∫ β α (β −α) ( ∂ (x,t) ∂x )2 dx ≤‖un‖ 2 b , ∀n, we have (32) d`un dx → 0 in l2 (0, 1) . now the uniqueness of the limit in l2 (0, 1) implies that φ ≡ 0. � theorem 1 is valid for strong solution, i.e., we have the inequality (33) ‖u‖b ≤ c ∥∥lu∥∥ f , ∀u ∈ d(l). hence we obtain corollary 3. a strong solution of the problem (2.1)−(2.5) is unique if it exists, and depends continuously on f = (f,φ) ∈ f. corollary 4. the range r(l) of the operator l is closed in f , and r(l) = r(l). 112 eddine and abdelfatah 4. existence of solutions to show the existance of solutions, we prove that r(l) is dense in f for all u ∈ d (l) and for arbitrary f = (f,φ) ∈ f. theorem 5. suppose the conditions of theorem 1 are satisfied. then the problem (2.1)−(2.5) admits a unique strong solution u = l −1 f = l−1f. proof. first we prove that r(l) is dense in f for the special case where d (l) is reduced to d0 (l) , where d0 (l) = {u, u ∈ d (l) : `u = 0} . � proposition 6. let the conditions of theorem 2 be satisfied, if, for ω ∈ l2 (ω) and for all u ∈ d0 (l) , we have (34) ∫ ω lu.ω dxdt = 0, then ω vanishes almost everywhere in ω. proof. the scalar product of f is defined by (lu,ω)f(35) = ∫ ω lu.ω dxdt + ∫ 1 0 ( ∂`u ∂x )( ∂ω0 ∂x ) dx. the equality (4.1) can be written as follows: (36) ∫ ω ∂u ∂t ωdxdt = ∫ ω ∂2 ∂x2 ( a(x,t) ∂u ∂x ) ωdxdt. if we put u = =t ( ec6tz ) = ∫ t 0 ec6tz (x,τ) dτ, where c6 is a constant such that c6c0−c2 ≥ 0, and z, ∂z∂x, ∂ ∂x ( a ∂=t(ec6tz) ∂x ) , ∂ 2 ∂x2 (a ∂=t(ec6tz) ∂x ) ∈ l2 (ω) , then, u satisfies the conditions (2.3) − (2.5). as a result of (4.3) , we obtain the equality (37) ∫ ω ec6tzωdxdt = ∫ ω ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x ) ωdxdt. in terms of the given function ω, and from the equality (4.4) we give the function ω in terms of z as follows: (38) ω =   ω1 = (1 −x) ∫α x zdξ − 2 ∫α x ∫α ξ zdηdξ 0 ≤ x ≤ α ω2 = −(β −x) ∫x α zdξ − (x−α) ∫β x zdξ α ≤ x ≤ β ω3 = −(1 −x) ∫x β zdξ − 2 ∫x β ∫ ξ β zdηdξ β ≤ x ≤ 1 so, ω ∈ l2 (ω) , and z satisfies the same conditions of the function u and (39) ∂2z ∂x2 ∣∣∣∣ x=α = 0, ∂2z ∂x2 ∣∣∣∣ x=β = 0. replacing ω in (4.4) by its representation (4.5) and integrating by parts each term of (4.4) with the use of conditions of z, we obtain 1) on the interval ωα = (0,α) × (0,τ) , we obtain (40) ∫ ωα ec6tzω1dxdt = ∫ ω ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x ) ω1dxdt. mixed problem with integral conditions 113 integrating by parts each term of (4.7) with respect to x and t by taking the conditions of the function z yields ∫ ωα ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x )( (1 −x) ∫ α x zdξ − 2 ∫ α x ∫ α ξ zdηdξ ) dxdt = ∫ t 0 ∂ ∂x ( a(x,t) ∂=t (ec6tz) ∂x )( (1 −x) ∫ α x zdξ − 2 ∫ α x ∫ α ξ zdηdξ )∣∣∣∣x=α x=0 dt − ∫ ωα ∂ ∂x ( a ∂=t (ec6tz) ∂x )( −(1 −x)z + ∫ α x zdξ ) dxdt = ∫ t 0 ( a(x,t) ∂=t (ec6tz) ∂x )( −(1 −x)z + ∫ α x zdξ )∣∣∣∣x=α x=0 dt + ∫ ωα ( a ∂=t (ec6tz) ∂x )( −(1 −x)z + ∫ α x zdξ ) dxdt = − ∫ ωα a (x,t) ∂=t (ec6tz) ∂x (1 −x) ∂z ∂x dxdt = − 1 2 ∫ α 0 e−c6t(1 −x)a (x,t) ( ∂=t (ec6tz) ∂x )2∣∣∣∣∣ t=t t=0 dx − 1 2 ∫ ωα e−c6t(1 −x) ( c6a (x,t) − ∂a (x,t) ∂t )( ∂=t (ec6tz) ∂x )2 dxdt by using the conditions of z, we obtain (41) − 1 2 (c6c0 − c2) ∫ ωα e−c6t(1 −x) ( ∂=t (ec6tz) ∂x )2 dxdt ≤ 0. and ∫ ωα ec6tzω1dxdt(42) = 3 2 ∫ ωα ec6t (∫ α x zdξ )2 dxdt + 1 2 ∫ t 0 ec6t (∫ α 0 zdx )2 dt −2 ∫ t 0 ec6t (∫ α 0 zdx )(∫ α 0 ξzdξ ) dt. 2) on the interval ωβ = (β, 1) × (0,τ) ,we obtain (43) ∫ ωβ ec6tzω3dxdt = ∫ ωβ ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x ) ω3dxdt. 114 eddine and abdelfatah integrating by parts each term of (4.10) with respect to x and t by taking the conditions of the function z yields ∫ ωβ ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x )[ −(1 −x) ∫ x β zdξ − 2 ∫ x β ∫ ξ β zdηdξ ] dxdt = ∫ t 0 ∂ ∂x ( a(x,t) ∂=t (ec6tz) ∂x )( −(1 −x) ∫ x β zdξ − 2 ∫ x β ∫ ξ β zdηdξ )∣∣∣∣∣ x=1 x=β dt − ∫ ωβ ∂ ∂x ( a ∂=t (ec6tz) ∂x )[ −(1 −x)z − ∫ x β zdξ ] dxdt = ∫ t 0 ( a(x,t) ∂=t (ec6tz) ∂x )( −(1 −x)z − ∫ x β zdξ )∣∣∣∣x=1 x=β dt + ∫ ωβ ( a ∂=t (ec6tz) ∂x )[ −(1 −x)z − ∫ x β zdξ ] dxdt = − ∫ ωβ a (x,t) ∂=t (ec6tz) ∂x (1 −x) ∂z ∂x dxdt = − 1 2 ∫ 1 β e−c6t(1 −x)a (x,t) ( ∂=t (ec6tz) ∂x )2∣∣∣∣∣ t=t t=0 dx − 1 2 ∫ ωβ e−c6t(1 −x) ( c6a (x,t) − ∂a (x,t) ∂t )( ∂=t (ec6tz) ∂x )2 dxdt by using the conditions of z, we obtain (44) − 1 2 (c6c0 − c2) ∫ ωβ e−c6t(1 −x) ( ∂=t (ec6tz) ∂x )2 dxdt ≤ 0. and ∫ ωβ ec6tzω3dxdt(45) = 3 2 ∫ ωβ ec6t (∫ 1 β zdx )2 dxdt −2 ∫ t 0 ec6t (∫ 1 β zdx )((∫ 1 β zdx ) − (∫ 1 β xzdx )) dt = 3 2 ∫ ωβ ec6t (∫ x β zdξ )2 dxdt +2ec6t (∫ 1 β zdx )(∫ 1 β xzdx ) dt. 3) on the interval ωα,β = (α,β) × (0,τ) ,we obtain (46) ∫ ωα,β ec6tzω2dxdt = ∫ ωα,β ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x ) ω2dxdt mixed problem with integral conditions 115 integrating by parts each term of (4.13) with respect to x and t by taking the conditions of the function z yields ∫ ωα,β ∂2 ∂x2 ( a ∂=t (ec6tz) ∂x )[ −(β −x) ∫ x α zdξ − (x−α) ∫ β x zdξ ] dxdt = ∫ t 0 ∂ ∂x ( a(x,t) ∂=t (ec6tz) ∂x )( −(β −x) ∫ x α zdξ − (x−α) ∫ β x zdξ )∣∣∣∣∣ x=β x=α dt + ∫ ωα,β ∂ ∂x ( a ∂=t (ec6tz) ∂x )( (β −x)z − ∫ x α zdξ ) dxdt + ∫ ωα,β ∂ ∂x ( a ∂=t (ec6tz) ∂x )( −(x−α)z + ∫ β x zdξ ) dxdt =   ∫t 0 ( a(x,t) ∂=t(ec6tz) ∂x )( (β −x)z − ∫x α zdξ )∣∣∣∣x=β x=α dt + ∫t 0 ( a(x,t) ∂=t(ec6tz) ∂x )( −(x−α)z + ∫β x zdξ )∣∣∣∣x=β x=α dt   −   ∫ ωα,β ( a ∂=t(ec6tz) ∂x )( (β −x) ∂z ∂x + 2z ) dxdt + ∫ ωα,β ( a ∂=t(ec6tz) ∂x )( (x−α) ∂z ∂x − 2z ) dxdt   = − ∫ ωα,β a (x,t) ∂=t (ec6tz) ∂x (β −α) ∂z ∂x dxdt = − 1 2 ∫ β α e−c6t(β −α)a (x,t) ( ∂=t (ec6tz) ∂x )2∣∣∣∣∣ t=t t=0 dx − 1 2 ∫ ωα,β e−c6t(β −α) ( c6a (x,t) − ∂a (x,t) ∂t )( ∂=t (ec6tz) ∂x )2 dxdt by using the conditions of z, we obtain (47) − 1 2 (c6c0 − c2) ∫ ωα,β e−c6t(β −α) ( ∂=t (ec6tz) ∂x )2 dxdt ≤ 0. and ∫ ωα,β ec6tzω2dxdt(48) = 1 2 ∫ ωα,β ec6t (∫ x α zdξ )2 dxdt + 1 2 ∫ ωα,β ec6t (∫ β x zdξ )2 dxdt 116 eddine and abdelfatah putting and using the results of (4.8),(4.9) , (4.11),(4.12) and (4.14),(4.15) into (4.4) , we obtain 3 2 ∫ ωα ec6t (∫ α x zdξ )2 dxdt + 3 2 ∫ ωβ ec6t (∫ x β zdξ )2 dxdt + 1 2 ∫ ωα,β ec6t (∫ x α zdξ )2 dxdt + 1 2 ∫ ωα,β ec6t (∫ β x zdξ )2 dxdt ≤ − 1 2 (c6c0 − c2) ∫ ωα e−c6t(1 −x) ( ∂=t (ec6tz) ∂x )2 dxdt − 1 2 (c6c0 − c2) ∫ ωβ e−c6t(1 −x) ( ∂=t (ec6tz) ∂x )2 dxdt − 1 2 (c6c0 − c2) ∫ ωα,β e−c6t(β −α) ( ∂=t (ec6tz) ∂x )2 dxdt ≤ 0. and thus z = 0 in ω, then ω = 0 in ω. this proves proposition 2. � we return to the proof of theorem 2. we have already noted that it is sufficient to prove that the set r(l) dense in f. suppose that, for some w = (ω,ω0) ∈ r(l)⊥ and for all u ∈ d(l), it holds (49) (lu,ω)f = ∫ ω lu.ωdxdt + ∫ 1 0 ( ∂`u ∂x )( ∂ω0 ∂x ) dx = 0. then we must prove that w = 0. putting u ∈ d0(l) in (4.16) , we have∫ ω lu.ω dxdt = 0, u ∈ d0(l). hence proposition 2 implies that ω = 0. thus (4.16) takes the form (50) ∫ 1 0 ( ∂`u ∂x )( ∂ω0 ∂x ) dx = 0, u ∈ d(l). since the range of the trace operator ` is dense in the hilbert f space with the norm(∫ 1 0 ( ∂`u ∂x )2 dx )1 2 , the equality (4.17) implies that ω0 = 0 (we recall satisfies a compatibility conditions). hence w = 0. this completes the proof of theorem 2. references [1] a. bouziani and n.-e. benouar, mixed problem with integral conditions for a third order parabolic equation, kobe j. math. 15 (1998), no. 1, 47–58. [2] n.e. benouar and n.i. yurchuk, mixed problem with an integral condition for parabolic equations with the bessel operator, differentsial’nye uravneniya, 27 (1991), 2094-2098. [3] a. bouziani, mixed problem for certain nonclassical equations with a small parameter, bulletin de la classe des sciences, académie royale de belgique, 5 (1994), 389-400. [4] a. bouziani, solution forte d’un problème de transmission parabolique-hyperbolique pour une structure pluridemensionnelle, bulletin de la classe des sciences, académie royale de belgique, 7 (1996), 369-386. [5] a. bouziani, mixed problem with integral conditions for a certain parabolic equation, j. of appl. math. and stoch. anal. 9 (1996), 323-330. [6] a. bouziani, mixed problem with nonlocal condition for certain pluriparabolic equations, hiroshima math. j. 27 (1997), 373-390. [7] a. bouziani and n.e. benouar, problème mixte avec conditions intégrales pour une classe d’équations paraboliques, comptes rendus de l’académie des sciences, paris t.321, série i, (1995), 1177-1182. [8] a. bouziani and n.e. benouar, problèmes aux limites pour use classe d’équations de type non classique pour use structure pluri-dimensionnelle, bull. of the polish acad. of sciencesmathematics 43,(1995), 317-328. mixed problem with integral conditions 117 [9] b. cahlon, d.m. kulkarni and p.shi, stewise stability for the heat equation with a nonlocal constraint, siam l. nurner. anal., 32 (1995), 571-593. [10] j. r. cannon, the solution of the heat equation subject to the specification of energy, quart. appl. math. 21 (1963), 155–160. [11] j. r. cannon, s. pérez esteva, and j. van der hoek, a galerkin procedure for the diffusionequation subject to the specification of mass, siam j., numer. anal. 24 (1987), no. 3, 499–515. [12] j.r. cannon and j. van der hoek, the existence and the continuous dependence for the solution of the heat equation subject to the specification of energy, boll. uni. math. ital. suppl. 1 (1981), 253-282. [13] j.r. cannon and j. van der hoek, an implicit finite difference scheme for the diffusion of mass in a portion of the domain, numer. solutions of pdes (ed. by j. noye), north-holland, amsterdam (1982), 527 539. [14] l. garding, , cauchy’s problem for hyperbolic equations, univ. of chicago lecture notes 1957. [15] n.i. ionkin, solution of boundary value problem in heat conduction theory with nonlocl boundary conditions, differentsial’nye uravneniya, 13 (1977), 294-304. [16] s. jones, b. jumarhon, s. mckee, and j.a.scott, a mathematical model of biosensor, j. eng. math. 30 (1996), 312-337. [17] b. jumarhon, and s. mckee, on the heat equation with nonlinear and nonlocal boundary conditions, j. math. anal. appl. 190 (1995), 806-820. [18] b. jumarhon, and s. mckee, product integration methods for solving a system of nonlinear volterra integral equations, j. comput. appl. math. 69 (1996), 285-301. [19] n.i. kamynin, a boundary value problem in the theory of the heat conduction with non-classical boundary condition, th. vychisl. mat. mat. fiz. 43 (1964),1006-1024. [20] k. rektorys, variational methods in mathematics, sciences and engineering, 2nd ed., dordrecht, boston, reidel 1979. [21] p. shi, weak solution to an evolution problem with a nonlocal constraint, siam. j. math. anal. 24 (1993), 46-58. [22] n.i. yurchuk, mixed problem with an integral condition for certain parabolic equations, differentsial’nye uravneniya 22 (1986), 2117-2126. [23] g. w. batten, jr., second order correct boundary conditions of the numerical solution of the mixed boundary problem for parabolic equations, math. compt., 17 (1963), 405–413. [24] n.i. yurchuk, mixed problems for linearized kortweg-de-vries equations degenerating in time into parabolic equation, soviet math., 33 (1986), 435-437. [25] a.a. samarskii, some problems in differential equations theory, differents. uravn. 16 (1980), 1925-1935. department of mathematics and informatics; the larbi ben m’hidi university, oum el bouaghi, algeria ∗corresponding author: taki maths@live.fr int. j. anal. appl. (2023), 21:89 numerical solution for fractional-order mathematical model of immune-chemotherapeutic treatment for breast cancer using modified fractional formula mamon abu hammad1, iqbal h. jebril1, shameseddin alshorm1,∗, iqbal m. batiha1,2, nancy abu hammad3 1department of mathematics, al zaytoonah university, amman 11733, jordan 2nonlinear dynamics research center (ndrc), ajman university, ajman 346, uae 3department of hematopathology, princess iman research and laboratory sciences center, amman, jordan ∗corresponding author: alshormanshams@gmail.com abstract. cancer is a complex and diverse group of diseases characterized by the uncontrolled growth and spread of abnormal cells in the body. tumors, which are commonly associated with cancer, refer to abnormal masses of tissue that can develop in various organs or tissues. cancer can arise from almost any cell type in the body and can affect different organs and systems. the disease occurs when the normal processes of cell division and growth go awry, leading to the formation of malignant tumors. these tumors have the potential to invade nearby tissues and spread to distant parts of the body through a process known as metastasis. in this paper, we aim to present a numerical solution for a recent fractional-order model related to immune-chemotherapeutic treatment for breast cancer (ict) using a novel numerical scheme called the modified fractional euler method (mfem). we will also compare our proposed scheme with the traditional numerical scheme, fractional euler method (fem), through numerical simulations. 1. introduction mathematical modeling plays a crucial role in understanding and studying cancer tumors. mathematical models are used to describe and simulate the growth, development, and behavior of tumors, received: jun. 24, 2023. 2020 mathematics subject classification. 26a33, 34a08, 34k37. key words and phrases. fractional calculus; caputo fractional derivative; fractional euler method (fem); modified fractional euler method (mfem). https://doi.org/10.28924/2291-8639-21-2023-89 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-89 2 int. j. anal. appl. (2023), 21:89 as well as their interactions with the surrounding tissues and the immune system. these models help researchers and clinicians gain insights into the underlying mechanisms of tumor growth, predict treatment outcomes, and optimize therapeutic strategies see [1,3]. there are several mathematical models used in cancer research, such as growth models: these models describe the growth dynamics of tumors over time. they incorporate factors such as cell proliferation, cell death, nutrient supply, and oxygen levels to simulate tumor growth patterns. pharmacokinetic models: these models focus on how drugs are absorbed, distributed, metabolized, and eliminated in the body. they help in understanding drug concentrations at tumor sites and predicting drug effectiveness. spatial models: tumors often exhibit spatial heterogeneity, meaning that their characteristics vary in different regions. spatial models consider the spatial distribution of tumor cells, nutrients, oxygen, and other factors, allowing researchers to study the impact of spatial organization on tumor growth and treatment response. immune response models: these models capture the interactions between the tumor and the immune system. they help investigate how the immune system recognizes and responds to tumor cells, and how immunotherapies can be used to enhance immune responses against cancer. evolutionary models: tumors can undergo genetic and phenotypic changes, leading to the development of drug resistance and disease progression. evolutionary models simulate the evolutionary dynamics within a tumor population, helping to understand the emergence and spread of resistant cell populations see [5,7]. these mathematical models are typically based on differential equations, cellular automata, agentbased models, or other mathematical frameworks. they are often calibrated and validated using experimental data and can provide valuable insights into tumor behavior, treatment response, and the effectiveness of various therapeutic interventions. it’s important to note that mathematical models are simplifications of the complex biological reality, and their predictions should be interpreted with caution. however, they serve as powerful tools for hypothesis generation, guiding experimental design, and aiding in clinical decision-making in the field of cancer research and treatment see [9,11]. overall, the mathematical system divides the fractional-order breast cancer mathematical model among four manifestations: normal cell population s, tumor cells r, immune response class n and estrogen compartment z, (srnz). the fractional-order ict model represents a promising approach for improving the diagnosis and treatment of this devastating disease. however, further research is needed to fully validate the model and to translate its findings into clinical practice. the objective of this paper is to develop a numerical solution for the fractional-order differential system associated with ict. we will employ a numerical scheme, the modified fractional euler method (mfem), to solve this system and obtain numerical results see [2,6,8,10]. anyhow, the ict model can be described in the following manner: dµs(t) = s(t)ρ1(k1 − ς1s(t) −γ1r(t)) − (1 −k)ψ1s(t)z(t), s(0) = s0, (1.1) int. j. anal. appl. (2023), 21:89 3 dµr(t) = r(t)ρ2(k2d − ς2r(t) −γ2n(t)) −δr(t) + (1 −k)ψ1s(t)r(t)z(t), r(0) = r0, dµn(t) = ηλ + n(t)ρ1(k3 − ς3 −γ3r(t)) − (1 −k)ψ2n(t)z(t), n(0) = n0, dµz(t) = γ4z(t) + (1 −k)v, z(0) = z0. in which the parameters are positive real values. this model 1.1 explains the relation between normal cell population (s), tumor cells (r), immune response class (n), and estrogen compartment (z). where caputo fractional derivative dµ with a fractional order µ, where 0 < µ < 1, is applied to the temporal derivative in the system. the system involves several positive constants as parameters, including ρ1,ρ2,γ1,γ2,γ3,γ4,ς1,ς2,ψ1,ψ2,δ,k,v. these parameters govern various aspects of the model such as the fractional cell kill, carrying capacity, competition term, death rate, and per capita growth rate respectively. overall, the system involves various parameters that govern the dynamics of different cell populations in the context of breast cancer, and the numerical solution using the mfem will provide insights into the behavior of the system under different conditions. 2. modified fractional euler method we introduce fundamental definitions and theorems related to fractional calculus, including riemann-liouville integral and derivative, caputo derivative and other relevant concepts [4]. definition 2.1. the fractional riemann-liouville integral of a function f (t) of order 0 < µ ≤ 1 is initially defined by jµf (t) = 1 γ(µ) ∫ t 0 f (τ)(t −τ)µ−1 dτ, t > 0, µ > 0. (2.1) some of the properties of the riemann-liouville integral are given below for completeness: j0f (t) = f (t). (2.2) jµ(t −a)γ = γ(γ + 1) γ(µ + γ + 1) (t −a)µ+γ, γ ≥−,a ∈r. (2.3) jµjβf (t) = jβjµf (t), µ,β ≥ 0. (2.4) jµjβf (t) = jµ+βf (t), µ,β ≥ 0. (2.5) definition 2.2. the caputo fractional derivative of a real-valued function f (t) of order 0 < α ≤ 0 is defined as cdα∗ f (t) = 1 γ(1 −α) ∫ t 0 f ′(τ) (t −τ)α dτ, (2.6) where 0 < α < 1 and t > 0. 4 int. j. anal. appl. (2023), 21:89 some of the characteristics of the caputo derivative are listed below: • cdα∗c = 0, where c is constant. • for a ∈r we have cdα∗ (t −a) ρ = { γ(ρ+1) γ(ρ−α+1) (t −a) ρ−α, ρ > α− 1, 0, otherwise. • cdα∗ is a linear operator, i.e., cdα∗ (µf (t) + ωk(t)) = µ cdα∗ (f (t)) + ω cdα∗ (k(t)), where µ and ω are constants. in addition, we need to recall the following basic property for their significance: jα cdα∗ f (t) = f (t) − n∑ i=1 f i (0+) ti i! , t > 0, (2.7) where m− 1 < α ≤ m such that m ∈n. definition 2.3. the caputo fractional derivative operator cdα∗ can be defined in terms of the riemann-liouville fractional integral operator as follows: cdα∗ f = j m−αdmf . (2.8) where α ∈r+ and m = dαe. definition 2.4. [4] the mittag-leffler function of two parameters α and β is outlined by the following series: eα,β(t) = ∞∑ k=0 tk γ(αk + β) , where α,β > 0 and t ∈c. theorem 2.1. [4] (generalized taylor’s formula) suppose that cdkα∗ f (x) ∈ c(0,b] for k = 0, 1, 2, · · · ,n + 1, where 0 < α ≤ 1. then we can expand the function f about the node x0 as follows: f (x) = n∑ i=0 (x −x0)iα γ(iα + 1) (cdiα∗ f )(x0) + (x −x0)(n+1)α γ((n + 1)α + 1) (cd (n+1)α ∗ f )(ξ), (2.9) with 0 < ξ < x, ∀x ∈ (0,b]. this section will recall some existing numerical methods to deal with the fractional initial value ivp problem formulated in the sense of caputo fractional differentiator [4]. such a problem has the form: dαy(t) = f (t,y(t)), (2.10) int. j. anal. appl. (2023), 21:89 5 with the initial condition: y(0) = y0, (2.11) where 0 < α ≤ 1. the authors in [12] developed a generalization of the classical euler method called the fractional euler method (fem) by proposing a general formula for solving fractional ivps (2.10-2.11). this formula has the following form: w0 = y0 wi+1 = wi + hα γ(α + 1) f (ti,wi ) , (2.12) for i = 0, 1, · · · ,k − 1. note that wi denotes the numerical solution of problem (2.10-2.11) at ti. more recently, the authors in [2] have successfully developed a new further modification for the fem, called mfem for solving fractional ivp (2.10-2.11). this formula has the form: w0 = y0 wi+1 = wi + hα γ(α + 1) f ( ti + hα 2γ(α + 1) ,wi + hα 2γ(α + 1) f (ti,wi ) ) , (2.13) for i = 0, 1, 2, · · · ,k − 1. 3. solving fractional-order cancer tumor disease model in this section, we intend to employ mfem to obtain a numerical solution of the fractional-order ict model 1.1. this method represents a fractional version of the traditional euler method. therefore, to obtain a full overview of this method, the reader may refer to [2]. from now, we endeavor to apply mfem to the fractional-order ict model 1.1. for this purpose, we can consider such a model again as follows: dµs(t) = d1(t,s(t),r(t),n(t),z(t)), dµr(t) = d2(t,s(t),r(t),n(t),z(t)), dµn(t) = d3(t,s(t),r(t),n(t),z(t)), dµz(t) = d4(t,s(t),r(t),n(t),z(t)), (3.1) where d1(t,s(t),r(t),n(t),z(t)) = s(t)ρ1(k1 − ς1s(t) −γ1r(t)) − (1 −k)ψ1s(t)z(t), d2(t,s(t),r(t),n(t),z(t)) = r(t)ρ2(k2d − ς2r(t) −γ2n(t)) −δr(t) + (1 −k)ψ1s(t)r(t)z(t), d3(t,s(t),r(t),n(t),z(t)) = ηλ + n(t)ρ1(k3 − ς3 −γ3r(t)) − (1 −k)ψ2n(t)z(t), d4(t,s(t),r(t),n(t),z(t)) = γ4z(t) + (1 −k)v. (3.2) 6 int. j. anal. appl. (2023), 21:89 for instance, to generate the set of points (tk,s(tk)) of the class s formulated in such a system, we have to assume that each of s(t), dµs(t) and d2µs(t) are continuous on (0,t ]. in this regard, if one supposes that: d1(t,s(t),r(t),n(t),z(t)) = ps(t) + r1s(t)(k1 −β1s(t)) −γs(t)r(t) −d1s(t) −τ1s(t)n(t), so that dµs(t) = d1(t,s(t),r(t),n(t),z(t)), then by using formula (2.13), we can have the following expression: s(ti+1) =s(ti ) + hµ γ(µ + 1) d1 ( ti + hµ 2γ(µ + 1) ,s(ti ) + hµ 2γ(µ + 1) d1(ti,s(ti ),r(ti ),n(ti ),z(ti )) ) , (3.3) where i = 0, 1, 2, · · · ,k − 1. similarly, the procedure outlined above can be used to get approximate numerical solutions for the remaining classes. ultimately, we can deduce the following recursive states that represent the whole approximate numerical solution of system (1.1): s(ti+1) =s(ti ) + hµ γ(µ + 1) d1 ( ti + hµ 2γ(µ + 1) ,s(ti ) + hµ 2γ(µ + 1) d1(t,s(ti ),r(ti ),n(ti ),z(ti )) ) , r(ti+1) =r(ti ) + hµ γ(µ + 1) d2 ( ti + hµ 2γ(µ + 1) ,e(ti ) + hµ 2γ(µ + 1) d2(t,s(ti ),r(ti ),n(ti ),z(ti )) ) , n(ti+1) =n(ti ) + hµ γ(µ + 1) d3 ( ti + hµ 2γ(µ + 1) , i(ti ) + hµ 2γ(µ + 1) d3(t,s(ti ),r(ti ),n(ti ),z(ti )) ) , z(ti+1) =z(ti ) + hµ γ(µ + 1) d4 ( ti + hµ 2γ(µ + 1) ,r(ti ) + hµ 2γ(µ + 1) d4(t,s(ti ),r(ti ),n(ti ),z(ti )) ) , (3.4) where d1,d2,d3,d4 are defined in (3.2) i = 0, 1, 2, · · · ,k − 1. 4. numerical results in this part, we offer some numerical findings using the technique described in the preceding section. we examine an ict interaction’s fractional-order differential system and for more, we recommend [15–19]. for this purpose, we consider the values of the parameters m = 20,l = 15,ψ1 = 0.1,ψ2 = 0.1,ψ3 = 0.1,ρ1 = 0.3,ρ2 = 0.4,d1 = 0.5,γ1 = 6 ∗ 10−8,γ2 = 3 ∗ 10−7,γ4 = 0.97,v = 1,λ = 0.01,δ = 2,ς1 = 0.2,ς2 = 0.002,k = 0.5,η = 1.3 ∗ 102,k1 = 1.232,k2 = 1.75,k3 = 0.15. for the given parameter values, we plot the numerical solution of the considered model in figures 1-2-3 and 4. this solution represents the numerical solution curves of the fractional-order differential system of ict. in particular, in figure 1 we use the fem and mfem to show the size of the normal cells with µ = 0.8, 0.9, 1. in figure 2 we use the fem and mfem to show the size of the tumor cells, with µ = 0.8, 0.9, 1. in figure 3 we use the fem and mfem to show the size of the int. j. anal. appl. (2023), 21:89 7 immune cells, with µ = 0.8, 0.9, 1. in figure 4 we use the fem and mfem to show the size of the chemotherapeutic drugs with µ = 0.8, 0.9, 1. 0 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 time s iz e o f n o rm a l c e ll s size of the normal cells with ρ=0.8,0.9,1 fmem with ρ=0.8 fem with ρ=0.8 fmem with ρ=0.9 fem with ρ=0.9 fmem with ρ=1 fem with ρ=1 figure 1. size of the normal cells with ρ = 0.8, 0.9, 1 0 0.5 1 1.5 2 2.5 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time s iz e o f tu m e r c e ll s size of the tumer cells with ρ=0.8,0.9,1 fmem with ρ=0.8 fem with ρ=0.8 fmem with ρ=0.9 fem with ρ=0.9 fmem with ρ=1 fem with ρ=1 figure 2. size of the tumor cells with ρ = 0.8, 0.9, 1 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 time s iz e o f im m u n e r e s p o n s e c e ll s size of the immune response cells with ρ=0.8,0.9,1 fmem with ρ=0.8 fem with ρ=0.8 fmem with ρ=0.9 fem with ρ=0.9 fmem with ρ=1 fem with ρ=1 figure 3. size of the immune cells, with ρ = 0.8, 0.9, 1 8 int. j. anal. appl. (2023), 21:89 0 0.5 1 1.5 2 2.5 3 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 time s iz e o f e s tr o g e n size of the estrogen with ρ=0.8,0.9,1 fmem with ρ=0.8 fem with ρ=0.8 fmem with ρ=0.9 fem with ρ=0.9 fmem with ρ=1 fem with ρ=1 figure 4. size of the chemotherapeutic drugs with ρ = 0.8, 0.9, 1 one might observe that the above numerical simulations illustrate the system’s dynamic behavior and stability around equilibria. based on these simulations it can be determined that the fractionalorder version of the differential system of the ict carried out by the two numerical methods (fem and mfem) are completely considered. for more about fractional calculus and its applications, the reader may refer to [6,8,10,13,14]. 5. conclusion the modified fractional euler method (mfem) has been employed in this paper to investigate a fractional-order version of the differential system of ict. the utilization of this powerful numerical scheme has yielded impressive results, demonstrating the high efficiency of the algorithm. the obtained numerical results open up possibilities for conducting further research in this field and showcasing additional findings in the future. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. abaid ur rehman, j. ahmad, a. hassan, j. awrejcewicz, w. pawlowski, h. karamti, f.m. alharbi, the dynamics of a fractional-order mathematical model of cancer tumor disease, symmetry. 14 (2022), 1694. https://doi.org/10.3390/sym14081694. [2] i.m. batiha, a. bataihah, abeer a. al-nana, s. alshorm, i.h. jebril, a. zraiqat, a numerical scheme for dealing with fractional initial value problem, int. j. innov. comput. inform. control. 19 (2023), 763-774. https://doi. org/10.24507/ijicic.19.03.763. [3] z. sabir, m. munawar, m.a. abdelkawy, m.a.z. raja, c. ünlü, m.b. jeelani, a.s. alnahdi, numerical investigations of the fractional-order mathematical model underlying immune-chemotherapeutic treatment for breast cancer using the neural networks, fractal fract. 6 (2022), 184. https://doi.org/10.3390/fractalfract6040184. [4] i.m. batiha, s. alshorm, i. jebril, a. zraiqat, zaid momani, s. momani, modified 5-point fractional formula with richardson extrapolation, aims math. 8 (2023), 9520-9534. https://doi.org/10.3934/math.2023480. https://doi.org/10.3390/sym14081694 https://doi.org/10.24507/ijicic.19.03.763 https://doi.org/10.24507/ijicic.19.03.763 https://doi.org/10.3390/fractalfract6040184 https://doi.org/10.3934/math.2023480 int. j. anal. appl. (2023), 21:89 9 [5] f. özköse, m.t. şenel, r. habbireeh, fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy, math. model. numer. simul. appl. 1 (2021), 67-83. https: //doi.org/10.53391/mmnsa.2021.01.007. [6] i.m. batiha, s. alshorm, a. al-husban, r. saadeh, g. gharib, s. momani, the n-point composite fractional formula for approximating riemann-liouville integrator, symmetry. 15 (2023), 938. https://doi.org/10.3390/ sym15040938. [7] e. ucar, n. ozdemir, e. altun, fractional order model of immune cells influenced by cancer cells, math. model. nat. phenom. 14 (2019), 308. https://doi.org/10.1051/mmnp/2019002. [8] m.m. al-shomrani, m.a. abdelkawy, numerical simulation for fractional-order differential system of a glioblastoma multiforme and immune system, adv. differ. equ. 2020 (2020), 516. https://doi.org/10.1186/ s13662-020-02978-2. [9] h. yasmin, m. abu hammad, r. shah, b.m. alotaibi, sherif.m.e. ismaeel, s.a. el-tantawy, on the solutions of the fractional-order sawada-kotera-ito equation and modeling nonlinear structures in fluid mediums, symmetry. 15 (2023), 605. https://doi.org/10.3390/sym15030605. [10] i.m. batiha, l.b. aoua, t.e. oussaeif, a. ouannas, s. alshorman, i.h. jebril, s. momani, common fixed point theorem in non-archimedean menger pm-spaces using clr property with application to functional equations, iaeng int. j. appl. math. 53 (2023), 1-9. [11] m. mhailan, m. abu hammad, m. al horani, r. khalil, on fractional vector analysis, j. math. comput. sci. 10 (2020), 2320-2326. https://doi.org/10.28919/jmcs/4863. [12] i.m. batiha, z. chebana, t.e. oussaeif, a. ouannas, s. alshorm, a. zraiqat, solvability and dynamics of superlinear reaction diffusion problem with integral condition, iaeng int. j. appl. math. 53 (2023), 1-9. [13] a. farah1, a. dababneh, a. zraiqat, on non-parametric criteria for random communication and processes relationship, int. j. adv. sci. eng. 7 (2020), 1675-1690. [14] i.m. batiha, a. obeidat, s. alshorm, a. alotaibi, h. alsubaie, s. momani, m. albdareen, f. zouidi, s.m. eldin, h. jahanshahi, a numerical confirmation of a fractional-order covid-19 model’s efficiency, symmetry. 14 (2022), 2583. https://doi.org/10.3390/sym14122583. [15] a. dababneh, a. zraiqat, a. farah, h. al-zoubi, m.a. hammad, numerical methods for finding periodic solutions of ordinary differential equations with strong nonlinearity, j. math. comput. sci. 11 (2021), 6910-6922. https: //doi.org/10.28919/jmcs/6477. [16] s. noor, m.a. hammad, r. shah, a.w. alrowaily, s.a. el-tantawy, numerical investigation of fractional-order fornberg–whitham equations in the framework of aboodh transformation, symmetry. 15 (2023), 1353. https: //doi.org/10.3390/sym15071353. [17] m.a. hammad, i. jebril, r. khalil, large fractional linear type differential equations, int. j. anal. appl. 21 (2023), 65. https://doi.org/10.28924/2291-8639-21-2023-65. [18] a.h. salas, m. abu hammad, b.m. alotaibi, l.s. el-sherif, s.a. el-tantawy, analytical and numerical approximations to some coupled forced damped duffing oscillators, symmetry. 14 (2022), 2286. https://doi.org/10. 3390/sym14112286. [19] m.a. hammad, conformable fractional martingales and some convergence theorems, mathematics. 10 (2021), 6. https://doi.org/10.3390/math10010006. https://doi.org/10.53391/mmnsa.2021.01.007 https://doi.org/10.53391/mmnsa.2021.01.007 https://doi.org/10.3390/sym15040938 https://doi.org/10.3390/sym15040938 https://doi.org/10.1051/mmnp/2019002 https://doi.org/10.1186/s13662-020-02978-2 https://doi.org/10.1186/s13662-020-02978-2 https://doi.org/10.3390/sym15030605 https://doi.org/10.28919/jmcs/4863 https://doi.org/10.3390/sym14122583 https://doi.org/10.28919/jmcs/6477 https://doi.org/10.28919/jmcs/6477 https://doi.org/10.3390/sym15071353 https://doi.org/10.3390/sym15071353 https://doi.org/10.28924/2291-8639-21-2023-65 https://doi.org/10.3390/sym14112286 https://doi.org/10.3390/sym14112286 https://doi.org/10.3390/math10010006 1. introduction 2. modified fractional euler method 3. solving fractional-order cancer tumor disease model 4. numerical results 5. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 54-63 http://www.etamaths.com a generalized iterative algorithm for hierarchical fixed points problems and variational inequalities vahid dadashi∗ and somayeh amjadi abstract. in this paper we propose a method for approximating of the common fixed point in ∞⋂ n=1 f(tn) where {tn} is a countable family of nonexpansive mappings on a closed convex subset c of a real hilbert space h. then, we prove strong convergence theorems with less control conditions for {tn} which solves some variational inequality. the main results improve and extend the corresponding results of ”f. cianciaruso, g. marino, l. muglia, and y. yao, on a two-step algorithm for hierarchical fixed point problems and variational inequalities, j. inequal. appl., 2009 (2009), article id 208692” and ”y. yao, y.j. cho, and y.c. liou, iterative algorithms for hierarchical fixed points problems and variational inequalities, mathematical and computer modelling, 52(9) (2010), 1697–1705”. 1. introduction let c be a nonempty closed convex subset of a real hilbert space h with the inner product 〈., .〉 and norm ‖.‖, respectively. recall that a mapping t : c → c is called nonexpansive if ‖tx−ty‖≤‖x−y‖ for all x,y ∈ c and a nonself-mapping f : c → h is called a ρ−contraction on c if there exists a constant ρ ∈ [0, 1) such that ‖f(x) − f(y)‖ ≤ ρ‖x − y‖ for all x,y ∈ c. the set of all fixed points of t is denoted by f(t), that is f(t) = {x ∈ c | x = tx}. note that each ρ−contraction f has a unique fixed point in c, and for any fixed element x0 ∈ c, picard’s iteration xn+1 = fn(x0) converges strongly to a unique fixed point of f. however, a simple example shows that picard’s iteration cannot be used in the case of nonexpansive mappings. one method in [6] used for nonexpansive mappings is to employ a halpern-type iterative scheme which produces a sequence {xn} as follows:{ x1 = x ∈ c xn+1 = βnu + (1 −βn)txn, n ≥ 1, (1.1) where u ∈ c is arbitrary and {βn}⊂ [0, 1]. in this paper, we consider the following variational inequalities problem: find x∗ ∈ f(t) such that 〈(i −s)x∗,x−x∗〉≥ 0, ∀x ∈ f(t), (1.2) where t and s are nonexpansive mappings such that f(t) is nonempty. it is easy to see that x∗ is a solution of the variational inequalities (1.2) if and only if it is a fixed point of the nonexpansive mapping pf(t)s, where pf(t) stands for the metric projection on the closed convex set f(t). in 2000, moudafi [8] introduced a viscosity approximation method for a nonexpansive mapping as follows: { x1 = x ∈ c xn+1 = βnf(xn) + (1 −βn)txn, n ≥ 1, (1.3) where f is a contractive mapping and {βn}⊂ [0, 1]. in a real hilbert space and under certain control conditions, he proved the sequence {xn} defined by (1.3) converges strongly to a fixed point of t which is the unique solution to the variational inequality 〈(i −f)x∗,x−x∗〉≥ 0 for all x ∈ f(t). received 19th july, 2016; accepted 20th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 47h09, 47h10. key words and phrases. fixed points; iterative algorithms; nonexpansive mappings; variational inequalities; ρ−contraction. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 54 a generalized iterative algorithm for hierarchical fixed points problems 55 mainge and moudafi [7] introduced an iterative scheme for approximating a specific solution of a fixed point problem as follows:   x1 = x ∈ c yn = αnsxn + (1 −αn)txn, n ≥ 1, xn+1 = βnf(xn) + (1 −βn)yn, n ≥ 1, (1.4) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and s and t are nonexpansive mappings. they proved that if the sequence {xn} given by scheme (1.4) is bounded, then {xn} strongly convergence to the fixed point of a nonexpansive mapping t with respect to a nonexpansive mapping s under some control conditions on {αn} and {βn}. recently, alimohammady and dadashi [1] studied the iterative scheme (1.5) for a countable family of nonexpansive mappings {tn} as follows:  x1 = x ∈ c yn = αnsxn + (1 −αn)tnxn, n ≥ 1, xn+1 = βnf(xn) + (1 −βn)yn, n ≥ 1, (1.5) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and {tn} is a sequence of nonexpansive mappings. they proved that the iterative scheme (1.5) strongly convergence to a common fixed point of {tn} with respect to a nonexpansive mapping s. on the other hand, cianciaruso et al. in [2] studied the sequence generated by the algorithm  x1 = x ∈ c yn = βnsxn + (1 −βn)xn, n ≥ 1, xn+1 = αnf(xn) + (1 −αn)tyn, n ≥ 1, (1.6) where f is a contractive mapping, {αn},{βn}⊂ [0, 1] and s and t are nonexpansive mappings. they proved the sequence {xn} generated by (1.6) strongly converges to the fixed point of a nonexpansive mapping t with respect to a nonexpansive mapping s under some control conditions on {αn} and {βn}. also, they show that this fixed point is a unique solution of a variational inequality. another results about fixed point and variational inequality problems can be found in [3, 4, 9] and the references therein. very recently, yao et al. in [10] introduced another iterative algorithm and proved some strong convergence results for solving the hierarchical fixed point problem (1.2). in this paper, inspired and motivated by the above iterative schemes, we introduced and studied a new composite iterative scheme for countable family of nonexpansive mappings tk (k ∈ n) with respect to a finite family of nonexpansive mapping sk(k ∈{1, 2, ...,n} for some n ∈ n) as follows:{ yn = βnsnxn + (1 −βn)xn, xn+1 = pc(αnf(xn) + (1 −αn)tnyn), ∀n ≥ 1, (1.7) where f is a contractive mapping, {αn},{βn} ⊂ [0, 1] and sn = sn mod n . in particular, if we take f ≡ 0, then it is reduced to the iterative scheme:{ yn = βnsnxn + (1 −βn)xn, xn+1 = pc((1 −αn)tnyn), ∀n ≥ 1, (1.8) the main results improve and extend the corresponding results of [2, 10]. in particular, it should be noticed that we prove strong convergence theorems with less control conditions for {tn} which solves some variational inequality. 2. preliminaries in this section, we recall the well known results and give some useful lemmas that will be used in the next section. let c be a nonempty closed convex subset of a real hilbert space h. for every point x ∈h, there exists a unique nearest point in c, denoted by pc(x), such that ‖x−pc(x)‖≤‖x−y‖, ∀y ∈ c. pc is called the metric projection of h onto c. recall that, pc is characterized by the following lemma 56 dadashi and amjadi lemma 2.1. let x ∈ h and z ∈ c be any points. then z = pc(x) if and only if 〈x− z,y − z〉 ≤ 0, ∀y ∈ c. lemma 2.2. [5] let c be a nonempty closed convex subset of a real hilbert space h and let t : c → c be a nonexpansive mapping with f(t) 6= ∅. if {xn} is a sequence in c weakly converging to x and if {(i −t)xn} converges strongly to y, then (i −t)x = y; in particular, if y = 0, then x ∈ f(t). lemma 2.3. let f : c → h be a contraction with coefficient ρ ∈ [0, 1) and t : c → c be a nonexpansive mapping. then, (i) the mapping (i −f) is strongly monotone with coefficient (1 −ρ) i.e. 〈x−y, (i −f)x− (i −f)y〉≥ (1 −ρ)‖x−y‖2, ∀x,y ∈ c; (ii) the mapping (i −t) is monotone that is 〈x−y, (i −t)x− (i −t)y〉≥ 0, ∀x,y ∈ c. lemma 2.4. [11] assume that {αn} is a sequence of nonnegative numbers such that αn+1 ≤ (1 −γn)αn + δn, ∀n ≥ 0, where {γn} is a subsequence in (0, 1) and {δn} is a sequence in r such that (i) ∞∑ n=1 γn = ∞, (ii) lim sup n→∞ δn γn ≤ 0 or ∞∑ n=1 |δn| < ∞. then lim n→∞ αn = 0. 3. main results in this section, we prove several strong convergence theorems of the iterative scheme (1.7). throughout this section, c is a nonempty closed convex subset of a real hilbert space h, tn for each n ∈ n and sn for each n = 1, 2, ...,n are nonexpansive mappings of c into itself such that f := ∞⋂ n=1 f(tn) is nonempty and f : c →h be a ρ−contraction (possibly nonself) with ρ ∈ [0, 1). theorem 3.1. suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (c1) lim n→∞ αn = 0, ∞∑ n=1 αn = ∞, (c2) lim n→∞ βn αn = 0. then the sequence {xn} generated by (1.7) converges strongly to a point z ∈ f , which is the unique solution of the variational inequality: 〈(i −f)z,x−z〉≥ 0, ∀x ∈ f. (3.1) in particular, if f = 0, then {xn} generated by (1.8) converges in norm to the minimum norm common fixed point z of tn, n ∈ n, namely, the point z is the unique solution to the quadratic minimization problem: z = arg min x∈ ∞⋂ n=1 f(tn) ‖x‖2. (3.2) a generalized iterative algorithm for hierarchical fixed points problems 57 proof. first, we claim that {xn} is bounded. indeed, take an arbitrary fixed u ∈ f = ∞⋂ n=1 f(tn) and using (c2), we can assume, without loss of generality, that βn ≤ αn for all n ≥ 1. from (1.7), we have ‖xn+1 −u‖ = ‖pc(αnf(xn) + (1 −αn)tnyn) −pc(u)‖ ≤ αn‖f(xn) −f(u)‖ + αn‖f(u) −u‖ + (1 −αn)‖tnyn −u‖ ≤ αnρ‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)‖yn −u‖ ≤ αnρ‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)βn‖snxn −u‖ + (1 −αn)(1 −βn)‖xn −u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + (1 −ρ)αn { ‖f(u) −u‖ + ‖snu−u‖ 1 −ρ } ≤ max { ‖xn −u‖, ‖f(u) −u‖ + ‖snu−u‖ 1 −ρ } ≤ max 1≤k≤n { ‖x1 −u‖, ‖f(u) −u‖ + ‖sku−u‖ 1 −ρ } , which implies that the sequence {xn} is bounded and so are the sequences {f(xn)}, {yn}, {tnxn}, {tnyn} and {snxn}. now, we prove that xn → z where, z = pff(z). from lemma 2.1 and set un := αnf(xn) + (1 −αn)tnyn, we get ‖xn+1 −z‖2 = 〈pc(un) −un,pc(un) −z〉 + 〈un −z,xn+1 −z〉 ≤ 〈un −z,xn+1 −z〉 = αn〈f(xn) −f(z),xn+1 −z〉 + (1 −αn)〈tnyn −z,xn+1 −z〉 + αn〈f(z) −z,xn+1 −z〉 ≤ αnρ‖xn −z‖‖xn+1 −z‖ + (1 −αn)‖tnyn −z‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉, (3.3) and hence by the definition of {yn}, we have ‖tnyn −z‖ ≤ ‖tnyn −tnxn‖ + ‖tnxn −xn‖ + ‖xn −z‖ ≤‖yn −xn‖ + ‖tnxn −xn‖ + ‖xn −z‖ ≤ βn‖snxn −xn‖ + ‖tnxn −xn‖ + ‖xn −z‖. (3.4) also, we have ‖tnxn −xn‖≤‖tnxn −tnz‖ + ‖z −xn‖≤ 2‖xn −z‖. (3.5) substituting (3.4) and (3.5) into (3.3) to obtain ‖xn+1 −z‖2 ≤ (αnρ + 3(1 −αn))‖xn −z‖‖xn+1 −z‖ + (1 −αn)βn‖snxn −xn‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉 ≤ αnρ + 3(1 −αn) 2 ( ‖xn −z‖2 + ‖xn+1 −z‖2 ) + (1 −αn)βn‖snxn −xn‖‖xn+1 −z‖ +αn〈f(z) −z,xn+1 −z〉 so ‖xn+1 −z‖2 ≤ ( 1 − 2αn(ρ− 3) + 4 αn(ρ− 3) + 1 ) ‖xn −z‖2 + 2(1 −αn)βn αn(3 −ρ) − 1 ‖snxn −xn‖‖xn+1 −z‖ + 2αn αn(3 −ρ) − 1 〈f(z) −z,xn+1 −z〉 = (1 −γn)‖xn −z‖2 + δn which γn = 2αn(ρ−3)+4 αn(ρ−3)+1 and δn = 2(1−αn)βn αn(3−ρ)−1 ‖snxn −xn‖‖xn+1 − z‖ + 2αnαn(3−ρ)−1〈f(z) − z,xn+1 − z〉. then, lemma 2.4 implies that xn → z as n →∞. in particular, if f = 0, then {xn} generated by (1.8) converges strongly to z ∈ ∞⋂ n=1 f(tn) such that z is the unique solution of the variational inequality 〈z,x−z〉≥ 0, ∀x ∈ f, 58 dadashi and amjadi and hence, for each x ∈ ∞⋂ n=1 f(tn) ‖z‖2 ≤〈z,x〉≤ ‖z‖‖x‖. then for each x ∈ ∞⋂ n=1 f(tn), ‖z‖2 ≤‖x‖2, that is, z is the unique solution to the quadratic minimization problem (3.2). � corollary 3.2. let s , t be nonexpansive mapping of c with f(t) 6= ∅. suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (c1) and (c2). then the sequence {xn} generated by { yn = βnsxn + (1 −βn)xn, xn+1 = pc(αnf(xn) + (1 −αn)tyn), ∀n ≥ 1, (3.6) converges strongly to a point z ∈ f(t), which is the unique solution of the variational inequality: 〈(i −f)z,x−z〉≥ 0, ∀x ∈ f(t). (3.7) in particular, if f = 0, then {xn} generated by (1.8) converges in norm to the minimum norm fixed point z of t , namely, the point z is the unique solution to the quadratic minimization problem: z = arg min x∈f(t) ‖x‖2. proof. it is sufficient that assume sn = s and tn = t in theorem 3.1. � remark 3.3. it is worth to mention that yao et al. in [10] proved that the sequence {xn} generated by (3.6) converges strongly to a point z ∈ f(t), which is the unique solution of the variational inequality (3.7) under control conditions (c1), (c2) and the following conditions lim n→∞ |αn −αn−1| αn = 0, lim n→∞ |βn −βn−1| βn = 0 ∈ (0,∞); (3.8) or ∞∑ n=1 |αn −αn−1| < ∞, ∞∑ n=1 |βn −βn−1| < ∞; (3.9) but corollary 3.2 proves that the sequence {xn} converges strongly under control conditions (c1) and (c2) and it does not require conditions (3.8) and (3.9) for convergence. theorem 3.4. suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (c1), (c2′) lim n→∞ βn αn = τ ∈ (0,∞); (c3) lim n→∞ |βn −βn−1| + |αn −αn−1| αnβn = 0; (c4) there exist a constant k > 0 such that 1 αn | 1 βn − 1 βn−1 | ≤ k; (c5) ∞∑ n=1 sup { ‖tnz − tn−1z‖‖ αnβn , z ∈ b } < ∞ for any bounded subset b of c. let t be a mapping of c into itself defined by tz = lim n→∞ tnz for all z ∈ c and suppose that f(t) = ∞⋂ n=1 f(tn). then the sequence {xn} generated by{ yn = βnsxn + (1 −βn)xn, xn+1 = pc(αnf(xn) + (1 −αn)tnyn), ∀n ≥ 1, (3.10) converges strongly to a point x∗ ∈ f , which is the unique solution of the variational inequality 〈 1 τ (i −f)x∗ + (i −s)x∗,y −x∗〉≥ 0, ∀y ∈ f. (3.11) a generalized iterative algorithm for hierarchical fixed points problems 59 proof. at first, we show that uniqueness of the solution to the variational inequality (3.11) in f(t). in fact, suppose that x∗ and x̃ satisfy in (3.11).then, since x̃ satisfy in (3.11), for y = x∗, it follows that 〈(i −f)x̃, x̃−x∗〉≤ τ〈(i −s)x̃,x∗ − x̃〉. (3.12) similarly, we have 〈(i −f)x∗,x∗ − x̃〉≤ τ〈(i −s)x∗, x̃−x∗〉. (3.13) by (3.12), (3.13) and lemma 2.3, we get (1 −ρ)‖x̃−x∗‖2 ≤ 〈(i −f)x̃− (i −f)x∗, x̃−x∗〉 = 〈(i −f)x̃, x̃−x∗〉−〈(i −f)x∗, x̃−x∗〉 ≤ τ〈(i −s)x̃,x∗ − x̃〉 + τ〈(i −s)x∗, x̃−x∗〉 = −τ〈(i −s)x̃− (i −s)x∗, x̃−x∗〉 ≤ 0. hence, x∗ = x̃. we can assume from (c2′), without loss of generality, that βn ≤ (τ + 1)αn for all n ≥ 1. by a similar argument as that of theorem 3.1, we have ‖xn+1 −u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + αn‖f(u) −u‖ + (1 −αn)βn‖su−u‖ ≤ (1 − (1 −ρ)αn)‖xn −u‖ + αn‖f(u) −u‖ + αn(τ + 1)‖su−u‖ = (1 − (1 −ρ)αn)‖xn −u‖ + (1 −ρ)αn [ ‖f(u) −u‖ 1 −ρ + (τ + 1)‖su−u‖ 1 −ρ ] ≤ max { ‖xn −u‖, ‖f(u) −u‖ 1 −ρ + (τ + 1)‖su−u‖ 1 −ρ } , which implies that the sequence {xn} is bounded. set un = αnf(xn) + (1 −αn)tnyn, then we have ‖xn+1 −xn‖ = ‖pc(un) −pc(un−1)‖≤‖un −un−1‖ ≤ αn‖f(xn) −f(xn−1)‖ + |αn −αn−1|‖f(xn−1) −tn−1yn−1‖ +(1 −αn)‖tnyn −tn−1yn−1‖ ≤ αnρ‖xn −xn−1‖ + |αn −αn−1|‖f(xn−1) −tn−1yn−1‖ + (1 −αn)‖yn −yn−1‖ +‖tnyn−1 −tn−1yn−1‖ (3.14) also by definition of {yn}, we get ‖yn −yn−1‖ ≤ βn‖sxn −sxn−1‖ + (1 −βn)‖xn −xn−1‖ + |βn −βn−1|‖sxn−1 −xn−1‖ ≤ ‖xn −xn−1‖ + |βn −βn−1|‖sxn−1 −xn−1‖. (3.15) set, m = max{sup‖f(xn−1)−tn−1yn−1‖,‖sxn−1−xn−1‖} and substituting (3.15) in (3.14) we have ‖xn+1 −xn‖ ≤ ‖un −un−1‖ ≤ (1 − (1 −ρ)αn)‖xn −xn−1‖ + m [|αn −αn−1| + |βn −βn−1|] (3.16) +‖tnyn−1 −tn−1yn−1‖ ≤ (1 − (1 −ρ)αn)‖xn −xn−1‖ + m(τ + 1)αn [ |αn −αn−1| αn + |βn −βn−1| βn ] +αn [ sup { ‖tnz −tn−1z‖ αn ,z ∈ b }] . 60 dadashi and amjadi from (c1), (c3), (c5) and lemma 2.4, we can deduce that ‖xn+1 −xn‖→ 0. by (3.16) and (c4) we have, ‖xn+1 −xn‖ βn ≤ ‖un −un−1‖ βn ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn + m [ |αn −αn−1| βn + |βn −βn−1| βn ] + [ ‖tnyn−1 −tn−1yn−1‖ βn ] = (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + (1 − (1 −ρ)αn) [ 1 βn − 1 βn−1 ] ‖xn −xn−1‖ +mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn [ ‖tnyn−1 −tn−1yn−1‖ αnβn ] ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + αn [ 1 βn − 1 βn−1 ] 1 αn ‖xn −xn−1‖ +mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn sup z∈b { ‖tnz −tn−1z‖ αnβn } ≤ (1 − (1 −ρ)αn) ‖xn −xn−1‖ βn−1 + αnk‖xn −xn−1‖ +mαn [ |αn −αn−1| + |βn −βn−1| αnβn ] + αn sup z∈b { ‖tnz −tn−1z‖ αnβn } . again, (c1), (c3), (c5) and lemma 2.4 imply that lim n→∞ ‖xn+1 −xn‖ βn = 0, lim n→∞ ‖un −un−1‖ βn = 0, and hence by (c2′) we get lim n→∞ ‖un −un−1‖ αn = 0. it follows from (c1) and (c2′) that βn → 0 and by (3.10), ‖yn − xn‖ → 0 and ‖xn+1 − tnyn‖ → 0. then, ‖xn −tnxn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −tnyn‖ + ‖tnyn −tnxn‖ ≤ ‖xn −xn+1‖ + ‖xn+1 −tnyn‖ + ‖yn −xn‖→ 0 as n →∞, therefore, we have ‖xn −txn‖ ≤ ‖xn −tnxn‖ + ‖tnxn −txn‖ ≤ ‖xn −tnxn‖ + sup{‖tnz −tz‖, z ∈{xn}}→ 0 as n →∞. by demiclosedness principle, lemma 2.2, we obtain ww(xn) ⊆ f(t) = ∞⋂ n=1 f(tn). also ‖yn −tnyn‖≤‖yn −xn‖ + ‖xn −xn+1‖ + ‖xn+1 −tnyn‖→ 0. from (3.10), we have xn+1 = pc(un) −un + αnf(xn) + (1 −αn)(tnyn −yn) + (1 −αn)[βnsxn + (1 −βn)xn], and hence xn −xn+1 = xn −pc(un) + un −αnf(xn) − (1 −αn)(tnyn −yn) − (1 −αn)βnsxn − (1 −αn)(1 −βn)xn = un −pc(un) + αn(i −f)xn + (1 −αn)(i −tn)yn + (1 −αn)βn(i −s)xn. set vn = xn−xn+1 (1−αn)βn . hence, we obtain vn = 1 (1 −αn)βn (un −pc[un]) + αn (1 −αn)βn (i −f)xn + 1 βn (i −tn)yn + (i −s)xn. a generalized iterative algorithm for hierarchical fixed points problems 61 for any z ∈ ∞⋂ n=1 f(tn) we have 〈vn,xn −z〉 = 1 (1 −αn)βn 〈un −pc(un),xn −z〉 + αn (1 −αn)βn 〈(i −f)xn,xn −z〉 (3.17) + 1 βn 〈(i −tn)yn,xn −z〉 + 〈(i −s)xn,xn −z〉 = 1 (1 −αn)βn 〈un −pc(un),pc(un−1) −pc(un) + pc(un) −z〉 + αn (1 −αn)βn 〈(i −f)xn − (i −f)z + (i −f)z,xn −z〉 + 1 βn 〈(i −tn)yn − (i −tn)z,xn −z〉 + 〈(i −s)xn − (i −s)z + (i −s)z,xn −z〉 = 1 (1 −αn)βn 〈un −pc(un),pc(un) −z〉 + 1 (1 −αn)βn 〈un −pc(un),pc(un−1) −pc(un)〉 + αn (1 −αn)βn 〈(i −f)xn − (i −f)z,xn −z〉 + αn (1 −αn)βn 〈(i −f)z,xn −z〉 + 1 βn 〈(i −tn)yn − (i −tn)z,xn −yn〉 + 1 βn 〈(i −tn)yn − (i −tn)z,yn −z〉 +〈(i −s)xn − (i −s)z,xn −z〉 + 〈(i −s)z,xn −z〉 by lemma 2.3, we obtain 〈vn,xn −z〉 ≥ 1 (1 −αn)βn 〈un −pc(un),pc(un−1) −pc(un)〉 + αn(1 −ρ) (1 −αn)βn ‖xn −z‖2 + αn (1 −αn)βn 〈(i −f)z,xn −z〉 + 1 βn 〈(i −tn)yn − (i −tn)z,xn −yn〉 + 〈(i −sn)z,xn −z〉 ≥ 1 (1 −αn)βn 〈un −pc(un),pc(un−1) −pc(un)〉 + αn(1 −ρ) (1 −αn)βn ‖xn −z‖2 + αn (1 −αn)βn 〈(i −f)z,xn −z〉 +〈(i −tn)yn,xn −sxn〉 + 〈(i −s)z,xn −z〉 then it follows that ‖xn −z‖2 ≤ (1 −αn)βn αn(1 −ρ) [〈vn,xn −z〉− 1 (1 −αn)βn 〈un −pc(un),pc(un−1) −pc(un)〉 − αn (1 −αn)βn 〈(i −f)z,xn −z〉−〈(i −tn)yn,xn −sxn〉−〈(i −s)z,xn −z〉] ≤ (1 −αn)βn αn(1 −ρ) [〈vn,xn −z〉−〈(i −tn)yn,xn −sxn〉−〈(i −s)z,xn −z〉] + ‖un −un−1‖ αn(1 −ρ) ‖un −pc(un)‖− 1 (1 −ρ) 〈(i −f)z,xn −z〉 since vn → 0, (i − tn)yn → 0, ‖un−un−1‖ αn → 0 and ωw(xn) ⊆ f(t) = ∞⋂ n=1 f(tn), then every weak cluster point of {xn} is also a strong cluster point. it follows from the boundedness of the sequence {xn} that there exists a subsequence {xnk} converging to a point x ′ ∈h. for all z ∈ f(t), it follows 62 dadashi and amjadi from (3.17) that 〈(i −f)xnk,xnk −z〉 = (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(i −s)xnk,xnk −z〉 − 1 αnk 〈unk −pc(unk ),pc(unk−1 ) −z〉− (1 −αnk ) αnk 〈(i −tnk )ynk,xnk −z〉 ≤ (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(i −s)z,xnk −z〉 − 1 αnk 〈unk −pc(unk ),pc(unk−1 ) −pc(unk )〉 − (1 −αnk ) αnk 〈(i −tnk )ynk − (i −tnk )z,xnk −ynk〉 ≤ (1 −αnk )βnk αnk 〈vnk,xnk −z〉− (1 −αnk )βnk αnk 〈(i −s)z,xnk −z〉 + ‖unk −unk−1‖ αnk ‖unk −pc(unk )‖− (1 −αnk )βnk αnk 〈(i −tnk )ynk,xnk −sxnk〉. letting k →∞, we obtain 〈(i −f)x′,x′ −z〉 ≤ −τ〈(i −s)z,x′ −z〉. thus x′ is a solution of the variational inequality (3.11) and since (3.11)has the unique solution, it follows that ωw(xn) = ωs(xn) = {x∗} and this ensures that xn → x∗ as n →∞. � corollary 3.5. suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (c1), (c3), (c4), (c5) and (c2′) lim n→∞ βn αn = 1; then the sequence {xn} defined by (1.8) converges strongly to a point x∗ ∈ f, which is the unique solution of the variational inequality 〈(2i −s)x∗,y −x∗〉≥ 0, ∀y ∈ f. (3.18) proof. it is sufficient that assume f = 0 and τ = 1 in theorem 3.4. � corollary 3.6. let s,t : c → c be two nonexpansive mappings with f(t) 6= ∅. suppose that {αn} and {βn} are sequences in (0, 1) which satisfy in conditions (c1), (c2′′), (c3) and (c4). then the sequence {xn} defined by (3.6) converges strongly to a point x∗ ∈ f(t), which is the unique solution of the variational inequality 〈 1 τ (i −f)x∗ + (i −s)x∗,y −x∗〉≥ 0, ∀y ∈ f(t). proof. it is sufficient that assume tn = t in theorem 3.4. � acknowledgment vahid dadashi and somayeh amjadi are supported by sari branch, islamic azad university. references [1] m. alimohammady and v. dadashi, convergence of a generalized iterations for a countable family of nonexpansive mappings, tjmm. 4(1) (2012), 15–24. [2] f. cianciaruso, g. marino, l. muglia, and y. yao, on a two-step algorithm for hierarchical fixed point problems and variational inequalities, j. inequal. appl., 2009 (2009), article id 208692. [3] v. dadashi, s. ghafari, convergence theorems of iterative approximation for finding zeros of accretive operator and fixed points problems, int. j. nonlinear anal. appl., 4(2) (2013) , 53–61. [4] y. dong and x. zhang, new step lengths in projection method for variational inequality problems, appl. math. comput. 220(1) (2013), 239–245. a generalized iterative algorithm for hierarchical fixed points problems 63 [5] k. goebel and w.a. kirk, topics in metric fixed point theory, cambridge studies in advanced mathematics, vol. 28, cambridge university press, 1990. [6] b. halpern, fixed points of nonexpanding maps, bull. amer. math. soc. 220(1) (1967), 957–961. [7] p.e. mainge and a. moudafi, strong convergence of an iterative method for hierarchical fixed point problems, pacific j. optim. 3 (2007), 529–538. [8] a. moudafi, viscosity approximation methods for fixed point problems, j. math. anal. appl., 241 (2000), 46–55. [9] x.qin, m. shang, h. zhou, strong convergence of a general iterative method for variational inequality problems and fixed point problems in hilbert spaces, appl. math. comput., 200(1) (2008), 242–253. [10] y. yao, y.j. cho, and y.c. liou, iterative algorithms for hierarchical fixed points problems and variational inequalities, mathematical and computer modelling, 52(9) (2010), 1697–1705. [11] h. k. xu, iterative algorithm for nonlinear operators, j. lond. math. soc., 2 (2002), 1–17. department of mathematics, sari branch, islamic azad university, sari, iran ∗corresponding author: vahid.dadashi@iausari.ac.ir 1. introduction 2. preliminaries 3. main results acknowledgment references polynomial approximation on unbounded subsets and the markov moment problem  2 l  1 l ,rx        .,exp ! exp ! ... !1 1exp 0            mdttt m x m xx x m xm  xexp rx  ,cz   k,0           .0, !12!2!2!1 1exp ,0, !2!2!1 1exp 12122222 2222              t k t k ttt t t k ttt t kkkk kk           ,,0,exp  ktkttk   .; kk       ,0,   tttpp lll  lplim ),0[ k  ,),0[ 1 l  ).,0[   r),0[:     rtt lim  llh kk ,       ll hlttth lim,,0, ).,0[      lllll plhpp ~ lim,, ~ , ~ ).,0[    .0,,exp  tntn n ra   ,a    ,0  ac  mmp ,a   mm pp ,  . 1 al   a a m ddp ,lim  p   al 1  p  . 1 al     .2,1,0,,,, 2 2121,  ltkjttttx l kj kj r       ,2,1,0, 2 2, 2 1,  lttpttptp llllllll h 21 , aa ,h   .2,1, ja j   21 , aayy      .; ,2,1,; 11 1 yttuutyuy jtatahty jj   x ,),0[: 2 rx  ,),0[ 2   kjx ,     .0,, 2121,  j kj kj tttttx      ., 121 kcxaak       .2,, yb zkjkj     yxlf ,1                        ;1,, ,,0,, 121 21 21 ,121 21 21 2 ,,         fxdedettf xxdedettxxfzkjbxf aa aa aa aa kjkj      zjj 21 ,     ,, 21 jkkjj j           .0 ,0 ,0 ,0 1 2 1 1 ,,, 1,1 ,,, 1 21 ,,, 1, ,,, 2 1 1 ,,, ,1 ,,, 21 ,,, , ,,, 2121 2121 2121 2121                         lkji lkji jlkjji lkjilkji jlkjji lkji lkji jlkjji lkjilkji jlkjji lkji lkji jlkjji lkjilkji jlkjji lkji lkji jlkjji lkjilkji jlkjji aab aab aab aab     x    .21 aa   x       .),0[),0[   cc cc        .,, 212211 aababa     2,1,, jba jj 2,1,0,21  jxxx j ,, 21 tt ,2,1,,,,0 ,,   jmxpzmxp jjmjjm ),,0[        .,2,1,2 , 2 ,,   zmjtrttqtp jjmjjjmjjm 21 p            ., 2 ,, , , , ,            zkjbxfbxf kjkj kj kjkj kj kjkj  .21 pp               .2,1,,0 21 21 221121     jtrpdedetptpppf jjaa aa      211 aacx   yxf 1: .f                   .,, lim limlim ,121 21 21 21 21 1 ,2,,1, 1 ,2,,1, 1 ,2,,1,                                              xxdedettx dedepp ppfppfxf aa aa aa aa mk j jmjmm mk j jmjmm mk j jmjmm                 .1, , 1 21 21 21       fxf dedettfffx aa aa    )()( ab  □ v v p .v x y 0 u xs  xa  v    avs  a s a    yslf jj j ,   ,0\~   yy    yxlf jj j ,  jsj ff | ., ~ | jjyf aj  v       , ~ ..0,|..0 ,,, 011 00 uytsptsr avsuusvf av j           .,,1 01 jjxxuxpxf vj   ., jjf j  hx   ,1 1    n j jzd  ,0,...,0 .d x     1,,,..., 1 1 1 11     n k k n kk nj n j nj jjjjzzzz  nka k ,...,1,  ,h .,...,1,1 nka k      .;,,...,1,; 111 yuuvvuyvynkuauahuy kk  y   .,...,1,1, 1 nkbyb k n kk     1,   ju n j j ,y   .1,,...,, 1 1 1 1 1  jjjjbbaau n n nj n j nj n j j       .1,10,...,0,, ~ n jjn j j jxyb      yxbf ,               .,1/1/: ,1/1/ ~ 2 ,, ~ ,1, 11 0 0 1 11 xibau ubabf jbfjuf n k k n k k n k k n k k n jjj                                                                                   .;,1,; nj n j jcoajjsps    0,,0  j          .11,0 10,...,00,...,0 |     avv mjmj ppabs      ,,1,0 00               jj jj jj jj ufsfbss  s ju          .1,0, 11 000 11 1 1 1 1 1 1 1 1 1 1 00 bssusfuui ba ibbaa bbaas uusf n k k n k k nj nj n j j nj nj n j j n j n j nj n jnj n j j jj j jj jj                                                                                                   . ~ 1/1/ ~~ 0 1 11 ubbabb n k k n k k                                  □ x y     yyxx jj j jj j   ,  yxlff ,, 21   yxlf ,         ;,21 jjyxfxxxfxfxf jj   jj 0   , 0 r jj j       .,, 1122 00 2112  ffyxx jj jj jj jj                 yxbf ,         ; ,,,...,,...,0,, 11      f xbbaafjuf nn n jj   .,...,,0 1 1 1 1 1 n n nj n j nj n j j jjjbbaau       ba           .,,0 ,...,,...,,0 1 1 1 1 11 nnj n jnj n j njnjjjj jbbaa bbaafux                  .0; ,0:,,...,,..., ,0,, 00 111221212 1 1 1 1 0 0 1 0 2              j nn nj n jnj n j n j jj n j j jj jj jj jj n mmm n m mm n m m jj jj jjj fffbbaa bbaauuu m       ,, yxlf        .,,...,,...,0 11  xbbaaf nn  ,x                    . ,,...,,..., ,...,,..., 11 11        fx fbbaa bbaafff nn nn □ n r international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 44-52 http://www.etamaths.com on the meromorphic solutions of certain nonlinear difference equations veena l. pujari abstract. in this article, we investigate the meromorphic solutions of certain non-linear difference equations using tumura-clunie theorem and also provide examples which satisfy our results. 1. introduction and main results meromorphic solutions of complex differential equations and complex difference equations plays a prominent role in the field of complex analysis. solutions of such equations admits several ways of approach, but recently solutions of complex differential or difference equations by nevanlinna theory techniques has become a subject of great interest. the clunie lemma and tumura-clunie type theorems were efficient tool in finding the solutions of complex differential or difference equations. in this article, we solve certain complex non-linear difference equations using tumura-clunie type theorems. we assume that the reader is familiar with the basic notions of nevanlinna’s value distribution theory [see [8],[9]]. in [7], anupama j.patil proved the following the result theorem a. no trancendental meromorphic function f with n(r,f) = s(r,f) will satisfy an equation of the form a1(z)p(f)π(f) + a2(z)π(f) + a3(z) ≡ 0 where a1(z)( 6≡ 0), a2(z) and a3(z) are small functions of f, p(f) = bnf n + bn−1f n−1 + . . . + b1f + b0 where n is a positive integer, bn(6≡ 0),bn−1, . . . ,b0 are small functions of f and π(f) is a differential polynomial in f i.e, π(f) = n∑ i=1 αi(z)f ni0 (f′)ni1 (f′′)ni2 . . . (f(m))nim in this paper, we obtain two main results by considering difference function f(z +c) and difference polynomial in place of π(f) in theorem a. theorem 1.1 no transcendental meromorphic function f of finite order ρ with n(r,f) = o(rρ−1+�) + s(r,f) will satisfy the non-linear difference equation of the form a1(z)p(f)f(z + c) + a2(z)f(z + c) + a3(z) ≡ 0 2010 mathematics subject classification. 30d35. key words and phrases. meromorphic function, difference polynomial, difference equation and tumura-clunie theorem. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 44 nonlinear difference equations 45 where c ∈ c, a1(z)( 6= 0), a2(z) and a3(z)( 6= 0) are small functions in the sense of t(r,ai) = o(r ρ−1+�) + s(r,f), i = 1, 2, 3 and p(f) = bnf n + bn−1f n−1 + . . . + b1f + b0 where n is a positive integer, bn( 6≡ 0),bn−1, . . . ,b0 are small functions in the sense of t(r,bj) = o(r ρ−1+�) + s(r,f),j = 0, 1, 2, . . . ,n. theorem 1.2 no transcendental meromorphic function f of finite order ρ with n(r,f) +n(r, 1/f) = o(rρ−1+�) +s(r,f) will satisfy the difference equation of the form a1(z)p(f)π(f) + a2(z)π(f) + a3(z) ≡ 0 where n ≥ 1, a1(z)( 6= 0), a2(z) and a3(z)( 6= 0) are small functions in the sense of t(r,ai) = o(r ρ−1+�) + s(r,f), i = 1, 2, 3 and π(f) = ∑ λ aλ(z)f(z) l0f(z + c1) l1 . . .f(z + cλ) lλ is a difference polynomial of degree n where n = max ∑λ j=1 lj and c1,c2, . . . ,cλ are distinct values in c and t(r,aλ) = s(r,f) 2. some lemmas lemma 2.1 [2]. let f be a meromorphic function of finite order ρ, and suppose that ψ(z) = an(z)f(z) n + · · · + a0(z) has small meromorphic coefficients aj(z), an 6= 0 in the sense of t(r,aj) = o(rρ−1+�)+ s(r,f). moreover , assume that n(r, 1 ψ ) + n(r,f) = o(rρ−1+�) + s(r,f). then ψ = an ( f + an−1 nan )n . lemma 2.2 [8]. suppose f(z) is a meromorphic function in the complex plane and p(z) = a0f n +a1f n−1 +· · ·+an, where a0(6≡ 0),a1, · · · ,an are small functions of f(z). then t(r,p(f)) = nt(r,f) + s(r,f) lemma 2.3 [1]. let f be a meromorphic function with exponent of convergence of poles λ( 1 f ) = λ < +∞, η 6= 0 be fixed, then for each � > 0, n(r,f(z + η)) = n(r,f) + o(rρ−1+�) + o(logr). lemma 2.4 [3,4]. let f(z) be a meromorphic function of finite order σ and let c be a fixed non-zero complex constant. then for each � > 0, we have m ( r, f(z + c) f(z) ) + m ( r, f(z) f(z + c) ) = o(rσ−1+�) lemma 2.5 [1]. let f(z) be a meromorphic function of finite order σ, and let η be a fixed non-zero complex number, then for each � > 0, we have t(r,f(z + η)) = t(r,f) + o(rσ−1+�) + o(logr). 46 veena l. pujari lemma 2.6 [3]. let f(z) be a non-constant meromorphic solution of f(z)np(z,f) = q(z,f), where p(z,f) and q(z,f) are difference polynomials in f(z), and let δ < 1 and � > 0. if the degree of q(z,f) as a polynomial in f(z) and its shifts is at most n, then m(r,p(z,f)) = o ( t(r + |c|,f)1+� rδ ) + o(t(r,f)) for all r outside of a possible exceptional set with finite logarithmic measure. proof of theorems proof of theorem 1.1 we prove this theorem by contradiction. we first consider the case n ≥ 2. suppose there exists a transcendental meromorphic function f(z) of finite order ρ with (1) n(r,f) = o(rρ−1+�) + s(r,f) satisfying (2) a1(z)p(f)f(z + c) + a2(z)f(z + c) + a3(z) ≡ 0 i.e a1 [ bnf n + bn−1f n−1 + · · · + b1f + b0 ] f(z + c) + a2f(z + c) + a3 ≡ 0 (3) =⇒ a1bnfnf(z + c) + p1(f)f(z + c) + a3 ≡ 0 where p1(f) = a1bn−1f n−1 + · · · + a1b1f + a1b0 + a2 by our assumption (1) and lemma 2.3, we have (4) n(r,f(z + c)) = o(rρ−1+�) + s(r,f) now (3) can be written as a1bnf n + p1(f) ≡− a3 f(z + c) consider h(z) ≡ fn + p1(f) a1bn ≡− a3 a1bnf(z + c) (5) from (4) and (5), we write n ( r, 1 h ) ≤ n ( r, 1 h ) = n ( r, −a1bnf(z + c) a3 ) = o(rρ−1+�) + s(r,f) with this and by our assumption, we have n ( r, 1 h ) + n(r,f) = o(rρ−1+�) + s(r,f) now applying lemma 2.1, we get h(z) = ( f(z) + bn−1 nbn )n (6) nonlinear difference equations 47 from (5) and (6), we have( f(z) + bn−1 nbn )n ≡− a3 a1bnf(z + c) =⇒ ( f(z) + bn−1 nbn )n f(z + c) ≡− a3 a1bn thus t ( r, ( f(z) + bn−1 nbn )n f(z + c) ) = t ( r, a3 a1bn ) =⇒ t ( r, ( f(z) + bn−1 nbn )n f(z + c) ) = o(rρ−1+�) + s(r,f)(7) using (4), we write n ( r, f(z + c) f(z) ) ≤ n(r,f(z + c)) + n ( r, 1 f(z) ) = n ( r, 1 f(z) ) + o(rρ−1+�) + s(r,f)(8) now, using lemma 2.4 and (8), we get t ( r, f(z + c) f(z) ) = m ( r, f(z + c) f(z) ) + n ( r, f(z + c) f(z) ) = n ( r, 1 f(z) ) + o(rρ−1+�) + s(r,f) ≤ t(r,f) + o(rρ−1+�) + s(r,f)(9) now by the first fundamental theorem of nevanlinna and from (7) and (9), we have t ( r,f ( f(z) + bn−1 nbn )n) = t  r, 1 f ( f(z) + bn−1 nbn )n   + o(1) ≤ t ( r, f(z + c) f(z) ) + t  r, 1 f(z + c) ( f(z) + bn−1 nbn )n   + o(1) ≤ t(r,f) + o(rρ−1+�) + s(r,f)(10) on the other hand, using lemma 2.2, we write t ( r,f ( f(z) + bn−1 nbn )n) = (n + 1)t(r,f) + s(r,f)(11) thus from (10) and (11), we get (n + 1)t(r,f) + s(r,f) ≤ t(r,f) + o(rρ−1+�) + s(r,f) nt(r,f) ≤ o(rρ−1+�) + s(r,f) 48 veena l. pujari which is contradiction. thus our assumption is false. next, we shall consider the case n = 1. if n = 1, then (2) becomes a1(z)(b1(z)f(z) + b0)f(z + c) + a2(z)f(z + c) + a3(z) ≡ 0 =⇒ a1b1f(z)f(z + c) + a1b0f(z + c) + a2f(z + c) ≡−a3 =⇒ [a1b1f(z) + (a1b0 + a2)] f(z + c) ≡−a3 =⇒ [ f(z) + (a1b0 + a2) a1b1 ] f(z + c) ≡− a3 a1b1 (12) degree of − a3 a1b1 is zero and the degree of the term [ f(z) + (a1b0+a2) a1b1 ] is one. hence by lemma 2.6, we get m(r,f(z + c)) = o ( t(r + |c|,f)1+� rδ ) + s(r,f), where δ < 1 and � > 0, which holds for all r outside of a possible exceptional set with finite logarithmic measure. thus using (4), we write t(r,f(z + c)) ≤ o(rρ−1+�) + s(r,f) now, we write (12) as[ f(z) + (a1b0 + a2) a1b1 ] ≡− a3 a1b1f(z + c) thus t ( r,f(z) + (a1b0 + a2) a1b1 ) ≡ t ( r,− a3 a1b1f(z + c) ) t(r,f) ≡ o(rρ−1+�) + s(r,f), which is again a contradiction. thus our assumption is false. hence the theorem. example : let f(z) = 2z2+1 and ρ(f(z)) = 0 (finite order) with n(r,f) = s(r,f). consider p(f) = f(z) + 1. then (2) becomes a1(z)(f(z) + 1)f(z + c) + a2(z)f(z + c)a3(z) ≡ 0 =⇒ f(z) + ( a1(z) + a2(z) a1(z) ) ≡− a3(z) a1(z)f(z + c) =⇒ t ( r,f(z) + ( a1(z) + a2(z) a1(z) )) = t ( r,− a3(z) a1(z)f(z + c) ) applying nevanlinna’s first fundamental theorem and using lemma 2.2 and 2.5, we obtain t(r,f) + s(r,f) = t(r,f(z + c)) + o(rρ−1+�) + s(r,f) =⇒ t(r,f) + s(r,f) = t(r,f) + o(rρ−1+�) + s(r,f) remarks: 1. in theorem 1.1, a2(z) may or may not be zero. if a2(z) 6= 0, we can proceed as in the proof of theorem 1.1. if a2(z) = 0, then (2) becomes a1(z)p(f)f(z+c)+a3(z) ≡ 0 =⇒ a1(z)p(f) ≡ −a3(z) f(z+c) we can proceed as in the proof of theorem 1.1. so, in both the cases, we obtain a non-transcendental meromorphic solution f(z) nonlinear difference equations 49 of finite order ρ with n(r,f) = o(rρ−1+�) + s(r,f) will satisfy the non-linear difference equation of the form a1(z)p(f)f(z + c) + a2(z)f(z + c) + a3(z) ≡ 0. 2. in theorem 1.1, a3(z) 6= 0. if a3(z) = 0, then (2) becomes a1(z)p(f)f(z + c) + a2(z)f(z + c) ≡ 0 =⇒ a1(z)p(f) ≡−a2(z) =⇒ p(f) ≡ −a2(z) a1(z) thus t(r,p(f)) = t ( r, −a2(z) a1(z) ) using lemma 2.2, we get nt(r,f) + s(r,f) = o(rρ−1+�) + s(r,f), which is contradiction. similarly,, if a1(z) = 0 we obtain a contradiction. hence a1(z) 6= 0. proof of theorem 1.2 we prove this theorem also by contradiction method. we first consider the case n ≥ 2. suppose, there exists a transcendental meromorphic function f(z) of finite order ρ with (13) n(r,f) = o(rρ−1+�) + s(r,f) satisfying the equation (14) a1(z)p(f)π(f) + a2(z)π(f) + a3(z) ≡ 0 i.e a1 [ bnf n + bn−1f n−1 + · · · + b1f + b0 ] π(f) + a2π(f) + a3 ≡ 0 (15) =⇒ a1bnfnπ(f) + p1(f)π(f) + a3 ≡ 0 where p1(f) = a1bn−1f n−1 + · · · + a1b1f + a1b0 + a2 we have difference polynomial as π(f) = ∑ λ aλ(z)f(z) l0f(z + c1) l1 . . .f(z + cλ) lλ = f(z)n ∑ λ aλ(z)f(z) l0f(z + c1) l1 . . .f(z + cλ) lλ f(z)n = f(z)n ∑ λ aλ(z) ( f(z + c1) f(z) )l1 (f(z + c2) f(z) )l2 . . . ( f(z + cλ) f(z) )lλ by lemma 2.3 and (13), we get n ( r, f(z + ci) f(z) ) ≤ n (r,f(z + ci)) + n ( r, 1 f(z) ) , i = 1, 2, . . .λ = o(rρ−1+�) + s(r,f)(16) combining this with the assumption that t(r,aλ) = s(r,f), we obtain that n (r, π(f)) = o(rρ−1+�) + s(r,f)(17) 50 veena l. pujari now (15) can be written as f(z)n + p1(f) a1bn ≡− a3 a1bnπ(f) ≡ ψ(z) (say)(18) from (17) and (18), we have n ( r, 1 ψ(z) ) ≡ n ( r,− a1bnπ(f) a3 ) = o(rρ−1+�) + s(r,f)(19) since ψ(z) = f(z)n + bn−1 bn f(z)n−1 + bn−2 bn f(z)n−2 + . . . + b0 bn + a2 bn by assumption and (19), we write n ( r, 1 ψ ) + n(r,f) ≤ n ( r, 1 ψ ) + n(r,f) = o(rρ−1+�) + s(r,f) then applying the lemma 2.1, we get ψ(z) = [ f(z) + bn−1 nbn ]n (20) from (18) and (20), we have[ f(z) + bn−1 nbn ]n ≡− a3 a1bnπ(f) =⇒ [ f(z) + bn−1 nbn ]n π(f) ≡− a3 a1bn thus t ( r, [ f(z) + bn−1 nbn ]n π(f) ) = t ( r,− a3 a1bn ) = o(rρ−1+�) + s(r,f)(21) consider π(f) f(z)n = ∑ λ aλ [ f(z + c1) f(z) ]l1 [f(z + c2) f(z) ]l2 · · · · · · [ f(z + cλ) f(z) ]lλ so m ( r, π(f) f(z)n ) = ∑ λ [ m(r,aλ) + λ∑ i=1 lim ( r, f(z + ci) f(z) )] using lemma 2.4 and m(r,aλ) = s(r,f), we have m ( r, π(f) f(z)n ) = o(rρ−1+�) + s(r,f) using (16) with this, we obtain t ( r, π(f) f(z)n ) = o(rρ−1+�) + s(r,f)(22) nonlinear difference equations 51 now, by the first fundamental theorem of nevanlinna and from (21) and (22), we have t ( r,f(z)n [ f(z) + bn−1 nbn ]n) = t  r, 1 f(z)n [ f(z) + bn−1 nbn ]n   + o(1) ≤ t ( r, π(f) f(z)n ) + t  r, 1 π(f) [ f(z) + bn−1 nbn ]n   + o(1) = o(rρ−1+�) + s(r,f)(23) by lemma 2.2, we have t ( r,f(z)n [ f(z) + bn−1 nbn ]n) = 2nt(r,f) + s(r,f)(24) thus from (23) and (24), we get 2nt(r,f) + s(r,f) ≤ o(rρ−1+�) + s(r,f) which is contradiction. hence, our assumption is false. now we shall consider the case when n = 1. if n = 1, then (14) becomes a1(z)(b1f + b0)π(f) + a2(z)π(f) + a3(z) ≡ 0[ f(z) + a1b0 + a2 a1b1 ] π(f) ≡− a3 a1b1 (25) the degree of − a3 a1b1 is zero and the degree of the term [ f(z) + a1b0+a2 a1b1 ] is one. hence applying lemma 2.6 to (25), we write m(r, π(f)) = o ( t(r + |c|,f)1+� rδ ) + s(r,f),(26) where δ < 1 and � > 0, which holds for all r outside of a possible exceptional set with finite logarithmic measure. thus adding (17) and (26), we write t(r, π(f)) = o(rρ−1+�) + s(r,f)(27) now (25)can be written as[ f(z) + (a1b0 + a2) a1b1 ] ≡− a3 a1b1π(f) thus by (27), we have t ( r,f(z) + (a1b0 + a2) a1b1 ) ≡ t ( r,− a3 a1b1π(f) ) =⇒ t(r,f) = o(rρ−1+�) + s(r,f) which is again a contradiction. thus our assumption is false. hence the theorem. example: let f(z) = 2z2 + z + 1 and ρ(f(z)) = 0 with n(r,f) + n(r, 1/f) = 52 veena l. pujari s(r,f). consider p(f) = f2(z) + 1 and π(f) = f(z)f(z + c). then (2) becomes a1(z)(f 2(z) + 1)(f(z)f(z + c)) + a2(z)(f(z)f(z + c)) + a3(z) ≡ 0 =⇒ f2(z) + ( a1(z) + a2(z) a1(z) ) ≡− a3(z) a1(z)f(z)f(z + c) =⇒ t ( r,f2(z) + ( a1(z) + a2(z) a1(z) )) = t ( r,− a3(z) a1(z)f(z)f(z + c) ) applying nevanlinna’s first fundamental theorem and using lemma 2.2 and 2.5, we obtain 2t(r,f) + s(r,f) = t(r,f) + t(r,f(z + c)) + o(rρ−1+�) + s(r,f) =⇒ 2t(r,f) + s(r,f) = 2t(r,f) + o(rρ−1+�) + s(r,f) remarks: 1. in theorem 1.2, a2(z) may or may not be zero. in both the cases, we obtain a non-transcendental meromorphic solution. 2. if a1(z) = 0 and a3(z) = 0, we obtain a contradiction. hence a1(z) 6= 0 and a3(z) 6= 0. 3. lemma 2.3, 2.4 and 2.5 fails for meromorphic function of infinite order. thus theorem 1.1 and 1.2 are not true for infinite order. references [1] yik-man chang and shao-ji feng, on the nevanlinna characteristic of f(z + η) and difference equations in the complex plane, ramanujan journal 16(2008), 105-129. [2] i.laine and c.c.yang , value distribution of difference polynomials, proc. japan acad., 83, ser.a(2007), 148-151. [3] r.g.halburd and r.korhonen, difference analogue of the lemma on the logarithmic derivative with application to difference equations, j.math.anal.appl., 314(2006), 477-487. [4] r.g.halburd and r.korhonen, nevanlinna theory for the difference operator, ann.acad.sci.fenn.math. 31(2006) 463-478. [5] x.luo and w.c.lin,value sharing results for shifts of meromorphic functions, j.math.math.anal.appl. 377(2011) 441-449. [6] e.mues and n.steinmetz,the theorem of tumura-clunie for meromorphic functions, j.london math. soc, 23(2) (1981), 113-122. [7] anupama j.patil, nevanlinna theory investigations and some applications, ph.d thesis, karnatak university, dharwad. karanataka-state, india, 2007. [8] c.c.yang and h.x.yi, uniqueness of meromorphic functions, kluwer academic publishers group, dordredt, 2003. [9] w.k.hayman, meromorphic functions, clarendon press, oxford, 1964. post graduate department of mathematics,vijaya college, r.v.road, basavanagudi, bangalore-560004, india international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 53-62 http://www.etamaths.com on estimates for the dunkl transform in the space l2,α(r) mohamed el hamma∗ and radouan daher abstract. in this paper, we study two estimates useful in applications are proved for the dunkl transform in a hilbert space l2,α(r) = l2(r, |x|2α+1dx), α > −1 2 as applied to some classes of functions characterized by a generalized modulus of continuity. 1. introduction and preliminaries dunkl operators are differential-difference operators introduced in 1989, by dunkl [5]. on the real line, these operators, which are denote by dα, depend on a real parameter α > −1 2 and they are associated with the reflection group z2 on r . for α > −1 2 , dunkl kernel eα is defined as the unique solution of a differentialdifference equation related to dα and satisfying eα(0) = 1. this kernel is used to define dunkl transform which was introduced by dunkl in [6]. more complete results concerning this transform were later obtained by de jeu [7]. rösler in [8] shows that dunkl kernels verify a product formula. this allows us to define dunkl translation operators th, h ∈ r. the dunkl operator on r of index (α + 1 2 ) is defined in [5] by df(x) = dαf(x) = df(x) dx + (α + 1 2 ) f(x) −f(−x) x , α > −1 2 . these operators are very important in mathematics and physics. in this paper, we prove two useful estimates in certain classes of functions characterized by a generalized continuity modulus and connected with the dunkl transform in l2,α(r), for this purpose, we use a translation operator in [4]. we point out that similar results have been established in the context of fourier transform in real line (see [2]). assume that l2,α(r), is stand for the hilbert space which consists of measurable functions f(x) is defined on r with the finite norm ‖f‖ = ‖f‖2,α = (∫ +∞ −∞ |f(x)|2|x|2α+1dx )1 2 , 2010 mathematics subject classification. 44a15; 33c52. key words and phrases. dunkl operator; generalized modulus of continuity; dunkl transform. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 53 54 hamma and daher given a function f ∈ l2,α(r), the dunkl transform [4] of order α is defined as f̂(λ) = ∫ +∞ −∞ f(x)eα(λx)|x|2α+1dx, λ ∈ r, where eα(x) dunkl kernel is defined by (1) eα(x) = jα(x) + i(2α + 2) −1xjα+1(x). the function y = eα(x) satisfies the equation dy = iy with the initial condition y(0) = 1, jα(x) is a normalized bessel function of the first kind, i.e (2) jα(x) = 2αγ(α + 1)jα(x) xα , where jα(x) is a bessel function of the first kind ([3], chap7) the function jα is infinitely differentiable and even, in addition, jα(0) = 1. from formula (1), we have (3) |1 − jα(x)| ≤ |1 −eα(x)| the inverse dunkl transform is defined by the formula f(x) = 1 (2α+1γ(α + 1))2 ∫ +∞ −∞ f̂(λ)eα(−λx)|λ|2α+1dλ. in l2,α(r), we define the operator of the generalized dunkl translation (see [9]) thf(x) = c( ∫ π 0 fe(g(x,h,ϕ))h e( x ,h,ϕ)sin2αϕdϕ + ∫ π 0 f0(g(x,h,ϕ))h 0(x,h,ϕ)sin2αϕdϕ) where c = γ(α + 1) γ( 1 2 )γ(α + 1 2 ) , g(x,h,ϕ) = √ x2 + h2 − 2|xh|cosϕ he(x,h,ϕ) = 1 −sgn(xh)cosϕ and { h0(x,h,ϕ) = (x+h)he(x,h,ϕ) g(x,h,ϕ) for (x,h) 6= (0, 0) h0(x,h,ϕ) = 0 for (x,h) = (0, 0) fe(x) = 1 2 (f(x) + f(−x)), f0(x) = 1 2 (f(x) −f(−x)). lemma 1.1. [4] let f ∈ l2,α(r), then the following equality is true for any h ∈ r (̂thf)(λ) = eα(λh)f̂(λ). dunkl transform in the space l2,α(r) 55 the first-and higher order finite differences of f(x) are defined as follows ∆hf(x) = thf(x) −f(x) = (th − i)f(x), where i is the identity operator in l2,α(r). ∆khf(x) = ∆h(∆ k−1 h f(x)) = (th − i) kf(x) = k∑ i=0 (−1)k−i(ki )t i hf(x), where t0hf(x) = f(x), t i hf(x) = th(t i−1 h f(x)) for i=1,2,....,k and k=1,2,..... the kth order generalized modulus of continuity of function f ∈ l2,α(r) such that ωk(f,δ) = sup 0<h≤δ ‖∆khf(x)‖. let wm2,α be the sobolev space constructed by the operator d such that wm2,α = {f ∈ l2,α(r), d jf ∈ l2,α(r), j = 1, 2, .....,m} w m,k 2,φ (d) denote the class of functions f ∈ w m 2,α, satistying the estimate ωk(d mf,δ) = o(φ(δm)), where φ(t) is any nonnegative function given on [0,∞) and φ(0) = 0, for the dunkl operator d, we have d0f = f, dmf = d(dm−1f),m = 1, 2, ... from lemma 1.1, we have thf(x) = 1 (2α+1γ(α + 1))2 ∫ +∞ −∞ eα(λh)f̂(λ)eα(−λx)|λ|2α+1dλ therefore, combining the relation thf(x) −f(x) = 1 (2α+1γ(α + 1))2 ∫ +∞ −∞ (eα(λh) − 1)f̂(λ)eα(−λx)|λ|2α+1dλ parseval’s identity gives (see [4]) ‖thf(x) −f(x)‖2 = a ∫ +∞ −∞ |eα(λh) − 1|2|f̂(λ)|2|λ|2α+1dλ, where a = (2α+1γ(α + 1))−2. hence, for any function f ∈ wm,k2,φ (d), we obtain (4) ‖∆khd mf(x)‖2 = a ∫ +∞ −∞ |eα(λh) − 1|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ. 56 hamma and daher 2. estimates for the dunkl transform taking into account what was said in section 1, for some classes of functions characterized by the generalized modulus of continuity, we can prove two estimates for the integral ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ, which are useful in applications. lemma 2.1. for x ∈ r the following inequalities are fulfilled: (1) |eα(x)| ≤ 1, (2) |1 −eα(x)| ≤ 2|x|. proof. (see[4]) lemma 2.2. the following inequalities are fulfilled: (1) 1 − jp(x) = o(1), x ≥ 1, (2) 1 − jp(x) = o(x2), 0 ≤ x ≤ 1, (3) √ hxjp(hx) = o(1), hx ≥ 0. proof. (see [1]) theorem 2.3. for f(x) ∈ l2,α(r) sup w m,k 2,φ (d) √∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o(r−mφ( c r )k), where m=0,1,....; k=1,2....; c > 0 is a fixed constant, and φ(t) is any nonnegative function defined on [0,∞). proof. let f ∈ wm,k2,φ (d). taking into account the hölder inequality yields dunkl transform in the space l2,α(r) 57 ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ− ∫ |λ|≥r jα(λh)|f̂(λ)|2|λ|2α+1dλ = ∫ |λ|≥r (1 − jα(λh))|f̂(λ)|2|λ|2α+1dλ = ∫ |λ|≥r (1 − jα(λh))(|f̂(λ)||λ|α+ 1 2 )2dλ = ∫ |λ|≥r (1 − jα(λh))(|f̂(λ)||λ|α+ 1 2 )2− 1 k (|f̂(λ)||λ|α+ 1 2 ) 1 k dλ ≤ ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ( ∫ |λ|≥r |1 − jα(λh)|2k|f̂(λ)|2|λ|2α+1dλ) 1 2k ≤ ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ( ∫ |λ|≥r |λ|−2m|1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ) 1 2k ≤ r −m k ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ( ∫ |λ|≥r |1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ) 1 2k in view of (4), we have ∫ |λ|≥r |λ|2m|1 −eα(λh)|2k|f̂(λ)|2|λ|2α+1dλ ≤ 1 a ‖∆khd mf(x)‖2. therefore ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ ≤ ∫ |λ|≥r jα(λh)|f̂(λ)|2|λ|2α+1dλ + 1 a r −m k ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k .‖∆khd mf(x)‖ 1 k in view of formulas (2) and (2) in lemma 2.2, jα(λh) = o((|λh|)−α− 1 2 ). consequently 58 hamma and daher ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o( ∫ |λ|≥r |hλ|−α− 1 2 |f̂(λ)|2|λ|2α+1dλ) +r −m k ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ‖∆khd mf(x)‖ 1 k = o((rh)−α− 1 2 ) ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ +r −m k ( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ‖∆khd mf(x)‖ 1 k or (1 −o(rh)−α− 1 2 ) ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o(r −m k )( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k ‖∆khd mf(x)‖ 1 k setting h = c r in the last inequality and choosing c > 0 such that 1 −o(c−α− 1 2 ) ≥ 1 2 we obtain ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o(r −m k )( ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ) 2k−1 2k φ 1 k (( c r )k) we have ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o(r−2mφ2(( c r )k)). the theorem is proved. theorem 2.4. let φ(t) = tν, then  ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ   1 2 = o(r−m−kν) ⇐⇒ f ∈ wm,k2,φ (d) where m=0,1,...; k=1,2,....; 0 < ν < 2. proof. sufficiency by theorem 2.3 let f ∈ wm,k2,tν (d) we have   ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ   1 2 = o(r−m−kν) dunkl transform in the space l2,α(r) 59 necessity: let √√√√ ∫ |λ|≥r |f̂(λ)|2|λ|2α+1dλ = o(r−m−kν) that is ∫ |λ|≥r |f̂(ξ)|2|λ|2α+1dλ = o(r−2m−2kν) it is easy to prove, that there exists a function f ∈ l2,α(r) such that dmf ∈ l2,α(r) and dmf(x) = (−i)m (2α+1γ(α + 1))2 ∫ +∞ −∞ λmf̂(λ)eα(−λx)|λ|2α+1dλ then, we have the equality ‖∆khd mf(x)‖2 = a ∫ +∞ −∞ |1 −eα(λh)|2k|λ|2m|f̂(λ)|2|λ|2α+1dλ this integral is divided into two: ∫ +∞ −∞ = ∫ |λ|<r + ∫ |λ|≥r = i1 + i2 where r = [h−1] . and estimate each of them. from (1) in lemma 2.1, we have 60 hamma and daher i2 = ∫ |λ|≥r |1 −eα(λh)|2k|f̂(λ)|2|λ|2α+2m+1dλ = o   ∫ |λ|≥r |f̂(λ)|2|λ|2α+2m+1dλ   = o ( ∞∑ n=r ∫ n+1 n |f̂(λ)|2|λ|2α+2m+1dλ ) = o ( ∞∑ n=r n2m ∫ n+1 n |f̂(λ)|2|λ|2α+1dλ ) = o ( ∞∑ n=r n2m [∫ ∞ n |f̂(λ)|2|λ|2α+1dλ− ∫ ∞ n+1 |f̂(λ)|2|λ|2α+1dλ ]) = o ( ∞∑ n=r n2m ∫ ∞ n |f̂(λ)|2|λ|2α+1dλ− ∞∑ n=r n2m ∫ ∞ n+1 |f̂(λ)|2|λ|2α+1dλ ) = o ( r2m ∫ ∞ r |f̂(λ)|2|λ|2α+1dλ + ∞∑ n=r+1 n2m ∫ ∞ n |f̂(λ)|2|λ|2α+1dλ− ∞∑ n=r n2m ∫ ∞ n+1 |f̂(λ)|2|λ|2α+1dλ ) = o(r2m ∫ ∞ r |f̂(λ)|2|λ|2α+1dλ + ∞∑ n=r ((n + 1)2m −n2m) ∫ ∞ n |f̂(λ)|2|λ|2α+1dλ) = o(r2m ∫ ∞ r |f̂(λ)|2|λ|2α+1dλ + ∞∑ n=r n2m−1 ∫ ∞ n |f̂(λ)|2|λ|2α+1dλ) = o(r2mr−2m−2kν) + o( ∞∑ n=r n2m−1n−2m−2kν) = o(r−2kν) + o(r−2kν) = o(h2kν) i.e i2 = o(h 2kν). we estimate i1, since (2) in lemma 2.1, we have dunkl transform in the space l2,α(r) 61 i1 = ∫ |λ|<r |f̂(λ)|2|λ|2α+2m+1|1 −eα(λh)|2kdλ = o(h2k) ∫ |λ|<r |f̂(λ)|2|λ|2α+2m+2k+1dλ = o(h2k) r∑ n=0 ∫ n≤|λ|<n+1 |f̂(λ)|2|λ|2α+2m+2k+1dλ = o(h2k) r∑ n=0 (n + 1)2m+2k ∫ n≤|λ|<n+1 |f̂(λ)|2|λ|2α+1dλ = o(h2k) r∑ n=0 (n + 1)2m+2k   ∫ |λ|≥n |f̂(λ)|2|λ|2α+1dλ− ∫ |λ|≥n+1 |f̂(λ)|2|λ|2α+1dλ   = o(h2k)  1 + r∑ n=0 ((n + 1)2m+2k −n2m+2k) ∫ |λ|≥n |f̂(λ)|2|λ|2α+1dλ   = o(h2k)  1 + r∑ n=0 n2m+2k−1 ∫ |λ|≥n |f̂(λ)|2|λ|2α+1dλ   = o(h2k) [ 1 + r∑ n=0 n2m+2k−1n−2m−2kν ] = o(h2k) [ 1 + r∑ n=0 n2k−2kν−1 ] = o(h2k)o(r2k−2kν) = o(h2kν) that is i1 = o(h 2kν) combining the estimates for i1 and i2 gives ‖∆khd mf(x)‖ = o(hkν) the necessity is proved references [1] v. a. abilov and f. v. abilova, approximation of functions by fourier-bessel sums, izv. vyssh. uchebn. zaved. mat., 8(2001), 3-9 . [2] v. a. abilov, f. v. abilova, and m.k. kerimov some remarks concerning the fourier transform in the space l2(r), zh. vychisl. mat. mat. fiz., 48 (2008), 2113-2120. [3] h. bateman and a.erdélyi, higher transcendental functions, vol ii(mc graw-hill, new york, 1953; nauka, moscow, 1971). 62 hamma and daher [4] e.s. belkina and s.s. platonov, equivalence of k-functionnals and modulus of smoothness constructed by generalized dunkl translations, izv. vyssh. uchebn. zaved. mat., 8 (2008), 3-15. [5] c.f. dunkl, differential-difference operators associated to reflexion groups, trans. amer. math. soc., 311(1989), 167-183. [6] c.f. dunkl, hankel transforms associated to finite reflection groups inproceedings of special session on hypergeometric functions on domains of positivity, jack polynomials and applications, proceedings, tampa 1991, contemp. math. 138(1992) 123-138. [7] m.f.e. jeu, the dunkl transform , inv. math. 113 (1993) 147-162. [8] m.rösler, bessel-type signed hypergroup on r in: h.heyer, a.mukherjea (eds.)probability measure on groups and related structures, (proc. conf. oberwolfach, 1994), world scientific, singapore, 1995, 292-304. [9] n.b. salem and s. kallel, mean-periodic function associated with the dunkl operators, adv. integral transforms spec. funct. 15(2)(2004), 155-179. department of mathematics, faculty of sciences äın chock, university of hassan ii, casablanca, morocco ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 207-214 http://www.etamaths.com on comparison theorems for conformable fractional differential equations mehmet zeki sarikaya∗ and fuat usta abstract. in this paper the more general comparison theorems for conformable fractional differential equations is proposed and tested. thus we prove some inequalities for conformable integrals by using the generalization of sturm’s separation and sturm’s comparison theorems. the results presented here would provide generalizations of those given in earlier works. the numerical example is also presented to verify the proposed theorem. 1. introduction an important point is that the fractional derivative at a point x is a local property only when a is an integer; in non-integer cases we cannot say that the fractional derivative at x of a function f depends only on values of f very near x, in the way that integer-power derivatives certainly do. therefore it is expected that the theory involves some sort of boundary conditions, involving information on the function further out. to use a metaphor, the fractional derivative requires some peripheral vision. as far as the existence of such a theory is concerned, the foundations of the subject were laid by liouville in a paper from 1832. the fractional derivative of a function to order a is often now defined by means of the fourier or mellin integral transforms. various types of fractional derivatives were introduced: riemann-liouville, caputo, hadamard, erdelyi-kober, grunwald-letnikov, marchaud and riesz are just a few to name [9]-[14]. recently a new local, limit-based definition of a so-called conformable derivative has been formulated in [1], [6] as follows dα (f) (t) = lim ε→0 f ( t + εt1−α ) −f (t) ε provied the limits exits. note that if f is fully differentiable at t, then the derivative is dα (f) (t) = t1−αf′(t). the reader interested on the subject of conformable calculus is referred in [1]-[8]. the aim of this paper is to establish the comparison theorems for conformable fractional differential equations which are based on conditions of mixed type, point-wise and integral inequalities. thus we provide the generalization for comparison of the solution of linear second order conformable differential equations investigated in [13]. the remaining part of the paper proceeds as follows: the second section of this paper will review the basic tools of conformable fractional calculus. the third section begins by laying out the main results and looks at how can be applied it to the conformable fractional differential equations. the fourth section concludes this study with some remarks. 2. preliminary here we recall basic notions, and provide results helpful for the main section. the basic definition is from [1]-[3]. definition 1. (conformable fractional derivative) given a function f : [0,∞) → r. then the “conformable fractional derivative” of f of order α is defined by (2.1) dα (f) (t) = lim �→0 f ( t + �t1−α ) −f (t) � 2010 mathematics subject classification. 26d15, 26a51, 26a33, 26a42. key words and phrases. sturm’s theorems; conformable fractional integrals. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 207 208 sarikaya and usta for all t > 0, α ∈ (0, 1) . if f is α−differentiable in some (0,a) , α > 0, lim t→0+ f(α) (t) exist, then define (2.2) f(α) (0) = lim t→0+ f(α) (t) . we can write f(α) (t) for dα (f) (t) to denote the conformable fractional derivatives of f of order α. in addition, if the conformable fractional derivative of f of order α exists, then we simply say f is α−differentiable. for 2 ≤ n ∈ n, we denote dnα (f) (t) = dαdn−1α (f) (t) (t) theorem 1. let α ∈ (0, 1] and f,g be α−differentiable at a point t > 0. then i. dα (af + bg) = adα (f) + bdα (g) , for all a,b ∈ r, ii. dα (λ) = 0, for all constant functions f (t) = λ, iii. dα (fg) = fdα (g) + gdα (f) , iv. dα ( f g ) = fdα (g) −gdα (f) g2 . if f is differentiable, then (2.3) dα (f) (t) = t 1−αdf dt (t) . theorem 2 (mean value theorem for conformable fractional differentiable functions). let α ∈ (0, 1] and f : [a,b] → r be a continuous on [a,b] and an α-fractional differentiable mapping on (a,b) with 0 ≤ a < b. then, there exists c ∈ (a,b), such that dα (f) (c) = f(b) −f(a) bα α − aα α . definition 2 (conformable fractional integral). let α ∈ (0, 1] and 0 ≤ a < b. a function f : [a,b] → r is α-fractional integrable on [a,b] if the integral (2.4) ∫ b a f (x) dαx := ∫ b a f (x) xα−1dx exists and is finite. remark 1. iaα (f) (t) = i a 1 ( tα−1f ) = ∫ t a f (x) x1−α dx, where the integral is the usual riemann improper integral, and α ∈ (0, 1]. theorem 3. let f : (a,b) → r be differentiable and 0 < α ≤ 1. then, for all t > a we have (2.5) iaαd a αf (t) = f (t) −f (a) . theorem 4. (integration by parts) let f,g : [a,b] → r be two functions such that fg is differentiable. then (2.6) ∫ b a f (x) daα (g) (x) dαx = fg| b a − ∫ b a g (x) daα (f) (x) dαx. theorem 5. assume that f : [a,∞) → r such that f(n)(t) is continuous and α ∈ (n,n + 1]. then, for all t > a we have daαf (t) i a α = f (t) . definition 3. for two functions y1 and y2 satisfying the α-conformable fractional equation and α ∈ (0, 1], we set wα (y1,y2) = ∣∣∣∣ y1 y2dαy1 dαy2 ∣∣∣∣ . in this paper, we establish the following comparison theorems for conformable fractional differential equations are based in conditions of a mixed type, point-wise and integral inequalities, and generalizes the results in [12] and [13]. the results presented here would provide generalizations of those given in earlier works. on comparison theorems for conformable fractional differential equations 209 3. main results in [13], pospisil and skripkova give the sturm’s comparison theorems for conformable fractional differential equations as follows: theorem 6 (sturm separation theorem). let x(t) and y(t) be linearly independent solutions of (3.1) d2αx(t) + p(t)dαx(t) + q(t)x(t) = 0 where p(t) and q(t) are continuous functions, on an open interval (a,b) and 0 < α ≤ 1. then x(t) has a zero between any two successive zeros of y(t). thus the zeros of x and y occur alternately. theorem 7 (sturm comparison theorem). let x(t) and y(t) be non-trivial solutions of (3.2) d2αx(t) + r(t)x(t) = 0 (3.3) d2αy(t) + r1(t)y(t) = 0. respectively, where r(t) ≥ r1(t) for t > a are given continuous functions. then exactly one of the following conditions holds: (1) x(t) has at least one zero between any two zeros of y(t), (2) r(t) = r1(t) for all t > a, and x(t) is a constant multiple of y(t). the sturm’s comparison theorem for conformable fractional differential equations deals with functions x(t) and y(t) satisfying equations (3.2) and (3.3). if r1(t) ≥ r(t), then solutions of (3.3) oscillate more rapidly than solutions of (3.2). more precisely, if x(t) is a non-trivial solution of (3.2) for which x(t1) = x(t2), t1 < t2, and r1(t) ≥ r(t) for t1 ≤ t ≤ t2, then y(t) has a zero in (t1, t2]. now, we give the following riccati equations for conformable fractional differential equations; (3.4) dαu(t) = u 2(t) + r(t) (3.5) dαv(t) = v 2(t) + r1(t). assume that r(t), r1(t) are given continuous functions on (τ1,τ2) . by the substitutions u(t) = − dαx(t) x(t) , v(t) = −dαy(t) y(t) in (3.4) and (3.5), respectively, we obtain the equations (3.2) and (3.3). in the following theorems we give the comparison theorems for conformable fractional differential equations. theorem 8. let x and y be non-trivial solutions of (3.2) and (3.3), respectively, such that x(t) does not vanish on [τ1,τ2] , y(τ1) 6= 0 and the inequality (3.6) − dαx(τ1) x(τ1) + t∫ τ1 r(s)dαs > ∣∣∣∣∣∣−dαy(τ1)y(τ1) + t∫ τ1 r1(s)dαs ∣∣∣∣∣∣ holds for all t on [τ1,τ2] . then y(t) does not vanish on [τ1,τ2] and (3.7) − dαx(t) x(t) > ∣∣∣∣dαy(t)y(t) ∣∣∣∣ , τ1 ≤ t ≤ τ2. proof. since x(t) does not vanish on [τ1,τ2] , u(t) = − dαx(t) x(t) is continuous on [τ1,τ2] and satisfies the riccati equation (3.4), which is equivalent to the integral equation. then by using the (2.5) we have (3.8) u(t) = u(τ1) + t∫ τ1 u2(s)dαs + t∫ τ1 r(s)dαs. by the hypothesis (3.6), we get (3.9) u(t) ≥− dαx(τ1) x(τ1) + t∫ τ1 r(s)dαs > 0. 210 sarikaya and usta since y(τ1) 6= 0, v(t) = − dαy(t) y(t) is continuous on some interval [τ1,δ] , τ1 < δ ≤ τ2. on this interval, the equation (3.4) is well defined and implies the following integral equation (3.10) v(t) = v(τ1) + t∫ τ1 v2(s)dαs + t∫ τ1 r1(s)dαs. therefore, by using (3.6) and (3.9) in (3.10), we have v(t) ≥ v(τ1) + t∫ τ1 r1(s)dαs ≥ −u(τ1) − t∫ τ1 r(s)dαs ≥−u(t) and consequently, u(t) ≥−v(t). in order to obtain (3.11) |v(t)| ≤ u(t) on τ1 ≤ x ≤ δ, it is sufficient to show that u(t) ≥ v(t) on this interval. suppose to the contrary that there exists a point t0 on [τ1,δ] such that u(t0) < v(t0). thus, since |v(τ1)| ≤ u(τ1) from (3.6) (with t = τ1) and u and v are continuous on [τ1,δ] , there exists t1 in τ1 < t1 ≤ t0 such that v(t1) = u(t1) and v(t) ≤ u(t) for τ1 < t ≤ t1. because u(t) ≥−v(t) was establishes, it follows that |v(t)| ≤ u(t) for τ1 < t ≤ t1, and consequently t∫ τ1 v2(s)dαs ≤ t∫ τ1 u2(s)dαs. by using (3.6),(3.8), (3.10), it follows that v(t1) = v(τ1) + t1∫ τ1 v2(s)dαs + t1∫ τ1 r1(s)dαs < u(τ1) + t1∫ τ1 u2(s)dαs + t1∫ τ1 r(s)dαs = u(t1), this is a contradiction with the fact that v(t1) = u(t1). hence (3.11) holds on any interval [τ1,δ] of continuity of v, τ1 < δ ≤ τ2, but this implies that v is continuous on the entire interval [τ1,τ2] , because u(t) is bounded and v(t) has only poles at its points of discontinuity. thus (3.11) holds on all of the interval [τ1,τ2]. this result proves (3.7), and since the left member is bounded on [τ1,τ2] , y(t) cannot have a zero on this interval. the proof is complete. � theorem 9. let x and y be non-trivial solutions of (3.2) and (3.3), respectively, such that x(t) does not vanish on [τ1,τ2] , y(τ1) 6= 0 and the inequality (3.12) dαx(τ2) x(τ2) + τ2∫ t r(s)dαs > ∣∣∣∣∣∣dαy(τ2)y(τ2) + τ2∫ t r1(s)dαs ∣∣∣∣∣∣ holds for all t on [τ1,τ2] . then y(t) does not vanish on [τ1,τ2] and (3.13) dαx(t) x(t) > ∣∣∣∣dαy(t)y(t) ∣∣∣∣ , τ1 ≤ t ≤ τ2. proof. let new functions x1,y1,λ,λ1 be defined on τ1 ≤ t ≤ τ2 by the following equations x1(t) = x(τ1 + τ2 − t) y1(t) = y(τ1 + τ2 − t) λ(t) = r(τ1 + τ2 − t) λ1(t) = r1(τ1 + τ2 − t). on comparison theorems for conformable fractional differential equations 211 then x1(t) does not vanish on [τ1,τ2] , y1(τ1) = y(τ2) 6= 0 and − dαx(τ1) x(τ1) + τ1+τ2−t∫ τ1 λ(s)dαs = dαx(τ2) x(τ2) + τ2∫ t r(s)dαs − dαy(τ1) y(τ1) + τ1+τ2−t∫ τ1 λ1(s)dαs = dαy(τ2) y(τ2) + τ2∫ t r1(s)dαs. thus the hypothesis (3.12) is equivalent to the hypothesis (3.6). since t ∈ [τ1,τ2] if and only if τ1 + τ2 − t ∈ [τ1,τ2], and the conclusion (3.13) follows from theorem 8. � theorem 10. let x(t) and y(t) be non-trivial solutions of the equations (3.14) d2αx(t) − 2b(t)dαx(t) + r(t)x(t) = 0, t > 0, (3.15) d2αy(t) − 2c(t)dαy(t) + r1(t)y(t) = 0, t > 0, respectively, where r and r1 are continuous functions such that r(t) ≤ r1(t) with the initial conditions: (3.16) dαx(t1) + σx(t1) = 0 (3.17) dαy(t1) + τy(t1) = 0 where σ and τ are constants. if b(t)dαy(t) > c(t)dαx(t), then between any two consecutive zeroes τ1 and τ2 of x(t), there exists at least one zero of y(t) unless r(t) ≡ r1(t) on [τ1,τ2]. proof. let τ1 and τ2 with 0 < τ1 < τ2 be consecutive zeroes of x(t). assume x(t) > 0 on [τ1,τ2] (if not, consider −x(t) or −y(t) which have these properties). consequently, by arguments in the proof of theorem 6, dαx(τ1) > 0, and adαx(τ2) < 0. suppose that y(t) does not have a zero on [τ1,τ2]. let y(t) > 0 on [τ1,τ2]. multiplying the equation satisfied by x(t), with y(t), and vice versa, and then subtract the two equations we get [ y(t)d2αx(t) −x(t)d 2 αy(t) ] + 2 [cx(t)dαy(t) − by(t)dαx(t)] + x(t)y(t)(r −r1) = 0. rewriting the last equation as dα [y(t)dαx(t) −x(t)dαy(t)] = −2 [cx(t)dαy(t) − by(t)dαx(t)] −x(t)y(t)(r −r1) integrating on both sides of the last equation from τ1 to τ2, we obtain y(t)dαx(t) −x(t)dαy(t)|τ2τ1 = −2iα [cx(t)dαy(t) − by(t)dαx(t)] − iα(x(t)y(t)(r −r1)) the left hand side of equation (7) is non-positive. the right hand side is strictly positive unless r(t) ≡ r1(t) on [τ1,τ2]. thus, if r(t) 6= r(t) on [τ1,τ2], we arrive at a contradiction. this finishes the proof of theorem. � now in order to test the result of theorem 6, we present a numerical example of equations 3.14 and 3.15. example 1. let us consider the coupled of equations (3.18) d2αx(t) + 3 2 cot ( √ t + π)dαx(t) + x(t) = 0, t > 0, (3.19) d2αy(t) + 3 2 ε1 cot ( √ t + π)dαy(t) + ε1y(t) = 0, t > 0, 212 sarikaya and usta where ε1 ∈ r. and x(t) and y(t) be non-trivial solutions of the equations such that (3.20) x(t) = cos( √ t + π) (3.21) y(t) = cos( √ ε1t) according to figure 1 we can say that the solution y(t) oscillates faster than the solution x(t) whenever 50 100 150 200 250 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 figure 1. function x(t) (black), function y(t) with ε1 = 0.3 (red) and ε1 = 5 (blue). ε1 > 1. otherwise the solution y(t) oscillates slower than the solution x(t) with ε1 < 1 and one of its zeros will not lie between two consecutive zeros of x(t). by means of the transformation u(t) = − dαx(t) x(t) , v(t) = − dαy(t) y(t) equations (3.14) and (3.15) are transformed into riccati equations (3.22) dαu(t) = u 2(t) + 2b(t)u(t) + r(t) (3.23) dαv(t) = v 2(t) + 2e(t)v(t) + r1(t). and the initial conditions − dαx(τ1) x(τ1) = σ , − dαy(τ1) y(τ1) = τ for (3.14) and (3.15), become initial values (3.24) u(τ1) = σ, v(τ1) = τ for (3.22) and (3.23). the differential equations (3.14) and (3.15) subject to (3.24) can be written as equivalent integral equations (3.25) u(t) = σ + t∫ τ1 u2(s)dαs + t∫ τ1 2b(s)u(s)dαs + t∫ τ1 r(s)dαs on comparison theorems for conformable fractional differential equations 213 (3.26) v(t) = τ + t∫ τ1 v2(s)dαs + t∫ τ1 2e(s)v(s)dαs + t∫ τ1 r1(s)dαs. it is obvious from these equations that if τ ≥ σ ≥ 0, e(t) ≥ b(t) ≥ 0 and t∫ τ1 r1(s)dαs ≥ t∫ τ1 r(s)dαs on an interval [τ1,τ2] , then v(t) ≥ u(t) ≥ 0 as long as v(t) can be continued on [τ1,τ2] . since the singularities of u(t) and v(t) correspond to the zeros of x(t) and y(t), respectively, these observations lead to the following comparison theorem for (3.14) and (3.15). theorem 11. suppose x is a non-trivial solutions of (3.14) satisfying −dαx(τ1) x(τ1) = σ ≥ 0, x(τ2) = 0. if i) e(t) ≥ b(t) ≥ 0 for τ1 ≤ t ≤ τ2 ii) t∫ τ1 r1(s)dαs ≥ t∫ τ1 r(s)dαs ≥ 0, for τ1 ≤ t ≤ τ2, then every solution of (3.15) satisfying −dαy(τ1) y(τ1) = σ has a zero in (τ1,τ2]. we note that the integral equations (3.25) and (3.26) can be written as follows u(t) = σ + t∫ τ1 (u(s) + b(s)) 2 dαs + t∫ τ1 ( r(s) − b2(s) ) dαs v(t) = τ + t∫ τ1 (v(s) + e(s)) 2 dαs + t∫ τ1 ( r1(s) −e2(s) ) dαs. this formulation shows that the condition ii) of theorem 11 can be replaced by t∫ τ1 ( r1(s) −e2(s) ) dαs ≥ t∫ τ1 ( r(s) − b2(s) ) dαs ≥ 0. 4. concluding remarks in this investigation, the aim was to present some inequalities for conformable fractional integrals through the instrument of the sturm’s comparison and separation theorems. since the obtained results are general forms of earlier works they would help for the future studies. references [1] t. abdeljawad, on conformable fractional calculus, journal of computational and applied mathematics 279 (2015) 57–66. [2] m. abu hammad, r. khalil, conformable fractional heat differential equations, international journal of differential equations and applications 13(3), 2014, 177-183. [3] m. abu hammad, r. khalil, abel’s formula and wronskian for conformable fractional differential equations, international journal of differential equations and applications 13(3), 2014, 177-183. [4] d. r. anderson, taylor’s formula and integral inequalities for conformable fractional derivatives, contributions in mathematics and engineering, in honor of constantin caratheodory, springer, pp. 25-43, 2016. [5] o.s. iyiola and e.r.nwaeze, some new results on the new conformable fractional calculus with application using d’alambert approach, progr. fract. differ. appl., 2(2), 115-122, 2016. [6] r. khalil, m. al horani, a. yousef and m. sababheh, a new definition of fractional derivative, journal of computational apllied mathematics, 264 (2014), 65-70. [7] t. khaniyev and m. merdan, on the fractional riccati differential equation, int. j. of pure and app. math., 107(1) 2016, 145-160. [8] u.n. katugampola, a new fractional derivative with classical properties, arxiv:1410.6535v2 [math.ca] [9] u.n. katugampola, new approach to a generalized fractional integral, appl. math. comput., 218(3) (2011), 860– 865. [10] u.n. katugampola, new approach to generalized fractional derivatives, b. math. anal. app., 6(4) (2014), 1–15. 214 sarikaya and usta [11] a. a. kilbas, h.m. srivastava and j.j. trujillo, theory and applications of fractional differential equations, elsevier b.v., amsterdam, netherlands, 2006. [12] b. g. pachpatte, mathematical inequalities, north-holland mathematical library, elsevier, 2005. [13] m. pospisil and l. p. skripkova, sturm’s theorems for conformable fractional differential equations, math. commun. 21 (2016), 273–281. [14] s.g. samko, a.a. kilbas and o.i. marichev, fractional integrals and derivatives: theory and applications, gordonand breach, yverdon et alibi, 1993. department of mathematics, faculty of science and arts, düzce university, düzce-turkey ∗corresponding author: sarikayamz@gmail.com international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 174-184 http://www.etamaths.com some coupled coincidence points results of monotone mappings in partially ordered metric spaces stojan radenović abstract. in this paper, we introduce the concepts of a monotone mappings and monotone mapping with respect to other mapping to obtain some coupled coincidence point results in partially ordered metric spaces. our results generalize, extend and complement various comparable results in the existing literature. 1. introduction and preliminaries the existence of fixed points in partially ordered metric spaces was first investigated in 2004 by ran and reurings [19], and then by nieto and lopez [13]. further results in this direction were proved, e.g ([2], [3], [4], [7], [9], [17], [20]. results on weak contractive mappings in such spaces, together with applications to differential equations, were obtained by harjani and sadarangani in [10]. the notion of a coupled fixed point was introduced and studied by opoitsev ([14][16]) and then by guo and lakshmikantham [8]. bhashkar and lakshmikantham in [5] introduced the concept of a coupled fixed point of a mapping f : x×x → x and investigated some coupled fixed point theorems in partially ordered complete metric spaces. they also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. choudhury and kundu [6] obtained coupled coincidence point results in partially ordered metric spaces for compatible mappings. recently, abbas et al. [1] proved coupled coincidence and coupled common fixed point results in cone metric spaces for w− compatible mappings (see also, [11]). the aim of this paper is to prove some coupled coincidence points results for socalled monotone mappings or monotone mappings with respect some other mapping in partially ordered metric spaces. the results presented in this paper generalize, extend and complement various comparable results in the existing literature ([5, 6, 11, 12, 18]). we start with the following. definition 1.1. [12] let (x,�) be a partially ordered set. a mapping f : x ×x → x is a monotone with respect to g : x → x, if for any x,y ∈ x, x1,x2 ∈ x, gx1 � gx2 implies f(x1,y) � f(x2,y), 2010 mathematics subject classification. 47h10, 34b15. key words and phrases. coupled coincidence point; monotone mappings; coupled coincidence point; partially ordered metric space; regular space. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 174 some coupled coincidence points results 175 and y1,y2 ∈ x, gy1 � gy2 implies f(x,y1) � f(x,y2). if we take g = ix (an identity mapping on x ), then f is a monotone mapping on x. ([5]). definition 1.2. [5] an element (x,y) ∈ x ×x is called a coupled fixed point of mapping f : x ×x → x if x = f(x,y) and y = f(y,x). definition 1.3. [1] an element (x,y) ∈ x ×x is called: a coupled coincidence point of mappings f : x×x → x and g : x → x if g(x) = f(x,y) and g(y) = f(y,x), and (gx,gy) is called coupled point of coincidence; a common coupled fixed point of mappings f : x ×x → x and g : x → x if x = g(x) = f(x,y) and y = g(y) = f(y,x). definition 1.4. [18] let (x,�) be an ordered set and d be a metric on x. we say that (x,d,�) is regular if it has the following properties: (i) if for non-decreasing sequence {xn} holds d (xn,x) → 0, then xn � x for all n, (ii) if for non-increasing sequence {yn}holds d (xn,x) → 0 , then yn � y for all n. 2. main results all the results in [2], [5], [6], [7], [9], [10], [18] are obtained for mixed monotone mappings, that is., for mappings f : x×x → x which are increasing with respect to the first variable and decreasing with respect to the second variable. it is our main aim in this paper to consider coupled coincidence points of mappings which are of the same monotonicity with respect to both variables. now, we start with the following result. theorem 2.1. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and d(f(x,y),f(u,v)) + d (f (y,x) ,f (v,u)) ≤ φ(max{ d(gx,gu) + d(gy,gv) 2 , d(f(x,y),gx) + d(f(x,y),gu) 2 , (2.1) d(gy,gv) + d(f(x,y),gx) 2 , d(gy,gv) + d(f(x,y),gu) 2 }) for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), where φ : [0,∞) → [0,∞) is continuous, nondecreasing function such that φ(t) < t for all t > 0. if f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . proof. let x0,y0 ∈ x be such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0). set gx1 = f(x0,y0) and gy1 = f(y0,x0), this can be done as f(x ×x) ⊆ g(x). similarly, g(x2) = f(x1,y1) and g(y2) = f(y1,x1) because f(x × x) ⊆ g(x). continuing this process we can construct sequences {xn} and {yn} in x such that (2.2) g(xn+1) = f(xn,yn) and g(yn+1) = f(yn,xn) for all n ≥ 0. we shall show that g(xn) � g(xn+1) and g(yn) � g(yn+1) for all n ≥ 0. 176 radenović by induction, let n = 0. since gx0 � f(x0,y0) and gy0 � f(y0,x0) also gx1 = f(x0,y0) and gy1 = f(y0,x0), so that gx0 � gx1 and gy0 � gy1. now, let it holds for some fixed n ≥ 0. since gxn � gxn+1 and gyn � gyn+1, and as f is monotone mapping with respect to g, so that gxn+1 = f(xn,yn) � f(xn+1,yn) � f(xn+1,yn+2) = gxn+2 and gyn+1 = f(yn,xn) � f(yn+1,xn) � f (yn+1,xn+1) = gyn+2. hence gxn+1 � gxn+2 and gyn+1 � gyn+2. thus by the mathematical induction we conclude that for all n ≥ 0, gx0 � gx1 � ... � gxn � gxn+1 � ..., and gy0 � gy1 � ... � gyn � gyn+1 � .... we will suppose that d(gxn,gxn+1) > 0 and d(gyn,gyn+1) > 0 for all n, since if gxn = gxn+1 and gyn = gyn+1 for some n, then by (2.2), gxn = f(xn,yn) and gyn = f(yn,xn), that is, f and g have a coupled coincidence point (xn,yn), and so we have finished the proof. now from (2.1), we have d(gxn,gxn+1) + d (gyn,gyn+1) = d(f(xn−1,yn−1),f(xn,yn)) + d(f(yn−1,xn−1),f(yn,xn)) ≤ φ(max{ d(gxn−1,gxn) + d(gyn−1,gyn) 2 , d(f(xn−1,yn−1),gxn−1) + d(f(xn−1,yn−1),gxn) 2 , d(gyn−1,gyn) + d(f(xn−1,yn−1),gxn−1) 2 , d(gyn−1,gyn) + d(f(xn−1,yn−1),gxn) 2 }) = φ(max{ d(gxn−1,gxn) + d(gyn−1,gyn) 2 , d(gxn,gxn−1) 2 , d(gyn−1,gyn) 2 }), and hence d(gxn,gxn+1) + d (gyn,gyn+1) ≤ φ( d(gxn−1,gxn) + d(gyn−1,gyn) 2 ) (2.3) < φ (d(gxn−1,gxn) + d(gyn−1,gyn)) now d(gxn,gxn+1) + d(gyn,gyn+1) ≤ φ(d(gxn−1,gxn) + d(gyn−1,gyn)) ≤ φ2(d(gxn−2,gxn−1) + d(gyn−2,gyn−1)) ≤ ... ≤ φn(d(gx0,gx1) + d(gy0,gy1)). since lim n→∞ φn(d(gx0,gx1) + d(gy0,gy1)) = 0, then for a given ε > 0, there is a positive integer n0 such that for all n ≥ n0, (2.4) φn(d(gx0,gx1) + d(gy0,gy1)) < ε−φ(ε) 2 . hence (2.5) d(gxn,gxn+1) + d(gyn,gyn+1) < ε−φ(ε), some coupled coincidence points results 177 for all n ≥ n0. that is, (2.6) d(gxn,gxn+1) < ε−φ(ε) and d(gyn,gyn+1) < ε−φ(ε). now, for any m,n ∈ n with m > n ≥ n0, we claim that (2.7) d(gxn,gxm) < ε (2.8) and d(gyn,gym) < ε. we prove the inequality (2.7) and (2.8) by induction on m. the inequality (2.7) and (2.8) hold for m = n + 1 by using (2.6). assume that (2.7) and (2.8) hold for m = k. since gxn � gxk and gyn � gyk, so that for m = k + 1, we have d(gxn,gxm) + d(gyn,gym) = d(gxn,gxk+1) + d(gyn,gyk+1) ≤ d(gxn,gxn+1) + d(gxn+1,gxk+1) + d(gyn,gyn+1) + d(gyn+1,gyk+1) < ε−φ(ε) + d(gxn+1,gxk+1) + d(gyn+1,gyk+1) = ε−φ(ε) + d(f(xn,yn),f(xk,yk)) + d (f (yn,xn) ,f (yk,xk)) ≤ ε−φ(ε) + φ(max{ d(gxn,gxk) + d(gyn,gyk) 2 , d(f(xn,yn),gxn) + d(f(xn,yn),gxk) 2 , d(gyn,gyk) + d(f(xn,yn),gxn) 2 , d(gyn,gyk) + d(f(xn,yn),gxk) 2 }) = ε−φ(ε) + φ(max{ d(gxn,gxk) + d(gyn,gyk) 2 , d(gxn+1,gxn) + d(gxn+1,gxk) 2 , d(gyn,gyk) + d(gxn+1,gxn) 2 , d(gyn,gyk) + d(gxn+1,gxk) 2 }) ≤ ε−φ(ε) + φ(max{ ε + ε 2 , ε−φ(ε) + ε 2 , ε + ε−φ(ε) 2 , ε + ε 2 }) = ε−φ(ε) + φ(ε) = ε. by induction on m, we conclude that (2.7) and (2.8) hold for m > n ≥ n0. hence {gxn} and {gyn} are cauchy sequences in g(x), so there exists x and y in x such that {gxn} and {gyn} converges to gx and gy respectively. now, we prove that f(x,y) = gx and f(y,x) = gy. since gxn � gx and gyn � gy for all n ≥ 0, so that we have d(f(x,y),gx) + d (f (y,x) ,gy) ≤ d(f(x,y),gxn+1) + d(gxn+1,gx) + d (f (y,x) ,gyn+1) + d (gyn+1,gy) = d(f(xn,yn),f(x,y)) + d (f (yn,xn) ,f (y,x)) + d(gxn+1,gx) + d (gyn+1,gy) ≤ φ(max{ d(gxn,gx) + d(gyn,gy) 2 , d(f(xn,yn),gxn) + d(f(xn,yn),gx) 2 , d(gyn,gy) + d(f(xn,yn),gxn) 2 , d(gyn,gy) + d(f(xn,yn),gx) 2 }) +d(gxn+1,gx) + d (gyn+1,gy) = φ(max{ d(gxn,gx) + d(gyn,gy) 2 , d(gxn+1,gxn) + d(gxn+1,gx) 2 , d(gyn,gy) + d(gxn+1,gxn) 2 , d(gyn,gy) + d(gxn+1,gx) 2 }) + d(gxn+1,gx) + d (gyn+1,gy) . on taking limit as n →∞, we obtain that (2.9) d(f(x,y),gx) + d (f (y,x) ,gy) ≤ 0, 178 radenović that is., f(x,y) = gx and f (y,x) = gy . hence (x,y) is a coupled coincidence point and (gx,gy) is coupled point of coincidence of mappings f and g. � following example support theorem 2.1. example 2.2. let x = [0, 1] be an ordered set with the natural ordering of real numbers and d a usual metric on x. let f : x × x → x, g : x → x and φ : [0,∞) → [0,∞) be defined by (2.10) f(x,y) = 2x + y + 1 18 ,g(x) = 3x 4 for all x,y ∈ x, and φ(t) = 8 9 t, for t ∈ [0,∞). note that f(x ×x) ⊆ g(x) and φ is nondecreasing, continuous with φ(t) < t for all t > 0. now for g(x) � g(u) and g(y) � g(v) , we obtain d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) = 1 18 |2x + y − 2u−v| + 1 18 |2y + x− 2v −u| ≤ 1 18 |2(x−u) + (y −v)| + 1 18 |2(y −v) + (x−u)| ≤ 6 18 (|x−u| + |y −v|) = 2 3 ∣∣3 4 x− 3 4 u ∣∣ + ∣∣3 4 y − 3 4 v ∣∣ 2 · 4 3 = 8 9 ∣∣3 4 x− 3 4 u ∣∣ + ∣∣3 4 y − 3 4 v ∣∣ 2 = φ( d(gx,gu) + d(gy,gv) 2 ) ≤ φ(max{ d(gx,gu) + d(gy,gv) 2 , d(f(x,y),gx) + d(f(x,y),gu) 2 , d(gy,gv) + d(f(x,y),gx) 2 , d(gy,gv) + d(f(x,y),gu) 2 }). thus (2.1) is satisfied and f and g have coupled coincidence points. here, ( 2 21 , 2 21 ) is a coupled coincidence point and ( g ( 2 21 ) ,g ( 2 21 )) = ( 1 14 , 1 14 ) is coupled point of coincidence of mappings f and g. � remark 2.3. since f has not a mixed monotone property with respect to g, it follows that a coupled coincidence point ( 2 21 , 2 21 ) cannot be obtained by theorem 2.1. from [2]. corollary 2.4. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) ≤ k max{d(gx,gu) + d(gy,gv),d(f(x,y),gx) + d(f(x,y),gu), (2.11) d(gy,gv) + d(f(x,y),gx),d(gy,gv) + d(f(x,y),gu)} for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), where k ∈ [0, 1 2 ). if f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . proof. taking φ(t) = kt with k ∈ [0, 1 2 ) in theorem 2.1, we obtain corollary 2.1. � some coupled coincidence points results 179 corollary 2.5. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone and d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) ≤ φ(max{ d(x,u) + d(y,v) 2 , d(f(x,y),x) + d(f(x,y),u) 2 , (2.12) d(y,v) + d(f(x,y),x) 2 , d(y,v) + d(f(x,y),u) 2 }) for all x,y,u,v ∈ x, for which x � u and y � v. if (x,d,�) is a complete and regular and if there exist x0,y0 ∈ x such that x0 � f(x0,y0) and y0 � f(y0,x0), then there exist x,y ∈ x such that x = f (x,y) and y = f (y,x) . proof. the result follows by taking g = i (identity mapping) in theorem 2.1. � corollary 2.6. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and (2.13) d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) ≤ φ( d(gx,gu) + d(gy,gv) 2 ) for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), where φ : [0,∞) → [0,∞) is continuous, nondecreasing function such that φ(t) < t for all t > 0. if f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . as φ(max{a,b}) = max{φ(a),φ(b)} for all a,b ∈ [0,∞) if φ : [0,∞) → [0,∞) is nondecreasing map, then we obtain following equivalent form of theorem 2.1. theorem 2.7. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) ≤ max{φ( d(gx,gu) + d(gy,gv) 2 ),φ( d(f(x,y),gx) + d(f(x,y),gu) 2 ), (2.14) φ( d(gy,gv) + d(f(x,y),gx) 2 ),φ( d(gy,gv) + d(f(x,y),gu) 2 )} for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), where φ : [0,∞) → [0,∞) is continuous, nondecreasing function such that φ(t) < t for all t > 0. if f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . theorem 2.8. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and (2.15) d(f(x,y),f(u,v)) ≤ φ(d(f(x,y),gx)) + φ(d(f(u,v),gu)) 2 for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), where φ : [0,∞) → [0,∞) is continuous, nondecreasing function such that φ(t) < t for all t > 0. if 180 radenović f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . proof. let x0,y0 ∈ x be such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0). using the similar arguments to those given in theorem 2.1, we construct sequences {xn} and {yn} in x such that g(xn+1) = f(xn,yn) and g(yn+1) = f(yn,xn) for all n ≥ 0 and for all n ≥ 0, gx0 � gx1 � ... � gxn � gxn+1 � ..., and gy0 � gy1 � ... � gyn � gyn+1 � .... now we will suppose that d(gxn,gxn+1) > 0 and d(gyn,gyn+1) > 0 for all n, otherwise, f and g have a coupled coincidence point at (xn,yn), and so we have finished the proof. from (2.15), d(gxn,gxn+1) = d(f(xn−1,yn−1),f(xn,yn)) ≤ φ(d(f(xn−1,yn−1),gxn−1) + φ(d(f(xn,yn),gxn)) 2 = φ(d(gxn,gxn−1)) + φ(d(gxn+1,gxn)) 2 ≤ φ(d(gxn,gxn−1)) + d(gxn+1,gxn) 2 , that is., (2.16) d(gxn,gxn+1) ≤ φ(d(gxn−1,gxn)). similarly, (2.17) d(gyn,gyn+1) ≤ φ(d(gyn−1,gyn)). from (2.16) and (2.17), we obtain (2.18) d(gxn,gxn+1) + d(gyn,gyn+1) ≤ φ(d(gxn−1,gxn)) + φ(d(gyn−1,gyn)). now d(gxn,gxn+1) + d(gyn,gyn+1) ≤ φ(d(gxn−1,gxn)) + φ(d(gyn−1,gyn)) ≤ φ2(d(gxn−2,gxn−1)) + φ2(d(gyn−2,gyn−1)) ≤ ... ≤ φn(d(gx0,gx1)) + φn(d(gy0,gy1)). for a given ε > 0, since lim n→∞ [φn(d(gx0,gx1)) + φ n(d(gy0,gy1))] = 0 and φ(ε) < ε, there is a positive integer n0 such that for all n ≥ n0, (2.19) φn(d(gx0,gx1)) + φ n(d(gy0,gy1)) < ε−φ(ε). hence d(gxn,gxn+1) + d(gyn,gyn+1) < ε−φ(ε), that is., (2.20) d(gxn,gxn+1) < ε−φ(ε) and d(gyn,gyn+1) < ε−φ(ε). now, for any m,n ∈ n with m > n, we claim that (2.21) d(gxn,gxm) < ε some coupled coincidence points results 181 and (2.22) d(gyn,gym) < ε. we prove the inequalities (2.21) by induction on m. the inequalities (2.21) holds for m = n+1 by using (2.20). assume that (2.21) holds for m = k. since gxn � gxk and gyn � gyk, so that for m = k + 1, we have d(gxn,gxm) = d(gxn,gxk+1) ≤ d(gxn,gxn+1) + d(gxn+1,gxk+1) ≤ ε−φ(ε) + d(gxn+1,gxk+1) = ε−φ(ε) + d(f(xn,yn),f(xk,yk)) ≤ ε−φ(ε) + φ(d(f(xn,yn),gxn)) + φ(d(f(xk,yk),gxk)) 2 = ε−φ(ε) + φ(d(gxn+1,gxn)) + φ(d(gxk+1,gxk)) 2 ≤ ε−φ(ε) + φ(ε−φ(ε)) + φ(ε−φ(ε)) 2 = ε−φ(ε) + φ(ε−φ(ε)) ≤ ε−φ(ε) + φ(ε) = ε. similarly, we obtain d(gyn,gym) < ε. by induction on m, we conclude that (2.21) and (2.22) holds for m > n ≥ n0. hence {gxn} and {gyn} are cauchy sequences in g(x), so there exists x and y in x such that {gxn} and {gyn} converges to gx and gy respectively. now, we prove that f(x,y) = gx and f(y,x) = gy. since gxn � gx and gyn � gy for all n ≥ 0, so that we have d(f(x,y),gx) ≤ d(f(x,y),gxn+1) + d(gxn+1,gx) = d(f(xn,yn),f(x,y)) + d(gxn+1,gx) ≤ φ(d(f(xn,yn),gxn)) + φ(d(f(x,y),gx)) 2 + d(gxn+1,gx) = φ(d(gxn+1,gxn)) + φ(d(gx,gx)) 2 + d(gxn+1,gx). on taking limit as n →∞, we obtain (2.23) d(f(x,y),gx) ≤ φ(0) = 0, and f(x,y) = gx. similarly, it can be shown that f(y,x) = gy. hence (x,y) is a coupled coincidence point and (gx,gy) is coupled point of coincidence of mappings f and g. � corollary 2.9. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone and (2.24) d(f(x,y),f(u,v)) ≤ φ(d(f(x,y),x) + d(f(u,v),u)) 2 , for all x,y,u,v ∈ x, for which x � u and y � v. if (x,d,�) is a complete and regular and if there exist x0,y0 ∈ x such that x0 � f(x0,y0) and y0 � f(y0,x0), then there exist x,y ∈ x such that x = f (x,y) and y = f (y,x) . proof. the results follows by taking g = i (identity mapping) in theorem 2.7. � 182 radenović theorem 2.10. let (x,d,�) be a partially ordered metric space. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and d(f(x,y),f(u,v)) + d (f (y,x) ,f (v,u)) ≤ a1d(gx,gu) + a2d(gy,gv) + a3d(f(x,y),gx) (2.25) +a4d(f(u,v),gu) + a5d(f(x,y),gu) for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v), with nonnegative real numbers ai, i = 1, 2, ..., 5 and ∑5 i=1ai < 1. if f(x×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . proof. let x0,y0 ∈ x be such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0). using the similar arguments to those given in theorem 2.1, we construct sequences {xn} and {yn} in x such that g(xn+1) = f(xn,yn) and g(yn+1) = f(yn,xn) for all n ≥ 0, and for all n ≥ 0, gx0 � gx1 � ... � gxn � gxn+1 � ..., and gy0 � gy1 � ... � gyn � gyn+1 � .... now we will suppose that d(gxn,gxn+1) > 0 and d(gyn,gyn+1) > 0 for all n, otherwise, f and g have a coupled coincidence point at (xn,yn), and so we have finished the proof. from (2.25), we have d(gxn,gxn+1) + d (gyn.gyn+1) = d(f(xn−1,yn−1),f(xn,yn)) + d (f (yn−1,xn−1) ,f (yn,xn)) ≤ a1d(gxn−1,gxn) + a2d(gyn−1,gyn) + a3d(f(xn−1,yn−1),gxn−1) +a4d(f(xn,yn),gxn) + a5d(f(xn−1,yn−1),gxn) = a1d(gxn−1,gxn) + a2d(gyn−1,gyn) + a3d(gxn,gxn−1) +a4d(gxn+1,gxn) + a5d(gxn,gxn) = (a1 + a3)d(gxn−1,gxn) + a2d(gyn−1,gyn) + a4d(gxn+1,gxn), from which it follows (2.26) d(gxn,gxn+1) +d (gyn.gyn+1) ≤ 1 1 −a4 [(a1 + a3) d(gxn−1,gxn) +a2d(gyn−1,gyn)]. from (2.26), we obtain d(gxn,gxn+1) + d(gyn,gyn+1) ≤ a1 + a3 1 −a4 [d(gyn−1,gyn) + d(gxn−1,gxn)], that is., (2.27) d(gxn,gxn+1) + d(gyn,gyn+1) ≤ λ[d(gxn−1,gxn) + d(gyn−1,gyn)], some coupled coincidence points results 183 where λ = a1 + a3 1 −a4 . obviously, 0 ≤ λ < 1. now d(gxn,gxn+1) + d(gyn,gyn+1) ≤ λ[d(gxn−1,gxn) + d(gyn−1,gyn)] ≤ λ2[d(gxn−2,gxn−1) + d(gyn−2,gyn−1)] ≤ ... ≤ λn[d(gx0,gx1) + d(gy0,gy1)]. then, for all n,m ∈ n, m > n, we have d(gxn,gxm) + d(gyn,gym) ≤ d(xn,xn+1) + d(yn,yn+1) + d(xn+1,xx+2) +d(yn+1,yx+2) + ... + d(xm−1,xm) + d(ym−1,ym) ≤ λn 1 −λ [d(gx0,gx1) + d(gy0,gy1)], which implies that d(gxn,gxm)+d(gyn,gym) → 0, as n,m →∞, that is., d(gxn,gxm) → 0 and d(gyn,gym) → 0 as n,m →∞. hence {gxn} and {gyn} are cauchy sequences in g(x), so there exists x and y in x such that {gxn} and {gyn} converges to gx and gy respectively. now, we prove that f(x,y) = gx and f(y,x) = gy. since gxn � gx and gyn � gy for all n ≥ 0, so that we have d(f(x,y),gx) + d (f (y,x) ,gy) ≤ d(f(x,y),gxn+1) + d (f (y,x) ,gyn+1) + d(gxn+1,gx) + d (gyn+1,gy) = d(f(xn,yn),f(x,y)) + d (f (yn,xn) ,f (y,x)) + d(gxn+1,gx) + d (gyn+1,gy) ≤ a1d(gxn,gxn) + a2d(gyn,gyn) + a3d(f(xn,yn),gxn) + a4d(f(x,y),gx) a5d(f(xn,yn),gx) + d(gxn+1,gx) + d (gyn+1,gy) = a3d(gxn+1,gxn) + a4d(f(x,y),gx) + a5d(gxn+1,gx) + d(gxn+1,gx) + d (gyn+1,gy) on taking the limit as n →∞, we obtain that d(f(x,y),gx) + d (f (y,x) ,gy) ≤ a4d(f(x,y),gx). since a4 < 1, so that f(x,y) = gx and f(y,x) = gy. hence (x,y) is a coupled coincidence point and (gx,gy) is coupled point of coincidence of mappings f and g. � corollary 2.11. let (x,d,�) be a partially ordered set and d a metric on x. suppose that a mapping f : x ×x → x is a monotone with respect to g : x → x and (2.28) d(f(x,y),f(u,v)) + d(f(y,x),f(v,u)) ≤ kd(f(x,y),gx) + ld(f(u,v),gu)] for all x,y,u,v ∈ x, for which g(x) � g(u) and g(y) � g(v) and k,l ≥ 0 with k + l < 1. if f(x ×x) is contained in a complete set g(x), (x,d �) is a regular and if there exist x0,y0 ∈ x such that g(x0) � f(x0,y0) and g(y0) � f(y0,x0), then there exist x,y ∈ x such that g (x) = f (x,y) and g (y) = f (y,x) . remarks 2.12. also, almost all known results from several papers on partially ordered metric spaces can be considered with monotone mappings instead with mappings which have a mixed monotone property. we note that the concept of coupled coincidence point for monotone mappings is essentially different of the corresponding one for mixed monotone mappings (for tripled case see [12]). 184 radenović references [1] m. abbas, m. a. khan and s. radenović, common coupled fixed point theorem in cone metric space for w−compatible mappings, appl. math.comput. 217 (2010) 195-202. [2] m. abbas, t. nazir, s. radenović, common coupled fixed points of generalized contractive mappings in partially ordered metric spaces, positivity (2013) 17:1021-1041. [3] r. p. agarval, m. a. el-gebeily and d. o’regan, generalized contractions in partially ordered metric spaces, applicable analysis 87 (1) (2008) 109-116. [4] i. altun and h. simsek, some fixed point theorems on ordered metric spaces and application, fixed point theory appl. volume 2010, article id 621469. [5] t. g. bhashkar and v. lakshmikantham, fixed point theorems in partially ordered cone metric spaces and applications, nonlinear anal. 65 (7) (2006) 825-832. [6] b. s. choudhury and a. kundu, a coupled coincidence point result in partially ordered metric spaces for compatible mappings, nonlinear anal. 73 (2010) 2524-2531. [7] h. s. ding, lu li and s. radenović, coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces, fixed point theory appl. 2012, 2012:96. [8] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal. tma 11 (1987) 623-632. [9] a. a. harandi and h. emami, a fixed point theorem for contractive type maps in partially ordered metric spaces and application to ordinary differential equations, nonlinear anal. tma 72 (2010) 2238-2242. [10] j. harjani and k. sadarangani, fixed point theorems for weakly contractive mappings in partially ordered sets, nonlinear anal. 71 (2009) 3403-3410. [11] e. karapinar, coupled fixed point theorems for nonlinear contractions in cone metric spaces, comput. math. appl. 59 (2010) 3656-3668. [12] m. borcut, tripled fixed point theorems for monotone mappings in partially ordered metric spaces, carpanthian j. math. 28 (2012), no. 2, 207-214. [13] j. j. nieto and r. r. lopez, contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22 (2005) 223-239. [14] v.i. opoitsev, heterogenic and combined-concave operators, syber. math. j. 16 (1975) 781792 (in russian). [15] v.i. opoitsev, dynamics of collective behavior. iii. heterogenic systems. avtomat. i telemekh. 36 (1975), 124-138 (in russian). [16] v.i. opoitsev, t.a. khurodze, nonlinear operators in space with a cone. tbilis. gos. univ. tbilisi (1984) 271 (in russian). [17] s. radenović and z. kadelburg, generalized weak contractions in partially ordered metric spaces, comput. math. appl. 60 (2010) 1776-1783. [18] s. radenović, remarks on some coupled coincidence point results in partially ordered metric spaces, arab j. math. sci. 20 (1) (2014), 29-39. [19] a. c. m. ran and m. c. b. reurings, a fixed point theorem in partially ordered sets and some application to matrix equations, proc. amer. math. soc. 132 (2004) 1435-1443. [20] d. o’regan and a. petrusel, fixed point theorems for generalized contractions in ordered metric spaces, j. math. anal.appl., 341 (2008) 1241-1252. faculty of mechanical engineering,university of belgrade, kraljice marije 16, 11 120 beograd, serbia int. j. anal. appl. (2023), 21:56 frictional contact problem with wear for thermo-viscoelastic materials with damage safa gherian1, abdelaziz azeb ahmed1, fares yazid2,∗, fatima siham djeradi2 1laboratory of operator theory and pde: foundations and applications, faculty of exact sciences, university of el oued, el oued 39000, algeria 2laboratory of pure and applied mathematics, amar telidji university of laghouat, laghouat 03000, algeria ∗corresponding author: f.yazid@lagh-univ.dz abstract. we consider a mathematical model which describes a dynamic frictional contact problem for thermo-viscoelastic materials with long memory and damage. the contact is modeled by the normal compliance condition and wear between surfaces are taken into account. we establish a variational formulation for the model and prove the existence and uniqueness of the weak solution. the proof is based on arguments of hyperbolic nonlinear differential equations, parabolic variational inequalities and banach fixed point. 1. introduction contact problems involving deformable bodies arise naturally in many situations and industrial processes as well as in everyday life and play an important role in mechanical and structural systems. that is way have been widely studied in the last years. the aim of this paper is to model and establish the variational analysis of a frictional contact problem for a dynamic thermo-viscoelastic body with wear and damage. the contact is modelled with normal compliance and wear. thermoviscoelastic contact problems by taking into account the evolution of the temperature parameter could be found in [5,6,13]. general models for thermoelastic frictional contact, derived from thermodynamical principles, have been obtained in [14, 23, 24]. quasistatic contact problems with normal compliance and friction have been considered in [1] and [16]. dynamic problems with normal compliance were received: mar. 8, 2023. 2020 mathematics subject classification. 74m15, 74m10, 74f15, 49j40. key words and phrases. thermoviscoelastic; damage; wear; dynamic contact problem. https://doi.org/10.28924/2291-8639-21-2023-56 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-56 2 int. j. anal. appl. (2023), 21:56 first considered in [17]. the existence of weak solutions to dynamic thermoelastic contact problems with frictional heat generation have been proven in [2] and when wear is taken into account in [3,9]. recently contact problems with wear were studied in [6, 8, 10, 18, 19]. the damage of the material caused by growth, temperature and various other external factors. the evolution of the microscopiccracks responsible for the damage is determined by a parabolic inclusion with a constitutive function describing the source of damage in the system which results from tension or compression. using the subdifferential of indicator function of the interval [0, 1] guarantees that the damage function ς, which measures the decrease in the load bearing capacity of the material, varies between 0 and 1; when ς = 1 the material has its full capacity; wvhen ς = 0 it is completely damaged, and if ς = 1, the material is partially damaged. because of the importance of the subject, three-dimensional problems which include this approach to material damage have been investigated recently in [7,12,13] this paper is organized as follows. in section 2, we present the notation and some preliminaries. in section 3 we present the original model and list the assumptions on the problem’s data and we derive the variational formulation. in section 4 we present our main result stated in theorem 4.1 and its proof which is based on arguments of time-dependent variational inequalities, parabolic inequalities, differential equations and fixed point. 2. notations and preliminaries first we will introduce some notations and preliminaries that we will use later. we denote by sd the space of second order symmetric tensors on rd(d = 1, 2, 3). let ” : ” and ” · ” represent the inner product on sd and rd, respectively, and ‖ ·‖ denotes the euclidean norm on sd and rd. thus, for all u,v ∈ rd, u · v = uivi,‖v‖ = (v · v) 1 2 and for all σ,ζ ∈ sd, σ : ζ = σijζij,‖ζ‖ = (ζ : ζ) 1 2 , 1 ≤ i, j ≤ d. also, an index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. ui;j = ∂ui ∂xj . we denote by t the time variable and a dot superscript represents the time derivative with respect to the time variable t, e.g. u̇ = ∂u ∂t , ü = ∂ 2u ∂t2 . in what follows, we use the standard notation for lebesgue and sobolev spaces associated to ω and γ introduce the spaces h = {u = (ui ) : ui ∈ l2(ω)}, h = {σ = (σij) : σij = σji ∈ l2(ω)}, h1 = {u = (ui ) : ε(u) ∈ h}, h1 = {σ ∈h : divœ ∈ h}. here ε and div are the deformation and divergence operators, respectively, defined by ε(u) = (εij(u)), εij(u) = 1 2 (ui,j + uj,i ), div σ = (σij,j). int. j. anal. appl. (2023), 21:56 3 the spaces h, h, h1 and h1 are real hilbert spaces endowed with the canonical inner products given by (u,v)h = ∫ ω u.vdx ∀u,v∈ h, (ζ,σ)h = ∫ ω ζ : σdx ∀ζ,σ ∈h, (u,v)h1 = (u,v)h + (ε(u),ε(v))h ∀u,v∈ h1, (ζ,σ)h1 = (ζ,σ)h + (divı, divœ)h ∀ı, œ ∈h1. the associated norms on the spaces h, h, h1 and h1 are denoted by ‖ · ‖h, ‖ · ‖h, ‖ · ‖h1 and ‖ · ‖h1, respectively. for every element v ∈ h1 we also use the notation v for the trace of v on γ and we denote by vν and vτ the normal and the tangential components of v on γ given by vν = v ·ν, vτ = v−vνν. we also denote by σν and στ the normal and the tangential traces of a function σ ∈h1, we recall that when σ is a regular function then σν = σν ·ν, στ = σν−σνν, and the following green’s formula holds (σ,ε(v))h + (divœ,v)h = ∫ γ œ˚.˚da ∀v ∈ h1. (2.1) for a real banach space (x,‖ · ‖x) we use the usual notation for the spaces lp(0,t ; x) and wk,p(0,t ; x) where k ∈ n and 1 ≤ p ≤ ∞; we also denote by c(0,t ; x) and c1([0,t ]; x) the spaces of continuous and continuously differentiable functions on [0,t ] with values in x, with the respective norms ‖x‖c([0,t ];x) = max t∈[0,t ] ‖x(t)‖x, ‖x‖c1([0,t ];x) = max t∈[0,t ] ‖x(t)‖x + max t∈[0,t ] ‖ẋ(t)‖x. we end this section by giving an existence, uniqueness and regularity result concerning evolution problems, taken from [4, p.268]. theorem 2.1. let v and h be two real hilbert spaces such that v ⊆ h and the inclusion mapping of v into h is continuous and densely defined. we suppose that v is endowed with the norm ‖·‖ induced by the inner product (·, ·〉) and h is endowed with the norm | · |. we denote by v ′ the dual space of v , by 〈·, ·〉v ′×v the duality pairing between an element of v and an element of v ′, and h is identified with its own dual h′. we assume that m is a maximal monotone set in v ′ × v and a is a linear, continuous and symmetric operator from v to v ′ satisfying the following coerciveness condition: 〈av,v〉v ′×v + λ|v|2 ≥ ω‖v‖2 ∀v∈ v (2.2) where λ ∈r and ω > 0. let f ∈ w 1,1(0,t ; h) be given in w 1,1(0,t ; h) and u0,v0 be given with u0 ∈ v, v0 ∈ d(m), {au0 + mv0}∩h 6= ∅. (2.3) 4 int. j. anal. appl. (2023), 21:56 then there exists a unique solution u to the following problem:  d2u dt2 + au + m( du dt ) 3 f(t) a.e on (0,t ), u(0) = u0, du dt (0) = v0. which satisfies u∈ w 1,∞(0,t ; v ) ∩w 2,∞(0,t ; h). 3. mechanical and variational formulations the physical setting is the following. a thermo-viscoelastic body occupies a bounded domain ω ⊂ rd (d = 2, 3) with outer surface γ = ∂ω, assumed to be sufficiently smooth and decomposed into three disjoint measurable parts γ1, γ2, and γ3, such that meas(γ1) > 0. let us denote by [0,t ], t > 0 the time interval of interest. the body is clamped on γ1, so the displacement field vanishes there. a surface traction of density f2 act on γ2. moreover, the body is submitted to the action of body forces of density f0 and a heat source of constant strength q. the body could come in sliding frictional contact with a moving obstacle made of a hard perfectly rigid material, and assume that the contact surface of the body γ3 is covered by a layer of soft material. this layer is deformable and the foundation may penetrate it, and could deteriorate over time as a result of frictional contact with the foundation. problem p. find a displacement field u : ω × [0,t ] −→ rd, a stress field σ : ω × [0,t ] −→ sd, a damage field ς : ω × [0,t ] −→ r, a temperature field θ : ω × [0,t ] −→ r and a wear field w : γ3 × [0,t ] −→r such that σ = aε(u̇) + gε(u) + ∫ t 0 b(t − s,ε(u(s)),ς(s),θ(s))ds in ω × (0,t ), (3.1) ς̇ − ∆ς + ∂ψk(ς) 3s(ε(u),ς,θ) in ω × (0,t ), (3.2) θ̇−µ0∆θ = φ(ε(u),ς,θ) + q in ω × (0,t ), (3.3) div σ + f0 = ρü in ω × (0,t ), (3.4) u = 0 on γ1 × (0,t ), (3.5) σν = f2 on γ2 × (0,t ), (3.6)  −σν = pν(uν −w),‖στ‖≤ pτ (uν −w), ‖στ‖ < pτ (uν −w) =⇒ u̇τ = v∗, ‖στ‖ = pτ (uν −w) =⇒ u̇τ = v∗ −λστ, λ > 0, ẇ = kω‖v∗‖pν(uν −w), on γ3 × (0,t ), (3.7) int. j. anal. appl. (2023), 21:56 5 µ0 ∂θ ∂ν + µ1θ = 0 on γ × (0,t ), (3.8) ∂ς ∂ν = 0 on γ × (0,t ), (3.9) w(0) = 0 on γ3, (3.10) u(0) = u0, u̇(0) = u̇0, θ(0) = θ0, ς(0) = ς0 on ω. (3.11) we now describe problem (3.1)−(3.11). equation (3.1) represents the thermo-viscoelastic constitutive with long memory and damage, a and g denote the linear viscosity operator and the elastic operator, respectively and b is the relaxation tensor depending on the damage ς and the temperature θ. equation (3.2) describes the evolution of the damage field, governed by the source damage function s and ∂ψk is the subdifferential of indicator function of the set of admissible damage functions. equation (3.3) represents the evolution of the temperature field θ where φ is a nonlinear constitutive function which represents the heat generated by the work of internal forces, q represents the density of volume heat sources and µ0 is a strictly positive constant. equation (3.4) represents the equilibrium equation for the stress displacement fields. equations (3.5) and (3.6) are the displacement and traction boundary conditions, respectively. equation (3.7) describes the condition with normal compliance, wear and the coulomb’s friction law. the wear function w which measures the wear accumulated of the surface. the evolution of the wear of the contacting surface is governed by the differential form of archard’s law (see, eg., [2,21,23,24]), where v∗ is a constant vector which represents the displacement of the foundation, kw > 0 is a wear coefficient, pν and pτ are prescribed functions of the normal compliance and friction bound, respectively. (3.8) is a fourier boundary condition for the temperature θ where µ1 > 0 and it represents a conduction coefficient of γ. (3.9) is a homogeneous niemann boundary condition for the damage ς, where ∂ς ∂ν is the normal derivative of ς. (3.10) represents the initial condition for the wear function, which shows that at the initial moment the foundation is new. finally the functions u0, u̇0, θ0 and ς0 in (3.11) are the initial data. we now turn to the variational formulation of problem p. we introduce the following space for the temperature field denoted by e = h1(ω). the following friedrichs-poincaré inequality holds on e is ‖∇ϑ‖l2(ω)d ≥ cf ‖ϑ‖e, ∀ϑ ∈ e. (3.12) l2(ω) is identified with its dual and with a subspace of the dual e′ of e, i.e., e ⊂ l2(ω) ⊂ e′, and we say that the inclusions above define a gelfand triple. we use the notation 〈., .〉e′×e to represent the duality pairing between e′ and e. 〈ω,ϑ〉e′×e = (ω,ϑ)l2(ω), ∀ω,ϑ ∈ l 2(ω). (3.13) 6 int. j. anal. appl. (2023), 21:56 we define the admissible space v = {v∈ (h1(ω))d : v = 0 on γ1}, also, the admissible damage set k = {ς ∈ h1(ω) : 0 ≤ ς ≤ 1 a.e in ω}. since meas(γ1) > 0, korn’s inequality holds and there exists a constant c0 > 0, that depends only on ω and γ1, such that c0‖v ‖h≤‖ε(v) ‖h, ∀ v ∈ v. on the space v , we consider the inner product and the associated norm given by (u,v)v = (ε(u),ε(v))h, ‖u‖v = ‖ε(u)‖h, ∀u,v ∈ v. (3.14) it follows that ‖ · ‖h1 and ‖ · ‖v are equivalent norms on v and therefore (v,‖ · ‖v ) is a real hilbert space. moreover, by the sobolev trace theorem and (3.14), there exists a constant c1 > 0, depending only on ω, γ1 and γ3, such that ‖v‖l2(γ3)d ≤ c1‖v‖v , ∀ v ∈ v. (3.15) in the study of the mechanical problem (3.1)-(3.11), we still need to assume that the operators a,g,b and the functions s, φ,pr (for r = ν, τ) satisfy the following conditions   (1) a : ω ×sd →sd; (2) there exists ma > 0 such that (a(x,ε1) −a(x,ε2)).(ε1 −ε2) ≥ ma‖ε1 −ε2‖2, for any ε1,ε2 ∈sd a.e x ∈ ω; (3) there exists m1,m2 > 0 such that ‖(a(x,ε)‖≤ m1‖ε‖ + m2, for any ε ∈sd a.e x ∈ ω; (4) the mapping x 7→a(x,ε) is continuous on sd, a.e. x ∈ ω; (5) the mapping x 7→a(x,ε)is lebesgue measurable on ω for any ε ∈sd. (3.16)   (1) g : ω ×sd →sd; (2) there exists mg > 0 such that g(x,ε) ·ε ≥ mg‖ε‖2, for any ε ∈sd a.e x ∈ ω; (3) g(x,ε1) ·ε2 = ε1 · g(x,ε2) for all ε1,ε2 ∈sd, a.e. x ∈ ω; (4) gijkl ∈ l∞(ω),∀i, j,k, l = 1, ..,d (3.17) int. j. anal. appl. (2023), 21:56 7  (1) b : ω × [0,t ] ×sd ×r×r→ sd; (2) there exists lb > 0 such that ‖b(x,t,ε1,ζ1,θ1) −b(x,ε2,ζ2,θ2)‖≤ lb(‖ε1 −ε2‖ + |ζ1 −ζ2| + |θ1 −θ2|) for all t ∈ [0,t ],ε1,ε2 ∈sd,ζ1,ζ2,θ1,θ2 ∈r, a.e. x ∈ ω; (3) the mapping x 7→b(x,t,ε,ζ,θ) is lebesgue measurable on ω for any t ∈ [0,t ],ε ∈sd,ζ,θ ∈r; (4) the mapping t 7→b(x,t,ε,ζ,θ) is continuous measurable on [0,t ] for any ε ∈sd,ζ ∈,θ ∈r, a.e. x ∈ ω; (5) the mapping x 7→b(x,t,0, 0, 0) ∈h. (3.18)   (1) s : ω ×sd ×r×r−→r; (2) there exists ls > 0 such that |s(x,ε1,ζ1,θ1) −s(x,ε2,ζ2,θ2)| ≤ ls(‖ε1 −ε2‖ + |ζ1 −ζ2| + |θ1 −θ2|) ∀x ∈ ω,∀ε1,ε2 ∈sd, ∀ ζ1,ζ2,θ1,θ2 ∈ r; (3) the function x −→s(x,ε,ζ,θ) is lebesgue measurable on ω ∀ ε ∈sd,∀ζ ∈r (4) the function x −→s(x,0, 0, 0) ∈ l2(ω). (3.19)   (1) φ : ω ×sd ×r×r−→r; (2) there exists lφ > 0 such that |φ(x,ε1,ς1,θ1) − φ(x,ε2,ς2,θ2)| ≤ lφ(‖ε1 −ε2‖ + |ς1 − ς2| + |θ1 −θ2|) ∀x ∈ ω,∀ε1,ε2 ∈sd ∀ ς1,ς2,θ1,θ2 ∈ r; (3) the mappingx −→ φ(x,ε, ∀ς,θ) is lebesgue mesurable on ω ∀ ε ∈sd,ς,θ ∈r; (4) the function x −→ φ(x,0, 0, 0) ∈ l2(ω). (3.20)   (1) pr : γ3 ×r−→r+, (r = ν,τ); (2) there exists lr > 0 such that ‖pr (x,a1) −pr (x,a2)‖≤ lr|a1 −a2| ∀x ∈ ω,∀a1,a2 ∈r,a.e. x ∈ γ3; (3) pr (x,a) ≤ 0 ∀a ≤ 0, a.e. x ∈ γ3; (4) the mapping x −→ pr (x,a) is lebesgue measurable on γ3 ∀a ∈r. (3.21) we suppose that the mass density satisfies, for ρ∗ > 0 ρ ∈ l∞(ω), ρ(x) ≥ ρ∗, a.e.x ∈ ω. (3.22) we also suppose the mechanical and heat forces satisfy q ∈ l2(0,t ; l2(ω)), f0 ∈ w 1,1(0,t ; h), f2 ∈ w 1,1(0,t ; l2(γ2)d). (3.23) 8 int. j. anal. appl. (2023), 21:56 next, we define the elements f (t) ∈ v by (f(t),v)v = ∫ ω f0 ·vdx + ∫ γ2 f2 ·vda, ∀v∈ v. let us define j : v ×v ×l2(γ3) →r be the functional j(u,v,w) = ∫ γ3 pν(uν −w)vνda + ∫ γ3 pτ (uν −w)‖vτ‖da, (3.24) the functional j satisfies for all u∈ v and w ∈ l2(γ3) j : v→ j(u,v,w) is proper, convex and lower semicontinuous on v. (3.25) taking into account assumptions (3.21) combined with (3.15), we get j(u1,v2,w) − j(u1,v1,w) + j(u2,v1,w) − j(u2,v2,w) ≤ c21 (lν + lτ )‖u1 −u2‖v‖v1 −v2‖v , ∀u1,u2,v1,v2 ∈ v, ∀w ∈ l2(γ3). (3.26) we note that condition (3.23) implies f ∈ w 1,1(0,t ; v ) (3.27) we suppose that the initial data satisfy θ0 ∈ l2(ω), ς0 ∈ k, w0 ∈ l2(γ3), u0 ∈ v, u̇0 ∈ d(∂2j), (3.28) where ∂2j denotes the partial subdifferential with respect to the second argument of the operator j and d(∂2j) represent its domain. there exists g∈ h such that( aε(u̇0) + bε(u0),ε(v) −ε(u̇0) ) h + j(u0,v, 0) − j(u0, u̇0, 0) ≥ (g,v− u̇0), ∀v∈ v. (3.29) we introduce the following continuous functionals a : v → v ′ and b : v → v ′ defined by ∀u ∈ v , ∀v∈ v 〈au,v〉 v ′×v = (gε(u),ε(v))h, (3.30) 〈bu,v〉 v ′×v = (aε(u),ε(v))h. (3.31) we define the bilinear forms a : e ×e →r and b : h1(ω) ×h1(ω) →r defined by a(ξ,κ) = µ0 ∫ ω ∇ξ ·∇κdx + µ1 ∫ γ ξκda, ∀ ξ, κ ∈ e, (3.32) b(ς,ζ) = ∫ ω ∇ς ·∇ζdx, ∀ ς, ζ ∈ h1(ω). (3.33) int. j. anal. appl. (2023), 21:56 9 we will use a modified inner product on the hilbert space h given by ((u,v))h = (ρu,v)h, ∀u,v∈ h, (3.34) that is, it is weighted with ρ, and we let ||| · |||h be the associated norm, i.e., |||v|||h = (ρv,v) 1 2 h , ∀v∈ h. (3.35) it follows from assumptions (3.22) that ||| · |||h and ‖ · ‖h are equivalent norms on h, and also the inclusion mapping of (v,‖ · ‖h) into (h, ||| · |||h) is continuous and dense. we denote by v ′ the dual space of v , and by identifying h with its own dual, write v ⊂ h = h ⊂ v ′ . we use the notation 〈·, ·〉 v ′×v to represent the duality pairing between v ′ and v and recall that 〈u,v〉 v ′×v = ((u,v))h, ∀u,v∈ h. (3.36) using the above notation and a standard procedure based on integrals by parts, we have the following variational formulation of the problem thermo-mechanical (3.1)-(3.11). problem pv. find a displacement field u : ω×[0,t ] −→ v , a damage field ς : ω×[0,t ] −→ l2(ω), a temperature field θ : ω × [0,t ] −→ e and a wear field w : γ3 × [0,t ] −→ l2(γ3) such that (aε(u̇) + gε(u) + ∫ t 0 b ( ε(u(s)),ς(s),θ(s) ) ds,ε(v) −ε(u̇))h + ((ü,v− u̇))h + j(u,v,w) − j(u, u̇,w) ≥ (f(t),v− u̇)v , ∀ v ∈ v, (3.37) (ς̇,β − ς(t))l2(ω) + b(ς(t),β − ς(t)) ≥ (s(ε(u(t)),ς(t),θ(t)),β − ς(t))l2(ω), ∀ β ∈ k, (3.38) 〈θ̇(t),α〉 e ′×e + a(θ(t),α) = 〈φ(ς(t),ε(u(t)),θ(t)),α〉e′×e + (q(t),α)l2(ω), ∀ α ∈ e, a.e t ∈ (0,t ), (3.39) ẇ = kω‖v∗‖pν(uν −w), a.e. t ∈ (0,t ). (3.40) to study problem (3.37)-(3.40), we need the following smallness assumption lν + lτ < 2ma c21 + c1 , (3.41) where ma, c1 and lr, (r = ν,τ) are given in (3.16), (3.15) and (3.17), respectively. our main existence and uniqueness result is stated and proved in the next section. 10 int. j. anal. appl. (2023), 21:56 4. existence and uniqueness result theorem 4.1. let the assumptions (3.16)-(3.29) and (3.41). then there exists an unique solution (u,ς,θ,w) to problem pv . moreover, the solution satisfies u∈ w 1,∞(0,t ; v ) ∩w 2,∞(0,t ; h), (4.1) ς ∈ h1(0,t ; l2(ω)) ∩l2(0,t ; h1(ω)), (4.2) θ ∈ c(0,t ; l2(ω)) ∩l2(0,t ; e), θ̇ ∈ l1(0,t ; e ′ ) (4.3) w ∈ c1(0,t ; l2(γ3)). (4.4) the functions u,σ,ς,θ and w which satisfy (3.1) and (3.37)-(3.40) are called a weak solution of the contact problem p. we conclude that, under the assumptions (3.16)-(3.29) and (3.41), the mechanical problem (3.1)(3.11) has a unique weak solution satisfying (4.1)-(4.4). the regularity of the weak solution is given by (4.1)-(4.4) and, in term of stress, σ ∈ w 1,∞(0,t ;h1). (4.5) indeed, it follows from (3.1), (3.16)(3), (3.17)(3), (3.17)(4), (3.18)(2) and the regularity (4.1)-(4.3) implies σ ∈ w 1,∞(0,t ;h). let t ∈ [0,t ] and we choose as a test function v = u̇(t) ± z where z∈d(ω)d in (3.37) and using (2.1), (3.34) and (3.36) to obtain div σ + f0 = ρü in h. it now follows from (3.23) and (4.1) that div σ ∈ w 1,∞(0,t ; h) which show (4.5). the proof of theorem 4.1 is carried out in several steps that we prove in what follows, everywhere in this section we suppose that assumptions of theorem 4.1 hold, and we consider that c is a generic positive constant which may depend on the problem’s data but it is independent on time, and whose value may change from place to place. let w ∈ c(0,t ; l2(γ3)) and η ∈ l2(0,t ;h). in the first step we consider the following variational problem problem pvwη. find uwη : [0,t ] → v such that ((üwη(t),v− u̇wη(t)))h + (auwη(t),v− u̇wη(t)) + (bu̇wη(t),v− u̇wη(t)) + j(uwη(t),v,w(t)) − j(uwη(t), u̇wη(t),w(t)) ≥ (f(t),v− u̇wη(t))v , ∀ v ∈ v, uwη(0) = u0, u̇wη(0) = u̇0, (4.6) where (f (t),v)v = (f (t),v)v − (η(t),ε(v))h, ∀v∈ v. in the study of problem pvwη we have the following result. int. j. anal. appl. (2023), 21:56 11 lemma 4.1. the problem pvwη has a unique solution which satisfies uwη ∈ w 1,∞(0,t ; v ) ∩ w 2,∞(0,t ; h) proof. by assumptions (3.30), (3.14), (3.17)(3), we see that the operator a is linear, continuous, and symmetric from v to v ′ and satisfies the condition (2.2) with λ = 0 and ω = mg. after, we define the set-valued operator ψ : v → v ′ by ψ = b + ∂2j. (4.7) from (3.16)(2), we deduce that the operator b defined by (3.31), is monotone. using (3.31) and (3.14), we have ‖bu−bv‖ v ′ ≤‖aε(u) −aε(v)‖h, ∀u,v∈ v, keeping in mind (3.16)(2), (3.16)(3), (3.16)(4) and krasnoselski’s theorem (see [15, p.60]), we find that b : v → v ′ is a continuous operator. using again (3.31) and (3.16)(2), we find that b is bounded. from (3.24) and (3.25), we deduce that j is maximal monotone. consequently, since b is monotone, bounded and hemicontinuous from v to v ′ , we conclude (see [4, p.39]) that ψ = b + ∂2j is maximal monotone. moreover, the initial data u0 and u̇0 satisfy (2.3) due to (3.28) and (3.29). thus, all the requirements of theorem 2.1, with a defined by (3.30), m = ψ given in (4.7) and f = f, are satisfied, it follows that there exists a unique solution uwη to problem pvwη satisfying the regularity expressed in (4.1). � let η ∈ l2(0,t ;h). in the second step, we consider the operator χ : c(0,t ; l2(γ3)) → c(0,t ; l2(γ3)) defined by χw(t) = kw‖v∗‖ ∫ t 0 pν(uwην(s) −w(s))ds ∀t ∈ [0,t ]. (4.8) lemma 4.2. the operator χ has a unique fixed point w∗ ∈ c(0,t ; l2(γ3)). proof. let w1,w2 ∈ c(0,t ; l2(γ3)) and denote by ui, i = 1, 2 the solutions to the problem pvwη, for w = wi i.e. ui = uwiη and vi = u̇i = uwiη. from the definition (4.8) of χ, we can write |χw1(t) −χw2(t)|l2(γ3) ≤ kw‖v ∗‖ ∫ t 0 |(pν(u1ν(t) −w1(t)) −pν(u2ν(t) −w2(t)))|ds using (3.21)(2) and (3.15), we get ‖χw1(t) −χw2(t)‖2l2(γ3) ≤ c (∫ t 0 ‖w1(s) −w2(s)‖2l2(γ3)ds + ∫ t 0 ‖u1(s) −u2(s)‖2v ) ds (4.9) 12 int. j. anal. appl. (2023), 21:56 using the relation (4.6), we find ((v̇1(t) − v̇2(t),v1(t) −v2(t)))h + (au1(t) −au2(t),v1(t) −v2(t))v + (bv1(t) −bv2(t),v1(t) −v2(t))v ≤ j(u1(t),v2(t),w1(t)) − j(u2(t),v2(t),w2(t)) + j(u2(t),v1(t),w2(t)) − j(u1(t),v1,w1(t)). by virtue of (3.30), (3.31), (3.16)(2), (3.17)(2), (3.21) and (3.14)(2), this inequality becomes 1 2 d dt |‖v1(t) −v2(t)|‖2h + mg 2 d dt ‖u1(t) −u2(t)‖2v + ma‖v1(t) −v2(t)‖ 2 v ≤ lν ∫ γ3 (|u1ν −u2ν| + |w1 −w2|)|v2ν|da−lν ∫ γ3 (|u1ν −u2ν| + |w1 −w2|)|v1ν|da + lτ ∫ γ3 (|u1ν −u2ν| + |w1 −w2|)‖v2τ‖da−lτ ∫ γ3 (|u1ν −u2ν| + |w1 −w2|)‖v1τ‖da. integrating this inequality over the interval time variable (0,t), using (3.15) and the inequality 2ab ≤ a2 + b2 leads to |‖v1(t) −v2(t)|‖2h + mg 2 ‖u1(t) −u2(t)‖2v + ma ∫ t 0 ‖v1(s) −v2(s)‖2v ds ≤ (lν + lτ ) c1 2 ∫ t 0 ( c1‖u1(s) −u2(s)‖2v + (c1 + 1)‖v1(s) −v2(s)‖ 2 v + ‖w1(s) −w2(s)‖2l2(γ3) ) ds, and keeping in mind (3.41), we obtain ∫ t 0 ‖v1(s) −v2(s)‖2v ds ≤ c ∫ t 0 ( ‖u1(s) −u2(s)‖2v + ‖w1(s) −w2(s)‖ 2 l2(γ3) ) ds. (4.10) on the other hand, since ui (t) = u0 + ∫ t 0 vi (s) ds, we have ‖u1(t) −u2(t)‖2v ≤ ∫ t 0 ‖v1(s) −v2(s)‖2v ds, (4.11) and using this inequality in (4.10) yields ‖u1(t) −u2(t)‖2v ≤ c ∫ t 0 ( ‖w1(t) −w2(t)‖2v + ∫ t 0 ‖v1(s) −v2(s)‖2v ds ) . it follows now from a gronwall-type argument that ‖u1(t) −v2(t)‖2v ds ≤ c ∫ t 0 ‖w1(s) −w2(s)‖2v ds. int. j. anal. appl. (2023), 21:56 13 which implies for s ≤ t ≤ t∫ t 0 ‖u1(s) −u2(s)‖2v ds ≤c ∫ t 0 ∫ s 0 ‖w1(r) −w2(r)‖2l2(γ3)drds ≤c ∫ t 0 ∫ t 0 ‖w1(r) −w2(r)‖2l2(γ3)drds ≤c ∫ t 0 ‖w1(r) −w2(r)‖2l2(γ3)dr ∫ t 0 ds. then ∫ t 0 ‖u1(s) −u2(s)‖2v ds ≤ ct ∫ t 0 ‖w1(s) −w2(s)‖2l2(γ3)ds. (4.12) from (4.8) and (4.12), we deduce that ‖χw1(s) −χw2(s)‖2l2(γ3) ≤ ct ∫ t 0 ‖w1(s) −w2(s)‖2l2(γ3)ds. (4.13) reiterating the last inequality n times, we infer that ‖χnw1 −χnw2‖c(0,t ;l2(γ3)) ≤ ( cntn+1 n! )1 2 ‖w1 −w2‖c(0,t ;l2(γ3)). thus, for n sufficiently large, a power χn of χ is a contraction in the banach space c(0,t ; l2(γ3)). which implies that the operator χ has a unique fixed point w∗ ∈ c(0,t ; l2(γ3)). � in the third step, let γ ∈ l2(0,t ; l2(ω)) be given and consider the following variational problem for the damage field. problem pvγ. find a damage field ςγ : [0,t ] → h1(ω) such that ςγ ∈ k, (ς̇γ,β − ςγ(t))l2(ω) + b(ςγ(t),β − ςγ(t)) ≥ (γ(t),β − ςγ(t))l2(ω) ∀ β ∈ k, ςγ(0) = ς0. (4.14) lemma 4.3. the problem pvγ has a unique solution ςγ satisfying ςγ ∈ h1(0,t ; l2(ω)) ∩l2(0,t ; h1(ω)) (4.15) proof. using (3.33) and (3.28), after some algebraic computations and from a classical existence and uniqueness result of parabolic equations (see for example the reference [22, p.60]), we find that the problem pvγ has a unique solution ςγ ∈ h1(0,t ; l2(ω)) ∩l2(0,t ; h1(ω)). � in the forth step, let ϕ ∈ l2(0,t ; e ′ ), we consider the following variational problem 14 int. j. anal. appl. (2023), 21:56 problem pvϕ. find a temperature θϕ : ω × (0,t ) →r such that 〈θ̇ϕ(t),α〉e′×e + a(θϕ(t),α) = 〈ϕ(t) + q(t),α〉e′×e ∀ α ∈ e a.e t ∈ (0,t ), θϕ(0) = θ0. (4.16) lemma 4.4. the problem pvϕ has a unique solution θϕ satisfies the regularity (4.3). proof. from the friedrichs-poincaré inequality, we can find that there exists a constant µ2 > 0 such that ∫ ω ‖∇(ξ)‖2dx + µ1 µ0 ∫ γ ‖ξ‖2da ≥ µ2 ∫ ω ‖ξ‖2dx. thus, we obtain a(ξ,ξ) ≥ µ3‖ξ‖2e, (4.17) where µ3 = µ0min(1,µ2) 2 , which implies that a is elliptic on e. consequently, based on a classical arguments of functional analysis concerning parabolic equations (see [4, p.140]), we conclude that the problem pvϕ has a unique solution θϕ which satisfies the regularity (4.3). � let us now consider the operator λ : l2(0,t ;h×l2(ω) ×e) → l2(0,t ;h×l2(ω) ×e) λ(η,γ,ϕ)(t) = (λ1(η,γ,ϕ)(t), λ2(η,γ,ϕ)(t), λ3(η,γ,ϕ)(t)), (4.18) defined by λ1(η,γ,ϕ)(t) = ∫ t 0 b ( ε(uw∗η)(s),ςγ(s),θϕ(s) ) ds, (4.19) λ2(η,γ,ϕ)(t) = s(ε(uw∗η(t)),ςγ(t),θϕ(t)), (4.20) λ3(η,γ,ϕ)(t) = ψ(ε(uw∗η(t)),ςγ(t),θϕ(t)). (4.21) here, w∗ be the fixed point of the operator χ. for every (η,γ,ϕ) ∈ l2(0,t ;h×l2(ω)×e ′ ), uw∗η,ςγ and θϕ represent the displacement, the damage and the temperature obtained in lemma 4.1, lemma 4.3 and lemma 4.4 respectively. the last step in the proof of theorem 4.1 is the next result. lemma 4.5. the operator λ has a unique fixed point (η∗,γ∗,ϕ∗) ∈ l2(0,t ;h×l2(ω) ×e′). proof. let (η1,γ1,ϕ1), (η2,γ2,ϕ2) ∈ l2(0,t ;h×l2(ω) ×e ′ ). we use the notation uw∗ηi = ui, u̇w∗ηi = vi,ςγi = ςi and θϕi = θi for i = 1, 2. using (4.19)-(4.21) and from (3.18)-(3.20) and (3.14), we get ‖λ(η1,γ1,ϕ1)(t) − λ(η2,γ2,ϕ2)‖h×l2(ω)×e′ ≤ lb ∫ t 0 ( ‖u1(s) −u2(s)‖v + ‖ς1(s) − ς2(s)‖l2(ω) + ‖θ1(s) −θ2(s)‖e ) ds + (ls + lφ) ( ‖u1(t) −u2(t)‖v + ‖ς1(t) − ς2(t)‖l2(ω) + ‖θ1(t) −θ2(t)‖l2(ω) ) int. j. anal. appl. (2023), 21:56 15 employing hölder’s and young’s inequalities, we deduce that ‖λ(η1,γ1,ϕ1)(t) − λ(η2,γ2,ϕ2)‖2h×l2(ω)×e′ ≤ c ∫ t 0 ( ‖u1(s) −u2(s)‖2v + ‖ς1(s) − ς2(s)‖ 2 l2(ω) + ‖θ1(s) −θ2(s)‖2e ) ds + c ( ‖u1(t) −u2(t)‖2v + ‖ς1(t) − ς2(t)‖2l2(ω) + ‖θ1(t) −θ2(t)‖ 2 l2(ω) ) (4.22) using the relation (4.6), we obtain (aε(v1) −aε(v2),ε(v1 −v2)v + ((v̇1 − v̇2,v1 −v2))h ≤ (gε(u1) −gε(u2),ε(v1 −v2))h + j(u1,v2,w) − j(u1,v1,w) + j(u2,v1,w) − j(u2,v2,w) − (η1 −η2, ,ε(v1) −ε(v2))h we use similar arguments that those used in the proof of the relation (4.12) to obtain that∫ t 0 ‖u1(s) −u2(s)‖2v ds ≤ c ∫ t 0 ‖η1(s) −η2(s)‖2hds. (4.23) from (4.14), we get (ς̇1 − ς̇1,ς1 − ς2)l2(ω) + b(ς1 − ς2,ς1 − ς2) ≤ (ς1 − ς2,ς1 − ς2)l2(ω) a.e. t ∈ (0,t ). integrating the previous inequality with respect to time, using the initial conditions ς1(0) = ς1(0) = ς0 and the inequality b(ς1 − ς2,ς1 − ς2) ≥ 0, we find 1 2 ‖ς1(t) − ς2(t)‖2l2(ω) ≤ ∫ t 0 (ς1(s) − ς2(s),ς1(s) − ς2(s))l2(ω)ds, which implies that ‖ς1(t) − ς2(t)‖2l2(ω) ≤ ∫ t 0 ( ‖γ1(s) −γ2(s)‖2l2(ω) + ‖ς1(s) − ς2(s)‖ 2 l2(ω) ) ds, this inequality combined with gronwall’s inequality leads to ‖ς1(t) − ς2(t)‖2l2(ω) ≤ c ∫ t 0 ‖γ1(s) −γ2(s)‖2l2(ω)ds, ∀t ∈ [0,t ]. (4.24) in order words from (4.16), it follows (θ̇1 − θ̇1,θ1 −θ2)e′×e + a(θ1 −θ2,θ1 −θ2) = (ϕ1 −ϕ2,θ1 −θ2)e′×e a.e. t ∈ (0,t ). we integrate the previous equality, using (3.13), the initial conditions θ1(0) = θ1(0) = θ0 and as a is e-elliptic, we get ‖θ1(t) −θ2(t)‖2l2(ω) + µ3 ∫ t 0 ‖θ1(t) −θ2(t)‖2e ≤ ∫ t 0 ‖ϕ1(s) −ϕ2(s)‖e′‖θ1(s) −θ2(s)‖e a.e. t ∈ (0,t ), 16 int. j. anal. appl. (2023), 21:56 employing young’s and hölder’s inequalities, we get ‖θ1(t) −θ2(t)‖2l2(ω) + ∫ t 0 ‖θ1(s) −θ2(s)‖2eds ≤ c ∫ t 0 ‖ϕ1(s) −ϕ2(s)‖2e′ds a.e. t ∈ (0,t ). (4.25) now, we combine (4.23), (4.24) and (4.25), we find that ‖λ(η1,γ1,ϕ1)(t) − λ(η2,γ2,ϕ2)‖2h×l2(ω)×e′ ≤ c ∫ t 0 ‖(η1,γ1,ϕ1)(t) − (η2,γ2,ϕ2)‖2h×l2(ω)×e′. reiterating this inequality n times we are led to ‖λ(η1,γ1,ϕ1)(t) − λ(η2,γ2,ϕ2)(t)‖2h×l2(ω)×e′ ≤ cn ∫ t 0 ∫ s 0 ... ∫ m 0︸ ︷︷ ︸ n integrals ∫ t 0 ‖(η1,γ1,ϕ1)(r) − (η2,γ2,ϕ2)(r)‖2h×l2(ω)×e′ dr...ds, which implies that ‖λn(η1,γ1,ϕ1) − λn(η2,γ2,ϕ2)‖2l2(h×l2(ω)×e′) ≤ cn tn n! ‖(η1,γ1,ϕ1) − (η2,γ2,ϕ2)‖2l2(h×l2(ω)×e′). (4.26) since lim n→∞ cn tn n! = 0, it follows that there exists a positive integer n such that c n tn n! < 1 and, therefore, (4.26) shows that the operator λn is a contraction on the banach space l2(h×l2(ω) × e ′ ) and, so, there exists a unique fixed point (η∗,γ∗,ϕ∗) ∈ l2(0,t ;h × l2(ω) × e ′ ) such that λ(η∗,γ∗,ϕ∗) = (η∗,γ∗,ϕ∗). � we have now all the ingredient to prove theorem 4.1 which we complete now. existence. let w∗ be the fixed point of the operator χ given by (4.8) and (η∗,γ∗,ϕ∗) be the fixed point of the operator λ given by (4.18)-(4.21) and denote u∗ = uw∗η∗, ς∗ = ςγ∗, θ∗ = θϕ∗. (4.27) it follows from (4.19)-(4.21) that η∗(t) = ∫ t 0 b ( ε(u∗(s)),ς∗(s),θ∗(s) ) ds, γ∗(t) = s(ε(u∗(t)),ς∗(t),θ∗(t)), ϕ∗(t) = ψ(ε(u∗(t)),ς∗(t),θ∗(t)), and, therefore, (4.6), (4.14), (4.16) and (4.8) imply that (u∗,ς∗,θ∗,w∗) is a solution of problem pv . int. j. anal. appl. (2023), 21:56 17 uniqueness. the uniqueness of the solution follows from the uniqueness of the fixed point of the operators χ and λ defined by (4.8) and (4.18)-(4.21) respectively. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] l.e. andersson, a quasistatic frictional problem with normal compliance, nonlinear anal.: theory methods appl. 16 (1991), 347–369. https://doi.org/10.1016/0362-546x(91)90035-y. [2] k.t. andrews, m. shillor, s. wright, a. klarbring, a dynamic thermoviscoelastic contact problem with friction and wear, int. j. eng. sci. 35 (1997), 1291–1309. https://doi.org/10.1016/s0020-7225(97)87426-5. [3] k.t. andrews, k.l. kuttler, m. shillor, on the dynamic behaviour of a thermoviscoelastic body in frictional contact with a rigid obstacle, eur. j. appl. math. 8 (1997), 417–436. https://doi.org/10.1017/ s0956792597003173. [4] v. barbu, nonlinear semigroups and differential equations in banach spaces, springer, dordrecht, 1976. [5] m. bouallala, e.h. essoufi, analysis results for dynamic contact problem with friction in thermo-viscoelasticity, methods funct. anal. topol. 26 (2020), 317–326. https://doi.org/10.31392/mfat-npu26_4.2020.03. [6] i. boukaroura, s. djabi, analysis of a quasistatic contact problem with wear and damage for thermo-viscoelastic materials, malaya j. mat. 06 (2018), 299–309. https://doi.org/10.26637/mjm0602/0001. [7] o. chau, a. petrov, a. heibig, m.m. marques, a frictional dynamic thermal contact problem with normal compliance and damage, in: t.m. rassias, p.m. pardalos (eds.), nonlinear analysis and global optimization, springer international publishing, cham, 2021: pp. 71–107. https://doi.org/10.1007/978-3-030-61732-5_4. [8] j. chen, w. han, m. sofonea, numerical analysis of a quasistatic problem of sliding frictional contact with wear, methods appl. anal. 7 (2000), 687–704. https://doi.org/10.4310/maa.2000.v7.n4.a5. [9] c. ciulcu, t.v. hoarau-mante, m. sofonea, viscoelastic sliding contact problems with wear, math. computer model. 36 (2002), 861–874. https://doi.org/10.1016/s0895-7177(02)00233-9. [10] l. chouchane, l. selmani, a history-dependent frictional contact problem with wear for thermoviscoelastic materials, math. model. anal. 24 (2019), 351–371. https://doi.org/10.3846/mma.2019.022. [11] a. djabi, a. merouani, bilateral contact problem with friction and wear for an electro elastic-viscoplastic materials with damage, taiwan. j. math. 19 (2015), 1161-1182. https://doi.org/10.11650/tjm.19.2015. 5453. [12] j.r. fernández, m. sofonea, variational and numerical analysis of the signorini’s contact problem in viscoplasticity with damage, j. appl. math. 2003 (2003), 87–114. https://doi.org/10.1155/s1110757x03202023. [13] l. gasiński, a. ochal, dynamic thermoviscoelastic problem with friction and damage, nonlinear anal.: real world appl. 21 (2015), 63–75. https://doi.org/10.1016/j.nonrwa.2014.06.004. [14] l. johansson, a. klarbring, thermoelastic frictional contact problems: modelling, finite element approximation and numerical realization, computer methods appl. mech. eng. 105 (1993), 181–210. https://doi.org/10. 1016/0045-7825(93)90122-e. [15] o. kavian, introduction à la théorie des points critiques et applications aux équations elliptiques, springer, berlin, 1993. [16] a. klarbring, a. mikelić, m. shillor, a global existence result for the quasistatic frictional contact problem with normal compliance, in: g. del piero, f. maceri (eds.), unilateral problems in structural analysis iv, birkhäuser basel, basel, 1991: pp. 85–111. https://doi.org/10.1007/978-3-0348-7303-1_8. https://doi.org/10.1016/0362-546x(91)90035-y https://doi.org/10.1016/s0020-7225(97)87426-5 https://doi.org/10.1017/s0956792597003173 https://doi.org/10.1017/s0956792597003173 https://doi.org/10.31392/mfat-npu26_4.2020.03 https://doi.org/10.26637/mjm0602/0001 https://doi.org/10.1007/978-3-030-61732-5_4 https://doi.org/10.4310/maa.2000.v7.n4.a5 https://doi.org/10.1016/s0895-7177(02)00233-9 https://doi.org/10.3846/mma.2019.022 https://doi.org/10.11650/tjm.19.2015.5453 https://doi.org/10.11650/tjm.19.2015.5453 https://doi.org/10.1155/s1110757x03202023 https://doi.org/10.1016/j.nonrwa.2014.06.004 https://doi.org/10.1016/0045-7825(93)90122-e https://doi.org/10.1016/0045-7825(93)90122-e https://doi.org/10.1007/978-3-0348-7303-1_8 18 int. j. anal. appl. (2023), 21:56 [17] j.a.c. martins, j.t. oden, existence and uniqueness results for dynamic contact problems with nonlinear normal and friction interface laws, nonlinear anal.: theory methods appl. 11 (1987), 407–428. https://doi.org/10. 1016/0362-546x(87)90055-1. [18] m.s. mesai aoun, m. selmani, a.a. ahmed, variational analysis of a frictional contact problem with wear and damage, math. model. anal. 26 (2021), 170–187. https://doi.org/10.3846/mma.2021.11942. [19] k. rimi, t.h. ammar, variational analysis of an electro-elasto-viscoplastic contact problem with friction and wear, commun. math. appl. 12 (2021), 145–159. https://doi.org/10.26713/cma.v12i1.1461. [20] m. selmani, frictional contact problem with wear for electro-viscoelastic materials with long memory, bull. belg. math. soc. simon stevin. 20 (2013), 461-479. https://doi.org/10.36045/bbms/1378314510. [21] m. shillor, m. sofonea, j.j. telega, quasistatic viscoelastic contact with friction and wear diffusion, quart. appl. math. 62 (2004), 379–399. https://doi.org/10.1090/qam/2054605. [22] m. sofonea, w. han, m. shillor, analysis and approximation of contact problems with adhesion or damage. pure and applied mathematics 276. chapman-hall/crc press, new york 2006. http://doi.org/10.1201/ 9781420034837. [23] n. stromberg, continuum thermodynamics of contact, friction and wear. ph.d. thesis, linkoping university, sweeden, 1995. [24] n. strömberg, l. johansson, a. klarbring, derivation and analysis of a generalized standard model for contact, friction and wear, int. j. solids struct. 33 (1996), 1817–1836. https://doi.org/10.1016/0020-7683(95) 00140-9. https://doi.org/10.1016/0362-546x(87)90055-1 https://doi.org/10.1016/0362-546x(87)90055-1 https://doi.org/10.3846/mma.2021.11942 https://doi.org/10.26713/cma.v12i1.1461 https://doi.org/10.36045/bbms/1378314510 https://doi.org/10.1090/qam/2054605 http://doi.org/10.1201/9781420034837 http://doi.org/10.1201/9781420034837 https://doi.org/10.1016/0020-7683(95)00140-9 https://doi.org/10.1016/0020-7683(95)00140-9 1. introduction 2. notations and preliminaries 3. mechanical and variational formulations problem p problem pv 4. existence and uniqueness result problem pvw problem pv problem pv existence uniqueness references international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 70-80 http://www.etamaths.com alpha convex functions associated with conic domains khalida inayat noor1, nasir khan1 and krzysztof piejko2,∗ abstract. in this paper we define a new class k − umα [a,b] of janowski type k−uniformly alpha convex functions. we use the method of differential subordinations theory to obtain some new results like sufficient condition, inclusion relations, coefficient estimate and covering properties. the results presented here include a number of well-known results as their special cases. 1. introduction let a denote the class of functions f (z) of the form (1.1) f (z) = z + ∞∑ n=2 anz n, which are analytic in the unit disk e = {z ∈ c : |z| < 1}. furthermore s represents class of all functions in a which are univalent in e. for two functions f(z) and g(z) analytic in a, we say that f(z) is subordinate to g(z) in e (and write f ≺ g or f(z) ≺ g(z)), if there exists an analytic function w(z) such that |w(z)| ≤ |z| and f(z) = g (w(z)) for z ∈ e. if g(z) is univalent in e then f(z) ≺ g(z) if and only if f (0) = g (0) and f (e) ⊂ g (e). the idea of subordination goes back to lindelöf [9]. subordination was more formally introduced and studied by littelwood [10] and later by rogosinski [20] and [19]. the concept of subordination was considered by miller [12] and further investigated by noor et al. [16] and many others see [9],[21]. definition 1. a function p (z) is said to be in the class p [a,b] , if it is analytic in e with p (0) = 1 and p (z) ≺ 1 + az 1 + bz , − 1 ≤ b < a ≤ 1. this class was presented by janowski [3] and explored by a few creators. kanas and wísniowska [4],[5] presented and examined the class k−st of k−starlike functions and the relating class k−ucv of k−uniformly convex functions. there were characterized subject to the conic region ωk, k ≥ 0, as ωk = { u + iv : u > k √ (u− 1)2 + v2 } . this domain represents the right half plane, a parabola, a hyperbola and an ellipse for k = 0,k = 1, 0 < k < 1 and k > 1 respectively. the extremal functions for these conic regions are (1.2) pk (z) =   1+z 1−z , k = 0, 1 + 2 π2 ( log 1+ √ z 1− √ z )2 , k = 1, 1 + 2 1−k2 sinh 2 {( 2 π arccos k ) arctan h √ z } , 0 < k < 1, 1 + 2 k2−1 sin ( π 2r(t) ∫ u(z)√ t 0 dx√ 1−x2 √ 1−(tx)2 ) + 1 k2−1, k > 1, where u(z) = z − √ t 1 − √ tx , (z ∈ e) , 2010 mathematics subject classification. 30c45, 30c50. key words and phrases. subordination; janowski functions; conic region. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 70 alpha convex functions 71 and t ∈ (0, 1) and z is chosen such that k = cosh ( πr′(t) 4r(t) ) . here r(t) is legendre’s complete elliptic integral of first kind and r′(t) is the complementary integral of r (t). if pk (z) = 1 + δkz + · · · , then it is shown in [5] that from (1.2), one can have (1.3) δk =   8(arccos k)2 π2(1−k2) , 0 ≤ k < 1, 8 π2 , k = 1, π2 4(k2−1) √ t(1+t)r2(t) , k > 1. using the concepts of janowski functions and the conic regions, noor et al. [16] gave the following definition 2. [16]a function p (z) is said to be in the class k −p [a,b] , if and only if p (z) ≺ (a + 1) pk (z) − (a− 1) (b + 1) pk (z) − (b − 1) , k ≥ 0, where pk (z) is defined in (1.2) and −1 ≤ b < a ≤ 1. geometrically, the function p (z) ∈ k − p [a,b], takes all values from the domain ωk[a,b], −1 ≤ b < a ≤ 1,k ≥ 0 which is defined as ωk[a,b] = { w : re ( (b − 1) w (z) − (a− 1) (b + 1) w (z) − (a + 1) ) > k ∣∣∣∣(b − 1) w (z) − (a− 1)(b + 1) w (z) − (a + 1) − 1 ∣∣∣∣ } . the domain ωk[a,b] retains the conic domain ωk inside the circular region defined by ω[a,b]. the impact of ω[a,b], on the conic domain ωk, changes the original shape of the conic regions. the ends of hyperbola and parabola get closer to one another but never meet anywhere and the ellipse gets the oval shape. when a → 1,b →−1 the radius of the circular disk defined by ω[a,b] tends to infinity, consequently the arm of the hyperbola and parabola expands to the oval terns into ellipse. we see that ωk[1,−1] = ωk, the conic domain defined by kanas and wísniowska [4]. now using janowski functions and the conic regions, we give the following definition 3. a function f (z) ∈ a is said to be in the class k−umα [a,b] , k ≥ 0, 0 ≤ α ≤ 1,−1 ≤ b < a ≤ 1, if and only if (1.4) j (α,f; z) ∈ k −p [a,b] , where j (α,f; z) = (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) . special cases: (i) k −um0 [a,b] = k −st [a,b], k −um1 [a,b] = k −ucv [a,b] , the classes introduced by noor et al. in [16]. (ii) k −um0 [1,−1] = k −st and k −um1 [1,−1] = k −ucv, we get the classes investigated by kanas and wisniowska [4], [5]. (iii) k −umα [1,−1] = k −umα, we have the class introduced and studied by kanas [7]. (iv) 0 −um0 [a,b] = s [a,b] and 0 −um1 [a,b] = c [a,b] , the well-known classes of janowski starlike and janowski convex functions, respectively, introduced by janowski [3]. definition 4. let ss∗ (β) denote the class of strongly starlike functions of order β, ss∗ (β) = { f ∈ a : ∣∣∣∣arg zf′(z)f (z) ∣∣∣∣ < βπ2 z ∈ e } , β ∈ (0, 1) , which was introduced in [24] and [1]. in this paper, several interesting subordination results are derived which yield sufficient condition, inclusion relations, coefficient estimate, covering result and order of strongly starlikeness in the class of uniformly alpha convex function. to avoid repetitions, it is admitted once that 0 ≤ α ≤ 1,k ≥ 0, and −1 ≤ b < a ≤ 1. 72 noor, khan and piejko 2. preliminary results to prove our main results we need the following lemmas. lemma 1. [19]let f(z) be subordinate to g(z), with f(z) = 1 + ∞∑ n=1 anz n, g(z) = 1 + ∞∑ n=1 bnz n. if g(z) is univalent in e and g(e) is convex, then |an| ≤ |b1|. lemma 2. [11]let f be analytic and convex in e. if f,g ∈ a and f,g ≺ f, then for t ∈ [0, 1] (1 − t)f + tg ≺ f. lemma 3. [14] let k ≥ 0 and let δ,σ be any complex numbers with δ 6= 0 and < (( 2k+1−a 2k+1−b ) δ + σ ) > 0. if p(z) is analytic in e and p(0) = 1 and satisfies (2.1a) p(z) + zp ′ (z) δp(z) + σ ≺ pk(a,b; z), where pk(a,b; z) = (a + 1) pk (z) − (a− 1) (b + 1) pk (z) − (b − 1) , and q(z) is an analytic solution of q(z) + zq(z) δq(z) + σ = pk(a,b; z) then function q(z) is univalent p(z) ≺ q(z) ≺ pk(a,b; z) and q(z) is the best dominant of (2.1a) and is given as q(z) = [ δ ∫ 1 0 ( tδ+σ−1 exp ∫ tz t pk(a,b,z) − 1 u du )δ dt ]−1 − σ δ . lemma 4. [18] let a function p (z) be analytic in e and has the form p(z) = 1 + ∞∑ n=m cnz n , cm 6= 0, with p (z) 6= 0 for |z| < 1. if there exists a point z◦, |z◦| < 1, such that |arg p(z)| < π 2 θ for |z| < |z◦| and |arg p(z◦)| = π 2 θ, for some θ > 0, then we have z◦p ′(z◦) p(z◦) = ilθ, where l ≥ m 2 (x + 1 x ) ≥ m when arg{p(z◦)} = π 2 θ and l ≤− m 2 (x + 1 x ) ≤−m when arg{p(z◦)} = − π 2 θ, where (p (z◦)) 1 θ = ±ix and x > 0. alpha convex functions 73 3. main results theorem 1. a function f (z) ∈ k −umα[a,b], if it satisfies the condition ∞∑ n=1 zn (k,α,a,b) < |b −a| , where (3.1) zn (k,α,a,b) = ∑∞ n=2[|{2 (k + 1) (1 −n) (1 −α (1 −n)) + ((a + 1) (n + 1) −(b + 1) ( 2n + α ( 1 −n2 )) )}| |an| + ∑n−1 j=2 |{2 (k + 1) ((1 − j) −α (n + 1 − 2j)) + ((a + 1) − (b + 1) ((1 − 2α) j + α (1 + n)))}(n + 1 − j) | |ajan+1−j|]. proof. assume that (3.1) holds, then it suffices to show that (3.2) k ∣∣∣∣(b − 1)j (α,f; z) − (a− 1)(b + 1)j (α,f; z) − (a + 1) − 1 ∣∣∣∣−re [ (b − 1)j (α,f; z) − (a− 1) (b + 1)j (α,f; z) − (a + 1) − 1 ] < 1. we have k ∣∣∣∣(b − 1)j (α,f; z) − (a− 1)(b + 1)j (α,f; z) − (a + 1) − 1 ∣∣∣∣−re [ (b − 1)j (α,f; z) − (a− 1) (b + 1)j (α,f; z) − (a + 1) − 1 ] ≤ (k + 1) ∣∣∣∣∣ (b − 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (a− 1)f (z) f′(z) (b + 1) ( (1 −α) zf′(z)f′ (z) + αf (z) (zf′(z))′ ) − (a + 1)f (z) f′(z) − 1 ∣∣∣∣∣ = 2 (k + 1) ∣∣∣∣∣ (1 −α) f (z) f ′(z) − (1 −α) zf′(z)f′(z) −αzf (z) f′′(z) (b + 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (a + 1)f (z) f′(z) ∣∣∣∣∣ . (3.3) now we have zf′(z)f′(z) = z ( ∞∑ n=0 nanz n−1 )( ∞∑ n=0 nanz n−1 ) (3.4) = 1 z ( ∞∑ n=0 nanz n )( ∞∑ n=0 nanz n ) = 1 z ∞∑ n=0   n∑ j=0 j (n− j) ajan−j  zn = ∞∑ n=0   n∑ j=0 j (n− j) ajan−j  zn−1 = z + ∞∑ n=3   n∑ j=0 j (n− j) ajan−j  zn−1 = z + ∞∑ n=2  n+1∑ j=0 j (n + 1 − j) ajan+1−j  zn = z + ∞∑ n=2  2nan + n−1∑ j=2 j (n + 1 − j) ajan+1−j  zn.(3.5) 74 noor, khan and piejko proceeding on the same way we have (3.6) f (z) f′(z) = z + ∞∑ n=2  (n + 1) an + n−1∑ j=2 (n + 1 − j) ajan+1−j  zn and (3.7) zf (z) f′′(z) = ∞∑ n=2  n (n− 1) an + n−1∑ j=2 (n + 1 − j) (n− j) ajan+1−j  zn. using the equalities (3.5),(3.6) and (3.7) in (3.3), the equation (3.3) in simplified form can be written as k ∣∣∣∣(b − 1)j (α,f; z) − (a− 1)(b + 1)j (α,f; z) − (a + 1) − 1 ∣∣∣∣−re [ (b − 1)j (α,f; z) − (a− 1) (b + 1)j (α,f; z) − (a + 1) − 1 ] ≤ 2 (k + 1) ∣∣∣∣∣ (1 −α) f (z) f ′(z) − (1 −α) zf′(z)f′ (z) −αzf (z) f′′(z) (b + 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (a + 1)f (z) f′(z) ∣∣∣∣∣ ≤ 2 (k + 1)   ∑∞ n=2[|(1 −n) (1 −α (1 −n))| |an| + ∑n−1 j=2 |[(1 − j) −α (n + 1 − 2j)] (n + 1 − j)| |ajan+1−j|]     |b −a|− ∑∞ n=2[ ∣∣((a + 1) (n + 1) − (b + 1) (2n + α(1 −n2)))∣∣ |an| − ∑n−1 j=2 |((a + 1) − (b + 1) ((1 − 2α) j + α (1 + n))) (n + 1 − j)| |ajan+1−j|]   . the last expression is bounded by 1, if∑∞ n=2[|{2 (k + 1) (1 −n) (1 −α (1 −n)) + ((a + 1) (n + 1) −(b + 1) ( 2n + α ( 1 −n2 )) )}| |an| + ∑n−1 j=2 |{2 (k + 1) ((1 − j) −α (n + 1 − 2j)) + ((a + 1) − (b + 1) ((1 − 2α) j + α (1 + n)))}(n + 1 − j) | |ajan+1−j|] < |b −a| . this completes the proof. � putting α = 0, in theorem 1, we have the result below which is comparable to the one obtained by noor and malik [15]. corollary 1. a function f ∈ k −st [a,b] , if it satisfies the condition ∞∑ n=2 {2 (k + 1) (n− 1) + |n (b + 1) + (a + 1)|} |an| < |b −a| . putting α = 0, a = 1 and b = −1 in theorem 1, we can obtain the following result which improves the result of kanas and wísniowska [4]. corollary 2. a function f ∈ k −st, if it satisfies the condition ∞∑ n=2 {n + k (n− 1)}|an| < 1. putting α = 0,a = 1 − 2β, b = −1 with 0 ≤ β < 1 in theorem 1, we have the result below which is comparable to the one obtained by shams et al. [22]. corollary 3. a function f (z) ∈ sd (k,β) , if it satisfies the condition ∞∑ n=2 {n (k + 1) − (k + β)}|an| < 1 −β. alpha convex functions 75 putting α = 0,a = 1 − 2β, b = −1 with 0 ≤ β < 1 and k = 0 in theorem 1, we get the following result proved by silverman [23]. corollary 4. a function f (z) ∈ s∗ (β) , if it satisfies the condition ∞∑ n=2 {n−β}|an| < 1 −β. putting α = 1, in theorem 1, we can obtain corollary 5, below which is comparable to the result obtained by noor and malik [15]. corollary 5. a function f ∈ k −ucv [a,b] , if it satisfies the condition ∞∑ n=2 n{2 (k + 1) (n− 1) + |n (b + 1) + (a + 1)|} |an| < |b −a| . the following is an inclusion result stating the fact that k −umα[a,b] ⊂ k −st[a,b]. theorem 2. let f(z) ∈ k −umα[a,b]. then f(z) ∈ k −st[a,b]. proof. let f(z) ∈ k −umα[a,b] and let (3.8) zf ′ (z) f(z) = p(z), where p(z) is analytic in e with p(0) = 1. differentiating logarithmically we have (3.9) (zf′(z)) ′ f′(z) = p(z) + zp ′ (z) p(z) . using (3.8) and (3.9), we have j (α,f; z) = p(z) + αzp ′ (z) p(z) . since f(z) ∈ k −umα[a,b], so we obtain j (α,f; z) = p(z) + zp ′ (z) 1 α p(z) ∈ k −umα[a,b]. since < (( 2k+1−a 2k+1−b ) 1 α ) > 0, z ∈ e, therefore applying lemma 3, with δ = 1 α and σ = 0, we have zf ′ (z) f(z) = p(z) ≺ pk(a,b,z), which implies that f(z) ∈ k −st[a,b]. � by giving special values to the parameters in theorem 2, we get the following well-known result proved by mocanu in [13]. corollary 6. let f(z) ∈ 0 −umα[1,−1]. then f(z) ∈ 0 −st[1,−1]. that is mα ⊂ s∗, α ≥ 0. 76 noor, khan and piejko theorem 3. if 0 ≤ α1 < α2, then k −umα2 [a,b] ⊂ k −umα1 [a,b]. proof. let f(z) ∈ k −umα2 [a,b]. then consider j (α1,f; z) = [ (1 −α1) zf ′ (z) f(z) + α1 ( 1 + zf ′′ (z) f ′ (z) )] = ( 1 − α1 α2 ) zf ′ (z) f(z) + α1 α2 [ (1 −α2) zf ′ (z) pf(z) + α2 ( 1 + zf ′′ (z) f ′ (z) )] = ( 1 − α1 α2 ) j (0,f; z) + α1 α2 (j (α2,f; z)) . now as f(z) ∈ k −umα2 [a,b] so j (α2,f; z) ∈ k −p[a,b], also from theorem 2, j (0,f; z) ∈ k −p[a,b]. using theses along with lemma 2, we have j (α1,f; z) ∈ k −p[a,b], which implies that f(z) ∈ k −umα1 [a,b]. � theorem 4. a function f (z) is in k −umα[a,b],α > 0, if and only if there exists a function g (z) belonging to the class k −st[a,b], such that (3.10) f (z) =   1 α z∫ 0 {g (z)} 1 α t−1dt  α . proof. let us set g (z) = f (z) { zf′(z) f (z) }α , so that (3.10) is satisfied. logarithmically differentiation gives us zg′(z) g (z) = (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) . hence f ∈ k −umα[a,b] if and only if g ∈ k −st[a,b]. � theorem 5. let the function f (z) ∈ k −umα[a,b]. then |a2| ≤ (a−b) δk 2 (1 + α) , where δk is given by (1.3). proof. let f (z) ∈ k −umα [a,b] . then (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) = p (z) z ∈ e, where p (z) ≺ (a + 1) pk (z) − (a− 1) (b + 1) pk (z) − (b − 1) = 1 + 1 2 (a−b) δkz + · · · , where pk (z) = 1 + δkz + · · · . alpha convex functions 77 now using the definition of subordination we can see that there exists a function ω (z) analytic in e with ω (0) = 0 and |ω (z)| < 1 such that (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) = 1 + 1 2 (a−b) δkω (z) + · · · 1 + (1 + α) a2z + ( 2 (1 + 2α) a3 − (1 + 3α) a22 ) z2 · · · = 1 + 1 2 (a−b) δk ( c1z + c2z 2 + · · · ) + · · · . comparing the coefficient of z both sides and using well known result due to janowski and lemma 1, we have |(1 + α) a2| ≤ 1 2 (a−b) δk. this gives |a2| ≤ (a−b) δk 2 (1 + α) and the proof is complete. � taking α = 0,a = 1, b = −1 in theorem 5, we can obtain the following result proved in [5]. corollary 7. let f ∈ k −st . then |a2| ≤ δk, where δk is given by (1.3). putting k = 0,δk = 2, α = 0,a = 1,b = −1 in theorem 5, we can obtain corollary 8 below which is the result obtained in [2]. corollary 8. let f ∈ s∗. then |a2| ≤ 2. putting α = 1,a = 1, b = −1 in theorem 5, we can obtain corollary 9 below which is comparable to the result obtained in [4]. corollary 9. let f ∈ k −ucv . then |a2| ≤ δk 2 , where δk is given by (1.3). putting k = 0,δk = 2, α = 0,a = 1, b = −1 in theorem 5, we can obtain corollary 10 below which is the result obtained in [2]. corollary 10. let f ∈c. then |a2| ≤ 1. theorem 6. the range of every univalent functions f ∈ k −umα[a,b], contains the unit disk rα,δk = 2 (1 + α) 4 (1 + α) + (a−b) δk , where δk is given by (1.3). proof. let ω◦ be any complex number such that f (z) 6= ω◦. then ω◦f (z) ω◦ −f (z) = z + ( a2 + 1 ω◦ ) z2 + · · · , is univalent in e so that ∣∣∣∣a2 + 1ω◦ ∣∣∣∣ ≤ 2. therefore ∣∣∣∣ 1ω◦ ∣∣∣∣ ≤ 4 (1 + α) + (a−b) δk2 (1 + α) . 78 noor, khan and piejko hence using theorem 5, we have |ω◦| ≤ 2 (1 + α) 4 (1 + α) + (a−b) δk = rα,δk. � putting α = 0,a = 1,b = −1 in theorem 6, we can obtain corollary 11. corollary 11. the range of every univalent functions f ∈ k −st contains the unit disk rδk = 1 2 + δk , where δk is given by (1.3). putting α = 1,a = 1,b = −1 in theorem 6, we can obtain corollary 12. corollary 12. the range of every univalent functions f ∈ k −ucv contains the unit disk rδk = 2 4 + δk , where δk is given by (1.3). letting k = 1,a = 1 and b = −1, we have the following theorem. theorem 7. let f ∈ umα and let it be of the form f (z) = z + ∞∑ n=m+1 anz n am+1 6= 0. then f (z) is strongly starlike of order θ◦, where (3.11) θ◦ = min θ∈(0,1) { 1 − 2xθ cos ( θπ 2 ) + ( αm ( x2 + 1 ) θ 2x + xθ sin ( θπ 2 )) ≥ 0 for all x > 0 } . proof. from the assumption we have (3.12) < { (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) } > ∣∣∣∣(1 −α) zf′(z)f (z) + α(zf ′(z)) ′ f′(z) − 1 ∣∣∣∣ . let p (z) = zf′(z) f(z) , then by p (z) has of the form p(z) = 1 + ∞∑ n=m cnz n, and (3.12) , becomes (3.13) < { p(z) + α z◦p ′(z◦) p(z) } > ∣∣∣∣p(z) + αz◦p′(z◦)p(z) − 1 ∣∣∣∣ . if there exists a point z◦, |z◦| < 1, such that |arg{p(z)}| < π 2 θ for |z| < |z◦| , and |arg p(z◦)| = π 2 θ. then, applying lemma 4, we have z◦p ′(z◦) p(z◦) = ilθ, where (p(z◦)) 1 θ = ±ix (x > 0), l ≥ m 2 (x + 1 x ) when arg{p(z◦)} = π 2 θ, and l ≤− m 2 (x + 1 x ) when arg{p(z◦)} = − π 2 θ. alpha convex functions 79 therefore, for the case arg{p(z◦)} = π2 θ, we have (3.14) < ( p(z◦) + α z◦p ′(z◦) p(z◦) ) = < { (ix) θ + αlθ } = xθ cos ( θπ 2 ) , and ∣∣∣∣p(z◦) + αz◦p′(z◦)p(z◦) − 1 ∣∣∣∣ = ∣∣∣(ix)θ + iαlθ − 1∣∣∣ = ∣∣∣∣xθ cos ( θπ 2 ) − 1 + i ( αlθ + xθ sin ( θπ 2 ))∣∣∣∣ = √( xθ cos ( θπ 2 ) − 1 )2 + ( αlθ + xθ sin ( θπ 2 ))2 . (3.15) from (3.11) and then from l ≥ m 2 (x + 1 x ) for θ ≥ θ◦, we have 0 ≤ 1 − 2xθ cos ( θπ 2 ) + ( αθm 2x (x2 + 1) + xθ sin ( θπ 2 ))2 ≤ 1 − 2xθ cos ( θπ 2 ) + ( αθl + xθ sin ( θπ 2 ))2 .(3.16) therefore, (3.17) 0 ≤ 1 − 2xθ◦ cos ( θ◦π 2 ) + ( αθ◦l + x θ◦ sin ( θ◦π 2 ))2 , by (3.14) and (3.15) is equivalent to the inequality < { p(z◦) + α z◦p ′(z◦) p(z◦) } ≤ ∣∣∣∣p(z◦) + αz◦p′(z◦)p(z◦) − 1 ∣∣∣∣ , which contradicts with (3.11). therefore, |arg{p(z◦)}| < π2 θ◦ for |z| < 1. for the case arg{p(z◦)} = −π2 θ◦, applying the same method as the above we will get a contradiction. in this way we have proved that f is strongly starlike of order θ◦. this completes the proof. � letting α = 1, in theorem 7, we have the result 13 below which is comparable to the one obtained in [18]. corollary 13. let f ∈ ucv and let it be of the form f (z) = z + ∞∑ n=m+1 anz n am+1 6= 0. then f (z) is strongly starlike of order θ◦, where θ◦ = min θ∈(0,1) { 1 − 2xθ cos ( θπ 2 ) + ( m ( x2 + 1 ) θ 2x + xθ sin ( θπ 2 )) ≥ 0 for all x > 0 } . acknowledgement the authors express deep gratitude to dr. s. m. junaid zaidi, rector, ciit, for his support and providing excellent research facilities. this research is carried out under the hec project grant no. nrpu no. 20-1966/r&d/11-2553. references [1] d. a. brannan., w. e. kirwan., on some classes of bounded univalent functions, j. london. math. soc. 1(2)(1969), 431-443. [2] p. l. duren., univalent functions grundlehren der math. wissenchanften, springer-verlag, new york-berlin (1983). [3] w. janowski., some external problem for certain families of analytic functions, i. ann. polon. math. 28(1973),298326. [4] s. kanas, a. wísniowska., conic regions and k-uniform convexity, j. comput. appl. math. 105(1999), 327-336. 80 noor, khan and piejko [5] s. kanas, a. wísniowska., conic domains and starlike functions, rev. roumaine math. pure. appl. 45(2000), 647-657. [6] s. kanas., coefficient estimates in subclasses of the caratheodory class related to conical domains, acta. math. univ. comenian. 74(2)(2005), 149-161. [7] s. kanas., alternative characterization of the class k − ucv and related classes of univalent functions, serdica math. j. 25(1999), 341-350. [8] s. kanas., techniques of the differential subordination for domains bounded by conic sections, int. j. math. math. sci. 38(2003), 2389-2400. [9] e. lindelöf., mémorie sur certaines inequalitiés dans la théorie des fonctions monogènes et sur quelques propiétés nouvelles de ces fonctions dans le voisinage d’un point singulier essentiel, acta soc. sci. fenn. 35(1908) 1-35. [10] j. e. littelwood., on inequalities in the theory of functions, proc. london. math. soc. 23(1925), 481-519. [11] m. s. liu., on certain subclass of analytic functions; j. south. china normal univ (in chinese). 4(2002), 15-20. [12] s. s. miller., p. t. mocanu., differential subordination and univalent functions, michigan. math. j. 28(2)(1981), 157-172. [13] p. t. mocanu., une propriete de convexite generlise dans la theorie de la representation conforme, mathematica (cluj). 11(1969), 127-133. [14] s. nawaz., certain subclasses of analytic functions associated with conic domains. ph.d thesis(2012), comsats institute of information technology islamabad, pakistan. [15] k. i. noor, s. n. malik., on coefficient inequalities of functions associated with conic domains, comput. math. appl. 62(2011), 2209-2217. [16] k. i. noor, s. n. malik., m. arif, m. raza., on bounded boundary and bounded radius rotation related with janowski function, world. appl. sci. j. 12(6) (2011), 895-902. [17] k. i. noor, m. arif, w. ul-haq., on k-uniformly close-to-convex functions of complex order, appl. math. comput. 215(2)(2009), 629-635. [18] m. nunokawa, j. sokol., on order of strongly starlikeness in the class of uniformly convex functions, math. nachr. 288(2015), 1003c1008. [19] w. rogosinski., on the coefficients of subordinate functions, proc. lond. math. soc. 48(2)(1943), 48-82. [20] w. rogosinski., on subordinate functions, proc. camb. phil. soc. 35(1939), 1-26. [21] m. a. rosihan., v. ravichandran., convolution and differential subordination for mulitivalent functions, bull. malays. math. sci. 32(3) (2009), 351-360. [22] s. shams, s. r. kulkarni, j. m. jahangiri., classes of uniformly starlike and convex functions, int. j. math. math. sci. 55(2004), 2959-2961. [23] h. silverman., univalent functions with negative coefficients, proc. amer. math. soc. 51(1975), 109-116. [24] j. stankiewicz., quelques probl‘emes extr‘emaux dans les classes des fonctions α−angulairement ‘etoil‘ees, ann. univ. mariae curie–sk lodowska sect. a 20(1966), 59-75. 1department of mathematics comsats institute of information technology, park road, islamabad, pakistan 2department of mathematics, rzeszów university of technology, al. powstańców warszawy 12, 35-959 rzeszów, poland ∗corresponding author: piejko@prz.edu.pl int. j. anal. appl. (2023), 21:64 best proximity point and existence of the positive definite solution for matrix equations satyendra kumar jain1, gopal meena2,∗, rashmi jain2 1department of mathematics, st. aloysius college jabalpur (m.p.), india 2department of applied mathematics, jabalpur engineering college jabalpur (m.p.), india ∗corresponding author: gmeena@jecjabalpur.ac.in abstract. in this research, α−ψ−θ contraction has been defined to find the best proximity point in partially ordered metric spaces. proper support for the result has been given in the form of a suitable example. the third part is fully devoted to the positive definite solution of matrix equations. 1. introduction and preliminaries the concept of the best proximity point was introduced by basha [5] with the help of the banach contraction principle. it may be impossible to find a fixed point for two non empty subsets l,m ⊆ w and a mapping s : l → m (for example, when l∩m = φ). however, it is very interesting to find a point x ∈ l, where x and sx are as close as possible; in other words, find an x ∈ l which minimizes %(x,sx). such optimal approximate solutions are called "best proximity points for s." letter on many mathematicians [1–3,6,9,10] established best proximity point results. in 2014, idea of θ contraction introduced by jleli et al. [8] and defined generalization of banach contraction. in this paper, we define α−ψ−θ contraction and establish the best proximity point in partially ordered metric spaces. moreover, as a consequence of the result, a fixed point result and the existence of a positive definite solution to matrix equations have been given. in the whole paper, complete metric space and the best proximity point are abbreviated as cms and bpp, respectively. the subsequent symbols used in our results are: received: apr. 5, 2023. 2020 mathematics subject classification. 55m20, 15b48, 54h25. key words and phrases. best proximity point; matrix equations; positive definite solution. https://doi.org/10.28924/2291-8639-21-2023-64 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-64 2 int. j. anal. appl. (2023), 21:64 let (w,%) be a metric space and c,d be non-empty subsets of w. % (c, d) = inf { % (u1,v1) : u1 ∈ c and v1 ∈ d} , c0 = {u1 ∈ c : % (u1,v1) = % (c,d) f or some v1 ∈ d} , d0 = {v1 ∈ d : % (u1,v1) = % (c,d) f or some u1 ∈ c}. in 2012, samet et al. [13] defined the following contraction: α(x1,y1)%(tx1,ty1) ≤ ψ(%(x1,y1)), where ψ : [0,∞) → [0,∞) satisfy the consequent conditions: (1) ψ is non decreasing, (2) ∑∞ m=1ψ m(t) < ∞ f orall t > 0, where ψm is the mth iterate of ψ and ψ(t) < t for any t > 0, t is αadmissible i.e. for all x1,y1 ∈ w, α(x1,y1) ≥ 1 ⇒ α(tx1,ty1) ≥ 1, where α : w ×w → [0,∞) is a mapping. jleli et al. [8] proposed θ contraction in 2014 as follows: definition 1.1. [8] let θ be the set of all functions θ : (0,∞) → (1,∞) satisfy the conditions: θ1. θ is non decreasing, θ2. for every sequence {αn} ⊂ (0,∞), lim n→∞ θ(αn) = 1 ⇔ lim n→∞ αn = 0 +, θ3. there exists s ∈ (0, 1) and l ∈ (0,∞) such that lim α→0 θ(α) − 1 αs = l and prove the following results: theorem 1.1. [8] let (v,%) be a cms and t : v → v be a mapping, if there exists θ ∈ θ and k ∈ (0, 1) such that for all u,v ∈ v, %(tu,tv) 6= 0 ⇒ θ(%(tu,tv)) ≤ [θ(%(u,v))]k. (1.1) then t has a unique fixed point. also, in 2017, ahmed et al. [4] used the subsequent weaker condition in place of condition (θ3) : (θ ′ 3) θ is continuous on (0,∞). in this order we denote ψ the set all functions θ satisfy θ1,θ2,θ ′ 3. in 2017, piri et al. [11] defined generalized khan contraction. int. j. anal. appl. (2023), 21:64 3 theorem 1.2. [11] let (w,%) be a cms and a : w → w be a mapping satisfies % (au, av) ≤ { k %(u,au)%(u,av)+%(v,av)%(v,au) max{%(u,av),%(au,v)} , if max{%(u,av),%(au,v)} 6= 0, 0, if max{%(u,av),%(au,v)} = 0, where k ∈ [0, 1) and u,v ∈ w, then a has a unique fixed point. however, the mappings involved in all results were self mappings. definition 1.2. [14] let (c,d) be a pair of non-empty subsets of a metric space w with c0 6= φ. then, the pair (c,d) is said to have the weak p −property if and only if % (u1, v1) = % (c, d) % (u2, v2) = %(c, d) } ⇒ % (u1, u2) ≤ % ( v1, v2) , where u1, u2 ∈ c and v1,v2 ∈ d. definition 1.3. [13] let c,d be the subsets of metric space (w,%). a non self mapping a : c → d is said to be α−proximal admissible if α(v1,v2) ≥ 1 % (u1, av1) = % (c, d) % (u2, av2) = %(c, d)   ⇒ α (u1, u2) ≥ 1, where u1, u2, v1,v2 ∈ c and α : c ×c → [0,∞) be a function. 2. main results let c,d be two subsets of a partially ordered cms (v,%,�) and α : c×c → [0,∞) be a function. a mapping t : c → d is said to be α−ψ − θ contraction, if for θ ∈ ψ, there exists κ ∈ (0, 1) and for every x,y ∈ c with α(x,y) ≥ 1, %(tx,ty) > 0, we have α(x,y)θ[%(tx,ty)] ≤ [ψ(θ(m(x,y)))]κ, (2.1) where m(x,y) = max{g(x,y),%(x,y)} and g(x,y) = { %(x,tx)%(x,ty)+%(y,ty)%(y,tx) max{%(x,ty),%(tx,y)} , if max{%(x,ty),%(tx,y)} 6= 0, 0, if max{%(x,ty),%(tx,y)} = 0. theorem 2.1. let (v,%,�) be a partially ordered cms and c, d are closed subsets of v and let t : c → d be a α−ψ −θ contraction satisfies (i) t is α−proximal admissible, (ii) t (c0) ⊆ d0 and the pair (c,d) satisfies week p −property, (iii) t is continuous, (iv) there exists x0,x1 ∈ c0,x0 � x1 with %(x1,tx0) = %(c,d) such that α(x0,x1) ≥ 1. then there exists x ∈ v such that %(x,tx) = %(c,d). 4 int. j. anal. appl. (2023), 21:64 proof. let xo ∈ c0, since t (c0) ⊆ d0, there exists an element x1 ∈ c0 such that %(x1,tx0) = %(c,d) and x0 � x1, by the assumption (iv), α(x0,x1) ≥ 1. again x1 ∈ c0 and t (c0) ⊆ d0, there exists x2 ∈ c0 such that %(x2,tx1) = %(c,d) and x1 � x2. by α−proximal admissibility of t, we have α(x1,x2) ≥ 1, continuing this process, we get %(xn+1,txn) = %(c,d) and α(xn,xn+1) ≥ 1 f or all n ∈ n, (2.2) where x0 � x1 � x2 � x3 · · · � xn � xn+1 � . . . now, if there exists n0 ∈ n such that xn0 = xn0+1, then we have %(xn0,txn0) = %(xn0+1,txn0) = %(c,d). then xn0 is the best proximity point (bpp)of t. therefore, we assume that xn 6= xn+1, that is %(xn,xn+1) > 0 for all n ∈ n ∪{0}. by the week p-property of the pair (c,d) and from 2.1, 2.2, we have for all n ∈ n, 1 < θ(%(xn+1,xn)) = θ(%(txn,txn−1)) ≤ α(xn,xn−1)θ(%(txn,txn−1) ≤ (ψ(θ(m(xn,xn−1))))κ, where m(xn,xn−1) = max{g(xn,xn−1),%(xn,xn−1)} = max{ %(xn−1,txn−1)%(xn−1,txn) + %(xn,txn−1)%(xn,txn) max{%(xn−1,txn),%(txn−1,xn)} ,%(xn,xn−1)} = max{ %(xn−1,xn)%(xn−1,xn+1) %(xn−1,xn+1) ,%(xn,xn−1)} = %(xn−1,xn), so, 1 < θ(%(xn,xn+1)) ≤ (ψ(θ(%(xn,xn−1))))κ ≤ (ψ(θ(%(xn−1,xn−2))))κ 2 ≤ (ψ(θ(%(xn−2,xn−3))))κ 3 int. j. anal. appl. (2023), 21:64 5 ≤ . . . . . . ≤ (ψ(θ(%(x0,x1))))κ n . taking n →∞ we get θ(%(xn,xn+1)) → 1, therefore, by θ2, we obtain lim n→∞ %(xn,xn+1) = 0. (2.3) now, we shall show that {xn} is a cauchy sequence in c. suppose, on the contrary that, if there exists � > 0, we can find the sequences {pn} and {qn} of natural numbers such that for pn > qn > n, we have %(xpn,xqn ) ≥ �. (2.4) then, %(xpn−1,xqn ) < � f or all n ∈ n. thus, by triangular inequality and 2.4, we get � ≤ %(xpn,xqn ) ≤ %(xpn,xpn−1) + %(xpn−1,xqn ) ≤ %(xpn,xpn−1) + �. taking limit n →∞ and using 2.3, we get lim n→∞ %(xpn,xqn ) = �. (2.5) again by triangular inequality, we have %(xpn,xqn ) ≤ %(xpn,xpn+1) + %(xpn+1,xqn+1) + %(xqn+1,xqn ) (2.6) and %(xpn+1,xqn+1) ≤ %(xpn+1,xpn ) + %(xpn,xqn ) + %(xqn,xqn+1). (2.7) taking limit n →∞ and from 2.3, 2.5, we have lim n→∞ %(xpn+1,xqn+1) = �, (2.8) so, equation 2.5 holds. then by assumption α(xpn,xqn ) ≥ 1, we get 1 ≤ θ(%(xpn+1,xqn+1)) ≤ θ(%(txpn,txqn )) ≤ α(xpn,xqn )θ(%(txpn,txqn )) ≤ (ψ(θ(m(xpn,xqn )))) κ < θ(m(xpn,xqn )), 6 int. j. anal. appl. (2023), 21:64 by taking limit as n →∞ in above inequality and using [θ ′ 3] in equation 2.3, we get lim n→∞ %(xpn,xqn ) = 0 < �, which is contraction. therefore, {xn} is a cauchy sequence. since {xn} ⊆ c and c is closed in a complete metric space, so we can find x ∈ c, such that xn → x. now, since t is continuous so, we have txn → tx. this implies that %(xn+1,txn) → %(x,tx), since the sequence {%(xn+1,txn)} is a constant sequence with the value %(c,d) . we deduce that %(c,d) = %(x,tx). so, x is the best proximity point. if we take c = d = v and α(x,y) = 1, we obtain the subsequent result: corollary 2.1. let (v,%,�) be a complete metric space and t : v → v be a mapping satisfying θ[%(tx,ty)] ≤ [ψ(θ(m(x,y)))]κ, where m(x,y) = max{g(x,y),%(x,y)} and g(x,y) = { %(x,tx)%(x,ty)+%(y,ty)%(y,tx) max{%(x,ty),%(tx,y)} , if max{%(x,ty),%(tx,y)} 6= 0, 0, if max{%(x,ty),%(tx,y)} = 0, for all x,y ∈ v with θ ∈ θ and κ ∈ (0, 1), suppose that (i) t is continuous, (ii) there exist x0 ∈ v such that x0 � tx0. then t has a unique fixed point. proof. by the theorem 2.1, {xn} is a cauchy sequence. since {xn} ⊆ v and v is a complete metric space, so we can find x ∈ v such that xn → x. now, we shall show that x is a fixed point of t. %(tx,x) = lim n→∞ %(txn,x) = lim n→∞ %(xn+1,x) = 0. then x is a fixed point of t. uniqueness: let, if possible there are two fixed points x1 and x2 such that x1 6= x2. since x1 and x2 int. j. anal. appl. (2023), 21:64 7 are fixed points, so tx1 = x1 and tx2 = x2. θ[%(tx1,tx2)] ≤ [ψ(θ(m(x1,x2)))]κ θ[%(x1,x2)] ≤ [ψ(θ(max{g(x1,x2),%(x1,x2)}))]κ ≤ [ψ(θ(max{ %(x1,tx1)%(x1,tx2) + %(x2,tx2)%(x2,tx1) max{%(x1,tx2),%(tx1,x2)} ,%(x1,x2)}))]κ ≤ [ψ(θ(max{ %(x1,x1)%(x1,x2) + %(x2,x2)%(x2,x1) max{%(x1,x2),%(x1,x2)} ,%(x1,x2)}))]κ ≤ [ψ(θ(%(x1,x2)))]κ ≤ [(θ(%(x1,x2)))]κ, which is contradiction, so x1 = x2. therefore, t has a unique fixed point. note. in this result, if ψ(t) = t and m(x,y) = %(x,y), then we get theorem 1.1. example 2.1. let w = { 0, 1, 2, 3} with the usual order ≤, be a partially ordered set and let % : w ×w → r be given as %(0, 0) = %(1, 1) = %((2, 2) = %(3, 3) = 0,%(0, 1) = %(1, 0) = 2, %(0, 2) = %(2, 0) = 3 2 ,%(0, 3) = %(3, 0) = 5 2 ,%(2, 3) = %(1, 3) = 5 2 ,%(1, 2) = 3. consider c = {0, 1},d = {2, 3} and t : c → d defined by t (0) = 2,t (1) = 3. so, %(c,d) = %(0, 2) = 3 2 . also, c0 = {0} and d0 = {2}. clearly t (c0) ⊆ d0 and % (u1, v1) = % (c, d) = 3 2 % (u2, v2) = %(c, d) = 3 2 } ⇒ % (u1, u2) ≤ % ( v1, v2) , where u1, u2 ∈ c and v1,v2 ∈ d. then, we have u1 = 0,v1 = 2 and u2 = 0,v2 = 2. in this case, %(0, 0) = 0 = %(2, 2), that is, the pair (c,d) has the weak p property. taking θ(u) = u + 1 and ψ(u) = 999 1000 u for all u ≥ 0 and define α : w ×w → [0,∞) as follows,{ α(u,v) = 1, if (u,v) ∈{(0, 0), (0, 1), (1, 1)}, α(u,v) = 0, if not. let u1,v1,u1 and u2 in c such that  α(u1,u2) ≥ 1, % (v1, tu1) = % (c, d) = 3 2 , % (v2, tu2) = %(c, d) = 3 2 . then we have u1 = v1 = u1 = u2 = 0. so, α(v1,v2) ≥ 1, 8 int. j. anal. appl. (2023), 21:64 that is, t is α− proximal admissible. by the symmetry of % and α, it suffices to study the cases (u = 0,v = 1) and (u = v = 0). if (u = 0,v = 1),u ≤ v, α(0, 1)θ(%(t 0,t 1)) = θ( 11 10 ) = 7 2 , m(u,v) = max{ %(u,tv)%(u,tv) + %(v,tv)%(v,tu) max{%(u,tv),%(tu,v)} ,%(u,v)} = max{ %(0,t 0)%(0,t 1) + %(1,t 1)%(1,t 0) max{%(0,t 1),%(t 0, 1)} ,%(0, 1)} = max{ %(0, 2)%(0, 3) + %(1, 3)%(1, 2) max{%(0, 3),%(2, 1)} ,%(0, 1)} = max{ 3 2 × 5 2 + 5 2 × 3 max{5 2 , 3)} , 2} = 15 4 . so, [ψ(θ(m(0, 1)]κ = ( 999 1000 × 19 4 )κ. therefore, for κ = .805, we have 7 2 = ( 999 1000 × 19 4 )κ. if (u = 0,v = 0), then α(0, 0)θ(%(t 0,t 0)) = 1, m(u,v) = max{ %(u,tv)%(u,tv) + %(v,tv)%(v,tu) max{%(u,tv),%(tu,v)} ,%(u,v)} = max{ %(0,t 0)%(0,t 0) + %(0,t 0)%(1,t 0) max{%(0,t 0),%(t 0, 0)} ,%(0, 1)} = max{ 3 2 × 3 2 + 3 2 × 3 2 max{3 2 , 3 2 )} , 0} = 3. so, [ψ(θ(m(0, 0)]κ = ( 999 1000 × 4)κ. therefore, for κ = .005, we have α(u,v)θ[%(tu,tv)] ≤ [ψ(θ(m(u,v)))]κ hence, all the conditions of the theorem 2.1 are fulfilled. so t has a best proximity point and it is u = 0. int. j. anal. appl. (2023), 21:64 9 3. application to matrix equations in this part, we will use the subsequent symbols: c(m) represents the collection of m ×m complex matrices, h(m) ⊂ c(m) represents the collection of the m×m hermitian matrices, ℘(m) ⊂ h(m) represents the collection of m×m positive definite matrices, h1(m) ⊂ h(m) is the set of positive semi definite matrices of m×m. in addition, u1,v1 ∈ c(m). so, if u1 ∈ ℘(m) this means that u1 � 0 and u1 � 0, means u1 ∈ h(m) . moreover, u1 � v1(u1 � v1) is replaced by u1 − v1 � 0(u1 − v1 � 0). the spectral norm of the matrix b is denoted by the notation ||.||, i.e., ||b|| = √ λ+(b∗b), where λ+(b∗b) is the largest eigenvalue of b∗b and b∗ is the traconjugate of b. we write ||b||y = m∑ j=1 sj(b), where sj(b) is the singular value of b ∈ c(m). for a given g ∈ ℘(m), we denoted the modified norm by ||b||y,g = ||g 1 2bg 1 2 ||y . the set h(m) equipped with the metric induced by ||.|| is cms. furthermore, h(m) is poset with partial order �, where u1 � v1 ⇔ v1 � u1. in this part, we use %(u1,v1) = ||v1 −u1||y,g = tr(g 1 2 (v1 −u1)g 1 2 ). we assume that the subsequent nonlinear matrix equation is u = g ± n∑ j=1 b∗j τ(u)bj. (3.1) where g ∈ ℘(m), bj, j = 1, 2, . . .n, are arbitrary m×m matrices and τ : h(m) → h(m) is continuous mapping, which maps ℘(m) into ℘(m). consider τ is order preserving, that is , if c,d ∈ h(m) ⇒ τ(c) � τ(d), where c � d. lemma 3.1. [12] let c � 0 and d � 0 be m×m matrices. then 0 ≤ tr(cd) ≤ ||c||.tr(d). theorem 3.1. let t : h(m) → h(m) be continuous (order preserving) mapping, which maps ℘(m) into ℘(m) and g ∈ ℘(m). consider that (i) for all u � v and m > 1, %(τ(u),τ(v )) ≤ %(t (u),t (v ))(θ(tr(m(u,v )))) 1 2 m 1 2θ(tr(t (u) −t (v ))) , 10 int. j. anal. appl. (2023), 21:64 where m(u,v ) = max{g(u,v ),%(u,v )} and g(u,v ) = { %(u,tu)%(u,tv )+%(v,tv )%(v,tu) max{%(u,tv ),%(tu,v )} , if max{%(u,tv ),%(tu,v )} 6= 0, 0, if max{%(u,tv ),%(tu,v )} = 0, (ii) 0 < ∑n j=1b ∗ j τ(g)bj ≤ g, hold. then 3.1 has a positive definite solution ū ∈ ℘(m). proof. define t : h(m) → h(m) by t (u) = g ± n∑ j=1 b∗j τ(u)bj, (3.2) and ψ(v) = v/m, then solution of 3.1 is a fixed point of t. let u,v ∈ h(m) with u � v, then t (u) � t (v ) . %(t (u),t (v )) = ||t (v ) −t (u)||y,g = tr(g 1 2 (t (v ) −t (u))g 1 2 ) = tr( n∑ j=1 b∗j g 1 2 (τ(v ) −τ(u))g 1 2bj) = n∑ j=1 tr(b∗j g 1 2 (τ(v ) −τ(u))g 1 2bj) = n∑ j=1 tr(b∗j gbj(τ(v ) −τ(u))) = n∑ j=1 tr(b∗j gbjg 1 2g −1 2 (τ(v ) −τ(u))g 1 2g −1 2 ) = n∑ j=1 tr(g −1 2 b∗j gbjg −1 2 g 1 2 (τ(v ) −τ(u))g 1 2 ) = tr( n∑ j=1 g −1 2 b∗j gbjg −1 2 )(g 1 2 (τ(v ) −τ(u))g 1 2 ), by lemma 3.1, we get %(t (u),t (v )) = || n∑ j=1 g −1 2 b∗j gbjg −1 2 ||.tr(g 1 2 (τ(v ) −τ(u))g 1 2 ) = || n∑ j=1 g −1 2 b∗j gbjg −1 2 ||.||τ(v ) −τ(u)||y,g %(t (u),t (v )) = || n∑ j=1 g −1 2 b∗j gbjg −1 2 ||.%(τ(v ),τ(u)). int. j. anal. appl. (2023), 21:64 11 so, by condition (i) and (ii), we get %(t (u),t (v )) ≤ %(t (u),t (v ))(θ(tr(m(u,v )))) 1 2 m 1 2θ(tr(t (u) −t (v ))) θ(tr(t (u) −t (v ))) ≤ (θ(tr(m(u,v )))) 1 2 m 1 2 θ(tr(t (u) −t (v ))) ≤ ( (θ(tr(m(u,v )))) m 1 2 ) 1 2 θ(tr(t (u) −t (v ))) ≤ (ψ(θ(m(u,v )))) 1 2 . hence, by corollary 2.1, t has a fixed point. therefore, matrix equation 3.1 has a unique solution ū ∈ ℘(m). numerical experiment: example 3.1. consider the matrix equation u = g + 2∑ j=1 b∗j τ(u)bj, (3.3) where g,b1 and b2 are given by g =   4 2 1 2 4 2 1 2 4   , b1 =   0.0241 0.047 0.047 0.047 0.0241 0.0241 0.047 0.0241 0.0241   , b2 =   0.58 0.0671 0.58 0.0671 0.58 0.0671 0.58 0.0671 0.58   . define θ(u) = u + 1 and t (u) = u 9 . then conditions (i) and (ii) of theorem 3.1 are satisfied for m = 2. by using the iteration un+1 = g + 2∑ j=1 b∗j unbj with u0 =   0 0 0 0 0 0 0 0 0   . after 15 iterations, we get the unique solution ū =   1005.154 237.819 1001.821 237.819 64.151 237.514 1001.821 237.514 1004.516   of the matrix equation 3.3. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 12 int. j. anal. appl. (2023), 21:64 references [1] a. abkar, m. gabeleh, best proximity points for cyclic mappings in ordered metric spaces, j. optim. theory appl. 150 (2011), 188-193. https://doi.org/10.1007/s10957-011-9810-x. [2] a. abkar, m. gabeleh, generalized cyclic contractions in partially ordered metric spaces, optim. lett. 6 (2011), 1819-1830. https://doi.org/10.1007/s11590-011-0379-y. [3] a. abkar, m. gabeleh, the existence of best proximity points for multivalued non-self-mappings, rev. r. acad. cienc. exactas fís. nat., ser. a mat., racsam. 107 (2012), 319-325. https://doi.org/10.1007/ s13398-012-0074-6. [4] j. ahmad, a.e. al-mazrooei, y.j. cho, et al. fixed point results for generalized theta-contractions, j. nonlinear sci. appl. 10 (2017), 2350-2358. https://doi.org/10.22436/jnsa.010.05.07. [5] s. sadiq basha, extensions of banach’s contraction principle, numer. funct. anal. optim. 31 (2010), 569–576. https://doi.org/10.1080/01630563.2010.485713. [6] s.s. basha, discrete optimization in partially ordered sets, j. glob. optim. 54 (2011), 511-517. https://doi. org/10.1007/s10898-011-9774-2. [7] m. jleli, b. samet, best proximity point for α−ψ contractive type mapping and applications, bull. sci. math. 137 (2013), 977-995. [8] m. jleli, b. samet, a new generalization of the banach contraction principle, j inequal appl. 2014 (2014), 38. https://doi.org/10.1186/1029-242x-2014-38. [9] v. pragadeeswarar, m. marudai, best proximity points: approximation and optimization in partially ordered metric spaces, optim. lett. 7 (2012), 1883-1892. https://doi.org/10.1007/s11590-012-0529-x. [10] v. pragadeeswarar, m. marudai, best proximity points for generalized proximal weak contractions in partially ordered metric spaces, optim. lett. 9 (2013), 105-118. https://doi.org/10.1007/s11590-013-0709-3. [11] h. piri, s. rahrovi, h. marasi, et al. a fixed point theorem for f-khan-contractions on complete metric spaces and application to integral equations, j. nonlinear sci. appl. 10 (2017), 4564-4573. https://doi.org/10.22436/ jnsa.010.09.02. [12] a.c.m. ran, m.c.b. reurings, a fixed point theorem in partially ordered sets and some applications to matrix equations, proc. amer. math. soc. 132 (2003), 1435–1443. [13] b. samet, c. vetro, p. vetro, fixed point theorem for α − ψ contractive mapping, nonlinear anal.: theory methods appl. 75 (2012), 2154-2165. https://doi.org/10.1016/j.na.2011.10.014. [14] j. zhang, y. su, q. cheng, a note on ’a best proximity point theorem for geraghty-contractions’, fixed point theory appl. 2013 (2013), 99. https://doi.org/10.1186/1687-1812-2013-99. https://doi.org/10.1007/s10957-011-9810-x https://doi.org/10.1007/s11590-011-0379-y https://doi.org/10.1007/s13398-012-0074-6 https://doi.org/10.1007/s13398-012-0074-6 https://doi.org/10.22436/jnsa.010.05.07 https://doi.org/10.1080/01630563.2010.485713 https://doi.org/10.1007/s10898-011-9774-2 https://doi.org/10.1007/s10898-011-9774-2 https://doi.org/10.1186/1029-242x-2014-38 https://doi.org/10.1007/s11590-012-0529-x https://doi.org/10.1007/s11590-013-0709-3 https://doi.org/10.22436/jnsa.010.09.02 https://doi.org/10.22436/jnsa.010.09.02 https://doi.org/10.1016/j.na.2011.10.014 https://doi.org/10.1186/1687-1812-2013-99 1. introduction and preliminaries 2. main results 3. application to matrix equations references int. j. anal. appl. (2023), 21:79 expectile-based capital allocation khalil said∗ national institute of statistics and applied economics, rabat, morocco ∗corresponding author: ksaid@insea.ac.ma abstract. this paper focuses on capital allocation using the euler principle with expectiles as risk measures. we delve into the allocation composition across various actuarial models, examining the influence of dependence through copulas, and studying the case of comonotonicity. additionally, we provide expressions for marginal contributions related to some of the models under investigation. introduction capital allocation is a crucial issue for insurance groups due to its significant impact on financial results. once the solvency capital is determined using risk aggregation methods, it needs to be allocated across different business lines. in the context of dependent risk processes x = (x1, . . . ,xd), the determination of solvency capital is based on studying the stochastic behavior of the aggregated claim amount s = x1 + · · · + xd. capital allocation involves determining the portion of the obtained economic capital that will be assigned to each risk xi, where i = 1, . . . ,d. the choices made for modeling dependence will inevitably influence the allocation contributions. this process typically follows a top-down approach. we assume a multivariate model for the risk vector x, select a risk measure to assess the solvency capital based on the distribution of the sum s and employ an allocation method to determine the marginal contribution of each risk to this capital. several methods for allocating economic capital have been proposed in the literature. one of the most well-known principles is euler’s method, also referred to as the gradient method. based on euler’s principle, allocation rules can be derived using any homogeneous risk measure. there is a wealth of literature on var and tvar allocation rules, which includes expressions for contributions in received: jun. 11, 2023. 2020 mathematics subject classification. 62h00, 62h05, 62p05, 91b05, 91b30, 91g05. key words and phrases. risk management; risk theory; dependence modeling; capital allocation; expectiles; elicitability; copulas. https://doi.org/10.28924/2291-8639-21-2023-79 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-79 2 int. j. anal. appl. (2023), 21:79 various cases. these studies examine the impact of dependence on allocation and provide concrete examples. for further reading, refer to tasche (2000) [20], bargès et al. (2009) [2], or cossette et al. (2012) [8]. the significance of var and tvar-based allocation rules stems directly from the practical interest in var and tvar as commonly used risk measures. however, var is a non-coherent measure with respect to coherence, as defined by artzner et al. (1999) [1], making var-based capital allocation subject to criticism for the same reason. on the other hand, the coherence of tvar naturally lends greater importance to allocation rules based on it. nevertheless, recent works in risk theory highlight the non-elicitability of tvar, making direct backtesting of tvar a challenging task (gneiting, 2011 [13]; bellini and bignozzi, 2015 [3]). this verdict inevitably affects the performance of an allocation constructed using the tvar rule. bellini and bignozzi (2015) [3] show that expectiles of level α ∈ [1/2, 1[ are the only law-invariant risk measures that are both elicitable and coherent. this property makes expectiles a perfect candidate for constructing capital allocation. emmer et al. (2015) [11] derived a general formula for contributions in a capital allocation based on expectiles. in this paper, we focus on expectile-based capital allocation. our main objective is to closely examine the allocation composition for some common risk models and analyze its differences with the tvar allocation rule. the paper is organized into 6 sections. section 1 provides a review of euler’s allocation principle and its application to derive allocation rules from homogeneous risk measures, particularly the wang measures family. it also provides a brief introduction to expectiles as risk measures and recalls the expectile-based allocation rule. we offer an economic interpretation of this rule and compare it to tvar allocation. in section 2, we examine capital allocation for various independent models. sections 3 and 4 focus on the allocation composition for exponential combinations and mixture models, respectively. section 5 examines the case of perfect dependence. the final section presents numerical illustrations. 1. expectile-based capital allocation this section is dedicated to the presentation of the allocation method. firstly, we provide a reminder of the euler allocation principle, followed by the presentation of the allocation rule derived from expectile risk measures. an economic interpretation of the resulting rule is provided, along with an initial comparison to the tvar-based allocation rule. euler’s capital allocation method is studied in tasche (2007) [22] and tasche (2008) [23]. this technique is based on the concept of allocating capital based on the infinitesimal marginal impact of each risk, which represents the reduction in overall risk resulting from an infinitely small decrement in risk xi. int. j. anal. appl. (2023), 21:79 3 we denote the contribution of risk xi to the overall risk as ρ(xi|s). this contribution can be obtained using euler’s principle: ρ(xi|s) = lim h→0 ρ(s) −ρ(s −hxi ) h . using euler’s allocation principle, it is possible to construct an allocation rule with any homogeneous risk measure. we provide a reminder of the definitions of the commonly used risk measures: value at risk (var) and tail value at risk (tvar). the var risk measure of level α is defined for any random variable x as: v arα(x) = inf{x ∈r : p(x ≤ x) ≥ α} = inf{x ∈r : fx(x) ≥ α} = f−1x (α), where fx denotes the cumulative distribution function (cdf) of x. this represents the quantile of the same level. the tvar of level α is defined as the mean of vars exceeding v arα(x): tv arα(x) = 1 1 −α ∫ 1 α v arµ(x)d¯. the well-known var-based and tvar-based allocations are examples of rules obtained from this method. the risk contribution of each risk in the overall risk using the var-based allocation rule is given by: v ar(xi|s) = e[xi|s = v arα(s)]. for continuous distributions, the expression for the risk contribution using the tvar risk measure and euler’s method is: tv ar (xi|s) = e[xi|s > v arα(s)] 1 −α . euler’s method has been extensively studied in the literature on capital allocation in the past decade. its properties, such as coherence and compatibility with risk-adjusted return on capital (rorac), have been analyzed in numerous works under various assumptions. examples include balog (2011) [26], tasche (2000) [20], and tasche (2004) [21]. the economic interpretation of euler’s method provides a relevant solution to the capital allocation problem and explains its popularity in actuarial practice. the composition of var-based capital allocation has been studied for several risk models in marceau (2013) [28]. the tvar-based allocation rule has been explored in bargès et al. (2009) [2] and cossette et al. (2012) [8]. elicitability is a desirable statistical property for risk measures. according to bellini and bignozzi (2015) [3], a risk measure ρ is said to be elicitable in respect to the class p if there exists a scoring function s : r2 →r+ such that ρ(p) = arg min x∈r ∫ s(x,y)dp(y), ∀p∈p. they demonstrate in the same paper that expectiles are the only risk measures that are both coherent and elicitable. 4 int. j. anal. appl. (2023), 21:79 expectiles were introduced in the context of statistical regression models by newey and powell (1987) [15]. for a random variable x with finite second moment, the expectile of level α is defined as follows: eα(x) = arg min x∈r e[α(x −x)2+ + (1 −α)(x −x) 2 +], (1.1) where (x)+ = max(x, 0). bellini et al. (2014) [5] introduced generalized quantile risk measures, which encompass expectiles and are defined as the minimizers of an asymmetric error given by: xα(x) = arg min x∈r {αe[φ+((x −x)+)] + (1 −α)e[φ−((x −x)−)]}, where φ+ and φ− are convex scoring functions. expectiles correspond to the case when φ+(x) = φ−(x) = x 2. maume-deschamps et al. (2017) [17] introduced multivariate extensions of expectile risk measures. expectiles are inherently elicitable. they are coherent for all α > 1/2. expectiles can also be defined equivalently for any random variable with a finite first-order moment as the unique solution to the following equation: αe[(x −x)+] = (1 −α)e[(x −x)+]. (1.2) the properties of expectile risk measures have been studied in several papers, including [11] and [4]. the asymptotic behavior of expectiles is examined in [4], and the second-order behavior is analyzed in [16]. extremes for multivariate expectiles are investigated in [18]. in this paper, our focus is on euler’s capital allocation rule based on expectiles. emmer et al. (2015) [11] showed that the contribution of risk xi to the sum s = ∑d `=1 x` is given by definition 1.1. definition 1.1 (expectile-based capital allocation). the marginal contribution of a risk xi to an aggregated risk s = ∑d `=1 x` using expectiles is given by eα(xi|s) = αe [ xi 11{s>eα(s)} ] + (1 −α)e [ xi 11{s<eα(s)} ] αp(s > eα(s)) + (1 −α)p(s < eα(s)) , (1.3) for α ∈ [1/2, 1[. to provide an economic interpretation of the capital allocation rule defined in definition 1.1, let αs denote the percentage given by αs = αp(s > eα(s)) αp(s > eα(s)) + (1 −α)p(s < eα(s)) . the contribution eα(xi|s) can then be expressed as eα(xi|s) = αs e [xi|s > eα(s)]︸ ︷︷ ︸ t− +(1 −αs)e [xi|s < eα(s)]︸ ︷︷ ︸ t+ . hence, the allocation can be interpreted as a linear combination of the marginal contribution in exceeding the overall expectile in a ruin scenario (t−) and the marginal contribution in achieving int. j. anal. appl. (2023), 21:79 5 overall solvency from an expectile perspective (t+). this allocation rule takes into account not only the marginal participation in negative global scenarios, as in the case of tvar allocation, but also the participation in overall performance. in order to clarify the relationship between the expectile-based allocation and the tvar-based allocation, we can express equation (1.3) in the following form: eα(xi|s) = (2α− 1)(1 −β) (2α− 1)(1 −β) + (1 −α) tv arβ (xi|s) + (1 −α) (2α− 1)(1 −β) + (1 −α) e [xi ] , where β = fs(eα(s)). the allocation using expectiles can be seen as a transformation of the tvar-based rule with a safety margin percentage. this transformation involves adjusting the tvar level (eα −→ tv arβ) as well as the composition, using a linear convex combination between the contribution based on tv arβ and the contribution based on e [xi ]. in a financial context, when the random variables represent p&l (profit and loss), an economic interpretation of the allocation contributions can be derived from the following expression: eα(xi|s) = α(1 −β) α(1 −β) + (1 −α)β tv arβ (xi|s) − (1 −α)β α(1 −β) + (1 −α)β tv ar1−β (−xi|−s) . this expression corresponds to a linear combination of the marginal participation in the global profits, measured by tv arβ, and the marginal participation in the global losses, measured by tv ar1−β. thus, the allocation rule considers both the positive and negative aspects of the p&l, taking into account the contributions to the overall profitability and loss. we can also express the contribution eα(xi|s) as follows: eα(xi|s) = αe [ xi 11{s>eα(s)} ] + (1 −α)e [ xi 11{s<eα(s)} ] αe [ s11{s>eα(s)} ] + (1 −α)e [ s11{s<eα(s)} ] eα(s), (1.4) since the expectile eα(s) satisfies eα(s) = αe [ s11{s>eα(s)} ] + (1 −α)e [ s11{s<eα(s)} ] αp(s > eα(s)) + (1 −α)p(s < eα(s)) . the allocation percentage eα(xi|s)/eα(s) can be directly obtained from (1.4). the expectile capital allocation is trivially additive, as stated in (1.4): d∑ i=1 eα(xi|s) = eα ( d∑ i=1 xi ) . moreover, it is a neutral allocation, since ∃ci ∈r, xi = ci a.s ⇒ eα(xi|s) = ci. the allocation is sub-additive, as for any subsets a ⊆ 1, . . . ,d, we have eα( ∑ `∈a x`|s) = ∑ `∈a eα(x`|s), which is also the case for var and tvar-based allocation rules. 6 int. j. anal. appl. (2023), 21:79 in the rest of this article, we will focus on analyzing the behavior of the contributions provided by the expectile-based allocation rule. we will examine the impact of dependence using different families of models. 2. some bivariate independent models this section presents a study of the expectile-based allocation rule in the case of independence. the main objective of this part is to highlight the impact of the nature of the marginal distributions on the allocation contributions. 2.1. bivariate independent exponential model. we consider a bivariate independent exponential random vector (x1,x2) with xi ∈ e(βi ), i ∈ {1, 2}. we denote by s the aggregated sum of risks x1 + x2. in the case where β1 = β2, the allocation is trivial eα(x1|s) = eα(x2|s) = eα(s)/2. proposition 2.1 provides the expressions for the allocation contributions. proposition 2.1 (expectile-based allocation, ei model). according to the expectile allocation rule, the contribution from the risk xi is eα (xi|s) = (2α− 1)βiξ (s∗; βi,β3−i ) + (1 −α) (2α− 1)h̄ (s∗; β1,β2) + (1 −α) 1 βi , where s∗ is the unique solution to the following equation (2α− 1) [ ζ (s; β1,β2) − sh̄ (s; β1,β2) ] = (1 −α) [ s − 1 β1 − 1 β2 ] , where h̄,ζ,ξ are defined as follows h̄ ( x; βi,βj ) =   e−βx ∑2−1 `=0 (βx) ` `! , βi = βj = β∑2 k=1 ( 2∏ `=1,` 6=k β` β`−βk ) e−βkx, βi 6= βj , ζ (x; β1,β2) =   2 β ( e−βx ∑2 `=0 (βx) ` `! ) , βi = βj = β∑2 k=1 ( 2∏ `=1,` 6=k β` β`−βk )( xe−βkx + e −βkx βk ) , βi 6= βj , and ξ ( x; βi,βj ) =   1 β h̄ (x; 3,β) , βi = βj = β βje −βix ( x+ 1 βi ) (βj−βi ) − ( βje −βix (βi−βj ) 2 − βie −βjx (βi−βj ) 2 ) , βi 6= βj . proof. from the expectile definition (1.2), eα(s) is the unique solution to the equation: αe[(s − s)+] = (1 −α)e[(s −s)+], which can be written as: (2α− 1)e[(s − s)+] = (1 −α) (s −e[s]) , int. j. anal. appl. (2023), 21:79 7 and from equation 1.3, the contribution eα (xi|s) can be written as: eα(xi|s) = (2α− 1)e [ xi 11{s>eα(s)} ] + (1 −α)e [xi ] (2α− 1)p(s > eα(s)) + (1 −α) , then, the expressions are obtained straightforwardly from their definition using fs (x) = h (x; β1,β2) =   1 − e−βx ∑2−1 j=0 (βx) j j! , β1 = β2 = β∑2 i=1 ( 2∏ j=1,j 6=i βj βj−βi )( 1 − e−βix ) , β1 6= β2 , e [ s × 11{s>x} ] = ζ (x; β1,β2) =   2 β ( e−βx ∑2 j=0 (βx) j j! ) , β1 = β2 = β∑2 i=1 ( 2∏ j=1,j 6=i βj βj−βi )( xe−βix + e −βix βi ) , β1 6= β2 , and e [ x1 × 11{s>x} ] = ξ (x; β1,β2) =   1 β h̄ (x; 3,β) , β1 = β2 = β β2e −β1x ( x+ 1 β1 ) (β2−β1) − ( β2e −β1x (β1−β2)2 − β1e −β2x (β1−β2)2 ) , β1 6= β2 . � in this model, the random variable s follows an erlang-2 distribution if β1 = β2 = β and a generalized erlang distribution if β1 6= β2. proposition 2.1 can be generalized in the case of higher dimension d > 2 and different distribution parameters. in fact, let x1,x2, . . . ,xd be independent exponential random variables with respective parameters 0 < β1 < β2 < · · · < βd. we denote by s the aggregated sum of risks xi, i = 1, . . . ,d. since xi ∼e(βi ) for all i ∈ 1, . . . ,d, we have f̄s(s) = h̄s(s,β1, . . . ,βd) = d∑ `=1   d∏ j=1,j 6=` βj βj −β`  e−β`s, ∀ s ∈r+, which represents the distribution function of the generalized erlang distribution. on the other hand, the sum’s expectile eα(s) is the unique solution to the equation: αe[(s − s)+] = (1 −α)e[(s − s)−], this equation can be rewritten as: s = e[s] + 2α− 1 1 −α e[(s − s)+]. since e[(s − s)+] = ∫ +∞ s f̄s(s)dt = d∑ `=1 a` β` e−β`s, 8 int. j. anal. appl. (2023), 21:79 where a` = d∏ j=1,j 6=` βj βj −β` , ∀` ∈ {1, . . . ,d}, we can express eα(s) as the unique solution to the equation: s = d∑ `=1 1 β` ( 1 + 2α− 1 1 −α a`e −β`s ) . (2.1) we observe that xi and s(−i) = ∑d `=1,` 6=1 x` are independent. since s (−i) is also the sum of exponentially independent random variables, its probability density function can be expressed as: f (−i) s (s) = d∑ `=1,` 6=i   d∏ j=1,j 6=`,j 6=i βj βj −β`  β`e−β`s = d∑ `=1,` 6=i a` β` βi (βi −β`)e−β`s, ∀ s ∈r+. this expression is used to calculate: e [ xi × 11{s>x} ] = ξi (x; β1, . . . ,βd) = d∑ `=1,` 6=i a` βi −β` [ e−β`x −e−βix ( 1 + x + 1 βi )] . finally, the allocation contributions in this case are given by eα (xi|s) = (2α− 1)βiξi (s∗; β1, . . . ,βd) + (1 −α) (2α− 1)h̄ (s∗; β1, . . . ,βd) + (1 −α) 1 βi , where s∗ is the unique solution to equation (2.1). note that in the particular case where β1 = β2 = · · · = βd, we have eα (xi|s) = eα(s)/d, ∀i ∈{1, . . . ,d}. 2.2. bivariate independent gamma model. in this subsection, we consider two random variables following the gamma distribution: xi ∼ gamma(αi,β) for i = 1, 2. the rate parameter is the same for both distributions. if x1 and x2 are independent, then the sum s = x1 + x2 follows the gamma distribution with parameters α1 + α2 and β. proposition 2.2 (expectile-allocation, ig-model). according to the expectile allocation rule, the contribution from the risk xi is given by: eα (xi|s) = (2α− 1)ḡ (s∗; α1 + α2 + 1,β) + (1 −α) 2α− 1)ḡ (s∗; α1 + α2,β) + (1 −α) αi β , i ∈{1, 2}, where s∗ is the unique solution to the following equation: (2α− 1) [ α1 + α2 β ḡ (s∗; α1 + α2 + 1,β) − s∗ḡ (s∗; α1 + α2,β) ] = (1 −α) [ s − α1 + α2 β ] , and ḡ (.,α,β) is the survival function of the gamma distribution with parameters α and β. int. j. anal. appl. (2023), 21:79 9 proposition 2.2 can be generalized for dimensions higher than 2. for d independent random variables following the gamma distribution, xi ∼ g(αi,β), where i = 1, . . . ,d, the allocation of xi given the sum s = ∑d `=1 x` is given by: eα ( xi ∣∣∣∣ s = d∑ `=1 x` ) = (2α− 1)ḡ ( s∗; 1 + ∑d `=1 α`,β ) + (1 −α) 2α− 1)ḡ ( s∗; ∑d `=1 α`,β ) + (1 −α) αi β , i ∈{1, . . . ,d}, where s∗ is the unique solution to the following equation: (2α− 1) [∑d `=1 α` β ḡ ( s∗; 1 + d∑ `=1 α`,β ) − s∗ḡ ( s∗; d∑ `=1 α`,β )] = (1 −α) [ s − ∑d `=1 α` β ] . proof. the result is obtained directly using equation 1.3. � 3. bivariate combinations of exponentials bivariate distributions with exponential marginals are well-known in the actuarial science literature, and extensive discussions can be found in kotz et al. (2004) [27] and balakrishnan and lai (2009) [24]. in cossette et al. (2015) [9], the tvar-based allocation rule was investigated for this family of bivariate models, providing explicit formulas for contributions. in this section, we begin by presenting the general expression for the expectile-based allocation contributions in the family of bivariate combinations of exponentials with exponential marginals. subsequently, we illustrate these expressions through several examples of models. 3.1. bivariate combinations of exponential distributions. a random vector (x1,x2) follows a bivariate combination of exponential distributions if its joint density can be expressed as: fx1,x2 (x1,x2) = m∑ i=1 m∑ j=1 ci,jγi e −γix1λje −λjx2, (3.1) where ci,j ∈r with ∑m i=1 ∑m j=1 ci,j = 1. we assume that 0 < γ1 < ... < γm and 0 < λ1 < ... < λm. we denote ci,∗ = ∑m j=1 ci,j and c∗,j = ∑m i=1 ci,j where { ci,j, i = 1, ...,m,j = 1, ...,m } are such that fx1,x2 (x1,x2) ≥ 0 for all (x1,x2) ∈r2. this class includes the family of bivariate mixed exponential distributions, where 0 ≤ ci,j ≤ 1. it is important to note that the class of bivariate combinations of exponential distributions is a subset within the family of bivariate matrix exponential distributions studied in bladt and nielsen (2010) [6].the marginal distributions are univariate combinations of exponentials, given by: fx1 (x1) = m∑ i=1 ci,∗ ( 1 − e−γix1 ) and fx2 (x2) = m∑ j=1 c∗,j ( 1 − e−λjx2 ) . proposition 3.1 provides the general expressions of marginal contributions in solvency capital using the expectile-based allocation method. 10 int. j. anal. appl. (2023), 21:79 proposition 3.1 (expectile-allocation for bivariate combinations of exponentials). let (x1,x2) follow a bivariate combination of exponentials. then, for s = x1 + x2, we have eα (x1|s) = (2α− 1) m∑ i=1 m∑ j=1 ci,jξ ( s∗; γi,λj ) + (1 −α) m∑ i=1 ci,∗ γi (2α− 1) m∑ i=1 m∑ j=1 ci,jh̄ ( s∗; γi,λj ) + 1 −α , where ξ,ζ and h̄ are the same functions defined in proposition 2.1, and s∗ is the unique solution to the following equation (2α− 1)   m∑ i=1 m∑ j=1 ci,j ( ζ ( s; γi,λj ) − sh̄ ( s; γi,λj )) = (1 −α) [ s − m∑ i=1 ( ci,∗ γi + c∗,i λi )] . (3.2) the contribution of x2 is given directly from eα (x2|s) = s∗ −eα (x1|s) , and it can also be obtained directly by eα (x2|s) = (2α− 1) m∑ i=1 m∑ j=1 cj,iξ ( s∗; λj,γi ) + (1 −α) m∑ i=1 c∗,j λj (2α− 1) m∑ i=1 m∑ j=1 ci,jh̄ ( s∗; γi,λj ) + 1 −α . proof. since the marginals are fx1 (x1) = ∑m i=1 ci,∗ ( 1 − e−γix1 ) and fx2 (x2) =∑m j=1 c∗,j ( 1 − e−λjx2 ) respectively, then e[x1] = m∑ i=1 ci,∗ γi and e[x2] = m∑ j=1 c∗,j λj . in this model, the joint distribution of (x1,x2) is a linear combination of m ×m terms. by a direct calculation, we get f̄s (s) = m∑ i=1 m∑ j=1 ci,jh̄ ( s; γi,λj ) , (3.3) e [ s × 11{s>s} ] = m∑ i=1 m∑ j=1 ci,jζ ( s; γi,λj ) , (3.4) and e [ x1 × 11{s>s} ] = m∑ i=1 m∑ j=1 ci,jξ ( s; γi,λj ) . (3.5) it also follows that e[(s − s)+] = m∑ i=1 m∑ j=1 ci,j ( ζ ( s; γi,λj ) − sh̄ ( s; γi,λj )) . (3.6) combining expressions (3.3), (3.5) and (3.6), we obtain the announced result. � int. j. anal. appl. (2023), 21:79 11 note that in this model, s follows a combination of erlang-2 and/or generalized erlang distributions. the value of eα (s) is obtained by solving equation 3.2 using numerical methods. subsequently, we compute eα (x1|s) and eα (x2|s). 3.2. specific models. we consider some well-known bivariate exponential distributions that belong to the class presented in the previous subsection. 3.2.1. bivariate fgm-exponential model. let the joint distribution of (x1,x2) be defined with a farlie-gumbel-morgenstern (fgm) copula, given by cθ (u1,u2) = u1u2 + θu1u2 (1 −u1) (1 −u2) , − 1 ≤ θ ≤ 1, (see e.g., nelsen (2007) [29], example 3.12, section 3.2.5). the marginal distributions are exponential with parameters β1 and β2, respectively. this leads to the joint cumulative distribution function: fx1,x2 (x1,x2) = ( 1 − e−β1x1 )( 1 − e−β2x2 ) + θ ( 1 − e−β1x1 )( 1 − e−β2x2 ) e−β1x1 e−β2x2. it is important to note that the fgm construction is considered a weak dependence model. the pearson correlation coefficient is ρp (x1,x2) = θ 4 , which implies ρp (x1,x2) ∈ [ −1 4 , 1 4 ] . the spearman’s correlation coefficient, denoted as ρs, is given by ρs = θ 3 ∈ [ −1 3 , 1 3 ] . we recall that spearman’s rho is a concordance measure defined for continuous bivariate distributions with copula c as the dependence structure. it can be calculated as: ρs = 12 ∫ ∫ [0,1]2 uvdc(u,v) − 3 = 12 ∫ ∫ [0,1]2 c(u,v)dudv − 3. the fgm construction is also considered as an asymptotic independent model since its upper tail dependence coefficient is λu = 0. we recall the definition of the upper tail dependence coefficient as presented in joe (1997) [25], for bivariate random variables (x,y ) of a continuous marginal distributions λu = lim u−→1− p(y > f−1 y (u)|x > f−1 x (u)). the upper tail dependence coefficient can be expressed in terms of copula as: λu = lim u−→1− 1 − 2u + c(u,u) 1 −u , when the limit exists. the joint density is given by: fx1,x2 (x1,x2) = β1e −β1x1β2e −β2x2 + θ 2∑ i=1 2∑ j=1 (−1)i+j × iβ1e−iβ1x1 × jβ2e−jβ2x2. (3.7) given (3.7), with m = 2, γi = iβ1 (i = 1, 2), and λj = jβ2 (j = 1, 2), the bivariate distribution defined with the fgm copula and exponential marginals is a bivariate combination of exponentials. the specific values for the coefficients are: c1,1 = 1 + θ, c1,2 = c2,1 = −θ, and c2,2 = θ. lemma 3.1 presents the expressions of marginal contributions obtained using the expectile-based 12 int. j. anal. appl. (2023), 21:79 allocation rule. the expressions for the contributions in the tvar allocation are given in bargès et al. (2009) [2]. lemma 3.1 (expectile-allocation, fgm model). let (x1,x2) follow a bivariate fgm model. then, for s = x1 + x2, we have for all (k,`) ∈{(1, 2), (2, 1)} eα (xk|s) = (2α− 1)βk  ξ (s∗; βk,β`) + θ 2∑ i=1 2∑ j=1 (−1)i+j ξ (s∗; iβk, jβ`)   + 1 −α (2α− 1) [ h̄ (s∗; β1,β2) + θ ∑2 i=1 ∑2 j=1 (−1) i+j h̄ (s∗; iβ1, jβ2) ] + 1 −α 1 βk , where s∗ is the unique solution to the following equation (2α− 1)  t (s; β1,β2) + θ 2∑ i=1 2∑ j=1 (−1)i+j t (s; iβ1, jβ2)   = (1 −α) [s −( 1 β1 + 1 β2 )] , and ξ,ζ and h̄ are the same function defined in proposition 2.1, and t is the function defined by t (s; iβ1, jβ2) = ζ (s; iβ1, jβ2) − sh̄ (s; iβ1, jβ2) , ∀sr+, ∀(i, j) ∈{1, 2}2. proof. the allocation contributions are directly obtained using 1.3 and the results of bargès et al. (2009) [2] without the constraints on β1 and β2 i.e. fs (x) = h (x; β1,β2) + θ 2∑ i=1 2∑ j=1 (−1)i+j h (x; iβ1, jβ2) , e [ s × 11{s>x} ] = ζ (x; β1,β2) + θ 2∑ i=1 2∑ j=1 (−1)i+j ζ (x; iβ1, jβ2) and e [ xk × 11{s>x} ] = ξ (x; βk,β3−k) + θ 2∑ i=1 2∑ j=1 (−1)i+j ξ1 (x; iβk, jβ3−k) . � 3.2.2. bivariate amh-exponential model. let the joint distribution of (x1,x2) be defined by a bivariate ali-mikhail-haq (amh) copula, given by cθ (u1,u2) = u1u2 1 −θ (1 −u1) (1 −u2) = u1u2 + u1u2 ∞∑ k=1 θk (1 −u1)k (1 −u2)k , with dependence parameter θ ∈ [−1, 1]. as a special case, c0 (u1,u2) = u1u2 represents the independence copula. the amh copula is also an archimedean copula associated with the following generator: φ(t) = ln (1 −θ(1 − t)) t . it introduces a moderate positive or negative dependence relation and is considered a perturbation of the independence copula. the first-degree approximation of the amh copula corresponds to the fgm int. j. anal. appl. (2023), 21:79 13 copula (see e.g., nelsen (2007) [29]). the pearson correlation coefficient is given by ρp (x1,x2) = ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j 1 (k + i)(k + j) ∈ [ 4 ln(2) − 3, π2 3 − 3 ] . the upper extremes are asymptotically independent since λu = 0. the joint density of (x1,x2) is given by fx1,x2 (x1,x2) = β1e −β1x1β2e −β2x2 + ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j (k + i) β1e−(k+i)β1x1 (k + j) β2e−(k+j)β2x2, which can be seen as a bivariate combination of exponentials by taking m = ∞, γi = iβ1 ( i ∈n+ ) , λj = jβ2 ( i ∈n+ ) , c1,1 = (1 + θ), c1,2 = c2,1 = −θ, c1,j = 0 for j = 2, 3, . . ., and ci,1 = 0 for i = 2, 3, . . .. additionally, ck,k = ck+1,k+1 = θ, ck,k+1 = ck+1,k = −θ, ck,j = ck+1,j = 0, ( j ∈n+\{k,k + 1} ) , ci,k = ci,k+1 = 0, ( i ∈n+\{k,k + 1} ) , for k = 2, 3, . . .. by proposition 3.1, we obtain the expressions of marginal contributions in expectile allocation as presented in lemma 3.2. lemma 3.2 (expectile-allocation, ahm model). let (x1,x2) follow a bivariate fgm model. then, for s = x1 + x2, we have for (k,`) ∈{(1, 2), (2, 1)} eα (xk|s) = (2α− 1)βk [ ξ (s∗; βk,β`) + ∑∞ k=1 θ k ∑1 i=0 ∑1 j=0 (−1) i+j ξ (s∗; (k + i) βk, (k + j) β`) ] + 1 −α (2α− 1) [ h̄ (s∗; β1,β2) + ∑∞ k=1 θ k ∑1 i=0 ∑1 j=0 (−1) i+j h̄ (s∗; (k + i) β1, (k + j) β2) ] + 1 −α 1 βk , where s∗ is the unique solution to the following equation (2α− 1)  t (x; β1,β2) + ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j t (x; (k + i) β1, (k + j) β2)   = (1 −α) [s −( 1 β1 + 1 β2 )] , and ξ,ζ and h̄ are the same function defined in proposition 2.1, and t is the function defined by t (s; a1,a2) = ζ (s; a1,a2) − sh̄ (s; a1,a2) . proof. the allocation contributions are obtained using 1.3 and the following expressions : fs (x) = h (x; β1,β2) + ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j h (x; (k + i) β1, (k + j) β2) , e [ s × 11{s>x} ] = ζ (x; β1,β2) + ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j ζ (x; (k + i) β1, (k + j) β2) , 14 int. j. anal. appl. (2023), 21:79 and e [ x` × 11{s>x} ] = ξ (x; β`,β3−`) + ∞∑ k=1 θk 1∑ i=0 1∑ j=0 (−1)i+j ξ (x; (k + i) β`, (k + j) β3−`) , ` ∈{1, 2}. � another interesting example is sarmanov’s bivariate exponential distribution introduced by sarmanov (1966) [19]. the bivariate density is given by fx1,x2 (x1,x2) = β1β2e −(β1x1 +β2x2 ) + θβ1β2 (β1 + 1)(β2 + 1) 1∑ i=0 1∑ j=0 (−1)i+j (β1 + i)e−(β1 +i)x1 (β2 + j)e−(β2 +j)x2. where −(1+β1)(1+β2) max(β1,β2,1) ≤ θ ≤ (1+β1)(1+β2) max(β1,β2) . the correlation coefficient is ρp (x1,x2) = θβ1β2 (1 + β1) 2(1 + β2) 2 ∈ [ − 1 4 , + 1 4 ] . the expectile-based allocation contributions can be found directly using proposition 3.1 by letting m = 3, γ1 = β1, λ1 = β2, γi = β1 + i − 2 (i = 2, 3), λj = β2 + i − 2 (j = 2, 3), c1,1 = 1, c1,2 = c1,3 = c2,1 = c3,1 = 0, c2,2 = c3,3 = θβ1β2 (β1+1)(β2+1) , and c2,3 = c3,2 = − θβ1β2(β1+1)(β2+1) . the main limitation of the three previous examples is the narrow range of correlation that is considered. to overcome this issue, bladt and nielsen (2010) [6] employed multivariate phase-type distributions to define a class of bivariate exponential distributions that encompass any feasible pearson correlation coefficient ρp (x1,x2) ∈ [ρmin,ρmax]. the joint density expression for bladt-nielsen’s bivariate exponential distribution, denoted by (x1,x2), is given by: fx1,x2 (x1,x2) = m∑ l=1 m∑ k=1 cl,klλe −lλx1kµe−kµx2, where cl,k = (−1)l+k−(m+1) m ( m l )( m k ) m∑ i=m+1−l k∑ j=1 pi,j(−1)−i−j ( l − 1 m− i )( k − 1 k − j ) and pi,j =   ρ ρ (m) max δi+j−n−1 + 1 m ( 1 − ρ ρ (m) max ) , ρ > 0 ρ ρ (m) min δi−j + 1 m ( 1 − ρ ρ (m) min ) , ρ < 0 , with δx = 1, if x = 0. from this expression and taking γi = iβ1 (i = 1, 2, ...,m) and λj = jβ2 (j = 1, 2, ...,m), this construction can be seen as a bivariate combination of exponentials. then, using proposition 3.1, we can find the expectile-based allocation contributions. int. j. anal. appl. (2023), 21:79 15 4. bivariate exponentials mixture models this section is devoted to stronger dependence models, in the sense of the presence of extreme dependence (λu > 0). the first subsection focuses on studying the marshall-olkin model. the second subsection presents the contributions made by expectile allocation in the case of a common mixture model. 4.1. marshall-olkin model. let yi ∼ exp(λi ), with i = 0, 1, 2, be three independent random variables. we construct two random variables with a common shock: xi = min(yi,y0) for i = 1, 2. the obtained random variables xi have exponential marginal distributions with parameters βi = λi + λ0 (see, e.g., nelsen [29], section 3.1.1). the joint distribution function is given by: f̄x1,x2(x1,x2) = p(x1 > x1,x2 > x2) = p(y1 > x1,y2 > x2,y0 > max(x1,x2)) = e−λ1x1e−λ2x2e−λ0 max(x1,x2) = e−(λ0+λ1)x1e−(λ0+λ2)x2eλ0 min(x1,x2) = f̄x1 (x1)f̄x2 (x2)e λ0 min(x1,x2). this construction leads to a copula given by: c(u1,u2) = min ( u 1−λ0/β1 1 u2,u1u 1−λ0/β2 2 ) . the joint density is: fx1,x2 (x1,x2) =   f 1x1,x2 (x1,x2) = β1e −β1x1 (β2 −λ0)e−(β2−λ0)x2 si x1 > x2 f 2x1,x2 (x1,x2) = (β1 −λ0)e −(β1−λ0)x1β2e −β2x2 si x1 < x2 f 0x1,x2 (x1,x2) = λ0e −β1xe−β2xeλ0x si x1 = x2 = x . this model has as pearson correlation coefficient ρp = λ0 λs , where λs = λ0 + λ1 + λ2. the spearman’s rho for marshall-olkin copulas is given by: ρs = 1 1 + 2 3 λ1+λ2 λ0 . since ρs ∈]0, 1[, the marshall-olkin copulas model only positive dependence. on the other hand, they have upper tail dependence, given by: λu = min ( λ0 β1 , λ0 β2 ) = λ0 max(λ1,λ2) + λ0 , 16 int. j. anal. appl. (2023), 21:79 the marshall-olkin model considers the presence of asymptotic dependence. the density of s = x1 + x2 can be expressed as follows: fs(s) = f 0 x1,x2 (s/2,s/2) + ∫ s/2 0 f 2x1,x2 (x,s −x)dx + ∫ s s/2 f 1x1,x2 (x,s −x)dx = λ0e −λs s2 + λ1β2 λ1 −β2 ( e−β2s −e−λs s 2 ) + λ2β1 λ2 −β1 ( e−β1s −e−λs s 2 ) = ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 ) e−λs s 2 + λ1β2 λ1 −β2 e−β2s + λ2β1 λ2 −β1 e−β1s. from this, we can deduce the cumulative distribution function: f̄mos (s,λ0,λ1,λ2) = 2 λs ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 ) e−λs s 2 + λ1 λ1 −β2 e−β2s + λ2 λ2 −β1 e−β1s. proposition 4.1 provides the allocation contributions based on expectiles for the marshall-olkin model. proposition 4.1 (expectile-allocation, mo model). let (x1,x2) follow a bivariate marshall-olkin model. then, for s = x1 + x2, we have for (k,`) ∈{(1, 2), (2, 1)} eα (xk|s) = (2α− 1)ξmo (s∗,λ0,λk,λ`) + (1 −α) 1λ0+λk (2α− 1)f̄mo s (s∗,λ0,λk,λ`) + (1 −α) , where s∗ is the unique solution to the following equation (1 −α)s = ( 2 λs )2 ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 )( (2α− 1)e−λs s 2 + 1 −α ) + λ1/β2 λ1 −β2 ( (2α− 1)e−β2s + 1 −α ) + λ2/β1 λ2 −β1 ( (2α− 1)e−β1s + 1 −α ) , and ξmo is defined by ξmo (s,λ0,λ1,λ2) = ( λ0 λs + λ1 λs β2 β2 −λ1 + λ2 λs β1 β1 −λ1 ) e−λs s 2 ( s + 2 λs ) + λ2 λ2 −β1 e−β1s ( s + 1 β1 ) + λ1β2 (λ1 −β2)2 ( 1 β2 e−β2s − 2 λs e−λs s 2 ) − λ2β1 (λ2 −β1)2 ( 1 β1 e−β1s − 2 λs e−λs s 2 ) . proof. using the expression of f̄mos , we obtain e[(s − s)+] = ( 2 λs )2 ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 ) e−λs s 2 + λ1/β2 λ1 −β2 e−β2s + λ2/β1 λ2 −β1 e−β1s, in particular e[s] = ( 2 λs )2 ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 ) + λ1 λ1 −β2 1 β2 + λ2 λ2 −β1 1 β1 . so, the expectile eα(s) is the unique solution to the following equation (1 −α)s = ( 2 λs )2 ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 )( (2α− 1)e−λs s 2 + 1 −α ) + λ1/β2 λ1 −β2 ( (2α− 1)e−β2s + 1 −α ) + λ2/β1 λ2 −β1 ( (2α− 1)e−β1s + 1 −α ) . int. j. anal. appl. (2023), 21:79 17 and using the bivariate distribution, we get e [ x1 × 11{s=s} ] = ( λ0 + λ1β2 β2 −λ1 + λ2β1 β1 −λ1 ) s 2 e−λs s 2 + β1λ2 λ2 −β1 se−β1s + λ1β2 (λ1 −β2)2 ( e−β2s −e−λs s 2 ) − λ2β1 (λ2 −β1)2 ( e−β1s −e−λs s 2 ) , and e [ x1 × 11{s>s} ] = ξmo (s,λ0,λ1,λ2) = ( λ0 λs + λ1 λs β2 β2 −λ1 + λ2 λs β1 β1 −λ1 ) e−λs s 2 ( s + 2 λs ) + λ2 λ2 −β1 e−β1s ( s + 1 β1 ) + λ1β2 (λ1 −β2)2 ( 1 β2 e−β2s − 2 λs e−λs s 2 ) − λ2β1 (λ2 −β1)2 ( 1 β1 e−β1s − 2 λs e−λs s 2 ) . that is sufficient to obtain the expressions for the allocation contributions. � note that in the marshall-olkin model, the dependence construction alters the marginal distributions, unlike the fgm model, for example, where the marginals remain the same throughout, and the dependence effect is confined to the copula. 4.2. common mixture model. this method of constructing multivariate models is presented in detail by joe (1997) [25]. it is based on choosing a random variable θ with support sθ and independent random variables yi to construct random variables xi that are conditionally independent given θ. this construction ensures that the conditional distribution function of xi given θ = θ is given by f̄xi|θ=θ(xi ) = (f̄yi (xi )) θ. this construction provides the marginal distributions and the joint distribution by integrating with respect to the law of θ, as described in marceau (2013) [28]. here, we are specifically interested in the case of a bivariate exponential mixture model. we assume that the moment-generating function of θ, denoted by mθ, exists. the joint density function of x1 and x2 is then given by: fx1,x2 (x1,x2) = ∫ θ∈sθ β1θe −β1θx1β2θe −β2θx2dfθ(θ) = β1β2 d2mθ(t) dt2 |t=−(β1x1+β2x2). let (x1,x2) be a pair of continuous random variables following a mixture of exponential distributions. for all i ∈ 1, 2, we have xi ∼ e(βiθ), with β1 < β2, and θ ∼ ga(γ,b). therefore, the survival functions of xi are given by: f̄xi (x) = ∫ ∞ 0 f̄xi|θ=θfθ(θ)dθ = ∫ ∞ 0 e−βiθxfθ(θ)dθ = ( 1 + βix b )−γ . 18 int. j. anal. appl. (2023), 21:79 consequently, xi follows a pareto distribution with parameters ( γ, b βi ) . the risks x1 and x2 are conditionally independent. the survival bivariate distribution is given by: f̄x1,x2 (x1,x2) = ( 1 1 + β1 b x1 + β2 b x2 )γ = ( f̄x1 (x1) −1/γ + f̄x1 (x1) −1/γ − 1 )−γ , which represents the survival clayton copula with a dependence parameter θ = 1/γ. therefore, the upper tail dependence coefficient is: λu = λ clayton l = 2−γ, where λclayton l is the lower tail dependence coefficient of the clayton copula. this dependence model exhibits upper tail dependence. the density of s is given by: fs(s) = β1β2γ (β1 −β2)b  ( 1 1 + β2 b s )γ+1 − ( 1 1 + β1 b s )γ+1 , and its distribution function is given by: f̄cms (s) = β1 β1 −β2 ( 1 1 + β2 b s )γ + β2 β2 −β1 ( 1 1 + β1 b s )γ . proposition 4.2 (expectile-allocation, cm model). let (x1,x2) follow a bivariate common gamma mixture model. then, for s = x1 + x2, we have for (k,`) ∈{(1, 2), (2, 1)} eα (xk|s) = (2α− 1)ξcm (s∗,βk,β`,γ,b) + (1 −α) b(γ−1)βk (2α− 1)f̄cm s (s∗,β1,β2,γ,b) + (1 −α) , where s∗ is the unique solution to the following equation (2α−1)   β1/β2 (β1 −β2) ( 1 1 + β2 b s )γ−1 + β2/β1 (β2 −β1) ( 1 1 + β1 b s )γ−1 = (1−α) [s b (γ − 1) − 1 β1 − 1 β1 ] , and ξcm is defined by ξcm (s ∗,βk,β`,γ,b) = β`b (β` −βk)βk(γ − 1) ( 1 1 + βk b s )γ ( 1 + γ βk b s ) + 1 (βk −β`)2(γ − 1)  βkb ( 1 1 + β` b s )γ−1 −β`b ( 1 1 + βk b s )γ−1 , ∀(k,`) ∈{(1, 2), (2, 1)}. proof. firstly, we have e[(s − s)+] = β1 β2 b (β1 −β2)(γ − 1) ( 1 1 + β2 b s )γ−1 + β2 β1 b (β2 −β1)(γ − 1) ( 1 1 + β1 b s )γ−1 , int. j. anal. appl. (2023), 21:79 19 the expectile eα(s) is then the unique solution to the following equation (2α−1)   β1/β2 (β1 −β2) ( 1 1 + β2 b s )γ−1 + β2/β1 (β2 −β1) ( 1 1 + β1 b s )γ−1 = (1−α) [s b (γ − 1) − 1 β1 − 1 β1 ] . now, using e [ x1 × 11{s=s} ] = β1β2γ (β2 −β1)b s ( 1 1 + β1 b s )γ+1 + β1β2 (β1 −β2)2 [( 1 1 + β2 b s )γ − ( 1 1 + β1 b s )γ] , we calculate the truncated expectation e [ x1 × 11{s>s} ] e [ x1 × 11{s≥s} ] = β2b (β2 −β1)β1(γ − 1) ( 1 1 + β1 b s )γ ( 1 + γ β1 b s ) + 1 (β1 −β2)2(γ − 1)  β1b ( 1 1 + β2 b s )γ−1 −β2b ( 1 1 + β1 b s )γ−1 , which gives us the announced expressions of the allocation contributions. � remark: computations can also be performed by conditioning on the random variable θ and then integrating the formulas derived for the case of independent exponential distributions. 5. comonotonic case for positive distributions in this section, we investigate the case of comonotonic risks, which correspond to perfect dependence. the concept of comonotonic random variables is related to the studies of hoeffding (1940) [14] and fréchet (1951) [12]. here, we adopt the definition of comonotonic risks as first introduced in the actuarial literature by borch (1962) [7]. a vector of random variables (x1,x2, . . . ,xn) is said to be comonotonic if and only if there exists a random variable y and non-decreasing functions ϕ1, . . . ,ϕn such that: (x1, . . . ,xn) d = (ϕ1(y ), . . . ,ϕn(y )). in the case where the risks x1, . . . ,xd are comonotonic, there exists a uniform random variable u such that xi = f −1 xi (u) for all i ∈ 1, . . . ,d, and s = ∑d i=1 f −1 xi (u) = ϕ(u), where ϕ(t) =∑d i=1 f −1 xi (t) and ϕ is a non-decreasing function. proposition 5.1 provides a general expression for marginal contributions using the expectile-based capital allocation rule for comonotonic risk vectors. two applications in the case of exponential and pareto distributions are presented respectively in lemmas 5.1 and 5.2. proposition 5.1 (expectile-based allocation for comonotonic risks). let x1, . . . ,xd be continuous risks with increasing distribution functions and comonotonicity. the marginal contributions using the 20 int. j. anal. appl. (2023), 21:79 expectile allocation rule are given by: eα (xi|s) = (2α− 1) [ (1 −ϕ−1(s∗))f−1xi ( ϕ−1(s∗) ) + e [( xi −f−1xi ( ϕ−1(s∗) )) + ]] + (1 −α)e[xi ] (2α− 1)(1 −ϕ−1(s∗)) + 1 −α , for all i ∈{1, . . . ,d}, where s∗ is the unique solution to the following equation (2α− 1) ( d∑ `=1 e [( x` −f−1x` ( ϕ−1(s) )) + ]) = (1 −α) ( s − d∑ `=1 e[x`] ) . proof. since the risks x1, . . . ,xd are comonotonic, xi and s are also comonotonic for all i ∈ 1, . . . ,d. assuming that the distributions are positive and continuous, we have: e [ xi × 11{s>s} ] = ∫ +∞ 0 min ( f̄xi (t), f̄s(s) ) dt = ∫ f−1 xi (fs(s)) 0 f̄s(s)dt + ∫ +∞ f−1 xi (fs(s)) f̄xi (t)dt = f̄s(s) ×f−1xi (fs(s)) + e [( xi −f−1xi (fs(s)) ) + ] . from equation 1.3, we directly obtain the corresponding contributions. the equation satisfied by the sum’s expectile is rewritten using theorem 7 of dhaene et al. (2002) [10]. � lemma 5.1 (comonotonic exponential distributions ). let x1, . . . ,xd be comonotonic risks with exponential marginal distributions, where xi ∼ e(βi ) for i = 1, . . . ,d. the marginal contributions using the expectile allocation rule are given by: eα (xi|s) = βs βi s∗, for all i ∈{1, . . . ,d}, where s∗ is the unique solution to the following equation (2α− 1) e−βss βs = (1 −α) ( s − 1 βs ) , and βs = 1/ ∑d `=1 1 β` . proof. in this case, s ∼ e (βs), where βs = 1/ ∑d `=1 1 β` . according to proposition 5.1, the marginal contributions are given by: eα (xi|s) = (2α− 1)e−βss (2α− 1)e−βss + 1 −α βs βi eα(s) + 1 βi , which directly yields the expressions for the obtained contributions. � remark: in this case, the allocation percentages can be written as follows: eα (xi|s) /eα(s) = e[xi ] e[s] , ∀i ∈{1, . . . ,d}. the allocation is proportional to the risk level, where the proportion is determined by the ratio of the expected values of xi and s. int. j. anal. appl. (2023), 21:79 21 lemma 5.2 (comonotonic pareto distributions). let x1, . . . ,xd be comonotonic risks following pareto marginal distributions with the same shape parameter, i.e., xi ∼ pa(β,λi ) for i = 1, . . . ,d, where β > 1. the marginal contributions using the expectile allocation rule are given by: eα (xi|s) = λi∑d `=1 λ` s∗, ∀i ∈{1, . . . ,d}, where s∗ is the unique solution to the following equation: s = ∑d `=1 λ` β − 1  2α− 1 1 −α ( ∑d `=1 λ`∑d `=1 λ` + s )β−1 + 1   . proof. we remark that in this case s ∼ pa(β, ∑d i=1 λi ). by proposition 5.1, we obtain the expressions for the marginal contributions as stated. � it is worth noting that in this case as well, the allocation is proportional to the risk level. in fact, the allocation percentages can be expressed as follows: eα (xi|s) /eα(s) = e[xi ] e[s] , ∀i ∈{1, . . . ,d}. 6. numerical illustrations in this section, we provide numerical illustrations to highlight the differences between contributions to the aggregate risk obtained from tvar (tail value at risk) and expectiles-based capital allocations. specifically, we focus on a bivariate scenario with exponential marginal distributions. our analysis involves evaluating the allocation amounts and their respective percentages in the aggregate risk. additionally, we investigate the influence of dependence on capital allocation using the fgm (farliegumbel-morgenstern) model. 6.1. case of independence. we consider a bivariate exponential model where x1 represents a riskier business line compared to x2 (β1 < β2). the expressions for the marginal contributions to the global risk can be found in proposition 2.1. figure 1 displays the contribution amount of x1 (left) and its percentage contribution to the aggregated risk (right). similarly, figure 2 presents the corresponding quantities for x2. 22 int. j. anal. appl. (2023), 21:79 figure 1. tvar allocation vs expectile allocation, exponential independent model (x1 ∼e(β1 = 0.10), x2 ∼e(β2 = 0.25)) x1 contribution. figure 2. tvar allocation vs expectile allocation, exponential independent model (x1 ∼e(β1 = 0.10), x2 ∼e(β2 = 0.25)) x2 contribution. the comparison between expectile-based and tvar-based allocations reveals that the contributions obtained from expectile-based allocation are consistently smaller for both risks. this discrepancy can be attributed to the nature of the expectile risk measure, which incorporates performance considerations in its quantification of risk. by examining the percentage allocations assigned to each risk, we gain further insights into the differences between the two methods. notably, the expectile-based allocation assigns a relatively smaller amount of capital to the riskier branch (x1), while maintaining an increasing allocation percentage for x1 as the level α increases. conversely, the allocation percentage for x2 symmetrically decreases with increasing α, mirroring the behavior observed with tvar allocation. int. j. anal. appl. (2023), 21:79 23 6.2. fgm model. for the given marginal distributions, we now introduce a dependence structure modeled using an fgm copula with a parameter of θ = 1. as a result, the correlation coefficient ρs is equal to 1/3, indicating a positive dependence within the model. the expressions for marginal contributions derived from the expectile-based allocation rule can be found in lemma 3.1. in figure 3, we present the contribution amount (left) and the corresponding percentage (right) of x1 to the aggregated risk. additionally, figure 4 illustrates the variation of both the contribution amount (left) and percentage (right) of x2 as a function of α. figure 3. tvar allocation vs expectile allocation, fgm model ( x1 ∼e(β1 = 0.10), x2 ∼e(β2 = 0.25), θ = 1) x1 contribution. figure 4. tvar allocation vs expectile allocation, fgm model ( x1 ∼e(β1 = 0.10), x2 ∼e(β2 = 0.25), θ = 1) x2 contribution. 24 int. j. anal. appl. (2023), 21:79 the inclusion of positive dependence between x1 and x2 resulted in an increase in the contribution of x2. this observation aligns with the reduction in diversification gain, indicating a stronger interdependence between the two risks. 6.3. fgm model, impact of dependence. to further analyze the influence of dependence on the allocation composition, we fix the level α and vary the dependency parameter θ of the fgm copula. the outcomes for the contribution (left) and its percentage (right) of x1 and x2 are illustrated in figures 5 and 6 respectively. figure 5. impact of dependence, fgm model ( x1 ∼ e(β1 = 0.10), x2 ∼ e(β2 = 0.25), α = 0.99) x1 contribution. figure 6. impact of dependence, fgm model ( x1 ∼ e(β1 = 0.10), x2 ∼ e(β2 = 0.25), α = 0.99) x2 contribution. int. j. anal. appl. (2023), 21:79 25 as the parameter θ increases, the bivariate dependence in the fgm copula, which belongs to the family of parametric copulas, also increases. in this context, the allocation percentage assigned to the riskier branch, represented by x1, decreases. hence, the dependence level has a direct impact on the participation of the less risky branch, denoted as x2, in the overall risk. specifically, higher dependence leads to an increased involvement of x2 in the aggregated risk. conclusion the main objective of this paper was to demonstrate, using various multivariate risk models, a practical approach to constructing capital allocation based on expectile risk measures. as expectiles are the only law-invariant risk measures that are both elicitable and coherent, it is natural to focus on marginal contributions in the sum’s expectile. the constructed allocation can be backtested using the elicitable nature of expectiles and it satisfies desirable properties derived from their coherence. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] p. artzner, f. delbaen, j.m. eber, d. heath, coherent measures of risk, math. finance. 9 (1999), 203-228. https://doi.org/10.1111/1467-9965.00068. [2] m. bargès, h. cossette, é. marceau, tvar-based capital allocation with copulas, insurance: math. econ. 45 (2009), 348-361. https://doi.org/10.1016/j.insmatheco.2009.08.002. [3] f. bellini, v. bignozzi, on elicitable risk measures, quant. finance. 15 (2015), 725-733. https://doi.org/10. 1080/14697688.2014.946955. [4] f. bellini, e. di bernardino, risk management with expectiles, eur. j. finance. 23 (2015), 487-506. https: //doi.org/10.1080/1351847x.2015.1052150. [5] f. bellini, b. klar, a. müller, e.r. gianin, generalized quantiles as risk measures, insurance: math. econ. 54 (2014), 41-48. https://doi.org/10.1016/j.insmatheco.2013.10.015. [6] m. bladt, b.f. nielsen, on the construction of bivariate exponential distributions with an arbitrary correlation coefficient, stoch. models. 26 (2010), 295-308. https://doi.org/10.1080/15326341003756486. [7] k. borch, equilibrium in a reinsurance market, econometrica. 30 (1962), 424-444. [8] h. cossette, m. mailhot, é. marceau, tvar-based capital allocation for multivariate compound distributions with positive continuous claim amounts, insurance: math. econ. 50 (2012), 247-256. https://doi.org/10. 1016/j.insmatheco.2011.11.006. [9] h. cossette, e. marceau, s. perreault, on two families of bivariate distributions with exponential marginals: aggregation and capital allocation, insurance: math. econ. 64 (2015), 214-224. https://doi.org/10.1016/j. insmatheco.2015.05.007. [10] j. dhaene, m. denuit, m.j. goovaerts, r. kaas, d. vyncke, the concept of comonotonicity in actuarial science and finance: theory, insurance: math. econ. 31 (2002), 3-33. https://doi.org/10.1016/s0167-6687(02) 00134-8. [11] s. emmer, m. kratz, d. tasche, what is the best risk measure in practice? a comparison of standard measures, j. risk. 18 (2015), 31-60. https://doi.org/10.21314/jor.2015.318. https://doi.org/10.1111/1467-9965.00068 https://doi.org/10.1016/j.insmatheco.2009.08.002 https://doi.org/10.1080/14697688.2014.946955 https://doi.org/10.1080/14697688.2014.946955 https://doi.org/10.1080/1351847x.2015.1052150 https://doi.org/10.1080/1351847x.2015.1052150 https://doi.org/10.1016/j.insmatheco.2013.10.015 https://doi.org/10.1080/15326341003756486 https://doi.org/10.1016/j.insmatheco.2011.11.006 https://doi.org/10.1016/j.insmatheco.2011.11.006 https://doi.org/10.1016/j.insmatheco.2015.05.007 https://doi.org/10.1016/j.insmatheco.2015.05.007 https://doi.org/10.1016/s0167-6687(02)00134-8 https://doi.org/10.1016/s0167-6687(02)00134-8 https://doi.org/10.21314/jor.2015.318 26 int. j. anal. appl. (2023), 21:79 [12] m. fréchet, sur les tableaux de corrélation dont les marges sont données, ann. l’univ. lyon, ser. 3, sect. a 14 (1951), 53-77. [13] t. gneiting, making and evaluating point forecasts, j. amer. stat. assoc. 106 (2011), 746-762. https://doi. org/10.1198/jasa.2011.r10138. [14] w. hoeffding, masstabinvariante korrelationstheorie, schriften des mathematischen instituts und des instituts für angewandte mathematik der universität berlin. 5 (1940), 179-233. [15] w.k. newey, j.l. powell, asymmetric least squares estimation and testing, econometrica. 55 (1987), 819-847. https://doi.org/10.2307/1911031. [16] t. mao, f. yang, risk concentration based on expectiles for extreme risks under fgm copula, insurance: math. econ. 64 (2015), 429-439. https://doi.org/10.1016/j.insmatheco.2015.06.009. [17] v. maume-deschamps, d. rullière, k. said, multivariate extensions of expectiles risk measures, dependence model. 5 (2017), 20-44. https://doi.org/10.1515/demo-2017-0002. [18] v. maume-deschamps, d. rullière, k. said, extremes for multivariate expectiles, stat. risk model. 35 (2018), 111-140. https://doi.org/10.1515/strm-2017-0014. [19] ov sarmanov, generalized normal correlation and 2-dimensional frechet-classes, dokl. akad. nauk sssr. 168 (1966), 32-35. [20] d. tasche, conditional expectation as quantile derivative, technical report, tu munchen, germany. (2000). [21] d. tasche, allocating portfolio economic capital to sub-portfolios, economic capital: a practitioner guide, risk books, risk books, 275-302, (2004). [22] d. tasche, euler allocation: theory and practice, technical report, (2007). [23] d. tasche, capital allocation to business units and sub-portfolios: the euler principle, (2008). https://doi. org/10.48550/arxiv.0708.2542. [24] c.d. lai, n. balakrishnan, continuous bivariate distributions, springer, new york, 2009. https://doi.org/10. 1007/b101765. [25] h. joe, multivariate models and dependence concepts, chapman & hall/crc, boca raton, 1997. [26] d. balog, capital allocation in financial institutions: the euler method, iehas discussion papers no. mt-dp 2011/26, institute of economics, hungarian academy of sciences, 2011. [27] s. kotz, n.l. johnson, n. balakrishnan, n.l. johnson, continuous multivariate distributions, wiley, new york, 2004. [28] é. marceau, modélisation et évaluation quantitative des risques en actuariat: modèles sur une période, springerverlag france, 2013. [29] r. nelsen, an introduction to copulas, springer, new york, 2007. https://doi.org/10.1198/jasa.2011.r10138 https://doi.org/10.1198/jasa.2011.r10138 https://doi.org/10.2307/1911031 https://doi.org/10.1016/j.insmatheco.2015.06.009 https://doi.org/10.1515/demo-2017-0002 https://doi.org/10.1515/strm-2017-0014 https://doi.org/10.48550/arxiv.0708.2542 https://doi.org/10.48550/arxiv.0708.2542 https://doi.org/10.1007/b101765 https://doi.org/10.1007/b101765 introduction 1. expectile-based capital allocation 2. some bivariate independent models 2.1. bivariate independent exponential model 2.2. bivariate independent gamma model 3. bivariate combinations of exponentials 3.1. bivariate combinations of exponential distributions 3.2. specific models 4. bivariate exponentials mixture models 4.1. marshall-olkin model 4.2. common mixture model 5. comonotonic case for positive distributions 6. numerical illustrations 6.1. case of independence 6.2. fgm model 6.3. fgm model, impact of dependence conclusion references int. j. anal. appl. (2023), 21:73 received: may 31, 2023. 2020 mathematics subject classification. 47h10, 54h25. key words and phrases. common fixed point; coincident point; graphical structures; neutrosophic b-metric space; unique solution, computational techniques. https://doi.org/10.28924/2291-8639-21-2023-73 © 2023 the author(s) issn: 2291-8639 1 new fixed point results in neutrosophic b-metric spaces with application muhammad saeed1, umar ishtiaq2,*, doha a. kattan3, khaleel ahmad4, salvatore sessa5,* 1department of mathematics, university of management and technology, lahore, pakistan 2office of research, innovation and commercialization, university of management and technology, lahore, pakistan 3department of mathematics, faculty of sciences and arts, king abdulaziz university, rabigh, saudi arabia 4department of mathematics and statistics, university of management and technology, lahore, pakistan 5università di napoli federico ii, dipartimento di architettura, via toledo 403, 80121 napoli, italy *corresponding authors: umarishtiaq000@gmail.com, salvasessa@gmail.com abstract. in this manuscript, we establish the notion of neutrosophic b-metric spaces as a generalization of fuzzy b-metric spaces, intuitionistic fuzzy b-metric spaces and neutrosophic metric spaces in which three symmetric properties plays an important role for membership, non-membership and neutral functions as well we derive some common fixed point and coincident point results for contraction mappings. also, we provide several non-trivial examples with graphical views of neutrosophic b-metric spaces and contraction mappings by using computational techniques. our results are more generalized with respect to the existing ones in the literature. at the end of the paper, we provide an application to test the validity of the main result. 1. introduction and prelimainaries fixed-point theorems in metric spaces (and their different generalizations) have made exquisite theoretical progress and have a variety of practical applications. these advancements over the last three decades were fantastic. the majority of scholars based their reference findings on banach’s https://doi.org/10.28924/2291-8639-21-2023-73 2 int. j. anal. appl. (2023), 21:73 contraction theorem [1]. by applying nonlinear equations to similar fixed-point (fp) applications, numerous problems in engineering and economics can be resolved. a fixed point 𝐹𝑥 = 𝑥 can be established for an operator sum 𝐺𝑥 = 0, where 𝐹 is a self-mapping in some relevant disciplines. for resolving issues brought on by a variety of mathematical inspection origins, such as split feasibility concerns, supporting problems, equilibrium problems, and matching and selection issues, fp theory has a number of key modes. studying the notion of fps is both exciting and interesting. this idea has already been demonstrated to be an amazing attempt to condense nonlinear analysis into a short timeframe. the idea of fuzzy sets (fss), that is utilized to characterize/manipulate information and data having non-statistical uncertainty, was first introduced by zadeh [2] in 1965. the idea of fss seeks to address issues where errors and a high degree of uncertainty are present by providing logical and set hypothetical tools. later, in 1986, atanassov [3] proposed the concept of intuitionistic fss. this set theory, which is a broader version of fs theory, defines both the degree of membership and the degree of non-membership. many authors used this idea in various branches of mathematics. this theory has been applied to groups and its properties by gulzar et al. [4-6]. akber [7] established the intuitionistic fuzzy mappings and developed standard fps for particular types of mappings. the important sign that the notion of a distance function plays in approximation theory has led to the application of fss to the fundamental notion of metric as well. a number of publications [8–10] have taken steps in this direction by introducing the applications of metric spaces to fuzzy circumstances. kramosil and michalek [11] introduced the idea of fuzzy metric spaces (fmss) in 1975, and in 1994, george and veeramani [12] established a hausdorff topology utilizing fuzzy metric. in fuzzy cone metric spaces, rehmanand and aydi [13] established their findings. the idea of b-metric space, which is a broader category than metric space, was initially put forth by bakhtin [14]. later, saleem et al. [15] coined the concept graphical of fmss. the concept of fuzzy b-metric space (fbms), was put up by nadaban [16] in 2016, generalization of fmss. ishtiaq et al. [17] derived several fp results in generalizations of fmss. in fuzzy strong b-metric spaces, shazia et al. [18] identified fps for a number of nonlinear contraction mappings. in 2004, park [19] used the idea of intuitionistic fss, continuous t-norm (ctn), and continuous t-conorm (ctcn) to established intuitionistic fuzzy metric spaces (ifmss) as a generalization of fmss. banach's contraction principle was improved by jungck [20] in 1976 by looking into coincidence and common fps in commuter mappings. in 1986, jungck [21] introduced the idea of common fps as well as compatible maps for a pair of self3 int. j. anal. appl. (2023), 21:73 mappings. jungck's common fp theorems were generalized by turkoglu et al. [22] in ifmss in 2006. weakly compatible (w-compatible) mappings were first described by jungck and rhoades [23] in 2006. since any pair of compatible mappings is w-compatible, but the converse is not true in general, weakly compatible mappings are more general. grabiec [24] derived the banach’s fp results in the context of fmss in 1988. schweizer and sklar [25] introduced statistical metric spaces. kanwal et al. [26] derived some new fp results in intuitionistic fuzzy b-metric spaces (ifbms). in 2005, smarandache [27] proposed the concept of neutrosophic sets (nss), as a generalization of ifss. in 2019, kirişci and şimşek [28] established the notion of neutrosophic metric spaces (nmss) and discussed a topological structure. in nmss, membership (𝑀), non-membership (𝑁) and neutral functions (𝑂) are used and they establish the following three symmetric properties for these functions: 𝑀(𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, 𝑁(𝜔, 𝜐, 𝜄) = 𝑁(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, 𝑂(𝜔, 𝜐, 𝜄) = 𝑂(𝜔, 𝜐, 𝜄), for all 𝜄 > 0. şimşek and kirişci [29] derived numerous fp results for contraction mappings in the context of nms. ishtiaq et al. [30] generalized the notion of nms and introduced the notion orthogonal nmss and proved some new types of fp theorems for contraction mappings. debnath [31] worked on a mathematical model using fixed point theorem for two-choice behavior of rhesus monkeys in a noncontingent environment. authors in [32] and [33] did amazing work in the direction of fixed point theory. in this manuscript, we aim to introduce the notion of neutrosophic b-metric space (nbms) as a generalization of nms and we use the above defined three symmetric properties to introduce the notion of nbms. we established some coincident point (c-point) and common fp results in which symmetric properties of nbms plays a very significant role. we coined several non-trivial examples and graphical views via computational techniques. also, we provide an application to support our main result. we start with some definitions that are helpful for readers to understand the main results. definition 1.1 [25] a binary operation ∗: [0,1] × [0,1] → [0,1] is called ctn if the below circumstances are fulfilled: (a1) ∗ is associative and commutative, (a2) ∗ is continuous, (a3) 𝜆 ∗ 1 = 𝜆, for all 𝜆 ∈ [0,1], 4 int. j. anal. appl. (2023), 21:73 (a4) if 𝜆 ≤ 𝑘 and 𝑎 ≤ 𝑙 with 𝜆, 𝑎, 𝑘, 𝑙 ∈ [0,1], then 𝜆 ∗ 𝑎 ≤ 𝑘 ∗ 𝑙. definition 1.2 [3] a binary operation ∘: [0,1] × [0,1] → [0,1] is called ctcn if the below circumstances are fulfilled: (b1) ∗ is associative and commutative, (b2) ∘ is continuous, (b3) 𝜆 ∘ 0 = 𝜆, for all 𝜆 ∈ [0,1], (b4) if 𝜆 ≤ 𝑘 and 𝑎 ≤ 𝑙 with 𝜆, 𝑎, 𝑘, 𝑙 ∈ [0,1], then 𝜆 ∗ 𝑎 ≤ 𝑘 ∗ 𝑙. definition 1.3 [16] suppose 𝜁 be a non-empty set. let 𝑠 ∈ ℝ, s ≥ 1 and ∗ be ctn. a fs 𝑀 on 𝜁 × 𝜁 × [0, +∞) is known as fuzzy b-metric if, for all 𝜔, 𝜐, 𝑧 ∈ 𝜁 the below conditions are verified: (bm1) 𝑀 (𝜔, 𝜐, 0) = 0, (bm2) 𝑀 (𝜔, 𝜐, 𝜄) = 1, for all 𝜄 ≥ 0 ⟺ 𝑆𝜔 = 𝜐, (bm3) 𝑀 (𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄), for all 𝜄 ≥ 0, (bm4) 𝑀 (𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≥ 𝑀(𝜔, 𝜐, 𝜄) ∗ 𝑀(𝜐, 𝑧, 𝜃)𝑑, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜄, 𝜃 ≥ 0, (bm5) lim 𝜄→+∞ 𝑀 (𝜔, 𝜐, 𝜄) = 1 and 𝑀(𝜔, 𝜐, . ): [0, +∞] → [0,1] is left continuous. then (𝜁, 𝑀,∗, 𝑠) is called a fuzzy b-metric space. definition 1.4 [27] let a set 𝜁 ≠ ∅ and 𝜗 ∈ 𝑋. a neutrosophic set (ns) 𝐺 in 𝜁 is categorized by three components (i) truth-membership function 𝑀(𝜗), (ii) indeterminacy-membership function 𝑁(𝜗), (iii) falsity-membership function 𝑂(𝜗). the functions 𝑀(𝜗), 𝑁(𝜗) and 𝑂(𝜗) are real standard or non-standard subsets of ]0−, 1+[, that is 𝑀(𝜗): 𝜁 →]0−, 1+[, 𝑁(𝜗): 𝜁 →]0−, 1+[ and 𝑂(𝜗): 𝜁 →]0−, 1+[ 𝑠uch that 0− ≤ sup 𝑀(𝜗) + sup 𝑁(𝜗) + sup 𝑂(𝜗) ≤ 3+. definition 1.5 [28] a 6-tuple (𝜁, 𝑀, 𝑁, 𝑂,∗,∘) is known as a nms if 𝜁 is an arbitrary set, ∗ and ∘ are ctn and ctcn respectively, 𝑀, 𝑁, 𝑂 are nss on 𝜁2 × [0, +∞) verifying the bellow circumstances 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜔, 𝜐, 𝑧 ∈ 𝜁, (n1) 𝑀(𝜔, 𝜐, 𝜄) + 𝑁(𝜔, 𝜐, 𝜄) + 𝑂(𝜔, 𝜐, 𝜄) ≤ 3, (n2) 𝑀(𝜔, 𝜐, 0) = 0, (n3) 𝑀(𝜔, 𝜐, 𝜄) = 1, for all 𝜄 > 0 iff 𝜔 = 𝜐, (n4) 𝑀(𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, 5 int. j. anal. appl. (2023), 21:73 (n5) 𝑀(𝜔, 𝑧, 𝜄 + 𝜃) ≥ 𝑀(𝜔, 𝜐, 𝜄) ∗ 𝑀(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (n6) 𝑀(𝜔, 𝜐, . ): [0, +∞) → [0,1] is left continuous and lim 𝜄→+∞ 𝑀(𝜔, 𝜐, 𝜄) = 1, (n7) 𝑁(𝜔, 𝜐, 0) = 1, (n8) 𝑁(𝜔, 𝜐, 𝜄) = 0, for all 𝜄 > 0 iff 𝜔 = 𝜐, (n9) 𝑁(𝜔, 𝜐, 𝜄) = 𝑁(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, (n10) 𝑁(𝜔, 𝑧, 𝜄 + 𝜃) ≤ 𝑁(𝜔, 𝜐, 𝜄) ∘ 𝑁(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (n11) lim 𝜄→+∞ 𝑁(𝜔, 𝜐, 𝜄) = 0 and 𝑁(𝜔, 𝜐, . ) : [0, +∞) → [0,1] is right continuous, (n12) 𝑂(𝜔, 𝜐, 0) = 1, (n13) 𝑂(𝜔, 𝜐, 𝜄) = 0, for all 𝜄 > 0 iff 𝜔 = 𝜐, (n14) 𝑂(𝜔, 𝜐, 𝜄) = 𝑂(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, (n15) 𝑂(𝜔, 𝑧, 𝜄 + 𝜃) ≤ 𝑂(𝜔, 𝜐, 𝜄) ∘ 𝑂(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (n16) lim 𝜄→+∞ 𝑂(𝜔, 𝜐, 𝜄) = 0 and 𝑂(𝜔, 𝜐, . ) : [0, +∞) → [0,1] is right continuous. then (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) said to be a nms. definition 1.6 [26] a 6-tuple (𝜁, 𝑀, 𝑁,∗,∘, 𝑠) is known as an ifbms if 𝜁 ≠ 𝜙, 𝑠 ≥ 1 is a given real number, ∗ and ∘ are ctn and ctcn, respectively, 𝑀 and 𝑁 are fss on 𝜁2 × [0, +∞) verifying the below circumstances 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜔, 𝜐, 𝑧 ∈ 𝜁, (ifb1) 𝑀(𝜔, 𝜐, 𝜄) + 𝑁(𝜔, 𝜐, 𝜄) ≤ 1, (ifb2) 𝑀(𝜔, 𝜐, 0) = 0, (ifb3) 𝑀(𝜔, 𝜐, 𝜄) = 1, for all 𝜄 > 0 iff 𝜔 = 𝜐, (ifb4) 𝑀(𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, (ifb5) (𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≥ 𝑀(𝜔, 𝜐, 𝜄) ∗ 𝑀(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (ifb6) 𝑀(𝜔, 𝜐, . ): [0, +∞) → [0,1] is left continuous and lim 𝜄→+∞ 𝑀(𝜔, 𝜐, 𝜄) = 1, (ifb7) 𝑁(𝜔, 𝜐, 0) = 1, (ifb8) 𝑁(𝜔, 𝜐, 𝜄) = 0, for all 𝜄 > 0 iff 𝜔 = 𝜐, (ifb9) 𝑁(𝜔, 𝜐, 𝜄) = 𝑁(𝜔, 𝜐, 𝜄), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜄 > 0, (ifb10) 𝑁(𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≤ 𝑁(𝜔, 𝜐, 𝜄) ∘ 𝑁(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (ifb11) lim 𝜄→+∞ 𝑁(𝜔, 𝜐, 𝜄) = 0, (ifb12) 𝑁(𝜔, 𝜐, . ) : [0, +∞) → [0,1] is right continuous. 6 int. j. anal. appl. (2023), 21:73 2. neutrosophic b-metric spaces in this section, we will establish the notion of nbms and several non-trivial examples with their graphical structures. definition 2.1 a 7-tuple (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) known to be an nbms if 𝜁 ≠ 𝜙, 𝑠 ≥ 1 is a given real number, ∗ and ∘ are ctn and ctcn, respectively, and 𝑀, 𝑁, 𝑂 are nss on 𝜁2 × [0, +∞) verifying the below circumstances 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜔, 𝜐, 𝑧 ∈ 𝜁, (nbm1) 𝑀(𝜔, 𝜐, 𝜄) + 𝑁(𝜔, 𝜐, 𝜄) + 𝑂(𝜔, 𝜐, 𝜄) ≤ 3, (nbm2) 𝑀(𝜔, 𝜐, 0) = 0, (nbm3) 𝑀(𝜔, 𝜐, 𝜄) = 1, for all𝜄 > 0 iff 𝜔 = 𝜐, (nbm4) 𝑀(𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, (nbm5) 𝑀(𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≥ 𝑀(𝜔, 𝜐, 𝜄) ∗ 𝑀(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (nbm6) 𝑀(𝜔, 𝜐, . ): [0, +∞) → [0,1] is left continuous and lim 𝜄→+∞ 𝑀(𝜔, 𝜐, 𝜄) = 1, (nbm7) 𝑁(𝜔, 𝜐, 0) = 1, (nbm8) 𝑁(𝜔, 𝜐, 𝜄) = 0, for all 𝜄 > 0 iff 𝜔 = 𝜐, (nbm9) 𝑁(𝜔, 𝜐, 𝜄) = 𝑁(𝜔, 𝜐, 𝜄), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝜄 > 0, (nbm10) 𝑁(𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≤ 𝑁(𝜔, 𝜐, 𝜄) ∘ 𝑁(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (nbm11) lim 𝜄→+∞ 𝑁(𝜔, 𝜐, 𝜄) = 0 and 𝑁(𝜔, 𝜐, . ) : [0, +∞) → [0,1] is right continuous, (nbm12) 𝑂(𝜔, 𝜐, 0) = 1, (nbm13) 𝑂(𝜔, 𝜐, 𝜄) = 0, for all 𝜄 > 0 iff 𝜔 = 𝜐, (nbm14) 𝑂(𝜔, 𝜐, 𝜄) = 𝑂(𝜔, 𝜐, 𝜄), for all 𝜄 > 0, (nbm15) 𝑂(𝜔, 𝑧, 𝑠(𝜄 + 𝜃)) ≤ 𝑂(𝜔, 𝜐, 𝜄) ∘ 𝑁(𝜐, 𝑧, 𝜃), for all 𝜄, 𝜃 > 0, (nbm16) lim 𝜄→+∞ 𝑂(𝜔, 𝜐, 𝜄) = 0 and 𝑂(𝜔, 𝜐, . ) : [0, +∞) → [0,1] is right continuous. then (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) said to be a nbms. remark 2.1 if, we let 𝑠 = 1 in the above definition, then it will become nms. so, every nms is an nbms, but the converse is not generally true. example 2.1 suppose (𝜁, ϖ, s) be a b-metric space and 𝑎 ∗ b = min{𝑎, 𝑏} , 𝑎 ∘ 𝑏 = max{𝑎, 𝑏}, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑎, 𝑏 ∈ [0,1], and let 𝑀𝜛, 𝑁𝜛 and 𝑂𝜛 be nss on 𝜁 2 × [0, +∞), defined as follows: 𝑀𝜛(𝜔, 𝜐, 𝜄) = { 𝜄 𝜄 + ϖ(𝜔, 𝜐) , if 𝜄 > 0, 0, if 𝜄 = 0 7 int. j. anal. appl. (2023), 21:73 𝑁𝜛 (𝜔, 𝜐, 𝜄) = { ϖ(𝜔, 𝜐) 𝜄 + ϖ(𝜔, 𝜐) , if 𝜄 > 0, 1, if 𝜄 = 0 and 𝑂𝜛(𝜔, 𝜐, 𝜄) = { ϖ(𝜔, 𝜐) 𝜄 , if 𝜄 > 0, 1, if 𝜄 = 0. we verify the axioms (nbm5), (nbm10) and (nbm15) of definition 2.1 others; are obvious. let 𝜔, 𝜐, 𝑧 ∈ 𝜁 and 𝜄, 𝜎 > 0. without loss of the generality, we suppose that 𝑀𝜛(𝜔, 𝜐, 𝜄) ≤ 𝑀𝜛(𝜐, 𝑧, 𝜎) 𝑁𝜛 (𝜔, 𝜐, 𝜄) ≥ 𝑁𝜛 (𝜐, 𝑧, 𝜎), and 𝑂𝜛(𝜔, 𝜐, 𝜄) ≥ 𝑂𝜛(𝜐, 𝑧, 𝜎), thus, 𝜄 𝜄 + 𝜛(𝜔, 𝜐) ≤ 𝜎 𝜎 + 𝜛(𝜔, 𝜐) 𝜛 (𝜔, 𝜐) 𝜄 + 𝜛(𝜔, 𝜐) ≥ 𝜛(𝜐, 𝑧) 𝜎 + 𝜛(𝜔, 𝜐) 𝜄𝜛(𝜐, 𝑧) ≤ 𝜎𝜛(𝜐, 𝑧) on the contrary, 𝑀𝜛(𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) = 𝑠(𝜄 + 𝜎) 𝑠(𝜄 + 𝜎) + 𝜛(𝜔, 𝑧) ≥ 𝑠(𝜄 + 𝜎) 𝑠(𝜄 + 𝜎) + 𝑠[𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧)] = 𝜄 + 𝜎 𝜄 + 𝜎 + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) . also, 𝑁𝜛(𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) = 𝜛(𝜔 + 𝑧) 𝑠(𝜄 + 𝜎) + 𝜛(𝜔, 𝑧) ≤ 𝑠[𝜛(𝜔, 𝜐) + 𝜔(𝜐, 𝑧)] 𝑠(𝜄 + 𝜎) + 𝑠[𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧)] , = 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) ≤ 𝜛(𝜔, 𝜐) 𝜄 + 𝜛(𝜔, 𝜐) and 8 int. j. anal. appl. (2023), 21:73 𝑂𝜛(𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) = 𝜛(𝜔 + 𝑧) 𝑠(𝜄 + 𝜎) ≤ 𝑠[𝜛(𝜔, 𝜐) + 𝜔(𝜐, 𝑧)] 𝑠(𝜄 + 𝜎) , = 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 ≤ 𝜛(𝜔, 𝜐) 𝜄 . hence, 𝑀𝜛(𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) ≥ 𝑀𝜛(𝜔, 𝜐, 𝜄) = 𝑀𝜛(𝜔, 𝜐, 𝜄) ∗ 𝑀𝜛(𝜐, 𝑧, 𝜎) 𝑁𝜛 (𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) ≤ 𝑁𝜛 (𝜔, 𝜐, 𝜄) = 𝑁𝜛 (𝜔, 𝜐, 𝜄) ∘ 𝑁𝜛 (𝜐, 𝑧, 𝜎), and 𝑂𝜛(𝜔, 𝑧, 𝑠(𝜄 + 𝜎)) ≤ 𝑂𝜛(𝜔, 𝜐, 𝜄) = 𝑂𝜛(𝜔, 𝜐, 𝜄) ∘ 𝑂𝜛(𝜐, 𝑧, 𝜎). now 𝜄 + 𝜎 𝜄 + 𝜎 + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) ≥ 𝜄 𝜄 + 𝜛(𝜔, 𝜐) ⇔ 𝜄2 + 𝜎𝜄 + 𝜄𝜛(𝜔, 𝜐) + 𝜎𝜛(𝜔, 𝜐) ≥ 𝜄2 + 𝜎𝜄 + 𝜄𝜛(𝜔, 𝜐) + 𝜄𝜛(𝜐, 𝑧) ⇔ 𝜎𝜛(𝜔, 𝜐) ≥ 𝜄𝜛(𝜐, 𝑧), which is true. also 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) ≤ 𝜛(𝜔, 𝜐) 𝜄 + 𝜛(𝜔, 𝜐) ⇔ 𝜄𝜛(𝜔, 𝜐) + 𝜄𝜛(𝜐, 𝑧) + 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) + (𝜛(𝜔, 𝜐)) 2 ≤ 𝜄𝜛(𝜔, 𝜐) + 𝜎𝜛(𝜔, 𝜐) + 𝜛(𝑥, 𝜐)𝜛(𝜐, 𝑧) + (𝜛(𝜔, 𝜐)) 2 ⇔ 𝜄𝜛(𝜐, 𝑧) ≤ 𝜎𝜛(𝜔, 𝜐), and 𝜛(𝜔, 𝜐) + 𝜛(𝜐, 𝑧) 𝜄 + 𝜎 ≤ 𝜛(𝜔, 𝜐) 𝜄 ⇔ 𝜄𝜛(𝜔, 𝜐) + 𝜄𝜛(𝜐, 𝑧) ≤ 𝜄𝜛(𝜔, 𝜐) + 𝜎𝜛(𝜔, 𝜐) ⇔ 𝜄𝜛(𝜐, 𝑧) ≤ 𝜎𝜛(𝜔, 𝜐), which is true. hence, (𝜁, 𝑀𝜛,𝑁𝜛, 𝑂𝜛,∗,∘, 𝑠) is an nbms. remark 2.2 let 𝜁 = [0,1] and 𝛼(𝜔, 𝜐) = |𝜔 − 𝜐|𝑠 with 𝑠 ≥ 1 be a b-metric space. consider the above example, we have the graphical views for 𝑀𝜛 in figure 1, 𝑁𝜛 in figure 2 and 𝑂𝜛 in figure 3. 9 int. j. anal. appl. (2023), 21:73 figure 1 shows the graphical behavior of 𝑀𝜛 for 𝑠 = 1, 𝑠 = 2, 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5. figure 2 shows the graphical behavior of 𝑁𝜛 for 𝑠 = 1, 𝑠 = 2, 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5. 10 int. j. anal. appl. (2023), 21:73 figure 3 shows the graphical behavior of 𝑂𝜛 for 𝑠 = 1, 𝑠 = 2, 𝑠 = 3, 𝑠 = 4 and 𝑠 = 5. definition 2.2 let 𝑠 ≥ 1 be a given real number. a function 𝑄: 𝑅 → 𝑅 is said to be an s-non-decreasing if 𝜄 < 𝜎 implies that 𝑄(𝜄) ≤ 𝑄(𝑠𝜎) and 𝑄 is said to be s-non-increasing if 𝜄 < 𝜎 implies that 𝑄(𝜄) ≥ 𝑄(𝑠𝜎). proposition 2.1 in an nbms (𝜁, 𝑀𝑏,𝑁𝑏 , 𝑂𝑏 ,∗,∘, 𝑠), 𝑀(𝜔, 𝜐, . ): [0, +∞) → [0,1] is s-non-decreasing, 𝑁(𝜔, 𝜐, . ): [0, +∞) → [0,1] is s-non-increasing and 𝑂(𝜔, 𝜐, . ): [0, +∞) → [0,1] is s-non-increasing, for all 𝜔, 𝜐 ∈ 𝜁. proof: for 0 < 𝜄 < 𝜎, we get 𝑀(𝜔, 𝜐, 𝑠𝜎) = 𝑀(𝜔, 𝜐, 𝑠(𝜎 − 𝜄 + 𝜄)) ≥ 𝑀(𝜔, 𝜔, 𝜎 − 𝜄) ∗ 𝑀(𝜔, 𝜐, 𝜄) = 1 ∗ 𝑀(𝜔, 𝜐, 𝜄) = 𝑀(𝜔, 𝜐, 𝜄). also 𝑁(𝜔, 𝜐, 𝑠𝜎) = 𝑁(𝜔, 𝜐, 𝑠(𝜄 − 𝜄 + 𝜄)) ≤ 𝑁(𝜔, 𝜔, 𝜎 − 𝜄) ∘ 𝑁(𝜔, 𝜐, 𝜄) = 0 ∘ 𝑁(𝜔, 𝜐, 𝜄) = 𝑁(𝜔, 𝜐, 𝜄) and 𝑂(𝜔, 𝜐, 𝑠𝜎) = 𝑂(𝜔, 𝜐, 𝑠(𝜄 − 𝜄 + 𝜄)) ≤ 𝑂(𝜔, 𝜔, 𝜎 − 𝜄) ∘ 𝑂(𝜔, 𝜐, 𝜄) 11 int. j. anal. appl. (2023), 21:73 = 0 ∘ 𝑂(𝜔, 𝜐, 𝜄) = 𝑂(𝜔, 𝜐, 𝜄). definition 2.3 suppose (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be an nbms. an open ball 𝐵(𝜔, 𝑟, 𝜄) with the center 𝜔 ∈ 𝜁 and radius 𝑟, 0 < 𝑟 < 1, and 𝜄 > 0 is defined as 𝐵(𝜔, 𝑟, 𝜄) = {𝜐 ∈ 𝜁: 𝑀(𝜔, 𝑟, 𝜄) > 1 − 𝑟, 𝑁(𝜔, 𝑟, 𝜄) < 𝑟 and 𝑂(𝜔, 𝑟, 𝜄) < 𝑟}. definition 2.4 let (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be an nbms and a subset 𝐴 of 𝜁. if for each 𝜔 ∈ 𝐴, there is an open ball 𝐵(𝜔, 𝑟, 𝜄) contained in 𝐴, then 𝐴 is called an open in 𝜁. definition 2.5 suppose (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be an nbms. define 𝜏𝑀,𝑁,𝑂 as 𝜏𝑀,𝑁,𝑂 = {𝐴 ∈ 𝑃(𝜁): 𝜔 ∈ 𝐴 iff there exists 𝜄 > 0 and 𝑟 ∈ (0,1): 𝐵(𝜔, 𝑟, 𝜄 ⊂ 𝐴)} then 𝜏𝑀,𝑁,𝑂 is a topology on 𝜁, where 𝑃(𝜁) is the power set of 𝜁. definition 2.6 suppose (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be an nbms. (a) any sequence 𝜔𝑛 in 𝜁 is said to be convergent if there exist 𝜔 ∈ 𝜁 such that lim 𝑛→+∞ 𝑀(𝜔𝑛 , 𝜔, 𝜄) = 1 , lim 𝑛→+∞ 𝑁(𝜔𝑛 , 𝜔, 𝜄) = 0 and lim 𝑛→+∞ 𝑂(𝜔𝑛 , 𝜔, 𝜄) = 0 𝑓𝑜𝑟 𝑎𝑙𝑙𝜄 > 0. a point ω is said to be the limit of the sequence 𝜔𝑛 and it is described as lim 𝑛→+∞ 𝜔𝑛 = 𝜔, or 𝜔𝑛 → 𝜔. (b) any sequence 𝜔𝑛 in (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) is said to be a cauchy sequence if, for every 𝜀 in (0,1), there is 𝑛0 ∈ 𝑁 such that 𝑀(𝜔𝑛 , 𝜔𝑚 , 𝜄) > 1 − 𝜀, 𝑁(𝜔𝑛 , 𝜔𝑚 , 𝜄) < 𝜀 and 𝑂(𝜔𝑛 , 𝜔𝑚 , 𝜄) < 𝜀 for all 𝑚, 𝑛 ≥ 𝑛0 and 𝜄 > 0. (c) 𝜁 is known to be complete if every cauchy sequence in 𝜁 is convergent in 𝜁. 3. main results in this section, we will derive several coincident point and common fp results in the context of nbms. definition 3.1 suppose 𝜁 ≠ ϕ and ∆, 𝜎: 𝜁 → 𝜁 be two mappings on 𝜁. (i) an element 𝜔 ∈ 𝜁 is said to be a c-point of ∆ and 𝜎 if ∆(ω) = 𝜎(ω). (ii) an element 𝜐 ∈ 𝜁 is said to be a c-point of ∆ and 𝜎 if there exists 𝜔 ∈ 𝜁 such that if 𝜐 = ∆(𝜔) = 𝜎(𝜔). (iii) an element 𝑧 ∈ 𝜁 is called a common fp of ∆ and 𝜎 if 𝑧 = ∆(z) = 𝜎(z). definition 3.2 two self-mappings ∆, 𝜎: 𝜁 → 𝜁 are called w-compatible if ∆𝜎(𝜔) = 𝜎∆(𝜔) when ∆(𝜔) = 𝜎(𝜔). theorem 3.1 suppose 𝜁 ≠ 𝜙, 𝑌 ≠ 𝜙, and (𝑌, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be an nbms and ∆, 𝜎: 𝜁 ⟶ 𝑌 be mappings verifying the below circumstances: 12 int. j. anal. appl. (2023), 21:73 (i) 𝜎(𝜁) ⊆ ∆(𝜔); (ii) there is 𝑘, such that 0 ≤ 𝑘 ≤ 1, for all 𝜔, 𝜐 ∈ 𝜁 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≥ 𝑀(∆(𝜔), ∆( 𝜐), 𝜄) 𝑁(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑁 (∆(𝜔), ∆( 𝜐), 𝜄). and 𝑂(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑂(∆(𝜔), ∆( 𝜐), 𝜄). if ∆(𝜁) or 𝜎(𝜁) is complete, then there exists an element 𝑧 ∈ 𝜁 such that ∆(𝜁) =𝜎(𝜁). furthermore, ∆ and 𝜎 have a unique c-point. proof: suppose 𝜔0 ∈ 𝜁. applying (i), we can deduce 𝜔1 ∈ 𝜁 such that ∆(𝜔1) = 𝜎(𝜔0), for 𝑘 = 0, we have 𝑀(𝜎( 𝜔0), 𝜎(𝜔1), 0𝜄) ≥ 𝑀(∆(𝜔0), ∆(𝜔1), 𝜄 ), 𝑁(𝜎(𝜔0), 𝜎(𝜔1), 0𝜄) ≤ 𝑁(∆(𝜔0), ∆(𝜔1), 𝜄 ) and 𝑂(𝜎(𝜔0), 𝜎(𝜔1), 0𝜄) ≤ 𝑂(∆(𝜔0), ∆(𝜔1), 𝜄 ) 𝑀(𝜎(𝜔0), 𝜎(𝜔1), 0𝜄) = 1, 𝑁(𝜎(𝜔0), 𝜎(𝜔1), 0𝜄) = 0 and 𝑂(𝜎(𝜔0), 𝜎(𝜔1), 0𝜄) = 0. that is, 𝜎(𝜔0) = 𝜎(𝜔1) ∆(𝜔1) = 𝜎(𝜔1). hence, 𝜔1 is the c-point of ∆ and 𝜎. for 𝑘 ≠ 0, by induction, we have a sequence {𝜔𝑛 } in 𝜁, such that ∆(𝜔𝑛 ) = 𝜎(𝜔𝑛−1): 𝑀(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄) = 𝑀(𝜎(𝜔𝑛−1), 𝜎(𝜔𝑛 ), 𝜄) ≥ 𝑀(∆(𝜔𝑛−1), ∆(𝜔𝑛 ), 𝜄 𝑘⁄ ) ≥ ⋯ ≥ 𝑀(∆(𝜔0), ∆(𝜔1), 𝜄 𝑘 𝑛⁄ ). clearly, 1 ≥ 𝑀(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄) ≥ 𝑀(∆(𝜔0), ∆(𝜔1), 𝜄 𝑘 𝑛⁄ ) → 1, when 𝑛 → +∞. thus, lim 𝑛→+∞ 𝑁(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄 ) = 𝑁(𝜎(𝜔𝑛−1), 𝜎(𝜔𝑛 ), 𝜄) ≤ 𝑁(∆(𝜔𝑛−1), ∆(𝜔𝑛 ), 𝜄 𝑘⁄ ) ≤ ⋯ ≤ 𝑁(∆(𝜔0), ∆(𝜔1), 𝜄 𝑘 𝑛⁄ ). clearly, 13 int. j. anal. appl. (2023), 21:73 0 ≤ 𝑁(∆(ωn), ∆(ωn+1), 𝜄) ≤ 𝑁(∆(ω0), ∆(ω1), 𝜄 𝑘 𝑛⁄ ) → 0, when 𝑛 → +∞. that is, lim 𝑛→+∞ 𝑁(∆(𝜔𝑛), ∆(𝜔𝑛+1), 𝜄) = 0. also, lim 𝑛→+∞ 𝑂(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄) = 𝑂(𝜎(𝜔𝑛−1), 𝜎(𝜔𝑛 ), 𝜄) ≤ 𝑂(∆(𝜔𝑛−1), ∆(𝜔𝑛 ), 𝜄 𝑘⁄ ) ≤ ⋯ ≤ 𝑂(∆(𝜔0), ∆(𝜔1), 𝜄 𝑘 𝑛⁄ ). clearly, 0 ≤ 𝑂(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄) ≤ 𝑂(∆(𝜔0), ∆(𝜔1), 𝜄 𝑘 𝑛⁄ ) → 0, when 𝑛 → +∞. that is, lim 𝑛→+∞ 𝑂(∆(𝜔𝑛), ∆(𝜔𝑛+1), 𝜄) = 0. let 𝜏𝑛 (𝜄) = 𝑀(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄 ), 𝜇𝑛 (𝜄) = 𝑁(∆(𝜔𝑛 ), ∆(𝜔𝑛+1), 𝜄 ), ℎ𝑛 (𝜄) = 𝑂(∆(𝜔𝑛), ∆(𝜔𝑛+1), 𝜄 ) for all 𝑛 ∈ ℕ ∪ {0}, 𝜄 > 0. to show that ∆(𝜔𝑛) is a cauchy sequence, assume it is not, then there exists 0 < 𝜀 < 1 and two sequences 𝑝(𝜂) and 𝑞(𝜂) such that for every 𝜂 ∈ ℕ ∪ {0}, 𝜄 > 0, 𝑝(𝜂) > 𝑞(𝜂) ≥ 𝜂, 𝑀 (∆(𝜔𝑝(𝜂), 𝜔𝑞 (𝜂), 𝜄 )) ≤ 1 − 𝜀, 𝑁 (∆𝜔𝑝(𝜂), (∆𝜔𝑞 (𝜂), 𝜄)) ≥ 𝜀, 𝑂 (∆𝜔𝑝(𝜂), (∆𝜔𝑞 (𝜂), 𝜄)) ≥ 𝜀. then 𝑀(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)−1), 𝜄) > 1 − 𝜀 𝑀(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄) > 1 − 𝜀, 𝑁(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)−1), 𝜄) < 𝜀 𝑁(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄) < 𝜀, and 𝑂(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)−1), 𝜄) < 𝜀 𝑂(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄) < 𝜀. now, 14 int. j. anal. appl. (2023), 21:73 1 − 𝜀 ≥ 𝑀(∆(𝜔𝑝(𝜂)), ∆(𝜔𝑞(𝜂)), 𝜄) ≥ 𝑀(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑝(𝜂)), 𝜄 2𝑠⁄ ) ∗ 𝑀(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄 2𝑠⁄ ) > 𝜏𝑝(𝜂)−1(𝜄 2𝑠⁄ ) ∗ 1 − 𝜀 𝜀 ≤ 𝑁(∆(𝜔𝑝(𝜂)), ∆(𝜔𝑞(𝜂)), 𝜄) ≤ 𝑁(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑝(𝜂)), 𝜄 2𝑠⁄ ) ∘ 𝑁(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄 2𝑠⁄ ) < 𝜇𝑝(𝜂)−1(𝜄 2𝑠⁄ ) ∘ 𝜀 and 𝜀 ≤ 𝑂(∆(𝜔𝑝(𝜂)), ∆(𝜔𝑞(𝜂)), 𝜄) ≤ 𝑂(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑝(𝜂)), 𝜄 2𝑠⁄ ) ∘ 𝑂(∆(𝜔𝑝(𝜂)−1), ∆(𝜔𝑞(𝜂)), 𝜄 2𝑠⁄ ) < ℎ𝑝(𝜂)−1(𝜄 2𝑠⁄ ) ∘ 𝜀. since, 𝜏𝑝(𝜂)−1(𝜄 2𝑠⁄ ) → 1 𝑎𝑠 𝜂 → +∞, 𝜇𝑝(𝜂)−1(𝜄 2𝑠⁄ ) → 0 as 𝜂 → +∞ and ℎ𝑝(𝜂)−1(𝜄 2𝑠⁄ ) → 0 as 𝜂 → +∞ for every 𝜄, supposing that 𝜂 → +∞, we have 1 − 𝜀 ≥ 𝑀(∆(𝜔𝑝(𝜂)), ∆(𝜔𝑞(𝜂)), 𝜄) < 𝜀. hence, it is a contradiction. that is, ∆(𝜔𝑛 ) is a cauchy sequence in ∆(𝜁). case1: let ∆(𝜁) is complete. then, there exists an element 𝜐 ∈ ∆(𝜁) such that lim 𝑛→+∞ ∆(𝜔𝑛 ) = 𝜐. this shows that there exists 𝑧 ∈ 𝜁 such that 𝜐 = ∆(𝑧). 𝑀(∆(𝑧), 𝜎(𝑧), 𝜄) ≥ 𝑀(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∗ 𝑀(∆(𝜔𝑛 ), 𝜎(𝑧), 𝜄 2𝑠⁄ ) = 𝑀(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∗ 𝑀(𝜎(𝜔𝑛−1), 𝜎(𝑧), 𝜄 2𝑠⁄ ) ≥ 𝑀(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∗ 𝑀(∆(𝜔𝑛−1), ∆(𝑧), 𝜄 2𝑠𝑘⁄ ) ≥ 1 ∗ 1 = 1 as 𝑛 → +∞, 𝑁(∆(𝑧), 𝜎(𝑧), 𝜄) ≤ 𝑁(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∘ 𝑁(∆(𝜔𝑛 ), 𝜎(𝑧), 𝜄 2𝑠⁄ ) = 𝑁(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∘ 𝑁(𝜎(𝜔𝑛−1), 𝜎(𝑧), 𝜄 2𝑠⁄ ) ≤ 𝑁(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∘ 𝑁(∆(𝜔𝑛−1), ∆(𝑧), 𝜄 2𝑠𝑘⁄ ) ≤ 0 ∘ 0 = 0 as 𝑛 → +∞ and 𝑂(∆(𝑧), 𝜎(𝑧), 𝜄) ≤ 𝑂(∆(𝑧), ∆(𝜔𝑛), 𝜄 2𝑠⁄ ) ∘ 𝑂(∆(𝜔𝑛 ), 𝜎(𝑧), 𝜄 2𝑠⁄ ) = 𝑂(∆(𝑧), ∆(𝜔𝑛), 𝜄 2𝑠⁄ ) ∘ 𝑂(𝜎(𝜔𝑛−1), 𝜎(𝑧), 𝜄 2𝑠⁄ ) ≤ 𝑂(∆(𝑧), ∆(𝜔𝑛 ), 𝜄 2𝑠⁄ ) ∘ 𝑂(∆(𝜔𝑛−1), ∆(𝑧), 𝜄 2𝑠𝑘⁄ ) ≤ 0 ∘ 0 = 0 as 𝑛 → +∞. by definition 2.1, it follows that ∆(𝑧) = 𝜎(𝑧). case 2: suppose that 𝜎(𝜁) is complete; then there exists an element 𝜐 ∈ 𝜎(𝜁) such that lim 𝑛→+∞ ∆(𝜔𝑛) = 𝜐. since, 𝜎(𝜁) ∈ ∆(𝜁), so there exists an element 𝑧 ∈ 𝜁 such that 𝜐 = ∆(𝜁). the existence of a coincident point is obvious from case 1. now, we examine the uniqueness of a 15 int. j. anal. appl. (2023), 21:73 coincident point of ∆ and 𝜎. suppose 𝜐1 be another point of coincidence of ∆ and 𝜎. then, 𝜐1 = ∆(𝑧1) = 𝜎(𝑧1) for some 𝑧1 in 𝜁 1 ≥ 𝑀(𝜐, 𝜐1, 𝜄) = 𝑀( 𝜎(𝑧), 𝜎(𝑧1), 𝜄) ≥ 𝑀(∆(𝑧), ∆(𝑧1), 𝜄 𝑘⁄ ) = 𝑀(𝜐, 𝜐1, 𝜄 𝑘⁄ ) ≥ ⋯ ≥ 𝑀(𝜐, 𝜐1, 𝜄 𝑘 𝑛⁄ ), 0 ≤ 𝑁(𝜐, 𝜐1, 𝜄) = 𝑁( 𝜎(𝑧), 𝜎(𝑧1), 𝜄) ≤ 𝑁(∆(𝑧), ∆(𝑧1), 𝜄 𝑘⁄ ) = 𝑁(𝜐, 𝜐1, 𝜄 𝑘⁄ ) ≤ ⋯ ≤ 𝑁(𝜐, 𝜐1, 𝜄 𝑘 𝑛⁄ ) and 0 ≤ 𝑂(𝜐, 𝜐1, 𝜄) = 𝑂( 𝜎(𝑧), 𝜎(𝑧1), 𝜄) ≤ 𝑂(∆(𝑧), ∆(𝑧1), 𝜄 𝑘⁄ ) = 𝑂(𝜐, 𝜐1, 𝜄 𝑘⁄ ) ≤ ⋯ ≤ 𝑂(𝜐, 𝜐1, 𝜄 𝑘 𝑛⁄ ). thus, by definition 2.1, lim 𝑛→+∞ 𝑀(𝜐, 𝜐1, 𝜄 𝑘 𝑛 ⁄ ) = 1, lim 𝑛→+∞ 𝑁(𝜐, 𝜐1, 𝜄 𝑘 𝑛⁄ ) = 0 and lim 𝑛→+∞ 𝑂(𝜐, 𝜐1, 𝜄 𝑘 𝑛 ⁄ ) = 0. it follows that 1 ≥ 𝑀(𝜐, 𝜐1, 𝜄) ≥ 1, 0 ≤ 𝑁(𝜐, 𝜐1, 𝜄) ≤ 0 and 0 ≤ 𝑂(𝜐, 𝜐1, 𝜄) ≤ 0, which implies that 𝜐 = 𝜐1, also by the definition 2.1. lim 𝑛→+∞ 𝑀(𝜐, 𝜐1, 𝜄 𝑘 𝑛 ⁄ ) = 1, lim 𝑛→+∞ 𝑁(𝜐, 𝜐1, 𝜄 𝑘 𝑛 ⁄ ) = 0 and lim 𝑛→+∞ 𝑂(𝜐, 𝜐1, 𝜄 𝑘 𝑛⁄ ) = 0. it follows that 1 ≥ 𝑀(𝜐, 𝜐1, 𝜄) ≥ 1, 0 ≤ 𝑁(𝜐, 𝜐1, 𝜄) ≤ 0 and 0 ≤ 𝑂(𝜐, 𝜐1, 𝜄) ≤ 0. which implies that 𝜐 = 𝜐1. remark 3.1 if ∆ or 𝜎 is a bijective, a unique coincident point must be exist. theorem 3.2 suppose (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) be a complete nbms and ∆, 𝜎: 𝜁 → 𝜁 are verifying the following circumstances: (1) 𝜎(𝜁) ⊆ ∆(𝜁), (2) there exists 𝑘, 0 ≤ k < 1, such that, for all 𝜔, 𝜐 ∈ 𝜁, 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≥ 𝑀(∆(𝜔), ∆(𝜐), 𝜄), 𝑁(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑁(∆(𝜔), ∆(𝜐), 𝜄), 𝑂(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑂(∆(𝜔), ∆(𝜐), 𝜄), (3) ∆ and 𝜎 are w-compatible. then, ∆ and 𝜎 have a unique-common fp in 𝜁. 16 int. j. anal. appl. (2023), 21:73 proof: by utilizing theorem 3.1, there exists a unique coincidence point of ∆ and 𝜎 in 𝜁. therefore, we have, 𝜐 in 𝜁 such that 𝜐 = ∆(𝜎(𝑧)) = ∆(𝜐).let 𝜎 = ∆(𝜐) = 𝜎(𝜐), then 𝜎 is a coincidence point of ∆ and 𝜎, therefore, the coincidence point is unique, this shows that 𝜎 = 𝜐 ⇒ 𝜐 = ∆(𝜐) = 𝜎(𝜐). hence, 𝜐 is a unique common fp of ∆ and 𝜎. corollary 3.1 suppose (𝜁, 𝑀, 𝑁, 𝑂,∗,∘) be a complete nms and ∆, 𝜎: 𝜁 → 𝜁 be mappings verifying the below circumstances: (1) 𝜎(𝜁) ⊆ ∆(𝜁), (2) there exists 𝑘, such that 0 ≤ 𝑘 < 1, for all 𝜔, 𝜐 ∈ 𝜁, 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≥ 𝑀(∆(𝜔), ∆(𝜐), 𝜄), 𝑁(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑁(∆(𝜔), ∆(𝜐), 𝜄) and 𝑂(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑂(∆(𝜔), ∆(𝜐), 𝜄), (3) ∆ and 𝜎 are w-compatible. then, ∆ and 𝜎 have a unique-common fp in 𝜁. proof: by taking 𝑠 = 1 in theorem 3.2, it is obvious. corollary 3.2 suppose (𝜁, 𝑀,∗) be a complete fuzzy b-metric space and ∆, 𝜎: 𝜁 → 𝜁 are mappings verifying the below circumstances: (1) 𝜎(𝜁) ⊆ ∆(𝜁), (2) there exists 𝑘 ∈ [0,1) such that, for all 𝜐 ∈ 𝜁, 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄 ) ≥ 𝑀(∆(𝜔), ∆(𝜐), 𝜄 ), (3) ∆ and 𝜎 are w-compatible. then, ∆ and 𝜎 have a unique-common fp in 𝜁. proof: by taking 𝑁 = 0 = 𝑂 (i.e., 𝑁 and 𝑂 are zero functions) in theorem 3.2, it is obvious. corollary 3.3 suppose (𝜁, 𝑀,∗) be a complete fms and ∆, 𝜎: 𝜁 → 𝜁 are mappings verifying the below circumstances: (1) 𝜎(𝜁) ⊆ ∆(𝜁), (2) there exists 𝑘 ∈ [0,1) such that, for all 𝜐 ∈ 𝜁, 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄 ) ≥ 𝑀(∆(𝜔), ∆(𝜐), 𝜄 ), (3) ∆ and 𝜎 are w-compatible. 17 int. j. anal. appl. (2023), 21:73 then, ∆ and 𝜎 have a unique-common fp in 𝜁. proof: by taking 𝑁 = 0 = 𝑂 (i.e., 𝑁 and 𝑂 are zero functions) and 𝑠 = 1 in theorem 3.2, it is obvious. example 3.1 suppose 𝜁 = [0,1] and ∆: 𝜁 → 𝜁 be a self mapping on 𝜁 defined as ∆(𝜔) = 3𝜔, for all 𝜔 ∈ 𝜁. define 𝑀, 𝑁, 𝑂: 𝜁2 × [0, +∞) → [0,1] by 𝑀(𝜔, 𝜐, 𝜄) = { 𝜄 𝜄 + |𝜔 − 𝜐|2 , if 𝜄 > 0, 0, if 𝜄 = 0, 𝑁(𝜔, 𝜐, 𝜄) = { |𝜔 − 𝜐|2 𝜄 + |𝜔 − 𝜐|2 , if 𝜄 > 0, 1, if 𝜄 = 0, and 𝑂(𝜔, 𝜐, 𝜄) = { |𝜔 − 𝜐|2 𝜄 , if 𝜄 > 0, 1, if 𝜄 = 0. it is clear that (𝜁, 𝑀, 𝑁, 𝑂,∗,∘, 𝑠) is a complete nbms but not a nms, where 𝑎 ∗ 𝑏 = min{𝑎, 𝑏} , 𝑎 ∘ 𝑏 = max{𝑎, 𝑏}, and 𝑓𝑜𝑟 𝑎𝑙𝑙𝑎, 𝑏 ∈ [0,1]. now, define 𝜎: 𝜁 → 𝜁 as 𝜎(𝜔) = 2𝜔, 𝑓𝑜𝑟 𝑎𝑙𝑙𝜔 ∈ 𝜁. it is obvious that 𝜎(𝜁) ⊆ ∆(𝜁) and ∆ and 𝜎 are weakly compatible. then 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) = 𝑘𝜄 𝑘𝜄 + |2𝜔 − 2𝜐|2 = 𝑘𝜄 𝑘𝜄 + 4|𝜔 − 𝜐|2 ≥ 𝜄 𝜄 + 9|𝜔 − 𝜐|2 = 𝑀(∆(𝜔), ∆(𝜐), 𝜄), 𝑁(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) = |2𝜔 − 2𝜐|2 𝑘𝜄 + |2𝜔 − 2𝜐|2 . = 4|2𝜔 − 2𝜐|2 𝑘𝜄 + 4|𝜔 − 𝜐|2 ≤ 9|𝜔 − 𝜐|2 𝜄 + 9|𝜔 − 𝜐|2 = 𝑁(∆(𝜔), ∆(𝜐), 𝜄) and 𝑂(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) = |2𝜔 − 2𝜐|2 𝑘𝜄 = 4|2𝜔 − 2𝜐|2 𝑘𝜄 + ≤ 9|𝜔 − 𝜐|2 𝜄 = 𝑂(∆(𝜔), ∆(𝜐), 𝜄). thus, all the circumstances of theorem 3.2 are fulfilled for 𝑘 = [0, 1 4 ). that is, ∆ and 𝜎 have a unique common fp 0. as we can see that the behavior of contractions in figure 4, figure 5 and figure 6. also, it is easy to see in figure 7 that 0 is unique common fp. 18 int. j. anal. appl. (2023), 21:73 figure 4 shows the graphical behavior of the contraction mapping 𝑀(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≥ 𝑀(∆(𝜔), ∆(𝜐), 𝜄) for 𝑘 = 1 10 and 𝜄 = 1. figure 5 shows the graphical behavior of the contraction mapping 𝑁(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑁(∆(𝜔), ∆(𝜐), 𝜄) for 𝑘 = 1 10 and 𝜄 = 1. 19 int. j. anal. appl. (2023), 21:73 figure 6 shows the graphical behavior of the contraction mapping 𝑂(𝜎(𝜔), 𝜎(𝜐), 𝑘𝜄) ≤ 𝑂(∆(𝜔), ∆(𝜐), 𝜄) for 𝑘 = 1 10 and 𝜄 = 1. figure 7 shows that 0 is a common fp, i.e., 0 = ∆(0) = 𝜎(0). 20 int. j. anal. appl. (2023), 21:73 4. application now, we will establish an application to show the validity of theorem 3.1. theorem 4.1 suppose continuous mappings 𝐹, 𝐺: ℝ × 𝐼 → ℝ and 𝑄: ℝ → ℝ such that 𝐺(𝜔, 𝜎) = 𝐹(𝜔, 𝜎) + 𝑄(𝜔), where, 𝐼 = {𝜎 ∈ ℝ; 𝑎 ≤ 𝜎 ≤ 𝑏, 𝑎, 𝑏 ∈ ℝ}. suppose 𝐶(𝐼) be the collection of all continuous functions defined from 𝐼 into ℝ. assume that, for each 𝜔 ∈ 𝐶(𝐼), there exist 𝜐 ∈ 𝐶(𝐼) such that (𝑄𝜐)(𝜎) = 𝐺(𝜔(𝜎), 𝜎) and {𝑄𝜔: 𝜔 ∈ 𝐶(𝐼)} is complete. if there exist 𝑘 ∈ [0,1] such that 𝑧, for all 𝜔1, 𝜔2 ∈ 𝐶(𝐼) and 𝜎 ∈ 𝐼, then the equation, 𝐹(𝜔, 𝜎) = 0, defines a continuous function 𝜔 in terms of 𝜎. proof: suppose 𝜁 = 𝑌 = 𝐶(𝐼). define 𝑀, 𝑁, 𝑂: 𝜁2 × [0, +∞) → [0,1] as 𝑀𝜔(𝜔, 𝜐, 𝜄) = { 𝜄 𝜄 + max𝜎∈𝐼 |𝜔(𝜎) − 𝜐(𝜎)| , if 𝜄 > 0, 0, if 𝜄 = 0, 𝑁𝜔 (𝜔, 𝜐, 𝜄) = { max𝜎∈𝐼 |𝜔(𝜎) − 𝜐(𝜎)| 𝜄 + max𝜎∈𝐼 |𝜔(𝜎) − 𝜐(𝜎)| , if 𝜄 > 0, 1, if 𝜄 = 0, and 𝑂𝜔 (𝜔, 𝜐, 𝜄) = { max𝜎∈𝐼 |𝜔(𝜎) − 𝜐(𝜎)| 𝜄 , if 𝜄 > 0, 1, if 𝜄 = 0. define a mapping 𝜎: 𝜁 → 𝜁 as follows: 𝜎(𝜔(𝜎)) = 𝐺(𝜔(𝜎), 𝜎). then, by assumption, 𝑄(𝜁) = {𝑄𝜔: 𝜔 ∈ 𝜁} is complete. let 𝜔∗ ∈ 𝜎(𝜁); then, 𝜔∗ = 𝜎𝜔 for 𝜔 ∈ 𝜁 and 𝜔∗(𝜎) = 𝜎𝜔(𝜎) = 𝐺(𝜔(𝜎), 𝜎). by assumptions, there exists 𝜐 ∈ 𝜁 such that 𝜎𝜔(𝜎) = 𝐺(𝜔(𝜎), 𝜎) = 𝑄𝜐(𝜎). hence, 𝜎(𝜁)⊆ 𝑄(𝜁). since |(𝜎 𝜔)(𝜎) − (𝜎 𝜐)(𝜎)| = |𝐺(𝜔(𝜎), 𝜎) − 𝐺(𝜐(𝜎), 𝜎)| ≤ k|(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)| ≤ 𝑘(max𝜎∈𝐼 |((𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎))|). that is, max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| ≤ 𝑘(max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) ⇒ max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| 𝑘𝜄 ≤ (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) 𝜄 ⇒ 𝑘𝜄 max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| ≥ 𝜄 (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) 21 int. j. anal. appl. (2023), 21:73 ⇒ 𝑘𝜄 𝑘𝜄 + max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| ≥ 𝜄 𝜄 + (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) ⇒ 𝑀(𝜎𝜔, 𝜎𝜐, 𝑘𝜄) ≥ 𝑀(𝑄𝜔, 𝑄𝜐, 𝜄). also, max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| 𝑘𝜄 ≤ (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) 𝜄 ⇒ max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| 𝑘𝜄 + max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| ≤ (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) 𝜄 + (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) ⇒ 𝑁(𝜎𝜔, 𝜎𝜐, 𝑘𝜄) ≤ 𝑁(𝑄𝜔, 𝑄𝜐, 𝜄). and max𝜎∈𝐼 |(𝜎𝜔)(𝜎) − (𝜎𝜐)(𝜎)| 𝑘𝜄 ≤ (max𝜎∈𝐼 |(𝑄𝜔)(𝜎) − (𝑄𝜐)(𝜎)|) 𝜄 ⇒ 𝑂(𝜎𝜔, 𝜎𝜐, 𝑘𝜄) ≤ 𝑂(𝑄𝜔, 𝑄𝜐, 𝜄). therefore, all the circumstances of theorem 3.1 are fulfilled to get a continuous function 𝑧: 𝐼 → ℝ such that 𝜎𝑧 = 𝑄𝑧. then, 𝐺(𝑧(𝜎), 𝜎) − 𝑄(𝑧(𝜎)) = 0, where 𝑧 will be a solution of the equation 𝐹(𝑧, 𝜎) = 0. example 4.1 if, we let an implicit form 𝐹(𝜔, 𝜎) = 10𝜔5(𝜎 − 1) + 𝜎, then by assumptions 𝐺(𝜔, 𝜎) = 10𝜔5(𝜎 − 1) + 𝜎 + 90𝜔5 and 𝑄(𝜔(𝜎)) = 90𝜔5 in theorem 4.1, we deduce the explicit representation as 𝜔 = √[5] 𝜎 10⁄ (1 − 𝜎). suppose the implicit equation, 𝜎 + sin(8𝜔5𝜎) − 𝜔5 = 0, in the space 𝐶 ([− 1 9 , 𝜄 1 9 ]). let 𝐹(𝜔, 𝜎) = 𝜎 + sin(8𝜔5𝜎) − 𝜔5, 𝑄(𝜔) = 5𝜔5 − 5, where 𝐹: ℝ × 𝐶 ([− 1 9 , 𝜄 1 9 ]) → ℝ and 𝑄: ℝ → ℝ. let 𝐺(𝜔, 𝜎) = 𝜎 + sin(8𝜔5𝜎) + 4𝜔5 − 5. here, 𝑄(𝜔) = 5𝜔5 − 5 implies that 𝑄(ℝ) = ℝ. now, |𝜎𝜔1 − 𝜎𝜔2| = |𝐺(𝜔1, 𝜎) − 𝐺(𝜔2, 𝜎)| = |𝜎 + sin(8𝜔1 5𝜎) + 4𝜔1 5 − 5 − 𝜎 − sin(8𝜔2 5𝜎) − 4𝜔2 5 + 5| ≤ sin(8𝜔1 5𝜎) − sin(8𝜔2 5𝜎) + 4𝜔1 5 − 4𝜔2 5 ≤ |sin(8𝜔1 5𝜎) − sin(8𝜔2 5𝜎)| + |4𝜔1 5 − 4𝜔2 5| ≤ 8|𝜎|𝜔1 5 − 𝜔2 5|𝜔1 5 − 𝜔2 5| 22 int. j. anal. appl. (2023), 21:73 ≤ 44 45 |5𝜔1 5 − 5 − 𝜔2 5 + 5|. therefore, all the circumstances of theorem 4.1 are fulfilled to apply theorem 3.1, choose an initial guess 𝜔0(𝜎) = 0, 𝜎(𝜔0(𝜎)) = 𝐺(𝜔0(𝜎), 𝜎) = 𝜎 − 5 = 𝑄(𝜔1(𝜎)) = 5𝜔1 5 − 5. this shows that 𝜔1(𝜎) = √[5] 𝜎 5.⁄ 𝜎(𝜔1(𝜎)) = 𝐺(𝜔1(𝜎), 𝜎) = 𝜎 + sin(8𝜔1 5𝜎) + 5𝜔1 5 − 5 = 𝜎 + sin (8 𝜎2 5 ) + 4 ( 𝜎 5 ) − 5 𝑄(𝜔) = 𝜎 + sin (8 𝜎2 5 ) + 4 ( 𝜎 5 ) − 5, 5𝜔2 5(𝜎) = 𝜎 + sin (8 𝜎 2 5 ) + 4 ( 𝜎 5 ) − 5, ⇒ 𝜔(𝜎) = √ 𝜎 + sin (8 𝜎2 5 ) + 4 ( 𝜎 5 ) − 5 5 5 𝜎(𝜔2(𝜎)) = 𝐺(𝑤2(𝜎), 𝜎) = 𝜎 + sin(8𝜔1 5𝜎) + 5𝜔1 5 − 5, 𝑄(𝜔3) = 𝜎 + sin 8 ( 𝜎 + sin8 ( 𝜎2 5 ) + 9 ( 𝜎2 5 ) − 5 5 ) + 4 ( 𝜎 + sin8 ( 𝜎2 5 ) + 9 ( 𝜎2 5 ) − 5 5 ) − 5, ⇒ 𝜔3 = √ 𝜎 + sin 8 (𝜎 + sin8 ( 𝜎2 5 ) + 9 ( 𝜎2 5 ) − 5 5⁄ ) + 4 (𝜎 + sin8 ( 𝜎2 5 ) + 9 ( 𝜎 5 ) − 5) 5⁄ 5 5 approximates the explicit form of 𝐹(𝜔, 𝜎). 5. conclusion in this manuscript, we established the notion of nbms that generalized the notions of fuzzy b-metric space, ifbms and nms. we provided numerous non-trivial examples and their graphical views via computational techniques. also, we derived several coincident points and common fixed-point results for contraction mappings in the context of nbms, as well, we presented a graphical view of defined contractions. at the end, we provided a novel application to support the validity of our main result. contributions: all the authors contributed equally. all authors read and approved the manuscript. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. 23 int. j. anal. appl. (2023), 21:73 references [1] s. banach, sur les oprations dans les ensembles abstraits et leur application aux quations intgrales, fund. math. 3 (1922), 133-181. [2] l. zadeh, fuzzy sets, inform. control, 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65)90241-x. [3] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/s01650114(86)80034-3. [4] s. bhunia, g. ghorai, m.a. kutbi, m. gulzar, m.a. alam, on the algebraic characteristics of fuzzy sub e-groups, j. funct. spaces. 2021 (2021), 5253346. https://doi.org/10.1155/2021/5253346. [5] m. gulzar, d. alghazzawi, m.h. mateen, n. kausar, a certain class of t-intuitionistic fuzzy subgroups, ieee access. 8 (2020), 163260–163268. https://doi.org/10.1109/access.2020.3020366. [6] m. gulzar, m.h. mateen, d. alghazzawi, n. kausar, a novel applications of complex intuitionistic fuzzy sets in group theory, ieee access. 8 (2020), 196075–196085. https://doi.org/10.1109/access.2020.3034626. [7] s. kanwal, a. azam, common fixed points of intuitionistic fuzzy maps for meir-keeler type contractions, adv. fuzzy syst. 2018 (2018), 1989423. https://doi.org/10.1155/2018/1989423. [8] z. deng, fuzzy pseudo-metric spaces, j. math. anal. appl. 86 (1982), 74–95. https://doi.org/10.1016/0022247x(82)90255-4. [9] m.a. erceg, metric spaces in fuzzy set theory, j. math. anal. appl. 69 (1979), 205–230. https://doi.org/10.1016/0022-247x(79)90189-6. [10] o. kaleva, s. seikkala, on fuzzy metric spaces, fuzzy sets syst. 12 (1984), 215–229. https://doi.org/10.1016/01650114(84)90069-1. [11] i. kramosil, j. michalek, fuzzy metric and statistical metric spaces, kybernetica, 11 (1975), 326–334. http://dml.cz/dmlcz/125556. [12] a. george, p. veeramani, on some results in fuzzy metric spaces, fuzzy sets syst. 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7. [13] s.u. rehman, h. aydi, rational fuzzy cone contractions on fuzzy cone metric spaces with an application to fredholm integral equations, j. funct. spaces. 2021 (2021), 5527864. https://doi.org/10.1155/2021/5527864. [14] i.a. bakhtin, the contraction mapping principle in quasi metric spaces, funct. anal. unianowsk gos. ped. inst. 30 (1989), 26–37. [15] n. saleem, u. ishtiaq, l. guran, m.f. bota, on graphical fuzzy metric spaces with application to fractional differential equations, fractal fract. 6 (2022), 238. https://doi.org/10.3390/fractalfract6050238. [16] s. nadaban, fuzzy b-metric spaces, int. j. computers, commun. control, 11 (2016), 273–281. [17] u. ishtiaq, a. hussain, h. al sulami, certain new aspects in fuzzy fixed point theory, aims math. 7 (2022), 8558– 8573. https://doi.org/10.3934/math.2022477. [18] s. kanwal, d. kattan, s. perveen, s. islam, m.s. shagari, existence of fixed points in fuzzy strong b-metric spaces, math. probl. eng. 2022 (2022), 2582192. https://doi.org/10.1155/2022/2582192. [19] j.h. park, intuitionistic fuzzy metric spaces, chaos solitons fractals. 22 (2004), 1039–1046. https://doi.org/10.1016/j.chaos.2004.02.051. [20] g. jungck, commuting mappings and fixed points, amer. math. mon. 83 (1976), 261–263. https://doi.org/10.1080/00029890.1976.11994093. https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1155/2021/5253346 https://doi.org/10.1109/access.2020.3020366 https://doi.org/10.1109/access.2020.3034626 https://doi.org/10.1155/2018/1989423 https://doi.org/10.1016/0022-247x(82)90255-4 https://doi.org/10.1016/0022-247x(82)90255-4 https://doi.org/10.1016/0022-247x(79)90189-6 https://doi.org/10.1016/0165-0114(84)90069-1 https://doi.org/10.1016/0165-0114(84)90069-1 http://dml.cz/dmlcz/125556 https://doi.org/10.1016/0165-0114(94)90162-7 https://doi.org/10.1155/2021/5527864 https://doi.org/10.3390/fractalfract6050238 https://doi.org/10.3934/math.2022477 https://doi.org/10.1155/2022/2582192 https://doi.org/10.1016/j.chaos.2004.02.051 https://doi.org/10.1080/00029890.1976.11994093 24 int. j. anal. appl. (2023), 21:73 [21] g. jungck, compatible mappings and common fixed points, int. j. math. math. sci. 9 (1986), 771–779. https://doi.org/10.1155/s0161171286000935. [22] d. turkoglu, c. alaca, y.j. cho, c. yildiz, common fixed point theorems in intuitionistic fuzzy metric spaces, j. appl. math. comput. 22 (2006), 411–424. https://doi.org/10.1007/bf02896489. [23] g. jungck, b. e. rhoades, fixed-point theorems for occasionally weakly compatible mappings, fixed point theory, 7 (2006), 286–296. [24] m. grabiec, fixed points in fuzzy metric spaces, fuzzy sets syst. 27 (1988), 385–389. https://doi.org/10.1016/01650114(88)90064-4. [25] b. schweizer, a. sklar, statistical metric spaces, pac. j. math. 10 (1960), 14–34. [26] s. kanwal, a. azam, f.a. shami, on coincidence theorem in intuitionistic fuzzy b-metric spaces with application, j. funct. spaces. 2022 (2022), 5616824. https://doi.org/10.1155/2022/5616824. [27] f. smarandache, neutrosophic set, a generalisation of the intuitionistic fuzzy sets, int. j. pure appl. math. 24 (2005), 287–297. [28] m. kirişci, n. şimşek, neutrosophic metric spaces, math sci. 14 (2020), 241–248. https://doi.org/10.1007/s40096020-00335-8. [29] n. şimşek, and m. kirişci, fixed point theorems in neutrosophic metric spaces, sigma j. eng. nat. sci. 10 (2019), 221-230. [30] u. ishtiaq, k. javed, f. uddin, m. de la sen, k. ahmed, m.u. ali, fixed point results in orthogonal neutrosophic metric spaces, complexity. 2021 (2021), 2809657. https://doi.org/10.1155/2021/2809657. [31] p. debnath, a mathematical model using fixed point theorem for two-choice behavior of rhesus monkeys in a noncontingent environment, in: p. debnath, n. konwar, s. radenović (eds.), metric fixed point theory, springer nature singapore, singapore, 2021: pp. 345–353. https://doi.org/10.1007/978-981-16-4896-0_15. [32] v. todorčević, subharmonic behavior and quasiconformal mappings, anal. math. phys. 9 (2019), 1211–1225. https://doi.org/10.1007/s13324-019-00308-8. [33] s. aleksić, z.d. mitrović, s. radenović, picard sequences in b-metric spaces, fixed point theory. 21 (2020), 35–46. https://doi.org/10.24193/fpt-ro.2020.1.03. https://doi.org/10.1155/s0161171286000935 https://doi.org/10.1007/bf02896489 https://doi.org/10.1016/0165-0114(88)90064-4 https://doi.org/10.1016/0165-0114(88)90064-4 https://doi.org/10.1155/2022/5616824 https://doi.org/10.1007/s40096-020-00335-8 https://doi.org/10.1007/s40096-020-00335-8 https://doi.org/10.1155/2021/2809657 https://doi.org/10.1007/978-981-16-4896-0_15 https://doi.org/10.1007/s13324-019-00308-8 https://doi.org/10.24193/fpt-ro.2020.1.03 international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 66-70 http://www.etamaths.com on generalized absolute matrix summability methods hi̇kmet seyhan özarslan∗ abstract. in this paper, we prove a general theorem dealing with absolute matrix summability methods of infinite series. this theorem also includes some new and known results. 1. introduction let ∑ an be a given infinite series with the partial sums (sn). let (pn) be a sequence of positive numbers such that pn = n∑ v=0 pv →∞ as n →∞, (p−i = p−i = 0, i ≥ 1) .(1) the sequence-to-sequence transformation σn = 1 pn n∑ v=0 pvsv(2) defines the sequence (σn) of the ( n̄,pn ) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [5]). the series ∑ an is said to be summable ∣∣n̄,pn∣∣k ,k ≥ 1, if (see [1]) ∞∑ n=1 ( pn pn )k−1 |σn −σn−1| k < ∞.(3) let a = (anv) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. then a defines the sequence-to-sequence transformation, mapping the sequence s = (sn) to as = (an(s)), where an(s) = n∑ v=0 anvsv, n = 0, 1, ...(4) the series ∑ an is said to be summable |a,pn|k ,k ≥ 1, if (see [6]) ∞∑ n=1 ( pn pn )k−1 ∣∣∆̄an(s)∣∣k < ∞,(5) where ∆̄an(s) = an(s) −an−1(s). let (ϕn) be any sequence of positive real numbers. the series ∑ an is summable ϕ−|a,pn|k, k ≥ 1, if ∞∑ n=1 ϕk−1n |∆̄an(s)| k < ∞.(6) if we take ϕn = pn pn , then ϕ−|a,pn|k summability reduces to |a,pn|k summability. if we set ϕn = n for all n, ϕ−|a,pn|k summability is the same as |a|k summability (see [7]). also, if we take ϕn = pn pn 2010 mathematics subject classification. 26d15, 40d15, 40f05, 40g99. key words and phrases. riesz mean; summability factors; absolute matrix summability; infinite series, hölder inequality; minkowski inequality. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 66 on generalized absolute matrix summability methods 67 and anv = pv pn , then we get |n̄,pn|k summability. if we take ϕn = n and anv = pvpn , then we get |r,pn|k summability (see [2]). furthermore, if we take ϕn = n, anv = pv pn and pn = 1 for all values of n, then ϕ−|a,pn|k summability reduces to |c, 1|k summability (see [4]). before stating the main theorem we must first introduce some further notations. given a normal matrix a = (anv), we associate two lower semimatrices ā = (ānv) and â = (ânv) as follows: ānv = n∑ i=v ani, n,v = 0, 1, ...(7) and â00 = ā00 = a00, ânv = ānv − ān−1,v, n = 1, 2, ...(8) it may be noted that ā and â are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. then, we have an (s) = n∑ v=0 anvsv = n∑ v=0 ānvav(9) and ∆̄an (s) = n∑ v=0 ânvav.(10) 2. known result bor [3] has proved the following theorem for ∣∣n̄,pn∣∣k summability method. theorem 1. let (pn) be a sequence of positive numbers such that pn = o(npn) as n →∞.(11) if (xn) is a positive monotonic non-decreasing sequence such that |λm|xm = o(1) as m →∞,(12) m∑ n=1 nxn|∆2λn| = o(1) as m →∞(13) and m∑ n=1 pn pn |tn|k = o(xm) as m →∞,(14) where tn = 1 n + 1 n∑ v=1 vav, then the series ∑ anλn is summable |n̄,pn|k, k ≥ 1. 3. main result the aim of this paper is to generalize theorem 1 to ϕ−|a,pn|k summability. now we shall prove the following theorem. theorem 2. let a = (anv) be a positive normal matrix such that ān0 = 1, n = 0, 1, ...,(15) an−1,v ≥ anv, for n ≥ v + 1,(16) ann = o ( pn pn ) ,(17) |ân,v+1| = o (v |∆vânv|) .(18) 68 özarslan let (xn) be a positive monotonic non-decreasing sequence and ( ϕnpn pn ) be a non-increasing sequence. if conditions (12)-(13) of theorem 1 and m∑ n=1 ϕk−1n ( pn pn )k |tn|k = o(xm) as m →∞,(19) are satisfied, then the series ∑ anλn is summable ϕ−|a,pn|k, k ≥ 1. it should be noted that if we take ϕn = pn pn and anv = pv pn in theorem 2, then we get theorem 1. in this case, condition (19) reduces to condition (14), condition (18) reduces to condition (11). also, the condition “ ( ϕnpn pn ) is a non-increasing sequence” and the conditions (15)-(17) are automatically satisfied. we require the following lemma for the proof of theorem 2. lemma 1 ([3]). under the conditions of theorem 2, we have that nxn|∆λn| = o(1) as n →∞,(20) ∞∑ n=1 xn|∆λn| < ∞.(21) 4. proof of theorem 2 let (in) denotes a-transform of the series ∑ anλn. then, by (9) and (10), we have ∆̄in = n∑ v=0 ânvavλv = n∑ v=1 ânvλv v vav. applying abel’s transformation to this sum, we get that ∆̄in = n−1∑ v=1 ∆v ( ânvλv v ) v∑ r=1 rar + ânnλn n n∑ r=1 rar = n + 1 n annλntn + n−1∑ v=1 v + 1 v ∆v (ânv) λvtv + n−1∑ v=1 v + 1 v ân,v+1∆λvtv + n−1∑ v=1 ân,v+1λv+1 tv v = in,1 + in,2 + in,3 + in,4. to complete the proof of theorem 2, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 ϕk−1n | in,r | k< ∞, for r = 1, 2, 3, 4.(22) first, by using abel’s transformation, we have that m∑ n=1 ϕk−1n |in,1| k = o(1) m∑ n=1 ϕk−1n a k nn|λn| k|tn|k = o(1) m∑ n=1 ϕk−1n ( pn pn )k |λn|k−1|λn||tn|k = o(1) m∑ n=1 ϕk−1n ( pn pn )k |λn||tn|k = o(1) m−1∑ n=1 ∆|λn| n∑ v=1 ϕk−1v ( pv pv )k |tv|k + o(1)|λm| m∑ n=1 ϕk−1n ( pn pn )k |tn|k = o(1) m−1∑ n=1 |∆λn|xn + o(1)|λm|xm = o(1) as m →∞, on generalized absolute matrix summability methods 69 by virtue of the hypotheses of theorem 2 and lemma 1. now, applying hölder’s inequality with indices k and k′, where k > 1 and 1 k + 1 k′ = 1, as in in,1, we have that m+1∑ n=2 ϕk−1n |in,2| k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |∆v(ânv)| |λv| |tv| )k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |∆v(ânv)| |λv| k |tv| k ) × ( n−1∑ v=1 |∆v(ânv)| )k−1 = o(1) m+1∑ n=2 ( ϕnpn pn )k−1 (n−1∑ v=1 |∆v(ânv)| |λv| k |t v |k ) = o(1) m∑ v=1 |λv|k|tv|k m+1∑ n=v+1 ( ϕnpn pn )k−1 |∆v(ânv)| = o(1) m∑ v=1 |λv|k|tv|k ( ϕvpv pv )k−1 m+1∑ n=v+1 |∆v(ânv)| = o(1) m∑ v=1 |λv|k−1|λv||tv|kavv ( ϕvpv pv )k−1 = o(1) m∑ v=1 ϕk−1v ( pv pv )k |λv| |tv| k = o(1) as m →∞, by virtue of the hypotheses of theorem 2 and lemma 1. now, using hölder’s inequality we have that m+1∑ n=2 ϕk−1n |in,3| k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |ân,v+1||∆λv||tv| )k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 v|∆v(ânv)||∆λv||tv| )k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 (v|∆λv|) k |tv|k|∆v(ânv)| ) × ( n−1∑ v=1 |∆v(ânv)| )k−1 = o(1) m+1∑ n=2 ( ϕnpn pn )k−1 (n−1∑ v=1 (v|∆λv|) k |tv|k|∆v(ânv)| ) = o(1) m∑ v=1 (v|∆λv|) k |tv|k m+1∑ n=v+1 ( ϕnpn pn )k−1 |∆v(ânv)| = o(1) m∑ v=1 (v|∆λv|) k−1 (v|∆λv|) |tv|k ( ϕvpv pv )k−1 m+1∑ n=v+1 |∆v(ânv)| = o(1) m∑ v=1 ϕk−1v ( pv pv )k v|∆λv||tv|k = o(1) m−1∑ v=1 ∆(v|∆λv|) v∑ r=1 ϕk−1r ( pr pr )k |tr|k + o(1)m|∆λm| m∑ v=1 ϕk−1v ( pv pv )k |tv|k = o(1) m−1∑ v=1 vxv|∆2λv| + o(1) m−1∑ v=1 |∆λv|xv + o(1)m|∆λm|xm = o(1) as m →∞, 70 özarslan by virtue of the hypotheses of theorem 2 and lemma 1. finally by using (18), as in in,1, we have that m+1∑ n=2 ϕk−1n |in,4| k ≤ m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |ân,v+1||λv+1| |tv| v )k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |∆v(ânv)||λv+1||tv| )k = o(1) m+1∑ n=2 ϕk−1n ( n−1∑ v=1 |∆v(ânv)||λv+1|k|tv|k ) × ( n−1∑ v=1 |∆v(ânv)| )k−1 = o(1) m+1∑ n=2 ( ϕnpn pn )k−1 (n−1∑ v=1 |∆v(ânv)||λv+1|k|tv|k ) = o(1) m∑ v=1 |λv+1|k−1|λv+1||tv|k m+1∑ n=v+1 ( ϕnpn pn )k−1 |∆v(ânv)| = o(1) m∑ v=1 |λv+1||tv|k ( ϕvpv pv )k−1 m+1∑ n=v+1 |∆v(ânv)| = o(1) m∑ v=1 |λv+1||tv|kavv ( ϕvpv pv )k−1 = o(1) m∑ v=1 ϕk−1v ( pv pv )k |λv+1| |tv| k = o(1) as m →∞, by virtue of hypotheses of theorem 2 and lemma 1. this completes the proof of theorem 2. 5. conclusions it should be noted that, if we take ϕn = pn pn , then we get a theorem dealing with |a,pn|k summability. also, if we take anv = pv pn , then we have a result dealing with ϕ−|n̄,pn|k summability. furthermore, if we take anv = pv pn and pn = 1 for all values of n, then we get another result dealing with ϕ−|c, 1|k summability. when we take ϕn = n, anv = pv pn and pn = 1 for all values of n, then we get a result for |c, 1|k summability. finally, if we take k = 1 and anv = pvpn , then we get a result for ∣∣n̄,pn∣∣ summability and in this case the condition “ ( ϕnpn pn ) is a non-increasing sequence” is not needed. references [1] h. bor, on two summability methods, math. proc. cambridge philos. soc. 97 (1985), 147-149. [2] h. bor, on the relative strength of two absolute summability methods, proc. amer. math. soc. 113 (1991), 10091012. [3] h. bor, on absolute summability factors, proc. amer. math. soc. 118 (1993), 71-75. [4] t. m. flett, on an extension of absolute summability and some theorems of littlewood and paley, proc. london math. soc. 7 (1957), 113-141. [5] g. h. hardy, divergent series, oxford university press, oxford, 1949. [6] w. t. sulaiman, inclusion theorems for absolute matrix summability methods of an infinite series. iv, indian j. pure appl. math. 34 (11) (2003), 1547-1557. [7] n. tanovic̆-miller, on strong summability, glas. mat. ser. iii 14 (34) (1979), 87-97. department of mathematics, erciyes university, 38039 kayseri, turkey ∗corresponding author: seyhan@erciyes.edu.tr int. j. anal. appl. (2023), 21:30 generalized result on global existence of weak solutions for parabolic reaction-diffusion systems abdelhamid bennoui1,∗, nabila barrouk2, mounir redjouh1 1department of mathematics and computer science, university center of barika, algeria 2faculty of science and technology, department of mathematics and informatics, mohamed cherif messaadia university, p.o. box 1553, souk ahras 41000, algeria laboratory of mathematics, algeria ∗corresponding author: abdelhamid.bennoui@cu-barika.dz abstract. in this paper, we study global existence of weak solutions for 2 × 2 parabolic reactiondiffusion systems with a full matrix of diffusion coefficients on a bounded domain, such as, we treat the main properties related: the positivity of the solutions and the total mass of the components are preserved with time. moreover, we suppose that the non-linearities have critical growth with respect to the gradient. the technique used is based on compact semigroup methods and some estimates. our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of non-linearities. 1. introduction the study of reaction-diffusion systems (or systems of parabolic partial differential equations) was extensively developed in the literature, see for example in [10,13,26,29,30]. the question on the existence of solution for reaction-diffusion systems have long been a subject of active research like their global existence, their positivity, and some other qualitative properties. in the present paper, we study a mathematical model of reaction-diffusion system  ∂u ∂t −a∆u −b∆v = f (t,x,u,v,∇u,∇v) , in qt , ∂v ∂t −c∆u −d∆v = −f (t,x,u,v,∇u,∇v) , in qt , u = v = 0, or ∂u ∂η = ∂v ∂η = 0, in σt , u (0,x) = u0 (x) , v (0,x) = v0 (x) , in ω, (1.1) received: feb. 9, 2023. 2020 mathematics subject classification. 35k57, 35k40, 35j20, 35j25, 35j57. key words and phrases. semigroups; local solution; global solution; reaction-diffusion systems. https://doi.org/10.28924/2291-8639-21-2023-30 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-30 2 int. j. anal. appl. (2023), 21:30 where ω is an open bounded subset of rn, with smooth boundary ∂ω, qt = ]0,t [×ω, σt = ]0,t [× ∂ω, t > 0, and ∆ denotes the laplacian operator on l1 (ω) with dirichlet or neumann boundary conditions, the constants a, b, c and d are supposed to be positives, a ≤ d, and (b + c)2 ≤ 4ad which reflects the parabolicity of the system and implies at the same time that the matrix of diffusion a = ( a b c d ) , is positive definite, that is the eigenvalues λ1 and λ2 (λ1 < λ2) of its transposed are positives. we consider the problem (1.1) where we suppose the following hypotheses  ( 1 + a−λ1 c )( u − a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ qt , (1.2)   ( 1 + a−λ1 c ) f ( t,x, a−λ1 c v,v, a−λ1 c s,s ) ≤ 0,( 1 + a−λ2 c ) f ( t,x, a−λ2 c v,v, a−λ2 c s,s ) ≥ 0, for all ∀v ≥ 0, ∀s ∈rn, a.e. (t,x) ∈ qt , (1.3)   λ1−λ2 c f (t,x,u,v,∇u,∇v) ≤ l1 ( λ2−λ1 c v + 1 ) ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ qt , (1.4) where l1 is a positive constant. now, we condider the following hypotheses:{ f : ]0,t [ × ω ×r2 ×r2n →r is measurable, f : r2 ×r2n →r is function locally lipschitz continuous. (1.5) − ( 1 + a−λ1 c ) f (t,x,u,v,∇u,∇v) ≤ c1 (∣∣∣−u (t,x) + a−λ1c v (t,x)∣∣∣) × ( f1 (t,x) + ∣∣∣−∇u (t,x) + a−λ1c ∇v (t,x)∣∣∣2 + ∣∣∣∇u (t,x) − a−λ2c ∇v (t,x)∣∣∣α ) , (1.6) ( 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) ≤ c2 (∣∣∣−u (t,x) + a−λ1c v (t,x)∣∣∣ ,∣∣∣u (t,x) − a−λ2c v (t,x)∣∣∣) × ( g1 (t,x) + ∣∣∣−∇u (t,x) + a−λ1c ∇v (t,x)∣∣∣2 + ∣∣∣∇u (t,x) − a−λ2c ∇v (t,x)∣∣∣α ) , (1.7) where c1 : [0,∞) → [0,∞) , c2 : [0,∞)2 → [0,∞) are non-decreasing, f1,g1 ∈ l1 (qt ) and 1 ≤ α < 2. in the diagonal case (i.e. when b = c = 0), alikakos [5] established global existence and l∞−bounds of solutions for positive initial data for f (u,v) = −uvσ, and 1 < σ < n + 2 n . masuda [23] showed that the solutions to this system exist globally for every σ > 1 and converge to a constant vector as t → +∞. int. j. anal. appl. (2023), 21:30 3 haraux and youkana [14] have generalized the method of masuda to non-linearities f (u,v) = −uψ (v) satisfying lim v→+∞ [log (1 + ψ (v))] v = 0. in [24, 25], moumeni and barrouk have obtained a global existence result of solutions for reactiondiffusion systems with a diagonal and triangular matrix of diffusion coefficents. by combining the compact semigroup methods and some l1 estimates, we show that global solutions exist for a large class of the function f . recently, kouachi and youkana [18] have generalized the method of haraux and youkana to the triangular case, i.e. when b = 0. in the same direction, kouachi [17] has proved the global existence of solutions for two-component reaction-diffusion systems with a general full matrix of diffusion coefficients, non-homogeneous boundary conditions and polynomial growth conditions on non-linear terms and he obtained in [18] the global existence of solutions for the same system with homogeneous neumann boundary conditions. rebiai and benachour [28] have treated the case of a general full matrix of diffusion coefficients with homogeneous boundary conditions and non-linearities of exponential growth. this article is a continuation of [3] where c,b 6= 0. in that article the calculations were relatively simple since the system can be regarded as a perturbation of the simple and trivial case where b = c = 0; for which non-negative solutions exist globally in time. in the present paper, to show global existence result for reaction-diffusion system with critical growth with respect to the gradient (m = 2), we truncate the system (1.1) then we give suitable estimates. to that end, we show the convergence of the approximating problem by using a technique introduced by boccardo et al. [7] and dall’aglio and orsina [11]. 2. existence multiplying second equation of (1.1) one time through by a−λ1 c and subtracting first equation of (1.1) and another time by −a−λ2 c and adding first equation of (1.1) we get,  ∂w ∂t −λ1∆w = f (t,x,w,z,∇w,∇z) , in qt , ∂z ∂t −λ2∆z = g (t,x,w,z,∇w,∇z) , in qt , w = z = 0 or ∂w ∂η = ∂z ∂η = 0, in σt , w (0,x) = w0 (x) ≥ 0, z (0,x) = z0 (x) ≥ 0, in ω, (2.1) where w (t,x) = −u (t,x) + a−λ1 c v (t,x) , z (t,x) = u (t,x) − a−λ2 c v (t,x) , (2.2) and   f (t,x,w,z,∇w,∇z) = − ( 1 + a−λ1 c ) f (t,x,u,v,∇u,∇v) , g (t,x,w,z,∇w,∇z) = ( 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) . (2.3) 4 int. j. anal. appl. (2023), 21:30 2.1. assumptions. suppose that the hypotheses (1.2)-(1.7) are satisfied, then the problem (2.1) true the following hypotheses: • the non-linearities f,g have critical growth with respect to |∇w| , |∇z|. with respect to w,z. we assume that the hypotheses (1.2) are satisfied and we obtain,  ( 1 + a−λ1 c )( u − a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀a−λ2 c v ≤ u ≤ a−λ1 c v, a.e. (t,x) ∈ qt , i.e.   − ( 1 + a−λ1 c )( −u + a−λ1 c v ) f (t,x,u,v,∇u,∇v) ≤ 0, ∀−u + a−λ1 c v ≥ 0, ∀u − a−λ2 c v ≥ 0, a.e. (t,x) ∈ qt , by (2.2)-(2.3), then f satisfies the sign condition wf (t,x,w,z,∇w,∇z) ≤ 0, ∀w,z ≥ 0, a.e. (t,x) ∈ q. (2.4) moreover, the following properties hold: w (t,x) = −u (t,x) + a−λ1 c v (t,x) , z (t,x) = u (t,x) − a−λ2 c v (t,x) , (2.5) • by (2.2){ w (t,x) = 0, if u (t,x) = a−λ1 c v (t,x) , and in the case z (t,x) = λ2−λ1 c v (t,x) , z (t,x) = 0, if u (t,x) = a−λ2 c v (t,x) , and in the case w (t,x) = λ2−λ1 c v (t,x) , from (1.3), we get that  − ( 1 + a−λ1 c ) f ( t,x, a−λ1 c v,v, a−λ1 c s,s ) ≥ 0,( 1 + a−λ2 c ) f ( t,x, a−λ2 c v,v, a−λ2 c s,s ) ≥ 0. for all ∀v ≥ 0, ∀s ∈rn, a.e. (t,x) ∈ qt ,{ f (t,x, 0,z, 0,s) ≥ 0, g (t,x,w, 0, r, 0) ≥ 0, ∀w,z ≥ 0, ∀r,s ∈rn, a.e. (t,x) ∈ qt . (2.6) • from (2.3), we obtain that f + g = ( −1 − a−λ1 c + 1 + a−λ2 c ) f (t,x,u,v,∇u,∇v) = λ1 −λ2 c f (t,x,u,v,∇u,∇v) and (2.2) is given w (t,x) + z (t,x) = −u (t,x) + a−λ1 c v (t,x) + u (t,x) − a−λ2 c v (t,x) = λ2 −λ1 c v, int. j. anal. appl. (2023), 21:30 5 then by the hypotheses (1.4), we obtain that f + g ≤ l1 (w + z + 1) , ∀w,z ≥ 0, a.e. (t,x) ∈ qt , (2.7) where l1 is a positive constant. • let us, now by (2.3) and (1.5) introduce for f and g the hypotheses f,g : ]0,t [ × ω ×r2 ×r2n →r are measurable. (2.8) f,g : r2 ×r2n →r are locally lipschitz continuous, (2.9) namely |f (t,x,w,z,p,q) −f (t,x,ŵ, ẑ, p̂, q̂)| + |g (t,x,w,z,p,q) −g (t,x,ŵ, ẑ, p̂, q̂)| ≤ k (r) (|w − ŵ| + |z − ẑ| + ‖p− p̂‖ + ‖q − q̂‖) for a.e. (t,x) ∈ qt and for all 0 ≤ |w| , |ŵ| , |z| , |ẑ| ,‖p‖ ,‖p̂‖ ,‖q‖ ,‖q̂‖≤ r. • by (2.2)-(2.3) and the hypotheses (1.6)-(1.7), we obtain that f (t,x,w,z,∇w,∇z) ≤ c1 (|w|) ( f1 (t,x) + |∇w|2 + |∇z|α ) , (2.10) g (t,x,w,z,∇w,∇z) ≤ c2 (|w| , |z|) ( g1 (t,x) + |∇w|2 + |∇z|α ) , (2.11) where c1 : [0,∞) → [0,∞) , c2 : [0,∞)2 → [0,∞) are non-decreasing function, f1,g1 ∈ l1 (qt ) and 1 ≤ α < 2. let us now point out that if the non-linearities f and g do not depend on the gradient (system (2.1) is semi-linear), the existence of global positive solutions has been obtained by haraux and youkana [14], hollis et al. [15], hollis and morgan [16], and martin and pierre [22]. one can see that in all of these works, the triangular structure, namely hypotheses (2.7) and f ≤ l2 (w + z + 1) , ∀w,z ≥ 0, a.e. (t,x) ∈ qt , (2.12) plays an important role in the study of semi-linear systems (in our case, hypothesis (2.12) is satisfied since by (2.4), f ≤ 0 whenever w,z ≥ 0). indeed, if (2.7) or (2.12) does not hold, pierre and schmitt [27] have proved blow up in finite time of the solutions to some semi-linear reaction-diffusion systems. when f and g are depend on the gradient, boudiba [8] has solved the case where the triangular structure is satisfied and the growth of f and g with respect to |∇w|, |∇z| is sub-quadratic{ ∃ 1 ≤ m < 2, c : [0,∞)2 → [0,∞) non-decreasing such that |f (w,z,∇w,∇z)| + |g (w,z,∇w,∇z)| ≤ c (|w| , |z|) [ 1 + |∇w|m + |∇z|m ] . about the critical growth with respect to the gradient (m = 2), we recall that for the case of a single equation (λ1 = λ2 and f = g), existence results have been proved for the elliptic case in [4,6,20]. the corresponding parabolic equations have also been studied by many authors, see for instance [1,7,11,21]. 6 int. j. anal. appl. (2023), 21:30 3. statement of the result first, we have to clarify in which sense we want to solve problem (2.1). definition 3.1. we say that (w,z) is a solution of (2.1) if  w,z ∈ c ( [0,t ] ; l1 (ω) ) ∩l1 ( 0,t ; w 1,10 (ω) ) f (t,x,w,z,∇w,∇z) and g (t,x,w,z,∇w,∇z) ∈ l1 (qt ) w (t) = sλ1 (t) w0 + ∫ t 0 sλ1 (t − s) f (s, .,w (s) ,z (s) ,∇w (s) ,∇z (s)) ds, ∀t ≥ 0 z (t) = sλ2 (t) z0 + ∫ t 0 sλ2 (t − s) g (s, .,w (s) ,z (s) ,∇w (s) ,∇z (s)) ds, ∀t ≥ 0 (3.1) where sλ1 (t) and sλ2 (t) denote the semigroups in l 1 (ω) generated by−λ1∆ and−λ2∆ with dirichlet or neumann boundary conditions. example 3.1. a typical example where the result of this paper can be applied is  ∂w ∂t −λ1∆w = −wϕ (z) |∇w|2 in qt ∂z ∂t −λ2∆z = wϕ (z) |∇w|2 in qt w = z = 0 or ∂w ∂η = ∂z ∂η = 0 on σt w (0,x) = w0 (x) , z (0,x) = z0 (x) in ω, where ϕ is a bounded function. 3.1. main result. theorem 3.1. assume that (2.4)-(2.11) hold. if w0,z0 ∈ l2 (ω), then there exists a positive global solution (w,z) of system (2.1). moreover, w,z ∈ l2 ( 0,t ; h10 (ω) ) . before giving the proof of the above theorem, let us define the following functions. given a real positive number k, we set tk (s) = max{−k, min (k,s)} , and gk (s) = s −tk (s) . we note that { tk (s) = s for 0 ≤ s ≤ k, tk (s) = k for s > k. 0 ≤ s ≤ k, tk (s) = s and tk (s) = k s > k. 4. proof of theorem 3.1 to prove theorem 3.1, we will use the results which we will present in this section. int. j. anal. appl. (2023), 21:30 7 4.1. preliminaries. theorem 4.1. let ω is an open bounded domain in rn, and x = l1 (ω) ∩h2 (ω). the operator a defined by   d (a) = { u ∈ l1 (ω) ∩h2 (ω) , ∂u ∂η = 0 or u = 0 on ∂ω } au = ∆u , for all u ∈ d (a) is m−dissipative in l1 (ω) ∩h2 (ω). an important result of functional analysis which ensures the local existence of the solution is the following lemma: lemma 4.1. let a be a m−dissipative operator of the dense domain in the banach space x and s (t) a semigroup engendered by a, f a function locally lipchitz. then for any w0 ∈ x it exists t (w0) = tmax such that the problem  w ∈ c ([0,t ] ,d (a)) ∩c1 ([0,t ] ,x) , dw dt −aw = f (s, .,w (s) ,∇w (s)) , w (0) = w0, admits a unique solution w verifying w (t) = s (t) w0 + ∫ t 0 s (t − s) f (s, .,w (s) ,∇w (s)) ds, ∀t ∈ [0,tmax] . 4.1.1. compactness result. in this subsection we will give a compactness result of the operator l defining the solution of the problem (2.1) where the initial value is equal to zero, i.e. l (f ) (t) = w (t) = ∫ t 0 s (t − s) f (s, .,w (s) ,∇w (s)) ds, ∀t ∈ [0,t ] . theorem 4.2. for all t > 0, if the operators s (t) are compact, then l are compact of l1 ([0,t ] ,x) in l1 ([0,t ] ,x) . proof. step 1: we show that s (λ) l : f → s (λ) l (f ) is compact in l1 ([0,t ] ,x), i.e. show that the set {s (λ) l (f ) (t) , ‖f‖1 ≤ 1} is relatively compact in l 1 ([0,t ] ,x). since s (t) is compact then, the application t 7→ s (t) is continuous of ]0, +∞[ in l(x), therefore ∀ε > 0, ∀δ > 0, ∃η > 0, ∀0 ≤ h ≤ η, ∀t ≥ δ, ‖s (t + h) −s (t)‖l(x) ≤ ε we choose λ = δ, we have for 0 ≤ t ≤ t −h s (λ) w (t + h) −s (λ) w (t) = ∫ t+h 0 s (λ + t + h− s) f (s, .,w (s) ,∇w (s)) ds 8 int. j. anal. appl. (2023), 21:30 − ∫ t 0 s (λ + t − s) f (s, .,w (s) ,∇w (s)) ds = ∫ t+h t s (λ + t + h− s) f (s, .,w (s) ,∇w (s)) ds + ∫ t 0 (s (λ + t + h− s) −s (λ + t − s)) f (s, .,w (s) ,∇w (s)) ds, from where ‖s (λ) w (t + h) −s (λ) w (t)‖x ≤ ∫ t+h t ‖f (s, .,w (s) ,∇w (s))‖x ds + ε ∫ t 0 ‖f (s, .,w (s) ,∇w (s))‖x ds. we define z (t) by z (t) = { w (t) if 0 ≤ t ≤ t 0 if not therefore ‖s (λ) z (t + h) −s (λ) z (t)‖1 ≤ (h + εt )‖f (s, .,w (s) ,∇w (s))‖1 , which implies that all {s (λ) z, ‖f‖1 ≤ 1} is equi-integrable, then it is conventional that all {s (λ) l (f ) (t) , ‖f‖1 ≤ 1} is relatively compact in l 1 ([0,t ] ,x), this way s (λ) l is compact. step 2: we show that s (λ) l → l when λ → 0, in l1 ([0,t ] ,x). we have s (λ) w (t) −w (t) = ∫ t 0 s (λ + t − s) f (s, .,w (s) ,∇w (s)) ds − ∫ t 0 s (t − s) f (s, .,w (s) ,∇w (s)) ds. so for t ≥ δ, we have ‖s (λ) w (t) −w (t)‖ ≤ ∫ t δ ‖s (λ + s) −s (s)‖l(x) ‖f (s, .,w (s) ,∇w (s))‖ds +2 ∫ t t−δ ‖f (s, .,w (s) ,∇w (s))‖ds. we choose 0 < λ < η, then ‖s (λ) w (t) −w (t)‖≤ ε ∫ t δ ‖f (s, .,w (s) ,∇w (s))‖ds + 2 ∫ t t−δ ‖f (s, .,w (s) ,∇w (s))‖ds, and for 0 ≤ t < δ, we have ‖s (λ) w (t) −w (t)‖≤ 2 ∫ t 0 ‖f (s, .,w (s) ,∇w (s))‖ds. since f ∈ l1 (0,t,x), from where ‖s (λ) w (t) −w (t)‖≤ (εt + 2δ)‖f (s, .,w (s) ,∇w (s))‖1 . therefore, if λ → 0 then s (λ) w → w into l1 ([0,t ] ,x), where the operator l is a uniform limit with compact linear operator between two banach spaces, then l is compact in l1 ([0,t ] ,x). � int. j. anal. appl. (2023), 21:30 9 remark 4.1. the semigroup s (t) generated by the operator ∆ is compact in l1 (ω). 4.2. approximating scheme. for every function h defined from r+ × ω ×r2 ×r2n into r, we associate ĥ such that ĥ (t,x,w,z,p,q) =   h (t,x,w,z,p,q) if w,z ≥ 0 h (t,x,w, 0,p,q) if w ≥ 0,z ≤ 0 h (t,x, 0,z,p,q) if u ≤ 0,z ≥ 0 h (t,x, 0, 0,p,q) if w,z ≤ 0, and consider the system   ∂w ∂t −λ1∆w = f̂ (t,x,w,z,∇w,∇z) in qt ∂z ∂t −λ2∆z = ĝ (t,x,w,z,∇w,∇z) in qt w = z = 0 or ∂w ∂η = ∂z ∂η = 0, on σt w (0,x) = w0 (x) , z (0,x) = z0 (x) in ω. (4.1) it is obviously seen, by the structure of f̂ and ĝ, that systems (2.1) and (4.1) are equivalent on the set where w,z ≥ 0. consequently, to prove theorem 3.1, we have to show that problem (4.1) has a weak solution which is positive. to this end, we consider the truncated function ψn in c ∞ c (r) such that 0 ≤ ψn ≤ 1 and ψn (r) = { 1 if |r| ≤ n 0 if |r| ≥ n + 1, and the mollification with respect to (t,x) is defined as follows. let ρ ∈ c∞c ( r×rn ) such that suppρ ⊂ b (0, 1) , ∫ ρ = 1, ρ ≥ 0 on r × rn and ρn (y) = nnρ (ny) . one can see that ρn ∈ c∞c ( r×rn ) , suppρn ⊂ b ( 0, 1 n ) , ∫ ρn = 1 and ρn ≥ 0 on r×rn. for all n > 0, we define the functions wn0 and zn0 by wn0 = min{w0,n}∈ c ∞ c (ω) and zn0 = min{z0,n}∈ c ∞ c (ω) it is clear that wn0 and zn0 are non-negative sequences and wn0 → w0, zn0 → z0, in l 2 (ω) , and define for all (t,x,w,z,p,q) in r+ × ω ×r2 ×r2n; fn (t,x,w,z,p,q) = [ψn (|w| + |z| + ‖p‖ + ‖q‖) f (., .,w,z,p,q)] ∗ρn (t,x) gn (t,x,w,z,p,q) = [ψn (|w| + |z| + ‖p‖ + ‖q‖) g (., .,w,z,p,q)] ∗ρn (t,x) 10 int. j. anal. appl. (2023), 21:30 note that these functions enjoy the same properties as f and g, moreover they are hölder continuous with respect to t,x and |fn| , |gn| ≤ mn, where mn is a constant depending only on n (these estimates can be derived from (2.9), the properties of the convolution product, and the fact that ∫ ρn = 1. let us now consider the truncated system  ∂wn ∂t −λ1∆wn = fn (t,x,wn,zn,∇wn,∇zn) in qt ∂zn ∂t −λ2∆zn = gn (t,x,wn,zn,∇wn,∇zn) in qt wn = zn = 0 or ∂wn ∂η = ∂zn ∂η = 0, on σt wn (0,x) = wn0 (x) , zn (0,x) = zn0 (x) in ω. (4.2) 4.2.1. local existence of the solution of problem (4.2). we transform the system (4.2) into a first order system in the banach space x = l1 (ω) ×l1 (ω), we obtain  ∂ωn ∂t = aωn + ψ (t,x,ωn,∇ωn) , t > 0 ωn (0) = ωn0 = (wn0,zn0 ) ∈ x. (4.3) here ωn = col(wn,zn), the operator a is defined as follows a = ( λ1∆ 0 0 λ2∆ ) where d (a) := {ωn = col (wn,zn) ∈ x : col (∆wn, ∆zn) ∈ x} and the function ψ is defined by ψ (t,x,ωn,∇ωn) = col (fn (t,x,ωn,∇ωn) ,gn (t,x,ωn,∇ωn)) with dirichlet (wn = zn = 0) or neumann ( ∂wn ∂η = ∂zn ∂η = 0) boundary conditions. theorem 4.3. there exist tm > 0 and (wn,zn) a local solution of (4.3) for all t ∈ [0,tm] . proof. we know that sλ1 (t) , sλ2 (t) are contraction semigroups and that ψ is locally lipschitz in ωn, then there exists tm > 0 such that (wn,zn) is a local solution of (4.3) on [0,tm] . � it remains to show the positivity of the solutions 4.2.2. positivity of the solution of problem (4.2). the positivity of the solution is preserved with time, which is ensured by 2.6. lemma 4.2. let (wn,zn) be a classical solution of (4.2) and suppose that wn0,zn0 ≥ 0. then wn,zn ≥ 0. int. j. anal. appl. (2023), 21:30 11 proof. let w̄n = e−σtwn and z̄n = e−σtzn σ > 0. then ∂wn ∂t = eσt ( ∂w̄n ∂t + σw̄n ) ∂zn ∂t = eσt ( ∂z̄n ∂t + σz̄n ) consequently by the problem (4.2), we have (w̄n, z̄n) is a solution of the system  ∂w̄n ∂t + σw̄n −λ1∆w̄n = e−σtfn (t,x,w̄n, z̄n,∇w̄n,∇z̄n) in qt ∂z̄n ∂t + σz̄n −λ2∆z̄n = e−σtgn (t,x,w̄n, z̄n,∇w̄n,∇z̄n) in qt w̄n = z̄n = 0 or ∂w̄n ∂η = ∂z̄n ∂η = 0, on σt w̄n (0,x) = wn0 (x) , z̄n (0,x) = zn0 (x) in ω, (4.4) let u0 = (t0,x0) be the minimum of w̄n on qt . we will show that w̄n (u0) ≥ 0 which will imply that w̄n ≥ 0 on qt and then wn ≥ 0 on qt . suppose the contrary, namely w̄n (u0) < 0. by the properties of the minimum, we can ensure that u0 ∈ ]0,t ] × ω and ∂w̄n ∂t (u0) = 0, ∇w̄n (u0) = 0, ∆w̄n (u0) ≥ 0 if 0 < t0 < t ∂w̄n ∂t (u0) ≤ 0, ∇w̄n (u0) = 0, ∆w̄n (u0) ≥ 0 if t0 = t. hence the first equation in (4.4) yields σw̄n (u0) = − ∂w̄n ∂t (u0) + λ1∆w̄n (u0) + e −σt0fn (u0, w̄n (u0) , z̄n (u0) , 0,∇z̄n (u0)) ≥ e−σt0fn (u0, w̄n (u0) , z̄n (u0) , 0,∇z̄n (u0)) . now we use the structure of w̄n (u0) and hypothesis (2.6) to write fn (u0, w̄n (u0) , z̄n (u0) , 0,∇z̄n (u0)) = fn (u0, 0, z̄n (u0) , 0,∇z̄n (u0)) ≥ 0. this implies that w̄n (u0) ≥ 0 which is impossible by the hypotheses. arguing in the same way for the second component z̄n, we obtain the positivity of (wn,zn). � 4.2.3. global existence of the solution of problem (4.2). the total mass of the components w,z is controlled with time, which is ensured by the following lemma. lemma 4.3. there exists a constant m depending on ‖w0‖l1(ω) , ‖z0‖l1(ω) , l1, t and |ω| such that ‖wn (t) + zn (t)‖l1(ω) ≤ m, ∀t ∈ [0,t ] . (4.5) proof. of the first and second equation of (4.2) with: ∂ ∂t (wn + zn) − ∆ (λ1wn + λ2zn) = fn + gn. 12 int. j. anal. appl. (2023), 21:30 the hypothesis (2.7) allowed the following estimate ∂ ∂t (wn + zn) − ∆ (λ1wn + λ2zn) ≤ l1 (wn + zn + 1) . let us integrate on ω and apply the formula of green, then∫ ω ∆wn = 0, and ∫ ω ∆zn = 0, (4.6) we find ∫ ω ∂ ∂t (wn + zn) ≤ l1 ∫ ω (wn + zn + 1) . so ∂ ∂t ∫ ω (wn + zn) dx∫ ω (wn + zn + 1) dx ≤ l1, integrating this inequality on [0,t] , ∀t ∈ ]0,t ] yields ln ∫ ω (wn + zn + 1) dx ∣∣∣∣t 0 ≤ l1t, thus ln ∫ ω (wn (t) + zn (t) + 1) dx∫ ω (wn0 + zn0 + 1) dx ≤ l1t, which implies ∫ ω (wn (t) + zn (t) + 1) dx∫ ω (wn0 + zn0 + 1) dx ≤ exp (l1t) then we have ∫ ω (wn (t) + zn (t) + 1) dx ≤ exp (l1t) ∫ ω (wn0 + zn0 + 1) dx also ∫ ω (wn + zn) (t) dx ≤ ∫ ω (wn (t) + zn (t) + 1) dx ≤ exp (l1t) ∫ ω (wn0 + zn0 + 1) dx = exp (l1t) [∫ ω (wn0 + zn0 ) dx + |ω| ] ≤ exp (l1t ) [∫ ω (w0 + z0) dx + |ω| ] as if wn0 ≤ w0, zn0 ≤ z0 ≤ exp (l1t ) [ ‖w0‖l1(ω) + ‖z0‖l1(ω) + |ω| ] . this ends the proof of the lemma. � we can conclude from this estimate that the solution (wn,zn) given by the theorem 4.3 is a global solution. int. j. anal. appl. (2023), 21:30 13 lemma 4.4. there exists a constant r1 depending on t, ‖w0‖l1(ω) , ‖z0‖l1(ω) , l1, l2 and |ω| such that ∫ qt |fn (t,x,wn,zn,∇wn,∇zn)| + |gn (t,x,wn,zn,∇wn,∇zn)| ≤ r1 proof. considering the equations satisfied by wn and zn, we can write −fn = − ∂wn ∂t + λ1∆wn, and −gn = − ∂zn ∂t + λ2∆zn. integrating on qt and using (4.6), the positivity of the solutions yield − ∫ qt fn ≤ ∫ ω wn0. hence by hypothesis (2.4) ∫ qt |fn| = − ∫ qt fn ≤ ∫ ω w0. (4.7) similarly, we get − ∫ qt gn ≤ ∫ ω z0. (4.8) integrating on qt and by hypothesis (2.7) we get∫ qt gn ≤− ∫ qt fn + ∫ qt l1 (wn + zn + 1) moreover, by (4.5) and (4.7) we have∫ qt gn ≤ l1t (m + |ω|) + ∫ ω w0. (4.9) by (4.8) and (4.9) we conclude that∫ qt |gn| ≤ l1t (m + |ω|) + ∫ ω w0. (4.10) by (4.7) and (4.10) we get∫ qt |fn (t,x,wn,zn,∇wn,∇zn)| + |gn (t,x,wn,zn,∇wn,∇zn)| ≤ r1 let us put: r1 = l1t (m + |ω|) + 2‖w0‖l1(ω) � lemma 4.5. (i) there exists a constant r2 depending on λ1, ‖w0‖l2(ω) such that∫ qt |∇wn|2 ≤ r2, ∫ qt |∇tk (wn)| 2 ≤ r2. (ii) there exists a constant r3 depending on λ1, λ2, l1, ‖w0‖l2(ω) , ‖z0‖l2(ω) , |ω| such that∫ qt |∇zn|2 ≤ r3, ∫ qt |∇tk (zn)| 2 ≤ r3. 14 int. j. anal. appl. (2023), 21:30 (iii) there exists a constant r4 depending on λ1, λ2, t, ‖w0‖l2(ω) , ‖z0‖l2(ω) , l1, |ω| such that∫ qt (2wn + zn) (|fn (t,x,wn,zn,∇wn,∇zn)| + |gn (t,x,wn,zn,∇wn,∇zn)|) ≤ r4. proof. (i) we multiply the first equation in the truncated problem by wn and we integrate on qt . we obtain ∫ qt wn ∂wn ∂t −λ1 ∫ qt wn∆wn = ∫ qt fnwn. since, by hypothesis (2.4), wnfn ≤ 0, we have∫ qt |∇wn|2 ≤ 1 λ1 ∫ ω (wn0 ) 2 ≤ 1 λ1 ∫ ω (w0) 2 . then ∫ qt |∇wn|2 ≤ r2, where r2 ≥ 1 λ1 ‖w0‖2l2(ω) . we have ∫ qt |∇wn|2 = ∫ [wn<k] |∇wn|2 + ∫ [wn≥k] |∇wn|2 ≤ r2 then, since ∫ [wn≥k] |∇wn| 2 ≥ 0 ∫ qt |∇tk (wn)| 2 ≤ r2 (4.11) (ii) of the first and second equation of (4.2) we obtain ∂ ∂t (wn + zn) − ∆ (λ1wn + λ2zn) = fn + gn. i.e ∂ (wn + zn) ∂t − ∆ (λ1wn + λ2zn) −λ2∆wn + λ2∆wn = fn + gn. we use hypothesis (2.7). we get ∂ (wn + zn) ∂t −λ2∆ (wn + zn) + (λ2 −λ1) ∆wn ≤ l1 (wn + zn) + l1. (4.12) multiply (4.12) by exp (−l1t). we obtain exp (−l1t) ∂ (wn + zn) ∂t −l1 exp (−l1t) (wn + zn) −λ2 exp (−l1t) ∆ (wn + zn) + (λ2 −λ1) exp (−l1t) ∆wn ≤ l1 exp (−l1t) . set xn = exp (−l1t) (wn + zn) we get ∂xn ∂t −λ2∆xn + (λ2 −λ1) exp (−l1t) ∆wn ≤ l1 exp (−l1t) ≤ l1. now, multiply by xn and integrate on qt and by simple use of green’s formula we have∫ qt xn ∂xn ∂t + λ2 ∫ qt |∇xn|2 + (λ1 −λ2) ∫ qt exp (−2l1t)∇wn∇(wn + zn) ≤ l1 ∫ qt xn. hence, by lemma 4.3 λ2 ∫ qt |∇xn|2 + (λ1 −λ2) ∫ qt exp (−2l1t)∇wn∇(wn + zn) ≤ 1 2 ∫ ω x2n (0) + l1mt. int. j. anal. appl. (2023), 21:30 15 using young’s inequality |ab| ≤ a p p + b q q (where a = exp (−l1t) |∇wn| , b = |∇xn| = exp (−l1t) |∇(wn + zn)| , p = q = 2), we have λ2 ∫ qt |∇xn|2 ≤ λ2 −λ1 2 ∫ qt |∇xn|2 + λ2 −λ1 2 ∫ qt exp (−2l1t) |∇wn|2 + 1 2 ∫ ω (wn + zn) 2 (0) + l1mt. then, by (i) (λ2 + λ1) ∫ qt |∇xn|2 ≤ (λ2 −λ1) r2 + ∫ ω (w0 + z0) 2 + 2l1mt. now, using young’s inequality another time and the fact that exp (−2l1t) ≥ exp (−2l1t ) , for all t ∈ [0,t ] , we end the proof of (ii). similarly, by the same method of proof 4.11 given∫ qt |∇tk (zn)| 2 ≤ r3 (4.13) (iii) set rn = l1 + l1 (wn + zn) −fn −gn ≥ 0, (4.14) by hypothesis (2.7). combining the equations of system (4.2), we have ∂ (2wn + zn) ∂t − ∆ (2λ1wn + λ2zn) + |fn| + rn = l1 + l1 (wn + zn) multiplying by (2wn + zn) and integrating on qt and by simple use of green’s formula we have 1 2 ∫ ω (2wn + zn) 2 (t ) + ∫ qt ∇(2λ1wn + λ2zn)∇(2wn + zn) + ∫ qt (2wn + zn) (|fn| + rn) = i + j, where i = l1 ∫ qt (2wn + zn) + l1 ∫ qt (2wn + zn) (wn + zn) and j = 1 2 ∫ ω (2wn + zn) 2 (0) we have by lemma 4.3 l1 ∫ qt (2wn + zn) + l1 ∫ qt (2wn + zn) (wn + zn) ≤ 2l1 ∫ qt (wn + zn) + 2l1 ∫ qt (wn + zn) 2 ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 , then i ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 (4.15) 16 int. j. anal. appl. (2023), 21:30 and j ≤ 1 2 ∫ ω (2w0 + z0) 2 (4.16) since ∫ qt |∇wn|2 and ∫ qt |∇zn|2 are uniformly bounded with respect to n by (i) and (ii), then ∫ qt |wn|2 ,∫ qt |zn|2 are bounded too. now, we investigate the second term of the inequality∫ qt ∇(2λ1wn + λ2zn)∇(2wn + zn) = 4λ1 ∫ qt |∇wn|2 + λ2 ∫ qt |∇zn|2 + 2 (λ1 + λ2) ∫ qt ∇wn∇zn hence ∫ qt (2wn + zn) (|fn| + rn) = i + j − 1 2 ∫ ω (2wn + zn) 2 (t ) − ∫ qt ∇(2λ1wn + λ2zn)∇(2wn + zn), = i + j − 1 2 ∫ ω (2wn + zn) 2 (t ) − 4λ1 ∫ qt |∇wn|2 −λ2 ∫ qt |∇zn|2 − 2 (λ1 + λ2) ∫ qt ∇wn∇zn using (4.15) and (4.16) yields∫ qt (2wn + zn) (|fn| + rn) ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 + 1 2 ∫ ω (2w0 + z0) 2 − 2 (λ1 + λ2) ∫ qt ∇wn∇zn, i.e. ∫ qt (2wn + zn) (|fn| + rn) ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 + 1 2 ∫ ω (2w0 + z0) 2 + 2 (λ1 + λ2) ∫ qt |∇wn∇zn| using young’s inequality (where a = |∇wn| , b = |∇zn| , p = q = 2) we conclude that∫ qt (2wn + zn) (|fn| + rn) ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 + 1 2 ∫ ω (2w0 + z0) 2 + (λ1 + λ2) [∫ qt |∇wn|2 + ∫ qt |∇zn|2 ] and by (i), (ii) we obtain∫ qt (2wn + zn) (|fn| + rn) ≤ 2l1mt + 2l1 ∫ qt (wn + zn) 2 + 1 2 ∫ ω (2w0 + z0) 2 + (λ1 + λ2) (r1 + r2) (4.17) by (4.14) we have gn = l1 + l1 (wn + zn) −fn −rn. int. j. anal. appl. (2023), 21:30 17 then |fn| + |gn| ≤ rn + 2 |fn| + l1 + l1 (wn + zn) now, multiply by (2wn + zn) and integrate on qt using (4.17) yields∫ qt (2wn + zn) (|fn| + |gn|) ≤ ∫ qt (2wn + zn) (rn + 2 |fn| + l1 + l1 (wn + zn)) = ∫ qt (2wn + zn) (rn + 2 |fn|) + l1 ∫ qt (2wn + zn) + l1 ∫ qt (2wn + zn) (wn + zn) ≤ 2 ∫ qt (2wn + zn) (rn + |fn|) + l1 ∫ qt (2wn + zn) + 2l1 ∫ qt (wn + zn) (wn + zn) ≤ 4l1mt + 4l1 ∫ qt (wn + zn) 2 + ∫ ω (2w0 + z0) 2 + 2 (λ1 + λ2) (r1 + r2) +2l1mt + 2l1 ∫ qt (wn + zn) 2 = 6l1mt + 6l1 ∫ qt (wn + zn) 2 + ∫ ω (2w0 + z0) 2 + 2 (λ1 + λ2) (r1 + r2) since ∫ qt (wn + zn) 2 is uniformly bounded with respect to n, we get the result where r4 ≥ ∫ ω (2w0 + z0) 2 + 2 (λ1 + λ2) (r1 + r2) + 6l1mt + 6l1 ∫ qt (wn + zn) 2 . � remark 4.2. the estimates (4.11) and (4.13) enable us to study the convergence of the truncated problem. 4.3. convergence. our objective is to show that (wn,zn) converges to some (w,z) solution of the problem (3.1). the sequences wn0 and zn0 are uniformly bounded in l 1 (ω) (since they converge in l2 (ω) , and by lemma 4.4, the non-linearities fn and gn are uniformly bounded in l1 (qt ) . we define the application l by l : (w0,h) 7→ sd (t) w0 + ∫ t 0 sd (t − s) h (s) ds where sd (t) is the contraction semigroup generated by the operator d∆. according to the previous theorem 4.2 and as sd (t) is compact, then the application l is the addition of two compact applications in l1 (qt ) , which shows that l is also compact from l1 (qt ) ×l1 (qt ) in l1 (qt ). then the applications (wn0,fn) → wn, and (zn0,gn) → zn are compact from l1 (ω) ×l1 (qt ) into l1 ( 0,t ; w 1,10 (ω) ) 18 int. j. anal. appl. (2023), 21:30 therefore, we can extract a subsequence, still denoted by (wn,zn) , such that (wn,zn) → (w,z) in l1 ( 0,t ; w 1,10 (ω) ) (wn,zn) → (w,z) a.e. in qt (∇wn,∇zn) → (∇w,∇z) a.e.in qt . since fn and gn are continuous, we have fn (t,x,wn,zn,∇wn,∇zn) → f (t,x,w,z,∇w,∇z) a.e. in qt gn (t,x,wn,zn,∇wn,∇zn) → g (t,x,w,z,∇w,∇z) a.e. in qt this is not sufficient to ensure that (w,z) is a solution of (3.1). in fact, we have to prove that the previous convergences are in l1 (qt ) . in view of the vitali theorem, to show that fn (t,x,wn,zn,∇wn,∇zn) (respectively gn (t,x,wn,zn,∇wn,∇zn)) converges to f (t,x,w,z,∇w,∇z) (respectively to g (t,x,w,z,∇w,∇z)) in l1 (qt ) , is equivalent to proving that (fn (t,x,wn,zn,∇wn,∇zn))n and (gn (t,x,wn,zn,∇wn,∇zn))n are equi-integrable in l 1 (qt ). lemma 4.6. (fn (t,x,wn,zn,∇wn,∇zn))n∈n and (gn (t,x,wn,zn,∇wn,∇zn))n∈n are equi-integrable in l1 (qt ). the proof of this lemma requires the following result based on some properties of two timeregularizations denoted by wγ and wσ(γ,σ > 0) which we define for a function w ∈ l2 ( 0,t ; h10 (ω) ) such that w (0) = w0 ∈ l2 (ω). in the following we will denote by ω (ε) a quantity that tends to zero as ε tends to zero, and ωσ (ε) a quantity that tends to zero for every fixed σ as ε tends to zero. lemma 4.7. let (wn)n be a sequence in l 2 ( 0,t ; h10 (ω) ) ∩c([0,t ]) such that wn (0) = wn0 ∈ l 2 (ω) and ∂wn ∂t = ρ1,n + ρ2,n with ρ1,n ∈ l2 ( 0,t ; h−1 (ω) ) and ρ2,n ∈ l1 (qt ) . moreover assume that wn converges to w in l2 (qt ) , and wn0 converges to w (0) in l 2 (ω) . let ψ be a function in c1([0,t ]) such that ψ ≥ 0, ψ′ ≤ 0, ψ(t ) = 0. let ϕ be a lipschitz increasing function in c0(r) such that ϕ (0) = 0. then for all k,γ > 0, 〈 ρ1,n, ψϕ ( tk (wn) −tk (wm)γ )〉 + ∫ qt ρ2,nψϕ ( tk (wn) −tk (wm)γ ) ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ∫ ω ψ (0) φ ( tk (w) −tk (w)γ ) (0) dx − ∫ ω gk (w) (0) ψ (0) ϕ ( tk (w) −tk (w)γ ) (0) dx where φ(t) = ∫ t 0 ϕ(s)ds and gk(s) = s −tk(s). proof. see n. alaa and i. mounir [3]. � int. j. anal. appl. (2023), 21:30 19 proof of lemma 4.6. let k be a measurable subset of qt . we have∫ k |fn (t,x,wn,zn,∇wn,∇zn)| = ∫ k∩[wn>k] |fn| + ∫ k∩[wn≤k,zn>k] |fn| + ∫ k∩[wn≤k,zn≤k] |fn| = i1 + i2 + i3 using lemma 4.5, we obtain for k large enough i1≤ r4 k ≤ ε 3 i2≤ r4 k ≤ ε 3 now, using hypothesis (2.10), we write i3 ≤ ∫ k∩[wn≤k,zn≤k] c1 (|wn|) [ f1 (t,x) + |∇wn|2 + |∇zn|α ] then i3 ≤ c1(k) [∫ k f1 (t,x) + ∫ k∩[wn≤k,zn≤k] |∇wn|2 + ∫ k∩[wn≤k,zn≤k] |∇zn|α ] the third integral can be controlled by using hölder’s inequality for α < 2∫ k∩[wn≤k,zn≤k] |∇zn|α ≤ [∫ k |∇zn|2 α 2 |k| 2−α 2 ≤ r α 2 3 |k| 2−α 2 ] , where in the last inequality we used lemma 4.5. therefore i3 ≤ c1(k) [∫ k f1 (t,x) + r α 2 3 |k| 2−α 2 + ∫ k |∇tk (wn)| 2 ] similarly by hypothesis (2.11), we get∫ k |gn| ≤ 1 k ∫ qt wn |gn| + 1 k ∫ qt zn |gn| + ∫ k∩[wn≤k,zn≤k] |gn| ≤ ε 3 + ε 3 + c2(k,k) [∫ k g1 (t,x) + r α 2 3 |k| 2−α 2 + ∫ k |∇tk (wn)| 2 ] . for the remaining term, we must prove that ( |∇tk (wn)| 2 ) n is equi-integrable in l1 (qt ) . to do this we will show that tk (wn) converges to tk (w) in l2 ( 0,t ; h10 (ω) ) ; more precisely we will show that lim n→∞ ∫ qt |∇tk (wn) −∇tk (w)| 2 = 0 let k and γ be positive real numbers, let m ∈ n, and choose ψ a test function as in lemma 4.7, define ϕ by ϕ(s) = s exp ( βs2 ) , with β to be fixed later. we will use a technique introduced by boccardo et al. [7], we will multiply the first equation in the truncated problem (4.2) by the function test ψϕ ( tk (wn) −tk (wm)γ ) , then we will integrate on qt . finally we will use lemma 4.7 to get the result. 20 int. j. anal. appl. (2023), 21:30 since ∂wn ∂t = ρ1,n +ρ2,n, where ρ1,n = λ1∆wn ∈ l2 ( 0,t ; h−1 (ω) ) and ρ2,n = fn ∈ l1 (qt ) , we have by lemma 4.7 ∫ qt ∂wn ∂t ψϕ ( tk (wn) −tk (wm)γ ) ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) − ∫ ω ψ (0) φ ( tk (w) −tk (w)γ ) dx − ∫ ω gk (w) (0) ψ (0) ϕ ( tk (w) −tk (w)γ ) (0) dx hence λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (wm)γ ) ∇ ( tk (wn) −tk (wm)γ ) − ∫ qt fnψϕ ( tk (wn) −tk (wm)γ ) ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ∫ ω ψ (0) φ ( tk (w) −tk (w)γ ) + ∫ ω gk (w) (0) ψ (0) ϕ ( tk (w) −tk (w)γ ) (0) ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) , since tk (w)γ → tk (w) strongly in l 2 ( 0,t ; h10 (ω) ) . we have i = λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (wm)γ ) ∇ ( tk (wn) −tk (wm)γ ) j = − ∫ qt fnψϕ ( tk (wn) −tk (wm)γ ) . the term i can be written as i = λ1 ∫ qt ∇tk (wn) ψϕ′ ( tk (wn) −tk (wm)γ ) ∇ ( tk (wn) −tk (wm)γ ) +λ1 ∫ [wn≥k] ∇wnψϕ′ ( tk (wn) −tk (wm)γ ) ∇ ( tk (wn) −tk (wm)γ ) = i1 + i2 for i2, we have i2 = −λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (wm)γ ) ∇ ( tk (wm)γ ) χ[wn≥k] = ωγ,n ( 1 m ) −λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (w)γ ) ∇ ( tk (w)γ ) χ[wn≥k] = ωγ,n ( 1 m ) −λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (w)γ ) χ[wn≥k]x[χ≥k] −λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (w)γ ) ∇ ( tk (w)γ ) χ[wn≥k]χ[w<k] = ωγ,n ( 1 m ) + i2.1 + i2.2. int. j. anal. appl. (2023), 21:30 21 for i2.1, we have by hölder’s inequality |i2.1| ≤ λ1 ∥∥∇wnϕ′(tk (wn) −tk (w)γ)∥∥l2(qt ) ∥∥∇(tk (w)γ)χ[w≥k]∥∥l2(qt ) using the fact that ϕ′ ( tk (wn) −tk (w)γ ) ≤ ϕ′(2k), and lemma 4.5, we obtain |i2.1| ≤ λ1c ∥∥∇(tk (w)γ)χ[w≥k]∥∥l2(qt ) = ω ( 1 γ ) since tk (w)γ → tk (w) in l 2 ( 0,t ; h10 (ω) ) , and ∇tk (w) χ[w≥k] = 0 a.e. in qt . now we study the term i2.2 i2.2 = −λ1 ∫ qt ∇wnψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (w)γ ) χ[wn≥k]χ[w<k] = ω γ ( 1 n ) since χ[un≥k]χ[u<k] → 0 a.e. in qt . thus i2 ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) we investigate i1 i1 = ω γ,n ( 1 m ) + λ1 ∫ qt ∇tk (wn) ψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (wn) −tk (w)γ ) = ωγ,n ( 1 m ) + λ1 ∫ qt ∇(tk (wn) −tk (w)) ψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (wn) −tk (w)γ ) +λ1 ∫ qt ∇tk (w) ψϕ′ ( tk (wn) −tk (w)γ ) ∇ ( tk (wn) −tk (w)γ ) = ωγ,n ( 1 m ) + ωγ ( 1 n ) + λ1 ∫ qt ∇tk (w) ψϕ′ ( tk (w) −tk (w)γ ) ×∇ ( tk (w) −tk (w)γ ) +λ1 ∫ qt ∇(tk (wn) −tk (w)) ψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (wn) −tk (w)γ ) = ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) +λ1 ∫ qt |∇(tk (wn) −tk (w))| 2 ψϕ′ ( tk (wn) −tk (w)γ ) +λ1 ∫ qt ∇(tk (wn) −tk (w)) ψϕ′ ( tk (wn) −tk (w)γ ) ×∇ ( tk (w) −tk (w)γ ) = ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) +λ ∫ qt |∇(tk (wn) −tk (w))| 2 ψϕ′ ( tk (wn) −tk (w)γ ) 22 int. j. anal. appl. (2023), 21:30 hence i ≥ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) +λ1 ∫ qt |∇(tk (wn) −tk (w))| 2 ψϕ′ ( tk (wn) −tk (w)γ ) for j, we have j = ωγ,n ( 1 m ) − ∫ qt fnψϕ ( tk (wn) −tk (w)γ ) = ωγ,n ( 1 m ) − ∫ [wn>k] fnψϕ ( tk (wn) −tk (w)γ ) − ∫ [wn≤k] fnψϕ ( tk (wn) −tk (w)γ ) then j ≥ ωγ,n ( 1 m ) − ∫ [wn≤k] fnψϕ ( tk (wn) −tk (w)γ ) since ϕ ( tk (wn) −tk (w)γ ) ≥ 0 on [wn > k] , ψ ≥ 0 and −fn ≥ 0 by hypotheses (2.4). on the other hand ∣∣∣∫[wn≤k] fnψϕ(tk (wn) −tk (w)γ)∣∣∣ ≤ c1 (k) ∫ [wn≤k] f1 (t,x) ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ +c1 (k) ∫ [wn≤k] |∇zn| α ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ +c1 (k) ∫ [wn≤k] |∇tk (wn)| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ we set j1 = c1(k) ∫ [wn≤k] f1 (t,x) ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) since α < 2, we have j2 = c1(k) ∫ [wn≤k] |∇zn|α ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) and j3 = c1(k) ∫ [wn≤k] |∇tk (wn)| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ = c1(k) ∫ [wn≤k] |∇(tk (wn) −tk (w))| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ +2c1(k) ∫ [wn≤k] ∇tk (wn)∇tk (w) ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ −c1(k) ∫ [wn≤k] |∇tk (w)| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ = ωγ ( 1 n ) + ω ( 1 γ ) +c1(k) ∫ [wn≤k] |∇(tk (wn) −tk (w))| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ . int. j. anal. appl. (2023), 21:30 23 thus − ∫ [wn≤k] fnψϕ ( tk (wn) −tk (w)γ ) ≥ ωγ ( 1 n ) + ω ( 1 γ ) −c1(k) ∫ [wn≤k] |∇(tk (wn) −tk (w))| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ hence j ≥ ωγ ( 1 n ) + ω ( 1 γ ) −c1(k) ∫ [wn≤k] |∇(tk (wn) −tk (w))| 2 ψ ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ then i + j ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) we conclude that ∫ qt ψ |∇(tk (wn) −tk (w))| 2 λ1ϕ ′(tk (wn) −tk (w)γ) −c1(k) ∣∣ϕ(tk (wn) −tk (w)γ)∣∣ ≤ ωγ,n ( 1 m ) + ωγ ( 1 n ) + ω ( 1 γ ) . now, choose β such that β ≥ c21 (k)/4λ 2 1. then we have λ1ϕ ′(s) −c1(k)|ϕ(s)| > λ1 2 , and this ends the proof. � consequently by (2.2), we proved the existence of global solutions to a reaction-diffusion system (1.1). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. alaa, solutions faibles d’équations paraboliques quasi-linéaires avec données initiales mesures, ann. math. blaise pascal. 3 (1996), 1-15. http://www.numdam.org/item?id=ambp_1996__3_2_1_0. [2] n. alaa, s. mesbahi, existence result for triangular reaction diffusion systems with l1 data and critical growth with respect to the gradient, mediterr. j. math. 10 (2013), 255-275. https://doi.org/10.1007/ s00009-012-0238-9. [3] n. alaa, i. mounir, global existence for reaction-diffusion systems with mass control and critical growth with respect to the gradient, j. math. anal. appl. 253 (2001), 532-557. https://doi.org/10.1006/jmaa.2000.7163. [4] n.e. alaa, m. pierre, weak solutions of some quasilinear elliptic equations with data measures, siam j. math. anal. 24 (1993), 23-35. https://doi.org/10.1137/0524002. [5] n.d. alikakos, lp bounds of solutions of reaction-diffusion equations, commun. part. differ. equ. 4 (1979), 827-868. https://doi.org/10.1080/03605307908820113. [6] a. bensoussan, l. boccardo, f. murat, on a non-linear p.d.e. having natural growth terms and unbounded solutions, ann. inst. h. poincaré, anal. non linéaire. 5 (1988), 347-364. http://www.numdam.org/item?id=ambp_1996__3_2_1_0 https://doi.org/10.1007/s00009-012-0238-9 https://doi.org/10.1007/s00009-012-0238-9 https://doi.org/10.1006/jmaa.2000.7163 https://doi.org/10.1137/0524002 https://doi.org/10.1080/03605307908820113 24 int. j. anal. appl. (2023), 21:30 [7] l. boccardo, f. murat, j.p. puel, existence results for some quasilinear parabolic equations, nonlinear anal.: theory meth. appl. 13 (1989), 373-392. https://doi.org/10.1016/0362-546x(89)90045-x. [8] n. boudiba, existence globale pour des systèmes de réaction-diffusion paraboliques quasilinéaires, thèse de troisième cycle, université des sciences et de la technologie houari boumediene d’alger, (1995). [9] h. brezis, w. strauss, semi-linear second order elliptic equations in l1, j. math. soc. japan. 25 (1973), 565-590. https://doi.org/10.2969/jmsj/02540565. [10] n.f. britton, reaction-diffusion equations and their applications to biology, academic press, london, (1986). [11] a. dall’aglio, l. orsina, nonlinear parabolic equations with natural growth conditions and l1 data, nonlinear anal.: theory meth. appl. 27 (1996), 59-73. https://doi.org/10.1016/0362-546x(94)00363-m. [12] t. diagana, some remarks on some strongly coupled reaction-diffusion equations, (2003). https://doi.org/ 10.48550/arxiv.math/0305152. [13] p.c. fife, mathematical aspects of reacting and diffusing systems, springer berlin heidelberg, berlin, heidelberg, 1979. https://doi.org/10.1007/978-3-642-93111-6. [14] a. haraux, a. youkana, on a result of k. masuda concerning reaction-diffusion equations, tohoku math. j. (2). 40 (1988), 159-163. https://doi.org/10.2748/tmj/1178228084. [15] s.l. hollis, r.h. martin, jr., m. pierre, global existence and boundedness in reaction-diffusion systems, siam j. math. anal. 18 (1987), 744-761. https://doi.org/10.1137/0518057. [16] s.l. hollis, j. morgan, interior estimates for a class of reaction-diffusion systems from l1 a priori estimates, j. differ. equ. 98 (1992), 260–276. https://doi.org/10.1016/0022-0396(92)90093-3. [17] s. kouachi, invariant regions and global existence of solutions for reaction-diffusion systems with full matrix of diffusion coefficients and nonhomogeneous boundary conditions, georgian math. j. 11 (2004), 349-359. https://doi.org/10.1515/gmj.2004.349. [18] s. kouachi, a. youkana, global existence for a class of reaction-diffusion systems, bull. polish. acad. sci. math. 49 (2001), 1-6. [19] o.a. ladyzenskaya, v.a. solonnikov, n.n. uralceva, linear and quasilinear equations of parabolic type, trans. math. monographs. vol. 23, american mathematical society, providence, ri, (1968). [20] r. landes, solvability of perturbed elliptic equations with critical growth exponent for the gradient, j. math. anal. appl. 139 (1989), 63-77. https://doi.org/10.1016/0022-247x(89)90230-8. [21] r. landes, v. mustonen, on parabolic initial-boundary value problems with critical growth for the gradient, ann. inst. h. poincaré anal. non linéaire. 11 (1994), 135-158. https://doi.org/10.1016/s0294-1449(16)30189-5. [22] r.h. martin, m. pierre, nonlinear reaction-diffusion systems, in: nonlinear equations in the applied sciences, math. sci. eng. academic press, new york (1991). [23] k. masuda, on the global existence and asymptotic behavior of solutions of reaction-diffusion equations, hokkaido math. j. 12 (1983), 360-370. https://doi.org/10.14492/hokmj/1470081012. [24] a. moumeni, n. barrouk, existence of global solutions for systems of reaction-diffusion with compact result, int. j. pure appl. math. 102 (2015), 169-186. https://doi.org/10.12732/ijpam.v102i2.1. [25] a. moumeni, n. barrouk, triangular reaction-diffusion systems with compact result, glob. j. pure appl. math. 11 (2015), 4729-4747. [26] j.d. murray, mathematical biology, springer-verlag, new york, (1993). [27] m. pierre, d. schmitt, existence globale ou explosion pour les systèmes de réaction-diffusion avec contrôle de masse, thèse de doctorat, université henri poincaré, nancy i, (1995). [28] b. rebiai, s. benachour, global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth, j. evol. equ. 10 (2010), 511-527. https://doi.org/10.1007/s00028-010-0059-x. [29] j. smoller, shock waves and reaction-difussion systems, springer-verlag, new york, (1983). https://doi.org/10.1016/0362-546x(89)90045-x https://doi.org/10.2969/jmsj/02540565 https://doi.org/10.1016/0362-546x(94)00363-m https://doi.org/10.48550/arxiv.math/0305152 https://doi.org/10.48550/arxiv.math/0305152 https://doi.org/10.1007/978-3-642-93111-6 https://doi.org/10.2748/tmj/1178228084 https://doi.org/10.1137/0518057 https://doi.org/10.1515/gmj.2004.349 https://doi.org/10.1016/0022-247x(89)90230-8 https://doi.org/10.1016/s0294-1449(16)30189-5 https://doi.org/10.14492/hokmj/1470081012 https://doi.org/10.12732/ijpam.v102i2.1 https://doi.org/10.1007/s00028-010-0059-x int. j. anal. appl. (2023), 21:30 25 [30] a.i. volpert, v.a. volpert, traveling wave solutions of parabolic systems, american mathematical society, providence, ri, (1994). 1. introduction 2. existence 2.1. assumptions 3. statement of the result 3.1. main result 4. proof of theorem 3.1 4.1. preliminaries 4.2. approximating scheme 4.3. convergence references international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 115-122 http://www.etamaths.com new integral inequalities through invexity with applications shahid qaisar1,∗, chuanjiang he1 and sabir hussain2 abstract. in this paper, we obtain some inequalities of simpson’s inequality type for functions whose derivatives absolute values are quasi-preinvex function. applications to some special means are considered. 1. introduction the simpson’s inequality is very important and the well-known in the literature: this inequality is stated that: if f : [a,b] → r be a four times continuously differentiable mapping on (a,b) and ∥∥f(4)∥∥∞ = sup x∈(a,b) ∣∣f(4) (x)∣∣ < ∞.then the following inequality holds:∣∣∣∣∣∣13 [ f(a) + f(b) 2 + 2f ( a + b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣∣ ≤ 12880 ∥∥∥f(4)∥∥∥ ∞ (b−a)4 recently, many others [5-7], [1] developed and discussed error estimates of the simpson’s type inequality interms of refinement, counterparts, generalizations and new simpson’s type inequalities. in [1],dragomir et.al. proved the following recent developments on simpson’s inequality for which the reminder is expressed in terms of lower derivatives than the fourth. theorem 1. suppose f : [a,b] → r is a differentiable mapping whose derivative is continuous on (a,b) and f′ ∈ l [a,b] .then the following inequality (1.1) ∣∣∣∣∣13 [ f(a)+f(b) 2 + 2f ( a+b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣ ≤ b−a3 ‖f′‖1 holds, where ‖f′‖1 = b∫ a |f′ (x)|dx. the bound of (1.1) for l-lipschitian mapping was given in [1] by 5 36 l (b−a). theorem 2. suppose f : [a,b] → r is an absolutely continuous mapping on [a,b] whose derivative belongs to lp [a,b] .then the following inequality holds, (1.2) ∣∣∣∣∣13 [ f(a)+f(b) 2 + 2f ( a+b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣ ≤ 1 6 [ 2q+1+1 3(q+1) ]1 q (b−a) 1 q ‖f′‖p 2010 mathematics subject classification. 26d07; 26d10; 26d15. key words and phrases. simpson’s inequality; quasi-convex function; power-mean inequality; holder’s integral inequality. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 115 116 qaisar, he and hussain where 1 p + 1 q = 1. in [2], kirmaci established the following hermite-hadamard inequalities for different convex functions as: theorem 3. let f : i ⊂ r → r be a differentiable function on i0 interior of i0a,b ∈ i with a < b.if the mapping |f′| is convex on [a,b] , and the following inequality holds (1.3) ∣∣∣ 1b−a ∫ ba f (x) dx−f (a+b2 )∣∣∣ ≤ b−a4 [|f′ (a)| + |f′ (b)|] . let k be a closed set rnand let f : k → rand η : k × k → rbe continuous functions. let x ∈ k,then the set k is said to be invex at xwith respect to η (., .) , if x + tη (y,x) ∈ k,∀x,y ∈ k,t ∈ [0, 1] . kis said to be invex set with respect to η if kis invex at each x ∈ k.the invex set kis also called a η-connected set. definition 5[3]. the function f on the invex set kis said to be preinvex with respect to η,if f (u + tη (v,u)) ≤ + (1 − t) f (u) + tf (v) ,∀u,v ∈ k,t ∈ [0, 1] . the function f is said to be preconcave if and only if −fis preinvex. it is to be noted that every convex function is preinvex with respect to the map η (x,y) = x−y but the converse is not true. definition 6[4]. the function f on the invex set kis said to be preinvex with respect to η,if f (u + tη (v,u)) ≤ max{f (u) ,f (v)} ,∀u,v ∈ k,t ∈ [0, 1] . also every quasi-convex function is a prequasiinvex with respect to the map η (u,v) but the converse does not hold, see for example [8]. the main aim of this paper is to establish new simpson’s type inequalities for the class of functions whose derivatives in absolute values are quasi-preinvex . . 2. main results before proceeding towards our main theorem regarding generalization of the simpson’s type inequality using prequasiinvex . we begin with the following lemma. lemma 2.1. let k ⊆ r be an open invex subset with respect to η : k ×k → rand a,b ∈ kwith a < a + η (b,a)suppose f : k → ris a differentiable mapping on ksuch that f′ ∈ l ([a,a + η (b,a)]) .then for every a,b ∈ k with η (b,a) 6= 0the following inequality holds: ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ = η(b,a) 2 [ 1∫ 0 ( λ 2 − 1 3 ) f′ ( a + ( 1+λ 2 ) η (b,a) ) dλ + 1∫ 0 ( 1 3 − λ 2 ) f′ ( a + ( 1−λ 2 ) η (b,a) ) dλ ] . new integral inequalities through invexity with applications 117 proof. integrating by parts, we have i1 = 1∫ 0 ( λ 2 − 1 3 ) f′ ( a + ( 1+λ 2 ) η (b,a) ) dλ = 2( λ2 − 1 3 )f(a+( 1+λ 2 )η(b,a)) η(b,a) ∣∣∣∣1 0 − 1 η(b,a) 1∫ 0 f ( a + ( 1+λ 2 ) η (b,a) ) = 2 6η(b,a) f (a + η (b,a)) + 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 1 η(b,a) 1∫ 0 f ( a + ( 1+λ 2 ) η (b,a) ) dλ setting x = a+ ( 1+λ 2 ) η (b,a) and dx = η(b,a) 2 dλ which gives i1 = 2 6η(b,a) f (a + η (b,a))+ 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 2 (η(b,a))2 a+η(b,a)∫ a+ 1 2 η(b,a) f (x) dx similarly we can show that i2 = 1∫ 0 ( 1 3 − λ 2 ) f′ ( a + ( 1−λ 2 ) η (b,a) ) dλ = 2 6η(b,a) f (a) + 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 2 (η(b,a))2 a+ 1 2 η(b,a)∫ a f (x) dx therefore η(b,a) 2 [i1 + i2] = [ f(a)+f(a+η(b,a)) 6 + 2 3 f ( 2a+η(b,a) 2 ) − 1 η(b,a) a+η(b,a)∫ a f(x)dx ] which completes the proof. in the following theorem, we shall propose some new upper bound for the righthand side of simpson’s type inequality for functions whose derivatives absolute values are prequasiinvex. theorem 2.2. let k ⊆ [0,∞) be an open invex subset with respect to η : k×k → r and a,b ∈ kwith a < a + η (b,a) suppose f : k → r is a differentiable mapping on k such that f′ ∈ l ([a,a + η (b,a)]) . if |f′| is preinvex on k, then for every a,b ∈ k with η (b,a) 6= 0 the following inequality holds: (2.1) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ sup { |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣} + sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|} ] . 118 qaisar, he and hussain proof. from lemma 2.1, and since |f′| is prequasiinvex, then we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 [ 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ + 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ] ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ sup {|f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣}dλ + η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|}dλ ≤ η(b,a) 2 sup { |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣} 1∫ 0 ∣∣λ 2 − 1 3 ∣∣dλ + η(b,a) 2 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|} 1∫ 0 ∣∣λ 2 − 1 3 ∣∣dλ = 5η(b,a) 72 sup { |f′′ (a)| , ∣∣f′′(a + 1 2 η (b,a) )∣∣} + 5η(b,a) 72 sup {∣∣f′′(a + 1 2 η (b,a) )∣∣ , |f′′ (a + η (b,a))|} . which completes the proof. the upper bound for the midpoint inequality for the first derivative is presented as corollary 2.3. let f as in theorem 2.2, if in addition (1) |f′| is increasing, then we have (2.2) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ |f′ (a + η (b,a))| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . (2) |f′| is decreasing, then we have (2.3) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . proof. it follows directly by theorem 2.2. remark 2.4. we note that the inequalities (2.2) and (2.3), are two new refinements of the trapezoid inequality for prequasiinvex functions, and thus for convex functions. the corresponding version for powers of the absolute value of the first derivative is incorporated in the following result: theorem 2.5. let k ⊆ [0,∞) be an open invex subset with respect to η : k×k → r and a,b ∈ k with a < a+η (b,a) suppose f : k → r is a differentiable mapping on k such that f′ ∈ l ([a,a + η (b,a)]) . if |f′|p is preinvex on k,from some p > 1, then for every a,b ∈ k with η (b,a) 6= 0 the following inequality holds: (2.4) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p ( sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 })p−1p + η(b,a) 12 ( 1+2p+1 3(p+1) )1/p ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 })p−1p new integral inequalities through invexity with applications 119 where q = p/(p− 1). proof . from lemma 2.1, and using the well known holder integral inequality, we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ + η(b,a) 2 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 )p)1p ( 1∫ 0 ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣ pp−1 dλ)p−1p + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 )p)1p ( 1∫ 0 ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣ pp−1 dλ)p−1p ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 )p)1p ( 1∫ 0 sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 }dλ)p−1p + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 )p)1p ( 1∫ 0 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 }dλ)p−1p = η(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( 1∫ 0 sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 }dλ)p−1p + η(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( 1∫ 0 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 }dλ)p−1p which completes the proof. corollary 2.6. let f as in theorem 2.5, if in addition (1) |f′|p/(p−1) is increasing, then we have (2.5) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p [ |f′ (a + η (b,a))| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . (2) |f′|p/(p−1) is decreasing, then we have (2.6) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p [ |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . an improvement of constants in theorem 2.5 and a consolidation of this result with theorem 2.2. are given in the following theorem. theorem 2.7. let k ⊆ [0,∞) be an open invex subset with respect to η : k×k → r and a,b ∈ k with a < a+η (b,a) suppose f : k → r is a differentiable mapping on k such that f′ ∈ l ([a,a + η (b,a)]) . if |f′|q is preinvex on k, q ≥ 1, 120 qaisar, he and hussain then for every a,b ∈ k with η (b,a) 6= 0 the following inequality holds: (2.7) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 η (b,a) )∣∣q})1q + 5η(b,a) 72 ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣q , |f′ (a + η (b,a))|q})1q . proof . suppose thatq ≥ 1. from lemma 2.1 and using the well known power mean inequality, we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ + η(b,a) 2 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 ) dλ )1−1 q ( 1∫ 0 ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣q dλ)1q + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 ) dλ )1−1 q ( 1∫ 0 ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣q dλ)1q since |f′|q is quasi-preinvexity , we have∣∣∣∣f′ ( a + ( 1 + λ 2 ) η (b,a) )∣∣∣∣q ≤ sup ( |f′ (a)|q , ∣∣∣∣f′ ( a + 1 2 η (b,a) )∣∣∣∣q ) and∣∣∣∣f′ ( a + ( 1 −λ 2 ) η (b,a) )∣∣∣∣q ≤ sup (∣∣∣∣f′ ( a + 1 2 η (b,a) )∣∣∣∣q , |f′ (a + η (b,a))|q ) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 η (b,a) )∣∣q})1q + 5η(b,a) 72 ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣q , |f′ (a + η (b,a))|q})1q . which completes the proof. 3. application to some special means in what follows we give certain generalization of some notions for a positive valued function of a positive variable. definition 3[9]. a function m : r → r,is called a mean function if it has the following properties: (1) homogeneity: m (ax,ay) = am (x,y) ,for all a > 0, (2) symmetry: m (x,y) = m (x,y) , (3) reflexivity: m (x,x) = x, (4) monotonicity: if x ≤ x′ and y ≤ y′, then m (x,y) = m (x′,y′) , (5) internality: min{x,y}≤ m (x,y) ≤ max{x,y} . new integral inequalities through invexity with applications 121 we consider some means for arbitrary positive real numbers a,b (see for instance [9]). we now consider the applications of our theorem to the special means. the arithmetic mean; a := a (a,b) = a + b 2 the geometic mean; g := g(a,b) = √ ab the power mean; pr := pr(a,b) = ( ar + br 2 )1 r , r ≥ 1, the indentric mean: i = i (a,b) = { 1 e ( bb aa ) , ifa 6= b a , ifa = b the harmonic mean: h := h (a,b) = 2ab a + b , the logarithmic mean: l = l (a,b) = a− b ln |a|− ln |b| , |a| 6= |b| the plogarithmic mean: lp ≡ lp (a,b) = [ bp+1 −ap+1 (p + 1) (b−a) ] , a 6= b p ∈ < \ {– 1, 0}: a, b > 0. it is well known that lp is monotonic nondecreasing over p ∈ r with l−1 := land l0 := i. in particular, we have the following inequalities h ≤ g ≤ l ≤ i ≤ a. now let a and bbe positive real numbers such that a < b.consider the function a < b. m : m (b,a) : [a,a + η (b,a)] × [a,a + η (b,a)] → r, which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows: η (b,a) = m (b,a)in (2.1), (2.4) and (2.7), one can obtain the following interesting inequalities involving means: ∣∣∣∣∣16 [ f (a) + 4f ( 2a+m(b,a) 2 ) + f (a + m (b,a)) ] ? − 1 m(b,a) a+m(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5m(b,a) 72 [ sup { |f′ (a)| , ∣∣f′(a + 1 2 m (b,a) )∣∣} + sup {∣∣f′(a + 1 2 m (b,a) )∣∣ , |f′ (a + m (b,a))|} ] . 122 qaisar, he and hussain (3.1) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+m(b,a) 2 ) + f (a + m (b,a)) ] − 1 m(b,a) a+m(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ m(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 m (b,a) )∣∣ pp−1 })p−1p + m(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( sup {∣∣f′(a + 1 2 m (b,a) )∣∣ pp−1 , |f′ (a + m (b,a))| pp−1 })p−1p (3.2) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+m(b,a) 2 ) + f (a + m (b,a)) ] − 1 η(b,a) a+m(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5m(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 m (b,a) )∣∣q})1q + 5m(b,a) 72 ( sup {∣∣f′(a + 1 2 m (b,a) )∣∣q , |f′ (a + m (b,a))|q})1q . for q ≥ 1.letting m = a,g,pr,i,h,l,lp in (3.1), (3.2) and (3.3), we can get the required inequalities, and the details are left to the interested reader. references [1] s. s. dragomir, r. p. agarwal, and p. cerone, on simpson’s inequality and applications, journal of inequalities and applications, 5 (2000), 33–579. [2] u.s. kirmaci, inequalities for differentiable mappings and applicatios to special means of real numbers to midpoint formula, appl. math. comp., 147 (2004), 137–146. [3] t. weir, and b. mond, preinvex functions in multiple bjective optimization, journal of mathematical analysis and applications, 136 (1998) 29-38. [4] s. r. mohan and s. k. neogy, on invex sets and preinvex function, j. math. anal. appl. 189 (1995), 901-908. [5] a. barani, s. barani and s.s. dragomir, simpson’s type inequalities for functions whose third derivatives in the absolute values are p-convex , rgmia res. rep. coll., 14 (2011). [6] m.z. sarikaya, e. set and m.e. ¨ozdemir, on new inequalities of simpson’s type for s-convex functions, comput. math. appl. 60 (2010) 2191–2199. [7] m. alomari and m. darus, “on some inequalities of simpson-type via quasi-convex functions and applications,” transylvanian journal of mathematics and mechanics, 2 (2010), 15–24. [8] x.m. yang, x.q. yang and k.l. teo, characterizations and applications of prequasiinvex functions, properties of preinvex functions, j. optim. theo. appl. 110 (2001) 645-668. [9] p.s. bullen, handbook of means and their inequalities, kluwer academic publishers, dordrecht, 2003. 1college of mathematics and statistics, chongqing university, chongqing, 401331, p. r. china 2department of mathematics, college of science, qassim university, p.o. box 6644, buraydah 51482, saudi arabia ∗corresponding author international journal of analysis and applications issn 2291-8639 volume 3, number 2 (2013), 119-130 http://www.etamaths.com cyclic contraction on smetric space animesh gupta abstract. in this paper we introduced the concepts of cyclic contraction on smetric space and proved some fixed point theorems on smetric space. our presented results are proper generalization of sedghi et al. [14]. we also give an example in support of our theorem. 1. introduction and preliminaries metric space is one of the most useful and important space in mathematics. its wide area provides a powerful tool to the study of variational inequalities, optimization and approximation theory, computer sciences and so many. recently the study of fixed point theory in metric space is very interesting and attract many researchers to investigated different results on it. on the other hand, some authors are interested and have tried to give generalizations of metric spaces in different ways. in 1963 gahler [3] gave the concepts of 2− metric space further in 1992 dhage [2] modified the concept of 2− metric space and introduced the concepts of d− metric space but in 2005 mustafa and sims [4] pointed out that these attempts are not valid and introduced the concepts of g− metric space and proved fixed point theorems in g− metric space. many authors proved different fixed point theorems in g− metric space in different ways see in [13] and references theirin. sedghi et al. [12] modified the concepts of d− metric space and introduced the concepts of d∗metric space also proved a common fixed point theorems in d∗metric space. recently, sedghi et al [14] introduced the concept of smetric space which is different from other space and proved fixed point theorems in s-metric space. they also gives some examples of smetric spaces which shows that smetric space is different form other spaces. in fact they gives following concepts of smetric space. definition 1. let x be a nonempty set. an smetric space on x is a function s : x3 → [0,∞) that satisfies the following conditions, for each x,y,z,a ∈ x, (1) s(x,y,z) ≥ 0, (2) s(x,y,z) = 0 if and only if x = y = z, (3) s(x,y,z) ≤ s(x,x,a) + s(y,y,a) + s(z,z,a). the pair (x,s) is called an smetric space. 2010 mathematics subject classification. 45h10; 54h25. key words and phrases. smetric space, fixed point, cyclic contraction, generalized cyclic contraction. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 119 120 gupta examples of such s metric space are as follows, example 2. let x = rn and ‖ . ‖ a norm on x, then s(x,y,z) =‖ y + z − 2x ‖ + ‖ y −z ‖ is an smetric on x. example 3. let x = rn and ‖ . ‖ a norm on x, then s(x,y,z) =‖ x− z ‖ + ‖ y −z ‖ is an smetric on x. example 4. let x be a nonempty set, d is ordinary metric on x, then s(x,y,z) = d(x,z) + d(y,z) is an smetric on x. lemma 5. let (x,s) be an smetric space, then we have, s(x,x,y) = s(y,y,x) . proof. by the third condition of smetric, we have s(x,x,y) ≤ s(x,x,x) + s(x,x,x) + s(y,y,x) and similarly s(y,y,x) ≤ s(y,y,y) + s(y,y,y) + s(x,x,y) which implies that s(x,x,y) = s(y,y,x) . � definition 6. let (x,s) be an smetric space. (1) a sequence {xn} in x is said to be converges to x if and only if s(xn,xn,x) → 0 as n → ∞. that is for each � > 0 there exists n0 ∈ n such that for all n ≥ n0, s(xn,xn,x) < � and we denote this by limn→∞xn = x. (2) a sequence {xn} in x is said to be cauchy sequence if and only if s(xn,xm,x) → 0 as n,m → ∞. that is for each � > 0 there exists n0 ∈ n such that for all n,m ≥ n0, s(xn,xm,x) < �. definition 7. the smetric space (x,s) is said to be complete if every cauchy sequence is convergent. every smetric on x defines a metric ds on x by ds(x,y) = s(x,x,y) + s(y,y,x) ∀x,y ∈ x.(1.1) let τ be the set of all a ⊂ x with x ∈ a if and only if there exists r > 0 such that bs(x,r) ⊂ a. then τ is a topology on x. also, nonempty subset a in the smetric space (x,s) is sclosed if ā = a. lemma 8. let (x,s) be a smetric space and a is a nonempty subset of x. a is said sclosed iof for any sequence {xn} is a such that xn → x as n →∞, then x ∈ a. cyclic contraction on smetric space 121 2. main results in this article we introduce the concept of cyclic contraction in smetric space and proved some fixed point theorems in smetric space. definition 9. denote by φ the set of functions φ : [0,∞) → [0,∞) satisfying, (1) φ is non-decreasing, (2) there exist k0 ∈ n, a ∈ (0, 1) and a convergent series of nonnegative terms σ∞k=1vk such that φk+1(t) ≤ aφk(t) + vk for k ≥ k0 and any t > 0. then φ ∈ φ is called a (c)comparison function. lemma 10. if φ ∈ φ, then the following properties hold: (1) (φn(t))n∈n converges to 0 as n →∞, for all t > 0, (2) φ(t) < t for any t > 0, (3) φ is continuous at 0, (4) the series σ∞k=0φ k(t) converges for any t > 0. lemma 11. if φ ∈ φ, then the function p : (0,∞) → (0,∞) defined by p(t) = σ∞k=0φ k(t), t > 0,(2.1) is non decreasing and continuous at 0. first, we consider the picard iteration {xn} defined by xn+1 = txn, ∀n ≥ 0.(2.2) our first result is the following. theorem 12. let (x,s) be a scomplete smetric space. let {ai}mic1 be a family of non empty sclosed subsets of x, m a positive integer and y = ∪mi=iai . let t : y → y be a mapping such that t(ai) ⊆ ai+1 ∀ i = 1, 2....m with ai+1 = ai(2.3) suppose also that there exists φ ∈ φ such that s(tx,ty,tz) ≤ φ(s(x,y,z))(2.4) for all (x,y,z) ∈ ai ×ai ×ai+1 for all i = 1, 2....m. then (i) t has a unique fixed point, say u, that belongs to ∩mi=iai, (ii) the following estimates hold: s(xn,xn,u) ≤ p(φn(s(x0,x0,x1))), n ≥ 1,(2.5) s(xn,xn,u) ≤ p(s(xn,xn,xn+1)), n ≥ 1,(2.6) (iii) for any x ∈ y , s(x,x,u) ≤ p(s(x,x,tx)),(2.7) where p is given in 2.1 in lemma 11. 122 gupta proof. let x0 ∈ y = ∪mi=iai,. without loss of generality, let x0 ∈ a1. consider the picard iteration {xn} defined by 2.2 and starting from x0. if for some integer k, xk = xk+1, so {xn} is constant for any n ≥ k then {xn} is s-cauchy in (x,s). suppose that xn 6= xn+1 for all n ≥ 0. for any n ≥ 0, there us in ∈ {1, 2....m} such that xn ∈ ain and xn+1 ∈ ain+1 . by 2.4, we have s(xn+1,xn+1,xn+2) = s(txn,txn,txn+1) ≤ φ(s(xn,xn,xn+1))(2.8) the function φ is non decreasing, so by induction s(xn,xn,xn+1) ≤ φn(s(x0,x0,x1))∀ n ≥ 0.(2.9) by rectangle inequality and 2.9 , for r ≥ 1 s(xn,xn,xn+r) ≤ s(xn,xn,xn+1) + s(xn+1,xn+1,xn+2) + ........ + s(xn+r−1,xn+r−1,xn+r) ≤ φn(s(x0,x0,x1)) + φn+1(s(x0,x0,x1)) + ........... + φn+r−1(s(x0,x0,x1)) denote δn = σ n k=0φ k(s(x0,x0,x1)), n ≥ 0 therefore s(xn,xn,xn+r) ≤ δn+p−1 − δn−1(2.10) since the function φ ∈ φ and s(x0,x0,x1) > 0, so by (4) of lemma 10, we get that σ∞k=0φ k(s(x0,x0,x1)) < ∞, which implies that there exists a positive real s such that limn→∞δn = δ. thus, from 2.10 we have limn→∞s(xn,xn,xn+r) = 0 this yields that {xn} is s-cauchy sequence in (x,s). since (x,s) is scomplete, hence there exists u ∈ x such that limn→∞xn = u(2.11) we shall prove that u ∈∩mi=iai.(2.12) since x0 ∈ a1, we have {xnp}n≥0 ∈ a1. since a1 is sclosed and 2.11, by lemma 8, we have u ∈ a1. again, {xnp+1}n≥0 ∈ a2. since a2 is sclosed and cyclic contraction on smetric space 123 2.11, by lemma 8, we have u ∈ a2. continuing this process, we obtain 2.12. we claim that u is a fixed point of t . we have that for any n ≥ 0 there exists us in ∈{1, 2....m} such that xn ∈ ain . also form 2.12, u ∈ ain+1, so applying 2.4 for x = y = xn and z = u, we get that s(xn+1,xn+1,u) = s(txn,txn,tu) ≤ φ(s(xn,xn,u))(2.13) since φ is continuous at 0 and limn→∞s(xn,xn,u) = 0, so limn→∞s(xn+1,xn+1,u) ≤ φ(0). but, since φ(t) < t for all t > 0 and again φ is continuous at 0, hence we get that φ(0) = 0. we deduce from the above inequality, xn+1 → tu as n → ∞. by uniqueness of limit, it follows that tu = u. now, we prove that u is the unique fixed point of t . assume that v us another fixed point of t , that is, tv = v. we have v ∈ ∩mi=1ai. suppose that u 6= v, so s(u,u,v) > 0. taking x = y = u and z = v in 2.4, we get that 0 < s(u,u,v) = s(tu,tu,tv) ≤ φ(s(u,u,v)) ≤ s(u,u,v), which is a contradiction. we deduce u is the unique fixed point of t. this completes the proof of (i). we shall prove (ii). from 2.10, we have s(xn,xn,xn+r) ≤ σn+r−1k=n φ k(s(x0,x0,x1)) letting r →∞ in above inequality, we get the estimate 2.5. for n ≥ 0 and k ≥ 1, we have s(xn+k,xn+k,xn+k+1) = s(txn+k−1,txn+k−1,txn+k) ≤ φ(s(xn+k−1,xn+k−1,xn+k))(2.14) and for k ≥ 2, s(xn+k−1,xn+k−1,xn+k) = s(txn+k−2,txn+k−2,txn+k−1) ≤ φ(s(xn+k−2,xn+k−2,xn+k−1))(2.15) by monotonicity of φ, 2.14 and 2.15 imply that s(xn+k,xn+k,xn+k+1) ≤ φ2(s(xn+k−2,xn+k−2,xn+k−1)), n ≥ 0, k ≥ 2.(2.16) by induction we get that s(xn+k,xn+k,xn+k+1) ≤ φk(s(xn,xn,xn+1)), n ≥ 0, k ≥ 0.(2.17) but by rectangle inequality s(xn,xn,xn+r) ≤ s(xn,xn,xn+1) + s(xn+1,xn+1,xn+2) + ........ + s(xn+r−1,xn+r−1,xn+r) hence, form 2.17, we have 124 gupta s(xn,xn,xn+r) ≤ σn+r−1k=0 φ k(s(xn,xn,xn+1)) letting r →∞ in the above inequality, we get that s(xn,xn,u) ≤ σ∞k=0φ k(s(xn,xn,xn+1)) = p(s(xn,xn,xn+1))(2.18) this yields (ii). now we will prove (iii). let x ∈ y . form 2.18, for x0 = x, we have s(x,x,u) ≤ σ∞k=0φ k(s(x,x,tx)) = p(s(x,x,tx)) which is the estimate 2.7. � as consequences of theorem 12, we have the following results. theorem 13. let t : y → y be defined as theorem 12. then σ∞n=0s(t nx,tnx,tn+1) < ∞, ∀x ∈ y,(2.19) that is,t is a good picard operator. proof. let x = x0 ∈ y . if for some integer k, tkx0 = tk+1x0 so the sequence {tnx0} is constant for all n ≥ k, hence obiviosly 2.19 holds. otherwise, assume that tkx0 6= tk+1x0 for all n ≥ 0. by 2.9 in the proof of theorem 12, we know that s(tnx,tnx,tn+1x) = s(xn,xn,xn+1) ≤ φ(s(x0,x0,x1)), ∀n ≥ 0. then σ∞n=0s(t nx,tnx,tn+1x) ≤ σ∞n=0φ(s(x0,x0,x1)) = p(s(x0,x0,x1)). by lemma 11, it follows that σ∞n=0s(t nx,tnx,tn+1x) < ∞, so t is a good picard operator. � theorem 14. let t : y → y be defined as in theorem 12. then s(tnx,tnx,u) = s(xn,xn,xn+1) ≤ φ(s(x0,x0,x1)), ∀n ≥ 0.(2.20) that is, t is a special picard operator. proof. if x = u, then cleary 2.20 is true. suppose x 6= u and x ∈ y . we rewrite 2.13 with tu = u. s(tn+1x,tn+1x,u) = s(tn+1x,tn+1x,tu) ≤ φ(s(xn,xn,u)) by induction and considering the monotonicity of φ, we obtain s(tnx,tnx,u) ≤ φn(s(x,x,u)), ∀n ≥ 0. therefore cyclic contraction on smetric space 125 σ∞n=0s(t nx,tnx,u) ≤ σ∞n=0φ n(s(x,x,u)) = p(s(x,x,u)), consequently, σ∞n=0s(t nx,tnx,u) ≤∞, so t is a special picard operator. � definition 15. let x be a nonempty set. a fixed point problem of a given mapping f : x → x on x is called well-posed if f(f) is a singleton and for any sequence {an} in x with x∗ ∈ f(f) and limn→inftys(an,an,fan) implies x∗ = limn→∞an. theorem 16. let f : y → y be defined as in theorem 12. then the fixed point problem for t is well posed that is, assuming that there exists {zn} ∈ y , n ∈ n such that limn→∞s(zn,zn,fzn) implies z = limn→inftyzn. proof. let {zn} ∈ y , n ∈ n such that limn→∞s(zn,zn,tzn) = 0. applying 2.7 for x = zn and u = z then we have s(zn,zn,z) ≤ p(s(zn,zn,tzn)).(2.21) having the mind from lemma 11 that p is continuous at 0, so letting n →∞ in 2.21, we have limn→∞s(zn,zn,z) = 0, so z = limn→∞zn. hence the fixed point problem for t is well posed. � theorem 17. let t : y → y be defined as in theorem 12. let f : y → y such that (1) f has at least one fixed point, say zf ∈ f(f) (2) there esists v > 0 such that s(fx,fx,tx) ≤ y, ∀x ∈ y.(2.22) then s(zf,zf,zt ) ≤ s(v) where f(t) = zt . proof. assume zf 6= zy . otherwise the proof is completed. we apply 2.7 from theorem 12 for x = xf to have s(zf,zf,zt ) ≤ p(s(zf,zf,tzf ) = p(s(fzf,fzf,tzf )(2.23) by lemma 11, then function p is non decreasing, so by ??ith x = zf , it follows that s(zf,zf,zt ) ≤ s(v).(2.24) � 3. cyclic (ψ −φ) contraction on smetric space denote by ψ the set of functions ψ : [0,∞) → [0,∞) satisfying (ψ1) ψ is continuous, (ψ2) ψ is non decreasing, (ψ3) ψ(t) = 0 if and only if t = 0. also, denote by φ the set of functions φ : [0,∞) → [0,∞) satisfying (φ1) φ is lower semicontinuous, 126 gupta (φ2) φ(t) = 0 if and only if t = 0 the object of this section is to give some more general classes of mappings involving cyclic (ψ − φ)contractions. note that, in our result the monotony property of the function φ is omitted and the continuity property of φ is replaced by lower semi-continuity. the main result of this section is the following. theorem 18. let (x,s) be a scomplete smetric space. let {ai}mic1 be a family of non empty sclosed subsets of x, m a positive integer and y = ∪mi=iai . let t : y → y be a mapping such that t(ai) ⊆ ai+1 ∀i = 1, 2....m withai+1 = ai(3.1) suppose also that there exists φ ∈ φ such that ψ(s(tx,ty,tz)) ≤ ψ(s(x,y,z)) −φ(s(x,y,z)), ∀(x,y,z) ∈ ai ×ai ×ai+1(3.2) for i = 1, 2......m. then t has a unique fixed point that belongs to ∩mi=1ai. proof. let x0 ∈ a1. consider the picard iteration {xn} defined by xn+1 = txn for all n ≥ 0. if for some integer k, xk = xk+1, so {xn} is constant for any n ≥ k, then {xn} is scauchy sequence in (x,s). suppose that xn 6= xn+1 for all n ≥ 0. for any n ≥ 0, there is in ∈{1, 2, .....m} such that xn ∈ ain and xn+1 ∈ ain+1 . by 3.2, we have ψ(s(xn+1,xn+1,xn+2)) = ψ(s(txn,txn,txn+1)) ≤ ψ(s(xn,xn,xn+1)) −φ(s(xn,xn,xn+1)) ψ(s(xn+1,xn+1,xn+2)) ≤ ψ(s(xn,xn,xn+1))(3.3) the function ψ is non-decreasing, so we have s(xn+1,xn+1,xn+2) ≤ s(xn,xn,xn+1), ∀n ≥ 0.(3.4) therefore the sequence {s(xn,xn,xn+1)} is non-increasing, so it converges to some real r ≥ 0. letting n → ∞ in 3.3, using the continuity of ψ and the lower semi-continuity of φ, we get that ψ(r) ≤ ψ(r) −φ(r). which implies that φ(r) = 0. by (φ2), we have r = 0, that is, limn→∞s(xn,xn,xn+1) = 0.(3.5) since s(x,x,y) = s(y,y,x) for all x,y ∈ x, hence by 3.5, we have limn→∞s(xn+1,xn+1,xn) = 0.(3.6) cyclic contraction on smetric space 127 now, we prove that {xn} is a scauchy sequence. we argue by contradiction. assume that for {xn} is not a scauchy sequence. then, following definition 6, there exists � > 0 for which we can find subsequences {xm(k)} and {xn(k)} of {xn} with n(k) > m(k) > k such that s(xn(k),xn(k),xm(k)) ≥ �(3.7) further corresponding to m(k), we can choose n(k) in such a way that it is the smallest integer with n(k) > m(k) > k and satisfying 3.7. then s(xn(k)−1,xn(k)−1,xm(k)) < �(3.8) using 3.8 and property of smetric space we have � ≤ s(xn(k),xn(k),xm(k)) ≤ s(xn(k),xn(k),xn(k)−1) + s(xn(k),xn(k),x(k)−1) + s(xm(k),xm(k),xn(k)−1)(3.9) � ≤ s(xn(k),xn(k),xm(k)) ≤ � + 2s(xn(k),xn(k),xn(k)−1) letting k →∞ in 3.9 and using 3.6, we find limk→∞s(xn(k),xn(k),xn(k)−1) = �(3.10) on the other hand, for all k, there exists j(k), 0 ≤ j(k) ≤ m, such that n(k) − m(k) + j(k) = 1(q). then xm(k)cj(k) (for k large enough, m(k) > jck)) and xn(k) lie in different adjacently labeled sets ai and ai+1 for certaing i = 1, 2, ....m. from 3.2, we have ψ(s(xn(k)+1,xn(k)+1,xm(k)cj(k)+1)) = ψ(s(txn(k),txn(k),txm(k)cj(k))) ψ(s(txn(k),txn(k),txm(k)cj(k))) ≤ ψ(s(xn(k),xn(k),xm(k)−j(k))) −φ(s(xn(k),xn(k),xm(k)−j(k)))(3.11) by using the property of smetric space and as n →∞ we have limk→∞s(xn(k),xn(k),xm(k)−j(k)) = �.(3.12) similarly by using the property of smetric space , 3.6, 3.7, 3.12 and as k →∞ we find limk→∞s(xn(k)+1,xn(k)+1,xm(k)cj(k)+1) = �.(3.13) now letting k →∞ in 3.11 and using 3.12, 3.13 we get that ψ(�) ≤ ψ(�) −φ(�)(3.14) which yields that � = 0, a contradiction. this shows that {xn} is scauchy sequence in (x,s). since (x,s) is s-complete, hence there exists u ∈ x such that 128 gupta limk→∞xn = u.(3.15) we shall prove that u ∈∩mi=1ai(3.16) since x0 ∈ a1, we have {xnl}n≥0a1. the fact that a1 is sclosed and 2.11 yield that u ∈ a1. again, {xnl+1}n≥0a2. since a2 is sclosed and 3.15 yield that u ∈ a2. continuing this process, we obtain 3.16. we claim that u is a fixed point of t . we have in mind that for any n ≥ 0, there exists in ∈{1, 2, ....m} such that xn ∈ ain . also, form 3.16, u ∈ ain+1 so applying 3.2 for x = y = xn and z = u, we get that ψ(s(xn+1,xn+1,tu)) = ψ(s(txn,txn,tu)) ≤ ψ(s(xn,xn,u)) −ψ(s(xn,xn,u)) letting n →∞ in above inequality, we obtain ψ(s(u,u,tu)) ≤ ψ(o) −φ(o) which implies that ψ(s(u,u,tu)) = 0, so s(u,u,tu) = 0. it follows that tu = u. now, we prove that u is the unique fixed point of t . assume that v is another fixed point of t , that is tv = v. we have v ∈∩mi=1ai. taking x = y = u and z = v in 3.2, we get that ψ(s(tu,tu,tv)) ≤ ψ(s(u,u,v)) −φ(s(u,u,v)),(3.17) so that φ(s(u,u,v)) = 0 that is u = v. � example 19. let x = [0,∞) be equipped with the smetric space s given as follows s(x,y,z) =| x−z | + | y −z | (x.s) is scomplete metric space. consider a1 = {0, 1}, a2 = {1, 4} and y = a1 ∪a2. it is obvious that a1 and a2 are sclosed subsets of (x,s). we define t : y → y by t0 = 1,t1 = 1 and t4 = 0 we have t(a1) ⊆ a2 and t(a2) ⊆ a1. define ψ(t) = t and φ = 23t. we shall prove that (x,y,z) ×a1 ×a1 ×a2 and (x,y,z) ×a2 ×a2 ×a1. to check this we have following conditions: (1) if (x,y,z) ×a1 ×a1 ×a2 then, case 1: if x = y = 0 and z = 1 in this case s(tx,ty,tz) = 0. cyclic contraction on smetric space 129 case 2: if x = 0,y = 1 and z = 4 or x = 1,y = 0 and z = 4 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 2 s(x,y,z) which is true. case 3: if x = y = z = 1 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 0. case 4: if x = y = 0 and z = 4 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 3 s(x,y,z) case 5: if x = y = 1 and z = 4 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 3 s(x,y,z) case 6: if x = y = 4 and z = 1 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 3 s(x,y,z) (2) if (x,y,z) ×a2 ×a2 ×a1 then, case 7: if x = y = 1 and z = 0 in this case s(tx,ty,tz) = 0 < 2 = s(x,y,z). case 8: if x = 1,y = 4 and z = 0 or x = 4,y = 1 and z = 0 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 1 = 1 5 s(x,y,z) which is true. case 9: if x = y = 4 and z = 0 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 4 s(x,y,z) case 10: if x = y = 4 and z = 1 in this case 18 true and from 3.2 we have s(tx,ty,tz) = 2 ≤ 1 3 s(x,y,z) 4. acknowledgements the author are grateful for the reviewers for the careful reading of the article and for the suggestions which improved the quality of this work. 130 gupta references [1] r. chugh, t. kadian, a. rani, b.e. rhoades, property p in g-metric spaces, fixed point theory appl. vol. 2010, article id 401684. [2] b.c. dhage, generalized metric spaces mappings with fixed point, bull. calcutta math. soc. 84 (1992), 329-336. [3] s. gahler, 2-metrische raume und iher topoloische struktur, math. nachr. 26 (1963), 115148. [4] z. mustafa, b. sims, a new approach to generalized metric spaces, j. nonlinear convex anal. 7 (2006), 289c297. [5] z. mustafa, h. obiedat, f. awawdeh, some common fixed point theorems for mapping on complete g-metric spaces, fixed point theory appl. vol. 2008, article id 189870. [6] z. mustafa, a new structure for generalized metric spaces with applications to fixed point theory, ph. d. thesis, the university of newcastle, callaghan, australia, 2005. [7] z. mustafa, b. sims, some results concerning d-metric spaces, proc. internat. conf. fixed point theory and applications, pp. 189-198, valencia, spain, 2003. [8] s.v.r. naidu , k.p.r. rao, n. srinivasa rao, on the topology of d-metric spaces and the generation of d-metric spaces from metric spaces, internat. j. math. math. sci. 2004 (2004), no. 51, 2719-2740. [9] s.v.r. naidu, k.p.r. rao, n. srinivasa rao, on the concepts of balls in a d-metric space, internat. j. math. math. sci. 2005 (2005), 133-141. [10] s.v.r. naidu, k.p.r. rao, n. srinivasa rao, on convergent sequences and fixed point theorems in d-metric spaces, internat. j. math. math. sci. 2005 (2005), 1969-1988. [11] s. sedghi, k.p.r. rao, n. shobe, common fixed point theorems for six weakly compatible mappings in d∗-metric spaces, internat. j. math. math. sci. 6 (2007), 225-237. [12] s. sedghi, n. shobe, h. zhou, a common fixed point theorem in d∗-metric spaces, fixed point theory appl. vol. 2007, article id 27906, 13 pages. [13] w. shatanawi, fixed point theory for contractive mappings satisfying ψ-maps in g-metric spaces, fixed point theory appl. vol. 2010, article id 181650. [14] s. sedghi, n. shobe, a. aliouche, a generalization of fixed point theorem in s-metric spaces,mat. vesnik 64 (2012), 258-266. department of mathematics, sagar institute of engineering, technology and research, ratibad bhopal (m.p.), india international journal of analysis and applications issn 2291-8639 volume 11, number 2 (2016), 110-123 http://www.etamaths.com stability in totally nonlinear neutral dynamic equations on time scales malik belaid2, abdelouaheb ardjouni1,2,∗ and ahcene djoudi2 abstract. let t be a time scale which is unbounded above and below and such that 0 ∈ t. let id− τ : [0,∞) ∩t → t be such that (id− τ) ([0,∞) ∩t) is a time scale. we use the krasnoselskiiburton’s fixed point theorem to obtain stability results about the zero solution for the following totally nonlinear neutral dynamic equation with variable delay x4 (t) = −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) g (x (t) ,x (t− τ (t))) , t ∈ [0,∞) ∩t, where f4 is the 4-derivative on t and f4̃ is the 4-derivative on (id− τ) (t). the results obtained here extend the work of ardjouni, derrardjia and djoudi [2]. 1. introduction the concept of time scales analysis is a fairly new idea. in 1988, it was introduced by the german mathematician stefan hilger in his ph.d. thesis [12]. it combines the traditional areas of continuous and discrete analysis into one theory. after the publication of two textbooks in this area by bohner and peterson [6] and [7], more and more researchers were getting involved in this fast-growing field of mathematics. the study of dynamic equations brings together the traditional research areas of differential and difference equations. it allows one to handle these two research areas at the same time, hence shedding light on the reasons for their seeming discrepancies. in fact, many new results for the continuous and discrete cases have been obtained by studying the more general time scales case (see [1, 3, 4, 13] and the references therein). there is no doubt that the lyapunov method have been used successfully to investigate stability properties of wide variety of ordinary, functional and partial equations. nevertheless, the application of this method to problem of stability in differential equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded term. it has been noticed that some of theses difficulties vanish by using the fixed point technic. other advantages of fixed point theory over lyapunov’s method is that the conditions of the former are average while those of the latter are pointwise (see [2, 5, 8, 9, 10, 11] and references therein). in paper, we consider the following neutral nonlinear dynamic equations with variable delay given by (1.1) x4 (t) = −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) g (x (t) ,x (t− τ (t))) , t ∈ [0,∞) ∩t, with an assumed initial function x (t) = ψ (t) , t ∈ [m0, 0] ∩t, where t is an unbounded above and below time scale and such that 0 ∈ t, ψ : [m0, 0] ∩ t → r is rd-continuous and m0 = inf {t− τ (t) : t ∈ [0,∞) ∩t}. throughout this paper, we assume that a,b : [0,∞) ∩ t → r are rd-continuous, h : r → r is continuous with h (0) = 0 and c : [0,∞) ∩ t → r is continuously delta-differentiable. in order for the function x (t− τ (t)) to be well-defined and 2010 mathematics subject classification. 34k20, 34k30, 34k40. key words and phrases. fixed points, neutral dynamic equations, stability, time scales. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 110 stability in totally nonlinear neutral dynamic equations 111 differentiable over [0,∞)∩t, we assume that τ : [0,∞)∩t → t is positive and twice continuously deltadifferentiable, and that id−τ : [0,∞)∩t → t is an increasing mapping such that (id− τ) ([0,∞) ∩t) is closed where id is the identity function. our purpose here is to use a modification of krasnoselskii’s fixed point theorem due to burton (see [8] theorem 3) to show the asymptotic stability and the stability of the zero solution for equation (1.1). clearly, the present problem is totally nonlinear so that the variation of parameters can not be applied directly. then, we resort to the idea of adding and subtracting a linear term. as noted by burton in [8], the added term destroys a contraction already present in part of the equation but it replaces it with the so called a large contraction mapping which is suitable for fixed point theory. during the process we have to transform (1.1) into an integral equation written as a sum of two mapping; one is a large contraction and the other is compact. after that, we use a variant of krasnoselskii fixed point theorem, to show the asymptotic stability and the stability of the zero for equation (1.1). in the special case t = r, ardjouni, derrardjia and djoudi [2] show the zero solution of (1.1) is asymptotically stable by using krasnoselskii-burton’s fixed point theorem. in section 2, we present some preliminary material that we will need through the remainder of the paper. we will state some facts about the exponential function on a time scale as well as the modification of krasnoselskii’s fixed point theorem established by burton (see ([8] theorem 3) and [10]). for details on krasnoselskii’s theorem we refer the reader to [14]. we present our main results on stability in section 3 and 4. the results presented in this paper extend the main results in [2]. 2. preliminaries in this section, we consider some advanced topics in the theory of dynamic equations on a time scales. again, we remind that for a review of this topic we direct the reader to the monographs of bohner and peterson [6] and [7]. a time scale t is a closed nonempty subset of r. for t ∈ t the forward jump operator σ, and the backward jump operator ρ, respectively, are defined as σ (t) = inf {s ∈ t : s > t} and ρ (t) = sup{t ∈ t : s < t}. these operators allow elements in the time scale to be classified as follows. we say t is right scattered if σ (t) > t and right dense if σ (t) = t. we say t is left scattered if ρ (t) < t and left dense if ρ (t) = t. the graininess function µ : t → [0,∞), is defined by µ (t) = σ (t) − t and gives the distance between an element and its successor. we set inf ∅ = sup t and sup∅ = inf t. if t has a left scattered maximum m, we define tk = t�{m}. otherwise, we define tk = t. if t has a right scattered minimum m, we define tk = t�{m}. otherwise, we define tk = t. let t ∈ tk and let f : t → r. the delta derivative of f (t), denoted f4 (t), is defined to be the number (when it exists), with the property that, for each � > 0, there is a neighborhood u of t such that ∣∣f (σ (t)) −f (s) −f4 (t) [σ (t) −s]∣∣ ≤ � |σ (t) −s| , for all s ∈ u. if t = r then f4 (t) = f′ (t) is the usual derivative. if t = z then f4 (t) = 4f (t) = f (t + 1) −f (t) is the forward difference of f at t. a function f is right dense continuous (rd-continuous), f ∈ crd = crd (t,r), if it is continuous at every right dense point t ∈ t and its left-hand limits exist at each left dense point t ∈ t. the function f : t → r is differentiable on tk provided f4 (t) exists for all t ∈ tk. we are now ready to state some properties of the delta-derivative of f. note fσ (t) = f (σ (t)). theorem 1 ([6, theorem 1.20]). assume f,g : t → r are differentiable at t ∈ tk and let α be a scalar. (i) (f + g) 4 (t) = g4 (t) + f4 (t). (ii) (αf) 4 (t) = αf4 (t). (iii) the product rules (fg) 4 (t) = f4 (t) g (t) + fσ (t) g4 (t) , (fg) 4 (t) = f (t) g4 (t) + f4 (t) gσ (t) . 112 belaid, ardjouni and djoudi (iv) if g (t) gσ (t) 6= 0 then ( f g )4 (t) = f4 (t) g (t) −f (t) g4 (t) g (t) gσ (t) . the next theorem is the chain rule on time scales ([6, theorem 1.93], theorem 1.93). theorem 2 (chain rule). assume ν : t → r is strictly increasing and t̃ := ν (t) is a time scale. let ω : t̃ → r. if ν4 (t) and ω4̃ (ν (t)) exist for t ∈ tk, then (ω ◦ν)4 = ( ω4̃ ◦ν ) ν4. in the sequel we will need to differentiate and integrate functions of the form f (t− τ (t)) = f (ν (t)) where, ν (t) := t−τ (t). our next theorem is the substitution rule ([6, theorem 1.98], theorem 1.98). theorem 3 (substitution). assume ν : t → r is strictly increasing and t̃ := ν (t) is a time scale. if f : t → r is rd-continuous function and ν is differentiable with rd-continuous derivative, then for a,b ∈ t , ∫ b a f (t) ν4 (t)4t = ∫ ν(b) ν(a) ( f ◦ν−1 ) (s)4̃s. a function p : t → r is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ tk. the set of all regressive rd-continuous function f : t → r is denoted by r. the set of all positively regressive functions r+, is given by r+ = {f ∈r : 1 + µ (t) f (t) > 0 for all t ∈ t}. let p ∈r and µ (t) 6= 0 for all t ∈ t. the exponential function on t is defined by ep (t,s) = exp (∫ t s 1 µ (z) log (1 + µ (z) p (z)) ∆z ) . it is well known that if p ∈ r+, then ep (t,s) > 0 for all t ∈ t. also, the exponential function y (t) = ep (t,s) is the solution to the initial value problem y 4 = p (t) y, y (s) = 1. other properties of the exponential function are given by the following lemma. lemma 1 ([6, theorem 2.36]). let p,q ∈r. then (i) e0 (t,s) = 1 and ep (t,t) = 1, (ii) ep (σ (t) ,s) = (1 + µ (t) p (t)) ep (t,s), (iii) 1 ep(t,s) = e p (t,s), where p (t) = − p(t) 1+µ(t)p(t) , (iv) ep (t,s) = 1 ep(s,t) = e p (s,t), (v) ep (t,s) ep (s,r) = ep (t,r), (vi) e4p (.,s) = pep (.,s) and ( 1 ep(.,s) )4 = − p(t) eσp(.,s) . lemma 2 ([1]). if p ∈r+, then 0 < ep (t,s) ≤ exp (∫ t s p (u)4u ) , ∀t ∈ t. corollary 1 ([1]). if p ∈r+ and p (t) < 0 for all t ∈ t, then for all s ∈ t with s ≤ t we have 0 < ep (t,s) ≤ exp (∫ t s p (u)4u ) < 1. 3. the inversion and the fixed point theorem in addition to the conditions mentioned in section 1, we assume that a ∈ r+ is rd-continuous and a (t) > 0 for all t ∈ [0,∞) ∩t, c is continuously delta-differentiable and τ is twice continuously delta-differentiable with (3.1) τ4 (t) 6= 1, t ∈ [0,∞) ∩t. we also assume that g (x,y) is locally lipschitz continuous in x and y. that is, there are positive constants n1 and n2 so that if |x| , |y| ≤ √ 3/3, then (3.2) |g (x,y) −g (z,w)| ≤ n1 ‖x−z‖ + n2 ‖y −w‖ and g (0, 0) = 0. stability in totally nonlinear neutral dynamic equations 113 one crucial step in the investigation of an equation using fixed point theory involves the construction of a suitable fixed point mapping. for that end we must invert (1.1) to obtain an equivalent integral equation from which we derive the needed mapping. during the process, an integration by parts has to be performed on the neutral term x4̃ (t− τ (t)). unfortunately, when doing this, a derivative τ4 (t) of the delay appears on the way, and so we have to support it. lemma 3. suppose (3.1) holds. x is a solution of equation (1.1) if and only if x (t) = [ ψ (0) − c (0) 1 − τ4 (0) ψ (−τ (0)) ] e a(t, 0) + ∫ t 0 a (s) (hx) (s) e a (t,s) ∆s + c (t) 1 − τ4 (t) x (t− τ (t)) − ∫ t 0 µ (s) xσ (s− τ (s)) e a (t,s) ∆s + ∫ t 0 b (s) g (x (s) ,x (s− τ (s))) e a (t,s) ∆s,(3.3) where (3.4) µ (t) = ( c4 (t) + a (t) cσ (t) )( 1 − τ4 (t) ) + c (t) τ44 (t) (1 − τ4 (t)) (1 − τ4 (σ (t))) , and (3.5) (hx) (t) = xσ (t) −h (xσ (t)) . proof. let x be a solution of equation (1.1). rewrite (1.1) as x4 (t) + a (t) xσ (t) = a (t) xσ (t) −a (t) h (xσ (t)) + c (t) x4̃ (t− τ (t)) + b (t) g (t,x (t) ,x (t− τ (t))) . multiply both sides of the above equation by ea (t, 0) and integrate from 0 to t to obtain x (t) = ψ (0) e a (t, 0) + ∫ t 0 a (s) (hx) (s) e a (t,s) ∆s + ∫ t 0 c (s) x4̃ (s− τ (s)) e a (t,s) ∆s + ∫ t 0 b (s) g (x (s) ,x (s− τ (s))) e a (t,s) ∆s,(3.6) letting ∫ t 0 c (s) x4̃ (s− τ (s)) e a (t,s) ∆s = ∫ t 0 c (s) (1 − τ4 (s)) ( 1 − τ4 (s) ) x4̃ (s− τ (s)) e a (t,s) ∆s. by performing an integration by parts, we obtain∫ t 0 c (s) x4̃ (s− τ (s)) e a (t,s) ∆s = c (t) 1 − τ4 (t) x (t− τ (t)) − c (0) 1 − τ4 (0) ψ (−τ (0)) e a (t, 0) − ∫ t 0 µ (s) xσ (s− τ (s)) e a (t,s) ∆s,(3.7) where µ (s) is given by (3.4). we obtain (3.3) by substituting (3.7) in (3.6). since each step is reversible, the converse follows easily. this completes the proof. � burton [8] observed that krasnoselskii result can be more interesting in applications with certain changes and formulated in theorem 5 below (see [8] for the proof). 114 belaid, ardjouni and djoudi definition 1. let (m,d) be a metric space and f : m → m. f is said to be a large contraction if ϕ,ψ ∈ m with ϕ 6= ψ, then d (fϕ,fψ) < d (ϕ,ψ), and if for all � > 0, there exists η < 1 such that [ϕ,ψ ∈ m, d (ϕ,ψ) ≥ �] =⇒ d (fϕ,fψ) ≤ ηd (ϕ,ψ) . theorem 4 (burton). let (m,d) be a complete metric space and f be a large contraction. suppose there is x ∈ m and ρ > 0 such that d (x,fnx) ≤ ρ for all n ≥ 1. then f has a unique fixed point in m. below, we state krasnoselskii-burton’s hybrid fixed point theorem which enables us to establish a stability result of the trivial solution of (1.1). for more details on krasnoselskii’s captivating theorem we refer to smart [14] or [10]. theorem 5 (krasnoselskii-burton). let m be a closed bounded convex non-empty subset of a banach space (s,‖.‖). suppose that a, b map m into m and that (i) for all x,y ∈ m =⇒ ax + by ∈ m, (ii) a is continuous and am is contained in a compact subset of m, (iii) b is a large contraction. then there is z ∈ m with z = az + bz. here we manipulate function spaces defined on infinite t-intervals. so for compactness, we need an extension of arzela-ascoli theorem. this extension is taken from [10, theorem 1.2.2, p. 20] and is as follows. theorem 6. let q : [0,∞) ∩t −→ r+ be a rd-continuous function such that q (t) −→ 0 as t −→∞. if {ϕn (t)} is an equicontinuous sequence of rm-valued functions on [0,∞)∩t with |ϕn (t)| ≤ q (t) for t ∈ [0,∞) ∩t, then there is a subsequence that converges uniformly on [0,∞) ∩t to a rd-continuous function ϕ (t) with |ϕ (t)| ≤ q (t) for t ∈ [0,∞) ∩t, where |.| denotes the euclidean norm on rm. 4. stability by krasnoselskii-burton’s theorem from the existence theory which can be found in [10], we conclude that for each continuous initial function ψ : [m0, 0] ∩t → r, there exists a continuous solution x (t, 0,ψ) which satisfies (1.1) on an interval [0,σ) ∩t for some σ > 0 and x (t, 0,ψ) = ψ (t), t ∈ [m0, 0] ∩t. we need the following stability definitions taken from [10]. definition 2. the zero solution of (1.1) is said to be stable at t = 0 if for each � > 0, there exists δ > 0 such that ψ : [m0, 0] ∩t → (−δ,δ) implies that |x (t)| < � for t ≥ m0. definition 3. the zero solution of (1.1) is said to be asymptotically stable if it is stable at t = 0 and δ > 0 exists such that for any rd-continuous function ψ : [m0, 0] ∩t → (−δ,δ), the solution x with x (t) = ψ (t) on [m0, 0] ∩t tends to zero as t →∞. to apply theorem 5, we have to choose carefully a banach space depending on the initial function ψ and construct two mappings, a large contraction and a compact operator which obey the conditions of the theorem. so let s be the banach space of rd-continuous bounded functions ϕ : [m0,∞)∩t → r with the supremum norm ‖.‖. let l > 0 and define the set sψ = {ϕ ∈ s | ϕ is lipschitzian |ϕ (t)| ≤ l, t ∈ [m0,∞) ∩t, ϕ (t) = ψ (t) if t ∈ [m0, 0] ∩t and ϕ (t) → 0 as t →∞} . clearly, if {ϕn} is a sequence of k-lipschitzian functions converging to a function ϕ, then |ϕ (u) −ϕ (v)| ≤ |ϕ (u) −ϕn (u)| + |ϕn (u) −ϕn (v)| + |ϕn (v) −ϕ (v)| ≤ ‖ϕ−ϕn‖ + k |u−v| + ‖ϕ−ϕn‖ . consequently, as n → ∞, we see that ϕ is k-lipschitzian. it is clear that sψ is convex, bounded and complete endowed with ‖.‖. stability in totally nonlinear neutral dynamic equations 115 for ϕ ∈ sψ and t ∈ [0,∞) ∩t, define the maps a, b and c on sψ as follows (aϕ) (t) := c (t) 1 − τ4 (t) ϕ (t− τ (t)) + ∫ t 0 b (s) g (ϕ (s) ,ϕ (s− τ (t))) e a (t,s) ∆s − ∫ t 0 µ (s) ϕσ (s− τ (s)) e a (t,s) ∆s,(4.1) (bϕ) (t) := [ ψ (0) − c (0) 1 − τ4 (0) ψ (−τ (0)) ] e a (t, 0) + ∫ t 0 a (s) (hx) (s) e a(t,s)∆s,(4.2) and (4.3) (cϕ) (t) := (aϕ) (t) + (bϕ) (t) . if we are able to prove that c possesses a fixed point ϕ on the set sψ, then x (t, 0,ψ) = ϕ (t) for t ∈ [0,∞) ∩t, x (t, 0,ψ) = ψ (t) on [m0, 0] ∩t, x (t, 0,ψ) satisfies (1.1) when its derivative exists and x (t, 0,ψ) → 0 as t →∞. let α (t) = c(t) 1−τ4(t) and assume that there are constants k1,k2,k3 > 0 such that for 0 ≤ t1 < t2, (4.4) ∣∣∣∣ ∫ t2 t1 a (u) ∆u ∣∣∣∣ ≤ k1 |t2 − t1| , (4.5) |τ (t2) − τ (t1)| ≤ k2 |t2 − t1| , and (4.6) |α (t2) −α (t1)| ≤ k3 |t2 − t1| . suppose that for t ∈ [0,∞) ∩t, (4.7) |µ (t)| ≤ δa (t) , (4.8) (n1 + n2) |b (t)| ≤ βa (t) , (4.9) sup t≥0 |α (t)| = α0, and that (4.10) j (α0 + β + δ) < 1, (4.11) max (|h (−l)| , |h (l)|) ≤ 2l j , where α0, β, δ and j are constants with j > 3. choose γ > 0 small enough and such that (4.12) ( 1 + ∣∣∣∣ c (0)1 − τ4 (0) ∣∣∣∣ ) γ + 3l j ≤ l. the chosen γ in the relation (4.12) is used below in lemma 5 to show that if � = l and if ‖ψ‖ < γ, then the solutions satisfy |x (t, 0,ψ)| < �. assume further that (4.13) t− τ (t) →∞ as t →∞ and ∫ t 0 a (u) ∆u →∞ as t →∞, (4.14) α (t) → 0 as t →∞, (4.15) µ (t) a (t) → 0 as t →∞, 116 belaid, ardjouni and djoudi and (4.16) b (t) a (t) → 0 as t →∞. we begin with the following theorem (see [1]) and for convenience we present its proof below. in the next theorem, we prove that for a well chosen function h, the mapping h given by (3.5) is a large contraction on the set sψ. so let us make the following assumptions on the function h : r → r. (h1) h : r → r is continuous on [−l,l] and differentiable on (−l,l), (h2) the function h is strictly increasing on [−l,l], (h3) sup t∈(−l,l) h′ (t) ≤ 1. theorem 7. let h : r → r be a function satisfying (h1) − (h3). then the mapping h in (3.5) is a large contraction on the set sψ. proof. let φ,ϕ ∈ sψ with φσ 6= ϕσ. then φσ (t) 6= ϕσ (t) for some t ∈ t. let us denote the set of all such t by d (φ,ϕ), i.e., d (φ,ϕ) = {t ∈ t : φσ (t) 6= ϕσ (t)} . for all t ∈ d (φ,ϕ), we have |(hφ) (t) − (hϕ) (t)| = |φσ (t) −h (φσ (t)) −ϕσ (t) + h (ϕσ (t))| = |φσ (t) −ϕσ (t)| ∣∣∣∣1 − ( h (φσ (t)) −h (ϕσ (t)) φσ (t) −ϕσ (t) )∣∣∣∣ .(4.17) since h is a strictly increasing function, we have (4.18) h (φσ (t)) −h (ϕσ (t)) φσ (t) −ϕσ (t) > 0 for all t ∈ d (φ,ϕ) . for each fixed t ∈ d (φ,ϕ), define the interval ut ⊂ [−l,l] by ut = { (ϕσ (t) ,φσ (t)) if φσ (t) > ϕσ (t) , (φσ (t) ,ϕσ (t)) if ϕσ (t) > φσ (t) . the mean value theorem implies that for each fixed t ∈ d (φ,ϕ), there exists a real number ct ∈ ut such that h (φσ (t)) −h (ϕσ (t)) φσ (t) −ϕσ (t) = h′ (ct) . by (h2) and (h3), we have (4.19) 0 ≤ inf u∈(−l,l) h′ (u) ≤ inf u∈ut h′ (u) ≤ h′ (ct) ≤ sup u∈ut h′ (u) ≤ sup u∈(−l,l) h′ (u) ≤ 1. hence, by (4.17)-(4.19), we obtain (4.20) |(hφ) (t) − (hϕ) (t)| ≤ ∣∣∣∣1 − inf u∈(−l,l) h′ (u) ∣∣∣∣ |φσ (t) −ϕσ (t)| , for all t ∈ d (φ,ϕ). then by (h3), we have ‖hφ−hϕ‖≤‖φ−ϕ‖ . now, choose a fixed � ∈ (0, 1) and assume that φ and ϕ are two functions in sψ satisfying � ≤ sup t∈d(φ,ϕ) |φ (t) −ϕ (t)| = ‖φ−ϕ‖ . if |φσ (t) −ϕσ (t)| ≤ � 2 for some t ∈ d (φ,ϕ), then by (4.19) and (4.20), we get (4.21) |(hφ) (t) − (hϕ) (t)| ≤ |φσ (t) −ϕσ (t)| ≤ 1 2 ‖φ−ϕ‖ . since h is continuous and strictly increasing, the function h ( u + � 2 ) − h (u) attains its minimum on the closed and bounded interval [−l,l]. thus, if � 2 ≤ |φσ (t) −ϕσ (t)| for some t ∈ d (φ,ϕ), then by (h2) and (h3), we conclude that 1 ≥ h (φσ (t)) −h (ϕσ (t)) φσ (t) −ϕσ (t) > λ, stability in totally nonlinear neutral dynamic equations 117 where λ := 1 2l min { h ( u + � 2 ) −h (t) , u ∈ [−l,l] } > 0. hence, (4.17) implies (4.22) |(hφ) (t) − (hϕ) (t)| ≤ (1 −λ)‖φ−ϕ‖ . consequently, combining (4.21) and (4.22), we obtain |(hφ) (t) − (hϕ) (t)| ≤ η‖φ−ϕ‖ , where η = max { 1 2 , 1 −λ } < 1. the proof is complete. � by step we will prove the fulfillment of (i), (ii) and (iii) in theorem 5. lemma 4. suppose that (3.1), (3.2), (4.7)-(4.10) and (4.13) are true. for a defined in (4.1), if ϕ ∈ sψ, then |(aϕ) (t)| ≤ l/j < l. moreover, (aϕ) (t) → 0 as t →∞. proof. using the conditions (4.7)-(4.10) and the expression (4.1) of the map a, we get |(aϕ) (t)| ≤ ∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t)) ∣∣∣∣ + ∫ t 0 |b (t)| |g (ϕ (s) ,ϕ (s− τ (s)))|e a (t,s) ∆s + ∫ t 0 |µ (s)| |ϕσ (s− τ (s))|e a (t,s) ∆s ≤ α0l + ∫ t 0 (n1 + n2) |b (t)|le a (t,s) ∆s + l ∫ t 0 |µ (s)|e a (t,s) ∆s ≤ l { α0 + ∫ t 0 βa (s) e a (t,s) ∆s + ∫ t 0 δa (s) e a (t,s) ∆s } ≤ l (α0 + β + δ) ≤ l j < l. so asψ is bounded by l as required. let ϕ ∈ sψ be fixed. we will prove that (aϕ) (t) → 0 as t → 0. due to the conditions t−τ (t) →∞ as t → ∞ in (4.13) and (4.9), it is obvious that the first term on the right hand side of a tends to 0 as t →∞. that is ∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t)) ∣∣∣∣ ≤ α0 |ϕ (t− τ (t))|→ 0 as t →∞. it is left to show that the two remaining integral terms of a go to zero as t →∞. let � > 0 be given. find t such that |ϕσ (t− τ (t))| < � for t ≥ t. then we have∣∣∣∣ ∫ t 0 µ (s) ϕσ (s− τ (s)) e a (t,s) ∆s ∣∣∣∣ ≤ ∫ t 0 |µ (s) ϕσ (s− τ (s))|e a (t,s) ∆s + ∫ t t |µ (s)| |ϕσ (s− τ (s))|e a (t,s) ∆s ≤ le a (t,t) ∫ t 0 |µ (s)|e a (t,s) ∆s + � ∫ t t |µ (s)|e a (t,s) ∆s ≤ lδe a (t,t) + �δ. the term lδe a (t,t) is arbitrarily small as t → ∞, because of (4.13). the remaining integral term in a goes to zero by just a similar argument. this ends the proof. � lemma 5. let (h1) − (h3), (3.1), (3.2), (4.7)-(4.11) and (4.13) hold. for a, b defined in (4.1) and (4.2), if φ,ϕ ∈ sψ are arbitrary, then ‖bϕ + aφ‖≤ l. moreover, b is a large contraction on sψ with a unique fixed point in sψ and bϕ (t) → 0 as t →∞. 118 belaid, ardjouni and djoudi proof. using the definitions (4.1), (4.2) of a and b and applying (4.7)-(4.11), we obtain |(bϕ) (t) + (aφ) (t)| ≤ ( 1 + ∣∣∣∣ c (0)1 − τ4 (0) ∣∣∣∣ ) ‖ψ‖e a (t, 0) + α0l + l ∫ t t |µ (s)|e a (t,s) ∆s + ∫ t 0 (n1 + n2) |b (s)|le a (t,s) ∆s + 2l j ∫ t 0 a (s) e a (t,s) ∆s ≤ ( 1 + ∣∣∣∣ c (0)1 − τ4 (0) ∣∣∣∣ ) ‖ψ‖ + (α0 + β + δ) l + 2l j ≤ ( 1 + ∣∣∣∣ c (0)1 − τ4 (0) ∣∣∣∣ ) ‖ψ‖ + l j + 2l j , by the monotonicity of the mapping h. so from the above inequality, by choosing the initial function ψ having small norm, say ‖ψ‖ < γ, then, and referring to (4.12), we obtain |(bϕ) (t) + (aφ) (t)| ≤ ( 1 + ∣∣∣∣ c (0)1 − τ4 (0) ∣∣∣∣ ) γ + 3l j ≤ l. since 0 ∈ sψ, we have also proved that |(bϕ) (t)| ≤ l. the proof that bϕ is lipschitzian is similar to that of the map aϕ below. to see that b is a large contraction on sψ with a unique fixed point, we know from theorem 7 that h (ϕ) = ϕσ −h (ϕσ) is a large contraction within the integrand. thus, for any �, from the proof of that theorem 7, we have found η < 1 such that |(bϕ) (t) − (aφ) (t)| ≤ ∫ t 0 a (s) |(hφ) (s) − (hϕ) (s)|e a (t,s) ∆s ≤ η ∫ t 0 a (s)‖ϕ−φ‖e a (t,s) ∆s ≤ η‖ϕ−φ‖ . to prove that (bϕ) (t) → 0 as t → ∞, we use (4.13) for the first term, and for the second term, we argue as above for the map a. � lemma 6. suppose (3.1), (3.2), (4.7)-(4.10) hold. then the mapping a is continuous on sψ. proof. let ϕ,φ ∈ sψ, then |(aϕ) (t) − (aφ) (t)| ≤ {α0 |ϕ (t− τ (t)) −φ (t− τ (t))| + ∣∣∣∣ ∫ t 0 b (s) [g (ϕ (s) ,ϕ (s− τ (s))) −g(φ(s),φ(s− τ(s)))] e a (t,s) ∆s ∣∣∣∣ + ∣∣∣∣ ∫ t 0 µ (s) [ϕσ (s− τ (s)) −φσ (s− τ (s))] e a (t,s) ∆s ∣∣∣∣ } ≤ α0 ‖ϕ−φ‖ + ∫ t 0 (n1 + n2) |b (s)|‖ϕ−φ‖e a (t,s) ∆s + ‖ϕ−φ‖ ∫ t 0 |µ (s)|e a (t,s) ∆s ≤ (α0 + β + δ)‖ϕ−φ‖ ∫ t 0 a (s) e a (t,s) ∆s ≤ (α0 + β + δ)‖ϕ−φ‖≤ (1/j)‖ϕ−φ‖ . let � > 0 be arbitrary. define η = �j. then for ‖ϕ−φ‖≤ η, we obtain ‖aϕ−aφ‖≤ 1 j ‖ϕ−φ‖≤ �. therefore, a is continuous. � stability in totally nonlinear neutral dynamic equations 119 lemma 7. let (3.1), (3.2), (4.4)-(4.9) and (4.14)-(4.16) hold. the function aϕ is lipschitzian and the operator a maps sψ into a compact subset of sψ proof. let ϕ ∈ sψ and let 0 ≤ t1 < t2. then |(aϕ) (t2) − (aϕ) (t1)| ≤ ∣∣∣∣ c (t2)1 − τ4 (t2)ϕ (t2 − τ (t2)) − c (t1)1 − τ4 (t1)ϕ (t1 − τ (t1)) ∣∣∣∣ + ∣∣∣∣ ∫ t2 0 µ (s) ϕσ (s− τ (s)) e a (t2,s) ∆s− ∫ t1 0 µ (s) ϕσ (s− τ (s)) e a (t1,s) ∆s ∣∣∣∣ + ∣∣∣∣ ∫ t2 0 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t2,s) ∆s − ∫ t1 0 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t1,s) ∆s ∣∣∣∣ .(4.23) by hypotheses (4.5)-(4.6), we have |α (t2) ϕ (t2 − τ (t2)) −α (t1) ϕ (t1 − τ (t1))| ≤ |α (t2)| |ϕ (t2 − τ (t2)) −ϕ (t1 − τ (t1))| + |ϕ (t1 − τ (t1))| |α (t2) −α (t1)| ≤ α0k |(t2 − t1) − (τ (t2) − τ (t1))| + lk3 |t2 − t1| ≤ (α0k + α0kk2 + lk3) |t2 − t1| ,(4.24) where k is the lipschitz constant of ϕ. by hypotheses (4.4) and (4.7), we have ∣∣∣∣ ∫ t2 0 µ (s) ϕσ (s− τ (s)) e a (t2,s) ∆s− ∫ t1 0 µ (s) ϕσ (s− τ (s)) e a (t1,s) ∆s ∣∣∣∣ = ∣∣∣∣ ∫ t1 0 µ (s) ϕσ (s− τ (s)) e a (t1,s) (e a (t2, t1) − 1) ∆s + ∫ t2 t1 µ (s) ϕσ (s− τ (s)) e a (t2,s) ∆s ∣∣∣∣ ≤ l |e a (t2, t1) − 1| ∫ t1 0 δa (s) e a (t1,s) ∆s + l ∫ t2 t1 |µ (s)|e a (t2,s) ∆s ≤ lδ ∫ t2 t1 a (s) ∆s + l ∫ t2 t1 e a (t2,s) (∫ s t1 |µ (v)|∆v )4 4s ≤ lδ ∫ t2 t1 a (s) ∆s + l {[ e a (t2,s) ∫ s t1 |µ (v)|∆v ]t2 t1 + ∫ t2 t1 a (s) e a (t2,s) ∫ s t1 |µ (v)|∆v∆s } ≤ lδ ∫ t2 t1 a (s) ∆s + l ∫ t2 t1 |µ (s)|∆s ( 1 + ∫ t2 t1 a (s) e a (t2,s) ∆s ) ≤ lδ ∫ t2 t1 a (s) ∆s + 2l ∫ t2 t1 |µ (s)|∆s ≤ lδ ∫ t2 t1 a (s) ∆s + 2lδ ∫ t2 t1 a (s) ∆s ≤ 3lδk1 |t2 − t1| .(4.25) 120 belaid, ardjouni and djoudi similarly, by (4.4) and (4.8), we deduce∣∣∣∣ ∫ t2 0 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t2,s) ∆s − ∫ t1 0 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t1,s) ∆s ∣∣∣∣ = ∣∣∣∣ ∫ t1 0 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t1,s) (e a (t2, t1) − 1) ∆s + ∫ t2 t1 b (s) g (ϕ (s) ,ϕ (s− τ (s))) e a (t2,s) ∆s ∣∣∣∣∣ ≤ l |e a (t2, t1) − 1| ∫ t1 0 βa (s) e a (t1,s) ∆s + (n1 + n2) l ∫ t2 t1 |b (s)|e a (t2,s) ∆s ≤ lβ ∫ t2 t1 a (u) ∆u + (n1 + n2) l ∫ t2 t1 e a (t2,s) (∫ s t1 |b (v)|∆v )4 4s ≤ lβ ∫ t2 t1 a (u) ∆u + (n1 + n2) l   [ e a (t2,s) ∫ s t1 |b (v)|∆v ]t2 t1 + ∫ t2 t1 a (s) e a (t2,s) ∫ s t1 |b (v)|∆v∆s } ≤ lβ ∫ t2 t1 a (u) ∆u + (n1 + n2) l ∫ t2 t1 |b (s)|∆s ( 1 + ∫ t2 t1 a (s) e a (t2,s) ∆s ) ≤ lβ ∫ t2 t1 a (u) ∆u + 2 (n1 + n2) l ∫ t2 t1 |b (s)|∆s ≤ lβ ∫ t2 t1 a (u) ∆u + 2lβ ∫ t2 t1 a (s) ∆s ≤ 3lβk1 |t2 − t1| .(4.26) thus, by substituting (4.24)-(4.26) in (4.23), we obtain |(aϕ) (t2) − (aϕ) (t1)| ≤ (α0k + α0kk2 + lk3) |t2 − t1| + 3lδk1 |t2 − t1| + 3lβk1 |t2 − t1| ≤ k |t2 − t1| ,(4.27) for a constant k > 0. this shows that aϕ is lipschitzian if ϕ is and that asψ is equicontinuous. next, we notice that for arbitrary ϕ ∈ sψ, we have |aϕ (t)| ≤ ∣∣∣∣ c (t)1 − τ4 (t)ϕ (t− τ (t)) ∣∣∣∣ + ∫ t 0 |b (s)| |g (ϕ (s) ,ϕ (s− τ (s)))|e a (t,s) ∆s + ∫ t 0 |µ (s) ϕ (s− τ (s))|e a (t,s) ∆s ≤ l |α (t)| + (n1 + n2) l ∫ t 0 a (s) [|b (s)|/a (s)] e a (t,s) ∆s + l ∫ t 0 a (s) [|µ (s)|/a (s)] e a (t,s) ∆s = q (t) , because of (4.14)-(4.16). using a method like the one used for the map a, we see that q (t) → 0 as t →∞. by theorem 6, we conclude that the set asψ resides in a compact set. � stability in totally nonlinear neutral dynamic equations 121 theorem 8. let l > 0. suppose that the conditions (h1)−(h3), (3.1), (3.2) and (4.14)-(4.16) hold. if ψ is a given initial function which is sufficiently small, then there is a solution x (t, 0,ψ) of (1.1) with |x (t, 0,ψ)| ≤ l and x (t, 0,ψ) → 0 as t →∞. proof. from lemmas 4 and 7 we have a is bounded by l, lipschitzian and (aφ) (t) → 0 as t →∞. so a maps sψ into sψ. from lemmas 5 and 7 for arbitrary, we have φ,ϕ ∈ sψ, bϕ+aφ ∈ sψ since both aφ and bϕ are lipschitzian bounded by l and (bϕ) (t) → 0 as t →∞. from lemmas 6 and 7, we have proved that a is continuous and asψ resides in a compact set. thus, all the conditions of theorem 5 are satisfied. therefore, there exists a solution of (1.1) with |x (t, 0,ψ)| ≤ l and x (t, 0,ψ) → 0 as t →∞. � 5. stability and compactness referring to burton [10], except for the fixed point method, we know of no other way proving that solutions of (1.1) converge to zero. nevertheless, if all we need is stability and not asymptotic stability, then we can avoid conditions (4.14)-(4.16) and still use krasnoselskii-burton’s theorem on a banach space endowed with a weighted norm. let g : [m0,∞)∩t → [1,∞) be any strictly increasing and rd-continuous function with g (m0) = 1, g (s) →∞ as s →∞. let ( s, |.|g ) be the banach space of rd-continuous function ϕ : [m0,∞)∩t → r for which |ϕ|g := sup t≥m0 ∣∣∣∣ϕ (t)g (t) ∣∣∣∣ < ∞, exists. we continue to use ‖.‖ as the supremum norm of any ϕ ∈ s provided ϕ bounded. also, we use ‖ψ‖ as the bound of the initial function. further, in a similar way as theorem 7, we can prove that the function h (ϕ) = ϕσ −h (ϕσ) is still a large contraction with the norm |.|g. theorem 9. if the conditions of theorem 8 hold, except for (4.14)-(4.16), then the zero solution of (1.1) is stable. proof. we prove the stability starting at t0 = 0. let � > 0 be given such that 0 < � < l, then for |x| ≤ � find γ∗ with |xσ −h (xσ)| ≤ γ∗ and choose a number γ such that (5.1) γ + γ∗ + � j ≤ �. in fact, since xσ −h (xσ) is increasing on (−l,l), we may take γ∗ = 2� j . thus, inequality (5.1) allows γ > 0. now, remove the condition ϕ (t) → 0 as t → 0 from sψ defined previously and consider the set mψ = {ϕ ∈ s | ϕ lipschitzian, |ϕ (t)| ≤ �, t ∈ [m0,∞) ∩t and ϕ (t) = ψ (t) if t ∈ [m0, 0] ∩t} . define a, b on mψ as before by (4.1), (4.2). we easily check that if ϕ ∈ mψ, then |(aϕ) (t)| < �, and b is a large contraction on mψ. also, by choosing ‖ψ‖ < γ and referring to (5.1), we verify that for ϕ,φ ∈ mψ |(bϕ) (t) + (aφ) (t)| ≤ � and |(bϕ) (t)| ≤ �. amψ is an equicontinuous set. according to [10, theorem 4.0.1], in the space ( s, |.|g ) the set amψ resides in a compact subset of mψ. moreover, 122 belaid, ardjouni and djoudi the operator a : mψ → mψ is continuous. indeed, for ϕ,φ ∈ sψ, |(aϕ) (t) − (aφ) (t)| g (t) ≤ 1 g (t) {α0 |ϕ (t− τ (t)) −φ (t− τ (t))| + ∣∣∣∣ ∫ t 0 b (s) [g (ϕ (s) ,ϕ (s− τ (s))) −g (φ (s− τ (s)))] e a (t,s) ∆s ∣∣∣∣ + ∣∣∣∣ ∫ t 0 µ(s) [ϕσ (s− τ (s)) −φσ (s− τ (s))] e a (t,s) ∆s ∣∣∣∣ } ≤ α0 |ϕ−φ|g + ∫ t 0 |b(s)| ( n1 |ϕ (s) −φ (s)| g (t) + n2 |ϕ (s− τ (s)) −φ (s− τ (s))| g (t) ) e a (t,s) ∆s + ∫ t 0 |µ (s)| |ϕσ (s− τ (s)) −φσ (s− τ (s))| g (t) e a (t,s) ∆s ≤ α0 |ϕ−φ|g + ∫ t 0 |b (s)| [ n1 |ϕ (s) −φ (s)| g (s) g (s) g (t) ] e a (t,s) ∆s + ∫ t 0 |b (s)| [ n2 |ϕ (s− τ (s)) −φ (s− τ (s))| g (s− τ (s)) g (s− τ (s)) g (t) ] e a (t,s) ∆s + ∫ t 0 |µ (s)| [ |ϕσ (s− τ (s)) −φσ (s− τ (s))| gσ (s− τ (s)) gσ (s− τ (s)) g (t) ] e a (t,s) ∆s ≤ α0 |ϕ−φ|g + |ϕ−φ|g ∫ t 0 |b(s)| [ n1g(s) + n2g(s− τ(s)) g(t) ] e a(t,s)∆s + δ |ϕ−φ|g ∫ t 0 a (s) gσ (s− τ (s)) g (t) e a (t,s) ∆s ≤ α0 |ϕ−φ|g + β |ϕ−φ|g ∫ t 0 a (s) e a (t,s) ∆s + δ |ϕ−φ|g ∫ t 0 a (s) e a (t,s) ∆s ≤ 1 j |ϕ−φ|g . the conditions of theorem 5 are satisfied on mψ, and so there exists a fixed point lying in mψ and solving (1.1). � references [1] m. adıvar, y. n. raffoul, existence of periodic solutions in totally nonlinear delay dynamic equations. electron. j. qual. theory differ. equ., spec. ed. 1 (2009), 1–20. [2] a. ardjouni, i. derrardjia and a. djoudi, stability in totally nonlinear neutral differential equations with variable delay, acta math. univ. comenianae, 83 (2014), 119-134. [3] a. ardjouni, a djoudi, existence of periodic solutions for nonlinear neutral dynamic equations with functional delay on a time scale, acta univ. palacki. olomnc., fac. rer. nat., mathematica 52, 1 (2013) 5-19. [4] a. ardjouni, a djoudi, stability in neutral nonlinear dynamic equations on time scale with unbounded delay, stud. univ. babeç-bolyai math. 57(2012), no. 4, 481-496. [5] a. ardjouni, a djoudi, fixed points and stability in linear neutral differential equations with variable delays, nonlinear analysis 74 (2011), 2062-2070. [6] m. bohner, a. peterson, dynamic equations on time scales, an introduction with applications, birkhauser, boston, 2001. [7] m. bohner, a. peterson, advances in dynamic equations on time scales, birkhäuser, boston, 2003. [8] t. a. burton, liapunov functionals, fixed points and stability by krasnoselskii’s theorem, nonlinear stud. 9 (2001), 181–190. [9] t. a. burton, stability by fixed point theory or liapunov theory: a comparaison, fixed point theory, 4(2003), 15-32. [10] t. a. burton, stability by fixed point theory for functional differential equations, dover publications, new york, 2006. stability in totally nonlinear neutral dynamic equations 123 [11] i. derrardjia, a. ardjouni and a. djoudi, stability by krasnoselskii’s theorem in totally nonlinear neutral differential equations, opuscula math. 33(2) (2013), 255-272. [12] s. hilger, ein maβkettenkalkül mit anwendung auf zentrumsmannigfaltigkeiten, ph. d. thesis, universität würzburg, würzburg, 1988. [13] e. r. kaufmann, y. n. raffoul, stability in neutral nonlinear dynamic equations on a time scale with functional delay, dynamic systems and applications 16 (2007) 561-570. [14] d. r. smart, fixed point theorems, cambridge tracts in mathematics, no. 66, cambridge university press, london– new york, 1974. 1department of mathematics and informatics, university of souk ahras, p.o. box 1553, souk ahras, 41000, algeria 2applied mathematics lab, faculty of sciences, department of mathematics, university of annaba, p.o. box 12, annaba 23000, algeria ∗corresponding author: abd ardjouni@yahoo.fr international journal of analysis and applications issn 2291-8639 volume 5, number 2 (2014), 185-190 http://www.etamaths.com generalization of integral inequalities for product of convex functions m. a. latif abstract. in this paper, generalizations of some inequalities for product of convex functions are given. 1. introduction a function f : [a,b] → r, with [a,b] ⊂ r, is said to be convex if whenever, x, y ∈ [a,b] and t ∈ [0, 1] the following inequality holds: f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f (y) . this definition has its origin in jensen’s result from [1] and has opened up a very useful and multi-disciplinary domain of mathematics, namely, convex analysis. a largely applied inequality for convex functions, due to its geometrical significance, is the hermite-hadamard’s inequality which has generated a wide range of directions for extensions and rich mathematical literature. hermite-hadamrd’s inequality is stated as follows: a convex function satisfies: (1.1) f ( a + b 2 ) ≤ 1 b−a ∫ b a f (x) dx ≤ f (a) + f (b) 2 . in a recent paper, pachpatte [2] established the following inequalities for product of convex functions which can be derived from hermite-hadamard’s inequality: theorem 1. [2] let f and g be real valued, nonnegative and convex functions on [a,b]. then (1.2) 3 2 · 1 (b−a)2 ∫ b a ∫ b a ∫ 1 0 f (tx + (1 − t) y) g (tx + (1 − t) y) dtdxdy ≤ 1 b−a ∫ b a f (x) g (x) dx + 1 8 [ m (a,b) + n (a,b) (b−a)2 ] 2010 mathematics subject classification. primary 46c05, 46c99; secondary 26d15, 26d20. key words and phrases. convex function, hermite-hadamard’s inequality, pachpatte’s inequalities, jensen’s inequality. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 185 186 m. a. latif and (1.3) 3 b−a ∫ b a ∫ 1 0 f ( tx + (1 − t) a + b 2 ) g ( tx + (1 − t) a + b 2 ) dtdx ≤ 1 b−a ∫ b a f (x) g (x) dx + 1 4 · 1 + b−a b−a [m (a,b) + n (a,b)] , where m (a,b) = f (a) g (a) + f (b) g (b) n (a,b) = f (a) g (b) + f (b) g (a) . the inequalities (1.2) and (1.3) are valid when the length of the interval [a,b] does not exceed 1. the inequality (1.2) is sharp for linear functions defined on [0, 1], while the inequality (1.3) does not have the same property. in [3], cristescu improved these inequalities by eliminating the condition b−a ≤ 1 and derived the inequalities which are sharp for the whole class of linear functions defined on [0, 1]. the main result from [3] is the following: theorem 2. [3] let f and g be real valued, nonnegative and convex functions on [a,b]. then (1.4) 3 2 · 1 (b−a)2 ∫ b a ∫ b a ∫ 1 0 f (tx + (1 − t) y) g (tx + (1 − t) y) dtdxdy ≤ 1 b−a ∫ b a f (x) g (x) dx + 1 8 [m (a,b) + n (a,b)] and (1.5) 3 b−a ∫ b a ∫ 1 0 f ( tx + (1 − t) a + b 2 ) g ( tx + (1 − t) a + b 2 ) dtdx ≤ 1 b−a ∫ b a f (x) g (x) dx + 1 2 [m (a,b) + n (a,b)] , where m (a,b) and n (a,b) are defined in theorem 1. the main aim of this paper is to generalize the inequalities (1.4) and (1.5). 2. main results let i be an interval of r and let f : i → r be a convex functions on i, h : [a,b] → r be continuous function such that h ([a,b]) ⊂ i and p : [a,b] → r be a positive integrable function a, b ∈ r with a < b. then the jenesen’s inequality f (∫ b a p (x) h (x) dx∫ b a p (x) dx ) ≤ ∫ b a p (x) f (h (x)) dx∫ b a p (x) dx holds. assume that f and p are as above. let us denote p = ∫ b a p(x)dx, h̄ = 1 p ∫ b a p(x)h(x)dx. generalization of integral inequalities 187 the following is the hermite-hdamard type inequality in this case: (2.1) f ( h̄ ) ≤ 1 p ∫ b a f (h(x))p(x)dx ≤ f (h(a)) + f (h(b)) 2 . we now state the following lemma which is very useful in this section: lemma 1. let [a,b] ⊂ r and f : [a,b] → r be a function and h : [a,b] → r be a continuous function such that h ([a,b]) ⊂ [a,b]. then the following statements are equivalent (1) function f is convex on [a,b] (2) for every x, y ∈ [a,b], the function γ : [0, 1] → r defined by γ (t) = f (th (x) + (1 − t) h (y)) is convex on [0, 1] for any positive real number λ. proof. it is a direct consequence of the convexity of the function f. � now we state and prove the main result of this section which will generalize the theorem 2. theorem 3. let f and g be real valued, nonnegative and convex functions on [a,b]. let h : [a,b] → r be continuous function such that h ([a,b]) ⊂ [a,b]and p : [a,b] → r be a positive integrable function. then (2.2) 3 2p2 ∫ b a ∫ b a ∫ 1 0 p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy ≤ 1 p ∫ b a f (h (x)) g (h (x)) p (x) dx + 1 8 [ m ′ (a,b) + n ′ (a,b) ] , where m ′ (a,b) = f(h(a))g(h(a)) + f(h(b))g(h(b)) and n ′ (a,b) = f(h(a))g(h(b)) + f(h(b))g(h(a)). proof. since both functions f and g are convex, for every two points x,y ∈ [a,b] and t ∈ [0, 1], the following inequalities are valid f (th (x) + (1 − t) h (y)) ≤ tf (h (x)) + (1 − t) f (h (y)) and g (th (x) + (1 − t) h (y)) ≤ tg (h (x)) + (1 − t) g (h (y)) multiplying these inequalities side by side, we obtain (2.3) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) ≤ t2f (h (x)) g (h (x)) + (1 − t)2 f (h (y)) g (h (y)) + t (1 − t) [f (h (x)) g (h (y)) + f (h (y)) g (h (x))] . due to lemma 1 and known properties of convex functions, both sides of the inequality (2.3) are integrable. multiplying both sides of (2.3) by p (x) p (y) and 188 m. a. latif integrating both sides of the inequality (2.3) with respect to t over [0, 1], with respect to x and y over [a,b], we get (2.4)∫ b a ∫ b a ∫ 1 0 p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy ≤ 2 3 p ∫ b a f (h (x)) g (h (x)) p(x)dx + 1 3 (∫ b a f (h (x)) p(x)dx )(∫ b a g (h (x)) p(x)dx ) . the convexity property of f and g allow us to use right side of the inequality (2.1) and thus the above inequality (2.4) takes the form: (2.5)∫ b a ∫ b a ∫ 1 0 p (x) p (y) f (th (x) + (1 − t) h (y)) g (th (x) + (1 − t) h (y)) dtdxdy ≤ 2 3 p ∫ b a f (h (x)) g (h (x)) p(x)dx + p2 12 [f (h(a)) + f (h(b))] [g (h(a)) + g (h(b))] = 2 3 p ∫ b a f (h (x)) g (h (x)) p(x)dx + p2 12 [ m ′ (a,b) + n ′ (a,b) ] . multiplying both sides of the inequality (2.5) by 3 2p 2 , we get the desired result. this completes the proof of the theorem. � theorem 4. let f and g be real valued, nonnegative and convex functions on [a,b]. let h : [a,b] → r be continuous function such that h ([a,b]) ⊂ [a,b]and p : [a,b] → r be a positive integrable function. then (2.6) 3 p ∫ b a ∫ 1 0 p(x)f ( th (x) + (1 − t) h̄ ) g ( th (x) + (1 − t) h̄ ) ≤ 1 p ∫ b a f(h(x))g(h(x))p(x)dx + 1 2 [ m ′ (a,b) + n ′ (a,b) ] , where m ′ (a,b) = f(h(a))g(h(a)) + f(h(b))g(h(b)) and n ′ (a,b) = f(h(a))g(h(b)) + f(h(b))g(h(a)). proof. again by the convexity of the functions f and g, we have f ( th (x) + (1 − t) h̄ ) ≤ tf(h(x)) + (1 − t) f ( h̄ ) and g ( th (x) + (1 − t) h̄ ) ≤ tg(h(x)) + (1 − t) g ( h̄ ) generalization of integral inequalities 189 multiplying the above two inequalities side by side, we get (2.7) f ( th (x) + (1 − t) h̄ ) g ( th (x) + (1 − t) h̄ ) ≤ t2f(h(x))g(h(x)) + t (1 − t) [ f(h(x))g(h̄) + g(h(x))f(h̄) ] + (1 − t)2 f(h̄)g(h̄). multiplying both sides of (2.7), by similar arguments as in obtaining (2.4) and using the jensen’s inequality, we have (2.8) ∫ b a ∫ 1 0 p(x)f ( th (x) + (1 − t) h̄ ) g ( th (x) + (1 − t) h̄ ) ≤ 1 3 ∫ b a f(h(x))g(h(x))p(x)dx + 1 6 ∫ b a [ f(h(x))g(h̄) + g(h(x))f(h̄) ] p(x)dx + 1 3 ∫ b a f(h̄)g(h̄)p(x)dx ≤ 1 3 ∫ b a f(h(x))g(h(x))p(x)dx+ 2 3p (∫ b a f(h(x))p(x)dx )(∫ b a g(h(x))p(x)dx ) . an application of the inequality (2.1), gives us (2.9) ∫ b a ∫ 1 0 p(x)f ( th (x) + (1 − t) h̄ ) g ( th (x) + (1 − t) h̄ ) ≤ 1 3 ∫ b a f(h(x))g(h(x))p(x)dx + p 6 [f(h(a)) + f(h(b))] [g(h(a)) + g(h(b))] = 1 3 ∫ b a f(h(x))g(h(x))p(x)dx + p 6 [ m ′ (a,b) + n ′ (a,b) ] . multiplying both sides of (2.9) by 3 p , we get the desired inequality and hence the proof of the theorem is complete. � remark 1. if in theorem 3 and theorem 4, p(x) = 1 and h(x) = x, x ∈ [a,b], then p = b − a, h̄ = a+b 2 , m ′ (a,b) = m(a,b) and n ′ (a,b) = n(a,b). then the inequalities (2.2) and (2.6) reduce to the inequalities (1.4) and (1.5) respectively. this also shows that our results generalize those results proved in theorem 2. references [1] j.l.w.v. jensen, on konvexe funktioner og uligheder mellem middlvaerdier, nyt. tidsskr. math. b., 16 (1905), 49–69. [2] b.g. pachpatte, on some inequalities for convex functions, rgmia research report collection, 6(e) (2003), [online: http://rgmia.vu.edu.au/v6(e).html]. [3] g. cristescu, improved integral inequalities for product of convex functions, journal of inequalities in pure and applied mathematics, 6(2005), issue 2, article 35. [4] g. cristescu and l. lupşa, non-connected convexities and applications, kluwer academic publishers, dordrecht / boston / london, 2002. [5] y. j. cho, m. matic, j. pecaric, two mappings in connections to jensen’s inequality, panamerican mathematical journal, 12(2002), number 1, 43-50. [6] s.s. dragomir and c.e.m. pearce, selected topics on hermite-hadamard inequalities and applications, rgmia monographs, victoria university, 2000. [online: http://rgmia.vu.edu.au/monographs/] 190 m. a. latif [7] j. hadamard, étude sur les propriétés des fonctions entières et en particulier d’une fonction considerée par riemann, j. math pures appl., 58 (1893), 171–215. [8] d.s. mitrinivić, analytic inequalities, springer verlag, berlin/new york, 1970. [9] r. webster, convexity, oxford university press, oxford new york tokyo, 1994. college of science, department of mathematics, university of hail, hail 2440, saudi arabia int. j. anal. appl. (2023), 21:55 existence of solutions via c-class functions in ab-metric spaces with applications n. mangapathi1,3,∗, b. srinuvasa rao2, k.r.k. rao3, m.i. pasha1,3 1department of mathematics, b v raju institute of technology, narsapur, medak-502313,telangana, india 2department of mathematics, dr.b.r.ambedkar university, srikakulam, etcherla-532410, andhra pradesh, india 3department of mathematics, gitam school of science, gitam deemed to be university, hyderabad, rudraram-502329, telangana, india ∗corresponding author: nmp.maths@gmail.com abstract. using c-class functions, we demonstrate a few popular common coupled fixed point theorems on ab-metric spaces and discuss some implications of the main findings. additionally, we provide examples to illustrate the findings and their applications to both homotopy theory and integral equations. 1. introduction fixed point theory plays a significant role in many parts of the development of nonlinear analysis. it has been applied to various fields of engineering and research. this research was inspired by recent work on the extension of the banach contraction principle on ab metric spaces, which was carried out by m. ughade et al. [1] and studied various fixed point results on these spaces. in the further, n. mlaiki et al. [2] and k. ravibabu et al. [3], [4] and p. naresh et al. [5] succeeded in deriving unique coupled common fixed point theorems in ab-metric spaces. sessa [6] began researching common fixed point theorems for weakly commuting pair of mappings in 1982. later, in 1986, jungck [7] expanded the idea of weakly commuting mappings to compatible mappings in metric spaces and proved compatible pair mappings commute on the sets of coincidence point of the involved mappings. when they commute at their coincidence sites, jungck and rhoades [8] received: mar. 23, 2023. 2020 mathematics subject classification. 54h25, 47h10, 54e50. key words and phrases. c-class function; ω-compatible mapping; ab-completeness; coupled fixed points. https://doi.org/10.28924/2291-8639-21-2023-55 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-55 2 int. j. anal. appl. (2023), 21:55 introduced the idea of weak compatibility in 1998 and demonstrated that compatible mappings are weakly compatible but the reverse is not true. however, khan et al. [9] first proposed the idea of modifying distance function, which is a control function that modifies the distance between two locations in a metric space. ansari [10] presented the idea of c-class functions in 2014 and proved the unique fixed point theorems for certain contractive mappings with regard to the c-class functions, which started a lot of work in this field (see. [11], [12], [13], [14], [15], [16], [17]) the idea of coupled fixed point was first developed by guo and lakshmikantham [18] in 1987 . later, employing a weak contractivity type assumption, bhaskar and lakshmikantham [19] developed a novel fixed point theorem for a mixed monotone mapping in a metric space driven with partial ordering. see study results in ( [20], [21], [22], [23], [24]) and related references for additional results on coupled fixed point outcomes. in the framework of ab-metric spaces, the goal of the current study is to establish original common coupled fixed point theorems using c-class functions. additionally, we may provide relevant applications for homotopy, integral equations, and appropriate examples. first we recall some basic results. 2. preliminaries definition 2.1. ( [1]) let = be a non-empty set and b ≥ 1 be given real number. a mapping ab : =n → [0,∞) is called an ab-metric on = if and only if for all υi,a ∈ = i = 1, 2, 3, ..n; the following conditions hold : (ab1) ab(υ1, υ2, ........, υn−1, υn) ≥ 0, (ab2) ab(υ1, υ2, ........, υn−1, υn) = 0 ⇔ υ1 = υ2 = · · · · · · = υn−1 = υn, (ab3) ab(υ1, υ2, ........, υn−1, υn) ≤ b   ab (υ1, υ1, ........, (υ1)n−1,a) +ab (υ2, υ2, ........, (υ2)n−1,a) + · · · · · · + ab (υn−1, υn−1, ........, (υn−1)n−1,a) +ab (υn, υn, ........, (υn)n−1,a)   then the pair (=,ab) is called an ab-metric space. remark 2.1. ( [1]) the class of ab-metric spaces is actually larger than that of a-metric spaces, it should be emphasised. each a-metric space is, in fact, a ab-metric space with b = 1. the opposite isn’t always true, though. a n-dimensional sb-metric space is also a ab-metric space. as a result, a ab-metric with n = 3 is a particular instance of a sb-metric. the example below demonstrates that an ab-metric on = need not be an a-metric on =. example 2.1. ( [1]) let = = [0, +∞), define ab : =n → [0, +∞) as ab (υ1, υ2, ........, υn−1, υn) = ∑n i=1 ∑ i<j |υi − υj| 2 for all υi ∈ =, i = 1, 2, · · · . then (=,ab) is an ab-metric space with b = 2 > 1. int. j. anal. appl. (2023), 21:55 3 definition 2.2. ( [1]) let (=,ab) be a ab-metric space. then, for υ ∈=, r > 0 we defined the open ball bab (υ, r) and closed ball bab [υ, r] with center υ and radius r as follows respectively: bab (υ, r) = {f∈= : ab(f,f, · · · , (f)n−1, υ) < r}, and bab [υ, r] = {f∈= : ab(f,f, · · · , (f)n−1, υ) ≤ r}. lemma 2.1. ( [1]) in a ab-metric space, we have (1) ab(υ, υ, · · · , (υ)n−1,f) ≤ bab(f,f, · · · , (f)n−1, υ); (2) ab(υ, υ, · · · , (υ)n−1,ð) ≤ b(n− 1)ab(υ, υ, · · · , (υ)n−1,f) + b2ab(f,f, · · · , (f)n−1,ð). definition 2.3. ( [1]) if (=,ab) be a ab-metric space. a sequence {υk} in = is said to be: (1) ab-cauchy sequence if, for each � > 0, there exists n0 ∈ n such that ab(υk, υk, · · · · · ·(υk)n−1, υm) < � for each m,k ≥ n0. (2) ab-convergent to a point υ ∈= if, for each � > 0, there exists a positive integer n0 such that ab(υk, υk, · · · · · ·(υk)n−1, υ) < � for all n ≥ n0 and we denote by lim k→∞ υk = υ. (3) if every ab-cauchy sequence is ab-convergent in =, then the ab-metric space (=,ab) is said to be complete. definition 2.4. [10] a continuous mapping γ : [0, +∞) × [0. + ∞) → r is called a c-class function if for all s∗,t∗ ∈ [0,∞), (a) γ(s∗,t∗) ≤ s; (b) γ(s∗,t∗) = s∗ implies that either s∗ = 0 or t∗ = 0. the family of all c-class functions is denoted by c. example 2.2. [10] each of the functions γ : [0, +∞) × [0. + ∞) → r defined below are elements of c. (a) γ(s∗,t∗) = s? − t?; (b) γ(s∗,t∗) = ms∗ where m ∈ (0, 1). (c) γ(s∗,t∗) = s ∗ (1+t?)r where r ∈ (0,∞). (d) γ(s∗,t∗) = s?η(s?) where η : [0,∞) → [0,∞) is continuous function. (e) γ(s∗,t∗) = s? −ϕ(s?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function ϕ : [0,∞) → [0,∞) such that ϕ(s?) = 0 ⇔ s? = 0. (f ) γ(s∗,t∗) = sω(s?,t?) for all s∗,t∗ ∈ [0, +∞) where, the continuous function ω : [0,∞)2 → [0,∞) such that ω(s?,t?) < 1. definition 2.5. [9] a function θ? : [0,∞) → [0,∞) is called an altering distance function if the following properties are satisfied: (a) θ? is nondecreasing and continuous; 4 int. j. anal. appl. (2023), 21:55 (b) θ?(t) = 0 if and only if t = 0. here θ represents the family of all altering distance functions. we must take the following into consideration in order to get our outcomes. 3. main results definition 3.1. let (=,ab) be a ab-metric spaces and suppose ω : =2 → = be a mapping. if ω (℘,$) = ℘, ω ($,℘) = $ for ℘,$ ∈= then (℘,$) is called a coupled fixed point of ω. definition 3.2. let (=,ab) be a ab-metric spaces and suppose ω : =2 → = and λ : = → = be two mappings. an element (℘,$) is said to be a coupled coincident point of ω and λ if f (℘,$) = λ℘, ω ($,℘) = λ$ definition 3.3. let (=,ab) be a ab-metric spaces and suppose ω : =2 → =, λ : = → = be two mappings. an element (℘,$) is said to be a coupled common point of ω and λ if ω (℘,$) = λ℘ = ℘, ω ($,℘) = λ$ = $, definition 3.4. let (=,ab) be a ab-metric space. (a) a pair (ω, λ) is called weakly compatible if λ(ω(℘,$)) = ω(λ℘, λ$) whenever for all ℘,$ ∈= such that f (℘,$) = λ℘, ω ($,℘) = λ$ (b) a pair (ω, λ) is called compatible if lim p→∞ ab(λω(ıp, p), λω(ıp, p) · · · , ω(λıp, λp)) = lim p→∞ ab(λω(p, ıp), λω(p, ıp) · · · , ω(λp, λıp)) = 0 wherever {ıp},{p} are sequences in =, such that lim p→∞ ω(ıp, p) = λıp = ı and lim p→∞ ω(p, ıp) = λp = . lemma 3.1. if the pair (ω, λ) of mappings on the ab-metric space (=,ab) is compatible, then it is weakly compatible. the converse does not hold. proof. let ω(i, j) = λi and ω(j, i) = λj for some i, j ∈=. we have to prove that λω(i, j) = ω(λi, λj) and λω(j, i) = ω(λj, λi). put ıp = i and p = j for every p ∈ n. we have ω(ıp, p) = λıp → λi and ω(p, ıp) = λp → λj. since (ω, λ) is compatible lim p→∞ ab(λω(ıp, p), λω(ıp, p) · · · , ω(λıp, λp)) = lim p→∞ ab(λω(p, ıp), λω(p, ıp) · · · , ω(λp, λıp)) = 0 therefore, λω(i, j) = ω(λi, λj) and λω(j, i) = ω(λj, λi) and hence the pair (ω, λ) is weakly compatible. however, the opposite need not be the case. for example, let = = [0, +∞), define ab : =n → [0, +∞) as ab (υ1, υ2, ........, υn−1, υn) = ∑n i=1 ∑ i<j |υi − υj| 2 for all υi ∈ =, i = 1, 2, · · · . then (=,ab) is an ab-metric space with b = 2 > 1. int. j. anal. appl. (2023), 21:55 5 define two mappings ω : =2 → = by ω(i, j) = { 6i−3j+6n−3 6n for i, j ∈ [0, 1 2 ) n 4 for i, j ∈ [1 2 ,∞) and λ : = → = by λ(i) = { 9i+6n−6 6n for i ∈ [0, 1 2 ) n 4 for i ∈ [1 2 ,∞) now we define the two sequences {ıp},{p} as ıp = 1p and p = 1 + 1 p , then ω(ıp, p) = 3 p +6n−6 6n → n−1 n as p → ∞ and λ(ıp) = 9 p +6n−6 6n → n−1 n as p → ∞, also ω(p, ıp) = 3 p +6n+3 6n → 2n+1 2n as p → ∞ and λ(p) = 9 p +6n+3 6n → 2n+1 2n as p →∞. but lim p→∞ ab(λω(ıp, p), λω(ıp, p) · · · , ω(λıp, λp)) = lim p→∞ ab( 27 p + 36n2 + 18n− 54 36n2 , 27 p + 36n2 + 18n− 54 36n2 · · · , 27 p + 36n2 − 45 36n2 ) = (n− 1)| 2n− 1 4n2 |2 6= 0, if n = 2. and lim p→∞ ab(λω(p, ıp), λω(p, ıp) · · · , ω(λp, λıp)) = lim p→∞ ab( 27 p + 36n2 + 18n + 27 36n2 , 27 p + 36n2 + 18n + 27 36n2 · · · , 27 p + 36n2 + 36 36n2 ) = (n− 1)| 2n− 1 4n2 |2 6= 0, if n = 2. thus the pair (ω, λ) is not compatible. also,then for any i, j ∈ [1 2 ,∞), (n 4 , n 4 )is a coupled coincidence point of ω and λ it is namely that i = j = n 4 , ω(i, j) = 7 8 = λi and ω(j, i) = 7 8 = λj for n = 2 and λω(i, j) = ω(λi, λj), λω(j, i) = ω(λj, λi), showing that ω, λ are weakly compatible maps on =. � theorem 3.1. let (=,ab) be a complete ab-metric space. suppose t : =2 →= and f : =→= be a two mappings satisfying the following: η? ( 2b2ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) ) ≤ γ (η? (ab(f ı, f ı, · · · , fð)) ,θ? (ab(f , f , · · · , ff))) (3.1) for all ı, ,ð,f∈=, where η?,θ? ∈ θ and γ ∈ c a) t (=2) ⊆ f (=); b) pair (t,f ) is compatible; c) f is continuous. then there is a unique common coupled fixed point of t and f in =. proof. let ı0, 0 ∈= be arbitrary, and from (a), we construct the sequences {ıp} ,{p} , in = as t (ıp, p) = f ıp+1 = ℵp, t (p, ıp) = f p+1 = υp, where p = 0, 1, 2, . . . . 6 int. j. anal. appl. (2023), 21:55 now from (3.1), we have η? ( 2b2ab(ℵ1,ℵ1, · · · ,ℵ2) ) = η? ( 2b2ab(t (ı1, 1),t (ı1, 1), · · · ,t (ı2, 2)) ) ≤ γ (η? (ab(f ı1, f ı1, · · · , f ı2)) ,θ? (ab(f 1, f 1, · · · , f 2))) ≤ η? (ab(f ı1, f ı1, · · · , f ı2)) ≤ η? (ab(ℵ0,ℵ0, · · · ,ℵ1)) by the definition of η?, we have that ab(ℵ1,ℵ1, · · · ,ℵ2) ≤ 1 2b2 ab(ℵ0,ℵ0, · · · ,ℵ1). also η? ( 2b2ab(ℵ2,ℵ2, · · · ,ℵ3) ) = η? ( 2b2ab(t (ı2, 2),t (2, 2), · · · ,t (ı3, 3)) ) ≤ γ (η? (ab(f ı2, f ı2, · · · , f ı3)) ,θ? (ab(f 2, f 2, · · · , f 3))) ≤ η? (ab(f ı2, f ı2, · · · , f ı3)) ≤ η? (ab(ℵ1,ℵ1, · · · ,ℵ2)) by the definition of η?, we have that ab(ℵ2,ℵ2, · · · ,ℵ3) ≤ 1 2b2 ab(ℵ1,ℵ1, · · · ,ℵ2) ≤ 1 (4b2)2 ab(ℵ0,ℵ0, · · · ,ℵ1) continuing this process, we can conclude that ab(ℵp,ℵp, · · · ,ℵp+1) ≤ 1 (2b2)p ab(ℵ0,ℵ0, · · · ,ℵ1) → 0 as p →∞. that is lim p→∞ ab(ℵp,ℵp, · · · ,ℵp+1) = 0. similarly, we can prove that lim p→∞ ab(υp, υp, · · · , υp+1) = 0. now for q > p, by use of (ab3), we have ab (ℵp,ℵp, · · · ,ℵn−1,ℵq) ≤ b   ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) + · · · · · · + ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +ab (ℵq,ℵq, ........, (ℵq)n−1,ℵp+1)   int. j. anal. appl. (2023), 21:55 7 ≤ b(n− 1)ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +bab (ℵq,ℵq, ........, (ℵq)n−1,ℵp+1) ≤ b(n− 1)ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +b2ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵq) ≤ b(n− 1)ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +b3(n− 1)ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵp+2) +b4ab (ℵp+2,ℵp+2, ........, (ℵp+2)n−1,ℵq) ≤ b(n− 1)ab (ℵp,ℵp, ........, (ℵp)n−1,ℵp+1) +b3(n− 1)ab (ℵp+1,ℵp+1, ........, (ℵp+1)n−1,ℵp+2) +b5(n− 1)ab (ℵp+2,ℵp+2, ........, (ℵp+2)n−1,ℵp+3) +b7(n− 1)ab (ℵp+3,ℵp+3, ........, (ℵp+3)n−1,ℵp+4) + . . . + b2q−2p−2(n− 1)ab (ℵq−2,ℵq−2, ........, (ℵq−2)n−1,ℵq−1) +b2q−2p−3ab (ℵq−1,ℵq−1, ........, (ℵq−1)n−1,ℵq) ≤ (n− 1) ( b 1 (2b2)p + b3 1 (2b2)p+1 + b5 1 (2b2)p+2 + . . . + b2q−2p−2 1 (2b2)q−2 ) ab (ℵ0,ℵ0, · · · ,ℵ1) +b2q−2p−3 1 (2b2)q−1 ab (ℵ0,ℵ0, · · · ,ℵ1) ≤ (n− 1)b 1 (2b2)p ( 1 + b2 1 2b2 + b4( 1 2b2 )2 + . . . + b2q−2p−4( 1 2b2 )q−p−2 ) ab (ℵ0,ℵ0, · · · ,ℵ1) +b2q−2p−3( 1 2b2 )q−p−1ab (ℵ0,ℵ0, · · · ,ℵ1) ≤ (n− 1)b 1 (2b2)p ( 1 + 1 2 + 1 22 + . . . + 1 2q−p−2 + · · · ) ab (ℵ0,ℵ0, · · · ,ℵ1) ≤ 2(n− 1)b 1 (2b2)p ab (ℵ0,ℵ0, · · · ,ℵ1) → 0 as p,q →∞. hence {ℵp} is a cauchy sequence in = . we can also demonstrate that {υp}, is cauchy sequence in =. therefore, lim p,q→∞ ab(ℵp,ℵp, · · · ,ℵq) = 0, and lim p,q→∞ ab(υp, υp, · · · , υq) = 0. since (=,ab) is complete, there exist ℵ, υ ∈= such that lim p→∞ ℵp = lim p→∞ t (ıp, p) = lim p→∞ f ıp+1 = ℵ lim p→∞ υp = lim p→∞ t (p, ıp) = lim p→∞ f p+1 = υ. since f : =→= is continuous lim p→∞ f 2ıp+1 = fℵ and lim p→∞ f t (ıp, p) = fℵ 8 int. j. anal. appl. (2023), 21:55 lim p→∞ f 2p+1 = f υ and lim p→∞ f t (p, ıp) = f υ since {t,f} is compatible, we have f (f ıp, f p) → fℵ and f (f p, f ıp) → f υ lim p→∞ ab(t (f ıp, f p),t (f ıp, f p) · · · , f t (ıp, p)) = 0. (3.2) lim p→∞ ab(t (f p, f ıp),t (f p, f ıp) · · · , f t (p, ıp)) = 0. (3.3) now, we prove that fℵ = t (ℵ, υ) and f υ = t (υ,ℵ). for all p ≥ 0, we have ab (fℵ, fℵ, · · · , (fℵ)n−1,t (f ıp, f p)) ≤ b ( (n− 1)ab (fℵ, fℵ, ........, (fℵ)n−1, f t (ıp, p)) +ab (f t (ıp, p), f t (ıp, p), ........, (f t (ıp, p))n−1,t (f ıp, f p)) ) ≤ (n− 1)bab (fℵ, fℵ, ........, (fℵ)n−1, f t (ıp, p)) +b2ab (t (f ıp, f p),t (f ıp, f p), ........, (t (f ıp, f p))n−1, f t (ıp, p)) on taking limits as p →∞ and from (3.2 ) we get ab (fℵ, fℵ, · · · , (fℵ)n−1,t (ℵ, υ)) = 0. similarly it is easy to see that ab (f υ, f υ, · · · , (f υ)n−1,t (υ,ℵ)) = 0. thus, t (ℵ, υ) = fℵ and t (υ,ℵ) = f υ. hence (ℵ, υ) is a coupled coincidence point of t and f . now we prove that fℵ = ℵ and f υ = υ. now consider η? ( 2b2ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ) = η? ( 2b2ab (t (ℵ, υ),t (ℵ, υ), · · · , (t (ℵ, υ))n−1,t (ıp, p)) ) ≤ γ (η? (ab(fℵ, fℵ, · · · , f ıp)) ,θ? (ab(f υ, f υ, · · · , f ))) ≤ η? (ab(fℵ, fℵ, · · · , f ıp)) by the definition of η?, we have ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ≤ 1 2b2 ab(fℵ, fℵ, · · · , f ıp) letting p → ∞ , we get ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) ≤ 12b2ab(fℵ, fℵ, · · · ,ℵ) which implies that fℵ = ℵ. similarly, we can prove f υ = υ. therefore, t (ℵ, υ) = fℵ = ℵ and t (υ,ℵ) = f υ = υ. thus,(ℵ, υ) is a common coupled point of t and f . in order to demonstrate uniqueness, we first assume that (ℵ?, υ?) is a another coupled common fixed point of t and f . now η? ( 2b2ab (ℵ,ℵ, · · · , (ℵ)n−1,ℵ?) ) = η? ( 2b2ab (t (ℵ, υ),t (ℵ, υ), · · · , (t (ℵ, υ))n−1,t (ℵ?, υ?)) ) ≤ γ (η? (ab(fℵ, fℵ, · · · , fℵ?)) ,θ? (ab(f υ, f υ, · · · , f υ?))) ≤ η? (ab(fℵ, fℵ, · · · , fℵ?)) ≤ η? (ab(ℵ,ℵ, · · · ,ℵ?)) int. j. anal. appl. (2023), 21:55 9 by the definition of η?, we have ab(ℵ,ℵ, · · · ,ℵ?) ≤ 12b2ab(ℵ,ℵ, · · · ,ℵ ?) therefore, ab(ℵ,ℵ, · · · ,ℵ?) = 0 implies ℵ = ℵ?. similarly, we can shows that υ = υ?. thus,(ℵ, υ) is a unique common coupled point of t and f . finally, we will show ℵ = υ. η? ( 2b2ab (ℵ,ℵ, · · · , (ℵ)n−1, υ) ) = η? ( 2b2ab (t (ℵ, υ),t (ℵ, υ), · · · , (t (ℵ, υ))n−1,t (υ,ℵ)) ) ≤ γ (η? (ab(fℵ, fℵ, · · · , f υ)) ,θ? (ab(f υ, f υ, · · · , fℵ))) ≤ η? (ab(fℵ, fℵ, · · · , f υ)) ≤ η? (ab(ℵ,ℵ, · · · , υ)) by the definition of η?, we have ab(ℵ,ℵ, · · · , υ) ≤ 12b2ab(ℵ,ℵ, · · · , υ). therefore, ab(ℵ,ℵ, · · · , υ) = 0 implies ℵ = υ. thus,(ℵ,ℵ) is a common fixed point of t and f . � theorem 3.2. let (=,ab) be a complete ab-metric space. suppose t : =2 →= and f : =→= be a two mappings satisfying the following: η? ( 2b2ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) ) ≤ γ (η? (ab(f ı, f ı, · · · , fð)) ,θ? (ab(f , f , · · · , ff))) (3.4) for all ı, ,ð,f∈=, where η?,θ? ∈ θ and γ ∈ c a) t (=2) ⊆ f (=); b) pair (t,f ) is weakly compatible; c) f (=) is closed in =. then there is a unique common coupled fixed point of t and f in =. proof. let ı0, 0 ∈= and from theorem 3.1, we construct the sequences {ℵp} ,{υp} in = are cauchy sequences. since (=,ab) is complete, {ℵp} ,{υp} are convergent as follows lim p→∞ ℵp = lim p→∞ t (ıp, p) = lim p→∞ f ıp+1 = ℵ lim p→∞ υp = lim p→∞ t (p, ıp) = lim p→∞ f p+1 = υ. since f (=) is closed in (=,ab), so {ℵp} ,{υp} ⊆ f (=) are converges in the complete abmetric spaces (=,ab), therefore, there exist ℵ, υ ∈ f (=) with lim p→∞ ℵp+1 = ℵ lim p→∞ υp+1 = υ since f : = → = and ℵ, υ ∈ f (=), there exist f,℘ ∈ = such that ff = ℵ and f ℘ = υ. we claim that t (f,℘) = ℵ and t (℘,f) = υ. by using (3.4), we have η? ( 2b2ab (t (ıp, p),t (ıp, p), · · · , (t (ıp, p))n−1,t (f,℘)) ) ≤ γ (η? (ab(f ıp, f ıp, · · · , ff)) ,θ? (ab(f p, f p, · · · , f ℘))) ≤ η? (ab(f ıp, f ıp, · · · , ff)) by the definition of η?, ab (t (ıp, p),t (ıp, p), · · · , (t (ıp, p))n−1,t (f,℘)) ≤ 12b2ab(f ıp, f ıp, · · · , ff) 10 int. j. anal. appl. (2023), 21:55 letting p →∞, it yields that lim p→∞ ab (t (ıp, p),t (ıp, p), · · · , (t (ıp, p))n−1,t (f,℘)) ≤ lim p→∞ 1 2b2 ab(f ıp, f ıp, · · · , ff) = 0. it follows that ab (ℵ,ℵ, · · · , (ℵ)n−1,t (f,℘)) = 0 implies that t (f,℘) = ℵ. similarly, we can prove t (℘,f) = υ. hence, t (f,℘) = ℵ = ff and t (℘,f) = υ = f ℘. since {t,f} is weakly compatible pair, we have t (ℵ, υ) = fℵ and t (υ,ℵ) = f υ. now we shall prove that fℵ = ℵ and f υ = υ. by using (3.4), we have η? ( 2b2ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ) = η? ( 2b2ab (t (ℵ, υ),t (ℵ, υ), · · · , (t (ℵ, υ))n−1,t (ıp, p)) ) ≤ γ (η? (ab(fℵ, fℵ, · · · , f ıp)) ,θ? (ab(f υ, f υ, · · · , f p))) ≤ η? (ab(fℵ, fℵ, · · · , f ıp)) by the definition of η?, ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵp) ≤ 12b2ab(fℵ, fℵ, · · · , f ıp) letting p → ∞, it yields that ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) ≤ 12b2ab(fℵ, fℵ, · · · ,ℵ), which possible holds only, ab (fℵ, fℵ, · · · , (fℵ)n−1,ℵ) = 0 implies that fℵ = ℵ. similarly, we shall show f υ = υ. it follows that t (ℵ, υ) = fℵ = ℵ and t (υ,ℵ) = f υ = υ. therefore, the common coupled fixed point of t and f is (ℵ, υ). it is simple to demonstrate the connected fixed point’s uniqueness and the common fixed point’s uniqueness of t and f , just like in the proof of theorem 3.1. � corollary 3.1. let (=,ab) be a complete ab-metric space. suppose a mapping t : =2 → = be satisfies: η? ( 2b2ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) ) ≤ γ (η? (ab(ı, ı, · · · ,ð)) ,θ? (ab(, , · · · ,f))) for all ı, ,ð,f∈=, where η?,θ? ∈ θ and γ ∈ c. then t has a unique coupled fixed point in =. example 3.1. let = = [0, +∞), define ab : =n → [0, +∞) as ab (℘1,℘2, ........,℘n−1,℘n) = ∑n i=1 ∑ i<j |℘i −℘j| 2 for all ℘i ∈ =, i = 1, 2, · · · . then (=,ab) is an complete ab-metric space with b = 2. let t : =2 →= and f : =→= be given by t (℘,$) = sin ( 4℘−2$+32n−2 32n ) and f (℘) = 2℘+4n−2 4n and γ(s?,t?) = s ? 1+2s? ∀ s?,t? ∈ [0,∞). let η?(t?) = t ? 4 and θ?(t?) = t? 2 then obviously, t (=2) ⊆ f (=) and the pair (t,f ) are ω-compatible and clearly for all ı, ,ð,f∈=, where η?,θ? ∈ θ and γ ∈ c, we have γ (η? (ab(f ı, f ı, · · · , fð)) ,θ? (ab(f , f , · · · , ff))) −η? ( 2b2ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) ) = η? (ab(f ı, f ı, · · · , fð)) 1 + 2η? (ab(f ı, f ı, · · · , fð)) − 2b2 ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) 4 = 1 4 ab(f ı, f ı, · · · , fð) 1 + 1 2 ab(f ı, f ı, · · · , fð) − b2 2 (n− 1)|sin( 4ı − 2 + 32n− 2 32n ) − sin( 4ð− 2f + 32n− 2 32n )|2 int. j. anal. appl. (2023), 21:55 11 = 1 4 |2ı−2ð 4n |2 1 + 1 2 |2ı−2ð 4n |2 − 2b2(n− 1)|cos( 4ı − 2 + 64n− 4 + 4ð− 2f 64n ) sin( 4ı − 2 − 4ð + 2f 64n )|2 ≥ |2ı−2ð 4n |2 2 ( 2 + |2ı−2ð 4n |2 ) ≥ 0. then we have η? ( 2b2ab(t (ı, ),t (ı, ), · · · ,t (ð,f)) ) ≤ γ (η? (ab(f ı, f ı, · · · , fð)) ,θ? (ab(f , f , · · · , ff))). thus, all assumptions of theorem 3.1 are satisfied and (1, 1) is unique common coupled fixed point of t and f . 4. application to integral equations in this part, we examine the existence of a singular initial value solution with reference to corollary 3.1 theorem 4.1. consider the initial value problem ℘1(t) = t (t, (℘,$)(t)), t ∈ i = [0, 1], (℘,$)(0) = (℘0,$0) (4.1) where t : i × [℘0 4 ,∞) → [℘0 4 ,∞) and ℘0 ∈r then there exists unique solution in c ( i, [ ℘0 4 ,∞) ) for the initial value problem (4.1). proof. the integral equation for the initial value problem (4.1) is ℘(t) = ℘0 + 2b 2 √ n− 1 t∫ 0 t (s, (℘,$)(s))ds. let = = c ( i, [ ℘0 4 ,∞) ) and ab (℘1,℘2, ........,℘n−1,℘n) = ∑n i=1 ∑ i<j |℘i −℘j| 2 for all ℘i ∈=, i = 1, 2, · · · , define η?,θ? : [0,∞) → [0,∞) as η?(t) = t, θ?(t) = 3t5 and γ : [0,∞) → [0,∞) as γ(s?,t?) = ms? where m ∈ (0, 1) r : =2 →= by r(℘,$)(t) = ℘0 2b2 √ (n− 1) + t∫ 0 t (s, (℘,$)(s))ds. now for all ℘,$ ∈=, we have η? ( 2b2ab (r(℘,$)(t),r(℘,$)(t), · · · ,r(ð,f)(t)) ) = 2b2(n− 1)|r(℘,$)(t) −r(ð,f)(t)|2 = 2b2(n− 1)| ℘0 2b2 √ (n− 1) + t∫ 0 t (s, (℘,$)(s))ds − ð0 2b2 √ (n− 1) + t∫ 0 t (s, (ð,f)(s))ds|2 = 1 2b2 |℘(t) −ð(t)|2 ≤ 1 2 (n− 1)|℘(t) −ð(t)|2 ≤ 1 2 ab(℘,℘, · · · ,ð) ≤ γ (η? (ab(ı, ı, · · · ,ð)) ,θ? (ab(, , · · · ,f))) it follows from corollary 3.1, we conclude that r has a unique solution in =. � 12 int. j. anal. appl. (2023), 21:55 5. application to homotopy in this part, we examine the possibility that homotopy theory has a single solution. theorem 5.1. let (=,ab) be complete ab-metric space, v and v be an open and closed subset of = such that v ⊆ v . suppose hb : v 2 × [0, 1] → = be an operator with following conditions are satisfying, τ0) ℘ 6= hb(℘,$,s), $ 6= hb($,℘,s), for each ℘,$ ∈ υv and s ∈ [0, 1] (here υv is boundary of v in =); τ1) for all ℘,$,ı,  ∈ v , s ∈ [0, 1] and η?,θ? ∈ θ, γ ∈ c such that η? ( 2b2ab(hb(℘,$,s),hb(℘,$,s), · · · ,hb(ı, ,s)) ) ≤ γ (η? (ab(℘,℘, · · · , ı)) ,θ? (ab($,$, · · · , ))) . τ2) ∃ m ≥ 0 3 ab(hb(℘,$,s),hb(℘,$,s), · · · ,hb(℘,$,t)) ≤ m|s − t| for every ℘,$ ∈ v and s,t ∈ [0, 1]. then hb(., 0) has a coupled fixed point ⇐⇒ hb(., 1) has a coupled fixed point. proof. let the set = = { s ∈ [0, 1] : hb(℘,$,s) = ℘,hb($,℘,s) = $ for some ℘,$ ∈ v } . we have that (0, 0) in =2 if hb(., 0) has a coupled fixed point in v 2. therefore, the set = is not empty. we now demonstrate that = is both closed and open in [0, 1] and that = = [0, 1] is the result of this connectivity. as a result, v 2 has a coupled fixed point for hb(., 1). we begin by demonstrating that = closed in [0, 1]. observing this, let {sp}∞p=1 ⊆ = with sp → κ ∈ [0, 1] as p → ∞. we must show that s ∈ =. since sp ∈ = for p = 0, 1, 2, 3, · · · , there exists sequences {℘p} ,{$p} with ℘p = hb(℘p,$p,sp), $p = hb($p,℘p,sp). consider ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) = ab (hb(℘p,$p,sp),hb(℘p,$p,sp), · · · , (hb(℘p,$p,sp))n−1,hb(℘p+1,$p+1,sp+1)) ≤ b(n− 1)ab ( hb(℘p,$p,sp),hb(℘p,$p,sp), · · · ,hb(℘p+1,$p+1,sp) ) +b2ab ( hb(℘p+1,$p+1,sp),hb(℘p+1,$p+1,sp), · · · ,hb(℘p+1,$p+1,sp+1) ) ≤ b(n− 1)ab ( hb(℘p,$p,sp),hb(℘p,$p,sp), · · · ,hb(℘p+1,$p+1,sp) ) +b2m|sp − sp+1| letting p →∞, we get lim p→∞ ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) ≤ lim p→∞ b(n− 1)ab (hb(℘p,$p,sp),hb(℘p,$p,sp), · · · ,hb(℘p+1,$p+1,sp)) + 0 int. j. anal. appl. (2023), 21:55 13 since η?,θ? are continuous and non-decreasing, we obtain lim p→∞ η? ( 2b n− 1 ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) ) ≤ lim p→∞ η? ( 2b2ab (hb(℘p,$p,sp),hb(℘p,$p,sp), · · · ,hb(℘p+1,$p+1,sp)) ) ≤ lim p→∞ γ (η? (ab(℘p,℘p, · · · ,℘p+1)) ,θ? (ab($p,$p, · · · ,$p+1))) ≤ lim p→∞ η? (ab(℘p,℘p, · · · ,℘p+1)) . by the definition of η?, it follows that lim p→∞ ( 2b n− 1 − 1)ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) ≤ 0 so that lim p→∞ ab(℘p,℘p, · · · , (℘p)n−1,℘p+1) = 0 now for q > p, by use of (ab3), we have ab (℘p,℘p, · · · ,℘n−1,℘q) ≤ b(n− 1)ab (℘p,℘p, ........, (℘p)n−1,℘p+1) +b2ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘q) ≤ b(n− 1)ab (℘p,℘p, ........, (℘p)n−1,℘p+1) +b3(n− 1)ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘p+2) +b4ab (℘p+2,℘p+2, ........, (℘p+2)n−1,℘q) ≤ b(n− 1)ab (℘p,℘p, ........, (℘p)n−1,℘p+1) +b3(n− 1)ab (℘p+1,℘p+1, ........, (℘p+1)n−1,℘p+2) +b5(n− 1)ab (℘p+2,℘p+2, ........, (℘p+2)n−1,℘p+3) +b7(n− 1)ab (℘p+3,℘p+3, ........, (℘p+3)n−1,℘p+4) + . . . + b2q−2p−2(n− 1)ab (℘q−2,℘q−2, ........, (℘q−2)n−1,℘q−1) +b2q−2p−3ab (℘q−1,℘q−1, ........, (℘q−1)n−1,℘q) → 0 as p,q →∞. hence {℘p} is a cauchy sequence in ab metric spaces (=,ab).similarly, we may demonstrate that the cauchy sequence in (=,ab) is {$p} and by the fact that (=,ab) is complete, there exist u,v ∈= with lim p→∞ ℘p+1 = u lim p→∞ ℘p lim p→∞ $p+1 = v = lim p→∞ $p we have η? ( 2b2ab (u,u, · · · ,hb(u,v,s)) ) = lim p→∞ η? ( 2b2ab (hb(℘p,$p,s),hb(℘p,$p,s), · · · ,hb(u,v,s)) ) ≤ lim n→∞ γ (η? (ab(℘p,℘p, · · · ,u)) ,θ? (ab($p,$p, · · · ,v))) ≤ lim n→∞ η? (ab(℘p,℘p, · · · ,u)) = 0 14 int. j. anal. appl. (2023), 21:55 it follows thathb(u,v,s) = u. similarly, we can provehb(v,u,s) = v. thus s ∈=. hence= is closed in [0, 1]. let s0 ∈ =, then there exist ℘0,$0 ∈ v with ℘0 = hb(℘0,$0,s0), $0 = hb($0,℘0,s0). since v is open, then there exist r > 0 such that bab (℘0, r) ⊆ v . choose s ∈ (s0−�,s0+�) such that |s − s0| ≤ 1mp < � 2 , then for ℘ ∈ bab (℘0, r) = {℘ ∈=/ab(℘,℘, · · · ,℘0) ≤ r + ab(℘0,℘0, · · · ,℘0)}. now we have ab (hb(℘,$,s),hb(℘,$,s), · · · ,℘0) = ab (hb(℘,$,s),hb(℘,$,s), · · · ,hb(℘0,$0,s0)) ≤ (n− 1)bab (hb(℘,$,s),hb(℘,$,s), · · · ,hb(℘,$,s0)) +b2ab (hb(℘,$,s0),hb(℘,$,s0), · · · ,hb(℘0,$0,s0)) ≤ b(n− 1)m|s − s0| + b2ab (hb(℘,$,s0),hb(℘,$,s0), · · · ,hb(℘0,$0,s0)) ≤ b(n− 1) 1 mp−1 + b2ab (hb(℘,$,s0),hb(℘,$,s0), · · · ,hb(℘0,$0,s0)) letting p →∞, we obtain ab (hb(℘,$,s),hb(℘,$,s), · · · ,℘0) ≤ b2ab (hb(℘,$,s0),hb(℘,$,s0), · · · ,hb(℘0,$0,s0)) since η?,θ? are continuous and non-decreasing, we obtain η? (ab (hb(℘,$,s),hb(℘,$,s), · · · ,℘0)) ≤ η? ( 2b2ab (hb(℘,$,s0),hb(℘,$,s0), · · · ,hb(℘0,$0,s0)) ) ≤ γ (η?(ab(℘,℘, · · · ,℘0)),θ?(ab($,$, · · · ,$0))) ≤ η?(ab(℘,℘, · · · ,℘0)) since η? is non-decreasing, we have ab (hb(℘,$,s),hb(℘,$,s), · · · ,℘0) ≤ ab(℘,℘, · · · ,℘0) ≤ r + ab(℘0,℘0, · · · ,℘0). similarly, we can prove, ab (hb($,℘,s),hb($,℘,s), · · · ,$0) ≤ r + ab($0,$0, · · · ,$0). thus for each fixed s ∈ (s0 − �,s0 + �), hb(.,s) : bab (℘0, r) → bab (℘0, r), hb(.,s) : bab ($0, r) → bab ($0, r). all of the theorem 5.1’s requirements are then met. accordingly, we deduce that hb(.,s) has a coupled fixed point in v 2 . but it has to be in v 2. since (τ0) is true. the result is that s ∈ = for any s ∈ (s0 − �,s0 + �). because of this, (s0 − �,s0 + �) ⊆ =. in [0, 1], = is obviously open. we follow the same approach for the opposite inference. � conclusion: this paper wraps up a few applications to integral equations and homotopy theory using coupled fixed point theorems for c-class functions in the framework of ab-metric spaces. int. j. anal. appl. (2023), 21:55 15 conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. ughade, d. turkoglu, s.r. singh, et al. some fixed point theorems in ab-metric space, br. j. math. comput. sci. 19 (2016), 1-24. [2] n. mlaiki, y. rohen, some coupled fixed point theorems in partially ordered ab-metric space, j. nonlinear sci. appl. 10 (2017), 1731?1743. https://doi.org/10.22436/jnsa.010.04.35. [3] r. konchada, s.r. chindirala, r.n. chappa, a novel coupled fixed point results pertinent to ab-metric spaces with application to integral equations, math. anal. contemp. appl. 4 (2022), 63-83. https://doi.org/10. 30495/maca.2022.1949822.1046. [4] k. ravibabu, c.s. rao, c.r. naidu, applications to integral equations with coupled fixed point theorems in ab-metric space, thai j. math. spec. iss. (2018), 148-167. [5] p. naresh, g.u. reddy, b.s. rao, existence suzuki type fixed point results in ab-metric spaces with application, int. j. anal. appl. 20 (2022), 67. https://doi.org/10.28924/2291-8639-20-2022-67. [6] s. sessa, on a weak commutativity condition of mappings in fixed point considerations, publ. l’inst. math. 32 (1982), 149-153. http://eudml.org/doc/254762. [7] g. jungck, compatible mappings and common fixed points, int. j. math. math. sci. 9 (1986), 771-779. https: //doi.org/10.1155/s0161171286000935. [8] g. jungck, b.e. rhoades, fixed point for set valued functions without continuity, indian j. pure appl. math. 29 (1998), 227-238. [9] m.s. khan, m. swaleh, s. sessa, fixed point theorems by altering distances between the points, bull. austral. math. soc. 30 (1984), 1-9. https://doi.org/10.1017/s0004972700001659. [10] a.h. ansari, note on ϕ−ψ-contractive type mappings and related fixed point, in: the 2nd regional conference on mathematics and applications, payame noor university, 2014. [11] a.h. ansari, a. kaewcharoen, c-class functions and fixed point theorems for generalized ℵ−η −ψ −ϕ−fcontraction type mappings in ℵ − η-complete metric spaces, j. nonlinear sci. appl. 9 (2016), 4177-4190. https://doi.org/10.22436/jnsa.009.06.60. [12] h. huang, g. deng, s. radenovic, fixed point theorems for c-class functions in b-metric spaces and applications, j. nonlinear sci. appl. 10 (2017), 5853-5868. https://doi.org/10.22436/jnsa.010.11.23. [13] a.h. ansari, w. shatanawi, a. kurdi, g. maniu, best proximity points in complete metric spaces with (p)property via c-class functions, j. math. anal. 7 (2016), 54-67. [14] v. ozturk, a.h. ansari, common fixed point theorems for mappings satisfying (e.a)-property via c-class functions in b-metric spaces, appl. gen. topol. 18 (2017), 45-52. https://doi.org/10.4995/agt.2017.4573. [15] t. hamaizia, common fixed point theorems involving c-class functions in partial metric spaces, sohag j. math. 8 (2021), 23-28. https://doi.org/10.18576/sjm/080103. [16] g.s. saluja, common fixed point theorems on s-metric spaces via c-class functions, int. j. math. comb. 3 (2022), 21-37. [17] w. shatanawi, m. postolache, a.h. ansari, w. kassab, common fixed points of dominating and weak annihilators in ordered metric spaces via c-class functions, j. math. anal. 8 (2017), 54-68. [18] d. guo, v. lakshmikantham, coupled fixed points of nonlinear operators with applications, nonlinear anal.: theory methods appl. 11 (1987), 623-632. https://doi.org/10.1016/0362-546x(87)90077-0. [19] t.g. bhaskar, v. lakshmikantham, fixed point theorems in partially ordered metric spaces and applications, nonlinear anal.: theory methods appl. 65 (2006), 1379-1393. https://doi.org/10.1016/j.na.2005.10.017. https://doi.org/10.22436/jnsa.010.04.35 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.30495/maca.2022.1949822.1046 https://doi.org/10.28924/2291-8639-20-2022-67 http://eudml.org/doc/254762 https://doi.org/10.1155/s0161171286000935 https://doi.org/10.1155/s0161171286000935 https://doi.org/10.1017/s0004972700001659 https://doi.org/10.22436/jnsa.009.06.60 https://doi.org/10.22436/jnsa.010.11.23 https://doi.org/10.4995/agt.2017.4573 https://doi.org/10.18576/sjm/080103 https://doi.org/10.1016/0362-546x(87)90077-0 https://doi.org/10.1016/j.na.2005.10.017 16 int. j. anal. appl. (2023), 21:55 [20] m. abbas, m.a. khan, s. radenovic, common coupled fixed point theorems in cone metric spaces for ωcompatible mappings, appl. math. comput. 217 (2010), 195-202. https://doi.org/10.1016/j.amc.2010.05. 042. [21] a. aghajani, m. abbas, e.p. kallehbasti, coupled fixed point theorems in partially ordered metric spaces and application, math. commun. 17 (2012), 497-509. https://hrcak.srce.hr/93280. [22] e. karapinar, coupled fixed point on cone metric spaces, gazi univ. j. sci. 1 (2011), 51-58. [23] m. abbas, b. ali, y.i. suleiman, generalized coupled common fixed point results in partially ordered a-metric spaces, fixed point theory appl. 2015 (2015), 64. [24] w. long, b.e. rhoades, m. rajovic, coupled coincidence points for two mappings in metric spaces and cone metric spaces, fixed point theory appl. 2012 (2012), 66. https://doi.org/10.1186/1687-1812-2012-66. https://doi.org/10.1016/j.amc.2010.05.042 https://doi.org/10.1016/j.amc.2010.05.042 https://hrcak.srce.hr/93280 https://doi.org/10.1186/1687-1812-2012-66 1. introduction 2. preliminaries 3. main results 4. application to integral equations 5. application to homotopy references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 118-128 http://www.etamaths.com an implicit algorithm for a family of total asymptotically nonexpansive mappings in cat(0) spaces g. s. saluja∗ abstract. in this paper, we establish some strong convergence theorems of an implicit algorithm for a finite family of of total asymptotically nonexpansive mappings in the setting of cat(0) spaces. our results extend and generalize several recent results from the current existing literatures (see, e.g., [2, 9, 14, 16, 17, 25, 29]). 1. introduction and preliminaries a metric space (x,d) is said to be a length space if any two points of x are joined by a rectifiable path (i.e., a path of finite length), and the distance between any two points of x is taken to be the infimum of the lengths of all rectifiable paths joining them. in this case, d is said to be a length metric (otherwise known as an inner metric or intrinsic metric). in case no rectifiable path joins two points of the space, the distance between them is taken to be ∞. a geodesic path joining x ∈ x to y ∈ x (or, more briefly, a geodesic from x to y) is a mapping c from a closed interval [0, l] ⊂ r to x such that c(0) = x, c(l) = y, and let d(c(t),c(t′)) = |t− t′| for t,t′ ∈ [0, l]. in particular, c is an isometry, and d(x,y) = l. the image α of c is called a geodesic (or metric) segment joining x and y. we say that x is (i) a geodesic space if any two points of x are joined by a geodesic and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each x,y ∈ x, which we will denoted by [x,y], called the segment joining x to y. a geodesic triangle 4(x1,x2,x3) in a geodesic metric space (x,d) consists of three points in x (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). a comparison triangle for geodesic triangle 4(x1,x2,x3) in (x,d) is a triangle 4(x1,x2,x3) := 4(x1,x2,x3) in r2 such that dr2 (xi,xj) = d(xi,xj) for i,j ∈{1, 2, 3}. such a triangle always exists (see [3]). a geodesic metric space is said to be a cat(0) space if all geodesic triangles of appropriate size satisfy the following cat(0) inequality. let 4 be a geodesic triangle in x, and let 4⊂ r2 be a comparison triangle for 4. then 4 is said to satisfy the cat(0) inequality if for all x,y ∈4 and all comparison points x,y ∈4, d(x,y) ≤ dr2 (x,y).(1.1) complete cat(0) spaces are often called hadamard spaces (see [12]). if x,y1,y2 are points of a cat(0) space and y0 is the mid-point of the segment [y1,y2] which we will denote by (y1⊕y2)/2, then the cat(0) inequality implies d2 ( x, y1 ⊕y2 2 ) ≤ 1 2 d2(x,y1) + 1 2 d2(x,y2) − 1 4 d2(y1,y2).(1.2) 2010 mathematics subject classification. 54h25, 54e40. key words and phrases. total asymptotically nonexpansive mapping; strong convergence; implicit iteration process; common fixed point; cat(0) space. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 118 an implicit algorithm for a family of total. . . . . . 119 the inequality (1.2) is the (cn) inequality of bruhat and tits [4]. the above inequality has been extended in [6] as d2(z,αx⊕ (1 −α)y) ≤ αd2(z,x) + (1 −α)d2(z,y) −α(1 −α)d2(x,y)(1.3) for any α ∈ [0, 1] and x,y,z ∈ x. let us recall that a geodesic metric space is a cat(0) space if and only if it satisfies the (cn) inequality (see [[3], page 163]). moreover, if x is a cat(0) metric space and x,y ∈ x, then for any α ∈ [0, 1], there exists a unique point αx⊕ (1 −α)y ∈ [x,y] such that d(z,αx⊕ (1 −α)y) ≤ αd(z,x) + (1 −α)d(z,y),(1.4) for any z ∈ x and [x,y] = {αx⊕ (1 −α)y : α ∈ [0, 1]}. a subset c of a cat(0) space x is convex if for any x,y ∈ c, we have [x,y] ⊂ c. let t be a self mapping on a nonempty subset c of x. denote the set of fixed points of t by f(t) = {x ∈ c : t(x) = x}. we say that t is: (1) nonexpansive if d(tx,ty) ≤ d(x,y) for all x,y ∈ c; (2) asymptotically nonexpansive ([10]) if there exists a sequence {rn}⊂ [0,∞) with limn→∞rn = 0 such that d(tnx,tny) ≤ (1 + rn)d(x,y) for all x,y ∈ c and n ≥ 1; (3) uniformly l-lipschitzian if there exists a constant l > 0 such that d(tnx,tny) ≤ ld(x,y) for all x,y ∈ c and n ≥ 1; (4) semi-compact if for a sequence {xn} in c with limn→∞d(xn,txn) = 0, there exists a subsequence {xnk} of {xn} such that xnk → p ∈ c. remark 1.1. from the above definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with the constant sequence {kn} = {1}, ∀n ≥ 1 and an asymptotically nonexpansive mapping is a uniformly l-lipschitzian mapping with l = supn≥1{kn}. chang et al. [5] defined the concept of total asymptotically nonexpansive mapping as follows. definition 1.2. ([5] definition 2.1) let (x,d) be a metric space, k be its nonempty subset and let t : k → k be a mapping. t is said to be a total asymptotically nonexpansive mapping if there exist non-negative real sequences {µn}, {νn} with µn → 0, νn → 0 and a strictly increasing continuous function ψ : [0,∞) → [0,∞) with ψ(0) = 0 such that d(tnx,tny) ≤ d(x,y) + νnψ(d(x,y)) + µn for all x,y ∈ k and n ≥ 1. remark 1.3. from the above definition, it is clear that each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with µn = 0, νn = kn − 1 for all n ≥ 1, ψ(t) = t, t ≥ 0. recently, there are a lot of papers have appeared on the iterative approximation of fixed points of asymptotically nonexpansive mappings, asymptotically quasi-nonexpansive mappings, asymptotically nonexpansive mappings in the intermediate sense and their generalizations through ishikawa, s-iteration, modified s-iteration, noor iteration and implicit iterations in uniformly convex banach spaces, convex metric spaces and cat(0) spaces (see, e.g., [1, 2, 5, 8, 9, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24]). let e be a hilbert space, let k be a nonempty closed convex subset of e and let {ti}: k → k {i = 1, 2, . . . ,n} be nonexpansive mappings. in 2001, xu and ori [29] introduced the following implicit iteration process {xn} defined by xn = αnxn−1 + (1 −αn)tn(modn)xn for n ≥ 1,(1.5) 120 saluja where x0 ∈ k is an initial point, {αn} is a real sequence in (0, 1) and proved a weak convergence of the sequence {xn} defined by (1.5) to a common fixed point p ∈ f = ∩ni=1f(ti). in 2003, sun [27] introduced the following implicit iterative sequence {xn} xn = αnxn−1 + (1 −αn)t k(n) i(n) xn for n ≥ 1,(1.6) for a finite family of asymptotically quasi-nonexpansive self-mappings on a bounded closed convex subset k of a hilbert space e with {αn} a real sequence in (0, 1) and an initial point x0 ∈ k, where n = (k(n) − 1)n + i(n), 1(n) ∈ {1, 2, . . . ,n}, and proved a strong convergence of the sequence {xn} defined (1.6) to a common fixed point p ∈ f = ∩ni=1f(ti). the result of sun [27] generalized and extended the corresponding main result of wittmann [28] and xu and ori [29]. inspired and motivated by [28, 29], we now define a modified implicit iteration process for a finite family of total asymptotically nonexpansive mappings as below. modified implicit iterative process in cat(0) space let c be a nonempty closed convex subset of a cat(0) space x, and {t1,t2, . . . ,tn} be a finite family of n ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive self mappings on c. from an arbitrary x0 ∈ c, we define the sequence {xn} by: x1 = (1 −α1)x0 ⊕α1t1x1, x2 = (1 −α2)x1 ⊕α2t2x2, ... xn = (1 −αn )xn−1 ⊕αntnxn, xn+1 = (1 −αn+1)xn ⊕αn+1t21 xn+1,(1.7) ... x2n = (1 −α2n )x2n ⊕α2nt2nx2n, x2n+1 = (1 −α2n+1)x2n+1 ⊕α2n+1t31 x2n+1, ... where {αn} is an appropriate sequence in (0, 1). the above iteration can be written in the following compact form: xn = αnxn−1 ⊕ (1 −αn)t k(n) i(n) xn, for n ≥ 1(1.8) where n = ( k(n) − 1 ) n + i(n), k(n) > 1 is a positive integer such that k(n) →∞ as n →∞. let x be a cat(0) space. then, the following inequality holds: d(λx⊕ (1 −λ)z,λy ⊕ (1 −λ)w) ≤ λd(x,y) + (1 −λ)d(z,w),(1.9) for all x,y,z,w ∈ x (see [6]). let {ti : i ∈ i = {1, 2, . . . ,n}} be the set of n uniformly li (i = 1, 2, . . . ,n)-lipschitzian self mappings of c. we show that (1.8) exists. let x0 ∈ c and x1 = α1x0 ⊕ (1 − α1)t1x1. define w : c → c by w(x) = α1x0 ⊕ (1 −α1)t1x for x ∈ c. the existence of x1 is guaranteed if w has a fixed point. for any x,y ∈ c, we have d(wx,wy) ≤ (1 −α1)d(t1x,t1y) ≤ (1 −α1)l1 d(x,y) ≤ (1 −α1)ld(x,y)(1.10) where l = max{li : i ∈ i}. now, w is a contraction if (1−α1)l < 1 or l > 1/(1−α1). as α1 ∈ (0, 1), therefore w is a contraction if 1 < l < 2. by the banach contraction principle w has a unique fixed an implicit algorithm for a family of total. . . . . . 121 point. thus, the existence of x1 is established. thus, the implicit algorithm (1.8) is well defined. the goal of this paper is to study strong convergence of iterative algorithm (1.8) for the class of uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings (for i = 1, 2, . . . ,n) in the setting of cat(0) spaces. our results extend, improve and generalize several results from the current existing literature. we need the following useful notion and lemmas for the development of our main results. let {ti : i ∈ i} be the set of n self mappings of c. a mapping t : c → c is said to satisfy condition (a) if there exists a nondecreasing function f : [0,∞) → [0,∞) with f(0) = 0, f(r) > 0 for all r ∈ (0,∞) such that d(x.p) ≥ f(d(x,f(t ))) for x ∈ c where d(x,f(t)) = inf{d(x,p) : p ∈ f(t) 6= ∅}. condition (a) was introduced by senter and dotson [26]. lemma 1.4. ([6]) let x be a cat(0) space. (i) for x,y ∈ x and t ∈ [0, 1], there exists a unique point z ∈ [x,y] such that d(x,z) = td(x,y) and d(y,z) = (1 − t) d(x,y). (a) we use the notation (1 − t)x⊕ ty for the unique point z satisfying (a). (ii) for x,y,z ∈ x and t ∈ [0, 1], we have d((1 − t)x⊕ ty,z) ≤ (1 − t) d(x,z) + td(y,z). lemma 1.5. ([18]) suppose that {an}, {bn} and {rn} be sequences of nonnegative numbers such that an+1 ≤ (1 + bn)an + rn for all n ≥ 1. if ∑∞ n=1 bn < ∞ and ∑∞ n=1 rn < ∞, then limn→∞an exists. 2. main results in this section, we establish strong convergence theorems using implicit iteration scheme (1.8) for ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings (for i = 1, 2, . . . ,n) in the setting of cat(0) spaces. lemma 2.1. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) 6= ∅. suppose that the sequence {xn} is defined by the algorithm (1.8), where {αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then limn→∞d(xn,p) and limn→∞d(xn,f) exist for p ∈f. proof. let p ∈f. then, from (1.8) and lemma 1.4(ii), we have d(xn,p) = d(αnxn−1 ⊕ (1 −αn)t k(n) i(n) xn,p) ≤ αnd(xn−1,p) + (1 −αn)d(t k(n) i(n) xn,p) ≤ αnd(xn−1,p) + (1 −αn)[d(xn,p) + νi,k(n)ψ(d(xn,p)) + µi,k(n)] ≤ αnd(xn−1,p) + (1 −αn)[d(xn,p) + mνi,k(n) d(xn,p) + µi,k(n)] = αnd(xn−1,p) + (1 −αn)[(1 + mνi,k(n)) d(xn,p) + µi,k(n)](2.1) ≤ αnd(xn−1,p) + (1 −αn + mνi,k(n)) d(xn,p) + (1 −αn)µi,k(n). since αn ∈ [δ, 1 − δ], the above inequality gives that d(xn,p) ≤ d(xn−1,p) + mνi,k(n) δ d(xn,p) + (1 δ − 1 ) µi,k(n).(2.2) 122 saluja on simplification, we get that d(xn,p) ≤ ( δ δ −mνi,k(n) ) d(xn−1,p) + (1 δ − 1 )( δ δ −mνi,k(n) ) µi,k(n) = ( 1 + mνi,k(n) δ −mνi,k(n) ) d(xn−1,p) + (1 δ − 1 )( δ δ −mνi,k(n) ) µi,k(n) = (1 + ai,k(n))d(xn−1,p) + bi,k(n)(2.3) where ai,k(n) = mνi,k(n) δ−mνi,k(n) and bi,k(n) = ( 1 δ − 1 )( δ δ−mνi,k(n) ) µi,k(n). since ∑∞ k(n)=1 νi,k(n) < ∞ for i ∈ i therefore limk(n)→∞νi,k(n) = 0, and hence, there exists a natural number n1 such that νik(n) < δ/2 for k(n) ≥ n1/n + 1 or n > n1. then, we have that ∑∞ k(n)=1 ai,k(n) < ( 2m δ(2−m) )∑∞ k(n)=1 νik(n) < ∞. similarly, ∑∞ k(n)=1 wi,k(n) < ∞. similarly, ∑∞ k(n)=1 bi,k(n) < ∞. now, for any p ∈f, from (2.3), for k(n) ≥ n1/n + 1, we have d(xn,f) ≤ (1 + ai,k(n)) d(xn−1,f) + bi,k(n),(2.4) by lemma 1.5, (2.3) and (2.4), we obtain limn→∞d(xn,p) and limn→∞d(xn,f) both exist. this completes the proof. � theorem 2.2. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) is nonempty and closed. suppose that the sequence {xn} is defined by the algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then the sequence {xn} converges strongly to a common fixed point of {ti : i ∈ i} if and only if lim infn→∞d(xn,f) = 0. proof. if xn → p as n → ∞, then limn→∞d(xn,p) = 0. since 0 ≤ d(xn,f) ≤ d(xn,p), we have lim infn→∞d(xn,f) = 0. conversely, suppose that lim infn→∞d(xn,f) = 0. by lemma 1.5, we have that limn→∞d(xn,f) exists. further, by assumption lim infn→∞d(xn,f) = 0, we conclude that limn→∞d(xn,f) = 0. next, we show that {xn} is a cauchy sequence. since x ≤ exp(x− 1) for x ≥ 1, therefore from (2.3), we have d(xn+m,p) ≤ (1 + ai,k(n))d(xn−1,p) + bi,k(n) ≤ ( e ∑n i=1 ∑∞ k(n)=1 ai,k(n) ) d(xn,p) + n∑ i=1 ∞∑ k(n)=1 bi,k(n) < rd(xn,p) + r n∑ i=1 ∞∑ k(n)=1 bi,k(n)(2.5) for all natural numbers m,n, where r = ( e ∑n i=1 ∑∞ k(n)=1 ai,k(n) ) + 1 < ∞. since limn→∞d(xn,f) = 0, without loss of generality, we may assume that a subsequence {xnk} of {xn} and a sequence {pnk}⊂f an implicit algorithm for a family of total. . . . . . 123 such that d(xnk,pnk ) → 0 as k →∞. then for any ε > 0, there exists kε ∈ n such that d(xnk,pnk ) < ε 4r and n∑ i=1 ∞∑ j=nkε bi,j < ε 4r (2.6) for k ≥ kε. hence, for any m ∈ n and for n ≥ nkε , by (2.5) we have d(xn+m,xn) ≤ d(xn+m,pnk ) + d(xn,pnk ) ≤ rd(xn,pnk ) + r n∑ i=1 ∞∑ j=nkε bi,j +rd(xn,pnk ) + r n∑ i=1 ∞∑ j=nkε bi,j(2.7) = 2rd(xn,pnk ) + 2r n∑ i=1 ∞∑ j=nkε bi,j < 2r. ε 4r + 2r. ε 4r = ε. this implies that {xn} is a cauchy sequence in c. by the completeness of c, we can assume that limn→∞xn = q. we will prove that q is a common fixed point of {ti : i ∈ i}, that is, we will show that q ∈f. since c is closed, therefore q ∈ c. next, we show that q ∈f. since limn→∞d(xn,f) = 0, gives that d(q,f) = 0. since f is closed, q ∈f. thus q is a common fixed point of {ti : i ∈ i}. this completes the proof. � theorem 2.3. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) 6= ∅. suppose that the sequence {xn} defined by the algorithm (1.8), where {αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then lim infn→∞d(xn,f) = lim supn→∞d(xn,f) = 0 if {xn} converges to a unique point in f. proof. let p ∈f. since {xn} converges to p, limn→∞d(xn,p) = 0. so, for a given ε > 0, there exists n0 ∈ n such that d(xn,p) < ε for n ≥ n0. taking the infimum over p ∈f, we obtain that d(xn,f) < ε for n ≥ n0. this means that limn→∞d(xn,f) = 0. thus we obtain that lim inf n→∞ d(xn,f) = lim sup n→∞ d(xn,f) = 0. this completes the proof. � as shown in the preceding proof, the property needed to assure that p ∈ f is exactly the following one. given any sequence {un} of real numbers there is a subsequence {unj} of {un} such that limj→∞unj = lim infn→∞un. in general, if {umj} is a convergent subsequence of {un}, then lim infn→∞un ≤ limj→∞umj . this immediately gives the following result. 124 saluja corollary 2.4. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) 6= ∅. suppose that the sequence {xn} defined by the algorithm (1.8), where {αn}⊂ [δ, 1 −δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then {xn} converges strongly to a common fixed point of {ti : i ∈ i} if and only if there exists some subsequence {xnj} of {xn} which converges to p ∈f. corollary 2.5. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n asymptotically nonexpansive mappings of c with {ki,n}⊂ [1,∞) such that ∑∞ n=1(ki,n−1) < ∞ for all i ∈ i. suppose that f = ∩ni=1f(ti) is nonempty and closed. starting from arbitrary x0 ∈ c, define the sequence {xn} by the algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). then {xn} converges strongly to a common fixed point of {ti : i ∈ i} if and only if lim infn→∞d(xn,f) = 0. proof. follows from theorem 2.2 with µi,n = 0, νi,n = (ki,n−1) for all i ∈ i and ψ(t) = t, t ≥ 0. this completes the proof. � lemma 2.6. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) 6= ∅. suppose that the sequence {xn} is defined by the algorithm (1.8), where {αn}⊂ [δ, 1 −δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then limn→∞d(xn,tlxn) = 0 for each l ∈ i. proof. let l = max{li : i ∈ i}. note that {xn} is bounded as limn→∞d(xn,p) exists by lemma 2.1. so, there exists r′ > 0 and x0 ∈ x such that xn ∈ b′r(x0) = {x : d(x,x0) < r ′} for n ≥ 1. denote d(xn−1,t k(n) i(n) ) by ρn. we claim that limn→∞ρn = 0. for any p ∈f, apply (1.3) to (1.8), we have d2(xn,p) = d 2(αnxn−1 ⊕ (1 −αn)t k(n) i(n) xn,p) ≤ αnd2(xn−1,p) + (1 −αn)d2(t k(n) i(n) xn,p) −αn(1 −αn)d2(xn−1,t k(n) i(n) xn) ≤ αnd2(xn−1,p) + (1 −αn)[d(xn,p) + νi,k(n)ψ(d(xn,p)) + µi,k(n)]2 −αn(1 −αn)d2(xn−1,t k(n) i(n) xn)(2.8) ≤ αnd2(xn−1,p) + (1 −αn)[d(xn,p) + mνi,k(n)d(xn,p) + µi,k(n)]2 −αn(1 −αn)d2(xn−1,t k(n) i(n) xn) = αnd 2(xn−1,p) + (1 −αn)[(1 + mνi,k(n))d(xn,p) + µi,k(n)]2 −αn(1 −αn)d2(xn−1,t k(n) i(n) xn). an implicit algorithm for a family of total. . . . . . 125 now, using (2.3), we get αn(1 −αn)ρ2n ≤ αnd 2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + mνi,k(n)) ×{(1 + ai,k(n)) d(xn−1,p) + bi,k(n)} + µi,k(n)]2 = αnd 2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + mνi,k(n))(1 + ai,k(n)) × d(xn−1,p) + (1 + mνi,k(n))bi,k(n) + µi,k(n)] 2(2.9) = αnd 2(xn−1,p) −d2(xn,p) + (1 −αn)[(1 + fi,k(n))d(xn−1,p) × +gi,k(n)] 2 where fi,k(n) = mνi,k(n) + ai,k(n) + mai,k(n)νi,k(n) and gi,k(n) = (1 + mνi,k(n))bi,k(n) + µi,k(n). since∑∞ k(n)=1 µi,k(n) < ∞, ∑∞ k(n)=1 νi,k(n) < ∞ and ∑∞ k(n)=1 bi,k(n) < ∞, it follows that ∑∞ k(n)=1 fi,k(n) < ∞ and ∑∞ k(n)=1 gi,k(n) < ∞. again, note that αn(1 −αn)ρ2n ≤ αnd 2(xn−1,p) −d2(xn,p) + (1 −αn)[d(xn−1,p) + li,k(n)]2 = d2(xn−1,p) −d2(xn,p) + (1 −αn)qi,k(n),(2.10) where li,k(n) = fi,k(n)d(xn−1,p) + gi,k(n) and qi,k(n) = l 2 i,k(n) + 2li,k(n)d(xn−1,p). since {d(xn−1,p)} is convergent, ∑∞ k(n)=1 fi,k(n) < ∞ and ∑∞ k(n)=1 gi,k(n) < ∞, it follows that ∑∞ k(n)=1 li,k(n) < ∞ and∑∞ k(n)=1 qi,k(n) < ∞. this implies that ρ2n ≤ 1 αn(1 −αn) [d2(xn−1,p) −d2(xn,p)] + qi,k(n) αn ≤ 1 δ2 [d2(xn−1,p) −d2(xn,p)] + qi,k(n) δ .(2.11) since ∑∞ k(n)=1 qi,k(n) < ∞, {d(xn,p)} is convergent and δ > 0, therefore on taking limit as n →∞ in (2.11), we get lim n→∞ ρn = 0.(2.12) further, d(xn,xn−1) ≤ (1 −αn)d ( t k(n) i(n) xn,xn−1 ) = (1 −αn)ρn ≤ (1 −δ)ρn,(2.13) which implies that limn→∞d(xn,xn−1) = 0. for a fixed j ∈ i, we have d(xn+j,xn) ≤ d(xn+j,xn+j−1) + · · · + d(xn,xn−1), and hence lim n→∞ d(xn+j,xn) = 0 for j ∈ i.(2.14) for n > n, n = (n−n)(modn). also, n = (k(n)−1)n+i(n). hence, n−n = ((k(n)−1)−1)n+i(n) = (k(n−n))n + i(n−n). that is, k(n−n) = k(n) − 1 and i(n−n) = i(n). therefore, we have d(xn−1,tnxn) ≤ d ( xn−1,t k(n) i(n) xn ) + d ( t k(n) i(n) xn,tnxn ) ≤ ρn + ld ( t k(n)−1 i(n) xn,xn ) ≤ ρn + l2 d(xn,xn−n ) + ld ( t k(n−n) i(n−n) xn−n,x(n−n)−1 ) +ld(x(n−n)−1,xn)(2.15) ≤ ρn + l2 d(xn,xn−n ) + lρn−n +ld(x(n−n)−1,xn). using (2.12) and (2.14) in (2.15), we get lim n→∞ d(xn−1,tnxn) = 0.(2.16) 126 saluja since d(xn,tnxn) ≤ d(xn,xn−1) + d(xn−1,tnxn),(2.17) using (2.13) and (2.16) in (2.17), we have lim n→∞ d(xn,tnxn) = 0.(2.18) hence, for all l ∈ i, we have d(xn,tn+lxn) ≤ d(xn,xn+l) + d(xn+l,tn+lxn+l) +d(tn+lxn+l,tn+lxn) ≤ (1 + l) d(xn,xn+l) + d(xn+l,tn+lxn+l).(2.19) using (2.14) and (2.18) in (2.19), we obtain lim n→∞ d(xn,tn+lxn) = 0, ∀ l ∈ i.(2.20) thus, limn→∞d(xn,tlxn) = 0 for l ∈ i. this completes the proof. � as an application of theorem 2.2, we establish some strong convergence results as follows. theorem 2.7. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n uniformly li-lipschitzian and ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings with f = ∩ni=1f(ti) 6= ∅ and there exists one member t in {ti : i ∈ i} which is either semicompact or satisfies condition (a). suppose that the sequence {xn} is defined by the algorithm (1.8), where {αn}⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2). if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then {xn} converges strongly to a common fixed point of {ti : i ∈ i}. proof. by lemma 2.1, we see that lim n→∞ d(xn,x ∗) and lim n→∞ d(xn,f) exist. let one of t ′is, say, ts, s ∈ i is either semicompact or satisfies condition (a). if ts is semicompact, then there exists a subsequence {xnj} of {xn} such that xnj → z ∈ c as j → ∞. now, lemma 2.6 guarantees that limn→∞d(xnj,tsxnj ) = 0 for s ∈ i and so d(z,tsz) = 0 for s ∈ i. this implies that z ∈ f. therefore, lim infn→∞d(xn,f) = 0. if ts satisfies condition (a), then we also have lim infn→∞d(xn,f) = 0. now, theorem 2.2 implies that {xn} converges strongly to a point in f. this completes the proof. � theorem 2.8. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings. suppose that f = ∩ni=1f(ti) 6= ∅ (ti, i = 1, 2, . . . ,n, need not to be continuous). starting from arbitrary x0 ∈ c, define the sequence {xn} by the algorithm (1.8), where {αn}⊂ [δ, 1−δ] for some δ ∈ (0, 1/2). assume that (i′) limn→∞d(xn,xn+1) = 0 if the sequence {zn} in c satisfies (ii′) limn→∞d(zn,zn+1) = 0, then lim infn→∞d(zn,f) = 0 or lim supn→∞d(zn,f) = 0. if the following conditions are satisfied: (i) ∑∞ n=1 µi,n < ∞, ∑∞ n=1 νi,n < ∞ for all i ∈ i; (ii) there exists a constant m > 0 such that ψ(t) ≤ mt, t ≥ 0, where ψ(a) = max{ψi(a) : i ∈ i}, a ≥ 0. then {xn} converges to a unique point in f. an implicit algorithm for a family of total. . . . . . 127 proof. by hypothesis (i′) and (ii′), we have that lim inf n→∞ d(xn,f) = 0 or lim sup n→∞ d(xn,f) = 0. therefore, we obtain from theorem 2.2 that the sequence {xn} converges to a unique point in f. this completes the proof. � finally, we obtain the following result from theorem 2.7 as corollary. corollary 2.9. let c be a nonempty closed convex subset of a complete cat(0) space x. let {ti : i ∈ i} be n asymptotically nonexpansive mappings of c with {hi,n} ⊂ [1,∞) for i ∈ i such that ∑∞ n=1(hn − 1) < ∞, where hn = max{hi,n : i ∈ i}. suppose that f = ∩ n i=1f(ti) 6= ∅ and there exists one member t in {ti : i ∈ i} which is either semicompact or satisfies condition (a). from an arbitrary x0 ∈ c, define the sequence {xn} by algorithm (1.8), where {αn} ⊂ [δ, 1 − δ] for some δ ∈ (0, 1/2).. then {xn} converges strongly to a common fixed point of {ti : i ∈ i}. remark 2.10. our results extend, generalize and improve several corresponding approximation results from the current existing literature to the case of implicit iteration process and more general class of nonexpansive and asymptotically nonexpansive mappings considered in this paper (see, e.g., [2, 7, 14, 16, 17, 29] and many others). remark 2.11. our results also extend the corresponding results [25] to the case of finite family of mappings and implicit iteration process considered in this paper. example 2.12. ([11], example 3.1) let r be the real line with the usual norm ‖.‖ and c = [−1, 1]. define a mapping t : c → c by t(x) = { −2 sinx 2 , if x ∈ [0, 1], 2 sinx 2 , if x ∈ [−1, 0). then t is an asymptotically nonexpansive mapping with constant sequence {kn} = {1} for n ≥ 1 and uniformly l-lipschtzian mapping with l = supn≥1{kn} and hence it is a total asymptotically nonexpansive mapping by remark 1.3. also the fixed point of t , that is, f(t) = {0}. 3. conclusion in this paper, we establish some strong convergence theorems using implicit algorithm (1.8) for a finite family of ({µi,n},{νi,n},ψi)-total asymptotically nonexpansive mappings which is more general than the class of nonexpansive and asymptotically nonexpansive mappings in the framework of cat(0) spaces. references [1] m. abbas, b.s. thakur and d. thakur, fixed points of asymptotically nonexpansive mappings in the intermediate sense in cat(0) spaces, commun. korean math. soc. 28(4) (2013), 107-121. [2] m. başarir and a. şahin, on the strong and 4-convergence for total asymptotically nonexpansive mappings on a cat(0) space, carpathian math. pub. 5(2) (2013), 170-179. [3] m.r. bridson and a. haefliger, metric spaces of non-positive curvature, vol. 319 of grundlehren der mathematischen wissenschaften, springer, berlin, germany, 1999. [4] f. bruhat and j. tits, groups reductifs sur un corps local, institut des hautes etudes scientifiques. publications mathematiques, 41 (1972), 5-251. [5] s.s. chang, l. wang, h.w. joesph lee, c.k. chan and l. yang, demiclosed principle and 4-convergence theorems for total asymptotically nonexpansive mappings in cat(0) spaces, appl. math. comput. 219(5) (2012), 2611-2617. [6] s. dhompongsa and b. panyanak, on 4-convergence theorem in cat(0) spaces, comput. math. appl. 56(10) (2008), 2572-2579. [7] h. fukhar-ud-din and s.h. khan, convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings, int. j. math. math. sci. (2004), no. 37-40, 1965-1971. [8] h. fukhar-ud-din and a.r. khan, convergence of implicit iterates with errors for mappings with unbounded domain in banach spaces, int. j. math. math. sci. 2005:10, 1643-1653. [9] h. fukhar-ud-din and s.h. khan, convergence of iterates with errors of asymptotically quasi-nonexpansive and applications, j. math. anal. appl. 328 (2007), 821-829. [10] k. goebel and w.a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. amer. math. soc. 35(1) (1972), 171-174. 128 saluja [11] w.p. guo, y.j. cho and w. guo, convergence theorems for mixed type asymptotically nonexpansive mappings, fixed point theory appl. 2012 (2012) art. id 224. [12] m.a. khamsi and w.a. kirk, an introduction to metric spaces and fixed point theory, pure appl. math, wileyinterscience, new york, ny, usa, 2001. [13] s.h. khan and m. abbas, strong and 4-convergence of some iterative schemes in cat(0) spaces, comput. math. appl. 61(1) (2011), 109-116. [14] a.r. khan, m.a. khamsi and h. fukhar-ud-din, strong convergence of a general iteration scheme in cat(0) spaces, nonlinear anal.: theory, method and applications, 74(3) (2011), 783-791. [15] p. kumam. g.s. saluja and h.k. nashine, convergence of modified s-iteration process for two asymptotically nonexpansive mappings in the intermediate sense in cat(0) spaces, j. ineq. appl. 2014 (2014), art. id 368. [16] b. nanjaras and b. panyanak, demiclosed principle for asymptotically nonexpansive mappings in cat(0) spaces, fixed point theory appl. 2010 (2010), art. id 268780, 14 pages. [17] y. niwongsa and b. panyanak, noor iterations for asymptotically nonexpansive mappings in cat(0) spaces, int. j. math. anal. 4(13) (2010), 645-656. [18] m.o. osilike and d.i. igbokue, weak and strong convergence theorems for fixed points of pseudocontractions and solution of monotone type operator equations, comput. math. appl. 40 (2000), 559-569. [19] a. şahin and m. başarir, on the strong convergrnce of a modified s-iteration process for asymptotically quasinonexpansive mapping in cat(0) space, fixed point theory appl. 2013 (2013), art. id 12, 10 pages. [20] g.s. saluja and h.k. nashine, convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces, opuscula mathematica 30(3) (2010), 331-340. [21] g.s. saluja, weak and strong convergence theorems for four nonexpansive mappings in uniformly convex banach spaces, thai. j. math. 10(2) (2012), 305-319. [22] g.s. saluja, convergence theorems for asymptotically nonexpansive mappings in the intermediate sense in uniformly convex banach spaces, j. indian acad. math. 34(2) (2012), 451-467. [23] g.s. saluja and h.k. nashine, weak convergence theorems of two-step iteration process for two asymptotically quasi-nonexpansive mappings, indian j. math. 56(3) (2014), 291-311. [24] g.s. saluja and m. postolache, strong and ∆-convergence theorems for two asymptotically nonexpansive mappings in the intermediate sense in cat(0) spaces, fixed point theory appl. 2015 (2015), art. id 12. [25] g.s. saluja, strong and ∆-convergence theorems for two totally asymptotically nonexpansive mappings in cat(0) spaces, nonlinear anal. forum 20(2) (2015), 107-120. [26] h.f. senter and w.g. dotson, approximating fixed points of nonexpansive mappings, proc. amer. math. soc. 44 (1974), 375-380. [27] z. sun, strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, j. math. anal. appl. 286(1) (2003), 351-358. [28] r. wittmann, approximation of fixed points of nonexpansive mappings, arch. math. 58 (1992), 486-491. [29] h.k. xu, r.g. ori, an implicit iteration process for nonexpansive mappings, numer. funct. anal. optim. 22(5-6) (2001), 767-773. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india ∗corresponding author: saluja1963@gmail.com int. j. anal. appl. (2023), 21:45 the new dagum-x family of distributions: properties and applications amani s. alghamdi∗, huda alghamdi, aisha fayomi department of statistics, faculty of science, king abdulaziz university, jeddah, saudi arabia ∗corresponding author: amaalghamdi@kau.edu.sa abstract. various statistical distributions are still being used extensively over the previous decades for modeling data in numerous areas such as engineering, sciences, and finance. nonetheless, in a lot of applied areas, there is a continuous need for expanded forms of these distributions. however, many common distributions do not fit the data well. thus, new distributions have been constructed in literature. the purpose of this article is to present a new family of distributions using the dagum distribution as a generator and to study its properties such as hazard rate functions, moments, quantile function, ordered statistics and renyi entropy. moreover, one sub model called dagum-frechet distribution is discussed with some of its properties. the maximum likelihood estimation is employed to estimate the parameters of the proposed distribution, and the confidence intervals are obtained. finally, two real data sets are analyzed to illustrate the performance of the purposed distribution. 1. introduction statistical literature is abounding with many statistical distributions that are used for data modeling in various areas of applied life, such as engineering, actuarial sciences, education, demography, economics, finance, insurance, environmental, medical, and biological studies. the quality of statistical distribution is based on fitting the assumed probability distribution to the data. however, there are various issues where any of these distributions do not fit the data appropriately, especially in the areas of engineering, finance, medicine and environmental hazards. therefore, a significant effort has been made in developing different families of distributions. recently, there has been a growing interest of generating wide families of distributions from existing families of distributions by adding one or more additional parameter(s) to the baseline distribution. there are a lot of well-known family of distributions, such as beta-g by [1], kumaraswamy-g by [2], exponentiated generalized-g by [3], gamma-x received: feb. 19, 2023. 2020 mathematics subject classification. 62e10. key words and phrases. estimation; fréchet distribution; t-x family; hazard function; moment; maximum likelihood. https://doi.org/10.28924/2291-8639-21-2023-45 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-45 2 int. j. anal. appl. (2023), 21:45 family by [4] and logistic-x by [5]. moreover, [6] and [7] introduced the odd lomax-g family of distributions and the lomax gumbel distribution respectively. the zubair-g family of distributions was studied by [8] and the zubair-weibull distribution is obtained. [9] proposed the exponentiated gumbel family of distributions and evolved three separate models. the transformed transformer (t-x) method is considered as one of the most important ways to generalize distributions, which many have relied on in their researches. it is introduced by [10] for generating families of continuous distributions. let r(t) be the probability density function (pdf) of a random variable t, where t ∈ [a,b], for −∞ ≤ a ≤ b ≤∞. assume w(g(x)) be a function of the cumulative density function (cdf) g(x) of any random variable x, where w(g(x)) satisfies the following: i. w(g(x)) ∈ [a,b]. ii. w(g(x)) is differentiable and monotonically nondecreasing. iii. w(g(x)) → a as x→ −∞ and w(g(x)) → b as x→ ∞. the cdf and pdf of the t-x family of distributions are given respectively as: f (x) = ∫ w(g(x)) a r(t)dt, (1.1) f (x) = [ d dx w (g(x)) ] r[w (g(x))]. (1.2) the definition of w(g(x)) depends on the support of the random variable t as follows: (1) when the support of t is bounded: w(g(x)) can be defined as g(x) or g(x)α. (2) when the support of t is [a,∞), for a ≥ 0: w(g(x)) can be defined as −log(1 −g(x)) or g(x)/(1 −g(x)) or −log(1 −g(x)α). (3) when the support of t is (−∞,∞): w(g(x)) can be defined as log[−log(1 − g(x))] or log[g(x)/(1 −g(x))] ( [10]). the main purpose of this article is to introduce a new family of distributions, called the dagumx family of distributions that are more adaptable to data in a wide range of applications. this article is organized as follows: in section 2, dagum-x family of distribution is defined. also, its propability and cumulative distribution functions are introduced. some special models of this family are presented in section 3. section 4 shows some mathematical properties of the dagum-x family of distributions including survival, hazard function, rth moments, quantile function, renyi entropy and order statistics. the characterizations of one sub-model of this family are studied in section 5. the maximum likelihood method for parameter estimation is discussed in section 6. in section 7 simulation studies will be conducted to show the performance of the maximum likelihood estimation (mle) method. section 8 provides a real data application to show the flexibility of the dagum-x family. finally, concluding remarks are presented in section 9. int. j. anal. appl. (2023), 21:45 3 2. the dagum x family the importance of a statistical model lies in the fitness of the probability distribution to the data. thus, different families of probability distributions have been developed for fitting different types of data. however, there are still several constraints that affect on fitting the formed distributions to the data appropriately especially in particular applications. as mentioned earlier, dagum distribution has received interest from researchers because of its competition with other models. different extensions that include dagum distribution have been proposed and developed using different approaches in attempt to provide more flexibility in fitting data. using the kurtosis diagram provided by [11] and [12], [13] presented the log-dagum distribution and examined the changes in the kurtosis. more structural properties and parameter estimates for the log-dagum distribution were addressed by [14]. [15] proposed a new class of distributions called mc-dagum distribution. several distributions, including the beta-dagum, beta-burr iii, beta-fisk, dagum, burr iii, and fisk distributions, are included in this class of distributions as special cases. they obtained the properties of the model and the maximum likelihood estimates of the model parameters. [16] proposed a new class of weighted dagum and related distributions and discussed this class in detail. [17] studied a new five-parameter model called the extended dagum distribution and discussed the features of the model. the proposed model contains as special cases the log-logistic and burr iii distributions among others. [18] proposed a new four parameter distribution called the dagum-poisson (dp) distribution by compounding dagum and poisson distributions. the structural properties and the maximum likelihood estimates (mles) of the parameters are obtained. [19] proposed the exponentiated generalized exponential dagum distribution. there are several sub-models in this family of distributions, including the dagum distribution, burr iii distribution, exponentiated generalized dagum distribution, fisk distribution, and exponentiated generalized exponential burr iii distribution. [20] introduced a new model called a power log-dagum distribution. the model consists of many new sub-models such as: linear log-dagum, power logistic, log-dagum distributions and linear logistic among them. three distinct estimating procedures are given along with the model’s properties. the odd dagum-g family, which [21] introduced, is a new family of continuous distributions with three additional shape parameters. the properties of the suggested family and the model parameters estimates are attained. using the t-x approach, several new distributions have been introduced in the literature. we generalized the dagum distribution using t-x method by [10]. the new family of dagum distribution, called dagum-x, can be defined as follows: let f (t) and f (t) be the cdf and the pdf for a dagum random variable t ∈ [0,∞), given by 4 int. j. anal. appl. (2023), 21:45 f (t; λ,δ,β) = (1 + λt−δ) −β ,t > 0, λ,δ,β > 0, (2.1) and f (t; λ,δ,β) = βλδt−δ−1(1 + λt−δ) −β−1 ,t > 0, λ,δ,β > 0, (2.2) where ≥ is a scale parameter and ◦ and � are shape parameters. by replacing t in equation (2.1) by the w (g(x)) = g(x;θ) g(x;θ) , we obtained the cdf of a new family namely, dagum-x family, where g(x; θ) and g(x; θ)=1-g(x; θ) are the baseline cdf and survival function (sf) depending on a parameter vector θ. f (x; λ,δ,β,θ) = ( 1 + λ [ g(x; θ) g(x; θ) ]−δ)−β , (2.3) the pdf is obtained by differentiating equation (2.3) with respect to (w. r. t.) x as follows: f (x; λ,δ,β,θ) = βλδg(x) [g(x)] −δ−1 [1 −g(x)]−δ+1 [ 1 + λ ( g(x) 1 −g(x) )−δ]−β−1 , t > 0, λ,δ,β > 0. (2.4) 3. special models one of the main reasons for the desire to generate different families of distributions is to provide different extensions of appropriate distributions that are more flexible to use with data in various applications. in this section, some dagum-x special distributions are introduced, such as dagumweibull(d-w), dagum-exponential(d-exp), dagum-rayleigh(d-r) and dagum-fréchet(d-fr). 3.1. the dagumweibull distribution. the weibull distribution is one of the lifetime distributions that is most frequently used in different areas, such as economics, biology, hydrology and engineering sciences due to its simplicity and versatility. it generalizes the exponential model to include non constant failure rate functions. in particular, it encompasses both increasing and decreasing failure rate functions. as it is well known that weibull distribution (with scale and shape parameters a, b > 0) has cdf and pdf given by: g(x; a,b) = 1 −e−( x a ) b , x > 0, a,b > 0 (3.1) and g(x; a,b) = b ab xb−1e−( x a ) b , x > 0, a,b > 0 (3.2) the cdf and pdf of dagum-weibull distribution can be obtained by substituting equations (3.1) and equations (3.2) in equations (2.3) and equations (2.4) as follows: f (x; λ,δ,β,a,b) =  1 + λ [ 1 −e−( x a ) b e−( x a ) b ]−δ−β, x > 0, λ,δ,β,a,b > 0, (3.3) int. j. anal. appl. (2023), 21:45 5 and f (x; λ,δ,β,a,b) = ( βλδb ab ) xb−1 (1 −e−( x a ) b ) −δ−1 (e−( x a ) b ) −δ  1 + λ [ 1 −e−( x a ) b (e−( x a ) b ) ]−δ−β−1, (3.4) x > 0, λ,δ,β,a,b > 0. 3.2. the dagum-exponential distribution. the exponential distribution is one of the common distributions in reliability analysis. it is a particular case of the gamma distribution and often used to model the time elapsed between events. the exponential distribution (with parameter a > 0) has cdf and pdf given by: g(x; a) = 1 −e−( x a ), x > 0, a > 0 (3.5) and g(x; a) = 1 a e−( x a ), x > 0, a > 0 (3.6) the cdf and pdf of dagum-exponential distribution can be obtained by substituting equations (3.5) and equations (3.6) in equations (2.3) and equations (2.4) as follows: f (x; λ,δ,β,a) =  1 + λ [ 1 −e−( x a ) e−( x a ) ]−δ−β, x > 0, λ,δ,β,a > 0, (3.7) and f (x; λ,δ,β,a) = ( βλδ a ) (1 −e−( x a )) −δ−1 (e−( x a )) −δ  1 + λ [ 1 −e−( x a ) (e−( x a )) ]−δ−β−1, x > 0, λ,δ,β,a > 0. (3.8) 3.3. the dagum-rayleigh distribution. the rayleigh distribution is a continuous probability distribution introduced by [22]. it is a special case of the weibull distribution with a scale parameter of 2. it plays an essential role in modeling and analyzing lifetime data such as survival and reliability analysis, theory of communication, physical sciences, technology, diagnostic imaging and applied statistics. the rayleigh distribution (with scale parameter a > 0) has cdf and pdf given by: g(x; a) = 1 −e−( x a ) 2 , x > 0, a > 0 (3.9) and g(x; a) = 2 a2 xe−( x a ) 2 , x > 0, a > 0 (3.10) the cdf and pdf of dagum-rayleigh distribution can be obtained by substituting equations (3.9) and equations (3.10) in equations (2.3) and equations (2.4) as follows: f (x; λ,δ,β,a) =  1 + λ [ 1 −e−( x a ) 2 e−( x a ) 2 ]−δ−β, x > 0, λ,δ,β,a > 0 (3.11) 6 int. j. anal. appl. (2023), 21:45 and f (x; λ,δ,β,a) = ( 2βλδ a2 ) (x) (1 −e−( x a ) 2 ) −δ−1 (e−( x a ) 2 ) −δ  1 + λ [ 1 −e−( x a ) 2 (e−( x a ) 2 ) ]−δ−β−1, (3.12) x > 0, λ,δ,β,a > 0. 3.4. the dagum-fréchet distribution. the fréchet (fr) distribution was developed in the 1920s by french mathematician maurice rené fréchet to model maximum values in a data set that came from different phenomena such as flood analysis, horse racing, human lifespans, maximum rainfalls and river discharges in hydrology. it is considered as one of the extreme value distributions (ev ds), known as the ev d type ii. the fr distribution (with scale and shape parameters a,b > 0 ) has cdf and pdf given by: g(x; a,b) = e−( a x ) b , x > 0, a,b > 0 (3.13) and g(x; a,b) = babx−b−1e−( a x ) b , x > 0, a,b > 0 (3.14) the cdf and pdf of dagum-fréchet distribution can be obtained by substituting equations (3.13) and equations (3.14) in equations (2.3) and equations (2.4) as follows: f (x; λ,δ,β,a,b) =  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β, x > 0, λ,δ,β,a,b > 0, (3.15) and f (x; λ,δ,β,a,b) = βλδbabx−b−1 (e−( a x ) b ) −δ (1 −e−( a x ) b ) −δ+1  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β−1,x > 0, λ,δ,β,a,b > 0. 4. mathematical properties of the dagum-x family this section describes some of mathematical properties of the dagum-x family of distributions. 4.1. survival and hazard rate functions. let the random variable t be the time to failure of the dagum-x family of distributions. the survival and hazard rate functions of dagum-x family of distributions are, respectively, given by: s(t; θ) = 1 − ( 1 + λ [ g(t; θ) g(t; θ) ]−δ)−β , (4.1) and h(t; θ) = βλδg(t) [g(t)] −δ−1 [1−[g(t)]]−δ+1 [ 1 + λ( g(t) 1−g(t)) −δ]−β−1 1 − ( 1 + λ [ g(t;θ) g(t;θ) ]−δ)−β , (4.2) int. j. anal. appl. (2023), 21:45 7 where θ = (β,λ,δ)t is a vector of parameters of baseline distribution. 4.2. moments. let x be a random variable follows the dagum-x family with the density function given in equation (2.4). the rth moment of x is given by: e(xr ) = βλδ ∫ ∞ 0 xrg(x) [g(x)] −δ−1 [1 − [g(x)]]−δ+1 [ 1 + λ ( g(x) 1 −g(x) )−δ]−β−1 dx. using the expansion (see [23]): (1 + x) −(n+1) = ∞∑ k=0 ( n + k k ) (−1)kxk, (4.3) we have e(xr ) =βλδ ∫ ∞ 0 xrg(x) [g(x)] −δ−1 [1 − [g(x)]]−δ+1 ∞∑ k=0 (−1)kλk ( β + k k )( g(x) 1 −g(x) )−δk dx, =βδ ∫ ∞ 0 xrg(x) ∞∑ k=0 (−1)kλk+1 ( β + k k ) [g(x)] −δ(k+1)−1 [1 −g(x)]δ(k+1)−1dx, and using the following expansion (see [24]) (1 −x)n = ∞∑ k=0 ( n k ) (−1)kxk, (4.4) we have e(xr ) =βδ ∫ ∞ 0 xrg(x) ∞∑ k=0 (−1)kλk+1 ( β + k k ) [g(x)] −δ(k+1)−1 ∞∑ m=0 (−1)m ( δ(k + 1) − 1 m ) [g(x)] m dx, =βδ ∫ ∞ 0 xrg(x) ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) [g(x)] j dx therefore, e(xr ) = c1 ∫ ∞ 0 xrg(x)[g(x)] j dx. (4.5) where c1 = βδ ∑∞ k,m=0 (−1) k+m λk+1 ( β+k k )( δ(k+1)−1 m ) , j = m−δ(k + 1) − 1. 4.3. quantail function. let x be a random variable that has the cdf given in equation (2.3). the quantile function, q(u) of x can be derived as follows: let u = f (x) = ( 1 + λ [ g(x; θ) g(x; θ) ]−δ)−β , after simplification, the quantile function is expressed as q(u) = g−1   ( [u] −1 β −1 λ )−1 δ 1 + ( [u] −1 β −1 λ )−1 δ   , (4.6) 8 int. j. anal. appl. (2023), 21:45 where, u is a uniform random number on the interval (0, 1) and g−1(.) is the inverse function of g(.). in particular, q(0.5) is the median of the family and defined by substituting u = 0.5 in equation (4.6): q(0.5) = g−1   ( [0.5] −1 β −1 λ )−1 δ 1 + ( [0.5] −1 β −1 λ )−1 δ   . the first and third quartiles can be obtained also by substituting u = 0.25 and u = 0.75, respectively, in equation (4.6), as follows: q(0.25) = g−1   ( [0.25] −1 β −1 λ )−1 δ 1 + ( [0.25] −1 β −1 λ )−1 δ   , and q(0.75) = g−1   ( [0.75] −1 β −1 λ )−1 δ 1 + ( [0.75] −1 β −1 λ )−1 δ   . 4.4. rényi entropys. the entropy of a random variable x represents a measure of uncertainty variation. let x be a random variable that has the pdf given in equation (2.4), then the rényi entropy of the random variable x is defined as: rθ(x) = (1 −θ) −1 log [∫ ∞ 0 f (x) θ dx ] , θ > 0 and θ 6= 1. (4.7) therefore, by applying equation (2.4) into equation (4.7), we have: rθ(x) = 1 1 −θ log  ∫ ∞ 0 (βλδ) θ [g(x)] θ [ [g(x)] −θ(δ+1) [1 −g(x)]−θ(δ−1) ][ 1 + λ ( g(x) 1 −g(x) )−δ]−θ(β+1) dx   . using the expansions in equation (4.3) and equation (4.4), the rényi entropy of the dagum-x, rθ(x), can be written as: rθ(x) = 1 1 −θ log ∫ ∞ 0 (βλδ) θ [g(x)] θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) × (−1)r+kλk[g(x)]−θ(δ+1)−δk+rdx thus, rθ(x) = 1 1 −θ log [ c1 ∫ ∞ 0 [g(x)] θ [g(x)] j1dx ] (4.8) int. j. anal. appl. (2023), 21:45 9 where j1 and c1 are resbectivaly, defined as follow: j1 = −θ(δ + 1) −δk + r c1 =(βλδ) θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk 4.5. order statistics. let x1:n,x2:n, ...,xn:n be the order statistics obtained from the dagum-x with cdf f (x) and pdf f (x), respectively, given in equation (2.3) and equation (2.4). the pdf of the ith order statistics can be expressed as: fi:n(x) = n!βλδg(x) (i − 1)!(n− i)! [g(x)] −δ−1 [1 − [g(x)]]−δ+1 [ 1 + λ ( g(x) 1 −g(x) )−δ]−β−1 ×  [1 + λ( g(x) 1 −g(x) )−δ]−βi−1  1 − [ 1 + λ ( g(x) 1 −g(x) )−δ]−βn−i let u = [ 1 + λ ( g(x) 1 −g(x) )−δ]−β , (4.9) then fi:n(x) = n!βλδg(x) (i − 1)!(n− i)! [g(x)] −δ−1 [1 − [g(x)]]−δ+1 u (1+1 β ) [u] i−1 [1 −u]n−i, = n!βλδg(x) (i − 1)!(n− i)! [g(x)] −δ−1 [1 − [g(x)]]−δ+1 u (i+1 β ) [1 −u]n−i, by applying the expansion in equation (4.4), we have fi:n(x) = n!βδg(x) (i − 1)!(n− i)! ∞∑ k,m=0 ( n− i k )( βi + βk + m m ) (−1)k+m(λ)m+1g(x)−(δm+δ+1) × ∞∑ l=0 ( δm + δ − 1 l ) (−1)lg(x)l, = n!βδg(x) (i − 1)!(n− i)! ∞∑ k,m,l=0 ( n− i k )( βi + βk + m m )( δm + δ − 1 l ) (−1)k+m+l(λ)m+1g(x)l−(δm+δ+1). (4.10) 5. dagum-fréchet distribution and its properties the fréchet distribution is becoming increasingly a preferred distribution in extending new statistical models. [25] introduced a distribution that generalizes the fréchet distribution, known as the exponentiated fréchet distribution and included a detailed analysis of the mathematical properties of this new distribution. [26] introduced and studied three component mixtures of the fréchet distributions when 10 int. j. anal. appl. (2023), 21:45 the shape parameter is known under bayesian view point. [27] developed a new compound continuous distribution named the gompertz fréchet distribution which extends the frèchet distribution. [28] proposed a new four-parameter fréchet distribution called the odd lomax fréchet distribution. the new model can be expressed as a linear mixture of fréchet densities. the d-fr distribution is introduced briefly in (3.4) as a special model of the dagum-x family. the cdf and pdf of the distribution are given in equations (3.15) and (??), respectively. in this section, mathematical properties of the new distribution are presented and the maximum likelihood estimation is employed to estimate the parameters of the new distribution. monte carlo simulation by using r program to assess the performance of the maximum likelihood estimation is applied and discussed. finally, real data sets are analyzed to illustrate the performance of the proposed distribution. the plot of the pdf is presented using different values for the five parameters to study its behaviour, as shown in figure (1). 0 2 4 6 8 10 0 .0 0 .2 0 .4 0 .6 0 .8 x p d f parameters β=1.5, λ=0.5, δ=0.75, a=1.0, b=1.5 β=2.0, λ=1.0, δ=0.85, a=1.5, b=2.0 β=2.5, λ=1.5, δ=0.95, a=2.0, b=2.5 β=3.0, λ=2.0, δ=1.15, a=2.5, b=3.0 figure 1. the d-fr density function when all shape and scale parameters are changing. figure (1) displays the density function of the d-fr for different values of the shape and scale parameters. it is right skewed and has different levels of kurtosis which shows the flexibility of the distribution for modelling skew data. 5.1. survival and hazrd functions. the survival and hazard rate functions of d-fr are given by substituting equation (3.13) and (3.14) in equation (4.1) and (4.2) respectively as follows: s(x; θ) = 1 −  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β, (5.1) int. j. anal. appl. (2023), 21:45 11 and h(x; θ) = βλδbabx−b−1 ( (e−( a x ) b ) −δ (1−e−( a x ) b ) −δ+1 )( 1 + λ [ e−( a x ) b (1−e−( a x ) b ) ]−δ)−β−1 1 − ( 1 + λ [ e−( a x ) b (1−e−( a x ) b ) ]−δ)−β . (5.2) 0 2 4 6 8 10 0 .0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 x h (x ) parameters β=1.5, λ=0.5, δ=0.75, a=1.0, b=1.5 β=2.0, λ=1.0, δ=0.85, a=1.5, b=2.0 β=2.5, λ=1.5, δ=0.95, a=2.0, b=2.5 β=3.0, λ=2.0, δ=1.15, a=2.5, b=3.0 figure 2. the d-fr hazard rate function when all shape and scale parameters are changing. various curves of the hazard function of dagum-fréchet distribution are shown in figure (2). by assuming different values of the shape and scale parameters, the curves appear to be unimodal and positive skewed with different levels of skewness and kurtosis. 5.2. moments. the rth moment of the d-fr distribution is obtained by substituting fréchet distribution’s cdf and pdf in equations (3.13) and (3.14) into the rth moment of dagum-x in equation (4.5). as a result, the rth moment of d-fr distribution is given as e(xr ) = c1 ∫ ∞ 0 xrg(x)[g(x)] j dx, = c1 ∫ ∞ 0 xr ( babx−b−1e−( a x ) b )[ e−( a x ) b ]j dx, = c2 ∫ ∞ 0 xr−b−1 [ e−( a x ) b ]j2 dx, where, c1 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , c2 = c1ba b, 12 int. j. anal. appl. (2023), 21:45 and j2 = j + 1 = m−δ(k + 1) using integration by substitution, let u = j2 [ ( a x ) b ] then x = [ a( u j2 ) −(1 b ) ] and dx = ( −a bj2 )( u j2 ) (−1 b −1) du. hence, e(xr ) = c2 ∫ ∞ 0 xr−b−1 [ e−( a x ) b ]j2 dx = c2 ∫ 0 ∞ [ a( u j2 ) −(1 b ) ]r−b−1 e−u( −a bj2 )( u j2 ) (−1 b −1) du = c2 ar−b b ( 1 j2 )−r b +1∫ ∞ 0 u −r b e−udu = c2 ar−b b ( 1 j2 )−r b +1 γ(− r b + 1) = βδbab ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) ar−b b ( 1 j2 )−r b +1 γ(− r b + 1) = βδar ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m )( 1 j2 )−r b +1 γ(− r b + 1) then, the moment of dagum-fréche distribution is given as e(xr ) = c3 ( 1 j2 )−r b +1 ar γ(− r b + 1), [1 − r b ] > 0 (5.3) where, c3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , and j2 = m−δ(k + 1). 5.2.1. mean and variance. the mean of d-fr distribution can be obtained by setting (r = 1) in equation (5.3), which results in the following form: e(x) = c3 ( 1 j2 )−1 b +1 aγ(− 1 b + 1), (5.4) where, c3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , int. j. anal. appl. (2023), 21:45 13 and j2 = m−δ(k + 1). the 2nd moment e(x2) can be found by setting (r = 2) in equation (5.3), then the variance of d-fr distribution can be obtained as follows: v ar(x) = [c3 ( 1 j2 )−2 b +1 a2γ(− 2 b + 1)] − [c3 ( 1 j2 )−1 b +1 aγ(− 1 b + 1)] 2 . (5.5) where, c3 = βδ ∞∑ k,m=0 (−1)k+mλk+1 ( β + k k )( δ(k + 1) − 1 m ) , and j2 = m−δ(k + 1). 5.3. quantail function. the quantail function of the d-fr distribution, x = f−1(u), can be obtained by inverting the cdf in equation (3.15) as follows: x = q(u) = a −log [ (u) −1 β −1 λ ]−1 δ  1+ ( (u) −1 β −1 λ )−1 δ     1 b . (5.6) where, u is a uniform random number on the interval (0, 1). therefore, the median of the d-fr can be found by substituting u = 0.5 in equation (5.6) as follows: q(0.5) = a −log [ (0.5) −1 β −1 λ ]−1 δ  1+ ( (0.5) −1 β −1 λ )−1 δ     1 b . (5.7) the first and third quartiles can also be obtained by substituting u = 0.25 and u = 0.75 in equation (5.6), respectively as follows: q(0.25) = a −log [ (0.25) −1 β −1 λ ]−1 δ  1+ ( (0.25) −1 β −1 λ )−1 δ     1 b , 14 int. j. anal. appl. (2023), 21:45 and q(0.75) = a −log [ (0.75) −1 β −1 λ ]−1 δ  1+ ( (0.75) −1 β −1 λ )−1 δ     1 b . 5.4. rényi entropys. using the definition of the rényi entropy in equation (4.7), and applying the cdf and pdf of frechet distribution in equation (3.13) and (3.14), we have: rθ(x) = 1 1 −θ log [ c1 ∫ ∞ 0 (babx−b−1e−( a x ) b ) θ (e−( a x ) b ) j1 dx ] = 1 1 −θ log [ c2 ∫ ∞ 0 (x−θ(b+1))(e−( a x ) b ) θ (e−( a x ) b ) j1 dx ] = 1 1 −θ log [ c2 ∫ ∞ 0 x−θ(b+1)(e−( a x ) b j2)dx ] where, c1 =(βλδ) θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk, c2 =c1(ba b) θ , j1 =−θ(δ + 1) −δk + r, and j2 =θ + j1. using integration by substitution and after simplification, we get rθ(x) = 1 1 −θ log [ c3 ∫ ∞ 0 uθ+ θ b −1 b −1e−udu ] , rθ(x) = 1 (1 −θ) log [ c3γ(θ + θ b − 1 b ) ] , [θ + θ b − 1 b ] > 0 (5.8) where c3 and j2 are, respectively, as follows: c3 =(βδ) θ bθ−1a1−θ ∞∑ k,r=0 ( θ(β + 1) + k − 1 k )( θ(δ − 1) + δk r ) (−1)r+kλk+θ ( 1 j2 )(θ+θ−1 b ) , j2 =−δ(θ + k) + r. int. j. anal. appl. (2023), 21:45 15 5.5. order statistics. the order statistics of d-fr distribution is obtained by substituting the fr distribution’s cdf and pdf in equation (3.13) and (3.14) in the order statistics of dagum-x family, in equation (4.10), as follows: fi:n(x) = n! (i − 1)!(n− i)! βδbabx(−b−1) ∞∑ k,m,l=0 ( n− i k )( βi + βk + m m )( δm + δ − 1 l ) (−1)k+m+l × (λ)m+1[e−( a x ) b ] l−δm−δ . (5.9) 6. maximum likelihood estimation in this section, the mle method will be applied to estimate the unknown parameters of the d-fr distribution. assume that x1,x2, ...xn is a random sample of the d-fr distribution, then the likelihood function for the vector of parameters θ = (β,λ,δ,a,b)t is given by: l(θ) = n∏ i=1 βλδbabx−b−1 (e−( a x ) b ) −δ (1 −e−( a x ) b ) −δ+1  1 + λ [ e−( a x ) b (1 −e−( a x ) b ) ]−δ−β−1, (6.1) then the log likelihood function can be written as: l = logl = nlogβ + nlogλ + nlogδ + nlogb + nbloga− (b + 1) n∑ i=1 log[x] −δ n∑ i=1 log[e−( a x ) b ] +(δ − 1) n∑ i=1 log[1 −e−( a x ) b ] − (β + 1) n∑ i=1 log  1 + λ ( e−( a x ) b 1 −e−( a x ) b )−δ. (6.2) the first partial derivatives of the log likelihood function in equation (6.2) with respect to �, ≥, ◦, a and b are respectively given as follows: ∂l ∂β = n β − n∑ i=1 log  1 + λ   e−( axi )b 1 −e−( a xi ) b  −δ  , (6.3) ∂l ∂λ = n λ − (β + 1) n∑ i=1 ( e −( axi ) b 1−e −( axi ) b )−δ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ , (6.4) ∂l ∂δ = n δ − n∑ i=1 log[e −( a xi ) b ] − (β + 1) n∑ i=1 − ( e −( axi ) b 1−e −( axi ) b )−δ λlog ( e −( axi ) b 1−e −( axi ) b ) 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ + n∑ i=1 log[1 −e−( a xi ) b ], (6.5) 16 int. j. anal. appl. (2023), 21:45 ∂l ∂a = nb a − (β + 1) n∑ i=1  λ   be −2( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δ( a xi ) −1+b xi(1−e −( axi ) b )2 + be −( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δ( a xi ) −1+b x(1−e −( axi ) b )     [ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ] −δ ∑n i=1 b[ a xi ]−1+b xi + (−1 + δ) ∑n i=1 be −( axi ) b ( a xi ) −1+b xi[1−e −( axi ) b ] , (6.6) ∂l ∂b = n b + nloga− n∑ i=1 log[xi ] −(β + 1) n∑ i=1  λ   e −2( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δlog( a xi )( a xi ) b (1−e −( axi ) b )2 + e −( axi ) b   e−( axi )b 1−e −( axi ) b  −δ−1δlog( a xi )( a xi ) b 1−e −( axi ) b     [ 1 + λ ( e −( axi ) b 1−e −( axi ) b )−δ] −δ ∑n i=1−log( a xi )( a xi ) b + (−1 + δ) ∑n i=1 e −( axi ) b log( a xi )( a xi ) b 1−e −( axi ) b . (6.7) the mles β̂, λ̂, δ̂, â, b̂ of β,λ,δ,a,b can be obtained by equating the results to zero and solving the system of nonlinear equations numerically. for interval estimation of the model parameters, inverting fisher information matrix is required, but finding the expectation of the fisher information matrix is not easy. therefore, the 5x5 observed information matrix is used to generate confidence intervals for the model parameters. the observed information matrix is given as follows: i(θ̂)=   − ∂ 2l ∂β2 − ∂ 2l ∂β∂λ − ∂ 2l ∂β∂δ − ∂ 2l ∂β∂a − ∂ 2l ∂β∂b − ∂ 2l ∂λ∂β − ∂ 2l ∂λ2 − ∂ 2l ∂λ∂δ − ∂ 2l ∂λ∂a − ∂ 2l ∂λ∂b − ∂ 2l ∂δ∂β − ∂ 2l ∂δ∂λ −∂ 2l ∂δ2 − ∂ 2l ∂δ∂a − ∂ 2l ∂δ∂b − ∂ 2l ∂a∂β − ∂ 2l ∂a∂λ − ∂ 2l ∂a∂δ −∂ 2l ∂a2 − ∂ 2l ∂a∂b − ∂ 2l ∂b∂β − ∂ 2l ∂b∂λ − ∂ 2l ∂b∂δ − ∂ 2l ∂b∂a −∂ 2l ∂b2   the expectation of the observed information matrix can be solved iteratively using r software. therefore, the multivariate normal distribution n5(0, i−1) can be used to construct 100(1-�)% two sided approximate confidence intervals for the model parameters �, ≥, ◦, a and b where α is the significant level. 7. simulation study in this section, simulation studies have been performed using r program to evaluate the theoretical results of the estimation process. the performance of the mles of the parameters has been considered. furthermore, the approximate confidence intervals with confidence level 90% are obtained. the int. j. anal. appl. (2023), 21:45 17 algorithm for the simulation procedure is described below: step 1: 5000 random samples of size n=75, 100, 200, 300, 600 and 1000 are generated from the d-fr distribution. the true parameter values are assumed as ( �=0.75, ≥=0.2, ◦=0.1, a = 0.9 and b = 0.7). step 2: the parameters of the distribution are estimated using the mle method for each sample. step 3: the r function (nlminb) is used to solve the five nonlinear likelihood for �, ≥, ◦, a and b. step 4: for each simulation, the average biases (abs) and the mean sqare errors (mses) are calculated by: bias(ŷ) = ∑5000 i=1 1 5000 (ŷ −y), mse(ŷ) = ∑5000 i=1 1 5000 (ŷ −y)2. table 1. mles, abs, mse and 90% confidence limits of the parameters when n= 75, 100, 200, 300, 600 and 1000. sample parameter estimate bias mse lower limit upper limit length β 0.8054288 0.053428772 0.121979438 0.23019589 1.3706617 1.1404658 λ 0.2755135 0.075513528 0.254675089 0.54778984 1.0988169 0.5510270 n=75 δ 0.1209819 0.020981874 0.002012837 0.05554949 0.1864143 0.1308648 a 0.8423633 -0.057636675 0.153099671 0.20379481 1.4809318 1.2771370 b 0.7091435 0.009143525 0.009290203 0.55082434 0.8674627 0.3166384 β 0.8005119 0.050511859 0.084791214 0.33123212 1.2757916 0.9445595 λ 0.2197650 0.019764989 0.057678640 0.17516100 0.6146910 0.4398520 n=100 δ 0.1173703 0.017370314 0.001444903 0.06158238 0.1731582 0.1115759 a 0.8595773 -0.040422730 0.100288315 0.34132407 1.3778305 1.0365064 b 0.6978595 -0.008140522 0.004583588 0.58620669 0.8095123 0.2233056 β 0.7746440 0.024643973 0.0211146825 0.53835754 1.0109304 0.4725729 λ 0.1953267 -0.004673258 0.0038030341 0.09386590 0.2967876 0.2029217 n=200 δ 0.1090029 0.009002877 0.0007865224 0.06517773 0.1528280 0.0876503 a 0.8856043 -0.014395702 0.0190316874 0.65922062 1.1119880 0.4527674 b 0.6917052 -0.007294776 0.0010332734 0.64046296 0.7429475 0.1024845 β 0.7622825 0.012282484 0.0073377378 0.62240298 0.9021620 0.27975901 λ 0.1955416 -0.004458362 0.0008766267 0.14724566 0.2438376 0.09659195 n=300 δ 0.1051773 0.005177325 0.0001848952 0.08443121 0.1259234 0.04149223 a 0.8932748 -0.006725190 0.0063372406 0.76239317 1.0241565 0.26176328 b 0.6936197 -0.006380267 0.0004032663 0.66220213 0.7250373 0.06283520 β 0.7520984 0.002098386 1.419876e-03 0.69002082 0.8141759 0.12415513 λ 0.1987551 -0.001244911 1.054027e-04 0.18194023 0.2155699 0.03362971 n=600 δ 0.1014451 0.001445127 5.141023e-05 0.08985726 0.1130330 0.02317574 a 0.8981831 -0.001816934 8.178124e-04 0.85109266 0.9452735 0.09418082 b 0.6982087 -0.001791341 6.182002e-05 0.68557661 0.7108407 0.02526411 β 0.7501594 0.0001594468 6.794310e-05 0.73656144 0.7637575 0.027196020 λ 0.1998049 -0.0001951371 7.570256e-06 0.19527647 0.2043333 0.009056788 n=1000 δ 0.1001842 0.0001841564 2.657683e-06 0.09751148 0.1028568 0.005345358 a 0.8996841 -0.0003158548 7.301607e-05 0.88559462 0.9137737 0.028179044 b 0.6997525 -0.0002474826 4.987021e-06 0.69609049 0.7034145 0.007324047 18 int. j. anal. appl. (2023), 21:45 from table (1), it can be observed that, as the sample size increases, the mles approach to the initial values of the parameters. for each sample size n, the mles are evaluated using two accuracy measures which are abs and mse. as the sample size increases, the abs and mses of the estimated parameters decrease. this indicates that the maximum likelihood estimation method provides consistent estimators for the parameters and approaches the population parameters’ values as the sample size increases. it is also noted that the lengths of the confidence intervals for the estimated parameters decrease as the sample size increases. 8. application for more illustration, this section compares the efficiency of the goodness-of-fit for the d-fr distribution with some selected distributions in literature. in particular, two real data sets are used to compare the proposed model with four other distributions, namely, beta-fréchet(bf) by [29], gammaextended-fréchet(gef) by [30], exponentiated-exponential-fréchet(eef) by [31] and fréchet(f) distributions by [32] which is also sudied by [33]. the first data set in table (2) that is used in comparison is provided by cordeiro and silva [34]. the data represent the strengths of 1.5 cm glass fibers, measured at the national physical laboratory, england. the second data in table (3) represents breaking stress of carbon fibers of 50 mm length (gpa) and have been previously used by [35]. table 2. strength of 1.5 cm glass fibres data (data set 1). 0.55 0.74 0.77 0.81 0.84 1.24 0.93 1.04 1.11 1.13 1.30 1.25 1.27 1.28 1.29 1.48 1.36 1.39 1.42 1.48 1.51 1.49 1.49 1.50 1.50 1.55 1.52 1.53 1.54 1.55 1.61 1.58 1.59 1.60 1.61 1.63 1.61 1.61 1.62 1.62 1.67 1.64 1.66 1.66 1.66 1.70 1.68 1.68 1.69 1.70 1.78 1.73 1.76 1.76 1.77 1.89 1.81 1.82 1.84 1.84 2.00 2.01 2.24 table 3. breaking stress of carbon fibers of 50 mm length data (data set 2). 0.39 0.85 1.08 1.25 1.47 1.57 1.61 1.61 1.69 1.80 1.84 1.87 1.89 2.03 2.03 2.05 2.12 2.35 2.41 2.43 2.48 2.50 2.53 2.55 2.55 2.56 2.59 2.67 2.73 2.74 2.79 2.81 2.82 2.85 2.87 2.88 2.93 2.95 2.96 2.97 3.09 3.11 3.11 3.15 3.15 3.19 3.22 3.22 3.27 3.28 3.31 3.31 3.33 3.39 3.39 3.56 3.60 3.65 3.68 3.70 3.75 4.20 4.38 4.42 4.70 4.90 certain criteria are used in order to compare between the distributions. the distribution with best fit is the one that has the lowest value of the information criteria (aic, aicc, bic and hqic) that are defined as int. j. anal. appl. (2023), 21:45 19 aic = −2l(θ̂) + 2p, bic = −2l(θ̂) + plog(n), aicc = aic + 2p(p + 1) n−p− 1 , hqic = −2l(θ̂) + 2plog(log(n)). where l(θ̂) is denoted by the log likelihood function evaluated at the maximum likelihood estimates, p is the number of parameters in the model and n is the sample size. table 4. the log likelihood, aic, aicc, bic and hqic for the data set 1 distribution l̂ aic aicc bic hqic d-fr 12.7337 35.46739 36.52003 46.18307 39.68192 bf 30.22177 68.44353 69.13319 77.01607 71.81516 gef 30.66209 69.32417 70.01383 77.89671 72.69579 f 46.85336 97.70672 97.90672 101.993 99.39253 eef 47.63954 103.2791 103.9687 111.8516 106.6507 table 5. the log likelihood, aic, aicc, bic and hqic for the data set 2 distribution l̂ aic aicc bic hqic d-fr 85.27445 180.5489 181.5489 191.4972 184.8751 bf 100.276 208.552 209.2077 217.3106 212.0129 eef 100.4079 208.8158 209.4716 217.5744 212.2768 gef 101.1724 210.3448 211.0006 219.1035 213.8058 f 121.195 246.39 246.5805 250.7693 248.1205 table 6. the mles for the data sets 1 and 2 data distribution β λ δ a b d-fr 0.2231043 15.0981157 15.6678754 1.0476744 0.9305450 bf 0.6071596 1.9876964 ... 17.9057801 41.5883328 data 1 gef 0.7120245 1.7786321 ... 35.0796598 13.7818068 f ... ... ... 1.263794 2.887747 eef 33.4764197 4.7270858 ... 27.9266876 0.3634021 d-fr 0.2502454 0.7341762 22.7850195 1.1852932 0.3223076 bf 0.3798751 3.1899014 ... 20.5420435 40.0668395 data 2 gef 0.5395719 3.5837019 ... 23.1357231 8.5334421 f ... ... ... 2.034154 1.649719 eef 30.5722159 2.2994984 ... 34.5713778 0.4487714 20 int. j. anal. appl. (2023), 21:45 tables (4) and (5), demonstrate that the d-fr model has the lowest value of the information criteria which implies that the proposed model provides a better fit than the other comparative models. x1 d e n s it y 0 1 2 3 4 0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0 distributions dagum−frechet frechet beta−frechet gamma extended frechet exponentiated−exponential frechet distributions dagum−frechet frechet beta−frechet gamma extended frechet exponentiated−exponential frechet figure 3. fitted density curves to the first real data. x1 d e n s it y 0 1 2 3 4 5 6 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 distributions dagum−frechet frechet beta−frechet gamma extended frechet exponentiated−exponential frechet figure 4. fitted density curves to the second real data. plots of the fitted densities of the five distributions are shown in figures (3) and (4). the plots illustrate that the d-fr distribution provides a better fit to the data than other distributions. 9. conclusion the development of generalizing families of distributions have attracted the attention of both theoretical and applied statisticians. in this paper, a new family of distributions, called the dagum-x family of distribution is introduced. the mathematical properties of dagum-x family of distributions are discussed. a sub model called dagum-frechet distribution is presented with some of its properties. the maximum likelihood estimation method was employed for estimating the model parameters and int. j. anal. appl. (2023), 21:45 21 investigated through a simulation study. the simulation study indicates that the maximum likelihood estimation method provides consistent estimators for the parameters. the performance of the dagumfrechet distribution was compared to that of beta frechet, gamma-extended-frechet, exponentiatedexponential-frechet, and frechet distributions using two real-life data sets for demonstration purposes. the proposed distribution has better fit than other competing distributions. it is concluded that the dagum-fréchet distribution is a competitive model for modeling real-life data in different areas. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] n. eugene, c. lee, f. famoye, beta-normal distribution and its applications, commun. stat. theory methods. 31 (2002), 497-512. https://doi.org/10.1081/sta-120003130. [2] g.m. cordeiro, m. de castro, a new family of generalized distributions, j. stat. comput. simul. 81 (2011), 883-898. https://doi.org/10.1080/00949650903530745. [3] g.m. cordeiro, e.m. ortega, d.c. da cunha, the exponentiated generalized class of distributions, j. data sci. 11 (2013), 1-27. [4] a. alzaatreh, f. famoye, c. lee, the gamma-normal distribution: properties and applications, comput. stat. data anal. 69 (2014), 67-80. https://doi.org/10.1016/j.csda.2013.07.035. [5] m.h. tahir, g.m. cordeiro, a. alzaatreh, m. mansoor, m. zubair, the logistic-x family of distributions and its applications, commun. stat. theory methods. 45 (2016), 7326-7349. https://doi.org/10.1080/03610926. 2014.980516. [6] g.m. cordeiro, a.z. afify, e.m.m. ortega, a.k. suzuki, m.e. mead, the odd lomax generator of distributions: properties, estimation and applications, j. comput. appl. math. 347 (2019), 222-237. https://doi.org/10. 1016/j.cam.2018.08.008. [7] m. mahmoud, r. mandouh, r. abdelatty, lomax-gumbel {frechet}: a new distribution, j. adv. math. computer sci. 31 (2019), 1-19. https://doi.org/10.9734/jamcs/2019/v31i230108. [8] z. ahmad, the zubair-g family of distributions: properties and applications, ann. data. sci. 7 (2018), 195-208. https://doi.org/10.1007/s40745-018-0169-9. [9] a. fayomi, s. khan, m.h. tahir, a. algarni, f. jamal, r. abu-shanab, a new extended gumbel distribution: properties and application, plos one. 17 (2022), e0267142. https://doi.org/10.1371/journal.pone.0267142. [10] a. alzaatreh, c. lee, f. famoye, a new method for generating families of continuous distributions, metron. 71 (2013), 63-79. https://doi.org/10.1007/s40300-013-0007-y. [11] m. zenga, la curtosi, statistica. 56 (1996), 87-102. [12] m. polisicchio, m. zenga, kurtosis diagram for continuous random variables, metron. 55 (1997), 21-41. [13] f. domma, kurtosis diagram for the log-dagum distribution, stat. appl. 2 (2004), 3–23. [14] f. domma, p.f. perri, some developments on the log-dagum distribution, stat. methods appl. 18 (2008), 205-220. https://doi.org/10.1007/s10260-007-0091-3. [15] b.o. oluyede, s. rajasooriya, the mc-dagum distribution and its statistical properties with applications, asian j. math. appl. 1 (2013), ama085. [16] b.o. oluyede, y. ye, weighted dagum and related distributions, afr. mat. 25 (2013), 1125-1141. https://doi. org/10.1007/s13370-013-0176-0. https://doi.org/10.1081/sta-120003130 https://doi.org/10.1080/00949650903530745 https://doi.org/10.1016/j.csda.2013.07.035 https://doi.org/10.1080/03610926.2014.980516 https://doi.org/10.1080/03610926.2014.980516 https://doi.org/10.1016/j.cam.2018.08.008 https://doi.org/10.1016/j.cam.2018.08.008 https://doi.org/10.9734/jamcs/2019/v31i230108 https://doi.org/10.1007/s40745-018-0169-9 https://doi.org/10.1371/journal.pone.0267142 https://doi.org/10.1007/s40300-013-0007-y https://doi.org/10.1007/s10260-007-0091-3 https://doi.org/10.1007/s13370-013-0176-0 https://doi.org/10.1007/s13370-013-0176-0 22 int. j. anal. appl. (2023), 21:45 [17] a. de o. silva, l.c.m. da silva, g.m. cordeiro, the extended dagum distribution: properties and application, j. data sci. 13 (2021), 53-72. https://doi.org/10.6339/jds.201501_13(1).0004. [18] b.o. oluyede, g. motsewabagale, s. huang, g. warahena-liyanage, m. pararai, the dagum-poisson distribution: model, properties application, electron. j. appl. stat. anal. 9 (2016), 169-197. https://doi.org/10.1285/ i20705948v9n1p169. [19] s. nasiru, p.n. mwita, o. ngesa, exponentiated generalized exponential dagum distribution, j. king saud univ. sci. 31 (2019), 362-371. https://doi.org/10.1016/j.jksus.2017.09.009. [20] h.s. bakouch, m.n. khan, t. hussain, c. chesneau, a power log-dagum distribution: estimation and applications, j. appl. stat. 46 (2018), 874-892. https://doi.org/10.1080/02664763.2018.1523376. [21] a.z. afify, m. alizadeh, the odd dagum family of distributions: properties and applications, j. appl. probab. stat. 15 (2020), 45–72. [22] l. rayleigh, xii. on the resultant of a large number of vibrations of the same pitch and of arbitrary phase, london edinburgh dublin phil. mag. j. sci. 10 (1880), 73-78. https://doi.org/10.1080/14786448008626893. [23] a. alzaghal, f. famoye, c. lee, exponentiated t-x family of distributions with some applications, int. j. stat. probab. 2 (2013), 31-49. https://doi.org/10.5539/ijsp.v2n3p31. [24] a.s. hassan, m. elgarhy, z. ahmad, type ii generalized topp-leone family of distributions: properties and applications, j. data sci. 17 (2019), 638-658. [25] s. nadarajah and s. kotz, the exponentiated frechet distribution, interstat electron. j. 14 (2003), 1-7. [26] t. sultana, m. aslam, j. shabbir, bayesian analysis of the mixture of frechet distribution under different loss functions, pak. j. stat. oper. res. 13 (2017), 501-528. https://doi.org/10.18187/pjsor.v13i3.1703. [27] p.e. oguntunde, m.a. khaleel, m.t. ahmed, h.i. okagbue, the gompertz frechet distribution: properties and applications, cogent math. stat. 6 (2019), 1568662. https://doi.org/10.1080/25742558.2019.1568662. [28] m. hamed, f. aldossary, a.z. afify, the four-parameter frechet distribution: properties and applications, pak. j. stat. oper. res. 16 (2020), 249–264. https://doi.org/10.18187/pjsor.v16i2.3097. [29] s. nadarajah, a.k. gupta, the beta frechet distribution, far east j. theor. stat. 14 (2004), 15-24. [30] r.v. da silva, t.a.n. de andrade, d.b.m. maciel, r.p.s. campos, g.m. cordeiro, a new lifetime model: the gamma extended frechet distribution, j. stat. theory appl. 12 (2013), 39-54. https://doi.org/10.2991/jsta. 2013.12.1.4. [31] m. mansoor, m.h. tahir, a. alzaatreh, g.m. cordeiro, m. zubair, s.a. ghazali, an extended frechet distribution: properties and applications, j. data sci. 14 (2021), 167-188. https://doi.org/10.6339/jds.201601_14(1) .0010. [32] m. frechet, sur les ensembles compacts de fonctions mesurables, fundam. math. 9 (1927), 25–32. [33] p.l. ramos, f. louzada, e. ramos, s. dey, the frechet distribution: estimation and application an overview, j. stat. manage. syst. 23 (2019), 549-578. https://doi.org/10.1080/09720510.2019.1645400. [34] g.m. cordeiro, r.b. da silva, the complementary extended weibull power series class of distributions, ciência e natura, 36 (2014), 1-13. https://doi.org/10.5902/2179460x13194. [35] l. tomy, j. gillariose, a generalized rayleigh distribution and its application, biometrics biostat. int. j. 8 (2019), 139-143. https://doi.org/10.15406/bbij.2019.08.00282. https://doi.org/10.6339/jds.201501_13(1).0004 https://doi.org/10.1285/i20705948v9n1p169 https://doi.org/10.1285/i20705948v9n1p169 https://doi.org/10.1016/j.jksus.2017.09.009 https://doi.org/10.1080/02664763.2018.1523376 https://doi.org/10.1080/14786448008626893 https://doi.org/10.5539/ijsp.v2n3p31 https://doi.org/10.18187/pjsor.v13i3.1703 https://doi.org/10.1080/25742558.2019.1568662 https://doi.org/10.18187/pjsor.v16i2.3097 https://doi.org/10.2991/jsta.2013.12.1.4 https://doi.org/10.2991/jsta.2013.12.1.4 https://doi.org/10.6339/jds.201601_14(1).0010 https://doi.org/10.6339/jds.201601_14(1).0010 https://doi.org/10.1080/09720510.2019.1645400 https://doi.org/10.5902/2179460x13194 https://doi.org/10.15406/bbij.2019.08.00282 1. introduction 2. the dagum x family 3. special models 3.1. the dagumweibull distribution 3.2. the dagum-exponential distribution 3.3. the dagum-rayleigh distribution 3.4. the dagum-fréchet distribution 4. mathematical properties of the dagum-x family 4.1. survival and hazard rate functions 4.2. moments 4.3. quantail function 4.4. rényi entropys 4.5. order statistics 5. dagum-fréchet distribution and its properties 5.1. survival and hazrd functions 5.2. moments 5.3. quantail function 5.4. rényi entropys 5.5. order statistics 6. maximum likelihood estimation 7. simulation study 8. application 9. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 32-40 http://www.etamaths.com some results on fixed point theorems in banach algebras dipankar das1, nilakshi goswami1 and vishnu narayan mishra2,3,∗ abstract. let x be a banach algebra and d be a nonempty subset of x. let (t1, t2) be a pair of self mappings on d satisfying some specific conditions. here we discuss different situations for existence of solution of the operator equation u = t1ut2u in d. similar results are established in case of reflexive banach algebra x with the subset d. again, considering a bounded, open and convex subset b in a uniformly convex banach algebra x with three self mappings t1, t2, t3 on b, we derive the conditions for existence of solution of the operator equation u = t1ut2u + t3u in b. application of some of these results to the tensor product is also shown here with some examples. 1. introduction in 1988, dhage initiated application of fixed point theorems in banach algebras. many papers ( [4], [5], [6]) of dhage deals with the study of non-linear integral equations via fixed point theorems in banach algebras. in 2010, amar et al. [1], introduced a class of banach algebras satisfying certain sequential conditions and gave applications of non-linear integral equations using fixed point theorems under certain conditions. in 2012, pathak and deepmala [24], defined p-lipschitzian maps and derived some fixed points theorem of dhage on a banach algebra with examples. in [12], kilbas et al. gave many applications in the field of integral equations. in [20] different application of convergent sequence can be seen. many researchers viz., mishra et al. ( [13], [14], [15], [16], [17], [18]), deepmala ( [8], [9]), mishra [19] etc., proved some results concerning the existence of solutions for some nonlinear functionalintegral equations in banach algebra and some interesting results. in 1982, hadzic [11] proved a generalization of rzepecki fixed point theorem for the sum of operators in hausdorff topological vector space. in ( [25], [26], [27]), vijayaraju proved the existance of fixed points for asymptotic 1-set contraction mappings in real banach spaces and also for the sum of two mappings in reflexive banach spaces. in this paper, for a banach algebra x with a subset d, we take a pair of self-mappings (t1,t2) on d and study the conditions under which the operator equation u = t1ut2u has a solution in d. an application of the results to the tensor product of banach algebras is also discussed here with some suitablle examples. also, give an apllication for nonlinear functional-integral equation. preliminaries def. 1 [3] let x be a banach space and f be a continuous (not necessarily linear) mapping of x into itself. the mapping f is said to be completely continuous if the image under f of each bounded set of x is contained in a compact set. def. 2 let x be a banach algebra and t1,t2 be two self mappings on x. then t1,t2 are said to satisfy the nonvacuous condition if for every sequence {xn}⊂ x the operator equation limn→∞t1(u)t2(xn) = u, u ∈ x has one and only one solution (xn)0 in x. def. 3 [21] t is demiclosed if {xn}⊂ d(t), xn → x and t(xn) → y (weakly) implies x ∈ d(t) and tx = y. def. 4 [21] t is closed if if {xn}⊂ d(t), xn → x and t(xn) → y implies x ∈ d(t) and tx = y. received 13th july, 2016; accepted 19th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 47b48; 46b28; 47a80; 47h10. key words and phrases. banach algebras; fixed points; projective tensor product. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 32 some results on fixed point theorems in banach algebras 33 def. 5 [22] t is said to be demicompact at a if for any bounded sequence {xn} in d such that xn − txn → a as n → ∞, there exists a subsequence xni and a point b in d such that xni → b as i →∞ and b−t(b) = a def. 6 ( [10], [23]) let t : d → d be a mapping. (1) t is said to be uniformly l-lipschitzian if there exists l > 0 such that, for any x,y ∈ d ‖tnx−tny‖6 l‖x−y‖ ∀ n ∈ n (2) t is said to be asymptotically nonexpansive if there exists a sequence bn ⊂ [1,∞) with bn → 1 such that, for any x,y ∈ d ‖tnx−tny‖6 bn‖x−y‖ ∀ n ∈ n algebric tensor product: [2] let x,y be normed spaces over f with dual spaces x∗ and y ∗ respectively. given x ∈ x,y ∈ y , let x⊗y be the element of bl(x∗,y ∗; f) (which is the set of all bounded bilinear forms from x∗ ×y ∗ to f), defined by x⊗y(f,g) = f(x)g(y), (f ∈ x∗,g ∈ y ∗) the algebraic tensor product of x and y , x⊗y is defined to be the linear span of {x⊗y : x ∈ x,y ∈ y} in bl(x∗,y ∗; f). projective tensor norm: [2] given normed spaces x and y , the projective tensor norm γ on x⊗y is defined by ‖u‖γ = inf{ ∑ i ‖xi‖‖yi‖ : u = ∑ i xi ⊗yi} where the infimum is taken over all (finite) representations of u. the completion of (x ⊗ y,γ) is called projective tensor product of x and y and it is denoted by x ⊗γ y . lemma 1: [28] let x and y be banach spaces. then γ is a cross norm on x⊗y and ‖x⊗y‖γ = ‖x‖‖y‖ for every x ∈ x,y ∈ y . lemma 2: [2] x ⊗γ y can be represented as a linear subspace of bl(x∗,y ∗; f) consisting of all elements of the form u = ∑ i xi ⊗ yi where ∑ i‖xi‖‖yi‖ < ∞. moreover, ‖u‖γ = inf{ ∑ i‖xi‖‖yi‖} over all such representations of u. lemma 3: [2] let x and y be normed algebras over f. there exists a unique product on x ⊗ y with respect to which x ⊗y is an algebra and (a⊗ b)(c⊗d) = ac⊗ bd (a,c ∈ x,b,d ∈ y ) lemma 4: [2] let x and y be normed algebras over f. then projective tensor norm on x ⊗y is an algebra norm. clearly, we can conclude that if x and y are banach algebras over f then x ⊗γ y becomes a banach algebra. 2. main results theorem 1: let d be a non-empty compact convex subset of a banach algebra x and let (t1,t2) be a pair of self-mappings on d such that (a) t1 and t2 are continuous, (b) t1ut2u ∈ d for all u ∈ d then the operator equation u = t1ut2u has a solution in d. proof. we define j : d → d by j(u) = t1ut2u. let {qn} be a sequence in d converging to a point q. so, q ∈ d as d is closed. now, ‖j(u) −j(v)‖ = ‖t1ut2u−t1vt2v‖ 6 ‖t1u−t1v‖‖t2u‖ + ‖t1v‖‖t2u−t2v‖ since t1 and t2 are continuous so, j is continuous. by an application of schauder’s fixed point theorem we have fixed point for j. hence the operator equation u = t1ut2u has a solution. � 34 das, goswami and mishra corollary 1: let dx, dy and dx ⊗dy be closed, convex and bounded subsets of banach algebras x, y and x ⊗γ y respectively. let (t1,t2) be a pair of self mappings on dx ⊗dy such that (a) t1 and t2 are completely continuous (b) t1ut2u ∈ dx ⊗dy for all u ∈ dx ⊗dy then the operator equation u = t1ut2u has a solution in dx ⊗dy . example 1: let dl1 , dk and dl1 ⊗dk be subsets of banach algebras l1, k and l1 ⊗γ k respectively. define dl1 = {x ∈ dl1 : ‖x‖6 m1} and dk = {y ∈ dk : ‖y‖6 m2} then clearly dl1 , dk and dl1 ⊗dk are closed, convex and bounded. we define t1,t2 : dl1 ⊗γ dk → dl1 ⊗γ dk by t1( ∑ i ai ⊗xi) = ∑ i{ ainxi n }n = t2( ∑ i ai ⊗xi), where ai = {ain}n. [ l1 ⊗γ x = l1(x) by [28]]. to show that t1 is compact: let t1m : dl1 ⊗γ dk → dl1 ⊗γ dk be defined by t1m( ∑ i ai ⊗xi) = ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , 0, 0, 0, ....} then each t1m is linear, bounded and compact [7]. also, ‖(t1m −t1)( ∑ i ai ⊗xi)‖ = ‖ ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , 0, 0, 0, ....} − ∑ i {ai1xi, ai2xi 2 , ai3xi 3 , ......, aimxi m , aim+1xi m + 1 , ....}‖ = ‖ ∑ i {0, 0, ......, 0, aim+1xi m + 1 , aim+2xi m + 2 , ....}‖ 6 ∑ i ∞∑ j=m+1 1 j |aij|.|xi| < 1 m + 1 ∑ i ∞∑ j=m+1 |aij|.|xi| 6 1 m + 1 ∑ i ∞∑ j=1 |aij|.|xi| = 1 m + 1 ∑ i ‖ai‖.|xi| so, taking the projective tensor norm, ‖(t1m −t1)( ∑ i ai ⊗xi)‖ < 1 m + 1 ‖ ∑ i ai ⊗xi‖ therefore, t1m → t1 and so, t1 is compact. similarly, t2 is compact. since every compact operator in banach space is completely continuous, so t1 and t2 are completely continuous. then, by corollary 1, the operator equation has a solution. theorem 2: let x be a non-empty banach algebra and let t1,t2 be three self mappings on x such that (a) s is a homomorphism and it has a unique fixed point (b) t1s = st1 and t2s = st2 then the unique fixed point of s is a solution of the operator equation u = t1ut2u in x. proof. defne j : x → x by j(u) = t1ut2u. let a be the unique fixed point of s. now, j(s(u)) = t1(s(u))t2(s(u)) = s(t1(u))s(t2(u)) = s(t1ut2u) = s(ju) hence, s(ja) = j(s(a)) = ja so ja = a as s has unique fixed point. hence the operator equation u = t1ut2u has a solution. � some results on fixed point theorems in banach algebras 35 example 2: given a closed and bounded interval i = [ 1 10 , 10 10 ] in r+ the set of real numbers, consider the nonlinear functional integral equation (in short fie) x(s) = [x(α(s))]2[q(s) + ∫ s 0 g(t,x(β(t)))dt]2 (2.1) for all s ∈ i , where α,β : i → i,q : i → r+ and g : i ×r+ → r+ are continuous. by a solution of the fie (1) we mean a continuous function x : i → r+ that satisfies fie (1) on i. let x = c(i,r+) be a banach algebra of all continuous real-valued functions on i with the norm ‖x‖ = sups∈i |x(s)|. we shall obtain the solution of fie (1) under some suitable conditions on the functions involved in (1). suppose that the function g satisfy the condition |g(s,x)|6 1 −q,‖q‖ < 1 for all s ∈ i and x ∈ r+. consider the two mappings t1,t2 : x → x defined by t1x(s) = [x(α(s))] 2, s ∈ i and t2x(s) = [q(s) + ∫ s 0 g(t,x(β(t)))dt]2,s ∈ i then the fie (1) is equivalent to the operator equation x(s) = t1x(s)t2x(s), s ∈ i. let s : x → x defined by s(y) = √ y, y ∈ x, where √ y(t) = √ y(t), (positive squareroot) t ∈ i . clearly, s is a homomorphism and it has a unique fixed point 1, where 1(s) = 1, s ∈ i. it is obvious that t1s = st1 and t2s = st2. so, 1 is a solution of fie(1). theorem 3: let d be a non-empty compact convex subset of a banach algebra x and let t1,t2 : d → d be two continuous self maps such that t1 and t2 satisfies nonvacuous condition, then there exists a solution of the operator equation u = t1ut2u in d. proof. we define j : d → d by j(xn) = (xn)0. first we show that j is continuous. let {yn}n be a sequence in d such that yn → y as n → ∞. since, t1 and t2 satisfies nonvacuous condition so we have j(yn) = (yn) 0 = lim n→∞ t1(yn) 0t2(yn) ⇒ lim n→∞ j(yn) = lim n→∞ t1( lim n→∞ j(yn))t2(yn) so, limn→∞j(yn) is a solution of the equation limn→∞t1(u)t2(xn) = u, u ∈ x. now, lim n→∞ j(yn) = ( lim n→∞ yn) 0 = (y)0 = j(y) therefore, j is continuous. for u ∈ d, ju = u0 = t1(u0)t2(u). clearly, we get j has a fixed point by schauder’s theorem, say α in d. therefore, α = j(α) = t1αt2α. thus, α is a solution of the equation u = t1ut2u ∈ d � theorem 4: let d be a nonempty closed bounded and convex subset of a weakly compact banach algebra x. let t1 : d → d and t2 : d → x be two mappings such that (a) t1 satisfies asymptotically nonexpansive mapping and limn→∞[sup‖t1x−tn1 x‖ : x ∈ d] = 0 (b) t2 is completely continuous and m = ‖t2(d)‖ < 1 (c) i −t1 �t2 is demiclosed and tn1 ut2v ∈ d for u,v ∈ d and n ∈ n then there exists a solution of the operator equation u = t1ut2u(= (t1 �t2)u) in d. proof. first we show that i − t1 � t2 is closed. let c ∈ i −t1 �t2. then there exists a sequence {cn}⊆ i −t1 �t2 such that cn → c as n →∞. since cn ∈ i −t1 �t2 so cn = (i −t1 �t2)zn for some zn ∈ x. since x is weakly compact so for every sequence {zn} in d there exists weakly convergence subsequence {zni} i.e., zni → z as n →∞. now, zni −t1 �t2zni → c as n →∞ since i −t1 �t2 is demiclosed so c = (i −t1 �t2)z. therefore c ∈ i −t1 �t2. hence i −t1 �t2 is closed. 36 das, goswami and mishra for u,v ∈ d, we define jn : d → d by jn(u) = qntn1 ut2v. where qn = (1 − 1 n ) bn and {bn}→ 1 as n → ∞. now, ‖jn(u) −jn(p)‖ = ‖qntn1 ut2v −qnt n 1 pt2v‖ = qn‖t2v‖‖t n 1 u−t n 1 v‖ 6 qnbnm‖u−p‖ = (1 − 1 n )m‖u−p‖6 m‖u−p‖ since jn is contraction and so it has unique fixed point kn(v) ∈ d (say), where kn(v) = jn(kn(v)) = qnt n 1 (kn(v))t2v. now, for any v,y ∈ d we have ‖kn(v) −kn(y)‖ = ‖qntn1 (knv)t2v −qnt n 1 (kny)t2y‖ 6 qn‖tn1 (knv) −t n 1 (kny)‖‖t2v‖ + qn‖t n 1 (kny)‖‖t2v −t2y‖ (2.2) for fixed a ∈ d, we have ‖tn1 (u)‖ = ‖t n 1 (u) −t n 1 (a) + t n 1 (a)‖ 6 bn‖u−a‖ + ‖tn1 (a)‖ = d(say) < ∞ from equation (2), we have ‖kn(v) −kn(y)‖6 dqn 1 −m ‖t2v −t2y‖ so, kn is completely continuous as t2 is completely continuous. by schauder’s fixed point theorem kn has a fixed point xn, say in d. hence xn = knxn = jn(xn) = qnt n 1 (xn)t2xn. now, xn −tn1 xnt2xn = (qn − 1)t n 1 xnt2xn → 0 as n →∞ (2.3) ‖xn −t1xnt2xn‖6 ‖xn −tn1 xnt2xn‖ + ‖t n 1 xnt2xn −t1xnt2xn‖ = ‖xn −tn1 xnt2xn‖ + ‖t2xn‖‖t n 1 xn −t1xn‖ → 0 as n →∞ (by (3) and condition (a)) so, 0 ∈ i −t1 �t2 as i −t1 �t2 is closed. hence there exists a point r such that 0 = (i −t1 �t2)r. hence the theorem follows. � theorem 5: let d be a nonempty closed bounded and convex subset of a reflexive banach algebra x. let t1 : d → d and t2 : d → x be two mappings such that (a) t1 satisfies uniformly l−lipschitzian mapping and limn→∞[sup‖t1x−tn1 x‖ : x ∈ d] = 0 (b) t2 is completely continuous and m = ‖t2(d)‖ such that lm < 1 (c) i −t1 �t2 is demiclosed and tn1 ut2v ∈ d for u,v ∈ d and n ∈ n then there exists a solution of the operator equation u = t1ut2u(= (t1 �t2)u) in d. theorem 6: let dx, dy and dx ⊗dy be closed bounded and convex subsets of a banach algebras x, y and x ⊗γ y respectively. let (t1,t2) be a pair of self mappings on dx ⊗dy such that (a) t1 satisfies uniformly l−lipschitzian mapping and limn→∞[sup‖t1x − tn1 x‖ : x ∈ dx ⊗ dy ] = 0 (b) t2 is completely continuous and m = ‖t2(dx ⊗dy )‖ such that lm < 1 (c) if {xn}⊂ dx ⊗dy with xn −t1xnt2xn → 0 as n →∞ then there exists b ∈ dx ⊗dy such that 0 = (i −t1 �t2)b and tn1 ut2v ∈ dx ⊗dy for u,v ∈ dx ⊗dy and n ∈ n then there exists a solution of the operator equation u = t1ut2u in dx ⊗dy . example 3: let dl1 , dr and dl1 ⊗dr be subsets of banach algebras l1, r and l1 ⊗γ r respectively. define dl1 = {x ∈ dl1 : ‖x‖6 1} and dr = {y ∈ dr : ‖y‖6 1} then clearly dl1 , dr and dl1 ⊗dr are bounded closed and convex. some results on fixed point theorems in banach algebras 37 we define t1 : dl1 ⊗γ dr → dl1 ⊗γ dr is defined by t1( ∑ i ai ⊗xi) = t1( ∑ i {(ain )xi}n) = t1(u), (say) = −u where if u = {y1,y2, ...} then −u = {−y1,−y2, ...}. it is easy to see that t1 satisfies uniformly l−lipschitzian (where l = 1) whether n is odd or even. but lim n→∞ [sup‖t1x−tn1 x‖ : x ∈ dl1 ⊗dr] = 0 only when n is odd. hence condition (a) of theorem 6 is satisfied. now, let t2 : dl1⊗γdr → dl1⊗γdr be defined by t2( ∑ i ai⊗xi) = 1 2 ∑ i{ ainxi n }n, where ai = {ain}n. clearly, condition (b) of theorem 6 is satisfied with m = ‖t2(dl1 ⊗ dr)‖ 6 1 2 hence lm < 1. proceeding as in theorem 4, we have, for {xn}⊂ dl1 ⊗dr xn −t1xnt2xn → 0 as n →∞. now we can take b as the constant sequence {0, 0, 0, ...} for which 0 = (i − t1 � t2)b and tn1 ut2v ∈ dl1 ⊗dr for u,v ∈ dl1 ⊗dr. so, the condition (c) of theorem 6 is satisfied. hence the operator equation u = t1ut2u has a solution. theorem 7: let b be the bounded, open and convex subset with 0 ∈ b in a uniformly convex banach algebra x. let (t1,t2,t3) be three self mappings on b such that (a) t1 satisfies uniformly l−lipschitzian mapping on b and limn→∞[sup‖t1x−tn1 x‖ : x ∈ b] = 0 (b) t1 is demicompact on b and m = ‖t2(b)‖ such that lm < 1 (c) t2, t3 are completely continuous and t n 1 ut2v + t3v ∈ b for u,v ∈ b and n ∈ n then there exists a solution of the operator equation u = t1ut2u + t3u(= (t1 �t2)u + t3u) in b. proof. since t2 is a completely continuous, it is demicompact on b. also t1 is demicompact by (b). so for a sequence {cn} ∈ b such that cn −t1cn → a, cn −t2cn → b as n → ∞ in b, there exists subsequence {cnk} such that cnk → c as k →∞, where c ∈ b. since t1, t2 and t3 are continuous so t1cnk → t1c, t2cnk → t2c and t3cnk → t3c. now we show that i −t1 �t2 −t3 is closed. let z ∈ i −t1 �t2 −t3. then for {zn}⊆ (i −t1 �t2 −t3)cn such that zn → z as n →∞. we have as in theorem 4, cnk −t1 �t2cnk −t3cnk → z as n →∞ since i −t1 �t2 −t3 is continuous so c ∈ i −t1 �t2 −t3. hence i −t1 �t2 −t3 is closed. define jn : b → b by jn(u) = qn(tn1 ut2v + t3v), where {qn}→ 1 as n →∞. now, ‖jn(u) −jn(p)‖6 qnlm‖u−p‖ since jn is contraction and so it has unique fixed point knv ∈ b (say) knv = jn(knv) = qn(t n 1 (knv)t2v + t3v). now, for any v,y ∈ b we have ‖kn(v) −kn(y)‖6 qn‖tn1 (knv) −t n 1 (kny)‖‖t2v‖ + qn‖t n 1 (kny)‖‖t2v −t2y‖ + ‖t3v −t3y‖ (2.4) for fixed a ∈ b, we have ‖tn1 (u)‖6 l‖u−a‖ + ‖t n 1 (a)‖ = d(say) < ∞ from equation (4), we have ‖kn(v) −kn(y)‖6 dqn 1 −lm ‖t2v −t2y‖ + qn 1 −lm ‖t3v −t3y‖ 38 das, goswami and mishra so, kn is completely continuous as t2 and t3 are completely continuous. by schauder’s fixed point theorem kn has a fixed point xn, say in b. hence xn = knxn = jn(xn) = qn(t n 1 (xn)t2xn + t3(xn)). now, xn −tn1 xnt2xn −t3xn = (qn − 1)(t n 1 xnt2xn + t3xn) → 0 as n →∞ (2.5) ‖xn −t1xnt2xn −t3xn‖6 ‖xn −tn1 xnt2xn −t3xn‖ + ‖t2xn‖‖t n 1 xn −t1xn‖ → 0 as n →∞ (by (5) and condition (a)) since, 0 ∈ i − t1 � t2 − t3 and i − t1 � t2 − t3 is closed. hence there exists a point r such that 0 = (i −t1 �t2 −t3)r. hence the theorem follows. � if 0 /∈ b in the above theorem 7. theorem 8: let b be the bounded, open and convex subset in a uniformly convex banach algebra x. let (t1,t2,t3) be three self mappings on b such that (a) there exists r ∈ b such that r = t1c + t2c for some c ∈ b (b) all the conditions of above theorem 7 then there exists a solution of the operator equation u = t1ut2u + t3u(= (t1 �t2)u + t3u) in b. proof. suppose that k = b − r = {x − r : x ∈ b}. since b is open and bounded, so is k, and k = b −r and 0 ∈ k. define (t1,t2,t3) are three self maps on k by t1(c−r) = t1c−r, t2(c−r) = t2c−r and t3(c−r) = t3c−r. hence (t1,t2,t3) are three continuous self mappings in k and i −t1 �t2 −t3 is closed in k. then (i) t1 satisfies uniformly l−lipschitzian mapping and lim n→∞ [sup‖t1(x−r) −tn1 (x−r)‖ : x−r ∈ k] = 0 (ii) since t1 is demicompact in b, so t1 is demicompact in k. also, lm < 1. similarly, since t2 and t3 are completely continuous in b, so t2 and t3 are completely continuous in k. (iii) clearly tn1 (u−r)t2(v −r) + t3(v −r) ∈ k for u−r,v −r ∈ k and n ∈ n. hence all the conditions of theorem 7 satisfied so, there exists a solution m−r such that m−r = t1(m−r)t2(m−r) + t3(m−r) (2.6) t1(a−r)t2(a−r) + t3(a−r) = [(t1(a) −r][t2(a) −r] + t3(a) −r = t1at2a + t3a−r − [r(t1a + t2a) −r2] without loss of generality if r = t1m + t2m, m ∈ b we have t1(m−r)t2(m−r) + t3(m−r) = t1(m)t2(m) + t3(m) −r then from equation (6) we have a solution of the equation u = t1ut2u + t3u. � theorem 9: let b be the bounded, open and convex subset in a uniformly convex banach algebra x. let (t1,t2,t3) be three self mappings on b such that (a) t1 satisfies uniformly l−lipschitzian mapping on b, there exists r ∈ d such that limn→∞[sup‖t1(x) −tn1 (x)‖ : x ∈ b] = 0 and r = t1c + t2c for some c ∈ b (b) t2 and t3 are completely continuous m = ‖t2(b)‖ such that lm < 1 and tn1 ut2v + t3v ∈ b for u,v ∈ b and n ∈ n (c) if {xn} ∈ b with xn − t1xnt2xn − t3xn → 0 as n → ∞ then there exists b ∈ b such that 0 = (i −t1 �t2 −t3)b. then there exists a solution of the operator equation u = t1ut2u + t3u(= (t1 �t2)u + t3u) in b. some results on fixed point theorems in banach algebras 39 acknowledgement the authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. the corresponding author vnm acknowledges that this project was supported by the cumulative professional development allowance (cpda), svnit, surat, gujarat, india. all the authors carried out the proof of theorems in this manuscript. vishnu narayan mishra conceived the study and participated in its design and coordination. references [1] afif ben amar, soufiene chouayekh, aref jeribi, new fixed point theorems in banach algebras under weak topology features and applications to nonlinear integral equations, journal of functional analysis 259 (2010), 2215-2237. [2] f. f. bonsal and j. duncan, complete normed algebras, springer-verlag, berlin heidelberg new york, 1973. [3] felix. e. browder on a generalization of the schauder fixed point theorem, duke mathematical journal, 26 (2) (1959), 291-303. [4] b.c. dhage on a fixed point theorem in banach algebras with applications, applied mathematics letters 18 (2005), 273-280. [5] b.c. dhage, on some variants of schauders fixed point principle and applications to nonlinear integral equations, j. math. phys. sci. 25 (1988) 603-611. [6] b.c. dhage, on existance theorems for nonlinear integral equations in banach algebra via fixed point techniques, east asian math. j. 17(2001), 33-45. [7] d. das, n. goswami, some fixed point theorems on the sum and product of operators in tensor product spaces, ijpam, 109(2016) 651-663. [8] deepmala and h.k. pathak, on solutions of some functional-integral equations in banach algebra, research j. science and tech. 5 (3) (2013), 358-362. [9] deepmala, a study on fixed point theorems for nonlinear contractions and its applications, ph.d. thesis, pt. ravishankar shukla university, raipur 492 010, chhatisgarh, india, 2014. [10] k. goebeli and w. a. kirk, a fixed point theorem for asymptotically nonexpansive mappings, proc. of the american math. soc. 35 (1) (1972), 171-174. [11] olga hadzic, a fixed point theorem for the sum of two mappings, proc. amer. math. soc. 85 (1) (1982), 37-41, [12] a.a. kilbas, h.m. srivastava, j.j. trujillo, theory and applications of fractional differential equations, northholland math. std., vol. 204, elsevier (north-holland) science publishers, amsterdam, london and new york, 2006. [13] l.n. mishra, h.m. srivastava, m. sen, on existence results for some nonlinear functional-integral equations in banach algebra with applications, int. j. anal. appl., 11 (1) (2016), 1-10. [14] l.n. mishra, s.k. tiwari, v.n. mishra, fixed point theorems for generalized weakly s-contractive mappings in partial metric spaces, j. of app. anal. and comp., 5 (4) 2015, 600-612. [15] l.n. mishra, m. sen, on the concept of existence and local attractivity of solutions for some quadratic volterra integral equation of fractional order, applied mathematics and computation 285 (2016), 174-183. [16] l. n. mishra, r. p. agarwal, m. sen, solvability and asymptotic behavior for some nonlinear quadratic integral equation involving erdélyi-kober fractional integrals on the unbounded interval, progress in fractional differentiation and applications 2 (3) (2016), 153-168. [17] l.n. mishra, m. sen, r.n. mohapatra, on existence theorems for some generalized nonlinear functional-integral equations with applications, filomat, in press. [18] l.n. mishra, s.k. tiwari, v.n. mishra, i.a. khan, unique fixed point theorems for generalized contractive mappings in partial metric spaces, journal of function spaces, 2015 (2015), article id 960827, 8 pages. [19] v.n. mishra, some problems on approximations of functions in banach spaces, ph.d. thesis, indian institute of technology, roorkee 247 667, uttarakhand, india, 2007. [20] m. mursaleen, h.m. srivastava, s.k. sharma, generalized statistically convergent sequences of fuzzy numbers, j. intelligent fuzzy systems 30 (2016), 1511-1518. [21] w. v. petryshyn and t. s. tucker, on the functional equations involving nonlinear generalized p-compact operators, transactions of the american mathematical society, 135 (1969), 343-373 [22] w. v. petryshyn and t. e. williamson, jr. strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, j. of math. ana. and appl. 43 (1973), 459-497. [23] j. schu, iterative construction of fixed points of asymptotically nonexpansive mappings, j. of math. ana. and appl. 158 (1991) , 407-413 [24] h.k. pathak and deepmala, remarks on some fixed point theorems of dhage, applied mathematics letters, 25 (11) (2012), 1969-1975. [25] p. vijayaraju, a fixed point theorem for a sum of two mappings in reflexive banach spaces, math. j. toyoma univ. 14 (1991), 41-50. [26] p. vijayaraju, iterative construction of fixed points of asymptotic 1-set contraction in banach spaces, taiwanese j. of math. 1 (3) (1997), 315-325. 40 das, goswami and mishra [27] p. vijayraju, fixed point theorems for a sum of two mappings in locally convex spaces, int. j. math. and math. sci., 17(4) (1994), 681-686. [28] a. raymond ryan, introduction to tensor product of banach spaces, london, springer -verlag, 2002. 1department of mathematics, gauhati university, guwahati-781014, assam, india 2applied mathematics and humanities department, sardar vallabhbhai national institute of technology, ichchhanath mahadev dumas road, surat 395 007, gujarat, india 3l. 1627 awadh puri colony beniganj, phase -iii, opposite industrial training institute (i.t.i.), ayodhya main road faizabad 224 001, uttar pradesh, india ∗corresponding author: vishnunarayanmishra@gmail.com, vishnu narayanmishra@yahoo.co.in 1. introduction preliminaries 2. main results acknowledgement references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 163-179 http://www.etamaths.com intuitionistic fuzzy topological polygroups n. abbasizadeh and b. davvaz∗ abstract. the notion of intuitionistic fuzzy set was introduced by atanassov as a generalization of the notion of fuzzy set. intuitionistic fuzzy topological spaces were introduced by coker. this paper provides a new connection between algebraic hyperstructures and intuitionistic fuzzy sets. in this paper, we introduce and study the concept of intuitionistic fuzzy subpolygroup and intuitionistic fuzzy topological polygroup. we also investigate some interesting properties of an intuitionistic fuzzy subpolygroup and intuitionistic fuzzy normal subpolygroup. 1. introduction the hyperstructure theory was born in 1934 when marty introduced the notion of hypergroup [24]. the concept of intuitionistic fuzzy sets was introduced by atanassov [5]. coker [7] has introduced the notions of intuitionistic fuzzy topological spaces. biswas [6] introduced the concept of intuitionistic fuzzy subgroup and some other concepts. the concepts of quasi-coincidence for intuitionistic fuzzy point was introduced and developed by gallego lupianez [14]. on the other hand, in the last few decades, many connections between hyperstructures and intuitionistic fuzzy sets has been established and investigated. in [18], heidari et, al introduced the notion of topological polygroups. then in [1, 2, 3] abbasizadeh et, al investigated to notion of fuzzy topological polygroups. we recall some basic definitions and results to be used in the sequel. let h be a non-empty set. then a mapping ◦ : h ×h −→ p∗(h) is called a hyperoperation, where p∗(h) is the family of non-empty subsets of h. the couple (h,◦) is called a hypergroupoid. in the above definition, if a and b are two non-empty subsets of h and x ∈ h, then we define a◦b = ⋃ a∈a b∈b a◦ b, x◦a = {x}◦a and a◦x = a◦{x}. a hypergroupoid (h,◦) is called a semihypergroup if for every x,y,z ∈ h, we have x◦(y◦z)=(x◦y)◦z and is called a quasihypergroup if for every x ∈ h, we have x ◦ h = h = h ◦ x. this condition is called the reproduction axiom. the couple (h,◦) is called a hypergroup if it is a semihypergroup and a quasihypergroup [9, 24]. for all n > 1, we define the relation βn on a semihypergroup h, as follows: a βn b ⇔∃ (x1, . . . ,xn) ∈ hn : {a,b}⊆ n∏ i=1 xi, and β = ⋃ βn, where β1 = {(x,x) | x ∈ h} is the diagonal relation on h. this relation was introduced by koskas [21] and studied mainly by corsini, davvaz, freni, leoreanu, vougiouklis and many others. suppose that β∗ is the smallest equivalence relation on a hypergroup (semihypergroup) h such that the qoutient h/β∗ is a group (semigroup). if h is a hypergroup, then β = β∗ [13]. the relation β∗ is called the fundamental relation on h and h/β∗ is called the fundamental groups. a special subclass of hypergroups is the class of polygroups. we recall the following definition from [8]. a polygroup is a system p = 〈p,◦,e,−1 〉, where ◦ : p × p −→ p∗(p), e ∈ p , −1 is a unitary operation p and the following axioms hold for all x,y,z ∈ p: (1) (x◦y) ◦z =x◦ (y ◦z), (2) e◦x = x =x◦e, 2010 mathematics subject classification. 20n20, 20n25, 03e72. key words and phrases. intuitionistic fuzzy set; polygroup; topological polygroup. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 163 164 abbasizadeh and davvaz (3) x ∈ y ◦z implies y ∈ x◦z−1 and z ∈ y−1 ◦x. the following elementary facts about polygroups follow easily from the axioms: e ∈ x◦x−1 ∩x−1 ◦x, e−1 = e, (x−1)−1 = x, and (x◦y)−1=y−1 ◦x−1. a non-empty subset k of a polygroup p is a subpolygroup of p if and only if a,b ∈ k implies a ◦ b ⊆ k and a ∈ k implies a−1 ∈ k. the subpolygroup n of p is normal in p if and only if a−1 ◦n ◦a ⊆ n for all a ∈ p . for a subpolygroup k of p and x ∈ p , denote the right coset of k by k ◦x and let p/k be the set of all right cosets of k in p . if n is a normal subpolygroup of p, then (p/n,�,n,−1 ) is a polygroup, where n ◦x�n ◦y={n ◦z|z ∈ n ◦x◦y} and (n ◦x)−1 = n ◦x−1. for more details about polygroups we refer to [11, 12, 15]. let p = 〈p,◦,e,−1 〉 be a polygroup and (p,t ) be a topological space. then, the system p = 〈p,◦,e,−1 ,t 〉 is called a topological polygroup if the mapping ◦ : p ×p −→ ℘∗(p) and −1 : p −→ p are continuous (see [18]). let p = 〈p,◦,e,−1 〉 be a polygroup and (p,t ) be a fuzzy topological space. a triad (p,◦,t ) is called a fuzzy topological polygroup or ftp for short, if (see [1, 2, 3]): (i) for all x,y ∈ p and any fuzzy open q-neighborhood w of any fuzzy point zλ of x◦y, there are fuzzy open q-neighborhood u of xλ and v of yλ such that: u •v ≤ w . (ii) for all x ∈ p and any fuzzy open q-neighborhood v of x−1λ , there exists a fuzzy open qneighborhood u of xλ such that: u−1 ≤ v . 2. preliminaries for the sake of convenience and completeness of our study, in this section some basic definition and results of [4, 5, 7, 16, 17, 22, 23], which will be needed in the sequel are recalled here. let x be a non-empty set and i be the closed interval [0, 1]. a complex mapping a = (µa,νa) : x −→ i × i is called an intuitionistic fuzzy set (in short, ifs) on x if µa(x) + νa(x) ≤ 1 for each x ∈ x, where the mapping µa : x −→ i and νa : x −→ i denote the degree of membership (namely µa(x)) and the degree of nonmembership (namely νa(x)) of each x ∈ x, respectively. in particular, 0∼ and 1∼ denote the intuitionistic fuzzy empty set and the intuitionistic fuzzy whole set in x defined by 0∼(x) = (0, 1) and 1∼(x) = (1, 0) for each x ∈ x, respectively. we will denote the set of all ifss in x as ifs(x) (see [5, 7]). let x be a non-empty set and let a = (µa,νa) and b = (µb,νb) be ifss on x. then (see [5]): (1) a ⊂ b iff µa ≤ µb and νa ≥ νb, (2) a = b iff a ⊂ b and b ⊂ a, (3) ac = (νa,µa), (4) a∩b = (µa ∧µb,νa ∨νb), (5) a∪b = (µa ∨µb,νa ∧νb). let {ai}i∈j be an arbitrary family of ifss in x, where ai = (µai,νai) for each i ∈ j. then (see [7]): (1) ⋂ ai = ( ∧ µai, ∨ νai). (2) ⋃ ai = ( ∨ µai, ∧ νai). let x and y be non-empty sets and let f : x −→ y a mapping. let a = (µa,νa) be an ifs in x and b = (µb,νb) be ifs on y . then (see [7]): (1) the preimage of b under f, denoted by f−1(b), is the ifs in x defined by: f−1(b) = (f−1(µb),f −1(νb)), where f−1(µb)(x) = µb(f(x)) and f −1(νb)(x) = νb(f(x)). (2) the image of a under f, denoted by f(a), is the ifs in y defined by: f(a) = (f(µa),f(νa)), where f(µa)(y) = { ∨ x∈f−1(y) µa(x) if f −1(y) 6= ∅ 0 otherwise, intuitionistic fuzzy topological polygroups 165 and f(νa)(y) = { ∧ x∈f−1(y) νa(x) if f −1(y) 6= ∅ 1 otherwise. throughout this paper, the symbol i will denote the unit interval [0, 1]. a intuitionistic fuzzy topology (in short, ift) in coker’s sense on a non-empty set x is a family t on ifss in x satisfying the following axioms: (1) 0∼, 1∼ ∈t . (2) for all a,b ∈t , then a∩b ∈t . (3) for all (aj)j∈j, then ⋃ j∈j aj ∈t . in this case, the pair (x,t ) is called an intuitionistic fuzzy topological space (in short, ifts) in the sense of coker, and each elements of t is called an intuitionistic fuzzy open set (in short, ifos) in x. the complement ac of an ifos a in x is called an intuitionistic fuzzy closed set (in short, ifcs) in x. we will denote the set of all the ifts on a set x as ift(x), and the set of all ifoss and the set of all ifcss in an ifts(x) as ifo(x) and ifc(x), respectively (see [7]). example 1. (1) the family t ={0∼,1∼} is an intuitionistic fuzzy topology on x. (2) the family of all intuitionistic fuzzy sets in x is an intuitionistic fuzzy topology on x. example 2. let x = {a,b,c} and a =< x, ( a 0.4 , b 0.3 , c 0.2 ), ( a 0.6 , b 0.5 , c 0.7 ) >. then, the family t = {0∼,a, 1∼} of ifs’s in x is an ift on x. example 3. let x = [0, 1] and consider the ifs a = (µa,νa) as follows: µa(x) = { −3 2 x + 1 if 0 ≤ x ≤ 2 3 0 if 2 3 ≤ x ≤ 1, and νa(x) = { x 4 if 0 ≤ x ≤ 2 3 1 6 if 2 3 ≤ x ≤ 1. then, t = {0∼,ac, 1∼} is an intuitionistic fuzzy topology on x. let x,y be non-empty sets and a = (µa,νa), b = (µb,νb) ifss of x and y , respectively. then a×b is an ifs of x ×y defined by (see [17]): (a×b)(x,y) = (µa(x) ∧µb(y),νa(x) ∨νb(y)). let x be a non-empty set. an intuitionistic fuzzy point, (in short, ifp) in x denoted by xr,s is an intuitionistic fuzzy set a = (µa,νa) such that µa(y) = { r if y = x 0 if y 6= x, and νa(y) = { s if y = x 1 if y 6= x, where x ∈ x is a fixed point, the constants r ∈ i0, s ∈ i1 and r + s ≤ 1. the intuitionistic fuzzy point xr,s is said to be contained in an intuitionistic fuzzy set a, denoted by xr,s ∈ a, if and only if µa(x) ≥ r and νa(x) ≤ s. in particular xr,s ⊆ ym,n ⇔ x = y and r ≤ m, s ≥ n. the intuitionistic fuzzy characteristic mapping of a subset a of a set x is denoted by χa is defined as χa(x) = { (1, 0) if x ∈ a (0, 1) otherwise. obviously an intuitionistic characteristic function χa is also an intuitionistic fuzzy set on x and for any non-empty subsets a and b of a set x, we have a ⊆ b if and only if χa ⊆ χb (see [22]). let xr,s be an ifp in x and let a = (µa,νa) be an ifs in x. we say that xr,s is quasi-coincident with a, written xr,s q a, if µa(x) + r > 1 and νa(x) + s < 1 (see [16]). 166 abbasizadeh and davvaz let (x,t ) be an ifts, and let p be an ifp of x. say that an ifs n of x is a q-neighbourhood of p if there exists an ifos a of (x,t ) such that p q a and a ⊆ n (see [23]). let x,y be two non-empty sets, let f : x −→ y be a map, let t be an ift in x and let σ be an ift in y . then, f : (x,t ) −→ (y,σ) is continuous if and only if, for each ifp p of x and for each q-neighbourhood v of f(p), there exists a q-neighbourhood u of p such that f(u) ⊆ v (see [23]). 3. intuitionistic fuzzy subpolygroups definition 3.1. let p be a polygroup and a ∈ ifs(p). then a is called intuitionistic fuzzy subpolygroup (in short, ifsp) of p if it satisfies the following conditions: (1) µa(z) ≥ µa(x) ∧µa(y) and νa(z) ≤ νa(x) ∨νa(y) for each z ∈ x◦y and x,y ∈ p . (2) µa(x −1) ≥ µa(x) and νa(x −1) ≤ νa(x) for each x ∈ p. we will denote the set of all ifsps of p as ifsp(p). example 4. let p = {e,a,b}. then, p together with the following hyperoperation ◦ e a b e e a b a a e b b b b {e,a} is a polygroup. let a =< x, ( e 0.7 , a 0.5 , b 0.3 ), ( e 0.1 , a 0.3 , b 0.5 ) >. then, a is an ifsp of p . definition 3.2. [28] let p be a polygroup. a fuzzy subset µ of p is called a fuzzy subpolygroup if (1) min{µ(x),µ(y)}≤ µ(z), for all x,y ∈ p and for all z ∈ x◦y, (2) µ(x) ≤ µ(x−1), for all x ∈ p . the following elementary facts about fuzzy subpolygroups follow easily from the axioms: µ(x) = µ(x−1) and µ(x) ≤ µ(e), for all x ∈ p . proposition 3.3. let a be an ifsp of a polygroup p . then a(x−1) = a(x), that is, µa(x) = µa(x −1), νa(x) = νa(x −1) and µa(x) ≤ µa(e), νa(x) ≥ νa(e) for each x ∈ p , where e is the identity element of p . proof. by definition 3.2, we have µa(x) = µa(x −1) and µa(x) ≤ µa(e) for each x ∈ p . thus it is enough to show that νa(x) = νa(x −1) and νa(x) ≥ νa(e) for each x ∈ p . let x ∈ p . then, νa(x) = νa((x −1)−1) ≤ νa(x−1) ≤ νa(x). on the other hand, for each z ∈ x◦x−1, we have νa(z) ≤ νa(x)∨νa(x−1). since e ∈ x◦x−1∩x−1◦x, so νa(e) ≤ νa(x) ∨νa(x−1) = νa(x). thus νa(e) ≤ νa(x) for each x ∈ p . this complete the proof. � proposition 3.4. let p be a polygroup. (1) if µa is a fuzzy subpolygroup of p , then a = (µa,µac) ∈ ifsp(p). (2) if a ∈ ifsp(p), then µa and νac are fuzzy subpolygroups of p . (3) a = (χt ,χtc) ∈ ifsp(p) if and only if t is a subpolygroup of p . proof. it is straightforward. � proposition 3.5. let {aα}α∈j ⊂ ifsp(p). then ⋂ α∈j aα ∈ ifsp(p). proof. it is straightforward. � proposition 3.6. if a be an ifsp of a polygroup p then, pa = {x ∈ p : a(x) = a(e), that is, µa(x) = µa(e) and νa(x) = νa(e)} is a subpolygroup of p . proof. we have to show that: (1) x◦y ⊆ pa for each x,y ∈ pa. intuitionistic fuzzy topological polygroups 167 (2) if x ∈ pa then, x−1 ∈ pa. let x,y ∈ pa and z ∈ x◦y. since x,y ∈ pa then, µa(x) = µa(e), νa(x) = νa(e) and µa(y) = µa(e), νa(y) = νa(e). since a ∈ ifsp(p) and z ∈ x◦y, µa(z) ≥ µa(x) ∧µa(y) = µa(e) ∧µa(e) = µa(e), and νa(z) ≤ νa(x) ∨νa(y) = νa(e) ∨νa(e) = νa(e). so µa(z) ≥ µa(e) and νa(z) ≤ νa(e). then µa(z) = µa(e), νa(z) = νa(e) and z ∈ pa, that is, x◦y ⊆ pa. now, if x ∈ pa then, µa(x) = µa(e) and νa(x) = νa(e). since µa(x−1) = µa(x) = µa(e) and νa(x −1) = νa(x) = νa(e), that is, x −1 ∈ pa. hence pa is a subpolygroup of p . � definition 3.7. [26] let a be an ifs in a set x and let α,β ∈ i with α + β ≤ 1. then the set cα,β(a) = {x ∈ x : µa(x) ≥ α and νa(x) ≤ β} is called a (α,β)cut set of a. proposition 3.8. let a be an ifsp of a polygroup p . then for each (α,β) ∈ i×i with (α,β) ≤ a(e), that is, α ≤ µa(e), β ≥ νa(e), cα,β(a) is a subpolygroup of p , where e is the identity of p . proof. clearly, cα,β(a) 6= ∅. let x,y ∈ cα,β(a). we show that x◦y ⊆ cα,β(a). let z ∈ x ◦ y. since x,y ∈ cα,β(a), then a(x) ≥ (α,β) and a(y) ≥ (α,β), that is, µa(x) ≥ α, νa(x) ≤ β and µa(y) ≥ α, νa(y) ≤ β. since a ∈ ifsp(p), µa(z) ≥ µa(x) ∧ µa(y) ≥ α and νa(z) ≤ νa(x) ∨νa(y) ≤ β. thus a(z) ≥ (α,β). so z ∈ cα,β(a) and x◦y ⊆ cα,β(a). on the other hand, µa(x −1) ≥ µa(x) ≥ α and νa(x−1) ≤ νa(x) ≤ β. thus a(x−1) ≥ (α,β). so x−1 ∈ cα,β(a). hence cα,β(a) is a subpolygroup of p . � proposition 3.9. let a be an ifs in a polygroup p such that cα,β(a) is a subpolygroup of p for each (α,β) ∈ i × i with (α,β) ≤ a(e). then a is an ifsp of p . proof. for any x,y ∈ p , let a(x) = (t1,s1) and a(y) = (t2,s2). then clearly x ∈ ct1,s1 (a) and y ∈ ct2,s2 (a). suppose t1 < t2 and s1 > s2, then ct1,s1 (a) ⊂ ct2,s2 (a). thus y ∈ ct1,s1 (a). since ct1,s1 (a) is a subpolygroup of p, x ◦ y ⊆ ct1,s1 (a). then for each z ∈ x ◦ y, a(z) = (t1,s1), that is, µa(z) ≥ t1 and νa(z) ≤ s1. so µa(z) ≥ µa(x) ∧µa(y) and νa(z) ≤ νa(x) ∨νa(y). on the other hand, for each x ∈ p , let a(x) = (α,β). then x ∈ cα,β(a). since cα,β(a) is a subpolygroup of p , x−1 ∈ cα,β(a). so a(x−1) ≥ (α,β), that is, µa(x−1) ≥ µa(x) and νa(x−1) ≤ νa(x). hence a is an ifsp of p . � theorem 3.10. let a and b be two ifsp’s of a polygroup p . then a∩b is ifsp of polygroup p . proof. by theorems 3.8 and 3.9, a ∩ b is ifsp of polygroup p if and only if cα,β(a ∩ b) is a subpolygroup of p . clearly, cα,β(a ∩ b) = cα,β(a) ∩ cα,β(b) and both cα,β(a) and cα,β(b) are subpolygroups of p and intersection of two subpolygroups of a polygroup is a subpolygroup of p implies that cα,β(a∩b) is a subpolygroup of p and hence a∩b is ifsp of polygroup p. � theorem 3.11. let a and b ifsp of polygroups p1 and p2 respectively. then a×b is also ifsp of polygroup p1 ×p2. proof. let a and b be ifsp of polygroups p1 and p2 respectively, then, cα,β(a) and cα,β(b) are subpolygroups of polygroups p1 and p2 respectively, for all α,β ∈ i with α + β ≤ 1. so cα,β(a) × cα,β(b) is subpolygroup of polygroup p1 × p2. hence cα,β(a × b) is subpolygroup of polygroup p1 ×p2. therefore a×b is an ifsp of polygroup p1 ×p2. � proposition 3.12. let a and b ifs of the polygroups p1 and p2 respectively such that µa(x) ≤ µb(e2) and νa(x) ≥ νb(e2) hold for x ∈ p1, e2 being the identity element of p2. if a × b is an ifsp of p1 ×p2, then, a is ifsp of polygroup p1. 168 abbasizadeh and davvaz proof. let x,y ∈ p1 and z ∈ x◦y, we have: µa(z) = µa(z) ∧µb(e2) = µa×b(z,e2) ≥ µa×b(x,e2) ∧µa×b(y,e2) = [µa(x) ∧µb(e2)] ∧ [µa(y) ∧µb(e2)] = µa(x) ∧µa(y). then, µa(z) ≥ µa(x) ∧µa(y). also, νa(z) = νa(z) ∨νb(e2) = νa×b(z,e2) ≤ νa×b(x,e2) ∨νa×b(y,e2) = [νa(x) ∨νb(e2)] ∨ [νa(y) ∨νb(e2)] = νa(x) ∨νa(y). hence, νa(z) ≤ νa(x) ∨νa(y). therefore a is an ifsp of p1. � proposition 3.13. let a and b ifs of the polygroups p1 and p2 respectively such that µb(y) ≤ µa(e1) and νb(y) ≥ νa(e1) hold for y ∈ p2, e1 being the identity element of p1. if a × b is an ifsp of p1 ×p2, then, b is ifsp of polygroup p2. proof. the proof is similar to the proof of proposition 3.12. � corollary 3.14. let a and b ifs of the polygroups p1 and p2 respectively. if a×b is an ifsp of p1 ×p2, then, either a is ifsp of p1 or b is ifsp of polygroup p2. definition 3.15. [12] let 〈p, ·,e,−1 〉 and 〈p ′,∗,e′,−i 〉 be two polygroups. let f be a mapping from p to p ′ such that f(e) = e′. then, f is called a strong homomorphism or a good homomorphism if f(x ·y) = f(x) ∗f(y), for all x,y ∈ p . definition 3.16. [19] let a ∈ ifs(p). then a is said to have the sup property if for any t ∈p∗(p), there exists a t0 ∈ t such that a(t0) = ⋃ t∈t a(t), that is, µa(t0) = ∨ t∈t µa(t) and νa(t0) = ∧ t∈t νa(t). proposition 3.17. let f : p −→ p ′ be a strong polygroup homomorphism and a ∈ ifsp(p), b ∈ ifsp(p ′). then, the following hold: (1) if a has the sup property then, f(a) ∈ ifsp(p ′). (2) f−1(b) ∈ ifsp(p). proof. (1) first, we suppose that a is an ifsp of p . in order to prove that f(a) is an ifsp of p ′, by proposition 3.8, it is sufficient to show that each non-empty (α,β)-cut set cα,β(f(a)) is an subpolygroup of p ′. so, suppose that cα,β(f(a)) is non-empty set for some (α,β) ∈ i × i with (α,β) ≤ a(e). let y1,y2 ∈ cα,β(f(a)). we show that y1 ∗y2 ⊆ cα,β(f(a)). we have µf(a)(y1) ≥ α, νf(a)(y1) ≤ β and µf(a)(y2) ≥ α, νf(a)(y2) ≤ β, which implies that∨ x∈f−1(y1) µa(x) ≥ α, ∧ x∈f−1(y1) νa(x) ≤ β and ∨ x∈f−1(y2) µa(x) ≥ α, ∧ x∈f−1(y2) νa(x) ≤ β. since a has the sup property, it follows that there exist x1 ∈ f−1(y1) and x2 ∈ f−1(y2) such that µa(x1) ≥ α, νa(x1) ≤ β and µa(x2) ≥ α, νa(x2) ≤ β. since f is strong homomorphism, it follows that y1 ∗y2 = f(x1) ∗f(x2) = f(x1.x2). let z ∈ y1 ∗y2. then, there exists x′ ∈ x1.x2 such that f(x′) = z. thus, f(x) = f(x′) = z. since a is an ifsp of p , we have µa(x ′) ≥ µa(x1) ∧µa(x2) ≥ α and νa(x′) ≤ νa(x1) ∨νa(x2) ≤ β. therefore, we obtain µf(a)(z) = ∨ x∈f−1(z) µa(x) ≥ α and νf(a)(z) = ∧ x∈f−1(z) νa(x) ≤ β. hence z ∈ cα,β(f(a)). thus, y1 ∗y2 ⊆ cα,β(f(a)). next, for y ∈ cα,β(f(a)), we have µf(a)(y) ≥ α and νf(a)(y) ≤ β. thus,∨ x∈f−1(y) µa(x) ≥ α and ∧ x∈f−1(y) νa(x) ≤ β. intuitionistic fuzzy topological polygroups 169 hence, we have ∨ x−1∈f−1(y−1) µa(x −1) ≥ α and ∧ x−1∈f−1(y−1) νa(x −1) ≤ β. therefore, µf(a)(y −1) ≥ α and νf(a)(y−1) ≤ β implies that y−1 ∈ cα,β(f(a)). (2) let x,y ∈ p and z ∈ x.y. then, µf−1(b)(z) = µb(f(z)) ≥ µb(f(x)) ∧µb(f(y)) = µf−1(b)(x) ∧µf−1(b)(y), and νf−1(b)(z) = νb(f(z)) ≤ νb(f(x)) ∨νb(f(y)) = νf−1(b)(x) ∨νf−1(b)(y). also, for all x ∈ p µf−1(b)(x −1) = µb(f(x −1)) = µb(f(x) −1) ≥ µb(f(x)) = µf−1(b)(x), and νf−1(b)(x −1) = νb(f(x −1)) = νb(f(x) −1) ≤ νb(f(x)) = νf−1(b)(x). hence f−1(b) ∈ ifsp(p). � definition 3.18. let a be an ifsp in a polygroup p. then a is called an intuitionistic fuzzy normal subpolygroup (in short, ifnsp) of p if for all x,y ∈ p a(z) = a(z′) i.e µa(z) = µa(z ′) and νa(z) = νa(z ′), for all z ∈ x◦y and z′ ∈ y ◦x. it is obvious that if a is an intuitionistic fuzzy normal subpolygroup of p , then for all x,y ∈ p , a(z) = a(z′) for all z,z′ ∈ x◦y. theorem 3.19. let a be an intuitionistic fuzzy subpolygroup of p . then a is an intuitionistic fuzzy normal subpolygroup if and only if (1) µa(z) = µa(y) and (2) νa(z) = νa(y), for all x,y ∈ p and for all z ∈ x◦y ◦x−1. proof. it is straightforward. � theorem 3.20. let a be an intuitionistic fuzzy normal subpolygroup of polygroup p . then cα,β(a) is normal subpolygroup of polygroup p , where µa(e) ≥ α, νa(e) ≤ β and e is the identity element of p . proof. let y ∈ cα,β(a) and x ∈ p be any element. then µa(y) ≥ α, νa(y) ≤ β. since a be intuitionistic fuzzy normal subpolygroup of polygroup p , so µa(z) = µa(y) and νa(z) = νa(y) for all x,y ∈ p and for all z ∈ x ◦ y ◦ x−1. therefore µa(z) = µa(y) ≥ α and νa(z) = νa(y) ≤ β implies that µa(z) ≥ α, νa(z) ≤ β and so z ∈ cα,β(a), x ◦ y ◦ x−1 ⊆ cα,β(a). hence cα,β(a) is normal subpolygroup of p . � proposition 3.21. if a is an ifnsp of p then, pa is a normal subpolygroup of p . proof. it is straightforward. � theorem 3.22. let a and b ifnsp of polygroups p1 and p2 respectively. then a×b is also ifnsp of polygroup p1 ×p2. proof. the proof is similar to the proof of theorem 3.11. � 170 abbasizadeh and davvaz 4. t-intuitionistic fuzzy subpolygroups and t-intuitionistic fuzzy quotient polygroups in this section, the notion of t-intuitionistic fuzzy (normal) subpolygroup, t-intuitionistic fuzzy cosets of an intuitionistic fuzzy normal subpolygroup and t-intuitionistic fuzzy quotient polygroup are defined and discussed. definition 4.1. [27] let a be an ifs of a polygroup p . then the ifs at of p is called the tintuitionistic fuzzy subset of p and is defined as at = (µat,νat), where µat(x) = µa(x) ∧ t and νat(x) = νa(x) ∨ (1 − t), for all x ∈ p and t ∈ [0, 1]. definition 4.2. let a be an ifs of a polygroup p. then a is called the t-intuitionistic fuzzy subpolygroup (in short t-ifsp) of p if at is ifsp of p , i.e, the following conditions hold: (1) µat(z) ≥ µat(x) ∧µat(y) and νat(z) ≤ νat(x) ∨νat(y) for each z ∈ x◦y and x,y ∈ p . (2) µat(x −1) ≥ µat(x) and νat(x −1) ≤ νat(x) for each x ∈ p . proposition 4.3. if a is ifsp of a polygroup p , then a is also t-ifsp of p . proof. let x,y ∈ p and z ∈ x◦y. then, we have: µat(z) = µa(z) ∧ t ≥ (µa(x) ∧µa(y)) ∧ t = (µa(x) ∧ t) ∧ (µa(y) ∧ t) = µat(x) ∧µat(y). thus µat(z) ≥ µat(x) ∧µat(y). similarly we can show that νat(z) ≤ νat(x) ∨νat(y). also, µat(x −1) = µa(x −1) ∧ t = µa(x) ∧ t = µat(x). similarly, we can show that νat(x −1) = νat(x). hence a is t-ifsp of p . � definition 4.4. let p be a polygroup, a be an intuitionistic fuzzy subpolygroup of p and t ∈ [0, 1]. then, the intuitionistic fuzzy subset ata of p which is defined by ata(x) = (µata(x),νata(x)), where µata(x) = ( ∧ z∈x◦a µa(z)) ∧ t and νata (x) = ( ∧ z∈x◦a µa(z)) ∨ (1 − t), is called the tintuitionistic fuzzy right coset of a. the tintuitionistic fuzzy left coset aa t of a is defined similarly. proposition 4.5. let a be an intuitionistic fuzzy normal subpolygroup of p and a an arbitrary element of p . then, the tintuitionistic fuzzy right coset ata is same as the tintuitionistic fuzzy left coset aa t. proof. let a be an intuitionistic fuzzy normal subpolygroup of p , a ∈ p and t ∈ [0, 1]. then, for any x ∈ p and z ∈ x◦a, z′ ∈ a◦x, µa(z) = µa(z′), νa(z) = νa(z′). so, µata(x) = ( ∧ z∈x◦a µa(z)) ∧ t = ( ∧ z′∈a◦x µa(z ′)) ∧ t = µ aat(x), and νata(x) = ( ∧ z∈x◦a νa(z)) ∨ (1 − t) = ( ∧ z′∈a◦x νa(z ′)) ∨ (1 − t) = ν aat(x). hence ata =a a t. � intuitionistic fuzzy topological polygroups 171 definition 4.6. let a be t-ifsp of a polygroup p . then a is called t-intuitionistic fuzzy normal subpolygroup (in short t-ifnsp) of p if and only if ata =a a t for all a ∈ p . lemma 4.7. let a be t-ifnsp of a polygroup p . then ata = a t b ⇔ na = nb for all a,b ∈ p , where n = ct,1−t(a). proof. it is straightforward. � consider the set p/at = {ata | a ∈ p} of all tintuitionistic fuzzy right coset of a. theorem 4.8. let a be an intuitionistic fuzzy normal subpolygroup of a polygroup p and n = ct,1−t(a). then, there is a bijection between p/a t and p/n. proof. the proof is similar to the proof of theorem 2.3.8 in [10]. � corollary 4.9. let p be a polygroup, a be an intuitionistic fuzzy normal subpolygroup of p and a ∈ p . then, atz = ata for all z ∈ na, where n = ct,1−t(a) and t ∈ [0, 1]. proposition 4.10. let p be a polygroup, a be an intuitionistic fuzzy normal subpolygroup of a polygroup p . then, (p/at,⊗) is a polygroup (called the polygroup of tintuitionistic fuzzy coset induced by a and t), where the hyperoperation ⊗ is defined as follows: ⊗ : p/at ×p/at −→p∗(p/at) (ata , a t b) 7→ {a t c | c ∈ n.a.b} and −1 on p/at is defined by (ata) −1 = at a−1 . proof. the proof is similar to the proof of theorem 2.3.10 in [10]. � definition 4.11. let p be a polygroup, a be an intuitionistic fuzzy subset of p and β∗ the fundamental relation on p . define the intuitionistic fuzzy subset aβ∗ on p/β ∗ as follows: aβ∗ : p/β ∗ −→ i × i aβ∗(β ∗(x)) = (µaβ∗ (β ∗(x)),νaβ∗ (β ∗(x))) = ( ∨ a∈β∗(x) µa(a), ∧ a∈β∗(x) νa(a)). theorem 4.12. let p be a polygroup, a be an intuitionistic fuzzy subpolygroup of p . then aβ∗ is an intuitionistic fuzzy subgroup of the group p/β∗. proof. we have νaβ∗ (β ∗(x)) ∨νaβ∗ (β ∗(y)) = ( ∧ a∈β∗(x) νa(a)) ∨ ( ∧ b∈β∗(y) νa(b)) = ∧ a∈β∗(x) b∈β∗(y) [νa(a) ∨νa(b)] ≥ ∧ a∈β∗(x) b∈β∗(y) ( ∨ z∈a◦b νa(z)) ≥ ∧ a∈β∗(x) b∈β∗(y) ( ∧ z∈a◦b νa(z)) ≥ ∧ a∈β∗(x) b∈β∗(y) ( ∧ z∈β∗(a.b) νa(z)) = ∧ a∈β∗(x) b∈β∗(y) (νaβ∗ (β ∗(a.b))) = ∧ a∈β∗(x) b∈β∗(y) (νaβ∗ (β ∗(a) �β∗(a))) = νaβ∗ (β ∗(x) �β∗(y)). similarly we can show that µaβ∗ (β ∗(x)) ∧µaβ∗ (β ∗(y)) ≤ µaβ∗ (β ∗(x) �β∗(y)). 172 abbasizadeh and davvaz now, suppose that β∗(x) is an arbitrary element of p/β∗. then, νaβ∗ (β ∗(x)−1) = νaβ∗ (β ∗(x−1)) = ∧ a∈β∗(x−1) νa(a) = ∧ a∈β∗(x−1) νa(a −1) = ∧ b∈β∗(x) νa(b) = νaβ∗ (β ∗(x)). similarly, we can show that µaβ∗ (β ∗(x)−1) = µaβ∗ (β ∗(x)). thus, the proof is complete. � 5. intuitionistic fuzzy topological polygroups in this section, we define and study the concept of intuitionistic fuzzy topological polygroups, and we prove some properties in this respect. definition 5.1. let (p,t ) be a polygroup and a = (µa,νa), b = (µb,νb) are two intuitionistic fuzzy sets in p . we define ab and a−1 by the respective formula: (1) µab(x) = { ∨ (x1,x2)∈x×x [µa(x1) ∧µb(x2)] if x ∈ x1 ◦x2. 0 otherwise, and νab(x) = { ∧ (x1,x2)∈x×x [νa(x1) ∨νb(x2)] if x ∈ x1 ◦x2. 1 otherwise. (2) µa−1 (x) = µa(x −1) and νa−1 (x) = νa(x −1). definition 5.2. let (p,◦) be a polygroup and (p,t ) be an intuitionistic fuzzy topological space. let u = (µu,νu ), v = (µv ,νv ) and w = (µw ,νw ) be an intuitionistic fuzzy sets in p. (p,◦,t ) is called an intuitionistic fuzzy topological polygroup or iftp for short, if and only if: (1) for all x,y ∈ p and any fuzzy open q-neighborhood w of any intuitionistic fuzzy point zr,s of x◦y, there are fuzzy open q-neighborhoods u of xr,s and v of yr,s such that: uv ⊆ w . (2) for all x ∈ p and any fuzzy open q-neighborhood v of an intuitionistic fuzzy point (x−1)r,s, there exists a fuzzy open q-neighborhood u of xr,s such that: u−1 ⊆ v . evidently, every intuitionistic fuzzy topological group is an intuitionistic fuzzy topological polygroup. we give some other examples. example 5. let p be a polygroup and t is the collection of all constant intuitionistic fuzzy sets in p. then (p,t ) is an intuitionistic fuzzy topological polygroup. example 6. let p = {e,a,b}. then, p together with the following hyperoperation ◦ e a b e e a b a a e b b b b {e,a} is a polygroup. it is clear that a−1 = a, b−1 = b. consider on p the fuzzy topology t = {0∼,a, 1∼}, where a =< x, ( e 0.7 , a 0.5 , b 0.3 ), ( e 0.3 , a 0.5 , b 0.7 ) >. then, (p,t ) is an intuitionistic fuzzy topological polygroup. definition 5.3. [25] let (x,t ) be an intuitionistic fuzzy topological space. let α,β ∈ [0, 1]. an intuitionistic fuzzy set (αβ)∗ = (µ(αβ)∗,ν(αβ)∗), where µ(αβ)∗(x) = α and ν(αβ)∗(x) = β, for every x ∈ x such that µ(αβ)∗(x) + ν(αβ)∗(x) = 1. then (x,t ) is called a fully stratified space if for every α,β ∈ [0, 1], (αβ)∗ ∈t . intuitionistic fuzzy topological polygroups 173 proposition 5.4. suppose (p,t ) is a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup. then the mapping f : x −→ x−1 is intuitionistic fuzzy homeomorphic function of (p,t ) onto itself. proof. it is seen that f is invertible. hence the only thing which needs to be proved that f is intuitionistic fuzzy continuous. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and v be a fuzzy open q-neighbourhood of intuitionistic fuzzy point (x−1)r,s. then, there exists a fuzzy open q-neighbourhood u of xr,s such that u −1 ⊆ v . since µu−1 (x) + r = µu (x −1) + r > µu (x) + r > 1, and νu−1 (x) + s = νu (x −1) + s < νu (x) + s < 1. this implies that xr,sq u −1. hence u−1 is a fuzzy open q-neighbourhood of xr,s. thus f(u) = u−1 ⊆ v . then f is an intuitionistic fuzzy continuous function at the intuitionistic fuzzy point xr,s. therefore, f is an intuitionistic fuzzy continuous function. � proposition 5.5. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup. (1) if u is an intuitionistic fuzzy compact subset of p then, u−1 is an intuitionistic fuzzy compact. (2) if u is an intuitionistic fuzzy open set in t then, u−1 is an intuitionistic fuzzy open set in t . proof. it is straightforward. � definition 5.6. [7] let (x,t ) be an ifts and a an ifs in x. then the fuzzy closure is defined by cl(a) = ∩{f| a ⊆ f, fc ∈t}, and the fuzzy interior is defined by int(a) = ∪{g| a ⊇ g, g ∈t}. definition 5.7. let p be a polygroup and a be ifsp of polygroup p . let a ∈ p be a fixed element. then the set aa =< µaa,νaa > where µaa(x) = ∨ z∈a−1◦x µa(z) for all x ∈ p, and νaa(x) = ∧ z∈a−1◦x νa(z) for all x ∈ p, is called intuitionistic fuzzy left coset of p determined by a and a. similarly, the set aa =< µaa,νaa > where µaa(x) = ∨ z∈x◦a−1 µa(z) for all x ∈ p, and νaa(x) = ∧ z∈x◦a−1 νa(z) for all x ∈ p, is called intuitionistic fuzzy right coset of p determined by a and a. if a is an intuitionistic fuzzy normal subpolygroup of p and a an arbitrary element of p , then the intuitionistic fuzzy right coset aa is same as the intuitionistic fuzzy left coset aa. consider the set p/a = {aa | a ∈ p} of all intuitionistic fuzzy right cosets of a. now we give a structure on p/a by defining the operation ⊗ between two intuitionistic fuzzy right cosets as aa⊗ab = {ac | c ∈ a◦ b}. if a is an intuitionistic fuzzy normal subpolygroup of a polygroup p , then the operation ⊗ defined on p/a is well defined. then, (p/a,⊗) becomes a polygroup and is called the intuitionistic fuzzy quotient polygroup relative to the intuitionistic fuzzy normal subpolygroup a. 174 abbasizadeh and davvaz proposition 5.8. let (p,t ) be an intuitionistic fuzzy topological polygroup. then, the family b = {ã ∈ ifs(p∗(p)) | a ∈ t}, where µã(x) = ∨ x∈x µa(x) and νã(x) = ∧ x∈x νa(x) is a base for an intuitionistic fuzzy topology t ∗ on p∗(p). proof. b is a base for an intuitionistic fuzzy topology on p∗(p) because: (1) for any ã1, ã2 ∈b, with a1,a2 ∈t , it follows that ã1 ∩ ã2 ∈b, because ã1 ∩ ã2 = ã1 ∩a2 and a1 ∩a2 ∈t . indeed, for any x ∈p∗(p), we have µ ã1∩a2 (x) = ∨ x∈x µ(a1∩a2)(x) = ∨ x∈x (µa1 (x) ∧µa2 (x)) = ( ∨ x∈x µa1 (x)) ∧ ( ∨ x∈x µa2 (x)) = µã1 (x) ∧µã2 (x) = µ(ã1∩ã2)(x), and ν ã1∩a2 (x) = ∧ x∈x ν(a1∩a2)(x) = ∧ x∈x (νa1 (x) ∨νa2 (x)) = ( ∧ x∈x νa1 (x)) ∨ ( ∧ x∈x νa2 (x)) = νã1 (x) ∨νã2 (x) = ν(ã1∩ã2)(x), (2) since 1∼ ∈t it follows that 1̃∼(x) = 1 for any x ∈p∗(p) and thus⋃̃ a∈b = 1. � lemma 5.9. let u be an intuitionistic fuzzy open subset of an intuitionistic fuzzy topological polygroup p . then, au and ua are intuitionistic fuzzy open subsets of p for every a ∈ p . proof. suppose that u be an intuitionistic fuzzy open subset of p . then, µ( a−1φ −1(ũ))(z) = µũ (a−1φ(z)) = µũ (a −1 ◦z) = ∨ t∈a−1◦z µu (t) = µau (z), and ν( a−1φ −1(ũ))(z) = νũ (a−1φ(z)) = νũ (a −1 ◦z) = ∧ t∈a−1◦z νu (t) = νau (z). since the mapping a−1φ −1 : p −→ p∗(p),x 7→ a−1 ◦x, is intuitionistic fuzzy continuous, thus au is intuitionistic fuzzy open. similarly, we can prove that ua is intuitionistic fuzzy open. � proposition 5.10. let (p,t ) be a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and u = (µu,νu ) be an intuitionistic fuzzy set of p . if ifcl(u) is an intuitionistic fuzzy closed set , then aifcl(u), ifcl(u)a are intuitionistic fuzzy closed sets, where a ∈ p is a definite point. proof. it is straightforward. � proposition 5.11. let a be an ifsp of polygroup p . then for each (r,s) ∈ i ×i with (r,s) ≥ a(e), xr,sa = xa, where x ∈ p and e is the identity of p . proof. we have (xr,sa)(t) = (µxr,sa(t),νxr,sa(t), where µxr,sa(t) = ∨ t∈t1◦t2 [µxr,s(t1) ∧µa(t2)] = { ∨ t∈t1◦t2 [r ∧µa(t2)] if t1 = x 0 if t1 6= x = ∨ t2∈x−1◦t µa(t2) = µxa(t), intuitionistic fuzzy topological polygroups 175 and νxr,sa(t) = ∧ t∈t1◦t2 [νxr,s(t1) ∨νa(t2)] = { ∧ t∈t1◦t2 [s∨νa(t2)] if t1 = x 1 if t1 6= x = ∧ t2∈x−1◦t νa(t2) = νxa(t). hence xr,sa = xa. � proposition 5.12. let (p,t ) be a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and u = (µu,νu ) be an intuitionistic fuzzy set of p . if ifcl(u) is an intuitionistic fuzzy closed set , then ar,sifcl(u), ifcl(u)ar,s and ifcl(u) −1 are intuitionistic fuzzy closed sets. theorem 5.13. in an intuitionistic fuzzy topological polygroup p , v is a q-neighbourhood of er,s if and only if v −1 is a q-neighbourhood of er,s. proof. let v be a q-neighbourhood of er,s. then there exists intuitionistic fuzzy open set a such that er,sqa ⊆ v that is, µa(e) + r > 1, a ⊆ v, νa(e) + s < 1, a ⊆ v. for all x ∈ p , µa(x−1) ≤ µv (x−1) and νa(x−1) ≥ νv (x−1), so µa−1 (x) ≤ µv−1 (x) and νa−1 (x) ≥ νv−1 (x) then, a −1 ⊆ v −1. now, µa−1 (e) + µer,s(e) = µa−1 (e) + r > 1, νa−1 (e) + νer,s(e) = νa−1 (e) + s < 1. hence, er,sqa −1 and a−1 ⊆ v −1. therefore, v −1 is a q-neighbourhood of er,s. conversely, let v −1 be a q-neighbourhood of er,s. then there exists intuitionistic fuzzy open set a such that er,sqa ⊆ v −1. as above, a−1 ⊆ v and er,sqa−1. that is, v is a q-neighbourhood of er,s. � proposition 5.14. let (p,t ) be a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and u = (µu,νu ) be an intuitionistic fuzzy set of p . if u is a q-neighbourhood of er,s, then x1,0u is a q-neighbourhood of xr,s. proof. since u is a q-neighbourhood of er,s, there exists an intuitionistic fuzzy open set a such that r + µa(e) > 1 and s + νa(e) < 1, a ⊆ u. so, µx1,0a(x) = ∨ x∈xy [µx1,0 (x) ∧µa(y)] ≥ 1 ∧µa(e) = µa(e), and r + µx1,0a(x) ≥ r + µa(e) > 1. also νx1,0a(x) = ∧ x∈xy [νx1,0 (x) ∨νa(y)] ≥ 0 ∨νa(e) = νa(e), and s + νx1,0a(x) ≤ s + νa(e) < 1. thus, for all z ∈ p , x1,0u(z) = xu(z) = ( ∨ t∈x−1◦z µu (t), ∧ t∈x−1◦z νu (t)) ⊇ ( ∨ t∈x−1◦z µa(t), ∧ t∈x−1◦z νa(t)) = x1,0a(z). 176 abbasizadeh and davvaz hence xr,sq x1,0a ⊆ x1,0u and since x1,0a is an intuitionistic fuzzy open set, therefore x1,0u is a q-neighbourhood of xr,s. � proposition 5.15. [25] an intuitionistic fuzzy point xr,s ⊆ ifcl(a) if and only if each q-neighbourhood of xr,s is quasi-coincident with a. proposition 5.16. let (p,t ) be a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and a = (µa,νa) be an intuitionistic fuzzy subset of p . if xr,s ⊆ ifcla, then ( ⋂ c∈{c} ac)(x) = ( ⋂ c∈{c} ca)(x) ⊃ 0, where {c} is the system of all qneighbourhood of ea,b in p with a ≤ r and b ≥ s. proof. since xr,s ⊆ ifcla then, each qneighbourhood of xr,s is quasi-coincident with a. for any c ∈{c}, there exists ifos b such that ea,b q b ⊆ c, that is, µb(e) + a > 1, b ⊆ c, νb(e) + b < 1, b ⊆ c. hence, x1,0b −1 is an ifos. moreover, we have µx1,0b−1 (x) = ∨ x∈xy [µx1,0 (x) ∧µb−1 (y)] ≥ 1 ∧µb−1 (e) = 1 ∧µb(e−1) = µb(e), and µx1,0b−1 (x) + r ≥ µb(e) + r ≥ µb(e) + a > 1. also, we have νx1,0b−1 (x) = ∧ x∈xy [νx1,0 (x) ∨νb−1 (y)] ≥ 0 ∨νb−1 (e) = 0 ∨νb(e−1) = νb(e), and νx1,0b−1 (x) + s ≤ νb(e) + s ≤ νb(e) + b < 1. hence, we conclude that x1,0c −1(z) = xc−1(z) = ( ∨ t∈x−1◦z µc−1 (t), ∧ t∈x−1◦z νc−1 (t)) ⊇ ( ∨ t∈x−1◦z µb−1 (t), ∧ t∈x−1◦z νb−1 (t)) = xb−1(z) = x1,0b −1(z). this implies that xr,s q x1,0b −1 ⊆ x1,0c−1. since x1,0c−1 and a are quasi-coincident, there exists y ∈ p such that µx1,0c−1 (y) + µa(y) > 1 and νx1,0c−1 (y) + νa(y) < 1. also µx1,0c−1 (y) = ∨ y∈xz [µx1,0 (x) ∧µc−1 (z)] = ∨ y∈xz [1 ∧µc−1 (z)] = ∨ z∈x−1y µc−1 (z) = µxc−1 (y), intuitionistic fuzzy topological polygroups 177 and νx1,0c−1 (y) = ∧ y∈xz [νx1,0 (x) ∨νc−1 (z)] = ∧ y∈xz [0 ∨νc−1 (z)] = ∧ z∈x−1y νc−1 (z) = νxc−1 (y). thus µac(x) = ∨ x∈t1t2 [µa(t1) ∧µc(t2)] ≥ µa(y) ∧ ( ∨ z∈y−1x µc(z)) = µa(y) ∧ ( ∨ z−1∈x−1y µc−1 (z)) = µa(y) ∧µx1,0c−1 (y) > 0, and νac(x) = ∧ x∈t1t2 [νa(t1) ∨νc(t2)] ≤ νa(y) ∨ ( ∧ z∈y−1x νc(z)) = νa(y) ∨ ( ∧ z−1∈x−1y νc−1 (z)) = νa(y) ∨νx1,0c−1 (y) < 1. that is, ac(x) ⊃ 0 for every c ∈ {c}. hence (∩ac)(x) = ∧ c∈{c} ac(x) ⊃ 0. it is easy to prove ∩ac = ∩ca. � proposition 5.17. let (p,t ) be a fully stratified space. let (p,◦,t ) be an intuitionistic fuzzy topological polygroup and a = (µa,νa) be an intuitionistic fuzzy subset of p . if xr,s ⊆ ⋂ c∈{c} ac = ⋂ c∈{c} ca and r > 0.5, s < 0.5, then xr,s ⊆ ifcl(a), where {c} is the system of all qneighbourhood of ea,b in p with a ≤ r and b ≥ s. proof. let xr,s ⊆ ac for each c ∈ {c}. then µac(x) ≥ r and νac(x) ≤ s. let d be an arbitrary q-neighbourhood of xr,s. then there exists an ifos b such that xr,s q b ⊆ d. that is, µb(x) + r > 1, b ⊆ d, νb(x) + s < 1, b ⊆ d. since µb(x) + r > 1, r > 0.5 and νb(x) + s < 1, s < 0.5, thus d(x) ⊇ b(x) ⊇ 0. hence, b−1x1,0 is an ifos. moreover we have µb−1x1,0 (e) = ∨ e∈yx [µb−1 (y) ∧µx1,0 (x)] ≥ µb−1 (x−1) ∧ 1 = µb−1 (x −1) = µb(x). and µb−1x1,0 (e) + a ≥ µb(x) + a ≥ µb(x) + r > 1. similarly, we have νb−1x1,0 (e) = ∧ e∈yx [νb−1 (y) ∨νx1,0 (x)] ≤ νb−1 (x−1) ∨ 0 = νb−1 (x −1) = νb(x). and νb−1x1,0 (e) + b ≤ νb(x) + b ≤ νb(x) + s < 1. 178 abbasizadeh and davvaz hence b−1x1,0(z) = b −1x(z) = ( ∨ t∈z◦x−1 µb−1 (t), ∧ t∈z◦x−1 νb−1 (t)) ⊆ ( ∨ t∈z◦x−1 µd−1 (t), ∧ t∈z◦x−1 νd−1 (t)) = d−1x1,0(z). this implies that ea,b q b −1x1,0 ⊆ d−1x1,0. so d−1x1,0 ∈ {c}. thus µad−1x1,0 (x) ≥ r and νad−1x1,0 (x) ≤ s. moreover, we have µad−1x1,0 (x) = ∨ x∈yx [µad−1 (y) ∧µx1,0 (x)] ≥ µad−1 (e) ∧ 1 = µad−1 (e), and νad−1x1,0 (x) = ∧ x∈yx [νad−1 (y) ∨νx1,0 (x)] ≤ νad−1 (e) ∨ 0 = νad−1 (e). µad−1 (e) = ∨ e∈t1t2 [µa(t1) ∧µd−1 (t2)] ≥ µa(k) ∧µd−1 (k−1) = µa(k) ∧µd(k), and νad−1 (e) = ∧ e∈t1t2 [νa(t1) ∨νd−1 (t2)] ≤ νa(k) ∨νd−1 (k−1) = νa(k) ∨νd(k). thus there exists z ∈ p such that µad−1 (e) ≥ µa(z) ∧µd(z) and νad−1 (e) ≤ νa(z) ∨νd(z). since µad−1 (e) ≥ r and νad−1 (e) ≤ s then, µa(z) ≥ r, νa(x) ≤ s and µd(z) ≥ r,νd(x) ≤ s. since r > 0.5 and s < 0.5, µa(z) + µd(z) ≥ r + r = 2r > 1 and νa(z) + νd(z) ≤ s + s = 2s < 1. that is, d is quasi-coincident with a. therefore xr,s ⊆ ifcl(a). � references [1] n. abbasizadeh and b. davvaz, topological polygroups in the framework on fuzzy sets, j. intell. fuzzy syst, 30 (2016) 2811-2820. [2] n. abbasizadeh, b. davvaz and v. leoreanu-fotea, studies on fuzzy topological polygroups, j. intell. fuzzy syst, in press. [3] n. abbasizadeh and b. davvaz, category of fuzzy topological polygroups, international journal of analysis and applications 11 (2016), 124-136. [4] t. c. ahn, k. hur and k. w. jang, intuitionistic fuzzy subgroups and level subgroups, international journal of fuzzy logic and intelligent system 6(3) (2006), 240-246. [5] k. atanassov, intuitionistic fuzzy sets, fuzzy sets and systems, 20 (1986), 87-96. [6] r. biswas intuitionistic fuzzy subgroups, mathematical forum 10 (1996), 39-44. [7] d. coker, an introduction to intuitionistic fuzzy topological spaces, fuzzy sets and systems 88 (1997), 81-89. [8] s. d. comer, polygroups derived from cogroups, j. algebra 89 (1984), 397-405. [9] p. corsini, prolegomena of hypergroup theory, aviani editore, 1993. [10] b. davvaz and i. cristea, fuzzy algebraic hyperstructures, springer, 2015. [11] b. davvaz, isomorphism theorems of polygroups, bull. math. sci. soc., 33(3) (2010), 11-19. [12] b. davvaz, polygroup theory and related system, world scientific publishing co. pte. ltd., hackensack, nj, 2013. [13] d. freni, une note sur le coeur dun hypergroupe et sur la cloture transitive β∗ de β, riv. di mat. pura appl. 8 (1991), 153-156. [14] f. gallego, hausdorffness in intuitionistic fuzzy topological space, mathware and soft computing 10 (2003), 17-22. [15] m. jafarpour, h. aghabozorgi and b. davvaz, on nilpotent and solvable polygroups, bulletin of iranian mathematical society 39 (2013), 487-499. [16] y. b. jun, on (φ, ψ)intuitionistic fuzzy mappings, int. j. pure appl. math. 12(2) (2004), 133-139. intuitionistic fuzzy topological polygroups 179 [17] i. m. hanafy, completely continuous functions in intuitionistic fuzzy topological spaces, czechoslovak math. j. 53 (2003), 793-803. [18] d. heidari, b. davvaz and s. m. s. modarres, topological hypergroups in thr sense of marty, comm. algebra 42 (2014), 4712-4721. [19] k. hur, s. y. jang and h. w. kang, intuitionistic fuzzy subgroupoids, international journal of fuzzy logic and intelligent system 3 (2003), 72-77. [20] k. hur, s. y. jang and h. w. kang, intuitionistic fuzzy subgroups and cosets, honam mathematical j. 26 (2004), 17-41. [21] m. koskas, groupoides, demi-hypergroupes et hypergroupes, j. math. pure appl. 49(9) (1970), 155-192. [22] s. j. lee and p. e. lee, the category of intuitionistic fuzzy topological spaces, bull. korean. math. soc. 37 (2000), 63-76. [23] f. g. lupianez, quasicoincident for intuitionistic fuzzy points, int. j. math. sci. 10 (2005), 1539-1542. [24] f. marty, sur une generalization de la notion de groupe, 8iem congress math. scandinaves, stockholm, (1934), 45-90. [25] s. padmapriya, m. k. uma and e. roja, a study on intuitionistic fuzzy topological groups, ann. fuzzy math. inform. 76 (2014), 991-1004. [26] p. k. sharma, (α,β)-cut of intuitionistic fuzzy groups, international mathematics forum 6 (2011), 2605-2614. [27] p. k. sharma, tintuitionistic fuzzy subgroups, international journal of fuzzy mathematics and systems 3 (2012), 233-243. [28] m.m. zahedi, m. bolurian and a. hasankhani, on polygroups and fuzzy subpolygroups, j. fuzzy math. 3 (1995), 1-15. department of mathematics, yazd university, yazd, iran ∗corresponding author: davvaz@yazd.ac.ir international journal of analysis and applications issn 2291-8639 volume 3, number 1 (2013), 14-24 http://www.etamaths.com fixed point theorems for t-ciric quasi-contractive operator in cat(0) spaces g. s. saluja abstract. the purpose of this paper to study a three-step iterative algorithm for t-ciric quasi-contractive (tcqc) operator in the framework of cat(0) spaces and establish strong convergence theorems for above said scheme and operator. our results improve and extend the recent corresponding results from the existing literature (see, e.g., [28, 29, 30] and some others). 1. introduction a metric space x is a cat(0) space if it is geodesically connected and if every geodesic triangle in x is at least as ”thin” as its comparison triangle in the euclidean plane. the precise definition is given below. it is well known that any complete, simply connected riemannian manifold having nonpositive sectional curvature is a cat(0) space. other examples include pre-hilbert spaces (see [3]), r-trees (see [18]), euclidean buildings (see [4]), the complex hilbert ball with a hyperbolic metric (see [12]), and many others. for a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to bridson and haefliger [3]. fixed point theory in cat(0) spaces was first studied by kirk (see [19, 20]). he showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete cat(0) space always has a fixed point. since, then the fixed point theory for single-valued and multi-valued mappings in cat(0) spaces has been rapidly developed, and many papers have appeared (see, e.g., [1], [7], [9]-[11], [13], [16]-[17], [21]-[22], [27], [31]-[32] and references therein). it is worth mentioning that the results in cat(0) spaces can be applied to any cat(k) space with k ≤ 0 since any cat(k) space is a cat(k′) space for every k′ ≥ k (see,e.g., [3]). let (x, d) be a metric space. a geodesic path joining x ∈ x to y ∈ x (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂r to x such that c(0) = x, c(l) = y, and let d(c(t), c(t′)) = |t−t′| for all t, t′ ∈ [0, l]. in particular, c is an isometry, and d(x, y) = l. the image α of c is called a geodesic (or metric) segment joining x and y. we say x is (i) a geodesic space if any two points of x are joined by 2010 mathematics subject classification. 54h25, 54e40. key words and phrases. t-ciric quasi-contractive operator, three-step iteration process, fixed point, strong convergence, cat(0) space. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 14 fixed point theorems 15 a geodesic and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ x, which we will denoted by [x, y], called the segment joining x to y. a geodesic triangle 4(x1, x2, x3) in a geodesic metric space (x, d) consists of three points in x (the vertices of 4) and a geodesic segment between each pair of vertices (the edges of 4). a comparison triangle for geodesic triangle 4(x1, x2, x3) in (x, d) is a triangle 4(x1, x2, x3) :=4(x1, x2, x3) in r2 such that dr2 (xi, x j) = d(xi, x j) for i, j ∈ {1, 2, 3}. such a triangle always exists (see [3]). a geodesic metric space is said to be a cat(0) space if all geodesic triangles of appropriate size satisfy the following cat(0) comparison axiom. cat(0) space. let 4 be a geodesic triangle in x, and let 4⊂r2 be a comparison triangle for 4. then 4 is said to satisfy the cat(0) inequality if for all x, y ∈4 and all comparison points x, y ∈4, d(x, y) ≤ d(x, y).(1.1) complete cat(0) spaces are often called hadamard spaces (see [15]). if x, y1, y2 are points of a cat(0) space and y0 is the mid point of the segment [y1, y2] which we will denote by (y1 ⊕ y2)/2, then the cat(0) inequality implies d2 ( x, y1 ⊕ y2 2 ) ≤ 1 2 d2(x, y1) + 1 2 d2(x, y2) − 1 4 d2(y1, y2).(1.2) the inequality (1.2) is the (cn) inequality of bruhat and titz [5]. the above inequality has been extended in [10] as d2(z,αx ⊕ (1 −α)y) ≤ αd2(z, x) + (1 −α)d2(z, y) −α(1 −α)d2(x, y).(1.3) for any α ∈ [0, 1] and x, y, z ∈ x. let us recall that a geodesic metric space is a cat(0) space if and only if it satisfies the (cn) inequality (see [3, page 163]). moreover, if x is a cat(0) metric space and x, y ∈ x, then for any α ∈ [0, 1], there exists a unique point αx ⊕ (1 − α)y ∈ [x, y] such that d(z,αx ⊕ (1 −α)y) ≤ αd(z, x) + (1 −α)d(z, y),(1.4) for any z ∈ x and [x, y] = {αx ⊕ (1 −α)y : α ∈ [0, 1]}. a subset c of a cat(0) space x is convex if for any x, y ∈ c, we have [x, y] ⊂ c. we recall the following definitions in a metric space (x, d). a mapping t : x → x is called an a-contraction if d(tx, ty) ≤ a d(x, y), for all x, y ∈ x,(1.5) where a ∈ (0, 1). the mapping t is called kannan mapping [14] if there exists b ∈ (0, 12 ) such that d(tx, ty) ≤ b [d(x, tx) + d(y, ty)], for all x, y ∈ x.(1.6) 16 saluja a similar definition is due to chatterjea [8]: there exists c ∈ (0, 12 ) such that d(tx, ty) ≤ c [d(x, ty) + d(y, tx)], for all x, y ∈ x.(1.7) combining these three definitions, zamfirescu [34] proved the following important result. theorem z. let (x, d) be a complete metric space and t : x → x a mapping for which there exist real number a, b and c satisfying a ∈ (0, 1), b, c ∈ (0, 12 ) such that for any pair x, y ∈ x, at least one of the following conditions holds: (z1) d(tx, ty) ≤ a d(x, y), (z2) d(tx, ty) ≤ b [d(x, tx) + d(y, ty)], (z3) d(tx, ty) ≤ c [d(x, ty) + d(y, tx)]. then t has a unique fixed point p and the picard iteration {xn}∞n=0 defined by xn+1 = txn, n = 0, 1, 2, . . . converges to p for any arbitrary but fixed x0 ∈ x. the conditions (z1) − (z3) can be written in the following equivalent form d(tx, ty) ≤ h max { d(x, y), d(x, tx) + d(y, ty) 2 , d(x, ty) + d(y, tx) 2 } , (∗) ∀x, y ∈ x; 0 < h < 1, has been obtained by ciric [6] in 1974. a mapping satisfying (∗) is called ciric quasi-contraction. it is obvious that each of the conditions (z1) − (z3) implies (∗). an operator t satisfying the contractive conditions (z1) − (z3) in the theorem z is called z-operator. in 2009, beiranvand et al. [2] introduced the concept of t-banach contraction and t-contractive mappings and they extended banach’s contraction principle and edelstein fixed point theorem. followed by this, moradi [23] introduced t-kannan contractive mappings, extending in the way, the well-known kannan fixed point theorem [14]. fixed point theorems 17 recently, morales and rojas [25], [26] have extended the concept of t-contraction mappings to cone metric space by proving fixed point theorems for t-kannan, tzamfirescu and t-weakly contraction mappings. in [24], they studied the existence of fixed point for t-zamfirescu operators in complete metric spaces and proved a convergence theorem of t-picard iteration for the class of t-zamfirescu operators. the result is as follows: theorem 1.1. (see [24]) let (m, d) be a complete metric space and t, s : m → m be two mappings such that t is continuous, one-to-one and subsequentially convergent. if s is a tz operator, s has a unique fixed point. moreover, if t is sequentially convergent, then for every x0 ∈ m the t-picard iteration associated to s, tsnx0 converges to tx∗, where x∗ is the fixed point of s. here we recall the definitions of the following classes of generalized t-contraction type mappings as given by morales and rojas [24]. definition 1.2. (see [24]) let (x, d) be a metric space and s, t : x → x be two mappings. a mapping s is said be t-contraction, if there exists a real number a ∈ [0, 1) such that for all x, y ∈ x, d(tsx, tsy) ≤ a d(tx, ty). if we take t = i, the identity map, in the definition 1.2, then we obtain the definition of banach’s contraction. the following example shows that a t-contraction mapping need not be a contraction mapping. example 1. let x = [1,∞) be with the usual metric. define two mappings t, s : x → x as tx = 12x +2 and sx = 3x. obviously, s is not contraction but s is t-contraction which is seen from the following: |tsx − tsy|= ∣∣∣∣ 16x − 16y ∣∣∣∣ = 13 |tx − ty|. definition 1.3. (see [24]) let (x, d) be a metric space and s, t : x → x be two mappings. a mapping s is said be t-kannan contraction, if there exists a real number b ∈ [0, 12 ) such that for all x, y ∈ x, d(tsx, tsy) ≤ b [d(tx, tsx) + d(ty, tsy)]. if we take t = i, the identity map, in the definition 1.3, then we obtain the definition of kannan operator [14]. definition 1.4. (see [24]) let (x, d) be a metric space and s, t : x → x be two mappings. a mapping s is said be t-chatterjea contraction, if there exists a real number c ∈ [0, 12 ) such that for all x, y ∈ x, d(tsx, tsy) ≤ c [d(tx, tsy) + d(ty, tsx)]. if we take t = i, the identity map, in the definition 1.4, then we obtain the definition of chatterjea operator [8]. 18 saluja definition 1.5. (see [24]) let (x, d) be a metric space and s, t : x → x be two mappings. a mapping s is said be t-zamfirescu operator (tz-operator), if there are real numbers 0 ≤ a < 1, 0 ≤ b < 12 , 0 ≤ c < 1 2 such that for all x, y ∈ x at least one of the following conditions holds: (tz1) d(tsx, tsy) ≤ a d(tx, ty), (tz2) d(tsx, tsy) ≤ b [d(tx, tsx) + d(ty, tsy)], (tz3) d(tsx, tsy) ≤ c [d(tx, tsy) + d(ty, tsx)]. if we take t = i, the identity map, in the definition 1.5, then we obtain the definition of zamfirescu operator [34]. definition 1.6. (see [29]) let (x, d) be a metric space and t : x → x be a mapping. (1) a mapping t is said to be sequentially convergent, if we have for every sequence {yn}, {tyn} is convergent implies that {yn} is also convergent. (2) a mapping t is said to be subsequentially convergent, if we have for every sequence {yn}, {tyn} is convergent implies that {yn} has a convergent subsequence. in 2002, xu and noor [33] introduced a three-step iterative scheme as follows: x0 ∈ c, xn+1 = (1 −αn)xn +αntn yn, yn = (1 −βn)xn +βntnzn, zn = (1 −γn)xn +γntnxn, n ≥ 0 where {αn}, {βn} and {γn} are real sequences in [0, 1]. recently, y. niwongsa and b. panyanak [27] studied the noor iteration scheme in cat(0) spaces and they proved some 4 and strong convergence theorems for asymptotically nonexpansive mappings which extend and improve some recent results from the literature. in this paper, inspired and motivated by [24, 28, 33, 34], we study a three-step iteration scheme and prove strong convergence theorem to approximate the fixed point for t-ciric quasi-contractive (tcqc) operator in the setting of cat(0) spaces. three-step iteration scheme in cat(0) space let c be a nonempty closed convex subset of a complete cat(0) space x. let t : c → c and let s : c → c be a t-contractive operator. then for a given fixed point theorems 19 x1 = x0 ∈ c, compute the sequence {xn} by the iterative scheme as follows: tzn = γntsxn ⊕ (1 −γn)txn, tyn = βntszn ⊕ (1 −βn)txn, txn+1 = αnstyn ⊕ (1 −αn)txn, n ≥ 0,(1.8) where {αn}∞n=0, {βn} ∞ n=0, {γn} ∞ n=0 are appropriate sequences in [0,1]. if we take t = i, the identity map, then (1.8) reduces to noor [33] iteration scheme in cat(0) space: zn = γnsxn ⊕ (1 −γn)xn, yn = βnszn ⊕ (1 −βn)xn, xn+1 = αnsyn ⊕ (1 −αn)xn, n ≥ 0,(1.9) where {αn}∞n=0, {βn} ∞ n=0, {γn} ∞ n=0 are appropriate sequences in [0,1]. if γn = 0 for all n ≥ 0, then (1.9) reduces to ishikawa iteration scheme in cat(0) space: yn = βntxn ⊕ (1 −βn)xn, xn+1 = αntyn ⊕ (1 −αn)xn, n ≥ 0,(1.10) where {αn}∞n=0 and {βn} ∞ n=0 are appropriate sequences in [0,1]. we note that if βn = 0 for all n ≥ 0, then (1.10) reduces to mann iteration scheme in cat(0) space: xn+1 = αntxn ⊕ (1 −αn)xn, n ≥ 0,(1.11) where {αn}∞n=0 is a sequence in (0,1). we need the following useful lemmas to prove our main results in this paper. lemma 1.7. (see [27]) let (x, d) be a cat(0) space. (i) for x, y ∈ x and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z) = t d(x, y) and d(y, z) = (1 − t) d(x, y). (a) we use the notation (1 − t)x ⊕ ty for the unique point z satisfying (a). (ii) for x, y ∈ x and t ∈ [0, 1], we have d((1 − t)x ⊕ ty, z) ≤ (1 − t)d(x, z) + td(y, z). lemma 1.8. (see [28]) let {pn}, {qn}, {rn} and {sn} be sequences of nonnegative numbers satisfying the following conditions: pn+1 ≤ (1 − qn)pn + qnrn + sn, n ≥ 1. if ∑ ∞ n=1 qn =∞, limn→∞ rn = 0 and ∑ ∞ n=1 sn < ∞ hold, then limn→∞ pn = 0. 20 saluja t-ciric quasi contraction mapping let x be a cat(0) space and s, t : x → x be two mappings. then s is called t-ciric quasi contraction mapping if it satisfies the following condition: d(tsx, tsy) ≤ h max { d(tx, ty), d(tx, tsx) + d(ty, tsy) 2 , d(tx, tsy) + d(ty, tsx) 2 } (tcqc)(1.12) for all x, y ∈ x and 0 < h < 1. remark 1.9. if we take t = i, then (1.3) reduces to quasi contraction mapping introduced by ciric [6] in 1974 (proc. amer. math. soc. 45 (1974), 727-730). remark 1.10. a mapping satisfying (tcqc) is called t-ciric quasi-contraction mapping. it is obvious that each of the conditions (tz1) − (tz3) implies (tcqc). 2. strong convergence theorems in cat(0) spaces in this section, we establish some strong convergence results of a three-step iteration scheme to converge to a fixed point for t-ciric quasi-contractive operator in the framework of cat(0) spaces. theorem 2.1. let c be a nonempty closed convex subset of a complete cat(0) space. let s, t : c → c be two commuting mappings such that t is continuous, one-to-one, subsequentially convergent and s : c → c is a t-ciric quasi-contractive operator satisfying (1.12) with 0 < h < 1. let {xn} be defined by the iteration scheme (1.8). if ∑ ∞ n=1 αn = ∞,∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, then {txn} converges strongly to tu, where u is the fixed point of the operator s in c. proof. from theorem 1.1, we get that s has a unique fixed point in c, say u. consider x, y ∈ c. since s is a t-ciric quasi-contractive operator satisfying (1.12), then if d(tsx, tsy) ≤ h 2 [d(tx, tsx) + d(ty, tsy)] ≤ h 2 [d(tx, tsx) + d(ty, tx) + d(tx, tsx) + d(tsx, tsy)], implies ( 1 − h 2 ) d(tsx, tsy) ≤ h 2 d(tx, ty) + h d(tx, tsx), which yields (using the fact that 0 < h < 1) d(tsx, tsy) ≤ ( h/2 1 − h/2 ) d(tx, ty) + ( h 1 − h/2 ) d(tx, tsx). fixed point theorems 21 if d(tsx, tsy) ≤ h 2 [d(tx, tsy) + d(ty, tsx)] ≤ h 2 [d(tx, tsx) + d(tsx, tsy) + d(ty, tx) + d(tx, tsx)], implies ( 1 − h 2 ) d(tsx, tsy) ≤ h 2 d(tx, ty) + h d(tx, tsx), which also yields (using the fact that 0 < h < 1) d(tsx, tsy) ≤ ( h/2 1 − h/2 ) d(tx, ty) + ( h 1 − h/2 ) d(tx, tsx).(2.1) denote δ = max { h, h/2 1 − h/2 } = h, l = h 1 − h/2 . thus, in all cases, d(tsx, tsy) ≤ δ d(tx, ty) + l d(tx, tsx) = h d(tx, ty) + ( h 1 − h/2 ) d(tx, tsx).(2.2) holds for all x, y ∈ c. also from (tcqc) with y = u = su, we have d(tsx, tsu) ≤ h max { d(tx, tu), d(tx, tsx) 2 , d(tx, tsu) + d(tu, tsx) 2 } ≤ h max { d(tx, tu), d(tx, tu) + d(tu, tsx) 2 , d(tx, tu) + d(tu, tsx) 2 } = h max { d(tx, tu), d(tx, tu) + d(tu, tsx) 2 } ≤ h d(tx, tu).(2.3) now (2.3) gives d(tsxn, tu) ≤ h d(txn, tu).(2.4) d(tsyn, tu) ≤ h d(tyn, tu).(2.5) 22 saluja and d(tszn, tu) ≤ h d(tzn, tu).(2.6) using (1.8), (2.6) and lemma 1.1 (ii), we have d(tzn, tu) = d(γntsxn ⊕ (1 −γn)txn, tu) ≤ γnd(tsxn, tu) + (1 −γn)d(txn, tu) ≤ γnhd(txn, tu) + (1 −γn)d(txn, tu) ≤ [1 − (1 − h)γn]d(txn, tu).(2.7) again using (1.8), (2.5), (2.7) and lemma 1.1 (ii), we have d(tyn, tu) = d(βntszn ⊕ (1 −βn)txn, tu) ≤ βnd(tszn, u) + (1 −βn)d(txn, tu) ≤ βnhd(tzn, tu) + (1 −βn)d(txn, tu) ≤ βnh[1 − (1 − h)γn]d(txn, tu) + (1 −βn)d(txn, tu) ≤ [1 − (1 − h)βn − h(1 − h)βnγn]d(txn, tu).(2.8) now using (1.8), (2.4), (2.8), ts = st (by assumption of the theorem) and lemma 1.7(ii), we have d(txn+1, tu) = d(αnstyn ⊕ (1 −αn)txn, tu) ≤ αnd(styn, tu) + (1 −αn)d(txn, tu) ≤ αnd(tsyn, tu) + (1 −αn)d(txn, tu) ≤ αnhd(tyn, tu) + (1 −αn)d(txn, tu) ≤ αnh[1 − (1 − h)βn − h(1 − h)βnγn]d(txn, tu) +(1 −αn)d(txn, tu) ≤ [1 −{(1 − h)αn − h(1 − h)αnβn +h2(1 − h)αnβnγn}]d(txn, tu) = (1 − bn)d(txn, tu),(2.9) where bn = {(1 − h)αn − h(1 − h)αnβn + h2(1 − h)αnβnγn}, since 0 < h < 1 and by assumption of the theorem ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, it follows that ∑ ∞ n=1 bn =∞, therefore by lemma 1.8, we get that limn→∞ d(txn, tu) = 0. therefore {txn} converges strongly to tu, where u is the fixed point of the operator s in c. this completes the proof. � since t-kannan contraction and t-chatterjea contraction are both included in the t-ciric quasi-contractive operator, by theorem 2.1, we obtain the corresponding convergence result of the iteration process defined by (1.8) for the above said class of operators as corollary: corollary 2.2. let c be a nonempty closed convex subset of a complete cat(0) space. let s, t : c → c be two commuting mappings such that t is continuous, one-to-one, fixed point theorems 23 subsequentially convergent and s : c → c is a t-kannan contractive operator satisfying the condition d(tsx, tsy) ≤ b [ d(tx, tsx) + d(ty, tsy) 2 ] , ∀x, y ∈ x; b ∈ (0, 12 ). let {txn} be defined by the iteration scheme (1.8). if ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn =∞ and ∑ ∞ n=1 αnβnγn =∞, then {txn} converges strongly to tu, where u is the fixed point of the operator s in c. corollary 2.3. let c be a nonempty closed convex subset of a complete cat(0) space. let s, t : c → c be two commuting mappings such that t is continuous, one-to-one, subsequentially convergent and s : c → c is a t-chatterjea contractive operator satisfying the condition d(tsx, tsy) ≤ c [ d(tx, tsy) + d(ty, tsx) 2 ] , ∀x, y ∈ x; c ∈ (0, 12 ). let {txn} be defined by the iteration scheme (1.8). if ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn =∞ and ∑ ∞ n=1 αnβnγn =∞, then {txn} converges strongly to tu, where u is the fixed point of the operator s in c. if we take t = i, the identity map, in equation (1.12) and (2.2), then we obtain the following results as corollary: corollary 2.4. let c be a nonempty closed convex subset of a complete cat(0) space and let s : c → c be an operator satisfying (2.2). let {xn} be defined by the iteration scheme (1.9). if ∑ ∞ n=1 αn = ∞, ∑ ∞ n=1 αnβn = ∞ and ∑ ∞ n=1 αnβnγn = ∞, then {xn} converges strongly to the unique fixed point of s. corollary 2.5. let c be a nonempty closed convex subset of a complete cat(0) space and let s : c → c be an operator satisfying (2.2). let {xn} be defined by the iteration scheme (1.10). if ∑ ∞ n=1 αn = ∞ and ∑ ∞ n=1 αnβn = ∞, then {xn} converges strongly to the unique fixed point of s. corollary 2.6. let c be a nonempty closed convex subset of a complete cat(0) space and let s : c → c be an operator satisfying (2.2). let {xn} be defined by the iteration scheme (1.11). if ∑ ∞ n=1 αn =∞, then {xn} converges strongly to the unique fixed point of s. remark 2.7. our results extend the corresponding results of rafiq [28], raphael and pulickakunnel [29] and rhoades [30] to the case of three-step iteration and from banach space or convex metric space or uniformly convex banach space to the setting of cat(0) spaces and by using t-ciric quasi-contractive operators. references [1] a. akbar and m. eslamian, common fixed point results in cat(0) spaces, nonlinear anal.: theory, method and applications, vol. 74(5) (2011), 1835-1840. [2] a. beiranvand, s. moradi, m. omid and h. pazandeh, two fixed point theorems for spcial mappings, arxiv:0903.1504v1 [math.fa]. [3] m.r. bridson and a. haefliger, metric spaces of non-positive curvature, vol. 319 of grundlehren der mathematischen wissenschaften, springer, berlin, germany, 1999. [4] k.s. brown, buildings, springer, new york, ny, usa, 1989. [5] f. bruhat and j. tits, ”groups reductifs sur un corps local”, institut des hautes etudes scientifiques. publications mathematiques, 41 (1972), 5-251. [6] l.b. ciric, a generalization of banach principle, proc. amer. math. soc. 45 (1974), 727-730. [7] p. chaoha and a. phon-on, a note on fixed point sets in cat(0) spaces, j. math. anal. appl. 320(2) (2006), 983-987. 24 saluja [8] s.k. chatterjee, fixed point theorems compactes, rend. acad. bulgare sci. 25 (1972), 727-730. [9] s. dhompongsa, a. kaewkho and b. panyanak, lim’s theorems for multivalued mappings in cat(0) spaces, j. math. anal. appl. 312(2) (2005), 478-487. [10] s. dhompongsa and b. panyanak, on 4-convergence theorem in cat(0) spaces, comput. math. appl. 56(10) (2008), 2572-2579. [11] r. espinola and a. fernandez-leon, cat(k)-spaces, weak convergence and fixed point, j. math. anal. appl. 353(1) (2009), 410-427. [12] k. goebel and s. reich, uniform convexity, hyperbolic geometry, and nonexpansive mappings, vol. 83 of monograph and textbooks in pure and applied mathematics, marcel dekker inc., new york, ny, usa, 1984. [13] n. hussain and m.a. khamsi, on asymptotic pointwise contractions in metric spaces, nonlinear anal.: theory, method and applications, 71(10) (2009), 4423-4429. [14] r. kannan, some results on fixed point theorems, bull. calcutta math. soc. 10 (1968), 71-76. [15] m.a. khamsi and w.a. kirk, an introduction to metric spaces and fixed point theory, pure appl. math, wiley-interscience, new york, ny, usa, 2001. [16] s.h. khan and m. abbas, strong and4-convergence of some iterative schemes in cat(0) spaces, comput. math. appl. vol. 61(1) (2011), 109-116. [17] a.r. khan, m.a. khamsi and h. fukhar-ud-din, strong convergence of a general iteration scheme in cat(0) spaces, nonlinear anal.: theory, method and applications, vol. 74(3) (2011), 783-791. [18] w.a. kirk, fixed point theory in cat(0) spaces and r-trees, fixed point and applications, 2004(4) (2004), 309-316. [19] w.a. kirk, geodesic geometry and fixed point theory, in seminar of mathematical analysis (malaga/seville, 2002/2003), vol. 64 of coleccion abierta, 195-225, university of seville secretary of publications, seville, spain, 2003. [20] w.a. kirk, geodesic geometry and fixed point theory ii, in international conference on fixed point theory and applications, 113-142, yokohama publishers, yokohama, japan, 2004. [21] w. laowang and b. panyanak, strong and 4 convergence theorems for multivalued mappings in cat(0) spaces, j. inequal. appl. vol. 2009, article id 730132, 16 pages. [22] l. leustean, a quadratic rate of asymptotic regularity for cat(0)-spaces, j. math. anal. appl. 325(1) (2007), 386-399. [23] s. moradi, kannan fixed point theorem on complete metric spaces and on generalized metric spaces depended on another function, arxiv:0903.1577v1 [math.fa]. [24] j. morales and e. rojas, some results on t-zamfirescu operators, revista notas de matematics, 5(1) (2009), 64-71. [25] j. morales and e. rojas, cone metric spaces and fixed point theorems of t-kannan contractive mappings, int. j. math. anal. 4(4) (2010), 175-184. [26] j. morales and e. rojas, t-zamfirescu and t-weak contraction mappings on cone metric spaces, arxiv:0909.1255v1 [math.fa]. [27] y. niwongsa and b. panyanak, noor iterations for asymptotically nonexpansive mappings in cat(0) spaces, int. j. math. anal. 4(13) (2010), 645-656. [28] a. rafiq, fixed points of ciric quasi-contractive operators in generalized convex metric spaces, general math. 14(3) (2006), 79-90. [29] p. raphael and s. pulickakunnel, fixed point theorems for t-zamfirescu operators, kragujevac j. math. 36(2) (2012), 199-206. [30] b.e. rhoades, fixed point iteration using infinite matrices, trans. amer. math. soc. 196 (1974), 161-176. [31] s. saejung, halpern’s iteration in cat(0) spaces, fixed point theory and applications, vol. 2010, article id 471781, 13 pages, 2010. [32] n. shahzad, fixed point results for multimaps in cat(0) spaces, topology and its applications, vol. 156(5) (2009), 997-1001. [33] b.l. xu and m.a. noor, fixed point iterations for asymptotically nonexpansive mappings in banach spaces, j. math. anal. appl. 267(2) (2002), 444-453. [34] t. zamfirescu, fixed point theorems in metric space, arch. math. (basel), 23 (1972), 292-298. department of mathematics, govt. nagarjuna p.g. college of science, raipur 492010 (c.g.), india international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 64-69 http://www.etamaths.com on the generalized ostrowski type integral inequality for double integrals mustafa kemal yildiz1,∗ and mehmet zeki sarikaya2 abstract. in this paper, we establish a new generalized ostrowski type inequality for double integrals involving functions of two independent variables by using fairly elementary analysis. 1. introduction in 1938, the classical integral inequality was established by ostrowski [5] as follows: theorem 1.1. let f : [a, b]→ r be a differentiable mapping on (a, b) whose derivative f ′ : (a, b)→ r is bounded on (a, b), i.e., ‖f′‖∞ = sup t∈(a,b) |f′(t)| < ∞. then, the inequality holds: ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ (1.1) for all x ∈ [a, b]. the constant 1 4 is the best possible. in a recent paper [3], barnett and dragomir proved the following ostrowski type inequality for double integrals: theorem 1.2. let f : [a, b]×[c, d]→ r be continuous on [a, b]×[c, d], f′′x,y = ∂2f ∂x∂y exists on (a, b)×(c, d) and is bounded, i.e., ∥∥f′′x,y∥∥∞ = sup (x,y)∈(a,b)×(c,d) ∣∣∣∣∂2f(x, y)∂x∂y ∣∣∣∣ < ∞. then, we have the inequality:∣∣∣∣∣∣ b∫ a d∫ c f(s, t)dtds− (d− c)(b−a)f(x, y) −  (b−a) d∫ c f(x, t)dt + (d− c) b∫ a f(s, y)ds   ∣∣∣∣∣∣ (1.2) ≤ [ 1 4 (b−a)2 + (x− a + b 2 )2 ][ 1 4 (d− c)2 + (y − d + c 2 )2 ]∥∥f′′x,y∥∥∞ for all (x, y) ∈ [a, b] × [c, d]. in [3], the inequality (1.2) is established by the use of integral identity involving peano kernels. in [7], pachpatte obtained an inequality in the view (1.2) by using elementary analysis. the interested reader is also refered to ( [3], [4], [6][13]) for ostrowski type inequalities in several independent variables and for recent weighted version of these type inequalities see [1], [2], [9] and [11]. received 22nd july, 2016; accepted 19th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d07, 26d15. key words and phrases. integral inequality; ostrowski’s inequality. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 64 on the generalized ostrowski type integral inequality 65 meanwhile, in [11] sarikaya and ogunmez gave the following interesting identity and by using this indentity they establised some interesting integral inequalities: lemma 1.1. let f : [a, b] × [c, d]→ r be an absolutely continuous function such that the partial derivative of order ∂2f(t,s) ∂t∂s exists for all (t, s) ∈ [a, b]× [c, d] and the weight function w : [a, b] → [0,∞) is integrable, nonnegative and m(a, b) = b∫ a w(t)dt < ∞. (1.3) then, we have f(x, y) = 1 m(a, b) b∫ a w(t)f(t, y)dt + 1 m(c, d) d∫ c w(s)f(x, s)ds − 1 m(a, b)m(c, d)   b∫ a d∫ c w(t)w(s)f(t, s)dsdt− b∫ a d∫ c p(x, t)q(y, s) ∂2f(t, s) ∂t∂s dsdt   (1.4) where p(x, t) =   p1(a, t) = t∫ a w(u)du, a ≤ t < x p2(b, t) = t∫ b w(u)du, x ≤ t ≤ b and q(y, s) =   q1(c, s) = s∫ c w(v)dv, c ≤ s < y q2(d, s) = s∫ d w(v)dv, y ≤ s ≤ d. the main aim of this paper is to establish a new generalized ostrowski type inequality for double integrals involving functions of two independent variables and their partial derivatives. 2. main result we begin with the following important result: lemma 2.1. let f : [a, b] × [c, d]→ r be an absolutely continuous function such that the partial derivative of order ∂2f(t,s) ∂t∂s exists for all (t, s) ∈ [a, b]× [c, d], and the function p : [a, b]× [c, d] → [0,∞) is integrable. then, we have  b∫ a d∫ c p(u, v)dvdu  f(x, y) − b∫ a d∫ c p(t, v)f(t, y)dvdt (2.1) − b∫ a d∫ c p(u, s)f(x, s)dsdu + b∫ a d∫ c p(t, s)f(t, s)dsdt = b∫ a d∫ c p (x, t; y, s) ∂2f(t, s) ∂t∂s dsdt 66 yildiz and sarikaya where p (x, t; y, s) =   t∫ a s∫ c p(u, v)dvdu, a ≤ t < x, c ≤ s < y t∫ a s∫ d p(u, v)dvdu, a ≤ t < x, y ≤ s ≤ d t∫ b s∫ c p(u, v)dvdu, x ≤ t ≤ b, c ≤ s < y t∫ b s∫ d p(u, v)dvdu, x ≤ t ≤ b, y ≤ s ≤ d. proof. by definitions of p (x, t; y, s), we have b∫ a d∫ c p (x, t; y, s) ∂2f(t, s) ∂t∂s dsdt = x∫ a y∫ c [ t∫ a s∫ c p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt + x∫ a d∫ y [ t∫ a s∫ d p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt + b∫ x y∫ c [ t∫ b s∫ c p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt + b∫ x d∫ y [ t∫ b s∫ d p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt. integrating by parts, we can state: x∫ a y∫ c [ t∫ a s∫ c p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt = x∫ a [( t∫ a y∫ c p(u, v)dvdu ) ∂f(t, y) ∂t − y∫ c ( t∫ a p(u, s)du ) ∂f(t, s) ∂t ds ] dt = ( x∫ a y∫ c p(u, v)dvdu ) f(x, y) − x∫ a ( y∫ c p(t, v)dv ) f(t, y)dt − y∫ c ( x∫ a p(u, s)du ) f(x, s)ds + x∫ a y∫ c p(t, s)f(t, s)dsdt, (2.2) x∫ a d∫ y [ t∫ a s∫ d p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt = − x∫ a [( t∫ a y∫ d p(u, v)dvdu ) ∂f(t, y) ∂t + d∫ y ( t∫ a p(u, s)du ) ∂f(t, s) ∂t ds ] dt = ( x∫ a d∫ y p(u, v)dvdu ) f(x, y) − x∫ a ( d∫ y p(t, v)dv ) f(t, y)dt − d∫ y ( x∫ a p(u, s)du ) f(x, s)ds + x∫ a d∫ y p(t, s)f(t, s)dsdt, (2.3) b∫ x y∫ c [ t∫ b s∫ c p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt = − b∫ x [( b∫ t y∫ c p(u, v)dvdu ) ∂f(t, y) ∂t − y∫ c ( b∫ t p(u, s)du ) ∂f(t, s) ∂t ds ] dt = ( b∫ x y∫ c p(u, v)dvdu ) f(x, y) − b∫ x ( y∫ c p(t, v)dv ) f(t, y)dt − y∫ c ( b∫ x p(u, s)du ) f(x, s)ds + b∫ x y∫ c p(t, s)f(t, s)dsdt, (2.4) on the generalized ostrowski type integral inequality 67 b∫ x d∫ y [ t∫ b s∫ d p(u, v)dvdu ] ∂2f(t, s) ∂t∂s dsdt = b∫ x [( b∫ t d∫ y p(u, v)dvdu ) ∂f(t, y) ∂t + d∫ y ( b∫ t p(u, s)du ) ∂f(t, s) ∂t ds ] dt = ( b∫ x d∫ y p(u, v)dvdu ) f(x, y) − b∫ x ( d∫ y p(t, v)dv ) f(t, y)dt − d∫ y ( b∫ x p(u, s)du ) f(x, s)ds + b∫ x d∫ y p(t, s)f(t, s)dsdt. (2.5) adding (2.2)-(2.5) and rewriting, we easily deduce required identity (2.1) which completes the proof. � remark 2.1. if take p(., .) ≡ 1 in lemma 2.1, we get f(x, y) − 1 (b−a) b∫ a f(t, y)dt − 1 (d− c) d∫ c f(x, s)dsdu + 1 (b−a) (d− c) b∫ a d∫ c f(t, s)dsdt = 1 (b−a) (d− c) b∫ a d∫ c p (x, t; y, s) ∂2f(t, s) ∂t∂s dsdt where p (x, t; y, s) =   (t−a) (s− c) , a ≤ t < x, c ≤ s < y (t−a) (s−d) , a ≤ t < x, y ≤ s ≤ d (t− b) (s− c) , x ≤ t ≤ b, c ≤ s < y (t− b) (s−d) , x ≤ t ≤ b, y ≤ s ≤ d. which is given by barnett and dragomir in [3]. remark 2.2. if take p(u, v) = w(u)w(v) in lemma 2.1, then the lemma 2.1 reduces to the lemma 1.1 which is proved by sarikaya and ogunmez in [11]. theorem 2.1. let f : [a, b] × [c, d]→ r be an absolutely continuous function such that the partial derivative of order ∂2f(t,s) ∂t∂s exists and is bounded, i.e.,∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞ = sup (t,s)∈(a,b)×(c,d) ∣∣∣∣∂2f(t, s)∂t∂s ∣∣∣∣ < ∞ for all (t, s) ∈ [a, b] × [c, d], the function p : [a, b] × [c, d] → [0,∞) is integrable. then, we have∣∣∣∣∣∣   b∫ a d∫ c p(u, v)dvdu  f(x, y) − b∫ a d∫ c p(t, v)f(t, y)dvdt − b∫ a d∫ c p(u, s)f(x, s)dsdu + b∫ a d∫ c p(t, s)f(t, s)dsdt ∣∣∣∣∣∣ (2.6) ≤ ∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞ x∫ a (x−u)a(u, y)du + b∫ x (u−x)a(u, y)du where a(u, y) = y∫ c (y −v) |p(u, v)|dv + d∫ y (v −y) |p(u, v)|dv. 68 yildiz and sarikaya proof. from lemma 2.1 and using the properties of modulus, we observe that∣∣∣∣∣∣   b∫ a d∫ c p(u, v)dvdu  f(x, y) − b∫ a d∫ c p(t, v)f(t, y)dvdt − b∫ a d∫ c p(u, s)f(x, s)dsdu + b∫ a d∫ c p(t, s)f(t, s)dsdt ∣∣∣∣∣∣ ≤ b∫ a d∫ c |p (x, t; y, s)| ∣∣∣∣∂2f(t, s)∂t∂s ∣∣∣∣dsdt (2.7) ≤ ∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞ b∫ a d∫ c |p (x, t; y, s)|dsdt ≤ ∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞   x∫ a y∫ c   t∫ a s∫ c |p(u, v)|dvdu  dsdt + x∫ a d∫ y   t∫ a d∫ s |p(u, v)|dvdu  dsdt + b∫ x y∫ c   b∫ t s∫ c |p(u, v)|dvdu  dsdt + b∫ x d∫ y   b∫ t d∫ s |p(u, v)|dvdu  dsdt   ≤ ∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞ {j1 + j2 + j3 + j4} . now, using the change of order of integration we get j1 = x∫ a y∫ c   t∫ a s∫ c |p(u, v)|dvdu  dsdt = x∫ a t∫ a   y∫ c s∫ c |p(u, v)|dvds  dudt = x∫ a t∫ a   y∫ c (y −v) |p(u, v)|dv  dudt = y∫ c   x∫ a t∫ a (y −v) |p(u, v)|dudt  dv (2.8) = x∫ a y∫ c (x−u) (y −v) |p(u, v)|dvdu and similarly, j2 = x∫ a d∫ y (x−u) (v −y) |p(u, v)|dvdu, (2.9) j3 = b∫ x y∫ c (u−x) (y −v) |p(u, v)|dvdu, (2.10) on the generalized ostrowski type integral inequality 69 j4 = b∫ x d∫ y (u−x) (v −y) |p(u, v)|dvdu. (2.11) thus, using (2.8), (2.9), (2.10) and (2.11) in (2.7), we obtain the inequality (2.6) and the proof is completed. � remark 2.3. if we choose p(., .) ≡ 1 in theorem 2.1, then the inequality (2.6) reduces the inequality (1.2) which is proved by barnett and dragomir in [3]. remark 2.4. if take p(u, v) = w(u)w(v) in theorem 2.1, then the inequality (2.6) reduces∣∣∣∣∣∣f(x, y) − 1m(a, b) b∫ a w(t)f(t, y)dt − 1 m(c, d) d∫ c w(s)f(x, s)ds + 1 m(a, b)m(c, d) b∫ a d∫ c w(s)w(t)f(t, s)dsdt ∣∣∣∣∣∣ ≤ ∥∥∥∥∂2f(t, s)∂t∂s ∥∥∥∥ ∞ x∫ a (x−u)a(u, y)du + b∫ x (u−x)a(u, y)du where a(u, y) = y∫ c (y −v)w(u)w(v)dv + d∫ y (v −y)w(u)w(v)dv. which is proved by sarikaya and ogunmez in [11]. references [1] f. ahmad, n. s. barnett and s. s. dragomir, new weighted ostrowski and cebysev type inequalities, nonlinear analysis: theory, methods & appl., 71 (12) (2009), 1408-1412. [2] f. ahmad, a. rafiq, n. a. mir, weighted ostrowski type inequality for twice differentiable mappings, global journal of research in pure and applied math., 2 (2) (2006), 147-154. [3] n. s. barnett and s. s. dragomir, an ostrowski type inequality for double integrals and applications for cubature formulae, soochow j. math., 27(1) (2001), 109-114. [4] s. s. dragomir, n. s. barnett and p. cerone, an n-dimensional version of ostrowski’s inequality for mappings of hölder type, rgmia res. pep. coll., 2(2) (1999), 169-180. [5] a. m. ostrowski, über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, comment. math. helv. 10 (1938), 226-227. [6] b. g. pachpatte, on an inequality of ostrowski type in three independent variables, j. math.anal. appl., 249 (2000), 583-591. [7] b. g. pachpatte, on a new ostrowski type inequality in two independent variables, tamkang j. math., 32 (1) (2001), 45-49 [8] b. g. pachpatte, a new ostrowski type inequality for double integrals, soochow j. math., 32 (2) (2006), 317-322. [9] a. rafiq and f. ahmad, another weighted ostrowski-grüss type inequality for twice differentiable mappings, kragujevac journal of mathematics, 31 (2008), 43-51. [10] m. z. sarikaya, on the ostrowski type integral inequality, acta math. univ. comenianae, 79 (1) (2010), 129-134. [11] m. z. sarikaya and h. ogunmez, on the weighted ostrowski type integral inequality for double integrals, the arabian journal for science and engineering (ajse)-mathematics, 36 (2011), 1153-1160. [12] m. z. sarikaya and h. yildirim, new inequalities for local fractional integrals, iranian journal of science and technology (sciences), in press. [13] n. ujević, some double integral inequalities and applications, appl. math. e-notes, 7 (2007), 93-101. 1department of mathematics, faculty of science and arts, afyon kocatepe university, afyon-turkey 2department of mathematics, faculty of science and arts, düzce university, düzce-turkey ∗corresponding author: myildiz@aku.edu.tr 1. introduction 2. main result references international journal of analysis and applications issn 2291-8639 volume 12, number 1 (2016), 10-14 http://www.etamaths.com some implicit methods for solving harmonic variational inequalities muhammad aslam noor∗, khalida inayat noor abstract. in this paper, we use the auxiliary principle technique to suggest an implicit method for solving the harmonic variational inequalities. it is shown that the convergence of the proposed method only needs pseudo monotonicity of the operator, which is a weaker condition than monotonicity. 1. introduction variational inequalities, which were introduced and investigated by stampacchia [18] in 1964, constitutes a significant extension and generalization of the variational principles. variational inequality theory describes a broad spectrum of very interesting developments involving a link among various fields of mathematics, physics, economics, regional and engineering sciences. the techniques and ideas of variational inequalities are being applied in a variety of diverse areas of pure and applied sciences and prove to productive. the variational inequality is related to simple fact that the minimum of a differentiable convex function on a convex set in a norm space can be characterized by the variational inequality. however, it is remarkable that this theory allows many diversified applications. for the applications, formulation, numerical methods and other aspects of variational inequalities, see [2, 5, 7, 9, 10, 11, 12, 13] and the references therein. we would like to mention that the harmonic means have applications in electrical circuits. to be more precise, the total resistance of a set of parallel resistors is obtained by adding up the reciprocals of the individual resistance values, and then taking the reciprocal of their total. more precisely, if r1 and r2 are the resistances of two parallel resistors, then the total resistance is computed by the formula: 1 r1 + 1 r2 = r1r2 r1+r2 , which is half the harmonic means. the harmonic mean has been used in developing the parallel algorithms for solving various problems. noor [13] used the harmonic mean to suggest some iterative methods for solving nonlinear equations. using the weighted harmonic means, one usually defines the harmonic convex set, which can be viewed as another extension of the convex set. this motivated to introduce the concept of the harmonic convex functions, which is a significant generalization of the convex functions, see [8]. anderson et al [1] have investigated several aspects of the harmonic convex functions. iscan [6] and noor et al [17] have derived several hermitehadmard type integral inequalities for the harmonic convex functions and their variant forms. for recent developments, see[15, 16] and the references therein. to the best our knowledge, no one has studied the properties of the differentiable harmonic convex functions. this fact motivated noor and noor [14] to investigate the characterizations of the differentiable harmonic convex functions. they have shown that the minimum of the differentiable harmonic convex function can be characterized by a class of variational inequalities, which is called harmonic variational inequality. we here use the auxiliary principle technique[4] to suggest an implicit method for solving the harmonic variational inequalities. convergence of the proposed implicit method is considered under the pseudomonotonicity of the operator. the ideas and techniques of this paper may be starting point for a wide range of novel and innovative applications of harmonic variational inequalities in various fields. development of efficient and implementable numerical methods for solving the harmonic variational inequalities is an interesting problem, which needs further efforts. 2010 mathematics subject classification. 26d15, 49j40, 90c23. key words and phrases. harmonic convex functions; variational inequalities; auxiliary principle technique. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 10 some implicit methods for solving harmonic variational inequalities 11 2. preliminaries let c be a nonempty closed and harmonic convex set in the real hilbert space h. we denote by 〈., .〉 and ‖.‖ be the inner product and norm, respectively. for a given nonlinear operator t, consider the problem of finding η ∈c, such that 〈tη, ηξ η − ξ 〉≥ 0, ∀ξ ∈c.(1) the inequality (1) is called the harmonic variational inequality. it has been shown [11,12] that the minimum of a differentiable harmonic convex function can be characterized by harmonic variational inequality of type (1). for the sake of completeness and to convey the main ideas, we include the relevant details. definitions 2.1[1,6]. the set c is said to be a harmonic convex, if ηξ ξ + λ(η − ξ) ∈c, ∀η,ξ ∈c, λ ∈ [0, 1]. definition 2.2[1,6]. the function φ on the harmonic convex set c is said to be harmonic convex, if φ( ηξ ξ + t(η − ξ) ) ≤ (1 −λ)φ(η) + tφ(ξ), ∀η,ξ ∈c λ ∈ [0, 1]. the function φ is said to be harmonic concave if and only if −φ is harmonic convex. we now show that the minimum of a differentiable harmonic convex function on the harmonic convex set c can be characterized by the harmonic variational inequality (1). this result is mainly due to noor and noor [14]. theorem 2.1 [13, 14]. let φ be a differentiable harmonic convex function on the harmonic convex set c. then η ∈c is a minimum of φ, if and only if, η ∈c is the solution of the inequality 〈φ′(η), ηξ η − ξ 〉≥ 0, ∀ξ ∈c,(2) which is called the harmonic variational inequality. we would like to mention that theorem 2.1 implies that harmonic convex programming problem can be studied via the harmonic variational inequality (1). theorem 2.2.[13, 14] let φ be a differentiable harmonic convex functions on the harmonic convex set c. then (i). φ(ξ) −φ(η) ≥〈φ′(η), ηξ η − ξ 〉, ∀η,ξ ∈c. (ii). 〈φ′(η) −φ′(ξ), ηξ ξ −η 〉≥ 0, ∀η,ξ ∈c, where φ′(η) is the differential of φ at η in the direction ηξ ξ−η . using theorem 2.2, we can introduce some new concepts. definition 2.3. an operator t is said to be a harmonic monottone operator, if and only if, 〈tη − tξ, ηξ ξ −η 〉≥ 0, ∀η,ξ ∈ h. definition 2.4. an operator t is said to a harmonic pseudomonotone operator, if 〈tη, ηξ η − ξ 〉≥ 0, implies −〈tξ, ηξ η − ξ 〉≥ 0, ∀η,ξ ∈ h. an harmonic monotone operator is a harmonic pseudomonotone operator, but the converse is not true. 12 m.a. noor and k.i. noor 3. main results in this section, we consider use the auxiliary principle technique to suggest an implicit method for solving harmonic variational inequality. this technique is mainly due to glowinski et al [4]. for a given η ∈c satisfying (1), consider the problem of finding w ∈c such that 〈ρtw, ξw w − ξ 〉 + 〈w −η,ξ −w〉≥ 0, ∀ξ ∈c,(3) which is called the auxiliary harmonic variational inequality. we remark that, if w = η, then w is a solution of the harmonic variational inequality (1). this observation is used to suggest and analyze an implicit method for solving (1). algorithm 3.1. for a given η0 ∈c, compute the approximated solution ηn+1 by the iterative scheme 〈ρtηn+1, ξξn+1 ηn+1 − ξ) 〉 + 〈ηn+1 −ηn,ξ −ηn+1〉≥ 0, ∀ξ ∈c,(4) which is called the proximal-point (implicit) method. to implement algorithm 3.1, one usually uses the predictor-corrector technique. consequently, algorithm 3.1 can be rewritten in the following equivalent form; algorithm 3.2. for a given η0 ∈c, compute the approximated solution ηn+1 by the iterative schemes 〈ρtηn, ξyn yn − ξ) 〉 + 〈yn −ηn,ξ −yn〉≥ 0, ∀ξ ∈c, 〈ρtyn, ξηn+1 ηn+1 − ξ) 〉 + 〈ηn+1 −ηn,ξ −ηn+1〉≥ 0, ∀ξ ∈c, algorithm 3.2 can be viewed as koperlevich method for solving the harmonic variational inequalities. we would also like to point out that algorithm 3.1 and algorithm 3.2 are equivalent. this equivalence is used to study the convergence analysis of algorithm 3.2. we now study the convergence of the proposed algorithm 3.1. theorem 3.1. let η ∈ c be a solution of (1) and let ηn+1 be the approximated solution obtained from algorithm 3.1. if the operator t is harmonic pseudomonotone, then ‖η −ηn+1‖2 ≤‖η −ηn‖2 −‖ηn −ηn+1‖2.(5) proof. let η ∈c be a solution of (1). then 〈tη, ηξ η − ξ 〉≥ 0, ∀ξ ∈c, which implies that 〈tξ, ηξ ξ −η 〉≥ 0, ∀ξ ∈c,(6) since t is harmonic pseudomonotone. taking ξ = ηn+1 in (6) and ξ = η in (4), we have 〈tηn+1, ηηn+1 ηn+1 −η 〉≥ 0.(7) and 〈ρtηn+1, ηηn+1 η −ηn+1 〉 + 〈ηn+1 −ηn,η −ηn+1〉≥ 0.(8) from (7) and (8), we have 〈ηn+1 −ηn,η −ηn+1〉≥ 〈ρtηn+1, ηηn+1 η −ηn+1 〉≥ 0.(9) from (9) and using the inequality 2〈η,ξ〉 = ‖η + ξ‖2 −‖η‖2 −‖ξ‖2 ∀η,ξ ∈ h, some implicit methods for solving harmonic variational inequalities 13 we obtain ‖η −ηn+1‖2 ≤‖η −ηn‖2 −‖ηn −ηn+1‖2, the required (5). � theorem 3.2. let h be a finite dimensional hilbert space and let t be harmonic pseudo monotone operator. if ηn+1 is the approximate solution obtained from algorithm 3.1 and η ∈ c is a solution of problem (1), then limn−→∞ηn = η. proof. let η ∈c be a solution of (1). from (5), we see that the sequence {‖η−ηn‖} is nondecreasing and consequently, the sequence {ηn} is bounded. also from (5), we obtain ∞∑ n=0 ‖ηn −ηn+1‖2 ≤‖η −η0‖2 which implies that ‖ηn −ηn+1‖ = 0.(10) let η̂ be the cluster point of {ηn} and the subsequent {ηnj} of the sequence converges to η̂ ∈ c. replacing ηn by ηnj in (4), taking the limit as nj −→∞ and using (10), we obtain 〈tη̂, ξη̂ η̂ − ξ 〉≥ 0, ∀ξ ∈c, which shows that η̂ is a solution of the harmonic variational inequality (1) and consequently ‖η̂ −ηn+1‖2 ≤‖η̂ −ηn‖2. using the above inequality, one can easily sow that the sequence {ηn} has exactly one cluster point and limn−→∞ηn = η̂, the required result. � acknowledgement. the authors would like to express their sincere gratitude to dr. s. m. junaid zaidi (h.i., s.i. ), rector, comsats institute of information technology, pakistan for providing excellent research facilities and environment. authors would like to thank the referees for their constructive and valuable comments. references [1] g. d. anderson, m. k. vamanamurthy and m. vuorinen, generalized convexity and inequalities, j. math. anal. appl., 335(2007), 1294-1308. [2] c. baiocchi and a. capelo, variational and quasi variational inequalities, john wiley, new york, 1984. [3] g. cristescu and l. lupsa, non-connected convexities and applications, kluwer academic publisher, dordrechet, holland, (2002). [4] r. glowinski, j. l. lions and r. tremolieres, numerical analysis of variational inequalities, north-holland, amsterdam, holland, (1981). [5] f. giannessi and a. maugeri, variational inequalities and network equilibrium problems, plenum press, new york, (1995). [6] i. iscan, hermite-hadamard type inequalities for harmonically convex functions. hacettepe, j. math. stats., 43(6)(2014), 935-942. [7] j. l. lionns and stampacchi, variational inequalities, commun. pure appl. math. 20(1967), 491-512. [8] c. p. niculescu and l. e. persson, convex functions and their applications, springer-verlag, new york, (2006). [9] m. a. noor, general variational inequalities, appl. math. letters, 1(1988), 119-121. [10] m. a. noor, new approximation schemes for general variational inequalities, j. math. anal. appl. 251(2000), 217-229. [11] m. a. noor, some developments in general variational inequalities, appl. math. comput. 152(2004), 199-277. [12] m. a. noor, extended general variational inequalities, appl. math. letters, 22(2009), 182-186. [13] m. a. noor, variational inequalitie and applications, lecture notes, comsats institute of information technology, islamabad, pakistan, 2008-2016. [14] m. a. noor and k. i. noor, harmonic variational inequalities, appl. math. inform. sci. in press. [15] m. a. noor, k. i. noor and s. iftikhar, integral inequalities for differentiable relative harmonic preinvex functions (survey), twms j. pure appl. math. 7(1)(2016), 3-19. [16] m. a. noor, k. i. noor and s. iftikhar, strongly generalized harmonic convex functions and integral inequalities, j. math. anal. in press. [17] m. a. noor, k. i. noor, m. u. awan and s. costache, some integral inequalities for harmonically h-convex functions, u.p.b. sci. bull. series a, 77(1)(2015), 5-16. [18] g. stampacchia, formes bilineaires coercivities sur les sensembles convexes, c. r. acad. sci. paris, 258(1964), 4413-4416. 14 m.a. noor and k.i. noor mathematics department, comsats institute of information technology, islamabad, pakistan ∗corresponding author: noormaslam@gmail.com international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 1-17 http://www.etamaths.com a new entropy formula and gradient estimates for the linear heat equation on static manifold abimbola abolarinwa abstract. in this paper we prove a new monotonicity formula for the heat equation via a generalized family of entropy functionals. this family of entropy formulas generalizes both perelman’s entropy for evolving metric and ni’s entropy on static manifold. we show that this entropy satisfies a pointwise differential inequality for heat kernel. the consequences of which are various gradient and harnack estimates for all positive solutions to the heat equation on compact manifold. 1. introduction and preliminaries we study the heat equation defined on a compact riemannian manifold m with static metric g (1.1) ( ∂ ∂t − ∆g ) u(x,t) = 0, where ∆g is the usual laplace-beltrami operator acting on functions in space with respect to metric g. throughout, m will be taken to be a closed manifold (i.e., compact without boundary) except when otherwise indicated. most of our calculations are done in local coordinates, where {xi} is fixed in a neighbourhood of every point x ∈ m. the riemannian metric g(x) at any point x ∈ m is a bilinear symmetric positive definite matrix denoted in local coordinates by gij = ds 2 = gijdx idxj the laplace-beltrami operator acting on a smooth function f on m is defined as the product of divergence and gradient of f written as ∆gf := div grad f = 1√ |g| ∂ ∂xi (√ |g|gij ∂ ∂xj f ) , where |g| = det(gij) and the inverse metric gij = (gij)−1. by the above we note that (grad f)i = (∇f)i = gij ∂ ∂xj f and divf = 1√ |g| ∂ ∂xi ( √ |g|fi). also we have the metric norm |∇f|2g = g ij∇if∇jf 2010 mathematics subject classification. 53c21, 58d17, 58j35. key words and phrases. riemannian manifold, heat equation, entropy, monotonicity formulas. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 1 2 abimbola abolarinwa and application of cauchy-schwarz inequality on the expression ∆f = gij∇i∇jf = trhessf yields the following inequality (hessf)2 ≥ 1 n (∆f)2. the riemann structure allows us to define riemannian volume measure dv (x) on m dv (x) = √ |gij(x)|dxi. by the divergence theorem we have the following integration by parts formulas for functions f,h ∈ c2(m)∫ m f∆gh dv = − ∫ m 〈∇f,∇h〉gdv = ∫ m ∆gf hdv. for any smooth function f on m, we have the bochner identity defined as ∆(|∇f|2) = 2|∇∇f|2 + 2〈∇f,∇∆f〉 + 2rc(∇f,∇f), where rc is the ricci curvature of m whose tensor components will be written in local coordinates as rij. we switch between coordinates to allow calculations to be explicit and we write in local coordinates ∇f = fi, ∇∇f = ∇i∇jf = fij and ∂ ∂xi = ∂i. also we write time derivative ∂ ∂t f = ∂tf = ft. we adopt summation convention with repeated indices summed up. any function 0 < u ∈ c∞(m × [0,t]) which satisfies (1.1) is called a positive solution. if u tends to a dirac-delta δ-function as t goes to zero, u will be called the heat kernel, that is the unique minimal positive solution on m. we are interested in the behaviours of all positive solutions, in particular, the heat kernel. we derive gradient estimates and differential harnack inequalities via the monotone property of a new family of entropy functionals. it is well known that entropy monotonicity formulas are closely related to the gradient estimate for the heat equation. the importance of gradient estimates as well as those of harnack inequalities can not be overemphasised in the fields of differential geometry and analysis among their numerous applications. differential harnack inequalities are used to study the behaviours of solutions to the heat equation in space-time. li and yau’s paper [15] can be said to mark the beginning of rigorous applications of these concepts. they derived gradient estimates for positive solutions to the heat operator defined on a complete manifold with static metrics, from which they obtained harnack inequalities. these inequalities were in turn used to establish various lower and upper bounds on the heat kernel. precisely, li and yau’s results for static metrics are the following; theorem a (li-yau [15]). let (m,g) be an n-dimensional complete riemannian manifold. suppose there exist some nonnegative constant k such that the ricci curvature rij(g) ≥−k. let u ∈ c2,1(m × [0,t]) be any smooth positive solution to the heat equation (1.1) in the geodesic ball b2ρ×[0,t]. then, the following estimate holds (1.2) sup x∈bρ {|∇u|2 u2 −α ut u } ≤ nα2 2t + cα2 ρ2 ( α2 α2 − 1 + √ kρ ) + nα2k 2(α− 1) . linear heat equation on static manifold 3 for all (x,t) ∈b2ρ,t , t > 0 and some constants c depending only on n and α > 1. moreover, the following estimate (1.3) sup x∈bρ {|∇u|2 u2 −α ut u } ≤ nα2 2t + nα2k 2(α− 1) holds for complete noncompact manifold by letting ρ →∞. the above results have been improved by davies [7, section 5.3] as follows (1.4) sup x∈bρ {|∇u|2 u2 −α ut u } ≤ nα2 2t + nα2k 4(α− 1) . as α → 1, the second terms in both (1.3) and (1.4) blow up and we obtain a sharp estimate |∇u|2 u2 − ut u ≤ n 2t . note that α can be chosen as a constant function of time only such in a way that it goes to 1 as t → 0, see for instance hamilton [11], huang, huang and li [12] and li and xu [14]. li and yau derived their gradient estimates using the maximum principle, but by now it is known how to use monotonicity formulas derived from classical entropies of shannon (from statistical thermodynamics) and fisher’s information (from information theory). let u > 0 be a positive solution to (1.1) with the normalization condition ∫ m udv (x) = 1, then, the classical shannon entropy is defined by (1.5) s0(u(t)) = ∫ m u(x,t) log u(x,t)dv (x) and the fisher information defined by (1.6) f0(u(t)) = ∫ m |∇u(x,t)|2 u(x,t) dv (x). a straightforward computation shows that d dt s0(u(t)) = − ∫ m |∇ log u(x,t)|2u(x,t)dv (x) = −f0(u(t)) and d2 dt2 s0(u(t)) = − d dt f0(u(t)) = 2 ∫ m ( |hess log u|2 + rc(∇ log u,∇ log u) ) udv (x), where rc is the ricci curvature of m. we now define normalised versions of s0 and f0 by s(u(t)) := s0(u(t)) + n 2 log(4πt) + n 2 = ∫ m ( log u + n 2 log(4πt) + n 2 ) udv (x) f(u(t)) := tf0(u(t)) − n 2 = ∫ m ( t|∇ log u|2 − n 2 ) udv (x). 4 abimbola abolarinwa here, the normalisation is done so that the entropies remain identically zero for all time when u is the heat kernel. it easily shown that s and f are identically zero on m = rn, the euclidean space, for u = h(x,y,t) = (4πt)− n 2 exp ( − |x−y|2 4t ) . by the above calculation, shannon entropy s0 for a positive solution to the heat equation on static manifold is seen to be monotone decreasing while its derivative is monotone nondecreasing on the condition that the ricci curvature of m is nonnegative. thus, the shannon entropy is convex in this case. we can now define another entropy w(u,t) based on the above (1.7) w(u,t) = f(u,t) −s(u,t) = − d dt ( ts(u,t) ) . obviously, the entropy w(u,t) reads w(u,t) = ∫ m ( t |∇u|2 u2 − log u− n 2 log(4πt) −n ) udv (x). let u = (4πt)− n 2 e−f be a positive solution to the heat equation, where f is a smooth function. here we have f = − log u− n 2 log(4πt), ∫ m (4πt)− n 2 e−f = 1, (1.8) w(f,t) = ∫ m (t|∇f|2 + f −n)(4πt)− n 2 e−fdv (x) and d dt w = − d dt ( ts ) = −2t ∫ m (∣∣∣∇∇f − 1 2t g ∣∣∣2 + rc(∇f,∇f)) e−f (4πt)− n 2 . this is exactly ni’s result in [17] which states that w(f,t) is monotone nonincreasing on a closed manifold with nonnegative ricci curvature. in the case the manifold is ricci flat this is indeed perelman’s entropy monotonicity formula [20] on a metric evolving by the ricci flow. notice that by application of integration by parts f(u(t)) can be written as (1.9) f(u(t)) = ∫ m − ( t∆ log u + n 2 ) udv (x). this has a surprising connection to the li-yau gradient estimate in theorem a above. clearly, the quantity under the integral is equivalent to the harnack quantity of li-yau − ( t∆ log u + n 2 ) u = − (∆u u − |∇u|2 u2 + n 2 ) u. li-yau gradient estimate [15] says f(u) ≤ 0 when rc ≥ 0, which implies ut u − ∇u|2 u2 + n 2 ≥ 0. this is in turn equivalent to (1.10) t∆f − n 2 ≤ 0, linear heat equation on static manifold 5 which can be viewed as a generalized laplacian comparison theorem. indeed, the laplacian comparison theorem on m is a consequence of (1.10) by applying inequality to the heat kernel and letting t tends to zero. one can also see that limt→0 s(u(t)) = 0 for the heat kernel and hence s(u(t)) is monotone increasing on nonnegative ricci curvature manifold. therefore, we have w(f,t) ≥ 0 for the heat kernel for some t > 0 if and only if m is isometric to rn. note that on rn we have f = |x|2/4t. lei ni also showed that these results hold for all complete manifolds with rc ≥ 0. let m be a complete riemannian manifold with nonnegative ricci curvature, then at t = 1/2, w ≥ 0 holds on m if and only if m is isometric to rn, (see also weissler [22]). this is indeed equivalent to gross logarithmic sobolev inequalities [9] on rn. thus, there is a strong relation between the log-sobolev inequality and the geometry of the manifold which was originally discovered by bakry, concordet and ledoux [3] (see also [4]). that is, (1.11) ∫ rn (1 2 |∇f|2 + f −n ) e−f (4πt)− n 2 ≥ 0 implies (1.12) ∫ rn u log u dx ≤ n 2 log ( 1 2nπe ∫ rn |∇u|2 u ) with equality on any gaussian with ∫ rn udµ. to get (1.12) from (1.11) one uses the monotone property of w on rn and asymptotic behaviour of the positive solution to the heat equation, noting that solution on rn converges after rescaling at infinity to constant multiples of the usual gaussian. the remarkable papers [17] and [18] have shown a desirable interpolation between entropy formula of ni on static manifolds and that of perelman [20] on evolving manifolds. the new w�(f,t) discussed in this paper (see section 2) is an example of such a family of entropies connecting both ni’s and perelman’s entropies. we demonstrated this in [2, chapter 3] and have applied it on manifold evolving by the ricci-harmonic map flow in [1]. we remark that estimates and bounds on parabolic equations behave in similar way whether the metric is static or moving. this can be justified by the fact that heat diffusion on a bounded geometry with either static or evolving metric behaves like heat diffusion in euclidean space, many a times, their estimates even coincide. in this paper however, we prove the monotonicity formulas for a family of entropy functionals w�(f,t) and discuss some of its analytic and geometric consequences. the plan of the rest of the paper is as follows: in section 2 we introduce a new family of entropy functionals and prove its monotonicity for a positive solution to the heat equation. the monotonicity derived here is used in section 3 to derive pointwise differential harnack inequalities and gradient estimates for the heat equation. as a consequence we obtain harnack estimates for the fundamental solution which also holds for all positive solutions in section 4. we give li-yau-hamilton type gradient estimates for bounded solutions in the last section. 2. a new entropy monotonicity formula we emphasize that the volume is kept fixed throughout the time of evolution for the heat equation on a closed n-dimensional manifold (m,g). we also impose the condition of nonnegativity on the ricci curvature of the underlying manifold m. 6 abimbola abolarinwa let u = u(x,t) be a positive solution to the heat equation (2.1) �u = ( ∂ ∂t − ∆ ) u(x,t) = 0. let f : m×(0,t] → r be smoothly defined as u = (4πt)− n 2 e−f with normalization condition ∫ m u(x,t)dv (x) = 1. we introduce a generalized family of entropy by (2.2) w�(f,t) = ∫ m [�2t 4π |∇f|2 + f + n 2 ln (4π �2 ) − n�2 4π ] e−f (4πt) n 2 dv (x), where 0 < �2 ≤ 4π. we remark that if �2 = 4π, we recover the perelman’s entropy as in the special case considered by ni in [17]. from this entropy formula we later derive the corresponding differential inequality and gradient estimate for the fundamental solution, which in fact, holds for all positive solutions to the heat equation. the same entropy is used by the author in his phd thesis [2] to examine the surprising relation that exists between the entropy formula for heat equation and the conjugate heat equation under the ricci flow. we have also used its monotonicity properties combined with some sobolev-type inequalities to derive sharp upper bound for conjugate heat kernel along ricci-harmonic map heat flow in [1]. lemma 2.1. let u = (4πt)− n 2 e−f be a positive solution to the heat equation �u = 0 on a closed riemannian manifold m. then (2.3) (∂t − ∆)|∇f|2 = −2f2ij − 2〈∇f,∇|∇f| 2〉− 2rijfifj and (2.4) (∂t − ∆)(∆f) = −2f2ij − 2〈∇f,∇|∇f| 2〉− 2〈∇f,∇∂tf〉− 2rijfifj. moreover, if w = 2∆f −|∇f|2, then (2.5) (∂t − ∆)w = −2f2ij − 2rijfifj − 2〈∇w,∇f〉. proof. since u = (4πt)− n 2 e−f , f = − log u− n 2 log(4πt) and ∂ ∂t f = ∆f−|∇f|2− n 2t . (1) ∂ ∂t |∇f|2 = ∂ ∂t (gij∂if∂jf) = 2g ij∂if∂jf∂tf = 2〈∇f,∇∂tf〉.(2.6) by bochner identity ∆(|∇f|2) = 2f2ij + 2〈∇f,∇∆f〉 + 2rijfifj = 2f2ij + 2〈∇f,∇(∂tf + |∇f| 2〉 + 2rijfifj = 2f2ij + 2〈∇f,∇|∇f| 2〉 + 2〈∇f,∇∂tf〉 + 2rijfifj. adding the last equality to (2.6) proves (2.3). (2) (∂t − ∆)(∆f) = ∆(∆f −|∇f|2) − ∆(∆f) = −∆|∇f|2 = −2f2ij − 2〈∇f,∇(∂tf + |∇f| 2〉− 2rijfifj = −2f2ij − 2〈∇f,∇|∇f| 2〉−∂t(|∇f|2) − 2rijfifj. (3) (∂t − ∆)w = 2(∂t − ∆)∆f − (∂t − ∆)|∇f|2 = −2f2ij − 2rijfifj − 2〈∇f,∇(|∇f| 2 + 2∂tf).〉 linear heat equation on static manifold 7 this ends the proof of the lemma. � we are now set to establish the monotone property of the w�(f,t)-entropy. by the monotonicity formula for this entropy functional, we will derive gradient estimates and the corresponding differential harnack inequalities for the fundamental solution to the heat equation on a static manifold. proposition 2.2. let m be any closed manifold and u = (4πt)− n 2 e−f be any positive solution to the heat equation �u = (∂t − ∆)u = 0 on m × (0.t]. denoting (2.7) p� = �2t 4π ( 2∆f −|∇f|2 ) + f + n 2 ln (4π �2 ) − n�2 4π , where 0 < �2 ≤ 4π. then (2.8) (∂t−∆)p� ≤− �2t 2π (∣∣∣fij−√π �t gij ∣∣∣2 +rijfifj)−2〈∇p�,∇f〉−(1− �2 4π ) |∇f|2. proof. here we write p̃� = �2t 4π w + f̃ + n 2 ln ( 1 �2t ) − n�2 4π . since f = − ln u− n 2 ln(4πt), taking u = e−f̃ implies f = f̃ − n 2 ln(4πt). we notice also that ∇f̃ = ∇f, ∆f̃ = ∆f and f̃ij = fij, then (∂t − ∆)f̃ = −|∇f̃|2 − n2t. now by direct differentiation and application of lemma 2.1, we have the following computation (∂t − ∆)p� = �2t 4π (∂t − ∆)w + �2 4π w + (∂t − ∆)f̃ + ∂ ∂t (n 2 ln ( 1 �2t ) − n�2 4π ) = �2t 4π ( − 2f2ij − 2rijfifj − 2〈∇w,∇f〉 ) + �2 4π (2∆f −|∇f|2) −|∇f|2 − n 2t = �2t 4π ( − 2f2ij − 2π �2 n t2 − 2rijfifj ) + �2 4π (2∆f −|∇f|2) − 2〈 �2t 4π ∇w,∇f〉− |∇f|2. notice that 2〈 �2t 4π ∇w,∇f〉 = 2〈(∇p� − f̃),∇f〉 = 2〈∇p�,∇f〉− 2|∇f|2 and (2∆f −|∇f|2) = (2∂tf + |∇f|2) then we have (∂t − ∆)p� ≤−2 �2t 4π ( f2ij + π �2 n t2 − 2 √ π �t ∆f + rijfifj ) − 2〈∇p�,∇f〉 + �2 4π |∇f|2 −|∇f|2 = − 2�2t 4π (∣∣∣fij − √π �t gij ∣∣∣2 + rijfifj)− 2〈∇p�,∇f〉−(1 − �2 4π ) |∇f|2. � theorem 2.3. let m be a closed riemannian manifold. assume that u = (4πt)− n 2 e−f is a positive solution to the heat equation (∂t−∆)u = 0, then, we have the following monotonicity formula for w�(f,t) defined in (2.2) (2.9) d dt w�(f,t) = − ∫ m [ �2t 2π (∣∣∣fij−√π �t gij ∣∣∣2 +rijfifj ) + ( 1− �2 4π ) |∇f|2 ] e−f (4πt) n 2 dv (x) 8 abimbola abolarinwa with (f,t) satisfying (2.10) ∫ m e−f (4πt) n 2 dv (x) = 1 and 0 < �2 ≤ 4π. proof. combining proposition 2.2 with the fact that �u = 0 and u∇f = −∇u, we have (∂t − ∆)(p�u) = (∂t − ∆)p� ·u + p�(∂t − ∆)u− 2〈∇p�,∇u〉 = − �2t 2π (∣∣∣fij − √π �t gij ∣∣∣2 + rijfifj)u− 2〈∇p�,∇f〉u − ( 1 − �2 4π ) |∇f|2u− 2〈∇p�,∇u〉. integrating over m, we have∫ m p�udv (x) = ∫ m [�2t 4π ( 2∆f −|∇f|2 ) + f + n 2 ln (4π �2 ) − n�2 4π ] udv (x) = ∫ m [�2t 4π |∇f|2 + f + n 2 ln (4π �2 ) − n�2 4π ] udv (x) + 2�2t 4π ∫ m (∆f −|∇f|2)udv (x) = w�(f,t), in the sense that the second integral in the rhs vanishes on a closed manifold since (∆f −|∇f|2)u = −∆u. therefore d dt w�(f,t) = ∂ ∂t ∫ m p�u dv (x) = ∫ m ( d dt p� u + p� ∂ ∂t u ) dv (x) = ∫ m [ (∂t − ∆)p� u + p�(∂t − ∆)u ] dv (x) = ∫ m (∂t − ∆)p� udv (x), where we have used integration by parts and �u = 0. using the evolution (∂t − ∆)p� from proposition 2.2, we get the desired result. moreover, if the manifold has nonnegative ricci curvature, i.e, rij ≥ 0, it becomes obvious from (2.9) that dw�/dt ≤ 0. � we remark that kuang and zhang [13] have a result in this direction, it is stated as follows; let m be a closed riemannian manifold with nonnegative ricci curvature. let u be the fundamental solution to the heat equation with f = − ln u− n 2 ln(4πt), we have (2.11) t(α∆f −|∇f|2) + f −α n 2 ≤ 0 for any constant α ≥ 1. indeed, if α = 2, this is exactly the differential inequality t(2∆f −|∇f|2) + f −n ≤ 0 linear heat equation on static manifold 9 proved in [17]. dividing through by α · t, with α ≥ 1 and t ≥ 0, we obtain ∆f − |∇f|2 α + f αt − n 2t ≤ 0 as t →∞, which is precisely the li-yau gradient estimate. for α > 2, the gradient estimate is an interpolation of perelman’s estimate and li-yau estimate. for 1 ≤ α ≤ 2, it is considered in [13]. in euclidean space rn, if u is the fundamental solution to the heat equation then (2.11) becomes an equality. 3. gradient estimates for heat equation on static manifold the monotonicity formula in the last section may be viewed as a local version of the perelman’s w-entropy formula in [20]. in what follows, we want to show that the local entropy satisfies a pointwise differential inequality for the heat kernel. we have the following fashioned after [17, theorem 1.2] with the proof follows from the argument of [16, proposition 3.6]. theorem 3.1. let m be a closed manifold with nonnegative ricci curvature and h(x,y,t) = h = (4πt)− n 2 e−f be the heat kernel, where h tends to a δ-function as t → 0 and satisfies ∫ m hdv (x) = 1. then for all t > 0, we have (3.1) p� = �2t 4π ( 2∆f −|∇f|2 ) + f + n 2 ln (4π �2 ) − n�2 4π ≤ 0. proof. let h be any compactly supported smooth function for all t0 > 0. suppose h(·, t) is a positive solution to the backward heat equation (∂t + ∆)h = 0, (this is perelman’s argument in [20, corollary 9.3]), then, it is clear that ∂ ∂t ∫ m hhdv = 0 and we have by direct calculation that ∂ ∂t ∫ m hp�hdv (x) = ∫ m [ ∂th(p�h) + h∂t(p�h) ] dv (x) = ∫ m [ (∂t + ∆)h(p�h) + h(∂t − ∆)p�h) ] dv (x) = ∫ m h(∂t − ∆)p�hdv (x) ≤ 0. the inequality is due to theorem 2.3 since rij ≥ 0. we are left to showing that for everywhere positive function h(·, t), the limit of ∫ m hp�hdv (x) is nonpositive as t → 0. we assume the claim apriori (i.e, limt→0 ∫ m hp�hdv = 0) and conclude the result. for completeness, we devote the next effort to justifying the claim (3.2) lim t→0 ∫ m hp�hdv ≤ 0. our argument follows from [16], for detail see [17, 19, 20], the calculation in [13] is also similar. if h tends to a dirac δ-function, say at a point p ∈ m, for t → 0, then f satisfies f(x,t) → d 2(p, x) 4t . this is in relation to llength of perelman. this yields (3.3) lim t→0 ∫ m fhhdv ≤ lim sup t→0 ∫ m d2(p,x) 4t hhdv = n 2 h(p, 0). 10 abimbola abolarinwa meanwhile, by the strong maximum principle we have h(x, 0) > 0 and limt→0 ∫ m hhdv = h(x, 0), hence by scaling argument, we assume that h(x, 0) = 1. all these will soon become clearer. rewriting p� and using integrating by parts methods we have∫ m p�hhdv = ∫ m �2t 4π (|∇f|2 − n 2t )hhdv − �2t 2π ∫ m 〈∇f,∇h〉hdv + ∫ m fhhdv + n 2 [ ln (4π �2 ) − �2 4π ]∫ m hh dv. though, the h appearing in the last equation is actually the heat kernel on an evolving manifold in ni’s result [19] while h satisfies the forward heat equation, his argument still holds in our case, we only need the asymptotic behaviour of heat kernel on a fixed metric. we should also note that since h(·, t0) is compactly supported and by strong maximum principle we have h(·, t0), |∇h(·, t0)| and |∆h(·, t0)| bounded on m. this implies that there exists a bounded solution h(·, t0). it turns out that we need to show that there exists a constant b ≥ 0 which may depend on the geometry of the underlying manifold and independent of t as t → 0, such that ∫ m p�hhdv ≤ b(n). now we claim that the first two terms on the right hand side of the last equation vanish as t → 0, we can see this in the following argument. by integration by parts and the fact that ∇h = −h∇f, we have −t ∫ m 〈∇f,∇h〉hdv = t ∫ m 〈∇h,∇h〉dv = −t ∫ m h∆hdv is bounded since |∆h| is bounded as stated earlier. thus, the second term is bounded and goes to zero as t → 0. we need a bound of li-yau type to obtain a bound for the first term |∇f|2. see lemma 3.2 below for the statement of the result ([5]) see also [6, corollary 16.23] and souplet and zhang [21]). by this we have for the heat kernel in the present case that (3.4) t ∫ m |∇f|2 ≤ 2 ( b(n,δ) + d2(x,y) (4 − δ)t ) , which is also clearly seen to be bounded from above as t → 0 by the justification of asymptotic behaviour of the heat kernel. we have now reduced the analysis to (3.5) lim t→0 ∫ m p�hhdv ≤ lim sup t→0 ∫ m ( f + nq 2 ) hhdv, where q = ln( 4π �2 ) − � 2 4π . for simplicity, we can choose � such that �2 → 4π as t → 0 so that the whole problem is reduced to finding (3.6) lim t→0 ∫ m ( f − n 2 ) hhdv. using the asymptotic behaviour of the heat kernel, i.e, f ≈ d 2 4t as t → 0. recall (cf. [8, 19]) as t → 0 h(x,y,t) ∼ (4πt)− n 2 exp (d2(x,y) 4t ) ∞∑ j=o uj(x,y,t)t j := wk(x,y,t) where d2(x,y) is the distance function and wk(x,y,t) satisfies uniformly for all x,y ∈ m wk(x,y,t) = o ( tk+1− n 2 exp (δd2(x,y) 4t )) linear heat equation on static manifold 11 and δ is just a number depending only on the geometry of (m,g). the function can be chosen such that u0(x,y, 0) = 1. though, the above asymptotic result does not require any curvature assumption, a result due to cheeger and yau [5] states that on manifold with nonnegative ricci curvature (which is our case), the heat kernel satisfies h(x,y,t) ≥ (4πt)− n 2 exp (d2(x,y) 4t ) which implies f(x,t) ≤ d2(x,y) 4t . therefore lim t→0 ∫ m ( f − n 2 ) hhdv ≤ lim t→0 ∫ m (d2(x,y) 4t − n 2 ) h(y,t)h(x,y,t)dv (y) = lim t→0 ∫ m (d2(x,y) 4t − n 2 )e−d2(x,y)/4t (4πt) n 2 h(y,t)dv (y). it is easy to see that for any δ > 0, the integration of the above integrand in the domain d(x,y) ≤ δ converges to zero exponentially fast. therefore (3.7) lim t→0 ∫ m ( f − n 2 ) hhdv ≤ lim t→0 ∫ d(x,y)≤δ (d2(x,y) 4t − n 2 )e−d2(x,y)4t (4πt) n 2 h(y,t)dv (y). whenever δ is chosen sufficiently small, d(x,y) is asymptotically sufficiently close to the euclidean distance. by standard approximation, we have (3.8) lim t→0 ∫ m ( f − n 2 ) hhdv ≤ lim t→0 ∫ d(x,y)≤δ (|x−y|2 4t − n 2 )e−|x−y|24t (4πt) n 2 hp(y)dv (y), where hp is the pullback of h(·, 0) to the euclidean space from the region d(x,y) ≤ δ. splitting the last integrand as in [13] we are left with lim t→0 ∫ m ( f − n 2 ) hhdv ≤ hp(x) lim t→0 ∫ rn (|x−y|2 4t − n 2 )e−|x−y|24t (4πt) n 2 dv (y) = hp(·) lim t→0 ∫ rn (|y|2 4t e− |y|2 4t (4πt) n 2 ) dv (y) − n 2 hp(·). the last equality is due to convolution properties of the heat kernel. lastly we show that the rhs evaluates to 0. recall, using standard gauss integral, that∫ rn |y|2e−α|y| 2 dy = n (∫ ∞ −∞ y2e−αy 2 dy )(∫ ∞ −∞ e−αy 2 dy )n−1 = n 2 √ π α3 · (√π α )n−1 = n 2α (√π α )n , so that we have∫ rn (|y|2 4t e− |y|2 4t (4πt) n 2 ) dv (y) = 1 (4πt) n 2 · n 4t (∫ ∞ −∞ y2e− 1 4t y2dy )(∫ ∞ −∞ e− 1 4t y2dy )n−1 = n 2 , by taking α = 1/4t in the above. we can then conclude the claim. � 12 abimbola abolarinwa lemma 3.2. on a complete riemannian manifold (m,g) with nonnegative ricci curvature, the following estimate holds for the gradient of the heat kernel h(x,y,t) and all δ > 0, (3.9) |∇h|2 h ≤ 2h t ( b(n,δ) + d2(x,y) (4 −δ)t ) for all x,y in m and t > 0. 4. harnack estimates for the heat kernel the following differential harnack quantity for linear heat equation on static manifold follows immediately as an application of the reuslts in the last subsection. corollary 4.1. let m be a closed manifold with curvature bounded from below by rc ≥ 0. then we have (4.1) �2t 4π ( 2∆f −|∇f|2 ) + f + n 2 ( ln (4π �2 ) − �2 2π ) ≤ 0, where f = − ln(4πt) n 2 h and h is the positive minimal solution to the heat equation( ∂ ∂t − ∆x ) h(x,y,t) = 0. remark 4.2. note that the quantity 2∆f −|∇f|2 can be expressed as |∇u| 2 u2 − 2ut u in terms of u, which is similar to li-yau gradient estimate [15] on a manifold with nonnegative ricci curvature, ut u − |∇u| 2 u2 + n 2t ≥ 0. this is equivalent to the differential harnack inequality 2t∆f ≤ n, where f = − ln(4πt) n 2 u, which can be regarded as a generalized laplacian comparison theorem in space for heat kernel on m. however, we have from (4.1) that f ≤ n 2 [ �2 2π − ln 4π �2 ] − �2t 4π (2∆f −|∇f|2) ≤ n 2 [ �2 2π − ln 4π �2 ] − �2n 8π = n 2 [ �2 4π − ln 4π �2 ] . define (4.2) q(x,t) = �2 π tf(x,t) (4.3) (∂t − ∆)q(x,t) = �2 π f(x,t) + �2 π t(∂t − ∆)f ≤ n�2 2π [ �2 4π − ln 4π �2 ] . still as � = 2 √ π we recover ni’s generalized laplacian. from corollary 4.1, we have the differential harnack inequality as follows �2t 4π ( 2∆f −|∇f|2 ) + f + n 2 ( ln (4π �2 ) − �2 2π ) ≤ 0. multiplying through by −2π �2t , we have −∆f + 1 2 |∇f|2 − 2π �2t f − nπ �2t ( ln (4π �2 ) − �2 2π ) ≥ 0 −∆f + 1 2 |∇f|2 − 2π �2t f + n 2t − nπ �2t ln (4π �2 ) ≥ 0. linear heat equation on static manifold 13 recall that (∂t − ∆)h = 0 implies ∆f = ∂tf + |∇f|2 + n2t, then we have −∂tf − 1 2 |∇f|2 − 2π �2t f ≥ nπ �2t ln (4π �2 ) ∂tf + 1 2 |∇f|2 ≤− 2π �2t f − nπ �2t ln (4π �2 ) = − 2π �2t ( f + n 2 ln (4π �2 )) . by the young’s inequality we have on the path γ(t), (γ(t) : [t1, t2] → m is a minimizing geodesic connecting points x1 and x2 such that γ(t1) = x1 and γ(t2) = x2.) d dt f(γ(t), t) = ∂tf + 〈∇f,γ′(t)〉 ≤ ∂tf + 1 2 |∇f|2 + 1 2 |γ′(t)|2 = 1 2 |γ′(t)|2 − 2π �2t ( f + n 2 ln (4π �2 )) since we have from (4.1) that f ≤ n 2 ( �2 4π − ln 4π �2 ) , inserting this quantity in the above inequality gives the following harnack estimates (4.4) d dt f(γ(t), t) ≤ 1 2 |γ′(t)|2 − n 4t . after the usual integration of (4.4) and exponentiation we have the following corollary 4.3. with the notation and assumption of corollary 4.1, we have the following differential harnack estimates (4.5) u(x2, t2) u(x1, t1) ≤ (t1 t2 )n 4 exp [1 2 ∫ t2 t1 |γ′(t)|2dt ] . remark 4.4. if m is a closed manifold with nonnegative ricci curvature and u = (4πt)− n 2 e−f is the heat kernel on m. then w�(f,t0) ≥ 0 for some t0 > 0, if and only if m is isometric to euclidean space rn. recall that we have obtained that d dt w�(f,t) ≤ 0 and w�(f,t) ≤ 0 which in turn imply that we must have w�(f,t) ≡ 0 for 0 ≤ t ≤ t0. for instance, in the case � = 2 √ π, we have |fij − 1 2t gij|2 = 0 and fij − 1 2t gij = 0. taking the trace of the above yields (4.6) t∆f − n 2 = 0. because f(x,t) ≈ f̃(x,t) = d 2(p,x) 4t for t small, we have limt→0 4tf = d 2(p,x). hence (4.6) implies that (4.7) ∆d2(p,x) = 2n so that we can get for the area ap(r) of ∂bp(r) and the volume vp(r) of the ball bp(r), the following quotient ap(r) vp(r) = n r . 14 abimbola abolarinwa this shows that vp(r) is exactly the same as the volume function of euclidean balls. this argument shows that the li-yau harnack inequality, which is equivalent to 2t∆f −n ≤ 0 for u = (4πt)− n 2 e−f becomes an equality if and only if the manifold m with rc ≥ 0 is isometric to rn and u is precisely the heat kernel. if t = 1 2 and m = rn, the inequality w�(f,t0) ≥ 0 for �2 = 4π, is equivalent to (4.8) ∫ rn ( 1 2 |∇f|2 + f −n)(2π)− n 2 e−fdv ≥ 0 for all f with the condition ∫ m (2π)− n 2 e−fdv = 1. the above implies a sharp (gross) logarithmic sobolev inequality on rn. for details about logarithmic-sobolev inequalities see for instance [9, 10, 22]. in the same vein our dual entropy also yields a version of logarithmic sobolev inequality. (this will not be discussed here). remark 4.5. note that fij − √ π �t gij ≥ 0 =⇒ ∆f ≥ n √ π �t which in turns =⇒ |∇u|2 u2 − ut u ≥ n √ π �t . it turns out that w�(f,t) being finite with u being the heat kernel, also has strong topological and geometric consequences. for instance, in the case m has nonnegative curvature, it implies that m has finite fundamental group. in fact one can show that m is of maximum volume growth if and only if the entropy w�(f,t) is uniformly bounded for all t ≥ 0, where u is the heat kernel. this analogy was originally discovered in [20] for ancient solution to the ricci flow with bounded nonnegative curvature, where perelman claims that ancient solution to the ricci flow with nonnegative curvature operator is κ-noncollapsed if and only if the entropy is uniformly bounded for any fundamental solution to the conjugate heat equation. lastly, in this subsection we make some comment to show how sharp the dual entropy for the heat equation. recall (4.9) w�(f,t) = ∫ m [�2t 4π |∇f|2 + f + n 2 ln (4π �2 ) − n�2 4π ] hdv with f = − ln(4πt) n 2 h and ∫ m hdv = 1 and 0 < �2 ≤ 4π. rewrite w�(f,t) as (4.10) w�(f,t) = �2 4π ∫ m (t|∇f|2 + f −n)hdv + (1 − �2 4π ) ∫ m fhdv + n 2 ln 4π �2 ∫ m hdv. hence, we have the following proposition 4.6. for 0 < �2 ≤ 4π, f = − ln(4πt) n 2 h with ∫ m hdv = 1, we have the following monotonicity formula on a manifold with nonnegative ricci curvature; (4.11) d dt w�(f,t) ≤− �2 2π t ∫ m ( |fij − 1 2t gij|2 + rijfifj ) hdv. proof. the proof follows from a straight forward computation on w� using the idea of [17, theorem 1.1]. (4.12) d dt w�(f,t) = �2 4π ∂ ∂t (∫ m t|∇f|2 + f −n ) hdv + (1 − �2 4π ) ∂ ∂t (∫ m fhdv ) . linear heat equation on static manifold 15 we are only left to justify the non-positivity of ∂ ∂t (∫ m fhdv ) . then we have by integration by parts ∂ ∂t (∫ m fhdv ) = ∫ m ( ∂ ∂t f h + f ∂ ∂t h ) dv = ∫ m ( ∂ ∂t f h + f∆h + f( ∂ ∂t − ∆)h ) dv = ∫ m ( ∂ ∂t + ∆ ) fhdv = ∫ m (2∆f −|∇f|2 − n 2t )hdv, where we have used the facts that ( ∂ ∂t −∆)h = 0 and ∂ ∂t f = ∆f−|∇f|2− n 2t . taking f = − ln((4πt) n 2 h, then the integrand in the rhs of the last equality becomes (4.13) 2∆f −|∇f|2 − n 2t = |∇h|2 h2 − 2∆h h − n 2t ≤ 0, which is precisely the li-yau harnack inequality since we are on nonnegative ricci curvature manifold. hence our claim. � 5. lyh gradient estimates for positive solutions in the next we give useful estimates found by hamilton [11]. he was inspired by the results of li and yau [15], hence the estimates are popularly referred to as li-yau-hamilton (lyh) estimates. we state and prove the result for bounded solutions on a closed manifold. as an application of this lyh-type estimates we can obtain a sharp upper bound on the heat kernel. theorem 5.1. let (m,g) be a closed riemannian manifold with rij ≥ −kgij, where k ≥ 0. suppose u is a positive solution to the heat equation with u ≤ m < ∞. then (5.1) t |∇u|2 u2 ≤ (1 + 2kt) log (m u ) . proof. let f = log u so that |∇f|2 = |∇ log u|2 = |∇u| 2 u2 and ( ∂ ∂t − ∆ ) f = |∇f|2. define a heat type operator l := ( ∂ ∂t − ∆ −〈∇f,∇·〉 ) . the idea to this proof is to apply the heat-type operator l on the quantity t |∇u|2 u2 − (1 + 2kt) log (m u ) and then use weak maximum principle. recall from the calculation in lemma 2.1 and the bochner identity that ∂ ∂t |∇f|2 = 2fifti & ∆|∇f|2 = 2|fij|2 + 2fjfjji + 2rijfifj. hence ( ∂ ∂t − ∆ ) |∇f|2 = −2|fij|2 − 2rijfifj + 2〈∇f,∇|∇f|2〉 16 abimbola abolarinwa and ( ∂ ∂t − ∆ ) (t|∇f|2) = |∇f|2 + t( ∂ ∂t − ∆)|∇f|2 = |∇f|2 − 2t|fij|2 − 2trijfifj + 2t〈∇f,∇|∇f|2〉. using the condition that rij ≥−kgij, we have (5.2) ( ∂ ∂t − ∆ ) (t|∇f|2) = (1 + 2kt)|∇f|2 + 2〈∇f,∇(t|∇f|2)〉. on the other hand l ( (1 + 2kt) log (m u )) = 2k log (m u ) + (1 + 2kt)l ( log (m u )) = 2k log (m u ) + ( ∂ ∂t − ∆ ) log (m u ) − 2(1 + 2kt)〈f,∇ log (m u ) 〉. computing ( ∂ ∂t − ∆ ) log (m u ) = ( ∂ ∂t − ∆ ) log m − ( ∂ ∂t − ∆ ) log u = −|∇f|2 = −2〈∇f,∇ log u ) 〉 + |∇f|2 = 2〈∇f,∇ log (m u ) 〉 + |∇f|2. then l ( (1 + 2kt) log (m u )) = 2k log (m u ) + 2〈∇f,∇ log (m u ) 〉 + |∇f|2 − 2(1 + 2kt)〈f,∇ log (m u ) 〉 = 2k log (m u ) + 2〈∇f,∇((1 + 2kt) log (m u ) 〉(5.3) + (1 + 2kt)|∇f|2.(5.4) combining the expressions in (5.2) and (5.3) we arrive at l ( t|∇f|2 − (1 + 2kt) log (a u )) ≤−2k log (m u ) (5.5) since k ≥ 0 and 0 ≤ log m u < ∞. note that at t = 0, − log (a u ) ≤ 0 and t|∇f|2 − (1 + 2kt) log (a u ) ≤ 0. hence, by the weak maximum principle we have t|∇f|2 − (1 + 2kt) log (a u ) ≤ 0 for all t ≥ 0. this completes the proof. � linear heat equation on static manifold 17 the above result can be extended to the case of complete noncompact manifold, although, a little more effort will be required. the idea here is to use �-regularization method by supposing that the solution u ≥ �, replacing u by u� = u + � for a sufficiently small � > 0 and letting � go to zero after the analysis for u� is completed. an application of this result shows we can bound the maximum of a positive solution by its integral (see [11]). furthermore, the estimate yields sharp lower and upper bounds for the fundamental solution, see [6]. references [1] a. abolarinwa, differential harnack and logarithmic sobolev inequalities along ricciharmonic map flow, to appear [2] a. abolarinwa, analysis of eigenvalues and conjugate heat kernel under the ricci flow, phd thesis, university of sussex, (2014). [3] d. bakry, d. concordet and m. ledoux, optimal heat kernel bounds under logarithmic sobolev inequalities, esaim probab. statist., 1,(1995), 391-407. [4] d. bakry and m. ledoux, a logarithmic sobolev form of the li-yau parabolic inequality, revist. mat. iberoamericana 22, (2006), 683-702. [5] j. cheeger and s-t. yau, a lower bound for the heat kernel, comm. pure appl. math. 34(4)(1981), 465-480. [6] b. chow, s. chu, d. glickenstein, c. guenther, j. idenberd. t. ivey, d. knopf, p. lu, f. luo and l. ni, the ricci flow: techniques and applications. part ii, analytic aspect, ams, providence, ri, (2008). [7] e. b. davies, heat kernel and spectral theory. cambridge university press (1989). [8] n. garofalo and e. lanconelli, asymptotic behaviour of fundamental solutions and potential theory of parabolic operators with variable coefficients, math. ann. 283(2)(1989), 211-239. [9] l. gross logarithmic sobolev inequalities, america j. math 97(1)(1975), 1061-1083. [10] l. gross logarithmic sobolev inequalities and contractivity properties of semigroups, dirichlet form, lecture notes in mathematics volume 1563, (1993), 54-88. [11] r. hamilton, a matrix harnack estimate for the heat equation, commun. anal. geom., 1, (1993), 113-126. [12] g. huang, z. huang, h. li, gradient estimates and differential harnack inequalities for a nonlinear parabolic equation on riemannian manifolds, ann. glob. anal. geom., 23(3) (1993), 209-232. [13] s. kuang, qi s. zhang, a gradient estimate for all positive solutions of the conjugate heat equation under ricci flow, j. funct. anal., 255(4)( 2008), 1008-1023. [14] j. li, x. xu, differential harnack inequalities on riemannian manifolds i: linear heat equation, advances in math., 226 (2011), 4456-4491. [15] p. li, s-t. yau, on the parabolic kernel of the schrödinger operator, acta math. 156 (1986), 153-201 [16] r. müller, differential harnack inequalities and the ricci flow. european mathematics society, (2006). [17] l. ni, the entropy formula for linear heat equation, journal of geom. analysis 14(2)(2004), 86-96 [18] l. ni, addenda to ”the entropy formula for linear heat equation”, journal of geom. analysis 14(2)(2004), 229-334. [19] l. ni, a note on perelman’s li-yau-hamilton inequality, comm. anal. geom 14(2006), 883-905. [20] g. perelman, the entropy formula for the ricci flow and its geometric application, arxiv:math.dg/0211159v1 (2002). [21] p. souplet, qi s. zhang, sharp gradient estimate and yaus liouville theorem for the heat equation on noncompact manifolds, bull. london math. soc. 38(2006), 1045-1053. [22] f. b. weissler, logarithmic sobolev inequalities for the heat-diffusion semigroup, trans am math. soc., 237(1978), 255-269. department of mathematics, university of sussex, brighton, bn1 9qh, uk int. j. anal. appl. (2023), 21:81 composition operators on nk(p,q)-type spaces on the unit ball h. gissyr, m. a. bakhit∗ department of mathematics, faculty of science, jazan university, jazan 45142, saudi arabia ∗corresponding author: mabakhit2020@hotmail.com abstract. we describe the boundedness and compactness of the composition operators cϕ acting in nk(p,q) on the open unit ball b. 1. introduction for the unit ball b of cn, hol(b) denotes the class of all holomorphic functions on b while h∞ = h∞(b) denotes the class of all functions that are holomorphic u ∈hol(b) equipped with the norm ‖u‖∞ = sup ζ∈b |u(ζ)|. for any d > 0, the weighted banach space h∞d = h ∞ d (b) consists of all functions u ∈hol(b) such that ‖u‖∞d := sup ζ∈b (1 −|ζ|)d|u(ζ)| < ∞. the space h∞d,0 = h ∞ d,0(b) indicate the closed subspace of h ∞ d such that lim |ζ|→1 |u(ζ)|(1 −|ζ|)d = 0. for further details about the properties of h∞d spaces see [10]). for ζ ∈b, we let dv be the lebesgue measure on b with v (b) = ∫ b dv (ζ) = 1. received: jun. 27, 2023. 2020 mathematics subject classification. 32a36, 46g20, 58b12, 47b33. key words and phrases. bergman spaces; nk(p,q)-type spaces; several complex variables; composition operators. https://doi.org/10.28924/2291-8639-21-2023-81 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-81 2 int. j. anal. appl. (2023), 21:81 in addition, we let dω be the surface measure on s, normalized so that ω(s) ≡ 1. if u is a nonnegative lebesgue measurable function on b, then the measures v and ω are related by∫ b u(ζ)dv (ζ) = 2n ∫ 1 0 t2n−1dt ∫ s u(tζ)dω(ζ). moreover, the formulas for integration on s (see, [11]) as:∫ s udω = ∫ s dω(ζ) 1 2π ∫ 2π 0 u(eiθζ)dθ, for all 0 ≤ θ ≤ 2π. for any ψ ∈ aut(b),u ∈ l1(b), the möbius invariant on b (see e.g., [5]) such that∫ b u(ζ)dλ(ζ) = ∫ b u ◦ψ(ζ) dv (1 −|ζ|2)n+1 . the inner product of ζ = (ζ1, . . . ,ζn) and η = (η1, . . . ,ηn) in cn, is given by 〈ζ,η〉 = n∑ i=1 ζiηi. for any ζ ∈b, we define the complex gradient and the radial derivative of the function u ∈hol(b) respectively as follows: ∇u(ζ) = ( ∂u ∂ζ1 (ζ), · · · , ∂u ∂ζn (ζ) ) , ru(ζ) = 〈∇u(ζ),ζ〉 = n∑ i=1 ζi ∂u ∂ζ1 (ζ). we know the bloch space bd = bd(b) is the banach space of functions u ∈ hol(b) such that ru ∈ h∞d which has the norm ‖u‖bd := |f (0)| + ‖ru‖ ∞ d . the involution automorphisms ψb (the möbius transformation of b) is define for ζ ∈ b and b ∈b−{0} as ψb(ζ) = b− 〈ζ,b〉b|b|2 − √ 1 −|b|2 ( ζ − 〈ζ,b〉b|b|2 ) 1 −〈ζ,b〉 , where ψ0(ζ) = −ζ, ψb(0) = b, ψb(b) = 0 and ψb = ψ−1b . it is well known that for any ζ ∈b 1 −|ψb(ζ)|2 = (1 −|b|2)(1 −|ζ|2) |1 −〈b,ζ〉|2 . the bergman metric and the bergman metric ball on b, for ζ,η ∈b and m > 0 as follows: β(ζ,η) = 1 2 log 1 + |ψζ(η)| 1 −|ψζ(η)| , d(ζ,m) = {η ∈b : β(ζ,η) < m}. int. j. anal. appl. (2023), 21:81 3 let rc+ denote the set of all right-continuous nondecreasing functions k 6= 0 and k : [0,∞) → [0,∞). for k ∈ rc+ and p,q > 0, the weighted banach type spaces nk(p,q) = nk(p,q)(b) consists of functions u ∈hol(b) such that nk(p,q) := {u ∈ h(b) : ‖u‖ p nk(p,q) < ∞}, where ‖u‖pnk(p,q) = sup b∈b ∫ b |u(ζ)|p(1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ). this space was introduced first by bakhit and aljuaid in [1] who study several fundamental properties of nk(p,q)-type spaces and its closed subspaces nk,0(p,q), which are banach spaces of functions that are analytic and their norms determined by a weighted function k ∈rc+, together with a möbius transformation. also in [1] the authors show that the norm of nk(p,q)-type space is equivalent to the norm ‖u‖pnk(p,q) = sup b∈b ∫ b |u(ζ)|p(1 −|ζ|2)qk ( g(b,ζ) ) dv (ζ) < ∞, where g(b,ζ) = log 1|ψb(ζ)|. we set the integral jk,q(t) with q > n as: jk,q(t) = ∫ 1 0 t2n−1 (1 − t2)n+1−q k ( (1 − t2)n ) dt. (1.1) throughout the paper, we suppose that jk,q(t) < ∞, then nk(p,q) contain all the polynomials, otherwise nk(p,q) consists only of zero functions. let x and y be two function spaces on b and consider ϕ be a holomorphic self-map of b. we define the composition operator cϕ : x →y by cϕ(u)(ζ) = u ◦ϕ, ∀u ∈x . recall that, for any two normed linear spaces x and y , the linear operator t : x −→ y is said to be bounded if there exists c > 0 such that ‖tu‖y ≤ c‖u‖x,∀u ∈ x. furthermore, a linear operator t : x −→ y is said to be compact if it maps every bounded set in x to a relatively compact set in y (i.e., a set whose closure is compact) (see e.g., [12]). studying the composition operators acting in different spaces is a quite classical topic since they arise in different problems; see the excellent monographs [2], [3] and [4]. some of the earlier study on this topic is reflected in [9] descriptions of bounded and compact composition operators on f (p,q,s) spaces were provided [8]. this paper is organized as follows: in section 2 we shortly give the preliminaries and background information. in section 3 we establish proving our main results respectively. we use the notation a . b in what follows to mean that there is a constant c > 0 with a ≤ cb. and the notation a � b means that a . b and b . a. 4 int. j. anal. appl. (2023), 21:81 2. preliminaries for 0 < t < ∞, we use the auxiliary function φk(t) = sup s∈(0,1] k(st) k(s) (see e.g., [6], [7]). the following constraints on φk(t) play a significant role in the study of any class of nk(p,q) spaces: jk(t) = ∫ 1 0 φk(t) dt t < ∞, (2.1) and ∫ ∞ 1 φk(t) dt t2 < ∞, (2.2) and more generally, ∫ ∞ 1 φk(t) dt t1+% < ∞, % > 0. (2.3) in the case that k satisfies condition (2.1), then k(2t) . k(t) ∀ 0 ≤ 2t ≤ 1. if we started with the property that k(t) = k(1) for t ≥ 1, then k(2t) ≈ k(t) for t > 0 (see, [6]). the following results will have an important role in the subsequent. the following lemma was proven in [1]. lemma 2.1. let k ∈rc+, p ≥ 1 and q > 0 then • nk(p,q) ⊆ h∞q/p(b). • nk(p,q) = h∞q/p(b) if ik(t) = ∫ 1 0 t2n−1 (1 − t2)n+1 k ( (1 − t2)n ) dt < ∞. (2.4) we can find the subsequent result in [11]. lemma 2.2. let δ ∈ (0, 1] then there is a sequence {ni}∈b such that • limi→∞ |ni| = 1. • b = ⋃∞ i=1 d(mi,δ). • let n > 0 be an integer, then ζ ∈ ⋂n+1 k=1 d(mik, 4δ) and mik ∈ d(ζ, 4δ) for each ζ ∈ b, 1 ≤ k ≤ n + 1. lemma 2.3. for any k ∈ rc+,δ > 0, let p,q > 0 and ζ,b ∈ b. then there is a positive constant c, such that |u(ζ)|p ≤ (1 −|z|2)−q−n−1 k ( (1 −|ψb(ζ)|2)n ) ∫ d(z,2δ) |u(w)|p(1 −|w|2)qk(1 −|ψb(w)|2)dv (w), for all ζ ∈ d(z,δ) and u ∈hol(b). proof. by the result in lemma 2.24 in [5], we obtain |u(ζ)|p ≤ 1 (1 −|ζ|2)n+1 ∫ d(ζ,δ) |u(w)|pdv (w), for all ζ ∈b and u ∈hol(b). int. j. anal. appl. (2023), 21:81 5 now let ζ ∈ d(z,δ) and w ∈ d(ζ,δ), then obtain β(w,z) ≤ β(w,ζ) < 2δ. thus, d(ζ,δ) ⊂ d(z, 2δ). from some results in [5], we obtain 1 −|ζ|2 � 1 −|z|2 � 1 −|w|2, |1 −〈b,w〉| � |1 −〈b,z〉|. thus, |u(ζ)|p ≤ (1 −|z|2)−q−n−1 k ( (1 −|ψb(ζ)|2)n ) ∫ d(z,2δ) |u(w)|p(1 −|w|2)qk ( (1 −|ψb(w)|2)n ) dv (w). � lemma 2.4. let φ be a holomorphic self-map of b and b ∈ b. if u is a nonnegative lebesgue measurable function on b, then∫ b u(ζ) dλk,q,ϕ(ζ) = ∫ b u(ϕ(ζ)) (1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ), where λk,q,ϕ = ∫ ϕ−1(e) (1 −|ζ|2)q k ( (1 −|ψb(ζ)|2)n ) dv (ζ), for any borel measurable set e ⊆b. proof. let u be a nonnegative simple lebesgue measurable function. assume that u(ζ) = n∑ i=1 bi κei, where ei is the measurable set on b. then,∫ b u(ζ)dλk,q,ϕ(ζ) = n∑ i=1 biλk,q,ϕ(ei ) = n∑ i=1 bi ∫ ei dλk,q,ϕ(ζ) = n∑ i=1 bi ∫ φ−1(ei ) (1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) = ∫ b ( n∑ i=1 biκφ−1(ei )∩b ) (1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) = ∫ b u(ϕ(ζ))(1 −|ζ|2)q k ( (1 −|ψb(ζ)|2)n ) dv (ζ). if u is a nonnegative lebesgue measurable function, for ζ ∈ b then there is a monotone increasing simple measurable function sequence {uj} such that lim j→∞ uj(ζ) = u(ζ). thus, lim j→∞ ∫ b uj(ζ) dλk,q,ϕ(ζ) = ∫ b u(ζ) dλk,q,ϕ(ζ). 6 int. j. anal. appl. (2023), 21:81 now let the function sequence {uj(k,q,φ)} = {uj(ϕ(ζ))(1 − |ζ|2)q k ( (1 − |ψb(ζ)|2)n ) }, then {uj(k,q,φ)} is a monotone increasing measurable function. moreover, lim j→∞ uj(k,q,φ) = u(ϕ(ζ))(1 −|ζ|2)q k ( (1 −|ψb(ζ)|2)n ) , which implies that ∫ b u(ζ) dλk,q,ϕ(ζ) = lim j→∞ ∫ b uj(ζ)dλk,q,ϕ(ζ) = lim j→∞ ∫ b uj(ϕ(ζ))(1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) = lim j→∞ ∫ b u(ϕ(ζ))(1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ). this completes the proof. � lemma 2.5. for k ∈rc+ and p > 0,q + n + 1 > 0. if (2.4) holds, then uw (ζ) ∈nk(p,q), where uw (ζ) = (1 −|w|2) (1 −〈ζ,w〉) q+n+1 p +1 . proof. firstly, suppose that (2.4) holds, to show that uw (ζ) ∈nk(p,q), it suffices to show that there is ε > 0, such that sup b∈b ∫ b (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p k ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ ε, ∀ z ∈b. now we let 1√ 2 < |ψb(ζ)| < 1, by the fact that (1 −|ζ|) ≤ |1 −〈ζ,b〉| and theorem 1.4.10 in [5], therefore ∫ 1√ 2 <|ψb(ζ)|<1 (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p k ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ ε ∫ b (1 −|ζ|2)−n−1k ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ ε ∫ 1 0 t2n−1 (1 − t2)n+1 k ( (1 − t2)n ) dt < ε. (2.5) at the same time, ∫ |ψb(ζ)|≤ 1√2 (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p k ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ ∫ |w|≤1 2 (1 −|z|2)p(1 −|ψb(w)|2)q(1 −|b|2)n+1 |1 −〈ψb(w),z〉|n+1+q+p|1 −〈w,b〉|2n+2 k ( (1 −|w|2)n ) dv (w) int. j. anal. appl. (2023), 21:81 7 ≤ ε ∫ |w|≤1 2 (1 −|b|2)n+1 (1 −|ψb(w)|2)n+1|1 −〈w,b〉|2n+2 k ( (1 −|w|2)n ) dv (w) ≤ ε ∫ |w|≤1 2 k ( (1 −|w|2)n ) dv (w) |1 −〈w,b〉|n+1 ≤ ε ∫ |w|≤1 2 k ( (1 −|w|2)n ) dv (w) (1 −|w|2)n+1 ≤ ε ∫ b k ( (1 −|w|2)n ) dv (w) < ε. (2.6) combining (2.5) and (2.6), it follows that sup b∈b ∫ b (1 −|z|2)p(1 −|ζ|2)q |1 −〈ζ,z〉|n+1+q+p k ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ ε, ∀ z ∈b. � 3. main results 3.1. boundedness. theorem 3.1. let k ∈ rc+ and 0 < p,q < ∞. then the operator cϕ is bounded on nk(p,q) if and only if sup w,b∈b (1 −|w|2)p (∫ b (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 k(1 −|ψb(ζ)|2)dv (ζ) ) < ∞. (3.1) proof. let cϕ be the bounded operator in nk(p,q). consider the function uw (ζ) = (1 −|w|2) (1 −〈ζ,w〉) q+n+1 p +1 . then by lemma 2.5, we obtain∫ b |uw (ζ)|p(1 −|ζ|2)qk(1 −|ψb(ζ)|2) dv (ζ) ≤ ∫ b (1 −|w|2)p(1 −|ζ|2)q |1 −〈ζ,w〉|p+q+n+1 k(1 −|ψb(ζ)|2)dv (ζ) ≤ ε, which exactly ‖cϕ(uw )‖nk(p,q) ≤‖cϕ‖‖uw‖nk(p,q) ≤ ε 1 p‖cϕ‖. that is sup w,b∈b (1 −|w|2)p ∫ b (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 k(1 −|ψb(ζ)|2)dv (ζ) ≤ ε‖cϕ‖p. conversely, suppose that (3.1) holds, then by lemma (2.3), there exists a constant ε such that (1 −|w|2)p k(1 −|ψb(w)|2) ∫ b dλk,q,ϕ(ζ) |1 −〈ζ,w〉|q+n+1 ≤ ε, ∀ w,b ∈b, 8 int. j. anal. appl. (2023), 21:81 where λk,q,ϕ = ∫ ϕ−1(e) (1 −|ζ|2)q k ( (1 −|ψb(ζ)|2)n ) dv (ζ), ∀ e ∈b. fixed δ > 0, so that (1 −|w|2)p k(1 −|ψb(w)|2) ∫ d(w,δ) dλk,q,ϕ(ζ) |1 −〈ζ,w〉|q+n+1 ≤ ε, ∀ w,b ∈b. then, we have λk,q,ϕ(d(w,δ)) . (1 −|w|2)q+n+1k(1 −|ψb(w)|2). if u ∈nk(p,q), then∫ b |u(ϕ(ζ))|p(1 −|ζ|2)qk(1 −|ψb(ζ)|2)dv (ζ) = ∫ b |u(ζ)|pdλk,q,ϕ(ζ) ≤ ∞∑ j=1 ∫ d(wj,δ) |u(ζ)|pdλk,q,ϕ(ζ) ≤ ∞∑ j=1 sup ζ∈d(wj,δ) |u(ζ)|p ∫ d(wj,δ) dλk,q,ϕ(ζ) . ∞∑ j=1 sup ζ∈d(wj,δ) |u(ζ)|p{(1 −|wj|2)q+n+1k(1 −|ψb(wj)|2)} . ∞∑ j=1 ∫ d(wj,4δ) |u(ζ)|p(1 −|ζ|2)qk(1 −|ψb(ζ)|2)dv (ζ) . ‖u‖qnk(p,q). � 3.2. compactness. theorem 3.2. let k ∈ rc+ and 0 < p,q < ∞. then the operator cϕ is compact on nk(p,q) if and only if lim |w|→1− sup b∈b (1 −|w|2)p (∫ b (1 −|ζ|2)q |1 −〈ϕ(ζ),w〉|q+n+1 k(1 −|ψb(ζ)|2)dv (ζ) ) = 0. (3.2) proof. let cϕ be compact on nk(p,q). then, for any sequence {ξj}⊂b with limj→∞ |ξj| = 1. take hj(ζ) = (1 −|ξj|) (1 −〈ζ,ξj〉) q+n+1 p . since {hj} is bounded on nk(p,q) and converges uniformly to 0 on any compact subset of b. so, by the compactness of cϕ, we obtain (1 −|w|2)p ∫ b (1 −|ζ|2)qk(1 −|ψb(ζ)|2) dv (ζ) |1 −〈ϕ(ζ),w〉|q+p+1 = ‖cϕ(hj)‖ p nk(p,q) → 0, as j →∞. int. j. anal. appl. (2023), 21:81 9 conversely, assume that (3.2) holds. then, we can choose the sequence {wi} ∈ b from lemma (2.2), such that sup b∈b (1 −|wi|2)p k(1 −|ψb(wi )|2) ∫ b dλk,q,ϕ(ζ) |1 −〈ζ,wi〉|q+n+p+1 → 0, as i → 0. thus, for any � > 0, there exists a positive integer n0 such that sup b∈b (1 −|wi|2)p k(1 −|ψb(wi )|2) ∫ b dλk,q,ϕ(ζ) |1 −〈ζ,wi〉|q+n+p+1 < �, when i > n0. (3.3) in this case, by (3.3) for all a ∈b when j > n0, we have λk,q,ϕ(d(wi,δ) . � p(1 −|w|2)q+n+p+1k(1 −|ψb(ζ)|n). (3.4) now we let {uj} be any sequence that converges to 0 uniformly on any compact subset of b with ‖uj‖nk(p,q) ≤ c. then, the sequence {uj} converges to 0 uniformly on m = ⋃n0 k=1 d(wk,δ). thus, there exists a positive integer n0 such that sup ζ∈m |uj(ζ)| < � when j > n0. (3.5) otherwise, λk,q,ϕ(b) ≤ ∫ b (1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) ≤ c. (3.6) therefore, when j > n0, by lemma 2.2-2.4, (3.4)-(3.6), for all a ∈b we have∫ b |uj(ϕ(ζ))|p(1 −|w|2)qk(1 −|ψb(ζ)|n)dv (ζ) = ∫ b|uj(ζ)|pdλk,q,ϕ ≤ ∞∑ k=1 ∫ d(wk,δ) |uj(ζ)|pdλk,q,ϕ ≤ n0∑ k=1 ∫ d(wk,δ) |uj(ζ)|pdλk,q,ϕ + ∞∑ k=n0+1 sup ζ∈d(wk,δ) |uj(ζ)|pλk,q,ϕ . n0 � p λk,q,ϕ(b) + �p ∞∑ k=n0+1 sup ζ∈d(wk,δ) |uj(ζ)|p(1 −|ζ|2)q+n+1k ( (1 −|ψb(ζ)|2)n ) . n0 � p λk,q,ϕ(b) + �p ∫ d(wk,4δ) |uj|p(1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) . n0 � p λk,q,ϕ(b) + �p ∫ b |uj|p(1 −|ζ|2)qk ( (1 −|ψb(ζ)|2)n ) dv (ζ) . n0 � p λk,q,ϕ(b) + �p‖uj‖nk(p,q) . � p, which exactly lim k→∞ ‖cϕ(uj)‖nk(p,q) = 0. in this case, the operator cϕ is compact on nk(p,q), which completed the proof. � 10 int. j. anal. appl. (2023), 21:81 data availability: the research conducted in this paper does not make use of separate data. acknowledgments: the authors extend their appreciation to the deanship of scientific research, jazan university, for supporting this research work through the research units support program, support number: rup2-02. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m.a. bakhit, m. aljuaid, basic properties of nk(p,q) type spaces with hadamard gap series in the unit ball of cn, res. math. 9 (2022), 2091724. https://doi.org/10.1080/27684830.2022.2091724. [2] j. dai, c. ouyang, composition operators between bloch type spaces in the unit ball, acta math. sci. 34 (2014), 73-81. https://doi.org/10.1016/s0252-9602(13)60127-7. [3] j.m. fan, y.f. lu, y.x. yang, difference of composition operators on some analytic function spaces, acta. math. sin.-english ser. 37 (2021), 1384-1400. https://doi.org/10.1007/s10114-021-0480-9. [4] j.h. shapiro, composition operators and classical function theory, springer new york, 1993. https://doi.org/ 10.1007/978-1-4612-0887-7. [5] w. rudin, function theory in the unit ball of cn, springer, berlin, 2008. https://doi.org/10.1007/ 978-3-540-68276-9. [6] h. wulan, j. zhou, qk type spaces of analytic functions, j. funct. spaces appl. 4 (2006), 73-84. https: //doi.org/10.1155/2006/910813. [7] h. wulan, and j. zhou, the higher order derivatives of qk type spaces, j. math. anal. appl. 332 (2007), 1216-1228. https://doi.org/10.1016/j.jmaa.2006.10.082. [8] s. xu, x. zhang, composition operator on f(p,q,s) spaces in the unit ball of cn, complex var. elliptic equ. (2022). https://doi.org/10.1080/17476933.2022.2142783. [9] x. zhang, s. xu, s. li and j. xiao, composition operators on f(p,q,s) type spaces in the unit ball of cn, complex anal. oper. theory. 12 (2016), 141-154. https://doi.org/10.1007/s11785-016-0610-z. [10] x. zhu, products of differentiation, composition and multiplication from bergman type spaces to bers type spaces, integral transforms spec. funct. 18 (2007), 223-231. https://doi.org/10.1080/10652460701210250. [11] k. zhu, spaces of holomorphic functions in the unit ball, springer, new york, 2005. https://doi.org/10.1007/ 0-387-27539-8. [12] k. zhu, operator theory in function spaces, second edition, mathematical surveys and monographs, vol. 138, american mathematical society, providence, rhode island, 2007. https://doi.org/10.1080/27684830.2022.2091724 https://doi.org/10.1016/s0252-9602(13)60127-7 https://doi.org/10.1007/s10114-021-0480-9 https://doi.org/10.1007/978-1-4612-0887-7 https://doi.org/10.1007/978-1-4612-0887-7 https://doi.org/10.1007/978-3-540-68276-9 https://doi.org/10.1007/978-3-540-68276-9 https://doi.org/10.1155/2006/910813 https://doi.org/10.1155/2006/910813 https://doi.org/10.1016/j.jmaa.2006.10.082 https://doi.org/10.1080/17476933.2022.2142783 https://doi.org/10.1007/s11785-016-0610-z https://doi.org/10.1080/10652460701210250 https://doi.org/10.1007/0-387-27539-8 https://doi.org/10.1007/0-387-27539-8 1. introduction 2. preliminaries 3. main results 3.1. boundedness 3.2. compactness references international journal of analysis and applications issn 2291-8639 volume 6, number 1 (2014), 63-81 http://www.etamaths.com existence of multiple positive solutions for p-laplacian fractional order boundary value problems k. r. prasad1 and b. m. b. krushna2,∗ abstract. this paper deals with the existence of at least one and multiple positive solutions for p-laplacian fractional order two-point boundary value problems, d q2 0+ ( φp ( d q1 0+ y(t) )) = f(t,y(t)), t ∈ (0, 1), y(j)(0) = 0, j = 0, 1, 2, · · ·,n− 2, y(q4)(1) = 0, φp ( d q1 0+ y(0) ) = 0 = d q3 0+ ( φp ( d q1 0+ y(1) )) , where q2 ∈ (1, 2],q1 ∈ (n − 1,n],n ≥ 2,q3 ∈ (0, 1], q4 ∈ [1,q1 − 1] is a fixed integer, φp(y) = |y|p−2y, p > 1, φ−1p = φq, 1/p + 1/q = 1, by applying krasnosel’skii and five functionals fixed point theorems. 1. introduction the goal of differential equations is to understand the phenomena of nature by developing mathematical models. fractional calculus is the field of mathematical analysis, which deals with investigation and applications of derivatives and integrals of an arbitrary order. among all, a class of differential equations governed by nonlinear differential operators appears frequently and generated great deal of interest in studying such problems. in this theory, the most applicable operator is the classical p-laplacian, given by φp(y) = |y|p−2y, p > 1. these types of problems arise in applied fields such as turbulent flow of a gas in a porous medium, modeling of viscoelastic flows, biophysics, plasma physics and material science. the existence of positive solutions of boundary value problems (bvps) associated with integer order differential equations were studied by many authors [11, 1, 9, 2, 19] and extended to p-laplacian boundary value problems [17, 14, 4, 23]. later, these results are further extended to fractional order boundary value problems [6, 10, 5, 7, 8, 20, 21, 22] by utilizing various fixed point theorems on cones. in recent years, researchers are concentrating on the theory of fractional order boundary value problems related with p-laplacian operator. 2010 mathematics subject classification. 26a33, 34b18, 35j05. key words and phrases. fractional derivative, p-laplacian, boundary value problem, twopoint, green’s function, positive solution. c©2014 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 63 64 prasad and krushna in 2012, chai [7] investigated the existence and multiplicity of positive solutions for a class of boundary value problem of fractional differential equation with plaplacian operator, d β 0+(φp(d α 0+u(t))) + f(t,u(t)) = 0, 0 < t < 1, u(0) = 0,u(1) + σd γ 0+u(1) = 0,d α 0+u(0) = 0, where 1 < α ≤ 2, 0 < β,γ ≤ 1, 0 ≤ α−γ−1,σ is a positive number, dα 0+ ,d β 0+ ,d γ 0+ are the standard riemann-liouville fractional order derivatives. the purpose of this paper is to establish the existence of at least one and multiple positive solutions for p-laplacian fractional order boundary value problems, (1.1) d q2 0+ ( φp ( d q1 0+ y(t) )) = f(t,y(t)), t ∈ (0, 1), (1.2) y(j)(0) = 0, j = 0, 1, 2, · · ·,n− 2, y(q4)(1) = 0, φp ( d q1 0+ y(0) ) = 0 = d q3 0+ ( φp ( d q1 0+ y(1) )) ,   where q2 ∈ (1, 2],q1 ∈ (n− 1,n],n ≥ 2,q3 ∈ (0, 1], q4 ∈ [1,q1 − 1] is a fixed integer, φp(y) = |y|p−2y, p > 1, φ−1p = φq, 1/p + 1/q = 1, f : [0, 1] × r + → r+ is continuous and d qi 0+ , for i = 1, 2, 3 are the standard riemann-liouville fractional order derivatives. define the nonnegative extended real numbers f0,f 0,f∞ and f ∞ by f0 = lim y→0+ min t∈[0,1] f(t,y) φp(y) , f0 = lim y→0+ max t∈[0,1] f(t,y) φp(y) , f∞ = lim y→∞ min t∈[0,1] f(t,y) φp(y) and f∞ = lim y→∞ max t∈[0,1] f(t,y) φp(y) , and also assume that they will exist. the rest of the paper is organized as follows. in section 2, we construct the green functions for the homogeneous bvps corresponding to (1.1)-(1.2) and estimate the bounds for the green functions. in section 3, we establish criteria for the existence of at least one positive solution for p-laplacian fractional order bvp (1.1)-(1.2), by using krasnosel’skii fixed point theorem. in section 4, we derive sufficient conditions for the existence of at least three positive solutions for the p-laplacian fractional order bvp (1.1)-(1.2), by applying five functionals fixed point theorem. we also establish the existence of at least 2k−1 positive solutions for an arbitrary positive integer k. in section 5, as an application, the results are demonstrated with examples. 2. green functions and bounds in this section, we construct the green functions for the homogeneous bvps and estimate the bounds for the green functions, which are essential to establish the main results. let g(t,s) be the green’s function for the homogeneous bvp, (2.1) −dq1 0+ y(t) = 0, t ∈ (0, 1), (2.2) y(j)(0) = 0, j = 0, 1, · · ·,n− 2, y(q4)(1) = 0. p-laplacian fractional order boundary value problems 65 lemma 2.1. let h(t) ∈ c[0, 1]. then the fractional order differential equation, (2.3) d q1 0+ y(t) + h(t) = 0, t ∈ (0, 1), satisfying (2.2) has a unique solution, y(t) = ∫ 1 0 g(t,s)h(s)ds, where (2.4) g(t,s) = { g1(t,s), 0 ≤ t ≤ s ≤ 1, g2(t,s), 0 ≤ s ≤ t ≤ 1, g1(t,s) = tq1−1(1 −s)q1−q4−1 γ(q1) , g2(t,s) = tq1−1(1 −s)q1−q4−1 − (t−s)q1−1 γ(q1) . proof. let y(t) ∈ c[q1]+1[0, 1] be the solution of fractional order differential equation (2.3) satisfying (2.2). then y(t) = −1 γ(q1) ∫ t 0 (t−s)q1−1h(s)ds + c1tq1−1 + c2tq1−2 + · · · + cntq1−n. from (2.2), ci = 0, i = 2, 3, · · ·,n and c1 = ∫ 1 0 (1−s)q1−q4−1 γ(q1) h(s)ds. thus, the unique solution of (2.3) with (2.2) is y(t) = −1 γ(q1) ∫ t 0 (t−s)q1−1h(s)ds + tq1−1 γ(q1) ∫ 1 0 (1 −s)q1−q4−1h(s)ds = ∫ 1 0 g(t,s)h(s)ds, where g(t,s) is the green’s function and given in (2.4). � lemma 2.2. let z(t) ∈ c[0, 1]. then the fractional order differential equation, (2.5) d q2 0+ ( φp ( d q1 0+ y(t) )) = z(t), t ∈ (0, 1), satisfying (2.6) φp ( d q1 0+ y(0) ) = 0,d q3 0+ ( φp ( d q1 0+ y(1) )) = 0, has a unique solution, y(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)z(τ)dτ ) ds, where (2.7) h(t,s) = { tq2−1(1−s)q2−q3−1 γ(q2) , 0 ≤ t ≤ s ≤ 1, tq2−1(1−s)q2−q3−1−(t−s)q2−1 γ(q2) , 0 ≤ s ≤ t ≤ 1. here h(t,s) is the green’s function for −dq2 0+ ( φp(x(t)) ) = 0, t ∈ (0, 1), φp(x(0)) = 0, d q3 0+ ( φp(x(1)) ) = 0. 66 prasad and krushna proof. an equivalent integral equation for (2.5) is given by φp ( d q1 0+ y(t) ) = 1 γ(q2) ∫ t 0 (t− τ)q2−1z(τ)dτ + c1tq2−1 + c2tq2−2. by (2.6), one can determine c2 = 0 and c1 = −1 γ(q2) ∫ 1 0 (1 − τ)q2−q3−1z(τ)dτ. then, φp ( d q1 0+ y(t) ) = 1 γ(q2) ∫ t 0 (t− τ)q2−1z(τ)dτ − tq2−1 γ(q2) ∫ 1 0 (1 − τ)q2−q3−1z(τ)dτ = − ∫ 1 0 h(t,τ)z(τ)dτ. therefore, φ−1p ( φp ( d q1 0+ y(t) )) = −φ−1p (∫ 1 0 h(t,τ)z(τ)dτ ) . consequently, d q1 0+ y(t) + φq (∫ 1 0 h(t,τ)z(τ)dτ ) = 0. hence, y(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)z(τ)dτ ) ds is the solution of fractional order bvp (2.5), (1.2). � lemma 2.3. for t ∈ i = [ 1 4 , 3 4 ] , the green’s function g(t,s) given in (2.4) satisfies the following inequalities (i) g(t,s) ≥ 0, for all (t,s) ∈ [0, 1] × [0, 1], (ii) g(t,s) ≤ g(1,s), for all (t,s) ∈ [0, 1] × [0, 1], (iii) g(t,s) ≥ ξg(1,s), for all (t,s) ∈ i × [0, 1], where ξ = ( 1 4 )q1−1 . proof. the green’s function g(t,s) of (2.1), (2.2) is given in (2.4). for 0 ≤ t ≤ s ≤ 1, g1(t,s) = 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 ] ≥ 0. for 0 ≤ s ≤ t ≤ 1, g2(t,s) = 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 − (t−s)q1−1 ] ≥ 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 − (t− ts)q1−1 ] = tq1−1 γ(q1) [( 1 + δs + q4(q4 + 1) 2 s2 + · · · ) − 1 ] (1 −s)q1−1 = tq1−1 γ(q1) [ q4s + o(s 2) ] (1 −s)q1−1 ≥ 0. hence the green’s function g(t,s) is nonnegative. for 0 ≤ t ≤ s ≤ 1, ∂g1(t,s) ∂t = 1 γ(q1) [ (q1 − 1)tq1−2(1 −s)q1−q4−1 ] ≥ 0. p-laplacian fractional order boundary value problems 67 therefore, g1(t,s) is increasing in t, which implies g1(t,s) ≤ g1(1,s). now, for 0 ≤ s ≤ t ≤ 1, ∂g2(t,s) ∂t = 1 γ(q1) [ (q1 − 1)tq1−2(1 −s)q1−q4−1 − (q1 − 1)(t−s)q1−2 ] ≥ 1 γ(q1) [ (q1 − 1)tq1−2(1 −s)q1−q4−1 − (q1 − 1)(t− ts)q1−2 ] = (q1 − 1)tq1−2 γ(q1) [ 1 − ( 1 − (q4 − 1)s + (q4 − 1)(q4 − 2) 2 s2 + · · · )] (1 −s)q1−q4−1 = (q1 − 1)tq1−2 γ(q1) [ (q4 − 1)s + o(s2) ] (1 −s)q1−q4−1 ≥ 0. therefore, g2(t,s) is increasing in t, which implies g2(t,s) ≤ g2(1,s). let 0 ≤ t ≤ s ≤ 1 and t ∈ i. then g1(t,s) = 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 ] =tq1−1 1 γ(q1) [ (1 −s)q1−q4−1 ] =tq1−1g1(1,s) ≥ (1 4 )q1−1 g1(1,s). let 0 ≤ s ≤ t ≤ 1 and t ∈ i. then g2(t,s) = 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 − (t−s)q1−1 ] ≥ 1 γ(q1) [ tq1−1(1 −s)q1−q4−1 − (t− ts)q1−1 ] =tq1−1g2(1,s) ≥ (1 4 )q1−1 g2(1,s). hence the result. � lemma 2.4. for t,s ∈ [0, 1], the green’s function h(t,s) given in (2.7) satisfies the following inequalities (i) h(t,s) ≥ 0, (ii) h(t,s) ≤ h(s,s). proof. the green’s function h(t,s) is given in (2.7). for 0 ≤ t ≤ s ≤ 1, h(t,s) = 1 γ(q2) [ tq2−1(1 −s)q2−q3−1 ] ≥ 0. 68 prasad and krushna for 0 ≤ s ≤ t ≤ 1, h(t,s) = 1 γ(q2) [ tq2−1(1 −s)q2−q3−1 − (t−s)q2−1 ] ≥ 1 γ(q2) [ tq2−1(1 −s)q2−q3−1 − (t− ts)q2−1 ] = tq2−1 γ(q2) [ (1 −s)−q3 − 1 ] (1 −s)q2−1 = tq2−1 γ(q2) [( 1 + q3s + q3(q3 + 1) 2 s2 + · · · ) − 1 ] (1 −s)q2−1 = tq2−1 γ(q2) [ q3s + o(s 2) ] (1 −s)q2−1 ≥ 0. for 0 ≤ t ≤ s ≤ 1, ∂h(t,s) ∂t = 1 γ(q2) [(q2 − 1)tq2−2(1 −s)q2−q3−1] ≥ 0. therefore, h(t,s) is increasing in t, for s ∈ [0, 1], which implies h(t,s) ≤ h(s,s). now, for 0 ≤ s ≤ t ≤ 1, ∂h(t,s) ∂t = 1 γ(q2) [ (q2 − 1)tq2−2(1 −s)q2−q3−1 − (q2 − 1)(t−s)q2−2 ] ≤ (q2 − 1) γ(q2) [ (1 −s)q2−q3−1 − (1 −s)q2−2 ] = (q2 − 1) γ(q2) [ (1 −s)−q3+1 − 1 ] (1 −s)q2−2 = (q2 − 1) γ(q2) [( 1 − (1 −q3)s + (1 −q3)(−q3) 2 s2 + · · · ) − 1 ] (1 −s)q2−2 = (q2 − 1) γ(q2) [ (q3 − 1)s + o(s2) ] (1 −s)q2−2 ≤ 0. therefore, h(t,s) is decreasing in t, for s ∈ [0, 1] which implies h(t,s) ≤ h(s,s). hence the result. � lemma 2.5. let µ ∈ ( 1 4 , 3 4 ). then the green’s function h(t,s) holds the inequality, (2.8) min t∈i h(t,s) ≥ δ∗(s)h(s,s), for 0 < s < 1, where (2.9) δ∗(s) = { ( 3 4 )q2−1(1−s)q2−q3−1−( 3 4 −s)q2−1 sq2−1(1−s)q2−q3−1 , s ∈ (0,µ], 1 (4s)q2−1 , s ∈ [µ, 1). proof. for s ∈ (0, 1), h(t,s) is increasing in t for t ≤ s and decreasing in t for s ≤ t. p-laplacian fractional order boundary value problems 69 we define h1(t,s) = [tq2−1(1 −s)q2−q3−1 − (t−s)q2−1] γ(q2) h2(t,s) = [tq2−1(1 −s)q2−q3−1] γ(q2) , and h(s,s) = 1 γ(q2) [ sq2−1(1 −s)q2−q3−1 ] . then, min t∈i h(t,s) =   h1( 3 4 ,s), s ∈ (0, 1 4 ], min{h1( 34,s),h2( 1 4 ,s)}, s ∈ [ 1 4 , 3 4 ], h2( 1 4 ,s), s ∈ [ 3 4 , 1), = { h1( 3 4 ,s), s ∈ (0,µ], h2( 1 4 ,s), s ∈ [µ, 1), =   ( 3 4 )q2−1(1−s)q2−q3−1−( 3 4 −s)q2−1 γ(q2) , s ∈ (0,µ], 1 γ(q2) (1−s)q2−q3−1 4q2−1 , s ∈ [µ, 1), ≥ { ( 3 4 )q2−1(1−s)q2−q3−1−( 3 4 −s)q2−1 sq2−1(1−s)q2−q3−1 h(s,s), s ∈ (0,µ], 1 (4s)q2−1 h(s,s), s ∈ [µ, 1), = δ∗(s)h(s,s). � let b = {y : y ∈ c[0, 1]} be the real banach space equipped with the norm ‖y‖ = max t∈[0,1] |y(t)|. define a cone p ⊂ b by p = { y ∈ b : y(t) ≥ 0, t ∈ [0, 1] and min t∈i y(t) ≥ ξ‖y‖ } . let k = 1∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds and l = 1∫ s∈i ξg(1,s)φq (∫ τ∈i δ ∗(τ)h(τ,τ)dτ ) ds . let t : p → b be the operator defined by (2.10) ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds. if y ∈ p is a fixed point of t, then y satisfies (2.10) and hence y is a positive solution of the p-laplacian fractional order bvp (1.1)-(1.2). lemma 2.6. the operator t defined by (2.10) is a self map on p . 70 prasad and krushna proof. let y ∈ p . clearly, ty(t) ≥ 0, for all t ∈ [0, 1], and ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds so that ‖ty‖≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds. next, if y ∈ p , then by the above inequality we have min t∈i ty(t) = min t∈i ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ‖ty‖. hence, ty ∈ p and so t : p → p. standard arguments involving the arzela-ascoli theorem shows that t is completely continuous. � 3. existence of at least one positive solution in this section, we establish criteria for the existence of at least one positive solution of the p-laplacian fractional order bvp (1.1)-(1.2) by using krasnosel’skii fixed point theorem. to establish the existence of at least one positive solution for p-laplacian fractional order bvp (1.1)-(1.2) by employing the following krasnosel’skii fixed point theorem. theorem 3.1. [15] let x be a banach space, p ⊆ x be a cone and suppose that ω1, ω2 are open subsets of x with 0 ∈ ω1 and ω1 ⊂ ω2. suppose further that t : p ∩ (ω2\ω1) → p is completely continuous operator such that either (i) ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω2, or (ii) ‖ tu ‖≥‖ u ‖, u ∈ p ∩∂ω1 and ‖ tu ‖≤‖ u ‖, u ∈ p ∩∂ω2 holds. then t has a fixed point in p ∩ (ω2\ω1). theorem 3.2. if f0 = 0 and f∞ = ∞, then the p-laplacian fractional order bvp (1.1)-(1.2) has at least one positive solution that lies in p . proof. let t be the cone preserving, completely continuous operator defined by (2.10). since f0 = 0, we may choose h1 > 0 so that max t∈[0,1] f(t,y) φp(y) ≤ η1, for 0 < y ≤ h1, where η1 > 0 satisfies η q−1 1 ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds = η q−1 1 1 k ≤ 1. p-laplacian fractional order boundary value problems 71 thus, if y ∈ p and ‖y‖ = h1, then we have ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)η1φp(y)dτ ) ds = ∫ 1 0 g(1,s)η q−1 1 φq (∫ 1 0 h(τ,τ)dτ ) yds ≤ ηq−11 ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds‖y‖ = η q−1 1 1 k ‖y‖≤‖y‖. therefore, ‖ty‖≤‖y‖. now, if we let ω1 = {y ∈ b : ‖y‖ < h1}, then (3.1) ‖ty‖≤‖y‖, for y ∈ p ∩∂ω1. further, since f∞ = ∞, there exists h 2 > 0 such that min t∈[0,1] f(t,y) φp(y) ≥ η2, for y ≥ h 2 , where η2 > 0 is chosen such that η q−1 2 ξ 2 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ds = η q−1 2 ξ 2 1 l ≥ 1. let h2 = max { 2h1, 1 ξ h 2 } , and define ω2 = {y ∈ b : ‖y‖ < h2}. if y ∈ p ∩∂ω2 and ‖y‖ = h2, then min t∈i y(t) ≥ ξ‖y‖≥ h 2 , 72 prasad and krushna and so ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≥ ∫ s∈i ξg(1,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≥ ξ ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)η2φp(y)dτ ) ds = ξη q−1 2 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) yds ≥ ξηq−12 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ξ‖y‖ds = ξ2η q−1 2 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ds‖y‖ = η q−1 2 ξ 2 1 l ‖y‖ ≥‖y‖. hence, (3.2) ‖ty‖≥‖y‖, for y ∈ p ∩∂ω2. an application of theorem 3.1 to (3.1) and (3.2) yields a fixed point of t that lies in p ∩ (ω2\ω1). this fixed point is the solution of the p-laplacian fractional order bvp (1.1)-(1.2). � theorem 3.3. if f0 = ∞ and f∞ = 0, then the p-laplacian fractional order bvp (1.1)-(1.2) has at least one positive solution that lies in p . proof. let t be the cone preserving, completely continuous operator defined by (2.10). since f0 = ∞, we choose j1 > 0 such that min t∈[0,1] f(t,y) φp(y) ≥ η1, for 0 < y ≤ j 1, where η1 > 0 satisfies (η1) q−1ξ2 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ds = (η1) q−1ξ2 1 l ≥ 1. in this case, we define ω1 = {y ∈ b : ‖y‖ < j1}, p-laplacian fractional order boundary value problems 73 we have f(τ,y) ≥ η1φp(y),τ ∈ i, and moreover y(t) ≥ ξ‖y‖, t ∈ i and so ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≥ ∫ s∈i ξg(1,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≥ ξ ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)η1φp(y)dτ ) ds = ξ(η1) q−1 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) yds ≥ ξ(η1) q−1 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ξ‖y‖ds ≥ ξ2(η1) q−1 ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ds‖y‖ = (η1) q−1ξ2 1 l ‖y‖ ≥‖y‖. thus, (3.3) ‖ty‖≥‖y‖, for y ∈ p ∩∂ω1. now, since f∞ = 0, there exists j 2 > 0 such that max t∈[0,1] f(t,y) φp(y) ≤ η2, for y ≥ j 2 , where η2 > 0 satisfies (η2) q−1 ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds = (η2) q−1 1 k ≤ 1. it follows that f(t,y) ≤ η2φp(y), for y ≥ j 2 . we establish the result in two subcases. case(i) : f is bounded. suppose n > 0 is such that maxt∈[0,1] f(t,y) ≤ n, for all 0 < y < ∞. in this case, we may choose j2 = max { 2j1,nq−1 1 k } , and let ω2 = {y ∈ b : ‖y‖ < j2}. 74 prasad and krushna then for y ∈ p ∩∂ω2 and ‖y‖ = j2, we have ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)ndτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) nq−1ds ≤ nq−1 ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds = nq−1 1 k = j2 = ‖y‖, and therefore (3.4) ‖ty‖≤‖y‖, for y ∈ p ∩∂ω2. case(ii) : f is unbounded. let j2 > max{2j1,j 2 } be such that f(t,y) ≤ f(t,j2), for 0 < y ≤ j2, and let ω2 = {y ∈ b : ‖y‖ < j2}. choosing y ∈ p ∩ ∂ω2 with ‖y‖ = j2, we have ty(t) = ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y)dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)f(τ,j2)dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)η2φp(j 2)dτ ) ds = (η2) q−1 ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) dsj2 = (η2) q−1 1 k j2 ≤ j2 = ‖y‖. and so (3.5) ‖ty‖≤‖y‖, for y ∈ p ∩∂ω2. an application of theorem 3.1 to (3.3), (3.4) and (3.5) yields a fixed point of t that lies in p ∩ (ω2\ω1). this fixed point is the solution of the p-laplacian fractional order bvp (1.1)-(1.2). � 4. existence of multiple positive solutions in this section, we derive sufficient conditions for the existence of at least three positive solutions for the p-laplacian fractional order bvp (1.1)-(1.2), by applying five functionals fixed point theorem. we also establish the existence of at least p-laplacian fractional order boundary value problems 75 2k − 1 positive solutions for an arbitrary positive integer k. let γ,β,θ be nonnegative continuous convex functionals on p and α,ψ be nonnegative continuous concave functionals on p , then for nonnegative numbers h′,a′,b′,d′ and c′, convex sets are defined. p(γ,c′) = {y ∈ p : γ(y) < c′}, p(γ,α,a′,c′) = {y ∈ p : a′ ≤ α(y); γ(y) ≤ c′}, q(γ,β,d′,c′) = {y ∈ p : β(y) ≤ d′; γ(y) ≤ c′}, p(γ,θ,α,a′,b′,c′) = {y ∈ p : a′ ≤ α(y); θ(y) ≤ b′; γ(y) ≤ c′}, q(γ,β,ψ,h′,d′,c′) = {y ∈ p : h′ ≤ ψ(y); β(y) ≤ d′; γ(y) ≤ c′}. in obtaining multiple positive solutions for the p-laplacian fractional order bvp (1.1)-(1.2), the following so called five functionals fixed point theorem is fundamental. theorem 4.1. [3] let p be a cone in the real banach space b. suppose α and ψ are nonnegative continuous concave functionals on p and γ,β,θ are nonnegative continuous convex functionals on p, such that for some positive numbers c′ and e′, α(y) ≤ β(y) and ‖ y ‖≤ e′γ(y), for all y ∈ p(γ,c′). suppose further that t : p(γ,c′) → p(γ,c′) is completely continuous and there exist constants h′,d′,a′ and b′ ≥ 0 with 0 < d′ < a′ such that each of the following is satisfied. (d1) {y ∈ p(γ,θ,α,a′,b′,c′) : α(y) > a′} 6= ∅ and α(ty) > a′ for y ∈ p(γ,θ,α,a′,b′,c′), (d2) {y ∈ q(γ,β,ψ,h′,d′,c′) : β(y) > d′} 6= ∅ and β(ty) > d′ for y ∈ q(γ,β,ψ,h′,d′,c′), (d3) α(ty) > a′ provided y ∈ p(γ,α,a′,c′) with θ(ty) > b′, (d4) β(ty) < d′ provided y ∈ q(γ,β,ψ,h′,d′,c′) with ψ(ty) < h′. then t has at least three fixed points y1,y2,y3 ∈ p(γ,c′) such that β(y1) < d′,a′ < α(y2) and d ′ < β(y3) with α(y3) < a ′. define the nonnegative continuous concave functionals α,ψ and the nonnegative continuous convex functionals β,θ,γ on p by α(y) = min t∈i y(t),ψ(y) = min t∈i1 y(t), γ(y) = max t∈[0,1] y(t),β(y) = max t∈i1 y(t),θ(y) = max t∈i y(t), where i1 = [ 1 3 , 2 3 ] . for any y ∈ p , (4.1) α(y) = min t∈i y(t) ≤ max t∈i1 y(t) = β(y) and (4.2) ‖y‖≤ 1 ξ min t∈i y(t) ≤ 1 ξ max t∈[0,1] y(t) = 1 ξ γ(y). 76 prasad and krushna theorem 4.2. suppose there exist 0 < a′ < b′ < b ′ ξ ≤ c′ such that f satisfies the following conditions: (c1) f(t,y(t)) < φp ( a′k ) , t ∈ [0, 1] and y ∈ [ξa′,a′], (c2) f(t,y(t)) > φp ( b′l ) , t ∈ i and y ∈ [ b′, b′ ξ ] , (c3) f(t,y(t)) < φp ( c′k ) , t ∈ [0, 1] and y ∈ [0,c′]. then the p-laplacian fractional order bvp (1.1)-(1.2) has at least three positive solutions y1, y2 and y3 such that β(y1) < a ′, b′ < α(y2) and a ′ < β(y3) with α(y3) < b ′. proof. we seek three fixed points y1,y2,y3 ∈ p of t defined by (2.10). from lemma 2.6, (4.1) and (4.2), for each y ∈ p , α(y) ≤ β(y) and ‖y‖ ≤ 1 ξ γ(y). we show that t : p(γ,c′) → p(γ,c′). let y ∈ p(γ,c′). then 0 ≤ y ≤ c′. we may use the condition (c3) to obtain γ(ty) = max t∈[0,1] ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)f(τ,y(τ))dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)φp ( c′k ) dτ ) ds < c′k ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds = c′. therefore t : p(γ,c′) → p(γ,c′). now we verify the conditions (d1) and (d2) of theorem 4.1 are satisfied. it is obvious that b′ + b ′ ξ 2 ∈ { y ∈ p ( γ,θ,α,b′, b′ ξ ,c′ ) : α(y) > b′ } 6= ∅ and ξa′ + a′ 2 ∈ { y ∈ q ( γ,β,ψ,ξa′,a′,c′ ) : β(y) < a′ } 6= ∅. next, let y ∈ p(γ,θ,α,b′, b ′ ξ ,c′) or y ∈ q(γ,β,ψ,ξa′,a′,c′). then, b′ ≤ y ≤ b ′ ξ and ηa′ ≤ y ≤ a′. now, we may apply the condition (c2) to get α(ty) = min t∈i ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ ∫ s∈i g(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)φp ( b′l ) dτ ) ds > b′l ∫ s∈i ξg(1,s)φq (∫ τ∈i δ∗(τ)h(τ,τ)dτ ) ds = b′. p-laplacian fractional order boundary value problems 77 clearly, by the condition (c1), we have β(ty) = max t∈i1 ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)φp ( a′k ) dτ ) ds < a′k ∫ 1 0 g(1,s)φq (∫ 1 0 h(τ,τ)dτ ) ds = a′. to see that (d3) is satisfied, let y ∈ p(γ,α,b′,c′) with θ(ty) > b ′ ξ . then α(ty) = min t∈i ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ max t∈[0,1] ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≥ ξ max t∈i ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds = ξθ(ty) > b′. finally, we show that (d4) holds. let y ∈ q(γ,β,a′,c′) with ψ(ty) < ξa′. then, we have β(ty) = max t∈i1 ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ max t∈[0,1] ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds = 1 ξ [ ξ ∫ 1 0 g(1,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ )] ≤ 1 ξ min t∈i ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds ≤ 1 ξ min t∈i1 ∫ 1 0 g(t,s)φq (∫ 1 0 h(s,τ)f(τ,y(τ))dτ ) ds = 1 ξ ψ(ty) < a′. we have proved that all the conditions of theorem 4.1 are satisfied. therefore, the p-laplacian fractional order bvp (1.1)-(1.2) has at least three positive solutions y1,y2 and y3 such that β(y1) < a ′, b′ < α(y2) and a ′ < β(y3) with α(y3) < b ′. this completes the proof. � 78 prasad and krushna theorem 4.3. let k be an arbitrary positive integer. assume that there exist numbers ar(r = 1, 2, · · ·,k) and bs(s = 1, 2, · · ·,k − 1) with 0 < a1 < b1 < b1ξ < a2 < b2 < b2 ξ < · · · < ak−1 < bk−1 < bk−1 ξ < ak such that f satisfies the following conditions: (c4) f(t,y(t)) < φp ( ark ) , t ∈ [0, 1] and y ∈ [ξar,ar],r = 1, 2, · · ·,k, (c5) f(t,y(t)) > φp ( bsl ) , t ∈ i and y ∈ [ bs, bs ξ ] ,s = 1, 2, · · ·,k − 1. then the p-laplacian fractional order bvp (1.1)-(1.2) has at least 2k − 1 positive solutions in pak. proof. we use induction on k. first, for k = 1, we know from the condition (c4) that t : pa1 → pa1, then it follows from the schauder fixed point theorem that the p-laplacian fractional order bvp (1.1)-(1.2) has at least one positive solution in pa1. next, we assume that this conclusion holds for k = l. in order to prove that this conclusion holds for k = l + 1. we suppose that there exist numbers ar(r = 1, 2, · · ·, l + 1) and bs(s = 1, 2, · · ·, l) with 0 < a1 < b1 < b1ξ < a2 < b2 < b2 ξ < · · · < al < bl < blξ < al+1 such that f satisfies the following conditions: (4.3) f(t,y(t)) < φp ( ark ) , t ∈ [0, 1] and y ∈ [ξar,ar],r = 1, 2, · · ·, l + 1, (4.4) f(t,y(t)) > φp ( bsl ) , t ∈ i and y ∈ [ bs, bs ξ ] ,s = 1, 2, · · ·, l. by assumption, the fractional order bvp (1.1)-(1.2) has at least 2l − 1 positive solutions y∗i (i = 1, 2, · · ·, 2l− 1) in pal. at the same time, it follows from theorem 4.2, (4.3) and (4.4) that the p-laplacian fractional order bvp (1.1)-(1.2) has at least three positive solutions y1, y2 and y3 in pal+1 such that β(y1) < al, bl < α(y2) and al < β(y3) with α(y3) < bl. obviously y2 and y3 are distinct from y ∗ i (i = 1, 2, · · ·, 2l− 1) in pal. therefore, the p-laplacian fractional order bvp (1.1)-(1.2) has at least 2l + 1 positive solutions in pal+1, which shows that this conclusion also holds for k = l + 1. this completes the proof. � 5. examples in this section, as an application, the results of the previous sections are demonstrated with examples. example 5.1 consider the p-laplacian fractional order bvp, (5.1) d1.70+ ( φp ( d2.60+ y(t) )) = f(t,y(t)), t ∈ (0, 1), (5.2) y(0) = y′(0) = y′′(1) = 0,φp ( d2.60+ y(0) ) = 0 = d0.60+ ( φp ( d2.60+ y(1) )) , where f(t,y) = y2(650 − 649e−3y) 5 . p-laplacian fractional order boundary value problems 79 then the green’s function g(t,s) for the homogeneous bvp, −d2.60+ y(t) = 0, t ∈ (0, 1), y(0) = y′(0) = y′′(1) = 0, and is given by g(t,s) = { t1.6(1−s)−0.6 γ(2.6) , t ≤ s, t1.6(1−s)−0.6−(t−s)1.6 γ(2.6) , s ≤ t. the green’s function h(t,s) for the bvp, −d1.70+ ( φp ( d2.60+ y(t) )) = 0, t ∈ (0, 1), φp ( d2.60+ y(0) ) = 0 = d0.60+ ( φp ( d2.60+ y(1) )) , and is given by h(t,s) = { t0.7(1−s)0.1 γ(1.7) , t ≤ s, t0.7(1−s)0.1−(t−s)0.7 γ(1.7) , s ≤ t. clearly, the green functions g(t,s) and h(t,s) are positive. let p = 2. by direct calculations, ξ = 0.1088, k = 1.0077, l = 27.1209, f0 = 0 and f∞ = ∞. then, all the conditions of theorem 3.2 are satisfied. thus by theorem 3.2, the p-laplacian fractional order bvp (5.1)-(5.2) has at least one positive solution. example 5.2 consider the p-laplacian fractional order bvp, (5.3) d1.80+ ( φp ( d2.70+ y(t) )) = f(t,y(t)), t ∈ (0, 1), (5.4) y(0) = y′(0) = y′′(1) = 0,φp ( d2.70+ y(0) ) = 0 = d0.70+ ( φp ( d2.70+ y(1) )) , where f(t,y) = { t 100 + 15 32 y7, 0 ≤ y ≤ 2, y + t 100 + 116 2 , y > 2. then the green’s function g(t,s) for the homogeneous bvp, −d2.70+ y(t) = 0, t ∈ (0, 1), y(0) = y′(0) = y′′(1) = 0, and is given by g(t,s) = { t1.7(1−s)−0.7 γ(2.7) , t ≤ s, t1.7(1−s)−0.7−(t−s)1.7 γ(2.7) , s ≤ t, the green’s function h(t,s) for the bvp, −d1.80+ ( φp ( d2.70+ y(t) )) = 0, t ∈ (0, 1), φp ( d2.70+ y(0) ) = 0 = d0.70+ ( φp ( d2.70+ y(1) )) , and is given by h(t,s) = { t0.8(1−s)0.1 γ(1.8) , t ≤ s, t0.8(1−s)0.1−(t−s)0.8 γ(1.8) , s ≤ t. 80 prasad and krushna clearly, the green functions g(t,s) and h(t,s) are positive and f is continuous and increasing on [0,∞). let p = 2. by direct calculations, ξ = 0.0947, k = 1.8901 and l = 29.6238. choosing a′ = 1,b′ = 2 and c′ = 100, then 0 < a′ < b′ < b ′ ξ ≤ c′ and f satisfies (i) f(t,y) < 1.8901 = φp ( a′k ) , t ∈ [0, 1] and y ∈ [0.0947, 1], (ii) f(t,y) > 59.25 = φp ( b′l ) , t ∈ [1 4 , 3 4 ] and y ∈ [2, 21.12], (iii) f(t,y) < 189.01 = φp ( c′k ) , t ∈ [0, 1] and y ∈ [0, 100]. then, all the conditions of theorem 4.2 are satisfied. thus by theorem 4.2, the p-laplacian fractional order bvp (5.3)-(5.4) has at least three positive solutions y1,y2 and y3 satisfying max t∈[ 1 3 , 2 3 ] y1(t) < 1, 2 < min t∈[ 1 4 , 3 4 ] y2(t), 1 < max t∈[ 1 3 , 2 3 ] y3(t), min t∈[ 1 4 , 3 4 ] y3(t) < 2. references [1] r. p. agarwal, d. o’regan and p. j. y. wong, positive solutions of differential, difference and integral equations, kluwer academic publishers, dordrecht, the netherlands, 1999. [2] d. r. anderson and j. m. davis, multiple positive solutions and eigenvalues for third order right focal boundary value problems, j. math. anal. appl., 267(2002), 135–157. [3] r. i. avery, a generalization of the leggett-williams fixed point theorem, math. sci. res. hot-line, 3(1999), 9–14. [4] r. i. avery, j. henderson, existence of three positive pseudo-symmetric solutions for a onedimensional p-laplacian, j. math. anal. appl., 277(2003), 395–404. [5] c. bai, existence of positive solutions for boundary value problems of fractional functional differential equations, electronic journal of qualitative theory of differential equations, 30(2010), 1–14. [6] z. bai and h. lü, positive solutions for boundary value problem of nonlinear fractional differential equation, j. math. anal. appl., 311(2005), 495–505. [7] g. chai, positive solutions for boundary value problem of fractional differential equation with p-laplacian operator, boundary value problems, 2012(2012), 1–18. [8] t. chen and w. liu, an anti-periodic boundary value problem for the fractional differential equation with a p-laplacian operator, appl. math. lett., 25(2012), 1671–1675. [9] j. m. davis, j. henderson, k. r. prasad and w. yin, eigenvalue intervals for non-linear right focal problems, appl. anal., 74(2000), 215–231. [10] r. dehghani and k. ghanbari, triple positive solutions for boundary value problem of a nonlinear fractional differential equation, bulletin of the iranian mathematical society, 33(2007), 1–14. [11] l. h. erbe and h. wang, on the existence of positive solutions of ordinary differential equations, proc. amer. math. soc., 120(1994), 743–748. [12] d. guo and v. lakshmikantham, nonlinear problems in abstract cones, acadamic press, san diego, 1988. [13] a. a. kilbas, h. m. srivasthava and j. j. trujillo, theory and applications of fractional differential equations, north-holland mathematics studies, vol. 204, elsevier science, amsterdam, 2006. [14] l. kong and j. wang, multiple positive solutions for the one-dimensional p-laplacian, nonlinear analysis, 42(2000), 1327–1333. [15] m. a. krasnosel’skii, positive solutions of operator equations, noordhoff, groningen, 1964. [16] r. w. leggett and l. r. williams, multiple positive fixed points of nonlinear operator on order banach spaces, indiana univer. math. j., 28(1979), 673–688. p-laplacian fractional order boundary value problems 81 [17] d. o’regan, some general existence principles and results for (φ(y′))′ = qf(t,y,y′), 0 < t < 1, siam j. math. appl., 24(1993), 648–668. [18] i. podulbny, fractional diffrential equations, academic press, san diego, 1999. [19] k. r. prasad and p. murali, eigenvalue intervals of two-point boundary value problems for general nth order differential equations, nonlinear studies, vol. 18, no.2(2011), 167–176. [20] k. r. prasad and b. m. b. krushna, multiple positive solutions for a coupled system of riemann-liouville fractional order two-point boundary value problems, nonlinear studies, vol.20, no.4(2013), 501–511. [21] k. r. prasad, b. m. b. krushna and n. sreedhar, eigenvalues for iterative systems of (n,p)type fractional order boundary value problems, international journal of analysis and applications, vol. 5, no. 2(2014), 136–146. [22] k. r. prasad and b. m. b. krushna, eigenvalues for iterative systems of sturm-liouville fractional order two-point boundary value problems, fract. calc. appl. anal., vol. 17, no.3(2014), 638–653, doi: 10.2478/s13540-014-0190-4. [23] c. yang and j. yan, positive solutions for third-order sturm-liouville boundary value problems with p-laplacian, comput. math. appl., 59(2010), 2059–2066. 1department of applied mathematics, andhra university, visakhapatnam, 530 003, india, 2department of mathematics, mvgr college of engineering, vizianagaram, 535 005, india, ∗corresponding author int. j. anal. appl. (2023), 21:52 well-posedness and exponential stability of the von kármán beam with infinite memory abdelkader dibes1, lamine bouzettouta2,∗, manel abdelli2, salah zitouni1 1mohamed-cherif messaadia university, souk ahras, algeria 2laboratory of applied mathematics and history and didactics of mathematics (lamahis) university of 20 august 1955, skikda, algeria ∗corresponding author: lami_750000@yahoo.fr, l.bouzettouta@univ-skikda.dz abstract. in the present work, we consider a one-dimensional von kármán beam with infinite memory, we establish the well-posedness of the system using semigroup theory and prove the exponential stability under some conditions on the kernel of the infinite memory term. 1. introduction many studies have looked at nonlinear dynamical elasticity systems modeled by von kármán equations, wich is one of the basic equations in mathematical models of physics (see [14,20–22]). their importance stems from the fact that many physical phenomena connected to the theory of oscillation are described by non-linear dynamic elastic models. this nonlinear elastic systems incorporating wave equations govern the propagation of waves, oscillations, and vibrations of membranes, plates, and shells. the entire von kármán’s model in contrast to other fundamental models like the euler-bernoulli, raleigh, or timoshenko is appropriate for considering both transverse and longitudinal displacements for vibrating slender bodies with large deflection (for more discussion see [5–7]). in [18,19] lagnese et al investigated a one-dimensional von kármán system listed below{ ρautt − [ eaux + 1 2 w2x ] x , in (0,l) ×r+, ρawtt −ei (wxx)xx − [ ea ( ux + 1 2 w2x ) wx ] x , in (0,l) ×r+. (1.1) received: apr. 9, 2023. 2020 mathematics subject classification. 35b40, 93d23, 74f05, 93d15. key words and phrases. von kármán beam; exponential stability; semi-group; lyapunov functional; infinite memory. https://doi.org/10.28924/2291-8639-21-2023-52 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-52 2 int. j. anal. appl. (2023), 21:52 here e is the young’s modulus, a is the cross-sectional area of the beam, l is the beam length, ρa represents the weight per unit length and ei is the beam stiffness or flexural rigidity. a substantial range of literature use this type of model, where they addressing the problems of existence, uniqueness and asymptotic behavior in time when some damping effects are considered , (see refs. [4, 17, 25] and the references therein for more information). in (1998) the system (1.1) was stabilized by benabdallah and teniou [14] by fusing the system using two heat equations: both the longitudinal and transverse components, respectively, they showed the unique solution decay exponentially by using lyapunov functions. many articles have looked into the stabilization of systems using boundary damping , see favini et al [10], puel and tucsnak [24], and the references therein (see [3,8,15,26]). in [9] djebabla and tatar (2013) by linking the system, take into account the following full von kármán beam in one dimension. (namely, the longitudinal component) with only one heat equation according to green and naghdi’s theory [11–13]  utt −d1 [( ux + 1 2 (wx) 2 )] x + γθtx = 0, in (0,l) ×r+, wtt −d1 [( ux + 1 2 (wx) 2 ) wx ] x + d2wxxxx + δwt = 0, in (0,l) ×r+, θtt −θxx + µ1θt + γutx = 0, in (0,l) ×r+, where d1, d2, δ, µ1 and γ are positive constants, with the boundary conditions{ u = 0, w = 0, θx = 0, x = 0,l, t > 0, wx = 0, x = 0,l, t > 0, (1.2) and the initial data{ u (0, .) = u0, ut (0, .) = u1, w (0, .) = w0, wt (0, .) = w1, θ (0, .) = θ0, θt (0, .) = θ1, (1.3) they succeeded an exponential decay result by using lyapunov functions. a natural weak damping term can be thought of as the integral in the infinite memory term. it appears as a memory component in the form of a convolution in the fundamental equation between constraint and deformation. in this context, khochemane et all in [16] think about infinite memory in a porous-elastic system and a nonlinear damping term:{ ρutt −µuxx −bφx = 0, x ∈ (0, 1) ,t > 0, jφtt −δφxx + bux + aφ + ∫∞ 0 g(s)φxx(t − s)ds + α(t)f (φ) = 0, x ∈ (0, 1) ,t > 0, where φ and u represent, respectively,the volume fraction and displacement of the solid elastic material, the parameter ρ is the mass density, and j equals the product of the equilibrated inertia by the mass density, and the function g is the relaxation function, the term α(t)f (φ) is the nonlinear damping term. int. j. anal. appl. (2023), 21:52 3 in this paper, we study the following von kármán system with infinite memory  wtt −d1 [( ux + 1 2 (wx) 2 ) wx ] x + d2wxxxx + µwt (x,t) = 0, utt −d1 [( ux + 1 2 (wx) 2 )] x − ∫∞ 0 g(s)uxx(t − s)ds = 0. (1.4) in ω ×(0,∞), where ω = [0,l] and d1,d2,d and µ are positive constants and the function g is called the relaxation function. we complement system (1.4) with boundary conditions{ w(0,t) = w(1,t) = ux(0,t) = ux(1,t), t > 0 wx = 0 at x = 0, l f or any t > 0, (1.5) and the initial data   u(0, .) = u0, ut(0, .) = u1, w(0, .) = w0, wt(0, .) = w1, wt(x,t) = f0(x,t) in ∈ (0,l), (1.6) 2. preliminaries first, we introduce the following hypothesis that has been considered in many works such that [1,2] (h1) g : r+ → r+ is a c1 function satisfying g(0) > 0, δ − ∫ ∞ 0 g(s)ds = l > 0, ∫ ∞ 0 g(s)ds = g0. (h2) there exists a non-increasing differentiable function α,ξ : r+ → r+ such that g ′ (t) ≤−ξ(t)g(t), ∀t ≥ 0. now, to prove that systems (1.4), (1.5), (1.6) are well posed using the semigroup theory we introduce the following new variable:  ηt(x,s) = u(x,t) + u(x; t − s) uxx(x,t − s) = ηtxx(x,s) −uxx(x,t) ηtt + η t s = ut therefore, the problem (1.4) is equivalent to  wtt −d1 [( ux + 1 2 (wx) 2 ) wx ] x + d2wxxxx + µwt (x,t) = 0, utt −d1 [( ux + 1 2 (wx) 2 )] x + g0uxx − ∫∞ 0 g(s)ηtxxds = 0, ηtt + η t s −ut = 0. (2.1) with g0 = ∫ ∞ 0 g(s)ds. for any regular solution of (2.1), the energy e (t), defined by e (t) = 1 2 ∫ l 0 { w2t + u 2 t + d2w 2 xx + d1 ( ux + 1 2 (wx) 2 )2} dx + 1 2 (g ◦ux)(t). (2.2) 4 int. j. anal. appl. (2023), 21:52 3. well-posedness of the problem in this section, we prove the existence and uniqueness of solutions for (2.1) using the semigroup theory [23]. first, we introduce the vector function let u = ( w,wt,u,ut,η t )t , then ut = ( wt,wtt,ut,utt,η t t )t . introducing the vector function v = wt, and ψ = ut, system (2.1) can be written as{ ut = au + f (u) , u (0) = (w0,w1,u0,u1,θ0,q0, f0) , (3.1) and the linear operator a is defined by: a   w v u ψ ηt   =   v −d2wxxxx −µv ψ luxx + ∫∞ 0 g(s)ηtxxds ψ −ηts   , (3.2) and f (u) =   0 d1 [( ux + 1 2 (wx) 2 ) wx ] x 0 d1 2 (wx) 2 x 0   , u0 =   w0 w1 u0 u1 θ0   . (3.3) it is clear that f (u) is a continuous and uniformly lipschitz operator. and h is the energy space given by h := {[ h4 (0,l) ∩h20 (0,l) ] ×h10 (0,l) × [ h2 (0,l) ∩h20 (0,l) ] ×h10 (0,l) ×lg } . lg = { ϕ : r+ → h10 (0,l) , ∫ l 0 ∫ ∞ 0 g(s)ϕ2xdsdx ≺∞ } , we equip the space lg with the inner product through 〈u,v〉lg = ∫ l 0 ∫ ∞ 0 g(s)ux(s)vx(s)dsdx, and we define on the hilbert space h the inner product, for u = ( w,v,u,ψ,ηt )t , ũ =( w̃, ṽ, ũ, ψ̃, η̃t )t 〈 u,ũ 〉 = ∫ l 0 vṽdx + ∫ l 0 ψψ̃dx + d2 ∫ l 0 wxxw̃xxdx + l ∫ l 0 uxũxdx + 〈 ηt, η̃t 〉 . int. j. anal. appl. (2023), 21:52 5 the domain of a is given by d (a) = { ( w,v,u,ψ,ηt )t ∈ [h4 (0,l) ∩h20 (0,l)]×h10 (0,l) × [ h2 (0,l) ∩h20 (0,l) ] ×h10 (0,l) ×lg } . (3.4) clearly, d (a) is dense in h. the next result is our first main goal in this paper theorem 3.1. let u ∈ h, for any initial datum u0 ∈ h there exists a unique solution u ∈ c ([0,∞) ,h) for problem. moreover, if u0 ∈ d (a) , then u ∈ c ([0,∞) ,d (a)) ∩c1 ([0,∞) ,h) . proof. we will show that the operator a generates a c0-semigroup in h. in this step, we prove that the operator a is dissipative. let u = ( w,v,u,ψ,ηt )t firstly we have: 〈au,u〉 = 〈   v −d2wxxxx −µv ψ luxx + ∫∞ 0 g(s)ηtxxds ψ −ηts   ,   w v u ψ ηt   〉 = ∫ l 0 v (−d2wxxxx −µv) dx −d2 ∫ l 0 ϕwxxxxdx + d1 ∫ l 0 ψuxxdx (3.5) + ∫ l 0 ψ ( luxx + ∫ ∞ 0 g(s)ηtxxds ) dx + d2 ∫ l 0 wxxvxxdx + l ∫ l 0 ψxuxdx + 〈 ηt,ψ−ηts 〉 . using integration by parts, we obtain 〈au,u〉 = −d2 ∫ l 0 vwxxxxdx −µ ∫ l 0 v2dx + l ∫ l 0 ψuxxdx + ∫ l 0 ψ ∫ ∞ 0 g(s)ηtxxdsdx + d2 ∫ l 0 vxxwxxdx + l ∫ l 0 ψxuxdx + ∫ l 0 ∫ ∞ 0 g(s)ηtx ( ψ −ηtsx ) dsdx, thus, 〈au,u〉 = −µ ∫ l 0 v2dx + 1 2 (g ′ ◦ux)(t). consequently, the operator a is dissipative. now, we will prove that the operator λi−a is surjective. 6 int. j. anal. appl. (2023), 21:52 for this purpose, let (f1, f2, f3, f4, f5) t ∈ h, we seek u = ( w,v,u,ψ,ηt )t ∈ d (a) , solution of the following system of equations  w −v = f1, v + d2wxxxx + µv = f2, u −ψ = f3, ψ − luxx − ∫∞ 0 g(s)ηtxxds = f4, ηt + ηts −ψ = f5, (3.6) we obtain   ηt = e−s ∫l 0 eς (ψ + f5(ς)) dς, v = w − f1, ψ = u − f3,{ w − f1 + d2wxxxx + µw −µf1 = f2, u − f3 − luxx − ∫∞ 0 g(s) ( e−s ∫ s 0 eς (ψ + f5(ς)) dς ) xx ds = f4, then − ∫ ∞ 0 g(s) ( e−s ∫ l 0 eς (ψ + f5(ς)) dς ) xx ds = − ∫ ∞ 0 g(s) ( e−s ∫ s 0 eς (u − f3 + f5(ς)) dς ) xx ds = − ∫ ∞ 0 g(s)e−s ∫ s 0 eς (uxx + (−f3 + f5)xx) dςds = − ∫ ∞ 0 g(s)e−s ∫ s 0 eςuxxdςds − ∫ ∞ 0 g(s)e−s ∫ s 0 eς (−f3 + f5)xx dςds = −uxx ∫ ∞ 0 g(s) ( 1 −e−s ) ds − ∫ ∞ 0 g(s)e−s ∫ s 0 eς (−f3 + f5)xx dςds, we obtain{ (µ + 1) w + d2wxxxx = f2 + (µ + 1) f1, u − [ l + ∫∞ 0 g(s) ( 1 −e−s ) ds ] uxx = f3 + f4 + ∫∞ 0 g(s)e−s ∫ s 0 eς (−f3 + f5)xx dςds, where { h1 = f2 + (µ + 1) f1, h2 = f3 + f4 + ∫∞ 0 g(s)e−s ∫ s 0 eς (−f3 + f5)xx dςds. now, we consider the following variational formulation b ((w,u) ; (w1,u1)) = l (w1,u1) , int. j. anal. appl. (2023), 21:52 7 where b : [ h20 (0, 1) ×h 1 0 (0, 1) ]2 → r is the bilinear form defined by b ((w,u) ; (w1,u1)) = (µ + 1) ∫ l 0 ww1dx + d2 ∫ l 0 wxxw1xxdx + ∫ l 0 uu 1 dx + [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ]∫ l 0 uxu1xdx, and l : h20 (0, 1) ×h 1 0 (0, 1) → r is the linear functional given by l (w1,u1) = ∫ l 0 h1w1dx + ∫ l 0 h2u1dx, now for v = h20 (0, 1) ×h 1 0 (0, 1) equipped with the norm ‖(w,u)‖2v = ‖w‖ 2 2 + ‖u‖ 2 2 + ‖wxx‖ 2 2 + ‖ux‖ 2 2 , we have b ((w,u) ; (w,u)) = ∫ l 0 w2dx + ∫ l 0 w2xxdx + ∫ l 0 u2dx + [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ]∫ l 0 u2xdx ≥‖w‖22 + ‖wxx‖ 2+ 2 ‖u‖ 2 2 + [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ] ‖ux‖22 ≥ m0 ( ‖w‖22 + ‖u‖ 2 2 + ‖wxx‖ 2 2 + ‖ux‖ 2 2 ) = m0‖(w,u)‖2v , thus b is coercive. on the other hand, by using cauchy-schwarz and poincaré inequalities, we obtain |b ((w,u) ; (w1,u1))| ≤ ‖w‖2‖w1‖2 + d2‖wxx‖2‖w1xx‖2 + ‖u‖2‖u1‖2 + [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ] ‖ux‖2‖u1x‖2 ≤ ξ (‖w‖2 + ‖u‖2 + ‖wxx‖2 + ‖ux‖2) × (‖w1‖2 + ‖u1‖2 + ‖w1xx‖2 + ‖u1x‖2) ≤ ξ‖(w,u)‖v ‖(w1,u1)‖v . similarly, we can show that |l (w1,u1)| ≤ κ‖(w1,u1)‖v , consequently, by the lax-milgram lemma the system has unique solution (w,u) ∈ h20 (0, 1) ×h 1 0 (0, 1) , satisfying b ((w,u) ; (w1,u1)) = l (w1,u1) ,∀(w1,u1) ∈ v, the substitution of w and u yields (v,ψ) ∈ h20 (0, 1) ×h 1 0 (0, 1) . 8 int. j. anal. appl. (2023), 21:52 similarly, now by taking u1 = 0 ∈ h20 (0, 1)∫ l 0 u1udx + [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ]∫ l 0 uxu1xdx = ∫ l 0 h2u1dx, hence, we obtain [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ]∫ l 0 uxu1xdx = ∫ l 0 (h2 −u) u1dx, by noting that (h2 −u) ∈ l2(0, 1), we obtain u ∈ h2 (0, 1) ∩h10 (0, 1) and consequentely the form∫ l 0 [ − [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ] uxx −h2 + u ] u1dx = 0,∀u1 ∈ h10 (0, 1) , therefore, we obtain − [ l + ∫ ∞ 0 g(s) ( 1 −e−s ) ds ] uxx + u = h2, similarly, if we take w1 = 0 ∈ h20 (0, 1) using twice integration by parts we obtain (µ + 1) ∫ l 0 ww1dx + d2 ∫ l 0 wxxw1xxdx = ∫ l 0 h1w1dx d2 ∫ l 0 wxxw1xxdx = ∫ l 0 (h1 − (µ + 1) w) w1dx∫ l 0 (d2wxxxx + (µ + 1) w −h1) w1dx = 0,∀w1 ∈ h20 (0, 1) . therefore, we obtain d2wxxxx + (µ + 1) w = h1 ∈ l2(0, 1). consequentely, we get w ∈ h4 (0, 1) ∩h20 (0, 1). hence, there exists a unique u ∈ d(a) such that λi −a is satisfied. therefore, the operator a is dissipative. from where, we conclude that a is a maximal monotone operator. now, we prove that the operator f is locally lipschitz in h. in fact, if u = ( w,v,u,ψ,ηt )t , ũ = ( w̃, ṽ, ũ, ψ̃, η̃ t )t belong to h, we have ∥∥∥f (u) −f(ũ)∥∥∥2 h = d1 ( |h|2 + |g|2 ) , (3.7) where h = [( ux + 1 2 w2x ) wx − ( ũx + 1 2 w̃2x ) w̃x ] x and g = 1 2 ( w2x − w̃2x ) x . adding and subtracting the term ( ux + 1 2 w2x ) w̃x inside the norm |h| , we gets |h| ≤ ‖wx − w̃x‖l∞(0,l) ∣∣∣∣ux + 12w2x ∣∣∣∣ + ‖w̃x‖l∞(0,l) |ux − ũx| + 1 2 ‖w̃x‖l∞(0,l) |wx + w̃x|‖wx − w̃x‖l∞(0,l) . (3.8) using the embedding of h1 (0,l) into l∞ (0,l) one has from (3.8) that |h| ≤ k ( ‖u‖h , ∥∥∥ũ∥∥∥ h )∥∥∥u − ũ∥∥∥ h (3.9) int. j. anal. appl. (2023), 21:52 9 using once again the embedding of h 1(0,l) into l∞ (0,l), one also sees that |g| ≤ k ( ‖u‖h , ∥∥∥ũ∥∥∥ h )∥∥∥u − ũ∥∥∥ h . (3.10) combining (3.7), (3.9) and (3.10), it follows that f(u) is locally lipschitz continuous in h. finally, by using the regularity theory to solve linear elliptic equations ensures the presence of unique solution u ∈ d (a). consequently, a is a dissipative operator. hence, the result of theorem 3.1 follows from the lumer-phillips theorem. � 4. stability result in this section, we use the multiplier technique, to study the stability result for the energy of solution of the system (2.1). first, we state and prove the following lemma. lemma 4.1. let ( w,v,u,ψ,ηt ) be the solution of (2.1). then the energy functional e (t), defined by (2.2) satisfies e ′ (t) ≤−µ ∫ l 0 w2t dx + 1 2 (g ′ ◦ux)(t). (4.1) proof. multiplying the first equation by wt , the second equation by ut, integrating over (0, 1), and summing them up we obtain d 2dt ∫ l 0 w2t dx + d1 ( ux + 1 2 (wx) 2 ) wxwxtdx + d2d 2dt ∫ l 0 (wxx) 2 dx + µ ∫ l 0 w2t dx = 0 d 2dt ∫ l 0 u2tdx + d1 ( ux + 1 2 (wx) 2 ) uxtdx − ∫ l 0 ut ∫ ∞ 0 g(s)ηtxxdsdx = 0 we estimate the last term as follows: − ∫ l 0 ut ∫ ∞ 0 g(s)ηtxxdsdx = − ∫ l 0 ( ηtt + η t s )∫ ∞ 0 g(s)ηtxxdsdx = − ∫ ∞ 0 g(s) ∫ l 0 ηtxx ( ηtt + η t s ) dxds = − ∫ ∞ 0 g(s) ∫ l 0 ηtxxη t tdxds − ∫ ∞ 0 g(s) ∫ l 0 ηtxxη t sdxds = ∫ ∞ 0 g(s) ∫ l 0 ηtxη t txdxds + ∫ ∞ 0 g(s) ∫ l 0 ηtxη t sxdxds = d 2dt ∫ l 0 ∫ ∞ 0 g(s) ( ηtx )2 dsdx + 1 2 ∫ l 0 ∫ ∞ 0 g(s) d ds ( ηtx )2 dsdx = d 2dt ∫ l 0 ∫ ∞ 0 g(s) [ux(t) −ux(t − s)] 2 dsdx − 1 2 ∫ l 0 ∫ ∞ 0 g ′ (s) ( ηtx )2 dsdx = d 2dt ∫ l 0 ∫ ∞ 0 g(s) [ux(t) −ux(t − s)] 2 dsdx − 1 2 ∫ l 0 ∫ ∞ 0 g ′ (s) [ux(t) −ux(t − s)] 2 dsdx = d 2dt (g ◦ux)(t) − 1 2 (g ′ ◦ux)(t). 10 int. j. anal. appl. (2023), 21:52 � lemma 4.2. let ( w,v,u,ψ,ηt ) be the solution of (2.1). then the functional i1(t) = ∫ l 0 ( utu + 1 2 wtw + µ 4 w2 ) dx, t ≥ 0. (4.2) satisfies, the estimate i′1(t) = −d1 ∫ l 0 ( ux + 1 2 (wx) 2 )2 dx + ∫ l 0 u2tdx − d2 2 ∫ l 0 w2xxdx + g0 ∫ l 0 u2xdx + 1 2 ∫ l 0 w2t dx + ∫ l 0 u ∫ ∞ 0 g(s)ηtxxdsdx (4.3) proof. using young’s inequalitie we get∫ l 0 u ∫ ∞ 0 g(s)ηtxxdsdx = − ∫ l 0 ux ∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx ≤ ε1 ∫ l 0 u2xdx + 1 4ε1 ∫ l 0 (∫ ∞ 0 g(s)(ux(t) −ux(t − s)) )2 dsdx ≤ ε1 ∫ l 0 u2xdx + g0 4ε1 (g ◦ux)(t), (4.4) i′1(t) ≤−d1 ∫ l 0 ( ux + 1 2 (wx) 2 )2 dx − d2 2 ∫ l 0 (wxx) 2 dx + (g0 + ε1) ∫ l 0 u2xdx + ∫ l 0 u2tdx + 1 2 ∫ l 0 w2t dx. + g0 4ε1 (g ◦ux)(t), � lemma 4.3. let ( w,v,u,ψ,ηt ) be the solution of (2.1). then the functional i2(t) := − ∫ l 0 ut ∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx t ≥ 0, (4.5) satisfies, the estimate i′2(t) ≤− g0 2 ∫ l 0 u2tdx + d1 2 ∫ l 0 ( ux + 1 2 (wx) 2 )2 dx + (g1 + d1) (g ◦ux(t)) + g0 2 ∫ l 0 u2xdx t ≥ 0, + g(0) 2g0 (g ′ ◦ux)(t). (4.6) int. j. anal. appl. (2023), 21:52 11 proof. firste we note that d dt ∫ ∞ 0 g(s)(u(t) −u(t − s))ds = d dt ∫ t −∞ g(t − s)(u(t) −u(s))ds = ∫ t −∞ g ′ (t − s)(u(t) −u(s))ds + ∫ t −∞ g(t − s)u(t)ds = g0ut(t) + ∫ ∞ 0 g ′ (s)(u(t) −u(t − s))ds, i′2(t) = − ∫ l 0 utt ∫ ∞ 0 g(s)(u(t) −u(t − s))dsdx − ∫ l 0 ut d dt ∫ ∞ 0 g(s)(u(t) −u(t − s))dsdx = − ∫ l 0 ( d1 [( ux + 1 2 (wx) 2 )] x −g0uxx + ∫ ∞ 0 g(s)ηtxxds )(∫ ∞ 0 g(s)(u(t) −u(t − s))ds ) dx −g0 ∫ l 0 u2tdx − ∫ l 0 ut ∫ ∞ 0 g ′ (s)(u(t) −u(t − s))dsdx = −d1 ∫ l 0 [( ux + 1 2 (wx) 2 )] x ∫ ∞ 0 g(s)(u(t) −u(t − s))dsdx + g0 ∫ l 0 uxx ∫ ∞ 0 g(s)(u(t) −u(t − s))dsdx − ∫ l 0 (∫ ∞ 0 g(s)ηtxxds )(∫ ∞ 0 g(s)(u(t) −u(t − s))ds ) dx −g0 ∫ l 0 u2tdx − ∫ l 0 ut ∫ ∞ 0 g ′ (s)(u(t) −u(t − s))dsdx = d1 ∫ l 0 ( ux + 1 2 (wx) 2 )∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx −g0 ∫ l 0 ux ∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx + ∫ l 0 (∫ ∞ 0 g(s)ηtxds )(∫ ∞ 0 g(s)(ux(t) −ux(t − s))ds ) dx −g0 ∫ l 0 u2tdx − ∫ l 0 ut ∫ ∞ 0 g ′ (s)(u(t) −u(t − s))dsdx. by recalling young’s inequality, we get for any ε2 > 0, d1 ∫ l 0 ( ux + 1 2 (wx) 2 )∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx ≤ d1ε2 ∫ l 0 ( ux + 1 2 (wx) 2 )2 dx + d1 4ε2 ∫ l 0 (∫ ∞ 0 g(s)(ux(t) −ux(t − s))ds )2 dx ≤ d1ε2 ∫ l 0 ( ux + 1 2 (wx) 2 )2 dx + g0d1 4ε2 (g ◦ux(t)) , 12 int. j. anal. appl. (2023), 21:52 −g0 ∫ l 0 ux ∫ ∞ 0 g(s)(ux(t) −ux(t − s))dsdx ≤ g0ε3 ∫ l 0 u2xdx + g0 4ε3 (g ◦ux)(t), − ∫ l 0 ut ∫ ∞ 0 g ′ (s)(u(t) −u(t − s))dsdx = − ∫ l 0 √ g0ut ∫ ∞ 0 (√ 1 g0 g ′ (s)(u(t) −u(t − s)) ) dsdx ≤ g0 2 ∫ l 0 u2tdx + g(0) 2g0 (g ′ ◦ux)(t), ∫ l 0 (∫ ∞ 0 g(s)ηtxds )(∫ ∞ 0 g(s)(ux(t) −ux(t − s))ds ) dx = ∫ l 0 (∫ ∞ 0 g(s)(ux(t) −ux(t − s))ds )2 dx ≤ d3(g ◦ux)(t). wich complete the proof. � now, we define the lyapunov functional l(t)by l(t) = ne(t) + n1f1(t) + n2f2(t), wheren,n1and n2 are positive constants. lemma 4.4. let (u,w) be the solution of then, there exist two positive constants c1 and c2 such that the lyapunov functional l (t) satisfies c1e (t) ≤ l (t) ≤ c2e (t) , ∀t ≥ 0. (4.7) in other words, the functions e and l are equivalent. proof. firstly we note that ∫ l 0 u2xdx ≤ 2 ∫ l 0 ( ux + 1 2 ( w2x ))2 dx + 1 2 ∫ l 0 w2xdx ≤ 2 ∫ l 0 ( ux + 1 2 ( w2x ))2 dx + 1 4 ∫ l 0 w4xdx, ≤ 2 ∫ l 0 ( ux + 1 2 ( w2x ))2 dx + l 4 ∫ l 0 w2xxdx, we get |l(t) −ne(t)| ≤ n1 ∫ l 0 ∣∣∣∣utu + 12wtw + µ4 w2 ∣∣∣∣dx + n2 ∫ l 0 ∣∣∣∣ut ∫ ∞ 0 g(s)(u(t) −u(t − s))ds ∣∣∣∣dx. int. j. anal. appl. (2023), 21:52 13 by using young’s inequality, cauchy–schwarz inequality, and poincaré’s inequality, we obtain |l(t) −ne(t)| ≤ [n1ε1 + n2ε3] ∫ l 0 u2tdx + n1ε2 4 ∫ l 0 w2t dx + n1ε1 2 ∫ l 0 ( ux + 1 2 ( w2x ))2 dx + [ n1l 16ε1 + n1 4ε2 + n1µ 4 ]∫ l 0 w2xxdx + d3 4ε3 (g ◦ux)(t). so |l(t) −ne(t)| ≤ c ∫ l 0 { w2t + u 2 t + d2w 2 xx + d1 ( ux + 1 2 (wx) 2 )2} dx + c(g ◦ux)(t) ≤ ce(t). then (n −c)e(t) ≤ l(t) ≤ (n + c)e(t). consequently, by choosing n large enough, we obtain the estimate (4.7). � now, we are ready to state and prove the main result of this section. theorem 4.1. let ( w,v,u,ψ,ηt ) be the solution of (2.1). then the energy functional (2.2) satisfies, e (t) ≤ k0e−k1t, ∀t ≥ 0, (4.8) where k0 and k1 are two positive constants. proof. l′(t) ≤− [ n2g0 2 −n1 ]∫ l 0 u2tdx − [ nµ− n1 2 ]∫ l 0 w2t dx − [ d1n1 − n2d1 2 − 2n1 (g0 + ε1) −n2g0 ]∫ l 0 ( ux + 1 2 ( w2x ))2 dx − [ n1d2 2 − n1l 4 (g0 + ε1) − n2g0l 8 ]∫ l 0 w2xxdx − [ n1g0 4ε1 + (n2g0 + d3) ] (g ◦ux)(t) − [ n 2 + n2g(0) 2g0 ] (g′ ◦ux)(t). 14 int. j. anal. appl. (2023), 21:52 by setting ε1 = 1 n1 , we obtain l′(t) ≤− [ n2g0 2 −n1 ]∫ l 0 u2tdx − [ nµ− n1 2 ]∫ l 0 w2t dx − [ d1n1 − n2d1 2 − 2n1g0 − 1 −n2g0 ]∫ l 0 ( ux + 1 2 ( w2x ))2 dx − [ n1d2 2 − n1l 4 g0 − l 4 − n2g0l 8 ]∫ l 0 w2xxdx − [ n21g0 4 + (n2g0 + d3) ] (g ◦ux)(t) − [ n 2 + n2g(0) 2g0 ] (g′ ◦ux)(t), next, we carefully choose our constants so that the terms inside the brackets are positive. we choose n1 large enough such that α1 = nµ− n1 2 > 0, then we choose n2 large enough such that α2 = n2g0 2 −n1 > 0, α3 = d1n1 − n2d1 2 − 2n1g0 − 1 −n2g0 > 0, α4 = n1d2 2 − n1l 4 g0 − l 4 − n2g0l 8 > 0, α5 = n21g0 4 + (n2g0 + d3) > 0, finally, once n1 and n2, is fixed, we choose n large enough so that α6 = n 2 + n2g(0) 2g0 > 0, we obtain l ′ (t) ≤− 1 2 ∫ 1 0 { α1w 2 t + α2u 2 t + α3 ( ux + 1 2 ( w2x ))2 + α4w 2 xx } − α5 2 (g ◦ux)(t), by (2.2), we obtain l ′ (t) ≤−σ0e (t) , ∀t ≥ 0, (4.9) for some σ0 > 0. a combination of (4.7) and (4.9) gives l ′ (t) ≤−k1l (t) , ∀t ≥ 0, (4.10) int. j. anal. appl. (2023), 21:52 15 a simple integration of (4.10) over (0,t) yields l (t) ≤ l (0) e−k1t, ∀t ≥ 0. (4.11) finally, by combining (4.7) and (4.11) we obtain (4.8), which completes the proof. � conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] t.a. apalara, general decay of solutions in one-dimensional porous-elastic system with memory, j. math. anal. appl. 469 (2019), 457–471. https://doi.org/10.1016/j.jmaa.2017.08.007. [2] t.a. apalara, on the stabilization of a memory-type porous thermoelastic system, bull. malays. math. sci. soc. 43 (2019), 1433–1448. https://doi.org/10.1007/s40840-019-00748-2. [3] s. baibeche, l. bouzettouta, a. guesmia, m. abdelli, well-posedness and exponential stability of swelling porous elastic soils with a second sound and distributed delay term, j. math. comput. sci. 12 (2022), 82. https: //doi.org/10.28919/jmcs/7106. [4] a. benabdallah, d. teniou, exponential stability of a von karman model with thermal effects, electron. j. differ. equ. 1998 (1998), 7. [5] l. bouzettouta, a. djebabla, exponential stabilization of the full von kármán beam by a thermal effect and a frictional damping and distributed delay, j. math. phys. 60 (2019), 041506. https://doi.org/10.1063/1. 5043615. [6] l. bouzettouta, f. hebhoub, k. ghennam, s. benferdi, exponential stability for a nonlinear timoshenko system with distributed delay, int. j. anal. appl. 19 (2021), 77-90. https://doi.org/10.28924/ 2291-8639-19-2021-77. [7] s. zitouni, k. zennir, l. bouzettouta, uniform decay for a viscoelastic wave equation with density and timevarying delay in rn, filomat. 33 (2019), 961–970. https://doi.org/10.2298/fil1903961z. [8] l. bouzettouta, s. zitouni, kh. zennir, h. sissaoui, well-posedness and decay of solutions to bresse system with internal distributed delay, int. j. appl. math. stat. 56 (2017), 153–168. [9] a. djebabla, n-e. tatar, exponential stabilization of the full von kármán beam by a thermal effect and a frictional damping, georgian math. j. 20 (2013), 427–438. https://doi.org/10.1515/gmj-2013-0019. [10] a. favini, m.a. horn, i. lasiecka, d. tataru, global existence, uniqueness and regularity of solutions to a von karman system with nonlinear boundary dissipation, differ. integral equ. 9 (1996), 267–294. [11] a.e. green, p.m. naghdi, a re-examination of the basic postulates of thermomechanics, proc. r. soc. lond. a. 432 (1991), 171–194. https://doi.org/10.1098/rspa.1991.0012. [12] a.e. green, p.m. naghdi, on undamped heat waves in an elastic solid, j. thermal stresses. 15 (1992), 253–264. https://doi.org/10.1080/01495739208946136. [13] a.e. green, p.m. naghdi, thermoelasticity without energy dissipation, j. elasticity. 31 (1993), 189–208. https: //doi.org/10.1007/bf00044969. [14] m.a. horn, i. lasiecka, uniform decay of weak solutions to a von kármán plate with nonlinear boundary dissipation, differ. integral equ. 7 (1994), 885–908. https://doi.org/10.57262/die/1370267712. [15] h.e. khochemane, l. bouzettouta, a. guerouah, exponential decay and well-posedness for a one-dimensional porous-elastic system with distributed delay, appl. anal. 100 (2019), 2950–2964. https://doi.org/10.1080/ 00036811.2019.1703958. https://doi.org/10.1016/j.jmaa.2017.08.007 https://doi.org/10.1007/s40840-019-00748-2 https://doi.org/10.28919/jmcs/7106 https://doi.org/10.28919/jmcs/7106 https://doi.org/10.1063/1.5043615 https://doi.org/10.1063/1.5043615 https://doi.org/10.28924/2291-8639-19-2021-77 https://doi.org/10.28924/2291-8639-19-2021-77 https://doi.org/10.2298/fil1903961z https://doi.org/10.1515/gmj-2013-0019 https://doi.org/10.1098/rspa.1991.0012 https://doi.org/10.1080/01495739208946136 https://doi.org/10.1007/bf00044969 https://doi.org/10.1007/bf00044969 https://doi.org/10.57262/die/1370267712 https://doi.org/10.1080/00036811.2019.1703958 https://doi.org/10.1080/00036811.2019.1703958 16 int. j. anal. appl. (2023), 21:52 [16] h.e. khochemane, a. djebabla, s. zitouni, l. bouzettouta, well-posedness and general decay of a nonlinear damping porous-elastic system with infinite memory, j. math. phys. 61 (2020), 021505. https://doi.org/10. 1063/1.5131031. [17] h. e. khochemane, s. zitouni, l. bouzettouta, stability result for a nonlinear damping porous-elastic system with delay term, nonlinear stud. 27 (2020), 487–503. [18] j.e. lagnese, modelling and stabilization of nonlinear plates, in: estimation and control of distributed parameter systems (vorau, 1990), international series of numerical mathematics, birkhäuser, basel, vol. 100, pp. 247–264, (1991). [19] j.e. lagnese, uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary, nonlinear anal.: theory methods appl. 16 (1991), 35–54. https: //doi.org/10.1016/0362-546x(91)90129-o. [20] j.e. lagnese, g. leugering, uniform stabilization of a nonlinear beam by nonlinear boundary feedback, j. differ. equ. 91 (1991), 355–388. https://doi.org/10.1016/0022-0396(91)90145-y. [21] w. liu, k. chen, j. yu, existence and general decay for the full von kármán beam with a thermo-viscoelastic damping, frictional dampings and a delay term, ima j. math. control inform. 34 (2017), 521-542. https: //doi.org/10.1093/imamci/dnv056. [22] g.p. menzala, e. zuazua, timoshenko’s beam equation as limit of a nonlinear one-dimensional von kármán system, proc. r. soc. edinburgh: sect. a math. 130 (2000), 855–875. https://doi.org/10.1017/ s0308210500000470. [23] a. pazy, semigroups of linear operators and applications to partial differential equations, new york: springer, applied mathematical sciences, 44, (1983). [24] j.p. puel, m. tucsnak, boundary stabilization for the von kármán equations, siam j. control optim. 33 (1995), 255–273. https://doi.org/10.1137/s0363012992228350. [25] f.d. araruna, p. braz e silva, e. zuazua, asymptotics and stabilization for dynamic models of nonlinear beams, proc. estonian acad. sci. 59 (2010), 150–155. https://doi.org/10.3176/proc.2010.2.14. [26] s. zitouni, l. bouzettouta, k. zennir, d. ouchenane. exponential decay of thermo-elastic bresse system with distributed delay term, hacettepe j. math. stat. 47 (2018), 1216–1230. https://doi.org/10.1063/1.5131031 https://doi.org/10.1063/1.5131031 https://doi.org/10.1016/0362-546x(91)90129-o https://doi.org/10.1016/0362-546x(91)90129-o https://doi.org/10.1016/0022-0396(91)90145-y https://doi.org/10.1093/imamci/dnv056 https://doi.org/10.1093/imamci/dnv056 https://doi.org/10.1017/s0308210500000470 https://doi.org/10.1017/s0308210500000470 https://doi.org/10.1137/s0363012992228350 https://doi.org/10.3176/proc.2010.2.14 1. introduction 2. preliminaries 3. well-posedness of the problem 4. stability result references int. j. anal. appl. (2023), 21:80 received: jun. 5, 2023. 2020 mathematics subject classification. 60e05. key words and phrases. failure rate; age distribution; randomly right censored data. https://doi.org/10.28924/2291-8639-21-2023-80 © 2023 the author(s) issn: 2291-8639 1 a class of tests for testing better failure rate at specific age distribution with randomly right censored data gamal r. elkahlout* faculty of business school, arab open university, riyadh, saudi arabia *corresponding author: g.elkahlout@arabou.edu.sa abstract: a device has a better failure rate at specific age 𝑡0 property, denoted by 𝐵𝐹𝑅 − 𝑡0 if its failure rate r(t) increases for 𝑡 ≤ 𝑡0 and for 𝑡 > 𝑡0, r(t) is not less than its value at 𝑡0. a test statistic is proposed to test exponentiality versus 𝐵𝐹𝑅 − 𝑡0 based on a randomly right censored sample of size n. kaplan-meier estimator is used to estimate the empirical life distribution. properties of the test are measured by power estimates, estimated risks, and test of normality. the efficiency loss due to censoring is investigated by using tests for censored sample data. 1. introduction the concept of ageing for engineering devices, biological organs or their corresponding systems has been characterized by various life distribution classes. the increasing failure rate (ifr) class of life distributions is the most used and have all other notions of ageing in reliability literature. among these notions are the increasing failure rate average (ifra), new better than used (nbu), new better than used failure rate (nbufr), new better than used in average failure rate (nbafr), decreasing mean remaining life (dmrl), new better than used in expectation (nbue) and harmonic new better than used in expectation (hnbue). see ([14], [23], [21], [6]) for definitions, properties and interrelationships of these classes of life distributions. in many practical situations, it is familiar that properties of life distributions may not be completely observed after a specific time. this arises in data collection in companies for their commodities with guarantee for some fixed time 𝑡0, say. https://doi.org/10.28924/2291-8639-21-2023-80 2 int. j. anal. appl. (2023), 21:80 in study of [15] for the new better than used at specific age 𝑡0 (𝑁𝐵𝑈 − 𝑡0) have considered the hypothetical cancer patients problem. it is interesting to investigate the problem of testing whether new diagnosed cancer patient has smaller chance of survival than a patient with similar initial diagnosis and survived, on treatment for a certain year. the decreasing mean remaining life at a specific age 𝑡0 (𝐷𝑀𝑅𝐿 − 𝑡0) is defined in similar line by [10]. in a study by [2] introduced the class of better failure rate at a specific age 𝑡0 (𝐵𝐹𝑅 − 𝑡0) and its dual class (𝑊𝐹𝑅 − 𝑡0). its closure properties under some reliability operations are studied. test statistic for testing exponentiality against 𝐵𝐹𝑅 − 𝑡0 or its dual class 𝑊𝐹𝑅 − 𝑡0 is also proposed for the complete sample. different classes of life distributions based on a random censored samples, have been studied in references such as ([26], [27]) for testing nbu and ifr classes of life distributions, [20] for testing ifra class of life distributions and ([3], [4]) for testing nbru and nbrue classes of life distributions and their dual classes. in research for [12] defined classes of life distributions ifra-to and nbu-t0. the properties of these classes are studied, and a nonparametric test is proposed which is designed to test the hypothesis whether nbu-t0 element is strictly new better than used after time 𝑡0. a paper for [19] about nbu class made by [15] to investigate the testing of new better than used at specified age (nbu-t0) based on a u-statistic whose kernel depends on sub-sample minima. a member of the class of tests proposed by [17] for this problem belongs to the class of tests are covered and distribution properties of the class of test statistics are studied. the performances of a few members of the proposed class of tests are studied in terms of pitman asymptotic relative efficiency. in a paper for [9] he introduced some properties of the new better than used in convex ordering at age t0 (nbuc− t0) and new better than used of second order (2) at age t0 (nbu (2)− t0) classes of life distributions, where the survival probability at age 0 is greater than or equal to the conditional survival probability at specified age t0> 0. preservation properties of the two classes under various reliability operations and shock model are arriving according to homogeneous poisson process are established. researchers [13] have defined two classes of life distributions, namely new better than used in expectation at specific age 𝑡0 (𝑁𝐵𝑈𝐸 − 𝑡0) and harmonic new better than used in expectation at specific age t0 (𝐻𝑁𝐵𝑈𝐸 − 𝑡0) and their dual classes (𝑁𝑊𝑈𝐸 − 𝑡0) and (𝐻𝑁𝑊𝑈𝐸 − 𝑡0). the closure 3 int. j. anal. appl. (2023), 21:80 properties under various reliability operations such as convolution, mixture, mixing and the homogeneous poisson shock model of these classes are studied. nonparametric tests are proposed to test exponentiality versus the nbue-t0 and hnbue-t0 classes. the critical values and the powers of these tests are calculated to assess the performance of the tests. they show that the proposed tests have high efficiencies for some commonly used distributions in reliability. a test statistic has been built by [7] for two classes of life distributions defined earlier by [1] namely new better than used renewal failure rate (nburfr) and new better than average renewal failure rate (nbarfr). these two classes include many other classes of life distributions. test statistics for testing of exponentiality as a null hypothesis against these two renewal ageing criteria, and their duals are derived in the case of randomly censored samples. percentiles tables, power estimates, estimated risks are calculated. the normality of their test statistics is also studied. 2. the 𝑩𝑭𝑹 − 𝒕𝟎 and 𝑾𝑭𝑹 − 𝒕𝟎 classes let 𝑇 be a life length of a device with continuous distribution f, survival function �̄�(𝑡) = 1 − 𝐹(𝑡) and failure rate 𝑟(𝑡), then it is called better failure rate at time 𝑡0 𝐵𝐹𝑅 − 𝑡0 (𝑊𝐹𝑅 − 𝑡0) if 𝑟(𝑠) ≤ (≥)𝑟(𝑡) ∀ 𝑠 < 𝑡 < 𝑡0 & 𝑡 ∈ [0, 𝑡0], (2.1) and 𝑟(𝑡0) ≤ (≥)𝑟(𝑡) ∀ 𝑡 ≥ 𝑡0 (2.2) this means that any device of age 𝑡0 or less has smaller failure rate than an older device, whereas a device of age 𝑡0 or more cannot have a failure rate less than 𝑟(𝑡0). 3. testing exponentiality versus 𝑩𝑭𝑹 − 𝒕𝟎 and 𝑾𝑭𝑹 − 𝒕𝟎 classes in this section we consider the problem of testing: 𝐻0: 𝐹(𝑡) = 1 − 𝑒 −𝜆𝑡0 , i.e. 𝑟(𝑡) = 𝑟(𝑡 + 𝑥)∀𝑥 ≥ 0, 0 ≤ 𝑡 ≤ 𝑡0, versus 𝐻1: 𝐹is 𝐵𝐹𝑅 − 𝑡0 i.e. 𝑟(𝑡) is increasing for 𝑡 ≤ 𝑡0 and 𝑟(𝑡) ≥ 𝑟(𝑡0) for 𝑡 ≥ 𝑡0. this test is based on randomly right-censored data by using kaplan-meier estimator [11] for the empirical survival function ( , ),( ) ( )z i ni i  1  . let {𝑇𝑖 }𝑖 = 1,2, . . . , 𝑛 be independent and identically distributed non-negative continuous random variables having a common distribution 𝐹 and survival function �̄�(𝑡) = 1 − 𝐹(𝑡). let {𝑌𝑖 }𝑖 = 1,2, . . . , 𝑛 be independent and identically distributed random variables according to a continuous censoring distribution 𝐻. {𝑇𝑖 } and {𝑌𝑖 } are independent of each other. the censoring 4 int. j. anal. appl. (2023), 21:80 distribution 𝐻 is usually, but not necessary, unknown and is considered as nuisance parameter. the pairs (𝑇1, 𝑌1), . . . . . , (𝑇𝑛 , 𝑌𝑛 ) are defined on a common probability space. in the censored situations of sample size n, the 𝑇1, . . . . . . . . . 𝑇𝑛 are not completely observed but the pairs (𝑍1, 𝛿1), . . . . . . , (𝑍𝑛 , 𝛿𝑛 ) are observed data, where 𝑍𝑖 = 𝑚𝑖𝑛(𝑇𝑖 , 𝑌𝑖 ) , ∀𝑖 = 1,2, . . . . , 𝑛 and 𝛿𝑖 = { 1 if 𝑇𝑖 ≤ 𝑌𝑖 0 if 𝑇𝑖 > 𝑌𝑖 4. test statistics for 𝑩𝑭𝑹 − 𝒕𝟎 in the 𝐵𝐹𝑅 − 𝑡0 the required test can be based on the estimation of the parameter m(f) = ∬ f̄(s)f̄(t){r(s)-r(t)}df(s) 0<z<t≤t0 df(t) + ∫ f̄(t0)f̄(t){r(t0)-r(t)}df(t) ∞ t0 (4.1) the statistic 𝑀(𝐹) = 0 under 𝐻0 whereas 𝑀(𝐹) < 0 under 𝐻1 where strict inequality is due to the continuity of the underlying distribution f. also the statistic 𝑀(𝐹) gives a measure of deviation from the exponentiality towards the 𝐵𝐹𝑅 − 𝑡0 property. for the present of randomly censored data, we base our test on the kaplan-meier estimator of �̄�𝑛 , see [11], which is given by )( 0 1 )( )( )( : nn zt in in tf i i tzi        +− − =          (4.2) where 0 = 𝑧(0) ≤ 𝑧(1) ≤. . . . . . . ≤ 𝑧(𝑛) , denote ordered sample of 𝑧1, 𝑧2, . . . . . . , 𝑧𝑛 and 𝛿(1), 𝛿(2), . . . . . , 𝛿(𝑛) are the corresponding 𝛿𝑖 for the ordered 𝑧(𝑖) for 𝑖 = 1,2, . . . . , 𝑛 . note that 𝛿(𝑛) is taken as one, that is 𝑧(𝑛) is treated as uncensored observation whether it is the case or not. the corresponding density function and failure rate of �̄̂�𝑛 (𝑡) are given in the following. 𝑓𝑛 (𝑡) = �̄̂�𝑖−1,𝑛(𝑡)−�̄̂�𝑖,𝑛(𝑡) 𝑧(𝑖)−𝑧(𝑖−1) (4.3) where �̄̂�𝑛 (𝑡) = 1 − �̂�𝑛 (𝑡), hence 𝑓𝑛 (𝑡) = (𝑧(𝑖) − 𝑧(𝑖−1)) −1 [∏ { 𝑛−𝑖 𝑛−𝑖+1 }{𝑖−1:𝑧(𝑖)≤𝑡} 𝛿(𝑖) − ∏ { 𝑛−𝑖 𝑛−𝑖+1 }{𝑖:𝑧(𝑖)≤𝑡} 𝛿(𝑖) ] = (𝑧(𝑖) − 𝑧(𝑖−1)) −1 [∏ { 𝑛−𝑗 𝑛−𝑗+1 }𝑖−1𝑗=1 𝑍(𝑖)≤𝑡 𝛿(𝑗) − ∏ { 𝑛−𝑗 𝑛−𝑗+1 }𝑖𝑗=1 𝛿(𝑗) ] = (𝑧(𝑖) − 𝑧(𝑖−1)) −1 ∏ { 𝑛−𝑗 𝑛−𝑗+1 }𝑖−1𝑗=1 𝛿(𝑗) [1 − { 𝑛−𝑖 𝑛−𝑖+1 } 𝛿(𝑖) ] ni ,....,2,1= 5 int. j. anal. appl. (2023), 21:80 = (𝑧(𝑖) − 𝑧(𝑖−1)) −1 [[1 − { 𝑛−𝑖 𝑛−𝑖+1 } 𝛿(𝑖) ] ∏ { 𝑛−𝑗 𝑛−𝑗+1 }𝑖−1𝑗=1 𝛿(𝑗) ] (4.4) it is reasonable to base a test of exponentiality versus 𝐵𝐹𝑅 − 𝑡0 property on a consistent estimator𝑀(𝐹), given by (3) and then exploit 𝑀(�̄̂�𝑛 ),w here �̄̂�𝑛 the kaplan-meier estimator. in fact the kaplan-meier estimator has been shown to be weakly convergent by ([5], [16], [22]) and strongly consistent by ([24], [18]). the test statistic in (3) can by simplified in the following. for easy calculation we write 𝑀(𝐹) = 𝐴 + 𝐵, where 𝐴 = ∬ �̄�(𝑠)�̄�(𝑡){𝑟(𝑠) − 𝑟(𝑡)}𝑑𝐹(𝑠)𝑑𝐹(𝑡) 0<𝑠<𝑡<𝑡0 (4.5) and 𝐵 = ∫ �̄�(𝑡0)�̄�(𝑡){𝑟((𝑡0) − 𝑟(𝑡))}𝑑𝐹(𝑡) ∞ 𝑡0 (4.6) now ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( ) ( ) ( )  ( )  ( )  ( ) ( ) ( )( ) ( )  ( )  ( ) ( ) ( )   ( )  dttftftdftfdssf dttf sf tdftfdssf dttfsdfsftdftfdssf dtsdftf tf tf tfsftdfdssf sf sf tfsf tdfsdftrtfsftdfsdfsrtfsfa tt t t t t t tt t t tt t t tt t 2 0 2 0 0 2 2 0 2 0 0 0 2 2 00 0 2 0 00 0 0 00 0 00 00 00 0 00 1 2 1 2                −−      = −      = −      = −= −= (4.7) and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )  ( ) ( )( )  ( ) ( )  ( ) ( )( ) ( ) ( )            −−= −= −= −= −= 0 0 0 00 00 00 2 0 2 00 2 00 2 00 0 0 0 0 000 2 1 2 1 2 t t t tt tt tt dttftftftf dttftftftf dttftftdftftf tdf tf tf tftftdf tf tf tftf tdftrtftftdftrtftfb (4.8) 6 int. j. anal. appl. (2023), 21:80 hence 𝑀(𝐹) = ∫ [∫ {𝑓(𝑠)}2𝑑𝑠 𝑡 0 ] 𝑡0 0 �̄�(𝑡)𝑑𝐹(𝑡) − 1 2 ∫ [{�̄�(𝑡)}2 − 1] 𝑡0 0 {𝑓(𝑡)}2𝑑𝑡 − 1 2 𝑓(𝑡0)(�̄�(𝑡0)) 2 − �̄�(𝑡0) ∫ {𝑓(𝑡)} 2𝑑𝑡 ∞ 𝑡0 (4.9) or 𝑀(𝐹) = ∫ [∫ {𝑓(𝑠)}2𝑑𝑠 𝑡 0 ] 𝑡0 0 �̄�(𝑡)𝑓(𝑡)𝑑𝑡 − 1 2 ∫ {�̄�(𝑡)}2 𝑡0 0 {𝑓(𝑡)}2𝑑𝑡 + 1 2 ∫ {𝑓(𝑡)}2𝑑𝑡 𝑡0 0 −�̄�(𝑡0) ∫ {𝑓(𝑡)} 2𝑑𝑡 ∞ 𝑡0 − 1 2 𝑓(𝑡0)(�̄�(𝑡0)) 2 (4.10) now substituting the empirical values of �̄̂�(𝑡) and 𝑓𝑛 (𝑡)in (12), then the statistics 𝑀𝑛(𝐹) in the case of randomly censored data defined earlier will be 𝑀𝑛 (𝐹) = ∑ [∑{𝑓𝑛(𝑍(𝑗))} 2 (𝑍(𝑗) − 𝑍(𝑗−1)) 𝑖 𝑗=1 ] 𝑚 𝑖=1 �̄�𝑛 (𝑍(𝑖))𝑓𝑛 (𝑍(𝑖))(𝑍(𝑖) − 𝑍(𝑖−1)) + 1 2 ∑{𝑓𝑛 (𝑍(𝑖))} 𝑚 𝑖=1 2 (𝑍(𝑖) − 𝑍(𝑖−1)) − 1 2 ∑{�̄�𝑛 (𝑍(𝑖))} 2 𝑚 𝑖=1 {𝑓𝑛 (𝑍(𝑖))} 2 (𝑍(𝑖) − 𝑍(𝑖−1)) −�⃑�𝑛 (𝑍(𝑚)) ∑ {𝑓𝑛(𝑍(𝑖))} 𝑛 𝑖=𝑚 2 (𝑍(𝑖) − 𝑍(𝑖−1)) − 1 2 𝑓𝑛 (𝑍(𝑚0)){�̄�𝑛(𝑍(𝑚0))} 2 (4.11) f zn i( )( ) is the kaplan-meier, product limit estimator of fn and given by [11] and defined as  f z f zn j n j k j n k n k k ( ) ( ) ( ) ( ) ( ) = − = − − + = 1 1 1  (4.12) also, z i( ) and ( )i are defined earlier, and  ( ) ( )d z zi i i= − −1 , here 𝑍(𝑛) is considered as an actual observation, whether or not it is censored, and in this case 𝜕𝑛 is taken as one to avoid appearance of indefinite values in the empirical calculation of the statistic 𝑀𝑛 (𝐹). for a specific significance level  = 0 and a sample size n n= 0 , we reject 𝐻0 in favor of 𝐻1, that f has the 𝐵𝐹𝑅 − 𝑡0 property, for small negative values of 𝑀𝑛 (𝐹). in otherwards, we reject 𝐻0 in favor of 𝐻1 if the observed or calculated values of the statistic is less than or equal the tabulated value of the statistic for the same value of 𝛼0 and 𝑛0. for testing 𝐻0 against 𝐻1 that f has the 𝑊𝐹𝑅 − 𝑡0 property, we reject 𝐻0 for small negative values of −𝑀𝑛(𝐹), or for large positive values of 𝑀𝑛 (𝐹). 7 int. j. anal. appl. (2023), 21:80 the asymptotic normality of the sequence, such as √𝑛[𝑀𝑛(𝐹) − 𝑀(𝐹)] , has been dealt with in the literature under the following crucial assumption.      ) sup g t h t t( ) ( ) , ,1 1 0− −     for some 0 1  (4.13) where g and h are the distribution functions of the actual values of x and the censored values y respectively. here g and h are assumed to have support on  )0, . the above sequence, or stochastic process, converges weakly as n → to a gaussian process with zero mean, for this result and the details of the corresponding variance function see [22]. a simulation of a small sample is calculated. in practice, simulated percentiles for small samples are commonly used by applied statisticians and reliability analysts. based on test statistics in (4.1), empirical evaluation in (4.11) using kaplan-meier, product limit estimator in (4.12), the lower percentiles in the 0.01, 0.05, and 0.10 regions and upper percentiles in the 0.90, 0.95, and 0.99 regions for the sample sizes n=10(2),30(5) and 50 are presented in table 1 for the 𝐵𝐹𝑅 − 𝑡0 and 𝑊𝐹𝑅 − 𝑡0 test statistics. 5. properties of the test in the following, we will cover three properties of the test, namely, power estimates, estimated risks, and test of normality. 5.1 power estimates the power estimates of the test statistic 𝑀𝑛 (𝐹) are considered for the significance level =0.05 and for commonly used distributions in reliability modeling. these distributions are gamma, weibull, pareto, and rayleigh with the following survival functions and failure rates. gamma distribution: f t x e dx t t x ( ) ( ) , , ,=   − −   1 0 0 1     (5.1) ( ) r t ut e u du − = + − −  1 1 1 0 ( ) ( )  weibull distribution: f t t t( ) exp( ) , ,= −     0 0 (5.2) r t t( ) = −   1 . pareto distribution: 8 int. j. anal. appl. (2023), 21:80 ( )f t t t( ) , , ,= +   − 1 0 0 1   (5.3) ( )r t t t( ) , ,= +   − 1 0 0 1   . rayleigh distribution: f t t t( ) exp( ) ,= −  1 2 2 0 (5.4) r t t( ) = t  0 for further references about these distributions, one can refer to [25]. table 2 contains the power estimates of 𝑀𝑛 (𝐹) test statistic with respect to these distributions. the estimates are based on the average of five runs each with 1000 simulated samples of sizes n=10,20,30,40 and 50 with significance level =0.05. the power estimates of 𝐵𝐹𝑅 − 𝑡0 statistic for gamma distribution increases more rapidly as  increases than the other distributions, namely, weibull, pareto, and rayleigh. this indicates that the power estimate increases as the class tends to be more 𝐵𝐹𝑅 − 𝑡0. the same pattern for the power estimates can be noted when the sample size increases. this table shows that the percentage of  = 1 ( i.e. the percentage of the uncensored random variables in the sample) is approximately fixed when the sample size increases for the same parameter and decreases as the parameter of the distribution increases for all the distributions, namely, gamma, weibull, pareto and rayleigh. this study shows that the percentage of censored values is dependent on the parameter values of the distribution more than the sample size. the resulting estimates indicate that the proposed statistic is suitable for different small sample sizes in reliability applications. 5.2 estimated risks the estimated risks (er) of the statistic 𝑀𝑛(𝐹) are given by ; 𝐸𝑅(𝑀𝑛(𝐹)) = 1 𝑚 ∑ (𝑋𝑘 − �̅�𝑀(𝐹)) 2𝑚 𝑘=1 (5.5) where x da ~ is the mean value, xi is the statistic value corresponding to the distribution under study, bfr-t0, and m is the sample size. estimated risks are summarized in table 3. it is noted that as the sample size increases, the estimated risks decrease, and the mean value is increasing. more precisely, er/n decreases as n increases. for a large sample size er/n approaches to zero. this indicates that our test is a powerful one. 5.3 a test of normality the kolmogorov-smirnov (ks) test is applied to check how well the underlying statistic 𝑀𝑛 (𝐹) tends to normality with unspecified mean and variance. for testing normality let s be the 9 int. j. anal. appl. (2023), 21:80 empirical distribution function based on the random sample x x x n1 2 , ,....., . the test statistics d is defined as the greatest vertical distance between standardized version of 𝑀𝑛 (𝐹) , denoted by , and s. symbolically, d s= −sup  (5.6) here we utilize the modified kolmogorov-smirnov test of normality proposed by [8] which accommodates the sample estimated normal parameters. the d values are given in table 4. by comparing the calculated ks value with the tabulated one, one accepts the hypothesis approach those to normality for the considered sample sizes. table 1: critical values for 𝑀𝑛 (𝐹) statistic for testing 𝐵𝐹𝑅 − 𝑡0 and 𝑊𝐹𝑅 − 𝑡0 ageing properties n 0.01 0.05 0.1 0.9 0.95 0.99 10 -3.6457 -2.431 -1.9567 0.2251 0.6764 1.7745 12 -3.2025 -2.1203 -1.7422 0.398 0.8059 1.8555 14 -2.9597 -1.9717 -1.617 0.4558 0.9362 2.177 16 -2.5969 -1.8157 -1.4816 0.601 1.1027 2.3588 18 -2.5062 -1.7031 -1.3924 0.6615 1.1451 2.4567 20 -2.2232 -1.6249 -1.3085 0.7522 1.1715 2.328 22 -2.1796 -1.5047 -1.242 0.7844 1.3103 2.4974 24 -1.9614 -1.4286 -1.1464 0.8456 1.3761 2.773 26 -1.8858 -1.317 -1.0815 0.963 1.4602 3.0735 28 -1.7508 -1.261 -1.0356 1.0441 1.6226 2.8913 30 -1.6563 -1.197 -0.9633 1.007 1.4842 2.8249 35 -1.5217 -1.0807 -0.8538 1.1309 1.6884 3.1401 40 -1.3643 -0.926 -0.7355 1.2301 1.7746 3.0854 45 -1.2075 -0.8482 -0.6432 1.307 1.8649 3.0055 50 -1.1091 -0.7636 -0.5834 1.373 1.8875 3.2362 10 int. j. anal. appl. (2023), 21:80 table 2 : power estimates for 𝑀𝑛 (𝐹) statistic for testing 𝐵𝐹𝑅 − 𝑡0 and 𝑊𝐹𝑅 − 𝑡0 ageing properties. distribution par. sample size 10 20 30 40 50 f (gamma) 2 0.009 0.012 0.014 0.016 0.021 percentage of delt=1 62 62 63 62 62 f (gamma) 3 0.162 0.223 0.271 0.272 0.347 percentage of delt=1 50 50 50 50 50 f (gamma) 4 0.562 0.613 0.652 0.665 0.691 percentage of delt=1 40 41 40 40 41 f (gamma) 5 0.834 0.855 0.862 0.954 1 percentage of delt=1 32 32 32 32 32 f (gamma) 6 0.999 1 1 1 1 percentage of delt=1 26 26 26 26 26 f (weibull) 0.25 0.08 0.12 0.18 0.3 0.46 percentage of delt=1 32 35 38 36 37 f (weibull) 0.5 0 0.02 0.08 0.1 0.18 percentage of delt=1 40 39 38 38 40 f (weibull) 2 0.1 0.14 0.2 0.32 0.48 percentage of delt=1 49 49 51 52 51 f (weibull) 3 0.11 0.12 0.13 0.14 0.16 percentage of delt=1 38 42 41 41 40 f (weibull) 4 0.15 0.19 0.25 0.28 0.35 percentage of delt=1 35 34 35 35 35 f (weibull) 5 0.17 0.23 0.47 0.53 0.6 percentage of delt=1 30 29 29 30 29 f (weibull) 6 0.26 0.47 0.64 0.72 0.82 percentage of delt=1 23 23 25 25 25 f (pareto) 0.25 0.093 0.118 0.094 0.082 0.087 percentage of delt=1 83 83 83 83 83 f (pareto) 0.5 0.217 0.238 0.25 0.216 0.228 percentage of delt=1 81 80 80 80 80 f (pareto) 2 0.385 0.505 0.569 0.582 0.618 percentage of delt=1 63 63 63 63 63 f (pareto) 3 0.325 0.436 0.459 0.506 0.53 percentage of delt=1 54 55 55 55 54 f (pareto) 4 0.275 0.321 0.303 0.341 0.371 11 int. j. anal. appl. (2023), 21:80 percentage of delt=1 48 48 48 48 48 f (pareto) 5 0.229 0.219 0.227 0.2 0.2 percentage of delt=1 43 43 43 43 43 f (pareto) 6 0.156 0.123 0.116 0.107 0.127 percentage of delt=1 39 38 38 38 39 f (rayleigh) 0.013 0.121 0.147 0.153 0.16 percentage of delt=1 58 58 58 58 58 table 3 : mean and estimated risks for 𝑀𝑛(𝐹) statistic for testing 𝐵𝐹𝑅 − 𝑡0 and 𝑊𝐹𝑅 − 𝑡0 ageing properties. n mean er er/n n mean er er/n 10 -0.3321 0.9927 0.0993 31 0.5667 0.8982 0.0290 11 -0.2782 0.9567 0.0870 32 0.5036 0.7378 0.0231 12 -0.1888 0.9000 0.0750 33 0.5329 0.7943 0.0241 13 -0.2058 0.7048 0.0542 34 0.5706 0.8068 0.0237 14 -0.0598 0.9599 0.0686 35 0.6364 0.8101 0.0231 15 -0.0485 0.7991 0.0533 36 0.6030 0.7587 0.0211 16 0.0075 0.9237 0.0577 37 0.6182 0.7333 0.0198 17 0.0579 0.8699 0.0512 38 0.6362 0.7163 0.0188 18 0.0768 0.7815 0.0434 39 0.6821 0.7759 0.0199 19 0.1144 0.8101 0.0426 40 0.6886 0.9857 0.0246 20 0.1872 0.9038 0.0452 41 0.6787 0.7548 0.0184 21 0.2494 0.8292 0.0395 42 0.7043 0.8144 0.0194 22 0.2207 0.7853 0.0357 43 0.7457 0.7930 0.0184 23 0.2370 0.7266 0.0316 44 0.7248 0.7257 0.0165 24 0.3127 0.8435 0.0351 45 0.7847 0.8133 0.0181 25 0.3467 0.8647 0.0346 46 0.7854 0.7610 0.0165 26 0.3367 0.7591 0.0292 47 0.8035 0.8464 0.0180 27 0.3529 0.7320 0.0271 48 0.8510 0.8627 0.0180 28 0.4842 0.9700 0.0346 49 0.8423 0.8315 0.0170 29 0.4765 0.9224 0.0318 50 0.8303 0.8485 0.0170 30 0.4503 0.8198 0.0273 12 int. j. anal. appl. (2023), 21:80 table 4 : test of normality for𝑀𝑛 (𝐹) statistic for testing 𝐵𝐹𝑅 − 𝑡0 and 𝑊𝐹𝑅 − 𝑡0 ageing properties n d n d 10 0.073 31 0.093 11 0.0835 32 0.1153 12 0.0853 33 0.1037 13 0.0874 34 0.1158 14 0.0794 35 0.0976 15 0.0882 36 0.1144 16 0.0892 37 0.1209 17 0.0894 38 0.1044 18 0.0803 39 0.0915 19 0.0907 40 0.1147 20 0.0774 41 0.0883 21 0.0913 42 0.0891 22 0.0925 43 0.0925 23 0.0917 44 0.113 24 0.099 45 0.1095 25 0.0951 46 0.1096 26 0.0806 47 0.1116 27 0.1002 48 0.0987 28 0.1153 49 0.1232 29 0.0963 50 0.0809 30 0.0784 acknowledgement: the author extends his appreciation to the arab open university for funding this work through aou research fund no. (aourg-2023-016). conflicts of interest: the author declares that there are no conflicts of interest regarding the publication of this paper. 13 int. j. anal. appl. (2023), 21:80 references [1] a.m. abouammoh, a.n. ahmed, on renewal failure rate classes of life distributions, stat. prob. lett. 14 (1992), 211–217. https://doi.org/10.1016/0167-7152(92)90024-y. [2] a. m. abouammoh, and a. n. ahmed, the class of better failure rate at specific age, derasat, 17 (1989), 27-37. [3] a.m. abouammoh, g.r. elkahlout, tests of new better than renewal used with randomly censored samples, parishankhyan samikha, 6 (1999), 47-63. [4] a.m. abouammoh, g.r. elkahlout, tests of randomly censored samples from new better than renewal used in expectation class of life distribution. derasat, 27 (2000), 32-54. [5] b. efron, the two-sample problem with censored data, in: proceedings of the fifth berkeley symposium on mathematical statistics and probability, vol. iv. university of california press, berkeley, 831-853, (1967). [6] b. klefsjö, the hnbue and hnwue classes of life distributions, naval res. log. quart. 29 (1982), 331-344. https://doi.org/10.1002/nav.3800290213. [7] g. r. elkahlout, tests for renewal failure rate properties with random censoring, far east j. theor. appl. sci. 1 (2022), 5-24. [8] h. w. lilliefors, on the kolmogorov-smirnov test for normality with mean and variance unknown. j. amer. stat. assoc. 62 (1967), 399-402. [9] i. elbatal, some aging classes of life distributions at specific age, int. math. forum, 2 (2007), 1445– 1456. [10] k.b. kulasekera, d.h. park, the class of better mean residual life at age t0, microelectronics reliability. 27 (1987), 725-735. https://doi.org/10.1016/0026-2714(87)90019-9. [11] l. kaplan, p. meier, nonparametric estimation from incomplete observations, j. amer. stat. assoc. 53 (1958), 457-481. [12] l. zehui, l. xiaohu, {ifr*t0} and {nbu*t0} classes of life distributions, j. stat. plan. inference. 70 (1998), 191-200. https://doi.org/10.1016/s0378-3758(97)00188-2. [13] m.a.w. mahmoud, m.e. moshref, a.m. gadallah, a.i. shawky, new classes at specific age: properties and testing hypotheses, j. stat. theory appl. 12 (2013), 106-109. https://doi.org/10.2991/jsta.2013.12.1.9. [14] m.c. bryson, m.m. siddiqui, some criteria for aging, j. amer. stat. assoc. 64 (1969), 1472-1483. [15] m. hollander, d.h. park, f. proschan, a class of life distributions for aging, j. amer. stat. assoc. 81 (1986), 91-95. [16] n. breslow, j. crowley, a large sample study of the life table and product limit estimates under random censorship, ann. stat. 2 (1974), 437-453. https://www.jstor.org/stable/2958131. https://doi.org/10.1016/0167-7152(92)90024-y https://doi.org/10.1002/nav.3800290213 https://doi.org/10.1016/0026-2714(87)90019-9 https://doi.org/10.1016/s0378-3758(97)00188-2 https://doi.org/10.2991/jsta.2013.12.1.9 https://www.jstor.org/stable/2958131 14 int. j. anal. appl. (2023), 21:80 [17] n. ebrahimi, m. habibullah, testing whether the survival distribution is new better than used of specified age, biometrika. 77 (1990), 212–215. https://doi.org/10.1093/biomet/77.1.212. [18] n. langberg, f. proschan, a. quinzi, estimating dependent life lengths, with applications to the theory of competing risks, ann. stat. 9 (1981), 157-167. https://www.jstor.org/stable/2240879. [19] p.v. pandit, m.p. anuradha, on testing exponentiality against new better than used of specified age, stat. methodol. 4 (2007), 13-21. https://doi.org/10.1016/j.stamet.2006.02.001. [20] t. ram c, s.r. jammalamadaka, j.n. zalkikar, testing an increasing failure rate average distribution with censored data, statistics. 20 (1989). 279-286. https://doi.org/10.1080/02331888908802170. [21] r.e. barlow, f. proschan, statistical theory of reliability and life testing: probability models, holt, reinhart and winston inc., new york, 1981. [22] r. gill, large sample behaviour of the product-limit estimator on the whole line, ann. stat. 11 (1983), 49-58. https://doi.org/10.1214/aos/1176346055. [23] t. rolski, mean residual life, int. stat. inst. 46 (1975), 266-270. [24] v. peterson, nonparametric estimation in the competing risks problem, technical report no. 13, division of biostatistics, stanford university, california, 1975. [25] w. feller, an introduction to probability theory and its applications, vol. 2. john wiley & sons, new york, 1966. [26] y. kumazawa, on testing whether new is better than used using randomly censored data, ann. stat. 15 (1987), 420-426. https://doi.org/10.1214/aos/1176350276. [27] y. kumazawa, tests for increasing failure rate with randomly censored data, statistics. 23 (1992), 17-25. https://doi.org/10.1080/02331889208802349. [28] y.y. chen, m. hollander, n.a. langberg, testing whether new is better than used with randomly censored data, ann. stat. 11 (1983), 267-274. https://www.jstor.org/stable/224048. https://doi.org/10.1093/biomet/77.1.212 https://www.jstor.org/stable/2240879 https://doi.org/10.1016/j.stamet.2006.02.001 https://doi.org/10.1080/02331888908802170 https://doi.org/10.1214/aos/1176346055 https://doi.org/10.1214/aos/1176350276 https://www.jstor.org/stable/224048. international journal of analysis and applications issn 2291-8639 volume 3, number 2 (2013), 112-118 http://www.etamaths.com common fixed point theorem in cone metric space for rational contractions r. uthayakumar and g. arockia prabakar∗ abstract. in this paper we prove the common fixed point theorem in cone metric space for rational expression in normal cone setting. our results generalize the main result of jaggi [10] and dass, gupta [11]. 1. introduction the banach contraction principle with rational expressions have been expanded and some fixed and common fixed point theorems have been obtained in [1], [2]. huang and zhang [3] initiated cone metric spaces, which is a generalization of metric spaces, by substituting the real numbers with ordered banach spaces. they have considered convergence in cone metric spaces, introduced completeness of cone metric spaces, and proved a banach contraction mapping theorem, and some other fixed point theorems involving contractive type mappings in cone metric spaces using the normality condition. later, various authors have proved some common fixed point theorems with normal and non-normal cones in these spaces [4], [5], [6], [7], [8]. quite recently muhammad arshad et al.[9] have introduced almost jaggi and gupta contraction in partially ordered metric space to prove the fixed point theorem. in this paper we prove the common fixed point theorem in cone metric space for rational expression in normal cone setting. our results generalize the main result of jaggi [10] and dass, gupta [11]. 1.1. basic facts and definitions. let e be a real banach space and p a subset of e. p is called a cone if and only if (i) p is closed, nonempty, and p 6= {0}; (ii) a,b ∈ r, a,b ≥ 0, x,y ∈ p ⇒ ax + by ∈ p (iii) p ∩ (−p) = {0} given a cone p ⊂ e, we define a partial ordering ≤ with respect to p by x ≤ y if and only if y −x ∈ p . we shall write x < y indicate that x ≤ y but x 6= y, while x � y will stand for y −x ∈ intp , intp denotes the interior of p . the cone p is called normal if there is a number m > 0 such that for all x,y ∈ e, 0 ≤ x ≤ y implies ‖x‖≤ m ‖y‖. the least positive number satisfying above is called the normal constant of p . in the following we always suppose e is a banach space, p is a cone in e with intp 6= ∅ and ≤ is partial ordering with respect to p . 2010 mathematics subject classification. 54e35, 54e50, 37c25, 54h25. key words and phrases. cone metric space; rational expression; fixed point. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 112 common fixed point theorem in cone metric space 113 definition 1.1. [3] let x be a nonempty set. suppose the mapping d : x×x → e satisfies (i) 0 ≤ d (x,y) for all x,y ∈ x and d (x,y) = 0 if and only if x = y; (ii) d (x,y) = d (y,x) for all x,y ∈ x; (iii) d (x,y) ≤ d (x,z) + d (z,y) for all x,y,z ∈ x. then d is called a cone metric on x, and (x,d) is called a cone metric space. example 1.1. [3] let e = r2, p = {(x,y) ∈ e|x,y ≥ 0} ⊂ r2, x = r and d : x ×x → e such that d (x,y) = (|x−y| ,α |x−y|), where α ≥ 0 is a constant. then (x,d) is a cone metric space. definition 1.2. [3] let (x,d) be a cone metric space, x ∈ x and {xn} be a sequence in x. then (i) {xn} converges to x whenever for every c ∈ e with 0 � c there is a natural number n such that d (xn,x) � c for all n ≥ n. (ii) {xn} is a cauchy sequence whenever for every c ∈ e with 0 � c there is a natural number n such that d (xn,xm) � c for all n,m ≥ n. definition 1.3. [3] let (x,d) is said to be a complete cone metric space, if every cauchy sequence is convergent in x. 2. main results definition 2.1. [9] let (x,d) be a cone metric space. a self mapping t on x is called an almost jaggi contraction if it satisfies the following condition: (1) d (tx,ty) ≤ αd (x,tx) d (y,ty) d (x,y) + βd (x,y) + lmin{d (x,ty) ,d (y,tx)} for all x,y ∈ x, where l ≥ 0 and α,β ∈ [0, 1) with α + β < 1. theorem 2.1. let (x,d) be a complete cone metric space and p a normal cone with normal constant m. let t : x → x be an almost jaggi contraction, for all x,y ∈ x where l ≥ 0 and α,β ∈ [0, 1) with α + β < 1. then t has a unique fixed point in x. proof : choose x0 ∈ x. set x1 = tx0, xn = txn−1 d (xn,xn+1) = d (txn−1,txn) ≤ [ αd (xn−1,txn−1) d (xn,txn) d (xn−1,xn) + βd (xn−1,xn) +lmin{d (xn−1,txn) ,d (xn,txn−1)}] ≤ [ αd (xn−1,xn) d (xn,xn+1) d (xn−1,xn) + βd (xn−1,xn) +lmin{d (xn−1,xn+1) ,d (xn,xn)}] ≤ d (xn,xn+1) + βd (xn−1,xn) (1 −α) d (xn,xn+1) ≤ βd (xn−1,xn) d (xn,xn+1) ≤ β 1 −α d (xn−1,xn) 114 uthayakumar and prabakar k = β 1−α, α + β < 1 0 < k < 1 and by induction, d (xn,xn+1) ≤ kd (xn−1,xn) . . ≤ knd (x0,x1) d (xn,xm) ≤ d (xn,xn+1) + d (xn+1,xn+2) + ...... + d (xn+m−1,xm) ≤ ( kn + kn+1 + ..... + kn+m−1 ) d (x0,x1) ≤ kn 1 −k d (x0,x1) we get ‖d (xn,xm)‖ ≤ m k n 1−k ‖d (x0,x1)‖ which implies that d (xn,xm) → 0 as n → ∞. hence xn is a cauchy sequence, so by completeness of x this sequence must be convergent in x. d (u,tu) ≤ d (u,xn+1) + d (xn+1,tu) ≤ d (u,xn+1) + d (txn,tu) ≤ d (u,xn+1) + αd (xn,txn) d (u,tu) d (xn,u) + βd (xn,u) +lmin{d (xn,tu) ,d (u,txn)} ≤ d (u,xn+1) + αd (xn,xn+1) d (u,u) d (xn,u) + βd (xn,u) +lmin{d (xn,u) ,d (u,xn+1)} ≤ d (u,xn+1) + βd (xn,u) + lmin{d (xn,u) ,d (u,xn+1)} so using the condition of normality of cone ‖d (u,t (u))‖≤ m (‖d (u,xn+1)‖ + β‖d (xn,u)‖ + lmin‖d (xn,u) ,d (u,xn+1)‖) as n → 0 we have ‖d (u,t (u))‖≤ 0. hence we get u = tu, u is a fixed point of t. definition 2.2. [10] let (x,d) be a cone metric space. a self mapping t on x is called jaggi contraction if it satisfies the following condition: (2) d (tx,ty) ≤ αd (x,tx) d (y,ty) d (x,y) + βd (x,y) for all x,y ∈ x and α,β ∈ [0, 1) with α + β < 1. corollary 2.1. let (x,d) be a complete cone metric space and p a normal cone with normal constant m. let t : x → x be a jaggi contraction (3) d (tx,ty) ≤ αd (x,tx) d (y,ty) d (x,y) + βd (x,y) for all x,y ∈ x and α,β ∈ [0, 1) with α + β < 1. then t has a unique fixed point in x. proof: set l = 0 in theorem 2.1. common fixed point theorem in cone metric space 115 theorem 2.2. let (x,d) be a complete cone metric space and p a normal cone with normal constant m. suppose the mappings s,t is called an almost jaggi contraction if it satisfies the following condition: (4) d (sx,ty) ≤ αd (x,sx) d (y,ty) d (x,y) + βd (x,y) + lmin{d (x,ty) ,d (y,sx)} for all x,y ∈ x where l ≥ 0 and α,β ∈ [0, 1) with α + β < 1. then each of s,t has a unique fixed point and these two fixed points coincide. proof : let x1 ∈ s (xo) and x2 = t (x1) such that x2n+1 = s (x2n), x2n+2 = t (x2n+1) d (x2n+1,x2n+2) = d (sx2n,tx2n+1) ≤ [ αd (x2n,sx2n) d (x2n+1,tx2n+1) d (x2n,x2n+1) + βd (x2n,x2n+1) +lmin{d (x2n,tx2n+1) ,d (x2n+1,sx2n)}] ≤ [ αd (x2n,x2n+1) d (x2n+1,x2n+2) d (x2n,x2n+1) + βd (x2n,x2n+1) +lmin{d (x2n,x2n+2) ,d (x2n+1,x2n+1)}] d (x2n+1,x2n+2) ≤ αd (x2n+1,x2n+2) + βd (x2n,x2n+1) (1 −α) d (x2n+1,x2n+2) ≤ βd (x2n,x2n+1) d (x2n+1,x2n+2) ≤ β 1 −α d (x2n,x2n+1) d (x2n+1,x2n+2) ≤ kd (x2n,x2n+1)(5) where k = β 1−α, α + β < 1 d (x2n+3,x2n+2) = d (s (x2n+2) ,t (x2n+1)) ≤ [ αd (x2n+2,sx2n+2) d (x2n+1,tx2n+1) d (x2n+2,x2n+1) + βd (x2n+2,x2n+1)(6) +lmin{d (x2n+2,tx2n+1) ,d (x2n+1,sx2n+2)}] ≤ [ αd (x2n+2,x2n+3) d (x2n+1,x2n+2) d (x2n+2,x2n+1) + βd (x2n+2,x2n+1) +lmin{d (x2n+2,x2n+2) ,d (x2n+1,x2n+2)}] d (x2n+3,x2n+2) ≤ αd (x2n+2,x2n+3) + βd (x2n+2,x2n+1) (1 −α) d (x2n+3,x2n+2) ≤ βd (x2n+2,x2n+1) d (x2n+3,x2n+2) ≤ β 1 −α d (x2n+2,x2n+1) d (x2n+3,x2n+2) ≤ kd (x2n+2,x2n+1) k = β 1−α, α + β < 1 add equation (6) and (7) we get ∞∑ n=1 d (xn,xn+1) ≤ ∞∑ n=1 knd (x0,x1)(7) = k 1 −k d (x0,x1) 116 uthayakumar and prabakar we get ‖d (xn,xn+1)‖≤ m k1−k ‖d (x0,x1)‖ which implies that d (xn,xn+1) → 0 as n → ∞. hence {xn} is a cauchy sequence, so by completeness of x this sequence must be convergent in x. we shall prove that u is a common fixed point of s and t . d (u,tu) ≤ d (u,x2n+1) + d (x2n+1,tu) ≤ d (u,x2n+1) + d (sx2n,tu) ≤ d (u,x2n+1) + [ αd (x2n,sx2n) d (u,tu) d (x2n,u) + βd (x2n,u) +lmin{d (x2n,tu) ,d (u,sx2n)}] ≤ d (u,x2n+1) + [ αd (x2n,x2n+1) d (u,u) d (x2n,u) + βd (x2n,u) +lmin{d (x2n,u) ,d (u,x2n+1)}] ≤ d (u,x2n+1) + βd (x2n,u) + lmin{d (x2n,u) ,d (u,x2n+1)} so using the condition of normality of cone ‖d (u,t (u))‖≤ m (‖d (u,x2n+1)‖ + β‖d (x2n,u)‖ + lmin‖d (x2n,u) ,d (u,x2n+1)‖) as n → 0 we have ‖d (u,t (u))‖≤ 0. hence we get u = tu, u is a fixed point of t . similarly d (u,s (u)) ≤ d (u,x2n+2) + d (x2n+2,su) ≤ d (u,x2n+2) + d (su,tx2n+1) ≤ d (u,x2n+2) + [ αd (u,su) d (x2n+1,tx2n+1) d (u,x2n+1) + βd (u,x2n+1) +lmin{d (u,tx2n+1) ,d (x2n+1,su)}] ≤ d (u,x2n+2) + [ αd (u,u) d (x2n+1,x2n+2) d (u,x2n+1) + βd (u,x2n+1) +lmin{d (u,x2n+2) ,d (x2n+1,u)}] ≤ d (u,x2n+2) + βd (u,x2n+1) + lmin{d (u,x2n+2) ,d (x2n+1,u)} so using the condition normality of cone ‖d (u,s (u))‖≤ m (‖d (u,x2n+1)‖ + β‖d (u,x2n+1)‖ + lmin‖d (u,x2n+2) ,d (x2n+1,u)‖) as n → 0 we have ‖d (u,s (u))‖≤ 0. hence we get u = su, u is a fixed point of s. definition 2.3. [9] let (x,d) be a cone metric space. a self mapping t on x is called dass and gupta contraction if it satisfies the following condition: (8) d (tx,ty) ≤ αd (y,ty) [1 + d (x,tx)] 1 + d (x,y) +βd (x,y)+lmin{d (x,tx) ,d (x,ty) ,d (y,tx)} for all x,y ∈ x, where l ≥ 0 and α,β ∈ [0, 1) with α + β < 1. theorem 2.3. let (x,d) be a complete cone metric space and p a normal cone with normal constant m. let t : x → x be a dass and gupta contraction, for all x,y ∈ x where l ≥ 0 and α,β ∈ [0, 1) with α + β < 1. then t has a unique fixed point in x. common fixed point theorem in cone metric space 117 proof : choose x0 ∈ x. set x1 = tx0, xn = txn−1 d (xn,xn+1) = d (txn−1,txn) ≤ [ αd (xn,txn) [1 + d (xn−1,txn−1)] 1 + d (xn−1,xn) + βd (xn−1,xn) +lmin{d (xn−1,txn−1) ,d (xn−1,txn) ,d (xn,txn−1)}] ≤ [ αd (xn,xn+1) [1 + d (xn−1,xn)] 1 + d (xn−1,xn) + βd (xn−1,xn) +lmin{d (xn−1,xn) ,d (xn−1,xn+1) ,d (xn,xn)}] (1 −α) d (xn,xn+1) ≤ βd (xn−1,xn) d (xn,xn+1) ≤ β 1 −α d (xn−1,xn) k = β 1−α, α + β < 1 0 < k < 1 and by induction, d (xn,xn+1) ≤ kd (xn−1,xn) . . ≤ knd (x0,x1) d (xn,xm) ≤ d (xn,xn+1) + d (xn+1,xn+2) + ...... + d (xn+m−1,xm) ≤ ( kn + kn+1 + ..... + kn+m−1 ) d (x0,x1) ≤ kn 1 −k d (x0,x1) we get ‖d (xn,xm)‖ ≤ m k n 1−k ‖d (x0,x1)‖ which implies that d (xn,xm) → 0 as n → ∞. hence xn is a cauchy sequence, so by completeness of x this sequence must be convergent in x. d (u,t (u)) ≤ d (u,xn+1) + d (xn+1,tu) ≤ d (u,xn+1) + d (txn,tu) ≤ d (u,xn+1) + αd (u,tu) [1 + d (xn,txn)] 1 + d (xn,u) + βd (xn,u) +lmin{d (xn,txn) ,d (xn,tu) ,d (u,txn)} ≤ d (u,xn+1) + αd (u,u) [1 + d (xn,xn+1)] 1 + d (xn,u) + βd (xn,u) +lmin{d (xn,xn+1) d (xn,u) ,d (u,xn+1)} ≤ d (u,xn+1) + βd (xn,u) + lmin{d (xn,xn+1) ,d (xn,u) ,d (u,xn+1)} so using the condition normality of cone ‖d (u,t (u))‖≤ m (‖d (u,xn+1)‖ + β‖d (xn,u)‖ + lmin‖d (xn,xn+1) ,d (xn,u) ,d (u,xn+1)‖) as n → 0 we have ‖d (u,t (u))‖≤ 0. hence we get u = tu, u is a fixed point of t . corollary 2.2. let (x,d) be a complete cone metric space and p a normal cone with normal constant m. let t : x → x a dass, gupta rationl contraction (9) d (tx,ty) ≤ αd (y,ty) [1 + d (x,tx)] 1 + d (x,y) + βd (x,y) 118 uthayakumar and prabakar for all x,y ∈ x and α,β ∈ [0, 1) with α + β < 1. then t has a unique fixed point in x. proof: set l = 0 in theorem 2.4. acknowledegement this research work has been supported by university grants commission (ugc sap ii) new delhi, india. references [1] b.fisher, common fixed points and constant mappings satisfying rational inequality, (math. sem. notes (univ kobe) (1978). [2] b.fisher, m.s khan, fixed points, common fixed points and constant mappings, studia sci. math. hungar. 11 (1978) 467-470. [3] l.g.huang and x.zhang, cone metric spaces and fixed point theorems of contractive mappings, journal of mathematical analysis and applications, 332 (2) (2007) 1468-1476. [4] s. rezapour, r. hamlbarani, some note on the paper cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 345 (2008) 719-724. [5] j.o.olaleru, some generalizations of fixed point theorems in cone metric spaces, fixed point theory and applications, (2009) article id 657914. [6] xiaoyan sun, yian zhao, guotao wang, new common fixed point theorems for maps on cone metric spaces, applied mathematics letters 23 (2010) 1033-1037. [7] mehdi asadi, s. mansour vaezpour, vladimir rakocevic, billy e. rhoades, fixed point theorems for contractive mapping in cone metric spaces, math. commun. 16 (2011) 147-155. [8] mahpeyker ozturkon, metin basarr, some common fixed point theorems with rational expressions on cone metric spaces over a banach algebra, hacettepe journal of mathematics and statistics, 41 (2) (2012) 211-222. [9] muhammad arshad, erdal karapinar jamshaid ahmad, some unique fixed point theorems for rational contractions in partially ordered metric spaces, journal of inequalities and applications 2013, 2013:248 doi:10.1186/1029-242x-2013-248. [10] d.s.jaggi, some unique fixed point theorems, indian j. pure appl. math. 8 (1977) 223-230. [11] dass, b.k., gupta, s, an extension of banach contraction principle through rational expressions, indian j. pure appl. math. 6, (1975) 1455-1458. department of mathematics, the gandhigram rural institute deemed university, gandhigram 624 302, dindigul, tamil nadu, india ∗corresponding author int. j. anal. appl. (2023), 21:84 on m∗-irresolute topological rings shallu sharma1,∗, naresh digra1, pooja saproo1, tsering landol2 1department of mathematics, university of jammu, jammu, india 2department of mathematics, cluster university of jammu, jammu, india ∗corresponding author: shallujamwal09@gmail.com abstract. the main aim of this paper is to introduce and study the new notions namely m∗-irresolute topological rings and m∗-irresolute topological r-modules by virtue of m∗-open sets. examples of an m∗-irresolute topological ring and module have been put forth. further, we provide several fundamental properties and characterizations of m∗-irresolute topological rings and m∗-irresolute topological rmodules. in addition, we shall define boundedness in these two structures and present several results on them. 1. introduction although topological rings are useful in many branches of mathematics, they are also fascinating on their own. since the 1940s, the theory of topological rings has been extensively developed, but primarily within the broader concept of a topological module. l.s. pontryagin obtained one of the first fundamental results in the theory of topological rings in the classification of locally compact skew fields, which was included in his famous book [14] on topological groups. some topological ring and module properties have also been noted in books [2, 11]. in-depth study has also been done in the last 50 years in the area of normed and banach algebras, which constitute one of the most significant classes of topological rings (see, for example [5–8,13]). besides, the theory of topological rings have been thoroughly investigated in a number of review papers and monographs (see, for example [1,9,10,15–18]). in 2016, a. devika and a. thilagavathi [4] introduced a new class of sets in topological spaces called m∗-open sets and studied some of its properties. by continuing the study of m∗-open sets and topological rings, in this paper we introduce received: jun. 12, 2023. 2020 mathematics subject classification. 54h13, 16w80, 16w99. key words and phrases. m∗-open sets; m∗-irresolute topological rings; m∗-irresolute topological r-modules. https://doi.org/10.28924/2291-8639-21-2023-84 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-84 2 int. j. anal. appl. (2023), 21:84 a new class of topological rings and modules called m∗-irresolute topological rings and m∗-irresolute topological r-modules. 2. preliminaries this section covers some fundamental definitions that will be utilized in the next sections. throughout this section, a and a′ will represent two topological spaces with topologies τ and τ′ respectively, on which no separation axioms are imposed. the notations int(u) and cl(u) stand for the interior and closure of a subset u of topological space a, respectively. definition 2.1. [3] let (a,τ) be a topological space and u ⊆ a. a point x ∈ a is said to be a θ-interior point of u if there exists an open set o containing x such that o ⊆ cl(o) ⊆ u. the set of all θ-interior points of u is said to be the θ-interior of u and is denoted by intθ(u). lemma 2.1. [3] for a subset u of a topological space (a,τ), the following statements are true: (1) intθ(u) is the union of all open sets of a whose closures are contained in u. (2) u is θ-open if and only if u = intθ(u). definition 2.2. [4] let (a,τ) be a topological space. then a subset u of a space (a,τ) is said to be, (1) an m∗-open set if u ⊆ int(cl(intθ(u))). (2) an m∗-closed set if u ⊇ cl(int(clθ(u))). the complement of an m∗-open set is called an m∗-closed. we denote the family of all m∗-open subsets (closed subsets) of a is denoted by τm∗ = m∗-o(a) (m∗-c(a)). lemma 2.2. [4] let (a,τ) be a topological space. then the following assertions are true: (1) the arbitrary union of m∗-open sets is m∗-open. (2) the arbitrary intersection of m∗-closed sets is m∗-closed. lemma 2.3. [4] for a topological space (a,τ) the family of all m∗-open sets of a forms a topology for a. definition 2.3. [4] let u be a subset of a space (a,τ). then, (1) the intersection of all m∗-closed sets containing u is called the m∗-closure of u and is denoted by clm∗(u) (sometimes by 〈u〉a). (2) the union of all m∗-open sets contained in u is called the m∗-interior of u and is denoted by intm∗(u). theorem 2.1. [4] let u be a subset of a topological space (a,τ). then the following statements hold: int. j. anal. appl. (2023), 21:84 3 (1) u is an m∗-open set if and only if u = intm∗(u). (2) u is an m∗-closed set if and only if u = clm∗(u). theorem 2.2. [4] let u and v be subsets of a topological space (a,τ). then the following statements are true: (1) clm∗(a−u) = a− intm∗(u). (2) intm∗(a−u) = a−clm∗(u). (3) if u ⊆ v then clm∗(u) ⊆ clm∗(v ) and intm∗(u) ⊆ intm∗(v ). (4) clm∗(clm∗(u)) = clm∗(u) and intm∗(intm∗(u)) = intm∗(u). definition 2.4. [4] a subset u of a topological space (a,τ) is said to be a m∗-neighborhood (briefly m∗-nbd) of a point x ∈ a if there exists an m∗-open set v such that x ∈ v ⊆ u. the family of m∗-neighborhoods of x ∈a is called the m∗-neighborhood system of x and denoted by m∗-nx. theorem 2.3. [4] a subset u of a topological space (a,τ) is said to be m∗-open if and only if it is m∗-neighborhood for every point x ∈ u. definition 2.5. [12] a mapping g : (a,τ) → (a′,τ′) is called m∗-irresolute at element x ∈a if for any m∗-open set u in a′ containing g(x) there exists an m∗-open set v in a containing x such that g(v ) ⊆ u. theorem 2.4. [12] let g : a→a′ be a function. then the following statements are equivalent. (1) g is m∗-irresolute. (2) for each x ∈a and each m∗-neighborhood u of g(x) in a′, there is an m∗-neighborhood v of x in a such that g(v ) ⊆ u. (3) the inverse image of every m∗-closed subset of a′ is an m∗-closed subset of a. (4) the inverse image of every m∗-open subset of a′ is an m∗-open subset of a. definition 2.6. [12] consider two topological spaces (a,τ) and (a′,τ′). a function g : a → a′ is called: (1) m∗-continuous if g−1(u) is m∗-open set in a for every open set u in a′. (2) pre-m∗-open if g(u) is m∗-open set in a′ for every m∗-open set u in a. (3) m∗-homeomorphism if g is bijective, m∗-irresolute and pre-m∗-open. definition 2.7. [12] let (a,τ) be a topological space. then a is said to be m∗-compact if each of its cover by m∗-open sets has finite subcover. a subset s ⊆ a is said to be m∗-compact if each cover of s by m∗-open sets of a has a finite subcover. definition 2.8. [1] let r be a ring and e be an r-module, s ⊆ e and q ⊆ r. if q.s ⊆ s, then the subset s is called q-stable. 4 int. j. anal. appl. (2023), 21:84 3. m∗-irresolute topological rings definition 3.1. an m∗-irresolute topological ring (r, +, ·,τ) is a ring (r, +, ·) endowed with some topology τ such that the following conditions are satisfied: (c1) for each r1, r2 ∈ r and m∗-open set u in r containing r1 + r2, there exist m∗-open sets u1 and u2 in r containing r1 and r2 respectively, such that u1 + u2 ⊆ u; (c2) for each r ∈ r and m∗-open set u in r containing −r, there exists m∗-open set u1 in r containing r such that −u1 ⊆ u; (c3) for each r1, r2 ∈r and m∗-open set u in r containing r1.r2, there exist m∗-open sets u1 and u2 in r containing r1 and r2 respectively, such that u1.u2 ⊆ u. remark 3.1. it is easy to verify that conditions (c1) and (c2) are equivalent to the following condition: for each r1, r2 ∈r and each m∗-open set u in r containing r1 −r2, there exist m∗-open sets u1 and u2 in r containing r1 and r2, respectively, such that u1 −u2 ⊆ u. example 3.1. consider z4, the ring of integers modulo 4. define topology τ on z4 as τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},z4}, τm∗ = {∅,{0, 2},{1, 3},z4}. then, (z4,⊕4,�4,τ) is an m∗-irresolute topological ring. example 3.2. consider the ring r = (z3,⊕3,�3). let τ = {∅,{0, 1},z3}, then τm∗ = {∅,z3}. thus, (z3,⊕3,�3,τ) is an m∗-irresolute topological ring. example 3.3. let (r, +, ·) be any ring and suppose τ be a discrete or indiscrete topology on r . then (r, +, ·,τ) is always an m∗-irresolute topological ring. definition 3.2. let (r, +, ·,τ) be an m∗-irresolute topological ring. a left r-module e is called an m∗-irresolute topological left r-module if on e is specified a topology such that the following conditions are satified: (m1) for every m,n ∈ e and m∗-open set u in e containing m + n, there exist m∗-open sets u1 and u2 in e containing m and n respectively, such that u1 + u2 ⊆ u; (m2) for every m ∈ e and m∗-open set u in e containing −m, there exists an m∗-open set u1 in e containing m such that −u1 ⊆ u; (m3) for every r ∈r and m ∈e and m∗-open set u in e containing r.m, there exist m∗-open set u1 in r containing r and m∗-open set u2 in e containing m such that u1.u2 ⊆ u. remark 3.2. in a similar manner, m∗-irresolute topological right r-modules over an m∗-irresolute topological ring can be investigated. any m∗-irresolute topological ring r is both an m∗-irresolute topological left r-module and an m∗-irresolute topological right r-module. remark 3.3. hereafter, by an m∗-irresolute topological r-module (unless otherwise asserted), we mean an m∗-irresolute topological left r-module. int. j. anal. appl. (2023), 21:84 5 example 3.4. consider the ring r = (z4,⊕4,�4) with topology τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},z4}, then τm∗ = {∅,{0, 2},{1, 3},z4}. therefore, r = (z4,⊕4,�4,τ) is an m∗-irresolute topological ring (see example 3.1). let τ′ be the indiscrete topology on the ring ({0, 2},⊕4,�4), τ′m∗ = {∅,{0, 2}}. then, left r-module {0, 2} with τ′ is an m∗-irresolute topological left r-module. note 3.1. by i(r), we denote the set of all invertible elements in an m∗-irresolute topological ring r. theorem 3.1. consider an m∗-irresolute topological ring r, an m∗-irresolute topological r-module e, a ∈r, m ∈e, and s be a subset in r, t a subset in e. then the following statements are true: (1) the mapping ωa : e → e, where ωa(x) = a.x, x ∈ e, is an m∗-irresolute mapping of the topological space e into itself. (2) the mapping ωm : r → e, where ωm(x) = x.m, x ∈ r is an m∗-irresolute mapping of the topological space r to the topological space e. (3) 〈s.t〉e ⊇〈s〉r.〈t〉e. proof. (1) let u be any m∗-open set in e containing ωa(x) = a.x, where x ∈ e is arbitrary. then, by definition 3.2, there exist m∗-open set u1 in r containing a and m∗-open set u2 in e containing x, such that u1.u2 ⊆ u. thus, ωa(u2) = a.u2 ⊆ u1.u2 ⊆ u. this proves that ωa is an m∗-irresolute mapping. (2) let x ∈ r and let u be any m∗-open set in e containing ωm(x) = x.m. then, by definition 3.2, there exist m∗-open set u1 in r containing x and m∗-open set u2 in e containing m, such that u1.u2 ⊆ u. thus, ωm(u1) = u1.m ⊆ u1.u2 ⊆ u. this proves that ωm is an m∗-irresolute mapping. (3) let x ∈ 〈s〉r.〈t〉e and let u be an m∗-open set in e containing x. then x = s.t, where s ∈ 〈s〉r and t ∈ 〈t〉e, and hence, there exist m∗-open set u1 in r containing s and m∗-open set u2 in e containing t, such that u1.u2 ⊆ u. by virtue of the fact that u1 ∩s 6= ∅ and u2 ∩t 6= ∅, elements s1 ∈ u1 ∩s and t1 ∈ u2 ∩t can be found. thus, s1.t1 ∈ s.t and s1.t1 ∈ u1.u2 ⊆ u, that is (s.t ) ∩u 6= ∅. consequently, 〈s.t〉e ⊇〈s〉r.〈t〉e. � theorem 3.2. let r be an m∗-irresolute topological ring, a ∈r, and let s and t be subsets in r. then the following statements are true: (1) the mappings σa : r→r and σ′a : r→r, where σa(x) = x.a and σ′a(x) = a.x for x ∈r, are m∗-irresolute mappings of the topological space r into itself; (2) the mappings γa : r → r and γ′a : r → r, where γa(x) = x + a and γ′a(x) = a + x, for x ∈r, are m∗-irresolute mappings of the topological space r into itself; (3) 〈s.t〉r ⊇〈s〉r.〈t〉r. 6 int. j. anal. appl. (2023), 21:84 proof. the proof of statements (1) and (2) directly follows from the definition of an m∗-irresolute topological ring r (see definition 3.1). statement (3) is proved in the same manner as statement (3) of theorem 3.1. � theorem 3.3. let s be an m∗-open set in an m∗-irresolute topological ring r. then s.a(respectively a.s) is m∗-open set in r for every a ∈i(r). proof. let x ∈ s.a be an arbitrary element. then x = s.a for some s ∈ s. now s = (s.a).a−1 = x.a−1 = σa−1 (x). by theorem 3.2(1), σa−1 : r→r is m∗-irresolute. thus for the m∗-open set s containing s = ∑ a−1 (x), there exists an m ∗-open set ux in r containing x such that σa−1 (ux ) ⊆ s. this results in ux.a−1 ⊆ s and hence ux ⊆ s.a. thus, s.a is an m∗-open set in r. � corollary 3.1. let s be an m∗-open set in an m∗-irresolute topological ring r. then s+a(respectively a + s) is m∗-open set in r for every a ∈r. remark 3.4. theorem 3.3 and corollary 3.1 do not hold for the choice of openness of s instead of m∗-openness of s. for instance, letr = (z4,⊕4,�4) with τ = {∅,{0},{1},{0, 1},{1, 3},{0, 1, 3},{0, 2},{0, 1, 2},z4}. then, by example 3.1, r is an m∗-irresolute topological ring. now, consider s = {0, 1, 2} ∈ τ and 3 ∈r. then s �4 3 = {0, 2, 3} /∈ τm∗. similarly, s ⊕4 3 = {0, 1, 3} /∈ τm∗. theorem 3.4. let h be an m∗-closed set in an m∗-irresolute topological ring r. then h.a (respectively a.h) is m∗-closed in r for every a ∈i(r). proof. let x ∈ 〈h.a〉r and u be an m∗-open set in r containing y, where y = x.a−1. then, by definition 3.1, there exist m∗-open sets u1 and u2 in r containing x and a−1 respectively such that u1.u2 ⊆ u. since x ∈ 〈h.a〉r, h.a∩u1 6= ∅. consider r ∈ h.a∩u1, then r.a−1 ∈ h∩(u1.u2) ⊆ h∩u. consequently, h ∩ u 6= ∅, and, hence y ∈ 〈h〉r. further, since h is m∗-closed, we have y ∈ h. therefore x ∈ h.a. thus 〈h.a〉r ⊆ h.a and since h.a ⊆ 〈h.a〉r is obvious. hence h.a is m∗closed. � corollary 3.2. let h be an m∗-closed set in an m∗-irresolute topological ring r. then h + a (respectively a + h) is m∗-closed in r for every a ∈r. remark 3.5. theorem 3.4 and corollary 3.2 do not hold for the choice of closedness of h instead of m∗-closedness of h. for example, let r = (z3,⊕3,�3) with τ = {∅,{0, 1},z3}. then, by example 3.2, r is an m∗irresolute topological ring. now, consider a closed set h = {2} in r and 2 ∈ r. then h �3 2 = h ⊕3 2 = {1} /∈ m∗-c(r). corollary 3.3. let s be an m∗-open set in an m∗-irresolute topological ring r. then −s ∈ τm∗. int. j. anal. appl. (2023), 21:84 7 theorem 3.5. let r be an m∗-irresolute topological ring with unity and a ∈ i(r). let e be an m∗-irresolute topological r-module, then (1) mapping ωa : e → e (see theorem 3.1(1)) is m∗-homeomorphism of the topological space e to itself; (2) mappings σa : r → r and σ′a : r → r (see theorem 3.2(1)) are m∗-homeomorphism of the topological space r into itself. proof. it is clearly evident that mappings ωa, σa and σ′a are bijective mappings. now, consider an m∗-open subset u of e, and x ∈ ωa(u). then x = ωa(u) = a.u for some u ∈ u. from u = a−1.x and definition 3.2, follows the existence of an m∗-open set u1 in e containing x such that a−1.u1 ⊆ u. then, u1 ⊆ a.u = ωa(u). this shows that ωa(u) is an m∗-neighborhood of each of its points and, hence, ωa(u) is an m∗-open subset of e. thus, ωa is pre-m∗-open mapping. next, consider an m∗-open set u in r. then σa(u) = u.a. by theorem 3.3, u.a is an m∗-open set in r. hence, σa is an pre-m∗-open mapping. in a similar manner, it can be proved that σ′a is also an pre-m∗-open mapping. in view of the fact that mappings ωa, σa and σ′a are m ∗-irresolute (see theorem 3.1 and theorem 3.2), the mappings ωa, σa and σ′a are m ∗-homeomorphism. � corollary 3.4. let r be an m∗-irresolute topological ring and a ∈r. then, the following statements are true: (1) the mappings γa : r → r and γ′a : r → r (see theorem 3.2) are m∗-homeomorhisms of the topological space r into itself. (2) the mapping γ : r → r, where γ(x) = −x, are m∗-homeomorphisms of the topological space r into itself. theorem 3.6. for any subset s of an m∗-irresolute topological ring r, 〈a.s〉r = a.〈s〉r for every a ∈i(r). proof. consider x ∈ 〈a.s〉r and y = a−1.x. let u be an m∗-open set in r containing y. then, there exist m∗-open sets u1 and u2 in r containing a−1 and x respectively such that u1.u2 ⊆ u. since x ∈ 〈a.s〉r, a.s∩u2 6= ∅. suppose r ∈ a.s ∩u2, then a−1.r ∈ s∩(u1.u2) ⊆ s∩u. thus each m∗open set containing y intersects s. hence, y ∈ 〈s〉r and so x ∈ a.〈s〉r. conversely, let x ∈ a.〈s〉r, then x = a.y for some y ∈ 〈s〉r. consider an m∗-open set u in r containing x. by hypothesis, there exist m∗-open sets u1 and u2 in r containing a and y respectively such that u1.u2 ⊆ u. as y ∈ 〈s〉r, there exists some element r ∈ s ∩ u2. then a.r ∈ (a.s) ∩ (u1.u2) ⊆ (a.s) ∩ u. thus, (a.s) ∩ u is non empty which implies each m∗-open set containing x intersects a.s. therefore, x ∈ 〈a.s〉r. hence 〈a.s〉r = a.〈s〉r. � corollary 3.5. for any subset s of an m∗-irresolute topological ring r, 〈a + s〉r = a + 〈s〉r for every a ∈r. 8 int. j. anal. appl. (2023), 21:84 lemma 3.1. let (a,τ) and (a′,τ′) be topological spaces, let g : a → a′ be an m∗-homeomorphism, and let u ⊆ a. then g(〈u〉a) = 〈g(u)〉a′. theorem 3.7. let s and t be subsets of an m∗-irresolute topological ring r. then the following statements are true: (1) 〈s + t〉r ⊇〈s〉r + 〈t〉r; (2) 〈−s〉r = −〈s〉r; (3) 〈s −t〉r ⊇〈s〉r −〈t〉r. proof. (1) let x ∈ 〈s〉r + 〈t〉r and let u be an m∗-open set in r containing x. then x = s + t, where s ∈ 〈s〉r and t ∈ 〈t〉r, and, hence, there exist m∗-open sets u1 and u2 in r containing s and t respectively, such that u1 + u2 ⊆ u. by virtue of the fact that u1 ∩s 6= ∅ and u2 ∩t 6= ∅, elements s1 ∈ u1 ∩s and t1 ∈ u2 ∩b can be found. then, s1 + t1 ∈ s + t and s1 + t1 ∈ u1 + u2 ⊆ u, that is (s + t ) ∩u 6= ∅. consequently, 〈s + t〉r ⊇〈s〉r + 〈t〉r. (2) since the mapping x 7→−x is an m∗-homeomorphism of the topological space r onto itself (see corollary 3.4), and in view of lemma 3.1, 〈−s〉r = −〈s〉r is valid. (3) inclusion 〈s −t〉r ⊇〈s〉r −〈t〉r results from (1) and (2). � 4. characterizations of m∗-irresolute topological rings theorem 4.1. let r be an m∗-irresolute topological ring with unity, a ∈i(r) and x ∈r. then the following statements are equivalent: (1) v is an m∗-neighborhood of element x of r; (2) v.a is an m∗-neighborhood of element x.a of r; (3) a.v is an m∗-neighborhood of element a.x of r. proof. consider the m∗-homeomorphism σa : r → r (see theorem 3.5). since x.a = σa(x) and v.a = σa(v ), then (1) ⇒ (2). the mapping ψa : r → r, where ψa(y) = a.(y.a−1) for y ∈ r, is the composition of the m∗-homeomorphisms σa−1 : r → r and σ′a : r → r (see theorem 3.5), hence, it is m∗homeomorphism. since ψa(x.a) = a.x and ψa(v.a) = a.v , then (2) ⇒ (3) by taking into consideration the m∗-homeomorphism σ′ a−1 : r→r, equalities σ′ a−1 (a.x) = x and σ′ a−1 (a.v ) = v are obtained. then from theorem 3.5, it follows that v is an m∗-neighborhood of the element x. thus, (3) ⇒ (1). � corollary 4.1. let r be an m∗-irresolute topological ring with unity and a ∈i(r). then the following statements are equivalent: (1) v is an m∗-neighborhood of element 0 of r; (2) v.a is an m∗-neighborhood of element 0 of r; (3) a.v is an m∗-neighborhood of element 0 of r. int. j. anal. appl. (2023), 21:84 9 corollary 4.2. let r be an m∗-irresolute topological ring and a ∈ r. then a subset v ⊆ r is an m∗-neighborhood of a if and only if v −a is an m∗-neighborhood of 0. proof. consider an m∗-neighborhood v of a. then there exists v ′ ∈ τm∗ such that a ∈ v ′ ⊆ v . since the mapping γ−a : r → r is an m∗-homeomorphism (see corollary 3.4(1)), then γ−a(v ′) = v ′ −a is an m∗-open set containing zero. furthermore, clearly v ′ − a ⊆ v − a. hence v − a is an m∗neighborhood of zero. by virtue of the fact that the mapping γa : r→r, where γa(x) = x +a, is an m∗-homeomorphism (see corollary 3.4(1)), the converse can be proved similarly. � definition 4.1. a collection ur of subsets of an m∗-irresolute topological ring r is called a basis of m∗-neighborhoods of r ∈r if any subset of ur is an m∗-neighborhood of r and any m∗-neighborhood of the element r contains some subset from ur. theorem 4.2. let u0 be a basis of m∗-neighborhoods of zero of an m∗-irresolute topological ring r. then the following conditions are satisfied: (n1) 0 ∈ ⋂ v∈u0 v ; (n2) for any subsets u and w from u0 there exists a subset v ∈ u0 such that v ⊆ u ∩w; (n3) for any subset v ∈ u0 there exists a subset w ∈ u0 such that w + w ⊆ v ; (n4) for any subset v ∈ u0 there exists a subset w ∈ u0 such that −w ⊆ v ; (n5) for any subset v ∈ u0 there exists a subset w ∈ u0 such that w.w ⊆ v ; (n6) for any subset v ∈ u0 and any element a ∈ r there exists a subset w ∈ u0 such that a.w ⊆ v and w.a ⊆ v . besides, if a ∈r, then ua = {a + u | u ∈ u0} is a basis of m∗-neighborhoods of the element a. proof. the definition of the basis of m∗-neighborhoods of an element in a topological space results in the fulfillment of conditions (n1) and (n2). consider an m∗-neighborhood v of 0 = 0 + 0. since r is an m∗-irresolute topological ring, then there are m∗-open sets v1 and v2 containing zero such that v1 + v2 ⊆ v . suppose that w = v1 ∩v2, then w is m∗-neighborhood of zero and w + w ⊆ v1 + v2 ⊆ v . in a similar way, the fulfillment of conditions (n4)-(n6) results from the fact that u0 is a basis of m∗-neighborhoods of zero in an m∗-irresolute topological ring r, from conditions (c2) and (c3) (see definition 3.1) with regard to −0 = 0, 0.0 = 0 and 0.a = 0 for any a ∈r. if a ∈ r, then the mapping γ′a : r → r is an m∗-homeomorphism in view of corollary 3.4, and, hence ua = {a + u | u ∈ u0} = {γ′a(u) | u ∈ u0} is a basis of m∗-neighborhoods of the element a. � theorem 4.3. let r be an m∗-irresolute topological ring and u0 be a basis of m∗-neighborhoods of zero of an m∗-irresolute topological r-module e. then conditions (n1) to (n4) of theorem 4.2, are satisfied together with the following conditions: 10 int. j. anal. appl. (2023), 21:84 (n5′) for any subset v ∈ u0 there exist a subset w ∈ u0 and an m∗-neighborhood u of zero in r such that u.w ⊆ v ; (n6′) for any subset v ∈ u0 and any element r ∈ r there exists a subset w ∈ u0 such that r.w ⊆ v ; (n7′) for any subset v ∈ u0 and any element a ∈ e there exists an m∗-neighborhood u of zero in r such that u.a ⊆ v . proof. to prove these conditions, it is necessary to use theorem 4.2, condition (m3) (see definition 3.2), and to take account of 0.a = r.0 = 0 for any r ∈r and a ∈e. � theorem 4.4. let s be a subset of an m∗-irresolute topological ring r and u0 be a basis of m∗neighborhoods of zero. then 〈s〉r = ⋂ u∈u0 (s + u). proof. let x ∈ 〈s〉r and u ∈ u0. thus, by condition n3 (see theorem 4.2), there exists w ∈ u0 such that −w ⊆ u. since x ∈ 〈s〉r, then (x + w ) ∩ s 6= ∅. this gives x ∈ s − w ⊆ s + u. therefore, 〈s〉r ⊆ s + u, and, hence 〈s〉r ⊆ ⋂ u∈u0 (s + u). now, let y ∈ ⋂ u∈u0 (s + u) and let u1 be an m∗-neighborhood of zero in r. let’s choose an m∗neighborhood u ∈ u0 of zero such that−u ⊆ u1. since y ∈ s + u, then s∩(y+u1) ⊇ s∩(y−u) 6= ∅. thus ⋂ u∈u0 (s + u) ⊆〈s〉r. � corollary 4.3. let u0 be a basis of m∗-neighborhoods of zero of an m∗-irresolute topological ring r. then ⋂ u∈u0 u is an m∗-closed set. proof. by theorem 4.4, ⋂ u∈u0 u is an m∗-closure of a subset {0} in r, and hence is m∗-closed set in r. � corollary 4.4. let u and w be m∗-neighborhoods of zero of an m∗-irresolute topological ring r such that w + w ⊆ u. then 〈w〉r ⊆ u. proof. it is obvious that the family u0 of all m∗-neighborhoods of zero in r is a basis of m∗neighborhoods of zero of r. by definition 4.1, for m∗-neighborhood w of zero, there exists v ∈ u0 such that v ⊆ w. in the view of theorem 4.4, 〈w〉r = ⋂ v∈u0 (w + v ) ⊆ w + v ⊆ w + w ⊆ u. thus, 〈w〉r ⊆ u. � corollary 4.5. letrbe an m∗-irresolute topological ring, e be an m∗-irresolute topologicalr-module, n ⊆ e and let u0(e) be a basis of m∗-neighborhoods of zero in e. then following statements are true: int. j. anal. appl. (2023), 21:84 11 (1) 〈n〉e = ⋂ u∈u0(e) (n + u). (2) ⋂ u∈u0(e) u is m∗-closed set in e. theorem 4.5. let r be an m∗-irresolute topological ring with unity, and let s be an m∗-compact subset in r. then the following assertions hold: (1) a.s is m∗-compact for each a ∈i(r); (2) a + s is m∗-compact for each a ∈r; proof. let λ be an indexing set and let {uβ : β ∈ λ} be an m∗-open cover of a.s. then, a.s ⊆ ⋃ β∈λ uβ implies that s ⊆ ⋃ β∈λ (a−1.uβ). since each uβ is m∗-open subset of an m∗-irresolute topological ring r, then by theorem 3.3, a−1.uβ ∈ m∗-o(r) for each β ∈ λ. further, since s is m∗-compact, therefore, s ⊆ ⋃ β∈λ′ (a−1.uβ) for some finite subset λ′ ⊆ λ. consequently, a.s ⊆ ⋃ β∈λ′ uβ. hence a.s is m∗-compact. analogously, the second part of the theorem is proved. � definition 4.2. let r and s be an m∗-irresolute topological rings. a mapping ω : r → s is said to be an m∗-irresolute (pre-m∗-open) homomorphism if ω is a homomorphism of rings and an m∗irresolute (pre-m∗-open) mapping of the topological spaces. a homomorphism of rings, which is at the same time m∗-irresolute and pre-m∗-open, is called an m∗-topological homomorphism. proposition 4.1. letrands be an m∗-irresolute topological rings, and ω : r→s be a homomorphic mapping of r to s. then (1) ω is an m∗-irresolute homomorphism if and only if ω−1(v ) is an m∗-neighborhood of zero in r for any m∗-neighborhood v of zero in s; (2) ω is a pre-m∗-open homomorphism if and only if ω(w ) is an m∗-neighborhood of zero in s for any m∗-neighborhood w of zero in r; (3) ω is a m∗-topological homomorphism if and only if for any m∗-neighborhoods u and u1 of zero in r and s, correspondingly, ω(u) and ω−1(u1) are m∗-neighborhoods of zero in s and r respectively. proof. (1) let v be an m∗-neighborhood of zero in s. then, there exists m∗-open set v ′ in s containing zero such that 0 ∈ v ′ ⊆ v . since ω is an m∗-irresolute homomorphism, then it is, in particular, m∗-irresolute at 0 ∈ r. further since ω(0) = 0, then by definition 2.5, there exists m∗-open set u in r containing zero such that 0 ∈ ω(u) ⊆ v ′. consequently, 0 ∈ u ⊆ ω−1(v ). this proves that ω−1(v ) is an m∗-neighborhood of zero in r. now let a ∈r and u′ be an m∗-neighborhood of ω(a) in s. then, due to corollary 4.2, u′−ω(a) is an m∗-neighborhood of zero in s. therefore, w = ω−1(u′−ω(a)) is an m∗-neighborhood of zero in r. consequently, w + a is an m∗-neighborhood of a ∈r, besides, 12 int. j. anal. appl. (2023), 21:84 ω(w + a) = ω(w ) + ω(a) = ω ( ω−1(u′ − ω(a)) ) + ω(a) ⊆ u′ − ω(a) + ω(a) ⊆ u′. therefore, the mapping ω : r→s is m∗-irresolute. (2) let w be an m∗-neighborhood of zero in r. then, there exists an m∗-open set u in r contaning zero such that 0 ∈ u ⊆ w. since ω is pre-m∗-open mapping, then ω(u) is an m∗-open set in s. besides, 0 = ω(0) ∈ ω(u) ⊆ ω(w ). thus, ω(w ) is an m∗-neighborhood of zero in s. conversely, consider an m∗-open subset v of r, and v ∈ ω(v ). then there exists an element u ∈ v such that v = ω(u). since v is m∗-neighborhood of point u in r, then v −u is m∗-neighborhood of zero in r. by hypothesis, ω(v −u) is an m∗-neighborhood of zero in s and ω(v −u) = ω(v ) − ω(u) = ω(v ) −v. thus, ω(v ) = ω(v −u) + v is an m∗-neighborhood of v in s. hence, ω(v ) is an m∗-neighborhood of each of its points. therefore, ω(v ) is an m∗-open set in s. this proves that ω is pre-m∗-open mapping. (3) follows from statements (1), (2) and definition 4.2. � definition 4.3. let r be an m∗-irresolute topological ring, e be an m∗-irresolute topological rmodule. a subset k of e is called bounded if for any m∗-neighbourhood u of zero in e there exists an m∗-neighborhood u1 of zero in r such that u1.k ⊆ u. an m∗-irresolute topological r-module e is called bounded if e is bounded subset of the module e. definition 4.4. let r be an m∗-irresolute topological ring. a subset k ⊆r is called bounded from left(right) if for any m∗-neighborhood u of zero in r there exists an m∗-neighborhood u1 of zero in r such that u1.k ⊆ u (correspondingly, k.u1 ⊆ u). a subset k of an m∗-irresolute topological ring is called bounded, if it bounded from left and from right. theorem 4.6. let r be an m∗-irresolute topological ring, e and e′ be an m∗-irrresolute topological r-modules. suppose ω : e →e′ is an m∗-irresolute homomorphism and a subset k is bounded in e. then the subset ω(k) is bounded in e′. proof. let u be an m∗-neighborhood of zero in m∗-irresolute topological r-module e′. then, due to proposition 4.1, ω−1(u) is an m∗-neighborhood of zero in m∗-irresolute topological r-module e. by boundedness of k, there exists an m∗-neighborhood u1 of zero in r such that u1.k ⊆ ω−1(u). then u1.ω(k) = ω(u1.k) ⊆ ω(ω−1(u)) ⊆ u, i.e ω(k) is a bounded subset in e′. � int. j. anal. appl. (2023), 21:84 13 theorem 4.7. let r and s be m∗-irresolute topological rings, ω : r → s be an m∗-topological homomorphism. let a subset k be bounded from left (bounded from right, bounded) in the ring r, then the subset ω(k) is bounded from left (bounded from right, bounded) in the ring s. proof. let u be an m∗-neighborhood of zero in m∗-irresolute topological ring s. then, due to proposition 4.1, ω−1(u) is an m∗-neighborhood of zero in m∗-irresolute topological ring r. since k is bounded from left in r, then there exists an m∗-neighborhood u1 of zero in r such that u1.k ⊆ ω−1(u). by proposition 4.1, ω(u1) is an m∗-neighborhood of zero in s. then ω(u1).ω(k) = ω(u1.k) ⊆ ω(ω−1(u)) ⊆ u, i.e the subset ω(k) is bounded from left in s. � when the subset k of the ring r is bounded from right or bounded, the proof is analogous. theorem 4.8. every m∗-compact set in an m∗-irresolute topological r-module e is bounded. proof. consider an m∗-compact subset k of e. let u be an m∗-neighborhood of zero in e. then, by condition (m3) (see definition 3.2), for any element n ∈ k there exist an m∗-open set vn in r containing zero and an m∗-open set wn in e containing n such that vn.wn ⊆ u. since {wn | n ∈ k} is an m∗-open cover of k, then there exist elements n1,n2, . . . ,ni ∈ k such that k ⊆ i⋃ j=1 wnj . then v = i⋂ j=1 vnj is an m ∗-neighborhood of zero in r and v.k ⊆ v.( i⋃ j=1 wnj ) ⊆ i⋃ j=1 (vnj.wnj ) ⊆ u. thus, k is bounded subset of m∗-irresolute topological r-module e. � theorem 4.9. m∗-closure of any bounded subset of an m∗-irresolute topological r-module e is bounded. proof. let k be any bounded subset of e and u be an m∗-neighbourhood of zero in e. then there exists m∗-closed neighborhood u1 of zero in e such that u1 ⊆ u. since k is bounded subset of e, then there exists an m∗-neighborhood u2 of zero in r such that u2.k ⊆ u1. then u2.〈k〉e ⊆〈u2〉r.〈k〉e ⊆〈u2.k〉e ⊆〈u1〉e = u1 ⊆ u. thus, 〈k〉e is a bounded set in e. � theorem 4.10. if k is a bounded from left subset of an m∗-irresolute topological ring r and n is a bounded subset of an m∗-irresolute topological r-module e, then k.n is bounded subset of e. 14 int. j. anal. appl. (2023), 21:84 proof. consider an m∗-neighborhood u of zero in e and m∗-neighborhood w of zero in r such that w.n ⊆ u. since k is bounded from left subset of r, then there exists m∗-neighborhood v of zero in r such that v.k ⊆ w. then v.(k.n) = (v.k).n ⊆ w.n ⊆ u. thus, k.n is bounded subset of e. � definition 4.5. let (r, +, ·,τ) be an m∗-irresolute topological ring. a subset s of r is called a subring of an m∗-irresolute topological ring r if s is a subring of r and s is endowed with the topology τ|s = {v ∩s | v ∈ τ}, induced by the topology τ. definition 4.6. let e be an m∗-irresolute topological r-module. a subset n of e is called a submodule of the m∗-irresolute topological r-module e if n is submodule of r-module e and r-module n is endowed with the topology induced by the topology of r-module e. theorem 4.11. every subring of an m∗-irresolute topological ring r is an m∗-irresolute topological ring. proof. consider a subring s of an m∗-irresolute topological ring r. let r1, r2 ∈ s and u be an m∗-open set in s containing r1 − r2. then u = v ∩ s, where v is m∗-open set in r containing r1 − r2. by definition 3.1, there exist m∗-open sets u1 and u2 in r containing r1 and r2 respectively such that u1 −u2 ⊆ v . then w1 = u1 ∩s and w2 = u2 ∩s are m∗-open sets in s containing r1 and r2 such that w1 −w2 ⊆ (u1 −u2) ∩s ⊆ v ∩s = u. now, suppose r1, r2 ∈ s and u be an m∗-open set in s containing r1.r2. then u = v ∩s, where v is m∗-open set in r containing r1.r2. by hypothesis, there exist m∗-open sets u1 and u2 in r containing r1 and r2 respectively such that u1.u2 ⊆ v . then w1 = u1 ∩ s and w2 = u2 ∩ s are m∗-open sets in s containing r1 and r2 respectively, besides w1.w2 ⊆ (u1.u2) ∩s ⊆ v ∩s = u. hence, s is an m∗-irresolute topological ring. � remark 4.1. a submodule n of an m∗-irresolute topological r-module e is an m∗-irresolute topological r-module. theorem 4.12. let q be a subset of an m∗-irresolute topological ring r, and s be a q-stable subset of an m∗-irresolute topological r-module e, then 〈s〉e is a 〈q〉r-stable subset. proof. since s is q-stable subset of e, then q.s ⊆ s (see definition 2.8). from theorem 3.1(3), 〈q〉r.〈s〉e ⊆〈q.s〉e ⊆〈s〉e. thus, 〈s〉e is a 〈q〉r-stable subset. � int. j. anal. appl. (2023), 21:84 15 proposition 4.2. let r be an m∗-irresolute topological ring and e be an m∗-irresolute topological r-module. let q be a subring of r, and n be a q-submodule of r-module e, then (1) 〈q〉r is a subring r; (2) 〈n〉e is a 〈q〉r-module. proof. let q be a subring of an r. thus q−q ⊆ q and q.q ⊆ q. by theorem 3.7(3), 〈q〉r − 〈q〉r ⊆〈q−q〉r ⊆〈q〉r. since q is a q-stable subset of the m-irresolute topological r-module r, then, due to theorem 4.12, 〈q〉r is a 〈q〉r-stable subset of the m∗-irresolute topological r-module r, that is 〈q〉r.〈q〉r ⊆〈q〉r. hence 〈q〉r is a subring r. since n is q-submodule of r-module e, then by theorem 3.7(1) 〈n〉e +〈n〉e ⊆〈n +n〉e ⊆〈n〉e. also n is q-stable subset of an m∗-irresolute topological r-module e, then by theorem 4.12, 〈n〉e is a 〈q〉r-stable subset of the m∗-irresolute topological r-module e, and since 〈q〉r is a subring r, then 〈n〉e is a 〈q〉r-module. � corollary 4.6. let q be a subring of an m∗-irresolute topological ring r with 〈q〉r = r and n be a q-submodule of an m∗-irresolute topological r-module e then, 〈n〉e is a submodule of the m∗-irresolute topological r-module e. in particular, the m∗-closure of any submodule of an m∗-irresolute topological r-module is also an m∗-irresolute topological r-module. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] v.i. arnautov, m.i. vodichar, a.v. mikhalev, introduction to the theory of topological rings and modules, kishinev, stiintsa, 1981. [2] n. bourbaki, éléments de mathématique, topologie générale, hermann, paris, (1965). [3] m. caldas, s. jafari, m.m. kovar, some properties of θ-open sets, divul. mat. 12 (2004), 161-169. [4] a. devika, a. thilagavathi, m∗-open sets in topological spaces, int. j. math. appl. 4 (2016), 1-8. [5] i.m. gelfand, d.a. raikov, g.e. shilov, commutative normed rings, physmatizdt, 1960. [6] a.ya. helemskii, topologies in banach and topological algebras, moscow university, moscow, 1986. [7] a.ya. helemskii, homology in banach and topological algebras, moscow university, moscow, 1986. [8] a.ya. helemskii, banach and normed algebras, nauka, 1989. [9] i. kaplansky, topological rings, amer. j. math. 69 (1947), 153-183. https://doi.org/10.2307/2371662. [10] i. kaplansky, topological rings, bull. amer. math. soc. 54 (1948), 809-826. https://doi.org/10.1090/ s0002-9904-1948-09058-9. [11] a.g. kurosh, lectures on general algebra, nauka, 1968. [12] r.m. latif, m–star–irresolute topological vector spaces, int. j. pure math. 7 (2021), 20-36. https://doi. org/10.46300/91019.2020.7.4. [13] m.a. najmark, normed rings, nauka, 1968. [14] l.s. pontryagin, continuous groups, nauka, 1973. https://doi.org/10.2307/2371662 https://doi.org/10.1090/s0002-9904-1948-09058-9 https://doi.org/10.1090/s0002-9904-1948-09058-9 https://doi.org/10.46300/91019.2020.7.4 https://doi.org/10.46300/91019.2020.7.4 16 int. j. anal. appl. (2023), 21:84 [15] f. szasz, on topological algebras and rings, i, mat. zapok. 13 (1962), 256-278. [16] f. szasz, on topological algebras and rings, ii, mat. zapok. 14 (1963), 74-87. [17] m. l. ursul, compact rings and their generalizations, kishinev, stiinsta, 1991. [18] s. warner, topological rings, north-holland mathematics studies, vol. 178, north-holland, (1993). 1. introduction 2. preliminaries 3. m*-irresolute topological rings 4. characterizations of m*-irresolute topological rings references int. j. anal. appl. (2023), 21:39 derivations of hilbert algebras aiyared iampan1,∗, r. alayakkaniamuthu2, p. gomathi sundari2, n. rajesh2 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2department of mathematics, rajah serfoji government college (affiliated to bharathidasan university), thanjavur 613005, tamilnadu, india ∗corresponding author: aiyared.ia@up.ac.th abstract. in this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations, and derivations of hilbert algebras and investigate some related properties. in addition, we define two subsets for a derivation d of a hilbert algebra x, kerd(x) and fixd(x), and we also take a look at some of their characteristics. 1. introduction and preliminaries logic algebras are a significant class of algebras among several other algebraic structures. the concept of hilbert algebras was introduced in early 50-ties by henkin [9] for some investigations of implication in intuitionistic and other non-classical logics. in 60-ties, these algebras were studied especially by diego [7] from algebraic point of view. diego [7] proved that hilbert algebras form a variety which is locally finite. hilbert algebras were treated by busneag [4,5] and jun [12] and some of their filters forming deductive systems were recognized. the study of derivations has continued, for example, in 2021, muangkarn et al. [14] studied fqderivations, and bantaojai et al. [3] studied derivations induced by an endomorphism of b-algebras. in 2022, bantaojai et al. [1, 2] studied derivations on d-algebras and b-algebras, and muangkarn et al. [13,15] studied derivations induced by an endomorphism of bg-algebras and d-algebras. iampan et al. [10,16,17] studied derivations on up-algebras. received: feb. 21, 2023. 2020 mathematics subject classification. 03g25. key words and phrases. hilbert algebra; (l, r)-derivation; (r, l)-derivation; derivation. https://doi.org/10.28924/2291-8639-21-2023-39 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-39 2 int. j. anal. appl. (2023), 21:39 the concepts of (l, r)-derivations, (r, l)-derivations, and derivations of hilbert algebras are introduced in this work along with several related features. in addition, we define two subsets for a derivation d of a hilbert algebra x, kerd(x) and fixd(x), and we also take a look at some of their characteristics. let’s go through the idea of hilbert algebras as it was introduced by diego [7] in 1966 before we start. definition 1.1. [7] a hilbert algebra is a triplet with the formula x =(x, ·,1), where x is a nonempty set, · is a binary operation, and 1 is a fixed member of x that is true according to the axioms stated below: (1) (∀x,y ∈x)(x · (y ·x)=1), (2) (∀x,y,z ∈x)((x · (y ·z)) · ((x ·y) · (x ·z))=1), (3) (∀x,y ∈x)(x ·y =1,y ·x =1⇒ x = y). in [8], the following conclusion was established. lemma 1.1. let x =(x, ·,1) be a hilbert algebra. then (1) (∀x ∈x)(x ·x =1), (2) (∀x ∈x)(1 ·x = x), (3) (∀x ∈x)(x ·1=1), (4) (∀x,y,z ∈x)(x · (y ·z)= y · (x ·z)). in a hilbert algebra x =(x, ·,1), the binary relation ≤ is defined by (∀x,y ∈x)(x ≤ y ⇔ x ·y =1), which is a partial order on x with 1 as the largest element. definition 1.2. [18] a nonempty subset d of a hilbert algebra x = (x, ·,1) is called a subalgebra of x if x ·y ∈d for all x,y ∈d. definition 1.3. [6] a nonempty subset d of a hilbert algebra x =(x, ·,1) is called an ideal of x if the following conditions hold: (1) 1∈d, (2) (∀x,y ∈x)(y ∈d ⇒ x ·y ∈d), (3) (∀x,y1,y2 ∈x)(y1,y2 ∈d ⇒ (y1 · (y2 ·x)) ·x ∈d). for any x,y in a hilbert algebra x =(x, ·,1), we define x ∨y by (y ·x) ·x. note that x ∨y is an upper bound of x and y for all x,y ∈x. a hilbert algebra x =(x, ·,1) is said to be commutative [11] int. j. anal. appl. (2023), 21:39 3 if for all x,y ∈x, (y ·x) ·x =(x ·y) ·y, that is, x ∨y = y ∨x. from [11], we know that (∀x ∈x)(x ∨x = x), (∀x ∈x)(x ∨1=1∨x =1). 2. main results in this section, we introduce the notions of an (l, r)-derivation, an (r, l)-derivation and a derivation of a hilbert algebra and study some of their basic properties. finally, we define two subsets kerd(x) and fixd(x) for a derivation d of a hilbert algebra x, and we consider some properties of these as well. definition 2.1. let x = (x, ·,1) be a hilbert algebra. a self-map d : x → x is called an (l, r)derivation of x if it satisfies the identity d(x ·y)= (d(x)·y)∨(x ·d(y)) for all x,y ∈x. similarly, a selfmap d :x →x is called an (r, l)-derivation of x if it satisfies the identity d(x·y)= (x·d(y))∨(d(x)·y) for all x,y ∈x. moreover, if d is both an (l, r)-derivation and an (r, l)-derivation of x, it is called a derivation of x. example 2.1. let x = {1,2,3,4} be a hilbert algebra with a fixed element 1 and a binary operation · defined by the following cayley table: · 1 2 3 4 1 1 2 3 4 2 1 1 3 4 3 1 2 1 4 4 1 2 3 1 define a self-map d :x →x by for any x ∈x, d(x)= { 1 if x 6=2 2 if x =2. then d is a derivation of x. definition 2.2. an (l, r)-derivation (resp., (r, l)-derivation, derivation) d of a hilbert algebra x = (x, ·,1) is said to be regular if d(1)=1. theorem 2.1. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) every (l, r)-derivation of x is regular, (2) every (r, l)-derivation of x is regular. proof. (1) assume that d is an (l, r)-derivation of x. then d(1)= d(1 ·1)= (d(1) ·1)∨(1 ·d(1))= 1∨d(1)=1. hence d is regular. 4 int. j. anal. appl. (2023), 21:39 (2) assume that d is an (r, l)-derivation of x. then d(1) = d(1 ·1) = (1 ·d(1))∨ (d(1) ·1) = d(1)∨1=1. hence d is regular. � corollary 2.1. every derivation of a hilbert algebra x =(x, ·,1) is regular. theorem 2.2. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then d(x)= x ∨d(x) for all x ∈x, (2) if d is an (r, l)-derivation of x, then d(x)= d(x)∨x for all x ∈x. proof. (1) assume that d is an (l, r)-derivation of x. then for all x ∈ x, d(x) = d(1 · x) = (d(1) ·x)∨ (1 ·d(x))= (1 ·x)∨d(x)= x ∨d(x). (2) assume that d is an (r, l)-derivation of x. then for all x ∈x, d(x)= d(1 ·x)= (1 ·d(x))∨ (d(1) ·x)= d(x)∨ (1 ·x)= d(x)∨x. � corollary 2.2. if d is a derivation of a hilbert algebra x =(x, ·,1), then d(x)∨x = x ∨d(x) for all x ∈x. definition 2.3. let d be an (l, r)-derivation (resp., (r, l)-derivation, derivation) of a hilbert algebra x =(x, ·,1). we define a subset kerd(x) of x by kerd(x)= {x ∈x : d(x)=1}. proposition 2.1. let d be an (l, r)-derivation of a hilbert algebra x = (x, ·,1). then the following properties hold: for any x,y ∈x, (1) x ≤ d(x), (2) d(x) ·y ≤ d(x ·y), (3) d(x ·d(x))=1, (4) d(d(x) ·x)=1, (5) x ≤ d(d(x)). proof. (1) for all x ∈x, x ·d(x)= x · (x ∨d(x))= x · ((d(x) ·x) ·x)=1. hence x ≤ d(x). (2) for all x,y ∈x, (d(x) ·y) ·d(x ·y)= (d(x) ·y) · ((d(x) ·y)∨ (x ·d(y)))= (d(x) ·y) · (((x · d(y)) · (d(x) ·y)) · (d(x) ·y))=1. hence d(x) ·y ≤ d(x ·y). (3) for all x ∈x, d(x ·d(x))= (d(x) ·d(x))∨ (x ·d(d(x)))=1∨ (x ·d(d(x)))=1. (4) for all x ∈x, d(d(x) ·x)= (d(d(x)) ·x)∨ (d(x) ·d(x))= (d(d(x)) ·x)∨1=1. (5) for all x ∈ x, d(d(x)) = d(x ∨ d(x)) = d((d(x) · x) · x) = (d(d(x) · x) · x)∨ ((d(x) · x) · d(x)) = (1 · x)∨ ((d(x) · x) · d(x)) = x ∨ ((d(x) · x) · d(x)) = (((d(x) · x) · d(x)) · x) · x. thus x ·d(d(x))= x · ((((d(x) ·x) ·d(x)) ·x) ·x)=1. hence x ≤ d(d(x)). � proposition 2.2. let d be an (r, l)-derivation of a hilbert algebra x = (x, ·,1). then the following properties hold: for any x,y ∈x, (1) x ·d(y)≤ d(x ·y), (2) d(x ·d(x))=1, int. j. anal. appl. (2023), 21:39 5 (3) d(d(x) ·x)=1. proof. (1) for all x,y ∈ x, (x ·d(y)) ·d(x · y) = (x ·d(y)) · ((x ·d(y))∨ (d(x) · y)) = (x ·d(y)) · (((d(x) ·y) · (x ·d(y))) · (x ·d(y)))=1. hence x ·d(y)≤ d(x ·y). (2) for all x ∈x, d(x ·d(x))= (x ·d(d(x)))∨ (d(x) ·d(x))= (x ·d(d(x)))∨1=1. (3) for all x ∈x, d(d(x) ·x)= (d(x) ·d(x))∨ (d(d(x)) ·x)=1∨ (d(d(x)) ·x)=1. � theorem 2.3. let d1,d2, . . . ,dn be (l, r)-derivations of a hilbert algebra x =(x, ·,1) for all n ∈n. then x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all x ∈ x. in particular, if d is an (l, r)-derivation of x, then x ≤ dn(x) for all n ∈n and x ∈x. proof. for n = 1, it follows from proposition 2.1 (1) that x ≤ d1(x) for all x ∈ x. let n ∈ n and assume that x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all x ∈x. let dn = dn(dn−1(. . .(d2(d1(x))) . . .)). then dn+1(dn)= dn+1(1 ·dn) = (dn+1(1) ·dn)∨ (1 ·dn+1(dn)) = (1 ·dn)∨ (1 ·dn+1(dn)) =dn ∨dn+1(dn) = (dn+1(dn) ·dn) ·dn. thus dn ·dn+1(dn)=dn · ((dn+1(dn) ·dn) ·dn)=1. therefore, dn ≤ dn+1(dn). by assumption, we get x ≤dn ≤ dn+1(dn)= dn+1(dn(dn−1(. . .(d2(d1(x))) . . .))) for all x ∈ x. hence x ≤ dn(dn−1(. . .(d2(d1(x))) . . .)) for all n ∈ n and x ∈ x. in particular, put d = dn for all n ∈n. hence x ≤ dn(dn−1(. . .(d2(d1(x))) . . .))= dn(x) for all n ∈n and x ∈x. � definition 2.4. an ideal d of a hilbert algebra x = (x, ·,1) is said to be invariant (with respect to an (l, r)-derivation (resp., (r, l)-derivation, derivation) d of x) if d(d)⊆d. theorem 2.4. every ideal of a hilbert algebra x = (x, ·,1) is invariant with respect to any (l, r)derivation of x. proof. let d be an ideal of x and d an (l, r)-derivation of x. let y ∈ d(d). then y = d(x) for some x ∈d. it follows that y ·x = d(x) ·x =1∈d, which implies y ∈d. thus d(d)⊆d. hence d is invariant with respect to an (l, r)-derivation d of x. � corollary 2.3. every ideal of a hilbert algebra x =(x, ·,1) is invariant with respect to any derivation of x. 6 int. j. anal. appl. (2023), 21:39 theorem 2.5. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then y ∨x ∈kerd(x) for all y ∈kerd(x) and x ∈x, (2) if d is an (r, l)-derivation of x, then y ∨x ∈kerd(x) for all y ∈kerd(x) and x ∈x. proof. (1) assume that d is an (l, r)-derivation of x. let y ∈kerd(x) and x ∈x. then d(y)=1. thusd(y∨x)= d((x·y)·y)= (d(x·y)·y)∨((x·y)·d(y))= (d(x·y)·y)∨((x·y)·1)= (d(x·y)·y)∨1=1. hence y ∨x ∈kerd(x). (2) assume that d is an (r, l)-derivation of x. let y ∈kerd(x) and x ∈x. then d(y)=1. thus d(y∨x)= d((x ·y)·y)= ((x ·y)·d(y))∨(d(x ·y)·y)= ((x ·y)·1)∨(d(x ·y)·y)=1∨(d(x ·y)·y)=1. hence y ∨x ∈kerd(x). � corollary 2.4. if d is a derivation of a hilbert algebra x = (x, ·,1), then y ∨ x ∈ kerd(x) for all y ∈kerd(x) and x ∈x. theorem 2.6. in a commutative hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x and for any x,y ∈ x is such that y ≤ x and y ∈ kerd(x), then x ∈kerd(x), (2) if d is an (r, l)-derivation of x and for any x,y ∈ x is such that y ≤ x and y ∈ kerd(x), then x ∈kerd(x). proof. (1) assume that d is an (l, r)-derivation of x. let x,y ∈ x be such that y ≤ x and y ∈kerd(x). then y ·x =1 and d(y)=1. thus d(x)= d(1 ·x)= d((y ·x) ·x)= d((x ·y) ·y)= (d(x ·y)·y)∨((x ·y)·d(y))= (d(x ·y)·y)∨((x ·y)·1)= (d(x ·y)·y)∨1=1. hence x ∈kerd(x). (2) assume that d is an (r, l)-derivation of x. let x,y ∈x be such that y ≤ x and y ∈kerd(x). then y ·x =1 and d(y)=1. thus d(x)= d(1 ·x)= d((y ·x) ·x)= d((x ·y) ·y)= ((x ·y) ·d(y))∨ (d(x ·y) ·y)= ((x ·y) ·1)∨ (d(x ·y) ·y)=1∨ (d(x ·y) ·y)=1. hence x ∈kerd(x). � corollary 2.5. if d is a derivation of a commutative hilbert algebra x =(x, ·,1) and for any x,y ∈x is such that y ≤ x and y ∈kerd(x), then x ∈kerd(x). theorem 2.7. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then y ·x ∈kerd(x) for all x ∈kerd(x) and y ∈x, (2) if d is an (r, l)-derivation of x, then y ·x ∈kerd(x) for all x ∈kerd(x) and y ∈x. proof. (1) assume that d is an (l, r)-derivation of x. let x ∈kerd(x) and y ∈x. then d(x)=1. thus d(y ·x)= (d(y)·x)∨(y ·d(x))= (d(y)·x)∨(y ·1)= (d(y)·x)∨1=1. hence y ·x ∈kerd(x). (2) assume that d is an (r, l)-derivation of x. let x ∈kerd(x) and y ∈x. then d(x)=1. thus d(y ·x)= (y ·d(x))∨(d(y)·x)= (y ·1)∨(d(y)·x)=1∨(d(y)·x)=1. hence y ·x ∈kerd(x). � corollary 2.6. if d is a derivation of a hilbert algebra x = (x, ·,1), then y · x ∈ kerd(x) for all x ∈kerd(x) and y ∈x. int. j. anal. appl. (2023), 21:39 7 theorem 2.8. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then kerd(x) is a subalgebra of x, (2) if d is an (r, l)-derivation of x, then kerd(x) is a subalgebra of x. proof. (1) assume that d is an (l, r)-derivation of x. by theorem 2.1 (1), we have d(1) = 1 and so 1 ∈ kerd(x) 6= ∅. let x,y ∈ kerd(x). then d(x) = 1 and d(y) = 1. thus d(x · y) = (d(x) · y)∨ (x · d(y)) = (1 · y)∨ (x · 1) = y ∨ 1 = 1. hence x · y ∈ kerd(x), so kerd(x) is a subalgebra of x. (2) assume that d is an (r, l)-derivation of x. by theorem 2.1 (2), we have d(1) = 1 and so 1 ∈ kerd(x) 6= ∅. let x,y ∈ kerd(x). then d(x) = 1 and d(y) = 1. thus d(x · y) = (x · d(y))∨ (d(x) · y) = (x · 1)∨ (1 · y) = 1∨ y = 1. hence x · y ∈ kerd(x), so kerd(x) is a subalgebra of x. � corollary 2.7. if d is a derivation of a hilbert algebra x =(x, ·,1), then kerd(x) is a subalgebra of x. definition 2.5. let d be an (l, r)-derivation (resp., (r, l)-derivation, derivation) of a hilbert algebra x =(x, ·,1). we define a subset fixd(x) of x by fixd(x)= {x ∈x : d(x)= x}. theorem 2.9. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then fixd(x) is a subalgebra of x, (2) if d is an (r, l)-derivation of x, then fixd(x) is a subalgebra of x. proof. (1) assume that d is an (l, r)-derivation of x. by theorem 2.1 (1), we have d(1) = 1 and so 1 ∈ fixd(x) 6= ∅. let x,y ∈ fixd(x). then d(x) = x and d(y) = y. thus d(x · y) = (d(x) ·y)∨ (x ·d(y))= (x ·y)∨ (x ·y)= x ·y. hence x ·y ∈fixd(x), so fixd(x) is a subalgebra of x. (2) assume that d is an (r, l)-derivation of x. by theorem 2.1 (2), we have d(1) = 1 and so 1 ∈ fixd(x) 6= ∅. let x,y ∈ fixd(x). then d(x) = x and d(y) = y. thus d(x · y) = (x ·d(y))∨ (d(x) ·y)= (x ·y)∨ (x ·y)= x ·y. hence x ·y ∈fixd(x), so fixd(x) is a subalgebra of x. � corollary 2.8. if d is a derivation of a hilbert algebra x =(x, ·,1), then fixd(x) is a subalgebra of x. theorem 2.10. in a hilbert algebra x =(x, ·,1), the following statements hold: (1) if d is an (l, r)-derivation of x, then x ∨y ∈fixd(x) for all x,y ∈fixd(x), (2) if d is an (r, l)-derivation of x, then x ∨y ∈fixd(x) for all x,y ∈fixd(x). proof. (1) assume that d is an (l, r)-derivation of x. let x,y ∈ fixd(x). then d(x) = x and d(y) = y. by theorem 2.9 (1), we get d(y · x) = y · x. thus d(x ∨ y) = d((y · x) · x) = (d(y ·x) ·x)∨((y ·x) ·d(x))= ((y ·x) ·x)∨((y ·x) ·x)= (y ·x) ·x = x∨y. hence x∨y ∈fixd(x). 8 int. j. anal. appl. (2023), 21:39 (2) assume that d is an (r, l)-derivation of x. let x,y ∈fixd(x). then d(x)= x and d(y)= y. by theorem 2.9 (2), we get d(y ·x)= y ·x. thus d(x∨y)= d((y ·x)·x)= ((y·x)·d(x))∨(d(y·x)·x)= ((y ·x) ·x)∨ ((y ·x) ·x)= (y ·x) ·x = x ∨y. hence x ∨y ∈fixd(x). � corollary 2.9. if d is a derivation of a hilbert algebra x = (x, ·,1), then x ∨ y ∈ fixd(x) for all x,y ∈fixd(x). 3. conclusion in this article, we introduced the ideas of (l, r)-derivations, (r, l)-derivations, and derivations of hilbert algebras, and deduced their significant features. additionally, two subsets kerd(x) and fixd(x) for a derivation d of a hilbert algebra x are defined. as a result, we have found that kerd(x) and fixd(x) are subalgebras of x. acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-uoe017). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] t. bantaojai, c. suanoom, j. phuto, a. iampan, a bi-endomorphism induces a new type of derivations on b-algebras, italian j. pure appl. math. 48 (2022), 336-348. [2] t. bantaojai, c. suanoom, j. phuto, a. iampan, a novel derivation induced by some binary operations on d-algebras, int. j. math. comput. sci. 17 (2022), 173-182. [3] t. bantaojai, c. suanoom, j. phuto, a. iampan, new derivations utilizing bi-endomorphisms on b-algebras, j. math. comput. sci. 11 (2021), 6420-6432. https://doi.org/10.28919/jmcs/6376. [4] d. busneag, a note on deductive systems of a hilbert algebra, kobe j. math. 2 (1985), 29-35. https://cir. nii.ac.jp/crid/1570854175360486400. [5] d. busneag, hilbert algebras of fractions and maximal hilbert algebras of quotients, kobe j. math. 5 (1988), 161-172. https://cir.nii.ac.jp/crid/1570572702603831808. [6] i. chajda, r. halas, congruences and ideals in hilbert algebras, kyungpook math. j. 39 (1999), 429-429. [7] a. diego, sur les algébres de hilbert, collect. logique math. ser. a (ed. hermann, paris), 21 (1966), 1-52. [8] w.a. dudek, on fuzzification in hilbert algebras, contrib. gen. algebra, 11 (1999), 77-83. [9] l. henkin, an algebraic characterization of quantifiers, fund. math. 37 (1950), 63-74. [10] a. iampan, derivations of up-algebras by means of up-endomorphisms, algebr. struct. appl. 3 (2016), 1-20. [11] y.b. jun, commutative hilbert algebras, soochow j. math. 22 (1996), 477-484. [12] y.b. jun, deductive systems of hilbert algebras, math. japon. 43 (1996), 51-54. https://cir.nii.ac.jp/crid/ 1571417124616097792. [13] p. muangkarn, c. suanoom, a. iampan, new derivations of d-algebras based on endomorphisms, int. j. math. comput. sci. 17 (2022), 1025-1032. [14] p. muangkarn, c. suanoom, p. pengyim, a. iampan, fq-derivations of b-algebras, j. math. comput. sci. 11 (2021), 2047-2057. https://doi.org/10.28919/jmcs/5472. [15] p. muangkarn, c. suanoom, a. yodkheeree, a. iampan, derivations induced by an endomorphism of bg-algebras, int. j. math. comput. sci. 17 (2022), 847-852. https://doi.org/10.28919/jmcs/6376 https://cir.nii.ac.jp/crid/1570854175360486400 https://cir.nii.ac.jp/crid/1570854175360486400 https://cir.nii.ac.jp/crid/1570572702603831808 https://cir.nii.ac.jp/crid/1571417124616097792 https://cir.nii.ac.jp/crid/1571417124616097792 https://doi.org/10.28919/jmcs/5472 int. j. anal. appl. (2023), 21:39 9 [16] k. sawika, r. intasan, a. kaewwasri, a. iampan, derivations of up-algebras, korean j. math. 24 (2016), 345–367. https://doi.org/10.11568/kjm.2016.24.3.345. [17] t. tippanya, n. iam-art, p. moonfong, a. iampan, a new derivation of up-algebras by means of upendomorphisms, algebra lett. 2017 (2017), 4. [18] j. zhan, z. tan, intuitionistic fuzzy deductive systems in hilbert algebra, southeast asian bull. math. 29 (2005), 813-826. https://doi.org/10.11568/kjm.2016.24.3.345 1. introduction and preliminaries 2. main results 3. conclusion references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 93-97 http://www.etamaths.com some equivalence theorems on absolute summability methods hi̇kmet seyhan özarslan∗ abstract. in this paper, we obtained necessary and sufficient conditions for the equivalence of two general summability methods. some new and known results are also obtained. 1. introduction let ∑ an be a given infinite series with partial sums (sn). let (pn) be a sequence of positive numbers such that pn = n∑ v=0 pv →∞ as n →∞, (p−i = p−i = 0, i ≥ 1). (1.1) the sequence-to-sequence transformation tn = 1 pn n∑ v=0 pvsv (1.2) defines the sequence (tn) of the riesz mean or simply the (n̄,pn) mean of the sequence (sn), generated by the sequence of coefficients (pn) (see [7]). the series ∑ an is said to be summable | n̄,pn; δ |k, k ≥ 1 and δ ≥ 0, if (see [5]) ∞∑ n=1 ( pn pn )δk+k−1 | ∆tn−1 |k< ∞, (1.3) where ∆tn−1 = − pn pnpn−1 n∑ v=1 pv−1av, n ≥ 1. (1.4) if we set δ = 0, then we obtain | n̄,pn |k summability (see [1]). in the special case pn = 1 for all values of n, then | n̄,pn; δ |k summability is the same as | c, 1; δ |k summability (see [6]). also if we take δ = 0 and k = 1, then we get | n̄,pn | summability. let (ϕn) be any sequence of positive real numbers. the series ∑ an is summable ϕ−|n̄,pn; δ|k, k ≥ 1, if (see [8]) ∞∑ n=1 ϕδk+k−1n |∆tn−1| k < ∞. (1.5) if we take ϕn = pn pn , then ϕ−|n̄,pn; δ|k summability reduces to |n̄,pn; δ|k summability. also, if we take δ = 0 and ϕn = pn pn , then ϕ−|n̄,pn; δ|k summability reduces to |n̄,pn|k summability. received 22nd july, 2016; accepted 18th september, 2016; published 3rd january, 2017. 2010 mathematics subject classification. 26d15, 40f05, 40g05, 40g99, 46a45. key words and phrases. riesz mean; absolute summability; hölder inequality; equivalence theorem; minkowski inequality; infinite series; sequence space. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 93 94 özarslan 2. known results we say that two summability methods are equivalent if they sum the same set of series (not necessarily to the same sums). bor and thorpe proved the following theorems about the | n̄,pn |k and | n̄,qn |k summability methods. theorem 2.1 ( [2]). let (pn) and (qn) be positive sequences and k ≥ 1. in order that | n̄,pn |k should be equivalent to | n̄,qn |k it is sufficient that qnpn qnpn = o(1) (2.1) and pnqn pnqn = o(1) (2.2) hold. theorem 2.2 ( [4]). let (pn) and (qn) be positive sequences and k ≥ 1. in order that every | n̄,pn |k summable series be | n̄,qn |k summable it is necessary that (2.1) holds. if (2.2) holds then (2.1) is also sufficient for the conclusion. theorem 2.3 ( [4]). let (pn) and (qn) be positive sequences and k ≥ 1. in order that ∣∣n̄,pn∣∣k be equivalent to | n̄,qn |k it is necessary and sufficient that (2.1) and (2.2) hold. 3. main results the aim of this paper is to generalize theorem 2.2 and theorem 2.3 for the general summability methods. now, we shall prove the following theorems. theorem 3.1. let k ≥ 1 and 0 ≤ δ < 1/k. (ϕn), (pn) and (qn) be sequences of positive numbers, and let m+1∑ n=v+1 ϕδk+k−1n q k n qknqn−1 = o { ϕδk+k−1v qk−1v qkv } as m →∞. (3.1) in order that every ϕ− | n̄,pn; δ |k summable series be ϕ− | n̄,qn; δ |k summable it is necessary that (2.1) holds. if (2.2) holds then (2.1) is also sufficient for the conclusion. it should be noted that if we take ϕn = pn pn , δ = 0 for ϕ− | n̄,pn; δ |k and ϕn = qnqn , δ = 0 for ϕ− | n̄,qn; δ |k, then theorem 3.1 reduces to theorem 2.2. in this case condition (8) reduces to m+1∑ n=v+1 qn qnqn−1 = o ( 1 qv ) as m →∞, (3.2) which always exists. it is also remarked that if we take ϕn = qn qn and qn = 1 for all values of n, then the condition (8) fulfils. we need the following lemma for the proof of theorem 3.1. lemma 3.2 ( [3]). let k ≥ 1 and a = (anv) be an infinite matrix. in order that a ∈ (lk, lk) it is necessary that anv = o(1) for all n,v ≥ 0. (3.3) 4. proof of theorem 3.1. firstly we prove sufficiency. let (tn) denote (n̄,pn) mean of the series ∑ an. then, by definition, we have tn = 1 pn n∑ v=0 pvsv = 1 pn n∑ v=0 (pn −pv−1)av. (4.1) if the series ∑ an is summable ϕ− ∣∣n̄,pn; δ∣∣k, then ∞∑ n=1 ϕδk+k−1n |∆tn−1| k < ∞. (4.2) some equivalence theorems on absolute summability methods 95 since, ∆tn−1 = ( − 1 pn−1 + 1 pn ) n∑ v=0 pv−1av = − pn pnpn−1 n∑ v=1 pv−1av, n ≥ 1, (p−1 = 0), (4.3) we have pn−1an = − pnpn−1 pn ∆tn−1 + pn−1pn−2 pn−1 ∆tn−2. (4.4) that is an = − pn pn ∆tn−1 + pn−2 pn−1 ∆tn−2. (4.5) if (tn) denotes the (n̄,qn) mean of the series ∑ an, similarly we have that tn = 1 qn n∑ v=0 qvsv = 1 qn n∑ v=0 (qn −qv−1)av. (4.6) hence ∆tn−1 = − qn qnqn−1 n∑ v=1 qv−1av, n ≥ 1, (q−1 = 0). (4.7) since av = − pv pv ∆tv−1 + pv−2 pv−1 ∆tv−2, by (15), we have that ∆tn−1 = − qn qnqn−1 n∑ v=1 qv−1 ( − pv pv ∆tv−1 + pv−2 pv−1 ∆tv−2 ) = qnpn pnqn ∆tn−1 + qn qnqn−1 n−1∑ v=1 qv−1 pv pv ∆tv−1 − qn qnqn−1 n−1∑ v=1 qv pv−1 pv ∆tv−1 = qnpn pnqn ∆tn−1 + qn qnqn−1 n−1∑ v=1 ∆tv−1 pv (qv−1pv −qvpv−1) . also, qv−1pv −qvpv−1 = qv−1pv −qv (pv −pv) = qv−1pv −qvpv + pvqv = (qv−1 −qv) pv + pvqv = −qvpv + pvqv, so that ∆tn−1 = qnpn qnpn ∆tn−1 − qn qnqn−1 n−1∑ v=1 pv pv qv∆tv−1 + qn qnqn−1 n−1∑ v=1 qv∆tv−1 = tn,1 + tn,2 + tn,3. to complete the proof of theorem 3.1, by minkowski’s inequality, it is sufficient to show that ∞∑ n=1 ϕδk+k−1n |tn,r| k < ∞, for r = 1, 2, 3. (4.8) 96 özarslan firstly, by using (6) and (12), we have m∑ n=1 ϕδk+k−1n |tn,1| k = m∑ n=1 ϕδk+k−1n ∣∣∣∣ qnpnqnpn ∆tn−1 ∣∣∣∣k = o(1) m∑ n=1 ϕδk+k−1n |∆tn−1| k = o(1) as m →∞. now, applying hölder’s inequality with indices k and k′, where k > 1 and 1 k + 1 k′ = 1, we have that m+1∑ n=2 ϕδk+k−1n |tn,2| k = m+1∑ n=2 ϕδk+k−1n ∣∣∣∣∣ qnqnqn−1 n−1∑ v=1 pv pv qv∆tv−1 ∣∣∣∣∣ k ≤ m+1∑ n=2 ϕδk+k−1n qkn qknq k n−1 { n−1∑ v=1 pv pv qv |∆tv−1| }k ≤ m+1∑ n=2 ϕδk+k−1n qkn qknqn−1 { n−1∑ v=1 ( pv pv )k qv |∆tv−1| k } × { 1 qn−1 n−1∑ v=1 qv }k−1 = o(1) m∑ v=1 ( pv pv )k qv |∆tv−1| k m+1∑ n=v+1 ϕδk+k−1n q k n qknqn−1 = o(1) m∑ v=1 ( pv pv )k qv |∆tv−1| k ϕδk+k−1v qk−1v qkv = o(1) m∑ v=1 ϕδk+k−1v |∆tv−1| k = o(1) as m →∞, by virtue of the hypotheses of theorem 3.1. finally, as in tn,2, we have that m+1∑ n=2 ϕδk+k−1n |tn,3| k = m+1∑ n=2 ϕδk+k−1n ∣∣∣∣∣ qnqnqn−1 n−1∑ v=1 qv∆tv−1 ∣∣∣∣∣ k ≤ m+1∑ n=2 ϕδk+k−1n qkn qknq k n−1 { n−1∑ v=1 qv qv qv |∆tv−1| }k ≤ m+1∑ n=2 ϕδk+k−1n qkn qknqn−1 { n−1∑ v=1 ( qv qv )k qv |∆tv−1| k } × { 1 qn−1 n−1∑ v=1 qv }k−1 = o(1) m∑ v=1 ( qv qv )k qv |∆tv−1| k m+1∑ n=v+1 ϕδk+k−1n q k n qknqn−1 = o(1) m∑ v=1 ( qv qv )k qv |∆tv−1| k ϕδk+k−1v qk−1v qkv = o(1) m∑ v=1 ϕδk+k−1v |∆tv−1| k = o(1) as m →∞, some equivalence theorems on absolute summability methods 97 by virtue of the hypotheses of theorem 3.1. therefore, we get ∞∑ n=1 ϕδk+k−1n |tn,r| k = o(1) as m →∞, for r = 1, 2, 3. this completes the proof of sufficiency of theorem 3.1. for the proof of the necessity, we consider the series to series version of (2) i.e. for n ≥ 1, let bn = pn pnpn−1 n∑ v=1 pv−1av, cn = qn qnqn−1 n∑ v=1 qv−1av. a simple calculation shows that for n ≥ 1 cn = qn qnqn−1 n−1∑ v=1 bv pv (qv−1pv −qvpv−1) + qnpn pnqn bn. from this we can write down at once the matrix a that transforms ( ϕ δk+k−1 k n bn ) into ( ϕ δk+k−1 k n cn ) . thus every ϕ− ∣∣n̄,pn; δ∣∣k summable series is ϕ− ∣∣n̄,qn; δ∣∣k summable if and only if a ∈ (lk, lk). by lemma 3.2, it is necessary that the diagonal terms of a must be bounded, which gives that (6) must hold. theorem 3.2. let (pn) and (qn) be positive sequences satisfying the condition (8), k ≥ 1 and 0 ≤ δ < 1/k. in order that ϕ− | n̄,pn; δ |k be equivalent to ϕ− | n̄,qn; δ |k it is necessary and sufficient that (6) and (7) hold. it should be remarked that if we set ϕn = pn pn , δ = 0 for ϕ− | n̄,pn; δ |k and ϕn = qnqn , δ = 0 for ϕ− | n̄,qn; δ |k, then theorem 3.2 reduces to theorem 2.3. proof of theorem 3.2. interchange the roles of (pn) and (qn) in theorem 3.1. references [1] h. bor, on two summability methods, math. proc. camb. philos. soc. 97 (1985), 147–149. [2] h. bor and b. thorpe, on some absolute summability methods, analysis 7 (1987), 145–152. [3] h. bor, on the relative strength of two absolute summability methods, proc. amer. math. soc. 113 (1991), 1009– 1012. [4] h. bor and b. thorpe, a note on two absolute summability methods, analysis 12 (1992), 1–3. [5] h. bor, on local property of | n̄,pn; δ |k summability of factored fourier series, j. math. anal. appl. 179 (2) (1993), 646–649. [6] t. m. flett, some more theorems concerning the absolute summability of fourier series and power series, proc. london math. soc. 8 (3) (1958), 357-387. [7] g. h. hardy, divergent series, oxford university press, oxford, 1949. [8] h. seyhan, on the local property of ϕ− | n̄,pn; δ |k summability of factored fourier series, bull. inst. math. acad. sinica 25 (1997), 311-316. department of mathematics, erciyes university, 38039 kayseri, turkey ∗corresponding author: seyhan@erciyes.edu.tr; hseyhan38@gmail.com 1. introduction 2. known results 3. main results 4. proof of theorem 3.1. references international journal of analysis and applications issn 2291-8639 volume 13, number 1 (2017), 22-31 http://www.etamaths.com peter-weyl theorem for homogeneous spaces of compact groups arash ghaani farashahi∗ abstract. this paper presents a structured formalism for a constructive generalization of the peterweyl theorem over homogeneous spaces of compact groups. let h be a closed subgroup of a compact group g and µ be the normalized g-invariant measure on the compact left coset space g/h. we then present an abstract th-version of the peter-weyl theorem for the hilbert function space l 2(g/h,µ). 1. introduction the abstract aspects of harmonic analysis over homogeneous spaces of compact non-abelian groups or precisely left coset (resp. right coset) spaces of non-normal subgroups of compact non-abelian groups is placed as building blocks for classical harmonic analysis [5, 7], coherent states analysis [8, 11], theoretical and particle physics [1]. over the last decades, abstract and computational aspects of plancherel formulas over symmetric spaces have achieved significant popularity in geometric analysis, mathematical physics and scientific computing (computational engineering), see [2–4, 6, 12, 13] and references therein. the peter-weyl theorem is a fundamental result in the theory of classical harmonic analysis, applying to compact topological groups that are not necessarily abelian. it was initially proved by hermann weyl and fritz peter, in the setting of a compact topological groups [15]. the theorem is a collection of results generalizing the significant facts about the decomposition of the regular representations of finite groups, as presented by f. g. frobenius and issai schur, see [1, 9, 10] and classical references therein. the theorem has three parts. the first part states that the matrix coefficients of irreducible representations of a compact groups g are dense in the space c(g) of continuous complex-valued functions on g, and thus also in the space l2(g) of square-integrable functions. the second part asserts the complete reducibility of unitary representations of g. the final part then asserts that the regular representation of g on l2(g) decomposes as the direct sum of all irreducible unitary representations. moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of l2(g). let g be a compact group and h be a closed subgroup of g. also, let g/h be the left coset space of h in g and ĝ/h be the abstract dual space of g/h. let µ be the normalized g-invariant measure over the homogeneous space g/h with respect to the probability measures of h and g, associated to the weil’s formula. then we present a structured formalism for a constructive generalization of the peter-weyl theorem for the hilbert function space l2(g/h,µ). the paper is organized as follows. section 2 is devoted to fixing notations and a brief summary on the non-abelian fourier analysis of compact groups, general formalism of the peter-weyl theorem, and preliminaries and classical results on harmonic analysis of compact homogeneous spaces. then we present a systematic study of abstract harmonic analysis over the hilbert function space l2(g/h,µ). in section 4, using the abstract notion of the dual space ĝ/h of the homogeneous space g/h, we prove that the hilbert function space l2(g/h,µ) satisfies a canonical decomposition into a direct sum of some closed and mutually orthogonal subspaces. this decomposition coincides with the peter-weyl received 31st june, 2016; accepted 10th august, 2016; published 3rd january, 2017. 2010 mathematics subject classification. primary 47a67, 43a85. key words and phrases. compact group; homogeneous space; g-invariant measure; peter-weyl theorem. c©2017 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 22 peter-weyl theorem for homogeneous spaces of compact groups 23 decomposition, when h is a normal subgroup of g. this result can be considered as a generalization of the peter-weyl theorem for homogeneous spaces of compact groups. 2. preliminaries and notations let h be a separable hilbert space. an operator t ∈ b(h) is called a hilbert-schmidt operator if for one, hence for any orthonormal basis {ek} of h we have ∑ k ‖tek‖ 2 < ∞. the set of all hilbert-schmidt operators on h is denoted by hs(h) and for t ∈ hs(h) the hilbert-schmidt norm of t is ‖t‖2hs = ∑ k ‖tek‖ 2. the set hs(h) is a self adjoint two sided ideal in b(h) and if h is finite-dimensional we have hs(hπ) = b(h). an operator t ∈ b(h) called trace-class, whenever ‖t‖tr = tr[|t |] < ∞, where tr[t] = ∑ k〈tek,ek〉 and |t| = (tt ∗)1/2, see [14]. let g be a compact group with the haar measure dx, h be a closed subgroup of g with the left haar measure dh. the left coset space g/h is considered as a locally compact homogeneous space that g acts on it from the left and q : g → g/h given by x 7→ q(x) := xh is the surjective canonical mapping. the function space c(g/h) consists of all functions th(f), where (see proposition 2.48 of [1]) f ∈c(g) and th(f)(xh) = ∫ h f(xh)dh. (2.1) let µ be a radon measure on g/h and x ∈ g. the translation µx of µ is defined by µx(e) = µ(xe), for borel subsets e of g/h. the measure µ is called g-invariant if µx = µ, for x ∈ g. if g is compact, the homogeneous space g/h has a g-invariant measure µ, which satisfies the following weil’s formula, for f ∈ l1(g) (see [1]) ∫ g/h th(f)(xh)dµ(xh) = ∫ g f(x)dx. (2.2) if µ is a g-invariant measure on the homogeneous space g/h and p ≥ 1, the notation lp(g/h,µ) stands for the banach space of all equivalence classes of µ-measurable complex valued functions φ : g/h → c such that ‖φ‖lp(g/h,µ) < ∞. each irreducible representation of g is finite dimensional and every unitary representation of g is a direct sum of irreducible representations, see [1, 9]. the set of of all unitary equivalence classes of irreducible unitary representations of g is denoted by ĝ. this definition of ĝ is in essential agreement with the classical definition when g is abelian, since each character of an abelian group is a one dimensional representation of g. if π is any unitary representation of g, for u,v ∈ hπ the functions πu,v(x) = 〈π(x)u,v〉 are called matrix elements of π. if {ej} is an orthonormal basis for hπ, then πij means πei,ej . the notation eπ is used for the linear span of the matrix elements of π and the notation e is used for the linear span of ⋃ [π]∈ĝeπ. the peter-weyl theorem (see [1, 9]) guarantees that if g is a compact group, e is uniformly dense in c(g), l2(g) = ⊕ [π]∈ĝeπ, and {d −1/2 π πij : i,j = 1...dπ, [π] ∈ ĝ} is an orthonormal basis for l2(g). using the peter-weyl theorem, for f ∈ l2(g) we have f = ∑ [π]∈ĝ dπ∑ i,j=1 cπij(f)πij, (2.3) where cπi,j(f) = dπ〈f,πij〉l2(g). 3. abstract harmonic analysis over homogeneous spaces of compact groups throughout this article we assume that h is a closed subgroup of a compact group g with normalized haar measures dh and dx respectively. we start this section with an extension of the linear map th : c(g) → c(g/h) for other function spaces related to the homogeneous space g/h. if p = 1, it is easy to check that ‖th(f)‖l1(g/h,µ) ≤ ‖f‖l1(g). proposition 3.1. let h be a closed subgroup of a compact group g. the linear map th : c(g) → c(g/h) is a uniformly continuous. 24 a. ghaani farashahi proof. let f ∈c(g) and x ∈ g. then we have |th(f)(xh)| = ∣∣∣∣ ∫ h f(xh)dh ∣∣∣∣ ≤ ∫ h |f(xh)|dh ≤‖f‖sup (∫ h dh ) = ‖f‖sup, which implies ‖th(f)‖sup ≤‖f‖sup. � next we prove that the linear map th is norm-decreasing in l 2-spaces. theorem 3.1. let h be a closed subgroup of a compact group g, µ be the normalized g-invariant measure on g/h associated to the weil’s formula. the linear map th : c(g) →c(g/h) has a unique extension to a bounded linear map from l2(g) onto l2(g/h,µ). proof. we shall show that, each f ∈c(g) satisfies ‖th(f)‖l2(g/h,µ) ≤‖f‖l2(g). let f ∈c(g). using compactness of h and the weil’s formula we have ‖th(f)‖2l2(g/h,µ) = ∫ g/h |th(f)(xh)|2dµ(xh) = ∫ g/h ∣∣∣∣ ∫ h f(xh)dh ∣∣∣∣2 dµ(xh) ≤ ∫ g/h (∫ h |f(xh)|dh )2 dµ(xh) ≤ ∫ g/h ∫ h |f(xh)|2dhdµ(xh) = ∫ g/h ∫ h |f|2(xh)dhdµ(xh) = ∫ g/h th(|f|2)(xh)dµ(xh) = ∫ g |f(x)|2dx = ‖f‖2l2(g). thus, we can extend th to a bounded linear operator from l 2(g) onto l2(g/h,µ), which we still denote it by th. � let jp(g,h) := {f ∈ lp(g) : th(f) = 0}. then, j2(g,h)⊥ is the orthogonal completion of the closed subspace j2(g,h) in l2(g). as an immediate consequence of proposition 3.1 we deduce the following corollary. corollary 3.1. let h be a closed subgroup of a compact group g and µ be a g-invariant measure on g/h. the linear map th : l 2(g) → l2(g/h,µ) is partial isometric. proof. let ϕ ∈ l2(g/h,µ) and ϕq := ϕ◦q. then, we have ϕq ∈ l2(g). indeed, ‖ϕq‖2l2(g) = ∫ g |ϕq(x)|2dx = ∫ g/h th ( |ϕq|2 ) (xh)dµ(xh) = ∫ g/h (∫ h |ϕq(xh)|2dh ) dµ(xh) = ∫ g/h (∫ h |ϕ(xhh)|2dh ) dµ(xh) = ∫ g/h (∫ h |ϕ(xh)|2dh ) dµ(xh) = ∫ g/h |ϕ(xh)|2 (∫ h dh ) dµ(xh) = ∫ g/h |ϕ(xh)|2dµ(xh) = ‖ϕ‖2l2(g/h,µ). peter-weyl theorem for homogeneous spaces of compact groups 25 also t∗h(ϕ) = ϕq and tht ∗ h(ϕ) = ϕ. because using the weil’s formula, for all f ∈ l 2(g) we achieve 〈t∗h(ϕ),f〉l2(g) = 〈ϕ,th(f)〉l2(g/h,µ) = ∫ g/h ϕ(xh)th(f)(xh)dµ(xh) = ∫ g/h ϕ(xh)th(f)(xh)dµ(xh) = ∫ g/h th(ϕq.f)(xh)dµ(xh) = ∫ g ϕq(x)f(x)dx = 〈ϕq,f〉l2(g). now a straightforward calculation implies th = tht ∗ hth. then by theorem 2.3.3 of [14], th is a partial isometric operator. � we can conclude the following corollaries as well. corollary 3.2. let h be a closed subgroup of a compact group g. let pj2(g,h) and pj2(g,h)⊥ be the orthogonal projections onto the closed subspaces j2(g,h) and j2(g,h)⊥ respectively. then, for each f ∈ l2(g) and a.e. x ∈ g we have pj2(g,h)⊥(f)(x) = th(f)(xh), pj2(g,h)(f)(x) = f(x) −th(f)(xh). (3.1) corollary 3.3. let h be a compact subgroup of a locally compact group g and µ be a g-invariant measure on g/h. the following statements hold. (1) j2(g,h)⊥ = {ψq : ψ ∈ l2(g/h,µ)}. (2) for all f ∈j2(g,h)⊥ and each h ∈ h we have rhf = f. (3) for all ψ ∈ l2(g/h,µ) we have ‖ψq‖l2(g) = ‖ψ‖l2(g/h,µ). (4) for all f,g ∈j2(g,h)⊥ we have 〈th(f),th(g)〉l2(g/h,µ) = 〈f,g〉l2(g). remark 3.1. invoking corollary 3.3 one can regard l2(g/h,µ) as a closed subspace of l2(g), that is the subspace consists of all f ∈ l2(g) which satisfies rhf = f for all h ∈ h. then theorem 3.1 and proposition 3.1 guarantees that the linear map th : l 2(g) → l2(g/h,µ) ⊂ l2(g) is an orthogonal projection. 4. peter-weyl theorem for homogeneous spaces of compact groups for a closed subgroup h of g, define h⊥ := { [π] ∈ ĝ : π(h) = i for all h ∈ h } , (4.1) if g is abelian, each closed subgroup h of g is normal and the locally compact group g/h is abelian and so ĝ/h is precisely the set of all characters (one dimensional irreducible representations) of g which are constant on h, that is precisely h⊥. if g is a non-abelian group and h is a closed normal subgroup of g, then the dual space ĝ/h which is the set of all unitary equivalence classes of unitary representations of g/h, has meaning and it is well-defined. indeed, g/h is a non-abelian group. in this case, the map φ : ĝ/h → h⊥ defined by σ 7→ φ(σ) := σ ◦ q is a borel isomorphism and ĝ/h = h⊥, see [1]. thus if h is normal, h⊥ coincides with the classic definitions of the dual space either when g is abelian or non-abelian. for a closed subgroup h of g and a continuous unitary representation (π,hπ) of g, define tπh := ∫ h π(h)dh, (4.2) where the operator valued integral (4.2) is considered in the weak sense. in other words, 〈tπhζ,ξ〉 = ∫ h 〈π(h)ζ,ξ〉dh, for ζ,ξ ∈hπ. (4.3) 26 a. ghaani farashahi the function h 7→ 〈π(h)ζ,ξ〉 is bounded and continuous on h. since h is compact, the right integral is the ordinary integral of a function in l1(h). hence, tπh defines a bounded linear operator on hπ with ‖tπh‖≤ 1. remark 4.1. let (π,hπ) be a continuous unitary representation of g with tπh 6= 0. let (σ,hσ) be a continuous unitary representation of g such that [π] = [σ]. let s : hπ → hσ be the unitary operator which satisfies σ(x)s = sπ(x) for all x ∈ g. then we have stπh = s (∫ h π(h)dh ) = ∫ h sπ(h)dh = ∫ h σ(h)sdh = (∫ h σ(h)dh ) s = tσhs, which implies that tσh 6= 0 as well. thus we deduce that the non-zero property of t π h depends only on [π], that is the unitary equivalence class of (π,hπ). let khπ := {ζ ∈hπ : π(h)ζ = ζ ∀h ∈ h} . (4.4) then, khπ is a closed subspace of hπ and r(tπh) = k h π , where r(tπh) = {t π hζ : ζ ∈hπ}. it is easy to see that [π] ∈ h⊥ if and only if khπ = hπ. proposition 4.1. let h be a closed subgroup of a compact group g and (π,hπ) be a continuous unitary representation of g. then, (1) the operator tπh is an orthogonal projection onto k h π . (2) the operator tπh is unitary if and only if [π] ∈ h ⊥. proof. (1) using compactness of h, it can be easily checked that (tπh) ∗ = tπh. as well as we achieve that tπht π h = (∫ h π(h)dh )(∫ h π(t)dt ) = ∫ h π(h) (∫ h π(t)dt ) dh = ∫ h (∫ h π(h)π(t)dt ) dh = ∫ h (∫ h π(ht)dt ) dh = ∫ h tπhdt = t π h. (2) since tπh is a projection, the operator t π h is unitary if and only if t π h = i. the operator th is identity if and only if π(h) = i for all h ∈ h. thus, tπh is unitary if and only if [π] ∈ h ⊥. � definition 4.1. let h be a closed subgroup of a compact group g. then we define the dual space of g/h, as the subset of ĝ which is given by ĝ/h := { [π] ∈ ĝ : tπh 6= 0 } = { [π] ∈ ĝ : ∫ h π(h)dh 6= 0 } . (4.5) evidently, any closed subgroup h of g satisfies h⊥ ⊂ ĝ/h. (4.6) next we shall show that the reverse inclusion of (4.6) holds, if and only if h is a normal subgroup of g. theorem 4.1. let h be a closed normal subgroup of a compact group g. then ĝ/h = h⊥. peter-weyl theorem for homogeneous spaces of compact groups 27 proof. let h be a closed normal subgroup of a compact group g. it is sufficient to show that ĝ/h ⊂ h⊥. let [π] ∈ ĝ/h be given. due to the normality of h in g, for all x ∈ g the map τx : h → h given by h 7→ τx(h) := x−1hx belongs to aut(h) and x−1hx = h. invoking compactness of g we have d (τx(h)) = dh, for x ∈ g. now, for x ∈ g we get∫ h π(h)dh = ∫ xhx−1 π(τx(h))d (τx(h)) = ∫ h π(τx(h))dh = ∫ h π(x)∗π(h)π(x)dh = π(x)∗ (∫ h π(h)dh ) π(x) = π(x)∗tπhπ(x). therefore π(x)tπh = t π hπ(x) for x ∈ g, which implies t π h ∈ c(π). irreducibility of π guarantees that tπh = αi for some non-zero α ∈ c with |α| ≤ 1. thus, for t ∈ h we can write π(t) = α−1π(t)αi = α−1π(t)tπh = α−1 ∫ h π(th)dh = α−1 ∫ h π(h)dh = α−1tπh = i, which implies [π] ∈ h⊥. � let (π,hπ) be a continuous unitary representation of g such that tπh 6= 0. then the functions πhζ,ξ : g/h → c defined by πhζ,ξ(xh) := 〈π(x)t π hζ,ξ〉 for xh ∈ g/h, (4.7) for ξ,ζ ∈hπ are called h-matrix elements of (π,hπ). for xh ∈ g/h and ζ,ξ ∈hπ, we have |πhζ,ξ(xh)| = |〈π(x)t π hζ,ξ〉| ≤ ‖π(x)tπhζ‖‖ξ‖≤‖t π hζ‖‖ξ‖≤‖ζ‖‖ξ‖. also we can write πhζ,ξ(xh) = 〈π(x)t π hζ,ξ〉 = πtπhζ,ξ(x). (4.8) invoking definition of the linear map th and also t π h we have th(πζ,ξ)(xh) = ∫ h πζ,ξ(xh)dh = ∫ h 〈π(xh)ζ,ξ〉dh = ∫ h 〈π(x)π(h)ζ,ξ〉dh = 〈π(x)tπhζ,ξ〉, which implies that th(πζ,ξ) = π h ζ,ξ. (4.9) theorem 4.2. let h be a closed subgroup of a compact group g, µ be the normalized g-invariant measure and (π,hπ) be a continuous unitary representation of g such that tπh 6= 0. then (1) the subspace eπ(g/h) depends on the unitary equivalence class of π. (2) the subspace eπ(g/h) is a closed left invariant subspace of l1(g/h,µ). 28 a. ghaani farashahi proof. (1) let (σ,hσ) be a continuous unitary representation of g such that [π] = [σ]. let s : hπ → hσ be the unitary operator which satisfies σ(x)s = sπ(x) for all x ∈ g. remark 4.1 guarantees that stπh = t σ hs and also t σ h 6= 0. thus for x ∈ g and ζ,ξ ∈hπ we can write πhζ,ξ(xh) = 〈π(x)t π hζ,ξ〉hπ = 〈s−1σ(x)stπhζ,ξ〉hπ = 〈σ(x)stπhζ,sξ〉hσ = 〈σ(x)tσhsζ,sξ〉hσ = σ h sζ,sξ(xh), which implies that eπ(g/h) = eσ(g/h). (2) it is straightforward. � if ζ,ξ belongs to an orthonormal basis {ei} for hπ, h-matrix elements of [π] with respect to an orthonormal basis {ej} changes in the form πhij (xh) = π h ej,ei (xh) = 〈π(x)tπhej,ei〉, for xh ∈ g/h. (4.10) the linear span of the h-matrix elements of a continuous unitary representation (π,hπ) satisfying tπh 6= 0, is denoted by eπ(g/h) which is a subspace of c(g/h). definition 4.2. let h be a closed subgroup of a compact group g and [π] ∈ ĝ/h. an ordered orthonormal basis b = {e` : 1 ≤ ` ≤ dπ} of the hilbert space hπ is called h-admissible, if it is an extension of an orthonormal basis {e` : 1 ≤ ` ≤ dπ,h} of the closed subspace khπ , which equivalently means that dπ,h-first elements of b be an orthogonal basis of khπ . let [π] ∈ ĝ/h and bπ = {e` : 1 ≤ ` ≤ dπ} be an h-admissible basis for the representation space hπ. then, each πi` with 1 ≤ i ≤ dπ and 1 ≤ ` ≤ dπ,h, is a well-defined continuous function over g/h. let e`π(g/h) be the subspace of c(g/h) spanned by the set b`π := { √ dππi` : 1 ≤ i ≤ dπ}. proposition 4.2. let [π] ∈ ĝ/h, bπ be an h-admissible basis for the representation space hπ, and 1 ≤ ` 6= `′ ≤ dπ,h. then (1) dime`π(g/h) = dπ and b`π is an orthonormal basis for e`π(g/h). (2) e`π(g/h) is a closed left translation invariant subspace of c(g/h). (3) e` ′ π (g/h) ⊥e`π(g/h). proof. (1) let 1 ≤ i, i′ ≤ dπ. then by theorem 27.19 of [10] we get 〈πi`,πi′`〉l2(g/h,µ) = 〈πi`,πi′`〉l2(g) = d−1π δii′. since dime`π(g/h) ≤ dπ we achieve that b`π is an orthonormal basis for e`π(g/h) and hence dime`π(g/h) = dπ. (2) it is straightforward. (3) let 1 ≤ i, i′ ≤ dπ. applying theorem 27.19 of [10] we get 〈πi`,πi′`′〉l2(g/h,µ) = 〈πi`,πi′`′〉l2(g) = d−1π δii′δ``′, which completes the proof. � the following theorem shows that h-admissible bases lead to orthogonal decompositions of the subspace eπ(g/h). theorem 4.3. let h be a closed subgroup of a compact group g. let [π] ∈ ĝ/h and bπ = {e`,π : 1 ≤ ` ≤ dπ} be an h-admissible basis for the representation space hπ. then bπ(g/h) := { √ dππi` : 1 ≤ i ≤ dπ, 1 ≤ ` ≤ dπ,h} is an orthonormal basis for the hilbert space eπ(g/h) and hence it satisfies the following direct sum decomposition eπ(g/h) = dπ,h⊕ `=1 e`π(g/h). (4.11) peter-weyl theorem for homogeneous spaces of compact groups 29 proof. it is straightforward to check that bπ(g/h) spans the subspace eπ(g/h). then proposition 4.2 guarantees that bπ(g/h) is an orthonormal set in eπ(g/h). since dimeπ(g/h) ≤ dπ,hdπ we deduce that it is an orthonormal basis for eπ(g/h), which automatically implies the decomposition (4.11). � next proposition lists basic properties of h-matrix elements. proposition 4.3. let h be a closed subgroup of a compact group g, µ be the normalized g-invariant measure on g/h, and (π,hπ) be a continuous unitary representation of g. then, (1) tπh = 0 if and only if eπ(g) ⊆j 2(g,h). (2) if tπh 6= 0 then th(eπ(g)) = eπ(g/h) and t ∗ h(eπ(g/h)) ⊆eπ(g). (3) eπ(g) ⊆j2(g,h)⊥ if and only if π(h) = i for all h ∈ h. then we can prove the following orthogonality relation concerning the functions in e(g/h). theorem 4.4. let h be a closed subgroup of a compact group g, µ be a normalized g-invariant measure on g/h and [π] 6= [σ] ∈ ĝ/h. the closed subspaces eπ(g/h) and eσ(g/h) are orthogonal to each other as subspaces of the hilbert space l2(g/h,µ). proof. let ψ ∈ eπ(g/h) and ϕ ∈ eσ(g/h). then we have ψq ∈ eπ(g) and also ϕq ∈ eσ(g). using proposition 4.3, corollary 3.3, and theorem 27.15 of [10], we get 〈ϕ,ψ〉l2(g/h,µ) = 〈ϕq,ψq〉l2(g) = 0. which completes the proof. � we can define e(g/h) := the linear span of ⋃ [π]∈ĝ/h eπ(g/h). (4.12) next theorem presents some analytic aspects of the function space e(g/h). theorem 4.5. let h be a closed subgroup of a compact group g and µ be the normalized g-invariant measure on g/h associated to the weil’s formula. then, (1) the linear operator th maps e(g) onto e(g/h). (2) e(g/h) is ‖.‖l2(g/h,µ)-dense in l2(g/h,µ). (3) e(g/h) is ‖.‖sup-dense in c(g/h). proof. (1) it is straightforward. (2) let φ ∈ l2(g/h,µ) and also f ∈ l2(g) with th(f) = φ. then by ‖.‖l2(g)-density of e(g) in l2(g) we can pick a sequence {fn} in e(g) such that f = ‖.‖l2(g) − limn fn. by proposition 4.3 we have {th(fn)}⊆e(g/h). then continuity of the linear map th : l2(g) → l2(g/h,µ) implies φ = th(f) = ‖.‖l2(g/h,µ) − lim n th(fn), which completes the proof. (3) invoking uniformly boundedness of th, uniformly density of e(g) in c(g), and the same argument as used in (1), we get ‖.‖sup-density of e(g/h) in c(g/h). � the following theorem can be considered as an abstract extension of the peter-weyl theorem for homogeneous spaces of compact groups. theorem 4.6. let h be a closed subgroup of a compact group g and µ be the normalized g-invariant measure on g/h. the hilbert space l2(g/h,µ) satisfies the following orthogonality decomposition l2(g/h,µ) = ⊕ [π]∈ĝ/h eπ(g/h). (4.13) proof. using peter-weyl theorem, proposition 4.3, and since the bounded linear map th : l 2(g) → l2(g/h,µ) is surjective we achieve that each ϕ ∈ l2(g/h,µ) has a decomposition to elements of eπ(g/h) with [π] ∈ ĝ/h, namely ϕ = ∑ [π]∈ĝ/h cπϕπ, (4.14) 30 a. ghaani farashahi with ϕπ ∈ eπ(g/h) for all [π] ∈ ĝ/h. since the subspaces eπ(g/h) with [π] ∈ ĝ/h are mutually orthogonal we conclude that decomposition (4.14) is unique for each ϕ, which guarantees (4.13). � we immediately deduce the following corollaries. corollary 4.1. let h be a closed subgroup of a compact group g and µ be the normalized g-invariant measure on g/h. for each [π] ∈ ĝ/h, let bπ = {e`,π : 1 ≤ ` ≤ dπ} be an h-admissible basis for the representation space hπ. then we have the following statements. (1) the hilbert space l2(g/h,µ) satisfies the following direct sum decomposition l2(g/h,µ) = ⊕ [π]∈ĝ/h dπ,h⊕ `=1 e`π(g/h), (4.15) (2) the set b(g/h) := {πi` : 1 ≤ i ≤ dπ, 1 ≤ ` ≤ dπ,h} constitutes an orthonormal basis for the hilbert space l2(g/h,µ). (3) each ϕ ∈ l2(g/h,µ) decomposes as the following ϕ = ∑ [π]∈ĝ/h dπ dπ,h∑ `=1 dπ∑ i=1 〈ϕ,πi`〉l2(g/h,µ)πi`, (4.16) where the series is converges in l2(g/h,µ). remark 4.2. let h be a closed normal subgroup of a compact group g. also, let µ be the normalized g-invariant measure over g/h associated to the weil’s formula. then g/h is a compact group and the normalized g-invariant measure µ is a haar measure of the quotient compact group g/h. by theorem 4.1, we deduce that ĝ/h = h⊥, and for each [π] ∈ ĝ/h we get tπh = i and dπ,h = dπ. thus we obtain l2(g/h) = ⊕ [π]∈h⊥ eπ(g/h), which precisely coincides with the decomposition associated to applying the peter-weyl theorem to the compact quotient group g/h. references [1] g.b. folland, a course in abstract harmonic analysis, crc press, 1995. [2] a. ghaani farashahi, abstract operator-valued fourier transforms over homogeneous spaces of compact groups, groups, geometry, dynamics, in press. [3] a. ghaani farashahi, abstract plancherel (trace) formulas over homogeneous spaces of compact groups, canadian mathematical bulletin, doi:10.4153/cmb-2016-037-6. [4] a. ghaani farashahi, abstract harmonic analysis of wave packet transforms over locally compact abelian groups, banach j. math. anal. 11 (1) (2017), 50-71. [5] a. ghaani farashahi, abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups, j. aust. math. soc., 101 (2) (2016) 171-187. [6] a. ghaani farashahi, abstract relative fourier transforms over canonical homogeneous spaces of semi-direct product groups with abelian normal factor, j. korean math. soc., 101 (2) (2016), 171-187. [7] a. ghaani farashahi, convolution and involution on function spaces of homogeneous spaces, bull. malays. math. sci. soc. (36) (4) (2013), 1109-1122. [8] a. ghaani farashahi, abstract non-commutative harmonic analysis of coherent state transforms, ph.d. thesis, ferdowsi university of mashhad (fum), mashhad 2012. [9] e. hewitt and k.a. ross, abstract harmonic analysis, vol 1, 1963. [10] e. hewitt and k.a. ross, abstract harmonic analysis, vol 2, 1970. [11] v. kisil, calculus of operators: covariant transform and relative convolutions, banach j. math. anal. 8 (2) (2014), 156-184. [12] v. kisil, geometry of möbius transformations. elliptic, parabolic and hyperbolic actions of sl2(r), imperial college press, london, 2012. [13] v. kisil, relative convolutions. i. properties and applications, adv. math. 147 (1) (1999), 35-73. [14] g.j. murphy, c*-algebras and operator theory, academic press, inc, 1990. [15] f. peter and h. weyl, die vollstndigkeit der primitiven darstellungen einer geschlossenen kontinuierlichen gruppe, math. ann., 97 (1927) 737-755. peter-weyl theorem for homogeneous spaces of compact groups 31 numerical harmonic analysis group (nuhag), faculty of mathematics, university of vienna, austria ∗corresponding author: arash.ghaani.farashahi@univie.ac.at, ghaanifarashahi@hotmail.com 1. introduction 2. preliminaries and notations 3. abstract harmonic analysis over homogeneous spaces of compact groups 4. peter-weyl theorem for homogeneous spaces of compact groups references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 180-187 http://www.etamaths.com on chebyshev functional and ostrowski-grüs type inequalities for two coordinates atiq ur rehman∗ and ghulam farid abstract. in this paper, we construct chebyshev functional and grüss inequality on two coordinates. also we establish ostrowski-grüss type inequality on two coordinates. related mean value theorems of lagrange and cauchy type are also given. 1. introduction let f,g : [a,b] → r be integrable functions. we consider (1) t(f,g) := 1 b−a ∫ b a f(x)dx 1 b−a ∫ b a g(x)dx− 1 b−a ∫ b a f(x)g(x)dx. if f and g are monotonic in same direction on [a,b], then (2) t(f,g) ≥ 0. if f and g are monotonic in opposite directions on interval [a,b], then the reverse of the inequality (2) is valid. the chebyshev functional (1) has a long history and an extensive repertoire of applications in many fields including numerical quadrature, transform theory, probability and statistical problems and special functions. it is worthwhile noting that a number of identities relating to the chebyshev functional already exist. in [11, chapter ix and x], one can see lots of results related to the chebyshev functional. one of them is famous as korkine’s identity (see [11, p. 243 ]) given by (3) t(f,g) = 1 2(b−a)2 ∫ b a ∫ b a (f(x) −f(y)) (g(x) −g(y)) dxdy. this identity is often used to prove an inequality due to grüss for functions bounded above and below (see in [6]). in literature it is known as grüss inequality. theorem 1.1. let f,g : [a,b] → r be integrable functions such that φ ≤ f(x) ≤ ϕ and γ ≤ g(x) ≤ γ for all x ∈ [a,b], where φ, ϕ, γ and γ are real constants. then (4) |t(f,g)| ≤ 1 4 (ϕ−φ)(γ −γ). an other celebrated inequality in this respect is by ostrowski sated in the following theorem (see [12]). theorem 1.2. let f : i → r, where i is an interval in r, be a mapping differentiable in io the interior of i and a, b ∈ io, a < b, if ∣∣∣f′ (t)∣∣∣ ≤ m, for all t ∈ [a,b], then we have (5) ∣∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ [ 1 4 + (x− a+b 2 )2 (b−a)2 ] (b−a)m, for all x ∈ [a,b]. 2000 mathematics subject classification. 26b15, 26a51, 34l15. key words and phrases. chebyshev inequality; chebyshev functional; grüss inequality; ostrowski-grüss inequality; mean value theorems. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 180 on chebyshev functional and ostrowski-grüs... 181 inequality in (5) is well known as ostrowski inequality and has interesting consequences in numerical integration (see [3]). it has been improved by dragomir and wang in [4] using grüss inequality in terms of the lower and upper bounds of the first derivative. that ostrowski-grüss type inequality is stated in the following theorem. theorem 1.3. let f : [a,b] → r be continuous on [a,b] and differentiable on (a,b) and its derivative satisfy the condition γ ≤ f′(x) ≤ γ for all x ∈ [a,b], then we have the inequality∣∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt− ( f(b) −f(a) b−a )( x− a + b 2 )∣∣∣∣∣ ≤ 14 (b−a)(γ −γ),(6) for all x ∈ [a,b]. in [2], barnett et al. pointed out a similar result to the above for twice differentiable mappings in terms of the upper and lower bounds of the second derivative. theorem 1.4. let f : [a,b] → r be continuous on [a,b] and twice differentiable on (a,b) and assume that the second derivative f′′ : (a,b) → r satisfies the condition γ ≤ f′′(x) ≤ γ for all x ∈ [a,b]. then, for all x ∈ [a,b], we have inequality∣∣∣∣∣f(x) − ( x− a + b 2 ) f′(x) + [ (b−a)2 24 + 1 2 ( x− a + b 2 )2]( f′(b) −f′(a) b−a ) − 1 b−a ∫ b a f(t)dt ∣∣∣∣∣ ≤ 18 (γ −γ) [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ]2 .(7) many authors considered different generalization of chebyshev functional on two coordinates and found the bounds of these functional, for example see [1, 14] and references there in. in this paper we give the chebyshev functional and grüss inequality on two coordinates and establish the ostrowski-grüss type inequality on two coordinates in terms of lower and upper bounds of first and second order partial derivatives. also we give lagrange and cauchy type mean value theorems for the chebyshev functional, as given in [5]. 2. main results let ∆ = [a,b]×[c,d] be a bi-dimensional interval in r2 and f : ∆ → r be a mapping. if x = (x1,x2) and y = (y1,y2), then we say x ≤ y if x1 ≤ y1 and x2 ≤ y2. also we say f is monotonically increasing on ∆ if for all x,y ∈ ∆ f(x) ≤ f(y) when x ≤ y. if we take f(x) = ∫ b a f(x,t)dt, provided that the integral exists, then one can note that f(x) = ∫ d c f(x,t)dt ≤ ∫ d c f(y,t)dt = f(y) for x < y, that is f is monotonically increasing on [a,b]. in the following theorem, we introduce chebyshev functional on two coordinates and generalize the chebyshev inequality on a rectangle from the plane. theorem 2.1. let f,g : ∆ → r be integrable functions. we consider (8) a(f; ∆) = 1 (b−a)(d− c) ∫ b a ∫ d c f(x,y)dydx, and t(f,g; ∆) = a(f; ∆)a(g; ∆) −a(fg; ∆)(9) if f and g are monotonic in same direction on ∆, then (10) t(f,g; ∆) ≥ 0. 182 rehman and farid proof. considering the monotonicity of f and g on second coordinate and using (2), we get 1 d− c ∫ d c f(x,y)g(x,y)dy ≤ 1 (d− c)2 ∫ d c f(x,y)dy ∫ d c g(x,y)dy. integrating above inequality over [a,b], we have 1 d− c ∫ b a ∫ d c f(x,y)g(x,y)dydx ≤ 1 (d− c)2 ∫ b a (∫ d c f(x,y)dy ∫ d c g(x,y)dy ) dx.(11) now if we take f(x) = ∫d c f(x,y)dy, then f is monotonic on [a,b] by considering monotonicity of f on first coordinate. similarly, we take g(x) = ∫d c g(x,y)dy, then g is monotonic on [a,b]. if f and g are monotone in same direction so are f and g, then using the chebyshev inequality, one has 1 (b−a) ∫ b a f(x)g(x)dx ≤ 1 (b−a)2 ∫ b a f(x)dx ∫ b a g(x)dx.(12) using the above inequality in (11), we get a(fg; ∆) ≤ a(f; ∆)a(g; ∆), which is equivalent to required result. � it is easy to find that t (f,g; ∆) = 1 2(b−a)2(d− c)2 ∫ b a ∫ d c ∫ b a ∫ d c (f(x,y) −f(u,v))(g(x,y) −g(u,v))dxdydudv. this identity can be considered as korkine’s identity in two coordinates. using this identity one can prove the following result similar to the proof of theorem 1.1 (see also [11, p. 296]) theorem 2.2. let f,g : ∆ → r be integrable functions such that ϕ ≤ f(x,y) ≤ φ and γ ≤ g(x,y) ≤ γ, for all x,y ∈ ∆ where φ, ϕ, γ and γ are constants. then (13) |t(f,g : ∆)| ≤ 1 4 (φ−ϕ)(γ −γ). in [9], an important result related to grüss inequality is given, we can have a similar result related to grüss inequality on two coordinates. if f,g : ∆ → r be integrable functions, then t (f,f; ∆) ≥ 0 and a following inequality holds: t 2(f,g; ∆) ≤ t (f,f; ∆)t (g,g; ∆).(14) by the combination of inequalities (13) and (14), we obtain the following result. theorem 2.3. let f,g : ∆ → r be two integrable functions. if ϕ ≤ f(x,y) ≤ φ, for all x ∈ [a,b] and y ∈ [c,d], where φ and ϕ are some constants, then (15) |t(f,g; ∆)| ≤ 1 2 (φ−ϕ) √ t(g,g; ∆). proof. setting g = f in (13), we get (16) t (f,f; ∆) = |t (f,f; ∆)| ≤ 1 4 (φ−ϕ)2. combining (16) with (14) we get t 2(f,g; ∆) ≤ 1 4 (φ−ϕ)2t (g,g; ∆), this is equivalent to required result (15). � in the following result we construct ostrowski-grüss type inequality on two coordinates in terms of the lower and upper bounds of the first order partial derivatives. on chebyshev functional and ostrowski-grüs... 183 theorem 2.4. let f : ∆ → r be continuous on ∆ and its partial derivative satisfy the condition γ1 ≤ ∂f∂x ≤ γ1 and γ2 ≤ ∂f ∂y ≤ γ2 on ∆. then we have ∣∣∣∣∣ ∫ b a f(x,c) + f(x,d) 2 dx + ∫ d c f(a,y) + f(b,y) 2 dy − ( 1 b−a + 1 d− c ) ∫ b a ∫ d c f(x,y)dxdy ∣∣∣∣∣ ≤ (b−a)(d− c)4 [(γ2 −γ2) + (γ1 −γ1)] .(17) proof. for all (x,y) ∈ ∆, consider two mappings fy : [a,b] → r and fx : [c,d] → r defined by fy(t) = f(t,y) and fx(t) = f(x,t) respectively. applying (6) for mapping fy at x = b, we have∣∣∣∣∣f(b,y) − 1b−a ∫ b a f(t,y)dt− f(b,y) −f(a,y) 2 ∣∣∣∣∣ ≤ 14 (b−a)(γ1 −γ1). on integrating over [c,d], we have∣∣∣∣∣ ∫ d c f(b,y)dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx− (d− c)(f(b,y) −f(a,y)) 2 ∣∣∣∣∣ ≤ 1 4 (b−a)(d− c)(γ1 −γ1).(18) applying (6) for mapping fy at x = a and then integrating over [c,d], we get∣∣∣∣∣ ∫ d c f(a,y)dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx + (d− c)(f(b,y) −f(a,y)) 2 ∣∣∣∣∣ ≤ 1 4 (b−a)(d− c)(γ1 −γ1).(19) addition of (18) and (19) lead us to∣∣∣∣∣ ∫ d c f(a,y) + f(b,y) 2 dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 1 4 (b−a)(d− c)(γ1 −γ1).(20) similarly using inequalities getting after applying (6) for mapping fx first at y = c then at y = d and integrating over [a,b], we can have∣∣∣∣∣ ∫ b a f(x,c) + f(x,d) 2 dx− 1 d− c ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 14 (b−a)(d− c)(γ2 −γ2).(21) using (20) and (21), we have (17). � in the following we establish the similar result to the theorem 2.4 for twice differentiable mappings in terms of the lower and upper bounds of the second order partial derivative. theorem 2.5. let f : ∆2 → r be continuous on ∆2 and differentiable for all x ∈ (a,b) and y ∈ (c,d) and assume that the second order partial derivative satisfies the condition γ2 ≤ ∂ 2f ∂x2 ≤ γ2 for all 184 rehman and farid x ∈ [a,b] and γ1 ≤ ∂ 2f ∂y2 ≤ γ1 for all y ∈ [c,d], then we have (22) ∣∣∣∣∣12 [∫ b a (f(x,c) + f(x,d)) dx + ∫ d c (f(a,y) + f(b,y)) dy ] + 1 12[ (b−a) ∫ d c ( ∂f(a,y) ∂x − ∂f(b,y) ∂x ) dy + (d− c) ∫ b a ( ∂f(x,c) ∂y − ∂f(x,d) ∂y ) dx ] − ( 1 b−a + 1 d− c )∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 18 (d− c)(b−a) ((γ1 −γ1)(b−a) +(γ2 −γ2)(d− c)) . proof. for all (x,y) ∈ ∆, consider two mappings fy : [a,b] → r and fx : [c,d] → r defined by fy(t) = f(t,y) and fx(t) = f(x,t) respectively. applying (7) for mapping fy at x = b, we have∣∣∣∣∣f(b,y) − (b−a)6 ( ∂f(a,y) ∂x + 2 ∂f(b,y) ∂x ) − 1 b−a ∫ b a f(t,y)dt ∣∣∣∣∣ ≤ 1 8 (γ1 −γ1)(b−a)2. integrating over [c,d], we have∣∣∣∣∣ ∫ d c f(b,y)dy − (b−a) 6 ∫ d c ( ∂f(a,y) ∂x + 2 ∂f(b,y) ∂x ) dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 18 (d− c)(γ1 −γ1)(b−a)2.(23) applying (7) for mapping fy at x = a and integrating over [c,d], we get∣∣∣∣∣ ∫ d c f(a,y)dy + (b−a) 6 ∫ d c ( ∂f(b,y) ∂x + 2 ∂f(a,y) ∂x ) dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 18 (d− c)(γ1 −γ1)(b−a)2.(24) using (23) and (24), we get (25) ∣∣∣∣∣12 ∫ d c (f(a,y)dy + f(b,y))dy + (b−a) 12 ∫ d c ( ∂f(a,y) ∂x − ∂f(b,y) ∂x ) dy − 1 b−a ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 18 (d− c)(γ1 −γ1)(b−a)2. similarly using inequalities getting after applying (7) for mapping fx first at y = c then at y = d and integrating over [a,b], we have (26) ∣∣∣∣∣12 ∫ b a (f(x,c)dy + f(x,d))dx + (d− c) 12 ∫ b a ( ∂f(x,c) ∂y − ∂f(x,d) ∂y ) dx − 1 d− c ∫ b a ∫ d c f(x,y)dydx ∣∣∣∣∣ ≤ 18 (b−a)(γ2 −γ2)(d− c)2. using (25) and (26), we get (22). � on chebyshev functional and ostrowski-grüs... 185 3. mean value theorems in this section, we give mean value theorems of lagrange and cauchy type for chebyshev functional on two coordinates. before presenting our main results, one can note: if a function f : ∆ → r has non-negative first order partial derivatives on ∆, then it is increasing on ∆. lemma 3.1. let f : ∆ → r be an integrable function and also monotonically increasing on coordinates, such that m1 ≤ ∂f(x,y) ∂x ≤ m1 and m2 ≤ ∂f(x,y) ∂y ≤ m2 for all interior points (x,y) in ∆. consider the functions h,k : ∆ → r defined as h(x,y) = max{m1,m2}(x + y) −f(x,y) and k(x,y) = f(x,y) − min{m1,m2}(x + y). then h and k are monotonically increasing on ∆. proof. since (27) ∂h(x,y) ∂x = max{m1,m2}− ∂f(x,y) ∂x ≥ 0 and (28) ∂h(x,y) ∂y = ∂f(x,y) ∂y − min{m1,m2}≥ 0 for all interior points (x,y) in ∆, h is monotonically increasing on coordinates. similarly it can also be proved that k is monotonically increasing on coordinates on ∆. � theorem 3.2. let f,g : ∆ → r be functions such that f has continuous partial derivatives of first order in ∆ and g is increasing on ∆. then there exists (ξ1,η1) and (ξ2,η2) in the interior of ∆ such that (29) t(f,g; ∆) = ∂f(ξ1,η1) ∂x t(r,g; ∆) and (30) t(f,g; ∆) = ∂f(ξ2,η2) ∂y t(r,g; ∆), where r(x,y) = x + y and t(r,g; ∆) 6= 0. proof. since f has continuous partial derivatives of first order in ∆, there exist real numbers m1, m2, m1 and m2, such that m1 ≤ ∂f(x,y) ∂x ≤ m1 and m2 ≤ ∂f(x,y) ∂y ≤ m2 for all (x,y) ∈ ∆. now consider function h defined in lemma 3.1. as h is increasing on coordinates in ∆, therefore t (h,g; ∆) ≥ 0, that is t (max{m1,m2}r −f,g; ∆) ≥ 0. this gives us (31) t (f,g; ∆) ≤ max{m1,m2}t (r,g; ∆). on the other hand for the function k defined in lemma 3.1, one has (32) min{m1,m2}t (r,g; ∆) ≤ t (f,g; ∆). as t (r,g; ∆) 6= 0, combining above inequalities (31) and (32), we get min{m1,m2}≤ t (f,g; ∆) t (r,g; ∆) ≤ max{m1,m2}. then there exist (ξ1,η1) and (ξ2,η2) in the interior of ∆, such that t (f,g; ∆) t (r,g; ∆) = ∂f(ξ1,η1) ∂x 186 rehman and farid and t (f,g; ∆) t (r,g; ∆) = ∂f(ξ2,η2) ∂y . hence the required results are proved. � theorem 3.3. let f,g : ∆ → r be functions having partial derivatives in ∆ and g is increasing on ∆. then there exists (ξi,ηi), i = 1, 2, 3, 4 in the interior of ∆ such that (33) t(f,g; ∆) = ∂f(ξ1,η1) ∂x ∂g(ξ3,η3) ∂x t(r,r; ∆) and (34) t(f,g; ∆) = ∂f(ξ2,η2) ∂y ∂g(ξ4,η4) ∂y t(r,r; ∆), where r(x,y) = x + y. proof. since t(r,g; ∆) = t(g,r; ∆) and r(x,y) = x+y is increasing on ∆, by theorem 3.2 there exists (ξ3,η3) in the interior of ∆ such that (35) t (r,g; ∆) = ∂g(ξ3,η3) ∂x t (r,r; ∆) using above expression in (29) gives us (33). in a similar way, one can deduce (34). � in [15], pečarić gave many interesting result related to chebyshev functional. a similar result is also valid for chebyshev functional on two coordinates. namely, the following corollary. corollary 3.4. let f,g : ∆ → r be functions, such that g is increasing on ∆ and f has partial derivatives of first order in ∆ with ∣∣∣∂f∂x∣∣∣ ≤ m1, ∣∣∣∂f∂y∣∣∣ ≤ m2, ∣∣∣∂g∂x∣∣∣ ≤ n1 and ∣∣∣∂f∂x∣∣∣ ≤ n2. then one has (36) t(f,g; ∆) ≤ minit(r,r; ∆), i = 1, 2, where r(x,y) = x + y. theorem 3.5. let f1,f2,g : ∆ → r be functions, such that f has partial derivatives of first order in ∆ and g is increasing on ∆. then there exists (ξ1,η1) and (ξ2,η2) in the interior of ∆ such that t(f1,g; ∆) t(f2,g; ∆) = ∂f1(ξ1,η1) ∂x ∂f2(ξ1,η1) ∂x and t(f1,g; ∆) t(f2,g; ∆) = ∂f1(ξ1,η1) ∂y ∂f2(ξ1,η1) ∂y . proof. we define the function h : ∆ → r, such that h = c1f1 − c2f2, where c1 = t (f2,g; ∆) and c2 = t (f1,g; ∆). now, using theorem 3.2 with f = h, we have 0 = ( c1 ∂f1(ξ1,η1) ∂x − c2 ∂f2(ξ1,η1) ∂x ) t (r,g; ∆) and 0 = ( c1 ∂f1(ξ2,η2) ∂y − c2 ∂f2(ξ2,η2) ∂y ) t (r,g; ∆). since t (r,g; ∆) 6= 0, we have c2 c1 = ∂f1(ξ1,η1) ∂x ∂f2(ξ1,η1) ∂x on chebyshev functional and ostrowski-grüs... 187 and c2 c1 = ∂f1(ξ1,η1) ∂y ∂f2(ξ1,η1) ∂y , this complete the proof. � references [1] m.w. alomari, new grüss type inequalities for double integrals, appl. math. comput. 228 (2014), 102-107. [2] n. barnett, p. cerone, s.s. dragomir, j. roumeliotis and a. sofo, a survey on ostrowski type inequalities for twice differentiable mappings and applications. in: inequality theory and applications, nova science publishers, huntington, 1 (2001) 24–30. [3] s.s. dragomir and th.m. rassias (eds.), ostrowski type inequalities and applications in numerical integration, kluwer academic publishers, dordrecht, 2002. [4] s.s. dragomir and s. wang, an inequality of ostrowski-grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, comput. math. 33 (1997), 15–22. [5] g. farid, m. marwan, and a. u. rehman, fejer-hadamard inequality for convex functions on the coordinates in a rectangle from the plane, int. j. analysis appl. 10(1) (2016), 40-47. [6] g. grüss, ber das maximum des absoluten betrages von 1 (b−a)2 ∫ b a f(x)dx ∫ b a g(x)dx, math. z. 39 (1935), 215–226. [7] s. hussain and a. qayyum , a generalized ostrowski-grüss type inequality for bounded differentiable mappings and its applications, j. inequal. appl. 2013 (2013) art. id 1. [8] j. jakšetić, j. pečarić and atiq ur rehman, cauchy means involving chebychev functional, proc. a. razmadze math. inst. 151 (2009), 43–54. [9] m. matic , j. pečarić and n. ujević , on new estimation of the remainder in generalized taylor’s formula, math. inequal. appl. 2(3) (1999), 343-361. [10] a.mcd. mercer and p. mercer, new proofs of the grüss inequality, aust. j. math. anal. appls. 1(2) (2004) art. id 12. [11] d.s. mitrinović, j.e. pečarić, and a.m. fink, classical and new inequalities in analysis, mathematics and its applications (east european series), vol. 61, kluwer academic, dordrecht, 1993. [12] a. ostrowski, uber die absolutabweichung einer differentiebaren funktion von ihrem integralmittelwert comment. math. helv. 10 (1938) 226–227. [13] a. m. ostrowski, on an integral inequality. aequationes math. 4 (1970) 358-373. [14] b.g. pachpatte, on grüss type inequalities for double integrals, j. math. anal. appl. 267 (2002) 454-459. [15] j. e. pečarić, on the ostrowski generalization of čebyšev’s inequality, j. math. anal. appl., 102 (1984) 479–487. [16] a. rafiq, n.a. mir and f. zafar, a generalized ostrowski-grüss type inequality for twice differentiable mappings and applications, jipam. j. inequal. pure appl. math, 7(4) (2006), art. id 124. comsats institute of information technology, attock campus, pakistan ∗corresponding author: atiq@mathcity.org international journal of analysis and applications issn 2291-8639 volume 2, number 2 (2013), 137-146 http://www.etamaths.com fixed point and common fixed point theorems for α−property in cone ball-metric spaces rajesh shrivastava1, rajendra kumar dubey2, pankaj tiwari3, animesh gupta4,∗ abstract. in this paper, we define a new cone ball-metric and get fixed points and common fixed points for the α − property in cone ball-metric spaces. 1. introduction and preliminaries throughout this paper, by r+, we denote the set of all non-negative numbers, while n is the set of all natural numbers. let ♦ : r+ ×r+ → r+ be a binary operation satisfying the following conditions : (i) ♦ is associative and commutative, (ii) ♦ is continuous. five typical examples of ♦ are : (i) a♦b = max{a,b}, (ii) a♦b = a + b, (iii) a♦b = ab, (iv) a♦b = ab + a + b, (v) a♦b = ab max{a,b,1}. definition 1.1. the binary operation ♦ is said to satisfy α − property if there exists a positive real number α such that a♦b ≤ α max{a,b} for all a,b ∈ r+. following five examples are stand for α−property. example 1.2. if a♦b = a + b, for each a,b ∈ r+, then for α ≥ 2, we have a♦b ≤ α max{a,b}. example 1.3. if a♦b = ab max{a,b,1}, for each a,b ∈ r +, then for α ≥ 1, we have a♦b ≤ α max{a,b}. example 1.4. if a♦b = ab, for each a,b ∈ r+, then for α ≥ a, we have a♦b ≤ α max{a,b}. 2010 mathematics subject classification. primary 47h10; secondary 54h25, 55m20. key words and phrases. α − property, cone ball-metric space; common fixed point theorem; fixed point theorem. c©2013 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 137 138 shrivatava, dubey, tiwari and gupta example 1.5. if a♦b = ab + a + b, for each a,b ∈ r+, then for α ≥ a + 2, we have a♦b ≤ α max{a,b}. example 1.6. if a♦b = max{a,b}, for each a,b ∈ r+, then for α > 1, we have a♦b ≤ α max{a,b}. huang and zhang [3] have introduced the concept of the cone metric space, replacing the set of real numbers by an ordered banach space, and they showed some fixed point theorems of contractive type mappings on cone metric spaces. in 2006, mustafa and sims [5] introduced a more appropriate generalization of metric spaces, g-metric spaces. recently, beg et al. [2] introduced the notion of generalized cone metric spaces, and proved some fixed point results for mappings satisfying certain contractive conditions. in this paper, we define a new cone ballmetric and get fixed point and common fixed results for the α−property in cone ball-metric spaces. we recall some definitions of the cone metric spaces and some of the properties [3], as follow: definition 1.7. [3] let e be a real banach space and p a subset of e. p is called a cone if and only if: (i) p is nonempty, closed, and p 6= {θ}, (ii) a,b ∈ r+, x,y ∈ p ⇒ ax + by ∈ p , (iii) x ∈ p and −x ∈ p ⇒ x = θ. for given a cone p ⊂ e, we can define a partial ordering with respect to p by x 4 y or x < y if and only if y − x ∈ p for all x,y ∈ e. the real banach space e equipped with the partial ordered induced by p is denoted by (e,4). we shall write x ≺ y to indicate that x 4 y but x 6= y, while x � y will stand for y −x ∈ intp , where intp denotes the interior of p . the cone p is called normal if there exists a real number k > 0 such that for all x,y ∈ e, θ 4 x 4 y ⇒ ‖x‖≤ k‖y‖. the least positive number k satisfying above is called the normal constant of p. the cone p is called regular if every non-decreasing sequence which is bounded from above is convergent, that is, if {xn} is a sequence such that x1 4 x2 4 · · ·4 xn 4 · · ·4 y, for some y ∈ e, then there is x ∈ e such that ‖xn−x‖→ 0 as n →∞. equivalently, the cone p is regular if and only if every non-increasing sequence which is bounded from below is convergent. it is well known that a regular cone is a normal cone. moreover, p is called stronger minihedral if every subset of e which is bounded above has a supremum [1]. in the following we always suppose that e is a real banach space with a stronger minihedral regular cone p and intp 6= φ, and 4 is a partial ordering with respect to p . metric spaces are playing an important role in mathematics and the applied sciences. in 2003, mustafa and sims [5] introduced a more appropriate and robust notion of a generalized metric space as follows. definition 1.8. [5] let x be a nonempty set, and let g : x ×x ×x → [0,∞) be a function satisfying the following axioms: fixed point and common fixed point theorems 139 (1) g(x,y,z) = 0 if and only if x = y = z; (2) g(x,x,y) > 0, for all x 6= y; (3) g(x,y,z) ≥ g(x,x,y), for all x,y,z ∈ x; (4) g(x,y,z) = g(x,z,y) = g(z,y,x) = · · · (symmetric in all three variables); (5) g(x,y,z) ≤ g(x,w,w) + g(w,y,z), for all x,y,z,w ∈ x. then the function g is called a generalized metric, or, more specifically a g-metric on x, and the pair (x,g) is called a g-metric space. this research subject is interesting and widespread. but is too abstract makes the human difficulty with to understand. so we introduce the concept of cone ball-metric spaces and we prove fixed point results on such spaces for functions satisfying the contractions involving the α−property. in [6] chen and tsai introduce the following notion of the cone ball-metric b. definition 1.9. [6] let (x,d) be a cone metric space, b : x × x × x → e, x,y,z ∈ x and we denote δ(b) = sup{d(a,b) : a,b ∈ b}, and b(x,y,z) = δ(b), where b = ∩{f ⊂ x|f is a closed ball and {x,y,z} ⊂ f}. then we call b a ball-metric with respect to the cone metric d, and (x,b) a cone ball-metric space. it is clear that b(x,x,y) = d(x,y). remark 1.10. it is clear that the cone ball-metric b has the following properties: (1) b(x,y,z) = θ if and only if x = y = z; (2) b(x,x,y) � θ, for all x 6= y; (3) b(x,x,y) 4b(x,y,z), for all x,y,z ∈ x; (4) b(x,y,z) = b(x,z,y) = b(z,y,x) = · · · (symmetric in all three variables); (5) b(x,y,z) 4b(x,w,w) + b(w,y,z), for all x,y,z,w ∈ x; (6) b(x,y,z) 4b(x,w,w) + b(y,w,w) + b(z,w,w), for all x,y,z,w ∈ x. definition 1.11. [6] let (x,b) be a cone ball-metric space and {xn} be a sequence in x. we say that {xn} is (1) cauchy sequence if for every ε ∈ e with θ � ε, there exists n0 ∈ n such that for all n,m,l > n0, b(xn,xm,xl) � ε. (2) convergent sequence if for every ε ∈ e with θ � ε, there exists n0 ∈ n such that for all n,m > n0, b(xn,xm,x) � ε for some x ∈ x. here x is called the limit of the sequence {xn} and is denoted by limn→∞xn = x or xn → x as n →∞. definition 1.12. [6] let (x,b) be a cone ball-metric space. then x is said to be complete if every cauchy sequence is convergent in x. proposition 1.13. [6] let (x,b) be a cone ball-metric space and {xn} be a sequence in x. then the following are equivalent: (i) {xn} converges to x; (ii) b(xn,xn,x) → θ as n →∞; (iii) b(xn,x,x) → θ as n →∞; (iv) b(xn,xm,x) → θ as n,m →∞. 140 shrivatava, dubey, tiwari and gupta proposition 1.14. [6] let (x,b) be a cone ball-metric space and {xn} be a sequence in x, x,y ∈ x. if xn → x and xn → y as n →∞, then x = y. proof. let ε ∈ e with θ � ε be given. since xn → x and xn → y as n →∞, there exists n0 ∈ n such that for all m,n > n0, b(xn,xm,x) � ε 3 and b(xn,xm,y) � ε 3 . therefore, b(x,x,y) 4 b(x,xn,xn) + b(xn,x,y) = b(x,xn,xn) + b(y,xn,x) 4 b(x,xn,xn) + b(y,xm,xm) + b(xm,xn,x) � ε 3 + ε 3 + ε 3 = ε. hence, b(x,x,y) � ε α for all α ≥ 1, and so ε α −b(x,x,y) ∈ p for all α ≥ 1. since ε α → θ as α →∞ and p is closed, we have that −b(x,x,y) ∈ p . this implies that b(x,x,y) = θ, since b(x,x,y) ∈ p . so x = y. � proposition 1.15. [6] let (x,b) be a cone ball-metric space and {xn},{ym},{zl} be three sequences in x. if xn → x, ym → y, zl → z as n →∞, then b(xn,ym,zl) → b(x,y,z) as n →∞. proof. let ε ∈ e with θ � ε be given. since xn → x, ym → y, zl → z as n → ∞, there exists n0 ∈ n such that for all n,m,l > n0, b(xn,x,x) � ε 3 , b(ym,y,y) � ε 3 , b(zl,z,z) � ε 3 , therefore, b(xn,ym,zl) 4 b(xn,x,x) + b(x,ym,zl) 4 � b(xn,x,x) + b(ym,y,y) + b(y,x,zl) 4 b(xn,x,x) + b(ym,y,y) + b(zl,z,z) + b(z,x,y) � ε 3 + ε 3 + ε 3 + b(x,y,z), that is, b(xn,ym,zl) −b(x,y,z) � ε. similarly, b(x,y,z) −b(xn,ym,zl) � ε. therefore, for all α ≥ 1, we have b(xn,ym,zl) −b(x,y,z) � ε α , and b(x,y,z) −b(xn,ym,zl) � ε α . these imply that ε α −b(xn,ym,zl) + b(x,y,z) ∈ p, ε α + b(xn,ym,zl) −b(x,y,z) ∈ p. since p is closed and ε α → θ as α →∞, we have that lim n,m,l→∞ [−b(xn,ym,zl) + b(x,y,z)] ∈ p, fixed point and common fixed point theorems 141 lim n,m,l→∞ [b(xn,ym,zl) −b(x,y,z)] ∈ p. these show that lim n,m,l→∞ b(xn,ym,zl) = b(x,y,z). so we complete the proof. � 2. main results let (x,b) be a cone ball-metric space with p , and let ♦ : r+ ×r+ → r+. we now state our main common fixed point result for the α−property in a cone ball-metric space (x,b), as follows: theorem 2.1. let (x,b) be a complete cone ball-metric space, p be a regular cone in e and f,g be two self mappings of x such that f(x) ⊂ g(x). suppose that ♦ satisfies α−property with α > 0 such that b(fx,fy,fz) 4 k1[b(gx,gy,gz)♦b(gx,fx,fx)] + k2[b(gx,gy,gz)♦b(gy,fy,fy)] + k3[b(gx,gy,gz)♦b(gz,fz,fz)],(2.1) where k1,k2,k3 > 0 and 0 < α(k1 + k2 + k3) < 1. if g(x) is closed, then f and g have a coincidence point in x. moreover, if f and g commute at their coincidence points, then f and g have a unique common fixed point in x proof. given x0 ∈ x. since f(x) ⊂ g(x), we can choose x1 ∈ x such that gx1 = fx0. continuing this process, we define the sequence {xn} in x recursively as follows: fxn = gxn+1 for each n ∈ n∪{0}. in what follows we will suppose that fxn+1 6= fxn for all n ∈ n, since if fxn+1 = fxn for some n, then fxn+1 = gxn+1, that is , f,g have a coincidence point xn+1, and so we complete the proof. by (2.1), we have b(fxn,fxn+1,fxn+1) 4 k1[b(gxn,gxn+1,gxn+1)♦b(gxn,fxn,fxn)] + k2[b(gxn,gxn+1,gxn+1)♦b(gxn+1,fxn+1,fxn+1)] + k3[b(gxn,gxn+1,gxn+1)♦b(gxn+1,fxn+1,fxn+1)]. therefore, by the condition of α−property, we conclude that for each n ∈ n, b(fxn,fxn+1,fxn+1) 4 k1α max{b(fxn−1,fxn,fxn),b(fxn−1,fxn,fxn)} + k2α max{b(fxn−1,fxn,fxn),b(fxn,fxn+1,fxn+1)} + k3α max{b(fxn−1,fxn,fxn),b(fxn,fxn+1,fxn+1)}. if b(fxn,fxn+1,fxn+1) > b(fxn−1,fxn,fxn), we obtain b(fxn,fxn+1,fxn+1) � α(k1 + k2 + k3)b(fxn,fxn+1,fxn+1), which contradiction. hence b(fxn,fxn+1,fxn+1) ≤ b(fxn−1,fxn,fxn). similarly it is easy to see that b(fxn+1,fxn+2,fxn+2) ≤b(fxn,fxn+1,fxn+1) 142 shrivatava, dubey, tiwari and gupta b(fxn,fxn+1,fxn+1) � α(k1 + k2 + k3)b(fxn−1,fxn,fxn), and b(fxn,fxn+1,fxn+1) 4 δb(fxn−1,fxn,fxn) 4 · · · 4 δnb(fx0,fx1,fx1) where α(k1 + k2 + k3) = δ < 1. so we have b(fxn,fxn+1,fxn+1) 4 δnb(fx0,fx1,fx1) → θ as n →∞. next, we claim that the sequence {fxn} is a cauchy sequence. suppose that {fxn} is not a cauchy sequence. then there exists γ ∈ e with θ � γ such that for all k ∈ n, there are mk,nk ∈ n with mk > nk ≥ k satisfying: (1) mk is even and nk is odd, (2) b(fxnk,fxmk,fxmk ) < γ, and (3) mk is the smallest even number such that the conditions (1), (2) hold. since limn→∞b(fxn,fxn+1,fxn+1) = θ and by (2), (3), we have that γ 4 b(fxnk,fxmk,fxmk ) 4 b(fxnk,fmk−1,fmk−1) + b(fxmk−1,fxmk,fxmk ) 4 b(fxnk,fxmk−2,fxmk−2) + b(fxmk−2,fxmk−1,fxmk−1) +b(fxmk−1,fmk,fmk ) 4 γ + b(fxmk−2,fxmk−1,fxmk−1) + b(fxmk−1,fxmk,fxmk ). taking limk→∞, we deduce lim k→∞ b(fxnk,fxmk,fxmk ) = γ. since b(fxnk−1,fxmk−1,fxmk−1) 4 b(fxnk−1,fxnk,fxnk ) + b(fxnk,fxmk,fxmk ) +b(fxmk,fxmk−1,fxmk−1). taking limk→∞, we deduce lim k→∞ b(fxnk−1,fxmk−1,fxmk−1) 4 γ.(2.2) on the other hand, γ 4 b(fxnk,fxmk,fxmk ) 4 b(fxnk,fxnk−1,fxnk−1) + b(fxnk−1,fxmk,fxmk ) 4 b(fxnk,fxnk−1,fxnk−1) + b(fxnk−1,fxmk−1,fxmk−1) +b(fxmk−1,fxmk,fxmk ). taking limk→∞, we also deduce γ 4 lim k→∞ b(fxnk−1,fxmk−1,fxmk−1).(2.3) by (2.2) and (2.3), we get lim k→∞ b(fxnk−1,fxmk−1,fxmk−1) = γ. fixed point and common fixed point theorems 143 and, by (2.1), we have that b(fxnk,fxmk,fxmk ) 4 ψ(l(xnk,xmk,xmk )) where b(fxnk,fxmk,fxmk ) 4 k1[b(gxnk,gxmk,gxmk )♦b(gxnk,fxnk,fxnk )] + k2[b(gxnk,gxmk,gxmk )♦b(gxmk,fxmk,fxmk )] + k3[b(gxnk,gxmk,gxmk )♦b(gxmk,fxmk,fxmk )]. b(fxnk,fxmk,fxmk ) 4 k1[b(fxnk−1,fxmk−1,fxmk−1)♦b(fxnk−1,fxnk,fxnk )] + k2[b(fxnk−1,fxmk−1,fxmk−1)♦b(fxmk−1,fxmk,fxmk )] + k3[b(fxnk−1,fxmk−1,fxmk−1)♦b(fxmk−1,fxmk,fxmk )]. b(fxnk,fxmk,fxmk ) 4 αk1 max{b(fxnk−1,fxmk−1,fxmk−1),b(fxnk−1,fxnk,fxnk )} + αk2 max{b(fxnk−1,fxmk−1,fxmk−1),b(fxmk−1,fxmk,fxmk )} + αk3 max{b(fxnk−1,fxmk−1,fxmk−1),b(fxmk−1,fxmk,fxmk )}. (i) if b(fxnk−1,fxmk−1,fxmk−1), is maximum then taking limk→∞, we deduce lim k→∞ b(fxnk−1,fxmk−1,fxmk−1) = γ, and γ 4 lim k→∞ b(fxnk,fxmk,fxmk ) � γ, a contradiction. (ii) if b(fxnk−1,fxnk,fxnk ), or b(fxmk−1,fxmk,fxmk ), is maximum then taking limk→∞, we deduce lim k→∞ b(fxnk−1,fxnk,fxnk ) = θ, lim k→∞ b(fxmk−1,fxmk,fxmk ) = θ, and γ 4 lim k→∞ b(fxnk,fxmk,fxmk ) 4 θ, a contradiction. follow (i) and (ii), we get the sequence {fxn} is a cauchy sequence. since x is complete and g(x) is closed, there exist ν,µ ∈ x such that lim n→∞ g(xn) = lim n→∞ f(xn) = g(µ) = ν. we shall show that µ is a coincidence point of f and g, that is, we claim that b(gµ,fµ,fµ) = θ. 144 shrivatava, dubey, tiwari and gupta if not, assume that b(gµ,fµ,fµ) 6= θ, then by (2.1), we have b(gµ,fµ,fµ) 4 b(gµ,fxn,fxn) + b(fxn,fµ,fµ) 4 b(gµ,fxn,fxn) + k1[b(gxn,gµ,gµ)♦b(gxn,fxn,fxn)] +k2[b(gxn,gµ,gµ)♦b(gµ,fµ,fµ)] +k3[b(gxn,gµ,gµ)♦b(gµ,fµ,fµ)], 4 b(gµ,fxn,fxn) +α(k1 + k2 + k3) max{b(gxn,gµ,gµ),b(gxn,fxn,fxn),b(gµ,fµ,fµ)}, (iii) if max{b(gxn,gµ,gµ),b(gxn,fxn,fxn),b(gµ,fµ,fµ)} = b(gxn,gµ,gµ), then taking limn→∞, we deduce lim n→∞ b(gxn,gµ,gµ) = b(gµ,gµ,gµ) = θ, and b(gµ,fµ,fµ) = lim n→∞ b(gµ,fxn,fxn) + α(k1 + k2 + k3) lim n→∞ b(gxn,gµ,gµ) 4 θ, a contradiction. (iv) if max{b(gxn,gµ,gµ),b(gxn,fxn,fxn),b(gµ,fµ,fµ)} = b(gxn,fxn,fxn), then taking limn→∞, we deduce lim n→∞ b(gxn,fxn,fxn) = b(gµ,gµ,gµ) = θ, and b(gµ,fµ,fµ) = lim n→∞ b(gµ,fxn,fxn) + α(k1 + k2 + k3) lim n→∞ b(gxn,fxn,fxn) 4 θ, a contradiction. (v) if max{b(gxn,gµ,gµ),b(gxn,fxn,fxn),b(gµ,fµ,fµ)} = b(gµ,fµ,fµ), then b(gµ,fµ,fµ) = α(k1 + k2 + k3)b(gµ,fµ,fµ) �b(gµ,fµ,fµ), a contradiction. follow (iii)-(v), we obtain that b(gµ,fµ,fµ) = θ, that is, gµ = fµ = ν, and so µ is a coincidence point of f and g. suppose that f and g commute at µ. then fν = fgµ = gfµ = gν. later, we claim that b(fµ,fν,fν) = θ. by (2.1), we have fixed point and common fixed point theorems 145 b(fµ,fν,fν) 4 k1[b(gµ,gν,gν)♦b(gµ,fµ,fµ)] + k2[b(gµ,gν,gν)♦b(gν,fν,fν)] + k3[b(gµ,gν,gν)♦b(gν,fν,fν)] = α(k1 + k2 + k3) max{b(fµ,fν,fν),b(fµ,fµ,fµ),b(fν,fν,fν)} = α(k1 + k2 + k3) max{b(fµ,fν,fν),θ}. therefore, if b(fµ,fν,fν) 4 α(k1 + k2 + k3)b(fµ,fν,fν) �b(fµ,fν,fν), then we get a contradition, which implies that b(fµ,fν,fν) = θ, b(ν,fν,fν) = θ, that is, ν = fν = gν. so ν is a common fixed point of f and g. let ν be another common fixed point of f and g. by (2.1), b(ν,ν,ν) = b(fν,fν,fν), where b(fν,fν,fν) 4 k1[b(gν,gν,gν)♦b(gνfν,fν)] + k2[b(gν,gν,gν)♦b(gν,fν,fν)] + k3[b(gν,gν,gν)♦b(gν,fν,fν)] = α(k1 + k2 + k3) max{b(fν,fν,fν),b(fν,fν,fν),b(fν,fν,fν)} = α(k1 + k2 + k3) max{b(fν,fν,fν),θ} = α(k1 + k2 + k3) max{b(ν,ν,ν),θ} 4 b(ν,ν,ν). therefore, we also conclude that b(ν,ν,ν) = θ, that is ν = ν. so we show that ν is the unique common fixed point of g and f. � corollary 2.2. let (x,b) be a complete cone ball-metric space, p be a regular cone in e and f,g be two self mappings of x such that f(x) ⊂ g(x). suppose that b(fx,fy,fz) 4 α max{b(gx,gy,gz),b(gx,fx,fx), b(gy,fy,fy),b(gz,fz,fz)}(2.4) where α ∈ [0, 1) . if g(x) is closed, then f and g have a coincidence point in x. moreover, if f and g commute at their coincidence points, then f and g have a unique common fixed point in x proof. it is sufficient if we take a♦b = α max{a,b} for α ∈ [0, 1) in theorem 2.1 then we get the result. � corollary 2.3. let (x,b) be a complete cone ball-metric space, p be a regular cone in e and f be a self mapping of x. suppose that ♦ satisfies α−property with α > 0 such that b(fx,fy,fz) 4 k1[b(x,y,z)♦b(x,fx,fx)] + k2[b(x,y,z)♦b(y,fy,fy)] + k3[b(x,y,z)♦b(z,fz,fz)],(2.5) where k1,k2,k3 > 0 and 0 < α(k1 + k2 + k3) < 1. then f has a fixed point in x. 146 shrivatava, dubey, tiwari and gupta proof. it is sufficient if we take g = i (identity mapping) in theorem 2.1 then we get the result. � references [1] c.d. aliprantis and r. tourky, cones and duality, in: graduate studies in mathematics, amer. math. soc. 84 (2007). 215–240. [2] i. beg, m. abbas and t. nazir, generalized cone metric spaces, j. nonlinear sci. appl. 3 (2010), no. 1, 23–31. [3] l. g. huang and x. zhang, cone metric spaces and fixed point theorems of contractive mappings, j. math. anal. appl. 322 (2007), 1468–1476. [4] a. meir and e. keeler, a theorem on contraction mappings, j. math. anal. appl. 28 (1969), 326–329. [5] z. mustafa and b. sims, a new approach to generalized metric spaces, j. nonlinear and convex anal. 7 (2006), no. 2, 289–297. [6] c. chen and p. tsai, fixed point and common fixed point theorems for the meir-keeler type functions in cone ball-metric spaces, ann. funct. anal. 3 (2012), no. 2, 155–169. 1department of mathematics, govt. science & commerce college, benazir bhopalindia 2department of mathematics, govt. science p.g. college, reewa -india 3department of applied mathematics, vidhyapeeth institute of science & technology, bhopalindia 4department of applied mathematics, sagar institute of science technology & research, ratibad bhopalindia ∗corresponding author int. j. anal. appl. (2023), 21:90 upper and lower weakly α-?-continuous multifunctions chawalit boonpok, prapart pue-on∗ mathematics and applied mathematics research unit, department of mathematics, faculty of science, mahasarakham university, maha sarakham, 44150, thailand ∗corresponding author: prapart.p@msu.ac.th abstract. this paper deals with the concepts of upper and lower weakly α-?-continuous multifunctions. moreover, some characterizations of upper and lower weakly α-?-continuous multifunctions are investigated. furthermore, the relationships between almost α-?-continuity and weak α-?-continuity are discussed. 1. introduction topology is concerned with all questions directly or indirectly related to continuity. weaker and stronger forms of open sets play an important role in the researches of generalizations of continuity for functions and multifunctions. in 1965, njåstad [18] introduced a weak form of open sets called α-sets. mashhour et al. [17] defined a function to be α-continuous if the inverse image of each open set is an α-set and obtained several characterizations of such functions. noiri [20] investigated the relationships between α-continuous functions and several known functions, for example, almost continuous functions, η-continuous functions, δ-continuous functions or irresolute functions. in [21], the present author introduced the concept of almost α-continuity in topological spaces as a generalization of α-continuity and almost continuity. neubrunn [19] introduced the notion of upper (resp. lower) αcontinuous multifunctions. these multifunctions are further investigated by the present authors [24]. in 1996, popa and noiri [23] introduced the notion of upper (resp. lower) almost α-continuous multifunctions and investigated several characterizations and some basic properties concerning upper (resp. lower) almost α-continuous multifunctions. moreover, some characterizations of weakly α-continuous multifunctions were investigated in [11], [22] and [23]. topological ideals have played an important received: jun. 13, 2023. 2020 mathematics subject classification. 54c08, 54c60. key words and phrases. upper weakly α-?-continuous multifunction; lower weakly α-?-continuous multifunction. https://doi.org/10.28924/2291-8639-21-2023-90 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-90 2 int. j. anal. appl. (2023), 21:90 role in topology. kuratowski [16] and vaidyanathswamy [25] introduced and studied the concept of ideals in topological spaces. every topological space is an ideal topological space and all the results of ideal topological spaces are generalizations of the results established in topological spaces. in 1990, janković and hamlett [15] introduced the concept of i -open sets in ideal topological spaces. abd elmonsef et al. [1] further investigated i -open sets and i -continuous functions. later, several authors studied ideal topological spaces giving several convenient definitions. some authors obtained decompositions of continuity. for instance, açikgöz et al. [4] studied the concepts of α-i -continuity and α-i -openness in ideal topological spaces and obtained several characterizations of these functions. hatir and noiri [14] introduced the notions of semi-i -open sets, α-i -open sets and β-i -open sets via idealization and using these sets obtained new decompositions of continuity. moreover, açikgöz et al. [3] introduced and studied the notions of weakly-i -continuous and weak?-i -continuous functions in ideal topological spaces. in [10], the present author introduced and investigated the concepts of upper and lower weakly ?-continuous multifunctions. in this paper, we introduce the concepts of upper and lower weakly α-?-continuous multifunctions. in particular, several characterizations of upper and lower weakly α-?-continuous multifunctions are discussed. 2. preliminaries throughout the present paper, spaces (x,τ)and (y,σ) (or simply x and y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. let a be a subset of a topological space (x,τ). the closure of a and the interior of a are denoted by cl(a) and int(a), respectively. an ideal i on a topological space (x,τ) is a nonempty collection of subsets of x satisfying the following properties: (1) a ∈ i and b ⊆ a imply b ∈ i ; (2) a ∈ i and b ∈ i imply a ∪ b ∈ i . a topological space (x,τ) with an ideal i on x is called an ideal topological space and is denoted by (x,τ, i). for an ideal topological space (x,τ, i) and a subset a of x, a?(i) is defined as follows: a?(i)= {x ∈ x : u ∩a 6∈ i for every open neighbourhood u of x}. in case there is no chance for confusion, a?(i) is simply written as a?. in [16], a? is called the local function of a with respect to i and τ and cl?(a) = a? ∪a defines a kuratowski closure operator for a topology τ?(i) finer than τ. a subset a is said to be ?-closed [15] if a? ⊆ a. the interior of a subset a in (x,τ?(i)) is denoted by int?(a). a subset a of an ideal topological space (x,τ, i) is said to be semi?-i -open [13] (resp. semi-i open [14]) if a ⊆ cl(int?(a)) (resp. a ⊆ cl?(int(a))). the complement of a semi?-i -open (resp. semi-i -open) set is said to be semi?-i -closed [13] (resp. semi-i -closed [14]). for a subset a of an ideal topological space (x,τ, i), the intersection of all semi-i -closed (resp. semi?-i -closed) sets containing a is called the semi-i -closure [14] (resp. semi?-i -closure [12]) of a and is denoted by scli(a) (resp. s ?cli(a)). the union of all semi-i -open (resp. semi ?-i -open) sets contained int. j. anal. appl. (2023), 21:90 3 in a is called the semi-i -interior (resp. semi?-i -interior) of a and is denoted by sinti(a) (resp. s?inti(a)). lemma 2.1. [6] for a subset a of an ideal topological space (x,τ, i), the following properties hold: (1) if a is an open set, then s?cli(a)= int(cl ?(a)). (2) if a is a ?-open set, then scli(a)= int ?(cl(a)). recall that a subset a of an ideal topological space (x,τ, i) is said to be α-?-closed [2] if cl?(int(cl?(a)))⊆ a. the complement of an α-?-closed set is said to be α-?-open. for a subset a of an ideal topological space (x,τ, i), the intersection of all α-?-closed sets containing a is called the α-?-closure [6] of a and is denoted by ?αcl(a). the α-?-interior [6] of a is defined by the union of all α-?-open sets contained in a and is denoted by ?αint(a). lemma 2.2. [6] for a subset a of an ideal topological space (x,τ, i), the following properties hold: (1) ?αcl(a) is α-?-closed. (2) a is α-?-closed if and only if a = ?αcl(a). lemma 2.3. [6] for a subset a of an ideal topological space (x,τ, i), the following properties are equivalent: (1) a is α-?-open in x; (2) g ⊆ a ⊆ int?(cl(g)) for some ?-open set g; (3) g ⊆ a ⊆ scli(g) for some ?-open set g; (4) a ⊆ scli(int?(a)). lemma 2.4. [6] for a subset a of an ideal topological space (x,τ, i), the following properties hold: (1) a is α-?-closed in x if and only if sinti(cl ?(a))⊆ a. (2) sinti(cl ?(a))= cl?(int(cl?(a))). (3) ?αcl(a)= a∪cl?(int(cl?(a))). (4) ?αint(a)= a∩ int?(cl(int?(a))). by a multifunction f : x → y , we mean a point-to-set correspondence from x into y , and we always assume that f(x) 6= ∅ for all x ∈ x. for a multifunction f : x → y , following [5] we shall denote the upper and lower inverse of a set b of y by f+(b) and f−(b), respectively, that is, f+(b) = {x ∈ x | f(x) ⊆ b} and f−(b) = {x ∈ x | f(x)∩ b 6= ∅}. in particular, f−(y)= {x ∈ x | y ∈ f(x)} for each point y ∈ y . for each a ⊆ x, f(a)=∪x∈af(x). 3. upper and lower weakly α-?-continuous multifunctions we begin this section by introducing the concepts of upper and lower weakly α-?-continuous multifunctions. 4 int. j. anal. appl. (2023), 21:90 definition 3.1. a multifunction f : (x,τ, i)→ (y,σ, j) is said to be: (1) upper weakly α-?-continuous at a point x ∈ x if, for each ?-open set v of y such that f(x)⊆ v , there exists an α-?-open set u of x containing x such that f(u)⊆ cl?(v ); (2) lower weakly α-?-continuous at a point x ∈ x if, for each ?-open set v of y such that f(x)∩v 6= ∅, there exists an α-?-open set u of x containing x such that f(z)∩cl?(v ) 6= ∅ for every z ∈ u; (3) upper (resp. lower) weakly α-?-continuous if f is upper (resp. lower) weakly α-?-continuous at each point of x. theorem 3.1. for a multifunction f : (x,τ, i)→ (y,σ, j), the following properties are equivalent: (1) f is upper weakly α-?-continuous at x ∈ x; (2) x ∈ ?αint(f+(cl?(v ))) for every ?-open set v of y containing f(x); (3) x ∈ int?(cl(int?(f+(cl?(v ))))) for every ?-open set v of y containing f(x). proof. (1)⇒ (2): let v be any ?-open set of y containing f(x). then, there exists an α-?-open set u of x containing x such that f(u)⊆ cl?(v ); hence u ⊆ f+(cl?(v )). thus, x ∈ ?αint(f+(cl?(v ))). (2) ⇒ (3): let v be any ?-open set of y containing f(x). then by (2), we have x ∈ ?αint(f+(cl?(v ))) and by lemma 2.4(4), x ∈ int?(cl(int?(f+(cl?(v ))))). (3)⇒ (1): let v be any ?-open set of y containing f(x). by (3), x ∈ int?(cl(int?(f+(cl?(v ))))) and by lemma 2.4(4), x ∈ ?αint(f+(cl?(v ))). then, there exists an α-?-open set u of x containing x such that u ⊆ f+(cl?(v )); hence f(u) ⊆ cl?(v ). this shows that f is upper weakly α-?continuous at x. � theorem 3.2. for a multifunction f : (x,τ, i)→ (y,σ, j), the following properties are equivalent: (1) f is lower weakly α-?-continuous at x ∈ x; (2) x ∈ ?αint(f−(cl?(v ))) for every ?-open set v of y such that f(x)∩v 6= ∅; (3) x ∈ int?(cl(int?(f−(cl?(v ))))) for every ?-open set v of y such that f(x)∩v 6= ∅. proof. the proof is similar to that of theorem 3.1. � definition 3.2. [10] a subset a of an ideal topological space (x,τ, i) is said to be: (1) r-i ?-open if a = int?(cl?(a)); (2) r-i ?-closed if its complement is r-i ?-open. definition 3.3. [9] a point x in an ideal topological space (x,τ, i) is called a ?θ-cluster point of a if cl?(u)∩a 6= ∅ for every ?-open set u of x containing x. the set of all ?θ-cluster points of a is called the ?θ-closure of a and is denoted by ?θcl(a). definition 3.4. [9] a subset a of an ideal topological space (x,τ, i) is said to be: (1) ?θ-closed if ?θcl(a)= a; int. j. anal. appl. (2023), 21:90 5 (2) ?θ-open if its complement is ?θ-closed. lemma 3.1. [9] for a subset a of an ideal topological space (x,τ, i), the following properties hold: (1) if a is ?-open in x, then cl?(a)= ?θcl(a). (2) ?θcl(a) is ?-closed in x. theorem 3.3. for a multifunction f : (x,τ, i)→ (y,σ, j), the following properties are equivalent: (1) f is upper weakly α-?-continuous; (2) f+(v )⊆ int?(cl(int?(f+(cl?(v ))))) for every ?-open set v of y ; (3) cl?(int(cl?(f−(int?(k)))))⊆ f−(k) for every ?-closed set k of y ; (4) ?αcl(f−(int?(k)))⊆ f−(k) for every ?-closed set k of y ; (5) ?αcl(f−(int?(cl?(b))))⊆ f−(cl?(b)) for every subset b of y ; (6) f+(int?(b))⊆ ?αint(f+(cl?(int?(b)))) for every subset b of y ; (7) f+(v )⊆ ?αint(f+(cl?(v ))) for every ?-open set v of y ; (8) ?αcl(f−(int?(k)))⊆ f−(k) for every r-j ?-closed set k of y ; (9) ?αcl(f−(v ))⊆ f−(cl?(v )) for every ?-open set v of y ; (10) ?αcl(f−(int?(?θcl(b))))⊆ f−(?θcl(b)) for every subset b of y . proof. (1) ⇒ (2): let v be any ?-open set of y and x ∈ f+(v ). then, f(x) ⊆ v and there exists an α-?-open set u of x containing x such that f(u) ⊆ cl?(v ); hence u ⊆ f+(cl?(v )) and so x ∈ u ⊆ int?(cl(int?(f+(cl?(v ))))). thus, f+(v )⊆ int?(cl(int?(f+(cl?(v ))))). (2)⇒ (3): let k be any ?-closed set of y . then, y −k is ?-open in y and by (2), we have x −f−(k)= f+(y −k) ⊆ int?(cl(int?(f+(cl?(y −k))))) = int?(cl(int?(f+(y − int?(k))))) = int?(cl(int?(x −f−(int?(k))))) = int?(cl(x −cl?(f−(int?(k))))) = int?(x − int(cl?(f−(int?(k))))) = x −cl?(int(cl?(f−(int?(k))))) and hence cl?(int(cl?(f−(int?(k)))))⊆ f−(k). (3)⇒ (4): let k be any ?-closed set of y . by (3), we have cl?(int(cl?(f−(int?(k)))))⊆ f−(k) and hence ?αcl(f−(int?(k)))⊆ f−(k) by lemma 2.4(3). (4) ⇒ (5): let b be any subset of y . then, cl?(b) is ?-closed in y and by (4), ?αcl(f−(int?(cl?(b))))⊆ f−(cl?(b)). 6 int. j. anal. appl. (2023), 21:90 (5)⇒ (6): let b be any subset of y . by (5), f+(int?(b))= x −f−(cl?(y −b)) ⊆ x −?αcl(f−(int?(cl?(y −b)))) = x −?αcl(f−(y −cl?(int?(b)))) = x −?αcl(x −f+(cl?(int?(b)))) = ?αint(f+(cl?(int?(b)))). (6)⇒ (7): the proof is obvious. (7) ⇒ (1): let x ∈ x and v be any ?-open set of y containing f(x). it follows from lemma 2.4(4) that x ∈ f+(v ) ⊆ ?αint(f+(cl?(v ))) ⊆ int?(cl(int?(f+(cl?(v ))))) and hence f is upper weakly α-?-continuous at x by theorem 3.1. this shows that f is upper weakly α-?-continuous. (4)⇒ (8): the proof is obvious. (8)⇒ (9): let v be any ?-open set of y . then, we have cl?(v ) is r-j ?-closed in y and by (8), ?αcl(f−(v ))⊆ ?αcl(f−(int?(cl?(v ))))⊆ f−(cl?(v )). (9)⇒ (7): let v be any ?-open set of y . by (9), we have x −?αint(f+(cl?(v )))= ?αcl(x −f+(cl?(v ))) = ?αcl(f−(y −cl?(v ))) ⊆ f−(cl?(y −cl?(v ))) = x −f+(int?(cl?(v ))) and hence f+(v )⊆ f+(int?(cl?(v )))⊆ ?αint(f+(cl?(v ))). (9) ⇒ (10): let b be any subset of y . then, int?(?θcl(b)) is ?-open in y . thus, by (9) and lemma 3.1(1), ?αcl(f−(int?(?θcl(b))))⊆ f−(cl?(int?(?θcl(b)))) ⊆ f−(?θcl(int?(?θcl(b)))) ⊆ f−(?θcl(b)). (10)⇒ (8): let k be any r-j ?-closed set of y . by (10) and lemma 3.1(1), we have ?αcl(f−(int?(k)))= ?αcl(f−(int?(cl?(int?(k))))) = ?αcl(f−(int?(?θcl(int ?(k))))) ⊆ f−(?θcl(int?(k))) = f−(cl?(int?(v ))) = f−(k). int. j. anal. appl. (2023), 21:90 7 � theorem 3.4. for a multifunction f : (x,τ, i)→ (y,σ, j), the following properties are equivalent: (1) f is lower weakly α-?-continuous; (2) f−(v )⊆ int?(cl(int?(f−(cl?(v ))))) for every ?-open set v of y ; (3) cl?(int(cl?(f+(int?(k)))))⊆ f+(k) for every ?-closed set k of y ; (4) ?αcl(f+(int?(k)))⊆ f+(k) for every ?-closed set k of y ; (5) ?αcl(f+(int?(cl?(b))))⊆ f+(cl?(b)) for every subset b of y ; (6) f−(int?(b))⊆ αint?(f−(cl?(int?(b)))) for every subset b of y ; (7) f−(v )⊆ αint?(f−(cl?(v ))) for every ?-open set v of y ; (8) ?αcl(f+(int?(k)))⊆ f+(k) for every r-j ?-closed set k of y ; (9) ?αcl(f+(v ))⊆ f+(cl?(v )) for every ?-open set v of y ; (10) ?αcl(f+(int?(?θcl(b))))⊆ f+(?θcl(b)) for every subset b of y . proof. the proof is similar to that of theorem 3.3. � definition 3.5. [7] a multifunction f : (x,τ, i)→ (y,σ, j) is said to be: (1) upper almost α-?-continuous at a point x ∈ x if for each ?-open set v of y such that f(x)⊆ v , there exists an α-?-open set u of x containing x such that f(u)⊆ int?cl(v ); (2) lower almost α-?-continuous at a point x ∈ x if for each ?-open set v of y such that f(x)∩v 6= ∅, there exists an α-?-open set u of x containing x such that f(z)∩int?(cl(v )) 6= ∅ for every z ∈ u; (3) upper (resp. lower) almost α-?-continuous if f is upper (resp. lower) almost α-?-continuous at each point of x. remark 3.1. for a multifunction f : (x,τ, i)→ (y,σ, j), the following implication holds: upper almost α-?-continuity ⇒ upper weakly α-?-continuity. the converse of the implication is not true in general. we give an example for the implication as follows. example 3.1. let x = {1,2,3} with a topology τ = {∅,x} and an ideal i = {∅}. let y = {a,b,c} with a topology σ = {∅,{a},{b},{a,b},y} and an ideal j = {∅,{c}}. define f : (x,τ, i) → (y,σ, j) as follows: f(1) = {a}, f(2) = {b} and f(3) = {a,c}. then, f is upper weakly α-?continuous but f is not upper almost α-?-continuous. definition 3.6. a function f : (x,τ, i)→ (y,σ, j) is said to be weakly α-?-continuous if, for each x ∈ x and each ?-open set v of y containing f (x), there exists an α-?-open set u of x containing x such that f (u)⊆ cl?(v ). corollary 3.1. for a function f : (x,τ, i)→ (y,σ, j), the following properties are equivalent: 8 int. j. anal. appl. (2023), 21:90 (1) f is weakly α-?-continuous; (2) f−1(v )⊆ ?αint(f−1(cl?(v ))) for every ?-open set v of y ; (3) ?αcl(f−1(int?(k)))⊆ f−1(k) for every r-j ?-closed set k of y ; (4) ?αcl(f−1(v ))⊆ f−1(cl?(v )) for every ?-open set v of y ; (5) ?αcl(f−1(int?(?θcl(b))))⊆ f−1(?θcl(b)) for every subset b of y ; (6) cl?(int(cl?(f−1(v ))))⊆ f−1(cl?(v )) for every ?-open set v of y ; (7) f−1(v )⊆ int?(cl(int?(f−1(cl?(v ))))) for every ?-open set v of y ; (8) f (cl?(int(cl?(a))))⊆ ?θcl(f (a)) for every subset a of x; (9) cl?(int(cl?(f−1(b))))⊆ f−1(?θcl(b)) for every subset b of y . lemma 3.2. [8] let a be a subset of an ideal topological space (x,τ, i). if a is a ?-regular ?paracompact set of x and each ?-open set u containing a, then there exists a ?-open set v such that a ⊆ v ⊆ cl(v )⊆ u. theorem 3.5. for a multifunction f : (x,τ, i) → (y,σ, j) such that f(x) is a ?-regular ?paracompact set for each x ∈ x, the following properties are equivalent: (1) f is upper α-?-continuous; (2) f is upper almost α-?-continuous; (3) f is upper weakly α-?-continuous. proof. we prove only the implication (3) ⇒ (1). suppose that f is upper weakly α-?-continuous. let x ∈ x and v be any ?-open set of y such that f(x)⊆ v . since f(x) is ?-regular ?-paracompact, by lemma 3.2, there exists a ?-open set g such that f(x) ⊆ g ⊆ cl(g) ⊆ v . since f is upper weakly α-?-continuous, there exists an α-?-open set u of x containing x such that f(u) ⊆ cl?(g) and hence f(u)⊆ cl?(g)⊆ cl(g)⊆ v . this shows that f is upper α-?-continuous. � theorem 3.6. for a multifunction f : (x,τ, i)→ (y,σ, j) such that f(x) is ?-open in y for each x ∈ x, the following properties are equivalent: (1) f is lower α-?-continuous; (2) f is lower almost α-?-continuous; (3) f is lower weakly α-?-continuous. proof. we prove the only implication (3)⇒ (1). suppose that f is lower weakly α-?-continuous. let x ∈ x and v be any ?-open set of y such that f(x)∩v 6= ∅. since f is lower weakly α-?-continuous, there exists an α-?-open set u of x containing x such that f(z)∩cl?(v ) 6= ∅ for each z ∈ u. since f(z) is ?-open, we have f(z)∩v 6= ∅ for each z ∈ u and hence f is lower α-?-continuous. � acknowledgements: this research project was financially supported by mahasarakham university. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. int. j. anal. appl. (2023), 21:90 9 references [1] m.e. abd el-monsef, e.f. lashien, a.a. nasef, on i -open sets and i continuous functions, kyungpook math. j. 32 (1992), 21–30. [2] a. açikgöz, t. noiri, ş. yüksel, on α-operfect sets and α-?-closed sets, acta math. hungarica. 105 (2010), 146–153. [3] a. açikgöz, ş. yüksel, e. gursel, on a new concept of functions in ideal topological spaces, j. fac. sci. ege univ. 29 (2006), 30–35. [4] a. açikgöz, t. noiri, ş. yüksel, on α-i -continuous and α-i -open functions, acta math. hungarica. 105 (2004), 27–37. [5] c. berge, espaces topologiques fonctions multivoques, dunod, paris, 1959. [6] c. boonpok, j. khampakdee, upper and lower α-?-continuity, (accepted). [7] c. boonpok, n. srisarakham, almost α-?-continuity for multifunctions, (submitted). [8] c. boonpok, on some types of continuity for multifunctions in ideal topological spaces, adv. math., sci. j. 9 (2020), 859–886. https://doi.org/10.37418/amsj.9.3.13. [9] c. boonpok, weak quasi continuity for multifunctions in ideal topological spaces, adv. math., sci. j. 9 (2020), 339–355. https://doi.org/10.37418/amsj.9.1.28. [10] c. boonpok, on continuous multifunctions in ideal topological spaces, lobachevskii j. math. 40 (2019), 24–35. https://doi.org/10.1134/s1995080219010049. [11] j. cao, j. dontchev, on some weaker forms of continuity for multifunctions, real anal. exchange, 22 (1996/97), 842–852. [12] e. ekici, t. noiri, ?-hyperconnected ideal topological spaces, ann. alexandru ioan cuza univ. math. 58 (2012), 121–129. https://doi.org/10.2478/v10157-011-0045-9. [13] e. ekici, t. noiri, ?-extremally disconnected ideal topological spaces, acta math. hungarica. 122 (2009), 81–90. [14] hatir and t. noiri, on decompositions of continuity via idealization, acta math. hungarica. 96 (2002), 341–349. [15] d. janković and t. r. hamlett, new topologies from old via ideals, amer. math. mon. 97 (1990), 295–310. https://doi.org/10.1080/00029890.1990.11995593. [16] k. kuratowski, topology, vol. i, academic press, new york, 1966. [17] a.s. mashhour, i.a. hasanein, s.n. el-deeb, α-continuous and α-open mapping, acta math. hungarica. 41 (1983), 213–218. [18] o. njåstad, on some classes of nearly open sets, pac. j. math. 15 (1965), 961–970. https://doi.org/10. 2140/pjm.1965.15.961. [19] t. neubrunn, strongly quasi-continuous multivalued mapping, in: general topology and its relations to modern analysis and algebra vi (prague 1986), heldermann, berlin, 1988, pp. 351–359. [20] t. noiri, on αcontinuous functions, časopis pěst. mat. 109 (1984), 118–126. http://dml.cz/dmlcz/108508. [21] t. noiri, almost α-continuous functions, kyungpook math. j. 28 (1988), 71–77. [22] v. popa, t. noiri, on upper and lower weakly α-continuous multifunctions, novi sad j. math. 32 (2002), 7–24. [23] v. popa, t. noiri, on upper and lower almost α-continuous multifunctions, demonstr. math. 24 (1996), 381– 396. [24] v. popa, t. noiri, on upper and lower α-continuous multifunctions, math. slovaca, 43 (1993), 477–491. [25] v. vaidyanathswamy, the localization theory in set topology, proc. indian acad. sci. 20 (1945), 51–61. https://doi.org/10.37418/amsj.9.3.13 https://doi.org/10.37418/amsj.9.1.28 https://doi.org/10.1134/s1995080219010049 https://doi.org/10.2478/v10157-011-0045-9 https://doi.org/10.1080/00029890.1990.11995593 https://doi.org/10.2140/pjm.1965.15.961 https://doi.org/10.2140/pjm.1965.15.961 http://dml.cz/dmlcz/108508 1. introduction 2. preliminaries 3. upper and lower weakly –continuous multifunctions references int. j. anal. appl. (2023), 21:88 some properties of generalized (λ,α)-closed sets chawalit boonpok, montri thongmoon∗ mathematics and applied mathematics research unit, department of mathematics, faculty of science, mahasarakham university, maha sarakham, 44150, thailand ∗corresponding author: montri.t@msu.ac.th abstract. the aim of this paper is to introduce the concept of generalized (λ,α)-closed sets. moreover, we investigate some characterizations of λα-t 1 2 -spaces, (λ,α)-normal spaces and (λ,α)-regular spaces by utilizing generalized (λ,α)-closed sets. 1. introduction the concept of generalized closed sets was first introduced by levine [7]. moreover, levine defined a separation axiom called t1 2 between t0 and t1. dontchev and ganster [3] introduced the notion of t3 4 -spaces which are situated between t1 and t1 2 and showed that the digital line or the khalimsky line [5] (z,κ) lies between t1 and t3 4 . as a modification of generalized closed sets, palaniappan and rao [10] introduced and studied the notion of regular generalized closed sets. as the further modification of regular generalized closed sets, noiri and popa [9] introduced and investigated the concept of regular generalized α-closed sets. park et al. [11] obtained some characterizations of t3 4 spaces. dungthaisong et al. [4] characterized µ(m,n)-t1 2 spaces by utilizing the concept of µ(m,n)closed sets. torton et al. [12] introduced and studied the notions of µ(m,n)-regular spaces and µ(m,n)normal spaces. buadong et al. [1] introduced and investigated the notions of t1-gtms spaces and t2-gtms spaces. caldas et al. [2] by considering the concepts of α-open sets and α-closed sets, introduced and investigated λα-sets, (λ,α)-closed sets, (λ,α)-open sets and the (λ,α)-closure operator. khampakdee and boonpok [6] studied some properties of (λ,α)-open sets. in the present paper, we introduce the concept of generalized (λ,α)-closed sets. furthermore, some properties of received: jun. 13, 2023. 2020 mathematics subject classification. 54a05, 54d10. key words and phrases. generalized (λ,α)-closed set; λα-t 1 2 -space; (λ,α)-normal space; (λ,α)-regular space. https://doi.org/10.28924/2291-8639-21-2023-88 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-88 2 int. j. anal. appl. (2023), 21:88 generalized (λ,α)-closed sets are discussed. in particular, several characterizations of λα-t1 2 -spaces, (λ,α)-normal spaces and (λ,α)-regular spaces are established. 2. preliminaries let a be a subset of a topological space (x,τ). the closure of a and the interior of a are denoted by cl(a) and int(a), respectively. a subset a of a topological space (x,τ) is said to be α-open [8] if a ⊆ int(cl(int(a))). the complement of an α-open set is called α-closed. the family of all α-open sets in a topological space (x,τ) is denoted by α(x,τ). a subset λα(a) [2] is defined as follows: λα(a) = ∩{o ∈ α(x,τ)|a ⊆ o}. lemma 2.1. [2] for subsets a, b and ai (i ∈ i) of a topological space (x,τ), the following properties hold: (1) a ⊆ λα(a). (2) if a ⊆ b, then λα(a) ⊆ λα(b). (3) λα(λα(a)) = λα(a). (4) λα(∩{ai|i ∈ i}) ⊆∩{λα(ai )|i ∈ i}. (5) λα(∪{ai|i ∈ i}) = ∪{λα(ai )|i ∈ i}. recall that a subset a of a topological space (x,τ) is said to be a λα-set [2] if a = λα(a). lemma 2.2. [2] for subsets a and ai (i ∈ i) of a topological space (x,τ), the following properties hold: (1) λα(a) is a λα-set. (2) if a is α-open, then a is a λα-set. (3) if ai is a λα-set for each i ∈ i, then ∩i∈iai is a λα-set. (4) if ai is a λα-set for each i ∈ i, then ∪i∈iai is a λα-set. a subset a of a topological space (x,τ) is called (λ,α)-closed [2] if a = t∩c, where t is a λα-set and c is an α-closed set. the complement of a (λ,α)-closed set is called (λ,α)-open. the collection of all (λ,α)-open (resp. (λ,α)-closed) sets in a topological space (x,τ) is denoted by λαo(x,τ) (resp. λαc(x,τ)). let a be a subset of a topological space (x,τ). a point x ∈ x is called a (λ,α)-cluster point of a [2] if for every (λ,α)-open set u of x containing x we have a∩u 6= ∅. the set of all (λ,α)-cluster points of a is called the (λ,α)-closure of a and is denoted by a(λ,α). lemma 2.3. [2] let a and b be subsets of a topological space (x,τ). for the (λ,α)-closure, the following properties hold: (1) a ⊆ a(λ,α) and [a(λ,α)](λ,α) = a(λ,α). (2) a(λ,α) = ∩{f |a ⊆ f and f is (λ,α)-closed}. (3) if a ⊆ b, then a(λ,α) ⊆ b(λ,α). int. j. anal. appl. (2023), 21:88 3 (4) a is (λ,α)-closed if and only if a = a(λ,α). (5) a(λ,α) is (λ,α)-closed. definition 2.1. [6] let a be a subset of a topological space (x,τ). the union of all (λ,α)-open sets of x contained in a is called the (λ,α)-interior of a and is denoted by a(λ,α). lemma 2.4. [6] let a and b be subsets of a topological space (x,τ). for the (λ,α)-interior, the following properties hold: (1) a(λ,α) ⊆ a and [a(λ,α)](λ,α) = a(λ,α). (2) if a ⊆ b, then a(λ,α) ⊆ b(λ,α). (3) a is (λ,α)-open if and only if a(λ,α) = a. (4) a(λ,α) is (λ,α)-open. (5) [x −a](λ,α) = x −a(λ,α). (6) [x −a](λ,α) = x −a(λ,α). 3. generalized (λ,α)-closed sets in this section, we introduce the notion of generalized (λ,α)-closed sets. moreover, some properties of generalized (λ,α)-closed sets are discussed. definition 3.1. a subset a of a topological space (x,τ) is said to be generalized (λ,α)-closed (briefly g-(λ,α)-closed) if a(λ,α) ⊆ u and u is (λ,α)-open in (x,τ). the complement of a generalized (λ,α)closed set is said to be generalized (λ,α)-open (briefly g-(λ,α)-open). definition 3.2. a topological space (x,τ) is said to be λα-symmetric if for x and y in x, x ∈{y}(λ,α) implies y ∈{x}(λ,α). theorem 3.1. a topological space (x,τ) is λα-symmetric if and only if {x} is g-(λ,α)-closed for each x ∈ x. proof. assume that x ∈ {y}(λ,α) but y 6∈ {x}(λ,α). this implies that the complement of {x}(λ,α) contains y. therefore, the set {y} is a subset of the complement of {x}(λ,α). this implies that {y}(λ,α) is a subset of the complement of {x}(λ,α). now the complement of {x}(λ,α) contains x which is a contradiction. conversely, suppose that {x} ⊆ v ∈ λαo(x,τ), but {x}(λ,α) is not a subset of v . this means that {x}(λ,α) and the complement of v are not disjoint. let y belongs to their intersection. now, we have x ∈{y}(λ,α) which is a subset of the complement of v and x 6∈ v . this is a contradiction. � theorem 3.2. a subset a of a topological space (x,τ) is g-(λ,α)-closed if and only if a(λ,α) − a contains no nonempty (λ,α)-closed set. 4 int. j. anal. appl. (2023), 21:88 proof. let f be a (λ,α)-closed subset of a(λ,α)−a. now, a ⊆ x−f and since a is g-(λ,α)-closed, we have a(λ,α) ⊆ x −f or f ⊆ x −a(λ,α). thus, f ⊆ a(λ,α) ∩ [x −a(λ,α)] = ∅ and hence f is empty. conversely, suppose that a ⊆ u and u is (λ,α)-open. if a(λ,α) * u, then a(λ,α) ∩ (x −u) is a nonempty (λ,α)-closed subset of a(λ,α) −a. � definition 3.3. let a be a subset of a topological space (x,τ). the (λ,α)-frontier of a, λαfr(a), is defined as follows: λαfr(a) = a(λ,α) ∩ [x −a](λ,α). theorem 3.3. let a be a subset of a topological space (x,τ). if a is g-(λ,α)-closed and a ⊆ v ∈ λαo(x,τ), then λαfr(v ) ⊆ [x −a](λ,α). proof. let a be g-(λ,α)-closed and a ⊆ v ∈ λαo(x,τ). then, a(λ,α) ⊆ v . suppose that x ∈ λαfr(v ). since v ∈ λαo(x,τ), λαfr(v ) = v (λ,α) −v . therefore, x 6∈ v and x 6∈ a(λ,α). thus, x ∈ [x −a](λ,α) and hence λαfr(v ) ⊆ [x −a](λ,α). � theorem 3.4. let (x,τ) be a topological space. for each x ∈ x, either {x} is (λ,α)-closed or g-(λ,α)-open. proof. suppose that {x} is not (λ,α)-closed. then, x−{x} is not (λ,α)-open and the only (λ,α)open set containing x−{x} is x itself. thus, [x−{x}](λ,α) ⊆ x and hence x−{x} is g-(λ,α)-closed. therefore, {x} is g-(λ,α)-open. � theorem 3.5. let a be a subset of a topological space (x,τ). then, a is g-(λ,α)-open if and only if f ⊆ a(λ,α) whenever f ⊆ a and f is (λ,α)-closed. proof. suppose that a is g-(λ,α)-open. let f ⊆ a and f be (λ,α)-closed. then, we have x −a ⊆ x −f ∈ λαo(x,τ) and x −a is g-(λ,α)-closed. thus, x −a(λ,α) = [x −a](λ,α) ⊆ x −f and hence f ⊆ a(λ,α). conversely, let x −a ⊆ u and u ∈ λαo(x,τ). then, x −u ⊆ a and x −u is (λ,α)-closed. by the hypothesis, x −u ⊆ a(λ,α) and hence [x −a](λ,α) = x −a(λ,α) ⊆ u. this shows that x −a is g-(λ,α)-closed. thus, a is g-(λ,α)-open. � theorem 3.6. a subset a of a topological space (x,τ) is g-(λ,α)-closed if and only if a∩{x}(λ,α) 6= ∅ for every x ∈ a(λ,α). proof. let a be a g-(λ,α)-closed set and suppose that there exists x ∈ a(λ,α) such that a∩{x}(λ,α) = ∅. therefore, a ⊆ x − {x}(λ,α) and so a(λ,α) ⊆ x − {x}(λ,α). hence x 6∈ a(λ,α), which is a contradiction. int. j. anal. appl. (2023), 21:88 5 conversely, suppose that the condition of the theorem holds and let u be any (λ,α)-open set containing a. let x ∈ a(λ,α). then, by the hypothesis a∩a(λ,α) 6= ∅, so there exists y ∈ a∩{x}(λ,α) and so y ∈ a ⊆ u. thus, {x}∩u 6= ∅. hence x ∈ u, which implies that a(λ,α) ⊆ u. this shows that a is g-(λ,α)-closed. � definition 3.4. a subset a of a topological space (x,τ) is said to be locally (λ,α)-closed if a = u∩f, where u ∈ λαo(x,τ) and f is a (λ,α)-closed set. theorem 3.7. for a subset a of a topological space (x,τ), the following properties are equivalent: (1) a is locally (λ,α)-closed; (2) a = u ∩a(λ,α) for some u ∈ λαo(x,τ); (3) a(λ,α) −a is (λ,α)-closed; (4) a∪ [x −a(λ,α)] ∈ λαo(x,τ); (5) a ⊆ [a∪ [x −a(λ,α)]](λ,α). proof. (1) ⇒ (2): suppose that a = u ∩ f, where u ∈ λαo(x,τ) and f is a (λ,α)-closed set. since a ⊆ f, we have a(λ,α) ⊆ f (λ,α) = f. since a ⊆ u, a ⊆ u ∩ a(λ,α) ⊆ u ∩ f = a. thus, a = u ∩a(λ,α) for some u ∈ λαo(x,τ). (2) ⇒ (3): suppose that a = u ∩a(λ,α) for some u ∈ λαo(x,τ). then, we have a(λ,α) −a = [x −u ∩a(λ,α)] ∩a(λ,α) = (x −u) ∩a(λ,α). since (x −u) ∩a(λ,α) is (λ,α)-closed, a(λ,α) −a is (λ,α)-closed. (3) ⇒ (4): since x − [a(λ,α) −a] = [x −a(λ,α)] ∪a and by (3), a∪ [x −a(λ,α)] ∈ λαo(x,τ). (4) ⇒ (5): by (4), we obtain a ⊆ a∪ [x −a(λ,α)] = [a∪ [x −a(λ,α)]](λ,α). (5) ⇒ (1): we put u = [a∪ [x −a(λ,α)]](λ,α). then, u ∈ λαo(x,τ) and a = a∩u ⊆ u ∩a(λ,α) ⊆ [a∪ [x −a(λ,α)]](λ,α) ∩a (λ,α) = a∩a(λ,α) = a. thus, a = u ∩a(λ,α), where u ∈ λαo(x,τ) and a(λ,α) is a (λ,α)-closed set. this shows that a is locally (λ,α)-closed. � theorem 3.8. a subset a of a topological space (x,τ) is (λ,α)-closed if and only if a is locally (λ,α)-closed and g-(λ,α)-closed. proof. let a be (λ,α)-closed. then, a is g-(λ,α)-closed. since x ∈ λαo(x,τ) and a = x ∩a, a is locally (λ,α)-closed. conversely, suppose that a is locally (λ,α)-closed and g-(λ,α)-closed. since a is locally (λ,α)closed, by theorem 3.7, a ⊆ [a∪ [x −a(λ,α)]](λ,α). since [a∪ [x −a(λ,α)]](λ,α) ∈ λαo(x,τ) and a is g-(λ,α)-closed, a(λ,α) ⊆ [a∪ [x−a(λ,α)]](λ,α) ⊆ a∪ [x−a(λ,α)] and hence a(λ,α) = a. thus, by lemma 2.3, a is (λ,α)-closed. � 6 int. j. anal. appl. (2023), 21:88 4. applications of generalized (λ,α)-closed sets we begin this section by introducing the concept of λα-t1 2 -spaces. definition 4.1. a topological space (x,τ) is called a λα-t1 2 -space if every g-(λ,α)-closed set of x is (λ,α)-closed. lemma 4.1. let (x,τ) be a topological space. for each x ∈ x, the singleton {x} is (λ,α)-closed or x −{x} is g-(λ,α)-closed. proof. let x ∈ x and the singleton {x} be not (λ,α)-closed. then, x −{x} is not (λ,α)-open and x is the only (λ,α)-open set which contains x −{x} and x −{x} is g-(λ,α)-closed. � let a be a subset of a topological space (x,τ). a subset λ(λ,α)(a) [6] is defined as follows: λ(λ,α)(a) = ∩{u | a ⊆ u,u ∈ λαo(x,τ)}. lemma 4.2. [6] for subsets a,b of a topological space (x,τ), the following properties hold: (1) a ⊆ λ(λ,α)(a). (2) if a ⊆ b, then λ(λ,α)(a) ⊆ λ(λ,α)(b). (3) λ(λ,α)[λ(λ,α)(a)] = λ(λ,α)(a). (4) if a is (λ,α)-open, λ(λ,α)(a) = a. a subset a of a topological space (x,τ) is called a λ(λ,α)-set if a = λ(λ,α)(a). the family of all λ(λ,α)-sets of (x,τ) is denoted by λ(λ,α)(x,τ) (or simply λ(λ,α)). definition 4.2. a subset a of a topological space (x,τ) is called a generalized λ(λ,α)-set (briefly g-λ(λ,α)-set) if λ(λ,α)(a) ⊆ f whenever a ⊆ f and f is (λ,α)-closed. lemma 4.3. let (x,τ) be a topological space. for each x ∈ x, the singleton {x} is (λ,α)-open or x −{x} is g-λ(λ,α)-set. proof. let x ∈ x and the singleton {x} be not (λ,α)-open. then, x −{x} is not (λ,α)-closed and x is the only (λ,α)-closed set which contains x −{x} and x −{x} is g-λ(λ,α)-set. � theorem 4.1. for a topological space (x,τ), the following properties are equivalent: (1) (x,τ) is a λα-t1 2 -space. (2) for each x ∈ x, the singleton {x} is (λ,α)-open or (λ,α)-closed. (3) every g-λ(λ,α)-set is a λ(λ,α)-set. proof. (1) ⇒ (2): by lemma 4.1, for each x ∈ x, the singleton {x} is (λ,α)-closed or x −{x} is g-(λ,α)-closed. since (x,τ) is a λα-t1 2 -space, we have x −{x} is (λ,α)-closed and hence {x} is (λ,α)-open in the latter case. thus, the singleton {x} is (λ,α)-open or (λ,α)-closed. int. j. anal. appl. (2023), 21:88 7 (2) ⇒ (3): suppose that there exists a g-λ(λ,α)-set a which is not a λ(λ,α)-set. then, there exists x ∈ λ(λ,α)(a) such that x 6∈ a. in case the singleton {x} is (λ,α)-open, a ⊆ x −{x} and x −{x} is (λ,α)-closed. since a is a g-λ(λ,α)-set, λ(λ,α)(a) ⊆ x −{x}. this is a contradiction. in case the singleton {x} is (λ,α)-closed, a ⊆ x −{x} and x −{x} is (λ,α)-open. by lemma 4.2, λ(λ,α)(a) ⊆ λ(λ,α)(x −{x}) = x −{x}. this is a contradiction. therefore, every g-λ(λ,α)-set is a λ(λ,α)-set. (3) ⇒ (1): suppose that (x,τ) is not a λα-t1 2 -space. there exists a g-(λ,α)-closed set a which is not (λ,α)-closed. since a is not (λ,α)-closed, there exists a point x ∈ a(λ,α) such that x 6∈ a. by lemma 4.3, the singleton {x} is (λ,α)-open or x −{x} is a g-λ(λ,α)-set. (a) in case {x} is (λ,α)-open, since x ∈ a(λ,α), {x}∩a 6= ∅ and x ∈ a. this is a contradiction. (b) in case x−{x} is a λ(λ,α)-set, if {x} is not (λ,α)-closed, x−{x} is not (λ,α)-open and λ(λ,α)(x−{x}) = x. thus, x −{x} is not a λ(λ,α)-set. this contradicts (3). if {x} is (λ,α)-closed, a ⊆ x −{x}∈ λαo(x,τ) and a is g-(λ,α)-closed. thus, a(λ,α) ⊆ x −{x}. this contradicts that x ∈ a(λ,α). therefore, (x,τ) is a λα-t1 2 -space. � definition 4.3. a topological space (x,τ) is said to be (λ,α)-normal if for any pair of disjoint (λ,α)closed sets f and h, there exist disjoint (λ,α)-open sets u and v such that f ⊆ u and h ⊆ v . lemma 4.4. let (x,τ) be a topological space. if u is a (λ,α)-open set, then u(λ,α)∩a ⊆ [u∩a](λ,α) for every subset a of x. theorem 4.2. for a topological space (x,τ), the following properties are equivalent: (1) (x,τ) is (λ,α)-normal. (2) for every pair of (λ,α)-open sets u and v whose union is x, there exist (λ,α)-closed sets f and h such that f ⊆ u, h ⊆ v and f ∪h = x. (3) for every (λ,α)-closed set f and every (λ,α)-open set g containing f , there exists a (λ,α)open set u such that f ⊆ u ⊆ u(λ,α) ⊆ g. (4) for every pair of disjoint (λ,α)-closed sets f and h, there exist disjoint (λ,α)-open sets u and v such that f ⊆ u and h ⊆ v and u(λ,α) ∩v (λ,α) = ∅. proof. (1) ⇒ (2): let u and v be a pair of (λ,α)-open sets such that x = u∪v . then, x−u and x −v are disjoint (λ,α)-closed sets. since (x,τ) is (λ,α)-normal, there exist disjoint (λ,α)-open sets g and w such that x −u ⊆ g and x −v ⊆ w. put f = x −g and h = x −w. then, f and h are (λ,α)-closed sets such that f ⊆ u, h ⊆ v and f ∪h = x. (2) ⇒ (3): let f be a (λ,α)-closed set and g be a (λ,α)-open set containing f. then, x −f and g are (λ,α)-open sets whose union is x. then by (2), there exist (λ,α)-closed sets m and n such that m ⊆ x − f, n ⊆ g and m ∪ n = x. then, f ⊆ x − m, x − g ⊆ x − n and (x −m) ∩ (x −n) = ∅. put u = x −m and v = x −n. then u and v are disjoint (λ,α)-open 8 int. j. anal. appl. (2023), 21:88 sets such that f ⊆ u ⊆ x −v ⊆ g. as x −v is a (λ,α)-closed set, we have u(λ,α) ⊆ x −v and hence f ⊆ u ⊆ u(λ,α) ⊆ g. (3) ⇒ (4): let f and h be two disjoint (λ,α)-closed sets of x. then, f ⊆ x −h and x −h is (λ,α)-open and hence there exists a (λ,α)-open set u of x such that f ⊆ u ⊆ u(λ,α) ⊆ x −h. put v = x −u(λ,α). then, u and v are disjoint (λ,α)-open sets of x such that f ⊆ u, h ⊆ v and u(λ,α) ∩v (λ,α) = ∅. (4) ⇒ (1): the proof is obvious. � theorem 4.3. for a topological space (x,τ), the following properties are equivalent: (1) (x,τ) is (λ,α)-normal. (2) for every pair of disjoint (λ,α)-closed sets f and h of x, there exist disjoint g-(λ,α)-open sets u and v of x such that f ⊆ u and h ⊆ v . (3) for each (λ,α)-closed set f and each (λ,α)-open set g containing f, there exists a g-(λ,α)open set u such that f ⊆ u ⊆ u(λ,α) ⊆ g. (4) for each (λ,α)-closed set f and each g-(λ,α)-open set g containing f, there exists a (λ,α)open set u such that f ⊆ u ⊆ u(λ,α) ⊆ g(λ,α). (5) for each (λ,α)-closed set f and each g-(λ,α)-open set g containing f, there exists a g(λ,α)-open set u such that f ⊆ u ⊆ u(λ,α) ⊆ g(λ,α). (6) for each g-(λ,α)-closed set f and each (λ,α)-open set g containing f, there exists a (λ,α)open set u such that f (λ,α) ⊆ u ⊆ u(λ,α) ⊆ g. (7) for each g-(λ,α)-closed set f and each (λ,α)-open set g containing f, there exists a g(λ,α)-open set u such that f (λ,α) ⊆ u ⊆ u(λ,α) ⊆ g. proof. (1) ⇒ (2): the proof is obvious. (2) ⇒ (3): let f be a (λ,α)-closed set and g be a (λ,α)-open set containing f. then, we have f and x−g are two disjoint (λ,α)-closed sets. hence by (2), there exist disjoint g-(λ,α)-open sets u and v of x such that f ⊆ u and x−g ⊆ v . since v is g-(λ,α)-open and x−g is (λ,α)-closed, by theorem 3.5, x−g ⊆ v(λ,α). thus, [x−v ](λ,α) = x−v(λ,α) ⊆ g and hence f ⊆ u ⊆ u(λ,α) ⊆ g. (3) ⇒ (1): let f and h be two disjoint (λ,α)-closed sets of x. then, f is a (λ,α)-closed set and x −h is a (λ,α)-open set containing f. thus by (3), there exists a g-(λ,α)-open set u such that f ⊆ u ⊆ u(λ,α) ⊆ x − h. by theorem 3.5, f ⊆ u(λ,α), h ⊆ x − u(λ,α), where u(λ,α) and x −u(λ,α) are two disjoint (λ,α)-open sets. (4) ⇒ (5) and (5) ⇒ (2): the proofs are obvious. (6) ⇒ (7) and (7) ⇒ (3): the proofs are obvious. (3) ⇒ (5): let f be a (λ,α)-closed set and g be a g-(λ,α)-open set containing f. since g is g-(λ,α)-open and f is (λ,α)-closed, by theorem 3.5, f ⊆ g(λ,α) and by (3), there exists a g-(λ,α)-open set u such that f ⊆ u ⊆ u(λ,α) ⊆ g(λ,α). int. j. anal. appl. (2023), 21:88 9 (5) ⇒ (6): let f be a g-(λ,α)-closed set and g be a (λ,α)-open set containing f . then, f (λ,α) ⊆ g. since g is g-(λ,α)-open, by (6), there exists a g-(λ,α)-open set u such that f (λ,α) ⊆ u ⊆ u(λ,α) ⊆ g. since u is g-(λ,α)-open and f (λ,α) ⊆ u, by theorem 3.5, f (λ,α) ⊆ u(λ,α). put v = u(λ,α). then, v is (λ,α)-open and f (λ,α) ⊆ v ⊆ v (λ,α) = [u(λ,α)](λ,α) ⊆ u(λ,α) ⊆ g. (6) ⇒ (4): let f be a (λ,α)-closed set and g be a g-(λ,α)-open set containing f. then by theorem 3.5, f (λ,α) = f ⊆ g(λ,α). since f is g-(λ,α)-closed and g(λ,α) is (λ,α)-open, by (6), there exists a (λ,α)-open set u such that f (λ,α) = f ⊆ u ⊆ u(λ,α) ⊆ g(λ,α). � definition 4.4. a topological space (x,τ) is said to be (λ,α)-regular if for each (λ,α)-closed set f of x not containing x, there exist disjoint (λ,α)-open sets u and v such that x ∈ u and f ⊆ v . theorem 4.4. for a topological space (x,τ), the following properties are equivalent: (1) (x,τ) is (λ,α)-regular. (2) for each x ∈ x and each u ∈ λαo(x,τ) with x ∈ u, there exists v ∈ λαo(x,τ) such that x ∈ v ⊆ v (λ,α) ⊆ u. (3) for each (λ,α)-closed set f of x, ∩{v (λ,α) | f ⊆ v ∈ λαo(x,τ)} = f. (4) for each subset a of x and each u ∈ λαo(x,τ) with a∩u 6= ∅, there exists v ∈ λαo(x,τ) such that a∩v 6= ∅ and v (λ,α) ⊆ u. (5) for each nonempty subset a of x and each (λ,α)-closed set f of x with a∩f = ∅, there exist v,w ∈ λαo(x,τ) such that a∩v 6= ∅, f ⊆ w and v ∩w = ∅. (6) for each (λ,α)-closed set f of x and x 6∈ f, there exist u ∈ λαo(x,τ) and a g-(λ,α)-open set v such that x ∈ u, f ⊆ v and u ∩v = ∅. (7) for each subset a of x and each (λ,α)-closed set f with a∩f = ∅, there exist u ∈ λαo(x,τ) and a g-(λ,α)-open set v such that a∩u 6= ∅, f ⊆ v and u ∩v = ∅. proof. (1) ⇒ (2): let g ∈ λαo(x,τ) and x 6∈ x −g. then, there exist disjoint u,v ∈ λαo(x,τ) such that x −g ⊆ u and x ∈ v . thus, v ⊆ x −u and so x ∈ v ⊆ v (λ,α) ⊆ x −u ⊆ g. (2) ⇒ (3): let x −f ∈ λαo(x,τ) with x ∈ x −f. then by (2), there exists u ∈ λαo(x,τ) such that x ∈ u ⊆ u(λ,α) ⊆ x −f. thus, f ⊆ x −u(λ,α) = v ∈ λαo(x,τ) and hence u ∩v = ∅. then, we have x 6∈ v (λ,α). this shows that f ⊇∩{v (λ,α) | f ⊆ v ∈ λαo(x,τ)}. (3) ⇒ (4): let a be a subset of x and u ∈ λαo(x,τ) such that a∩u 6= ∅. let x ∈ a∩u. then, x 6∈ x −u. hence by (3), there exists w ∈ λαo(x,τ) such that x −u ⊆ w and x 6∈ w (λ,α). put v = x −w (λ,b) which is a (λ,α)-open set containing x and a∩v 6= ∅. now, v ⊆ x −w and so v (λ,α) ⊆ x −w ⊆ u. (4) ⇒ (5): let a be a nonempty subset of x and f be a (λ,α)-closed set such that a∩f = ∅. then, x −f ∈ λαo(x,τ) with a∩ (x −f ) 6= ∅ and hence by (4), there exists v ∈ λαo(x,τ) such that a∩v 6= ∅ and v (λ,α) ⊆ x −f. if we put w = x −v (λ,α), then f ⊆ w and w ∩v = ∅. 10 int. j. anal. appl. (2023), 21:88 (5) ⇒ (1): let f be a (λ,α)-closed set not containing x. then, f ∩{x} = ∅. thus by (5), there exist v,w ∈ λαo(x,τ) such that x ∈ v,f ⊆ w and v ∩w = ∅. (1) ⇒ (6): the proof is obvious. (6) ⇒ (7): let a be a subset of x and f be a (λ,α)-closed set such that a∩f = ∅. then, for x ∈ a, x 6∈ f and by (6), there exist u ∈ λαo(x,τ) and a g-(λ,α)-open set v such that x ∈ u, f ⊆ v and u ∩v = ∅. thus, a∩u 6= ∅, f ⊆ v and u ∩v = ∅. (7) ⇒ (1): let f be a (λ,α)-closed set such that x 6∈ f. since {x}∩f = ∅, by (7), there exist u ∈ λαo(x,τ) and a g-(λ,α)-open set w such that x ∈ u, f ⊆ w and u ∩ w = ∅. since w is g-(λ,α)-open, by theorem 3.5, we have f ⊆ w(λ,α) = v ∈ λαo(x,τ) and hence u ∩v = ∅. this shows that (x,τ) is (λ,α)-regular. � acknowledgements: this research project was financially supported by mahasarakham university. conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] s. buadong, c. viriyapong, c. boonpok, on generalized topology and minimal structure spaces, int. j. math. anal. 5 (2011), 1507–1516. [2] m. caldas, d. n. georgiou, s. jafari, study of (λ,α)-closed sets and the related notions in topological spaces, bull. malays. math. sci. soc. (2), 30 (2007), 23–36. [3] j. dontchev, m. ganster, on δ-generalized closed sets and t 3 4 -spaces, mem. fac. sci. kochi univ. ser. a math. 17 (1996), 15–31. [4] w. dungthaisong, c. boonpok, c. viriyapong, generalized closed sets in bigeneralized topological spaces, int. j. math. anal. 5 (2011), 1175–1184. [5] e. khalimsky, r. kopperman, p. r. meyer, computer graphics and connected topologies on finite ordered sets, topol. appl. 36 (1990), 1–17. https://doi.org/10.1016/0166-8641(90)90031-v. [6] j. khampakdee, c. boonpok, some properties of (λ,α)-open sets, wseas trans. math. 22 (2023), 13–31. [7] n. levine, generalized closed sets in topology, rend. circ. mat. palermo (2), 19 (1970), 89–96. [8] o. njåstad, on some classes of nearly open sets, pac. j. math. 15 (1965), 961–970. https://doi.org/10. 2140/pjm.1965.15.961. [9] t. noiri, v. popa, a note on modifications of rg-closed sets in topological spaces, cubo. 15 (2013), 65–70. https://doi.org/10.4067/s0719-06462013000200006. [10] n. palaniappan, k. c. rao, regular generalized closed sets, kyungpook math. j. 33 (1993), 211–219. [11] j.h. park, d.s. song, r. saadati, on generalized δ-semiclosed sets in topological spaces, chaos solitons fractals, 33 (2007), 1329–1338. https://doi.org/10.1016/j.chaos.2006.01.086. [12] p. torton, c. viriyapong, c. boonpok, some separation axioms in bigeneralized topological spaces, int. j. math. anal. 6 (2012), 2789–2796. https://doi.org/10.1016/0166-8641(90)90031-v https://doi.org/10.2140/pjm.1965.15.961 https://doi.org/10.2140/pjm.1965.15.961 https://doi.org/10.4067/s0719-06462013000200006 https://doi.org/10.1016/j.chaos.2006.01.086 1. introduction 2. preliminaries 3. generalized (,)-closed sets 4. applications of generalized (,)-closed sets references international journal of analysis and applications issn 2291-8639 volume 12, number 2 (2016), 129-141 http://www.etamaths.com c-class functions on shorter proofs of some even-tupled coincidence theorems in ordered metric spaces anupam sharma∗ abstract. the purpose of this paper is to prove some even tupled coincidence theorems for mappings with one variable in ordered complete metric spaces by using the concept of c-class functions. our results generalize and improve several results in the literature. 1. introduction ran and reurings [30] extended the banach contraction principle on ordered metric spaces for continuous monotone mappings with some applications to matrix equations. thereafter nieto and lópez [25] modified ran and reurings’ fixed point theorem for an increasing mapping not necessarily continuous by assuming an another hypothesis on the ordered metric space and proved some fixed point theorems besides giving some applications to ordinary differential equations. in the same development, nieto and lópez [26] analogously proved a fixed point theorem for a decreasing mapping on ordered metric space. in recent years, nieto and lópez’s [25] fixed point theorems were extended and refined by many authors ([1, 2, 7], [11]-[13], [18, 19, 24, 27]). the idea of a coupled fixed point was introduced by guo and lakshmikantham [10] which was well followed by bhaskar and lakshmikantham [5] where the authors introduced the notion of mixed monotone property and proved some coupled fixed point theorems for weakly linear contractions enjoying mixed monotone property in ordered complete metric spaces. in [23], lakshmikantham and ćirić generalized these results for nonlinear contraction mappings by introducing the notion of coupled coincidence point and mixed g-monotone property. recently, berzig and samet [6] extended and generalized some fixed point results to higher dimensions. however, they used permutations of variables and distinguished between the first and last variables. further, roldán et al. [31] proved some existence and uniqueness theorems for nonlinear mappings of any number of arguments, not necessarily permuted or ordered. for more details see ([20, 31, 32, 33]). recently, imdad et al. [16] extended the idea of mixed g-monotone property to the mapping f : xn → x (where n is even natural number) and proved an even-tupled coincidence point theorem for nonlinear contraction mappings satisfying mixed g-monotone property. basically their results are true for only even n but not for odd ones (for details see [14]-[17]). very recently, samet et al. [36] have shown that the coupled (analogously n-tupled) fixed results can be more easily obtained by using well known fixed point theorems on ordered metric spaces (see also [9, 28, 29]). the concept of c-class functions was introduced by ansari [3] which actually covers a large class of contractive conditions. in this paper, we generalize the results of sharma et al. [37] by using the concept of c-class functions. 2. preliminaries with a view to make, our presentation self-contained, we collect some basic definitions and needed results which will be used frequently in the text later. 2010 mathematics subject classification. 47h10; 54h25. key words and phrases. partially ordered set; compatible mapping; mixed g-monotone property; n-tupled coincidence point; c-class function. c©2016 authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the creative commons attribution license. 129 130 sharma definition 2.1. let x be a non-empty set. a relation ‘ �’ on x is said to be a partial order if the following properties are satisfied: (i) reflexive: x � x for all x ∈ x, (ii) anti-symmetric: x � y and y � x implies x = y, (iii) transitive: x � y and y � z implies x � z for all x,y,z ∈ x. a non-empty set x together with a partial order ‘ �’ is said to be an ordered set and we denote it by (x,�). definition 2.2. let (x,�) be an ordered set. any two elements x and y are said to be comparable elements in x if either x � y or y � x. definition 2.3. ([27]) a triplet (x,d,�) is called an ordered metric space if (x,d) is a metric space and (x,�) is an ordered set. moreover, if d is a complete metric on x, then we say that (x,d,�) is an ordered complete metric space. recently, kutbi et al. [22] introduced the concept of regular map. definition 2.4. ([22]) an ordered metric space (x,d,�) is said to be nondecreasing regular (resp. nonincreasing regular) if it satisfies the following property: if {xm} is a nondecreasing (resp. nonincreasing) sequence and xm → x, then xm � x (resp. x � xm) ∀m ∈ n∪{0}. definition 2.5. ([22]) an ordered metric space (x,d,�) is said to be regular if it is both nondecreasing regular and nonincreasing regular. definition 2.6. let (x,d,�) be an ordered metric space and g : x → x be a mapping. then x is said to be nondecreasing g-regular (resp. nonincreasing g-regular) if it satisfies the following property: if {xm} is a nondecreasing (resp. nonincreasing) sequence and xm → x, then gxm � gx (resp. gx � gxm) ∀m ∈ n∪{0}. definition 2.7. an ordered metric space (x,d,�) is said to be g-regular if it is both nondecreasing g-regular and nonincreasing g-regular. notice that, on setting g = i (identity mapping on x), definitions 2.6 and 2.7 reduce to definitions 2.4 and 2.5 respectively. throughout the paper, n stands for a general even natural number. let us denote by xn the product space x ×x × . . .×x of n identical copies of x. definition 2.8. ([16]) let (x,�) be an ordered set and f : xn → x and g : x → x two mappings. then f is said to have the mixed g-monotone property if f is g-nondecreasing in its odd position arguments and g-nonincreasing in its even position arguments, that is, for x1,x2,x3, ...,xn ∈ x, if for all x11,x 1 2 ∈ x, gx11 � gx12 ⇒ f(x11,x2,x3, ...,xn) � f(x12,x2,x3, ...,xn) for all x21,x 2 2 ∈ x, gx21 � gx22 ⇒ f(x1,x22,x3, ...,xn) � f(x1,x21,x3, ...,xn) for all x31,x 3 2 ∈ x, gx31 � gx32 ⇒ f(x1,x2,x31, ...,xn) � f(x1,x2,x32, ...,xn) ... for all xn1 ,x n 2 ∈ x, gxn1 � gxn2 ⇒ f(x1,x2,x3, ...,xn2 ) � f(x1,x2,x3, ...,xn1 ). for g = i (identity mapping), definition 2.8 reduces to mixed monotone property (for details see [16]). c-class functions on shorter proofs of some even-tupled coincidence theorems 131 definition 2.9. ([34]) an element (x1,x2, ...,xn) ∈ xn is called an n-tupled fixed point of the mapping f : xn → x if   f(x1,x2,x3, ...,xn) = x1 f(x2,x3, ...,xn,x1) = x2 f(x3, ...,xn,x1,x2) = x3 ... f(xn,x1,x2, ...,xn−1) = xn. definition 2.10. ([16]) an element (x1,x2, ...,xn) ∈ xn is called an n-tupled coincidence point of mappings f : xn → x and g : x → x if  f(x1,x2,x3, ...,xn) = g(x1) f(x2,x3, ...,xn,x1) = g(x2) f(x3, ...,xn,x1,x2) = g(x3) ... f(xn,x1,x2, ...,xn−1) = g(xn). remark 2.1. for n = 2, definitions 2.9 and 2.10 yield the definitions of coupled fixed point and coupled coincidence point respectively while on the other hand, for n = 4 these definitions yield the definitions of quadrupled fixed point and quadrupled coincidence point respectively. definition 2.11. an element (x1,x2, ...,xn) ∈ xn is called an n-tupled common fixed point of mappings f : xn → x and g : x → x if  f(x1,x2,x3, ...,xn) = g(x1) = x1 f(x2,x3, ...,xn,x1) = g(x2) = x2 f(x3, ...,xn,x1,x2) = g(x3) = x3 ... f(xn,x1,x2, ...,xn−1) = g(xn) = xn. definition 2.12. ([14]) let x be a non-empty set. then the mappings f : xn → x and g : x → x are said to be compatible if  lim m→∞ d(g(f(x1m,x 2 m, ...,x n m)),f(gx 1 m,gx 2 m, ...,gx n m)) = 0 lim m→∞ d(g(f(x2m, ...,x n m,x 1 m)),f(gx 2 m, ...,gx n m,gx 1 m)) = 0 ... lim m→∞ d(g(f(xnm,x 1 m, ...,x n−1 m )),f(gx n m,gx 1 m, ...,gx n−1 m )) = 0, where {x1m},{x2m}, ...,{xnm} are sequences in x such that  lim m→∞ f(x1m,x 2 m, ...,x n m) = lim m→∞ g(x1m) = x 1 lim m→∞ f(x2m, ...,x n m,x 1 m) = lim m→∞ g(x2m) = x 2 ... lim m→∞ f(xnm,x 1 m, ...,x n−1 m ) = lim m→∞ g(xnm) = x n, for some x1,x2, ...,xn ∈ x are satisfied. the following families of control functions are indicated in choudhury et al. [8]. (1) = := {ζ : [0,∞) → [0,∞) : ζ is continuous and ζ(t) = 0 if and only if t = 0} (2) ω := {ϕ : [0,∞) → [0,∞) : ϕ is continuous and monotone nondecreasing and ϕ(t) = 0 if and only if t = 0} (3) =u := {ζ : [0,∞) → [0,∞) : ζ is continuous and ζ(t) > 0 , t > 0 and ζ(0) ≥ 0}. 132 sharma notice that members of ω are called altering distance functions (cf. [21]). ansari [3] introduced the concept of c-class functions which covers a large class of contractive conditions (see example 2.1 (1),(2),(9),(15)). definition 2.13. ([3]) a continuous function f : [0,∞)2 → r is called a c-function if f is continuous and satisfies the following: (1) f(s,t) ≤ s; (2) f(s,t) = s implies that either s = 0 or t = 0 for all s,t ∈ [0,∞). an extra condition on f is that f(0, 0) = 0 could be imposed in some cases if required. the letter c denotes the class of all c-functions. example 2.1. ([3]) define f : [0,∞)2 → r by (1) f(s,t) = s− t, f(s,t) = s ⇒ t = 0; (2) f(s,t) = ms, 0<m<1, f(s,t) = s ⇒ s = 0; (3) f(s,t) = s (1+t)r ; r ∈ (0,∞), f(s,t) = s ⇒ s = 0 or t = 0; (4) f(s,t) = log(t + as)/(1 + t), a > 1, f(s,t) = s ⇒ s = 0 or t = 0; (5) f(s,t) = ln(1 + as)/2, a > e, f(s, 1) = s ⇒ s = 0; (6) f(s,t) = (s + l)(1/(1+t) r) − l, l > 1,r ∈ (0,∞), f(s,t) = s ⇒ t = 0; (7) f(s,t) = s logt+a a, a > 1, f(s,t) = s ⇒ s = 0 or t = 0; (8) f(s,t) = s− ( 1+s 2+s )( t 1+t ), f(s,t) = s ⇒ t = 0; (9) f(s,t) = sβ(s), β : [0,∞) → [0, 1), and is continuous, f(s,t) = s ⇒ s = 0; (10) f(s,t) = s− t k+t , f(s,t) = s ⇒ t = 0; (11) f(s,t) = s − ϕ(s), f(s,t) = s ⇒ s = 0, where ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 ⇔ t = 0; (12) f(s,t) = sh(s,t), f(s,t) = s ⇒ s = 0, where h : [0,∞) × [0,∞) → [0,∞) is a continuous function such that h(s,t) < 1 for all t,s > 0; (13) f(s,t) = s− ( 2+t 1+t )t, f(s,t) = s ⇒ t = 0; (14) f(s,t) = n √ ln(1 + sn), f(s,t) = s ⇒ s = 0; (15) f(s,t) = φ(s),f(s,t) = s ⇒ s = 0, where φ : [0,∞) → [0,∞) is an upper semi-continuous function such that φ(0) = 0, and φ(t) < t for t > 0, (16) f(s,t) = s (1+s)r ; r ∈ (0,∞), f(s,t) = s ⇒ s = 0 ; (17) f(s,t) = ϑ(s); ϑ : r+×r+ → r is a generalized mizoguchi-takahashi type function, f(s,t) = s ⇒ s = 0; (18) f(s,t) = s γ(1/2) ∫∞ 0 e−x√ x+t dx, where γ is the euler gamma function; for all s,t ∈ [0,∞). then f is an element of c. 3. main results (a) let (x,�) be an ordered set. define the following partial order v on the product space xn, for u = (x1,x2, . . . ,xn), v = (y1,y2, . . . ,yn) ∈ xn u v v ⇔ x1 � y1, y2 � x2, x3 � y3, . . . ,yn � xn. (b) let (x,d) be a metric space. define the following metric d̃ on the product space xn, for u = (x1,x2, . . . ,xn), v = (y1,y2, . . . ,yn) ∈ xn, d̃(u,v ) = max 1≤i≤n d(xi,yi). the proof of the following lemmas are immediately. we note the same idea here, but in the case of coupled and tripled fixed point theorems, we have been first used in ([4], [28], [35]). lemma 3.1. let (x,d,�) be an ordered complete metric space. then (xn,d̃,v) is an ordered complete metric space. c-class functions on shorter proofs of some even-tupled coincidence theorems 133 lemma 3.2. let (x,d,�) be an ordered metric space and f : xn → x and g : x → x be two mappings. define mappings tf : x n → xn and tg : xn → xn by tf (x 1,x2, . . . ,xn) = (f(x1,x2, . . . ,xn),f(x2, . . . ,xn,x1), . . . ,f(xn,x1, . . . ,xn−1)) and tg(x 1,x2, . . . ,xn) = (gx1,gx2, . . . ,gxn). then the following hold: (1) if f has the mixed g-monotone property, then tf is monotone tg-nondecreasing with respect to v . (2) if f and g are compatible, then tf and tg are compatible. (3) if g is continuous, then tg is continuous. (4) if f is continuous, then tf is continuous. (5) if (x,d,�) is g-regular, then (xn,d̃,v) is nondecreasing g-regular. (6) a point (x1,x2, . . . ,xn) ∈ xn is an n-tupled coincidence point of f and g iff (x1,x2, . . . ,xn) is a coincidence point of tf and tg. the following lemma is crucial for our main result. lemma 3.3. let (x,d,�) be an ordered complete metric space and f and g be two self-mappings on x. suppose that the following conditions are satisfied: (i) f(x) ⊆ g(x), (ii) f is monotone g-nondecreasing, (iii) f and g are compatible, (iv) g is continuous, (v) either f is continuous or x is nondecreasing g-regular, (vi) there exists x0 ∈ x such that g(x0) � f(x0), (vii) there exist ϕ ∈ ω and ζ ∈=u and f ∈c such that for all x,y ∈ x, ϕ(d(f(x),f(y))) ≤f(ϕ(d(g(x),g(y))),ζ(d(g(x),g(y)))), with g(x) � g(y). (3.1) then f and g have a coincidence point. proof. in view of assumption (vi), if g(x0) = f(x0), then x0 is a coincidence point of f and g and hence proof is finished. on the other hand if g(x0) 6= f(x0), then we have g(x0) ≺ f(x0). so according to assumption (i), that is, f(x) ⊆ g(x), we can choose x1 ∈ x such that g(x1) = f(x0). again from f(x) ⊆ g(x), we can choose x2 ∈ x such that g(x2) = f(x1). continuing this process, we define a sequence {xm}⊂ x of joint iterates such that g(xm+1) = f(xm) ∀m ∈ n∪{0}. (3.2) now, we assert that {g(xm)} is a non-decreasing sequence, that is g(xm) � g(xm+1) ∀m ∈ n∪{0}. (3.3) we prove this fact by mathematical induction. on using (3.2) for m = 0 and assumption (vi), we have g(x0) � f(x0) = g(x1). thus, (3.3) holds for m = 0. suppose that (3.3) holds for m = r > 0, that is, g(xr) � g(xr+1). (3.4) then we have to show that (3.3) holds for m = r + 1. to accomplish this we use (3.2), (3.4) and assumption (ii) so that g(xr+1) = f(xr) � f(xr+1) = g(xr+2). thus, by induction, (3.3) holds for all m ∈ n∪{0}. if g(xm) = g(xm+1) for some m ∈ n, then by using (3.2), we have g(xm) = f(xm), that is, xm is a coincidence point of f and g and hence proof is finished. on the other hand if g(xm) 6= g(xm+1) for each m ∈ n∪{0}, we can define a sequence δm := d(g(xm),g(xm+1)), m ∈ n∪{0}. (3.5) 134 sharma on using (3.2), (3.3), (3.5) and assumption (vii), we obtain ϕ(δm+1) = ϕ(d(g(xm+1),g(xm+2))) = ϕ(d(f(xm),f(xm+1))) ≤ f(ϕ(d(g(xm),g(xm+1))),ζ(d(g(xm),g(xm+1)))) = f(ϕ(δm),ζ(δm)) ≤ ϕ(δm). (3.6) on using the property of ϕ, we have ϕ(δm+1) ≤ ϕ(δm), which implies that δm+1 ≤ δm. therefore {δm} is a monotone decreasing sequence of nonnegative real numbers. hence there exists δ ≥ 0 such that δm → δ as m → ∞. taking limit as m → ∞ in (3.6) and using the continuities of ϕ and ζ, we have ϕ(δ) ≤f(ϕ(δ),ζ(δ)), so ϕ(δ) = 0, or ,ζ(δ) = 0, therefore δ = 0 , which is a contradiction . therefore lim m→∞ δm = lim m→∞ d(g(xm),g(xm+1)) = 0. (3.7) now, we show that {g(xm)} is a cauchy sequence. on contrary suppose that {g(xm)} is not a cauchy sequence. then there exists an � > 0 and sequences of positive integers {m(k)} and {t(k)} such that for all positive integers k, t(k) > m(k) > k, such that ηk = d(g(xm(k)),g(xt(k))) ≥ �, and d(g(xm(k)),g(xt(k)−1)) < �. now, � ≤ ηk = d(g(xm(k)),g(xt(k))) ≤ d(g(xm(k)),g(xt(k)−1)) + d(g(xt(k)−1),g(xt(k))) < � + δt(k)−1 that is, � ≤ ηk < � + δt(k)−1. letting k →∞ in above inequality and using (3.7), we get lim k→∞ ηk = �. (3.8) again, ηk+1 = d(g(xm(k)+1),g(xt(k)+1)) ≤ d(g(xm(k)+1),g(xm(k))) + d(g(xm(k)),g(xt(k))) + d(g(xt(k)),g(xt(k)+1)) < δm(k)+1 + ηk + δt(k)+1 ⇒ ηk+1 < δm(k)+1 + ηk + δt(k)+1. letting k →∞ in above inequality and using (3.7) and (3.8), we get lim k→∞ ηk+1 = �. (3.9) since t(k) > m(k), hence by (3.3), we get g(xm(k)) ≤ g(xt(k)). therefore, owing to (3.1) and assumption (vii), we get ϕ(ηk+1) = ϕ(d(g(xm(k)+1),g(xt(k)+1))) = ϕ(d(f(xm(k)),f(xt(k)))) ≤ f(ϕ(d(g(xm(k)),g(xt(k)))),ζ(d(g(xm(k)),g(xt(k))))) = f(ϕ(ηk),ζ(ηk)) that is, ϕ(ηk+1) ≤f(ϕ(ηk),ζ(ηk)). letting k →∞ in above inequality and using (3.8), (3.9) and continuities of ϕ and ζ, we get ϕ(�) ≤f(ϕ(�),ζ(�)) so ϕ(�) = 0, or ,ζ(�) = 0 thus � = 0 which is a contradiction . therefore the sequence {g(xm)} is cauchy. from the completeness of x, there exists x ∈ x such that lim m→∞ f(xm) = lim m→∞ g(xm) = x. (3.10) since f and g are compatible, we have from (3.10), lim m→∞ d(f(gxm),g(fxm)) = 0. (3.11) c-class functions on shorter proofs of some even-tupled coincidence theorems 135 now, we use assumption (v). firstly, we assume that f is continuous. then for all m ∈ n ∪{0}, we have d(g(x),f(gxm)) ≤ d(g(x),g(fxm)) + d(g(fxm),f(gxm)). taking k → ∞ in above inequality and using (3.10), (3.11) and continuities of f and g, we get d(g(x),f(x)) = 0, that is, g(x) = f(x). hence the element x ∈ x is a coincidence point of f and g. next, we suppose that x is nondecreasing g-regular. from (3.3) and (3.10), we get g(gxm) � g(x). (3.12) since f and g are compatible and g is continuous by (3.10) and (3.11), we have lim m→∞ g(gxm) = g(x) = lim m→∞ g(fxm) = lim m→∞ f(gxm). (3.13) now, using triangle inequality, we have d(f(x),g(x)) ≤ d(f(x),g(gxm+1)) + d(g(gxm+1),g(x)) = d(f(x),g(fxm)) + d(g(gxm+1),g(x)). taking k →∞ in above inequality and using (3.13), we have d(f(x),g(x)) ≤ lim m→∞ d(f(x),g(fxm)) + lim m→∞ d(g(gxm+1),g(x)) = lim m→∞ d(f(x),f(gxm)). since ϕ is continuous and monotone nondecreasing, from the above inequality we have ϕ(d(f(x),g(x))) ≤ ϕ( lim m→∞ d(f(x),f(gxm))) = lim m→∞ ϕ(d(f(x),f(gxm))). by (3.12) and assumption (vii), we get ϕ(d(f(x),g(x))) ≤ lim m→∞ ϕ(d(f(x),f(gxm))) ≤ lim m→∞ f(ϕd(g(x),g(gxm))),ζ(d(g(x),g(gxm)))) = f( lim m→∞ ϕ(d(g(x),g(gxm))), lim m→∞ ζ(d(g(x),g(gxm)))) = f(ϕ(d(f(x),g(x))),ζ(d(f(x),g(x))). so ϕ(d(f(x),g(x))) = 0, or ,ζ(d(f(x),g(x)) = 0 ,which implies that d(f(x),g(x)) = 0, that is, g(x) = f(x). hence x ∈ x is a coincidence point of f and g. lemma 3.4. in addition to the hypotheses of lemma 3.3, suppose that for real x,y ∈ x there exists, z ∈ x such that f(z) is comparable to f(x) and f(y). then f and g have a unique common fixed point. proof. the set of coincidence points of f and g is non-empty due to lemma 3.3. assume now, x and y are two coincidence points of f and g, that is, f(x) = g(x) and f(y) = g(y). now we will show that g(x) = g(y). by assumption, there exists z ∈ x such that f(z) is comparable to f(x) and f(y). put z10 = z and choose z1 ∈ x such that g(z1) = f(z0). further define sequence {g(zm)} such that g(zm+1) = f(zm). further set x0 = x and y0 = y. in the same way, define the sequences {g(xm)} and {g(ym)}. then it is easy to show that g(xm+1) = f(xm) and g(ym+1) = f(ym). since f(x) = g(x1) = g(x) and f(z) = g(z1) are comparable, we have g(x) � g(z1). it is easy to show that g(x) and g(zm) are comparable, that is, for all m ∈ n, g(x) � g(zm). 136 sharma thus from (3.1) we have ϕ(d(g(x),g(zm+1))) = ϕ(d(f(x),f(zm))) ≤ f(ϕ(d(g(x),g(zm))),ζ(d(g(x),g(zm)))). let rm = d(g(x),g(zm+1)). then ϕ(rm) ≤f(ϕ(rm−1),ζ(rm−1)). (3.14) using the property of ϕ, we have ϕ(rm) ≤ ϕ(rm−1), which implies that rm ≤ rm−1 (by the property of ϕ). therefore {rm} is a monotone decreasing sequence of nonnegative real numbers. hence there exists r ≥ 0 such that rm → r as m → ∞. taking the limit as m → ∞ in (3.14) and using the continuities of ϕ and ζ, we have ϕ(r) ≤ f(ϕ(r),ζ(r)),so ϕ(r) = 0, ,ζ(r) = 0 thus r = 0 which is a contradiction . therefore rm → 0 as m →∞, that is, lim m→∞ d(g(x),g(zm+1)) = 0. similarly we can prove that lim m→∞ d(g(y),g(zm+1)) = 0. therefore by triangle inequality d(g(x),g(y)) ≤ d(g(x),g(zm+1)) + d(g(zm+1),g(y)) → 0 as m →∞. hence g(x) = g(y). (3.15) since g(x) = f(x) and f and g are compatible, we have gg(x) = f(gx). write g(x) = a, then we have g(a) = f(a). (3.16) thus a is the coincidence point of f and g. then owing to (3.15) with y = a, it follows that g(x) = g(a), that is, g(a) = a. (3.17) using (3.16) and (3.17), we have a = g(a) = f(a). thus a is the common fixed point of f and g. to prove the uniqueness, assume that b is another common fixed point of f and g. then by (3.15), we have b = g(b) = g(a) = a. this completes the proof of lemma. theorem 3.1. let (x,d,�) be an ordered complete metric space and f : xn → x and g : x → x be two mappings. suppose that the following conditions are satisfied: (i) f(xn) ⊆ g(x), (ii) f and g are compatible, (iii) f has the mixed g-monotone property, (iv) g is continuous, (v) either f is continuous or x is g-regular, (vi) there exist x10,x 2 0,x 3 0, ...,x n 0 ∈ x such that  gx10 � f(x10,x20,x30, ...,xn0 ) f(x20,x 3 0, ...,x n 0 ,x 1 0) � gx20 gx30 � f(x30, ...,xn0 ,x10,x20) ... f(xn0 ,x 1 0,x 2 0, ...,x n−1 0 ) � gx n 0 , (3.18) (vii) there exist ϕ ∈ ω and ζ ∈=u and f a c-function such that ϕ(d(fu,fv )) ≤f(ϕ( max d(gxi,gyi)),ζ(max d(gxi,gyi))) for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn � gyn. then f and g have an n-tupled coincidence point. c-class functions on shorter proofs of some even-tupled coincidence theorems 137 proof. consider the product space y = xn equipped with the metric d̃ (given by (b)) and the partial order v (given by (a)). then by lemma 3.1, (y,d̃,v) is an ordered complete metric space. also f and g induce mappings tf : y → y and tg : y → y (defined in lemma 3.2). clearly, • (i) implies that tf (y ) ⊆ tg(y ), • (ii) implies that tf is monotone tg-nondecreasing (by item (1) of lemma 3.2), • (iii) implies that tf and tg are compatible (by item (2) of lemma 3.2), • (iv) implies that tg is continuous (by item (3) of lemma 3.2), • (v) implies that either tf is continuous (by item (4) of lemma 3.2) or (y,d̃,v) is nondecreasing g-regular (by item (5) of lemma 3.2), • (vi) is equivalent to the condition: there exists u0 = (x10,x20, . . . ,xn0 ) ∈ y such that tg(u0) ⊆ tf (u0). now, in view of (vii), for given u,v ∈ y such that tg(u) v tg(v ) implies that (gx1,gx2, . . . ,gxn) v (gy1,gy2, . . . ,gyn). it follows that for odd i, (gxi,gxi+1, . . . ,gxn,gx1, . . . ,gxi−1) v (gyi,gyi+1, . . . ,gyn,gy1, . . . ,gyi−1), (3.19) and for even i, (gyi,gyi+1, . . . ,gyn,gy1, . . . ,gyi−1) v (gxi,gxi+1, . . . ,gxn,gx1, . . . ,gxi−1). (3.20) if i is odd, then by using (3.19) and (vii), we get d(f(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),f(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1)) ≤ ϕ(max{d(gxi,gyi),d(gxi+1,gyi+1), . . . ,d(gxn,gyn),d(gx1,gy1), f(d(gx2,gy2), . . . ,d(gxi−1,gyi−1)}),ζ(max{d(gxi,gyi),d(gxi+1,gyi+1), . . . , d(gxn,gyn),d(gx1,gy1),d(gx2,gy2), . . . ,d(gxi−1,gyi−1)})) = f(ϕ( max 1≤i≤n d(gxi,gyi)),ζ( max 1≤i≤n d(gxi,gyi))). if i is even, then by using (3.20) and (vii), we get d(f(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),f(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1)) = d(f(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1),f(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1)) ≤ ϕ(max{d(gyi,gxi),d(gyi+1,gxi+1), . . . ,d(gyn,gxn),d(gy1,gx1), d(gy2,gx2), . . . ,d(gyi−1,gxi−1)}) − ζ(max{d(gyi,gxi),d(gyi+1,gxi+1), . . . , d(gyn,gxn),d(gy1,gx1),d(gy2,gx2), . . . ,d(gyi−1,gxi−1)}) = ϕ( max 1≤i≤n d(gxi,gyi)) − ζ( max 1≤i≤n d(gxi,gyi)). hence, in both the cases, for each i (1 ≤ i ≤ n), we have d(f(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),f(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1)) ≤ ϕ( max 1≤i≤n d(gxi,gyi)) − ζ( max 1≤i≤n d(gxi,gyi)). (3.21) 138 sharma hence by using (3.21), we have d̃(tf (u),tf (v )) = max 1≤i≤n d(f(xi,xi+1, . . . ,xn,x1,x2, . . . ,xi−1),f(yi,yi+1, . . . ,yn,y1,y2, . . . ,yi−1)) ≤f( max 1≤i≤n [ϕ( max 1≤i≤n d(gxi,gyi)),ζ( max 1≤i≤n d(gxi,gyi))]) = f(ϕ( max 1≤i≤n d(gxi,gyi)),ζ( max 1≤i≤n d(gxi,gyi))) = f(ϕ(d̃(tg(u),tg(v )),ζ(d̃(tg(u),tg(v ))). thus all conditions of lemma 3.3 are satisfied for ordered complete metric space (y,d̃,v) and mappings tf : y → y and tg : y → y. therefore tf and tg have a coincidence point in y = xn. according to item (6) of lemma 3.2, the mappings f and g have an n-tupled coincidence point. corollary 3.1. ([37]) let (x,d,�) be an ordered complete metric space and f : xn → x and g : x → x be two mappings. suppose that the following conditions are satisfied: (i) f(xn) ⊆ g(x), (ii) f and g are compatible, (iii) f has the mixed g-monotone property, (iv) g is continuous, (v) either f is continuous or x is g-regular, (vi) there exist x10,x 2 0,x 3 0, ...,x n 0 ∈ x such that (3.18) holds, (vii) there exist ϕ ∈ ω and ζ ∈= such that ϕ(d(fu,fv )) ≤ ϕ( max 1≤i≤n d(gxi,gyi)) − ζ( max 1≤i≤n d(gxi,gyi)), for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn � gyn. then f and g have an n-tupled coincidence point. proof. it is sufficient to take f(s,t) = s− t in theorem 3.1. corollary 3.2. corollary 3.1 remains true if condition (vii) is replaced by the following: (vii)’ there exist ϕ ∈ ω such that ϕ(d(fu,fv )) ≤ kϕ( max d(gxi,gyi)), 0 < k < 1, for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn � gyn. proof. it is sufficient to take f(s,t) = ks, 0 < k < 1 in theorem 3.1. corollary 3.3. corollary 3.1 remains true if condition (vii) is replaced by the following: (vii)” there exist ϕ ∈ ω and β : [0,∞) → [0, 1) which is semi-continuous such that ϕ(d(fu,fv )) ≤ ϕ( max d(gxi,gyi))β(ϕ( max d(gxi,gyi))) for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn � gyn. proof. it is sufficient to take f(s,t) = sβ(s) (where β : [0,∞) → [0, 1) and semi-continuous) in theorem 3.1. corollary 3.4. corollary 3.1 remains true if condition (vii) is replaced by the following: (vii)”’ there exist ϕ ∈ ω and φ : [0,∞) → [0,∞) which is an upper semi-continuous function such that c-class functions on shorter proofs of some even-tupled coincidence theorems 139 φ(0) = 0 and φ(t) < t for t > 0 such that ϕ(d(fu,fv )) ≤ φ(ϕ( max d(gxi,gyi))) for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with gy1 � gx1,gx2 � gy2,gy3 � gx3, . . . ,gxn � gyn. proof. it is sufficient to take f(s,t) = φ(s) (where φ : [0,∞) → [0,∞) is an upper semi-continuous function such that φ(0) = 0 and φ(t) < t for t > 0) in theorem 3.1. corollary 3.5. let (x,d,�) be an ordered complete metric space and f : xn → x be a mapping. suppose that the following conditions are satisfied: (i) f has the mixed monotone property, (ii) either f is continuous or x is regular, (iii) there exist x10,x 2 0,x 3 0, ...,x n 0 ∈ x such that  x10 � f(x10,x20,x30, ...,xn0 ) f(x20,x 3 0, ...,x n 0 ,x 1 0) � x20 x30 � f(x30, ...,xn0 ,x10,x20) ... f(xn0 ,x 1 0,x 2 0, ...,x n−1 0 ) � x n 0 , (iv) there exist ϕ ∈ ω and ζ ∈=u and f a c-function such that ϕ(d(fu,fv )) ≤f(ϕ( max d(xi,yi)),ζ(max d(xi,yi))) for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn. then f has an n-tupled fixed point. proof. it is sufficient to take g = i (identity mapping) in theorem 3.1. corollary 3.6. corollary 3.5 remains true if condition (iv) is replaced by the following: (iv)’ there exists ζ ∈=u such that d(fu,fv ) ≤f( max 1≤i≤n d(xi,yi),ζ( max 1≤i≤n d(xi,yi))), for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn. proof. it is sufficient to take ϕ and g to be identity mappings in theorem 3.1. corollary 3.7. corollary 3.1 remains true if condition (iv) is replaced by the following: (iv)” there exists k ∈ (0, 1) such that d(fu,fv ) ≤ k max 1≤i≤n d(xi,yi), for all u = (x1,x2, ...,xn), v = (y1,y2, ...,yn) ∈ xn with x1 � y1,y2 � x2,x3 � y3, . . . ,yn � xn. proof. it is sufficient to take ϕ and g to be identity mappings and ζ(t) = (1 − k)t, k ∈ (0, 1) in theorem 3.1. now we shall prove the uniqueness of n-tupled fixed point. theorem 3.2. in addition to the hypotheses of theorem 3.1, suppose that for real (x1,x2, ...,xn) and (y1,y2, ...,yn) ∈ xn there exists, (z1,z2, ...,zn) ∈ xn such that (f(z1,z2, ...,zn),f(z2, ...,zn,z1), ...,f(zn, z1, ...,zn−1)) is comparable to (f(x1,x2, ...,xn), f(x2, ...,xn,x1), ...,f(xn,x1, ...,xn−1)) and (f(y1,y2, ..., yn),f(y2, ...,yn,y1), ...,f(yn,y1, ...,yn−1)). then f and g have a unique n-tupled common fixed point. 140 sharma proof. set u = (x1,x2, . . . ,xn), v = (y1,y2, . . . ,yn) and w = (z1,z2, . . . ,zn). then we have tf (w) v tf (u) or tf (u) v tf (w) and tf (w) v tf (v ) or tf (v ) v tf (w). hence by using lemma 3.4, tf and tg have a unique n-tupled common fixed point. references [1] agarwal, r. p., el-gebeily, m. a., oregan, d: generalized contractions in partially ordered metric spaces. appl. anal. 87(1) (2008), 109-116. [2] altun, i. and simsek, h: some fixed point theorems on ordered metric spaces and application. fixed point theory appl. 2010 (2010), article id 621469. [3] ansari, a. h.: note on (ϕ,ψ)-contractive type mappings and related fixed point. the 2nd regional conference on mathematics and applications, pnu, (2014) , 377-380. [4] berinde, v. and borcut, m., tripled fixed point theorems for contractive type mappings partially ordered metric spaces. nonlinear anal., 75 (15) (2011), 4889-4897. [5] bhaskar, t. g., lakshmikantham, v.: fixed points theorems in partially ordered metric spaces and applications. nonlinear anal. tma 65 (2006), 1379-1393. [6] berzig, m. and samet, b.: an extension of coupled fixed point’s concept in higher dimension and applications. comput. math. appl., 63 (2012), 1319-1334. [7] caballero, j., harjani, j. and sadarangani, k: contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations. fixed point theory appl. 2010 (2010), article id 916064. [8] choudhury, b. s., metiya, n. and kundu, a.: coupled coincidence point theorems in ordered metric spaces. ann. univ. ferrara 57 (2011), 1-16. [9] dalal, s., khan, l. a., masmali, i. and radenovic, s.: some remarks on multidimensional fixed point theorems in partially ordered metric spaces, j. adv. math. 7 (1) (2014), 1084-1094. [10] guo, d. j. and lakshmikantham, v.: coupled fixed points of nonlinear operators with applications. nonlinear anal. 11 (1987), no. 5, 623-632. [11] harandi, a. a. and emami, h: a fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. nonlinear anal. 72(5) (2010), 2238-2242. [12] harjani, j.and sadarangani, k: fixed point theorems for weakly contractive mappings in partially ordered sets. nonlinear anal. 71(7-8) (2009), 3403-3410. [13] harjani, j. and sadarangani, k: generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. nonlinear anal. 72 (2010), 1188-1197. [14] imdad, m., sharma, a. and rao, k. p. r.: n-tupled coincidence and common fixed point results for weakly contractive mappings in complete metric spaces. bull. math. anal. appl., 5(4) (2013), 19-39. [15] imdad, m., sharma, a. and rao, k. p. r.: generalized n-tupled fixed point theorems for nonlinear contraction mapping. afrika matematika, 26 (2015), 443455. [16] imdad, m., soliman, a. h., choudhury, b. s. and das, p.: on n-tupled coincidence and common fixed points results in metric spaces. jour. of operators, 2013 (2013), article id 532867, 9 pages. [17] imdad, m., alam, a. and soliman, a. h.: remarks on a recent general even-tupled coincidence theorem. j. adv. math. 9 (1) (2014), 1787-1805. [18] jachymski, j: equivalent conditions for generalized contractions on (ordered) metric spaces. nonlinear anal. 74(3) (2011), 768-774. [19] jleli, m., rajić, v. ć., samet, b. and vetro, c.: fixed point theorems on ordered metric spaces and applications to nonlinear elastic beam equations. j. fixed point theory appl. 12 (2012), 175-192. [20] karapınar, e., roldan, a., martinez-moreno, j. and roldan, c.: meir-keeler type multidimensional fixed point theorems in partially ordered metric spaces, abstract and applied analysis, 2013 (2013), article id 406026. [21] khan, m. s., swaleh, m., sessa, s.: fixed point theorems by altering distance functions between the points. bull. aust. math. soc. 30 (1984), 1-9. [22] kutbi, m. a., roldán, a., sintunavarat, w., moreno, j. m. and roldán, c.: f -closed sets and coupled fixed point theorems without the mixed monotone property. fixed point theory and applications, 2013 (2013), article id 330. [23] lakshmikantham, v. and ćirić, lj. b.: coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. nonlinear anal. 70 (2009), 4341-4349. [24] nashine, h. k. and altun, i: a common fixed point theorem on ordered metric spaces. bull. iran. math. soc. 38(4) (2012), 925-934. [25] nieto, j. j. and lópez, r. r.: contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, order 22, 223-239, (2005). [26] nieto, j. j. and lópez, r. r.: existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. acta math. sinica, engl. ser. 23 (12) (2007), 2205-2212. [27] o’regan, d. and petrusel a.: fixed point theorems for generalized contractions in ordered metric spaces. j. math. anal. appl. 341 (2008), 1241-1252. c-class functions on shorter proofs of some even-tupled coincidence theorems 141 [28] radenovic, s.: remarks on some coupled coincidence point in partially ordered metric spaces. arab jour. math. sci., 20(1) (2014), 29-39. [29] radenovic, s.: a note on tripled coincidence and tripled common fixed point theorems in partially ordered metric spaces. app. math. comp. 236 (2014), 367-372. [30] ran, a. c. m., reurings, m. c. b.: a fixed point theorem in partially ordered sets and some applications to matrix equations. proc. amer. math. soc. 132 (2004), 1435-1443. [31] roldán, a., mart́ınez-moreno, j. and roldán, c.: multidimensional fixed point theorems in partially ordered metric spaces. journal of mathematical analysis and applications, 396 (2012), 536-545. [32] roldán, a., mart́ınez-moreno, j., roldán, c., karapınar, e.: multidimensional fixed-point theorems in partially ordered completely partial metric spaces under (ϕ,ψ)-contractivity conditions. abst. appl. anal. 2013 (2013). article id 634371. [33] roldán, a., mart́ınez-moreno, j., roldán, c., cho, y.j.: multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces. fuzzy sets syst. (2013). [34] samet, b., vetro, c.: coupled fixed point, f-invariant set and fixed point of n-order. ann. funct. anal. 1 (2) (2010), 4656-4662. [35] samet, b., vetro, c. and vetro, f.: from metric spaces to partial metric spaces. fixed point theory appl. 2013 (2013), art. id 5. [36] samet, b., karapınar, e., aydi, h. and rajic, v. c.: discussion on some coupled fixed point theorems. fixed point theory appl., 2013 (2013), art. id 50. [37] sharma, a., imdad, m., alam, a.: shorter proofs of some recent even-tupled coincidence theorems for weak contractions in ordered metric spaces. math. sci., 8 (2014), 131-138. department of mathematics and statistics,indian institute of technology, kanpur 208 016, india ∗corresponding author: annusharma241@gmail.com int. j. anal. appl. (2023), 21:75 characteristic picture fuzzy sets and level subsets in up (bcc)-algebras pimwaree kankaew1, sunisa yuphaphin1, nattacha lapo1, ronnason chinram2, pongpun julatha3, aiyared iampan1,∗ 1fuzzy algebras and decision-making problems research unit, department of mathematics, school of science, university of phayao, mae ka, mueang, phayao 56000, thailand 2division of computational science, faculty of science, prince of songkla university, hat yai, songkhla 90110, thailand 3department of mathematics, faculty of science and technology, pibulsongkram rajabhat university, phitsanulok 65000, thailand ∗corresponding author: aiyared.ia@up.ac.th abstract. the eight new concepts of picture fuzzy sets in up (bcc)-algebras are introduced by kankaew et al. in 2022. this idea is extended to the lower and upper level subsets of picture fuzzy sets in up (bcc)-algebras. moreover, we define a picture fuzzy set in the same way as a characteristic function and study its characterizations from the related subset. 1. introduction among many algebraic structures, algebras of logic form important class of algebras. examples of these are bck-algebras [16], bci-algebras [17], be-algebras [23], up-algebras [11], fully upsemigroups [12], topological up-algebras [28], up-hyperalgebras [14], extension of ku/up-algebras [27] and others. they are strongly connected with logic. for example, bci-algebras introduced by iséki [17] in 1966 have connections with bci-logic being the bci-system in combinatory logic which has application in the language of functional programming. bck and bci-algebras are two classes of logical algebras. they were introduced by imai and iséki [16, 17] in 1966 and have been extensively investigated by many researchers. received: jun. 13, 2023. 2020 mathematics subject classification. 06f35, 08a72, 03e72. key words and phrases. up (bcc)-algebra; picture fuzzy set; characteristic picture fuzzy set; lower and upper level subset. https://doi.org/10.28924/2291-8639-21-2023-75 issn: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-75 2 int. j. anal. appl. (2023), 21:75 the concept of fuzzy sets was first considered by zadeh [38] in 1965. the fuzzy set theories developed by zadeh and others have found many applications in the domain of mathematics and elsewhere. after the introduction of the concept of fuzzy sets by zadeh [38], atanassov [3,4] defined a new concept called an intuitionistic fuzzy set which is a generalization of fuzzy set. the concept of picture fuzzy sets was first considered by cuong and kreinovich [6] in 2013, which is direct extensions of the fuzzy sets and the intuitionistic fuzzy sets. the picture fuzzy set is characterized by three functions expressing the degree of membership, the degree of neutral membership, and the degree of nonmembership. the only constraint is that the sum of the three degrees must not exceed 1. cuong [5] presented the concept of picture fuzzy sets in the journal of computer science and cybernetics in 2014. some operations on picture fuzzy sets with some properties are considered. the zadeh extension principle, picture fuzzy relations, and picture fuzzy soft sets are studied. several researches were conducted on the generalizations of the concept of picture fuzzy sets in a variety of different fields and its application to a decision-making problem. in 2015, singh [33] presented a geometrical interpretation of picture fuzzy sets. the author proposed correlation coefficients for picture fuzzy sets which considers the degree of positive membership, degree of neutral membership, degree of negative membership and the degree of refusal membership. in 2017, wei [35] presented another form of eight similarity measures between picture fuzzy sets based on the cosine function between picture fuzzy sets by considering the degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership in picture fuzzy sets. the author applied these weighted cosine function similarity measures between picture fuzzy sets to strategic decision making. in 2018, wei and gao [37] presented some novel dice similarity measures of picture fuzzy sets and the generalized dice similarity measures of picture fuzzy sets and indicate that the dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized dice similarity measures in some parameter values. wei [36] presented some novel process to measure the similarity between picture fuzzy sets. the author applied these similarity measures between picture fuzzy sets to building material recognition and minerals field recognition. in 2020, ganie et al. [8] introduced two correlation coefficients of picture fuzzy sets. these correlation coefficients of picture fuzzy sets are better than existing ones and effective in expressing the nature of correlation (positive or negative correlation). in 2022, jun et al. [18] have shown that the concept of up-algebras (see [11]) and the concept of bcc-algebras (see [24]) are the same concept. therefore, in this article and future research, our research team will use the name bcc instead of up in honor of komori, who first defined it in 1984. in this paper, we applied the concept of picture fuzzy sets in bcc-algebras to introduce the eight new concepts of picture fuzzy sets: picture fuzzy bcc-subalgebras, picture fuzzy near bcc-filters, picture fuzzy bcc-filters, picture fuzzy implicative bcc-filters, picture fuzzy comparative bcc-filters, picture fuzzy shift bcc-filters, picture fuzzy bcc-ideals, and picture fuzzy strong bcc-ideals. also, int. j. anal. appl. (2023), 21:75 3 we discuss the relationship between the eight new concepts of picture fuzzy sets in bcc-algebras. this idea is extended to the lower and upper level subsets of picture fuzzy sets in bcc-algebras. moreover, we define a picture fuzzy set in the same way as a characteristic function and study its characterizations from the related subset. 2. basic results on bcc-algebras the concept of bcc-algebras (see [24]) can be redefined without the condition (2.6) as follows: definition 2.1. [10] an algebra x = (x, ·,0) of type (2,0) is called a bcc-algebra, where x is a nonempty set, · is a binary operation on x, and 0 is a fixed element of x (i.e., a nullary operation) if it satisfies the following axioms: (∀x,y,z ∈ x)((y ·z) · ((x ·y) · (x ·z))=0), (2.1) (∀x ∈ x)(0 ·x = x), (2.2) (∀x ∈ x)(x ·0=0), (2.3) (∀x,y ∈ x)(x ·y =0,y ·x =0⇒ x = y). (2.4) from [11], we know that the concept of bcc-algebras is a generalization of ku-algebras (see [26]). the binary relation ≤ on a bcc-algebra x =(x, ·,0) is defined as follows: (∀x,y ∈ x)(x ≤ y ⇔ x ·y =0) (2.5) and the following assertions are valid (see [11,12]). (∀x ∈ x)(x ≤ x), (2.6) (∀x,y,z ∈ x)(x ≤ y,y ≤ z ⇒ x ≤ z), (2.7) (∀x,y,z ∈ x)(x ≤ y ⇒ z ·x ≤ z ·y), (2.8) (∀x,y,z ∈ x)(x ≤ y ⇒ y ·z ≤ x ·z), (2.9) (∀x,y,z ∈ x)(x ≤ y ·x, in particular, y ·z ≤ x · (y ·z)), (2.10) (∀x,y ∈ x)(y ·x ≤ x ⇔ x = y ·x), (2.11) (∀x,y ∈ x)(x ≤ y ·y), (2.12) (∀a,x,y,z ∈ x)(x · (y ·z)≤ x · ((a ·y) · (a ·z))), (2.13) (∀a,x,y,z ∈ x)(((a ·x) · (a ·y)) ·z ≤ (x ·y) ·z), (2.14) (∀x,y,z ∈ x)((x ·y) ·z ≤ y ·z), (2.15) (∀x,y,z ∈ x)(x ≤ y ⇒ x ≤ z ·y), (2.16) (∀x,y,z ∈ x)((x ·y) ·z ≤ x · (y ·z)), (2.17) (∀a,x,y,z ∈ x)((x ·y) ·z ≤ y · (a ·z)). (2.18) 4 int. j. anal. appl. (2023), 21:75 example 2.1. [30] let u be a nonempty set and let x ∈ p(u) where p(u) means the power set of u. let px(u) = {a ∈ p(u) | x ⊆ a}. define a binary operation m on px(u) by putting a m b = b ∩ (ac ∪x) for all a,b ∈px(u) where ac means the complement of a subset a. then (px(u),m,x) is a bcc-algebra. let px(u)= {a ∈p(u) | a ⊆ x}. define a binary operation n on px(u) by putting anb = b∪(ac∩x) for all a,b ∈px(u). then (px(u),n,x) is a bcc-algebra. example 2.2. [7] let z∗ be the set of all nonnegative integers. define two binary operations ◦ and ? on z∗ by: (∀m,n ∈ z∗) ( m◦n = { n if m < n, 0 otherwise ) and (∀m,n ∈ z∗) ( m ? n = { n if m > n or m =0, 0 otherwise ) . then (z∗,◦,0) and (z∗,?,0) are bcc-algebras. for more examples of bcc-algebras, see [1,2,12,15,25,29–32]. for a nonempty subset s of a bcc-algebra x =(x, ·,0) which satisfies the following condition: (∀x,y ∈ x)(y ∈ s ⇒ x ·y ∈ s). (2.19) then the constant 0 of x is in s. indeed, let x ∈ s. by (2.6) and (2.19), we have 0= x ·x ∈ s. definition 2.2. [9,11,13,19–21,34] a nonempty subset s of a bcc-algebra x =(x, ·,0) is called (1) a bcc-subalgebra of x if it satisfies the following condition: (∀x,y ∈ s)(x ·y ∈ s), (2.20) (2) a near bcc-filter of x if it satisfies the condition (2.19), (3) a bcc-filter of x if it satisfies the following conditions: the constant 0 of x is in s, (2.21) (∀x,y ∈ x)(x ·y ∈ s,x ∈ s ⇒ y ∈ s), (2.22) (4) an implicative bcc-filter of x if it satisfies the condition (2.21) and the following condition: (∀x,y,z ∈ x)(x · (y ·z)∈ s,x ·y ∈ s ⇒ x ·z ∈ s), (2.23) (5) a comparative bcc-filter of x if it satisfies the condition (2.21) and the following condition: (∀x,y,z ∈ x)(x · ((y ·z) ·y)∈ s,x ∈ s ⇒ y ∈ s), (2.24) (6) a shift bcc-filter of x if it satisfies the condition (2.21) and the following condition: (∀x,y,z ∈ x)(x · (y ·z)∈ s,x ∈ s ⇒ ((z ·y) ·y) ·z ∈ s), (2.25) int. j. anal. appl. (2023), 21:75 5 (7) a bcc-ideal of x if it satisfies the condition (2.21) and the following condition: (∀x,y,z ∈ x)(x · (y ·z)∈ s,y ∈ s ⇒ x ·z ∈ s), (2.26) (8) a strong bcc-ideal of x if it satisfies the condition (2.21) and the following condition: (∀x,y,z ∈ x)((z ·y) · (z ·x)∈ s,y ∈ s ⇒ x ∈ s). (2.27) guntasow et al. [9] proved that the only strong bcc-ideal of a bcc-algebra x is x. the following theorem is easy to verify. theorem 2.1. let f be a nonempty family of bcc-subalgebras (resp., near bcc-filters, bcc-filters, implicative bcc-filters, comparative bcc-filters, shift bcc-filters, bcc-ideals, strong bcc-ideals) of a bcc-algebra x = (x, ·,0). then ⋂ f is a bcc-subalgebra (resp., near bcc-filter, bcc-filter, implicative bcc-filter, comparative bcc-filter, shift bcc-filter, bcc-ideal, strong bcc-ideal) of x. 3. pfss in bcc-algebras in 2013, cuong and kreinovich [6] introduced the concept of picture fuzzy sets as the following definition. a picture fuzzy set (briefly, pfs) in a nonempty set x is a structure of the form: p = {(x,rp(x),gp(x),bp(x)) | x ∈ x}, where rp : x → [0,1] is a positive membership, gp : x → [0,1] is a neutral membership, and bp : x → [0,1] is a negative membership satisfy the following condition: (∀x ∈ x)(rp(x)+gp(x)+bp(x)≤ 1). for our convenience, we will denote a pfs as p =(x,rp ,gp ,bp). a pfs p in x is said to be constant if p is a constant function from x to [0,1]3. that is, rp ,gp , and bp are constant functions from x to [0,1]. in what follows, let x denote a bcc-algebra (x, ·,0) unless otherwise specified. kankaew et al. [22] introduced the eight new concepts of pfss in bcc-algebras: picture fuzzy bcc-subalgebras, picture fuzzy near bcc-filters, picture fuzzy bcc-filters, picture fuzzy implicative bcc-filters, picture fuzzy comparative bcc-filters, picture fuzzy shift bcc-filters, picture fuzzy bccideals, and picture fuzzy strong bcc-ideals. definition 3.1. a pfs p in x is called 6 int. j. anal. appl. (2023), 21:75 (1) a picture fuzzy bcc-subalgebra of x if it satisfies the following conditions: (∀x,y ∈ x)(rp(x ·y)≥min{rp(x), rp(y)}), (3.1) (∀x,y ∈ x)(gp(x ·y)≥min{gp(x),gp(y)}), (3.2) (∀x,y ∈ x)(bp(x ·y)≤max{bp(x),bp(y)}), (3.3) (2) a picture fuzzy near bcc-filter of x if it satisfies the following conditions: (∀x,y ∈ x)(rp(x ·y)≥ rp(y)), (3.4) (∀x,y ∈ x)(gp(x ·y)≥ gp(y)), (3.5) (∀x,y ∈ x)(bp(x ·y)≤ bp(y)), (3.6) (3) a picture fuzzy bcc-filter of x if it satisfies the following conditions: (∀x ∈ x)(rp(0)≥ rp(x)), (3.7) (∀x ∈ x)(gp(0)≥ gp(x)), (3.8) (∀x ∈ x)(bp(0)≤ bp(x)), (3.9) (∀x,y ∈ x)(rp(y)≥min{rp(x ·y), rp(x)}), (3.10) (∀x,y ∈ x)(gp(y)≥min{gp(x ·y),gp(x)}), (3.11) (∀x,y ∈ x)(bp(y)≤max{bp(x ·y),bp(x)}), (3.12) (4) a picture fuzzy implicative bcc-filter of x if it satisfies the following conditions: (3.7), (3.8), (3.9), and (∀x,y,z ∈ x)(rp(x ·z)≥min{rp(x · (y ·z)), rp(x ·y)}), (3.13) (∀x,y,z ∈ x)(gp(x ·z)≥min{gp(x · (y ·z)),gp(x ·y)}), (3.14) (∀x,y,z ∈ x)(bp(x ·z)≤max{bp(x · (y ·z)),bp(x ·y)}), (3.15) (5) a picture fuzzy comparative bcc-filter of x if it satisfies the following conditions: (3.7), (3.8), (3.9), and (∀x,y,z ∈ x)(rp(y)≥min{rp(x · ((y ·z) ·y)), rp(x)}), (3.16) (∀x,y,z ∈ x)(gp(y)≥min{gp(x · ((y ·z) ·y)),gp(x)}), (3.17) (∀x,y,z ∈ x)(bp(y)≤max{bp(x · ((y ·z) ·y)),bp(x)}), (3.18) int. j. anal. appl. (2023), 21:75 7 (6) a picture fuzzy shift bcc-filter of x if it satisfies the following conditions: (3.7), (3.8), (3.9), and (∀x,y,z ∈ x)(rp(((z ·y) ·y) ·z)≥min{rp(x · (y ·z)), rp(x)}), (3.19) (∀x,y,z ∈ x(gp(((z ·y) ·y) ·z)≥min{gp(x · (y ·z)),gp(x)}), (3.20) (∀x,y,z ∈ x)(bp(((z ·y) ·y) ·z)≤max{bp(x · (y ·z)),bp(x)}), (3.21) (7) a picture fuzzy bcc-ideal of x if it satisfies the following conditions: (3.7), (3.8), (3.9), and (∀x,y,z ∈ x)(rp(x ·z)≥min{rp(x · (y ·z)), rp(y)}), (3.22) (∀x,y,z ∈ x)(gp(x ·z)≥min{gp(x · (y ·z)),gp(y)}), (3.23) (∀x,y,z ∈ x)(bp(x ·z)≤max{bp(x · (y ·z)),bp(y)}), (3.24) (8) a picture fuzzy strong bcc-ideal of x if it satisfies the following conditions: (3.7), (3.8), (3.9), and (∀x,y,z ∈ x)(rp(x)≥min{rp((z ·y) · (z ·x)), rp(y)}), (3.25) (∀x,y,z ∈ x)(gp(x)≥min{gp((z ·y) · (z ·x)),gp(y)}), (3.26) (∀x,y,z ∈ x)(bp(x)≤max{bp((z ·y) · (z ·x)),bp(y)}). (3.27) kankaew et al. [22] proved the generalization that the concept of picture fuzzy bcc-subalgebras is a generalization of picture fuzzy near bcc-filters, picture fuzzy near bcc-filters is a generalization of picture fuzzy bcc-filters, picture fuzzy bcc-filters is a generalization of picture fuzzy comparative bcc-filters, picture fuzzy bcc-filters is a generalization of picture fuzzy shift bcc-filters, picture fuzzy bcc-filters is a generalization of picture fuzzy bcc-ideals, picture fuzzy bcc-ideals is a generalization of picture fuzzy implicative bcc-filters, and picture fuzzy implicative bcc-filters, picture fuzzy comparative bcc-filters, and picture fuzzy shift bcc-filters is a generalization of picture fuzzy strong bcc-ideals. moreover, they proved that picture fuzzy strong bcc-ideals and constant pfss coincide. in this part, we define a pfs in the same way as a characteristic function and study its characterizations from the related subset. for any fixed numbers r+, r−,g+,g−,b+,b− ∈ [0,1] such that r+ > r−,g+ > g−,b+ > b− and a nonempty subset g of x, a pfs pg[r +,g+,b− r−,g−,b+ ] = (x,rgp [ r+ r− ],ggp [ g+ g− ], bgp [ b− b+ ]) in x where rgp [ r+ r− ],ggp [ g+ g− ], and bgp [ b− b+ ] are functions on x which are given as follows: rgp [ r+ r−](x)=  r + if x ∈ g, r− otherwise, 8 int. j. anal. appl. (2023), 21:75 ggp [ g+ g− ](x)=  g + if x ∈ g, g− otherwise, bgp [ b− b+](x)=  b − if x ∈ g, b+ otherwise. lemma 3.1. if the constant 0 of x is in a nonempty subset g of x, then the pfs pg[r +,g+,b− r−,g−,b+ ] in x satisfies the conditions (3.7), (3.8), and (3.9). proof. if 0∈ g, then rgp [ r+ r− ](0)= r+,ggp [ g+ g− ](0)= g+, and bgp [ b− b+ ](0)= b−. thus (∀x ∈ x)   rgp [ r+ r− ](0)= r+ ≥ rgp [ r+ r− ](x) ggp [ g+ g− ](0)= g+ ≥ ggp [ g+ g− ](x) bgp [ b− b+ ](0)= b− ≤ bgp [ b− b+ ](x)   . hence, pg[r +,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). � lemma 3.2. if the pfs pg[r +,g+,b− r−,g−,b+ ] in x satisfies the condition (3.7) (resp., (3.8), (3.9)), then the constant 0 of x is in a nonempty subset g of x. proof. assume that the pfs pg[r +,g+,b− r−,g−,b+ ] in x satisfies the condition (3.7). then rgp [ r+ r− ](0) ≥ rgp [ r+ r− ](x) for all x ∈ x. since g is nonempty, there exists g ∈ g. thus rgp [ r+ r− ](g) = r+ and so rgp [ r+ r− ](0)≥ rgp [ r+ r− ](g)= r+ ≥ rgp [ r+ r− ](0), that is, rgp [ r+ r− ](0)= r+. hence, 0∈ g. � theorem 3.1. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy bcc-subalgebra of x if and only if a nonempty subset g of x is a bcc-subalgebra of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-subalgebra of x. let x,y ∈ g. then rgp [ r+ r− ](x)= r+ = rgp [ r+ r− ](y). by (3.1), we have rgp [ r+ r−](x ·y)≥min{r g p [ r+ r−](x), r g p [ r+ r−](y)}=min{r +, r+}= r+ ≥ rgp [ r+ r−](x ·y) and so rgp [ r+ r− ](x ·y)= r+. thus x ·y ∈ g. hence, g is a bcc-subalgebra of x. conversely, assume that g is a bcc-subalgebra of x. let x,y ∈ x. case 1: x,y ∈ g. then rgp [ r+ r−](x)= r + = rgp [ r+ r−](y), ggp [ g+ g− ](x)= g+ = ggp [ g+ g− ](y), bgp [ b− b+](x)= b − = bgp [ b− b+](y). int. j. anal. appl. (2023), 21:75 9 thus min{rgp [ r+ r−](x), r g p [ r+ r−](y)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x),ggp [ g+ g− ](y)}=min{g+,g+}= g+, max{bgp [ b− b+](x),b g p [ b− b+](y)}=max{b −,b−}= b−. since g is a bcc-subalgebra of x, we have x ·y ∈ g and so rgp [ r+ r− ](x ·y)= r+,ggp [ g+ g− ](x · y)= g+, and bgp [ b− b+ ](x ·y)= b−. hence, rgp [ r+ r−](x ·y)= r + ≥ r+ =min{rgp [ r+ r−](x), r g p [ r+ r−](y)}, ggp [ g+ g− ](x ·y)= g+ ≥ g+ =min{ggp [ g+ g− ](x),ggp [ g+ g− ](y)}, bgp [ b− b+](x ·y)= b − ≤ b− =max{bgp [ b− b+](x),b g p [ b− b+](y)}. case 2: x 6∈ g or y 6∈ g. then rgp [ r+ r−](x)= r − or rgp [ r+ r−](y)= r −, ggp [ g+ g− ](x)= g− or ggp [ g+ g− ](y)= g−, bgp [ b− b+](x)= b + or bgp [ b− b+](y)= b +. thus min{rgp [ r+ r−](x), r g p [ r+ r−](y)}= r −, min{ggp [ g+ g− ](x),ggp [ g+ g− ](y)}= g−, max{bgp [ b− b+](x),b g p [ b− b+](y)}= b +. therefore, rgp [ r+ r−](x ·y)≥ r − =min{rgp [ r+ r−](x), r g p [ r+ r−](y)}, ggp [ g+ g− ](x ·y)≥ g− =min{ggp [ g+ g− ](x),ggp [ g+ g− ](y)}, bgp [ b− b+](x ·y)≤ b + =max{bgp [ b− b+](x),b g p [ b− b+](y)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-subalgebra of x. � theorem 3.2. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy near bcc-filter of x if and only if a nonempty subset g of x is a near bcc-filter of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is picture fuzzy near bcc-filter of x. let x ∈ x and y ∈ g. then rgp [ r+ r− ](y)= r+. by (3.4), we have rgp [ r+ r−](x ·y)≥ r g p [ r+ r−](y)= r + ≥ rgp [ r+ r−](x ·y) and so rgp [ r+ r− ](x ·y)= r+. thus x ·y ∈ g. hence, g is a near bcc-filter of x. 10 int. j. anal. appl. (2023), 21:75 conversely, assume that g is a near bcc-filter of x. let x,y ∈ x. case 1: y ∈ g. then rgp [ r+ r− ](y) = r+,ggp [ g+ g− ](y) = g+, and bgp [ b− b+ ](y) = b−. since g is a near bcc-filter of x, we have x ·y ∈ g and so rgp [ r+ r− ](x ·y)= r+,ggp [ g+ g− ](x ·y)= g+, and bgp [ b− b+ ](x ·y)= b−. thus rgp [ r+ r−](x ·y)= r + ≥ r+ = rgp [ r+ r−](y), ggp [ g+ g− ](x ·y)= g+ ≥ g+ = ggp [ g+ g− ](y), bgp [ b− b+](x ·y)= b − ≤ b− = bgp [ b− b+](y). case 2: y 6∈ g. then rgp [ r+ r− ](y)= r−,ggp [ g+ g− ](y)= g−, and bgp [ b− b+ ](y)= b+. thus rgp [ r+ r−](x ·y)≥ r − = rgp [ r+ r−](y), ggp [ g+ g− ](x ·y)≥ g− = ggp [ g+ g− ](y), bgp [ b− b+](x ·y)≤ b + = bgp [ b− b+](y). hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy near bcc-filter of x. � theorem 3.3. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy bcc-filter of x if and only if a nonempty subset g of x is a bcc-filter of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-filter of x. since pg[r +,g+,b− r−,g−,b+ ] satisfies the condition (3.7), it follows from lemma 3.2 that 0∈ g. next, let x,y ∈ x be such that x ·y ∈ g and x ∈ g. then rgp [ r+ r− ](x ·y)= r+ = rgp [ r+ r− ](x). by (3.10), we have rgp [ r+ r−](y)≥min{r g p [ r+ r−](x ·y), r g p [ r+ r−](x)}=min{r +, r+}= r+ ≥ rgp [ r+ r−](y) and so rgp [ r+ r− ](y)= r+. thus y ∈ g. hence, g is a bcc-filter of x. conversely, assume that g is a bcc-filter of x. since 0 ∈ g, it follows from lemma 3.1 that pg[ r+,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). next, let x,y ∈ x. case 1: x ·y ∈ g and x ∈ g. then rgp [ r+ r−](x ·y)= r + = rgp [ r+ r−](x), ggp [ g+ g− ](x ·y)= g+ = ggp [ g+ g− ](x), bgp [ b− b+](x ·y)= b − = bgp [ b− b+](x). thus min{rgp [ r+ r−](x ·y), r g p [ r+ r−](x)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x ·y),ggp [ g+ g− ](x)}=min{g+,g+}= g+, max{bgp [ b− b+](x ·y),b g p [ b− b+](x)}=max{b −,b−}= b−. int. j. anal. appl. (2023), 21:75 11 since g is a bcc-filter of x, we have y ∈ g and so rgp [ r+ r− ](y)= r+,ggp [ g+ g− ](y)= g+, and bgp [ b− b+ ](y)= b−. thus rgp [ r+ r−](y)= r + ≥ r+ =min{rgp [ r+ r−](x ·y), r g p [ r+ r−](x)}, ggp [ g+ g− ](y)= g+ ≥ g+ =min{ggp [ g+ g− ](x ·y),ggp [ g+ g− ](x)}, bgp [ b− b+](y)= b − ≤ b− =max{bgp [ b− b+](x ·y),b g p [ b− b+](x)}. case 2: x ·y 6∈ g or x 6∈ g. then rgp [ r+ r−](x ·y)= r − or rgp [ r+ r−](x)= r −, ggp [ g+ g− ](x ·y)= g− or ggp [ g+ g− ](x)= g−, bgp [ b− b+](x ·y)= b + or bgp [ b− b+](x)= b +. thus min{rgp [ r+ r−](x ·y), r g p [ r+ r−](x)}= r −, min{ggp [ g+ g− ](x ·y),ggp [ g+ g− ](x)}= g−, max{bgp [ b− b+](x ·y),b g p [ b− b+](x)}= b +. therefore, rgp [ r+ r−](y)≥ r − =min{rgp [ r+ r−](x ·y), r g p [ r+ r−](x)}, ggp [ g+ g− ](y)≥ g− =min{ggp [ g+ g− ](x ·y),ggp [ g+ g− ](x)}, bgp [ b− b+](y)≤ b + =max{bgp [ b− b+](x ·y),b g p [ b− b+](x)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-filter of x. � theorem 3.4. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy implicative bcc-filter of x if and only if a nonempty subset g of x is an implicative bcc-filter of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy implicative bcc-filter of x. since pg[r +,g+,b− r−,g−,b+ ] satisfies the condition (3.7), it follows from lemma 3.2 that 0 ∈ g. next, let x,y,z ∈ x be such that x · (y ·z)∈ g and x ·y ∈ g. then rgp [ r+ r− ](x · (y ·z))= r+ = rgp [ r+ r− ](x ·y). by (3.13), we have rgp [ r+ r−](x ·z)≥min{r g p [ r+ r−](x · (y ·z)), r g p [ r+ r−](x ·y)} =min{r+, r+}= r+ ≥ rgp [ r+ r−](x ·z) and so rgp [ r+ r− ](x ·z)= r+. thus x ·z ∈ g. hence, g is an implicative bcc-filter of x. conversely, assume that g is an implicative bcc-filter of x. since 0∈ g, it follows from lemma 3.1 that pg[r +,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). next, let x,y,z ∈ x. 12 int. j. anal. appl. (2023), 21:75 case 1: x · (y ·z)∈ g and x ·y ∈ g. then rgp [ r+ r−](x · (y ·z))= r + = rgp [ r+ r−](x ·y), ggp [ g+ g− ](x · (y ·z))= g+ = ggp [ g+ g− ](x ·y), bgp [ b− b+](x · (y ·z))= b − = bgp [ b− b+](x ·y). thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x ·y)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x ·y)}=min{g+,g+}= g+, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x ·y)}=max{b −,b−}= b−. since g is an implicative bcc-filter of x, we have x·z ∈ g and so rgp [ r+ r− ](x·z)= r+,ggp [ g+ g− ](x·z)= g+, and bgp [ b− b+ ](x ·z)= b−. thus rgp [ r+ r−](x ·z)= r + ≥ r+ =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x ·y)}, ggp [ g+ g− ](x ·z)= g+ ≥ g+ =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x ·y)}, bgp [ b− b+](x ·z)= b − ≤ b− =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x ·y)}. case 2: x · (y ·z) 6∈ g or x ·y 6∈ g. then rgp [ r+ r−](x · (y ·z))= r − or rgp [ r+ r−](x ·y)= r −, ggp [ g+ g− ](x · (y ·z))= g− or ggp [ g+ g− ](x ·y)= g−, bgp [ b− b+](x · (y ·z))= b + or bgp [ b− b+](x ·y)= b +. thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x ·y)}= r −, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x ·y)}= g−, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x ·y)}= b +. therefore, rgp [ r+ r−](x ·z)≥ r − =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x ·y)}, ggp [ g+ g− ](x ·z)≥ g− =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x ·y)}, bgp [ b− b+](x ·z)≤ b + =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x ·y)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy implicative bcc-filter of x. � theorem 3.5. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy comparative bcc-filter of x if and only if a nonempty subset g of x is a comparative bcc-filter of x. int. j. anal. appl. (2023), 21:75 13 proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy comparative bcc-filter of x. since pg[r +,g+,b− r−,g−,b+ ] satisfies the condition (3.7), it follows from lemma 3.2 that 0 ∈ g. next, let x,y,z ∈ x be such that x · ((y · z) · y) ∈ g and x ∈ g. then rgp [ r+ r− ](x · ((y · z) · y)) = r+ = rgp [ r+ r− ](x). by (3.16), we have rgp [ r+ r−](y)≥min{r g p [ r+ r−](x · ((y ·z) ·y)), r g p [ r+ r−](x)}=min{r +, r+}= r+ ≥ rgp [ r+ r−](y) and so rgp [ r+ r− ](y)= r+. thus y ∈ g. hence, g is a comparative bcc-filter of x. conversely, assume that g is a comparative bcc-filter of x. since 0∈ g, it follows from lemma 3.1 that pg[r +,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). next, let x,y,z ∈ x. case 1: x · ((y ·z) ·y)∈ g and x ∈ g. then rgp [ r+ r−](x · ((y ·z) ·y))= r + = rgp [ r+ r−](x), ggp [ g+ g− ](x · ((y ·z) ·y))= g+ = ggp [ g+ g− ](x), bgp [ b− b+](x · ((y ·z) ·y))= b − = bgp [ b− b+](x). thus min{rgp [ r+ r−](x · ((y ·z) ·y)), r g p [ r+ r−](x)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x · ((y ·z) ·y)),ggp [ g+ g− ](x)}=min{g+,g+}= g+, max{bgp [ b− b+](x · ((y ·z) ·y)),b g p [ b− b+](x)}=max{b −,b−}= b−. since g is a comparative bcc-filter of x, we have y ∈ g and so rgp [ r+ r− ](y) = r+,ggp [ g+ g− ](y) = g+, and bgp [ b− b+ ](y)= b−. thus rgp [ r+ r−](y)= r + ≥ r+ =min{rgp [ r+ r−](x · ((y ·z) ·y)), r g p [ r+ r−](x)}, ggp [ g+ g− ](y)= g+ ≥ g+ =min{ggp [ g+ g− ](x · ((y ·z) ·y)),ggp [ g+ g− ](x)}, bgp [ b− b+](y)= b − ≤ b− =max{bgp [ b− b+](x · ((y ·z) ·y)),b g p [ b− b+](x)}. case 2: x · ((y ·z) ·y) 6∈ g or x 6∈ g. then rgp [ r+ r−](x · ((y ·z) ·y))= r − or rgp [ r+ r−](x)= r −, ggp [ g+ g− ](x · ((y ·z) ·y))= g− or ggp [ g+ g− ](x)= g−, bgp [ b− b+](x · ((y ·z) ·y))= b + or bgp [ b− b+](x)= b +. thus min{rgp [ r+ r−](x · ((y ·z) ·y)), r g p [ r+ r−](x)}= r −, min{ggp [ g+ g− ](x · ((y ·z) ·y)),ggp [ g+ g− ](x)}= g−, max{bgp [ b− b+](x · ((y ·z) ·y)),b g p [ b− b+](x)}= b +. 14 int. j. anal. appl. (2023), 21:75 therefore, rgp [ r+ r−](y)≥ r − =min{rgp [ r+ r−](x · ((y ·z) ·y)), r g p [ r+ r−](x)}, ggp [ g+ g− ](y)≥ g− =min{ggp [ g+ g− ](x · ((y ·z) ·y)),ggp [ g+ g− ](x)}, bgp [ b− b+](y)≤ b + =max{bgp [ b− b+](x · ((y ·z) ·y)),b g p [ b− b+](x)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy comparative bcc-filter of x. � theorem 3.6. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy shift bcc-filter of x if and only if a nonempty subset g of x is a shift bcc-filter of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy shift bcc-filter of x. since pg[r +,g+,b− r−,g−,b+ ] satisfies the condition (3.7), it follows from lemma 3.2 that 0 ∈ g. next, let x,y,z ∈ x be such that x · (y ·z)∈ g and x ∈ g. then rgp [ r+ r− ](x · (y ·z))= r+ = rgp [ r+ r− ](x). by (3.19), we have rgp [ r+ r−](((z ·y) ·y) ·z)≥min{r g p [ r+ r−](x · (y ·z)), r g p [ r+ r−](x)} =min{r+, r+}= r+ ≥ rgp [ r+ r−](((z ·y) ·y) ·z) and so rgp [ r+ r− ](((z ·y) ·y) ·z)= r+. thus ((z ·y) ·y) ·z ∈ g. hence, g is a shift bcc-filter of x. conversely, assume that g is a shift bcc-filter of x. since 0∈ g, it follows from lemma 3.1 that pg[ r+,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). next, let x,y,z ∈ x. case 1: x · (y ·z)∈ g and x ∈ g. then rgp [ r+ r−](x · (y ·z))= r + = rgp [ r+ r−](x), ggp [ g+ g− ](x · (y ·z))= g+ = ggp [ g+ g− ](x), bgp [ b− b+](x · (y ·z))= b − = bgp [ b− b+](x). thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x)}=min{g+,g+}= g+, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x)}=max{b −,b−}= b−. since g is a shift bcc-filter of x, we have ((z · y) · y) · z ∈ g and so rgp [ r+ r− ](((z · y) · y) · z) = r+,ggp [ g+ g− ](((z ·y) ·y) ·z)= g+, and bgp [ b− b+ ](((z ·y) ·y) ·z)= b−. thus rgp [ r+ r−](((z ·y) ·y) ·z)= r + ≥ r+ =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x)}, ggp [ g+ g− ](((z ·y) ·y) ·z)= g+ ≥ g+ =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x)}, bgp [ b− b+](((z ·y) ·y) ·z)= b − ≤ b− =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x)}. int. j. anal. appl. (2023), 21:75 15 case 2: x · (y ·z) 6∈ g or x 6∈ g. then rgp [ r+ r−](x · (y ·z))= r − or rgp [ r+ r−](x)= r −, ggp [ g+ g− ](x · (y ·z))= g− or ggp [ g+ g− ](x)= g−, bgp [ b− b+](x · (y ·z))= b + or bgp [ b− b+](x)= b +. thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x)}= r −, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x)}= g−, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x)}= b +. therefore, rgp [ r+ r−](((z ·y) ·y) ·z)≥ r − =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](x)}, ggp [ g+ g− ](((z ·y) ·y) ·z)≥ g− =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](x)}, bgp [ b− b+](((z ·y) ·y) ·z)≤ b + =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](x)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy shift bcc-filter of x. � theorem 3.7. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy bcc-ideal of x if and only if a nonempty subset g of x is a bcc-ideal of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-ideal of x. since pg[r +,g+,b− r−,g−,b+ ] satisfies the condition (3.7), it follows from lemma 3.2 that 0∈ g. next, let x,y,z ∈ x be such that x ·(y ·z)∈ g and y ∈ g. then rgp [ r+ r− ](x · (y ·z))= r+ = rgp [ r+ r− ](y). by (3.22), we have rgp [ r+ r−](x ·z)≥min{r g p [ r+ r−](x · (y ·z)), r g p [ r+ r−](y)}=min{r +, r+}= r+ ≥ rgp [ r+ r−](x ·z) and so rgp [ r+ r− ](x ·z)= r+. thus x ·z ∈ g. hence, g is a bcc-ideal of x. conversely, assume that g is a bcc-ideal of x. since 0 ∈ g, it follows from lemma 3.1 that pg[ r+,g+,b− r−,g−,b+ ] satisfies the conditions (3.7), (3.8), and (3.9). next, let x,y,z ∈ x. case 1: x · (y ·z)∈ g and y ∈ g. then rgp [ r+ r−](x · (y ·z))= r + = rgp [ r+ r−](y), ggp [ g+ g− ](x · (y ·z))= g+ = ggp [ g+ g− ](y), bgp [ b− b+](x · (y ·z))= b − = bgp [ b− b+](y). 16 int. j. anal. appl. (2023), 21:75 thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](y)}=min{r +, r+}= r+, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](y)}=min{g+,g+}= g+, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](y)}=max{b −,b−}= b−. since g is a bcc-ideal of x, we have x · z ∈ g and so rgp [ r+ r− ](x · z) = r+,ggp [ g+ g− ](x · z) = g+, and bgp [ b− b+ ](x ·z)= b−. thus rgp [ r+ r−](x ·z)= r + ≥ r+ =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](y)}, ggp [ g+ g− ](x ·z)= g+ ≥ g+ =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](y)}, bgp [ b− b+](x ·z)= b − ≤ b− =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](y)}. case 2: x · (y ·z) 6∈ g or y 6∈ g. then rgp [ r+ r−](x · (y ·z))= r − or rgp [ r+ r−](y)= r −, ggp [ g+ g− ](x · (y ·z))= g− or ggp [ g+ g− ](y)= g−, bgp [ b− b+](x · (y ·z))= b + or bgp [ b− b+](y)= b +. thus min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](y)}= r −, min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](y)}= g−, max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](y)}= b +. therefore, rgp [ r+ r−](x ·z)≥ r − =min{rgp [ r+ r−](x · (y ·z)), r g p [ r+ r−](y)}, ggp [ g+ g− ](x ·z)≥ g− =min{ggp [ g+ g− ](x · (y ·z)),ggp [ g+ g− ](y)}, bgp [ b− b+](x ·z)≤ b + =max{bgp [ b− b+](x · (y ·z)),b g p [ b− b+](y)}. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy bcc-ideal of x. � theorem 3.8. the pfs pg[r +,g+,b− r−,g−,b+ ] in x is a picture fuzzy strong bcc-ideal of x if and only if a nonempty subset g of x is a strong bcc-ideal of x. proof. assume that pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy strong bcc-ideal of x. then pg[r +,g+,b− r−,g−,b+ ] is constant, that is, rgp [ r+ r− ] is constant. since g is nonempty, we have rgp [ r+ r− ](x) = r+ for all x ∈ x. thus g = x. hence, g is a strong bcc-ideal of x. int. j. anal. appl. (2023), 21:75 17 conversely, assume that g is a strong bcc-ideal of x. then g = x, so (∀x ∈ x)   rgp [ r+ r− ](x)= r+ ggp [ g+ g− ](x)= g+ bgp [ b− b+ ](x)= b−   . thus rgp [ r+ r− ],ggp [ g+ g− ], and bgp [ b− b+ ] are constant, that is, pg[r +,g+,b− r−,g−,b+ ] is constant. hence, pg[r +,g+,b− r−,g−,b+ ] is a picture fuzzy strong bcc-ideal of x. � 4. level subsets of a pfs in this section, we discuss the relationships between picture fuzzy bcc-subalgebras (resp., picture fuzzy near bcc-filters, picture fuzzy bcc-filters, picture fuzzy implicative bcc-filters, picture fuzzy comparative bcc-filters, picture fuzzy shift bcc-filters, picture fuzzy bcc-ideals, and picture fuzzy strong bcc-ideals) of bcc-algebras and their level subsets. definition 4.1. [34] let f be a fuzzy set in x. for any t ∈ [0,1], the sets u(f ;t)= {x ∈ x | f (x)≥ t}, u+(f ;t)= {x ∈ x | f (x) > t}, l(f ;t)= {x ∈ x | f (x)≤ t}, l−(f ;t)= {x ∈ x | f (x) < t}, e(f ;t)= {x ∈ x | f (x)= t} are called an upper t-level subset, an upper t-strong level subset, a lower t-level subset, a lower t-strong level subset and an equal t-level subset of f , respectively. theorem 4.1. a pfs p in x is a picture fuzzy bcc-subalgebra of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-subalgebras of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-subalgebra of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x,y ∈ u(rp ;t). then rp(x) ≥ t and rp(y) ≥ t, so t is a lower bound of {rp(x), rp(y)}. by (3.1), we have rp(x ·y)≥min{rp(x), rp(y)}≥ t. thus x ·y ∈ u(rp ;t). let x,y ∈ u(gp ;t). then gp(x)≥ t and gp(y)≥ t, so t is a lower bound of {gp(x),gp(y)}. by (3.2), we have gp(x ·y)≥min{gp(x),gp(y)}≥ t. thus x ·y ∈ u(gp ;t). let x,y ∈ l(bp ;t). then bp(x) ≤ t and bp(y) ≤ t, so t is an upper bound of {bp(x),bp(y)}. by (3.3), we have bp(x ·y)≤max{bp(x),bp(y)}≤ t. thus x ·y ∈ l(bp ;t). hence, u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-subalgebras of x. 18 int. j. anal. appl. (2023), 21:75 conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bccsubalgebras of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x,y ∈ x. then rp(x), rp(y) ∈ [0,1]. choose t = min{rp(x), rp(y)}. thus rp(x) ≥ t and rp(y)≥ t, so x,y ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a bcc-subalgebra of x and so x ·y ∈ u(rp ;t). thus rp(x ·y)≥ t =min{rp(x), rp(y)}. let x,y ∈ x. then gp(x),gp(y) ∈ [0,1]. choose t = min{gp(x),gp(y)}. thus gp(x) ≥ t and gp(y) ≥ t, so x,y ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a bcc-subalgebra of x and so x ·y ∈ u(gp ;t). thus gp(x ·y)≥ t =min{gp(x),gp(y)}. let x,y ∈ x. then bp(x),bp(y) ∈ [0,1]. choose t = max{bp(x),bp(y)}. thus bp(x) ≤ t and bp(y) ≤ t, so x,y ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a bcc-subalgebra of x and so x ·y ∈ l(bp ;t). thus bp(x ·y)≤ t =max{bp(x),bp(y)}. therefore, p is a picture fuzzy bcc-subalgebra of x. � theorem 4.2. if p is a picture fuzzy bcc-subalgebra of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are bcc-subalgebras of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-subalgebra of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x,y ∈ u+(rp ;t). then rp(x) > t and rp(y) > t, so t is a lower bound of {rp(x), rp(y)}. by (3.1), we have rp(x ·y)≥min{rp(x), rp(y)} > t. thus x ·y ∈ u+(rp ;t). let x,y ∈ u+(gp ;t). then gp(x) > t and gp(y) > t, so t is a lower bound of {gp(x),gp(y)}. by (3.2), we have gp(x ·y)≥min{gp(x),gp(y)} > t. thus x ·y ∈ u+(gp ;t). let x,y ∈ l−(bp ;t). then bp(x) < t and bp(y) < t, so t is an upper bound of {bp(x),bp(y)}. by (3.3), we have bp(x ·y)≤max{bp(x),bp(y)} < t. thus x ·y ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are bcc-subalgebras of x. � theorem 4.3. a pfs p in x is a picture fuzzy near bcc-filter of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are near bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy near bcc-filter of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x and y ∈ u(rp ;t). then rp(y) ≥ t. by (3.4), we have rp(x · y) ≥ rp(y) ≥ t. thus x ·y ∈ u(rp ;t). let x ∈ x and y ∈ u(gp ;t). then gp(y) ≥ t. by (3.5), we have gp(x · y) ≥ gp(y) ≥ t. thus x ·y ∈ u(gp ;t). let x ∈ x and y ∈ l(bp ;t). then bp(y) ≤ t. by (3.6), we have bp(x · y) ≤ bp(y) ≤ t. thus x ·y ∈ l(bp ;t). int. j. anal. appl. (2023), 21:75 19 hence, u(rp ;t),u(gp ;t), and l(bp ;t) are near bcc-filters of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are near bccfilters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x,y ∈ x. then rp(y) ∈ [0,1]. choose t = rp(y). thus rp(y) ≥ t, so y ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a near bcc-filter of x and so x ·y ∈ u(rp ;t). thus rp(x ·y)≥ t = rp(y). let x,y ∈ x. then gp(y)∈ [0,1]. choose t = gp(y). thus gp(y)≥ t, so y ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a near bcc-filter of x and so x ·y ∈ u(gp ;t). thus gp(x ·y)≥ t = gp(y). let x,y ∈ x. then bp(y)∈ [0,1]. choose t = bp(y). thus bp(y)≤ t, so y ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a near bcc-filter of x and so x ·y ∈ l(bp ;t). thus bp(x ·y)≤ t = bp(y). therefore, p is a picture fuzzy near bcc-filter of x. � theorem 4.4. if p is a picture fuzzy near bcc-filter of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are near bcc-filters of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy near bcc-filter of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ x and y ∈ u+(rp ;t). then rp(y) > t. by (3.4), we have rp(x · y) ≥ rp(y) > t. thus x ·y ∈ u+(rp ;t). let x ∈ x and y ∈ u+(gp ;t). then gp(y) > t. by (3.5), we have gp(x · y) ≥ gp(y) > t. thus x ·y ∈ u+(gp ;t). let x ∈ x and y ∈ l−(bp ;t). then bp(y) < t. by (3.6), we have bp(x · y) ≤ bp(y) < t. thus x ·y ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are near bcc-filters of x. � theorem 4.5. a pfs p in x is a picture fuzzy bcc-filter of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-filter of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ u(rp ;t). then rp(x) ≥ t. by (3.7), we have rp(0) ≥ rp(x) ≥ t. thus 0 ∈ u(rp ;t). next, let x,y ∈ x be such that x ·y ∈ u(rp ;t) and x ∈ u(rp ;t). then rp(x ·y)≥ t and rp(x)≥ t, so t is a lower bound of {rp(x · y), rp(x)}. by (3.10), we have rp(y) ≥ min{rp(x · y), rp(x)} ≥ t. thus y ∈ u(rp ;t). let x ∈ u(gp ;t). then gp(x) ≥ t. by (3.8), we have gp(0) ≥ gp(x) ≥ t. thus 0 ∈ u(gp ;t). next, let x,y ∈ x be such that x ·y ∈ u(gp ;t) and x ∈ u(gp ;t). then gp(x ·y)≥ t and gp(x)≥ t, 20 int. j. anal. appl. (2023), 21:75 so t is a lower bound of {gp(x ·y),gp(x)}. by (3.11), we have gp(y)≥min{gp(x ·y),gp(x)}≥ t. thus y ∈ u(gp ;t). let x ∈ l(bp ;t). then bp(x) ≤ t. by (3.9), we have bp(0) ≤ bp(x) ≤ t. thus 0 ∈ l(bp ;t). next, let x,y ∈ x be such that x ·y ∈ l(bp ;t) and x ∈ l(bp ;t). then bp(x ·y)≤ t and bp(x)≤ t, so t is an upper bound of {bp(x ·y),bp(x)}. by (3.12), we have bp(y)≤max{bp(x ·y),bp(x)}≤ t. thus y ∈ l(bp ;t). hence, u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-filters of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x. then rp(x) ∈ [0,1]. choose t = rp(x). thus rp(x) ≥ t, so x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a bcc-filter of x and so 0 ∈ u(rp ;t). thus rp(0) ≥ t = rp(x). next, let x,y ∈ x. then rp(x ·y), rp(x)∈ [0,1]. choose t =min{rp(x ·y), rp(x)}. thus rp(x ·y)≥ t and rp(x)≥ t, so x · y,x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a bcc-filter of x and so y ∈ u(rp ;t). thus rp(y)≥ t =min{rp(x ·y), rp(x)}. let x ∈ x. then gp(x) ∈ [0,1]. choose t = gp(x). thus gp(x) ≥ t, so x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a bcc-filter of x and so 0 ∈ u(gp ;t). thus gp(0) ≥ t = gp(x). next, let x,y ∈ x. then gp(x · y),gp(x) ∈ [0,1]. choose t = min{gp(x · y),gp(x)}. thus gp(x · y) ≥ t and gp(x) ≥ t, so x · y,x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a bcc-filter of x and so y ∈ u(gp ;t). thus gp(y)≥ t =min{gp(x ·y),gp(x)}. let x ∈ x. then bp(x) ∈ [0,1]. choose t = bp(x). thus bp(x) ≤ t, so x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a bcc-filter of x and so 0 ∈ l(bp ;t). thus bp(0) ≤ t = bp(x). next, let x,y ∈ x. then bp(x · y),bp(x) ∈ [0,1]. choose t = max{bp(x · y),bp(x)}. thus bp(x · y) ≤ t and bp(x) ≤ t, so x · y,x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a bcc-filter of x and so y ∈ l(bp ;t). thus bp(y)≤ t =max{bp(x ·y),bp(x)}. therefore, p is a picture fuzzy bcc-filter of x. � theorem 4.6. if p is a picture fuzzy bcc-filter of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are bcc-filters of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-filter of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ u+(rp ;t). then rp(x) > t. by (3.7), we have rp(0) ≥ rp(x) > t. thus 0 ∈ u+(rp ;t). next, let x,y ∈ x be such that x ·y ∈ u+(rp ;t) and x ∈ u+(rp ;t). then rp(x ·y) > t and rp(x) > t, so t is a lower bound of {rp(x · y), rp(x)}. by (3.10), we have rp(y) ≥ min{rp(x · y), rp(x)} > t. thus y ∈ u+(rp ;t). let x ∈ u+(gp ;t). then gp(x) > t. by (3.8), we have gp(0)≥ gp(x) > t. thus 0∈ u+(gp ;t). next, let x,y ∈ x be such that x·y ∈ u+(gp ;t) and x ∈ u+(gp ;t). then gp(x·y) > t and gp(x) > t, int. j. anal. appl. (2023), 21:75 21 so t is a lower bound of {gp(x ·y),gp(x)}. by (3.11), we have gp(y)≥min{gp(x ·y),gp(x)} > t. thus y ∈ u+(gp ;t). let x ∈ l−(bp ;t). then bp(x) < t. by (3.9), we have bp(0)≤ bp(x) < t. thus 0∈ l−(bp ;t). next, let x,y ∈ x be such that x ·y ∈ l−(bp ;t) and x ∈ l−(bp ;t). then bp(x ·y) < t and bp(x) < t, so t is an upper bound of {bp(x ·y),bp(x)}. by (3.12), we have bp(y)≤max{bp(x ·y),bp(x)} < t. thus y ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are bcc-filters of x. � theorem 4.7. a pfs p in x is a picture fuzzy implicative bcc-filter of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are implicative bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy implicative bcc-filter of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ u(rp ;t). then rp(x) ≥ t. by (3.7), we have rp(0) ≥ rp(x) ≥ t. thus 0 ∈ u(rp ;t). next, let x,y,z ∈ x be such that x · (y · z)∈ u(rp ;t) and x ·y ∈ u(rp ;t). then rp(x · (y · z))≥ t and rp(x ·y)≥ t, so t is a lower bound of {rp(x · (y ·z)), rp(x ·y)}. by (3.13), we have rp(x ·z)≥ min{rp(x · (y ·z)), rp(x ·y)}≥ t. thus x ·z ∈ u(rp ;t). let x ∈ u(gp ;t). then gp(x) ≥ t. by (3.8), we have gp(0) ≥ gp(x) ≥ t. thus 0 ∈ u(gp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u(gp ;t) and x ·y ∈ u(gp ;t). then gp(x ·(y ·z))≥ t and gp(x ·y)≥ t, so t is a lower bound of {gp(x ·(y ·z)),gp(x ·y)}. by (3.14), we have gp(x ·z)≥ min{gp(x · (y ·z)),gp(x ·y)}≥ t. thus x ·z ∈ u(gp ;t). let x ∈ l(bp ;t). then bp(x) ≤ t. by (3.9), we have bp(0) ≤ bp(x) ≤ t. thus 0 ∈ l(bp ;t). next, let x,y,z ∈ x be such that x · (y ·z)∈ l(bp ;t) and x ·y ∈ l(bp ;t). then bp(x · (y ·z))≤ t and bp(x · y) ≤ t, so t is an upper bound of {bp(x · (y · z)),bp(x · y)}. by (3.15), we have bp(x ·z)≤max{bp(x · (y ·z)),bp(x ·y)}≤ t. thus x ·z ∈ l(bp ;t). hence, u(rp ;t),u(gp ;t), and l(bp ;t) are implicative bcc-filters of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are implicative bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x. then rp(x) ∈ [0,1]. choose t = rp(x). thus rp(x) ≥ t, so x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is an implicative bcc-filter of x and so 0 ∈ u(rp ;t). thus rp(0) ≥ t = rp(x). next, let x,y,z ∈ x. then rp(x · (y · z)), rp(x · y) ∈ [0,1]. choose t = min{rp(x ·(y ·z)), rp(x ·y)}. thus rp(x ·(y ·z))≥ t and rp(x ·y)≥ t, so x ·(y ·z),x ·y ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is an implicative bcc-filter of x and so x · z ∈ u(rp ;t). thus rp(x ·z)≥ t =min{rp(x · (y ·z)), rp(x ·y)}. let x ∈ x. then gp(x) ∈ [0,1]. choose t = gp(x). thus gp(x) ≥ t, so x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is an implicative bcc-filter of x and so 0 ∈ u(gp ;t). thus 22 int. j. anal. appl. (2023), 21:75 gp(0) ≥ t = gp(x). next, let x,y,z ∈ x. then gp(x · (y · z)),gp(x · y) ∈ [0,1]. choose t = min{gp(x ·(y ·z)),gp(x ·y)}. thus gp(x ·(y ·z))≥ t and gp(x ·y)≥ t, so x ·(y ·z),x ·y ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is an implicative bcc-filter of x and so x · z ∈ u(gp ;t). thus gp(x ·z)≥ t =min{gp(x · (y ·z)),gp(x ·y)}. let x ∈ x. then bp(x) ∈ [0,1]. choose t = bp(x). thus bp(x) ≤ t, so x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is an implicative bcc-filter of x and so 0 ∈ l(bp ;t). thus bp(0) ≤ t = bp(x). next, let x,y,z ∈ x. then bp(x · (y · z)),bp(x · y) ∈ [0,1]. choose t = max{bp(x · (y · z)),bp(x · y)}. thus bp(x · (y · z)) ≤ t and bp(x · y) ≤ t, so x · (y · z),x · y ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is an implicative bcc-filter of x and so x·z ∈ l(bp ;t). thus bp(x ·z)≤ t =max{bp(x · (y ·z)),bp(x ·y)}. therefore, p is a picture fuzzy implicative bcc-filter of x. � theorem 4.8. if p is a picture fuzzy implicative bcc-filter of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are implicative bcc-filters of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy implicative bcc-filter of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ u+(rp ;t). then rp(x) > t. by (3.7), we have rp(0) ≥ rp(x) > t. thus 0 ∈ u+(rp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u+(rp ;t) and x ·y ∈ u+(rp ;t). then rp(x ·(y ·z)) > t and rp(x ·y) > t, so t is a lower bound of {rp(x · (y ·z)), rp(x ·y)}. by (3.13), we have rp(x ·z)≥ min{rp(x · (y ·z)), rp(x ·y)} > t. thus x ·z ∈ u+(rp ;t). let x ∈ u+(gp ;t). then gp(x) > t. by (3.8), we have gp(0)≥ gp(x) > t. thus 0∈ u+(gp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u+(gp ;t) and x ·y ∈ u+(gp ;t). then gp(x ·(y ·z)) > t and gp(x ·y) > t, so t is a lower bound of {gp(x ·(y ·z)),gp(x ·y)}. by (3.14), we have gp(x ·z)≥ min{gp(x · (y ·z)),gp(x ·y)} > t. thus x ·z ∈ u+(gp ;t). let x ∈ l−(bp ;t). then bp(x) < t. by (3.9), we have bp(0)≤ bp(x) < t. thus 0∈ l−(bp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ l−(bp ;t) and x ·y ∈ l−(bp ;t). then bp(x ·(y ·z)) < t and bp(x · y) < t, so t is an upper bound of {bp(x · (y · z)),bp(x · y)}. by (3.15), we have bp(x ·z)≤max{bp(x · (y ·z)),bp(x ·y)} < t. thus x ·z ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are implicative bcc-filters of x. � theorem 4.9. a pfs p in x is a picture fuzzy comparative bcc-filter of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are comparative bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy comparative bcc-filter of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. int. j. anal. appl. (2023), 21:75 23 let x ∈ u(rp ;t). then rp(x)≥ t. by (3.7), we have rp(0)≥ rp(x)≥ t. thus 0∈ u(rp ;t). next, let x,y,z ∈ x be such that x · ((y ·z) ·y)∈ u(rp ;t) and x ∈ u(rp ;t). then rp(x · ((y ·z) ·y))≥ t and rp(x) ≥ t, so t is a lower bound of {rp(x · ((y · z) · y)), rp(x)}. by (3.16), we have rp(y) ≥ min{rp(x · ((y ·z) ·y)), rp(x)}≥ t. thus y ∈ u(rp ;t). let x ∈ u(gp ;t). then gp(x) ≥ t. by (3.8), we have gp(0) ≥ gp(x) ≥ t. thus 0 ∈ u(gp ;t). next, let x,y,z ∈ x be such that x·((y ·z)·y)∈ u(gp ;t) and x ∈ u(gp ;t). then gp(x·((y ·z)·y))≥ t and gp(x) ≥ t, so t is a lower bound of {gp(x · ((y · z) · y)),gp(x)}. by (3.17), we have gp(y) ≥ min{gp(x · ((y ·z) ·y)),gp(x)}≥ t. thus y ∈ u(gp ;t). let x ∈ l(bp ;t). then bp(x) ≤ t. by (3.9), we have bp(0) ≤ bp(x) ≤ t. thus 0 ∈ l(bp ;t). next, let x,y,z ∈ x be such that x·((y ·z)·y)∈ l(bp ;t) and x ∈ l(bp ;t). then bp(x·((y ·z)·y))≤ t and bp(x)≤ t, so t is an upper bound of {bp(x · ((y · z) ·y)),bp(x)}. by (3.18), we have bp(y)≤ max{bp(x · ((y ·z) ·y)),bp(x)}≤ t. thus y ∈ l(bp ;t). hence, u(rp ;t),u(gp ;t), and l(bp ;t) are comparative bcc-filters of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are comparative bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x. then rp(x) ∈ [0,1]. choose t = rp(x). thus rp(x) ≥ t, so x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a comparative bcc-filter of x and so 0 ∈ u(rp ;t). thus rp(0) ≥ t = rp(x). next, let x,y,z ∈ x. then rp(x · ((y · z) · y)), rp(x) ∈ [0,1]. choose t =min{rp(x ·((y ·z) ·y)), rp(x)}. thus rp(x ·((y ·z) ·y))≥ t and rp(x)≥ t, so x ·((y ·z) ·y),x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a comparative bcc-filter of x and so y ∈ u(rp ;t). thus rp(y)≥ t =min{rp(x · ((y ·z) ·y)), rp(x)}. let x ∈ x. then gp(x) ∈ [0,1]. choose t = gp(x). thus gp(x) ≥ t, so x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a comparative bcc-filter of x and so 0 ∈ u(gp ;t). thus gp(0) ≥ t = gp(x). next, let x,y,z ∈ x. then gp(x · ((y · z) · y)),gp(x) ∈ [0,1]. choose t =min{gp(x ·((y ·z)·y)),gp(x)}. thus gp(x ·((y ·z)·y))≥ t and gp(x)≥ t, so x ·((y ·z)·y),x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a comparative bcc-filter of x and so y ∈ u(gp ;t). thus gp(y)≥ t =min{gp(x · ((y ·z) ·y)),gp(x)}. let x ∈ x. then bp(x) ∈ [0,1]. choose t = bp(x). thus bp(x) ≤ t, so x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a comparative bcc-filter of x and so 0 ∈ l(bp ;t). thus bp(0) ≤ t = bp(x). next, let x,y,z ∈ x. then bp(x · ((y · z) · y)),bp(x) ∈ [0,1]. choose t =max{bp(x ·((y ·z)·y)),bp(x)}. thus bp(x ·((y ·z)·y))≤ t and bp(x)≤ t, so x ·((y ·z)·y),x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a comparative bcc-filter of x and so y ∈ l(bp ;t). thus bp(y)≤ t =max{bp(x · ((y ·z) ·y)),bp(x)}. therefore, p is a picture fuzzy comparative bcc-filter of x. � 24 int. j. anal. appl. (2023), 21:75 theorem 4.10. if p is a picture fuzzy comparative bcc-filter of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are comparative bcc-filters of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy comparative bcc-filter of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ u+(rp ;t). then rp(x) > t. by (3.7), we have rp(0) ≥ rp(x) > t. thus 0 ∈ u+(rp ;t). next, let x,y,z ∈ x be such that x · ((y · z) · y) ∈ u+(rp ;t) and x ∈ u+(rp ;t). then rp(x · ((y · z) ·y)) > t and rp(x) > t, so t is a lower bound of {rp(x · ((y ·z) ·y)), rp(x)}. by (3.16), we have rp(y)≥min{rp(x · ((y ·z) ·y)), rp(x)} > t. thus y ∈ u+(rp ;t). let x ∈ u+(gp ;t). then gp(x) > t. by (3.8), we have gp(0) ≥ gp(x) > t. thus 0 ∈ u+(gp ;t). next, let x,y,z ∈ x be such that x · ((y · z) · y) ∈ u+(gp ;t) and x ∈ u+(gp ;t). then gp(x · ((y · z) · y)) > t and gp(x) > t, so t is a lower bound of {gp(x · ((y · z) · y)),gp(x)}. by (3.17), we have gp(y)≥min{gp(x · ((y ·z) ·y)),gp(x)} > t. thus y ∈ u+(gp ;t). let x ∈ l−(bp ;t). then bp(x) < t. by (3.9), we have bp(0) ≤ bp(x) < t. thus 0 ∈ l−(bp ;t). next, let x,y,z ∈ x be such that x · ((y · z) · y) ∈ l−(bp ;t) and x ∈ l−(bp ;t). then bp(x · ((y · z) · y)) < t and bp(x) < t, so t is an upper bound of {bp(x · ((y · z) · y)),bp(x)}. by (3.18), we have bp(y)≤max{bp(x · ((y ·z) ·y)),bp(x)} < t. thus y ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are comparative bcc-filters of x. � theorem 4.11. a pfs p in x is a picture fuzzy shift bcc-filter of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are shift bcc-filters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy shift bcc-filter of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ u(rp ;t). then rp(x) ≥ t. by (3.7), we have rp(0) ≥ rp(x) ≥ t. thus 0 ∈ u(rp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u(rp ;t) and x ∈ u(rp ;t). then rp(x ·(y ·z))≥ t and rp(x)≥ t, so t is a lower bound of {rp(x · (y ·z)), rp(x)}. by (3.19), we have rp(((z ·y) ·y) ·z)≥ min{rp(x · (y ·z)), rp(x)}≥ t. thus ((z ·y) ·y) ·z ∈ u(rp ;t). let x ∈ u(gp ;t). then gp(x) ≥ t. by (3.8), we have gp(0) ≥ gp(x) ≥ t. thus 0 ∈ u(gp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u(gp ;t) and x ∈ u(gp ;t). then gp(x ·(y ·z))≥ t and gp(x)≥ t, so t is a lower bound of {gp(x ·(y ·z)),gp(x)}. by (3.20), we have gp(((z ·y) ·y) ·z)≥ min{gp(x · (y ·z)),gp(x)}≥ t. thus ((z ·y) ·y) ·z ∈ u(gp ;t). let x ∈ l(bp ;t). then bp(x) ≤ t. by (3.9), we have bp(0) ≤ bp(x) ≤ t. thus 0 ∈ l(bp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ l(bp ;t) and x ∈ l(bp ;t). then bp(x ·(y ·z))≤ t and bp(x)≤ t, so t is an upper bound of {bp(x ·(y ·z)),bp(x)}. by (3.21), we have bp(((z ·y)·y) ·z)≤ max{bp(x · (y ·z)),bp(x)}≤ t. thus ((z ·y) ·y) ·z ∈ l(bp ;t). int. j. anal. appl. (2023), 21:75 25 hence, u(rp ;t),u(gp ;t), and l(bp ;t) are shift bcc-filters of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are shift bccfilters of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x. then rp(x) ∈ [0,1]. choose t = rp(x). thus rp(x) ≥ t, so x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a shift bcc-filter of x and so 0∈ u(rp ;t). thus rp(0)≥ t = rp(x). next, let x,y,z ∈ x. then rp(x ·(y ·z)), rp(x)∈ [0,1]. choose t =min{rp(x ·(y ·z)), rp(x)}. thus rp(x·(y·z))≥ t and rp(x)≥ t, so x·(y·z),x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a shift bcc-filter of x and so ((z ·y)·y)·z ∈ u(rp ;t). thus rp(((z ·y)·y)·z)≥ t =min{rp(x ·(y ·z)), rp(x)}. let x ∈ x. then gp(x) ∈ [0,1]. choose t = gp(x). thus gp(x) ≥ t, so x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a shift bcc-filter of x and so 0∈ u(gp ;t). thus gp(0)≥ t = gp(x). next, let x,y,z ∈ x. then gp(x · (y · z)),gp(x) ∈ [0,1]. choose t = min{gp(x · (y · z)),gp(x)}. thus gp(x · (y · z)) ≥ t and gp(x) ≥ t, so x · (y · z),x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a shift bcc-filter of x and so ((z ·y) ·y) ·z ∈ u(gp ;t). thus gp(((z ·y) ·y) ·z)≥ t = min{gp(x · (y ·z)),gp(x)}. let x ∈ x. then bp(x) ∈ [0,1]. choose t = bp(x). thus bp(x) ≤ t, so x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a shift bcc-filter of x and so 0∈ l(bp ;t). thus bp(0)≤ t = bp(x). next, let x,y,z ∈ x. then bp(x · (y · z)),bp(x) ∈ [0,1]. choose t = max{bp(x · (y · z)),bp(x)}. thus bp(x · (y · z)) ≤ t and bp(x) ≤ t, so x · (y · z),x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a shift bcc-filter of x and so ((z ·y) ·y) · z ∈ l(bp ;t). thus bp(((z ·y) ·y) · z)≤ t = max{bp(x · (y ·z)),bp(x)}. therefore, p is a picture fuzzy shift bcc-filter of x. � theorem 4.12. if p is a picture fuzzy shift bcc-filter of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are shift bcc-filters of x if u+(rp ;t), u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy shift bcc-filter of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ u+(rp ;t). then rp(x) > t. by (3.7), we have rp(0) ≥ rp(x) > t. thus 0 ∈ u+(rp ;t). next, let x,y,z ∈ x be such that x · (y · z)∈ u+(rp ;t) and x ∈ u+(rp ;t). then rp(x · (y · z)) > t and rp(x) > t, so t is a lower bound of {rp(x ·(y ·z)), rp(x)}. by (3.19), we have rp(((z ·y)·y)·z)≥ min{rp(x · (y ·z)), rp(x)} > t. thus ((z ·y) ·y) ·z ∈ u+(rp ;t). let x ∈ u+(gp ;t). then gp(x) > t. by (3.8), we have gp(0)≥ gp(x) > t. thus 0∈ u+(gp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u+(gp ;t) and x ∈ u+(gp ;t). then gp(x ·(y ·z)) > t and gp(x) > t, so t is a lower bound of {gp(x ·(y ·z)),gp(x)}. by (3.20), we have gp(((z ·y)·y)·z)≥ min{gp(x · (y ·z)),gp(x)} > t. thus ((z ·y) ·y) ·z ∈ u+(gp ;t). let x ∈ l−(bp ;t). then bp(x) < t. by (3.9), we have bp(0)≤ bp(x) < t. thus 0∈ l−(bp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ l−(bp ;t) and x ∈ l−(bp ;t). then bp(x ·(y ·z)) < t and 26 int. j. anal. appl. (2023), 21:75 bp(x) < t, so t is an upper bound of {bp(x ·(y ·z)),bp(x)}. by (3.21), we have bp(((z ·y) ·y) ·z)≤ max{bp(x · (y ·z)),bp(x)} < t. thus ((z ·y) ·y) ·z ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are shift bcc-filters of x. � theorem 4.13. a pfs p in x is a picture fuzzy bcc-ideal of x if and only if for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-ideals of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-ideal of x. let t ∈ [0,1] be such that u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ u(rp ;t). then rp(x) ≥ t. by (3.7), we have rp(0) ≥ rp(x) ≥ t. thus 0 ∈ u(rp ;t). next, let x,y,z ∈ x be such that x · (y · z) ∈ u(rp ;t) and y ∈ u(rp ;t). then rp(x · (y · z)) ≥ t and rp(y) ≥ t, so t is a lower bound of {rp(x · (y · z)), rp(y)}. by (3.22), we have rp(x · z) ≥ min{rp(x · (y ·z)), rp(y)}≥ t. thus x ·z ∈ u(rp ;t). let x ∈ u(gp ;t). then gp(x) ≥ t. by (3.8), we have gp(0) ≥ gp(x) ≥ t. thus 0 ∈ u(gp ;t). next, let x,y,z ∈ x be such that x · (y · z) ∈ u(gp ;t) and y ∈ u(gp ;t). then gp(x · (y · z)) ≥ t and gp(y) ≥ t, so t is a lower bound of {gp(x · (y · z)),gp(y)}. by (3.23), we have gp(x · z) ≥ min{gp(x · (y ·z)),gp(y)}≥ t. thus x ·z ∈ u(gp ;t). let x ∈ l(bp ;t). then bp(x) ≤ t. by (3.9), we have bp(0) ≤ bp(x) ≤ t. thus 0 ∈ l(bp ;t). next, let x,y,z ∈ x be such that x · (y · z) ∈ l(bp ;t) and y ∈ l(bp ;t). then bp(x · (y · z)) ≤ t and bp(y) ≤ t, so t is an upper bound of {bp(x · (y · z)),bp(y)}. by (3.24), we have bp(x · z) ≤ max{bp(x · (y ·z)),bp(y)}≤ t. thus x ·z ∈ l(bp ;t). hence, u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-ideals of x. conversely, assume that for all t ∈ [0,1], the sets u(rp ;t),u(gp ;t), and l(bp ;t) are bcc-ideals of x if u(rp ;t),u(gp ;t), and l(bp ;t) are nonempty. let x ∈ x. then rp(x) ∈ [0,1]. choose t = rp(x). thus rp(x) ≥ t, so x ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a bcc-ideal of x and so 0 ∈ u(rp ;t). thus rp(0) ≥ t = rp(x). next, let x,y,z ∈ x. then rp(x ·(y ·z)), rp(y)∈ [0,1]. choose t =min{rp(x ·(y ·z)), rp(y)}. thus rp(x · (y ·z))≥ t and rp(y)≥ t, so x · (y ·z),y ∈ u(rp ;t) 6= ∅. by assumption, we have u(rp ;t) is a bcc-ideal of x and so x ·z ∈ u(rp ;t). thus rp(x ·z)≥ t =min{rp(x · (y ·z)), rp(y)}. let x ∈ x. then gp(x) ∈ [0,1]. choose t = gp(x). thus gp(x) ≥ t, so x ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a bcc-ideal of x and so 0 ∈ u(gp ;t). thus gp(0) ≥ t = gp(x). next, let x,y,z ∈ x. then gp(x · (y · z)),gp(y) ∈ [0,1]. choose t = min{gp(x · (y · z)),gp(y)}. thus gp(x · (y · z)) ≥ t and gp(y) ≥ t, so x · (y · z),y ∈ u(gp ;t) 6= ∅. by assumption, we have u(gp ;t) is a bcc-ideal of x and so x ·z ∈ u(gp ;t). thus gp(x ·z)≥ t =min{gp(x ·(y ·z)),gp(y)}. let x ∈ x. then bp(x) ∈ [0,1]. choose t = bp(x). thus bp(x) ≤ t, so x ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a bcc-ideal of x and so 0 ∈ l(bp ;t). thus bp(0) ≤ t = bp(x). next, let x,y,z ∈ x. then bp(x · (y · z)),bp(y) ∈ [0,1]. choose t = max{bp(x · (y · z)),bp(y)}. int. j. anal. appl. (2023), 21:75 27 thus bp(x · (y · z)) ≤ t and bp(y) ≤ t, so x · (y · z),y ∈ l(bp ;t) 6= ∅. by assumption, we have l(bp ;t) is a bcc-ideal of x and so x ·z ∈ l(bp ;t). thus bp(x ·z)≤ t =max{bp(x ·(y ·z)),bp(y)}. therefore, p is a picture fuzzy bcc-ideal of x. � theorem 4.14. if p in x is a picture fuzzy bcc-ideal of x, then for all t ∈ [0,1], the sets u+(rp ;t),u +(gp ;t), and l−(bp ;t) are bcc-ideals of x if u+(rp ;t),u+(gp ;t), and l−(bp ;t) are nonempty. proof. assume that p is a picture fuzzy bcc-ideal of x. let t ∈ [0,1] be such that u+(rp ;t),u +(gp ;t), and l−(bp ;t) are nonempty. let x ∈ u+(rp ;t). then rp(x) > t. by (3.7), we have rp(0) ≥ rp(x) > t. thus 0 ∈ u+(rp ;t). next, let x,y,z ∈ x be such that x · (y · z)∈ u+(rp ;t) and y ∈ u+(rp ;t). then rp(x · (y · z)) > t and rp(y) > t, so t is a lower bound of {rp(x · (y · z)), rp(y)}. by (3.22), we have rp(x · z) ≥ min{rp(x · (y ·z)), rp(y)} > t. thus x ·z ∈ u+(rp ;t). let x ∈ u+(gp ;t). then gp(x) > t. by (3.8), we have gp(0)≥ gp(x) > t. thus 0∈ u+(gp ;t). next, let x,y,z ∈ x be such that x ·(y ·z)∈ u+(gp ;t) and y ∈ u+(gp ;t). then gp(x ·(y ·z)) > t and gp(y) > t, so t is a lower bound of {gp(x · (y · z)),gp(y)}. by (3.23), we have gp(x · z) ≥ min{gp(x · (y ·z)),gp(y)} > t. thus x ·z ∈ u+(gp ;t). let x ∈ l−(bp ;t). then bp(x) < t. by (3.9), we have bp(0)≤ bp(x) < t. thus 0∈ l−(bp ;t). next, let x,y,z ∈ x be such that x · (y ·z)∈ l−(bp ;t) and y ∈ l−(bp ;t). then bp(x · (y ·z)) < t and bp(y) < t, so t is an upper bound of {bp(x · (y · z)),bp(y)}. by (3.24), we have bp(x · z) ≤ max{bp(x · (y ·z)),bp(y)} < t. thus x ·z ∈ l−(bp ;t). hence, u+(rp ;t),u+(gp ;t), and l−(bp ;t) are bcc-ideals of x. � theorem 4.15. a pfs p in x is a picture fuzzy strong bcc-ideal of x if and only if the sets e(rp ;rp(0)),e(gp ;gp(0)), and e(bp ;bp(0)) are strong bcc-ideals of x. proof. assume that p is a picture fuzzy strong bcc-ideal of x. then p is constant, that is, rp ,gp , and bp are constant. thus (∀x ∈ x)   rp(x)= rp(0) gp(x)= gp(0) bp(x)= bp(0)   . hence, e(rp ;rp(0)) = x,e(gp ;gp(0)) = x, and e(bp ;bp(0)) = x and so e(rp ; rp(0)),e(gp ;gp(0)), and e(bp ;bp(0)) are strong bcc-ideals of x. 28 int. j. anal. appl. (2023), 21:75 conversely, assume that e(rp ;rp(0)),e(gp ;gp(0)), and e(bp ;bp(0)) are strong bcc-ideals of x. then e(rp ;rp(0))= x,e(gp ;gp(0))= x, and e(bp ;bp(0))= x and so (∀x ∈ x)   rp(x)= rp(0) gp(x)= gp(0) bp(x)= bp(0)   . thus rp ,gp , and bp are constant, that is, p is constant. hence, p is a picture fuzzy strong bcc-ideal of x. � definition 4.2. let p be a pfs in x. for any α,β,γ ∈ [0,1], the sets uulp(α,β,γ)= {x ∈ x | rp(x)≥ α,gp(x)≥ β,bp(x)≤ γ}, llup(α,β,γ)= {x ∈ x | rp(x)≤ α,gp(x)≤ β,bp(x)≥ γ}, ep(α,β,γ)= {x ∈ x | rp(x)= α,gp(x)= β,bp(x)= γ} are called a uul-(α,β,γ)-level subset, a llu-(α,β,γ)-level subset, and an e-(α,β,γ)-level subset of p, respectively. then we see that uulp(α,β,γ)= u(rp ;α)∩u(gp ;β)∩l(bp ;γ), llup(α,β,γ)= l(rp ;α)∩l(gp ;β)∩u(bp ;γ), ep(α,β,γ)= e(rp ;α)∩e(gp ;β)∩e(bp ;γ). corollary 4.1. a pfs p in x is a picture fuzzy bcc-subalgebra of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a bcc-subalgebra of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.1 and 2.1. � corollary 4.2. a pfs p in x is a picture fuzzy near bcc-filter of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a near bcc-filter of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.3 and 2.1. � corollary 4.3. a pfs p in x is a picture fuzzy bcc-filter of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a bcc-filter of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.5 and 2.1. � corollary 4.4. a pfs p in x is a picture fuzzy implicative bcc-filter of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a implicative bcc-filter of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.7 and 2.1. � corollary 4.5. a pfs p in x is a picture fuzzy comparative bcc-filter of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a comparative bcc-filter of x if uulp(α,β,γ) is nonempty. int. j. anal. appl. (2023), 21:75 29 proof. it is straightforward by theorems 4.9 and 2.1. � corollary 4.6. a pfs p in x is a picture fuzzy shift bcc-filter of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a shift bcc-filter of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.11 and 2.1. � corollary 4.7. a pfs p in x is a picture fuzzy bcc-ideal of x if and only if for all α,β,γ ∈ [0,1],uulp(α,β,γ) is a bcc-ideal of x if uulp(α,β,γ) is nonempty. proof. it is straightforward by theorems 4.13 and 2.1. � corollary 4.8. a pfs p in x is a picture fuzzy strong bcc-ideal of x if and only if ep(rp(0),gp(0),bp(0)) is a strong bcc-ideal of x, that is, e(rp , rp(0)) = x,e(gp ,gp(0)) = x, and e(bp ,bp(0))= x. proof. it is straightforward by theorems 4.15 and 2.1. � acknowledgment: this research project was supported by the thailand science research and innovation fund and the university of phayao (grant no. ff66-rim032). conflicts of interest: the authors declare that there are no conflicts of interest regarding the publication of this paper. references [1] m. ansari, a. haidar, a. koam, on a graph associated to up-algebras, math. comput. appl. 23 (2018), 61. https://doi.org/10.3390/mca23040061. [2] m.a. ansari, a.n.a. koam, a. haider, rough set theory applied to up-algebras, italian j. pure appl. math. 42 (2019), 388-402. [3] k.t. atanassov, intuitionistic fuzzy sets, fuzzy sets syst. 20 (1986), 87-96. https://doi.org/10.1016/ s0165-0114(86)80034-3. [4] k.t. atanassov, new operations defined over the intuitionistic fuzzy sets, fuzzy sets syst. 61 (1994), 137-142. [5] b.c. cuong, picture fuzzy sets, j. computer sci. cybern. 30 (2014), 409-420. https://doi.org/10.15625/ 1813-9663/30/4/5032. [6] b.c. cuong, v. kreinovich, picture fuzzy sets a new concept for computational intelligence problems, in: 2013 third world congress on information and communication technologies (wict 2013), ieee, hanoi, vietnam, 2013: pp. 1-6. https://doi.org/10.1109/wict.2013.7113099. [7] n. dokkhamdang, a. kesorn, a. iampan, generalized fuzzy sets in up-algebras, ann. fuzzy math. inform. 16 (2018), 171-190. https://doi.org/10.30948/afmi.2018.16.2.171. [8] a.h. ganie, s. singh, p.k. bhatia, some new correlation coefficients of picture fuzzy sets with applications, neural comput. appl. 32 (2020), 12609-12625. https://doi.org/10.1007/s00521-020-04715-y. [9] t. guntasow, s. sajak, a. jomkham, et al. fuzzy translations of a fuzzy set in up-algebras, j. indones. math. soc. 23 (2017), 1-19. [10] y. huang, bci-algebra, science press, beijing, china, 2006. [11] a. iampan, a new branch of the logical algebra: up-algebras, j. algebra related topics. 5 (2017), 35-54. https: //doi.org/10.22124/jart.2017.2403. https://doi.org/10.3390/mca23040061 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.1016/s0165-0114(86)80034-3 https://doi.org/10.15625/1813-9663/30/4/5032 https://doi.org/10.15625/1813-9663/30/4/5032 https://doi.org/10.1109/wict.2013.7113099 https://doi.org/10.30948/afmi.2018.16.2.171 https://doi.org/10.1007/s00521-020-04715-y https://doi.org/10.22124/jart.2017.2403 https://doi.org/10.22124/jart.2017.2403 30 int. j. anal. appl. (2023), 21:75 [12] a. iampan, introducing fully up-semigroups, discuss. math. gen. algebra appl. 38 (2018), 297-306. [13] a. iampan, multipliers and near up-filters of up-algebras, j. discr. math. sci. cryptogr. 24 (2019), 667-680. https://doi.org/10.1080/09720529.2019.1649027. [14] a. iampan, a. satirad, m. songsaeng, a note on up-hyperalgebras, j. algebr. hyperstruct. log. algebr. 1 (2020), 77-95. https://doi.org/10.29252/hatef.jahla.1.2.7. [15] a. iampan, m. songsaeng, g. muhiuddin, fuzzy duplex up-algebras, eur. j. pure appl. math. 13 (2020), 459-471. https://doi.org/10.29020/nybg.ejpam.v13i3.3752. [16] y. imai, k. iséki, on axiom systems of propositional calculi, xiv, proc. japan acad. 4 (1966), 19-22. https: //doi.org/10.3792/pja/1195522169. [17] k. iséki, an algebra related with a propositional calculus, proc. japan acad. 42 (1966), 26-29. https://cir.nii. ac.jp/crid/1570009749777934080. [18] y.b. jun, b. brundha, n. rajesh, et al. (3,2)-fuzzy up (bcc)-subalgebras and (3,2)-fuzzy up (bcc)-filters, j. mahani math. res. 11 (2022), 1-14. https://doi.org/10.22103/jmmrc.2022.18786.1191. [19] y.b. jun, a. iampan, comparative and allied up-filters, lobachevskii j. math. 40 (2019), 60-66. https://doi. org/10.1134/s1995080219010086. [20] y.b. jun, a. iampan, implicative up-filters, afr. mat. 30 (2019), 1093-1101. https://doi.org/10.1007/ s13370-019-00704-0. [21] y.b. jun, a. iampan, shift up-filters and decompositions of up-filters in up-algebras, missouri j. math. sci. 31 (2019), 36-45. https://doi.org/10.35834/mjms/1559181624. [22] p. kankaew, s. yuphaphin, n. lapo, et al. picture fuzzy set theory applied to up-algebras, missouri j. math. sci. 34 (2022), 94-120. https://doi.org/10.35834/2022/3401094. [23] h.s. kim, y.h. kim, on be-algebras, sci. math. japon. 66 (2007), 113-116. https://doi.org/10.32219/isms. 66.1_113. [24] y. komori, the class of bcc-algebras is not a variety, math. japon. 29 (1984), 391-394. [25] p. mosrijai, a. iampan, a new branch of bialgebraic structures: up-bialgebras, j. taibah univ. sci. 13 (2019), 450-459. https://doi.org/10.1080/16583655.2019.1592932. [26] c. prabpayak, u. leerawat, on ideals and congruences in ku-algebras, sci. magna, 5 (2009), 54-57. [27] a. satirad, r. chinram, a. iampan, four new concepts of extensions of ku/up-algebras, missouri j. math. sci. 32 (2020), 138-157. https://doi.org/10.35834/2020/3202138. [28] a. iampan, a. satirad, topological up-algebras, discuss. math. gen. algebra appl. 39 (2019), 231-250. https: //doi.org/10.7151/dmgaa.1317. [29] a. satirad, p. mosrijai, a. iampan, formulas for finding up-algebras, int. j. math. computer sci. 14 (2019), 403-409. [30] a. satirad, p. mosrijai, a. iampan, generalized power up-algebras, int. j. math. computer sci. 14 (2019), 17-25. [31] t. senapati, y.b. jun, k.p. shum, cubic set structure applied in up-algebras, discr. math. algorithm. appl. 10 (2018), 1850049. https://doi.org/10.1142/s1793830918500490. [32] t. senapati, g. muhiuddin, k.p. shum, representation of up-algebras in interval-valued intuitionistic fuzzy environment, italian j. pure appl. math. 38 (2017), 497-517. [33] p. singh, correlation coefficients for picture fuzzy sets, j. intell. fuzzy syst. 28 (2015), 591-604. https://doi. org/10.3233/ifs-141338. [34] j. somjanta, n. thuekaew, p. kumpeangkeaw, a. iampan, fuzzy sets in up-algebras, ann. fuzzy math. inform. 12 (2016), 739-756. [35] g. wei, some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making, informatica, 28, no. 3, (2017), 547-564. https://doi.org/10.1080/09720529.2019.1649027 https://doi.org/10.29252/hatef.jahla.1.2.7 https://doi.org/10.29020/nybg.ejpam.v13i3.3752 https://doi.org/10.3792/pja/1195522169 https://doi.org/10.3792/pja/1195522169 https://cir.nii.ac.jp/crid/1570009749777934080 https://cir.nii.ac.jp/crid/1570009749777934080 https://doi.org/10.22103/jmmrc.2022.18786.1191 https://doi.org/10.1134/s1995080219010086 https://doi.org/10.1134/s1995080219010086 https://doi.org/10.1007/s13370-019-00704-0 https://doi.org/10.1007/s13370-019-00704-0 https://doi.org/10.35834/mjms/1559181624 https://doi.org/10.35834/2022/3401094 https://doi.org/10.32219/isms.66.1_113 https://doi.org/10.32219/isms.66.1_113 https://doi.org/10.1080/16583655.2019.1592932 https://doi.org/10.35834/2020/3202138 https://doi.org/10.7151/dmgaa.1317 https://doi.org/10.7151/dmgaa.1317 https://doi.org/10.1142/s1793830918500490 https://doi.org/10.3233/ifs-141338 https://doi.org/10.3233/ifs-141338 int. j. anal. appl. (2023), 21:75 31 [36] g. wei, some similarity measures for picture fuzzy sets and their applications, iran. j. fuzzy syst. 15 (2018), 77-89. [37] g. wei, h. gao, the generalized dice similarity measures for picture fuzzy sets and their applications, informatica, 29 (2018), 107-124. [38] l.a. zadeh, fuzzy sets, inform. control. 8 (1965), 338-353. https://doi.org/10.1016/s0019-9958(65) 90241-x. https://doi.org/10.1016/s0019-9958(65)90241-x https://doi.org/10.1016/s0019-9958(65)90241-x 1. introduction 2. basic results on bcc-algebras 3. pfss in bcc-algebras 4. level subsets of a pfs references